LMFDB - Embedded newform 38.8.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [38,8,Mod(1,38)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(38, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("38.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 38 = 2 \cdot 19 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	38.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$11.8706309684$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9097x^{2} - 110520x + 10368000 $ x^4 - 9097x^2 - 110520x + 10368000
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$29.4051$ of defining polynomial
Character	$\chi$	$=$	38.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.00000 q^{2} -26.4051 q^{3} +64.0000 q^{4} +139.358 q^{5} -211.240 q^{6} +458.899 q^{7} +512.000 q^{8} -1489.77 q^{9} +O(q^{10})$	\(q+8.00000 q^{2} -26.4051 q^{3} +64.0000 q^{4} +139.358 q^{5} -211.240 q^{6} +458.899 q^{7} +512.000 q^{8} -1489.77 q^{9} +1114.87 q^{10} +6029.39 q^{11} -1689.92 q^{12} +9191.95 q^{13} +3671.19 q^{14} -3679.77 q^{15} +4096.00 q^{16} +37435.7 q^{17} -11918.2 q^{18} +6859.00 q^{19} +8918.94 q^{20} -12117.3 q^{21} +48235.1 q^{22} -89730.1 q^{23} -13519.4 q^{24} -58704.2 q^{25} +73535.6 q^{26} +97085.4 q^{27} +29369.5 q^{28} +47796.8 q^{29} -29438.1 q^{30} +116461. q^{31} +32768.0 q^{32} -159206. q^{33} +299485. q^{34} +63951.4 q^{35} -95345.5 q^{36} -573531. q^{37} +54872.0 q^{38} -242714. q^{39} +71351.5 q^{40} -273810. q^{41} -96938.0 q^{42} +321102. q^{43} +385881. q^{44} -207612. q^{45} -717841. q^{46} -766472. q^{47} -108155. q^{48} -612955. q^{49} -469634. q^{50} -988491. q^{51} +588285. q^{52} +218381. q^{53} +776683. q^{54} +840246. q^{55} +234956. q^{56} -181112. q^{57} +382374. q^{58} +878690. q^{59} -235505. q^{60} +967679. q^{61} +931689. q^{62} -683655. q^{63} +262144. q^{64} +1.28098e6 q^{65} -1.27365e6 q^{66} +1.50171e6 q^{67} +2.39588e6 q^{68} +2.36933e6 q^{69} +511612. q^{70} +4.32518e6 q^{71} -762764. q^{72} -5.52112e6 q^{73} -4.58825e6 q^{74} +1.55009e6 q^{75} +438976. q^{76} +2.76688e6 q^{77} -1.94171e6 q^{78} +2.01493e6 q^{79} +570812. q^{80} +694589. q^{81} -2.19048e6 q^{82} -2.16645e6 q^{83} -775504. q^{84} +5.21698e6 q^{85} +2.56881e6 q^{86} -1.26208e6 q^{87} +3.08705e6 q^{88} -1.00604e7 q^{89} -1.66090e6 q^{90} +4.21818e6 q^{91} -5.74273e6 q^{92} -3.07516e6 q^{93} -6.13178e6 q^{94} +955859. q^{95} -865241. q^{96} +5.88299e6 q^{97} -4.90364e6 q^{98} -8.98242e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 32 q^{2} + 12 q^{3} + 256 q^{4} - 279 q^{5} + 96 q^{6} + 2485 q^{7} + 2048 q^{8} + 9482 q^{9}+O(q^{10}) $ 4 * q + 32 * q^2 + 12 * q^3 + 256 * q^4 - 279 * q^5 + 96 * q^6 + 2485 * q^7 + 2048 * q^8 + 9482 * q^9	\( 4 q + 32 q^{2} + 12 q^{3} + 256 q^{4} - 279 q^{5} + 96 q^{6} + 2485 q^{7} + 2048 q^{8} + 9482 q^{9} - 2232 q^{10} + 5269 q^{11} + 768 q^{12} + 5406 q^{13} + 19880 q^{14} + 26658 q^{15} + 16384 q^{16} + 22885 q^{17} + 75856 q^{18} + 27436 q^{19} - 17856 q^{20} + 2854 q^{21} + 42152 q^{22} + 3364 q^{23} + 6144 q^{24} + 112561 q^{25} + 43248 q^{26} - 220194 q^{27} + 159040 q^{28} - 122136 q^{29} + 213264 q^{30} + 225480 q^{31} + 131072 q^{32} + 176138 q^{33} + 183080 q^{34} - 785781 q^{35} + 606848 q^{36} + 154096 q^{37} + 219488 q^{38} - 1749220 q^{39} - 142848 q^{40} - 1054628 q^{41} + 22832 q^{42} - 840795 q^{43} + 337216 q^{44} - 4162563 q^{45} + 26912 q^{46} - 1021877 q^{47} + 49152 q^{48} - 621441 q^{49} + 900488 q^{50} + 724892 q^{51} + 345984 q^{52} - 326842 q^{53} - 1761552 q^{54} - 221553 q^{55} + 1272320 q^{56} + 82308 q^{57} - 977088 q^{58} + 421384 q^{59} + 1706112 q^{60} + 116825 q^{61} + 1803840 q^{62} + 10245825 q^{63} + 1048576 q^{64} + 4477428 q^{65} + 1409104 q^{66} + 5794566 q^{67} + 1464640 q^{68} - 2472196 q^{69} - 6286248 q^{70} + 10590626 q^{71} + 4854784 q^{72} + 3971389 q^{73} + 1232768 q^{74} - 3690042 q^{75} + 1755904 q^{76} + 5806573 q^{77} - 13993760 q^{78} + 5597800 q^{79} - 1142784 q^{80} + 20567744 q^{81} - 8437024 q^{82} + 4665800 q^{83} + 182656 q^{84} - 2014461 q^{85} - 6726360 q^{86} - 14449584 q^{87} + 2697728 q^{88} - 2794214 q^{89} - 33300504 q^{90} - 8827314 q^{91} + 215296 q^{92} - 43981204 q^{93} - 8175016 q^{94} - 1913661 q^{95} + 393216 q^{96} - 14445130 q^{97} - 4971528 q^{98} - 7940315 q^{99}+O(q^{100}) \) 4 * q + 32 * q^2 + 12 * q^3 + 256 * q^4 - 279 * q^5 + 96 * q^6 + 2485 * q^7 + 2048 * q^8 + 9482 * q^9 - 2232 * q^10 + 5269 * q^11 + 768 * q^12 + 5406 * q^13 + 19880 * q^14 + 26658 * q^15 + 16384 * q^16 + 22885 * q^17 + 75856 * q^18 + 27436 * q^19 - 17856 * q^20 + 2854 * q^21 + 42152 * q^22 + 3364 * q^23 + 6144 * q^24 + 112561 * q^25 + 43248 * q^26 - 220194 * q^27 + 159040 * q^28 - 122136 * q^29 + 213264 * q^30 + 225480 * q^31 + 131072 * q^32 + 176138 * q^33 + 183080 * q^34 - 785781 * q^35 + 606848 * q^36 + 154096 * q^37 + 219488 * q^38 - 1749220 * q^39 - 142848 * q^40 - 1054628 * q^41 + 22832 * q^42 - 840795 * q^43 + 337216 * q^44 - 4162563 * q^45 + 26912 * q^46 - 1021877 * q^47 + 49152 * q^48 - 621441 * q^49 + 900488 * q^50 + 724892 * q^51 + 345984 * q^52 - 326842 * q^53 - 1761552 * q^54 - 221553 * q^55 + 1272320 * q^56 + 82308 * q^57 - 977088 * q^58 + 421384 * q^59 + 1706112 * q^60 + 116825 * q^61 + 1803840 * q^62 + 10245825 * q^63 + 1048576 * q^64 + 4477428 * q^65 + 1409104 * q^66 + 5794566 * q^67 + 1464640 * q^68 - 2472196 * q^69 - 6286248 * q^70 + 10590626 * q^71 + 4854784 * q^72 + 3971389 * q^73 + 1232768 * q^74 - 3690042 * q^75 + 1755904 * q^76 + 5806573 * q^77 - 13993760 * q^78 + 5597800 * q^79 - 1142784 * q^80 + 20567744 * q^81 - 8437024 * q^82 + 4665800 * q^83 + 182656 * q^84 - 2014461 * q^85 - 6726360 * q^86 - 14449584 * q^87 + 2697728 * q^88 - 2794214 * q^89 - 33300504 * q^90 - 8827314 * q^91 + 215296 * q^92 - 43981204 * q^93 - 8175016 * q^94 - 1913661 * q^95 + 393216 * q^96 - 14445130 * q^97 - 4971528 * q^98 - 7940315 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	8.00000	0.707107
$2$	8.00000	0.707107
$3$	−26.4051	−0.564628	−0.282314	−	0.959322i	$-0.591102\pi$
$3$	−26.4051	−0.564628	−0.282314	+	0.959322i	$0.591102\pi$
$4$	64.0000	0.500000
$5$	139.358	0.498584	0.249292	−	0.968428i	$-0.419802\pi$
$5$	139.358	0.498584	0.249292	+	0.968428i	$0.419802\pi$
$6$	−211.240	−0.399252
$7$	458.899	0.505678	0.252839	−	0.967508i	$-0.418636\pi$
$7$	458.899	0.505678	0.252839	+	0.967508i	$0.418636\pi$
$8$	512.000	0.353553
$9$	−1489.77	−0.681195
$10$	1114.87	0.352552
$11$	6029.39	1.36584	0.682919	−	0.730494i	$-0.260710\pi$
$11$	6029.39	1.36584	0.682919	+	0.730494i	$0.260710\pi$
$12$	−1689.92	−0.282314
$13$	9191.95	1.16039	0.580197	−	0.814476i	$-0.302975\pi$
$13$	9191.95	1.16039	0.580197	+	0.814476i	$0.302975\pi$
$14$	3671.19	0.357568
$15$	−3679.77	−0.281515
$16$	4096.00	0.250000
$17$	37435.7	1.84805	0.924027	−	0.382328i	$-0.124878\pi$
$17$	37435.7	1.84805	0.924027	+	0.382328i	$0.124878\pi$
$18$	−11918.2	−0.481678
$19$	6859.00	0.229416
$19$	6859.00	0.229416
$20$	8918.94	0.249292
$21$	−12117.3	−0.285520
$22$	48235.1	0.965793
$23$	−89730.1	−1.53777	−0.768884	−	0.639388i	$-0.779188\pi$
$23$	−89730.1	−1.53777	−0.768884	+	0.639388i	$0.779188\pi$
$24$	−13519.4	−0.199626
$25$	−58704.2	−0.751414
$26$	73535.6	0.820523
$27$	97085.4	0.949250
$28$	29369.5	0.252839
$29$	47796.8	0.363920	0.181960	−	0.983306i	$-0.441756\pi$
$29$	47796.8	0.363920	0.181960	+	0.983306i	$0.441756\pi$
$30$	−29438.1	−0.199061
$31$	116461.	0.702126	0.351063	−	0.936352i	$-0.385820\pi$
$31$	116461.	0.702126	0.351063	+	0.936352i	$0.385820\pi$
$32$	32768.0	0.176777
$33$	−159206.	−0.771190
$34$	299485.	1.30677
$35$	63951.4	0.252123
$36$	−95345.5	−0.340597
$37$	−573531.	−1.86145	−0.930724	−	0.365722i	$-0.880822\pi$
$37$	−573531.	−1.86145	−0.930724	+	0.365722i	$0.880822\pi$
$38$	54872.0	0.162221
$39$	−242714.	−0.655192
$40$	71351.5	0.176276
$41$	−273810.	−0.620450	−0.310225	−	0.950663i	$-0.600404\pi$
$41$	−273810.	−0.620450	−0.310225	+	0.950663i	$0.600404\pi$
$42$	−96938.0	−0.201893
$43$	321102.	0.615890	0.307945	−	0.951404i	$-0.400359\pi$
$43$	321102.	0.615890	0.307945	+	0.951404i	$0.400359\pi$
$44$	385881.	0.682919
$45$	−207612.	−0.339633
$46$	−717841.	−1.08737
$47$	−766472.	−1.07685	−0.538423	−	0.842674i	$-0.680980\pi$
$47$	−766472.	−1.07685	−0.538423	+	0.842674i	$0.680980\pi$
$48$	−108155.	−0.141157
$49$	−612955.	−0.744290
$50$	−469634.	−0.531330
$51$	−988491.	−1.04346
$52$	588285.	0.580197
$53$	218381.	0.201488	0.100744	−	0.994912i	$-0.467878\pi$
$53$	218381.	0.201488	0.100744	+	0.994912i	$0.467878\pi$
$54$	776683.	0.671221
$55$	840246.	0.680984
$56$	234956.	0.178784
$57$	−181112.	−0.129535
$58$	382374.	0.257330
$59$	878690.	0.556998	0.278499	−	0.960437i	$-0.410163\pi$
$59$	878690.	0.556998	0.278499	+	0.960437i	$0.410163\pi$
$60$	−235505.	−0.140757
$61$	967679.	0.545854	0.272927	−	0.962035i	$-0.412008\pi$
$61$	967679.	0.545854	0.272927	+	0.962035i	$0.412008\pi$
$62$	931689.	0.496478
$63$	−683655.	−0.344465
$64$	262144.	0.125000
$65$	1.28098e6	0.578554
$66$	−1.27365e6	−0.545314
$67$	1.50171e6	0.609993	0.304996	−	0.952353i	$-0.401345\pi$
$67$	1.50171e6	0.609993	0.304996	+	0.952353i	$0.401345\pi$
$68$	2.39588e6	0.924027
$69$	2.36933e6	0.868267
$70$	511612.	0.178278
$71$	4.32518e6	1.43417	0.717083	−	0.696988i	$-0.245477\pi$
$71$	4.32518e6	1.43417	0.717083	+	0.696988i	$0.245477\pi$
$72$	−762764.	−0.240839
$73$	−5.52112e6	−1.66110	−0.830552	−	0.556941i	$-0.811975\pi$
$73$	−5.52112e6	−1.66110	−0.830552	+	0.556941i	$0.811975\pi$
$74$	−4.58825e6	−1.31624
$75$	1.55009e6	0.424270
$76$	438976.	0.114708
$77$	2.76688e6	0.690674
$78$	−1.94171e6	−0.463291
$79$	2.01493e6	0.459797	0.229898	−	0.973215i	$-0.426161\pi$
$79$	2.01493e6	0.459797	0.229898	+	0.973215i	$0.426161\pi$
$80$	570812.	0.124646
$81$	694589.	0.145221
$82$	−2.19048e6	−0.438724
$83$	−2.16645e6	−0.415887	−0.207943	−	0.978141i	$-0.566677\pi$
$83$	−2.16645e6	−0.415887	−0.207943	+	0.978141i	$0.566677\pi$
$84$	−775504.	−0.142760
$85$	5.21698e6	0.921410
$86$	2.56881e6	0.435500
$87$	−1.26208e6	−0.205479
$88$	3.08705e6	0.482896
$89$	−1.00604e7	−1.51270	−0.756349	−	0.654169i	$-0.773019\pi$
$89$	−1.00604e7	−1.51270	−0.756349	+	0.654169i	$0.773019\pi$
$90$	−1.66090e6	−0.240157
$91$	4.21818e6	0.586786
$92$	−5.74273e6	−0.768884
$93$	−3.07516e6	−0.396440
$94$	−6.13178e6	−0.761446
$95$	955859.	0.114383
$96$	−865241.	−0.0998131
$97$	5.88299e6	0.654481	0.327241	−	0.944941i	$-0.393881\pi$
$97$	5.88299e6	0.654481	0.327241	+	0.944941i	$0.393881\pi$
$98$	−4.90364e6	−0.526292
$99$	−8.98242e6	−0.930401
$100$	−3.75707e6	−0.375707
$101$	981082.	0.0947502	0.0473751	−	0.998877i	$-0.484914\pi$
$101$	981082.	0.0947502	0.0473751	+	0.998877i	$0.484914\pi$
$102$	−7.90793e6	−0.737840
$103$	−8.90617e6	−0.803084	−0.401542	−	0.915841i	$-0.631526\pi$
$103$	−8.90617e6	−0.803084	−0.401542	+	0.915841i	$0.631526\pi$
$104$	4.70628e6	0.410262
$105$	−1.68864e6	−0.142356
$106$	1.74705e6	0.142474
$107$	1.39635e7	1.10192	0.550962	−	0.834530i	$-0.314261\pi$
$107$	1.39635e7	1.10192	0.550962	+	0.834530i	$0.314261\pi$
$108$	6.21346e6	0.474625
$109$	−1.93012e7	−1.42755	−0.713776	−	0.700374i	$-0.753016\pi$
$109$	−1.93012e7	−1.42755	−0.713776	+	0.700374i	$0.753016\pi$
$110$	6.72197e6	0.481529
$111$	1.51441e7	1.05103
$112$	1.87965e6	0.126419
$113$	3.25129e6	0.211973	0.105987	−	0.994368i	$-0.466200\pi$
$113$	3.25129e6	0.211973	0.105987	+	0.994368i	$0.466200\pi$
$114$	−1.44890e6	−0.0915948
$115$	−1.25046e7	−0.766706
$116$	3.05900e6	0.181960
$117$	−1.36939e7	−0.790455
$118$	7.02952e6	0.393857
$119$	1.71792e7	0.934520
$120$	−1.88404e6	−0.0995304
$121$	1.68664e7	0.865511
$122$	7.74143e6	0.385977
$123$	7.22998e6	0.350323
$124$	7.45351e6	0.351063
$125$	−1.90683e7	−0.873227
$126$	−5.46924e6	−0.243574
$127$	1.90185e7	0.823877	0.411939	−	0.911212i	$-0.364852\pi$
$127$	1.90185e7	0.823877	0.411939	+	0.911212i	$0.364852\pi$
$128$	2.09715e6	0.0883883
$129$	−8.47871e6	−0.347749
$130$	1.02478e7	0.409100
$131$	−4.87038e7	−1.89284	−0.946419	−	0.322942i	$-0.895328\pi$
$131$	−4.87038e7	−1.89284	−0.946419	+	0.322942i	$0.895328\pi$
$132$	−1.01892e7	−0.385595
$133$	3.14759e6	0.116010
$134$	1.20137e7	0.431330
$135$	1.35297e7	0.473281
$136$	1.91671e7	0.653385
$137$	−1.59773e7	−0.530863	−0.265431	−	0.964130i	$-0.585514\pi$
$137$	−1.59773e7	−0.530863	−0.265431	+	0.964130i	$0.585514\pi$
$138$	1.89546e7	0.613958
$139$	−2.41980e6	−0.0764235	−0.0382118	−	0.999270i	$-0.512166\pi$
$139$	−2.41980e6	−0.0764235	−0.0382118	+	0.999270i	$0.512166\pi$
$140$	4.09289e6	0.126061
$141$	2.02387e7	0.608018
$142$	3.46014e7	1.01411
$143$	5.54218e7	1.58491
$144$	−6.10211e6	−0.170299
$145$	6.66089e6	0.181445
$146$	−4.41689e7	−1.17458
$147$	1.61851e7	0.420247
$148$	−3.67060e7	−0.930724
$149$	−6.08533e7	−1.50707	−0.753533	−	0.657410i	$-0.771652\pi$
$149$	−6.08533e7	−1.50707	−0.753533	+	0.657410i	$0.771652\pi$
$150$	1.24007e7	0.300004
$151$	−4.73792e6	−0.111987	−0.0559937	−	0.998431i	$-0.517833\pi$
$151$	−4.73792e6	−0.111987	−0.0559937	+	0.998431i	$0.517833\pi$
$152$	3.51181e6	0.0811107
$153$	−5.57707e7	−1.25888
$154$	2.21350e7	0.488380
$155$	1.62298e7	0.350069
$156$	−1.55337e7	−0.327596
$157$	7.82369e7	1.61348	0.806738	−	0.590909i	$-0.201231\pi$
$157$	7.82369e7	1.61348	0.806738	+	0.590909i	$0.201231\pi$
$158$	1.61195e7	0.325125
$159$	−5.76637e6	−0.113766
$160$	4.56650e6	0.0881380
$161$	−4.11771e7	−0.777615
$162$	5.55671e6	0.102687
$163$	6.82942e7	1.23517	0.617586	−	0.786503i	$-0.288111\pi$
$163$	6.82942e7	1.23517	0.617586	+	0.786503i	$0.288111\pi$
$164$	−1.75239e7	−0.310225
$165$	−2.21867e7	−0.384503
$166$	−1.73316e7	−0.294076
$167$	−6.21748e7	−1.03302	−0.516508	−	0.856282i	$-0.672768\pi$
$167$	−6.21748e7	−1.03302	−0.516508	+	0.856282i	$0.672768\pi$
$168$	−6.20403e6	−0.100947
$169$	2.17434e7	0.346516
$170$	4.17358e7	0.651535
$171$	−1.02184e7	−0.156277
$172$	2.05505e7	0.307945
$173$	−6.22173e7	−0.913586	−0.456793	−	0.889573i	$-0.651002\pi$
$173$	−6.22173e7	−0.913586	−0.456793	+	0.889573i	$0.651002\pi$
$174$	−1.00966e7	−0.145296
$175$	−2.69393e7	−0.379974
$176$	2.46964e7	0.341459
$177$	−2.32019e7	−0.314497
$178$	−8.04835e7	−1.06964
$179$	−9.38512e7	−1.22308	−0.611539	−	0.791214i	$-0.709449\pi$
$179$	−9.38512e7	−1.22308	−0.611539	+	0.791214i	$0.709449\pi$
$180$	−1.32872e7	−0.169816
$181$	4.63487e7	0.580982	0.290491	−	0.956878i	$-0.406181\pi$
$181$	4.63487e7	0.580982	0.290491	+	0.956878i	$0.406181\pi$
$182$	3.37454e7	0.414920
$183$	−2.55516e7	−0.308205
$184$	−4.59418e7	−0.543683
$185$	−7.99264e7	−0.928088
$186$	−2.46013e7	−0.280325
$187$	2.25714e8	2.52414
$188$	−4.90542e7	−0.538423
$189$	4.45524e7	0.480015
$190$	7.64688e6	0.0808810
$191$	3.48040e7	0.361421	0.180710	−	0.983536i	$-0.442160\pi$
$191$	3.48040e7	0.361421	0.180710	+	0.983536i	$0.442160\pi$
$192$	−6.92193e6	−0.0705785
$193$	−7.51827e7	−0.752779	−0.376389	−	0.926462i	$-0.622835\pi$
$193$	−7.51827e7	−0.752779	−0.376389	+	0.926462i	$0.622835\pi$
$194$	4.70639e7	0.462788
$195$	−3.38242e7	−0.326668
$196$	−3.92291e7	−0.372145
$197$	−1.44970e8	−1.35097	−0.675487	−	0.737372i	$-0.736067\pi$
$197$	−1.44970e8	−1.35097	−0.675487	+	0.737372i	$0.736067\pi$
$198$	−7.18594e7	−0.657893
$199$	1.66896e8	1.50128	0.750638	−	0.660714i	$-0.229746\pi$
$199$	1.66896e8	1.50128	0.750638	+	0.660714i	$0.229746\pi$
$200$	−3.00566e7	−0.265665
$201$	−3.96528e7	−0.344419
$202$	7.84865e6	0.0669985
$203$	2.19339e7	0.184026
$204$	−6.32634e7	−0.521732
$205$	−3.81578e7	−0.309346
$206$	−7.12494e7	−0.567866
$207$	1.33678e8	1.04752
$208$	3.76502e7	0.290099
$209$	4.13556e7	0.313345
$210$	−1.35091e7	−0.100661
$211$	1.38279e8	1.01337	0.506686	−	0.862131i	$-0.330870\pi$
$211$	1.38279e8	1.01337	0.506686	+	0.862131i	$0.330870\pi$
$212$	1.39764e7	0.100744
$213$	−1.14207e8	−0.809771
$214$	1.11708e8	0.779179
$215$	4.47482e7	0.307073
$216$	4.97077e7	0.335611
$217$	5.34439e7	0.355049
$218$	−1.54410e8	−1.00943
$219$	1.45785e8	0.937906
$220$	5.37758e7	0.340492
$221$	3.44107e8	2.14447
$222$	1.21153e8	0.743188
$223$	8.57258e7	0.517660	0.258830	−	0.965923i	$-0.416663\pi$
$223$	8.57258e7	0.517660	0.258830	+	0.965923i	$0.416663\pi$
$224$	1.50372e7	0.0893921
$225$	8.74560e7	0.511859
$226$	2.60103e7	0.149888
$227$	−1.34003e7	−0.0760370	−0.0380185	−	0.999277i	$-0.512105\pi$
$227$	−1.34003e7	−0.0760370	−0.0380185	+	0.999277i	$0.512105\pi$
$228$	−1.15912e7	−0.0647673
$229$	−1.55852e7	−0.0857609	−0.0428804	−	0.999080i	$-0.513653\pi$
$229$	−1.55852e7	−0.0857609	−0.0428804	+	0.999080i	$0.513653\pi$
$230$	−1.00037e8	−0.542143
$231$	−7.30596e7	−0.389974
$232$	2.44720e7	0.128665
$233$	2.22498e8	1.15234	0.576169	−	0.817331i	$-0.304547\pi$
$233$	2.22498e8	1.15234	0.576169	+	0.817331i	$0.304547\pi$
$234$	−1.09551e8	−0.558936
$235$	−1.06814e8	−0.536898
$236$	5.62362e7	0.278499
$237$	−5.32044e7	−0.259614
$238$	1.37434e8	0.660805
$239$	−2.32571e8	−1.10195	−0.550975	−	0.834522i	$-0.685744\pi$
$239$	−2.32571e8	−1.10195	−0.550975	+	0.834522i	$0.685744\pi$
$240$	−1.50723e7	−0.0703786
$241$	1.44168e8	0.663451	0.331726	−	0.943376i	$-0.392369\pi$
$241$	1.44168e8	0.663451	0.331726	+	0.943376i	$0.392369\pi$
$242$	1.34931e8	0.612009
$243$	−2.30666e8	−1.03125
$244$	6.19314e7	0.272927
$245$	−8.54204e7	−0.371091
$246$	5.78398e7	0.247716
$247$	6.30476e7	0.266213
$248$	5.96281e7	0.248239
$249$	5.72052e7	0.234821
$250$	−1.52546e8	−0.617465
$251$	−4.01006e8	−1.60064	−0.800318	−	0.599576i	$-0.795336\pi$
$251$	−4.01006e8	−1.60064	−0.800318	+	0.599576i	$0.795336\pi$
$252$	−4.37539e7	−0.172233
$253$	−5.41018e8	−2.10034
$254$	1.52148e8	0.582569
$255$	−1.37755e8	−0.520254
$256$	1.67772e7	0.0625000
$257$	−2.01579e8	−0.740763	−0.370381	−	0.928880i	$-0.620773\pi$
$257$	−2.01579e8	−0.740763	−0.370381	+	0.928880i	$0.620773\pi$
$258$	−6.78297e7	−0.245896
$259$	−2.63193e8	−0.941293
$260$	8.19824e7	0.289277
$261$	−7.12064e7	−0.247900
$262$	−3.89630e8	−1.33844
$263$	2.33213e8	0.790512	0.395256	−	0.918571i	$-0.370656\pi$
$263$	2.33213e8	0.790512	0.395256	+	0.918571i	$0.370656\pi$
$264$	−8.15137e7	−0.272657
$265$	3.04333e7	0.100459
$266$	2.51807e7	0.0820318
$267$	2.65646e8	0.854112
$268$	9.61096e7	0.304996
$269$	2.97370e8	0.931461	0.465730	−	0.884927i	$-0.345792\pi$
$269$	2.97370e8	0.931461	0.465730	+	0.884927i	$0.345792\pi$
$270$	1.08237e8	0.334660
$271$	2.13782e6	0.00652497	0.00326248	−	0.999995i	$-0.498962\pi$
$271$	2.13782e6	0.00652497	0.00326248	+	0.999995i	$0.498962\pi$
$272$	1.53337e8	0.462013
$273$	−1.11381e8	−0.331316
$274$	−1.27819e8	−0.375377
$275$	−3.53951e8	−1.02631
$276$	1.51637e8	0.434134
$277$	−4.59774e8	−1.29977	−0.649883	−	0.760034i	$-0.725182\pi$
$277$	−4.59774e8	−1.29977	−0.649883	+	0.760034i	$0.725182\pi$
$278$	−1.93584e7	−0.0540396
$279$	−1.73501e8	−0.478284
$280$	3.27431e7	0.0891389
$281$	−2.91914e8	−0.784843	−0.392422	−	0.919785i	$-0.628363\pi$
$281$	−2.91914e8	−0.784843	−0.392422	+	0.919785i	$0.628363\pi$
$282$	1.61910e8	0.429934
$283$	7.90928e7	0.207436	0.103718	−	0.994607i	$-0.466926\pi$
$283$	7.90928e7	0.207436	0.103718	+	0.994607i	$0.466926\pi$
$284$	2.76811e8	0.717083
$285$	−2.52395e7	−0.0645839
$286$	4.43375e8	1.12070
$287$	−1.25651e8	−0.313748
$288$	−4.88169e7	−0.120419
$289$	9.91091e8	2.41530
$290$	5.32871e7	0.128301
$291$	−1.55341e8	−0.369539
$292$	−3.53351e8	−0.830552
$293$	1.22634e8	0.284821	0.142411	−	0.989808i	$-0.454515\pi$
$293$	1.22634e8	0.284821	0.142411	+	0.989808i	$0.454515\pi$
$294$	1.29481e8	0.297160
$295$	1.22453e8	0.277710
$296$	−2.93648e8	−0.658121
$297$	5.85366e8	1.29652
$298$	−4.86827e8	−1.06566
$299$	−8.24795e8	−1.78442
$300$	9.92056e7	0.212135
$301$	1.47353e8	0.311442
$302$	−3.79034e7	−0.0791870
$303$	−2.59055e7	−0.0534987
$304$	2.80945e7	0.0573539
$305$	1.34854e8	0.272154
$306$	−4.46165e8	−0.890166
$307$	7.28289e8	1.43655	0.718273	−	0.695762i	$-0.244933\pi$
$307$	7.28289e8	1.43655	0.718273	+	0.695762i	$0.244933\pi$
$308$	1.77080e8	0.345337
$309$	2.35168e8	0.453444
$310$	1.29839e8	0.247536
$311$	−4.02524e8	−0.758806	−0.379403	−	0.925232i	$-0.623871\pi$
$311$	−4.02524e8	−0.758806	−0.379403	+	0.925232i	$0.623871\pi$
$312$	−1.24269e8	−0.231645
$313$	−4.27061e8	−0.787200	−0.393600	−	0.919282i	$-0.628771\pi$
$313$	−4.27061e8	−0.787200	−0.393600	+	0.919282i	$0.628771\pi$
$314$	6.25895e8	1.14090
$315$	−9.52731e7	−0.171745
$316$	1.28956e8	0.229898
$317$	9.76559e8	1.72183	0.860917	−	0.508745i	$-0.169891\pi$
$317$	9.76559e8	1.72183	0.860917	+	0.508745i	$0.169891\pi$
$318$	−4.61310e7	−0.0804447
$319$	2.88186e8	0.497055
$320$	3.65320e7	0.0623230
$321$	−3.68708e8	−0.622178
$322$	−3.29416e8	−0.549857
$323$	2.56771e8	0.423972
$324$	4.44537e7	0.0726107
$325$	−5.39606e8	−0.871937
$326$	5.46354e8	0.873398
$327$	5.09649e8	0.806036
$328$	−1.40191e8	−0.219362
$329$	−3.51733e8	−0.544538
$330$	−1.77494e8	−0.271885
$331$	−9.56625e8	−1.44992	−0.724960	−	0.688791i	$-0.758142\pi$
$331$	−9.56625e8	−1.44992	−0.724960	+	0.688791i	$0.758142\pi$
$332$	−1.38653e8	−0.207943
$333$	8.54432e8	1.26801
$334$	−4.97399e8	−0.730453
$335$	2.09276e8	0.304133
$336$	−4.96323e7	−0.0713800
$337$	1.14440e9	1.62883	0.814413	−	0.580286i	$-0.197059\pi$
$337$	1.14440e9	1.62883	0.814413	+	0.580286i	$0.197059\pi$
$338$	1.73947e8	0.245024
$339$	−8.58505e7	−0.119686
$340$	3.33887e8	0.460705
$341$	7.02189e8	0.958989
$342$	−8.17468e7	−0.110504
$343$	−6.59207e8	−0.882049
$344$	1.64404e8	0.217750
$345$	3.30186e8	0.432904
$346$	−4.97738e8	−0.646003
$347$	5.04160e8	0.647761	0.323881	−	0.946098i	$-0.395012\pi$
$347$	5.04160e8	0.647761	0.323881	+	0.946098i	$0.395012\pi$
$348$	−8.07729e7	−0.102740
$349$	−5.90912e8	−0.744104	−0.372052	−	0.928212i	$-0.621346\pi$
$349$	−5.90912e8	−0.744104	−0.372052	+	0.928212i	$0.621346\pi$
$350$	−2.15514e8	−0.268682
$351$	8.92404e8	1.10151
$352$	1.97571e8	0.241448
$353$	−5.62804e8	−0.680998	−0.340499	−	0.940245i	$-0.610596\pi$
$353$	−5.62804e8	−0.680998	−0.340499	+	0.940245i	$0.610596\pi$
$354$	−1.85615e8	−0.222383
$355$	6.02750e8	0.715052
$356$	−6.43868e8	−0.756349
$357$	−4.53618e8	−0.527656
$358$	−7.50809e8	−0.864847
$359$	5.65283e8	0.644816	0.322408	−	0.946601i	$-0.395508\pi$
$359$	5.65283e8	0.644816	0.322408	+	0.946601i	$0.395508\pi$
$360$	−1.06298e8	−0.120078
$361$	4.70459e7	0.0526316
$362$	3.70790e8	0.410816
$363$	−4.45357e8	−0.488692
$364$	2.69963e8	0.293393
$365$	−7.69414e8	−0.828200
$366$	−2.04413e8	−0.217934
$367$	−1.50886e9	−1.59337	−0.796685	−	0.604395i	$-0.793415\pi$
$367$	−1.50886e9	−1.59337	−0.796685	+	0.604395i	$0.793415\pi$
$368$	−3.67535e8	−0.384442
$369$	4.07915e8	0.422647
$370$	−6.39411e8	−0.656257
$371$	1.00215e8	0.101888
$372$	−1.96810e8	−0.198220
$373$	−6.89477e8	−0.687921	−0.343961	−	0.938984i	$-0.611769\pi$
$373$	−6.89477e8	−0.687921	−0.343961	+	0.938984i	$0.611769\pi$
$374$	1.80571e9	1.78484
$375$	5.03500e8	0.493049
$376$	−3.92434e8	−0.380723
$377$	4.39346e8	0.422291
$378$	3.56419e8	0.339422
$379$	9.41203e7	0.0888068	0.0444034	−	0.999014i	$-0.485861\pi$
$379$	9.41203e7	0.0888068	0.0444034	+	0.999014i	$0.485861\pi$
$380$	6.11750e7	0.0571915
$381$	−5.02184e8	−0.465184
$382$	2.78432e8	0.255563
$383$	1.10869e9	1.00836	0.504179	−	0.863599i	$-0.331795\pi$
$383$	1.10869e9	1.00836	0.504179	+	0.863599i	$0.331795\pi$
$384$	−5.53754e7	−0.0499066
$385$	3.85588e8	0.344359
$386$	−6.01462e8	−0.532295
$387$	−4.78369e8	−0.419541
$388$	3.76512e8	0.327241
$389$	−6.58076e8	−0.566830	−0.283415	−	0.958997i	$-0.591467\pi$
$389$	−6.58076e8	−0.566830	−0.283415	+	0.958997i	$0.591467\pi$
$390$	−2.70594e8	−0.230989
$391$	−3.35911e9	−2.84188
$392$	−3.13833e8	−0.263146
$393$	1.28603e9	1.06875
$394$	−1.15976e9	−0.955283
$395$	2.80798e8	0.229247
$396$	−5.74875e8	−0.465201
$397$	−6.11564e7	−0.0490541	−0.0245270	−	0.999699i	$-0.507808\pi$
$397$	−6.11564e7	−0.0490541	−0.0245270	+	0.999699i	$0.507808\pi$
$398$	1.33517e9	1.06156
$399$	−8.31122e7	−0.0655028
$400$	−2.40453e8	−0.187854
$401$	8.40800e8	0.651160	0.325580	−	0.945515i	$-0.394440\pi$
$401$	8.40800e8	0.651160	0.325580	+	0.945515i	$0.394440\pi$
$402$	−3.17222e8	−0.243541
$403$	1.07050e9	0.814743
$404$	6.27892e7	0.0473751
$405$	9.67969e7	0.0724050
$406$	1.75471e8	0.130126
$407$	−3.45804e9	−2.54244
$408$	−5.06107e8	−0.368920
$409$	1.33833e9	0.967234	0.483617	−	0.875280i	$-0.339323\pi$
$409$	1.33833e9	0.967234	0.483617	+	0.875280i	$0.339323\pi$
$410$	−3.05262e8	−0.218741
$411$	4.21882e8	0.299740
$412$	−5.69995e8	−0.401542
$413$	4.03230e8	0.281662
$414$	1.06942e9	0.740708
$415$	−3.01913e8	−0.207354
$416$	3.01202e8	0.205131
$417$	6.38949e7	0.0431509
$418$	3.30845e8	0.221568
$419$	−4.66880e7	−0.0310067	−0.0155034	−	0.999880i	$-0.504935\pi$
$419$	−4.66880e7	−0.0310067	−0.0155034	+	0.999880i	$0.504935\pi$
$420$	−1.08073e8	−0.0711778
$421$	1.14687e9	0.749077	0.374539	−	0.927211i	$-0.377801\pi$
$421$	1.14687e9	0.749077	0.374539	+	0.927211i	$0.377801\pi$
$422$	1.10624e9	0.716562
$423$	1.14187e9	0.733543
$424$	1.11811e8	0.0712369
$425$	−2.19763e9	−1.38865
$426$	−9.13652e8	−0.572595
$427$	4.44067e8	0.276027
$428$	8.93666e8	0.550962
$429$	−1.46342e9	−0.894885
$430$	3.57986e8	0.217133
$431$	−3.17063e8	−0.190754	−0.0953772	−	0.995441i	$-0.530406\pi$
$431$	−3.17063e8	−0.190754	−0.0953772	+	0.995441i	$0.530406\pi$
$432$	3.97662e8	0.237313
$433$	1.27700e9	0.755936	0.377968	−	0.925819i	$-0.376623\pi$
$433$	1.27700e9	0.755936	0.377968	+	0.925819i	$0.376623\pi$
$434$	4.27551e8	0.251058
$435$	−1.75881e8	−0.102449
$436$	−1.23528e9	−0.713776
$437$	−6.15459e8	−0.352788
$438$	1.16628e9	0.663200
$439$	1.91425e7	0.0107988	0.00539938	−	0.999985i	$-0.498281\pi$
$439$	1.91425e7	0.0107988	0.00539938	+	0.999985i	$0.498281\pi$
$440$	4.30206e8	0.240764
$441$	9.13164e8	0.507006
$442$	2.75285e9	1.51637
$443$	−2.13064e9	−1.16439	−0.582194	−	0.813050i	$-0.697806\pi$
$443$	−2.13064e9	−1.16439	−0.582194	+	0.813050i	$0.697806\pi$
$444$	9.69224e8	0.525513
$445$	−1.40201e9	−0.754206
$446$	6.85807e8	0.366041
$447$	1.60684e9	0.850932
$448$	1.20298e8	0.0632097
$449$	−1.66049e9	−0.865715	−0.432857	−	0.901462i	$-0.642495\pi$
$449$	−1.66049e9	−0.865715	−0.432857	+	0.901462i	$0.642495\pi$
$450$	6.99648e8	0.361939
$451$	−1.65091e9	−0.847433
$452$	2.08083e8	0.105987
$453$	1.25105e8	0.0632312
$454$	−1.07203e8	−0.0537663
$455$	5.87838e8	0.292562
$456$	−9.27295e7	−0.0457974
$457$	3.14683e9	1.54229	0.771146	−	0.636658i	$-0.219684\pi$
$457$	3.14683e9	1.54229	0.771146	+	0.636658i	$0.219684\pi$
$458$	−1.24682e8	−0.0606421
$459$	3.63446e9	1.75426
$460$	−8.00297e8	−0.383353
$461$	−1.38484e9	−0.658332	−0.329166	−	0.944272i	$-0.606768\pi$
$461$	−1.38484e9	−0.658332	−0.329166	+	0.944272i	$0.606768\pi$
$462$	−5.84477e8	−0.275753
$463$	−1.19885e9	−0.561345	−0.280673	−	0.959804i	$-0.590558\pi$
$463$	−1.19885e9	−0.561345	−0.280673	+	0.959804i	$0.590558\pi$
$464$	1.95776e8	0.0909800
$465$	−4.28550e8	−0.197659
$466$	1.77998e9	0.814825
$467$	4.00489e9	1.81962	0.909812	−	0.415020i	$-0.136225\pi$
$467$	4.00489e9	1.81962	0.909812	+	0.415020i	$0.136225\pi$
$468$	−8.76411e8	−0.395228
$469$	6.89134e8	0.308460
$470$	−8.54515e8	−0.379645
$471$	−2.06585e9	−0.911014
$472$	4.49889e8	0.196929
$473$	1.93605e9	0.841205
$474$	−4.25635e8	−0.183575
$475$	−4.02652e8	−0.172386
$476$	1.09947e9	0.467260
$477$	−3.25339e8	−0.137253
$478$	−1.86056e9	−0.779197
$479$	1.88332e9	0.782980	0.391490	−	0.920182i	$-0.371960\pi$
$479$	1.88332e9	0.782980	0.391490	+	0.920182i	$0.371960\pi$
$480$	−1.20579e8	−0.0497652
$481$	−5.27187e9	−2.16002
$482$	1.15334e9	0.469131
$483$	1.08728e9	0.439064
$484$	1.07945e9	0.432756
$485$	8.19845e8	0.326314
$486$	−1.84533e9	−0.729201
$487$	4.10893e9	1.61205	0.806024	−	0.591883i	$-0.201615\pi$
$487$	4.10893e9	1.61205	0.806024	+	0.591883i	$0.201615\pi$
$488$	4.95452e8	0.192989
$489$	−1.80331e9	−0.697413
$490$	−6.83363e8	−0.262401
$491$	−6.03281e8	−0.230003	−0.115002	−	0.993365i	$-0.536687\pi$
$491$	−6.03281e8	−0.230003	−0.115002	+	0.993365i	$0.536687\pi$
$492$	4.62719e8	0.175162
$493$	1.78931e9	0.672543
$494$	5.04381e8	0.188241
$495$	−1.25178e9	−0.463883
$496$	4.77025e8	0.175531
$497$	1.98482e9	0.725226
$498$	4.57641e8	0.166044
$499$	−3.14142e9	−1.13181	−0.565906	−	0.824470i	$-0.691473\pi$
$499$	−3.14142e9	−1.13181	−0.565906	+	0.824470i	$0.691473\pi$
$500$	−1.22037e9	−0.436613
$501$	1.64173e9	0.583270
$502$	−3.20805e9	−1.13182
$503$	−3.26721e9	−1.14469	−0.572347	−	0.820012i	$-0.693967\pi$
$503$	−3.26721e9	−1.14469	−0.572347	+	0.820012i	$0.693967\pi$
$504$	−3.50032e8	−0.121787
$505$	1.36722e8	0.0472409
$506$	−4.32814e9	−1.48517
$507$	−5.74135e8	−0.195653
$508$	1.21718e9	0.411939
$509$	−4.74552e9	−1.59504	−0.797519	−	0.603293i	$-0.793855\pi$
$509$	−4.74552e9	−1.59504	−0.797519	+	0.603293i	$0.793855\pi$
$510$	−1.10204e9	−0.367875
$511$	−2.53363e9	−0.839984
$512$	1.34218e8	0.0441942
$513$	6.65909e8	0.217773
$514$	−1.61263e9	−0.523798
$515$	−1.24115e9	−0.400405
$516$	−5.42637e8	−0.173874
$517$	−4.62136e9	−1.47080
$518$	−2.10554e9	−0.665595
$519$	1.64285e9	0.515837
$520$	6.55859e8	0.204550
$521$	−3.01235e9	−0.933196	−0.466598	−	0.884469i	$-0.654521\pi$
$521$	−3.01235e9	−0.933196	−0.466598	+	0.884469i	$0.654521\pi$
$522$	−5.69651e8	−0.175292
$523$	2.87061e9	0.877442	0.438721	−	0.898623i	$-0.355432\pi$
$523$	2.87061e9	0.877442	0.438721	+	0.898623i	$0.355432\pi$
$524$	−3.11704e9	−0.946419
$525$	7.11334e8	0.214544
$526$	1.86571e9	0.558976
$527$	4.35980e9	1.29757
$528$	−6.52109e8	−0.192798
$529$	4.64667e9	1.36473
$530$	2.43466e8	0.0710351
$531$	−1.30905e9	−0.379424
$532$	2.01446e8	0.0580052
$533$	−2.51685e9	−0.719966
$534$	2.12517e9	0.603948
$535$	1.94594e9	0.549402
$536$	7.68877e8	0.215665
$537$	2.47814e9	0.690584
$538$	2.37896e9	0.658642
$539$	−3.69574e9	−1.01658
$540$	8.65899e8	0.236640
$541$	5.39646e8	0.146527	0.0732636	−	0.997313i	$-0.476659\pi$
$541$	5.39646e8	0.146527	0.0732636	+	0.997313i	$0.476659\pi$
$542$	1.71025e7	0.00461385
$543$	−1.22384e9	−0.328039
$544$	1.22669e9	0.326693
$545$	−2.68979e9	−0.711754
$546$	−8.91049e8	−0.234276
$547$	5.82950e9	1.52292	0.761458	−	0.648214i	$-0.224484\pi$
$547$	5.82950e9	1.52292	0.761458	+	0.648214i	$0.224484\pi$
$548$	−1.02255e9	−0.265431
$549$	−1.44162e9	−0.371833
$550$	−2.83161e9	−0.725710
$551$	3.27838e8	0.0834890
$552$	1.21310e9	0.306979
$553$	9.24651e8	0.232509
$554$	−3.67819e9	−0.919073
$555$	2.11046e9	0.524025
$556$	−1.54867e8	−0.0382118
$557$	1.77149e8	0.0434357	0.0217178	−	0.999764i	$-0.493086\pi$
$557$	1.77149e8	0.0434357	0.0217178	+	0.999764i	$0.493086\pi$
$558$	−1.38800e9	−0.338198
$559$	2.95155e9	0.714676
$560$	2.61945e8	0.0630307
$561$	−5.96000e9	−1.42520
$562$	−2.33531e9	−0.554968
$563$	6.34767e9	1.49912	0.749558	−	0.661939i	$-0.230266\pi$
$563$	6.34767e9	1.49912	0.749558	+	0.661939i	$0.230266\pi$
$564$	1.29528e9	0.304009
$565$	4.53095e8	0.105686
$566$	6.32742e8	0.146680
$567$	3.18746e8	0.0734352
$568$	2.21449e9	0.507054
$569$	5.96846e9	1.35822	0.679109	−	0.734038i	$-0.262366\pi$
$569$	5.96846e9	1.35822	0.679109	+	0.734038i	$0.262366\pi$
$570$	−2.01916e8	−0.0456677
$571$	−3.09454e9	−0.695616	−0.347808	−	0.937566i	$-0.613074\pi$
$571$	−3.09454e9	−0.695616	−0.347808	+	0.937566i	$0.613074\pi$
$572$	3.54700e9	0.792455
$573$	−9.19002e8	−0.204068
$574$	−1.00521e9	−0.221853
$575$	5.26754e9	1.15550
$576$	−3.90535e8	−0.0851494
$577$	−1.61147e8	−0.0349226	−0.0174613	−	0.999848i	$-0.505558\pi$
$577$	−1.61147e8	−0.0349226	−0.0174613	+	0.999848i	$0.505558\pi$
$578$	7.92873e9	1.70788
$579$	1.98520e9	0.425040
$580$	4.26297e8	0.0907223
$581$	−9.94181e8	−0.210305
$582$	−1.24273e9	−0.261303
$583$	1.31671e9	0.275200
$584$	−2.82681e9	−0.587289
$585$	−1.90836e9	−0.394108
$586$	9.81069e8	0.201399
$587$	−1.39327e8	−0.0284316	−0.0142158	−	0.999899i	$-0.504525\pi$
$587$	−1.39327e8	−0.0284316	−0.0142158	+	0.999899i	$0.504525\pi$
$588$	1.03585e9	0.210124
$589$	7.98806e8	0.161079
$590$	9.79623e8	0.196371
$591$	3.82795e9	0.762798
$592$	−2.34918e9	−0.465362
$593$	4.95725e9	0.976225	0.488112	−	0.872781i	$-0.337686\pi$
$593$	4.95725e9	0.976225	0.488112	+	0.872781i	$0.337686\pi$
$594$	4.68293e9	0.916779
$595$	2.39407e9	0.465936
$596$	−3.89461e9	−0.753533
$597$	−4.40690e9	−0.847662
$598$	−6.59836e9	−1.26177
$599$	1.28406e9	0.244113	0.122057	−	0.992523i	$-0.461051\pi$
$599$	1.28406e9	0.244113	0.122057	+	0.992523i	$0.461051\pi$
$600$	7.93645e8	0.150002
$601$	−3.91849e8	−0.0736305	−0.0368152	−	0.999322i	$-0.511721\pi$
$601$	−3.91849e8	−0.0736305	−0.0368152	+	0.999322i	$0.511721\pi$
$602$	1.17883e9	0.220223
$603$	−2.23721e9	−0.415524
$604$	−3.03227e8	−0.0559937
$605$	2.35047e9	0.431530
$606$	−2.07244e8	−0.0378293
$607$	2.53376e9	0.459838	0.229919	−	0.973210i	$-0.426154\pi$
$607$	2.53376e9	0.459838	0.229919	+	0.973210i	$0.426154\pi$
$608$	2.24756e8	0.0405554
$609$	−5.79166e8	−0.103906
$610$	1.07883e9	0.192442
$611$	−7.04537e9	−1.24957
$612$	−3.56932e9	−0.629442
$613$	−6.77289e9	−1.18758	−0.593789	−	0.804621i	$-0.702369\pi$
$613$	−6.77289e9	−1.18758	−0.593789	+	0.804621i	$0.702369\pi$
$614$	5.82631e9	1.01579
$615$	1.00756e9	0.174666
$616$	1.41664e9	0.244190
$617$	9.94599e9	1.70471	0.852354	−	0.522965i	$-0.175174\pi$
$617$	9.94599e9	1.70471	0.852354	+	0.522965i	$0.175174\pi$
$618$	1.88134e9	0.320633
$619$	7.89957e9	1.33871	0.669354	−	0.742944i	$-0.266571\pi$
$619$	7.89957e9	1.33871	0.669354	+	0.742944i	$0.266571\pi$
$620$	1.03871e9	0.175034
$621$	−8.71148e9	−1.45973
$622$	−3.22019e9	−0.536557
$623$	−4.61673e9	−0.764938
$624$	−9.94156e8	−0.163798
$625$	1.92894e9	0.316037
$626$	−3.41649e9	−0.556635
$627$	−1.09200e9	−0.176923
$628$	5.00716e9	0.806738
$629$	−2.14705e10	−3.44006
$630$	−7.62185e8	−0.121442
$631$	−3.92434e9	−0.621819	−0.310909	−	0.950440i	$-0.600634\pi$
$631$	−3.92434e9	−0.621819	−0.310909	+	0.950440i	$0.600634\pi$
$632$	1.03165e9	0.162563
$633$	−3.65128e9	−0.572179
$634$	7.81248e9	1.21752
$635$	2.65038e9	0.410772
$636$	−3.69048e8	−0.0568830
$637$	−5.63425e9	−0.863670
$638$	2.30548e9	0.351471
$639$	−6.44353e9	−0.976947
$640$	2.92256e8	0.0440690
$641$	9.32921e9	1.39908	0.699538	−	0.714595i	$-0.253389\pi$
$641$	9.32921e9	1.39908	0.699538	+	0.714595i	$0.253389\pi$
$642$	−2.94966e9	−0.439946
$643$	4.47191e9	0.663369	0.331684	−	0.943390i	$-0.392383\pi$
$643$	4.47191e9	0.663369	0.331684	+	0.943390i	$0.392383\pi$
$644$	−2.63533e9	−0.388808
$645$	−1.18158e9	−0.173382
$646$	2.05417e9	0.299794
$647$	−8.34699e8	−0.121162	−0.0605808	−	0.998163i	$-0.519295\pi$
$647$	−8.34699e8	−0.121162	−0.0605808	+	0.998163i	$0.519295\pi$
$648$	3.55630e8	0.0513435
$649$	5.29796e9	0.760769
$650$	−4.31685e9	−0.616553
$651$	−1.41119e9	−0.200471
$652$	4.37083e9	0.617586
$653$	−3.33630e9	−0.468887	−0.234444	−	0.972130i	$-0.575327\pi$
$653$	−3.33630e9	−0.468887	−0.234444	+	0.972130i	$0.575327\pi$
$654$	4.07719e9	0.569953
$655$	−6.78728e9	−0.943738
$656$	−1.12153e9	−0.155112
$657$	8.22521e9	1.13154
$658$	−2.81387e9	−0.385046
$659$	−2.37169e9	−0.322819	−0.161410	−	0.986887i	$-0.551604\pi$
$659$	−2.37169e9	−0.322819	−0.161410	+	0.986887i	$0.551604\pi$
$660$	−1.41995e9	−0.192252
$661$	−1.77417e9	−0.238941	−0.119471	−	0.992838i	$-0.538120\pi$
$661$	−1.77417e9	−0.238941	−0.119471	+	0.992838i	$0.538120\pi$
$662$	−7.65300e9	−1.02525
$663$	−9.08616e9	−1.21083
$664$	−1.10922e9	−0.147038
$665$	4.38643e8	0.0578409
$666$	6.83545e9	0.896618
$667$	−4.28881e9	−0.559624
$668$	−3.97919e9	−0.516508
$669$	−2.26359e9	−0.292286
$670$	1.67421e9	0.215054
$671$	5.83451e9	0.745548
$672$	−3.97058e8	−0.0504733
$673$	1.46957e10	1.85839	0.929197	−	0.369584i	$-0.120500\pi$
$673$	1.46957e10	1.85839	0.929197	+	0.369584i	$0.120500\pi$
$674$	9.15523e9	1.15175
$675$	−5.69932e9	−0.713280
$676$	1.39158e9	0.173258
$677$	1.10015e10	1.36268	0.681339	−	0.731968i	$-0.261398\pi$
$677$	1.10015e10	1.36268	0.681339	+	0.731968i	$0.261398\pi$
$678$	−6.86804e8	−0.0846309
$679$	2.69970e9	0.330957
$680$	2.67109e9	0.325767
$681$	3.53837e8	0.0429327
$682$	5.61751e9	0.678108
$683$	−1.10176e10	−1.32317	−0.661585	−	0.749870i	$-0.730116\pi$
$683$	−1.10176e10	−1.32317	−0.661585	+	0.749870i	$0.730116\pi$
$684$	−6.53975e8	−0.0781384
$685$	−2.22658e9	−0.264680
$686$	−5.27366e9	−0.623703
$687$	4.11529e8	0.0484230
$688$	1.31523e9	0.153972
$689$	2.00735e9	0.233806
$690$	2.64149e9	0.306109
$691$	−3.13667e9	−0.361656	−0.180828	−	0.983515i	$-0.557878\pi$
$691$	−3.13667e9	−0.361656	−0.180828	+	0.983515i	$0.557878\pi$
$692$	−3.98191e9	−0.456793
$693$	−4.12202e9	−0.470483
$694$	4.03328e9	0.458036
$695$	−3.37219e8	−0.0381035
$696$	−6.46183e8	−0.0726480
$697$	−1.02503e10	−1.14662
$698$	−4.72730e9	−0.526161
$699$	−5.87506e9	−0.650642
$700$	−1.72412e9	−0.189987
$701$	−5.37036e9	−0.588831	−0.294415	−	0.955678i	$-0.595125\pi$
$701$	−5.37036e9	−0.588831	−0.294415	+	0.955678i	$0.595125\pi$
$702$	7.13923e9	0.778882
$703$	−3.93385e9	−0.427046
$704$	1.58057e9	0.170730
$705$	2.82044e9	0.303148
$706$	−4.50244e9	−0.481539
$707$	4.50217e8	0.0479131
$708$	−1.48492e9	−0.157248
$709$	6.70949e9	0.707013	0.353507	−	0.935432i	$-0.384989\pi$
$709$	6.70949e9	0.707013	0.353507	+	0.935432i	$0.384989\pi$
$710$	4.82200e9	0.505618
$711$	−3.00179e9	−0.313211
$712$	−5.15095e9	−0.534819
$713$	−1.04501e10	−1.07971
$714$	−3.62894e9	−0.373109
$715$	7.72350e9	0.790211
$716$	−6.00647e9	−0.611539
$717$	6.14104e9	0.622192
$718$	4.52227e9	0.455954
$719$	1.00930e9	0.101267	0.0506337	−	0.998717i	$-0.483876\pi$
$719$	1.00930e9	0.101267	0.0506337	+	0.998717i	$0.483876\pi$
$720$	−8.50381e8	−0.0849082
$721$	−4.08703e9	−0.406102
$722$	3.76367e8	0.0372161
$723$	−3.80676e9	−0.374603
$724$	2.96632e9	0.290491
$725$	−2.80587e9	−0.273455
$726$	−3.56286e9	−0.345558
$727$	1.40159e10	1.35285	0.676426	−	0.736510i	$-0.263528\pi$
$727$	1.40159e10	1.35285	0.676426	+	0.736510i	$0.263528\pi$
$728$	2.15971e9	0.207460
$729$	4.57169e9	0.437049
$730$	−6.15531e9	−0.585626
$731$	1.20207e10	1.13820
$732$	−1.63530e9	−0.154102
$733$	1.41404e10	1.32616	0.663081	−	0.748548i	$-0.269249\pi$
$733$	1.41404e10	1.32616	0.663081	+	0.748548i	$0.269249\pi$
$734$	−1.20708e10	−1.12668
$735$	2.25553e9	0.209528
$736$	−2.94028e9	−0.271842
$737$	9.05441e9	0.833151
$738$	3.26332e9	0.298857
$739$	−5.48432e9	−0.499882	−0.249941	−	0.968261i	$-0.580411\pi$
$739$	−5.48432e9	−0.499882	−0.249941	+	0.968261i	$0.580411\pi$
$740$	−5.11529e9	−0.464044
$741$	−1.66477e9	−0.150311
$742$	8.01720e8	0.0720458
$743$	2.22905e10	1.99369	0.996847	−	0.0793449i	$-0.0252828\pi$
$743$	2.22905e10	1.99369	0.996847	+	0.0793449i	$0.0252828\pi$
$744$	−1.57448e9	−0.140163
$745$	−8.48042e9	−0.751399
$746$	−5.51581e9	−0.486434
$747$	3.22752e9	0.283300
$748$	1.44457e10	1.26207
$749$	6.40785e9	0.557219
$750$	4.02800e9	0.348638
$751$	−1.79825e10	−1.54921	−0.774607	−	0.632443i	$-0.782052\pi$
$751$	−1.79825e10	−1.54921	−0.774607	+	0.632443i	$0.782052\pi$
$752$	−3.13947e9	−0.269212
$753$	1.05886e10	0.903764
$754$	3.51477e9	0.298605
$755$	−6.60270e8	−0.0558351
$756$	2.85135e9	0.240007
$757$	9.03342e9	0.756862	0.378431	−	0.925629i	$-0.376464\pi$
$757$	9.03342e9	0.756862	0.378431	+	0.925629i	$0.376464\pi$
$758$	7.52963e8	0.0627959
$759$	1.42856e10	1.18591
$760$	4.89400e8	0.0404405
$761$	−1.58415e10	−1.30301	−0.651506	−	0.758643i	$-0.725863\pi$
$761$	−1.58415e10	−1.30301	−0.651506	+	0.758643i	$0.725863\pi$
$762$	−4.01747e9	−0.328935
$763$	−8.85730e9	−0.721881
$764$	2.22746e9	0.180710
$765$	−7.77211e9	−0.627659
$766$	8.86953e9	0.713017
$767$	8.07687e9	0.646338
$768$	−4.43003e8	−0.0352893
$769$	1.07736e10	0.854316	0.427158	−	0.904177i	$-0.359515\pi$
$769$	1.07736e10	0.854316	0.427158	+	0.904177i	$0.359515\pi$
$770$	3.08471e9	0.243498
$771$	5.32270e9	0.418256
$772$	−4.81169e9	−0.376389
$773$	9.38402e9	0.730736	0.365368	−	0.930863i	$-0.380943\pi$
$773$	9.38402e9	0.730736	0.365368	+	0.930863i	$0.380943\pi$
$774$	−3.82695e9	−0.296660
$775$	−6.83676e9	−0.527587
$776$	3.01209e9	0.231394
$777$	6.94962e9	0.531481
$778$	−5.26461e9	−0.400809
$779$	−1.87807e9	−0.142341
$780$	−2.16475e9	−0.163334
$781$	2.60782e10	1.95884
$782$	−2.68729e10	−2.00951
$783$	4.64037e9	0.345451
$784$	−2.51066e9	−0.186072
$785$	1.09030e10	0.804453
$786$	1.02882e10	0.755720
$787$	8.04728e9	0.588488	0.294244	−	0.955730i	$-0.404932\pi$
$787$	8.04728e9	0.588488	0.294244	+	0.955730i	$0.404932\pi$
$788$	−9.27810e9	−0.675487
$789$	−6.15801e9	−0.446345
$790$	2.24638e9	0.162102
$791$	1.49201e9	0.107190
$792$	−4.59900e9	−0.328947
$793$	8.89485e9	0.633407
$794$	−4.89251e8	−0.0346865
$795$	−8.03592e8	−0.0567219
$796$	1.06813e10	0.750638
$797$	−1.28778e10	−0.901028	−0.450514	−	0.892769i	$-0.648759\pi$
$797$	−1.28778e10	−0.901028	−0.450514	+	0.892769i	$0.648759\pi$
$798$	−6.64898e8	−0.0463175
$799$	−2.86934e10	−1.99007
$800$	−1.92362e9	−0.132833
$801$	1.49878e10	1.03044
$802$	6.72640e9	0.460440
$803$	−3.32890e10	−2.26880
$804$	−2.53778e9	−0.172210
$805$	−5.73837e9	−0.387706
$806$	8.56403e9	0.576110
$807$	−7.85208e9	−0.525929
$808$	5.02314e8	0.0334993
$809$	2.19480e10	1.45739	0.728693	−	0.684841i	$-0.240128\pi$
$809$	2.19480e10	1.45739	0.728693	+	0.684841i	$0.240128\pi$
$810$	7.74375e8	0.0511981
$811$	−1.52243e10	−1.00222	−0.501112	−	0.865383i	$-0.667075\pi$
$811$	−1.52243e10	−1.00222	−0.501112	+	0.865383i	$0.667075\pi$
$812$	1.40377e9	0.0920131
$813$	−5.64492e7	−0.00368418
$814$	−2.76643e10	−1.79777
$815$	9.51738e9	0.615837
$816$	−4.04886e9	−0.260866
$817$	2.20244e9	0.141295
$818$	1.07066e10	0.683937
$819$	−6.28412e9	−0.399716
$820$	−2.44210e9	−0.154673
$821$	−9.51841e9	−0.600293	−0.300146	−	0.953893i	$-0.597036\pi$
$821$	−9.51841e9	−0.600293	−0.300146	+	0.953893i	$0.597036\pi$
$822$	3.37506e9	0.211948
$823$	−2.78727e10	−1.74293	−0.871463	−	0.490461i	$-0.836828\pi$
$823$	−2.78727e10	−1.74293	−0.871463	+	0.490461i	$0.836828\pi$
$824$	−4.55996e9	−0.283933
$825$	9.34609e9	0.579483
$826$	3.22584e9	0.199165
$827$	−1.16769e10	−0.717892	−0.358946	−	0.933358i	$-0.616864\pi$
$827$	−1.16769e10	−0.717892	−0.358946	+	0.933358i	$0.616864\pi$
$828$	8.55536e9	0.523760
$829$	−1.84728e10	−1.12614	−0.563068	−	0.826411i	$-0.690379\pi$
$829$	−1.84728e10	−1.12614	−0.563068	+	0.826411i	$0.690379\pi$
$830$	−2.41530e9	−0.146622
$831$	1.21404e10	0.733885
$832$	2.40961e9	0.145049
$833$	−2.29464e10	−1.37549
$834$	5.11159e8	0.0305123
$835$	−8.66459e9	−0.515045
$836$	2.64676e9	0.156672
$837$	1.13067e10	0.666493
$838$	−3.73504e8	−0.0219251
$839$	1.20774e10	0.706006	0.353003	−	0.935622i	$-0.385161\pi$
$839$	1.20774e10	0.706006	0.353003	+	0.935622i	$0.385161\pi$
$840$	−8.64584e8	−0.0503303
$841$	−1.49653e10	−0.867562
$842$	9.17495e9	0.529678
$843$	7.70801e9	0.443145
$844$	8.84988e9	0.506686
$845$	3.03012e9	0.172767
$846$	9.13496e9	0.518693
$847$	7.73996e9	0.437670
$848$	8.94490e8	0.0503721
$849$	−2.08845e9	−0.117124
$850$	−1.75811e10	−0.981926
$851$	5.14630e10	2.86248
$852$	−7.30922e9	−0.404886
$853$	−2.56649e10	−1.41585	−0.707925	−	0.706287i	$-0.750369\pi$
$853$	−2.56649e10	−1.41585	−0.707925	+	0.706287i	$0.750369\pi$
$854$	3.55253e9	0.195180
$855$	−1.42401e9	−0.0779171
$856$	7.14933e9	0.389589
$857$	−1.51698e10	−0.823280	−0.411640	−	0.911347i	$-0.635044\pi$
$857$	−1.51698e10	−0.823280	−0.411640	+	0.911347i	$0.635044\pi$
$858$	−1.17073e10	−0.632779
$859$	−1.64255e10	−0.884186	−0.442093	−	0.896969i	$-0.645764\pi$
$859$	−1.64255e10	−0.884186	−0.442093	+	0.896969i	$0.645764\pi$
$860$	2.86389e9	0.153536
$861$	3.31783e9	0.177151
$862$	−2.53650e9	−0.134884
$863$	−3.33682e10	−1.76724	−0.883619	−	0.468206i	$-0.844901\pi$
$863$	−3.33682e10	−1.76724	−0.883619	+	0.468206i	$0.844901\pi$
$864$	3.18129e9	0.167805
$865$	−8.67050e9	−0.455499
$866$	1.02160e10	0.534527
$867$	−2.61698e10	−1.36375
$868$	3.42041e9	0.177525
$869$	1.21488e10	0.628007
$870$	−1.40705e9	−0.0724422
$871$	1.38037e10	0.707833
$872$	−9.88222e9	−0.504716
$873$	−8.76433e9	−0.445829
$874$	−4.92367e9	−0.249459
$875$	−8.75043e9	−0.441571
$876$	9.33026e9	0.468953
$877$	−6.13596e9	−0.307174	−0.153587	−	0.988135i	$-0.549082\pi$
$877$	−6.13596e9	−0.307174	−0.153587	+	0.988135i	$0.549082\pi$
$878$	1.53140e8	0.00763587
$879$	−3.23815e9	−0.160818
$880$	3.44165e9	0.170246
$881$	2.76439e10	1.36202	0.681011	−	0.732273i	$-0.261541\pi$
$881$	2.76439e10	1.36202	0.681011	+	0.732273i	$0.261541\pi$
$882$	7.30531e9	0.358508
$883$	2.94957e10	1.44177	0.720884	−	0.693055i	$-0.243736\pi$
$883$	2.94957e10	1.44177	0.720884	+	0.693055i	$0.243736\pi$
$884$	2.20228e10	1.07224
$885$	−3.23337e9	−0.156803
$886$	−1.70451e10	−0.823346
$887$	−3.73921e10	−1.79906	−0.899532	−	0.436854i	$-0.856093\pi$
$887$	−3.73921e10	−1.79906	−0.899532	+	0.436854i	$0.856093\pi$
$888$	7.75379e9	0.371594
$889$	8.72755e9	0.416616
$890$	−1.12161e10	−0.533304
$891$	4.18795e9	0.198349
$892$	5.48645e9	0.258830
$893$	−5.25723e9	−0.247046
$894$	1.28547e10	0.601700
$895$	−1.30790e10	−0.609807
$896$	9.62381e8	0.0446960
$897$	2.17787e10	1.00753
$898$	−1.32839e10	−0.612153
$899$	5.56647e9	0.255517
$900$	5.59718e9	0.255930
$901$	8.17525e9	0.372361
$902$	−1.32073e10	−0.599226
$903$	−3.89087e9	−0.175849
$904$	1.66466e9	0.0749439
$905$	6.45908e9	0.289668
$906$	1.00084e9	0.0447112
$907$	−8.53150e9	−0.379664	−0.189832	−	0.981817i	$-0.560794\pi$
$907$	−8.53150e9	−0.379664	−0.189832	+	0.981817i	$0.560794\pi$
$908$	−8.57621e8	−0.0380185
$909$	−1.46159e9	−0.0645434
$910$	4.70271e9	0.206873
$911$	3.79368e10	1.66244	0.831221	−	0.555943i	$-0.187643\pi$
$911$	3.79368e10	1.66244	0.831221	+	0.555943i	$0.187643\pi$
$912$	−7.41836e8	−0.0323837
$913$	−1.30624e10	−0.568033
$914$	2.51746e10	1.09057
$915$	−3.56083e9	−0.153666
$916$	−9.97455e8	−0.0428804
$917$	−2.23501e10	−0.957166
$918$	2.90757e10	1.24045
$919$	−6.78946e9	−0.288556	−0.144278	−	0.989537i	$-0.546086\pi$
$919$	−6.78946e9	−0.288556	−0.144278	+	0.989537i	$0.546086\pi$
$920$	−6.40238e9	−0.271072
$921$	−1.92305e10	−0.811114
$922$	−1.10787e10	−0.465511
$923$	3.97568e10	1.66420
$924$	−4.67582e9	−0.194987
$925$	3.36687e10	1.39872
$926$	−9.59077e9	−0.396931
$927$	1.32682e10	0.547057
$928$	1.56621e9	0.0643326
$929$	1.41246e10	0.577991	0.288995	−	0.957330i	$-0.406679\pi$
$929$	1.41246e10	0.577991	0.288995	+	0.957330i	$0.406679\pi$
$930$	−3.42840e9	−0.139766
$931$	−4.20426e9	−0.170752
$932$	1.42398e10	0.576169
$933$	1.06287e10	0.428443
$934$	3.20391e10	1.28667
$935$	3.14552e10	1.25850
$936$	−7.01129e9	−0.279468
$937$	1.06701e9	0.0423719	0.0211860	−	0.999776i	$-0.493256\pi$
$937$	1.06701e9	0.0423719	0.0211860	+	0.999776i	$0.493256\pi$
$938$	5.51307e9	0.218114
$939$	1.12766e10	0.444476
$940$	−6.83612e9	−0.268449
$941$	7.26515e9	0.284237	0.142119	−	0.989850i	$-0.454609\pi$
$941$	7.26515e9	0.284237	0.142119	+	0.989850i	$0.454609\pi$
$942$	−1.65268e10	−0.644184
$943$	2.45690e10	0.954107
$944$	3.59911e9	0.139249
$945$	6.20875e9	0.239328
$946$	1.54884e10	0.594822
$947$	2.55705e10	0.978396	0.489198	−	0.872173i	$-0.337290\pi$
$947$	2.55705e10	0.978396	0.489198	+	0.872173i	$0.337290\pi$
$948$	−3.40508e9	−0.129807
$949$	−5.07498e10	−1.92754
$950$	−3.22122e9	−0.121895
$951$	−2.57861e10	−0.972196
$952$	8.79575e9	0.330403
$953$	3.15041e9	0.117908	0.0589538	−	0.998261i	$-0.481224\pi$
$953$	3.15041e9	0.117908	0.0589538	+	0.998261i	$0.481224\pi$
$954$	−2.60271e9	−0.0970524
$955$	4.85024e9	0.180198
$956$	−1.48845e10	−0.550975
$957$	−7.60955e9	−0.280652
$958$	1.50666e10	0.553650
$959$	−7.33198e9	−0.268446
$960$	−9.64629e8	−0.0351893
$961$	−1.39494e10	−0.507020
$962$	−4.21750e10	−1.52736
$963$	−2.08025e10	−0.750626
$964$	9.22675e9	0.331726
$965$	−1.04773e10	−0.375323
$966$	8.69826e9	0.310465
$967$	5.30314e9	0.188599	0.0942997	−	0.995544i	$-0.469939\pi$
$967$	5.30314e9	0.188599	0.0942997	+	0.995544i	$0.469939\pi$
$968$	8.63558e9	0.306004
$969$	−6.78006e9	−0.239387
$970$	6.55876e9	0.230739
$971$	−2.06978e9	−0.0725532	−0.0362766	−	0.999342i	$-0.511550\pi$
$971$	−2.06978e9	−0.0725532	−0.0362766	+	0.999342i	$0.511550\pi$
$972$	−1.47627e10	−0.515623
$973$	−1.11044e9	−0.0386457
$974$	3.28715e10	1.13989
$975$	1.42483e10	0.492320
$976$	3.96361e9	0.136464
$977$	9.60317e9	0.329446	0.164723	−	0.986340i	$-0.447327\pi$
$977$	9.60317e9	0.329446	0.164723	+	0.986340i	$0.447327\pi$
$978$	−1.44265e10	−0.493145
$979$	−6.06583e10	−2.06610
$980$	−5.46691e9	−0.185545
$981$	2.87544e10	0.972441
$982$	−4.82625e9	−0.162637
$983$	−7.75193e9	−0.260299	−0.130150	−	0.991494i	$-0.541546\pi$
$983$	−7.75193e9	−0.260299	−0.130150	+	0.991494i	$0.541546\pi$
$984$	3.70175e9	0.123858
$985$	−2.02028e10	−0.673574
$986$	1.43144e10	0.475560
$987$	9.28754e9	0.307461
$988$	4.03504e9	0.133106
$989$	−2.88125e10	−0.947096
$990$	−1.00142e10	−0.328015
$991$	6.42831e9	0.209816	0.104908	−	0.994482i	$-0.466545\pi$
$991$	6.42831e9	0.209816	0.104908	+	0.994482i	$0.466545\pi$
$992$	3.81620e9	0.124119
$993$	2.52597e10	0.818666
$994$	1.58786e10	0.512812
$995$	2.32584e10	0.748512
$996$	3.66113e9	0.117411
$997$	9.58448e9	0.306292	0.153146	−	0.988204i	$-0.451060\pi$
$997$	9.58448e9	0.306292	0.153146	+	0.988204i	$0.451060\pi$
$998$	−2.51314e10	−0.800311
$999$	−5.56815e10	−1.76698

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.8.a.e.1.2	✓	4
3.2	odd	2		342.8.a.o.1.2		4
4.3	odd	2		304.8.a.e.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.a.e.1.2	✓	4	1.1	even	1	trivial
304.8.a.e.1.3		4	4.3	odd	2
342.8.a.o.1.2		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	38.a (trivial)

\(f(q)\)	\(=\)	\(q+8.00000 q^{2} -26.4051 q^{3} +64.0000 q^{4} +139.358 q^{5} -211.240 q^{6} +458.899 q^{7} +512.000 q^{8} -1489.77 q^{9} +O(q^{10})\)	\(q+8.00000 q^{2} -26.4051 q^{3} +64.0000 q^{4} +139.358 q^{5} -211.240 q^{6} +458.899 q^{7} +512.000 q^{8} -1489.77 q^{9} +1114.87 q^{10} +6029.39 q^{11} -1689.92 q^{12} +9191.95 q^{13} +3671.19 q^{14} -3679.77 q^{15} +4096.00 q^{16} +37435.7 q^{17} -11918.2 q^{18} +6859.00 q^{19} +8918.94 q^{20} -12117.3 q^{21} +48235.1 q^{22} -89730.1 q^{23} -13519.4 q^{24} -58704.2 q^{25} +73535.6 q^{26} +97085.4 q^{27} +29369.5 q^{28} +47796.8 q^{29} -29438.1 q^{30} +116461. q^{31} +32768.0 q^{32} -159206. q^{33} +299485. q^{34} +63951.4 q^{35} -95345.5 q^{36} -573531. q^{37} +54872.0 q^{38} -242714. q^{39} +71351.5 q^{40} -273810. q^{41} -96938.0 q^{42} +321102. q^{43} +385881. q^{44} -207612. q^{45} -717841. q^{46} -766472. q^{47} -108155. q^{48} -612955. q^{49} -469634. q^{50} -988491. q^{51} +588285. q^{52} +218381. q^{53} +776683. q^{54} +840246. q^{55} +234956. q^{56} -181112. q^{57} +382374. q^{58} +878690. q^{59} -235505. q^{60} +967679. q^{61} +931689. q^{62} -683655. q^{63} +262144. q^{64} +1.28098e6 q^{65} -1.27365e6 q^{66} +1.50171e6 q^{67} +2.39588e6 q^{68} +2.36933e6 q^{69} +511612. q^{70} +4.32518e6 q^{71} -762764. q^{72} -5.52112e6 q^{73} -4.58825e6 q^{74} +1.55009e6 q^{75} +438976. q^{76} +2.76688e6 q^{77} -1.94171e6 q^{78} +2.01493e6 q^{79} +570812. q^{80} +694589. q^{81} -2.19048e6 q^{82} -2.16645e6 q^{83} -775504. q^{84} +5.21698e6 q^{85} +2.56881e6 q^{86} -1.26208e6 q^{87} +3.08705e6 q^{88} -1.00604e7 q^{89} -1.66090e6 q^{90} +4.21818e6 q^{91} -5.74273e6 q^{92} -3.07516e6 q^{93} -6.13178e6 q^{94} +955859. q^{95} -865241. q^{96} +5.88299e6 q^{97} -4.90364e6 q^{98} -8.98242e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 32 q^{2} + 12 q^{3} + 256 q^{4} - 279 q^{5} + 96 q^{6} + 2485 q^{7} + 2048 q^{8} + 9482 q^{9}+O(q^{10}) \) 4 * q + 32 * q^2 + 12 * q^3 + 256 * q^4 - 279 * q^5 + 96 * q^6 + 2485 * q^7 + 2048 * q^8 + 9482 * q^9	\( 4 q + 32 q^{2} + 12 q^{3} + 256 q^{4} - 279 q^{5} + 96 q^{6} + 2485 q^{7} + 2048 q^{8} + 9482 q^{9} - 2232 q^{10} + 5269 q^{11} + 768 q^{12} + 5406 q^{13} + 19880 q^{14} + 26658 q^{15} + 16384 q^{16} + 22885 q^{17} + 75856 q^{18} + 27436 q^{19} - 17856 q^{20} + 2854 q^{21} + 42152 q^{22} + 3364 q^{23} + 6144 q^{24} + 112561 q^{25} + 43248 q^{26} - 220194 q^{27} + 159040 q^{28} - 122136 q^{29} + 213264 q^{30} + 225480 q^{31} + 131072 q^{32} + 176138 q^{33} + 183080 q^{34} - 785781 q^{35} + 606848 q^{36} + 154096 q^{37} + 219488 q^{38} - 1749220 q^{39} - 142848 q^{40} - 1054628 q^{41} + 22832 q^{42} - 840795 q^{43} + 337216 q^{44} - 4162563 q^{45} + 26912 q^{46} - 1021877 q^{47} + 49152 q^{48} - 621441 q^{49} + 900488 q^{50} + 724892 q^{51} + 345984 q^{52} - 326842 q^{53} - 1761552 q^{54} - 221553 q^{55} + 1272320 q^{56} + 82308 q^{57} - 977088 q^{58} + 421384 q^{59} + 1706112 q^{60} + 116825 q^{61} + 1803840 q^{62} + 10245825 q^{63} + 1048576 q^{64} + 4477428 q^{65} + 1409104 q^{66} + 5794566 q^{67} + 1464640 q^{68} - 2472196 q^{69} - 6286248 q^{70} + 10590626 q^{71} + 4854784 q^{72} + 3971389 q^{73} + 1232768 q^{74} - 3690042 q^{75} + 1755904 q^{76} + 5806573 q^{77} - 13993760 q^{78} + 5597800 q^{79} - 1142784 q^{80} + 20567744 q^{81} - 8437024 q^{82} + 4665800 q^{83} + 182656 q^{84} - 2014461 q^{85} - 6726360 q^{86} - 14449584 q^{87} + 2697728 q^{88} - 2794214 q^{89} - 33300504 q^{90} - 8827314 q^{91} + 215296 q^{92} - 43981204 q^{93} - 8175016 q^{94} - 1913661 q^{95} + 393216 q^{96} - 14445130 q^{97} - 4971528 q^{98} - 7940315 q^{99}+O(q^{100}) \) 4 * q + 32 * q^2 + 12 * q^3 + 256 * q^4 - 279 * q^5 + 96 * q^6 + 2485 * q^7 + 2048 * q^8 + 9482 * q^9 - 2232 * q^10 + 5269 * q^11 + 768 * q^12 + 5406 * q^13 + 19880 * q^14 + 26658 * q^15 + 16384 * q^16 + 22885 * q^17 + 75856 * q^18 + 27436 * q^19 - 17856 * q^20 + 2854 * q^21 + 42152 * q^22 + 3364 * q^23 + 6144 * q^24 + 112561 * q^25 + 43248 * q^26 - 220194 * q^27 + 159040 * q^28 - 122136 * q^29 + 213264 * q^30 + 225480 * q^31 + 131072 * q^32 + 176138 * q^33 + 183080 * q^34 - 785781 * q^35 + 606848 * q^36 + 154096 * q^37 + 219488 * q^38 - 1749220 * q^39 - 142848 * q^40 - 1054628 * q^41 + 22832 * q^42 - 840795 * q^43 + 337216 * q^44 - 4162563 * q^45 + 26912 * q^46 - 1021877 * q^47 + 49152 * q^48 - 621441 * q^49 + 900488 * q^50 + 724892 * q^51 + 345984 * q^52 - 326842 * q^53 - 1761552 * q^54 - 221553 * q^55 + 1272320 * q^56 + 82308 * q^57 - 977088 * q^58 + 421384 * q^59 + 1706112 * q^60 + 116825 * q^61 + 1803840 * q^62 + 10245825 * q^63 + 1048576 * q^64 + 4477428 * q^65 + 1409104 * q^66 + 5794566 * q^67 + 1464640 * q^68 - 2472196 * q^69 - 6286248 * q^70 + 10590626 * q^71 + 4854784 * q^72 + 3971389 * q^73 + 1232768 * q^74 - 3690042 * q^75 + 1755904 * q^76 + 5806573 * q^77 - 13993760 * q^78 + 5597800 * q^79 - 1142784 * q^80 + 20567744 * q^81 - 8437024 * q^82 + 4665800 * q^83 + 182656 * q^84 - 2014461 * q^85 - 6726360 * q^86 - 14449584 * q^87 + 2697728 * q^88 - 2794214 * q^89 - 33300504 * q^90 - 8827314 * q^91 + 215296 * q^92 - 43981204 * q^93 - 8175016 * q^94 - 1913661 * q^95 + 393216 * q^96 - 14445130 * q^97 - 4971528 * q^98 - 7940315 * q^99

Introduction

L-functions

Modular forms

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