LMFDB - Embedded newform 354.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [354,4,Mod(1,354)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(354, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("354.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 354 = 2 \cdot 3 \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	354.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:	$20.8866761420$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	354.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -13.6056 q^{5} +6.00000 q^{6} +4.72498 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -13.6056 q^{5} +6.00000 q^{6} +4.72498 q^{7} +8.00000 q^{8} +9.00000 q^{9} -27.2111 q^{10} -58.0278 q^{11} +12.0000 q^{12} -87.0555 q^{13} +9.44996 q^{14} -40.8167 q^{15} +16.0000 q^{16} -17.9083 q^{17} +18.0000 q^{18} -29.3028 q^{19} -54.4222 q^{20} +14.1749 q^{21} -116.056 q^{22} +172.369 q^{23} +24.0000 q^{24} +60.1110 q^{25} -174.111 q^{26} +27.0000 q^{27} +18.8999 q^{28} +41.9916 q^{29} -81.6333 q^{30} -287.902 q^{31} +32.0000 q^{32} -174.083 q^{33} -35.8167 q^{34} -64.2860 q^{35} +36.0000 q^{36} -22.1640 q^{37} -58.6056 q^{38} -261.167 q^{39} -108.844 q^{40} +321.463 q^{41} +28.3499 q^{42} -423.461 q^{43} -232.111 q^{44} -122.450 q^{45} +344.738 q^{46} -7.95798 q^{47} +48.0000 q^{48} -320.675 q^{49} +120.222 q^{50} -53.7250 q^{51} -348.222 q^{52} -177.005 q^{53} +54.0000 q^{54} +789.500 q^{55} +37.7998 q^{56} -87.9083 q^{57} +83.9832 q^{58} +59.0000 q^{59} -163.267 q^{60} +217.919 q^{61} -575.805 q^{62} +42.5248 q^{63} +64.0000 q^{64} +1184.44 q^{65} -348.167 q^{66} +561.394 q^{67} -71.6333 q^{68} +517.108 q^{69} -128.572 q^{70} +572.472 q^{71} +72.0000 q^{72} -242.636 q^{73} -44.3280 q^{74} +180.333 q^{75} -117.211 q^{76} -274.180 q^{77} -522.333 q^{78} -587.994 q^{79} -217.689 q^{80} +81.0000 q^{81} +642.927 q^{82} -887.596 q^{83} +56.6998 q^{84} +243.653 q^{85} -846.922 q^{86} +125.975 q^{87} -464.222 q^{88} -428.281 q^{89} -244.900 q^{90} -411.336 q^{91} +689.477 q^{92} -863.707 q^{93} -15.9160 q^{94} +398.680 q^{95} +96.0000 q^{96} +1051.67 q^{97} -641.349 q^{98} -522.250 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} - 20 q^{5} + 12 q^{6} - 23 q^{7} + 16 q^{8} + 18 q^{9}+O(q^{10}) $ 2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 - 20 * q^5 + 12 * q^6 - 23 * q^7 + 16 * q^8 + 18 * q^9	\( 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} - 20 q^{5} + 12 q^{6} - 23 q^{7} + 16 q^{8} + 18 q^{9} - 40 q^{10} - 80 q^{11} + 24 q^{12} - 102 q^{13} - 46 q^{14} - 60 q^{15} + 32 q^{16} - 25 q^{17} + 36 q^{18} - 55 q^{19} - 80 q^{20} - 69 q^{21} - 160 q^{22} - 5 q^{23} + 48 q^{24} - 24 q^{25} - 204 q^{26} + 54 q^{27} - 92 q^{28} - 35 q^{29} - 120 q^{30} - 53 q^{31} + 64 q^{32} - 240 q^{33} - 50 q^{34} + 113 q^{35} + 72 q^{36} - 221 q^{37} - 110 q^{38} - 306 q^{39} - 160 q^{40} - 89 q^{41} - 138 q^{42} - 508 q^{43} - 320 q^{44} - 180 q^{45} - 10 q^{46} + 579 q^{47} + 96 q^{48} + 105 q^{49} - 48 q^{50} - 75 q^{51} - 408 q^{52} + 432 q^{53} + 108 q^{54} + 930 q^{55} - 184 q^{56} - 165 q^{57} - 70 q^{58} + 118 q^{59} - 240 q^{60} + 151 q^{61} - 106 q^{62} - 207 q^{63} + 128 q^{64} + 1280 q^{65} - 480 q^{66} - 168 q^{67} - 100 q^{68} - 15 q^{69} + 226 q^{70} + 532 q^{71} + 144 q^{72} - 49 q^{73} - 442 q^{74} - 72 q^{75} - 220 q^{76} + 335 q^{77} - 612 q^{78} - 664 q^{79} - 320 q^{80} + 162 q^{81} - 178 q^{82} - 351 q^{83} - 276 q^{84} + 289 q^{85} - 1016 q^{86} - 105 q^{87} - 640 q^{88} - 1401 q^{89} - 360 q^{90} + 3 q^{91} - 20 q^{92} - 159 q^{93} + 1158 q^{94} + 563 q^{95} + 192 q^{96} + 452 q^{97} + 210 q^{98} - 720 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 - 20 * q^5 + 12 * q^6 - 23 * q^7 + 16 * q^8 + 18 * q^9 - 40 * q^10 - 80 * q^11 + 24 * q^12 - 102 * q^13 - 46 * q^14 - 60 * q^15 + 32 * q^16 - 25 * q^17 + 36 * q^18 - 55 * q^19 - 80 * q^20 - 69 * q^21 - 160 * q^22 - 5 * q^23 + 48 * q^24 - 24 * q^25 - 204 * q^26 + 54 * q^27 - 92 * q^28 - 35 * q^29 - 120 * q^30 - 53 * q^31 + 64 * q^32 - 240 * q^33 - 50 * q^34 + 113 * q^35 + 72 * q^36 - 221 * q^37 - 110 * q^38 - 306 * q^39 - 160 * q^40 - 89 * q^41 - 138 * q^42 - 508 * q^43 - 320 * q^44 - 180 * q^45 - 10 * q^46 + 579 * q^47 + 96 * q^48 + 105 * q^49 - 48 * q^50 - 75 * q^51 - 408 * q^52 + 432 * q^53 + 108 * q^54 + 930 * q^55 - 184 * q^56 - 165 * q^57 - 70 * q^58 + 118 * q^59 - 240 * q^60 + 151 * q^61 - 106 * q^62 - 207 * q^63 + 128 * q^64 + 1280 * q^65 - 480 * q^66 - 168 * q^67 - 100 * q^68 - 15 * q^69 + 226 * q^70 + 532 * q^71 + 144 * q^72 - 49 * q^73 - 442 * q^74 - 72 * q^75 - 220 * q^76 + 335 * q^77 - 612 * q^78 - 664 * q^79 - 320 * q^80 + 162 * q^81 - 178 * q^82 - 351 * q^83 - 276 * q^84 + 289 * q^85 - 1016 * q^86 - 105 * q^87 - 640 * q^88 - 1401 * q^89 - 360 * q^90 + 3 * q^91 - 20 * q^92 - 159 * q^93 + 1158 * q^94 + 563 * q^95 + 192 * q^96 + 452 * q^97 + 210 * q^98 - 720 * 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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.00000	0.500000
$5$	−13.6056	−1.21692	−0.608459	−	0.793586i	$-0.708212\pi$
$5$	−13.6056	−1.21692	−0.608459	+	0.793586i	$0.708212\pi$
$6$	6.00000	0.408248
$7$	4.72498	0.255125	0.127562	−	0.991831i	$-0.459285\pi$
$7$	4.72498	0.255125	0.127562	+	0.991831i	$0.459285\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−27.2111	−0.860491
$11$	−58.0278	−1.59055	−0.795274	−	0.606250i	$-0.792673\pi$
$11$	−58.0278	−1.59055	−0.795274	+	0.606250i	$0.792673\pi$
$12$	12.0000	0.288675
$13$	−87.0555	−1.85730	−0.928648	−	0.370961i	$-0.879028\pi$
$13$	−87.0555	−1.85730	−0.928648	+	0.370961i	$0.879028\pi$
$14$	9.44996	0.180401
$15$	−40.8167	−0.702588
$16$	16.0000	0.250000
$17$	−17.9083	−0.255495	−0.127747	−	0.991807i	$-0.540775\pi$
$17$	−17.9083	−0.255495	−0.127747	+	0.991807i	$0.540775\pi$
$18$	18.0000	0.235702
$19$	−29.3028	−0.353817	−0.176908	−	0.984227i	$-0.556610\pi$
$19$	−29.3028	−0.353817	−0.176908	+	0.984227i	$0.556610\pi$
$20$	−54.4222	−0.608459
$21$	14.1749	0.147296
$22$	−116.056	−1.12469
$23$	172.369	1.56267	0.781336	−	0.624110i	$-0.214538\pi$
$23$	172.369	1.56267	0.781336	+	0.624110i	$0.214538\pi$
$24$	24.0000	0.204124
$25$	60.1110	0.480888
$26$	−174.111	−1.31331
$27$	27.0000	0.192450
$28$	18.8999	0.127562
$29$	41.9916	0.268884	0.134442	−	0.990921i	$-0.457076\pi$
$29$	41.9916	0.268884	0.134442	+	0.990921i	$0.457076\pi$
$30$	−81.6333	−0.496804
$31$	−287.902	−1.66803	−0.834013	−	0.551745i	$-0.813962\pi$
$31$	−287.902	−1.66803	−0.834013	+	0.551745i	$0.813962\pi$
$32$	32.0000	0.176777
$33$	−174.083	−0.918303
$34$	−35.8167	−0.180662
$35$	−64.2860	−0.310466
$36$	36.0000	0.166667
$37$	−22.1640	−0.0984794	−0.0492397	−	0.998787i	$-0.515680\pi$
$37$	−22.1640	−0.0984794	−0.0492397	+	0.998787i	$0.515680\pi$
$38$	−58.6056	−0.250186
$39$	−261.167	−1.07231
$40$	−108.844	−0.430245
$41$	321.463	1.22449	0.612246	−	0.790667i	$-0.290266\pi$
$41$	321.463	1.22449	0.612246	+	0.790667i	$0.290266\pi$
$42$	28.3499	0.104154
$43$	−423.461	−1.50179	−0.750897	−	0.660419i	$-0.770379\pi$
$43$	−423.461	−1.50179	−0.750897	+	0.660419i	$0.770379\pi$
$44$	−232.111	−0.795274
$45$	−122.450	−0.405639
$46$	344.738	1.10498
$47$	−7.95798	−0.0246977	−0.0123488	−	0.999924i	$-0.503931\pi$
$47$	−7.95798	−0.0246977	−0.0123488	+	0.999924i	$0.503931\pi$
$48$	48.0000	0.144338
$49$	−320.675	−0.934911
$50$	120.222	0.340039
$51$	−53.7250	−0.147510
$52$	−348.222	−0.928648
$53$	−177.005	−0.458746	−0.229373	−	0.973339i	$-0.573667\pi$
$53$	−177.005	−0.458746	−0.229373	+	0.973339i	$0.573667\pi$
$54$	54.0000	0.136083
$55$	789.500	1.93557
$56$	37.7998	0.0902003
$57$	−87.9083	−0.204276
$58$	83.9832	0.190130
$59$	59.0000	0.130189
$59$	59.0000	0.130189
$60$	−163.267	−0.351294
$61$	217.919	0.457405	0.228702	−	0.973496i	$-0.426552\pi$
$61$	217.919	0.457405	0.228702	+	0.973496i	$0.426552\pi$
$62$	−575.805	−1.17947
$63$	42.5248	0.0850417
$64$	64.0000	0.125000
$65$	1184.44	2.26018
$66$	−348.167	−0.649338
$67$	561.394	1.02366	0.511830	−	0.859087i	$-0.328968\pi$
$67$	561.394	1.02366	0.511830	+	0.859087i	$0.328968\pi$
$68$	−71.6333	−0.127747
$69$	517.108	0.902209
$70$	−128.572	−0.219533
$71$	572.472	0.956900	0.478450	−	0.878115i	$-0.341199\pi$
$71$	572.472	0.956900	0.478450	+	0.878115i	$0.341199\pi$
$72$	72.0000	0.117851
$73$	−242.636	−0.389019	−0.194509	−	0.980901i	$-0.562311\pi$
$73$	−242.636	−0.389019	−0.194509	+	0.980901i	$0.562311\pi$
$74$	−44.3280	−0.0696355
$75$	180.333	0.277641
$76$	−117.211	−0.176908
$77$	−274.180	−0.405788
$78$	−522.333	−0.758238
$79$	−587.994	−0.837399	−0.418699	−	0.908125i	$-0.637514\pi$
$79$	−587.994	−0.837399	−0.418699	+	0.908125i	$0.637514\pi$
$80$	−217.689	−0.304229
$81$	81.0000	0.111111
$82$	642.927	0.865846
$83$	−887.596	−1.17381	−0.586906	−	0.809655i	$-0.699654\pi$
$83$	−887.596	−1.17381	−0.586906	+	0.809655i	$0.699654\pi$
$84$	56.6998	0.0736482
$85$	243.653	0.310916
$86$	−846.922	−1.06193
$87$	125.975	0.155240
$88$	−464.222	−0.562344
$89$	−428.281	−0.510086	−0.255043	−	0.966930i	$-0.582090\pi$
$89$	−428.281	−0.510086	−0.255043	+	0.966930i	$0.582090\pi$
$90$	−244.900	−0.286830
$91$	−411.336	−0.473843
$92$	689.477	0.781336
$93$	−863.707	−0.963035
$94$	−15.9160	−0.0174639
$95$	398.680	0.430566
$96$	96.0000	0.102062
$97$	1051.67	1.10084	0.550418	−	0.834889i	$-0.314468\pi$
$97$	1051.67	1.10084	0.550418	+	0.834889i	$0.314468\pi$
$98$	−641.349	−0.661082
$99$	−522.250	−0.530183
$100$	240.444	0.240444
$101$	−219.016	−0.215771	−0.107886	−	0.994163i	$-0.534408\pi$
$101$	−219.016	−0.215771	−0.107886	+	0.994163i	$0.534408\pi$
$102$	−107.450	−0.104305
$103$	−758.705	−0.725800	−0.362900	−	0.931828i	$-0.618213\pi$
$103$	−758.705	−0.725800	−0.362900	+	0.931828i	$0.618213\pi$
$104$	−696.444	−0.656653
$105$	−192.858	−0.179248
$106$	−354.010	−0.324382
$107$	153.819	0.138974	0.0694872	−	0.997583i	$-0.477864\pi$
$107$	153.819	0.138974	0.0694872	+	0.997583i	$0.477864\pi$
$108$	108.000	0.0962250
$109$	2252.12	1.97903	0.989515	−	0.144428i	$-0.0461343\pi$
$109$	2252.12	1.97903	0.989515	+	0.144428i	$0.0461343\pi$
$110$	1579.00	1.36865
$111$	−66.4920	−0.0568571
$112$	75.5997	0.0637812
$113$	687.949	0.572715	0.286357	−	0.958123i	$-0.407556\pi$
$113$	687.949	0.572715	0.286357	+	0.958123i	$0.407556\pi$
$114$	−175.817	−0.144445
$115$	−2345.18	−1.90164
$116$	167.966	0.134442
$117$	−783.500	−0.619099
$118$	118.000	0.0920575
$119$	−84.6165	−0.0651830
$120$	−326.533	−0.248402
$121$	2036.22	1.52984
$122$	435.839	0.323434
$123$	964.390	0.706961
$124$	−1151.61	−0.834013
$125$	882.850	0.631716
$126$	85.0497	0.0601335
$127$	−301.638	−0.210756	−0.105378	−	0.994432i	$-0.533605\pi$
$127$	−301.638	−0.210756	−0.105378	+	0.994432i	$0.533605\pi$
$128$	128.000	0.0883883
$129$	−1270.38	−0.867062
$130$	2368.88	1.59819
$131$	−1527.79	−1.01896	−0.509481	−	0.860482i	$-0.670162\pi$
$131$	−1527.79	−1.01896	−0.509481	+	0.860482i	$0.670162\pi$
$132$	−696.333	−0.459152
$133$	−138.455	−0.0902675
$134$	1122.79	0.723836
$135$	−367.350	−0.234196
$136$	−143.267	−0.0903310
$137$	1138.68	0.710105	0.355052	−	0.934846i	$-0.384463\pi$
$137$	1138.68	0.710105	0.355052	+	0.934846i	$0.384463\pi$
$138$	1034.22	0.637958
$139$	28.8655	0.0176140	0.00880699	−	0.999961i	$-0.497197\pi$
$139$	28.8655	0.0176140	0.00880699	+	0.999961i	$0.497197\pi$
$140$	−257.144	−0.155233
$141$	−23.8739	−0.0142592
$142$	1144.94	0.676631
$143$	5051.64	2.95412
$144$	144.000	0.0833333
$145$	−571.319	−0.327210
$146$	−485.272	−0.275078
$147$	−962.024	−0.539771
$148$	−88.6560	−0.0492397
$149$	−1443.88	−0.793877	−0.396938	−	0.917845i	$-0.629927\pi$
$149$	−1443.88	−0.793877	−0.396938	+	0.917845i	$0.629927\pi$
$150$	360.666	0.196322
$151$	1895.65	1.02163	0.510813	−	0.859692i	$-0.329344\pi$
$151$	1895.65	1.02163	0.510813	+	0.859692i	$0.329344\pi$
$152$	−234.422	−0.125093
$153$	−161.175	−0.0851648
$154$	−548.360	−0.286936
$155$	3917.07	2.02985
$156$	−1044.67	−0.536155
$157$	−2336.87	−1.18792	−0.593958	−	0.804496i	$-0.702435\pi$
$157$	−2336.87	−1.18792	−0.593958	+	0.804496i	$0.702435\pi$
$158$	−1175.99	−0.592130
$159$	−531.015	−0.264857
$160$	−435.378	−0.215123
$161$	814.441	0.398677
$162$	162.000	0.0785674
$163$	−1665.86	−0.800492	−0.400246	−	0.916408i	$-0.631075\pi$
$163$	−1665.86	−0.800492	−0.400246	+	0.916408i	$0.631075\pi$
$164$	1285.85	0.612246
$165$	2368.50	1.11750
$166$	−1775.19	−0.830010
$167$	3101.34	1.43706	0.718529	−	0.695497i	$-0.244816\pi$
$167$	3101.34	1.43706	0.718529	+	0.695497i	$0.244816\pi$
$168$	113.400	0.0520772
$169$	5381.66	2.44955
$170$	487.305	0.219851
$171$	−263.725	−0.117939
$172$	−1693.84	−0.750897
$173$	713.460	0.313545	0.156773	−	0.987635i	$-0.449891\pi$
$173$	713.460	0.313545	0.156773	+	0.987635i	$0.449891\pi$
$174$	251.950	0.109772
$175$	284.023	0.122687
$176$	−928.444	−0.397637
$177$	177.000	0.0751646
$178$	−856.562	−0.360685
$179$	−3075.23	−1.28410	−0.642049	−	0.766664i	$-0.721915\pi$
$179$	−3075.23	−1.28410	−0.642049	+	0.766664i	$0.721915\pi$
$180$	−489.800	−0.202820
$181$	−3322.19	−1.36429	−0.682145	−	0.731217i	$-0.738953\pi$
$181$	−3322.19	−1.36429	−0.682145	+	0.731217i	$0.738953\pi$
$182$	−822.671	−0.335057
$183$	653.758	0.264083
$184$	1378.95	0.552488
$185$	301.553	0.119841
$186$	−1727.41	−0.680969
$187$	1039.18	0.406376
$188$	−31.8319	−0.0123488
$189$	127.574	0.0490988
$190$	797.361	0.304456
$191$	−2067.04	−0.783068	−0.391534	−	0.920164i	$-0.628055\pi$
$191$	−2067.04	−0.783068	−0.391534	+	0.920164i	$0.628055\pi$
$192$	192.000	0.0721688
$193$	−4684.78	−1.74724	−0.873622	−	0.486605i	$-0.838235\pi$
$193$	−4684.78	−1.74724	−0.873622	+	0.486605i	$0.838235\pi$
$194$	2103.34	0.778408
$195$	3553.31	1.30491
$196$	−1282.70	−0.467456
$197$	2878.53	1.04105	0.520524	−	0.853847i	$-0.325736\pi$
$197$	2878.53	1.04105	0.520524	+	0.853847i	$0.325736\pi$
$198$	−1044.50	−0.374896
$199$	2303.53	0.820566	0.410283	−	0.911958i	$-0.365430\pi$
$199$	2303.53	0.820566	0.410283	+	0.911958i	$0.365430\pi$
$200$	480.888	0.170020
$201$	1684.18	0.591010
$202$	−438.032	−0.152573
$203$	198.409	0.0685991
$204$	−214.900	−0.0737549
$205$	−4373.69	−1.49011
$206$	−1517.41	−0.513218
$207$	1551.32	0.520891
$208$	−1392.89	−0.464324
$209$	1700.37	0.562762
$210$	−385.716	−0.126747
$211$	−1659.58	−0.541470	−0.270735	−	0.962654i	$-0.587267\pi$
$211$	−1659.58	−0.541470	−0.270735	+	0.962654i	$0.587267\pi$
$212$	−708.020	−0.229373
$213$	1717.42	0.552467
$214$	307.638	0.0982698
$215$	5761.42	1.82756
$216$	216.000	0.0680414
$217$	−1360.33	−0.425555
$218$	4504.25	1.39939
$219$	−727.908	−0.224600
$220$	3158.00	0.967783
$221$	1559.02	0.474529
$222$	−132.984	−0.0402041
$223$	−5229.78	−1.57046	−0.785229	−	0.619205i	$-0.787455\pi$
$223$	−5229.78	−1.57046	−0.785229	+	0.619205i	$0.787455\pi$
$224$	151.199	0.0451002
$225$	540.999	0.160296
$226$	1375.90	0.404971
$227$	−4636.74	−1.35573	−0.677866	−	0.735186i	$-0.737095\pi$
$227$	−4636.74	−1.35573	−0.677866	+	0.735186i	$0.737095\pi$
$228$	−351.633	−0.102138
$229$	−2717.57	−0.784201	−0.392101	−	0.919922i	$-0.628252\pi$
$229$	−2717.57	−0.784201	−0.392101	+	0.919922i	$0.628252\pi$
$230$	−4690.36	−1.34467
$231$	−822.540	−0.234282
$232$	335.933	0.0950649
$233$	267.329	0.0751643	0.0375822	−	0.999294i	$-0.488034\pi$
$233$	267.329	0.0751643	0.0375822	+	0.999294i	$0.488034\pi$
$234$	−1567.00	−0.437769
$235$	108.273	0.0300550
$236$	236.000	0.0650945
$237$	−1763.98	−0.483472
$238$	−169.233	−0.0460914
$239$	4266.26	1.15465	0.577325	−	0.816514i	$-0.304097\pi$
$239$	4266.26	1.15465	0.577325	+	0.816514i	$0.304097\pi$
$240$	−653.066	−0.175647
$241$	−4854.52	−1.29754	−0.648770	−	0.760984i	$-0.724716\pi$
$241$	−4854.52	−1.29754	−0.648770	+	0.760984i	$0.724716\pi$
$242$	4072.44	1.08176
$243$	243.000	0.0641500
$244$	871.677	0.228702
$245$	4362.95	1.13771
$246$	1928.78	0.499897
$247$	2550.97	0.657143
$248$	−2303.22	−0.589736
$249$	−2662.79	−0.677700
$250$	1765.70	0.446691
$251$	5264.44	1.32386	0.661929	−	0.749566i	$-0.269738\pi$
$251$	5264.44	1.32386	0.661929	+	0.749566i	$0.269738\pi$
$252$	170.099	0.0425208
$253$	−10002.2	−2.48551
$254$	−603.275	−0.149027
$255$	730.958	0.179507
$256$	256.000	0.0625000
$257$	−7090.26	−1.72093	−0.860464	−	0.509511i	$-0.829826\pi$
$257$	−7090.26	−1.72093	−0.860464	+	0.509511i	$0.829826\pi$
$258$	−2540.77	−0.613105
$259$	−104.724	−0.0251246
$260$	4737.75	1.13009
$261$	377.924	0.0896281
$262$	−3055.59	−0.720514
$263$	2055.72	0.481981	0.240990	−	0.970528i	$-0.422528\pi$
$263$	2055.72	0.481981	0.240990	+	0.970528i	$0.422528\pi$
$264$	−1392.67	−0.324669
$265$	2408.25	0.558256
$266$	−276.910	−0.0638288
$267$	−1284.84	−0.294498
$268$	2245.57	0.511830
$269$	104.612	0.0237111	0.0118556	−	0.999930i	$-0.496226\pi$
$269$	104.612	0.0237111	0.0118556	+	0.999930i	$0.496226\pi$
$270$	−734.700	−0.165601
$271$	8080.34	1.81124	0.905619	−	0.424092i	$-0.139407\pi$
$271$	8080.34	1.81124	0.905619	+	0.424092i	$0.139407\pi$
$272$	−286.533	−0.0638736
$273$	−1234.01	−0.273573
$274$	2277.37	0.502120
$275$	−3488.11	−0.764876
$276$	2068.43	0.451105
$277$	1291.07	0.280046	0.140023	−	0.990148i	$-0.455282\pi$
$277$	1291.07	0.280046	0.140023	+	0.990148i	$0.455282\pi$
$278$	57.7311	0.0124550
$279$	−2591.12	−0.556009
$280$	−514.288	−0.109766
$281$	−1521.86	−0.323084	−0.161542	−	0.986866i	$-0.551647\pi$
$281$	−1521.86	−0.323084	−0.161542	+	0.986866i	$0.551647\pi$
$282$	−47.7479	−0.0100828
$283$	7997.12	1.67979	0.839893	−	0.542752i	$-0.182618\pi$
$283$	7997.12	1.67979	0.839893	+	0.542752i	$0.182618\pi$
$284$	2289.89	0.478450
$285$	1196.04	0.248587
$286$	10103.3	2.08888
$287$	1518.91	0.312398
$288$	288.000	0.0589256
$289$	−4592.29	−0.934723
$290$	−1142.64	−0.231372
$291$	3155.01	0.635568
$292$	−970.543	−0.194509
$293$	−2053.55	−0.409453	−0.204726	−	0.978819i	$-0.565630\pi$
$293$	−2053.55	−0.409453	−0.204726	+	0.978819i	$0.565630\pi$
$294$	−1924.05	−0.381676
$295$	−802.728	−0.158429
$296$	−177.312	−0.0348177
$297$	−1566.75	−0.306101
$298$	−2887.77	−0.561356
$299$	−15005.7	−2.90235
$300$	721.332	0.138820
$301$	−2000.84	−0.383145
$302$	3791.29	0.722399
$303$	−657.048	−0.124576
$304$	−468.844	−0.0884542
$305$	−2964.91	−0.556624
$306$	−322.350	−0.0602206
$307$	5619.12	1.04463	0.522313	−	0.852754i	$-0.325069\pi$
$307$	5619.12	1.04463	0.522313	+	0.852754i	$0.325069\pi$
$308$	−1096.72	−0.202894
$309$	−2276.11	−0.419041
$310$	7834.14	1.43532
$311$	−4740.42	−0.864323	−0.432162	−	0.901796i	$-0.642249\pi$
$311$	−4740.42	−0.864323	−0.432162	+	0.901796i	$0.642249\pi$
$312$	−2089.33	−0.379119
$313$	−3551.37	−0.641326	−0.320663	−	0.947193i	$-0.603906\pi$
$313$	−3551.37	−0.641326	−0.320663	+	0.947193i	$0.603906\pi$
$314$	−4673.74	−0.839983
$315$	−578.574	−0.103489
$316$	−2351.98	−0.418699
$317$	−2650.14	−0.469548	−0.234774	−	0.972050i	$-0.575435\pi$
$317$	−2650.14	−0.469548	−0.234774	+	0.972050i	$0.575435\pi$
$318$	−1062.03	−0.187282
$319$	−2436.68	−0.427673
$320$	−870.755	−0.152115
$321$	461.458	0.0802369
$322$	1628.88	0.281907
$323$	524.764	0.0903982
$324$	324.000	0.0555556
$325$	−5233.00	−0.893152
$326$	−3331.72	−0.566033
$327$	6756.37	1.14259
$328$	2571.71	0.432923
$329$	−37.6013	−0.00630099
$330$	4737.00	0.790191
$331$	2546.22	0.422818	0.211409	−	0.977398i	$-0.432195\pi$
$331$	2546.22	0.422818	0.211409	+	0.977398i	$0.432195\pi$
$332$	−3550.39	−0.586906
$333$	−199.476	−0.0328265
$334$	6202.68	1.01615
$335$	−7638.07	−1.24571
$336$	226.799	0.0368241
$337$	−11381.2	−1.83968	−0.919840	−	0.392295i	$-0.871681\pi$
$337$	−11381.2	−1.83968	−0.919840	+	0.392295i	$0.871681\pi$
$338$	10763.3	1.73209
$339$	2063.85	0.330657
$340$	974.611	0.155458
$341$	16706.3	2.65308
$342$	−527.450	−0.0833954
$343$	−3135.85	−0.493644
$344$	−3387.69	−0.530965
$345$	−7035.54	−1.09791
$346$	1426.92	0.221710
$347$	−11752.8	−1.81822	−0.909109	−	0.416559i	$-0.863236\pi$
$347$	−11752.8	−1.81822	−0.909109	+	0.416559i	$0.863236\pi$
$348$	503.899	0.0776202
$349$	−9009.73	−1.38189	−0.690945	−	0.722907i	$-0.742805\pi$
$349$	−9009.73	−1.38189	−0.690945	+	0.722907i	$0.742805\pi$
$350$	568.047	0.0867525
$351$	−2350.50	−0.357437
$352$	−1856.89	−0.281172
$353$	−7604.09	−1.14653	−0.573265	−	0.819370i	$-0.694323\pi$
$353$	−7604.09	−1.14653	−0.573265	+	0.819370i	$0.694323\pi$
$354$	354.000	0.0531494
$355$	−7788.80	−1.16447
$356$	−1713.12	−0.255043
$357$	−253.849	−0.0376334
$358$	−6150.46	−0.907994
$359$	−649.281	−0.0954534	−0.0477267	−	0.998860i	$-0.515198\pi$
$359$	−649.281	−0.0954534	−0.0477267	+	0.998860i	$0.515198\pi$
$360$	−979.600	−0.143415
$361$	−6000.35	−0.874814
$362$	−6644.38	−0.964699
$363$	6108.66	0.883255
$364$	−1645.34	−0.236921
$365$	3301.19	0.473404
$366$	1307.52	0.186735
$367$	−6877.12	−0.978154	−0.489077	−	0.872241i	$-0.662666\pi$
$367$	−6877.12	−0.978154	−0.489077	+	0.872241i	$0.662666\pi$
$368$	2757.91	0.390668
$369$	2893.17	0.408164
$370$	603.107	0.0847406
$371$	−836.346	−0.117037
$372$	−3454.83	−0.481518
$373$	−8814.31	−1.22356	−0.611780	−	0.791028i	$-0.709546\pi$
$373$	−8814.31	−1.22356	−0.611780	+	0.791028i	$0.709546\pi$
$374$	2078.36	0.287351
$375$	2648.55	0.364722
$376$	−63.6638	−0.00873195
$377$	−3655.60	−0.499398
$378$	255.149	0.0347181
$379$	−955.646	−0.129520	−0.0647602	−	0.997901i	$-0.520628\pi$
$379$	−955.646	−0.129520	−0.0647602	+	0.997901i	$0.520628\pi$
$380$	1594.72	0.215283
$381$	−904.913	−0.121680
$382$	−4134.08	−0.553712
$383$	13706.4	1.82863	0.914313	−	0.405008i	$-0.132731\pi$
$383$	13706.4	1.82863	0.914313	+	0.405008i	$0.132731\pi$
$384$	384.000	0.0510310
$385$	3730.37	0.493811
$386$	−9369.57	−1.23549
$387$	−3811.15	−0.500598
$388$	4206.68	0.550418
$389$	−8827.03	−1.15051	−0.575255	−	0.817974i	$-0.695097\pi$
$389$	−8827.03	−1.15051	−0.575255	+	0.817974i	$0.695097\pi$
$390$	7106.63	0.922713
$391$	−3086.84	−0.399254
$392$	−2565.40	−0.330541
$393$	−4583.38	−0.588298
$394$	5757.06	0.736133
$395$	7999.98	1.01905
$396$	−2089.00	−0.265091
$397$	7761.96	0.981263	0.490632	−	0.871367i	$-0.336766\pi$
$397$	7761.96	0.981263	0.490632	+	0.871367i	$0.336766\pi$
$398$	4607.05	0.580228
$399$	−415.365	−0.0521160
$400$	961.776	0.120222
$401$	784.697	0.0977205	0.0488602	−	0.998806i	$-0.484441\pi$
$401$	784.697	0.0977205	0.0488602	+	0.998806i	$0.484441\pi$
$402$	3368.36	0.417907
$403$	25063.5	3.09802
$404$	−876.064	−0.107886
$405$	−1102.05	−0.135213
$406$	396.819	0.0485069
$407$	1286.13	0.156636
$408$	−429.800	−0.0521526
$409$	12030.7	1.45448	0.727238	−	0.686385i	$-0.240803\pi$
$409$	12030.7	1.45448	0.727238	+	0.686385i	$0.240803\pi$
$410$	−8747.38	−1.05366
$411$	3416.05	0.409979
$412$	−3034.82	−0.362900
$413$	278.774	0.0332144
$414$	3102.65	0.368325
$415$	12076.2	1.42843
$416$	−2785.78	−0.328327
$417$	86.5966	0.0101694
$418$	3400.75	0.397933
$419$	−12845.8	−1.49775	−0.748877	−	0.662709i	$-0.769407\pi$
$419$	−12845.8	−1.49775	−0.748877	+	0.662709i	$0.769407\pi$
$420$	−771.432	−0.0896238
$421$	−5313.07	−0.615067	−0.307534	−	0.951537i	$-0.599504\pi$
$421$	−5313.07	−0.615067	−0.307534	+	0.951537i	$0.599504\pi$
$422$	−3319.16	−0.382877
$423$	−71.6218	−0.00823256
$424$	−1416.04	−0.162191
$425$	−1076.49	−0.122864
$426$	3434.83	0.390653
$427$	1029.66	0.116695
$428$	615.277	0.0694872
$429$	15154.9	1.70556
$430$	11522.8	1.29228
$431$	−6032.27	−0.674163	−0.337082	−	0.941475i	$-0.609440\pi$
$431$	−6032.27	−0.674163	−0.337082	+	0.941475i	$0.609440\pi$
$432$	432.000	0.0481125
$433$	−9215.28	−1.02277	−0.511384	−	0.859353i	$-0.670867\pi$
$433$	−9215.28	−1.02277	−0.511384	+	0.859353i	$0.670867\pi$
$434$	−2720.67	−0.300913
$435$	−1713.96	−0.188915
$436$	9008.49	0.989515
$437$	−5050.90	−0.552900
$438$	−1455.82	−0.158816
$439$	12796.0	1.39116	0.695579	−	0.718450i	$-0.255148\pi$
$439$	12796.0	1.39116	0.695579	+	0.718450i	$0.255148\pi$
$440$	6316.00	0.684326
$441$	−2886.07	−0.311637
$442$	3118.04	0.335543
$443$	9073.27	0.973102	0.486551	−	0.873652i	$-0.338255\pi$
$443$	9073.27	0.973102	0.486551	+	0.873652i	$0.338255\pi$
$444$	−265.968	−0.0284286
$445$	5827.00	0.620733
$446$	−10459.6	−1.11048
$447$	−4331.65	−0.458345
$448$	302.399	0.0318906
$449$	13988.0	1.47023	0.735116	−	0.677942i	$-0.237128\pi$
$449$	13988.0	1.47023	0.735116	+	0.677942i	$0.237128\pi$
$450$	1082.00	0.113346
$451$	−18653.8	−1.94761
$452$	2751.80	0.286357
$453$	5686.94	0.589836
$454$	−9273.48	−0.958647
$455$	5596.45	0.576628
$456$	−703.267	−0.0722225
$457$	4797.02	0.491017	0.245509	−	0.969394i	$-0.421045\pi$
$457$	4797.02	0.491017	0.245509	+	0.969394i	$0.421045\pi$
$458$	−5435.14	−0.554514
$459$	−483.525	−0.0491699
$460$	−9380.71	−0.950822
$461$	3968.21	0.400907	0.200453	−	0.979703i	$-0.435758\pi$
$461$	3968.21	0.400907	0.200453	+	0.979703i	$0.435758\pi$
$462$	−1645.08	−0.165662
$463$	273.878	0.0274907	0.0137454	−	0.999906i	$-0.495625\pi$
$463$	273.878	0.0274907	0.0137454	+	0.999906i	$0.495625\pi$
$464$	671.866	0.0672211
$465$	11751.2	1.17193
$466$	534.658	0.0531492
$467$	−8354.39	−0.827827	−0.413913	−	0.910316i	$-0.635838\pi$
$467$	−8354.39	−0.827827	−0.413913	+	0.910316i	$0.635838\pi$
$468$	−3134.00	−0.309549
$469$	2652.57	0.261161
$470$	216.545	0.0212521
$471$	−7010.62	−0.685843
$472$	472.000	0.0460287
$473$	24572.5	2.38868
$474$	−3527.96	−0.341867
$475$	−1761.42	−0.170146
$476$	−338.466	−0.0325915
$477$	−1593.05	−0.152915
$478$	8532.52	0.816461
$479$	−12651.8	−1.20684	−0.603421	−	0.797423i	$-0.706196\pi$
$479$	−12651.8	−1.20684	−0.603421	+	0.797423i	$0.706196\pi$
$480$	−1306.13	−0.124201
$481$	1929.50	0.182905
$482$	−9709.05	−0.917500
$483$	2443.32	0.230176
$484$	8144.88	0.764921
$485$	−14308.6	−1.33963
$486$	486.000	0.0453609
$487$	1104.17	0.102741	0.0513703	−	0.998680i	$-0.483641\pi$
$487$	1104.17	0.102741	0.0513703	+	0.998680i	$0.483641\pi$
$488$	1743.35	0.161717
$489$	−4997.58	−0.462164
$490$	8725.91	0.804482
$491$	−5451.06	−0.501024	−0.250512	−	0.968113i	$-0.580599\pi$
$491$	−5451.06	−0.501024	−0.250512	+	0.968113i	$0.580599\pi$
$492$	3857.56	0.353480
$493$	−751.999	−0.0686985
$494$	5101.94	0.464670
$495$	7105.50	0.645189
$496$	−4606.44	−0.417007
$497$	2704.92	0.244129
$498$	−5325.58	−0.479207
$499$	−2432.45	−0.218219	−0.109110	−	0.994030i	$-0.534800\pi$
$499$	−2432.45	−0.218219	−0.109110	+	0.994030i	$0.534800\pi$
$500$	3531.40	0.315858
$501$	9304.02	0.829686
$502$	10528.9	0.936109
$503$	12095.7	1.07221	0.536105	−	0.844151i	$-0.319895\pi$
$503$	12095.7	1.07221	0.536105	+	0.844151i	$0.319895\pi$
$504$	340.199	0.0300668
$505$	2979.83	0.262576
$506$	−20004.4	−1.75752
$507$	16145.0	1.41425
$508$	−1206.55	−0.105378
$509$	3966.74	0.345428	0.172714	−	0.984972i	$-0.444746\pi$
$509$	3966.74	0.345428	0.172714	+	0.984972i	$0.444746\pi$
$510$	1461.92	0.126931
$511$	−1146.45	−0.0992484
$512$	512.000	0.0441942
$513$	−791.175	−0.0680921
$514$	−14180.5	−1.21688
$515$	10322.6	0.883239
$516$	−5081.53	−0.433531
$517$	461.784	0.0392828
$518$	−209.449	−0.0177657
$519$	2140.38	0.181026
$520$	9475.51	0.799093
$521$	−20833.0	−1.75184	−0.875922	−	0.482452i	$-0.839746\pi$
$521$	−20833.0	−1.75184	−0.875922	+	0.482452i	$0.839746\pi$
$522$	755.849	0.0633766
$523$	15056.1	1.25881	0.629406	−	0.777077i	$-0.283298\pi$
$523$	15056.1	1.25881	0.629406	+	0.777077i	$0.283298\pi$
$524$	−6111.17	−0.509481
$525$	852.070	0.0708331
$526$	4111.43	0.340812
$527$	5155.85	0.426172
$528$	−2785.33	−0.229576
$529$	17544.2	1.44195
$530$	4816.50	0.394746
$531$	531.000	0.0433963
$532$	−553.820	−0.0451337
$533$	−27985.2	−2.27424
$534$	−2569.69	−0.208242
$535$	−2092.79	−0.169120
$536$	4491.15	0.361918
$537$	−9225.69	−0.741374
$538$	209.224	0.0167663
$539$	18608.0	1.48702
$540$	−1469.40	−0.117098
$541$	22579.0	1.79436	0.897179	−	0.441667i	$-0.145613\pi$
$541$	22579.0	1.79436	0.897179	+	0.441667i	$0.145613\pi$
$542$	16160.7	1.28074
$543$	−9966.58	−0.787673
$544$	−573.066	−0.0451655
$545$	−30641.4	−2.40832
$546$	−2468.01	−0.193445
$547$	−13199.8	−1.03178	−0.515888	−	0.856656i	$-0.672538\pi$
$547$	−13199.8	−1.03178	−0.515888	+	0.856656i	$0.672538\pi$
$548$	4554.73	0.355052
$549$	1961.27	0.152468
$550$	−6976.22	−0.540849
$551$	−1230.47	−0.0951357
$552$	4136.86	0.318979
$553$	−2778.26	−0.213641
$554$	2582.13	0.198022
$555$	904.660	0.0691904
$556$	115.462	0.00880699
$557$	23025.8	1.75159	0.875793	−	0.482686i	$-0.160339\pi$
$557$	23025.8	1.75159	0.875793	+	0.482686i	$0.160339\pi$
$558$	−5182.24	−0.393158
$559$	36864.6	2.78928
$560$	−1028.58	−0.0776165
$561$	3117.54	0.234621
$562$	−3043.72	−0.228455
$563$	−9920.84	−0.742653	−0.371327	−	0.928502i	$-0.621097\pi$
$563$	−9920.84	−0.742653	−0.371327	+	0.928502i	$0.621097\pi$
$564$	−95.4958	−0.00712960
$565$	−9359.92	−0.696947
$566$	15994.2	1.18779
$567$	382.723	0.0283472
$568$	4579.77	0.338315
$569$	5899.99	0.434693	0.217347	−	0.976094i	$-0.430260\pi$
$569$	5899.99	0.434693	0.217347	+	0.976094i	$0.430260\pi$
$570$	2392.08	0.175778
$571$	3789.68	0.277746	0.138873	−	0.990310i	$-0.455652\pi$
$571$	3789.68	0.277746	0.138873	+	0.990310i	$0.455652\pi$
$572$	20206.5	1.47706
$573$	−6201.13	−0.452104
$574$	3037.82	0.220899
$575$	10361.3	0.751471
$576$	576.000	0.0416667
$577$	−11381.2	−0.821152	−0.410576	−	0.911826i	$-0.634672\pi$
$577$	−11381.2	−0.821152	−0.410576	+	0.911826i	$0.634672\pi$
$578$	−9184.58	−0.660949
$579$	−14054.4	−1.00877
$580$	−2285.28	−0.163605
$581$	−4193.88	−0.299469
$582$	6310.03	0.449414
$583$	10271.2	0.729657
$584$	−1941.09	−0.137539
$585$	10659.9	0.753392
$586$	−4107.10	−0.289527
$587$	18352.4	1.29043	0.645217	−	0.763999i	$-0.276767\pi$
$587$	18352.4	1.29043	0.645217	+	0.763999i	$0.276767\pi$
$588$	−3848.09	−0.269886
$589$	8436.34	0.590176
$590$	−1605.46	−0.112026
$591$	8635.58	0.601050
$592$	−354.624	−0.0246199
$593$	−13770.3	−0.953587	−0.476794	−	0.879015i	$-0.658201\pi$
$593$	−13770.3	−0.953587	−0.476794	+	0.879015i	$0.658201\pi$
$594$	−3133.50	−0.216446
$595$	1151.25	0.0793224
$596$	−5775.54	−0.396938
$597$	6910.58	0.473754
$598$	−30011.4	−2.05227
$599$	23090.9	1.57507	0.787535	−	0.616270i	$-0.211357\pi$
$599$	23090.9	1.57507	0.787535	+	0.616270i	$0.211357\pi$
$600$	1442.66	0.0981609
$601$	−10235.9	−0.694727	−0.347364	−	0.937731i	$-0.612923\pi$
$601$	−10235.9	−0.694727	−0.347364	+	0.937731i	$0.612923\pi$
$602$	−4001.69	−0.270925
$603$	5052.54	0.341220
$604$	7582.59	0.510813
$605$	−27703.9	−1.86169
$606$	−1314.10	−0.0880883
$607$	8035.03	0.537285	0.268642	−	0.963240i	$-0.413425\pi$
$607$	8035.03	0.537285	0.268642	+	0.963240i	$0.413425\pi$
$608$	−937.689	−0.0625466
$609$	595.228	0.0396057
$610$	−5929.82	−0.393593
$611$	692.786	0.0458709
$612$	−644.700	−0.0425824
$613$	8185.59	0.539336	0.269668	−	0.962953i	$-0.413086\pi$
$613$	8185.59	0.539336	0.269668	+	0.962953i	$0.413086\pi$
$614$	11238.2	0.738662
$615$	−13121.1	−0.860313
$616$	−2193.44	−0.143468
$617$	−5211.31	−0.340032	−0.170016	−	0.985441i	$-0.554382\pi$
$617$	−5211.31	−0.340032	−0.170016	+	0.985441i	$0.554382\pi$
$618$	−4552.23	−0.296307
$619$	−25902.5	−1.68192	−0.840959	−	0.541098i	$-0.818009\pi$
$619$	−25902.5	−1.68192	−0.840959	+	0.541098i	$0.818009\pi$
$620$	15668.3	1.01493
$621$	4653.97	0.300736
$622$	−9480.84	−0.611169
$623$	−2023.62	−0.130136
$624$	−4178.66	−0.268078
$625$	−19525.5	−1.24963
$626$	−7102.73	−0.453486
$627$	5101.12	0.324911
$628$	−9347.49	−0.593958
$629$	396.920	0.0251610
$630$	−1157.15	−0.0731776
$631$	8605.56	0.542919	0.271460	−	0.962450i	$-0.412494\pi$
$631$	8605.56	0.542919	0.271460	+	0.962450i	$0.412494\pi$
$632$	−4703.95	−0.296065
$633$	−4978.74	−0.312618
$634$	−5300.28	−0.332021
$635$	4103.95	0.256473
$636$	−2124.06	−0.132428
$637$	27916.5	1.73641
$638$	−4873.36	−0.302411
$639$	5152.25	0.318967
$640$	−1741.51	−0.107561
$641$	12273.5	0.756280	0.378140	−	0.925748i	$-0.376564\pi$
$641$	12273.5	0.756280	0.378140	+	0.925748i	$0.376564\pi$
$642$	922.915	0.0567361
$643$	26543.5	1.62795	0.813977	−	0.580897i	$-0.197298\pi$
$643$	26543.5	1.62795	0.813977	+	0.580897i	$0.197298\pi$
$644$	3257.77	0.199338
$645$	17284.3	1.05514
$646$	1049.53	0.0639212
$647$	−22171.2	−1.34720	−0.673602	−	0.739094i	$-0.735254\pi$
$647$	−22171.2	−1.34720	−0.673602	+	0.739094i	$0.735254\pi$
$648$	648.000	0.0392837
$649$	−3423.64	−0.207072
$650$	−10466.0	−0.631554
$651$	−4081.00	−0.245694
$652$	−6663.43	−0.400246
$653$	8278.40	0.496108	0.248054	−	0.968746i	$-0.420209\pi$
$653$	8278.40	0.496108	0.248054	+	0.968746i	$0.420209\pi$
$654$	13512.7	0.807936
$655$	20786.5	1.23999
$656$	5143.42	0.306123
$657$	−2183.72	−0.129673
$658$	−75.2026	−0.00445548
$659$	4052.25	0.239534	0.119767	−	0.992802i	$-0.461785\pi$
$659$	4052.25	0.239534	0.119767	+	0.992802i	$0.461785\pi$
$660$	9474.00	0.558750
$661$	8782.11	0.516770	0.258385	−	0.966042i	$-0.416810\pi$
$661$	8782.11	0.516770	0.258385	+	0.966042i	$0.416810\pi$
$662$	5092.44	0.298978
$663$	4677.06	0.273970
$664$	−7100.77	−0.415005
$665$	1883.76	0.109848
$666$	−398.952	−0.0232118
$667$	7238.06	0.420178
$668$	12405.4	0.718529
$669$	−15689.4	−0.906705
$670$	−15276.1	−0.880849
$671$	−12645.4	−0.727525
$672$	453.598	0.0260386
$673$	−20005.2	−1.14583	−0.572916	−	0.819614i	$-0.694188\pi$
$673$	−20005.2	−1.14583	−0.572916	+	0.819614i	$0.694188\pi$
$674$	−22762.3	−1.30085
$675$	1623.00	0.0925470
$676$	21526.6	1.22478
$677$	−6373.35	−0.361814	−0.180907	−	0.983500i	$-0.557903\pi$
$677$	−6373.35	−0.361814	−0.180907	+	0.983500i	$0.557903\pi$
$678$	4127.69	0.233810
$679$	4969.13	0.280851
$680$	1949.22	0.109925
$681$	−13910.2	−0.782732
$682$	33412.7	1.87601
$683$	−8074.61	−0.452367	−0.226183	−	0.974085i	$-0.572625\pi$
$683$	−8074.61	−0.452367	−0.226183	+	0.974085i	$0.572625\pi$
$684$	−1054.90	−0.0589695
$685$	−15492.4	−0.864139
$686$	−6271.70	−0.349059
$687$	−8152.71	−0.452759
$688$	−6775.37	−0.375449
$689$	15409.3	0.852027
$690$	−14071.1	−0.776343
$691$	−21122.1	−1.16284	−0.581421	−	0.813603i	$-0.697503\pi$
$691$	−21122.1	−1.16284	−0.581421	+	0.813603i	$0.697503\pi$
$692$	2853.84	0.156773
$693$	−2467.62	−0.135263
$694$	−23505.5	−1.28567
$695$	−392.732	−0.0214348
$696$	1007.80	0.0548858
$697$	−5756.87	−0.312851
$698$	−18019.5	−0.977144
$699$	801.986	0.0433961
$700$	1136.09	0.0613433
$701$	−5540.89	−0.298540	−0.149270	−	0.988796i	$-0.547692\pi$
$701$	−5540.89	−0.298540	−0.149270	+	0.988796i	$0.547692\pi$
$702$	−4701.00	−0.252746
$703$	649.467	0.0348437
$704$	−3713.78	−0.198818
$705$	324.818	0.0173523
$706$	−15208.2	−0.810719
$707$	−1034.85	−0.0550487
$708$	708.000	0.0375823
$709$	26323.6	1.39436	0.697182	−	0.716894i	$-0.254437\pi$
$709$	26323.6	1.39436	0.697182	+	0.716894i	$0.254437\pi$
$710$	−15577.6	−0.823404
$711$	−5291.95	−0.279133
$712$	−3426.25	−0.180343
$713$	−49625.5	−2.60658
$714$	−507.699	−0.0266109
$715$	−68730.3	−3.59492
$716$	−12300.9	−0.642049
$717$	12798.8	0.666637
$718$	−1298.56	−0.0674957
$719$	22465.8	1.16528	0.582638	−	0.812732i	$-0.302021\pi$
$719$	22465.8	1.16528	0.582638	+	0.812732i	$0.302021\pi$
$720$	−1959.20	−0.101410
$721$	−3584.87	−0.185170
$722$	−12000.7	−0.618587
$723$	−14563.6	−0.749135
$724$	−13288.8	−0.682145
$725$	2524.16	0.129303
$726$	12217.3	0.624556
$727$	−2309.36	−0.117812	−0.0589061	−	0.998264i	$-0.518761\pi$
$727$	−2309.36	−0.117812	−0.0589061	+	0.998264i	$0.518761\pi$
$728$	−3290.68	−0.167529
$729$	729.000	0.0370370
$730$	6602.39	0.334747
$731$	7583.48	0.383700
$732$	2615.03	0.132041
$733$	2759.39	0.139046	0.0695228	−	0.997580i	$-0.477852\pi$
$733$	2759.39	0.139046	0.0695228	+	0.997580i	$0.477852\pi$
$734$	−13754.2	−0.691660
$735$	13088.9	0.656857
$736$	5515.82	0.276244
$737$	−32576.4	−1.62818
$738$	5786.34	0.288615
$739$	35484.2	1.76632	0.883159	−	0.469075i	$-0.155412\pi$
$739$	35484.2	1.76632	0.883159	+	0.469075i	$0.155412\pi$
$740$	1206.21	0.0599207
$741$	7652.90	0.379401
$742$	−1672.69	−0.0827580
$743$	−24662.3	−1.21773	−0.608864	−	0.793275i	$-0.708374\pi$
$743$	−24662.3	−1.21773	−0.608864	+	0.793275i	$0.708374\pi$
$744$	−6909.66	−0.340484
$745$	19644.8	0.966083
$746$	−17628.6	−0.865187
$747$	−7988.37	−0.391271
$748$	4156.72	0.203188
$749$	726.793	0.0354558
$750$	5297.10	0.257897
$751$	−24929.0	−1.21128	−0.605641	−	0.795738i	$-0.707083\pi$
$751$	−24929.0	−1.21128	−0.605641	+	0.795738i	$0.707083\pi$
$752$	−127.328	−0.00617442
$753$	15793.3	0.764330
$754$	−7311.20	−0.353128
$755$	−25791.3	−1.24323
$756$	510.298	0.0245494
$757$	−37990.6	−1.82403	−0.912015	−	0.410156i	$-0.865474\pi$
$757$	−37990.6	−1.82403	−0.912015	+	0.410156i	$0.865474\pi$
$758$	−1911.29	−0.0915848
$759$	−30006.6	−1.43501
$760$	3189.44	0.152228
$761$	16711.8	0.796060	0.398030	−	0.917372i	$-0.369694\pi$
$761$	16711.8	0.796060	0.398030	+	0.917372i	$0.369694\pi$
$762$	−1809.83	−0.0860408
$763$	10641.2	0.504900
$764$	−8268.17	−0.391534
$765$	2192.87	0.103639
$766$	27412.8	1.29303
$767$	−5136.28	−0.241799
$768$	768.000	0.0360844
$769$	−10429.2	−0.489059	−0.244530	−	0.969642i	$-0.578634\pi$
$769$	−10429.2	−0.489059	−0.244530	+	0.969642i	$0.578634\pi$
$770$	7460.74	0.349177
$771$	−21270.8	−0.993578
$772$	−18739.1	−0.873622
$773$	13542.0	0.630104	0.315052	−	0.949074i	$-0.397978\pi$
$773$	13542.0	0.630104	0.315052	+	0.949074i	$0.397978\pi$
$774$	−7622.30	−0.353976
$775$	−17306.1	−0.802134
$776$	8413.37	0.389204
$777$	−314.173	−0.0145057
$778$	−17654.1	−0.813533
$779$	−9419.77	−0.433246
$780$	14213.3	0.652457
$781$	−33219.3	−1.52200
$782$	−6173.69	−0.282315
$783$	1133.77	0.0517468
$784$	−5130.79	−0.233728
$785$	31794.4	1.44559
$786$	−9166.76	−0.415989
$787$	−32391.3	−1.46712	−0.733561	−	0.679624i	$-0.762143\pi$
$787$	−32391.3	−1.46712	−0.733561	+	0.679624i	$0.762143\pi$
$788$	11514.1	0.520524
$789$	6167.15	0.278272
$790$	16000.0	0.720574
$791$	3250.54	0.146114
$792$	−4178.00	−0.187448
$793$	−18971.1	−0.849537
$794$	15523.9	0.693858
$795$	7224.76	0.322309
$796$	9214.10	0.410283
$797$	22715.8	1.00958	0.504789	−	0.863243i	$-0.331570\pi$
$797$	22715.8	1.00958	0.504789	+	0.863243i	$0.331570\pi$
$798$	−830.730	−0.0368515
$799$	142.514	0.00631012
$800$	1923.55	0.0850098
$801$	−3854.53	−0.170029
$802$	1569.39	0.0690988
$803$	14079.6	0.618753
$804$	6736.72	0.295505
$805$	−11080.9	−0.485157
$806$	50127.0	2.19063
$807$	313.836	0.0136896
$808$	−1752.13	−0.0762867
$809$	23607.8	1.02596	0.512982	−	0.858400i	$-0.328541\pi$
$809$	23607.8	1.02596	0.512982	+	0.858400i	$0.328541\pi$
$810$	−2204.10	−0.0956101
$811$	19669.5	0.851651	0.425825	−	0.904805i	$-0.359984\pi$
$811$	19669.5	0.851651	0.425825	+	0.904805i	$0.359984\pi$
$812$	793.638	0.0342995
$813$	24241.0	1.04572
$814$	2572.25	0.110759
$815$	22664.9	0.974132
$816$	−859.600	−0.0368775
$817$	12408.6	0.531360
$818$	24061.4	1.02847
$819$	−3702.02	−0.157948
$820$	−17494.8	−0.745053
$821$	32190.9	1.36842	0.684208	−	0.729287i	$-0.260148\pi$
$821$	32190.9	1.36842	0.684208	+	0.729287i	$0.260148\pi$
$822$	6832.10	0.289899
$823$	7078.32	0.299799	0.149899	−	0.988701i	$-0.452105\pi$
$823$	7078.32	0.299799	0.149899	+	0.988701i	$0.452105\pi$
$824$	−6069.64	−0.256609
$825$	−10464.3	−0.441601
$826$	557.548	0.0234862
$827$	34550.1	1.45275	0.726375	−	0.687298i	$-0.241203\pi$
$827$	34550.1	1.45275	0.726375	+	0.687298i	$0.241203\pi$
$828$	6205.29	0.260445
$829$	671.363	0.0281271	0.0140636	−	0.999901i	$-0.495523\pi$
$829$	671.363	0.0281271	0.0140636	+	0.999901i	$0.495523\pi$
$830$	24152.5	1.01005
$831$	3873.20	0.161684
$832$	−5571.55	−0.232162
$833$	5742.74	0.238865
$834$	173.193	0.00719088
$835$	−42195.4	−1.74878
$836$	6801.50	0.281381
$837$	−7773.37	−0.321012
$838$	−25691.6	−1.05907
$839$	23460.2	0.965360	0.482680	−	0.875797i	$-0.339663\pi$
$839$	23460.2	0.965360	0.482680	+	0.875797i	$0.339663\pi$
$840$	−1542.86	−0.0633736
$841$	−22625.7	−0.927701
$842$	−10626.1	−0.434918
$843$	−4565.58	−0.186533
$844$	−6638.32	−0.270735
$845$	−73220.5	−2.98090
$846$	−143.244	−0.00582130
$847$	9621.10	0.390301
$848$	−2832.08	−0.114686
$849$	23991.4	0.969825
$850$	−2152.98	−0.0868782
$851$	−3820.39	−0.153891
$852$	6869.66	0.276233
$853$	1994.35	0.0800529	0.0400264	−	0.999199i	$-0.487256\pi$
$853$	1994.35	0.0800529	0.0400264	+	0.999199i	$0.487256\pi$
$854$	2059.33	0.0825161
$855$	3588.12	0.143522
$856$	1230.55	0.0491349
$857$	7764.43	0.309484	0.154742	−	0.987955i	$-0.450545\pi$
$857$	7764.43	0.309484	0.154742	+	0.987955i	$0.450545\pi$
$858$	30309.8	1.20601
$859$	22883.2	0.908923	0.454461	−	0.890766i	$-0.349832\pi$
$859$	22883.2	0.908923	0.454461	+	0.890766i	$0.349832\pi$
$860$	23045.7	0.913780
$861$	4556.73	0.180363
$862$	−12064.5	−0.476705
$863$	18183.6	0.717238	0.358619	−	0.933484i	$-0.383248\pi$
$863$	18183.6	0.717238	0.358619	+	0.933484i	$0.383248\pi$
$864$	864.000	0.0340207
$865$	−9707.02	−0.381559
$866$	−18430.6	−0.723206
$867$	−13776.9	−0.539662
$868$	−5441.33	−0.212778
$869$	34120.0	1.33192
$870$	−3427.91	−0.133583
$871$	−48872.4	−1.90124
$872$	18017.0	0.699693
$873$	9465.04	0.366945
$874$	−10101.8	−0.390959
$875$	4171.45	0.161167
$876$	−2911.63	−0.112300
$877$	2607.13	0.100384	0.0501918	−	0.998740i	$-0.484017\pi$
$877$	2607.13	0.100384	0.0501918	+	0.998740i	$0.484017\pi$
$878$	25591.9	0.983697
$879$	−6160.65	−0.236398
$880$	12632.0	0.483891
$881$	−4802.17	−0.183643	−0.0918213	−	0.995776i	$-0.529269\pi$
$881$	−4802.17	−0.183643	−0.0918213	+	0.995776i	$0.529269\pi$
$882$	−5772.14	−0.220361
$883$	48441.6	1.84620	0.923098	−	0.384565i	$-0.125649\pi$
$883$	48441.6	1.84620	0.923098	+	0.384565i	$0.125649\pi$
$884$	6236.07	0.237265
$885$	−2408.18	−0.0914691
$886$	18146.5	0.688087
$887$	34835.5	1.31867	0.659335	−	0.751849i	$-0.270838\pi$
$887$	34835.5	1.31867	0.659335	+	0.751849i	$0.270838\pi$
$888$	−531.936	−0.0201020
$889$	−1425.23	−0.0537691
$890$	11654.0	0.438924
$891$	−4700.25	−0.176728
$892$	−20919.1	−0.785229
$893$	233.191	0.00873845
$894$	−8663.31	−0.324099
$895$	41840.2	1.56264
$896$	604.798	0.0225501
$897$	−45017.1	−1.67567
$898$	27976.0	1.03961
$899$	−12089.5	−0.448506
$900$	2164.00	0.0801480
$901$	3169.86	0.117207
$902$	−37307.6	−1.37717
$903$	−6002.53	−0.221209
$904$	5503.59	0.202485
$905$	45200.3	1.66023
$906$	11373.9	0.417077
$907$	14913.9	0.545986	0.272993	−	0.962016i	$-0.411986\pi$
$907$	14913.9	0.545986	0.272993	+	0.962016i	$0.411986\pi$
$908$	−18547.0	−0.677866
$909$	−1971.14	−0.0719238
$910$	11192.9	0.407737
$911$	−21396.5	−0.778153	−0.389076	−	0.921206i	$-0.627206\pi$
$911$	−21396.5	−0.778153	−0.389076	+	0.921206i	$0.627206\pi$
$912$	−1406.53	−0.0510690
$913$	51505.2	1.86700
$914$	9594.03	0.347202
$915$	−8894.74	−0.321367
$916$	−10870.3	−0.392101
$917$	−7218.79	−0.259962
$918$	−967.050	−0.0347684
$919$	−35338.7	−1.26846	−0.634231	−	0.773143i	$-0.718683\pi$
$919$	−35338.7	−1.26846	−0.634231	+	0.773143i	$0.718683\pi$
$920$	−18761.4	−0.672333
$921$	16857.4	0.603115
$922$	7936.42	0.283484
$923$	−49836.8	−1.77725
$924$	−3290.16	−0.117141
$925$	−1332.30	−0.0473576
$926$	547.757	0.0194389
$927$	−6828.34	−0.241933
$928$	1343.73	0.0475325
$929$	−6896.39	−0.243556	−0.121778	−	0.992557i	$-0.538860\pi$
$929$	−6896.39	−0.243556	−0.121778	+	0.992557i	$0.538860\pi$
$930$	23502.4	0.828683
$931$	9396.65	0.330787
$932$	1069.32	0.0375822
$933$	−14221.3	−0.499017
$934$	−16708.8	−0.585362
$935$	−14138.6	−0.494526
$936$	−6268.00	−0.218884
$937$	−55015.0	−1.91810	−0.959051	−	0.283233i	$-0.908593\pi$
$937$	−55015.0	−1.91810	−0.959051	+	0.283233i	$0.908593\pi$
$938$	5305.15	0.184669
$939$	−10654.1	−0.370270
$940$	433.091	0.0150275
$941$	−16723.6	−0.579356	−0.289678	−	0.957124i	$-0.593548\pi$
$941$	−16723.6	−0.579356	−0.289678	+	0.957124i	$0.593548\pi$
$942$	−14021.2	−0.484964
$943$	55410.4	1.91348
$944$	944.000	0.0325472
$945$	−1735.72	−0.0597492
$946$	49145.0	1.68905
$947$	2260.20	0.0775573	0.0387786	−	0.999248i	$-0.487653\pi$
$947$	2260.20	0.0775573	0.0387786	+	0.999248i	$0.487653\pi$
$948$	−7055.93	−0.241736
$949$	21122.8	0.722523
$950$	−3522.84	−0.120312
$951$	−7950.43	−0.271094
$952$	−676.932	−0.0230457
$953$	50880.7	1.72947	0.864736	−	0.502226i	$-0.167485\pi$
$953$	50880.7	1.72947	0.864736	+	0.502226i	$0.167485\pi$
$954$	−3186.09	−0.108127
$955$	28123.2	0.952929
$956$	17065.0	0.577325
$957$	−7310.03	−0.246917
$958$	−25303.7	−0.853366
$959$	5380.26	0.181165
$960$	−2612.27	−0.0878235
$961$	53096.8	1.78231
$962$	3859.00	0.129334
$963$	1384.37	0.0463248
$964$	−19418.1	−0.648770
$965$	63739.1	2.12625
$966$	4886.65	0.162759
$967$	−32911.8	−1.09449	−0.547246	−	0.836972i	$-0.684324\pi$
$967$	−32911.8	−1.09449	−0.547246	+	0.836972i	$0.684324\pi$
$968$	16289.8	0.540881
$969$	1574.29	0.0521915
$970$	−28617.1	−0.947259
$971$	−29583.7	−0.977742	−0.488871	−	0.872356i	$-0.662591\pi$
$971$	−29583.7	−0.977742	−0.488871	+	0.872356i	$0.662591\pi$
$972$	972.000	0.0320750
$973$	136.389	0.00449377
$974$	2208.34	0.0726486
$975$	−15699.0	−0.515662
$976$	3486.71	0.114351
$977$	35766.1	1.17120	0.585598	−	0.810602i	$-0.300860\pi$
$977$	35766.1	1.17120	0.585598	+	0.810602i	$0.300860\pi$
$978$	−9995.15	−0.326799
$979$	24852.2	0.811317
$980$	17451.8	0.568855
$981$	20269.1	0.659677
$982$	−10902.1	−0.354278
$983$	4150.15	0.134658	0.0673292	−	0.997731i	$-0.478552\pi$
$983$	4150.15	0.134658	0.0673292	+	0.997731i	$0.478552\pi$
$984$	7715.12	0.249948
$985$	−39164.0	−1.26687
$986$	−1504.00	−0.0485771
$987$	−112.804	−0.00363788
$988$	10203.9	0.328571
$989$	−72991.6	−2.34681
$990$	14211.0	0.456217
$991$	−4535.74	−0.145391	−0.0726955	−	0.997354i	$-0.523160\pi$
$991$	−4535.74	−0.145391	−0.0726955	+	0.997354i	$0.523160\pi$
$992$	−9212.88	−0.294868
$993$	7638.66	0.244114
$994$	5409.84	0.172625
$995$	−31340.7	−0.998561
$996$	−10651.2	−0.338850
$997$	25919.5	0.823349	0.411675	−	0.911331i	$-0.364944\pi$
$997$	25919.5	0.823349	0.411675	+	0.911331i	$0.364944\pi$
$998$	−4864.90	−0.154304
$999$	−598.428	−0.0189524

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.4.a.c.1.1	✓	2
3.2	odd	2		1062.4.a.f.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.4.a.c.1.1	✓	2	1.1	even	1	trivial
1062.4.a.f.1.2		2	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:	\( N \)	\(=\)	\( 354 = 2 \cdot 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	354.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -13.6056 q^{5} +6.00000 q^{6} +4.72498 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -13.6056 q^{5} +6.00000 q^{6} +4.72498 q^{7} +8.00000 q^{8} +9.00000 q^{9} -27.2111 q^{10} -58.0278 q^{11} +12.0000 q^{12} -87.0555 q^{13} +9.44996 q^{14} -40.8167 q^{15} +16.0000 q^{16} -17.9083 q^{17} +18.0000 q^{18} -29.3028 q^{19} -54.4222 q^{20} +14.1749 q^{21} -116.056 q^{22} +172.369 q^{23} +24.0000 q^{24} +60.1110 q^{25} -174.111 q^{26} +27.0000 q^{27} +18.8999 q^{28} +41.9916 q^{29} -81.6333 q^{30} -287.902 q^{31} +32.0000 q^{32} -174.083 q^{33} -35.8167 q^{34} -64.2860 q^{35} +36.0000 q^{36} -22.1640 q^{37} -58.6056 q^{38} -261.167 q^{39} -108.844 q^{40} +321.463 q^{41} +28.3499 q^{42} -423.461 q^{43} -232.111 q^{44} -122.450 q^{45} +344.738 q^{46} -7.95798 q^{47} +48.0000 q^{48} -320.675 q^{49} +120.222 q^{50} -53.7250 q^{51} -348.222 q^{52} -177.005 q^{53} +54.0000 q^{54} +789.500 q^{55} +37.7998 q^{56} -87.9083 q^{57} +83.9832 q^{58} +59.0000 q^{59} -163.267 q^{60} +217.919 q^{61} -575.805 q^{62} +42.5248 q^{63} +64.0000 q^{64} +1184.44 q^{65} -348.167 q^{66} +561.394 q^{67} -71.6333 q^{68} +517.108 q^{69} -128.572 q^{70} +572.472 q^{71} +72.0000 q^{72} -242.636 q^{73} -44.3280 q^{74} +180.333 q^{75} -117.211 q^{76} -274.180 q^{77} -522.333 q^{78} -587.994 q^{79} -217.689 q^{80} +81.0000 q^{81} +642.927 q^{82} -887.596 q^{83} +56.6998 q^{84} +243.653 q^{85} -846.922 q^{86} +125.975 q^{87} -464.222 q^{88} -428.281 q^{89} -244.900 q^{90} -411.336 q^{91} +689.477 q^{92} -863.707 q^{93} -15.9160 q^{94} +398.680 q^{95} +96.0000 q^{96} +1051.67 q^{97} -641.349 q^{98} -522.250 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} - 20 q^{5} + 12 q^{6} - 23 q^{7} + 16 q^{8} + 18 q^{9}+O(q^{10}) \) 2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 - 20 * q^5 + 12 * q^6 - 23 * q^7 + 16 * q^8 + 18 * q^9	\( 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} - 20 q^{5} + 12 q^{6} - 23 q^{7} + 16 q^{8} + 18 q^{9} - 40 q^{10} - 80 q^{11} + 24 q^{12} - 102 q^{13} - 46 q^{14} - 60 q^{15} + 32 q^{16} - 25 q^{17} + 36 q^{18} - 55 q^{19} - 80 q^{20} - 69 q^{21} - 160 q^{22} - 5 q^{23} + 48 q^{24} - 24 q^{25} - 204 q^{26} + 54 q^{27} - 92 q^{28} - 35 q^{29} - 120 q^{30} - 53 q^{31} + 64 q^{32} - 240 q^{33} - 50 q^{34} + 113 q^{35} + 72 q^{36} - 221 q^{37} - 110 q^{38} - 306 q^{39} - 160 q^{40} - 89 q^{41} - 138 q^{42} - 508 q^{43} - 320 q^{44} - 180 q^{45} - 10 q^{46} + 579 q^{47} + 96 q^{48} + 105 q^{49} - 48 q^{50} - 75 q^{51} - 408 q^{52} + 432 q^{53} + 108 q^{54} + 930 q^{55} - 184 q^{56} - 165 q^{57} - 70 q^{58} + 118 q^{59} - 240 q^{60} + 151 q^{61} - 106 q^{62} - 207 q^{63} + 128 q^{64} + 1280 q^{65} - 480 q^{66} - 168 q^{67} - 100 q^{68} - 15 q^{69} + 226 q^{70} + 532 q^{71} + 144 q^{72} - 49 q^{73} - 442 q^{74} - 72 q^{75} - 220 q^{76} + 335 q^{77} - 612 q^{78} - 664 q^{79} - 320 q^{80} + 162 q^{81} - 178 q^{82} - 351 q^{83} - 276 q^{84} + 289 q^{85} - 1016 q^{86} - 105 q^{87} - 640 q^{88} - 1401 q^{89} - 360 q^{90} + 3 q^{91} - 20 q^{92} - 159 q^{93} + 1158 q^{94} + 563 q^{95} + 192 q^{96} + 452 q^{97} + 210 q^{98} - 720 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 - 20 * q^5 + 12 * q^6 - 23 * q^7 + 16 * q^8 + 18 * q^9 - 40 * q^10 - 80 * q^11 + 24 * q^12 - 102 * q^13 - 46 * q^14 - 60 * q^15 + 32 * q^16 - 25 * q^17 + 36 * q^18 - 55 * q^19 - 80 * q^20 - 69 * q^21 - 160 * q^22 - 5 * q^23 + 48 * q^24 - 24 * q^25 - 204 * q^26 + 54 * q^27 - 92 * q^28 - 35 * q^29 - 120 * q^30 - 53 * q^31 + 64 * q^32 - 240 * q^33 - 50 * q^34 + 113 * q^35 + 72 * q^36 - 221 * q^37 - 110 * q^38 - 306 * q^39 - 160 * q^40 - 89 * q^41 - 138 * q^42 - 508 * q^43 - 320 * q^44 - 180 * q^45 - 10 * q^46 + 579 * q^47 + 96 * q^48 + 105 * q^49 - 48 * q^50 - 75 * q^51 - 408 * q^52 + 432 * q^53 + 108 * q^54 + 930 * q^55 - 184 * q^56 - 165 * q^57 - 70 * q^58 + 118 * q^59 - 240 * q^60 + 151 * q^61 - 106 * q^62 - 207 * q^63 + 128 * q^64 + 1280 * q^65 - 480 * q^66 - 168 * q^67 - 100 * q^68 - 15 * q^69 + 226 * q^70 + 532 * q^71 + 144 * q^72 - 49 * q^73 - 442 * q^74 - 72 * q^75 - 220 * q^76 + 335 * q^77 - 612 * q^78 - 664 * q^79 - 320 * q^80 + 162 * q^81 - 178 * q^82 - 351 * q^83 - 276 * q^84 + 289 * q^85 - 1016 * q^86 - 105 * q^87 - 640 * q^88 - 1401 * q^89 - 360 * q^90 + 3 * q^91 - 20 * q^92 - 159 * q^93 + 1158 * q^94 + 563 * q^95 + 192 * q^96 + 452 * q^97 + 210 * q^98 - 720 * q^99

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