LMFDB - Embedded newform 8034.2.a.ba.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8034,2,Mod(1,8034)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8034, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8034.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8034 = 2 \cdot 3 \cdot 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8034.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:	$64.1518129839$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - x^{13} - 44 x^{12} + 36 x^{11} + 722 x^{10} - 451 x^{9} - 5438 x^{8} + 2268 x^{7} + 18441 x^{6} + \cdots - 1492 $ x^14 - x^13 - 44x^12 + 36x^11 + 722x^10 - 451x^9 - 5438x^8 + 2268x^7 + 18441x^6 - 4126x^5 - 22759x^4 + 2997x^3 + 10504x^2 - 506x - 1492
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			1.13
Root			$-3.59906$ of defining polynomial
Character	$\chi$	$=$	8034.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.59906 q^{5} +1.00000 q^{6} +0.919823 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.59906 q^{5} +1.00000 q^{6} +0.919823 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.59906 q^{10} +6.28384 q^{11} -1.00000 q^{12} -1.00000 q^{13} -0.919823 q^{14} -3.59906 q^{15} +1.00000 q^{16} -0.964091 q^{17} -1.00000 q^{18} +5.87871 q^{19} +3.59906 q^{20} -0.919823 q^{21} -6.28384 q^{22} -1.55833 q^{23} +1.00000 q^{24} +7.95327 q^{25} +1.00000 q^{26} -1.00000 q^{27} +0.919823 q^{28} +0.377820 q^{29} +3.59906 q^{30} +3.05624 q^{31} -1.00000 q^{32} -6.28384 q^{33} +0.964091 q^{34} +3.31050 q^{35} +1.00000 q^{36} +5.17705 q^{37} -5.87871 q^{38} +1.00000 q^{39} -3.59906 q^{40} +5.29346 q^{41} +0.919823 q^{42} -9.33676 q^{43} +6.28384 q^{44} +3.59906 q^{45} +1.55833 q^{46} +6.01076 q^{47} -1.00000 q^{48} -6.15393 q^{49} -7.95327 q^{50} +0.964091 q^{51} -1.00000 q^{52} +2.69483 q^{53} +1.00000 q^{54} +22.6159 q^{55} -0.919823 q^{56} -5.87871 q^{57} -0.377820 q^{58} +0.838786 q^{59} -3.59906 q^{60} +0.508838 q^{61} -3.05624 q^{62} +0.919823 q^{63} +1.00000 q^{64} -3.59906 q^{65} +6.28384 q^{66} -14.4793 q^{67} -0.964091 q^{68} +1.55833 q^{69} -3.31050 q^{70} -3.24540 q^{71} -1.00000 q^{72} +5.27039 q^{73} -5.17705 q^{74} -7.95327 q^{75} +5.87871 q^{76} +5.78001 q^{77} -1.00000 q^{78} +9.98774 q^{79} +3.59906 q^{80} +1.00000 q^{81} -5.29346 q^{82} -11.4796 q^{83} -0.919823 q^{84} -3.46983 q^{85} +9.33676 q^{86} -0.377820 q^{87} -6.28384 q^{88} +17.3823 q^{89} -3.59906 q^{90} -0.919823 q^{91} -1.55833 q^{92} -3.05624 q^{93} -6.01076 q^{94} +21.1579 q^{95} +1.00000 q^{96} +3.40393 q^{97} +6.15393 q^{98} +6.28384 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{2} - 14 q^{3} + 14 q^{4} - q^{5} + 14 q^{6} + 5 q^{7} - 14 q^{8} + 14 q^{9}+O(q^{10}) $ 14 * q - 14 * q^2 - 14 * q^3 + 14 * q^4 - q^5 + 14 * q^6 + 5 * q^7 - 14 * q^8 + 14 * q^9	\( 14 q - 14 q^{2} - 14 q^{3} + 14 q^{4} - q^{5} + 14 q^{6} + 5 q^{7} - 14 q^{8} + 14 q^{9} + q^{10} - q^{11} - 14 q^{12} - 14 q^{13} - 5 q^{14} + q^{15} + 14 q^{16} - 12 q^{17} - 14 q^{18} + 10 q^{19} - q^{20} - 5 q^{21} + q^{22} - q^{23} + 14 q^{24} + 19 q^{25} + 14 q^{26} - 14 q^{27} + 5 q^{28} + 6 q^{29} - q^{30} + 20 q^{31} - 14 q^{32} + q^{33} + 12 q^{34} - 16 q^{35} + 14 q^{36} - 3 q^{37} - 10 q^{38} + 14 q^{39} + q^{40} + q^{41} + 5 q^{42} + 6 q^{43} - q^{44} - q^{45} + q^{46} - 13 q^{47} - 14 q^{48} + 9 q^{49} - 19 q^{50} + 12 q^{51} - 14 q^{52} - 27 q^{53} + 14 q^{54} + 10 q^{55} - 5 q^{56} - 10 q^{57} - 6 q^{58} - 6 q^{59} + q^{60} - 4 q^{61} - 20 q^{62} + 5 q^{63} + 14 q^{64} + q^{65} - q^{66} + 13 q^{67} - 12 q^{68} + q^{69} + 16 q^{70} + 18 q^{71} - 14 q^{72} + 11 q^{73} + 3 q^{74} - 19 q^{75} + 10 q^{76} - 15 q^{77} - 14 q^{78} + 33 q^{79} - q^{80} + 14 q^{81} - q^{82} - 25 q^{83} - 5 q^{84} + 25 q^{85} - 6 q^{86} - 6 q^{87} + q^{88} + 3 q^{89} + q^{90} - 5 q^{91} - q^{92} - 20 q^{93} + 13 q^{94} + 30 q^{95} + 14 q^{96} + 11 q^{97} - 9 q^{98} - q^{99}+O(q^{100}) \) 14 * q - 14 * q^2 - 14 * q^3 + 14 * q^4 - q^5 + 14 * q^6 + 5 * q^7 - 14 * q^8 + 14 * q^9 + q^10 - q^11 - 14 * q^12 - 14 * q^13 - 5 * q^14 + q^15 + 14 * q^16 - 12 * q^17 - 14 * q^18 + 10 * q^19 - q^20 - 5 * q^21 + q^22 - q^23 + 14 * q^24 + 19 * q^25 + 14 * q^26 - 14 * q^27 + 5 * q^28 + 6 * q^29 - q^30 + 20 * q^31 - 14 * q^32 + q^33 + 12 * q^34 - 16 * q^35 + 14 * q^36 - 3 * q^37 - 10 * q^38 + 14 * q^39 + q^40 + q^41 + 5 * q^42 + 6 * q^43 - q^44 - q^45 + q^46 - 13 * q^47 - 14 * q^48 + 9 * q^49 - 19 * q^50 + 12 * q^51 - 14 * q^52 - 27 * q^53 + 14 * q^54 + 10 * q^55 - 5 * q^56 - 10 * q^57 - 6 * q^58 - 6 * q^59 + q^60 - 4 * q^61 - 20 * q^62 + 5 * q^63 + 14 * q^64 + q^65 - q^66 + 13 * q^67 - 12 * q^68 + q^69 + 16 * q^70 + 18 * q^71 - 14 * q^72 + 11 * q^73 + 3 * q^74 - 19 * q^75 + 10 * q^76 - 15 * q^77 - 14 * q^78 + 33 * q^79 - q^80 + 14 * q^81 - q^82 - 25 * q^83 - 5 * q^84 + 25 * q^85 - 6 * q^86 - 6 * q^87 + q^88 + 3 * q^89 + q^90 - 5 * q^91 - q^92 - 20 * q^93 + 13 * q^94 + 30 * q^95 + 14 * q^96 + 11 * q^97 - 9 * q^98 - 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	3.59906	1.60955	0.804775	−	0.593579i	$-0.202286\pi$
$5$	3.59906	1.60955	0.804775	+	0.593579i	$0.202286\pi$
$6$	1.00000	0.408248
$7$	0.919823	0.347660	0.173830	−	0.984776i	$-0.444386\pi$
$7$	0.919823	0.347660	0.173830	+	0.984776i	$0.444386\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−3.59906	−1.13812
$11$	6.28384	1.89465	0.947324	−	0.320277i	$-0.103776\pi$
$11$	6.28384	1.89465	0.947324	+	0.320277i	$0.103776\pi$
$12$	−1.00000	−0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−0.919823	−0.245833
$15$	−3.59906	−0.929275
$16$	1.00000	0.250000
$17$	−0.964091	−0.233826	−0.116913	−	0.993142i	$-0.537300\pi$
$17$	−0.964091	−0.233826	−0.116913	+	0.993142i	$0.537300\pi$
$18$	−1.00000	−0.235702
$19$	5.87871	1.34867	0.674335	−	0.738426i	$-0.264431\pi$
$19$	5.87871	1.34867	0.674335	+	0.738426i	$0.264431\pi$
$20$	3.59906	0.804775
$21$	−0.919823	−0.200722
$22$	−6.28384	−1.33972
$23$	−1.55833	−0.324934	−0.162467	−	0.986714i	$-0.551945\pi$
$23$	−1.55833	−0.324934	−0.162467	+	0.986714i	$0.551945\pi$
$24$	1.00000	0.204124
$25$	7.95327	1.59065
$26$	1.00000	0.196116
$27$	−1.00000	−0.192450
$28$	0.919823	0.173830
$29$	0.377820	0.0701595	0.0350797	−	0.999385i	$-0.488831\pi$
$29$	0.377820	0.0701595	0.0350797	+	0.999385i	$0.488831\pi$
$30$	3.59906	0.657096
$31$	3.05624	0.548917	0.274458	−	0.961599i	$-0.411501\pi$
$31$	3.05624	0.548917	0.274458	+	0.961599i	$0.411501\pi$
$32$	−1.00000	−0.176777
$33$	−6.28384	−1.09388
$34$	0.964091	0.165340
$35$	3.31050	0.559577
$36$	1.00000	0.166667
$37$	5.17705	0.851101	0.425551	−	0.904935i	$-0.360080\pi$
$37$	5.17705	0.851101	0.425551	+	0.904935i	$0.360080\pi$
$38$	−5.87871	−0.953653
$39$	1.00000	0.160128
$40$	−3.59906	−0.569062
$41$	5.29346	0.826700	0.413350	−	0.910572i	$-0.364359\pi$
$41$	5.29346	0.826700	0.413350	+	0.910572i	$0.364359\pi$
$42$	0.919823	0.141932
$43$	−9.33676	−1.42384	−0.711921	−	0.702259i	$-0.752175\pi$
$43$	−9.33676	−1.42384	−0.711921	+	0.702259i	$0.752175\pi$
$44$	6.28384	0.947324
$45$	3.59906	0.536517
$46$	1.55833	0.229763
$47$	6.01076	0.876759	0.438379	−	0.898790i	$-0.355553\pi$
$47$	6.01076	0.876759	0.438379	+	0.898790i	$0.355553\pi$
$48$	−1.00000	−0.144338
$49$	−6.15393	−0.879132
$50$	−7.95327	−1.12476
$51$	0.964091	0.135000
$52$	−1.00000	−0.138675
$53$	2.69483	0.370163	0.185081	−	0.982723i	$-0.440745\pi$
$53$	2.69483	0.370163	0.185081	+	0.982723i	$0.440745\pi$
$54$	1.00000	0.136083
$55$	22.6159	3.04953
$56$	−0.919823	−0.122916
$57$	−5.87871	−0.778655
$58$	−0.377820	−0.0496102
$59$	0.838786	0.109201	0.0546003	−	0.998508i	$-0.482612\pi$
$59$	0.838786	0.109201	0.0546003	+	0.998508i	$0.482612\pi$
$60$	−3.59906	−0.464637
$61$	0.508838	0.0651501	0.0325750	−	0.999469i	$-0.489629\pi$
$61$	0.508838	0.0651501	0.0325750	+	0.999469i	$0.489629\pi$
$62$	−3.05624	−0.388143
$63$	0.919823	0.115887
$64$	1.00000	0.125000
$65$	−3.59906	−0.446409
$66$	6.28384	0.773487
$67$	−14.4793	−1.76893	−0.884465	−	0.466606i	$-0.845477\pi$
$67$	−14.4793	−1.76893	−0.884465	+	0.466606i	$0.845477\pi$
$68$	−0.964091	−0.116913
$69$	1.55833	0.187601
$70$	−3.31050	−0.395681
$71$	−3.24540	−0.385158	−0.192579	−	0.981282i	$-0.561685\pi$
$71$	−3.24540	−0.385158	−0.192579	+	0.981282i	$0.561685\pi$
$72$	−1.00000	−0.117851
$73$	5.27039	0.616852	0.308426	−	0.951248i	$-0.400198\pi$
$73$	5.27039	0.616852	0.308426	+	0.951248i	$0.400198\pi$
$74$	−5.17705	−0.601819
$75$	−7.95327	−0.918364
$76$	5.87871	0.674335
$77$	5.78001	0.658694
$78$	−1.00000	−0.113228
$79$	9.98774	1.12371	0.561854	−	0.827236i	$-0.310088\pi$
$79$	9.98774	1.12371	0.561854	+	0.827236i	$0.310088\pi$
$80$	3.59906	0.402388
$81$	1.00000	0.111111
$82$	−5.29346	−0.584565
$83$	−11.4796	−1.26005	−0.630024	−	0.776576i	$-0.716955\pi$
$83$	−11.4796	−1.26005	−0.630024	+	0.776576i	$0.716955\pi$
$84$	−0.919823	−0.100361
$85$	−3.46983	−0.376356
$86$	9.33676	1.00681
$87$	−0.377820	−0.0405066
$88$	−6.28384	−0.669859
$89$	17.3823	1.84252	0.921260	−	0.388946i	$-0.127161\pi$
$89$	17.3823	1.84252	0.921260	+	0.388946i	$0.127161\pi$
$90$	−3.59906	−0.379375
$91$	−0.919823	−0.0964236
$92$	−1.55833	−0.162467
$93$	−3.05624	−0.316917
$94$	−6.01076	−0.619962
$95$	21.1579	2.17075
$96$	1.00000	0.102062
$97$	3.40393	0.345617	0.172808	−	0.984955i	$-0.444716\pi$
$97$	3.40393	0.345617	0.172808	+	0.984955i	$0.444716\pi$
$98$	6.15393	0.621640
$99$	6.28384	0.631549
$100$	7.95327	0.795327
$101$	0.987205	0.0982306	0.0491153	−	0.998793i	$-0.484360\pi$
$101$	0.987205	0.0982306	0.0491153	+	0.998793i	$0.484360\pi$
$102$	−0.964091	−0.0954593
$103$	1.00000	0.0985329
$103$	1.00000	0.0985329
$104$	1.00000	0.0980581
$105$	−3.31050	−0.323072
$106$	−2.69483	−0.261745
$107$	17.9742	1.73763	0.868814	−	0.495138i	$-0.164883\pi$
$107$	17.9742	1.73763	0.868814	+	0.495138i	$0.164883\pi$
$108$	−1.00000	−0.0962250
$109$	−8.38673	−0.803303	−0.401651	−	0.915793i	$-0.631564\pi$
$109$	−8.38673	−0.803303	−0.401651	+	0.915793i	$0.631564\pi$
$110$	−22.6159	−2.15634
$111$	−5.17705	−0.491384
$112$	0.919823	0.0869151
$113$	5.12907	0.482502	0.241251	−	0.970463i	$-0.422442\pi$
$113$	5.12907	0.482502	0.241251	+	0.970463i	$0.422442\pi$
$114$	5.87871	0.550592
$115$	−5.60853	−0.522998
$116$	0.377820	0.0350797
$117$	−1.00000	−0.0924500
$118$	−0.838786	−0.0772165
$119$	−0.886793	−0.0812922
$120$	3.59906	0.328548
$121$	28.4866	2.58969
$122$	−0.508838	−0.0460681
$123$	−5.29346	−0.477295
$124$	3.05624	0.274458
$125$	10.6290	0.950687
$126$	−0.919823	−0.0819443
$127$	4.19542	0.372284	0.186142	−	0.982523i	$-0.440402\pi$
$127$	4.19542	0.372284	0.186142	+	0.982523i	$0.440402\pi$
$128$	−1.00000	−0.0883883
$129$	9.33676	0.822056
$130$	3.59906	0.315659
$131$	3.43244	0.299894	0.149947	−	0.988694i	$-0.452090\pi$
$131$	3.43244	0.299894	0.149947	+	0.988694i	$0.452090\pi$
$132$	−6.28384	−0.546938
$133$	5.40737	0.468879
$134$	14.4793	1.25082
$135$	−3.59906	−0.309758
$136$	0.964091	0.0826701
$137$	−17.0982	−1.46079	−0.730397	−	0.683023i	$-0.760665\pi$
$137$	−17.0982	−1.46079	−0.730397	+	0.683023i	$0.760665\pi$
$138$	−1.55833	−0.132654
$139$	−9.30098	−0.788899	−0.394449	−	0.918918i	$-0.629065\pi$
$139$	−9.30098	−0.788899	−0.394449	+	0.918918i	$0.629065\pi$
$140$	3.31050	0.279788
$141$	−6.01076	−0.506197
$142$	3.24540	0.272348
$143$	−6.28384	−0.525481
$144$	1.00000	0.0833333
$145$	1.35980	0.112925
$146$	−5.27039	−0.436180
$147$	6.15393	0.507567
$148$	5.17705	0.425551
$149$	−7.18823	−0.588883	−0.294442	−	0.955670i	$-0.595134\pi$
$149$	−7.18823	−0.588883	−0.294442	+	0.955670i	$0.595134\pi$
$150$	7.95327	0.649382
$151$	−12.2638	−0.998013	−0.499006	−	0.866598i	$-0.666302\pi$
$151$	−12.2638	−0.998013	−0.499006	+	0.866598i	$0.666302\pi$
$152$	−5.87871	−0.476827
$153$	−0.964091	−0.0779422
$154$	−5.78001	−0.465767
$155$	10.9996	0.883510
$156$	1.00000	0.0800641
$157$	−16.6452	−1.32843	−0.664216	−	0.747540i	$-0.731235\pi$
$157$	−16.6452	−1.32843	−0.664216	+	0.747540i	$0.731235\pi$
$158$	−9.98774	−0.794582
$159$	−2.69483	−0.213714
$160$	−3.59906	−0.284531
$161$	−1.43339	−0.112967
$162$	−1.00000	−0.0785674
$163$	19.0669	1.49344	0.746719	−	0.665140i	$-0.231628\pi$
$163$	19.0669	1.49344	0.746719	+	0.665140i	$0.231628\pi$
$164$	5.29346	0.413350
$165$	−22.6159	−1.76065
$166$	11.4796	0.890988
$167$	3.57968	0.277004	0.138502	−	0.990362i	$-0.455771\pi$
$167$	3.57968	0.277004	0.138502	+	0.990362i	$0.455771\pi$
$168$	0.919823	0.0709658
$169$	1.00000	0.0769231
$170$	3.46983	0.266124
$171$	5.87871	0.449556
$172$	−9.33676	−0.711921
$173$	−1.57490	−0.119738	−0.0598689	−	0.998206i	$-0.519068\pi$
$173$	−1.57490	−0.119738	−0.0598689	+	0.998206i	$0.519068\pi$
$174$	0.377820	0.0286425
$175$	7.31559	0.553007
$176$	6.28384	0.473662
$177$	−0.838786	−0.0630470
$178$	−17.3823	−1.30286
$179$	−8.00874	−0.598601	−0.299301	−	0.954159i	$-0.596753\pi$
$179$	−8.00874	−0.598601	−0.299301	+	0.954159i	$0.596753\pi$
$180$	3.59906	0.268258
$181$	6.76636	0.502939	0.251470	−	0.967865i	$-0.419086\pi$
$181$	6.76636	0.502939	0.251470	+	0.967865i	$0.419086\pi$
$182$	0.919823	0.0681818
$183$	−0.508838	−0.0376144
$184$	1.55833	0.114882
$185$	18.6325	1.36989
$186$	3.05624	0.224094
$187$	−6.05819	−0.443019
$188$	6.01076	0.438379
$189$	−0.919823	−0.0669072
$190$	−21.1579	−1.53495
$191$	3.62788	0.262504	0.131252	−	0.991349i	$-0.458100\pi$
$191$	3.62788	0.262504	0.131252	+	0.991349i	$0.458100\pi$
$192$	−1.00000	−0.0721688
$193$	−17.8569	−1.28537	−0.642684	−	0.766132i	$-0.722179\pi$
$193$	−17.8569	−1.28537	−0.642684	+	0.766132i	$0.722179\pi$
$194$	−3.40393	−0.244388
$195$	3.59906	0.257734
$196$	−6.15393	−0.439566
$197$	−1.10763	−0.0789151	−0.0394575	−	0.999221i	$-0.512563\pi$
$197$	−1.10763	−0.0789151	−0.0394575	+	0.999221i	$0.512563\pi$
$198$	−6.28384	−0.446573
$199$	−9.47297	−0.671521	−0.335761	−	0.941947i	$-0.608993\pi$
$199$	−9.47297	−0.671521	−0.335761	+	0.941947i	$0.608993\pi$
$200$	−7.95327	−0.562381
$201$	14.4793	1.02129
$202$	−0.987205	−0.0694595
$203$	0.347528	0.0243917
$204$	0.964091	0.0674999
$205$	19.0515	1.33062
$206$	−1.00000	−0.0696733
$207$	−1.55833	−0.108311
$208$	−1.00000	−0.0693375
$209$	36.9409	2.55525
$210$	3.31050	0.228446
$211$	16.5894	1.14206	0.571030	−	0.820929i	$-0.306544\pi$
$211$	16.5894	1.14206	0.571030	+	0.820929i	$0.306544\pi$
$212$	2.69483	0.185081
$213$	3.24540	0.222371
$214$	−17.9742	−1.22869
$215$	−33.6036	−2.29175
$216$	1.00000	0.0680414
$217$	2.81120	0.190837
$218$	8.38673	0.568021
$219$	−5.27039	−0.356140
$220$	22.6159	1.52477
$221$	0.964091	0.0648518
$222$	5.17705	0.347461
$223$	10.5549	0.706811	0.353406	−	0.935470i	$-0.385024\pi$
$223$	10.5549	0.706811	0.353406	+	0.935470i	$0.385024\pi$
$224$	−0.919823	−0.0614582
$225$	7.95327	0.530218
$226$	−5.12907	−0.341180
$227$	−6.01618	−0.399308	−0.199654	−	0.979866i	$-0.563982\pi$
$227$	−6.01618	−0.399308	−0.199654	+	0.979866i	$0.563982\pi$
$228$	−5.87871	−0.389327
$229$	−17.7551	−1.17329	−0.586645	−	0.809844i	$-0.699551\pi$
$229$	−17.7551	−1.17329	−0.586645	+	0.809844i	$0.699551\pi$
$230$	5.60853	0.369816
$231$	−5.78001	−0.380297
$232$	−0.377820	−0.0248051
$233$	−11.1388	−0.729726	−0.364863	−	0.931061i	$-0.618884\pi$
$233$	−11.1388	−0.729726	−0.364863	+	0.931061i	$0.618884\pi$
$234$	1.00000	0.0653720
$235$	21.6331	1.41119
$236$	0.838786	0.0546003
$237$	−9.98774	−0.648774
$238$	0.886793	0.0574822
$239$	−11.3486	−0.734080	−0.367040	−	0.930205i	$-0.619629\pi$
$239$	−11.3486	−0.734080	−0.367040	+	0.930205i	$0.619629\pi$
$240$	−3.59906	−0.232319
$241$	13.8214	0.890316	0.445158	−	0.895452i	$-0.353148\pi$
$241$	13.8214	0.890316	0.445158	+	0.895452i	$0.353148\pi$
$242$	−28.4866	−1.83119
$243$	−1.00000	−0.0641500
$244$	0.508838	0.0325750
$245$	−22.1484	−1.41501
$246$	5.29346	0.337499
$247$	−5.87871	−0.374054
$248$	−3.05624	−0.194071
$249$	11.4796	0.727489
$250$	−10.6290	−0.672237
$251$	16.6162	1.04880	0.524402	−	0.851471i	$-0.324289\pi$
$251$	16.6162	1.04880	0.524402	+	0.851471i	$0.324289\pi$
$252$	0.919823	0.0579434
$253$	−9.79230	−0.615636
$254$	−4.19542	−0.263244
$255$	3.46983	0.217289
$256$	1.00000	0.0625000
$257$	−15.8529	−0.988879	−0.494440	−	0.869212i	$-0.664627\pi$
$257$	−15.8529	−0.988879	−0.494440	+	0.869212i	$0.664627\pi$
$258$	−9.33676	−0.581281
$259$	4.76196	0.295894
$260$	−3.59906	−0.223205
$261$	0.377820	0.0233865
$262$	−3.43244	−0.212057
$263$	−14.0523	−0.866502	−0.433251	−	0.901273i	$-0.642634\pi$
$263$	−14.0523	−0.866502	−0.433251	+	0.901273i	$0.642634\pi$
$264$	6.28384	0.386743
$265$	9.69886	0.595796
$266$	−5.40737	−0.331547
$267$	−17.3823	−1.06378
$268$	−14.4793	−0.884465
$269$	−21.7523	−1.32626	−0.663130	−	0.748505i	$-0.730772\pi$
$269$	−21.7523	−1.32626	−0.663130	+	0.748505i	$0.730772\pi$
$270$	3.59906	0.219032
$271$	29.6801	1.80294	0.901469	−	0.432844i	$-0.142490\pi$
$271$	29.6801	1.80294	0.901469	+	0.432844i	$0.142490\pi$
$272$	−0.964091	−0.0584566
$273$	0.919823	0.0556702
$274$	17.0982	1.03294
$275$	49.9770	3.01373
$276$	1.55833	0.0938005
$277$	0.303986	0.0182647	0.00913237	−	0.999958i	$-0.497093\pi$
$277$	0.303986	0.0182647	0.00913237	+	0.999958i	$0.497093\pi$
$278$	9.30098	0.557836
$279$	3.05624	0.182972
$280$	−3.31050	−0.197840
$281$	−31.4890	−1.87848	−0.939240	−	0.343262i	$-0.888468\pi$
$281$	−31.4890	−1.87848	−0.939240	+	0.343262i	$0.888468\pi$
$282$	6.01076	0.357935
$283$	7.53378	0.447837	0.223918	−	0.974608i	$-0.428115\pi$
$283$	7.53378	0.447837	0.223918	+	0.974608i	$0.428115\pi$
$284$	−3.24540	−0.192579
$285$	−21.1579	−1.25328
$286$	6.28384	0.371571
$287$	4.86905	0.287411
$288$	−1.00000	−0.0589256
$289$	−16.0705	−0.945325
$290$	−1.35980	−0.0798502
$291$	−3.40393	−0.199542
$292$	5.27039	0.308426
$293$	28.1598	1.64511	0.822556	−	0.568683i	$-0.192547\pi$
$293$	28.1598	1.64511	0.822556	+	0.568683i	$0.192547\pi$
$294$	−6.15393	−0.358904
$295$	3.01885	0.175764
$296$	−5.17705	−0.300910
$297$	−6.28384	−0.364625
$298$	7.18823	0.416403
$299$	1.55833	0.0901206
$300$	−7.95327	−0.459182
$301$	−8.58816	−0.495013
$302$	12.2638	0.705701
$303$	−0.987205	−0.0567135
$304$	5.87871	0.337167
$305$	1.83134	0.104862
$306$	0.964091	0.0551134
$307$	3.43777	0.196204	0.0981021	−	0.995176i	$-0.468723\pi$
$307$	3.43777	0.196204	0.0981021	+	0.995176i	$0.468723\pi$
$308$	5.78001	0.329347
$309$	−1.00000	−0.0568880
$310$	−10.9996	−0.624736
$311$	−1.77167	−0.100462	−0.0502311	−	0.998738i	$-0.515996\pi$
$311$	−1.77167	−0.100462	−0.0502311	+	0.998738i	$0.515996\pi$
$312$	−1.00000	−0.0566139
$313$	−27.0983	−1.53169	−0.765843	−	0.643028i	$-0.777678\pi$
$313$	−27.0983	−1.53169	−0.765843	+	0.643028i	$0.777678\pi$
$314$	16.6452	0.939344
$315$	3.31050	0.186526
$316$	9.98774	0.561854
$317$	−26.5689	−1.49226	−0.746130	−	0.665800i	$-0.768090\pi$
$317$	−26.5689	−1.49226	−0.746130	+	0.665800i	$0.768090\pi$
$318$	2.69483	0.151118
$319$	2.37416	0.132928
$320$	3.59906	0.201194
$321$	−17.9742	−1.00322
$322$	1.43339	0.0798796
$323$	−5.66762	−0.315355
$324$	1.00000	0.0555556
$325$	−7.95327	−0.441168
$326$	−19.0669	−1.05602
$327$	8.38673	0.463787
$328$	−5.29346	−0.292283
$329$	5.52883	0.304814
$330$	22.6159	1.24497
$331$	−23.2926	−1.28028	−0.640139	−	0.768259i	$-0.721123\pi$
$331$	−23.2926	−1.28028	−0.640139	+	0.768259i	$0.721123\pi$
$332$	−11.4796	−0.630024
$333$	5.17705	0.283700
$334$	−3.57968	−0.195871
$335$	−52.1120	−2.84718
$336$	−0.919823	−0.0501804
$337$	15.3551	0.836446	0.418223	−	0.908344i	$-0.362653\pi$
$337$	15.3551	0.836446	0.418223	+	0.908344i	$0.362653\pi$
$338$	−1.00000	−0.0543928
$339$	−5.12907	−0.278573
$340$	−3.46983	−0.188178
$341$	19.2049	1.04000
$342$	−5.87871	−0.317884
$343$	−12.0993	−0.653300
$344$	9.33676	0.503404
$345$	5.60853	0.301953
$346$	1.57490	0.0846674
$347$	−21.2368	−1.14005	−0.570025	−	0.821628i	$-0.693066\pi$
$347$	−21.2368	−1.14005	−0.570025	+	0.821628i	$0.693066\pi$
$348$	−0.377820	−0.0202533
$349$	26.5667	1.42208	0.711041	−	0.703150i	$-0.248224\pi$
$349$	26.5667	1.42208	0.711041	+	0.703150i	$0.248224\pi$
$350$	−7.31559	−0.391035
$351$	1.00000	0.0533761
$352$	−6.28384	−0.334930
$353$	−22.3861	−1.19149	−0.595746	−	0.803173i	$-0.703143\pi$
$353$	−22.3861	−1.19149	−0.595746	+	0.803173i	$0.703143\pi$
$354$	0.838786	0.0445810
$355$	−11.6804	−0.619931
$356$	17.3823	0.921260
$357$	0.886793	0.0469341
$358$	8.00874	0.423275
$359$	−8.20847	−0.433227	−0.216613	−	0.976257i	$-0.569501\pi$
$359$	−8.20847	−0.433227	−0.216613	+	0.976257i	$0.569501\pi$
$360$	−3.59906	−0.189687
$361$	15.5593	0.818909
$362$	−6.76636	−0.355632
$363$	−28.4866	−1.49516
$364$	−0.919823	−0.0482118
$365$	18.9685	0.992855
$366$	0.508838	0.0265974
$367$	30.9920	1.61777	0.808883	−	0.587969i	$-0.200072\pi$
$367$	30.9920	1.61777	0.808883	+	0.587969i	$0.200072\pi$
$368$	−1.55833	−0.0812336
$369$	5.29346	0.275567
$370$	−18.6325	−0.968659
$371$	2.47876	0.128691
$372$	−3.05624	−0.158459
$373$	−6.06159	−0.313857	−0.156929	−	0.987610i	$-0.550159\pi$
$373$	−6.06159	−0.313857	−0.156929	+	0.987610i	$0.550159\pi$
$374$	6.05819	0.313262
$375$	−10.6290	−0.548879
$376$	−6.01076	−0.309981
$377$	−0.377820	−0.0194587
$378$	0.919823	0.0473106
$379$	6.33028	0.325164	0.162582	−	0.986695i	$-0.448018\pi$
$379$	6.33028	0.325164	0.162582	+	0.986695i	$0.448018\pi$
$380$	21.1579	1.08538
$381$	−4.19542	−0.214938
$382$	−3.62788	−0.185618
$383$	−7.01057	−0.358224	−0.179112	−	0.983829i	$-0.557322\pi$
$383$	−7.01057	−0.358224	−0.179112	+	0.983829i	$0.557322\pi$
$384$	1.00000	0.0510310
$385$	20.8026	1.06020
$386$	17.8569	0.908892
$387$	−9.33676	−0.474614
$388$	3.40393	0.172808
$389$	−5.80948	−0.294552	−0.147276	−	0.989095i	$-0.547051\pi$
$389$	−5.80948	−0.294552	−0.147276	+	0.989095i	$0.547051\pi$
$390$	−3.59906	−0.182246
$391$	1.50237	0.0759783
$392$	6.15393	0.310820
$393$	−3.43244	−0.173144
$394$	1.10763	0.0558014
$395$	35.9465	1.80867
$396$	6.28384	0.315775
$397$	−21.4430	−1.07620	−0.538098	−	0.842882i	$-0.680857\pi$
$397$	−21.4430	−1.07620	−0.538098	+	0.842882i	$0.680857\pi$
$398$	9.47297	0.474837
$399$	−5.40737	−0.270707
$400$	7.95327	0.397663
$401$	−4.30754	−0.215108	−0.107554	−	0.994199i	$-0.534302\pi$
$401$	−4.30754	−0.215108	−0.107554	+	0.994199i	$0.534302\pi$
$402$	−14.4793	−0.722163
$403$	−3.05624	−0.152242
$404$	0.987205	0.0491153
$405$	3.59906	0.178839
$406$	−0.347528	−0.0172475
$407$	32.5317	1.61254
$408$	−0.964091	−0.0477296
$409$	3.40095	0.168166	0.0840830	−	0.996459i	$-0.473204\pi$
$409$	3.40095	0.168166	0.0840830	+	0.996459i	$0.473204\pi$
$410$	−19.0515	−0.940887
$411$	17.0982	0.843390
$412$	1.00000	0.0492665
$413$	0.771535	0.0379647
$414$	1.55833	0.0765878
$415$	−41.3158	−2.02811
$416$	1.00000	0.0490290
$417$	9.30098	0.455471
$418$	−36.9409	−1.80684
$419$	18.2249	0.890345	0.445173	−	0.895445i	$-0.353142\pi$
$419$	18.2249	0.890345	0.445173	+	0.895445i	$0.353142\pi$
$420$	−3.31050	−0.161536
$421$	27.8199	1.35586	0.677930	−	0.735127i	$-0.262877\pi$
$421$	27.8199	1.35586	0.677930	+	0.735127i	$0.262877\pi$
$422$	−16.5894	−0.807558
$423$	6.01076	0.292253
$424$	−2.69483	−0.130872
$425$	−7.66768	−0.371937
$426$	−3.24540	−0.157240
$427$	0.468041	0.0226501
$428$	17.9742	0.868814
$429$	6.28384	0.303386
$430$	33.6036	1.62051
$431$	8.55540	0.412099	0.206050	−	0.978542i	$-0.433939\pi$
$431$	8.55540	0.412099	0.206050	+	0.978542i	$0.433939\pi$
$432$	−1.00000	−0.0481125
$433$	−12.5585	−0.603525	−0.301763	−	0.953383i	$-0.597575\pi$
$433$	−12.5585	−0.603525	−0.301763	+	0.953383i	$0.597575\pi$
$434$	−2.81120	−0.134942
$435$	−1.35980	−0.0651974
$436$	−8.38673	−0.401651
$437$	−9.16098	−0.438229
$438$	5.27039	0.251829
$439$	−8.35803	−0.398907	−0.199454	−	0.979907i	$-0.563917\pi$
$439$	−8.35803	−0.398907	−0.199454	+	0.979907i	$0.563917\pi$
$440$	−22.6159	−1.07817
$441$	−6.15393	−0.293044
$442$	−0.964091	−0.0458571
$443$	11.2701	0.535459	0.267730	−	0.963494i	$-0.413727\pi$
$443$	11.2701	0.535459	0.267730	+	0.963494i	$0.413727\pi$
$444$	−5.17705	−0.245692
$445$	62.5601	2.96563
$446$	−10.5549	−0.499791
$447$	7.18823	0.339992
$448$	0.919823	0.0434575
$449$	−16.6832	−0.787330	−0.393665	−	0.919254i	$-0.628793\pi$
$449$	−16.6832	−0.787330	−0.393665	+	0.919254i	$0.628793\pi$
$450$	−7.95327	−0.374921
$451$	33.2633	1.56631
$452$	5.12907	0.241251
$453$	12.2638	0.576203
$454$	6.01618	0.282353
$455$	−3.31050	−0.155199
$456$	5.87871	0.275296
$457$	32.0280	1.49821	0.749104	−	0.662452i	$-0.230484\pi$
$457$	32.0280	1.49821	0.749104	+	0.662452i	$0.230484\pi$
$458$	17.7551	0.829641
$459$	0.964091	0.0449999
$460$	−5.60853	−0.261499
$461$	−30.6252	−1.42636	−0.713179	−	0.700982i	$-0.752745\pi$
$461$	−30.6252	−1.42636	−0.713179	+	0.700982i	$0.752745\pi$
$462$	5.78001	0.268911
$463$	−11.1986	−0.520443	−0.260221	−	0.965549i	$-0.583796\pi$
$463$	−11.1986	−0.520443	−0.260221	+	0.965549i	$0.583796\pi$
$464$	0.377820	0.0175399
$465$	−10.9996	−0.510095
$466$	11.1388	0.515994
$467$	11.7793	0.545080	0.272540	−	0.962144i	$-0.412136\pi$
$467$	11.7793	0.545080	0.272540	+	0.962144i	$0.412136\pi$
$468$	−1.00000	−0.0462250
$469$	−13.3184	−0.614987
$470$	−21.6331	−0.997861
$471$	16.6452	0.766971
$472$	−0.838786	−0.0386083
$473$	−58.6707	−2.69768
$474$	9.98774	0.458752
$475$	46.7550	2.14527
$476$	−0.886793	−0.0406461
$477$	2.69483	0.123388
$478$	11.3486	0.519073
$479$	7.23414	0.330536	0.165268	−	0.986249i	$-0.447151\pi$
$479$	7.23414	0.330536	0.165268	+	0.986249i	$0.447151\pi$
$480$	3.59906	0.164274
$481$	−5.17705	−0.236053
$482$	−13.8214	−0.629548
$483$	1.43339	0.0652214
$484$	28.4866	1.29485
$485$	12.2510	0.556288
$486$	1.00000	0.0453609
$487$	14.2025	0.643578	0.321789	−	0.946811i	$-0.395716\pi$
$487$	14.2025	0.643578	0.321789	+	0.946811i	$0.395716\pi$
$488$	−0.508838	−0.0230340
$489$	−19.0669	−0.862237
$490$	22.1484	1.00056
$491$	20.5180	0.925964	0.462982	−	0.886368i	$-0.346779\pi$
$491$	20.5180	0.925964	0.462982	+	0.886368i	$0.346779\pi$
$492$	−5.29346	−0.238648
$493$	−0.364253	−0.0164051
$494$	5.87871	0.264496
$495$	22.6159	1.01651
$496$	3.05624	0.137229
$497$	−2.98519	−0.133904
$498$	−11.4796	−0.514412
$499$	41.1316	1.84130	0.920651	−	0.390387i	$-0.127659\pi$
$499$	41.1316	1.84130	0.920651	+	0.390387i	$0.127659\pi$
$500$	10.6290	0.475343
$501$	−3.57968	−0.159928
$502$	−16.6162	−0.741616
$503$	−24.2328	−1.08049	−0.540244	−	0.841508i	$-0.681668\pi$
$503$	−24.2328	−1.08049	−0.540244	+	0.841508i	$0.681668\pi$
$504$	−0.919823	−0.0409722
$505$	3.55302	0.158107
$506$	9.79230	0.435321
$507$	−1.00000	−0.0444116
$508$	4.19542	0.186142
$509$	−31.4345	−1.39331	−0.696655	−	0.717407i	$-0.745329\pi$
$509$	−31.4345	−1.39331	−0.696655	+	0.717407i	$0.745329\pi$
$510$	−3.46983	−0.153647
$511$	4.84782	0.214455
$512$	−1.00000	−0.0441942
$513$	−5.87871	−0.259552
$514$	15.8529	0.699243
$515$	3.59906	0.158594
$516$	9.33676	0.411028
$517$	37.7706	1.66115
$518$	−4.76196	−0.209229
$519$	1.57490	0.0691306
$520$	3.59906	0.157829
$521$	−7.44209	−0.326044	−0.163022	−	0.986622i	$-0.552124\pi$
$521$	−7.44209	−0.326044	−0.163022	+	0.986622i	$0.552124\pi$
$522$	−0.377820	−0.0165367
$523$	6.33874	0.277174	0.138587	−	0.990350i	$-0.455744\pi$
$523$	6.33874	0.277174	0.138587	+	0.990350i	$0.455744\pi$
$524$	3.43244	0.149947
$525$	−7.31559	−0.319279
$526$	14.0523	0.612709
$527$	−2.94649	−0.128351
$528$	−6.28384	−0.273469
$529$	−20.5716	−0.894418
$530$	−9.69886	−0.421291
$531$	0.838786	0.0364002
$532$	5.40737	0.234439
$533$	−5.29346	−0.229285
$534$	17.3823	0.752206
$535$	64.6902	2.79680
$536$	14.4793	0.625412
$537$	8.00874	0.345602
$538$	21.7523	0.937807
$539$	−38.6703	−1.66565
$540$	−3.59906	−0.154879
$541$	−33.7271	−1.45004	−0.725020	−	0.688728i	$-0.758169\pi$
$541$	−33.7271	−1.45004	−0.725020	+	0.688728i	$0.758169\pi$
$542$	−29.6801	−1.27487
$543$	−6.76636	−0.290372
$544$	0.964091	0.0413351
$545$	−30.1844	−1.29296
$546$	−0.919823	−0.0393648
$547$	24.8785	1.06373	0.531864	−	0.846830i	$-0.321492\pi$
$547$	24.8785	1.06373	0.531864	+	0.846830i	$0.321492\pi$
$548$	−17.0982	−0.730397
$549$	0.508838	0.0217167
$550$	−49.9770	−2.13103
$551$	2.22110	0.0946219
$552$	−1.55833	−0.0663270
$553$	9.18695	0.390669
$554$	−0.303986	−0.0129151
$555$	−18.6325	−0.790907
$556$	−9.30098	−0.394449
$557$	−23.3359	−0.988775	−0.494387	−	0.869242i	$-0.664608\pi$
$557$	−23.3359	−0.988775	−0.494387	+	0.869242i	$0.664608\pi$
$558$	−3.05624	−0.129381
$559$	9.33676	0.394903
$560$	3.31050	0.139894
$561$	6.05819	0.255777
$562$	31.4890	1.32829
$563$	−6.00586	−0.253117	−0.126558	−	0.991959i	$-0.540393\pi$
$563$	−6.00586	−0.253117	−0.126558	+	0.991959i	$0.540393\pi$
$564$	−6.01076	−0.253099
$565$	18.4598	0.776611
$566$	−7.53378	−0.316669
$567$	0.919823	0.0386289
$568$	3.24540	0.136174
$569$	11.7306	0.491772	0.245886	−	0.969299i	$-0.420921\pi$
$569$	11.7306	0.491772	0.245886	+	0.969299i	$0.420921\pi$
$570$	21.1579	0.886206
$571$	−16.8136	−0.703627	−0.351814	−	0.936070i	$-0.614435\pi$
$571$	−16.8136	−0.703627	−0.351814	+	0.936070i	$0.614435\pi$
$572$	−6.28384	−0.262740
$573$	−3.62788	−0.151557
$574$	−4.86905	−0.203230
$575$	−12.3938	−0.516858
$576$	1.00000	0.0416667
$577$	−33.4762	−1.39363	−0.696817	−	0.717249i	$-0.745401\pi$
$577$	−33.4762	−1.39363	−0.696817	+	0.717249i	$0.745401\pi$
$578$	16.0705	0.668446
$579$	17.8569	0.742107
$580$	1.35980	0.0564626
$581$	−10.5592	−0.438069
$582$	3.40393	0.141097
$583$	16.9339	0.701329
$584$	−5.27039	−0.218090
$585$	−3.59906	−0.148803
$586$	−28.1598	−1.16327
$587$	−29.1504	−1.20317	−0.601583	−	0.798810i	$-0.705463\pi$
$587$	−29.1504	−1.20317	−0.601583	+	0.798810i	$0.705463\pi$
$588$	6.15393	0.253784
$589$	17.9668	0.740307
$590$	−3.01885	−0.124284
$591$	1.10763	0.0455616
$592$	5.17705	0.212775
$593$	5.65412	0.232187	0.116093	−	0.993238i	$-0.462963\pi$
$593$	5.65412	0.232187	0.116093	+	0.993238i	$0.462963\pi$
$594$	6.28384	0.257829
$595$	−3.19163	−0.130844
$596$	−7.18823	−0.294442
$597$	9.47297	0.387703
$598$	−1.55833	−0.0637249
$599$	−34.9730	−1.42896	−0.714480	−	0.699656i	$-0.753336\pi$
$599$	−34.9730	−1.42896	−0.714480	+	0.699656i	$0.753336\pi$
$600$	7.95327	0.324691
$601$	32.1974	1.31336	0.656680	−	0.754170i	$-0.271960\pi$
$601$	32.1974	1.31336	0.656680	+	0.754170i	$0.271960\pi$
$602$	8.58816	0.350027
$603$	−14.4793	−0.589644
$604$	−12.2638	−0.499006
$605$	102.525	4.16824
$606$	0.987205	0.0401025
$607$	−1.13354	−0.0460088	−0.0230044	−	0.999735i	$-0.507323\pi$
$607$	−1.13354	−0.0460088	−0.0230044	+	0.999735i	$0.507323\pi$
$608$	−5.87871	−0.238413
$609$	−0.347528	−0.0140825
$610$	−1.83134	−0.0741489
$611$	−6.01076	−0.243169
$612$	−0.964091	−0.0389711
$613$	−6.05830	−0.244693	−0.122346	−	0.992487i	$-0.539042\pi$
$613$	−6.05830	−0.244693	−0.122346	+	0.992487i	$0.539042\pi$
$614$	−3.43777	−0.138737
$615$	−19.0515	−0.768231
$616$	−5.78001	−0.232883
$617$	24.6131	0.990885	0.495442	−	0.868641i	$-0.335006\pi$
$617$	24.6131	0.990885	0.495442	+	0.868641i	$0.335006\pi$
$618$	1.00000	0.0402259
$619$	30.9999	1.24599	0.622995	−	0.782226i	$-0.285916\pi$
$619$	30.9999	1.24599	0.622995	+	0.782226i	$0.285916\pi$
$620$	10.9996	0.441755
$621$	1.55833	0.0625337
$622$	1.77167	0.0710375
$623$	15.9886	0.640571
$624$	1.00000	0.0400320
$625$	−1.51187	−0.0604748
$626$	27.0983	1.08306
$627$	−36.9409	−1.47528
$628$	−16.6452	−0.664216
$629$	−4.99115	−0.199010
$630$	−3.31050	−0.131894
$631$	20.1742	0.803122	0.401561	−	0.915832i	$-0.368468\pi$
$631$	20.1742	0.803122	0.401561	+	0.915832i	$0.368468\pi$
$632$	−9.98774	−0.397291
$633$	−16.5894	−0.659368
$634$	26.5689	1.05519
$635$	15.0996	0.599209
$636$	−2.69483	−0.106857
$637$	6.15393	0.243827
$638$	−2.37416	−0.0939939
$639$	−3.24540	−0.128386
$640$	−3.59906	−0.142266
$641$	−9.99257	−0.394683	−0.197341	−	0.980335i	$-0.563231\pi$
$641$	−9.99257	−0.394683	−0.197341	+	0.980335i	$0.563231\pi$
$642$	17.9742	0.709384
$643$	−10.2068	−0.402515	−0.201257	−	0.979538i	$-0.564503\pi$
$643$	−10.2068	−0.402515	−0.201257	+	0.979538i	$0.564503\pi$
$644$	−1.43339	−0.0564834
$645$	33.6036	1.32314
$646$	5.66762	0.222989
$647$	−6.66966	−0.262211	−0.131106	−	0.991368i	$-0.541853\pi$
$647$	−6.66966	−0.262211	−0.131106	+	0.991368i	$0.541853\pi$
$648$	−1.00000	−0.0392837
$649$	5.27080	0.206897
$650$	7.95327	0.311953
$651$	−2.81120	−0.110180
$652$	19.0669	0.746719
$653$	2.46264	0.0963704	0.0481852	−	0.998838i	$-0.484656\pi$
$653$	2.46264	0.0963704	0.0481852	+	0.998838i	$0.484656\pi$
$654$	−8.38673	−0.327947
$655$	12.3536	0.482694
$656$	5.29346	0.206675
$657$	5.27039	0.205617
$658$	−5.52883	−0.215536
$659$	19.7091	0.767759	0.383880	−	0.923383i	$-0.374588\pi$
$659$	19.7091	0.767759	0.383880	+	0.923383i	$0.374588\pi$
$660$	−22.6159	−0.880324
$661$	43.5189	1.69269	0.846344	−	0.532637i	$-0.178799\pi$
$661$	43.5189	1.69269	0.846344	+	0.532637i	$0.178799\pi$
$662$	23.2926	0.905294
$663$	−0.964091	−0.0374422
$664$	11.4796	0.445494
$665$	19.4615	0.754684
$666$	−5.17705	−0.200606
$667$	−0.588769	−0.0227972
$668$	3.57968	0.138502
$669$	−10.5549	−0.408078
$670$	52.1120	2.01326
$671$	3.19746	0.123436
$672$	0.919823	0.0354829
$673$	30.2039	1.16428	0.582138	−	0.813090i	$-0.302216\pi$
$673$	30.2039	1.16428	0.582138	+	0.813090i	$0.302216\pi$
$674$	−15.3551	−0.591457
$675$	−7.95327	−0.306121
$676$	1.00000	0.0384615
$677$	36.4193	1.39971	0.699853	−	0.714287i	$-0.253249\pi$
$677$	36.4193	1.39971	0.699853	+	0.714287i	$0.253249\pi$
$678$	5.12907	0.196981
$679$	3.13101	0.120157
$680$	3.46983	0.133062
$681$	6.01618	0.230540
$682$	−19.2049	−0.735394
$683$	−15.1809	−0.580882	−0.290441	−	0.956893i	$-0.593802\pi$
$683$	−15.1809	−0.580882	−0.290441	+	0.956893i	$0.593802\pi$
$684$	5.87871	0.224778
$685$	−61.5374	−2.35122
$686$	12.0993	0.461953
$687$	17.7551	0.677399
$688$	−9.33676	−0.355961
$689$	−2.69483	−0.102665
$690$	−5.60853	−0.213513
$691$	−21.8484	−0.831153	−0.415577	−	0.909558i	$-0.636420\pi$
$691$	−21.8484	−0.831153	−0.415577	+	0.909558i	$0.636420\pi$
$692$	−1.57490	−0.0598689
$693$	5.78001	0.219565
$694$	21.2368	0.806137
$695$	−33.4748	−1.26977
$696$	0.377820	0.0143212
$697$	−5.10338	−0.193304
$698$	−26.5667	−1.00556
$699$	11.1388	0.421307
$700$	7.31559	0.276503
$701$	46.2024	1.74504	0.872521	−	0.488577i	$-0.162484\pi$
$701$	46.2024	1.74504	0.872521	+	0.488577i	$0.162484\pi$
$702$	−1.00000	−0.0377426
$703$	30.4344	1.14785
$704$	6.28384	0.236831
$705$	−21.6331	−0.814750
$706$	22.3861	0.842512
$707$	0.908054	0.0341509
$708$	−0.838786	−0.0315235
$709$	16.4636	0.618302	0.309151	−	0.951013i	$-0.399955\pi$
$709$	16.4636	0.618302	0.309151	+	0.951013i	$0.399955\pi$
$710$	11.6804	0.438357
$711$	9.98774	0.374570
$712$	−17.3823	−0.651429
$713$	−4.76263	−0.178362
$714$	−0.886793	−0.0331874
$715$	−22.6159	−0.845788
$716$	−8.00874	−0.299301
$717$	11.3486	0.423821
$718$	8.20847	0.306338
$719$	−7.66482	−0.285850	−0.142925	−	0.989734i	$-0.545651\pi$
$719$	−7.66482	−0.285850	−0.142925	+	0.989734i	$0.545651\pi$
$720$	3.59906	0.134129
$721$	0.919823	0.0342560
$722$	−15.5593	−0.579056
$723$	−13.8214	−0.514024
$724$	6.76636	0.251470
$725$	3.00491	0.111599
$726$	28.4866	1.05724
$727$	49.1257	1.82197	0.910985	−	0.412440i	$-0.135323\pi$
$727$	49.1257	1.82197	0.910985	+	0.412440i	$0.135323\pi$
$728$	0.919823	0.0340909
$729$	1.00000	0.0370370
$730$	−18.9685	−0.702055
$731$	9.00149	0.332932
$732$	−0.508838	−0.0188072
$733$	23.4568	0.866398	0.433199	−	0.901298i	$-0.357385\pi$
$733$	23.4568	0.866398	0.433199	+	0.901298i	$0.357385\pi$
$734$	−30.9920	−1.14393
$735$	22.1484	0.816955
$736$	1.55833	0.0574408
$737$	−90.9857	−3.35150
$738$	−5.29346	−0.194855
$739$	−18.6694	−0.686764	−0.343382	−	0.939196i	$-0.611573\pi$
$739$	−18.6694	−0.686764	−0.343382	+	0.939196i	$0.611573\pi$
$740$	18.6325	0.684945
$741$	5.87871	0.215960
$742$	−2.47876	−0.0909982
$743$	22.0970	0.810662	0.405331	−	0.914170i	$-0.367156\pi$
$743$	22.0970	0.810662	0.405331	+	0.914170i	$0.367156\pi$
$744$	3.05624	0.112047
$745$	−25.8709	−0.947837
$746$	6.06159	0.221930
$747$	−11.4796	−0.420016
$748$	−6.05819	−0.221509
$749$	16.5330	0.604104
$750$	10.6290	0.388116
$751$	−18.7085	−0.682683	−0.341341	−	0.939939i	$-0.610881\pi$
$751$	−18.7085	−0.682683	−0.341341	+	0.939939i	$0.610881\pi$
$752$	6.01076	0.219190
$753$	−16.6162	−0.605527
$754$	0.377820	0.0137594
$755$	−44.1382	−1.60635
$756$	−0.919823	−0.0334536
$757$	42.8940	1.55901	0.779504	−	0.626397i	$-0.215471\pi$
$757$	42.8940	1.55901	0.779504	+	0.626397i	$0.215471\pi$
$758$	−6.33028	−0.229926
$759$	9.79230	0.355438
$760$	−21.1579	−0.767477
$761$	−4.69061	−0.170035	−0.0850173	−	0.996379i	$-0.527095\pi$
$761$	−4.69061	−0.170035	−0.0850173	+	0.996379i	$0.527095\pi$
$762$	4.19542	0.151984
$763$	−7.71430	−0.279276
$764$	3.62788	0.131252
$765$	−3.46983	−0.125452
$766$	7.01057	0.253302
$767$	−0.838786	−0.0302868
$768$	−1.00000	−0.0360844
$769$	−8.30375	−0.299441	−0.149720	−	0.988728i	$-0.547837\pi$
$769$	−8.30375	−0.299441	−0.149720	+	0.988728i	$0.547837\pi$
$770$	−20.8026	−0.749675
$771$	15.8529	0.570930
$772$	−17.8569	−0.642684
$773$	21.0421	0.756831	0.378416	−	0.925636i	$-0.376469\pi$
$773$	21.0421	0.756831	0.378416	+	0.925636i	$0.376469\pi$
$774$	9.33676	0.335603
$775$	24.3071	0.873137
$776$	−3.40393	−0.122194
$777$	−4.76196	−0.170835
$778$	5.80948	0.208280
$779$	31.1187	1.11494
$780$	3.59906	0.128867
$781$	−20.3935	−0.729738
$782$	−1.50237	−0.0537248
$783$	−0.377820	−0.0135022
$784$	−6.15393	−0.219783
$785$	−59.9072	−2.13818
$786$	3.43244	0.122431
$787$	30.1420	1.07445	0.537224	−	0.843440i	$-0.319473\pi$
$787$	30.1420	1.07445	0.537224	+	0.843440i	$0.319473\pi$
$788$	−1.10763	−0.0394575
$789$	14.0523	0.500275
$790$	−35.9465	−1.27892
$791$	4.71783	0.167747
$792$	−6.28384	−0.223286
$793$	−0.508838	−0.0180694
$794$	21.4430	0.760985
$795$	−9.69886	−0.343983
$796$	−9.47297	−0.335761
$797$	31.5970	1.11922	0.559612	−	0.828755i	$-0.310950\pi$
$797$	31.5970	1.11922	0.559612	+	0.828755i	$0.310950\pi$
$798$	5.40737	0.191419
$799$	−5.79492	−0.205009
$800$	−7.95327	−0.281190
$801$	17.3823	0.614174
$802$	4.30754	0.152105
$803$	33.1183	1.16872
$804$	14.4793	0.510646
$805$	−5.15886	−0.181826
$806$	3.05624	0.107651
$807$	21.7523	0.765716
$808$	−0.987205	−0.0347298
$809$	17.1403	0.602622	0.301311	−	0.953526i	$-0.402576\pi$
$809$	17.1403	0.602622	0.301311	+	0.953526i	$0.402576\pi$
$810$	−3.59906	−0.126458
$811$	41.2262	1.44765	0.723824	−	0.689984i	$-0.242383\pi$
$811$	41.2262	1.44765	0.723824	+	0.689984i	$0.242383\pi$
$812$	0.347528	0.0121958
$813$	−29.6801	−1.04093
$814$	−32.5317	−1.14024
$815$	68.6232	2.40376
$816$	0.964091	0.0337499
$817$	−54.8881	−1.92029
$818$	−3.40095	−0.118911
$819$	−0.919823	−0.0321412
$820$	19.0515	0.665308
$821$	13.7131	0.478592	0.239296	−	0.970947i	$-0.423083\pi$
$821$	13.7131	0.478592	0.239296	+	0.970947i	$0.423083\pi$
$822$	−17.0982	−0.596367
$823$	49.1486	1.71321	0.856606	−	0.515972i	$-0.172569\pi$
$823$	49.1486	1.71321	0.856606	+	0.515972i	$0.172569\pi$
$824$	−1.00000	−0.0348367
$825$	−49.9770	−1.73998
$826$	−0.771535	−0.0268451
$827$	37.5446	1.30555	0.652777	−	0.757551i	$-0.273604\pi$
$827$	37.5446	1.30555	0.652777	+	0.757551i	$0.273604\pi$
$828$	−1.55833	−0.0541557
$829$	41.5322	1.44247	0.721236	−	0.692689i	$-0.243574\pi$
$829$	41.5322	1.44247	0.721236	+	0.692689i	$0.243574\pi$
$830$	41.3158	1.43409
$831$	−0.303986	−0.0105451
$832$	−1.00000	−0.0346688
$833$	5.93295	0.205564
$834$	−9.30098	−0.322067
$835$	12.8835	0.445852
$836$	36.9409	1.27763
$837$	−3.05624	−0.105639
$838$	−18.2249	−0.629569
$839$	−22.7145	−0.784192	−0.392096	−	0.919924i	$-0.628250\pi$
$839$	−22.7145	−0.784192	−0.392096	+	0.919924i	$0.628250\pi$
$840$	3.31050	0.114223
$841$	−28.8573	−0.995078
$842$	−27.8199	−0.958737
$843$	31.4890	1.08454
$844$	16.5894	0.571030
$845$	3.59906	0.123812
$846$	−6.01076	−0.206654
$847$	26.2026	0.900333
$848$	2.69483	0.0925407
$849$	−7.53378	−0.258559
$850$	7.66768	0.262999
$851$	−8.06755	−0.276552
$852$	3.24540	0.111185
$853$	1.68429	0.0576690	0.0288345	−	0.999584i	$-0.490820\pi$
$853$	1.68429	0.0576690	0.0288345	+	0.999584i	$0.490820\pi$
$854$	−0.468041	−0.0160160
$855$	21.1579	0.723584
$856$	−17.9742	−0.614344
$857$	1.23658	0.0422407	0.0211204	−	0.999777i	$-0.493277\pi$
$857$	1.23658	0.0422407	0.0211204	+	0.999777i	$0.493277\pi$
$858$	−6.28384	−0.214527
$859$	−3.53572	−0.120637	−0.0603185	−	0.998179i	$-0.519212\pi$
$859$	−3.53572	−0.120637	−0.0603185	+	0.998179i	$0.519212\pi$
$860$	−33.6036	−1.14587
$861$	−4.86905	−0.165937
$862$	−8.55540	−0.291398
$863$	3.59457	0.122360	0.0611802	−	0.998127i	$-0.480514\pi$
$863$	3.59457	0.122360	0.0611802	+	0.998127i	$0.480514\pi$
$864$	1.00000	0.0340207
$865$	−5.66818	−0.192724
$866$	12.5585	0.426757
$867$	16.0705	0.545784
$868$	2.81120	0.0954183
$869$	62.7613	2.12903
$870$	1.35980	0.0461015
$871$	14.4793	0.490613
$872$	8.38673	0.284010
$873$	3.40393	0.115206
$874$	9.16098	0.309875
$875$	9.77680	0.330516
$876$	−5.27039	−0.178070
$877$	−23.0893	−0.779670	−0.389835	−	0.920885i	$-0.627468\pi$
$877$	−23.0893	−0.779670	−0.389835	+	0.920885i	$0.627468\pi$
$878$	8.35803	0.282070
$879$	−28.1598	−0.949806
$880$	22.6159	0.762383
$881$	−18.8867	−0.636309	−0.318155	−	0.948039i	$-0.603063\pi$
$881$	−18.8867	−0.636309	−0.318155	+	0.948039i	$0.603063\pi$
$882$	6.15393	0.207213
$883$	1.12572	0.0378834	0.0189417	−	0.999821i	$-0.493970\pi$
$883$	1.12572	0.0378834	0.0189417	+	0.999821i	$0.493970\pi$
$884$	0.964091	0.0324259
$885$	−3.01885	−0.101477
$886$	−11.2701	−0.378627
$887$	35.9661	1.20762	0.603812	−	0.797127i	$-0.293648\pi$
$887$	35.9661	1.20762	0.603812	+	0.797127i	$0.293648\pi$
$888$	5.17705	0.173730
$889$	3.85905	0.129428
$890$	−62.5601	−2.09702
$891$	6.28384	0.210516
$892$	10.5549	0.353406
$893$	35.3355	1.18246
$894$	−7.18823	−0.240410
$895$	−28.8240	−0.963479
$896$	−0.919823	−0.0307291
$897$	−1.55833	−0.0520312
$898$	16.6832	0.556726
$899$	1.15471	0.0385117
$900$	7.95327	0.265109
$901$	−2.59806	−0.0865539
$902$	−33.2633	−1.10755
$903$	8.58816	0.285796
$904$	−5.12907	−0.170590
$905$	24.3526	0.809506
$906$	−12.2638	−0.407437
$907$	23.0319	0.764760	0.382380	−	0.924005i	$-0.375105\pi$
$907$	23.0319	0.764760	0.382380	+	0.924005i	$0.375105\pi$
$908$	−6.01618	−0.199654
$909$	0.987205	0.0327435
$910$	3.31050	0.109742
$911$	12.7623	0.422833	0.211416	−	0.977396i	$-0.432192\pi$
$911$	12.7623	0.422833	0.211416	+	0.977396i	$0.432192\pi$
$912$	−5.87871	−0.194664
$913$	−72.1358	−2.38735
$914$	−32.0280	−1.05939
$915$	−1.83134	−0.0605423
$916$	−17.7551	−0.586645
$917$	3.15723	0.104261
$918$	−0.964091	−0.0318198
$919$	12.2281	0.403368	0.201684	−	0.979451i	$-0.435359\pi$
$919$	12.2281	0.403368	0.201684	+	0.979451i	$0.435359\pi$
$920$	5.60853	0.184908
$921$	−3.43777	−0.113279
$922$	30.6252	1.00859
$923$	3.24540	0.106824
$924$	−5.78001	−0.190149
$925$	41.1744	1.35381
$926$	11.1986	0.368009
$927$	1.00000	0.0328443
$928$	−0.377820	−0.0124026
$929$	37.7516	1.23859	0.619296	−	0.785158i	$-0.287418\pi$
$929$	37.7516	1.23859	0.619296	+	0.785158i	$0.287418\pi$
$930$	10.9996	0.360691
$931$	−36.1772	−1.18566
$932$	−11.1388	−0.364863
$933$	1.77167	0.0580018
$934$	−11.7793	−0.385430
$935$	−21.8038	−0.713061
$936$	1.00000	0.0326860
$937$	14.9497	0.488385	0.244193	−	0.969727i	$-0.421477\pi$
$937$	14.9497	0.488385	0.244193	+	0.969727i	$0.421477\pi$
$938$	13.3184	0.434861
$939$	27.0983	0.884319
$940$	21.6331	0.705594
$941$	−37.2030	−1.21278	−0.606392	−	0.795166i	$-0.707384\pi$
$941$	−37.2030	−1.21278	−0.606392	+	0.795166i	$0.707384\pi$
$942$	−16.6452	−0.542330
$943$	−8.24897	−0.268623
$944$	0.838786	0.0273002
$945$	−3.31050	−0.107691
$946$	58.6707	1.90755
$947$	53.3472	1.73355	0.866775	−	0.498699i	$-0.166189\pi$
$947$	53.3472	1.73355	0.866775	+	0.498699i	$0.166189\pi$
$948$	−9.98774	−0.324387
$949$	−5.27039	−0.171084
$950$	−46.7550	−1.51693
$951$	26.5689	0.861557
$952$	0.886793	0.0287411
$953$	−47.1799	−1.52831	−0.764153	−	0.645035i	$-0.776843\pi$
$953$	−47.1799	−1.52831	−0.764153	+	0.645035i	$0.776843\pi$
$954$	−2.69483	−0.0872482
$955$	13.0570	0.422514
$956$	−11.3486	−0.367040
$957$	−2.37416	−0.0767457
$958$	−7.23414	−0.233724
$959$	−15.7273	−0.507860
$960$	−3.59906	−0.116159
$961$	−21.6594	−0.698690
$962$	5.17705	0.166915
$963$	17.9742	0.579209
$964$	13.8214	0.445158
$965$	−64.2681	−2.06886
$966$	−1.43339	−0.0461185
$967$	−16.0862	−0.517296	−0.258648	−	0.965972i	$-0.583277\pi$
$967$	−16.0862	−0.517296	−0.258648	+	0.965972i	$0.583277\pi$
$968$	−28.4866	−0.915594
$969$	5.66762	0.182070
$970$	−12.2510	−0.393355
$971$	54.1754	1.73857	0.869286	−	0.494310i	$-0.164579\pi$
$971$	54.1754	1.73857	0.869286	+	0.494310i	$0.164579\pi$
$972$	−1.00000	−0.0320750
$973$	−8.55525	−0.274269
$974$	−14.2025	−0.455078
$975$	7.95327	0.254708
$976$	0.508838	0.0162875
$977$	−50.8217	−1.62593	−0.812965	−	0.582312i	$-0.802148\pi$
$977$	−50.8217	−1.62593	−0.812965	+	0.582312i	$0.802148\pi$
$978$	19.0669	0.609694
$979$	109.228	3.49093
$980$	−22.1484	−0.707504
$981$	−8.38673	−0.267768
$982$	−20.5180	−0.654755
$983$	10.6021	0.338153	0.169077	−	0.985603i	$-0.445921\pi$
$983$	10.6021	0.338153	0.169077	+	0.985603i	$0.445921\pi$
$984$	5.29346	0.168749
$985$	−3.98642	−0.127018
$986$	0.364253	0.0116002
$987$	−5.52883	−0.175985
$988$	−5.87871	−0.187027
$989$	14.5498	0.462656
$990$	−22.6159	−0.718782
$991$	−11.6809	−0.371055	−0.185527	−	0.982639i	$-0.559399\pi$
$991$	−11.6809	−0.371055	−0.185527	+	0.982639i	$0.559399\pi$
$992$	−3.05624	−0.0970357
$993$	23.2926	0.739169
$994$	2.98519	0.0946844
$995$	−34.0938	−1.08085
$996$	11.4796	0.363744
$997$	−56.4318	−1.78721	−0.893607	−	0.448851i	$-0.851833\pi$
$997$	−56.4318	−1.78721	−0.893607	+	0.448851i	$0.851833\pi$
$998$	−41.1316	−1.30200
$999$	−5.17705	−0.163795

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8034.2.a.ba.1.13	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8034.2.a.ba.1.13	✓	14	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8034.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.59906 q^{5} +1.00000 q^{6} +0.919823 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.59906 q^{5} +1.00000 q^{6} +0.919823 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.59906 q^{10} +6.28384 q^{11} -1.00000 q^{12} -1.00000 q^{13} -0.919823 q^{14} -3.59906 q^{15} +1.00000 q^{16} -0.964091 q^{17} -1.00000 q^{18} +5.87871 q^{19} +3.59906 q^{20} -0.919823 q^{21} -6.28384 q^{22} -1.55833 q^{23} +1.00000 q^{24} +7.95327 q^{25} +1.00000 q^{26} -1.00000 q^{27} +0.919823 q^{28} +0.377820 q^{29} +3.59906 q^{30} +3.05624 q^{31} -1.00000 q^{32} -6.28384 q^{33} +0.964091 q^{34} +3.31050 q^{35} +1.00000 q^{36} +5.17705 q^{37} -5.87871 q^{38} +1.00000 q^{39} -3.59906 q^{40} +5.29346 q^{41} +0.919823 q^{42} -9.33676 q^{43} +6.28384 q^{44} +3.59906 q^{45} +1.55833 q^{46} +6.01076 q^{47} -1.00000 q^{48} -6.15393 q^{49} -7.95327 q^{50} +0.964091 q^{51} -1.00000 q^{52} +2.69483 q^{53} +1.00000 q^{54} +22.6159 q^{55} -0.919823 q^{56} -5.87871 q^{57} -0.377820 q^{58} +0.838786 q^{59} -3.59906 q^{60} +0.508838 q^{61} -3.05624 q^{62} +0.919823 q^{63} +1.00000 q^{64} -3.59906 q^{65} +6.28384 q^{66} -14.4793 q^{67} -0.964091 q^{68} +1.55833 q^{69} -3.31050 q^{70} -3.24540 q^{71} -1.00000 q^{72} +5.27039 q^{73} -5.17705 q^{74} -7.95327 q^{75} +5.87871 q^{76} +5.78001 q^{77} -1.00000 q^{78} +9.98774 q^{79} +3.59906 q^{80} +1.00000 q^{81} -5.29346 q^{82} -11.4796 q^{83} -0.919823 q^{84} -3.46983 q^{85} +9.33676 q^{86} -0.377820 q^{87} -6.28384 q^{88} +17.3823 q^{89} -3.59906 q^{90} -0.919823 q^{91} -1.55833 q^{92} -3.05624 q^{93} -6.01076 q^{94} +21.1579 q^{95} +1.00000 q^{96} +3.40393 q^{97} +6.15393 q^{98} +6.28384 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{2} - 14 q^{3} + 14 q^{4} - q^{5} + 14 q^{6} + 5 q^{7} - 14 q^{8} + 14 q^{9}+O(q^{10}) \) 14 * q - 14 * q^2 - 14 * q^3 + 14 * q^4 - q^5 + 14 * q^6 + 5 * q^7 - 14 * q^8 + 14 * q^9	\( 14 q - 14 q^{2} - 14 q^{3} + 14 q^{4} - q^{5} + 14 q^{6} + 5 q^{7} - 14 q^{8} + 14 q^{9} + q^{10} - q^{11} - 14 q^{12} - 14 q^{13} - 5 q^{14} + q^{15} + 14 q^{16} - 12 q^{17} - 14 q^{18} + 10 q^{19} - q^{20} - 5 q^{21} + q^{22} - q^{23} + 14 q^{24} + 19 q^{25} + 14 q^{26} - 14 q^{27} + 5 q^{28} + 6 q^{29} - q^{30} + 20 q^{31} - 14 q^{32} + q^{33} + 12 q^{34} - 16 q^{35} + 14 q^{36} - 3 q^{37} - 10 q^{38} + 14 q^{39} + q^{40} + q^{41} + 5 q^{42} + 6 q^{43} - q^{44} - q^{45} + q^{46} - 13 q^{47} - 14 q^{48} + 9 q^{49} - 19 q^{50} + 12 q^{51} - 14 q^{52} - 27 q^{53} + 14 q^{54} + 10 q^{55} - 5 q^{56} - 10 q^{57} - 6 q^{58} - 6 q^{59} + q^{60} - 4 q^{61} - 20 q^{62} + 5 q^{63} + 14 q^{64} + q^{65} - q^{66} + 13 q^{67} - 12 q^{68} + q^{69} + 16 q^{70} + 18 q^{71} - 14 q^{72} + 11 q^{73} + 3 q^{74} - 19 q^{75} + 10 q^{76} - 15 q^{77} - 14 q^{78} + 33 q^{79} - q^{80} + 14 q^{81} - q^{82} - 25 q^{83} - 5 q^{84} + 25 q^{85} - 6 q^{86} - 6 q^{87} + q^{88} + 3 q^{89} + q^{90} - 5 q^{91} - q^{92} - 20 q^{93} + 13 q^{94} + 30 q^{95} + 14 q^{96} + 11 q^{97} - 9 q^{98} - q^{99}+O(q^{100}) \) 14 * q - 14 * q^2 - 14 * q^3 + 14 * q^4 - q^5 + 14 * q^6 + 5 * q^7 - 14 * q^8 + 14 * q^9 + q^10 - q^11 - 14 * q^12 - 14 * q^13 - 5 * q^14 + q^15 + 14 * q^16 - 12 * q^17 - 14 * q^18 + 10 * q^19 - q^20 - 5 * q^21 + q^22 - q^23 + 14 * q^24 + 19 * q^25 + 14 * q^26 - 14 * q^27 + 5 * q^28 + 6 * q^29 - q^30 + 20 * q^31 - 14 * q^32 + q^33 + 12 * q^34 - 16 * q^35 + 14 * q^36 - 3 * q^37 - 10 * q^38 + 14 * q^39 + q^40 + q^41 + 5 * q^42 + 6 * q^43 - q^44 - q^45 + q^46 - 13 * q^47 - 14 * q^48 + 9 * q^49 - 19 * q^50 + 12 * q^51 - 14 * q^52 - 27 * q^53 + 14 * q^54 + 10 * q^55 - 5 * q^56 - 10 * q^57 - 6 * q^58 - 6 * q^59 + q^60 - 4 * q^61 - 20 * q^62 + 5 * q^63 + 14 * q^64 + q^65 - q^66 + 13 * q^67 - 12 * q^68 + q^69 + 16 * q^70 + 18 * q^71 - 14 * q^72 + 11 * q^73 + 3 * q^74 - 19 * q^75 + 10 * q^76 - 15 * q^77 - 14 * q^78 + 33 * q^79 - q^80 + 14 * q^81 - q^82 - 25 * q^83 - 5 * q^84 + 25 * q^85 - 6 * q^86 - 6 * q^87 + q^88 + 3 * q^89 + q^90 - 5 * q^91 - q^92 - 20 * q^93 + 13 * q^94 + 30 * q^95 + 14 * q^96 + 11 * q^97 - 9 * q^98 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more