LMFDB - Embedded newform 6422.2.a.v.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6422,2,Mod(1,6422)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6422, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6422 = 2 \cdot 13^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6422.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:	$51.2799281781$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	6422.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.24698 q^{3} +1.00000 q^{4} -3.80194 q^{5} +1.24698 q^{6} -1.10992 q^{7} +1.00000 q^{8} -1.44504 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.24698 q^{3} +1.00000 q^{4} -3.80194 q^{5} +1.24698 q^{6} -1.10992 q^{7} +1.00000 q^{8} -1.44504 q^{9} -3.80194 q^{10} +0.890084 q^{11} +1.24698 q^{12} -1.10992 q^{14} -4.74094 q^{15} +1.00000 q^{16} +2.66487 q^{17} -1.44504 q^{18} +1.00000 q^{19} -3.80194 q^{20} -1.38404 q^{21} +0.890084 q^{22} +6.31767 q^{23} +1.24698 q^{24} +9.45473 q^{25} -5.54288 q^{27} -1.10992 q^{28} +8.09783 q^{29} -4.74094 q^{30} +4.85086 q^{31} +1.00000 q^{32} +1.10992 q^{33} +2.66487 q^{34} +4.21983 q^{35} -1.44504 q^{36} -11.4819 q^{37} +1.00000 q^{38} -3.80194 q^{40} -12.0978 q^{41} -1.38404 q^{42} -4.98792 q^{43} +0.890084 q^{44} +5.49396 q^{45} +6.31767 q^{46} -6.98792 q^{47} +1.24698 q^{48} -5.76809 q^{49} +9.45473 q^{50} +3.32304 q^{51} -11.6039 q^{53} -5.54288 q^{54} -3.38404 q^{55} -1.10992 q^{56} +1.24698 q^{57} +8.09783 q^{58} +5.67994 q^{59} -4.74094 q^{60} -1.97823 q^{61} +4.85086 q^{62} +1.60388 q^{63} +1.00000 q^{64} +1.10992 q^{66} +7.74094 q^{67} +2.66487 q^{68} +7.87800 q^{69} +4.21983 q^{70} +2.84117 q^{71} -1.44504 q^{72} -2.66487 q^{73} -11.4819 q^{74} +11.7899 q^{75} +1.00000 q^{76} -0.987918 q^{77} -8.36658 q^{79} -3.80194 q^{80} -2.57673 q^{81} -12.0978 q^{82} -6.49396 q^{83} -1.38404 q^{84} -10.1317 q^{85} -4.98792 q^{86} +10.0978 q^{87} +0.890084 q^{88} -15.4276 q^{89} +5.49396 q^{90} +6.31767 q^{92} +6.04892 q^{93} -6.98792 q^{94} -3.80194 q^{95} +1.24698 q^{96} +7.92154 q^{97} -5.76809 q^{98} -1.28621 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - q^{3} + 3 q^{4} - 7 q^{5} - q^{6} - 4 q^{7} + 3 q^{8} - 4 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 - q^3 + 3 * q^4 - 7 * q^5 - q^6 - 4 * q^7 + 3 * q^8 - 4 * q^9	$ 3 q + 3 q^{2} - q^{3} + 3 q^{4} - 7 q^{5} - q^{6} - 4 q^{7} + 3 q^{8} - 4 q^{9} - 7 q^{10} + 2 q^{11} - q^{12} - 4 q^{14} + 3 q^{16} + 9 q^{17} - 4 q^{18} + 3 q^{19} - 7 q^{20} + 6 q^{21} + 2 q^{22} + 2 q^{23} - q^{24} + 6 q^{25} + 2 q^{27} - 4 q^{28} + 6 q^{29} + q^{31} + 3 q^{32} + 4 q^{33} + 9 q^{34} + 14 q^{35} - 4 q^{36} - 6 q^{37} + 3 q^{38} - 7 q^{40} - 18 q^{41} + 6 q^{42} + 4 q^{43} + 2 q^{44} + 7 q^{45} + 2 q^{46} - 2 q^{47} - q^{48} + 3 q^{49} + 6 q^{50} - 10 q^{51} - 26 q^{53} + 2 q^{54} - 4 q^{56} - q^{57} + 6 q^{58} - 7 q^{59} - 9 q^{61} + q^{62} - 4 q^{63} + 3 q^{64} + 4 q^{66} + 9 q^{67} + 9 q^{68} + 4 q^{69} + 14 q^{70} + 17 q^{71} - 4 q^{72} - 9 q^{73} - 6 q^{74} + 12 q^{75} + 3 q^{76} + 16 q^{77} + q^{79} - 7 q^{80} - 5 q^{81} - 18 q^{82} - 10 q^{83} + 6 q^{84} - 28 q^{85} + 4 q^{86} + 12 q^{87} + 2 q^{88} - 30 q^{89} + 7 q^{90} + 2 q^{92} + 9 q^{93} - 2 q^{94} - 7 q^{95} - q^{96} - 2 q^{97} + 3 q^{98} - 12 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - q^3 + 3 * q^4 - 7 * q^5 - q^6 - 4 * q^7 + 3 * q^8 - 4 * q^9 - 7 * q^10 + 2 * q^11 - q^12 - 4 * q^14 + 3 * q^16 + 9 * q^17 - 4 * q^18 + 3 * q^19 - 7 * q^20 + 6 * q^21 + 2 * q^22 + 2 * q^23 - q^24 + 6 * q^25 + 2 * q^27 - 4 * q^28 + 6 * q^29 + q^31 + 3 * q^32 + 4 * q^33 + 9 * q^34 + 14 * q^35 - 4 * q^36 - 6 * q^37 + 3 * q^38 - 7 * q^40 - 18 * q^41 + 6 * q^42 + 4 * q^43 + 2 * q^44 + 7 * q^45 + 2 * q^46 - 2 * q^47 - q^48 + 3 * q^49 + 6 * q^50 - 10 * q^51 - 26 * q^53 + 2 * q^54 - 4 * q^56 - q^57 + 6 * q^58 - 7 * q^59 - 9 * q^61 + q^62 - 4 * q^63 + 3 * q^64 + 4 * q^66 + 9 * q^67 + 9 * q^68 + 4 * q^69 + 14 * q^70 + 17 * q^71 - 4 * q^72 - 9 * q^73 - 6 * q^74 + 12 * q^75 + 3 * q^76 + 16 * q^77 + q^79 - 7 * q^80 - 5 * q^81 - 18 * q^82 - 10 * q^83 + 6 * q^84 - 28 * q^85 + 4 * q^86 + 12 * q^87 + 2 * q^88 - 30 * q^89 + 7 * q^90 + 2 * q^92 + 9 * q^93 - 2 * q^94 - 7 * q^95 - q^96 - 2 * q^97 + 3 * q^98 - 12 * 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
$3$	1.24698	0.719944	0.359972	−	0.932963i	$-0.382786\pi$
$3$	1.24698	0.719944	0.359972	+	0.932963i	$0.382786\pi$
$4$	1.00000	0.500000
$5$	−3.80194	−1.70028	−0.850139	−	0.526558i	$-0.823482\pi$
$5$	−3.80194	−1.70028	−0.850139	+	0.526558i	$0.823482\pi$
$6$	1.24698	0.509077
$7$	−1.10992	−0.419509	−0.209754	−	0.977754i	$-0.567266\pi$
$7$	−1.10992	−0.419509	−0.209754	+	0.977754i	$0.567266\pi$
$8$	1.00000	0.353553
$9$	−1.44504	−0.481681
$10$	−3.80194	−1.20228
$11$	0.890084	0.268370	0.134185	−	0.990956i	$-0.457158\pi$
$11$	0.890084	0.268370	0.134185	+	0.990956i	$0.457158\pi$
$12$	1.24698	0.359972
$13$	0	0
$13$	0	0
$14$	−1.10992	−0.296638
$15$	−4.74094	−1.22411
$16$	1.00000	0.250000
$17$	2.66487	0.646327	0.323163	−	0.946343i	$-0.395254\pi$
$17$	2.66487	0.646327	0.323163	+	0.946343i	$0.395254\pi$
$18$	−1.44504	−0.340600
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−3.80194	−0.850139
$21$	−1.38404	−0.302023
$22$	0.890084	0.189766
$23$	6.31767	1.31732	0.658662	−	0.752439i	$-0.271123\pi$
$23$	6.31767	1.31732	0.658662	+	0.752439i	$0.271123\pi$
$24$	1.24698	0.254539
$25$	9.45473	1.89095
$26$	0	0
$27$	−5.54288	−1.06673
$28$	−1.10992	−0.209754
$29$	8.09783	1.50373	0.751865	−	0.659317i	$-0.229154\pi$
$29$	8.09783	1.50373	0.751865	+	0.659317i	$0.229154\pi$
$30$	−4.74094	−0.865573
$31$	4.85086	0.871239	0.435620	−	0.900131i	$-0.356529\pi$
$31$	4.85086	0.871239	0.435620	+	0.900131i	$0.356529\pi$
$32$	1.00000	0.176777
$33$	1.10992	0.193212
$34$	2.66487	0.457022
$35$	4.21983	0.713282
$36$	−1.44504	−0.240840
$37$	−11.4819	−1.88761	−0.943805	−	0.330504i	$-0.892781\pi$
$37$	−11.4819	−1.88761	−0.943805	+	0.330504i	$0.892781\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	−3.80194	−0.601139
$41$	−12.0978	−1.88936	−0.944682	−	0.327987i	$-0.893630\pi$
$41$	−12.0978	−1.88936	−0.944682	+	0.327987i	$0.893630\pi$
$42$	−1.38404	−0.213562
$43$	−4.98792	−0.760650	−0.380325	−	0.924853i	$-0.624188\pi$
$43$	−4.98792	−0.760650	−0.380325	+	0.924853i	$0.624188\pi$
$44$	0.890084	0.134185
$45$	5.49396	0.818991
$46$	6.31767	0.931489
$47$	−6.98792	−1.01929	−0.509646	−	0.860384i	$-0.670224\pi$
$47$	−6.98792	−1.01929	−0.509646	+	0.860384i	$0.670224\pi$
$48$	1.24698	0.179986
$49$	−5.76809	−0.824012
$50$	9.45473	1.33710
$51$	3.32304	0.465319
$52$	0	0
$53$	−11.6039	−1.59391	−0.796957	−	0.604035i	$-0.793559\pi$
$53$	−11.6039	−1.59391	−0.796957	+	0.604035i	$0.793559\pi$
$54$	−5.54288	−0.754290
$55$	−3.38404	−0.456304
$56$	−1.10992	−0.148319
$57$	1.24698	0.165166
$58$	8.09783	1.06330
$59$	5.67994	0.739465	0.369733	−	0.929138i	$-0.379449\pi$
$59$	5.67994	0.739465	0.369733	+	0.929138i	$0.379449\pi$
$60$	−4.74094	−0.612053
$61$	−1.97823	−0.253286	−0.126643	−	0.991948i	$-0.540420\pi$
$61$	−1.97823	−0.253286	−0.126643	+	0.991948i	$0.540420\pi$
$62$	4.85086	0.616059
$63$	1.60388	0.202069
$64$	1.00000	0.125000
$65$	0	0
$66$	1.10992	0.136621
$67$	7.74094	0.945706	0.472853	−	0.881141i	$-0.343224\pi$
$67$	7.74094	0.945706	0.472853	+	0.881141i	$0.343224\pi$
$68$	2.66487	0.323163
$69$	7.87800	0.948400
$70$	4.21983	0.504366
$71$	2.84117	0.337184	0.168592	−	0.985686i	$-0.446078\pi$
$71$	2.84117	0.337184	0.168592	+	0.985686i	$0.446078\pi$
$72$	−1.44504	−0.170300
$73$	−2.66487	−0.311900	−0.155950	−	0.987765i	$-0.549844\pi$
$73$	−2.66487	−0.311900	−0.155950	+	0.987765i	$0.549844\pi$
$74$	−11.4819	−1.33474
$75$	11.7899	1.36138
$76$	1.00000	0.114708
$77$	−0.987918	−0.112584
$78$	0	0
$79$	−8.36658	−0.941314	−0.470657	−	0.882316i	$-0.655983\pi$
$79$	−8.36658	−0.941314	−0.470657	+	0.882316i	$0.655983\pi$
$80$	−3.80194	−0.425070
$81$	−2.57673	−0.286303
$82$	−12.0978	−1.33598
$83$	−6.49396	−0.712805	−0.356402	−	0.934333i	$-0.615997\pi$
$83$	−6.49396	−0.712805	−0.356402	+	0.934333i	$0.615997\pi$
$84$	−1.38404	−0.151011
$85$	−10.1317	−1.09894
$86$	−4.98792	−0.537861
$87$	10.0978	1.08260
$88$	0.890084	0.0948832
$89$	−15.4276	−1.63532	−0.817660	−	0.575701i	$-0.804729\pi$
$89$	−15.4276	−1.63532	−0.817660	+	0.575701i	$0.804729\pi$
$90$	5.49396	0.579114
$91$	0	0
$92$	6.31767	0.658662
$93$	6.04892	0.627244
$94$	−6.98792	−0.720749
$95$	−3.80194	−0.390071
$96$	1.24698	0.127269
$97$	7.92154	0.804311	0.402155	−	0.915571i	$-0.368261\pi$
$97$	7.92154	0.804311	0.402155	+	0.915571i	$0.368261\pi$
$98$	−5.76809	−0.582665
$99$	−1.28621	−0.129269
$100$	9.45473	0.945473
$101$	0.637727	0.0634562	0.0317281	−	0.999497i	$-0.489899\pi$
$101$	0.637727	0.0634562	0.0317281	+	0.999497i	$0.489899\pi$
$102$	3.32304	0.329030
$103$	8.47219	0.834790	0.417395	−	0.908725i	$-0.362943\pi$
$103$	8.47219	0.834790	0.417395	+	0.908725i	$0.362943\pi$
$104$	0	0
$105$	5.26205	0.513523
$106$	−11.6039	−1.12707
$107$	−9.65279	−0.933171	−0.466585	−	0.884476i	$-0.654516\pi$
$107$	−9.65279	−0.933171	−0.466585	+	0.884476i	$0.654516\pi$
$108$	−5.54288	−0.533364
$109$	−4.79225	−0.459014	−0.229507	−	0.973307i	$-0.573711\pi$
$109$	−4.79225	−0.459014	−0.229507	+	0.973307i	$0.573711\pi$
$110$	−3.38404	−0.322656
$111$	−14.3177	−1.35897
$112$	−1.10992	−0.104877
$113$	1.50604	0.141676	0.0708382	−	0.997488i	$-0.477433\pi$
$113$	1.50604	0.141676	0.0708382	+	0.997488i	$0.477433\pi$
$114$	1.24698	0.116790
$115$	−24.0194	−2.23982
$116$	8.09783	0.751865
$117$	0	0
$118$	5.67994	0.522881
$119$	−2.95779	−0.271140
$120$	−4.74094	−0.432787
$121$	−10.2078	−0.927977
$122$	−1.97823	−0.179101
$123$	−15.0858	−1.36024
$124$	4.85086	0.435620
$125$	−16.9366	−1.51486
$126$	1.60388	0.142885
$127$	−18.3569	−1.62891	−0.814456	−	0.580226i	$-0.802964\pi$
$127$	−18.3569	−1.62891	−0.814456	+	0.580226i	$0.802964\pi$
$128$	1.00000	0.0883883
$129$	−6.21983	−0.547626
$130$	0	0
$131$	−5.10992	−0.446455	−0.223228	−	0.974766i	$-0.571659\pi$
$131$	−5.10992	−0.446455	−0.223228	+	0.974766i	$0.571659\pi$
$132$	1.10992	0.0966058
$133$	−1.10992	−0.0962419
$134$	7.74094	0.668715
$135$	21.0737	1.81373
$136$	2.66487	0.228511
$137$	−0.570024	−0.0487004	−0.0243502	−	0.999703i	$-0.507752\pi$
$137$	−0.570024	−0.0487004	−0.0243502	+	0.999703i	$0.507752\pi$
$138$	7.87800	0.670620
$139$	6.14138	0.520905	0.260452	−	0.965487i	$-0.416128\pi$
$139$	6.14138	0.520905	0.260452	+	0.965487i	$0.416128\pi$
$140$	4.21983	0.356641
$141$	−8.71379	−0.733834
$142$	2.84117	0.238425
$143$	0	0
$144$	−1.44504	−0.120420
$145$	−30.7875	−2.55676
$146$	−2.66487	−0.220547
$147$	−7.19269	−0.593243
$148$	−11.4819	−0.943805
$149$	−4.40150	−0.360585	−0.180293	−	0.983613i	$-0.557704\pi$
$149$	−4.40150	−0.360585	−0.180293	+	0.983613i	$0.557704\pi$
$150$	11.7899	0.962638
$151$	−11.0368	−0.898165	−0.449082	−	0.893490i	$-0.648249\pi$
$151$	−11.0368	−0.898165	−0.449082	+	0.893490i	$0.648249\pi$
$152$	1.00000	0.0811107
$153$	−3.85086	−0.311323
$154$	−0.987918	−0.0796087
$155$	−18.4426	−1.48135
$156$	0	0
$157$	22.0248	1.75777	0.878883	−	0.477037i	$-0.158289\pi$
$157$	22.0248	1.75777	0.878883	+	0.477037i	$0.158289\pi$
$158$	−8.36658	−0.665610
$159$	−14.4698	−1.14753
$160$	−3.80194	−0.300570
$161$	−7.01208	−0.552629
$162$	−2.57673	−0.202447
$163$	−10.3177	−0.808142	−0.404071	−	0.914728i	$-0.632405\pi$
$163$	−10.3177	−0.808142	−0.404071	+	0.914728i	$0.632405\pi$
$164$	−12.0978	−0.944682
$165$	−4.21983	−0.328514
$166$	−6.49396	−0.504029
$167$	−16.7724	−1.29789	−0.648944	−	0.760837i	$-0.724789\pi$
$167$	−16.7724	−1.29789	−0.648944	+	0.760837i	$0.724789\pi$
$168$	−1.38404	−0.106781
$169$	0	0
$170$	−10.1317	−0.777065
$171$	−1.44504	−0.110505
$172$	−4.98792	−0.380325
$173$	−9.35988	−0.711618	−0.355809	−	0.934559i	$-0.615795\pi$
$173$	−9.35988	−0.711618	−0.355809	+	0.934559i	$0.615795\pi$
$174$	10.0978	0.765515
$175$	−10.4940	−0.793269
$176$	0.890084	0.0670926
$177$	7.08277	0.532374
$178$	−15.4276	−1.15635
$179$	−1.89679	−0.141773	−0.0708863	−	0.997484i	$-0.522583\pi$
$179$	−1.89679	−0.141773	−0.0708863	+	0.997484i	$0.522583\pi$
$180$	5.49396	0.409496
$181$	6.57242	0.488524	0.244262	−	0.969709i	$-0.421454\pi$
$181$	6.57242	0.488524	0.244262	+	0.969709i	$0.421454\pi$
$182$	0	0
$183$	−2.46681	−0.182352
$184$	6.31767	0.465745
$185$	43.6534	3.20946
$186$	6.04892	0.443528
$187$	2.37196	0.173455
$188$	−6.98792	−0.509646
$189$	6.15213	0.447502
$190$	−3.80194	−0.275822
$191$	−7.32975	−0.530362	−0.265181	−	0.964199i	$-0.585432\pi$
$191$	−7.32975	−0.530362	−0.265181	+	0.964199i	$0.585432\pi$
$192$	1.24698	0.0899930
$193$	−14.8224	−1.06694	−0.533469	−	0.845820i	$-0.679112\pi$
$193$	−14.8224	−1.06694	−0.533469	+	0.845820i	$0.679112\pi$
$194$	7.92154	0.568734
$195$	0	0
$196$	−5.76809	−0.412006
$197$	6.70841	0.477955	0.238977	−	0.971025i	$-0.423188\pi$
$197$	6.70841	0.477955	0.238977	+	0.971025i	$0.423188\pi$
$198$	−1.28621	−0.0914068
$199$	22.5676	1.59978	0.799888	−	0.600149i	$-0.204892\pi$
$199$	22.5676	1.59978	0.799888	+	0.600149i	$0.204892\pi$
$200$	9.45473	0.668550
$201$	9.65279	0.680856
$202$	0.637727	0.0448703
$203$	−8.98792	−0.630828
$204$	3.32304	0.232660
$205$	45.9952	3.21245
$206$	8.47219	0.590285
$207$	−9.12929	−0.634530
$208$	0	0
$209$	0.890084	0.0615684
$210$	5.26205	0.363116
$211$	−7.30127	−0.502640	−0.251320	−	0.967904i	$-0.580865\pi$
$211$	−7.30127	−0.502640	−0.251320	+	0.967904i	$0.580865\pi$
$212$	−11.6039	−0.796957
$213$	3.54288	0.242754
$214$	−9.65279	−0.659851
$215$	18.9638	1.29332
$216$	−5.54288	−0.377145
$217$	−5.38404	−0.365493
$218$	−4.79225	−0.324572
$219$	−3.32304	−0.224551
$220$	−3.38404	−0.228152
$221$	0	0
$222$	−14.3177	−0.960939
$223$	23.4282	1.56887	0.784433	−	0.620213i	$-0.212954\pi$
$223$	23.4282	1.56887	0.784433	+	0.620213i	$0.212954\pi$
$224$	−1.10992	−0.0741594
$225$	−13.6625	−0.910832
$226$	1.50604	0.100180
$227$	−13.2470	−0.879233	−0.439616	−	0.898186i	$-0.644886\pi$
$227$	−13.2470	−0.879233	−0.439616	+	0.898186i	$0.644886\pi$
$228$	1.24698	0.0825832
$229$	−6.41789	−0.424106	−0.212053	−	0.977258i	$-0.568015\pi$
$229$	−6.41789	−0.424106	−0.212053	+	0.977258i	$0.568015\pi$
$230$	−24.0194	−1.58379
$231$	−1.23191	−0.0810540
$232$	8.09783	0.531649
$233$	9.16182	0.600211	0.300105	−	0.953906i	$-0.402978\pi$
$233$	9.16182	0.600211	0.300105	+	0.953906i	$0.402978\pi$
$234$	0	0
$235$	26.5676	1.73308
$236$	5.67994	0.369733
$237$	−10.4330	−0.677694
$238$	−2.95779	−0.191725
$239$	10.7922	0.698093	0.349046	−	0.937105i	$-0.386506\pi$
$239$	10.7922	0.698093	0.349046	+	0.937105i	$0.386506\pi$
$240$	−4.74094	−0.306026
$241$	−24.8418	−1.60020	−0.800099	−	0.599868i	$-0.795220\pi$
$241$	−24.8418	−1.60020	−0.800099	+	0.599868i	$0.795220\pi$
$242$	−10.2078	−0.656179
$243$	13.4155	0.860605
$244$	−1.97823	−0.126643
$245$	21.9299	1.40105
$246$	−15.0858	−0.961832
$247$	0	0
$248$	4.85086	0.308030
$249$	−8.09783	−0.513179
$250$	−16.9366	−1.07117
$251$	26.9879	1.70346	0.851731	−	0.523979i	$-0.175553\pi$
$251$	26.9879	1.70346	0.851731	+	0.523979i	$0.175553\pi$
$252$	1.60388	0.101035
$253$	5.62325	0.353531
$254$	−18.3569	−1.15181
$255$	−12.6340	−0.791172
$256$	1.00000	0.0625000
$257$	−13.7259	−0.856196	−0.428098	−	0.903732i	$-0.640816\pi$
$257$	−13.7259	−0.856196	−0.428098	+	0.903732i	$0.640816\pi$
$258$	−6.21983	−0.387230
$259$	12.7439	0.791869
$260$	0	0
$261$	−11.7017	−0.724318
$262$	−5.10992	−0.315692
$263$	14.5676	0.898279	0.449139	−	0.893462i	$-0.351731\pi$
$263$	14.5676	0.898279	0.449139	+	0.893462i	$0.351731\pi$
$264$	1.10992	0.0683106
$265$	44.1172	2.71010
$266$	−1.10992	−0.0680533
$267$	−19.2379	−1.17734
$268$	7.74094	0.472853
$269$	−12.1521	−0.740928	−0.370464	−	0.928847i	$-0.620801\pi$
$269$	−12.1521	−0.740928	−0.370464	+	0.928847i	$0.620801\pi$
$270$	21.0737	1.28250
$271$	17.6474	1.07200	0.536002	−	0.844217i	$-0.319934\pi$
$271$	17.6474	1.07200	0.536002	+	0.844217i	$0.319934\pi$
$272$	2.66487	0.161582
$273$	0	0
$274$	−0.570024	−0.0344364
$275$	8.41550	0.507474
$276$	7.87800	0.474200
$277$	−28.9855	−1.74157	−0.870786	−	0.491663i	$-0.836389\pi$
$277$	−28.9855	−1.74157	−0.870786	+	0.491663i	$0.836389\pi$
$278$	6.14138	0.368335
$279$	−7.00969	−0.419659
$280$	4.21983	0.252183
$281$	5.18359	0.309227	0.154613	−	0.987975i	$-0.450587\pi$
$281$	5.18359	0.309227	0.154613	+	0.987975i	$0.450587\pi$
$282$	−8.71379	−0.518899
$283$	23.3250	1.38653	0.693263	−	0.720685i	$-0.256173\pi$
$283$	23.3250	1.38653	0.693263	+	0.720685i	$0.256173\pi$
$284$	2.84117	0.168592
$285$	−4.74094	−0.280829
$286$	0	0
$287$	13.4276	0.792605
$288$	−1.44504	−0.0851499
$289$	−9.89844	−0.582261
$290$	−30.7875	−1.80790
$291$	9.87800	0.579059
$292$	−2.66487	−0.155950
$293$	4.81163	0.281098	0.140549	−	0.990074i	$-0.455113\pi$
$293$	4.81163	0.281098	0.140549	+	0.990074i	$0.455113\pi$
$294$	−7.19269	−0.419486
$295$	−21.5948	−1.25730
$296$	−11.4819	−0.667371
$297$	−4.93362	−0.286278
$298$	−4.40150	−0.254972
$299$	0	0
$300$	11.7899	0.680688
$301$	5.53617	0.319100
$302$	−11.0368	−0.635099
$303$	0.795233	0.0456849
$304$	1.00000	0.0573539
$305$	7.52111	0.430657
$306$	−3.85086	−0.220139
$307$	−9.88231	−0.564013	−0.282007	−	0.959412i	$-0.591000\pi$
$307$	−9.88231	−0.564013	−0.282007	+	0.959412i	$0.591000\pi$
$308$	−0.987918	−0.0562919
$309$	10.5646	0.601002
$310$	−18.4426	−1.04747
$311$	15.4383	0.875428	0.437714	−	0.899114i	$-0.355788\pi$
$311$	15.4383	0.875428	0.437714	+	0.899114i	$0.355788\pi$
$312$	0	0
$313$	−14.9855	−0.847032	−0.423516	−	0.905889i	$-0.639204\pi$
$313$	−14.9855	−0.847032	−0.423516	+	0.905889i	$0.639204\pi$
$314$	22.0248	1.24293
$315$	−6.09783	−0.343574
$316$	−8.36658	−0.470657
$317$	−2.26337	−0.127124	−0.0635618	−	0.997978i	$-0.520246\pi$
$317$	−2.26337	−0.127124	−0.0635618	+	0.997978i	$0.520246\pi$
$318$	−14.4698	−0.811426
$319$	7.20775	0.403557
$320$	−3.80194	−0.212535
$321$	−12.0368	−0.671831
$322$	−7.01208	−0.390768
$323$	2.66487	0.148278
$324$	−2.57673	−0.143152
$325$	0	0
$326$	−10.3177	−0.571443
$327$	−5.97584	−0.330465
$328$	−12.0978	−0.667991
$329$	7.75600	0.427602
$330$	−4.21983	−0.232294
$331$	−1.44265	−0.0792952	−0.0396476	−	0.999214i	$-0.512624\pi$
$331$	−1.44265	−0.0792952	−0.0396476	+	0.999214i	$0.512624\pi$
$332$	−6.49396	−0.356402
$333$	16.5918	0.909225
$334$	−16.7724	−0.917745
$335$	−29.4306	−1.60796
$336$	−1.38404	−0.0755057
$337$	−30.5133	−1.66217	−0.831084	−	0.556147i	$-0.812279\pi$
$337$	−30.5133	−1.66217	−0.831084	+	0.556147i	$0.812279\pi$
$338$	0	0
$339$	1.87800	0.101999
$340$	−10.1317	−0.549468
$341$	4.31767	0.233815
$342$	−1.44504	−0.0781389
$343$	14.1715	0.765189
$344$	−4.98792	−0.268931
$345$	−29.9517	−1.61254
$346$	−9.35988	−0.503190
$347$	−20.6461	−1.10834	−0.554170	−	0.832403i	$-0.686964\pi$
$347$	−20.6461	−1.10834	−0.554170	+	0.832403i	$0.686964\pi$
$348$	10.0978	0.541301
$349$	−10.6310	−0.569066	−0.284533	−	0.958666i	$-0.591838\pi$
$349$	−10.6310	−0.569066	−0.284533	+	0.958666i	$0.591838\pi$
$350$	−10.4940	−0.560926
$351$	0	0
$352$	0.890084	0.0474416
$353$	−0.0935228	−0.00497772	−0.00248886	−	0.999997i	$-0.500792\pi$
$353$	−0.0935228	−0.00497772	−0.00248886	+	0.999997i	$0.500792\pi$
$354$	7.08277	0.376445
$355$	−10.8019	−0.573307
$356$	−15.4276	−0.817660
$357$	−3.68830	−0.195206
$358$	−1.89679	−0.100248
$359$	28.6703	1.51316	0.756579	−	0.653902i	$-0.226869\pi$
$359$	28.6703	1.51316	0.756579	+	0.653902i	$0.226869\pi$
$360$	5.49396	0.289557
$361$	1.00000	0.0526316
$362$	6.57242	0.345439
$363$	−12.7289	−0.668092
$364$	0	0
$365$	10.1317	0.530317
$366$	−2.46681	−0.128942
$367$	14.4940	0.756579	0.378289	−	0.925687i	$-0.376512\pi$
$367$	14.4940	0.756579	0.378289	+	0.925687i	$0.376512\pi$
$368$	6.31767	0.329331
$369$	17.4819	0.910070
$370$	43.6534	2.26943
$371$	12.8793	0.668662
$372$	6.04892	0.313622
$373$	−20.5241	−1.06270	−0.531349	−	0.847153i	$-0.678315\pi$
$373$	−20.5241	−1.06270	−0.531349	+	0.847153i	$0.678315\pi$
$374$	2.37196	0.122651
$375$	−21.1196	−1.09061
$376$	−6.98792	−0.360374
$377$	0	0
$378$	6.15213	0.316431
$379$	−1.34050	−0.0688570	−0.0344285	−	0.999407i	$-0.510961\pi$
$379$	−1.34050	−0.0688570	−0.0344285	+	0.999407i	$0.510961\pi$
$380$	−3.80194	−0.195035
$381$	−22.8907	−1.17272
$382$	−7.32975	−0.375023
$383$	0.667858	0.0341260	0.0170630	−	0.999854i	$-0.494568\pi$
$383$	0.667858	0.0341260	0.0170630	+	0.999854i	$0.494568\pi$
$384$	1.24698	0.0636347
$385$	3.75600	0.191424
$386$	−14.8224	−0.754439
$387$	7.20775	0.366391
$388$	7.92154	0.402155
$389$	34.0388	1.72583	0.862917	−	0.505346i	$-0.168635\pi$
$389$	34.0388	1.72583	0.862917	+	0.505346i	$0.168635\pi$
$390$	0	0
$391$	16.8358	0.851422
$392$	−5.76809	−0.291332
$393$	−6.37196	−0.321423
$394$	6.70841	0.337965
$395$	31.8092	1.60050
$396$	−1.28621	−0.0646344
$397$	11.6063	0.582502	0.291251	−	0.956647i	$-0.405929\pi$
$397$	11.6063	0.582502	0.291251	+	0.956647i	$0.405929\pi$
$398$	22.5676	1.13121
$399$	−1.38404	−0.0692888
$400$	9.45473	0.472737
$401$	−11.2948	−0.564037	−0.282018	−	0.959409i	$-0.591004\pi$
$401$	−11.2948	−0.564037	−0.282018	+	0.959409i	$0.591004\pi$
$402$	9.65279	0.481438
$403$	0	0
$404$	0.637727	0.0317281
$405$	9.79656	0.486795
$406$	−8.98792	−0.446063
$407$	−10.2198	−0.506578
$408$	3.32304	0.164515
$409$	−6.07846	−0.300560	−0.150280	−	0.988643i	$-0.548018\pi$
$409$	−6.07846	−0.300560	−0.150280	+	0.988643i	$0.548018\pi$
$410$	45.9952	2.27154
$411$	−0.710808	−0.0350616
$412$	8.47219	0.417395
$413$	−6.30426	−0.310212
$414$	−9.12929	−0.448680
$415$	24.6896	1.21197
$416$	0	0
$417$	7.65817	0.375022
$418$	0.890084	0.0435354
$419$	18.9202	0.924313	0.462156	−	0.886798i	$-0.347076\pi$
$419$	18.9202	0.924313	0.462156	+	0.886798i	$0.347076\pi$
$420$	5.26205	0.256762
$421$	−9.23191	−0.449936	−0.224968	−	0.974366i	$-0.572228\pi$
$421$	−9.23191	−0.449936	−0.224968	+	0.974366i	$0.572228\pi$
$422$	−7.30127	−0.355420
$423$	10.0978	0.490974
$424$	−11.6039	−0.563534
$425$	25.1957	1.22217
$426$	3.54288	0.171653
$427$	2.19567	0.106256
$428$	−9.65279	−0.466585
$429$	0	0
$430$	18.9638	0.914513
$431$	25.5646	1.23141	0.615703	−	0.787978i	$-0.288872\pi$
$431$	25.5646	1.23141	0.615703	+	0.787978i	$0.288872\pi$
$432$	−5.54288	−0.266682
$433$	−30.3129	−1.45674	−0.728372	−	0.685182i	$-0.759723\pi$
$433$	−30.3129	−1.45674	−0.728372	+	0.685182i	$0.759723\pi$
$434$	−5.38404	−0.258442
$435$	−38.3913	−1.84072
$436$	−4.79225	−0.229507
$437$	6.31767	0.302215
$438$	−3.32304	−0.158781
$439$	−0.576728	−0.0275257	−0.0137629	−	0.999905i	$-0.504381\pi$
$439$	−0.576728	−0.0275257	−0.0137629	+	0.999905i	$0.504381\pi$
$440$	−3.38404	−0.161328
$441$	8.33513	0.396911
$442$	0	0
$443$	−39.3250	−1.86839	−0.934193	−	0.356769i	$-0.883878\pi$
$443$	−39.3250	−1.86839	−0.934193	+	0.356769i	$0.883878\pi$
$444$	−14.3177	−0.679486
$445$	58.6547	2.78050
$446$	23.4282	1.10936
$447$	−5.48858	−0.259601
$448$	−1.10992	−0.0524386
$449$	5.92154	0.279455	0.139727	−	0.990190i	$-0.455377\pi$
$449$	5.92154	0.279455	0.139727	+	0.990190i	$0.455377\pi$
$450$	−13.6625	−0.644056
$451$	−10.7681	−0.507049
$452$	1.50604	0.0708382
$453$	−13.7627	−0.646628
$454$	−13.2470	−0.621712
$455$	0	0
$456$	1.24698	0.0583952
$457$	−27.6256	−1.29227	−0.646137	−	0.763222i	$-0.723616\pi$
$457$	−27.6256	−1.29227	−0.646137	+	0.763222i	$0.723616\pi$
$458$	−6.41789	−0.299889
$459$	−14.7711	−0.689454
$460$	−24.0194	−1.11991
$461$	−33.2325	−1.54779	−0.773896	−	0.633313i	$-0.781695\pi$
$461$	−33.2325	−1.54779	−0.773896	+	0.633313i	$0.781695\pi$
$462$	−1.23191	−0.0573138
$463$	−42.4784	−1.97414	−0.987070	−	0.160291i	$-0.948757\pi$
$463$	−42.4784	−1.97414	−0.987070	+	0.160291i	$0.948757\pi$
$464$	8.09783	0.375933
$465$	−22.9976	−1.06649
$466$	9.16182	0.424413
$467$	35.7318	1.65347	0.826736	−	0.562590i	$-0.190195\pi$
$467$	35.7318	1.65347	0.826736	+	0.562590i	$0.190195\pi$
$468$	0	0
$469$	−8.59179	−0.396732
$470$	26.5676	1.22547
$471$	27.4644	1.26549
$472$	5.67994	0.261440
$473$	−4.43967	−0.204136
$474$	−10.4330	−0.479202
$475$	9.45473	0.433813
$476$	−2.95779	−0.135570
$477$	16.7681	0.767758
$478$	10.7922	0.493626
$479$	8.43967	0.385618	0.192809	−	0.981236i	$-0.438240\pi$
$479$	8.43967	0.385618	0.192809	+	0.981236i	$0.438240\pi$
$480$	−4.74094	−0.216393
$481$	0	0
$482$	−24.8418	−1.13151
$483$	−8.74392	−0.397862
$484$	−10.2078	−0.463989
$485$	−30.1172	−1.36755
$486$	13.4155	0.608540
$487$	25.8689	1.17223	0.586116	−	0.810227i	$-0.300656\pi$
$487$	25.8689	1.17223	0.586116	+	0.810227i	$0.300656\pi$
$488$	−1.97823	−0.0895503
$489$	−12.8659	−0.581817
$490$	21.9299	0.990692
$491$	13.9323	0.628756	0.314378	−	0.949298i	$-0.398204\pi$
$491$	13.9323	0.628756	0.314378	+	0.949298i	$0.398204\pi$
$492$	−15.0858	−0.680118
$493$	21.5797	0.971901
$494$	0	0
$495$	4.89008	0.219793
$496$	4.85086	0.217810
$497$	−3.15346	−0.141452
$498$	−8.09783	−0.362873
$499$	32.2500	1.44371	0.721853	−	0.692046i	$-0.243291\pi$
$499$	32.2500	1.44371	0.721853	+	0.692046i	$0.243291\pi$
$500$	−16.9366	−0.757428
$501$	−20.9148	−0.934406
$502$	26.9879	1.20453
$503$	−9.41417	−0.419757	−0.209879	−	0.977727i	$-0.567307\pi$
$503$	−9.41417	−0.419757	−0.209879	+	0.977727i	$0.567307\pi$
$504$	1.60388	0.0714423
$505$	−2.42460	−0.107893
$506$	5.62325	0.249984
$507$	0	0
$508$	−18.3569	−0.814456
$509$	27.6039	1.22352	0.611760	−	0.791043i	$-0.290462\pi$
$509$	27.6039	1.22352	0.611760	+	0.791043i	$0.290462\pi$
$510$	−12.6340	−0.559443
$511$	2.95779	0.130845
$512$	1.00000	0.0441942
$513$	−5.54288	−0.244724
$514$	−13.7259	−0.605422
$515$	−32.2107	−1.41937
$516$	−6.21983	−0.273813
$517$	−6.21983	−0.273548
$518$	12.7439	0.559936
$519$	−11.6716	−0.512325
$520$	0	0
$521$	20.6461	0.904522	0.452261	−	0.891886i	$-0.350618\pi$
$521$	20.6461	0.904522	0.452261	+	0.891886i	$0.350618\pi$
$522$	−11.7017	−0.512170
$523$	7.20775	0.315173	0.157586	−	0.987505i	$-0.449629\pi$
$523$	7.20775	0.315173	0.157586	+	0.987505i	$0.449629\pi$
$524$	−5.10992	−0.223228
$525$	−13.0858	−0.571109
$526$	14.5676	0.635179
$527$	12.9269	0.563105
$528$	1.10992	0.0483029
$529$	16.9129	0.735344
$530$	44.1172	1.91633
$531$	−8.20775	−0.356186
$532$	−1.10992	−0.0481210
$533$	0	0
$534$	−19.2379	−0.832505
$535$	36.6993	1.58665
$536$	7.74094	0.334358
$537$	−2.36526	−0.102068
$538$	−12.1521	−0.523915
$539$	−5.13408	−0.221140
$540$	21.0737	0.906866
$541$	22.5827	0.970906	0.485453	−	0.874263i	$-0.338655\pi$
$541$	22.5827	0.970906	0.485453	+	0.874263i	$0.338655\pi$
$542$	17.6474	0.758021
$543$	8.19567	0.351710
$544$	2.66487	0.114256
$545$	18.2198	0.780452
$546$	0	0
$547$	−11.3351	−0.484655	−0.242327	−	0.970195i	$-0.577911\pi$
$547$	−11.3351	−0.484655	−0.242327	+	0.970195i	$0.577911\pi$
$548$	−0.570024	−0.0243502
$549$	2.85862	0.122003
$550$	8.41550	0.358838
$551$	8.09783	0.344979
$552$	7.87800	0.335310
$553$	9.28621	0.394890
$554$	−28.9855	−1.23148
$555$	54.4349	2.31063
$556$	6.14138	0.260452
$557$	17.9734	0.761559	0.380780	−	0.924666i	$-0.375656\pi$
$557$	17.9734	0.761559	0.380780	+	0.924666i	$0.375656\pi$
$558$	−7.00969	−0.296744
$559$	0	0
$560$	4.21983	0.178320
$561$	2.95779	0.124878
$562$	5.18359	0.218656
$563$	31.3013	1.31919	0.659596	−	0.751621i	$-0.270728\pi$
$563$	31.3013	1.31919	0.659596	+	0.751621i	$0.270728\pi$
$564$	−8.71379	−0.366917
$565$	−5.72587	−0.240889
$566$	23.3250	0.980421
$567$	2.85995	0.120107
$568$	2.84117	0.119213
$569$	−0.396125	−0.0166064	−0.00830320	−	0.999966i	$-0.502643\pi$
$569$	−0.396125	−0.0166064	−0.00830320	+	0.999966i	$0.502643\pi$
$570$	−4.74094	−0.198576
$571$	46.1232	1.93020	0.965098	−	0.261891i	$-0.0843460\pi$
$571$	46.1232	1.93020	0.965098	+	0.261891i	$0.0843460\pi$
$572$	0	0
$573$	−9.14005	−0.381831
$574$	13.4276	0.560457
$575$	59.7318	2.49099
$576$	−1.44504	−0.0602101
$577$	−12.6987	−0.528655	−0.264327	−	0.964433i	$-0.585150\pi$
$577$	−12.6987	−0.528655	−0.264327	+	0.964433i	$0.585150\pi$
$578$	−9.89844	−0.411721
$579$	−18.4832	−0.768136
$580$	−30.7875	−1.27838
$581$	7.20775	0.299028
$582$	9.87800	0.409456
$583$	−10.3284	−0.427759
$584$	−2.66487	−0.110273
$585$	0	0
$586$	4.81163	0.198766
$587$	15.1051	0.623455	0.311728	−	0.950171i	$-0.399092\pi$
$587$	15.1051	0.623455	0.311728	+	0.950171i	$0.399092\pi$
$588$	−7.19269	−0.296621
$589$	4.85086	0.199876
$590$	−21.5948	−0.889043
$591$	8.36526	0.344101
$592$	−11.4819	−0.471902
$593$	8.06292	0.331104	0.165552	−	0.986201i	$-0.447059\pi$
$593$	8.06292	0.331104	0.165552	+	0.986201i	$0.447059\pi$
$594$	−4.93362	−0.202429
$595$	11.2453	0.461013
$596$	−4.40150	−0.180293
$597$	28.1414	1.15175
$598$	0	0
$599$	−30.5526	−1.24834	−0.624172	−	0.781287i	$-0.714564\pi$
$599$	−30.5526	−1.24834	−0.624172	+	0.781287i	$0.714564\pi$
$600$	11.7899	0.481319
$601$	−28.9396	−1.18047	−0.590235	−	0.807231i	$-0.700965\pi$
$601$	−28.9396	−1.18047	−0.590235	+	0.807231i	$0.700965\pi$
$602$	5.53617	0.225638
$603$	−11.1860	−0.455528
$604$	−11.0368	−0.449082
$605$	38.8092	1.57782
$606$	0.795233	0.0323041
$607$	−30.1825	−1.22507	−0.612535	−	0.790443i	$-0.709850\pi$
$607$	−30.1825	−1.22507	−0.612535	+	0.790443i	$0.709850\pi$
$608$	1.00000	0.0405554
$609$	−11.2078	−0.454161
$610$	7.52111	0.304521
$611$	0	0
$612$	−3.85086	−0.155662
$613$	−37.9259	−1.53181	−0.765905	−	0.642953i	$-0.777709\pi$
$613$	−37.9259	−1.53181	−0.765905	+	0.642953i	$0.777709\pi$
$614$	−9.88231	−0.398818
$615$	57.3551	2.31278
$616$	−0.987918	−0.0398044
$617$	0.190293	0.00766089	0.00383044	−	0.999993i	$-0.498781\pi$
$617$	0.190293	0.00766089	0.00383044	+	0.999993i	$0.498781\pi$
$618$	10.5646	0.424972
$619$	40.8310	1.64114	0.820568	−	0.571548i	$-0.193657\pi$
$619$	40.8310	1.64114	0.820568	+	0.571548i	$0.193657\pi$
$620$	−18.4426	−0.740675
$621$	−35.0180	−1.40523
$622$	15.4383	0.619021
$623$	17.1233	0.686032
$624$	0	0
$625$	17.1183	0.684731
$626$	−14.9855	−0.598942
$627$	1.10992	0.0443258
$628$	22.0248	0.878883
$629$	−30.5978	−1.22001
$630$	−6.09783	−0.242944
$631$	−1.43237	−0.0570217	−0.0285109	−	0.999593i	$-0.509077\pi$
$631$	−1.43237	−0.0570217	−0.0285109	+	0.999593i	$0.509077\pi$
$632$	−8.36658	−0.332805
$633$	−9.10454	−0.361873
$634$	−2.26337	−0.0898900
$635$	69.7918	2.76960
$636$	−14.4698	−0.573765
$637$	0	0
$638$	7.20775	0.285358
$639$	−4.10560	−0.162415
$640$	−3.80194	−0.150285
$641$	−27.2814	−1.07755	−0.538776	−	0.842449i	$-0.681113\pi$
$641$	−27.2814	−1.07755	−0.538776	+	0.842449i	$0.681113\pi$
$642$	−12.0368	−0.475056
$643$	15.5555	0.613451	0.306725	−	0.951798i	$-0.400767\pi$
$643$	15.5555	0.613451	0.306725	+	0.951798i	$0.400767\pi$
$644$	−7.01208	−0.276315
$645$	23.6474	0.931116
$646$	2.66487	0.104848
$647$	−45.8625	−1.80304	−0.901520	−	0.432738i	$-0.857548\pi$
$647$	−45.8625	−1.80304	−0.901520	+	0.432738i	$0.857548\pi$
$648$	−2.57673	−0.101223
$649$	5.05562	0.198451
$650$	0	0
$651$	−6.71379	−0.263134
$652$	−10.3177	−0.404071
$653$	−16.6950	−0.653326	−0.326663	−	0.945141i	$-0.605924\pi$
$653$	−16.6950	−0.653326	−0.326663	+	0.945141i	$0.605924\pi$
$654$	−5.97584	−0.233674
$655$	19.4276	0.759099
$656$	−12.0978	−0.472341
$657$	3.85086	0.150236
$658$	7.75600	0.302361
$659$	−26.7162	−1.04071	−0.520357	−	0.853949i	$-0.674201\pi$
$659$	−26.7162	−1.04071	−0.520357	+	0.853949i	$0.674201\pi$
$660$	−4.21983	−0.164257
$661$	−7.31037	−0.284340	−0.142170	−	0.989842i	$-0.545408\pi$
$661$	−7.31037	−0.284340	−0.142170	+	0.989842i	$0.545408\pi$
$662$	−1.44265	−0.0560701
$663$	0	0
$664$	−6.49396	−0.252014
$665$	4.21983	0.163638
$666$	16.5918	0.642919
$667$	51.1594	1.98090
$668$	−16.7724	−0.648944
$669$	29.2145	1.12950
$670$	−29.4306	−1.13700
$671$	−1.76079	−0.0679745
$672$	−1.38404	−0.0533906
$673$	−17.2513	−0.664988	−0.332494	−	0.943105i	$-0.607890\pi$
$673$	−17.2513	−0.664988	−0.332494	+	0.943105i	$0.607890\pi$
$674$	−30.5133	−1.17533
$675$	−52.4064	−2.01712
$676$	0	0
$677$	15.9022	0.611170	0.305585	−	0.952165i	$-0.401148\pi$
$677$	15.9022	0.611170	0.305585	+	0.952165i	$0.401148\pi$
$678$	1.87800	0.0721242
$679$	−8.79225	−0.337416
$680$	−10.1317	−0.388532
$681$	−16.5187	−0.632998
$682$	4.31767	0.165332
$683$	45.5881	1.74438	0.872190	−	0.489168i	$-0.162700\pi$
$683$	45.5881	1.74438	0.872190	+	0.489168i	$0.162700\pi$
$684$	−1.44504	−0.0552526
$685$	2.16719	0.0828042
$686$	14.1715	0.541071
$687$	−8.00298	−0.305333
$688$	−4.98792	−0.190163
$689$	0	0
$690$	−29.9517	−1.14024
$691$	−45.5555	−1.73301	−0.866507	−	0.499164i	$-0.833640\pi$
$691$	−45.5555	−1.73301	−0.866507	+	0.499164i	$0.833640\pi$
$692$	−9.35988	−0.355809
$693$	1.42758	0.0542294
$694$	−20.6461	−0.783715
$695$	−23.3491	−0.885683
$696$	10.0978	0.382757
$697$	−32.2392	−1.22115
$698$	−10.6310	−0.402390
$699$	11.4246	0.432118
$700$	−10.4940	−0.396634
$701$	−19.3448	−0.730644	−0.365322	−	0.930881i	$-0.619041\pi$
$701$	−19.3448	−0.730644	−0.365322	+	0.930881i	$0.619041\pi$
$702$	0	0
$703$	−11.4819	−0.433047
$704$	0.890084	0.0335463
$705$	33.1293	1.24772
$706$	−0.0935228	−0.00351978
$707$	−0.707824	−0.0266205
$708$	7.08277	0.266187
$709$	5.53558	0.207893	0.103947	−	0.994583i	$-0.466853\pi$
$709$	5.53558	0.207893	0.103947	+	0.994583i	$0.466853\pi$
$710$	−10.8019	−0.405389
$711$	12.0901	0.453413
$712$	−15.4276	−0.578173
$713$	30.6461	1.14771
$714$	−3.68830	−0.138031
$715$	0	0
$716$	−1.89679	−0.0708863
$717$	13.4577	0.502588
$718$	28.6703	1.06996
$719$	1.53617	0.0572895	0.0286448	−	0.999590i	$-0.490881\pi$
$719$	1.53617	0.0572895	0.0286448	+	0.999590i	$0.490881\pi$
$720$	5.49396	0.204748
$721$	−9.40342	−0.350202
$722$	1.00000	0.0372161
$723$	−30.9772	−1.15205
$724$	6.57242	0.244262
$725$	76.5628	2.84347
$726$	−12.7289	−0.472412
$727$	17.5496	0.650878	0.325439	−	0.945563i	$-0.394488\pi$
$727$	17.5496	0.650878	0.325439	+	0.945563i	$0.394488\pi$
$728$	0	0
$729$	24.4590	0.905890
$730$	10.1317	0.374991
$731$	−13.2922	−0.491629
$732$	−2.46681	−0.0911760
$733$	−14.0140	−0.517619	−0.258809	−	0.965928i	$-0.583330\pi$
$733$	−14.0140	−0.517619	−0.258809	+	0.965928i	$0.583330\pi$
$734$	14.4940	0.534982
$735$	27.3461	1.00868
$736$	6.31767	0.232872
$737$	6.89008	0.253799
$738$	17.4819	0.643517
$739$	−17.6125	−0.647886	−0.323943	−	0.946077i	$-0.605009\pi$
$739$	−17.6125	−0.647886	−0.323943	+	0.946077i	$0.605009\pi$
$740$	43.6534	1.60473
$741$	0	0
$742$	12.8793	0.472815
$743$	−34.5827	−1.26872	−0.634358	−	0.773039i	$-0.718735\pi$
$743$	−34.5827	−1.26872	−0.634358	+	0.773039i	$0.718735\pi$
$744$	6.04892	0.221764
$745$	16.7342	0.613095
$746$	−20.5241	−0.751440
$747$	9.38404	0.343344
$748$	2.37196	0.0867275
$749$	10.7138	0.391473
$750$	−21.1196	−0.771179
$751$	−24.0388	−0.877187	−0.438593	−	0.898686i	$-0.644523\pi$
$751$	−24.0388	−0.877187	−0.438593	+	0.898686i	$0.644523\pi$
$752$	−6.98792	−0.254823
$753$	33.6534	1.22640
$754$	0	0
$755$	41.9614	1.52713
$756$	6.15213	0.223751
$757$	−7.49289	−0.272334	−0.136167	−	0.990686i	$-0.543478\pi$
$757$	−7.49289	−0.272334	−0.136167	+	0.990686i	$0.543478\pi$
$758$	−1.34050	−0.0486892
$759$	7.01208	0.254522
$760$	−3.80194	−0.137911
$761$	−34.0200	−1.23322	−0.616611	−	0.787268i	$-0.711495\pi$
$761$	−34.0200	−1.23322	−0.616611	+	0.787268i	$0.711495\pi$
$762$	−22.8907	−0.829242
$763$	5.31900	0.192561
$764$	−7.32975	−0.265181
$765$	14.6407	0.529336
$766$	0.667858	0.0241307
$767$	0	0
$768$	1.24698	0.0449965
$769$	−35.7342	−1.28861	−0.644305	−	0.764769i	$-0.722853\pi$
$769$	−35.7342	−1.28861	−0.644305	+	0.764769i	$0.722853\pi$
$770$	3.75600	0.135357
$771$	−17.1159	−0.616414
$772$	−14.8224	−0.533469
$773$	17.5797	0.632298	0.316149	−	0.948709i	$-0.397610\pi$
$773$	17.5797	0.632298	0.316149	+	0.948709i	$0.397610\pi$
$774$	7.20775	0.259077
$775$	45.8635	1.64747
$776$	7.92154	0.284367
$777$	15.8914	0.570101
$778$	34.0388	1.22035
$779$	−12.0978	−0.433450
$780$	0	0
$781$	2.52888	0.0904903
$782$	16.8358	0.602047
$783$	−44.8853	−1.60407
$784$	−5.76809	−0.206003
$785$	−83.7367	−2.98869
$786$	−6.37196	−0.227280
$787$	−35.5687	−1.26789	−0.633944	−	0.773379i	$-0.718565\pi$
$787$	−35.5687	−1.26789	−0.633944	+	0.773379i	$0.718565\pi$
$788$	6.70841	0.238977
$789$	18.1655	0.646710
$790$	31.8092	1.13172
$791$	−1.67158	−0.0594345
$792$	−1.28621	−0.0457034
$793$	0	0
$794$	11.6063	0.411891
$795$	55.0133	1.95112
$796$	22.5676	0.799888
$797$	50.8805	1.80228	0.901140	−	0.433528i	$-0.142731\pi$
$797$	50.8805	1.80228	0.901140	+	0.433528i	$0.142731\pi$
$798$	−1.38404	−0.0489946
$799$	−18.6219	−0.658796
$800$	9.45473	0.334275
$801$	22.2935	0.787702
$802$	−11.2948	−0.398834
$803$	−2.37196	−0.0837047
$804$	9.65279	0.340428
$805$	26.6595	0.939624
$806$	0	0
$807$	−15.1535	−0.533427
$808$	0.637727	0.0224352
$809$	−6.20477	−0.218148	−0.109074	−	0.994034i	$-0.534789\pi$
$809$	−6.20477	−0.218148	−0.109074	+	0.994034i	$0.534789\pi$
$810$	9.79656	0.344216
$811$	−39.3032	−1.38012	−0.690061	−	0.723751i	$-0.742416\pi$
$811$	−39.3032	−1.38012	−0.690061	+	0.723751i	$0.742416\pi$
$812$	−8.98792	−0.315414
$813$	22.0060	0.771783
$814$	−10.2198	−0.358205
$815$	39.2271	1.37407
$816$	3.32304	0.116330
$817$	−4.98792	−0.174505
$818$	−6.07846	−0.212528
$819$	0	0
$820$	45.9952	1.60622
$821$	−13.3817	−0.467023	−0.233511	−	0.972354i	$-0.575022\pi$
$821$	−13.3817	−0.467023	−0.233511	+	0.972354i	$0.575022\pi$
$822$	−0.710808	−0.0247923
$823$	−47.0180	−1.63895	−0.819473	−	0.573118i	$-0.805734\pi$
$823$	−47.0180	−1.63895	−0.819473	+	0.573118i	$0.805734\pi$
$824$	8.47219	0.295143
$825$	10.4940	0.365353
$826$	−6.30426	−0.219353
$827$	18.5321	0.644425	0.322213	−	0.946667i	$-0.395573\pi$
$827$	18.5321	0.644425	0.322213	+	0.946667i	$0.395573\pi$
$828$	−9.12929	−0.317265
$829$	−24.3236	−0.844795	−0.422397	−	0.906411i	$-0.638811\pi$
$829$	−24.3236	−0.844795	−0.422397	+	0.906411i	$0.638811\pi$
$830$	24.6896	0.856990
$831$	−36.1444	−1.25383
$832$	0	0
$833$	−15.3712	−0.532581
$834$	7.65817	0.265181
$835$	63.7676	2.20677
$836$	0.890084	0.0307842
$837$	−26.8877	−0.929375
$838$	18.9202	0.653588
$839$	−1.39506	−0.0481628	−0.0240814	−	0.999710i	$-0.507666\pi$
$839$	−1.39506	−0.0481628	−0.0240814	+	0.999710i	$0.507666\pi$
$840$	5.26205	0.181558
$841$	36.5749	1.26120
$842$	−9.23191	−0.318153
$843$	6.46383	0.222626
$844$	−7.30127	−0.251320
$845$	0	0
$846$	10.0978	0.347171
$847$	11.3297	0.389295
$848$	−11.6039	−0.398479
$849$	29.0858	0.998220
$850$	25.1957	0.864204
$851$	−72.5387	−2.48659
$852$	3.54288	0.121377
$853$	55.2006	1.89003	0.945016	−	0.327025i	$-0.106046\pi$
$853$	55.2006	1.89003	0.945016	+	0.327025i	$0.106046\pi$
$854$	2.19567	0.0751343
$855$	5.49396	0.187889
$856$	−9.65279	−0.329926
$857$	33.5797	1.14706	0.573531	−	0.819184i	$-0.305573\pi$
$857$	33.5797	1.14706	0.573531	+	0.819184i	$0.305573\pi$
$858$	0	0
$859$	−36.0581	−1.23029	−0.615144	−	0.788415i	$-0.710902\pi$
$859$	−36.0581	−1.23029	−0.615144	+	0.788415i	$0.710902\pi$
$860$	18.9638	0.646659
$861$	16.7439	0.570631
$862$	25.5646	0.870735
$863$	51.9517	1.76846	0.884228	−	0.467056i	$-0.154685\pi$
$863$	51.9517	1.76846	0.884228	+	0.467056i	$0.154685\pi$
$864$	−5.54288	−0.188572
$865$	35.5857	1.20995
$866$	−30.3129	−1.03007
$867$	−12.3432	−0.419196
$868$	−5.38404	−0.182746
$869$	−7.44696	−0.252621
$870$	−38.3913	−1.30159
$871$	0	0
$872$	−4.79225	−0.162286
$873$	−11.4470	−0.387421
$874$	6.31767	0.213698
$875$	18.7982	0.635496
$876$	−3.32304	−0.112275
$877$	−23.7017	−0.800350	−0.400175	−	0.916439i	$-0.631051\pi$
$877$	−23.7017	−0.800350	−0.400175	+	0.916439i	$0.631051\pi$
$878$	−0.576728	−0.0194636
$879$	6.00000	0.202375
$880$	−3.38404	−0.114076
$881$	−18.9573	−0.638688	−0.319344	−	0.947639i	$-0.603463\pi$
$881$	−18.9573	−0.638688	−0.319344	+	0.947639i	$0.603463\pi$
$882$	8.33513	0.280658
$883$	−26.6160	−0.895698	−0.447849	−	0.894109i	$-0.647810\pi$
$883$	−26.6160	−0.895698	−0.447849	+	0.894109i	$0.647810\pi$
$884$	0	0
$885$	−26.9282	−0.905183
$886$	−39.3250	−1.32115
$887$	6.19998	0.208175	0.104087	−	0.994568i	$-0.466808\pi$
$887$	6.19998	0.208175	0.104087	+	0.994568i	$0.466808\pi$
$888$	−14.3177	−0.480469
$889$	20.3746	0.683343
$890$	58.6547	1.96611
$891$	−2.29350	−0.0768353
$892$	23.4282	0.784433
$893$	−6.98792	−0.233842
$894$	−5.48858	−0.183566
$895$	7.21147	0.241053
$896$	−1.10992	−0.0370797
$897$	0	0
$898$	5.92154	0.197604
$899$	39.2814	1.31011
$900$	−13.6625	−0.455416
$901$	−30.9229	−1.03019
$902$	−10.7681	−0.358538
$903$	6.90349	0.229734
$904$	1.50604	0.0500902
$905$	−24.9879	−0.830627
$906$	−13.7627	−0.457235
$907$	28.1086	0.933330	0.466665	−	0.884434i	$-0.345455\pi$
$907$	28.1086	0.933330	0.466665	+	0.884434i	$0.345455\pi$
$908$	−13.2470	−0.439616
$909$	−0.921543	−0.0305656
$910$	0	0
$911$	−6.49635	−0.215234	−0.107617	−	0.994192i	$-0.534322\pi$
$911$	−6.49635	−0.215234	−0.107617	+	0.994192i	$0.534322\pi$
$912$	1.24698	0.0412916
$913$	−5.78017	−0.191296
$914$	−27.6256	−0.913775
$915$	9.37867	0.310049
$916$	−6.41789	−0.212053
$917$	5.67158	0.187292
$918$	−14.7711	−0.487518
$919$	27.7017	0.913795	0.456898	−	0.889519i	$-0.348961\pi$
$919$	27.7017	0.913795	0.456898	+	0.889519i	$0.348961\pi$
$920$	−24.0194	−0.791895
$921$	−12.3230	−0.406058
$922$	−33.2325	−1.09445
$923$	0	0
$924$	−1.23191	−0.0405270
$925$	−108.558	−3.56937
$926$	−42.4784	−1.39593
$927$	−12.2427	−0.402102
$928$	8.09783	0.265824
$929$	51.2529	1.68155	0.840777	−	0.541381i	$-0.182098\pi$
$929$	51.2529	1.68155	0.840777	+	0.541381i	$0.182098\pi$
$930$	−22.9976	−0.754121
$931$	−5.76809	−0.189041
$932$	9.16182	0.300105
$933$	19.2513	0.630259
$934$	35.7318	1.16918
$935$	−9.01805	−0.294922
$936$	0	0
$937$	−12.8461	−0.419663	−0.209831	−	0.977738i	$-0.567292\pi$
$937$	−12.8461	−0.419663	−0.209831	+	0.977738i	$0.567292\pi$
$938$	−8.59179	−0.280532
$939$	−18.6866	−0.609816
$940$	26.5676	0.866541
$941$	46.0844	1.50231	0.751155	−	0.660126i	$-0.229497\pi$
$941$	46.0844	1.50231	0.751155	+	0.660126i	$0.229497\pi$
$942$	27.4644	0.894839
$943$	−76.4301	−2.48891
$944$	5.67994	0.184866
$945$	−23.3900	−0.760877
$946$	−4.43967	−0.144346
$947$	−3.09054	−0.100429	−0.0502145	−	0.998738i	$-0.515990\pi$
$947$	−3.09054	−0.100429	−0.0502145	+	0.998738i	$0.515990\pi$
$948$	−10.4330	−0.338847
$949$	0	0
$950$	9.45473	0.306752
$951$	−2.82238	−0.0915219
$952$	−2.95779	−0.0958624
$953$	57.7512	1.87075	0.935373	−	0.353663i	$-0.115064\pi$
$953$	57.7512	1.87075	0.935373	+	0.353663i	$0.115064\pi$
$954$	16.7681	0.542887
$955$	27.8672	0.901763
$956$	10.7922	0.349046
$957$	8.98792	0.290538
$958$	8.43967	0.272673
$959$	0.632678	0.0204303
$960$	−4.74094	−0.153013
$961$	−7.46921	−0.240942
$962$	0	0
$963$	13.9487	0.449490
$964$	−24.8418	−0.800099
$965$	56.3538	1.81409
$966$	−8.74392	−0.281331
$967$	22.3177	0.717688	0.358844	−	0.933398i	$-0.383171\pi$
$967$	22.3177	0.717688	0.358844	+	0.933398i	$0.383171\pi$
$968$	−10.2078	−0.328090
$969$	3.32304	0.106752
$970$	−30.1172	−0.967005
$971$	−36.4064	−1.16834	−0.584169	−	0.811632i	$-0.698579\pi$
$971$	−36.4064	−1.16834	−0.584169	+	0.811632i	$0.698579\pi$
$972$	13.4155	0.430302
$973$	−6.81641	−0.218524
$974$	25.8689	0.828893
$975$	0	0
$976$	−1.97823	−0.0633216
$977$	39.2728	1.25645	0.628224	−	0.778032i	$-0.283782\pi$
$977$	39.2728	1.25645	0.628224	+	0.778032i	$0.283782\pi$
$978$	−12.8659	−0.411407
$979$	−13.7318	−0.438872
$980$	21.9299	0.700525
$981$	6.92500	0.221098
$982$	13.9323	0.444597
$983$	27.4174	0.874480	0.437240	−	0.899345i	$-0.355956\pi$
$983$	27.4174	0.874480	0.437240	+	0.899345i	$0.355956\pi$
$984$	−15.0858	−0.480916
$985$	−25.5050	−0.812656
$986$	21.5797	0.687238
$987$	9.67158	0.307850
$988$	0	0
$989$	−31.5120	−1.00202
$990$	4.89008	0.155417
$991$	−59.2838	−1.88321	−0.941606	−	0.336716i	$-0.890684\pi$
$991$	−59.2838	−1.88321	−0.941606	+	0.336716i	$0.890684\pi$
$992$	4.85086	0.154015
$993$	−1.79895	−0.0570881
$994$	−3.15346	−0.100022
$995$	−85.8007	−2.72007
$996$	−8.09783	−0.256590
$997$	44.0103	1.39382	0.696910	−	0.717159i	$-0.254558\pi$
$997$	44.0103	1.39382	0.696910	+	0.717159i	$0.254558\pi$
$998$	32.2500	1.02085
$999$	63.6426	2.01356

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.v.1.3	yes	3
13.12	even	2		6422.2.a.n.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6422.2.a.n.1.3	✓	3	13.12	even	2
6422.2.a.v.1.3	yes	3	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:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.24698 q^{3} +1.00000 q^{4} -3.80194 q^{5} +1.24698 q^{6} -1.10992 q^{7} +1.00000 q^{8} -1.44504 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.24698 q^{3} +1.00000 q^{4} -3.80194 q^{5} +1.24698 q^{6} -1.10992 q^{7} +1.00000 q^{8} -1.44504 q^{9} -3.80194 q^{10} +0.890084 q^{11} +1.24698 q^{12} -1.10992 q^{14} -4.74094 q^{15} +1.00000 q^{16} +2.66487 q^{17} -1.44504 q^{18} +1.00000 q^{19} -3.80194 q^{20} -1.38404 q^{21} +0.890084 q^{22} +6.31767 q^{23} +1.24698 q^{24} +9.45473 q^{25} -5.54288 q^{27} -1.10992 q^{28} +8.09783 q^{29} -4.74094 q^{30} +4.85086 q^{31} +1.00000 q^{32} +1.10992 q^{33} +2.66487 q^{34} +4.21983 q^{35} -1.44504 q^{36} -11.4819 q^{37} +1.00000 q^{38} -3.80194 q^{40} -12.0978 q^{41} -1.38404 q^{42} -4.98792 q^{43} +0.890084 q^{44} +5.49396 q^{45} +6.31767 q^{46} -6.98792 q^{47} +1.24698 q^{48} -5.76809 q^{49} +9.45473 q^{50} +3.32304 q^{51} -11.6039 q^{53} -5.54288 q^{54} -3.38404 q^{55} -1.10992 q^{56} +1.24698 q^{57} +8.09783 q^{58} +5.67994 q^{59} -4.74094 q^{60} -1.97823 q^{61} +4.85086 q^{62} +1.60388 q^{63} +1.00000 q^{64} +1.10992 q^{66} +7.74094 q^{67} +2.66487 q^{68} +7.87800 q^{69} +4.21983 q^{70} +2.84117 q^{71} -1.44504 q^{72} -2.66487 q^{73} -11.4819 q^{74} +11.7899 q^{75} +1.00000 q^{76} -0.987918 q^{77} -8.36658 q^{79} -3.80194 q^{80} -2.57673 q^{81} -12.0978 q^{82} -6.49396 q^{83} -1.38404 q^{84} -10.1317 q^{85} -4.98792 q^{86} +10.0978 q^{87} +0.890084 q^{88} -15.4276 q^{89} +5.49396 q^{90} +6.31767 q^{92} +6.04892 q^{93} -6.98792 q^{94} -3.80194 q^{95} +1.24698 q^{96} +7.92154 q^{97} -5.76809 q^{98} -1.28621 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - q^{3} + 3 q^{4} - 7 q^{5} - q^{6} - 4 q^{7} + 3 q^{8} - 4 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 - q^3 + 3 * q^4 - 7 * q^5 - q^6 - 4 * q^7 + 3 * q^8 - 4 * q^9	\( 3 q + 3 q^{2} - q^{3} + 3 q^{4} - 7 q^{5} - q^{6} - 4 q^{7} + 3 q^{8} - 4 q^{9} - 7 q^{10} + 2 q^{11} - q^{12} - 4 q^{14} + 3 q^{16} + 9 q^{17} - 4 q^{18} + 3 q^{19} - 7 q^{20} + 6 q^{21} + 2 q^{22} + 2 q^{23} - q^{24} + 6 q^{25} + 2 q^{27} - 4 q^{28} + 6 q^{29} + q^{31} + 3 q^{32} + 4 q^{33} + 9 q^{34} + 14 q^{35} - 4 q^{36} - 6 q^{37} + 3 q^{38} - 7 q^{40} - 18 q^{41} + 6 q^{42} + 4 q^{43} + 2 q^{44} + 7 q^{45} + 2 q^{46} - 2 q^{47} - q^{48} + 3 q^{49} + 6 q^{50} - 10 q^{51} - 26 q^{53} + 2 q^{54} - 4 q^{56} - q^{57} + 6 q^{58} - 7 q^{59} - 9 q^{61} + q^{62} - 4 q^{63} + 3 q^{64} + 4 q^{66} + 9 q^{67} + 9 q^{68} + 4 q^{69} + 14 q^{70} + 17 q^{71} - 4 q^{72} - 9 q^{73} - 6 q^{74} + 12 q^{75} + 3 q^{76} + 16 q^{77} + q^{79} - 7 q^{80} - 5 q^{81} - 18 q^{82} - 10 q^{83} + 6 q^{84} - 28 q^{85} + 4 q^{86} + 12 q^{87} + 2 q^{88} - 30 q^{89} + 7 q^{90} + 2 q^{92} + 9 q^{93} - 2 q^{94} - 7 q^{95} - q^{96} - 2 q^{97} + 3 q^{98} - 12 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - q^3 + 3 * q^4 - 7 * q^5 - q^6 - 4 * q^7 + 3 * q^8 - 4 * q^9 - 7 * q^10 + 2 * q^11 - q^12 - 4 * q^14 + 3 * q^16 + 9 * q^17 - 4 * q^18 + 3 * q^19 - 7 * q^20 + 6 * q^21 + 2 * q^22 + 2 * q^23 - q^24 + 6 * q^25 + 2 * q^27 - 4 * q^28 + 6 * q^29 + q^31 + 3 * q^32 + 4 * q^33 + 9 * q^34 + 14 * q^35 - 4 * q^36 - 6 * q^37 + 3 * q^38 - 7 * q^40 - 18 * q^41 + 6 * q^42 + 4 * q^43 + 2 * q^44 + 7 * q^45 + 2 * q^46 - 2 * q^47 - q^48 + 3 * q^49 + 6 * q^50 - 10 * q^51 - 26 * q^53 + 2 * q^54 - 4 * q^56 - q^57 + 6 * q^58 - 7 * q^59 - 9 * q^61 + q^62 - 4 * q^63 + 3 * q^64 + 4 * q^66 + 9 * q^67 + 9 * q^68 + 4 * q^69 + 14 * q^70 + 17 * q^71 - 4 * q^72 - 9 * q^73 - 6 * q^74 + 12 * q^75 + 3 * q^76 + 16 * q^77 + q^79 - 7 * q^80 - 5 * q^81 - 18 * q^82 - 10 * q^83 + 6 * q^84 - 28 * q^85 + 4 * q^86 + 12 * q^87 + 2 * q^88 - 30 * q^89 + 7 * q^90 + 2 * q^92 + 9 * q^93 - 2 * q^94 - 7 * q^95 - q^96 - 2 * q^97 + 3 * q^98 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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