LMFDB - Embedded newform 6811.2.a.p.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6811,2,Mod(1,6811)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6811, 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("6811.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6811 = 7^{2} \cdot 139 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6811.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:	$54.3861088167$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 11x^{5} + 8x^{4} + 35x^{3} - 10x^{2} - 32x - 8 $ x^7 - x^6 - 11x^5 + 8x^4 + 35x^3 - 10x^2 - 32*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 139)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.44228$ of defining polynomial
Character	$\chi$	$=$	6811.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.44228 q^{2} -1.01681 q^{3} +0.0801788 q^{4} -2.10270 q^{5} -1.46653 q^{6} -2.76892 q^{8} -1.96609 q^{9} +O(q^{10})$	\(q+1.44228 q^{2} -1.01681 q^{3} +0.0801788 q^{4} -2.10270 q^{5} -1.46653 q^{6} -2.76892 q^{8} -1.96609 q^{9} -3.03269 q^{10} -1.65710 q^{11} -0.0815268 q^{12} -2.72182 q^{13} +2.13805 q^{15} -4.15393 q^{16} -4.48334 q^{17} -2.83566 q^{18} -2.23411 q^{19} -0.168592 q^{20} -2.39001 q^{22} -3.67193 q^{23} +2.81548 q^{24} -0.578649 q^{25} -3.92563 q^{26} +5.04959 q^{27} -6.87982 q^{29} +3.08368 q^{30} -1.29710 q^{31} -0.453290 q^{32} +1.68496 q^{33} -6.46625 q^{34} -0.157639 q^{36} +9.38804 q^{37} -3.22221 q^{38} +2.76758 q^{39} +5.82222 q^{40} -10.2034 q^{41} -2.68294 q^{43} -0.132864 q^{44} +4.13410 q^{45} -5.29596 q^{46} -5.68799 q^{47} +4.22377 q^{48} -0.834575 q^{50} +4.55872 q^{51} -0.218232 q^{52} -0.660925 q^{53} +7.28293 q^{54} +3.48439 q^{55} +2.27167 q^{57} -9.92265 q^{58} -5.46888 q^{59} +0.171427 q^{60} +2.95008 q^{61} -1.87078 q^{62} +7.65409 q^{64} +5.72316 q^{65} +2.43019 q^{66} -9.39699 q^{67} -0.359469 q^{68} +3.73367 q^{69} +12.6029 q^{71} +5.44396 q^{72} -9.25327 q^{73} +13.5402 q^{74} +0.588378 q^{75} -0.179128 q^{76} +3.99163 q^{78} -10.6724 q^{79} +8.73447 q^{80} +0.763792 q^{81} -14.7162 q^{82} +2.94980 q^{83} +9.42713 q^{85} -3.86956 q^{86} +6.99549 q^{87} +4.58839 q^{88} +10.6226 q^{89} +5.96254 q^{90} -0.294411 q^{92} +1.31890 q^{93} -8.20369 q^{94} +4.69766 q^{95} +0.460911 q^{96} +4.39900 q^{97} +3.25801 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + q^{2} + 2 q^{3} + 9 q^{4} - 11 q^{5} + 7 q^{6} + 6 q^{8} + 13 q^{9}+O(q^{10}) $ 7 * q + q^2 + 2 * q^3 + 9 * q^4 - 11 * q^5 + 7 * q^6 + 6 * q^8 + 13 * q^9	$ 7 q + q^{2} + 2 q^{3} + 9 q^{4} - 11 q^{5} + 7 q^{6} + 6 q^{8} + 13 q^{9} + 4 q^{10} + 2 q^{11} + 8 q^{12} - 6 q^{13} - 3 q^{15} + 5 q^{16} - 5 q^{17} - 10 q^{18} + 10 q^{19} - 12 q^{20} - 18 q^{22} - q^{23} + 21 q^{24} + 14 q^{25} + 8 q^{26} + 11 q^{27} + 30 q^{29} - 41 q^{30} + 20 q^{31} - 12 q^{32} + 20 q^{33} + 17 q^{34} + 2 q^{36} + 6 q^{37} - 6 q^{38} + 11 q^{39} + 22 q^{40} - 19 q^{41} - 12 q^{43} + 25 q^{44} - 27 q^{45} + 22 q^{46} + 3 q^{47} - 15 q^{48} + 12 q^{50} + 23 q^{51} + 8 q^{52} + 38 q^{53} + 7 q^{54} - 7 q^{55} - 19 q^{57} - 21 q^{58} + 14 q^{59} - 8 q^{60} - 4 q^{61} + q^{62} - 16 q^{64} + 10 q^{65} - 18 q^{66} + 9 q^{67} + 25 q^{68} - 9 q^{69} + 24 q^{71} + 41 q^{72} + 5 q^{73} + 9 q^{74} + 21 q^{75} - 3 q^{76} + 20 q^{78} + 8 q^{79} - 11 q^{80} + 39 q^{81} - 56 q^{82} + 9 q^{83} - 22 q^{85} + 39 q^{86} + 25 q^{87} - 29 q^{88} - 10 q^{89} - 72 q^{90} + 29 q^{92} - 15 q^{93} + 36 q^{94} - 21 q^{95} + 11 q^{96} + 5 q^{97} - 31 q^{99}+O(q^{100}) $ 7 * q + q^2 + 2 * q^3 + 9 * q^4 - 11 * q^5 + 7 * q^6 + 6 * q^8 + 13 * q^9 + 4 * q^10 + 2 * q^11 + 8 * q^12 - 6 * q^13 - 3 * q^15 + 5 * q^16 - 5 * q^17 - 10 * q^18 + 10 * q^19 - 12 * q^20 - 18 * q^22 - q^23 + 21 * q^24 + 14 * q^25 + 8 * q^26 + 11 * q^27 + 30 * q^29 - 41 * q^30 + 20 * q^31 - 12 * q^32 + 20 * q^33 + 17 * q^34 + 2 * q^36 + 6 * q^37 - 6 * q^38 + 11 * q^39 + 22 * q^40 - 19 * q^41 - 12 * q^43 + 25 * q^44 - 27 * q^45 + 22 * q^46 + 3 * q^47 - 15 * q^48 + 12 * q^50 + 23 * q^51 + 8 * q^52 + 38 * q^53 + 7 * q^54 - 7 * q^55 - 19 * q^57 - 21 * q^58 + 14 * q^59 - 8 * q^60 - 4 * q^61 + q^62 - 16 * q^64 + 10 * q^65 - 18 * q^66 + 9 * q^67 + 25 * q^68 - 9 * q^69 + 24 * q^71 + 41 * q^72 + 5 * q^73 + 9 * q^74 + 21 * q^75 - 3 * q^76 + 20 * q^78 + 8 * q^79 - 11 * q^80 + 39 * q^81 - 56 * q^82 + 9 * q^83 - 22 * q^85 + 39 * q^86 + 25 * q^87 - 29 * q^88 - 10 * q^89 - 72 * q^90 + 29 * q^92 - 15 * q^93 + 36 * q^94 - 21 * q^95 + 11 * q^96 + 5 * q^97 - 31 * 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 $

$2$	1.44228	1.01985	0.509924	−	0.860220i	$-0.329674\pi$
$2$	1.44228	1.01985	0.509924	+	0.860220i	$0.329674\pi$
$3$	−1.01681	−0.587057	−0.293529	−	0.955950i	$-0.594830\pi$
$3$	−1.01681	−0.587057	−0.293529	+	0.955950i	$0.594830\pi$
$4$	0.0801788	0.0400894
$5$	−2.10270	−0.940356	−0.470178	−	0.882572i	$-0.655810\pi$
$5$	−2.10270	−0.940356	−0.470178	+	0.882572i	$0.655810\pi$
$6$	−1.46653	−0.598709
$7$	0	0
$7$	0	0
$8$	−2.76892	−0.978963
$9$	−1.96609	−0.655364
$10$	−3.03269	−0.959020
$11$	−1.65710	−0.499635	−0.249817	−	0.968293i	$-0.580371\pi$
$11$	−1.65710	−0.499635	−0.249817	+	0.968293i	$0.580371\pi$
$12$	−0.0815268	−0.0235348
$13$	−2.72182	−0.754896	−0.377448	−	0.926031i	$-0.623198\pi$
$13$	−2.72182	−0.754896	−0.377448	+	0.926031i	$0.623198\pi$
$14$	0	0
$15$	2.13805	0.552043
$16$	−4.15393	−1.03848
$17$	−4.48334	−1.08737	−0.543685	−	0.839289i	$-0.682971\pi$
$17$	−4.48334	−1.08737	−0.543685	+	0.839289i	$0.682971\pi$
$18$	−2.83566	−0.668371
$19$	−2.23411	−0.512539	−0.256270	−	0.966605i	$-0.582494\pi$
$19$	−2.23411	−0.512539	−0.256270	+	0.966605i	$0.582494\pi$
$20$	−0.168592	−0.0376983
$21$	0	0
$22$	−2.39001	−0.509551
$23$	−3.67193	−0.765651	−0.382825	−	0.923821i	$-0.625049\pi$
$23$	−3.67193	−0.765651	−0.382825	+	0.923821i	$0.625049\pi$
$24$	2.81548	0.574707
$25$	−0.578649	−0.115730
$26$	−3.92563	−0.769879
$27$	5.04959	0.971793
$28$	0	0
$29$	−6.87982	−1.27755	−0.638776	−	0.769393i	$-0.720559\pi$
$29$	−6.87982	−1.27755	−0.638776	+	0.769393i	$0.720559\pi$
$30$	3.08368	0.563000
$31$	−1.29710	−0.232965	−0.116483	−	0.993193i	$-0.537162\pi$
$31$	−1.29710	−0.232965	−0.116483	+	0.993193i	$0.537162\pi$
$32$	−0.453290	−0.0801311
$33$	1.68496	0.293314
$34$	−6.46625	−1.10895
$35$	0	0
$36$	−0.157639	−0.0262731
$37$	9.38804	1.54338	0.771692	−	0.635997i	$-0.219411\pi$
$37$	9.38804	1.54338	0.771692	+	0.635997i	$0.219411\pi$
$38$	−3.22221	−0.522712
$39$	2.76758	0.443167
$40$	5.82222	0.920574
$41$	−10.2034	−1.59350	−0.796750	−	0.604309i	$-0.793449\pi$
$41$	−10.2034	−1.59350	−0.796750	+	0.604309i	$0.793449\pi$
$42$	0	0
$43$	−2.68294	−0.409145	−0.204572	−	0.978851i	$-0.565580\pi$
$43$	−2.68294	−0.409145	−0.204572	+	0.978851i	$0.565580\pi$
$44$	−0.132864	−0.0200301
$45$	4.13410	0.616276
$46$	−5.29596	−0.780847
$47$	−5.68799	−0.829679	−0.414840	−	0.909895i	$-0.636162\pi$
$47$	−5.68799	−0.829679	−0.414840	+	0.909895i	$0.636162\pi$
$48$	4.22377	0.609648
$49$	0	0
$50$	−0.834575	−0.118027
$51$	4.55872	0.638349
$52$	−0.218232	−0.0302633
$53$	−0.660925	−0.0907851	−0.0453925	−	0.998969i	$-0.514454\pi$
$53$	−0.660925	−0.0907851	−0.0453925	+	0.998969i	$0.514454\pi$
$54$	7.28293	0.991081
$55$	3.48439	0.469835
$56$	0	0
$57$	2.27167	0.300890
$58$	−9.92265	−1.30291
$59$	−5.46888	−0.711988	−0.355994	−	0.934488i	$-0.615858\pi$
$59$	−5.46888	−0.711988	−0.355994	+	0.934488i	$0.615858\pi$
$60$	0.171427	0.0221311
$61$	2.95008	0.377719	0.188859	−	0.982004i	$-0.439521\pi$
$61$	2.95008	0.377719	0.188859	+	0.982004i	$0.439521\pi$
$62$	−1.87078	−0.237589
$63$	0	0
$64$	7.65409	0.956761
$65$	5.72316	0.709871
$66$	2.43019	0.299136
$67$	−9.39699	−1.14802	−0.574012	−	0.818847i	$-0.694614\pi$
$67$	−9.39699	−1.14802	−0.574012	+	0.818847i	$0.694614\pi$
$68$	−0.359469	−0.0435920
$69$	3.73367	0.449481
$70$	0	0
$71$	12.6029	1.49568	0.747842	−	0.663877i	$-0.231090\pi$
$71$	12.6029	1.49568	0.747842	+	0.663877i	$0.231090\pi$
$72$	5.44396	0.641577
$73$	−9.25327	−1.08301	−0.541507	−	0.840696i	$-0.682146\pi$
$73$	−9.25327	−1.08301	−0.541507	+	0.840696i	$0.682146\pi$
$74$	13.5402	1.57402
$75$	0.588378	0.0679400
$76$	−0.179128	−0.0205474
$77$	0	0
$78$	3.99163	0.451963
$79$	−10.6724	−1.20073	−0.600367	−	0.799725i	$-0.704979\pi$
$79$	−10.6724	−1.20073	−0.600367	+	0.799725i	$0.704979\pi$
$80$	8.73447	0.976543
$81$	0.763792	0.0848658
$82$	−14.7162	−1.62513
$83$	2.94980	0.323783	0.161891	−	0.986809i	$-0.448241\pi$
$83$	2.94980	0.323783	0.161891	+	0.986809i	$0.448241\pi$
$84$	0	0
$85$	9.42713	1.02252
$86$	−3.86956	−0.417265
$87$	6.99549	0.749995
$88$	4.58839	0.489124
$89$	10.6226	1.12599	0.562994	−	0.826461i	$-0.309649\pi$
$89$	10.6226	1.12599	0.562994	+	0.826461i	$0.309649\pi$
$90$	5.96254	0.628507
$91$	0	0
$92$	−0.294411	−0.0306945
$93$	1.31890	0.136764
$94$	−8.20369	−0.846146
$95$	4.69766	0.481970
$96$	0.460911	0.0470415
$97$	4.39900	0.446651	0.223325	−	0.974744i	$-0.428309\pi$
$97$	4.39900	0.446651	0.223325	+	0.974744i	$0.428309\pi$
$98$	0	0
$99$	3.25801	0.327443
$100$	−0.0463954	−0.00463954
$101$	−18.0072	−1.79178	−0.895892	−	0.444272i	$-0.853462\pi$
$101$	−18.0072	−1.79178	−0.895892	+	0.444272i	$0.853462\pi$
$102$	6.57496	0.651018
$103$	4.12106	0.406060	0.203030	−	0.979173i	$-0.434921\pi$
$103$	4.12106	0.406060	0.203030	+	0.979173i	$0.434921\pi$
$104$	7.53650	0.739015
$105$	0	0
$106$	−0.953241	−0.0925870
$107$	−4.15932	−0.402097	−0.201048	−	0.979581i	$-0.564435\pi$
$107$	−4.15932	−0.402097	−0.201048	+	0.979581i	$0.564435\pi$
$108$	0.404870	0.0389586
$109$	−15.3002	−1.46550	−0.732748	−	0.680500i	$-0.761762\pi$
$109$	−15.3002	−1.46550	−0.732748	+	0.680500i	$0.761762\pi$
$110$	5.02547	0.479160
$111$	−9.54588	−0.906054
$112$	0	0
$113$	13.3298	1.25396	0.626980	−	0.779036i	$-0.284291\pi$
$113$	13.3298	1.25396	0.626980	+	0.779036i	$0.284291\pi$
$114$	3.27639	0.306862
$115$	7.72098	0.719985
$116$	−0.551616	−0.0512162
$117$	5.35134	0.494731
$118$	−7.88767	−0.726119
$119$	0	0
$120$	−5.92011	−0.540429
$121$	−8.25402	−0.750365
$122$	4.25485	0.385216
$123$	10.3749	0.935476
$124$	−0.104000	−0.00933943
$125$	11.7302	1.04918
$126$	0	0
$127$	0.581539	0.0516032	0.0258016	−	0.999667i	$-0.491786\pi$
$127$	0.581539	0.0516032	0.0258016	+	0.999667i	$0.491786\pi$
$128$	11.9459	1.05588
$129$	2.72805	0.240191
$130$	8.25442	0.723960
$131$	20.8834	1.82459	0.912296	−	0.409532i	$-0.134308\pi$
$131$	20.8834	1.82459	0.912296	+	0.409532i	$0.134308\pi$
$132$	0.135098	0.0117588
$133$	0	0
$134$	−13.5531	−1.17081
$135$	−10.6178	−0.913832
$136$	12.4140	1.06450
$137$	8.21826	0.702134	0.351067	−	0.936350i	$-0.385819\pi$
$137$	8.21826	0.702134	0.351067	+	0.936350i	$0.385819\pi$
$138$	5.38500	0.458402
$139$	−1.00000	−0.0848189
$139$	−1.00000	−0.0848189
$140$	0	0
$141$	5.78362	0.487069
$142$	18.1769	1.52537
$143$	4.51032	0.377172
$144$	8.16701	0.680584
$145$	14.4662	1.20135
$146$	−13.3458	−1.10451
$147$	0	0
$148$	0.752721	0.0618733
$149$	6.64525	0.544400	0.272200	−	0.962241i	$-0.412249\pi$
$149$	6.64525	0.544400	0.272200	+	0.962241i	$0.412249\pi$
$150$	0.848607	0.0692885
$151$	−7.46212	−0.607259	−0.303629	−	0.952790i	$-0.598198\pi$
$151$	−7.46212	−0.607259	−0.303629	+	0.952790i	$0.598198\pi$
$152$	6.18608	0.501757
$153$	8.81467	0.712624
$154$	0	0
$155$	2.72740	0.219070
$156$	0.221901	0.0177663
$157$	0.686050	0.0547527	0.0273764	−	0.999625i	$-0.491285\pi$
$157$	0.686050	0.0547527	0.0273764	+	0.999625i	$0.491285\pi$
$158$	−15.3926	−1.22457
$159$	0.672037	0.0532960
$160$	0.953133	0.0753518
$161$	0	0
$162$	1.10160	0.0865502
$163$	−15.6818	−1.22829	−0.614147	−	0.789191i	$-0.710500\pi$
$163$	−15.6818	−1.22829	−0.614147	+	0.789191i	$0.710500\pi$
$164$	−0.818095	−0.0638825
$165$	−3.54297	−0.275820
$166$	4.25445	0.330209
$167$	9.59823	0.742733	0.371367	−	0.928486i	$-0.378889\pi$
$167$	9.59823	0.742733	0.371367	+	0.928486i	$0.378889\pi$
$168$	0	0
$169$	−5.59172	−0.430132
$170$	13.5966	1.04281
$171$	4.39246	0.335900
$172$	−0.215115	−0.0164024
$173$	−7.08082	−0.538345	−0.269172	−	0.963092i	$-0.586750\pi$
$173$	−7.08082	−0.538345	−0.269172	+	0.963092i	$0.586750\pi$
$174$	10.0895	0.764881
$175$	0	0
$176$	6.88348	0.518862
$177$	5.56083	0.417978
$178$	15.3207	1.14834
$179$	−3.92939	−0.293696	−0.146848	−	0.989159i	$-0.546913\pi$
$179$	−3.92939	−0.293696	−0.146848	+	0.989159i	$0.546913\pi$
$180$	0.331467	0.0247061
$181$	−21.6810	−1.61154	−0.805769	−	0.592230i	$-0.798248\pi$
$181$	−21.6810	−1.61154	−0.805769	+	0.592230i	$0.798248\pi$
$182$	0	0
$183$	−2.99968	−0.221742
$184$	10.1673	0.749544
$185$	−19.7402	−1.45133
$186$	1.90223	0.139478
$187$	7.42935	0.543288
$188$	−0.456056	−0.0332613
$189$	0	0
$190$	6.77535	0.491536
$191$	17.1191	1.23870	0.619349	−	0.785116i	$-0.287397\pi$
$191$	17.1191	1.23870	0.619349	+	0.785116i	$0.287397\pi$
$192$	−7.78277	−0.561673
$193$	−25.1710	−1.81185	−0.905923	−	0.423442i	$-0.860822\pi$
$193$	−25.1710	−1.81185	−0.905923	+	0.423442i	$0.860822\pi$
$194$	6.34460	0.455516
$195$	−5.81939	−0.416735
$196$	0	0
$197$	8.92502	0.635881	0.317941	−	0.948111i	$-0.397009\pi$
$197$	8.92502	0.635881	0.317941	+	0.948111i	$0.397009\pi$
$198$	4.69898	0.333942
$199$	−10.9052	−0.773049	−0.386525	−	0.922279i	$-0.626325\pi$
$199$	−10.9052	−0.773049	−0.386525	+	0.922279i	$0.626325\pi$
$200$	1.60224	0.113295
$201$	9.55498	0.673956
$202$	−25.9715	−1.82735
$203$	0	0
$204$	0.365513	0.0255910
$205$	21.4547	1.49846
$206$	5.94373	0.414119
$207$	7.21936	0.501780
$208$	11.3062	0.783946
$209$	3.70214	0.256083
$210$	0	0
$211$	3.88066	0.267155	0.133578	−	0.991038i	$-0.457353\pi$
$211$	3.88066	0.267155	0.133578	+	0.991038i	$0.457353\pi$
$212$	−0.0529922	−0.00363952
$213$	−12.8147	−0.878052
$214$	−5.99892	−0.410078
$215$	5.64142	0.384742
$216$	−13.9819	−0.951349
$217$	0	0
$218$	−22.0672	−1.49458
$219$	9.40885	0.635791
$220$	0.279374	0.0188354
$221$	12.2028	0.820851
$222$	−13.7678	−0.924038
$223$	10.3790	0.695032	0.347516	−	0.937674i	$-0.387025\pi$
$223$	10.3790	0.695032	0.347516	+	0.937674i	$0.387025\pi$
$224$	0	0
$225$	1.13768	0.0758451
$226$	19.2253	1.27885
$227$	28.4485	1.88819	0.944097	−	0.329667i	$-0.106936\pi$
$227$	28.4485	1.88819	0.944097	+	0.329667i	$0.106936\pi$
$228$	0.182140	0.0120625
$229$	−23.3247	−1.54134	−0.770671	−	0.637233i	$-0.780079\pi$
$229$	−23.3247	−1.54134	−0.770671	+	0.637233i	$0.780079\pi$
$230$	11.1358	0.734275
$231$	0	0
$232$	19.0497	1.25067
$233$	−13.8555	−0.907705	−0.453852	−	0.891077i	$-0.649951\pi$
$233$	−13.8555	−0.907705	−0.453852	+	0.891077i	$0.649951\pi$
$234$	7.71814	0.504551
$235$	11.9601	0.780194
$236$	−0.438488	−0.0285432
$237$	10.8518	0.704899
$238$	0	0
$239$	21.0113	1.35911	0.679555	−	0.733625i	$-0.262173\pi$
$239$	21.0113	1.35911	0.679555	+	0.733625i	$0.262173\pi$
$240$	−8.88132	−0.573287
$241$	−3.16874	−0.204116	−0.102058	−	0.994778i	$-0.532543\pi$
$241$	−3.16874	−0.204116	−0.102058	+	0.994778i	$0.532543\pi$
$242$	−11.9046	−0.765258
$243$	−15.9254	−1.02161
$244$	0.236534	0.0151425
$245$	0	0
$246$	14.9636	0.954043
$247$	6.08083	0.386914
$248$	3.59156	0.228064
$249$	−2.99940	−0.190079
$250$	16.9183	1.07001
$251$	−5.34519	−0.337385	−0.168693	−	0.985669i	$-0.553955\pi$
$251$	−5.34519	−0.337385	−0.168693	+	0.985669i	$0.553955\pi$
$252$	0	0
$253$	6.08476	0.382546
$254$	0.838743	0.0526274
$255$	−9.58563	−0.600275
$256$	1.92124	0.120077
$257$	9.77662	0.609849	0.304925	−	0.952377i	$-0.401369\pi$
$257$	9.77662	0.609849	0.304925	+	0.952377i	$0.401369\pi$
$258$	3.93462	0.244958
$259$	0	0
$260$	0.458876	0.0284583
$261$	13.5264	0.837261
$262$	30.1198	1.86081
$263$	−21.2549	−1.31063	−0.655316	−	0.755355i	$-0.727464\pi$
$263$	−21.2549	−1.31063	−0.655316	+	0.755355i	$0.727464\pi$
$264$	−4.66553	−0.287144
$265$	1.38973	0.0853703
$266$	0	0
$267$	−10.8012	−0.661020
$268$	−0.753439	−0.0460236
$269$	−10.6372	−0.648563	−0.324281	−	0.945961i	$-0.605122\pi$
$269$	−10.6372	−0.648563	−0.324281	+	0.945961i	$0.605122\pi$
$270$	−15.3138	−0.931969
$271$	15.6524	0.950816	0.475408	−	0.879766i	$-0.342301\pi$
$271$	15.6524	0.950816	0.475408	+	0.879766i	$0.342301\pi$
$272$	18.6235	1.12922
$273$	0	0
$274$	11.8531	0.716069
$275$	0.958880	0.0578226
$276$	0.299361	0.0180194
$277$	−20.6725	−1.24209	−0.621044	−	0.783776i	$-0.713291\pi$
$277$	−20.6725	−1.24209	−0.621044	+	0.783776i	$0.713291\pi$
$278$	−1.44228	−0.0865024
$279$	2.55021	0.152677
$280$	0	0
$281$	3.32086	0.198106	0.0990530	−	0.995082i	$-0.468419\pi$
$281$	3.32086	0.198106	0.0990530	+	0.995082i	$0.468419\pi$
$282$	8.34162	0.496736
$283$	14.6513	0.870930	0.435465	−	0.900206i	$-0.356584\pi$
$283$	14.6513	0.870930	0.435465	+	0.900206i	$0.356584\pi$
$284$	1.01048	0.0599611
$285$	−4.77664	−0.282944
$286$	6.50516	0.384658
$287$	0	0
$288$	0.891210	0.0525150
$289$	3.10037	0.182375
$290$	20.8644	1.22520
$291$	−4.47296	−0.262209
$292$	−0.741916	−0.0434174
$293$	0.927136	0.0541639	0.0270819	−	0.999633i	$-0.491379\pi$
$293$	0.927136	0.0541639	0.0270819	+	0.999633i	$0.491379\pi$
$294$	0	0
$295$	11.4994	0.669522
$296$	−25.9948	−1.51092
$297$	−8.36767	−0.485542
$298$	9.58433	0.555205
$299$	9.99432	0.577987
$300$	0.0471754	0.00272367
$301$	0	0
$302$	−10.7625	−0.619311
$303$	18.3100	1.05188
$304$	9.28032	0.532263
$305$	−6.20313	−0.355190
$306$	12.7132	0.726767
$307$	−32.2115	−1.83841	−0.919204	−	0.393782i	$-0.871167\pi$
$307$	−32.2115	−1.83841	−0.919204	+	0.393782i	$0.871167\pi$
$308$	0	0
$309$	−4.19034	−0.238380
$310$	3.93369	0.223418
$311$	−32.0918	−1.81976	−0.909881	−	0.414870i	$-0.863827\pi$
$311$	−32.0918	−1.81976	−0.909881	+	0.414870i	$0.863827\pi$
$312$	−7.66321	−0.433844
$313$	−6.61799	−0.374071	−0.187035	−	0.982353i	$-0.559888\pi$
$313$	−6.61799	−0.374071	−0.187035	+	0.982353i	$0.559888\pi$
$314$	0.989477	0.0558394
$315$	0	0
$316$	−0.855697	−0.0481367
$317$	5.92720	0.332905	0.166452	−	0.986049i	$-0.446769\pi$
$317$	5.92720	0.332905	0.166452	+	0.986049i	$0.446769\pi$
$318$	0.969268	0.0543538
$319$	11.4006	0.638309
$320$	−16.0943	−0.899696
$321$	4.22925	0.236054
$322$	0	0
$323$	10.0163	0.557320
$324$	0.0612400	0.00340222
$325$	1.57498	0.0873639
$326$	−22.6176	−1.25267
$327$	15.5575	0.860330
$328$	28.2524	1.55998
$329$	0	0
$330$	−5.10996	−0.281294
$331$	32.6437	1.79426	0.897131	−	0.441764i	$-0.145647\pi$
$331$	32.6437	1.79426	0.897131	+	0.441764i	$0.145647\pi$
$332$	0.236512	0.0129803
$333$	−18.4577	−1.01148
$334$	13.8434	0.757475
$335$	19.7591	1.07955
$336$	0	0
$337$	5.63098	0.306739	0.153369	−	0.988169i	$-0.450988\pi$
$337$	5.63098	0.306739	0.153369	+	0.988169i	$0.450988\pi$
$338$	−8.06484	−0.438670
$339$	−13.5539	−0.736146
$340$	0.755856	0.0409920
$341$	2.14942	0.116398
$342$	6.33517	0.342567
$343$	0	0
$344$	7.42886	0.400537
$345$	−7.85079	−0.422672
$346$	−10.2125	−0.549030
$347$	19.6551	1.05514	0.527571	−	0.849511i	$-0.323103\pi$
$347$	19.6551	1.05514	0.527571	+	0.849511i	$0.323103\pi$
$348$	0.560890	0.0300669
$349$	−2.21624	−0.118632	−0.0593162	−	0.998239i	$-0.518892\pi$
$349$	−2.21624	−0.118632	−0.0593162	+	0.998239i	$0.518892\pi$
$350$	0	0
$351$	−13.7440	−0.733603
$352$	0.751147	0.0400363
$353$	28.6867	1.52684	0.763419	−	0.645903i	$-0.223519\pi$
$353$	28.6867	1.52684	0.763419	+	0.645903i	$0.223519\pi$
$354$	8.02029	0.426274
$355$	−26.5000	−1.40648
$356$	0.851704	0.0451402
$357$	0	0
$358$	−5.66729	−0.299526
$359$	9.67172	0.510454	0.255227	−	0.966881i	$-0.417850\pi$
$359$	9.67172	0.510454	0.255227	+	0.966881i	$0.417850\pi$
$360$	−11.4470	−0.603311
$361$	−14.0088	−0.737303
$362$	−31.2702	−1.64352
$363$	8.39279	0.440507
$364$	0	0
$365$	19.4569	1.01842
$366$	−4.32638	−0.226144
$367$	−25.4918	−1.33066	−0.665330	−	0.746549i	$-0.731709\pi$
$367$	−25.4918	−1.33066	−0.665330	+	0.746549i	$0.731709\pi$
$368$	15.2529	0.795115
$369$	20.0608	1.04432
$370$	−28.4710	−1.48014
$371$	0	0
$372$	0.105748	0.00548278
$373$	3.86874	0.200316	0.100158	−	0.994972i	$-0.468065\pi$
$373$	3.86874	0.200316	0.100158	+	0.994972i	$0.468065\pi$
$374$	10.7152	0.554071
$375$	−11.9274	−0.615931
$376$	15.7496	0.812225
$377$	18.7256	0.964418
$378$	0	0
$379$	28.5264	1.46530	0.732651	−	0.680604i	$-0.238283\pi$
$379$	28.5264	1.46530	0.732651	+	0.680604i	$0.238283\pi$
$380$	0.376653	0.0193219
$381$	−0.591316	−0.0302940
$382$	24.6906	1.26328
$383$	9.01261	0.460523	0.230261	−	0.973129i	$-0.426042\pi$
$383$	9.01261	0.460523	0.230261	+	0.973129i	$0.426042\pi$
$384$	−12.1468	−0.619863
$385$	0	0
$386$	−36.3037	−1.84781
$387$	5.27491	0.268139
$388$	0.352706	0.0179059
$389$	35.7699	1.81360	0.906802	−	0.421557i	$-0.138516\pi$
$389$	35.7699	1.81360	0.906802	+	0.421557i	$0.138516\pi$
$390$	−8.39320	−0.425006
$391$	16.4625	0.832546
$392$	0	0
$393$	−21.2345	−1.07114
$394$	12.8724	0.648502
$395$	22.4408	1.12912
$396$	0.261224	0.0131270
$397$	18.1594	0.911397	0.455698	−	0.890134i	$-0.349390\pi$
$397$	18.1594	0.911397	0.455698	+	0.890134i	$0.349390\pi$
$398$	−15.7284	−0.788393
$399$	0	0
$400$	2.40367	0.120183
$401$	−17.8319	−0.890483	−0.445241	−	0.895411i	$-0.646882\pi$
$401$	−17.8319	−0.890483	−0.445241	+	0.895411i	$0.646882\pi$
$402$	13.7810	0.687333
$403$	3.53045	0.175864
$404$	−1.44380	−0.0718315
$405$	−1.60603	−0.0798041
$406$	0	0
$407$	−15.5569	−0.771128
$408$	−12.6228	−0.624920
$409$	12.7017	0.628060	0.314030	−	0.949413i	$-0.398321\pi$
$409$	12.7017	0.628060	0.314030	+	0.949413i	$0.398321\pi$
$410$	30.9437	1.52820
$411$	−8.35643	−0.412192
$412$	0.330421	0.0162787
$413$	0	0
$414$	10.4124	0.511739
$415$	−6.20255	−0.304471
$416$	1.23377	0.0604906
$417$	1.01681	0.0497935
$418$	5.33954	0.261165
$419$	−38.2854	−1.87037	−0.935183	−	0.354164i	$-0.884765\pi$
$419$	−38.2854	−1.87037	−0.935183	+	0.354164i	$0.884765\pi$
$420$	0	0
$421$	−34.9446	−1.70310	−0.851548	−	0.524277i	$-0.824336\pi$
$421$	−34.9446	−1.70310	−0.851548	+	0.524277i	$0.824336\pi$
$422$	5.59700	0.272458
$423$	11.1831	0.543742
$424$	1.83005	0.0888752
$425$	2.59428	0.125841
$426$	−18.4825	−0.895479
$427$	0	0
$428$	−0.333490	−0.0161198
$429$	−4.58615	−0.221422
$430$	8.13652	0.392378
$431$	8.75076	0.421509	0.210755	−	0.977539i	$-0.432408\pi$
$431$	8.75076	0.421509	0.210755	+	0.977539i	$0.432408\pi$
$432$	−20.9756	−1.00919
$433$	−30.1605	−1.44942	−0.724710	−	0.689054i	$-0.758026\pi$
$433$	−30.1605	−1.44942	−0.724710	+	0.689054i	$0.758026\pi$
$434$	0	0
$435$	−14.7094	−0.705263
$436$	−1.22675	−0.0587508
$437$	8.20349	0.392426
$438$	13.5702	0.648410
$439$	21.7996	1.04044	0.520220	−	0.854032i	$-0.325850\pi$
$439$	21.7996	1.04044	0.520220	+	0.854032i	$0.325850\pi$
$440$	−9.64801	−0.459951
$441$	0	0
$442$	17.5999	0.837143
$443$	−35.9082	−1.70605	−0.853026	−	0.521868i	$-0.825235\pi$
$443$	−35.9082	−1.70605	−0.853026	+	0.521868i	$0.825235\pi$
$444$	−0.765377	−0.0363232
$445$	−22.3361	−1.05883
$446$	14.9695	0.708827
$447$	−6.75697	−0.319594
$448$	0	0
$449$	−8.49151	−0.400739	−0.200369	−	0.979720i	$-0.564214\pi$
$449$	−8.49151	−0.400739	−0.200369	+	0.979720i	$0.564214\pi$
$450$	1.64085	0.0773505
$451$	16.9080	0.796168
$452$	1.06876	0.0502705
$453$	7.58758	0.356496
$454$	41.0308	1.92567
$455$	0	0
$456$	−6.29008	−0.294560
$457$	−22.6167	−1.05796	−0.528982	−	0.848633i	$-0.677426\pi$
$457$	−22.6167	−1.05796	−0.528982	+	0.848633i	$0.677426\pi$
$458$	−33.6409	−1.57193
$459$	−22.6390	−1.05670
$460$	0.619059	0.0288638
$461$	−11.0838	−0.516223	−0.258111	−	0.966115i	$-0.583100\pi$
$461$	−11.0838	−0.516223	−0.258111	+	0.966115i	$0.583100\pi$
$462$	0	0
$463$	38.6313	1.79535	0.897675	−	0.440659i	$-0.145255\pi$
$463$	38.6313	1.79535	0.897675	+	0.440659i	$0.145255\pi$
$464$	28.5783	1.32671
$465$	−2.77326	−0.128607
$466$	−19.9836	−0.925721
$467$	−24.0609	−1.11340	−0.556702	−	0.830712i	$-0.687934\pi$
$467$	−24.0609	−1.11340	−0.556702	+	0.830712i	$0.687934\pi$
$468$	0.429064	0.0198335
$469$	0	0
$470$	17.2499	0.795679
$471$	−0.697584	−0.0321430
$472$	15.1429	0.697010
$473$	4.44590	0.204423
$474$	15.6513	0.718890
$475$	1.29276	0.0593161
$476$	0	0
$477$	1.29944	0.0594973
$478$	30.3043	1.38609
$479$	−29.1169	−1.33038	−0.665192	−	0.746672i	$-0.731650\pi$
$479$	−29.1169	−1.33038	−0.665192	+	0.746672i	$0.731650\pi$
$480$	−0.969158	−0.0442358
$481$	−25.5525	−1.16509
$482$	−4.57021	−0.208168
$483$	0	0
$484$	−0.661797	−0.0300817
$485$	−9.24978	−0.420011
$486$	−22.9689	−1.04189
$487$	−28.2034	−1.27802	−0.639008	−	0.769200i	$-0.720655\pi$
$487$	−28.2034	−1.27802	−0.639008	+	0.769200i	$0.720655\pi$
$488$	−8.16854	−0.369773
$489$	15.9455	0.721079
$490$	0	0
$491$	−12.4287	−0.560900	−0.280450	−	0.959869i	$-0.590484\pi$
$491$	−12.4287	−0.560900	−0.280450	+	0.959869i	$0.590484\pi$
$492$	0.831849	0.0375027
$493$	30.8446	1.38917
$494$	8.77027	0.394593
$495$	−6.85063	−0.307913
$496$	5.38804	0.241930
$497$	0	0
$498$	−4.32598	−0.193852
$499$	39.4279	1.76503	0.882517	−	0.470281i	$-0.155848\pi$
$499$	39.4279	1.76503	0.882517	+	0.470281i	$0.155848\pi$
$500$	0.940516	0.0420611
$501$	−9.75960	−0.436027
$502$	−7.70927	−0.344082
$503$	−40.2997	−1.79687	−0.898437	−	0.439102i	$-0.855297\pi$
$503$	−40.2997	−1.79687	−0.898437	+	0.439102i	$0.855297\pi$
$504$	0	0
$505$	37.8638	1.68492
$506$	8.77595	0.390139
$507$	5.68573	0.252512
$508$	0.0466271	0.00206874
$509$	−34.6004	−1.53364	−0.766819	−	0.641863i	$-0.778162\pi$
$509$	−34.6004	−1.53364	−0.766819	+	0.641863i	$0.778162\pi$
$510$	−13.8252	−0.612189
$511$	0	0
$512$	−21.1209	−0.933421
$513$	−11.2813	−0.498082
$514$	14.1007	0.621953
$515$	−8.66535	−0.381841
$516$	0.218732	0.00962912
$517$	9.42558	0.414537
$518$	0	0
$519$	7.19987	0.316039
$520$	−15.8470	−0.694937
$521$	7.86638	0.344632	0.172316	−	0.985042i	$-0.444875\pi$
$521$	7.86638	0.344632	0.172316	+	0.985042i	$0.444875\pi$
$522$	19.5088	0.853879
$523$	7.66833	0.335313	0.167656	−	0.985846i	$-0.446380\pi$
$523$	7.66833	0.335313	0.167656	+	0.985846i	$0.446380\pi$
$524$	1.67441	0.0731468
$525$	0	0
$526$	−30.6555	−1.33664
$527$	5.81532	0.253319
$528$	−6.99921	−0.304602
$529$	−9.51691	−0.413779
$530$	2.00438	0.0870648
$531$	10.7523	0.466611
$532$	0	0
$533$	27.7717	1.20293
$534$	−15.5783	−0.674140
$535$	8.74581	0.378114
$536$	26.0196	1.12387
$537$	3.99545	0.172417
$538$	−15.3419	−0.661435
$539$	0	0
$540$	−0.851320	−0.0366350
$541$	−15.0954	−0.649004	−0.324502	−	0.945885i	$-0.605197\pi$
$541$	−15.0954	−0.649004	−0.324502	+	0.945885i	$0.605197\pi$
$542$	22.5752	0.969687
$543$	22.0455	0.946065
$544$	2.03225	0.0871322
$545$	32.1718	1.37809
$546$	0	0
$547$	2.88384	0.123304	0.0616521	−	0.998098i	$-0.480363\pi$
$547$	2.88384	0.123304	0.0616521	+	0.998098i	$0.480363\pi$
$548$	0.658930	0.0281481
$549$	−5.80012	−0.247543
$550$	1.38298	0.0589703
$551$	15.3703	0.654795
$552$	−10.3382	−0.440025
$553$	0	0
$554$	−29.8155	−1.26674
$555$	20.0721	0.852014
$556$	−0.0801788	−0.00340034
$557$	−4.00819	−0.169832	−0.0849162	−	0.996388i	$-0.527062\pi$
$557$	−4.00819	−0.169832	−0.0849162	+	0.996388i	$0.527062\pi$
$558$	3.67812	0.155707
$559$	7.30247	0.308861
$560$	0	0
$561$	−7.55426	−0.318941
$562$	4.78962	0.202038
$563$	−16.1490	−0.680599	−0.340299	−	0.940317i	$-0.610528\pi$
$563$	−16.1490	−0.680599	−0.340299	+	0.940317i	$0.610528\pi$
$564$	0.463724	0.0195263
$565$	−28.0285	−1.17917
$566$	21.1313	0.888216
$567$	0	0
$568$	−34.8964	−1.46422
$569$	−19.4554	−0.815613	−0.407807	−	0.913068i	$-0.633706\pi$
$569$	−19.4554	−0.815613	−0.407807	+	0.913068i	$0.633706\pi$
$570$	−6.88927	−0.288560
$571$	−33.8282	−1.41567	−0.707833	−	0.706380i	$-0.750327\pi$
$571$	−33.8282	−1.41567	−0.707833	+	0.706380i	$0.750327\pi$
$572$	0.361632	0.0151206
$573$	−17.4070	−0.727186
$574$	0	0
$575$	2.12476	0.0886086
$576$	−15.0486	−0.627026
$577$	42.0018	1.74856	0.874278	−	0.485425i	$-0.161335\pi$
$577$	42.0018	1.74856	0.874278	+	0.485425i	$0.161335\pi$
$578$	4.47161	0.185995
$579$	25.5942	1.06366
$580$	1.15988	0.0481615
$581$	0	0
$582$	−6.45127	−0.267414
$583$	1.09522	0.0453594
$584$	25.6216	1.06023
$585$	−11.2523	−0.465224
$586$	1.33719	0.0552389
$587$	37.3166	1.54022	0.770110	−	0.637911i	$-0.220201\pi$
$587$	37.3166	1.54022	0.770110	+	0.637911i	$0.220201\pi$
$588$	0	0
$589$	2.89785	0.119404
$590$	16.5854	0.682811
$591$	−9.07507	−0.373299
$592$	−38.9972	−1.60278
$593$	1.45651	0.0598115	0.0299057	−	0.999553i	$-0.490479\pi$
$593$	1.45651	0.0598115	0.0299057	+	0.999553i	$0.490479\pi$
$594$	−12.0686	−0.495179
$595$	0	0
$596$	0.532808	0.0218247
$597$	11.0886	0.453824
$598$	14.4146	0.589458
$599$	−31.2287	−1.27597	−0.637985	−	0.770048i	$-0.720232\pi$
$599$	−31.2287	−1.27597	−0.637985	+	0.770048i	$0.720232\pi$
$600$	−1.62917	−0.0665107
$601$	−24.4099	−0.995699	−0.497849	−	0.867263i	$-0.665877\pi$
$601$	−24.4099	−0.995699	−0.497849	+	0.867263i	$0.665877\pi$
$602$	0	0
$603$	18.4753	0.752374
$604$	−0.598304	−0.0243446
$605$	17.3557	0.705611
$606$	26.4081	1.07276
$607$	−16.7206	−0.678667	−0.339334	−	0.940666i	$-0.610202\pi$
$607$	−16.7206	−0.678667	−0.339334	+	0.940666i	$0.610202\pi$
$608$	1.01270	0.0410704
$609$	0	0
$610$	−8.94667	−0.362240
$611$	15.4817	0.626321
$612$	0.706749	0.0285686
$613$	−13.3978	−0.541132	−0.270566	−	0.962701i	$-0.587211\pi$
$613$	−13.3978	−0.541132	−0.270566	+	0.962701i	$0.587211\pi$
$614$	−46.4581	−1.87490
$615$	−21.8154	−0.879681
$616$	0	0
$617$	29.1852	1.17495	0.587475	−	0.809242i	$-0.300122\pi$
$617$	29.1852	1.17495	0.587475	+	0.809242i	$0.300122\pi$
$618$	−6.04366	−0.243112
$619$	−7.18553	−0.288811	−0.144405	−	0.989519i	$-0.546127\pi$
$619$	−7.18553	−0.288811	−0.144405	+	0.989519i	$0.546127\pi$
$620$	0.218680	0.00878240
$621$	−18.5417	−0.744054
$622$	−46.2855	−1.85588
$623$	0	0
$624$	−11.4963	−0.460221
$625$	−21.7719	−0.870877
$626$	−9.54501	−0.381495
$627$	−3.76439	−0.150335
$628$	0.0550066	0.00219500
$629$	−42.0898	−1.67823
$630$	0	0
$631$	−23.3041	−0.927720	−0.463860	−	0.885909i	$-0.653536\pi$
$631$	−23.3041	−0.927720	−0.463860	+	0.885909i	$0.653536\pi$
$632$	29.5509	1.17547
$633$	−3.94590	−0.156836
$634$	8.54870	0.339512
$635$	−1.22280	−0.0485254
$636$	0.0538832	0.00213661
$637$	0	0
$638$	16.4428	0.650978
$639$	−24.7784	−0.980217
$640$	−25.1187	−0.992905
$641$	45.0737	1.78031	0.890153	−	0.455662i	$-0.150598\pi$
$641$	45.0737	1.78031	0.890153	+	0.455662i	$0.150598\pi$
$642$	6.09978	0.240739
$643$	−7.19585	−0.283777	−0.141888	−	0.989883i	$-0.545317\pi$
$643$	−7.19585	−0.283777	−0.141888	+	0.989883i	$0.545317\pi$
$644$	0	0
$645$	−5.73627	−0.225865
$646$	14.4463	0.568382
$647$	26.2361	1.03145	0.515723	−	0.856755i	$-0.327523\pi$
$647$	26.2361	1.03145	0.515723	+	0.856755i	$0.327523\pi$
$648$	−2.11488	−0.0830805
$649$	9.06249	0.355734
$650$	2.27156	0.0890979
$651$	0	0
$652$	−1.25735	−0.0492416
$653$	6.64240	0.259937	0.129969	−	0.991518i	$-0.458512\pi$
$653$	6.64240	0.259937	0.129969	+	0.991518i	$0.458512\pi$
$654$	22.4383	0.877405
$655$	−43.9116	−1.71577
$656$	42.3841	1.65482
$657$	18.1928	0.709768
$658$	0	0
$659$	11.1361	0.433802	0.216901	−	0.976194i	$-0.430405\pi$
$659$	11.1361	0.433802	0.216901	+	0.976194i	$0.430405\pi$
$660$	−0.284071	−0.0110575
$661$	4.92047	0.191384	0.0956920	−	0.995411i	$-0.469494\pi$
$661$	4.92047	0.191384	0.0956920	+	0.995411i	$0.469494\pi$
$662$	47.0815	1.82987
$663$	−12.4080	−0.481887
$664$	−8.16778	−0.316971
$665$	0	0
$666$	−26.6213	−1.03155
$667$	25.2622	0.978158
$668$	0.769574	0.0297757
$669$	−10.5535	−0.408024
$670$	28.4981	1.10098
$671$	−4.88858	−0.188721
$672$	0	0
$673$	−30.4163	−1.17246	−0.586231	−	0.810144i	$-0.699389\pi$
$673$	−30.4163	−1.17246	−0.586231	+	0.810144i	$0.699389\pi$
$674$	8.12146	0.312827
$675$	−2.92194	−0.112465
$676$	−0.448337	−0.0172437
$677$	16.4659	0.632834	0.316417	−	0.948620i	$-0.397520\pi$
$677$	16.4659	0.632834	0.316417	+	0.948620i	$0.397520\pi$
$678$	−19.5485	−0.750757
$679$	0	0
$680$	−26.1030	−1.00100
$681$	−28.9268	−1.10848
$682$	3.10007	0.118708
$683$	32.9941	1.26248	0.631242	−	0.775586i	$-0.282546\pi$
$683$	32.9941	1.26248	0.631242	+	0.775586i	$0.282546\pi$
$684$	0.352182	0.0134660
$685$	−17.2805	−0.660256
$686$	0	0
$687$	23.7169	0.904856
$688$	11.1447	0.424889
$689$	1.79892	0.0685333
$690$	−11.3231	−0.431061
$691$	1.55254	0.0590614	0.0295307	−	0.999564i	$-0.490599\pi$
$691$	1.55254	0.0590614	0.0295307	+	0.999564i	$0.490599\pi$
$692$	−0.567732	−0.0215819
$693$	0	0
$694$	28.3482	1.07608
$695$	2.10270	0.0797600
$696$	−19.3700	−0.734218
$697$	45.7453	1.73273
$698$	−3.19644	−0.120987
$699$	14.0885	0.532875
$700$	0	0
$701$	32.4721	1.22645	0.613227	−	0.789907i	$-0.289871\pi$
$701$	32.4721	1.22645	0.613227	+	0.789907i	$0.289871\pi$
$702$	−19.8228	−0.748163
$703$	−20.9739	−0.791045
$704$	−12.6836	−0.478031
$705$	−12.1612	−0.458019
$706$	41.3743	1.55714
$707$	0	0
$708$	0.445861	0.0167565
$709$	−25.8801	−0.971947	−0.485974	−	0.873973i	$-0.661535\pi$
$709$	−25.8801	−0.971947	−0.485974	+	0.873973i	$0.661535\pi$
$710$	−38.2205	−1.43439
$711$	20.9828	0.786918
$712$	−29.4131	−1.10230
$713$	4.76285	0.178370
$714$	0	0
$715$	−9.48386	−0.354676
$716$	−0.315054	−0.0117741
$717$	−21.3646	−0.797875
$718$	13.9494	0.520585
$719$	−14.6407	−0.546005	−0.273002	−	0.962013i	$-0.588017\pi$
$719$	−14.6407	−0.546005	−0.273002	+	0.962013i	$0.588017\pi$
$720$	−17.1728	−0.639991
$721$	0	0
$722$	−20.2046	−0.751937
$723$	3.22201	0.119828
$724$	−1.73836	−0.0646056
$725$	3.98100	0.147851
$726$	12.1048	0.449250
$727$	−27.5259	−1.02088	−0.510440	−	0.859913i	$-0.670517\pi$
$727$	−27.5259	−1.02088	−0.510440	+	0.859913i	$0.670517\pi$
$728$	0	0
$729$	13.9018	0.514880
$730$	28.0623	1.03863
$731$	12.0285	0.444892
$732$	−0.240510	−0.00888952
$733$	−24.7837	−0.915407	−0.457703	−	0.889105i	$-0.651328\pi$
$733$	−24.7837	−0.915407	−0.457703	+	0.889105i	$0.651328\pi$
$734$	−36.7664	−1.35707
$735$	0	0
$736$	1.66445	0.0613525
$737$	15.5718	0.573593
$738$	28.9333	1.06505
$739$	−16.8509	−0.619870	−0.309935	−	0.950758i	$-0.600307\pi$
$739$	−16.8509	−0.619870	−0.309935	+	0.950758i	$0.600307\pi$
$740$	−1.58275	−0.0581830
$741$	−6.18306	−0.227141
$742$	0	0
$743$	−19.8239	−0.727270	−0.363635	−	0.931542i	$-0.618464\pi$
$743$	−19.8239	−0.727270	−0.363635	+	0.931542i	$0.618464\pi$
$744$	−3.65194	−0.133887
$745$	−13.9730	−0.511930
$746$	5.57982	0.204292
$747$	−5.79958	−0.212196
$748$	0.595677	0.0217801
$749$	0	0
$750$	−17.2027	−0.628156
$751$	−11.5453	−0.421294	−0.210647	−	0.977562i	$-0.567557\pi$
$751$	−11.5453	−0.421294	−0.210647	+	0.977562i	$0.567557\pi$
$752$	23.6275	0.861607
$753$	5.43506	0.198064
$754$	27.0076	0.983559
$755$	15.6906	0.571040
$756$	0	0
$757$	−38.2106	−1.38879	−0.694393	−	0.719596i	$-0.744327\pi$
$757$	−38.2106	−1.38879	−0.694393	+	0.719596i	$0.744327\pi$
$758$	41.1431	1.49439
$759$	−6.18707	−0.224576
$760$	−13.0075	−0.471830
$761$	7.74559	0.280777	0.140389	−	0.990096i	$-0.455165\pi$
$761$	7.74559	0.280777	0.140389	+	0.990096i	$0.455165\pi$
$762$	−0.852845	−0.0308953
$763$	0	0
$764$	1.37259	0.0496586
$765$	−18.5346	−0.670120
$766$	12.9987	0.469663
$767$	14.8853	0.537477
$768$	−1.95354	−0.0704923
$769$	3.48152	0.125547	0.0627734	−	0.998028i	$-0.480005\pi$
$769$	3.48152	0.125547	0.0627734	+	0.998028i	$0.480005\pi$
$770$	0	0
$771$	−9.94100	−0.358016
$772$	−2.01818	−0.0726358
$773$	4.93133	0.177367	0.0886837	−	0.996060i	$-0.471734\pi$
$773$	4.93133	0.177367	0.0886837	+	0.996060i	$0.471734\pi$
$774$	7.60791	0.273461
$775$	0.750563	0.0269610
$776$	−12.1805	−0.437254
$777$	0	0
$778$	51.5902	1.84960
$779$	22.7955	0.816732
$780$	−0.466591	−0.0167066
$781$	−20.8842	−0.747296
$782$	23.7436	0.849071
$783$	−34.7403	−1.24152
$784$	0	0
$785$	−1.44256	−0.0514871
$786$	−30.6262	−1.09240
$787$	9.56773	0.341053	0.170526	−	0.985353i	$-0.445453\pi$
$787$	9.56773	0.341053	0.170526	+	0.985353i	$0.445453\pi$
$788$	0.715597	0.0254921
$789$	21.6122	0.769415
$790$	32.3659	1.15153
$791$	0	0
$792$	−9.02119	−0.320554
$793$	−8.02957	−0.285138
$794$	26.1911	0.929486
$795$	−1.41309	−0.0501173
$796$	−0.874366	−0.0309911
$797$	−15.0726	−0.533898	−0.266949	−	0.963711i	$-0.586016\pi$
$797$	−15.0726	−0.533898	−0.266949	+	0.963711i	$0.586016\pi$
$798$	0	0
$799$	25.5012	0.902169
$800$	0.262296	0.00927356
$801$	−20.8849	−0.737933
$802$	−25.7186	−0.908157
$803$	15.3336	0.541111
$804$	0.766107	0.0270185
$805$	0	0
$806$	5.09191	0.179355
$807$	10.8161	0.380743
$808$	49.8606	1.75409
$809$	−23.5580	−0.828254	−0.414127	−	0.910219i	$-0.635913\pi$
$809$	−23.5580	−0.828254	−0.414127	+	0.910219i	$0.635913\pi$
$810$	−2.31634	−0.0813881
$811$	−47.7098	−1.67532	−0.837658	−	0.546195i	$-0.816076\pi$
$811$	−47.7098	−1.67532	−0.837658	+	0.546195i	$0.816076\pi$
$812$	0	0
$813$	−15.9156	−0.558183
$814$	−22.4375	−0.786433
$815$	32.9742	1.15503
$816$	−18.9366	−0.662914
$817$	5.99398	0.209703
$818$	18.3195	0.640526
$819$	0	0
$820$	1.72021	0.0600723
$821$	4.87978	0.170306	0.0851528	−	0.996368i	$-0.472862\pi$
$821$	4.87978	0.170306	0.0851528	+	0.996368i	$0.472862\pi$
$822$	−12.0523	−0.420374
$823$	27.1236	0.945469	0.472735	−	0.881205i	$-0.343267\pi$
$823$	27.1236	0.945469	0.472735	+	0.881205i	$0.343267\pi$
$824$	−11.4109	−0.397517
$825$	−0.975002	−0.0339452
$826$	0	0
$827$	−52.5172	−1.82620	−0.913101	−	0.407733i	$-0.866319\pi$
$827$	−52.5172	−1.82620	−0.913101	+	0.407733i	$0.866319\pi$
$828$	0.578839	0.0201161
$829$	−5.14938	−0.178845	−0.0894226	−	0.995994i	$-0.528502\pi$
$829$	−5.14938	−0.178845	−0.0894226	+	0.995994i	$0.528502\pi$
$830$	−8.94583	−0.310514
$831$	21.0200	0.729177
$832$	−20.8330	−0.722255
$833$	0	0
$834$	1.46653	0.0507818
$835$	−20.1822	−0.698434
$836$	0.296833	0.0102662
$837$	−6.54979	−0.226394
$838$	−55.2184	−1.90749
$839$	6.54789	0.226058	0.113029	−	0.993592i	$-0.463945\pi$
$839$	6.54789	0.226058	0.113029	+	0.993592i	$0.463945\pi$
$840$	0	0
$841$	18.3320	0.632137
$842$	−50.4000	−1.73690
$843$	−3.37669	−0.116299
$844$	0.311146	0.0107101
$845$	11.7577	0.404478
$846$	16.1292	0.554534
$847$	0	0
$848$	2.74544	0.0942787
$849$	−14.8976	−0.511286
$850$	3.74169	0.128339
$851$	−34.4722	−1.18169
$852$	−1.02747	−0.0352006
$853$	17.0529	0.583880	0.291940	−	0.956437i	$-0.405699\pi$
$853$	17.0529	0.583880	0.291940	+	0.956437i	$0.405699\pi$
$854$	0	0
$855$	−9.23603	−0.315866
$856$	11.5169	0.393638
$857$	27.6000	0.942798	0.471399	−	0.881920i	$-0.343749\pi$
$857$	27.6000	0.942798	0.471399	+	0.881920i	$0.343749\pi$
$858$	−6.61453	−0.225816
$859$	−4.93570	−0.168404	−0.0842019	−	0.996449i	$-0.526834\pi$
$859$	−4.93570	−0.168404	−0.0842019	+	0.996449i	$0.526834\pi$
$860$	0.452322	0.0154241
$861$	0	0
$862$	12.6211	0.429875
$863$	−36.0972	−1.22876	−0.614381	−	0.789010i	$-0.710594\pi$
$863$	−36.0972	−1.22876	−0.614381	+	0.789010i	$0.710594\pi$
$864$	−2.28893	−0.0778709
$865$	14.8888	0.506236
$866$	−43.4999	−1.47819
$867$	−3.15250	−0.107064
$868$	0	0
$869$	17.6852	0.599928
$870$	−21.2151	−0.719261
$871$	25.5769	0.866639
$872$	42.3652	1.43467
$873$	−8.64883	−0.292719
$874$	11.8318	0.400215
$875$	0	0
$876$	0.754390	0.0254885
$877$	44.8926	1.51592	0.757958	−	0.652303i	$-0.226197\pi$
$877$	44.8926	1.51592	0.757958	+	0.652303i	$0.226197\pi$
$878$	31.4413	1.06109
$879$	−0.942724	−0.0317973
$880$	−14.4739	−0.487915
$881$	−23.9726	−0.807658	−0.403829	−	0.914835i	$-0.632321\pi$
$881$	−23.9726	−0.807658	−0.403829	+	0.914835i	$0.632321\pi$
$882$	0	0
$883$	−3.36329	−0.113184	−0.0565918	−	0.998397i	$-0.518023\pi$
$883$	−3.36329	−0.113184	−0.0565918	+	0.998397i	$0.518023\pi$
$884$	0.978409	0.0329074
$885$	−11.6928	−0.393048
$886$	−51.7898	−1.73991
$887$	11.3959	0.382636	0.191318	−	0.981528i	$-0.438724\pi$
$887$	11.3959	0.382636	0.191318	+	0.981528i	$0.438724\pi$
$888$	26.4318	0.886993
$889$	0	0
$890$	−32.2149	−1.07985
$891$	−1.26568	−0.0424019
$892$	0.832179	0.0278634
$893$	12.7076	0.425243
$894$	−9.74546	−0.325937
$895$	8.26233	0.276179
$896$	0	0
$897$	−10.1624	−0.339311
$898$	−12.2472	−0.408693
$899$	8.92378	0.297625
$900$	0.0912176	0.00304059
$901$	2.96316	0.0987171
$902$	24.3862	0.811971
$903$	0	0
$904$	−36.9091	−1.22758
$905$	45.5887	1.51542
$906$	10.9434	0.363571
$907$	14.5687	0.483745	0.241873	−	0.970308i	$-0.422238\pi$
$907$	14.5687	0.483745	0.241873	+	0.970308i	$0.422238\pi$
$908$	2.28097	0.0756966
$909$	35.4038	1.17427
$910$	0	0
$911$	−11.3880	−0.377301	−0.188651	−	0.982044i	$-0.560411\pi$
$911$	−11.3880	−0.377301	−0.188651	+	0.982044i	$0.560411\pi$
$912$	−9.43635	−0.312469
$913$	−4.88812	−0.161773
$914$	−32.6196	−1.07896
$915$	6.30742	0.208517
$916$	−1.87015	−0.0617915
$917$	0	0
$918$	−32.6519	−1.07767
$919$	−8.63177	−0.284736	−0.142368	−	0.989814i	$-0.545472\pi$
$919$	−8.63177	−0.284736	−0.142368	+	0.989814i	$0.545472\pi$
$920$	−21.3788	−0.704838
$921$	32.7531	1.07925
$922$	−15.9859	−0.526469
$923$	−34.3026	−1.12909
$924$	0	0
$925$	−5.43238	−0.178616
$926$	55.7173	1.83098
$927$	−8.10237	−0.266117
$928$	3.11855	0.102372
$929$	−21.2022	−0.695623	−0.347812	−	0.937564i	$-0.613075\pi$
$929$	−21.2022	−0.695623	−0.347812	+	0.937564i	$0.613075\pi$
$930$	−3.99982	−0.131159
$931$	0	0
$932$	−1.11092	−0.0363893
$933$	32.6314	1.06830
$934$	−34.7026	−1.13550
$935$	−15.6217	−0.510885
$936$	−14.8175	−0.484324
$937$	39.4766	1.28964	0.644822	−	0.764333i	$-0.276932\pi$
$937$	39.4766	1.28964	0.644822	+	0.764333i	$0.276932\pi$
$938$	0	0
$939$	6.72926	0.219601
$940$	0.958950	0.0312775
$941$	−8.73780	−0.284844	−0.142422	−	0.989806i	$-0.545489\pi$
$941$	−8.73780	−0.284844	−0.142422	+	0.989806i	$0.545489\pi$
$942$	−1.00611	−0.0327809
$943$	37.4661	1.22007
$944$	22.7174	0.739387
$945$	0	0
$946$	6.41225	0.208480
$947$	−9.63654	−0.313145	−0.156573	−	0.987666i	$-0.550045\pi$
$947$	−9.63654	−0.313145	−0.156573	+	0.987666i	$0.550045\pi$
$948$	0.870083	0.0282590
$949$	25.1857	0.817562
$950$	1.86453	0.0604934
$951$	−6.02686	−0.195434
$952$	0	0
$953$	19.1476	0.620251	0.310125	−	0.950696i	$-0.399629\pi$
$953$	19.1476	0.620251	0.310125	+	0.950696i	$0.399629\pi$
$954$	1.87416	0.0606782
$955$	−35.9964	−1.16482
$956$	1.68466	0.0544859
$957$	−11.5922	−0.374724
$958$	−41.9948	−1.35679
$959$	0	0
$960$	16.3648	0.528173
$961$	−29.3175	−0.945727
$962$	−36.8539	−1.18822
$963$	8.17761	0.263520
$964$	−0.254066	−0.00818290
$965$	52.9270	1.70378
$966$	0	0
$967$	−36.4956	−1.17362	−0.586810	−	0.809725i	$-0.699616\pi$
$967$	−36.4956	−1.17362	−0.586810	+	0.809725i	$0.699616\pi$
$968$	22.8547	0.734579
$969$	−10.1847	−0.327179
$970$	−13.3408	−0.428347
$971$	12.8734	0.413128	0.206564	−	0.978433i	$-0.433772\pi$
$971$	12.8734	0.413128	0.206564	+	0.978433i	$0.433772\pi$
$972$	−1.27688	−0.0409559
$973$	0	0
$974$	−40.6772	−1.30338
$975$	−1.60146	−0.0512876
$976$	−12.2544	−0.392254
$977$	−27.8335	−0.890473	−0.445236	−	0.895413i	$-0.646880\pi$
$977$	−27.8335	−0.890473	−0.445236	+	0.895413i	$0.646880\pi$
$978$	22.9979	0.735391
$979$	−17.6027	−0.562583
$980$	0	0
$981$	30.0816	0.960433
$982$	−17.9257	−0.572033
$983$	−27.1809	−0.866936	−0.433468	−	0.901169i	$-0.642710\pi$
$983$	−27.1809	−0.866936	−0.433468	+	0.901169i	$0.642710\pi$
$984$	−28.7274	−0.915796
$985$	−18.7666	−0.597955
$986$	44.4866	1.41674
$987$	0	0
$988$	0.487553	0.0155111
$989$	9.85158	0.313262
$990$	−9.88054	−0.314024
$991$	27.9355	0.887399	0.443700	−	0.896176i	$-0.353666\pi$
$991$	27.9355	0.887399	0.443700	+	0.896176i	$0.353666\pi$
$992$	0.587960	0.0186678
$993$	−33.1926	−1.05333
$994$	0	0
$995$	22.9304	0.726942
$996$	−0.240488	−0.00762015
$997$	−26.9778	−0.854397	−0.427198	−	0.904158i	$-0.640499\pi$
$997$	−26.9778	−0.854397	−0.427198	+	0.904158i	$0.640499\pi$
$998$	56.8661	1.80007
$999$	47.4057	1.49985

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6811.2.a.p.1.5		7
7.6	odd	2		139.2.a.c.1.5	✓	7
21.20	even	2		1251.2.a.k.1.3		7
28.27	even	2		2224.2.a.o.1.3		7
35.34	odd	2		3475.2.a.e.1.3		7
56.13	odd	2		8896.2.a.be.1.3		7
56.27	even	2		8896.2.a.bd.1.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
139.2.a.c.1.5	✓	7	7.6	odd	2
1251.2.a.k.1.3		7	21.20	even	2
2224.2.a.o.1.3		7	28.27	even	2
3475.2.a.e.1.3		7	35.34	odd	2
6811.2.a.p.1.5		7	1.1	even	1	trivial
8896.2.a.bd.1.5		7	56.27	even	2
8896.2.a.be.1.3		7	56.13	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6811 = 7^{2} \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6811.a (trivial)

\(f(q)\)	\(=\)	\(q+1.44228 q^{2} -1.01681 q^{3} +0.0801788 q^{4} -2.10270 q^{5} -1.46653 q^{6} -2.76892 q^{8} -1.96609 q^{9} +O(q^{10})\)	\(q+1.44228 q^{2} -1.01681 q^{3} +0.0801788 q^{4} -2.10270 q^{5} -1.46653 q^{6} -2.76892 q^{8} -1.96609 q^{9} -3.03269 q^{10} -1.65710 q^{11} -0.0815268 q^{12} -2.72182 q^{13} +2.13805 q^{15} -4.15393 q^{16} -4.48334 q^{17} -2.83566 q^{18} -2.23411 q^{19} -0.168592 q^{20} -2.39001 q^{22} -3.67193 q^{23} +2.81548 q^{24} -0.578649 q^{25} -3.92563 q^{26} +5.04959 q^{27} -6.87982 q^{29} +3.08368 q^{30} -1.29710 q^{31} -0.453290 q^{32} +1.68496 q^{33} -6.46625 q^{34} -0.157639 q^{36} +9.38804 q^{37} -3.22221 q^{38} +2.76758 q^{39} +5.82222 q^{40} -10.2034 q^{41} -2.68294 q^{43} -0.132864 q^{44} +4.13410 q^{45} -5.29596 q^{46} -5.68799 q^{47} +4.22377 q^{48} -0.834575 q^{50} +4.55872 q^{51} -0.218232 q^{52} -0.660925 q^{53} +7.28293 q^{54} +3.48439 q^{55} +2.27167 q^{57} -9.92265 q^{58} -5.46888 q^{59} +0.171427 q^{60} +2.95008 q^{61} -1.87078 q^{62} +7.65409 q^{64} +5.72316 q^{65} +2.43019 q^{66} -9.39699 q^{67} -0.359469 q^{68} +3.73367 q^{69} +12.6029 q^{71} +5.44396 q^{72} -9.25327 q^{73} +13.5402 q^{74} +0.588378 q^{75} -0.179128 q^{76} +3.99163 q^{78} -10.6724 q^{79} +8.73447 q^{80} +0.763792 q^{81} -14.7162 q^{82} +2.94980 q^{83} +9.42713 q^{85} -3.86956 q^{86} +6.99549 q^{87} +4.58839 q^{88} +10.6226 q^{89} +5.96254 q^{90} -0.294411 q^{92} +1.31890 q^{93} -8.20369 q^{94} +4.69766 q^{95} +0.460911 q^{96} +4.39900 q^{97} +3.25801 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{2} + 2 q^{3} + 9 q^{4} - 11 q^{5} + 7 q^{6} + 6 q^{8} + 13 q^{9}+O(q^{10}) \) 7 * q + q^2 + 2 * q^3 + 9 * q^4 - 11 * q^5 + 7 * q^6 + 6 * q^8 + 13 * q^9	\( 7 q + q^{2} + 2 q^{3} + 9 q^{4} - 11 q^{5} + 7 q^{6} + 6 q^{8} + 13 q^{9} + 4 q^{10} + 2 q^{11} + 8 q^{12} - 6 q^{13} - 3 q^{15} + 5 q^{16} - 5 q^{17} - 10 q^{18} + 10 q^{19} - 12 q^{20} - 18 q^{22} - q^{23} + 21 q^{24} + 14 q^{25} + 8 q^{26} + 11 q^{27} + 30 q^{29} - 41 q^{30} + 20 q^{31} - 12 q^{32} + 20 q^{33} + 17 q^{34} + 2 q^{36} + 6 q^{37} - 6 q^{38} + 11 q^{39} + 22 q^{40} - 19 q^{41} - 12 q^{43} + 25 q^{44} - 27 q^{45} + 22 q^{46} + 3 q^{47} - 15 q^{48} + 12 q^{50} + 23 q^{51} + 8 q^{52} + 38 q^{53} + 7 q^{54} - 7 q^{55} - 19 q^{57} - 21 q^{58} + 14 q^{59} - 8 q^{60} - 4 q^{61} + q^{62} - 16 q^{64} + 10 q^{65} - 18 q^{66} + 9 q^{67} + 25 q^{68} - 9 q^{69} + 24 q^{71} + 41 q^{72} + 5 q^{73} + 9 q^{74} + 21 q^{75} - 3 q^{76} + 20 q^{78} + 8 q^{79} - 11 q^{80} + 39 q^{81} - 56 q^{82} + 9 q^{83} - 22 q^{85} + 39 q^{86} + 25 q^{87} - 29 q^{88} - 10 q^{89} - 72 q^{90} + 29 q^{92} - 15 q^{93} + 36 q^{94} - 21 q^{95} + 11 q^{96} + 5 q^{97} - 31 q^{99}+O(q^{100}) \) 7 * q + q^2 + 2 * q^3 + 9 * q^4 - 11 * q^5 + 7 * q^6 + 6 * q^8 + 13 * q^9 + 4 * q^10 + 2 * q^11 + 8 * q^12 - 6 * q^13 - 3 * q^15 + 5 * q^16 - 5 * q^17 - 10 * q^18 + 10 * q^19 - 12 * q^20 - 18 * q^22 - q^23 + 21 * q^24 + 14 * q^25 + 8 * q^26 + 11 * q^27 + 30 * q^29 - 41 * q^30 + 20 * q^31 - 12 * q^32 + 20 * q^33 + 17 * q^34 + 2 * q^36 + 6 * q^37 - 6 * q^38 + 11 * q^39 + 22 * q^40 - 19 * q^41 - 12 * q^43 + 25 * q^44 - 27 * q^45 + 22 * q^46 + 3 * q^47 - 15 * q^48 + 12 * q^50 + 23 * q^51 + 8 * q^52 + 38 * q^53 + 7 * q^54 - 7 * q^55 - 19 * q^57 - 21 * q^58 + 14 * q^59 - 8 * q^60 - 4 * q^61 + q^62 - 16 * q^64 + 10 * q^65 - 18 * q^66 + 9 * q^67 + 25 * q^68 - 9 * q^69 + 24 * q^71 + 41 * q^72 + 5 * q^73 + 9 * q^74 + 21 * q^75 - 3 * q^76 + 20 * q^78 + 8 * q^79 - 11 * q^80 + 39 * q^81 - 56 * q^82 + 9 * q^83 - 22 * q^85 + 39 * q^86 + 25 * q^87 - 29 * q^88 - 10 * q^89 - 72 * q^90 + 29 * q^92 - 15 * q^93 + 36 * q^94 - 21 * q^95 + 11 * q^96 + 5 * q^97 - 31 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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