LMFDB - Embedded newform 431.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [431,2,Mod(1,431)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(431, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("431.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 431 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	431.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:	$3.44155232712$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.725.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.737640$ of defining polynomial
Character	$\chi$	$=$	431.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.737640 q^{2} -2.35567 q^{3} -1.45589 q^{4} +0.355674 q^{5} +1.73764 q^{6} +2.19353 q^{7} +2.54920 q^{8} +2.54920 q^{9} -0.262360 q^{10} +1.31313 q^{11} +3.42960 q^{12} -0.219819 q^{13} -1.61803 q^{14} -0.837853 q^{15} +1.03138 q^{16} -4.72333 q^{17} -1.88039 q^{18} -3.76902 q^{19} -0.517822 q^{20} -5.16724 q^{21} -0.968620 q^{22} -0.150986 q^{23} -6.00509 q^{24} -4.87350 q^{25} +0.162147 q^{26} +1.06193 q^{27} -3.19353 q^{28} -1.01939 q^{29} +0.618034 q^{30} -4.64433 q^{31} -5.85919 q^{32} -3.09331 q^{33} +3.48412 q^{34} +0.780181 q^{35} -3.71135 q^{36} -1.24723 q^{37} +2.78018 q^{38} +0.517822 q^{39} +0.906685 q^{40} +0.961212 q^{41} +3.81156 q^{42} -2.70626 q^{43} -1.91177 q^{44} +0.906685 q^{45} +0.111374 q^{46} -0.791342 q^{47} -2.42960 q^{48} -2.18844 q^{49} +3.59489 q^{50} +11.1266 q^{51} +0.320031 q^{52} -4.41446 q^{53} -0.783326 q^{54} +0.467048 q^{55} +5.59174 q^{56} +8.87858 q^{57} +0.751946 q^{58} -4.46097 q^{59} +1.21982 q^{60} -3.10155 q^{61} +3.42584 q^{62} +5.59174 q^{63} +2.25922 q^{64} -0.0781839 q^{65} +2.28175 q^{66} +9.44584 q^{67} +6.87664 q^{68} +0.355674 q^{69} -0.575493 q^{70} +2.60980 q^{71} +6.49843 q^{72} -10.8065 q^{73} +0.920006 q^{74} +11.4804 q^{75} +5.48727 q^{76} +2.88039 q^{77} -0.381966 q^{78} -4.27546 q^{79} +0.366835 q^{80} -10.1492 q^{81} -0.709029 q^{82} +5.53916 q^{83} +7.52291 q^{84} -1.67997 q^{85} +1.99625 q^{86} +2.40136 q^{87} +3.34744 q^{88} -14.6037 q^{89} -0.668808 q^{90} -0.482178 q^{91} +0.219819 q^{92} +10.9405 q^{93} +0.583726 q^{94} -1.34054 q^{95} +13.8023 q^{96} -9.16806 q^{97} +1.61428 q^{98} +3.34744 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} - 3 q^{3} - q^{4} - 5 q^{5} + 5 q^{6} + 2 q^{7} - 3 q^{8} - 3 q^{9} - 3 q^{10} + q^{11} - 2 q^{12} - 5 q^{13} - 2 q^{14} - 3 q^{15} - 3 q^{16} + 2 q^{17} - 5 q^{18} - 6 q^{19} + 4 q^{20}+ \cdots + 11 q^{99}+O(q^{100}) $ 4 * q - q^2 - 3 * q^3 - q^4 - 5 * q^5 + 5 * q^6 + 2 * q^7 - 3 * q^8 - 3 * q^9 - 3 * q^10 + q^11 - 2 * q^12 - 5 * q^13 - 2 * q^14 - 3 * q^15 - 3 * q^16 + 2 * q^17 - 5 * q^18 - 6 * q^19 + 4 * q^20 - 3 * q^21 - 11 * q^22 + 4 * q^23 - 6 * q^24 - 7 * q^25 + q^26 + 3 * q^27 - 6 * q^28 - 5 * q^29 - 2 * q^30 - 25 * q^31 + 8 * q^32 - 4 * q^33 - 12 * q^34 - q^35 - 2 * q^36 - q^37 + 7 * q^38 - 4 * q^39 + 12 * q^40 + 2 * q^41 + 4 * q^42 - 16 * q^43 + 2 * q^44 + 12 * q^45 + 7 * q^46 - 4 * q^47 + 6 * q^48 - 20 * q^49 + 13 * q^50 - 3 * q^51 + 7 * q^52 + 4 * q^53 - 13 * q^54 + 2 * q^55 + 7 * q^56 + 5 * q^57 + 20 * q^58 + 5 * q^59 + 9 * q^60 + 2 * q^62 + 7 * q^63 - 3 * q^64 + 14 * q^65 + 12 * q^66 + 9 * q^67 + 29 * q^68 - 5 * q^69 + 10 * q^71 + 19 * q^72 - 50 * q^73 - 10 * q^74 + 24 * q^75 + 10 * q^76 + 9 * q^77 - 6 * q^78 + 8 * q^79 - 8 * q^81 + 37 * q^82 - 15 * q^83 + 6 * q^84 - q^85 + 12 * q^86 + 15 * q^87 + 11 * q^88 - 35 * q^89 + 8 * q^90 - 8 * q^91 + 5 * q^92 + 12 * q^93 - 4 * q^94 + 7 * q^95 + 7 * q^96 - 6 * q^97 + 6 * q^98 + 11 * q^99

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 $

$2$	−0.737640	−0.521590	−0.260795	−	0.965394i	$-0.583985\pi$
$2$	−0.737640	−0.521590	−0.260795	+	0.965394i	$0.583985\pi$
$3$	−2.35567	−1.36005	−0.680025	−	0.733189i	$-0.738031\pi$
$3$	−2.35567	−1.36005	−0.680025	+	0.733189i	$0.738031\pi$
$4$	−1.45589	−0.727943
$5$	0.355674	0.159062	0.0795312	−	0.996832i	$-0.474658\pi$
$5$	0.355674	0.159062	0.0795312	+	0.996832i	$0.474658\pi$
$6$	1.73764	0.709389
$7$	2.19353	0.829075	0.414538	−	0.910032i	$-0.363943\pi$
$7$	2.19353	0.829075	0.414538	+	0.910032i	$0.363943\pi$
$8$	2.54920	0.901279
$9$	2.54920	0.849734
$10$	−0.262360	−0.0829654
$11$	1.31313	0.395925	0.197962	−	0.980210i	$-0.436568\pi$
$11$	1.31313	0.395925	0.197962	+	0.980210i	$0.436568\pi$
$12$	3.42960	0.990039
$13$	−0.219819	−0.0609668	−0.0304834	−	0.999535i	$-0.509705\pi$
$13$	−0.219819	−0.0609668	−0.0304834	+	0.999535i	$0.509705\pi$
$14$	−1.61803	−0.432438
$15$	−0.837853	−0.216333
$16$	1.03138	0.257845
$17$	−4.72333	−1.14558	−0.572788	−	0.819703i	$-0.694138\pi$
$17$	−4.72333	−1.14558	−0.572788	+	0.819703i	$0.694138\pi$
$18$	−1.88039	−0.443213
$19$	−3.76902	−0.864673	−0.432336	−	0.901712i	$-0.642311\pi$
$19$	−3.76902	−0.864673	−0.432336	+	0.901712i	$0.642311\pi$
$20$	−0.517822	−0.115788
$21$	−5.16724	−1.12758
$22$	−0.968620	−0.206511
$23$	−0.150986	−0.0314828	−0.0157414	−	0.999876i	$-0.505011\pi$
$23$	−0.150986	−0.0314828	−0.0157414	+	0.999876i	$0.505011\pi$
$24$	−6.00509	−1.22578
$25$	−4.87350	−0.974699
$26$	0.162147	0.0317997
$27$	1.06193	0.204369
$28$	−3.19353	−0.603520
$29$	−1.01939	−0.189297	−0.0946483	−	0.995511i	$-0.530173\pi$
$29$	−1.01939	−0.189297	−0.0946483	+	0.995511i	$0.530173\pi$
$30$	0.618034	0.112837
$31$	−4.64433	−0.834146	−0.417073	−	0.908873i	$-0.636944\pi$
$31$	−4.64433	−0.834146	−0.417073	+	0.908873i	$0.636944\pi$
$32$	−5.85919	−1.03577
$33$	−3.09331	−0.538477
$34$	3.48412	0.597522
$35$	0.780181	0.131875
$36$	−3.71135	−0.618558
$37$	−1.24723	−0.205043	−0.102522	−	0.994731i	$-0.532691\pi$
$37$	−1.24723	−0.205043	−0.102522	+	0.994731i	$0.532691\pi$
$38$	2.78018	0.451005
$39$	0.517822	0.0829178
$40$	0.906685	0.143360
$41$	0.961212	0.150116	0.0750581	−	0.997179i	$-0.476086\pi$
$41$	0.961212	0.150116	0.0750581	+	0.997179i	$0.476086\pi$
$42$	3.81156	0.588137
$43$	−2.70626	−0.412701	−0.206350	−	0.978478i	$-0.566159\pi$
$43$	−2.70626	−0.412701	−0.206350	+	0.978478i	$0.566159\pi$
$44$	−1.91177	−0.288211
$45$	0.906685	0.135161
$46$	0.111374	0.0164211
$47$	−0.791342	−0.115429	−0.0577146	−	0.998333i	$-0.518381\pi$
$47$	−0.791342	−0.115429	−0.0577146	+	0.998333i	$0.518381\pi$
$48$	−2.42960	−0.350682
$49$	−2.18844	−0.312634
$50$	3.59489	0.508394
$51$	11.1266	1.55804
$52$	0.320031	0.0443804
$53$	−4.41446	−0.606373	−0.303187	−	0.952931i	$-0.598051\pi$
$53$	−4.41446	−0.606373	−0.303187	+	0.952931i	$0.598051\pi$
$54$	−0.783326	−0.106597
$55$	0.467048	0.0629767
$56$	5.59174	0.747228
$57$	8.87858	1.17600
$58$	0.751946	0.0987353
$59$	−4.46097	−0.580769	−0.290385	−	0.956910i	$-0.593783\pi$
$59$	−4.46097	−0.580769	−0.290385	+	0.956910i	$0.593783\pi$
$60$	1.21982	0.157478
$61$	−3.10155	−0.397112	−0.198556	−	0.980089i	$-0.563625\pi$
$61$	−3.10155	−0.397112	−0.198556	+	0.980089i	$0.563625\pi$
$62$	3.42584	0.435082
$63$	5.59174	0.704493
$64$	2.25922	0.282402
$65$	−0.0781839	−0.00969752
$66$	2.28175	0.280864
$67$	9.44584	1.15399	0.576997	−	0.816746i	$-0.304225\pi$
$67$	9.44584	1.15399	0.576997	+	0.816746i	$0.304225\pi$
$68$	6.87664	0.833915
$69$	0.355674	0.0428182
$70$	−0.575493	−0.0687846
$71$	2.60980	0.309726	0.154863	−	0.987936i	$-0.450506\pi$
$71$	2.60980	0.309726	0.154863	+	0.987936i	$0.450506\pi$
$72$	6.49843	0.765847
$73$	−10.8065	−1.26480	−0.632401	−	0.774641i	$-0.717930\pi$
$73$	−10.8065	−1.26480	−0.632401	+	0.774641i	$0.717930\pi$
$74$	0.920006	0.106949
$75$	11.4804	1.32564
$76$	5.48727	0.629433
$77$	2.88039	0.328251
$78$	−0.381966	−0.0432491
$79$	−4.27546	−0.481027	−0.240514	−	0.970646i	$-0.577316\pi$
$79$	−4.27546	−0.481027	−0.240514	+	0.970646i	$0.577316\pi$
$80$	0.366835	0.0410134
$81$	−10.1492	−1.12769
$82$	−0.709029	−0.0782992
$83$	5.53916	0.608002	0.304001	−	0.952672i	$-0.401677\pi$
$83$	5.53916	0.608002	0.304001	+	0.952672i	$0.401677\pi$
$84$	7.52291	0.820817
$85$	−1.67997	−0.182218
$86$	1.99625	0.215261
$87$	2.40136	0.257453
$88$	3.34744	0.356838
$89$	−14.6037	−1.54799	−0.773996	−	0.633190i	$-0.781745\pi$
$89$	−14.6037	−1.54799	−0.773996	+	0.633190i	$0.781745\pi$
$90$	−0.668808	−0.0704985
$91$	−0.482178	−0.0505460
$92$	0.219819	0.0229177
$93$	10.9405	1.13448
$94$	0.583726	0.0602067
$95$	−1.34054	−0.137537
$96$	13.8023	1.40870
$97$	−9.16806	−0.930875	−0.465438	−	0.885081i	$-0.654103\pi$
$97$	−9.16806	−0.930875	−0.465438	+	0.885081i	$0.654103\pi$
$98$	1.61428	0.163067
$99$	3.34744	0.336431
$100$	7.09526	0.709526
$101$	5.98194	0.595225	0.297613	−	0.954687i	$-0.403810\pi$
$101$	5.98194	0.595225	0.297613	+	0.954687i	$0.403810\pi$
$102$	−8.20746	−0.812659
$103$	−10.5400	−1.03854	−0.519268	−	0.854612i	$-0.673795\pi$
$103$	−10.5400	−1.03854	−0.519268	+	0.854612i	$0.673795\pi$
$104$	−0.560362	−0.0549481
$105$	−1.83785	−0.179356
$106$	3.25629	0.316279
$107$	10.8015	1.04422	0.522111	−	0.852877i	$-0.325145\pi$
$107$	10.8015	1.04422	0.522111	+	0.852877i	$0.325145\pi$
$108$	−1.54606	−0.148769
$109$	−19.3739	−1.85568	−0.927840	−	0.372979i	$-0.878336\pi$
$109$	−19.3739	−1.85568	−0.927840	+	0.372979i	$0.878336\pi$
$110$	−0.344513	−0.0328481
$111$	2.93807	0.278869
$112$	2.26236	0.213773
$113$	11.7803	1.10820	0.554099	−	0.832451i	$-0.313063\pi$
$113$	11.7803	1.10820	0.554099	+	0.832451i	$0.313063\pi$
$114$	−6.54920	−0.613389
$115$	−0.0537019	−0.00500773
$116$	1.48412	0.137797
$117$	−0.560362	−0.0518055
$118$	3.29059	0.302924
$119$	−10.3608	−0.949770
$120$	−2.13586	−0.194976
$121$	−9.27568	−0.843244
$122$	2.28783	0.207130
$123$	−2.26430	−0.204165
$124$	6.76161	0.607211
$125$	−3.51175	−0.314100
$126$	−4.12469	−0.367457
$127$	11.0823	0.983394	0.491697	−	0.870766i	$-0.336377\pi$
$127$	11.0823	0.983394	0.491697	+	0.870766i	$0.336377\pi$
$128$	10.0519	0.888470
$129$	6.37507	0.561293
$130$	0.0576716	0.00505813
$131$	9.41103	0.822245	0.411123	−	0.911580i	$-0.365137\pi$
$131$	9.41103	0.822245	0.411123	+	0.911580i	$0.365137\pi$
$132$	4.50352	0.391981
$133$	−8.26745	−0.716879
$134$	−6.96764	−0.601912
$135$	0.377703	0.0325075
$136$	−12.0407	−1.03248
$137$	14.4355	1.23331	0.616654	−	0.787234i	$-0.288488\pi$
$137$	14.4355	1.23331	0.616654	+	0.787234i	$0.288488\pi$
$138$	−0.262360	−0.0223335
$139$	−6.29486	−0.533923	−0.266961	−	0.963707i	$-0.586020\pi$
$139$	−6.29486	−0.533923	−0.266961	+	0.963707i	$0.586020\pi$
$140$	−1.13586	−0.0959973
$141$	1.86414	0.156989
$142$	−1.92509	−0.161550
$143$	−0.288651	−0.0241382
$144$	2.62920	0.219100
$145$	−0.362572	−0.0301100
$146$	7.97129	0.659709
$147$	5.15525	0.425198
$148$	1.81582	0.149260
$149$	1.27486	0.104440	0.0522201	−	0.998636i	$-0.483370\pi$
$149$	1.27486	0.104440	0.0522201	+	0.998636i	$0.483370\pi$
$150$	−8.46838	−0.691441
$151$	−13.7654	−1.12021	−0.560106	−	0.828421i	$-0.689240\pi$
$151$	−13.7654	−1.12021	−0.560106	+	0.828421i	$0.689240\pi$
$152$	−9.60799	−0.779311
$153$	−12.0407	−0.973435
$154$	−2.12469	−0.171213
$155$	−1.65187	−0.132681
$156$	−0.753889	−0.0603595
$157$	0.224691	0.0179323	0.00896613	−	0.999960i	$-0.497146\pi$
$157$	0.224691	0.0179323	0.00896613	+	0.999960i	$0.497146\pi$
$158$	3.15375	0.250899
$159$	10.3990	0.824698
$160$	−2.08396	−0.164752
$161$	−0.331192	−0.0261016
$162$	7.48644	0.588190
$163$	10.7653	0.843201	0.421600	−	0.906782i	$-0.361469\pi$
$163$	10.7653	0.843201	0.421600	+	0.906782i	$0.361469\pi$
$164$	−1.39942	−0.109276
$165$	−1.10021	−0.0856514
$166$	−4.08591	−0.317128
$167$	5.99953	0.464257	0.232129	−	0.972685i	$-0.425431\pi$
$167$	5.99953	0.464257	0.232129	+	0.972685i	$0.425431\pi$
$168$	−13.1723	−1.01627
$169$	−12.9517	−0.996283
$170$	1.23921	0.0950433
$171$	−9.60799	−0.734741
$172$	3.94001	0.300423
$173$	−3.24856	−0.246984	−0.123492	−	0.992346i	$-0.539409\pi$
$173$	−3.24856	−0.246984	−0.123492	+	0.992346i	$0.539409\pi$
$174$	−1.77134	−0.134285
$175$	−10.6901	−0.808099
$176$	1.35434	0.102087
$177$	10.5086	0.789875
$178$	10.7723	0.807418
$179$	−4.62709	−0.345845	−0.172923	−	0.984935i	$-0.555321\pi$
$179$	−4.62709	−0.345845	−0.172923	+	0.984935i	$0.555321\pi$
$180$	−1.32003	−0.0983893
$181$	−9.89238	−0.735295	−0.367647	−	0.929965i	$-0.619837\pi$
$181$	−9.89238	−0.735295	−0.367647	+	0.929965i	$0.619837\pi$
$182$	0.355674	0.0263643
$183$	7.30624	0.540092
$184$	−0.384894	−0.0283748
$185$	−0.443607	−0.0326147
$186$	−8.07017	−0.591733
$187$	−6.20237	−0.453562
$188$	1.15210	0.0840259
$189$	2.32938	0.169438
$190$	0.988839	0.0717379
$191$	8.84323	0.639874	0.319937	−	0.947439i	$-0.396338\pi$
$191$	8.84323	0.639874	0.319937	+	0.947439i	$0.396338\pi$
$192$	−5.32197	−0.384080
$193$	9.17608	0.660508	0.330254	−	0.943892i	$-0.392866\pi$
$193$	9.17608	0.660508	0.330254	+	0.943892i	$0.392866\pi$
$194$	6.76273	0.485536
$195$	0.184176	0.0131891
$196$	3.18612	0.227580
$197$	−0.248564	−0.0177094	−0.00885472	−	0.999961i	$-0.502819\pi$
$197$	−0.248564	−0.0177094	−0.00885472	+	0.999961i	$0.502819\pi$
$198$	−2.46921	−0.175479
$199$	5.15802	0.365642	0.182821	−	0.983146i	$-0.441477\pi$
$199$	5.15802	0.365642	0.182821	+	0.983146i	$0.441477\pi$
$200$	−12.4235	−0.878476
$201$	−22.2513	−1.56949
$202$	−4.41252	−0.310464
$203$	−2.23607	−0.156941
$204$	−16.1991	−1.13417
$205$	0.341879	0.0238778
$206$	7.77472	0.541690
$207$	−0.384894	−0.0267520
$208$	−0.226717	−0.0157200
$209$	−4.94923	−0.342345
$210$	1.35567	0.0935504
$211$	17.5140	1.20571	0.602856	−	0.797850i	$-0.294029\pi$
$211$	17.5140	1.20571	0.602856	+	0.797850i	$0.294029\pi$
$212$	6.42696	0.441405
$213$	−6.14784	−0.421243
$214$	−7.96764	−0.544656
$215$	−0.962547	−0.0656452
$216$	2.70709	0.184194
$217$	−10.1875	−0.691569
$218$	14.2909	0.967905
$219$	25.4565	1.72019
$220$	−0.679969	−0.0458435
$221$	1.03828	0.0698421
$222$	−2.16724	−0.145455
$223$	−18.8858	−1.26469	−0.632344	−	0.774687i	$-0.717907\pi$
$223$	−18.8858	−1.26469	−0.632344	+	0.774687i	$0.717907\pi$
$224$	−12.8523	−0.858730
$225$	−12.4235	−0.828235
$226$	−8.68964	−0.578026
$227$	4.29688	0.285194	0.142597	−	0.989781i	$-0.454455\pi$
$227$	4.29688	0.285194	0.142597	+	0.989781i	$0.454455\pi$
$228$	−12.9262	−0.856059
$229$	5.94217	0.392670	0.196335	−	0.980537i	$-0.437096\pi$
$229$	5.94217	0.392670	0.196335	+	0.980537i	$0.437096\pi$
$230$	0.0396127	0.00261198
$231$	−6.78527	−0.446438
$232$	−2.59864	−0.170609
$233$	2.48037	0.162494	0.0812472	−	0.996694i	$-0.474110\pi$
$233$	2.48037	0.162494	0.0812472	+	0.996694i	$0.474110\pi$
$234$	0.413346	0.0270213
$235$	−0.281460	−0.0183604
$236$	6.49467	0.422767
$237$	10.0716	0.654221
$238$	7.64252	0.495391
$239$	−20.0791	−1.29881	−0.649406	−	0.760442i	$-0.724982\pi$
$239$	−20.0791	−1.29881	−0.649406	+	0.760442i	$0.724982\pi$
$240$	−0.864145	−0.0557803
$241$	17.7876	1.14580	0.572899	−	0.819626i	$-0.305819\pi$
$241$	17.7876	1.14580	0.572899	+	0.819626i	$0.305819\pi$
$242$	6.84212	0.439828
$243$	20.7223	1.32934
$244$	4.51550	0.289075
$245$	−0.778371	−0.0497283
$246$	1.67024	0.106491
$247$	0.828502	0.0527163
$248$	−11.8393	−0.751798
$249$	−13.0485	−0.826912
$250$	2.59041	0.163832
$251$	−7.46524	−0.471202	−0.235601	−	0.971850i	$-0.575706\pi$
$251$	−7.46524	−0.471202	−0.235601	+	0.971850i	$0.575706\pi$
$252$	−8.14094	−0.512831
$253$	−0.198265	−0.0124648
$254$	−8.17474	−0.512929
$255$	3.95746	0.247826
$256$	−11.9331	−0.745819
$257$	−3.16460	−0.197402	−0.0987012	−	0.995117i	$-0.531469\pi$
$257$	−3.16460	−0.197402	−0.0987012	+	0.995117i	$0.531469\pi$
$258$	−4.70251	−0.292765
$259$	−2.73583	−0.169996
$260$	0.113827	0.00705924
$261$	−2.59864	−0.160852
$262$	−6.94195	−0.428875
$263$	−7.20962	−0.444564	−0.222282	−	0.974982i	$-0.571351\pi$
$263$	−7.20962	−0.444564	−0.222282	+	0.974982i	$0.571351\pi$
$264$	−7.88548	−0.485318
$265$	−1.57011	−0.0964512
$266$	6.09840	0.373917
$267$	34.4016	2.10535
$268$	−13.7521	−0.840042
$269$	1.81673	0.110768	0.0553840	−	0.998465i	$-0.482362\pi$
$269$	1.81673	0.110768	0.0553840	+	0.998465i	$0.482362\pi$
$270$	−0.278609	−0.0169556
$271$	15.4315	0.937399	0.468700	−	0.883358i	$-0.344723\pi$
$271$	15.4315	0.937399	0.468700	+	0.883358i	$0.344723\pi$
$272$	−4.87155	−0.295381
$273$	1.13586	0.0687451
$274$	−10.6482	−0.643282
$275$	−6.39955	−0.385907
$276$	−0.517822	−0.0311692
$277$	−3.68852	−0.221621	−0.110811	−	0.993842i	$-0.535345\pi$
$277$	−3.68852	−0.221621	−0.110811	+	0.993842i	$0.535345\pi$
$278$	4.64334	0.278489
$279$	−11.8393	−0.708802
$280$	1.98884	0.118856
$281$	−28.8380	−1.72033	−0.860165	−	0.510016i	$-0.829640\pi$
$281$	−28.8380	−1.72033	−0.860165	+	0.510016i	$0.829640\pi$
$282$	−1.37507	−0.0818841
$283$	−19.5214	−1.16043	−0.580214	−	0.814464i	$-0.697031\pi$
$283$	−19.5214	−1.16043	−0.580214	+	0.814464i	$0.697031\pi$
$284$	−3.79958	−0.225463
$285$	3.15788	0.187057
$286$	0.212921	0.0125903
$287$	2.10845	0.124458
$288$	−14.9363	−0.880127
$289$	5.30989	0.312346
$290$	0.267448	0.0157051
$291$	21.5970	1.26604
$292$	15.7330	0.920704
$293$	19.2030	1.12185	0.560926	−	0.827866i	$-0.310445\pi$
$293$	19.2030	1.12185	0.560926	+	0.827866i	$0.310445\pi$
$294$	−3.80272	−0.221779
$295$	−1.58665	−0.0923786
$296$	−3.17944	−0.184801
$297$	1.39446	0.0809149
$298$	−0.940385	−0.0544750
$299$	0.0331896	0.00191940
$300$	−16.7141	−0.964990
$301$	−5.93626	−0.342160
$302$	10.1539	0.584292
$303$	−14.0915	−0.809536
$304$	−3.88729	−0.222951
$305$	−1.10314	−0.0631657
$306$	8.88173	0.507735
$307$	8.44718	0.482106	0.241053	−	0.970512i	$-0.422507\pi$
$307$	8.44718	0.482106	0.241053	+	0.970512i	$0.422507\pi$
$308$	−4.19353	−0.238948
$309$	24.8288	1.41246
$310$	1.21848	0.0692052
$311$	24.4490	1.38637	0.693187	−	0.720758i	$-0.256206\pi$
$311$	24.4490	1.38637	0.693187	+	0.720758i	$0.256206\pi$
$312$	1.32003	0.0747321
$313$	−4.86660	−0.275076	−0.137538	−	0.990496i	$-0.543919\pi$
$313$	−4.86660	−0.275076	−0.137538	+	0.990496i	$0.543919\pi$
$314$	−0.165741	−0.00935329
$315$	1.98884	0.112058
$316$	6.22459	0.350161
$317$	34.5036	1.93792	0.968959	−	0.247221i	$-0.0795175\pi$
$317$	34.5036	1.93792	0.968959	+	0.247221i	$0.0795175\pi$
$318$	−7.67075	−0.430154
$319$	−1.33860	−0.0749472
$320$	0.803545	0.0449195
$321$	−25.4449	−1.42019
$322$	0.244301	0.0136144
$323$	17.8023	0.990549
$324$	14.7761	0.820892
$325$	1.07129	0.0594243
$326$	−7.94089	−0.439805
$327$	45.6385	2.52382
$328$	2.45032	0.135296
$329$	−1.73583	−0.0956994
$330$	0.811561	0.0446750
$331$	−5.74204	−0.315611	−0.157805	−	0.987470i	$-0.550442\pi$
$331$	−5.74204	−0.315611	−0.157805	+	0.987470i	$0.550442\pi$
$332$	−8.06439	−0.442591
$333$	−3.17944	−0.174232
$334$	−4.42549	−0.242152
$335$	3.35964	0.183557
$336$	−5.32938	−0.290742
$337$	−12.3242	−0.671343	−0.335671	−	0.941979i	$-0.608963\pi$
$337$	−12.3242	−0.671343	−0.335671	+	0.941979i	$0.608963\pi$
$338$	9.55368	0.519652
$339$	−27.7506	−1.50720
$340$	2.44584	0.132645
$341$	−6.09862	−0.330259
$342$	7.08724	0.383234
$343$	−20.1551	−1.08827
$344$	−6.89880	−0.371959
$345$	0.126504	0.00681076
$346$	2.39627	0.128824
$347$	11.8379	0.635489	0.317745	−	0.948176i	$-0.397075\pi$
$347$	11.8379	0.635489	0.317745	+	0.948176i	$0.397075\pi$
$348$	−3.49611	−0.187411
$349$	0.350720	0.0187736	0.00938680	−	0.999956i	$-0.497012\pi$
$349$	0.350720	0.0187736	0.00938680	+	0.999956i	$0.497012\pi$
$350$	7.88548	0.421497
$351$	−0.233433	−0.0124597
$352$	−7.69390	−0.410086
$353$	7.12371	0.379157	0.189578	−	0.981866i	$-0.439288\pi$
$353$	7.12371	0.379157	0.189578	+	0.981866i	$0.439288\pi$
$354$	−7.75157	−0.411991
$355$	0.928239	0.0492658
$356$	21.2614	1.12685
$357$	24.4066	1.29173
$358$	3.41313	0.180389
$359$	−19.6384	−1.03647	−0.518237	−	0.855237i	$-0.673412\pi$
$359$	−19.6384	−1.03647	−0.518237	+	0.855237i	$0.673412\pi$
$360$	2.31132	0.121817
$361$	−4.79449	−0.252341
$362$	7.29702	0.383523
$363$	21.8505	1.14685
$364$	0.701997	0.0367947
$365$	−3.84358	−0.201182
$366$	−5.38937	−0.281707
$367$	−1.92083	−0.100267	−0.0501333	−	0.998743i	$-0.515965\pi$
$367$	−1.92083	−0.100267	−0.0501333	+	0.998743i	$0.515965\pi$
$368$	−0.155724	−0.00811768
$369$	2.45032	0.127559
$370$	0.327223	0.0170115
$371$	−9.68325	−0.502729
$372$	−15.9282	−0.825836
$373$	38.1400	1.97481	0.987406	−	0.158206i	$-0.0505710\pi$
$373$	38.1400	1.97481	0.987406	+	0.158206i	$0.0505710\pi$
$374$	4.57512	0.236574
$375$	8.27254	0.427192
$376$	−2.01729	−0.104034
$377$	0.224082	0.0115408
$378$	−1.71825	−0.0883771
$379$	37.7074	1.93690	0.968448	−	0.249215i	$-0.0801724\pi$
$379$	37.7074	1.93690	0.968448	+	0.249215i	$0.0801724\pi$
$380$	1.95168	0.100119
$381$	−26.1063	−1.33746
$382$	−6.52313	−0.333752
$383$	7.78393	0.397740	0.198870	−	0.980026i	$-0.436273\pi$
$383$	7.78393	0.397740	0.198870	+	0.980026i	$0.436273\pi$
$384$	−23.6790	−1.20836
$385$	1.02448	0.0522124
$386$	−6.76864	−0.344515
$387$	−6.89880	−0.350686
$388$	13.3477	0.677625
$389$	−15.3669	−0.779132	−0.389566	−	0.920998i	$-0.627375\pi$
$389$	−15.3669	−0.779132	−0.389566	+	0.920998i	$0.627375\pi$
$390$	−0.135855	−0.00687931
$391$	0.713158	0.0360660
$392$	−5.57877	−0.281771
$393$	−22.1693	−1.11829
$394$	0.183351	0.00923708
$395$	−1.52067	−0.0765133
$396$	−4.87350	−0.244902
$397$	8.41124	0.422148	0.211074	−	0.977470i	$-0.432304\pi$
$397$	8.41124	0.422148	0.211074	+	0.977470i	$0.432304\pi$
$398$	−3.80476	−0.190715
$399$	19.4754	0.974990
$400$	−5.02643	−0.251321
$401$	35.1299	1.75430	0.877152	−	0.480213i	$-0.159440\pi$
$401$	35.1299	1.75430	0.877152	+	0.480213i	$0.159440\pi$
$402$	16.4135	0.818630
$403$	1.02091	0.0508552
$404$	−8.70903	−0.433290
$405$	−3.60980	−0.179372
$406$	1.64941	0.0818590
$407$	−1.63778	−0.0811816
$408$	28.3640	1.40423
$409$	30.9050	1.52816	0.764078	−	0.645124i	$-0.223194\pi$
$409$	30.9050	1.52816	0.764078	+	0.645124i	$0.223194\pi$
$410$	−0.252183	−0.0124544
$411$	−34.0054	−1.67736
$412$	15.3450	0.755995
$413$	−9.78527	−0.481502
$414$	0.283913	0.0139536
$415$	1.97014	0.0967102
$416$	1.28796	0.0631474
$417$	14.8286	0.726161
$418$	3.65075	0.178564
$419$	34.1501	1.66834	0.834172	−	0.551505i	$-0.185946\pi$
$419$	34.1501	1.66834	0.834172	+	0.551505i	$0.185946\pi$
$420$	2.67571	0.130561
$421$	−10.1575	−0.495048	−0.247524	−	0.968882i	$-0.579617\pi$
$421$	−10.1575	−0.495048	−0.247524	+	0.968882i	$0.579617\pi$
$422$	−12.9190	−0.628888
$423$	−2.01729	−0.0980840
$424$	−11.2534	−0.546511
$425$	23.0192	1.11659
$426$	4.53490	0.219716
$427$	−6.80333	−0.329236
$428$	−15.7258	−0.760135
$429$	0.679969	0.0328292
$430$	0.710014	0.0342399
$431$	−1.00000	−0.0481683
$431$	−1.00000	−0.0481683
$432$	1.09526	0.0526956
$433$	5.06771	0.243539	0.121769	−	0.992558i	$-0.461143\pi$
$433$	5.06771	0.243539	0.121769	+	0.992558i	$0.461143\pi$
$434$	7.51468	0.360716
$435$	0.854102	0.0409511
$436$	28.2062	1.35083
$437$	0.569070	0.0272223
$438$	−18.7778	−0.897236
$439$	−21.0411	−1.00424	−0.502118	−	0.864799i	$-0.667446\pi$
$439$	−21.0411	−1.00424	−0.502118	+	0.864799i	$0.667446\pi$
$440$	1.19060	0.0567596
$441$	−5.57877	−0.265656
$442$	−0.765876	−0.0364290
$443$	16.4981	0.783846	0.391923	−	0.919998i	$-0.371810\pi$
$443$	16.4981	0.783846	0.391923	+	0.919998i	$0.371810\pi$
$444$	−4.27749	−0.203001
$445$	−5.19417	−0.246227
$446$	13.9310	0.659650
$447$	−3.00314	−0.142044
$448$	4.95565	0.234132
$449$	30.4997	1.43937	0.719684	−	0.694301i	$-0.244286\pi$
$449$	30.4997	1.43937	0.719684	+	0.694301i	$0.244286\pi$
$450$	9.16409	0.431999
$451$	1.26220	0.0594347
$452$	−17.1508	−0.806706
$453$	32.4268	1.52354
$454$	−3.16955	−0.148755
$455$	−0.171498	−0.00803997
$456$	22.6333	1.05990
$457$	11.8416	0.553929	0.276964	−	0.960880i	$-0.410672\pi$
$457$	11.8416	0.553929	0.276964	+	0.960880i	$0.410672\pi$
$458$	−4.38318	−0.204813
$459$	−5.01587	−0.234121
$460$	0.0781839	0.00364534
$461$	−11.9837	−0.558138	−0.279069	−	0.960271i	$-0.590026\pi$
$461$	−11.9837	−0.558138	−0.279069	+	0.960271i	$0.590026\pi$
$462$	5.00509	0.232858
$463$	4.46877	0.207682	0.103841	−	0.994594i	$-0.466887\pi$
$463$	4.46877	0.207682	0.103841	+	0.994594i	$0.466887\pi$
$464$	−1.05138	−0.0488092
$465$	3.89126	0.180453
$466$	−1.82962	−0.0847555
$467$	−29.9185	−1.38446	−0.692231	−	0.721676i	$-0.743372\pi$
$467$	−29.9185	−1.38446	−0.692231	+	0.721676i	$0.743372\pi$
$468$	0.815824	0.0377115
$469$	20.7197	0.956748
$470$	0.207616	0.00957663
$471$	−0.529298	−0.0243887
$472$	−11.3719	−0.523435
$473$	−3.55368	−0.163398
$474$	−7.42922	−0.341235
$475$	18.3683	0.842796
$476$	15.0841	0.691378
$477$	−11.2534	−0.515256
$478$	14.8112	0.677448
$479$	−33.6205	−1.53616	−0.768080	−	0.640354i	$-0.778788\pi$
$479$	−33.6205	−1.53616	−0.768080	+	0.640354i	$0.778788\pi$
$480$	4.90914	0.224071
$481$	0.274164	0.0125008
$482$	−13.1208	−0.597637
$483$	0.780181	0.0354995
$484$	13.5043	0.613834
$485$	−3.26084	−0.148067
$486$	−15.2856	−0.693371
$487$	−7.66808	−0.347474	−0.173737	−	0.984792i	$-0.555584\pi$
$487$	−7.66808	−0.347474	−0.173737	+	0.984792i	$0.555584\pi$
$488$	−7.90647	−0.357909
$489$	−25.3595	−1.14679
$490$	0.574158	0.0259378
$491$	−23.7671	−1.07259	−0.536296	−	0.844030i	$-0.680177\pi$
$491$	−23.7671	−1.07259	−0.536296	+	0.844030i	$0.680177\pi$
$492$	3.29657	0.148621
$493$	4.81494	0.216854
$494$	−0.611136	−0.0274963
$495$	1.19060	0.0535134
$496$	−4.79006	−0.215080
$497$	5.72467	0.256787
$498$	9.62507	0.431310
$499$	29.6770	1.32853	0.664263	−	0.747499i	$-0.268746\pi$
$499$	29.6770	1.32853	0.664263	+	0.747499i	$0.268746\pi$
$500$	5.11271	0.228647
$501$	−14.1329	−0.631413
$502$	5.50666	0.245774
$503$	−36.5805	−1.63105	−0.815523	−	0.578725i	$-0.803550\pi$
$503$	−36.5805	−1.63105	−0.815523	+	0.578725i	$0.803550\pi$
$504$	14.2545	0.634945
$505$	2.12762	0.0946780
$506$	0.146248	0.00650153
$507$	30.5099	1.35499
$508$	−16.1346	−0.715855
$509$	−38.2366	−1.69481	−0.847403	−	0.530951i	$-0.821835\pi$
$509$	−38.2366	−1.69481	−0.847403	+	0.530951i	$0.821835\pi$
$510$	−2.91918	−0.129264
$511$	−23.7043	−1.04862
$512$	−11.3014	−0.499458
$513$	−4.00245	−0.176713
$514$	2.33434	0.102963
$515$	−3.74880	−0.165192
$516$	−9.28138	−0.408590
$517$	−1.03914	−0.0457012
$518$	2.01806	0.0886684
$519$	7.65256	0.335910
$520$	−0.199306	−0.00874017
$521$	−31.7558	−1.39125	−0.695624	−	0.718406i	$-0.744872\pi$
$521$	−31.7558	−1.39125	−0.695624	+	0.718406i	$0.744872\pi$
$522$	1.91686	0.0838988
$523$	4.30013	0.188031	0.0940157	−	0.995571i	$-0.470030\pi$
$523$	4.30013	0.188031	0.0940157	+	0.995571i	$0.470030\pi$
$524$	−13.7014	−0.598548
$525$	25.1825	1.09905
$526$	5.31810	0.231880
$527$	21.9367	0.955578
$528$	−3.19038	−0.138844
$529$	−22.9772	−0.999009
$530$	1.15818	0.0503080
$531$	−11.3719	−0.493499
$532$	12.0365	0.521847
$533$	−0.211293	−0.00915210
$534$	−25.3760	−1.09813
$535$	3.84182	0.166096
$536$	24.0794	1.04007
$537$	10.8999	0.470366
$538$	−1.34009	−0.0577756
$539$	−2.87371	−0.123780
$540$	−0.549893	−0.0236636
$541$	−25.1363	−1.08069	−0.540347	−	0.841443i	$-0.681707\pi$
$541$	−25.1363	−1.08069	−0.540347	+	0.841443i	$0.681707\pi$
$542$	−11.3829	−0.488938
$543$	23.3032	1.00004
$544$	27.6749	1.18655
$545$	−6.89079	−0.295169
$546$	−0.837853	−0.0358568
$547$	−12.8712	−0.550333	−0.275167	−	0.961397i	$-0.588733\pi$
$547$	−12.8712	−0.550333	−0.275167	+	0.961397i	$0.588733\pi$
$548$	−21.0165	−0.897779
$549$	−7.90647	−0.337440
$550$	4.72057	0.201286
$551$	3.84212	0.163680
$552$	0.906685	0.0385911
$553$	−9.37835	−0.398808
$554$	2.72080	0.115596
$555$	1.04499	0.0443575
$556$	9.16460	0.388666
$557$	−3.93164	−0.166589	−0.0832945	−	0.996525i	$-0.526544\pi$
$557$	−3.93164	−0.166589	−0.0832945	+	0.996525i	$0.526544\pi$
$558$	8.73316	0.369704
$559$	0.594887	0.0251610
$560$	0.804663	0.0340032
$561$	14.6108	0.616867
$562$	21.2721	0.897308
$563$	−17.0326	−0.717838	−0.358919	−	0.933369i	$-0.616855\pi$
$563$	−17.0326	−0.717838	−0.358919	+	0.933369i	$0.616855\pi$
$564$	−2.71398	−0.114279
$565$	4.18996	0.176273
$566$	14.3998	0.605268
$567$	−22.2625	−0.934937
$568$	6.65291	0.279150
$569$	18.0132	0.755154	0.377577	−	0.925978i	$-0.376757\pi$
$569$	18.0132	0.755154	0.377577	+	0.925978i	$0.376757\pi$
$570$	−2.32938	−0.0975671
$571$	−30.4440	−1.27404	−0.637020	−	0.770847i	$-0.719833\pi$
$571$	−30.4440	−1.27404	−0.637020	+	0.770847i	$0.719833\pi$
$572$	0.420244	0.0175713
$573$	−20.8318	−0.870260
$574$	−1.55527	−0.0649159
$575$	0.735831	0.0306863
$576$	5.75919	0.239966
$577$	34.3629	1.43055	0.715273	−	0.698846i	$-0.246303\pi$
$577$	34.3629	1.43055	0.715273	+	0.698846i	$0.246303\pi$
$578$	−3.91679	−0.162917
$579$	−21.6158	−0.898324
$580$	0.527864	0.0219184
$581$	12.1503	0.504079
$582$	−15.9308	−0.660353
$583$	−5.79678	−0.240078
$584$	−27.5479	−1.13994
$585$	−0.199306	−0.00824031
$586$	−14.1649	−0.585147
$587$	−35.3347	−1.45842	−0.729209	−	0.684291i	$-0.760112\pi$
$587$	−35.3347	−1.45842	−0.729209	+	0.684291i	$0.760112\pi$
$588$	−7.50546	−0.309520
$589$	17.5046	0.721263
$590$	1.17038	0.0481838
$591$	0.585536	0.0240857
$592$	−1.28637	−0.0528694
$593$	−9.31148	−0.382377	−0.191188	−	0.981553i	$-0.561234\pi$
$593$	−9.31148	−0.382377	−0.191188	+	0.981553i	$0.561234\pi$
$594$	−1.02861	−0.0422044
$595$	−3.68506	−0.151073
$596$	−1.85605	−0.0760266
$597$	−12.1506	−0.497291
$598$	−0.0244820	−0.00100114
$599$	15.9577	0.652013	0.326006	−	0.945368i	$-0.394297\pi$
$599$	15.9577	0.652013	0.326006	+	0.945368i	$0.394297\pi$
$600$	29.2658	1.19477
$601$	−40.6737	−1.65912	−0.829558	−	0.558421i	$-0.811407\pi$
$601$	−40.6737	−1.65912	−0.829558	+	0.558421i	$0.811407\pi$
$602$	4.37882	0.178467
$603$	24.0794	0.980587
$604$	20.0409	0.815451
$605$	−3.29912	−0.134128
$606$	10.3945	0.422246
$607$	−30.2347	−1.22719	−0.613594	−	0.789621i	$-0.710277\pi$
$607$	−30.2347	−1.22719	−0.613594	+	0.789621i	$0.710277\pi$
$608$	22.0834	0.895600
$609$	5.26745	0.213448
$610$	0.813721	0.0329466
$611$	0.173952	0.00703734
$612$	17.5299	0.708606
$613$	13.2408	0.534789	0.267395	−	0.963587i	$-0.413837\pi$
$613$	13.2408	0.534789	0.267395	+	0.963587i	$0.413837\pi$
$614$	−6.23098	−0.251462
$615$	−0.805354	−0.0324750
$616$	7.34270	0.295846
$617$	4.31606	0.173758	0.0868790	−	0.996219i	$-0.472311\pi$
$617$	4.31606	0.173758	0.0868790	+	0.996219i	$0.472311\pi$
$618$	−18.3147	−0.736725
$619$	−8.65208	−0.347757	−0.173878	−	0.984767i	$-0.555630\pi$
$619$	−8.65208	−0.347757	−0.173878	+	0.984767i	$0.555630\pi$
$620$	2.40493	0.0965844
$621$	−0.160337	−0.00643412
$622$	−18.0345	−0.723119
$623$	−32.0337	−1.28340
$624$	0.534071	0.0213799
$625$	23.1184	0.924738
$626$	3.58980	0.143477
$627$	11.6588	0.465606
$628$	−0.327124	−0.0130537
$629$	5.89108	0.234893
$630$	−1.46705	−0.0584486
$631$	4.03074	0.160461	0.0802305	−	0.996776i	$-0.474434\pi$
$631$	4.03074	0.160461	0.0802305	+	0.996776i	$0.474434\pi$
$632$	−10.8990	−0.433540
$633$	−41.2572	−1.63983
$634$	−25.4513	−1.01080
$635$	3.94168	0.156421
$636$	−15.1398	−0.600333
$637$	0.481060	0.0190603
$638$	0.987405	0.0390918
$639$	6.65291	0.263185
$640$	3.57520	0.141322
$641$	−42.7002	−1.68656	−0.843279	−	0.537476i	$-0.819378\pi$
$641$	−42.7002	−1.68656	−0.843279	+	0.537476i	$0.819378\pi$
$642$	18.7692	0.740759
$643$	−2.51260	−0.0990873	−0.0495436	−	0.998772i	$-0.515777\pi$
$643$	−2.51260	−0.0990873	−0.0495436	+	0.998772i	$0.515777\pi$
$644$	0.482178	0.0190005
$645$	2.26745	0.0892807
$646$	−13.1317	−0.516661
$647$	−5.08862	−0.200054	−0.100027	−	0.994985i	$-0.531893\pi$
$647$	−5.08862	−0.200054	−0.100027	+	0.994985i	$0.531893\pi$
$648$	−25.8723	−1.01636
$649$	−5.85786	−0.229941
$650$	−0.790224	−0.0309951
$651$	23.9983	0.940568
$652$	−15.6730	−0.613802
$653$	−43.4653	−1.70093	−0.850465	−	0.526031i	$-0.823679\pi$
$653$	−43.4653	−1.70093	−0.850465	+	0.526031i	$0.823679\pi$
$654$	−33.6648	−1.31640
$655$	3.34726	0.130788
$656$	0.991375	0.0387067
$657$	−27.5479	−1.07474
$658$	1.28042	0.0499159
$659$	0.206667	0.00805059	0.00402530	−	0.999992i	$-0.498719\pi$
$659$	0.206667	0.00805059	0.00402530	+	0.999992i	$0.498719\pi$
$660$	1.60178	0.0623494
$661$	35.9230	1.39724	0.698621	−	0.715492i	$-0.253797\pi$
$661$	35.9230	1.39724	0.698621	+	0.715492i	$0.253797\pi$
$662$	4.23556	0.164620
$663$	−2.44584	−0.0949887
$664$	14.1204	0.547979
$665$	−2.94052	−0.114028
$666$	2.34528	0.0908778
$667$	0.153914	0.00595959
$668$	−8.73463	−0.337953
$669$	44.4889	1.72004
$670$	−2.47821	−0.0957416
$671$	−4.07275	−0.157227
$672$	30.2758	1.16791
$673$	35.1183	1.35371	0.676855	−	0.736116i	$-0.263342\pi$
$673$	35.1183	1.35371	0.676855	+	0.736116i	$0.263342\pi$
$674$	9.09084	0.350166
$675$	−5.17533	−0.199199
$676$	18.8562	0.725238
$677$	−5.23428	−0.201170	−0.100585	−	0.994928i	$-0.532071\pi$
$677$	−5.23428	−0.201170	−0.100585	+	0.994928i	$0.532071\pi$
$678$	20.4700	0.786144
$679$	−20.1104	−0.771766
$680$	−4.28258	−0.164229
$681$	−10.1221	−0.387878
$682$	4.49859	0.172260
$683$	20.7013	0.792113	0.396057	−	0.918226i	$-0.370378\pi$
$683$	20.7013	0.792113	0.396057	+	0.918226i	$0.370378\pi$
$684$	13.9881	0.534850
$685$	5.13434	0.196173
$686$	14.8672	0.567633
$687$	−13.9978	−0.534050
$688$	−2.79118	−0.106413
$689$	0.970382	0.0369686
$690$	−0.0933146	−0.00355243
$691$	14.7800	0.562258	0.281129	−	0.959670i	$-0.409291\pi$
$691$	14.7800	0.562258	0.281129	+	0.959670i	$0.409291\pi$
$692$	4.72954	0.179790
$693$	7.34270	0.278926
$694$	−8.73208	−0.331465
$695$	−2.23892	−0.0849270
$696$	6.12155	0.232037
$697$	−4.54013	−0.171970
$698$	−0.258705	−0.00979213
$699$	−5.84294	−0.221000
$700$	15.5636	0.588250
$701$	34.1735	1.29072	0.645358	−	0.763881i	$-0.276708\pi$
$701$	34.1735	1.29072	0.645358	+	0.763881i	$0.276708\pi$
$702$	0.172190	0.00649888
$703$	4.70083	0.177295
$704$	2.96665	0.111810
$705$	0.663028	0.0249711
$706$	−5.25474	−0.197765
$707$	13.1215	0.493487
$708$	−15.2993	−0.574984
$709$	8.21388	0.308479	0.154239	−	0.988034i	$-0.450707\pi$
$709$	8.21388	0.308479	0.154239	+	0.988034i	$0.450707\pi$
$710$	−0.684707	−0.0256966
$711$	−10.8990	−0.408745
$712$	−37.2278	−1.39517
$713$	0.701229	0.0262612
$714$	−18.0033	−0.673756
$715$	−0.102666	−0.00383949
$716$	6.73652	0.251756
$717$	47.2999	1.76645
$718$	14.4861	0.540615
$719$	−33.7298	−1.25791	−0.628954	−	0.777443i	$-0.716517\pi$
$719$	−33.7298	−1.25791	−0.628954	+	0.777443i	$0.716517\pi$
$720$	0.935137	0.0348505
$721$	−23.1197	−0.861024
$722$	3.53661	0.131619
$723$	−41.9017	−1.55834
$724$	14.4022	0.535253
$725$	4.96801	0.184507
$726$	−16.1178	−0.598188
$727$	−21.4570	−0.795798	−0.397899	−	0.917429i	$-0.630260\pi$
$727$	−21.4570	−0.795798	−0.397899	+	0.917429i	$0.630260\pi$
$728$	−1.22917	−0.0455561
$729$	−18.3676	−0.680281
$730$	2.83518	0.104935
$731$	12.7826	0.472781
$732$	−10.6371	−0.393157
$733$	−25.9785	−0.959539	−0.479770	−	0.877395i	$-0.659280\pi$
$733$	−25.9785	−0.959539	−0.479770	+	0.877395i	$0.659280\pi$
$734$	1.41688	0.0522981
$735$	1.83359	0.0676330
$736$	0.884657	0.0326089
$737$	12.4037	0.456894
$738$	−1.80746	−0.0665334
$739$	34.0810	1.25369	0.626845	−	0.779144i	$-0.284346\pi$
$739$	34.0810	1.25369	0.626845	+	0.779144i	$0.284346\pi$
$740$	0.645842	0.0237416
$741$	−1.95168	−0.0716967
$742$	7.14275	0.262219
$743$	8.04036	0.294972	0.147486	−	0.989064i	$-0.452882\pi$
$743$	8.04036	0.294972	0.147486	+	0.989064i	$0.452882\pi$
$744$	27.8896	1.02248
$745$	0.453433	0.0166125
$746$	−28.1336	−1.03004
$747$	14.1204	0.516640
$748$	9.02995	0.330168
$749$	23.6934	0.865739
$750$	−6.10216	−0.222819
$751$	36.3502	1.32644	0.663220	−	0.748425i	$-0.269190\pi$
$751$	36.3502	1.32644	0.663220	+	0.748425i	$0.269190\pi$
$752$	−0.816174	−0.0297628
$753$	17.5857	0.640857
$754$	−0.165292	−0.00601957
$755$	−4.89600	−0.178184
$756$	−3.39132	−0.123341
$757$	−6.55362	−0.238195	−0.119098	−	0.992883i	$-0.538000\pi$
$757$	−6.55362	−0.238195	−0.119098	+	0.992883i	$0.538000\pi$
$758$	−27.8145	−1.01027
$759$	0.467048	0.0169528
$760$	−3.41732	−0.123959
$761$	−28.9735	−1.05029	−0.525144	−	0.851014i	$-0.675988\pi$
$761$	−28.9735	−1.05029	−0.525144	+	0.851014i	$0.675988\pi$
$762$	19.2570	0.697609
$763$	−42.4971	−1.53850
$764$	−12.8747	−0.465792
$765$	−4.28258	−0.154837
$766$	−5.74174	−0.207458
$767$	0.980606	0.0354076
$768$	28.1105	1.01435
$769$	−20.7996	−0.750052	−0.375026	−	0.927014i	$-0.622366\pi$
$769$	−20.7996	−0.750052	−0.375026	+	0.927014i	$0.622366\pi$
$770$	−0.755699	−0.0272335
$771$	7.45477	0.268477
$772$	−13.3593	−0.480813
$773$	34.7699	1.25059	0.625293	−	0.780390i	$-0.284980\pi$
$773$	34.7699	1.25059	0.625293	+	0.780390i	$0.284980\pi$
$774$	5.08884	0.182914
$775$	22.6341	0.813041
$776$	−23.3712	−0.838978
$777$	6.44473	0.231203
$778$	11.3352	0.406388
$779$	−3.62283	−0.129801
$780$	−0.268139	−0.00960092
$781$	3.42702	0.122628
$782$	−0.526054	−0.0188117
$783$	−1.08253	−0.0386865
$784$	−2.25711	−0.0806111
$785$	0.0799166	0.00285235
$786$	16.3530	0.583291
$787$	18.6268	0.663973	0.331986	−	0.943284i	$-0.392281\pi$
$787$	18.6268	0.663973	0.331986	+	0.943284i	$0.392281\pi$
$788$	0.361881	0.0128915
$789$	16.9835	0.604629
$790$	1.12171	0.0399086
$791$	25.8404	0.918780
$792$	8.53330	0.303218
$793$	0.681778	0.0242107
$794$	−6.20447	−0.220188
$795$	3.69867	0.131178
$796$	−7.50949	−0.266167
$797$	−8.55696	−0.303103	−0.151552	−	0.988449i	$-0.548427\pi$
$797$	−8.55696	−0.303103	−0.151552	+	0.988449i	$0.548427\pi$
$798$	−14.3659	−0.508546
$799$	3.73777	0.132233
$800$	28.5547	1.00956
$801$	−37.2278	−1.31538
$802$	−25.9132	−0.915028
$803$	−14.1903	−0.500766
$804$	32.3954	1.14250
$805$	−0.117797	−0.00415178
$806$	−0.753064	−0.0265256
$807$	−4.27963	−0.150650
$808$	15.2492	0.536464
$809$	36.5255	1.28417	0.642084	−	0.766634i	$-0.278070\pi$
$809$	36.5255	1.28417	0.642084	+	0.766634i	$0.278070\pi$
$810$	2.66273	0.0935590
$811$	0.291636	0.0102407	0.00512037	−	0.999987i	$-0.498370\pi$
$811$	0.291636	0.0102407	0.00512037	+	0.999987i	$0.498370\pi$
$812$	3.25546	0.114244
$813$	−36.3517	−1.27491
$814$	1.20809	0.0423436
$815$	3.82893	0.134122
$816$	11.4758	0.401733
$817$	10.2000	0.356851
$818$	−22.7968	−0.797072
$819$	−1.22917	−0.0429507
$820$	−0.497736	−0.0173817
$821$	−24.0196	−0.838291	−0.419145	−	0.907919i	$-0.637670\pi$
$821$	−24.0196	−0.838291	−0.419145	+	0.907919i	$0.637670\pi$
$822$	25.0837	0.874895
$823$	−37.3060	−1.30040	−0.650202	−	0.759761i	$-0.725316\pi$
$823$	−37.3060	−1.30040	−0.650202	+	0.759761i	$0.725316\pi$
$824$	−26.8685	−0.936010
$825$	15.0753	0.524853
$826$	7.21801	0.251147
$827$	2.15858	0.0750610	0.0375305	−	0.999295i	$-0.488051\pi$
$827$	2.15858	0.0750610	0.0375305	+	0.999295i	$0.488051\pi$
$828$	0.560362	0.0194739
$829$	39.2679	1.36383	0.681915	−	0.731431i	$-0.261147\pi$
$829$	39.2679	1.36383	0.681915	+	0.731431i	$0.261147\pi$
$830$	−1.45325	−0.0504431
$831$	8.68894	0.301416
$832$	−0.496618	−0.0172171
$833$	10.3367	0.358146
$834$	−10.9382	−0.378759
$835$	2.13388	0.0738459
$836$	7.20551	0.249208
$837$	−4.93197	−0.170474
$838$	−25.1905	−0.870192
$839$	19.4462	0.671358	0.335679	−	0.941976i	$-0.391034\pi$
$839$	19.4462	0.671358	0.335679	+	0.941976i	$0.391034\pi$
$840$	−4.68506	−0.161650
$841$	−27.9608	−0.964167
$842$	7.49261	0.258213
$843$	67.9329	2.33973
$844$	−25.4984	−0.877690
$845$	−4.60658	−0.158471
$846$	1.48803	0.0511597
$847$	−20.3465	−0.699113
$848$	−4.55299	−0.156350
$849$	45.9861	1.57824
$850$	−16.9799	−0.582404
$851$	0.188314	0.00645533
$852$	8.95056	0.306641
$853$	−31.5937	−1.08175	−0.540874	−	0.841104i	$-0.681906\pi$
$853$	−31.5937	−1.08175	−0.540874	+	0.841104i	$0.681906\pi$
$854$	5.01841	0.171726
$855$	−3.41732	−0.116870
$856$	27.5352	0.941135
$857$	−38.8076	−1.32564	−0.662822	−	0.748777i	$-0.730641\pi$
$857$	−38.8076	−1.32564	−0.662822	+	0.748777i	$0.730641\pi$
$858$	−0.501572	−0.0171234
$859$	−30.1896	−1.03005	−0.515027	−	0.857174i	$-0.672218\pi$
$859$	−30.1896	−1.03005	−0.515027	+	0.857174i	$0.672218\pi$
$860$	1.40136	0.0477860
$861$	−4.96681	−0.169268
$862$	0.737640	0.0251241
$863$	11.4522	0.389837	0.194918	−	0.980819i	$-0.437556\pi$
$863$	11.4522	0.389837	0.194918	+	0.980819i	$0.437556\pi$
$864$	−6.22208	−0.211679
$865$	−1.15543	−0.0392858
$866$	−3.73815	−0.127028
$867$	−12.5084	−0.424807
$868$	14.8318	0.503423
$869$	−5.61425	−0.190451
$870$	−0.630020	−0.0213597
$871$	−2.07637	−0.0703553
$872$	−49.3879	−1.67248
$873$	−23.3712	−0.790996
$874$	−0.419769	−0.0141989
$875$	−7.70312	−0.260413
$876$	−37.0618	−1.25220
$877$	−37.2682	−1.25846	−0.629229	−	0.777220i	$-0.716629\pi$
$877$	−37.2682	−1.25846	−0.629229	+	0.777220i	$0.716629\pi$
$878$	15.5207	0.523800
$879$	−45.2360	−1.52577
$880$	0.481704	0.0162382
$881$	−7.29814	−0.245880	−0.122940	−	0.992414i	$-0.539232\pi$
$881$	−7.29814	−0.245880	−0.122940	+	0.992414i	$0.539232\pi$
$882$	4.11513	0.138564
$883$	−19.7549	−0.664805	−0.332402	−	0.943138i	$-0.607859\pi$
$883$	−19.7549	−0.664805	−0.332402	+	0.943138i	$0.607859\pi$
$884$	−1.51161	−0.0508411
$885$	3.73764	0.125639
$886$	−12.1696	−0.408847
$887$	48.7214	1.63591	0.817953	−	0.575285i	$-0.195109\pi$
$887$	48.7214	1.63591	0.817953	+	0.575285i	$0.195109\pi$
$888$	7.48972	0.251339
$889$	24.3093	0.815308
$890$	3.83143	0.128430
$891$	−13.3272	−0.446479
$892$	27.4956	0.920622
$893$	2.98258	0.0998084
$894$	2.21524	0.0740887
$895$	−1.64574	−0.0550109
$896$	22.0491	0.736609
$897$	−0.0781839	−0.00261048
$898$	−22.4978	−0.750761
$899$	4.73440	0.157901
$900$	18.0872	0.602908
$901$	20.8510	0.694647
$902$	−0.931050	−0.0310006
$903$	13.9839	0.465355
$904$	30.0304	0.998796
$905$	−3.51847	−0.116958
$906$	−23.9193	−0.794666
$907$	7.89607	0.262185	0.131092	−	0.991370i	$-0.458152\pi$
$907$	7.89607	0.262185	0.131092	+	0.991370i	$0.458152\pi$
$908$	−6.25578	−0.207605
$909$	15.2492	0.505783
$910$	0.126504	0.00419357
$911$	41.9892	1.39117	0.695583	−	0.718446i	$-0.255146\pi$
$911$	41.9892	1.39117	0.695583	+	0.718446i	$0.255146\pi$
$912$	9.15719	0.303225
$913$	7.27365	0.240723
$914$	−8.73487	−0.288924
$915$	2.59864	0.0859084
$916$	−8.65112	−0.285841
$917$	20.6433	0.681703
$918$	3.69991	0.122115
$919$	47.6809	1.57285	0.786424	−	0.617687i	$-0.211930\pi$
$919$	47.6809	1.57285	0.786424	+	0.617687i	$0.211930\pi$
$920$	−0.136897	−0.00451336
$921$	−19.8988	−0.655688
$922$	8.83968	0.291119
$923$	−0.573683	−0.0188830
$924$	9.87858	0.324982
$925$	6.07837	0.199855
$926$	−3.29635	−0.108325
$927$	−26.8685	−0.882479
$928$	5.97282	0.196067
$929$	9.33964	0.306424	0.153212	−	0.988193i	$-0.451038\pi$
$929$	9.33964	0.306424	0.153212	+	0.988193i	$0.451038\pi$
$930$	−2.87035	−0.0941225
$931$	8.24827	0.270326
$932$	−3.61114	−0.118287
$933$	−57.5938	−1.88554
$934$	22.0691	0.722122
$935$	−2.20602	−0.0721447
$936$	−1.42848	−0.0466912
$937$	34.6393	1.13162	0.565809	−	0.824536i	$-0.308564\pi$
$937$	34.6393	1.13162	0.565809	+	0.824536i	$0.308564\pi$
$938$	−15.2837	−0.499030
$939$	11.4641	0.374117
$940$	0.409774	0.0133654
$941$	48.6872	1.58716	0.793579	−	0.608467i	$-0.208215\pi$
$941$	48.6872	1.58716	0.793579	+	0.608467i	$0.208215\pi$
$942$	0.390431	0.0127209
$943$	−0.145130	−0.00472608
$944$	−4.60096	−0.149748
$945$	0.828502	0.0269512
$946$	2.62134	0.0852271
$947$	−35.0995	−1.14058	−0.570290	−	0.821444i	$-0.693169\pi$
$947$	−35.0995	−1.14058	−0.570290	+	0.821444i	$0.693169\pi$
$948$	−14.6631	−0.476236
$949$	2.37547	0.0771109
$950$	−13.5492	−0.439594
$951$	−81.2794	−2.63566
$952$	−26.4117	−0.856007
$953$	41.5286	1.34524	0.672622	−	0.739986i	$-0.265168\pi$
$953$	41.5286	1.34524	0.672622	+	0.739986i	$0.265168\pi$
$954$	8.30093	0.268753
$955$	3.14531	0.101780
$956$	29.2330	0.945461
$957$	3.15331	0.101932
$958$	24.7998	0.801246
$959$	31.6647	1.02251
$960$	−1.89289	−0.0610928
$961$	−9.43024	−0.304201
$962$	−0.202235	−0.00652031
$963$	27.5352	0.887311
$964$	−25.8967	−0.834076
$965$	3.26369	0.105062
$966$	−0.575493	−0.0185162
$967$	25.9307	0.833875	0.416937	−	0.908935i	$-0.363103\pi$
$967$	25.9307	0.833875	0.416937	+	0.908935i	$0.363103\pi$
$968$	−23.6456	−0.759998
$969$	−41.9365	−1.34720
$970$	2.40533	0.0772305
$971$	−34.0711	−1.09339	−0.546697	−	0.837331i	$-0.684115\pi$
$971$	−34.0711	−1.09339	−0.546697	+	0.837331i	$0.684115\pi$
$972$	−30.1694	−0.967684
$973$	−13.8079	−0.442662
$974$	5.65629	0.181239
$975$	−2.52360	−0.0808199
$976$	−3.19887	−0.102393
$977$	57.9989	1.85555	0.927775	−	0.373139i	$-0.121719\pi$
$977$	57.9989	1.85555	0.927775	+	0.373139i	$0.121719\pi$
$978$	18.7062	0.598157
$979$	−19.1766	−0.612888
$980$	1.13322	0.0361994
$981$	−49.3879	−1.57683
$982$	17.5315	0.559454
$983$	−34.3211	−1.09467	−0.547337	−	0.836912i	$-0.684359\pi$
$983$	−34.3211	−1.09467	−0.547337	+	0.836912i	$0.684359\pi$
$984$	−5.77216	−0.184010
$985$	−0.0884078	−0.00281691
$986$	−3.55169	−0.113109
$987$	4.08905	0.130156
$988$	−1.20620	−0.0383745
$989$	0.408608	0.0129930
$990$	−0.878234	−0.0279121
$991$	49.4157	1.56974	0.784870	−	0.619660i	$-0.212730\pi$
$991$	49.4157	1.56974	0.784870	+	0.619660i	$0.212730\pi$
$992$	27.2120	0.863981
$993$	13.5264	0.429246
$994$	−4.22275	−0.133937
$995$	1.83457	0.0581599
$996$	18.9971	0.601945
$997$	−18.9248	−0.599354	−0.299677	−	0.954041i	$-0.596879\pi$
$997$	−18.9248	−0.599354	−0.299677	+	0.954041i	$0.596879\pi$
$998$	−21.8910	−0.692946
$999$	−1.32448	−0.0419046

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	431.2.a.e.1.2	✓	4
3.2	odd	2		3879.2.a.m.1.3		4
4.3	odd	2		6896.2.a.o.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
431.2.a.e.1.2	✓	4	1.1	even	1	trivial
3879.2.a.m.1.3		4	3.2	odd	2
6896.2.a.o.1.4		4	4.3	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:	\( N \)	\(=\)	\( 431 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	431.a (trivial)

\(f(q)\)	\(=\)	\(q-0.737640 q^{2} -2.35567 q^{3} -1.45589 q^{4} +0.355674 q^{5} +1.73764 q^{6} +2.19353 q^{7} +2.54920 q^{8} +2.54920 q^{9} -0.262360 q^{10} +1.31313 q^{11} +3.42960 q^{12} -0.219819 q^{13} -1.61803 q^{14} -0.837853 q^{15} +1.03138 q^{16} -4.72333 q^{17} -1.88039 q^{18} -3.76902 q^{19} -0.517822 q^{20} -5.16724 q^{21} -0.968620 q^{22} -0.150986 q^{23} -6.00509 q^{24} -4.87350 q^{25} +0.162147 q^{26} +1.06193 q^{27} -3.19353 q^{28} -1.01939 q^{29} +0.618034 q^{30} -4.64433 q^{31} -5.85919 q^{32} -3.09331 q^{33} +3.48412 q^{34} +0.780181 q^{35} -3.71135 q^{36} -1.24723 q^{37} +2.78018 q^{38} +0.517822 q^{39} +0.906685 q^{40} +0.961212 q^{41} +3.81156 q^{42} -2.70626 q^{43} -1.91177 q^{44} +0.906685 q^{45} +0.111374 q^{46} -0.791342 q^{47} -2.42960 q^{48} -2.18844 q^{49} +3.59489 q^{50} +11.1266 q^{51} +0.320031 q^{52} -4.41446 q^{53} -0.783326 q^{54} +0.467048 q^{55} +5.59174 q^{56} +8.87858 q^{57} +0.751946 q^{58} -4.46097 q^{59} +1.21982 q^{60} -3.10155 q^{61} +3.42584 q^{62} +5.59174 q^{63} +2.25922 q^{64} -0.0781839 q^{65} +2.28175 q^{66} +9.44584 q^{67} +6.87664 q^{68} +0.355674 q^{69} -0.575493 q^{70} +2.60980 q^{71} +6.49843 q^{72} -10.8065 q^{73} +0.920006 q^{74} +11.4804 q^{75} +5.48727 q^{76} +2.88039 q^{77} -0.381966 q^{78} -4.27546 q^{79} +0.366835 q^{80} -10.1492 q^{81} -0.709029 q^{82} +5.53916 q^{83} +7.52291 q^{84} -1.67997 q^{85} +1.99625 q^{86} +2.40136 q^{87} +3.34744 q^{88} -14.6037 q^{89} -0.668808 q^{90} -0.482178 q^{91} +0.219819 q^{92} +10.9405 q^{93} +0.583726 q^{94} -1.34054 q^{95} +13.8023 q^{96} -9.16806 q^{97} +1.61428 q^{98} +3.34744 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} - 3 q^{3} - q^{4} - 5 q^{5} + 5 q^{6} + 2 q^{7} - 3 q^{8} - 3 q^{9} - 3 q^{10} + q^{11} - 2 q^{12} - 5 q^{13} - 2 q^{14} - 3 q^{15} - 3 q^{16} + 2 q^{17} - 5 q^{18} - 6 q^{19} + 4 q^{20}+ \cdots + 11 q^{99}+O(q^{100}) \) 4 * q - q^2 - 3 * q^3 - q^4 - 5 * q^5 + 5 * q^6 + 2 * q^7 - 3 * q^8 - 3 * q^9 - 3 * q^10 + q^11 - 2 * q^12 - 5 * q^13 - 2 * q^14 - 3 * q^15 - 3 * q^16 + 2 * q^17 - 5 * q^18 - 6 * q^19 + 4 * q^20 - 3 * q^21 - 11 * q^22 + 4 * q^23 - 6 * q^24 - 7 * q^25 + q^26 + 3 * q^27 - 6 * q^28 - 5 * q^29 - 2 * q^30 - 25 * q^31 + 8 * q^32 - 4 * q^33 - 12 * q^34 - q^35 - 2 * q^36 - q^37 + 7 * q^38 - 4 * q^39 + 12 * q^40 + 2 * q^41 + 4 * q^42 - 16 * q^43 + 2 * q^44 + 12 * q^45 + 7 * q^46 - 4 * q^47 + 6 * q^48 - 20 * q^49 + 13 * q^50 - 3 * q^51 + 7 * q^52 + 4 * q^53 - 13 * q^54 + 2 * q^55 + 7 * q^56 + 5 * q^57 + 20 * q^58 + 5 * q^59 + 9 * q^60 + 2 * q^62 + 7 * q^63 - 3 * q^64 + 14 * q^65 + 12 * q^66 + 9 * q^67 + 29 * q^68 - 5 * q^69 + 10 * q^71 + 19 * q^72 - 50 * q^73 - 10 * q^74 + 24 * q^75 + 10 * q^76 + 9 * q^77 - 6 * q^78 + 8 * q^79 - 8 * q^81 + 37 * q^82 - 15 * q^83 + 6 * q^84 - q^85 + 12 * q^86 + 15 * q^87 + 11 * q^88 - 35 * q^89 + 8 * q^90 - 8 * q^91 + 5 * q^92 + 12 * q^93 - 4 * q^94 + 7 * q^95 + 7 * q^96 - 6 * q^97 + 6 * q^98 + 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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