LMFDB - Embedded newform 6171.2.a.bs.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6171,2,Mod(1,6171)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6171, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6171.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6171 = 3 \cdot 11^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6171.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:	$49.2756830873$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 3 x^{19} - 28 x^{18} + 85 x^{17} + 320 x^{16} - 989 x^{15} - 1923 x^{14} + 6124 x^{13} + \cdots + 16 $ x^20 - 3x^19 - 28x^18 + 85x^17 + 320x^16 - 989x^15 - 1923x^14 + 6124x^13 + 6496x^12 - 21860x^11 - 12084x^10 + 45502x^9 + 10780x^8 - 53060x^7 - 2295x^6 + 31131x^5 - 1511x^4 - 7302x^3 + 356x^2 + 264*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 561)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.30762$ of defining polynomial
Character	$\chi$	$=$	6171.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.30762 q^{2} -1.00000 q^{3} +3.32509 q^{4} +3.77481 q^{5} +2.30762 q^{6} -4.19225 q^{7} -3.05780 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.30762 q^{2} -1.00000 q^{3} +3.32509 q^{4} +3.77481 q^{5} +2.30762 q^{6} -4.19225 q^{7} -3.05780 q^{8} +1.00000 q^{9} -8.71082 q^{10} -3.32509 q^{12} +4.38311 q^{13} +9.67410 q^{14} -3.77481 q^{15} +0.406040 q^{16} +1.00000 q^{17} -2.30762 q^{18} -0.378355 q^{19} +12.5516 q^{20} +4.19225 q^{21} +6.45853 q^{23} +3.05780 q^{24} +9.24921 q^{25} -10.1145 q^{26} -1.00000 q^{27} -13.9396 q^{28} +8.96391 q^{29} +8.71082 q^{30} +3.68119 q^{31} +5.17861 q^{32} -2.30762 q^{34} -15.8250 q^{35} +3.32509 q^{36} +7.02632 q^{37} +0.873098 q^{38} -4.38311 q^{39} -11.5426 q^{40} +8.81102 q^{41} -9.67410 q^{42} -10.9795 q^{43} +3.77481 q^{45} -14.9038 q^{46} -2.64757 q^{47} -0.406040 q^{48} +10.5750 q^{49} -21.3436 q^{50} -1.00000 q^{51} +14.5742 q^{52} +2.15107 q^{53} +2.30762 q^{54} +12.8190 q^{56} +0.378355 q^{57} -20.6853 q^{58} +6.17697 q^{59} -12.5516 q^{60} -8.61172 q^{61} -8.49477 q^{62} -4.19225 q^{63} -12.7623 q^{64} +16.5454 q^{65} +5.18683 q^{67} +3.32509 q^{68} -6.45853 q^{69} +36.5179 q^{70} -7.19531 q^{71} -3.05780 q^{72} -1.88661 q^{73} -16.2140 q^{74} -9.24921 q^{75} -1.25806 q^{76} +10.1145 q^{78} -1.40591 q^{79} +1.53273 q^{80} +1.00000 q^{81} -20.3324 q^{82} -9.21071 q^{83} +13.9396 q^{84} +3.77481 q^{85} +25.3364 q^{86} -8.96391 q^{87} +0.222644 q^{89} -8.71082 q^{90} -18.3751 q^{91} +21.4752 q^{92} -3.68119 q^{93} +6.10958 q^{94} -1.42822 q^{95} -5.17861 q^{96} +3.50534 q^{97} -24.4029 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 3 q^{2} - 20 q^{3} + 25 q^{4} + 7 q^{5} - 3 q^{6} - 5 q^{7} + 12 q^{8} + 20 q^{9}+O(q^{10}) $ 20 * q + 3 * q^2 - 20 * q^3 + 25 * q^4 + 7 * q^5 - 3 * q^6 - 5 * q^7 + 12 * q^8 + 20 * q^9	$ 20 q + 3 q^{2} - 20 q^{3} + 25 q^{4} + 7 q^{5} - 3 q^{6} - 5 q^{7} + 12 q^{8} + 20 q^{9} + 4 q^{10} - 25 q^{12} + 8 q^{13} + 3 q^{14} - 7 q^{15} + 47 q^{16} + 20 q^{17} + 3 q^{18} + 6 q^{19} + 26 q^{20} + 5 q^{21} - 16 q^{23} - 12 q^{24} + 31 q^{25} + 15 q^{26} - 20 q^{27} - 15 q^{28} + 4 q^{29} - 4 q^{30} + 27 q^{31} + 47 q^{32} + 3 q^{34} - 2 q^{35} + 25 q^{36} + 36 q^{37} - 8 q^{39} - 31 q^{40} + 6 q^{41} - 3 q^{42} + 46 q^{43} + 7 q^{45} - 25 q^{46} - 24 q^{47} - 47 q^{48} + 73 q^{49} - 20 q^{51} + 8 q^{52} + 6 q^{53} - 3 q^{54} + 39 q^{56} - 6 q^{57} - 25 q^{58} + 4 q^{59} - 26 q^{60} - 38 q^{61} - 9 q^{62} - 5 q^{63} + 100 q^{64} + 26 q^{65} + 21 q^{67} + 25 q^{68} + 16 q^{69} + 9 q^{70} - 24 q^{71} + 12 q^{72} + 5 q^{73} - 94 q^{74} - 31 q^{75} + 81 q^{76} - 15 q^{78} - 47 q^{79} + 17 q^{80} + 20 q^{81} + 75 q^{82} + 6 q^{83} + 15 q^{84} + 7 q^{85} + 54 q^{86} - 4 q^{87} + 44 q^{89} + 4 q^{90} + 37 q^{91} - 28 q^{92} - 27 q^{93} - 36 q^{94} + 18 q^{95} - 47 q^{96} + 19 q^{97} + 81 q^{98}+O(q^{100}) $ 20 * q + 3 * q^2 - 20 * q^3 + 25 * q^4 + 7 * q^5 - 3 * q^6 - 5 * q^7 + 12 * q^8 + 20 * q^9 + 4 * q^10 - 25 * q^12 + 8 * q^13 + 3 * q^14 - 7 * q^15 + 47 * q^16 + 20 * q^17 + 3 * q^18 + 6 * q^19 + 26 * q^20 + 5 * q^21 - 16 * q^23 - 12 * q^24 + 31 * q^25 + 15 * q^26 - 20 * q^27 - 15 * q^28 + 4 * q^29 - 4 * q^30 + 27 * q^31 + 47 * q^32 + 3 * q^34 - 2 * q^35 + 25 * q^36 + 36 * q^37 - 8 * q^39 - 31 * q^40 + 6 * q^41 - 3 * q^42 + 46 * q^43 + 7 * q^45 - 25 * q^46 - 24 * q^47 - 47 * q^48 + 73 * q^49 - 20 * q^51 + 8 * q^52 + 6 * q^53 - 3 * q^54 + 39 * q^56 - 6 * q^57 - 25 * q^58 + 4 * q^59 - 26 * q^60 - 38 * q^61 - 9 * q^62 - 5 * q^63 + 100 * q^64 + 26 * q^65 + 21 * q^67 + 25 * q^68 + 16 * q^69 + 9 * q^70 - 24 * q^71 + 12 * q^72 + 5 * q^73 - 94 * q^74 - 31 * q^75 + 81 * q^76 - 15 * q^78 - 47 * q^79 + 17 * q^80 + 20 * q^81 + 75 * q^82 + 6 * q^83 + 15 * q^84 + 7 * q^85 + 54 * q^86 - 4 * q^87 + 44 * q^89 + 4 * q^90 + 37 * q^91 - 28 * q^92 - 27 * q^93 - 36 * q^94 + 18 * q^95 - 47 * q^96 + 19 * q^97 + 81 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.30762	−1.63173	−0.815865	−	0.578242i	$-0.803739\pi$
$2$	−2.30762	−1.63173	−0.815865	+	0.578242i	$0.803739\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	3.32509	1.66254
$5$	3.77481	1.68815	0.844074	−	0.536227i	$-0.180151\pi$
$5$	3.77481	1.68815	0.844074	+	0.536227i	$0.180151\pi$
$6$	2.30762	0.942080
$7$	−4.19225	−1.58452	−0.792261	−	0.610183i	$-0.791096\pi$
$7$	−4.19225	−1.58452	−0.792261	+	0.610183i	$0.791096\pi$
$8$	−3.05780	−1.08109
$9$	1.00000	0.333333
$10$	−8.71082	−2.75460
$11$	0	0
$11$	0	0
$12$	−3.32509	−0.959871
$13$	4.38311	1.21566	0.607828	−	0.794069i	$-0.292041\pi$
$13$	4.38311	1.21566	0.607828	+	0.794069i	$0.292041\pi$
$14$	9.67410	2.58551
$15$	−3.77481	−0.974653
$16$	0.406040	0.101510
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−2.30762	−0.543910
$19$	−0.378355	−0.0868006	−0.0434003	−	0.999058i	$-0.513819\pi$
$19$	−0.378355	−0.0868006	−0.0434003	+	0.999058i	$0.513819\pi$
$20$	12.5516	2.80662
$21$	4.19225	0.914824
$22$	0	0
$23$	6.45853	1.34670	0.673348	−	0.739326i	$-0.264856\pi$
$23$	6.45853	1.34670	0.673348	+	0.739326i	$0.264856\pi$
$24$	3.05780	0.624170
$25$	9.24921	1.84984
$26$	−10.1145	−1.98362
$27$	−1.00000	−0.192450
$28$	−13.9396	−2.63434
$29$	8.96391	1.66456	0.832278	−	0.554358i	$-0.187036\pi$
$29$	8.96391	1.66456	0.832278	+	0.554358i	$0.187036\pi$
$30$	8.71082	1.59037
$31$	3.68119	0.661161	0.330580	−	0.943778i	$-0.392756\pi$
$31$	3.68119	0.661161	0.330580	+	0.943778i	$0.392756\pi$
$32$	5.17861	0.915457
$33$	0	0
$34$	−2.30762	−0.395753
$35$	−15.8250	−2.67491
$36$	3.32509	0.554182
$37$	7.02632	1.15512	0.577560	−	0.816348i	$-0.304005\pi$
$37$	7.02632	1.15512	0.577560	+	0.816348i	$0.304005\pi$
$38$	0.873098	0.141635
$39$	−4.38311	−0.701860
$40$	−11.5426	−1.82505
$41$	8.81102	1.37605	0.688025	−	0.725687i	$-0.258478\pi$
$41$	8.81102	1.37605	0.688025	+	0.725687i	$0.258478\pi$
$42$	−9.67410	−1.49275
$43$	−10.9795	−1.67435	−0.837176	−	0.546933i	$-0.815795\pi$
$43$	−10.9795	−1.67435	−0.837176	+	0.546933i	$0.815795\pi$
$44$	0	0
$45$	3.77481	0.562716
$46$	−14.9038	−2.19744
$47$	−2.64757	−0.386188	−0.193094	−	0.981180i	$-0.561852\pi$
$47$	−2.64757	−0.386188	−0.193094	+	0.981180i	$0.561852\pi$
$48$	−0.406040	−0.0586069
$49$	10.5750	1.51071
$50$	−21.3436	−3.01844
$51$	−1.00000	−0.140028
$52$	14.5742	2.02108
$53$	2.15107	0.295473	0.147736	−	0.989027i	$-0.452801\pi$
$53$	2.15107	0.295473	0.147736	+	0.989027i	$0.452801\pi$
$54$	2.30762	0.314027
$55$	0	0
$56$	12.8190	1.71302
$57$	0.378355	0.0501144
$58$	−20.6853	−2.71611
$59$	6.17697	0.804173	0.402087	−	0.915602i	$-0.368285\pi$
$59$	6.17697	0.804173	0.402087	+	0.915602i	$0.368285\pi$
$60$	−12.5516	−1.62040
$61$	−8.61172	−1.10262	−0.551309	−	0.834301i	$-0.685871\pi$
$61$	−8.61172	−1.10262	−0.551309	+	0.834301i	$0.685871\pi$
$62$	−8.49477	−1.07884
$63$	−4.19225	−0.528174
$64$	−12.7623	−1.59529
$65$	16.5454	2.05221
$66$	0	0
$67$	5.18683	0.633672	0.316836	−	0.948480i	$-0.397379\pi$
$67$	5.18683	0.633672	0.316836	+	0.948480i	$0.397379\pi$
$68$	3.32509	0.403226
$69$	−6.45853	−0.777515
$70$	36.5179	4.36473
$71$	−7.19531	−0.853927	−0.426963	−	0.904269i	$-0.640417\pi$
$71$	−7.19531	−0.853927	−0.426963	+	0.904269i	$0.640417\pi$
$72$	−3.05780	−0.360365
$73$	−1.88661	−0.220811	−0.110406	−	0.993887i	$-0.535215\pi$
$73$	−1.88661	−0.220811	−0.110406	+	0.993887i	$0.535215\pi$
$74$	−16.2140	−1.88484
$75$	−9.24921	−1.06801
$76$	−1.25806	−0.144310
$77$	0	0
$78$	10.1145	1.14525
$79$	−1.40591	−0.158177	−0.0790884	−	0.996868i	$-0.525201\pi$
$79$	−1.40591	−0.158177	−0.0790884	+	0.996868i	$0.525201\pi$
$80$	1.53273	0.171364
$81$	1.00000	0.111111
$82$	−20.3324	−2.24534
$83$	−9.21071	−1.01101	−0.505504	−	0.862824i	$-0.668693\pi$
$83$	−9.21071	−1.01101	−0.505504	+	0.862824i	$0.668693\pi$
$84$	13.9396	1.52094
$85$	3.77481	0.409436
$86$	25.3364	2.73209
$87$	−8.96391	−0.961032
$88$	0	0
$89$	0.222644	0.0236002	0.0118001	−	0.999930i	$-0.496244\pi$
$89$	0.222644	0.0236002	0.0118001	+	0.999930i	$0.496244\pi$
$90$	−8.71082	−0.918201
$91$	−18.3751	−1.92623
$92$	21.4752	2.23894
$93$	−3.68119	−0.381721
$94$	6.10958	0.630155
$95$	−1.42822	−0.146532
$96$	−5.17861	−0.528540
$97$	3.50534	0.355914	0.177957	−	0.984038i	$-0.443051\pi$
$97$	3.50534	0.355914	0.177957	+	0.984038i	$0.443051\pi$
$98$	−24.4029	−2.46507
$99$	0	0
$100$	30.7545	3.07545
$101$	−14.4196	−1.43480	−0.717401	−	0.696660i	$-0.754669\pi$
$101$	−14.4196	−1.43480	−0.717401	+	0.696660i	$0.754669\pi$
$102$	2.30762	0.228488
$103$	−1.89164	−0.186388	−0.0931942	−	0.995648i	$-0.529708\pi$
$103$	−1.89164	−0.186388	−0.0931942	+	0.995648i	$0.529708\pi$
$104$	−13.4027	−1.31424
$105$	15.8250	1.54436
$106$	−4.96385	−0.482132
$107$	10.8400	1.04794	0.523972	−	0.851735i	$-0.324449\pi$
$107$	10.8400	1.04794	0.523972	+	0.851735i	$0.324449\pi$
$108$	−3.32509	−0.319957
$109$	17.6444	1.69003	0.845013	−	0.534746i	$-0.179593\pi$
$109$	17.6444	1.69003	0.845013	+	0.534746i	$0.179593\pi$
$110$	0	0
$111$	−7.02632	−0.666909
$112$	−1.70222	−0.160845
$113$	−8.50226	−0.799826	−0.399913	−	0.916553i	$-0.630960\pi$
$113$	−8.50226	−0.799826	−0.399913	+	0.916553i	$0.630960\pi$
$114$	−0.873098	−0.0817731
$115$	24.3797	2.27342
$116$	29.8058	2.76740
$117$	4.38311	0.405219
$118$	−14.2541	−1.31219
$119$	−4.19225	−0.384303
$120$	11.5426	1.05369
$121$	0	0
$122$	19.8725	1.79917
$123$	−8.81102	−0.794463
$124$	12.2403	1.09921
$125$	16.0400	1.43466
$126$	9.67410	0.861837
$127$	−1.54971	−0.137515	−0.0687573	−	0.997633i	$-0.521903\pi$
$127$	−1.54971	−0.137515	−0.0687573	+	0.997633i	$0.521903\pi$
$128$	19.0933	1.68763
$129$	10.9795	0.966688
$130$	−38.1805	−3.34865
$131$	2.44615	0.213721	0.106861	−	0.994274i	$-0.465920\pi$
$131$	2.44615	0.213721	0.106861	+	0.994274i	$0.465920\pi$
$132$	0	0
$133$	1.58616	0.137537
$134$	−11.9692	−1.03398
$135$	−3.77481	−0.324884
$136$	−3.05780	−0.262204
$137$	−6.12285	−0.523111	−0.261555	−	0.965189i	$-0.584235\pi$
$137$	−6.12285	−0.523111	−0.261555	+	0.965189i	$0.584235\pi$
$138$	14.9038	1.26870
$139$	12.8105	1.08657	0.543285	−	0.839549i	$-0.317180\pi$
$139$	12.8105	1.08657	0.543285	+	0.839549i	$0.317180\pi$
$140$	−52.6194	−4.44715
$141$	2.64757	0.222966
$142$	16.6040	1.39338
$143$	0	0
$144$	0.406040	0.0338367
$145$	33.8371	2.81002
$146$	4.35358	0.360305
$147$	−10.5750	−0.872208
$148$	23.3631	1.92044
$149$	15.9008	1.30265	0.651323	−	0.758801i	$-0.274214\pi$
$149$	15.9008	1.30265	0.651323	+	0.758801i	$0.274214\pi$
$150$	21.3436	1.74270
$151$	18.6015	1.51377	0.756884	−	0.653549i	$-0.226721\pi$
$151$	18.6015	1.51377	0.756884	+	0.653549i	$0.226721\pi$
$152$	1.15693	0.0938397
$153$	1.00000	0.0808452
$154$	0	0
$155$	13.8958	1.11614
$156$	−14.5742	−1.16687
$157$	−19.9080	−1.58883	−0.794417	−	0.607373i	$-0.792223\pi$
$157$	−19.9080	−1.58883	−0.794417	+	0.607373i	$0.792223\pi$
$158$	3.24429	0.258102
$159$	−2.15107	−0.170591
$160$	19.5483	1.54543
$161$	−27.0758	−2.13387
$162$	−2.30762	−0.181303
$163$	−1.37845	−0.107968	−0.0539841	−	0.998542i	$-0.517192\pi$
$163$	−1.37845	−0.107968	−0.0539841	+	0.998542i	$0.517192\pi$
$164$	29.2974	2.28774
$165$	0	0
$166$	21.2548	1.64969
$167$	−7.03313	−0.544240	−0.272120	−	0.962263i	$-0.587725\pi$
$167$	−7.03313	−0.544240	−0.272120	+	0.962263i	$0.587725\pi$
$168$	−12.8190	−0.989011
$169$	6.21167	0.477821
$170$	−8.71082	−0.668089
$171$	−0.378355	−0.0289335
$172$	−36.5077	−2.78369
$173$	10.5534	0.802360	0.401180	−	0.915999i	$-0.368600\pi$
$173$	10.5534	0.802360	0.401180	+	0.915999i	$0.368600\pi$
$174$	20.6853	1.56815
$175$	−38.7750	−2.93112
$176$	0	0
$177$	−6.17697	−0.464290
$178$	−0.513777	−0.0385092
$179$	−9.24990	−0.691370	−0.345685	−	0.938351i	$-0.612353\pi$
$179$	−9.24990	−0.691370	−0.345685	+	0.938351i	$0.612353\pi$
$180$	12.5516	0.935540
$181$	2.10014	0.156102	0.0780510	−	0.996949i	$-0.475130\pi$
$181$	2.10014	0.156102	0.0780510	+	0.996949i	$0.475130\pi$
$182$	42.4027	3.14309
$183$	8.61172	0.636596
$184$	−19.7489	−1.45591
$185$	26.5230	1.95001
$186$	8.49477	0.622866
$187$	0	0
$188$	−8.80341	−0.642055
$189$	4.19225	0.304941
$190$	3.29578	0.239101
$191$	−19.9133	−1.44088	−0.720439	−	0.693518i	$-0.756060\pi$
$191$	−19.9133	−1.44088	−0.720439	+	0.693518i	$0.756060\pi$
$192$	12.7623	0.921041
$193$	−14.0315	−1.01001	−0.505006	−	0.863116i	$-0.668510\pi$
$193$	−14.0315	−1.01001	−0.505006	+	0.863116i	$0.668510\pi$
$194$	−8.08899	−0.580756
$195$	−16.5454	−1.18484
$196$	35.1627	2.51162
$197$	−5.96274	−0.424828	−0.212414	−	0.977180i	$-0.568133\pi$
$197$	−5.96274	−0.424828	−0.212414	+	0.977180i	$0.568133\pi$
$198$	0	0
$199$	−6.97968	−0.494776	−0.247388	−	0.968916i	$-0.579572\pi$
$199$	−6.97968	−0.494776	−0.247388	+	0.968916i	$0.579572\pi$
$200$	−28.2822	−1.99985
$201$	−5.18683	−0.365851
$202$	33.2749	2.34121
$203$	−37.5790	−2.63753
$204$	−3.32509	−0.232803
$205$	33.2600	2.32298
$206$	4.36517	0.304136
$207$	6.45853	0.448899
$208$	1.77972	0.123401
$209$	0	0
$210$	−36.5179	−2.51998
$211$	4.94243	0.340251	0.170126	−	0.985422i	$-0.445583\pi$
$211$	4.94243	0.340251	0.170126	+	0.985422i	$0.445583\pi$
$212$	7.15251	0.491237
$213$	7.19531	0.493015
$214$	−25.0146	−1.70996
$215$	−41.4454	−2.82655
$216$	3.05780	0.208057
$217$	−15.4325	−1.04762
$218$	−40.7165	−2.75767
$219$	1.88661	0.127485
$220$	0	0
$221$	4.38311	0.294840
$222$	16.2140	1.08822
$223$	24.1387	1.61645	0.808223	−	0.588876i	$-0.200429\pi$
$223$	24.1387	1.61645	0.808223	+	0.588876i	$0.200429\pi$
$224$	−21.7100	−1.45056
$225$	9.24921	0.616614
$226$	19.6200	1.30510
$227$	0.270342	0.0179432	0.00897162	−	0.999960i	$-0.497144\pi$
$227$	0.270342	0.0179432	0.00897162	+	0.999960i	$0.497144\pi$
$228$	1.25806	0.0833174
$229$	10.1079	0.667946	0.333973	−	0.942583i	$-0.391611\pi$
$229$	10.1079	0.667946	0.333973	+	0.942583i	$0.391611\pi$
$230$	−56.2590	−3.70961
$231$	0	0
$232$	−27.4098	−1.79954
$233$	−19.1273	−1.25307	−0.626537	−	0.779392i	$-0.715528\pi$
$233$	−19.1273	−1.25307	−0.626537	+	0.779392i	$0.715528\pi$
$234$	−10.1145	−0.661208
$235$	−9.99409	−0.651942
$236$	20.5390	1.33697
$237$	1.40591	0.0913234
$238$	9.67410	0.627079
$239$	20.6828	1.33786	0.668930	−	0.743326i	$-0.266753\pi$
$239$	20.6828	1.33786	0.668930	+	0.743326i	$0.266753\pi$
$240$	−1.53273	−0.0989371
$241$	26.5845	1.71246	0.856228	−	0.516598i	$-0.172802\pi$
$241$	26.5845	1.71246	0.856228	+	0.516598i	$0.172802\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	−28.6347	−1.83315
$245$	39.9185	2.55030
$246$	20.3324	1.29635
$247$	−1.65837	−0.105520
$248$	−11.2563	−0.714777
$249$	9.21071	0.583705
$250$	−37.0141	−2.34098
$251$	−6.95728	−0.439140	−0.219570	−	0.975597i	$-0.570465\pi$
$251$	−6.95728	−0.439140	−0.219570	+	0.975597i	$0.570465\pi$
$252$	−13.9396	−0.878112
$253$	0	0
$254$	3.57614	0.224387
$255$	−3.77481	−0.236388
$256$	−18.5354	−1.15846
$257$	−12.0320	−0.750533	−0.375267	−	0.926917i	$-0.622449\pi$
$257$	−12.0320	−0.750533	−0.375267	+	0.926917i	$0.622449\pi$
$258$	−25.3364	−1.57737
$259$	−29.4561	−1.83031
$260$	55.0150	3.41189
$261$	8.96391	0.554852
$262$	−5.64478	−0.348736
$263$	3.34144	0.206042	0.103021	−	0.994679i	$-0.467149\pi$
$263$	3.34144	0.206042	0.103021	+	0.994679i	$0.467149\pi$
$264$	0	0
$265$	8.11990	0.498802
$266$	−3.66025	−0.224424
$267$	−0.222644	−0.0136256
$268$	17.2467	1.05351
$269$	7.45332	0.454437	0.227218	−	0.973844i	$-0.427037\pi$
$269$	7.45332	0.454437	0.227218	+	0.973844i	$0.427037\pi$
$270$	8.71082	0.530123
$271$	−5.71421	−0.347113	−0.173557	−	0.984824i	$-0.555526\pi$
$271$	−5.71421	−0.347113	−0.173557	+	0.984824i	$0.555526\pi$
$272$	0.406040	0.0246198
$273$	18.3751	1.11211
$274$	14.1292	0.853575
$275$	0	0
$276$	−21.4752	−1.29265
$277$	−10.1322	−0.608785	−0.304392	−	0.952547i	$-0.598453\pi$
$277$	−10.1322	−0.608785	−0.304392	+	0.952547i	$0.598453\pi$
$278$	−29.5616	−1.77299
$279$	3.68119	0.220387
$280$	48.3895	2.89183
$281$	−4.57540	−0.272945	−0.136473	−	0.990644i	$-0.543577\pi$
$281$	−4.57540	−0.272945	−0.136473	+	0.990644i	$0.543577\pi$
$282$	−6.10958	−0.363820
$283$	5.10455	0.303434	0.151717	−	0.988424i	$-0.451520\pi$
$283$	5.10455	0.303434	0.151717	+	0.988424i	$0.451520\pi$
$284$	−23.9251	−1.41969
$285$	1.42822	0.0846004
$286$	0	0
$287$	−36.9380	−2.18038
$288$	5.17861	0.305152
$289$	1.00000	0.0588235
$290$	−78.0830	−4.58519
$291$	−3.50534	−0.205487
$292$	−6.27315	−0.367109
$293$	−19.3486	−1.13036	−0.565180	−	0.824968i	$-0.691193\pi$
$293$	−19.3486	−1.13036	−0.565180	+	0.824968i	$0.691193\pi$
$294$	24.4029	1.42321
$295$	23.3169	1.35756
$296$	−21.4850	−1.24879
$297$	0	0
$298$	−36.6930	−2.12557
$299$	28.3084	1.63712
$300$	−30.7545	−1.77561
$301$	46.0287	2.65305
$302$	−42.9251	−2.47006
$303$	14.4196	0.828384
$304$	−0.153627	−0.00881114
$305$	−32.5076	−1.86138
$306$	−2.30762	−0.131918
$307$	4.48483	0.255962	0.127981	−	0.991777i	$-0.459150\pi$
$307$	4.48483	0.255962	0.127981	+	0.991777i	$0.459150\pi$
$308$	0	0
$309$	1.89164	0.107611
$310$	−32.0662	−1.82124
$311$	8.77102	0.497359	0.248679	−	0.968586i	$-0.420003\pi$
$311$	8.77102	0.497359	0.248679	+	0.968586i	$0.420003\pi$
$312$	13.4027	0.758776
$313$	−12.6164	−0.713119	−0.356560	−	0.934273i	$-0.616050\pi$
$313$	−12.6164	−0.713119	−0.356560	+	0.934273i	$0.616050\pi$
$314$	45.9401	2.59255
$315$	−15.8250	−0.891635
$316$	−4.67476	−0.262976
$317$	−23.6082	−1.32597	−0.662985	−	0.748633i	$-0.730711\pi$
$317$	−23.6082	−1.32597	−0.662985	+	0.748633i	$0.730711\pi$
$318$	4.96385	0.278359
$319$	0	0
$320$	−48.1754	−2.69308
$321$	−10.8400	−0.605031
$322$	62.4804	3.48190
$323$	−0.378355	−0.0210522
$324$	3.32509	0.184727
$325$	40.5403	2.24877
$326$	3.18092	0.176175
$327$	−17.6444	−0.975737
$328$	−26.9423	−1.48764
$329$	11.0993	0.611923
$330$	0	0
$331$	27.8926	1.53311	0.766557	−	0.642176i	$-0.221968\pi$
$331$	27.8926	1.53311	0.766557	+	0.642176i	$0.221968\pi$
$332$	−30.6264	−1.68084
$333$	7.02632	0.385040
$334$	16.2297	0.888052
$335$	19.5793	1.06973
$336$	1.70222	0.0928638
$337$	8.71340	0.474649	0.237325	−	0.971430i	$-0.423730\pi$
$337$	8.71340	0.474649	0.237325	+	0.971430i	$0.423730\pi$
$338$	−14.3341	−0.779675
$339$	8.50226	0.461780
$340$	12.5516	0.680706
$341$	0	0
$342$	0.873098	0.0472117
$343$	−14.9871	−0.809227
$344$	33.5730	1.81013
$345$	−24.3797	−1.31256
$346$	−24.3532	−1.30923
$347$	−21.8345	−1.17213	−0.586067	−	0.810262i	$-0.699325\pi$
$347$	−21.8345	−1.17213	−0.586067	+	0.810262i	$0.699325\pi$
$348$	−29.8058	−1.59776
$349$	−8.11375	−0.434319	−0.217160	−	0.976136i	$-0.569679\pi$
$349$	−8.11375	−0.434319	−0.217160	+	0.976136i	$0.569679\pi$
$350$	89.4778	4.78279
$351$	−4.38311	−0.233953
$352$	0	0
$353$	−24.0745	−1.28136	−0.640678	−	0.767809i	$-0.721347\pi$
$353$	−24.0745	−1.28136	−0.640678	+	0.767809i	$0.721347\pi$
$354$	14.2541	0.757595
$355$	−27.1610	−1.44155
$356$	0.740311	0.0392364
$357$	4.19225	0.221877
$358$	21.3452	1.12813
$359$	2.70253	0.142634	0.0713170	−	0.997454i	$-0.477280\pi$
$359$	2.70253	0.142634	0.0713170	+	0.997454i	$0.477280\pi$
$360$	−11.5426	−0.608349
$361$	−18.8568	−0.992466
$362$	−4.84631	−0.254717
$363$	0	0
$364$	−61.0988	−3.20245
$365$	−7.12161	−0.372762
$366$	−19.8725	−1.03875
$367$	17.6147	0.919481	0.459740	−	0.888053i	$-0.347942\pi$
$367$	17.6147	0.919481	0.459740	+	0.888053i	$0.347942\pi$
$368$	2.62242	0.136703
$369$	8.81102	0.458683
$370$	−61.2050	−3.18189
$371$	−9.01784	−0.468183
$372$	−12.2403	−0.634629
$373$	−7.99006	−0.413710	−0.206855	−	0.978372i	$-0.566323\pi$
$373$	−7.99006	−0.413710	−0.206855	+	0.978372i	$0.566323\pi$
$374$	0	0
$375$	−16.0400	−0.828301
$376$	8.09574	0.417506
$377$	39.2898	2.02353
$378$	−9.67410	−0.497582
$379$	−14.7115	−0.755679	−0.377839	−	0.925871i	$-0.623333\pi$
$379$	−14.7115	−0.755679	−0.377839	+	0.925871i	$0.623333\pi$
$380$	−4.74896	−0.243616
$381$	1.54971	0.0793941
$382$	45.9523	2.35113
$383$	6.18886	0.316236	0.158118	−	0.987420i	$-0.449457\pi$
$383$	6.18886	0.316236	0.158118	+	0.987420i	$0.449457\pi$
$384$	−19.0933	−0.974351
$385$	0	0
$386$	32.3794	1.64807
$387$	−10.9795	−0.558118
$388$	11.6556	0.591723
$389$	−10.4049	−0.527551	−0.263776	−	0.964584i	$-0.584968\pi$
$389$	−10.4049	−0.527551	−0.263776	+	0.964584i	$0.584968\pi$
$390$	38.1805	1.93334
$391$	6.45853	0.326622
$392$	−32.3361	−1.63322
$393$	−2.44615	−0.123392
$394$	13.7597	0.693205
$395$	−5.30703	−0.267026
$396$	0	0
$397$	−0.899353	−0.0451372	−0.0225686	−	0.999745i	$-0.507184\pi$
$397$	−0.899353	−0.0451372	−0.0225686	+	0.999745i	$0.507184\pi$
$398$	16.1064	0.807342
$399$	−1.58616	−0.0794073
$400$	3.75555	0.187778
$401$	24.8744	1.24217	0.621083	−	0.783745i	$-0.286693\pi$
$401$	24.8744	1.24217	0.621083	+	0.783745i	$0.286693\pi$
$402$	11.9692	0.596970
$403$	16.1351	0.803745
$404$	−47.9464	−2.38542
$405$	3.77481	0.187572
$406$	86.7178	4.30373
$407$	0	0
$408$	3.05780	0.151383
$409$	−10.9166	−0.539791	−0.269895	−	0.962890i	$-0.586989\pi$
$409$	−10.9166	−0.539791	−0.269895	+	0.962890i	$0.586989\pi$
$410$	−76.7512	−3.79047
$411$	6.12285	0.302018
$412$	−6.28986	−0.309879
$413$	−25.8954	−1.27423
$414$	−14.9038	−0.732482
$415$	−34.7687	−1.70673
$416$	22.6984	1.11288
$417$	−12.8105	−0.627331
$418$	0	0
$419$	12.1272	0.592451	0.296226	−	0.955118i	$-0.404272\pi$
$419$	12.1272	0.592451	0.296226	+	0.955118i	$0.404272\pi$
$420$	52.6194	2.56756
$421$	−0.304244	−0.0148280	−0.00741398	−	0.999973i	$-0.502360\pi$
$421$	−0.304244	−0.0148280	−0.00741398	+	0.999973i	$0.502360\pi$
$422$	−11.4052	−0.555198
$423$	−2.64757	−0.128729
$424$	−6.57755	−0.319434
$425$	9.24921	0.448653
$426$	−16.6040	−0.804467
$427$	36.1025	1.74712
$428$	36.0441	1.74226
$429$	0	0
$430$	95.6401	4.61218
$431$	−17.9336	−0.863832	−0.431916	−	0.901914i	$-0.642162\pi$
$431$	−17.9336	−0.863832	−0.431916	+	0.901914i	$0.642162\pi$
$432$	−0.406040	−0.0195356
$433$	−31.1775	−1.49829	−0.749147	−	0.662404i	$-0.769536\pi$
$433$	−31.1775	−1.49829	−0.749147	+	0.662404i	$0.769536\pi$
$434$	35.6122	1.70944
$435$	−33.8371	−1.62236
$436$	58.6692	2.80974
$437$	−2.44362	−0.116894
$438$	−4.35358	−0.208022
$439$	20.4934	0.978095	0.489048	−	0.872257i	$-0.337344\pi$
$439$	20.4934	0.978095	0.489048	+	0.872257i	$0.337344\pi$
$440$	0	0
$441$	10.5750	0.503569
$442$	−10.1145	−0.481099
$443$	4.40610	0.209340	0.104670	−	0.994507i	$-0.466621\pi$
$443$	4.40610	0.209340	0.104670	+	0.994507i	$0.466621\pi$
$444$	−23.3631	−1.10877
$445$	0.840440	0.0398407
$446$	−55.7028	−2.63761
$447$	−15.9008	−0.752083
$448$	53.5028	2.52777
$449$	−7.04531	−0.332489	−0.166244	−	0.986085i	$-0.553164\pi$
$449$	−7.04531	−0.332489	−0.166244	+	0.986085i	$0.553164\pi$
$450$	−21.3436	−1.00615
$451$	0	0
$452$	−28.2708	−1.32975
$453$	−18.6015	−0.873975
$454$	−0.623846	−0.0292785
$455$	−69.3626	−3.25177
$456$	−1.15693	−0.0541784
$457$	−8.14319	−0.380923	−0.190461	−	0.981695i	$-0.560998\pi$
$457$	−8.14319	−0.380923	−0.190461	+	0.981695i	$0.560998\pi$
$458$	−23.3251	−1.08991
$459$	−1.00000	−0.0466760
$460$	81.0648	3.77966
$461$	−13.8855	−0.646713	−0.323357	−	0.946277i	$-0.604811\pi$
$461$	−13.8855	−0.646713	−0.323357	+	0.946277i	$0.604811\pi$
$462$	0	0
$463$	14.1521	0.657703	0.328852	−	0.944382i	$-0.393338\pi$
$463$	14.1521	0.657703	0.328852	+	0.944382i	$0.393338\pi$
$464$	3.63971	0.168969
$465$	−13.8958	−0.644402
$466$	44.1386	2.04468
$467$	31.6541	1.46478	0.732388	−	0.680888i	$-0.238406\pi$
$467$	31.6541	1.46478	0.732388	+	0.680888i	$0.238406\pi$
$468$	14.5742	0.673694
$469$	−21.7445	−1.00407
$470$	23.0625	1.06379
$471$	19.9080	0.917314
$472$	−18.8879	−0.869387
$473$	0	0
$474$	−3.24429	−0.149015
$475$	−3.49949	−0.160567
$476$	−13.9396	−0.638921
$477$	2.15107	0.0984909
$478$	−47.7280	−2.18303
$479$	12.3499	0.564284	0.282142	−	0.959373i	$-0.408955\pi$
$479$	12.3499	0.564284	0.282142	+	0.959373i	$0.408955\pi$
$480$	−19.5483	−0.892253
$481$	30.7971	1.40423
$482$	−61.3467	−2.79427
$483$	27.0758	1.23199
$484$	0	0
$485$	13.2320	0.600835
$486$	2.30762	0.104676
$487$	19.8077	0.897574	0.448787	−	0.893639i	$-0.351856\pi$
$487$	19.8077	0.897574	0.448787	+	0.893639i	$0.351856\pi$
$488$	26.3329	1.19203
$489$	1.37845	0.0623355
$490$	−92.1165	−4.16140
$491$	41.6442	1.87938	0.939688	−	0.342033i	$-0.111116\pi$
$491$	41.6442	1.87938	0.939688	+	0.342033i	$0.111116\pi$
$492$	−29.2974	−1.32083
$493$	8.96391	0.403714
$494$	3.82689	0.172180
$495$	0	0
$496$	1.49471	0.0671145
$497$	30.1646	1.35307
$498$	−21.2548	−0.952450
$499$	−36.4304	−1.63085	−0.815424	−	0.578865i	$-0.803496\pi$
$499$	−36.4304	−1.63085	−0.815424	+	0.578865i	$0.803496\pi$
$500$	53.3344	2.38519
$501$	7.03313	0.314217
$502$	16.0547	0.716558
$503$	32.2176	1.43651	0.718255	−	0.695780i	$-0.244941\pi$
$503$	32.2176	1.43651	0.718255	+	0.695780i	$0.244941\pi$
$504$	12.8190	0.571006
$505$	−54.4312	−2.42216
$506$	0	0
$507$	−6.21167	−0.275870
$508$	−5.15293	−0.228624
$509$	−0.707558	−0.0313620	−0.0156810	−	0.999877i	$-0.504992\pi$
$509$	−0.707558	−0.0313620	−0.0156810	+	0.999877i	$0.504992\pi$
$510$	8.71082	0.385721
$511$	7.90915	0.349880
$512$	4.58590	0.202670
$513$	0.378355	0.0167048
$514$	27.7652	1.22467
$515$	−7.14057	−0.314651
$516$	36.5077	1.60716
$517$	0	0
$518$	67.9733	2.98658
$519$	−10.5534	−0.463243
$520$	−50.5926	−2.21863
$521$	42.5840	1.86564	0.932819	−	0.360346i	$-0.117341\pi$
$521$	42.5840	1.86564	0.932819	+	0.360346i	$0.117341\pi$
$522$	−20.6853	−0.905369
$523$	39.1747	1.71299	0.856494	−	0.516157i	$-0.172638\pi$
$523$	39.1747	1.71299	0.856494	+	0.516157i	$0.172638\pi$
$524$	8.13368	0.355322
$525$	38.7750	1.69228
$526$	−7.71077	−0.336205
$527$	3.68119	0.160355
$528$	0	0
$529$	18.7126	0.813590
$530$	−18.7376	−0.813910
$531$	6.17697	0.268058
$532$	5.27412	0.228662
$533$	38.6197	1.67280
$534$	0.513777	0.0222333
$535$	40.9191	1.76909
$536$	−15.8603	−0.685060
$537$	9.24990	0.399163
$538$	−17.1994	−0.741518
$539$	0	0
$540$	−12.5516	−0.540134
$541$	15.8954	0.683395	0.341698	−	0.939810i	$-0.388998\pi$
$541$	15.8954	0.683395	0.341698	+	0.939810i	$0.388998\pi$
$542$	13.1862	0.566396
$543$	−2.10014	−0.0901256
$544$	5.17861	0.222031
$545$	66.6043	2.85301
$546$	−42.4027	−1.81467
$547$	0.180199	0.00770474	0.00385237	−	0.999993i	$-0.498774\pi$
$547$	0.180199	0.00770474	0.00385237	+	0.999993i	$0.498774\pi$
$548$	−20.3590	−0.869695
$549$	−8.61172	−0.367539
$550$	0	0
$551$	−3.39154	−0.144485
$552$	19.7489	0.840567
$553$	5.89391	0.250634
$554$	23.3812	0.993373
$555$	−26.5230	−1.12584
$556$	42.5959	1.80647
$557$	15.6066	0.661274	0.330637	−	0.943758i	$-0.392736\pi$
$557$	15.6066	0.661274	0.330637	+	0.943758i	$0.392736\pi$
$558$	−8.49477	−0.359612
$559$	−48.1242	−2.03544
$560$	−6.42557	−0.271530
$561$	0	0
$562$	10.5583	0.445373
$563$	44.9651	1.89505	0.947527	−	0.319675i	$-0.103574\pi$
$563$	44.9651	1.89505	0.947527	+	0.319675i	$0.103574\pi$
$564$	8.80341	0.370691
$565$	−32.0945	−1.35022
$566$	−11.7793	−0.495123
$567$	−4.19225	−0.176058
$568$	22.0018	0.923175
$569$	−41.1582	−1.72544	−0.862720	−	0.505682i	$-0.831241\pi$
$569$	−41.1582	−1.72544	−0.862720	+	0.505682i	$0.831241\pi$
$570$	−3.29578	−0.138045
$571$	21.9293	0.917714	0.458857	−	0.888510i	$-0.348259\pi$
$571$	21.9293	0.917714	0.458857	+	0.888510i	$0.348259\pi$
$572$	0	0
$573$	19.9133	0.831892
$574$	85.2387	3.55779
$575$	59.7363	2.49118
$576$	−12.7623	−0.531763
$577$	27.1251	1.12923	0.564617	−	0.825353i	$-0.309024\pi$
$577$	27.1251	1.12923	0.564617	+	0.825353i	$0.309024\pi$
$578$	−2.30762	−0.0959842
$579$	14.0315	0.583131
$580$	112.511	4.67178
$581$	38.6136	1.60196
$582$	8.08899	0.335299
$583$	0	0
$584$	5.76888	0.238718
$585$	16.5454	0.684069
$586$	44.6492	1.84444
$587$	25.4529	1.05055	0.525276	−	0.850932i	$-0.323962\pi$
$587$	25.4529	1.05055	0.525276	+	0.850932i	$0.323962\pi$
$588$	−35.1627	−1.45008
$589$	−1.39280	−0.0573892
$590$	−53.8065	−2.21518
$591$	5.96274	0.245275
$592$	2.85297	0.117256
$593$	−16.2842	−0.668712	−0.334356	−	0.942447i	$-0.608519\pi$
$593$	−16.2842	−0.668712	−0.334356	+	0.942447i	$0.608519\pi$
$594$	0	0
$595$	−15.8250	−0.648760
$596$	52.8716	2.16571
$597$	6.97968	0.285659
$598$	−65.3250	−2.67134
$599$	−29.1752	−1.19207	−0.596034	−	0.802959i	$-0.703258\pi$
$599$	−29.1752	−1.19207	−0.596034	+	0.802959i	$0.703258\pi$
$600$	28.2822	1.15462
$601$	7.07831	0.288730	0.144365	−	0.989524i	$-0.453886\pi$
$601$	7.07831	0.288730	0.144365	+	0.989524i	$0.453886\pi$
$602$	−106.216	−4.32906
$603$	5.18683	0.211224
$604$	61.8516	2.51671
$605$	0	0
$606$	−33.2749	−1.35170
$607$	9.95503	0.404062	0.202031	−	0.979379i	$-0.435246\pi$
$607$	9.95503	0.404062	0.202031	+	0.979379i	$0.435246\pi$
$608$	−1.95935	−0.0794623
$609$	37.5790	1.52278
$610$	75.0151	3.03727
$611$	−11.6046	−0.469472
$612$	3.32509	0.134409
$613$	−6.37682	−0.257557	−0.128779	−	0.991673i	$-0.541106\pi$
$613$	−6.37682	−0.257557	−0.128779	+	0.991673i	$0.541106\pi$
$614$	−10.3493	−0.417662
$615$	−33.2600	−1.34117
$616$	0	0
$617$	33.7189	1.35747	0.678735	−	0.734383i	$-0.262528\pi$
$617$	33.7189	1.35747	0.678735	+	0.734383i	$0.262528\pi$
$618$	−4.36517	−0.175593
$619$	−1.37979	−0.0554585	−0.0277293	−	0.999615i	$-0.508828\pi$
$619$	−1.37979	−0.0554585	−0.0277293	+	0.999615i	$0.508828\pi$
$620$	46.2048	1.85563
$621$	−6.45853	−0.259172
$622$	−20.2401	−0.811555
$623$	−0.933379	−0.0373951
$624$	−1.77972	−0.0712458
$625$	14.3019	0.572075
$626$	29.1137	1.16362
$627$	0	0
$628$	−66.1960	−2.64151
$629$	7.02632	0.280158
$630$	36.5179	1.45491
$631$	26.1401	1.04062	0.520310	−	0.853977i	$-0.325816\pi$
$631$	26.1401	1.04062	0.520310	+	0.853977i	$0.325816\pi$
$632$	4.29897	0.171004
$633$	−4.94243	−0.196444
$634$	54.4787	2.16363
$635$	−5.84987	−0.232145
$636$	−7.15251	−0.283616
$637$	46.3512	1.83650
$638$	0	0
$639$	−7.19531	−0.284642
$640$	72.0737	2.84896
$641$	−17.0446	−0.673220	−0.336610	−	0.941644i	$-0.609280\pi$
$641$	−17.0446	−0.673220	−0.336610	+	0.941644i	$0.609280\pi$
$642$	25.0146	0.987248
$643$	5.29221	0.208705	0.104352	−	0.994540i	$-0.466723\pi$
$643$	5.29221	0.208705	0.104352	+	0.994540i	$0.466723\pi$
$644$	−90.0293	−3.54765
$645$	41.4454	1.63191
$646$	0.873098	0.0343516
$647$	−32.6612	−1.28404	−0.642022	−	0.766686i	$-0.721904\pi$
$647$	−32.6612	−1.28404	−0.642022	+	0.766686i	$0.721904\pi$
$648$	−3.05780	−0.120122
$649$	0	0
$650$	−93.5515	−3.66939
$651$	15.4325	0.604846
$652$	−4.58346	−0.179502
$653$	8.40982	0.329102	0.164551	−	0.986369i	$-0.447383\pi$
$653$	8.40982	0.329102	0.164551	+	0.986369i	$0.447383\pi$
$654$	40.7165	1.59214
$655$	9.23378	0.360793
$656$	3.57763	0.139683
$657$	−1.88661	−0.0736038
$658$	−25.6129	−0.998494
$659$	27.2003	1.05957	0.529786	−	0.848131i	$-0.322272\pi$
$659$	27.2003	1.05957	0.529786	+	0.848131i	$0.322272\pi$
$660$	0	0
$661$	−16.1211	−0.627040	−0.313520	−	0.949582i	$-0.601508\pi$
$661$	−16.1211	−0.627040	−0.313520	+	0.949582i	$0.601508\pi$
$662$	−64.3654	−2.50163
$663$	−4.38311	−0.170226
$664$	28.1645	1.09299
$665$	5.98745	0.232183
$666$	−16.2140	−0.628281
$667$	57.8937	2.24165
$668$	−23.3858	−0.904823
$669$	−24.1387	−0.933256
$670$	−45.1815	−1.74552
$671$	0	0
$672$	21.7100	0.837482
$673$	−15.8443	−0.610754	−0.305377	−	0.952232i	$-0.598782\pi$
$673$	−15.8443	−0.610754	−0.305377	+	0.952232i	$0.598782\pi$
$674$	−20.1072	−0.774499
$675$	−9.24921	−0.356002
$676$	20.6544	0.794398
$677$	−39.9379	−1.53494	−0.767469	−	0.641086i	$-0.778484\pi$
$677$	−39.9379	−1.53494	−0.767469	+	0.641086i	$0.778484\pi$
$678$	−19.6200	−0.753500
$679$	−14.6953	−0.563953
$680$	−11.5426	−0.442639
$681$	−0.270342	−0.0103595
$682$	0	0
$683$	2.64273	0.101121	0.0505606	−	0.998721i	$-0.483899\pi$
$683$	2.64273	0.101121	0.0505606	+	0.998721i	$0.483899\pi$
$684$	−1.25806	−0.0481033
$685$	−23.1126	−0.883088
$686$	34.5845	1.32044
$687$	−10.1079	−0.385639
$688$	−4.45811	−0.169964
$689$	9.42840	0.359193
$690$	56.2590	2.14175
$691$	−2.18207	−0.0830100	−0.0415050	−	0.999138i	$-0.513215\pi$
$691$	−2.18207	−0.0830100	−0.0415050	+	0.999138i	$0.513215\pi$
$692$	35.0910	1.33396
$693$	0	0
$694$	50.3855	1.91261
$695$	48.3571	1.83429
$696$	27.4098	1.03897
$697$	8.81102	0.333741
$698$	18.7234	0.708692
$699$	19.1273	0.723463
$700$	−128.930	−4.87311
$701$	39.3321	1.48555	0.742776	−	0.669539i	$-0.233509\pi$
$701$	39.3321	1.48555	0.742776	+	0.669539i	$0.233509\pi$
$702$	10.1145	0.381749
$703$	−2.65844	−0.100265
$704$	0	0
$705$	9.99409	0.376399
$706$	55.5547	2.09083
$707$	60.4505	2.27348
$708$	−20.5390	−0.771902
$709$	31.4504	1.18114	0.590571	−	0.806985i	$-0.298902\pi$
$709$	31.4504	1.18114	0.590571	+	0.806985i	$0.298902\pi$
$710$	62.6771	2.35223
$711$	−1.40591	−0.0527256
$712$	−0.680800	−0.0255141
$713$	23.7750	0.890383
$714$	−9.67410	−0.362044
$715$	0	0
$716$	−30.7568	−1.14943
$717$	−20.6828	−0.772414
$718$	−6.23639	−0.232740
$719$	−40.4589	−1.50886	−0.754431	−	0.656380i	$-0.772087\pi$
$719$	−40.4589	−1.50886	−0.754431	+	0.656380i	$0.772087\pi$
$720$	1.53273	0.0571213
$721$	7.93021	0.295336
$722$	43.5144	1.61944
$723$	−26.5845	−0.988687
$724$	6.98315	0.259527
$725$	82.9092	3.07917
$726$	0	0
$727$	−16.3981	−0.608171	−0.304085	−	0.952645i	$-0.598351\pi$
$727$	−16.3981	−0.608171	−0.304085	+	0.952645i	$0.598351\pi$
$728$	56.1873	2.08244
$729$	1.00000	0.0370370
$730$	16.4339	0.608247
$731$	−10.9795	−0.406090
$732$	28.6347	1.05837
$733$	−51.2912	−1.89448	−0.947242	−	0.320521i	$-0.896142\pi$
$733$	−51.2912	−1.89448	−0.947242	+	0.320521i	$0.896142\pi$
$734$	−40.6480	−1.50034
$735$	−39.9185	−1.47242
$736$	33.4462	1.23284
$737$	0	0
$738$	−20.3324	−0.748448
$739$	30.0661	1.10600	0.553000	−	0.833181i	$-0.313483\pi$
$739$	30.0661	1.10600	0.553000	+	0.833181i	$0.313483\pi$
$740$	88.1915	3.24198
$741$	1.65837	0.0609218
$742$	20.8097	0.763948
$743$	−38.7046	−1.41993	−0.709967	−	0.704235i	$-0.751290\pi$
$743$	−38.7046	−1.41993	−0.709967	+	0.704235i	$0.751290\pi$
$744$	11.2563	0.412677
$745$	60.0226	2.19906
$746$	18.4380	0.675063
$747$	−9.21071	−0.337002
$748$	0	0
$749$	−45.4441	−1.66049
$750$	37.0141	1.35156
$751$	−31.7069	−1.15700	−0.578501	−	0.815682i	$-0.696362\pi$
$751$	−31.7069	−1.15700	−0.578501	+	0.815682i	$0.696362\pi$
$752$	−1.07502	−0.0392020
$753$	6.95728	0.253537
$754$	−90.6658	−3.30185
$755$	70.2172	2.55546
$756$	13.9396	0.506978
$757$	29.9790	1.08961	0.544803	−	0.838564i	$-0.316604\pi$
$757$	29.9790	1.08961	0.544803	+	0.838564i	$0.316604\pi$
$758$	33.9485	1.23306
$759$	0	0
$760$	4.36721	0.158415
$761$	44.9637	1.62993	0.814967	−	0.579507i	$-0.196755\pi$
$761$	44.9637	1.62993	0.814967	+	0.579507i	$0.196755\pi$
$762$	−3.57614	−0.129550
$763$	−73.9697	−2.67788
$764$	−66.2136	−2.39553
$765$	3.77481	0.136479
$766$	−14.2815	−0.516012
$767$	27.0744	0.977598
$768$	18.5354	0.668838
$769$	19.1560	0.690782	0.345391	−	0.938459i	$-0.387746\pi$
$769$	19.1560	0.690782	0.345391	+	0.938459i	$0.387746\pi$
$770$	0	0
$771$	12.0320	0.433321
$772$	−46.6561	−1.67919
$773$	−9.54239	−0.343216	−0.171608	−	0.985165i	$-0.554896\pi$
$773$	−9.54239	−0.343216	−0.171608	+	0.985165i	$0.554896\pi$
$774$	25.3364	0.910697
$775$	34.0481	1.22304
$776$	−10.7186	−0.384776
$777$	29.4561	1.05673
$778$	24.0106	0.860822
$779$	−3.33369	−0.119442
$780$	−55.0150	−1.96985
$781$	0	0
$782$	−14.9038	−0.532959
$783$	−8.96391	−0.320344
$784$	4.29386	0.153352
$785$	−75.1491	−2.68219
$786$	5.64478	0.201343
$787$	−23.5501	−0.839470	−0.419735	−	0.907647i	$-0.637877\pi$
$787$	−23.5501	−0.839470	−0.419735	+	0.907647i	$0.637877\pi$
$788$	−19.8267	−0.706295
$789$	−3.34144	−0.118959
$790$	12.2466	0.435714
$791$	35.6436	1.26734
$792$	0	0
$793$	−37.7461	−1.34040
$794$	2.07536	0.0736518
$795$	−8.11990	−0.287983
$796$	−23.2081	−0.822588
$797$	1.01449	0.0359349	0.0179675	−	0.999839i	$-0.494280\pi$
$797$	1.01449	0.0359349	0.0179675	+	0.999839i	$0.494280\pi$
$798$	3.66025	0.129571
$799$	−2.64757	−0.0936644
$800$	47.8981	1.69345
$801$	0.222644	0.00786674
$802$	−57.4004	−2.02688
$803$	0	0
$804$	−17.2467	−0.608243
$805$	−102.206	−3.60229
$806$	−37.2335	−1.31149
$807$	−7.45332	−0.262369
$808$	44.0922	1.55116
$809$	44.1095	1.55081	0.775403	−	0.631466i	$-0.217547\pi$
$809$	44.1095	1.55081	0.775403	+	0.631466i	$0.217547\pi$
$810$	−8.71082	−0.306067
$811$	−4.04781	−0.142138	−0.0710689	−	0.997471i	$-0.522641\pi$
$811$	−4.04781	−0.142138	−0.0710689	+	0.997471i	$0.522641\pi$
$812$	−124.953	−4.38500
$813$	5.71421	0.200406
$814$	0	0
$815$	−5.20338	−0.182266
$816$	−0.406040	−0.0142143
$817$	4.15414	0.145335
$818$	25.1913	0.880793
$819$	−18.3751	−0.642078
$820$	110.592	3.86205
$821$	−17.0067	−0.593537	−0.296769	−	0.954949i	$-0.595909\pi$
$821$	−17.0067	−0.593537	−0.296769	+	0.954949i	$0.595909\pi$
$822$	−14.1292	−0.492812
$823$	6.46639	0.225404	0.112702	−	0.993629i	$-0.464049\pi$
$823$	6.46639	0.225404	0.112702	+	0.993629i	$0.464049\pi$
$824$	5.78424	0.201503
$825$	0	0
$826$	59.7566	2.07920
$827$	−9.79185	−0.340496	−0.170248	−	0.985401i	$-0.554457\pi$
$827$	−9.79185	−0.340496	−0.170248	+	0.985401i	$0.554457\pi$
$828$	21.4752	0.746314
$829$	37.3656	1.29776	0.648880	−	0.760891i	$-0.275238\pi$
$829$	37.3656	1.29776	0.648880	+	0.760891i	$0.275238\pi$
$830$	80.2328	2.78492
$831$	10.1322	0.351482
$832$	−55.9387	−1.93932
$833$	10.5750	0.366400
$834$	29.5616	1.02364
$835$	−26.5487	−0.918757
$836$	0	0
$837$	−3.68119	−0.127240
$838$	−27.9849	−0.966721
$839$	20.7159	0.715191	0.357595	−	0.933877i	$-0.383597\pi$
$839$	20.7159	0.715191	0.357595	+	0.933877i	$0.383597\pi$
$840$	−48.3895	−1.66960
$841$	51.3517	1.77075
$842$	0.702079	0.0241952
$843$	4.57540	0.157585
$844$	16.4340	0.565683
$845$	23.4479	0.806632
$846$	6.10958	0.210052
$847$	0	0
$848$	0.873423	0.0299935
$849$	−5.10455	−0.175188
$850$	−21.3436	−0.732080
$851$	45.3797	1.55559
$852$	23.9251	0.819659
$853$	−9.36207	−0.320551	−0.160276	−	0.987072i	$-0.551238\pi$
$853$	−9.36207	−0.320551	−0.160276	+	0.987072i	$0.551238\pi$
$854$	−83.3106	−2.85083
$855$	−1.42822	−0.0488441
$856$	−33.1466	−1.13293
$857$	57.1849	1.95340	0.976700	−	0.214611i	$-0.0688482\pi$
$857$	57.1849	1.95340	0.976700	+	0.214611i	$0.0688482\pi$
$858$	0	0
$859$	10.1796	0.347323	0.173662	−	0.984805i	$-0.444440\pi$
$859$	10.1796	0.347323	0.173662	+	0.984805i	$0.444440\pi$
$860$	−137.810	−4.69927
$861$	36.9380	1.25884
$862$	41.3839	1.40954
$863$	−35.8000	−1.21865	−0.609323	−	0.792922i	$-0.708559\pi$
$863$	−35.8000	−1.21865	−0.609323	+	0.792922i	$0.708559\pi$
$864$	−5.17861	−0.176180
$865$	39.8371	1.35450
$866$	71.9456	2.44481
$867$	−1.00000	−0.0339618
$868$	−51.3143	−1.74172
$869$	0	0
$870$	78.0830	2.64726
$871$	22.7345	0.770328
$872$	−53.9529	−1.82708
$873$	3.50534	0.118638
$874$	5.63893	0.190740
$875$	−67.2436	−2.27325
$876$	6.27315	0.211950
$877$	8.75159	0.295520	0.147760	−	0.989023i	$-0.452794\pi$
$877$	8.75159	0.295520	0.147760	+	0.989023i	$0.452794\pi$
$878$	−47.2908	−1.59599
$879$	19.3486	0.652613
$880$	0	0
$881$	29.8492	1.00564	0.502822	−	0.864390i	$-0.332295\pi$
$881$	29.8492	1.00564	0.502822	+	0.864390i	$0.332295\pi$
$882$	−24.4029	−0.821689
$883$	45.1703	1.52010	0.760051	−	0.649863i	$-0.225174\pi$
$883$	45.1703	1.52010	0.760051	+	0.649863i	$0.225174\pi$
$884$	14.5742	0.490185
$885$	−23.3169	−0.783789
$886$	−10.1676	−0.341587
$887$	3.84205	0.129003	0.0645017	−	0.997918i	$-0.479454\pi$
$887$	3.84205	0.129003	0.0645017	+	0.997918i	$0.479454\pi$
$888$	21.4850	0.720991
$889$	6.49678	0.217895
$890$	−1.93941	−0.0650092
$891$	0	0
$892$	80.2633	2.68742
$893$	1.00172	0.0335214
$894$	36.6930	1.22720
$895$	−34.9167	−1.16714
$896$	−80.0439	−2.67408
$897$	−28.3084	−0.945191
$898$	16.2579	0.542532
$899$	32.9978	1.10054
$900$	30.7545	1.02515
$901$	2.15107	0.0716627
$902$	0	0
$903$	−46.0287	−1.53174
$904$	25.9982	0.864687
$905$	7.92763	0.263523
$906$	42.9251	1.42609
$907$	14.4707	0.480491	0.240245	−	0.970712i	$-0.422772\pi$
$907$	14.4707	0.480491	0.240245	+	0.970712i	$0.422772\pi$
$908$	0.898912	0.0298315
$909$	−14.4196	−0.478267
$910$	160.062	5.30601
$911$	−4.05630	−0.134391	−0.0671956	−	0.997740i	$-0.521405\pi$
$911$	−4.05630	−0.134391	−0.0671956	+	0.997740i	$0.521405\pi$
$912$	0.153627	0.00508711
$913$	0	0
$914$	18.7914	0.621563
$915$	32.5076	1.07467
$916$	33.6095	1.11049
$917$	−10.2549	−0.338646
$918$	2.30762	0.0761627
$919$	19.4210	0.640640	0.320320	−	0.947309i	$-0.396210\pi$
$919$	19.4210	0.640640	0.320320	+	0.947309i	$0.396210\pi$
$920$	−74.5483	−2.45778
$921$	−4.48483	−0.147780
$922$	32.0424	1.05526
$923$	−31.5379	−1.03808
$924$	0	0
$925$	64.9879	2.13679
$926$	−32.6576	−1.07319
$927$	−1.89164	−0.0621295
$928$	46.4206	1.52383
$929$	−29.3816	−0.963979	−0.481990	−	0.876177i	$-0.660086\pi$
$929$	−29.3816	−0.963979	−0.481990	+	0.876177i	$0.660086\pi$
$930$	32.0662	1.05149
$931$	−4.00109	−0.131130
$932$	−63.6001	−2.08329
$933$	−8.77102	−0.287150
$934$	−73.0454	−2.39012
$935$	0	0
$936$	−13.4027	−0.438080
$937$	−14.8303	−0.484485	−0.242242	−	0.970216i	$-0.577883\pi$
$937$	−14.8303	−0.484485	−0.242242	+	0.970216i	$0.577883\pi$
$938$	50.1779	1.63837
$939$	12.6164	0.411720
$940$	−33.2312	−1.08388
$941$	−49.1972	−1.60378	−0.801892	−	0.597469i	$-0.796173\pi$
$941$	−49.1972	−1.60378	−0.801892	+	0.597469i	$0.796173\pi$
$942$	−45.9401	−1.49681
$943$	56.9062	1.85312
$944$	2.50810	0.0816317
$945$	15.8250	0.514786
$946$	0	0
$947$	−34.9524	−1.13580	−0.567900	−	0.823098i	$-0.692244\pi$
$947$	−34.9524	−1.13580	−0.567900	+	0.823098i	$0.692244\pi$
$948$	4.67476	0.151829
$949$	−8.26923	−0.268431
$950$	8.07547	0.262003
$951$	23.6082	0.765549
$952$	12.8190	0.415468
$953$	33.3124	1.07909	0.539547	−	0.841955i	$-0.318595\pi$
$953$	33.3124	1.07909	0.539547	+	0.841955i	$0.318595\pi$
$954$	−4.96385	−0.160711
$955$	−75.1691	−2.43242
$956$	68.7722	2.22425
$957$	0	0
$958$	−28.4989	−0.920759
$959$	25.6685	0.828880
$960$	48.1754	1.55485
$961$	−17.4489	−0.562866
$962$	−71.0679	−2.29132
$963$	10.8400	0.349315
$964$	88.3957	2.84703
$965$	−52.9665	−1.70505
$966$	−62.4804	−2.01027
$967$	−33.5204	−1.07794	−0.538971	−	0.842324i	$-0.681187\pi$
$967$	−33.5204	−1.07794	−0.538971	+	0.842324i	$0.681187\pi$
$968$	0	0
$969$	0.378355	0.0121545
$970$	−30.5344	−0.980401
$971$	11.9695	0.384120	0.192060	−	0.981383i	$-0.438483\pi$
$971$	11.9695	0.384120	0.192060	+	0.981383i	$0.438483\pi$
$972$	−3.32509	−0.106652
$973$	−53.7047	−1.72169
$974$	−45.7087	−1.46460
$975$	−40.5403	−1.29833
$976$	−3.49670	−0.111927
$977$	−5.57729	−0.178433	−0.0892166	−	0.996012i	$-0.528436\pi$
$977$	−5.57729	−0.178433	−0.0892166	+	0.996012i	$0.528436\pi$
$978$	−3.18092	−0.101715
$979$	0	0
$980$	132.733	4.23998
$981$	17.6444	0.563342
$982$	−96.0988	−3.06663
$983$	−6.08929	−0.194218	−0.0971090	−	0.995274i	$-0.530960\pi$
$983$	−6.08929	−0.194218	−0.0971090	+	0.995274i	$0.530960\pi$
$984$	26.9423	0.858889
$985$	−22.5082	−0.717172
$986$	−20.6853	−0.658753
$987$	−11.0993	−0.353294
$988$	−5.51424	−0.175431
$989$	−70.9112	−2.25484
$990$	0	0
$991$	30.0580	0.954822	0.477411	−	0.878680i	$-0.341575\pi$
$991$	30.0580	0.954822	0.477411	+	0.878680i	$0.341575\pi$
$992$	19.0634	0.605265
$993$	−27.8926	−0.885144
$994$	−69.6082	−2.20784
$995$	−26.3470	−0.835256
$996$	30.6264	0.970436
$997$	−19.5309	−0.618549	−0.309275	−	0.950973i	$-0.600086\pi$
$997$	−19.5309	−0.618549	−0.309275	+	0.950973i	$0.600086\pi$
$998$	84.0673	2.66110
$999$	−7.02632	−0.222303

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6171.2.a.bs.1.3		20
11.3	even	5		561.2.m.e.460.1	✓	40
11.4	even	5		561.2.m.e.511.1	yes	40
11.10	odd	2		6171.2.a.bp.1.18		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
561.2.m.e.460.1	✓	40	11.3	even	5
561.2.m.e.511.1	yes	40	11.4	even	5
6171.2.a.bp.1.18		20	11.10	odd	2
6171.2.a.bs.1.3		20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6171 = 3 \cdot 11^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6171.a (trivial)

\(f(q)\)	\(=\)	\(q-2.30762 q^{2} -1.00000 q^{3} +3.32509 q^{4} +3.77481 q^{5} +2.30762 q^{6} -4.19225 q^{7} -3.05780 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.30762 q^{2} -1.00000 q^{3} +3.32509 q^{4} +3.77481 q^{5} +2.30762 q^{6} -4.19225 q^{7} -3.05780 q^{8} +1.00000 q^{9} -8.71082 q^{10} -3.32509 q^{12} +4.38311 q^{13} +9.67410 q^{14} -3.77481 q^{15} +0.406040 q^{16} +1.00000 q^{17} -2.30762 q^{18} -0.378355 q^{19} +12.5516 q^{20} +4.19225 q^{21} +6.45853 q^{23} +3.05780 q^{24} +9.24921 q^{25} -10.1145 q^{26} -1.00000 q^{27} -13.9396 q^{28} +8.96391 q^{29} +8.71082 q^{30} +3.68119 q^{31} +5.17861 q^{32} -2.30762 q^{34} -15.8250 q^{35} +3.32509 q^{36} +7.02632 q^{37} +0.873098 q^{38} -4.38311 q^{39} -11.5426 q^{40} +8.81102 q^{41} -9.67410 q^{42} -10.9795 q^{43} +3.77481 q^{45} -14.9038 q^{46} -2.64757 q^{47} -0.406040 q^{48} +10.5750 q^{49} -21.3436 q^{50} -1.00000 q^{51} +14.5742 q^{52} +2.15107 q^{53} +2.30762 q^{54} +12.8190 q^{56} +0.378355 q^{57} -20.6853 q^{58} +6.17697 q^{59} -12.5516 q^{60} -8.61172 q^{61} -8.49477 q^{62} -4.19225 q^{63} -12.7623 q^{64} +16.5454 q^{65} +5.18683 q^{67} +3.32509 q^{68} -6.45853 q^{69} +36.5179 q^{70} -7.19531 q^{71} -3.05780 q^{72} -1.88661 q^{73} -16.2140 q^{74} -9.24921 q^{75} -1.25806 q^{76} +10.1145 q^{78} -1.40591 q^{79} +1.53273 q^{80} +1.00000 q^{81} -20.3324 q^{82} -9.21071 q^{83} +13.9396 q^{84} +3.77481 q^{85} +25.3364 q^{86} -8.96391 q^{87} +0.222644 q^{89} -8.71082 q^{90} -18.3751 q^{91} +21.4752 q^{92} -3.68119 q^{93} +6.10958 q^{94} -1.42822 q^{95} -5.17861 q^{96} +3.50534 q^{97} -24.4029 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 3 q^{2} - 20 q^{3} + 25 q^{4} + 7 q^{5} - 3 q^{6} - 5 q^{7} + 12 q^{8} + 20 q^{9}+O(q^{10}) \) 20 * q + 3 * q^2 - 20 * q^3 + 25 * q^4 + 7 * q^5 - 3 * q^6 - 5 * q^7 + 12 * q^8 + 20 * q^9	\( 20 q + 3 q^{2} - 20 q^{3} + 25 q^{4} + 7 q^{5} - 3 q^{6} - 5 q^{7} + 12 q^{8} + 20 q^{9} + 4 q^{10} - 25 q^{12} + 8 q^{13} + 3 q^{14} - 7 q^{15} + 47 q^{16} + 20 q^{17} + 3 q^{18} + 6 q^{19} + 26 q^{20} + 5 q^{21} - 16 q^{23} - 12 q^{24} + 31 q^{25} + 15 q^{26} - 20 q^{27} - 15 q^{28} + 4 q^{29} - 4 q^{30} + 27 q^{31} + 47 q^{32} + 3 q^{34} - 2 q^{35} + 25 q^{36} + 36 q^{37} - 8 q^{39} - 31 q^{40} + 6 q^{41} - 3 q^{42} + 46 q^{43} + 7 q^{45} - 25 q^{46} - 24 q^{47} - 47 q^{48} + 73 q^{49} - 20 q^{51} + 8 q^{52} + 6 q^{53} - 3 q^{54} + 39 q^{56} - 6 q^{57} - 25 q^{58} + 4 q^{59} - 26 q^{60} - 38 q^{61} - 9 q^{62} - 5 q^{63} + 100 q^{64} + 26 q^{65} + 21 q^{67} + 25 q^{68} + 16 q^{69} + 9 q^{70} - 24 q^{71} + 12 q^{72} + 5 q^{73} - 94 q^{74} - 31 q^{75} + 81 q^{76} - 15 q^{78} - 47 q^{79} + 17 q^{80} + 20 q^{81} + 75 q^{82} + 6 q^{83} + 15 q^{84} + 7 q^{85} + 54 q^{86} - 4 q^{87} + 44 q^{89} + 4 q^{90} + 37 q^{91} - 28 q^{92} - 27 q^{93} - 36 q^{94} + 18 q^{95} - 47 q^{96} + 19 q^{97} + 81 q^{98}+O(q^{100}) \) 20 * q + 3 * q^2 - 20 * q^3 + 25 * q^4 + 7 * q^5 - 3 * q^6 - 5 * q^7 + 12 * q^8 + 20 * q^9 + 4 * q^10 - 25 * q^12 + 8 * q^13 + 3 * q^14 - 7 * q^15 + 47 * q^16 + 20 * q^17 + 3 * q^18 + 6 * q^19 + 26 * q^20 + 5 * q^21 - 16 * q^23 - 12 * q^24 + 31 * q^25 + 15 * q^26 - 20 * q^27 - 15 * q^28 + 4 * q^29 - 4 * q^30 + 27 * q^31 + 47 * q^32 + 3 * q^34 - 2 * q^35 + 25 * q^36 + 36 * q^37 - 8 * q^39 - 31 * q^40 + 6 * q^41 - 3 * q^42 + 46 * q^43 + 7 * q^45 - 25 * q^46 - 24 * q^47 - 47 * q^48 + 73 * q^49 - 20 * q^51 + 8 * q^52 + 6 * q^53 - 3 * q^54 + 39 * q^56 - 6 * q^57 - 25 * q^58 + 4 * q^59 - 26 * q^60 - 38 * q^61 - 9 * q^62 - 5 * q^63 + 100 * q^64 + 26 * q^65 + 21 * q^67 + 25 * q^68 + 16 * q^69 + 9 * q^70 - 24 * q^71 + 12 * q^72 + 5 * q^73 - 94 * q^74 - 31 * q^75 + 81 * q^76 - 15 * q^78 - 47 * q^79 + 17 * q^80 + 20 * q^81 + 75 * q^82 + 6 * q^83 + 15 * q^84 + 7 * q^85 + 54 * q^86 - 4 * q^87 + 44 * q^89 + 4 * q^90 + 37 * q^91 - 28 * q^92 - 27 * q^93 - 36 * q^94 + 18 * q^95 - 47 * q^96 + 19 * q^97 + 81 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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