LMFDB - Embedded newform 961.2.a.h.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [961,2,Mod(1,961)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("961.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(961, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 961 = 31^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	961.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,6,0,6,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$7.67362363425$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 4 $ x^4 - 6*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.874032$ of defining polynomial
Character	$\chi$	$=$	961.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.61803 q^{2} -0.874032 q^{3} +4.85410 q^{4} +2.23607 q^{5} -2.28825 q^{6} -1.00000 q^{7} +7.47214 q^{8} -2.23607 q^{9} +5.85410 q^{10} +4.24264 q^{11} -4.24264 q^{12} -2.62210 q^{13} -2.61803 q^{14} -1.95440 q^{15} +9.85410 q^{16} +3.70246 q^{17} -5.85410 q^{18} +1.00000 q^{19} +10.8541 q^{20} +0.874032 q^{21} +11.1074 q^{22} -2.62210 q^{23} -6.53089 q^{24} -6.86474 q^{26} +4.57649 q^{27} -4.85410 q^{28} -0.540182 q^{29} -5.11667 q^{30} +10.8541 q^{32} -3.70820 q^{33} +9.69316 q^{34} -2.23607 q^{35} -10.8541 q^{36} +4.24264 q^{37} +2.61803 q^{38} +2.29180 q^{39} +16.7082 q^{40} -1.47214 q^{41} +2.28825 q^{42} -9.69316 q^{43} +20.5942 q^{44} -5.00000 q^{45} -6.86474 q^{46} -9.70820 q^{47} -8.61280 q^{48} -6.00000 q^{49} -3.23607 q^{51} -12.7279 q^{52} -13.7295 q^{53} +11.9814 q^{54} +9.48683 q^{55} -7.47214 q^{56} -0.874032 q^{57} -1.41421 q^{58} +11.9443 q^{59} -9.48683 q^{60} +13.9358 q^{61} +2.23607 q^{63} +8.70820 q^{64} -5.86319 q^{65} -9.70820 q^{66} +6.00000 q^{67} +17.9721 q^{68} +2.29180 q^{69} -5.85410 q^{70} -1.47214 q^{71} -16.7082 q^{72} +4.24264 q^{73} +11.1074 q^{74} +4.85410 q^{76} -4.24264 q^{77} +6.00000 q^{78} -1.62054 q^{79} +22.0344 q^{80} +2.70820 q^{81} -3.85410 q^{82} -3.16228 q^{83} +4.24264 q^{84} +8.27895 q^{85} -25.3770 q^{86} +0.472136 q^{87} +31.7016 q^{88} -15.3500 q^{89} -13.0902 q^{90} +2.62210 q^{91} -12.7279 q^{92} -25.4164 q^{94} +2.23607 q^{95} -9.48683 q^{96} -7.00000 q^{97} -15.7082 q^{98} -9.48683 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{2} + 6 q^{4} - 4 q^{7} + 12 q^{8} + 10 q^{10} - 6 q^{14} + 26 q^{16} - 10 q^{18} + 4 q^{19} + 30 q^{20} - 6 q^{28} + 30 q^{32} + 12 q^{33} - 30 q^{36} + 6 q^{38} + 36 q^{39} + 40 q^{40} + 12 q^{41}+ \cdots - 36 q^{98}+O(q^{100}) $ 4 * q + 6 * q^2 + 6 * q^4 - 4 * q^7 + 12 * q^8 + 10 * q^10 - 6 * q^14 + 26 * q^16 - 10 * q^18 + 4 * q^19 + 30 * q^20 - 6 * q^28 + 30 * q^32 + 12 * q^33 - 30 * q^36 + 6 * q^38 + 36 * q^39 + 40 * q^40 + 12 * q^41 - 20 * q^45 - 12 * q^47 - 24 * q^49 - 4 * q^51 - 12 * q^56 + 12 * q^59 + 8 * q^64 - 12 * q^66 + 24 * q^67 + 36 * q^69 - 10 * q^70 + 12 * q^71 - 40 * q^72 + 6 * q^76 + 24 * q^78 + 30 * q^80 - 16 * q^81 - 2 * q^82 - 16 * q^87 - 30 * q^90 - 48 * q^94 - 28 * q^97 - 36 * q^98

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 $

$2$	2.61803	1.85123	0.925615	−	0.378467i	$-0.123549\pi$
$2$	2.61803	1.85123	0.925615	+	0.378467i	$0.123549\pi$
$3$	−0.874032	−0.504623	−0.252311	−	0.967646i	$-0.581191\pi$
$3$	−0.874032	−0.504623	−0.252311	+	0.967646i	$0.581191\pi$
$4$	4.85410	2.42705
$5$	2.23607	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$5$	2.23607	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$6$	−2.28825	−0.934172
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	7.47214	2.64180
$9$	−2.23607	−0.745356
$10$	5.85410	1.85123
$11$	4.24264	1.27920	0.639602	−	0.768706i	$-0.279099\pi$
$11$	4.24264	1.27920	0.639602	+	0.768706i	$0.279099\pi$
$12$	−4.24264	−1.22474
$13$	−2.62210	−0.727239	−0.363619	−	0.931548i	$-0.618459\pi$
$13$	−2.62210	−0.727239	−0.363619	+	0.931548i	$0.618459\pi$
$14$	−2.61803	−0.699699
$15$	−1.95440	−0.504623
$16$	9.85410	2.46353
$17$	3.70246	0.897978	0.448989	−	0.893537i	$-0.351784\pi$
$17$	3.70246	0.897978	0.448989	+	0.893537i	$0.351784\pi$
$18$	−5.85410	−1.37983
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	10.8541	2.42705
$21$	0.874032	0.190729
$22$	11.1074	2.36810
$23$	−2.62210	−0.546745	−0.273372	−	0.961908i	$-0.588139\pi$
$23$	−2.62210	−0.546745	−0.273372	+	0.961908i	$0.588139\pi$
$24$	−6.53089	−1.33311
$25$	0	0
$26$	−6.86474	−1.34629
$27$	4.57649	0.880746
$28$	−4.85410	−0.917339
$29$	−0.540182	−0.100309	−0.0501546	−	0.998741i	$-0.515971\pi$
$29$	−0.540182	−0.100309	−0.0501546	+	0.998741i	$0.515971\pi$
$30$	−5.11667	−0.934172
$31$	0	0
$31$	0	0
$32$	10.8541	1.91875
$33$	−3.70820	−0.645515
$34$	9.69316	1.66236
$35$	−2.23607	−0.377964
$36$	−10.8541	−1.80902
$37$	4.24264	0.697486	0.348743	−	0.937218i	$-0.386609\pi$
$37$	4.24264	0.697486	0.348743	+	0.937218i	$0.386609\pi$
$38$	2.61803	0.424701
$39$	2.29180	0.366981
$40$	16.7082	2.64180
$41$	−1.47214	−0.229909	−0.114955	−	0.993371i	$-0.536672\pi$
$41$	−1.47214	−0.229909	−0.114955	+	0.993371i	$0.536672\pi$
$42$	2.28825	0.353084
$43$	−9.69316	−1.47819	−0.739097	−	0.673599i	$-0.764747\pi$
$43$	−9.69316	−1.47819	−0.739097	+	0.673599i	$0.764747\pi$
$44$	20.5942	3.10469
$45$	−5.00000	−0.745356
$46$	−6.86474	−1.01215
$47$	−9.70820	−1.41609	−0.708044	−	0.706169i	$-0.750422\pi$
$47$	−9.70820	−1.41609	−0.708044	+	0.706169i	$0.750422\pi$
$48$	−8.61280	−1.24315
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−3.23607	−0.453140
$52$	−12.7279	−1.76505
$53$	−13.7295	−1.88589	−0.942944	−	0.332951i	$-0.891956\pi$
$53$	−13.7295	−1.88589	−0.942944	+	0.332951i	$0.891956\pi$
$54$	11.9814	1.63046
$55$	9.48683	1.27920
$56$	−7.47214	−0.998506
$57$	−0.874032	−0.115768
$58$	−1.41421	−0.185695
$59$	11.9443	1.55501	0.777506	−	0.628876i	$-0.216485\pi$
$59$	11.9443	1.55501	0.777506	+	0.628876i	$0.216485\pi$
$60$	−9.48683	−1.22474
$61$	13.9358	1.78430	0.892148	−	0.451742i	$-0.149197\pi$
$61$	13.9358	1.78430	0.892148	+	0.451742i	$0.149197\pi$
$62$	0	0
$63$	2.23607	0.281718
$64$	8.70820	1.08853
$65$	−5.86319	−0.727239
$66$	−9.70820	−1.19500
$67$	6.00000	0.733017	0.366508	−	0.930415i	$-0.380553\pi$
$67$	6.00000	0.733017	0.366508	+	0.930415i	$0.380553\pi$
$68$	17.9721	2.17944
$69$	2.29180	0.275900
$70$	−5.85410	−0.699699
$71$	−1.47214	−0.174710	−0.0873552	−	0.996177i	$-0.527842\pi$
$71$	−1.47214	−0.174710	−0.0873552	+	0.996177i	$0.527842\pi$
$72$	−16.7082	−1.96908
$73$	4.24264	0.496564	0.248282	−	0.968688i	$-0.420134\pi$
$73$	4.24264	0.496564	0.248282	+	0.968688i	$0.420134\pi$
$74$	11.1074	1.29121
$75$	0	0
$76$	4.85410	0.556804
$77$	−4.24264	−0.483494
$78$	6.00000	0.679366
$79$	−1.62054	−0.182326	−0.0911628	−	0.995836i	$-0.529058\pi$
$79$	−1.62054	−0.182326	−0.0911628	+	0.995836i	$0.529058\pi$
$80$	22.0344	2.46353
$81$	2.70820	0.300912
$82$	−3.85410	−0.425614
$83$	−3.16228	−0.347105	−0.173553	−	0.984825i	$-0.555525\pi$
$83$	−3.16228	−0.347105	−0.173553	+	0.984825i	$0.555525\pi$
$84$	4.24264	0.462910
$85$	8.27895	0.897978
$86$	−25.3770	−2.73648
$87$	0.472136	0.0506183
$88$	31.7016	3.37940
$89$	−15.3500	−1.62710	−0.813549	−	0.581496i	$-0.802468\pi$
$89$	−15.3500	−1.62710	−0.813549	+	0.581496i	$0.802468\pi$
$90$	−13.0902	−1.37983
$91$	2.62210	0.274870
$92$	−12.7279	−1.32698
$93$	0	0
$94$	−25.4164	−2.62150
$95$	2.23607	0.229416
$96$	−9.48683	−0.968246
$97$	−7.00000	−0.710742	−0.355371	−	0.934725i	$-0.615646\pi$
$97$	−7.00000	−0.710742	−0.355371	+	0.934725i	$0.615646\pi$
$98$	−15.7082	−1.58677
$99$	−9.48683	−0.953463
$100$	0	0
$101$	2.23607	0.222497	0.111249	−	0.993793i	$-0.464515\pi$
$101$	2.23607	0.222497	0.111249	+	0.993793i	$0.464515\pi$
$102$	−8.47214	−0.838866
$103$	10.7082	1.05511	0.527555	−	0.849521i	$-0.323109\pi$
$103$	10.7082	1.05511	0.527555	+	0.849521i	$0.323109\pi$
$104$	−19.5927	−1.92122
$105$	1.95440	0.190729
$106$	−35.9442	−3.49121
$107$	1.47214	0.142317	0.0711584	−	0.997465i	$-0.477330\pi$
$107$	1.47214	0.142317	0.0711584	+	0.997465i	$0.477330\pi$
$108$	22.2148	2.13762
$109$	−1.29180	−0.123732	−0.0618658	−	0.998084i	$-0.519705\pi$
$109$	−1.29180	−0.123732	−0.0618658	+	0.998084i	$0.519705\pi$
$110$	24.8369	2.36810
$111$	−3.70820	−0.351967
$112$	−9.85410	−0.931125
$113$	−9.76393	−0.918513	−0.459257	−	0.888304i	$-0.651884\pi$
$113$	−9.76393	−0.918513	−0.459257	+	0.888304i	$0.651884\pi$
$114$	−2.28825	−0.214314
$115$	−5.86319	−0.546745
$116$	−2.62210	−0.243456
$117$	5.86319	0.542052
$118$	31.2705	2.87868
$119$	−3.70246	−0.339404
$120$	−14.6035	−1.33311
$121$	7.00000	0.636364
$122$	36.4844	3.30314
$123$	1.28669	0.116017
$124$	0	0
$125$	−11.1803	−1.00000
$126$	5.85410	0.521525
$127$	−9.48683	−0.841820	−0.420910	−	0.907102i	$-0.638289\pi$
$127$	−9.48683	−0.841820	−0.420910	+	0.907102i	$0.638289\pi$
$128$	1.09017	0.0963583
$129$	8.47214	0.745930
$130$	−15.3500	−1.34629
$131$	7.41641	0.647975	0.323987	−	0.946061i	$-0.394976\pi$
$131$	7.41641	0.647975	0.323987	+	0.946061i	$0.394976\pi$
$132$	−18.0000	−1.56670
$133$	−1.00000	−0.0867110
$134$	15.7082	1.35698
$135$	10.2333	0.880746
$136$	27.6653	2.37228
$137$	−2.62210	−0.224021	−0.112010	−	0.993707i	$-0.535729\pi$
$137$	−2.62210	−0.224021	−0.112010	+	0.993707i	$0.535729\pi$
$138$	6.00000	0.510754
$139$	−17.9721	−1.52437	−0.762187	−	0.647356i	$-0.775875\pi$
$139$	−17.9721	−1.52437	−0.762187	+	0.647356i	$0.775875\pi$
$140$	−10.8541	−0.917339
$141$	8.48528	0.714590
$142$	−3.85410	−0.323429
$143$	−11.1246	−0.930287
$144$	−22.0344	−1.83620
$145$	−1.20788	−0.100309
$146$	11.1074	0.919253
$147$	5.24419	0.432534
$148$	20.5942	1.69283
$149$	−13.4164	−1.09911	−0.549557	−	0.835456i	$-0.685204\pi$
$149$	−13.4164	−1.09911	−0.549557	+	0.835456i	$0.685204\pi$
$150$	0	0
$151$	12.5216	1.01899	0.509496	−	0.860473i	$-0.329832\pi$
$151$	12.5216	1.01899	0.509496	+	0.860473i	$0.329832\pi$
$152$	7.47214	0.606070
$153$	−8.27895	−0.669313
$154$	−11.1074	−0.895058
$155$	0	0
$156$	11.1246	0.890682
$157$	8.70820	0.694990	0.347495	−	0.937682i	$-0.387032\pi$
$157$	8.70820	0.694990	0.347495	+	0.937682i	$0.387032\pi$
$158$	−4.24264	−0.337526
$159$	12.0000	0.951662
$160$	24.2705	1.91875
$161$	2.62210	0.206650
$162$	7.09017	0.557056
$163$	14.4164	1.12918	0.564590	−	0.825371i	$-0.309034\pi$
$163$	14.4164	1.12918	0.564590	+	0.825371i	$0.309034\pi$
$164$	−7.14590	−0.558001
$165$	−8.29180	−0.645515
$166$	−8.27895	−0.642571
$167$	−1.08036	−0.0836010	−0.0418005	−	0.999126i	$-0.513309\pi$
$167$	−1.08036	−0.0836010	−0.0418005	+	0.999126i	$0.513309\pi$
$168$	6.53089	0.503869
$169$	−6.12461	−0.471124
$170$	21.6746	1.66236
$171$	−2.23607	−0.170996
$172$	−47.0516	−3.58765
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	1.23607	0.0937061
$175$	0	0
$176$	41.8074	3.15135
$177$	−10.4397	−0.784694
$178$	−40.1869	−3.01213
$179$	13.2681	0.991705	0.495852	−	0.868407i	$-0.334856\pi$
$179$	13.2681	0.991705	0.495852	+	0.868407i	$0.334856\pi$
$180$	−24.2705	−1.80902
$181$	8.69161	0.646042	0.323021	−	0.946392i	$-0.395301\pi$
$181$	8.69161	0.646042	0.323021	+	0.946392i	$0.395301\pi$
$182$	6.86474	0.508848
$183$	−12.1803	−0.900397
$184$	−19.5927	−1.44439
$185$	9.48683	0.697486
$186$	0	0
$187$	15.7082	1.14870
$188$	−47.1246	−3.43692
$189$	−4.57649	−0.332891
$190$	5.85410	0.424701
$191$	9.76393	0.706493	0.353247	−	0.935530i	$-0.385078\pi$
$191$	9.76393	0.706493	0.353247	+	0.935530i	$0.385078\pi$
$192$	−7.61125	−0.549295
$193$	−20.7082	−1.49061	−0.745305	−	0.666724i	$-0.767696\pi$
$193$	−20.7082	−1.49061	−0.745305	+	0.666724i	$0.767696\pi$
$194$	−18.3262	−1.31575
$195$	5.12461	0.366981
$196$	−29.1246	−2.08033
$197$	−1.08036	−0.0769727	−0.0384863	−	0.999259i	$-0.512254\pi$
$197$	−1.08036	−0.0769727	−0.0384863	+	0.999259i	$0.512254\pi$
$198$	−24.8369	−1.76508
$199$	23.2163	1.64576	0.822880	−	0.568215i	$-0.192366\pi$
$199$	23.2163	1.64576	0.822880	+	0.568215i	$0.192366\pi$
$200$	0	0
$201$	−5.24419	−0.369897
$202$	5.85410	0.411893
$203$	0.540182	0.0379133
$204$	−15.7082	−1.09979
$205$	−3.29180	−0.229909
$206$	28.0344	1.95325
$207$	5.86319	0.407520
$208$	−25.8384	−1.79157
$209$	4.24264	0.293470
$210$	5.11667	0.353084
$211$	−5.00000	−0.344214	−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000	−0.344214	−0.172107	+	0.985078i	$0.555058\pi$
$212$	−66.6443	−4.57715
$213$	1.28669	0.0881628
$214$	3.85410	0.263461
$215$	−21.6746	−1.47819
$216$	34.1962	2.32675
$217$	0	0
$218$	−3.38197	−0.229056
$219$	−3.70820	−0.250577
$220$	46.0501	3.10469
$221$	−9.70820	−0.653044
$222$	−9.70820	−0.651572
$223$	16.1452	1.08117	0.540583	−	0.841291i	$-0.318204\pi$
$223$	16.1452	1.08117	0.540583	+	0.841291i	$0.318204\pi$
$224$	−10.8541	−0.725220
$225$	0	0
$226$	−25.5623	−1.70038
$227$	2.29180	0.152112	0.0760559	−	0.997104i	$-0.475767\pi$
$227$	2.29180	0.152112	0.0760559	+	0.997104i	$0.475767\pi$
$228$	−4.24264	−0.280976
$229$	4.24264	0.280362	0.140181	−	0.990126i	$-0.455232\pi$
$229$	4.24264	0.280362	0.140181	+	0.990126i	$0.455232\pi$
$230$	−15.3500	−1.01215
$231$	3.70820	0.243982
$232$	−4.03631	−0.264997
$233$	−13.4721	−0.882589	−0.441294	−	0.897362i	$-0.645481\pi$
$233$	−13.4721	−0.882589	−0.441294	+	0.897362i	$0.645481\pi$
$234$	15.3500	1.00346
$235$	−21.7082	−1.41609
$236$	57.9787	3.77409
$237$	1.41641	0.0920056
$238$	−9.69316	−0.628314
$239$	12.1089	0.783262	0.391631	−	0.920122i	$-0.371911\pi$
$239$	12.1089	0.783262	0.391631	+	0.920122i	$0.371911\pi$
$240$	−19.2588	−1.24315
$241$	−23.2163	−1.49549	−0.747747	−	0.663984i	$-0.768864\pi$
$241$	−23.2163	−1.49549	−0.747747	+	0.663984i	$0.768864\pi$
$242$	18.3262	1.17806
$243$	−16.0965	−1.03259
$244$	67.6458	4.33058
$245$	−13.4164	−0.857143
$246$	3.36861	0.214775
$247$	−2.62210	−0.166840
$248$	0	0
$249$	2.76393	0.175157
$250$	−29.2705	−1.85123
$251$	11.0286	0.696117	0.348058	−	0.937473i	$-0.386841\pi$
$251$	11.0286	0.696117	0.348058	+	0.937473i	$0.386841\pi$
$252$	10.8541	0.683744
$253$	−11.1246	−0.699398
$254$	−24.8369	−1.55840
$255$	−7.23607	−0.453140
$256$	−14.5623	−0.910144
$257$	6.05573	0.377746	0.188873	−	0.982002i	$-0.439517\pi$
$257$	6.05573	0.377746	0.188873	+	0.982002i	$0.439517\pi$
$258$	22.1803	1.38089
$259$	−4.24264	−0.263625
$260$	−28.4605	−1.76505
$261$	1.20788	0.0747661
$262$	19.4164	1.19955
$263$	18.5123	1.14152	0.570759	−	0.821118i	$-0.306649\pi$
$263$	18.5123	1.14152	0.570759	+	0.821118i	$0.306649\pi$
$264$	−27.7082	−1.70532
$265$	−30.7000	−1.88589
$266$	−2.61803	−0.160522
$267$	13.4164	0.821071
$268$	29.1246	1.77907
$269$	2.16073	0.131742	0.0658709	−	0.997828i	$-0.479017\pi$
$269$	2.16073	0.131742	0.0658709	+	0.997828i	$0.479017\pi$
$270$	26.7912	1.63046
$271$	−18.5911	−1.12933	−0.564665	−	0.825320i	$-0.690994\pi$
$271$	−18.5911	−1.12933	−0.564665	+	0.825320i	$0.690994\pi$
$272$	36.4844	2.21219
$273$	−2.29180	−0.138706
$274$	−6.86474	−0.414714
$275$	0	0
$276$	11.1246	0.669623
$277$	10.1058	0.607200	0.303600	−	0.952800i	$-0.401811\pi$
$277$	10.1058	0.607200	0.303600	+	0.952800i	$0.401811\pi$
$278$	−47.0516	−2.82197
$279$	0	0
$280$	−16.7082	−0.998506
$281$	13.3607	0.797031	0.398516	−	0.917162i	$-0.369525\pi$
$281$	13.3607	0.797031	0.398516	+	0.917162i	$0.369525\pi$
$282$	22.2148	1.32287
$283$	24.0000	1.42665	0.713326	−	0.700832i	$-0.247188\pi$
$283$	24.0000	1.42665	0.713326	+	0.700832i	$0.247188\pi$
$284$	−7.14590	−0.424031
$285$	−1.95440	−0.115768
$286$	−29.1246	−1.72217
$287$	1.47214	0.0868974
$288$	−24.2705	−1.43015
$289$	−3.29180	−0.193635
$290$	−3.16228	−0.185695
$291$	6.11822	0.358657
$292$	20.5942	1.20519
$293$	−2.29180	−0.133888	−0.0669441	−	0.997757i	$-0.521325\pi$
$293$	−2.29180	−0.133888	−0.0669441	+	0.997757i	$0.521325\pi$
$294$	13.7295	0.800719
$295$	26.7082	1.55501
$296$	31.7016	1.84262
$297$	19.4164	1.12665
$298$	−35.1246	−2.03471
$299$	6.87539	0.397614
$300$	0	0
$301$	9.69316	0.558705
$302$	32.7820	1.88639
$303$	−1.95440	−0.112277
$304$	9.85410	0.565172
$305$	31.1614	1.78430
$306$	−21.6746	−1.23905
$307$	24.1246	1.37686	0.688432	−	0.725301i	$-0.258299\pi$
$307$	24.1246	1.37686	0.688432	+	0.725301i	$0.258299\pi$
$308$	−20.5942	−1.17346
$309$	−9.35931	−0.532433
$310$	0	0
$311$	25.4721	1.44439	0.722196	−	0.691688i	$-0.243133\pi$
$311$	25.4721	1.44439	0.722196	+	0.691688i	$0.243133\pi$
$312$	17.1246	0.969490
$313$	−11.5200	−0.651151	−0.325576	−	0.945516i	$-0.605558\pi$
$313$	−11.5200	−0.651151	−0.325576	+	0.945516i	$0.605558\pi$
$314$	22.7984	1.28659
$315$	5.00000	0.281718
$316$	−7.86629	−0.442513
$317$	26.2361	1.47356	0.736782	−	0.676130i	$-0.236344\pi$
$317$	26.2361	1.47356	0.736782	+	0.676130i	$0.236344\pi$
$318$	31.4164	1.76174
$319$	−2.29180	−0.128316
$320$	19.4721	1.08853
$321$	−1.28669	−0.0718163
$322$	6.86474	0.382557
$323$	3.70246	0.206010
$324$	13.1459	0.730328
$325$	0	0
$326$	37.7426	2.09037
$327$	1.12907	0.0624378
$328$	−11.0000	−0.607373
$329$	9.70820	0.535231
$330$	−21.7082	−1.19500
$331$	31.2889	1.71979	0.859897	−	0.510467i	$-0.170527\pi$
$331$	31.2889	1.71979	0.859897	+	0.510467i	$0.170527\pi$
$332$	−15.3500	−0.842442
$333$	−9.48683	−0.519875
$334$	−2.82843	−0.154765
$335$	13.4164	0.733017
$336$	8.61280	0.469867
$337$	−15.3500	−0.836169	−0.418084	−	0.908408i	$-0.637298\pi$
$337$	−15.3500	−0.836169	−0.418084	+	0.908408i	$0.637298\pi$
$338$	−16.0344	−0.872159
$339$	8.53399	0.463503
$340$	40.1869	2.17944
$341$	0	0
$342$	−5.85410	−0.316554
$343$	13.0000	0.701934
$344$	−72.4286	−3.90509
$345$	5.12461	0.275900
$346$	47.1246	2.53343
$347$	−21.5958	−1.15932	−0.579661	−	0.814858i	$-0.696815\pi$
$347$	−21.5958	−1.15932	−0.579661	+	0.814858i	$0.696815\pi$
$348$	2.29180	0.122853
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	0	0
$351$	−12.0000	−0.640513
$352$	46.0501	2.45448
$353$	9.56564	0.509128	0.254564	−	0.967056i	$-0.418068\pi$
$353$	9.56564	0.509128	0.254564	+	0.967056i	$0.418068\pi$
$354$	−27.3314	−1.45265
$355$	−3.29180	−0.174710
$356$	−74.5106	−3.94905
$357$	3.23607	0.171271
$358$	34.7363	1.83587
$359$	14.2361	0.751351	0.375675	−	0.926751i	$-0.377411\pi$
$359$	14.2361	0.751351	0.375675	+	0.926751i	$0.377411\pi$
$360$	−37.3607	−1.96908
$361$	−18.0000	−0.947368
$362$	22.7549	1.19597
$363$	−6.11822	−0.321123
$364$	12.7279	0.667124
$365$	9.48683	0.496564
$366$	−31.8885	−1.66684
$367$	6.65841	0.347566	0.173783	−	0.984784i	$-0.444401\pi$
$367$	6.65841	0.347566	0.173783	+	0.984784i	$0.444401\pi$
$368$	−25.8384	−1.34692
$369$	3.29180	0.171364
$370$	24.8369	1.29121
$371$	13.7295	0.712799
$372$	0	0
$373$	−9.29180	−0.481111	−0.240555	−	0.970635i	$-0.577330\pi$
$373$	−9.29180	−0.481111	−0.240555	+	0.970635i	$0.577330\pi$
$374$	41.1246	2.12650
$375$	9.77198	0.504623
$376$	−72.5410	−3.74102
$377$	1.41641	0.0729487
$378$	−11.9814	−0.616257
$379$	7.41641	0.380955	0.190478	−	0.981692i	$-0.438996\pi$
$379$	7.41641	0.380955	0.190478	+	0.981692i	$0.438996\pi$
$380$	10.8541	0.556804
$381$	8.29180	0.424802
$382$	25.5623	1.30788
$383$	−6.40337	−0.327197	−0.163598	−	0.986527i	$-0.552310\pi$
$383$	−6.40337	−0.327197	−0.163598	+	0.986527i	$0.552310\pi$
$384$	−0.952843	−0.0486246
$385$	−9.48683	−0.483494
$386$	−54.2148	−2.75946
$387$	21.6746	1.10178
$388$	−33.9787	−1.72501
$389$	−16.8129	−0.852450	−0.426225	−	0.904617i	$-0.640157\pi$
$389$	−16.8129	−0.852450	−0.426225	+	0.904617i	$0.640157\pi$
$390$	13.4164	0.679366
$391$	−9.70820	−0.490965
$392$	−44.8328	−2.26440
$393$	−6.48218	−0.326983
$394$	−2.82843	−0.142494
$395$	−3.62365	−0.182326
$396$	−46.0501	−2.31410
$397$	14.7082	0.738184	0.369092	−	0.929393i	$-0.379669\pi$
$397$	14.7082	0.738184	0.369092	+	0.929393i	$0.379669\pi$
$398$	60.7811	3.04668
$399$	0.874032	0.0437563
$400$	0	0
$401$	−0.540182	−0.0269754	−0.0134877	−	0.999909i	$-0.504293\pi$
$401$	−0.540182	−0.0269754	−0.0134877	+	0.999909i	$0.504293\pi$
$402$	−13.7295	−0.684764
$403$	0	0
$404$	10.8541	0.540012
$405$	6.05573	0.300912
$406$	1.41421	0.0701862
$407$	18.0000	0.892227
$408$	−24.1803	−1.19711
$409$	30.0810	1.48741	0.743706	−	0.668507i	$-0.233066\pi$
$409$	30.0810	1.48741	0.743706	+	0.668507i	$0.233066\pi$
$410$	−8.61803	−0.425614
$411$	2.29180	0.113046
$412$	51.9787	2.56081
$413$	−11.9443	−0.587739
$414$	15.3500	0.754412
$415$	−7.07107	−0.347105
$416$	−28.4605	−1.39539
$417$	15.7082	0.769234
$418$	11.1074	0.543280
$419$	30.5967	1.49475	0.747374	−	0.664403i	$-0.231314\pi$
$419$	30.5967	1.49475	0.747374	+	0.664403i	$0.231314\pi$
$420$	9.48683	0.462910
$421$	26.4164	1.28746	0.643728	−	0.765254i	$-0.277387\pi$
$421$	26.4164	1.28746	0.643728	+	0.765254i	$0.277387\pi$
$422$	−13.0902	−0.637220
$423$	21.7082	1.05549
$424$	−102.588	−4.98214
$425$	0	0
$426$	3.36861	0.163210
$427$	−13.9358	−0.674401
$428$	7.14590	0.345410
$429$	9.72327	0.469444
$430$	−56.7448	−2.73648
$431$	−35.8885	−1.72869	−0.864345	−	0.502899i	$-0.832267\pi$
$431$	−35.8885	−1.72869	−0.864345	+	0.502899i	$0.832267\pi$
$432$	45.0972	2.16974
$433$	−25.8384	−1.24171	−0.620857	−	0.783924i	$-0.713215\pi$
$433$	−25.8384	−1.24171	−0.620857	+	0.783924i	$0.713215\pi$
$434$	0	0
$435$	1.05573	0.0506183
$436$	−6.27051	−0.300303
$437$	−2.62210	−0.125432
$438$	−9.70820	−0.463876
$439$	−25.0000	−1.19318	−0.596592	−	0.802544i	$-0.703479\pi$
$439$	−25.0000	−1.19318	−0.596592	+	0.802544i	$0.703479\pi$
$440$	70.8869	3.37940
$441$	13.4164	0.638877
$442$	−25.4164	−1.20894
$443$	6.05573	0.287716	0.143858	−	0.989598i	$-0.454049\pi$
$443$	6.05573	0.287716	0.143858	+	0.989598i	$0.454049\pi$
$444$	−18.0000	−0.854242
$445$	−34.3237	−1.62710
$446$	42.2688	2.00148
$447$	11.7264	0.554638
$448$	−8.70820	−0.411424
$449$	33.7047	1.59062	0.795311	−	0.606201i	$-0.207307\pi$
$449$	33.7047	1.59062	0.795311	+	0.606201i	$0.207307\pi$
$450$	0	0
$451$	−6.24574	−0.294101
$452$	−47.3951	−2.22928
$453$	−10.9443	−0.514207
$454$	6.00000	0.281594
$455$	5.86319	0.274870
$456$	−6.53089	−0.305837
$457$	11.3137	0.529233	0.264616	−	0.964354i	$-0.414755\pi$
$457$	11.3137	0.529233	0.264616	+	0.964354i	$0.414755\pi$
$458$	11.1074	0.519014
$459$	16.9443	0.790891
$460$	−28.4605	−1.32698
$461$	5.16538	0.240576	0.120288	−	0.992739i	$-0.461618\pi$
$461$	5.16538	0.240576	0.120288	+	0.992739i	$0.461618\pi$
$462$	9.70820	0.451667
$463$	−3.82998	−0.177994	−0.0889971	−	0.996032i	$-0.528366\pi$
$463$	−3.82998	−0.177994	−0.0889971	+	0.996032i	$0.528366\pi$
$464$	−5.32300	−0.247114
$465$	0	0
$466$	−35.2705	−1.63387
$467$	−15.6525	−0.724310	−0.362155	−	0.932118i	$-0.617959\pi$
$467$	−15.6525	−0.724310	−0.362155	+	0.932118i	$0.617959\pi$
$468$	28.4605	1.31559
$469$	−6.00000	−0.277054
$470$	−56.8328	−2.62150
$471$	−7.61125	−0.350708
$472$	89.2492	4.10803
$473$	−41.1246	−1.89091
$474$	3.70820	0.170323
$475$	0	0
$476$	−17.9721	−0.823751
$477$	30.7000	1.40566
$478$	31.7016	1.45000
$479$	16.5279	0.755177	0.377589	−	0.925973i	$-0.376753\pi$
$479$	16.5279	0.755177	0.377589	+	0.925973i	$0.376753\pi$
$480$	−21.2132	−0.968246
$481$	−11.1246	−0.507239
$482$	−60.7811	−2.76850
$483$	−2.29180	−0.104280
$484$	33.9787	1.54449
$485$	−15.6525	−0.710742
$486$	−42.1413	−1.91157
$487$	28.0779	1.27233	0.636166	−	0.771552i	$-0.280519\pi$
$487$	28.0779	1.27233	0.636166	+	0.771552i	$0.280519\pi$
$488$	104.130	4.71375
$489$	−12.6004	−0.569810
$490$	−35.1246	−1.58677
$491$	−15.8902	−0.717115	−0.358557	−	0.933508i	$-0.616731\pi$
$491$	−15.8902	−0.717115	−0.358557	+	0.933508i	$0.616731\pi$
$492$	6.24574	0.281580
$493$	−2.00000	−0.0900755
$494$	−6.86474	−0.308859
$495$	−21.2132	−0.953463
$496$	0	0
$497$	1.47214	0.0660343
$498$	7.23607	0.324256
$499$	18.3848	0.823016	0.411508	−	0.911406i	$-0.365002\pi$
$499$	18.3848	0.823016	0.411508	+	0.911406i	$0.365002\pi$
$500$	−54.2705	−2.42705
$501$	0.944272	0.0421870
$502$	28.8732	1.28867
$503$	−32.8885	−1.46643	−0.733214	−	0.679998i	$-0.761981\pi$
$503$	−32.8885	−1.46643	−0.733214	+	0.679998i	$0.761981\pi$
$504$	16.7082	0.744243
$505$	5.00000	0.222497
$506$	−29.1246	−1.29475
$507$	5.35311	0.237740
$508$	−46.0501	−2.04314
$509$	−7.94510	−0.352160	−0.176080	−	0.984376i	$-0.556342\pi$
$509$	−7.94510	−0.352160	−0.176080	+	0.984376i	$0.556342\pi$
$510$	−18.9443	−0.838866
$511$	−4.24264	−0.187683
$512$	−40.3050	−1.78124
$513$	4.57649	0.202057
$514$	15.8541	0.699294
$515$	23.9443	1.05511
$516$	41.1246	1.81041
$517$	−41.1884	−1.81146
$518$	−11.1074	−0.488030
$519$	−15.7326	−0.690583
$520$	−43.8105	−1.92122
$521$	−13.4164	−0.587784	−0.293892	−	0.955839i	$-0.594951\pi$
$521$	−13.4164	−0.587784	−0.293892	+	0.955839i	$0.594951\pi$
$522$	3.16228	0.138409
$523$	0.618993	0.0270667	0.0135333	−	0.999908i	$-0.495692\pi$
$523$	0.618993	0.0270667	0.0135333	+	0.999908i	$0.495692\pi$
$524$	36.0000	1.57267
$525$	0	0
$526$	48.4658	2.11321
$527$	0	0
$528$	−36.5410	−1.59024
$529$	−16.1246	−0.701070
$530$	−80.3737	−3.49121
$531$	−26.7082	−1.15904
$532$	−4.85410	−0.210452
$533$	3.86008	0.167199
$534$	35.1246	1.51999
$535$	3.29180	0.142317
$536$	44.8328	1.93648
$537$	−11.5967	−0.500437
$538$	5.65685	0.243884
$539$	−25.4558	−1.09646
$540$	49.6737	2.13762
$541$	25.8328	1.11064	0.555320	−	0.831637i	$-0.312596\pi$
$541$	25.8328	1.11064	0.555320	+	0.831637i	$0.312596\pi$
$542$	−48.6722	−2.09065
$543$	−7.59675	−0.326008
$544$	40.1869	1.72300
$545$	−2.88854	−0.123732
$546$	−6.00000	−0.256776
$547$	2.41641	0.103318	0.0516591	−	0.998665i	$-0.483549\pi$
$547$	2.41641	0.103318	0.0516591	+	0.998665i	$0.483549\pi$
$548$	−12.7279	−0.543710
$549$	−31.1614	−1.32994
$550$	0	0
$551$	−0.540182	−0.0230125
$552$	17.1246	0.728872
$553$	1.62054	0.0689126
$554$	26.4574	1.12407
$555$	−8.29180	−0.351967
$556$	−87.2385	−3.69974
$557$	−10.6460	−0.451086	−0.225543	−	0.974233i	$-0.572416\pi$
$557$	−10.6460	−0.451086	−0.225543	+	0.974233i	$0.572416\pi$
$558$	0	0
$559$	25.4164	1.07500
$560$	−22.0344	−0.931125
$561$	−13.7295	−0.579659
$562$	34.9787	1.47549
$563$	9.76393	0.411501	0.205750	−	0.978605i	$-0.434037\pi$
$563$	9.76393	0.411501	0.205750	+	0.978605i	$0.434037\pi$
$564$	41.1884	1.73435
$565$	−21.8328	−0.918513
$566$	62.8328	2.64106
$567$	−2.70820	−0.113734
$568$	−11.0000	−0.461550
$569$	−11.7264	−0.491595	−0.245798	−	0.969321i	$-0.579050\pi$
$569$	−11.7264	−0.491595	−0.245798	+	0.969321i	$0.579050\pi$
$570$	−5.11667	−0.214314
$571$	−2.62210	−0.109731	−0.0548657	−	0.998494i	$-0.517473\pi$
$571$	−2.62210	−0.109731	−0.0548657	+	0.998494i	$0.517473\pi$
$572$	−54.0000	−2.25785
$573$	−8.53399	−0.356513
$574$	3.85410	0.160867
$575$	0	0
$576$	−19.4721	−0.811339
$577$	−30.0000	−1.24892	−0.624458	−	0.781058i	$-0.714680\pi$
$577$	−30.0000	−1.24892	−0.624458	+	0.781058i	$0.714680\pi$
$578$	−8.61803	−0.358463
$579$	18.0996	0.752195
$580$	−5.86319	−0.243456
$581$	3.16228	0.131193
$582$	16.0177	0.663956
$583$	−58.2492	−2.41244
$584$	31.7016	1.31182
$585$	13.1105	0.542052
$586$	−6.00000	−0.247858
$587$	−15.7326	−0.649353	−0.324676	−	0.945825i	$-0.605255\pi$
$587$	−15.7326	−0.649353	−0.324676	+	0.945825i	$0.605255\pi$
$588$	25.4558	1.04978
$589$	0	0
$590$	69.9230	2.87868
$591$	0.944272	0.0388422
$592$	41.8074	1.71827
$593$	5.18034	0.212731	0.106366	−	0.994327i	$-0.466079\pi$
$593$	5.18034	0.212731	0.106366	+	0.994327i	$0.466079\pi$
$594$	50.8328	2.08570
$595$	−8.27895	−0.339404
$596$	−65.1246	−2.66761
$597$	−20.2918	−0.830488
$598$	18.0000	0.736075
$599$	6.05573	0.247430	0.123715	−	0.992318i	$-0.460519\pi$
$599$	6.05573	0.247430	0.123715	+	0.992318i	$0.460519\pi$
$600$	0	0
$601$	18.1784	0.741514	0.370757	−	0.928730i	$-0.379098\pi$
$601$	18.1784	0.741514	0.370757	+	0.928730i	$0.379098\pi$
$602$	25.3770	1.03429
$603$	−13.4164	−0.546358
$604$	60.7811	2.47315
$605$	15.6525	0.636364
$606$	−5.11667	−0.207851
$607$	15.1246	0.613889	0.306945	−	0.951727i	$-0.400693\pi$
$607$	15.1246	0.613889	0.306945	+	0.951727i	$0.400693\pi$
$608$	10.8541	0.440192
$609$	−0.472136	−0.0191319
$610$	81.5816	3.30314
$611$	25.4558	1.02983
$612$	−40.1869	−1.62446
$613$	0.588890	0.0237850	0.0118925	−	0.999929i	$-0.496214\pi$
$613$	0.588890	0.0237850	0.0118925	+	0.999929i	$0.496214\pi$
$614$	63.1591	2.54889
$615$	2.87714	0.116017
$616$	−31.7016	−1.27729
$617$	−6.87539	−0.276793	−0.138396	−	0.990377i	$-0.544195\pi$
$617$	−6.87539	−0.276793	−0.138396	+	0.990377i	$0.544195\pi$
$618$	−24.5030	−0.985655
$619$	25.8384	1.03853	0.519267	−	0.854612i	$-0.326205\pi$
$619$	25.8384	1.03853	0.519267	+	0.854612i	$0.326205\pi$
$620$	0	0
$621$	−12.0000	−0.481543
$622$	66.6869	2.67390
$623$	15.3500	0.614985
$624$	22.5836	0.904067
$625$	−25.0000	−1.00000
$626$	−30.1599	−1.20543
$627$	−3.70820	−0.148091
$628$	42.2705	1.68678
$629$	15.7082	0.626327
$630$	13.0902	0.521525
$631$	−23.6290	−0.940654	−0.470327	−	0.882492i	$-0.655864\pi$
$631$	−23.6290	−0.940654	−0.470327	+	0.882492i	$0.655864\pi$
$632$	−12.1089	−0.481667
$633$	4.37016	0.173698
$634$	68.6869	2.72791
$635$	−21.2132	−0.841820
$636$	58.2492	2.30973
$637$	15.7326	0.623347
$638$	−6.00000	−0.237542
$639$	3.29180	0.130221
$640$	2.43769	0.0963583
$641$	−8.48528	−0.335148	−0.167574	−	0.985859i	$-0.553593\pi$
$641$	−8.48528	−0.335148	−0.167574	+	0.985859i	$0.553593\pi$
$642$	−3.36861	−0.132948
$643$	−4.65530	−0.183587	−0.0917936	−	0.995778i	$-0.529260\pi$
$643$	−4.65530	−0.183587	−0.0917936	+	0.995778i	$0.529260\pi$
$644$	12.7279	0.501550
$645$	18.9443	0.745930
$646$	9.69316	0.381372
$647$	31.7804	1.24942	0.624708	−	0.780858i	$-0.285218\pi$
$647$	31.7804	1.24942	0.624708	+	0.780858i	$0.285218\pi$
$648$	20.2361	0.794948
$649$	50.6753	1.98918
$650$	0	0
$651$	0	0
$652$	69.9787	2.74058
$653$	24.6525	0.964726	0.482363	−	0.875971i	$-0.339779\pi$
$653$	24.6525	0.964726	0.482363	+	0.875971i	$0.339779\pi$
$654$	2.95595	0.115587
$655$	16.5836	0.647975
$656$	−14.5066	−0.566387
$657$	−9.48683	−0.370117
$658$	25.4164	0.990835
$659$	24.8197	0.966837	0.483418	−	0.875389i	$-0.339395\pi$
$659$	24.8197	0.966837	0.483418	+	0.875389i	$0.339395\pi$
$660$	−40.2492	−1.56670
$661$	7.83282	0.304661	0.152331	−	0.988330i	$-0.451322\pi$
$661$	7.83282	0.304661	0.152331	+	0.988330i	$0.451322\pi$
$662$	81.9155	3.18374
$663$	8.48528	0.329541
$664$	−23.6290	−0.916982
$665$	−2.23607	−0.0867110
$666$	−24.8369	−0.962408
$667$	1.41641	0.0548435
$668$	−5.24419	−0.202904
$669$	−14.1115	−0.545580
$670$	35.1246	1.35698
$671$	59.1246	2.28248
$672$	9.48683	0.365963
$673$	39.9805	1.54114	0.770568	−	0.637357i	$-0.219973\pi$
$673$	39.9805	1.54114	0.770568	+	0.637357i	$0.219973\pi$
$674$	−40.1869	−1.54794
$675$	0	0
$676$	−29.7295	−1.14344
$677$	−21.2132	−0.815290	−0.407645	−	0.913141i	$-0.633650\pi$
$677$	−21.2132	−0.815290	−0.407645	+	0.913141i	$0.633650\pi$
$678$	22.3423	0.858050
$679$	7.00000	0.268635
$680$	61.8614	2.37228
$681$	−2.00310	−0.0767591
$682$	0	0
$683$	−41.1803	−1.57572	−0.787861	−	0.615853i	$-0.788811\pi$
$683$	−41.1803	−1.57572	−0.787861	+	0.615853i	$0.788811\pi$
$684$	−10.8541	−0.415017
$685$	−5.86319	−0.224021
$686$	34.0344	1.29944
$687$	−3.70820	−0.141477
$688$	−95.5174	−3.64157
$689$	36.0000	1.37149
$690$	13.4164	0.510754
$691$	−48.4164	−1.84185	−0.920923	−	0.389743i	$-0.872564\pi$
$691$	−48.4164	−1.84185	−0.920923	+	0.389743i	$0.872564\pi$
$692$	87.3738	3.32145
$693$	9.48683	0.360375
$694$	−56.5384	−2.14617
$695$	−40.1869	−1.52437
$696$	3.52786	0.133723
$697$	−5.45052	−0.206453
$698$	15.7082	0.594564
$699$	11.7751	0.445374
$700$	0	0
$701$	−3.65248	−0.137952	−0.0689761	−	0.997618i	$-0.521973\pi$
$701$	−3.65248	−0.137952	−0.0689761	+	0.997618i	$0.521973\pi$
$702$	−31.4164	−1.18574
$703$	4.24264	0.160014
$704$	36.9458	1.39245
$705$	18.9737	0.714590
$706$	25.0432	0.942513
$707$	−2.23607	−0.0840960
$708$	−50.6753	−1.90449
$709$	−41.1884	−1.54686	−0.773432	−	0.633880i	$-0.781462\pi$
$709$	−41.1884	−1.54686	−0.773432	+	0.633880i	$0.781462\pi$
$710$	−8.61803	−0.323429
$711$	3.62365	0.135897
$712$	−114.697	−4.29847
$713$	0	0
$714$	8.47214	0.317062
$715$	−24.8754	−0.930287
$716$	64.4047	2.40692
$717$	−10.5836	−0.395251
$718$	37.2705	1.39092
$719$	−37.5648	−1.40093	−0.700465	−	0.713687i	$-0.747024\pi$
$719$	−37.5648	−1.40093	−0.700465	+	0.713687i	$0.747024\pi$
$720$	−49.2705	−1.83620
$721$	−10.7082	−0.398794
$722$	−47.1246	−1.75380
$723$	20.2918	0.754660
$724$	42.1900	1.56798
$725$	0	0
$726$	−16.0177	−0.594473
$727$	−30.4164	−1.12808	−0.564041	−	0.825747i	$-0.690754\pi$
$727$	−30.4164	−1.12808	−0.564041	+	0.825747i	$0.690754\pi$
$728$	19.5927	0.726152
$729$	5.94427	0.220158
$730$	24.8369	0.919253
$731$	−35.8885	−1.32739
$732$	−59.1246	−2.18531
$733$	−14.4164	−0.532482	−0.266241	−	0.963906i	$-0.585782\pi$
$733$	−14.4164	−0.532482	−0.266241	+	0.963906i	$0.585782\pi$
$734$	17.4319	0.643424
$735$	11.7264	0.432534
$736$	−28.4605	−1.04907
$737$	25.4558	0.937678
$738$	8.61803	0.317234
$739$	−35.1490	−1.29298	−0.646489	−	0.762924i	$-0.723763\pi$
$739$	−35.1490	−1.29298	−0.646489	+	0.762924i	$0.723763\pi$
$740$	46.0501	1.69283
$741$	2.29180	0.0841912
$742$	35.9442	1.31955
$743$	−36.4844	−1.33848	−0.669242	−	0.743045i	$-0.733381\pi$
$743$	−36.4844	−1.33848	−0.669242	+	0.743045i	$0.733381\pi$
$744$	0	0
$745$	−30.0000	−1.09911
$746$	−24.3262	−0.890647
$747$	7.07107	0.258717
$748$	76.2492	2.78795
$749$	−1.47214	−0.0537907
$750$	25.5834	0.934172
$751$	−37.5410	−1.36989	−0.684946	−	0.728594i	$-0.740174\pi$
$751$	−37.5410	−1.36989	−0.684946	+	0.728594i	$0.740174\pi$
$752$	−95.6656	−3.48857
$753$	−9.63932	−0.351276
$754$	3.70820	0.135045
$755$	27.9991	1.01899
$756$	−22.2148	−0.807943
$757$	9.28050	0.337306	0.168653	−	0.985676i	$-0.446058\pi$
$757$	9.28050	0.337306	0.168653	+	0.985676i	$0.446058\pi$
$758$	19.4164	0.705236
$759$	9.72327	0.352932
$760$	16.7082	0.606070
$761$	40.1869	1.45677	0.728386	−	0.685167i	$-0.240271\pi$
$761$	40.1869	1.45677	0.728386	+	0.685167i	$0.240271\pi$
$762$	21.7082	0.786405
$763$	1.29180	0.0467662
$764$	47.3951	1.71470
$765$	−18.5123	−0.669313
$766$	−16.7642	−0.605716
$767$	−31.3190	−1.13086
$768$	12.7279	0.459279
$769$	54.1246	1.95178	0.975892	−	0.218255i	$-0.0700365\pi$
$769$	54.1246	1.95178	0.975892	+	0.218255i	$0.0700365\pi$
$770$	−24.8369	−0.895058
$771$	−5.29290	−0.190619
$772$	−100.520	−3.61778
$773$	−11.1862	−0.402339	−0.201170	−	0.979556i	$-0.564474\pi$
$773$	−11.1862	−0.402339	−0.201170	+	0.979556i	$0.564474\pi$
$774$	56.7448	2.03965
$775$	0	0
$776$	−52.3050	−1.87764
$777$	3.70820	0.133031
$778$	−44.0168	−1.57808
$779$	−1.47214	−0.0527447
$780$	24.8754	0.890682
$781$	−6.24574	−0.223490
$782$	−25.4164	−0.908889
$783$	−2.47214	−0.0883469
$784$	−59.1246	−2.11159
$785$	19.4721	0.694990
$786$	−16.9706	−0.605320
$787$	−37.5648	−1.33904	−0.669520	−	0.742794i	$-0.733500\pi$
$787$	−37.5648	−1.33904	−0.669520	+	0.742794i	$0.733500\pi$
$788$	−5.24419	−0.186817
$789$	−16.1803	−0.576035
$790$	−9.48683	−0.337526
$791$	9.76393	0.347165
$792$	−70.8869	−2.51886
$793$	−36.5410	−1.29761
$794$	38.5066	1.36655
$795$	26.8328	0.951662
$796$	112.694	3.99434
$797$	−13.7295	−0.486323	−0.243161	−	0.969986i	$-0.578184\pi$
$797$	−13.7295	−0.486323	−0.243161	+	0.969986i	$0.578184\pi$
$798$	2.28825	0.0810030
$799$	−35.9442	−1.27162
$800$	0	0
$801$	34.3237	1.21277
$802$	−1.41421	−0.0499376
$803$	18.0000	0.635206
$804$	−25.4558	−0.897758
$805$	5.86319	0.206650
$806$	0	0
$807$	−1.88854	−0.0664799
$808$	16.7082	0.587793
$809$	47.5130	1.67047	0.835234	−	0.549895i	$-0.185332\pi$
$809$	47.5130	1.67047	0.835234	+	0.549895i	$0.185332\pi$
$810$	15.8541	0.557056
$811$	−16.5836	−0.582329	−0.291164	−	0.956673i	$-0.594043\pi$
$811$	−16.5836	−0.582329	−0.291164	+	0.956673i	$0.594043\pi$
$812$	2.62210	0.0920175
$813$	16.2492	0.569885
$814$	47.1246	1.65172
$815$	32.2361	1.12918
$816$	−31.8885	−1.11632
$817$	−9.69316	−0.339121
$818$	78.7532	2.75354
$819$	−5.86319	−0.204876
$820$	−15.9787	−0.558001
$821$	23.8353	0.831858	0.415929	−	0.909397i	$-0.363457\pi$
$821$	23.8353	0.831858	0.415929	+	0.909397i	$0.363457\pi$
$822$	6.00000	0.209274
$823$	−13.5231	−0.471387	−0.235694	−	0.971827i	$-0.575736\pi$
$823$	−13.5231	−0.471387	−0.235694	+	0.971827i	$0.575736\pi$
$824$	80.0132	2.78739
$825$	0	0
$826$	−31.2705	−1.08804
$827$	−18.3547	−0.638255	−0.319127	−	0.947712i	$-0.603390\pi$
$827$	−18.3547	−0.638255	−0.319127	+	0.947712i	$0.603390\pi$
$828$	28.4605	0.989071
$829$	51.0879	1.77436	0.887178	−	0.461427i	$-0.152662\pi$
$829$	51.0879	1.77436	0.887178	+	0.461427i	$0.152662\pi$
$830$	−18.5123	−0.642571
$831$	−8.83282	−0.306407
$832$	−22.8337	−0.791618
$833$	−22.2148	−0.769696
$834$	41.1246	1.42403
$835$	−2.41577	−0.0836010
$836$	20.5942	0.712266
$837$	0	0
$838$	80.1033	2.76712
$839$	−19.4164	−0.670329	−0.335164	−	0.942160i	$-0.608792\pi$
$839$	−19.4164	−0.670329	−0.335164	+	0.942160i	$0.608792\pi$
$840$	14.6035	0.503869
$841$	−28.7082	−0.989938
$842$	69.1591	2.38338
$843$	−11.6777	−0.402200
$844$	−24.2705	−0.835425
$845$	−13.6950	−0.471124
$846$	56.8328	1.95395
$847$	−7.00000	−0.240523
$848$	−135.292	−4.64593
$849$	−20.9768	−0.719921
$850$	0	0
$851$	−11.1246	−0.381347
$852$	6.24574	0.213976
$853$	−32.2918	−1.10565	−0.552825	−	0.833297i	$-0.686450\pi$
$853$	−32.2918	−1.10565	−0.552825	+	0.833297i	$0.686450\pi$
$854$	−36.4844	−1.24847
$855$	−5.00000	−0.170996
$856$	11.0000	0.375972
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	25.4558	0.869048
$859$	−41.6011	−1.41941	−0.709705	−	0.704499i	$-0.751172\pi$
$859$	−41.6011	−1.41941	−0.709705	+	0.704499i	$0.751172\pi$
$860$	−105.211	−3.58765
$861$	−1.28669	−0.0438504
$862$	−93.9574	−3.20020
$863$	−31.0826	−1.05806	−0.529032	−	0.848602i	$-0.677445\pi$
$863$	−31.0826	−1.05806	−0.529032	+	0.848602i	$0.677445\pi$
$864$	49.6737	1.68993
$865$	40.2492	1.36851
$866$	−67.6458	−2.29870
$867$	2.87714	0.0977126
$868$	0	0
$869$	−6.87539	−0.233232
$870$	2.76393	0.0937061
$871$	−15.7326	−0.533078
$872$	−9.65248	−0.326874
$873$	15.6525	0.529756
$874$	−6.86474	−0.232203
$875$	11.1803	0.377964
$876$	−18.0000	−0.608164
$877$	−28.7082	−0.969407	−0.484704	−	0.874678i	$-0.661073\pi$
$877$	−28.7082	−0.969407	−0.484704	+	0.874678i	$0.661073\pi$
$878$	−65.4508	−2.20886
$879$	2.00310	0.0675630
$880$	93.4842	3.15135
$881$	7.24730	0.244168	0.122084	−	0.992520i	$-0.461042\pi$
$881$	7.24730	0.244168	0.122084	+	0.992520i	$0.461042\pi$
$882$	35.1246	1.18271
$883$	−33.9411	−1.14221	−0.571105	−	0.820877i	$-0.693485\pi$
$883$	−33.9411	−1.14221	−0.571105	+	0.820877i	$0.693485\pi$
$884$	−47.1246	−1.58497
$885$	−23.3438	−0.784694
$886$	15.8541	0.532629
$887$	25.3607	0.851528	0.425764	−	0.904834i	$-0.360005\pi$
$887$	25.3607	0.851528	0.425764	+	0.904834i	$0.360005\pi$
$888$	−27.7082	−0.929826
$889$	9.48683	0.318178
$890$	−89.8606	−3.01213
$891$	11.4899	0.384927
$892$	78.3706	2.62404
$893$	−9.70820	−0.324873
$894$	30.7000	1.02676
$895$	29.6684	0.991705
$896$	−1.09017	−0.0364200
$897$	−6.00931	−0.200645
$898$	88.2400	2.94461
$899$	0	0
$900$	0	0
$901$	−50.8328	−1.69349
$902$	−16.3516	−0.544448
$903$	−8.47214	−0.281935
$904$	−72.9574	−2.42653
$905$	19.4350	0.646042
$906$	−28.6525	−0.951915
$907$	11.5410	0.383213	0.191607	−	0.981472i	$-0.438630\pi$
$907$	11.5410	0.383213	0.191607	+	0.981472i	$0.438630\pi$
$908$	11.1246	0.369183
$909$	−5.00000	−0.165840
$910$	15.3500	0.508848
$911$	−26.9188	−0.891859	−0.445929	−	0.895068i	$-0.647127\pi$
$911$	−26.9188	−0.891859	−0.445929	+	0.895068i	$0.647127\pi$
$912$	−8.61280	−0.285198
$913$	−13.4164	−0.444018
$914$	29.6197	0.979732
$915$	−27.2361	−0.900397
$916$	20.5942	0.680452
$917$	−7.41641	−0.244911
$918$	44.3607	1.46412
$919$	−42.5410	−1.40330	−0.701649	−	0.712522i	$-0.747553\pi$
$919$	−42.5410	−1.40330	−0.701649	+	0.712522i	$0.747553\pi$
$920$	−43.8105	−1.44439
$921$	−21.0857	−0.694797
$922$	13.5231	0.445361
$923$	3.86008	0.127056
$924$	18.0000	0.592157
$925$	0	0
$926$	−10.0270	−0.329508
$927$	−23.9443	−0.786433
$928$	−5.86319	−0.192468
$929$	10.6460	0.349284	0.174642	−	0.984632i	$-0.444123\pi$
$929$	10.6460	0.349284	0.174642	+	0.984632i	$0.444123\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	−65.3951	−2.14209
$933$	−22.2635	−0.728873
$934$	−40.9787	−1.34086
$935$	35.1246	1.14870
$936$	43.8105	1.43199
$937$	24.2492	0.792188	0.396094	−	0.918210i	$-0.370366\pi$
$937$	24.2492	0.792188	0.396094	+	0.918210i	$0.370366\pi$
$938$	−15.7082	−0.512891
$939$	10.0689	0.328586
$940$	−105.374	−3.43692
$941$	25.5347	0.832406	0.416203	−	0.909272i	$-0.363361\pi$
$941$	25.5347	0.832406	0.416203	+	0.909272i	$0.363361\pi$
$942$	−19.9265	−0.649241
$943$	3.86008	0.125702
$944$	117.700	3.83081
$945$	−10.2333	−0.332891
$946$	−107.666	−3.50051
$947$	40.1869	1.30590	0.652949	−	0.757402i	$-0.273532\pi$
$947$	40.1869	1.30590	0.652949	+	0.757402i	$0.273532\pi$
$948$	6.87539	0.223302
$949$	−11.1246	−0.361120
$950$	0	0
$951$	−22.9312	−0.743594
$952$	−27.6653	−0.896637
$953$	18.6699	0.604778	0.302389	−	0.953185i	$-0.402216\pi$
$953$	18.6699	0.604778	0.302389	+	0.953185i	$0.402216\pi$
$954$	80.3737	2.60220
$955$	21.8328	0.706493
$956$	58.7780	1.90102
$957$	2.00310	0.0647511
$958$	43.2705	1.39801
$959$	2.62210	0.0846719
$960$	−17.0193	−0.549295
$961$	0	0
$962$	−29.1246	−0.939015
$963$	−3.29180	−0.106077
$964$	−112.694	−3.62964
$965$	−46.3050	−1.49061
$966$	−6.00000	−0.193047
$967$	44.0168	1.41549	0.707743	−	0.706470i	$-0.249713\pi$
$967$	44.0168	1.41549	0.707743	+	0.706470i	$0.249713\pi$
$968$	52.3050	1.68114
$969$	−3.23607	−0.103957
$970$	−40.9787	−1.31575
$971$	53.8885	1.72937	0.864683	−	0.502318i	$-0.167519\pi$
$971$	53.8885	1.72937	0.864683	+	0.502318i	$0.167519\pi$
$972$	−78.1342	−2.50616
$973$	17.9721	0.576160
$974$	73.5090	2.35538
$975$	0	0
$976$	137.325	4.39566
$977$	−42.4853	−1.35922	−0.679612	−	0.733571i	$-0.737852\pi$
$977$	−42.4853	−1.35922	−0.679612	+	0.733571i	$0.737852\pi$
$978$	−32.9883	−1.05485
$979$	−65.1246	−2.08139
$980$	−65.1246	−2.08033
$981$	2.88854	0.0922241
$982$	−41.6011	−1.32754
$983$	53.9163	1.71966	0.859832	−	0.510577i	$-0.170568\pi$
$983$	53.9163	1.71966	0.859832	+	0.510577i	$0.170568\pi$
$984$	9.61435	0.306494
$985$	−2.41577	−0.0769727
$986$	−5.23607	−0.166750
$987$	−8.48528	−0.270089
$988$	−12.7279	−0.404929
$989$	25.4164	0.808195
$990$	−55.5369	−1.76508
$991$	−13.9358	−0.442685	−0.221343	−	0.975196i	$-0.571044\pi$
$991$	−13.9358	−0.442685	−0.221343	+	0.975196i	$0.571044\pi$
$992$	0	0
$993$	−27.3475	−0.867847
$994$	3.85410	0.122245
$995$	51.9132	1.64576
$996$	13.4164	0.425115
$997$	−58.6656	−1.85796	−0.928980	−	0.370131i	$-0.879313\pi$
$997$	−58.6656	−1.85796	−0.928980	+	0.370131i	$0.879313\pi$
$998$	48.1320	1.52359
$999$	19.4164	0.614308

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	961.2.a.h.1.3	✓	4
3.2	odd	2		8649.2.a.r.1.1		4
31.2	even	5		961.2.d.j.531.1		8
31.3	odd	30		961.2.g.p.846.1		16
31.4	even	5		961.2.d.h.388.2		8
31.5	even	3		961.2.c.h.521.4		8
31.6	odd	6		961.2.c.h.439.3		8
31.7	even	15		961.2.g.i.235.1		16
31.8	even	5		961.2.d.h.374.2		8
31.9	even	15		961.2.g.i.732.1		16
31.10	even	15		961.2.g.p.844.2		16
31.11	odd	30		961.2.g.i.338.1		16
31.12	odd	30		961.2.g.p.547.2		16
31.13	odd	30		961.2.g.p.448.2		16
31.14	even	15		961.2.g.i.816.2		16
31.15	odd	10		961.2.d.j.628.2		8
31.16	even	5		961.2.d.j.628.1		8
31.17	odd	30		961.2.g.i.816.1		16
31.18	even	15		961.2.g.p.448.1		16
31.19	even	15		961.2.g.p.547.1		16
31.20	even	15		961.2.g.i.338.2		16
31.21	odd	30		961.2.g.p.844.1		16
31.22	odd	30		961.2.g.i.732.2		16
31.23	odd	10		961.2.d.h.374.1		8
31.24	odd	30		961.2.g.i.235.2		16
31.25	even	3		961.2.c.h.439.4		8
31.26	odd	6		961.2.c.h.521.3		8
31.27	odd	10		961.2.d.h.388.1		8
31.28	even	15		961.2.g.p.846.2		16
31.29	odd	10		961.2.d.j.531.2		8
31.30	odd	2	inner	961.2.a.h.1.4	yes	4
93.92	even	2		8649.2.a.r.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
961.2.a.h.1.3	✓	4	1.1	even	1	trivial
961.2.a.h.1.4	yes	4	31.30	odd	2	inner
961.2.c.h.439.3		8	31.6	odd	6
961.2.c.h.439.4		8	31.25	even	3
961.2.c.h.521.3		8	31.26	odd	6
961.2.c.h.521.4		8	31.5	even	3
961.2.d.h.374.1		8	31.23	odd	10
961.2.d.h.374.2		8	31.8	even	5
961.2.d.h.388.1		8	31.27	odd	10
961.2.d.h.388.2		8	31.4	even	5
961.2.d.j.531.1		8	31.2	even	5
961.2.d.j.531.2		8	31.29	odd	10
961.2.d.j.628.1		8	31.16	even	5
961.2.d.j.628.2		8	31.15	odd	10
961.2.g.i.235.1		16	31.7	even	15
961.2.g.i.235.2		16	31.24	odd	30
961.2.g.i.338.1		16	31.11	odd	30
961.2.g.i.338.2		16	31.20	even	15
961.2.g.i.732.1		16	31.9	even	15
961.2.g.i.732.2		16	31.22	odd	30
961.2.g.i.816.1		16	31.17	odd	30
961.2.g.i.816.2		16	31.14	even	15
961.2.g.p.448.1		16	31.18	even	15
961.2.g.p.448.2		16	31.13	odd	30
961.2.g.p.547.1		16	31.19	even	15
961.2.g.p.547.2		16	31.12	odd	30
961.2.g.p.844.1		16	31.21	odd	30
961.2.g.p.844.2		16	31.10	even	15
961.2.g.p.846.1		16	31.3	odd	30
961.2.g.p.846.2		16	31.28	even	15
8649.2.a.r.1.1		4	3.2	odd	2
8649.2.a.r.1.2		4	93.92	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 961 = 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	961.a (trivial)

\(f(q)\)	\(=\)	\(q+2.61803 q^{2} -0.874032 q^{3} +4.85410 q^{4} +2.23607 q^{5} -2.28825 q^{6} -1.00000 q^{7} +7.47214 q^{8} -2.23607 q^{9} +5.85410 q^{10} +4.24264 q^{11} -4.24264 q^{12} -2.62210 q^{13} -2.61803 q^{14} -1.95440 q^{15} +9.85410 q^{16} +3.70246 q^{17} -5.85410 q^{18} +1.00000 q^{19} +10.8541 q^{20} +0.874032 q^{21} +11.1074 q^{22} -2.62210 q^{23} -6.53089 q^{24} -6.86474 q^{26} +4.57649 q^{27} -4.85410 q^{28} -0.540182 q^{29} -5.11667 q^{30} +10.8541 q^{32} -3.70820 q^{33} +9.69316 q^{34} -2.23607 q^{35} -10.8541 q^{36} +4.24264 q^{37} +2.61803 q^{38} +2.29180 q^{39} +16.7082 q^{40} -1.47214 q^{41} +2.28825 q^{42} -9.69316 q^{43} +20.5942 q^{44} -5.00000 q^{45} -6.86474 q^{46} -9.70820 q^{47} -8.61280 q^{48} -6.00000 q^{49} -3.23607 q^{51} -12.7279 q^{52} -13.7295 q^{53} +11.9814 q^{54} +9.48683 q^{55} -7.47214 q^{56} -0.874032 q^{57} -1.41421 q^{58} +11.9443 q^{59} -9.48683 q^{60} +13.9358 q^{61} +2.23607 q^{63} +8.70820 q^{64} -5.86319 q^{65} -9.70820 q^{66} +6.00000 q^{67} +17.9721 q^{68} +2.29180 q^{69} -5.85410 q^{70} -1.47214 q^{71} -16.7082 q^{72} +4.24264 q^{73} +11.1074 q^{74} +4.85410 q^{76} -4.24264 q^{77} +6.00000 q^{78} -1.62054 q^{79} +22.0344 q^{80} +2.70820 q^{81} -3.85410 q^{82} -3.16228 q^{83} +4.24264 q^{84} +8.27895 q^{85} -25.3770 q^{86} +0.472136 q^{87} +31.7016 q^{88} -15.3500 q^{89} -13.0902 q^{90} +2.62210 q^{91} -12.7279 q^{92} -25.4164 q^{94} +2.23607 q^{95} -9.48683 q^{96} -7.00000 q^{97} -15.7082 q^{98} -9.48683 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{2} + 6 q^{4} - 4 q^{7} + 12 q^{8} + 10 q^{10} - 6 q^{14} + 26 q^{16} - 10 q^{18} + 4 q^{19} + 30 q^{20} - 6 q^{28} + 30 q^{32} + 12 q^{33} - 30 q^{36} + 6 q^{38} + 36 q^{39} + 40 q^{40} + 12 q^{41}+ \cdots - 36 q^{98}+O(q^{100}) \) 4 * q + 6 * q^2 + 6 * q^4 - 4 * q^7 + 12 * q^8 + 10 * q^10 - 6 * q^14 + 26 * q^16 - 10 * q^18 + 4 * q^19 + 30 * q^20 - 6 * q^28 + 30 * q^32 + 12 * q^33 - 30 * q^36 + 6 * q^38 + 36 * q^39 + 40 * q^40 + 12 * q^41 - 20 * q^45 - 12 * q^47 - 24 * q^49 - 4 * q^51 - 12 * q^56 + 12 * q^59 + 8 * q^64 - 12 * q^66 + 24 * q^67 + 36 * q^69 - 10 * q^70 + 12 * q^71 - 40 * q^72 + 6 * q^76 + 24 * q^78 + 30 * q^80 - 16 * q^81 - 2 * q^82 - 16 * q^87 - 30 * q^90 - 48 * q^94 - 28 * q^97 - 36 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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