LMFDB - Embedded newform 4334.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4334,2,Mod(1,4334)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4334, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4334.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4334 = 2 \cdot 11 \cdot 197 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4334.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:	$34.6071642360$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 6 x^{14} - 8 x^{13} + 94 x^{12} - 13 x^{11} - 582 x^{10} + 295 x^{9} + 1814 x^{8} - 1056 x^{7} + \cdots - 45 $ x^15 - 6x^14 - 8x^13 + 94x^12 - 13x^11 - 582x^10 + 295x^9 + 1814x^8 - 1056x^7 - 2966x^6 + 1621x^5 + 2328x^4 - 1160x^3 - 589x^2 + 364x - 45
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.22455$ of defining polynomial
Character	$\chi$	$=$	4334.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -3.22455 q^{3} +1.00000 q^{4} +1.46351 q^{5} -3.22455 q^{6} -3.27885 q^{7} +1.00000 q^{8} +7.39772 q^{9} +1.46351 q^{10} +1.00000 q^{11} -3.22455 q^{12} +1.88840 q^{13} -3.27885 q^{14} -4.71917 q^{15} +1.00000 q^{16} -3.22716 q^{17} +7.39772 q^{18} -3.16213 q^{19} +1.46351 q^{20} +10.5728 q^{21} +1.00000 q^{22} +3.24393 q^{23} -3.22455 q^{24} -2.85813 q^{25} +1.88840 q^{26} -14.1807 q^{27} -3.27885 q^{28} +3.66625 q^{29} -4.71917 q^{30} -9.41043 q^{31} +1.00000 q^{32} -3.22455 q^{33} -3.22716 q^{34} -4.79864 q^{35} +7.39772 q^{36} +7.27939 q^{37} -3.16213 q^{38} -6.08923 q^{39} +1.46351 q^{40} -2.13599 q^{41} +10.5728 q^{42} +11.5417 q^{43} +1.00000 q^{44} +10.8267 q^{45} +3.24393 q^{46} -6.85798 q^{47} -3.22455 q^{48} +3.75083 q^{49} -2.85813 q^{50} +10.4061 q^{51} +1.88840 q^{52} +6.68351 q^{53} -14.1807 q^{54} +1.46351 q^{55} -3.27885 q^{56} +10.1964 q^{57} +3.66625 q^{58} -3.07002 q^{59} -4.71917 q^{60} -3.79452 q^{61} -9.41043 q^{62} -24.2560 q^{63} +1.00000 q^{64} +2.76369 q^{65} -3.22455 q^{66} -1.54211 q^{67} -3.22716 q^{68} -10.4602 q^{69} -4.79864 q^{70} -4.99095 q^{71} +7.39772 q^{72} +2.72767 q^{73} +7.27939 q^{74} +9.21617 q^{75} -3.16213 q^{76} -3.27885 q^{77} -6.08923 q^{78} +4.37047 q^{79} +1.46351 q^{80} +23.5331 q^{81} -2.13599 q^{82} +13.3456 q^{83} +10.5728 q^{84} -4.72299 q^{85} +11.5417 q^{86} -11.8220 q^{87} +1.00000 q^{88} -1.75396 q^{89} +10.8267 q^{90} -6.19176 q^{91} +3.24393 q^{92} +30.3444 q^{93} -6.85798 q^{94} -4.62782 q^{95} -3.22455 q^{96} -2.05832 q^{97} +3.75083 q^{98} +7.39772 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 15 q^{2} - 9 q^{3} + 15 q^{4} - 11 q^{5} - 9 q^{6} - 11 q^{7} + 15 q^{8} + 10 q^{9} - 11 q^{10} + 15 q^{11} - 9 q^{12} - 21 q^{13} - 11 q^{14} - 2 q^{15} + 15 q^{16} - 4 q^{17} + 10 q^{18} - 22 q^{19}+ \cdots + 10 q^{99}+O(q^{100}) $ 15 * q + 15 * q^2 - 9 * q^3 + 15 * q^4 - 11 * q^5 - 9 * q^6 - 11 * q^7 + 15 * q^8 + 10 * q^9 - 11 * q^10 + 15 * q^11 - 9 * q^12 - 21 * q^13 - 11 * q^14 - 2 * q^15 + 15 * q^16 - 4 * q^17 + 10 * q^18 - 22 * q^19 - 11 * q^20 - 13 * q^21 + 15 * q^22 - 16 * q^23 - 9 * q^24 + 6 * q^25 - 21 * q^26 - 21 * q^27 - 11 * q^28 - 8 * q^29 - 2 * q^30 - 33 * q^31 + 15 * q^32 - 9 * q^33 - 4 * q^34 - 2 * q^35 + 10 * q^36 - q^37 - 22 * q^38 + q^39 - 11 * q^40 - 10 * q^41 - 13 * q^42 - 8 * q^43 + 15 * q^44 - 10 * q^45 - 16 * q^46 - 31 * q^47 - 9 * q^48 + 2 * q^49 + 6 * q^50 + 2 * q^51 - 21 * q^52 - 18 * q^53 - 21 * q^54 - 11 * q^55 - 11 * q^56 + 16 * q^57 - 8 * q^58 - 37 * q^59 - 2 * q^60 - 31 * q^61 - 33 * q^62 - 20 * q^63 + 15 * q^64 - 13 * q^65 - 9 * q^66 + q^67 - 4 * q^68 - 25 * q^69 - 2 * q^70 - 28 * q^71 + 10 * q^72 - 20 * q^73 - q^74 - 9 * q^75 - 22 * q^76 - 11 * q^77 + q^78 - 6 * q^79 - 11 * q^80 + 3 * q^81 - 10 * q^82 - 15 * q^83 - 13 * q^84 - 31 * q^85 - 8 * q^86 - 16 * q^87 + 15 * q^88 - 17 * q^89 - 10 * q^90 - 21 * q^91 - 16 * q^92 + 10 * q^93 - 31 * q^94 - 3 * q^95 - 9 * q^96 - 9 * q^97 + 2 * q^98 + 10 * q^99

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−3.22455	−1.86169	−0.930847	−	0.365408i	$-0.880929\pi$
$3$	−3.22455	−1.86169	−0.930847	+	0.365408i	$0.880929\pi$
$4$	1.00000	0.500000
$5$	1.46351	0.654503	0.327252	−	0.944937i	$-0.393878\pi$
$5$	1.46351	0.654503	0.327252	+	0.944937i	$0.393878\pi$
$6$	−3.22455	−1.31642
$7$	−3.27885	−1.23929	−0.619644	−	0.784883i	$-0.712723\pi$
$7$	−3.27885	−1.23929	−0.619644	+	0.784883i	$0.712723\pi$
$8$	1.00000	0.353553
$9$	7.39772	2.46591
$10$	1.46351	0.462804
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−3.22455	−0.930847
$13$	1.88840	0.523747	0.261873	−	0.965102i	$-0.415660\pi$
$13$	1.88840	0.523747	0.261873	+	0.965102i	$0.415660\pi$
$14$	−3.27885	−0.876308
$15$	−4.71917	−1.21849
$16$	1.00000	0.250000
$17$	−3.22716	−0.782701	−0.391351	−	0.920242i	$-0.627992\pi$
$17$	−3.22716	−0.782701	−0.391351	+	0.920242i	$0.627992\pi$
$18$	7.39772	1.74366
$19$	−3.16213	−0.725442	−0.362721	−	0.931898i	$-0.618152\pi$
$19$	−3.16213	−0.725442	−0.362721	+	0.931898i	$0.618152\pi$
$20$	1.46351	0.327252
$21$	10.5728	2.30717
$22$	1.00000	0.213201
$23$	3.24393	0.676407	0.338203	−	0.941073i	$-0.390181\pi$
$23$	3.24393	0.676407	0.338203	+	0.941073i	$0.390181\pi$
$24$	−3.22455	−0.658209
$25$	−2.85813	−0.571625
$26$	1.88840	0.370345
$27$	−14.1807	−2.72907
$28$	−3.27885	−0.619644
$29$	3.66625	0.680806	0.340403	−	0.940280i	$-0.389437\pi$
$29$	3.66625	0.680806	0.340403	+	0.940280i	$0.389437\pi$
$30$	−4.71917	−0.861599
$31$	−9.41043	−1.69016	−0.845081	−	0.534638i	$-0.820448\pi$
$31$	−9.41043	−1.69016	−0.845081	+	0.534638i	$0.820448\pi$
$32$	1.00000	0.176777
$33$	−3.22455	−0.561322
$34$	−3.22716	−0.553453
$35$	−4.79864	−0.811118
$36$	7.39772	1.23295
$37$	7.27939	1.19672	0.598362	−	0.801226i	$-0.295818\pi$
$37$	7.27939	1.19672	0.598362	+	0.801226i	$0.295818\pi$
$38$	−3.16213	−0.512965
$39$	−6.08923	−0.975056
$40$	1.46351	0.231402
$41$	−2.13599	−0.333585	−0.166792	−	0.985992i	$-0.553341\pi$
$41$	−2.13599	−0.333585	−0.166792	+	0.985992i	$0.553341\pi$
$42$	10.5728	1.63142
$43$	11.5417	1.76010	0.880049	−	0.474882i	$-0.157509\pi$
$43$	11.5417	1.76010	0.880049	+	0.474882i	$0.157509\pi$
$44$	1.00000	0.150756
$45$	10.8267	1.61394
$46$	3.24393	0.478292
$47$	−6.85798	−1.00034	−0.500169	−	0.865928i	$-0.666729\pi$
$47$	−6.85798	−1.00034	−0.500169	+	0.865928i	$0.666729\pi$
$48$	−3.22455	−0.465424
$49$	3.75083	0.535833
$50$	−2.85813	−0.404200
$51$	10.4061	1.45715
$52$	1.88840	0.261873
$53$	6.68351	0.918051	0.459026	−	0.888423i	$-0.348199\pi$
$53$	6.68351	0.918051	0.459026	+	0.888423i	$0.348199\pi$
$54$	−14.1807	−1.92975
$55$	1.46351	0.197340
$56$	−3.27885	−0.438154
$57$	10.1964	1.35055
$58$	3.66625	0.481402
$59$	−3.07002	−0.399682	−0.199841	−	0.979828i	$-0.564043\pi$
$59$	−3.07002	−0.399682	−0.199841	+	0.979828i	$0.564043\pi$
$60$	−4.71917	−0.609243
$61$	−3.79452	−0.485839	−0.242919	−	0.970046i	$-0.578105\pi$
$61$	−3.79452	−0.485839	−0.242919	+	0.970046i	$0.578105\pi$
$62$	−9.41043	−1.19513
$63$	−24.2560	−3.05597
$64$	1.00000	0.125000
$65$	2.76369	0.342794
$66$	−3.22455	−0.396915
$67$	−1.54211	−0.188399	−0.0941994	−	0.995553i	$-0.530029\pi$
$67$	−1.54211	−0.188399	−0.0941994	+	0.995553i	$0.530029\pi$
$68$	−3.22716	−0.391351
$69$	−10.4602	−1.25926
$70$	−4.79864	−0.573547
$71$	−4.99095	−0.592316	−0.296158	−	0.955139i	$-0.595706\pi$
$71$	−4.99095	−0.592316	−0.296158	+	0.955139i	$0.595706\pi$
$72$	7.39772	0.871830
$73$	2.72767	0.319249	0.159625	−	0.987178i	$-0.448972\pi$
$73$	2.72767	0.319249	0.159625	+	0.987178i	$0.448972\pi$
$74$	7.27939	0.846212
$75$	9.21617	1.06419
$76$	−3.16213	−0.362721
$77$	−3.27885	−0.373659
$78$	−6.08923	−0.689469
$79$	4.37047	0.491716	0.245858	−	0.969306i	$-0.420930\pi$
$79$	4.37047	0.491716	0.245858	+	0.969306i	$0.420930\pi$
$80$	1.46351	0.163626
$81$	23.5331	2.61479
$82$	−2.13599	−0.235880
$83$	13.3456	1.46487	0.732435	−	0.680837i	$-0.238384\pi$
$83$	13.3456	1.46487	0.732435	+	0.680837i	$0.238384\pi$
$84$	10.5728	1.15359
$85$	−4.72299	−0.512280
$86$	11.5417	1.24458
$87$	−11.8220	−1.26745
$88$	1.00000	0.106600
$89$	−1.75396	−0.185920	−0.0929598	−	0.995670i	$-0.529633\pi$
$89$	−1.75396	−0.185920	−0.0929598	+	0.995670i	$0.529633\pi$
$90$	10.8267	1.14123
$91$	−6.19176	−0.649073
$92$	3.24393	0.338203
$93$	30.3444	3.14657
$94$	−6.85798	−0.707346
$95$	−4.62782	−0.474804
$96$	−3.22455	−0.329104
$97$	−2.05832	−0.208991	−0.104495	−	0.994525i	$-0.533323\pi$
$97$	−2.05832	−0.208991	−0.104495	+	0.994525i	$0.533323\pi$
$98$	3.75083	0.378891
$99$	7.39772	0.743499
$100$	−2.85813	−0.285813
$101$	−11.7968	−1.17383	−0.586913	−	0.809650i	$-0.699657\pi$
$101$	−11.7968	−1.17383	−0.586913	+	0.809650i	$0.699657\pi$
$102$	10.4061	1.03036
$103$	−13.3040	−1.31089	−0.655443	−	0.755245i	$-0.727518\pi$
$103$	−13.3040	−1.31089	−0.655443	+	0.755245i	$0.727518\pi$
$104$	1.88840	0.185172
$105$	15.4734	1.51005
$106$	6.68351	0.649160
$107$	−9.05843	−0.875711	−0.437856	−	0.899045i	$-0.644262\pi$
$107$	−9.05843	−0.875711	−0.437856	+	0.899045i	$0.644262\pi$
$108$	−14.1807	−1.36454
$109$	−9.51916	−0.911770	−0.455885	−	0.890039i	$-0.650677\pi$
$109$	−9.51916	−0.911770	−0.455885	+	0.890039i	$0.650677\pi$
$110$	1.46351	0.139541
$111$	−23.4728	−2.22794
$112$	−3.27885	−0.309822
$113$	16.8112	1.58146	0.790731	−	0.612163i	$-0.209700\pi$
$113$	16.8112	1.58146	0.790731	+	0.612163i	$0.209700\pi$
$114$	10.1964	0.954984
$115$	4.74754	0.442710
$116$	3.66625	0.340403
$117$	13.9698	1.29151
$118$	−3.07002	−0.282618
$119$	10.5814	0.969992
$120$	−4.71917	−0.430800
$121$	1.00000	0.0909091
$122$	−3.79452	−0.343540
$123$	6.88759	0.621033
$124$	−9.41043	−0.845081
$125$	−11.5005	−1.02863
$126$	−24.2560	−2.16090
$127$	−9.94431	−0.882415	−0.441207	−	0.897405i	$-0.645450\pi$
$127$	−9.94431	−0.882415	−0.441207	+	0.897405i	$0.645450\pi$
$128$	1.00000	0.0883883
$129$	−37.2169	−3.27677
$130$	2.76369	0.242392
$131$	−9.47450	−0.827791	−0.413895	−	0.910324i	$-0.635832\pi$
$131$	−9.47450	−0.827791	−0.413895	+	0.910324i	$0.635832\pi$
$132$	−3.22455	−0.280661
$133$	10.3681	0.899031
$134$	−1.54211	−0.133218
$135$	−20.7536	−1.78619
$136$	−3.22716	−0.276727
$137$	−10.9738	−0.937554	−0.468777	−	0.883317i	$-0.655305\pi$
$137$	−10.9738	−0.937554	−0.468777	+	0.883317i	$0.655305\pi$
$138$	−10.4602	−0.890433
$139$	19.3355	1.64002	0.820009	−	0.572351i	$-0.193969\pi$
$139$	19.3355	1.64002	0.820009	+	0.572351i	$0.193969\pi$
$140$	−4.79864	−0.405559
$141$	22.1139	1.86233
$142$	−4.99095	−0.418831
$143$	1.88840	0.157916
$144$	7.39772	0.616477
$145$	5.36561	0.445590
$146$	2.72767	0.225743
$147$	−12.0947	−0.997558
$148$	7.27939	0.598362
$149$	−19.0019	−1.55670	−0.778349	−	0.627832i	$-0.783942\pi$
$149$	−19.0019	−1.55670	−0.778349	+	0.627832i	$0.783942\pi$
$150$	9.21617	0.752497
$151$	−22.0145	−1.79151	−0.895756	−	0.444547i	$-0.853365\pi$
$151$	−22.0145	−1.79151	−0.895756	+	0.444547i	$0.853365\pi$
$152$	−3.16213	−0.256482
$153$	−23.8736	−1.93007
$154$	−3.27885	−0.264217
$155$	−13.7723	−1.10622
$156$	−6.08923	−0.487528
$157$	−22.1762	−1.76985	−0.884927	−	0.465729i	$-0.845792\pi$
$157$	−22.1762	−1.76985	−0.884927	+	0.465729i	$0.845792\pi$
$158$	4.37047	0.347696
$159$	−21.5513	−1.70913
$160$	1.46351	0.115701
$161$	−10.6364	−0.838262
$162$	23.5331	1.84894
$163$	6.30258	0.493657	0.246828	−	0.969059i	$-0.420612\pi$
$163$	6.30258	0.493657	0.246828	+	0.969059i	$0.420612\pi$
$164$	−2.13599	−0.166792
$165$	−4.71917	−0.367387
$166$	13.3456	1.03582
$167$	−22.6563	−1.75319	−0.876597	−	0.481225i	$-0.840192\pi$
$167$	−22.6563	−1.75319	−0.876597	+	0.481225i	$0.840192\pi$
$168$	10.5728	0.815709
$169$	−9.43396	−0.725689
$170$	−4.72299	−0.362237
$171$	−23.3925	−1.78887
$172$	11.5417	0.880049
$173$	5.82155	0.442605	0.221302	−	0.975205i	$-0.428969\pi$
$173$	5.82155	0.442605	0.221302	+	0.975205i	$0.428969\pi$
$174$	−11.8220	−0.896224
$175$	9.37136	0.708408
$176$	1.00000	0.0753778
$177$	9.89942	0.744086
$178$	−1.75396	−0.131465
$179$	0.756193	0.0565205	0.0282603	−	0.999601i	$-0.491003\pi$
$179$	0.756193	0.0565205	0.0282603	+	0.999601i	$0.491003\pi$
$180$	10.8267	0.806972
$181$	9.38516	0.697593	0.348797	−	0.937198i	$-0.386590\pi$
$181$	9.38516	0.697593	0.348797	+	0.937198i	$0.386590\pi$
$182$	−6.19176	−0.458964
$183$	12.2356	0.904483
$184$	3.24393	0.239146
$185$	10.6535	0.783260
$186$	30.3444	2.22496
$187$	−3.22716	−0.235993
$188$	−6.85798	−0.500169
$189$	46.4963	3.38211
$190$	−4.62782	−0.335737
$191$	−24.7871	−1.79353	−0.896764	−	0.442509i	$-0.854088\pi$
$191$	−24.7871	−1.79353	−0.896764	+	0.442509i	$0.854088\pi$
$192$	−3.22455	−0.232712
$193$	7.29045	0.524778	0.262389	−	0.964962i	$-0.415490\pi$
$193$	7.29045	0.524778	0.262389	+	0.964962i	$0.415490\pi$
$194$	−2.05832	−0.147779
$195$	−8.91167	−0.638178
$196$	3.75083	0.267917
$197$	−1.00000	−0.0712470
$197$	−1.00000	−0.0712470
$198$	7.39772	0.525733
$199$	24.8930	1.76462	0.882310	−	0.470669i	$-0.155987\pi$
$199$	24.8930	1.76462	0.882310	+	0.470669i	$0.155987\pi$
$200$	−2.85813	−0.202100
$201$	4.97261	0.350741
$202$	−11.7968	−0.830020
$203$	−12.0211	−0.843714
$204$	10.4061	0.728575
$205$	−3.12604	−0.218332
$206$	−13.3040	−0.926936
$207$	23.9977	1.66796
$208$	1.88840	0.130937
$209$	−3.16213	−0.218729
$210$	15.4734	1.06777
$211$	−9.43436	−0.649488	−0.324744	−	0.945802i	$-0.605278\pi$
$211$	−9.43436	−0.649488	−0.324744	+	0.945802i	$0.605278\pi$
$212$	6.68351	0.459026
$213$	16.0936	1.10271
$214$	−9.05843	−0.619221
$215$	16.8915	1.15199
$216$	−14.1807	−0.964873
$217$	30.8553	2.09460
$218$	−9.51916	−0.644719
$219$	−8.79550	−0.594345
$220$	1.46351	0.0986701
$221$	−6.09415	−0.409937
$222$	−23.4728	−1.57539
$223$	20.3599	1.36340	0.681700	−	0.731631i	$-0.261241\pi$
$223$	20.3599	1.36340	0.681700	+	0.731631i	$0.261241\pi$
$224$	−3.27885	−0.219077
$225$	−21.1436	−1.40958
$226$	16.8112	1.11826
$227$	7.21042	0.478573	0.239286	−	0.970949i	$-0.423087\pi$
$227$	7.21042	0.478573	0.239286	+	0.970949i	$0.423087\pi$
$228$	10.1964	0.675276
$229$	−6.97072	−0.460638	−0.230319	−	0.973115i	$-0.573977\pi$
$229$	−6.97072	−0.460638	−0.230319	+	0.973115i	$0.573977\pi$
$230$	4.74754	0.313044
$231$	10.5728	0.695639
$232$	3.66625	0.240701
$233$	−6.59616	−0.432129	−0.216065	−	0.976379i	$-0.569322\pi$
$233$	−6.59616	−0.432129	−0.216065	+	0.976379i	$0.569322\pi$
$234$	13.9698	0.913236
$235$	−10.0367	−0.654725
$236$	−3.07002	−0.199841
$237$	−14.0928	−0.915425
$238$	10.5814	0.685888
$239$	−0.968041	−0.0626174	−0.0313087	−	0.999510i	$-0.509967\pi$
$239$	−0.968041	−0.0626174	−0.0313087	+	0.999510i	$0.509967\pi$
$240$	−4.71917	−0.304621
$241$	−5.82666	−0.375328	−0.187664	−	0.982233i	$-0.560092\pi$
$241$	−5.82666	−0.375328	−0.187664	+	0.982233i	$0.560092\pi$
$242$	1.00000	0.0642824
$243$	−33.3417	−2.13887
$244$	−3.79452	−0.242919
$245$	5.48939	0.350705
$246$	6.88759	0.439137
$247$	−5.97135	−0.379948
$248$	−9.41043	−0.597563
$249$	−43.0336	−2.72714
$250$	−11.5005	−0.727354
$251$	−20.7747	−1.31129	−0.655643	−	0.755071i	$-0.727603\pi$
$251$	−20.7747	−1.31129	−0.655643	+	0.755071i	$0.727603\pi$
$252$	−24.2560	−1.52798
$253$	3.24393	0.203944
$254$	−9.94431	−0.623962
$255$	15.2295	0.953710
$256$	1.00000	0.0625000
$257$	5.54090	0.345632	0.172816	−	0.984954i	$-0.444713\pi$
$257$	5.54090	0.345632	0.172816	+	0.984954i	$0.444713\pi$
$258$	−37.2169	−2.31702
$259$	−23.8680	−1.48309
$260$	2.76369	0.171397
$261$	27.1219	1.67880
$262$	−9.47450	−0.585336
$263$	−9.53867	−0.588180	−0.294090	−	0.955778i	$-0.595016\pi$
$263$	−9.53867	−0.588180	−0.294090	+	0.955778i	$0.595016\pi$
$264$	−3.22455	−0.198457
$265$	9.78141	0.600867
$266$	10.3681	0.635711
$267$	5.65574	0.346125
$268$	−1.54211	−0.0941994
$269$	−5.08353	−0.309948	−0.154974	−	0.987919i	$-0.549529\pi$
$269$	−5.08353	−0.309948	−0.154974	+	0.987919i	$0.549529\pi$
$270$	−20.7536	−1.26302
$271$	3.85481	0.234163	0.117081	−	0.993122i	$-0.462646\pi$
$271$	3.85481	0.234163	0.117081	+	0.993122i	$0.462646\pi$
$272$	−3.22716	−0.195675
$273$	19.9656	1.20838
$274$	−10.9738	−0.662951
$275$	−2.85813	−0.172352
$276$	−10.4602	−0.629632
$277$	−23.2911	−1.39942	−0.699712	−	0.714425i	$-0.746689\pi$
$277$	−23.2911	−1.39942	−0.699712	+	0.714425i	$0.746689\pi$
$278$	19.3355	1.15967
$279$	−69.6157	−4.16778
$280$	−4.79864	−0.286773
$281$	−14.6493	−0.873906	−0.436953	−	0.899484i	$-0.643942\pi$
$281$	−14.6493	−0.873906	−0.436953	+	0.899484i	$0.643942\pi$
$282$	22.1139	1.31686
$283$	−15.2123	−0.904279	−0.452140	−	0.891947i	$-0.649339\pi$
$283$	−15.2123	−0.904279	−0.452140	+	0.891947i	$0.649339\pi$
$284$	−4.99095	−0.296158
$285$	14.9226	0.883940
$286$	1.88840	0.111663
$287$	7.00357	0.413408
$288$	7.39772	0.435915
$289$	−6.58544	−0.387379
$290$	5.36561	0.315079
$291$	6.63715	0.389077
$292$	2.72767	0.159625
$293$	−16.7396	−0.977939	−0.488969	−	0.872301i	$-0.662627\pi$
$293$	−16.7396	−0.977939	−0.488969	+	0.872301i	$0.662627\pi$
$294$	−12.0947	−0.705380
$295$	−4.49301	−0.261593
$296$	7.27939	0.423106
$297$	−14.1807	−0.822846
$298$	−19.0019	−1.10075
$299$	6.12583	0.354266
$300$	9.21617	0.532096
$301$	−37.8436	−2.18127
$302$	−22.0145	−1.26679
$303$	38.0394	2.18531
$304$	−3.16213	−0.181360
$305$	−5.55333	−0.317983
$306$	−23.8736	−1.36476
$307$	−14.9984	−0.856003	−0.428001	−	0.903778i	$-0.640782\pi$
$307$	−14.9984	−0.856003	−0.428001	+	0.903778i	$0.640782\pi$
$308$	−3.27885	−0.186830
$309$	42.8995	2.44047
$310$	−13.7723	−0.782214
$311$	7.32819	0.415544	0.207772	−	0.978177i	$-0.433379\pi$
$311$	7.32819	0.415544	0.207772	+	0.978177i	$0.433379\pi$
$312$	−6.08923	−0.344735
$313$	−31.2656	−1.76723	−0.883617	−	0.468211i	$-0.844899\pi$
$313$	−31.2656	−1.76723	−0.883617	+	0.468211i	$0.844899\pi$
$314$	−22.1762	−1.25148
$315$	−35.4990	−2.00014
$316$	4.37047	0.245858
$317$	13.0410	0.732458	0.366229	−	0.930525i	$-0.380649\pi$
$317$	13.0410	0.732458	0.366229	+	0.930525i	$0.380649\pi$
$318$	−21.5513	−1.20854
$319$	3.66625	0.205271
$320$	1.46351	0.0818129
$321$	29.2093	1.63031
$322$	−10.6364	−0.592741
$323$	10.2047	0.567804
$324$	23.5331	1.30740
$325$	−5.39727	−0.299387
$326$	6.30258	0.349068
$327$	30.6950	1.69744
$328$	−2.13599	−0.117940
$329$	22.4862	1.23971
$330$	−4.71917	−0.259782
$331$	11.0037	0.604818	0.302409	−	0.953178i	$-0.402209\pi$
$331$	11.0037	0.604818	0.302409	+	0.953178i	$0.402209\pi$
$332$	13.3456	0.732435
$333$	53.8509	2.95101
$334$	−22.6563	−1.23970
$335$	−2.25690	−0.123308
$336$	10.5728	0.576794
$337$	19.9921	1.08904	0.544520	−	0.838748i	$-0.316712\pi$
$337$	19.9921	1.08904	0.544520	+	0.838748i	$0.316712\pi$
$338$	−9.43396	−0.513140
$339$	−54.2085	−2.94420
$340$	−4.72299	−0.256140
$341$	−9.41043	−0.509603
$342$	−23.3925	−1.26492
$343$	10.6535	0.575236
$344$	11.5417	0.622289
$345$	−15.3087	−0.824192
$346$	5.82155	0.312969
$347$	−3.51582	−0.188739	−0.0943696	−	0.995537i	$-0.530084\pi$
$347$	−3.51582	−0.188739	−0.0943696	+	0.995537i	$0.530084\pi$
$348$	−11.8220	−0.633726
$349$	17.4947	0.936471	0.468236	−	0.883604i	$-0.344890\pi$
$349$	17.4947	0.936471	0.468236	+	0.883604i	$0.344890\pi$
$350$	9.37136	0.500920
$351$	−26.7787	−1.42934
$352$	1.00000	0.0533002
$353$	0.557129	0.0296530	0.0148265	−	0.999890i	$-0.495280\pi$
$353$	0.557129	0.0296530	0.0148265	+	0.999890i	$0.495280\pi$
$354$	9.89942	0.526148
$355$	−7.30432	−0.387673
$356$	−1.75396	−0.0929598
$357$	−34.1201	−1.80583
$358$	0.756193	0.0399660
$359$	21.8054	1.15084	0.575422	−	0.817856i	$-0.304838\pi$
$359$	21.8054	1.15084	0.575422	+	0.817856i	$0.304838\pi$
$360$	10.8267	0.570616
$361$	−9.00095	−0.473734
$362$	9.38516	0.493273
$363$	−3.22455	−0.169245
$364$	−6.19176	−0.324536
$365$	3.99198	0.208950
$366$	12.2356	0.639566
$367$	28.5855	1.49215	0.746076	−	0.665861i	$-0.231936\pi$
$367$	28.5855	1.49215	0.746076	+	0.665861i	$0.231936\pi$
$368$	3.24393	0.169102
$369$	−15.8014	−0.822590
$370$	10.6535	0.553849
$371$	−21.9142	−1.13773
$372$	30.3444	1.57328
$373$	−17.7695	−0.920068	−0.460034	−	0.887901i	$-0.652163\pi$
$373$	−17.7695	−0.920068	−0.460034	+	0.887901i	$0.652163\pi$
$374$	−3.22716	−0.166872
$375$	37.0839	1.91500
$376$	−6.85798	−0.353673
$377$	6.92333	0.356570
$378$	46.4963	2.39151
$379$	−30.3399	−1.55846	−0.779229	−	0.626739i	$-0.784389\pi$
$379$	−30.3399	−1.55846	−0.779229	+	0.626739i	$0.784389\pi$
$380$	−4.62782	−0.237402
$381$	32.0659	1.64279
$382$	−24.7871	−1.26822
$383$	−4.71362	−0.240855	−0.120427	−	0.992722i	$-0.538426\pi$
$383$	−4.71362	−0.240855	−0.120427	+	0.992722i	$0.538426\pi$
$384$	−3.22455	−0.164552
$385$	−4.79864	−0.244561
$386$	7.29045	0.371074
$387$	85.3826	4.34024
$388$	−2.05832	−0.104495
$389$	−32.2092	−1.63307	−0.816534	−	0.577297i	$-0.804108\pi$
$389$	−32.2092	−1.63307	−0.816534	+	0.577297i	$0.804108\pi$
$390$	−8.91167	−0.451260
$391$	−10.4687	−0.529424
$392$	3.75083	0.189446
$393$	30.5510	1.54109
$394$	−1.00000	−0.0503793
$395$	6.39624	0.321830
$396$	7.39772	0.371750
$397$	7.50691	0.376761	0.188380	−	0.982096i	$-0.439676\pi$
$397$	7.50691	0.376761	0.188380	+	0.982096i	$0.439676\pi$
$398$	24.8930	1.24777
$399$	−33.4326	−1.67372
$400$	−2.85813	−0.142906
$401$	−3.75518	−0.187525	−0.0937624	−	0.995595i	$-0.529889\pi$
$401$	−3.75518	−0.187525	−0.0937624	+	0.995595i	$0.529889\pi$
$402$	4.97261	0.248011
$403$	−17.7706	−0.885217
$404$	−11.7968	−0.586913
$405$	34.4411	1.71139
$406$	−12.0211	−0.596596
$407$	7.27939	0.360826
$408$	10.4061	0.515181
$409$	5.89350	0.291415	0.145707	−	0.989328i	$-0.453454\pi$
$409$	5.89350	0.291415	0.145707	+	0.989328i	$0.453454\pi$
$410$	−3.12604	−0.154384
$411$	35.3855	1.74544
$412$	−13.3040	−0.655443
$413$	10.0661	0.495321
$414$	23.9977	1.17942
$415$	19.5315	0.958763
$416$	1.88840	0.0925862
$417$	−62.3483	−3.05321
$418$	−3.16213	−0.154665
$419$	−9.91227	−0.484246	−0.242123	−	0.970246i	$-0.577844\pi$
$419$	−9.91227	−0.484246	−0.242123	+	0.970246i	$0.577844\pi$
$420$	15.4734	0.755027
$421$	18.9747	0.924770	0.462385	−	0.886679i	$-0.346994\pi$
$421$	18.9747	0.924770	0.462385	+	0.886679i	$0.346994\pi$
$422$	−9.43436	−0.459258
$423$	−50.7334	−2.46674
$424$	6.68351	0.324580
$425$	9.22363	0.447412
$426$	16.0936	0.779736
$427$	12.4417	0.602094
$428$	−9.05843	−0.437856
$429$	−6.08923	−0.293991
$430$	16.8915	0.814580
$431$	−34.2926	−1.65182	−0.825908	−	0.563805i	$-0.809337\pi$
$431$	−34.2926	−1.65182	−0.825908	+	0.563805i	$0.809337\pi$
$432$	−14.1807	−0.682268
$433$	−7.01812	−0.337270	−0.168635	−	0.985679i	$-0.553936\pi$
$433$	−7.01812	−0.337270	−0.168635	+	0.985679i	$0.553936\pi$
$434$	30.8553	1.48110
$435$	−17.3017	−0.829552
$436$	−9.51916	−0.455885
$437$	−10.2577	−0.490694
$438$	−8.79550	−0.420265
$439$	18.2788	0.872397	0.436199	−	0.899850i	$-0.356325\pi$
$439$	18.2788	0.872397	0.436199	+	0.899850i	$0.356325\pi$
$440$	1.46351	0.0697703
$441$	27.7476	1.32132
$442$	−6.09415	−0.289869
$443$	−35.3941	−1.68163	−0.840813	−	0.541326i	$-0.817923\pi$
$443$	−35.3941	−1.68163	−0.840813	+	0.541326i	$0.817923\pi$
$444$	−23.4728	−1.11397
$445$	−2.56695	−0.121685
$446$	20.3599	0.964070
$447$	61.2726	2.89810
$448$	−3.27885	−0.154911
$449$	23.6620	1.11668	0.558340	−	0.829612i	$-0.311439\pi$
$449$	23.6620	1.11668	0.558340	+	0.829612i	$0.311439\pi$
$450$	−21.1436	−0.996720
$451$	−2.13599	−0.100580
$452$	16.8112	0.790731
$453$	70.9867	3.33525
$454$	7.21042	0.338402
$455$	−9.06172	−0.424820
$456$	10.1964	0.477492
$457$	−9.08643	−0.425045	−0.212523	−	0.977156i	$-0.568168\pi$
$457$	−9.08643	−0.425045	−0.212523	+	0.977156i	$0.568168\pi$
$458$	−6.97072	−0.325721
$459$	45.7633	2.13605
$460$	4.74754	0.221355
$461$	35.1534	1.63726	0.818629	−	0.574323i	$-0.194735\pi$
$461$	35.1534	1.63726	0.818629	+	0.574323i	$0.194735\pi$
$462$	10.5728	0.491891
$463$	12.1423	0.564299	0.282150	−	0.959370i	$-0.408953\pi$
$463$	12.1423	0.564299	0.282150	+	0.959370i	$0.408953\pi$
$464$	3.66625	0.170201
$465$	44.4094	2.05944
$466$	−6.59616	−0.305561
$467$	17.0844	0.790572	0.395286	−	0.918558i	$-0.370646\pi$
$467$	17.0844	0.790572	0.395286	+	0.918558i	$0.370646\pi$
$468$	13.9698	0.645755
$469$	5.05634	0.233480
$470$	−10.0367	−0.462960
$471$	71.5083	3.29493
$472$	−3.07002	−0.141309
$473$	11.5417	0.530690
$474$	−14.0928	−0.647303
$475$	9.03776	0.414681
$476$	10.5814	0.484996
$477$	49.4428	2.26383
$478$	−0.968041	−0.0442772
$479$	−22.9729	−1.04966	−0.524828	−	0.851208i	$-0.675870\pi$
$479$	−22.9729	−1.04966	−0.524828	+	0.851208i	$0.675870\pi$
$480$	−4.71917	−0.215400
$481$	13.7464	0.626781
$482$	−5.82666	−0.265397
$483$	34.2975	1.56059
$484$	1.00000	0.0454545
$485$	−3.01238	−0.136785
$486$	−33.3417	−1.51241
$487$	−8.84084	−0.400617	−0.200308	−	0.979733i	$-0.564194\pi$
$487$	−8.84084	−0.400617	−0.200308	+	0.979733i	$0.564194\pi$
$488$	−3.79452	−0.171770
$489$	−20.3230	−0.919038
$490$	5.48939	0.247986
$491$	21.9229	0.989369	0.494684	−	0.869073i	$-0.335284\pi$
$491$	21.9229	0.989369	0.494684	+	0.869073i	$0.335284\pi$
$492$	6.88759	0.310517
$493$	−11.8316	−0.532867
$494$	−5.97135	−0.268664
$495$	10.8267	0.486623
$496$	−9.41043	−0.422541
$497$	16.3645	0.734050
$498$	−43.0336	−1.92838
$499$	−34.5442	−1.54641	−0.773205	−	0.634156i	$-0.781348\pi$
$499$	−34.5442	−1.54641	−0.773205	+	0.634156i	$0.781348\pi$
$500$	−11.5005	−0.514317
$501$	73.0562	3.26391
$502$	−20.7747	−0.927220
$503$	14.8199	0.660787	0.330393	−	0.943843i	$-0.392819\pi$
$503$	14.8199	0.660787	0.330393	+	0.943843i	$0.392819\pi$
$504$	−24.2560	−1.08045
$505$	−17.2648	−0.768273
$506$	3.24393	0.144210
$507$	30.4203	1.35101
$508$	−9.94431	−0.441207
$509$	8.37425	0.371182	0.185591	−	0.982627i	$-0.440580\pi$
$509$	8.37425	0.371182	0.185591	+	0.982627i	$0.440580\pi$
$510$	15.2295	0.674375
$511$	−8.94360	−0.395642
$512$	1.00000	0.0441942
$513$	44.8411	1.97978
$514$	5.54090	0.244399
$515$	−19.4706	−0.857979
$516$	−37.2169	−1.63838
$517$	−6.85798	−0.301613
$518$	−23.8680	−1.04870
$519$	−18.7719	−0.823995
$520$	2.76369	0.121196
$521$	−16.4563	−0.720965	−0.360483	−	0.932766i	$-0.617388\pi$
$521$	−16.4563	−0.720965	−0.360483	+	0.932766i	$0.617388\pi$
$522$	27.1219	1.18709
$523$	−33.4814	−1.46404	−0.732019	−	0.681284i	$-0.761422\pi$
$523$	−33.4814	−1.46404	−0.732019	+	0.681284i	$0.761422\pi$
$524$	−9.47450	−0.413895
$525$	−30.2184	−1.31884
$526$	−9.53867	−0.415906
$527$	30.3689	1.32289
$528$	−3.22455	−0.140331
$529$	−12.4769	−0.542474
$530$	9.78141	0.424877
$531$	−22.7111	−0.985579
$532$	10.3681	0.449516
$533$	−4.03359	−0.174714
$534$	5.65574	0.244748
$535$	−13.2571	−0.573156
$536$	−1.54211	−0.0666090
$537$	−2.43838	−0.105224
$538$	−5.08353	−0.219167
$539$	3.75083	0.161560
$540$	−20.7536	−0.893094
$541$	34.1327	1.46748	0.733739	−	0.679431i	$-0.237774\pi$
$541$	34.1327	1.46748	0.733739	+	0.679431i	$0.237774\pi$
$542$	3.85481	0.165578
$543$	−30.2629	−1.29871
$544$	−3.22716	−0.138363
$545$	−13.9314	−0.596756
$546$	19.9656	0.854450
$547$	32.3174	1.38179	0.690896	−	0.722954i	$-0.257216\pi$
$547$	32.3174	1.38179	0.690896	+	0.722954i	$0.257216\pi$
$548$	−10.9738	−0.468777
$549$	−28.0708	−1.19803
$550$	−2.85813	−0.121871
$551$	−11.5932	−0.493885
$552$	−10.4602	−0.445217
$553$	−14.3301	−0.609377
$554$	−23.2911	−0.989543
$555$	−34.3527	−1.45819
$556$	19.3355	0.820009
$557$	4.11467	0.174344	0.0871721	−	0.996193i	$-0.472217\pi$
$557$	4.11467	0.174344	0.0871721	+	0.996193i	$0.472217\pi$
$558$	−69.6157	−2.94707
$559$	21.7954	0.921846
$560$	−4.79864	−0.202779
$561$	10.4061	0.439347
$562$	−14.6493	−0.617945
$563$	34.7699	1.46538	0.732688	−	0.680565i	$-0.238265\pi$
$563$	34.7699	1.46538	0.732688	+	0.680565i	$0.238265\pi$
$564$	22.1139	0.931163
$565$	24.6034	1.03507
$566$	−15.2123	−0.639422
$567$	−77.1615	−3.24048
$568$	−4.99095	−0.209416
$569$	5.14857	0.215839	0.107920	−	0.994160i	$-0.465581\pi$
$569$	5.14857	0.215839	0.107920	+	0.994160i	$0.465581\pi$
$570$	14.9226	0.625040
$571$	6.69170	0.280039	0.140019	−	0.990149i	$-0.455283\pi$
$571$	6.69170	0.280039	0.140019	+	0.990149i	$0.455283\pi$
$572$	1.88840	0.0789578
$573$	79.9271	3.33900
$574$	7.00357	0.292323
$575$	−9.27157	−0.386651
$576$	7.39772	0.308238
$577$	21.4450	0.892766	0.446383	−	0.894842i	$-0.352712\pi$
$577$	21.4450	0.892766	0.446383	+	0.894842i	$0.352712\pi$
$578$	−6.58544	−0.273918
$579$	−23.5084	−0.976976
$580$	5.36561	0.222795
$581$	−43.7582	−1.81540
$582$	6.63715	0.275119
$583$	6.68351	0.276803
$584$	2.72767	0.112872
$585$	20.4450	0.845298
$586$	−16.7396	−0.691507
$587$	−10.6716	−0.440463	−0.220231	−	0.975448i	$-0.570681\pi$
$587$	−10.6716	−0.440463	−0.220231	+	0.975448i	$0.570681\pi$
$588$	−12.0947	−0.498779
$589$	29.7570	1.22611
$590$	−4.49301	−0.184974
$591$	3.22455	0.132640
$592$	7.27939	0.299181
$593$	22.1441	0.909351	0.454675	−	0.890657i	$-0.349755\pi$
$593$	22.1441	0.909351	0.454675	+	0.890657i	$0.349755\pi$
$594$	−14.1807	−0.581840
$595$	15.4860	0.634863
$596$	−19.0019	−0.778349
$597$	−80.2688	−3.28518
$598$	6.12583	0.250504
$599$	−21.1143	−0.862706	−0.431353	−	0.902183i	$-0.641964\pi$
$599$	−21.1143	−0.862706	−0.431353	+	0.902183i	$0.641964\pi$
$600$	9.21617	0.376249
$601$	12.8157	0.522764	0.261382	−	0.965235i	$-0.415822\pi$
$601$	12.8157	0.522764	0.261382	+	0.965235i	$0.415822\pi$
$602$	−37.8436	−1.54239
$603$	−11.4081	−0.464574
$604$	−22.0145	−0.895756
$605$	1.46351	0.0595003
$606$	38.0394	1.54524
$607$	38.7370	1.57229	0.786143	−	0.618045i	$-0.212075\pi$
$607$	38.7370	1.57229	0.786143	+	0.618045i	$0.212075\pi$
$608$	−3.16213	−0.128241
$609$	38.7625	1.57074
$610$	−5.55333	−0.224848
$611$	−12.9506	−0.523924
$612$	−23.8736	−0.965034
$613$	35.9280	1.45112	0.725558	−	0.688161i	$-0.241582\pi$
$613$	35.9280	1.45112	0.725558	+	0.688161i	$0.241582\pi$
$614$	−14.9984	−0.605285
$615$	10.0801	0.406468
$616$	−3.27885	−0.132108
$617$	27.3421	1.10075	0.550376	−	0.834917i	$-0.314485\pi$
$617$	27.3421	1.10075	0.550376	+	0.834917i	$0.314485\pi$
$618$	42.8995	1.72567
$619$	−10.8855	−0.437524	−0.218762	−	0.975778i	$-0.570202\pi$
$619$	−10.8855	−0.437524	−0.218762	+	0.975778i	$0.570202\pi$
$620$	−13.7723	−0.553108
$621$	−46.0012	−1.84596
$622$	7.32819	0.293834
$623$	5.75097	0.230408
$624$	−6.08923	−0.243764
$625$	−2.54047	−0.101619
$626$	−31.2656	−1.24962
$627$	10.1964	0.407207
$628$	−22.1762	−0.884927
$629$	−23.4918	−0.936678
$630$	−35.4990	−1.41431
$631$	−24.4297	−0.972532	−0.486266	−	0.873811i	$-0.661641\pi$
$631$	−24.4297	−0.972532	−0.486266	+	0.873811i	$0.661641\pi$
$632$	4.37047	0.173848
$633$	30.4216	1.20915
$634$	13.0410	0.517926
$635$	−14.5536	−0.577543
$636$	−21.5513	−0.854565
$637$	7.08305	0.280641
$638$	3.66625	0.145148
$639$	−36.9216	−1.46060
$640$	1.46351	0.0578505
$641$	21.1143	0.833965	0.416983	−	0.908914i	$-0.363088\pi$
$641$	21.1143	0.833965	0.416983	+	0.908914i	$0.363088\pi$
$642$	29.2093	1.15280
$643$	28.0916	1.10783	0.553913	−	0.832574i	$-0.313134\pi$
$643$	28.0916	1.10783	0.553913	+	0.832574i	$0.313134\pi$
$644$	−10.6364	−0.419131
$645$	−54.4675	−2.14465
$646$	10.2047	0.401498
$647$	−22.0156	−0.865522	−0.432761	−	0.901509i	$-0.642461\pi$
$647$	−22.0156	−0.865522	−0.432761	+	0.901509i	$0.642461\pi$
$648$	23.5331	0.924469
$649$	−3.07002	−0.120509
$650$	−5.39727	−0.211699
$651$	−99.4946	−3.89950
$652$	6.30258	0.246828
$653$	−13.0495	−0.510666	−0.255333	−	0.966853i	$-0.582185\pi$
$653$	−13.0495	−0.510666	−0.255333	+	0.966853i	$0.582185\pi$
$654$	30.6950	1.20027
$655$	−13.8661	−0.541792
$656$	−2.13599	−0.0833962
$657$	20.1785	0.787240
$658$	22.4862	0.876605
$659$	24.0695	0.937616	0.468808	−	0.883300i	$-0.344684\pi$
$659$	24.0695	0.937616	0.468808	+	0.883300i	$0.344684\pi$
$660$	−4.71917	−0.183694
$661$	−25.8769	−1.00649	−0.503247	−	0.864142i	$-0.667862\pi$
$661$	−25.8769	−1.00649	−0.503247	+	0.864142i	$0.667862\pi$
$662$	11.0037	0.427671
$663$	19.6509	0.763178
$664$	13.3456	0.517910
$665$	15.1739	0.588419
$666$	53.8509	2.08668
$667$	11.8931	0.460502
$668$	−22.6563	−0.876597
$669$	−65.6516	−2.53824
$670$	−2.25690	−0.0871917
$671$	−3.79452	−0.146486
$672$	10.5728	0.407855
$673$	−44.8651	−1.72942	−0.864711	−	0.502269i	$-0.832499\pi$
$673$	−44.8651	−1.72942	−0.864711	+	0.502269i	$0.832499\pi$
$674$	19.9921	0.770067
$675$	40.5302	1.56001
$676$	−9.43396	−0.362845
$677$	26.3140	1.01133	0.505664	−	0.862730i	$-0.331247\pi$
$677$	26.3140	1.01133	0.505664	+	0.862730i	$0.331247\pi$
$678$	−54.2085	−2.08186
$679$	6.74891	0.258999
$680$	−4.72299	−0.181118
$681$	−23.2504	−0.890956
$682$	−9.41043	−0.360344
$683$	−27.5525	−1.05427	−0.527133	−	0.849783i	$-0.676733\pi$
$683$	−27.5525	−1.05427	−0.527133	+	0.849783i	$0.676733\pi$
$684$	−23.3925	−0.894436
$685$	−16.0603	−0.613632
$686$	10.6535	0.406753
$687$	22.4774	0.857568
$688$	11.5417	0.440025
$689$	12.6211	0.480826
$690$	−15.3087	−0.582792
$691$	−6.48019	−0.246518	−0.123259	−	0.992375i	$-0.539335\pi$
$691$	−6.48019	−0.246518	−0.123259	+	0.992375i	$0.539335\pi$
$692$	5.82155	0.221302
$693$	−24.2560	−0.921409
$694$	−3.51582	−0.133459
$695$	28.2978	1.07340
$696$	−11.8220	−0.448112
$697$	6.89317	0.261097
$698$	17.4947	0.662185
$699$	21.2697	0.804492
$700$	9.37136	0.354204
$701$	−30.4377	−1.14962	−0.574809	−	0.818288i	$-0.694923\pi$
$701$	−30.4377	−1.14962	−0.574809	+	0.818288i	$0.694923\pi$
$702$	−26.7787	−1.01070
$703$	−23.0184	−0.868154
$704$	1.00000	0.0376889
$705$	32.3640	1.21890
$706$	0.557129	0.0209678
$707$	38.6799	1.45471
$708$	9.89942	0.372043
$709$	−18.4969	−0.694664	−0.347332	−	0.937742i	$-0.612912\pi$
$709$	−18.4969	−0.694664	−0.347332	+	0.937742i	$0.612912\pi$
$710$	−7.30432	−0.274126
$711$	32.3315	1.21253
$712$	−1.75396	−0.0657325
$713$	−30.5268	−1.14324
$714$	−34.1201	−1.27691
$715$	2.76369	0.103356
$716$	0.756193	0.0282603
$717$	3.12150	0.116574
$718$	21.8054	0.813770
$719$	6.55112	0.244316	0.122158	−	0.992511i	$-0.461019\pi$
$719$	6.55112	0.244316	0.122158	+	0.992511i	$0.461019\pi$
$720$	10.8267	0.403486
$721$	43.6219	1.62456
$722$	−9.00095	−0.334981
$723$	18.7884	0.698746
$724$	9.38516	0.348797
$725$	−10.4786	−0.389166
$726$	−3.22455	−0.119674
$727$	2.79126	0.103522	0.0517610	−	0.998660i	$-0.483517\pi$
$727$	2.79126	0.103522	0.0517610	+	0.998660i	$0.483517\pi$
$728$	−6.19176	−0.229482
$729$	36.9127	1.36714
$730$	3.99198	0.147750
$731$	−37.2470	−1.37763
$732$	12.2356	0.452242
$733$	−3.49792	−0.129199	−0.0645994	−	0.997911i	$-0.520577\pi$
$733$	−3.49792	−0.129199	−0.0645994	+	0.997911i	$0.520577\pi$
$734$	28.5855	1.05511
$735$	−17.7008	−0.652905
$736$	3.24393	0.119573
$737$	−1.54211	−0.0568044
$738$	−15.8014	−0.581659
$739$	2.88651	0.106182	0.0530910	−	0.998590i	$-0.483093\pi$
$739$	2.88651	0.106182	0.0530910	+	0.998590i	$0.483093\pi$
$740$	10.6535	0.391630
$741$	19.2549	0.707347
$742$	−21.9142	−0.804496
$743$	15.2975	0.561210	0.280605	−	0.959823i	$-0.409465\pi$
$743$	15.2975	0.561210	0.280605	+	0.959823i	$0.409465\pi$
$744$	30.3444	1.11248
$745$	−27.8096	−1.01886
$746$	−17.7695	−0.650586
$747$	98.7271	3.61224
$748$	−3.22716	−0.117997
$749$	29.7012	1.08526
$750$	37.0839	1.35411
$751$	35.3957	1.29161	0.645803	−	0.763504i	$-0.276523\pi$
$751$	35.3957	1.29161	0.645803	+	0.763504i	$0.276523\pi$
$752$	−6.85798	−0.250085
$753$	66.9890	2.44122
$754$	6.92333	0.252133
$755$	−32.2185	−1.17255
$756$	46.4963	1.69105
$757$	7.07738	0.257232	0.128616	−	0.991694i	$-0.458947\pi$
$757$	7.07738	0.257232	0.128616	+	0.991694i	$0.458947\pi$
$758$	−30.3399	−1.10200
$759$	−10.4602	−0.379682
$760$	−4.62782	−0.167869
$761$	−9.52475	−0.345272	−0.172636	−	0.984986i	$-0.555228\pi$
$761$	−9.52475	−0.345272	−0.172636	+	0.984986i	$0.555228\pi$
$762$	32.0659	1.16163
$763$	31.2118	1.12994
$764$	−24.7871	−0.896764
$765$	−34.9394	−1.26324
$766$	−4.71362	−0.170310
$767$	−5.79740	−0.209332
$768$	−3.22455	−0.116356
$769$	−8.23684	−0.297028	−0.148514	−	0.988910i	$-0.547449\pi$
$769$	−8.23684	−0.297028	−0.148514	+	0.988910i	$0.547449\pi$
$770$	−4.79864	−0.172931
$771$	−17.8669	−0.643461
$772$	7.29045	0.262389
$773$	−34.3972	−1.23718	−0.618591	−	0.785713i	$-0.712296\pi$
$773$	−34.3972	−1.23718	−0.618591	+	0.785713i	$0.712296\pi$
$774$	85.3826	3.06901
$775$	26.8962	0.966140
$776$	−2.05832	−0.0738893
$777$	76.9636	2.76105
$778$	−32.2092	−1.15475
$779$	6.75426	0.241996
$780$	−8.91167	−0.319089
$781$	−4.99095	−0.178590
$782$	−10.4687	−0.374360
$783$	−51.9899	−1.85797
$784$	3.75083	0.133958
$785$	−32.4552	−1.15838
$786$	30.5510	1.08972
$787$	−0.588943	−0.0209935	−0.0104968	−	0.999945i	$-0.503341\pi$
$787$	−0.588943	−0.0209935	−0.0104968	+	0.999945i	$0.503341\pi$
$788$	−1.00000	−0.0356235
$789$	30.7579	1.09501
$790$	6.39624	0.227568
$791$	−55.1213	−1.95989
$792$	7.39772	0.262867
$793$	−7.16556	−0.254456
$794$	7.50691	0.266410
$795$	−31.5407	−1.11863
$796$	24.8930	0.882310
$797$	29.9710	1.06163	0.530813	−	0.847489i	$-0.321887\pi$
$797$	29.9710	1.06163	0.530813	+	0.847489i	$0.321887\pi$
$798$	−33.4326	−1.18350
$799$	22.1318	0.782966
$800$	−2.85813	−0.101050
$801$	−12.9753	−0.458460
$802$	−3.75518	−0.132600
$803$	2.72767	0.0962573
$804$	4.97261	0.175371
$805$	−15.5665	−0.548645
$806$	−17.7706	−0.625943
$807$	16.3921	0.577029
$808$	−11.7968	−0.415010
$809$	−31.6484	−1.11270	−0.556349	−	0.830948i	$-0.687798\pi$
$809$	−31.6484	−1.11270	−0.556349	+	0.830948i	$0.687798\pi$
$810$	34.4411	1.21014
$811$	2.85210	0.100151	0.0500754	−	0.998745i	$-0.484054\pi$
$811$	2.85210	0.100151	0.0500754	+	0.998745i	$0.484054\pi$
$812$	−12.0211	−0.421857
$813$	−12.4300	−0.435940
$814$	7.27939	0.255143
$815$	9.22392	0.323100
$816$	10.4061	0.364288
$817$	−36.4965	−1.27685
$818$	5.89350	0.206061
$819$	−45.8049	−1.60055
$820$	−3.12604	−0.109166
$821$	−3.88605	−0.135624	−0.0678120	−	0.997698i	$-0.521602\pi$
$821$	−3.88605	−0.135624	−0.0678120	+	0.997698i	$0.521602\pi$
$822$	35.3855	1.23421
$823$	44.3840	1.54713	0.773565	−	0.633717i	$-0.218472\pi$
$823$	44.3840	1.54713	0.773565	+	0.633717i	$0.218472\pi$
$824$	−13.3040	−0.463468
$825$	9.21617	0.320866
$826$	10.0661	0.350245
$827$	3.19638	0.111149	0.0555746	−	0.998455i	$-0.482301\pi$
$827$	3.19638	0.111149	0.0555746	+	0.998455i	$0.482301\pi$
$828$	23.9977	0.833978
$829$	−12.8995	−0.448019	−0.224009	−	0.974587i	$-0.571915\pi$
$829$	−12.8995	−0.448019	−0.224009	+	0.974587i	$0.571915\pi$
$830$	19.5315	0.677947
$831$	75.1032	2.60530
$832$	1.88840	0.0654683
$833$	−12.1045	−0.419397
$834$	−62.3483	−2.15895
$835$	−33.1577	−1.14747
$836$	−3.16213	−0.109364
$837$	133.446	4.61258
$838$	−9.91227	−0.342414
$839$	−36.1362	−1.24756	−0.623779	−	0.781600i	$-0.714404\pi$
$839$	−36.1362	−1.24756	−0.623779	+	0.781600i	$0.714404\pi$
$840$	15.4734	0.533885
$841$	−15.5586	−0.536504
$842$	18.9747	0.653911
$843$	47.2375	1.62695
$844$	−9.43436	−0.324744
$845$	−13.8067	−0.474966
$846$	−50.7334	−1.74425
$847$	−3.27885	−0.112662
$848$	6.68351	0.229513
$849$	49.0529	1.68349
$850$	9.22363	0.316368
$851$	23.6139	0.809473
$852$	16.0936	0.551356
$853$	49.0426	1.67919	0.839593	−	0.543215i	$-0.182793\pi$
$853$	49.0426	1.67919	0.839593	+	0.543215i	$0.182793\pi$
$854$	12.4417	0.425745
$855$	−34.2353	−1.17082
$856$	−9.05843	−0.309611
$857$	38.2707	1.30730	0.653652	−	0.756796i	$-0.273236\pi$
$857$	38.2707	1.30730	0.653652	+	0.756796i	$0.273236\pi$
$858$	−6.08923	−0.207883
$859$	−8.50638	−0.290234	−0.145117	−	0.989415i	$-0.546356\pi$
$859$	−8.50638	−0.290234	−0.145117	+	0.989415i	$0.546356\pi$
$860$	16.8915	0.575995
$861$	−22.5834	−0.769639
$862$	−34.2926	−1.16801
$863$	−24.0893	−0.820009	−0.410004	−	0.912084i	$-0.634473\pi$
$863$	−24.0893	−0.820009	−0.410004	+	0.912084i	$0.634473\pi$
$864$	−14.1807	−0.482436
$865$	8.51993	0.289686
$866$	−7.01812	−0.238486
$867$	21.2351	0.721181
$868$	30.8553	1.04730
$869$	4.37047	0.148258
$870$	−17.3017	−0.586582
$871$	−2.91211	−0.0986732
$872$	−9.51916	−0.322359
$873$	−15.2269	−0.515351
$874$	−10.2577	−0.346973
$875$	37.7083	1.27477
$876$	−8.79550	−0.297172
$877$	−46.3089	−1.56374	−0.781870	−	0.623441i	$-0.785734\pi$
$877$	−46.3089	−1.56374	−0.781870	+	0.623441i	$0.785734\pi$
$878$	18.2788	0.616878
$879$	53.9777	1.82062
$880$	1.46351	0.0493350
$881$	44.7813	1.50872	0.754360	−	0.656461i	$-0.227947\pi$
$881$	44.7813	1.50872	0.754360	+	0.656461i	$0.227947\pi$
$882$	27.7476	0.934311
$883$	−11.4396	−0.384975	−0.192487	−	0.981299i	$-0.561655\pi$
$883$	−11.4396	−0.384975	−0.192487	+	0.981299i	$0.561655\pi$
$884$	−6.09415	−0.204969
$885$	14.4879	0.487007
$886$	−35.3941	−1.18909
$887$	10.5955	0.355762	0.177881	−	0.984052i	$-0.443076\pi$
$887$	10.5955	0.355762	0.177881	+	0.984052i	$0.443076\pi$
$888$	−23.4728	−0.787695
$889$	32.6059	1.09357
$890$	−2.56695	−0.0860443
$891$	23.5331	0.788390
$892$	20.3599	0.681700
$893$	21.6858	0.725688
$894$	61.2726	2.04926
$895$	1.10670	0.0369929
$896$	−3.27885	−0.109539
$897$	−19.7530	−0.659535
$898$	23.6620	0.789612
$899$	−34.5010	−1.15067
$900$	−21.1436	−0.704788
$901$	−21.5688	−0.718560
$902$	−2.13599	−0.0711205
$903$	122.029	4.06086
$904$	16.8112	0.559132
$905$	13.7353	0.456577
$906$	70.9867	2.35838
$907$	−7.14551	−0.237263	−0.118631	−	0.992938i	$-0.537851\pi$
$907$	−7.14551	−0.237263	−0.118631	+	0.992938i	$0.537851\pi$
$908$	7.21042	0.239286
$909$	−87.2695	−2.89455
$910$	−9.06172	−0.300393
$911$	35.8735	1.18854	0.594271	−	0.804265i	$-0.297441\pi$
$911$	35.8735	1.18854	0.594271	+	0.804265i	$0.297441\pi$
$912$	10.1964	0.337638
$913$	13.3456	0.441675
$914$	−9.08643	−0.300552
$915$	17.9070	0.591987
$916$	−6.97072	−0.230319
$917$	31.0654	1.02587
$918$	45.7633	1.51041
$919$	−24.6221	−0.812207	−0.406103	−	0.913827i	$-0.633113\pi$
$919$	−24.6221	−0.812207	−0.406103	+	0.913827i	$0.633113\pi$
$920$	4.74754	0.156522
$921$	48.3630	1.59362
$922$	35.1534	1.15772
$923$	−9.42488	−0.310224
$924$	10.5728	0.347820
$925$	−20.8054	−0.684078
$926$	12.1423	0.399020
$927$	−98.4196	−3.23252
$928$	3.66625	0.120351
$929$	59.9353	1.96641	0.983207	−	0.182495i	$-0.0584172\pi$
$929$	59.9353	1.96641	0.983207	+	0.182495i	$0.0584172\pi$
$930$	44.4094	1.45624
$931$	−11.8606	−0.388716
$932$	−6.59616	−0.216065
$933$	−23.6301	−0.773616
$934$	17.0844	0.559019
$935$	−4.72299	−0.154458
$936$	13.9698	0.456618
$937$	22.2126	0.725655	0.362828	−	0.931856i	$-0.381811\pi$
$937$	22.2126	0.725655	0.362828	+	0.931856i	$0.381811\pi$
$938$	5.05634	0.165095
$939$	100.817	3.29005
$940$	−10.0367	−0.327362
$941$	52.6719	1.71705	0.858527	−	0.512768i	$-0.171380\pi$
$941$	52.6719	1.71705	0.858527	+	0.512768i	$0.171380\pi$
$942$	71.5083	2.32987
$943$	−6.92899	−0.225639
$944$	−3.07002	−0.0999205
$945$	68.0479	2.21360
$946$	11.5417	0.375254
$947$	8.80457	0.286110	0.143055	−	0.989715i	$-0.454307\pi$
$947$	8.80457	0.286110	0.143055	+	0.989715i	$0.454307\pi$
$948$	−14.0928	−0.457713
$949$	5.15092	0.167206
$950$	9.03776	0.293224
$951$	−42.0515	−1.36361
$952$	10.5814	0.342944
$953$	−27.7991	−0.900500	−0.450250	−	0.892902i	$-0.648665\pi$
$953$	−27.7991	−0.900500	−0.450250	+	0.892902i	$0.648665\pi$
$954$	49.4428	1.60077
$955$	−36.2762	−1.17387
$956$	−0.968041	−0.0313087
$957$	−11.8220	−0.382151
$958$	−22.9729	−0.742219
$959$	35.9813	1.16190
$960$	−4.71917	−0.152311
$961$	57.5561	1.85665
$962$	13.7464	0.443201
$963$	−67.0117	−2.15942
$964$	−5.82666	−0.187664
$965$	10.6697	0.343469
$966$	34.2975	1.10350
$967$	52.9129	1.70156	0.850782	−	0.525519i	$-0.176129\pi$
$967$	52.9129	1.70156	0.850782	+	0.525519i	$0.176129\pi$
$968$	1.00000	0.0321412
$969$	−32.9055	−1.05708
$970$	−3.01238	−0.0967216
$971$	−15.3774	−0.493484	−0.246742	−	0.969081i	$-0.579360\pi$
$971$	−15.3774	−0.493484	−0.246742	+	0.969081i	$0.579360\pi$
$972$	−33.3417	−1.06944
$973$	−63.3982	−2.03245
$974$	−8.84084	−0.283279
$975$	17.4038	0.557367
$976$	−3.79452	−0.121460
$977$	−28.5120	−0.912180	−0.456090	−	0.889934i	$-0.650751\pi$
$977$	−28.5120	−0.912180	−0.456090	+	0.889934i	$0.650751\pi$
$978$	−20.3230	−0.649858
$979$	−1.75396	−0.0560569
$980$	5.48939	0.175352
$981$	−70.4201	−2.24834
$982$	21.9229	0.699589
$983$	−41.7092	−1.33032	−0.665158	−	0.746703i	$-0.731636\pi$
$983$	−41.7092	−1.33032	−0.665158	+	0.746703i	$0.731636\pi$
$984$	6.88759	0.219568
$985$	−1.46351	−0.0466314
$986$	−11.8316	−0.376794
$987$	−72.5080	−2.30796
$988$	−5.97135	−0.189974
$989$	37.4406	1.19054
$990$	10.8267	0.344094
$991$	26.7159	0.848658	0.424329	−	0.905508i	$-0.360510\pi$
$991$	26.7159	0.848658	0.424329	+	0.905508i	$0.360510\pi$
$992$	−9.41043	−0.298781
$993$	−35.4820	−1.12599
$994$	16.3645	0.519052
$995$	36.4313	1.15495
$996$	−43.0336	−1.36357
$997$	−7.94451	−0.251605	−0.125803	−	0.992055i	$-0.540151\pi$
$997$	−7.94451	−0.251605	−0.125803	+	0.992055i	$0.540151\pi$
$998$	−34.5442	−1.09348
$999$	−103.227	−3.26595

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4334.2.a.b.1.1	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4334.2.a.b.1.1	✓	15	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:	\( N \)	\(=\)	\( 4334 = 2 \cdot 11 \cdot 197 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4334.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.22455 q^{3} +1.00000 q^{4} +1.46351 q^{5} -3.22455 q^{6} -3.27885 q^{7} +1.00000 q^{8} +7.39772 q^{9} +1.46351 q^{10} +1.00000 q^{11} -3.22455 q^{12} +1.88840 q^{13} -3.27885 q^{14} -4.71917 q^{15} +1.00000 q^{16} -3.22716 q^{17} +7.39772 q^{18} -3.16213 q^{19} +1.46351 q^{20} +10.5728 q^{21} +1.00000 q^{22} +3.24393 q^{23} -3.22455 q^{24} -2.85813 q^{25} +1.88840 q^{26} -14.1807 q^{27} -3.27885 q^{28} +3.66625 q^{29} -4.71917 q^{30} -9.41043 q^{31} +1.00000 q^{32} -3.22455 q^{33} -3.22716 q^{34} -4.79864 q^{35} +7.39772 q^{36} +7.27939 q^{37} -3.16213 q^{38} -6.08923 q^{39} +1.46351 q^{40} -2.13599 q^{41} +10.5728 q^{42} +11.5417 q^{43} +1.00000 q^{44} +10.8267 q^{45} +3.24393 q^{46} -6.85798 q^{47} -3.22455 q^{48} +3.75083 q^{49} -2.85813 q^{50} +10.4061 q^{51} +1.88840 q^{52} +6.68351 q^{53} -14.1807 q^{54} +1.46351 q^{55} -3.27885 q^{56} +10.1964 q^{57} +3.66625 q^{58} -3.07002 q^{59} -4.71917 q^{60} -3.79452 q^{61} -9.41043 q^{62} -24.2560 q^{63} +1.00000 q^{64} +2.76369 q^{65} -3.22455 q^{66} -1.54211 q^{67} -3.22716 q^{68} -10.4602 q^{69} -4.79864 q^{70} -4.99095 q^{71} +7.39772 q^{72} +2.72767 q^{73} +7.27939 q^{74} +9.21617 q^{75} -3.16213 q^{76} -3.27885 q^{77} -6.08923 q^{78} +4.37047 q^{79} +1.46351 q^{80} +23.5331 q^{81} -2.13599 q^{82} +13.3456 q^{83} +10.5728 q^{84} -4.72299 q^{85} +11.5417 q^{86} -11.8220 q^{87} +1.00000 q^{88} -1.75396 q^{89} +10.8267 q^{90} -6.19176 q^{91} +3.24393 q^{92} +30.3444 q^{93} -6.85798 q^{94} -4.62782 q^{95} -3.22455 q^{96} -2.05832 q^{97} +3.75083 q^{98} +7.39772 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 15 q^{2} - 9 q^{3} + 15 q^{4} - 11 q^{5} - 9 q^{6} - 11 q^{7} + 15 q^{8} + 10 q^{9} - 11 q^{10} + 15 q^{11} - 9 q^{12} - 21 q^{13} - 11 q^{14} - 2 q^{15} + 15 q^{16} - 4 q^{17} + 10 q^{18} - 22 q^{19}+ \cdots + 10 q^{99}+O(q^{100}) \) 15 * q + 15 * q^2 - 9 * q^3 + 15 * q^4 - 11 * q^5 - 9 * q^6 - 11 * q^7 + 15 * q^8 + 10 * q^9 - 11 * q^10 + 15 * q^11 - 9 * q^12 - 21 * q^13 - 11 * q^14 - 2 * q^15 + 15 * q^16 - 4 * q^17 + 10 * q^18 - 22 * q^19 - 11 * q^20 - 13 * q^21 + 15 * q^22 - 16 * q^23 - 9 * q^24 + 6 * q^25 - 21 * q^26 - 21 * q^27 - 11 * q^28 - 8 * q^29 - 2 * q^30 - 33 * q^31 + 15 * q^32 - 9 * q^33 - 4 * q^34 - 2 * q^35 + 10 * q^36 - q^37 - 22 * q^38 + q^39 - 11 * q^40 - 10 * q^41 - 13 * q^42 - 8 * q^43 + 15 * q^44 - 10 * q^45 - 16 * q^46 - 31 * q^47 - 9 * q^48 + 2 * q^49 + 6 * q^50 + 2 * q^51 - 21 * q^52 - 18 * q^53 - 21 * q^54 - 11 * q^55 - 11 * q^56 + 16 * q^57 - 8 * q^58 - 37 * q^59 - 2 * q^60 - 31 * q^61 - 33 * q^62 - 20 * q^63 + 15 * q^64 - 13 * q^65 - 9 * q^66 + q^67 - 4 * q^68 - 25 * q^69 - 2 * q^70 - 28 * q^71 + 10 * q^72 - 20 * q^73 - q^74 - 9 * q^75 - 22 * q^76 - 11 * q^77 + q^78 - 6 * q^79 - 11 * q^80 + 3 * q^81 - 10 * q^82 - 15 * q^83 - 13 * q^84 - 31 * q^85 - 8 * q^86 - 16 * q^87 + 15 * q^88 - 17 * q^89 - 10 * q^90 - 21 * q^91 - 16 * q^92 + 10 * q^93 - 31 * q^94 - 3 * q^95 - 9 * q^96 - 9 * q^97 + 2 * q^98 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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