LMFDB - Embedded newform 3850.2.c.ba.1849.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3850,2,Mod(1849,3850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3850, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("3850.1849");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3850.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$30.7424047782$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.399424.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 $ x^6 - 2x^5 + 3x^4 - 6x^3 + 6x^2 - 8*x + 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 770)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.2
Root			$-0.671462 + 1.24464i$ of defining polynomial
Character	$\chi$	$=$	3850.1849
Dual form			3850.2.c.ba.1849.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.14637i q^{3} -1.00000 q^{4} +1.14637 q^{6} +1.00000i q^{7} +1.00000i q^{8} +1.68585 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +1.14637i q^{3} -1.00000 q^{4} +1.14637 q^{6} +1.00000i q^{7} +1.00000i q^{8} +1.68585 q^{9} +1.00000 q^{11} -1.14637i q^{12} +4.68585i q^{13} +1.00000 q^{14} +1.00000 q^{16} +0.292731i q^{17} -1.68585i q^{18} -6.51806 q^{19} -1.14637 q^{21} -1.00000i q^{22} +2.85363i q^{23} -1.14637 q^{24} +4.68585 q^{26} +5.37169i q^{27} -1.00000i q^{28} +1.43910 q^{29} +0.978577 q^{31} -1.00000i q^{32} +1.14637i q^{33} +0.292731 q^{34} -1.68585 q^{36} -0.853635i q^{37} +6.51806i q^{38} -5.37169 q^{39} -6.22533 q^{41} +1.14637i q^{42} -10.3503i q^{43} -1.00000 q^{44} +2.85363 q^{46} +9.95715i q^{47} +1.14637i q^{48} -1.00000 q^{49} -0.335577 q^{51} -4.68585i q^{52} +5.43910i q^{53} +5.37169 q^{54} -1.00000 q^{56} -7.47208i q^{57} -1.43910i q^{58} +9.37169 q^{59} -11.9572 q^{61} -0.978577i q^{62} +1.68585i q^{63} -1.00000 q^{64} +1.14637 q^{66} +0.585462i q^{67} -0.292731i q^{68} -3.27131 q^{69} -0.335577 q^{71} +1.68585i q^{72} -3.70727i q^{73} -0.853635 q^{74} +6.51806 q^{76} +1.00000i q^{77} +5.37169i q^{78} +2.51806 q^{79} -1.10038 q^{81} +6.22533i q^{82} -1.70727i q^{83} +1.14637 q^{84} -10.3503 q^{86} +1.64973i q^{87} +1.00000i q^{88} -13.0790 q^{89} -4.68585 q^{91} -2.85363i q^{92} +1.12181i q^{93} +9.95715 q^{94} +1.14637 q^{96} -9.10352i q^{97} +1.00000i q^{98} +1.68585 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{4} + 4 q^{6} - 14 q^{9}+O(q^{10}) $ 6 * q - 6 * q^4 + 4 * q^6 - 14 * q^9	$ 6 q - 6 q^{4} + 4 q^{6} - 14 q^{9} + 6 q^{11} + 6 q^{14} + 6 q^{16} + 12 q^{19} - 4 q^{21} - 4 q^{24} + 4 q^{26} - 24 q^{31} - 4 q^{34} + 14 q^{36} + 16 q^{39} + 8 q^{41} - 6 q^{44} + 20 q^{46} - 6 q^{49} - 56 q^{51} - 16 q^{54} - 6 q^{56} + 8 q^{59} - 12 q^{61} - 6 q^{64} + 4 q^{66} + 16 q^{69} - 56 q^{71} - 8 q^{74} - 12 q^{76} - 36 q^{79} + 6 q^{81} + 4 q^{84} + 16 q^{86} - 36 q^{89} - 4 q^{91} + 4 q^{96} - 14 q^{99}+O(q^{100}) $ 6 * q - 6 * q^4 + 4 * q^6 - 14 * q^9 + 6 * q^11 + 6 * q^14 + 6 * q^16 + 12 * q^19 - 4 * q^21 - 4 * q^24 + 4 * q^26 - 24 * q^31 - 4 * q^34 + 14 * q^36 + 16 * q^39 + 8 * q^41 - 6 * q^44 + 20 * q^46 - 6 * q^49 - 56 * q^51 - 16 * q^54 - 6 * q^56 + 8 * q^59 - 12 * q^61 - 6 * q^64 + 4 * q^66 + 16 * q^69 - 56 * q^71 - 8 * q^74 - 12 * q^76 - 36 * q^79 + 6 * q^81 + 4 * q^84 + 16 * q^86 - 36 * q^89 - 4 * q^91 + 4 * q^96 - 14 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3850\mathbb{Z}\right)^\times$.

$n$	$1751$	$2201$	$2927$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	1.14637i	0.661854i	0.943656	+	0.330927i	$0.107361\pi$
$3$	1.14637i	0.661854i	−0.943656	+	0.330927i	$0.892639\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	1.14637	0.468002
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	1.00000i	0.353553i
$9$	1.68585	0.561949
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	− 1.14637i	− 0.330927i
$13$	4.68585i	1.29962i	0.760097	+	0.649810i	$0.225152\pi$
$13$	4.68585i	1.29962i	−0.760097	+	0.649810i	$0.774848\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	0.292731i	0.0709977i	0.999370	+	0.0354988i	$0.0113020\pi$
$17$	0.292731i	0.0709977i	−0.999370	+	0.0354988i	$0.988698\pi$
$18$	− 1.68585i	− 0.397358i
$19$	−6.51806	−1.49535	−0.747673	−	0.664068i	$-0.768829\pi$
$19$	−6.51806	−1.49535	−0.747673	+	0.664068i	$0.768829\pi$
$20$	0	0
$21$	−1.14637	−0.250157
$22$	− 1.00000i	− 0.213201i
$23$	2.85363i	0.595024i	0.954718	+	0.297512i	$0.0961568\pi$
$23$	2.85363i	0.595024i	−0.954718	+	0.297512i	$0.903843\pi$
$24$	−1.14637	−0.234001
$25$	0	0
$26$	4.68585	0.918970
$27$	5.37169i	1.03378i
$28$	− 1.00000i	− 0.188982i
$29$	1.43910	0.267234	0.133617	−	0.991033i	$-0.457341\pi$
$29$	1.43910	0.267234	0.133617	+	0.991033i	$0.457341\pi$
$30$	0	0
$31$	0.978577	0.175758	0.0878788	−	0.996131i	$-0.471991\pi$
$31$	0.978577	0.175758	0.0878788	+	0.996131i	$0.471991\pi$
$32$	− 1.00000i	− 0.176777i
$33$	1.14637i	0.199557i
$34$	0.292731	0.0502029
$35$	0	0
$36$	−1.68585	−0.280974
$37$	− 0.853635i	− 0.140337i	−0.997535	−	0.0701683i	$-0.977646\pi$
$37$	− 0.853635i	− 0.140337i	0.997535	−	0.0701683i	$-0.0223536\pi$
$38$	6.51806i	1.05737i
$39$	−5.37169	−0.860159
$40$	0	0
$41$	−6.22533	−0.972233	−0.486116	−	0.873894i	$-0.661587\pi$
$41$	−6.22533	−0.972233	−0.486116	+	0.873894i	$0.661587\pi$
$42$	1.14637i	0.176888i
$43$	− 10.3503i	− 1.57840i	−0.614135	−	0.789201i	$-0.710495\pi$
$43$	− 10.3503i	− 1.57840i	0.614135	−	0.789201i	$-0.289505\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	2.85363	0.420745
$47$	9.95715i	1.45240i	0.687483	+	0.726200i	$0.258715\pi$
$47$	9.95715i	1.45240i	−0.687483	+	0.726200i	$0.741285\pi$
$48$	1.14637i	0.165464i
$49$	−1.00000	−0.142857
$50$	0	0
$51$	−0.335577	−0.0469901
$52$	− 4.68585i	− 0.649810i
$53$	5.43910i	0.747117i	0.927607	+	0.373559i	$0.121863\pi$
$53$	5.43910i	0.747117i	−0.927607	+	0.373559i	$0.878137\pi$
$54$	5.37169	0.730995
$55$	0	0
$56$	−1.00000	−0.133631
$57$	− 7.47208i	− 0.989701i
$58$	− 1.43910i	− 0.188963i
$59$	9.37169	1.22009	0.610045	−	0.792367i	$-0.291151\pi$
$59$	9.37169	1.22009	0.610045	+	0.792367i	$0.291151\pi$
$60$	0	0
$61$	−11.9572	−1.53096	−0.765478	−	0.643462i	$-0.777498\pi$
$61$	−11.9572	−1.53096	−0.765478	+	0.643462i	$0.777498\pi$
$62$	− 0.978577i	− 0.124279i
$63$	1.68585i	0.212397i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	1.14637	0.141108
$67$	0.585462i	0.0715256i	0.999360	+	0.0357628i	$0.0113861\pi$
$67$	0.585462i	0.0715256i	−0.999360	+	0.0357628i	$0.988614\pi$
$68$	− 0.292731i	− 0.0354988i
$69$	−3.27131	−0.393819
$70$	0	0
$71$	−0.335577	−0.0398256	−0.0199128	−	0.999802i	$-0.506339\pi$
$71$	−0.335577	−0.0398256	−0.0199128	+	0.999802i	$0.506339\pi$
$72$	1.68585i	0.198679i
$73$	− 3.70727i	− 0.433903i	−0.976182	−	0.216952i	$-0.930389\pi$
$73$	− 3.70727i	− 0.433903i	0.976182	−	0.216952i	$-0.0696114\pi$
$74$	−0.853635	−0.0992330
$75$	0	0
$76$	6.51806	0.747673
$77$	1.00000i	0.113961i
$78$	5.37169i	0.608224i
$79$	2.51806	0.283304	0.141652	−	0.989917i	$-0.454759\pi$
$79$	2.51806	0.283304	0.141652	+	0.989917i	$0.454759\pi$
$80$	0	0
$81$	−1.10038	−0.122265
$82$	6.22533i	0.687472i
$83$	− 1.70727i	− 0.187397i	−0.995601	−	0.0936986i	$-0.970131\pi$
$83$	− 1.70727i	− 0.187397i	0.995601	−	0.0936986i	$-0.0298690\pi$
$84$	1.14637	0.125079
$85$	0	0
$86$	−10.3503	−1.11610
$87$	1.64973i	0.176870i
$88$	1.00000i	0.106600i
$89$	−13.0790	−1.38637	−0.693184	−	0.720761i	$-0.743792\pi$
$89$	−13.0790	−1.38637	−0.693184	+	0.720761i	$0.743792\pi$
$90$	0	0
$91$	−4.68585	−0.491210
$92$	− 2.85363i	− 0.297512i
$93$	1.12181i	0.116326i
$94$	9.95715	1.02700
$95$	0	0
$96$	1.14637	0.117000
$97$	− 9.10352i	− 0.924322i	−0.886796	−	0.462161i	$-0.847074\pi$
$97$	− 9.10352i	− 0.924322i	0.886796	−	0.462161i	$-0.152926\pi$
$98$	1.00000i	0.101015i
$99$	1.68585	0.169434
$100$	0	0
$101$	−10.7862	−1.07327	−0.536635	−	0.843814i	$-0.680305\pi$
$101$	−10.7862	−1.07327	−0.536635	+	0.843814i	$0.680305\pi$
$102$	0.335577i	0.0332270i
$103$	7.66442i	0.755198i	0.925969	+	0.377599i	$0.123250\pi$
$103$	7.66442i	0.755198i	−0.925969	+	0.377599i	$0.876750\pi$
$104$	−4.68585	−0.459485
$105$	0	0
$106$	5.43910	0.528292
$107$	− 9.56404i	− 0.924591i	−0.886726	−	0.462295i	$-0.847026\pi$
$107$	− 9.56404i	− 0.924591i	0.886726	−	0.462295i	$-0.152974\pi$
$108$	− 5.37169i	− 0.516891i
$109$	−3.14637	−0.301367	−0.150684	−	0.988582i	$-0.548147\pi$
$109$	−3.14637	−0.301367	−0.150684	+	0.988582i	$0.548147\pi$
$110$	0	0
$111$	0.978577	0.0928824
$112$	1.00000i	0.0944911i
$113$	− 2.58546i	− 0.243220i	−0.992578	−	0.121610i	$-0.961194\pi$
$113$	− 2.58546i	− 0.243220i	0.992578	−	0.121610i	$-0.0388057\pi$
$114$	−7.47208	−0.699824
$115$	0	0
$116$	−1.43910	−0.133617
$117$	7.89962i	0.730320i
$118$	− 9.37169i	− 0.862734i
$119$	−0.292731	−0.0268346
$120$	0	0
$121$	1.00000	0.0909091
$122$	11.9572i	1.08255i
$123$	− 7.13650i	− 0.643477i
$124$	−0.978577	−0.0878788
$125$	0	0
$126$	1.68585	0.150187
$127$	13.3717i	1.18655i	0.805001	+	0.593273i	$0.202164\pi$
$127$	13.3717i	1.18655i	−0.805001	+	0.593273i	$0.797836\pi$
$128$	1.00000i	0.0883883i
$129$	11.8652	1.04467
$130$	0	0
$131$	−11.4391	−0.999438	−0.499719	−	0.866187i	$-0.666563\pi$
$131$	−11.4391	−0.999438	−0.499719	+	0.866187i	$0.666563\pi$
$132$	− 1.14637i	− 0.0997783i
$133$	− 6.51806i	− 0.565187i
$134$	0.585462	0.0505762
$135$	0	0
$136$	−0.292731	−0.0251015
$137$	− 10.0000i	− 0.854358i	−0.904167	−	0.427179i	$-0.859507\pi$
$137$	− 10.0000i	− 0.854358i	0.904167	−	0.427179i	$-0.140493\pi$
$138$	3.27131i	0.278472i
$139$	14.1825	1.20294	0.601471	−	0.798895i	$-0.294581\pi$
$139$	14.1825	1.20294	0.601471	+	0.798895i	$0.294581\pi$
$140$	0	0
$141$	−11.4145	−0.961278
$142$	0.335577i	0.0281610i
$143$	4.68585i	0.391850i
$144$	1.68585	0.140487
$145$	0	0
$146$	−3.70727	−0.306816
$147$	− 1.14637i	− 0.0945506i
$148$	0.853635i	0.0701683i
$149$	−11.5970	−0.950065	−0.475032	−	0.879968i	$-0.657564\pi$
$149$	−11.5970	−0.950065	−0.475032	+	0.879968i	$0.657564\pi$
$150$	0	0
$151$	1.73183	0.140934	0.0704671	−	0.997514i	$-0.477551\pi$
$151$	1.73183	0.140934	0.0704671	+	0.997514i	$0.477551\pi$
$152$	− 6.51806i	− 0.528684i
$153$	0.493499i	0.0398971i
$154$	1.00000	0.0805823
$155$	0	0
$156$	5.37169	0.430080
$157$	12.2927i	0.981067i	0.871422	+	0.490533i	$0.163198\pi$
$157$	12.2927i	0.981067i	−0.871422	+	0.490533i	$0.836802\pi$
$158$	− 2.51806i	− 0.200326i
$159$	−6.23519	−0.494483
$160$	0	0
$161$	−2.85363	−0.224898
$162$	1.10038i	0.0864543i
$163$	23.9143i	1.87311i	0.350516	+	0.936557i	$0.386006\pi$
$163$	23.9143i	1.87311i	−0.350516	+	0.936557i	$0.613994\pi$
$164$	6.22533	0.486116
$165$	0	0
$166$	−1.70727	−0.132510
$167$	8.00000i	0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$	8.00000i	0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$	− 1.14637i	− 0.0884440i
$169$	−8.95715	−0.689012
$170$	0	0
$171$	−10.9884	−0.840307
$172$	10.3503i	0.789201i
$173$	− 7.37169i	− 0.560459i	−0.959933	−	0.280230i	$-0.909589\pi$
$173$	− 7.37169i	− 0.560459i	0.959933	−	0.280230i	$-0.0904106\pi$
$174$	1.64973	0.125066
$175$	0	0
$176$	1.00000	0.0753778
$177$	10.7434i	0.807522i
$178$	13.0790i	0.980310i
$179$	−1.56404	−0.116902	−0.0584509	−	0.998290i	$-0.518616\pi$
$179$	−1.56404	−0.116902	−0.0584509	+	0.998290i	$0.518616\pi$
$180$	0	0
$181$	14.7862	1.09905	0.549526	−	0.835477i	$-0.314808\pi$
$181$	14.7862	1.09905	0.549526	+	0.835477i	$0.314808\pi$
$182$	4.68585i	0.347338i
$183$	− 13.7073i	− 1.01327i
$184$	−2.85363	−0.210373
$185$	0	0
$186$	1.12181	0.0822549
$187$	0.292731i	0.0214066i
$188$	− 9.95715i	− 0.726200i
$189$	−5.37169	−0.390733
$190$	0	0
$191$	−17.9572	−1.29933	−0.649667	−	0.760219i	$-0.725092\pi$
$191$	−17.9572	−1.29933	−0.649667	+	0.760219i	$0.725092\pi$
$192$	− 1.14637i	− 0.0827318i
$193$	18.6430i	1.34195i	0.741479	+	0.670976i	$0.234125\pi$
$193$	18.6430i	1.34195i	−0.741479	+	0.670976i	$0.765875\pi$
$194$	−9.10352	−0.653595
$195$	0	0
$196$	1.00000	0.0714286
$197$	15.0361i	1.07128i	0.844447	+	0.535639i	$0.179929\pi$
$197$	15.0361i	1.07128i	−0.844447	+	0.535639i	$0.820071\pi$
$198$	− 1.68585i	− 0.119808i
$199$	−15.3288	−1.08663	−0.543317	−	0.839528i	$-0.682832\pi$
$199$	−15.3288	−1.08663	−0.543317	+	0.839528i	$0.682832\pi$
$200$	0	0
$201$	−0.671153	−0.0473395
$202$	10.7862i	0.758917i
$203$	1.43910i	0.101005i
$204$	0.335577	0.0234951
$205$	0	0
$206$	7.66442	0.534006
$207$	4.81079i	0.334373i
$208$	4.68585i	0.324905i
$209$	−6.51806	−0.450863
$210$	0	0
$211$	2.04285	0.140635	0.0703176	−	0.997525i	$-0.477599\pi$
$211$	2.04285	0.140635	0.0703176	+	0.997525i	$0.477599\pi$
$212$	− 5.43910i	− 0.373559i
$213$	− 0.384694i	− 0.0263588i
$214$	−9.56404	−0.653784
$215$	0	0
$216$	−5.37169	−0.365497
$217$	0.978577i	0.0664301i
$218$	3.14637i	0.213099i
$219$	4.24989	0.287181
$220$	0	0
$221$	−1.37169	−0.0922700
$222$	− 0.978577i	− 0.0656778i
$223$	11.0790i	0.741902i	0.928652	+	0.370951i	$0.120968\pi$
$223$	11.0790i	0.741902i	−0.928652	+	0.370951i	$0.879032\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−2.58546	−0.171982
$227$	18.5426i	1.23072i	0.788248	+	0.615358i	$0.210989\pi$
$227$	18.5426i	1.23072i	−0.788248	+	0.615358i	$0.789011\pi$
$228$	7.47208i	0.494850i
$229$	8.01469	0.529626	0.264813	−	0.964300i	$-0.414690\pi$
$229$	8.01469	0.529626	0.264813	+	0.964300i	$0.414690\pi$
$230$	0	0
$231$	−1.14637	−0.0754253
$232$	1.43910i	0.0944813i
$233$	27.9572i	1.83153i	0.401710	+	0.915767i	$0.368416\pi$
$233$	27.9572i	1.83153i	−0.401710	+	0.915767i	$0.631584\pi$
$234$	7.89962	0.516414
$235$	0	0
$236$	−9.37169	−0.610045
$237$	2.88661i	0.187506i
$238$	0.292731i	0.0189749i
$239$	28.8108	1.86362	0.931808	−	0.362953i	$-0.118231\pi$
$239$	28.8108	1.86362	0.931808	+	0.362953i	$0.118231\pi$
$240$	0	0
$241$	−8.51806	−0.548696	−0.274348	−	0.961630i	$-0.588462\pi$
$241$	−8.51806	−0.548696	−0.274348	+	0.961630i	$0.588462\pi$
$242$	− 1.00000i	− 0.0642824i
$243$	14.8536i	0.952861i
$244$	11.9572	0.765478
$245$	0	0
$246$	−7.13650	−0.455007
$247$	− 30.5426i	− 1.94338i
$248$	0.978577i	0.0621397i
$249$	1.95715	0.124030
$250$	0	0
$251$	−23.1281	−1.45983	−0.729916	−	0.683537i	$-0.760441\pi$
$251$	−23.1281	−1.45983	−0.729916	+	0.683537i	$0.760441\pi$
$252$	− 1.68585i	− 0.106198i
$253$	2.85363i	0.179406i
$254$	13.3717	0.839015
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 14.8108i	− 0.923872i	−0.886913	−	0.461936i	$-0.847155\pi$
$257$	− 14.8108i	− 0.923872i	0.886913	−	0.461936i	$-0.152845\pi$
$258$	− 11.8652i	− 0.738695i
$259$	0.853635	0.0530423
$260$	0	0
$261$	2.42610	0.150172
$262$	11.4391i	0.706710i
$263$	18.7434i	1.15577i	0.816119	+	0.577883i	$0.196121\pi$
$263$	18.7434i	1.15577i	−0.816119	+	0.577883i	$0.803879\pi$
$264$	−1.14637	−0.0705539
$265$	0	0
$266$	−6.51806	−0.399648
$267$	− 14.9933i	− 0.917573i
$268$	− 0.585462i	− 0.0357628i
$269$	−12.6858	−0.773470	−0.386735	−	0.922191i	$-0.626397\pi$
$269$	−12.6858	−0.773470	−0.386735	+	0.922191i	$0.626397\pi$
$270$	0	0
$271$	1.12181	0.0681449	0.0340725	−	0.999419i	$-0.489152\pi$
$271$	1.12181	0.0681449	0.0340725	+	0.999419i	$0.489152\pi$
$272$	0.292731i	0.0177494i
$273$	− 5.37169i	− 0.325110i
$274$	−10.0000	−0.604122
$275$	0	0
$276$	3.27131	0.196910
$277$	− 30.1151i	− 1.80944i	−0.426007	−	0.904720i	$-0.640080\pi$
$277$	− 30.1151i	− 1.80944i	0.426007	−	0.904720i	$-0.359920\pi$
$278$	− 14.1825i	− 0.850609i
$279$	1.64973	0.0987668
$280$	0	0
$281$	−14.3356	−0.855189	−0.427594	−	0.903971i	$-0.640639\pi$
$281$	−14.3356	−0.855189	−0.427594	+	0.903971i	$0.640639\pi$
$282$	11.4145i	0.679726i
$283$	3.21377i	0.191039i	0.995428	+	0.0955194i	$0.0304512\pi$
$283$	3.21377i	0.191039i	−0.995428	+	0.0955194i	$0.969549\pi$
$284$	0.335577	0.0199128
$285$	0	0
$286$	4.68585	0.277080
$287$	− 6.22533i	− 0.367469i
$288$	− 1.68585i	− 0.0993394i
$289$	16.9143	0.994959
$290$	0	0
$291$	10.4360	0.611767
$292$	3.70727i	0.216952i
$293$	− 16.6858i	− 0.974798i	−0.873179	−	0.487399i	$-0.837946\pi$
$293$	− 16.6858i	− 0.974798i	0.873179	−	0.487399i	$-0.162054\pi$
$294$	−1.14637	−0.0668574
$295$	0	0
$296$	0.853635	0.0496165
$297$	5.37169i	0.311697i
$298$	11.5970i	0.671797i
$299$	−13.3717	−0.773305
$300$	0	0
$301$	10.3503	0.596580
$302$	− 1.73183i	− 0.0996555i
$303$	− 12.3650i	− 0.710349i
$304$	−6.51806	−0.373836
$305$	0	0
$306$	0.493499	0.0282115
$307$	6.29273i	0.359145i	0.983745	+	0.179573i	$0.0574715\pi$
$307$	6.29273i	0.359145i	−0.983745	+	0.179573i	$0.942529\pi$
$308$	− 1.00000i	− 0.0569803i
$309$	−8.78623	−0.499831
$310$	0	0
$311$	−20.3931	−1.15639	−0.578194	−	0.815900i	$-0.696242\pi$
$311$	−20.3931	−1.15639	−0.578194	+	0.815900i	$0.696242\pi$
$312$	− 5.37169i	− 0.304112i
$313$	12.0674i	0.682090i	0.940047	+	0.341045i	$0.110781\pi$
$313$	12.0674i	0.682090i	−0.940047	+	0.341045i	$0.889219\pi$
$314$	12.2927	0.693719
$315$	0	0
$316$	−2.51806	−0.141652
$317$	− 28.7679i	− 1.61577i	−0.589341	−	0.807884i	$-0.700613\pi$
$317$	− 28.7679i	− 1.61577i	0.589341	−	0.807884i	$-0.299387\pi$
$318$	6.23519i	0.349652i
$319$	1.43910	0.0805739
$320$	0	0
$321$	10.9639	0.611944
$322$	2.85363i	0.159027i
$323$	− 1.90804i	− 0.106166i
$324$	1.10038	0.0611325
$325$	0	0
$326$	23.9143	1.32449
$327$	− 3.60688i	− 0.199461i
$328$	− 6.22533i	− 0.343736i
$329$	−9.95715	−0.548956
$330$	0	0
$331$	−3.80765	−0.209288	−0.104644	−	0.994510i	$-0.533370\pi$
$331$	−3.80765	−0.209288	−0.104644	+	0.994510i	$0.533370\pi$
$332$	1.70727i	0.0936986i
$333$	− 1.43910i	− 0.0788620i
$334$	8.00000	0.437741
$335$	0	0
$336$	−1.14637	−0.0625394
$337$	− 15.3717i	− 0.837349i	−0.908136	−	0.418675i	$-0.862495\pi$
$337$	− 15.3717i	− 0.837349i	0.908136	−	0.418675i	$-0.137505\pi$
$338$	8.95715i	0.487205i
$339$	2.96388	0.160976
$340$	0	0
$341$	0.978577	0.0529929
$342$	10.9884i	0.594187i
$343$	− 1.00000i	− 0.0539949i
$344$	10.3503	0.558049
$345$	0	0
$346$	−7.37169	−0.396305
$347$	− 3.02142i	− 0.162198i	−0.996706	−	0.0810992i	$-0.974157\pi$
$347$	− 3.02142i	− 0.162198i	0.996706	−	0.0810992i	$-0.0258431\pi$
$348$	− 1.64973i	− 0.0884348i
$349$	−11.0361	−0.590750	−0.295375	−	0.955381i	$-0.595445\pi$
$349$	−11.0361	−0.590750	−0.295375	+	0.955381i	$0.595445\pi$
$350$	0	0
$351$	−25.1709	−1.34352
$352$	− 1.00000i	− 0.0533002i
$353$	23.9817i	1.27642i	0.769863	+	0.638209i	$0.220324\pi$
$353$	23.9817i	1.27642i	−0.769863	+	0.638209i	$0.779676\pi$
$354$	10.7434	0.571004
$355$	0	0
$356$	13.0790	0.693184
$357$	− 0.335577i	− 0.0177606i
$358$	1.56404i	0.0826620i
$359$	10.5181	0.555122	0.277561	−	0.960708i	$-0.410474\pi$
$359$	10.5181	0.555122	0.277561	+	0.960708i	$0.410474\pi$
$360$	0	0
$361$	23.4851	1.23606
$362$	− 14.7862i	− 0.777147i
$363$	1.14637i	0.0601686i
$364$	4.68585	0.245605
$365$	0	0
$366$	−13.7073	−0.716490
$367$	35.5787i	1.85719i	0.371089	+	0.928597i	$0.378985\pi$
$367$	35.5787i	1.85719i	−0.371089	+	0.928597i	$0.621015\pi$
$368$	2.85363i	0.148756i
$369$	−10.4949	−0.546345
$370$	0	0
$371$	−5.43910	−0.282384
$372$	− 1.12181i	− 0.0581630i
$373$	7.50650i	0.388672i	0.980935	+	0.194336i	$0.0622552\pi$
$373$	7.50650i	0.388672i	−0.980935	+	0.194336i	$0.937745\pi$
$374$	0.292731	0.0151368
$375$	0	0
$376$	−9.95715	−0.513501
$377$	6.74338i	0.347302i
$378$	5.37169i	0.276290i
$379$	−33.4868	−1.72010	−0.860050	−	0.510210i	$-0.829568\pi$
$379$	−33.4868	−1.72010	−0.860050	+	0.510210i	$0.829568\pi$
$380$	0	0
$381$	−15.3288	−0.785321
$382$	17.9572i	0.918768i
$383$	15.6644i	0.800415i	0.916425	+	0.400207i	$0.131062\pi$
$383$	15.6644i	0.800415i	−0.916425	+	0.400207i	$0.868938\pi$
$384$	−1.14637	−0.0585002
$385$	0	0
$386$	18.6430	0.948904
$387$	− 17.4490i	− 0.886981i
$388$	9.10352i	0.462161i
$389$	19.6216	0.994853	0.497427	−	0.867506i	$-0.334278\pi$
$389$	19.6216	0.994853	0.497427	+	0.867506i	$0.334278\pi$
$390$	0	0
$391$	−0.835347	−0.0422453
$392$	− 1.00000i	− 0.0505076i
$393$	− 13.1134i	− 0.661483i
$394$	15.0361	0.757509
$395$	0	0
$396$	−1.68585	−0.0847170
$397$	− 10.2499i	− 0.514427i	−0.966355	−	0.257213i	$-0.917196\pi$
$397$	− 10.2499i	− 0.514427i	0.966355	−	0.257213i	$-0.0828044\pi$
$398$	15.3288i	0.768366i
$399$	7.47208	0.374072
$400$	0	0
$401$	−16.8866	−0.843277	−0.421639	−	0.906764i	$-0.638545\pi$
$401$	−16.8866	−0.843277	−0.421639	+	0.906764i	$0.638545\pi$
$402$	0.671153i	0.0334741i
$403$	4.58546i	0.228418i
$404$	10.7862	0.536635
$405$	0	0
$406$	1.43910	0.0714212
$407$	− 0.853635i	− 0.0423131i
$408$	− 0.335577i	− 0.0166135i
$409$	−18.2744	−0.903613	−0.451807	−	0.892116i	$-0.649220\pi$
$409$	−18.2744	−0.903613	−0.451807	+	0.892116i	$0.649220\pi$
$410$	0	0
$411$	11.4637	0.565460
$412$	− 7.66442i	− 0.377599i
$413$	9.37169i	0.461151i
$414$	4.81079	0.236437
$415$	0	0
$416$	4.68585	0.229743
$417$	16.2583i	0.796173i
$418$	6.51806i	0.318809i
$419$	37.2860	1.82154	0.910770	−	0.412914i	$-0.135489\pi$
$419$	37.2860	1.82154	0.910770	+	0.412914i	$0.135489\pi$
$420$	0	0
$421$	−2.78623	−0.135793	−0.0678963	−	0.997692i	$-0.521629\pi$
$421$	−2.78623	−0.135793	−0.0678963	+	0.997692i	$0.521629\pi$
$422$	− 2.04285i	− 0.0994442i
$423$	16.7862i	0.816174i
$424$	−5.43910	−0.264146
$425$	0	0
$426$	−0.384694	−0.0186385
$427$	− 11.9572i	− 0.578647i
$428$	9.56404i	0.462295i
$429$	−5.37169	−0.259348
$430$	0	0
$431$	38.0477	1.83269	0.916346	−	0.400387i	$-0.131124\pi$
$431$	38.0477	1.83269	0.916346	+	0.400387i	$0.131124\pi$
$432$	5.37169i	0.258446i
$433$	− 10.8108i	− 0.519533i	−0.965671	−	0.259767i	$-0.916354\pi$
$433$	− 10.8108i	− 0.519533i	0.965671	−	0.259767i	$-0.0836457\pi$
$434$	0.978577	0.0469732
$435$	0	0
$436$	3.14637	0.150684
$437$	− 18.6002i	− 0.889766i
$438$	− 4.24989i	− 0.203067i
$439$	−30.9933	−1.47923	−0.739614	−	0.673031i	$-0.764992\pi$
$439$	−30.9933	−1.47923	−0.739614	+	0.673031i	$0.764992\pi$
$440$	0	0
$441$	−1.68585	−0.0802784
$442$	1.37169i	0.0652448i
$443$	25.3717i	1.20545i	0.797951	+	0.602723i	$0.205917\pi$
$443$	25.3717i	1.20545i	−0.797951	+	0.602723i	$0.794083\pi$
$444$	−0.978577	−0.0464412
$445$	0	0
$446$	11.0790	0.524604
$447$	− 13.2944i	− 0.628805i
$448$	− 1.00000i	− 0.0472456i
$449$	0.886615	0.0418419	0.0209210	−	0.999781i	$-0.493340\pi$
$449$	0.886615	0.0418419	0.0209210	+	0.999781i	$0.493340\pi$
$450$	0	0
$451$	−6.22533	−0.293139
$452$	2.58546i	0.121610i
$453$	1.98531i	0.0932779i
$454$	18.5426	0.870248
$455$	0	0
$456$	7.47208	0.349912
$457$	− 29.9143i	− 1.39933i	−0.714470	−	0.699666i	$-0.753332\pi$
$457$	− 29.9143i	− 1.39933i	0.714470	−	0.699666i	$-0.246668\pi$
$458$	− 8.01469i	− 0.374502i
$459$	−1.57246	−0.0733962
$460$	0	0
$461$	−2.33558	−0.108779	−0.0543893	−	0.998520i	$-0.517321\pi$
$461$	−2.33558	−0.108779	−0.0543893	+	0.998520i	$0.517321\pi$
$462$	1.14637i	0.0533337i
$463$	0.110250i	0.00512374i	0.999997	+	0.00256187i	$0.000815470\pi$
$463$	0.110250i	0.00512374i	−0.999997	+	0.00256187i	$0.999185\pi$
$464$	1.43910	0.0668084
$465$	0	0
$466$	27.9572	1.29509
$467$	− 36.6760i	− 1.69716i	−0.529066	−	0.848581i	$-0.677457\pi$
$467$	− 36.6760i	− 1.69716i	0.529066	−	0.848581i	$-0.322543\pi$
$468$	− 7.89962i	− 0.365160i
$469$	−0.585462	−0.0270341
$470$	0	0
$471$	−14.0920	−0.649323
$472$	9.37169i	0.431367i
$473$	− 10.3503i	− 0.475906i
$474$	2.88661	0.132587
$475$	0	0
$476$	0.292731	0.0134173
$477$	9.16948i	0.419842i
$478$	− 28.8108i	− 1.31777i
$479$	42.7434	1.95300	0.976498	−	0.215529i	$-0.0691474\pi$
$479$	42.7434	1.95300	0.976498	+	0.215529i	$0.0691474\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	8.51806i	0.387987i
$483$	− 3.27131i	− 0.148850i
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	14.8536	0.673775
$487$	36.1396i	1.63764i	0.574048	+	0.818822i	$0.305372\pi$
$487$	36.1396i	1.63764i	−0.574048	+	0.818822i	$0.694628\pi$
$488$	− 11.9572i	− 0.541275i
$489$	−27.4145	−1.23973
$490$	0	0
$491$	2.54262	0.114747	0.0573733	−	0.998353i	$-0.481727\pi$
$491$	2.54262	0.114747	0.0573733	+	0.998353i	$0.481727\pi$
$492$	7.13650i	0.321738i
$493$	0.421268i	0.0189730i
$494$	−30.5426	−1.37418
$495$	0	0
$496$	0.978577	0.0439394
$497$	− 0.335577i	− 0.0150527i
$498$	− 1.95715i	− 0.0877022i
$499$	−3.13650	−0.140409	−0.0702045	−	0.997533i	$-0.522365\pi$
$499$	−3.13650	−0.140409	−0.0702045	+	0.997533i	$0.522365\pi$
$500$	0	0
$501$	−9.17092	−0.409727
$502$	23.1281i	1.03226i
$503$	− 28.5855i	− 1.27456i	−0.770631	−	0.637281i	$-0.780059\pi$
$503$	− 28.5855i	− 1.27456i	0.770631	−	0.637281i	$-0.219941\pi$
$504$	−1.68585	−0.0750936
$505$	0	0
$506$	2.85363	0.126860
$507$	− 10.2682i	− 0.456026i
$508$	− 13.3717i	− 0.593273i
$509$	−2.20077	−0.0975473	−0.0487737	−	0.998810i	$-0.515531\pi$
$509$	−2.20077	−0.0975473	−0.0487737	+	0.998810i	$0.515531\pi$
$510$	0	0
$511$	3.70727	0.164000
$512$	− 1.00000i	− 0.0441942i
$513$	− 35.0130i	− 1.54586i
$514$	−14.8108	−0.653276
$515$	0	0
$516$	−11.8652	−0.522336
$517$	9.95715i	0.437915i
$518$	− 0.853635i	− 0.0375065i
$519$	8.45065	0.370943
$520$	0	0
$521$	37.1940	1.62950	0.814750	−	0.579812i	$-0.196874\pi$
$521$	37.1940	1.62950	0.814750	+	0.579812i	$0.196874\pi$
$522$	− 2.42610i	− 0.106187i
$523$	30.4078i	1.32964i	0.747003	+	0.664820i	$0.231492\pi$
$523$	30.4078i	1.32964i	−0.747003	+	0.664820i	$0.768508\pi$
$524$	11.4391	0.499719
$525$	0	0
$526$	18.7434	0.817250
$527$	0.286460i	0.0124784i
$528$	1.14637i	0.0498892i
$529$	14.8568	0.645947
$530$	0	0
$531$	15.7992	0.685628
$532$	6.51806i	0.282594i
$533$	− 29.1709i	− 1.26353i
$534$	−14.9933	−0.648822
$535$	0	0
$536$	−0.585462	−0.0252881
$537$	− 1.79296i	− 0.0773720i
$538$	12.6858i	0.546926i
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	39.0607	1.67935	0.839675	−	0.543090i	$-0.182746\pi$
$541$	39.0607	1.67935	0.839675	+	0.543090i	$0.182746\pi$
$542$	− 1.12181i	− 0.0481857i
$543$	16.9504i	0.727412i
$544$	0.292731	0.0125507
$545$	0	0
$546$	−5.37169	−0.229887
$547$	0.585462i	0.0250325i	0.999922	+	0.0125163i	$0.00398416\pi$
$547$	0.585462i	0.0250325i	−0.999922	+	0.0125163i	$0.996016\pi$
$548$	10.0000i	0.427179i
$549$	−20.1579	−0.860319
$550$	0	0
$551$	−9.38011	−0.399606
$552$	− 3.27131i	− 0.139236i
$553$	2.51806i	0.107079i
$554$	−30.1151	−1.27947
$555$	0	0
$556$	−14.1825	−0.601471
$557$	− 17.4637i	− 0.739959i	−0.929040	−	0.369979i	$-0.879365\pi$
$557$	− 17.4637i	− 0.739959i	0.929040	−	0.369979i	$-0.120635\pi$
$558$	− 1.64973i	− 0.0698387i
$559$	48.4998	2.05132
$560$	0	0
$561$	−0.335577	−0.0141681
$562$	14.3356i	0.604710i
$563$	− 18.1579i	− 0.765265i	−0.923901	−	0.382633i	$-0.875018\pi$
$563$	− 18.1579i	− 0.765265i	0.923901	−	0.382633i	$-0.124982\pi$
$564$	11.4145	0.480639
$565$	0	0
$566$	3.21377	0.135085
$567$	− 1.10038i	− 0.0462118i
$568$	− 0.335577i	− 0.0140805i
$569$	22.0000	0.922288	0.461144	−	0.887325i	$-0.347439\pi$
$569$	22.0000	0.922288	0.461144	+	0.887325i	$0.347439\pi$
$570$	0	0
$571$	−9.03612	−0.378150	−0.189075	−	0.981963i	$-0.560549\pi$
$571$	−9.03612	−0.378150	−0.189075	+	0.981963i	$0.560549\pi$
$572$	− 4.68585i	− 0.195925i
$573$	− 20.5855i	− 0.859970i
$574$	−6.22533	−0.259840
$575$	0	0
$576$	−1.68585	−0.0702436
$577$	− 19.3963i	− 0.807476i	−0.914875	−	0.403738i	$-0.867711\pi$
$577$	− 19.3963i	− 0.807476i	0.914875	−	0.403738i	$-0.132289\pi$
$578$	− 16.9143i	− 0.703542i
$579$	−21.3717	−0.888177
$580$	0	0
$581$	1.70727	0.0708295
$582$	− 10.4360i	− 0.432584i
$583$	5.43910i	0.225264i
$584$	3.70727	0.153408
$585$	0	0
$586$	−16.6858	−0.689286
$587$	− 17.2614i	− 0.712456i	−0.934399	−	0.356228i	$-0.884063\pi$
$587$	− 17.2614i	− 0.712456i	0.934399	−	0.356228i	$-0.115937\pi$
$588$	1.14637i	0.0472753i
$589$	−6.37842	−0.262818
$590$	0	0
$591$	−17.2369	−0.709031
$592$	− 0.853635i	− 0.0350842i
$593$	− 11.0361i	− 0.453199i	−0.973988	−	0.226599i	$-0.927239\pi$
$593$	− 11.0361i	− 0.453199i	0.973988	−	0.226599i	$-0.0727608\pi$
$594$	5.37169	0.220403
$595$	0	0
$596$	11.5970	0.475032
$597$	− 17.5725i	− 0.719193i
$598$	13.3717i	0.546809i
$599$	41.7367	1.70531	0.852657	−	0.522472i	$-0.174990\pi$
$599$	41.7367	1.70531	0.852657	+	0.522472i	$0.174990\pi$
$600$	0	0
$601$	39.3106	1.60351	0.801756	−	0.597652i	$-0.203900\pi$
$601$	39.3106	1.60351	0.801756	+	0.597652i	$0.203900\pi$
$602$	− 10.3503i	− 0.421845i
$603$	0.986999i	0.0401937i
$604$	−1.73183	−0.0704671
$605$	0	0
$606$	−12.3650	−0.502292
$607$	6.35027i	0.257749i	0.991661	+	0.128875i	$0.0411365\pi$
$607$	6.35027i	0.257749i	−0.991661	+	0.128875i	$0.958863\pi$
$608$	6.51806i	0.264342i
$609$	−1.64973	−0.0668505
$610$	0	0
$611$	−46.6577	−1.88757
$612$	− 0.493499i	− 0.0199485i
$613$	− 42.4507i	− 1.71457i	−0.514846	−	0.857283i	$-0.672151\pi$
$613$	− 42.4507i	− 1.71457i	0.514846	−	0.857283i	$-0.327849\pi$
$614$	6.29273	0.253954
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	0.677425i	0.0272721i	0.999907	+	0.0136360i	$0.00434062\pi$
$617$	0.677425i	0.0272721i	−0.999907	+	0.0136360i	$0.995659\pi$
$618$	8.78623i	0.353434i
$619$	18.5426	0.745291	0.372645	−	0.927974i	$-0.378451\pi$
$619$	18.5426	0.745291	0.372645	+	0.927974i	$0.378451\pi$
$620$	0	0
$621$	−15.3288	−0.615125
$622$	20.3931i	0.817689i
$623$	− 13.0790i	− 0.523998i
$624$	−5.37169	−0.215040
$625$	0	0
$626$	12.0674	0.482310
$627$	− 7.47208i	− 0.298406i
$628$	− 12.2927i	− 0.490533i
$629$	0.249885	0.00996358
$630$	0	0
$631$	21.3717	0.850794	0.425397	−	0.905007i	$-0.360135\pi$
$631$	21.3717	0.850794	0.425397	+	0.905007i	$0.360135\pi$
$632$	2.51806i	0.100163i
$633$	2.34185i	0.0930801i
$634$	−28.7679	−1.14252
$635$	0	0
$636$	6.23519	0.247241
$637$	− 4.68585i	− 0.185660i
$638$	− 1.43910i	− 0.0569744i
$639$	−0.565731	−0.0223800
$640$	0	0
$641$	23.8715	0.942866	0.471433	−	0.881902i	$-0.343737\pi$
$641$	23.8715	0.942866	0.471433	+	0.881902i	$0.343737\pi$
$642$	− 10.9639i	− 0.432710i
$643$	0.311018i	0.0122654i	0.999981	+	0.00613268i	$0.00195211\pi$
$643$	0.311018i	0.0122654i	−0.999981	+	0.00613268i	$0.998048\pi$
$644$	2.85363	0.112449
$645$	0	0
$646$	−1.90804	−0.0750707
$647$	9.62158i	0.378263i	0.981952	+	0.189132i	$0.0605673\pi$
$647$	9.62158i	0.378263i	−0.981952	+	0.189132i	$0.939433\pi$
$648$	− 1.10038i	− 0.0432272i
$649$	9.37169	0.367871
$650$	0	0
$651$	−1.12181	−0.0439671
$652$	− 23.9143i	− 0.936557i
$653$	− 27.0607i	− 1.05897i	−0.848321	−	0.529483i	$-0.822386\pi$
$653$	− 27.0607i	− 1.05897i	0.848321	−	0.529483i	$-0.177614\pi$
$654$	−3.60688	−0.141040
$655$	0	0
$656$	−6.22533	−0.243058
$657$	− 6.24989i	− 0.243831i
$658$	9.95715i	0.388170i
$659$	26.4935	1.03204	0.516020	−	0.856576i	$-0.327413\pi$
$659$	26.4935	1.03204	0.516020	+	0.856576i	$0.327413\pi$
$660$	0	0
$661$	35.0852	1.36466	0.682329	−	0.731046i	$-0.260967\pi$
$661$	35.0852	1.36466	0.682329	+	0.731046i	$0.260967\pi$
$662$	3.80765i	0.147989i
$663$	− 1.57246i	− 0.0610693i
$664$	1.70727	0.0662549
$665$	0	0
$666$	−1.43910	−0.0557639
$667$	4.10666i	0.159010i
$668$	− 8.00000i	− 0.309529i
$669$	−12.7005	−0.491031
$670$	0	0
$671$	−11.9572	−0.461601
$672$	1.14637i	0.0442220i
$673$	32.6577i	1.25886i	0.777057	+	0.629431i	$0.216712\pi$
$673$	32.6577i	1.25886i	−0.777057	+	0.629431i	$0.783288\pi$
$674$	−15.3717	−0.592095
$675$	0	0
$676$	8.95715	0.344506
$677$	− 9.12808i	− 0.350821i	−0.984495	−	0.175410i	$-0.943875\pi$
$677$	− 9.12808i	− 0.350821i	0.984495	−	0.175410i	$-0.0561252\pi$
$678$	− 2.96388i	− 0.113827i
$679$	9.10352	0.349361
$680$	0	0
$681$	−21.2566	−0.814555
$682$	− 0.978577i	− 0.0374717i
$683$	10.4935i	0.401523i	0.979640	+	0.200761i	$0.0643416\pi$
$683$	10.4935i	0.401523i	−0.979640	+	0.200761i	$0.935658\pi$
$684$	10.9884	0.420154
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	9.18777i	0.350535i
$688$	− 10.3503i	− 0.394600i
$689$	−25.4868	−0.970969
$690$	0	0
$691$	1.70727	0.0649476	0.0324738	−	0.999473i	$-0.489661\pi$
$691$	1.70727	0.0649476	0.0324738	+	0.999473i	$0.489661\pi$
$692$	7.37169i	0.280230i
$693$	1.68585i	0.0640400i
$694$	−3.02142	−0.114692
$695$	0	0
$696$	−1.64973	−0.0625329
$697$	− 1.82235i	− 0.0690263i
$698$	11.0361i	0.417723i
$699$	−32.0491	−1.21221
$700$	0	0
$701$	−37.3534	−1.41082	−0.705409	−	0.708800i	$-0.749237\pi$
$701$	−37.3534	−1.41082	−0.705409	+	0.708800i	$0.749237\pi$
$702$	25.1709i	0.950015i
$703$	5.56404i	0.209852i
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	23.9817	0.902564
$707$	− 10.7862i	− 0.405658i
$708$	− 10.7434i	− 0.403761i
$709$	8.20704	0.308222	0.154111	−	0.988054i	$-0.450749\pi$
$709$	8.20704	0.308222	0.154111	+	0.988054i	$0.450749\pi$
$710$	0	0
$711$	4.24506	0.159202
$712$	− 13.0790i	− 0.490155i
$713$	2.79250i	0.104580i
$714$	−0.335577	−0.0125586
$715$	0	0
$716$	1.56404	0.0584509
$717$	33.0277i	1.23344i
$718$	− 10.5181i	− 0.392530i
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	0	0
$721$	−7.66442	−0.285438
$722$	− 23.4851i	− 0.874024i
$723$	− 9.76481i	− 0.363157i
$724$	−14.7862	−0.549526
$725$	0	0
$726$	1.14637	0.0425456
$727$	46.9442i	1.74106i	0.492113	+	0.870531i	$0.336225\pi$
$727$	46.9442i	1.74106i	−0.492113	+	0.870531i	$0.663775\pi$
$728$	− 4.68585i	− 0.173669i
$729$	−20.3288	−0.752920
$730$	0	0
$731$	3.02984	0.112063
$732$	13.7073i	0.506635i
$733$	− 42.0000i	− 1.55131i	−0.631160	−	0.775653i	$-0.717421\pi$
$733$	− 42.0000i	− 1.55131i	0.631160	−	0.775653i	$-0.282579\pi$
$734$	35.5787	1.31323
$735$	0	0
$736$	2.85363	0.105186
$737$	0.585462i	0.0215658i
$738$	10.4949i	0.386324i
$739$	0.871922	0.0320742	0.0160371	−	0.999871i	$-0.494895\pi$
$739$	0.871922	0.0320742	0.0160371	+	0.999871i	$0.494895\pi$
$740$	0	0
$741$	35.0130	1.28623
$742$	5.43910i	0.199676i
$743$	− 24.6712i	− 0.905097i	−0.891740	−	0.452548i	$-0.850515\pi$
$743$	− 24.6712i	− 0.905097i	0.891740	−	0.452548i	$-0.149485\pi$
$744$	−1.12181	−0.0411274
$745$	0	0
$746$	7.50650	0.274833
$747$	− 2.87819i	− 0.105308i
$748$	− 0.292731i	− 0.0107033i
$749$	9.56404	0.349462
$750$	0	0
$751$	−3.75011	−0.136844	−0.0684218	−	0.997656i	$-0.521796\pi$
$751$	−3.75011	−0.136844	−0.0684218	+	0.997656i	$0.521796\pi$
$752$	9.95715i	0.363100i
$753$	− 26.5132i	− 0.966196i
$754$	6.74338	0.245580
$755$	0	0
$756$	5.37169	0.195367
$757$	− 15.7318i	− 0.571783i	−0.958262	−	0.285891i	$-0.907710\pi$
$757$	− 15.7318i	− 0.571783i	0.958262	−	0.285891i	$-0.0922897\pi$
$758$	33.4868i	1.21629i
$759$	−3.27131	−0.118741
$760$	0	0
$761$	15.2614	0.553227	0.276613	−	0.960981i	$-0.410788\pi$
$761$	15.2614	0.553227	0.276613	+	0.960981i	$0.410788\pi$
$762$	15.3288i	0.555306i
$763$	− 3.14637i	− 0.113906i
$764$	17.9572	0.649667
$765$	0	0
$766$	15.6644	0.565979
$767$	43.9143i	1.58565i
$768$	1.14637i	0.0413659i
$769$	9.63986	0.347622	0.173811	−	0.984779i	$-0.444392\pi$
$769$	9.63986	0.347622	0.173811	+	0.984779i	$0.444392\pi$
$770$	0	0
$771$	16.9786	0.611469
$772$	− 18.6430i	− 0.670976i
$773$	12.8291i	0.461430i	0.973021	+	0.230715i	$0.0741065\pi$
$773$	12.8291i	0.461430i	−0.973021	+	0.230715i	$0.925894\pi$
$774$	−17.4490	−0.627190
$775$	0	0
$776$	9.10352	0.326797
$777$	0.978577i	0.0351063i
$778$	− 19.6216i	− 0.703468i
$779$	40.5770	1.45382
$780$	0	0
$781$	−0.335577	−0.0120079
$782$	0.835347i	0.0298720i
$783$	7.73038i	0.276261i
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	−13.1134	−0.467739
$787$	− 10.8291i	− 0.386015i	−0.981197	−	0.193007i	$-0.938176\pi$
$787$	− 10.8291i	− 0.386015i	0.981197	−	0.193007i	$-0.0618242\pi$
$788$	− 15.0361i	− 0.535639i
$789$	−21.4868	−0.764949
$790$	0	0
$791$	2.58546	0.0919284
$792$	1.68585i	0.0599039i
$793$	− 56.0294i	− 1.98966i
$794$	−10.2499	−0.363755
$795$	0	0
$796$	15.3288	0.543317
$797$	21.0790i	0.746655i	0.927700	+	0.373328i	$0.121783\pi$
$797$	21.0790i	0.746655i	−0.927700	+	0.373328i	$0.878217\pi$
$798$	− 7.47208i	− 0.264509i
$799$	−2.91477	−0.103117
$800$	0	0
$801$	−22.0491	−0.779067
$802$	16.8866i	0.596287i
$803$	− 3.70727i	− 0.130827i
$804$	0.671153	0.0236698
$805$	0	0
$806$	4.58546	0.161516
$807$	− 14.5426i	− 0.511924i
$808$	− 10.7862i	− 0.379458i
$809$	−3.62158	−0.127328	−0.0636639	−	0.997971i	$-0.520279\pi$
$809$	−3.62158	−0.127328	−0.0636639	+	0.997971i	$0.520279\pi$
$810$	0	0
$811$	34.7188	1.21914	0.609571	−	0.792731i	$-0.291342\pi$
$811$	34.7188	1.21914	0.609571	+	0.792731i	$0.291342\pi$
$812$	− 1.43910i	− 0.0505024i
$813$	1.28600i	0.0451020i
$814$	−0.853635	−0.0299199
$815$	0	0
$816$	−0.335577	−0.0117475
$817$	67.4637i	2.36025i
$818$	18.2744i	0.638951i
$819$	−7.89962	−0.276035
$820$	0	0
$821$	−37.7549	−1.31766	−0.658828	−	0.752293i	$-0.728948\pi$
$821$	−37.7549	−1.31766	−0.658828	+	0.752293i	$0.728948\pi$
$822$	− 11.4637i	− 0.399841i
$823$	11.1892i	0.390031i	0.980800	+	0.195016i	$0.0624758\pi$
$823$	11.1892i	0.390031i	−0.980800	+	0.195016i	$0.937524\pi$
$824$	−7.66442	−0.267003
$825$	0	0
$826$	9.37169	0.326083
$827$	7.91431i	0.275207i	0.990487	+	0.137604i	$0.0439400\pi$
$827$	7.91431i	0.275207i	−0.990487	+	0.137604i	$0.956060\pi$
$828$	− 4.81079i	− 0.167186i
$829$	44.7152	1.55302	0.776512	−	0.630102i	$-0.216987\pi$
$829$	44.7152	1.55302	0.776512	+	0.630102i	$0.216987\pi$
$830$	0	0
$831$	34.5229	1.19759
$832$	− 4.68585i	− 0.162452i
$833$	− 0.292731i	− 0.0101425i
$834$	16.2583	0.562979
$835$	0	0
$836$	6.51806	0.225432
$837$	5.25662i	0.181695i
$838$	− 37.2860i	− 1.28802i
$839$	−21.6791	−0.748446	−0.374223	−	0.927339i	$-0.622091\pi$
$839$	−21.6791	−0.748446	−0.374223	+	0.927339i	$0.622091\pi$
$840$	0	0
$841$	−26.9290	−0.928586
$842$	2.78623i	0.0960198i
$843$	− 16.4338i	− 0.566010i
$844$	−2.04285	−0.0703176
$845$	0	0
$846$	16.7862	0.577122
$847$	1.00000i	0.0343604i
$848$	5.43910i	0.186779i
$849$	−3.68415	−0.126440
$850$	0	0
$851$	2.43596	0.0835037
$852$	0.384694i	0.0131794i
$853$	9.41454i	0.322348i	0.986926	+	0.161174i	$0.0515280\pi$
$853$	9.41454i	0.322348i	−0.986926	+	0.161174i	$0.948472\pi$
$854$	−11.9572	−0.409165
$855$	0	0
$856$	9.56404	0.326892
$857$	25.8652i	0.883538i	0.897129	+	0.441769i	$0.145649\pi$
$857$	25.8652i	0.883538i	−0.897129	+	0.441769i	$0.854351\pi$
$858$	5.37169i	0.183387i
$859$	29.2860	0.999225	0.499613	−	0.866249i	$-0.333476\pi$
$859$	29.2860	0.999225	0.499613	+	0.866249i	$0.333476\pi$
$860$	0	0
$861$	7.13650	0.243211
$862$	− 38.0477i	− 1.29591i
$863$	− 20.3110i	− 0.691395i	−0.938346	−	0.345698i	$-0.887642\pi$
$863$	− 20.3110i	− 0.691395i	0.938346	−	0.345698i	$-0.112358\pi$
$864$	5.37169	0.182749
$865$	0	0
$866$	−10.8108	−0.367366
$867$	19.3900i	0.658518i
$868$	− 0.978577i	− 0.0332151i
$869$	2.51806	0.0854193
$870$	0	0
$871$	−2.74338	−0.0929560
$872$	− 3.14637i	− 0.106549i
$873$	− 15.3471i	− 0.519422i
$874$	−18.6002	−0.629160
$875$	0	0
$876$	−4.24989	−0.143590
$877$	38.3650i	1.29549i	0.761856	+	0.647746i	$0.224288\pi$
$877$	38.3650i	1.29549i	−0.761856	+	0.647746i	$0.775712\pi$
$878$	30.9933i	1.04597i
$879$	19.1281	0.645174
$880$	0	0
$881$	40.5426	1.36592	0.682958	−	0.730458i	$-0.260693\pi$
$881$	40.5426	1.36592	0.682958	+	0.730458i	$0.260693\pi$
$882$	1.68585i	0.0567654i
$883$	28.0491i	0.943928i	0.881618	+	0.471964i	$0.156455\pi$
$883$	28.0491i	0.943928i	−0.881618	+	0.471964i	$0.843545\pi$
$884$	1.37169	0.0461350
$885$	0	0
$886$	25.3717	0.852379
$887$	6.65769i	0.223543i	0.993734	+	0.111772i	$0.0356525\pi$
$887$	6.65769i	0.223543i	−0.993734	+	0.111772i	$0.964347\pi$
$888$	0.978577i	0.0328389i
$889$	−13.3717	−0.448472
$890$	0	0
$891$	−1.10038	−0.0368643
$892$	− 11.0790i	− 0.370951i
$893$	− 64.9013i	− 2.17184i
$894$	−13.2944	−0.444632
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	− 15.3288i	− 0.511815i
$898$	− 0.886615i	− 0.0295867i
$899$	1.40827	0.0469683
$900$	0	0
$901$	−1.59219	−0.0530436
$902$	6.22533i	0.207281i
$903$	11.8652i	0.394849i
$904$	2.58546	0.0859912
$905$	0	0
$906$	1.98531	0.0659574
$907$	− 39.5787i	− 1.31419i	−0.753808	−	0.657095i	$-0.771785\pi$
$907$	− 39.5787i	− 1.31419i	0.753808	−	0.657095i	$-0.228215\pi$
$908$	− 18.5426i	− 0.615358i
$909$	−18.1839	−0.603123
$910$	0	0
$911$	−24.3356	−0.806274	−0.403137	−	0.915140i	$-0.632080\pi$
$911$	−24.3356	−0.806274	−0.403137	+	0.915140i	$0.632080\pi$
$912$	− 7.47208i	− 0.247425i
$913$	− 1.70727i	− 0.0565024i
$914$	−29.9143	−0.989477
$915$	0	0
$916$	−8.01469	−0.264813
$917$	− 11.4391i	− 0.377752i
$918$	1.57246i	0.0518989i
$919$	−50.6331	−1.67023	−0.835117	−	0.550073i	$-0.814600\pi$
$919$	−50.6331	−1.67023	−0.835117	+	0.550073i	$0.814600\pi$
$920$	0	0
$921$	−7.21377	−0.237702
$922$	2.33558i	0.0769181i
$923$	− 1.57246i	− 0.0517582i
$924$	1.14637	0.0377127
$925$	0	0
$926$	0.110250	0.00362303
$927$	12.9210i	0.424383i
$928$	− 1.43910i	− 0.0472407i
$929$	−47.1512	−1.54698	−0.773490	−	0.633808i	$-0.781491\pi$
$929$	−47.1512	−1.54698	−0.773490	+	0.633808i	$0.781491\pi$
$930$	0	0
$931$	6.51806	0.213621
$932$	− 27.9572i	− 0.915767i
$933$	− 23.3780i	− 0.765360i
$934$	−36.6760	−1.20007
$935$	0	0
$936$	−7.89962	−0.258207
$937$	42.0294i	1.37304i	0.727111	+	0.686520i	$0.240863\pi$
$937$	42.0294i	1.37304i	−0.727111	+	0.686520i	$0.759137\pi$
$938$	0.585462i	0.0191160i
$939$	−13.8337	−0.451444
$940$	0	0
$941$	−36.7434	−1.19780	−0.598900	−	0.800824i	$-0.704395\pi$
$941$	−36.7434	−1.19780	−0.598900	+	0.800824i	$0.704395\pi$
$942$	14.0920i	0.459141i
$943$	− 17.7648i	− 0.578502i
$944$	9.37169	0.305023
$945$	0	0
$946$	−10.3503	−0.336516
$947$	− 18.8782i	− 0.613459i	−0.951797	−	0.306729i	$-0.900765\pi$
$947$	− 18.8782i	− 0.613459i	0.951797	−	0.306729i	$-0.0992347\pi$
$948$	− 2.88661i	− 0.0937529i
$949$	17.3717	0.563909
$950$	0	0
$951$	32.9786	1.06940
$952$	− 0.292731i	− 0.00948747i
$953$	− 43.2285i	− 1.40031i	−0.713992	−	0.700154i	$-0.753115\pi$
$953$	− 43.2285i	− 1.40031i	0.713992	−	0.700154i	$-0.246885\pi$
$954$	9.16948	0.296873
$955$	0	0
$956$	−28.8108	−0.931808
$957$	1.64973i	0.0533282i
$958$	− 42.7434i	− 1.38098i
$959$	10.0000	0.322917
$960$	0	0
$961$	−30.0424	−0.969109
$962$	− 4.00000i	− 0.128965i
$963$	− 16.1235i	− 0.519572i
$964$	8.51806	0.274348
$965$	0	0
$966$	−3.27131	−0.105253
$967$	− 48.7299i	− 1.56705i	−0.621361	−	0.783524i	$-0.713420\pi$
$967$	− 48.7299i	− 1.56705i	0.621361	−	0.783524i	$-0.286580\pi$
$968$	1.00000i	0.0321412i
$969$	2.18731	0.0702665
$970$	0	0
$971$	−25.9865	−0.833948	−0.416974	−	0.908918i	$-0.636909\pi$
$971$	−25.9865	−0.833948	−0.416974	+	0.908918i	$0.636909\pi$
$972$	− 14.8536i	− 0.476431i
$973$	14.1825i	0.454669i
$974$	36.1396	1.15799
$975$	0	0
$976$	−11.9572	−0.382739
$977$	− 2.67115i	− 0.0854578i	−0.999087	−	0.0427289i	$-0.986395\pi$
$977$	− 2.67115i	− 0.0854578i	0.999087	−	0.0427289i	$-0.0136052\pi$
$978$	27.4145i	0.876620i
$979$	−13.0790	−0.418005
$980$	0	0
$981$	−5.30429	−0.169353
$982$	− 2.54262i	− 0.0811381i
$983$	− 8.33558i	− 0.265864i	−0.991125	−	0.132932i	$-0.957561\pi$
$983$	− 8.33558i	− 0.265864i	0.991125	−	0.132932i	$-0.0424391\pi$
$984$	7.13650	0.227503
$985$	0	0
$986$	0.421268	0.0134159
$987$	− 11.4145i	− 0.363329i
$988$	30.5426i	0.971690i
$989$	29.5359	0.939187
$990$	0	0
$991$	46.2730	1.46991	0.734955	−	0.678116i	$-0.237203\pi$
$991$	46.2730	1.46991	0.734955	+	0.678116i	$0.237203\pi$
$992$	− 0.978577i	− 0.0310699i
$993$	− 4.36496i	− 0.138518i
$994$	−0.335577	−0.0106438
$995$	0	0
$996$	−1.95715	−0.0620148
$997$	− 35.8715i	− 1.13606i	−0.823008	−	0.568030i	$-0.807706\pi$
$997$	− 35.8715i	− 1.13606i	0.823008	−	0.568030i	$-0.192294\pi$
$998$	3.13650i	0.0992842i
$999$	4.58546	0.145078

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3850.2.c.ba.1849.2		6
5.2	odd	4		770.2.a.m.1.2	✓	3
5.3	odd	4		3850.2.a.bt.1.2		3
5.4	even	2	inner	3850.2.c.ba.1849.5		6
15.2	even	4		6930.2.a.ce.1.3		3
20.7	even	4		6160.2.a.bf.1.2		3
35.27	even	4		5390.2.a.ca.1.2		3
55.32	even	4		8470.2.a.ci.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.m.1.2	✓	3	5.2	odd	4
3850.2.a.bt.1.2		3	5.3	odd	4
3850.2.c.ba.1849.2		6	1.1	even	1	trivial
3850.2.c.ba.1849.5		6	5.4	even	2	inner
5390.2.a.ca.1.2		3	35.27	even	4
6160.2.a.bf.1.2		3	20.7	even	4
6930.2.a.ce.1.3		3	15.2	even	4
8470.2.a.ci.1.2		3	55.32	even	4

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:	\( N \)	\(=\)	\( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3850.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.14637i q^{3} -1.00000 q^{4} +1.14637 q^{6} +1.00000i q^{7} +1.00000i q^{8} +1.68585 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +1.14637i q^{3} -1.00000 q^{4} +1.14637 q^{6} +1.00000i q^{7} +1.00000i q^{8} +1.68585 q^{9} +1.00000 q^{11} -1.14637i q^{12} +4.68585i q^{13} +1.00000 q^{14} +1.00000 q^{16} +0.292731i q^{17} -1.68585i q^{18} -6.51806 q^{19} -1.14637 q^{21} -1.00000i q^{22} +2.85363i q^{23} -1.14637 q^{24} +4.68585 q^{26} +5.37169i q^{27} -1.00000i q^{28} +1.43910 q^{29} +0.978577 q^{31} -1.00000i q^{32} +1.14637i q^{33} +0.292731 q^{34} -1.68585 q^{36} -0.853635i q^{37} +6.51806i q^{38} -5.37169 q^{39} -6.22533 q^{41} +1.14637i q^{42} -10.3503i q^{43} -1.00000 q^{44} +2.85363 q^{46} +9.95715i q^{47} +1.14637i q^{48} -1.00000 q^{49} -0.335577 q^{51} -4.68585i q^{52} +5.43910i q^{53} +5.37169 q^{54} -1.00000 q^{56} -7.47208i q^{57} -1.43910i q^{58} +9.37169 q^{59} -11.9572 q^{61} -0.978577i q^{62} +1.68585i q^{63} -1.00000 q^{64} +1.14637 q^{66} +0.585462i q^{67} -0.292731i q^{68} -3.27131 q^{69} -0.335577 q^{71} +1.68585i q^{72} -3.70727i q^{73} -0.853635 q^{74} +6.51806 q^{76} +1.00000i q^{77} +5.37169i q^{78} +2.51806 q^{79} -1.10038 q^{81} +6.22533i q^{82} -1.70727i q^{83} +1.14637 q^{84} -10.3503 q^{86} +1.64973i q^{87} +1.00000i q^{88} -13.0790 q^{89} -4.68585 q^{91} -2.85363i q^{92} +1.12181i q^{93} +9.95715 q^{94} +1.14637 q^{96} -9.10352i q^{97} +1.00000i q^{98} +1.68585 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{4} + 4 q^{6} - 14 q^{9}+O(q^{10}) \) 6 * q - 6 * q^4 + 4 * q^6 - 14 * q^9	\( 6 q - 6 q^{4} + 4 q^{6} - 14 q^{9} + 6 q^{11} + 6 q^{14} + 6 q^{16} + 12 q^{19} - 4 q^{21} - 4 q^{24} + 4 q^{26} - 24 q^{31} - 4 q^{34} + 14 q^{36} + 16 q^{39} + 8 q^{41} - 6 q^{44} + 20 q^{46} - 6 q^{49} - 56 q^{51} - 16 q^{54} - 6 q^{56} + 8 q^{59} - 12 q^{61} - 6 q^{64} + 4 q^{66} + 16 q^{69} - 56 q^{71} - 8 q^{74} - 12 q^{76} - 36 q^{79} + 6 q^{81} + 4 q^{84} + 16 q^{86} - 36 q^{89} - 4 q^{91} + 4 q^{96} - 14 q^{99}+O(q^{100}) \) 6 * q - 6 * q^4 + 4 * q^6 - 14 * q^9 + 6 * q^11 + 6 * q^14 + 6 * q^16 + 12 * q^19 - 4 * q^21 - 4 * q^24 + 4 * q^26 - 24 * q^31 - 4 * q^34 + 14 * q^36 + 16 * q^39 + 8 * q^41 - 6 * q^44 + 20 * q^46 - 6 * q^49 - 56 * q^51 - 16 * q^54 - 6 * q^56 + 8 * q^59 - 12 * q^61 - 6 * q^64 + 4 * q^66 + 16 * q^69 - 56 * q^71 - 8 * q^74 - 12 * q^76 - 36 * q^79 + 6 * q^81 + 4 * q^84 + 16 * q^86 - 36 * q^89 - 4 * q^91 + 4 * q^96 - 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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