LMFDB - Embedded newform 9280.2.a.cs.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9280,2,Mod(1,9280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9280, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("9280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9280 = 2^{6} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9280.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:	$74.1011730757$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 15x^{6} + 57x^{4} - 40x^{2} + 4 $ x^8 - 15x^6 + 57x^4 - 40*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 4640)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.346147$ of defining polynomial
Character	$\chi$	$=$	9280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.346147 q^{3} -1.00000 q^{5} -1.09203 q^{7} -2.88018 q^{9} +O(q^{10})$	$q-0.346147 q^{3} -1.00000 q^{5} -1.09203 q^{7} -2.88018 q^{9} -5.77789 q^{11} -0.497822 q^{13} +0.346147 q^{15} +4.15482 q^{17} +4.48837 q^{19} +0.378004 q^{21} +0.346147 q^{23} +1.00000 q^{25} +2.03541 q^{27} +1.00000 q^{29} +9.77145 q^{31} +2.00000 q^{33} +1.09203 q^{35} -2.27464 q^{37} +0.172320 q^{39} +0.446365 q^{41} +5.25791 q^{43} +2.88018 q^{45} -2.20924 q^{47} -5.80746 q^{49} -1.43818 q^{51} -13.1942 q^{53} +5.77789 q^{55} -1.55364 q^{57} -1.26586 q^{59} +3.81182 q^{61} +3.14525 q^{63} +0.497822 q^{65} +11.1860 q^{67} -0.119818 q^{69} +4.16588 q^{71} +13.4688 q^{73} -0.346147 q^{75} +6.30964 q^{77} -3.22251 q^{79} +7.93600 q^{81} +1.26435 q^{83} -4.15482 q^{85} -0.346147 q^{87} +13.6236 q^{89} +0.543638 q^{91} -3.38236 q^{93} -4.48837 q^{95} -7.09082 q^{97} +16.6414 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{5} + 6 q^{9}+O(q^{10}) $ 8 * q - 8 * q^5 + 6 * q^9	$ 8 q - 8 q^{5} + 6 q^{9} - 6 q^{13} + 2 q^{17} - 24 q^{21} + 8 q^{25} + 8 q^{29} + 16 q^{33} - 16 q^{37} + 12 q^{41} - 6 q^{45} + 14 q^{49} - 10 q^{53} - 4 q^{57} - 34 q^{61} + 6 q^{65} - 30 q^{69} + 10 q^{73} - 12 q^{77} + 24 q^{81} - 2 q^{85} - 20 q^{89} + 4 q^{93} + 14 q^{97}+O(q^{100}) $ 8 * q - 8 * q^5 + 6 * q^9 - 6 * q^13 + 2 * q^17 - 24 * q^21 + 8 * q^25 + 8 * q^29 + 16 * q^33 - 16 * q^37 + 12 * q^41 - 6 * q^45 + 14 * q^49 - 10 * q^53 - 4 * q^57 - 34 * q^61 + 6 * q^65 - 30 * q^69 + 10 * q^73 - 12 * q^77 + 24 * q^81 - 2 * q^85 - 20 * q^89 + 4 * q^93 + 14 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−0.346147	−0.199848	−0.0999241	−	0.994995i	$-0.531860\pi$
$3$	−0.346147	−0.199848	−0.0999241	+	0.994995i	$0.531860\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.09203	−0.412749	−0.206375	−	0.978473i	$-0.566167\pi$
$7$	−1.09203	−0.412749	−0.206375	+	0.978473i	$0.566167\pi$
$8$	0	0
$9$	−2.88018	−0.960061
$10$	0	0
$11$	−5.77789	−1.74210	−0.871050	−	0.491195i	$-0.836560\pi$
$11$	−5.77789	−1.74210	−0.871050	+	0.491195i	$0.836560\pi$
$12$	0	0
$13$	−0.497822	−0.138071	−0.0690355	−	0.997614i	$-0.521992\pi$
$13$	−0.497822	−0.138071	−0.0690355	+	0.997614i	$0.521992\pi$
$14$	0	0
$15$	0.346147	0.0893748
$16$	0	0
$17$	4.15482	1.00769	0.503846	−	0.863793i	$-0.331918\pi$
$17$	4.15482	1.00769	0.503846	+	0.863793i	$0.331918\pi$
$18$	0	0
$19$	4.48837	1.02970	0.514851	−	0.857280i	$-0.327847\pi$
$19$	4.48837	1.02970	0.514851	+	0.857280i	$0.327847\pi$
$20$	0	0
$21$	0.378004	0.0824872
$22$	0	0
$23$	0.346147	0.0721767	0.0360883	−	0.999349i	$-0.488510\pi$
$23$	0.346147	0.0721767	0.0360883	+	0.999349i	$0.488510\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	2.03541	0.391715
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	9.77145	1.75500	0.877502	−	0.479572i	$-0.159208\pi$
$31$	9.77145	1.75500	0.877502	+	0.479572i	$0.159208\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	1.09203	0.184587
$36$	0	0
$37$	−2.27464	−0.373948	−0.186974	−	0.982365i	$-0.559868\pi$
$37$	−2.27464	−0.373948	−0.186974	+	0.982365i	$0.559868\pi$
$38$	0	0
$39$	0.172320	0.0275932
$40$	0	0
$41$	0.446365	0.0697105	0.0348552	−	0.999392i	$-0.488903\pi$
$41$	0.446365	0.0697105	0.0348552	+	0.999392i	$0.488903\pi$
$42$	0	0
$43$	5.25791	0.801824	0.400912	−	0.916116i	$-0.368693\pi$
$43$	5.25791	0.801824	0.400912	+	0.916116i	$0.368693\pi$
$44$	0	0
$45$	2.88018	0.429352
$46$	0	0
$47$	−2.20924	−0.322250	−0.161125	−	0.986934i	$-0.551512\pi$
$47$	−2.20924	−0.322250	−0.161125	+	0.986934i	$0.551512\pi$
$48$	0	0
$49$	−5.80746	−0.829638
$50$	0	0
$51$	−1.43818	−0.201385
$52$	0	0
$53$	−13.1942	−1.81236	−0.906180	−	0.422892i	$-0.861015\pi$
$53$	−13.1942	−1.81236	−0.906180	+	0.422892i	$0.861015\pi$
$54$	0	0
$55$	5.77789	0.779090
$56$	0	0
$57$	−1.55364	−0.205784
$58$	0	0
$59$	−1.26586	−0.164801	−0.0824005	−	0.996599i	$-0.526259\pi$
$59$	−1.26586	−0.164801	−0.0824005	+	0.996599i	$0.526259\pi$
$60$	0	0
$61$	3.81182	0.488054	0.244027	−	0.969768i	$-0.421531\pi$
$61$	3.81182	0.488054	0.244027	+	0.969768i	$0.421531\pi$
$62$	0	0
$63$	3.14525	0.396265
$64$	0	0
$65$	0.497822	0.0617472
$66$	0	0
$67$	11.1860	1.36658	0.683292	−	0.730145i	$-0.260548\pi$
$67$	11.1860	1.36658	0.683292	+	0.730145i	$0.260548\pi$
$68$	0	0
$69$	−0.119818	−0.0144244
$70$	0	0
$71$	4.16588	0.494399	0.247200	−	0.968965i	$-0.420490\pi$
$71$	4.16588	0.494399	0.247200	+	0.968965i	$0.420490\pi$
$72$	0	0
$73$	13.4688	1.57641	0.788203	−	0.615415i	$-0.211012\pi$
$73$	13.4688	1.57641	0.788203	+	0.615415i	$0.211012\pi$
$74$	0	0
$75$	−0.346147	−0.0399696
$76$	0	0
$77$	6.30964	0.719051
$78$	0	0
$79$	−3.22251	−0.362560	−0.181280	−	0.983431i	$-0.558024\pi$
$79$	−3.22251	−0.362560	−0.181280	+	0.983431i	$0.558024\pi$
$80$	0	0
$81$	7.93600	0.881777
$82$	0	0
$83$	1.26435	0.138781	0.0693903	−	0.997590i	$-0.477895\pi$
$83$	1.26435	0.138781	0.0693903	+	0.997590i	$0.477895\pi$
$84$	0	0
$85$	−4.15482	−0.450654
$86$	0	0
$87$	−0.346147	−0.0371109
$88$	0	0
$89$	13.6236	1.44410	0.722052	−	0.691839i	$-0.243199\pi$
$89$	13.6236	1.44410	0.722052	+	0.691839i	$0.243199\pi$
$90$	0	0
$91$	0.543638	0.0569887
$92$	0	0
$93$	−3.38236	−0.350734
$94$	0	0
$95$	−4.48837	−0.460497
$96$	0	0
$97$	−7.09082	−0.719963	−0.359982	−	0.932959i	$-0.617217\pi$
$97$	−7.09082	−0.719963	−0.359982	+	0.932959i	$0.617217\pi$
$98$	0	0
$99$	16.6414	1.67252
$100$	0	0
$101$	−8.50653	−0.846432	−0.423216	−	0.906029i	$-0.639099\pi$
$101$	−8.50653	−0.846432	−0.423216	+	0.906029i	$0.639099\pi$
$102$	0	0
$103$	3.59382	0.354110	0.177055	−	0.984201i	$-0.443343\pi$
$103$	3.59382	0.354110	0.177055	+	0.984201i	$0.443343\pi$
$104$	0	0
$105$	−0.378004	−0.0368894
$106$	0	0
$107$	−12.8201	−1.23937	−0.619684	−	0.784851i	$-0.712739\pi$
$107$	−12.8201	−1.23937	−0.619684	+	0.784851i	$0.712739\pi$
$108$	0	0
$109$	−4.99564	−0.478496	−0.239248	−	0.970959i	$-0.576901\pi$
$109$	−4.99564	−0.478496	−0.239248	+	0.970959i	$0.576901\pi$
$110$	0	0
$111$	0.787360	0.0747329
$112$	0	0
$113$	−7.09082	−0.667048	−0.333524	−	0.942742i	$-0.608238\pi$
$113$	−7.09082	−0.667048	−0.333524	+	0.942742i	$0.608238\pi$
$114$	0	0
$115$	−0.346147	−0.0322784
$116$	0	0
$117$	1.43382	0.132556
$118$	0	0
$119$	−4.53720	−0.415924
$120$	0	0
$121$	22.3840	2.03491
$122$	0	0
$123$	−0.154508	−0.0139315
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−2.20924	−0.196038	−0.0980190	−	0.995185i	$-0.531251\pi$
$127$	−2.20924	−0.196038	−0.0980190	+	0.995185i	$0.531251\pi$
$128$	0	0
$129$	−1.82001	−0.160243
$130$	0	0
$131$	−6.67243	−0.582973	−0.291487	−	0.956575i	$-0.594150\pi$
$131$	−6.67243	−0.582973	−0.291487	+	0.956575i	$0.594150\pi$
$132$	0	0
$133$	−4.90144	−0.425009
$134$	0	0
$135$	−2.03541	−0.175180
$136$	0	0
$137$	10.8425	0.926335	0.463167	−	0.886271i	$-0.346713\pi$
$137$	10.8425	0.926335	0.463167	+	0.886271i	$0.346713\pi$
$138$	0	0
$139$	6.96499	0.590763	0.295381	−	0.955379i	$-0.404553\pi$
$139$	6.96499	0.590763	0.295381	+	0.955379i	$0.404553\pi$
$140$	0	0
$141$	0.764721	0.0644011
$142$	0	0
$143$	2.87636	0.240533
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	2.01024	0.165802
$148$	0	0
$149$	−6.44636	−0.528107	−0.264053	−	0.964508i	$-0.585060\pi$
$149$	−6.44636	−0.528107	−0.264053	+	0.964508i	$0.585060\pi$
$150$	0	0
$151$	−18.2534	−1.48544	−0.742720	−	0.669603i	$-0.766464\pi$
$151$	−18.2534	−1.48544	−0.742720	+	0.669603i	$0.766464\pi$
$152$	0	0
$153$	−11.9666	−0.967446
$154$	0	0
$155$	−9.77145	−0.784862
$156$	0	0
$157$	−15.8983	−1.26882	−0.634411	−	0.772996i	$-0.718757\pi$
$157$	−15.8983	−1.26882	−0.634411	+	0.772996i	$0.718757\pi$
$158$	0	0
$159$	4.56713	0.362197
$160$	0	0
$161$	−0.378004	−0.0297909
$162$	0	0
$163$	18.3233	1.43519	0.717595	−	0.696461i	$-0.245243\pi$
$163$	18.3233	1.43519	0.717595	+	0.696461i	$0.245243\pi$
$164$	0	0
$165$	−2.00000	−0.155700
$166$	0	0
$167$	−9.67639	−0.748781	−0.374391	−	0.927271i	$-0.622148\pi$
$167$	−9.67639	−0.748781	−0.374391	+	0.927271i	$0.622148\pi$
$168$	0	0
$169$	−12.7522	−0.980936
$170$	0	0
$171$	−12.9273	−0.988576
$172$	0	0
$173$	−15.5722	−1.18393	−0.591966	−	0.805963i	$-0.701648\pi$
$173$	−15.5722	−1.18393	−0.591966	+	0.805963i	$0.701648\pi$
$174$	0	0
$175$	−1.09203	−0.0825499
$176$	0	0
$177$	0.438174	0.0329352
$178$	0	0
$179$	−20.6634	−1.54445	−0.772226	−	0.635348i	$-0.780857\pi$
$179$	−20.6634	−1.54445	−0.772226	+	0.635348i	$0.780857\pi$
$180$	0	0
$181$	−21.0657	−1.56580	−0.782899	−	0.622149i	$-0.786260\pi$
$181$	−21.0657	−1.56580	−0.782899	+	0.622149i	$0.786260\pi$
$182$	0	0
$183$	−1.31945	−0.0975366
$184$	0	0
$185$	2.27464	0.167235
$186$	0	0
$187$	−24.0061	−1.75550
$188$	0	0
$189$	−2.22273	−0.161680
$190$	0	0
$191$	7.14467	0.516970	0.258485	−	0.966015i	$-0.416777\pi$
$191$	7.14467	0.516970	0.258485	+	0.966015i	$0.416777\pi$
$192$	0	0
$193$	−1.67390	−0.120490	−0.0602451	−	0.998184i	$-0.519188\pi$
$193$	−1.67390	−0.120490	−0.0602451	+	0.998184i	$0.519188\pi$
$194$	0	0
$195$	−0.172320	−0.0123401
$196$	0	0
$197$	−8.97874	−0.639709	−0.319854	−	0.947467i	$-0.603634\pi$
$197$	−8.97874	−0.639709	−0.319854	+	0.947467i	$0.603634\pi$
$198$	0	0
$199$	12.1530	0.861504	0.430752	−	0.902470i	$-0.358248\pi$
$199$	12.1530	0.861504	0.430752	+	0.902470i	$0.358248\pi$
$200$	0	0
$201$	−3.87199	−0.273109
$202$	0	0
$203$	−1.09203	−0.0766457
$204$	0	0
$205$	−0.446365	−0.0311755
$206$	0	0
$207$	−0.996967	−0.0692940
$208$	0	0
$209$	−25.9333	−1.79384
$210$	0	0
$211$	−6.42286	−0.442168	−0.221084	−	0.975255i	$-0.570959\pi$
$211$	−6.42286	−0.442168	−0.221084	+	0.975255i	$0.570959\pi$
$212$	0	0
$213$	−1.44201	−0.0988048
$214$	0	0
$215$	−5.25791	−0.358587
$216$	0	0
$217$	−10.6707	−0.724377
$218$	0	0
$219$	−4.66219	−0.315042
$220$	0	0
$221$	−2.06836	−0.139133
$222$	0	0
$223$	−10.0152	−0.670666	−0.335333	−	0.942100i	$-0.608849\pi$
$223$	−10.0152	−0.670666	−0.335333	+	0.942100i	$0.608849\pi$
$224$	0	0
$225$	−2.88018	−0.192012
$226$	0	0
$227$	−15.4973	−1.02859	−0.514295	−	0.857614i	$-0.671946\pi$
$227$	−15.4973	−1.02859	−0.514295	+	0.857614i	$0.671946\pi$
$228$	0	0
$229$	−8.87583	−0.586531	−0.293266	−	0.956031i	$-0.594742\pi$
$229$	−8.87583	−0.586531	−0.293266	+	0.956031i	$0.594742\pi$
$230$	0	0
$231$	−2.18407	−0.143701
$232$	0	0
$233$	−1.45072	−0.0950399	−0.0475199	−	0.998870i	$-0.515132\pi$
$233$	−1.45072	−0.0950399	−0.0475199	+	0.998870i	$0.515132\pi$
$234$	0	0
$235$	2.20924	0.144115
$236$	0	0
$237$	1.11546	0.0724570
$238$	0	0
$239$	29.2240	1.89035	0.945173	−	0.326569i	$-0.105893\pi$
$239$	29.2240	1.89035	0.945173	+	0.326569i	$0.105893\pi$
$240$	0	0
$241$	4.04274	0.260416	0.130208	−	0.991487i	$-0.458435\pi$
$241$	4.04274	0.260416	0.130208	+	0.991487i	$0.458435\pi$
$242$	0	0
$243$	−8.85325	−0.567936
$244$	0	0
$245$	5.80746	0.371025
$246$	0	0
$247$	−2.23441	−0.142172
$248$	0	0
$249$	−0.437652	−0.0277351
$250$	0	0
$251$	5.67071	0.357932	0.178966	−	0.983855i	$-0.442725\pi$
$251$	5.67071	0.357932	0.178966	+	0.983855i	$0.442725\pi$
$252$	0	0
$253$	−2.00000	−0.125739
$254$	0	0
$255$	1.43818	0.0900623
$256$	0	0
$257$	2.11163	0.131720	0.0658598	−	0.997829i	$-0.479021\pi$
$257$	2.11163	0.131720	0.0658598	+	0.997829i	$0.479021\pi$
$258$	0	0
$259$	2.48398	0.154347
$260$	0	0
$261$	−2.88018	−0.178279
$262$	0	0
$263$	12.3804	0.763410	0.381705	−	0.924284i	$-0.375337\pi$
$263$	12.3804	0.763410	0.381705	+	0.924284i	$0.375337\pi$
$264$	0	0
$265$	13.1942	0.810512
$266$	0	0
$267$	−4.71579	−0.288601
$268$	0	0
$269$	9.95019	0.606674	0.303337	−	0.952883i	$-0.401899\pi$
$269$	9.95019	0.606674	0.303337	+	0.952883i	$0.401899\pi$
$270$	0	0
$271$	6.81784	0.414154	0.207077	−	0.978325i	$-0.433605\pi$
$271$	6.81784	0.414154	0.207077	+	0.978325i	$0.433605\pi$
$272$	0	0
$273$	−0.188179	−0.0113891
$274$	0	0
$275$	−5.77789	−0.348420
$276$	0	0
$277$	1.30529	0.0784271	0.0392135	−	0.999231i	$-0.487515\pi$
$277$	1.30529	0.0784271	0.0392135	+	0.999231i	$0.487515\pi$
$278$	0	0
$279$	−28.1436	−1.68491
$280$	0	0
$281$	3.88889	0.231992	0.115996	−	0.993250i	$-0.462994\pi$
$281$	3.88889	0.231992	0.115996	+	0.993250i	$0.462994\pi$
$282$	0	0
$283$	15.8419	0.941703	0.470852	−	0.882212i	$-0.343947\pi$
$283$	15.8419	0.941703	0.470852	+	0.882212i	$0.343947\pi$
$284$	0	0
$285$	1.55364	0.0920294
$286$	0	0
$287$	−0.487445	−0.0287730
$288$	0	0
$289$	0.262543	0.0154437
$290$	0	0
$291$	2.45447	0.143883
$292$	0	0
$293$	−19.5886	−1.14438	−0.572190	−	0.820121i	$-0.693906\pi$
$293$	−19.5886	−1.14438	−0.572190	+	0.820121i	$0.693906\pi$
$294$	0	0
$295$	1.26586	0.0737012
$296$	0	0
$297$	−11.7604	−0.682406
$298$	0	0
$299$	−0.172320	−0.00996550
$300$	0	0
$301$	−5.74181	−0.330953
$302$	0	0
$303$	2.94451	0.169158
$304$	0	0
$305$	−3.81182	−0.218264
$306$	0	0
$307$	20.1627	1.15075	0.575373	−	0.817891i	$-0.304857\pi$
$307$	20.1627	1.15075	0.575373	+	0.817891i	$0.304857\pi$
$308$	0	0
$309$	−1.24399	−0.0707682
$310$	0	0
$311$	−25.1990	−1.42891	−0.714453	−	0.699683i	$-0.753325\pi$
$311$	−25.1990	−1.42891	−0.714453	+	0.699683i	$0.753325\pi$
$312$	0	0
$313$	−27.4540	−1.55179	−0.775896	−	0.630860i	$-0.782702\pi$
$313$	−27.4540	−1.55179	−0.775896	+	0.630860i	$0.782702\pi$
$314$	0	0
$315$	−3.14525	−0.177215
$316$	0	0
$317$	26.2079	1.47198	0.735992	−	0.676990i	$-0.236716\pi$
$317$	26.2079	1.47198	0.735992	+	0.676990i	$0.236716\pi$
$318$	0	0
$319$	−5.77789	−0.323500
$320$	0	0
$321$	4.43765	0.247686
$322$	0	0
$323$	18.6484	1.03762
$324$	0	0
$325$	−0.497822	−0.0276142
$326$	0	0
$327$	1.72923	0.0956265
$328$	0	0
$329$	2.41256	0.133009
$330$	0	0
$331$	27.4424	1.50837	0.754185	−	0.656662i	$-0.228032\pi$
$331$	27.4424	1.50837	0.754185	+	0.656662i	$0.228032\pi$
$332$	0	0
$333$	6.55138	0.359013
$334$	0	0
$335$	−11.1860	−0.611155
$336$	0	0
$337$	−9.81227	−0.534508	−0.267254	−	0.963626i	$-0.586116\pi$
$337$	−9.81227	−0.534508	−0.267254	+	0.963626i	$0.586116\pi$
$338$	0	0
$339$	2.45447	0.133308
$340$	0	0
$341$	−56.4584	−3.05739
$342$	0	0
$343$	13.9862	0.755182
$344$	0	0
$345$	0.119818	0.00645078
$346$	0	0
$347$	13.2750	0.712638	0.356319	−	0.934364i	$-0.384032\pi$
$347$	13.2750	0.712638	0.356319	+	0.934364i	$0.384032\pi$
$348$	0	0
$349$	−25.3840	−1.35877	−0.679387	−	0.733780i	$-0.737754\pi$
$349$	−25.3840	−1.35877	−0.679387	+	0.733780i	$0.737754\pi$
$350$	0	0
$351$	−1.01327	−0.0540844
$352$	0	0
$353$	−25.1860	−1.34052	−0.670258	−	0.742129i	$-0.733816\pi$
$353$	−25.1860	−1.34052	−0.670258	+	0.742129i	$0.733816\pi$
$354$	0	0
$355$	−4.16588	−0.221102
$356$	0	0
$357$	1.57054	0.0831217
$358$	0	0
$359$	13.5959	0.717567	0.358783	−	0.933421i	$-0.383192\pi$
$359$	13.5959	0.717567	0.358783	+	0.933421i	$0.383192\pi$
$360$	0	0
$361$	1.14543	0.0602860
$362$	0	0
$363$	−7.74816	−0.406673
$364$	0	0
$365$	−13.4688	−0.704990
$366$	0	0
$367$	−25.9627	−1.35524	−0.677622	−	0.735410i	$-0.736989\pi$
$367$	−25.9627	−1.35524	−0.677622	+	0.735410i	$0.736989\pi$
$368$	0	0
$369$	−1.28561	−0.0669263
$370$	0	0
$371$	14.4085	0.748051
$372$	0	0
$373$	−16.4469	−0.851588	−0.425794	−	0.904820i	$-0.640005\pi$
$373$	−16.4469	−0.851588	−0.425794	+	0.904820i	$0.640005\pi$
$374$	0	0
$375$	0.346147	0.0178750
$376$	0	0
$377$	−0.497822	−0.0256391
$378$	0	0
$379$	28.8894	1.48395	0.741976	−	0.670427i	$-0.233889\pi$
$379$	28.8894	1.48395	0.741976	+	0.670427i	$0.233889\pi$
$380$	0	0
$381$	0.764721	0.0391778
$382$	0	0
$383$	−28.3574	−1.44899	−0.724497	−	0.689278i	$-0.757928\pi$
$383$	−28.3574	−1.44899	−0.724497	+	0.689278i	$0.757928\pi$
$384$	0	0
$385$	−6.30964	−0.321569
$386$	0	0
$387$	−15.1438	−0.769800
$388$	0	0
$389$	−20.2067	−1.02452	−0.512261	−	0.858830i	$-0.671192\pi$
$389$	−20.2067	−1.02452	−0.512261	+	0.858830i	$0.671192\pi$
$390$	0	0
$391$	1.43818	0.0727319
$392$	0	0
$393$	2.30964	0.116506
$394$	0	0
$395$	3.22251	0.162142
$396$	0	0
$397$	−16.2598	−0.816058	−0.408029	−	0.912969i	$-0.633784\pi$
$397$	−16.2598	−0.816058	−0.408029	+	0.912969i	$0.633784\pi$
$398$	0	0
$399$	1.69662	0.0849373
$400$	0	0
$401$	18.8862	0.943131	0.471566	−	0.881831i	$-0.343689\pi$
$401$	18.8862	0.943131	0.471566	+	0.881831i	$0.343689\pi$
$402$	0	0
$403$	−4.86444	−0.242315
$404$	0	0
$405$	−7.93600	−0.394343
$406$	0	0
$407$	13.1426	0.651455
$408$	0	0
$409$	−6.00871	−0.297112	−0.148556	−	0.988904i	$-0.547462\pi$
$409$	−6.00871	−0.297112	−0.148556	+	0.988904i	$0.547462\pi$
$410$	0	0
$411$	−3.75309	−0.185126
$412$	0	0
$413$	1.38236	0.0680215
$414$	0	0
$415$	−1.26435	−0.0620646
$416$	0	0
$417$	−2.41091	−0.118063
$418$	0	0
$419$	8.70633	0.425332	0.212666	−	0.977125i	$-0.431785\pi$
$419$	8.70633	0.425332	0.212666	+	0.977125i	$0.431785\pi$
$420$	0	0
$421$	19.4540	0.948131	0.474065	−	0.880490i	$-0.342786\pi$
$421$	19.4540	0.948131	0.474065	+	0.880490i	$0.342786\pi$
$422$	0	0
$423$	6.36300	0.309380
$424$	0	0
$425$	4.15482	0.201538
$426$	0	0
$427$	−4.16263	−0.201444
$428$	0	0
$429$	−0.995644	−0.0480701
$430$	0	0
$431$	19.1952	0.924603	0.462301	−	0.886723i	$-0.347024\pi$
$431$	19.1952	0.924603	0.462301	+	0.886723i	$0.347024\pi$
$432$	0	0
$433$	27.2210	1.30816	0.654079	−	0.756426i	$-0.273056\pi$
$433$	27.2210	1.30816	0.654079	+	0.756426i	$0.273056\pi$
$434$	0	0
$435$	0.346147	0.0165965
$436$	0	0
$437$	1.55364	0.0743205
$438$	0	0
$439$	20.4374	0.975426	0.487713	−	0.873004i	$-0.337831\pi$
$439$	20.4374	0.975426	0.487713	+	0.873004i	$0.337831\pi$
$440$	0	0
$441$	16.7266	0.796503
$442$	0	0
$443$	−19.9962	−0.950050	−0.475025	−	0.879972i	$-0.657561\pi$
$443$	−19.9962	−0.950050	−0.475025	+	0.879972i	$0.657561\pi$
$444$	0	0
$445$	−13.6236	−0.645823
$446$	0	0
$447$	2.23139	0.105541
$448$	0	0
$449$	17.1357	0.808682	0.404341	−	0.914608i	$-0.367501\pi$
$449$	17.1357	0.808682	0.404341	+	0.914608i	$0.367501\pi$
$450$	0	0
$451$	−2.57905	−0.121443
$452$	0	0
$453$	6.31836	0.296862
$454$	0	0
$455$	−0.543638	−0.0254861
$456$	0	0
$457$	26.8347	1.25528	0.627638	−	0.778505i	$-0.284022\pi$
$457$	26.8347	1.25528	0.627638	+	0.778505i	$0.284022\pi$
$458$	0	0
$459$	8.45676	0.394728
$460$	0	0
$461$	−6.81453	−0.317384	−0.158692	−	0.987328i	$-0.550728\pi$
$461$	−6.81453	−0.317384	−0.158692	+	0.987328i	$0.550728\pi$
$462$	0	0
$463$	−26.5153	−1.23227	−0.616134	−	0.787641i	$-0.711302\pi$
$463$	−26.5153	−1.23227	−0.616134	+	0.787641i	$0.711302\pi$
$464$	0	0
$465$	3.38236	0.156853
$466$	0	0
$467$	−19.1879	−0.887910	−0.443955	−	0.896049i	$-0.646425\pi$
$467$	−19.1879	−0.887910	−0.443955	+	0.896049i	$0.646425\pi$
$468$	0	0
$469$	−12.2154	−0.564057
$470$	0	0
$471$	5.50315	0.253572
$472$	0	0
$473$	−30.3797	−1.39686
$474$	0	0
$475$	4.48837	0.205940
$476$	0	0
$477$	38.0016	1.73998
$478$	0	0
$479$	−7.72978	−0.353183	−0.176591	−	0.984284i	$-0.556507\pi$
$479$	−7.72978	−0.353183	−0.176591	+	0.984284i	$0.556507\pi$
$480$	0	0
$481$	1.13237	0.0516314
$482$	0	0
$483$	0.130845	0.00595365
$484$	0	0
$485$	7.09082	0.321977
$486$	0	0
$487$	18.0500	0.817925	0.408963	−	0.912551i	$-0.365891\pi$
$487$	18.0500	0.817925	0.408963	+	0.912551i	$0.365891\pi$
$488$	0	0
$489$	−6.34255	−0.286820
$490$	0	0
$491$	38.4634	1.73583	0.867915	−	0.496713i	$-0.165460\pi$
$491$	38.4634	1.73583	0.867915	+	0.496713i	$0.165460\pi$
$492$	0	0
$493$	4.15482	0.187124
$494$	0	0
$495$	−16.6414	−0.747974
$496$	0	0
$497$	−4.54928	−0.204063
$498$	0	0
$499$	−29.2356	−1.30876	−0.654382	−	0.756164i	$-0.727071\pi$
$499$	−29.2356	−1.30876	−0.654382	+	0.756164i	$0.727071\pi$
$500$	0	0
$501$	3.34945	0.149643
$502$	0	0
$503$	−12.6209	−0.562738	−0.281369	−	0.959600i	$-0.590789\pi$
$503$	−12.6209	−0.562738	−0.281369	+	0.959600i	$0.590789\pi$
$504$	0	0
$505$	8.50653	0.378536
$506$	0	0
$507$	4.41413	0.196038
$508$	0	0
$509$	−27.1444	−1.20315	−0.601577	−	0.798815i	$-0.705461\pi$
$509$	−27.1444	−1.20315	−0.601577	+	0.798815i	$0.705461\pi$
$510$	0	0
$511$	−14.7084	−0.650661
$512$	0	0
$513$	9.13566	0.403349
$514$	0	0
$515$	−3.59382	−0.158363
$516$	0	0
$517$	12.7647	0.561392
$518$	0	0
$519$	5.39027	0.236606
$520$	0	0
$521$	−16.7478	−0.733735	−0.366868	−	0.930273i	$-0.619570\pi$
$521$	−16.7478	−0.733735	−0.366868	+	0.930273i	$0.619570\pi$
$522$	0	0
$523$	13.8123	0.603971	0.301986	−	0.953312i	$-0.402350\pi$
$523$	13.8123	0.603971	0.301986	+	0.953312i	$0.402350\pi$
$524$	0	0
$525$	0.378004	0.0164974
$526$	0	0
$527$	40.5986	1.76850
$528$	0	0
$529$	−22.8802	−0.994791
$530$	0	0
$531$	3.64591	0.158219
$532$	0	0
$533$	−0.222210	−0.00962499
$534$	0	0
$535$	12.8201	0.554263
$536$	0	0
$537$	7.15256	0.308656
$538$	0	0
$539$	33.5549	1.44531
$540$	0	0
$541$	−32.6024	−1.40169	−0.700843	−	0.713316i	$-0.747193\pi$
$541$	−32.6024	−1.40169	−0.700843	+	0.713316i	$0.747193\pi$
$542$	0	0
$543$	7.29182	0.312922
$544$	0	0
$545$	4.99564	0.213990
$546$	0	0
$547$	−9.78625	−0.418430	−0.209215	−	0.977870i	$-0.567091\pi$
$547$	−9.78625	−0.418430	−0.209215	+	0.977870i	$0.567091\pi$
$548$	0	0
$549$	−10.9787	−0.468561
$550$	0	0
$551$	4.48837	0.191211
$552$	0	0
$553$	3.51908	0.149647
$554$	0	0
$555$	−0.787360	−0.0334216
$556$	0	0
$557$	35.8731	1.51999	0.759996	−	0.649927i	$-0.225201\pi$
$557$	35.8731	1.51999	0.759996	+	0.649927i	$0.225201\pi$
$558$	0	0
$559$	−2.61750	−0.110709
$560$	0	0
$561$	8.30964	0.350833
$562$	0	0
$563$	18.3469	0.773231	0.386616	−	0.922241i	$-0.373644\pi$
$563$	18.3469	0.773231	0.386616	+	0.922241i	$0.373644\pi$
$564$	0	0
$565$	7.09082	0.298313
$566$	0	0
$567$	−8.66637	−0.363953
$568$	0	0
$569$	−30.0787	−1.26097	−0.630483	−	0.776203i	$-0.717143\pi$
$569$	−30.0787	−1.26097	−0.630483	+	0.776203i	$0.717143\pi$
$570$	0	0
$571$	−37.4727	−1.56818	−0.784092	−	0.620645i	$-0.786871\pi$
$571$	−37.4727	−1.56818	−0.784092	+	0.620645i	$0.786871\pi$
$572$	0	0
$573$	−2.47311	−0.103315
$574$	0	0
$575$	0.346147	0.0144353
$576$	0	0
$577$	11.2447	0.468122	0.234061	−	0.972222i	$-0.424799\pi$
$577$	11.2447	0.468122	0.234061	+	0.972222i	$0.424799\pi$
$578$	0	0
$579$	0.579417	0.0240797
$580$	0	0
$581$	−1.38071	−0.0572817
$582$	0	0
$583$	76.2345	3.15731
$584$	0	0
$585$	−1.43382	−0.0592811
$586$	0	0
$587$	20.3645	0.840534	0.420267	−	0.907400i	$-0.361936\pi$
$587$	20.3645	0.840534	0.420267	+	0.907400i	$0.361936\pi$
$588$	0	0
$589$	43.8579	1.80713
$590$	0	0
$591$	3.10797	0.127845
$592$	0	0
$593$	−28.8589	−1.18509	−0.592547	−	0.805536i	$-0.701878\pi$
$593$	−28.8589	−1.18509	−0.592547	+	0.805536i	$0.701878\pi$
$594$	0	0
$595$	4.53720	0.186007
$596$	0	0
$597$	−4.20673	−0.172170
$598$	0	0
$599$	−20.9201	−0.854773	−0.427387	−	0.904069i	$-0.640566\pi$
$599$	−20.9201	−0.854773	−0.427387	+	0.904069i	$0.640566\pi$
$600$	0	0
$601$	5.69907	0.232470	0.116235	−	0.993222i	$-0.462917\pi$
$601$	5.69907	0.232470	0.116235	+	0.993222i	$0.462917\pi$
$602$	0	0
$603$	−32.2176	−1.31200
$604$	0	0
$605$	−22.3840	−0.910039
$606$	0	0
$607$	34.1074	1.38438	0.692188	−	0.721717i	$-0.256647\pi$
$607$	34.1074	1.38438	0.692188	+	0.721717i	$0.256647\pi$
$608$	0	0
$609$	0.378004	0.0153175
$610$	0	0
$611$	1.09981	0.0444934
$612$	0	0
$613$	−12.9787	−0.524206	−0.262103	−	0.965040i	$-0.584416\pi$
$613$	−12.9787	−0.524206	−0.262103	+	0.965040i	$0.584416\pi$
$614$	0	0
$615$	0.154508	0.00623036
$616$	0	0
$617$	9.22919	0.371553	0.185777	−	0.982592i	$-0.440520\pi$
$617$	9.22919	0.371553	0.185777	+	0.982592i	$0.440520\pi$
$618$	0	0
$619$	−43.7166	−1.75712	−0.878559	−	0.477635i	$-0.841494\pi$
$619$	−43.7166	−1.75712	−0.878559	+	0.477635i	$0.841494\pi$
$620$	0	0
$621$	0.704551	0.0282726
$622$	0	0
$623$	−14.8775	−0.596053
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	8.97673	0.358496
$628$	0	0
$629$	−9.45072	−0.376825
$630$	0	0
$631$	−37.6835	−1.50016	−0.750078	−	0.661349i	$-0.769984\pi$
$631$	−37.6835	−1.50016	−0.750078	+	0.661349i	$0.769984\pi$
$632$	0	0
$633$	2.22325	0.0883664
$634$	0	0
$635$	2.20924	0.0876709
$636$	0	0
$637$	2.89108	0.114549
$638$	0	0
$639$	−11.9985	−0.474653
$640$	0	0
$641$	−28.5864	−1.12909	−0.564547	−	0.825401i	$-0.690949\pi$
$641$	−28.5864	−1.12909	−0.564547	+	0.825401i	$0.690949\pi$
$642$	0	0
$643$	−12.7251	−0.501828	−0.250914	−	0.968009i	$-0.580731\pi$
$643$	−12.7251	−0.501828	−0.250914	+	0.968009i	$0.580731\pi$
$644$	0	0
$645$	1.82001	0.0716629
$646$	0	0
$647$	33.4503	1.31507	0.657534	−	0.753425i	$-0.271600\pi$
$647$	33.4503	1.31507	0.657534	+	0.753425i	$0.271600\pi$
$648$	0	0
$649$	7.31400	0.287100
$650$	0	0
$651$	3.69365	0.144765
$652$	0	0
$653$	11.7954	0.461588	0.230794	−	0.973003i	$-0.425868\pi$
$653$	11.7954	0.461588	0.230794	+	0.973003i	$0.425868\pi$
$654$	0	0
$655$	6.67243	0.260713
$656$	0	0
$657$	−38.7927	−1.51345
$658$	0	0
$659$	−11.2780	−0.439329	−0.219665	−	0.975575i	$-0.570496\pi$
$659$	−11.2780	−0.439329	−0.219665	+	0.975575i	$0.570496\pi$
$660$	0	0
$661$	−16.6531	−0.647730	−0.323865	−	0.946103i	$-0.604982\pi$
$661$	−16.6531	−0.647730	−0.323865	+	0.946103i	$0.604982\pi$
$662$	0	0
$663$	0.715957	0.0278055
$664$	0	0
$665$	4.90144	0.190070
$666$	0	0
$667$	0.346147	0.0134029
$668$	0	0
$669$	3.46672	0.134031
$670$	0	0
$671$	−22.0243	−0.850238
$672$	0	0
$673$	−10.6531	−0.410647	−0.205323	−	0.978694i	$-0.565825\pi$
$673$	−10.6531	−0.410647	−0.205323	+	0.978694i	$0.565825\pi$
$674$	0	0
$675$	2.03541	0.0783429
$676$	0	0
$677$	39.5557	1.52025	0.760125	−	0.649777i	$-0.225138\pi$
$677$	39.5557	1.52025	0.760125	+	0.649777i	$0.225138\pi$
$678$	0	0
$679$	7.74340	0.297165
$680$	0	0
$681$	5.36433	0.205562
$682$	0	0
$683$	−39.1678	−1.49871	−0.749357	−	0.662166i	$-0.769638\pi$
$683$	−39.1678	−1.49871	−0.749357	+	0.662166i	$0.769638\pi$
$684$	0	0
$685$	−10.8425	−0.414269
$686$	0	0
$687$	3.07234	0.117217
$688$	0	0
$689$	6.56835	0.250234
$690$	0	0
$691$	−16.0841	−0.611868	−0.305934	−	0.952053i	$-0.598969\pi$
$691$	−16.0841	−0.611868	−0.305934	+	0.952053i	$0.598969\pi$
$692$	0	0
$693$	−18.1729	−0.690332
$694$	0	0
$695$	−6.96499	−0.264197
$696$	0	0
$697$	1.85457	0.0702467
$698$	0	0
$699$	0.502163	0.0189935
$700$	0	0
$701$	−8.85892	−0.334597	−0.167298	−	0.985906i	$-0.553504\pi$
$701$	−8.85892	−0.334597	−0.167298	+	0.985906i	$0.553504\pi$
$702$	0	0
$703$	−10.2094	−0.385055
$704$	0	0
$705$	−0.764721	−0.0288010
$706$	0	0
$707$	9.28941	0.349364
$708$	0	0
$709$	6.66181	0.250189	0.125095	−	0.992145i	$-0.460077\pi$
$709$	6.66181	0.250189	0.125095	+	0.992145i	$0.460077\pi$
$710$	0	0
$711$	9.28141	0.348080
$712$	0	0
$713$	3.38236	0.126670
$714$	0	0
$715$	−2.87636	−0.107570
$716$	0	0
$717$	−10.1158	−0.377782
$718$	0	0
$719$	−33.2736	−1.24090	−0.620449	−	0.784247i	$-0.713050\pi$
$719$	−33.2736	−1.24090	−0.620449	+	0.784247i	$0.713050\pi$
$720$	0	0
$721$	−3.92457	−0.146159
$722$	0	0
$723$	−1.39938	−0.0520437
$724$	0	0
$725$	1.00000	0.0371391
$726$	0	0
$727$	10.1964	0.378162	0.189081	−	0.981961i	$-0.439449\pi$
$727$	10.1964	0.378162	0.189081	+	0.981961i	$0.439449\pi$
$728$	0	0
$729$	−20.7435	−0.768276
$730$	0	0
$731$	21.8457	0.807992
$732$	0	0
$733$	48.8697	1.80504	0.902522	−	0.430643i	$-0.141713\pi$
$733$	48.8697	1.80504	0.902522	+	0.430643i	$0.141713\pi$
$734$	0	0
$735$	−2.01024	−0.0741487
$736$	0	0
$737$	−64.6313	−2.38072
$738$	0	0
$739$	−45.6511	−1.67930	−0.839651	−	0.543127i	$-0.817240\pi$
$739$	−45.6511	−1.67930	−0.839651	+	0.543127i	$0.817240\pi$
$740$	0	0
$741$	0.773434	0.0284128
$742$	0	0
$743$	−3.35335	−0.123023	−0.0615113	−	0.998106i	$-0.519592\pi$
$743$	−3.35335	−0.123023	−0.0615113	+	0.998106i	$0.519592\pi$
$744$	0	0
$745$	6.44636	0.236177
$746$	0	0
$747$	−3.64156	−0.133238
$748$	0	0
$749$	14.0000	0.511549
$750$	0	0
$751$	−17.9812	−0.656145	−0.328072	−	0.944653i	$-0.606399\pi$
$751$	−17.9812	−0.656145	−0.328072	+	0.944653i	$0.606399\pi$
$752$	0	0
$753$	−1.96290	−0.0715320
$754$	0	0
$755$	18.2534	0.664309
$756$	0	0
$757$	25.1181	0.912932	0.456466	−	0.889741i	$-0.349115\pi$
$757$	25.1181	0.912932	0.456466	+	0.889741i	$0.349115\pi$
$758$	0	0
$759$	0.692294	0.0251287
$760$	0	0
$761$	−42.1656	−1.52850	−0.764251	−	0.644918i	$-0.776891\pi$
$761$	−42.1656	−1.52850	−0.764251	+	0.644918i	$0.776891\pi$
$762$	0	0
$763$	5.45541	0.197499
$764$	0	0
$765$	11.9666	0.432655
$766$	0	0
$767$	0.630173	0.0227542
$768$	0	0
$769$	0.181635	0.00654991	0.00327495	−	0.999995i	$-0.498958\pi$
$769$	0.181635	0.00654991	0.00327495	+	0.999995i	$0.498958\pi$
$770$	0	0
$771$	−0.730934	−0.0263239
$772$	0	0
$773$	17.5557	0.631436	0.315718	−	0.948853i	$-0.397755\pi$
$773$	17.5557	0.631436	0.315718	+	0.948853i	$0.397755\pi$
$774$	0	0
$775$	9.77145	0.351001
$776$	0	0
$777$	−0.859823	−0.0308460
$778$	0	0
$779$	2.00345	0.0717810
$780$	0	0
$781$	−24.0700	−0.861293
$782$	0	0
$783$	2.03541	0.0727396
$784$	0	0
$785$	15.8983	0.567434
$786$	0	0
$787$	12.3210	0.439196	0.219598	−	0.975590i	$-0.429525\pi$
$787$	12.3210	0.439196	0.219598	+	0.975590i	$0.429525\pi$
$788$	0	0
$789$	−4.28545	−0.152566
$790$	0	0
$791$	7.74340	0.275324
$792$	0	0
$793$	−1.89761	−0.0673860
$794$	0	0
$795$	−4.56713	−0.161979
$796$	0	0
$797$	−2.66300	−0.0943284	−0.0471642	−	0.998887i	$-0.515018\pi$
$797$	−2.66300	−0.0943284	−0.0471642	+	0.998887i	$0.515018\pi$
$798$	0	0
$799$	−9.17898	−0.324729
$800$	0	0
$801$	−39.2386	−1.38643
$802$	0	0
$803$	−77.8214	−2.74626
$804$	0	0
$805$	0.378004	0.0133229
$806$	0	0
$807$	−3.44423	−0.121243
$808$	0	0
$809$	−26.4826	−0.931077	−0.465539	−	0.885028i	$-0.654139\pi$
$809$	−26.4826	−0.931077	−0.465539	+	0.885028i	$0.654139\pi$
$810$	0	0
$811$	20.9695	0.736340	0.368170	−	0.929758i	$-0.379984\pi$
$811$	20.9695	0.736340	0.368170	+	0.929758i	$0.379984\pi$
$812$	0	0
$813$	−2.35998	−0.0827680
$814$	0	0
$815$	−18.3233	−0.641836
$816$	0	0
$817$	23.5994	0.825640
$818$	0	0
$819$	−1.56578	−0.0547126
$820$	0	0
$821$	−6.44636	−0.224980	−0.112490	−	0.993653i	$-0.535883\pi$
$821$	−6.44636	−0.224980	−0.112490	+	0.993653i	$0.535883\pi$
$822$	0	0
$823$	−26.3577	−0.918773	−0.459386	−	0.888237i	$-0.651931\pi$
$823$	−26.3577	−0.918773	−0.459386	+	0.888237i	$0.651931\pi$
$824$	0	0
$825$	2.00000	0.0696311
$826$	0	0
$827$	35.7116	1.24182	0.620908	−	0.783884i	$-0.286764\pi$
$827$	35.7116	1.24182	0.620908	+	0.783884i	$0.286764\pi$
$828$	0	0
$829$	5.31348	0.184545	0.0922724	−	0.995734i	$-0.470587\pi$
$829$	5.31348	0.184545	0.0922724	+	0.995734i	$0.470587\pi$
$830$	0	0
$831$	−0.451821	−0.0156735
$832$	0	0
$833$	−24.1290	−0.836020
$834$	0	0
$835$	9.67639	0.334865
$836$	0	0
$837$	19.8889	0.687461
$838$	0	0
$839$	−23.9865	−0.828108	−0.414054	−	0.910252i	$-0.635887\pi$
$839$	−23.9865	−0.828108	−0.414054	+	0.910252i	$0.635887\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−1.34613	−0.0463632
$844$	0	0
$845$	12.7522	0.438688
$846$	0	0
$847$	−24.4441	−0.839908
$848$	0	0
$849$	−5.48363	−0.188198
$850$	0	0
$851$	−0.787360	−0.0269904
$852$	0	0
$853$	−16.6968	−0.571688	−0.285844	−	0.958276i	$-0.592274\pi$
$853$	−16.6968	−0.571688	−0.285844	+	0.958276i	$0.592274\pi$
$854$	0	0
$855$	12.9273	0.442105
$856$	0	0
$857$	−28.8589	−0.985802	−0.492901	−	0.870085i	$-0.664064\pi$
$857$	−28.8589	−0.985802	−0.492901	+	0.870085i	$0.664064\pi$
$858$	0	0
$859$	−2.39635	−0.0817625	−0.0408812	−	0.999164i	$-0.513017\pi$
$859$	−2.39635	−0.0817625	−0.0408812	+	0.999164i	$0.513017\pi$
$860$	0	0
$861$	0.168728	0.00575022
$862$	0	0
$863$	−11.6493	−0.396548	−0.198274	−	0.980147i	$-0.563534\pi$
$863$	−11.6493	−0.396548	−0.198274	+	0.980147i	$0.563534\pi$
$864$	0	0
$865$	15.5722	0.529470
$866$	0	0
$867$	−0.0908783	−0.00308639
$868$	0	0
$869$	18.6193	0.631616
$870$	0	0
$871$	−5.56862	−0.188685
$872$	0	0
$873$	20.4228	0.691209
$874$	0	0
$875$	1.09203	0.0369174
$876$	0	0
$877$	42.2331	1.42611	0.713055	−	0.701108i	$-0.247311\pi$
$877$	42.2331	1.42611	0.713055	+	0.701108i	$0.247311\pi$
$878$	0	0
$879$	6.78055	0.228702
$880$	0	0
$881$	8.49895	0.286337	0.143168	−	0.989698i	$-0.454271\pi$
$881$	8.49895	0.286337	0.143168	+	0.989698i	$0.454271\pi$
$882$	0	0
$883$	52.5509	1.76848	0.884239	−	0.467034i	$-0.154677\pi$
$883$	52.5509	1.76848	0.884239	+	0.467034i	$0.154677\pi$
$884$	0	0
$885$	−0.438174	−0.0147291
$886$	0	0
$887$	−18.1331	−0.608851	−0.304426	−	0.952536i	$-0.598465\pi$
$887$	−18.1331	−0.608851	−0.304426	+	0.952536i	$0.598465\pi$
$888$	0	0
$889$	2.41256	0.0809146
$890$	0	0
$891$	−45.8533	−1.53614
$892$	0	0
$893$	−9.91586	−0.331822
$894$	0	0
$895$	20.6634	0.690700
$896$	0	0
$897$	0.0596479	0.00199159
$898$	0	0
$899$	9.77145	0.325896
$900$	0	0
$901$	−54.8195	−1.82630
$902$	0	0
$903$	1.98751	0.0661403
$904$	0	0
$905$	21.0657	0.700246
$906$	0	0
$907$	19.8329	0.658539	0.329270	−	0.944236i	$-0.393197\pi$
$907$	19.8329	0.658539	0.329270	+	0.944236i	$0.393197\pi$
$908$	0	0
$909$	24.5004	0.812626
$910$	0	0
$911$	33.2248	1.10079	0.550394	−	0.834905i	$-0.314478\pi$
$911$	33.2248	1.10079	0.550394	+	0.834905i	$0.314478\pi$
$912$	0	0
$913$	−7.30529	−0.241770
$914$	0	0
$915$	1.31945	0.0436197
$916$	0	0
$917$	7.28651	0.240622
$918$	0	0
$919$	−54.8075	−1.80793	−0.903966	−	0.427605i	$-0.859358\pi$
$919$	−54.8075	−1.80793	−0.903966	+	0.427605i	$0.859358\pi$
$920$	0	0
$921$	−6.97926	−0.229974
$922$	0	0
$923$	−2.07387	−0.0682622
$924$	0	0
$925$	−2.27464	−0.0747897
$926$	0	0
$927$	−10.3509	−0.339967
$928$	0	0
$929$	−26.5749	−0.871894	−0.435947	−	0.899972i	$-0.643586\pi$
$929$	−26.5749	−0.871894	−0.435947	+	0.899972i	$0.643586\pi$
$930$	0	0
$931$	−26.0660	−0.854280
$932$	0	0
$933$	8.72258	0.285564
$934$	0	0
$935$	24.0061	0.785083
$936$	0	0
$937$	1.47491	0.0481834	0.0240917	−	0.999710i	$-0.492331\pi$
$937$	1.47491	0.0481834	0.0240917	+	0.999710i	$0.492331\pi$
$938$	0	0
$939$	9.50313	0.310123
$940$	0	0
$941$	−16.9047	−0.551079	−0.275539	−	0.961290i	$-0.588856\pi$
$941$	−16.9047	−0.551079	−0.275539	+	0.961290i	$0.588856\pi$
$942$	0	0
$943$	0.154508	0.00503147
$944$	0	0
$945$	2.22273	0.0723055
$946$	0	0
$947$	12.4544	0.404715	0.202357	−	0.979312i	$-0.435140\pi$
$947$	12.4544	0.404715	0.202357	+	0.979312i	$0.435140\pi$
$948$	0	0
$949$	−6.70507	−0.217656
$950$	0	0
$951$	−9.07180	−0.294173
$952$	0	0
$953$	−21.5033	−0.696560	−0.348280	−	0.937390i	$-0.613234\pi$
$953$	−21.5033	−0.696560	−0.348280	+	0.937390i	$0.613234\pi$
$954$	0	0
$955$	−7.14467	−0.231196
$956$	0	0
$957$	2.00000	0.0646508
$958$	0	0
$959$	−11.8403	−0.382344
$960$	0	0
$961$	64.4813	2.08004
$962$	0	0
$963$	36.9243	1.18987
$964$	0	0
$965$	1.67390	0.0538849
$966$	0	0
$967$	−2.45922	−0.0790833	−0.0395416	−	0.999218i	$-0.512590\pi$
$967$	−2.45922	−0.0790833	−0.0395416	+	0.999218i	$0.512590\pi$
$968$	0	0
$969$	−6.45508	−0.207367
$970$	0	0
$971$	46.0525	1.47790	0.738948	−	0.673762i	$-0.235323\pi$
$971$	46.0525	1.47790	0.738948	+	0.673762i	$0.235323\pi$
$972$	0	0
$973$	−7.60599	−0.243837
$974$	0	0
$975$	0.172320	0.00551864
$976$	0	0
$977$	−61.4124	−1.96476	−0.982378	−	0.186903i	$-0.940155\pi$
$977$	−61.4124	−1.96476	−0.982378	+	0.186903i	$0.940155\pi$
$978$	0	0
$979$	−78.7159	−2.51577
$980$	0	0
$981$	14.3884	0.459385
$982$	0	0
$983$	−17.6457	−0.562810	−0.281405	−	0.959589i	$-0.590801\pi$
$983$	−17.6457	−0.562810	−0.281405	+	0.959589i	$0.590801\pi$
$984$	0	0
$985$	8.97874	0.286086
$986$	0	0
$987$	−0.835100	−0.0265815
$988$	0	0
$989$	1.82001	0.0578730
$990$	0	0
$991$	48.4193	1.53809	0.769045	−	0.639195i	$-0.220732\pi$
$991$	48.4193	1.53809	0.769045	+	0.639195i	$0.220732\pi$
$992$	0	0
$993$	−9.49911	−0.301445
$994$	0	0
$995$	−12.1530	−0.385276
$996$	0	0
$997$	28.3972	0.899349	0.449675	−	0.893192i	$-0.351540\pi$
$997$	28.3972	0.899349	0.449675	+	0.893192i	$0.351540\pi$
$998$	0	0
$999$	−4.62982	−0.146481

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9280.2.a.cs.1.4		8
4.3	odd	2	inner	9280.2.a.cs.1.5		8
8.3	odd	2		4640.2.a.y.1.4	✓	8
8.5	even	2		4640.2.a.y.1.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4640.2.a.y.1.4	✓	8	8.3	odd	2
4640.2.a.y.1.5	yes	8	8.5	even	2
9280.2.a.cs.1.4		8	1.1	even	1	trivial
9280.2.a.cs.1.5		8	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 9280 = 2^{6} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9280.a (trivial)

\(f(q)\)	\(=\)	\(q-0.346147 q^{3} -1.00000 q^{5} -1.09203 q^{7} -2.88018 q^{9} +O(q^{10})\)	\(q-0.346147 q^{3} -1.00000 q^{5} -1.09203 q^{7} -2.88018 q^{9} -5.77789 q^{11} -0.497822 q^{13} +0.346147 q^{15} +4.15482 q^{17} +4.48837 q^{19} +0.378004 q^{21} +0.346147 q^{23} +1.00000 q^{25} +2.03541 q^{27} +1.00000 q^{29} +9.77145 q^{31} +2.00000 q^{33} +1.09203 q^{35} -2.27464 q^{37} +0.172320 q^{39} +0.446365 q^{41} +5.25791 q^{43} +2.88018 q^{45} -2.20924 q^{47} -5.80746 q^{49} -1.43818 q^{51} -13.1942 q^{53} +5.77789 q^{55} -1.55364 q^{57} -1.26586 q^{59} +3.81182 q^{61} +3.14525 q^{63} +0.497822 q^{65} +11.1860 q^{67} -0.119818 q^{69} +4.16588 q^{71} +13.4688 q^{73} -0.346147 q^{75} +6.30964 q^{77} -3.22251 q^{79} +7.93600 q^{81} +1.26435 q^{83} -4.15482 q^{85} -0.346147 q^{87} +13.6236 q^{89} +0.543638 q^{91} -3.38236 q^{93} -4.48837 q^{95} -7.09082 q^{97} +16.6414 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5} + 6 q^{9}+O(q^{10}) \) 8 * q - 8 * q^5 + 6 * q^9	\( 8 q - 8 q^{5} + 6 q^{9} - 6 q^{13} + 2 q^{17} - 24 q^{21} + 8 q^{25} + 8 q^{29} + 16 q^{33} - 16 q^{37} + 12 q^{41} - 6 q^{45} + 14 q^{49} - 10 q^{53} - 4 q^{57} - 34 q^{61} + 6 q^{65} - 30 q^{69} + 10 q^{73} - 12 q^{77} + 24 q^{81} - 2 q^{85} - 20 q^{89} + 4 q^{93} + 14 q^{97}+O(q^{100}) \) 8 * q - 8 * q^5 + 6 * q^9 - 6 * q^13 + 2 * q^17 - 24 * q^21 + 8 * q^25 + 8 * q^29 + 16 * q^33 - 16 * q^37 + 12 * q^41 - 6 * q^45 + 14 * q^49 - 10 * q^53 - 4 * q^57 - 34 * q^61 + 6 * q^65 - 30 * q^69 + 10 * q^73 - 12 * q^77 + 24 * q^81 - 2 * q^85 - 20 * q^89 + 4 * q^93 + 14 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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