LMFDB - Embedded newform 7168.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7168,2,Mod(1,7168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7168, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7168.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7168 = 2^{10} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7168.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:	$57.2367681689$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 112)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	7168.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843 q^{5} -1.00000 q^{7} -3.00000 q^{9} +O(q^{10})$	$q-2.82843 q^{5} -1.00000 q^{7} -3.00000 q^{9} +1.41421 q^{11} +2.00000 q^{17} -2.82843 q^{19} +6.00000 q^{23} +3.00000 q^{25} +9.89949 q^{29} -8.00000 q^{31} +2.82843 q^{35} +7.07107 q^{37} -10.0000 q^{41} -1.41421 q^{43} +8.48528 q^{45} +12.0000 q^{47} +1.00000 q^{49} -1.41421 q^{53} -4.00000 q^{55} -11.3137 q^{59} +8.48528 q^{61} +3.00000 q^{63} +4.24264 q^{67} +6.00000 q^{73} -1.41421 q^{77} -10.0000 q^{79} +9.00000 q^{81} +14.1421 q^{83} -5.65685 q^{85} -14.0000 q^{89} +8.00000 q^{95} -2.00000 q^{97} -4.24264 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{7} - 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^7 - 6 * q^9	$ 2 q - 2 q^{7} - 6 q^{9} + 4 q^{17} + 12 q^{23} + 6 q^{25} - 16 q^{31} - 20 q^{41} + 24 q^{47} + 2 q^{49} - 8 q^{55} + 6 q^{63} + 12 q^{73} - 20 q^{79} + 18 q^{81} - 28 q^{89} + 16 q^{95} - 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^7 - 6 * q^9 + 4 * q^17 + 12 * q^23 + 6 * q^25 - 16 * q^31 - 20 * q^41 + 24 * q^47 + 2 * q^49 - 8 * q^55 + 6 * q^63 + 12 * q^73 - 20 * q^79 + 18 * q^81 - 28 * q^89 + 16 * q^95 - 4 * 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	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	0	0
$5$	−2.82843	−1.26491	−0.632456	−	0.774597i	$-0.717953\pi$
$5$	−2.82843	−1.26491	−0.632456	+	0.774597i	$0.717953\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−3.00000	−1.00000
$10$	0	0
$11$	1.41421	0.426401	0.213201	−	0.977008i	$-0.431611\pi$
$11$	1.41421	0.426401	0.213201	+	0.977008i	$0.431611\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	9.89949	1.83829	0.919145	−	0.393919i	$-0.128881\pi$
$29$	9.89949	1.83829	0.919145	+	0.393919i	$0.128881\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.82843	0.478091
$36$	0	0
$37$	7.07107	1.16248	0.581238	−	0.813733i	$-0.302568\pi$
$37$	7.07107	1.16248	0.581238	+	0.813733i	$0.302568\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	−1.41421	−0.215666	−0.107833	−	0.994169i	$-0.534391\pi$
$43$	−1.41421	−0.215666	−0.107833	+	0.994169i	$0.534391\pi$
$44$	0	0
$45$	8.48528	1.26491
$46$	0	0
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.41421	−0.194257	−0.0971286	−	0.995272i	$-0.530966\pi$
$53$	−1.41421	−0.194257	−0.0971286	+	0.995272i	$0.530966\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−11.3137	−1.47292	−0.736460	−	0.676481i	$-0.763504\pi$
$59$	−11.3137	−1.47292	−0.736460	+	0.676481i	$0.763504\pi$
$60$	0	0
$61$	8.48528	1.08643	0.543214	−	0.839594i	$-0.317207\pi$
$61$	8.48528	1.08643	0.543214	+	0.839594i	$0.317207\pi$
$62$	0	0
$63$	3.00000	0.377964
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.24264	0.518321	0.259161	−	0.965834i	$-0.416554\pi$
$67$	4.24264	0.518321	0.259161	+	0.965834i	$0.416554\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.41421	−0.161165
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	14.1421	1.55230	0.776151	−	0.630548i	$-0.217170\pi$
$83$	14.1421	1.55230	0.776151	+	0.630548i	$0.217170\pi$
$84$	0	0
$85$	−5.65685	−0.613572
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8.00000	0.820783
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	−4.24264	−0.426401
$100$	0	0
$101$	−8.48528	−0.844317	−0.422159	−	0.906522i	$-0.638727\pi$
$101$	−8.48528	−0.844317	−0.422159	+	0.906522i	$0.638727\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.07107	0.683586	0.341793	−	0.939775i	$-0.388966\pi$
$107$	7.07107	0.683586	0.341793	+	0.939775i	$0.388966\pi$
$108$	0	0
$109$	4.24264	0.406371	0.203186	−	0.979140i	$-0.434871\pi$
$109$	4.24264	0.406371	0.203186	+	0.979140i	$0.434871\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−4.00000	−0.376288	−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000	−0.376288	−0.188144	+	0.982141i	$0.560247\pi$
$114$	0	0
$115$	−16.9706	−1.58251
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.00000	−0.183340
$120$	0	0
$121$	−9.00000	−0.818182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	5.65685	0.505964
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−19.7990	−1.72985	−0.864923	−	0.501905i	$-0.832633\pi$
$131$	−19.7990	−1.72985	−0.864923	+	0.501905i	$0.832633\pi$
$132$	0	0
$133$	2.82843	0.245256
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	−2.82843	−0.239904	−0.119952	−	0.992780i	$-0.538274\pi$
$139$	−2.82843	−0.239904	−0.119952	+	0.992780i	$0.538274\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−28.0000	−2.32527
$146$	0	0
$147$	0	0
$148$	0	0
$149$	4.24264	0.347571	0.173785	−	0.984784i	$-0.444400\pi$
$149$	4.24264	0.347571	0.173785	+	0.984784i	$0.444400\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	0	0
$155$	22.6274	1.81748
$156$	0	0
$157$	−16.9706	−1.35440	−0.677199	−	0.735800i	$-0.736806\pi$
$157$	−16.9706	−1.35440	−0.677199	+	0.735800i	$0.736806\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−6.00000	−0.472866
$162$	0	0
$163$	21.2132	1.66155	0.830773	−	0.556611i	$-0.187899\pi$
$163$	21.2132	1.66155	0.830773	+	0.556611i	$0.187899\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	8.48528	0.648886
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	−3.00000	−0.226779
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.24264	0.317110	0.158555	−	0.987350i	$-0.449317\pi$
$179$	4.24264	0.317110	0.158555	+	0.987350i	$0.449317\pi$
$180$	0	0
$181$	−5.65685	−0.420471	−0.210235	−	0.977651i	$-0.567423\pi$
$181$	−5.65685	−0.420471	−0.210235	+	0.977651i	$0.567423\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−20.0000	−1.47043
$186$	0	0
$187$	2.82843	0.206835
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	0	0
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−7.07107	−0.503793	−0.251896	−	0.967754i	$-0.581054\pi$
$197$	−7.07107	−0.503793	−0.251896	+	0.967754i	$0.581054\pi$
$198$	0	0
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−9.89949	−0.694808
$204$	0	0
$205$	28.2843	1.97546
$206$	0	0
$207$	−18.0000	−1.25109
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	12.7279	0.876226	0.438113	−	0.898920i	$-0.355647\pi$
$211$	12.7279	0.876226	0.438113	+	0.898920i	$0.355647\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.00000	0.272798
$216$	0	0
$217$	8.00000	0.543075
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	0	0
$225$	−9.00000	−0.600000
$226$	0	0
$227$	−2.82843	−0.187729	−0.0938647	−	0.995585i	$-0.529922\pi$
$227$	−2.82843	−0.187729	−0.0938647	+	0.995585i	$0.529922\pi$
$228$	0	0
$229$	−11.3137	−0.747631	−0.373815	−	0.927503i	$-0.621951\pi$
$229$	−11.3137	−0.747631	−0.373815	+	0.927503i	$0.621951\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	16.0000	1.04819	0.524097	−	0.851658i	$-0.324403\pi$
$233$	16.0000	1.04819	0.524097	+	0.851658i	$0.324403\pi$
$234$	0	0
$235$	−33.9411	−2.21407
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.82843	−0.180702
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	19.7990	1.24970	0.624851	−	0.780744i	$-0.285160\pi$
$251$	19.7990	1.24970	0.624851	+	0.780744i	$0.285160\pi$
$252$	0	0
$253$	8.48528	0.533465
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	0	0
$259$	−7.07107	−0.439375
$260$	0	0
$261$	−29.6985	−1.83829
$262$	0	0
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	0	0
$265$	4.00000	0.245718
$266$	0	0
$267$	0	0
$268$	0	0
$269$	11.3137	0.689809	0.344904	−	0.938638i	$-0.387911\pi$
$269$	11.3137	0.689809	0.344904	+	0.938638i	$0.387911\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.24264	0.255841
$276$	0	0
$277$	−21.2132	−1.27458	−0.637289	−	0.770625i	$-0.719944\pi$
$277$	−21.2132	−1.27458	−0.637289	+	0.770625i	$0.719944\pi$
$278$	0	0
$279$	24.0000	1.43684
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	−5.65685	−0.336265	−0.168133	−	0.985764i	$-0.553774\pi$
$283$	−5.65685	−0.336265	−0.168133	+	0.985764i	$0.553774\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10.0000	0.590281
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−19.7990	−1.15667	−0.578335	−	0.815800i	$-0.696297\pi$
$293$	−19.7990	−1.15667	−0.578335	+	0.815800i	$0.696297\pi$
$294$	0	0
$295$	32.0000	1.86311
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	1.41421	0.0815139
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−24.0000	−1.37424
$306$	0	0
$307$	−25.4558	−1.45284	−0.726421	−	0.687250i	$-0.758818\pi$
$307$	−25.4558	−1.45284	−0.726421	+	0.687250i	$0.758818\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−20.0000	−1.13410	−0.567048	−	0.823685i	$-0.691915\pi$
$311$	−20.0000	−1.13410	−0.567048	+	0.823685i	$0.691915\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	−8.48528	−0.478091
$316$	0	0
$317$	−9.89949	−0.556011	−0.278006	−	0.960579i	$-0.589673\pi$
$317$	−9.89949	−0.556011	−0.278006	+	0.960579i	$0.589673\pi$
$318$	0	0
$319$	14.0000	0.783850
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−5.65685	−0.314756
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−12.0000	−0.661581
$330$	0	0
$331$	29.6985	1.63238	0.816188	−	0.577786i	$-0.196083\pi$
$331$	29.6985	1.63238	0.816188	+	0.577786i	$0.196083\pi$
$332$	0	0
$333$	−21.2132	−1.16248
$334$	0	0
$335$	−12.0000	−0.655630
$336$	0	0
$337$	−8.00000	−0.435788	−0.217894	−	0.975972i	$-0.569919\pi$
$337$	−8.00000	−0.435788	−0.217894	+	0.975972i	$0.569919\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−11.3137	−0.612672
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−21.2132	−1.13878	−0.569392	−	0.822066i	$-0.692821\pi$
$347$	−21.2132	−1.13878	−0.569392	+	0.822066i	$0.692821\pi$
$348$	0	0
$349$	16.9706	0.908413	0.454207	−	0.890896i	$-0.349923\pi$
$349$	16.9706	0.908413	0.454207	+	0.890896i	$0.349923\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−16.9706	−0.888280
$366$	0	0
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	0	0
$369$	30.0000	1.56174
$370$	0	0
$371$	1.41421	0.0734223
$372$	0	0
$373$	−15.5563	−0.805477	−0.402739	−	0.915315i	$-0.631942\pi$
$373$	−15.5563	−0.805477	−0.402739	+	0.915315i	$0.631942\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−18.3848	−0.944363	−0.472181	−	0.881501i	$-0.656533\pi$
$379$	−18.3848	−0.944363	−0.472181	+	0.881501i	$0.656533\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	4.00000	0.204390	0.102195	−	0.994764i	$-0.467413\pi$
$383$	4.00000	0.204390	0.102195	+	0.994764i	$0.467413\pi$
$384$	0	0
$385$	4.00000	0.203859
$386$	0	0
$387$	4.24264	0.215666
$388$	0	0
$389$	−4.24264	−0.215110	−0.107555	−	0.994199i	$-0.534302\pi$
$389$	−4.24264	−0.215110	−0.107555	+	0.994199i	$0.534302\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	0	0
$393$	0	0
$394$	0	0
$395$	28.2843	1.42314
$396$	0	0
$397$	16.9706	0.851728	0.425864	−	0.904787i	$-0.359970\pi$
$397$	16.9706	0.851728	0.425864	+	0.904787i	$0.359970\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−25.4558	−1.26491
$406$	0	0
$407$	10.0000	0.495682
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	11.3137	0.556711
$414$	0	0
$415$	−40.0000	−1.96352
$416$	0	0
$417$	0	0
$418$	0	0
$419$	16.9706	0.829066	0.414533	−	0.910034i	$-0.363945\pi$
$419$	16.9706	0.829066	0.414533	+	0.910034i	$0.363945\pi$
$420$	0	0
$421$	12.7279	0.620321	0.310160	−	0.950684i	$-0.399617\pi$
$421$	12.7279	0.620321	0.310160	+	0.950684i	$0.399617\pi$
$422$	0	0
$423$	−36.0000	−1.75038
$424$	0	0
$425$	6.00000	0.291043
$426$	0	0
$427$	−8.48528	−0.410632
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	0	0
$433$	6.00000	0.288342	0.144171	−	0.989553i	$-0.453949\pi$
$433$	6.00000	0.288342	0.144171	+	0.989553i	$0.453949\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−16.9706	−0.811812
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	1.41421	0.0671913	0.0335957	−	0.999436i	$-0.489304\pi$
$443$	1.41421	0.0671913	0.0335957	+	0.999436i	$0.489304\pi$
$444$	0	0
$445$	39.5980	1.87712
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	−14.1421	−0.665927
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−22.6274	−1.05386	−0.526932	−	0.849907i	$-0.676658\pi$
$461$	−22.6274	−1.05386	−0.526932	+	0.849907i	$0.676658\pi$
$462$	0	0
$463$	−14.0000	−0.650635	−0.325318	−	0.945605i	$-0.605471\pi$
$463$	−14.0000	−0.650635	−0.325318	+	0.945605i	$0.605471\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−25.4558	−1.17796	−0.588978	−	0.808149i	$-0.700470\pi$
$467$	−25.4558	−1.17796	−0.588978	+	0.808149i	$0.700470\pi$
$468$	0	0
$469$	−4.24264	−0.195907
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−2.00000	−0.0919601
$474$	0	0
$475$	−8.48528	−0.389331
$476$	0	0
$477$	4.24264	0.194257
$478$	0	0
$479$	−20.0000	−0.913823	−0.456912	−	0.889512i	$-0.651044\pi$
$479$	−20.0000	−0.913823	−0.456912	+	0.889512i	$0.651044\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.65685	0.256865
$486$	0	0
$487$	22.0000	0.996915	0.498458	−	0.866914i	$-0.333900\pi$
$487$	22.0000	0.996915	0.498458	+	0.866914i	$0.333900\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−26.8701	−1.21263	−0.606314	−	0.795225i	$-0.707353\pi$
$491$	−26.8701	−1.21263	−0.606314	+	0.795225i	$0.707353\pi$
$492$	0	0
$493$	19.7990	0.891702
$494$	0	0
$495$	12.0000	0.539360
$496$	0	0
$497$	0	0
$498$	0	0
$499$	32.5269	1.45610	0.728052	−	0.685522i	$-0.240426\pi$
$499$	32.5269	1.45610	0.728052	+	0.685522i	$0.240426\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	0	0
$505$	24.0000	1.06799
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−11.3137	−0.501471	−0.250736	−	0.968056i	$-0.580672\pi$
$509$	−11.3137	−0.501471	−0.250736	+	0.968056i	$0.580672\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−11.3137	−0.498542
$516$	0	0
$517$	16.9706	0.746364
$518$	0	0
$519$	0	0
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	0	0
$523$	33.9411	1.48414	0.742071	−	0.670321i	$-0.233844\pi$
$523$	33.9411	1.48414	0.742071	+	0.670321i	$0.233844\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−16.0000	−0.696971
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	33.9411	1.47292
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−20.0000	−0.864675
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.41421	0.0609145
$540$	0	0
$541$	15.5563	0.668820	0.334410	−	0.942428i	$-0.391463\pi$
$541$	15.5563	0.668820	0.334410	+	0.942428i	$0.391463\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−12.0000	−0.514024
$546$	0	0
$547$	32.5269	1.39075	0.695375	−	0.718647i	$-0.255238\pi$
$547$	32.5269	1.39075	0.695375	+	0.718647i	$0.255238\pi$
$548$	0	0
$549$	−25.4558	−1.08643
$550$	0	0
$551$	−28.0000	−1.19284
$552$	0	0
$553$	10.0000	0.425243
$554$	0	0
$555$	0	0
$556$	0	0
$557$	24.0416	1.01868	0.509338	−	0.860566i	$-0.329890\pi$
$557$	24.0416	1.01868	0.509338	+	0.860566i	$0.329890\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−28.2843	−1.19204	−0.596020	−	0.802970i	$-0.703252\pi$
$563$	−28.2843	−1.19204	−0.596020	+	0.802970i	$0.703252\pi$
$564$	0	0
$565$	11.3137	0.475971
$566$	0	0
$567$	−9.00000	−0.377964
$568$	0	0
$569$	−14.0000	−0.586911	−0.293455	−	0.955973i	$-0.594805\pi$
$569$	−14.0000	−0.586911	−0.293455	+	0.955973i	$0.594805\pi$
$570$	0	0
$571$	12.7279	0.532647	0.266323	−	0.963884i	$-0.414191\pi$
$571$	12.7279	0.532647	0.266323	+	0.963884i	$0.414191\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	18.0000	0.750652
$576$	0	0
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−14.1421	−0.586715
$582$	0	0
$583$	−2.00000	−0.0828315
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.1421	0.583708	0.291854	−	0.956463i	$-0.405728\pi$
$587$	14.1421	0.583708	0.291854	+	0.956463i	$0.405728\pi$
$588$	0	0
$589$	22.6274	0.932346
$590$	0	0
$591$	0	0
$592$	0	0
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	5.65685	0.231908
$596$	0	0
$597$	0	0
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	−12.7279	−0.518321
$604$	0	0
$605$	25.4558	1.03493
$606$	0	0
$607$	28.0000	1.13648	0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000	1.13648	0.568242	+	0.822861i	$0.307624\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−12.7279	−0.514076	−0.257038	−	0.966401i	$-0.582747\pi$
$613$	−12.7279	−0.514076	−0.257038	+	0.966401i	$0.582747\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−42.0000	−1.69086	−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000	−1.69086	−0.845428	+	0.534089i	$0.820655\pi$
$618$	0	0
$619$	−16.9706	−0.682105	−0.341052	−	0.940044i	$-0.610783\pi$
$619$	−16.9706	−0.682105	−0.341052	+	0.940044i	$0.610783\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	14.0000	0.560898
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	14.1421	0.563884
$630$	0	0
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−22.6274	−0.897942
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	12.0000	0.473972	0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000	0.473972	0.236986	+	0.971513i	$0.423841\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	−16.0000	−0.628055
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−35.3553	−1.38356	−0.691781	−	0.722108i	$-0.743173\pi$
$653$	−35.3553	−1.38356	−0.691781	+	0.722108i	$0.743173\pi$
$654$	0	0
$655$	56.0000	2.18810
$656$	0	0
$657$	−18.0000	−0.702247
$658$	0	0
$659$	4.24264	0.165270	0.0826349	−	0.996580i	$-0.473666\pi$
$659$	4.24264	0.165270	0.0826349	+	0.996580i	$0.473666\pi$
$660$	0	0
$661$	36.7696	1.43017	0.715085	−	0.699038i	$-0.246388\pi$
$661$	36.7696	1.43017	0.715085	+	0.699038i	$0.246388\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−8.00000	−0.310227
$666$	0	0
$667$	59.3970	2.29986
$668$	0	0
$669$	0	0
$670$	0	0
$671$	12.0000	0.463255
$672$	0	0
$673$	34.0000	1.31060	0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000	1.31060	0.655302	+	0.755367i	$0.272541\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	28.2843	1.08705	0.543526	−	0.839392i	$-0.317089\pi$
$677$	28.2843	1.08705	0.543526	+	0.839392i	$0.317089\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−15.5563	−0.595247	−0.297624	−	0.954683i	$-0.596194\pi$
$683$	−15.5563	−0.595247	−0.297624	+	0.954683i	$0.596194\pi$
$684$	0	0
$685$	33.9411	1.29682
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−36.7696	−1.39878	−0.699390	−	0.714740i	$-0.746545\pi$
$691$	−36.7696	−1.39878	−0.699390	+	0.714740i	$0.746545\pi$
$692$	0	0
$693$	4.24264	0.161165
$694$	0	0
$695$	8.00000	0.303457
$696$	0	0
$697$	−20.0000	−0.757554
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−26.8701	−1.01487	−0.507434	−	0.861691i	$-0.669406\pi$
$701$	−26.8701	−1.01487	−0.507434	+	0.861691i	$0.669406\pi$
$702$	0	0
$703$	−20.0000	−0.754314
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8.48528	0.319122
$708$	0	0
$709$	9.89949	0.371783	0.185892	−	0.982570i	$-0.440483\pi$
$709$	9.89949	0.371783	0.185892	+	0.982570i	$0.440483\pi$
$710$	0	0
$711$	30.0000	1.12509
$712$	0	0
$713$	−48.0000	−1.79761
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−20.0000	−0.745874	−0.372937	−	0.927857i	$-0.621649\pi$
$719$	−20.0000	−0.745874	−0.372937	+	0.927857i	$0.621649\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	0	0
$723$	0	0
$724$	0	0
$725$	29.6985	1.10297
$726$	0	0
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−2.82843	−0.104613
$732$	0	0
$733$	−14.1421	−0.522352	−0.261176	−	0.965291i	$-0.584110\pi$
$733$	−14.1421	−0.522352	−0.261176	+	0.965291i	$0.584110\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.00000	0.221013
$738$	0	0
$739$	9.89949	0.364159	0.182079	−	0.983284i	$-0.441717\pi$
$739$	9.89949	0.364159	0.182079	+	0.983284i	$0.441717\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−6.00000	−0.220119	−0.110059	−	0.993925i	$-0.535104\pi$
$743$	−6.00000	−0.220119	−0.110059	+	0.993925i	$0.535104\pi$
$744$	0	0
$745$	−12.0000	−0.439646
$746$	0	0
$747$	−42.4264	−1.55230
$748$	0	0
$749$	−7.07107	−0.258371
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	28.2843	1.02937
$756$	0	0
$757$	−21.2132	−0.771007	−0.385503	−	0.922706i	$-0.625972\pi$
$757$	−21.2132	−0.771007	−0.385503	+	0.922706i	$0.625972\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	−4.24264	−0.153594
$764$	0	0
$765$	16.9706	0.613572
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	8.48528	0.305194	0.152597	−	0.988288i	$-0.451236\pi$
$773$	8.48528	0.305194	0.152597	+	0.988288i	$0.451236\pi$
$774$	0	0
$775$	−24.0000	−0.862105
$776$	0	0
$777$	0	0
$778$	0	0
$779$	28.2843	1.01339
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	48.0000	1.71319
$786$	0	0
$787$	−25.4558	−0.907403	−0.453701	−	0.891154i	$-0.649897\pi$
$787$	−25.4558	−0.907403	−0.453701	+	0.891154i	$0.649897\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	4.00000	0.142224
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−45.2548	−1.60301	−0.801504	−	0.597989i	$-0.795967\pi$
$797$	−45.2548	−1.60301	−0.801504	+	0.597989i	$0.795967\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	0	0
$801$	42.0000	1.48400
$802$	0	0
$803$	8.48528	0.299439
$804$	0	0
$805$	16.9706	0.598134
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	−33.9411	−1.19183	−0.595917	−	0.803046i	$-0.703211\pi$
$811$	−33.9411	−1.19183	−0.595917	+	0.803046i	$0.703211\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−60.0000	−2.10171
$816$	0	0
$817$	4.00000	0.139942
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.41421	0.0493564	0.0246782	−	0.999695i	$-0.492144\pi$
$821$	1.41421	0.0493564	0.0246782	+	0.999695i	$0.492144\pi$
$822$	0	0
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7.07107	0.245885	0.122943	−	0.992414i	$-0.460767\pi$
$827$	7.07107	0.245885	0.122943	+	0.992414i	$0.460767\pi$
$828$	0	0
$829$	2.82843	0.0982353	0.0491177	−	0.998793i	$-0.484359\pi$
$829$	2.82843	0.0982353	0.0491177	+	0.998793i	$0.484359\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.00000	0.0692959
$834$	0	0
$835$	−33.9411	−1.17458
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−4.00000	−0.138095	−0.0690477	−	0.997613i	$-0.521996\pi$
$839$	−4.00000	−0.138095	−0.0690477	+	0.997613i	$0.521996\pi$
$840$	0	0
$841$	69.0000	2.37931
$842$	0	0
$843$	0	0
$844$	0	0
$845$	36.7696	1.26491
$846$	0	0
$847$	9.00000	0.309244
$848$	0	0
$849$	0	0
$850$	0	0
$851$	42.4264	1.45436
$852$	0	0
$853$	33.9411	1.16212	0.581061	−	0.813860i	$-0.302638\pi$
$853$	33.9411	1.16212	0.581061	+	0.813860i	$0.302638\pi$
$854$	0	0
$855$	−24.0000	−0.820783
$856$	0	0
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	2.82843	0.0965047	0.0482523	−	0.998835i	$-0.484635\pi$
$859$	2.82843	0.0965047	0.0482523	+	0.998835i	$0.484635\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	14.0000	0.476566	0.238283	−	0.971196i	$-0.423415\pi$
$863$	14.0000	0.476566	0.238283	+	0.971196i	$0.423415\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−14.1421	−0.479739
$870$	0	0
$871$	0	0
$872$	0	0
$873$	6.00000	0.203069
$874$	0	0
$875$	−5.65685	−0.191237
$876$	0	0
$877$	−4.24264	−0.143264	−0.0716319	−	0.997431i	$-0.522821\pi$
$877$	−4.24264	−0.143264	−0.0716319	+	0.997431i	$0.522821\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	35.3553	1.18980	0.594901	−	0.803799i	$-0.297191\pi$
$883$	35.3553	1.18980	0.594901	+	0.803799i	$0.297191\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−12.0000	−0.402921	−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000	−0.402921	−0.201460	+	0.979497i	$0.564569\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	12.7279	0.426401
$892$	0	0
$893$	−33.9411	−1.13580
$894$	0	0
$895$	−12.0000	−0.401116
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−79.1960	−2.64133
$900$	0	0
$901$	−2.82843	−0.0942286
$902$	0	0
$903$	0	0
$904$	0	0
$905$	16.0000	0.531858
$906$	0	0
$907$	21.2132	0.704373	0.352186	−	0.935930i	$-0.385438\pi$
$907$	21.2132	0.704373	0.352186	+	0.935930i	$0.385438\pi$
$908$	0	0
$909$	25.4558	0.844317
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	0	0
$913$	20.0000	0.661903
$914$	0	0
$915$	0	0
$916$	0	0
$917$	19.7990	0.653820
$918$	0	0
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	21.2132	0.697486
$926$	0	0
$927$	−12.0000	−0.394132
$928$	0	0
$929$	−50.0000	−1.64045	−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000	−1.64045	−0.820223	+	0.572043i	$0.806151\pi$
$930$	0	0
$931$	−2.82843	−0.0926980
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−8.00000	−0.261628
$936$	0	0
$937$	58.0000	1.89478	0.947389	−	0.320085i	$-0.103712\pi$
$937$	58.0000	1.89478	0.947389	+	0.320085i	$0.103712\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−36.7696	−1.19865	−0.599327	−	0.800505i	$-0.704565\pi$
$941$	−36.7696	−1.19865	−0.599327	+	0.800505i	$0.704565\pi$
$942$	0	0
$943$	−60.0000	−1.95387
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−46.6690	−1.51654	−0.758270	−	0.651940i	$-0.773955\pi$
$947$	−46.6690	−1.51654	−0.758270	+	0.651940i	$0.773955\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−16.0000	−0.518291	−0.259145	−	0.965838i	$-0.583441\pi$
$953$	−16.0000	−0.518291	−0.259145	+	0.965838i	$0.583441\pi$
$954$	0	0
$955$	50.9117	1.64746
$956$	0	0
$957$	0	0
$958$	0	0
$959$	12.0000	0.387500
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	−21.2132	−0.683586
$964$	0	0
$965$	45.2548	1.45680
$966$	0	0
$967$	2.00000	0.0643157	0.0321578	−	0.999483i	$-0.489762\pi$
$967$	2.00000	0.0643157	0.0321578	+	0.999483i	$0.489762\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−5.65685	−0.181537	−0.0907685	−	0.995872i	$-0.528932\pi$
$971$	−5.65685	−0.181537	−0.0907685	+	0.995872i	$0.528932\pi$
$972$	0	0
$973$	2.82843	0.0906752
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−38.0000	−1.21573	−0.607864	−	0.794041i	$-0.707973\pi$
$977$	−38.0000	−1.21573	−0.607864	+	0.794041i	$0.707973\pi$
$978$	0	0
$979$	−19.7990	−0.632778
$980$	0	0
$981$	−12.7279	−0.406371
$982$	0	0
$983$	−44.0000	−1.40338	−0.701691	−	0.712481i	$-0.747571\pi$
$983$	−44.0000	−1.40338	−0.701691	+	0.712481i	$0.747571\pi$
$984$	0	0
$985$	20.0000	0.637253
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.48528	−0.269816
$990$	0	0
$991$	22.0000	0.698853	0.349427	−	0.936964i	$-0.386376\pi$
$991$	22.0000	0.698853	0.349427	+	0.936964i	$0.386376\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	67.8823	2.15201
$996$	0	0
$997$	42.4264	1.34366	0.671829	−	0.740706i	$-0.265509\pi$
$997$	42.4264	1.34366	0.671829	+	0.740706i	$0.265509\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7168.2.a.b.1.1		2
4.3	odd	2		7168.2.a.k.1.1		2
8.3	odd	2		7168.2.a.k.1.2		2
8.5	even	2	inner	7168.2.a.b.1.2		2
32.3	odd	8		448.2.m.a.337.1		2
32.5	even	8		896.2.m.b.225.1		2
32.11	odd	8		448.2.m.a.113.1		2
32.13	even	8		896.2.m.b.673.1		2
32.19	odd	8		896.2.m.c.673.1		2
32.21	even	8		112.2.m.b.85.1	yes	2
32.27	odd	8		896.2.m.c.225.1		2
32.29	even	8		112.2.m.b.29.1	✓	2
224.53	even	24		784.2.x.d.373.1		4
224.61	odd	24		784.2.x.e.557.1		4
224.93	even	24		784.2.x.d.557.1		4
224.117	odd	24		784.2.x.e.165.1		4
224.125	odd	8		784.2.m.a.589.1		2
224.149	even	24		784.2.x.d.165.1		4
224.157	odd	24		784.2.x.e.765.1		4
224.181	odd	8		784.2.m.a.197.1		2
224.213	odd	24		784.2.x.e.373.1		4
224.221	even	24		784.2.x.d.765.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.m.b.29.1	✓	2	32.29	even	8
112.2.m.b.85.1	yes	2	32.21	even	8
448.2.m.a.113.1		2	32.11	odd	8
448.2.m.a.337.1		2	32.3	odd	8
784.2.m.a.197.1		2	224.181	odd	8
784.2.m.a.589.1		2	224.125	odd	8
784.2.x.d.165.1		4	224.149	even	24
784.2.x.d.373.1		4	224.53	even	24
784.2.x.d.557.1		4	224.93	even	24
784.2.x.d.765.1		4	224.221	even	24
784.2.x.e.165.1		4	224.117	odd	24
784.2.x.e.373.1		4	224.213	odd	24
784.2.x.e.557.1		4	224.61	odd	24
784.2.x.e.765.1		4	224.157	odd	24
896.2.m.b.225.1		2	32.5	even	8
896.2.m.b.673.1		2	32.13	even	8
896.2.m.c.225.1		2	32.27	odd	8
896.2.m.c.673.1		2	32.19	odd	8
7168.2.a.b.1.1		2	1.1	even	1	trivial
7168.2.a.b.1.2		2	8.5	even	2	inner
7168.2.a.k.1.1		2	4.3	odd	2
7168.2.a.k.1.2		2	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7168 = 2^{10} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7168.a (trivial)

\(f(q)\)	\(=\)	\(q-2.82843 q^{5} -1.00000 q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-2.82843 q^{5} -1.00000 q^{7} -3.00000 q^{9} +1.41421 q^{11} +2.00000 q^{17} -2.82843 q^{19} +6.00000 q^{23} +3.00000 q^{25} +9.89949 q^{29} -8.00000 q^{31} +2.82843 q^{35} +7.07107 q^{37} -10.0000 q^{41} -1.41421 q^{43} +8.48528 q^{45} +12.0000 q^{47} +1.00000 q^{49} -1.41421 q^{53} -4.00000 q^{55} -11.3137 q^{59} +8.48528 q^{61} +3.00000 q^{63} +4.24264 q^{67} +6.00000 q^{73} -1.41421 q^{77} -10.0000 q^{79} +9.00000 q^{81} +14.1421 q^{83} -5.65685 q^{85} -14.0000 q^{89} +8.00000 q^{95} -2.00000 q^{97} -4.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{7} - 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^7 - 6 * q^9	\( 2 q - 2 q^{7} - 6 q^{9} + 4 q^{17} + 12 q^{23} + 6 q^{25} - 16 q^{31} - 20 q^{41} + 24 q^{47} + 2 q^{49} - 8 q^{55} + 6 q^{63} + 12 q^{73} - 20 q^{79} + 18 q^{81} - 28 q^{89} + 16 q^{95} - 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^7 - 6 * q^9 + 4 * q^17 + 12 * q^23 + 6 * q^25 - 16 * q^31 - 20 * q^41 + 24 * q^47 + 2 * q^49 - 8 * q^55 + 6 * q^63 + 12 * q^73 - 20 * q^79 + 18 * q^81 - 28 * q^89 + 16 * q^95 - 4 * q^97

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