LMFDB - Embedded newform 352.2.g.b.175.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [352,2,Mod(175,352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(352, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

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

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

chi := DirichletCharacter("352.175");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 352 = 2^{5} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	352.g (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:	$2.81073415115$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.10070523904.6
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 6x^{4} + 16 $ x^8 + 6*x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 88)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			175.2
Root			$-1.16342 + 0.804019i$ of defining polynomial
Character	$\chi$	$=$	352.175
Dual form			352.2.g.b.175.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.414214 q^{3} -2.64575i q^{5} +3.29066 q^{7} -2.82843 q^{9} +(-2.41421 - 2.27411i) q^{11} +1.36303 q^{13} +1.09591i q^{15} -5.49019i q^{17} +2.27411i q^{19} -1.36303 q^{21} -6.38741i q^{23} -2.00000 q^{25} +2.41421 q^{27} +6.58132 q^{29} -1.09591i q^{31} +(1.00000 + 0.941967i) q^{33} -8.70626i q^{35} -4.83756i q^{37} -0.564588 q^{39} +10.0384i q^{41} +6.43215i q^{43} +7.48331i q^{45} +3.82843 q^{49} +2.27411i q^{51} +7.48331i q^{53} +(-6.01673 + 6.38741i) q^{55} -0.941967i q^{57} -3.24264 q^{59} -6.58132 q^{61} -9.30739 q^{63} -3.60625i q^{65} +6.07107 q^{67} +2.64575i q^{69} +6.38741i q^{71} -0.941967i q^{73} +0.828427 q^{75} +(-7.94435 - 7.48331i) q^{77} +9.30739 q^{79} +7.48528 q^{81} +4.54822i q^{83} -14.5257 q^{85} -2.72607 q^{87} -10.6569 q^{89} +4.48528 q^{91} +0.453939i q^{93} +6.01673 q^{95} +7.48528 q^{97} +(6.82843 + 6.43215i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} - 8 q^{11} - 16 q^{25} + 8 q^{27} + 8 q^{33} + 8 q^{49} + 8 q^{59} - 8 q^{67} - 16 q^{75} - 8 q^{81} - 40 q^{89} - 32 q^{91} - 8 q^{97} + 32 q^{99}+O(q^{100}) $ 8 * q + 8 * q^3 - 8 * q^11 - 16 * q^25 + 8 * q^27 + 8 * q^33 + 8 * q^49 + 8 * q^59 - 8 * q^67 - 16 * q^75 - 8 * q^81 - 40 * q^89 - 32 * q^91 - 8 * q^97 + 32 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/352\mathbb{Z}\right)^\times$.

$n$	$133$	$287$	$321$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	−0.414214			−0.239146			−0.119573	−	0.992825i	$-0.538153\pi$
$3$	−0.414214			−0.239146			−0.119573	+	0.992825i	$0.538153\pi$
$4$		0			0
$5$		−	2.64575i		−	1.18322i	−0.806226	−	0.591608i	$-0.798493\pi$
$5$		−	2.64575i		−	1.18322i	0.806226	−	0.591608i	$-0.201507\pi$
$6$		0			0
$7$	3.29066			1.24375			0.621876	−	0.783116i	$-0.286371\pi$
$7$	3.29066			1.24375			0.621876	+	0.783116i	$0.286371\pi$
$8$		0			0
$9$	−2.82843			−0.942809
$10$		0			0
$11$	−2.41421	−	2.27411i	−0.727913	−	0.685670i
$11$	−2.41421	−	2.27411i	−0.727913	−	0.685670i
$12$		0			0
$13$	1.36303			0.378038			0.189019	−	0.981973i	$-0.439469\pi$
$13$	1.36303			0.378038			0.189019	+	0.981973i	$0.439469\pi$
$14$		0			0
$15$			1.09591i			0.282962i
$16$		0			0
$17$		−	5.49019i		−	1.33157i	−0.746146	−	0.665783i	$-0.768098\pi$
$17$		−	5.49019i		−	1.33157i	0.746146	−	0.665783i	$-0.231902\pi$
$18$		0			0
$19$			2.27411i			0.521716i	0.965377	+	0.260858i	$0.0840055\pi$
$19$			2.27411i			0.521716i	−0.965377	+	0.260858i	$0.915994\pi$
$20$		0			0
$21$	−1.36303			−0.297439
$22$		0			0
$23$		−	6.38741i		−	1.33187i	−0.746011	−	0.665933i	$-0.768033\pi$
$23$		−	6.38741i		−	1.33187i	0.746011	−	0.665933i	$-0.231967\pi$
$24$		0			0
$25$	−2.00000			−0.400000
$26$		0			0
$27$	2.41421			0.464616
$28$		0			0
$29$	6.58132			1.22212			0.611060	−	0.791584i	$-0.290743\pi$
$29$	6.58132			1.22212			0.611060	+	0.791584i	$0.290743\pi$
$30$		0			0
$31$		−	1.09591i		−	0.196831i	−0.995145	−	0.0984153i	$-0.968623\pi$
$31$		−	1.09591i		−	0.196831i	0.995145	−	0.0984153i	$-0.0313773\pi$
$32$		0			0
$33$	1.00000	+	0.941967i	0.174078	+	0.163975i
$34$		0			0
$35$		−	8.70626i		−	1.47163i
$36$		0			0
$37$		−	4.83756i		−	0.795291i	−0.917539	−	0.397645i	$-0.869827\pi$
$37$		−	4.83756i		−	0.795291i	0.917539	−	0.397645i	$-0.130173\pi$
$38$		0			0
$39$	−0.564588			−0.0904064
$40$		0			0
$41$			10.0384i			1.56774i	0.620928	+	0.783868i	$0.286756\pi$
$41$			10.0384i			1.56774i	−0.620928	+	0.783868i	$0.713244\pi$
$42$		0			0
$43$			6.43215i			0.980894i	0.871471	+	0.490447i	$0.163166\pi$
$43$			6.43215i			0.980894i	−0.871471	+	0.490447i	$0.836834\pi$
$44$		0			0
$45$			7.48331i			1.11555i
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	3.82843			0.546918
$50$		0			0
$51$			2.27411i			0.318439i
$52$		0			0
$53$			7.48331i			1.02791i	0.857816	+	0.513956i	$0.171821\pi$
$53$			7.48331i			1.02791i	−0.857816	+	0.513956i	$0.828179\pi$
$54$		0			0
$55$	−6.01673	+	6.38741i	−0.811295	+	0.861278i
$56$		0			0
$57$		−	0.941967i		−	0.124767i
$58$		0			0
$59$	−3.24264			−0.422156			−0.211078	−	0.977469i	$-0.567697\pi$
$59$	−3.24264			−0.422156			−0.211078	+	0.977469i	$0.567697\pi$
$60$		0			0
$61$	−6.58132			−0.842651			−0.421326	−	0.906909i	$-0.638435\pi$
$61$	−6.58132			−0.842651			−0.421326	+	0.906909i	$0.638435\pi$
$62$		0			0
$63$	−9.30739			−1.17262
$64$		0			0
$65$		−	3.60625i		−	0.447300i
$66$		0			0
$67$	6.07107			0.741699			0.370849	−	0.928693i	$-0.379067\pi$
$67$	6.07107			0.741699			0.370849	+	0.928693i	$0.379067\pi$
$68$		0			0
$69$			2.64575i			0.318511i
$70$		0			0
$71$			6.38741i			0.758046i	0.925387	+	0.379023i	$0.123740\pi$
$71$			6.38741i			0.758046i	−0.925387	+	0.379023i	$0.876260\pi$
$72$		0			0
$73$		−	0.941967i		−	0.110249i	−0.998479	−	0.0551244i	$-0.982444\pi$
$73$		−	0.941967i		−	0.110249i	0.998479	−	0.0551244i	$-0.0175556\pi$
$74$		0			0
$75$	0.828427			0.0956585
$76$		0			0
$77$	−7.94435	−	7.48331i	−0.905343	−	0.852803i
$78$		0			0
$79$	9.30739			1.04716			0.523581	−	0.851976i	$-0.324596\pi$
$79$	9.30739			1.04716			0.523581	+	0.851976i	$0.324596\pi$
$80$		0			0
$81$	7.48528			0.831698
$82$		0			0
$83$			4.54822i			0.499232i	0.968345	+	0.249616i	$0.0803044\pi$
$83$			4.54822i			0.499232i	−0.968345	+	0.249616i	$0.919696\pi$
$84$		0			0
$85$	−14.5257			−1.57553
$86$		0			0
$87$	−2.72607			−0.292265
$88$		0			0
$89$	−10.6569			−1.12962			−0.564812	−	0.825220i	$-0.691051\pi$
$89$	−10.6569			−1.12962			−0.564812	+	0.825220i	$0.691051\pi$
$90$		0			0
$91$	4.48528			0.470185
$92$		0			0
$93$			0.453939i			0.0470713i
$94$		0			0
$95$	6.01673			0.617303
$96$		0			0
$97$	7.48528			0.760015			0.380008	−	0.924983i	$-0.375921\pi$
$97$	7.48528			0.760015			0.380008	+	0.924983i	$0.375921\pi$
$98$		0			0
$99$	6.82843	+	6.43215i	0.686283	+	0.646456i
$100$		0			0
$101$	17.2517			1.71661			0.858306	−	0.513138i	$-0.171517\pi$
$101$	17.2517			1.71661			0.858306	+	0.513138i	$0.171517\pi$
$102$		0			0
$103$			18.0663i			1.78013i	0.455836	+	0.890064i	$0.349340\pi$
$103$			18.0663i			1.78013i	−0.455836	+	0.890064i	$0.650660\pi$
$104$		0			0
$105$			3.60625i			0.351934i
$106$		0			0
$107$			8.70626i			0.841666i	0.907138	+	0.420833i	$0.138262\pi$
$107$			8.70626i			0.841666i	−0.907138	+	0.420833i	$0.861738\pi$
$108$		0			0
$109$	−2.72607			−0.261110			−0.130555	−	0.991441i	$-0.541676\pi$
$109$	−2.72607			−0.261110			−0.130555	+	0.991441i	$0.541676\pi$
$110$		0			0
$111$			2.00378i			0.190191i
$112$		0			0
$113$	6.17157			0.580573			0.290286	−	0.956940i	$-0.406249\pi$
$113$	6.17157			0.580573			0.290286	+	0.956940i	$0.406249\pi$
$114$		0			0
$115$	−16.8995			−1.57589
$116$		0			0
$117$	−3.85525			−0.356418
$118$		0			0
$119$		−	18.0663i		−	1.65614i
$120$		0			0
$121$	0.656854	+	10.9804i	0.0597140	+	0.998216i
$122$		0			0
$123$		−	4.15804i		−	0.374918i
$124$		0			0
$125$		−	7.93725i		−	0.709930i
$126$		0			0
$127$	0.564588			0.0500990			0.0250495	−	0.999686i	$-0.492026\pi$
$127$	0.564588			0.0500990			0.0250495	+	0.999686i	$0.492026\pi$
$128$		0			0
$129$		−	2.66428i		−	0.234577i
$130$		0			0
$131$		−	6.82233i		−	0.596070i	−0.954555	−	0.298035i	$-0.903669\pi$
$131$		−	6.82233i		−	0.596070i	0.954555	−	0.298035i	$-0.0963311\pi$
$132$		0			0
$133$			7.48331i			0.648886i
$134$		0			0
$135$		−	6.38741i		−	0.549741i
$136$		0			0
$137$	−8.65685			−0.739605			−0.369802	−	0.929110i	$-0.620575\pi$
$137$	−8.65685			−0.739605			−0.369802	+	0.929110i	$0.620575\pi$
$138$		0			0
$139$		−	19.6866i		−	1.66980i	−0.550403	−	0.834899i	$-0.685526\pi$
$139$		−	19.6866i		−	1.66980i	0.550403	−	0.834899i	$-0.314474\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.29066	−	3.09969i	−0.275179	−	0.259209i
$144$		0			0
$145$		−	17.4125i		−	1.44603i
$146$		0			0
$147$	−1.58579			−0.130793
$148$		0			0
$149$	21.1070			1.72915			0.864576	−	0.502503i	$-0.167587\pi$
$149$	21.1070			1.72915			0.864576	+	0.502503i	$0.167587\pi$
$150$		0			0
$151$	−13.1626			−1.07116			−0.535580	−	0.844485i	$-0.679907\pi$
$151$	−13.1626			−1.07116			−0.535580	+	0.844485i	$0.679907\pi$
$152$		0			0
$153$			15.5286i			1.25541i
$154$		0			0
$155$	−2.89949			−0.232893
$156$		0			0
$157$		−	0.453939i		−	0.0362283i	−0.999836	−	0.0181141i	$-0.994234\pi$
$157$		−	0.453939i		−	0.0362283i	0.999836	−	0.0181141i	$-0.00576623\pi$
$158$		0			0
$159$		−	3.09969i		−	0.245821i
$160$		0			0
$161$		−	21.0188i		−	1.65651i
$162$		0			0
$163$	12.8284			1.00480			0.502400	−	0.864635i	$-0.332451\pi$
$163$	12.8284			1.00480			0.502400	+	0.864635i	$0.332451\pi$
$164$		0			0
$165$	2.49221	−	2.64575i	0.194018	−	0.205971i
$166$		0			0
$167$	21.9054			1.69509			0.847546	−	0.530721i	$-0.178079\pi$
$167$	21.9054			1.69509			0.847546	+	0.530721i	$0.178079\pi$
$168$		0			0
$169$	−11.1421			−0.857087
$170$		0			0
$171$		−	6.43215i		−	0.491879i
$172$		0			0
$173$	−5.45214			−0.414519			−0.207259	−	0.978286i	$-0.566454\pi$
$173$	−5.45214			−0.414519			−0.207259	+	0.978286i	$0.566454\pi$
$174$		0			0
$175$	−6.58132			−0.497501
$176$		0			0
$177$	1.34315			0.100957
$178$		0			0
$179$	−9.72792			−0.727099			−0.363550	−	0.931575i	$-0.618435\pi$
$179$	−9.72792			−0.727099			−0.363550	+	0.931575i	$0.618435\pi$
$180$		0			0
$181$			2.64575i			0.196657i	0.995154	+	0.0983286i	$0.0313496\pi$
$181$			2.64575i			0.196657i	−0.995154	+	0.0983286i	$0.968650\pi$
$182$		0			0
$183$	2.72607			0.201517
$184$		0			0
$185$	−12.7990			−0.941000
$186$		0			0
$187$	−12.4853	+	13.2545i	−0.913014	+	0.969264i
$188$		0			0
$189$	7.94435			0.577866
$190$		0			0
$191$		−	16.9704i		−	1.22794i	−0.789331	−	0.613968i	$-0.789572\pi$
$191$		−	16.9704i		−	1.22794i	0.789331	−	0.613968i	$-0.210428\pi$
$192$		0			0
$193$			12.8643i			0.925993i	0.886360	+	0.462996i	$0.153226\pi$
$193$			12.8643i			0.925993i	−0.886360	+	0.462996i	$0.846774\pi$
$194$		0			0
$195$			1.49376i			0.106970i
$196$		0			0
$197$	−10.4366			−0.743574			−0.371787	−	0.928318i	$-0.621255\pi$
$197$	−10.4366			−0.743574			−0.371787	+	0.928318i	$0.621255\pi$
$198$		0			0
$199$		−	21.1660i		−	1.50042i	−0.661200	−	0.750209i	$-0.729953\pi$
$199$		−	21.1660i		−	1.50042i	0.661200	−	0.750209i	$-0.270047\pi$
$200$		0			0
$201$	−2.51472			−0.177375
$202$		0			0
$203$	21.6569			1.52001
$204$		0			0
$205$	26.5591			1.85497
$206$		0			0
$207$			18.0663i			1.25570i
$208$		0			0
$209$	5.17157	−	5.49019i	0.357725	−	0.379764i
$210$		0			0
$211$			9.09644i			0.626225i	0.949716	+	0.313112i	$0.101372\pi$
$211$			9.09644i			0.626225i	−0.949716	+	0.313112i	$0.898628\pi$
$212$		0			0
$213$		−	2.64575i		−	0.181284i
$214$		0			0
$215$	17.0179			1.16061
$216$		0			0
$217$		−	3.60625i		−	0.244808i
$218$		0			0
$219$			0.390175i			0.0263656i
$220$		0			0
$221$		−	7.48331i		−	0.503382i
$222$		0			0
$223$			4.19560i			0.280958i	0.990084	+	0.140479i	$0.0448642\pi$
$223$			4.19560i			0.280958i	−0.990084	+	0.140479i	$0.955136\pi$
$224$		0			0
$225$	5.65685			0.377124
$226$		0			0
$227$		−	13.2545i		−	0.879731i	−0.898064	−	0.439865i	$-0.855026\pi$
$227$		−	13.2545i		−	0.879731i	0.898064	−	0.439865i	$-0.144974\pi$
$228$		0			0
$229$		−	10.1291i		−	0.669348i	−0.942334	−	0.334674i	$-0.891374\pi$
$229$		−	10.1291i		−	0.669348i	0.942334	−	0.334674i	$-0.108626\pi$
$230$		0			0
$231$	3.29066	+	3.09969i	0.216509	+	0.203945i
$232$		0			0
$233$		−	8.31609i		−	0.544805i	−0.962183	−	0.272402i	$-0.912182\pi$
$233$		−	8.31609i		−	0.544805i	0.962183	−	0.272402i	$-0.0878182\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−3.85525			−0.250425
$238$		0			0
$239$	−5.45214			−0.352670			−0.176335	−	0.984330i	$-0.556424\pi$
$239$	−5.45214			−0.352670			−0.176335	+	0.984330i	$0.556424\pi$
$240$		0			0
$241$			4.54822i			0.292977i	0.989212	+	0.146488i	$0.0467971\pi$
$241$			4.54822i			0.292977i	−0.989212	+	0.146488i	$0.953203\pi$
$242$		0			0
$243$	−10.3431			−0.663513
$244$		0			0
$245$		−	10.1291i		−	0.647122i
$246$		0			0
$247$			3.09969i			0.197229i
$248$		0			0
$249$		−	1.88393i		−	0.119389i
$250$		0			0
$251$	−18.4142			−1.16229			−0.581147	−	0.813798i	$-0.697396\pi$
$251$	−18.4142			−1.16229			−0.581147	+	0.813798i	$0.697396\pi$
$252$		0			0
$253$	−14.5257	+	15.4206i	−0.913221	+	0.969483i
$254$		0			0
$255$	6.01673			0.376782
$256$		0			0
$257$	6.48528			0.404541			0.202270	−	0.979330i	$-0.435168\pi$
$257$	6.48528			0.404541			0.202270	+	0.979330i	$0.435168\pi$
$258$		0			0
$259$		−	15.9188i		−	0.989144i
$260$		0			0
$261$	−18.6148			−1.15223
$262$		0			0
$263$	−19.1794			−1.18265			−0.591325	−	0.806433i	$-0.701395\pi$
$263$	−19.1794			−1.18265			−0.591325	+	0.806433i	$0.701395\pi$
$264$		0			0
$265$	19.7990			1.21624
$266$		0			0
$267$	4.41421			0.270145
$268$		0			0
$269$		−	13.6827i		−	0.834249i	−0.908849	−	0.417124i	$-0.863038\pi$
$269$		−	13.6827i		−	0.834249i	0.908849	−	0.417124i	$-0.136962\pi$
$270$		0			0
$271$	12.5980			0.765277			0.382638	−	0.923898i	$-0.375016\pi$
$271$	12.5980			0.765277			0.382638	+	0.923898i	$0.375016\pi$
$272$		0			0
$273$	−1.85786			−0.112443
$274$		0			0
$275$	4.82843	+	4.54822i	0.291165	+	0.274268i
$276$		0			0
$277$	3.85525			0.231639			0.115820	−	0.993270i	$-0.463051\pi$
$277$	3.85525			0.231639			0.115820	+	0.993270i	$0.463051\pi$
$278$		0			0
$279$			3.09969i			0.185574i
$280$		0			0
$281$			6.43215i			0.383710i	0.981423	+	0.191855i	$0.0614503\pi$
$281$			6.43215i			0.383710i	−0.981423	+	0.191855i	$0.938550\pi$
$282$		0			0
$283$			1.88393i			0.111988i	0.998431	+	0.0559941i	$0.0178328\pi$
$283$			1.88393i			0.111988i	−0.998431	+	0.0559941i	$0.982167\pi$
$284$		0			0
$285$	−2.49221			−0.147626
$286$		0			0
$287$			33.0329i			1.94987i
$288$		0			0
$289$	−13.1421			−0.773067
$290$		0			0
$291$	−3.10051			−0.181755
$292$		0			0
$293$	−14.5257			−0.848598			−0.424299	−	0.905522i	$-0.639480\pi$
$293$	−14.5257			−0.848598			−0.424299	+	0.905522i	$0.639480\pi$
$294$		0			0
$295$			8.57922i			0.499502i
$296$		0			0
$297$	−5.82843	−	5.49019i	−0.338200	−	0.318573i
$298$		0			0
$299$		−	8.70626i		−	0.503496i
$300$		0			0
$301$			21.1660i			1.21999i
$302$		0			0
$303$	−7.14590			−0.410521
$304$		0			0
$305$			17.4125i			0.997038i
$306$		0			0
$307$			22.3509i			1.27563i	0.770188	+	0.637817i	$0.220162\pi$
$307$			22.3509i			1.27563i	−0.770188	+	0.637817i	$0.779838\pi$
$308$		0			0
$309$		−	7.48331i		−	0.425711i
$310$		0			0
$311$			14.9666i			0.848680i	0.905503	+	0.424340i	$0.139494\pi$
$311$			14.9666i			0.848680i	−0.905503	+	0.424340i	$0.860506\pi$
$312$		0			0
$313$	7.82843			0.442489			0.221244	−	0.975218i	$-0.428988\pi$
$313$	7.82843			0.442489			0.221244	+	0.975218i	$0.428988\pi$
$314$		0			0
$315$			24.6250i			1.38746i
$316$		0			0
$317$			15.4206i			0.866105i	0.901369	+	0.433053i	$0.142564\pi$
$317$			15.4206i			0.866105i	−0.901369	+	0.433053i	$0.857436\pi$
$318$		0			0
$319$	−15.8887	−	14.9666i	−0.889596	−	0.837970i
$320$		0			0
$321$		−	3.60625i		−	0.201281i
$322$		0			0
$323$	12.4853			0.694700
$324$		0			0
$325$	−2.72607			−0.151215
$326$		0			0
$327$	1.12918			0.0624435
$328$		0			0
$329$		0			0
$330$		0			0
$331$	18.0711			0.993276			0.496638	−	0.867958i	$-0.334568\pi$
$331$	18.0711			0.993276			0.496638	+	0.867958i	$0.334568\pi$
$332$		0			0
$333$			13.6827i			0.749807i
$334$		0			0
$335$		−	16.0625i		−	0.877590i
$336$		0			0
$337$			19.1348i			1.04234i	0.853453	+	0.521171i	$0.174504\pi$
$337$			19.1348i			1.04234i	−0.853453	+	0.521171i	$0.825496\pi$
$338$		0			0
$339$	−2.55635			−0.138842
$340$		0			0
$341$	−2.49221	+	2.64575i	−0.134961	+	0.143275i
$342$		0			0
$343$	−10.4366			−0.563521
$344$		0			0
$345$	7.00000			0.376867
$346$		0			0
$347$			4.15804i			0.223215i	0.993752	+	0.111608i	$0.0356000\pi$
$347$			4.15804i			0.223215i	−0.993752	+	0.111608i	$0.964400\pi$
$348$		0			0
$349$	14.5257			0.777541			0.388771	−	0.921335i	$-0.372900\pi$
$349$	14.5257			0.777541			0.388771	+	0.921335i	$0.372900\pi$
$350$		0			0
$351$	3.29066			0.175642
$352$		0			0
$353$	1.82843			0.0973174			0.0486587	−	0.998815i	$-0.484505\pi$
$353$	1.82843			0.0973174			0.0486587	+	0.998815i	$0.484505\pi$
$354$		0			0
$355$	16.8995			0.896932
$356$		0			0
$357$			7.48331i			0.396059i
$358$		0			0
$359$	−3.29066			−0.173674			−0.0868371	−	0.996223i	$-0.527676\pi$
$359$	−3.29066			−0.173674			−0.0868371	+	0.996223i	$0.527676\pi$
$360$		0			0
$361$	13.8284			0.727812
$362$		0			0
$363$	−0.272078	−	4.54822i	−0.0142804	−	0.238720i
$364$		0			0
$365$	−2.49221			−0.130448
$366$		0			0
$367$		−	22.2619i		−	1.16206i	−0.813881	−	0.581031i	$-0.802649\pi$
$367$		−	22.2619i		−	1.16206i	0.813881	−	0.581031i	$-0.197351\pi$
$368$		0			0
$369$		−	28.3929i		−	1.47808i
$370$		0			0
$371$			24.6250i			1.27847i
$372$		0			0
$373$	−0.233860			−0.0121088			−0.00605440	−	0.999982i	$-0.501927\pi$
$373$	−0.233860			−0.0121088			−0.00605440	+	0.999982i	$0.501927\pi$
$374$		0			0
$375$			3.28772i			0.169777i
$376$		0			0
$377$	8.97056			0.462007
$378$		0			0
$379$	16.0711			0.825515			0.412758	−	0.910841i	$-0.364566\pi$
$379$	16.0711			0.825515			0.412758	+	0.910841i	$0.364566\pi$
$380$		0			0
$381$	−0.233860			−0.0119810
$382$		0			0
$383$			7.29529i			0.372772i	0.982477	+	0.186386i	$0.0596775\pi$
$383$			7.29529i			0.372772i	−0.982477	+	0.186386i	$0.940323\pi$
$384$		0			0
$385$	−19.7990	+	21.0188i	−1.00905	+	1.07122i
$386$		0			0
$387$		−	18.1929i		−	0.924796i
$388$		0			0
$389$			32.5790i			1.65182i	0.563801	+	0.825911i	$0.309339\pi$
$389$			32.5790i			1.65182i	−0.563801	+	0.825911i	$0.690661\pi$
$390$		0			0
$391$	−35.0681			−1.77347
$392$		0			0
$393$			2.82590i			0.142548i
$394$		0			0
$395$		−	24.6250i		−	1.23902i
$396$		0			0
$397$		−	22.4499i		−	1.12673i	−0.826208	−	0.563365i	$-0.809506\pi$
$397$		−	22.4499i		−	1.12673i	0.826208	−	0.563365i	$-0.190494\pi$
$398$		0			0
$399$		−	3.09969i		−	0.155179i
$400$		0			0
$401$	−4.82843			−0.241120			−0.120560	−	0.992706i	$-0.538469\pi$
$401$	−4.82843			−0.241120			−0.120560	+	0.992706i	$0.538469\pi$
$402$		0			0
$403$		−	1.49376i		−	0.0744094i
$404$		0			0
$405$		−	19.8042i		−	0.984078i
$406$		0			0
$407$	−11.0011	+	11.6789i	−0.545307	+	0.578902i
$408$		0			0
$409$			1.88393i			0.0931545i	0.998915	+	0.0465773i	$0.0148314\pi$
$409$			1.88393i			0.0931545i	−0.998915	+	0.0465773i	$0.985169\pi$
$410$		0			0
$411$	3.58579			0.176874
$412$		0			0
$413$	−10.6704			−0.525057
$414$		0			0
$415$	12.0335			0.590699
$416$		0			0
$417$			8.15447i			0.399326i
$418$		0			0
$419$	−16.1421			−0.788595			−0.394297	−	0.918983i	$-0.629012\pi$
$419$	−16.1421			−0.788595			−0.394297	+	0.918983i	$0.629012\pi$
$420$		0			0
$421$			7.48331i			0.364714i	0.983232	+	0.182357i	$0.0583727\pi$
$421$			7.48331i			0.364714i	−0.983232	+	0.182357i	$0.941627\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			10.9804i			0.532626i
$426$		0			0
$427$	−21.6569			−1.04805
$428$		0			0
$429$	1.36303	+	1.28393i	0.0658079	+	0.0619889i
$430$		0			0
$431$	−2.16148			−0.104115			−0.0520575	−	0.998644i	$-0.516578\pi$
$431$	−2.16148			−0.104115			−0.0520575	+	0.998644i	$0.516578\pi$
$432$		0			0
$433$	−17.3431			−0.833458			−0.416729	−	0.909031i	$-0.636824\pi$
$433$	−17.3431			−0.833458			−0.416729	+	0.909031i	$0.636824\pi$
$434$		0			0
$435$			7.21250i			0.345813i
$436$		0			0
$437$	14.5257			0.694857
$438$		0			0
$439$	34.5035			1.64676			0.823380	−	0.567490i	$-0.192085\pi$
$439$	34.5035			1.64676			0.823380	+	0.567490i	$0.192085\pi$
$440$		0			0
$441$	−10.8284			−0.515639
$442$		0			0
$443$	−26.4142			−1.25498			−0.627489	−	0.778626i	$-0.715917\pi$
$443$	−26.4142			−1.25498			−0.627489	+	0.778626i	$0.715917\pi$
$444$		0			0
$445$			28.1954i			1.33659i
$446$		0			0
$447$	−8.74280			−0.413520
$448$		0			0
$449$	15.9706			0.753697			0.376849	−	0.926275i	$-0.377008\pi$
$449$	15.9706			0.753697			0.376849	+	0.926275i	$0.377008\pi$
$450$		0			0
$451$	22.8284	−	24.2349i	1.07495	−	1.14117i
$452$		0			0
$453$	5.45214			0.256164
$454$		0			0
$455$		−	11.8669i		−	0.556331i
$456$		0			0
$457$		−	22.7411i		−	1.06378i	−0.846812	−	0.531892i	$-0.821481\pi$
$457$		−	22.7411i		−	1.06378i	0.846812	−	0.531892i	$-0.178519\pi$
$458$		0			0
$459$		−	13.2545i		−	0.618666i
$460$		0			0
$461$	3.85525			0.179557			0.0897783	−	0.995962i	$-0.471384\pi$
$461$	3.85525			0.179557			0.0897783	+	0.995962i	$0.471384\pi$
$462$		0			0
$463$			22.2619i			1.03460i	0.855804	+	0.517300i	$0.173063\pi$
$463$			22.2619i			1.03460i	−0.855804	+	0.517300i	$0.826937\pi$
$464$		0			0
$465$	1.20101			0.0556955
$466$		0			0
$467$	−36.0711			−1.66917			−0.834585	−	0.550879i	$-0.814292\pi$
$467$	−36.0711			−1.66917			−0.834585	+	0.550879i	$0.814292\pi$
$468$		0			0
$469$	19.9778			0.922489
$470$		0			0
$471$			0.188028i			0.00866386i
$472$		0			0
$473$	14.6274	−	15.5286i	0.672569	−	0.714005i
$474$		0			0
$475$		−	4.54822i		−	0.208687i
$476$		0			0
$477$		−	21.1660i		−	0.969125i
$478$		0			0
$479$	27.3576			1.25000			0.625000	−	0.780625i	$-0.285099\pi$
$479$	27.3576			1.25000			0.625000	+	0.780625i	$0.285099\pi$
$480$		0			0
$481$		−	6.59377i		−	0.300650i
$482$		0			0
$483$			8.70626i			0.396149i
$484$		0			0
$485$		−	19.8042i		−	0.899262i
$486$		0			0
$487$		−	26.6455i		−	1.20742i	−0.797202	−	0.603712i	$-0.793688\pi$
$487$		−	26.6455i		−	1.20742i	0.797202	−	0.603712i	$-0.206312\pi$
$488$		0			0
$489$	−5.31371			−0.240294
$490$		0			0
$491$			25.7286i			1.16112i	0.814219	+	0.580558i	$0.197166\pi$
$491$			25.7286i			1.16112i	−0.814219	+	0.580558i	$0.802834\pi$
$492$		0			0
$493$		−	36.1326i		−	1.62733i
$494$		0			0
$495$	17.0179	−	18.0663i	0.764897	−	0.812021i
$496$		0			0
$497$			21.0188i			0.942821i
$498$		0			0
$499$	−18.4853			−0.827515			−0.413757	−	0.910387i	$-0.635784\pi$
$499$	−18.4853			−0.827515			−0.413757	+	0.910387i	$0.635784\pi$
$500$		0			0
$501$	−9.07353			−0.405375
$502$		0			0
$503$	3.85525			0.171897			0.0859484	−	0.996300i	$-0.472608\pi$
$503$	3.85525			0.171897			0.0859484	+	0.996300i	$0.472608\pi$
$504$		0			0
$505$		−	45.6438i		−	2.03112i
$506$		0			0
$507$	4.61522			0.204969
$508$		0			0
$509$		−	35.6787i		−	1.58143i	−0.612184	−	0.790715i	$-0.709709\pi$
$509$		−	35.6787i		−	1.58143i	0.612184	−	0.790715i	$-0.290291\pi$
$510$		0			0
$511$		−	3.09969i		−	0.137122i
$512$		0			0
$513$			5.49019i			0.242398i
$514$		0			0
$515$	47.7990			2.10628
$516$		0			0
$517$		0			0
$518$		0			0
$519$	2.25835			0.0991306
$520$		0			0
$521$	−20.1127			−0.881153			−0.440577	−	0.897715i	$-0.645226\pi$
$521$	−20.1127			−0.881153			−0.440577	+	0.897715i	$0.645226\pi$
$522$		0			0
$523$			17.0223i			0.744335i	0.928166	+	0.372168i	$0.121385\pi$
$523$			17.0223i			0.744335i	−0.928166	+	0.372168i	$0.878615\pi$
$524$		0			0
$525$	2.72607			0.118975
$526$		0			0
$527$	−6.01673			−0.262093
$528$		0			0
$529$	−17.7990			−0.773869
$530$		0			0
$531$	9.17157			0.398012
$532$		0			0
$533$			13.6827i			0.592663i
$534$		0			0
$535$	23.0346			0.995873
$536$		0			0
$537$	4.02944			0.173883
$538$		0			0
$539$	−9.24264	−	8.70626i	−0.398109	−	0.375005i
$540$		0			0
$541$	−17.2517			−0.741710			−0.370855	−	0.928691i	$-0.620935\pi$
$541$	−17.2517			−0.741710			−0.370855	+	0.928691i	$0.620935\pi$
$542$		0			0
$543$		−	1.09591i		−	0.0470298i
$544$		0			0
$545$			7.21250i			0.308950i
$546$		0			0
$547$		−	41.2572i		−	1.76403i	−0.471221	−	0.882015i	$-0.656187\pi$
$547$		−	41.2572i		−	1.76403i	0.471221	−	0.882015i	$-0.343813\pi$
$548$		0			0
$549$	18.6148			0.794459
$550$		0			0
$551$			14.9666i			0.637600i
$552$		0			0
$553$	30.6274			1.30241
$554$		0			0
$555$	5.30152			0.225037
$556$		0			0
$557$	−23.8331			−1.00984			−0.504919	−	0.863167i	$-0.668478\pi$
$557$	−23.8331			−1.00984			−0.504919	+	0.863167i	$0.668478\pi$
$558$		0			0
$559$			8.76725i			0.370815i
$560$		0			0
$561$	5.17157	−	5.49019i	0.218344	−	0.231796i
$562$		0			0
$563$			11.7607i			0.495655i	0.968804	+	0.247828i	$0.0797167\pi$
$563$			11.7607i			0.495655i	−0.968804	+	0.247828i	$0.920283\pi$
$564$		0			0
$565$		−	16.3284i		−	0.686943i
$566$		0			0
$567$	24.6315			1.03443
$568$		0			0
$569$			2.66428i			0.111693i	0.998439	+	0.0558463i	$0.0177857\pi$
$569$			2.66428i			0.111693i	−0.998439	+	0.0558463i	$0.982214\pi$
$570$		0			0
$571$			21.9607i			0.919028i	0.888170	+	0.459514i	$0.151976\pi$
$571$			21.9607i			0.919028i	−0.888170	+	0.459514i	$0.848024\pi$
$572$		0			0
$573$			7.02938i			0.293656i
$574$		0			0
$575$			12.7748i			0.532747i
$576$		0			0
$577$	−31.6274			−1.31667			−0.658333	−	0.752727i	$-0.728738\pi$
$577$	−31.6274			−1.31667			−0.658333	+	0.752727i	$0.728738\pi$
$578$		0			0
$579$		−	5.32857i		−	0.221448i
$580$		0			0
$581$			14.9666i			0.620920i
$582$		0			0
$583$	17.0179	−	18.0663i	0.704808	−	0.748231i
$584$		0			0
$585$			10.2000i			0.421719i
$586$		0			0
$587$	30.2843			1.24997			0.624983	−	0.780639i	$-0.285106\pi$
$587$	30.2843			1.24997			0.624983	+	0.780639i	$0.285106\pi$
$588$		0			0
$589$	2.49221			0.102690
$590$		0			0
$591$	4.32296			0.177823
$592$		0			0
$593$			1.88393i			0.0773639i	0.999252	+	0.0386819i	$0.0123159\pi$
$593$			1.88393i			0.0773639i	−0.999252	+	0.0386819i	$0.987684\pi$
$594$		0			0
$595$	−47.7990			−1.95957
$596$		0			0
$597$			8.76725i			0.358820i
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$		−	32.7795i		−	1.33710i	−0.743666	−	0.668552i	$-0.766915\pi$
$601$		−	32.7795i		−	1.33710i	0.743666	−	0.668552i	$-0.233085\pi$
$602$		0			0
$603$	−17.1716			−0.699281
$604$		0			0
$605$	29.0513	−	1.73787i	1.18110	−	0.0706546i
$606$		0			0
$607$	−29.0513			−1.17916			−0.589579	−	0.807711i	$-0.700706\pi$
$607$	−29.0513			−1.17916			−0.589579	+	0.807711i	$0.700706\pi$
$608$		0			0
$609$	−8.97056			−0.363506
$610$		0			0
$611$		0			0
$612$		0			0
$613$	26.7930			1.08216			0.541079	−	0.840972i	$-0.318016\pi$
$613$	26.7930			1.08216			0.541079	+	0.840972i	$0.318016\pi$
$614$		0			0
$615$	−11.0011			−0.443609
$616$		0			0
$617$	33.1127			1.33307			0.666534	−	0.745475i	$-0.267777\pi$
$617$	33.1127			1.33307			0.666534	+	0.745475i	$0.267777\pi$
$618$		0			0
$619$	−31.5269			−1.26717			−0.633587	−	0.773672i	$-0.718418\pi$
$619$	−31.5269			−1.26717			−0.633587	+	0.773672i	$0.718418\pi$
$620$		0			0
$621$		−	15.4206i		−	0.618806i
$622$		0			0
$623$	−35.0681			−1.40497
$624$		0			0
$625$	−31.0000			−1.24000
$626$		0			0
$627$	−2.14214	+	2.27411i	−0.0855487	+	0.0908192i
$628$		0			0
$629$	−26.5591			−1.05898
$630$		0			0
$631$			11.6789i			0.464930i	0.972605	+	0.232465i	$0.0746791\pi$
$631$			11.6789i			0.464930i	−0.972605	+	0.232465i	$0.925321\pi$
$632$		0			0
$633$		−	3.76787i		−	0.149759i
$634$		0			0
$635$		−	1.49376i		−	0.0592780i
$636$		0			0
$637$	5.21828			0.206756
$638$		0			0
$639$		−	18.0663i		−	0.714693i
$640$		0			0
$641$	−17.8284			−0.704181			−0.352090	−	0.935966i	$-0.614529\pi$
$641$	−17.8284			−0.704181			−0.352090	+	0.935966i	$0.614529\pi$
$642$		0			0
$643$	−0.615224			−0.0242621			−0.0121310	−	0.999926i	$-0.503862\pi$
$643$	−0.615224			−0.0242621			−0.0121310	+	0.999926i	$0.503862\pi$
$644$		0			0
$645$	−7.04903			−0.277555
$646$		0			0
$647$		−	23.5459i		−	0.925683i	−0.886441	−	0.462841i	$-0.846830\pi$
$647$		−	23.5459i		−	0.925683i	0.886441	−	0.462841i	$-0.153170\pi$
$648$		0			0
$649$	7.82843	+	7.37412i	0.307293	+	0.289459i
$650$		0			0
$651$			1.49376i			0.0585450i
$652$		0			0
$653$			36.5866i			1.43174i	0.698232	+	0.715872i	$0.253970\pi$
$653$			36.5866i			1.43174i	−0.698232	+	0.715872i	$0.746030\pi$
$654$		0			0
$655$	−18.0502			−0.705279
$656$		0			0
$657$			2.66428i			0.103944i
$658$		0			0
$659$		−	10.9804i		−	0.427735i	−0.976863	−	0.213867i	$-0.931394\pi$
$659$		−	10.9804i		−	0.427735i	0.976863	−	0.213867i	$-0.0686060\pi$
$660$		0			0
$661$			12.3209i			0.479227i	0.970868	+	0.239613i	$0.0770207\pi$
$661$			12.3209i			0.479227i	−0.970868	+	0.239613i	$0.922979\pi$
$662$		0			0
$663$			3.09969i			0.120382i
$664$		0			0
$665$	19.7990			0.767772
$666$		0			0
$667$		−	42.0375i		−	1.62770i
$668$		0			0
$669$		−	1.73787i		−	0.0671901i
$670$		0			0
$671$	15.8887	+	14.9666i	0.613376	+	0.577780i
$672$		0			0
$673$			38.2697i			1.47519i	0.675245	+	0.737594i	$0.264038\pi$
$673$			38.2697i			1.47519i	−0.675245	+	0.737594i	$0.735962\pi$
$674$		0			0
$675$	−4.82843			−0.185846
$676$		0			0
$677$	2.49221			0.0957834			0.0478917	−	0.998853i	$-0.484750\pi$
$677$	2.49221			0.0957834			0.0478917	+	0.998853i	$0.484750\pi$
$678$		0			0
$679$	24.6315			0.945270
$680$		0			0
$681$			5.49019i			0.210384i
$682$		0			0
$683$	36.4264			1.39382			0.696909	−	0.717160i	$-0.254558\pi$
$683$	36.4264			1.39382			0.696909	+	0.717160i	$0.254558\pi$
$684$		0			0
$685$			22.9039i			0.875112i
$686$		0			0
$687$			4.19560i			0.160072i
$688$		0			0
$689$			10.2000i			0.388590i
$690$		0			0
$691$	8.75736			0.333146			0.166573	−	0.986029i	$-0.446730\pi$
$691$	8.75736			0.333146			0.166573	+	0.986029i	$0.446730\pi$
$692$		0			0
$693$	22.4700	+	21.1660i	0.853565	+	0.804030i
$694$		0			0
$695$	−52.0859			−1.97573
$696$		0			0
$697$	55.1127			2.08754
$698$		0			0
$699$			3.44464i			0.130288i
$700$		0			0
$701$	−26.0914			−0.985459			−0.492729	−	0.870183i	$-0.664001\pi$
$701$	−26.0914			−0.985459			−0.492729	+	0.870183i	$0.664001\pi$
$702$		0			0
$703$	11.0011			0.414916
$704$		0			0
$705$		0			0
$706$		0			0
$707$	56.7696			2.13504
$708$		0			0
$709$		−	46.2617i		−	1.73739i	−0.495343	−	0.868697i	$-0.664958\pi$
$709$		−	46.2617i		−	1.73739i	0.495343	−	0.868697i	$-0.335042\pi$
$710$		0			0
$711$	−26.3253			−0.987274
$712$		0			0
$713$	−7.00000			−0.262152
$714$		0			0
$715$	−8.20101	+	8.70626i	−0.306700	+	0.325596i
$716$		0			0
$717$	2.25835			0.0843396
$718$		0			0
$719$			5.10347i			0.190327i	0.995462	+	0.0951637i	$0.0303375\pi$
$719$			5.10347i			0.190327i	−0.995462	+	0.0951637i	$0.969663\pi$
$720$		0			0
$721$			59.4501i			2.21404i
$722$		0			0
$723$		−	1.88393i		−	0.0700643i
$724$		0			0
$725$	−13.1626			−0.488848
$726$		0			0
$727$		−	8.57922i		−	0.318186i	−0.987264	−	0.159093i	$-0.949143\pi$
$727$		−	8.57922i		−	0.318186i	0.987264	−	0.159093i	$-0.0508569\pi$
$728$		0			0
$729$	−18.1716			−0.673021
$730$		0			0
$731$	35.3137			1.30612
$732$		0			0
$733$	−27.4544			−1.01405			−0.507026	−	0.861931i	$-0.669255\pi$
$733$	−27.4544			−1.01405			−0.507026	+	0.861931i	$0.669255\pi$
$734$		0			0
$735$			4.19560i			0.154757i
$736$		0			0
$737$	−14.6569	−	13.8063i	−0.539892	−	0.508561i
$738$		0			0
$739$			47.6894i			1.75428i	0.480234	+	0.877141i	$0.340552\pi$
$739$			47.6894i			1.75428i	−0.480234	+	0.877141i	$0.659448\pi$
$740$		0			0
$741$		−	1.28393i		−	0.0471665i
$742$		0			0
$743$	−25.1961			−0.924355			−0.462177	−	0.886788i	$-0.652932\pi$
$743$	−25.1961			−0.924355			−0.462177	+	0.886788i	$0.652932\pi$
$744$		0			0
$745$		−	55.8438i		−	2.04596i
$746$		0			0
$747$		−	12.8643i		−	0.470680i
$748$		0			0
$749$			28.6493i			1.04682i
$750$		0			0
$751$			7.29529i			0.266209i	0.991102	+	0.133104i	$0.0424946\pi$
$751$			7.29529i			0.266209i	−0.991102	+	0.133104i	$0.957505\pi$
$752$		0			0
$753$	7.62742			0.277959
$754$		0			0
$755$			34.8250i			1.26741i
$756$		0			0
$757$			28.6493i			1.04128i	0.853777	+	0.520639i	$0.174306\pi$
$757$			28.6493i			1.04128i	−0.853777	+	0.520639i	$0.825694\pi$
$758$		0			0
$759$	6.01673	−	6.38741i	0.218393	−	0.231848i
$760$		0			0
$761$			7.37412i			0.267312i	0.991028	+	0.133656i	$0.0426717\pi$
$761$			7.37412i			0.267312i	−0.991028	+	0.133656i	$0.957328\pi$
$762$		0			0
$763$	−8.97056			−0.324756
$764$		0			0
$765$	41.0848			1.48542
$766$		0			0
$767$	−4.41983			−0.159591
$768$		0			0
$769$		−	2.82590i		−	0.101905i	−0.998701	−	0.0509523i	$-0.983774\pi$
$769$		−	2.82590i		−	0.101905i	0.998701	−	0.0509523i	$-0.0162256\pi$
$770$		0			0
$771$	−2.68629			−0.0967444
$772$		0			0
$773$		−	8.76725i		−	0.315336i	−0.987492	−	0.157668i	$-0.949602\pi$
$773$		−	8.76725i		−	0.315336i	0.987492	−	0.157668i	$-0.0503976\pi$
$774$		0			0
$775$			2.19181i			0.0787322i
$776$		0			0
$777$			6.59377i			0.236550i
$778$		0			0
$779$	−22.8284			−0.817913
$780$		0			0
$781$	14.5257	−	15.4206i	0.519769	−	0.551791i
$782$		0			0
$783$	15.8887			0.567816
$784$		0			0
$785$	−1.20101			−0.0428659
$786$		0			0
$787$			32.9411i			1.17422i	0.809506	+	0.587112i	$0.199735\pi$
$787$			32.9411i			1.17422i	−0.809506	+	0.587112i	$0.800265\pi$
$788$		0			0
$789$	7.94435			0.282826
$790$		0			0
$791$	20.3085			0.722088
$792$		0			0
$793$	−8.97056			−0.318554
$794$		0			0
$795$	−8.20101			−0.290860
$796$		0			0
$797$			44.0699i			1.56104i	0.625133	+	0.780518i	$0.285045\pi$
$797$			44.0699i			1.56104i	−0.625133	+	0.780518i	$0.714955\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	30.1421			1.06502
$802$		0			0
$803$	−2.14214	+	2.27411i	−0.0755943	+	0.0802516i
$804$		0			0
$805$	−55.6105			−1.96001
$806$		0			0
$807$			5.66756i			0.199507i
$808$		0			0
$809$		−	0.941967i		−	0.0331178i	−0.999863	−	0.0165589i	$-0.994729\pi$
$809$		−	0.941967i		−	0.0331178i	0.999863	−	0.0165589i	$-0.00527110\pi$
$810$		0			0
$811$		−	5.65180i		−	0.198462i	−0.995064	−	0.0992308i	$-0.968362\pi$
$811$		−	5.65180i		−	0.198462i	0.995064	−	0.0992308i	$-0.0316382\pi$
$812$		0			0
$813$	−5.21828			−0.183013
$814$		0			0
$815$		−	33.9408i		−	1.18890i
$816$		0			0
$817$	−14.6274			−0.511749
$818$		0			0
$819$	−12.6863			−0.443295
$820$		0			0
$821$	−21.3408			−0.744800			−0.372400	−	0.928072i	$-0.621465\pi$
$821$	−21.3408			−0.744800			−0.372400	+	0.928072i	$0.621465\pi$
$822$		0			0
$823$			35.9446i			1.25295i	0.779441	+	0.626475i	$0.215503\pi$
$823$			35.9446i			1.25295i	−0.779441	+	0.626475i	$0.784497\pi$
$824$		0			0
$825$	−2.00000	−	1.88393i	−0.0696311	−	0.0655902i
$826$		0			0
$827$			50.7438i			1.76454i	0.470748	+	0.882268i	$0.343984\pi$
$827$			50.7438i			1.76454i	−0.470748	+	0.882268i	$0.656016\pi$
$828$		0			0
$829$		−	41.8781i		−	1.45449i	−0.686380	−	0.727243i	$-0.740801\pi$
$829$		−	41.8781i		−	1.45449i	0.686380	−	0.727243i	$-0.259199\pi$
$830$		0			0
$831$	−1.59689			−0.0553957
$832$		0			0
$833$		−	21.0188i		−	0.728257i
$834$		0			0
$835$		−	57.9563i		−	2.00566i
$836$		0			0
$837$		−	2.64575i		−	0.0914505i
$838$		0			0
$839$			54.3870i			1.87765i	0.344398	+	0.938824i	$0.388083\pi$
$839$			54.3870i			1.87765i	−0.344398	+	0.938824i	$0.611917\pi$
$840$		0			0
$841$	14.3137			0.493576
$842$		0			0
$843$		−	2.66428i		−	0.0917628i
$844$		0			0
$845$			29.4793i			1.01412i
$846$		0			0
$847$	2.16148	+	36.1326i	0.0742694	+	1.24153i
$848$		0			0
$849$		−	0.780351i		−	0.0267816i
$850$		0			0
$851$	−30.8995			−1.05922
$852$		0			0
$853$	−58.1027			−1.98940			−0.994699	−	0.102833i	$-0.967209\pi$
$853$	−58.1027			−1.98940			−0.994699	+	0.102833i	$0.967209\pi$
$854$		0			0
$855$	−17.0179			−0.581999
$856$		0			0
$857$			21.9607i			0.750165i	0.926992	+	0.375082i	$0.122386\pi$
$857$			21.9607i			0.750165i	−0.926992	+	0.375082i	$0.877614\pi$
$858$		0			0
$859$	49.2426			1.68014			0.840069	−	0.542480i	$-0.182515\pi$
$859$	49.2426			1.68014			0.840069	+	0.542480i	$0.182515\pi$
$860$		0			0
$861$		−	13.6827i		−	0.466305i
$862$		0			0
$863$			14.9666i			0.509470i	0.967011	+	0.254735i	$0.0819882\pi$
$863$			14.9666i			0.509470i	−0.967011	+	0.254735i	$0.918012\pi$
$864$		0			0
$865$			14.4250i			0.490465i
$866$		0			0
$867$	5.44365			0.184876
$868$		0			0
$869$	−22.4700	−	21.1660i	−0.762243	−	0.718008i
$870$		0			0
$871$	8.27508			0.280390
$872$		0			0
$873$	−21.1716			−0.716549
$874$		0			0
$875$		−	26.1188i		−	0.882976i
$876$		0			0
$877$	29.2852			0.988891			0.494445	−	0.869209i	$-0.335371\pi$
$877$	29.2852			0.988891			0.494445	+	0.869209i	$0.335371\pi$
$878$		0			0
$879$	6.01673			0.202939
$880$		0			0
$881$	6.51472			0.219486			0.109743	−	0.993960i	$-0.464997\pi$
$881$	6.51472			0.219486			0.109743	+	0.993960i	$0.464997\pi$
$882$		0			0
$883$	4.34315			0.146158			0.0730792	−	0.997326i	$-0.476717\pi$
$883$	4.34315			0.146158			0.0730792	+	0.997326i	$0.476717\pi$
$884$		0			0
$885$		−	3.55363i		−	0.119454i
$886$		0			0
$887$	−46.5369			−1.56256			−0.781279	−	0.624183i	$-0.785432\pi$
$887$	−46.5369			−1.56256			−0.781279	+	0.624183i	$0.785432\pi$
$888$		0			0
$889$	1.85786			0.0623108
$890$		0			0
$891$	−18.0711	−	17.0223i	−0.605404	−	0.570270i
$892$		0			0
$893$		0			0
$894$		0			0
$895$			25.7377i			0.860315i
$896$		0			0
$897$			3.60625i			0.120409i
$898$		0			0
$899$		−	7.21250i		−	0.240550i
$900$		0			0
$901$	41.0848			1.36873
$902$		0			0
$903$		−	8.76725i		−	0.291756i
$904$		0			0
$905$	7.00000			0.232688
$906$		0			0
$907$	33.5147			1.11284			0.556419	−	0.830902i	$-0.312175\pi$
$907$	33.5147			1.11284			0.556419	+	0.830902i	$0.312175\pi$
$908$		0			0
$909$	−48.7953			−1.61844
$910$		0			0
$911$		−	44.8999i		−	1.48760i	−0.668402	−	0.743800i	$-0.733022\pi$
$911$		−	44.8999i		−	1.48760i	0.668402	−	0.743800i	$-0.266978\pi$
$912$		0			0
$913$	10.3431	−	10.9804i	0.342308	−	0.363397i
$914$		0			0
$915$		−	7.21250i		−	0.238438i
$916$		0			0
$917$		−	22.4499i		−	0.741362i
$918$		0			0
$919$	23.5992			0.778465			0.389233	−	0.921139i	$-0.372740\pi$
$919$	23.5992			0.778465			0.389233	+	0.921139i	$0.372740\pi$
$920$		0			0
$921$		−	9.25805i		−	0.305063i
$922$		0			0
$923$			8.70626i			0.286570i
$924$		0			0
$925$			9.67513i			0.318116i
$926$		0			0
$927$		−	51.0993i		−	1.67832i
$928$		0			0
$929$	34.4853			1.13143			0.565713	−	0.824602i	$-0.308601\pi$
$929$	34.4853			1.13143			0.565713	+	0.824602i	$0.308601\pi$
$930$		0			0
$931$			8.70626i			0.285336i
$932$		0			0
$933$		−	6.19938i		−	0.202959i
$934$		0			0
$935$	35.0681	+	33.0329i	1.14685	+	1.08029i
$936$		0			0
$937$		−	21.7991i		−	0.712146i	−0.934458	−	0.356073i	$-0.884115\pi$
$937$		−	21.7991i		−	0.712146i	0.934458	−	0.356073i	$-0.115885\pi$
$938$		0			0
$939$	−3.24264			−0.105820
$940$		0			0
$941$	14.7595			0.481147			0.240573	−	0.970631i	$-0.422665\pi$
$941$	14.7595			0.481147			0.240573	+	0.970631i	$0.422665\pi$
$942$		0			0
$943$	64.1194			2.08801
$944$		0			0
$945$		−	21.0188i		−	0.683741i
$946$		0			0
$947$	−24.4142			−0.793355			−0.396678	−	0.917958i	$-0.629837\pi$
$947$	−24.4142			−0.793355			−0.396678	+	0.917958i	$0.629837\pi$
$948$		0			0
$949$		−	1.28393i		−	0.0416783i
$950$		0			0
$951$		−	6.38741i		−	0.207126i
$952$		0			0
$953$			36.5474i			1.18389i	0.805980	+	0.591943i	$0.201639\pi$
$953$			36.5474i			1.18389i	−0.805980	+	0.591943i	$0.798361\pi$
$954$		0			0
$955$	−44.8995			−1.45291
$956$		0			0
$957$	6.58132	+	6.19938i	0.212744	+	0.200398i
$958$		0			0
$959$	−28.4867			−0.919885
$960$		0			0
$961$	29.7990			0.961258
$962$		0			0
$963$		−	24.6250i		−	0.793530i
$964$		0			0
$965$	34.0358			1.09565
$966$		0			0
$967$	−3.85525			−0.123976			−0.0619882	−	0.998077i	$-0.519744\pi$
$967$	−3.85525			−0.123976			−0.0619882	+	0.998077i	$0.519744\pi$
$968$		0			0
$969$	−5.17157			−0.166135
$970$		0			0
$971$	−32.8406			−1.05391			−0.526953	−	0.849895i	$-0.676666\pi$
$971$	−32.8406			−1.05391			−0.526953	+	0.849895i	$0.676666\pi$
$972$		0			0
$973$		−	64.7820i		−	2.07681i
$974$		0			0
$975$	1.12918			0.0361625
$976$		0			0
$977$	−45.6274			−1.45975			−0.729875	−	0.683580i	$-0.760422\pi$
$977$	−45.6274			−1.45975			−0.729875	+	0.683580i	$0.760422\pi$
$978$		0			0
$979$	25.7279	+	24.2349i	0.822268	+	0.774549i
$980$		0			0
$981$	7.71049			0.246177
$982$		0			0
$983$		−	17.8783i		−	0.570229i	−0.958493	−	0.285114i	$-0.907968\pi$
$983$		−	17.8783i		−	0.570229i	0.958493	−	0.285114i	$-0.0920316\pi$
$984$		0			0
$985$			27.6125i			0.879809i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	41.0848			1.30642
$990$		0			0
$991$		−	29.9333i		−	0.950861i	−0.879753	−	0.475431i	$-0.842292\pi$
$991$		−	29.9333i		−	0.950861i	0.879753	−	0.475431i	$-0.157708\pi$
$992$		0			0
$993$	−7.48528			−0.237538
$994$		0			0
$995$	−56.0000			−1.77532
$996$		0			0
$997$	55.6105			1.76120			0.880600	−	0.473860i	$-0.157140\pi$
$997$	55.6105			1.76120			0.880600	+	0.473860i	$0.157140\pi$
$998$		0			0
$999$		−	11.6789i		−	0.369504i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	352.2.g.b.175.2		8
3.2	odd	2		3168.2.h.g.2287.8		8
4.3	odd	2		88.2.g.b.43.8	yes	8
8.3	odd	2	inner	352.2.g.b.175.3		8
8.5	even	2		88.2.g.b.43.2	yes	8
11.10	odd	2	inner	352.2.g.b.175.1		8
12.11	even	2		792.2.h.g.307.1		8
16.3	odd	4		2816.2.e.o.2815.10		16
16.5	even	4		2816.2.e.o.2815.11		16
16.11	odd	4		2816.2.e.o.2815.8		16
16.13	even	4		2816.2.e.o.2815.5		16
24.5	odd	2		792.2.h.g.307.7		8
24.11	even	2		3168.2.h.g.2287.1		8
33.32	even	2		3168.2.h.g.2287.5		8
44.3	odd	10		968.2.k.g.475.7		32
44.7	even	10		968.2.k.g.699.5		32
44.15	odd	10		968.2.k.g.699.4		32
44.19	even	10		968.2.k.g.475.2		32
44.27	odd	10		968.2.k.g.723.1		32
44.31	odd	10		968.2.k.g.403.3		32
44.35	even	10		968.2.k.g.403.6		32
44.39	even	10		968.2.k.g.723.8		32
44.43	even	2		88.2.g.b.43.1	✓	8
88.5	even	10		968.2.k.g.723.6		32
88.13	odd	10		968.2.k.g.403.1		32
88.21	odd	2		88.2.g.b.43.7	yes	8
88.29	odd	10		968.2.k.g.699.7		32
88.37	even	10		968.2.k.g.699.2		32
88.43	even	2	inner	352.2.g.b.175.4		8
88.53	even	10		968.2.k.g.403.8		32
88.61	odd	10		968.2.k.g.723.3		32
88.69	even	10		968.2.k.g.475.5		32
88.85	odd	10		968.2.k.g.475.4		32
132.131	odd	2		792.2.h.g.307.8		8
176.21	odd	4		2816.2.e.o.2815.12		16
176.43	even	4		2816.2.e.o.2815.7		16
176.109	odd	4		2816.2.e.o.2815.6		16
176.131	even	4		2816.2.e.o.2815.9		16
264.131	odd	2		3168.2.h.g.2287.4		8
264.197	even	2		792.2.h.g.307.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.g.b.43.1	✓	8	44.43	even	2
88.2.g.b.43.2	yes	8	8.5	even	2
88.2.g.b.43.7	yes	8	88.21	odd	2
88.2.g.b.43.8	yes	8	4.3	odd	2
352.2.g.b.175.1		8	11.10	odd	2	inner
352.2.g.b.175.2		8	1.1	even	1	trivial
352.2.g.b.175.3		8	8.3	odd	2	inner
352.2.g.b.175.4		8	88.43	even	2	inner
792.2.h.g.307.1		8	12.11	even	2
792.2.h.g.307.2		8	264.197	even	2
792.2.h.g.307.7		8	24.5	odd	2
792.2.h.g.307.8		8	132.131	odd	2
968.2.k.g.403.1		32	88.13	odd	10
968.2.k.g.403.3		32	44.31	odd	10
968.2.k.g.403.6		32	44.35	even	10
968.2.k.g.403.8		32	88.53	even	10
968.2.k.g.475.2		32	44.19	even	10
968.2.k.g.475.4		32	88.85	odd	10
968.2.k.g.475.5		32	88.69	even	10
968.2.k.g.475.7		32	44.3	odd	10
968.2.k.g.699.2		32	88.37	even	10
968.2.k.g.699.4		32	44.15	odd	10
968.2.k.g.699.5		32	44.7	even	10
968.2.k.g.699.7		32	88.29	odd	10
968.2.k.g.723.1		32	44.27	odd	10
968.2.k.g.723.3		32	88.61	odd	10
968.2.k.g.723.6		32	88.5	even	10
968.2.k.g.723.8		32	44.39	even	10
2816.2.e.o.2815.5		16	16.13	even	4
2816.2.e.o.2815.6		16	176.109	odd	4
2816.2.e.o.2815.7		16	176.43	even	4
2816.2.e.o.2815.8		16	16.11	odd	4
2816.2.e.o.2815.9		16	176.131	even	4
2816.2.e.o.2815.10		16	16.3	odd	4
2816.2.e.o.2815.11		16	16.5	even	4
2816.2.e.o.2815.12		16	176.21	odd	4
3168.2.h.g.2287.1		8	24.11	even	2
3168.2.h.g.2287.4		8	264.131	odd	2
3168.2.h.g.2287.5		8	33.32	even	2
3168.2.h.g.2287.8		8	3.2	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:	\( N \)	\(=\)	\( 352 = 2^{5} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	352.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.414214 q^{3} -2.64575i q^{5} +3.29066 q^{7} -2.82843 q^{9} +(-2.41421 - 2.27411i) q^{11} +1.36303 q^{13} +1.09591i q^{15} -5.49019i q^{17} +2.27411i q^{19} -1.36303 q^{21} -6.38741i q^{23} -2.00000 q^{25} +2.41421 q^{27} +6.58132 q^{29} -1.09591i q^{31} +(1.00000 + 0.941967i) q^{33} -8.70626i q^{35} -4.83756i q^{37} -0.564588 q^{39} +10.0384i q^{41} +6.43215i q^{43} +7.48331i q^{45} +3.82843 q^{49} +2.27411i q^{51} +7.48331i q^{53} +(-6.01673 + 6.38741i) q^{55} -0.941967i q^{57} -3.24264 q^{59} -6.58132 q^{61} -9.30739 q^{63} -3.60625i q^{65} +6.07107 q^{67} +2.64575i q^{69} +6.38741i q^{71} -0.941967i q^{73} +0.828427 q^{75} +(-7.94435 - 7.48331i) q^{77} +9.30739 q^{79} +7.48528 q^{81} +4.54822i q^{83} -14.5257 q^{85} -2.72607 q^{87} -10.6569 q^{89} +4.48528 q^{91} +0.453939i q^{93} +6.01673 q^{95} +7.48528 q^{97} +(6.82843 + 6.43215i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} - 8 q^{11} - 16 q^{25} + 8 q^{27} + 8 q^{33} + 8 q^{49} + 8 q^{59} - 8 q^{67} - 16 q^{75} - 8 q^{81} - 40 q^{89} - 32 q^{91} - 8 q^{97} + 32 q^{99}+O(q^{100}) \) 8 * q + 8 * q^3 - 8 * q^11 - 16 * q^25 + 8 * q^27 + 8 * q^33 + 8 * q^49 + 8 * q^59 - 8 * q^67 - 16 * q^75 - 8 * q^81 - 40 * q^89 - 32 * q^91 - 8 * q^97 + 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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