LMFDB - Embedded newform 416.2.f.e.129.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [416,2,Mod(129,416)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("416.129"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(416, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$N$	$=$	$416 = 2^{5} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	416.f (of order $2$ , degree $1$ , minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$3.32177672409$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 3x^{2} + 1$ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2^{2}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.4
Root			$-0.618034i$ of defining polynomial
Character	$\chi$	$=$	416.129
Dual form			416.2.f.e.129.3

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607 q^{3} +1.00000i q^{5} +2.23607i q^{7} +2.00000 q^{9} +(2.00000 + 3.00000i) q^{13} +2.23607i q^{15} +3.00000 q^{17} -4.47214i q^{19} +5.00000i q^{21} -4.47214 q^{23} +4.00000 q^{25} -2.23607 q^{27} -8.94427i q^{31} -2.23607 q^{35} +3.00000i q^{37} +(4.47214 + 6.70820i) q^{39} -8.00000i q^{41} -6.70820 q^{43} +2.00000i q^{45} -6.70820i q^{47} +2.00000 q^{49} +6.70820 q^{51} -6.00000 q^{53} -10.0000i q^{57} +13.4164i q^{59} +4.47214i q^{63} +(-3.00000 + 2.00000i) q^{65} +8.94427i q^{67} -10.0000 q^{69} -6.70820i q^{71} +6.00000i q^{73} +8.94427 q^{75} +13.4164 q^{79} -11.0000 q^{81} -13.4164i q^{83} +3.00000i q^{85} -14.0000i q^{89} +(-6.70820 + 4.47214i) q^{91} -20.0000i q^{93} +4.47214 q^{95} +12.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 8 q^{9} + 8 q^{13} + 12 q^{17} + 16 q^{25} + 8 q^{49} - 24 q^{53} - 12 q^{65} - 40 q^{69} - 44 q^{81}+O(q^{100})$ 4 * q + 8 * q^9 + 8 * q^13 + 12 * q^17 + 16 * q^25 + 8 * q^49 - 24 * q^53 - 12 * q^65 - 40 * q^69 - 44 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$ .

$n$	$261$	$287$	$353$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	2.23607			1.29099			0.645497	−	0.763763i	$-0.276650\pi$
$3$	2.23607			1.29099			0.645497	+	0.763763i	$0.276650\pi$
$4$		0			0
$5$			1.00000i			0.447214i	0.974679	+	0.223607i	$0.0717831\pi$
$5$			1.00000i			0.447214i	−0.974679	+	0.223607i	$0.928217\pi$
$6$		0			0
$7$			2.23607i			0.845154i	0.906327	+	0.422577i	$0.138874\pi$
$7$			2.23607i			0.845154i	−0.906327	+	0.422577i	$0.861126\pi$
$8$		0			0
$9$	2.00000			0.666667
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	2.00000	+	3.00000i	0.554700	+	0.832050i
$13$	2.00000	+	3.00000i	0.554700	+	0.832050i
$14$		0			0
$15$			2.23607i			0.577350i
$16$		0			0
$17$	3.00000			0.727607			0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000			0.727607			0.363803	+	0.931476i	$0.381478\pi$
$18$		0			0
$19$		−	4.47214i		−	1.02598i	−0.858395	−	0.512989i	$-0.828538\pi$
$19$		−	4.47214i		−	1.02598i	0.858395	−	0.512989i	$-0.171462\pi$
$20$		0			0
$21$			5.00000i			1.09109i
$22$		0			0
$23$	−4.47214			−0.932505			−0.466252	−	0.884652i	$-0.654396\pi$
$23$	−4.47214			−0.932505			−0.466252	+	0.884652i	$0.654396\pi$
$24$		0			0
$25$	4.00000			0.800000
$26$		0			0
$27$	−2.23607			−0.430331
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$		−	8.94427i		−	1.60644i	−0.595683	−	0.803219i	$-0.703119\pi$
$31$		−	8.94427i		−	1.60644i	0.595683	−	0.803219i	$-0.296881\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.23607			−0.377964
$36$		0			0
$37$			3.00000i			0.493197i	0.969118	+	0.246598i	$0.0793129\pi$
$37$			3.00000i			0.493197i	−0.969118	+	0.246598i	$0.920687\pi$
$38$		0			0
$39$	4.47214	+	6.70820i	0.716115	+	1.07417i
$40$		0			0
$41$		−	8.00000i		−	1.24939i	−0.780869	−	0.624695i	$-0.785223\pi$
$41$		−	8.00000i		−	1.24939i	0.780869	−	0.624695i	$-0.214777\pi$
$42$		0			0
$43$	−6.70820			−1.02299			−0.511496	−	0.859286i	$-0.670908\pi$
$43$	−6.70820			−1.02299			−0.511496	+	0.859286i	$0.670908\pi$
$44$		0			0
$45$			2.00000i			0.298142i
$46$		0			0
$47$		−	6.70820i		−	0.978492i	−0.872146	−	0.489246i	$-0.837272\pi$
$47$		−	6.70820i		−	0.978492i	0.872146	−	0.489246i	$-0.162728\pi$
$48$		0			0
$49$	2.00000			0.285714
$50$		0			0
$51$	6.70820			0.939336
$52$		0			0
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	10.0000i		−	1.32453i
$58$		0			0
$59$			13.4164i			1.74667i	0.487122	+	0.873334i	$0.338047\pi$
$59$			13.4164i			1.74667i	−0.487122	+	0.873334i	$0.661953\pi$
$60$		0			0
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$			4.47214i			0.563436i
$64$		0			0
$65$	−3.00000	+	2.00000i	−0.372104	+	0.248069i
$66$		0			0
$67$			8.94427i			1.09272i	0.837552	+	0.546358i	$0.183986\pi$
$67$			8.94427i			1.09272i	−0.837552	+	0.546358i	$0.816014\pi$
$68$		0			0
$69$	−10.0000			−1.20386
$70$		0			0
$71$		−	6.70820i		−	0.796117i	−0.917360	−	0.398059i	$-0.869684\pi$
$71$		−	6.70820i		−	0.796117i	0.917360	−	0.398059i	$-0.130316\pi$
$72$		0			0
$73$			6.00000i			0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$			6.00000i			0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$		0			0
$75$	8.94427			1.03280
$76$		0			0
$77$		0			0
$78$		0			0
$79$	13.4164			1.50946			0.754732	−	0.656033i	$-0.227767\pi$
$79$	13.4164			1.50946			0.754732	+	0.656033i	$0.227767\pi$
$80$		0			0
$81$	−11.0000			−1.22222
$82$		0			0
$83$		−	13.4164i		−	1.47264i	−0.676632	−	0.736321i	$-0.736561\pi$
$83$		−	13.4164i		−	1.47264i	0.676632	−	0.736321i	$-0.263439\pi$
$84$		0			0
$85$			3.00000i			0.325396i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	14.0000i		−	1.48400i	−0.670402	−	0.741999i	$-0.733878\pi$
$89$		−	14.0000i		−	1.48400i	0.670402	−	0.741999i	$-0.266122\pi$
$90$		0			0
$91$	−6.70820	+	4.47214i	−0.703211	+	0.468807i
$92$		0			0
$93$		−	20.0000i		−	2.07390i
$94$		0			0
$95$	4.47214			0.458831
$96$		0			0
$97$			12.0000i			1.21842i	0.793011	+	0.609208i	$0.208512\pi$
$97$			12.0000i			1.21842i	−0.793011	+	0.609208i	$0.791488\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−12.0000			−1.19404			−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000			−1.19404			−0.597022	+	0.802225i	$0.703650\pi$
$102$		0			0
$103$	−13.4164			−1.32196			−0.660979	−	0.750404i	$-0.729859\pi$
$103$	−13.4164			−1.32196			−0.660979	+	0.750404i	$0.729859\pi$
$104$		0			0
$105$	−5.00000			−0.487950
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$		−	15.0000i		−	1.43674i	−0.695662	−	0.718370i	$-0.744889\pi$
$109$		−	15.0000i		−	1.43674i	0.695662	−	0.718370i	$-0.255111\pi$
$110$		0			0
$111$			6.70820i			0.636715i
$112$		0			0
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$		−	4.47214i		−	0.417029i
$116$		0			0
$117$	4.00000	+	6.00000i	0.369800	+	0.554700i
$118$		0			0
$119$			6.70820i			0.614940i
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$		−	17.8885i		−	1.61296i
$124$		0			0
$125$			9.00000i			0.804984i
$126$		0			0
$127$	−13.4164			−1.19051			−0.595257	−	0.803535i	$-0.702950\pi$
$127$	−13.4164			−1.19051			−0.595257	+	0.803535i	$0.702950\pi$
$128$		0			0
$129$	−15.0000			−1.32068
$130$		0			0
$131$	−20.1246			−1.75830			−0.879148	−	0.476549i	$-0.841887\pi$
$131$	−20.1246			−1.75830			−0.879148	+	0.476549i	$0.841887\pi$
$132$		0			0
$133$	10.0000			0.867110
$134$		0			0
$135$		−	2.23607i		−	0.192450i
$136$		0			0
$137$			2.00000i			0.170872i	0.996344	+	0.0854358i	$0.0272282\pi$
$137$			2.00000i			0.170872i	−0.996344	+	0.0854358i	$0.972772\pi$
$138$		0			0
$139$	6.70820			0.568982			0.284491	−	0.958679i	$-0.408175\pi$
$139$	6.70820			0.568982			0.284491	+	0.958679i	$0.408175\pi$
$140$		0			0
$141$		−	15.0000i		−	1.26323i
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	4.47214			0.368856
$148$		0			0
$149$			10.0000i			0.819232i	0.912258	+	0.409616i	$0.134337\pi$
$149$			10.0000i			0.819232i	−0.912258	+	0.409616i	$0.865663\pi$
$150$		0			0
$151$		−	2.23607i		−	0.181969i	−0.995852	−	0.0909843i	$-0.970999\pi$
$151$		−	2.23607i		−	0.181969i	0.995852	−	0.0909843i	$-0.0290013\pi$
$152$		0			0
$153$	6.00000			0.485071
$154$		0			0
$155$	8.94427			0.718421
$156$		0			0
$157$	18.0000			1.43656			0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000			1.43656			0.718278	+	0.695756i	$0.244931\pi$
$158$		0			0
$159$	−13.4164			−1.06399
$160$		0			0
$161$		−	10.0000i		−	0.788110i
$162$		0			0
$163$			4.47214i			0.350285i	0.984543	+	0.175142i	$0.0560386\pi$
$163$			4.47214i			0.350285i	−0.984543	+	0.175142i	$0.943961\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−5.00000	+	12.0000i	−0.384615	+	0.923077i
$170$		0			0
$171$		−	8.94427i		−	0.683986i
$172$		0			0
$173$	−6.00000			−0.456172			−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000			−0.456172			−0.228086	+	0.973641i	$0.573247\pi$
$174$		0			0
$175$			8.94427i			0.676123i
$176$		0			0
$177$			30.0000i			2.25494i
$178$		0			0
$179$	20.1246			1.50418			0.752092	−	0.659058i	$-0.229045\pi$
$179$	20.1246			1.50418			0.752092	+	0.659058i	$0.229045\pi$
$180$		0			0
$181$	18.0000			1.33793			0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000			1.33793			0.668965	+	0.743294i	$0.266738\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.00000			−0.220564
$186$		0			0
$187$		0			0
$188$		0			0
$189$		−	5.00000i		−	0.363696i
$190$		0			0
$191$	8.94427			0.647185			0.323592	−	0.946197i	$-0.395109\pi$
$191$	8.94427			0.647185			0.323592	+	0.946197i	$0.395109\pi$
$192$		0			0
$193$			6.00000i			0.431889i	0.976406	+	0.215945i	$0.0692831\pi$
$193$			6.00000i			0.431889i	−0.976406	+	0.215945i	$0.930717\pi$
$194$		0			0
$195$	−6.70820	+	4.47214i	−0.480384	+	0.320256i
$196$		0			0
$197$		−	13.0000i		−	0.926212i	−0.886303	−	0.463106i	$-0.846735\pi$
$197$		−	13.0000i		−	0.926212i	0.886303	−	0.463106i	$-0.153265\pi$
$198$		0			0
$199$	26.8328			1.90213			0.951064	−	0.308994i	$-0.0999924\pi$
$199$	26.8328			1.90213			0.951064	+	0.308994i	$0.0999924\pi$
$200$		0			0
$201$			20.0000i			1.41069i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	8.00000			0.558744
$206$		0			0
$207$	−8.94427			−0.621670
$208$		0			0
$209$		0			0
$210$		0			0
$211$	6.70820			0.461812			0.230906	−	0.972976i	$-0.425831\pi$
$211$	6.70820			0.461812			0.230906	+	0.972976i	$0.425831\pi$
$212$		0			0
$213$		−	15.0000i		−	1.02778i
$214$		0			0
$215$		−	6.70820i		−	0.457496i
$216$		0			0
$217$	20.0000			1.35769
$218$		0			0
$219$			13.4164i			0.906597i
$220$		0			0
$221$	6.00000	+	9.00000i	0.403604	+	0.605406i
$222$		0			0
$223$			2.23607i			0.149738i	0.997193	+	0.0748691i	$0.0238539\pi$
$223$			2.23607i			0.149738i	−0.997193	+	0.0748691i	$0.976146\pi$
$224$		0			0
$225$	8.00000			0.533333
$226$		0			0
$227$			26.8328i			1.78096i	0.455026	+	0.890478i	$0.349630\pi$
$227$			26.8328i			1.78096i	−0.455026	+	0.890478i	$0.650370\pi$
$228$		0			0
$229$			21.0000i			1.38772i	0.720110	+	0.693860i	$0.244091\pi$
$229$			21.0000i			1.38772i	−0.720110	+	0.693860i	$0.755909\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−9.00000			−0.589610			−0.294805	−	0.955557i	$-0.595255\pi$
$233$	−9.00000			−0.589610			−0.294805	+	0.955557i	$0.595255\pi$
$234$		0			0
$235$	6.70820			0.437595
$236$		0			0
$237$	30.0000			1.94871
$238$		0			0
$239$			6.70820i			0.433918i	0.976181	+	0.216959i	$0.0696137\pi$
$239$			6.70820i			0.433918i	−0.976181	+	0.216959i	$0.930386\pi$
$240$		0			0
$241$			18.0000i			1.15948i	0.814801	+	0.579741i	$0.196846\pi$
$241$			18.0000i			1.15948i	−0.814801	+	0.579741i	$0.803154\pi$
$242$		0			0
$243$	−17.8885			−1.14755
$244$		0			0
$245$			2.00000i			0.127775i
$246$		0			0
$247$	13.4164	−	8.94427i	0.853666	−	0.569110i
$248$		0			0
$249$		−	30.0000i		−	1.90117i
$250$		0			0
$251$	−17.8885			−1.12911			−0.564557	−	0.825394i	$-0.690953\pi$
$251$	−17.8885			−1.12911			−0.564557	+	0.825394i	$0.690953\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$			6.70820i			0.420084i
$256$		0			0
$257$	27.0000			1.68421			0.842107	−	0.539311i	$-0.181315\pi$
$257$	27.0000			1.68421			0.842107	+	0.539311i	$0.181315\pi$
$258$		0			0
$259$	−6.70820			−0.416828
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		−	6.00000i		−	0.368577i
$266$		0			0
$267$		−	31.3050i		−	1.91583i
$268$		0			0
$269$	−24.0000			−1.46331			−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000			−1.46331			−0.731653	+	0.681677i	$0.761251\pi$
$270$		0			0
$271$		−	15.6525i		−	0.950820i	−0.879764	−	0.475410i	$-0.842300\pi$
$271$		−	15.6525i		−	0.950820i	0.879764	−	0.475410i	$-0.157700\pi$
$272$		0			0
$273$	−15.0000	+	10.0000i	−0.907841	+	0.605228i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−8.00000			−0.480673			−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000			−0.480673			−0.240337	+	0.970690i	$0.577258\pi$
$278$		0			0
$279$		−	17.8885i		−	1.07096i
$280$		0			0
$281$			2.00000i			0.119310i	0.998219	+	0.0596550i	$0.0190001\pi$
$281$			2.00000i			0.119310i	−0.998219	+	0.0596550i	$0.981000\pi$
$282$		0			0
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$		0			0
$285$	10.0000			0.592349
$286$		0			0
$287$	17.8885			1.05593
$288$		0			0
$289$	−8.00000			−0.470588
$290$		0			0
$291$			26.8328i			1.57297i
$292$		0			0
$293$		−	1.00000i		−	0.0584206i	−0.999573	−	0.0292103i	$-0.990701\pi$
$293$		−	1.00000i		−	0.0584206i	0.999573	−	0.0292103i	$-0.00929925\pi$
$294$		0			0
$295$	−13.4164			−0.781133
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−8.94427	−	13.4164i	−0.517261	−	0.775891i
$300$		0			0
$301$		−	15.0000i		−	0.864586i
$302$		0			0
$303$	−26.8328			−1.54150
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	4.47214i		−	0.255238i	−0.991823	−	0.127619i	$-0.959266\pi$
$307$		−	4.47214i		−	0.255238i	0.991823	−	0.127619i	$-0.0407335\pi$
$308$		0			0
$309$	−30.0000			−1.70664
$310$		0			0
$311$	4.47214			0.253592			0.126796	−	0.991929i	$-0.459531\pi$
$311$	4.47214			0.253592			0.126796	+	0.991929i	$0.459531\pi$
$312$		0			0
$313$	−1.00000			−0.0565233			−0.0282617	−	0.999601i	$-0.508997\pi$
$313$	−1.00000			−0.0565233			−0.0282617	+	0.999601i	$0.508997\pi$
$314$		0			0
$315$	−4.47214			−0.251976
$316$		0			0
$317$			2.00000i			0.112331i	0.998421	+	0.0561656i	$0.0178875\pi$
$317$			2.00000i			0.112331i	−0.998421	+	0.0561656i	$0.982113\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	13.4164i		−	0.746509i
$324$		0			0
$325$	8.00000	+	12.0000i	0.443760	+	0.665640i
$326$		0			0
$327$		−	33.5410i		−	1.85482i
$328$		0			0
$329$	15.0000			0.826977
$330$		0			0
$331$			22.3607i			1.22905i	0.788896	+	0.614527i	$0.210653\pi$
$331$			22.3607i			1.22905i	−0.788896	+	0.614527i	$0.789347\pi$
$332$		0			0
$333$			6.00000i			0.328798i
$334$		0			0
$335$	−8.94427			−0.488678
$336$		0			0
$337$	−23.0000			−1.25289			−0.626445	−	0.779466i	$-0.715491\pi$
$337$	−23.0000			−1.25289			−0.626445	+	0.779466i	$0.715491\pi$
$338$		0			0
$339$	−13.4164			−0.728679
$340$		0			0
$341$		0			0
$342$		0			0
$343$			20.1246i			1.08663i
$344$		0			0
$345$		−	10.0000i		−	0.538382i
$346$		0			0
$347$	15.6525			0.840269			0.420134	−	0.907462i	$-0.361983\pi$
$347$	15.6525			0.840269			0.420134	+	0.907462i	$0.361983\pi$
$348$		0			0
$349$		−	9.00000i		−	0.481759i	−0.970555	−	0.240879i	$-0.922564\pi$
$349$		−	9.00000i		−	0.481759i	0.970555	−	0.240879i	$-0.0774359\pi$
$350$		0			0
$351$	−4.47214	−	6.70820i	−0.238705	−	0.358057i
$352$		0			0
$353$		−	34.0000i		−	1.80964i	−0.425797	−	0.904819i	$-0.640006\pi$
$353$		−	34.0000i		−	1.80964i	0.425797	−	0.904819i	$-0.359994\pi$
$354$		0			0
$355$	6.70820			0.356034
$356$		0			0
$357$			15.0000i			0.793884i
$358$		0			0
$359$			26.8328i			1.41618i	0.706121	+	0.708091i	$0.250443\pi$
$359$			26.8328i			1.41618i	−0.706121	+	0.708091i	$0.749557\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$		0			0
$363$	24.5967			1.29099
$364$		0			0
$365$	−6.00000			−0.314054
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		0			0
$369$		−	16.0000i		−	0.832927i
$370$		0			0
$371$		−	13.4164i		−	0.696545i
$372$		0			0
$373$	−4.00000			−0.207112			−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000			−0.207112			−0.103556	+	0.994624i	$0.533022\pi$
$374$		0			0
$375$			20.1246i			1.03923i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	4.47214i		−	0.229718i	−0.993382	−	0.114859i	$-0.963358\pi$
$379$		−	4.47214i		−	0.229718i	0.993382	−	0.114859i	$-0.0366417\pi$
$380$		0			0
$381$	−30.0000			−1.53695
$382$		0			0
$383$			20.1246i			1.02832i	0.857694	+	0.514160i	$0.171896\pi$
$383$			20.1246i			1.02832i	−0.857694	+	0.514160i	$0.828104\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−13.4164			−0.681994
$388$		0			0
$389$	−6.00000			−0.304212			−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000			−0.304212			−0.152106	+	0.988364i	$0.548606\pi$
$390$		0			0
$391$	−13.4164			−0.678497
$392$		0			0
$393$	−45.0000			−2.26995
$394$		0			0
$395$			13.4164i			0.675053i
$396$		0			0
$397$			18.0000i			0.903394i	0.892171	+	0.451697i	$0.149181\pi$
$397$			18.0000i			0.903394i	−0.892171	+	0.451697i	$0.850819\pi$
$398$		0			0
$399$	22.3607			1.11943
$400$		0			0
$401$		−	10.0000i		−	0.499376i	−0.968326	−	0.249688i	$-0.919672\pi$
$401$		−	10.0000i		−	0.499376i	0.968326	−	0.249688i	$-0.0803281\pi$
$402$		0			0
$403$	26.8328	−	17.8885i	1.33664	−	0.891092i
$404$		0			0
$405$		−	11.0000i		−	0.546594i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	30.0000i		−	1.48340i	−0.670729	−	0.741702i	$-0.734019\pi$
$409$		−	30.0000i		−	1.48340i	0.670729	−	0.741702i	$-0.265981\pi$
$410$		0			0
$411$			4.47214i			0.220594i
$412$		0			0
$413$	−30.0000			−1.47620
$414$		0			0
$415$	13.4164			0.658586
$416$		0			0
$417$	15.0000			0.734553
$418$		0			0
$419$	20.1246			0.983152			0.491576	−	0.870835i	$-0.336421\pi$
$419$	20.1246			0.983152			0.491576	+	0.870835i	$0.336421\pi$
$420$		0			0
$421$		−	27.0000i		−	1.31590i	−0.753062	−	0.657950i	$-0.771424\pi$
$421$		−	27.0000i		−	1.31590i	0.753062	−	0.657950i	$-0.228576\pi$
$422$		0			0
$423$		−	13.4164i		−	0.652328i
$424$		0			0
$425$	12.0000			0.582086
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$			6.70820i			0.323123i	0.986863	+	0.161561i	$0.0516530\pi$
$431$			6.70820i			0.323123i	−0.986863	+	0.161561i	$0.948347\pi$
$432$		0			0
$433$	9.00000			0.432512			0.216256	−	0.976337i	$-0.430615\pi$
$433$	9.00000			0.432512			0.216256	+	0.976337i	$0.430615\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			20.0000i			0.956730i
$438$		0			0
$439$	−13.4164			−0.640330			−0.320165	−	0.947362i	$-0.603738\pi$
$439$	−13.4164			−0.640330			−0.320165	+	0.947362i	$0.603738\pi$
$440$		0			0
$441$	4.00000			0.190476
$442$		0			0
$443$	20.1246			0.956149			0.478075	−	0.878319i	$-0.341335\pi$
$443$	20.1246			0.956149			0.478075	+	0.878319i	$0.341335\pi$
$444$		0			0
$445$	14.0000			0.663664
$446$		0			0
$447$			22.3607i			1.05762i
$448$		0			0
$449$		−	40.0000i		−	1.88772i	−0.330350	−	0.943858i	$-0.607167\pi$
$449$		−	40.0000i		−	1.88772i	0.330350	−	0.943858i	$-0.392833\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		−	5.00000i		−	0.234920i
$454$		0			0
$455$	−4.47214	−	6.70820i	−0.209657	−	0.314485i
$456$		0			0
$457$			18.0000i			0.842004i	0.907060	+	0.421002i	$0.138322\pi$
$457$			18.0000i			0.842004i	−0.907060	+	0.421002i	$0.861678\pi$
$458$		0			0
$459$	−6.70820			−0.313112
$460$		0			0
$461$			25.0000i			1.16437i	0.813058	+	0.582183i	$0.197801\pi$
$461$			25.0000i			1.16437i	−0.813058	+	0.582183i	$0.802199\pi$
$462$		0			0
$463$			8.94427i			0.415676i	0.978163	+	0.207838i	$0.0666426\pi$
$463$			8.94427i			0.415676i	−0.978163	+	0.207838i	$0.933357\pi$
$464$		0			0
$465$	20.0000			0.927478
$466$		0			0
$467$	−17.8885			−0.827783			−0.413892	−	0.910326i	$-0.635831\pi$
$467$	−17.8885			−0.827783			−0.413892	+	0.910326i	$0.635831\pi$
$468$		0			0
$469$	−20.0000			−0.923514
$470$		0			0
$471$	40.2492			1.85459
$472$		0			0
$473$		0			0
$474$		0			0
$475$		−	17.8885i		−	0.820783i
$476$		0			0
$477$	−12.0000			−0.549442
$478$		0			0
$479$		−	33.5410i		−	1.53253i	−0.642526	−	0.766264i	$-0.722113\pi$
$479$		−	33.5410i		−	1.53253i	0.642526	−	0.766264i	$-0.277887\pi$
$480$		0			0
$481$	−9.00000	+	6.00000i	−0.410365	+	0.273576i
$482$		0			0
$483$		−	22.3607i		−	1.01745i
$484$		0			0
$485$	−12.0000			−0.544892
$486$		0			0
$487$			35.7771i			1.62121i	0.585590	+	0.810607i	$0.300863\pi$
$487$			35.7771i			1.62121i	−0.585590	+	0.810607i	$0.699137\pi$
$488$		0			0
$489$			10.0000i			0.452216i
$490$		0			0
$491$	−15.6525			−0.706386			−0.353193	−	0.935550i	$-0.614904\pi$
$491$	−15.6525			−0.706386			−0.353193	+	0.935550i	$0.614904\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	15.0000			0.672842
$498$		0			0
$499$		−	8.94427i		−	0.400401i	−0.979755	−	0.200200i	$-0.935841\pi$
$499$		−	8.94427i		−	0.400401i	0.979755	−	0.200200i	$-0.0641593\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−40.2492			−1.79462			−0.897312	−	0.441397i	$-0.854483\pi$
$503$	−40.2492			−1.79462			−0.897312	+	0.441397i	$0.854483\pi$
$504$		0			0
$505$		−	12.0000i		−	0.533993i
$506$		0			0
$507$	−11.1803	+	26.8328i	−0.496536	+	1.19169i
$508$		0			0
$509$			26.0000i			1.15243i	0.817298	+	0.576215i	$0.195471\pi$
$509$			26.0000i			1.15243i	−0.817298	+	0.576215i	$0.804529\pi$
$510$		0			0
$511$	−13.4164			−0.593507
$512$		0			0
$513$			10.0000i			0.441511i
$514$		0			0
$515$		−	13.4164i		−	0.591198i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−13.4164			−0.588915
$520$		0			0
$521$	−33.0000			−1.44576			−0.722878	−	0.690976i	$-0.757181\pi$
$521$	−33.0000			−1.44576			−0.722878	+	0.690976i	$0.757181\pi$
$522$		0			0
$523$	−26.8328			−1.17332			−0.586659	−	0.809834i	$-0.699557\pi$
$523$	−26.8328			−1.17332			−0.586659	+	0.809834i	$0.699557\pi$
$524$		0			0
$525$			20.0000i			0.872872i
$526$		0			0
$527$		−	26.8328i		−	1.16886i
$528$		0			0
$529$	−3.00000			−0.130435
$530$		0			0
$531$			26.8328i			1.16445i
$532$		0			0
$533$	24.0000	−	16.0000i	1.03956	−	0.693037i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	45.0000			1.94189
$538$		0			0
$539$		0			0
$540$		0			0
$541$		−	15.0000i		−	0.644900i	−0.946586	−	0.322450i	$-0.895494\pi$
$541$		−	15.0000i		−	0.644900i	0.946586	−	0.322450i	$-0.104506\pi$
$542$		0			0
$543$	40.2492			1.72726
$544$		0			0
$545$	15.0000			0.642529
$546$		0			0
$547$	−6.70820			−0.286822			−0.143411	−	0.989663i	$-0.545807\pi$
$547$	−6.70820			−0.286822			−0.143411	+	0.989663i	$0.545807\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$			30.0000i			1.27573i
$554$		0			0
$555$	−6.70820			−0.284747
$556$		0			0
$557$			13.0000i			0.550828i	0.961326	+	0.275414i	$0.0888149\pi$
$557$			13.0000i			0.550828i	−0.961326	+	0.275414i	$0.911185\pi$
$558$		0			0
$559$	−13.4164	−	20.1246i	−0.567454	−	0.851180i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	20.1246			0.848151			0.424076	−	0.905627i	$-0.360599\pi$
$563$	20.1246			0.848151			0.424076	+	0.905627i	$0.360599\pi$
$564$		0			0
$565$		−	6.00000i		−	0.252422i
$566$		0			0
$567$		−	24.5967i		−	1.03297i
$568$		0			0
$569$	21.0000			0.880366			0.440183	−	0.897908i	$-0.354914\pi$
$569$	21.0000			0.880366			0.440183	+	0.897908i	$0.354914\pi$
$570$		0			0
$571$	−20.1246			−0.842189			−0.421094	−	0.907017i	$-0.638354\pi$
$571$	−20.1246			−0.842189			−0.421094	+	0.907017i	$0.638354\pi$
$572$		0			0
$573$	20.0000			0.835512
$574$		0			0
$575$	−17.8885			−0.746004
$576$		0			0
$577$		−	18.0000i		−	0.749350i	−0.927156	−	0.374675i	$-0.877754\pi$
$577$		−	18.0000i		−	0.749350i	0.927156	−	0.374675i	$-0.122246\pi$
$578$		0			0
$579$			13.4164i			0.557567i
$580$		0			0
$581$	30.0000			1.24461
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−6.00000	+	4.00000i	−0.248069	+	0.165380i
$586$		0			0
$587$			13.4164i			0.553754i	0.960905	+	0.276877i	$0.0892995\pi$
$587$			13.4164i			0.553754i	−0.960905	+	0.276877i	$0.910700\pi$
$588$		0			0
$589$	−40.0000			−1.64817
$590$		0			0
$591$		−	29.0689i		−	1.19573i
$592$		0			0
$593$			4.00000i			0.164260i	0.996622	+	0.0821302i	$0.0261723\pi$
$593$			4.00000i			0.164260i	−0.996622	+	0.0821302i	$0.973828\pi$
$594$		0			0
$595$	−6.70820			−0.275010
$596$		0			0
$597$	60.0000			2.45564
$598$		0			0
$599$	−40.2492			−1.64454			−0.822269	−	0.569099i	$-0.807292\pi$
$599$	−40.2492			−1.64454			−0.822269	+	0.569099i	$0.807292\pi$
$600$		0			0
$601$	−35.0000			−1.42768			−0.713840	−	0.700309i	$-0.753046\pi$
$601$	−35.0000			−1.42768			−0.713840	+	0.700309i	$0.753046\pi$
$602$		0			0
$603$			17.8885i			0.728478i
$604$		0			0
$605$			11.0000i			0.447214i
$606$		0			0
$607$	13.4164			0.544555			0.272278	−	0.962219i	$-0.412223\pi$
$607$	13.4164			0.544555			0.272278	+	0.962219i	$0.412223\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	20.1246	−	13.4164i	0.814155	−	0.542770i
$612$		0			0
$613$			6.00000i			0.242338i	0.992632	+	0.121169i	$0.0386643\pi$
$613$			6.00000i			0.242338i	−0.992632	+	0.121169i	$0.961336\pi$
$614$		0			0
$615$	17.8885			0.721336
$616$		0			0
$617$			28.0000i			1.12724i	0.826035	+	0.563619i	$0.190591\pi$
$617$			28.0000i			1.12724i	−0.826035	+	0.563619i	$0.809409\pi$
$618$		0			0
$619$			17.8885i			0.719001i	0.933145	+	0.359501i	$0.117053\pi$
$619$			17.8885i			0.719001i	−0.933145	+	0.359501i	$0.882947\pi$
$620$		0			0
$621$	10.0000			0.401286
$622$		0			0
$623$	31.3050			1.25421
$624$		0			0
$625$	11.0000			0.440000
$626$		0			0
$627$		0			0
$628$		0			0
$629$			9.00000i			0.358854i
$630$		0			0
$631$			15.6525i			0.623115i	0.950227	+	0.311558i	$0.100851\pi$
$631$			15.6525i			0.623115i	−0.950227	+	0.311558i	$0.899149\pi$
$632$		0			0
$633$	15.0000			0.596196
$634$		0			0
$635$		−	13.4164i		−	0.532414i
$636$		0			0
$637$	4.00000	+	6.00000i	0.158486	+	0.237729i
$638$		0			0
$639$		−	13.4164i		−	0.530745i
$640$		0			0
$641$	30.0000			1.18493			0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000			1.18493			0.592464	+	0.805597i	$0.298155\pi$
$642$		0			0
$643$		−	8.94427i		−	0.352728i	−0.984325	−	0.176364i	$-0.943566\pi$
$643$		−	8.94427i		−	0.352728i	0.984325	−	0.176364i	$-0.0564335\pi$
$644$		0			0
$645$		−	15.0000i		−	0.590624i
$646$		0			0
$647$	40.2492			1.58236			0.791180	−	0.611583i	$-0.209467\pi$
$647$	40.2492			1.58236			0.791180	+	0.611583i	$0.209467\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	44.7214			1.75277
$652$		0			0
$653$	36.0000			1.40879			0.704394	−	0.709809i	$-0.251219\pi$
$653$	36.0000			1.40879			0.704394	+	0.709809i	$0.251219\pi$
$654$		0			0
$655$		−	20.1246i		−	0.786334i
$656$		0			0
$657$			12.0000i			0.468165i
$658$		0			0
$659$	−8.94427			−0.348419			−0.174210	−	0.984709i	$-0.555737\pi$
$659$	−8.94427			−0.348419			−0.174210	+	0.984709i	$0.555737\pi$
$660$		0			0
$661$		−	18.0000i		−	0.700119i	−0.936727	−	0.350059i	$-0.886161\pi$
$661$		−	18.0000i		−	0.700119i	0.936727	−	0.350059i	$-0.113839\pi$
$662$		0			0
$663$	13.4164	+	20.1246i	0.521050	+	0.781575i
$664$		0			0
$665$			10.0000i			0.387783i
$666$		0			0
$667$		0			0
$668$		0			0
$669$			5.00000i			0.193311i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−9.00000			−0.346925			−0.173462	−	0.984841i	$-0.555495\pi$
$673$	−9.00000			−0.346925			−0.173462	+	0.984841i	$0.555495\pi$
$674$		0			0
$675$	−8.94427			−0.344265
$676$		0			0
$677$	18.0000			0.691796			0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000			0.691796			0.345898	+	0.938272i	$0.387574\pi$
$678$		0			0
$679$	−26.8328			−1.02975
$680$		0			0
$681$			60.0000i			2.29920i
$682$		0			0
$683$			40.2492i			1.54009i	0.637987	+	0.770047i	$0.279767\pi$
$683$			40.2492i			1.54009i	−0.637987	+	0.770047i	$0.720233\pi$
$684$		0			0
$685$	−2.00000			−0.0764161
$686$		0			0
$687$			46.9574i			1.79154i
$688$		0			0
$689$	−12.0000	−	18.0000i	−0.457164	−	0.685745i
$690$		0			0
$691$			17.8885i			0.680512i	0.940333	+	0.340256i	$0.110514\pi$
$691$			17.8885i			0.680512i	−0.940333	+	0.340256i	$0.889486\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			6.70820i			0.254457i
$696$		0			0
$697$		−	24.0000i		−	0.909065i
$698$		0			0
$699$	−20.1246			−0.761183
$700$		0			0
$701$	−30.0000			−1.13308			−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000			−1.13308			−0.566542	+	0.824033i	$0.691719\pi$
$702$		0			0
$703$	13.4164			0.506009
$704$		0			0
$705$	15.0000			0.564933
$706$		0			0
$707$		−	26.8328i		−	1.00915i
$708$		0			0
$709$		−	6.00000i		−	0.225335i	−0.993633	−	0.112667i	$-0.964061\pi$
$709$		−	6.00000i		−	0.225335i	0.993633	−	0.112667i	$-0.0359394\pi$
$710$		0			0
$711$	26.8328			1.00631
$712$		0			0
$713$			40.0000i			1.49801i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			15.0000i			0.560185i
$718$		0			0
$719$	−17.8885			−0.667130			−0.333565	−	0.942727i	$-0.608252\pi$
$719$	−17.8885			−0.667130			−0.333565	+	0.942727i	$0.608252\pi$
$720$		0			0
$721$		−	30.0000i		−	1.11726i
$722$		0			0
$723$			40.2492i			1.49688i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	26.8328			0.995174			0.497587	−	0.867414i	$-0.334220\pi$
$727$	26.8328			0.995174			0.497587	+	0.867414i	$0.334220\pi$
$728$		0			0
$729$	−7.00000			−0.259259
$730$		0			0
$731$	−20.1246			−0.744336
$732$		0			0
$733$		−	21.0000i		−	0.775653i	−0.921732	−	0.387826i	$-0.873226\pi$
$733$		−	21.0000i		−	0.775653i	0.921732	−	0.387826i	$-0.126774\pi$
$734$		0			0
$735$			4.47214i			0.164957i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		−	44.7214i		−	1.64510i	−0.568691	−	0.822551i	$-0.692550\pi$
$739$		−	44.7214i		−	1.64510i	0.568691	−	0.822551i	$-0.307450\pi$
$740$		0			0
$741$	30.0000	−	20.0000i	1.10208	−	0.734718i
$742$		0			0
$743$			33.5410i			1.23050i	0.788332	+	0.615250i	$0.210945\pi$
$743$			33.5410i			1.23050i	−0.788332	+	0.615250i	$0.789055\pi$
$744$		0			0
$745$	−10.0000			−0.366372
$746$		0			0
$747$		−	26.8328i		−	0.981761i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$	−40.0000			−1.45768
$754$		0			0
$755$	2.23607			0.0813788
$756$		0			0
$757$	−18.0000			−0.654221			−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000			−0.654221			−0.327111	+	0.944986i	$0.606075\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			10.0000i			0.362500i	0.983437	+	0.181250i	$0.0580143\pi$
$761$			10.0000i			0.362500i	−0.983437	+	0.181250i	$0.941986\pi$
$762$		0			0
$763$	33.5410			1.21427
$764$		0			0
$765$			6.00000i			0.216930i
$766$		0			0
$767$	−40.2492	+	26.8328i	−1.45332	+	0.968877i
$768$		0			0
$769$			24.0000i			0.865462i	0.901523	+	0.432731i	$0.142450\pi$
$769$			24.0000i			0.865462i	−0.901523	+	0.432731i	$0.857550\pi$
$770$		0			0
$771$	60.3738			2.17431
$772$		0			0
$773$		−	31.0000i		−	1.11499i	−0.830179	−	0.557496i	$-0.811762\pi$
$773$		−	31.0000i		−	1.11499i	0.830179	−	0.557496i	$-0.188238\pi$
$774$		0			0
$775$		−	35.7771i		−	1.28515i
$776$		0			0
$777$	−15.0000			−0.538122
$778$		0			0
$779$	−35.7771			−1.28185
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$			18.0000i			0.642448i
$786$		0			0
$787$			8.94427i			0.318829i	0.987212	+	0.159414i	$0.0509606\pi$
$787$			8.94427i			0.318829i	−0.987212	+	0.159414i	$0.949039\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	13.4164i		−	0.477033i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		−	13.4164i		−	0.475831i
$796$		0			0
$797$	18.0000			0.637593			0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000			0.637593			0.318796	+	0.947823i	$0.396721\pi$
$798$		0			0
$799$		−	20.1246i		−	0.711958i
$800$		0			0
$801$		−	28.0000i		−	0.989331i
$802$		0			0
$803$		0			0
$804$		0			0
$805$	10.0000			0.352454
$806$		0			0
$807$	−53.6656			−1.88912
$808$		0			0
$809$	−45.0000			−1.58212			−0.791058	−	0.611741i	$-0.790469\pi$
$809$	−45.0000			−1.58212			−0.791058	+	0.611741i	$0.790469\pi$
$810$		0			0
$811$		−	35.7771i		−	1.25630i	−0.778091	−	0.628152i	$-0.783812\pi$
$811$		−	35.7771i		−	1.25630i	0.778091	−	0.628152i	$-0.216188\pi$
$812$		0			0
$813$		−	35.0000i		−	1.22750i
$814$		0			0
$815$	−4.47214			−0.156652
$816$		0			0
$817$			30.0000i			1.04957i
$818$		0			0
$819$	−13.4164	+	8.94427i	−0.468807	+	0.312538i
$820$		0			0
$821$		−	7.00000i		−	0.244302i	−0.992512	−	0.122151i	$-0.961021\pi$
$821$		−	7.00000i		−	0.244302i	0.992512	−	0.122151i	$-0.0389792\pi$
$822$		0			0
$823$	−13.4164			−0.467667			−0.233833	−	0.972277i	$-0.575127\pi$
$823$	−13.4164			−0.467667			−0.233833	+	0.972277i	$0.575127\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	40.2492i		−	1.39960i	−0.714338	−	0.699801i	$-0.753272\pi$
$827$		−	40.2492i		−	1.39960i	0.714338	−	0.699801i	$-0.246728\pi$
$828$		0			0
$829$	−36.0000			−1.25033			−0.625166	−	0.780492i	$-0.714969\pi$
$829$	−36.0000			−1.25033			−0.625166	+	0.780492i	$0.714969\pi$
$830$		0			0
$831$	−17.8885			−0.620547
$832$		0			0
$833$	6.00000			0.207888
$834$		0			0
$835$		0			0
$836$		0			0
$837$			20.0000i			0.691301i
$838$		0			0
$839$			26.8328i			0.926372i	0.886261	+	0.463186i	$0.153294\pi$
$839$			26.8328i			0.926372i	−0.886261	+	0.463186i	$0.846706\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$			4.47214i			0.154029i
$844$		0			0
$845$	−12.0000	−	5.00000i	−0.412813	−	0.172005i
$846$		0			0
$847$			24.5967i			0.845154i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	13.4164i		−	0.459909i
$852$		0			0
$853$		−	51.0000i		−	1.74621i	−0.487535	−	0.873103i	$-0.662104\pi$
$853$		−	51.0000i		−	1.74621i	0.487535	−	0.873103i	$-0.337896\pi$
$854$		0			0
$855$	8.94427			0.305888
$856$		0			0
$857$	18.0000			0.614868			0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000			0.614868			0.307434	+	0.951569i	$0.400530\pi$
$858$		0			0
$859$	26.8328			0.915524			0.457762	−	0.889075i	$-0.348651\pi$
$859$	26.8328			0.915524			0.457762	+	0.889075i	$0.348651\pi$
$860$		0			0
$861$	40.0000			1.36320
$862$		0			0
$863$		−	46.9574i		−	1.59845i	−0.601032	−	0.799225i	$-0.705244\pi$
$863$		−	46.9574i		−	1.59845i	0.601032	−	0.799225i	$-0.294756\pi$
$864$		0			0
$865$		−	6.00000i		−	0.204006i
$866$		0			0
$867$	−17.8885			−0.607527
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−26.8328	+	17.8885i	−0.909195	+	0.606130i
$872$		0			0
$873$			24.0000i			0.812277i
$874$		0			0
$875$	−20.1246			−0.680336
$876$		0			0
$877$		−	57.0000i		−	1.92475i	−0.271719	−	0.962377i	$-0.587592\pi$
$877$		−	57.0000i		−	1.92475i	0.271719	−	0.962377i	$-0.412408\pi$
$878$		0			0
$879$		−	2.23607i		−	0.0754207i
$880$		0			0
$881$	15.0000			0.505363			0.252681	−	0.967550i	$-0.418688\pi$
$881$	15.0000			0.505363			0.252681	+	0.967550i	$0.418688\pi$
$882$		0			0
$883$	−20.1246			−0.677247			−0.338624	−	0.940922i	$-0.609961\pi$
$883$	−20.1246			−0.677247			−0.338624	+	0.940922i	$0.609961\pi$
$884$		0			0
$885$	−30.0000			−1.00844
$886$		0			0
$887$	44.7214			1.50160			0.750798	−	0.660532i	$-0.229669\pi$
$887$	44.7214			1.50160			0.750798	+	0.660532i	$0.229669\pi$
$888$		0			0
$889$		−	30.0000i		−	1.00617i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−30.0000			−1.00391
$894$		0			0
$895$			20.1246i			0.672692i
$896$		0			0
$897$	−20.0000	−	30.0000i	−0.667781	−	1.00167i
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−18.0000			−0.599667
$902$		0			0
$903$		−	33.5410i		−	1.11618i
$904$		0			0
$905$			18.0000i			0.598340i
$906$		0			0
$907$	−6.70820			−0.222742			−0.111371	−	0.993779i	$-0.535524\pi$
$907$	−6.70820			−0.222742			−0.111371	+	0.993779i	$0.535524\pi$
$908$		0			0
$909$	−24.0000			−0.796030
$910$		0			0
$911$	8.94427			0.296337			0.148168	−	0.988962i	$-0.452662\pi$
$911$	8.94427			0.296337			0.148168	+	0.988962i	$0.452662\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	45.0000i		−	1.48603i
$918$		0			0
$919$	−26.8328			−0.885133			−0.442566	−	0.896736i	$-0.645932\pi$
$919$	−26.8328			−0.885133			−0.442566	+	0.896736i	$0.645932\pi$
$920$		0			0
$921$		−	10.0000i		−	0.329511i
$922$		0			0
$923$	20.1246	−	13.4164i	0.662410	−	0.441606i
$924$		0			0
$925$			12.0000i			0.394558i
$926$		0			0
$927$	−26.8328			−0.881305
$928$		0			0
$929$		−	20.0000i		−	0.656179i	−0.944647	−	0.328089i	$-0.893595\pi$
$929$		−	20.0000i		−	0.656179i	0.944647	−	0.328089i	$-0.106405\pi$
$930$		0			0
$931$		−	8.94427i		−	0.293137i
$932$		0			0
$933$	10.0000			0.327385
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−18.0000			−0.588034			−0.294017	−	0.955800i	$-0.594992\pi$
$937$	−18.0000			−0.588034			−0.294017	+	0.955800i	$0.594992\pi$
$938$		0			0
$939$	−2.23607			−0.0729713
$940$		0			0
$941$			55.0000i			1.79295i	0.443096	+	0.896474i	$0.353880\pi$
$941$			55.0000i			1.79295i	−0.443096	+	0.896474i	$0.646120\pi$
$942$		0			0
$943$			35.7771i			1.16506i
$944$		0			0
$945$	5.00000			0.162650
$946$		0			0
$947$			53.6656i			1.74390i	0.489596	+	0.871949i	$0.337144\pi$
$947$			53.6656i			1.74390i	−0.489596	+	0.871949i	$0.662856\pi$
$948$		0			0
$949$	−18.0000	+	12.0000i	−0.584305	+	0.389536i
$950$		0			0
$951$			4.47214i			0.145019i
$952$		0			0
$953$	−39.0000			−1.26333			−0.631667	−	0.775240i	$-0.717629\pi$
$953$	−39.0000			−1.26333			−0.631667	+	0.775240i	$0.717629\pi$
$954$		0			0
$955$			8.94427i			0.289430i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−4.47214			−0.144413
$960$		0			0
$961$	−49.0000			−1.58065
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−6.00000			−0.193147
$966$		0			0
$967$		−	51.4296i		−	1.65386i	−0.562302	−	0.826932i	$-0.690084\pi$
$967$		−	51.4296i		−	1.65386i	0.562302	−	0.826932i	$-0.309916\pi$
$968$		0			0
$969$		−	30.0000i		−	0.963739i
$970$		0			0
$971$	38.0132			1.21990			0.609950	−	0.792440i	$-0.291189\pi$
$971$	38.0132			1.21990			0.609950	+	0.792440i	$0.291189\pi$
$972$		0			0
$973$			15.0000i			0.480878i
$974$		0			0
$975$	17.8885	+	26.8328i	0.572892	+	0.859338i
$976$		0			0
$977$			28.0000i			0.895799i	0.894084	+	0.447900i	$0.147828\pi$
$977$			28.0000i			0.895799i	−0.894084	+	0.447900i	$0.852172\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		−	30.0000i		−	0.957826i
$982$		0			0
$983$		−	6.70820i		−	0.213958i	−0.994261	−	0.106979i	$-0.965882\pi$
$983$		−	6.70820i		−	0.213958i	0.994261	−	0.106979i	$-0.0341178\pi$
$984$		0			0
$985$	13.0000			0.414214
$986$		0			0
$987$	33.5410			1.06762
$988$		0			0
$989$	30.0000			0.953945
$990$		0			0
$991$	13.4164			0.426186			0.213093	−	0.977032i	$-0.431646\pi$
$991$	13.4164			0.426186			0.213093	+	0.977032i	$0.431646\pi$
$992$		0			0
$993$			50.0000i			1.58670i
$994$		0			0
$995$			26.8328i			0.850657i
$996$		0			0
$997$	−8.00000			−0.253363			−0.126681	−	0.991943i	$-0.540433\pi$
$997$	−8.00000			−0.253363			−0.126681	+	0.991943i	$0.540433\pi$
$998$		0			0
$999$		−	6.70820i		−	0.212238i

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.2.f.e.129.4	yes	4
3.2	odd	2		3744.2.c.k.3457.2		4
4.3	odd	2	inner	416.2.f.e.129.2	yes	4
8.3	odd	2		832.2.f.h.129.3		4
8.5	even	2		832.2.f.h.129.1		4
12.11	even	2		3744.2.c.k.3457.1		4
13.5	odd	4		5408.2.a.bb.1.2		2
13.8	odd	4		5408.2.a.t.1.2		2
13.12	even	2	inner	416.2.f.e.129.3	yes	4
39.38	odd	2		3744.2.c.k.3457.3		4
52.31	even	4		5408.2.a.bb.1.1		2
52.47	even	4		5408.2.a.t.1.1		2
52.51	odd	2	inner	416.2.f.e.129.1	✓	4
104.51	odd	2		832.2.f.h.129.4		4
104.77	even	2		832.2.f.h.129.2		4
156.155	even	2		3744.2.c.k.3457.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.2.f.e.129.1	✓	4	52.51	odd	2	inner
416.2.f.e.129.2	yes	4	4.3	odd	2	inner
416.2.f.e.129.3	yes	4	13.12	even	2	inner
416.2.f.e.129.4	yes	4	1.1	even	1	trivial
832.2.f.h.129.1		4	8.5	even	2
832.2.f.h.129.2		4	104.77	even	2
832.2.f.h.129.3		4	8.3	odd	2
832.2.f.h.129.4		4	104.51	odd	2
3744.2.c.k.3457.1		4	12.11	even	2
3744.2.c.k.3457.2		4	3.2	odd	2
3744.2.c.k.3457.3		4	39.38	odd	2
3744.2.c.k.3457.4		4	156.155	even	2
5408.2.a.t.1.1		2	52.47	even	4
5408.2.a.t.1.2		2	13.8	odd	4
5408.2.a.bb.1.1		2	52.31	even	4
5408.2.a.bb.1.2		2	13.5	odd	4

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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	416.2.f.e.129.4

Level	$416$
Weight	$2$
Character	416.129
Analytic conductor	$3.322$
Analytic rank	$0$
Dimension	$4$
Inner twists	$4$