LMFDB - Embedded newform 3168.2.o.f.703.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3168,2,Mod(703,3168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3168, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))

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

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

chi := DirichletCharacter("3168.703");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3168 = 2^{5} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3168.o (of order $2$, degree $1$, 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:	$25.2966073603$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			703.12
Character	$\chi$	$=$	3168.703
Dual form			3168.2.o.f.703.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.09364 q^{5} -0.221727 q^{7} +(-2.10394 + 2.56387i) q^{11} -3.84759i q^{13} +3.39691i q^{17} +5.17250 q^{19} -0.780070i q^{23} -3.80395 q^{25} -3.63940i q^{29} -3.14688i q^{31} +0.242490 q^{35} -3.14688 q^{37} +0.810970i q^{41} +2.07922 q^{43} +6.53495i q^{47} -6.95084 q^{49} +11.9763 q^{53} +(2.30095 - 2.80395i) q^{55} -5.75488i q^{59} -0.754311i q^{61} +4.20788i q^{65} -12.7548i q^{67} -8.72223i q^{71} -9.90155i q^{73} +(0.466500 - 0.568480i) q^{77} +7.47345 q^{79} +9.62224 q^{83} -3.71499i q^{85} -11.8156 q^{89} +0.853115i q^{91} -5.65685 q^{95} -5.95084 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 40 q^{25} + 40 q^{49} + 64 q^{97}+O(q^{100}) $ 24 * q + 40 * q^25 + 40 * q^49 + 64 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3168\mathbb{Z}\right)^\times$.

$n$	$353$	$991$	$1189$	$1729$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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			0
$3$		0			0
$4$		0			0
$5$	−1.09364			−0.489090			−0.244545	−	0.969638i	$-0.578639\pi$
$5$	−1.09364			−0.489090			−0.244545	+	0.969638i	$0.578639\pi$
$6$		0			0
$7$	−0.221727			−0.0838050			−0.0419025	−	0.999122i	$-0.513342\pi$
$7$	−0.221727			−0.0838050			−0.0419025	+	0.999122i	$0.513342\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.10394	+	2.56387i	−0.634361	+	0.773037i
$11$	−2.10394	+	2.56387i	−0.634361	+	0.773037i
$12$		0			0
$13$		−	3.84759i		−	1.06713i	−0.845759	−	0.533565i	$-0.820852\pi$
$13$		−	3.84759i		−	1.06713i	0.845759	−	0.533565i	$-0.179148\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			3.39691i			0.823871i	0.911213	+	0.411936i	$0.135147\pi$
$17$			3.39691i			0.823871i	−0.911213	+	0.411936i	$0.864853\pi$
$18$		0			0
$19$	5.17250			1.18665			0.593327	−	0.804962i	$-0.297814\pi$
$19$	5.17250			1.18665			0.593327	+	0.804962i	$0.297814\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	0.780070i		−	0.162656i	−0.996687	−	0.0813279i	$-0.974084\pi$
$23$		−	0.780070i		−	0.162656i	0.996687	−	0.0813279i	$-0.0259161\pi$
$24$		0			0
$25$	−3.80395			−0.760790
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	3.63940i		−	0.675819i	−0.941179	−	0.337910i	$-0.890280\pi$
$29$		−	3.63940i		−	0.675819i	0.941179	−	0.337910i	$-0.109720\pi$
$30$		0			0
$31$		−	3.14688i		−	0.565197i	−0.959238	−	0.282599i	$-0.908804\pi$
$31$		−	3.14688i		−	0.565197i	0.959238	−	0.282599i	$-0.0911964\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.242490			0.0409882
$36$		0			0
$37$	−3.14688			−0.517345			−0.258672	−	0.965965i	$-0.583285\pi$
$37$	−3.14688			−0.517345			−0.258672	+	0.965965i	$0.583285\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			0.810970i			0.126652i	0.997993	+	0.0633261i	$0.0201708\pi$
$41$			0.810970i			0.126652i	−0.997993	+	0.0633261i	$0.979829\pi$
$42$		0			0
$43$	2.07922			0.317079			0.158539	−	0.987353i	$-0.449322\pi$
$43$	2.07922			0.317079			0.158539	+	0.987353i	$0.449322\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			6.53495i			0.953221i	0.879115	+	0.476611i	$0.158135\pi$
$47$			6.53495i			0.953221i	−0.879115	+	0.476611i	$0.841865\pi$
$48$		0			0
$49$	−6.95084			−0.992977
$50$		0			0
$51$		0			0
$52$		0			0
$53$	11.9763			1.64507			0.822534	−	0.568717i	$-0.192560\pi$
$53$	11.9763			1.64507			0.822534	+	0.568717i	$0.192560\pi$
$54$		0			0
$55$	2.30095	−	2.80395i	0.310260	−	0.378085i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	5.75488i		−	0.749222i	−0.927182	−	0.374611i	$-0.877776\pi$
$59$		−	5.75488i		−	0.749222i	0.927182	−	0.374611i	$-0.122224\pi$
$60$		0			0
$61$		−	0.754311i		−	0.0965796i	−0.998833	−	0.0482898i	$-0.984623\pi$
$61$		−	0.754311i		−	0.0965796i	0.998833	−	0.0482898i	$-0.0153771\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			4.20788i			0.521923i
$66$		0			0
$67$		−	12.7548i		−	1.55825i	−0.626871	−	0.779123i	$-0.715665\pi$
$67$		−	12.7548i		−	1.55825i	0.626871	−	0.779123i	$-0.284335\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	8.72223i		−	1.03514i	−0.855641	−	0.517569i	$-0.826837\pi$
$71$		−	8.72223i		−	1.03514i	0.855641	−	0.517569i	$-0.173163\pi$
$72$		0			0
$73$		−	9.90155i		−	1.15889i	−0.815012	−	0.579445i	$-0.803270\pi$
$73$		−	9.90155i		−	1.15889i	0.815012	−	0.579445i	$-0.196730\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.466500	−	0.568480i	0.0531627	−	0.0647843i
$78$		0			0
$79$	7.47345			0.840829			0.420415	−	0.907332i	$-0.361885\pi$
$79$	7.47345			0.840829			0.420415	+	0.907332i	$0.361885\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	9.62224			1.05618			0.528089	−	0.849189i	$-0.322909\pi$
$83$	9.62224			1.05618			0.528089	+	0.849189i	$0.322909\pi$
$84$		0			0
$85$		−	3.71499i		−	0.402947i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−11.8156			−1.25245			−0.626227	−	0.779641i	$-0.715402\pi$
$89$	−11.8156			−1.25245			−0.626227	+	0.779641i	$0.715402\pi$
$90$		0			0
$91$			0.853115i			0.0894308i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−5.65685			−0.580381
$96$		0			0
$97$	−5.95084			−0.604216			−0.302108	−	0.953274i	$-0.597690\pi$
$97$	−5.95084			−0.604216			−0.302108	+	0.953274i	$0.597690\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	4.77636i		−	0.475265i	−0.971355	−	0.237633i	$-0.923629\pi$
$101$		−	4.77636i		−	0.475265i	0.971355	−	0.237633i	$-0.0763714\pi$
$102$		0			0
$103$		−	8.75479i		−	0.862635i	−0.902200	−	0.431318i	$-0.858049\pi$
$103$		−	8.75479i		−	0.862635i	0.902200	−	0.431318i	$-0.141951\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−9.86473			−0.953660			−0.476830	−	0.878996i	$-0.658214\pi$
$107$	−9.86473			−0.953660			−0.476830	+	0.878996i	$0.658214\pi$
$108$		0			0
$109$		−	16.8424i		−	1.61321i	−0.591090	−	0.806606i	$-0.701302\pi$
$109$		−	16.8424i		−	1.61321i	0.591090	−	0.806606i	$-0.298698\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−8.69535			−0.817990			−0.408995	−	0.912537i	$-0.634121\pi$
$113$	−8.69535			−0.817990			−0.408995	+	0.912537i	$0.634121\pi$
$114$		0			0
$115$			0.853115i			0.0795534i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	0.753187i		−	0.0690445i
$120$		0			0
$121$	−2.14688	−	10.7885i	−0.195171	−	0.980769i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9.62835			0.861186
$126$		0			0
$127$	18.2619			1.62048			0.810241	−	0.586096i	$-0.199336\pi$
$127$	18.2619			1.62048			0.810241	+	0.586096i	$0.199336\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3.96539			0.346458			0.173229	−	0.984882i	$-0.444580\pi$
$131$	3.96539			0.346458			0.173229	+	0.984882i	$0.444580\pi$
$132$		0			0
$133$	−1.14688			−0.0994475
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−10.8826			−0.929766			−0.464883	−	0.885372i	$-0.653904\pi$
$137$	−10.8826			−0.929766			−0.464883	+	0.885372i	$0.653904\pi$
$138$		0			0
$139$	17.0261			1.44414			0.722069	−	0.691821i	$-0.243191\pi$
$139$	17.0261			1.44414			0.722069	+	0.691821i	$0.243191\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	9.86473	+	8.09509i	0.824930	+	0.676946i
$144$		0			0
$145$			3.98019i			0.330537i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			0.880497i			0.0721331i	0.999349	+	0.0360666i	$0.0114828\pi$
$149$			0.880497i			0.0721331i	−0.999349	+	0.0360666i	$0.988517\pi$
$150$		0			0
$151$	5.52138			0.449323			0.224662	−	0.974437i	$-0.427872\pi$
$151$	5.52138			0.449323			0.224662	+	0.974437i	$0.427872\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			3.44156i			0.276433i
$156$		0			0
$157$	−13.9017			−1.10947			−0.554737	−	0.832026i	$-0.687181\pi$
$157$	−13.9017			−1.10947			−0.554737	+	0.832026i	$0.687181\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			0.172963i			0.0136314i
$162$		0			0
$163$		−	5.60790i		−	0.439245i	−0.975585	−	0.219623i	$-0.929517\pi$
$163$		−	5.60790i		−	0.439245i	0.975585	−	0.219623i	$-0.0704826\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	5.17187			0.400212			0.200106	−	0.979774i	$-0.435871\pi$
$167$	5.17187			0.400212			0.200106	+	0.979774i	$0.435871\pi$
$168$		0			0
$169$	−1.80395			−0.138766
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	17.7120i		−	1.34662i	−0.739361	−	0.673309i	$-0.764872\pi$
$173$		−	17.7120i		−	1.34662i	0.739361	−	0.673309i	$-0.235128\pi$
$174$		0			0
$175$	0.843440			0.0637580
$176$		0			0
$177$		0			0
$178$		0			0
$179$			8.69535i			0.649921i	0.945728	+	0.324960i	$0.105351\pi$
$179$			8.69535i			0.649921i	−0.945728	+	0.324960i	$0.894649\pi$
$180$		0			0
$181$	−7.60790			−0.565491			−0.282746	−	0.959195i	$-0.591245\pi$
$181$	−7.60790			−0.565491			−0.282746	+	0.959195i	$0.591245\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.44156			0.253028
$186$		0			0
$187$	−8.70924	−	7.14688i	−0.636883	−	0.522632i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	7.78923i		−	0.563609i	−0.959472	−	0.281805i	$-0.909067\pi$
$191$		−	7.78923i		−	0.563609i	0.959472	−	0.281805i	$-0.0909330\pi$
$192$		0			0
$193$		−	11.4102i		−	0.821322i	−0.911788	−	0.410661i	$-0.865298\pi$
$193$		−	11.4102i		−	0.821322i	0.911788	−	0.410661i	$-0.134702\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	12.0552i		−	0.858894i	−0.903092	−	0.429447i	$-0.858709\pi$
$197$		−	12.0552i		−	0.858894i	0.903092	−	0.429447i	$-0.141291\pi$
$198$		0			0
$199$			4.85312i			0.344028i	0.985094	+	0.172014i	$0.0550275\pi$
$199$			4.85312i			0.344028i	−0.985094	+	0.172014i	$0.944973\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			0.806953i			0.0566370i
$204$		0			0
$205$		−	0.886909i		−	0.0619444i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−10.8826	+	13.2616i	−0.752767	+	0.917327i
$210$		0			0
$211$	4.72905			0.325561			0.162781	−	0.986662i	$-0.447954\pi$
$211$	4.72905			0.325561			0.162781	+	0.986662i	$0.447954\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2.27392			−0.155080
$216$		0			0
$217$			0.697750i			0.0473663i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	13.0699			0.879177
$222$		0			0
$223$		−	11.1469i		−	0.746451i	−0.927741	−	0.373225i	$-0.878252\pi$
$223$		−	11.1469i		−	0.746451i	0.927741	−	0.373225i	$-0.121748\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−7.10583			−0.471631			−0.235815	−	0.971798i	$-0.575776\pi$
$227$	−7.10583			−0.471631			−0.235815	+	0.971798i	$0.575776\pi$
$228$		0			0
$229$	−8.75479			−0.578533			−0.289266	−	0.957249i	$-0.593411\pi$
$229$	−8.75479			−0.578533			−0.289266	+	0.957249i	$0.593411\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			21.6774i			1.42013i	0.704134	+	0.710067i	$0.251335\pi$
$233$			21.6774i			1.42013i	−0.704134	+	0.710067i	$0.748665\pi$
$234$		0			0
$235$		−	7.14688i		−	0.466211i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	2.75890			0.178458			0.0892292	−	0.996011i	$-0.471560\pi$
$239$	2.75890			0.178458			0.0892292	+	0.996011i	$0.471560\pi$
$240$		0			0
$241$		−	3.98019i		−	0.256386i	−0.991749	−	0.128193i	$-0.959082\pi$
$241$		−	3.98019i		−	0.256386i	0.991749	−	0.128193i	$-0.0409178\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	7.60171			0.485655
$246$		0			0
$247$		−	19.9017i		−	1.26631i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	18.3775i		−	1.15998i	−0.814625	−	0.579988i	$-0.803057\pi$
$251$		−	18.3775i		−	1.15998i	0.814625	−	0.579988i	$-0.196943\pi$
$252$		0			0
$253$	2.00000	+	1.64122i	0.125739	+	0.103183i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−13.0699			−0.815278			−0.407639	−	0.913143i	$-0.633648\pi$
$257$	−13.0699			−0.815278			−0.407639	+	0.913143i	$0.633648\pi$
$258$		0			0
$259$	0.697750			0.0433561
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−13.5876			−0.837849			−0.418925	−	0.908021i	$-0.637593\pi$
$263$	−13.5876			−0.837849			−0.418925	+	0.908021i	$0.637593\pi$
$264$		0			0
$265$	−13.0977			−0.804587
$266$		0			0
$267$		0			0
$268$		0			0
$269$	21.6046			1.31726			0.658628	−	0.752468i	$-0.271137\pi$
$269$	21.6046			1.31726			0.658628	+	0.752468i	$0.271137\pi$
$270$		0			0
$271$	−21.3552			−1.29724			−0.648618	−	0.761114i	$-0.724653\pi$
$271$	−21.3552			−1.29724			−0.648618	+	0.761114i	$0.724653\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	8.00328	−	9.75285i	0.482616	−	0.588119i
$276$		0			0
$277$		−	15.5230i		−	0.932684i	−0.884604	−	0.466342i	$-0.845572\pi$
$277$		−	15.5230i		−	0.932684i	0.884604	−	0.466342i	$-0.154428\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			23.6113i			1.40854i	0.709935	+	0.704268i	$0.248724\pi$
$281$			23.6113i			1.40854i	−0.709935	+	0.704268i	$0.751276\pi$
$282$		0			0
$283$	27.1929			1.61645			0.808224	−	0.588875i	$-0.200429\pi$
$283$	27.1929			1.61645			0.808224	+	0.588875i	$0.200429\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	0.179814i		−	0.0106141i
$288$		0			0
$289$	5.46102			0.321236
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	28.5407i		−	1.66737i	−0.552242	−	0.833684i	$-0.686228\pi$
$293$		−	28.5407i		−	1.66737i	0.552242	−	0.833684i	$-0.313772\pi$
$294$		0			0
$295$			6.29377i			0.366437i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.00139			−0.173575
$300$		0			0
$301$	−0.461020			−0.0265728
$302$		0			0
$303$		0			0
$304$		0			0
$305$			0.824944i			0.0472362i
$306$		0			0
$307$	−3.66388			−0.209109			−0.104554	−	0.994519i	$-0.533342\pi$
$307$	−3.66388			−0.209109			−0.104554	+	0.994519i	$0.533342\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	7.28814i		−	0.413273i	−0.978418	−	0.206636i	$-0.933748\pi$
$311$		−	7.28814i		−	0.413273i	0.978418	−	0.206636i	$-0.0662517\pi$
$312$		0			0
$313$	12.7548			0.720943			0.360472	−	0.932770i	$-0.382616\pi$
$313$	12.7548			0.720943			0.360472	+	0.932770i	$0.382616\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−5.46820			−0.307125			−0.153562	−	0.988139i	$-0.549075\pi$
$317$	−5.46820			−0.307125			−0.153562	+	0.988139i	$0.549075\pi$
$318$		0			0
$319$	9.33095	+	7.65707i	0.522433	+	0.428713i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			17.5705i			0.977650i
$324$		0			0
$325$			14.6361i			0.811862i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	1.44898i		−	0.0798847i
$330$		0			0
$331$			3.90167i			0.214455i	0.994234	+	0.107228i	$0.0341974\pi$
$331$			3.90167i			0.214455i	−0.994234	+	0.107228i	$0.965803\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			13.9491i			0.762123i
$336$		0			0
$337$		−	9.90155i		−	0.539372i	−0.962948	−	0.269686i	$-0.913080\pi$
$337$		−	9.90155i		−	0.539372i	0.962948	−	0.269686i	$-0.0869199\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	8.06821	+	6.62085i	0.436918	+	0.358539i
$342$		0			0
$343$	3.09328			0.167021
$344$		0			0
$345$		0			0
$346$		0			0
$347$	13.0052			0.698154			0.349077	−	0.937094i	$-0.386495\pi$
$347$	13.0052			0.698154			0.349077	+	0.937094i	$0.386495\pi$
$348$		0			0
$349$			18.6162i			0.996504i	0.867032	+	0.498252i	$0.166025\pi$
$349$			18.6162i			0.996504i	−0.867032	+	0.498252i	$0.833975\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	5.62884			0.299593			0.149796	−	0.988717i	$-0.452138\pi$
$353$	5.62884			0.299593			0.149796	+	0.988717i	$0.452138\pi$
$354$		0			0
$355$			9.53898i			0.506276i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−25.3863			−1.33984			−0.669919	−	0.742434i	$-0.733671\pi$
$359$	−25.3863			−1.33984			−0.669919	+	0.742434i	$0.733671\pi$
$360$		0			0
$361$	7.75479			0.408147
$362$		0			0
$363$		0			0
$364$		0			0
$365$			10.8287i			0.566802i
$366$		0			0
$367$			16.7548i			0.874593i	0.899317	+	0.437296i	$0.144064\pi$
$367$			16.7548i			0.874593i	−0.899317	+	0.437296i	$0.855936\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−2.65546			−0.137865
$372$		0			0
$373$		−	1.45206i		−	0.0751848i	−0.999293	−	0.0375924i	$-0.988031\pi$
$373$		−	1.45206i		−	0.0751848i	0.999293	−	0.0375924i	$-0.0119689\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−14.0029			−0.721187
$378$		0			0
$379$			14.2938i			0.734222i	0.930177	+	0.367111i	$0.119653\pi$
$379$			14.2938i			0.734222i	−0.930177	+	0.367111i	$0.880347\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			17.2915i			0.883556i	0.897124	+	0.441778i	$0.145652\pi$
$383$			17.2915i			0.883556i	−0.897124	+	0.441778i	$0.854348\pi$
$384$		0			0
$385$	−0.510183	+	0.621712i	−0.0260013	+	0.0316854i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	12.9093			0.654526			0.327263	−	0.944933i	$-0.393874\pi$
$389$	12.9093			0.654526			0.327263	+	0.944933i	$0.393874\pi$
$390$		0			0
$391$	2.64983			0.134007
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−8.17326			−0.411242
$396$		0			0
$397$	−10.4610			−0.525024			−0.262512	−	0.964929i	$-0.584551\pi$
$397$	−10.4610			−0.525024			−0.262512	+	0.964929i	$0.584551\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2.18728			0.109228			0.0546138	−	0.998508i	$-0.482607\pi$
$401$	2.18728			0.109228			0.0546138	+	0.998508i	$0.482607\pi$
$402$		0			0
$403$	−12.1079			−0.603139
$404$		0			0
$405$		0			0
$406$		0			0
$407$	6.62085	−	8.06821i	0.328183	−	0.399926i
$408$		0			0
$409$		−	5.92136i		−	0.292793i	−0.989226	−	0.146396i	$-0.953233\pi$
$409$		−	5.92136i		−	0.292793i	0.989226	−	0.146396i	$-0.0467675\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			1.27601i			0.0627886i
$414$		0			0
$415$	−10.5233			−0.516567
$416$		0			0
$417$		0			0
$418$		0			0
$419$			22.7520i			1.11151i	0.831346	+	0.555755i	$0.187571\pi$
$419$			22.7520i			1.11151i	−0.831346	+	0.555755i	$0.812429\pi$
$420$		0			0
$421$	29.8033			1.45253			0.726263	−	0.687417i	$-0.241255\pi$
$421$	29.8033			1.45253			0.726263	+	0.687417i	$0.241255\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	12.9217i		−	0.626793i
$426$		0			0
$427$			0.167251i			0.00809385i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	28.6302			1.37907			0.689534	−	0.724253i	$-0.257815\pi$
$431$	28.6302			1.37907			0.689534	+	0.724253i	$0.257815\pi$
$432$		0			0
$433$	−1.53898			−0.0739586			−0.0369793	−	0.999316i	$-0.511774\pi$
$433$	−1.53898			−0.0739586			−0.0369793	+	0.999316i	$0.511774\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	4.03491i		−	0.193016i
$438$		0			0
$439$	−16.9316			−0.808099			−0.404049	−	0.914737i	$-0.632398\pi$
$439$	−16.9316			−0.808099			−0.404049	+	0.914737i	$0.632398\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	32.8277i		−	1.55969i	−0.625972	−	0.779846i	$-0.715298\pi$
$443$		−	32.8277i		−	1.55969i	0.625972	−	0.779846i	$-0.284702\pi$
$444$		0			0
$445$	12.9220			0.612563
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−40.1427			−1.89445			−0.947226	−	0.320566i	$-0.896127\pi$
$449$	−40.1427			−1.89445			−0.947226	+	0.320566i	$0.896127\pi$
$450$		0			0
$451$	−2.07922	−	1.70623i	−0.0979068	−	0.0803433i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	0.933001i		−	0.0437398i
$456$		0			0
$457$			29.2721i			1.36929i	0.728876	+	0.684646i	$0.240043\pi$
$457$			29.2721i			1.36929i	−0.728876	+	0.684646i	$0.759957\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	9.94823i		−	0.463335i	−0.972795	−	0.231668i	$-0.925582\pi$
$461$		−	9.94823i		−	0.463335i	0.972795	−	0.231668i	$-0.0744182\pi$
$462$		0			0
$463$			19.1469i			0.889831i	0.895573	+	0.444916i	$0.146766\pi$
$463$			19.1469i			0.889831i	−0.895573	+	0.444916i	$0.853234\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			28.5069i			1.31914i	0.751642	+	0.659571i	$0.229262\pi$
$467$			28.5069i			1.31914i	−0.751642	+	0.659571i	$0.770738\pi$
$468$		0			0
$469$			2.82808i			0.130589i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−4.37456	+	5.33086i	−0.201142	+	0.245113i
$474$		0			0
$475$	−19.6760			−0.902795
$476$		0			0
$477$		0			0
$478$		0			0
$479$	30.5582			1.39624			0.698120	−	0.715981i	$-0.254020\pi$
$479$	30.5582			1.39624			0.698120	+	0.715981i	$0.254020\pi$
$480$		0			0
$481$			12.1079i			0.552074i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	6.50807			0.295516
$486$		0			0
$487$		−	26.4610i		−	1.19906i	−0.800351	−	0.599532i	$-0.795354\pi$
$487$		−	26.4610i		−	1.19906i	0.800351	−	0.599532i	$-0.204646\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−40.9079			−1.84615			−0.923074	−	0.384623i	$-0.874332\pi$
$491$	−40.9079			−1.84615			−0.923074	+	0.384623i	$0.874332\pi$
$492$		0			0
$493$	12.3627			0.556788
$494$		0			0
$495$		0			0
$496$		0			0
$497$			1.93396i			0.0867498i
$498$		0			0
$499$		−	8.00000i		−	0.358129i	−0.983837	−	0.179065i	$-0.942693\pi$
$499$		−	8.00000i		−	0.358129i	0.983837	−	0.179065i	$-0.0573071\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−39.9439			−1.78101			−0.890505	−	0.454973i	$-0.849649\pi$
$503$	−39.9439			−1.78101			−0.890505	+	0.454973i	$0.849649\pi$
$504$		0			0
$505$			5.22361i			0.232448i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	17.2838			0.766092			0.383046	−	0.923729i	$-0.374875\pi$
$509$	17.2838			0.766092			0.383046	+	0.923729i	$0.374875\pi$
$510$		0			0
$511$			2.19544i			0.0971207i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			9.57458i			0.421907i
$516$		0			0
$517$	−16.7548	−	13.7491i	−0.736875	−	0.604687i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−15.2034			−0.666074			−0.333037	−	0.942914i	$-0.608073\pi$
$521$	−15.2034			−0.666074			−0.333037	+	0.942914i	$0.608073\pi$
$522$		0			0
$523$	−21.8823			−0.956847			−0.478424	−	0.878129i	$-0.658792\pi$
$523$	−21.8823			−0.956847			−0.478424	+	0.878129i	$0.658792\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	10.6897			0.465650
$528$		0			0
$529$	22.3915			0.973543
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3.12028			0.135154
$534$		0			0
$535$	10.7885			0.466426
$536$		0			0
$537$		0			0
$538$		0			0
$539$	14.6241	−	17.8211i	0.629906	−	0.767607i
$540$		0			0
$541$			38.1541i			1.64037i	0.572095	+	0.820187i	$0.306131\pi$
$541$			38.1541i			1.64037i	−0.572095	+	0.820187i	$0.693869\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			18.4195i			0.789006i
$546$		0			0
$547$	−25.6082			−1.09493			−0.547464	−	0.836829i	$-0.684407\pi$
$547$	−25.6082			−1.09493			−0.547464	+	0.836829i	$0.684407\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	18.8248i		−	0.801963i
$552$		0			0
$553$	−1.65707			−0.0704657
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	20.6100i		−	0.873272i	−0.899638	−	0.436636i	$-0.856170\pi$
$557$		−	20.6100i		−	0.873272i	0.899638	−	0.436636i	$-0.143830\pi$
$558$		0			0
$559$		−	8.00000i		−	0.338364i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	23.9373			1.00884			0.504419	−	0.863459i	$-0.331707\pi$
$563$	23.9373			1.00884			0.504419	+	0.863459i	$0.331707\pi$
$564$		0			0
$565$	9.50958			0.400071
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	19.8825i		−	0.833517i	−0.909017	−	0.416759i	$-0.863166\pi$
$569$		−	19.8825i		−	0.833517i	0.909017	−	0.416759i	$-0.136834\pi$
$570$		0			0
$571$	−13.7546			−0.575612			−0.287806	−	0.957689i	$-0.592926\pi$
$571$	−13.7546			−0.575612			−0.287806	+	0.957689i	$0.592926\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			2.96735i			0.123747i
$576$		0			0
$577$	−2.51018			−0.104500			−0.0522501	−	0.998634i	$-0.516639\pi$
$577$	−2.51018			−0.104500			−0.0522501	+	0.998634i	$0.516639\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2.13351			−0.0885130
$582$		0			0
$583$	−25.1973	+	30.7056i	−1.04357	+	1.27170i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	11.6358i		−	0.480262i	−0.970741	−	0.240131i	$-0.922810\pi$
$587$		−	11.6358i		−	0.480262i	0.970741	−	0.240131i	$-0.0771903\pi$
$588$		0			0
$589$		−	16.2773i		−	0.670693i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			19.0915i			0.783992i	0.919967	+	0.391996i	$0.128215\pi$
$593$			19.0915i			0.783992i	−0.919967	+	0.391996i	$0.871785\pi$
$594$		0			0
$595$			0.823715i			0.0337690i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	32.2274i		−	1.31678i	−0.752678	−	0.658389i	$-0.771238\pi$
$599$		−	32.2274i		−	1.31678i	0.752678	−	0.658389i	$-0.228762\pi$
$600$		0			0
$601$			40.6823i			1.65946i	0.558162	+	0.829732i	$0.311507\pi$
$601$			40.6823i			1.65946i	−0.558162	+	0.829732i	$0.688493\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.34792	+	11.7987i	0.0954564	+	0.479685i
$606$		0			0
$607$	0.221727			0.00899963			0.00449981	−	0.999990i	$-0.498568\pi$
$607$	0.221727			0.00899963			0.00449981	+	0.999990i	$0.498568\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	25.1438			1.01721
$612$		0			0
$613$			6.94087i			0.280339i	0.990128	+	0.140170i	$0.0447648\pi$
$613$			6.94087i			0.280339i	−0.990128	+	0.140170i	$0.955235\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	10.5614			0.425184			0.212592	−	0.977141i	$-0.431809\pi$
$617$	10.5614			0.425184			0.212592	+	0.977141i	$0.431809\pi$
$618$		0			0
$619$			35.0486i			1.40872i	0.709843	+	0.704360i	$0.248766\pi$
$619$			35.0486i			1.40872i	−0.709843	+	0.704360i	$0.751234\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	2.61985			0.104962
$624$		0			0
$625$	8.48982			0.339593
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	10.6897i		−	0.426225i
$630$		0			0
$631$			42.2644i			1.68252i	0.540632	+	0.841259i	$0.318185\pi$
$631$			42.2644i			1.68252i	−0.540632	+	0.841259i	$0.681815\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−19.9720			−0.792563
$636$		0			0
$637$			26.7440i			1.05963i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	25.2068			0.995609			0.497805	−	0.867289i	$-0.334140\pi$
$641$	25.2068			0.995609			0.497805	+	0.867289i	$0.334140\pi$
$642$		0			0
$643$			3.24521i			0.127979i	0.997951	+	0.0639893i	$0.0203824\pi$
$643$			3.24521i			0.127979i	−0.997951	+	0.0639893i	$0.979618\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	1.35344i		−	0.0532093i	−0.999646	−	0.0266047i	$-0.991530\pi$
$647$		−	1.35344i		−	0.0532093i	0.999646	−	0.0266047i	$-0.00846953\pi$
$648$		0			0
$649$	14.7548	+	12.1079i	0.579176	+	0.475278i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−29.0995			−1.13875			−0.569375	−	0.822078i	$-0.692815\pi$
$653$	−29.0995			−1.13875			−0.569375	+	0.822078i	$0.692815\pi$
$654$		0			0
$655$	−4.33670			−0.169449
$656$		0			0
$657$		0			0
$658$		0			0
$659$	6.96678			0.271387			0.135694	−	0.990751i	$-0.456674\pi$
$659$	6.96678			0.271387			0.135694	+	0.990751i	$0.456674\pi$
$660$		0			0
$661$	21.9017			0.851876			0.425938	−	0.904752i	$-0.359944\pi$
$661$	21.9017			0.851876			0.425938	+	0.904752i	$0.359944\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1.25428			0.0486388
$666$		0			0
$667$	−2.83898			−0.109926
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.93396	+	1.58702i	0.0746596	+	0.0612664i
$672$		0			0
$673$			32.2893i			1.24466i	0.782754	+	0.622331i	$0.213814\pi$
$673$			32.2893i			1.24466i	−0.782754	+	0.622331i	$0.786186\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			39.8544i			1.53173i	0.643001	+	0.765865i	$0.277689\pi$
$677$			39.8544i			1.53173i	−0.643001	+	0.765865i	$0.722311\pi$
$678$		0			0
$679$	1.31946			0.0506363
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	31.3936i		−	1.20124i	−0.799533	−	0.600622i	$-0.794920\pi$
$683$		−	31.3936i		−	1.20124i	0.799533	−	0.600622i	$-0.205080\pi$
$684$		0			0
$685$	11.9017			0.454740
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	46.0798i		−	1.75550i
$690$		0			0
$691$			36.5581i			1.39074i	0.718653	+	0.695369i	$0.244759\pi$
$691$			36.5581i			1.39074i	−0.718653	+	0.695369i	$0.755241\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−18.6205			−0.706314
$696$		0			0
$697$	−2.75479			−0.104345
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	46.6483i		−	1.76188i	−0.473228	−	0.880940i	$-0.656911\pi$
$701$		−	46.6483i		−	1.76188i	0.473228	−	0.880940i	$-0.343089\pi$
$702$		0			0
$703$	−16.2773			−0.613909
$704$		0			0
$705$		0			0
$706$		0			0
$707$			1.05905i			0.0398296i
$708$		0			0
$709$	7.04856			0.264714			0.132357	−	0.991202i	$-0.457745\pi$
$709$	7.04856			0.264714			0.132357	+	0.991202i	$0.457745\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−2.45479			−0.0919326
$714$		0			0
$715$	−10.7885	−	8.85312i	−0.403466	−	0.331088i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			50.9985i			1.90192i	0.309311	+	0.950961i	$0.399902\pi$
$719$			50.9985i			1.90192i	−0.309311	+	0.950961i	$0.600098\pi$
$720$		0			0
$721$			1.94117i			0.0722931i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			13.8441i			0.514157i
$726$		0			0
$727$			19.1469i			0.710119i	0.934844	+	0.355059i	$0.115539\pi$
$727$			19.1469i			0.710119i	−0.934844	+	0.355059i	$0.884461\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			7.06293i			0.261232i
$732$		0			0
$733$		−	16.8424i		−	0.622089i	−0.950395	−	0.311045i	$-0.899321\pi$
$733$		−	16.8424i		−	0.622089i	0.950395	−	0.311045i	$-0.100679\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	32.7017	+	26.8353i	1.20458	+	0.988491i
$738$		0			0
$739$	1.45751			0.0536154			0.0268077	−	0.999641i	$-0.491466\pi$
$739$	1.45751			0.0536154			0.0268077	+	0.999641i	$0.491466\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	33.4561			1.22739			0.613693	−	0.789544i	$-0.289683\pi$
$743$	33.4561			1.22739			0.613693	+	0.789544i	$0.289683\pi$
$744$		0			0
$745$		−	0.962946i		−	0.0352796i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	2.18728			0.0799214
$750$		0			0
$751$		−	15.2452i		−	0.556306i	−0.960537	−	0.278153i	$-0.910278\pi$
$751$		−	15.2452i		−	0.556306i	0.960537	−	0.278153i	$-0.0897222\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−6.03840			−0.219760
$756$		0			0
$757$	21.9017			0.796030			0.398015	−	0.917379i	$-0.369699\pi$
$757$	21.9017			0.796030			0.398015	+	0.917379i	$0.369699\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			11.4667i			0.415669i	0.978164	+	0.207834i	$0.0666415\pi$
$761$			11.4667i			0.415669i	−0.978164	+	0.207834i	$0.933358\pi$
$762$		0			0
$763$			3.73442i			0.135195i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−22.1424			−0.799517
$768$		0			0
$769$		−	32.9871i		−	1.18955i	−0.803894	−	0.594773i	$-0.797242\pi$
$769$		−	32.9871i		−	1.18955i	0.803894	−	0.594773i	$-0.202758\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−6.72248			−0.241791			−0.120895	−	0.992665i	$-0.538577\pi$
$773$	−6.72248			−0.241791			−0.120895	+	0.992665i	$0.538577\pi$
$774$		0			0
$775$			11.9706i			0.429997i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			4.19474i			0.150292i
$780$		0			0
$781$	22.3627	+	18.3510i	0.800200	+	0.656652i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	15.2034			0.542633
$786$		0			0
$787$	31.9730			1.13972			0.569858	−	0.821743i	$-0.306998\pi$
$787$	31.9730			1.13972			0.569858	+	0.821743i	$0.306998\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1.92800			0.0685516
$792$		0			0
$793$	−2.90228			−0.103063
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−35.6075			−1.26128			−0.630642	−	0.776074i	$-0.717208\pi$
$797$	−35.6075			−1.26128			−0.630642	+	0.776074i	$0.717208\pi$
$798$		0			0
$799$	−22.1986			−0.785331
$800$		0			0
$801$		0			0
$802$		0			0
$803$	25.3863	+	20.8323i	0.895864	+	0.735154i
$804$		0			0
$805$		−	0.189159i		−	0.00666697i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			29.7472i			1.04586i	0.852377	+	0.522928i	$0.175160\pi$
$809$			29.7472i			1.04586i	−0.852377	+	0.522928i	$0.824840\pi$
$810$		0			0
$811$	−15.6958			−0.551153			−0.275576	−	0.961279i	$-0.588869\pi$
$811$	−15.6958			−0.551153			−0.275576	+	0.961279i	$0.588869\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			6.13303i			0.214831i
$816$		0			0
$817$	10.7548			0.376262
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	29.1648i		−	1.01786i	−0.860809	−	0.508929i	$-0.830042\pi$
$821$		−	29.1648i		−	1.01786i	0.860809	−	0.508929i	$-0.169958\pi$
$822$		0			0
$823$		−	16.7548i		−	0.584035i	−0.956413	−	0.292018i	$-0.905673\pi$
$823$		−	16.7548i		−	0.584035i	0.956413	−	0.292018i	$-0.0943266\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.55241			−0.0539827			−0.0269913	−	0.999636i	$-0.508593\pi$
$827$	−1.55241			−0.0539827			−0.0269913	+	0.999636i	$0.508593\pi$
$828$		0			0
$829$	11.1469			0.387147			0.193574	−	0.981086i	$-0.437992\pi$
$829$	11.1469			0.387147			0.193574	+	0.981086i	$0.437992\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	23.6113i		−	0.818085i
$834$		0			0
$835$	−5.65617			−0.195740
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	2.28644i		−	0.0789368i	−0.999221	−	0.0394684i	$-0.987434\pi$
$839$		−	2.28644i		−	0.0789368i	0.999221	−	0.0394684i	$-0.0125664\pi$
$840$		0			0
$841$	15.7548			0.543269
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.97287			0.0678689
$846$		0			0
$847$	0.476023	+	2.39210i	0.0163563	+	0.0821934i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			2.45479i			0.0841491i
$852$		0			0
$853$			29.1395i			0.997718i	0.866683	+	0.498859i	$0.166247\pi$
$853$			29.1395i			0.997718i	−0.866683	+	0.498859i	$0.833753\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	36.2350i		−	1.23776i	−0.785484	−	0.618882i	$-0.787586\pi$
$857$		−	36.2350i		−	1.23776i	0.785484	−	0.618882i	$-0.212414\pi$
$858$		0			0
$859$		−	49.5096i		−	1.68925i	−0.535362	−	0.844623i	$-0.679825\pi$
$859$		−	49.5096i		−	1.68925i	0.535362	−	0.844623i	$-0.320175\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	21.0390i		−	0.716174i	−0.933688	−	0.358087i	$-0.883429\pi$
$863$		−	21.0390i		−	0.716174i	0.933688	−	0.358087i	$-0.116571\pi$
$864$		0			0
$865$			19.3706i			0.658618i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−15.7237	+	19.1610i	−0.533390	+	0.649992i
$870$		0			0
$871$	−49.0752			−1.66285
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.13487			−0.0721717
$876$		0			0
$877$		−	5.16705i		−	0.174479i	−0.996187	−	0.0872395i	$-0.972195\pi$
$877$		−	5.16705i		−	0.174479i	0.996187	−	0.0872395i	$-0.0278045\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−27.3941			−0.922931			−0.461465	−	0.887158i	$-0.652676\pi$
$881$	−27.3941			−0.922931			−0.461465	+	0.887158i	$0.652676\pi$
$882$		0			0
$883$			49.6768i			1.67176i	0.548913	+	0.835879i	$0.315042\pi$
$883$			49.6768i			1.67176i	−0.548913	+	0.835879i	$0.684958\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−50.2877			−1.68849			−0.844247	−	0.535954i	$-0.819952\pi$
$887$	−50.2877			−1.68849			−0.844247	+	0.535954i	$0.819952\pi$
$888$		0			0
$889$	−4.04916			−0.135805
$890$		0			0
$891$		0			0
$892$		0			0
$893$			33.8021i			1.13114i
$894$		0			0
$895$		−	9.50958i		−	0.317870i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−11.4528			−0.381971
$900$		0			0
$901$			40.6823i			1.35532i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	8.32031			0.276576
$906$		0			0
$907$			36.7548i			1.22042i	0.792239	+	0.610211i	$0.208915\pi$
$907$			36.7548i			1.22042i	−0.792239	+	0.610211i	$0.791085\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			46.3564i			1.53586i	0.640536	+	0.767928i	$0.278712\pi$
$911$			46.3564i			1.53586i	−0.640536	+	0.767928i	$0.721288\pi$
$912$		0			0
$913$	−20.2446	+	24.6702i	−0.669999	+	0.816465i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−0.879234			−0.0290349
$918$		0			0
$919$	−59.4637			−1.96153			−0.980763	−	0.195203i	$-0.937464\pi$
$919$	−59.4637			−1.96153			−0.980763	+	0.195203i	$0.937464\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−33.5596			−1.10463
$924$		0			0
$925$	11.9706			0.393591
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−22.7520			−0.746470			−0.373235	−	0.927737i	$-0.621751\pi$
$929$	−22.7520			−0.746470			−0.373235	+	0.927737i	$0.621751\pi$
$930$		0			0
$931$	−35.9532			−1.17832
$932$		0			0
$933$		0			0
$934$		0			0
$935$	9.52477	+	7.81612i	0.311493	+	0.255614i
$936$		0			0
$937$		−	46.1711i		−	1.50834i	−0.656677	−	0.754172i	$-0.728039\pi$
$937$		−	46.1711i		−	1.50834i	0.656677	−	0.754172i	$-0.271961\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			56.0340i			1.82666i	0.407226	+	0.913328i	$0.366496\pi$
$941$			56.0340i			1.82666i	−0.407226	+	0.913328i	$0.633504\pi$
$942$		0			0
$943$	0.632613			0.0206007
$944$		0			0
$945$		0			0
$946$		0			0
$947$			19.2567i			0.625759i	0.949793	+	0.312879i	$0.101294\pi$
$947$			19.2567i			0.625759i	−0.949793	+	0.312879i	$0.898706\pi$
$948$		0			0
$949$	−38.0971			−1.23668
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	12.9157i		−	0.418381i	−0.977875	−	0.209190i	$-0.932917\pi$
$953$		−	12.9157i		−	0.418381i	0.977875	−	0.209190i	$-0.0670829\pi$
$954$		0			0
$955$			8.51861i			0.275656i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	2.41297			0.0779190
$960$		0			0
$961$	21.0971			0.680552
$962$		0			0
$963$		0			0
$964$		0			0
$965$			12.4786i			0.401701i
$966$		0			0
$967$	45.3276			1.45764			0.728819	−	0.684706i	$-0.240069\pi$
$967$	45.3276			1.45764			0.728819	+	0.684706i	$0.240069\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			4.82188i			0.154742i	0.997002	+	0.0773708i	$0.0246525\pi$
$971$			4.82188i			0.154742i	−0.997002	+	0.0773708i	$0.975347\pi$
$972$		0			0
$973$	−3.77516			−0.121026
$974$		0			0
$975$		0			0
$976$		0			0
$977$	57.4797			1.83894			0.919469	−	0.393163i	$-0.128619\pi$
$977$	57.4797			1.83894			0.919469	+	0.393163i	$0.128619\pi$
$978$		0			0
$979$	24.8594	−	30.2938i	0.794509	−	0.968193i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	61.4338i		−	1.95943i	−0.200388	−	0.979717i	$-0.564220\pi$
$983$		−	61.4338i		−	1.95943i	0.200388	−	0.979717i	$-0.435780\pi$
$984$		0			0
$985$			13.1840i			0.420077i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	1.62194i		−	0.0515747i
$990$		0			0
$991$			41.4407i			1.31641i	0.752841	+	0.658203i	$0.228683\pi$
$991$			41.4407i			1.31641i	−0.752841	+	0.658203i	$0.771317\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	5.30756i		−	0.168261i
$996$		0			0
$997$		−	6.05396i		−	0.191731i	−0.995394	−	0.0958654i	$-0.969438\pi$
$997$		−	6.05396i		−	0.191731i	0.995394	−	0.0958654i	$-0.0305619\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3168.2.o.f.703.12	yes	24
3.2	odd	2	inner	3168.2.o.f.703.14	yes	24
4.3	odd	2	inner	3168.2.o.f.703.10	yes	24
11.10	odd	2	inner	3168.2.o.f.703.11	yes	24
12.11	even	2	inner	3168.2.o.f.703.16	yes	24
33.32	even	2	inner	3168.2.o.f.703.13	yes	24
44.43	even	2	inner	3168.2.o.f.703.9	✓	24
132.131	odd	2	inner	3168.2.o.f.703.15	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3168.2.o.f.703.9	✓	24	44.43	even	2	inner
3168.2.o.f.703.10	yes	24	4.3	odd	2	inner
3168.2.o.f.703.11	yes	24	11.10	odd	2	inner
3168.2.o.f.703.12	yes	24	1.1	even	1	trivial
3168.2.o.f.703.13	yes	24	33.32	even	2	inner
3168.2.o.f.703.14	yes	24	3.2	odd	2	inner
3168.2.o.f.703.15	yes	24	132.131	odd	2	inner
3168.2.o.f.703.16	yes	24	12.11	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3168 = 2^{5} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3168.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.09364 q^{5} -0.221727 q^{7} +(-2.10394 + 2.56387i) q^{11} -3.84759i q^{13} +3.39691i q^{17} +5.17250 q^{19} -0.780070i q^{23} -3.80395 q^{25} -3.63940i q^{29} -3.14688i q^{31} +0.242490 q^{35} -3.14688 q^{37} +0.810970i q^{41} +2.07922 q^{43} +6.53495i q^{47} -6.95084 q^{49} +11.9763 q^{53} +(2.30095 - 2.80395i) q^{55} -5.75488i q^{59} -0.754311i q^{61} +4.20788i q^{65} -12.7548i q^{67} -8.72223i q^{71} -9.90155i q^{73} +(0.466500 - 0.568480i) q^{77} +7.47345 q^{79} +9.62224 q^{83} -3.71499i q^{85} -11.8156 q^{89} +0.853115i q^{91} -5.65685 q^{95} -5.95084 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 40 q^{25} + 40 q^{49} + 64 q^{97}+O(q^{100}) \) 24 * q + 40 * q^25 + 40 * q^49 + 64 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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