LMFDB - Embedded newform 3276.2.bi.c.1945.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3276,2,Mod(1945,3276)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3276, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 2, 1]))

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

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

chi := DirichletCharacter("3276.1945");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$3276 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3276.bi (of order $4$ , degree $2$ , 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:	$26.1589917022$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} + 36x^{14} + 472x^{12} + 2912x^{10} + 8914x^{8} + 13164x^{6} + 8828x^{4} + 2648x^{2} + 289$ x^16 + 36x^14 + 472x^12 + 2912x^10 + 8914x^8 + 13164x^6 + 8828x^4 + 2648*x^2 + 289
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{2}$
Twist minimal:	no (minimal twist has level 364)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1945.6
Root			$-1.93284i$ of defining polynomial
Character	$\chi$	$=$	3276.1945
Dual form			3276.2.bi.c.2449.6

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.32637 - 1.32637i) q^{5} +(0.818817 + 2.51586i) q^{7} +(-1.71151 + 1.71151i) q^{11} +(-3.41311 - 1.16220i) q^{13} -2.93284 q^{17} +(1.56404 - 1.56404i) q^{19} +0.481482i q^{23} +1.48148i q^{25} -2.69704 q^{29} +(-4.25921 + 4.25921i) q^{31} +(4.42301 + 2.25090i) q^{35} +(-5.58707 + 5.58707i) q^{37} +(2.72624 - 2.72624i) q^{41} -10.4771i q^{43} +(-8.39025 - 8.39025i) q^{47} +(-5.65908 + 4.12005i) q^{49} -13.6846 q^{53} +4.54019i q^{55} +(6.30994 + 6.30994i) q^{59} +9.47895i q^{61} +(-6.06855 + 2.98553i) q^{65} +(2.42301 + 2.42301i) q^{67} +(-9.27601 - 9.27601i) q^{71} +(0.480268 + 0.480268i) q^{73} +(-5.70732 - 2.90450i) q^{77} +7.08302 q^{79} +(-2.89534 + 2.89534i) q^{83} +(-3.89003 + 3.89003i) q^{85} +(1.21194 + 1.21194i) q^{89} +(0.129223 - 9.53852i) q^{91} -4.14899i q^{95} +(7.75805 - 7.75805i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$16 q - 2 q^{7} - 12 q^{11} - 20 q^{29} + 40 q^{35} - 28 q^{53} + 20 q^{65} + 8 q^{67} + 4 q^{71} + 4 q^{79} + 4 q^{85} + 10 q^{91}+O(q^{100})$ 16 * q - 2 * q^7 - 12 * q^11 - 20 * q^29 + 40 * q^35 - 28 * q^53 + 20 * q^65 + 8 * q^67 + 4 * q^71 + 4 * q^79 + 4 * q^85 + 10 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3276\mathbb{Z}\right)^\times$ .

$n$	$1639$	$2017$	$2341$	$2549$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$	$-1$	$1$

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
$5$	1.32637	−	1.32637i	0.593171	−	0.593171i	−0.345316	−	0.938487i	$-0.612228\pi$
$5$	1.32637	−	1.32637i	0.593171	−	0.593171i	0.938487	+	0.345316i	$0.112228\pi$
$6$		0			0
$7$	0.818817	+	2.51586i	0.309484	+	0.950905i
$7$	0.818817	+	2.51586i	0.309484	+	0.950905i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.71151	+	1.71151i	−0.516039	+	0.516039i	−0.916370	−	0.400332i	$-0.868895\pi$
$11$	−1.71151	+	1.71151i	−0.516039	+	0.516039i	0.400332	+	0.916370i	$0.368895\pi$
$12$		0			0
$13$	−3.41311	−	1.16220i	−0.946625	−	0.322337i
$13$	−3.41311	−	1.16220i	−0.946625	−	0.322337i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.93284			−0.711318			−0.355659	−	0.934616i	$-0.615743\pi$
$17$	−2.93284			−0.711318			−0.355659	+	0.934616i	$0.615743\pi$
$18$		0			0
$19$	1.56404	−	1.56404i	0.358815	−	0.358815i	−0.504561	−	0.863376i	$-0.668346\pi$
$19$	1.56404	−	1.56404i	0.358815	−	0.358815i	0.863376	+	0.504561i	$0.168346\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			0.481482i			0.100396i	0.998739	+	0.0501980i	$0.0159852\pi$
$23$			0.481482i			0.100396i	−0.998739	+	0.0501980i	$0.984015\pi$
$24$		0			0
$25$			1.48148i			0.296296i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.69704			−0.500828			−0.250414	−	0.968139i	$-0.580567\pi$
$29$	−2.69704			−0.500828			−0.250414	+	0.968139i	$0.580567\pi$
$30$		0			0
$31$	−4.25921	+	4.25921i	−0.764976	+	0.764976i	−0.977217	−	0.212241i	$-0.931924\pi$
$31$	−4.25921	+	4.25921i	−0.764976	+	0.764976i	0.212241	+	0.977217i	$0.431924\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	4.42301	+	2.25090i	0.747626	+	0.380472i
$36$		0			0
$37$	−5.58707	+	5.58707i	−0.918509	+	0.918509i	−0.996921	−	0.0784124i	$-0.975015\pi$
$37$	−5.58707	+	5.58707i	−0.918509	+	0.918509i	0.0784124	+	0.996921i	$0.475015\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.72624	−	2.72624i	0.425767	−	0.425767i	−0.461417	−	0.887184i	$-0.652659\pi$
$41$	2.72624	−	2.72624i	0.425767	−	0.425767i	0.887184	+	0.461417i	$0.152659\pi$
$42$		0			0
$43$		−	10.4771i		−	1.59774i	−0.601502	−	0.798871i	$-0.705431\pi$
$43$		−	10.4771i		−	1.59774i	0.601502	−	0.798871i	$-0.294569\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−8.39025	−	8.39025i	−1.22384	−	1.22384i	−0.966255	−	0.257589i	$-0.917072\pi$
$47$	−8.39025	−	8.39025i	−1.22384	−	1.22384i	−0.257589	−	0.966255i	$-0.582928\pi$
$48$		0			0
$49$	−5.65908	+	4.12005i	−0.808440	+	0.588579i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−13.6846			−1.87972			−0.939859	−	0.341562i	$-0.889044\pi$
$53$	−13.6846			−1.87972			−0.939859	+	0.341562i	$0.889044\pi$
$54$		0			0
$55$			4.54019i			0.612198i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.30994	+	6.30994i	0.821484	+	0.821484i	0.986321	−	0.164837i	$-0.0527097\pi$
$59$	6.30994	+	6.30994i	0.821484	+	0.821484i	−0.164837	+	0.986321i	$0.552710\pi$
$60$		0			0
$61$			9.47895i			1.21366i	0.794833	+	0.606828i	$0.207558\pi$
$61$			9.47895i			1.21366i	−0.794833	+	0.606828i	$0.792442\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−6.06855	+	2.98553i	−0.752711	+	0.370310i
$66$		0			0
$67$	2.42301	+	2.42301i	0.296018	+	0.296018i	0.839452	−	0.543434i	$-0.182876\pi$
$67$	2.42301	+	2.42301i	0.296018	+	0.296018i	−0.543434	+	0.839452i	$0.682876\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−9.27601	−	9.27601i	−1.10086	−	1.10086i	−0.994307	−	0.106553i	$-0.966019\pi$
$71$	−9.27601	−	9.27601i	−1.10086	−	1.10086i	−0.106553	−	0.994307i	$-0.533981\pi$
$72$		0			0
$73$	0.480268	+	0.480268i	0.0562111	+	0.0562111i	0.734654	−	0.678442i	$-0.237345\pi$
$73$	0.480268	+	0.480268i	0.0562111	+	0.0562111i	−0.678442	+	0.734654i	$0.737345\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−5.70732	−	2.90450i	−0.650409	−	0.330998i
$78$		0			0
$79$	7.08302			0.796902			0.398451	−	0.917190i	$-0.369548\pi$
$79$	7.08302			0.796902			0.398451	+	0.917190i	$0.369548\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−2.89534	+	2.89534i	−0.317805	+	0.317805i	−0.847924	−	0.530119i	$-0.822147\pi$
$83$	−2.89534	+	2.89534i	−0.317805	+	0.317805i	0.530119	+	0.847924i	$0.322147\pi$
$84$		0			0
$85$	−3.89003	+	3.89003i	−0.421933	+	0.421933i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.21194	+	1.21194i	0.128465	+	0.128465i	0.768416	−	0.639951i	$-0.221045\pi$
$89$	1.21194	+	1.21194i	0.128465	+	0.128465i	−0.639951	+	0.768416i	$0.721045\pi$
$90$		0			0
$91$	0.129223	−	9.53852i	0.0135463	−	0.999908i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	4.14899i		−	0.425677i
$96$		0			0
$97$	7.75805	−	7.75805i	0.787711	−	0.787711i	−0.193408	−	0.981118i	$-0.561954\pi$
$97$	7.75805	−	7.75805i	0.787711	−	0.787711i	0.981118	+	0.193408i	$0.0619540\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.86459			0.683052			0.341526	−	0.939872i	$-0.389056\pi$
$101$	6.86459			0.683052			0.341526	+	0.939872i	$0.389056\pi$
$102$		0			0
$103$	4.64880			0.458060			0.229030	−	0.973419i	$-0.426445\pi$
$103$	4.64880			0.458060			0.229030	+	0.973419i	$0.426445\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.38970			−0.714389			−0.357195	−	0.934030i	$-0.616267\pi$
$107$	−7.38970			−0.714389			−0.357195	+	0.934030i	$0.616267\pi$
$108$		0			0
$109$	−9.52050	−	9.52050i	−0.911899	−	0.911899i	0.0845230	−	0.996422i	$-0.473063\pi$
$109$	−9.52050	−	9.52050i	−0.911899	−	0.911899i	−0.996422	+	0.0845230i	$0.973063\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−17.9247			−1.68621			−0.843105	−	0.537749i	$-0.819275\pi$
$113$	−17.9247			−1.68621			−0.843105	+	0.537749i	$0.819275\pi$
$114$		0			0
$115$	0.638624	+	0.638624i	0.0595520	+	0.0595520i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.40146	−	7.37860i	−0.220141	−	0.676395i
$120$		0			0
$121$			5.14149i			0.467408i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	8.59685	+	8.59685i	0.768925	+	0.768925i
$126$		0			0
$127$		−	19.8335i		−	1.75994i	−0.475027	−	0.879971i	$-0.657562\pi$
$127$		−	19.8335i		−	1.75994i	0.475027	−	0.879971i	$-0.342438\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			14.7026i			1.28457i	0.766466	+	0.642285i	$0.222013\pi$
$131$			14.7026i			1.28457i	−0.766466	+	0.642285i	$0.777987\pi$
$132$		0			0
$133$	5.21556	+	2.65424i	0.452246	+	0.230151i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	7.83156	−	7.83156i	0.669095	−	0.669095i	−0.288411	−	0.957507i	$-0.593127\pi$
$137$	7.83156	−	7.83156i	0.669095	−	0.669095i	0.957507	+	0.288411i	$0.0931270\pi$
$138$		0			0
$139$		−	3.56503i		−	0.302382i	−0.988505	−	0.151191i	$-0.951689\pi$
$139$		−	3.56503i		−	0.302382i	0.988505	−	0.151191i	$-0.0483109\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	7.83067	−	3.85244i	0.654833	−	0.322157i
$144$		0			0
$145$	−3.57727	+	3.57727i	−0.297077	+	0.297077i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	10.9812	+	10.9812i	0.899611	+	0.899611i	0.995402	−	0.0957906i	$-0.0305379\pi$
$149$	10.9812	+	10.9812i	0.899611	+	0.899611i	−0.0957906	+	0.995402i	$0.530538\pi$
$150$		0			0
$151$	−13.6116	+	13.6116i	−1.10770	+	1.10770i	−0.114245	+	0.993453i	$0.536445\pi$
$151$	−13.6116	+	13.6116i	−1.10770	+	1.10770i	−0.993453	+	0.114245i	$0.963555\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			11.2986i			0.907524i
$156$		0			0
$157$			22.7073i			1.81224i	0.423024	+	0.906119i	$0.360969\pi$
$157$			22.7073i			1.81224i	−0.423024	+	0.906119i	$0.639031\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1.21134	+	0.394246i	−0.0954670	+	0.0310709i
$162$		0			0
$163$	−3.45255	+	3.45255i	−0.270424	+	0.270424i	−0.829271	−	0.558847i	$-0.811244\pi$
$163$	−3.45255	+	3.45255i	−0.270424	+	0.270424i	0.558847	+	0.829271i	$0.311244\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	4.44863	+	4.44863i	0.344246	+	0.344246i	0.857961	−	0.513715i	$-0.171731\pi$
$167$	4.44863	+	4.44863i	0.344246	+	0.344246i	−0.513715	+	0.857961i	$0.671731\pi$
$168$		0			0
$169$	10.2986	+	7.93343i	0.792198	+	0.610264i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−7.49678			−0.569970			−0.284985	−	0.958532i	$-0.591989\pi$
$173$	−7.49678			−0.569970			−0.284985	+	0.958532i	$0.591989\pi$
$174$		0			0
$175$	−3.72720	+	1.21306i	−0.281750	+	0.0916989i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	17.2653i		−	1.29047i	−0.763986	−	0.645233i	$-0.776760\pi$
$179$		−	17.2653i		−	1.29047i	0.763986	−	0.645233i	$-0.223240\pi$
$180$		0			0
$181$	−9.59920			−0.713503			−0.356751	−	0.934199i	$-0.616116\pi$
$181$	−9.59920			−0.713503			−0.356751	+	0.934199i	$0.616116\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			14.8210i			1.08967i
$186$		0			0
$187$	5.01957	−	5.01957i	0.367067	−	0.367067i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−17.9956			−1.30212			−0.651059	−	0.759027i	$-0.725675\pi$
$191$	−17.9956			−1.30212			−0.651059	+	0.759027i	$0.725675\pi$
$192$		0			0
$193$	−3.43112	+	3.43112i	−0.246977	+	0.246977i	−0.819729	−	0.572752i	$-0.805876\pi$
$193$	−3.43112	+	3.43112i	−0.246977	+	0.246977i	0.572752	+	0.819729i	$0.305876\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	4.32751	+	4.32751i	0.308322	+	0.308322i	0.844259	−	0.535936i	$-0.180041\pi$
$197$	4.32751	+	4.32751i	0.308322	+	0.308322i	−0.535936	+	0.844259i	$0.680041\pi$
$198$		0			0
$199$	20.0447			1.42093			0.710464	−	0.703733i	$-0.248485\pi$
$199$	20.0447			1.42093			0.710464	+	0.703733i	$0.248485\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2.20838	−	6.78537i	−0.154998	−	0.476240i
$204$		0			0
$205$		−	7.23201i		−	0.505105i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			5.35372i			0.370325i
$210$		0			0
$211$	3.10361			0.213661			0.106831	−	0.994277i	$-0.465930\pi$
$211$	3.10361			0.213661			0.106831	+	0.994277i	$0.465930\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−13.8965	−	13.8965i	−0.947735	−	0.947735i
$216$		0			0
$217$	−14.2031	−	7.22805i	−0.964167	−	0.490672i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	10.0101	+	3.40855i	0.673351	+	0.229284i
$222$		0			0
$223$	11.0006	−	11.0006i	0.736652	−	0.736652i	−0.235276	−	0.971928i	$-0.575600\pi$
$223$	11.0006	−	11.0006i	0.736652	−	0.736652i	0.971928	+	0.235276i	$0.0755995\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−19.6911	+	19.6911i	−1.30694	+	1.30694i	−0.383331	+	0.923611i	$0.625223\pi$
$227$	−19.6911	+	19.6911i	−1.30694	+	1.30694i	−0.923611	+	0.383331i	$0.874777\pi$
$228$		0			0
$229$	−7.98929	−	7.98929i	−0.527947	−	0.527947i	0.392013	−	0.919960i	$-0.371779\pi$
$229$	−7.98929	−	7.98929i	−0.527947	−	0.527947i	−0.919960	+	0.392013i	$0.871779\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			7.26526i			0.475963i	0.971270	+	0.237982i	$0.0764858\pi$
$233$			7.26526i			0.475963i	−0.971270	+	0.237982i	$0.923514\pi$
$234$		0			0
$235$	−22.2572			−1.45190
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.57699	+	1.57699i	0.102007	+	0.102007i	0.756268	−	0.654262i	$-0.227020\pi$
$239$	1.57699	+	1.57699i	0.102007	+	0.102007i	−0.654262	+	0.756268i	$0.727020\pi$
$240$		0			0
$241$	−12.0489	−	12.0489i	−0.776141	−	0.776141i	0.203032	−	0.979172i	$-0.434921\pi$
$241$	−12.0489	−	12.0489i	−0.776141	−	0.776141i	−0.979172	+	0.203032i	$0.934921\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.04131	+	12.9708i	−0.130415	+	0.828671i
$246$		0			0
$247$	−7.15595	+	3.52050i	−0.455322	+	0.224004i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−28.4881			−1.79815			−0.899076	−	0.437792i	$-0.855760\pi$
$251$	−28.4881			−1.79815			−0.899076	+	0.437792i	$0.855760\pi$
$252$		0			0
$253$	−0.824060	−	0.824060i	−0.0518082	−	0.0518082i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	10.4907			0.654393			0.327197	−	0.944956i	$-0.393896\pi$
$257$	10.4907			0.654393			0.327197	+	0.944956i	$0.393896\pi$
$258$		0			0
$259$	−18.6311	−	9.48148i	−1.15768	−	0.589151i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	9.71787			0.599230			0.299615	−	0.954060i	$-0.403142\pi$
$263$	9.71787			0.599230			0.299615	+	0.954060i	$0.403142\pi$
$264$		0			0
$265$	−18.1508	+	18.1508i	−1.11499	+	1.11499i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			28.2800i			1.72426i	0.506686	+	0.862131i	$0.330870\pi$
$269$			28.2800i			1.72426i	−0.506686	+	0.862131i	$0.669130\pi$
$270$		0			0
$271$	−3.00634	−	3.00634i	−0.182622	−	0.182622i	0.609875	−	0.792497i	$-0.291219\pi$
$271$	−3.00634	−	3.00634i	−0.182622	−	0.182622i	−0.792497	+	0.609875i	$0.791219\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.53557	−	2.53557i	−0.152900	−	0.152900i
$276$		0			0
$277$			12.0460i			0.723773i	0.932222	+	0.361886i	$0.117867\pi$
$277$			12.0460i			0.723773i	−0.932222	+	0.361886i	$0.882133\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−12.7901	+	12.7901i	−0.762996	+	0.762996i	−0.976863	−	0.213867i	$-0.931394\pi$
$281$	−12.7901	+	12.7901i	−0.762996	+	0.762996i	0.213867	+	0.976863i	$0.431394\pi$
$282$		0			0
$283$	10.4296			0.619977			0.309988	−	0.950740i	$-0.399675\pi$
$283$	10.4296			0.619977			0.309988	+	0.950740i	$0.399675\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	9.09112	+	4.62654i	0.536632	+	0.273096i
$288$		0			0
$289$	−8.39846			−0.494027
$290$		0			0
$291$		0			0
$292$		0			0
$293$	6.47904	+	6.47904i	0.378510	+	0.378510i	0.870564	−	0.492055i	$-0.163754\pi$
$293$	6.47904	+	6.47904i	0.378510	+	0.378510i	−0.492055	+	0.870564i	$0.663754\pi$
$294$		0			0
$295$	16.7386			0.974561
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.559579	−	1.64335i	0.0323613	−	0.0950374i
$300$		0			0
$301$	26.3589	−	8.57883i	1.51930	−	0.494475i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	12.5726	+	12.5726i	0.719905	+	0.719905i
$306$		0			0
$307$	7.67725	+	7.67725i	0.438164	+	0.438164i	0.891394	−	0.453230i	$-0.149728\pi$
$307$	7.67725	+	7.67725i	0.438164	+	0.438164i	−0.453230	+	0.891394i	$0.649728\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	0.170758			0.00968278			0.00484139	−	0.999988i	$-0.498459\pi$
$311$	0.170758			0.00968278			0.00484139	+	0.999988i	$0.498459\pi$
$312$		0			0
$313$		−	18.7755i		−	1.06126i	−0.847605	−	0.530628i	$-0.821956\pi$
$313$		−	18.7755i		−	1.06126i	0.847605	−	0.530628i	$-0.178044\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−17.2112	−	17.2112i	−0.966676	−	0.966676i	0.0327865	−	0.999462i	$-0.489562\pi$
$317$	−17.2112	−	17.2112i	−0.966676	−	0.966676i	−0.999462	+	0.0327865i	$0.989562\pi$
$318$		0			0
$319$	4.61600	−	4.61600i	0.258447	−	0.258447i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−4.58707	+	4.58707i	−0.255231	+	0.255231i
$324$		0			0
$325$	1.72178	−	5.05645i	0.0955072	−	0.280482i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	14.2386	−	27.9787i	0.784999	−	1.54252i
$330$		0			0
$331$	1.50603	+	1.50603i	0.0827791	+	0.0827791i	0.747284	−	0.664505i	$-0.231358\pi$
$331$	1.50603	+	1.50603i	0.0827791	+	0.0827791i	−0.664505	+	0.747284i	$0.731358\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	6.42763			0.351179
$336$		0			0
$337$		−	33.5098i		−	1.82539i	−0.408639	−	0.912696i	$-0.633996\pi$
$337$		−	33.5098i		−	1.82539i	0.408639	−	0.912696i	$-0.366004\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	14.5793i		−	0.789515i
$342$		0			0
$343$	−14.9992	−	10.8639i	−0.809882	−	0.586593i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−19.5312			−1.04849			−0.524244	−	0.851568i	$-0.675652\pi$
$347$	−19.5312			−1.04849			−0.524244	+	0.851568i	$0.675652\pi$
$348$		0			0
$349$	4.37364	+	4.37364i	0.234116	+	0.234116i	0.814408	−	0.580293i	$-0.197062\pi$
$349$	4.37364	+	4.37364i	0.234116	+	0.234116i	−0.580293	+	0.814408i	$0.697062\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	18.3598	−	18.3598i	0.977192	−	0.977192i	−0.0225539	−	0.999746i	$-0.507180\pi$
$353$	18.3598	−	18.3598i	0.977192	−	0.977192i	0.999746	+	0.0225539i	$0.00717975\pi$
$354$		0			0
$355$	−24.6068			−1.30600
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−7.68696	+	7.68696i	−0.405702	+	0.405702i	−0.880237	−	0.474535i	$-0.842616\pi$
$359$	−7.68696	+	7.68696i	−0.405702	+	0.405702i	0.474535	+	0.880237i	$0.342616\pi$
$360$		0			0
$361$			14.1076i			0.742504i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1.27403			0.0666856
$366$		0			0
$367$			0.257377i			0.0134350i	0.999977	+	0.00671748i	$0.00213826\pi$
$367$			0.257377i			0.0134350i	−0.999977	+	0.00671748i	$0.997862\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−11.2051	−	34.4284i	−0.581742	−	1.78743i
$372$		0			0
$373$	7.32439			0.379243			0.189621	−	0.981857i	$-0.439274\pi$
$373$	7.32439			0.379243			0.189621	+	0.981857i	$0.439274\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	9.20528	+	3.13450i	0.474096	+	0.161435i
$378$		0			0
$379$	13.4004	+	13.4004i	0.688335	+	0.688335i	0.961864	−	0.273529i	$-0.0881910\pi$
$379$	13.4004	+	13.4004i	0.688335	+	0.688335i	−0.273529	+	0.961864i	$0.588191\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−0.316979	+	0.316979i	−0.0161969	+	0.0161969i	−0.715159	−	0.698962i	$-0.753646\pi$
$383$	−0.316979	+	0.316979i	−0.0161969	+	0.0161969i	0.698962	+	0.715159i	$0.253646\pi$
$384$		0			0
$385$	−11.4225	+	3.71758i	−0.582142	+	0.189466i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			15.1408i			0.767670i	0.923402	+	0.383835i	$0.125397\pi$
$389$			15.1408i			0.767670i	−0.923402	+	0.383835i	$0.874603\pi$
$390$		0			0
$391$		−	1.41211i		−	0.0714134i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	9.39471	−	9.39471i	0.472699	−	0.472699i
$396$		0			0
$397$	8.72158	+	8.72158i	0.437723	+	0.437723i	0.891245	−	0.453522i	$-0.149833\pi$
$397$	8.72158	+	8.72158i	0.437723	+	0.437723i	−0.453522	+	0.891245i	$0.649833\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.02893	+	1.02893i	−0.0513825	+	0.0513825i	−0.732331	−	0.680949i	$-0.761568\pi$
$401$	−1.02893	+	1.02893i	−0.0513825	+	0.0513825i	0.680949	+	0.732331i	$0.261568\pi$
$402$		0			0
$403$	19.4872	−	9.58707i	0.970726	−	0.477566i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	19.1246i		−	0.947972i
$408$		0			0
$409$	10.7275	−	10.7275i	0.530441	−	0.530441i	−0.390263	−	0.920704i	$-0.627616\pi$
$409$	10.7275	−	10.7275i	0.530441	−	0.530441i	0.920704	+	0.390263i	$0.127616\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−10.7082	+	21.0416i	−0.526917	+	1.03539i
$414$		0			0
$415$			7.68059i			0.377025i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			14.4828i			0.707533i	0.935334	+	0.353767i	$0.115099\pi$
$419$			14.4828i			0.707533i	−0.935334	+	0.353767i	$0.884901\pi$
$420$		0			0
$421$	−9.04340	−	9.04340i	−0.440748	−	0.440748i	0.451515	−	0.892263i	$-0.350884\pi$
$421$	−9.04340	−	9.04340i	−0.440748	−	0.440748i	−0.892263	+	0.451515i	$0.850884\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	4.34495i		−	0.210761i
$426$		0			0
$427$	−23.8477	+	7.76153i	−1.15407	+	0.375607i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−8.05409	−	8.05409i	−0.387952	−	0.387952i	0.486005	−	0.873956i	$-0.338454\pi$
$431$	−8.05409	−	8.05409i	−0.387952	−	0.387952i	−0.873956	+	0.486005i	$0.838454\pi$
$432$		0			0
$433$	1.91121			0.0918467			0.0459233	−	0.998945i	$-0.485377\pi$
$433$	1.91121			0.0918467			0.0459233	+	0.998945i	$0.485377\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.753057	+	0.753057i	0.0360236	+	0.0360236i
$438$		0			0
$439$	−16.2953			−0.777732			−0.388866	−	0.921294i	$-0.627133\pi$
$439$	−16.2953			−0.777732			−0.388866	+	0.921294i	$0.627133\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	30.1949			1.43460			0.717302	−	0.696763i	$-0.245377\pi$
$443$	30.1949			1.43460			0.717302	+	0.696763i	$0.245377\pi$
$444$		0			0
$445$	3.21496			0.152404
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−5.33561	+	5.33561i	−0.251803	+	0.251803i	−0.821710	−	0.569906i	$-0.806979\pi$
$449$	−5.33561	+	5.33561i	−0.251803	+	0.251803i	0.569906	+	0.821710i	$0.306979\pi$
$450$		0			0
$451$			9.33196i			0.439425i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−12.4802	−	12.8230i	−0.585081	−	0.601152i
$456$		0			0
$457$	21.7562	+	21.7562i	1.01771	+	1.01771i	0.999840	+	0.0178731i	$0.00568947\pi$
$457$	21.7562	+	21.7562i	1.01771	+	1.01771i	0.0178731	+	0.999840i	$0.494311\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	16.1834	−	16.1834i	0.753737	−	0.753737i	−0.221437	−	0.975175i	$-0.571075\pi$
$461$	16.1834	−	16.1834i	0.753737	−	0.753737i	0.975175	+	0.221437i	$0.0710749\pi$
$462$		0			0
$463$	−6.50843	+	6.50843i	−0.302473	+	0.302473i	−0.841981	−	0.539508i	$-0.818610\pi$
$463$	−6.50843	+	6.50843i	−0.302473	+	0.302473i	0.539508	+	0.841981i	$0.318610\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	10.0647			0.465737			0.232869	−	0.972508i	$-0.425189\pi$
$467$	10.0647			0.465737			0.232869	+	0.972508i	$0.425189\pi$
$468$		0			0
$469$	−4.11195	+	8.07996i	−0.189872	+	0.373098i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	17.9316	+	17.9316i	0.824497	+	0.824497i
$474$		0			0
$475$	2.31709	+	2.31709i	0.106316	+	0.106316i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	3.12421	+	3.12421i	0.142749	+	0.142749i	0.774870	−	0.632121i	$-0.217815\pi$
$479$	3.12421	+	3.12421i	0.142749	+	0.142749i	−0.632121	+	0.774870i	$0.717815\pi$
$480$		0			0
$481$	25.5626	−	12.5760i	1.16555	−	0.573414i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	20.5801i		−	0.934494i
$486$		0			0
$487$	2.75551	+	2.75551i	0.124864	+	0.124864i	0.766777	−	0.641913i	$-0.221859\pi$
$487$	2.75551	+	2.75551i	0.124864	+	0.124864i	−0.641913	+	0.766777i	$0.721859\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			1.39846i			0.0631118i	0.999502	+	0.0315559i	$0.0100462\pi$
$491$			1.39846i			0.0631118i	−0.999502	+	0.0315559i	$0.989954\pi$
$492$		0			0
$493$	7.90998			0.356248
$494$		0			0
$495$		0			0
$496$		0			0
$497$	15.7418	−	30.9325i	0.706114	−	1.38751i
$498$		0			0
$499$	−17.9718	−	17.9718i	−0.804528	−	0.804528i	0.179272	−	0.983800i	$-0.442626\pi$
$499$	−17.9718	−	17.9718i	−0.804528	−	0.804528i	−0.983800	+	0.179272i	$0.942626\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	12.7164i		−	0.566995i	−0.958973	−	0.283497i	$-0.908505\pi$
$503$		−	12.7164i		−	0.566995i	0.958973	−	0.283497i	$-0.0914947\pi$
$504$		0			0
$505$	9.10499	−	9.10499i	0.405167	−	0.405167i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5.38541	−	5.38541i	0.238704	−	0.238704i	−0.577609	−	0.816313i	$-0.696014\pi$
$509$	5.38541	−	5.38541i	0.238704	−	0.238704i	0.816313	+	0.577609i	$0.196014\pi$
$510$		0			0
$511$	−0.815034	+	1.60154i	−0.0360550	+	0.0708478i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	6.16604	−	6.16604i	0.271708	−	0.271708i
$516$		0			0
$517$	28.7199			1.26310
$518$		0			0
$519$		0			0
$520$		0			0
$521$			10.0983i			0.442414i	0.975227	+	0.221207i	$0.0709997\pi$
$521$			10.0983i			0.442414i	−0.975227	+	0.221207i	$0.929000\pi$
$522$		0			0
$523$		−	2.39342i		−	0.104657i	−0.998630	−	0.0523284i	$-0.983336\pi$
$523$		−	2.39342i		−	0.104657i	0.998630	−	0.0523284i	$-0.0166643\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	12.4916	−	12.4916i	0.544141	−	0.544141i
$528$		0			0
$529$	22.7682			0.989921
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−12.4734	+	6.13650i	−0.540282	+	0.265801i
$534$		0			0
$535$	−9.80148	+	9.80148i	−0.423755	+	0.423755i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.63405	−	16.7371i	0.113456	−	0.720916i
$540$		0			0
$541$	−7.29618	+	7.29618i	−0.313687	+	0.313687i	−0.846336	−	0.532649i	$-0.821197\pi$
$541$	−7.29618	+	7.29618i	−0.313687	+	0.313687i	0.532649	+	0.846336i	$0.321197\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−25.2554			−1.08182
$546$		0			0
$547$	31.7047			1.35560			0.677798	−	0.735248i	$-0.262934\pi$
$547$	31.7047			1.35560			0.677798	+	0.735248i	$0.262934\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−4.21827	+	4.21827i	−0.179704	+	0.179704i
$552$		0			0
$553$	5.79970	+	17.8199i	0.246628	+	0.757778i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	17.0416	−	17.0416i	0.722076	−	0.722076i	−0.246952	−	0.969028i	$-0.579429\pi$
$557$	17.0416	−	17.0416i	0.722076	−	0.722076i	0.969028	+	0.246952i	$0.0794290\pi$
$558$		0			0
$559$	−12.1765	+	35.7594i	−0.515011	+	1.51246i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	14.7716			0.622548			0.311274	−	0.950320i	$-0.399244\pi$
$563$	14.7716			0.622548			0.311274	+	0.950320i	$0.399244\pi$
$564$		0			0
$565$	−23.7747	+	23.7747i	−1.00021	+	1.00021i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			15.1321i			0.634371i	0.948363	+	0.317186i	$0.102738\pi$
$569$			15.1321i			0.634371i	−0.948363	+	0.317186i	$0.897262\pi$
$570$		0			0
$571$		−	7.85936i		−	0.328904i	−0.986385	−	0.164452i	$-0.947414\pi$
$571$		−	7.85936i		−	0.328904i	0.986385	−	0.164452i	$-0.0525856\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.713307			−0.0297470
$576$		0			0
$577$	−11.2929	+	11.2929i	−0.470129	+	0.470129i	−0.901956	−	0.431827i	$-0.857869\pi$
$577$	−11.2929	+	11.2929i	−0.470129	+	0.470129i	0.431827	+	0.901956i	$0.357869\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−9.65502	−	4.91351i	−0.400558	−	0.203847i
$582$		0			0
$583$	23.4212	−	23.4212i	0.970008	−	0.970008i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	7.36352	−	7.36352i	0.303925	−	0.303925i	−0.538622	−	0.842547i	$-0.681055\pi$
$587$	7.36352	−	7.36352i	0.303925	−	0.303925i	0.842547	+	0.538622i	$0.181055\pi$
$588$		0			0
$589$			13.3231i			0.548970i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−31.3736	−	31.3736i	−1.28836	−	1.28836i	−0.935784	−	0.352574i	$-0.885307\pi$
$593$	−31.3736	−	31.3736i	−1.28836	−	1.28836i	−0.352574	−	0.935784i	$-0.614693\pi$
$594$		0			0
$595$	−12.9720	−	6.60154i	−0.531799	−	0.270637i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	20.9284			0.855111			0.427555	−	0.903989i	$-0.359375\pi$
$599$	20.9284			0.855111			0.427555	+	0.903989i	$0.359375\pi$
$600$		0			0
$601$			12.0322i			0.490804i	0.969421	+	0.245402i	$0.0789200\pi$
$601$			12.0322i			0.490804i	−0.969421	+	0.245402i	$0.921080\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	6.81952	+	6.81952i	0.277253	+	0.277253i
$606$		0			0
$607$		−	10.7196i		−	0.435094i	−0.976050	−	0.217547i	$-0.930194\pi$
$607$		−	10.7196i		−	0.435094i	0.976050	−	0.217547i	$-0.0698056\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	18.8856	+	38.3880i	0.764031	+	1.55301i
$612$		0			0
$613$	−24.6802	−	24.6802i	−0.996823	−	0.996823i	0.00317218	−	0.999995i	$-0.498990\pi$
$613$	−24.6802	−	24.6802i	−0.996823	−	0.996823i	−0.999995	+	0.00317218i	$0.998990\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	15.2892	+	15.2892i	0.615520	+	0.615520i	0.944379	−	0.328859i	$-0.106664\pi$
$617$	15.2892	+	15.2892i	0.615520	+	0.615520i	−0.328859	+	0.944379i	$0.606664\pi$
$618$		0			0
$619$	2.27467	+	2.27467i	0.0914265	+	0.0914265i	0.751341	−	0.659914i	$-0.229407\pi$
$619$	2.27467	+	2.27467i	0.0914265	+	0.0914265i	−0.659914	+	0.751341i	$0.729407\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−2.05671	+	4.04142i	−0.0824002	+	0.161916i
$624$		0			0
$625$	15.3978			0.615912
$626$		0			0
$627$		0			0
$628$		0			0
$629$	16.3860	−	16.3860i	0.653351	−	0.653351i
$630$		0			0
$631$	16.3275	−	16.3275i	0.649988	−	0.649988i	−0.303002	−	0.952990i	$-0.597989\pi$
$631$	16.3275	−	16.3275i	0.649988	−	0.649988i	0.952990	+	0.303002i	$0.0979889\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−26.3066	−	26.3066i	−1.04395	−	1.04395i
$636$		0			0
$637$	24.1034	−	7.48519i	0.955010	−	0.296574i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			13.5558i			0.535422i	0.963499	+	0.267711i	$0.0862672\pi$
$641$			13.5558i			0.535422i	−0.963499	+	0.267711i	$0.913733\pi$
$642$		0			0
$643$	−5.02688	+	5.02688i	−0.198241	+	0.198241i	−0.799245	−	0.601005i	$-0.794767\pi$
$643$	−5.02688	+	5.02688i	−0.198241	+	0.198241i	0.601005	+	0.799245i	$0.294767\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	33.6597			1.32330			0.661651	−	0.749812i	$-0.269856\pi$
$647$	33.6597			1.32330			0.661651	+	0.749812i	$0.269856\pi$
$648$		0			0
$649$	−21.5990			−0.847836
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−0.435080			−0.0170260			−0.00851300	−	0.999964i	$-0.502710\pi$
$653$	−0.435080			−0.0170260			−0.00851300	+	0.999964i	$0.502710\pi$
$654$		0			0
$655$	19.5010	+	19.5010i	0.761969	+	0.761969i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	24.4854			0.953818			0.476909	−	0.878953i	$-0.341757\pi$
$659$	24.4854			0.953818			0.476909	+	0.878953i	$0.341757\pi$
$660$		0			0
$661$	27.2337	+	27.2337i	1.05927	+	1.05927i	0.998129	+	0.0611397i	$0.0194735\pi$
$661$	27.2337	+	27.2337i	1.05927	+	1.05927i	0.0611397	+	0.998129i	$0.480526\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	10.4383	−	3.39726i	0.404778	−	0.131740i
$666$		0			0
$667$		−	1.29858i		−	0.0502811i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−16.2233	−	16.2233i	−0.626293	−	0.626293i
$672$		0			0
$673$			11.0914i			0.427541i	0.976884	+	0.213770i	$0.0685744\pi$
$673$			11.0914i			0.427541i	−0.976884	+	0.213770i	$0.931426\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			34.6478i			1.33162i	0.746121	+	0.665811i	$0.231914\pi$
$677$			34.6478i			1.33162i	−0.746121	+	0.665811i	$0.768086\pi$
$678$		0			0
$679$	25.8706	+	13.1657i	0.992822	+	0.505254i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	11.5481	−	11.5481i	0.441874	−	0.441874i	−0.450767	−	0.892641i	$-0.648850\pi$
$683$	11.5481	−	11.5481i	0.441874	−	0.441874i	0.892641	+	0.450767i	$0.148850\pi$
$684$		0			0
$685$		−	20.7751i		−	0.793776i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	46.7068	+	15.9042i	1.77939	+	0.605902i
$690$		0			0
$691$	33.1181	−	33.1181i	1.25987	−	1.25987i	0.308716	−	0.951154i	$-0.400101\pi$
$691$	33.1181	−	33.1181i	1.25987	−	1.25987i	0.951154	−	0.308716i	$-0.0998994\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−4.72856	−	4.72856i	−0.179364	−	0.179364i
$696$		0			0
$697$	−7.99562	+	7.99562i	−0.302856	+	0.302856i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			22.7850i			0.860579i	0.902691	+	0.430290i	$0.141589\pi$
$701$			22.7850i			0.860579i	−0.902691	+	0.430290i	$0.858411\pi$
$702$		0			0
$703$			17.4768i			0.659149i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	5.62084	+	17.2703i	0.211394	+	0.649517i
$708$		0			0
$709$	−12.4878	+	12.4878i	−0.468991	+	0.468991i	−0.901588	−	0.432597i	$-0.857597\pi$
$709$	−12.4878	+	12.4878i	−0.468991	+	0.468991i	0.432597	+	0.901588i	$0.357597\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−2.05073	−	2.05073i	−0.0768006	−	0.0768006i
$714$		0			0
$715$	5.27661	−	15.4961i	0.197334	−	0.579522i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	24.3275			0.907262			0.453631	−	0.891190i	$-0.350128\pi$
$719$	24.3275			0.907262			0.453631	+	0.891190i	$0.350128\pi$
$720$		0			0
$721$	3.80652	+	11.6957i	0.141762	+	0.435572i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	3.99562i		−	0.148393i
$726$		0			0
$727$	−17.6018			−0.652814			−0.326407	−	0.945229i	$-0.605838\pi$
$727$	−17.6018			−0.652814			−0.326407	+	0.945229i	$0.605838\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			30.7276i			1.13650i
$732$		0			0
$733$	−11.9512	+	11.9512i	−0.441429	+	0.441429i	−0.892492	−	0.451063i	$-0.851045\pi$
$733$	−11.9512	+	11.9512i	−0.441429	+	0.441429i	0.451063	+	0.892492i	$0.351045\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−8.29401			−0.305514
$738$		0			0
$739$	−32.5326	+	32.5326i	−1.19673	+	1.19673i	−0.221590	+	0.975140i	$0.571125\pi$
$739$	−32.5326	+	32.5326i	−1.19673	+	1.19673i	−0.975140	+	0.221590i	$0.928875\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−24.8497	−	24.8497i	−0.911649	−	0.911649i	0.0847532	−	0.996402i	$-0.472990\pi$
$743$	−24.8497	−	24.8497i	−0.911649	−	0.911649i	−0.996402	+	0.0847532i	$0.972990\pi$
$744$		0			0
$745$	29.1301			1.06725
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−6.05081	−	18.5914i	−0.221092	−	0.679316i
$750$		0			0
$751$		−	27.4024i		−	0.999926i	−0.866047	−	0.499963i	$-0.833347\pi$
$751$		−	27.4024i		−	0.999926i	0.866047	−	0.499963i	$-0.166653\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			36.1081i			1.31411i
$756$		0			0
$757$	−16.3514			−0.594302			−0.297151	−	0.954831i	$-0.596036\pi$
$757$	−16.3514			−0.594302			−0.297151	+	0.954831i	$0.596036\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−30.1312	−	30.1312i	−1.09225	−	1.09225i	−0.995288	−	0.0969664i	$-0.969086\pi$
$761$	−30.1312	−	30.1312i	−1.09225	−	1.09225i	−0.0969664	−	0.995288i	$-0.530914\pi$
$762$		0			0
$763$	16.1567	−	31.7478i	0.584911	−	1.14935i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−14.2031	−	28.8699i	−0.512843	−	1.04243i
$768$		0			0
$769$	17.7676	−	17.7676i	0.640717	−	0.640717i	−0.310015	−	0.950732i	$-0.600334\pi$
$769$	17.7676	−	17.7676i	0.640717	−	0.640717i	0.950732	+	0.310015i	$0.100334\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−27.9630	+	27.9630i	−1.00576	+	1.00576i	−0.00577643	+	0.999983i	$0.501839\pi$
$773$	−27.9630	+	27.9630i	−1.00576	+	1.00576i	−0.999983	+	0.00577643i	$0.998161\pi$
$774$		0			0
$775$	−6.30994	−	6.30994i	−0.226660	−	0.226660i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	8.52788i		−	0.305543i
$780$		0			0
$781$	31.7519			1.13617
$782$		0			0
$783$		0			0
$784$		0			0
$785$	30.1183	+	30.1183i	1.07497	+	1.07497i
$786$		0			0
$787$	−5.45232	−	5.45232i	−0.194354	−	0.194354i	0.603220	−	0.797575i	$-0.293884\pi$
$787$	−5.45232	−	5.45232i	−0.194354	−	0.194354i	−0.797575	+	0.603220i	$0.793884\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−14.6770	−	45.0959i	−0.521855	−	1.60343i
$792$		0			0
$793$	11.0164	−	32.3527i	0.391206	−	1.14888i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	22.8078			0.807893			0.403947	−	0.914783i	$-0.367638\pi$
$797$	22.8078			0.807893			0.403947	+	0.914783i	$0.367638\pi$
$798$		0			0
$799$	24.6072	+	24.6072i	0.870541	+	0.870541i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.64396			−0.0580142
$804$		0			0
$805$	−1.08377	+	2.12960i	−0.0381979	+	0.0750586i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	49.6876			1.74692			0.873462	−	0.486893i	$-0.161870\pi$
$809$	49.6876			1.74692			0.873462	+	0.486893i	$0.161870\pi$
$810$		0			0
$811$	2.48452	−	2.48452i	0.0872433	−	0.0872433i	−0.662138	−	0.749382i	$-0.730351\pi$
$811$	2.48452	−	2.48452i	0.0872433	−	0.0872433i	0.749382	+	0.662138i	$0.230351\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			9.15872i			0.320816i
$816$		0			0
$817$	−16.3866	−	16.3866i	−0.573294	−	0.573294i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.71721	−	1.71721i	−0.0599310	−	0.0599310i	0.676506	−	0.736437i	$-0.263493\pi$
$821$	−1.71721	−	1.71721i	−0.0599310	−	0.0599310i	−0.736437	+	0.676506i	$0.763493\pi$
$822$		0			0
$823$			3.72657i			0.129900i	0.997889	+	0.0649501i	$0.0206888\pi$
$823$			3.72657i			0.129900i	−0.997889	+	0.0649501i	$0.979311\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	25.8675	−	25.8675i	0.899500	−	0.899500i	−0.0958920	−	0.995392i	$-0.530570\pi$
$827$	25.8675	−	25.8675i	0.899500	−	0.899500i	0.995392	+	0.0958920i	$0.0305703\pi$
$828$		0			0
$829$	−23.0228			−0.799613			−0.399807	−	0.916599i	$-0.630923\pi$
$829$	−23.0228			−0.799613			−0.399807	+	0.916599i	$0.630923\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	16.5972	−	12.0835i	0.575057	−	0.418667i
$834$		0			0
$835$	11.8011			0.408393
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−17.4869	−	17.4869i	−0.603715	−	0.603715i	0.337581	−	0.941296i	$-0.390391\pi$
$839$	−17.4869	−	17.4869i	−0.603715	−	0.603715i	−0.941296	+	0.337581i	$0.890391\pi$
$840$		0			0
$841$	−21.7260			−0.749172
$842$		0			0
$843$		0			0
$844$		0			0
$845$	24.1824	−	3.13706i	0.831900	−	0.107918i
$846$		0			0
$847$	−12.9352	+	4.20994i	−0.444460	+	0.144655i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2.69007	−	2.69007i	−0.0922146	−	0.0922146i
$852$		0			0
$853$	7.89801	+	7.89801i	0.270423	+	0.270423i	0.829270	−	0.558848i	$-0.188756\pi$
$853$	7.89801	+	7.89801i	0.270423	+	0.270423i	−0.558848	+	0.829270i	$0.688756\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−2.82721			−0.0965756			−0.0482878	−	0.998833i	$-0.515376\pi$
$857$	−2.82721			−0.0965756			−0.0482878	+	0.998833i	$0.515376\pi$
$858$		0			0
$859$		−	24.6521i		−	0.841119i	−0.907265	−	0.420559i	$-0.861834\pi$
$859$		−	24.6521i		−	0.841119i	0.907265	−	0.420559i	$-0.138166\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	30.0944	+	30.0944i	1.02443	+	1.02443i	0.999694	+	0.0247319i	$0.00787321\pi$
$863$	30.0944	+	30.0944i	1.02443	+	1.02443i	0.0247319	+	0.999694i	$0.492127\pi$
$864$		0			0
$865$	−9.94351	+	9.94351i	−0.338090	+	0.338090i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−12.1226	+	12.1226i	−0.411232	+	0.411232i
$870$		0			0
$871$	−5.45397	−	11.0860i	−0.184801	−	0.375636i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−14.5892	+	28.6677i	−0.493205	+	0.969145i
$876$		0			0
$877$	16.8598	+	16.8598i	0.569316	+	0.569316i	0.931937	−	0.362621i	$-0.118118\pi$
$877$	16.8598	+	16.8598i	0.569316	+	0.569316i	−0.362621	+	0.931937i	$0.618118\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	23.8583			0.803806			0.401903	−	0.915682i	$-0.368349\pi$
$881$	23.8583			0.803806			0.401903	+	0.915682i	$0.368349\pi$
$882$		0			0
$883$			51.7581i			1.74180i	0.491460	+	0.870900i	$0.336463\pi$
$883$			51.7581i			1.74180i	−0.491460	+	0.870900i	$0.663537\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	53.0170i		−	1.78014i	−0.455829	−	0.890068i	$-0.650657\pi$
$887$		−	53.0170i		−	1.78014i	0.455829	−	0.890068i	$-0.349343\pi$
$888$		0			0
$889$	49.8984	−	16.2400i	1.67354	−	0.544674i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−26.2453			−0.878267
$894$		0			0
$895$	−22.9001	−	22.9001i	−0.765467	−	0.765467i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	11.4873	−	11.4873i	0.383121	−	0.383121i
$900$		0			0
$901$	40.1346			1.33708
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−12.7321	+	12.7321i	−0.423229	+	0.423229i
$906$		0			0
$907$		−	31.0409i		−	1.03070i	−0.856981	−	0.515349i	$-0.827662\pi$
$907$		−	31.0409i		−	1.03070i	0.856981	−	0.515349i	$-0.172338\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−19.6438			−0.650828			−0.325414	−	0.945572i	$-0.605504\pi$
$911$	−19.6438			−0.650828			−0.325414	+	0.945572i	$0.605504\pi$
$912$		0			0
$913$		−	9.91079i		−	0.327999i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−36.9896	+	12.0387i	−1.22150	+	0.397553i
$918$		0			0
$919$	−0.734258			−0.0242209			−0.0121105	−	0.999927i	$-0.503855\pi$
$919$	−0.734258			−0.0242209			−0.0121105	+	0.999927i	$0.503855\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	20.8794	+	42.4406i	0.687254	+	1.39695i
$924$		0			0
$925$	−8.27714	−	8.27714i	−0.272151	−	0.272151i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−3.43281	+	3.43281i	−0.112627	+	0.112627i	−0.761174	−	0.648547i	$-0.775377\pi$
$929$	−3.43281	+	3.43281i	−0.112627	+	0.112627i	0.648547	+	0.761174i	$0.275377\pi$
$930$		0			0
$931$	−2.40709	+	15.2949i	−0.0788892	+	0.501271i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	13.3156i		−	0.435468i
$936$		0			0
$937$			14.3000i			0.467159i	0.972338	+	0.233580i	$0.0750439\pi$
$937$			14.3000i			0.467159i	−0.972338	+	0.233580i	$0.924956\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−2.44377	+	2.44377i	−0.0796646	+	0.0796646i	−0.745816	−	0.666152i	$-0.767940\pi$
$941$	−2.44377	+	2.44377i	−0.0796646	+	0.0796646i	0.666152	+	0.745816i	$0.267940\pi$
$942$		0			0
$943$	1.31264	+	1.31264i	0.0427453	+	0.0427453i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	27.8714	−	27.8714i	0.905700	−	0.905700i	−0.0902221	−	0.995922i	$-0.528758\pi$
$947$	27.8714	−	27.8714i	0.905700	−	0.905700i	0.995922	+	0.0902221i	$0.0287577\pi$
$948$		0			0
$949$	−1.08104	−	2.19737i	−0.0350919	−	0.0713297i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			10.7387i			0.347861i	0.984758	+	0.173930i	$0.0556468\pi$
$953$			10.7387i			0.347861i	−0.984758	+	0.173930i	$0.944353\pi$
$954$		0			0
$955$	−23.8689	+	23.8689i	−0.772378	+	0.772378i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	26.1157	+	13.2905i	0.843320	+	0.429172i
$960$		0			0
$961$		−	5.28171i		−	0.170378i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			9.10186i			0.292999i
$966$		0			0
$967$	−39.9183	−	39.9183i	−1.28369	−	1.28369i	−0.938554	−	0.345132i	$-0.887834\pi$
$967$	−39.9183	−	39.9183i	−1.28369	−	1.28369i	−0.345132	−	0.938554i	$-0.612166\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	49.5537i		−	1.59025i	−0.606444	−	0.795126i	$-0.707404\pi$
$971$		−	49.5537i		−	1.59025i	0.606444	−	0.795126i	$-0.292596\pi$
$972$		0			0
$973$	8.96912	−	2.91911i	0.287537	−	0.0935824i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	18.1970	+	18.1970i	0.582172	+	0.582172i	0.935500	−	0.353328i	$-0.114950\pi$
$977$	18.1970	+	18.1970i	0.582172	+	0.582172i	−0.353328	+	0.935500i	$0.614950\pi$
$978$		0			0
$979$	−4.14848			−0.132586
$980$		0			0
$981$		0			0
$982$		0			0
$983$	15.8103	+	15.8103i	0.504270	+	0.504270i	0.912762	−	0.408492i	$-0.133945\pi$
$983$	15.8103	+	15.8103i	0.504270	+	0.504270i	−0.408492	+	0.912762i	$0.633945\pi$
$984$		0			0
$985$	11.4798			0.365776
$986$		0			0
$987$		0			0
$988$		0			0
$989$	5.04454			0.160407
$990$		0			0
$991$	−27.6727			−0.879051			−0.439526	−	0.898230i	$-0.644853\pi$
$991$	−27.6727			−0.879051			−0.439526	+	0.898230i	$0.644853\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	26.5867	−	26.5867i	0.842854	−	0.842854i
$996$		0			0
$997$		−	27.8094i		−	0.880733i	−0.897818	−	0.440366i	$-0.854849\pi$
$997$		−	27.8094i		−	0.880733i	0.897818	−	0.440366i	$-0.145151\pi$
$998$		0			0
$999$		0			0

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	3276.2.bi.c.1945.6		16
3.2	odd	2		364.2.n.a.125.8	yes	16
7.6	odd	2	inner	3276.2.bi.c.1945.3		16
13.5	odd	4	inner	3276.2.bi.c.2449.3		16
21.20	even	2		364.2.n.a.125.1	✓	16
39.5	even	4		364.2.n.a.265.8	yes	16
91.83	even	4	inner	3276.2.bi.c.2449.6		16
273.83	odd	4		364.2.n.a.265.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
364.2.n.a.125.1	✓	16	21.20	even	2
364.2.n.a.125.8	yes	16	3.2	odd	2
364.2.n.a.265.1	yes	16	273.83	odd	4
364.2.n.a.265.8	yes	16	39.5	even	4
3276.2.bi.c.1945.3		16	7.6	odd	2	inner
3276.2.bi.c.1945.6		16	1.1	even	1	trivial
3276.2.bi.c.2449.3		16	13.5	odd	4	inner
3276.2.bi.c.2449.6		16	91.83	even	4	inner

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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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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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	3276.2.bi.c.1945.6

Level	$3276$
Weight	$2$
Character	3276.1945
Analytic conductor	$26.159$
Analytic rank	$0$
Dimension	$16$
Inner twists	$4$