LMFDB - Embedded newform 3276.2.bi.c.2449.7

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			2449.7
Root			$-1.36847i$ of defining polynomial
Character	$\chi$	$=$	3276.2449
Dual form			3276.2.bi.c.1945.7

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.64651 + 1.64651i) q^{5} +(-2.60386 - 0.468972i) q^{7} +(2.25755 + 2.25755i) q^{11} +(-2.84680 - 2.21262i) q^{13} +0.368467 q^{17} +(-3.97904 - 3.97904i) q^{19} +1.42196i q^{23} +0.421962i q^{25} -4.07283 q^{29} +(-1.27804 - 1.27804i) q^{31} +(-3.51510 - 5.05943i) q^{35} +(-2.46614 - 2.46614i) q^{37} +(-6.19166 - 6.19166i) q^{41} +2.85946i q^{43} +(-1.71457 + 1.71457i) q^{47} +(6.56013 + 2.44227i) q^{49} +5.15043 q^{53} +7.43413i q^{55} +(-0.539283 + 0.539283i) q^{59} -8.98662i q^{61} +(-1.04418 - 8.33038i) q^{65} +(-5.51510 + 5.51510i) q^{67} +(6.96570 - 6.96570i) q^{71} +(3.21527 - 3.21527i) q^{73} +(-4.81961 - 6.93706i) q^{77} -3.28619 q^{79} +(-12.1338 - 12.1338i) q^{83} +(0.606683 + 0.606683i) q^{85} +(-6.35280 + 6.35280i) q^{89} +(6.37501 + 7.09642i) q^{91} -13.1030i q^{95} +(3.00228 + 3.00228i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$16 q - 2 q^{7} - 12 q^{11} - 20 q^{29} + 40 q^{35} - 28 q^{53} + 20 q^{65} + 8 q^{67} + 4 q^{71} + 4 q^{79} + 4 q^{85} + 10 q^{91}+O(q^{100})$ 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 $\left(\mathbb{Z}/3276\mathbb{Z}\right)^\times$ .

$n$	$1639$	$2017$	$2341$	$2549$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$-1$	$1$

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
$5$	1.64651	+	1.64651i	0.736340	+	0.736340i	0.971868	−	0.235528i	$-0.0756819\pi$
$5$	1.64651	+	1.64651i	0.736340	+	0.736340i	−0.235528	+	0.971868i	$0.575682\pi$
$6$		0			0
$7$	−2.60386	−	0.468972i	−0.984165	−	0.177255i
$7$	−2.60386	−	0.468972i	−0.984165	−	0.177255i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.25755	+	2.25755i	0.680677	+	0.680677i	0.960153	−	0.279476i	$-0.0901607\pi$
$11$	2.25755	+	2.25755i	0.680677	+	0.680677i	−0.279476	+	0.960153i	$0.590161\pi$
$12$		0			0
$13$	−2.84680	−	2.21262i	−0.789562	−	0.613671i
$13$	−2.84680	−	2.21262i	−0.789562	−	0.613671i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.368467			0.0893664			0.0446832	−	0.999001i	$-0.485772\pi$
$17$	0.368467			0.0893664			0.0446832	+	0.999001i	$0.485772\pi$
$18$		0			0
$19$	−3.97904	−	3.97904i	−0.912854	−	0.912854i	0.0836415	−	0.996496i	$-0.473345\pi$
$19$	−3.97904	−	3.97904i	−0.912854	−	0.912854i	−0.996496	+	0.0836415i	$0.973345\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			1.42196i			0.296499i	0.988950	+	0.148250i	$0.0473639\pi$
$23$			1.42196i			0.296499i	−0.988950	+	0.148250i	$0.952636\pi$
$24$		0			0
$25$			0.421962i			0.0843923i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.07283			−0.756305			−0.378153	−	0.925743i	$-0.623441\pi$
$29$	−4.07283			−0.756305			−0.378153	+	0.925743i	$0.623441\pi$
$30$		0			0
$31$	−1.27804	−	1.27804i	−0.229543	−	0.229543i	0.582959	−	0.812502i	$-0.301895\pi$
$31$	−1.27804	−	1.27804i	−0.229543	−	0.229543i	−0.812502	+	0.582959i	$0.801895\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.51510	−	5.05943i	−0.594160	−	0.855199i
$36$		0			0
$37$	−2.46614	−	2.46614i	−0.405432	−	0.405432i	0.474710	−	0.880142i	$-0.342553\pi$
$37$	−2.46614	−	2.46614i	−0.405432	−	0.405432i	−0.880142	+	0.474710i	$0.842553\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−6.19166	−	6.19166i	−0.966975	−	0.966975i	0.0324964	−	0.999472i	$-0.489654\pi$
$41$	−6.19166	−	6.19166i	−0.966975	−	0.966975i	−0.999472	+	0.0324964i	$0.989654\pi$
$42$		0			0
$43$			2.85946i			0.436064i	0.975942	+	0.218032i	$0.0699637\pi$
$43$			2.85946i			0.436064i	−0.975942	+	0.218032i	$0.930036\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.71457	+	1.71457i	−0.250096	+	0.250096i	−0.821010	−	0.570914i	$-0.806589\pi$
$47$	−1.71457	+	1.71457i	−0.250096	+	0.250096i	0.570914	+	0.821010i	$0.306589\pi$
$48$		0			0
$49$	6.56013	+	2.44227i	0.937162	+	0.348896i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.15043			0.707465			0.353733	−	0.935347i	$-0.384912\pi$
$53$	5.15043			0.707465			0.353733	+	0.935347i	$0.384912\pi$
$54$		0			0
$55$			7.43413i			1.00242i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.539283	+	0.539283i	−0.0702087	+	0.0702087i	−0.741339	−	0.671131i	$-0.765809\pi$
$59$	−0.539283	+	0.539283i	−0.0702087	+	0.0702087i	0.671131	+	0.741339i	$0.265809\pi$
$60$		0			0
$61$		−	8.98662i		−	1.15062i	−0.817936	−	0.575309i	$-0.804882\pi$
$61$		−	8.98662i		−	1.15062i	0.817936	−	0.575309i	$-0.195118\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.04418	−	8.33038i	−0.129515	−	1.03326i
$66$		0			0
$67$	−5.51510	+	5.51510i	−0.673777	+	0.673777i	−0.958585	−	0.284808i	$-0.908070\pi$
$67$	−5.51510	+	5.51510i	−0.673777	+	0.673777i	0.284808	+	0.958585i	$0.408070\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	6.96570	−	6.96570i	0.826677	−	0.826677i	−0.160379	−	0.987056i	$-0.551272\pi$
$71$	6.96570	−	6.96570i	0.826677	−	0.826677i	0.987056	+	0.160379i	$0.0512715\pi$
$72$		0			0
$73$	3.21527	−	3.21527i	0.376319	−	0.376319i	−0.493453	−	0.869772i	$-0.664266\pi$
$73$	3.21527	−	3.21527i	0.376319	−	0.376319i	0.869772	+	0.493453i	$0.164266\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−4.81961	−	6.93706i	−0.549245	−	0.790551i
$78$		0			0
$79$	−3.28619			−0.369726			−0.184863	−	0.982764i	$-0.559184\pi$
$79$	−3.28619			−0.369726			−0.184863	+	0.982764i	$0.559184\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−12.1338	−	12.1338i	−1.33186	−	1.33186i	−0.903705	−	0.428156i	$-0.859163\pi$
$83$	−12.1338	−	12.1338i	−1.33186	−	1.33186i	−0.428156	−	0.903705i	$-0.640837\pi$
$84$		0			0
$85$	0.606683	+	0.606683i	0.0658040	+	0.0658040i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.35280	+	6.35280i	−0.673396	+	0.673396i	−0.958497	−	0.285101i	$-0.907973\pi$
$89$	−6.35280	+	6.35280i	−0.673396	+	0.673396i	0.285101	+	0.958497i	$0.407973\pi$
$90$		0			0
$91$	6.37501	+	7.09642i	0.668283	+	0.743907i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	13.1030i		−	1.34434i
$96$		0			0
$97$	3.00228	+	3.00228i	0.304836	+	0.304836i	0.842902	−	0.538067i	$-0.180845\pi$
$97$	3.00228	+	3.00228i	0.304836	+	0.304836i	−0.538067	+	0.842902i	$0.680845\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−11.8594			−1.18005			−0.590026	−	0.807384i	$-0.700883\pi$
$101$	−11.8594			−1.18005			−0.590026	+	0.807384i	$0.700883\pi$
$102$		0			0
$103$	−8.85049			−0.872065			−0.436033	−	0.899931i	$-0.643617\pi$
$103$	−8.85049			−0.872065			−0.436033	+	0.899931i	$0.643617\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4.42708			−0.427982			−0.213991	−	0.976836i	$-0.568646\pi$
$107$	−4.42708			−0.427982			−0.213991	+	0.976836i	$0.568646\pi$
$108$		0			0
$109$	14.1317	−	14.1317i	1.35357	−	1.35357i	0.471935	−	0.881633i	$-0.343556\pi$
$109$	14.1317	−	14.1317i	1.35357	−	1.35357i	0.881633	−	0.471935i	$-0.156444\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	14.0350			1.32030			0.660149	−	0.751134i	$-0.270493\pi$
$113$	14.0350			1.32030			0.660149	+	0.751134i	$0.270493\pi$
$114$		0			0
$115$	−2.34127	+	2.34127i	−0.218324	+	0.218324i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.959435	−	0.172801i	−0.0879512	−	0.0158406i
$120$		0			0
$121$		−	0.806943i		−	0.0733585i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	7.53777	−	7.53777i	0.674198	−	0.674198i
$126$		0			0
$127$		−	16.2535i		−	1.44226i	−0.692800	−	0.721130i	$-0.743623\pi$
$127$		−	16.2535i		−	1.44226i	0.692800	−	0.721130i	$-0.256377\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			19.3953i			1.69458i	0.531133	+	0.847289i	$0.321767\pi$
$131$			19.3953i			1.69458i	−0.531133	+	0.847289i	$0.678233\pi$
$132$		0			0
$133$	8.49479	+	12.2269i	0.736592	+	1.06021i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2.69982	−	2.69982i	−0.230661	−	0.230661i	0.582307	−	0.812969i	$-0.302150\pi$
$137$	−2.69982	−	2.69982i	−0.230661	−	0.230661i	−0.812969	+	0.582307i	$0.802150\pi$
$138$		0			0
$139$		−	1.65618i		−	0.140475i	−0.997530	−	0.0702377i	$-0.977624\pi$
$139$		−	1.65618i		−	0.140475i	0.997530	−	0.0702377i	$-0.0223758\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.43170	−	11.4219i	−0.119724	−	0.955148i
$144$		0			0
$145$	−6.70593	−	6.70593i	−0.556897	−	0.556897i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	10.6118	−	10.6118i	0.869353	−	0.869353i	−0.123048	−	0.992401i	$-0.539267\pi$
$149$	10.6118	−	10.6118i	0.869353	−	0.869353i	0.992401	+	0.123048i	$0.0392670\pi$
$150$		0			0
$151$	5.91319	+	5.91319i	0.481208	+	0.481208i	0.905517	−	0.424309i	$-0.139483\pi$
$151$	5.91319	+	5.91319i	0.481208	+	0.481208i	−0.424309	+	0.905517i	$0.639483\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	4.20860i		−	0.338043i
$156$		0			0
$157$		−	11.9648i		−	0.954898i	−0.878660	−	0.477449i	$-0.841562\pi$
$157$		−	11.9648i		−	0.954898i	0.878660	−	0.477449i	$-0.158438\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	0.666860	−	3.70258i	0.0525559	−	0.291804i
$162$		0			0
$163$	−12.2388	−	12.2388i	−0.958616	−	0.958616i	0.0405609	−	0.999177i	$-0.487086\pi$
$163$	−12.2388	−	12.2388i	−0.958616	−	0.958616i	−0.999177	+	0.0405609i	$0.987086\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−15.7272	+	15.7272i	−1.21701	+	1.21701i	−0.248337	+	0.968674i	$0.579884\pi$
$167$	−15.7272	+	15.7272i	−1.21701	+	1.21701i	−0.968674	+	0.248337i	$0.920116\pi$
$168$		0			0
$169$	3.20860	+	12.5978i	0.246815	+	0.969063i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	13.1471			0.999555			0.499778	−	0.866154i	$-0.333415\pi$
$173$	13.1471			0.999555			0.499778	+	0.866154i	$0.333415\pi$
$174$		0			0
$175$	0.197888	−	1.09873i	0.0149589	−	0.0830559i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			15.1508i			1.13242i	0.824260	+	0.566211i	$0.191591\pi$
$179$			15.1508i			1.13242i	−0.824260	+	0.566211i	$0.808409\pi$
$180$		0			0
$181$	−26.4016			−1.96242			−0.981208	−	0.192955i	$-0.938193\pi$
$181$	−26.4016			−1.96242			−0.981208	+	0.192955i	$0.938193\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	8.12104i		−	0.597071i
$186$		0			0
$187$	0.831832	+	0.831832i	0.0608296	+	0.0608296i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−12.2814			−0.888653			−0.444326	−	0.895865i	$-0.646557\pi$
$191$	−12.2814			−0.888653			−0.444326	+	0.895865i	$0.646557\pi$
$192$		0			0
$193$	−9.98958	−	9.98958i	−0.719066	−	0.719066i	0.249348	−	0.968414i	$-0.419784\pi$
$193$	−9.98958	−	9.98958i	−0.719066	−	0.719066i	−0.968414	+	0.249348i	$0.919784\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−13.4522	+	13.4522i	−0.958427	+	0.958427i	−0.999170	−	0.0407431i	$-0.987027\pi$
$197$	−13.4522	+	13.4522i	−0.958427	+	0.958427i	0.0407431	+	0.999170i	$0.487027\pi$
$198$		0			0
$199$	−20.2609			−1.43626			−0.718129	−	0.695910i	$-0.755001\pi$
$199$	−20.2609			−1.43626			−0.718129	+	0.695910i	$0.755001\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	10.6051	+	1.91004i	0.744329	+	0.134059i
$204$		0			0
$205$		−	20.3892i		−	1.42404i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	17.9658i		−	1.24272i
$210$		0			0
$211$	27.4417			1.88917			0.944584	−	0.328271i	$-0.106466\pi$
$211$	27.4417			1.88917			0.944584	+	0.328271i	$0.106466\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−4.70812	+	4.70812i	−0.321091	+	0.321091i
$216$		0			0
$217$	2.72846	+	3.92719i	0.185220	+	0.266595i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.04895	−	0.815279i	−0.0705602	−	0.0548416i
$222$		0			0
$223$	3.04351	+	3.04351i	0.203809	+	0.203809i	0.801630	−	0.597821i	$-0.203967\pi$
$223$	3.04351	+	3.04351i	0.203809	+	0.203809i	−0.597821	+	0.801630i	$0.703967\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	7.11475	+	7.11475i	0.472223	+	0.472223i	0.902633	−	0.430411i	$-0.141631\pi$
$227$	7.11475	+	7.11475i	0.472223	+	0.472223i	−0.430411	+	0.902633i	$0.641631\pi$
$228$		0			0
$229$	1.56971	−	1.56971i	0.103729	−	0.103729i	−0.653337	−	0.757067i	$-0.726632\pi$
$229$	1.56971	−	1.56971i	0.103729	−	0.103729i	0.757067	+	0.653337i	$0.226632\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	5.15077i		−	0.337438i	−0.985664	−	0.168719i	$-0.946037\pi$
$233$		−	5.15077i		−	0.337438i	0.985664	−	0.168719i	$-0.0539631\pi$
$234$		0			0
$235$	−5.64610			−0.368311
$236$		0			0
$237$		0			0
$238$		0			0
$239$	9.51510	−	9.51510i	0.615481	−	0.615481i	−0.328888	−	0.944369i	$-0.606674\pi$
$239$	9.51510	−	9.51510i	0.615481	−	0.615481i	0.944369	+	0.328888i	$0.106674\pi$
$240$		0			0
$241$	17.6376	−	17.6376i	1.13614	−	1.13614i	0.147005	−	0.989136i	$-0.453037\pi$
$241$	17.6376	−	17.6376i	1.13614	−	1.13614i	0.989136	−	0.147005i	$-0.0469632\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	6.78008	+	14.8225i	0.433163	+	0.946975i
$246$		0			0
$247$	2.52343	+	20.1317i	0.160562	+	1.28095i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−7.29977			−0.460757			−0.230379	−	0.973101i	$-0.573996\pi$
$251$	−7.29977			−0.460757			−0.230379	+	0.973101i	$0.573996\pi$
$252$		0			0
$253$	−3.21015	+	3.21015i	−0.201820	+	0.201820i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4.24293			−0.264667			−0.132333	−	0.991205i	$-0.542247\pi$
$257$	−4.24293			−0.264667			−0.132333	+	0.991205i	$0.542247\pi$
$258$		0			0
$259$	5.26493	+	7.57804i	0.327147	+	0.470876i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−14.0926			−0.868987			−0.434494	−	0.900675i	$-0.643073\pi$
$263$	−14.0926			−0.868987			−0.434494	+	0.900675i	$0.643073\pi$
$264$		0			0
$265$	8.48020	+	8.48020i	0.520935	+	0.520935i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	16.9965i		−	1.03629i	−0.855292	−	0.518147i	$-0.826622\pi$
$269$		−	16.9965i		−	1.03629i	0.855292	−	0.518147i	$-0.173378\pi$
$270$		0			0
$271$	9.85314	−	9.85314i	0.598536	−	0.598536i	−0.341387	−	0.939923i	$-0.610897\pi$
$271$	9.85314	−	9.85314i	0.598536	−	0.598536i	0.939923	+	0.341387i	$0.110897\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.952599	+	0.952599i	−0.0574439	+	0.0574439i
$276$		0			0
$277$			2.13012i			0.127986i	0.997950	+	0.0639932i	$0.0203836\pi$
$277$			2.13012i			0.127986i	−0.997950	+	0.0639932i	$0.979616\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	7.26232	+	7.26232i	0.433234	+	0.433234i	0.889727	−	0.456493i	$-0.150895\pi$
$281$	7.26232	+	7.26232i	0.433234	+	0.433234i	−0.456493	+	0.889727i	$0.650895\pi$
$282$		0			0
$283$	−13.5156			−0.803417			−0.401708	−	0.915768i	$-0.631583\pi$
$283$	−13.5156			−0.803417			−0.401708	+	0.915768i	$0.631583\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	13.2185	+	19.0259i	0.780262	+	1.12306i
$288$		0			0
$289$	−16.8642			−0.992014
$290$		0			0
$291$		0			0
$292$		0			0
$293$	17.7862	−	17.7862i	1.03908	−	1.03908i	0.0398769	−	0.999205i	$-0.487303\pi$
$293$	17.7862	−	17.7862i	1.03908	−	1.03908i	0.999205	−	0.0398769i	$-0.0126966\pi$
$294$		0			0
$295$	−1.77587			−0.103395
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3.14627	−	4.04805i	0.181953	−	0.234105i
$300$		0			0
$301$	1.34101	−	7.44563i	0.0772944	−	0.429159i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	14.7965	−	14.7965i	0.847246	−	0.847246i
$306$		0			0
$307$	18.5912	−	18.5912i	1.06106	−	1.06106i	0.0630456	−	0.998011i	$-0.479919\pi$
$307$	18.5912	−	18.5912i	1.06106	−	1.06106i	0.998011	−	0.0630456i	$-0.0200814\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−24.3138			−1.37871			−0.689356	−	0.724423i	$-0.742106\pi$
$311$	−24.3138			−1.37871			−0.689356	+	0.724423i	$0.742106\pi$
$312$		0			0
$313$			23.4557i			1.32580i	0.748709	+	0.662899i	$0.230674\pi$
$313$			23.4557i			1.32580i	−0.748709	+	0.662899i	$0.769326\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−14.7762	+	14.7762i	−0.829915	+	0.829915i	−0.987505	−	0.157590i	$-0.949628\pi$
$317$	−14.7762	+	14.7762i	−0.829915	+	0.829915i	0.157590	+	0.987505i	$0.449628\pi$
$318$		0			0
$319$	−9.19461	−	9.19461i	−0.514799	−	0.514799i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1.46614	−	1.46614i	−0.0815785	−	0.0815785i
$324$		0			0
$325$	0.933642	−	1.20124i	0.0517891	−	0.0666329i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	5.26858	−	3.66041i	0.290466	−	0.201805i
$330$		0			0
$331$	−16.8013	+	16.8013i	−0.923483	+	0.923483i	−0.997274	−	0.0737912i	$-0.976490\pi$
$331$	−16.8013	+	16.8013i	−0.923483	+	0.923483i	0.0737912	+	0.997274i	$0.476490\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−18.1613			−0.992257
$336$		0			0
$337$			23.9848i			1.30654i	0.757127	+	0.653268i	$0.226602\pi$
$337$			23.9848i			1.30654i	−0.757127	+	0.653268i	$0.773398\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	5.77047i		−	0.312488i
$342$		0			0
$343$	−15.9363	−	9.43584i	−0.860478	−	0.509487i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−12.2340			−0.656757			−0.328378	−	0.944546i	$-0.606502\pi$
$347$	−12.2340			−0.656757			−0.328378	+	0.944546i	$0.606502\pi$
$348$		0			0
$349$	9.27735	−	9.27735i	0.496605	−	0.496605i	−0.413774	−	0.910379i	$-0.635790\pi$
$349$	9.27735	−	9.27735i	0.496605	−	0.496605i	0.910379	+	0.413774i	$0.135790\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−23.2276	−	23.2276i	−1.23628	−	1.23628i	−0.961509	−	0.274772i	$-0.911398\pi$
$353$	−23.2276	−	23.2276i	−1.23628	−	1.23628i	−0.274772	−	0.961509i	$-0.588602\pi$
$354$		0			0
$355$	22.9381			1.21743
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−20.1218	−	20.1218i	−1.06199	−	1.06199i	−0.997947	−	0.0640396i	$-0.979602\pi$
$359$	−20.1218	−	20.1218i	−1.06199	−	1.06199i	−0.0640396	−	0.997947i	$-0.520398\pi$
$360$		0			0
$361$			12.6655i			0.666607i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	10.5879			0.554197
$366$		0			0
$367$			30.4871i			1.59141i	0.605682	+	0.795707i	$0.292900\pi$
$367$			30.4871i			1.59141i	−0.605682	+	0.795707i	$0.707100\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−13.4110	−	2.41541i	−0.696263	−	0.125402i
$372$		0			0
$373$	8.17639			0.423357			0.211679	−	0.977339i	$-0.432107\pi$
$373$	8.17639			0.423357			0.211679	+	0.977339i	$0.432107\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	11.5945	+	9.01163i	0.597149	+	0.464123i
$378$		0			0
$379$	−3.68940	+	3.68940i	−0.189512	+	0.189512i	−0.795485	−	0.605973i	$-0.792784\pi$
$379$	−3.68940	+	3.68940i	−0.189512	+	0.189512i	0.605973	+	0.795485i	$0.292784\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	5.69455	+	5.69455i	0.290978	+	0.290978i	0.837467	−	0.546489i	$-0.184036\pi$
$383$	5.69455	+	5.69455i	0.290978	+	0.290978i	−0.546489	+	0.837467i	$0.684036\pi$
$384$		0			0
$385$	3.48640	−	19.3574i	0.177683	−	0.986545i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	13.8745i		−	0.703463i	−0.936101	−	0.351732i	$-0.885593\pi$
$389$		−	13.8745i		−	0.703463i	0.936101	−	0.351732i	$-0.114407\pi$
$390$		0			0
$391$			0.523946i			0.0264971i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−5.41074	−	5.41074i	−0.272244	−	0.272244i
$396$		0			0
$397$	−21.6070	+	21.6070i	−1.08442	+	1.08442i	−0.0883336	+	0.996091i	$0.528154\pi$
$397$	−21.6070	+	21.6070i	−1.08442	+	1.08442i	−0.996091	+	0.0883336i	$0.971846\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	9.66075	+	9.66075i	0.482435	+	0.482435i	0.905909	−	0.423473i	$-0.139189\pi$
$401$	9.66075	+	9.66075i	0.482435	+	0.482435i	−0.423473	+	0.905909i	$0.639189\pi$
$402$		0			0
$403$	0.810508	+	6.46614i	0.0403743	+	0.322102i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	11.1349i		−	0.551936i
$408$		0			0
$409$	−4.81778	−	4.81778i	−0.238224	−	0.238224i	0.577890	−	0.816114i	$-0.303876\pi$
$409$	−4.81778	−	4.81778i	−0.238224	−	0.238224i	−0.816114	+	0.577890i	$0.803876\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	1.65712	−	1.15131i	0.0815417	−	0.0566521i
$414$		0			0
$415$		−	39.9568i		−	1.96140i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			28.5299i			1.39378i	0.717179	+	0.696889i	$0.245433\pi$
$419$			28.5299i			1.39378i	−0.717179	+	0.696889i	$0.754567\pi$
$420$		0			0
$421$	6.99113	−	6.99113i	0.340727	−	0.340727i	−0.515914	−	0.856641i	$-0.672547\pi$
$421$	6.99113	−	6.99113i	0.340727	−	0.340727i	0.856641	+	0.515914i	$0.172547\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			0.155479i			0.00754183i
$426$		0			0
$427$	−4.21447	+	23.3999i	−0.203953	+	1.13240i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−8.37456	+	8.37456i	−0.403388	+	0.403388i	−0.879425	−	0.476037i	$-0.842073\pi$
$431$	−8.37456	+	8.37456i	−0.403388	+	0.403388i	0.476037	+	0.879425i	$0.342073\pi$
$432$		0			0
$433$	−16.0716			−0.772354			−0.386177	−	0.922425i	$-0.626205\pi$
$433$	−16.0716			−0.772354			−0.386177	+	0.922425i	$0.626205\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	5.65804	−	5.65804i	0.270661	−	0.270661i
$438$		0			0
$439$	14.2525			0.680235			0.340117	−	0.940383i	$-0.389533\pi$
$439$	14.2525			0.680235			0.340117	+	0.940383i	$0.389533\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	29.2490			1.38966			0.694832	−	0.719172i	$-0.255479\pi$
$443$	29.2490			1.38966			0.694832	+	0.719172i	$0.255479\pi$
$444$		0			0
$445$	−20.9199			−0.991696
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−2.05252	−	2.05252i	−0.0968643	−	0.0968643i	0.657014	−	0.753878i	$-0.271819\pi$
$449$	−2.05252	−	2.05252i	−0.0968643	−	0.0968643i	−0.753878	+	0.657014i	$0.771819\pi$
$450$		0			0
$451$		−	27.9560i		−	1.31640i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.18781	+	22.1808i	−0.0556854	+	1.03985i
$456$		0			0
$457$	−20.7348	+	20.7348i	−0.969932	+	0.969932i	−0.999561	−	0.0296286i	$-0.990568\pi$
$457$	−20.7348	+	20.7348i	−0.969932	+	0.969932i	0.0296286	+	0.999561i	$0.490568\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	12.7852	+	12.7852i	0.595465	+	0.595465i	0.939102	−	0.343638i	$-0.111659\pi$
$461$	12.7852	+	12.7852i	0.595465	+	0.595465i	−0.343638	+	0.939102i	$0.611659\pi$
$462$		0			0
$463$	−19.4709	−	19.4709i	−0.904890	−	0.904890i	0.0909640	−	0.995854i	$-0.471005\pi$
$463$	−19.4709	−	19.4709i	−0.904890	−	0.904890i	−0.995854	+	0.0909640i	$0.971005\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−13.7822			−0.637765			−0.318883	−	0.947794i	$-0.603308\pi$
$467$	−13.7822			−0.637765			−0.318883	+	0.947794i	$0.603308\pi$
$468$		0			0
$469$	16.9469	−	11.7741i	0.782537	−	0.543677i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−6.45538	+	6.45538i	−0.296819	+	0.296819i
$474$		0			0
$475$	1.67900	−	1.67900i	0.0770379	−	0.0770379i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	28.1324	−	28.1324i	1.28540	−	1.28540i	0.347857	−	0.937548i	$-0.386910\pi$
$479$	28.1324	−	28.1324i	1.28540	−	1.28540i	0.937548	−	0.347857i	$-0.113090\pi$
$480$		0			0
$481$	1.56398	+	12.4773i	0.0713115	+	0.568915i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			9.88655i			0.448925i
$486$		0			0
$487$	10.1660	−	10.1660i	0.460664	−	0.460664i	−0.438209	−	0.898873i	$-0.644387\pi$
$487$	10.1660	−	10.1660i	0.460664	−	0.460664i	0.898873	+	0.438209i	$0.144387\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	9.86423i		−	0.445167i	−0.974914	−	0.222583i	$-0.928551\pi$
$491$		−	9.86423i		−	0.445167i	0.974914	−	0.222583i	$-0.0714489\pi$
$492$		0			0
$493$	−1.50070			−0.0675882
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−21.4044	+	14.8710i	−0.960119	+	0.667054i
$498$		0			0
$499$	21.2400	−	21.2400i	0.950833	−	0.950833i	−0.0480138	−	0.998847i	$-0.515289\pi$
$499$	21.2400	−	21.2400i	0.950833	−	0.950833i	0.998847	+	0.0480138i	$0.0152891\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			21.6809i			0.966704i	0.875426	+	0.483352i	$0.160581\pi$
$503$			21.6809i			0.966704i	−0.875426	+	0.483352i	$0.839419\pi$
$504$		0			0
$505$	−19.5265	−	19.5265i	−0.868920	−	0.868920i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−3.95221	−	3.95221i	−0.175178	−	0.175178i	0.614072	−	0.789250i	$-0.289531\pi$
$509$	−3.95221	−	3.95221i	−0.175178	−	0.175178i	−0.789250	+	0.614072i	$0.789531\pi$
$510$		0			0
$511$	−9.87998	+	6.86423i	−0.437064	+	0.303656i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−14.5724	−	14.5724i	−0.642136	−	0.642136i
$516$		0			0
$517$	−7.74145			−0.340469
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	9.80601i		−	0.429609i	−0.976657	−	0.214805i	$-0.931089\pi$
$521$		−	9.80601i		−	0.429609i	0.976657	−	0.214805i	$-0.0689115\pi$
$522$		0			0
$523$			32.6498i			1.42768i	0.700310	+	0.713839i	$0.253045\pi$
$523$			32.6498i			1.42768i	−0.700310	+	0.713839i	$0.746955\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−0.470915	−	0.470915i	−0.0205134	−	0.0205134i
$528$		0			0
$529$	20.9780			0.912088
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3.92664	+	31.3263i	0.170082	+	1.35689i
$534$		0			0
$535$	−7.28921	−	7.28921i	−0.315140	−	0.315140i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	9.29627	+	20.3234i	0.400419	+	0.875389i
$540$		0			0
$541$	31.0636	+	31.0636i	1.33553	+	1.33553i	0.900338	+	0.435191i	$0.143319\pi$
$541$	31.0636	+	31.0636i	1.33553	+	1.33553i	0.435191	+	0.900338i	$0.356681\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	46.5357			1.99337
$546$		0			0
$547$	−9.24833			−0.395430			−0.197715	−	0.980260i	$-0.563352\pi$
$547$	−9.24833			−0.395430			−0.197715	+	0.980260i	$0.563352\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	16.2059	+	16.2059i	0.690397	+	0.690397i
$552$		0			0
$553$	8.55677	+	1.54113i	0.363871	+	0.0655356i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2.84869	−	2.84869i	−0.120703	−	0.120703i	0.644175	−	0.764878i	$-0.277201\pi$
$557$	−2.84869	−	2.84869i	−0.120703	−	0.120703i	−0.764878	+	0.644175i	$0.777201\pi$
$558$		0			0
$559$	6.32691	−	8.14033i	0.267600	−	0.344299i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	17.6797			0.745113			0.372556	−	0.928010i	$-0.378481\pi$
$563$	17.6797			0.745113			0.372556	+	0.928010i	$0.378481\pi$
$564$		0			0
$565$	23.1087	+	23.1087i	0.972188	+	0.972188i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			28.0449i			1.17570i	0.808969	+	0.587851i	$0.200026\pi$
$569$			28.0449i			1.17570i	−0.808969	+	0.587851i	$0.799974\pi$
$570$		0			0
$571$		−	20.2857i		−	0.848928i	−0.905445	−	0.424464i	$-0.860462\pi$
$571$		−	20.2857i		−	0.848928i	0.905445	−	0.424464i	$-0.139538\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.600013			−0.0250223
$576$		0			0
$577$	−7.74415	−	7.74415i	−0.322393	−	0.322393i	0.527291	−	0.849685i	$-0.323208\pi$
$577$	−7.74415	−	7.74415i	−0.322393	−	0.322393i	−0.849685	+	0.527291i	$0.823208\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	25.9043	+	37.2852i	1.07469	+	1.54685i
$582$		0			0
$583$	11.6273	+	11.6273i	0.481555	+	0.481555i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.33555	−	1.33555i	−0.0551240	−	0.0551240i	0.679007	−	0.734131i	$-0.262410\pi$
$587$	−1.33555	−	1.33555i	−0.0551240	−	0.0551240i	−0.734131	+	0.679007i	$0.762410\pi$
$588$		0			0
$589$			10.1707i			0.419078i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	9.49913	−	9.49913i	0.390083	−	0.390083i	−0.484634	−	0.874717i	$-0.661047\pi$
$593$	9.49913	−	9.49913i	0.390083	−	0.390083i	0.874717	+	0.484634i	$0.161047\pi$
$594$		0			0
$595$	−1.29520	−	1.86423i	−0.0530979	−	0.0764261i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−2.24887			−0.0918862			−0.0459431	−	0.998944i	$-0.514629\pi$
$599$	−2.24887			−0.0918862			−0.0459431	+	0.998944i	$0.514629\pi$
$600$		0			0
$601$		−	20.5600i		−	0.838660i	−0.907834	−	0.419330i	$-0.862265\pi$
$601$		−	20.5600i		−	0.838660i	0.907834	−	0.419330i	$-0.137735\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.32864	−	1.32864i	0.0540168	−	0.0540168i
$606$		0			0
$607$			11.7557i			0.477149i	0.971124	+	0.238574i	$0.0766801\pi$
$607$			11.7557i			0.477149i	−0.971124	+	0.238574i	$0.923320\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	8.67474	−	1.08735i	0.350942	−	0.0439894i
$612$		0			0
$613$	−0.130998	+	0.130998i	−0.00529095	+	0.00529095i	−0.709747	−	0.704456i	$-0.751191\pi$
$613$	−0.130998	+	0.130998i	−0.00529095	+	0.00529095i	0.704456	+	0.709747i	$0.251191\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−30.6432	+	30.6432i	−1.23365	+	1.23365i	−0.271097	+	0.962552i	$0.587386\pi$
$617$	−30.6432	+	30.6432i	−1.23365	+	1.23365i	−0.962552	+	0.271097i	$0.912614\pi$
$618$		0			0
$619$	−0.285066	+	0.285066i	−0.0114578	+	0.0114578i	−0.712812	−	0.701355i	$-0.752579\pi$
$619$	−0.285066	+	0.285066i	−0.0114578	+	0.0114578i	0.701355	+	0.712812i	$0.252579\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	19.5211	−	13.5625i	0.782095	−	0.543370i
$624$		0			0
$625$	26.9318			1.07727
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−0.908693	−	0.908693i	−0.0362320	−	0.0362320i
$630$		0			0
$631$	−1.45216	−	1.45216i	−0.0578095	−	0.0578095i	0.677611	−	0.735421i	$-0.263015\pi$
$631$	−1.45216	−	1.45216i	−0.0578095	−	0.0578095i	−0.735421	+	0.677611i	$0.763015\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	26.7614	−	26.7614i	1.06199	−	1.06199i
$636$		0			0
$637$	−13.2716	−	21.4678i	−0.525839	−	0.850584i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			40.6275i			1.60469i	0.596861	+	0.802344i	$0.296414\pi$
$641$			40.6275i			1.60469i	−0.596861	+	0.802344i	$0.703586\pi$
$642$		0			0
$643$	5.27242	+	5.27242i	0.207924	+	0.207924i	0.803384	−	0.595461i	$-0.203030\pi$
$643$	5.27242	+	5.27242i	0.207924	+	0.207924i	−0.595461	+	0.803384i	$0.703030\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−23.4908			−0.923519			−0.461759	−	0.887005i	$-0.652782\pi$
$647$	−23.4908			−0.923519			−0.461759	+	0.887005i	$0.652782\pi$
$648$		0			0
$649$	−2.43492			−0.0955788
$650$		0			0
$651$		0			0
$652$		0			0
$653$	44.1177			1.72646			0.863229	−	0.504812i	$-0.168438\pi$
$653$	44.1177			1.72646			0.863229	+	0.504812i	$0.168438\pi$
$654$		0			0
$655$	−31.9345	+	31.9345i	−1.24778	+	1.24778i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−28.5292			−1.11134			−0.555670	−	0.831403i	$-0.687538\pi$
$659$	−28.5292			−1.11134			−0.555670	+	0.831403i	$0.687538\pi$
$660$		0			0
$661$	11.6885	−	11.6885i	0.454630	−	0.454630i	−0.442258	−	0.896888i	$-0.645822\pi$
$661$	11.6885	−	11.6885i	0.454630	−	0.454630i	0.896888	+	0.442258i	$0.145822\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−6.14495	+	34.1184i	−0.238291	+	1.32305i
$666$		0			0
$667$		−	5.79140i		−	0.224244i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	20.2877	−	20.2877i	0.783199	−	0.783199i
$672$		0			0
$673$			44.6749i			1.72209i	0.508529	+	0.861045i	$0.330190\pi$
$673$			44.6749i			1.72209i	−0.508529	+	0.861045i	$0.669810\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			31.3759i			1.20587i	0.797789	+	0.602936i	$0.206003\pi$
$677$			31.3759i			1.20587i	−0.797789	+	0.602936i	$0.793997\pi$
$678$		0			0
$679$	−6.40953	−	9.22550i	−0.245975	−	0.354042i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	30.1759	+	30.1759i	1.15465	+	1.15465i	0.985610	+	0.169037i	$0.0540658\pi$
$683$	30.1759	+	30.1759i	1.15465	+	1.15465i	0.169037	+	0.985610i	$0.445934\pi$
$684$		0			0
$685$		−	8.89054i		−	0.339690i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−14.6623	−	11.3960i	−0.558588	−	0.434151i
$690$		0			0
$691$	18.2601	+	18.2601i	0.694648	+	0.694648i	0.963251	−	0.268603i	$-0.0865619\pi$
$691$	18.2601	+	18.2601i	0.694648	+	0.694648i	−0.268603	+	0.963251i	$0.586562\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2.72691	−	2.72691i	0.103438	−	0.103438i
$696$		0			0
$697$	−2.28142	−	2.28142i	−0.0864151	−	0.0864151i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	46.9199i		−	1.77214i	−0.463553	−	0.886069i	$-0.653426\pi$
$701$		−	46.9199i		−	1.77214i	0.463553	−	0.886069i	$-0.346574\pi$
$702$		0			0
$703$			19.6258i			0.740200i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	30.8801	+	5.56172i	1.16137	+	0.209170i
$708$		0			0
$709$	9.25702	+	9.25702i	0.347655	+	0.347655i	0.859235	−	0.511581i	$-0.170940\pi$
$709$	9.25702	+	9.25702i	0.347655	+	0.347655i	−0.511581	+	0.859235i	$0.670940\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.81732	−	1.81732i	0.0680592	−	0.0680592i
$714$		0			0
$715$	16.4489	−	21.1635i	0.615155	−	0.791471i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	2.79206			0.104126			0.0520631	−	0.998644i	$-0.483420\pi$
$719$	2.79206			0.104126			0.0520631	+	0.998644i	$0.483420\pi$
$720$		0			0
$721$	23.0454	+	4.15063i	0.858256	+	0.154578i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	1.71858i		−	0.0638263i
$726$		0			0
$727$	43.3520			1.60784			0.803919	−	0.594739i	$-0.202745\pi$
$727$	43.3520			1.60784			0.803919	+	0.594739i	$0.202745\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			1.05362i			0.0389694i
$732$		0			0
$733$	19.4587	+	19.4587i	0.718722	+	0.718722i	0.968344	−	0.249621i	$-0.0803061\pi$
$733$	19.4587	+	19.4587i	0.718722	+	0.718722i	−0.249621	+	0.968344i	$0.580306\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−24.9012			−0.917248
$738$		0			0
$739$	27.7343	+	27.7343i	1.02022	+	1.02022i	0.999791	+	0.0204297i	$0.00650343\pi$
$739$	27.7343	+	27.7343i	1.02022	+	1.02022i	0.0204297	+	0.999791i	$0.493497\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−17.7559	+	17.7559i	−0.651401	+	0.651401i	−0.953330	−	0.301929i	$-0.902369\pi$
$743$	−17.7559	+	17.7559i	−0.651401	+	0.651401i	0.301929	+	0.953330i	$0.402369\pi$
$744$		0			0
$745$	34.9448			1.28028
$746$		0			0
$747$		0			0
$748$		0			0
$749$	11.5275	+	2.07618i	0.421205	+	0.0758618i
$750$		0			0
$751$			15.2391i			0.556084i	0.960569	+	0.278042i	$0.0896855\pi$
$751$			15.2391i			0.556084i	−0.960569	+	0.278042i	$0.910315\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			19.4722i			0.708665i
$756$		0			0
$757$	14.7640			0.536605			0.268303	−	0.963335i	$-0.413537\pi$
$757$	14.7640			0.536605			0.268303	+	0.963335i	$0.413537\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	2.16681	−	2.16681i	0.0785467	−	0.0785467i	−0.666742	−	0.745289i	$-0.732312\pi$
$761$	2.16681	−	2.16681i	0.0785467	−	0.0785467i	0.745289	+	0.666742i	$0.232312\pi$
$762$		0			0
$763$	−43.4242	+	30.1695i	−1.57206	+	1.09221i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	2.72846	−	0.342003i	0.0985191	−	0.0123490i
$768$		0			0
$769$	0.594774	+	0.594774i	0.0214481	+	0.0214481i	0.717750	−	0.696301i	$-0.245172\pi$
$769$	0.594774	+	0.594774i	0.0214481	+	0.0214481i	−0.696301	+	0.717750i	$0.745172\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−22.6910	−	22.6910i	−0.816140	−	0.816140i	0.169406	−	0.985546i	$-0.445815\pi$
$773$	−22.6910	−	22.6910i	−0.816140	−	0.816140i	−0.985546	+	0.169406i	$0.945815\pi$
$774$		0			0
$775$	0.539283	−	0.539283i	0.0193716	−	0.0193716i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			49.2738i			1.76542i
$780$		0			0
$781$	31.4508			1.12540
$782$		0			0
$783$		0			0
$784$		0			0
$785$	19.7002	−	19.7002i	0.703129	−	0.703129i
$786$		0			0
$787$	−14.7361	+	14.7361i	−0.525285	+	0.525285i	−0.919163	−	0.393878i	$-0.871133\pi$
$787$	−14.7361	+	14.7361i	−0.525285	+	0.525285i	0.393878	+	0.919163i	$0.371133\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−36.5450	−	6.58201i	−1.29939	−	0.234029i
$792$		0			0
$793$	−19.8840	+	25.5832i	−0.706102	+	0.908485i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−28.4856			−1.00901			−0.504507	−	0.863408i	$-0.668326\pi$
$797$	−28.4856			−1.00901			−0.504507	+	0.863408i	$0.668326\pi$
$798$		0			0
$799$	−0.631762	+	0.631762i	−0.0223501	+	0.0223501i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	14.5173			0.512303
$804$		0			0
$805$	7.19431	−	4.99833i	0.253566	−	0.176168i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	42.7032			1.50136			0.750682	−	0.660663i	$-0.229725\pi$
$809$	42.7032			1.50136			0.750682	+	0.660663i	$0.229725\pi$
$810$		0			0
$811$	−20.0832	−	20.0832i	−0.705216	−	0.705216i	0.260310	−	0.965525i	$-0.416175\pi$
$811$	−20.0832	−	20.0832i	−0.705216	−	0.705216i	−0.965525	+	0.260310i	$0.916175\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	40.3025i		−	1.41173i
$816$		0			0
$817$	11.3779	−	11.3779i	0.398063	−	0.398063i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	19.0251	−	19.0251i	0.663980	−	0.663980i	−0.292336	−	0.956316i	$-0.594433\pi$
$821$	19.0251	−	19.0251i	0.663980	−	0.663980i	0.956316	+	0.292336i	$0.0944325\pi$
$822$		0			0
$823$		−	21.8267i		−	0.760832i	−0.924816	−	0.380416i	$-0.875781\pi$
$823$		−	21.8267i		−	0.760832i	0.924816	−	0.380416i	$-0.124219\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	12.2190	+	12.2190i	0.424897	+	0.424897i	0.886886	−	0.461989i	$-0.152864\pi$
$827$	12.2190	+	12.2190i	0.424897	+	0.424897i	−0.461989	+	0.886886i	$0.652864\pi$
$828$		0			0
$829$	−21.7902			−0.756805			−0.378402	−	0.925641i	$-0.623526\pi$
$829$	−21.7902			−0.756805			−0.378402	+	0.925641i	$0.623526\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2.41719	+	0.899896i	0.0837507	+	0.0311795i
$834$		0			0
$835$	−51.7900			−1.79227
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−14.7255	+	14.7255i	−0.508380	+	0.508380i	−0.914029	−	0.405649i	$-0.867046\pi$
$839$	−14.7255	+	14.7255i	−0.508380	+	0.508380i	0.405649	+	0.914029i	$0.367046\pi$
$840$		0			0
$841$	−12.4121			−0.428003
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−15.4594	+	26.0253i	−0.531820	+	0.895299i
$846$		0			0
$847$	−0.378434	+	2.10116i	−0.0130031	+	0.0721968i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	3.50676	−	3.50676i	0.120210	−	0.120210i
$852$		0			0
$853$	−4.44429	+	4.44429i	−0.152170	+	0.152170i	−0.779086	−	0.626917i	$-0.784317\pi$
$853$	−4.44429	+	4.44429i	−0.152170	+	0.152170i	0.626917	+	0.779086i	$0.284317\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	29.9919			1.02451			0.512253	−	0.858835i	$-0.328811\pi$
$857$	29.9919			1.02451			0.512253	+	0.858835i	$0.328811\pi$
$858$		0			0
$859$			19.4814i			0.664698i	0.943157	+	0.332349i	$0.107841\pi$
$859$			19.4814i			0.664698i	−0.943157	+	0.332349i	$0.892159\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−13.8213	+	13.8213i	−0.470481	+	0.470481i	−0.902070	−	0.431589i	$-0.857953\pi$
$863$	−13.8213	+	13.8213i	−0.470481	+	0.470481i	0.431589	+	0.902070i	$0.357953\pi$
$864$		0			0
$865$	21.6468	+	21.6468i	0.736012	+	0.736012i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−7.41874	−	7.41874i	−0.251664	−	0.251664i
$870$		0			0
$871$	27.9032	−	3.49757i	0.945465	−	0.118511i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−23.1623	+	16.0923i	−0.783027	+	0.544018i
$876$		0			0
$877$	−1.29305	+	1.29305i	−0.0436632	+	0.0436632i	−0.728601	−	0.684938i	$-0.759829\pi$
$877$	−1.29305	+	1.29305i	−0.0436632	+	0.0436632i	0.684938	+	0.728601i	$0.259829\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	51.9470			1.75014			0.875069	−	0.483998i	$-0.160816\pi$
$881$	51.9470			1.75014			0.875069	+	0.483998i	$0.160816\pi$
$882$		0			0
$883$		−	14.1938i		−	0.477658i	−0.971062	−	0.238829i	$-0.923236\pi$
$883$		−	14.1938i		−	0.477658i	0.971062	−	0.238829i	$-0.0767635\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			40.4060i			1.35670i	0.734738	+	0.678351i	$0.237305\pi$
$887$			40.4060i			1.35670i	−0.734738	+	0.678351i	$0.762695\pi$
$888$		0			0
$889$	−7.62241	+	42.3216i	−0.255647	+	1.41942i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	13.6447			0.456602
$894$		0			0
$895$	−24.9458	+	24.9458i	−0.833847	+	0.833847i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	5.20523	+	5.20523i	0.173604	+	0.173604i
$900$		0			0
$901$	1.89776			0.0632236
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−43.4704	−	43.4704i	−1.44500	−	1.44500i
$906$		0			0
$907$			14.2188i			0.472129i	0.971737	+	0.236064i	$0.0758576\pi$
$907$			14.2188i			0.472129i	−0.971737	+	0.236064i	$0.924142\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	11.7804			0.390304			0.195152	−	0.980773i	$-0.437480\pi$
$911$	11.7804			0.390304			0.195152	+	0.980773i	$0.437480\pi$
$912$		0			0
$913$		−	54.7854i		−	1.81313i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	9.09586	−	50.5026i	0.300372	−	1.66774i
$918$		0			0
$919$	23.4944			0.775010			0.387505	−	0.921868i	$-0.373337\pi$
$919$	23.4944			0.775010			0.387505	+	0.921868i	$0.373337\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−35.2425	+	4.41752i	−1.16002	+	0.145404i
$924$		0			0
$925$	1.04062	−	1.04062i	0.0342153	−	0.0342153i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	22.0147	+	22.0147i	0.722280	+	0.722280i	0.969069	−	0.246789i	$-0.0793754\pi$
$929$	22.0147	+	22.0147i	0.722280	+	0.722280i	−0.246789	+	0.969069i	$0.579375\pi$
$930$		0			0
$931$	−16.3851	−	35.8209i	−0.537001	−	1.17398i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			2.73923i			0.0895825i
$936$		0			0
$937$		−	19.9597i		−	0.652056i	−0.945360	−	0.326028i	$-0.894290\pi$
$937$		−	19.9597i		−	0.652056i	0.945360	−	0.326028i	$-0.105710\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−18.0404	−	18.0404i	−0.588101	−	0.588101i	0.349016	−	0.937117i	$-0.386516\pi$
$941$	−18.0404	−	18.0404i	−0.588101	−	0.588101i	−0.937117	+	0.349016i	$0.886516\pi$
$942$		0			0
$943$	8.80431	−	8.80431i	0.286708	−	0.286708i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−36.8882	−	36.8882i	−1.19871	−	1.19871i	−0.974554	−	0.224153i	$-0.928039\pi$
$947$	−36.8882	−	36.8882i	−1.19871	−	1.19871i	−0.224153	−	0.974554i	$-0.571961\pi$
$948$		0			0
$949$	−16.2674	+	2.03907i	−0.528063	+	0.0661909i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			38.2580i			1.23930i	0.784879	+	0.619650i	$0.212725\pi$
$953$			38.2580i			1.23930i	−0.784879	+	0.619650i	$0.787275\pi$
$954$		0			0
$955$	−20.2214	−	20.2214i	−0.654350	−	0.654350i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	5.76380	+	8.29608i	0.186123	+	0.267894i
$960$		0			0
$961$		−	27.7332i		−	0.894620i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	32.8958i		−	1.05895i
$966$		0			0
$967$	−27.8001	+	27.8001i	−0.893991	+	0.893991i	−0.994896	−	0.100905i	$-0.967826\pi$
$967$	−27.8001	+	27.8001i	−0.893991	+	0.893991i	0.100905	+	0.994896i	$0.467826\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	19.3267i		−	0.620223i	−0.950700	−	0.310112i	$-0.899634\pi$
$971$		−	19.3267i		−	0.620223i	0.950700	−	0.310112i	$-0.100366\pi$
$972$		0			0
$973$	−0.776703	+	4.31246i	−0.0248999	+	0.138251i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−38.7868	+	38.7868i	−1.24090	+	1.24090i	−0.281270	+	0.959629i	$0.590756\pi$
$977$	−38.7868	+	38.7868i	−1.24090	+	1.24090i	−0.959629	+	0.281270i	$0.909244\pi$
$978$		0			0
$979$	−28.6835			−0.916730
$980$		0			0
$981$		0			0
$982$		0			0
$983$	23.6735	−	23.6735i	0.755066	−	0.755066i	−0.220354	−	0.975420i	$-0.570721\pi$
$983$	23.6735	−	23.6735i	0.755066	−	0.755066i	0.975420	+	0.220354i	$0.0707213\pi$
$984$		0			0
$985$	−44.2981			−1.41146
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−4.06604			−0.129293
$990$		0			0
$991$	−16.0410			−0.509558			−0.254779	−	0.966999i	$-0.582003\pi$
$991$	−16.0410			−0.509558			−0.254779	+	0.966999i	$0.582003\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−33.3597	−	33.3597i	−1.05757	−	1.05757i
$996$		0			0
$997$		−	12.8936i		−	0.408346i	−0.978935	−	0.204173i	$-0.934550\pi$
$997$		−	12.8936i		−	0.408346i	0.978935	−	0.204173i	$-0.0654505\pi$
$998$		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.2449.7		16
3.2	odd	2		364.2.n.a.265.5	yes	16
7.6	odd	2	inner	3276.2.bi.c.2449.2		16
13.8	odd	4	inner	3276.2.bi.c.1945.2		16
21.20	even	2		364.2.n.a.265.4	yes	16
39.8	even	4		364.2.n.a.125.5	yes	16
91.34	even	4	inner	3276.2.bi.c.1945.7		16
273.125	odd	4		364.2.n.a.125.4	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
364.2.n.a.125.4	✓	16	273.125	odd	4
364.2.n.a.125.5	yes	16	39.8	even	4
364.2.n.a.265.4	yes	16	21.20	even	2
364.2.n.a.265.5	yes	16	3.2	odd	2
3276.2.bi.c.1945.2		16	13.8	odd	4	inner
3276.2.bi.c.1945.7		16	91.34	even	4	inner
3276.2.bi.c.2449.2		16	7.6	odd	2	inner
3276.2.bi.c.2449.7		16	1.1	even	1	trivial

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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.2449.7

Level	$3276$
Weight	$2$
Character	3276.2449
Analytic conductor	$26.159$
Analytic rank	$0$
Dimension	$16$
Inner twists	$4$