LMFDB - Embedded newform 4680.2.l.g.2809.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4680,2,Mod(2809,4680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4680, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4680.2809");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4680.l (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:	$37.3699881460$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.57815240704.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 2x^{6} + 89x^{4} - 170x^{3} + 162x^{2} - 72x + 16 $ x^8 - 2x^7 + 2x^6 + 89x^4 - 170x^3 + 162x^2 - 72x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1560)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2809.1
Root			$-2.15569 - 2.15569i$ of defining polynomial
Character	$\chi$	$=$	4680.2809
Dual form			4680.2.l.g.2809.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.15569 - 0.594137i) q^{5} -0.633776i q^{7} +O(q^{10})$	$q+(-2.15569 - 0.594137i) q^{5} -0.633776i q^{7} +0.177949 q^{11} -1.00000i q^{13} -4.48933i q^{17} -6.48933i q^{23} +(4.29400 + 2.56155i) q^{25} -3.49966 q^{29} +(-0.376550 + 1.36622i) q^{35} +1.75688i q^{37} -7.67760 q^{41} +7.49966i q^{43} +3.18827i q^{47} +6.59833 q^{49} -1.75688i q^{53} +(-0.383604 - 0.105726i) q^{55} +3.93483 q^{59} +3.01033 q^{61} +(-0.594137 + 2.15569i) q^{65} -1.62345i q^{67} +4.41005 q^{71} +3.85555i q^{73} -0.112780i q^{77} -9.63309 q^{79} -1.72338i q^{83} +(-2.66728 + 9.67760i) q^{85} -0.589258 q^{89} -0.633776 q^{91} -4.45457i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{5}+O(q^{10}) $ 8 * q + 2 * q^5	$ 8 q + 2 q^{5} - 2 q^{11} + 16 q^{29} + 8 q^{35} - 14 q^{41} - 18 q^{49} + 10 q^{55} + 4 q^{59} + 22 q^{61} - 2 q^{65} - 30 q^{71} + 2 q^{79} + 24 q^{85} + 18 q^{89} - 14 q^{91}+O(q^{100}) $ 8 * q + 2 * q^5 - 2 * q^11 + 16 * q^29 + 8 * q^35 - 14 * q^41 - 18 * q^49 + 10 * q^55 + 4 * q^59 + 22 * q^61 - 2 * q^65 - 30 * q^71 + 2 * q^79 + 24 * q^85 + 18 * q^89 - 14 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4680\mathbb{Z}\right)^\times$.

$n$	$937$	$1081$	$2081$	$2341$	$3511$
$\chi(n)$	$-1$	$1$	$1$	$1$	$1$

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
$5$	−2.15569	−	0.594137i	−0.964054	−	0.265706i
$5$	−2.15569	−	0.594137i	−0.964054	−	0.265706i
$6$		0			0
$7$		−	0.633776i		−	0.239545i	−0.992801	−	0.119772i	$-0.961784\pi$
$7$		−	0.633776i		−	0.239545i	0.992801	−	0.119772i	$-0.0382165\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.177949			0.0536537			0.0268269	−	0.999640i	$-0.491460\pi$
$11$	0.177949			0.0536537			0.0268269	+	0.999640i	$0.491460\pi$
$12$		0			0
$13$		−	1.00000i		−	0.277350i
$13$		−	1.00000i		−	0.277350i
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	4.48933i		−	1.08882i	−0.838818	−	0.544411i	$-0.816753\pi$
$17$		−	4.48933i		−	1.08882i	0.838818	−	0.544411i	$-0.183247\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	6.48933i		−	1.35312i	−0.736388	−	0.676559i	$-0.763470\pi$
$23$		−	6.48933i		−	1.35312i	0.736388	−	0.676559i	$-0.236530\pi$
$24$		0			0
$25$	4.29400	+	2.56155i	0.858800	+	0.512311i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.49966			−0.649870			−0.324935	−	0.945736i	$-0.605342\pi$
$29$	−3.49966			−0.649870			−0.324935	+	0.945736i	$0.605342\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.376550	+	1.36622i	−0.0636486	+	0.230934i
$36$		0			0
$37$			1.75688i			0.288830i	0.989517	+	0.144415i	$0.0461299\pi$
$37$			1.75688i			0.288830i	−0.989517	+	0.144415i	$0.953870\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−7.67760			−1.19904			−0.599520	−	0.800360i	$-0.704642\pi$
$41$	−7.67760			−1.19904			−0.599520	+	0.800360i	$0.704642\pi$
$42$		0			0
$43$			7.49966i			1.14369i	0.820363	+	0.571843i	$0.193772\pi$
$43$			7.49966i			1.14369i	−0.820363	+	0.571843i	$0.806228\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			3.18827i			0.465058i	0.972589	+	0.232529i	$0.0747000\pi$
$47$			3.18827i			0.465058i	−0.972589	+	0.232529i	$0.925300\pi$
$48$		0			0
$49$	6.59833			0.942618
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	1.75688i		−	0.241326i	−0.992694	−	0.120663i	$-0.961498\pi$
$53$		−	1.75688i		−	0.241326i	0.992694	−	0.120663i	$-0.0385021\pi$
$54$		0			0
$55$	−0.383604	−	0.105726i	−0.0517251	−	0.0142561i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.93483			0.512271			0.256136	−	0.966641i	$-0.417551\pi$
$59$	3.93483			0.512271			0.256136	+	0.966641i	$0.417551\pi$
$60$		0			0
$61$	3.01033			0.385433			0.192716	−	0.981255i	$-0.438270\pi$
$61$	3.01033			0.385433			0.192716	+	0.981255i	$0.438270\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.594137	+	2.15569i	−0.0736937	+	0.267380i
$66$		0			0
$67$		−	1.62345i		−	0.198336i	−0.995071	−	0.0991680i	$-0.968382\pi$
$67$		−	1.62345i		−	0.198336i	0.995071	−	0.0991680i	$-0.0316181\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	4.41005			0.523377			0.261689	−	0.965152i	$-0.415721\pi$
$71$	4.41005			0.523377			0.261689	+	0.965152i	$0.415721\pi$
$72$		0			0
$73$			3.85555i			0.451258i	0.974213	+	0.225629i	$0.0724438\pi$
$73$			3.85555i			0.451258i	−0.974213	+	0.225629i	$0.927556\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	0.112780i		−	0.0128525i
$78$		0			0
$79$	−9.63309			−1.08381			−0.541903	−	0.840441i	$-0.682296\pi$
$79$	−9.63309			−1.08381			−0.541903	+	0.840441i	$0.682296\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	1.72338i		−	0.189165i	−0.995517	−	0.0945827i	$-0.969848\pi$
$83$		−	1.72338i		−	0.189165i	0.995517	−	0.0945827i	$-0.0301517\pi$
$84$		0			0
$85$	−2.66728	+	9.67760i	−0.289307	+	1.04968i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−0.589258			−0.0624612			−0.0312306	−	0.999512i	$-0.509943\pi$
$89$	−0.589258			−0.0624612			−0.0312306	+	0.999512i	$0.509943\pi$
$90$		0			0
$91$	−0.633776			−0.0664378
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	4.45457i		−	0.452293i	−0.974093	−	0.226147i	$-0.927387\pi$
$97$		−	4.45457i		−	0.452293i	0.974093	−	0.226147i	$-0.0726128\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−16.4783			−1.63965			−0.819827	−	0.572612i	$-0.805930\pi$
$101$	−16.4783			−1.63965			−0.819827	+	0.572612i	$0.805930\pi$
$102$		0			0
$103$		−	5.86966i		−	0.578355i	−0.957276	−	0.289177i	$-0.906618\pi$
$103$		−	5.86966i		−	0.578355i	0.957276	−	0.289177i	$-0.0933818\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	0.509981i		−	0.0493017i	−0.999696	−	0.0246509i	$-0.992153\pi$
$107$		−	0.509981i		−	0.0493017i	0.999696	−	0.0246509i	$-0.00784741\pi$
$108$		0			0
$109$	−6.35590			−0.608785			−0.304392	−	0.952547i	$-0.598453\pi$
$109$	−6.35590			−0.608785			−0.304392	+	0.952547i	$0.598453\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	9.71111i		−	0.913544i	−0.889584	−	0.456772i	$-0.849005\pi$
$113$		−	9.71111i		−	0.913544i	0.889584	−	0.456772i	$-0.150995\pi$
$114$		0			0
$115$	−3.85555	+	13.9890i	−0.359532	+	1.30448i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.84523			−0.260822
$120$		0			0
$121$	−10.9683			−0.997121
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−7.73462	−	8.07314i	−0.691806	−	0.722084i
$126$		0			0
$127$		−	11.1231i		−	0.987016i	−0.869741	−	0.493508i	$-0.835714\pi$
$127$		−	11.1231i		−	0.987016i	0.869741	−	0.493508i	$-0.164286\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−12.9439			−1.13091			−0.565457	−	0.824778i	$-0.691300\pi$
$131$	−12.9439			−1.13091			−0.565457	+	0.824778i	$0.691300\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$			11.7904i			1.00732i	0.863902	+	0.503660i	$0.168014\pi$
$137$			11.7904i			1.00732i	−0.863902	+	0.503660i	$0.831986\pi$
$138$		0			0
$139$	−0.112780			−0.00956587			−0.00478293	−	0.999989i	$-0.501522\pi$
$139$	−0.112780			−0.00956587			−0.00478293	+	0.999989i	$0.501522\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	0.177949i		−	0.0148809i
$144$		0			0
$145$	7.54417	+	2.07928i	0.626510	+	0.172675i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2.30829			0.189102			0.0945512	−	0.995520i	$-0.469858\pi$
$149$	2.30829			0.189102			0.0945512	+	0.995520i	$0.469858\pi$
$150$		0			0
$151$	−17.7318			−1.44299			−0.721495	−	0.692420i	$-0.756545\pi$
$151$	−17.7318			−1.44299			−0.721495	+	0.692420i	$0.756545\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$			12.8549i			1.02593i	0.858410	+	0.512965i	$0.171453\pi$
$157$			12.8549i			1.02593i	−0.858410	+	0.512965i	$0.828547\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−4.11278			−0.324132
$162$		0			0
$163$			14.3308i			1.12247i	0.827655	+	0.561237i	$0.189674\pi$
$163$			14.3308i			1.12247i	−0.827655	+	0.561237i	$0.810326\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	1.43449i		−	0.111004i	−0.998459	−	0.0555019i	$-0.982324\pi$
$167$		−	1.43449i		−	0.111004i	0.998459	−	0.0555019i	$-0.0176759\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	1.24690i		−	0.0948001i	−0.998876	−	0.0474000i	$-0.984906\pi$
$173$		−	1.24690i		−	0.0948001i	0.998876	−	0.0474000i	$-0.0150936\pi$
$174$		0			0
$175$	1.62345	−	2.72143i	0.122721	−	0.205721i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−7.87621			−0.588695			−0.294348	−	0.955698i	$-0.595102\pi$
$179$	−7.87621			−0.588695			−0.294348	+	0.955698i	$0.595102\pi$
$180$		0			0
$181$	10.3796			0.771513			0.385756	−	0.922601i	$-0.373941\pi$
$181$	10.3796			0.771513			0.385756	+	0.922601i	$0.373941\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.04383	−	3.78729i	0.0767438	−	0.278447i
$186$		0			0
$187$		−	0.798873i		−	0.0584194i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−13.7318			−0.993595			−0.496798	−	0.867866i	$-0.665491\pi$
$191$	−13.7318			−0.993595			−0.496798	+	0.867866i	$0.665491\pi$
$192$		0			0
$193$			6.34488i			0.456715i	0.973577	+	0.228357i	$0.0733355\pi$
$193$			6.34488i			0.456715i	−0.973577	+	0.228357i	$0.926665\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			21.2966i			1.51732i	0.651487	+	0.758659i	$0.274145\pi$
$197$			21.2966i			1.51732i	−0.651487	+	0.758659i	$0.725855\pi$
$198$		0			0
$199$	−11.2108			−0.794710			−0.397355	−	0.917665i	$-0.630072\pi$
$199$	−11.2108			−0.794710			−0.397355	+	0.917665i	$0.630072\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			2.21800i			0.155673i
$204$		0			0
$205$	16.5505	+	4.56155i	1.15594	+	0.318593i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−20.2810			−1.39620			−0.698100	−	0.716000i	$-0.745971\pi$
$211$	−20.2810			−1.39620			−0.698100	+	0.716000i	$0.745971\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.45583	−	16.1669i	0.303885	−	1.10258i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−4.48933			−0.301985
$222$		0			0
$223$		−	11.3900i		−	0.762729i	−0.924425	−	0.381364i	$-0.875454\pi$
$223$		−	11.3900i		−	0.762729i	0.924425	−	0.381364i	$-0.124546\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			14.7020i			0.975809i	0.872897	+	0.487904i	$0.162238\pi$
$227$			14.7020i			0.975809i	−0.872897	+	0.487904i	$0.837762\pi$
$228$		0			0
$229$	−9.84145			−0.650341			−0.325171	−	0.945655i	$-0.605422\pi$
$229$	−9.84145			−0.650341			−0.325171	+	0.945655i	$0.605422\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			22.8439i			1.49655i	0.663388	+	0.748275i	$0.269118\pi$
$233$			22.8439i			1.49655i	−0.663388	+	0.748275i	$0.730882\pi$
$234$		0			0
$235$	1.89427	−	6.87293i	0.123569	−	0.448341i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−24.1907			−1.56476			−0.782382	−	0.622798i	$-0.785996\pi$
$239$	−24.1907			−1.56476			−0.782382	+	0.622798i	$0.785996\pi$
$240$		0			0
$241$	−22.8896			−1.47445			−0.737225	−	0.675647i	$-0.763864\pi$
$241$	−22.8896			−1.47445			−0.737225	+	0.675647i	$0.763864\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−14.2240	−	3.92031i	−0.908735	−	0.250460i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	11.6763			0.737005			0.368502	−	0.929627i	$-0.379871\pi$
$251$	11.6763			0.737005			0.368502	+	0.929627i	$0.379871\pi$
$252$		0			0
$253$		−	1.15477i		−	0.0725999i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			24.9993i			1.55941i	0.626144	+	0.779707i	$0.284632\pi$
$257$			24.9993i			1.55941i	−0.626144	+	0.779707i	$0.715368\pi$
$258$		0			0
$259$	1.11347			0.0691876
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	7.12965i		−	0.439633i	−0.975541	−	0.219817i	$-0.929454\pi$
$263$		−	7.12965i		−	0.439633i	0.975541	−	0.219817i	$-0.0705459\pi$
$264$		0			0
$265$	−1.04383	+	3.78729i	−0.0641219	+	0.232652i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−8.47832			−0.516932			−0.258466	−	0.966020i	$-0.583217\pi$
$269$	−8.47832			−0.516932			−0.258466	+	0.966020i	$0.583217\pi$
$270$		0			0
$271$	5.60142			0.340262			0.170131	−	0.985421i	$-0.445581\pi$
$271$	5.60142			0.340262			0.170131	+	0.985421i	$0.445581\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.764114	+	0.455826i	0.0460778	+	0.0274874i
$276$		0			0
$277$		−	24.2991i		−	1.45999i	−0.683451	−	0.729996i	$-0.739522\pi$
$277$		−	24.2991i		−	1.45999i	0.683451	−	0.729996i	$-0.260478\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−11.6318			−0.693897			−0.346948	−	0.937884i	$-0.612782\pi$
$281$	−11.6318			−0.693897			−0.346948	+	0.937884i	$0.612782\pi$
$282$		0			0
$283$		−	0.197345i		−	0.0117310i	−0.999983	−	0.00586549i	$-0.998133\pi$
$283$		−	0.197345i		−	0.0117310i	0.999983	−	0.00586549i	$-0.00186705\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			4.86588i			0.287224i
$288$		0			0
$289$	−3.15408			−0.185534
$290$		0			0
$291$		0			0
$292$		0			0
$293$			29.3918i			1.71709i	0.512740	+	0.858544i	$0.328630\pi$
$293$			29.3918i			1.71709i	−0.512740	+	0.858544i	$0.671370\pi$
$294$		0			0
$295$	−8.48228	−	2.33783i	−0.493857	−	0.136114i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6.48933			−0.375288
$300$		0			0
$301$	4.75310			0.273964
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−6.48933	−	1.78855i	−0.371578	−	0.102412i
$306$		0			0
$307$		−	29.1997i		−	1.66652i	−0.552883	−	0.833259i	$-0.686472\pi$
$307$		−	29.1997i		−	1.66652i	0.552883	−	0.833259i	$-0.313528\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	13.3346			0.756133			0.378067	−	0.925778i	$-0.376589\pi$
$311$	13.3346			0.756133			0.378067	+	0.925778i	$0.376589\pi$
$312$		0			0
$313$			5.16441i			0.291910i	0.989291	+	0.145955i	$0.0466254\pi$
$313$			5.16441i			0.291910i	−0.989291	+	0.145955i	$0.953375\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			28.3461i			1.59208i	0.605245	+	0.796039i	$0.293075\pi$
$317$			28.3461i			1.59208i	−0.605245	+	0.796039i	$0.706925\pi$
$318$		0			0
$319$	−0.622761			−0.0348679
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	2.56155	−	4.29400i	0.142089	−	0.238188i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	2.02065			0.111402
$330$		0			0
$331$	−11.1296			−0.611741			−0.305870	−	0.952073i	$-0.598947\pi$
$331$	−11.1296			−0.611741			−0.305870	+	0.952073i	$0.598947\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−0.964553	+	3.49966i	−0.0526991	+	0.191207i
$336$		0			0
$337$			12.0529i			0.656563i	0.944580	+	0.328282i	$0.106470\pi$
$337$			12.0529i			0.656563i	−0.944580	+	0.328282i	$0.893530\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	8.61829i		−	0.465344i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	7.62654i		−	0.409414i	−0.978823	−	0.204707i	$-0.934376\pi$
$347$		−	7.62654i		−	0.409414i	0.978823	−	0.204707i	$-0.0656242\pi$
$348$		0			0
$349$	15.2455			0.816074			0.408037	−	0.912965i	$-0.366213\pi$
$349$	15.2455			0.816074			0.408037	+	0.912965i	$0.366213\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	4.83238i		−	0.257201i	−0.991696	−	0.128601i	$-0.958951\pi$
$353$		−	4.83238i		−	0.257201i	0.991696	−	0.128601i	$-0.0410486\pi$
$354$		0			0
$355$	−9.50671	−	2.62018i	−0.504564	−	0.139065i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	7.25528			0.382919			0.191460	−	0.981501i	$-0.438678\pi$
$359$	7.25528			0.382919			0.191460	+	0.981501i	$0.438678\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2.29073	−	8.31138i	0.119902	−	0.435038i
$366$		0			0
$367$		−	33.5808i		−	1.75290i	−0.481491	−	0.876451i	$-0.659905\pi$
$367$		−	33.5808i		−	1.75290i	0.481491	−	0.876451i	$-0.340095\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1.11347			−0.0578084
$372$		0			0
$373$		−	2.78131i		−	0.144011i	−0.997404	−	0.0720055i	$-0.977060\pi$
$373$		−	2.78131i		−	0.144011i	0.997404	−	0.0720055i	$-0.0229399\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			3.49966i			0.180241i
$378$		0			0
$379$	34.0268			1.74784			0.873921	−	0.486069i	$-0.161570\pi$
$379$	34.0268			1.74784			0.873921	+	0.486069i	$0.161570\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	8.45332i		−	0.431944i	−0.976400	−	0.215972i	$-0.930708\pi$
$383$		−	8.45332i		−	0.431944i	0.976400	−	0.215972i	$-0.0692920\pi$
$384$		0			0
$385$	−0.0670068	+	0.243119i	−0.00341498	+	0.0123905i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−14.2991			−0.724994			−0.362497	−	0.931985i	$-0.618076\pi$
$389$	−14.2991			−0.724994			−0.362497	+	0.931985i	$0.618076\pi$
$390$		0			0
$391$	−29.1327			−1.47331
$392$		0			0
$393$		0			0
$394$		0			0
$395$	20.7660	+	5.72338i	1.04485	+	0.287974i
$396$		0			0
$397$			3.97488i			0.199493i	0.995013	+	0.0997467i	$0.0318033\pi$
$397$			3.97488i			0.199493i	−0.995013	+	0.0997467i	$0.968197\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	10.2560			0.512159			0.256079	−	0.966656i	$-0.417569\pi$
$401$	10.2560			0.512159			0.256079	+	0.966656i	$0.417569\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$			0.312636i			0.0154968i
$408$		0			0
$409$	−19.6296			−0.970623			−0.485311	−	0.874341i	$-0.661294\pi$
$409$	−19.6296			−0.970623			−0.485311	+	0.874341i	$0.661294\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	2.49380i		−	0.122712i
$414$		0			0
$415$	−1.02392	+	3.71507i	−0.0502624	+	0.182366i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	23.7238			1.15899			0.579493	−	0.814977i	$-0.303251\pi$
$419$	23.7238			1.15899			0.579493	+	0.814977i	$0.303251\pi$
$420$		0			0
$421$	17.8697			0.870914			0.435457	−	0.900210i	$-0.356587\pi$
$421$	17.8697			0.870914			0.435457	+	0.900210i	$0.356587\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	11.4997	−	19.2772i	0.557815	−	0.935081i
$426$		0			0
$427$		−	1.90787i		−	0.0923284i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−35.9476			−1.73153			−0.865767	−	0.500448i	$-0.833169\pi$
$431$	−35.9476			−1.73153			−0.865767	+	0.500448i	$0.833169\pi$
$432$		0			0
$433$			8.21396i			0.394738i	0.980329	+	0.197369i	$0.0632397\pi$
$433$			8.21396i			0.394738i	−0.980329	+	0.197369i	$0.936760\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	25.0914			1.19755			0.598775	−	0.800918i	$-0.295655\pi$
$439$	25.0914			1.19755			0.598775	+	0.800918i	$0.295655\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			2.75619i			0.130951i	0.997854	+	0.0654753i	$0.0208564\pi$
$443$			2.75619i			0.130951i	−0.997854	+	0.0654753i	$0.979144\pi$
$444$		0			0
$445$	1.27026	+	0.350100i	0.0602160	+	0.0165963i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	39.7639			1.87657			0.938287	−	0.345858i	$-0.112412\pi$
$449$	39.7639			1.87657			0.938287	+	0.345858i	$0.112412\pi$
$450$		0			0
$451$	−1.36622			−0.0643330
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1.36622	+	0.376550i	0.0640496	+	0.0176529i
$456$		0			0
$457$			8.72143i			0.407971i	0.978974	+	0.203986i	$0.0653896\pi$
$457$			8.72143i			0.407971i	−0.978974	+	0.203986i	$0.934610\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−17.5332			−0.816601			−0.408300	−	0.912848i	$-0.633878\pi$
$461$	−17.5332			−0.816601			−0.408300	+	0.912848i	$0.633878\pi$
$462$		0			0
$463$		−	18.1049i		−	0.841404i	−0.907199	−	0.420702i	$-0.861784\pi$
$463$		−	18.1049i		−	0.841404i	0.907199	−	0.420702i	$-0.138216\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	11.0451i		−	0.511106i	−0.966795	−	0.255553i	$-0.917743\pi$
$467$		−	11.0451i		−	0.511106i	0.966795	−	0.255553i	$-0.0822575\pi$
$468$		0			0
$469$	−1.02890			−0.0475103
$470$		0			0
$471$		0			0
$472$		0			0
$473$			1.33456i			0.0613631i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−33.3887			−1.52557			−0.762785	−	0.646653i	$-0.776168\pi$
$479$	−33.3887			−1.52557			−0.762785	+	0.646653i	$0.776168\pi$
$480$		0			0
$481$	1.75688			0.0801069
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.64663	+	9.60268i	−0.120177	+	0.436035i
$486$		0			0
$487$		−	35.7414i		−	1.61960i	−0.586708	−	0.809799i	$-0.699576\pi$
$487$		−	35.7414i		−	1.61960i	0.586708	−	0.809799i	$-0.300424\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−36.9914			−1.66940			−0.834699	−	0.550706i	$-0.814358\pi$
$491$	−36.9914			−1.66940			−0.834699	+	0.550706i	$0.814358\pi$
$492$		0			0
$493$			15.7111i			0.707593i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	2.79498i		−	0.125372i
$498$		0			0
$499$	20.0670			0.898323			0.449161	−	0.893451i	$-0.351723\pi$
$499$	20.0670			0.898323			0.449161	+	0.893451i	$0.351723\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	15.3332i		−	0.683673i	−0.939759	−	0.341836i	$-0.888951\pi$
$503$		−	15.3332i		−	0.683673i	0.939759	−	0.341836i	$-0.111049\pi$
$504$		0			0
$505$	35.5221	+	9.79038i	1.58071	+	0.435666i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	30.0476			1.33184			0.665918	−	0.746025i	$-0.268040\pi$
$509$	30.0476			1.33184			0.665918	+	0.746025i	$0.268040\pi$
$510$		0			0
$511$	2.44356			0.108097
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−3.48739	+	12.6532i	−0.153673	+	0.557565i
$516$		0			0
$517$			0.567351i			0.0249521i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	30.8263			1.35052			0.675262	−	0.737578i	$-0.264030\pi$
$521$	30.8263			1.35052			0.675262	+	0.737578i	$0.264030\pi$
$522$		0			0
$523$			12.0141i			0.525340i	0.964886	+	0.262670i	$0.0846031\pi$
$523$			12.0141i			0.525340i	−0.964886	+	0.262670i	$0.915397\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−19.1114			−0.830931
$530$		0			0
$531$		0			0
$532$		0			0
$533$			7.67760i			0.332554i
$534$		0			0
$535$	−0.302999	+	1.09936i	−0.0130998	+	0.0475295i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.17417			0.0505750
$540$		0			0
$541$	16.3132			0.701360			0.350680	−	0.936495i	$-0.385950\pi$
$541$	16.3132			0.701360			0.350680	+	0.936495i	$0.385950\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	13.7013	+	3.77628i	0.586901	+	0.161758i
$546$		0			0
$547$		−	0.239667i		−	0.0102474i	−0.999987	−	0.00512371i	$-0.998369\pi$
$547$		−	0.239667i		−	0.0102474i	0.999987	−	0.00512371i	$-0.00163094\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$			6.10522i			0.259620i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			10.0304i			0.425002i	0.977161	+	0.212501i	$0.0681609\pi$
$557$			10.0304i			0.425002i	−0.977161	+	0.212501i	$0.931839\pi$
$558$		0			0
$559$	7.49966			0.317202
$560$		0			0
$561$		0			0
$562$		0			0
$563$			0.310125i			0.0130702i	0.999979	+	0.00653511i	$0.00208021\pi$
$563$			0.310125i			0.0130702i	−0.999979	+	0.00653511i	$0.997920\pi$
$564$		0			0
$565$	−5.76973	+	20.9341i	−0.242735	+	0.880706i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	12.2850			0.515014			0.257507	−	0.966276i	$-0.417099\pi$
$569$	12.2850			0.515014			0.257507	+	0.966276i	$0.417099\pi$
$570$		0			0
$571$	−31.6124			−1.32294			−0.661470	−	0.749972i	$-0.730067\pi$
$571$	−31.6124			−1.32294			−0.661470	+	0.749972i	$0.730067\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	16.6228	−	27.8652i	0.693217	−	1.16206i
$576$		0			0
$577$		−	6.36691i		−	0.265058i	−0.991179	−	0.132529i	$-0.957690\pi$
$577$		−	6.36691i		−	0.265058i	0.991179	−	0.132529i	$-0.0423098\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1.09224			−0.0453136
$582$		0			0
$583$		−	0.312636i		−	0.0129481i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			14.8103i			0.611288i	0.952146	+	0.305644i	$0.0988718\pi$
$587$			14.8103i			0.611288i	−0.952146	+	0.305644i	$0.901128\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	33.9864i		−	1.39565i	−0.716267	−	0.697826i	$-0.754151\pi$
$593$		−	33.9864i		−	1.39565i	0.716267	−	0.697826i	$-0.245849\pi$
$594$		0			0
$595$	6.13343	+	1.69046i	0.251446	+	0.0693020i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1.10900			−0.0453124			−0.0226562	−	0.999743i	$-0.507212\pi$
$599$	−1.10900			−0.0453124			−0.0226562	+	0.999743i	$0.507212\pi$
$600$		0			0
$601$	−47.9670			−1.95661			−0.978306	−	0.207163i	$-0.933577\pi$
$601$	−47.9670			−1.95661			−0.978306	+	0.207163i	$0.933577\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	23.6443	+	6.51670i	0.961279	+	0.264941i
$606$		0			0
$607$		−	35.5319i		−	1.44220i	−0.692833	−	0.721098i	$-0.743638\pi$
$607$		−	35.5319i		−	1.44220i	0.692833	−	0.721098i	$-0.256362\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	3.18827			0.128984
$612$		0			0
$613$			20.4955i			0.827806i	0.910321	+	0.413903i	$0.135835\pi$
$613$			20.4955i			0.827806i	−0.910321	+	0.413903i	$0.864165\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			10.6119i			0.427218i	0.976919	+	0.213609i	$0.0685218\pi$
$617$			10.6119i			0.427218i	−0.976919	+	0.213609i	$0.931478\pi$
$618$		0			0
$619$	28.5312			1.14677			0.573383	−	0.819287i	$-0.305631\pi$
$619$	28.5312			1.14677			0.573383	+	0.819287i	$0.305631\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			0.373457i			0.0149623i
$624$		0			0
$625$	11.8769	+	21.9986i	0.475076	+	0.879945i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	7.88722			0.314484
$630$		0			0
$631$	−7.71111			−0.306974			−0.153487	−	0.988151i	$-0.549050\pi$
$631$	−7.71111			−0.306974			−0.153487	+	0.988151i	$0.549050\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−6.60865	+	23.9780i	−0.262256	+	0.951537i
$636$		0			0
$637$		−	6.59833i		−	0.261435i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	27.5124			1.08667			0.543337	−	0.839515i	$-0.317161\pi$
$641$	27.5124			1.08667			0.543337	+	0.839515i	$0.317161\pi$
$642$		0			0
$643$		−	24.1798i		−	0.953558i	−0.879023	−	0.476779i	$-0.841804\pi$
$643$		−	24.1798i		−	0.953558i	0.879023	−	0.476779i	$-0.158196\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	27.9347i		−	1.09823i	−0.835748	−	0.549113i	$-0.814966\pi$
$647$		−	27.9347i		−	1.09823i	0.835748	−	0.549113i	$-0.185034\pi$
$648$		0			0
$649$	0.700200			0.0274853
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	27.2868i		−	1.06782i	−0.845543	−	0.533908i	$-0.820723\pi$
$653$		−	27.2868i		−	1.06782i	0.845543	−	0.533908i	$-0.179277\pi$
$654$		0			0
$655$	27.9030	+	7.69046i	1.09026	+	0.300491i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−30.3198			−1.18109			−0.590545	−	0.807005i	$-0.701087\pi$
$659$	−30.3198			−1.18109			−0.590545	+	0.807005i	$0.701087\pi$
$660$		0			0
$661$	16.7104			0.649960			0.324980	−	0.945721i	$-0.394642\pi$
$661$	16.7104			0.649960			0.324980	+	0.945721i	$0.394642\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$			22.7104i			0.879351i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	0.535685			0.0206799
$672$		0			0
$673$		−	0.950445i		−	0.0366370i	−0.999832	−	0.0183185i	$-0.994169\pi$
$673$		−	0.950445i		−	0.0366370i	0.999832	−	0.0183185i	$-0.00583128\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	34.1991i		−	1.31438i	−0.753726	−	0.657188i	$-0.771746\pi$
$677$		−	34.1991i		−	1.31438i	0.753726	−	0.657188i	$-0.228254\pi$
$678$		0			0
$679$	−2.82320			−0.108344
$680$		0			0
$681$		0			0
$682$		0			0
$683$			5.68208i			0.217419i	0.994074	+	0.108709i	$0.0346718\pi$
$683$			5.68208i			0.217419i	−0.994074	+	0.108709i	$0.965328\pi$
$684$		0			0
$685$	7.00511	−	25.4164i	0.267651	−	0.971111i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1.75688			−0.0669319
$690$		0			0
$691$	−13.7795			−0.524197			−0.262098	−	0.965041i	$-0.584414\pi$
$691$	−13.7795			−0.524197			−0.262098	+	0.965041i	$0.584414\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	0.243119	+	0.0670068i	0.00922201	+	0.00254171i
$696$		0			0
$697$			34.4673i			1.30554i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−20.4113			−0.770924			−0.385462	−	0.922724i	$-0.625958\pi$
$701$	−20.4113			−0.770924			−0.385462	+	0.922724i	$0.625958\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$			10.4436i			0.392770i
$708$		0			0
$709$	−3.17852			−0.119372			−0.0596858	−	0.998217i	$-0.519010\pi$
$709$	−3.17852			−0.119372			−0.0596858	+	0.998217i	$0.519010\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−0.105726	+	0.383604i	−0.00395394	+	0.0143460i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−40.5312			−1.51156			−0.755780	−	0.654826i	$-0.772742\pi$
$719$	−40.5312			−1.51156			−0.755780	+	0.654826i	$0.772742\pi$
$720$		0			0
$721$	−3.72005			−0.138542
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−15.0275	−	8.96455i	−0.558108	−	0.332935i
$726$		0			0
$727$		−	4.39066i		−	0.162840i	−0.996680	−	0.0814202i	$-0.974054\pi$
$727$		−	4.39066i		−	0.162840i	0.996680	−	0.0814202i	$-0.0259456\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	33.6684			1.24527
$732$		0			0
$733$			30.1823i			1.11481i	0.830241	+	0.557404i	$0.188203\pi$
$733$			30.1823i			1.11481i	−0.830241	+	0.557404i	$0.811797\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	0.288892i		−	0.0106415i
$738$		0			0
$739$	49.1565			1.80825			0.904125	−	0.427267i	$-0.140524\pi$
$739$	49.1565			1.80825			0.904125	+	0.427267i	$0.140524\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			21.8317i			0.800927i	0.916313	+	0.400463i	$0.131151\pi$
$743$			21.8317i			0.800927i	−0.916313	+	0.400463i	$0.868849\pi$
$744$		0			0
$745$	−4.97595	−	1.37144i	−0.182305	−	0.0502457i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−0.323214			−0.0118100
$750$		0			0
$751$	−22.8170			−0.832605			−0.416302	−	0.909226i	$-0.636674\pi$
$751$	−22.8170			−0.832605			−0.416302	+	0.909226i	$0.636674\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	38.2242	+	10.5351i	1.39112	+	0.383412i
$756$		0			0
$757$		−	11.4649i		−	0.416699i	−0.978054	−	0.208349i	$-0.933191\pi$
$757$		−	11.4649i		−	0.416699i	0.978054	−	0.208349i	$-0.0668091\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	26.9005			0.975143			0.487571	−	0.873083i	$-0.337883\pi$
$761$	26.9005			0.975143			0.487571	+	0.873083i	$0.337883\pi$
$762$		0			0
$763$			4.02821i			0.145831i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	3.93483i		−	0.142079i
$768$		0			0
$769$	−12.9298			−0.466260			−0.233130	−	0.972446i	$-0.574897\pi$
$769$	−12.9298			−0.466260			−0.233130	+	0.972446i	$0.574897\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	7.56231i		−	0.271998i	−0.990709	−	0.135999i	$-0.956576\pi$
$773$		−	7.56231i		−	0.271998i	0.990709	−	0.135999i	$-0.0434243\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	0.784766			0.0280811
$782$		0			0
$783$		0			0
$784$		0			0
$785$	7.63756	−	27.7111i	0.272596	−	0.989052i
$786$		0			0
$787$			53.3318i			1.90107i	0.310612	+	0.950537i	$0.399466\pi$
$787$			53.3318i			1.90107i	−0.310612	+	0.950537i	$0.600534\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−6.15466			−0.218835
$792$		0			0
$793$		−	3.01033i		−	0.106900i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			46.9391i			1.66267i	0.555774	+	0.831334i	$0.312422\pi$
$797$			46.9391i			1.66267i	−0.555774	+	0.831334i	$0.687578\pi$
$798$		0			0
$799$	14.3132			0.506365
$800$		0			0
$801$		0			0
$802$		0			0
$803$			0.686093i			0.0242117i
$804$		0			0
$805$	8.86588	+	2.44356i	0.312481	+	0.0861241i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	18.1365			0.637646			0.318823	−	0.947814i	$-0.396712\pi$
$809$	18.1365			0.637646			0.318823	+	0.947814i	$0.396712\pi$
$810$		0			0
$811$	50.6664			1.77914			0.889568	−	0.456802i	$-0.151005\pi$
$811$	50.6664			1.77914			0.889568	+	0.456802i	$0.151005\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	8.51445	−	30.8927i	0.298248	−	1.08212i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−36.1986			−1.26334			−0.631670	−	0.775237i	$-0.717630\pi$
$821$	−36.1986			−1.26334			−0.631670	+	0.775237i	$0.717630\pi$
$822$		0			0
$823$		−	0.307033i		−	0.0107025i	−0.999986	−	0.00535124i	$-0.998297\pi$
$823$		−	0.307033i		−	0.0107025i	0.999986	−	0.00535124i	$-0.00170336\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	55.9733i		−	1.94638i	−0.230000	−	0.973191i	$-0.573873\pi$
$827$		−	55.9733i		−	1.94638i	0.230000	−	0.973191i	$-0.426127\pi$
$828$		0			0
$829$	37.1912			1.29171			0.645853	−	0.763462i	$-0.276502\pi$
$829$	37.1912			1.29171			0.645853	+	0.763462i	$0.276502\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	29.6221i		−	1.02634i
$834$		0			0
$835$	−0.852282	+	3.09231i	−0.0294944	+	0.107014i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	31.2570			1.07911			0.539556	−	0.841950i	$-0.318592\pi$
$839$	31.2570			1.07911			0.539556	+	0.841950i	$0.318592\pi$
$840$		0			0
$841$	−16.7524			−0.577669
$842$		0			0
$843$		0			0
$844$		0			0
$845$	2.15569	+	0.594137i	0.0741580	+	0.0204390i
$846$		0			0
$847$			6.95146i			0.238855i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	11.4010			0.390821
$852$		0			0
$853$		−	37.7066i		−	1.29105i	−0.763739	−	0.645525i	$-0.776638\pi$
$853$		−	37.7066i		−	1.29105i	0.763739	−	0.645525i	$-0.223362\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			19.5364i			0.667350i	0.942688	+	0.333675i	$0.108289\pi$
$857$			19.5364i			0.667350i	−0.942688	+	0.333675i	$0.891711\pi$
$858$		0			0
$859$	−34.0340			−1.16122			−0.580612	−	0.814180i	$-0.697187\pi$
$859$	−34.0340			−1.16122			−0.580612	+	0.814180i	$0.697187\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	16.9482i		−	0.576925i	−0.957491	−	0.288463i	$-0.906856\pi$
$863$		−	16.9482i		−	0.576925i	0.957491	−	0.288463i	$-0.0931441\pi$
$864$		0			0
$865$	−0.740830	+	2.68793i	−0.0251890	+	0.0913924i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.71420			−0.0581503
$870$		0			0
$871$	−1.62345			−0.0550085
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−5.11656	+	4.90202i	−0.172971	+	0.165718i
$876$		0			0
$877$			30.3339i			1.02430i	0.858895	+	0.512151i	$0.171151\pi$
$877$			30.3339i			1.02430i	−0.858895	+	0.512151i	$0.828849\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	8.66544			0.291946			0.145973	−	0.989289i	$-0.453369\pi$
$881$	8.66544			0.291946			0.145973	+	0.989289i	$0.453369\pi$
$882$		0			0
$883$			12.6898i			0.427045i	0.976938	+	0.213522i	$0.0684936\pi$
$883$			12.6898i			0.427045i	−0.976938	+	0.213522i	$0.931506\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	41.8638i		−	1.40565i	−0.711363	−	0.702825i	$-0.751922\pi$
$887$		−	41.8638i		−	1.40565i	0.711363	−	0.702825i	$-0.248078\pi$
$888$		0			0
$889$	−7.04955			−0.236434
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	16.9787	+	4.67955i	0.567534	+	0.156420i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−7.88722			−0.262761
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−22.3753	−	6.16693i	−0.743780	−	0.204996i
$906$		0			0
$907$			19.2531i			0.639288i	0.947538	+	0.319644i	$0.103563\pi$
$907$			19.2531i			0.639288i	−0.947538	+	0.319644i	$0.896437\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	9.30885			0.308416			0.154208	−	0.988038i	$-0.450717\pi$
$911$	9.30885			0.308416			0.154208	+	0.988038i	$0.450717\pi$
$912$		0			0
$913$		−	0.306674i		−	0.0101494i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			8.20353i			0.270904i
$918$		0			0
$919$	13.9282			0.459448			0.229724	−	0.973256i	$-0.426218\pi$
$919$	13.9282			0.459448			0.229724	+	0.973256i	$0.426218\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	4.41005i		−	0.145159i
$924$		0			0
$925$	−4.50034	+	7.54405i	−0.147970	+	0.248047i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−10.4734			−0.343621			−0.171810	−	0.985130i	$-0.554962\pi$
$929$	−10.4734			−0.343621			−0.171810	+	0.985130i	$0.554962\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−0.474640	+	1.72212i	−0.0155224	+	0.0563194i
$936$		0			0
$937$			6.63825i			0.216862i	0.994104	+	0.108431i	$0.0345827\pi$
$937$			6.63825i			0.216862i	−0.994104	+	0.108431i	$0.965417\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−43.5538			−1.41981			−0.709907	−	0.704296i	$-0.751263\pi$
$941$	−43.5538			−1.41981			−0.709907	+	0.704296i	$0.751263\pi$
$942$		0			0
$943$			49.8225i			1.62244i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			29.8379i			0.969601i	0.874625	+	0.484800i	$0.161108\pi$
$947$			29.8379i			0.969601i	−0.874625	+	0.484800i	$0.838892\pi$
$948$		0			0
$949$	3.85555			0.125157
$950$		0			0
$951$		0			0
$952$		0			0
$953$			16.4285i			0.532172i	0.963949	+	0.266086i	$0.0857304\pi$
$953$			16.4285i			0.532172i	−0.963949	+	0.266086i	$0.914270\pi$
$954$		0			0
$955$	29.6014	+	8.15855i	0.957879	+	0.264005i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	7.47246			0.241298
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$	3.76973	−	13.6776i	0.121352	−	0.440298i
$966$		0			0
$967$		−	39.1011i		−	1.25741i	−0.777646	−	0.628703i	$-0.783586\pi$
$967$		−	39.1011i		−	1.25741i	0.777646	−	0.628703i	$-0.216414\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−45.0240			−1.44489			−0.722444	−	0.691429i	$-0.756981\pi$
$971$	−45.0240			−1.44489			−0.722444	+	0.691429i	$0.756981\pi$
$972$		0			0
$973$			0.0714772i			0.00229145i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	26.5616i		−	0.849782i	−0.905245	−	0.424891i	$-0.860312\pi$
$977$		−	26.5616i		−	0.849782i	0.905245	−	0.424891i	$-0.139688\pi$
$978$		0			0
$979$	−0.104858			−0.00335128
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	12.7420i		−	0.406405i	−0.979137	−	0.203203i	$-0.934865\pi$
$983$		−	12.7420i		−	0.406405i	0.979137	−	0.203203i	$-0.0651351\pi$
$984$		0			0
$985$	12.6531	−	45.9088i	0.403161	−	1.46278i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	48.6677			1.54754
$990$		0			0
$991$	−34.0639			−1.08208			−0.541038	−	0.840998i	$-0.681968\pi$
$991$	−34.0639			−1.08208			−0.541038	+	0.840998i	$0.681968\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	24.1669	+	6.66073i	0.766143	+	0.211159i
$996$		0			0
$997$			17.5887i			0.557039i	0.960431	+	0.278520i	$0.0898438\pi$
$997$			17.5887i			0.557039i	−0.960431	+	0.278520i	$0.910156\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4680.2.l.g.2809.1		8
3.2	odd	2		1560.2.l.d.1249.8	yes	8
5.4	even	2	inner	4680.2.l.g.2809.2		8
12.11	even	2		3120.2.l.n.1249.4		8
15.2	even	4		7800.2.a.by.1.2		4
15.8	even	4		7800.2.a.bt.1.3		4
15.14	odd	2		1560.2.l.d.1249.4	✓	8
60.59	even	2		3120.2.l.n.1249.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.l.d.1249.4	✓	8	15.14	odd	2
1560.2.l.d.1249.8	yes	8	3.2	odd	2
3120.2.l.n.1249.4		8	12.11	even	2
3120.2.l.n.1249.8		8	60.59	even	2
4680.2.l.g.2809.1		8	1.1	even	1	trivial
4680.2.l.g.2809.2		8	5.4	even	2	inner
7800.2.a.bt.1.3		4	15.8	even	4
7800.2.a.by.1.2		4	15.2	even	4

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:	\( N \)	\(=\)	\( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4680.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.15569 - 0.594137i) q^{5} -0.633776i q^{7} +O(q^{10})\)	\(q+(-2.15569 - 0.594137i) q^{5} -0.633776i q^{7} +0.177949 q^{11} -1.00000i q^{13} -4.48933i q^{17} -6.48933i q^{23} +(4.29400 + 2.56155i) q^{25} -3.49966 q^{29} +(-0.376550 + 1.36622i) q^{35} +1.75688i q^{37} -7.67760 q^{41} +7.49966i q^{43} +3.18827i q^{47} +6.59833 q^{49} -1.75688i q^{53} +(-0.383604 - 0.105726i) q^{55} +3.93483 q^{59} +3.01033 q^{61} +(-0.594137 + 2.15569i) q^{65} -1.62345i q^{67} +4.41005 q^{71} +3.85555i q^{73} -0.112780i q^{77} -9.63309 q^{79} -1.72338i q^{83} +(-2.66728 + 9.67760i) q^{85} -0.589258 q^{89} -0.633776 q^{91} -4.45457i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{5}+O(q^{10}) \) 8 * q + 2 * q^5	\( 8 q + 2 q^{5} - 2 q^{11} + 16 q^{29} + 8 q^{35} - 14 q^{41} - 18 q^{49} + 10 q^{55} + 4 q^{59} + 22 q^{61} - 2 q^{65} - 30 q^{71} + 2 q^{79} + 24 q^{85} + 18 q^{89} - 14 q^{91}+O(q^{100}) \) 8 * q + 2 * q^5 - 2 * q^11 + 16 * q^29 + 8 * q^35 - 14 * q^41 - 18 * q^49 + 10 * q^55 + 4 * q^59 + 22 * q^61 - 2 * q^65 - 30 * q^71 + 2 * q^79 + 24 * q^85 + 18 * q^89 - 14 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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