LMFDB - Embedded newform 720.3.bh.k.433.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [720,3,Mod(433,720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(720, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 0, 3]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("720.433");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	720.bh (of order $4$, degree $2$, not 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:	$19.6185790339$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			433.2
Root			$-1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	720.433
Dual form			720.3.bh.k.577.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.67423 + 1.77526i) q^{5} +(-3.44949 + 3.44949i) q^{7} +11.3485 q^{11} +(-5.55051 - 5.55051i) q^{13} +(17.3485 - 17.3485i) q^{17} +8.69694i q^{19} +(11.5505 + 11.5505i) q^{23} +(18.6969 + 16.5959i) q^{25} +35.1464i q^{29} -10.6969 q^{31} +(-22.2474 + 10.0000i) q^{35} +(-6.04541 + 6.04541i) q^{37} -0.696938 q^{41} +(26.4949 + 26.4949i) q^{43} +(44.2474 - 44.2474i) q^{47} +25.2020i q^{49} +(0.696938 + 0.696938i) q^{53} +(53.0454 + 20.1464i) q^{55} +39.9342i q^{59} +5.90918 q^{61} +(-16.0908 - 35.7980i) q^{65} +(45.1010 - 45.1010i) q^{67} -68.0000 q^{71} +(77.7878 + 77.7878i) q^{73} +(-39.1464 + 39.1464i) q^{77} -24.4949i q^{79} +(13.1464 + 13.1464i) q^{83} +(111.889 - 50.2929i) q^{85} +82.1816i q^{89} +38.2929 q^{91} +(-15.4393 + 40.6515i) q^{95} +(-24.5959 + 24.5959i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} - 4 q^{7} + 16 q^{11} - 32 q^{13} + 40 q^{17} + 56 q^{23} + 16 q^{25} + 16 q^{31} - 40 q^{35} + 64 q^{37} + 56 q^{41} + 8 q^{43} + 128 q^{47} - 56 q^{53} + 124 q^{55} + 200 q^{61} + 112 q^{65}+ \cdots - 20 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 - 4 * q^7 + 16 * q^11 - 32 * q^13 + 40 * q^17 + 56 * q^23 + 16 * q^25 + 16 * q^31 - 40 * q^35 + 64 * q^37 + 56 * q^41 + 8 * q^43 + 128 * q^47 - 56 * q^53 + 124 * q^55 + 200 * q^61 + 112 * q^65 + 200 * q^67 - 272 * q^71 + 76 * q^73 - 88 * q^77 - 16 * q^83 + 232 * q^85 + 16 * q^91 + 144 * q^95 - 20 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$.

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$	$1$

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
$5$	4.67423	+	1.77526i	0.934847	+	0.355051i
$5$	4.67423	+	1.77526i	0.934847	+	0.355051i
$6$		0			0
$7$	−3.44949	+	3.44949i	−0.492784	+	0.492784i	−0.909182	−	0.416398i	$-0.863292\pi$
$7$	−3.44949	+	3.44949i	−0.492784	+	0.492784i	0.416398	+	0.909182i	$0.363292\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	11.3485			1.03168			0.515840	−	0.856685i	$-0.327480\pi$
$11$	11.3485			1.03168			0.515840	+	0.856685i	$0.327480\pi$
$12$		0			0
$13$	−5.55051	−	5.55051i	−0.426962	−	0.426962i	0.460630	−	0.887592i	$-0.347624\pi$
$13$	−5.55051	−	5.55051i	−0.426962	−	0.426962i	−0.887592	+	0.460630i	$0.847624\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	17.3485	−	17.3485i	1.02050	−	1.02050i	0.0207127	−	0.999785i	$-0.493406\pi$
$17$	17.3485	−	17.3485i	1.02050	−	1.02050i	0.999785	−	0.0207127i	$-0.00659354\pi$
$18$		0			0
$19$			8.69694i			0.457734i	0.973458	+	0.228867i	$0.0735020\pi$
$19$			8.69694i			0.457734i	−0.973458	+	0.228867i	$0.926498\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	11.5505	+	11.5505i	0.502196	+	0.502196i	0.912120	−	0.409924i	$-0.134445\pi$
$23$	11.5505	+	11.5505i	0.502196	+	0.502196i	−0.409924	+	0.912120i	$0.634445\pi$
$24$		0			0
$25$	18.6969	+	16.5959i	0.747878	+	0.663837i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			35.1464i			1.21195i	0.795485	+	0.605973i	$0.207216\pi$
$29$			35.1464i			1.21195i	−0.795485	+	0.605973i	$0.792784\pi$
$30$		0			0
$31$	−10.6969			−0.345063			−0.172531	−	0.985004i	$-0.555195\pi$
$31$	−10.6969			−0.345063			−0.172531	+	0.985004i	$0.555195\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−22.2474	+	10.0000i	−0.635641	+	0.285714i
$36$		0			0
$37$	−6.04541	+	6.04541i	−0.163389	+	0.163389i	−0.784066	−	0.620677i	$-0.786858\pi$
$37$	−6.04541	+	6.04541i	−0.163389	+	0.163389i	0.620677	+	0.784066i	$0.286858\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.696938			−0.0169985			−0.00849925	−	0.999964i	$-0.502705\pi$
$41$	−0.696938			−0.0169985			−0.00849925	+	0.999964i	$0.502705\pi$
$42$		0			0
$43$	26.4949	+	26.4949i	0.616160	+	0.616160i	0.944544	−	0.328384i	$-0.106504\pi$
$43$	26.4949	+	26.4949i	0.616160	+	0.616160i	−0.328384	+	0.944544i	$0.606504\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	44.2474	−	44.2474i	0.941435	−	0.941435i	−0.0569424	−	0.998377i	$-0.518135\pi$
$47$	44.2474	−	44.2474i	0.941435	−	0.941435i	0.998377	+	0.0569424i	$0.0181351\pi$
$48$		0			0
$49$			25.2020i			0.514327i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.696938	+	0.696938i	0.0131498	+	0.0131498i	0.713651	−	0.700501i	$-0.247040\pi$
$53$	0.696938	+	0.696938i	0.0131498	+	0.0131498i	−0.700501	+	0.713651i	$0.747040\pi$
$54$		0			0
$55$	53.0454	+	20.1464i	0.964462	+	0.366299i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			39.9342i			0.676851i	0.940993	+	0.338425i	$0.109894\pi$
$59$			39.9342i			0.676851i	−0.940993	+	0.338425i	$0.890106\pi$
$60$		0			0
$61$	5.90918			0.0968719			0.0484359	−	0.998826i	$-0.484576\pi$
$61$	5.90918			0.0968719			0.0484359	+	0.998826i	$0.484576\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−16.0908	−	35.7980i	−0.247551	−	0.550738i
$66$		0			0
$67$	45.1010	−	45.1010i	0.673150	−	0.673150i	−0.285291	−	0.958441i	$-0.592090\pi$
$67$	45.1010	−	45.1010i	0.673150	−	0.673150i	0.958441	+	0.285291i	$0.0920903\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−68.0000			−0.957746			−0.478873	−	0.877884i	$-0.658955\pi$
$71$	−68.0000			−0.957746			−0.478873	+	0.877884i	$0.658955\pi$
$72$		0			0
$73$	77.7878	+	77.7878i	1.06559	+	1.06559i	0.997693	+	0.0678931i	$0.0216277\pi$
$73$	77.7878	+	77.7878i	1.06559	+	1.06559i	0.0678931	+	0.997693i	$0.478372\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−39.1464	+	39.1464i	−0.508395	+	0.508395i
$78$		0			0
$79$		−	24.4949i		−	0.310062i	−0.987910	−	0.155031i	$-0.950452\pi$
$79$		−	24.4949i		−	0.310062i	0.987910	−	0.155031i	$-0.0495477\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	13.1464	+	13.1464i	0.158391	+	0.158391i	0.781853	−	0.623463i	$-0.214275\pi$
$83$	13.1464	+	13.1464i	0.158391	+	0.158391i	−0.623463	+	0.781853i	$0.714275\pi$
$84$		0			0
$85$	111.889	−	50.2929i	1.31634	−	0.591681i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			82.1816i			0.923389i	0.887039	+	0.461695i	$0.152758\pi$
$89$			82.1816i			0.923389i	−0.887039	+	0.461695i	$0.847242\pi$
$90$		0			0
$91$	38.2929			0.420801
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−15.4393	+	40.6515i	−0.162519	+	0.427911i
$96$		0			0
$97$	−24.5959	+	24.5959i	−0.253566	+	0.253566i	−0.822431	−	0.568865i	$-0.807383\pi$
$97$	−24.5959	+	24.5959i	−0.253566	+	0.253566i	0.568865	+	0.822431i	$0.307383\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	105.621			1.04575			0.522876	−	0.852409i	$-0.324859\pi$
$101$	105.621			1.04575			0.522876	+	0.852409i	$0.324859\pi$
$102$		0			0
$103$	89.2474	+	89.2474i	0.866480	+	0.866480i	0.992081	−	0.125601i	$-0.0400858\pi$
$103$	89.2474	+	89.2474i	0.866480	+	0.866480i	−0.125601	+	0.992081i	$0.540086\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	68.7423	−	68.7423i	0.642452	−	0.642452i	−0.308706	−	0.951158i	$-0.599896\pi$
$107$	68.7423	−	68.7423i	0.642452	−	0.642452i	0.951158	+	0.308706i	$0.0998958\pi$
$108$		0			0
$109$		−	68.6969i		−	0.630247i	−0.949051	−	0.315124i	$-0.897954\pi$
$109$		−	68.6969i		−	0.630247i	0.949051	−	0.315124i	$-0.102046\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−97.6413	−	97.6413i	−0.864083	−	0.864083i	0.127727	−	0.991809i	$-0.459232\pi$
$113$	−97.6413	−	97.6413i	−0.864083	−	0.864083i	−0.991809	+	0.127727i	$0.959232\pi$
$114$		0			0
$115$	33.4847	+	74.4949i	0.291171	+	0.647782i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			119.687i			1.00577i
$120$		0			0
$121$	7.78775			0.0643616
$122$		0			0
$123$		0			0
$124$		0			0
$125$	57.9319	+	110.765i	0.463455	+	0.886120i
$126$		0			0
$127$	−164.621	+	164.621i	−1.29623	+	1.29623i	−0.365362	+	0.930865i	$0.619055\pi$
$127$	−164.621	+	164.621i	−1.29623	+	1.29623i	−0.930865	+	0.365362i	$0.880945\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	106.136			0.810200			0.405100	−	0.914272i	$-0.367237\pi$
$131$	106.136			0.810200			0.405100	+	0.914272i	$0.367237\pi$
$132$		0			0
$133$	−30.0000	−	30.0000i	−0.225564	−	0.225564i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	166.631	−	166.631i	1.21629	−	1.21629i	0.247363	−	0.968923i	$-0.420436\pi$
$137$	166.631	−	166.631i	1.21629	−	1.21629i	0.968923	−	0.247363i	$-0.0795639\pi$
$138$		0			0
$139$		−	191.171i		−	1.37533i	−0.726026	−	0.687667i	$-0.758635\pi$
$139$		−	191.171i		−	1.37533i	0.726026	−	0.687667i	$-0.241365\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−62.9898	−	62.9898i	−0.440488	−	0.440488i
$144$		0			0
$145$	−62.3939	+	164.283i	−0.430303	+	1.13298i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			84.8536i			0.569487i	0.958604	+	0.284744i	$0.0919084\pi$
$149$			84.8536i			0.569487i	−0.958604	+	0.284744i	$0.908092\pi$
$150$		0			0
$151$	−148.969			−0.986552			−0.493276	−	0.869873i	$-0.664201\pi$
$151$	−148.969			−0.986552			−0.493276	+	0.869873i	$0.664201\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−50.0000	−	18.9898i	−0.322581	−	0.122515i
$156$		0			0
$157$	16.8536	−	16.8536i	0.107348	−	0.107348i	−0.651393	−	0.758741i	$-0.725815\pi$
$157$	16.8536	−	16.8536i	0.107348	−	0.107348i	0.758741	+	0.651393i	$0.225815\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−79.6867			−0.494949
$162$		0			0
$163$	−130.606	−	130.606i	−0.801265	−	0.801265i	0.182029	−	0.983293i	$-0.441734\pi$
$163$	−130.606	−	130.606i	−0.801265	−	0.801265i	−0.983293	+	0.182029i	$0.941734\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−45.0352	+	45.0352i	−0.269672	+	0.269672i	−0.828968	−	0.559296i	$-0.811071\pi$
$167$	−45.0352	+	45.0352i	−0.269672	+	0.269672i	0.559296	+	0.828968i	$0.311071\pi$
$168$		0			0
$169$		−	107.384i		−	0.635406i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−146.631	−	146.631i	−0.847579	−	0.847579i	0.142252	−	0.989831i	$-0.454566\pi$
$173$	−146.631	−	146.631i	−0.847579	−	0.847579i	−0.989831	+	0.142252i	$0.954566\pi$
$174$		0			0
$175$	−121.742	+	7.24745i	−0.695671	+	0.0414140i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	183.712i		−	1.02632i	−0.858292	−	0.513161i	$-0.828474\pi$
$179$		−	183.712i		−	1.02632i	0.858292	−	0.513161i	$-0.171526\pi$
$180$		0			0
$181$	−286.272			−1.58162			−0.790808	−	0.612064i	$-0.790339\pi$
$181$	−286.272			−1.58162			−0.790808	+	0.612064i	$0.790339\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−38.9898	+	17.5255i	−0.210756	+	0.0947325i
$186$		0			0
$187$	196.879	−	196.879i	1.05283	−	1.05283i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	48.0908			0.251784			0.125892	−	0.992044i	$-0.459821\pi$
$191$	48.0908			0.251784			0.125892	+	0.992044i	$0.459821\pi$
$192$		0			0
$193$	−255.565	−	255.565i	−1.32417	−	1.32417i	−0.910364	−	0.413809i	$-0.864198\pi$
$193$	−255.565	−	255.565i	−1.32417	−	1.32417i	−0.413809	−	0.910364i	$-0.635802\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	96.6969	−	96.6969i	0.490847	−	0.490847i	−0.417726	−	0.908573i	$-0.637173\pi$
$197$	96.6969	−	96.6969i	0.490847	−	0.490847i	0.908573	+	0.417726i	$0.137173\pi$
$198$		0			0
$199$			192.606i			0.967870i	0.875104	+	0.483935i	$0.160793\pi$
$199$			192.606i			0.967870i	−0.875104	+	0.483935i	$0.839207\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−121.237	−	121.237i	−0.597228	−	0.597228i
$204$		0			0
$205$	−3.25765	−	1.23724i	−0.0158910	−	0.00603533i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			98.6969i			0.472234i
$210$		0			0
$211$	−147.212			−0.697688			−0.348844	−	0.937181i	$-0.613426\pi$
$211$	−147.212			−0.697688			−0.348844	+	0.937181i	$0.613426\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	76.8082	+	170.879i	0.357247	+	0.794784i
$216$		0			0
$217$	36.8990	−	36.8990i	0.170041	−	0.170041i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−192.586			−0.871429
$222$		0			0
$223$	−167.429	−	167.429i	−0.750803	−	0.750803i	0.223826	−	0.974629i	$-0.428145\pi$
$223$	−167.429	−	167.429i	−0.750803	−	0.750803i	−0.974629	+	0.223826i	$0.928145\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	253.171	−	253.171i	1.11529	−	1.11529i	0.122870	−	0.992423i	$-0.460790\pi$
$227$	253.171	−	253.171i	1.11529	−	1.11529i	0.992423	−	0.122870i	$-0.0392098\pi$
$228$		0			0
$229$			224.202i			0.979048i	0.871990	+	0.489524i	$0.162829\pi$
$229$			224.202i			0.979048i	−0.871990	+	0.489524i	$0.837171\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	205.712	+	205.712i	0.882883	+	0.882883i	0.993827	−	0.110944i	$-0.0353873\pi$
$233$	205.712	+	205.712i	0.882883	+	0.882883i	−0.110944	+	0.993827i	$0.535387\pi$
$234$		0			0
$235$	285.373	−	128.272i	1.21436	−	0.545840i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			345.798i			1.44685i	0.690401	+	0.723427i	$0.257434\pi$
$239$			345.798i			1.44685i	−0.690401	+	0.723427i	$0.742566\pi$
$240$		0			0
$241$	101.576			0.421475			0.210738	−	0.977543i	$-0.432413\pi$
$241$	101.576			0.421475			0.210738	+	0.977543i	$0.432413\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−44.7401	+	117.800i	−0.182612	+	0.480817i
$246$		0			0
$247$	48.2724	−	48.2724i	0.195435	−	0.195435i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−331.258			−1.31975			−0.659876	−	0.751375i	$-0.729391\pi$
$251$	−331.258			−1.31975			−0.659876	+	0.751375i	$0.729391\pi$
$252$		0			0
$253$	131.081	+	131.081i	0.518105	+	0.518105i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−33.2372	+	33.2372i	−0.129328	+	0.129328i	−0.768808	−	0.639480i	$-0.779150\pi$
$257$	−33.2372	+	33.2372i	−0.129328	+	0.129328i	0.639480	+	0.768808i	$0.279150\pi$
$258$		0			0
$259$		−	41.7071i		−	0.161031i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−278.157	−	278.157i	−1.05763	−	1.05763i	−0.998235	−	0.0593952i	$-0.981083\pi$
$263$	−278.157	−	278.157i	−1.05763	−	1.05763i	−0.0593952	−	0.998235i	$-0.518917\pi$
$264$		0			0
$265$	2.02041	+	4.49490i	0.00762419	+	0.0169619i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	488.499i		−	1.81598i	−0.418988	−	0.907992i	$-0.637615\pi$
$269$		−	488.499i		−	1.81598i	0.418988	−	0.907992i	$-0.362385\pi$
$270$		0			0
$271$	−131.576			−0.485518			−0.242759	−	0.970087i	$-0.578053\pi$
$271$	−131.576			−0.485518			−0.242759	+	0.970087i	$0.578053\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	212.182	+	188.338i	0.771570	+	0.684866i
$276$		0			0
$277$	101.510	−	101.510i	0.366461	−	0.366461i	−0.499724	−	0.866185i	$-0.666565\pi$
$277$	101.510	−	101.510i	0.366461	−	0.366461i	0.866185	+	0.499724i	$0.166565\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−343.303			−1.22172			−0.610860	−	0.791739i	$-0.709176\pi$
$281$	−343.303			−1.22172			−0.610860	+	0.791739i	$0.709176\pi$
$282$		0			0
$283$	−1.19184	−	1.19184i	−0.00421143	−	0.00421143i	0.704998	−	0.709209i	$-0.250948\pi$
$283$	−1.19184	−	1.19184i	−0.00421143	−	0.00421143i	−0.709209	+	0.704998i	$0.750948\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2.40408	−	2.40408i	0.00837659	−	0.00837659i
$288$		0			0
$289$		−	312.939i		−	1.08283i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−96.5653	−	96.5653i	−0.329574	−	0.329574i	0.522850	−	0.852425i	$-0.324869\pi$
$293$	−96.5653	−	96.5653i	−0.329574	−	0.329574i	−0.852425	+	0.522850i	$0.824869\pi$
$294$		0			0
$295$	−70.8934	+	186.662i	−0.240316	+	0.632752i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	128.222i		−	0.428838i
$300$		0			0
$301$	−182.788			−0.607268
$302$		0			0
$303$		0			0
$304$		0			0
$305$	27.6209	+	10.4903i	0.0905604	+	0.0343945i
$306$		0			0
$307$	124.969	−	124.969i	0.407066	−	0.407066i	−0.473648	−	0.880714i	$-0.657063\pi$
$307$	124.969	−	124.969i	0.407066	−	0.407066i	0.880714	+	0.473648i	$0.157063\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−586.302			−1.88522			−0.942608	−	0.333902i	$-0.891635\pi$
$311$	−586.302			−1.88522			−0.942608	+	0.333902i	$0.891635\pi$
$312$		0			0
$313$	−102.373	−	102.373i	−0.327072	−	0.327072i	0.524400	−	0.851472i	$-0.324290\pi$
$313$	−102.373	−	102.373i	−0.327072	−	0.327072i	−0.851472	+	0.524400i	$0.824290\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	108.783	−	108.783i	0.343165	−	0.343165i	−0.514391	−	0.857556i	$-0.671982\pi$
$317$	108.783	−	108.783i	0.343165	−	0.343165i	0.857556	+	0.514391i	$0.171982\pi$
$318$		0			0
$319$			398.858i			1.25034i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	150.879	+	150.879i	0.467116	+	0.467116i
$324$		0			0
$325$	−11.6617	−	195.893i	−0.0358823	−	0.602749i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			305.262i			0.927849i
$330$		0			0
$331$	245.423			0.741461			0.370730	−	0.928741i	$-0.379107\pi$
$331$	245.423			0.741461			0.370730	+	0.928741i	$0.379107\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	290.879	−	130.747i	0.868294	−	0.390289i
$336$		0			0
$337$	213.808	−	213.808i	0.634446	−	0.634446i	−0.314734	−	0.949180i	$-0.601915\pi$
$337$	213.808	−	213.808i	0.634446	−	0.634446i	0.949180	+	0.314734i	$0.101915\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−121.394			−0.355994
$342$		0			0
$343$	−255.959	−	255.959i	−0.746237	−	0.746237i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−160.050	+	160.050i	−0.461239	+	0.461239i	−0.899062	−	0.437822i	$-0.855750\pi$
$347$	−160.050	+	160.050i	−0.461239	+	0.461239i	0.437822	+	0.899062i	$0.355750\pi$
$348$		0			0
$349$			298.009i			0.853894i	0.904277	+	0.426947i	$0.140411\pi$
$349$			298.009i			0.853894i	−0.904277	+	0.426947i	$0.859589\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	22.5199	+	22.5199i	0.0637957	+	0.0637957i	0.738285	−	0.674489i	$-0.235636\pi$
$353$	22.5199	+	22.5199i	0.0637957	+	0.0637957i	−0.674489	+	0.738285i	$0.735636\pi$
$354$		0			0
$355$	−317.848	−	120.717i	−0.895346	−	0.340049i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	48.2724i		−	0.134464i	−0.997737	−	0.0672318i	$-0.978583\pi$
$359$		−	48.2724i		−	0.134464i	0.997737	−	0.0672318i	$-0.0214167\pi$
$360$		0			0
$361$	285.363			0.790480
$362$		0			0
$363$		0			0
$364$		0			0
$365$	225.505	+	501.691i	0.617822	+	1.37450i
$366$		0			0
$367$	−146.510	+	146.510i	−0.399209	+	0.399209i	−0.877954	−	0.478745i	$-0.841092\pi$
$367$	−146.510	+	146.510i	−0.399209	+	0.399209i	0.478745	+	0.877954i	$0.341092\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−4.80816			−0.0129600
$372$		0			0
$373$	86.2066	+	86.2066i	0.231117	+	0.231117i	0.813159	−	0.582042i	$-0.197746\pi$
$373$	86.2066	+	86.2066i	0.231117	+	0.231117i	−0.582042	+	0.813159i	$0.697746\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	195.081	−	195.081i	0.517455	−	0.517455i
$378$		0			0
$379$		−	210.000i		−	0.554090i	−0.960857	−	0.277045i	$-0.910645\pi$
$379$		−	210.000i		−	0.554090i	0.960857	−	0.277045i	$-0.0893551\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−10.6311	−	10.6311i	−0.0277575	−	0.0277575i	0.693092	−	0.720849i	$-0.256248\pi$
$383$	−10.6311	−	10.6311i	−0.0277575	−	0.0277575i	−0.720849	+	0.693092i	$0.756248\pi$
$384$		0			0
$385$	−252.474	+	113.485i	−0.655778	+	0.294765i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	535.337i		−	1.37619i	−0.725621	−	0.688094i	$-0.758448\pi$
$389$		−	535.337i		−	1.37619i	0.725621	−	0.688094i	$-0.241552\pi$
$390$		0			0
$391$	400.767			1.02498
$392$		0			0
$393$		0			0
$394$		0			0
$395$	43.4847	−	114.495i	0.110088	−	0.289860i
$396$		0			0
$397$	118.742	−	118.742i	0.299099	−	0.299099i	−0.541562	−	0.840661i	$-0.682167\pi$
$397$	118.742	−	118.742i	0.299099	−	0.299099i	0.840661	+	0.541562i	$0.182167\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−420.302			−1.04813			−0.524067	−	0.851677i	$-0.675586\pi$
$401$	−420.302			−1.04813			−0.524067	+	0.851677i	$0.675586\pi$
$402$		0			0
$403$	59.3735	+	59.3735i	0.147329	+	0.147329i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−68.6061	+	68.6061i	−0.168565	+	0.168565i
$408$		0			0
$409$			515.110i			1.25944i	0.776823	+	0.629719i	$0.216830\pi$
$409$			515.110i			1.25944i	−0.776823	+	0.629719i	$0.783170\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−137.753	−	137.753i	−0.333541	−	0.333541i
$414$		0			0
$415$	38.1112	+	84.7878i	0.0918343	+	0.204308i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			88.6015i			0.211460i	0.994395	+	0.105730i	$0.0337178\pi$
$419$			88.6015i			0.211460i	−0.994395	+	0.105730i	$0.966282\pi$
$420$		0			0
$421$	−257.151			−0.610810			−0.305405	−	0.952223i	$-0.598792\pi$
$421$	−257.151			−0.610810			−0.305405	+	0.952223i	$0.598792\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	612.277	−	36.4495i	1.44065	−	0.0857635i
$426$		0			0
$427$	−20.3837	+	20.3837i	−0.0477369	+	0.0477369i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	804.636			1.86690			0.933452	−	0.358702i	$-0.116781\pi$
$431$	804.636			1.86690			0.933452	+	0.358702i	$0.116781\pi$
$432$		0			0
$433$	−344.848	−	344.848i	−0.796416	−	0.796416i	0.186113	−	0.982528i	$-0.440411\pi$
$433$	−344.848	−	344.848i	−0.796416	−	0.796416i	−0.982528	+	0.186113i	$0.940411\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−100.454	+	100.454i	−0.229872	+	0.229872i
$438$		0			0
$439$			432.929i			0.986170i	0.869981	+	0.493085i	$0.164131\pi$
$439$			432.929i			0.986170i	−0.869981	+	0.493085i	$0.835869\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	245.131	+	245.131i	0.553342	+	0.553342i	0.927404	−	0.374062i	$-0.122035\pi$
$443$	245.131	+	245.131i	0.553342	+	0.553342i	−0.374062	+	0.927404i	$0.622035\pi$
$444$		0			0
$445$	−145.893	+	384.136i	−0.327850	+	0.863227i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			386.091i			0.859890i	0.902855	+	0.429945i	$0.141467\pi$
$449$			386.091i			0.859890i	−0.902855	+	0.429945i	$0.858533\pi$
$450$		0			0
$451$	−7.90918			−0.0175370
$452$		0			0
$453$		0			0
$454$		0			0
$455$	178.990	+	67.9796i	0.393384	+	0.149406i
$456$		0			0
$457$	−223.747	+	223.747i	−0.489599	+	0.489599i	−0.908180	−	0.418580i	$-0.862528\pi$
$457$	−223.747	+	223.747i	−0.489599	+	0.489599i	0.418580	+	0.908180i	$0.362528\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	722.620			1.56751			0.783753	−	0.621073i	$-0.213303\pi$
$461$	722.620			1.56751			0.783753	+	0.621073i	$0.213303\pi$
$462$		0			0
$463$	129.702	+	129.702i	0.280133	+	0.280133i	0.833162	−	0.553029i	$-0.186528\pi$
$463$	129.702	+	129.702i	0.280133	+	0.280133i	−0.553029	+	0.833162i	$0.686528\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	415.258	−	415.258i	0.889203	−	0.889203i	−0.105244	−	0.994446i	$-0.533562\pi$
$467$	415.258	−	415.258i	0.889203	−	0.889203i	0.994446	+	0.105244i	$0.0335623\pi$
$468$		0			0
$469$			311.151i			0.663435i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	300.677	+	300.677i	0.635680	+	0.635680i
$474$		0			0
$475$	−144.334	+	162.606i	−0.303860	+	0.342329i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	304.949i		−	0.636637i	−0.947984	−	0.318318i	$-0.896882\pi$
$479$		−	304.949i		−	0.636637i	0.947984	−	0.318318i	$-0.103118\pi$
$480$		0			0
$481$	67.1102			0.139522
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−158.631	+	71.3031i	−0.327074	+	0.147017i
$486$		0			0
$487$	429.318	−	429.318i	0.881556	−	0.881556i	−0.112137	−	0.993693i	$-0.535769\pi$
$487$	429.318	−	429.318i	0.881556	−	0.881556i	0.993693	+	0.112137i	$0.0357694\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−414.318			−0.843825			−0.421912	−	0.906637i	$-0.638641\pi$
$491$	−414.318			−0.843825			−0.421912	+	0.906637i	$0.638641\pi$
$492$		0			0
$493$	609.737	+	609.737i	1.23679	+	1.23679i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	234.565	−	234.565i	0.471962	−	0.471962i
$498$		0			0
$499$			367.585i			0.736643i	0.929699	+	0.368321i	$0.120067\pi$
$499$			367.585i			0.736643i	−0.929699	+	0.368321i	$0.879933\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−9.59133	−	9.59133i	−0.0190683	−	0.0190683i	0.697508	−	0.716577i	$-0.254292\pi$
$503$	−9.59133	−	9.59133i	−0.0190683	−	0.0190683i	−0.716577	+	0.697508i	$0.754292\pi$
$504$		0			0
$505$	493.697	+	187.504i	0.977618	+	0.371295i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			777.489i			1.52748i	0.645522	+	0.763742i	$0.276640\pi$
$509$			777.489i			1.52748i	−0.645522	+	0.763742i	$0.723360\pi$
$510$		0			0
$511$	−536.656			−1.05021
$512$		0			0
$513$		0			0
$514$		0			0
$515$	258.727	+	575.601i	0.502382	+	1.11767i
$516$		0			0
$517$	502.141	−	502.141i	0.971259	−	0.971259i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−321.605			−0.617284			−0.308642	−	0.951178i	$-0.599875\pi$
$521$	−321.605			−0.617284			−0.308642	+	0.951178i	$0.599875\pi$
$522$		0			0
$523$	582.454	+	582.454i	1.11368	+	1.11368i	0.992649	+	0.121030i	$0.0386198\pi$
$523$	582.454	+	582.454i	1.11368	+	1.11368i	0.121030	+	0.992649i	$0.461380\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−185.576	+	185.576i	−0.352136	+	0.352136i
$528$		0			0
$529$		−	262.171i		−	0.495598i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3.86836	+	3.86836i	0.00725772	+	0.00725772i
$534$		0			0
$535$	443.353	−	199.283i	0.828697	−	0.372491i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			286.005i			0.530621i
$540$		0			0
$541$	460.697			0.851566			0.425783	−	0.904825i	$-0.359999\pi$
$541$	460.697			0.851566			0.425783	+	0.904825i	$0.359999\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	121.955	−	321.106i	0.223770	−	0.589185i
$546$		0			0
$547$	661.778	−	661.778i	1.20983	−	1.20983i	0.238750	−	0.971081i	$-0.423262\pi$
$547$	661.778	−	661.778i	1.20983	−	1.20983i	0.971081	−	0.238750i	$-0.0767376\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−305.666			−0.554748
$552$		0			0
$553$	84.4949	+	84.4949i	0.152794	+	0.152794i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−125.909	+	125.909i	−0.226049	+	0.226049i	−0.811040	−	0.584991i	$-0.801098\pi$
$557$	−125.909	+	125.909i	−0.226049	+	0.226049i	0.584991	+	0.811040i	$0.301098\pi$
$558$		0			0
$559$		−	294.120i		−	0.526155i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−200.009	−	200.009i	−0.355256	−	0.355256i	0.506805	−	0.862061i	$-0.330826\pi$
$563$	−200.009	−	200.009i	−0.355256	−	0.355256i	−0.862061	+	0.506805i	$0.830826\pi$
$564$		0			0
$565$	−283.060	−	629.737i	−0.500992	−	1.11458i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	599.839i		−	1.05420i	−0.849804	−	0.527099i	$-0.823280\pi$
$569$		−	599.839i		−	1.05420i	0.849804	−	0.527099i	$-0.176720\pi$
$570$		0			0
$571$	247.970			0.434274			0.217137	−	0.976141i	$-0.430328\pi$
$571$	247.970			0.434274			0.217137	+	0.976141i	$0.430328\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	24.2679	+	407.650i	0.0422050	+	0.708957i
$576$		0			0
$577$	−292.121	+	292.121i	−0.506276	+	0.506276i	−0.913381	−	0.407105i	$-0.866538\pi$
$577$	−292.121	+	292.121i	−0.506276	+	0.506276i	0.407105	+	0.913381i	$0.366538\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−90.6969			−0.156105
$582$		0			0
$583$	7.90918	+	7.90918i	0.0135664	+	0.0135664i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	611.217	−	611.217i	1.04126	−	1.04126i	0.0421437	−	0.999112i	$-0.486581\pi$
$587$	611.217	−	611.217i	1.04126	−	1.04126i	0.999112	−	0.0421437i	$-0.0134187\pi$
$588$		0			0
$589$		−	93.0306i		−	0.157947i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−524.742	−	524.742i	−0.884894	−	0.884894i	0.109133	−	0.994027i	$-0.465193\pi$
$593$	−524.742	−	524.742i	−0.884894	−	0.884894i	−0.994027	+	0.109133i	$0.965193\pi$
$594$		0			0
$595$	−212.474	+	559.444i	−0.357100	+	0.940242i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			368.858i			0.615790i	0.951420	+	0.307895i	$0.0996245\pi$
$599$			368.858i			0.615790i	−0.951420	+	0.307895i	$0.900375\pi$
$600$		0			0
$601$	932.484			1.55155			0.775777	−	0.631007i	$-0.217358\pi$
$601$	932.484			1.55155			0.775777	+	0.631007i	$0.217358\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	36.4018	+	13.8252i	0.0601682	+	0.0228517i
$606$		0			0
$607$	−513.611	+	513.611i	−0.846146	+	0.846146i	−0.989650	−	0.143504i	$-0.954163\pi$
$607$	−513.611	+	513.611i	−0.846146	+	0.846146i	0.143504	+	0.989650i	$0.454163\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−491.192			−0.803915
$612$		0			0
$613$	−615.287	−	615.287i	−1.00373	−	1.00373i	−0.999993	−	0.00373821i	$-0.998810\pi$
$613$	−615.287	−	615.287i	−1.00373	−	1.00373i	−0.00373821	−	0.999993i	$-0.501190\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−546.752	+	546.752i	−0.886145	+	0.886145i	−0.994150	−	0.108005i	$-0.965554\pi$
$617$	−546.752	+	546.752i	−0.886145	+	0.886145i	0.108005	+	0.994150i	$0.465554\pi$
$618$		0			0
$619$			152.869i			0.246962i	0.992347	+	0.123481i	$0.0394058\pi$
$619$			152.869i			0.246962i	−0.992347	+	0.123481i	$0.960594\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−283.485	−	283.485i	−0.455032	−	0.455032i
$624$		0			0
$625$	74.1510	+	620.586i	0.118642	+	0.992937i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			209.757i			0.333477i
$630$		0			0
$631$	41.4847			0.0657444			0.0328722	−	0.999460i	$-0.489535\pi$
$631$	41.4847			0.0657444			0.0328722	+	0.999460i	$0.489535\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1061.72	+	477.233i	−1.67200	+	0.751547i
$636$		0			0
$637$	139.884	−	139.884i	0.219598	−	0.219598i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−47.2122			−0.0736541			−0.0368270	−	0.999322i	$-0.511725\pi$
$641$	−47.2122			−0.0736541			−0.0368270	+	0.999322i	$0.511725\pi$
$642$		0			0
$643$	−460.372	−	460.372i	−0.715976	−	0.715976i	0.251803	−	0.967779i	$-0.418977\pi$
$643$	−460.372	−	460.372i	−0.715976	−	0.715976i	−0.967779	+	0.251803i	$0.918977\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−281.287	+	281.287i	−0.434756	+	0.434756i	−0.890243	−	0.455487i	$-0.849465\pi$
$647$	−281.287	+	281.287i	−0.434756	+	0.434756i	0.455487	+	0.890243i	$0.349465\pi$
$648$		0			0
$649$			453.192i			0.698293i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−89.8230	−	89.8230i	−0.137554	−	0.137554i	0.634977	−	0.772531i	$-0.281010\pi$
$653$	−89.8230	−	89.8230i	−0.137554	−	0.137554i	−0.772531	+	0.634977i	$0.781010\pi$
$654$		0			0
$655$	496.106	+	188.419i	0.757413	+	0.287662i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	1081.24i		−	1.64072i	−0.571844	−	0.820362i	$-0.693772\pi$
$659$		−	1081.24i		−	1.64072i	0.571844	−	0.820362i	$-0.306228\pi$
$660$		0			0
$661$	−632.393			−0.956721			−0.478361	−	0.878163i	$-0.658769\pi$
$661$	−632.393			−0.956721			−0.478361	+	0.878163i	$0.658769\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−86.9694	−	193.485i	−0.130781	−	0.290954i
$666$		0			0
$667$	−405.959	+	405.959i	−0.608634	+	0.608634i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	67.0602			0.0999407
$672$		0			0
$673$	233.293	+	233.293i	0.346646	+	0.346646i	0.858859	−	0.512213i	$-0.171174\pi$
$673$	233.293	+	233.293i	0.346646	+	0.346646i	−0.512213	+	0.858859i	$0.671174\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	48.3883	−	48.3883i	0.0714745	−	0.0714745i	−0.670466	−	0.741940i	$-0.733906\pi$
$677$	48.3883	−	48.3883i	0.0714745	−	0.0714745i	0.741940	+	0.670466i	$0.233906\pi$
$678$		0			0
$679$		−	169.687i		−	0.249907i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	213.410	+	213.410i	0.312459	+	0.312459i	0.845862	−	0.533402i	$-0.179087\pi$
$683$	213.410	+	213.410i	0.312459	+	0.312459i	−0.533402	+	0.845862i	$0.679087\pi$
$684$		0			0
$685$	1074.69	−	483.060i	1.56888	−	0.705197i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	7.73673i		−	0.0112289i
$690$		0			0
$691$	−151.121			−0.218700			−0.109350	−	0.994003i	$-0.534877\pi$
$691$	−151.121			−0.218700			−0.109350	+	0.994003i	$0.534877\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	339.378	−	893.580i	0.488314	−	1.28573i
$696$		0			0
$697$	−12.0908	+	12.0908i	−0.0173469	+	0.0173469i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	745.680			1.06374			0.531869	−	0.846827i	$-0.321490\pi$
$701$	745.680			1.06374			0.531869	+	0.846827i	$0.321490\pi$
$702$		0			0
$703$	−52.5765	−	52.5765i	−0.0747888	−	0.0747888i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−364.338	+	364.338i	−0.515330	+	0.515330i
$708$		0			0
$709$		−	719.049i		−	1.01417i	−0.861895	−	0.507087i	$-0.830722\pi$
$709$		−	719.049i		−	1.01417i	0.861895	−	0.507087i	$-0.169278\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−123.555	−	123.555i	−0.173289	−	0.173289i
$714$		0			0
$715$	−182.606	−	406.252i	−0.255393	−	0.568185i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			605.271i			0.841824i	0.907101	+	0.420912i	$0.138290\pi$
$719$			605.271i			0.841824i	−0.907101	+	0.420912i	$0.861710\pi$
$720$		0			0
$721$	−615.716			−0.853975
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−583.287	+	657.131i	−0.804534	+	0.906387i
$726$		0			0
$727$	246.126	−	246.126i	0.338550	−	0.338550i	−0.517271	−	0.855821i	$-0.673052\pi$
$727$	246.126	−	246.126i	0.338550	−	0.338550i	0.855821	+	0.517271i	$0.173052\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	919.292			1.25758
$732$		0			0
$733$	−270.763	−	270.763i	−0.369390	−	0.369390i	0.497865	−	0.867255i	$-0.334118\pi$
$733$	−270.763	−	270.763i	−0.369390	−	0.369390i	−0.867255	+	0.497865i	$0.834118\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	511.828	−	511.828i	0.694474	−	0.694474i
$738$		0			0
$739$		−	515.666i		−	0.697789i	−0.937162	−	0.348895i	$-0.886557\pi$
$739$		−	515.666i		−	0.697789i	0.937162	−	0.348895i	$-0.113443\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	420.702	+	420.702i	0.566220	+	0.566220i	0.931067	−	0.364847i	$-0.118879\pi$
$743$	420.702	+	420.702i	0.566220	+	0.566220i	−0.364847	+	0.931067i	$0.618879\pi$
$744$		0			0
$745$	−150.637	+	396.626i	−0.202197	+	0.532383i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			474.252i			0.633180i
$750$		0			0
$751$	859.787			1.14486			0.572428	−	0.819955i	$-0.306002\pi$
$751$	859.787			1.14486			0.572428	+	0.819955i	$0.306002\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−696.318	−	264.459i	−0.922275	−	0.350276i
$756$		0			0
$757$	−956.075	+	956.075i	−1.26298	+	1.26298i	−0.313337	+	0.949642i	$0.601447\pi$
$757$	−956.075	+	956.075i	−1.26298	+	1.26298i	−0.949642	+	0.313337i	$0.898553\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	322.758			0.424124			0.212062	−	0.977256i	$-0.431982\pi$
$761$	322.758			0.424124			0.212062	+	0.977256i	$0.431982\pi$
$762$		0			0
$763$	236.969	+	236.969i	0.310576	+	0.310576i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	221.655	−	221.655i	0.288990	−	0.288990i
$768$		0			0
$769$		−	692.402i		−	0.900393i	−0.892930	−	0.450196i	$-0.851354\pi$
$769$		−	692.402i		−	0.900393i	0.892930	−	0.450196i	$-0.148646\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−375.226	−	375.226i	−0.485415	−	0.485415i	0.421441	−	0.906856i	$-0.361525\pi$
$773$	−375.226	−	375.226i	−0.485415	−	0.485415i	−0.906856	+	0.421441i	$0.861525\pi$
$774$		0			0
$775$	−200.000	−	177.526i	−0.258065	−	0.229065i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	6.06123i		−	0.00778078i
$780$		0			0
$781$	−771.696			−0.988087
$782$		0			0
$783$		0			0
$784$		0			0
$785$	108.697	−	48.8582i	0.138467	−	0.0622397i
$786$		0			0
$787$	−910.990	+	910.990i	−1.15755	+	1.15755i	−0.172546	+	0.985001i	$0.555199\pi$
$787$	−910.990	+	910.990i	−1.15755	+	1.15755i	−0.985001	+	0.172546i	$0.944801\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	673.626			0.851613
$792$		0			0
$793$	−32.7990	−	32.7990i	−0.0413606	−	0.0413606i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	7.21683	−	7.21683i	0.00905500	−	0.00905500i	−0.702565	−	0.711620i	$-0.747962\pi$
$797$	7.21683	−	7.21683i	0.00905500	−	0.00905500i	0.711620	+	0.702565i	$0.247962\pi$
$798$		0			0
$799$		−	1535.25i		−	1.92147i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	882.772	+	882.772i	1.09934	+	1.09934i
$804$		0			0
$805$	−372.474	−	141.464i	−0.462701	−	0.175732i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	150.000i		−	0.185414i	−0.995693	−	0.0927070i	$-0.970448\pi$
$809$		−	150.000i		−	0.185414i	0.995693	−	0.0927070i	$-0.0295520\pi$
$810$		0			0
$811$	−1336.85			−1.64839			−0.824197	−	0.566304i	$-0.808373\pi$
$811$	−1336.85			−1.64839			−0.824197	+	0.566304i	$0.808373\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−378.624	−	842.343i	−0.464570	−	1.03355i
$816$		0			0
$817$	−230.424	+	230.424i	−0.282037	+	0.282037i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	33.8934			0.0412830			0.0206415	−	0.999787i	$-0.493429\pi$
$821$	33.8934			0.0412830			0.0206415	+	0.999787i	$0.493429\pi$
$822$		0			0
$823$	−481.631	−	481.631i	−0.585214	−	0.585214i	0.351117	−	0.936331i	$-0.385802\pi$
$823$	−481.631	−	481.631i	−0.585214	−	0.585214i	−0.936331	+	0.351117i	$0.885802\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	350.756	−	350.756i	0.424131	−	0.424131i	−0.462492	−	0.886623i	$-0.653045\pi$
$827$	350.756	−	350.756i	0.424131	−	0.424131i	0.886623	+	0.462492i	$0.153045\pi$
$828$		0			0
$829$			697.423i			0.841283i	0.907227	+	0.420641i	$0.138195\pi$
$829$			697.423i			0.841283i	−0.907227	+	0.420641i	$0.861805\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	437.217	+	437.217i	0.524870	+	0.524870i
$834$		0			0
$835$	−290.454	+	130.556i	−0.347849	+	0.156355i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	72.3724i		−	0.0862604i	−0.999069	−	0.0431302i	$-0.986267\pi$
$839$		−	72.3724i		−	0.0862604i	0.999069	−	0.0431302i	$-0.0137330\pi$
$840$		0			0
$841$	−394.271			−0.468813
$842$		0			0
$843$		0			0
$844$		0			0
$845$	190.633	−	501.936i	0.225602	−	0.594008i
$846$		0			0
$847$	−26.8638	+	26.8638i	−0.0317164	+	0.0317164i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−139.655			−0.164107
$852$		0			0
$853$	−74.5699	−	74.5699i	−0.0874207	−	0.0874207i	0.662044	−	0.749465i	$-0.269689\pi$
$853$	−74.5699	−	74.5699i	−0.0874207	−	0.0874207i	−0.749465	+	0.662044i	$0.769689\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	293.176	−	293.176i	0.342096	−	0.342096i	−0.515059	−	0.857155i	$-0.672230\pi$
$857$	293.176	−	293.176i	0.342096	−	0.342096i	0.857155	+	0.515059i	$0.172230\pi$
$858$		0			0
$859$		−	786.867i		−	0.916027i	−0.888945	−	0.458014i	$-0.848561\pi$
$859$		−	786.867i		−	0.916027i	0.888945	−	0.458014i	$-0.151439\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1072.68	−	1072.68i	−1.24297	−	1.24297i	−0.958764	−	0.284204i	$-0.908271\pi$
$863$	−1072.68	−	1072.68i	−1.24297	−	1.24297i	−0.284204	−	0.958764i	$-0.591729\pi$
$864$		0			0
$865$	−425.081	−	945.696i	−0.491423	−	1.09329i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	277.980i		−	0.319884i
$870$		0			0
$871$	−500.667			−0.574819
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−581.918	−	182.247i	−0.665050	−	0.208283i
$876$		0			0
$877$	239.460	−	239.460i	0.273044	−	0.273044i	−0.557280	−	0.830324i	$-0.688155\pi$
$877$	239.460	−	239.460i	0.273044	−	0.273044i	0.830324	+	0.557280i	$0.188155\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−62.8490			−0.0713382			−0.0356691	−	0.999364i	$-0.511356\pi$
$881$	−62.8490			−0.0713382			−0.0356691	+	0.999364i	$0.511356\pi$
$882$		0			0
$883$	158.061	+	158.061i	0.179005	+	0.179005i	0.790922	−	0.611917i	$-0.209601\pi$
$883$	158.061	+	158.061i	0.179005	+	0.179005i	−0.611917	+	0.790922i	$0.709601\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	30.2066	−	30.2066i	0.0340548	−	0.0340548i	−0.689874	−	0.723929i	$-0.742334\pi$
$887$	30.2066	−	30.2066i	0.0340548	−	0.0340548i	0.723929	+	0.689874i	$0.242334\pi$
$888$		0			0
$889$		−	1135.72i		−	1.27752i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	384.817	+	384.817i	0.430926	+	0.430926i
$894$		0			0
$895$	326.135	−	858.712i	0.364397	−	0.959454i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	375.959i		−	0.418197i
$900$		0			0
$901$	24.1816			0.0268387
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1338.10	−	508.207i	−1.47857	−	0.561554i
$906$		0			0
$907$	−571.342	+	571.342i	−0.629925	+	0.629925i	−0.948049	−	0.318124i	$-0.896947\pi$
$907$	−571.342	+	571.342i	−0.629925	+	0.629925i	0.318124	+	0.948049i	$0.396947\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	173.362			0.190299			0.0951494	−	0.995463i	$-0.469667\pi$
$911$	173.362			0.190299			0.0951494	+	0.995463i	$0.469667\pi$
$912$		0			0
$913$	149.192	+	149.192i	0.163408	+	0.163408i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−366.116	+	366.116i	−0.399254	+	0.399254i
$918$		0			0
$919$			1147.42i			1.24856i	0.781202	+	0.624278i	$0.214607\pi$
$919$			1147.42i			1.24856i	−0.781202	+	0.624278i	$0.785393\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	377.435	+	377.435i	0.408922	+	0.408922i
$924$		0			0
$925$	−213.360	+	12.7015i	−0.230659	+	0.0137314i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	220.293i		−	0.237129i	−0.992946	−	0.118565i	$-0.962171\pi$
$929$		−	220.293i		−	0.237129i	0.992946	−	0.118565i	$-0.0378292\pi$
$930$		0			0
$931$	−219.181			−0.235425
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1269.77	−	570.747i	1.35804	−	0.610425i
$936$		0			0
$937$	−396.090	+	396.090i	−0.422721	+	0.422721i	−0.886140	−	0.463418i	$-0.846623\pi$
$937$	−396.090	+	396.090i	−0.422721	+	0.422721i	0.463418	+	0.886140i	$0.346623\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−185.771			−0.197419			−0.0987093	−	0.995116i	$-0.531471\pi$
$941$	−185.771			−0.197419			−0.0987093	+	0.995116i	$0.531471\pi$
$942$		0			0
$943$	−8.04999	−	8.04999i	−0.00853658	−	0.00853658i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−845.190	+	845.190i	−0.892492	+	0.892492i	−0.994757	−	0.102265i	$-0.967391\pi$
$947$	−845.190	+	845.190i	−0.892492	+	0.892492i	0.102265	+	0.994757i	$0.467391\pi$
$948$		0			0
$949$		−	863.523i		−	0.909930i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	630.499	+	630.499i	0.661594	+	0.661594i	0.955756	−	0.294161i	$-0.0950403\pi$
$953$	630.499	+	630.499i	0.661594	+	0.661594i	−0.294161	+	0.955756i	$0.595040\pi$
$954$		0			0
$955$	224.788	+	85.3735i	0.235380	+	0.0893963i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			1149.58i			1.19873i
$960$		0			0
$961$	−846.576			−0.880932
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−740.879	−	1648.27i	−0.767750	−	1.70805i
$966$		0			0
$967$	381.690	−	381.690i	0.394716	−	0.394716i	−0.481649	−	0.876364i	$-0.659962\pi$
$967$	381.690	−	381.690i	0.394716	−	0.394716i	0.876364	+	0.481649i	$0.159962\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1000.44			−1.03032			−0.515159	−	0.857095i	$-0.672267\pi$
$971$	−1000.44			−1.03032			−0.515159	+	0.857095i	$0.672267\pi$
$972$		0			0
$973$	659.444	+	659.444i	0.677743	+	0.677743i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−593.662	+	593.662i	−0.607637	+	0.607637i	−0.942328	−	0.334691i	$-0.891368\pi$
$977$	−593.662	+	593.662i	−0.607637	+	0.607637i	0.334691	+	0.942328i	$0.391368\pi$
$978$		0			0
$979$			932.636i			0.952641i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1217.34	−	1217.34i	−1.23839	−	1.23839i	−0.960659	−	0.277731i	$-0.910418\pi$
$983$	−1217.34	−	1217.34i	−1.23839	−	1.23839i	−0.277731	−	0.960659i	$-0.589582\pi$
$984$		0			0
$985$	623.646	−	280.322i	0.633143	−	0.284591i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			612.059i			0.618867i
$990$		0			0
$991$	544.061			0.549002			0.274501	−	0.961587i	$-0.411487\pi$
$991$	544.061			0.549002			0.274501	+	0.961587i	$0.411487\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−341.925	+	900.286i	−0.343643	+	0.904810i
$996$		0			0
$997$	−316.733	+	316.733i	−0.317686	+	0.317686i	−0.847878	−	0.530192i	$-0.822120\pi$
$997$	−316.733	+	316.733i	−0.317686	+	0.317686i	0.530192	+	0.847878i	$0.322120\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	720.3.bh.k.433.2		4
3.2	odd	2		240.3.bg.a.193.2		4
4.3	odd	2		45.3.g.b.28.1		4
5.2	odd	4	inner	720.3.bh.k.577.2		4
12.11	even	2		15.3.f.a.13.2	yes	4
15.2	even	4		240.3.bg.a.97.2		4
15.8	even	4		1200.3.bg.k.1057.1		4
15.14	odd	2		1200.3.bg.k.193.1		4
20.3	even	4		225.3.g.a.82.2		4
20.7	even	4		45.3.g.b.37.1		4
20.19	odd	2		225.3.g.a.118.2		4
24.5	odd	2		960.3.bg.h.193.1		4
24.11	even	2		960.3.bg.i.193.2		4
36.7	odd	6		405.3.l.f.28.2		8
36.11	even	6		405.3.l.h.28.1		8
36.23	even	6		405.3.l.h.298.2		8
36.31	odd	6		405.3.l.f.298.1		8
60.23	odd	4		75.3.f.c.7.1		4
60.47	odd	4		15.3.f.a.7.2	✓	4
60.59	even	2		75.3.f.c.43.1		4
120.77	even	4		960.3.bg.h.577.1		4
120.107	odd	4		960.3.bg.i.577.2		4
180.7	even	12		405.3.l.f.352.1		8
180.47	odd	12		405.3.l.h.352.2		8
180.67	even	12		405.3.l.f.217.2		8
180.167	odd	12		405.3.l.h.217.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.3.f.a.7.2	✓	4	60.47	odd	4
15.3.f.a.13.2	yes	4	12.11	even	2
45.3.g.b.28.1		4	4.3	odd	2
45.3.g.b.37.1		4	20.7	even	4
75.3.f.c.7.1		4	60.23	odd	4
75.3.f.c.43.1		4	60.59	even	2
225.3.g.a.82.2		4	20.3	even	4
225.3.g.a.118.2		4	20.19	odd	2
240.3.bg.a.97.2		4	15.2	even	4
240.3.bg.a.193.2		4	3.2	odd	2
405.3.l.f.28.2		8	36.7	odd	6
405.3.l.f.217.2		8	180.67	even	12
405.3.l.f.298.1		8	36.31	odd	6
405.3.l.f.352.1		8	180.7	even	12
405.3.l.h.28.1		8	36.11	even	6
405.3.l.h.217.1		8	180.167	odd	12
405.3.l.h.298.2		8	36.23	even	6
405.3.l.h.352.2		8	180.47	odd	12
720.3.bh.k.433.2		4	1.1	even	1	trivial
720.3.bh.k.577.2		4	5.2	odd	4	inner
960.3.bg.h.193.1		4	24.5	odd	2
960.3.bg.h.577.1		4	120.77	even	4
960.3.bg.i.193.2		4	24.11	even	2
960.3.bg.i.577.2		4	120.107	odd	4
1200.3.bg.k.193.1		4	15.14	odd	2
1200.3.bg.k.1057.1		4	15.8	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:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	720.bh (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(4.67423 + 1.77526i) q^{5} +(-3.44949 + 3.44949i) q^{7} +11.3485 q^{11} +(-5.55051 - 5.55051i) q^{13} +(17.3485 - 17.3485i) q^{17} +8.69694i q^{19} +(11.5505 + 11.5505i) q^{23} +(18.6969 + 16.5959i) q^{25} +35.1464i q^{29} -10.6969 q^{31} +(-22.2474 + 10.0000i) q^{35} +(-6.04541 + 6.04541i) q^{37} -0.696938 q^{41} +(26.4949 + 26.4949i) q^{43} +(44.2474 - 44.2474i) q^{47} +25.2020i q^{49} +(0.696938 + 0.696938i) q^{53} +(53.0454 + 20.1464i) q^{55} +39.9342i q^{59} +5.90918 q^{61} +(-16.0908 - 35.7980i) q^{65} +(45.1010 - 45.1010i) q^{67} -68.0000 q^{71} +(77.7878 + 77.7878i) q^{73} +(-39.1464 + 39.1464i) q^{77} -24.4949i q^{79} +(13.1464 + 13.1464i) q^{83} +(111.889 - 50.2929i) q^{85} +82.1816i q^{89} +38.2929 q^{91} +(-15.4393 + 40.6515i) q^{95} +(-24.5959 + 24.5959i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 4 q^{7} + 16 q^{11} - 32 q^{13} + 40 q^{17} + 56 q^{23} + 16 q^{25} + 16 q^{31} - 40 q^{35} + 64 q^{37} + 56 q^{41} + 8 q^{43} + 128 q^{47} - 56 q^{53} + 124 q^{55} + 200 q^{61} + 112 q^{65}+ \cdots - 20 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 - 4 * q^7 + 16 * q^11 - 32 * q^13 + 40 * q^17 + 56 * q^23 + 16 * q^25 + 16 * q^31 - 40 * q^35 + 64 * q^37 + 56 * q^41 + 8 * q^43 + 128 * q^47 - 56 * q^53 + 124 * q^55 + 200 * q^61 + 112 * q^65 + 200 * q^67 - 272 * q^71 + 76 * q^73 - 88 * q^77 - 16 * q^83 + 232 * q^85 + 16 * q^91 + 144 * q^95 - 20 * q^97

Introduction

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