LMFDB - Embedded newform 504.4.bl.a.17.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [504,4,Mod(17,504)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(504, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("504.17");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 504 = 2^{3} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	504.bl (of order $6$, 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:	$29.7369626429$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.1
Character	$\chi$	$=$	504.17
Dual form			504.4.bl.a.89.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-10.9253 - 18.9231i) q^{5} +(-12.1367 - 13.9893i) q^{7} +O(q^{10})$	$q+(-10.9253 - 18.9231i) q^{5} +(-12.1367 - 13.9893i) q^{7} +(-45.1734 - 26.0809i) q^{11} -54.9986i q^{13} +(-40.8239 + 70.7091i) q^{17} +(113.707 - 65.6488i) q^{19} +(-38.0795 + 21.9852i) q^{23} +(-176.223 + 305.227i) q^{25} -238.538i q^{29} +(174.225 + 100.589i) q^{31} +(-132.125 + 382.501i) q^{35} +(12.0321 + 20.8402i) q^{37} +102.220 q^{41} +119.740 q^{43} +(-20.2851 - 35.1348i) q^{47} +(-48.4020 + 339.568i) q^{49} +(297.988 + 172.043i) q^{53} +1139.76i q^{55} +(142.286 - 246.447i) q^{59} +(-386.826 + 223.334i) q^{61} +(-1040.75 + 600.874i) q^{65} +(113.537 - 196.651i) q^{67} -886.964i q^{71} +(6.46845 + 3.73456i) q^{73} +(183.401 + 948.480i) q^{77} +(-404.328 - 700.317i) q^{79} -943.208 q^{83} +1784.05 q^{85} +(-575.503 - 996.801i) q^{89} +(-769.393 + 667.501i) q^{91} +(-2484.56 - 1434.46i) q^{95} +1557.47i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 24 q^{7}+O(q^{10}) $ 48 * q - 24 * q^7	$ 48 q - 24 q^{7} + 540 q^{19} - 924 q^{25} + 648 q^{31} - 132 q^{37} - 792 q^{43} + 672 q^{49} - 12 q^{67} + 2412 q^{73} + 1680 q^{79} + 480 q^{85} + 1404 q^{91}+O(q^{100}) $ 48 * q - 24 * q^7 + 540 * q^19 - 924 * q^25 + 648 * q^31 - 132 * q^37 - 792 * q^43 + 672 * q^49 - 12 * q^67 + 2412 * q^73 + 1680 * q^79 + 480 * q^85 + 1404 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/504\mathbb{Z}\right)^\times$.

$n$	$73$	$127$	$253$	$281$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$	$1$	$-1$

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
$5$	−10.9253	−	18.9231i	−0.977185	−	1.69253i	−0.672528	−	0.740072i	$-0.734791\pi$
$5$	−10.9253	−	18.9231i	−0.977185	−	1.69253i	−0.304657	−	0.952462i	$-0.598542\pi$
$6$		0			0
$7$	−12.1367	−	13.9893i	−0.655319	−	0.755352i
$7$	−12.1367	−	13.9893i	−0.655319	−	0.755352i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−45.1734	−	26.0809i	−1.23821	−	0.714879i	−0.269480	−	0.963006i	$-0.586852\pi$
$11$	−45.1734	−	26.0809i	−1.23821	−	0.714879i	−0.968728	+	0.248127i	$0.920185\pi$
$12$		0			0
$13$		−	54.9986i		−	1.17338i	−0.809813	−	0.586688i	$-0.800432\pi$
$13$		−	54.9986i		−	1.17338i	0.809813	−	0.586688i	$-0.199568\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−40.8239	+	70.7091i	−0.582427	+	1.00879i	0.412764	+	0.910838i	$0.364563\pi$
$17$	−40.8239	+	70.7091i	−0.582427	+	1.00879i	−0.995191	+	0.0979551i	$0.968770\pi$
$18$		0			0
$19$	113.707	−	65.6488i	1.37296	−	0.792677i	0.381657	−	0.924304i	$-0.375354\pi$
$19$	113.707	−	65.6488i	1.37296	−	0.792677i	0.991299	+	0.131627i	$0.0420202\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−38.0795	+	21.9852i	−0.345223	+	0.199315i	−0.662579	−	0.748992i	$-0.730538\pi$
$23$	−38.0795	+	21.9852i	−0.345223	+	0.199315i	0.317356	+	0.948306i	$0.397205\pi$
$24$		0			0
$25$	−176.223	+	305.227i	−1.40978	+	2.44181i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	238.538i		−	1.52742i	−0.645557	−	0.763712i	$-0.723375\pi$
$29$		−	238.538i		−	1.52742i	0.645557	−	0.763712i	$-0.276625\pi$
$30$		0			0
$31$	174.225	+	100.589i	1.00941	+	0.582783i	0.911019	−	0.412365i	$-0.135297\pi$
$31$	174.225	+	100.589i	1.00941	+	0.582783i	0.0983905	+	0.995148i	$0.468631\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−132.125	+	382.501i	−0.638091	+	1.84727i
$36$		0			0
$37$	12.0321	+	20.8402i	0.0534613	+	0.0925977i	0.891518	−	0.452986i	$-0.149641\pi$
$37$	12.0321	+	20.8402i	0.0534613	+	0.0925977i	−0.838056	+	0.545584i	$0.816308\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	102.220			0.389369			0.194685	−	0.980866i	$-0.437632\pi$
$41$	102.220			0.389369			0.194685	+	0.980866i	$0.437632\pi$
$42$		0			0
$43$	119.740			0.424654			0.212327	−	0.977199i	$-0.431896\pi$
$43$	119.740			0.424654			0.212327	+	0.977199i	$0.431896\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−20.2851	−	35.1348i	−0.0629550	−	0.109041i	0.832830	−	0.553529i	$-0.186719\pi$
$47$	−20.2851	−	35.1348i	−0.0629550	−	0.109041i	−0.895785	+	0.444488i	$0.853386\pi$
$48$		0			0
$49$	−48.4020	+	339.568i	−0.141114	+	0.989993i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	297.988	+	172.043i	0.772298	+	0.445887i	0.833694	−	0.552227i	$-0.186222\pi$
$53$	297.988	+	172.043i	0.772298	+	0.445887i	−0.0613956	+	0.998114i	$0.519555\pi$
$54$		0			0
$55$			1139.76i			2.79428i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	142.286	−	246.447i	0.313968	−	0.543808i	−0.665250	−	0.746621i	$-0.731675\pi$
$59$	142.286	−	246.447i	0.313968	−	0.543808i	0.979218	+	0.202813i	$0.0650083\pi$
$60$		0			0
$61$	−386.826	+	223.334i	−0.811934	+	0.468770i	−0.847627	−	0.530593i	$-0.821969\pi$
$61$	−386.826	+	223.334i	−0.811934	+	0.468770i	0.0356931	+	0.999363i	$0.488636\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1040.75	+	600.874i	−1.98598	+	1.14660i
$66$		0			0
$67$	113.537	−	196.651i	0.207025	−	0.358578i	−0.743751	−	0.668457i	$-0.766955\pi$
$67$	113.537	−	196.651i	0.207025	−	0.358578i	0.950776	+	0.309879i	$0.100288\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	886.964i		−	1.48258i	−0.671184	−	0.741291i	$-0.734214\pi$
$71$		−	886.964i		−	1.48258i	0.671184	−	0.741291i	$-0.265786\pi$
$72$		0			0
$73$	6.46845	+	3.73456i	0.0103709	+	0.00598763i	0.505176	−	0.863016i	$-0.331427\pi$
$73$	6.46845	+	3.73456i	0.0103709	+	0.00598763i	−0.494806	+	0.869004i	$0.664761\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	183.401	+	948.480i	0.271435	+	1.40376i
$78$		0			0
$79$	−404.328	−	700.317i	−0.575829	−	0.997365i	−0.995951	−	0.0898973i	$-0.971346\pi$
$79$	−404.328	−	700.317i	−0.575829	−	0.997365i	0.420122	−	0.907468i	$-0.361987\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−943.208			−1.24736			−0.623678	−	0.781681i	$-0.714362\pi$
$83$	−943.208			−1.24736			−0.623678	+	0.781681i	$0.714362\pi$
$84$		0			0
$85$	1784.05			2.27656
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−575.503	−	996.801i	−0.685430	−	1.18720i	−0.973302	−	0.229530i	$-0.926281\pi$
$89$	−575.503	−	996.801i	−0.685430	−	1.18720i	0.287872	−	0.957669i	$-0.407052\pi$
$90$		0			0
$91$	−769.393	+	667.501i	−0.886312	+	0.768935i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2484.56	−	1434.46i	−2.68326	−	1.54918i
$96$		0			0
$97$			1557.47i			1.63028i	0.579264	+	0.815140i	$0.303340\pi$
$97$			1557.47i			1.63028i	−0.579264	+	0.815140i	$0.696660\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−758.342	+	1313.49i	−0.747107	+	1.29403i	0.202097	+	0.979366i	$0.435224\pi$
$101$	−758.342	+	1313.49i	−0.747107	+	1.29403i	−0.949204	+	0.314662i	$0.898109\pi$
$102$		0			0
$103$	870.518	−	502.594i	0.832763	−	0.480796i	−0.0220344	−	0.999757i	$-0.507014\pi$
$103$	870.518	−	502.594i	0.832763	−	0.480796i	0.854798	+	0.518961i	$0.173681\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−919.111	+	530.649i	−0.830410	+	0.479437i	−0.853993	−	0.520285i	$-0.825826\pi$
$107$	−919.111	+	530.649i	−0.830410	+	0.479437i	0.0235833	+	0.999722i	$0.492492\pi$
$108$		0			0
$109$	248.409	−	430.256i	0.218287	−	0.378083i	−0.735998	−	0.676984i	$-0.763287\pi$
$109$	248.409	−	430.256i	0.218287	−	0.378083i	0.954284	+	0.298901i	$0.0966199\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			975.345i			0.811971i	0.913880	+	0.405985i	$0.133072\pi$
$113$			975.345i			0.811971i	−0.913880	+	0.405985i	$0.866928\pi$
$114$		0			0
$115$	832.057	+	480.389i	0.674693	+	0.389534i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1484.64	−	287.075i	1.14367	−	0.221144i
$120$		0			0
$121$	694.922	+	1203.64i	0.522105	+	0.904313i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	4969.80			3.55610
$126$		0			0
$127$	573.808			0.400923			0.200461	−	0.979702i	$-0.435756\pi$
$127$	573.808			0.400923			0.200461	+	0.979702i	$0.435756\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	105.592	+	182.890i	0.0704244	+	0.121979i	0.899087	−	0.437769i	$-0.144231\pi$
$131$	105.592	+	182.890i	0.0704244	+	0.121979i	−0.828663	+	0.559748i	$0.810898\pi$
$132$		0			0
$133$	−2298.41	−	793.925i	−1.49847	−	0.517609i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	279.288	+	161.247i	0.174169	+	0.100557i	0.584550	−	0.811358i	$-0.301271\pi$
$137$	279.288	+	161.247i	0.174169	+	0.100557i	−0.410381	+	0.911914i	$0.634604\pi$
$138$		0			0
$139$			491.082i			0.299662i	0.988712	+	0.149831i	$0.0478730\pi$
$139$			491.082i			0.299662i	−0.988712	+	0.149831i	$0.952127\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1434.41	+	2484.47i	−0.838822	+	1.45288i
$144$		0			0
$145$	−4513.87	+	2606.09i	−2.58522	+	1.49258i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	184.226	−	106.363i	0.101291	−	0.0584805i	−0.448499	−	0.893784i	$-0.648041\pi$
$149$	184.226	−	106.363i	0.101291	−	0.0584805i	0.549790	+	0.835303i	$0.314708\pi$
$150$		0			0
$151$	−99.5733	+	172.466i	−0.0536633	+	0.0929476i	−0.891609	−	0.452806i	$-0.850423\pi$
$151$	−99.5733	+	172.466i	−0.0536633	+	0.0929476i	0.837946	+	0.545753i	$0.183756\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	4395.83i		−	2.27795i
$156$		0			0
$157$	−92.2932	−	53.2855i	−0.0469159	−	0.0270869i	0.476359	−	0.879251i	$-0.341956\pi$
$157$	−92.2932	−	53.2855i	−0.0469159	−	0.0270869i	−0.523275	+	0.852164i	$0.675290\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	769.717	+	265.879i	0.376784	+	0.130150i
$162$		0			0
$163$	−1806.95	−	3129.74i	−0.868292	−	1.50393i	−0.863741	−	0.503937i	$-0.831884\pi$
$163$	−1806.95	−	3129.74i	−0.868292	−	1.50393i	−0.00455163	−	0.999990i	$-0.501449\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2838.38			1.31521			0.657606	−	0.753362i	$-0.271569\pi$
$167$	2838.38			1.31521			0.657606	+	0.753362i	$0.271569\pi$
$168$		0			0
$169$	−827.851			−0.376810
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−74.3270	−	128.738i	−0.0326646	−	0.0565767i	0.849231	−	0.528022i	$-0.177066\pi$
$173$	−74.3270	−	128.738i	−0.0326646	−	0.0565767i	−0.881896	+	0.471445i	$0.843733\pi$
$174$		0			0
$175$	6408.67	−	1239.20i	2.76828	−	0.535285i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	94.4896	+	54.5536i	0.0394552	+	0.0227795i	0.519598	−	0.854411i	$-0.326082\pi$
$179$	94.4896	+	54.5536i	0.0394552	+	0.0227795i	−0.480143	+	0.877190i	$0.659415\pi$
$180$		0			0
$181$		−	321.177i		−	0.131894i	−0.997823	−	0.0659472i	$-0.978993\pi$
$181$		−	321.177i		−	0.131894i	0.997823	−	0.0659472i	$-0.0210069\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	262.908	−	455.370i	0.104483	−	0.180970i
$186$		0			0
$187$	3688.31	−	2129.45i	1.44233	−	0.832730i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1841.59	−	1063.24i	0.697658	−	0.402793i	−0.108816	−	0.994062i	$-0.534706\pi$
$191$	1841.59	−	1063.24i	0.697658	−	0.402793i	0.806475	+	0.591269i	$0.201373\pi$
$192$		0			0
$193$	1760.98	−	3050.11i	0.656779	−	1.13757i	−0.324666	−	0.945829i	$-0.605252\pi$
$193$	1760.98	−	3050.11i	0.656779	−	1.13757i	0.981445	−	0.191746i	$-0.0614149\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			466.612i			0.168755i	0.996434	+	0.0843775i	$0.0268902\pi$
$197$			466.612i			0.168755i	−0.996434	+	0.0843775i	$0.973110\pi$
$198$		0			0
$199$	−3708.65	−	2141.19i	−1.32110	−	0.762739i	−0.337198	−	0.941434i	$-0.609479\pi$
$199$	−3708.65	−	2141.19i	−1.32110	−	0.762739i	−0.983905	+	0.178695i	$0.942813\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−3336.98	+	2895.05i	−1.15374	+	1.00095i
$204$		0			0
$205$	−1116.78	−	1934.33i	−0.380486	−	0.659021i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6848.70			−2.26667
$210$		0			0
$211$	−451.133			−0.147191			−0.0735954	−	0.997288i	$-0.523447\pi$
$211$	−451.133			−0.147191			−0.0735954	+	0.997288i	$0.523447\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1308.19	−	2265.85i	−0.414966	−	0.718742i
$216$		0			0
$217$	−707.343	−	3658.10i	−0.221279	−	1.14437i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3888.91	+	2245.26i	1.18369	+	0.683405i
$222$		0			0
$223$		−	710.949i		−	0.213492i	−0.994286	−	0.106746i	$-0.965957\pi$
$223$		−	710.949i		−	0.213492i	0.994286	−	0.106746i	$-0.0340431\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1762.96	−	3053.54i	0.515470	−	0.892821i	−0.484368	−	0.874864i	$-0.660951\pi$
$227$	1762.96	−	3053.54i	0.515470	−	0.892821i	0.999839	−	0.0179567i	$-0.00571610\pi$
$228$		0			0
$229$	−2034.98	+	1174.90i	−0.587229	+	0.339037i	−0.764001	−	0.645215i	$-0.776768\pi$
$229$	−2034.98	+	1174.90i	−0.587229	+	0.339037i	0.176772	+	0.984252i	$0.443434\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	2648.53	−	1529.13i	0.744683	−	0.429943i	−0.0790864	−	0.996868i	$-0.525200\pi$
$233$	2648.53	−	1529.13i	0.744683	−	0.429943i	0.823770	+	0.566925i	$0.191867\pi$
$234$		0			0
$235$	−443.240	+	767.714i	−0.123037	+	0.213107i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			4374.81i			1.18403i	0.805927	+	0.592015i	$0.201667\pi$
$239$			4374.81i			1.18403i	−0.805927	+	0.592015i	$0.798333\pi$
$240$		0			0
$241$	1806.54	+	1043.01i	0.482861	+	0.278780i	0.721608	−	0.692302i	$-0.243403\pi$
$241$	1806.54	+	1043.01i	0.482861	+	0.278780i	−0.238747	+	0.971082i	$0.576737\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	6954.48	−	2793.95i	1.81349	−	0.728567i
$246$		0			0
$247$	−3610.59	−	6253.73i	−0.930107	−	1.61099i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−4161.17			−1.04642			−0.523209	−	0.852204i	$-0.675265\pi$
$251$	−4161.17			−1.04642			−0.523209	+	0.852204i	$0.675265\pi$
$252$		0			0
$253$	2293.57			0.569944
$254$		0			0
$255$		0			0
$256$		0			0
$257$	657.550	+	1138.91i	0.159599	+	0.276433i	0.934724	−	0.355374i	$-0.115647\pi$
$257$	657.550	+	1138.91i	0.159599	+	0.276433i	−0.775125	+	0.631808i	$0.782313\pi$
$258$		0			0
$259$	145.511	−	421.253i	0.0349096	−	0.101063i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2438.23	−	1407.71i	−0.571665	−	0.330051i	0.186149	−	0.982521i	$-0.440399\pi$
$263$	−2438.23	−	1407.71i	−0.571665	−	0.330051i	−0.757814	+	0.652471i	$0.773733\pi$
$264$		0			0
$265$		−	7518.48i		−	1.74285i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1025.86	−	1776.83i	0.232519	−	0.402734i	−0.726030	−	0.687663i	$-0.758637\pi$
$269$	1025.86	−	1776.83i	0.232519	−	0.402734i	0.958549	+	0.284929i	$0.0919700\pi$
$270$		0			0
$271$	−1176.68	+	679.356i	−0.263757	+	0.152280i	−0.626047	−	0.779785i	$-0.715328\pi$
$271$	−1176.68	+	679.356i	−0.263757	+	0.152280i	0.362290	+	0.932065i	$0.381995\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	15921.1	−	9192.07i	3.49120	−	2.01565i
$276$		0			0
$277$	−2683.21	+	4647.46i	−0.582017	+	1.00808i	0.413223	+	0.910630i	$0.364403\pi$
$277$	−2683.21	+	4647.46i	−0.582017	+	1.00808i	−0.995240	+	0.0974533i	$0.968930\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			6750.22i			1.43304i	0.697567	+	0.716520i	$0.254266\pi$
$281$			6750.22i			1.43304i	−0.697567	+	0.716520i	$0.745734\pi$
$282$		0			0
$283$	4812.98	+	2778.77i	1.01096	+	0.583678i	0.911473	−	0.411360i	$-0.134946\pi$
$283$	4812.98	+	2778.77i	1.01096	+	0.583678i	0.0994879	+	0.995039i	$0.468280\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1240.62	−	1429.99i	−0.255161	−	0.294111i
$288$		0			0
$289$	−876.688	−	1518.47i	−0.178442	−	0.309071i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3205.49			−0.639136			−0.319568	−	0.947563i	$-0.603538\pi$
$293$	−3205.49			−0.639136			−0.319568	+	0.947563i	$0.603538\pi$
$294$		0			0
$295$	−6218.06			−1.22722
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1209.16	+	2094.32i	0.233871	+	0.405076i
$300$		0			0
$301$	−1453.24	−	1675.08i	−0.278284	−	0.320763i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	8452.34	+	4879.96i	1.58682	+	0.916150i
$306$		0			0
$307$		−	2624.02i		−	0.487820i	−0.969798	−	0.243910i	$-0.921570\pi$
$307$		−	2624.02i		−	0.487820i	0.969798	−	0.243910i	$-0.0784301\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	378.873	−	656.227i	0.0690801	−	0.119650i	−0.829417	−	0.558631i	$-0.811327\pi$
$311$	378.873	−	656.227i	0.0690801	−	0.119650i	0.898497	+	0.438980i	$0.144660\pi$
$312$		0			0
$313$	1419.39	−	819.484i	0.256321	−	0.147987i	−0.366334	−	0.930483i	$-0.619387\pi$
$313$	1419.39	−	819.484i	0.256321	−	0.147987i	0.622655	+	0.782496i	$0.286054\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−784.941	+	453.186i	−0.139075	+	0.0802949i	−0.567923	−	0.823082i	$-0.692253\pi$
$317$	−784.941	+	453.186i	−0.139075	+	0.0802949i	0.428848	+	0.903377i	$0.358920\pi$
$318$		0			0
$319$	−6221.26	+	10775.5i	−1.09192	+	1.89127i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			10720.2i			1.84671i
$324$		0			0
$325$	16787.0	+	9692.00i	2.86516	+	1.65420i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−245.318	+	710.195i	−0.0411090	+	0.119010i
$330$		0			0
$331$	−2577.90	−	4465.05i	−0.428078	−	0.741453i	0.568624	−	0.822598i	$-0.307476\pi$
$331$	−2577.90	−	4465.05i	−0.428078	−	0.741453i	−0.996702	+	0.0811441i	$0.974143\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−4961.66			−0.809208
$336$		0			0
$337$	−7707.19			−1.24581			−0.622904	−	0.782298i	$-0.714047\pi$
$337$	−7707.19			−1.24581			−0.622904	+	0.782298i	$0.714047\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−5246.88	−	9087.86i	−0.833239	−	1.44321i
$342$		0			0
$343$	5337.76	−	3444.11i	0.840268	−	0.542171i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−634.026	−	366.055i	−0.0980872	−	0.0566307i	0.450154	−	0.892951i	$-0.351369\pi$
$347$	−634.026	−	366.055i	−0.0980872	−	0.0566307i	−0.548241	+	0.836320i	$0.684702\pi$
$348$		0			0
$349$			4863.32i			0.745925i	0.927846	+	0.372962i	$0.121658\pi$
$349$			4863.32i			0.745925i	−0.927846	+	0.372962i	$0.878342\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	994.452	−	1722.44i	0.149942	−	0.259706i	−0.781264	−	0.624201i	$-0.785425\pi$
$353$	994.452	−	1722.44i	0.149942	−	0.259706i	0.931206	+	0.364494i	$0.118758\pi$
$354$		0			0
$355$	−16784.1	+	9690.31i	−2.50932	+	1.44876i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2023.15	+	1168.07i	−0.297432	+	0.171722i	−0.641289	−	0.767300i	$-0.721600\pi$
$359$	−2023.15	+	1168.07i	−0.297432	+	0.171722i	0.343857	+	0.939022i	$0.388266\pi$
$360$		0			0
$361$	5190.02	−	8989.37i	0.756673	−	1.31060i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	163.204i		−	0.0234041i
$366$		0			0
$367$	3846.05	+	2220.52i	0.547037	+	0.315832i	0.747926	−	0.663782i	$-0.231050\pi$
$367$	3846.05	+	2220.52i	0.547037	+	0.315832i	−0.200889	+	0.979614i	$0.564383\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1209.82	−	6256.69i	−0.169300	−	0.875555i
$372$		0			0
$373$	3473.33	+	6015.99i	0.482151	+	0.835110i	0.999790	−	0.0204893i	$-0.00652240\pi$
$373$	3473.33	+	6015.99i	0.482151	+	0.835110i	−0.517639	+	0.855599i	$0.673189\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−13119.2			−1.79224
$378$		0			0
$379$	12964.3			1.75708			0.878539	−	0.477670i	$-0.158519\pi$
$379$	12964.3			1.75708			0.878539	+	0.477670i	$0.158519\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1355.12	+	2347.14i	0.180792	+	0.313141i	0.942151	−	0.335190i	$-0.108801\pi$
$383$	1355.12	+	2347.14i	0.180792	+	0.313141i	−0.761358	+	0.648331i	$0.775467\pi$
$384$		0			0
$385$	15944.5	−	13832.9i	2.11066	−	1.83114i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−9782.93	−	5648.18i	−1.27510	−	0.736180i	−0.299158	−	0.954204i	$-0.596706\pi$
$389$	−9782.93	−	5648.18i	−1.27510	−	0.736180i	−0.975944	+	0.218023i	$0.930039\pi$
$390$		0			0
$391$		−	3590.09i		−	0.464345i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−8834.78	+	15302.3i	−1.12538	+	1.94922i
$396$		0			0
$397$	−12029.8	+	6945.42i	−1.52080	+	0.878036i	−0.521105	+	0.853493i	$0.674480\pi$
$397$	−12029.8	+	6945.42i	−1.52080	+	0.878036i	−0.999699	+	0.0245435i	$0.992187\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−5046.46	+	2913.58i	−0.628450	+	0.362836i	−0.780151	−	0.625591i	$-0.784858\pi$
$401$	−5046.46	+	2913.58i	−0.628450	+	0.362836i	0.151702	+	0.988426i	$0.451525\pi$
$402$		0			0
$403$	5532.24	−	9582.12i	0.683823	−	1.18442i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	1255.23i		−	0.152874i
$408$		0			0
$409$	−9495.40	−	5482.17i	−1.14796	−	0.662777i	−0.199574	−	0.979883i	$-0.563956\pi$
$409$	−9495.40	−	5482.17i	−1.14796	−	0.662777i	−0.948390	+	0.317106i	$0.897289\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−5174.51	+	1000.56i	−0.616516	+	0.119212i
$414$		0			0
$415$	10304.8	+	17848.4i	1.21890	+	2.11119i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−4657.67			−0.543060			−0.271530	−	0.962430i	$-0.587530\pi$
$419$	−4657.67			−0.543060			−0.271530	+	0.962430i	$0.587530\pi$
$420$		0			0
$421$	−4001.54			−0.463238			−0.231619	−	0.972807i	$-0.574402\pi$
$421$	−4001.54			−0.463238			−0.231619	+	0.972807i	$0.574402\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−14388.2	−	24921.1i	−1.64219	−	2.84435i
$426$		0			0
$427$	7819.07	+	2700.90i	0.886162	+	0.306102i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	6306.30	+	3640.95i	0.704789	+	0.406910i	0.809128	−	0.587632i	$-0.199940\pi$
$431$	6306.30	+	3640.95i	0.704789	+	0.406910i	−0.104340	+	0.994542i	$0.533273\pi$
$432$		0			0
$433$			3622.55i			0.402052i	0.979586	+	0.201026i	$0.0644276\pi$
$433$			3622.55i			0.402052i	−0.979586	+	0.201026i	$0.935572\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2886.60	+	4999.75i	−0.315984	+	0.547300i
$438$		0			0
$439$	−4907.92	+	2833.59i	−0.533582	+	0.308064i	−0.742474	−	0.669875i	$-0.766348\pi$
$439$	−4907.92	+	2833.59i	−0.533582	+	0.308064i	0.208892	+	0.977939i	$0.433014\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4348.34	−	2510.52i	0.466357	−	0.269251i	−0.248357	−	0.968669i	$-0.579890\pi$
$443$	4348.34	−	2510.52i	0.466357	−	0.269251i	0.714713	+	0.699417i	$0.246557\pi$
$444$		0			0
$445$	−12575.0	+	21780.6i	−1.33958	+	2.32023i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	6341.85i		−	0.666572i	−0.942826	−	0.333286i	$-0.891843\pi$
$449$		−	6341.85i		−	0.666572i	0.942826	−	0.333286i	$-0.108157\pi$
$450$		0			0
$451$	−4617.64	−	2666.00i	−0.482120	−	0.278352i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	21037.0	+	7266.69i	2.16754	+	0.748720i
$456$		0			0
$457$	−6306.35	−	10922.9i	−0.645511	−	1.11806i	−0.984183	−	0.177153i	$-0.943311\pi$
$457$	−6306.35	−	10922.9i	−0.645511	−	1.11806i	0.338672	−	0.940904i	$-0.390022\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1495.67			0.151107			0.0755536	−	0.997142i	$-0.475928\pi$
$461$	1495.67			0.151107			0.0755536	+	0.997142i	$0.475928\pi$
$462$		0			0
$463$	−5614.86			−0.563595			−0.281798	−	0.959474i	$-0.590931\pi$
$463$	−5614.86			−0.563595			−0.281798	+	0.959474i	$0.590931\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	4405.48	+	7630.52i	0.436534	+	0.756100i	0.997419	−	0.0717941i	$-0.0228724\pi$
$467$	4405.48	+	7630.52i	0.436534	+	0.756100i	−0.560885	+	0.827894i	$0.689539\pi$
$468$		0			0
$469$	−4128.97	+	798.392i	−0.406521	+	0.0786063i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−5409.04	−	3122.91i	−0.525810	−	0.303577i
$474$		0			0
$475$			46275.2i			4.47000i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−6722.03	+	11642.9i	−0.641206	+	1.11060i	0.343958	+	0.938985i	$0.388232\pi$
$479$	−6722.03	+	11642.9i	−0.641206	+	1.11060i	−0.985164	+	0.171616i	$0.945101\pi$
$480$		0			0
$481$	1146.19	−	661.750i	0.108652	−	0.0627302i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	29472.2	−	17015.8i	2.75930	−	1.59308i
$486$		0			0
$487$	144.328	−	249.984i	0.0134295	−	0.0232605i	−0.859233	−	0.511585i	$-0.829058\pi$
$487$	144.328	−	249.984i	0.0134295	−	0.0232605i	0.872662	+	0.488325i	$0.162392\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			17459.7i			1.60478i	0.596803	+	0.802388i	$0.296437\pi$
$491$			17459.7i			1.60478i	−0.596803	+	0.802388i	$0.703563\pi$
$492$		0			0
$493$	16866.8	+	9738.04i	1.54086	+	0.889613i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−12408.0	+	10764.8i	−1.11987	+	0.971564i
$498$		0			0
$499$	−9639.96	−	16696.9i	−0.864817	−	1.49791i	−0.867229	−	0.497910i	$-0.834101\pi$
$499$	−9639.96	−	16696.9i	−0.864817	−	1.49791i	0.00241142	−	0.999997i	$-0.499232\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	20846.3			1.84789			0.923947	−	0.382520i	$-0.124944\pi$
$503$	20846.3			1.84789			0.923947	+	0.382520i	$0.124944\pi$
$504$		0			0
$505$	33140.3			2.92025
$506$		0			0
$507$		0			0
$508$		0			0
$509$	8277.31	+	14336.7i	0.720796	+	1.24846i	0.960681	+	0.277654i	$0.0895570\pi$
$509$	8277.31	+	14336.7i	0.720796	+	1.24846i	−0.239885	+	0.970801i	$0.577110\pi$
$510$		0			0
$511$	−26.2616	−	135.814i	−0.00227347	−	0.0117575i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−19021.3	−	10981.9i	−1.62753	−	0.939654i
$516$		0			0
$517$			2116.21i			0.180021i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−8720.17	+	15103.8i	−0.733278	+	1.27007i	0.222197	+	0.975002i	$0.428677\pi$
$521$	−8720.17	+	15103.8i	−0.733278	+	1.27007i	−0.955475	+	0.295073i	$0.904656\pi$
$522$		0			0
$523$	−19059.3	+	11003.9i	−1.59351	+	0.920014i	−0.600813	+	0.799390i	$0.705156\pi$
$523$	−19059.3	+	11003.9i	−1.59351	+	0.920014i	−0.992698	+	0.120625i	$0.961510\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−14225.1	+	8212.85i	−1.17581	+	0.678857i
$528$		0			0
$529$	−5116.80	+	8862.56i	−0.420547	+	0.728409i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	5621.98i		−	0.456876i
$534$		0			0
$535$	20083.1	+	11595.0i	1.62293	+	0.936998i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	11042.7	−	14077.1i	0.882454	−	1.12494i
$540$		0			0
$541$	−10144.2	−	17570.4i	−0.806165	−	1.39632i	−0.915502	−	0.402314i	$-0.868206\pi$
$541$	−10144.2	−	17570.4i	−0.806165	−	1.39632i	0.109336	−	0.994005i	$-0.465127\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−10855.7			−0.853225
$546$		0			0
$547$	−3726.23			−0.291265			−0.145633	−	0.989339i	$-0.546522\pi$
$547$	−3726.23			−0.291265			−0.145633	+	0.989339i	$0.546522\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−15659.7	−	27123.4i	−1.21075	−	2.09709i
$552$		0			0
$553$	−4889.75	+	14155.8i	−0.376010	+	1.08855i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−2400.90	−	1386.16i	−0.182638	−	0.105446i	0.405894	−	0.913920i	$-0.366960\pi$
$557$	−2400.90	−	1386.16i	−0.182638	−	0.105446i	−0.588531	+	0.808474i	$0.700294\pi$
$558$		0			0
$559$		−	6585.52i		−	0.498279i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6501.90	+	11261.6i	−0.486718	+	0.843021i	−0.999883	−	0.0152692i	$-0.995139\pi$
$563$	−6501.90	+	11261.6i	−0.486718	+	0.843021i	0.513165	+	0.858290i	$0.328473\pi$
$564$		0			0
$565$	18456.5	−	10655.9i	1.37429	−	0.793446i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−9698.81	+	5599.61i	−0.714579	+	0.412562i	−0.812754	−	0.582607i	$-0.802033\pi$
$569$	−9698.81	+	5599.61i	−0.714579	+	0.412562i	0.0981755	+	0.995169i	$0.468699\pi$
$570$		0			0
$571$	7242.52	−	12544.4i	0.530805	−	0.919382i	−0.468548	−	0.883438i	$-0.655223\pi$
$571$	7242.52	−	12544.4i	0.530805	−	0.919382i	0.999354	−	0.0359441i	$-0.0114438\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	15497.2i		−	1.12396i
$576$		0			0
$577$	10364.2	+	5983.80i	0.747780	+	0.431731i	0.824891	−	0.565291i	$-0.191236\pi$
$577$	10364.2	+	5983.80i	0.747780	+	0.431731i	−0.0771109	+	0.997023i	$0.524570\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	11447.4	+	13194.8i	0.817417	+	0.942193i
$582$		0			0
$583$	−8974.08	−	15543.6i	−0.637510	−	1.10420i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	12202.6			0.858017			0.429009	−	0.903300i	$-0.358863\pi$
$587$	12202.6			0.858017			0.429009	+	0.903300i	$0.358863\pi$
$588$		0			0
$589$	26414.1			1.84783
$590$		0			0
$591$		0			0
$592$		0			0
$593$	5152.12	+	8923.73i	0.356783	+	0.617966i	0.987421	−	0.158111i	$-0.0505403\pi$
$593$	5152.12	+	8923.73i	0.356783	+	0.617966i	−0.630639	+	0.776077i	$0.717207\pi$
$594$		0			0
$595$	−21652.4	−	24957.6i	−1.49187	−	1.71960i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1846.43	−	1066.04i	−0.125949	−	0.0727165i	0.435702	−	0.900091i	$-0.356500\pi$
$599$	−1846.43	−	1066.04i	−0.125949	−	0.0727165i	−0.561651	+	0.827374i	$0.689833\pi$
$600$		0			0
$601$			3844.71i			0.260947i	0.991452	+	0.130473i	$0.0416497\pi$
$601$			3844.71i			0.260947i	−0.991452	+	0.130473i	$0.958350\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	15184.4	−	26300.2i	1.02039	−	1.76736i
$606$		0			0
$607$	−5692.75	+	3286.71i	−0.380662	+	0.219775i	−0.678106	−	0.734964i	$-0.737199\pi$
$607$	−5692.75	+	3286.71i	−0.380662	+	0.219775i	0.297444	+	0.954739i	$0.403866\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1932.37	+	1115.65i	−0.127946	+	0.0738699i
$612$		0			0
$613$	4464.58	−	7732.88i	0.294164	−	0.509508i	−0.680626	−	0.732631i	$-0.738292\pi$
$613$	4464.58	−	7732.88i	0.294164	−	0.509508i	0.974790	+	0.223124i	$0.0716253\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	20869.2i		−	1.36169i	−0.732429	−	0.680843i	$-0.761613\pi$
$617$		−	20869.2i		−	1.36169i	0.732429	−	0.680843i	$-0.238387\pi$
$618$		0			0
$619$	−11798.1	−	6811.63i	−0.766083	−	0.442298i	0.0653928	−	0.997860i	$-0.479170\pi$
$619$	−11798.1	−	6811.63i	−0.766083	−	0.442298i	−0.831475	+	0.555562i	$0.812503\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−6959.86	+	20148.8i	−0.447578	+	1.29573i
$624$		0			0
$625$	−32268.5	−	55890.7i	−2.06518	−	3.57700i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1964.79			−0.124549
$630$		0			0
$631$	−20380.3			−1.28578			−0.642890	−	0.765959i	$-0.722265\pi$
$631$	−20380.3			−1.28578			−0.642890	+	0.765959i	$0.722265\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−6269.00	−	10858.2i	−0.391776	−	0.678576i
$636$		0			0
$637$	18675.8	+	2662.04i	1.16163	+	0.165579i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−22692.6	−	13101.6i	−1.39829	−	0.807302i	−0.404074	−	0.914726i	$-0.632406\pi$
$641$	−22692.6	−	13101.6i	−1.39829	−	0.807302i	−0.994213	+	0.107425i	$0.965740\pi$
$642$		0			0
$643$			21463.4i			1.31639i	0.752850	+	0.658193i	$0.228679\pi$
$643$			21463.4i			1.31639i	−0.752850	+	0.658193i	$0.771321\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	782.398	−	1355.15i	0.0475414	−	0.0823440i	−0.841275	−	0.540607i	$-0.818195\pi$
$647$	782.398	−	1355.15i	0.0475414	−	0.0823440i	0.888817	+	0.458263i	$0.151528\pi$
$648$		0			0
$649$	−12855.1	+	7421.90i	−0.777515	+	0.448898i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	2832.40	−	1635.29i	0.169740	−	0.0979996i	−0.412723	−	0.910857i	$-0.635422\pi$
$653$	2832.40	−	1635.29i	0.169740	−	0.0979996i	0.582463	+	0.812857i	$0.302089\pi$
$654$		0			0
$655$	2307.23	−	3996.25i	0.137635	−	0.238391i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	3271.53i		−	0.193385i	−0.995314	−	0.0966924i	$-0.969174\pi$
$659$		−	3271.53i		−	0.193385i	0.995314	−	0.0966924i	$-0.0308263\pi$
$660$		0			0
$661$	−8150.02	−	4705.41i	−0.479575	−	0.276883i	0.240665	−	0.970608i	$-0.422635\pi$
$661$	−8150.02	−	4705.41i	−0.479575	−	0.276883i	−0.720239	+	0.693726i	$0.755968\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	10087.2	+	52166.8i	0.588216	+	3.04202i
$666$		0			0
$667$	5244.30	+	9083.40i	0.304438	+	0.527302i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	23299.0			1.34046
$672$		0			0
$673$	−15000.7			−0.859187			−0.429594	−	0.903022i	$-0.641343\pi$
$673$	−15000.7			−0.859187			−0.429594	+	0.903022i	$0.641343\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−10894.9	−	18870.5i	−0.618501	−	1.07127i	−0.989759	−	0.142745i	$-0.954407\pi$
$677$	−10894.9	−	18870.5i	−0.618501	−	1.07127i	0.371259	−	0.928529i	$-0.378926\pi$
$678$		0			0
$679$	21787.9	−	18902.5i	1.23144	−	1.06835i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	11577.3	+	6684.18i	0.648601	+	0.374470i	0.787920	−	0.615778i	$-0.211158\pi$
$683$	11577.3	+	6684.18i	0.648601	+	0.374470i	−0.139319	+	0.990248i	$0.544491\pi$
$684$		0			0
$685$		−	7046.65i		−	0.393049i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	9462.16	−	16388.9i	0.523192	−	0.906196i
$690$		0			0
$691$	2603.71	−	1503.26i	0.143343	−	0.0827591i	−0.426613	−	0.904434i	$-0.640293\pi$
$691$	2603.71	−	1503.26i	0.143343	−	0.0827591i	0.569956	+	0.821675i	$0.306960\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	9292.81	−	5365.20i	0.507189	−	0.292826i
$696$		0			0
$697$	−4173.04	+	7227.92i	−0.226779	+	0.392793i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	5449.86i		−	0.293635i	−0.989164	−	0.146818i	$-0.953097\pi$
$701$		−	5449.86i		−	0.293635i	0.989164	−	0.146818i	$-0.0469030\pi$
$702$		0			0
$703$	2736.27	+	1579.79i	0.146800	+	0.0847551i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	27578.5	−	5332.68i	1.46704	−	0.283672i
$708$		0			0
$709$	−7610.90	−	13182.5i	−0.403150	−	0.698277i	0.590954	−	0.806705i	$-0.298751\pi$
$709$	−7610.90	−	13182.5i	−0.403150	−	0.698277i	−0.994104	+	0.108429i	$0.965418\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−8845.86			−0.464628
$714$		0			0
$715$	62685.3			3.27874
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1575.16	+	2728.27i	0.0817020	+	0.141512i	0.903981	−	0.427572i	$-0.140631\pi$
$719$	1575.16	+	2728.27i	0.0817020	+	0.141512i	−0.822279	+	0.569084i	$0.807298\pi$
$720$		0			0
$721$	−17596.1	−	6078.13i	−0.908896	−	0.313955i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	72808.0	+	42035.7i	3.72968	+	2.15333i
$726$		0			0
$727$		−	30799.4i		−	1.57123i	−0.618713	−	0.785617i	$-0.712345\pi$
$727$		−	30799.4i		−	1.57123i	0.618713	−	0.785617i	$-0.287655\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−4888.24	+	8466.69i	−0.247330	+	0.428388i
$732$		0			0
$733$	7439.96	−	4295.46i	0.374899	−	0.216448i	−0.300697	−	0.953720i	$-0.597219\pi$
$733$	7439.96	−	4295.46i	0.374899	−	0.216448i	0.675597	+	0.737271i	$0.263886\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−10257.7	+	5922.26i	−0.512681	+	0.295996i
$738$		0			0
$739$	15874.5	−	27495.4i	0.790192	−	1.36865i	−0.135655	−	0.990756i	$-0.543314\pi$
$739$	15874.5	−	27495.4i	0.790192	−	1.36865i	0.925847	−	0.377897i	$-0.123353\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	15722.6i		−	0.776323i	−0.921591	−	0.388161i	$-0.873110\pi$
$743$		−	15722.6i		−	0.776323i	0.921591	−	0.388161i	$-0.126890\pi$
$744$		0			0
$745$	−4025.44	−	2324.09i	−0.197961	−	0.114293i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	18578.4	+	6417.42i	0.906327	+	0.313067i
$750$		0			0
$751$	15181.8	+	26295.7i	0.737674	+	1.27769i	0.953540	+	0.301265i	$0.0974090\pi$
$751$	15181.8	+	26295.7i	0.737674	+	1.27769i	−0.215867	+	0.976423i	$0.569258\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	4351.46			0.209756
$756$		0			0
$757$	−28207.3			−1.35431			−0.677155	−	0.735841i	$-0.736787\pi$
$757$	−28207.3			−1.35431			−0.677155	+	0.735841i	$0.736787\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−13564.9	−	23495.1i	−0.646160	−	1.11918i	−0.984032	−	0.177990i	$-0.943041\pi$
$761$	−13564.9	−	23495.1i	−0.646160	−	1.11918i	0.337873	−	0.941192i	$-0.390293\pi$
$762$		0			0
$763$	−9033.85	+	1746.82i	−0.428633	+	0.0828821i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−13554.3	−	7825.56i	−0.638091	−	0.368402i
$768$		0			0
$769$			8577.68i			0.402235i	0.979567	+	0.201118i	$0.0644574\pi$
$769$			8577.68i			0.402235i	−0.979567	+	0.201118i	$0.935543\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−700.177	+	1212.74i	−0.0325790	+	0.0564286i	−0.881855	−	0.471520i	$-0.843705\pi$
$773$	−700.177	+	1212.74i	−0.0325790	+	0.0564286i	0.849276	+	0.527949i	$0.177039\pi$
$774$		0			0
$775$	−61404.7	+	35452.0i	−2.84609	+	1.64319i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	11623.2	−	6710.64i	0.534587	−	0.308644i
$780$		0			0
$781$	−23132.8	+	40067.2i	−1.05987	+	1.83574i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			2328.63i			0.105876i
$786$		0			0
$787$	−3002.03	−	1733.22i	−0.135973	−	0.0785041i	0.430470	−	0.902605i	$-0.358348\pi$
$787$	−3002.03	−	1733.22i	−0.135973	−	0.0785041i	−0.566443	+	0.824101i	$0.691681\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	13644.4	−	11837.4i	0.613324	−	0.532100i
$792$		0			0
$793$	12283.1	+	21274.9i	0.550043	+	0.952703i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	37737.6			1.67721			0.838604	−	0.544741i	$-0.183372\pi$
$797$	37737.6			1.67721			0.838604	+	0.544741i	$0.183372\pi$
$798$		0			0
$799$	3312.47			0.146667
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−194.801	−	337.405i	−0.00856087	−	0.0148279i
$804$		0			0
$805$	−3378.11	−	17470.2i	−0.147904	−	0.764900i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−13254.0	−	7652.19i	−0.576001	−	0.332554i	0.183541	−	0.983012i	$-0.441244\pi$
$809$	−13254.0	−	7652.19i	−0.576001	−	0.332554i	−0.759543	+	0.650458i	$0.774577\pi$
$810$		0			0
$811$		−	15665.8i		−	0.678299i	−0.940733	−	0.339149i	$-0.889861\pi$
$811$		−	15665.8i		−	0.678299i	0.940733	−	0.339149i	$-0.110139\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−39482.9	+	68386.4i	−1.69696	+	2.93923i
$816$		0			0
$817$	13615.2	−	7860.76i	0.583032	−	0.336613i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	30911.4	−	17846.7i	1.31403	−	0.758654i	0.331267	−	0.943537i	$-0.392524\pi$
$821$	30911.4	−	17846.7i	1.31403	−	0.758654i	0.982761	+	0.184883i	$0.0591907\pi$
$822$		0			0
$823$	20981.1	−	36340.3i	0.888645	−	1.53918i	0.0471660	−	0.998887i	$-0.484981\pi$
$823$	20981.1	−	36340.3i	0.888645	−	1.53918i	0.841479	−	0.540290i	$-0.181686\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			13652.3i			0.574048i	0.957923	+	0.287024i	$0.0926660\pi$
$827$			13652.3i			0.574048i	−0.957923	+	0.287024i	$0.907334\pi$
$828$		0			0
$829$	25866.9	+	14934.2i	1.08371	+	0.625679i	0.931894	−	0.362731i	$-0.118156\pi$
$829$	25866.9	+	14934.2i	1.08371	+	0.625679i	0.151813	+	0.988409i	$0.451489\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−22034.6	−	17285.0i	−0.916510	−	0.718953i
$834$		0			0
$835$	−31010.0	−	53711.0i	−1.28521	−	2.22604i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−4257.88			−0.175207			−0.0876033	−	0.996155i	$-0.527921\pi$
$839$	−4257.88			−0.175207			−0.0876033	+	0.996155i	$0.527921\pi$
$840$		0			0
$841$	−32511.2			−1.33303
$842$		0			0
$843$		0			0
$844$		0			0
$845$	9044.48	+	15665.5i	0.368213	+	0.637763i
$846$		0			0
$847$	8404.06	−	24329.7i	0.340929	−	0.986987i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−916.355	−	529.058i	−0.0369121	−	0.0213112i
$852$		0			0
$853$		−	4490.96i		−	0.180267i	−0.995930	−	0.0901334i	$-0.971271\pi$
$853$		−	4490.96i		−	0.180267i	0.995930	−	0.0901334i	$-0.0287293\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	2406.72	−	4168.55i	0.0959298	−	0.166155i	−0.814066	−	0.580772i	$-0.802751\pi$
$857$	2406.72	−	4168.55i	0.0959298	−	0.166155i	0.909996	+	0.414616i	$0.136084\pi$
$858$		0			0
$859$	38157.8	−	22030.4i	1.51563	−	0.875049i	0.515798	−	0.856710i	$-0.327496\pi$
$859$	38157.8	−	22030.4i	1.51563	−	0.875049i	0.999832	−	0.0183388i	$-0.00583775\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	14563.6	−	8408.30i	0.574450	−	0.331659i	−0.184475	−	0.982837i	$-0.559058\pi$
$863$	14563.6	−	8408.30i	0.574450	−	0.331659i	0.758925	+	0.651178i	$0.225725\pi$
$864$		0			0
$865$	−1624.08	+	2812.99i	−0.0638387	+	0.110572i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			42180.9i			1.64659i
$870$		0			0
$871$	−10815.5	−	6244.36i	−0.420747	−	0.242918i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−60316.8	−	69524.1i	−2.33038	−	2.68611i
$876$		0			0
$877$	−15444.7	−	26751.1i	−0.594677	−	1.03001i	−0.993592	−	0.113023i	$-0.963947\pi$
$877$	−15444.7	−	26751.1i	−0.594677	−	1.03001i	0.398916	−	0.916988i	$-0.369387\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−24183.7			−0.924824			−0.462412	−	0.886665i	$-0.653016\pi$
$881$	−24183.7			−0.924824			−0.462412	+	0.886665i	$0.653016\pi$
$882$		0			0
$883$	51420.0			1.95971			0.979853	−	0.199718i	$-0.0640028\pi$
$883$	51420.0			1.95971			0.979853	+	0.199718i	$0.0640028\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1984.25	−	3436.83i	−0.0751125	−	0.130099i	0.826023	−	0.563637i	$-0.190598\pi$
$887$	−1984.25	−	3436.83i	−0.0751125	−	0.130099i	−0.901135	+	0.433538i	$0.857265\pi$
$888$		0			0
$889$	−6964.12	−	8027.18i	−0.262732	−	0.302838i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−4613.12	−	2663.38i	−0.172869	−	0.0998060i
$894$		0			0
$895$		−	2384.05i		−	0.0890391i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	23994.2	−	41559.1i	0.890157	−	1.54180i
$900$		0			0
$901$	−24330.1	+	14047.0i	−0.899615	+	0.519393i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−6077.67	+	3508.94i	−0.223236	+	0.128885i
$906$		0			0
$907$	−13568.9	+	23501.9i	−0.496743	+	0.860385i	−0.999993	−	0.00375625i	$-0.998804\pi$
$907$	−13568.9	+	23501.9i	−0.496743	+	0.860385i	0.503249	+	0.864141i	$0.332138\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	27356.2i		−	0.994896i	−0.867494	−	0.497448i	$-0.834270\pi$
$911$		−	27356.2i		−	0.994896i	0.867494	−	0.497448i	$-0.165730\pi$
$912$		0			0
$913$	42607.9	+	24599.7i	1.54449	+	0.891709i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1276.98	−	3696.84i	0.0459863	−	0.133130i
$918$		0			0
$919$	20099.1	+	34812.6i	0.721445	+	1.24958i	0.960421	+	0.278553i	$0.0898549\pi$
$919$	20099.1	+	34812.6i	0.721445	+	1.24958i	−0.238976	+	0.971025i	$0.576812\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−48781.8			−1.73962
$924$		0			0
$925$	−8481.33			−0.301475
$926$		0			0
$927$		0			0
$928$		0			0
$929$	17817.9	+	30861.4i	0.629263	+	1.08992i	0.987700	+	0.156361i	$0.0499764\pi$
$929$	17817.9	+	30861.4i	0.629263	+	1.08992i	−0.358437	+	0.933554i	$0.616690\pi$
$930$		0			0
$931$	16788.6	+	41788.8i	0.591002	+	1.47108i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−80591.5	−	46529.5i	−2.81885	−	1.62746i
$936$		0			0
$937$		−	20780.5i		−	0.724514i	−0.932078	−	0.362257i	$-0.882006\pi$
$937$		−	20780.5i		−	0.724514i	0.932078	−	0.362257i	$-0.117994\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	11286.1	−	19548.1i	0.390983	−	0.677203i	−0.601596	−	0.798800i	$-0.705468\pi$
$941$	11286.1	−	19548.1i	0.390983	−	0.677203i	0.992580	+	0.121597i	$0.0388016\pi$
$942$		0			0
$943$	−3892.50	+	2247.34i	−0.134419	+	0.0776070i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	648.189	−	374.232i	0.0222421	−	0.0128415i	−0.488838	−	0.872375i	$-0.662579\pi$
$947$	648.189	−	374.232i	0.0222421	−	0.0128415i	0.511080	+	0.859533i	$0.329246\pi$
$948$		0			0
$949$	205.396	−	355.756i	0.00702574	−	0.0121689i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			54667.1i			1.85818i	0.369858	+	0.929088i	$0.379406\pi$
$953$			54667.1i			1.85818i	−0.369858	+	0.929088i	$0.620594\pi$
$954$		0			0
$955$	−40239.7	−	23232.4i	−1.36348	−	0.787207i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1133.89	−	5864.04i	−0.0381807	−	0.197456i
$960$		0			0
$961$	5340.66	+	9250.30i	0.179271	+	0.310507i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−76956.8			−2.56718
$966$		0			0
$967$	−52999.6			−1.76252			−0.881258	−	0.472635i	$-0.843303\pi$
$967$	−52999.6			−1.76252			−0.881258	+	0.472635i	$0.843303\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−3688.22	−	6388.18i	−0.121896	−	0.211129i	0.798620	−	0.601836i	$-0.205564\pi$
$971$	−3688.22	−	6388.18i	−0.121896	−	0.211129i	−0.920515	+	0.390707i	$0.872231\pi$
$972$		0			0
$973$	6869.91	−	5960.11i	0.226351	−	0.196375i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	45060.9	+	26015.9i	1.47556	+	0.851918i	0.999620	−	0.0275576i	$-0.00877298\pi$
$977$	45060.9	+	26015.9i	1.47556	+	0.851918i	0.475944	+	0.879475i	$0.342106\pi$
$978$		0			0
$979$			60038.5i			1.96000i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−16690.9	+	28909.5i	−0.541564	+	0.938016i	0.457250	+	0.889338i	$0.348834\pi$
$983$	−16690.9	+	28909.5i	−0.541564	+	0.938016i	−0.998814	+	0.0486785i	$0.984499\pi$
$984$		0			0
$985$	8829.75	−	5097.86i	0.285624	−	0.164905i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−4559.63	+	2632.50i	−0.146600	+	0.0846398i
$990$		0			0
$991$	13260.6	−	22968.1i	0.425063	−	0.736231i	−0.571363	−	0.820697i	$-0.693585\pi$
$991$	13260.6	−	22968.1i	0.425063	−	0.736231i	0.996426	+	0.0844661i	$0.0269185\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			93572.3i			2.98135i
$996$		0			0
$997$	13272.1	+	7662.66i	0.421597	+	0.243409i	0.695760	−	0.718274i	$-0.255068\pi$
$997$	13272.1	+	7662.66i	0.421597	+	0.243409i	−0.274163	+	0.961683i	$0.588401\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.4.bl.a.17.1	✓	48
3.2	odd	2	inner	504.4.bl.a.17.24	yes	48
4.3	odd	2		1008.4.bt.d.17.1		48
7.5	odd	6	inner	504.4.bl.a.89.24	yes	48
12.11	even	2		1008.4.bt.d.17.24		48
21.5	even	6	inner	504.4.bl.a.89.1	yes	48
28.19	even	6		1008.4.bt.d.593.24		48
84.47	odd	6		1008.4.bt.d.593.1		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.4.bl.a.17.1	✓	48	1.1	even	1	trivial
504.4.bl.a.17.24	yes	48	3.2	odd	2	inner
504.4.bl.a.89.1	yes	48	21.5	even	6	inner
504.4.bl.a.89.24	yes	48	7.5	odd	6	inner
1008.4.bt.d.17.1		48	4.3	odd	2
1008.4.bt.d.17.24		48	12.11	even	2
1008.4.bt.d.593.1		48	84.47	odd	6
1008.4.bt.d.593.24		48	28.19	even	6

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:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	504.bl (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-10.9253 - 18.9231i) q^{5} +(-12.1367 - 13.9893i) q^{7} +O(q^{10})\)	\(q+(-10.9253 - 18.9231i) q^{5} +(-12.1367 - 13.9893i) q^{7} +(-45.1734 - 26.0809i) q^{11} -54.9986i q^{13} +(-40.8239 + 70.7091i) q^{17} +(113.707 - 65.6488i) q^{19} +(-38.0795 + 21.9852i) q^{23} +(-176.223 + 305.227i) q^{25} -238.538i q^{29} +(174.225 + 100.589i) q^{31} +(-132.125 + 382.501i) q^{35} +(12.0321 + 20.8402i) q^{37} +102.220 q^{41} +119.740 q^{43} +(-20.2851 - 35.1348i) q^{47} +(-48.4020 + 339.568i) q^{49} +(297.988 + 172.043i) q^{53} +1139.76i q^{55} +(142.286 - 246.447i) q^{59} +(-386.826 + 223.334i) q^{61} +(-1040.75 + 600.874i) q^{65} +(113.537 - 196.651i) q^{67} -886.964i q^{71} +(6.46845 + 3.73456i) q^{73} +(183.401 + 948.480i) q^{77} +(-404.328 - 700.317i) q^{79} -943.208 q^{83} +1784.05 q^{85} +(-575.503 - 996.801i) q^{89} +(-769.393 + 667.501i) q^{91} +(-2484.56 - 1434.46i) q^{95} +1557.47i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 24 q^{7}+O(q^{10}) \) 48 * q - 24 * q^7	\( 48 q - 24 q^{7} + 540 q^{19} - 924 q^{25} + 648 q^{31} - 132 q^{37} - 792 q^{43} + 672 q^{49} - 12 q^{67} + 2412 q^{73} + 1680 q^{79} + 480 q^{85} + 1404 q^{91}+O(q^{100}) \) 48 * q - 24 * q^7 + 540 * q^19 - 924 * q^25 + 648 * q^31 - 132 * q^37 - 792 * q^43 + 672 * q^49 - 12 * q^67 + 2412 * q^73 + 1680 * q^79 + 480 * q^85 + 1404 * q^91

Introduction

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