LMFDB - Embedded newform 1620.3.b.a.809.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,3,Mod(809,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1620.809");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1620.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$44.1418028264$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			809.18
Character	$\chi$	$=$	1620.809
Dual form			1620.3.b.a.809.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.82619 + 4.12464i) q^{5} -7.52777i q^{7} +O(q^{10})$	$q+(2.82619 + 4.12464i) q^{5} -7.52777i q^{7} +6.67363i q^{11} -15.5692i q^{13} +18.9462 q^{17} -27.1064 q^{19} +19.6113 q^{23} +(-9.02534 + 23.3140i) q^{25} -6.97175i q^{29} +29.5191 q^{31} +(31.0494 - 21.2749i) q^{35} -52.2126i q^{37} -60.8539i q^{41} -10.6611i q^{43} -75.8093 q^{47} -7.66737 q^{49} -47.3234 q^{53} +(-27.5263 + 18.8609i) q^{55} +24.4204i q^{59} +40.7094 q^{61} +(64.2172 - 44.0013i) q^{65} +58.9100i q^{67} -33.9404i q^{71} -79.8213i q^{73} +50.2376 q^{77} +86.0506 q^{79} +153.076 q^{83} +(53.5456 + 78.1464i) q^{85} -83.3156i q^{89} -117.201 q^{91} +(-76.6077 - 111.804i) q^{95} -16.2175i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 48 q^{25} - 288 q^{49} - 36 q^{55} + 120 q^{61} + 480 q^{79} - 24 q^{85} + 48 q^{91}+O(q^{100}) $ 24 * q - 48 * q^25 - 288 * q^49 - 36 * q^55 + 120 * q^61 + 480 * q^79 - 24 * q^85 + 48 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	2.82619	+	4.12464i	0.565237	+	0.824928i
$5$	2.82619	+	4.12464i	0.565237	+	0.824928i
$6$		0			0
$7$		−	7.52777i		−	1.07540i	−0.843137	−	0.537698i	$-0.819294\pi$
$7$		−	7.52777i		−	1.07540i	0.843137	−	0.537698i	$-0.180706\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			6.67363i			0.606694i	0.952880	+	0.303347i	$0.0981041\pi$
$11$			6.67363i			0.606694i	−0.952880	+	0.303347i	$0.901896\pi$
$12$		0			0
$13$		−	15.5692i		−	1.19763i	−0.800888	−	0.598814i	$-0.795639\pi$
$13$		−	15.5692i		−	1.19763i	0.800888	−	0.598814i	$-0.204361\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	18.9462			1.11448			0.557242	−	0.830350i	$-0.311860\pi$
$17$	18.9462			1.11448			0.557242	+	0.830350i	$0.311860\pi$
$18$		0			0
$19$	−27.1064			−1.42665			−0.713326	−	0.700832i	$-0.752812\pi$
$19$	−27.1064			−1.42665			−0.713326	+	0.700832i	$0.752812\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	19.6113			0.852665			0.426333	−	0.904567i	$-0.359805\pi$
$23$	19.6113			0.852665			0.426333	+	0.904567i	$0.359805\pi$
$24$		0			0
$25$	−9.02534	+	23.3140i	−0.361014	+	0.932560i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	6.97175i		−	0.240405i	−0.992749	−	0.120203i	$-0.961646\pi$
$29$		−	6.97175i		−	0.240405i	0.992749	−	0.120203i	$-0.0383544\pi$
$30$		0			0
$31$	29.5191			0.952228			0.476114	−	0.879383i	$-0.342045\pi$
$31$	29.5191			0.952228			0.476114	+	0.879383i	$0.342045\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	31.0494	−	21.2749i	0.887125	−	0.607854i
$36$		0			0
$37$		−	52.2126i		−	1.41115i	−0.708634	−	0.705576i	$-0.750688\pi$
$37$		−	52.2126i		−	1.41115i	0.708634	−	0.705576i	$-0.249312\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	60.8539i		−	1.48424i	−0.670266	−	0.742121i	$-0.733820\pi$
$41$		−	60.8539i		−	1.48424i	0.670266	−	0.742121i	$-0.266180\pi$
$42$		0			0
$43$		−	10.6611i		−	0.247934i	−0.992286	−	0.123967i	$-0.960438\pi$
$43$		−	10.6611i		−	0.247934i	0.992286	−	0.123967i	$-0.0395616\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−75.8093			−1.61296			−0.806482	−	0.591259i	$-0.798631\pi$
$47$	−75.8093			−1.61296			−0.806482	+	0.591259i	$0.798631\pi$
$48$		0			0
$49$	−7.66737			−0.156477
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−47.3234			−0.892895			−0.446447	−	0.894810i	$-0.647311\pi$
$53$	−47.3234			−0.892895			−0.446447	+	0.894810i	$0.647311\pi$
$54$		0			0
$55$	−27.5263	+	18.8609i	−0.500479	+	0.342926i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			24.4204i			0.413905i	0.978351	+	0.206952i	$0.0663545\pi$
$59$			24.4204i			0.413905i	−0.978351	+	0.206952i	$0.933646\pi$
$60$		0			0
$61$	40.7094			0.667367			0.333683	−	0.942685i	$-0.391708\pi$
$61$	40.7094			0.667367			0.333683	+	0.942685i	$0.391708\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	64.2172	−	44.0013i	0.987957	−	0.676944i
$66$		0			0
$67$			58.9100i			0.879254i	0.898180	+	0.439627i	$0.144889\pi$
$67$			58.9100i			0.879254i	−0.898180	+	0.439627i	$0.855111\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	33.9404i		−	0.478034i	−0.971015	−	0.239017i	$-0.923175\pi$
$71$		−	33.9404i		−	0.478034i	0.971015	−	0.239017i	$-0.0768252\pi$
$72$		0			0
$73$		−	79.8213i		−	1.09344i	−0.837315	−	0.546722i	$-0.815876\pi$
$73$		−	79.8213i		−	1.09344i	0.837315	−	0.546722i	$-0.184124\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	50.2376			0.652436
$78$		0			0
$79$	86.0506			1.08925			0.544624	−	0.838680i	$-0.316672\pi$
$79$	86.0506			1.08925			0.544624	+	0.838680i	$0.316672\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	153.076			1.84429			0.922143	−	0.386850i	$-0.126437\pi$
$83$	153.076			1.84429			0.922143	+	0.386850i	$0.126437\pi$
$84$		0			0
$85$	53.5456	+	78.1464i	0.629948	+	0.919370i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	83.3156i		−	0.936130i	−0.883694	−	0.468065i	$-0.844951\pi$
$89$		−	83.3156i		−	0.936130i	0.883694	−	0.468065i	$-0.155049\pi$
$90$		0			0
$91$	−117.201			−1.28792
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−76.6077	−	111.804i	−0.806397	−	1.17689i
$96$		0			0
$97$		−	16.2175i		−	0.167190i	−0.996500	−	0.0835952i	$-0.973360\pi$
$97$		−	16.2175i		−	0.167190i	0.996500	−	0.0835952i	$-0.0266403\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	127.041i		−	1.25783i	−0.777473	−	0.628916i	$-0.783499\pi$
$101$		−	127.041i		−	1.25783i	0.777473	−	0.628916i	$-0.216501\pi$
$102$		0			0
$103$			131.249i			1.27426i	0.770755	+	0.637132i	$0.219879\pi$
$103$			131.249i			1.27426i	−0.770755	+	0.637132i	$0.780121\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	131.922			1.23292			0.616458	−	0.787388i	$-0.288567\pi$
$107$	131.922			1.23292			0.616458	+	0.787388i	$0.288567\pi$
$108$		0			0
$109$	−201.576			−1.84932			−0.924662	−	0.380790i	$-0.875652\pi$
$109$	−201.576			−1.84932			−0.924662	+	0.380790i	$0.875652\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	33.6972			0.298205			0.149102	−	0.988822i	$-0.452362\pi$
$113$	33.6972			0.298205			0.149102	+	0.988822i	$0.452362\pi$
$114$		0			0
$115$	55.4252	+	80.8896i	0.481958	+	0.703388i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	142.623i		−	1.19851i
$120$		0			0
$121$	76.4626			0.631923
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−121.669	+	28.6634i	−0.973354	+	0.229307i
$126$		0			0
$127$		−	118.081i		−	0.929768i	−0.885372	−	0.464884i	$-0.846096\pi$
$127$		−	118.081i		−	0.929768i	0.885372	−	0.464884i	$-0.153904\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			28.0634i			0.214224i	0.994247	+	0.107112i	$0.0341604\pi$
$131$			28.0634i			0.214224i	−0.994247	+	0.107112i	$0.965840\pi$
$132$		0			0
$133$			204.051i			1.53422i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	107.772			0.786656			0.393328	−	0.919398i	$-0.371324\pi$
$137$	107.772			0.786656			0.393328	+	0.919398i	$0.371324\pi$
$138$		0			0
$139$	170.944			1.22981			0.614907	−	0.788600i	$-0.289194\pi$
$139$	170.944			1.22981			0.614907	+	0.788600i	$0.289194\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	103.903			0.726593
$144$		0			0
$145$	28.7560	−	19.7035i	0.198317	−	0.135886i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	81.6965i		−	0.548299i	−0.961687	−	0.274149i	$-0.911604\pi$
$149$		−	81.6965i		−	0.548299i	0.961687	−	0.274149i	$-0.0883963\pi$
$150$		0			0
$151$	9.21306			0.0610136			0.0305068	−	0.999535i	$-0.490288\pi$
$151$	9.21306			0.0610136			0.0305068	+	0.999535i	$0.490288\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	83.4264	+	121.756i	0.538235	+	0.785520i
$156$		0			0
$157$		−	288.365i		−	1.83672i	−0.395749	−	0.918359i	$-0.629515\pi$
$157$		−	288.365i		−	1.83672i	0.395749	−	0.918359i	$-0.370485\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	147.629i		−	0.916953i
$162$		0			0
$163$		−	69.1070i		−	0.423970i	−0.977273	−	0.211985i	$-0.932007\pi$
$163$		−	69.1070i		−	0.423970i	0.977273	−	0.211985i	$-0.0679927\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	195.225			1.16901			0.584507	−	0.811389i	$-0.301288\pi$
$167$	195.225			1.16901			0.584507	+	0.811389i	$0.301288\pi$
$168$		0			0
$169$	−73.3987			−0.434312
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−59.3786			−0.343229			−0.171614	−	0.985164i	$-0.554898\pi$
$173$	−59.3786			−0.343229			−0.171614	+	0.985164i	$0.554898\pi$
$174$		0			0
$175$	175.503	+	67.9407i	1.00287	+	0.388233i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	82.1619i		−	0.459005i	−0.973308	−	0.229503i	$-0.926290\pi$
$179$		−	82.1619i		−	0.459005i	0.973308	−	0.229503i	$-0.0737099\pi$
$180$		0			0
$181$	−242.120			−1.33768			−0.668840	−	0.743407i	$-0.733209\pi$
$181$	−242.120			−1.33768			−0.668840	+	0.743407i	$0.733209\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	215.358	−	147.563i	1.16410	−	0.797636i
$186$		0			0
$187$			126.440i			0.676151i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			27.4015i			0.143463i	0.997424	+	0.0717317i	$0.0228525\pi$
$191$			27.4015i			0.143463i	−0.997424	+	0.0717317i	$0.977147\pi$
$192$		0			0
$193$			139.702i			0.723844i	0.932208	+	0.361922i	$0.117879\pi$
$193$			139.702i			0.723844i	−0.932208	+	0.361922i	$0.882121\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	125.291			0.635994			0.317997	−	0.948092i	$-0.396990\pi$
$197$	125.291			0.635994			0.317997	+	0.948092i	$0.396990\pi$
$198$		0			0
$199$	−242.646			−1.21933			−0.609664	−	0.792660i	$-0.708696\pi$
$199$	−242.646			−1.21933			−0.609664	+	0.792660i	$0.708696\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−52.4818			−0.258531
$204$		0			0
$205$	251.001	−	171.984i	1.22439	−	0.838949i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	180.898i		−	0.865541i
$210$		0			0
$211$	163.529			0.775018			0.387509	−	0.921866i	$-0.373336\pi$
$211$	163.529			0.775018			0.387509	+	0.921866i	$0.373336\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	43.9734	−	30.1304i	0.204528	−	0.140141i
$216$		0			0
$217$		−	222.213i		−	1.02402i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	294.977i		−	1.33474i
$222$		0			0
$223$		−	201.029i		−	0.901477i	−0.892656	−	0.450739i	$-0.851161\pi$
$223$		−	201.029i		−	0.901477i	0.892656	−	0.450739i	$-0.148839\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−155.168			−0.683559			−0.341780	−	0.939780i	$-0.611030\pi$
$227$	−155.168			−0.683559			−0.341780	+	0.939780i	$0.611030\pi$
$228$		0			0
$229$	−310.150			−1.35437			−0.677183	−	0.735814i	$-0.736800\pi$
$229$	−310.150			−1.35437			−0.677183	+	0.735814i	$0.736800\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	375.988			1.61368			0.806842	−	0.590768i	$-0.201175\pi$
$233$	375.988			1.61368			0.806842	+	0.590768i	$0.201175\pi$
$234$		0			0
$235$	−214.251	−	312.686i	−0.911707	−	1.33058i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			308.019i			1.28878i	0.764696	+	0.644391i	$0.222889\pi$
$239$			308.019i			1.28878i	−0.764696	+	0.644391i	$0.777111\pi$
$240$		0			0
$241$	300.232			1.24577			0.622887	−	0.782312i	$-0.285959\pi$
$241$	300.232			1.24577			0.622887	+	0.782312i	$0.285959\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−21.6694	−	31.6251i	−0.0884466	−	0.129082i
$246$		0			0
$247$			422.024i			1.70860i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			153.779i			0.612666i	0.951924	+	0.306333i	$0.0991021\pi$
$251$			153.779i			0.612666i	−0.951924	+	0.306333i	$0.900898\pi$
$252$		0			0
$253$			130.879i			0.517307i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	29.5779			0.115089			0.0575446	−	0.998343i	$-0.481673\pi$
$257$	29.5779			0.115089			0.0575446	+	0.998343i	$0.481673\pi$
$258$		0			0
$259$	−393.045			−1.51755
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−258.391			−0.982475			−0.491237	−	0.871026i	$-0.663455\pi$
$263$	−258.391			−0.982475			−0.491237	+	0.871026i	$0.663455\pi$
$264$		0			0
$265$	−133.745	−	195.192i	−0.504697	−	0.736574i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	443.969i		−	1.65044i	−0.564810	−	0.825221i	$-0.691051\pi$
$269$		−	443.969i		−	1.65044i	0.564810	−	0.825221i	$-0.308949\pi$
$270$		0			0
$271$	233.582			0.861928			0.430964	−	0.902369i	$-0.358174\pi$
$271$	233.582			0.861928			0.430964	+	0.902369i	$0.358174\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−155.589	−	60.2318i	−0.565779	−	0.219025i
$276$		0			0
$277$		−	199.108i		−	0.718802i	−0.933183	−	0.359401i	$-0.882981\pi$
$277$		−	199.108i		−	0.718802i	0.933183	−	0.359401i	$-0.117019\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			173.925i			0.618949i	0.950908	+	0.309474i	$0.100153\pi$
$281$			173.925i			0.618949i	−0.950908	+	0.309474i	$0.899847\pi$
$282$		0			0
$283$		−	221.755i		−	0.783587i	−0.920053	−	0.391793i	$-0.871855\pi$
$283$		−	221.755i		−	0.783587i	0.920053	−	0.391793i	$-0.128145\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−458.094			−1.59615
$288$		0			0
$289$	69.9596			0.242075
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−100.496			−0.342990			−0.171495	−	0.985185i	$-0.554860\pi$
$293$	−100.496			−0.342990			−0.171495	+	0.985185i	$0.554860\pi$
$294$		0			0
$295$	−100.725	+	69.0165i	−0.341442	+	0.233954i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	305.331i		−	1.02118i
$300$		0			0
$301$	−80.2547			−0.266627
$302$		0			0
$303$		0			0
$304$		0			0
$305$	115.052	+	167.912i	0.377221	+	0.550530i
$306$		0			0
$307$			214.703i			0.699360i	0.936869	+	0.349680i	$0.113710\pi$
$307$			214.703i			0.699360i	−0.936869	+	0.349680i	$0.886290\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	347.773i		−	1.11824i	−0.829087	−	0.559120i	$-0.811139\pi$
$311$		−	347.773i		−	1.11824i	0.829087	−	0.559120i	$-0.188861\pi$
$312$		0			0
$313$			254.365i			0.812668i	0.913725	+	0.406334i	$0.133193\pi$
$313$			254.365i			0.812668i	−0.913725	+	0.406334i	$0.866807\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	417.859			1.31817			0.659083	−	0.752070i	$-0.270944\pi$
$317$	417.859			1.31817			0.659083	+	0.752070i	$0.270944\pi$
$318$		0			0
$319$	46.5269			0.145852
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−513.564			−1.58998
$324$		0			0
$325$	362.980	+	140.517i	1.11686	+	0.432360i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			570.675i			1.73457i
$330$		0			0
$331$	38.0345			0.114908			0.0574540	−	0.998348i	$-0.481702\pi$
$331$	38.0345			0.114908			0.0574540	+	0.998348i	$0.481702\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−242.983	+	166.491i	−0.725322	+	0.496987i
$336$		0			0
$337$			342.230i			1.01552i	0.861499	+	0.507760i	$0.169526\pi$
$337$			342.230i			1.01552i	−0.861499	+	0.507760i	$0.830474\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			196.999i			0.577711i
$342$		0			0
$343$		−	311.143i		−	0.907122i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−590.327			−1.70123			−0.850615	−	0.525788i	$-0.823770\pi$
$347$	−590.327			−1.70123			−0.850615	+	0.525788i	$0.823770\pi$
$348$		0			0
$349$	45.5246			0.130443			0.0652215	−	0.997871i	$-0.479225\pi$
$349$	45.5246			0.130443			0.0652215	+	0.997871i	$0.479225\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	215.450			0.610339			0.305170	−	0.952298i	$-0.401287\pi$
$353$	215.450			0.610339			0.305170	+	0.952298i	$0.401287\pi$
$354$		0			0
$355$	139.992	−	95.9219i	0.394344	−	0.270203i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	69.0790i		−	0.192421i	−0.995361	−	0.0962104i	$-0.969328\pi$
$359$		−	69.0790i		−	0.192421i	0.995361	−	0.0962104i	$-0.0306722\pi$
$360$		0			0
$361$	373.756			1.03534
$362$		0			0
$363$		0			0
$364$		0			0
$365$	329.234	−	225.590i	0.902012	−	0.618055i
$366$		0			0
$367$			234.990i			0.640299i	0.947367	+	0.320150i	$0.103733\pi$
$367$			234.990i			0.640299i	−0.947367	+	0.320150i	$0.896267\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			356.240i			0.960216i
$372$		0			0
$373$		−	138.314i		−	0.370814i	−0.982662	−	0.185407i	$-0.940640\pi$
$373$		−	138.314i		−	0.370814i	0.982662	−	0.185407i	$-0.0593604\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−108.544			−0.287916
$378$		0			0
$379$	−412.330			−1.08794			−0.543971	−	0.839104i	$-0.683080\pi$
$379$	−412.330			−1.08794			−0.543971	+	0.839104i	$0.683080\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	448.998			1.17232			0.586159	−	0.810196i	$-0.300639\pi$
$383$	448.998			1.17232			0.586159	+	0.810196i	$0.300639\pi$
$384$		0			0
$385$	141.981	+	207.212i	0.368781	+	0.538213i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	361.573i		−	0.929494i	−0.885444	−	0.464747i	$-0.846145\pi$
$389$		−	361.573i		−	0.929494i	0.885444	−	0.464747i	$-0.153855\pi$
$390$		0			0
$391$	371.560			0.950282
$392$		0			0
$393$		0			0
$394$		0			0
$395$	243.195	+	354.928i	0.615684	+	0.898552i
$396$		0			0
$397$			673.227i			1.69579i	0.530168	+	0.847893i	$0.322129\pi$
$397$			673.227i			1.69579i	−0.530168	+	0.847893i	$0.677871\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			138.781i			0.346086i	0.984914	+	0.173043i	$0.0553600\pi$
$401$			138.781i			0.346086i	−0.984914	+	0.173043i	$0.944640\pi$
$402$		0			0
$403$		−	459.587i		−	1.14041i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	348.448			0.856137
$408$		0			0
$409$	−635.790			−1.55450			−0.777249	−	0.629193i	$-0.783386\pi$
$409$	−635.790			−1.55450			−0.777249	+	0.629193i	$0.783386\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	183.831			0.445111
$414$		0			0
$415$	432.620	+	631.382i	1.04246	+	1.52140i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			785.128i			1.87381i	0.349578	+	0.936907i	$0.386325\pi$
$419$			785.128i			1.87381i	−0.349578	+	0.936907i	$0.613675\pi$
$420$		0			0
$421$	−116.223			−0.276064			−0.138032	−	0.990428i	$-0.544078\pi$
$421$	−116.223			−0.276064			−0.138032	+	0.990428i	$0.544078\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−170.996	+	441.713i	−0.402344	+	1.03932i
$426$		0			0
$427$		−	306.451i		−	0.717684i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			606.430i			1.40703i	0.710680	+	0.703515i	$0.248387\pi$
$431$			606.430i			1.40703i	−0.710680	+	0.703515i	$0.751613\pi$
$432$		0			0
$433$			638.451i			1.47448i	0.675629	+	0.737242i	$0.263872\pi$
$433$			638.451i			1.47448i	−0.675629	+	0.737242i	$0.736128\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−531.591			−1.21646
$438$		0			0
$439$	−752.450			−1.71401			−0.857005	−	0.515309i	$-0.827677\pi$
$439$	−752.450			−1.71401			−0.857005	+	0.515309i	$0.827677\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	308.710			0.696862			0.348431	−	0.937334i	$-0.386715\pi$
$443$	308.710			0.696862			0.348431	+	0.937334i	$0.386715\pi$
$444$		0			0
$445$	343.647	−	235.465i	0.772240	−	0.529136i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			516.232i			1.14974i	0.818246	+	0.574869i	$0.194947\pi$
$449$			516.232i			1.14974i	−0.818246	+	0.574869i	$0.805053\pi$
$450$		0			0
$451$	406.117			0.900480
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−331.232	−	483.413i	−0.727983	−	1.06245i
$456$		0			0
$457$			481.476i			1.05356i	0.850002	+	0.526779i	$0.176600\pi$
$457$			481.476i			1.05356i	−0.850002	+	0.526779i	$0.823400\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	68.6414i		−	0.148897i	−0.997225	−	0.0744484i	$-0.976280\pi$
$461$		−	68.6414i		−	0.148897i	0.997225	−	0.0744484i	$-0.0237196\pi$
$462$		0			0
$463$		−	219.016i		−	0.473037i	−0.971627	−	0.236518i	$-0.923994\pi$
$463$		−	219.016i		−	0.473037i	0.971627	−	0.236518i	$-0.0760064\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−87.1086			−0.186528			−0.0932640	−	0.995641i	$-0.529730\pi$
$467$	−87.1086			−0.186528			−0.0932640	+	0.995641i	$0.529730\pi$
$468$		0			0
$469$	443.461			0.945546
$470$		0			0
$471$		0			0
$472$		0			0
$473$	71.1486			0.150420
$474$		0			0
$475$	244.644	−	631.959i	0.515041	−	1.33044i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			702.080i			1.46572i	0.680379	+	0.732860i	$0.261815\pi$
$479$			702.080i			1.46572i	−0.680379	+	0.732860i	$0.738185\pi$
$480$		0			0
$481$	−812.907			−1.69003
$482$		0			0
$483$		0			0
$484$		0			0
$485$	66.8912	−	45.8336i	0.137920	−	0.0945022i
$486$		0			0
$487$			272.680i			0.559918i	0.960012	+	0.279959i	$0.0903208\pi$
$487$			272.680i			0.559918i	−0.960012	+	0.279959i	$0.909679\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			764.337i			1.55669i	0.627834	+	0.778347i	$0.283942\pi$
$491$			764.337i			1.55669i	−0.627834	+	0.778347i	$0.716058\pi$
$492$		0			0
$493$		−	132.088i		−	0.267928i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−255.496			−0.514076
$498$		0			0
$499$	−873.142			−1.74978			−0.874891	−	0.484319i	$-0.839067\pi$
$499$	−873.142			−1.74978			−0.874891	+	0.484319i	$0.839067\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−966.017			−1.92051			−0.960255	−	0.279123i	$-0.909956\pi$
$503$	−966.017			−1.92051			−0.960255	+	0.279123i	$0.909956\pi$
$504$		0			0
$505$	523.999	−	359.042i	1.03762	−	0.710974i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			987.379i			1.93984i	0.243424	+	0.969920i	$0.421729\pi$
$509$			987.379i			1.93984i	−0.243424	+	0.969920i	$0.578271\pi$
$510$		0			0
$511$	−600.877			−1.17588
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−541.356	+	370.935i	−1.05118	+	0.720261i
$516$		0			0
$517$		−	505.923i		−	0.978575i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	952.317i		−	1.82786i	−0.405868	−	0.913932i	$-0.633031\pi$
$521$		−	952.317i		−	1.82786i	0.405868	−	0.913932i	$-0.366969\pi$
$522$		0			0
$523$		−	660.769i		−	1.26342i	−0.775205	−	0.631710i	$-0.782353\pi$
$523$		−	660.769i		−	1.26342i	0.775205	−	0.631710i	$-0.217647\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	559.275			1.06124
$528$		0			0
$529$	−144.397			−0.272962
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−947.444			−1.77757
$534$		0			0
$535$	372.836	+	544.131i	0.696890	+	1.01707i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	51.1692i		−	0.0949336i
$540$		0			0
$541$	520.946			0.962931			0.481466	−	0.876465i	$-0.340105\pi$
$541$	520.946			0.962931			0.481466	+	0.876465i	$0.340105\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−569.692	−	831.430i	−1.04531	−	1.52556i
$546$		0			0
$547$		−	788.066i		−	1.44071i	−0.693608	−	0.720353i	$-0.743980\pi$
$547$		−	788.066i		−	1.44071i	0.693608	−	0.720353i	$-0.256020\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			188.979i			0.342974i
$552$		0			0
$553$		−	647.769i		−	1.17137i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−142.413			−0.255678			−0.127839	−	0.991795i	$-0.540804\pi$
$557$	−142.413			−0.255678			−0.127839	+	0.991795i	$0.540804\pi$
$558$		0			0
$559$	−165.985			−0.296932
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−832.795			−1.47921			−0.739605	−	0.673041i	$-0.764988\pi$
$563$	−832.795			−1.47921			−0.739605	+	0.673041i	$0.764988\pi$
$564$		0			0
$565$	95.2345	+	138.989i	0.168557	+	0.245998i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	1017.03i		−	1.78740i	−0.448666	−	0.893699i	$-0.648101\pi$
$569$		−	1017.03i		−	1.78740i	0.448666	−	0.893699i	$-0.351899\pi$
$570$		0			0
$571$	356.274			0.623948			0.311974	−	0.950091i	$-0.399010\pi$
$571$	356.274			0.623948			0.311974	+	0.950091i	$0.399010\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−176.999	+	457.218i	−0.307824	+	0.795162i
$576$		0			0
$577$			269.258i			0.466651i	0.972399	+	0.233326i	$0.0749608\pi$
$577$			269.258i			0.466651i	−0.972399	+	0.233326i	$0.925039\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	1152.32i		−	1.98334i
$582$		0			0
$583$		−	315.819i		−	0.541714i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−716.019			−1.21979			−0.609897	−	0.792481i	$-0.708789\pi$
$587$	−716.019			−1.21979			−0.609897	+	0.792481i	$0.708789\pi$
$588$		0			0
$589$	−800.155			−1.35850
$590$		0			0
$591$		0			0
$592$		0			0
$593$	680.538			1.14762			0.573809	−	0.818989i	$-0.305465\pi$
$593$	680.538			1.14762			0.573809	+	0.818989i	$0.305465\pi$
$594$		0			0
$595$	588.268	−	403.079i	0.988687	−	0.677444i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			1006.72i			1.68066i	0.542073	+	0.840332i	$0.317640\pi$
$599$			1006.72i			1.68066i	−0.542073	+	0.840332i	$0.682360\pi$
$600$		0			0
$601$	165.880			0.276007			0.138004	−	0.990432i	$-0.455931\pi$
$601$	165.880			0.276007			0.138004	+	0.990432i	$0.455931\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	216.098	+	315.381i	0.357186	+	0.521291i
$606$		0			0
$607$			201.754i			0.332379i	0.986094	+	0.166190i	$0.0531464\pi$
$607$			201.754i			0.332379i	−0.986094	+	0.166190i	$0.946854\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			1180.29i			1.93173i
$612$		0			0
$613$			741.425i			1.20950i	0.796415	+	0.604751i	$0.206727\pi$
$613$			741.425i			1.20950i	−0.796415	+	0.604751i	$0.793273\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	695.522			1.12726			0.563632	−	0.826026i	$-0.309404\pi$
$617$	695.522			1.12726			0.563632	+	0.826026i	$0.309404\pi$
$618$		0			0
$619$	−465.817			−0.752532			−0.376266	−	0.926512i	$-0.622792\pi$
$619$	−465.817			−0.752532			−0.376266	+	0.926512i	$0.622792\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−627.181			−1.00671
$624$		0			0
$625$	−462.086	−	420.834i	−0.739338	−	0.673334i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	989.233i		−	1.57271i
$630$		0			0
$631$	340.458			0.539554			0.269777	−	0.962923i	$-0.413050\pi$
$631$	340.458			0.539554			0.269777	+	0.962923i	$0.413050\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	487.040	−	333.718i	0.766992	−	0.525540i
$636$		0			0
$637$			119.374i			0.187401i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			1060.95i			1.65515i	0.561359	+	0.827573i	$0.310279\pi$
$641$			1060.95i			1.65515i	−0.561359	+	0.827573i	$0.689721\pi$
$642$		0			0
$643$			969.459i			1.50771i	0.657039	+	0.753856i	$0.271809\pi$
$643$			969.459i			1.50771i	−0.657039	+	0.753856i	$0.728191\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	761.354			1.17674			0.588372	−	0.808590i	$-0.299769\pi$
$647$	761.354			1.17674			0.588372	+	0.808590i	$0.299769\pi$
$648$		0			0
$649$	−162.973			−0.251113
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−363.354			−0.556438			−0.278219	−	0.960518i	$-0.589744\pi$
$653$	−363.354			−0.556438			−0.278219	+	0.960518i	$0.589744\pi$
$654$		0			0
$655$	−115.751	+	79.3123i	−0.176720	+	0.121088i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			283.185i			0.429720i	0.976645	+	0.214860i	$0.0689295\pi$
$659$			283.185i			0.429720i	−0.976645	+	0.214860i	$0.931070\pi$
$660$		0			0
$661$	−703.455			−1.06423			−0.532115	−	0.846672i	$-0.678602\pi$
$661$	−703.455			−1.06423			−0.532115	+	0.846672i	$0.678602\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−841.636	+	576.685i	−1.26562	+	0.867196i
$666$		0			0
$667$		−	136.725i		−	0.204985i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			271.679i			0.404887i
$672$		0			0
$673$		−	450.802i		−	0.669839i	−0.942247	−	0.334920i	$-0.891291\pi$
$673$		−	450.802i		−	0.669839i	0.942247	−	0.334920i	$-0.108709\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−292.845			−0.432563			−0.216281	−	0.976331i	$-0.569393\pi$
$677$	−292.845			−0.432563			−0.216281	+	0.976331i	$0.569393\pi$
$678$		0			0
$679$	−122.081			−0.179796
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−161.354			−0.236243			−0.118121	−	0.992999i	$-0.537687\pi$
$683$	−161.354			−0.236243			−0.118121	+	0.992999i	$0.537687\pi$
$684$		0			0
$685$	304.583	+	444.520i	0.444647	+	0.648934i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			736.786i			1.06936i
$690$		0			0
$691$	611.834			0.885432			0.442716	−	0.896662i	$-0.354015\pi$
$691$	611.834			0.885432			0.442716	+	0.896662i	$0.354015\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	483.120	+	705.083i	0.695136	+	1.01451i
$696$		0			0
$697$		−	1152.95i		−	1.65416i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			702.946i			1.00278i	0.865222	+	0.501388i	$0.167177\pi$
$701$			702.946i			1.00278i	−0.865222	+	0.501388i	$0.832823\pi$
$702$		0			0
$703$			1415.30i			2.01322i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−956.336			−1.35267
$708$		0			0
$709$	−361.678			−0.510124			−0.255062	−	0.966925i	$-0.582096\pi$
$709$	−361.678			−0.510124			−0.255062	+	0.966925i	$0.582096\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	578.907			0.811932
$714$		0			0
$715$	293.649	+	428.562i	0.410698	+	0.599387i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	271.814i		−	0.378044i	−0.981973	−	0.189022i	$-0.939468\pi$
$719$		−	271.814i		−	0.378044i	0.981973	−	0.189022i	$-0.0605318\pi$
$720$		0			0
$721$	988.014			1.37034
$722$		0			0
$723$		0			0
$724$		0			0
$725$	162.539	+	62.9224i	0.224192	+	0.0867896i
$726$		0			0
$727$		−	754.003i		−	1.03714i	−0.855034	−	0.518572i	$-0.826464\pi$
$727$		−	754.003i		−	1.03714i	0.855034	−	0.518572i	$-0.173536\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	201.989i		−	0.276318i
$732$		0			0
$733$			16.9221i			0.0230860i	0.999933	+	0.0115430i	$0.00367434\pi$
$733$			16.9221i			0.0230860i	−0.999933	+	0.0115430i	$0.996326\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−393.144			−0.533438
$738$		0			0
$739$	−874.635			−1.18354			−0.591769	−	0.806107i	$-0.701570\pi$
$739$	−874.635			−1.18354			−0.591769	+	0.806107i	$0.701570\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	272.840			0.367213			0.183607	−	0.983000i	$-0.441223\pi$
$743$	272.840			0.367213			0.183607	+	0.983000i	$0.441223\pi$
$744$		0			0
$745$	336.969	−	230.890i	0.452307	−	0.309919i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	993.079i		−	1.32587i
$750$		0			0
$751$	1277.28			1.70077			0.850386	−	0.526160i	$-0.176369\pi$
$751$	1277.28			1.70077			0.850386	+	0.526160i	$0.176369\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	26.0378	+	38.0006i	0.0344872	+	0.0503319i
$756$		0			0
$757$		−	1087.23i		−	1.43624i	−0.695921	−	0.718118i	$-0.745004\pi$
$757$		−	1087.23i		−	1.43624i	0.695921	−	0.718118i	$-0.254996\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	1017.04i		−	1.33645i	−0.743958	−	0.668226i	$-0.767054\pi$
$761$		−	1017.04i		−	1.33645i	0.743958	−	0.668226i	$-0.232946\pi$
$762$		0			0
$763$			1517.42i			1.98876i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	380.205			0.495704
$768$		0			0
$769$	−205.301			−0.266971			−0.133486	−	0.991051i	$-0.542617\pi$
$769$	−205.301			−0.266971			−0.133486	+	0.991051i	$0.542617\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	254.726			0.329530			0.164765	−	0.986333i	$-0.447313\pi$
$773$	254.726			0.329530			0.164765	+	0.986333i	$0.447313\pi$
$774$		0			0
$775$	−266.420	+	688.208i	−0.343768	+	0.888010i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			1649.53i			2.11750i
$780$		0			0
$781$	226.506			0.290020
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1189.40	−	814.972i	1.51516	−	1.03818i
$786$		0			0
$787$		−	66.9905i		−	0.0851214i	−0.999094	−	0.0425607i	$-0.986448\pi$
$787$		−	66.9905i		−	0.0851214i	0.999094	−	0.0425607i	$-0.0135516\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	253.665i		−	0.320688i
$792$		0			0
$793$		−	633.811i		−	0.799257i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	642.644			0.806329			0.403164	−	0.915128i	$-0.367910\pi$
$797$	642.644			0.806329			0.403164	+	0.915128i	$0.367910\pi$
$798$		0			0
$799$	−1436.30			−1.79762
$800$		0			0
$801$		0			0
$802$		0			0
$803$	532.698			0.663385
$804$		0			0
$805$	608.918	−	417.228i	0.756420	−	0.518296i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	479.966i		−	0.593283i	−0.954989	−	0.296642i	$-0.904133\pi$
$809$		−	479.966i		−	0.593283i	0.954989	−	0.296642i	$-0.0958667\pi$
$810$		0			0
$811$	−478.009			−0.589407			−0.294703	−	0.955589i	$-0.595221\pi$
$811$	−478.009			−0.589407			−0.294703	+	0.955589i	$0.595221\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	285.042	−	195.309i	0.349745	−	0.239643i
$816$		0			0
$817$			288.985i			0.353715i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			894.509i			1.08954i	0.838587	+	0.544768i	$0.183382\pi$
$821$			894.509i			1.08954i	−0.838587	+	0.544768i	$0.816618\pi$
$822$		0			0
$823$		−	376.036i		−	0.456909i	−0.973555	−	0.228455i	$-0.926633\pi$
$823$		−	376.036i		−	0.456909i	0.973555	−	0.228455i	$-0.0733672\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−363.894			−0.440017			−0.220008	−	0.975498i	$-0.570608\pi$
$827$	−363.894			−0.440017			−0.220008	+	0.975498i	$0.570608\pi$
$828$		0			0
$829$	−398.314			−0.480475			−0.240238	−	0.970714i	$-0.577225\pi$
$829$	−398.314			−0.480475			−0.240238	+	0.970714i	$0.577225\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−145.268			−0.174391
$834$		0			0
$835$	551.743	+	805.234i	0.660770	+	0.964352i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	1499.60i		−	1.78736i	−0.448702	−	0.893682i	$-0.648113\pi$
$839$		−	1499.60i		−	1.78736i	0.448702	−	0.893682i	$-0.351887\pi$
$840$		0			0
$841$	792.395			0.942205
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−207.438	−	302.743i	−0.245489	−	0.358276i
$846$		0			0
$847$		−	575.593i		−	0.679567i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	1023.96i		−	1.20324i
$852$		0			0
$853$			1524.29i			1.78697i	0.449093	+	0.893485i	$0.351747\pi$
$853$			1524.29i			1.78697i	−0.449093	+	0.893485i	$0.648253\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−931.475			−1.08690			−0.543451	−	0.839441i	$-0.682883\pi$
$857$	−931.475			−1.08690			−0.543451	+	0.839441i	$0.682883\pi$
$858$		0			0
$859$	−102.105			−0.118865			−0.0594325	−	0.998232i	$-0.518929\pi$
$859$	−102.105			−0.118865			−0.0594325	+	0.998232i	$0.518929\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	971.646			1.12589			0.562947	−	0.826493i	$-0.309668\pi$
$863$	971.646			1.12589			0.562947	+	0.826493i	$0.309668\pi$
$864$		0			0
$865$	−167.815	−	244.915i	−0.194006	−	0.283139i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			574.270i			0.660840i
$870$		0			0
$871$	917.179			1.05302
$872$		0			0
$873$		0			0
$874$		0			0
$875$	215.772	+	915.899i	0.246596	+	1.04674i
$876$		0			0
$877$			771.593i			0.879810i	0.898044	+	0.439905i	$0.144988\pi$
$877$			771.593i			0.879810i	−0.898044	+	0.439905i	$0.855012\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			279.972i			0.317789i	0.987296	+	0.158895i	$0.0507930\pi$
$881$			279.972i			0.317789i	−0.987296	+	0.158895i	$0.949207\pi$
$882$		0			0
$883$			75.4516i			0.0854492i	0.999087	+	0.0427246i	$0.0136038\pi$
$883$			75.4516i			0.0854492i	−0.999087	+	0.0427246i	$0.986396\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−669.288			−0.754553			−0.377276	−	0.926101i	$-0.623139\pi$
$887$	−669.288			−0.754553			−0.377276	+	0.926101i	$0.623139\pi$
$888$		0			0
$889$	−888.884			−0.999869
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2054.92			2.30114
$894$		0			0
$895$	338.888	−	232.205i	0.378646	−	0.259447i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	205.800i		−	0.228921i
$900$		0			0
$901$	−896.601			−0.995117
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−684.276	−	998.658i	−0.756106	−	1.10349i
$906$		0			0
$907$		−	459.134i		−	0.506212i	−0.967439	−	0.253106i	$-0.918548\pi$
$907$		−	459.134i		−	0.506212i	0.967439	−	0.253106i	$-0.0814521\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	670.249i		−	0.735729i	−0.929879	−	0.367864i	$-0.880089\pi$
$911$		−	670.249i		−	0.735729i	0.929879	−	0.367864i	$-0.119911\pi$
$912$		0			0
$913$			1021.57i			1.11892i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	211.255			0.230376
$918$		0			0
$919$	−22.6103			−0.0246032			−0.0123016	−	0.999924i	$-0.503916\pi$
$919$	−22.6103			−0.0246032			−0.0123016	+	0.999924i	$0.503916\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−528.424			−0.572507
$924$		0			0
$925$	1217.29	+	471.237i	1.31598	+	0.509445i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	944.272i		−	1.01644i	−0.861228	−	0.508220i	$-0.830304\pi$
$929$		−	944.272i		−	1.01644i	0.861228	−	0.508220i	$-0.169696\pi$
$930$		0			0
$931$	207.835			0.223238
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−521.520	+	357.343i	−0.557776	+	0.382185i
$936$		0			0
$937$			1037.45i			1.10721i	0.832781	+	0.553603i	$0.186747\pi$
$937$			1037.45i			1.10721i	−0.832781	+	0.553603i	$0.813253\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			410.283i			0.436007i	0.975948	+	0.218004i	$0.0699544\pi$
$941$			410.283i			0.436007i	−0.975948	+	0.218004i	$0.930046\pi$
$942$		0			0
$943$		−	1193.42i		−	1.26556i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−753.302			−0.795462			−0.397731	−	0.917502i	$-0.630202\pi$
$947$	−753.302			−0.795462			−0.397731	+	0.917502i	$0.630202\pi$
$948$		0			0
$949$	−1242.75			−1.30954
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−661.356			−0.693973			−0.346986	−	0.937870i	$-0.612795\pi$
$953$	−661.356			−0.693973			−0.346986	+	0.937870i	$0.612795\pi$
$954$		0			0
$955$	−113.021	+	77.4418i	−0.118347	+	0.0810909i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	811.282i		−	0.845966i
$960$		0			0
$961$	−89.6241			−0.0932613
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−576.221	+	394.824i	−0.597120	+	0.409144i
$966$		0			0
$967$			1584.38i			1.63845i	0.573470	+	0.819227i	$0.305597\pi$
$967$			1584.38i			1.63845i	−0.573470	+	0.819227i	$0.694403\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	982.405i		−	1.01175i	−0.862608	−	0.505873i	$-0.831171\pi$
$971$		−	982.405i		−	1.01175i	0.862608	−	0.505873i	$-0.168829\pi$
$972$		0			0
$973$		−	1286.83i		−	1.32254i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1891.35			1.93588			0.967939	−	0.251185i	$-0.0808201\pi$
$977$	1891.35			1.93588			0.967939	+	0.251185i	$0.0808201\pi$
$978$		0			0
$979$	556.017			0.567944
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1431.06			−1.45581			−0.727906	−	0.685677i	$-0.759506\pi$
$983$	−1431.06			−1.45581			−0.727906	+	0.685677i	$0.759506\pi$
$984$		0			0
$985$	354.095	+	516.780i	0.359488	+	0.524650i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	209.079i		−	0.211404i
$990$		0			0
$991$	159.789			0.161240			0.0806200	−	0.996745i	$-0.474310\pi$
$991$	159.789			0.161240			0.0806200	+	0.996745i	$0.474310\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−685.764	−	1000.83i	−0.689210	−	1.00586i
$996$		0			0
$997$		−	651.900i		−	0.653861i	−0.945048	−	0.326931i	$-0.893986\pi$
$997$		−	651.900i		−	0.653861i	0.945048	−	0.326931i	$-0.106014\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.3.b.a.809.18	yes	24
3.2	odd	2	inner	1620.3.b.a.809.7	✓	24
5.4	even	2	inner	1620.3.b.a.809.8	yes	24
9.2	odd	6		1620.3.t.f.1349.23		48
9.4	even	3		1620.3.t.f.269.15		48
9.5	odd	6		1620.3.t.f.269.10		48
9.7	even	3		1620.3.t.f.1349.2		48
15.14	odd	2	inner	1620.3.b.a.809.17	yes	24
45.4	even	6		1620.3.t.f.269.23		48
45.14	odd	6		1620.3.t.f.269.2		48
45.29	odd	6		1620.3.t.f.1349.15		48
45.34	even	6		1620.3.t.f.1349.10		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.3.b.a.809.7	✓	24	3.2	odd	2	inner
1620.3.b.a.809.8	yes	24	5.4	even	2	inner
1620.3.b.a.809.17	yes	24	15.14	odd	2	inner
1620.3.b.a.809.18	yes	24	1.1	even	1	trivial
1620.3.t.f.269.2		48	45.14	odd	6
1620.3.t.f.269.10		48	9.5	odd	6
1620.3.t.f.269.15		48	9.4	even	3
1620.3.t.f.269.23		48	45.4	even	6
1620.3.t.f.1349.2		48	9.7	even	3
1620.3.t.f.1349.10		48	45.34	even	6
1620.3.t.f.1349.15		48	45.29	odd	6
1620.3.t.f.1349.23		48	9.2	odd	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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1620.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(2.82619 + 4.12464i) q^{5} -7.52777i q^{7} +O(q^{10})\)	\(q+(2.82619 + 4.12464i) q^{5} -7.52777i q^{7} +6.67363i q^{11} -15.5692i q^{13} +18.9462 q^{17} -27.1064 q^{19} +19.6113 q^{23} +(-9.02534 + 23.3140i) q^{25} -6.97175i q^{29} +29.5191 q^{31} +(31.0494 - 21.2749i) q^{35} -52.2126i q^{37} -60.8539i q^{41} -10.6611i q^{43} -75.8093 q^{47} -7.66737 q^{49} -47.3234 q^{53} +(-27.5263 + 18.8609i) q^{55} +24.4204i q^{59} +40.7094 q^{61} +(64.2172 - 44.0013i) q^{65} +58.9100i q^{67} -33.9404i q^{71} -79.8213i q^{73} +50.2376 q^{77} +86.0506 q^{79} +153.076 q^{83} +(53.5456 + 78.1464i) q^{85} -83.3156i q^{89} -117.201 q^{91} +(-76.6077 - 111.804i) q^{95} -16.2175i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 48 q^{25} - 288 q^{49} - 36 q^{55} + 120 q^{61} + 480 q^{79} - 24 q^{85} + 48 q^{91}+O(q^{100}) \) 24 * q - 48 * q^25 - 288 * q^49 - 36 * q^55 + 120 * q^61 + 480 * q^79 - 24 * q^85 + 48 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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