LMFDB - Embedded newform 1620.3.g.a.161.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,3,Mod(161,1620)]

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, 0]))

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

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

chi := DirichletCharacter("1620.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1620.g (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:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 25 $ x^4 - 5*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.4
Root			$-1.93649 - 1.11803i$ of defining polynomial
Character	$\chi$	$=$	1620.161
Dual form			1620.3.g.a.161.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607i q^{5} +3.74597 q^{7} +O(q^{10})$	$q+2.23607i q^{5} +3.74597 q^{7} -11.6844i q^{11} -2.25403 q^{13} +11.6844i q^{17} -26.7460 q^{19} +19.9046i q^{23} -5.00000 q^{25} -44.1533i q^{29} -52.2218 q^{31} +8.37624i q^{35} +14.0000 q^{37} +25.9808i q^{41} +41.9839 q^{43} +45.4733i q^{47} -34.9677 q^{49} -10.8323i q^{53} +26.1270 q^{55} +36.8131i q^{59} -45.2379 q^{61} -5.04017i q^{65} -99.9516 q^{67} +102.603i q^{71} -13.7621 q^{73} -43.7692i q^{77} +56.7621 q^{79} -90.0666i q^{83} -26.1270 q^{85} -95.2349i q^{89} -8.44353 q^{91} -59.8058i q^{95} -101.698 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{7}+O(q^{10}) $ 4 * q - 16 * q^7	$ 4 q - 16 q^{7} - 40 q^{13} - 76 q^{19} - 20 q^{25} + 8 q^{31} + 56 q^{37} + 44 q^{43} + 108 q^{49} + 120 q^{55} - 88 q^{61} - 28 q^{67} - 148 q^{73} + 320 q^{79} - 120 q^{85} + 400 q^{91} - 4 q^{97}+O(q^{100}) $ 4 * q - 16 * q^7 - 40 * q^13 - 76 * q^19 - 20 * q^25 + 8 * q^31 + 56 * q^37 + 44 * q^43 + 108 * q^49 + 120 * q^55 - 88 * q^61 - 28 * q^67 - 148 * q^73 + 320 * q^79 - 120 * q^85 + 400 * q^91 - 4 * q^97

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
$5$	2.23607i	0.447214i
$5$	2.23607i	0.447214i
$6$	0	0
$7$	3.74597	0.535138	0.267569	−	0.963539i	$-0.413780\pi$
$7$	3.74597	0.535138	0.267569	+	0.963539i	$0.413780\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 11.6844i	− 1.06221i	−0.847305	−	0.531107i	$-0.821776\pi$
$11$	− 11.6844i	− 1.06221i	0.847305	−	0.531107i	$-0.178224\pi$
$12$	0	0
$13$	−2.25403	−0.173387	−0.0866936	−	0.996235i	$-0.527630\pi$
$13$	−2.25403	−0.173387	−0.0866936	+	0.996235i	$0.527630\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	11.6844i	0.687315i	0.939095	+	0.343658i	$0.111666\pi$
$17$	11.6844i	0.687315i	−0.939095	+	0.343658i	$0.888334\pi$
$18$	0	0
$19$	−26.7460	−1.40768	−0.703841	−	0.710357i	$-0.748533\pi$
$19$	−26.7460	−1.40768	−0.703841	+	0.710357i	$0.748533\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	19.9046i	0.865418i	0.901534	+	0.432709i	$0.142442\pi$
$23$	19.9046i	0.865418i	−0.901534	+	0.432709i	$0.857558\pi$
$24$	0	0
$25$	−5.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 44.1533i	− 1.52253i	−0.648442	−	0.761264i	$-0.724579\pi$
$29$	− 44.1533i	− 1.52253i	0.648442	−	0.761264i	$-0.275421\pi$
$30$	0	0
$31$	−52.2218	−1.68457	−0.842287	−	0.539030i	$-0.818791\pi$
$31$	−52.2218	−1.68457	−0.842287	+	0.539030i	$0.818791\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	8.37624i	0.239321i
$36$	0	0
$37$	14.0000	0.378378	0.189189	−	0.981941i	$-0.439414\pi$
$37$	14.0000	0.378378	0.189189	+	0.981941i	$0.439414\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	25.9808i	0.633677i	0.948479	+	0.316839i	$0.102621\pi$
$41$	25.9808i	0.633677i	−0.948479	+	0.316839i	$0.897379\pi$
$42$	0	0
$43$	41.9839	0.976369	0.488184	−	0.872740i	$-0.337659\pi$
$43$	41.9839	0.976369	0.488184	+	0.872740i	$0.337659\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	45.4733i	0.967517i	0.875201	+	0.483759i	$0.160729\pi$
$47$	45.4733i	0.967517i	−0.875201	+	0.483759i	$0.839271\pi$
$48$	0	0
$49$	−34.9677	−0.713627
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 10.8323i	− 0.204383i	−0.994765	−	0.102192i	$-0.967415\pi$
$53$	− 10.8323i	− 0.204383i	0.994765	−	0.102192i	$-0.0325855\pi$
$54$	0	0
$55$	26.1270	0.475037
$56$	0	0
$57$	0	0
$58$	0	0
$59$	36.8131i	0.623950i	0.950090	+	0.311975i	$0.100991\pi$
$59$	36.8131i	0.623950i	−0.950090	+	0.311975i	$0.899009\pi$
$60$	0	0
$61$	−45.2379	−0.741605	−0.370802	−	0.928712i	$-0.620917\pi$
$61$	−45.2379	−0.741605	−0.370802	+	0.928712i	$0.620917\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 5.04017i	− 0.0775411i
$66$	0	0
$67$	−99.9516	−1.49181	−0.745907	−	0.666050i	$-0.767984\pi$
$67$	−99.9516	−1.49181	−0.745907	+	0.666050i	$0.767984\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	102.603i	1.44511i	0.691312	+	0.722557i	$0.257033\pi$
$71$	102.603i	1.44511i	−0.691312	+	0.722557i	$0.742967\pi$
$72$	0	0
$73$	−13.7621	−0.188522	−0.0942610	−	0.995548i	$-0.530049\pi$
$73$	−13.7621	−0.188522	−0.0942610	+	0.995548i	$0.530049\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 43.7692i	− 0.568431i
$78$	0	0
$79$	56.7621	0.718508	0.359254	−	0.933240i	$-0.383031\pi$
$79$	56.7621	0.718508	0.359254	+	0.933240i	$0.383031\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 90.0666i	− 1.08514i	−0.840011	−	0.542570i	$-0.817451\pi$
$83$	− 90.0666i	− 1.08514i	0.840011	−	0.542570i	$-0.182549\pi$
$84$	0	0
$85$	−26.1270	−0.307377
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 95.2349i	− 1.07005i	−0.844835	−	0.535027i	$-0.820301\pi$
$89$	− 95.2349i	− 1.07005i	0.844835	−	0.535027i	$-0.179699\pi$
$90$	0	0
$91$	−8.44353	−0.0927861
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 59.8058i	− 0.629535i
$96$	0	0
$97$	−101.698	−1.04843	−0.524214	−	0.851586i	$-0.675641\pi$
$97$	−101.698	−1.04843	−0.524214	+	0.851586i	$0.675641\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 101.339i	− 1.00336i	−0.865054	−	0.501678i	$-0.832716\pi$
$101$	− 101.339i	− 1.00336i	0.865054	−	0.501678i	$-0.167284\pi$
$102$	0	0
$103$	16.2863	0.158119	0.0790597	−	0.996870i	$-0.474808\pi$
$103$	16.2863	0.158119	0.0790597	+	0.996870i	$0.474808\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 84.8705i	− 0.793182i	−0.917995	−	0.396591i	$-0.870193\pi$
$107$	− 84.8705i	− 0.793182i	0.917995	−	0.396591i	$-0.129807\pi$
$108$	0	0
$109$	−161.968	−1.48594	−0.742971	−	0.669323i	$-0.766584\pi$
$109$	−161.968	−1.48594	−0.742971	+	0.669323i	$0.766584\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 6.92820i	− 0.0613115i	−0.999530	−	0.0306558i	$-0.990240\pi$
$113$	− 6.92820i	− 0.0613115i	0.999530	−	0.0306558i	$-0.00975956\pi$
$114$	0	0
$115$	−44.5081	−0.387027
$116$	0	0
$117$	0	0
$118$	0	0
$119$	43.7692i	0.367809i
$120$	0	0
$121$	−15.5242	−0.128299
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 11.1803i	− 0.0894427i
$126$	0	0
$127$	−225.903	−1.77877	−0.889383	−	0.457163i	$-0.848865\pi$
$127$	−225.903	−1.77877	−0.889383	+	0.457163i	$0.848865\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	94.3549i	0.720266i	0.932901	+	0.360133i	$0.117269\pi$
$131$	94.3549i	0.720266i	−0.932901	+	0.360133i	$0.882731\pi$
$132$	0	0
$133$	−100.190	−0.753305
$134$	0	0
$135$	0	0
$136$	0	0
$137$	207.378i	1.51371i	0.653583	+	0.756855i	$0.273265\pi$
$137$	207.378i	1.51371i	−0.653583	+	0.756855i	$0.726735\pi$
$138$	0	0
$139$	32.7137	0.235350	0.117675	−	0.993052i	$-0.462456\pi$
$139$	32.7137	0.235350	0.117675	+	0.993052i	$0.462456\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	26.3369i	0.184174i
$144$	0	0
$145$	98.7298	0.680895
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 21.1687i	− 0.142072i	−0.997474	−	0.0710360i	$-0.977369\pi$
$149$	− 21.1687i	− 0.142072i	0.997474	−	0.0710360i	$-0.0226305\pi$
$150$	0	0
$151$	−275.427	−1.82402	−0.912011	−	0.410165i	$-0.865471\pi$
$151$	−275.427	−1.82402	−0.912011	+	0.410165i	$0.865471\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 116.771i	− 0.753364i
$156$	0	0
$157$	179.935	1.14609	0.573043	−	0.819525i	$-0.305763\pi$
$157$	179.935	1.14609	0.573043	+	0.819525i	$0.305763\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	74.5620i	0.463118i
$162$	0	0
$163$	−264.411	−1.62216	−0.811078	−	0.584939i	$-0.801119\pi$
$163$	−264.411	−1.62216	−0.811078	+	0.584939i	$0.801119\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 172.653i	− 1.03385i	−0.856030	−	0.516926i	$-0.827076\pi$
$167$	− 172.653i	− 1.03385i	0.856030	−	0.516926i	$-0.172924\pi$
$168$	0	0
$169$	−163.919	−0.969937
$170$	0	0
$171$	0	0
$172$	0	0
$173$	294.337i	1.70137i	0.525676	+	0.850685i	$0.323812\pi$
$173$	294.337i	1.70137i	−0.525676	+	0.850685i	$0.676188\pi$
$174$	0	0
$175$	−18.7298	−0.107028
$176$	0	0
$177$	0	0
$178$	0	0
$179$	64.1138i	0.358178i	0.983833	+	0.179089i	$0.0573150\pi$
$179$	64.1138i	0.358178i	−0.983833	+	0.179089i	$0.942685\pi$
$180$	0	0
$181$	291.206	1.60887	0.804435	−	0.594040i	$-0.202468\pi$
$181$	291.206	1.60887	0.804435	+	0.594040i	$0.202468\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	31.3050i	0.169216i
$186$	0	0
$187$	136.524	0.730076
$188$	0	0
$189$	0	0
$190$	0	0
$191$	15.9446i	0.0834797i	0.999129	+	0.0417399i	$0.0132901\pi$
$191$	15.9446i	0.0834797i	−0.999129	+	0.0417399i	$0.986710\pi$
$192$	0	0
$193$	−244.714	−1.26795	−0.633973	−	0.773355i	$-0.718577\pi$
$193$	−244.714	−1.26795	−0.633973	+	0.773355i	$0.718577\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 210.374i	− 1.06789i	−0.845519	−	0.533945i	$-0.820709\pi$
$197$	− 210.374i	− 1.06789i	0.845519	−	0.533945i	$-0.179291\pi$
$198$	0	0
$199$	−102.730	−0.516230	−0.258115	−	0.966114i	$-0.583101\pi$
$199$	−102.730	−0.516230	−0.258115	+	0.966114i	$0.583101\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 165.397i	− 0.814763i
$204$	0	0
$205$	−58.0948	−0.283389
$206$	0	0
$207$	0	0
$208$	0	0
$209$	312.509i	1.49526i
$210$	0	0
$211$	−227.968	−1.08042	−0.540208	−	0.841532i	$-0.681654\pi$
$211$	−227.968	−1.08042	−0.540208	+	0.841532i	$0.681654\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	93.8788i	0.436645i
$216$	0	0
$217$	−195.621	−0.901479
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 26.3369i	− 0.119172i
$222$	0	0
$223$	−279.363	−1.25275	−0.626374	−	0.779523i	$-0.715462\pi$
$223$	−279.363	−1.25275	−0.626374	+	0.779523i	$0.715462\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	248.563i	1.09499i	0.836808	+	0.547496i	$0.184419\pi$
$227$	248.563i	1.09499i	−0.836808	+	0.547496i	$0.815581\pi$
$228$	0	0
$229$	157.778	0.688988	0.344494	−	0.938789i	$-0.388051\pi$
$229$	157.778	0.688988	0.344494	+	0.938789i	$0.388051\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 332.854i	− 1.42856i	−0.699861	−	0.714279i	$-0.746755\pi$
$233$	− 332.854i	− 1.42856i	0.699861	−	0.714279i	$-0.253245\pi$
$234$	0	0
$235$	−101.681	−0.432687
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 239.847i	− 1.00354i	−0.865000	−	0.501772i	$-0.832682\pi$
$239$	− 239.847i	− 1.00354i	0.865000	−	0.501772i	$-0.167318\pi$
$240$	0	0
$241$	−189.508	−0.786341	−0.393170	−	0.919466i	$-0.628622\pi$
$241$	−189.508	−0.786341	−0.393170	+	0.919466i	$0.628622\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 78.1902i	− 0.319144i
$246$	0	0
$247$	60.2863	0.244074
$248$	0	0
$249$	0	0
$250$	0	0
$251$	447.114i	1.78133i	0.454661	+	0.890664i	$0.349760\pi$
$251$	447.114i	1.78133i	−0.454661	+	0.890664i	$0.650240\pi$
$252$	0	0
$253$	232.573	0.919259
$254$	0	0
$255$	0	0
$256$	0	0
$257$	447.225i	1.74018i	0.492896	+	0.870088i	$0.335938\pi$
$257$	447.225i	1.74018i	−0.492896	+	0.870088i	$0.664062\pi$
$258$	0	0
$259$	52.4435	0.202485
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 354.547i	− 1.34809i	−0.738692	−	0.674043i	$-0.764556\pi$
$263$	− 354.547i	− 1.34809i	0.738692	−	0.674043i	$-0.235444\pi$
$264$	0	0
$265$	24.2218	0.0914029
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 227.367i	− 0.845229i	−0.906310	−	0.422614i	$-0.861112\pi$
$269$	− 227.367i	− 0.845229i	0.906310	−	0.422614i	$-0.138888\pi$
$270$	0	0
$271$	228.573	0.843441	0.421721	−	0.906726i	$-0.361426\pi$
$271$	228.573	0.843441	0.421721	+	0.906726i	$0.361426\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	58.4218i	0.212443i
$276$	0	0
$277$	440.444	1.59005	0.795024	−	0.606577i	$-0.207458\pi$
$277$	440.444	1.59005	0.795024	+	0.606577i	$0.207458\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 314.297i	− 1.11850i	−0.829000	−	0.559248i	$-0.811090\pi$
$281$	− 314.297i	− 1.11850i	0.829000	−	0.559248i	$-0.188910\pi$
$282$	0	0
$283$	267.968	0.946882	0.473441	−	0.880825i	$-0.343012\pi$
$283$	267.968	0.946882	0.473441	+	0.880825i	$0.343012\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	97.3231i	0.339105i
$288$	0	0
$289$	152.476	0.527598
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 310.065i	− 1.05824i	−0.848546	−	0.529121i	$-0.822522\pi$
$293$	− 310.065i	− 1.05824i	0.848546	−	0.529121i	$-0.177478\pi$
$294$	0	0
$295$	−82.3165	−0.279039
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 44.8657i	− 0.150052i
$300$	0	0
$301$	157.270	0.522492
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 101.155i	− 0.331656i
$306$	0	0
$307$	−495.952	−1.61548	−0.807739	−	0.589541i	$-0.799309\pi$
$307$	−495.952	−1.61548	−0.807739	+	0.589541i	$0.799309\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	226.047i	0.726838i	0.931626	+	0.363419i	$0.118391\pi$
$311$	226.047i	0.726838i	−0.931626	+	0.363419i	$0.881609\pi$
$312$	0	0
$313$	327.157	1.04523	0.522615	−	0.852569i	$-0.324956\pi$
$313$	327.157	1.04523	0.522615	+	0.852569i	$0.324956\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 8.19231i	− 0.0258433i	−0.999917	−	0.0129216i	$-0.995887\pi$
$317$	− 8.19231i	− 0.0258433i	0.999917	−	0.0129216i	$-0.00411320\pi$
$318$	0	0
$319$	−515.903	−1.61725
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 312.509i	− 0.967521i
$324$	0	0
$325$	11.2702	0.0346774
$326$	0	0
$327$	0	0
$328$	0	0
$329$	170.342i	0.517755i
$330$	0	0
$331$	277.806	0.839294	0.419647	−	0.907687i	$-0.362154\pi$
$331$	277.806	0.839294	0.419647	+	0.907687i	$0.362154\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 223.499i	− 0.667160i
$336$	0	0
$337$	55.8266	0.165658	0.0828288	−	0.996564i	$-0.473605\pi$
$337$	55.8266	0.165658	0.0828288	+	0.996564i	$0.473605\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	610.178i	1.78938i
$342$	0	0
$343$	−314.540	−0.917027
$344$	0	0
$345$	0	0
$346$	0	0
$347$	508.203i	1.46456i	0.681002	+	0.732281i	$0.261544\pi$
$347$	508.203i	1.46456i	−0.681002	+	0.732281i	$0.738456\pi$
$348$	0	0
$349$	138.476	0.396779	0.198389	−	0.980123i	$-0.436429\pi$
$349$	138.476	0.396779	0.198389	+	0.980123i	$0.436429\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 576.053i	− 1.63188i	−0.578137	−	0.815940i	$-0.696220\pi$
$353$	− 576.053i	− 1.63188i	0.578137	−	0.815940i	$-0.303780\pi$
$354$	0	0
$355$	−229.427	−0.646274
$356$	0	0
$357$	0	0
$358$	0	0
$359$	267.888i	0.746207i	0.927790	+	0.373103i	$0.121706\pi$
$359$	267.888i	0.746207i	−0.927790	+	0.373103i	$0.878294\pi$
$360$	0	0
$361$	354.347	0.981570
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 30.7730i	− 0.0843096i
$366$	0	0
$367$	60.4758	0.164784	0.0823921	−	0.996600i	$-0.473744\pi$
$367$	60.4758	0.164784	0.0823921	+	0.996600i	$0.473744\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 40.5774i	− 0.109373i
$372$	0	0
$373$	−216.508	−0.580451	−0.290225	−	0.956958i	$-0.593730\pi$
$373$	−216.508	−0.580451	−0.290225	+	0.956958i	$0.593730\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	99.5231i	0.263987i
$378$	0	0
$379$	285.254	0.752649	0.376325	−	0.926488i	$-0.377188\pi$
$379$	285.254	0.752649	0.376325	+	0.926488i	$0.377188\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 163.581i	− 0.427104i	−0.976932	−	0.213552i	$-0.931497\pi$
$383$	− 163.581i	− 0.427104i	0.976932	−	0.213552i	$-0.0685033\pi$
$384$	0	0
$385$	97.8709	0.254210
$386$	0	0
$387$	0	0
$388$	0	0
$389$	289.553i	0.744352i	0.928162	+	0.372176i	$0.121388\pi$
$389$	289.553i	0.744352i	−0.928162	+	0.372176i	$0.878612\pi$
$390$	0	0
$391$	−232.573	−0.594815
$392$	0	0
$393$	0	0
$394$	0	0
$395$	126.924i	0.321326i
$396$	0	0
$397$	−66.5081	−0.167527	−0.0837633	−	0.996486i	$-0.526694\pi$
$397$	−66.5081	−0.167527	−0.0837633	+	0.996486i	$0.526694\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	452.833i	1.12926i	0.825344	+	0.564630i	$0.190981\pi$
$401$	452.833i	1.12926i	−0.825344	+	0.564630i	$0.809019\pi$
$402$	0	0
$403$	117.710	0.292083
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 163.581i	− 0.401919i
$408$	0	0
$409$	300.871	0.735626	0.367813	−	0.929900i	$-0.380107\pi$
$409$	300.871	0.735626	0.367813	+	0.929900i	$0.380107\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	137.901i	0.333900i
$414$	0	0
$415$	201.395	0.485289
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 119.539i	− 0.285297i	−0.989773	−	0.142649i	$-0.954438\pi$
$419$	− 119.539i	− 0.285297i	0.989773	−	0.142649i	$-0.0455619\pi$
$420$	0	0
$421$	−162.286	−0.385478	−0.192739	−	0.981250i	$-0.561737\pi$
$421$	−162.286	−0.385478	−0.192739	+	0.981250i	$0.561737\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 58.4218i	− 0.137463i
$426$	0	0
$427$	−169.460	−0.396861
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 499.815i	− 1.15966i	−0.814736	−	0.579832i	$-0.803118\pi$
$431$	− 499.815i	− 1.15966i	0.814736	−	0.579832i	$-0.196882\pi$
$432$	0	0
$433$	739.883	1.70874	0.854368	−	0.519668i	$-0.173944\pi$
$433$	739.883	1.70874	0.854368	+	0.519668i	$0.173944\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 532.368i	− 1.21823i
$438$	0	0
$439$	248.665	0.566436	0.283218	−	0.959056i	$-0.408598\pi$
$439$	248.665	0.566436	0.283218	+	0.959056i	$0.408598\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 91.0864i	− 0.205613i	−0.994701	−	0.102806i	$-0.967218\pi$
$443$	− 91.0864i	− 0.205613i	0.994701	−	0.102806i	$-0.0327822\pi$
$444$	0	0
$445$	212.952	0.478543
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 93.6705i	− 0.208620i	−0.994545	−	0.104310i	$-0.966737\pi$
$449$	− 93.6705i	− 0.208620i	0.994545	−	0.104310i	$-0.0332634\pi$
$450$	0	0
$451$	303.569	0.673101
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 18.8803i	− 0.0414952i
$456$	0	0
$457$	402.996	0.881829	0.440915	−	0.897549i	$-0.354654\pi$
$457$	402.996	0.881829	0.440915	+	0.897549i	$0.354654\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 66.9144i	− 0.145150i	−0.997363	−	0.0725752i	$-0.976878\pi$
$461$	− 66.9144i	− 0.145150i	0.997363	−	0.0725752i	$-0.0231217\pi$
$462$	0	0
$463$	177.621	0.383631	0.191815	−	0.981431i	$-0.438563\pi$
$463$	177.621	0.383631	0.191815	+	0.981431i	$0.438563\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 620.647i	− 1.32901i	−0.747285	−	0.664504i	$-0.768643\pi$
$467$	− 620.647i	− 1.32901i	0.747285	−	0.664504i	$-0.231357\pi$
$468$	0	0
$469$	−374.415	−0.798327
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 490.554i	− 1.03711i
$474$	0	0
$475$	133.730	0.281536
$476$	0	0
$477$	0	0
$478$	0	0
$479$	168.637i	0.352061i	0.984385	+	0.176031i	$0.0563258\pi$
$479$	168.637i	0.352061i	−0.984385	+	0.176031i	$0.943674\pi$
$480$	0	0
$481$	−31.5565	−0.0656060
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 227.403i	− 0.468871i
$486$	0	0
$487$	401.935	0.825330	0.412665	−	0.910883i	$-0.364598\pi$
$487$	401.935	0.825330	0.412665	+	0.910883i	$0.364598\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	355.455i	0.723940i	0.932190	+	0.361970i	$0.117896\pi$
$491$	355.455i	0.723940i	−0.932190	+	0.361970i	$0.882104\pi$
$492$	0	0
$493$	515.903	1.04646
$494$	0	0
$495$	0	0
$496$	0	0
$497$	384.348i	0.773335i
$498$	0	0
$499$	−16.6169	−0.0333004	−0.0166502	−	0.999861i	$-0.505300\pi$
$499$	−16.6169	−0.0333004	−0.0166502	+	0.999861i	$0.505300\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 362.522i	− 0.720721i	−0.932813	−	0.360360i	$-0.882654\pi$
$503$	− 362.522i	− 0.720721i	0.932813	−	0.360360i	$-0.117346\pi$
$504$	0	0
$505$	226.601	0.448714
$506$	0	0
$507$	0	0
$508$	0	0
$509$	667.748i	1.31188i	0.754813	+	0.655941i	$0.227728\pi$
$509$	667.748i	1.31188i	−0.754813	+	0.655941i	$0.772272\pi$
$510$	0	0
$511$	−51.5524	−0.100885
$512$	0	0
$513$	0	0
$514$	0	0
$515$	36.4173i	0.0707132i
$516$	0	0
$517$	531.327	1.02771
$518$	0	0
$519$	0	0
$520$	0	0
$521$	676.352i	1.29818i	0.760712	+	0.649090i	$0.224850\pi$
$521$	676.352i	1.29818i	−0.760712	+	0.649090i	$0.775150\pi$
$522$	0	0
$523$	189.427	0.362194	0.181097	−	0.983465i	$-0.442035\pi$
$523$	189.427	0.362194	0.181097	+	0.983465i	$0.442035\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 610.178i	− 1.15783i
$528$	0	0
$529$	132.806	0.251052
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 58.5615i	− 0.109871i
$534$	0	0
$535$	189.776	0.354722
$536$	0	0
$537$	0	0
$538$	0	0
$539$	408.575i	0.758025i
$540$	0	0
$541$	−590.629	−1.09174	−0.545868	−	0.837871i	$-0.683800\pi$
$541$	−590.629	−1.09174	−0.545868	+	0.837871i	$0.683800\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 362.171i	− 0.664534i
$546$	0	0
$547$	−789.564	−1.44344	−0.721722	−	0.692183i	$-0.756649\pi$
$547$	−789.564	−1.44344	−0.721722	+	0.692183i	$0.756649\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1180.92i	2.14324i
$552$	0	0
$553$	212.629	0.384501
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1004.69i	1.80376i	0.431987	+	0.901880i	$0.357813\pi$
$557$	1004.69i	1.80376i	−0.431987	+	0.901880i	$0.642187\pi$
$558$	0	0
$559$	−94.6330	−0.169290
$560$	0	0
$561$	0	0
$562$	0	0
$563$	315.205i	0.559867i	0.960019	+	0.279934i	$0.0903125\pi$
$563$	315.205i	0.559867i	−0.960019	+	0.279934i	$0.909688\pi$
$564$	0	0
$565$	15.4919	0.0274194
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 951.385i	− 1.67203i	−0.548707	−	0.836015i	$-0.684880\pi$
$569$	− 951.385i	− 1.67203i	0.548707	−	0.836015i	$-0.315120\pi$
$570$	0	0
$571$	458.367	0.802744	0.401372	−	0.915915i	$-0.368533\pi$
$571$	458.367	0.802744	0.401372	+	0.915915i	$0.368533\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 99.5231i	− 0.173084i
$576$	0	0
$577$	991.190	1.71783	0.858916	−	0.512116i	$-0.171138\pi$
$577$	991.190	1.71783	0.858916	+	0.512116i	$0.171138\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 337.387i	− 0.580700i
$582$	0	0
$583$	−126.569	−0.217099
$584$	0	0
$585$	0	0
$586$	0	0
$587$	167.297i	0.285003i	0.989795	+	0.142501i	$0.0455145\pi$
$587$	167.297i	0.285003i	−0.989795	+	0.142501i	$0.954485\pi$
$588$	0	0
$589$	1396.72	2.37134
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 868.330i	− 1.46430i	−0.681143	−	0.732150i	$-0.738517\pi$
$593$	− 868.330i	− 1.46430i	0.681143	−	0.732150i	$-0.261483\pi$
$594$	0	0
$595$	−97.8709	−0.164489
$596$	0	0
$597$	0	0
$598$	0	0
$599$	196.686i	0.328357i	0.986431	+	0.164178i	$0.0524972\pi$
$599$	196.686i	0.328357i	−0.986431	+	0.164178i	$0.947503\pi$
$600$	0	0
$601$	664.363	1.10543	0.552715	−	0.833371i	$-0.313592\pi$
$601$	664.363	1.10543	0.552715	+	0.833371i	$0.313592\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 34.7132i	− 0.0573771i
$606$	0	0
$607$	−74.4758	−0.122695	−0.0613474	−	0.998116i	$-0.519540\pi$
$607$	−74.4758	−0.122695	−0.0613474	+	0.998116i	$0.519540\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 102.498i	− 0.167755i
$612$	0	0
$613$	−304.569	−0.496849	−0.248425	−	0.968651i	$-0.579913\pi$
$613$	−304.569	−0.496849	−0.248425	+	0.968651i	$0.579913\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	941.488i	1.52591i	0.646450	+	0.762956i	$0.276253\pi$
$617$	941.488i	1.52591i	−0.646450	+	0.762956i	$0.723747\pi$
$618$	0	0
$619$	−235.472	−0.380407	−0.190203	−	0.981745i	$-0.560915\pi$
$619$	−235.472	−0.380407	−0.190203	+	0.981745i	$0.560915\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 356.747i	− 0.572627i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	163.581i	0.260065i
$630$	0	0
$631$	−832.125	−1.31874	−0.659370	−	0.751819i	$-0.729177\pi$
$631$	−832.125	−1.31874	−0.659370	+	0.751819i	$0.729177\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 505.135i	− 0.795488i
$636$	0	0
$637$	78.8184	0.123734
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1054.65i	1.64532i	0.568532	+	0.822661i	$0.307511\pi$
$641$	1054.65i	1.64532i	−0.568532	+	0.822661i	$0.692489\pi$
$642$	0	0
$643$	987.185	1.53528	0.767640	−	0.640881i	$-0.221431\pi$
$643$	987.185	1.53528	0.767640	+	0.640881i	$0.221431\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	75.6097i	0.116862i	0.998291	+	0.0584310i	$0.0186098\pi$
$647$	75.6097i	0.116862i	−0.998291	+	0.0584310i	$0.981390\pi$
$648$	0	0
$649$	430.137	0.662769
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 1247.02i	− 1.90968i	−0.297121	−	0.954840i	$-0.596027\pi$
$653$	− 1247.02i	− 1.90968i	0.297121	−	0.954840i	$-0.403973\pi$
$654$	0	0
$655$	−210.984	−0.322113
$656$	0	0
$657$	0	0
$658$	0	0
$659$	629.139i	0.954688i	0.878717	+	0.477344i	$0.158400\pi$
$659$	629.139i	0.954688i	−0.878717	+	0.477344i	$0.841600\pi$
$660$	0	0
$661$	646.564	0.978161	0.489080	−	0.872239i	$-0.337332\pi$
$661$	646.564	0.978161	0.489080	+	0.872239i	$0.337332\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 224.031i	− 0.336888i
$666$	0	0
$667$	878.855	1.31762
$668$	0	0
$669$	0	0
$670$	0	0
$671$	528.576i	0.787743i
$672$	0	0
$673$	91.9032	0.136558	0.0682788	−	0.997666i	$-0.478249\pi$
$673$	91.9032	0.136558	0.0682788	+	0.997666i	$0.478249\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	731.526i	1.08054i	0.841492	+	0.540270i	$0.181678\pi$
$677$	731.526i	1.08054i	−0.841492	+	0.540270i	$0.818322\pi$
$678$	0	0
$679$	−380.956	−0.561054
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 682.177i	− 0.998794i	−0.866373	−	0.499397i	$-0.833555\pi$
$683$	− 682.177i	− 0.998794i	0.866373	−	0.499397i	$-0.166445\pi$
$684$	0	0
$685$	−463.712	−0.676951
$686$	0	0
$687$	0	0
$688$	0	0
$689$	24.4164i	0.0354374i
$690$	0	0
$691$	−420.661	−0.608772	−0.304386	−	0.952549i	$-0.598451\pi$
$691$	−420.661	−0.608772	−0.304386	+	0.952549i	$0.598451\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	73.1501i	0.105252i
$696$	0	0
$697$	−303.569	−0.435536
$698$	0	0
$699$	0	0
$700$	0	0
$701$	70.8185i	0.101025i	0.998723	+	0.0505125i	$0.0160855\pi$
$701$	70.8185i	0.101025i	−0.998723	+	0.0505125i	$0.983915\pi$
$702$	0	0
$703$	−374.444	−0.532637
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 379.612i	− 0.536934i
$708$	0	0
$709$	799.391	1.12749	0.563745	−	0.825949i	$-0.309360\pi$
$709$	799.391	1.12749	0.563745	+	0.825949i	$0.309360\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 1039.45i	− 1.45786i
$714$	0	0
$715$	−58.8912	−0.0823653
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 368.675i	− 0.512761i	−0.966576	−	0.256381i	$-0.917470\pi$
$719$	− 368.675i	− 0.512761i	0.966576	−	0.256381i	$-0.0825300\pi$
$720$	0	0
$721$	61.0079	0.0846157
$722$	0	0
$723$	0	0
$724$	0	0
$725$	220.767i	0.304506i
$726$	0	0
$727$	−906.190	−1.24648	−0.623239	−	0.782031i	$-0.714184\pi$
$727$	−906.190	−1.24648	−0.623239	+	0.782031i	$0.714184\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	490.554i	0.671073i
$732$	0	0
$733$	101.520	0.138499	0.0692497	−	0.997599i	$-0.477939\pi$
$733$	101.520	0.138499	0.0692497	+	0.997599i	$0.477939\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1167.87i	1.58463i
$738$	0	0
$739$	557.665	0.754622	0.377311	−	0.926087i	$-0.376849\pi$
$739$	557.665	0.754622	0.377311	+	0.926087i	$0.376849\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	356.579i	0.479918i	0.970783	+	0.239959i	$0.0771340\pi$
$743$	356.579i	0.479918i	−0.970783	+	0.239959i	$0.922866\pi$
$744$	0	0
$745$	47.3347	0.0635365
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 317.922i	− 0.424462i
$750$	0	0
$751$	489.621	0.651959	0.325979	−	0.945377i	$-0.394306\pi$
$751$	489.621	0.651959	0.325979	+	0.945377i	$0.394306\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 615.874i	− 0.815728i
$756$	0	0
$757$	198.379	0.262059	0.131030	−	0.991378i	$-0.458172\pi$
$757$	198.379	0.262059	0.131030	+	0.991378i	$0.458172\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	35.4092i	0.0465299i	0.999729	+	0.0232649i	$0.00740613\pi$
$761$	35.4092i	0.0465299i	−0.999729	+	0.0232649i	$0.992594\pi$
$762$	0	0
$763$	−606.726	−0.795184
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 82.9779i	− 0.108185i
$768$	0	0
$769$	−229.105	−0.297926	−0.148963	−	0.988843i	$-0.547593\pi$
$769$	−229.105	−0.297926	−0.148963	+	0.988843i	$0.547593\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	905.388i	1.17126i	0.810577	+	0.585632i	$0.199154\pi$
$773$	905.388i	1.17126i	−0.810577	+	0.585632i	$0.800846\pi$
$774$	0	0
$775$	261.109	0.336915
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 694.881i	− 0.892016i
$780$	0	0
$781$	1198.85	1.53502
$782$	0	0
$783$	0	0
$784$	0	0
$785$	402.348i	0.512545i
$786$	0	0
$787$	−148.290	−0.188425	−0.0942125	−	0.995552i	$-0.530033\pi$
$787$	−148.290	−0.188425	−0.0942125	+	0.995552i	$0.530033\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 25.9528i	− 0.0328101i
$792$	0	0
$793$	101.968	0.128585
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 245.728i	− 0.308316i	−0.988046	−	0.154158i	$-0.950734\pi$
$797$	− 245.728i	− 0.308316i	0.988046	−	0.154158i	$-0.0492664\pi$
$798$	0	0
$799$	−531.327	−0.664989
$800$	0	0
$801$	0	0
$802$	0	0
$803$	160.801i	0.200251i
$804$	0	0
$805$	−166.726	−0.207113
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 1551.44i	− 1.91773i	−0.283865	−	0.958864i	$-0.591617\pi$
$809$	− 1551.44i	− 1.91773i	0.283865	−	0.958864i	$-0.408383\pi$
$810$	0	0
$811$	−439.512	−0.541939	−0.270969	−	0.962588i	$-0.587344\pi$
$811$	−439.512	−0.541939	−0.270969	+	0.962588i	$0.587344\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 591.242i	− 0.725450i
$816$	0	0
$817$	−1122.90	−1.37442
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1087.94i	1.32515i	0.748998	+	0.662573i	$0.230535\pi$
$821$	1087.94i	1.32515i	−0.748998	+	0.662573i	$0.769465\pi$
$822$	0	0
$823$	−938.435	−1.14026	−0.570131	−	0.821554i	$-0.693107\pi$
$823$	−938.435	−1.14026	−0.570131	+	0.821554i	$0.693107\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	615.395i	0.744129i	0.928207	+	0.372065i	$0.121350\pi$
$827$	615.395i	0.744129i	−0.928207	+	0.372065i	$0.878650\pi$
$828$	0	0
$829$	−559.230	−0.674583	−0.337292	−	0.941400i	$-0.609511\pi$
$829$	−559.230	−0.674583	−0.337292	+	0.941400i	$0.609511\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 408.575i	− 0.490487i
$834$	0	0
$835$	386.065	0.462353
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1055.55i	− 1.25811i	−0.777362	−	0.629054i	$-0.783443\pi$
$839$	− 1055.55i	− 1.25811i	0.777362	−	0.629054i	$-0.216557\pi$
$840$	0	0
$841$	−1108.52	−1.31809
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 366.535i	− 0.433769i
$846$	0	0
$847$	−58.1531	−0.0686578
$848$	0	0
$849$	0	0
$850$	0	0
$851$	278.665i	0.327455i
$852$	0	0
$853$	−91.9073	−0.107746	−0.0538730	−	0.998548i	$-0.517157\pi$
$853$	−91.9073	−0.107746	−0.0538730	+	0.998548i	$0.517157\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1179.00i	− 1.37572i	−0.725841	−	0.687862i	$-0.758549\pi$
$857$	− 1179.00i	− 1.37572i	0.725841	−	0.687862i	$-0.241451\pi$
$858$	0	0
$859$	−1339.57	−1.55945	−0.779726	−	0.626121i	$-0.784641\pi$
$859$	−1339.57	−1.55945	−0.779726	+	0.626121i	$0.784641\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1332.42i	− 1.54393i	−0.635663	−	0.771967i	$-0.719273\pi$
$863$	− 1332.42i	− 1.54393i	0.635663	−	0.771967i	$-0.280727\pi$
$864$	0	0
$865$	−658.157	−0.760875
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 663.229i	− 0.763209i
$870$	0	0
$871$	225.294	0.258662
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 41.8812i	− 0.0478642i
$876$	0	0
$877$	−642.633	−0.732763	−0.366381	−	0.930465i	$-0.619403\pi$
$877$	−642.633	−0.732763	−0.366381	+	0.930465i	$0.619403\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 983.693i	− 1.11656i	−0.829651	−	0.558282i	$-0.811461\pi$
$881$	− 983.693i	− 1.11656i	0.829651	−	0.558282i	$-0.188539\pi$
$882$	0	0
$883$	197.540	0.223715	0.111857	−	0.993724i	$-0.464320\pi$
$883$	197.540	0.223715	0.111857	+	0.993724i	$0.464320\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1311.84i	1.47896i	0.673177	+	0.739481i	$0.264929\pi$
$887$	1311.84i	1.47896i	−0.673177	+	0.739481i	$0.735071\pi$
$888$	0	0
$889$	−846.226	−0.951885
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 1216.23i	− 1.36196i
$894$	0	0
$895$	−143.363	−0.160182
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2305.76i	2.56481i
$900$	0	0
$901$	126.569	0.140476
$902$	0	0
$903$	0	0
$904$	0	0
$905$	651.156i	0.719509i
$906$	0	0
$907$	1730.99	1.90848	0.954240	−	0.299041i	$-0.0966668\pi$
$907$	1730.99	1.90848	0.954240	+	0.299041i	$0.0966668\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	243.318i	0.267089i	0.991043	+	0.133545i	$0.0426360\pi$
$911$	243.318i	0.267089i	−0.991043	+	0.133545i	$0.957364\pi$
$912$	0	0
$913$	−1052.37	−1.15265
$914$	0	0
$915$	0	0
$916$	0	0
$917$	353.450i	0.385442i
$918$	0	0
$919$	449.145	0.488733	0.244366	−	0.969683i	$-0.421420\pi$
$919$	449.145	0.488733	0.244366	+	0.969683i	$0.421420\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 231.271i	− 0.250564i
$924$	0	0
$925$	−70.0000	−0.0756757
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 52.2410i	− 0.0562335i	−0.999605	−	0.0281168i	$-0.991049\pi$
$929$	− 52.2410i	− 0.0562335i	0.999605	−	0.0281168i	$-0.00895103\pi$
$930$	0	0
$931$	935.246	1.00456
$932$	0	0
$933$	0	0
$934$	0	0
$935$	305.277i	0.326500i
$936$	0	0
$937$	196.798	0.210030	0.105015	−	0.994471i	$-0.466511\pi$
$937$	196.798	0.210030	0.105015	+	0.994471i	$0.466511\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	949.597i	1.00914i	0.863372	+	0.504568i	$0.168348\pi$
$941$	949.597i	1.00914i	−0.863372	+	0.504568i	$0.831652\pi$
$942$	0	0
$943$	−517.137	−0.548396
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1628.76i	− 1.71991i	−0.510369	−	0.859956i	$-0.670491\pi$
$947$	− 1628.76i	− 1.71991i	0.510369	−	0.859956i	$-0.329509\pi$
$948$	0	0
$949$	31.0202	0.0326873
$950$	0	0
$951$	0	0
$952$	0	0
$953$	789.836i	0.828789i	0.910097	+	0.414395i	$0.136007\pi$
$953$	789.836i	0.828789i	−0.910097	+	0.414395i	$0.863993\pi$
$954$	0	0
$955$	−35.6533	−0.0373333
$956$	0	0
$957$	0	0
$958$	0	0
$959$	776.832i	0.810043i
$960$	0	0
$961$	1766.11	1.83779
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 547.196i	− 0.567043i
$966$	0	0
$967$	527.339	0.545335	0.272667	−	0.962108i	$-0.412094\pi$
$967$	527.339	0.545335	0.272667	+	0.962108i	$0.412094\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	179.700i	0.185067i	0.995710	+	0.0925337i	$0.0294966\pi$
$971$	179.700i	0.185067i	−0.995710	+	0.0925337i	$0.970503\pi$
$972$	0	0
$973$	122.544	0.125945
$974$	0	0
$975$	0	0
$976$	0	0
$977$	926.752i	0.948569i	0.880372	+	0.474284i	$0.157293\pi$
$977$	926.752i	0.948569i	−0.880372	+	0.474284i	$0.842707\pi$
$978$	0	0
$979$	−1112.76	−1.13663
$980$	0	0
$981$	0	0
$982$	0	0
$983$	564.984i	0.574755i	0.957817	+	0.287377i	$0.0927834\pi$
$983$	564.984i	0.574755i	−0.957817	+	0.287377i	$0.907217\pi$
$984$	0	0
$985$	470.411	0.477575
$986$	0	0
$987$	0	0
$988$	0	0
$989$	835.673i	0.844967i
$990$	0	0
$991$	−1815.42	−1.83191	−0.915953	−	0.401285i	$-0.868564\pi$
$991$	−1815.42	−1.83191	−0.915953	+	0.401285i	$0.868564\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 229.711i	− 0.230865i
$996$	0	0
$997$	682.383	0.684436	0.342218	−	0.939621i	$-0.388822\pi$
$997$	682.383	0.684436	0.342218	+	0.939621i	$0.388822\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.3.g.a.161.4		4
3.2	odd	2	inner	1620.3.g.a.161.2		4
9.2	odd	6		540.3.o.a.341.2		4
9.4	even	3		540.3.o.a.521.2		4
9.5	odd	6		180.3.o.a.101.1	yes	4
9.7	even	3		180.3.o.a.41.1	✓	4
36.7	odd	6		720.3.bs.a.401.1		4
36.11	even	6		2160.3.bs.a.881.2		4
36.23	even	6		720.3.bs.a.641.1		4
36.31	odd	6		2160.3.bs.a.1601.2		4
45.2	even	12		2700.3.u.a.449.3		8
45.4	even	6		2700.3.p.a.1601.2		4
45.7	odd	12		900.3.u.b.149.2		8
45.13	odd	12		2700.3.u.a.2249.3		8
45.14	odd	6		900.3.p.b.101.2		4
45.22	odd	12		2700.3.u.a.2249.2		8
45.23	even	12		900.3.u.b.749.2		8
45.29	odd	6		2700.3.p.a.2501.2		4
45.32	even	12		900.3.u.b.749.3		8
45.34	even	6		900.3.p.b.401.2		4
45.38	even	12		2700.3.u.a.449.2		8
45.43	odd	12		900.3.u.b.149.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.3.o.a.41.1	✓	4	9.7	even	3
180.3.o.a.101.1	yes	4	9.5	odd	6
540.3.o.a.341.2		4	9.2	odd	6
540.3.o.a.521.2		4	9.4	even	3
720.3.bs.a.401.1		4	36.7	odd	6
720.3.bs.a.641.1		4	36.23	even	6
900.3.p.b.101.2		4	45.14	odd	6
900.3.p.b.401.2		4	45.34	even	6
900.3.u.b.149.2		8	45.7	odd	12
900.3.u.b.149.3		8	45.43	odd	12
900.3.u.b.749.2		8	45.23	even	12
900.3.u.b.749.3		8	45.32	even	12
1620.3.g.a.161.2		4	3.2	odd	2	inner
1620.3.g.a.161.4		4	1.1	even	1	trivial
2160.3.bs.a.881.2		4	36.11	even	6
2160.3.bs.a.1601.2		4	36.31	odd	6
2700.3.p.a.1601.2		4	45.4	even	6
2700.3.p.a.2501.2		4	45.29	odd	6
2700.3.u.a.449.2		8	45.38	even	12
2700.3.u.a.449.3		8	45.2	even	12
2700.3.u.a.2249.2		8	45.22	odd	12
2700.3.u.a.2249.3		8	45.13	odd	12

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.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.23607i q^{5} +3.74597 q^{7} +O(q^{10})\)	\(q+2.23607i q^{5} +3.74597 q^{7} -11.6844i q^{11} -2.25403 q^{13} +11.6844i q^{17} -26.7460 q^{19} +19.9046i q^{23} -5.00000 q^{25} -44.1533i q^{29} -52.2218 q^{31} +8.37624i q^{35} +14.0000 q^{37} +25.9808i q^{41} +41.9839 q^{43} +45.4733i q^{47} -34.9677 q^{49} -10.8323i q^{53} +26.1270 q^{55} +36.8131i q^{59} -45.2379 q^{61} -5.04017i q^{65} -99.9516 q^{67} +102.603i q^{71} -13.7621 q^{73} -43.7692i q^{77} +56.7621 q^{79} -90.0666i q^{83} -26.1270 q^{85} -95.2349i q^{89} -8.44353 q^{91} -59.8058i q^{95} -101.698 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{7}+O(q^{10}) \) 4 * q - 16 * q^7	\( 4 q - 16 q^{7} - 40 q^{13} - 76 q^{19} - 20 q^{25} + 8 q^{31} + 56 q^{37} + 44 q^{43} + 108 q^{49} + 120 q^{55} - 88 q^{61} - 28 q^{67} - 148 q^{73} + 320 q^{79} - 120 q^{85} + 400 q^{91} - 4 q^{97}+O(q^{100}) \) 4 * q - 16 * q^7 - 40 * q^13 - 76 * q^19 - 20 * q^25 + 8 * q^31 + 56 * q^37 + 44 * q^43 + 108 * q^49 + 120 * q^55 - 88 * q^61 - 28 * q^67 - 148 * q^73 + 320 * q^79 - 120 * q^85 + 400 * q^91 - 4 * q^97

Introduction

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