LMFDB - Embedded newform 2160.3.c.q.1889.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2160,3,Mod(1889,2160)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2160, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2160.1889"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2160.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$58.8557371018$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 1080)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1889.12
Character	$\chi$	$=$	2160.1889
Dual form			2160.3.c.q.1889.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.322433 + 4.98959i) q^{5} +9.79740i q^{7} -20.4984i q^{11} -4.54977i q^{13} +12.0050 q^{17} +30.3719 q^{19} +20.6653 q^{23} +(-24.7921 - 3.21762i) q^{25} -29.8036i q^{29} -36.5502 q^{31} +(-48.8850 - 3.15901i) q^{35} -0.0295872i q^{37} -11.8553i q^{41} -81.6243i q^{43} +58.1176 q^{47} -46.9891 q^{49} -42.3523 q^{53} +(102.279 + 6.60937i) q^{55} -20.4866i q^{59} -62.6603 q^{61} +(22.7015 + 1.46700i) q^{65} -75.0895i q^{67} +28.6668i q^{71} -111.205i q^{73} +200.831 q^{77} +96.5016 q^{79} +130.230 q^{83} +(-3.87080 + 59.8999i) q^{85} +40.9592i q^{89} +44.5760 q^{91} +(-9.79290 + 151.543i) q^{95} -14.9075i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 72 q^{25} + 72 q^{31} - 408 q^{49} + 168 q^{55} - 240 q^{61} + 312 q^{79} - 96 q^{85}+O(q^{100}) $ 24 * q - 72 * q^25 + 72 * q^31 - 408 * q^49 + 168 * q^55 - 240 * q^61 + 312 * q^79 - 96 * q^85

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$1$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.322433	+	4.98959i	−0.0644867	+	0.997919i
$5$	−0.322433	+	4.98959i	−0.0644867	+	0.997919i
$6$		0			0
$7$			9.79740i			1.39963i	0.714325	+	0.699814i	$0.246734\pi$
$7$			9.79740i			1.39963i	−0.714325	+	0.699814i	$0.753266\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	20.4984i		−	1.86349i	−0.363112	−	0.931745i	$-0.618286\pi$
$11$		−	20.4984i		−	1.86349i	0.363112	−	0.931745i	$-0.381714\pi$
$12$		0			0
$13$		−	4.54977i		−	0.349982i	−0.984570	−	0.174991i	$-0.944010\pi$
$13$		−	4.54977i		−	0.349982i	0.984570	−	0.174991i	$-0.0559897\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	12.0050			0.706175			0.353087	−	0.935590i	$-0.385132\pi$
$17$	12.0050			0.706175			0.353087	+	0.935590i	$0.385132\pi$
$18$		0			0
$19$	30.3719			1.59852			0.799259	−	0.600986i	$-0.205225\pi$
$19$	30.3719			1.59852			0.799259	+	0.600986i	$0.205225\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	20.6653			0.898492			0.449246	−	0.893408i	$-0.351693\pi$
$23$	20.6653			0.898492			0.449246	+	0.893408i	$0.351693\pi$
$24$		0			0
$25$	−24.7921	−	3.21762i	−0.991683	−	0.128705i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	29.8036i		−	1.02771i	−0.857877	−	0.513855i	$-0.828217\pi$
$29$		−	29.8036i		−	1.02771i	0.857877	−	0.513855i	$-0.171783\pi$
$30$		0			0
$31$	−36.5502			−1.17904			−0.589520	−	0.807754i	$-0.700683\pi$
$31$	−36.5502			−1.17904			−0.589520	+	0.807754i	$0.700683\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−48.8850	−	3.15901i	−1.39672	−	0.0902574i
$36$		0			0
$37$		−	0.0295872i		−	0.000799655i	−1.00000		0.000399827i	$-0.999873\pi$
$37$		−	0.0295872i		−	0.000799655i	1.00000		0.000399827i	$-0.000127269\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	11.8553i		−	0.289155i	−0.989494	−	0.144577i	$-0.953818\pi$
$41$		−	11.8553i		−	0.289155i	0.989494	−	0.144577i	$-0.0461823\pi$
$42$		0			0
$43$		−	81.6243i		−	1.89824i	−0.314916	−	0.949119i	$-0.601976\pi$
$43$		−	81.6243i		−	1.89824i	0.314916	−	0.949119i	$-0.398024\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	58.1176			1.23654			0.618272	−	0.785964i	$-0.287833\pi$
$47$	58.1176			1.23654			0.618272	+	0.785964i	$0.287833\pi$
$48$		0			0
$49$	−46.9891			−0.958961
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−42.3523			−0.799099			−0.399550	−	0.916712i	$-0.630833\pi$
$53$	−42.3523			−0.799099			−0.399550	+	0.916712i	$0.630833\pi$
$54$		0			0
$55$	102.279	+	6.60937i	1.85961	+	0.120170i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	20.4866i		−	0.347230i	−0.984814	−	0.173615i	$-0.944455\pi$
$59$		−	20.4866i		−	0.347230i	0.984814	−	0.173615i	$-0.0555449\pi$
$60$		0			0
$61$	−62.6603			−1.02722			−0.513609	−	0.858024i	$-0.671692\pi$
$61$	−62.6603			−1.02722			−0.513609	+	0.858024i	$0.671692\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	22.7015	+	1.46700i	0.349254	+	0.0225692i
$66$		0			0
$67$		−	75.0895i		−	1.12074i	−0.828243	−	0.560369i	$-0.810659\pi$
$67$		−	75.0895i		−	1.12074i	0.828243	−	0.560369i	$-0.189341\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			28.6668i			0.403757i	0.979411	+	0.201879i	$0.0647047\pi$
$71$			28.6668i			0.403757i	−0.979411	+	0.201879i	$0.935295\pi$
$72$		0			0
$73$		−	111.205i		−	1.52335i	−0.647957	−	0.761677i	$-0.724377\pi$
$73$		−	111.205i		−	1.52335i	0.647957	−	0.761677i	$-0.275623\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	200.831			2.60820
$78$		0			0
$79$	96.5016			1.22154			0.610769	−	0.791809i	$-0.290860\pi$
$79$	96.5016			1.22154			0.610769	+	0.791809i	$0.290860\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	130.230			1.56904			0.784520	−	0.620103i	$-0.212909\pi$
$83$	130.230			1.56904			0.784520	+	0.620103i	$0.212909\pi$
$84$		0			0
$85$	−3.87080	+	59.8999i	−0.0455389	+	0.704705i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			40.9592i			0.460215i	0.973165	+	0.230108i	$0.0739079\pi$
$89$			40.9592i			0.460215i	−0.973165	+	0.230108i	$0.926092\pi$
$90$		0			0
$91$	44.5760			0.489846
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−9.79290	+	151.543i	−0.103083	+	1.59519i
$96$		0			0
$97$		−	14.9075i		−	0.153685i	−0.997043	−	0.0768426i	$-0.975516\pi$
$97$		−	14.9075i		−	0.153685i	0.997043	−	0.0768426i	$-0.0244839\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			66.2695i			0.656134i	0.944655	+	0.328067i	$0.106397\pi$
$101$			66.2695i			0.656134i	−0.944655	+	0.328067i	$0.893603\pi$
$102$		0			0
$103$			134.582i			1.30662i	0.757090	+	0.653310i	$0.226620\pi$
$103$			134.582i			1.30662i	−0.757090	+	0.653310i	$0.773380\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	127.716			1.19361			0.596803	−	0.802388i	$-0.296437\pi$
$107$	127.716			1.19361			0.596803	+	0.802388i	$0.296437\pi$
$108$		0			0
$109$	−51.6828			−0.474154			−0.237077	−	0.971491i	$-0.576189\pi$
$109$	−51.6828			−0.474154			−0.237077	+	0.971491i	$0.576189\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−38.9887			−0.345033			−0.172517	−	0.985007i	$-0.555190\pi$
$113$	−38.9887			−0.345033			−0.172517	+	0.985007i	$0.555190\pi$
$114$		0			0
$115$	−6.66319	+	103.112i	−0.0579408	+	0.896622i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			117.618i			0.988383i
$120$		0			0
$121$	−299.184			−2.47260
$122$		0			0
$123$		0			0
$124$		0			0
$125$	24.0484	−	122.665i	0.192387	−	0.981319i
$126$		0			0
$127$			127.914i			1.00720i	0.863937	+	0.503599i	$0.167991\pi$
$127$			127.914i			1.00720i	−0.863937	+	0.503599i	$0.832009\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	143.190i		−	1.09305i	−0.837443	−	0.546525i	$-0.815950\pi$
$131$		−	143.190i		−	1.09305i	0.837443	−	0.546525i	$-0.184050\pi$
$132$		0			0
$133$			297.565i			2.23733i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	100.407			0.732901			0.366450	−	0.930438i	$-0.380573\pi$
$137$	100.407			0.732901			0.366450	+	0.930438i	$0.380573\pi$
$138$		0			0
$139$	166.049			1.19460			0.597299	−	0.802018i	$-0.296240\pi$
$139$	166.049			1.19460			0.597299	+	0.802018i	$0.296240\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−93.2631			−0.652189
$144$		0			0
$145$	148.708	+	9.60968i	1.02557	+	0.0662736i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			143.836i			0.965345i	0.875801	+	0.482673i	$0.160334\pi$
$149$			143.836i			0.965345i	−0.875801	+	0.482673i	$0.839666\pi$
$150$		0			0
$151$	79.0229			0.523330			0.261665	−	0.965159i	$-0.415728\pi$
$151$	79.0229			0.523330			0.261665	+	0.965159i	$0.415728\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	11.7850	−	182.371i	0.0760323	−	1.17659i
$156$		0			0
$157$			40.4882i			0.257887i	0.991652	+	0.128943i	$0.0411585\pi$
$157$			40.4882i			0.257887i	−0.991652	+	0.128943i	$0.958841\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			202.466i			1.25756i
$162$		0			0
$163$			212.218i			1.30195i	0.759099	+	0.650975i	$0.225640\pi$
$163$			212.218i			1.30195i	−0.759099	+	0.650975i	$0.774360\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	216.468			1.29621			0.648107	−	0.761549i	$-0.275561\pi$
$167$	216.468			1.29621			0.648107	+	0.761549i	$0.275561\pi$
$168$		0			0
$169$	148.300			0.877512
$170$		0			0
$171$		0			0
$172$		0			0
$173$	283.865			1.64084			0.820419	−	0.571763i	$-0.193740\pi$
$173$	283.865			1.64084			0.820419	+	0.571763i	$0.193740\pi$
$174$		0			0
$175$	31.5243	−	242.898i	0.180139	−	1.38799i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	34.3092i		−	0.191672i	−0.995397	−	0.0958358i	$-0.969448\pi$
$179$		−	34.3092i		−	0.191672i	0.995397	−	0.0958358i	$-0.0305524\pi$
$180$		0			0
$181$	−224.998			−1.24308			−0.621541	−	0.783382i	$-0.713493\pi$
$181$	−224.998			−1.24308			−0.621541	+	0.783382i	$0.713493\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.147628	+	0.00953991i	0.000797990	+	5.15671e-5i
$186$		0			0
$187$		−	246.083i		−	1.31595i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	314.419i		−	1.64617i	−0.567916	−	0.823087i	$-0.692250\pi$
$191$		−	314.419i		−	1.64617i	0.567916	−	0.823087i	$-0.307750\pi$
$192$		0			0
$193$			188.604i			0.977225i	0.872501	+	0.488613i	$0.162497\pi$
$193$			188.604i			0.977225i	−0.872501	+	0.488613i	$0.837503\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	183.211			0.930003			0.465001	−	0.885310i	$-0.346054\pi$
$197$	183.211			0.930003			0.465001	+	0.885310i	$0.346054\pi$
$198$		0			0
$199$	−184.489			−0.927081			−0.463540	−	0.886076i	$-0.653421\pi$
$199$	−184.489			−0.927081			−0.463540	+	0.886076i	$0.653421\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	291.998			1.43841
$204$		0			0
$205$	59.1533	+	3.82256i	0.288553	+	0.0186466i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	622.574i		−	2.97882i
$210$		0			0
$211$	−115.761			−0.548630			−0.274315	−	0.961640i	$-0.588451\pi$
$211$	−115.761			−0.548630			−0.274315	+	0.961640i	$0.588451\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	407.272	+	26.3184i	1.89429	+	0.122411i
$216$		0			0
$217$		−	358.097i		−	1.65022i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	54.6199i		−	0.247149i
$222$		0			0
$223$		−	313.319i		−	1.40502i	−0.711675	−	0.702509i	$-0.752063\pi$
$223$		−	313.319i		−	1.40502i	0.711675	−	0.702509i	$-0.247937\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−84.8965			−0.373993			−0.186997	−	0.982361i	$-0.559875\pi$
$227$	−84.8965			−0.373993			−0.186997	+	0.982361i	$0.559875\pi$
$228$		0			0
$229$	137.919			0.602268			0.301134	−	0.953582i	$-0.402635\pi$
$229$	137.919			0.602268			0.301134	+	0.953582i	$0.402635\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	341.974			1.46770			0.733850	−	0.679312i	$-0.237722\pi$
$233$	341.974			1.46770			0.733850	+	0.679312i	$0.237722\pi$
$234$		0			0
$235$	−18.7391	+	289.983i	−0.0797406	+	1.23397i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			136.991i			0.573184i	0.958053	+	0.286592i	$0.0925225\pi$
$239$			136.991i			0.573184i	−0.958053	+	0.286592i	$0.907478\pi$
$240$		0			0
$241$	92.7885			0.385015			0.192507	−	0.981296i	$-0.438338\pi$
$241$	92.7885			0.385015			0.192507	+	0.981296i	$0.438338\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	15.1509	−	234.456i	0.0618402	−	0.956965i
$246$		0			0
$247$		−	138.185i		−	0.559454i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			140.501i			0.559765i	0.960034	+	0.279883i	$0.0902956\pi$
$251$			140.501i			0.559765i	−0.960034	+	0.279883i	$0.909704\pi$
$252$		0			0
$253$		−	423.606i		−	1.67433i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−448.630			−1.74564			−0.872821	−	0.488041i	$-0.837712\pi$
$257$	−448.630			−1.74564			−0.872821	+	0.488041i	$0.837712\pi$
$258$		0			0
$259$	0.289878			0.00111922
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−270.384			−1.02808			−0.514038	−	0.857767i	$-0.671851\pi$
$263$	−270.384			−1.02808			−0.514038	+	0.857767i	$0.671851\pi$
$264$		0			0
$265$	13.6558	−	211.320i	0.0515312	−	0.797436i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	96.8138i		−	0.359903i	−0.983676	−	0.179951i	$-0.942406\pi$
$269$		−	96.8138i		−	0.359903i	0.983676	−	0.179951i	$-0.0575940\pi$
$270$		0			0
$271$	−161.859			−0.597266			−0.298633	−	0.954368i	$-0.596531\pi$
$271$	−161.859			−0.597266			−0.298633	+	0.954368i	$0.596531\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−65.9561	+	508.198i	−0.239840	+	1.84799i
$276$		0			0
$277$			45.4093i			0.163933i	0.996635	+	0.0819663i	$0.0261200\pi$
$277$			45.4093i			0.163933i	−0.996635	+	0.0819663i	$0.973880\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			347.471i			1.23655i	0.785962	+	0.618275i	$0.212168\pi$
$281$			347.471i			1.23655i	−0.785962	+	0.618275i	$0.787832\pi$
$282$		0			0
$283$			133.573i			0.471991i	0.971754	+	0.235995i	$0.0758350\pi$
$283$			133.573i			0.471991i	−0.971754	+	0.235995i	$0.924165\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	116.152			0.404709
$288$		0			0
$289$	−144.881			−0.501317
$290$		0			0
$291$		0			0
$292$		0			0
$293$	510.677			1.74292			0.871462	−	0.490463i	$-0.163172\pi$
$293$	510.677			1.74292			0.871462	+	0.490463i	$0.163172\pi$
$294$		0			0
$295$	102.220	+	6.60556i	0.346508	+	0.0223917i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	94.0225i		−	0.314457i
$300$		0			0
$301$	799.706			2.65683
$302$		0			0
$303$		0			0
$304$		0			0
$305$	20.2038	−	312.649i	0.0662419	−	1.02508i
$306$		0			0
$307$			105.199i			0.342668i	0.985213	+	0.171334i	$0.0548078\pi$
$307$			105.199i			0.342668i	−0.985213	+	0.171334i	$0.945192\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	337.291i		−	1.08454i	−0.840205	−	0.542268i	$-0.817566\pi$
$311$		−	337.291i		−	1.08454i	0.840205	−	0.542268i	$-0.182434\pi$
$312$		0			0
$313$		−	169.006i		−	0.539956i	−0.962867	−	0.269978i	$-0.912984\pi$
$313$		−	169.006i		−	0.539956i	0.962867	−	0.269978i	$-0.0870164\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−198.794			−0.627111			−0.313555	−	0.949570i	$-0.601520\pi$
$317$	−198.794			−0.627111			−0.313555	+	0.949570i	$0.601520\pi$
$318$		0			0
$319$	−610.926			−1.91513
$320$		0			0
$321$		0			0
$322$		0			0
$323$	364.613			1.12883
$324$		0			0
$325$	−14.6394	+	112.798i	−0.0450445	+	0.347072i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			569.402i			1.73070i
$330$		0			0
$331$	−654.104			−1.97615			−0.988073	−	0.153985i	$-0.950789\pi$
$331$	−654.104			−1.97615			−0.988073	+	0.153985i	$0.950789\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	374.666	+	24.2114i	1.11841	+	0.0722727i
$336$		0			0
$337$		−	79.9305i		−	0.237183i	−0.992943	−	0.118591i	$-0.962162\pi$
$337$		−	79.9305i		−	0.237183i	0.992943	−	0.118591i	$-0.0378378\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			749.221i			2.19713i
$342$		0			0
$343$			19.7016i			0.0574392i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−340.308			−0.980716			−0.490358	−	0.871521i	$-0.663134\pi$
$347$	−340.308			−0.980716			−0.490358	+	0.871521i	$0.663134\pi$
$348$		0			0
$349$	−185.073			−0.530294			−0.265147	−	0.964208i	$-0.585421\pi$
$349$	−185.073			−0.530294			−0.265147	+	0.964208i	$0.585421\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−180.031			−0.510003			−0.255001	−	0.966941i	$-0.582076\pi$
$353$	−180.031			−0.510003			−0.255001	+	0.966941i	$0.582076\pi$
$354$		0			0
$355$	−143.035	−	9.24312i	−0.402917	−	0.0260370i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	62.2779i		−	0.173476i	−0.996231	−	0.0867380i	$-0.972356\pi$
$359$		−	62.2779i		−	0.173476i	0.996231	−	0.0867380i	$-0.0276443\pi$
$360$		0			0
$361$	561.450			1.55526
$362$		0			0
$363$		0			0
$364$		0			0
$365$	554.867	+	35.8561i	1.52018	+	0.0982360i
$366$		0			0
$367$		−	379.817i		−	1.03492i	−0.855706	−	0.517462i	$-0.826877\pi$
$367$		−	379.817i		−	1.03492i	0.855706	−	0.517462i	$-0.173123\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	414.942i		−	1.11844i
$372$		0			0
$373$			630.707i			1.69090i	0.534051	+	0.845452i	$0.320669\pi$
$373$			630.707i			1.69090i	−0.534051	+	0.845452i	$0.679331\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−135.600			−0.359681
$378$		0			0
$379$	336.706			0.888407			0.444204	−	0.895926i	$-0.353487\pi$
$379$	336.706			0.888407			0.444204	+	0.895926i	$0.353487\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−91.1717			−0.238046			−0.119023	−	0.992891i	$-0.537976\pi$
$383$	−91.1717			−0.238046			−0.119023	+	0.992891i	$0.537976\pi$
$384$		0			0
$385$	−64.7546	+	1002.07i	−0.168194	+	2.60277i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	245.537i		−	0.631201i	−0.948892	−	0.315601i	$-0.897794\pi$
$389$		−	245.537i		−	0.631201i	0.948892	−	0.315601i	$-0.102206\pi$
$390$		0			0
$391$	248.087			0.634492
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−31.1153	+	481.503i	−0.0787730	+	1.21900i
$396$		0			0
$397$		−	465.923i		−	1.17361i	−0.809728	−	0.586805i	$-0.800386\pi$
$397$		−	465.923i		−	1.17361i	0.809728	−	0.586805i	$-0.199614\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			215.443i			0.537264i	0.963243	+	0.268632i	$0.0865715\pi$
$401$			215.443i			0.537264i	−0.963243	+	0.268632i	$0.913428\pi$
$402$		0			0
$403$			166.295i			0.412643i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−0.606491			−0.00149015
$408$		0			0
$409$	349.453			0.854408			0.427204	−	0.904155i	$-0.359499\pi$
$409$	349.453			0.854408			0.427204	+	0.904155i	$0.359499\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	200.715			0.485994
$414$		0			0
$415$	−41.9906	+	649.796i	−0.101182	+	1.56577i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			740.404i			1.76707i	0.468361	+	0.883537i	$0.344845\pi$
$419$			740.404i			1.76707i	−0.468361	+	0.883537i	$0.655155\pi$
$420$		0			0
$421$	192.237			0.456621			0.228310	−	0.973588i	$-0.426680\pi$
$421$	192.237			0.456621			0.228310	+	0.973588i	$0.426680\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−297.628	−	38.6275i	−0.700301	−	0.0908881i
$426$		0			0
$427$		−	613.908i		−	1.43772i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	388.141i		−	0.900560i	−0.892887	−	0.450280i	$-0.851324\pi$
$431$		−	388.141i		−	0.900560i	0.892887	−	0.450280i	$-0.148676\pi$
$432$		0			0
$433$		−	333.349i		−	0.769859i	−0.922946	−	0.384930i	$-0.874226\pi$
$433$		−	333.349i		−	0.769859i	0.922946	−	0.384930i	$-0.125774\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	627.644			1.43626
$438$		0			0
$439$	−613.576			−1.39767			−0.698834	−	0.715284i	$-0.746297\pi$
$439$	−613.576			−1.39767			−0.698834	+	0.715284i	$0.746297\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	241.960			0.546186			0.273093	−	0.961988i	$-0.411953\pi$
$443$	241.960			0.546186			0.273093	+	0.961988i	$0.411953\pi$
$444$		0			0
$445$	−204.370	−	13.2066i	−0.459258	−	0.0296778i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	157.626i		−	0.351060i	−0.984474	−	0.175530i	$-0.943836\pi$
$449$		−	157.626i		−	0.351060i	0.984474	−	0.175530i	$-0.0561640\pi$
$450$		0			0
$451$	−243.016			−0.538837
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−14.3728	+	222.416i	−0.0315885	+	0.488826i
$456$		0			0
$457$			83.1829i			0.182019i	0.995850	+	0.0910097i	$0.0290094\pi$
$457$			83.1829i			0.182019i	−0.995850	+	0.0910097i	$0.970991\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	478.994i		−	1.03903i	−0.854461	−	0.519516i	$-0.826112\pi$
$461$		−	478.994i		−	1.03903i	0.854461	−	0.519516i	$-0.173888\pi$
$462$		0			0
$463$			14.1236i			0.0305045i	0.999884	+	0.0152523i	$0.00485514\pi$
$463$			14.1236i			0.0305045i	−0.999884	+	0.0152523i	$0.995145\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−363.313			−0.777973			−0.388986	−	0.921244i	$-0.627175\pi$
$467$	−363.313			−0.777973			−0.388986	+	0.921244i	$0.627175\pi$
$468$		0			0
$469$	735.682			1.56862
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1673.17			−3.53735
$474$		0			0
$475$	−752.981	−	97.7251i	−1.58522	−	0.205737i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	426.790i		−	0.891002i	−0.895282	−	0.445501i	$-0.853026\pi$
$479$		−	426.790i		−	0.891002i	0.895282	−	0.445501i	$-0.146974\pi$
$480$		0			0
$481$	−0.134615			−0.000279865
$482$		0			0
$483$		0			0
$484$		0			0
$485$	74.3822	+	4.80667i	0.153365	+	0.00991065i
$486$		0			0
$487$			416.702i			0.855651i	0.903861	+	0.427826i	$0.140720\pi$
$487$			416.702i			0.855651i	−0.903861	+	0.427826i	$0.859280\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	537.708i		−	1.09513i	−0.836764	−	0.547564i	$-0.815555\pi$
$491$		−	537.708i		−	1.09513i	0.836764	−	0.547564i	$-0.184445\pi$
$492$		0			0
$493$		−	357.792i		−	0.725743i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−280.860			−0.565110
$498$		0			0
$499$	513.498			1.02905			0.514527	−	0.857474i	$-0.327968\pi$
$499$	513.498			1.02905			0.514527	+	0.857474i	$0.327968\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−689.783			−1.37134			−0.685669	−	0.727913i	$-0.740490\pi$
$503$	−689.783			−1.37134			−0.685669	+	0.727913i	$0.740490\pi$
$504$		0			0
$505$	−330.658	−	21.3675i	−0.654768	−	0.0423119i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	475.164i		−	0.933524i	−0.884383	−	0.466762i	$-0.845420\pi$
$509$		−	475.164i		−	0.933524i	0.884383	−	0.466762i	$-0.154580\pi$
$510$		0			0
$511$	1089.52			2.13213
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−671.509	−	43.3937i	−1.30390	−	0.0842596i
$516$		0			0
$517$		−	1191.32i		−	2.30429i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			400.503i			0.768720i	0.923183	+	0.384360i	$0.125578\pi$
$521$			400.503i			0.768720i	−0.923183	+	0.384360i	$0.874422\pi$
$522$		0			0
$523$		−	325.250i		−	0.621892i	−0.950428	−	0.310946i	$-0.899354\pi$
$523$		−	325.250i		−	0.621892i	0.950428	−	0.310946i	$-0.100646\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−438.784			−0.832608
$528$		0			0
$529$	−101.945			−0.192712
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−53.9391			−0.101199
$534$		0			0
$535$	−41.1798	+	637.250i	−0.0769716	+	1.19112i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			963.201i			1.78702i
$540$		0			0
$541$	−558.262			−1.03191			−0.515953	−	0.856617i	$-0.672562\pi$
$541$	−558.262			−1.03191			−0.515953	+	0.856617i	$0.672562\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	16.6643	−	257.876i	0.0305766	−	0.473167i
$546$		0			0
$547$		−	188.007i		−	0.343706i	−0.985123	−	0.171853i	$-0.945025\pi$
$547$		−	188.007i		−	0.343706i	0.985123	−	0.171853i	$-0.0549755\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	905.191i		−	1.64281i
$552$		0			0
$553$			945.465i			1.70970i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−391.234			−0.702395			−0.351197	−	0.936301i	$-0.614225\pi$
$557$	−391.234			−0.702395			−0.351197	+	0.936301i	$0.614225\pi$
$558$		0			0
$559$	−371.372			−0.664350
$560$		0			0
$561$		0			0
$562$		0			0
$563$	888.504			1.57816			0.789080	−	0.614290i	$-0.210558\pi$
$563$	888.504			1.57816			0.789080	+	0.614290i	$0.210558\pi$
$564$		0			0
$565$	12.5713	−	194.538i	0.0222500	−	0.344315i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	576.466i		−	1.01312i	−0.862204	−	0.506561i	$-0.830917\pi$
$569$		−	576.466i		−	1.01312i	0.862204	−	0.506561i	$-0.169083\pi$
$570$		0			0
$571$	1043.32			1.82718			0.913591	−	0.406634i	$-0.133298\pi$
$571$	1043.32			1.82718			0.913591	+	0.406634i	$0.133298\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−512.336	−	66.4932i	−0.891019	−	0.115640i
$576$		0			0
$577$			473.407i			0.820463i	0.911981	+	0.410231i	$0.134552\pi$
$577$			473.407i			0.820463i	−0.911981	+	0.410231i	$0.865448\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			1275.92i			2.19607i
$582$		0			0
$583$			868.153i			1.48911i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	520.307			0.886383			0.443192	−	0.896427i	$-0.353846\pi$
$587$	520.307			0.886383			0.443192	+	0.896427i	$0.353846\pi$
$588$		0			0
$589$	−1110.10			−1.88472
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−181.237			−0.305627			−0.152814	−	0.988255i	$-0.548833\pi$
$593$	−181.237			−0.305627			−0.152814	+	0.988255i	$0.548833\pi$
$594$		0			0
$595$	−586.864	−	37.9238i	−0.986325	−	0.0637375i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			555.146i			0.926788i	0.886152	+	0.463394i	$0.153369\pi$
$599$			555.146i			0.926788i	−0.886152	+	0.463394i	$0.846631\pi$
$600$		0			0
$601$	254.752			0.423880			0.211940	−	0.977283i	$-0.432022\pi$
$601$	254.752			0.423880			0.211940	+	0.977283i	$0.432022\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	96.4670	−	1492.81i	0.159450	−	2.46745i
$606$		0			0
$607$		−	579.724i		−	0.955064i	−0.878614	−	0.477532i	$-0.841531\pi$
$607$		−	579.724i		−	0.955064i	0.878614	−	0.477532i	$-0.158469\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	264.422i		−	0.432769i
$612$		0			0
$613$			582.756i			0.950663i	0.879807	+	0.475331i	$0.157672\pi$
$613$			582.756i			0.950663i	−0.879807	+	0.475331i	$0.842328\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	322.513			0.522711			0.261355	−	0.965243i	$-0.415831\pi$
$617$	322.513			0.522711			0.261355	+	0.965243i	$0.415831\pi$
$618$		0			0
$619$	−172.742			−0.279067			−0.139533	−	0.990217i	$-0.544560\pi$
$619$	−172.742			−0.279067			−0.139533	+	0.990217i	$0.544560\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−401.294			−0.644131
$624$		0			0
$625$	604.294	+	159.543i	0.966870	+	0.255269i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	0.355194i		−	0.000564696i
$630$		0			0
$631$	55.0383			0.0872240			0.0436120	−	0.999049i	$-0.486113\pi$
$631$	55.0383			0.0872240			0.0436120	+	0.999049i	$0.486113\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−638.240	−	41.2438i	−1.00510	−	0.0649509i
$636$		0			0
$637$			213.790i			0.335620i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	129.694i		−	0.202331i	−0.994870	−	0.101166i	$-0.967743\pi$
$641$		−	129.694i		−	0.202331i	0.994870	−	0.101166i	$-0.0322572\pi$
$642$		0			0
$643$			635.666i			0.988595i	0.869293	+	0.494297i	$0.164575\pi$
$643$			635.666i			0.988595i	−0.869293	+	0.494297i	$0.835425\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	118.098			0.182532			0.0912659	−	0.995827i	$-0.470909\pi$
$647$	118.098			0.182532			0.0912659	+	0.995827i	$0.470909\pi$
$648$		0			0
$649$	−419.942			−0.647060
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−411.896			−0.630775			−0.315387	−	0.948963i	$-0.602134\pi$
$653$	−411.896			−0.630775			−0.315387	+	0.948963i	$0.602134\pi$
$654$		0			0
$655$	714.458	+	46.1691i	1.09078	+	0.0704872i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	1082.14i		−	1.64209i	−0.570864	−	0.821045i	$-0.693392\pi$
$659$		−	1082.14i		−	1.64209i	0.570864	−	0.821045i	$-0.306608\pi$
$660$		0			0
$661$	−350.371			−0.530062			−0.265031	−	0.964240i	$-0.585382\pi$
$661$	−350.371			−0.530062			−0.265031	+	0.964240i	$0.585382\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1484.73	−	95.9450i	−2.23268	−	0.144278i
$666$		0			0
$667$		−	615.901i		−	0.923390i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			1284.44i			1.91421i
$672$		0			0
$673$			170.817i			0.253814i	0.991915	+	0.126907i	$0.0405050\pi$
$673$			170.817i			0.253814i	−0.991915	+	0.126907i	$0.959495\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1080.49			1.59600			0.797999	−	0.602659i	$-0.205892\pi$
$677$	1080.49			1.59600			0.797999	+	0.602659i	$0.205892\pi$
$678$		0			0
$679$	146.055			0.215102
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−970.271			−1.42060			−0.710301	−	0.703898i	$-0.751441\pi$
$683$	−970.271			−1.42060			−0.710301	+	0.703898i	$0.751441\pi$
$684$		0			0
$685$	−32.3747	+	500.992i	−0.0472623	+	0.731375i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			192.693i			0.279671i
$690$		0			0
$691$	1134.57			1.64193			0.820963	−	0.570981i	$-0.193437\pi$
$691$	1134.57			1.64193			0.820963	+	0.570981i	$0.193437\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−53.5398	+	828.518i	−0.0770357	+	1.19211i
$696$		0			0
$697$		−	142.323i		−	0.204194i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			119.201i			0.170045i	0.996379	+	0.0850225i	$0.0270962\pi$
$701$			119.201i			0.170045i	−0.996379	+	0.0850225i	$0.972904\pi$
$702$		0			0
$703$		−	0.898619i		−	0.00127826i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−649.269			−0.918343
$708$		0			0
$709$	−375.731			−0.529945			−0.264973	−	0.964256i	$-0.585363\pi$
$709$	−375.731			−0.529945			−0.264973	+	0.964256i	$0.585363\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−755.322			−1.05936
$714$		0			0
$715$	30.0711	−	465.345i	0.0420575	−	0.650832i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			36.5907i			0.0508911i	0.999676	+	0.0254456i	$0.00810045\pi$
$719$			36.5907i			0.0508911i	−0.999676	+	0.0254456i	$0.991900\pi$
$720$		0			0
$721$	−1318.55			−1.82878
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−95.8968	+	738.893i	−0.132271	+	1.01916i
$726$		0			0
$727$		−	929.324i		−	1.27830i	−0.769082	−	0.639150i	$-0.779286\pi$
$727$		−	929.324i		−	1.27830i	0.769082	−	0.639150i	$-0.220714\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	979.897i		−	1.34049i
$732$		0			0
$733$			421.172i			0.574587i	0.957843	+	0.287293i	$0.0927555\pi$
$733$			421.172i			0.574587i	−0.957843	+	0.287293i	$0.907245\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1539.21			−2.08849
$738$		0			0
$739$	1371.75			1.85622			0.928112	−	0.372302i	$-0.121432\pi$
$739$	1371.75			1.85622			0.928112	+	0.372302i	$0.121432\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	395.236			0.531946			0.265973	−	0.963980i	$-0.414307\pi$
$743$	395.236			0.531946			0.265973	+	0.963980i	$0.414307\pi$
$744$		0			0
$745$	−717.685	−	46.3777i	−0.963336	−	0.0622519i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			1251.28i			1.67060i
$750$		0			0
$751$	−755.093			−1.00545			−0.502725	−	0.864446i	$-0.667669\pi$
$751$	−755.093			−1.00545			−0.502725	+	0.864446i	$0.667669\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−25.4796	+	394.292i	−0.0337478	+	0.522241i
$756$		0			0
$757$		−	1443.36i		−	1.90668i	−0.301899	−	0.953340i	$-0.597621\pi$
$757$		−	1443.36i		−	1.90668i	0.301899	−	0.953340i	$-0.402379\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			1162.57i			1.52769i	0.645402	+	0.763843i	$0.276690\pi$
$761$			1162.57i			1.52769i	−0.645402	+	0.763843i	$0.723310\pi$
$762$		0			0
$763$		−	506.357i		−	0.663640i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−93.2093			−0.121525
$768$		0			0
$769$	315.120			0.409779			0.204890	−	0.978785i	$-0.434316\pi$
$769$	315.120			0.409779			0.204890	+	0.978785i	$0.434316\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	505.575			0.654043			0.327021	−	0.945017i	$-0.393955\pi$
$773$	505.575			0.654043			0.327021	+	0.945017i	$0.393955\pi$
$774$		0			0
$775$	906.156	+	117.605i	1.16923	+	0.151748i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	360.069i		−	0.462219i
$780$		0			0
$781$	587.623			0.752398
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−202.020	−	13.0547i	−0.257350	−	0.0166303i
$786$		0			0
$787$			315.935i			0.401442i	0.979648	+	0.200721i	$0.0643285\pi$
$787$			315.935i			0.401442i	−0.979648	+	0.200721i	$0.935672\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	381.988i		−	0.482918i
$792$		0			0
$793$			285.090i			0.359508i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	143.948			0.180612			0.0903060	−	0.995914i	$-0.471216\pi$
$797$	143.948			0.180612			0.0903060	+	0.995914i	$0.471216\pi$
$798$		0			0
$799$	697.700			0.873217
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−2279.52			−2.83875
$804$		0			0
$805$	−1010.23	−	65.2819i	−1.25494	−	0.0810956i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			959.959i			1.18660i	0.804982	+	0.593300i	$0.202175\pi$
$809$			959.959i			1.18660i	−0.804982	+	0.593300i	$0.797825\pi$
$810$		0			0
$811$	−462.010			−0.569680			−0.284840	−	0.958575i	$-0.591940\pi$
$811$	−462.010			−0.569680			−0.284840	+	0.958575i	$0.591940\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1058.88	−	68.4261i	−1.29924	−	0.0839584i
$816$		0			0
$817$		−	2479.08i		−	3.03437i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	406.587i		−	0.495234i	−0.968858	−	0.247617i	$-0.920353\pi$
$821$		−	406.587i		−	0.495234i	0.968858	−	0.247617i	$-0.0796474\pi$
$822$		0			0
$823$			763.410i			0.927594i	0.885941	+	0.463797i	$0.153513\pi$
$823$			763.410i			0.927594i	−0.885941	+	0.463797i	$0.846487\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	709.076			0.857407			0.428704	−	0.903445i	$-0.358970\pi$
$827$	709.076			0.857407			0.428704	+	0.903445i	$0.358970\pi$
$828$		0			0
$829$	378.921			0.457082			0.228541	−	0.973534i	$-0.426605\pi$
$829$	378.921			0.457082			0.228541	+	0.973534i	$0.426605\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−564.103			−0.677194
$834$		0			0
$835$	−69.7964	+	1080.09i	−0.0835885	+	1.29352i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			186.622i			0.222434i	0.993796	+	0.111217i	$0.0354748\pi$
$839$			186.622i			0.222434i	−0.993796	+	0.111217i	$0.964525\pi$
$840$		0			0
$841$	−47.2555			−0.0561896
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−47.8167	+	739.954i	−0.0565878	+	0.875686i
$846$		0			0
$847$		−	2931.23i		−	3.46072i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	0.611430i		−	0.000718484i
$852$		0			0
$853$		−	374.949i		−	0.439565i	−0.975549	−	0.219782i	$-0.929465\pi$
$853$		−	374.949i		−	0.439565i	0.975549	−	0.219782i	$-0.0705348\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1448.08			−1.68971			−0.844855	−	0.534996i	$-0.820313\pi$
$857$	−1448.08			−1.68971			−0.844855	+	0.534996i	$0.820313\pi$
$858$		0			0
$859$	−205.005			−0.238655			−0.119328	−	0.992855i	$-0.538074\pi$
$859$	−205.005			−0.238655			−0.119328	+	0.992855i	$0.538074\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−416.605			−0.482740			−0.241370	−	0.970433i	$-0.577597\pi$
$863$	−416.605			−0.482740			−0.241370	+	0.970433i	$0.577597\pi$
$864$		0			0
$865$	−91.5275	+	1416.37i	−0.105812	+	1.63742i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	1978.13i		−	2.27633i
$870$		0			0
$871$	−341.640			−0.392239
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1201.80	+	235.612i	1.37348	+	0.269271i
$876$		0			0
$877$		−	617.847i		−	0.704500i	−0.935906	−	0.352250i	$-0.885417\pi$
$877$		−	617.847i		−	0.704500i	0.935906	−	0.352250i	$-0.114583\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			56.6963i			0.0643545i	0.999482	+	0.0321773i	$0.0102441\pi$
$881$			56.6963i			0.0643545i	−0.999482	+	0.0321773i	$0.989756\pi$
$882$		0			0
$883$			127.768i			0.144698i	0.997379	+	0.0723490i	$0.0230495\pi$
$883$			127.768i			0.144698i	−0.997379	+	0.0723490i	$0.976950\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1040.74			1.17333			0.586663	−	0.809831i	$-0.300441\pi$
$887$	1040.74			1.17333			0.586663	+	0.809831i	$0.300441\pi$
$888$		0			0
$889$	−1253.23			−1.40970
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1765.14			1.97664
$894$		0			0
$895$	171.189	+	11.0624i	0.191273	+	0.0123603i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			1089.33i			1.21171i
$900$		0			0
$901$	−508.438			−0.564304
$902$		0			0
$903$		0			0
$904$		0			0
$905$	72.5468	−	1122.65i	0.0801622	−	1.24049i
$906$		0			0
$907$			1194.54i			1.31702i	0.752572	+	0.658510i	$0.228813\pi$
$907$			1194.54i			1.31702i	−0.752572	+	0.658510i	$0.771187\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1177.78i			1.29285i	0.762979	+	0.646423i	$0.223736\pi$
$911$			1177.78i			1.29285i	−0.762979	+	0.646423i	$0.776264\pi$
$912$		0			0
$913$		−	2669.51i		−	2.92389i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1402.89			1.52987
$918$		0			0
$919$	92.3298			0.100468			0.0502339	−	0.998737i	$-0.484003\pi$
$919$	92.3298			0.100468			0.0502339	+	0.998737i	$0.484003\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	130.427			0.141308
$924$		0			0
$925$	−0.0952005	+	0.733529i	−0.000102919	+	0.000793004i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	1261.17i		−	1.35756i	−0.734341	−	0.678781i	$-0.762509\pi$
$929$		−	1261.17i		−	1.35756i	0.734341	−	0.678781i	$-0.237491\pi$
$930$		0			0
$931$	−1427.15			−1.53292
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1227.85	+	79.3453i	1.31321	+	0.0848612i
$936$		0			0
$937$			762.324i			0.813579i	0.913522	+	0.406790i	$0.133352\pi$
$937$			762.324i			0.813579i	−0.913522	+	0.406790i	$0.866648\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			1419.27i			1.50826i	0.656727	+	0.754129i	$0.271940\pi$
$941$			1419.27i			1.50826i	−0.656727	+	0.754129i	$0.728060\pi$
$942$		0			0
$943$		−	244.994i		−	0.259803i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1036.68			−1.09470			−0.547351	−	0.836903i	$-0.684364\pi$
$947$	−1036.68			−1.09470			−0.547351	+	0.836903i	$0.684364\pi$
$948$		0			0
$949$	−505.956			−0.533147
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−27.5762			−0.0289362			−0.0144681	−	0.999895i	$-0.504605\pi$
$953$	−27.5762			−0.0289362			−0.0144681	+	0.999895i	$0.504605\pi$
$954$		0			0
$955$	1568.82	+	101.379i	1.64275	+	0.106156i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			983.732i			1.02579i
$960$		0			0
$961$	374.920			0.390135
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−941.060	−	60.8124i	−0.975191	−	0.0630180i
$966$		0			0
$967$		−	964.705i		−	0.997627i	−0.866709	−	0.498813i	$-0.833769\pi$
$967$		−	964.705i		−	0.997627i	0.866709	−	0.498813i	$-0.166231\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			117.707i			0.121222i	0.998161	+	0.0606111i	$0.0193049\pi$
$971$			117.707i			0.121222i	−0.998161	+	0.0606111i	$0.980695\pi$
$972$		0			0
$973$			1626.85i			1.67199i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−932.883			−0.954845			−0.477422	−	0.878674i	$-0.658429\pi$
$977$	−932.883			−0.954845			−0.477422	+	0.878674i	$0.658429\pi$
$978$		0			0
$979$	839.598			0.857607
$980$		0			0
$981$		0			0
$982$		0			0
$983$	291.311			0.296349			0.148174	−	0.988961i	$-0.452660\pi$
$983$	291.311			0.296349			0.148174	+	0.988961i	$0.452660\pi$
$984$		0			0
$985$	−59.0732	+	914.146i	−0.0599728	+	0.928067i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	1686.79i		−	1.70555i
$990$		0			0
$991$	−100.973			−0.101890			−0.0509450	−	0.998701i	$-0.516223\pi$
$991$	−100.973			−0.101890			−0.0509450	+	0.998701i	$0.516223\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	59.4854	−	920.525i	0.0597843	−	0.925151i
$996$		0			0
$997$			326.771i			0.327755i	0.986481	+	0.163877i	$0.0524001\pi$
$997$			326.771i			0.327755i	−0.986481	+	0.163877i	$0.947600\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.3.c.q.1889.12		24
3.2	odd	2	inner	2160.3.c.q.1889.13		24
4.3	odd	2		1080.3.c.c.809.12	yes	24
5.4	even	2	inner	2160.3.c.q.1889.14		24
12.11	even	2		1080.3.c.c.809.13	yes	24
15.14	odd	2	inner	2160.3.c.q.1889.11		24
20.19	odd	2		1080.3.c.c.809.14	yes	24
60.59	even	2		1080.3.c.c.809.11	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.3.c.c.809.11	✓	24	60.59	even	2
1080.3.c.c.809.12	yes	24	4.3	odd	2
1080.3.c.c.809.13	yes	24	12.11	even	2
1080.3.c.c.809.14	yes	24	20.19	odd	2
2160.3.c.q.1889.11		24	15.14	odd	2	inner
2160.3.c.q.1889.12		24	1.1	even	1	trivial
2160.3.c.q.1889.13		24	3.2	odd	2	inner
2160.3.c.q.1889.14		24	5.4	even	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2160.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.322433 + 4.98959i) q^{5} +9.79740i q^{7} -20.4984i q^{11} -4.54977i q^{13} +12.0050 q^{17} +30.3719 q^{19} +20.6653 q^{23} +(-24.7921 - 3.21762i) q^{25} -29.8036i q^{29} -36.5502 q^{31} +(-48.8850 - 3.15901i) q^{35} -0.0295872i q^{37} -11.8553i q^{41} -81.6243i q^{43} +58.1176 q^{47} -46.9891 q^{49} -42.3523 q^{53} +(102.279 + 6.60937i) q^{55} -20.4866i q^{59} -62.6603 q^{61} +(22.7015 + 1.46700i) q^{65} -75.0895i q^{67} +28.6668i q^{71} -111.205i q^{73} +200.831 q^{77} +96.5016 q^{79} +130.230 q^{83} +(-3.87080 + 59.8999i) q^{85} +40.9592i q^{89} +44.5760 q^{91} +(-9.79290 + 151.543i) q^{95} -14.9075i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 72 q^{25} + 72 q^{31} - 408 q^{49} + 168 q^{55} - 240 q^{61} + 312 q^{79} - 96 q^{85}+O(q^{100}) \) 24 * q - 72 * q^25 + 72 * q^31 - 408 * q^49 + 168 * q^55 - 240 * q^61 + 312 * q^79 - 96 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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