LMFDB - Embedded newform 936.4.c.d.649.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [936,4,Mod(649,936)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(936, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("936.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 936 = 2^{3} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	936.c (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:	$55.2257877654$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 772 x^{18} + 254556 x^{16} + 46787036 x^{14} + 5240196398 x^{12} + 366523224996 x^{10} + \cdots + 24\!\cdots\!76 $ x^20 + 772x^18 + 254556x^16 + 46787036x^14 + 5240196398x^12 + 366523224996x^10 + 15714484512880x^8 + 385506177829060x^6 + 4590326443525329x^4 + 16656245475356664*x^2 + 2455943229260676
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{66} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.17
Root			$-8.41248i$ of defining polynomial
Character	$\chi$	$=$	936.649
Dual form			936.4.c.d.649.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+17.2883i q^{5} -1.33952i q^{7} -16.4827i q^{11} +(13.5060 - 44.8842i) q^{13} +99.2502 q^{17} -40.5282i q^{19} -125.857 q^{23} -173.887 q^{25} +51.7704 q^{29} -303.633i q^{31} +23.1580 q^{35} -85.9949i q^{37} -418.697i q^{41} -203.121 q^{43} +350.809i q^{47} +341.206 q^{49} +619.498 q^{53} +284.959 q^{55} -168.719i q^{59} +91.0490 q^{61} +(775.973 + 233.496i) q^{65} +864.165i q^{67} +198.574i q^{71} -87.1751i q^{73} -22.0789 q^{77} -740.668 q^{79} -1066.25i q^{83} +1715.87i q^{85} -382.555i q^{89} +(-60.1230 - 18.0915i) q^{91} +700.665 q^{95} +1020.69i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 84 q^{13} - 412 q^{25} - 288 q^{43} + 284 q^{49} + 576 q^{55} + 104 q^{61} + 432 q^{79} + 624 q^{91}+O(q^{100}) $ 20 * q - 84 * q^13 - 412 * q^25 - 288 * q^43 + 284 * q^49 + 576 * q^55 + 104 * q^61 + 432 * q^79 + 624 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/936\mathbb{Z}\right)^\times$.

$n$	$145$	$209$	$469$	$703$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$5$			17.2883i			1.54632i	0.634213	+	0.773158i	$0.281324\pi$
$5$			17.2883i			1.54632i	−0.634213	+	0.773158i	$0.718676\pi$
$6$		0			0
$7$		−	1.33952i		−	0.0723270i	−0.999346	−	0.0361635i	$-0.988486\pi$
$7$		−	1.33952i		−	0.0723270i	0.999346	−	0.0361635i	$-0.0115137\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	16.4827i		−	0.451794i	−0.974151	−	0.225897i	$-0.927469\pi$
$11$		−	16.4827i		−	0.451794i	0.974151	−	0.225897i	$-0.0725313\pi$
$12$		0			0
$13$	13.5060	−	44.8842i	0.288145	−	0.957587i
$13$	13.5060	−	44.8842i	0.288145	−	0.957587i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	99.2502			1.41598			0.707991	−	0.706221i	$-0.249602\pi$
$17$	99.2502			1.41598			0.707991	+	0.706221i	$0.249602\pi$
$18$		0			0
$19$		−	40.5282i		−	0.489358i	−0.969604	−	0.244679i	$-0.921317\pi$
$19$		−	40.5282i		−	0.489358i	0.969604	−	0.244679i	$-0.0786825\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−125.857			−1.14100			−0.570500	−	0.821297i	$-0.693251\pi$
$23$	−125.857			−1.14100			−0.570500	+	0.821297i	$0.693251\pi$
$24$		0			0
$25$	−173.887			−1.39110
$26$		0			0
$27$		0			0
$28$		0			0
$29$	51.7704			0.331501			0.165750	−	0.986168i	$-0.446995\pi$
$29$	51.7704			0.331501			0.165750	+	0.986168i	$0.446995\pi$
$30$		0			0
$31$		−	303.633i		−	1.75917i	−0.475745	−	0.879583i	$-0.657822\pi$
$31$		−	303.633i		−	1.75917i	0.475745	−	0.879583i	$-0.342178\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	23.1580			0.111840
$36$		0			0
$37$		−	85.9949i		−	0.382094i	−0.981581	−	0.191047i	$-0.938812\pi$
$37$		−	85.9949i		−	0.382094i	0.981581	−	0.191047i	$-0.0611883\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	418.697i		−	1.59487i	−0.603407	−	0.797434i	$-0.706190\pi$
$41$		−	418.697i		−	1.59487i	0.603407	−	0.797434i	$-0.293810\pi$
$42$		0			0
$43$	−203.121			−0.720365			−0.360183	−	0.932882i	$-0.617286\pi$
$43$	−203.121			−0.720365			−0.360183	+	0.932882i	$0.617286\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			350.809i			1.08874i	0.838845	+	0.544370i	$0.183231\pi$
$47$			350.809i			1.08874i	−0.838845	+	0.544370i	$0.816769\pi$
$48$		0			0
$49$	341.206			0.994769
$50$		0			0
$51$		0			0
$52$		0			0
$53$	619.498			1.60556			0.802779	−	0.596276i	$-0.203354\pi$
$53$	619.498			1.60556			0.802779	+	0.596276i	$0.203354\pi$
$54$		0			0
$55$	284.959			0.698617
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	168.719i		−	0.372293i	−0.982522	−	0.186146i	$-0.940400\pi$
$59$		−	168.719i		−	0.372293i	0.982522	−	0.186146i	$-0.0595999\pi$
$60$		0			0
$61$	91.0490			0.191109			0.0955543	−	0.995424i	$-0.469538\pi$
$61$	91.0490			0.191109			0.0955543	+	0.995424i	$0.469538\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	775.973	+	233.496i	1.48073	+	0.445564i
$66$		0			0
$67$			864.165i			1.57574i	0.615842	+	0.787869i	$0.288816\pi$
$67$			864.165i			1.57574i	−0.615842	+	0.787869i	$0.711184\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			198.574i			0.331920i	0.986132	+	0.165960i	$0.0530723\pi$
$71$			198.574i			0.331920i	−0.986132	+	0.165960i	$0.946928\pi$
$72$		0			0
$73$		−	87.1751i		−	0.139768i	−0.997555	−	0.0698840i	$-0.977737\pi$
$73$		−	87.1751i		−	0.139768i	0.997555	−	0.0698840i	$-0.0222629\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−22.0789			−0.0326769
$78$		0			0
$79$	−740.668			−1.05483			−0.527416	−	0.849607i	$-0.676839\pi$
$79$	−740.668			−1.05483			−0.527416	+	0.849607i	$0.676839\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	1066.25i		−	1.41007i	−0.709174	−	0.705034i	$-0.750932\pi$
$83$		−	1066.25i		−	1.41007i	0.709174	−	0.705034i	$-0.249068\pi$
$84$		0			0
$85$			1715.87i			2.18956i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	382.555i		−	0.455627i	−0.973705	−	0.227813i	$-0.926842\pi$
$89$		−	382.555i		−	0.455627i	0.973705	−	0.227813i	$-0.0731576\pi$
$90$		0			0
$91$	−60.1230	−	18.0915i	−0.0692594	−	0.0208407i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	700.665			0.756702
$96$		0			0
$97$			1020.69i			1.06840i	0.845357	+	0.534201i	$0.179388\pi$
$97$			1020.69i			1.06840i	−0.845357	+	0.534201i	$0.820612\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	442.610			0.436053			0.218026	−	0.975943i	$-0.430038\pi$
$101$	442.610			0.436053			0.218026	+	0.975943i	$0.430038\pi$
$102$		0			0
$103$	1350.29			1.29173			0.645866	−	0.763451i	$-0.276497\pi$
$103$	1350.29			1.29173			0.645866	+	0.763451i	$0.276497\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1538.49			1.39002			0.695008	−	0.719002i	$-0.255401\pi$
$107$	1538.49			1.39002			0.695008	+	0.719002i	$0.255401\pi$
$108$		0			0
$109$		−	745.005i		−	0.654666i	−0.944909	−	0.327333i	$-0.893850\pi$
$109$		−	745.005i		−	0.654666i	0.944909	−	0.327333i	$-0.106150\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1643.74			1.36841			0.684206	−	0.729289i	$-0.260149\pi$
$113$	1643.74			1.36841			0.684206	+	0.729289i	$0.260149\pi$
$114$		0			0
$115$		−	2175.86i		−	1.76435i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	132.947i		−	0.102414i
$120$		0			0
$121$	1059.32			0.795882
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	845.175i		−	0.604758i
$126$		0			0
$127$	229.512			0.160362			0.0801808	−	0.996780i	$-0.474450\pi$
$127$	229.512			0.160362			0.0801808	+	0.996780i	$0.474450\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	960.594			0.640668			0.320334	−	0.947305i	$-0.396205\pi$
$131$	960.594			0.640668			0.320334	+	0.947305i	$0.396205\pi$
$132$		0			0
$133$	−54.2881			−0.0353938
$134$		0			0
$135$		0			0
$136$		0			0
$137$			377.802i			0.235604i	0.993037	+	0.117802i	$0.0375849\pi$
$137$			377.802i			0.235604i	−0.993037	+	0.117802i	$0.962415\pi$
$138$		0			0
$139$	−1982.87			−1.20997			−0.604983	−	0.796239i	$-0.706820\pi$
$139$	−1982.87			−1.20997			−0.604983	+	0.796239i	$0.706820\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−739.814	−	222.616i	−0.432632	−	0.130182i
$144$		0			0
$145$			895.025i			0.512605i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	534.521i		−	0.293890i	−0.989145	−	0.146945i	$-0.953056\pi$
$149$		−	534.521i		−	0.293890i	0.989145	−	0.146945i	$-0.0469441\pi$
$150$		0			0
$151$		−	1479.80i		−	0.797511i	−0.917057	−	0.398756i	$-0.869442\pi$
$151$		−	1479.80i		−	0.797511i	0.917057	−	0.398756i	$-0.130558\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	5249.32			2.72023
$156$		0			0
$157$	−2873.65			−1.46078			−0.730389	−	0.683031i	$-0.760661\pi$
$157$	−2873.65			−1.46078			−0.730389	+	0.683031i	$0.760661\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			168.587i			0.0825252i
$162$		0			0
$163$		−	2978.08i		−	1.43105i	−0.698586	−	0.715526i	$-0.746187\pi$
$163$		−	2978.08i		−	1.43105i	0.698586	−	0.715526i	$-0.253813\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	751.862i		−	0.348388i	−0.984711	−	0.174194i	$-0.944268\pi$
$167$		−	751.862i		−	0.348388i	0.984711	−	0.174194i	$-0.0557320\pi$
$168$		0			0
$169$	−1832.18	−	1212.41i	−0.833945	−	0.551848i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−416.512			−0.183045			−0.0915227	−	0.995803i	$-0.529173\pi$
$173$	−416.512			−0.183045			−0.0915227	+	0.995803i	$0.529173\pi$
$174$		0			0
$175$			232.924i			0.100614i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	4391.66			1.83379			0.916894	−	0.399132i	$-0.130688\pi$
$179$	4391.66			1.83379			0.916894	+	0.399132i	$0.130688\pi$
$180$		0			0
$181$	2429.14			0.997550			0.498775	−	0.866731i	$-0.333783\pi$
$181$	2429.14			0.997550			0.498775	+	0.866731i	$0.333783\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1486.71			0.590838
$186$		0			0
$187$		−	1635.92i		−	0.639732i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1484.95			−0.562551			−0.281275	−	0.959627i	$-0.590757\pi$
$191$	−1484.95			−0.562551			−0.281275	+	0.959627i	$0.590757\pi$
$192$		0			0
$193$			2413.60i			0.900182i	0.892983	+	0.450091i	$0.148608\pi$
$193$			2413.60i			0.900182i	−0.892983	+	0.450091i	$0.851392\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			1217.73i			0.440403i	0.975454	+	0.220202i	$0.0706716\pi$
$197$			1217.73i			0.440403i	−0.975454	+	0.220202i	$0.929328\pi$
$198$		0			0
$199$	−3803.60			−1.35492			−0.677462	−	0.735558i	$-0.736920\pi$
$199$	−3803.60			−1.35492			−0.677462	+	0.735558i	$0.736920\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	69.3473i		−	0.0239765i
$204$		0			0
$205$	7238.59			2.46617
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−668.015			−0.221089
$210$		0			0
$211$	4203.36			1.37143			0.685713	−	0.727872i	$-0.259490\pi$
$211$	4203.36			1.37143			0.685713	+	0.727872i	$0.259490\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	3511.63i		−	1.11391i
$216$		0			0
$217$	−406.721			−0.127235
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1340.47	−	4454.76i	0.408008	−	1.35593i
$222$		0			0
$223$		−	3488.05i		−	1.04743i	−0.851893	−	0.523715i	$-0.824546\pi$
$223$		−	3488.05i		−	1.04743i	0.851893	−	0.523715i	$-0.175454\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	3099.32i		−	0.906208i	−0.891458	−	0.453104i	$-0.850317\pi$
$227$		−	3099.32i		−	0.906208i	0.891458	−	0.453104i	$-0.149683\pi$
$228$		0			0
$229$			5024.62i			1.44994i	0.688781	+	0.724970i	$0.258146\pi$
$229$			5024.62i			1.44994i	−0.688781	+	0.724970i	$0.741854\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−3763.69			−1.05823			−0.529115	−	0.848550i	$-0.677476\pi$
$233$	−3763.69			−1.05823			−0.529115	+	0.848550i	$0.677476\pi$
$234$		0			0
$235$	−6064.91			−1.68354
$236$		0			0
$237$		0			0
$238$		0			0
$239$			3728.62i			1.00914i	0.863371	+	0.504569i	$0.168349\pi$
$239$			3728.62i			1.00914i	−0.863371	+	0.504569i	$0.831651\pi$
$240$		0			0
$241$		−	718.392i		−	0.192015i	−0.995381	−	0.0960077i	$-0.969393\pi$
$241$		−	718.392i		−	0.192015i	0.995381	−	0.0960077i	$-0.0306073\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			5898.88i			1.53823i
$246$		0			0
$247$	−1819.07	−	547.373i	−0.468603	−	0.141006i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2207.13			0.555032			0.277516	−	0.960721i	$-0.410489\pi$
$251$	2207.13			0.555032			0.277516	+	0.960721i	$0.410489\pi$
$252$		0			0
$253$			2074.47i			0.515497i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1478.59			0.358879			0.179439	−	0.983769i	$-0.442572\pi$
$257$	1478.59			0.358879			0.179439	+	0.983769i	$0.442572\pi$
$258$		0			0
$259$	−115.191			−0.0276357
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2587.04			−0.606555			−0.303277	−	0.952902i	$-0.598081\pi$
$263$	−2587.04			−0.606555			−0.303277	+	0.952902i	$0.598081\pi$
$264$		0			0
$265$			10710.1i			2.48270i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	4495.23			1.01888			0.509440	−	0.860506i	$-0.329852\pi$
$269$	4495.23			1.01888			0.509440	+	0.860506i	$0.329852\pi$
$270$		0			0
$271$			3047.50i			0.683109i	0.939862	+	0.341555i	$0.110953\pi$
$271$			3047.50i			0.683109i	−0.939862	+	0.341555i	$0.889047\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			2866.13i			0.628489i
$276$		0			0
$277$	−4563.55			−0.989881			−0.494940	−	0.868927i	$-0.664810\pi$
$277$	−4563.55			−0.989881			−0.494940	+	0.868927i	$0.664810\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	3176.55i		−	0.674366i	−0.941439	−	0.337183i	$-0.890526\pi$
$281$		−	3176.55i		−	0.674366i	0.941439	−	0.337183i	$-0.109474\pi$
$282$		0			0
$283$	7429.65			1.56059			0.780295	−	0.625411i	$-0.215069\pi$
$283$	7429.65			1.56059			0.780295	+	0.625411i	$0.215069\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−560.851			−0.115352
$288$		0			0
$289$	4937.60			1.00501
$290$		0			0
$291$		0			0
$292$		0			0
$293$			1997.88i			0.398353i	0.979964	+	0.199176i	$0.0638267\pi$
$293$			1997.88i			0.398353i	−0.979964	+	0.199176i	$0.936173\pi$
$294$		0			0
$295$	2916.86			0.575683
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1699.82	+	5648.99i	−0.328774	+	1.09261i
$300$		0			0
$301$			272.084i			0.0521019i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			1574.09i			0.295514i
$306$		0			0
$307$		−	3896.95i		−	0.724464i	−0.932088	−	0.362232i	$-0.882015\pi$
$307$		−	3896.95i		−	0.724464i	0.932088	−	0.362232i	$-0.117985\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−5855.95			−1.06772			−0.533859	−	0.845573i	$-0.679259\pi$
$311$	−5855.95			−1.06772			−0.533859	+	0.845573i	$0.679259\pi$
$312$		0			0
$313$	−6573.33			−1.18705			−0.593525	−	0.804816i	$-0.702264\pi$
$313$	−6573.33			−1.18705			−0.593525	+	0.804816i	$0.702264\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	4104.55i		−	0.727238i	−0.931548	−	0.363619i	$-0.881541\pi$
$317$		−	4104.55i		−	0.727238i	0.931548	−	0.363619i	$-0.118459\pi$
$318$		0			0
$319$		−	853.318i		−	0.149770i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	4022.43i		−	0.692922i
$324$		0			0
$325$	−2348.51	+	7804.77i	−0.400837	+	1.33209i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	469.914			0.0787454
$330$		0			0
$331$		−	8447.81i		−	1.40282i	−0.712758	−	0.701410i	$-0.752554\pi$
$331$		−	8447.81i		−	1.40282i	0.712758	−	0.701410i	$-0.247446\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−14940.0			−2.43659
$336$		0			0
$337$	11278.4			1.82307			0.911534	−	0.411224i	$-0.134899\pi$
$337$	11278.4			1.82307			0.911534	+	0.411224i	$0.134899\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−5004.71			−0.794781
$342$		0			0
$343$		−	916.504i		−	0.144276i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6649.57			−1.02873			−0.514363	−	0.857573i	$-0.671972\pi$
$347$	−6649.57			−1.02873			−0.514363	+	0.857573i	$0.671972\pi$
$348$		0			0
$349$			279.964i			0.0429402i	0.999769	+	0.0214701i	$0.00683467\pi$
$349$			279.964i			0.0429402i	−0.999769	+	0.0214701i	$0.993165\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	3686.19i		−	0.555797i	−0.960611	−	0.277898i	$-0.910362\pi$
$353$		−	3686.19i		−	0.555797i	0.960611	−	0.277898i	$-0.0896378\pi$
$354$		0			0
$355$	−3433.01			−0.513254
$356$		0			0
$357$		0			0
$358$		0			0
$359$			10399.8i			1.52892i	0.644671	+	0.764460i	$0.276995\pi$
$359$			10399.8i			1.52892i	−0.644671	+	0.764460i	$0.723005\pi$
$360$		0			0
$361$	5216.47			0.760529
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1507.11			0.216126
$366$		0			0
$367$	−7144.35			−1.01616			−0.508081	−	0.861309i	$-0.669645\pi$
$367$	−7144.35			−1.01616			−0.508081	+	0.861309i	$0.669645\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	829.827i		−	0.116125i
$372$		0			0
$373$	−3431.34			−0.476322			−0.238161	−	0.971226i	$-0.576545\pi$
$373$	−3431.34			−0.476322			−0.238161	+	0.971226i	$0.576545\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	699.210	−	2323.67i	0.0955203	−	0.317441i
$378$		0			0
$379$		−	14342.7i		−	1.94390i	−0.235196	−	0.971948i	$-0.575573\pi$
$379$		−	14342.7i		−	1.94390i	0.235196	−	0.971948i	$-0.424427\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			5888.11i			0.785557i	0.919633	+	0.392778i	$0.128486\pi$
$383$			5888.11i			0.785557i	−0.919633	+	0.392778i	$0.871514\pi$
$384$		0			0
$385$		−	381.707i		−	0.0505288i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−3245.91			−0.423070			−0.211535	−	0.977370i	$-0.567846\pi$
$389$	−3245.91			−0.423070			−0.211535	+	0.977370i	$0.567846\pi$
$390$		0			0
$391$	−12491.3			−1.61564
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	12804.9i		−	1.63110i
$396$		0			0
$397$			413.285i			0.0522473i	0.999659	+	0.0261237i	$0.00831637\pi$
$397$			413.285i			0.0522473i	−0.999659	+	0.0261237i	$0.991684\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			8404.94i			1.04669i	0.852121	+	0.523345i	$0.175316\pi$
$401$			8404.94i			1.04669i	−0.852121	+	0.523345i	$0.824684\pi$
$402$		0			0
$403$	−13628.3	−	4100.87i	−1.68455	−	0.506895i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1417.43			−0.172628
$408$		0			0
$409$		−	11130.7i		−	1.34567i	−0.739795	−	0.672833i	$-0.765077\pi$
$409$		−	11130.7i		−	1.34567i	0.739795	−	0.672833i	$-0.234923\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−226.001			−0.0269268
$414$		0			0
$415$	18433.6			2.18041
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−10209.4			−1.19036			−0.595180	−	0.803592i	$-0.702919\pi$
$419$	−10209.4			−1.19036			−0.595180	+	0.803592i	$0.702919\pi$
$420$		0			0
$421$		−	8100.97i		−	0.937808i	−0.883249	−	0.468904i	$-0.844649\pi$
$421$		−	8100.97i		−	0.937808i	0.883249	−	0.468904i	$-0.155351\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−17258.3			−1.96977
$426$		0			0
$427$		−	121.961i		−	0.0138223i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			7383.64i			0.825191i	0.910914	+	0.412595i	$0.135378\pi$
$431$			7383.64i			0.825191i	−0.910914	+	0.412595i	$0.864622\pi$
$432$		0			0
$433$	−11312.3			−1.25551			−0.627756	−	0.778410i	$-0.716026\pi$
$433$	−11312.3			−1.25551			−0.627756	+	0.778410i	$0.716026\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			5100.76i			0.558357i
$438$		0			0
$439$	12699.3			1.38065			0.690325	−	0.723500i	$-0.257468\pi$
$439$	12699.3			1.38065			0.690325	+	0.723500i	$0.257468\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1394.40			−0.149549			−0.0747744	−	0.997200i	$-0.523824\pi$
$443$	−1394.40			−0.149549			−0.0747744	+	0.997200i	$0.523824\pi$
$444$		0			0
$445$	6613.75			0.704543
$446$		0			0
$447$		0			0
$448$		0			0
$449$			10523.6i			1.10610i	0.833147	+	0.553052i	$0.186537\pi$
$449$			10523.6i			1.10610i	−0.833147	+	0.553052i	$0.813463\pi$
$450$		0			0
$451$	−6901.28			−0.720551
$452$		0			0
$453$		0			0
$454$		0			0
$455$	312.772	−	1039.43i	0.0322263	−	0.107097i
$456$		0			0
$457$			14783.8i			1.51326i	0.653844	+	0.756629i	$0.273155\pi$
$457$			14783.8i			1.51326i	−0.653844	+	0.756629i	$0.726845\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			3483.79i			0.351965i	0.984393	+	0.175983i	$0.0563103\pi$
$461$			3483.79i			0.351965i	−0.984393	+	0.175983i	$0.943690\pi$
$462$		0			0
$463$			451.040i			0.0452734i	0.999744	+	0.0226367i	$0.00720610\pi$
$463$			451.040i			0.0452734i	−0.999744	+	0.0226367i	$0.992794\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−6717.97			−0.665676			−0.332838	−	0.942984i	$-0.608006\pi$
$467$	−6717.97			−0.665676			−0.332838	+	0.942984i	$0.608006\pi$
$468$		0			0
$469$	1157.56			0.113968
$470$		0			0
$471$		0			0
$472$		0			0
$473$			3348.00i			0.325457i
$474$		0			0
$475$			7047.32i			0.680743i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	1562.03i		−	0.149000i	−0.997221	−	0.0744999i	$-0.976264\pi$
$479$		−	1562.03i		−	0.149000i	0.997221	−	0.0744999i	$-0.0237361\pi$
$480$		0			0
$481$	−3859.81	−	1161.45i	−0.365888	−	0.110099i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−17646.0			−1.65209
$486$		0			0
$487$			3174.01i			0.295335i	0.989037	+	0.147668i	$0.0471766\pi$
$487$			3174.01i			0.295335i	−0.989037	+	0.147668i	$0.952823\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	3038.22			0.279253			0.139626	−	0.990204i	$-0.455410\pi$
$491$	3038.22			0.279253			0.139626	+	0.990204i	$0.455410\pi$
$492$		0			0
$493$	5138.22			0.469399
$494$		0			0
$495$		0			0
$496$		0			0
$497$	265.992			0.0240068
$498$		0			0
$499$		−	8223.09i		−	0.737707i	−0.929488	−	0.368854i	$-0.879750\pi$
$499$		−	8223.09i		−	0.737707i	0.929488	−	0.368854i	$-0.120250\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	18925.2			1.67760			0.838801	−	0.544438i	$-0.183257\pi$
$503$	18925.2			1.67760			0.838801	+	0.544438i	$0.183257\pi$
$504$		0			0
$505$			7651.99i			0.674276i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	16536.9i		−	1.44005i	−0.693947	−	0.720026i	$-0.744130\pi$
$509$		−	16536.9i		−	1.44005i	0.693947	−	0.720026i	$-0.255870\pi$
$510$		0			0
$511$	−116.772			−0.0101090
$512$		0			0
$513$		0			0
$514$		0			0
$515$			23344.3i			1.99743i
$516$		0			0
$517$	5782.30			0.491886
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−19979.0			−1.68003			−0.840014	−	0.542565i	$-0.817453\pi$
$521$	−19979.0			−1.68003			−0.840014	+	0.542565i	$0.817453\pi$
$522$		0			0
$523$	9944.23			0.831416			0.415708	−	0.909498i	$-0.363534\pi$
$523$	9944.23			0.831416			0.415708	+	0.909498i	$0.363534\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	30135.7i		−	2.49095i
$528$		0			0
$529$	3673.00			0.301882
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−18792.9	−	5654.92i	−1.52722	−	0.459553i
$534$		0			0
$535$			26598.0i			2.14941i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	5624.00i		−	0.449430i
$540$		0			0
$541$			6330.16i			0.503059i	0.967850	+	0.251529i	$0.0809335\pi$
$541$			6330.16i			0.503059i	−0.967850	+	0.251529i	$0.919066\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	12879.9			1.01232
$546$		0			0
$547$	3405.93			0.266229			0.133114	−	0.991101i	$-0.457502\pi$
$547$	3405.93			0.266229			0.133114	+	0.991101i	$0.457502\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	2098.16i		−	0.162223i
$552$		0			0
$553$			992.136i			0.0762928i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			17105.5i			1.30123i	0.759409	+	0.650613i	$0.225488\pi$
$557$			17105.5i			1.30123i	−0.759409	+	0.650613i	$0.774512\pi$
$558$		0			0
$559$	−2743.35	+	9116.93i	−0.207570	+	0.689812i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−16065.5			−1.20263			−0.601313	−	0.799013i	$-0.705356\pi$
$563$	−16065.5			−1.20263			−0.601313	+	0.799013i	$0.705356\pi$
$564$		0			0
$565$			28417.6i			2.11600i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−4798.25			−0.353521			−0.176760	−	0.984254i	$-0.556562\pi$
$569$	−4798.25			−0.353521			−0.176760	+	0.984254i	$0.556562\pi$
$570$		0			0
$571$	−14322.9			−1.04973			−0.524863	−	0.851187i	$-0.675883\pi$
$571$	−14322.9			−1.04973			−0.524863	+	0.851187i	$0.675883\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	21884.9			1.58724
$576$		0			0
$577$		−	7218.09i		−	0.520785i	−0.965503	−	0.260393i	$-0.916148\pi$
$577$		−	7218.09i		−	0.520785i	0.965503	−	0.260393i	$-0.0838520\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1428.25			−0.101986
$582$		0			0
$583$		−	10211.0i		−	0.725382i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			21791.8i			1.53227i	0.642678	+	0.766137i	$0.277823\pi$
$587$			21791.8i			1.53227i	−0.642678	+	0.766137i	$0.722177\pi$
$588$		0			0
$589$	−12305.7			−0.860862
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	18073.3i		−	1.25157i	−0.779995	−	0.625786i	$-0.784778\pi$
$593$		−	18073.3i		−	1.25157i	0.779995	−	0.625786i	$-0.215222\pi$
$594$		0			0
$595$	2298.44			0.158364
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−6806.01			−0.464250			−0.232125	−	0.972686i	$-0.574568\pi$
$599$	−6806.01			−0.464250			−0.232125	+	0.972686i	$0.574568\pi$
$600$		0			0
$601$	−4385.26			−0.297635			−0.148817	−	0.988865i	$-0.547547\pi$
$601$	−4385.26			−0.297635			−0.148817	+	0.988865i	$0.547547\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			18313.9i			1.23069i
$606$		0			0
$607$	20310.8			1.35814			0.679071	−	0.734073i	$-0.262383\pi$
$607$	20310.8			1.35814			0.679071	+	0.734073i	$0.262383\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	15745.8	+	4738.03i	1.04256	+	0.313715i
$612$		0			0
$613$		−	23747.4i		−	1.56468i	−0.622853	−	0.782339i	$-0.714026\pi$
$613$		−	23747.4i		−	1.56468i	0.622853	−	0.782339i	$-0.285974\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			6584.00i			0.429598i	0.976658	+	0.214799i	$0.0689096\pi$
$617$			6584.00i			0.429598i	−0.976658	+	0.214799i	$0.931090\pi$
$618$		0			0
$619$		−	4151.45i		−	0.269565i	−0.990875	−	0.134783i	$-0.956966\pi$
$619$		−	4151.45i		−	0.269565i	0.990875	−	0.134783i	$-0.0430336\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−512.439			−0.0329541
$624$		0			0
$625$	−7124.20			−0.455949
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	8535.01i		−	0.541039i
$630$		0			0
$631$			7471.18i			0.471352i	0.971832	+	0.235676i	$0.0757304\pi$
$631$			7471.18i			0.471352i	−0.971832	+	0.235676i	$0.924270\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			3967.89i			0.247970i
$636$		0			0
$637$	4608.32	−	15314.7i	0.286638	−	0.952577i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	20633.5			1.27141			0.635705	−	0.771932i	$-0.280710\pi$
$641$	20633.5			1.27141			0.635705	+	0.771932i	$0.280710\pi$
$642$		0			0
$643$			1331.30i			0.0816507i	0.999166	+	0.0408253i	$0.0129987\pi$
$643$			1331.30i			0.0816507i	−0.999166	+	0.0408253i	$0.987001\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	20430.6			1.24144			0.620718	−	0.784034i	$-0.286841\pi$
$647$	20430.6			1.24144			0.620718	+	0.784034i	$0.286841\pi$
$648$		0			0
$649$	−2780.94			−0.168200
$650$		0			0
$651$		0			0
$652$		0			0
$653$	4573.33			0.274071			0.137035	−	0.990566i	$-0.456243\pi$
$653$	4573.33			0.274071			0.137035	+	0.990566i	$0.456243\pi$
$654$		0			0
$655$			16607.1i			0.990675i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−23726.7			−1.40252			−0.701261	−	0.712905i	$-0.747379\pi$
$659$	−23726.7			−1.40252			−0.701261	+	0.712905i	$0.747379\pi$
$660$		0			0
$661$		−	24731.0i		−	1.45526i	−0.685970	−	0.727629i	$-0.740622\pi$
$661$		−	24731.0i		−	1.45526i	0.685970	−	0.727629i	$-0.259378\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	938.551i		−	0.0547300i
$666$		0			0
$667$	−6515.67			−0.378243
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	1500.74i		−	0.0863417i
$672$		0			0
$673$	13664.6			0.782660			0.391330	−	0.920250i	$-0.372015\pi$
$673$	13664.6			0.782660			0.391330	+	0.920250i	$0.372015\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	17331.5			0.983906			0.491953	−	0.870622i	$-0.336283\pi$
$677$	17331.5			0.983906			0.491953	+	0.870622i	$0.336283\pi$
$678$		0			0
$679$	1367.23			0.0772744
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	27747.6i		−	1.55451i	−0.629184	−	0.777257i	$-0.716611\pi$
$683$		−	27747.6i		−	1.55451i	0.629184	−	0.777257i	$-0.283389\pi$
$684$		0			0
$685$	−6531.57			−0.364319
$686$		0			0
$687$		0			0
$688$		0			0
$689$	8366.93	−	27805.7i	0.462634	−	1.53746i
$690$		0			0
$691$			11477.2i			0.631857i	0.948783	+	0.315929i	$0.102316\pi$
$691$			11477.2i			0.631857i	−0.948783	+	0.315929i	$0.897684\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	34280.6i		−	1.87099i
$696$		0			0
$697$		−	41555.8i		−	2.25830i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	29086.2			1.56715			0.783574	−	0.621299i	$-0.213395\pi$
$701$	29086.2			1.56715			0.783574	+	0.621299i	$0.213395\pi$
$702$		0			0
$703$	−3485.22			−0.186981
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	592.882i		−	0.0315384i
$708$		0			0
$709$		−	3149.07i		−	0.166806i	−0.996516	−	0.0834032i	$-0.973421\pi$
$709$		−	3149.07i		−	0.166806i	0.996516	−	0.0834032i	$-0.0265789\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			38214.4i			2.00721i
$714$		0			0
$715$	3848.66	−	12790.2i	0.201303	−	0.668986i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	12432.9			0.644880			0.322440	−	0.946590i	$-0.395497\pi$
$719$	12432.9			0.644880			0.322440	+	0.946590i	$0.395497\pi$
$720$		0			0
$721$		−	1808.74i		−	0.0934271i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−9002.20			−0.461149
$726$		0			0
$727$	4756.17			0.242636			0.121318	−	0.992614i	$-0.461288\pi$
$727$	4756.17			0.242636			0.121318	+	0.992614i	$0.461288\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−20159.8			−1.02003
$732$		0			0
$733$		−	1633.85i		−	0.0823294i	−0.999152	−	0.0411647i	$-0.986893\pi$
$733$		−	1633.85i		−	0.0823294i	0.999152	−	0.0411647i	$-0.0131068\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	14243.8			0.711909
$738$		0			0
$739$		−	18777.7i		−	0.934709i	−0.884070	−	0.467355i	$-0.845207\pi$
$739$		−	18777.7i		−	0.934709i	0.884070	−	0.467355i	$-0.154793\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			39073.1i			1.92928i	0.263577	+	0.964638i	$0.415098\pi$
$743$			39073.1i			1.92928i	−0.263577	+	0.964638i	$0.584902\pi$
$744$		0			0
$745$	9240.99			0.454448
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	2060.84i		−	0.100536i
$750$		0			0
$751$	−30055.7			−1.46038			−0.730192	−	0.683242i	$-0.760570\pi$
$751$	−30055.7			−1.46038			−0.730192	+	0.683242i	$0.760570\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	25583.3			1.23321
$756$		0			0
$757$	−27033.8			−1.29797			−0.648983	−	0.760802i	$-0.724806\pi$
$757$	−27033.8			−1.29797			−0.648983	+	0.760802i	$0.724806\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	6219.42i		−	0.296260i	−0.988968	−	0.148130i	$-0.952675\pi$
$761$		−	6219.42i		−	0.296260i	0.988968	−	0.148130i	$-0.0473254\pi$
$762$		0			0
$763$	−997.946			−0.0473500
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−7572.79	−	2278.71i	−0.356503	−	0.107274i
$768$		0			0
$769$		−	15898.2i		−	0.745518i	−0.927928	−	0.372759i	$-0.878412\pi$
$769$		−	15898.2i		−	0.745518i	0.927928	−	0.372759i	$-0.121588\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	5912.88i		−	0.275125i	−0.990493	−	0.137562i	$-0.956073\pi$
$773$		−	5912.88i		−	0.275125i	0.990493	−	0.137562i	$-0.0439268\pi$
$774$		0			0
$775$			52797.9i			2.44717i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−16969.0			−0.780461
$780$		0			0
$781$	3273.04			0.149960
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	49680.6i		−	2.25883i
$786$		0			0
$787$		−	33010.4i		−	1.49516i	−0.664171	−	0.747581i	$-0.731215\pi$
$787$		−	33010.4i		−	1.49516i	0.664171	−	0.747581i	$-0.268785\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	2201.82i		−	0.0989731i
$792$		0			0
$793$	1229.71	−	4086.66i	0.0550670	−	0.183003i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	37181.2			1.65248			0.826240	−	0.563318i	$-0.190476\pi$
$797$	37181.2			1.65248			0.826240	+	0.563318i	$0.190476\pi$
$798$		0			0
$799$			34817.9i			1.54164i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1436.88			−0.0631464
$804$		0			0
$805$	−2914.60			−0.127610
$806$		0			0
$807$		0			0
$808$		0			0
$809$	21539.0			0.936060			0.468030	−	0.883713i	$-0.344964\pi$
$809$	21539.0			0.936060			0.468030	+	0.883713i	$0.344964\pi$
$810$		0			0
$811$			228.372i			0.00988807i	0.999988	+	0.00494403i	$0.00157374\pi$
$811$			228.372i			0.00988807i	−0.999988	+	0.00494403i	$0.998426\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	51486.1			2.21286
$816$		0			0
$817$			8232.13i			0.352516i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	2897.96i		−	0.123190i	−0.998101	−	0.0615952i	$-0.980381\pi$
$821$		−	2897.96i		−	0.123190i	0.998101	−	0.0615952i	$-0.0196188\pi$
$822$		0			0
$823$	27421.6			1.16143			0.580714	−	0.814108i	$-0.302773\pi$
$823$	27421.6			1.16143			0.580714	+	0.814108i	$0.302773\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			26237.4i			1.10322i	0.834101	+	0.551611i	$0.185987\pi$
$827$			26237.4i			1.10322i	−0.834101	+	0.551611i	$0.814013\pi$
$828$		0			0
$829$	−24444.9			−1.02413			−0.512067	−	0.858946i	$-0.671120\pi$
$829$	−24444.9			−1.02413			−0.512067	+	0.858946i	$0.671120\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	33864.7			1.40858
$834$		0			0
$835$	12998.5			0.538719
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	7472.74i		−	0.307494i	−0.988110	−	0.153747i	$-0.950866\pi$
$839$		−	7472.74i		−	0.307494i	0.988110	−	0.153747i	$-0.0491341\pi$
$840$		0			0
$841$	−21708.8			−0.890107
$842$		0			0
$843$		0			0
$844$		0			0
$845$	20960.6	−	31675.3i	0.853332	−	1.28954i
$846$		0			0
$847$		−	1418.97i		−	0.0575638i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			10823.1i			0.435969i
$852$		0			0
$853$			5909.06i			0.237189i	0.992943	+	0.118595i	$0.0378389\pi$
$853$			5909.06i			0.237189i	−0.992943	+	0.118595i	$0.962161\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−28020.8			−1.11689			−0.558444	−	0.829542i	$-0.688601\pi$
$857$	−28020.8			−1.11689			−0.558444	+	0.829542i	$0.688601\pi$
$858$		0			0
$859$	33349.5			1.32464			0.662322	−	0.749219i	$-0.269571\pi$
$859$	33349.5			1.32464			0.662322	+	0.749219i	$0.269571\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	8614.68i		−	0.339800i	−0.985461	−	0.169900i	$-0.945656\pi$
$863$		−	8614.68i		−	0.339800i	0.985461	−	0.169900i	$-0.0543444\pi$
$864$		0			0
$865$		−	7200.81i		−	0.283046i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			12208.2i			0.476566i
$870$		0			0
$871$	38787.3	+	11671.4i	1.50891	+	0.454041i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1132.12			−0.0437403
$876$		0			0
$877$		−	33898.9i		−	1.30523i	−0.757691	−	0.652613i	$-0.773673\pi$
$877$		−	33898.9i		−	1.30523i	0.757691	−	0.652613i	$-0.226327\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	9645.86			0.368873			0.184437	−	0.982844i	$-0.440954\pi$
$881$	9645.86			0.368873			0.184437	+	0.982844i	$0.440954\pi$
$882$		0			0
$883$	−3982.78			−0.151791			−0.0758953	−	0.997116i	$-0.524181\pi$
$883$	−3982.78			−0.151791			−0.0758953	+	0.997116i	$0.524181\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	35181.8			1.33178			0.665891	−	0.746049i	$-0.268052\pi$
$887$	35181.8			1.33178			0.665891	+	0.746049i	$0.268052\pi$
$888$		0			0
$889$		−	307.435i		−	0.0115985i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	14217.7			0.532784
$894$		0			0
$895$			75924.5i			2.83562i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	15719.2i		−	0.583165i
$900$		0			0
$901$	61485.3			2.27344
$902$		0			0
$903$		0			0
$904$		0			0
$905$			41995.8i			1.54253i
$906$		0			0
$907$	−40259.6			−1.47387			−0.736934	−	0.675965i	$-0.763727\pi$
$907$	−40259.6			−1.47387			−0.736934	+	0.675965i	$0.763727\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−31942.1			−1.16168			−0.580839	−	0.814018i	$-0.697276\pi$
$911$	−31942.1			−1.16168			−0.580839	+	0.814018i	$0.697276\pi$
$912$		0			0
$913$	−17574.6			−0.637060
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	1286.73i		−	0.0463376i
$918$		0			0
$919$	−4761.17			−0.170899			−0.0854496	−	0.996342i	$-0.527233\pi$
$919$	−4761.17			−0.170899			−0.0854496	+	0.996342i	$0.527233\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	8912.81	+	2681.93i	0.317843	+	0.0956412i
$924$		0			0
$925$			14953.4i			0.531529i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			26291.6i			0.928526i	0.885697	+	0.464263i	$0.153681\pi$
$929$			26291.6i			0.928526i	−0.885697	+	0.464263i	$0.846319\pi$
$930$		0			0
$931$		−	13828.4i		−	0.486798i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	28282.3			0.989229
$936$		0			0
$937$	50820.0			1.77184			0.885921	−	0.463835i	$-0.153527\pi$
$937$	50820.0			1.77184			0.885921	+	0.463835i	$0.153527\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	18005.6i		−	0.623769i	−0.950120	−	0.311885i	$-0.899040\pi$
$941$		−	18005.6i		−	0.623769i	0.950120	−	0.311885i	$-0.100960\pi$
$942$		0			0
$943$			52696.0i			1.81974i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	13145.9i		−	0.451094i	−0.974232	−	0.225547i	$-0.927583\pi$
$947$		−	13145.9i		−	0.451094i	0.974232	−	0.225547i	$-0.0724169\pi$
$948$		0			0
$949$	−3912.78	−	1177.38i	−0.133840	−	0.0402735i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	9472.99			0.321994			0.160997	−	0.986955i	$-0.448529\pi$
$953$	9472.99			0.321994			0.160997	+	0.986955i	$0.448529\pi$
$954$		0			0
$955$		−	25672.3i		−	0.869882i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	506.072			0.0170406
$960$		0			0
$961$	−62402.2			−2.09467
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−41727.2			−1.39197
$966$		0			0
$967$			42074.2i			1.39919i	0.714541	+	0.699594i	$0.246636\pi$
$967$			42074.2i			1.39919i	−0.714541	+	0.699594i	$0.753364\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−139.025			−0.00459479			−0.00229739	−	0.999997i	$-0.500731\pi$
$971$	−139.025			−0.00459479			−0.00229739	+	0.999997i	$0.500731\pi$
$972$		0			0
$973$			2656.09i			0.0875132i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	51136.1i		−	1.67450i	−0.546818	−	0.837251i	$-0.684161\pi$
$977$		−	51136.1i		−	1.67450i	0.546818	−	0.837251i	$-0.315839\pi$
$978$		0			0
$979$	−6305.56			−0.205849
$980$		0			0
$981$		0			0
$982$		0			0
$983$			13373.1i			0.433911i	0.976182	+	0.216955i	$0.0696126\pi$
$983$			13373.1i			0.433911i	−0.976182	+	0.216955i	$0.930387\pi$
$984$		0			0
$985$	−21052.5			−0.681003
$986$		0			0
$987$		0			0
$988$		0			0
$989$	25564.3			0.821937
$990$		0			0
$991$	22640.2			0.725722			0.362861	−	0.931843i	$-0.381800\pi$
$991$	22640.2			0.725722			0.362861	+	0.931843i	$0.381800\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	65757.9i		−	2.09514i
$996$		0			0
$997$	−1503.11			−0.0477473			−0.0238737	−	0.999715i	$-0.507600\pi$
$997$	−1503.11			−0.0477473			−0.0238737	+	0.999715i	$0.507600\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	936.4.c.d.649.17	yes	20
3.2	odd	2	inner	936.4.c.d.649.3	✓	20
13.12	even	2	inner	936.4.c.d.649.4	yes	20
39.38	odd	2	inner	936.4.c.d.649.18	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
936.4.c.d.649.3	✓	20	3.2	odd	2	inner
936.4.c.d.649.4	yes	20	13.12	even	2	inner
936.4.c.d.649.17	yes	20	1.1	even	1	trivial
936.4.c.d.649.18	yes	20	39.38	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	936.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+17.2883i q^{5} -1.33952i q^{7} -16.4827i q^{11} +(13.5060 - 44.8842i) q^{13} +99.2502 q^{17} -40.5282i q^{19} -125.857 q^{23} -173.887 q^{25} +51.7704 q^{29} -303.633i q^{31} +23.1580 q^{35} -85.9949i q^{37} -418.697i q^{41} -203.121 q^{43} +350.809i q^{47} +341.206 q^{49} +619.498 q^{53} +284.959 q^{55} -168.719i q^{59} +91.0490 q^{61} +(775.973 + 233.496i) q^{65} +864.165i q^{67} +198.574i q^{71} -87.1751i q^{73} -22.0789 q^{77} -740.668 q^{79} -1066.25i q^{83} +1715.87i q^{85} -382.555i q^{89} +(-60.1230 - 18.0915i) q^{91} +700.665 q^{95} +1020.69i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 84 q^{13} - 412 q^{25} - 288 q^{43} + 284 q^{49} + 576 q^{55} + 104 q^{61} + 432 q^{79} + 624 q^{91}+O(q^{100}) \) 20 * q - 84 * q^13 - 412 * q^25 - 288 * q^43 + 284 * q^49 + 576 * q^55 + 104 * q^61 + 432 * q^79 + 624 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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