LMFDB - Embedded newform 2736.3.o.r.721.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,3,Mod(721,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2736.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2736.o (of order $2$, degree $1$, not 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:	$74.5506003290$
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} - 4 x^{19} + 80 x^{18} - 152 x^{17} + 4326 x^{16} - 10096 x^{15} + 70116 x^{14} - 93436 x^{13} + \cdots + 36100 $ x^20 - 4x^19 + 80x^18 - 152x^17 + 4326x^16 - 10096x^15 + 70116x^14 - 93436x^13 + 597327x^12 - 837564x^11 + 2838686x^10 - 2898276x^9 + 7652658x^8 - 6988600x^7 + 12843126x^6 - 6054124x^5 + 7434909x^4 - 621764x^3 + 3099194x^2 - 329080*x + 36100
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{37} $
Twist minimal:	no (minimal twist has level 456)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.1
Root			$0.0545091 - 0.0944125i$ of defining polynomial
Character	$\chi$	$=$	2736.721
Dual form			2736.3.o.r.721.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.51170 q^{5} +7.72974 q^{7} +O(q^{10})$	$q-7.51170 q^{5} +7.72974 q^{7} +9.01455 q^{11} -10.6419i q^{13} -31.2209 q^{17} +(8.20262 + 17.1382i) q^{19} -4.85819 q^{23} +31.4256 q^{25} +23.2161i q^{29} -7.74375i q^{31} -58.0635 q^{35} -21.4280i q^{37} -32.7421i q^{41} -7.82679 q^{43} +50.7042 q^{47} +10.7488 q^{49} +66.8345i q^{53} -67.7146 q^{55} -86.1944i q^{59} -3.46955 q^{61} +79.9386i q^{65} -14.0271i q^{67} +35.9438i q^{71} +79.1658 q^{73} +69.6801 q^{77} +61.9335i q^{79} -143.733 q^{83} +234.522 q^{85} -53.0951i q^{89} -82.2590i q^{91} +(-61.6156 - 128.737i) q^{95} +124.208i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 16 q^{7}+O(q^{10}) $ 20 * q + 16 * q^7	$ 20 q + 16 q^{7} + 16 q^{11} - 32 q^{17} - 40 q^{19} + 64 q^{23} + 68 q^{25} - 208 q^{35} - 64 q^{43} + 48 q^{47} + 20 q^{49} + 336 q^{55} + 184 q^{61} + 104 q^{73} - 88 q^{77} + 224 q^{83} - 136 q^{85} - 320 q^{95}+O(q^{100}) $ 20 * q + 16 * q^7 + 16 * q^11 - 32 * q^17 - 40 * q^19 + 64 * q^23 + 68 * q^25 - 208 * q^35 - 64 * q^43 + 48 * q^47 + 20 * q^49 + 336 * q^55 + 184 * q^61 + 104 * q^73 - 88 * q^77 + 224 * q^83 - 136 * q^85 - 320 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$5$	−7.51170			−1.50234			−0.751170	−	0.660109i	$-0.770510\pi$
$5$	−7.51170			−1.50234			−0.751170	+	0.660109i	$0.770510\pi$
$6$		0			0
$7$	7.72974			1.10425			0.552124	−	0.833762i	$-0.313817\pi$
$7$	7.72974			1.10425			0.552124	+	0.833762i	$0.313817\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	9.01455			0.819504			0.409752	−	0.912197i	$-0.365615\pi$
$11$	9.01455			0.819504			0.409752	+	0.912197i	$0.365615\pi$
$12$		0			0
$13$		−	10.6419i		−	0.818606i	−0.912398	−	0.409303i	$-0.865772\pi$
$13$		−	10.6419i		−	0.818606i	0.912398	−	0.409303i	$-0.134228\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−31.2209			−1.83652			−0.918262	−	0.395972i	$-0.870408\pi$
$17$	−31.2209			−1.83652			−0.918262	+	0.395972i	$0.870408\pi$
$18$		0			0
$19$	8.20262	+	17.1382i	0.431717	+	0.902009i
$19$	8.20262	+	17.1382i	0.431717	+	0.902009i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.85819			−0.211226			−0.105613	−	0.994407i	$-0.533680\pi$
$23$	−4.85819			−0.211226			−0.105613	+	0.994407i	$0.533680\pi$
$24$		0			0
$25$	31.4256			1.25703
$26$		0			0
$27$		0			0
$28$		0			0
$29$			23.2161i			0.800555i	0.916394	+	0.400277i	$0.131086\pi$
$29$			23.2161i			0.800555i	−0.916394	+	0.400277i	$0.868914\pi$
$30$		0			0
$31$		−	7.74375i		−	0.249798i	−0.992169	−	0.124899i	$-0.960139\pi$
$31$		−	7.74375i		−	0.249798i	0.992169	−	0.124899i	$-0.0398608\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−58.0635			−1.65896
$36$		0			0
$37$		−	21.4280i		−	0.579134i	−0.957158	−	0.289567i	$-0.906489\pi$
$37$		−	21.4280i		−	0.579134i	0.957158	−	0.289567i	$-0.0935114\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	32.7421i		−	0.798587i	−0.916823	−	0.399293i	$-0.869255\pi$
$41$		−	32.7421i		−	0.798587i	0.916823	−	0.399293i	$-0.130745\pi$
$42$		0			0
$43$	−7.82679			−0.182018			−0.0910091	−	0.995850i	$-0.529009\pi$
$43$	−7.82679			−0.182018			−0.0910091	+	0.995850i	$0.529009\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	50.7042			1.07881			0.539406	−	0.842046i	$-0.318649\pi$
$47$	50.7042			1.07881			0.539406	+	0.842046i	$0.318649\pi$
$48$		0			0
$49$	10.7488			0.219364
$50$		0			0
$51$		0			0
$52$		0			0
$53$			66.8345i			1.26103i	0.776177	+	0.630514i	$0.217156\pi$
$53$			66.8345i			1.26103i	−0.776177	+	0.630514i	$0.782844\pi$
$54$		0			0
$55$	−67.7146			−1.23117
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	86.1944i		−	1.46092i	−0.682954	−	0.730461i	$-0.739305\pi$
$59$		−	86.1944i		−	1.46092i	0.682954	−	0.730461i	$-0.260695\pi$
$60$		0			0
$61$	−3.46955			−0.0568779			−0.0284390	−	0.999596i	$-0.509054\pi$
$61$	−3.46955			−0.0568779			−0.0284390	+	0.999596i	$0.509054\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			79.9386i			1.22983i
$66$		0			0
$67$		−	14.0271i		−	0.209360i	−0.994506	−	0.104680i	$-0.966618\pi$
$67$		−	14.0271i		−	0.209360i	0.994506	−	0.104680i	$-0.0333818\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			35.9438i			0.506251i	0.967433	+	0.253126i	$0.0814586\pi$
$71$			35.9438i			0.506251i	−0.967433	+	0.253126i	$0.918541\pi$
$72$		0			0
$73$	79.1658			1.08446			0.542231	−	0.840229i	$-0.317580\pi$
$73$	79.1658			1.08446			0.542231	+	0.840229i	$0.317580\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	69.6801			0.904936
$78$		0			0
$79$			61.9335i			0.783968i	0.919972	+	0.391984i	$0.128211\pi$
$79$			61.9335i			0.783968i	−0.919972	+	0.391984i	$0.871789\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−143.733			−1.73172			−0.865859	−	0.500288i	$-0.833227\pi$
$83$	−143.733			−1.73172			−0.865859	+	0.500288i	$0.833227\pi$
$84$		0			0
$85$	234.522			2.75908
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	53.0951i		−	0.596574i	−0.954476	−	0.298287i	$-0.903585\pi$
$89$		−	53.0951i		−	0.596574i	0.954476	−	0.298287i	$-0.0964153\pi$
$90$		0			0
$91$		−	82.2590i		−	0.903945i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−61.6156	−	128.737i	−0.648585	−	1.35512i
$96$		0			0
$97$			124.208i			1.28049i	0.768171	+	0.640245i	$0.221167\pi$
$97$			124.208i			1.28049i	−0.768171	+	0.640245i	$0.778833\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−22.0427			−0.218245			−0.109122	−	0.994028i	$-0.534804\pi$
$101$	−22.0427			−0.218245			−0.109122	+	0.994028i	$0.534804\pi$
$102$		0			0
$103$		−	44.9662i		−	0.436565i	−0.975886	−	0.218283i	$-0.929955\pi$
$103$		−	44.9662i		−	0.436565i	0.975886	−	0.218283i	$-0.0700455\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			150.109i			1.40289i	0.712725	+	0.701444i	$0.247461\pi$
$107$			150.109i			1.40289i	−0.712725	+	0.701444i	$0.752539\pi$
$108$		0			0
$109$			153.420i			1.40752i	0.710438	+	0.703760i	$0.248497\pi$
$109$			153.420i			1.40752i	−0.710438	+	0.703760i	$0.751503\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	171.319i		−	1.51609i	−0.652200	−	0.758047i	$-0.726154\pi$
$113$		−	171.319i		−	1.51609i	0.652200	−	0.758047i	$-0.273846\pi$
$114$		0			0
$115$	36.4933			0.317333
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−241.330			−2.02798
$120$		0			0
$121$	−39.7380			−0.328413
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−48.2675			−0.386140
$126$		0			0
$127$		−	95.7983i		−	0.754317i	−0.926149	−	0.377159i	$-0.876901\pi$
$127$		−	95.7983i		−	0.754317i	0.926149	−	0.377159i	$-0.123099\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	124.774			0.952471			0.476236	−	0.879318i	$-0.342001\pi$
$131$	124.774			0.952471			0.476236	+	0.879318i	$0.342001\pi$
$132$		0			0
$133$	63.4041	+	132.474i	0.476722	+	0.996042i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	138.395			1.01018			0.505091	−	0.863066i	$-0.331459\pi$
$137$	138.395			1.01018			0.505091	+	0.863066i	$0.331459\pi$
$138$		0			0
$139$	11.2708			0.0810851			0.0405426	−	0.999178i	$-0.487091\pi$
$139$	11.2708			0.0810851			0.0405426	+	0.999178i	$0.487091\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	95.9318i		−	0.670851i
$144$		0			0
$145$		−	174.392i		−	1.20271i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−205.089			−1.37644			−0.688219	−	0.725503i	$-0.741607\pi$
$149$	−205.089			−1.37644			−0.688219	+	0.725503i	$0.741607\pi$
$150$		0			0
$151$			241.101i			1.59669i	0.602199	+	0.798346i	$0.294292\pi$
$151$			241.101i			1.59669i	−0.602199	+	0.798346i	$0.705708\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			58.1687i			0.375282i
$156$		0			0
$157$	−298.863			−1.90359			−0.951793	−	0.306740i	$-0.900762\pi$
$157$	−298.863			−1.90359			−0.951793	+	0.306740i	$0.900762\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−37.5525			−0.233245
$162$		0			0
$163$	−287.076			−1.76120			−0.880602	−	0.473857i	$-0.842861\pi$
$163$	−287.076			−1.76120			−0.880602	+	0.473857i	$0.842861\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			189.220i			1.13305i	0.824044	+	0.566526i	$0.191713\pi$
$167$			189.220i			1.13305i	−0.824044	+	0.566526i	$0.808287\pi$
$168$		0			0
$169$	55.7503			0.329884
$170$		0			0
$171$		0			0
$172$		0			0
$173$			223.168i			1.28999i	0.764186	+	0.644995i	$0.223141\pi$
$173$			223.168i			1.28999i	−0.764186	+	0.644995i	$0.776859\pi$
$174$		0			0
$175$	242.912			1.38807
$176$		0			0
$177$		0			0
$178$		0			0
$179$			256.021i			1.43028i	0.698980	+	0.715141i	$0.253638\pi$
$179$			256.021i			1.43028i	−0.698980	+	0.715141i	$0.746362\pi$
$180$		0			0
$181$			350.532i			1.93664i	0.249712	+	0.968320i	$0.419664\pi$
$181$			350.532i			1.93664i	−0.249712	+	0.968320i	$0.580336\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			160.961i			0.870057i
$186$		0			0
$187$	−281.442			−1.50504
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−248.432			−1.30069			−0.650346	−	0.759638i	$-0.725376\pi$
$191$	−248.432			−1.30069			−0.650346	+	0.759638i	$0.725376\pi$
$192$		0			0
$193$			166.976i			0.865160i	0.901596	+	0.432580i	$0.142397\pi$
$193$			166.976i			0.865160i	−0.901596	+	0.432580i	$0.857603\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−17.1673			−0.0871435			−0.0435718	−	0.999050i	$-0.513874\pi$
$197$	−17.1673			−0.0871435			−0.0435718	+	0.999050i	$0.513874\pi$
$198$		0			0
$199$	−54.3523			−0.273127			−0.136564	−	0.990631i	$-0.543606\pi$
$199$	−54.3523			−0.273127			−0.136564	+	0.990631i	$0.543606\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			179.454i			0.884011i
$204$		0			0
$205$			245.949i			1.19975i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	73.9429	+	154.493i	0.353794	+	0.739200i
$210$		0			0
$211$			349.068i			1.65435i	0.561942	+	0.827176i	$0.310054\pi$
$211$			349.068i			1.65435i	−0.561942	+	0.827176i	$0.689946\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	58.7925			0.273453
$216$		0			0
$217$		−	59.8572i		−	0.275839i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			332.249i			1.50339i
$222$		0			0
$223$		−	252.146i		−	1.13070i	−0.824851	−	0.565351i	$-0.808741\pi$
$223$		−	252.146i		−	1.13070i	0.824851	−	0.565351i	$-0.191259\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			435.646i			1.91915i	0.281460	+	0.959573i	$0.409181\pi$
$227$			435.646i			1.91915i	−0.281460	+	0.959573i	$0.590819\pi$
$228$		0			0
$229$	−388.938			−1.69842			−0.849211	−	0.528054i	$-0.822922\pi$
$229$	−388.938			−1.69842			−0.849211	+	0.528054i	$0.822922\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	42.6777			0.183166			0.0915830	−	0.995797i	$-0.470807\pi$
$233$	42.6777			0.183166			0.0915830	+	0.995797i	$0.470807\pi$
$234$		0			0
$235$	−380.874			−1.62074
$236$		0			0
$237$		0			0
$238$		0			0
$239$	22.2232			0.0929840			0.0464920	−	0.998919i	$-0.485196\pi$
$239$	22.2232			0.0929840			0.0464920	+	0.998919i	$0.485196\pi$
$240$		0			0
$241$		−	120.140i		−	0.498505i	−0.968439	−	0.249252i	$-0.919815\pi$
$241$		−	120.140i		−	0.498505i	0.968439	−	0.249252i	$-0.0801848\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−80.7420			−0.329559
$246$		0			0
$247$	182.382	−	87.2913i	0.738391	−	0.353406i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	363.427			1.44791			0.723957	−	0.689845i	$-0.242321\pi$
$251$	363.427			1.44791			0.723957	+	0.689845i	$0.242321\pi$
$252$		0			0
$253$	−43.7944			−0.173100
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	486.717i		−	1.89384i	−0.321469	−	0.946920i	$-0.604176\pi$
$257$		−	486.717i		−	1.89384i	0.321469	−	0.946920i	$-0.395824\pi$
$258$		0			0
$259$		−	165.633i		−	0.639508i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−126.177			−0.479762			−0.239881	−	0.970802i	$-0.577108\pi$
$263$	−126.177			−0.479762			−0.239881	+	0.970802i	$0.577108\pi$
$264$		0			0
$265$		−	502.041i		−	1.89449i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			230.390i			0.856468i	0.903668	+	0.428234i	$0.140864\pi$
$269$			230.390i			0.856468i	−0.903668	+	0.428234i	$0.859136\pi$
$270$		0			0
$271$	−40.5338			−0.149571			−0.0747857	−	0.997200i	$-0.523827\pi$
$271$	−40.5338			−0.149571			−0.0747857	+	0.997200i	$0.523827\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	283.288			1.03014
$276$		0			0
$277$	187.564			0.677128			0.338564	−	0.940943i	$-0.390059\pi$
$277$	187.564			0.677128			0.338564	+	0.940943i	$0.390059\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			377.684i			1.34407i	0.740518	+	0.672036i	$0.234580\pi$
$281$			377.684i			1.34407i	−0.740518	+	0.672036i	$0.765420\pi$
$282$		0			0
$283$	−56.2391			−0.198725			−0.0993624	−	0.995051i	$-0.531680\pi$
$283$	−56.2391			−0.198725			−0.0993624	+	0.995051i	$0.531680\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	253.087i		−	0.881838i
$288$		0			0
$289$	685.746			2.37282
$290$		0			0
$291$		0			0
$292$		0			0
$293$			261.701i			0.893179i	0.894739	+	0.446589i	$0.147362\pi$
$293$			261.701i			0.893179i	−0.894739	+	0.446589i	$0.852638\pi$
$294$		0			0
$295$			647.467i			2.19480i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			51.7003i			0.172911i
$300$		0			0
$301$	−60.4990			−0.200993
$302$		0			0
$303$		0			0
$304$		0			0
$305$	26.0623			0.0854500
$306$		0			0
$307$		−	145.251i		−	0.473131i	−0.971616	−	0.236566i	$-0.923978\pi$
$307$		−	145.251i		−	0.473131i	0.971616	−	0.236566i	$-0.0760218\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−65.2971			−0.209959			−0.104979	−	0.994474i	$-0.533478\pi$
$311$	−65.2971			−0.209959			−0.104979	+	0.994474i	$0.533478\pi$
$312$		0			0
$313$	−267.469			−0.854532			−0.427266	−	0.904126i	$-0.640523\pi$
$313$	−267.469			−0.854532			−0.427266	+	0.904126i	$0.640523\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			486.160i			1.53363i	0.641869	+	0.766814i	$0.278159\pi$
$317$			486.160i			1.53363i	−0.641869	+	0.766814i	$0.721841\pi$
$318$		0			0
$319$			209.283i			0.656058i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−256.093	−	535.070i	−0.792859	−	1.65656i
$324$		0			0
$325$		−	334.428i		−	1.02901i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	391.930			1.19128
$330$		0			0
$331$		−	101.043i		−	0.305266i	−0.988283	−	0.152633i	$-0.951225\pi$
$331$		−	101.043i		−	0.305266i	0.988283	−	0.152633i	$-0.0487752\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			105.367i			0.314530i
$336$		0			0
$337$		−	482.627i		−	1.43213i	−0.698035	−	0.716064i	$-0.745942\pi$
$337$		−	482.627i		−	1.43213i	0.698035	−	0.716064i	$-0.254058\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	69.8064i		−	0.204711i
$342$		0			0
$343$	−295.671			−0.862016
$344$		0			0
$345$		0			0
$346$		0			0
$347$	196.697			0.566849			0.283425	−	0.958994i	$-0.408529\pi$
$347$	196.697			0.566849			0.283425	+	0.958994i	$0.408529\pi$
$348$		0			0
$349$	−520.244			−1.49067			−0.745335	−	0.666691i	$-0.767710\pi$
$349$	−520.244			−1.49067			−0.745335	+	0.666691i	$0.767710\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−113.689			−0.322065			−0.161032	−	0.986949i	$-0.551482\pi$
$353$	−113.689			−0.322065			−0.161032	+	0.986949i	$0.551482\pi$
$354$		0			0
$355$		−	269.999i		−	0.760562i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−222.612			−0.620089			−0.310044	−	0.950722i	$-0.600344\pi$
$359$	−222.612			−0.620089			−0.310044	+	0.950722i	$0.600344\pi$
$360$		0			0
$361$	−226.434	+	281.156i	−0.627241	+	0.778825i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−594.670			−1.62923
$366$		0			0
$367$	340.842			0.928724			0.464362	−	0.885645i	$-0.346284\pi$
$367$	340.842			0.928724			0.464362	+	0.885645i	$0.346284\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			516.613i			1.39249i
$372$		0			0
$373$			256.214i			0.686901i	0.939171	+	0.343450i	$0.111596\pi$
$373$			256.214i			0.686901i	−0.939171	+	0.343450i	$0.888404\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	247.063			0.655339
$378$		0			0
$379$			251.751i			0.664250i	0.943235	+	0.332125i	$0.107766\pi$
$379$			251.751i			0.664250i	−0.943235	+	0.332125i	$0.892234\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			88.2656i			0.230458i	0.993339	+	0.115229i	$0.0367603\pi$
$383$			88.2656i			0.230458i	−0.993339	+	0.115229i	$0.963240\pi$
$384$		0			0
$385$	−523.416			−1.35952
$386$		0			0
$387$		0			0
$388$		0			0
$389$	274.636			0.706004			0.353002	−	0.935622i	$-0.385161\pi$
$389$	274.636			0.706004			0.353002	+	0.935622i	$0.385161\pi$
$390$		0			0
$391$	151.677			0.387921
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	465.226i		−	1.17779i
$396$		0			0
$397$	−311.724			−0.785199			−0.392599	−	0.919710i	$-0.628424\pi$
$397$	−311.724			−0.785199			−0.392599	+	0.919710i	$0.628424\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	475.804i		−	1.18654i	−0.805003	−	0.593271i	$-0.797836\pi$
$401$		−	475.804i		−	1.18654i	0.805003	−	0.593271i	$-0.202164\pi$
$402$		0			0
$403$	−82.4081			−0.204487
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	193.163i		−	0.474603i
$408$		0			0
$409$		−	133.969i		−	0.327553i	−0.986497	−	0.163776i	$-0.947632\pi$
$409$		−	133.969i		−	0.327553i	0.986497	−	0.163776i	$-0.0523676\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	666.260i		−	1.61322i
$414$		0			0
$415$	1079.68			2.60163
$416$		0			0
$417$		0			0
$418$		0			0
$419$	628.311			1.49955			0.749774	−	0.661694i	$-0.230162\pi$
$419$	628.311			1.49955			0.749774	+	0.661694i	$0.230162\pi$
$420$		0			0
$421$		−	505.724i		−	1.20125i	−0.799533	−	0.600623i	$-0.794919\pi$
$421$		−	505.724i		−	1.20125i	0.799533	−	0.600623i	$-0.205081\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−981.138			−2.30856
$426$		0			0
$427$	−26.8187			−0.0628074
$428$		0			0
$429$		0			0
$430$		0			0
$431$			200.343i			0.464832i	0.972616	+	0.232416i	$0.0746631\pi$
$431$			200.343i			0.464832i	−0.972616	+	0.232416i	$0.925337\pi$
$432$		0			0
$433$			54.4918i			0.125847i	0.998018	+	0.0629236i	$0.0200424\pi$
$433$			54.4918i			0.125847i	−0.998018	+	0.0629236i	$0.979958\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−39.8499	−	83.2605i	−0.0911896	−	0.190527i
$438$		0			0
$439$			152.838i			0.348151i	0.984732	+	0.174075i	$0.0556936\pi$
$439$			152.838i			0.348151i	−0.984732	+	0.174075i	$0.944306\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−199.742			−0.450884			−0.225442	−	0.974257i	$-0.572383\pi$
$443$	−199.742			−0.450884			−0.225442	+	0.974257i	$0.572383\pi$
$444$		0			0
$445$			398.834i			0.896257i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			516.091i			1.14942i	0.818356	+	0.574712i	$0.194886\pi$
$449$			516.091i			1.14942i	−0.818356	+	0.574712i	$0.805114\pi$
$450$		0			0
$451$		−	295.155i		−	0.654445i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			617.905i			1.35803i
$456$		0			0
$457$	236.491			0.517485			0.258742	−	0.965946i	$-0.416692\pi$
$457$	236.491			0.517485			0.258742	+	0.965946i	$0.416692\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	475.078			1.03054			0.515269	−	0.857029i	$-0.327692\pi$
$461$	475.078			1.03054			0.515269	+	0.857029i	$0.327692\pi$
$462$		0			0
$463$	−785.292			−1.69610			−0.848048	−	0.529920i	$-0.822222\pi$
$463$	−785.292			−1.69610			−0.848048	+	0.529920i	$0.822222\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	664.097			1.42205			0.711025	−	0.703167i	$-0.248231\pi$
$467$	664.097			1.42205			0.711025	+	0.703167i	$0.248231\pi$
$468$		0			0
$469$		−	108.426i		−	0.231185i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−70.5549			−0.149165
$474$		0			0
$475$	257.773	+	538.578i	0.542679	+	1.13385i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	648.759			1.35440			0.677202	−	0.735797i	$-0.263192\pi$
$479$	648.759			1.35440			0.677202	+	0.735797i	$0.263192\pi$
$480$		0			0
$481$	−228.034			−0.474083
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	933.010i		−	1.92373i
$486$		0			0
$487$		−	131.004i		−	0.269002i	−0.990913	−	0.134501i	$-0.957057\pi$
$487$		−	131.004i		−	0.269002i	0.990913	−	0.134501i	$-0.0429431\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−425.829			−0.867269			−0.433635	−	0.901089i	$-0.642769\pi$
$491$	−425.829			−0.867269			−0.433635	+	0.901089i	$0.642769\pi$
$492$		0			0
$493$		−	724.828i		−	1.47024i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			277.836i			0.559027i
$498$		0			0
$499$	394.155			0.789890			0.394945	−	0.918705i	$-0.370764\pi$
$499$	394.155			0.789890			0.394945	+	0.918705i	$0.370764\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	137.310			0.272982			0.136491	−	0.990641i	$-0.456417\pi$
$503$	137.310			0.272982			0.136491	+	0.990641i	$0.456417\pi$
$504$		0			0
$505$	165.578			0.327878
$506$		0			0
$507$		0			0
$508$		0			0
$509$			785.565i			1.54335i	0.636018	+	0.771675i	$0.280581\pi$
$509$			785.565i			1.54335i	−0.636018	+	0.771675i	$0.719419\pi$
$510$		0			0
$511$	611.931			1.19752
$512$		0			0
$513$		0			0
$514$		0			0
$515$			337.773i			0.655870i
$516$		0			0
$517$	457.075			0.884091
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	237.468i		−	0.455793i	−0.973685	−	0.227896i	$-0.926815\pi$
$521$		−	237.468i		−	0.455793i	0.973685	−	0.227896i	$-0.0731848\pi$
$522$		0			0
$523$			477.984i			0.913928i	0.889485	+	0.456964i	$0.151063\pi$
$523$			477.984i			0.913928i	−0.889485	+	0.456964i	$0.848937\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			241.767i			0.458761i
$528$		0			0
$529$	−505.398			−0.955384
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−348.437			−0.653728
$534$		0			0
$535$		−	1127.57i		−	2.10761i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	96.8958			0.179770
$540$		0			0
$541$	−80.4435			−0.148694			−0.0743470	−	0.997232i	$-0.523687\pi$
$541$	−80.4435			−0.148694			−0.0743470	+	0.997232i	$0.523687\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	1152.44i		−	2.11457i
$546$		0			0
$547$		−	983.581i		−	1.79814i	−0.437808	−	0.899069i	$-0.644245\pi$
$547$		−	983.581i		−	1.79814i	0.437808	−	0.899069i	$-0.355755\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−397.881	+	190.433i	−0.722108	+	0.345613i
$552$		0			0
$553$			478.729i			0.865695i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	769.474			1.38146			0.690731	−	0.723112i	$-0.257289\pi$
$557$	769.474			1.38146			0.690731	+	0.723112i	$0.257289\pi$
$558$		0			0
$559$			83.2918i			0.149001i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	517.229i		−	0.918702i	−0.888255	−	0.459351i	$-0.848082\pi$
$563$		−	517.229i		−	0.918702i	0.888255	−	0.459351i	$-0.151918\pi$
$564$		0			0
$565$			1286.89i			2.27769i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	443.895i		−	0.780132i	−0.920787	−	0.390066i	$-0.872452\pi$
$569$		−	443.895i		−	0.780132i	0.920787	−	0.390066i	$-0.127548\pi$
$570$		0			0
$571$	−215.655			−0.377680			−0.188840	−	0.982008i	$-0.560473\pi$
$571$	−215.655			−0.377680			−0.188840	+	0.982008i	$0.560473\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−152.672			−0.265516
$576$		0			0
$577$	−83.1806			−0.144161			−0.0720803	−	0.997399i	$-0.522964\pi$
$577$	−83.1806			−0.144161			−0.0720803	+	0.997399i	$0.522964\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1111.02			−1.91225
$582$		0			0
$583$			602.483i			1.03342i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1030.98			−1.75636			−0.878180	−	0.478331i	$-0.841242\pi$
$587$	−1030.98			−1.75636			−0.878180	+	0.478331i	$0.841242\pi$
$588$		0			0
$589$	132.714	−	63.5190i	0.225321	−	0.107842i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	583.289			0.983624			0.491812	−	0.870701i	$-0.336335\pi$
$593$	583.289			0.983624			0.491812	+	0.870701i	$0.336335\pi$
$594$		0			0
$595$	1812.80			3.04671
$596$		0			0
$597$		0			0
$598$		0			0
$599$			178.158i			0.297426i	0.988880	+	0.148713i	$0.0475131\pi$
$599$			178.158i			0.297426i	−0.988880	+	0.148713i	$0.952487\pi$
$600$		0			0
$601$			459.376i			0.764352i	0.924090	+	0.382176i	$0.124825\pi$
$601$			459.376i			0.764352i	−0.924090	+	0.382176i	$0.875175\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	298.500			0.493388
$606$		0			0
$607$			503.489i			0.829472i	0.909942	+	0.414736i	$0.136126\pi$
$607$			503.489i			0.829472i	−0.909942	+	0.414736i	$0.863874\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	539.588i		−	0.883122i
$612$		0			0
$613$	949.371			1.54873			0.774364	−	0.632740i	$-0.218070\pi$
$613$	949.371			1.54873			0.774364	+	0.632740i	$0.218070\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−665.653			−1.07885			−0.539427	−	0.842032i	$-0.681359\pi$
$617$	−665.653			−1.07885			−0.539427	+	0.842032i	$0.681359\pi$
$618$		0			0
$619$	−29.6403			−0.0478841			−0.0239420	−	0.999713i	$-0.507622\pi$
$619$	−29.6403			−0.0478841			−0.0239420	+	0.999713i	$0.507622\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	410.411i		−	0.658766i
$624$		0			0
$625$	−423.070			−0.676912
$626$		0			0
$627$		0			0
$628$		0			0
$629$			669.001i			1.06359i
$630$		0			0
$631$	363.987			0.576841			0.288420	−	0.957504i	$-0.406870\pi$
$631$	363.987			0.576841			0.288420	+	0.957504i	$0.406870\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			719.608i			1.13324i
$636$		0			0
$637$		−	114.388i		−	0.179573i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			458.061i			0.714604i	0.933989	+	0.357302i	$0.116303\pi$
$641$			458.061i			0.714604i	−0.933989	+	0.357302i	$0.883697\pi$
$642$		0			0
$643$	351.562			0.546753			0.273376	−	0.961907i	$-0.411860\pi$
$643$	351.562			0.546753			0.273376	+	0.961907i	$0.411860\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−669.353			−1.03455			−0.517275	−	0.855819i	$-0.673053\pi$
$647$	−669.353			−1.03455			−0.517275	+	0.855819i	$0.673053\pi$
$648$		0			0
$649$		−	777.004i		−	1.19723i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1204.76			−1.84496			−0.922478	−	0.386049i	$-0.873840\pi$
$653$	−1204.76			−1.84496			−0.922478	+	0.386049i	$0.873840\pi$
$654$		0			0
$655$	−937.263			−1.43094
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	674.070i		−	1.02287i	−0.859322	−	0.511434i	$-0.829114\pi$
$659$		−	674.070i		−	1.02287i	0.859322	−	0.511434i	$-0.170886\pi$
$660$		0			0
$661$		−	498.931i		−	0.754812i	−0.926048	−	0.377406i	$-0.876816\pi$
$661$		−	498.931i		−	0.754812i	0.926048	−	0.377406i	$-0.123184\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−476.272	−	995.102i	−0.716199	−	1.49639i
$666$		0			0
$667$		−	112.788i		−	0.169098i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−31.2765			−0.0466117
$672$		0			0
$673$			388.499i			0.577264i	0.957440	+	0.288632i	$0.0932005\pi$
$673$			388.499i			0.577264i	−0.957440	+	0.288632i	$0.906800\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	641.967i		−	0.948253i	−0.880457	−	0.474127i	$-0.842764\pi$
$677$		−	641.967i		−	0.948253i	0.880457	−	0.474127i	$-0.157236\pi$
$678$		0			0
$679$			960.091i			1.41398i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			625.904i			0.916404i	0.888848	+	0.458202i	$0.151506\pi$
$683$			625.904i			0.916404i	−0.888848	+	0.458202i	$0.848494\pi$
$684$		0			0
$685$	−1039.58			−1.51764
$686$		0			0
$687$		0			0
$688$		0			0
$689$	711.245			1.03229
$690$		0			0
$691$	−531.098			−0.768593			−0.384297	−	0.923210i	$-0.625556\pi$
$691$	−531.098			−0.768593			−0.384297	+	0.923210i	$0.625556\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−84.6631			−0.121817
$696$		0			0
$697$			1022.24i			1.46662i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	276.791			0.394852			0.197426	−	0.980318i	$-0.436742\pi$
$701$	276.791			0.394852			0.197426	+	0.980318i	$0.436742\pi$
$702$		0			0
$703$	367.236	−	175.766i	0.522385	−	0.250022i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−170.384			−0.240996
$708$		0			0
$709$	−148.429			−0.209350			−0.104675	−	0.994506i	$-0.533380\pi$
$709$	−148.429			−0.209350			−0.104675	+	0.994506i	$0.533380\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			37.6206i			0.0527638i
$714$		0			0
$715$			720.611i			1.00785i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−974.249			−1.35501			−0.677503	−	0.735520i	$-0.736938\pi$
$719$	−974.249			−1.35501			−0.677503	+	0.735520i	$0.736938\pi$
$720$		0			0
$721$		−	347.577i		−	0.482077i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			729.581i			1.00632i
$726$		0			0
$727$	218.182			0.300113			0.150057	−	0.988677i	$-0.452054\pi$
$727$	218.182			0.300113			0.150057	+	0.988677i	$0.452054\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	244.360			0.334281
$732$		0			0
$733$	1427.70			1.94775			0.973877	−	0.227077i	$-0.0729170\pi$
$733$	1427.70			1.94775			0.973877	+	0.227077i	$0.0729170\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	126.448i		−	0.171571i
$738$		0			0
$739$	−1278.66			−1.73026			−0.865131	−	0.501547i	$-0.832765\pi$
$739$	−1278.66			−1.73026			−0.865131	+	0.501547i	$0.832765\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	701.930i		−	0.944724i	−0.881405	−	0.472362i	$-0.843402\pi$
$743$		−	701.930i		−	0.944724i	0.881405	−	0.472362i	$-0.156598\pi$
$744$		0			0
$745$	1540.57			2.06788
$746$		0			0
$747$		0			0
$748$		0			0
$749$			1160.30i			1.54914i
$750$		0			0
$751$			765.047i			1.01870i	0.860558	+	0.509352i	$0.170115\pi$
$751$			765.047i			1.01870i	−0.860558	+	0.509352i	$0.829885\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	1811.08i		−	2.39878i
$756$		0			0
$757$	−818.796			−1.08163			−0.540817	−	0.841141i	$-0.681885\pi$
$757$	−818.796			−1.08163			−0.540817	+	0.841141i	$0.681885\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.578219			−0.000759815			−0.000379908	−	1.00000i	$-0.500121\pi$
$761$	−0.578219			−0.000759815			−0.000379908		1.00000i	$0.500121\pi$
$762$		0			0
$763$			1185.89i			1.55425i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−917.271			−1.19592
$768$		0			0
$769$	607.915			0.790526			0.395263	−	0.918568i	$-0.370653\pi$
$769$	607.915			0.790526			0.395263	+	0.918568i	$0.370653\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	517.166i		−	0.669038i	−0.942389	−	0.334519i	$-0.891426\pi$
$773$		−	517.166i		−	0.669038i	0.942389	−	0.334519i	$-0.108574\pi$
$774$		0			0
$775$		−	243.352i		−	0.314003i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	561.139	−	268.571i	0.720333	−	0.344763i
$780$		0			0
$781$			324.017i			0.414875i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	2244.97			2.85983
$786$		0			0
$787$		−	1116.29i		−	1.41841i	−0.705000	−	0.709207i	$-0.749053\pi$
$787$		−	1116.29i		−	1.41841i	0.705000	−	0.709207i	$-0.250947\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	1324.25i		−	1.67414i
$792$		0			0
$793$			36.9226i			0.0465606i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			520.198i			0.652696i	0.945250	+	0.326348i	$0.105818\pi$
$797$			520.198i			0.652696i	−0.945250	+	0.326348i	$0.894182\pi$
$798$		0			0
$799$	−1583.03			−1.98127
$800$		0			0
$801$		0			0
$802$		0			0
$803$	713.644			0.888722
$804$		0			0
$805$	282.083			0.350414
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−38.9064			−0.0480920			−0.0240460	−	0.999711i	$-0.507655\pi$
$809$	−38.9064			−0.0480920			−0.0240460	+	0.999711i	$0.507655\pi$
$810$		0			0
$811$			107.200i			0.132182i	0.997814	+	0.0660910i	$0.0210528\pi$
$811$			107.200i			0.132182i	−0.997814	+	0.0660910i	$0.978947\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2156.43			2.64593
$816$		0			0
$817$	−64.2001	−	134.137i	−0.0785804	−	0.164182i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−278.570			−0.339306			−0.169653	−	0.985504i	$-0.554265\pi$
$821$	−278.570			−0.339306			−0.169653	+	0.985504i	$0.554265\pi$
$822$		0			0
$823$	291.839			0.354603			0.177302	−	0.984157i	$-0.443263\pi$
$823$	291.839			0.354603			0.177302	+	0.984157i	$0.443263\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			835.380i			1.01013i	0.863080	+	0.505067i	$0.168532\pi$
$827$			835.380i			1.01013i	−0.863080	+	0.505067i	$0.831468\pi$
$828$		0			0
$829$			708.916i			0.855146i	0.903981	+	0.427573i	$0.140631\pi$
$829$			708.916i			0.855146i	−0.903981	+	0.427573i	$0.859369\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−335.588			−0.402867
$834$		0			0
$835$		−	1421.36i		−	1.70223i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	42.2231i		−	0.0503256i	−0.999683	−	0.0251628i	$-0.991990\pi$
$839$		−	42.2231i		−	0.0503256i	0.999683	−	0.0251628i	$-0.00801041\pi$
$840$		0			0
$841$	302.013			0.359112
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−418.780			−0.495597
$846$		0			0
$847$	−307.164			−0.362649
$848$		0			0
$849$		0			0
$850$		0			0
$851$			104.101i			0.122328i
$852$		0			0
$853$	755.300			0.885464			0.442732	−	0.896654i	$-0.354009\pi$
$853$	755.300			0.885464			0.442732	+	0.896654i	$0.354009\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	665.500i		−	0.776547i	−0.921544	−	0.388273i	$-0.873072\pi$
$857$		−	665.500i		−	0.776547i	0.921544	−	0.388273i	$-0.126928\pi$
$858$		0			0
$859$	−1555.43			−1.81075			−0.905374	−	0.424615i	$-0.860410\pi$
$859$	−1555.43			−1.81075			−0.905374	+	0.424615i	$0.860410\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			237.941i			0.275714i	0.990452	+	0.137857i	$0.0440214\pi$
$863$			237.941i			0.275714i	−0.990452	+	0.137857i	$0.955979\pi$
$864$		0			0
$865$		−	1676.37i		−	1.93801i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			558.302i			0.642465i
$870$		0			0
$871$	−149.275			−0.171383
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−373.095			−0.426394
$876$		0			0
$877$			1258.09i			1.43453i	0.696799	+	0.717266i	$0.254607\pi$
$877$			1258.09i			1.43453i	−0.696799	+	0.717266i	$0.745393\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	969.322			1.10025			0.550126	−	0.835082i	$-0.314580\pi$
$881$	969.322			1.10025			0.550126	+	0.835082i	$0.314580\pi$
$882$		0			0
$883$	−244.276			−0.276643			−0.138322	−	0.990387i	$-0.544171\pi$
$883$	−244.276			−0.276643			−0.138322	+	0.990387i	$0.544171\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	734.273i		−	0.827816i	−0.910319	−	0.413908i	$-0.864163\pi$
$887$		−	734.273i		−	0.827816i	0.910319	−	0.413908i	$-0.135837\pi$
$888$		0			0
$889$		−	740.495i		−	0.832953i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	415.907	+	868.977i	0.465741	+	0.973098i
$894$		0			0
$895$		−	1923.15i		−	2.14877i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	179.780			0.199977
$900$		0			0
$901$		−	2086.64i		−	2.31591i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	2633.09i		−	2.90949i
$906$		0			0
$907$		−	1538.77i		−	1.69654i	−0.529561	−	0.848272i	$-0.677643\pi$
$907$		−	1538.77i		−	1.69654i	0.529561	−	0.848272i	$-0.322357\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	7.35724i		−	0.00807601i	−0.999992	−	0.00403800i	$-0.998715\pi$
$911$		−	7.35724i		−	0.00807601i	0.999992	−	0.00403800i	$-0.00128534\pi$
$912$		0			0
$913$	−1295.68			−1.41915
$914$		0			0
$915$		0			0
$916$		0			0
$917$	964.468			1.05176
$918$		0			0
$919$	1135.22			1.23528			0.617638	−	0.786463i	$-0.288090\pi$
$919$	1135.22			1.23528			0.617638	+	0.786463i	$0.288090\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	382.510			0.414421
$924$		0			0
$925$		−	673.388i		−	0.727987i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−410.616			−0.441998			−0.220999	−	0.975274i	$-0.570932\pi$
$929$	−410.616			−0.441998			−0.220999	+	0.975274i	$0.570932\pi$
$930$		0			0
$931$	88.1685	+	184.215i	0.0947031	+	0.197868i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	2114.11			2.26108
$936$		0			0
$937$	1058.91			1.13011			0.565054	−	0.825054i	$-0.308855\pi$
$937$	1058.91			1.13011			0.565054	+	0.825054i	$0.308855\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	394.569i		−	0.419309i	−0.977776	−	0.209654i	$-0.932766\pi$
$941$		−	394.569i		−	0.419309i	0.977776	−	0.209654i	$-0.0672339\pi$
$942$		0			0
$943$			159.067i			0.168682i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−827.769			−0.874097			−0.437048	−	0.899438i	$-0.643976\pi$
$947$	−827.769			−0.874097			−0.437048	+	0.899438i	$0.643976\pi$
$948$		0			0
$949$		−	842.473i		−	0.887748i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1500.00i			1.57398i	0.616966	+	0.786990i	$0.288362\pi$
$953$			1500.00i			1.57398i	−0.616966	+	0.786990i	$0.711638\pi$
$954$		0			0
$955$	1866.15			1.95408
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1069.76			1.11549
$960$		0			0
$961$	901.034			0.937601
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	1254.27i		−	1.29976i
$966$		0			0
$967$	−1614.91			−1.67002			−0.835012	−	0.550231i	$-0.814539\pi$
$967$	−1614.91			−1.67002			−0.835012	+	0.550231i	$0.814539\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1299.27i		−	1.33807i	−0.743230	−	0.669036i	$-0.766707\pi$
$971$		−	1299.27i		−	1.33807i	0.743230	−	0.669036i	$-0.233293\pi$
$972$		0			0
$973$	87.1206			0.0895381
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1043.86i		−	1.06844i	−0.845347	−	0.534218i	$-0.820606\pi$
$977$		−	1043.86i		−	1.06844i	0.845347	−	0.534218i	$-0.179394\pi$
$978$		0			0
$979$		−	478.628i		−	0.488895i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	135.881i		−	0.138231i	−0.997609	−	0.0691156i	$-0.977982\pi$
$983$		−	135.881i		−	0.138231i	0.997609	−	0.0691156i	$-0.0220177\pi$
$984$		0			0
$985$	128.955			0.130919
$986$		0			0
$987$		0			0
$988$		0			0
$989$	38.0240			0.0384469
$990$		0			0
$991$		−	1788.07i		−	1.80431i	−0.431415	−	0.902153i	$-0.641986\pi$
$991$		−	1788.07i		−	1.80431i	0.431415	−	0.902153i	$-0.358014\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	408.278			0.410330
$996$		0			0
$997$	1038.20			1.04133			0.520664	−	0.853761i	$-0.325684\pi$
$997$	1038.20			1.04133			0.520664	+	0.853761i	$0.325684\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.3.o.r.721.1		20
3.2	odd	2		912.3.o.e.721.20		20
4.3	odd	2		1368.3.o.c.721.1		20
12.11	even	2		456.3.o.a.265.10	✓	20
19.18	odd	2	inner	2736.3.o.r.721.2		20
57.56	even	2		912.3.o.e.721.10		20
76.75	even	2		1368.3.o.c.721.2		20
228.227	odd	2		456.3.o.a.265.20	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.3.o.a.265.10	✓	20	12.11	even	2
456.3.o.a.265.20	yes	20	228.227	odd	2
912.3.o.e.721.10		20	57.56	even	2
912.3.o.e.721.20		20	3.2	odd	2
1368.3.o.c.721.1		20	4.3	odd	2
1368.3.o.c.721.2		20	76.75	even	2
2736.3.o.r.721.1		20	1.1	even	1	trivial
2736.3.o.r.721.2		20	19.18	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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2736.o (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-7.51170 q^{5} +7.72974 q^{7} +O(q^{10})\)	\(q-7.51170 q^{5} +7.72974 q^{7} +9.01455 q^{11} -10.6419i q^{13} -31.2209 q^{17} +(8.20262 + 17.1382i) q^{19} -4.85819 q^{23} +31.4256 q^{25} +23.2161i q^{29} -7.74375i q^{31} -58.0635 q^{35} -21.4280i q^{37} -32.7421i q^{41} -7.82679 q^{43} +50.7042 q^{47} +10.7488 q^{49} +66.8345i q^{53} -67.7146 q^{55} -86.1944i q^{59} -3.46955 q^{61} +79.9386i q^{65} -14.0271i q^{67} +35.9438i q^{71} +79.1658 q^{73} +69.6801 q^{77} +61.9335i q^{79} -143.733 q^{83} +234.522 q^{85} -53.0951i q^{89} -82.2590i q^{91} +(-61.6156 - 128.737i) q^{95} +124.208i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 16 q^{7}+O(q^{10}) \) 20 * q + 16 * q^7	\( 20 q + 16 q^{7} + 16 q^{11} - 32 q^{17} - 40 q^{19} + 64 q^{23} + 68 q^{25} - 208 q^{35} - 64 q^{43} + 48 q^{47} + 20 q^{49} + 336 q^{55} + 184 q^{61} + 104 q^{73} - 88 q^{77} + 224 q^{83} - 136 q^{85} - 320 q^{95}+O(q^{100}) \) 20 * q + 16 * q^7 + 16 * q^11 - 32 * q^17 - 40 * q^19 + 64 * q^23 + 68 * q^25 - 208 * q^35 - 64 * q^43 + 48 * q^47 + 20 * q^49 + 336 * q^55 + 184 * q^61 + 104 * q^73 - 88 * q^77 + 224 * q^83 - 136 * q^85 - 320 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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