LMFDB - Embedded newform 2736.3.o.r.721.20

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			721.20
Root			$4.05338 - 7.02066i$ of defining polynomial
Character	$\chi$	$=$	2736.721
Dual form			2736.3.o.r.721.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.20492 q^{5} +8.00860 q^{7} +O(q^{10})$	$q+8.20492 q^{5} +8.00860 q^{7} +7.74291 q^{11} +7.64699i q^{13} -11.4841 q^{17} +(-17.5525 + 7.27380i) q^{19} +22.9620 q^{23} +42.3207 q^{25} +17.5778i q^{29} -52.3102i q^{31} +65.7099 q^{35} +40.7090i q^{37} +36.2297i q^{41} +52.5407 q^{43} +3.90967 q^{47} +15.1376 q^{49} +49.8052i q^{53} +63.5299 q^{55} +46.6932i q^{59} +40.8276 q^{61} +62.7429i q^{65} -35.0230i q^{67} +106.718i q^{71} +113.791 q^{73} +62.0098 q^{77} -116.271i q^{79} +48.3546 q^{83} -94.2264 q^{85} -24.3444i q^{89} +61.2416i q^{91} +(-144.017 + 59.6809i) q^{95} +85.7882i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	8.20492			1.64098			0.820492	−	0.571658i	$-0.193700\pi$
$5$	8.20492			1.64098			0.820492	+	0.571658i	$0.193700\pi$
$6$		0			0
$7$	8.00860			1.14409			0.572043	−	0.820224i	$-0.306151\pi$
$7$	8.00860			1.14409			0.572043	+	0.820224i	$0.306151\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	7.74291			0.703901			0.351950	−	0.936019i	$-0.385519\pi$
$11$	7.74291			0.703901			0.351950	+	0.936019i	$0.385519\pi$
$12$		0			0
$13$			7.64699i			0.588230i	0.955770	+	0.294115i	$0.0950248\pi$
$13$			7.64699i			0.588230i	−0.955770	+	0.294115i	$0.904975\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−11.4841			−0.675537			−0.337769	−	0.941229i	$-0.609672\pi$
$17$	−11.4841			−0.675537			−0.337769	+	0.941229i	$0.609672\pi$
$18$		0			0
$19$	−17.5525	+	7.27380i	−0.923818	+	0.382831i
$19$	−17.5525	+	7.27380i	−0.923818	+	0.382831i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	22.9620			0.998347			0.499173	−	0.866502i	$-0.333637\pi$
$23$	22.9620			0.998347			0.499173	+	0.866502i	$0.333637\pi$
$24$		0			0
$25$	42.3207			1.69283
$26$		0			0
$27$		0			0
$28$		0			0
$29$			17.5778i			0.606132i	0.952970	+	0.303066i	$0.0980103\pi$
$29$			17.5778i			0.606132i	−0.952970	+	0.303066i	$0.901990\pi$
$30$		0			0
$31$		−	52.3102i		−	1.68742i	−0.536796	−	0.843712i	$-0.680365\pi$
$31$		−	52.3102i		−	1.68742i	0.536796	−	0.843712i	$-0.319635\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	65.7099			1.87743
$36$		0			0
$37$			40.7090i			1.10024i	0.835084	+	0.550122i	$0.185419\pi$
$37$			40.7090i			1.10024i	−0.835084	+	0.550122i	$0.814581\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			36.2297i			0.883650i	0.897101	+	0.441825i	$0.145669\pi$
$41$			36.2297i			0.883650i	−0.897101	+	0.441825i	$0.854331\pi$
$42$		0			0
$43$	52.5407			1.22188			0.610938	−	0.791678i	$-0.290792\pi$
$43$	52.5407			1.22188			0.610938	+	0.791678i	$0.290792\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.90967			0.0831844			0.0415922	−	0.999135i	$-0.486757\pi$
$47$	3.90967			0.0831844			0.0415922	+	0.999135i	$0.486757\pi$
$48$		0			0
$49$	15.1376			0.308931
$50$		0			0
$51$		0			0
$52$		0			0
$53$			49.8052i			0.939720i	0.882741	+	0.469860i	$0.155696\pi$
$53$			49.8052i			0.939720i	−0.882741	+	0.469860i	$0.844304\pi$
$54$		0			0
$55$	63.5299			1.15509
$56$		0			0
$57$		0			0
$58$		0			0
$59$			46.6932i			0.791410i	0.918378	+	0.395705i	$0.129500\pi$
$59$			46.6932i			0.791410i	−0.918378	+	0.395705i	$0.870500\pi$
$60$		0			0
$61$	40.8276			0.669305			0.334652	−	0.942342i	$-0.391381\pi$
$61$	40.8276			0.669305			0.334652	+	0.942342i	$0.391381\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			62.7429i			0.965276i
$66$		0			0
$67$		−	35.0230i		−	0.522731i	−0.965240	−	0.261365i	$-0.915827\pi$
$67$		−	35.0230i		−	0.522731i	0.965240	−	0.261365i	$-0.0841727\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			106.718i			1.50307i	0.659694	+	0.751535i	$0.270686\pi$
$71$			106.718i			1.50307i	−0.659694	+	0.751535i	$0.729314\pi$
$72$		0			0
$73$	113.791			1.55878			0.779392	−	0.626537i	$-0.215528\pi$
$73$	113.791			1.55878			0.779392	+	0.626537i	$0.215528\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	62.0098			0.805322
$78$		0			0
$79$		−	116.271i		−	1.47178i	−0.677099	−	0.735892i	$-0.736763\pi$
$79$		−	116.271i		−	1.47178i	0.677099	−	0.735892i	$-0.263237\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	48.3546			0.582585			0.291293	−	0.956634i	$-0.405915\pi$
$83$	48.3546			0.582585			0.291293	+	0.956634i	$0.405915\pi$
$84$		0			0
$85$	−94.2264			−1.10855
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	24.3444i		−	0.273532i	−0.990603	−	0.136766i	$-0.956329\pi$
$89$		−	24.3444i		−	0.273532i	0.990603	−	0.136766i	$-0.0436708\pi$
$90$		0			0
$91$			61.2416i			0.672985i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−144.017	+	59.6809i	−1.51597	+	0.628220i
$96$		0			0
$97$			85.7882i			0.884414i	0.896913	+	0.442207i	$0.145804\pi$
$97$			85.7882i			0.884414i	−0.896913	+	0.442207i	$0.854196\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−166.786			−1.65135			−0.825674	−	0.564148i	$-0.809205\pi$
$101$	−166.786			−1.65135			−0.825674	+	0.564148i	$0.809205\pi$
$102$		0			0
$103$		−	159.516i		−	1.54870i	−0.632758	−	0.774349i	$-0.718077\pi$
$103$		−	159.516i		−	1.54870i	0.632758	−	0.774349i	$-0.281923\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	99.9199i		−	0.933830i	−0.884302	−	0.466915i	$-0.845365\pi$
$107$		−	99.9199i		−	0.933830i	0.884302	−	0.466915i	$-0.154635\pi$
$108$		0			0
$109$		−	152.435i		−	1.39849i	−0.714883	−	0.699244i	$-0.753520\pi$
$109$		−	152.435i		−	1.39849i	0.714883	−	0.699244i	$-0.246480\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	58.3110i		−	0.516026i	−0.966141	−	0.258013i	$-0.916932\pi$
$113$		−	58.3110i		−	0.516026i	0.966141	−	0.258013i	$-0.0830678\pi$
$114$		0			0
$115$	188.401			1.63827
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−91.9718			−0.772872
$120$		0			0
$121$	−61.0474			−0.504524
$122$		0			0
$123$		0			0
$124$		0			0
$125$	142.115			1.13692
$126$		0			0
$127$			110.826i			0.872649i	0.899789	+	0.436325i	$0.143720\pi$
$127$			110.826i			0.872649i	−0.899789	+	0.436325i	$0.856280\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	93.5498			0.714120			0.357060	−	0.934081i	$-0.383779\pi$
$131$	93.5498			0.714120			0.357060	+	0.934081i	$0.383779\pi$
$132$		0			0
$133$	−140.571	+	58.2529i	−1.05693	+	0.437992i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	80.4035			0.586887			0.293443	−	0.955976i	$-0.405199\pi$
$137$	80.4035			0.586887			0.293443	+	0.955976i	$0.405199\pi$
$138$		0			0
$139$	−205.954			−1.48168			−0.740841	−	0.671680i	$-0.765573\pi$
$139$	−205.954			−1.48168			−0.740841	+	0.671680i	$0.765573\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			59.2099i			0.414055i
$144$		0			0
$145$			144.225i			0.994653i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−97.8024			−0.656392			−0.328196	−	0.944610i	$-0.606441\pi$
$149$	−97.8024			−0.656392			−0.328196	+	0.944610i	$0.606441\pi$
$150$		0			0
$151$		−	37.3870i		−	0.247596i	−0.992307	−	0.123798i	$-0.960492\pi$
$151$		−	37.3870i		−	0.247596i	0.992307	−	0.123798i	$-0.0395075\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	429.201i		−	2.76904i
$156$		0			0
$157$	−270.158			−1.72075			−0.860375	−	0.509662i	$-0.829770\pi$
$157$	−270.158			−1.72075			−0.860375	+	0.509662i	$0.829770\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	183.893			1.14219
$162$		0			0
$163$	67.0137			0.411127			0.205563	−	0.978644i	$-0.434097\pi$
$163$	67.0137			0.411127			0.205563	+	0.978644i	$0.434097\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	188.784i		−	1.13044i	−0.824939	−	0.565222i	$-0.808791\pi$
$167$		−	188.784i		−	1.13044i	0.824939	−	0.565222i	$-0.191209\pi$
$168$		0			0
$169$	110.524			0.653986
$170$		0			0
$171$		0			0
$172$		0			0
$173$			316.740i			1.83086i	0.402472	+	0.915432i	$0.368151\pi$
$173$			316.740i			1.83086i	−0.402472	+	0.915432i	$0.631849\pi$
$174$		0			0
$175$	338.930			1.93674
$176$		0			0
$177$		0			0
$178$		0			0
$179$			254.460i			1.42156i	0.703413	+	0.710781i	$0.251658\pi$
$179$			254.460i			1.42156i	−0.703413	+	0.710781i	$0.748342\pi$
$180$		0			0
$181$		−	199.320i		−	1.10122i	−0.834763	−	0.550609i	$-0.814396\pi$
$181$		−	199.320i		−	1.10122i	0.834763	−	0.550609i	$-0.185604\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			334.014i			1.80548i
$186$		0			0
$187$	−88.9205			−0.475511
$188$		0			0
$189$		0			0
$190$		0			0
$191$	103.088			0.539728			0.269864	−	0.962898i	$-0.413021\pi$
$191$	103.088			0.539728			0.269864	+	0.962898i	$0.413021\pi$
$192$		0			0
$193$			291.262i			1.50913i	0.656225	+	0.754565i	$0.272152\pi$
$193$			291.262i			1.50913i	−0.656225	+	0.754565i	$0.727848\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	246.923			1.25342			0.626709	−	0.779254i	$-0.284402\pi$
$197$	246.923			1.25342			0.626709	+	0.779254i	$0.284402\pi$
$198$		0			0
$199$	−17.5380			−0.0881307			−0.0440653	−	0.999029i	$-0.514031\pi$
$199$	−17.5380			−0.0881307			−0.0440653	+	0.999029i	$0.514031\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			140.774i			0.693467i
$204$		0			0
$205$			297.261i			1.45006i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−135.908	+	56.3203i	−0.650276	+	0.269475i
$210$		0			0
$211$		−	300.813i		−	1.42565i	−0.701341	−	0.712826i	$-0.747415\pi$
$211$		−	300.813i		−	1.42565i	0.701341	−	0.712826i	$-0.252585\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	431.092			2.00508
$216$		0			0
$217$		−	418.931i		−	1.93056i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	87.8190i		−	0.397371i
$222$		0			0
$223$		−	118.853i		−	0.532971i	−0.963839	−	0.266486i	$-0.914138\pi$
$223$		−	118.853i		−	0.532971i	0.963839	−	0.266486i	$-0.0858625\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	50.4628i		−	0.222303i	−0.993803	−	0.111152i	$-0.964546\pi$
$227$		−	50.4628i		−	0.222303i	0.993803	−	0.111152i	$-0.0354539\pi$
$228$		0			0
$229$	109.997			0.480337			0.240168	−	0.970731i	$-0.422797\pi$
$229$	109.997			0.480337			0.240168	+	0.970731i	$0.422797\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	18.8197			0.0807711			0.0403855	−	0.999184i	$-0.487141\pi$
$233$	18.8197			0.0807711			0.0403855	+	0.999184i	$0.487141\pi$
$234$		0			0
$235$	32.0785			0.136504
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−43.6319			−0.182560			−0.0912802	−	0.995825i	$-0.529096\pi$
$239$	−43.6319			−0.182560			−0.0912802	+	0.995825i	$0.529096\pi$
$240$		0			0
$241$			114.931i			0.476893i	0.971156	+	0.238447i	$0.0766382\pi$
$241$			114.931i			0.476893i	−0.971156	+	0.238447i	$0.923362\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	124.203			0.506951
$246$		0			0
$247$	−55.6226	−	134.224i	−0.225193	−	0.543417i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	120.802			0.481281			0.240641	−	0.970614i	$-0.422643\pi$
$251$	120.802			0.481281			0.240641	+	0.970614i	$0.422643\pi$
$252$		0			0
$253$	177.792			0.702737
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	463.010i		−	1.80160i	−0.434238	−	0.900798i	$-0.642982\pi$
$257$		−	463.010i		−	1.80160i	0.434238	−	0.900798i	$-0.357018\pi$
$258$		0			0
$259$			326.022i			1.25877i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	87.4806			0.332626			0.166313	−	0.986073i	$-0.446814\pi$
$263$	87.4806			0.332626			0.166313	+	0.986073i	$0.446814\pi$
$264$		0			0
$265$			408.648i			1.54207i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			396.728i			1.47482i	0.675443	+	0.737412i	$0.263952\pi$
$269$			396.728i			1.47482i	−0.675443	+	0.737412i	$0.736048\pi$
$270$		0			0
$271$	−328.002			−1.21034			−0.605169	−	0.796097i	$-0.706894\pi$
$271$	−328.002			−1.21034			−0.605169	+	0.796097i	$0.706894\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	327.685			1.19158
$276$		0			0
$277$	−332.247			−1.19945			−0.599724	−	0.800207i	$-0.704723\pi$
$277$	−332.247			−1.19945			−0.599724	+	0.800207i	$0.704723\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	0.472892i		−	0.00168289i	−1.00000		0.000841445i	$-0.999732\pi$
$281$		−	0.472892i		−	0.00168289i	1.00000		0.000841445i	$-0.000267840\pi$
$282$		0			0
$283$	56.6305			0.200108			0.100054	−	0.994982i	$-0.468098\pi$
$283$	56.6305			0.200108			0.100054	+	0.994982i	$0.468098\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			290.149i			1.01097i
$288$		0			0
$289$	−157.115			−0.543650
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	456.196i		−	1.55698i	−0.627656	−	0.778491i	$-0.715985\pi$
$293$		−	456.196i		−	1.55698i	0.627656	−	0.778491i	$-0.284015\pi$
$294$		0			0
$295$			383.114i			1.29869i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			175.590i			0.587257i
$300$		0			0
$301$	420.777			1.39793
$302$		0			0
$303$		0			0
$304$		0			0
$305$	334.987			1.09832
$306$		0			0
$307$			223.200i			0.727037i	0.931587	+	0.363519i	$0.118425\pi$
$307$			223.200i			0.727037i	−0.931587	+	0.363519i	$0.881575\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−423.534			−1.36185			−0.680924	−	0.732354i	$-0.738421\pi$
$311$	−423.534			−1.36185			−0.680924	+	0.732354i	$0.738421\pi$
$312$		0			0
$313$	276.738			0.884148			0.442074	−	0.896979i	$-0.354243\pi$
$313$	276.738			0.884148			0.442074	+	0.896979i	$0.354243\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	580.133i		−	1.83007i	−0.403370	−	0.915037i	$-0.632161\pi$
$317$		−	580.133i		−	1.83007i	0.403370	−	0.915037i	$-0.367839\pi$
$318$		0			0
$319$			136.104i			0.426657i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	201.576	−	83.5332i	0.624073	−	0.258617i
$324$		0			0
$325$			323.626i			0.995772i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	31.3109			0.0951700
$330$		0			0
$331$			387.001i			1.16919i	0.811326	+	0.584594i	$0.198746\pi$
$331$			387.001i			1.16919i	−0.811326	+	0.584594i	$0.801254\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	287.361i		−	0.857793i
$336$		0			0
$337$		−	318.289i		−	0.944478i	−0.881471	−	0.472239i	$-0.843446\pi$
$337$		−	318.289i		−	0.944478i	0.881471	−	0.472239i	$-0.156554\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	405.033i		−	1.18778i
$342$		0			0
$343$	−271.190			−0.790642
$344$		0			0
$345$		0			0
$346$		0			0
$347$	439.969			1.26792			0.633961	−	0.773365i	$-0.281428\pi$
$347$	439.969			1.26792			0.633961	+	0.773365i	$0.281428\pi$
$348$		0			0
$349$	350.244			1.00356			0.501782	−	0.864994i	$-0.332678\pi$
$349$	350.244			1.00356			0.501782	+	0.864994i	$0.332678\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−104.803			−0.296891			−0.148446	−	0.988921i	$-0.547427\pi$
$353$	−104.803			−0.296891			−0.148446	+	0.988921i	$0.547427\pi$
$354$		0			0
$355$			875.612i			2.46651i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	429.578			1.19660			0.598298	−	0.801273i	$-0.295844\pi$
$359$	429.578			1.19660			0.598298	+	0.801273i	$0.295844\pi$
$360$		0			0
$361$	255.184	−	255.347i	0.706880	−	0.707333i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	933.648			2.55794
$366$		0			0
$367$	83.5313			0.227606			0.113803	−	0.993503i	$-0.463697\pi$
$367$	83.5313			0.227606			0.113803	+	0.993503i	$0.463697\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			398.870i			1.07512i
$372$		0			0
$373$			306.430i			0.821528i	0.911742	+	0.410764i	$0.134738\pi$
$373$			306.430i			0.821528i	−0.911742	+	0.410764i	$0.865262\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−134.417			−0.356545
$378$		0			0
$379$		−	1.09504i		−	0.00288929i	−0.999999	−	0.00144465i	$-0.999540\pi$
$379$		−	1.09504i		−	0.00288929i	0.999999	−	0.00144465i	$-0.000459845\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			521.642i			1.36199i	0.732289	+	0.680994i	$0.238452\pi$
$383$			521.642i			1.36199i	−0.732289	+	0.680994i	$0.761548\pi$
$384$		0			0
$385$	508.786			1.32152
$386$		0			0
$387$		0			0
$388$		0			0
$389$	282.146			0.725311			0.362655	−	0.931923i	$-0.381870\pi$
$389$	282.146			0.725311			0.362655	+	0.931923i	$0.381870\pi$
$390$		0			0
$391$	−263.698			−0.674420
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	953.994i		−	2.41517i
$396$		0			0
$397$	541.473			1.36391			0.681955	−	0.731394i	$-0.261130\pi$
$397$	541.473			1.36391			0.681955	+	0.731394i	$0.261130\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			118.923i			0.296566i	0.988945	+	0.148283i	$0.0473746\pi$
$401$			118.923i			0.296566i	−0.988945	+	0.148283i	$0.952625\pi$
$402$		0			0
$403$	400.015			0.992593
$404$		0			0
$405$		0			0
$406$		0			0
$407$			315.206i			0.774462i
$408$		0			0
$409$		−	289.380i		−	0.707531i	−0.935334	−	0.353765i	$-0.884901\pi$
$409$		−	289.380i		−	0.707531i	0.935334	−	0.353765i	$-0.115099\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			373.947i			0.905440i
$414$		0			0
$415$	396.746			0.956013
$416$		0			0
$417$		0			0
$418$		0			0
$419$	478.095			1.14104			0.570519	−	0.821284i	$-0.306742\pi$
$419$	478.095			1.14104			0.570519	+	0.821284i	$0.306742\pi$
$420$		0			0
$421$		−	446.855i		−	1.06141i	−0.847555	−	0.530707i	$-0.821926\pi$
$421$		−	446.855i		−	1.06141i	0.847555	−	0.530707i	$-0.178074\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−486.017			−1.14357
$426$		0			0
$427$	326.972			0.765742
$428$		0			0
$429$		0			0
$430$		0			0
$431$			281.363i			0.652815i	0.945229	+	0.326408i	$0.105838\pi$
$431$			281.363i			0.652815i	−0.945229	+	0.326408i	$0.894162\pi$
$432$		0			0
$433$			273.831i			0.632405i	0.948692	+	0.316203i	$0.102408\pi$
$433$			273.831i			0.632405i	−0.948692	+	0.316203i	$0.897592\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−403.041	+	167.021i	−0.922291	+	0.382198i
$438$		0			0
$439$		−	746.505i		−	1.70047i	−0.526405	−	0.850234i	$-0.676460\pi$
$439$		−	746.505i		−	1.70047i	0.526405	−	0.850234i	$-0.323540\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−101.138			−0.228301			−0.114151	−	0.993463i	$-0.536415\pi$
$443$	−101.138			−0.228301			−0.114151	+	0.993463i	$0.536415\pi$
$444$		0			0
$445$		−	199.743i		−	0.448862i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	617.400i		−	1.37506i	−0.726158	−	0.687528i	$-0.758696\pi$
$449$		−	617.400i		−	1.37506i	0.726158	−	0.687528i	$-0.241304\pi$
$450$		0			0
$451$			280.523i			0.622002i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			502.483i			1.10436i
$456$		0			0
$457$	40.5778			0.0887917			0.0443959	−	0.999014i	$-0.485864\pi$
$457$	40.5778			0.0887917			0.0443959	+	0.999014i	$0.485864\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−409.282			−0.887814			−0.443907	−	0.896073i	$-0.646408\pi$
$461$	−409.282			−0.887814			−0.443907	+	0.896073i	$0.646408\pi$
$462$		0			0
$463$	642.547			1.38779			0.693895	−	0.720076i	$-0.255893\pi$
$463$	642.547			1.38779			0.693895	+	0.720076i	$0.255893\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−787.885			−1.68712			−0.843560	−	0.537035i	$-0.819544\pi$
$467$	−787.885			−1.68712			−0.843560	+	0.537035i	$0.819544\pi$
$468$		0			0
$469$		−	280.485i		−	0.598049i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	406.818			0.860080
$474$		0			0
$475$	−742.836	+	307.832i	−1.56387	+	0.648068i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−329.344			−0.687566			−0.343783	−	0.939049i	$-0.611708\pi$
$479$	−329.344			−0.687566			−0.343783	+	0.939049i	$0.611708\pi$
$480$		0			0
$481$	−311.301			−0.647196
$482$		0			0
$483$		0			0
$484$		0			0
$485$			703.885i			1.45131i
$486$		0			0
$487$			172.055i			0.353296i	0.984274	+	0.176648i	$0.0565254\pi$
$487$			172.055i			0.353296i	−0.984274	+	0.176648i	$0.943475\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−768.359			−1.56489			−0.782443	−	0.622723i	$-0.786026\pi$
$491$	−768.359			−1.56489			−0.782443	+	0.622723i	$0.786026\pi$
$492$		0			0
$493$		−	201.866i		−	0.409465i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			854.661i			1.71964i
$498$		0			0
$499$	−704.745			−1.41232			−0.706158	−	0.708055i	$-0.749573\pi$
$499$	−704.745			−1.41232			−0.706158	+	0.708055i	$0.749573\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−621.881			−1.23634			−0.618172	−	0.786043i	$-0.712126\pi$
$503$	−621.881			−1.23634			−0.618172	+	0.786043i	$0.712126\pi$
$504$		0			0
$505$	−1368.47			−2.70984
$506$		0			0
$507$		0			0
$508$		0			0
$509$			334.039i			0.656265i	0.944632	+	0.328133i	$0.106419\pi$
$509$			334.039i			0.656265i	−0.944632	+	0.328133i	$0.893581\pi$
$510$		0			0
$511$	911.308			1.78338
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	1308.82i		−	2.54139i
$516$		0			0
$517$	30.2722			0.0585535
$518$		0			0
$519$		0			0
$520$		0			0
$521$			439.593i			0.843749i	0.906654	+	0.421875i	$0.138628\pi$
$521$			439.593i			0.843749i	−0.906654	+	0.421875i	$0.861372\pi$
$522$		0			0
$523$		−	624.484i		−	1.19404i	−0.802226	−	0.597021i	$-0.796351\pi$
$523$		−	624.484i		−	1.19404i	0.802226	−	0.597021i	$-0.203649\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			600.737i			1.13992i
$528$		0			0
$529$	−1.74751			−0.00330342
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−277.048			−0.519789
$534$		0			0
$535$		−	819.834i		−	1.53240i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	117.209			0.217457
$540$		0			0
$541$	−641.553			−1.18586			−0.592932	−	0.805252i	$-0.702030\pi$
$541$	−641.553			−1.18586			−0.592932	+	0.805252i	$0.702030\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	1250.72i		−	2.29490i
$546$		0			0
$547$		−	118.198i		−	0.216084i	−0.994146	−	0.108042i	$-0.965542\pi$
$547$		−	118.198i		−	0.216084i	0.994146	−	0.108042i	$-0.0344581\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−127.858	−	308.536i	−0.232046	−	0.559956i
$552$		0			0
$553$		−	931.167i		−	1.68385i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	992.629			1.78210			0.891049	−	0.453907i	$-0.149970\pi$
$557$	992.629			1.78210			0.891049	+	0.453907i	$0.149970\pi$
$558$		0			0
$559$			401.778i			0.718744i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	171.881i		−	0.305295i	−0.988281	−	0.152648i	$-0.951220\pi$
$563$		−	171.881i		−	0.305295i	0.988281	−	0.152648i	$-0.0487800\pi$
$564$		0			0
$565$		−	478.437i		−	0.846791i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			353.453i			0.621182i	0.950544	+	0.310591i	$0.100527\pi$
$569$			353.453i			0.621182i	−0.950544	+	0.310591i	$0.899473\pi$
$570$		0			0
$571$	−178.222			−0.312122			−0.156061	−	0.987747i	$-0.549880\pi$
$571$	−178.222			−0.312122			−0.156061	+	0.987747i	$0.549880\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	971.767			1.69003
$576$		0			0
$577$	−988.498			−1.71317			−0.856584	−	0.516008i	$-0.827417\pi$
$577$	−988.498			−1.71317			−0.856584	+	0.516008i	$0.827417\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	387.252			0.666527
$582$		0			0
$583$			385.637i			0.661470i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−448.477			−0.764014			−0.382007	−	0.924159i	$-0.624767\pi$
$587$	−448.477			−0.764014			−0.382007	+	0.924159i	$0.624767\pi$
$588$		0			0
$589$	380.493	+	918.176i	0.645999	+	1.55887i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−869.085			−1.46557			−0.732787	−	0.680458i	$-0.761781\pi$
$593$	−869.085			−1.46557			−0.732787	+	0.680458i	$0.761781\pi$
$594$		0			0
$595$	−754.621			−1.26827
$596$		0			0
$597$		0			0
$598$		0			0
$599$			187.244i			0.312594i	0.987710	+	0.156297i	$0.0499556\pi$
$599$			187.244i			0.312594i	−0.987710	+	0.156297i	$0.950044\pi$
$600$		0			0
$601$		−	353.316i		−	0.587881i	−0.955824	−	0.293940i	$-0.905033\pi$
$601$		−	353.316i		−	0.587881i	0.955824	−	0.293940i	$-0.0949667\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−500.889			−0.827916
$606$		0			0
$607$		−	60.7690i		−	0.100114i	−0.998746	−	0.0500569i	$-0.984060\pi$
$607$		−	60.7690i		−	0.100114i	0.998746	−	0.0500569i	$-0.0159403\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			29.8972i			0.0489315i
$612$		0			0
$613$	774.683			1.26376			0.631878	−	0.775067i	$-0.282284\pi$
$613$	774.683			1.26376			0.631878	+	0.775067i	$0.282284\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	379.362			0.614849			0.307424	−	0.951573i	$-0.400533\pi$
$617$	379.362			0.614849			0.307424	+	0.951573i	$0.400533\pi$
$618$		0			0
$619$	−428.108			−0.691612			−0.345806	−	0.938306i	$-0.612394\pi$
$619$	−428.108			−0.691612			−0.345806	+	0.938306i	$0.612394\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	194.964i		−	0.312944i
$624$		0			0
$625$	108.025			0.172840
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	467.508i		−	0.743255i
$630$		0			0
$631$	−878.002			−1.39145			−0.695723	−	0.718310i	$-0.744916\pi$
$631$	−878.002			−1.39145			−0.695723	+	0.718310i	$0.744916\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			909.322i			1.43200i
$636$		0			0
$637$			115.757i			0.181722i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	1234.82i		−	1.92640i	−0.268785	−	0.963200i	$-0.586622\pi$
$641$		−	1234.82i		−	1.92640i	0.268785	−	0.963200i	$-0.413378\pi$
$642$		0			0
$643$	43.6391			0.0678680			0.0339340	−	0.999424i	$-0.489196\pi$
$643$	43.6391			0.0678680			0.0339340	+	0.999424i	$0.489196\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	0.0739040			0.000114226			5.71128e−5	−	1.00000i	$-0.499982\pi$
$647$	0.0739040			0.000114226			5.71128e−5		1.00000i	$0.499982\pi$
$648$		0			0
$649$			361.541i			0.557074i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	468.859			0.718008			0.359004	−	0.933336i	$-0.383116\pi$
$653$	468.859			0.718008			0.359004	+	0.933336i	$0.383116\pi$
$654$		0			0
$655$	767.568			1.17186
$656$		0			0
$657$		0			0
$658$		0			0
$659$			153.264i			0.232571i	0.993216	+	0.116286i	$0.0370988\pi$
$659$			153.264i			0.232571i	−0.993216	+	0.116286i	$0.962901\pi$
$660$		0			0
$661$		−	670.154i		−	1.01385i	−0.861991	−	0.506924i	$-0.830782\pi$
$661$		−	670.154i		−	1.01385i	0.861991	−	0.506924i	$-0.169218\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1153.38	+	477.960i	−1.73440	+	0.718737i
$666$		0			0
$667$			403.622i			0.605130i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	316.124			0.471124
$672$		0			0
$673$		−	819.498i		−	1.21768i	−0.793294	−	0.608839i	$-0.791635\pi$
$673$		−	819.498i		−	1.21768i	0.793294	−	0.608839i	$-0.208365\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			842.863i			1.24500i	0.782621	+	0.622498i	$0.213882\pi$
$677$			842.863i			1.24500i	−0.782621	+	0.622498i	$0.786118\pi$
$678$		0			0
$679$			687.043i			1.01185i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			1286.00i			1.88286i	0.337205	+	0.941431i	$0.390518\pi$
$683$			1286.00i			1.88286i	−0.337205	+	0.941431i	$0.609482\pi$
$684$		0			0
$685$	659.704			0.963071
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−380.860			−0.552771
$690$		0			0
$691$	−154.537			−0.223642			−0.111821	−	0.993728i	$-0.535668\pi$
$691$	−154.537			−0.223642			−0.111821	+	0.993728i	$0.535668\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1689.83			−2.43142
$696$		0			0
$697$		−	416.066i		−	0.596938i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	934.984			1.33379			0.666893	−	0.745153i	$-0.267624\pi$
$701$	934.984			1.33379			0.666893	+	0.745153i	$0.267624\pi$
$702$		0			0
$703$	−296.109	−	714.547i	−0.421208	−	1.01643i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1335.72			−1.88928
$708$		0			0
$709$	616.444			0.869455			0.434728	−	0.900562i	$-0.356845\pi$
$709$	616.444			0.869455			0.434728	+	0.900562i	$0.356845\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	1201.14i		−	1.68463i
$714$		0			0
$715$			485.813i			0.679458i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1116.28			1.55255			0.776273	−	0.630397i	$-0.217108\pi$
$719$	1116.28			1.55255			0.776273	+	0.630397i	$0.217108\pi$
$720$		0			0
$721$		−	1277.50i		−	1.77184i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			743.907i			1.02608i
$726$		0			0
$727$	−457.431			−0.629203			−0.314602	−	0.949224i	$-0.601871\pi$
$727$	−457.431			−0.629203			−0.314602	+	0.949224i	$0.601871\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−603.384			−0.825423
$732$		0			0
$733$	−747.874			−1.02029			−0.510146	−	0.860088i	$-0.670409\pi$
$733$	−747.874			−1.02029			−0.510146	+	0.860088i	$0.670409\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	271.180i		−	0.367950i
$738$		0			0
$739$	1211.75			1.63971			0.819855	−	0.572571i	$-0.194054\pi$
$739$	1211.75			1.63971			0.819855	+	0.572571i	$0.194054\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			354.724i			0.477422i	0.971091	+	0.238711i	$0.0767248\pi$
$743$			354.724i			0.477422i	−0.971091	+	0.238711i	$0.923275\pi$
$744$		0			0
$745$	−802.461			−1.07713
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	800.218i		−	1.06838i
$750$		0			0
$751$			946.440i			1.26024i	0.776498	+	0.630120i	$0.216994\pi$
$751$			946.440i			1.26024i	−0.776498	+	0.630120i	$0.783006\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	306.757i		−	0.406301i
$756$		0			0
$757$	−378.745			−0.500324			−0.250162	−	0.968204i	$-0.580484\pi$
$757$	−378.745			−0.500324			−0.250162	+	0.968204i	$0.580484\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−827.308			−1.08713			−0.543567	−	0.839366i	$-0.682926\pi$
$761$	−827.308			−1.08713			−0.543567	+	0.839366i	$0.682926\pi$
$762$		0			0
$763$		−	1220.79i		−	1.59999i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−357.062			−0.465531
$768$		0			0
$769$	−674.940			−0.877686			−0.438843	−	0.898564i	$-0.644612\pi$
$769$	−674.940			−0.877686			−0.438843	+	0.898564i	$0.644612\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			1468.47i			1.89971i	0.312693	+	0.949854i	$0.398769\pi$
$773$			1468.47i			1.89971i	−0.312693	+	0.949854i	$0.601231\pi$
$774$		0			0
$775$		−	2213.80i		−	2.85652i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−263.527	−	635.923i	−0.338289	−	0.816332i
$780$		0			0
$781$			826.307i			1.05801i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2216.62			−2.82372
$786$		0			0
$787$		−	1121.86i		−	1.42548i	−0.701427	−	0.712742i	$-0.747453\pi$
$787$		−	1121.86i		−	1.42548i	0.701427	−	0.712742i	$-0.252547\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	466.989i		−	0.590378i
$792$		0			0
$793$			312.208i			0.393705i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	161.200i		−	0.202258i	−0.994873	−	0.101129i	$-0.967755\pi$
$797$		−	161.200i		−	0.202258i	0.994873	−	0.101129i	$-0.0322455\pi$
$798$		0			0
$799$	−44.8991			−0.0561941
$800$		0			0
$801$		0			0
$802$		0			0
$803$	881.074			1.09723
$804$		0			0
$805$	1508.83			1.87432
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1304.40			−1.61236			−0.806181	−	0.591669i	$-0.798469\pi$
$809$	−1304.40			−1.61236			−0.806181	+	0.591669i	$0.798469\pi$
$810$		0			0
$811$		−	990.315i		−	1.22110i	−0.791976	−	0.610552i	$-0.790948\pi$
$811$		−	990.315i		−	1.22110i	0.791976	−	0.610552i	$-0.209052\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	549.842			0.674653
$816$		0			0
$817$	−922.223	+	382.170i	−1.12879	+	0.467773i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−11.1570			−0.0135896			−0.00679479	−	0.999977i	$-0.502163\pi$
$821$	−11.1570			−0.0135896			−0.00679479	+	0.999977i	$0.502163\pi$
$822$		0			0
$823$	−1108.14			−1.34646			−0.673232	−	0.739432i	$-0.735094\pi$
$823$	−1108.14			−1.34646			−0.673232	+	0.739432i	$0.735094\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	412.437i		−	0.498715i	−0.968412	−	0.249357i	$-0.919781\pi$
$827$		−	412.437i		−	0.498715i	0.968412	−	0.249357i	$-0.0802194\pi$
$828$		0			0
$829$			1123.73i			1.35552i	0.735282	+	0.677761i	$0.237050\pi$
$829$			1123.73i			1.35552i	−0.735282	+	0.677761i	$0.762950\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−173.842			−0.208694
$834$		0			0
$835$		−	1548.96i		−	1.85504i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	351.722i		−	0.419216i	−0.977785	−	0.209608i	$-0.932781\pi$
$839$		−	351.722i		−	0.419216i	0.977785	−	0.209608i	$-0.0672188\pi$
$840$		0			0
$841$	532.020			0.632604
$842$		0			0
$843$		0			0
$844$		0			0
$845$	906.837			1.07318
$846$		0			0
$847$	−488.904			−0.577218
$848$		0			0
$849$		0			0
$850$		0			0
$851$			934.760i			1.09842i
$852$		0			0
$853$	1698.58			1.99130			0.995649	−	0.0931811i	$-0.0297036\pi$
$853$	1698.58			1.99130			0.995649	+	0.0931811i	$0.0297036\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			1684.15i			1.96517i	0.185816	+	0.982585i	$0.440507\pi$
$857$			1684.15i			1.96517i	−0.185816	+	0.982585i	$0.559493\pi$
$858$		0			0
$859$	828.579			0.964585			0.482293	−	0.876010i	$-0.339804\pi$
$859$	828.579			0.964585			0.482293	+	0.876010i	$0.339804\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	841.313i		−	0.974870i	−0.873159	−	0.487435i	$-0.837933\pi$
$863$		−	841.313i		−	0.974870i	0.873159	−	0.487435i	$-0.162067\pi$
$864$		0			0
$865$			2598.82i			3.00442i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	900.275i		−	1.03599i
$870$		0			0
$871$	267.820			0.307486
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1138.14			1.30073
$876$		0			0
$877$		−	274.890i		−	0.313443i	−0.987643	−	0.156722i	$-0.949907\pi$
$877$		−	274.890i		−	0.313443i	0.987643	−	0.156722i	$-0.0500926\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1175.79			−1.33460			−0.667302	−	0.744787i	$-0.732551\pi$
$881$	−1175.79			−1.33460			−0.667302	+	0.744787i	$0.732551\pi$
$882$		0			0
$883$	213.071			0.241303			0.120652	−	0.992695i	$-0.461502\pi$
$883$	213.071			0.241303			0.120652	+	0.992695i	$0.461502\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	1091.62i		−	1.23069i	−0.788259	−	0.615344i	$-0.789017\pi$
$887$		−	1091.62i		−	1.23069i	0.788259	−	0.615344i	$-0.210983\pi$
$888$		0			0
$889$			887.564i			0.998385i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−68.6246	+	28.4381i	−0.0768472	+	0.0318456i
$894$		0			0
$895$			2087.82i			2.33276i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	919.499			1.02280
$900$		0			0
$901$		−	571.969i		−	0.634816i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	1635.41i		−	1.80708i
$906$		0			0
$907$			178.701i			0.197024i	0.995136	+	0.0985120i	$0.0314083\pi$
$907$			178.701i			0.197024i	−0.995136	+	0.0985120i	$0.968592\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	709.200i		−	0.778485i	−0.921135	−	0.389242i	$-0.872737\pi$
$911$		−	709.200i		−	0.778485i	0.921135	−	0.389242i	$-0.127263\pi$
$912$		0			0
$913$	374.405			0.410082
$914$		0			0
$915$		0			0
$916$		0			0
$917$	749.202			0.817014
$918$		0			0
$919$	−577.167			−0.628038			−0.314019	−	0.949417i	$-0.601676\pi$
$919$	−577.167			−0.628038			−0.314019	+	0.949417i	$0.601676\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−816.071			−0.884150
$924$		0			0
$925$			1722.83i			1.86252i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−474.299			−0.510547			−0.255274	−	0.966869i	$-0.582166\pi$
$929$	−474.299			−0.510547			−0.255274	+	0.966869i	$0.582166\pi$
$930$		0			0
$931$	−265.704	+	110.108i	−0.285396	+	0.118268i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−729.586			−0.780306
$936$		0			0
$937$	−895.322			−0.955520			−0.477760	−	0.878490i	$-0.658551\pi$
$937$	−895.322			−0.955520			−0.477760	+	0.878490i	$0.658551\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	702.877i		−	0.746947i	−0.927641	−	0.373473i	$-0.878167\pi$
$941$		−	702.877i		−	0.746947i	0.927641	−	0.373473i	$-0.121833\pi$
$942$		0			0
$943$			831.905i			0.882189i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	189.972			0.200604			0.100302	−	0.994957i	$-0.468019\pi$
$947$	189.972			0.200604			0.100302	+	0.994957i	$0.468019\pi$
$948$		0			0
$949$			870.160i			0.916923i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1018.27i		−	1.06849i	−0.845329	−	0.534247i	$-0.820595\pi$
$953$		−	1018.27i		−	1.06849i	0.845329	−	0.534247i	$-0.179405\pi$
$954$		0			0
$955$	845.830			0.885686
$956$		0			0
$957$		0			0
$958$		0			0
$959$	643.919			0.671448
$960$		0			0
$961$	−1775.35			−1.84740
$962$		0			0
$963$		0			0
$964$		0			0
$965$			2389.78i			2.47646i
$966$		0			0
$967$	569.922			0.589371			0.294686	−	0.955594i	$-0.404785\pi$
$967$	569.922			0.589371			0.294686	+	0.955594i	$0.404785\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			98.6776i			0.101625i	0.998708	+	0.0508123i	$0.0161810\pi$
$971$			98.6776i			0.101625i	−0.998708	+	0.0508123i	$0.983819\pi$
$972$		0			0
$973$	−1649.40			−1.69517
$974$		0			0
$975$		0			0
$976$		0			0
$977$			859.441i			0.879674i	0.898078	+	0.439837i	$0.144964\pi$
$977$			859.441i			0.879674i	−0.898078	+	0.439837i	$0.855036\pi$
$978$		0			0
$979$		−	188.496i		−	0.192539i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	1014.55i		−	1.03210i	−0.856559	−	0.516049i	$-0.827402\pi$
$983$		−	1014.55i		−	1.03210i	0.856559	−	0.516049i	$-0.172598\pi$
$984$		0			0
$985$	2025.99			2.05684
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1206.44			1.21986
$990$		0			0
$991$		−	803.940i		−	0.811241i	−0.914042	−	0.405621i	$-0.867055\pi$
$991$		−	803.940i		−	0.811241i	0.914042	−	0.405621i	$-0.132945\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−143.898			−0.144621
$996$		0			0
$997$	−865.263			−0.867867			−0.433933	−	0.900945i	$-0.642875\pi$
$997$	−865.263			−0.867867			−0.433933	+	0.900945i	$0.642875\pi$
$998$		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.721.20		20
3.2	odd	2		912.3.o.e.721.11		20
4.3	odd	2		1368.3.o.c.721.20		20
12.11	even	2		456.3.o.a.265.1	✓	20
19.18	odd	2	inner	2736.3.o.r.721.19		20
57.56	even	2		912.3.o.e.721.1		20
76.75	even	2		1368.3.o.c.721.19		20
228.227	odd	2		456.3.o.a.265.11	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.3.o.a.265.1	✓	20	12.11	even	2
456.3.o.a.265.11	yes	20	228.227	odd	2
912.3.o.e.721.1		20	57.56	even	2
912.3.o.e.721.11		20	3.2	odd	2
1368.3.o.c.721.19		20	76.75	even	2
1368.3.o.c.721.20		20	4.3	odd	2
2736.3.o.r.721.19		20	19.18	odd	2	inner
2736.3.o.r.721.20		20	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+8.20492 q^{5} +8.00860 q^{7} +O(q^{10})\)	\(q+8.20492 q^{5} +8.00860 q^{7} +7.74291 q^{11} +7.64699i q^{13} -11.4841 q^{17} +(-17.5525 + 7.27380i) q^{19} +22.9620 q^{23} +42.3207 q^{25} +17.5778i q^{29} -52.3102i q^{31} +65.7099 q^{35} +40.7090i q^{37} +36.2297i q^{41} +52.5407 q^{43} +3.90967 q^{47} +15.1376 q^{49} +49.8052i q^{53} +63.5299 q^{55} +46.6932i q^{59} +40.8276 q^{61} +62.7429i q^{65} -35.0230i q^{67} +106.718i q^{71} +113.791 q^{73} +62.0098 q^{77} -116.271i q^{79} +48.3546 q^{83} -94.2264 q^{85} -24.3444i q^{89} +61.2416i q^{91} +(-144.017 + 59.6809i) q^{95} +85.7882i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 16 q^{7}+O(q^{10}) \) 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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