LMFDB - Embedded newform 2736.3.o.r.721.11

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			721.11
Root			$1.76236 + 3.05250i$ of defining polynomial
Character	$\chi$	$=$	2736.721
Dual form			2736.3.o.r.721.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.816130 q^{5} +6.23331 q^{7} +O(q^{10})$	$q+0.816130 q^{5} +6.23331 q^{7} +7.41992 q^{11} -0.641191i q^{13} +26.4643 q^{17} +(-9.92451 + 16.2020i) q^{19} -14.1462 q^{23} -24.3339 q^{25} +45.5452i q^{29} +24.8536i q^{31} +5.08719 q^{35} +15.7228i q^{37} +23.7752i q^{41} +47.2892 q^{43} +89.4238 q^{47} -10.1459 q^{49} +16.7925i q^{53} +6.05561 q^{55} -4.08625i q^{59} -94.1995 q^{61} -0.523295i q^{65} -39.8269i q^{67} -124.430i q^{71} -68.5017 q^{73} +46.2506 q^{77} +34.2126i q^{79} +11.0883 q^{83} +21.5983 q^{85} -115.791i q^{89} -3.99674i q^{91} +(-8.09968 + 13.2229i) q^{95} +150.705i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	0.816130			0.163226			0.0816130	−	0.996664i	$-0.473993\pi$
$5$	0.816130			0.163226			0.0816130	+	0.996664i	$0.473993\pi$
$6$		0			0
$7$	6.23331			0.890472			0.445236	−	0.895413i	$-0.353120\pi$
$7$	6.23331			0.890472			0.445236	+	0.895413i	$0.353120\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	7.41992			0.674538			0.337269	−	0.941408i	$-0.390497\pi$
$11$	7.41992			0.674538			0.337269	+	0.941408i	$0.390497\pi$
$12$		0			0
$13$		−	0.641191i		−	0.0493224i	−0.999696	−	0.0246612i	$-0.992149\pi$
$13$		−	0.641191i		−	0.0493224i	0.999696	−	0.0246612i	$-0.00785070\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	26.4643			1.55672			0.778361	−	0.627817i	$-0.216051\pi$
$17$	26.4643			1.55672			0.778361	+	0.627817i	$0.216051\pi$
$18$		0			0
$19$	−9.92451	+	16.2020i	−0.522342	+	0.852736i
$19$	−9.92451	+	16.2020i	−0.522342	+	0.852736i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−14.1462			−0.615052			−0.307526	−	0.951540i	$-0.599501\pi$
$23$	−14.1462			−0.615052			−0.307526	+	0.951540i	$0.599501\pi$
$24$		0			0
$25$	−24.3339			−0.973357
$26$		0			0
$27$		0			0
$28$		0			0
$29$			45.5452i			1.57052i	0.619164	+	0.785262i	$0.287472\pi$
$29$			45.5452i			1.57052i	−0.619164	+	0.785262i	$0.712528\pi$
$30$		0			0
$31$			24.8536i			0.801730i	0.916137	+	0.400865i	$0.131290\pi$
$31$			24.8536i			0.801730i	−0.916137	+	0.400865i	$0.868710\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	5.08719			0.145348
$36$		0			0
$37$			15.7228i			0.424942i	0.977167	+	0.212471i	$0.0681511\pi$
$37$			15.7228i			0.424942i	−0.977167	+	0.212471i	$0.931849\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			23.7752i			0.579882i	0.957045	+	0.289941i	$0.0936356\pi$
$41$			23.7752i			0.579882i	−0.957045	+	0.289941i	$0.906364\pi$
$42$		0			0
$43$	47.2892			1.09975			0.549875	−	0.835247i	$-0.314675\pi$
$43$	47.2892			1.09975			0.549875	+	0.835247i	$0.314675\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	89.4238			1.90263			0.951317	−	0.308215i	$-0.0997316\pi$
$47$	89.4238			1.90263			0.951317	+	0.308215i	$0.0997316\pi$
$48$		0			0
$49$	−10.1459			−0.207059
$50$		0			0
$51$		0			0
$52$		0			0
$53$			16.7925i			0.316839i	0.987372	+	0.158419i	$0.0506398\pi$
$53$			16.7925i			0.316839i	−0.987372	+	0.158419i	$0.949360\pi$
$54$		0			0
$55$	6.05561			0.110102
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	4.08625i		−	0.0692585i	−0.999400	−	0.0346293i	$-0.988975\pi$
$59$		−	4.08625i		−	0.0692585i	0.999400	−	0.0346293i	$-0.0110250\pi$
$60$		0			0
$61$	−94.1995			−1.54425			−0.772127	−	0.635468i	$-0.780807\pi$
$61$	−94.1995			−1.54425			−0.772127	+	0.635468i	$0.780807\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	0.523295i		−	0.00805069i
$66$		0			0
$67$		−	39.8269i		−	0.594432i	−0.954810	−	0.297216i	$-0.903942\pi$
$67$		−	39.8269i		−	0.594432i	0.954810	−	0.297216i	$-0.0960581\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	124.430i		−	1.75254i	−0.481825	−	0.876268i	$-0.660026\pi$
$71$		−	124.430i		−	1.75254i	0.481825	−	0.876268i	$-0.339974\pi$
$72$		0			0
$73$	−68.5017			−0.938379			−0.469190	−	0.883098i	$-0.655454\pi$
$73$	−68.5017			−0.938379			−0.469190	+	0.883098i	$0.655454\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	46.2506			0.600657
$78$		0			0
$79$			34.2126i			0.433071i	0.976275	+	0.216535i	$0.0694757\pi$
$79$			34.2126i			0.433071i	−0.976275	+	0.216535i	$0.930524\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	11.0883			0.133594			0.0667968	−	0.997767i	$-0.478722\pi$
$83$	11.0883			0.133594			0.0667968	+	0.997767i	$0.478722\pi$
$84$		0			0
$85$	21.5983			0.254097
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	115.791i		−	1.30102i	−0.759498	−	0.650510i	$-0.774555\pi$
$89$		−	115.791i		−	1.30102i	0.759498	−	0.650510i	$-0.225445\pi$
$90$		0			0
$91$		−	3.99674i		−	0.0439202i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−8.09968	+	13.2229i	−0.0852598	+	0.139189i
$96$		0			0
$97$			150.705i			1.55366i	0.629710	+	0.776830i	$0.283174\pi$
$97$			150.705i			1.55366i	−0.629710	+	0.776830i	$0.716826\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	81.4618			0.806552			0.403276	−	0.915078i	$-0.367871\pi$
$101$	81.4618			0.806552			0.403276	+	0.915078i	$0.367871\pi$
$102$		0			0
$103$			3.64778i			0.0354154i	0.999843	+	0.0177077i	$0.00563683\pi$
$103$			3.64778i			0.0354154i	−0.999843	+	0.0177077i	$0.994363\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			138.034i			1.29004i	0.764167	+	0.645019i	$0.223150\pi$
$107$			138.034i			1.29004i	−0.764167	+	0.645019i	$0.776850\pi$
$108$		0			0
$109$			111.302i			1.02112i	0.859841	+	0.510562i	$0.170562\pi$
$109$			111.302i			1.02112i	−0.859841	+	0.510562i	$0.829438\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			50.1730i			0.444009i	0.975046	+	0.222005i	$0.0712600\pi$
$113$			50.1730i			0.444009i	−0.975046	+	0.222005i	$0.928740\pi$
$114$		0			0
$115$	−11.5451			−0.100392
$116$		0			0
$117$		0			0
$118$		0			0
$119$	164.960			1.38622
$120$		0			0
$121$	−65.9448			−0.544999
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−40.2629			−0.322103
$126$		0			0
$127$			65.4792i			0.515584i	0.966200	+	0.257792i	$0.0829950\pi$
$127$			65.4792i			0.515584i	−0.966200	+	0.257792i	$0.917005\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	116.672			0.890626			0.445313	−	0.895375i	$-0.353092\pi$
$131$	116.672			0.890626			0.445313	+	0.895375i	$0.353092\pi$
$132$		0			0
$133$	−61.8625	+	100.992i	−0.465131	+	0.759338i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−132.420			−0.966569			−0.483285	−	0.875463i	$-0.660556\pi$
$137$	−132.420			−0.966569			−0.483285	+	0.875463i	$0.660556\pi$
$138$		0			0
$139$	162.425			1.16852			0.584262	−	0.811565i	$-0.301384\pi$
$139$	162.425			1.16852			0.584262	+	0.811565i	$0.301384\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	4.75759i		−	0.0332698i
$144$		0			0
$145$			37.1708i			0.256350i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	37.4905			0.251614			0.125807	−	0.992055i	$-0.459848\pi$
$149$	37.4905			0.251614			0.125807	+	0.992055i	$0.459848\pi$
$150$		0			0
$151$		−	253.157i		−	1.67653i	−0.545260	−	0.838267i	$-0.683569\pi$
$151$		−	253.157i		−	1.67653i	0.545260	−	0.838267i	$-0.316431\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			20.2838i			0.130863i
$156$		0			0
$157$	58.4348			0.372196			0.186098	−	0.982531i	$-0.440416\pi$
$157$	58.4348			0.372196			0.186098	+	0.982531i	$0.440416\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−88.1775			−0.547687
$162$		0			0
$163$	12.9957			0.0797282			0.0398641	−	0.999205i	$-0.487308\pi$
$163$	12.9957			0.0797282			0.0398641	+	0.999205i	$0.487308\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			111.084i			0.665174i	0.943073	+	0.332587i	$0.107921\pi$
$167$			111.084i			0.665174i	−0.943073	+	0.332587i	$0.892079\pi$
$168$		0			0
$169$	168.589			0.997567
$170$		0			0
$171$		0			0
$172$		0			0
$173$			188.503i			1.08961i	0.838562	+	0.544806i	$0.183397\pi$
$173$			188.503i			1.08961i	−0.838562	+	0.544806i	$0.816603\pi$
$174$		0			0
$175$	−151.681			−0.866748
$176$		0			0
$177$		0			0
$178$		0			0
$179$			2.77763i			0.0155175i	0.999970	+	0.00775874i	$0.00246971\pi$
$179$			2.77763i			0.0155175i	−0.999970	+	0.00775874i	$0.997530\pi$
$180$		0			0
$181$		−	148.538i		−	0.820653i	−0.911939	−	0.410327i	$-0.865415\pi$
$181$		−	148.538i		−	0.820653i	0.911939	−	0.410327i	$-0.134585\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			12.8319i			0.0693615i
$186$		0			0
$187$	196.363			1.05007
$188$		0			0
$189$		0			0
$190$		0			0
$191$	92.5344			0.484473			0.242237	−	0.970217i	$-0.422119\pi$
$191$	92.5344			0.484473			0.242237	+	0.970217i	$0.422119\pi$
$192$		0			0
$193$			155.956i			0.808060i	0.914746	+	0.404030i	$0.132391\pi$
$193$			155.956i			0.808060i	−0.914746	+	0.404030i	$0.867609\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.90553			0.0147489			0.00737445	−	0.999973i	$-0.497653\pi$
$197$	2.90553			0.0147489			0.00737445	+	0.999973i	$0.497653\pi$
$198$		0			0
$199$	−67.0257			−0.336813			−0.168406	−	0.985718i	$-0.553862\pi$
$199$	−67.0257			−0.336813			−0.168406	+	0.985718i	$0.553862\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			283.897i			1.39851i
$204$		0			0
$205$			19.4036i			0.0946517i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−73.6390	+	120.217i	−0.352340	+	0.575203i
$210$		0			0
$211$		−	283.197i		−	1.34216i	−0.741383	−	0.671082i	$-0.765830\pi$
$211$		−	283.197i		−	1.34216i	0.741383	−	0.671082i	$-0.234170\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	38.5941			0.179508
$216$		0			0
$217$			154.920i			0.713919i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	16.9687i		−	0.0767813i
$222$		0			0
$223$			313.208i			1.40452i	0.711919	+	0.702261i	$0.247826\pi$
$223$			313.208i			1.40452i	−0.711919	+	0.702261i	$0.752174\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			57.3587i			0.252682i	0.991987	+	0.126341i	$0.0403233\pi$
$227$			57.3587i			0.252682i	−0.991987	+	0.126341i	$0.959677\pi$
$228$		0			0
$229$	−365.278			−1.59510			−0.797551	−	0.603251i	$-0.793872\pi$
$229$	−365.278			−1.59510			−0.797551	+	0.603251i	$0.793872\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	353.166			1.51573			0.757866	−	0.652410i	$-0.226242\pi$
$233$	353.166			1.51573			0.757866	+	0.652410i	$0.226242\pi$
$234$		0			0
$235$	72.9814			0.310559
$236$		0			0
$237$		0			0
$238$		0			0
$239$	130.909			0.547738			0.273869	−	0.961767i	$-0.411697\pi$
$239$	130.909			0.547738			0.273869	+	0.961767i	$0.411697\pi$
$240$		0			0
$241$		−	242.322i		−	1.00548i	−0.864436	−	0.502742i	$-0.832325\pi$
$241$		−	242.322i		−	1.00548i	0.864436	−	0.502742i	$-0.167675\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−8.28037			−0.0337974
$246$		0			0
$247$	10.3886	+	6.36351i	0.0420590	+	0.0257632i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	38.0581			0.151626			0.0758129	−	0.997122i	$-0.475845\pi$
$251$	38.0581			0.151626			0.0758129	+	0.997122i	$0.475845\pi$
$252$		0			0
$253$	−104.964			−0.414876
$254$		0			0
$255$		0			0
$256$		0			0
$257$			169.438i			0.659292i	0.944105	+	0.329646i	$0.106929\pi$
$257$			169.438i			0.659292i	−0.944105	+	0.329646i	$0.893071\pi$
$258$		0			0
$259$			98.0053i			0.378399i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−120.124			−0.456746			−0.228373	−	0.973574i	$-0.573341\pi$
$263$	−120.124			−0.456746			−0.228373	+	0.973574i	$0.573341\pi$
$264$		0			0
$265$			13.7048i			0.0517163i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			507.945i			1.88827i	0.329558	+	0.944135i	$0.393100\pi$
$269$			507.945i			1.88827i	−0.329558	+	0.944135i	$0.606900\pi$
$270$		0			0
$271$	278.413			1.02735			0.513677	−	0.857983i	$-0.328283\pi$
$271$	278.413			1.02735			0.513677	+	0.857983i	$0.328283\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−180.556			−0.656566
$276$		0			0
$277$	350.159			1.26411			0.632056	−	0.774923i	$-0.282211\pi$
$277$	350.159			1.26411			0.632056	+	0.774923i	$0.282211\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			206.689i			0.735549i	0.929915	+	0.367774i	$0.119880\pi$
$281$			206.689i			0.735549i	−0.929915	+	0.367774i	$0.880120\pi$
$282$		0			0
$283$	518.693			1.83284			0.916418	−	0.400223i	$-0.131067\pi$
$283$	518.693			1.83284			0.916418	+	0.400223i	$0.131067\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			148.198i			0.516369i
$288$		0			0
$289$	411.358			1.42338
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	162.315i		−	0.553978i	−0.960873	−	0.276989i	$-0.910663\pi$
$293$		−	162.315i		−	0.553978i	0.960873	−	0.276989i	$-0.0893365\pi$
$294$		0			0
$295$		−	3.33491i		−	0.0113048i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			9.07041i			0.0303358i
$300$		0			0
$301$	294.768			0.979297
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−76.8790			−0.252062
$306$		0			0
$307$		−	43.2158i		−	0.140768i	−0.997520	−	0.0703840i	$-0.977578\pi$
$307$		−	43.2158i		−	0.140768i	0.997520	−	0.0703840i	$-0.0224225\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	43.1673			0.138801			0.0694007	−	0.997589i	$-0.477891\pi$
$311$	43.1673			0.138801			0.0694007	+	0.997589i	$0.477891\pi$
$312$		0			0
$313$	−114.450			−0.365654			−0.182827	−	0.983145i	$-0.558525\pi$
$313$	−114.450			−0.365654			−0.182827	+	0.983145i	$0.558525\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			277.943i			0.876793i	0.898782	+	0.438397i	$0.144453\pi$
$317$			277.943i			0.876793i	−0.898782	+	0.438397i	$0.855547\pi$
$318$		0			0
$319$			337.941i			1.05938i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−262.645	+	428.774i	−0.813142	+	1.32747i
$324$		0			0
$325$			15.6027i			0.0480083i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	557.406			1.69424
$330$		0			0
$331$			575.160i			1.73764i	0.495124	+	0.868822i	$0.335122\pi$
$331$			575.160i			1.73764i	−0.495124	+	0.868822i	$0.664878\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	32.5039i		−	0.0970267i
$336$		0			0
$337$		−	433.213i		−	1.28550i	−0.766076	−	0.642750i	$-0.777794\pi$
$337$		−	433.213i		−	1.28550i	0.766076	−	0.642750i	$-0.222206\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			184.412i			0.540798i
$342$		0			0
$343$	−368.674			−1.07485
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−428.798			−1.23573			−0.617864	−	0.786285i	$-0.712002\pi$
$347$	−428.798			−1.23573			−0.617864	+	0.786285i	$0.712002\pi$
$348$		0			0
$349$	619.190			1.77418			0.887092	−	0.461593i	$-0.152722\pi$
$349$	619.190			1.77418			0.887092	+	0.461593i	$0.152722\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−384.359			−1.08884			−0.544418	−	0.838814i	$-0.683249\pi$
$353$	−384.359			−1.08884			−0.544418	+	0.838814i	$0.683249\pi$
$354$		0			0
$355$		−	101.551i		−	0.286059i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.81251			0.00504878			0.00252439	−	0.999997i	$-0.499196\pi$
$359$	1.81251			0.00504878			0.00252439	+	0.999997i	$0.499196\pi$
$360$		0			0
$361$	−164.008	−	321.593i	−0.454317	−	0.890840i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−55.9062			−0.153168
$366$		0			0
$367$	70.1902			0.191254			0.0956270	−	0.995417i	$-0.469514\pi$
$367$	70.1902			0.191254			0.0956270	+	0.995417i	$0.469514\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			104.673i			0.282136i
$372$		0			0
$373$			11.0710i			0.0296810i	0.999890	+	0.0148405i	$0.00472405\pi$
$373$			11.0710i			0.0296810i	−0.999890	+	0.0148405i	$0.995276\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	29.2032			0.0774620
$378$		0			0
$379$		−	62.8446i		−	0.165817i	−0.996557	−	0.0829085i	$-0.973579\pi$
$379$		−	62.8446i		−	0.165817i	0.996557	−	0.0829085i	$-0.0264209\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	445.898i		−	1.16423i	−0.813108	−	0.582113i	$-0.802226\pi$
$383$		−	445.898i		−	1.16423i	0.813108	−	0.582113i	$-0.197774\pi$
$384$		0			0
$385$	37.7465			0.0980428
$386$		0			0
$387$		0			0
$388$		0			0
$389$	265.155			0.681633			0.340816	−	0.940130i	$-0.389297\pi$
$389$	265.155			0.681633			0.340816	+	0.940130i	$0.389297\pi$
$390$		0			0
$391$	−374.369			−0.957465
$392$		0			0
$393$		0			0
$394$		0			0
$395$			27.9219i			0.0706884i
$396$		0			0
$397$	191.391			0.482094			0.241047	−	0.970513i	$-0.422509\pi$
$397$	191.391			0.482094			0.241047	+	0.970513i	$0.422509\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	528.591i		−	1.31818i	−0.752063	−	0.659091i	$-0.770941\pi$
$401$		−	528.591i		−	1.31818i	0.752063	−	0.659091i	$-0.229059\pi$
$402$		0			0
$403$	15.9359			0.0395433
$404$		0			0
$405$		0			0
$406$		0			0
$407$			116.662i			0.286639i
$408$		0			0
$409$			29.4744i			0.0720646i	0.999351	+	0.0360323i	$0.0114719\pi$
$409$			29.4744i			0.0720646i	−0.999351	+	0.0360323i	$0.988528\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	25.4709i		−	0.0616728i
$414$		0			0
$415$	9.04947			0.0218060
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−470.295			−1.12242			−0.561212	−	0.827672i	$-0.689665\pi$
$419$	−470.295			−1.12242			−0.561212	+	0.827672i	$0.689665\pi$
$420$		0			0
$421$			176.989i			0.420401i	0.977658	+	0.210201i	$0.0674117\pi$
$421$			176.989i			0.420401i	−0.977658	+	0.210201i	$0.932588\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−643.980			−1.51525
$426$		0			0
$427$	−587.175			−1.37512
$428$		0			0
$429$		0			0
$430$		0			0
$431$			120.187i			0.278857i	0.990232	+	0.139428i	$0.0445265\pi$
$431$			120.187i			0.278857i	−0.990232	+	0.139428i	$0.955473\pi$
$432$		0			0
$433$		−	249.639i		−	0.576535i	−0.957550	−	0.288267i	$-0.906921\pi$
$433$		−	249.639i		−	0.576535i	0.957550	−	0.288267i	$-0.0930792\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	140.394	−	229.196i	0.321268	−	0.524477i
$438$		0			0
$439$			557.627i			1.27022i	0.772422	+	0.635110i	$0.219045\pi$
$439$			557.627i			1.27022i	−0.772422	+	0.635110i	$0.780955\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	54.2347			0.122426			0.0612130	−	0.998125i	$-0.480503\pi$
$443$	54.2347			0.122426			0.0612130	+	0.998125i	$0.480503\pi$
$444$		0			0
$445$		−	94.5002i		−	0.212360i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			383.199i			0.853449i	0.904382	+	0.426725i	$0.140333\pi$
$449$			383.199i			0.853449i	−0.904382	+	0.426725i	$0.859667\pi$
$450$		0			0
$451$			176.410i			0.391152i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	3.26186i		−	0.00716892i
$456$		0			0
$457$	−138.486			−0.303032			−0.151516	−	0.988455i	$-0.548416\pi$
$457$	−138.486			−0.303032			−0.151516	+	0.988455i	$0.548416\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	499.010			1.08245			0.541226	−	0.840877i	$-0.317960\pi$
$461$	499.010			1.08245			0.541226	+	0.840877i	$0.317960\pi$
$462$		0			0
$463$	−357.129			−0.771337			−0.385669	−	0.922637i	$-0.626029\pi$
$463$	−357.129			−0.771337			−0.385669	+	0.922637i	$0.626029\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	378.433			0.810349			0.405174	−	0.914239i	$-0.367211\pi$
$467$	378.433			0.810349			0.405174	+	0.914239i	$0.367211\pi$
$468$		0			0
$469$		−	248.253i		−	0.529325i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	350.882			0.741823
$474$		0			0
$475$	241.502	−	394.258i	0.508426	−	0.830017i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	712.228			1.48691			0.743453	−	0.668789i	$-0.233187\pi$
$479$	712.228			1.48691			0.743453	+	0.668789i	$0.233187\pi$
$480$		0			0
$481$	10.0813			0.0209591
$482$		0			0
$483$		0			0
$484$		0			0
$485$			122.995i			0.253598i
$486$		0			0
$487$		−	542.116i		−	1.11317i	−0.830789	−	0.556587i	$-0.812110\pi$
$487$		−	542.116i		−	1.11317i	0.830789	−	0.556587i	$-0.187890\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−471.347			−0.959974			−0.479987	−	0.877275i	$-0.659359\pi$
$491$	−471.347			−0.959974			−0.479987	+	0.877275i	$0.659359\pi$
$492$		0			0
$493$			1205.32i			2.44487i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	775.610i		−	1.56058i
$498$		0			0
$499$	322.957			0.647209			0.323605	−	0.946192i	$-0.395105\pi$
$499$	322.957			0.647209			0.323605	+	0.946192i	$0.395105\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	831.376			1.65283			0.826417	−	0.563058i	$-0.190375\pi$
$503$	831.376			1.65283			0.826417	+	0.563058i	$0.190375\pi$
$504$		0			0
$505$	66.4834			0.131650
$506$		0			0
$507$		0			0
$508$		0			0
$509$			149.926i			0.294550i	0.989096	+	0.147275i	$0.0470502\pi$
$509$			149.926i			0.294550i	−0.989096	+	0.147275i	$0.952950\pi$
$510$		0			0
$511$	−426.992			−0.835601
$512$		0			0
$513$		0			0
$514$		0			0
$515$			2.97706i			0.00578071i
$516$		0			0
$517$	663.517			1.28340
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	830.352i		−	1.59377i	−0.604134	−	0.796883i	$-0.706481\pi$
$521$		−	830.352i		−	1.59377i	0.604134	−	0.796883i	$-0.293519\pi$
$522$		0			0
$523$			237.339i			0.453803i	0.973918	+	0.226901i	$0.0728595\pi$
$523$			237.339i			0.453803i	−0.973918	+	0.226901i	$0.927140\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			657.734i			1.24807i
$528$		0			0
$529$	−328.885			−0.621711
$530$		0			0
$531$		0			0
$532$		0			0
$533$	15.2444			0.0286012
$534$		0			0
$535$			112.654i			0.210568i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−75.2817			−0.139669
$540$		0			0
$541$	552.145			1.02060			0.510300	−	0.859996i	$-0.329534\pi$
$541$	552.145			1.02060			0.510300	+	0.859996i	$0.329534\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			90.8372i			0.166674i
$546$		0			0
$547$		−	828.591i		−	1.51479i	−0.652957	−	0.757395i	$-0.726472\pi$
$547$		−	828.591i		−	1.51479i	0.652957	−	0.757395i	$-0.273528\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−737.922	−	452.013i	−1.33924	−	0.820351i
$552$		0			0
$553$			213.258i			0.385638i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	207.953			0.373344			0.186672	−	0.982422i	$-0.440230\pi$
$557$	207.953			0.373344			0.186672	+	0.982422i	$0.440230\pi$
$558$		0			0
$559$		−	30.3214i		−	0.0542423i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			66.1493i			0.117494i	0.998273	+	0.0587472i	$0.0187106\pi$
$563$			66.1493i			0.117494i	−0.998273	+	0.0587472i	$0.981289\pi$
$564$		0			0
$565$			40.9477i			0.0724738i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	307.379i		−	0.540209i	−0.962831	−	0.270104i	$-0.912942\pi$
$569$		−	307.379i		−	0.540209i	0.962831	−	0.270104i	$-0.0870582\pi$
$570$		0			0
$571$	−790.403			−1.38424			−0.692122	−	0.721781i	$-0.743324\pi$
$571$	−790.403			−1.38424			−0.692122	+	0.721781i	$0.743324\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	344.232			0.598665
$576$		0			0
$577$	−390.153			−0.676176			−0.338088	−	0.941115i	$-0.609780\pi$
$577$	−390.153			−0.676176			−0.338088	+	0.941115i	$0.609780\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	69.1166			0.118961
$582$		0			0
$583$			124.599i			0.213720i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	36.1864			0.0616464			0.0308232	−	0.999525i	$-0.490187\pi$
$587$	36.1864			0.0616464			0.0308232	+	0.999525i	$0.490187\pi$
$588$		0			0
$589$	−402.678	−	246.660i	−0.683664	−	0.418778i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−318.118			−0.536455			−0.268228	−	0.963356i	$-0.586438\pi$
$593$	−318.118			−0.536455			−0.268228	+	0.963356i	$0.586438\pi$
$594$		0			0
$595$	134.629			0.226267
$596$		0			0
$597$		0			0
$598$		0			0
$599$			900.190i			1.50282i	0.659835	+	0.751411i	$0.270626\pi$
$599$			900.190i			1.50282i	−0.659835	+	0.751411i	$0.729374\pi$
$600$		0			0
$601$		−	728.173i		−	1.21160i	−0.795616	−	0.605801i	$-0.792853\pi$
$601$		−	728.173i		−	1.21160i	0.795616	−	0.605801i	$-0.207147\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−53.8195			−0.0889579
$606$		0			0
$607$			608.487i			1.00245i	0.865317	+	0.501225i	$0.167117\pi$
$607$			608.487i			1.00245i	−0.865317	+	0.501225i	$0.832883\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	57.3377i		−	0.0938425i
$612$		0			0
$613$	−860.320			−1.40346			−0.701729	−	0.712444i	$-0.747588\pi$
$613$	−860.320			−1.40346			−0.701729	+	0.712444i	$0.747588\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−522.213			−0.846374			−0.423187	−	0.906042i	$-0.639089\pi$
$617$	−522.213			−0.846374			−0.423187	+	0.906042i	$0.639089\pi$
$618$		0			0
$619$	−988.374			−1.59673			−0.798364	−	0.602176i	$-0.794301\pi$
$619$	−988.374			−1.59673			−0.798364	+	0.602176i	$0.794301\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	721.759i		−	1.15852i
$624$		0			0
$625$	575.489			0.920782
$626$		0			0
$627$		0			0
$628$		0			0
$629$			416.094i			0.661516i
$630$		0			0
$631$	−809.517			−1.28291			−0.641455	−	0.767160i	$-0.721669\pi$
$631$	−809.517			−1.28291			−0.641455	+	0.767160i	$0.721669\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			53.4395i			0.0841567i
$636$		0			0
$637$			6.50546i			0.0102127i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	782.395i		−	1.22059i	−0.792176	−	0.610293i	$-0.791052\pi$
$641$		−	782.395i		−	1.22059i	0.792176	−	0.610293i	$-0.208948\pi$
$642$		0			0
$643$	651.757			1.01362			0.506809	−	0.862058i	$-0.330825\pi$
$643$	651.757			1.01362			0.506809	+	0.862058i	$0.330825\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	452.705			0.699698			0.349849	−	0.936806i	$-0.386233\pi$
$647$	452.705			0.699698			0.349849	+	0.936806i	$0.386233\pi$
$648$		0			0
$649$		−	30.3196i		−	0.0467175i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	553.670			0.847887			0.423944	−	0.905689i	$-0.360645\pi$
$653$	553.670			0.847887			0.423944	+	0.905689i	$0.360645\pi$
$654$		0			0
$655$	95.2195			0.145373
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	349.034i		−	0.529642i	−0.964298	−	0.264821i	$-0.914687\pi$
$659$		−	349.034i		−	0.529642i	0.964298	−	0.264821i	$-0.0853129\pi$
$660$		0			0
$661$			658.000i			0.995462i	0.867331	+	0.497731i	$0.165833\pi$
$661$			658.000i			0.995462i	−0.867331	+	0.497731i	$0.834167\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−50.4878	+	82.4225i	−0.0759215	+	0.123944i
$666$		0			0
$667$		−	644.291i		−	0.965953i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−698.953			−1.04166
$672$		0			0
$673$		−	855.349i		−	1.27095i	−0.772122	−	0.635475i	$-0.780804\pi$
$673$		−	855.349i		−	1.27095i	0.772122	−	0.635475i	$-0.219196\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			338.539i			0.500058i	0.968238	+	0.250029i	$0.0804401\pi$
$677$			338.539i			0.500058i	−0.968238	+	0.250029i	$0.919560\pi$
$678$		0			0
$679$			939.391i			1.38349i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	1315.45i		−	1.92599i	−0.269521	−	0.962994i	$-0.586865\pi$
$683$		−	1315.45i		−	1.92599i	0.269521	−	0.962994i	$-0.413135\pi$
$684$		0			0
$685$	−108.072			−0.157769
$686$		0			0
$687$		0			0
$688$		0			0
$689$	10.7672			0.0156273
$690$		0			0
$691$	−570.218			−0.825207			−0.412603	−	0.910911i	$-0.635380\pi$
$691$	−570.218			−0.825207			−0.412603	+	0.910911i	$0.635380\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	132.560			0.190733
$696$		0			0
$697$			629.192i			0.902715i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	125.252			0.178676			0.0893381	−	0.996001i	$-0.471525\pi$
$701$	125.252			0.178676			0.0893381	+	0.996001i	$0.471525\pi$
$702$		0			0
$703$	−254.741	−	156.041i	−0.362363	−	0.221965i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	507.776			0.718212
$708$		0			0
$709$	−762.172			−1.07500			−0.537498	−	0.843265i	$-0.680630\pi$
$709$	−762.172			−1.07500			−0.537498	+	0.843265i	$0.680630\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	351.584i		−	0.493106i
$714$		0			0
$715$		−	3.88281i		−	0.00543050i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−1088.35			−1.51370			−0.756852	−	0.653586i	$-0.773264\pi$
$719$	−1088.35			−1.51370			−0.756852	+	0.653586i	$0.773264\pi$
$720$		0			0
$721$			22.7378i			0.0315364i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	1108.29i		−	1.52868i
$726$		0			0
$727$	−108.746			−0.149581			−0.0747907	−	0.997199i	$-0.523829\pi$
$727$	−108.746			−0.149581			−0.0747907	+	0.997199i	$0.523829\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1251.48			1.71200
$732$		0			0
$733$	97.4870			0.132997			0.0664987	−	0.997787i	$-0.478817\pi$
$733$	97.4870			0.132997			0.0664987	+	0.997787i	$0.478817\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	295.513i		−	0.400967i
$738$		0			0
$739$	83.4602			0.112937			0.0564683	−	0.998404i	$-0.482016\pi$
$739$	83.4602			0.112937			0.0564683	+	0.998404i	$0.482016\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	300.236i		−	0.404087i	−0.979377	−	0.202043i	$-0.935242\pi$
$743$		−	300.236i		−	0.404087i	0.979377	−	0.202043i	$-0.0647582\pi$
$744$		0			0
$745$	30.5971			0.0410700
$746$		0			0
$747$		0			0
$748$		0			0
$749$			860.408i			1.14874i
$750$		0			0
$751$		−	1086.94i		−	1.44732i	−0.690154	−	0.723662i	$-0.742457\pi$
$751$		−	1086.94i		−	1.44732i	0.690154	−	0.723662i	$-0.257543\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	206.609i		−	0.273654i
$756$		0			0
$757$	−551.348			−0.728333			−0.364167	−	0.931334i	$-0.618646\pi$
$757$	−551.348			−0.728333			−0.364167	+	0.931334i	$0.618646\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1350.96			1.77525			0.887624	−	0.460569i	$-0.152355\pi$
$761$	1350.96			1.77525			0.887624	+	0.460569i	$0.152355\pi$
$762$		0			0
$763$			693.782i			0.909282i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.62007			−0.00341600
$768$		0			0
$769$	170.142			0.221251			0.110625	−	0.993862i	$-0.464715\pi$
$769$	170.142			0.221251			0.110625	+	0.993862i	$0.464715\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			63.5741i			0.0822434i	0.999154	+	0.0411217i	$0.0130931\pi$
$773$			63.5741i			0.0822434i	−0.999154	+	0.0411217i	$0.986907\pi$
$774$		0			0
$775$		−	604.787i		−	0.780370i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−385.205	−	235.957i	−0.494486	−	0.302897i
$780$		0			0
$781$		−	923.260i		−	1.18215i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	47.6904			0.0607521
$786$		0			0
$787$			1008.74i			1.28175i	0.767646	+	0.640875i	$0.221428\pi$
$787$			1008.74i			1.28175i	−0.767646	+	0.640875i	$0.778572\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			312.744i			0.395378i
$792$		0			0
$793$			60.3999i			0.0761664i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	431.564i		−	0.541485i	−0.962652	−	0.270743i	$-0.912731\pi$
$797$		−	431.564i		−	0.541485i	0.962652	−	0.270743i	$-0.0872693\pi$
$798$		0			0
$799$	2366.54			2.96187
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−508.277			−0.632972
$804$		0			0
$805$	−71.9643			−0.0893966
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1194.98			−1.47711			−0.738555	−	0.674194i	$-0.764491\pi$
$809$	−1194.98			−1.47711			−0.738555	+	0.674194i	$0.764491\pi$
$810$		0			0
$811$			199.532i			0.246032i	0.992405	+	0.123016i	$0.0392566\pi$
$811$			199.532i			0.246032i	−0.992405	+	0.123016i	$0.960743\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	10.6062			0.0130137
$816$		0			0
$817$	−469.322	+	766.179i	−0.574446	+	0.937796i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−112.765			−0.137350			−0.0686751	−	0.997639i	$-0.521877\pi$
$821$	−112.765			−0.137350			−0.0686751	+	0.997639i	$0.521877\pi$
$822$		0			0
$823$	1426.67			1.73350			0.866752	−	0.498739i	$-0.166203\pi$
$823$	1426.67			1.73350			0.866752	+	0.498739i	$0.166203\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1134.80i		−	1.37219i	−0.727514	−	0.686093i	$-0.759324\pi$
$827$		−	1134.80i		−	1.37219i	0.727514	−	0.686093i	$-0.240676\pi$
$828$		0			0
$829$		−	1284.23i		−	1.54913i	−0.632493	−	0.774566i	$-0.717968\pi$
$829$		−	1284.23i		−	1.54913i	0.632493	−	0.774566i	$-0.282032\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−268.504			−0.322333
$834$		0			0
$835$			90.6589i			0.108574i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	1481.33i		−	1.76559i	−0.469758	−	0.882795i	$-0.655659\pi$
$839$		−	1481.33i		−	1.76559i	0.469758	−	0.882795i	$-0.344341\pi$
$840$		0			0
$841$	−1233.36			−1.46654
$842$		0			0
$843$		0			0
$844$		0			0
$845$	137.590			0.162829
$846$		0			0
$847$	−411.054			−0.485306
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	222.418i		−	0.261361i
$852$		0			0
$853$	−601.805			−0.705515			−0.352758	−	0.935715i	$-0.614756\pi$
$853$	−601.805			−0.705515			−0.352758	+	0.935715i	$0.614756\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			329.975i			0.385035i	0.981294	+	0.192517i	$0.0616652\pi$
$857$			329.975i			0.385035i	−0.981294	+	0.192517i	$0.938335\pi$
$858$		0			0
$859$	1528.04			1.77886			0.889431	−	0.457070i	$-0.151101\pi$
$859$	1528.04			1.77886			0.889431	+	0.457070i	$0.151101\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	194.767i		−	0.225686i	−0.993613	−	0.112843i	$-0.964004\pi$
$863$		−	194.767i		−	0.225686i	0.993613	−	0.112843i	$-0.0359957\pi$
$864$		0			0
$865$			153.843i			0.177853i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			253.855i			0.292123i
$870$		0			0
$871$	−25.5367			−0.0293188
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−250.971			−0.286824
$876$		0			0
$877$			1288.75i			1.46950i	0.678337	+	0.734751i	$0.262701\pi$
$877$			1288.75i			1.46950i	−0.678337	+	0.734751i	$0.737299\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1386.64			−1.57394			−0.786972	−	0.616989i	$-0.788352\pi$
$881$	−1386.64			−1.57394			−0.786972	+	0.616989i	$0.788352\pi$
$882$		0			0
$883$	−648.757			−0.734719			−0.367360	−	0.930079i	$-0.619738\pi$
$883$	−648.757			−0.734719			−0.367360	+	0.930079i	$0.619738\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			230.138i			0.259457i	0.991550	+	0.129728i	$0.0414105\pi$
$887$			230.138i			0.259457i	−0.991550	+	0.129728i	$0.958589\pi$
$888$		0			0
$889$			408.152i			0.459114i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−887.487	+	1448.84i	−0.993826	+	1.62244i
$894$		0			0
$895$			2.26691i			0.00253285i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1131.96			−1.25914
$900$		0			0
$901$			444.400i			0.493230i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	121.226i		−	0.133952i
$906$		0			0
$907$		−	1404.50i		−	1.54851i	−0.632875	−	0.774254i	$-0.718125\pi$
$907$		−	1404.50i		−	1.54851i	0.632875	−	0.774254i	$-0.281875\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			70.4530i			0.0773359i	0.999252	+	0.0386679i	$0.0123115\pi$
$911$			70.4530i			0.0773359i	−0.999252	+	0.0386679i	$0.987689\pi$
$912$		0			0
$913$	82.2741			0.0901140
$914$		0			0
$915$		0			0
$916$		0			0
$917$	727.252			0.793078
$918$		0			0
$919$	520.334			0.566196			0.283098	−	0.959091i	$-0.408638\pi$
$919$	520.334			0.566196			0.283098	+	0.959091i	$0.408638\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−79.7834			−0.0864392
$924$		0			0
$925$		−	382.599i		−	0.413620i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1733.34			−1.86581			−0.932904	−	0.360124i	$-0.882734\pi$
$929$	−1733.34			−1.86581			−0.932904	+	0.360124i	$0.882734\pi$
$930$		0			0
$931$	100.693	−	164.384i	0.108156	−	0.176567i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	160.257			0.171398
$936$		0			0
$937$	−1522.87			−1.62526			−0.812629	−	0.582781i	$-0.801964\pi$
$937$	−1522.87			−1.62526			−0.812629	+	0.582781i	$0.801964\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			1195.02i			1.26995i	0.772533	+	0.634975i	$0.218990\pi$
$941$			1195.02i			1.26995i	−0.772533	+	0.634975i	$0.781010\pi$
$942$		0			0
$943$		−	336.328i		−	0.356657i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1439.01			1.51955			0.759774	−	0.650188i	$-0.225310\pi$
$947$	1439.01			1.51955			0.759774	+	0.650188i	$0.225310\pi$
$948$		0			0
$949$			43.9227i			0.0462831i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	982.009i		−	1.03044i	−0.857058	−	0.515220i	$-0.827710\pi$
$953$		−	982.009i		−	1.03044i	0.857058	−	0.515220i	$-0.172290\pi$
$954$		0			0
$955$	75.5201			0.0790786
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−825.414			−0.860703
$960$		0			0
$961$	343.296			0.357228
$962$		0			0
$963$		0			0
$964$		0			0
$965$			127.280i			0.131896i
$966$		0			0
$967$	−65.0083			−0.0672268			−0.0336134	−	0.999435i	$-0.510701\pi$
$967$	−65.0083			−0.0672268			−0.0336134	+	0.999435i	$0.510701\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			127.496i			0.131304i	0.997843	+	0.0656522i	$0.0209128\pi$
$971$			127.496i			0.131304i	−0.997843	+	0.0656522i	$0.979087\pi$
$972$		0			0
$973$	1012.44			1.04054
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1883.39i		−	1.92773i	−0.266397	−	0.963863i	$-0.585833\pi$
$977$		−	1883.39i		−	1.92773i	0.266397	−	0.963863i	$-0.414167\pi$
$978$		0			0
$979$		−	859.158i		−	0.877587i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	953.914i		−	0.970411i	−0.874400	−	0.485206i	$-0.838745\pi$
$983$		−	953.914i		−	0.970411i	0.874400	−	0.485206i	$-0.161255\pi$
$984$		0			0
$985$	2.37129			0.00240740
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−668.962			−0.676403
$990$		0			0
$991$		−	151.179i		−	0.152552i	−0.997087	−	0.0762762i	$-0.975697\pi$
$991$		−	151.179i		−	0.152552i	0.997087	−	0.0762762i	$-0.0243031\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−54.7017			−0.0549766
$996$		0			0
$997$	727.613			0.729802			0.364901	−	0.931046i	$-0.381103\pi$
$997$	727.613			0.729802			0.364901	+	0.931046i	$0.381103\pi$
$998$		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.721.11		20
3.2	odd	2		912.3.o.e.721.5		20
4.3	odd	2		1368.3.o.c.721.11		20
12.11	even	2		456.3.o.a.265.15	yes	20
19.18	odd	2	inner	2736.3.o.r.721.12		20
57.56	even	2		912.3.o.e.721.15		20
76.75	even	2		1368.3.o.c.721.12		20
228.227	odd	2		456.3.o.a.265.5	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.3.o.a.265.5	✓	20	228.227	odd	2
456.3.o.a.265.15	yes	20	12.11	even	2
912.3.o.e.721.5		20	3.2	odd	2
912.3.o.e.721.15		20	57.56	even	2
1368.3.o.c.721.11		20	4.3	odd	2
1368.3.o.c.721.12		20	76.75	even	2
2736.3.o.r.721.11		20	1.1	even	1	trivial
2736.3.o.r.721.12		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)

\(f(q)\)	\(=\)	\(q+0.816130 q^{5} +6.23331 q^{7} +O(q^{10})\)	\(q+0.816130 q^{5} +6.23331 q^{7} +7.41992 q^{11} -0.641191i q^{13} +26.4643 q^{17} +(-9.92451 + 16.2020i) q^{19} -14.1462 q^{23} -24.3339 q^{25} +45.5452i q^{29} +24.8536i q^{31} +5.08719 q^{35} +15.7228i q^{37} +23.7752i q^{41} +47.2892 q^{43} +89.4238 q^{47} -10.1459 q^{49} +16.7925i q^{53} +6.05561 q^{55} -4.08625i q^{59} -94.1995 q^{61} -0.523295i q^{65} -39.8269i q^{67} -124.430i q^{71} -68.5017 q^{73} +46.2506 q^{77} +34.2126i q^{79} +11.0883 q^{83} +21.5983 q^{85} -115.791i q^{89} -3.99674i q^{91} +(-8.09968 + 13.2229i) q^{95} +150.705i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 16 q^{7}+O(q^{10}) \) 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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