LMFDB - Embedded newform 1728.3.b.g.1567.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(1567,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1728.1567");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$47.0845896815$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1567.4
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1728.1567
Dual form			1728.3.b.g.1567.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.48528i q^{5} +7.00000i q^{7} +O(q^{10})$	$q-8.48528i q^{5} +7.00000i q^{7} +8.48528 q^{11} +19.0526i q^{13} -14.6969 q^{17} -8.66025 q^{19} -14.6969i q^{23} -47.0000 q^{25} -50.9117i q^{29} -10.0000i q^{31} +59.3970 q^{35} -19.0526i q^{37} -58.7878 q^{41} +41.5692 q^{43} -73.4847i q^{47} +16.9706i q^{53} -72.0000i q^{55} +25.4558 q^{59} +19.0526i q^{61} +161.666 q^{65} -116.047 q^{67} +117.576i q^{71} -71.0000 q^{73} +59.3970i q^{77} -151.000i q^{79} -135.765 q^{83} +124.708i q^{85} -161.666 q^{89} -133.368 q^{91} +73.4847i q^{95} -25.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 376 q^{25} - 568 q^{73} - 200 q^{97}+O(q^{100}) $ 8 * q - 376 * q^25 - 568 * q^73 - 200 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$-1$	$-1$	$1$

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
$5$	− 8.48528i	− 1.69706i	−0.529150	−	0.848528i	$-0.677489\pi$
$5$	− 8.48528i	− 1.69706i	0.529150	−	0.848528i	$-0.322511\pi$
$6$	0	0
$7$	7.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$7$	7.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	8.48528	0.771389	0.385695	−	0.922627i	$-0.373962\pi$
$11$	8.48528	0.771389	0.385695	+	0.922627i	$0.373962\pi$
$12$	0	0
$13$	19.0526i	1.46558i	0.680454	+	0.732791i	$0.261783\pi$
$13$	19.0526i	1.46558i	−0.680454	+	0.732791i	$0.738217\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−14.6969	−0.864526	−0.432263	−	0.901748i	$-0.642285\pi$
$17$	−14.6969	−0.864526	−0.432263	+	0.901748i	$0.642285\pi$
$18$	0	0
$19$	−8.66025	−0.455803	−0.227901	−	0.973684i	$-0.573186\pi$
$19$	−8.66025	−0.455803	−0.227901	+	0.973684i	$0.573186\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 14.6969i	− 0.638997i	−0.947587	−	0.319499i	$-0.896486\pi$
$23$	− 14.6969i	− 0.638997i	0.947587	−	0.319499i	$-0.103514\pi$
$24$	0	0
$25$	−47.0000	−1.88000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 50.9117i	− 1.75558i	−0.479050	−	0.877788i	$-0.659019\pi$
$29$	− 50.9117i	− 1.75558i	0.479050	−	0.877788i	$-0.340981\pi$
$30$	0	0
$31$	− 10.0000i	− 0.322581i	−0.986907	−	0.161290i	$-0.948434\pi$
$31$	− 10.0000i	− 0.322581i	0.986907	−	0.161290i	$-0.0515656\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	59.3970	1.69706
$36$	0	0
$37$	− 19.0526i	− 0.514934i	−0.966287	−	0.257467i	$-0.917112\pi$
$37$	− 19.0526i	− 0.514934i	0.966287	−	0.257467i	$-0.0828879\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−58.7878	−1.43385	−0.716924	−	0.697152i	$-0.754450\pi$
$41$	−58.7878	−1.43385	−0.716924	+	0.697152i	$0.754450\pi$
$42$	0	0
$43$	41.5692	0.966726	0.483363	−	0.875420i	$-0.339415\pi$
$43$	41.5692	0.966726	0.483363	+	0.875420i	$0.339415\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 73.4847i	− 1.56350i	−0.623589	−	0.781752i	$-0.714326\pi$
$47$	− 73.4847i	− 1.56350i	0.623589	−	0.781752i	$-0.285674\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	16.9706i	0.320199i	0.987101	+	0.160100i	$0.0511816\pi$
$53$	16.9706i	0.320199i	−0.987101	+	0.160100i	$0.948818\pi$
$54$	0	0
$55$	− 72.0000i	− 1.30909i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	25.4558	0.431455	0.215727	−	0.976454i	$-0.430788\pi$
$59$	25.4558	0.431455	0.215727	+	0.976454i	$0.430788\pi$
$60$	0	0
$61$	19.0526i	0.312337i	0.987730	+	0.156169i	$0.0499143\pi$
$61$	19.0526i	0.312337i	−0.987730	+	0.156169i	$0.950086\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	161.666	2.48717
$66$	0	0
$67$	−116.047	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$67$	−116.047	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	117.576i	1.65599i	0.560733	+	0.827997i	$0.310519\pi$
$71$	117.576i	1.65599i	−0.560733	+	0.827997i	$0.689481\pi$
$72$	0	0
$73$	−71.0000	−0.972603	−0.486301	−	0.873791i	$-0.661654\pi$
$73$	−71.0000	−0.972603	−0.486301	+	0.873791i	$0.661654\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	59.3970i	0.771389i
$78$	0	0
$79$	− 151.000i	− 1.91139i	−0.294355	−	0.955696i	$-0.595105\pi$
$79$	− 151.000i	− 1.91139i	0.294355	−	0.955696i	$-0.404895\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−135.765	−1.63572	−0.817858	−	0.575419i	$-0.804839\pi$
$83$	−135.765	−1.63572	−0.817858	+	0.575419i	$0.804839\pi$
$84$	0	0
$85$	124.708i	1.46715i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−161.666	−1.81648	−0.908238	−	0.418454i	$-0.862572\pi$
$89$	−161.666	−1.81648	−0.908238	+	0.418454i	$0.862572\pi$
$90$	0	0
$91$	−133.368	−1.46558
$92$	0	0
$93$	0	0
$94$	0	0
$95$	73.4847i	0.773523i
$96$	0	0
$97$	−25.0000	−0.257732	−0.128866	−	0.991662i	$-0.541134\pi$
$97$	−25.0000	−0.257732	−0.128866	+	0.991662i	$0.541134\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.9706i	0.168025i	0.996465	+	0.0840127i	$0.0267736\pi$
$101$	16.9706i	0.168025i	−0.996465	+	0.0840127i	$0.973226\pi$
$102$	0	0
$103$	− 103.000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$103$	− 103.000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−110.309	−1.03092	−0.515461	−	0.856913i	$-0.672379\pi$
$107$	−110.309	−1.03092	−0.515461	+	0.856913i	$0.672379\pi$
$108$	0	0
$109$	41.5692i	0.381369i	0.981651	+	0.190684i	$0.0610707\pi$
$109$	41.5692i	0.381369i	−0.981651	+	0.190684i	$0.938929\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	161.666	1.43068	0.715338	−	0.698779i	$-0.246273\pi$
$113$	161.666	1.43068	0.715338	+	0.698779i	$0.246273\pi$
$114$	0	0
$115$	−124.708	−1.08441
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 102.879i	− 0.864526i
$120$	0	0
$121$	−49.0000	−0.404959
$122$	0	0
$123$	0	0
$124$	0	0
$125$	186.676i	1.49341i
$126$	0	0
$127$	34.0000i	0.267717i	0.991000	+	0.133858i	$0.0427367\pi$
$127$	34.0000i	0.267717i	−0.991000	+	0.133858i	$0.957263\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−118.794	−0.906824	−0.453412	−	0.891301i	$-0.649793\pi$
$131$	−118.794	−0.906824	−0.453412	+	0.891301i	$0.649793\pi$
$132$	0	0
$133$	− 60.6218i	− 0.455803i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.6969	0.107277	0.0536385	−	0.998560i	$-0.482918\pi$
$137$	14.6969	0.107277	0.0536385	+	0.998560i	$0.482918\pi$
$138$	0	0
$139$	216.506	1.55760	0.778800	−	0.627273i	$-0.215829\pi$
$139$	216.506	1.55760	0.778800	+	0.627273i	$0.215829\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	161.666i	1.13053i
$144$	0	0
$145$	−432.000	−2.97931
$146$	0	0
$147$	0	0
$148$	0	0
$149$	33.9411i	0.227793i	0.993493	+	0.113896i	$0.0363332\pi$
$149$	33.9411i	0.227793i	−0.993493	+	0.113896i	$0.963667\pi$
$150$	0	0
$151$	− 209.000i	− 1.38411i	−0.721847	−	0.692053i	$-0.756706\pi$
$151$	− 209.000i	− 1.38411i	0.721847	−	0.692053i	$-0.243294\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−84.8528	−0.547438
$156$	0	0
$157$	290.985i	1.85340i	0.375796	+	0.926702i	$0.377369\pi$
$157$	290.985i	1.85340i	−0.375796	+	0.926702i	$0.622631\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	102.879	0.638997
$162$	0	0
$163$	−74.4782	−0.456921	−0.228461	−	0.973553i	$-0.573369\pi$
$163$	−74.4782	−0.456921	−0.228461	+	0.973553i	$0.573369\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	44.0908i	0.264017i	0.991249	+	0.132008i	$0.0421426\pi$
$167$	44.0908i	0.264017i	−0.991249	+	0.132008i	$0.957857\pi$
$168$	0	0
$169$	−194.000	−1.14793
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 67.8823i	− 0.392383i	−0.980566	−	0.196191i	$-0.937143\pi$
$173$	− 67.8823i	− 0.392383i	0.980566	−	0.196191i	$-0.0628574\pi$
$174$	0	0
$175$	− 329.000i	− 1.88000i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	271.529	1.51692	0.758461	−	0.651719i	$-0.225952\pi$
$179$	271.529	1.51692	0.758461	+	0.651719i	$0.225952\pi$
$180$	0	0
$181$	− 143.760i	− 0.794255i	−0.917763	−	0.397128i	$-0.870007\pi$
$181$	− 143.760i	− 0.794255i	0.917763	−	0.397128i	$-0.129993\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−161.666	−0.873872
$186$	0	0
$187$	−124.708	−0.666886
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 132.272i	− 0.692526i	−0.938138	−	0.346263i	$-0.887451\pi$
$191$	− 132.272i	− 0.692526i	0.938138	−	0.346263i	$-0.112549\pi$
$192$	0	0
$193$	−265.000	−1.37306	−0.686528	−	0.727103i	$-0.740866\pi$
$193$	−265.000	−1.37306	−0.686528	+	0.727103i	$0.740866\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 144.250i	− 0.732232i	−0.930569	−	0.366116i	$-0.880687\pi$
$197$	− 144.250i	− 0.732232i	0.930569	−	0.366116i	$-0.119313\pi$
$198$	0	0
$199$	89.0000i	0.447236i	0.974677	+	0.223618i	$0.0717868\pi$
$199$	89.0000i	0.447236i	−0.974677	+	0.223618i	$0.928213\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	356.382	1.75558
$204$	0	0
$205$	498.831i	2.43332i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−73.4847	−0.351601
$210$	0	0
$211$	74.4782	0.352977	0.176489	−	0.984303i	$-0.443526\pi$
$211$	74.4782	0.352977	0.176489	+	0.984303i	$0.443526\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 352.727i	− 1.64059i
$216$	0	0
$217$	70.0000	0.322581
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 280.014i	− 1.26703i
$222$	0	0
$223$	202.000i	0.905830i	0.891554	+	0.452915i	$0.149616\pi$
$223$	202.000i	0.905830i	−0.891554	+	0.452915i	$0.850384\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	305.470	1.34568	0.672842	−	0.739786i	$-0.265074\pi$
$227$	305.470	1.34568	0.672842	+	0.739786i	$0.265074\pi$
$228$	0	0
$229$	− 166.277i	− 0.726100i	−0.931770	−	0.363050i	$-0.881735\pi$
$229$	− 166.277i	− 0.726100i	0.931770	−	0.363050i	$-0.118265\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−29.3939	−0.126154	−0.0630770	−	0.998009i	$-0.520091\pi$
$233$	−29.3939	−0.126154	−0.0630770	+	0.998009i	$0.520091\pi$
$234$	0	0
$235$	−623.538	−2.65335
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 29.3939i	− 0.122987i	−0.998107	−	0.0614935i	$-0.980414\pi$
$239$	− 29.3939i	− 0.122987i	0.998107	−	0.0614935i	$-0.0195863\pi$
$240$	0	0
$241$	169.000	0.701245	0.350622	−	0.936517i	$-0.385970\pi$
$241$	169.000	0.701245	0.350622	+	0.936517i	$0.385970\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 165.000i	− 0.668016i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−203.647	−0.811342	−0.405671	−	0.914019i	$-0.632962\pi$
$251$	−203.647	−0.811342	−0.405671	+	0.914019i	$0.632962\pi$
$252$	0	0
$253$	− 124.708i	− 0.492916i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−146.969	−0.571865	−0.285933	−	0.958250i	$-0.592303\pi$
$257$	−146.969	−0.571865	−0.285933	+	0.958250i	$0.592303\pi$
$258$	0	0
$259$	133.368	0.514934
$260$	0	0
$261$	0	0
$262$	0	0
$263$	323.333i	1.22940i	0.788760	+	0.614701i	$0.210723\pi$
$263$	323.333i	1.22940i	−0.788760	+	0.614701i	$0.789277\pi$
$264$	0	0
$265$	144.000	0.543396
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 330.926i	− 1.23021i	−0.788446	−	0.615104i	$-0.789114\pi$
$269$	− 330.926i	− 1.23021i	0.788446	−	0.615104i	$-0.210886\pi$
$270$	0	0
$271$	− 305.000i	− 1.12546i	−0.826640	−	0.562731i	$-0.809751\pi$
$271$	− 305.000i	− 1.12546i	0.826640	−	0.562731i	$-0.190249\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−398.808	−1.45021
$276$	0	0
$277$	207.846i	0.750347i	0.926955	+	0.375173i	$0.122417\pi$
$277$	207.846i	0.750347i	−0.926955	+	0.375173i	$0.877583\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−440.908	−1.56907	−0.784534	−	0.620086i	$-0.787098\pi$
$281$	−440.908	−1.56907	−0.784534	+	0.620086i	$0.787098\pi$
$282$	0	0
$283$	207.846	0.734439	0.367219	−	0.930134i	$-0.380310\pi$
$283$	207.846	0.734439	0.367219	+	0.930134i	$0.380310\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 411.514i	− 1.43385i
$288$	0	0
$289$	−73.0000	−0.252595
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 76.3675i	− 0.260640i	−0.991472	−	0.130320i	$-0.958400\pi$
$293$	− 76.3675i	− 0.260640i	0.991472	−	0.130320i	$-0.0416005\pi$
$294$	0	0
$295$	− 216.000i	− 0.732203i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	280.014	0.936503
$300$	0	0
$301$	290.985i	0.966726i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	161.666	0.530054
$306$	0	0
$307$	−374.123	−1.21864	−0.609321	−	0.792924i	$-0.708558\pi$
$307$	−374.123	−1.21864	−0.609321	+	0.792924i	$0.708558\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 367.423i	− 1.18143i	−0.806882	−	0.590713i	$-0.798847\pi$
$311$	− 367.423i	− 1.18143i	0.806882	−	0.590713i	$-0.201153\pi$
$312$	0	0
$313$	−335.000	−1.07029	−0.535144	−	0.844761i	$-0.679743\pi$
$313$	−335.000	−1.07029	−0.535144	+	0.844761i	$0.679743\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	135.765i	0.428279i	0.976803	+	0.214140i	$0.0686947\pi$
$317$	135.765i	0.428279i	−0.976803	+	0.214140i	$0.931305\pi$
$318$	0	0
$319$	− 432.000i	− 1.35423i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	127.279	0.394053
$324$	0	0
$325$	− 895.470i	− 2.75529i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	514.393	1.56350
$330$	0	0
$331$	−32.9090	−0.0994229	−0.0497114	−	0.998764i	$-0.515830\pi$
$331$	−32.9090	−0.0994229	−0.0497114	+	0.998764i	$0.515830\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	984.695i	2.93939i
$336$	0	0
$337$	599.000	1.77745	0.888724	−	0.458443i	$-0.151593\pi$
$337$	599.000	1.77745	0.888724	+	0.458443i	$0.151593\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 84.8528i	− 0.248835i
$342$	0	0
$343$	343.000i	1.00000i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	220.617	0.635785	0.317892	−	0.948127i	$-0.397025\pi$
$347$	220.617	0.635785	0.317892	+	0.948127i	$0.397025\pi$
$348$	0	0
$349$	− 226.899i	− 0.650139i	−0.945690	−	0.325070i	$-0.894612\pi$
$349$	− 226.899i	− 0.650139i	0.945690	−	0.325070i	$-0.105388\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−382.120	−1.08249	−0.541247	−	0.840864i	$-0.682048\pi$
$353$	−382.120	−1.08249	−0.541247	+	0.840864i	$0.682048\pi$
$354$	0	0
$355$	997.661	2.81031
$356$	0	0
$357$	0	0
$358$	0	0
$359$	367.423i	1.02346i	0.859145	+	0.511732i	$0.170996\pi$
$359$	367.423i	1.02346i	−0.859145	+	0.511732i	$0.829004\pi$
$360$	0	0
$361$	−286.000	−0.792244
$362$	0	0
$363$	0	0
$364$	0	0
$365$	602.455i	1.65056i
$366$	0	0
$367$	425.000i	1.15804i	0.815314	+	0.579019i	$0.196564\pi$
$367$	425.000i	1.15804i	−0.815314	+	0.579019i	$0.803436\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−118.794	−0.320199
$372$	0	0
$373$	230.363i	0.617595i	0.951128	+	0.308797i	$0.0999265\pi$
$373$	230.363i	0.617595i	−0.951128	+	0.308797i	$0.900074\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	969.998	2.57294
$378$	0	0
$379$	−74.4782	−0.196512	−0.0982562	−	0.995161i	$-0.531326\pi$
$379$	−74.4782	−0.196512	−0.0982562	+	0.995161i	$0.531326\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 382.120i	− 0.997703i	−0.866687	−	0.498852i	$-0.833755\pi$
$383$	− 382.120i	− 0.997703i	0.866687	−	0.498852i	$-0.166245\pi$
$384$	0	0
$385$	504.000	1.30909
$386$	0	0
$387$	0	0
$388$	0	0
$389$	721.249i	1.85411i	0.374925	+	0.927055i	$0.377668\pi$
$389$	721.249i	1.85411i	−0.374925	+	0.927055i	$0.622332\pi$
$390$	0	0
$391$	216.000i	0.552430i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1281.28	−3.24374
$396$	0	0
$397$	41.5692i	0.104708i	0.998629	+	0.0523542i	$0.0166725\pi$
$397$	41.5692i	0.104708i	−0.998629	+	0.0523542i	$0.983328\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−440.908	−1.09952	−0.549761	−	0.835322i	$-0.685281\pi$
$401$	−440.908	−1.09952	−0.549761	+	0.835322i	$0.685281\pi$
$402$	0	0
$403$	190.526	0.472768
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 161.666i	− 0.397215i
$408$	0	0
$409$	193.000	0.471883	0.235941	−	0.971767i	$-0.424183\pi$
$409$	193.000	0.471883	0.235941	+	0.971767i	$0.424183\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	178.191i	0.431455i
$414$	0	0
$415$	1152.00i	2.77590i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−636.396	−1.51885	−0.759423	−	0.650598i	$-0.774518\pi$
$419$	−636.396	−1.51885	−0.759423	+	0.650598i	$0.774518\pi$
$420$	0	0
$421$	− 646.055i	− 1.53457i	−0.641305	−	0.767286i	$-0.721607\pi$
$421$	− 646.055i	− 1.53457i	0.641305	−	0.767286i	$-0.278393\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	690.756	1.62531
$426$	0	0
$427$	−133.368	−0.312337
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 220.454i	− 0.511494i	−0.966744	−	0.255747i	$-0.917679\pi$
$431$	− 220.454i	− 0.511494i	0.966744	−	0.255747i	$-0.0823215\pi$
$432$	0	0
$433$	−362.000	−0.836028	−0.418014	−	0.908441i	$-0.637274\pi$
$433$	−362.000	−0.836028	−0.418014	+	0.908441i	$0.637274\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	127.279i	0.291257i
$438$	0	0
$439$	− 58.0000i	− 0.132118i	−0.997816	−	0.0660592i	$-0.978957\pi$
$439$	− 58.0000i	− 0.132118i	0.997816	−	0.0660592i	$-0.0210426\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	356.382	0.804474	0.402237	−	0.915536i	$-0.368233\pi$
$443$	356.382	0.804474	0.402237	+	0.915536i	$0.368233\pi$
$444$	0	0
$445$	1371.78i	3.08266i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−338.030	−0.752850	−0.376425	−	0.926447i	$-0.622847\pi$
$449$	−338.030	−0.752850	−0.376425	+	0.926447i	$0.622847\pi$
$450$	0	0
$451$	−498.831	−1.10605
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1131.66i	2.48717i
$456$	0	0
$457$	454.000	0.993435	0.496718	−	0.867912i	$-0.334538\pi$
$457$	454.000	0.993435	0.496718	+	0.867912i	$0.334538\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	93.3381i	0.202469i	0.994863	+	0.101234i	$0.0322792\pi$
$461$	93.3381i	0.202469i	−0.994863	+	0.101234i	$0.967721\pi$
$462$	0	0
$463$	391.000i	0.844492i	0.906481	+	0.422246i	$0.138758\pi$
$463$	391.000i	0.844492i	−0.906481	+	0.422246i	$0.861242\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	704.278	1.50809	0.754045	−	0.656822i	$-0.228100\pi$
$467$	704.278	1.50809	0.754045	+	0.656822i	$0.228100\pi$
$468$	0	0
$469$	− 812.332i	− 1.73205i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	352.727	0.745722
$474$	0	0
$475$	407.032	0.856909
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 499.696i	− 1.04321i	−0.853188	−	0.521603i	$-0.825334\pi$
$479$	− 499.696i	− 1.04321i	0.853188	−	0.521603i	$-0.174666\pi$
$480$	0	0
$481$	363.000	0.754678
$482$	0	0
$483$	0	0
$484$	0	0
$485$	212.132i	0.437386i
$486$	0	0
$487$	559.000i	1.14784i	0.818910	+	0.573922i	$0.194579\pi$
$487$	559.000i	1.14784i	−0.818910	+	0.573922i	$0.805421\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	364.867	0.743110	0.371555	−	0.928411i	$-0.378825\pi$
$491$	364.867	0.743110	0.371555	+	0.928411i	$0.378825\pi$
$492$	0	0
$493$	748.246i	1.51774i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−823.029	−1.65599
$498$	0	0
$499$	249.415	0.499830	0.249915	−	0.968268i	$-0.419597\pi$
$499$	249.415	0.499830	0.249915	+	0.968268i	$0.419597\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 514.393i	− 1.02265i	−0.859387	−	0.511325i	$-0.829155\pi$
$503$	− 514.393i	− 1.02265i	0.859387	−	0.511325i	$-0.170845\pi$
$504$	0	0
$505$	144.000	0.285149
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 313.955i	− 0.616808i	−0.951255	−	0.308404i	$-0.900205\pi$
$509$	− 313.955i	− 0.616808i	0.951255	−	0.308404i	$-0.0997949\pi$
$510$	0	0
$511$	− 497.000i	− 0.972603i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−873.984	−1.69706
$516$	0	0
$517$	− 623.538i	− 1.20607i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−367.423	−0.705227	−0.352614	−	0.935769i	$-0.614707\pi$
$521$	−367.423	−0.705227	−0.352614	+	0.935769i	$0.614707\pi$
$522$	0	0
$523$	715.337	1.36776	0.683879	−	0.729596i	$-0.260292\pi$
$523$	715.337	1.36776	0.683879	+	0.729596i	$0.260292\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	146.969i	0.278879i
$528$	0	0
$529$	313.000	0.591682
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 1120.06i	− 2.10142i
$534$	0	0
$535$	936.000i	1.74953i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 604.486i	− 1.11735i	−0.829387	−	0.558674i	$-0.811310\pi$
$541$	− 604.486i	− 1.11735i	0.829387	−	0.558674i	$-0.188690\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	352.727	0.647205
$546$	0	0
$547$	798.475	1.45974	0.729868	−	0.683588i	$-0.239582\pi$
$547$	798.475	1.45974	0.729868	+	0.683588i	$0.239582\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	440.908i	0.800196i
$552$	0	0
$553$	1057.00	1.91139
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 161.220i	− 0.289444i	−0.989472	−	0.144722i	$-0.953771\pi$
$557$	− 161.220i	− 0.289444i	0.989472	−	0.144722i	$-0.0462288\pi$
$558$	0	0
$559$	792.000i	1.41682i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	339.411	0.602862	0.301431	−	0.953488i	$-0.402536\pi$
$563$	339.411	0.602862	0.301431	+	0.953488i	$0.402536\pi$
$564$	0	0
$565$	− 1371.78i	− 2.42794i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	308.636	0.542418	0.271209	−	0.962521i	$-0.412577\pi$
$569$	308.636	0.542418	0.271209	+	0.962521i	$0.412577\pi$
$570$	0	0
$571$	282.324	0.494438	0.247219	−	0.968960i	$-0.420483\pi$
$571$	282.324	0.494438	0.247219	+	0.968960i	$0.420483\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	690.756i	1.20131i
$576$	0	0
$577$	145.000	0.251300	0.125650	−	0.992075i	$-0.459898\pi$
$577$	145.000	0.251300	0.125650	+	0.992075i	$0.459898\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 950.352i	− 1.63572i
$582$	0	0
$583$	144.000i	0.246998i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−8.48528	−0.0144553	−0.00722767	−	0.999974i	$-0.502301\pi$
$587$	−8.48528	−0.0144553	−0.00722767	+	0.999974i	$0.502301\pi$
$588$	0	0
$589$	86.6025i	0.147033i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−411.514	−0.693953	−0.346977	−	0.937874i	$-0.612792\pi$
$593$	−411.514	−0.693953	−0.346977	+	0.937874i	$0.612792\pi$
$594$	0	0
$595$	−872.954	−1.46715
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 1028.79i	− 1.71751i	−0.512390	−	0.858753i	$-0.671240\pi$
$599$	− 1028.79i	− 1.71751i	0.512390	−	0.858753i	$-0.328760\pi$
$600$	0	0
$601$	698.000	1.16140	0.580699	−	0.814118i	$-0.302779\pi$
$601$	698.000	1.16140	0.580699	+	0.814118i	$0.302779\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	415.779i	0.687238i
$606$	0	0
$607$	− 977.000i	− 1.60956i	−0.593576	−	0.804778i	$-0.702285\pi$
$607$	− 977.000i	− 1.60956i	0.593576	−	0.804778i	$-0.297715\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1400.07	2.29144
$612$	0	0
$613$	− 1061.75i	− 1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$613$	− 1061.75i	− 1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−749.544	−1.21482	−0.607410	−	0.794389i	$-0.707791\pi$
$617$	−749.544	−1.21482	−0.607410	+	0.794389i	$0.707791\pi$
$618$	0	0
$619$	573.309	0.926185	0.463093	−	0.886310i	$-0.346740\pi$
$619$	573.309	0.926185	0.463093	+	0.886310i	$0.346740\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 1131.66i	− 1.81648i
$624$	0	0
$625$	409.000	0.654400
$626$	0	0
$627$	0	0
$628$	0	0
$629$	280.014i	0.445174i
$630$	0	0
$631$	− 449.000i	− 0.711569i	−0.934568	−	0.355784i	$-0.884214\pi$
$631$	− 449.000i	− 0.711569i	0.934568	−	0.355784i	$-0.115786\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	288.500	0.454330
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	705.453	1.10055	0.550275	−	0.834983i	$-0.314523\pi$
$641$	705.453	1.10055	0.550275	+	0.834983i	$0.314523\pi$
$642$	0	0
$643$	374.123	0.581840	0.290920	−	0.956747i	$-0.406039\pi$
$643$	374.123	0.581840	0.290920	+	0.956747i	$0.406039\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	117.576i	0.181724i	0.995863	+	0.0908621i	$0.0289622\pi$
$647$	117.576i	0.181724i	−0.995863	+	0.0908621i	$0.971038\pi$
$648$	0	0
$649$	216.000	0.332820
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 780.646i	− 1.19548i	−0.801691	−	0.597738i	$-0.796066\pi$
$653$	− 780.646i	− 1.19548i	0.801691	−	0.597738i	$-0.203934\pi$
$654$	0	0
$655$	1008.00i	1.53893i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	729.734	1.10734	0.553668	−	0.832738i	$-0.313228\pi$
$659$	729.734	1.10734	0.553668	+	0.832738i	$0.313228\pi$
$660$	0	0
$661$	601.022i	0.909261i	0.890680	+	0.454631i	$0.150229\pi$
$661$	601.022i	0.909261i	−0.890680	+	0.454631i	$0.849771\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−514.393	−0.773523
$666$	0	0
$667$	−748.246	−1.12181
$668$	0	0
$669$	0	0
$670$	0	0
$671$	161.666i	0.240933i
$672$	0	0
$673$	−695.000	−1.03269	−0.516345	−	0.856381i	$-0.672708\pi$
$673$	−695.000	−1.03269	−0.516345	+	0.856381i	$0.672708\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 534.573i	− 0.789620i	−0.918763	−	0.394810i	$-0.870810\pi$
$677$	− 534.573i	− 0.789620i	0.918763	−	0.394810i	$-0.129190\pi$
$678$	0	0
$679$	− 175.000i	− 0.257732i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	661.852	0.969037	0.484518	−	0.874781i	$-0.338995\pi$
$683$	661.852	0.969037	0.484518	+	0.874781i	$0.338995\pi$
$684$	0	0
$685$	− 124.708i	− 0.182055i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−323.333	−0.469278
$690$	0	0
$691$	−706.677	−1.02269	−0.511344	−	0.859376i	$-0.670852\pi$
$691$	−706.677	−1.02269	−0.511344	+	0.859376i	$0.670852\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1837.12i	− 2.64333i
$696$	0	0
$697$	864.000	1.23960
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 670.337i	− 0.956259i	−0.878289	−	0.478129i	$-0.841315\pi$
$701$	− 670.337i	− 0.956259i	0.878289	−	0.478129i	$-0.158685\pi$
$702$	0	0
$703$	165.000i	0.234708i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−118.794	−0.168025
$708$	0	0
$709$	559.452i	0.789073i	0.918880	+	0.394536i	$0.129095\pi$
$709$	559.452i	0.789073i	−0.918880	+	0.394536i	$0.870905\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−146.969	−0.206128
$714$	0	0
$715$	1371.78	1.91858
$716$	0	0
$717$	0	0
$718$	0	0
$719$	881.816i	1.22645i	0.789909	+	0.613224i	$0.210128\pi$
$719$	881.816i	1.22645i	−0.789909	+	0.613224i	$0.789872\pi$
$720$	0	0
$721$	721.000	1.00000
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2392.85i	3.30048i
$726$	0	0
$727$	778.000i	1.07015i	0.844804	+	0.535076i	$0.179717\pi$
$727$	778.000i	1.07015i	−0.844804	+	0.535076i	$0.820283\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−610.940	−0.835760
$732$	0	0
$733$	540.400i	0.737244i	0.929579	+	0.368622i	$0.120170\pi$
$733$	540.400i	0.737244i	−0.929579	+	0.368622i	$0.879830\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−984.695	−1.33609
$738$	0	0
$739$	−540.400	−0.731258	−0.365629	−	0.930761i	$-0.619146\pi$
$739$	−540.400	−0.731258	−0.365629	+	0.930761i	$0.619146\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	426.211i	0.573636i	0.957985	+	0.286818i	$0.0925974\pi$
$743$	426.211i	0.573636i	−0.957985	+	0.286818i	$0.907403\pi$
$744$	0	0
$745$	288.000	0.386577
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 772.161i	− 1.03092i
$750$	0	0
$751$	113.000i	0.150466i	0.997166	+	0.0752330i	$0.0239701\pi$
$751$	113.000i	0.150466i	−0.997166	+	0.0752330i	$0.976030\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1773.42	−2.34891
$756$	0	0
$757$	− 937.039i	− 1.23783i	−0.785457	−	0.618916i	$-0.787572\pi$
$757$	− 937.039i	− 1.23783i	0.785457	−	0.618916i	$-0.212428\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	396.817	0.521442	0.260721	−	0.965414i	$-0.416040\pi$
$761$	396.817	0.521442	0.260721	+	0.965414i	$0.416040\pi$
$762$	0	0
$763$	−290.985	−0.381369
$764$	0	0
$765$	0	0
$766$	0	0
$767$	484.999i	0.632332i
$768$	0	0
$769$	383.000	0.498049	0.249025	−	0.968497i	$-0.419890\pi$
$769$	383.000	0.498049	0.249025	+	0.968497i	$0.419890\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	814.587i	1.05380i	0.849927	+	0.526900i	$0.176646\pi$
$773$	814.587i	1.05380i	−0.849927	+	0.526900i	$0.823354\pi$
$774$	0	0
$775$	470.000i	0.606452i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	509.117	0.653552
$780$	0	0
$781$	997.661i	1.27742i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2469.09	3.14533
$786$	0	0
$787$	240.755	0.305915	0.152957	−	0.988233i	$-0.451120\pi$
$787$	240.755	0.305915	0.152957	+	0.988233i	$0.451120\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1131.66i	1.43068i
$792$	0	0
$793$	−363.000	−0.457755
$794$	0	0
$795$	0	0
$796$	0	0
$797$	492.146i	0.617499i	0.951143	+	0.308749i	$0.0999104\pi$
$797$	492.146i	0.617499i	−0.951143	+	0.308749i	$0.900090\pi$
$798$	0	0
$799$	1080.00i	1.35169i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−602.455	−0.750255
$804$	0	0
$805$	− 872.954i	− 1.08441i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	540.400	0.666338	0.333169	−	0.942867i	$-0.391882\pi$
$811$	540.400	0.666338	0.333169	+	0.942867i	$0.391882\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	631.968i	0.775421i
$816$	0	0
$817$	−360.000	−0.440636
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 636.396i	− 0.775148i	−0.921839	−	0.387574i	$-0.873313\pi$
$821$	− 636.396i	− 0.775148i	0.921839	−	0.387574i	$-0.126687\pi$
$822$	0	0
$823$	967.000i	1.17497i	0.809235	+	0.587485i	$0.199882\pi$
$823$	967.000i	1.17497i	−0.809235	+	0.587485i	$0.800118\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	381.838	0.461714	0.230857	−	0.972988i	$-0.425847\pi$
$827$	381.838	0.461714	0.230857	+	0.972988i	$0.425847\pi$
$828$	0	0
$829$	1020.18i	1.23061i	0.788288	+	0.615306i	$0.210968\pi$
$829$	1020.18i	1.23061i	−0.788288	+	0.615306i	$0.789032\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	374.123	0.448051
$836$	0	0
$837$	0	0
$838$	0	0
$839$	440.908i	0.525516i	0.964862	+	0.262758i	$0.0846321\pi$
$839$	440.908i	0.525516i	−0.964862	+	0.262758i	$0.915368\pi$
$840$	0	0
$841$	−1751.00	−2.08205
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1646.14i	1.94810i
$846$	0	0
$847$	− 343.000i	− 0.404959i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−280.014	−0.329041
$852$	0	0
$853$	− 143.760i	− 0.168535i	−0.996443	−	0.0842674i	$-0.973145\pi$
$853$	− 143.760i	− 0.168535i	0.996443	−	0.0842674i	$-0.0268550\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	293.939	0.342986	0.171493	−	0.985185i	$-0.445141\pi$
$857$	293.939	0.342986	0.171493	+	0.985185i	$0.445141\pi$
$858$	0	0
$859$	−91.7987	−0.106867	−0.0534335	−	0.998571i	$-0.517017\pi$
$859$	−91.7987	−0.106867	−0.0534335	+	0.998571i	$0.517017\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 955.301i	− 1.10695i	−0.832865	−	0.553477i	$-0.813301\pi$
$863$	− 955.301i	− 1.10695i	0.832865	−	0.553477i	$-0.186699\pi$
$864$	0	0
$865$	−576.000	−0.665896
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 1281.28i	− 1.47443i
$870$	0	0
$871$	− 2211.00i	− 2.53846i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1306.73	−1.49341
$876$	0	0
$877$	− 434.745i	− 0.495718i	−0.968796	−	0.247859i	$-0.920273\pi$
$877$	− 434.745i	− 0.495718i	0.968796	−	0.247859i	$-0.0797270\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1043.48	1.18443	0.592215	−	0.805780i	$-0.298254\pi$
$881$	1043.48	1.18443	0.592215	+	0.805780i	$0.298254\pi$
$882$	0	0
$883$	133.368	0.151040	0.0755198	−	0.997144i	$-0.475938\pi$
$883$	133.368	0.151040	0.0755198	+	0.997144i	$0.475938\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 411.514i	− 0.463939i	−0.972723	−	0.231970i	$-0.925483\pi$
$887$	− 411.514i	− 0.463939i	0.972723	−	0.231970i	$-0.0745170\pi$
$888$	0	0
$889$	−238.000	−0.267717
$890$	0	0
$891$	0	0
$892$	0	0
$893$	636.396i	0.712650i
$894$	0	0
$895$	− 2304.00i	− 2.57430i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−509.117	−0.566315
$900$	0	0
$901$	− 249.415i	− 0.276821i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1219.85	−1.34790
$906$	0	0
$907$	465.922	0.513695	0.256848	−	0.966452i	$-0.417316\pi$
$907$	465.922	0.513695	0.256848	+	0.966452i	$0.417316\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	117.576i	0.129062i	0.997916	+	0.0645310i	$0.0205551\pi$
$911$	117.576i	0.129062i	−0.997916	+	0.0645310i	$0.979445\pi$
$912$	0	0
$913$	−1152.00	−1.26177
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 831.558i	− 0.906824i
$918$	0	0
$919$	394.000i	0.428727i	0.976754	+	0.214363i	$0.0687677\pi$
$919$	394.000i	0.428727i	−0.976754	+	0.214363i	$0.931232\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−2240.11	−2.42699
$924$	0	0
$925$	895.470i	0.968076i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−617.271	−0.664447	−0.332224	−	0.943201i	$-0.607799\pi$
$929$	−617.271	−0.664447	−0.332224	+	0.943201i	$0.607799\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1058.18i	1.13174i
$936$	0	0
$937$	−1439.00	−1.53575	−0.767876	−	0.640598i	$-0.778686\pi$
$937$	−1439.00	−1.53575	−0.767876	+	0.640598i	$0.778686\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1009.75i	1.07306i	0.843882	+	0.536529i	$0.180265\pi$
$941$	1009.75i	1.07306i	−0.843882	+	0.536529i	$0.819735\pi$
$942$	0	0
$943$	864.000i	0.916225i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	551.543	0.582411	0.291206	−	0.956661i	$-0.405944\pi$
$947$	551.543	0.582411	0.291206	+	0.956661i	$0.405944\pi$
$948$	0	0
$949$	− 1352.73i	− 1.42543i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1249.24	−1.31085	−0.655425	−	0.755260i	$-0.727510\pi$
$953$	−1249.24	−1.31085	−0.655425	+	0.755260i	$0.727510\pi$
$954$	0	0
$955$	−1122.37	−1.17526
$956$	0	0
$957$	0	0
$958$	0	0
$959$	102.879i	0.107277i
$960$	0	0
$961$	861.000	0.895942
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2248.60i	2.33015i
$966$	0	0
$967$	689.000i	0.712513i	0.934388	+	0.356256i	$0.115947\pi$
$967$	689.000i	0.712513i	−0.934388	+	0.356256i	$0.884053\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	568.514	0.585493	0.292747	−	0.956190i	$-0.405431\pi$
$971$	568.514	0.585493	0.292747	+	0.956190i	$0.405431\pi$
$972$	0	0
$973$	1515.54i	1.55760i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1881.21	1.92549	0.962747	−	0.270403i	$-0.0871569\pi$
$977$	1881.21	1.92549	0.962747	+	0.270403i	$0.0871569\pi$
$978$	0	0
$979$	−1371.78	−1.40121
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1807.72i	1.83899i	0.393106	+	0.919493i	$0.371400\pi$
$983$	1807.72i	1.83899i	−0.393106	+	0.919493i	$0.628600\pi$
$984$	0	0
$985$	−1224.00	−1.24264
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 610.940i	− 0.617735i
$990$	0	0
$991$	233.000i	0.235116i	0.993066	+	0.117558i	$0.0375066\pi$
$991$	233.000i	0.235116i	−0.993066	+	0.117558i	$0.962493\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	755.190	0.758985
$996$	0	0
$997$	− 374.123i	− 0.375249i	−0.982241	−	0.187624i	$-0.939921\pi$
$997$	− 374.123i	− 0.375249i	0.982241	−	0.187624i	$-0.0600788\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.b.g.1567.4	yes	8
3.2	odd	2	inner	1728.3.b.g.1567.8	yes	8
4.3	odd	2	inner	1728.3.b.g.1567.2	yes	8
8.3	odd	2	inner	1728.3.b.g.1567.5	yes	8
8.5	even	2	inner	1728.3.b.g.1567.7	yes	8
12.11	even	2	inner	1728.3.b.g.1567.6	yes	8
24.5	odd	2	inner	1728.3.b.g.1567.3	yes	8
24.11	even	2	inner	1728.3.b.g.1567.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.3.b.g.1567.1	✓	8	24.11	even	2	inner
1728.3.b.g.1567.2	yes	8	4.3	odd	2	inner
1728.3.b.g.1567.3	yes	8	24.5	odd	2	inner
1728.3.b.g.1567.4	yes	8	1.1	even	1	trivial
1728.3.b.g.1567.5	yes	8	8.3	odd	2	inner
1728.3.b.g.1567.6	yes	8	12.11	even	2	inner
1728.3.b.g.1567.7	yes	8	8.5	even	2	inner
1728.3.b.g.1567.8	yes	8	3.2	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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-8.48528i q^{5} +7.00000i q^{7} +O(q^{10})\)	\(q-8.48528i q^{5} +7.00000i q^{7} +8.48528 q^{11} +19.0526i q^{13} -14.6969 q^{17} -8.66025 q^{19} -14.6969i q^{23} -47.0000 q^{25} -50.9117i q^{29} -10.0000i q^{31} +59.3970 q^{35} -19.0526i q^{37} -58.7878 q^{41} +41.5692 q^{43} -73.4847i q^{47} +16.9706i q^{53} -72.0000i q^{55} +25.4558 q^{59} +19.0526i q^{61} +161.666 q^{65} -116.047 q^{67} +117.576i q^{71} -71.0000 q^{73} +59.3970i q^{77} -151.000i q^{79} -135.765 q^{83} +124.708i q^{85} -161.666 q^{89} -133.368 q^{91} +73.4847i q^{95} -25.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 376 q^{25} - 568 q^{73} - 200 q^{97}+O(q^{100}) \) 8 * q - 376 * q^25 - 568 * q^73 - 200 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more