LMFDB - Embedded newform 1728.3.g.n.703.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(703,1728)]

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, 0, 0]))

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

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

chi := DirichletCharacter("1728.703");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.g (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:	$47.0845896815$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.22581504.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 $ x^8 - 4x^7 + 5x^6 + 2x^5 - 11x^4 + 4x^3 + 20x^2 - 32*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 864)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			703.4
Root			$-1.27597 + 0.609843i$ of defining polynomial
Character	$\chi$	$=$	1728.703
Dual form			1728.3.g.n.703.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.22512 q^{5} +6.57221i q^{7} +O(q^{10})$	$q-3.22512 q^{5} +6.57221i q^{7} +13.8903i q^{11} +19.0865 q^{13} -23.0865 q^{17} +18.8425i q^{19} +13.5947i q^{23} -14.5986 q^{25} +0.752110 q^{29} -46.4433i q^{31} -21.1962i q^{35} +2.68178 q^{37} +34.1405 q^{41} +20.9127i q^{43} -15.4086i q^{47} +5.80609 q^{49} -46.8189 q^{53} -44.7980i q^{55} +40.4126i q^{59} +105.543 q^{61} -61.5562 q^{65} +27.9369i q^{67} +24.0130i q^{71} -120.117 q^{73} -91.2901 q^{77} -95.6394i q^{79} -115.446i q^{83} +74.4567 q^{85} -169.589 q^{89} +125.440i q^{91} -60.7694i q^{95} -93.1142 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{5}+O(q^{10}) $ 8 * q + 8 * q^5	$ 8 q + 8 q^{5} + 16 q^{13} - 48 q^{17} + 48 q^{25} - 32 q^{29} + 96 q^{37} + 128 q^{41} + 168 q^{53} - 32 q^{61} + 112 q^{65} + 24 q^{73} - 440 q^{77} - 144 q^{85} - 624 q^{89} - 136 q^{97}+O(q^{100}) $ 8 * q + 8 * q^5 + 16 * q^13 - 48 * q^17 + 48 * q^25 - 32 * q^29 + 96 * q^37 + 128 * q^41 + 168 * q^53 - 32 * q^61 + 112 * q^65 + 24 * q^73 - 440 * q^77 - 144 * q^85 - 624 * q^89 - 136 * 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
$5$	−3.22512	−0.645024	−0.322512	−	0.946565i	$-0.604527\pi$
$5$	−3.22512	−0.645024	−0.322512	+	0.946565i	$0.604527\pi$
$6$	0	0
$7$	6.57221i	0.938887i	0.882963	+	0.469443i	$0.155545\pi$
$7$	6.57221i	0.938887i	−0.882963	+	0.469443i	$0.844455\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	13.8903i	1.26276i	0.775475	+	0.631379i	$0.217511\pi$
$11$	13.8903i	1.26276i	−0.775475	+	0.631379i	$0.782489\pi$
$12$	0	0
$13$	19.0865	1.46819	0.734095	−	0.679046i	$-0.237606\pi$
$13$	19.0865	1.46819	0.734095	+	0.679046i	$0.237606\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−23.0865	−1.35803	−0.679014	−	0.734125i	$-0.737593\pi$
$17$	−23.0865	−1.35803	−0.679014	+	0.734125i	$0.737593\pi$
$18$	0	0
$19$	18.8425i	0.991713i	0.868405	+	0.495856i	$0.165146\pi$
$19$	18.8425i	0.991713i	−0.868405	+	0.495856i	$0.834854\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	13.5947i	0.591072i	0.955332	+	0.295536i	$0.0954982\pi$
$23$	13.5947i	0.591072i	−0.955332	+	0.295536i	$0.904502\pi$
$24$	0	0
$25$	−14.5986	−0.583944
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.752110	0.0259348	0.0129674	−	0.999916i	$-0.495872\pi$
$29$	0.752110	0.0259348	0.0129674	+	0.999916i	$0.495872\pi$
$30$	0	0
$31$	− 46.4433i	− 1.49817i	−0.662473	−	0.749085i	$-0.730493\pi$
$31$	− 46.4433i	− 1.49817i	0.662473	−	0.749085i	$-0.269507\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 21.1962i	− 0.605604i
$36$	0	0
$37$	2.68178	0.0724807	0.0362403	−	0.999343i	$-0.488462\pi$
$37$	2.68178	0.0724807	0.0362403	+	0.999343i	$0.488462\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	34.1405	0.832694	0.416347	−	0.909206i	$-0.363310\pi$
$41$	34.1405	0.832694	0.416347	+	0.909206i	$0.363310\pi$
$42$	0	0
$43$	20.9127i	0.486341i	0.969984	+	0.243171i	$0.0781875\pi$
$43$	20.9127i	0.486341i	−0.969984	+	0.243171i	$0.921812\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 15.4086i	− 0.327844i	−0.986473	−	0.163922i	$-0.947586\pi$
$47$	− 15.4086i	− 0.327844i	0.986473	−	0.163922i	$-0.0524145\pi$
$48$	0	0
$49$	5.80609	0.118492
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−46.8189	−0.883376	−0.441688	−	0.897169i	$-0.645620\pi$
$53$	−46.8189	−0.883376	−0.441688	+	0.897169i	$0.645620\pi$
$54$	0	0
$55$	− 44.7980i	− 0.814508i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	40.4126i	0.684959i	0.939525	+	0.342480i	$0.111267\pi$
$59$	40.4126i	0.684959i	−0.939525	+	0.342480i	$0.888733\pi$
$60$	0	0
$61$	105.543	1.73022	0.865108	−	0.501586i	$-0.167250\pi$
$61$	105.543	1.73022	0.865108	+	0.501586i	$0.167250\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−61.5562	−0.947018
$66$	0	0
$67$	27.9369i	0.416969i	0.978026	+	0.208485i	$0.0668531\pi$
$67$	27.9369i	0.416969i	−0.978026	+	0.208485i	$0.933147\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	24.0130i	0.338212i	0.985598	+	0.169106i	$0.0540880\pi$
$71$	24.0130i	0.338212i	−0.985598	+	0.169106i	$0.945912\pi$
$72$	0	0
$73$	−120.117	−1.64545	−0.822723	−	0.568443i	$-0.807546\pi$
$73$	−120.117	−1.64545	−0.822723	+	0.568443i	$0.807546\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−91.2901	−1.18559
$78$	0	0
$79$	− 95.6394i	− 1.21062i	−0.795988	−	0.605312i	$-0.793048\pi$
$79$	− 95.6394i	− 1.21062i	0.795988	−	0.605312i	$-0.206952\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 115.446i	− 1.39091i	−0.718568	−	0.695457i	$-0.755202\pi$
$83$	− 115.446i	− 1.39091i	0.718568	−	0.695457i	$-0.244798\pi$
$84$	0	0
$85$	74.4567	0.875961
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−169.589	−1.90549	−0.952745	−	0.303770i	$-0.901755\pi$
$89$	−169.589	−1.90549	−0.952745	+	0.303770i	$0.901755\pi$
$90$	0	0
$91$	125.440i	1.37846i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 60.7694i	− 0.639678i
$96$	0	0
$97$	−93.1142	−0.959940	−0.479970	−	0.877285i	$-0.659352\pi$
$97$	−93.1142	−0.959940	−0.479970	+	0.877285i	$0.659352\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−159.090	−1.57514	−0.787572	−	0.616222i	$-0.788662\pi$
$101$	−159.090	−1.57514	−0.787572	+	0.616222i	$0.788662\pi$
$102$	0	0
$103$	75.3674i	0.731722i	0.930669	+	0.365861i	$0.119226\pi$
$103$	75.3674i	0.731722i	−0.930669	+	0.365861i	$0.880774\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 69.4279i	− 0.648859i	−0.945910	−	0.324429i	$-0.894828\pi$
$107$	− 69.4279i	− 0.648859i	0.945910	−	0.324429i	$-0.105172\pi$
$108$	0	0
$109$	−60.4669	−0.554742	−0.277371	−	0.960763i	$-0.589463\pi$
$109$	−60.4669	−0.554742	−0.277371	+	0.960763i	$0.589463\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−140.698	−1.24512	−0.622559	−	0.782573i	$-0.713907\pi$
$113$	−140.698	−1.24512	−0.622559	+	0.782573i	$0.713907\pi$
$114$	0	0
$115$	− 43.8444i	− 0.381255i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 151.729i	− 1.27503i
$120$	0	0
$121$	−71.9412	−0.594556
$122$	0	0
$123$	0	0
$124$	0	0
$125$	127.710	1.02168
$126$	0	0
$127$	66.9726i	0.527343i	0.964612	+	0.263672i	$0.0849335\pi$
$127$	66.9726i	0.527343i	−0.964612	+	0.263672i	$0.915066\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	255.148i	1.94770i	0.227196	+	0.973849i	$0.427044\pi$
$131$	255.148i	1.94770i	−0.227196	+	0.973849i	$0.572956\pi$
$132$	0	0
$133$	−123.837	−0.931106
$134$	0	0
$135$	0	0
$136$	0	0
$137$	27.1946	0.198501	0.0992505	−	0.995062i	$-0.468355\pi$
$137$	27.1946	0.198501	0.0992505	+	0.995062i	$0.468355\pi$
$138$	0	0
$139$	203.956i	1.46731i	0.679521	+	0.733656i	$0.262188\pi$
$139$	203.956i	1.46731i	−0.679521	+	0.733656i	$0.737812\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	265.117i	1.85397i
$144$	0	0
$145$	−2.42564	−0.0167286
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−251.507	−1.68797	−0.843984	−	0.536369i	$-0.819796\pi$
$149$	−251.507	−1.68797	−0.843984	+	0.536369i	$0.819796\pi$
$150$	0	0
$151$	− 250.251i	− 1.65729i	−0.559774	−	0.828645i	$-0.689112\pi$
$151$	− 250.251i	− 1.65729i	0.559774	−	0.828645i	$-0.310888\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	149.785i	0.966356i
$156$	0	0
$157$	−18.1730	−0.115751	−0.0578757	−	0.998324i	$-0.518433\pi$
$157$	−18.1730	−0.115751	−0.0578757	+	0.998324i	$0.518433\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−89.3469	−0.554950
$162$	0	0
$163$	− 50.0180i	− 0.306859i	−0.988160	−	0.153429i	$-0.950968\pi$
$163$	− 50.0180i	− 0.306859i	0.988160	−	0.153429i	$-0.0490318\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 84.8389i	− 0.508017i	−0.967202	−	0.254009i	$-0.918251\pi$
$167$	− 84.8389i	− 0.508017i	0.967202	−	0.254009i	$-0.0817492\pi$
$168$	0	0
$169$	195.294	1.15558
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−144.703	−0.836433	−0.418216	−	0.908347i	$-0.637345\pi$
$173$	−144.703	−0.836433	−0.418216	+	0.908347i	$0.637345\pi$
$174$	0	0
$175$	− 95.9451i	− 0.548258i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 280.926i	− 1.56942i	−0.619864	−	0.784709i	$-0.712812\pi$
$179$	− 280.926i	− 1.56942i	0.619864	−	0.784709i	$-0.287188\pi$
$180$	0	0
$181$	295.508	1.63264	0.816320	−	0.577600i	$-0.196011\pi$
$181$	295.508	1.63264	0.816320	+	0.577600i	$0.196011\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−8.64907	−0.0467518
$186$	0	0
$187$	− 320.679i	− 1.71486i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	295.982i	1.54965i	0.632179	+	0.774823i	$0.282161\pi$
$191$	295.982i	1.54965i	−0.632179	+	0.774823i	$0.717839\pi$
$192$	0	0
$193$	−18.0202	−0.0933688	−0.0466844	−	0.998910i	$-0.514866\pi$
$193$	−18.0202	−0.0933688	−0.0466844	+	0.998910i	$0.514866\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	119.020	0.604162	0.302081	−	0.953282i	$-0.402319\pi$
$197$	119.020	0.604162	0.302081	+	0.953282i	$0.402319\pi$
$198$	0	0
$199$	304.912i	1.53222i	0.642708	+	0.766111i	$0.277811\pi$
$199$	304.912i	1.53222i	−0.642708	+	0.766111i	$0.722189\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4.94302i	0.0243499i
$204$	0	0
$205$	−110.107	−0.537108
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−261.729	−1.25229
$210$	0	0
$211$	− 282.143i	− 1.33717i	−0.743635	−	0.668585i	$-0.766900\pi$
$211$	− 282.143i	− 1.33717i	0.743635	−	0.668585i	$-0.233100\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 67.4459i	− 0.313702i
$216$	0	0
$217$	305.235	1.40661
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−440.640	−1.99384
$222$	0	0
$223$	91.9070i	0.412139i	0.978537	+	0.206070i	$0.0660673\pi$
$223$	91.9070i	0.412139i	−0.978537	+	0.206070i	$0.933933\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	31.5125i	0.138822i	0.997588	+	0.0694108i	$0.0221119\pi$
$227$	31.5125i	0.138822i	−0.997588	+	0.0694108i	$0.977888\pi$
$228$	0	0
$229$	−242.515	−1.05902	−0.529509	−	0.848304i	$-0.677624\pi$
$229$	−242.515	−1.05902	−0.529509	+	0.848304i	$0.677624\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−402.297	−1.72660	−0.863298	−	0.504695i	$-0.831605\pi$
$233$	−402.297	−1.72660	−0.863298	+	0.504695i	$0.831605\pi$
$234$	0	0
$235$	49.6947i	0.211467i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 167.034i	− 0.698886i	−0.936958	−	0.349443i	$-0.886371\pi$
$239$	− 167.034i	− 0.698886i	0.936958	−	0.349443i	$-0.113629\pi$
$240$	0	0
$241$	−21.4458	−0.0889868	−0.0444934	−	0.999010i	$-0.514167\pi$
$241$	−21.4458	−0.0889868	−0.0444934	+	0.999010i	$0.514167\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−18.7253	−0.0764299
$246$	0	0
$247$	359.638i	1.45602i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	22.4825i	0.0895718i	0.998997	+	0.0447859i	$0.0142606\pi$
$251$	22.4825i	0.0895718i	−0.998997	+	0.0447859i	$0.985739\pi$
$252$	0	0
$253$	−188.834	−0.746380
$254$	0	0
$255$	0	0
$256$	0	0
$257$	437.652	1.70293	0.851463	−	0.524415i	$-0.175716\pi$
$257$	437.652	1.70293	0.851463	+	0.524415i	$0.175716\pi$
$258$	0	0
$259$	17.6252i	0.0680511i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 266.194i	− 1.01214i	−0.862491	−	0.506072i	$-0.831097\pi$
$263$	− 266.194i	− 1.01214i	0.862491	−	0.506072i	$-0.168903\pi$
$264$	0	0
$265$	150.997	0.569799
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−220.671	−0.820338	−0.410169	−	0.912010i	$-0.634530\pi$
$269$	−220.671	−0.820338	−0.410169	+	0.912010i	$0.634530\pi$
$270$	0	0
$271$	224.295i	0.827656i	0.910355	+	0.413828i	$0.135808\pi$
$271$	224.295i	0.827656i	−0.910355	+	0.413828i	$0.864192\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 202.779i	− 0.737380i
$276$	0	0
$277$	51.4458	0.185725	0.0928625	−	0.995679i	$-0.470398\pi$
$277$	51.4458	0.185725	0.0928625	+	0.995679i	$0.470398\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−40.1606	−0.142920	−0.0714602	−	0.997443i	$-0.522766\pi$
$281$	−40.1606	−0.142920	−0.0714602	+	0.997443i	$0.522766\pi$
$282$	0	0
$283$	− 190.774i	− 0.674112i	−0.941485	−	0.337056i	$-0.890569\pi$
$283$	− 190.774i	− 0.674112i	0.941485	−	0.337056i	$-0.109431\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	224.378i	0.781806i
$288$	0	0
$289$	243.986	0.844241
$290$	0	0
$291$	0	0
$292$	0	0
$293$	115.872	0.395467	0.197733	−	0.980256i	$-0.436642\pi$
$293$	115.872	0.395467	0.197733	+	0.980256i	$0.436642\pi$
$294$	0	0
$295$	− 130.335i	− 0.441815i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	259.474i	0.867806i
$300$	0	0
$301$	−137.442	−0.456619
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−340.389	−1.11603
$306$	0	0
$307$	497.554i	1.62070i	0.585949	+	0.810348i	$0.300722\pi$
$307$	497.554i	1.62070i	−0.585949	+	0.810348i	$0.699278\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 358.447i	− 1.15256i	−0.817252	−	0.576281i	$-0.804503\pi$
$311$	− 358.447i	− 1.15256i	0.817252	−	0.576281i	$-0.195497\pi$
$312$	0	0
$313$	32.9519	0.105278	0.0526388	−	0.998614i	$-0.483237\pi$
$313$	32.9519	0.105278	0.0526388	+	0.998614i	$0.483237\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	216.101	0.681707	0.340853	−	0.940117i	$-0.389284\pi$
$317$	216.101	0.681707	0.340853	+	0.940117i	$0.389284\pi$
$318$	0	0
$319$	10.4471i	0.0327494i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 435.008i	− 1.34677i
$324$	0	0
$325$	−278.636	−0.857342
$326$	0	0
$327$	0	0
$328$	0	0
$329$	101.269	0.307808
$330$	0	0
$331$	− 578.213i	− 1.74687i	−0.486944	−	0.873433i	$-0.661888\pi$
$331$	− 578.213i	− 1.74687i	0.486944	−	0.873433i	$-0.338112\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 90.0999i	− 0.268955i
$336$	0	0
$337$	358.117	1.06266	0.531331	−	0.847164i	$-0.321692\pi$
$337$	358.117	1.06266	0.531331	+	0.847164i	$0.321692\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	645.113	1.89183
$342$	0	0
$343$	360.197i	1.05014i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	483.855i	1.39440i	0.716879	+	0.697198i	$0.245570\pi$
$347$	483.855i	1.39440i	−0.716879	+	0.697198i	$0.754430\pi$
$348$	0	0
$349$	−22.1213	−0.0633849	−0.0316924	−	0.999498i	$-0.510090\pi$
$349$	−22.1213	−0.0633849	−0.0316924	+	0.999498i	$0.510090\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	459.068	1.30048	0.650238	−	0.759731i	$-0.274669\pi$
$353$	459.068	1.30048	0.650238	+	0.759731i	$0.274669\pi$
$354$	0	0
$355$	− 77.4449i	− 0.218155i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	414.323i	1.15410i	0.816708	+	0.577052i	$0.195797\pi$
$359$	414.323i	1.15410i	−0.816708	+	0.577052i	$0.804203\pi$
$360$	0	0
$361$	5.95856	0.0165057
$362$	0	0
$363$	0	0
$364$	0	0
$365$	387.393	1.06135
$366$	0	0
$367$	− 90.5390i	− 0.246700i	−0.992363	−	0.123350i	$-0.960636\pi$
$367$	− 90.5390i	− 0.246700i	0.992363	−	0.123350i	$-0.0393638\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 307.704i	− 0.829390i
$372$	0	0
$373$	−223.011	−0.597884	−0.298942	−	0.954271i	$-0.596634\pi$
$373$	−223.011	−0.597884	−0.298942	+	0.954271i	$0.596634\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	14.3551	0.0380773
$378$	0	0
$379$	− 118.162i	− 0.311774i	−0.987775	−	0.155887i	$-0.950176\pi$
$379$	− 118.162i	− 0.311774i	0.987775	−	0.155887i	$-0.0498236\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 19.6418i	− 0.0512840i	−0.999671	−	0.0256420i	$-0.991837\pi$
$383$	− 19.6418i	− 0.0512840i	0.999671	−	0.0256420i	$-0.00816300\pi$
$384$	0	0
$385$	294.422	0.764731
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−352.392	−0.905892	−0.452946	−	0.891538i	$-0.649627\pi$
$389$	−352.392	−0.905892	−0.452946	+	0.891538i	$0.649627\pi$
$390$	0	0
$391$	− 313.853i	− 0.802692i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	308.448i	0.780882i
$396$	0	0
$397$	5.66047	0.0142581	0.00712906	−	0.999975i	$-0.497731\pi$
$397$	5.66047	0.0142581	0.00712906	+	0.999975i	$0.497731\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	250.097	0.623683	0.311842	−	0.950134i	$-0.399054\pi$
$401$	250.097	0.623683	0.311842	+	0.950134i	$0.399054\pi$
$402$	0	0
$403$	− 886.439i	− 2.19960i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	37.2509i	0.0915255i
$408$	0	0
$409$	−643.885	−1.57429	−0.787145	−	0.616767i	$-0.788442\pi$
$409$	−643.885	−1.57429	−0.787145	+	0.616767i	$0.788442\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−265.600	−0.643099
$414$	0	0
$415$	372.327i	0.897173i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 439.563i	− 1.04908i	−0.851387	−	0.524539i	$-0.824238\pi$
$419$	− 439.563i	− 1.04908i	0.851387	−	0.524539i	$-0.175762\pi$
$420$	0	0
$421$	−65.4392	−0.155438	−0.0777188	−	0.996975i	$-0.524764\pi$
$421$	−65.4392	−0.155438	−0.0777188	+	0.996975i	$0.524764\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	337.030	0.793013
$426$	0	0
$427$	693.651i	1.62448i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	299.824i	0.695646i	0.937560	+	0.347823i	$0.113079\pi$
$431$	299.824i	0.695646i	−0.937560	+	0.347823i	$0.886921\pi$
$432$	0	0
$433$	−699.217	−1.61482	−0.807410	−	0.589991i	$-0.799131\pi$
$433$	−699.217	−1.61482	−0.807410	+	0.589991i	$0.799131\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−256.158	−0.586174
$438$	0	0
$439$	− 744.832i	− 1.69666i	−0.529472	−	0.848328i	$-0.677610\pi$
$439$	− 744.832i	− 1.69666i	0.529472	−	0.848328i	$-0.322390\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	320.933i	0.724455i	0.932090	+	0.362227i	$0.117984\pi$
$443$	320.933i	0.724455i	−0.932090	+	0.362227i	$0.882016\pi$
$444$	0	0
$445$	546.944	1.22909
$446$	0	0
$447$	0	0
$448$	0	0
$449$	439.651	0.979179	0.489589	−	0.871953i	$-0.337147\pi$
$449$	439.651	0.979179	0.489589	+	0.871953i	$0.337147\pi$
$450$	0	0
$451$	474.222i	1.05149i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 404.560i	− 0.889143i
$456$	0	0
$457$	−732.044	−1.60185	−0.800923	−	0.598767i	$-0.795658\pi$
$457$	−732.044	−1.60185	−0.800923	+	0.598767i	$0.795658\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	381.075	0.826627	0.413314	−	0.910589i	$-0.364371\pi$
$461$	381.075	0.826627	0.413314	+	0.910589i	$0.364371\pi$
$462$	0	0
$463$	591.775i	1.27813i	0.769152	+	0.639066i	$0.220679\pi$
$463$	591.775i	1.27813i	−0.769152	+	0.639066i	$0.779321\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	50.8002i	0.108780i	0.998520	+	0.0543899i	$0.0173214\pi$
$467$	50.8002i	0.108780i	−0.998520	+	0.0543899i	$0.982679\pi$
$468$	0	0
$469$	−183.607	−0.391487
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−290.484	−0.614131
$474$	0	0
$475$	− 275.075i	− 0.579105i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	844.826i	1.76373i	0.471503	+	0.881865i	$0.343712\pi$
$479$	844.826i	1.76373i	−0.471503	+	0.881865i	$0.656288\pi$
$480$	0	0
$481$	51.1858	0.106415
$482$	0	0
$483$	0	0
$484$	0	0
$485$	300.304	0.619184
$486$	0	0
$487$	− 328.276i	− 0.674077i	−0.941491	−	0.337039i	$-0.890575\pi$
$487$	− 328.276i	− 0.674077i	0.941491	−	0.337039i	$-0.109425\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	558.319i	1.13711i	0.822647	+	0.568553i	$0.192496\pi$
$491$	558.319i	1.13711i	−0.822647	+	0.568553i	$0.807504\pi$
$492$	0	0
$493$	−17.3636	−0.0352202
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−157.819	−0.317543
$498$	0	0
$499$	507.059i	1.01615i	0.861313	+	0.508075i	$0.169643\pi$
$499$	507.059i	1.01615i	−0.861313	+	0.508075i	$0.830357\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 884.817i	− 1.75908i	−0.475825	−	0.879540i	$-0.657850\pi$
$503$	− 884.817i	− 1.75908i	0.475825	−	0.879540i	$-0.342150\pi$
$504$	0	0
$505$	513.083	1.01601
$506$	0	0
$507$	0	0
$508$	0	0
$509$	398.001	0.781928	0.390964	−	0.920406i	$-0.372142\pi$
$509$	398.001	0.781928	0.390964	+	0.920406i	$0.372142\pi$
$510$	0	0
$511$	− 789.437i	− 1.54489i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 243.069i	− 0.471978i
$516$	0	0
$517$	214.031	0.413987
$518$	0	0
$519$	0	0
$520$	0	0
$521$	716.672	1.37557	0.687785	−	0.725914i	$-0.258583\pi$
$521$	716.672	1.37557	0.687785	+	0.725914i	$0.258583\pi$
$522$	0	0
$523$	706.115i	1.35012i	0.737761	+	0.675062i	$0.235883\pi$
$523$	706.115i	1.35012i	−0.737761	+	0.675062i	$0.764117\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1072.21i	2.03456i
$528$	0	0
$529$	344.185	0.650634
$530$	0	0
$531$	0	0
$532$	0	0
$533$	651.621	1.22255
$534$	0	0
$535$	223.913i	0.418529i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	80.6485i	0.149626i
$540$	0	0
$541$	116.644	0.215609	0.107804	−	0.994172i	$-0.465618\pi$
$541$	116.644	0.215609	0.107804	+	0.994172i	$0.465618\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	195.013	0.357822
$546$	0	0
$547$	422.278i	0.771989i	0.922501	+	0.385995i	$0.126142\pi$
$547$	422.278i	0.771989i	−0.922501	+	0.385995i	$0.873858\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	14.1717i	0.0257199i
$552$	0	0
$553$	628.562	1.13664
$554$	0	0
$555$	0	0
$556$	0	0
$557$	686.288	1.23212	0.616058	−	0.787701i	$-0.288729\pi$
$557$	686.288	1.23212	0.616058	+	0.787701i	$0.288729\pi$
$558$	0	0
$559$	399.149i	0.714042i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 439.191i	− 0.780090i	−0.920796	−	0.390045i	$-0.872459\pi$
$563$	− 439.191i	− 0.780090i	0.920796	−	0.390045i	$-0.127541\pi$
$564$	0	0
$565$	453.769	0.803130
$566$	0	0
$567$	0	0
$568$	0	0
$569$	406.032	0.713589	0.356794	−	0.934183i	$-0.383870\pi$
$569$	406.032	0.713589	0.356794	+	0.934183i	$0.383870\pi$
$570$	0	0
$571$	− 166.816i	− 0.292148i	−0.989274	−	0.146074i	$-0.953336\pi$
$571$	− 166.816i	− 0.292148i	0.989274	−	0.146074i	$-0.0466637\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 198.463i	− 0.345153i
$576$	0	0
$577$	−897.895	−1.55614	−0.778072	−	0.628175i	$-0.783802\pi$
$577$	−897.895	−1.55614	−0.778072	+	0.628175i	$0.783802\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	758.734	1.30591
$582$	0	0
$583$	− 650.330i	− 1.11549i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 152.207i	− 0.259296i	−0.991560	−	0.129648i	$-0.958615\pi$
$587$	− 152.207i	− 0.259296i	0.991560	−	0.129648i	$-0.0413847\pi$
$588$	0	0
$589$	875.110	1.48576
$590$	0	0
$591$	0	0
$592$	0	0
$593$	964.580	1.62661	0.813305	−	0.581838i	$-0.197666\pi$
$593$	964.580	1.62661	0.813305	+	0.581838i	$0.197666\pi$
$594$	0	0
$595$	489.345i	0.822428i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 983.578i	− 1.64203i	−0.570905	−	0.821016i	$-0.693407\pi$
$599$	− 983.578i	− 1.64203i	0.570905	−	0.821016i	$-0.306593\pi$
$600$	0	0
$601$	−443.392	−0.737757	−0.368878	−	0.929478i	$-0.620258\pi$
$601$	−443.392	−0.737757	−0.368878	+	0.929478i	$0.620258\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	232.019	0.383503
$606$	0	0
$607$	− 487.520i	− 0.803163i	−0.915823	−	0.401582i	$-0.868461\pi$
$607$	− 487.520i	− 0.803163i	0.915823	−	0.401582i	$-0.131539\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 294.097i	− 0.481337i
$612$	0	0
$613$	909.815	1.48420	0.742100	−	0.670289i	$-0.233830\pi$
$613$	909.815	1.48420	0.742100	+	0.670289i	$0.233830\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	481.557	0.780481	0.390241	−	0.920713i	$-0.372392\pi$
$617$	481.557	0.780481	0.390241	+	0.920713i	$0.372392\pi$
$618$	0	0
$619$	350.103i	0.565594i	0.959180	+	0.282797i	$0.0912623\pi$
$619$	350.103i	0.565594i	−0.959180	+	0.282797i	$0.908738\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 1114.57i	− 1.78904i
$624$	0	0
$625$	−46.9156	−0.0750649
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−61.9130	−0.0984308
$630$	0	0
$631$	274.658i	0.435274i	0.976030	+	0.217637i	$0.0698350\pi$
$631$	274.658i	0.435274i	−0.976030	+	0.217637i	$0.930165\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 215.995i	− 0.340149i
$636$	0	0
$637$	110.818	0.173968
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−383.085	−0.597636	−0.298818	−	0.954310i	$-0.596592\pi$
$641$	−383.085	−0.597636	−0.298818	+	0.954310i	$0.596592\pi$
$642$	0	0
$643$	177.134i	0.275480i	0.990468	+	0.137740i	$0.0439838\pi$
$643$	177.134i	0.275480i	−0.990468	+	0.137740i	$0.956016\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 877.305i	− 1.35596i	−0.735081	−	0.677979i	$-0.762856\pi$
$647$	− 877.305i	− 1.35596i	0.735081	−	0.677979i	$-0.237144\pi$
$648$	0	0
$649$	−561.344	−0.864937
$650$	0	0
$651$	0	0
$652$	0	0
$653$	35.3934	0.0542012	0.0271006	−	0.999633i	$-0.491373\pi$
$653$	35.3934	0.0542012	0.0271006	+	0.999633i	$0.491373\pi$
$654$	0	0
$655$	− 822.884i	− 1.25631i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	556.337i	0.844214i	0.906546	+	0.422107i	$0.138709\pi$
$659$	556.337i	0.844214i	−0.906546	+	0.422107i	$0.861291\pi$
$660$	0	0
$661$	−318.875	−0.482413	−0.241206	−	0.970474i	$-0.577543\pi$
$661$	−318.875	−0.482413	−0.241206	+	0.970474i	$0.577543\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	399.389	0.600586
$666$	0	0
$667$	10.2247i	0.0153293i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1466.03i	2.18484i
$672$	0	0
$673$	135.295	0.201033	0.100517	−	0.994935i	$-0.467950\pi$
$673$	135.295	0.201033	0.100517	+	0.994935i	$0.467950\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−123.989	−0.183145	−0.0915723	−	0.995798i	$-0.529189\pi$
$677$	−123.989	−0.183145	−0.0915723	+	0.995798i	$0.529189\pi$
$678$	0	0
$679$	− 611.966i	− 0.901275i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	609.347i	0.892163i	0.894992	+	0.446081i	$0.147181\pi$
$683$	609.347i	0.892163i	−0.894992	+	0.446081i	$0.852819\pi$
$684$	0	0
$685$	−87.7059	−0.128038
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−893.609	−1.29696
$690$	0	0
$691$	− 353.682i	− 0.511840i	−0.966698	−	0.255920i	$-0.917622\pi$
$691$	− 353.682i	− 0.511840i	0.966698	−	0.255920i	$-0.0823784\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 657.784i	− 0.946452i
$696$	0	0
$697$	−788.183	−1.13082
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−85.2186	−0.121567	−0.0607836	−	0.998151i	$-0.519360\pi$
$701$	−85.2186	−0.121567	−0.0607836	+	0.998151i	$0.519360\pi$
$702$	0	0
$703$	50.5316i	0.0718800i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1045.57i	− 1.47888i
$708$	0	0
$709$	701.943	0.990046	0.495023	−	0.868880i	$-0.335160\pi$
$709$	701.943	0.990046	0.495023	+	0.868880i	$0.335160\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	631.381	0.885527
$714$	0	0
$715$	− 855.036i	− 1.19585i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	993.471i	1.38174i	0.722979	+	0.690870i	$0.242772\pi$
$719$	993.471i	1.38174i	−0.722979	+	0.690870i	$0.757228\pi$
$720$	0	0
$721$	−495.330	−0.687004
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−10.9798	−0.0151445
$726$	0	0
$727$	485.145i	0.667325i	0.942693	+	0.333662i	$0.108285\pi$
$727$	485.145i	0.667325i	−0.942693	+	0.333662i	$0.891715\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 482.800i	− 0.660465i
$732$	0	0
$733$	−706.167	−0.963394	−0.481697	−	0.876338i	$-0.659979\pi$
$733$	−706.167	−0.963394	−0.481697	+	0.876338i	$0.659979\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−388.053	−0.526531
$738$	0	0
$739$	− 414.250i	− 0.560554i	−0.959919	−	0.280277i	$-0.909574\pi$
$739$	− 414.250i	− 0.560554i	0.959919	−	0.280277i	$-0.0904264\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1386.38i	1.86592i	0.359978	+	0.932961i	$0.382784\pi$
$743$	1386.38i	1.86592i	−0.359978	+	0.932961i	$0.617216\pi$
$744$	0	0
$745$	811.141	1.08878
$746$	0	0
$747$	0	0
$748$	0	0
$749$	456.294	0.609205
$750$	0	0
$751$	− 601.365i	− 0.800752i	−0.916351	−	0.400376i	$-0.868879\pi$
$751$	− 601.365i	− 0.800752i	0.916351	−	0.400376i	$-0.131121\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	807.089i	1.06899i
$756$	0	0
$757$	254.294	0.335924	0.167962	−	0.985794i	$-0.446281\pi$
$757$	254.294	0.335924	0.167962	+	0.985794i	$0.446281\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	466.017	0.612375	0.306187	−	0.951971i	$-0.400947\pi$
$761$	466.017	0.612375	0.306187	+	0.951971i	$0.400947\pi$
$762$	0	0
$763$	− 397.401i	− 0.520840i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	771.334i	1.00565i
$768$	0	0
$769$	−365.822	−0.475712	−0.237856	−	0.971300i	$-0.576445\pi$
$769$	−365.822	−0.475712	−0.237856	+	0.971300i	$0.576445\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1426.70	−1.84566	−0.922830	−	0.385207i	$-0.874130\pi$
$773$	−1426.70	−1.84566	−0.922830	+	0.385207i	$0.874130\pi$
$774$	0	0
$775$	678.007i	0.874848i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	643.293i	0.825794i
$780$	0	0
$781$	−333.549	−0.427079
$782$	0	0
$783$	0	0
$784$	0	0
$785$	58.6100	0.0746624
$786$	0	0
$787$	410.337i	0.521394i	0.965421	+	0.260697i	$0.0839523\pi$
$787$	410.337i	0.521394i	−0.965421	+	0.260697i	$0.916048\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 924.698i	− 1.16902i
$792$	0	0
$793$	2014.45	2.54029
$794$	0	0
$795$	0	0
$796$	0	0
$797$	738.968	0.927187	0.463594	−	0.886048i	$-0.346560\pi$
$797$	738.968	0.927187	0.463594	+	0.886048i	$0.346560\pi$
$798$	0	0
$799$	355.731i	0.445221i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 1668.47i	− 2.07780i
$804$	0	0
$805$	288.154	0.357956
$806$	0	0
$807$	0	0
$808$	0	0
$809$	130.444	0.161241	0.0806204	−	0.996745i	$-0.474310\pi$
$809$	130.444	0.161241	0.0806204	+	0.996745i	$0.474310\pi$
$810$	0	0
$811$	1558.44i	1.92163i	0.277187	+	0.960816i	$0.410598\pi$
$811$	1558.44i	1.92163i	−0.277187	+	0.960816i	$0.589402\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	161.314i	0.197931i
$816$	0	0
$817$	−394.048	−0.482311
$818$	0	0
$819$	0	0
$820$	0	0
$821$	55.0345	0.0670335	0.0335168	−	0.999438i	$-0.489329\pi$
$821$	55.0345	0.0670335	0.0335168	+	0.999438i	$0.489329\pi$
$822$	0	0
$823$	293.058i	0.356085i	0.984023	+	0.178043i	$0.0569765\pi$
$823$	293.058i	0.356085i	−0.984023	+	0.178043i	$0.943023\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1118.55i	1.35253i	0.736657	+	0.676267i	$0.236403\pi$
$827$	1118.55i	1.35253i	−0.736657	+	0.676267i	$0.763597\pi$
$828$	0	0
$829$	−1069.30	−1.28986	−0.644932	−	0.764240i	$-0.723115\pi$
$829$	−1069.30	−1.28986	−0.644932	+	0.764240i	$0.723115\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−134.042	−0.160915
$834$	0	0
$835$	273.616i	0.327683i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	41.6050i	0.0495888i	0.999693	+	0.0247944i	$0.00789311\pi$
$839$	41.6050i	0.0495888i	−0.999693	+	0.0247944i	$0.992107\pi$
$840$	0	0
$841$	−840.434	−0.999327
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−629.846	−0.745379
$846$	0	0
$847$	− 472.813i	− 0.558220i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	36.4579i	0.0428413i
$852$	0	0
$853$	−1302.63	−1.52711	−0.763555	−	0.645743i	$-0.776548\pi$
$853$	−1302.63	−1.52711	−0.763555	+	0.645743i	$0.776548\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−734.683	−0.857273	−0.428636	−	0.903477i	$-0.641006\pi$
$857$	−734.683	−0.857273	−0.428636	+	0.903477i	$0.641006\pi$
$858$	0	0
$859$	445.296i	0.518388i	0.965825	+	0.259194i	$0.0834570\pi$
$859$	445.296i	0.518388i	−0.965825	+	0.259194i	$0.916543\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1338.23i	− 1.55067i	−0.631550	−	0.775335i	$-0.717581\pi$
$863$	− 1338.23i	− 1.55067i	0.631550	−	0.775335i	$-0.282419\pi$
$864$	0	0
$865$	466.684	0.539519
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1328.46	1.52873
$870$	0	0
$871$	533.218i	0.612190i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	839.338i	0.959244i
$876$	0	0
$877$	329.803	0.376058	0.188029	−	0.982163i	$-0.439790\pi$
$877$	329.803	0.376058	0.188029	+	0.982163i	$0.439790\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1173.32	1.33181	0.665903	−	0.746038i	$-0.268046\pi$
$881$	1173.32	1.33181	0.665903	+	0.746038i	$0.268046\pi$
$882$	0	0
$883$	− 410.459i	− 0.464846i	−0.972615	−	0.232423i	$-0.925335\pi$
$883$	− 410.459i	− 0.464846i	0.972615	−	0.232423i	$-0.0746654\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 604.280i	− 0.681263i	−0.940197	−	0.340631i	$-0.889359\pi$
$887$	− 604.280i	− 0.681263i	0.940197	−	0.340631i	$-0.110641\pi$
$888$	0	0
$889$	−440.158	−0.495116
$890$	0	0
$891$	0	0
$892$	0	0
$893$	290.338	0.325127
$894$	0	0
$895$	906.020i	1.01231i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 34.9305i	− 0.0388548i
$900$	0	0
$901$	1080.88	1.19965
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−953.048	−1.05309
$906$	0	0
$907$	− 735.914i	− 0.811371i	−0.914013	−	0.405686i	$-0.867033\pi$
$907$	− 735.914i	− 0.811371i	0.914013	−	0.405686i	$-0.132967\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	586.390i	0.643677i	0.946795	+	0.321838i	$0.104301\pi$
$911$	586.390i	0.643677i	−0.946795	+	0.321838i	$0.895699\pi$
$912$	0	0
$913$	1603.58	1.75639
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1676.89	−1.82867
$918$	0	0
$919$	− 158.999i	− 0.173013i	−0.996251	−	0.0865063i	$-0.972430\pi$
$919$	− 158.999i	− 0.173013i	0.996251	−	0.0865063i	$-0.0275703\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	458.324i	0.496560i
$924$	0	0
$925$	−39.1503	−0.0423247
$926$	0	0
$927$	0	0
$928$	0	0
$929$	540.670	0.581992	0.290996	−	0.956724i	$-0.406013\pi$
$929$	540.670	0.581992	0.290996	+	0.956724i	$0.406013\pi$
$930$	0	0
$931$	109.402i	0.117510i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1034.23i	1.10613i
$936$	0	0
$937$	384.947	0.410829	0.205415	−	0.978675i	$-0.434146\pi$
$937$	384.947	0.410829	0.205415	+	0.978675i	$0.434146\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	39.6287	0.0421134	0.0210567	−	0.999778i	$-0.493297\pi$
$941$	39.6287	0.0421134	0.0210567	+	0.999778i	$0.493297\pi$
$942$	0	0
$943$	464.128i	0.492182i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 449.648i	− 0.474813i	−0.971410	−	0.237407i	$-0.923703\pi$
$947$	− 449.648i	− 0.474813i	0.971410	−	0.237407i	$-0.0762974\pi$
$948$	0	0
$949$	−2292.62	−2.41583
$950$	0	0
$951$	0	0
$952$	0	0
$953$	21.5070	0.0225676	0.0112838	−	0.999936i	$-0.496408\pi$
$953$	21.5070	0.0225676	0.0112838	+	0.999936i	$0.496408\pi$
$954$	0	0
$955$	− 954.578i	− 0.999558i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	178.729i	0.186370i
$960$	0	0
$961$	−1195.98	−1.24452
$962$	0	0
$963$	0	0
$964$	0	0
$965$	58.1172	0.0602251
$966$	0	0
$967$	1355.49i	1.40175i	0.713283	+	0.700876i	$0.247207\pi$
$967$	1355.49i	1.40175i	−0.713283	+	0.700876i	$0.752793\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	938.488i	0.966517i	0.875478	+	0.483258i	$0.160547\pi$
$971$	938.488i	0.966517i	−0.875478	+	0.483258i	$0.839453\pi$
$972$	0	0
$973$	−1340.44	−1.37764
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1090.81	−1.11649	−0.558243	−	0.829678i	$-0.688524\pi$
$977$	−1090.81	−1.11649	−0.558243	+	0.829678i	$0.688524\pi$
$978$	0	0
$979$	− 2355.64i	− 2.40617i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	27.7492i	0.0282291i	0.999900	+	0.0141146i	$0.00449296\pi$
$983$	27.7492i	0.0282291i	−0.999900	+	0.0141146i	$0.995507\pi$
$984$	0	0
$985$	−383.853	−0.389699
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−284.301	−0.287463
$990$	0	0
$991$	272.858i	0.275336i	0.990478	+	0.137668i	$0.0439607\pi$
$991$	272.858i	0.275336i	−0.990478	+	0.137668i	$0.956039\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 983.378i	− 0.988320i
$996$	0	0
$997$	−1190.35	−1.19393	−0.596964	−	0.802268i	$-0.703626\pi$
$997$	−1190.35	−1.19393	−0.596964	+	0.802268i	$0.703626\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.g.n.703.4		8
3.2	odd	2		1728.3.g.k.703.6		8
4.3	odd	2	inner	1728.3.g.n.703.3		8
8.3	odd	2		864.3.g.a.703.5	✓	8
8.5	even	2		864.3.g.a.703.6	yes	8
12.11	even	2		1728.3.g.k.703.5		8
24.5	odd	2		864.3.g.c.703.4	yes	8
24.11	even	2		864.3.g.c.703.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.3.g.a.703.5	✓	8	8.3	odd	2
864.3.g.a.703.6	yes	8	8.5	even	2
864.3.g.c.703.3	yes	8	24.11	even	2
864.3.g.c.703.4	yes	8	24.5	odd	2
1728.3.g.k.703.5		8	12.11	even	2
1728.3.g.k.703.6		8	3.2	odd	2
1728.3.g.n.703.3		8	4.3	odd	2	inner
1728.3.g.n.703.4		8	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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.22512 q^{5} +6.57221i q^{7} +O(q^{10})\)	\(q-3.22512 q^{5} +6.57221i q^{7} +13.8903i q^{11} +19.0865 q^{13} -23.0865 q^{17} +18.8425i q^{19} +13.5947i q^{23} -14.5986 q^{25} +0.752110 q^{29} -46.4433i q^{31} -21.1962i q^{35} +2.68178 q^{37} +34.1405 q^{41} +20.9127i q^{43} -15.4086i q^{47} +5.80609 q^{49} -46.8189 q^{53} -44.7980i q^{55} +40.4126i q^{59} +105.543 q^{61} -61.5562 q^{65} +27.9369i q^{67} +24.0130i q^{71} -120.117 q^{73} -91.2901 q^{77} -95.6394i q^{79} -115.446i q^{83} +74.4567 q^{85} -169.589 q^{89} +125.440i q^{91} -60.7694i q^{95} -93.1142 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{5}+O(q^{10}) \) 8 * q + 8 * q^5	\( 8 q + 8 q^{5} + 16 q^{13} - 48 q^{17} + 48 q^{25} - 32 q^{29} + 96 q^{37} + 128 q^{41} + 168 q^{53} - 32 q^{61} + 112 q^{65} + 24 q^{73} - 440 q^{77} - 144 q^{85} - 624 q^{89} - 136 q^{97}+O(q^{100}) \) 8 * q + 8 * q^5 + 16 * q^13 - 48 * q^17 + 48 * q^25 - 32 * q^29 + 96 * q^37 + 128 * q^41 + 168 * q^53 - 32 * q^61 + 112 * q^65 + 24 * q^73 - 440 * q^77 - 144 * q^85 - 624 * q^89 - 136 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more