LMFDB - Embedded newform 756.4.t.d.269.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [756,4,Mod(269,756)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("756.269"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(756, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	756.t (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,50] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$44.6054439643$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 214 x^{14} + 30952 x^{12} - 2415192 x^{10} + 136176800 x^{8} - 4497757024 x^{6} + \cdots + 7263930548224 $ x^16 - 214x^14 + 30952x^12 - 2415192x^10 + 136176800x^8 - 4497757024x^6 + 104934553152x^4 - 1026082799616*x^2 + 7263930548224
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{16} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			269.5
Root			$2.90703 + 1.67838i$ of defining polynomial
Character	$\chi$	$=$	756.269
Dual form			756.4.t.d.593.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.90703 + 5.03513i) q^{5} +(-16.4502 + 8.50837i) q^{7} +(-30.5015 - 17.6100i) q^{11} -5.07550i q^{13} +(-39.0221 + 67.5883i) q^{17} +(-47.6118 + 27.4887i) q^{19} +(164.568 - 95.0136i) q^{23} +(45.5983 - 78.9786i) q^{25} -189.333i q^{29} +(71.5699 + 41.3209i) q^{31} +(-90.6618 - 58.0945i) q^{35} +(-116.269 - 201.384i) q^{37} -489.773 q^{41} +361.194 q^{43} +(262.717 + 455.039i) q^{47} +(198.215 - 279.928i) q^{49} +(278.692 + 160.903i) q^{53} -204.772i q^{55} +(172.411 - 298.625i) q^{59} +(673.393 - 388.784i) q^{61} +(25.5558 - 14.7547i) q^{65} +(198.380 - 343.604i) q^{67} +871.421i q^{71} +(-32.1813 - 18.5799i) q^{73} +(651.587 + 30.1702i) q^{77} +(-106.270 - 184.065i) q^{79} -376.941 q^{83} -453.754 q^{85} +(-770.458 - 1334.47i) q^{89} +(43.1843 + 83.4929i) q^{91} +(-276.818 - 159.821i) q^{95} -1440.86i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 50 q^{7} - 390 q^{19} - 284 q^{25} + 984 q^{31} - 182 q^{37} + 472 q^{43} + 130 q^{49} - 24 q^{61} - 1522 q^{67} + 4458 q^{73} + 434 q^{79} - 2616 q^{85} - 4560 q^{91}+O(q^{100}) $ 16 * q + 50 * q^7 - 390 * q^19 - 284 * q^25 + 984 * q^31 - 182 * q^37 + 472 * q^43 + 130 * q^49 - 24 * q^61 - 1522 * q^67 + 4458 * q^73 + 434 * q^79 - 2616 * q^85 - 4560 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$.

$n$	$29$	$325$	$379$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$	$1$

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
$5$	2.90703	+	5.03513i	0.260013	+	0.450355i	0.966245	−	0.257625i	$-0.0829399\pi$
$5$	2.90703	+	5.03513i	0.260013	+	0.450355i	−0.706232	+	0.707980i	$0.749607\pi$
$6$		0			0
$7$	−16.4502	+	8.50837i	−0.888225	+	0.459409i
$7$	−16.4502	+	8.50837i	−0.888225	+	0.459409i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−30.5015	−	17.6100i	−0.836050	−	0.482694i	0.0198698	−	0.999803i	$-0.493675\pi$
$11$	−30.5015	−	17.6100i	−0.836050	−	0.482694i	−0.855920	+	0.517109i	$0.827008\pi$
$12$		0			0
$13$		−	5.07550i		−	0.108284i	−0.998533	−	0.0541420i	$-0.982758\pi$
$13$		−	5.07550i		−	0.108284i	0.998533	−	0.0541420i	$-0.0172424\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−39.0221	+	67.5883i	−0.556721	+	0.964269i	0.441046	+	0.897484i	$0.354607\pi$
$17$	−39.0221	+	67.5883i	−0.556721	+	0.964269i	−0.997767	+	0.0667848i	$0.978726\pi$
$18$		0			0
$19$	−47.6118	+	27.4887i	−0.574889	+	0.331912i	−0.759100	−	0.650975i	$-0.774360\pi$
$19$	−47.6118	+	27.4887i	−0.574889	+	0.331912i	0.184211	+	0.982887i	$0.441027\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	164.568	−	95.0136i	1.49195	−	0.861378i	0.491993	−	0.870599i	$-0.336268\pi$
$23$	164.568	−	95.0136i	1.49195	−	0.861378i	0.999957	+	0.00922090i	$0.00293515\pi$
$24$		0			0
$25$	45.5983	−	78.9786i	0.364787	−	0.631829i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	189.333i		−	1.21235i	−0.795331	−	0.606176i	$-0.792703\pi$
$29$		−	189.333i		−	1.21235i	0.795331	−	0.606176i	$-0.207297\pi$
$30$		0			0
$31$	71.5699	+	41.3209i	0.414656	+	0.239402i	0.692788	−	0.721141i	$-0.256382\pi$
$31$	71.5699	+	41.3209i	0.414656	+	0.239402i	−0.278132	+	0.960543i	$0.589715\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−90.6618	−	58.0945i	−0.437847	−	0.280565i
$36$		0			0
$37$	−116.269	−	201.384i	−0.516608	−	0.894792i	−0.999814	−	0.0192850i	$-0.993861\pi$
$37$	−116.269	−	201.384i	−0.516608	−	0.894792i	0.483206	−	0.875507i	$-0.339472\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−489.773			−1.86560			−0.932801	−	0.360392i	$-0.882643\pi$
$41$	−489.773			−1.86560			−0.932801	+	0.360392i	$0.882643\pi$
$42$		0			0
$43$	361.194			1.28097			0.640483	−	0.767972i	$-0.278734\pi$
$43$	361.194			1.28097			0.640483	+	0.767972i	$0.278734\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	262.717	+	455.039i	0.815344	+	1.41222i	0.909081	+	0.416620i	$0.136785\pi$
$47$	262.717	+	455.039i	0.815344	+	1.41222i	−0.0937367	+	0.995597i	$0.529881\pi$
$48$		0			0
$49$	198.215	−	279.928i	0.577888	−	0.816116i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	278.692	+	160.903i	0.722289	+	0.417014i	0.815594	−	0.578624i	$-0.196410\pi$
$53$	278.692	+	160.903i	0.722289	+	0.417014i	−0.0933058	+	0.995637i	$0.529743\pi$
$54$		0			0
$55$		−	204.772i		−	0.502026i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	172.411	−	298.625i	0.380441	−	0.658943i	−0.610684	−	0.791874i	$-0.709106\pi$
$59$	172.411	−	298.625i	0.380441	−	0.658943i	0.991125	+	0.132931i	$0.0424389\pi$
$60$		0			0
$61$	673.393	−	388.784i	1.41343	−	0.816043i	0.417719	−	0.908576i	$-0.362830\pi$
$61$	673.393	−	388.784i	1.41343	−	0.816043i	0.995710	+	0.0925333i	$0.0294965\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	25.5558	−	14.7547i	0.0487663	−	0.0281552i
$66$		0			0
$67$	198.380	−	343.604i	0.361731	−	0.626536i	−0.626515	−	0.779409i	$-0.715519\pi$
$67$	198.380	−	343.604i	0.361731	−	0.626536i	0.988246	+	0.152873i	$0.0488526\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			871.421i			1.45660i	0.685258	+	0.728300i	$0.259689\pi$
$71$			871.421i			1.45660i	−0.685258	+	0.728300i	$0.740311\pi$
$72$		0			0
$73$	−32.1813	−	18.5799i	−0.0515963	−	0.0297892i	0.473980	−	0.880536i	$-0.342817\pi$
$73$	−32.1813	−	18.5799i	−0.0515963	−	0.0297892i	−0.525576	+	0.850746i	$0.676150\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	651.587	+	30.1702i	0.964354	+	0.0446521i
$78$		0			0
$79$	−106.270	−	184.065i	−0.151346	−	0.262139i	0.780377	−	0.625310i	$-0.215027\pi$
$79$	−106.270	−	184.065i	−0.151346	−	0.262139i	−0.931723	+	0.363171i	$0.881694\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−376.941			−0.498490			−0.249245	−	0.968440i	$-0.580182\pi$
$83$	−376.941			−0.498490			−0.249245	+	0.968440i	$0.580182\pi$
$84$		0			0
$85$	−453.754			−0.579018
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−770.458	−	1334.47i	−0.917622	−	1.58937i	−0.803017	−	0.595956i	$-0.796773\pi$
$89$	−770.458	−	1334.47i	−0.917622	−	1.58937i	−0.114605	−	0.993411i	$-0.536560\pi$
$90$		0			0
$91$	43.1843	+	83.4929i	0.0497466	+	0.0961805i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−276.818	−	159.821i	−0.298957	−	0.172603i
$96$		0			0
$97$		−	1440.86i		−	1.50822i	−0.656750	−	0.754108i	$-0.728069\pi$
$97$		−	1440.86i		−	1.50822i	0.656750	−	0.754108i	$-0.271931\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−494.128	+	855.854i	−0.486807	+	0.843175i	−0.999885	−	0.0151671i	$-0.995172\pi$
$101$	−494.128	+	855.854i	−0.486807	+	0.843175i	0.513078	+	0.858342i	$0.328505\pi$
$102$		0			0
$103$	384.982	−	222.270i	0.368286	−	0.212630i	−0.304424	−	0.952537i	$-0.598464\pi$
$103$	384.982	−	222.270i	0.368286	−	0.212630i	0.672709	+	0.739907i	$0.265131\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	694.572	−	401.011i	0.627541	−	0.362311i	−0.152258	−	0.988341i	$-0.548655\pi$
$107$	694.572	−	401.011i	0.627541	−	0.362311i	0.779799	+	0.626030i	$0.215321\pi$
$108$		0			0
$109$	841.538	−	1457.59i	0.739492	−	1.28084i	−0.213232	−	0.977002i	$-0.568399\pi$
$109$	841.538	−	1457.59i	0.739492	−	1.28084i	0.952724	−	0.303837i	$-0.0982678\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			698.403i			0.581418i	0.956812	+	0.290709i	$0.0938912\pi$
$113$			698.403i			0.581418i	−0.956812	+	0.290709i	$0.906109\pi$
$114$		0			0
$115$	956.810	+	552.415i	0.775852	+	0.447939i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	66.8542	−	1443.85i	0.0515001	−	1.11225i
$120$		0			0
$121$	−45.2726	−	78.4144i	−0.0340140	−	0.0589139i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1256.98			0.899422
$126$		0			0
$127$	884.682			0.618133			0.309066	−	0.951041i	$-0.399984\pi$
$127$	884.682			0.618133			0.309066	+	0.951041i	$0.399984\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−368.903	−	638.959i	−0.246040	−	0.426154i	0.716384	−	0.697707i	$-0.245796\pi$
$131$	−368.903	−	638.959i	−0.246040	−	0.426154i	−0.962423	+	0.271553i	$0.912463\pi$
$132$		0			0
$133$	549.337	−	857.291i	0.358147	−	0.558922i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	254.851	+	147.138i	0.158930	+	0.0917580i	0.577355	−	0.816493i	$-0.304085\pi$
$137$	254.851	+	147.138i	0.158930	+	0.0917580i	−0.418426	+	0.908251i	$0.637418\pi$
$138$		0			0
$139$		−	1388.68i		−	0.847385i	−0.905806	−	0.423693i	$-0.860734\pi$
$139$		−	1388.68i		−	0.847385i	0.905806	−	0.423693i	$-0.139266\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−89.3799	+	154.810i	−0.0522680	+	0.0905308i
$144$		0			0
$145$	953.314	−	550.396i	0.545989	−	0.315227i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−430.804	+	248.725i	−0.236865	+	0.136754i	−0.613735	−	0.789512i	$-0.710334\pi$
$149$	−430.804	+	248.725i	−0.236865	+	0.136754i	0.376870	+	0.926266i	$0.377000\pi$
$150$		0			0
$151$	1375.74	−	2382.84i	0.741429	−	1.28419i	−0.210415	−	0.977612i	$-0.567482\pi$
$151$	1375.74	−	2382.84i	0.741429	−	1.28419i	0.951845	−	0.306581i	$-0.0991850\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			480.484i			0.248990i
$156$		0			0
$157$	−1169.64	−	675.292i	−0.594570	−	0.343275i	0.172332	−	0.985039i	$-0.444870\pi$
$157$	−1169.64	−	675.292i	−0.594570	−	0.343275i	−0.766902	+	0.641764i	$0.778203\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1898.76	+	2963.20i	−0.929464	+	1.45051i
$162$		0			0
$163$	−174.965	−	303.048i	−0.0840755	−	0.145623i	0.820921	−	0.571041i	$-0.193460\pi$
$163$	−174.965	−	303.048i	−0.0840755	−	0.145623i	−0.904997	+	0.425418i	$0.860127\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−679.288			−0.314760			−0.157380	−	0.987538i	$-0.550305\pi$
$167$	−679.288			−0.314760			−0.157380	+	0.987538i	$0.550305\pi$
$168$		0			0
$169$	2171.24			0.988275
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−571.274	−	989.476i	−0.251059	−	0.434847i	0.712759	−	0.701409i	$-0.247445\pi$
$173$	−571.274	−	989.476i	−0.251059	−	0.434847i	−0.963818	+	0.266563i	$0.914112\pi$
$174$		0			0
$175$	−78.1208	+	1687.18i	−0.0337450	+	0.728793i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−789.128	−	455.603i	−0.329510	−	0.190242i	0.326114	−	0.945331i	$-0.394261\pi$
$179$	−789.128	−	455.603i	−0.329510	−	0.190242i	−0.655623	+	0.755088i	$0.727594\pi$
$180$		0			0
$181$			470.211i			0.193097i	0.995328	+	0.0965483i	$0.0307802\pi$
$181$			470.211i			0.193097i	−0.995328	+	0.0965483i	$0.969220\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	675.995	−	1170.86i	0.268650	−	0.465315i
$186$		0			0
$187$	2380.47	−	1374.36i	0.930893	−	0.537451i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2486.11	−	1435.36i	0.941826	−	0.543764i	0.0512941	−	0.998684i	$-0.483665\pi$
$191$	2486.11	−	1435.36i	0.941826	−	0.543764i	0.890532	+	0.454920i	$0.150332\pi$
$192$		0			0
$193$	−294.386	+	509.892i	−0.109795	+	0.190170i	−0.915687	−	0.401892i	$-0.868353\pi$
$193$	−294.386	+	509.892i	−0.109795	+	0.190170i	0.805892	+	0.592062i	$0.201686\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	4508.44i		−	1.63052i	−0.579094	−	0.815261i	$-0.696594\pi$
$197$		−	4508.44i		−	1.63052i	0.579094	−	0.815261i	$-0.303406\pi$
$198$		0			0
$199$	807.278	+	466.082i	0.287570	+	0.166029i	0.636845	−	0.770992i	$-0.280239\pi$
$199$	807.278	+	466.082i	0.287570	+	0.166029i	−0.349276	+	0.937020i	$0.613572\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1610.91	+	3114.55i	0.556965	+	1.07684i
$204$		0			0
$205$	−1423.78	−	2466.07i	−0.485080	−	0.840184i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1936.31			0.640847
$210$		0			0
$211$	−3175.72			−1.03614			−0.518070	−	0.855338i	$-0.673349\pi$
$211$	−3175.72			−1.03614			−0.518070	+	0.855338i	$0.673349\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1050.00	+	1818.66i	0.333068	+	0.576890i
$216$		0			0
$217$	−1528.91	−	70.7925i	−0.478291	−	0.0221461i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	343.045	+	198.057i	0.104415	+	0.0602840i
$222$		0			0
$223$			1875.86i			0.563305i	0.959517	+	0.281652i	$0.0908825\pi$
$223$			1875.86i			0.563305i	−0.959517	+	0.281652i	$0.909118\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2164.36	+	3748.78i	−0.632835	+	1.09610i	0.354135	+	0.935194i	$0.384775\pi$
$227$	−2164.36	+	3748.78i	−0.632835	+	1.09610i	−0.986969	+	0.160908i	$0.948558\pi$
$228$		0			0
$229$	−4665.03	+	2693.36i	−1.34617	+	0.777214i	−0.987705	−	0.156327i	$-0.950035\pi$
$229$	−4665.03	+	2693.36i	−1.34617	+	0.777214i	−0.358470	+	0.933541i	$0.616701\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−550.824	+	318.019i	−0.154874	+	0.0894167i	−0.575434	−	0.817848i	$-0.695167\pi$
$233$	−550.824	+	318.019i	−0.154874	+	0.0894167i	0.420560	+	0.907265i	$0.361834\pi$
$234$		0			0
$235$	−1527.45	+	2645.62i	−0.424000	+	0.734389i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			381.980i			0.103382i	0.998663	+	0.0516909i	$0.0164611\pi$
$239$			381.980i			0.103382i	−0.998663	+	0.0516909i	$0.983539\pi$
$240$		0			0
$241$	3996.63	+	2307.45i	1.06824	+	0.616748i	0.927700	−	0.373327i	$-0.121783\pi$
$241$	3996.63	+	2307.45i	1.06824	+	0.616748i	0.140539	+	0.990075i	$0.455117\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1985.69	+	184.280i	0.517800	+	0.0480541i
$246$		0			0
$247$	139.519	+	241.654i	0.0359408	+	0.0622512i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1890.20			−0.475332			−0.237666	−	0.971347i	$-0.576382\pi$
$251$	−1890.20			−0.475332			−0.237666	+	0.971347i	$0.576382\pi$
$252$		0			0
$253$	−6692.77			−1.66313
$254$		0			0
$255$		0			0
$256$		0			0
$257$	806.150	+	1396.29i	0.195666	+	0.338904i	0.947119	−	0.320883i	$-0.103980\pi$
$257$	806.150	+	1396.29i	0.195666	+	0.338904i	−0.751452	+	0.659787i	$0.770646\pi$
$258$		0			0
$259$	3626.09	+	2323.54i	0.869939	+	0.557442i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6367.98	−	3676.55i	−1.49303	−	0.862000i	−0.493060	−	0.869995i	$-0.664122\pi$
$263$	−6367.98	−	3676.55i	−1.49303	−	0.862000i	−0.999968	+	0.00799526i	$0.997455\pi$
$264$		0			0
$265$			1871.00i			0.433715i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−710.981	+	1231.46i	−0.161150	+	0.279119i	−0.935281	−	0.353905i	$-0.884854\pi$
$269$	−710.981	+	1231.46i	−0.161150	+	0.279119i	0.774132	+	0.633025i	$0.218187\pi$
$270$		0			0
$271$	−7126.20	+	4114.31i	−1.59736	+	0.922239i	−0.605372	+	0.795943i	$0.706976\pi$
$271$	−7126.20	+	4114.31i	−1.59736	+	0.922239i	−0.991993	+	0.126296i	$0.959691\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2781.64	+	1605.98i	−0.609960	+	0.352160i
$276$		0			0
$277$	4026.78	−	6974.60i	0.873451	−	1.51286i	0.0150480	−	0.999887i	$-0.495210\pi$
$277$	4026.78	−	6974.60i	0.873451	−	1.51286i	0.858403	−	0.512975i	$-0.171457\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			2085.65i			0.442774i	0.975186	+	0.221387i	$0.0710585\pi$
$281$			2085.65i			0.442774i	−0.975186	+	0.221387i	$0.928942\pi$
$282$		0			0
$283$	3472.02	+	2004.57i	0.729294	+	0.421058i	0.818164	−	0.574985i	$-0.194992\pi$
$283$	3472.02	+	2004.57i	0.729294	+	0.421058i	−0.0888698	+	0.996043i	$0.528326\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	8056.84	−	4167.17i	1.65707	−	0.857073i
$288$		0			0
$289$	−588.954	−	1020.10i	−0.119877	−	0.207632i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1664.93			−0.331968			−0.165984	−	0.986128i	$-0.553080\pi$
$293$	−1664.93			−0.331968			−0.165984	+	0.986128i	$0.553080\pi$
$294$		0			0
$295$	2004.82			0.395678
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−482.242	−	835.267i	−0.0932735	−	0.161554i
$300$		0			0
$301$	−5941.70	+	3073.17i	−1.13779	+	0.588487i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	3915.15	+	2260.41i	0.735019	+	0.424363i
$306$		0			0
$307$			2518.51i			0.468205i	0.972212	+	0.234103i	$0.0752152\pi$
$307$			2518.51i			0.468205i	−0.972212	+	0.234103i	$0.924785\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1925.18	−	3334.50i	0.351018	−	0.607982i	−0.635410	−	0.772175i	$-0.719169\pi$
$311$	1925.18	−	3334.50i	0.351018	−	0.607982i	0.986428	+	0.164193i	$0.0525021\pi$
$312$		0			0
$313$	2707.89	−	1563.40i	0.489007	−	0.282328i	−0.235156	−	0.971958i	$-0.575560\pi$
$313$	2707.89	−	1563.40i	0.489007	−	0.282328i	0.724162	+	0.689630i	$0.242227\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	245.222	−	141.579i	0.0434480	−	0.0250847i	−0.478119	−	0.878295i	$-0.658681\pi$
$317$	245.222	−	141.579i	0.0434480	−	0.0250847i	0.521567	+	0.853211i	$0.325348\pi$
$318$		0			0
$319$	−3334.16	+	5774.93i	−0.585194	+	1.01359i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	4290.66i		−	0.739130i
$324$		0			0
$325$	−400.856	−	231.435i	−0.0684170	−	0.0395006i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−8193.37	−	5250.17i	−1.37299	−	0.879791i
$330$		0			0
$331$	5254.85	+	9101.66i	0.872606	+	1.51140i	0.859292	+	0.511486i	$0.170905\pi$
$331$	5254.85	+	9101.66i	0.872606	+	1.51140i	0.0133139	+	0.999911i	$0.495762\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2306.79			0.376219
$336$		0			0
$337$	−8431.97			−1.36296			−0.681482	−	0.731835i	$-0.738664\pi$
$337$	−8431.97			−1.36296			−0.681482	+	0.731835i	$0.738664\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1455.33	−	2520.70i	−0.231115	−	0.400303i
$342$		0			0
$343$	−878.946	+	6291.35i	−0.138363	+	0.990382i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	7880.32	+	4549.70i	1.21913	+	0.703864i	0.964732	−	0.263236i	$-0.0847897\pi$
$347$	7880.32	+	4549.70i	1.21913	+	0.703864i	0.254397	+	0.967100i	$0.418123\pi$
$348$		0			0
$349$		−	8534.69i		−	1.30903i	−0.756049	−	0.654515i	$-0.772873\pi$
$349$		−	8534.69i		−	1.30903i	0.756049	−	0.654515i	$-0.227127\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1599.23	−	2769.94i	0.241128	−	0.417646i	−0.719908	−	0.694070i	$-0.755816\pi$
$353$	1599.23	−	2769.94i	0.241128	−	0.417646i	0.961036	+	0.276423i	$0.0891492\pi$
$354$		0			0
$355$	−4387.71	+	2533.25i	−0.655988	+	0.378735i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−11218.7	+	6477.11i	−1.64930	+	0.952225i	−0.671952	+	0.740594i	$0.734544\pi$
$359$	−11218.7	+	6477.11i	−1.64930	+	0.952225i	−0.977350	+	0.211631i	$0.932123\pi$
$360$		0			0
$361$	−1918.25	+	3322.50i	−0.279669	+	0.484400i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	216.049i		−	0.0309822i
$366$		0			0
$367$	5053.20	+	2917.47i	0.718733	+	0.414961i	0.814286	−	0.580464i	$-0.197129\pi$
$367$	5053.20	+	2917.47i	0.718733	+	0.414961i	−0.0955532	+	0.995424i	$0.530462\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−5953.55	−	275.665i	−0.833134	−	0.0385763i
$372$		0			0
$373$	−2536.02	−	4392.52i	−0.352038	−	0.609748i	0.634568	−	0.772867i	$-0.281178\pi$
$373$	−2536.02	−	4392.52i	−0.352038	−	0.609748i	−0.986606	+	0.163119i	$0.947845\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−960.959			−0.131278
$378$		0			0
$379$	−5750.48			−0.779373			−0.389686	−	0.920948i	$-0.627417\pi$
$379$	−5750.48			−0.779373			−0.389686	+	0.920948i	$0.627417\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−2763.34	−	4786.24i	−0.368668	−	0.638552i	0.620689	−	0.784056i	$-0.286853\pi$
$383$	−2763.34	−	4786.24i	−0.368668	−	0.638552i	−0.989358	+	0.145505i	$0.953519\pi$
$384$		0			0
$385$	1742.27	+	3368.53i	0.230635	+	0.445912i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	6559.00	+	3786.84i	0.854895	+	0.493574i	0.862300	−	0.506399i	$-0.169024\pi$
$389$	6559.00	+	3786.84i	0.854895	+	0.493574i	−0.00740420	+	0.999973i	$0.502357\pi$
$390$		0			0
$391$			14830.5i			1.91819i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	617.862	−	1070.17i	0.0787038	−	0.136319i
$396$		0			0
$397$	−8272.16	+	4775.94i	−1.04576	+	0.603772i	−0.921460	−	0.388473i	$-0.873003\pi$
$397$	−8272.16	+	4775.94i	−1.04576	+	0.603772i	−0.124303	+	0.992244i	$0.539669\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	12867.4	−	7428.97i	1.60241	−	0.925150i	0.611402	−	0.791320i	$-0.290606\pi$
$401$	12867.4	−	7428.97i	1.60241	−	0.925150i	0.991004	−	0.133830i	$-0.0427276\pi$
$402$		0			0
$403$	209.724	−	363.253i	0.0259234	−	0.0449006i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			8190.01i			0.997454i
$408$		0			0
$409$	−5030.51	−	2904.36i	−0.608172	−	0.351129i	0.164077	−	0.986447i	$-0.447535\pi$
$409$	−5030.51	−	2904.36i	−0.608172	−	0.351129i	−0.772250	+	0.635319i	$0.780869\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−295.381	+	6379.36i	−0.0351931	+	0.760068i
$414$		0			0
$415$	−1095.78	−	1897.95i	−0.129614	−	0.224498i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	481.473			0.0561372			0.0280686	−	0.999606i	$-0.491064\pi$
$419$	481.473			0.0561372			0.0280686	+	0.999606i	$0.491064\pi$
$420$		0			0
$421$	−4020.71			−0.465457			−0.232729	−	0.972542i	$-0.574765\pi$
$421$	−4020.71			−0.465457			−0.232729	+	0.972542i	$0.574765\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	3558.69	+	6163.83i	0.406169	+	0.703505i
$426$		0			0
$427$	−7769.51	+	12125.0i	−0.880545	+	1.37417i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−5832.70	−	3367.51i	−0.651859	−	0.376351i	0.137309	−	0.990528i	$-0.456155\pi$
$431$	−5832.70	−	3367.51i	−0.651859	−	0.376351i	−0.789168	+	0.614177i	$0.789488\pi$
$432$		0			0
$433$		−	1566.34i		−	0.173841i	−0.996215	−	0.0869207i	$-0.972297\pi$
$433$		−	1566.34i		−	0.173841i	0.996215	−	0.0869207i	$-0.0277027\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−5223.59	+	9047.53i	−0.571804	+	0.990393i
$438$		0			0
$439$	1356.35	−	783.086i	0.147460	−	0.0851359i	−0.424455	−	0.905449i	$-0.639534\pi$
$439$	1356.35	−	783.086i	0.147460	−	0.0851359i	0.571915	+	0.820313i	$0.306201\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3642.26	−	2102.86i	0.390630	−	0.225530i	−0.291803	−	0.956478i	$-0.594255\pi$
$443$	3642.26	−	2102.86i	0.390630	−	0.225530i	0.682433	+	0.730948i	$0.260922\pi$
$444$		0			0
$445$	4479.49	−	7758.70i	0.477187	−	0.826512i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			11447.3i			1.20319i	0.798801	+	0.601595i	$0.205468\pi$
$449$			11447.3i			1.20319i	−0.798801	+	0.601595i	$0.794532\pi$
$450$		0			0
$451$	14938.8	+	8624.92i	1.55974	+	0.900514i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−294.859	+	460.154i	−0.0303807	+	0.0474118i
$456$		0			0
$457$	−7976.27	−	13815.3i	−0.816442	−	1.41412i	−0.908288	−	0.418346i	$-0.862610\pi$
$457$	−7976.27	−	13815.3i	−0.816442	−	1.41412i	0.0918453	−	0.995773i	$-0.470723\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−8977.52			−0.906995			−0.453497	−	0.891258i	$-0.649824\pi$
$461$	−8977.52			−0.906995			−0.453497	+	0.891258i	$0.649824\pi$
$462$		0			0
$463$	−12198.9			−1.22447			−0.612235	−	0.790676i	$-0.709729\pi$
$463$	−12198.9			−1.22447			−0.612235	+	0.790676i	$0.709729\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	6134.37	+	10625.0i	0.607848	+	1.05282i	0.991594	+	0.129385i	$0.0413002\pi$
$467$	6134.37	+	10625.0i	0.607848	+	1.05282i	−0.383747	+	0.923438i	$0.625366\pi$
$468$		0			0
$469$	−339.872	+	7340.23i	−0.0334623	+	0.722687i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−11017.0	−	6360.64i	−1.07095	−	0.618314i
$474$		0			0
$475$			5013.75i			0.484309i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	9254.77	−	16029.7i	0.882801	−	1.52906i	0.0345865	−	0.999402i	$-0.488989\pi$
$479$	9254.77	−	16029.7i	0.882801	−	1.52906i	0.848214	−	0.529654i	$-0.177678\pi$
$480$		0			0
$481$	−1022.12	+	590.124i	−0.0968916	+	0.0559404i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	7254.90	−	4188.62i	0.679233	−	0.392156i
$486$		0			0
$487$	7936.33	−	13746.1i	0.738459	−	1.27905i	−0.214731	−	0.976673i	$-0.568887\pi$
$487$	7936.33	−	13746.1i	0.738459	−	1.27905i	0.953189	−	0.302374i	$-0.0977793\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	3988.61i		−	0.366605i	−0.983057	−	0.183303i	$-0.941321\pi$
$491$		−	3988.61i		−	0.366605i	0.983057	−	0.183303i	$-0.0586788\pi$
$492$		0			0
$493$	12796.7	+	7388.17i	1.16903	+	0.674942i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−7414.37	−	14335.0i	−0.669175	−	1.29379i
$498$		0			0
$499$	4207.05	+	7286.83i	0.377422	+	0.653714i	0.990686	−	0.136164i	$-0.0434774\pi$
$499$	4207.05	+	7286.83i	0.377422	+	0.653714i	−0.613265	+	0.789878i	$0.710144\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	12481.8			1.10643			0.553215	−	0.833038i	$-0.313401\pi$
$503$	12481.8			1.10643			0.553215	+	0.833038i	$0.313401\pi$
$504$		0			0
$505$	−5745.78			−0.506305
$506$		0			0
$507$		0			0
$508$		0			0
$509$	2203.68	+	3816.88i	0.191898	+	0.332378i	0.945879	−	0.324518i	$-0.105202\pi$
$509$	2203.68	+	3816.88i	0.191898	+	0.332378i	−0.753981	+	0.656896i	$0.771869\pi$
$510$		0			0
$511$	687.471	+	31.8317i	0.0595146	+	0.00275568i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2238.31	+	1292.29i	0.191518	+	0.110573i
$516$		0			0
$517$		−	18505.8i		−	1.57424i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	4627.22	−	8014.57i	0.389102	−	0.673944i	−0.603227	−	0.797569i	$-0.706119\pi$
$521$	4627.22	−	8014.57i	0.389102	−	0.673944i	0.992329	+	0.123625i	$0.0394521\pi$
$522$		0			0
$523$	6598.60	−	3809.70i	0.551695	−	0.318521i	−0.198110	−	0.980180i	$-0.563480\pi$
$523$	6598.60	−	3809.70i	0.551695	−	0.318521i	0.749805	+	0.661658i	$0.230147\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−5585.62	+	3224.86i	−0.461695	+	0.266560i
$528$		0			0
$529$	11971.7	−	20735.5i	0.983945	−	1.70424i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			2485.84i			0.202015i
$534$		0			0
$535$	4038.29	+	2331.51i	0.326337	+	0.188411i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−10975.4	+	5047.64i	−0.877077	+	0.403371i
$540$		0			0
$541$	−1602.70	−	2775.97i	−0.127367	−	0.220606i	0.795289	−	0.606231i	$-0.207319\pi$
$541$	−1602.70	−	2775.97i	−0.127367	−	0.220606i	−0.922656	+	0.385625i	$0.873986\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	9785.50			0.769110
$546$		0			0
$547$	10785.4			0.843057			0.421529	−	0.906815i	$-0.361494\pi$
$547$	10785.4			0.843057			0.421529	+	0.906815i	$0.361494\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	5204.50	+	9014.46i	0.402394	+	0.696967i
$552$		0			0
$553$	3314.26	+	2123.72i	0.254858	+	0.163309i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−16783.5	−	9689.97i	−1.27673	−	0.737122i	−0.300487	−	0.953786i	$-0.597149\pi$
$557$	−16783.5	−	9689.97i	−1.27673	−	0.737122i	−0.976246	+	0.216663i	$0.930483\pi$
$558$		0			0
$559$		−	1833.24i		−	0.138708i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1734.71	+	3004.61i	−0.129857	+	0.224918i	−0.923621	−	0.383307i	$-0.874785\pi$
$563$	−1734.71	+	3004.61i	−0.129857	+	0.224918i	0.793764	+	0.608226i	$0.208118\pi$
$564$		0			0
$565$	−3516.55	+	2030.28i	−0.261845	+	0.151176i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−19602.9	+	11317.7i	−1.44428	+	0.833857i	−0.998132	−	0.0611013i	$-0.980539\pi$
$569$	−19602.9	+	11317.7i	−1.44428	+	0.833857i	−0.446151	+	0.894958i	$0.647205\pi$
$570$		0			0
$571$	5989.94	−	10374.9i	0.439004	−	0.760378i	−0.558609	−	0.829431i	$-0.688665\pi$
$571$	5989.94	−	10374.9i	0.439004	−	0.760378i	0.997613	+	0.0690537i	$0.0219980\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	17329.8i		−	1.25688i
$576$		0			0
$577$	3498.26	+	2019.72i	0.252399	+	0.145723i	0.620862	−	0.783920i	$-0.286783\pi$
$577$	3498.26	+	2019.72i	0.252399	+	0.145723i	−0.368463	+	0.929642i	$0.620116\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	6200.74	−	3207.15i	0.442771	−	0.229011i
$582$		0			0
$583$	−5667.02	−	9815.56i	−0.402579	−	0.697288i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−19907.7			−1.39979			−0.699897	−	0.714244i	$-0.746771\pi$
$587$	−19907.7			−1.39979			−0.699897	+	0.714244i	$0.746771\pi$
$588$		0			0
$589$	−4543.42			−0.317841
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−7421.96	−	12855.2i	−0.513968	−	0.890220i	−0.999869	−	0.0162052i	$-0.994841\pi$
$593$	−7421.96	−	12855.2i	−0.513968	−	0.890220i	0.485900	−	0.874014i	$-0.338492\pi$
$594$		0			0
$595$	7464.33	−	3860.71i	0.514299	−	0.266006i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	15891.1	+	9174.74i	1.08396	+	0.625826i	0.931962	−	0.362555i	$-0.118095\pi$
$599$	15891.1	+	9174.74i	1.08396	+	0.625826i	0.151999	+	0.988381i	$0.451429\pi$
$600$		0			0
$601$		−	13467.0i		−	0.914027i	−0.889460	−	0.457014i	$-0.848919\pi$
$601$		−	13467.0i		−	0.914027i	0.889460	−	0.457014i	$-0.151081\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	263.218	−	455.906i	0.0176881	−	0.0306367i
$606$		0			0
$607$	−236.043	+	136.279i	−0.0157836	+	0.00911269i	−0.507871	−	0.861433i	$-0.669567\pi$
$607$	−236.043	+	136.279i	−0.0157836	+	0.00911269i	0.492087	+	0.870546i	$0.336234\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2309.55	−	1333.42i	0.152920	−	0.0882887i
$612$		0			0
$613$	5275.00	−	9136.57i	0.347562	−	0.601994i	−0.638254	−	0.769826i	$-0.720343\pi$
$613$	5275.00	−	9136.57i	0.347562	−	0.601994i	0.985816	+	0.167831i	$0.0536764\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			2035.13i			0.132790i	0.997793	+	0.0663949i	$0.0211497\pi$
$617$			2035.13i			0.132790i	−0.997793	+	0.0663949i	$0.978850\pi$
$618$		0			0
$619$	24269.7	+	14012.1i	1.57590	+	0.909846i	0.995423	+	0.0955690i	$0.0304671\pi$
$619$	24269.7	+	14012.1i	1.57590	+	0.909846i	0.580477	+	0.814277i	$0.302866\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	24028.3	+	15396.9i	1.54522	+	0.990153i
$624$		0			0
$625$	−2045.71	−	3543.27i	−0.130925	−	0.226770i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	18148.3			1.15043
$630$		0			0
$631$	−13605.8			−0.858383			−0.429191	−	0.903214i	$-0.641201\pi$
$631$	−13605.8			−0.858383			−0.429191	+	0.903214i	$0.641201\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2571.80	+	4454.48i	0.160722	+	0.278379i
$636$		0			0
$637$	−1420.78	−	1006.04i	−0.0883723	−	0.0625760i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	18842.0	+	10878.4i	1.16102	+	0.670315i	0.951548	−	0.307500i	$-0.0994925\pi$
$641$	18842.0	+	10878.4i	1.16102	+	0.670315i	0.209471	+	0.977815i	$0.432826\pi$
$642$		0			0
$643$		−	17522.8i		−	1.07470i	−0.843360	−	0.537349i	$-0.819426\pi$
$643$		−	17522.8i		−	1.07470i	0.843360	−	0.537349i	$-0.180574\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−5943.00	+	10293.6i	−0.361118	+	0.625475i	−0.988145	−	0.153522i	$-0.950938\pi$
$647$	−5943.00	+	10293.6i	−0.361118	+	0.625475i	0.627027	+	0.778998i	$0.284272\pi$
$648$		0			0
$649$	−10517.6	+	6072.34i	−0.636135	+	0.367273i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−18199.6	+	10507.5i	−1.09067	+	0.629696i	−0.933754	−	0.357917i	$-0.883487\pi$
$653$	−18199.6	+	10507.5i	−1.09067	+	0.629696i	−0.156912	+	0.987613i	$0.550154\pi$
$654$		0			0
$655$	2144.83	−	3714.95i	0.127947	−	0.221611i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	28701.9i		−	1.69661i	−0.529507	−	0.848305i	$-0.677623\pi$
$659$		−	28701.9i		−	1.69661i	0.529507	−	0.848305i	$-0.322377\pi$
$660$		0			0
$661$	9447.51	+	5454.52i	0.555923	+	0.320963i	0.751508	−	0.659724i	$-0.229327\pi$
$661$	9447.51	+	5454.52i	0.555923	+	0.320963i	−0.195584	+	0.980687i	$0.562660\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	5913.51	+	273.811i	0.344836	+	0.0159668i
$666$		0			0
$667$	−17989.2	−	31158.2i	−1.04429	−	1.80877i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−27386.0			−1.57559
$672$		0			0
$673$	−14602.3			−0.836371			−0.418186	−	0.908362i	$-0.637334\pi$
$673$	−14602.3			−0.836371			−0.418186	+	0.908362i	$0.637334\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−13120.5	−	22725.4i	−0.744848	−	1.29011i	−0.950266	−	0.311440i	$-0.899189\pi$
$677$	−13120.5	−	22725.4i	−0.744848	−	1.29011i	0.205418	−	0.978674i	$-0.434145\pi$
$678$		0			0
$679$	12259.4	+	23702.3i	0.692888	+	1.33964i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	9578.05	+	5529.89i	0.536594	+	0.309803i	0.743698	−	0.668516i	$-0.233070\pi$
$683$	9578.05	+	5529.89i	0.536594	+	0.309803i	−0.207103	+	0.978319i	$0.566404\pi$
$684$		0			0
$685$			1710.94i			0.0954331i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	816.664	−	1414.50i	0.0451559	−	0.0782123i
$690$		0			0
$691$	8017.12	−	4628.69i	0.441368	−	0.254824i	−0.262810	−	0.964848i	$-0.584649\pi$
$691$	8017.12	−	4628.69i	0.441368	−	0.254824i	0.704178	+	0.710024i	$0.251316\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	6992.19	−	4036.94i	0.381624	−	0.220331i
$696$		0			0
$697$	19112.0	−	33102.9i	1.03862	−	1.79894i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			3110.78i			0.167607i	0.996482	+	0.0838035i	$0.0267068\pi$
$701$			3110.78i			0.167607i	−0.996482	+	0.0838035i	$0.973293\pi$
$702$		0			0
$703$	11071.5	+	6392.16i	0.593985	+	0.342937i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	846.558	−	18283.2i	0.0450327	−	0.972573i
$708$		0			0
$709$	−1921.67	−	3328.43i	−0.101791	−	0.176307i	0.810632	−	0.585556i	$-0.199124\pi$
$709$	−1921.67	−	3328.43i	−0.101791	−	0.176307i	−0.912423	+	0.409249i	$0.865791\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	15704.2			0.824861
$714$		0			0
$715$	−1039.32			−0.0543614
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−5811.36	−	10065.6i	−0.301429	−	0.522090i	0.675031	−	0.737789i	$-0.264130\pi$
$719$	−5811.36	−	10065.6i	−0.301429	−	0.522090i	−0.976460	+	0.215699i	$0.930797\pi$
$720$		0			0
$721$	−4441.87	+	6931.94i	−0.229437	+	0.358057i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−14953.2	−	8633.26i	−0.765999	−	0.442250i
$726$		0			0
$727$		−	7551.88i		−	0.385259i	−0.981272	−	0.192630i	$-0.938298\pi$
$727$		−	7551.88i		−	0.385259i	0.981272	−	0.192630i	$-0.0617016\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−14094.6	+	24412.5i	−0.713141	+	1.23520i
$732$		0			0
$733$	25948.2	−	14981.2i	1.30753	−	0.754901i	0.325844	−	0.945423i	$-0.394351\pi$
$733$	25948.2	−	14981.2i	1.30753	−	0.754901i	0.981683	+	0.190522i	$0.0610181\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−12101.8	+	6986.96i	−0.604850	+	0.349210i
$738$		0			0
$739$	15610.1	−	27037.5i	0.777034	−	1.34586i	−0.156611	−	0.987660i	$-0.550057\pi$
$739$	15610.1	−	27037.5i	0.777034	−	1.34586i	0.933644	−	0.358201i	$-0.116610\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			9961.10i			0.491840i	0.969290	+	0.245920i	$0.0790901\pi$
$743$			9961.10i			0.491840i	−0.969290	+	0.245920i	$0.920910\pi$
$744$		0			0
$745$	−2504.72	−	1446.10i	−0.123176	−	0.0711155i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−8013.87	+	12506.4i	−0.390949	+	0.610111i
$750$		0			0
$751$	1321.30	+	2288.55i	0.0642008	+	0.111199i	0.896339	−	0.443369i	$-0.146217\pi$
$751$	1321.30	+	2288.55i	0.0642008	+	0.111199i	−0.832138	+	0.554568i	$0.812884\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	15997.2			0.771124
$756$		0			0
$757$	13097.5			0.628847			0.314424	−	0.949283i	$-0.398189\pi$
$757$	13097.5			0.628847			0.314424	+	0.949283i	$0.398189\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	14526.3	+	25160.2i	0.691954	+	1.19850i	0.971197	+	0.238279i	$0.0765833\pi$
$761$	14526.3	+	25160.2i	0.691954	+	1.19850i	−0.279243	+	0.960221i	$0.590083\pi$
$762$		0			0
$763$	−1441.75	+	31137.6i	−0.0684076	+	1.47740i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1515.67	−	875.073i	−0.0713530	−	0.0411957i
$768$		0			0
$769$		−	15912.2i		−	0.746175i	−0.927796	−	0.373087i	$-0.878299\pi$
$769$		−	15912.2i		−	0.746175i	0.927796	−	0.373087i	$-0.121701\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−8521.68	+	14760.0i	−0.396511	+	0.686778i	−0.993293	−	0.115626i	$-0.963113\pi$
$773$	−8521.68	+	14760.0i	−0.396511	+	0.686778i	0.596781	+	0.802404i	$0.296446\pi$
$774$		0			0
$775$	6526.93	−	3768.33i	0.302522	−	0.174661i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	23318.9	−	13463.2i	1.07251	−	0.619216i
$780$		0			0
$781$	15345.8	−	26579.6i	0.703092	−	1.21779i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	7852.38i		−	0.357024i
$786$		0			0
$787$	20787.2	+	12001.5i	0.941531	+	0.543593i	0.890440	−	0.455101i	$-0.150397\pi$
$787$	20787.2	+	12001.5i	0.941531	+	0.543593i	0.0510911	+	0.998694i	$0.483730\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−5942.27	−	11488.8i	−0.267108	−	0.516430i
$792$		0			0
$793$	−1973.27	−	3417.81i	−0.0883644	−	0.153052i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	34369.2			1.52750			0.763752	−	0.645510i	$-0.223355\pi$
$797$	34369.2			1.52750			0.763752	+	0.645510i	$0.223355\pi$
$798$		0			0
$799$	−41007.1			−1.81568
$800$		0			0
$801$		0			0
$802$		0			0
$803$	654.385	+	1133.43i	0.0287581	+	0.0498104i
$804$		0			0
$805$	−20439.8	−	946.417i	−0.894918	−	0.0414371i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	5944.55	+	3432.09i	0.258343	+	0.149154i	0.623578	−	0.781761i	$-0.285678\pi$
$809$	5944.55	+	3432.09i	0.258343	+	0.149154i	−0.365236	+	0.930915i	$0.619012\pi$
$810$		0			0
$811$		−	9243.29i		−	0.400217i	−0.979774	−	0.200108i	$-0.935871\pi$
$811$		−	9243.29i		−	0.400217i	0.979774	−	0.200108i	$-0.0641295\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1017.26	−	1761.94i	0.0437214	−	0.0757277i
$816$		0			0
$817$	−17197.1	+	9928.74i	−0.736413	+	0.425168i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	22160.0	−	12794.1i	0.942009	−	0.543869i	0.0514195	−	0.998677i	$-0.483625\pi$
$821$	22160.0	−	12794.1i	0.942009	−	0.543869i	0.890590	+	0.454808i	$0.150292\pi$
$822$		0			0
$823$	9771.65	−	16925.0i	0.413874	−	0.716851i	−0.581436	−	0.813592i	$-0.697509\pi$
$823$	9771.65	−	16925.0i	0.413874	−	0.716851i	0.995310	+	0.0967418i	$0.0308421\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			11905.9i			0.500617i	0.968166	+	0.250308i	$0.0805320\pi$
$827$			11905.9i			0.500617i	−0.968166	+	0.250308i	$0.919468\pi$
$828$		0			0
$829$	35422.5	+	20451.2i	1.48405	+	0.856815i	0.999836	−	0.0181356i	$-0.00577306\pi$
$829$	35422.5	+	20451.2i	1.48405	+	0.856815i	0.484212	+	0.874951i	$0.339106\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	11185.1	+	24320.4i	0.465234	+	1.01159i
$834$		0			0
$835$	−1974.71	−	3420.30i	−0.0818415	−	0.141754i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−19206.9			−0.790340			−0.395170	−	0.918608i	$-0.629314\pi$
$839$	−19206.9			−0.790340			−0.395170	+	0.918608i	$0.629314\pi$
$840$		0			0
$841$	−11457.9			−0.469797
$842$		0			0
$843$		0			0
$844$		0			0
$845$	6311.86	+	10932.5i	0.256964	+	0.445075i
$846$		0			0
$847$	1411.92	+	904.734i	0.0572776	+	0.0367025i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−38268.4	−	22094.3i	−1.54151	−	0.889990i
$852$		0			0
$853$			13311.1i			0.534308i	0.963654	+	0.267154i	$0.0860832\pi$
$853$			13311.1i			0.534308i	−0.963654	+	0.267154i	$0.913917\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	16128.9	−	27936.1i	0.642887	−	1.11351i	−0.341899	−	0.939737i	$-0.611070\pi$
$857$	16128.9	−	27936.1i	0.642887	−	1.11351i	0.984785	−	0.173775i	$-0.0555967\pi$
$858$		0			0
$859$	16105.5	−	9298.54i	0.639714	−	0.369339i	−0.144791	−	0.989462i	$-0.546251\pi$
$859$	16105.5	−	9298.54i	0.639714	−	0.369339i	0.784504	+	0.620123i	$0.212918\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−17760.4	+	10254.0i	−0.700547	+	0.404461i	−0.807551	−	0.589798i	$-0.799207\pi$
$863$	−17760.4	+	10254.0i	−0.700547	+	0.404461i	0.107004	+	0.994259i	$0.465874\pi$
$864$		0			0
$865$	3321.42	−	5752.88i	0.130557	−	0.226131i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			7485.69i			0.292215i
$870$		0			0
$871$	−1743.96	−	1006.88i	−0.0678438	−	0.0391697i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−20677.5	+	10694.9i	−0.798889	+	0.413202i
$876$		0			0
$877$	4892.74	+	8474.48i	0.188388	+	0.326297i	0.944713	−	0.327899i	$-0.106340\pi$
$877$	4892.74	+	8474.48i	0.188388	+	0.326297i	−0.756325	+	0.654196i	$0.773007\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−40977.8			−1.56706			−0.783528	−	0.621357i	$-0.786582\pi$
$881$	−40977.8			−1.56706			−0.783528	+	0.621357i	$0.786582\pi$
$882$		0			0
$883$	18185.4			0.693076			0.346538	−	0.938036i	$-0.387357\pi$
$883$	18185.4			0.693076			0.346538	+	0.938036i	$0.387357\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−13768.6	−	23847.9i	−0.521199	−	0.902743i	−0.999696	−	0.0246542i	$-0.992152\pi$
$887$	−13768.6	−	23847.9i	−0.521199	−	0.902743i	0.478497	−	0.878089i	$-0.341182\pi$
$888$		0			0
$889$	−14553.2	+	7527.20i	−0.549041	+	0.283975i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−25016.8	−	14443.5i	−0.937464	−	0.541245i
$894$		0			0
$895$		−	5297.81i		−	0.197862i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	7823.39	−	13550.5i	0.290239	−	0.502709i
$900$		0			0
$901$	−21750.3	+	12557.6i	−0.804227	+	0.464320i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2367.57	+	1366.92i	−0.0869621	+	0.0502076i
$906$		0			0
$907$	−13880.8	+	24042.2i	−0.508163	+	0.880164i	0.491792	+	0.870713i	$0.336342\pi$
$907$	−13880.8	+	24042.2i	−0.508163	+	0.880164i	−0.999955	+	0.00945166i	$0.996991\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			42947.4i			1.56192i	0.624579	+	0.780961i	$0.285270\pi$
$911$			42947.4i			1.56192i	−0.624579	+	0.780961i	$0.714730\pi$
$912$		0			0
$913$	11497.3	+	6637.95i	0.416762	+	0.240618i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	11505.0	+	7372.21i	0.414317	+	0.265487i
$918$		0			0
$919$	−6988.16	−	12103.8i	−0.250836	−	0.434460i	0.712920	−	0.701245i	$-0.247372\pi$
$919$	−6988.16	−	12103.8i	−0.250836	−	0.434460i	−0.963756	+	0.266785i	$0.914039\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	4422.90			0.157727
$924$		0			0
$925$	−21206.7			−0.753807
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1325.99	−	2296.69i	−0.0468293	−	0.0811107i	0.841661	−	0.540007i	$-0.181578\pi$
$929$	−1325.99	−	2296.69i	−0.0468293	−	0.0811107i	−0.888490	+	0.458896i	$0.848245\pi$
$930$		0			0
$931$	−1742.54	+	18776.5i	−0.0613421	+	0.660984i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	13840.2	+	7990.63i	0.484088	+	0.279488i
$936$		0			0
$937$			8574.24i			0.298942i	0.988766	+	0.149471i	$0.0477570\pi$
$937$			8574.24i			0.298942i	−0.988766	+	0.149471i	$0.952243\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	21095.7	−	36538.8i	0.730817	−	1.26581i	−0.225717	−	0.974193i	$-0.572473\pi$
$941$	21095.7	−	36538.8i	0.730817	−	1.26581i	0.956534	−	0.291619i	$-0.0941941\pi$
$942$		0			0
$943$	−80601.1	+	46535.1i	−2.78339	+	1.60699i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	31045.4	−	17924.1i	1.06530	−	0.615052i	0.138408	−	0.990375i	$-0.455801\pi$
$947$	31045.4	−	17924.1i	1.06530	−	0.615052i	0.926894	+	0.375323i	$0.122468\pi$
$948$		0			0
$949$	−94.3022	+	163.336i	−0.00322569	+	0.00558706i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	36867.2i		−	1.25314i	−0.779364	−	0.626572i	$-0.784458\pi$
$953$		−	36867.2i		−	1.25314i	0.779364	−	0.626572i	$-0.215542\pi$
$954$		0			0
$955$	14454.4	+	8345.26i	0.489774	+	0.282771i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−5444.24	−	252.082i	−0.183320	−	0.00848818i
$960$		0			0
$961$	−11480.7	−	19885.1i	−0.385374	−	0.667487i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−3423.16			−0.114192
$966$		0			0
$967$	−42645.2			−1.41818			−0.709088	−	0.705120i	$-0.750893\pi$
$967$	−42645.2			−1.41818			−0.709088	+	0.705120i	$0.750893\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	402.301	+	696.806i	0.0132961	+	0.0230294i	0.872597	−	0.488441i	$-0.162434\pi$
$971$	402.301	+	696.806i	0.0132961	+	0.0230294i	−0.859301	+	0.511471i	$0.829101\pi$
$972$		0			0
$973$	11815.4	+	22844.1i	0.389296	+	0.752669i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1806.36	+	1042.90i	0.0591511	+	0.0341509i	0.529284	−	0.848445i	$-0.322461\pi$
$977$	1806.36	+	1042.90i	0.0591511	+	0.0341509i	−0.470133	+	0.882596i	$0.655794\pi$
$978$		0			0
$979$			54271.2i			1.77172i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−2803.20	+	4855.29i	−0.0909546	+	0.157538i	−0.907913	−	0.419159i	$-0.862325\pi$
$983$	−2803.20	+	4855.29i	−0.0909546	+	0.157538i	0.816958	+	0.576696i	$0.195658\pi$
$984$		0			0
$985$	22700.5	−	13106.2i	0.734314	−	0.423956i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	59441.1	−	34318.3i	1.91114	−	1.10340i
$990$		0			0
$991$	4030.87	−	6981.68i	0.129208	−	0.223794i	−0.794162	−	0.607706i	$-0.792090\pi$
$991$	4030.87	−	6981.68i	0.129208	−	0.223794i	0.923370	+	0.383912i	$0.125423\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			5419.66i			0.172678i
$996$		0			0
$997$	−14812.9	−	8552.24i	−0.470541	−	0.271667i	0.245925	−	0.969289i	$-0.420908\pi$
$997$	−14812.9	−	8552.24i	−0.470541	−	0.271667i	−0.716466	+	0.697622i	$0.754242\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.4.t.d.269.5	yes	16
3.2	odd	2	inner	756.4.t.d.269.4	✓	16
7.5	odd	6	inner	756.4.t.d.593.4	yes	16
21.5	even	6	inner	756.4.t.d.593.5	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.4.t.d.269.4	✓	16	3.2	odd	2	inner
756.4.t.d.269.5	yes	16	1.1	even	1	trivial
756.4.t.d.593.4	yes	16	7.5	odd	6	inner
756.4.t.d.593.5	yes	16	21.5	even	6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	756.t (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.90703 + 5.03513i) q^{5} +(-16.4502 + 8.50837i) q^{7} +(-30.5015 - 17.6100i) q^{11} -5.07550i q^{13} +(-39.0221 + 67.5883i) q^{17} +(-47.6118 + 27.4887i) q^{19} +(164.568 - 95.0136i) q^{23} +(45.5983 - 78.9786i) q^{25} -189.333i q^{29} +(71.5699 + 41.3209i) q^{31} +(-90.6618 - 58.0945i) q^{35} +(-116.269 - 201.384i) q^{37} -489.773 q^{41} +361.194 q^{43} +(262.717 + 455.039i) q^{47} +(198.215 - 279.928i) q^{49} +(278.692 + 160.903i) q^{53} -204.772i q^{55} +(172.411 - 298.625i) q^{59} +(673.393 - 388.784i) q^{61} +(25.5558 - 14.7547i) q^{65} +(198.380 - 343.604i) q^{67} +871.421i q^{71} +(-32.1813 - 18.5799i) q^{73} +(651.587 + 30.1702i) q^{77} +(-106.270 - 184.065i) q^{79} -376.941 q^{83} -453.754 q^{85} +(-770.458 - 1334.47i) q^{89} +(43.1843 + 83.4929i) q^{91} +(-276.818 - 159.821i) q^{95} -1440.86i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 50 q^{7} - 390 q^{19} - 284 q^{25} + 984 q^{31} - 182 q^{37} + 472 q^{43} + 130 q^{49} - 24 q^{61} - 1522 q^{67} + 4458 q^{73} + 434 q^{79} - 2616 q^{85} - 4560 q^{91}+O(q^{100}) \) 16 * q + 50 * q^7 - 390 * q^19 - 284 * q^25 + 984 * q^31 - 182 * q^37 + 472 * q^43 + 130 * q^49 - 24 * q^61 - 1522 * q^67 + 4458 * q^73 + 434 * q^79 - 2616 * q^85 - 4560 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more