LMFDB - Embedded newform 756.3.p.a.577.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [756,3,Mod(397,756)] 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("756.397"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(756, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4, 5])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	756.p (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$20.5995079856$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			577.4
Character	$\chi$	$=$	756.577
Dual form			756.3.p.a.397.13

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-4.90249i q^{5} +(-6.05364 - 3.51475i) q^{7} -13.3736 q^{11} +(17.0699 + 9.85532i) q^{13} +(-17.0873 - 9.86533i) q^{17} +(-17.7784 + 10.2644i) q^{19} +6.02775 q^{23} +0.965576 q^{25} +(18.8027 + 32.5672i) q^{29} +(-24.6426 + 14.2274i) q^{31} +(-17.2310 + 29.6779i) q^{35} +(11.3781 + 19.7075i) q^{37} +(27.0533 + 15.6192i) q^{41} +(-9.51702 - 16.4840i) q^{43} +(54.7739 + 31.6237i) q^{47} +(24.2931 + 42.5540i) q^{49} +(-29.9936 + 51.9504i) q^{53} +65.5642i q^{55} +(-48.1779 + 27.8155i) q^{59} +(-75.0566 - 43.3340i) q^{61} +(48.3156 - 83.6851i) q^{65} +(-48.7632 - 84.4603i) q^{67} -3.00845 q^{71} +(39.7410 + 22.9445i) q^{73} +(80.9592 + 47.0050i) q^{77} +(2.66963 - 4.62393i) q^{79} +(-26.4913 + 15.2948i) q^{83} +(-48.3647 + 83.7701i) q^{85} +(-118.327 + 68.3162i) q^{89} +(-68.6961 - 119.657i) q^{91} +(50.3209 + 87.1584i) q^{95} +(-152.175 + 87.8581i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + q^{7} + 12 q^{11} + 15 q^{13} + 27 q^{17} - 30 q^{23} - 160 q^{25} - 24 q^{29} - 24 q^{31} - 141 q^{35} + 11 q^{37} + 90 q^{41} - 16 q^{43} - 108 q^{47} - 61 q^{49} - 54 q^{53} - 45 q^{59} - 165 q^{61}+ \cdots - 57 q^{97}+O(q^{100}) $ 32 * q + q^7 + 12 * q^11 + 15 * q^13 + 27 * q^17 - 30 * q^23 - 160 * q^25 - 24 * q^29 - 24 * q^31 - 141 * q^35 + 11 * q^37 + 90 * q^41 - 16 * q^43 - 108 * q^47 - 61 * q^49 - 54 * q^53 - 45 * q^59 - 165 * q^61 + 33 * q^65 - 35 * q^67 - 138 * q^71 + 123 * q^77 + 37 * q^79 - 378 * q^83 + 30 * q^85 + 261 * q^89 + 45 * q^91 + 48 * q^95 - 57 * q^97

Character values

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

$n$	$29$	$325$	$379$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{1}{6}\right)$	$1$

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
$5$		−	4.90249i		−	0.980498i	−0.871582	−	0.490249i	$-0.836906\pi$
$5$		−	4.90249i		−	0.980498i	0.871582	−	0.490249i	$-0.163094\pi$
$6$		0			0
$7$	−6.05364	−	3.51475i	−0.864806	−	0.502107i
$7$	−6.05364	−	3.51475i	−0.864806	−	0.502107i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−13.3736			−1.21579			−0.607893	−	0.794019i	$-0.707985\pi$
$11$	−13.3736			−1.21579			−0.607893	+	0.794019i	$0.707985\pi$
$12$		0			0
$13$	17.0699	+	9.85532i	1.31307	+	0.758101i	0.982603	−	0.185717i	$-0.0594606\pi$
$13$	17.0699	+	9.85532i	1.31307	+	0.758101i	0.330466	+	0.943818i	$0.392794\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−17.0873	−	9.86533i	−1.00513	−	0.580314i	−0.0953697	−	0.995442i	$-0.530403\pi$
$17$	−17.0873	−	9.86533i	−1.00513	−	0.580314i	−0.909763	+	0.415128i	$0.863737\pi$
$18$		0			0
$19$	−17.7784	+	10.2644i	−0.935705	+	0.540230i	−0.888611	−	0.458661i	$-0.848329\pi$
$19$	−17.7784	+	10.2644i	−0.935705	+	0.540230i	−0.0470936	+	0.998890i	$0.514996\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	6.02775			0.262076			0.131038	−	0.991377i	$-0.458169\pi$
$23$	6.02775			0.262076			0.131038	+	0.991377i	$0.458169\pi$
$24$		0			0
$25$	0.965576			0.0386230
$26$		0			0
$27$		0			0
$28$		0			0
$29$	18.8027	+	32.5672i	0.648367	+	1.12301i	0.983513	+	0.180839i	$0.0578813\pi$
$29$	18.8027	+	32.5672i	0.648367	+	1.12301i	−0.335145	+	0.942166i	$0.608785\pi$
$30$		0			0
$31$	−24.6426	+	14.2274i	−0.794922	+	0.458948i	−0.841692	−	0.539957i	$-0.818440\pi$
$31$	−24.6426	+	14.2274i	−0.794922	+	0.458948i	0.0467706	+	0.998906i	$0.485107\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−17.2310	+	29.6779i	−0.492315	+	0.847940i
$36$		0			0
$37$	11.3781	+	19.7075i	0.307517	+	0.532636i	0.977819	−	0.209453i	$-0.0671685\pi$
$37$	11.3781	+	19.7075i	0.307517	+	0.532636i	−0.670301	+	0.742089i	$0.733835\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	27.0533	+	15.6192i	0.659836	+	0.380957i	0.792215	−	0.610243i	$-0.208928\pi$
$41$	27.0533	+	15.6192i	0.659836	+	0.380957i	−0.132378	+	0.991199i	$0.542261\pi$
$42$		0			0
$43$	−9.51702	−	16.4840i	−0.221326	−	0.383348i	0.733885	−	0.679274i	$-0.237705\pi$
$43$	−9.51702	−	16.4840i	−0.221326	−	0.383348i	−0.955211	+	0.295926i	$0.904372\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	54.7739	+	31.6237i	1.16540	+	0.672845i	0.952593	−	0.304249i	$-0.0984053\pi$
$47$	54.7739	+	31.6237i	1.16540	+	0.672845i	0.212809	+	0.977094i	$0.431739\pi$
$48$		0			0
$49$	24.2931	+	42.5540i	0.495777	+	0.868450i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−29.9936	+	51.9504i	−0.565916	+	0.980196i	0.431048	+	0.902329i	$0.358144\pi$
$53$	−29.9936	+	51.9504i	−0.565916	+	0.980196i	−0.996964	+	0.0778664i	$0.975189\pi$
$54$		0			0
$55$			65.5642i			1.19208i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−48.1779	+	27.8155i	−0.816574	+	0.471449i	−0.849234	−	0.528017i	$-0.822936\pi$
$59$	−48.1779	+	27.8155i	−0.816574	+	0.471449i	0.0326596	+	0.999467i	$0.489602\pi$
$60$		0			0
$61$	−75.0566	−	43.3340i	−1.23044	−	0.710393i	−0.263315	−	0.964710i	$-0.584816\pi$
$61$	−75.0566	−	43.3340i	−1.23044	−	0.710393i	−0.967121	+	0.254317i	$0.918149\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	48.3156	−	83.6851i	0.743317	−	1.28746i
$66$		0			0
$67$	−48.7632	−	84.4603i	−0.727808	−	1.26060i	−0.957808	−	0.287410i	$-0.907206\pi$
$67$	−48.7632	−	84.4603i	−0.727808	−	1.26060i	0.229999	−	0.973191i	$-0.426128\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−3.00845			−0.0423725			−0.0211862	−	0.999776i	$-0.506744\pi$
$71$	−3.00845			−0.0423725			−0.0211862	+	0.999776i	$0.506744\pi$
$72$		0			0
$73$	39.7410	+	22.9445i	0.544397	+	0.314308i	0.746859	−	0.664982i	$-0.231561\pi$
$73$	39.7410	+	22.9445i	0.544397	+	0.314308i	−0.202462	+	0.979290i	$0.564894\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	80.9592	+	47.0050i	1.05142	+	0.610455i
$78$		0			0
$79$	2.66963	−	4.62393i	0.0337928	−	0.0585308i	−0.848634	−	0.528980i	$-0.822575\pi$
$79$	2.66963	−	4.62393i	0.0337928	−	0.0585308i	0.882427	+	0.470449i	$0.155908\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−26.4913	+	15.2948i	−0.319173	+	0.184274i	−0.651024	−	0.759057i	$-0.725660\pi$
$83$	−26.4913	+	15.2948i	−0.319173	+	0.184274i	0.331851	+	0.943332i	$0.392327\pi$
$84$		0			0
$85$	−48.3647	+	83.7701i	−0.568996	+	0.985531i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−118.327	+	68.3162i	−1.32952	+	0.767598i	−0.985225	−	0.171264i	$-0.945215\pi$
$89$	−118.327	+	68.3162i	−1.32952	+	0.767598i	−0.344293	+	0.938862i	$0.611881\pi$
$90$		0			0
$91$	−68.6961	−	119.657i	−0.754902	−	1.31491i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	50.3209	+	87.1584i	0.529694	+	0.917457i
$96$		0			0
$97$	−152.175	+	87.8581i	−1.56881	+	0.905753i	−0.572502	+	0.819903i	$0.694027\pi$
$97$	−152.175	+	87.8581i	−1.56881	+	0.905753i	−0.996308	+	0.0858502i	$0.972639\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	25.6545i		−	0.254005i	−0.991902	−	0.127003i	$-0.959464\pi$
$101$		−	25.6545i		−	0.254005i	0.991902	−	0.127003i	$-0.0405357\pi$
$102$		0			0
$103$		−	53.1171i		−	0.515700i	−0.966185	−	0.257850i	$-0.916986\pi$
$103$		−	53.1171i		−	0.515700i	0.966185	−	0.257850i	$-0.0830141\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	86.8236	+	150.383i	0.811436	+	1.40545i	0.911859	+	0.410503i	$0.134647\pi$
$107$	86.8236	+	150.383i	0.811436	+	1.40545i	−0.100424	+	0.994945i	$0.532020\pi$
$108$		0			0
$109$	−102.950	+	178.315i	−0.944500	+	1.63592i	−0.187750	+	0.982217i	$0.560119\pi$
$109$	−102.950	+	178.315i	−0.944500	+	1.63592i	−0.756750	+	0.653705i	$0.773214\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	52.4362	−	90.8222i	0.464037	−	0.803736i	−0.535120	−	0.844776i	$-0.679734\pi$
$113$	52.4362	−	90.8222i	0.464037	−	0.803736i	0.999158	+	0.0410399i	$0.0130671\pi$
$114$		0			0
$115$		−	29.5510i		−	0.256965i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	68.7659	+	119.779i	0.577865	+	1.00654i
$120$		0			0
$121$	57.8545			0.478136
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	127.296i		−	1.01837i
$126$		0			0
$127$	−183.067			−1.44147			−0.720737	−	0.693209i	$-0.756196\pi$
$127$	−183.067			−1.44147			−0.720737	+	0.693209i	$0.756196\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	146.682i		−	1.11971i	−0.828591	−	0.559854i	$-0.810857\pi$
$131$		−	146.682i		−	1.11971i	0.828591	−	0.559854i	$-0.189143\pi$
$132$		0			0
$133$	143.701	+	0.349839i	1.08046	+	0.00263037i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	30.8085			0.224880			0.112440	−	0.993659i	$-0.464133\pi$
$137$	30.8085			0.224880			0.112440	+	0.993659i	$0.464133\pi$
$138$		0			0
$139$	83.2406	+	48.0590i	0.598853	+	0.345748i	0.768590	−	0.639741i	$-0.220959\pi$
$139$	83.2406	+	48.0590i	0.598853	+	0.345748i	−0.169737	+	0.985489i	$0.554292\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−228.287	−	131.802i	−1.59641	−	0.921689i
$144$		0			0
$145$	159.660	−	92.1799i	1.10110	−	0.635723i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−50.5757			−0.339434			−0.169717	−	0.985493i	$-0.554285\pi$
$149$	−50.5757			−0.339434			−0.169717	+	0.985493i	$0.554285\pi$
$150$		0			0
$151$	26.6414			0.176433			0.0882166	−	0.996101i	$-0.471883\pi$
$151$	26.6414			0.176433			0.0882166	+	0.996101i	$0.471883\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	69.7497	+	120.810i	0.449998	+	0.779420i
$156$		0			0
$157$	180.504	−	104.214i	1.14971	−	0.663783i	0.200890	−	0.979614i	$-0.435617\pi$
$157$	180.504	−	104.214i	1.14971	−	0.663783i	0.948816	+	0.315831i	$0.102283\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−36.4898	−	21.1860i	−0.226645	−	0.131590i
$162$		0			0
$163$	22.1007	+	38.2796i	0.135587	+	0.234844i	0.925822	−	0.377961i	$-0.123375\pi$
$163$	22.1007	+	38.2796i	0.135587	+	0.234844i	−0.790234	+	0.612805i	$0.790041\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−270.401	−	156.116i	−1.61917	−	0.934827i	−0.987136	−	0.159885i	$-0.948888\pi$
$167$	−270.401	−	156.116i	−1.61917	−	0.934827i	−0.632032	−	0.774942i	$-0.717779\pi$
$168$		0			0
$169$	109.754	+	190.100i	0.649435	+	1.12485i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4.74990	−	2.74236i	−0.0274561	−	0.0158518i	0.486209	−	0.873842i	$-0.338379\pi$
$173$	−4.74990	−	2.74236i	−0.0274561	−	0.0158518i	−0.513665	+	0.857991i	$0.671713\pi$
$174$		0			0
$175$	−5.84525	−	3.39376i	−0.0334014	−	0.0193929i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−77.8089	+	134.769i	−0.434687	+	0.752899i	−0.997270	−	0.0738412i	$-0.976474\pi$
$179$	−77.8089	+	134.769i	−0.434687	+	0.752899i	0.562583	+	0.826741i	$0.309808\pi$
$180$		0			0
$181$		−	198.542i		−	1.09692i	−0.836177	−	0.548459i	$-0.815215\pi$
$181$		−	198.542i		−	1.09692i	0.836177	−	0.548459i	$-0.184785\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	96.6159	−	55.7812i	0.522248	−	0.301520i
$186$		0			0
$187$	228.519	+	131.935i	1.22203	+	0.705537i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−119.774	+	207.455i	−0.627090	+	1.08615i	0.361043	+	0.932549i	$0.382421\pi$
$191$	−119.774	+	207.455i	−0.627090	+	1.08615i	−0.988133	+	0.153602i	$0.950913\pi$
$192$		0			0
$193$	−68.5591	−	118.748i	−0.355228	−	0.615274i	0.631929	−	0.775027i	$-0.282264\pi$
$193$	−68.5591	−	118.748i	−0.355228	−	0.615274i	−0.987157	+	0.159753i	$0.948930\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	229.951			1.16727			0.583633	−	0.812018i	$-0.301631\pi$
$197$	229.951			1.16727			0.583633	+	0.812018i	$0.301631\pi$
$198$		0			0
$199$	57.1033	+	32.9686i	0.286951	+	0.165671i	0.636566	−	0.771222i	$-0.280354\pi$
$199$	57.1033	+	32.9686i	0.286951	+	0.165671i	−0.349615	+	0.936894i	$0.613688\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	0.640849	−	263.236i	0.00315689	−	1.29673i
$204$		0			0
$205$	76.5731	−	132.629i	0.373527	−	0.646969i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	237.762	−	137.272i	1.13762	−	0.656804i
$210$		0			0
$211$	−16.6716	+	28.8760i	−0.0790121	+	0.136853i	−0.902824	−	0.430010i	$-0.858510\pi$
$211$	−16.6716	+	28.8760i	−0.0790121	+	0.136853i	0.823812	+	0.566863i	$0.191843\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−80.8125	+	46.6571i	−0.375872	+	0.217010i
$216$		0			0
$217$	199.183	+	0.484911i	0.917894	+	0.00223461i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−194.452	−	336.801i	−0.879873	−	1.52398i
$222$		0			0
$223$	−92.4490	+	53.3754i	−0.414569	+	0.239352i	−0.692751	−	0.721177i	$-0.743602\pi$
$223$	−92.4490	+	53.3754i	−0.414569	+	0.239352i	0.278182	+	0.960528i	$0.410268\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			41.1468i			0.181263i	0.995884	+	0.0906316i	$0.0288886\pi$
$227$			41.1468i			0.181263i	−0.995884	+	0.0906316i	$0.971111\pi$
$228$		0			0
$229$			36.5274i			0.159508i	0.996815	+	0.0797542i	$0.0254135\pi$
$229$			36.5274i			0.159508i	−0.996815	+	0.0797542i	$0.974586\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1.30917	−	2.26755i	−0.00561876	−	0.00973198i	0.863202	−	0.504858i	$-0.168455\pi$
$233$	−1.30917	−	2.26755i	−0.00561876	−	0.00973198i	−0.868821	+	0.495126i	$0.835122\pi$
$234$		0			0
$235$	155.035	−	268.529i	0.659724	−	1.14267i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	160.707	−	278.353i	0.672415	−	1.16466i	−0.304802	−	0.952416i	$-0.598590\pi$
$239$	160.707	−	278.353i	0.672415	−	1.16466i	0.977217	−	0.212242i	$-0.0680765\pi$
$240$		0			0
$241$		−	426.134i		−	1.76819i	−0.467306	−	0.884095i	$-0.654775\pi$
$241$		−	426.134i		−	1.76819i	0.467306	−	0.884095i	$-0.345225\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	208.621	−	119.097i	0.851513	−	0.486109i
$246$		0			0
$247$	−404.634			−1.63819
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	80.6265i		−	0.321221i	−0.987018	−	0.160611i	$-0.948654\pi$
$251$		−	80.6265i		−	0.321221i	0.987018	−	0.160611i	$-0.0513463\pi$
$252$		0			0
$253$	−80.6130			−0.318628
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	46.0579i		−	0.179214i	−0.995977	−	0.0896068i	$-0.971439\pi$
$257$		−	46.0579i		−	0.179214i	0.995977	−	0.0896068i	$-0.0285610\pi$
$258$		0			0
$259$	0.387800	−	159.293i	0.00149730	−	0.615033i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−306.879			−1.16684			−0.583419	−	0.812171i	$-0.698286\pi$
$263$	−306.879			−1.16684			−0.583419	+	0.812171i	$0.698286\pi$
$264$		0			0
$265$	254.686	+	147.043i	0.961080	+	0.554880i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−360.761	−	208.286i	−1.34112	−	0.774296i	−0.354148	−	0.935189i	$-0.615229\pi$
$269$	−360.761	−	208.286i	−1.34112	−	0.774296i	−0.986972	+	0.160894i	$0.948562\pi$
$270$		0			0
$271$	164.266	−	94.8389i	0.606147	−	0.349959i	−0.165309	−	0.986242i	$-0.552862\pi$
$271$	164.266	−	94.8389i	0.606147	−	0.349959i	0.771456	+	0.636283i	$0.219529\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−12.9133			−0.0469574
$276$		0			0
$277$	−230.407			−0.831793			−0.415896	−	0.909412i	$-0.636532\pi$
$277$	−230.407			−0.831793			−0.415896	+	0.909412i	$0.636532\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	253.908	+	439.782i	0.903588	+	1.56506i	0.822801	+	0.568329i	$0.192410\pi$
$281$	253.908	+	439.782i	0.903588	+	1.56506i	0.0807870	+	0.996731i	$0.474257\pi$
$282$		0			0
$283$	−229.432	+	132.463i	−0.810713	+	0.468066i	−0.847204	−	0.531268i	$-0.821716\pi$
$283$	−229.432	+	132.463i	−0.810713	+	0.468066i	0.0364902	+	0.999334i	$0.488382\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−108.873	−	189.639i	−0.379349	−	0.660762i
$288$		0			0
$289$	50.1495	+	86.8614i	0.173528	+	0.300559i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	239.069	+	138.026i	0.815934	+	0.471079i	0.849012	−	0.528373i	$-0.177198\pi$
$293$	239.069	+	138.026i	0.815934	+	0.471079i	−0.0330786	+	0.999453i	$0.510531\pi$
$294$		0			0
$295$	136.365	+	236.192i	0.462255	+	0.800649i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	102.893	+	59.4053i	0.344124	+	0.198680i
$300$		0			0
$301$	−0.324368	+	133.238i	−0.00107763	+	0.442651i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−212.444	+	367.964i	−0.696539	+	1.20644i
$306$		0			0
$307$			45.6152i			0.148584i	0.997237	+	0.0742919i	$0.0236697\pi$
$307$			45.6152i			0.148584i	−0.997237	+	0.0742919i	$0.976330\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	137.483	−	79.3759i	0.442068	−	0.255228i	−0.262406	−	0.964957i	$-0.584516\pi$
$311$	137.483	−	79.3759i	0.442068	−	0.255228i	0.704474	+	0.709729i	$0.251183\pi$
$312$		0			0
$313$	240.928	+	139.100i	0.769737	+	0.444408i	0.832781	−	0.553603i	$-0.186747\pi$
$313$	240.928	+	139.100i	0.769737	+	0.444408i	−0.0630435	+	0.998011i	$0.520081\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	100.359	−	173.827i	0.316589	−	0.548349i	−0.663185	−	0.748456i	$-0.730796\pi$
$317$	100.359	−	173.827i	0.316589	−	0.548349i	0.979774	+	0.200107i	$0.0641289\pi$
$318$		0			0
$319$	−251.460	−	435.542i	−0.788276	−	1.36533i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	405.045			1.25401
$324$		0			0
$325$	16.4823	+	9.51606i	0.0507148	+	0.0292802i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−220.432	−	383.955i	−0.670006	−	1.16704i
$330$		0			0
$331$	−204.562	+	354.312i	−0.618012	+	1.07043i	0.371836	+	0.928298i	$0.378728\pi$
$331$	−204.562	+	354.312i	−0.618012	+	1.07043i	−0.989848	+	0.142130i	$0.954605\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−414.066	+	239.061i	−1.23602	+	0.713615i
$336$		0			0
$337$	106.717	−	184.839i	0.316667	−	0.548484i	−0.663123	−	0.748510i	$-0.730769\pi$
$337$	106.717	−	184.839i	0.316667	−	0.548484i	0.979790	+	0.200026i	$0.0641028\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	329.561	−	190.272i	0.966455	−	0.557983i
$342$		0			0
$343$	2.50507	−	342.991i	0.00730342	−	0.999973i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	160.213	+	277.497i	0.461709	+	0.799704i	0.999046	−	0.0436638i	$-0.0139030\pi$
$347$	160.213	+	277.497i	0.461709	+	0.799704i	−0.537337	+	0.843368i	$0.680570\pi$
$348$		0			0
$349$	79.0543	−	45.6420i	0.226517	−	0.130779i	−0.382447	−	0.923977i	$-0.624919\pi$
$349$	79.0543	−	45.6420i	0.226517	−	0.130779i	0.608964	+	0.793198i	$0.291585\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	249.332i		−	0.706324i	−0.935562	−	0.353162i	$-0.885106\pi$
$353$		−	249.332i		−	0.706324i	0.935562	−	0.353162i	$-0.114894\pi$
$354$		0			0
$355$			14.7489i			0.0415461i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−224.034	−	388.038i	−0.624049	−	1.08088i	−0.988724	−	0.149750i	$-0.952153\pi$
$359$	−224.034	−	388.038i	−0.624049	−	1.08088i	0.364675	−	0.931135i	$-0.381180\pi$
$360$		0			0
$361$	30.2142	−	52.3325i	0.0836958	−	0.144965i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	112.485	−	194.830i	0.308178	−	0.533780i
$366$		0			0
$367$		−	404.798i		−	1.10299i	−0.834178	−	0.551496i	$-0.814057\pi$
$367$		−	404.798i		−	1.10299i	0.834178	−	0.551496i	$-0.185943\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	364.163	−	209.069i	0.981570	−	0.563528i
$372$		0			0
$373$	−139.949			−0.375198			−0.187599	−	0.982246i	$-0.560071\pi$
$373$	−139.949			−0.375198			−0.187599	+	0.982246i	$0.560071\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			741.224i			1.96611i
$378$		0			0
$379$	293.943			0.775576			0.387788	−	0.921748i	$-0.373239\pi$
$379$	293.943			0.775576			0.387788	+	0.921748i	$0.373239\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			691.945i			1.80665i	0.428962	+	0.903323i	$0.358880\pi$
$383$			691.945i			1.80665i	−0.428962	+	0.903323i	$0.641120\pi$
$384$		0			0
$385$	230.442	−	396.902i	0.598550	−	1.03091i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	35.6572			0.0916636			0.0458318	−	0.998949i	$-0.485406\pi$
$389$	35.6572			0.0916636			0.0458318	+	0.998949i	$0.485406\pi$
$390$		0			0
$391$	−102.998	−	59.4657i	−0.263421	−	0.152086i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−22.6688	−	13.0878i	−0.0573894	−	0.0331338i
$396$		0			0
$397$	−334.306	+	193.012i	−0.842081	+	0.486176i	−0.857971	−	0.513698i	$-0.828275\pi$
$397$	−334.306	+	193.012i	−0.842081	+	0.486176i	0.0158903	+	0.999874i	$0.494942\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−328.642			−0.819555			−0.409777	−	0.912186i	$-0.634394\pi$
$401$	−328.642			−0.819555			−0.409777	+	0.912186i	$0.634394\pi$
$402$		0			0
$403$	−560.862			−1.39172
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−152.167	−	263.561i	−0.373875	−	0.647571i
$408$		0			0
$409$	681.725	−	393.594i	1.66681	−	0.962333i	0.697469	−	0.716615i	$-0.254310\pi$
$409$	681.725	−	393.594i	1.66681	−	0.962333i	0.969341	−	0.245718i	$-0.0790237\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	389.416	+	0.948033i	0.942896	+	0.00229548i
$414$		0			0
$415$	74.9825	+	129.874i	0.180681	+	0.312948i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−154.898	−	89.4302i	−0.369684	−	0.213437i	0.303636	−	0.952788i	$-0.401799\pi$
$419$	−154.898	−	89.4302i	−0.369684	−	0.213437i	−0.673320	+	0.739351i	$0.735133\pi$
$420$		0			0
$421$	237.634	+	411.595i	0.564452	+	0.977659i	0.997100	+	0.0760966i	$0.0242457\pi$
$421$	237.634	+	411.595i	0.564452	+	0.977659i	−0.432649	+	0.901563i	$0.642421\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−16.4990	−	9.52573i	−0.0388213	−	0.0224135i
$426$		0			0
$427$	302.058	+	526.133i	0.707395	+	1.23216i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−102.044	+	176.746i	−0.236762	+	0.410084i	−0.959783	−	0.280742i	$-0.909420\pi$
$431$	−102.044	+	176.746i	−0.236762	+	0.410084i	0.723021	+	0.690826i	$0.242753\pi$
$432$		0			0
$433$			20.3694i			0.0470424i	0.999723	+	0.0235212i	$0.00748772\pi$
$433$			20.3694i			0.0470424i	−0.999723	+	0.0235212i	$0.992512\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−107.164	+	61.8710i	−0.245226	+	0.141581i
$438$		0			0
$439$	118.916	+	68.6560i	0.270879	+	0.156392i	0.629287	−	0.777173i	$-0.283347\pi$
$439$	118.916	+	68.6560i	0.270879	+	0.156392i	−0.358408	+	0.933565i	$0.616680\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−224.089	+	388.134i	−0.505845	+	0.876150i	0.494132	+	0.869387i	$0.335486\pi$
$443$	−224.089	+	388.134i	−0.505845	+	0.876150i	−0.999977	+	0.00676284i	$0.997847\pi$
$444$		0			0
$445$	334.920	+	580.098i	0.752628	+	1.30359i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−688.878			−1.53425			−0.767125	−	0.641498i	$-0.778313\pi$
$449$	−688.878			−1.53425			−0.767125	+	0.641498i	$0.778313\pi$
$450$		0			0
$451$	−361.801	−	208.886i	−0.802220	−	0.463162i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−586.617	+	336.782i	−1.28927	+	0.740180i
$456$		0			0
$457$	−335.231	+	580.638i	−0.733548	+	1.27054i	0.221810	+	0.975090i	$0.428804\pi$
$457$	−335.231	+	580.638i	−0.733548	+	1.27054i	−0.955357	+	0.295452i	$0.904530\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	208.261	−	120.239i	0.451758	−	0.260823i	−0.256814	−	0.966461i	$-0.582673\pi$
$461$	208.261	−	120.239i	0.451758	−	0.260823i	0.708573	+	0.705638i	$0.249339\pi$
$462$		0			0
$463$	−194.444	+	336.788i	−0.419966	+	0.727403i	−0.995936	−	0.0900685i	$-0.971291\pi$
$463$	−194.444	+	336.788i	−0.419966	+	0.727403i	0.575969	+	0.817471i	$0.304625\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	594.959	−	343.500i	1.27400	−	0.735546i	0.298264	−	0.954483i	$-0.403592\pi$
$467$	594.959	−	343.500i	1.27400	−	0.735546i	0.975739	+	0.218937i	$0.0702590\pi$
$468$		0			0
$469$	−1.66199	+	682.682i	−0.00354369	+	1.45561i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	127.277	+	220.451i	0.269085	+	0.466069i
$474$		0			0
$475$	−17.1664	+	9.91102i	−0.0361398	+	0.0208653i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	178.852i		−	0.373386i	−0.982418	−	0.186693i	$-0.940223\pi$
$479$		−	178.852i		−	0.373386i	0.982418	−	0.186693i	$-0.0597770\pi$
$480$		0			0
$481$			448.541i			0.932517i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	430.723	+	746.035i	0.888089	+	1.53822i
$486$		0			0
$487$	−381.643	+	661.025i	−0.783661	+	1.35734i	0.146134	+	0.989265i	$0.453317\pi$
$487$	−381.643	+	661.025i	−0.783661	+	1.35734i	−0.929795	+	0.368077i	$0.880016\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−170.162	+	294.729i	−0.346561	+	0.600262i	−0.985636	−	0.168883i	$-0.945984\pi$
$491$	−170.162	+	294.729i	−0.346561	+	0.600262i	0.639075	+	0.769145i	$0.279317\pi$
$492$		0			0
$493$		−	741.978i		−	1.50503i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	18.2120	+	10.5739i	0.0366440	+	0.0212755i
$498$		0			0
$499$	−75.2336			−0.150769			−0.0753844	−	0.997155i	$-0.524018\pi$
$499$	−75.2336			−0.150769			−0.0753844	+	0.997155i	$0.524018\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	13.5435i		−	0.0269253i	−0.999909	−	0.0134627i	$-0.995715\pi$
$503$		−	13.5435i		−	0.0269253i	0.999909	−	0.0134627i	$-0.00428543\pi$
$504$		0			0
$505$	−125.771			−0.249052
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	90.2076i		−	0.177225i	−0.996066	−	0.0886126i	$-0.971757\pi$
$509$		−	90.2076i		−	0.177225i	0.996066	−	0.0886126i	$-0.0282433\pi$
$510$		0			0
$511$	−159.933	−	278.577i	−0.312981	−	0.545160i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−260.406			−0.505643
$516$		0			0
$517$	−732.527	−	422.925i	−1.41688	−	0.818036i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−786.918	−	454.327i	−1.51040	−	0.872029i	−0.999926	−	0.0121356i	$-0.996137\pi$
$521$	−786.918	−	454.327i	−1.51040	−	0.872029i	−0.510473	−	0.859894i	$-0.670530\pi$
$522$		0			0
$523$	−874.531	+	504.911i	−1.67214	+	0.965413i	−0.705708	+	0.708503i	$0.749371\pi$
$523$	−874.531	+	504.911i	−1.67214	+	0.965413i	−0.966435	+	0.256910i	$0.917296\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	561.432			1.06534
$528$		0			0
$529$	−492.666			−0.931316
$530$		0			0
$531$		0			0
$532$		0			0
$533$	307.865	+	533.238i	0.577608	+	1.00045i
$534$		0			0
$535$	737.251	−	425.652i	1.37804	−	0.795611i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−324.887	−	569.103i	−0.602759	−	1.05585i
$540$		0			0
$541$	144.207	+	249.775i	0.266557	+	0.461691i	0.967970	−	0.251064i	$-0.0807805\pi$
$541$	144.207	+	249.775i	0.266557	+	0.461691i	−0.701413	+	0.712755i	$0.747447\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	874.190	+	504.714i	1.60402	+	0.926080i
$546$		0			0
$547$	219.485	+	380.160i	0.401253	+	0.694991i	0.993877	−	0.110488i	$-0.0352415\pi$
$547$	219.485	+	380.160i	0.401253	+	0.694991i	−0.592624	+	0.805479i	$0.701908\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−668.562	−	385.994i	−1.21336	−	0.700534i
$552$		0			0
$553$	−32.4129	+	18.6086i	−0.0586129	+	0.0336502i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−527.053	+	912.883i	−0.946235	+	1.63893i	−0.192976	+	0.981203i	$0.561814\pi$
$557$	−527.053	+	912.883i	−0.946235	+	1.63893i	−0.753259	+	0.657724i	$0.771519\pi$
$558$		0			0
$559$		−	375.173i		−	0.671150i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	607.074	−	350.494i	1.07828	−	0.622547i	0.147851	−	0.989010i	$-0.452764\pi$
$563$	607.074	−	350.494i	1.07828	−	0.622547i	0.930433	+	0.366462i	$0.119431\pi$
$564$		0			0
$565$	−445.255	−	257.068i	−0.788062	−	0.454988i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	215.419	−	373.117i	0.378592	−	0.655741i	−0.612265	−	0.790652i	$-0.709742\pi$
$569$	215.419	−	373.117i	0.378592	−	0.655741i	0.990858	+	0.134911i	$0.0430749\pi$
$570$		0			0
$571$	−253.326	−	438.773i	−0.443653	−	0.768429i	0.554305	−	0.832314i	$-0.312984\pi$
$571$	−253.326	−	438.773i	−0.443653	−	0.768429i	−0.997957	+	0.0638850i	$0.979651\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	5.82025			0.0101222
$576$		0			0
$577$	−524.574	−	302.863i	−0.909140	−	0.524892i	−0.0289855	−	0.999580i	$-0.509228\pi$
$577$	−524.574	−	302.863i	−0.909140	−	0.524892i	−0.880154	+	0.474688i	$0.842561\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	214.126	+	0.521290i	0.368548	+	0.000897229i
$582$		0			0
$583$	401.123	−	694.766i	0.688033	−	1.19171i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−653.922	+	377.542i	−1.11401	+	0.643172i	−0.939864	−	0.341549i	$-0.889049\pi$
$587$	−653.922	+	377.542i	−1.11401	+	0.643172i	−0.174142	+	0.984721i	$0.555715\pi$
$588$		0			0
$589$	292.070	−	505.881i	0.495875	−	0.858880i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	536.697	−	309.862i	0.905053	−	0.522533i	0.0262169	−	0.999656i	$-0.491654\pi$
$593$	536.697	−	309.862i	0.905053	−	0.522533i	0.878836	+	0.477124i	$0.158321\pi$
$594$		0			0
$595$	587.213	−	337.124i	0.986913	−	0.566595i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	205.020	+	355.105i	0.342271	+	0.592830i	0.984854	−	0.173385i	$-0.0554706\pi$
$599$	205.020	+	355.105i	0.342271	+	0.592830i	−0.642583	+	0.766216i	$0.722137\pi$
$600$		0			0
$601$	−130.531	+	75.3624i	−0.217190	+	0.125395i	−0.604649	−	0.796492i	$-0.706686\pi$
$601$	−130.531	+	75.3624i	−0.217190	+	0.125395i	0.387458	+	0.921887i	$0.373353\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	283.631i		−	0.468812i
$606$		0			0
$607$			43.2527i			0.0712565i	0.999365	+	0.0356282i	$0.0113432\pi$
$607$			43.2527i			0.0712565i	−0.999365	+	0.0356282i	$0.988657\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	623.324	+	1079.63i	1.02017	+	1.76699i
$612$		0			0
$613$	−303.509	+	525.692i	−0.495120	+	0.857573i	−0.999984	−	0.00562585i	$-0.998209\pi$
$613$	−303.509	+	525.692i	−0.495120	+	0.857573i	0.504864	+	0.863199i	$0.331543\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−315.594	+	546.624i	−0.511497	+	0.885939i	0.488414	+	0.872612i	$0.337576\pi$
$617$	−315.594	+	546.624i	−0.511497	+	0.885939i	−0.999911	+	0.0133268i	$0.995758\pi$
$618$		0			0
$619$			997.283i			1.61112i	0.592514	+	0.805560i	$0.298135\pi$
$619$			997.283i			1.61112i	−0.592514	+	0.805560i	$0.701865\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	956.424	+	2.32841i	1.53519	+	0.00373742i
$624$		0			0
$625$	−599.928			−0.959885
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	448.996i		−	0.713826i
$630$		0			0
$631$	61.2563			0.0970782			0.0485391	−	0.998821i	$-0.484543\pi$
$631$	61.2563			0.0970782			0.0485391	+	0.998821i	$0.484543\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			897.485i			1.41336i
$636$		0			0
$637$	−4.70255	+	965.809i	−0.00738234	+	1.51618i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−434.394			−0.677682			−0.338841	−	0.940844i	$-0.610035\pi$
$641$	−434.394			−0.677682			−0.338841	+	0.940844i	$0.610035\pi$
$642$		0			0
$643$	−107.554	−	62.0966i	−0.167270	−	0.0965733i	0.414028	−	0.910264i	$-0.364122\pi$
$643$	−107.554	−	62.0966i	−0.167270	−	0.0965733i	−0.581298	+	0.813691i	$0.697455\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−219.777	−	126.888i	−0.339686	−	0.196118i	0.320447	−	0.947266i	$-0.396167\pi$
$647$	−219.777	−	126.888i	−0.339686	−	0.196118i	−0.660133	+	0.751149i	$0.729500\pi$
$648$		0			0
$649$	644.314	−	371.995i	0.992780	−	0.573182i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−582.566			−0.892138			−0.446069	−	0.894999i	$-0.647176\pi$
$653$	−582.566			−0.892138			−0.446069	+	0.894999i	$0.647176\pi$
$654$		0			0
$655$	−719.106			−1.09787
$656$		0			0
$657$		0			0
$658$		0			0
$659$	95.9048	+	166.112i	0.145531	+	0.252067i	0.929571	−	0.368643i	$-0.120178\pi$
$659$	95.9048	+	166.112i	0.145531	+	0.252067i	−0.784040	+	0.620710i	$0.786844\pi$
$660$		0			0
$661$	−556.748	+	321.439i	−0.842282	+	0.486292i	−0.858039	−	0.513584i	$-0.828317\pi$
$661$	−556.748	+	321.439i	−0.842282	+	0.486292i	0.0157572	+	0.999876i	$0.494984\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1.71508	−	704.491i	0.00257907	−	1.05939i
$666$		0			0
$667$	113.338	+	196.307i	0.169921	+	0.294313i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1003.78	+	579.533i	1.49595	+	0.863686i
$672$		0			0
$673$	215.745	+	373.682i	0.320573	+	0.555248i	0.980606	−	0.195988i	$-0.0627913\pi$
$673$	215.745	+	373.682i	0.320573	+	0.555248i	−0.660034	+	0.751236i	$0.729458\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	40.0771	+	23.1385i	0.0591980	+	0.0341780i	0.529307	−	0.848430i	$-0.322452\pi$
$677$	40.0771	+	23.1385i	0.0591980	+	0.0341780i	−0.470109	+	0.882608i	$0.655785\pi$
$678$		0			0
$679$	1230.01	+	2.99446i	1.81150	+	0.00441010i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	370.500	−	641.724i	0.542459	−	0.939567i	−0.456303	−	0.889825i	$-0.650827\pi$
$683$	370.500	−	641.724i	0.542459	−	0.939567i	0.998762	−	0.0497424i	$-0.0158400\pi$
$684$		0			0
$685$		−	151.039i		−	0.220494i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1023.97	+	591.192i	−1.48617	+	0.858043i
$690$		0			0
$691$	−223.604	−	129.098i	−0.323595	−	0.186828i	0.329399	−	0.944191i	$-0.393154\pi$
$691$	−223.604	−	129.098i	−0.323595	−	0.186828i	−0.652994	+	0.757363i	$0.726487\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	235.609	−	408.086i	0.339005	−	0.587174i
$696$		0			0
$697$	−308.178	−	533.779i	−0.442149	−	0.765824i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	416.749			0.594507			0.297253	−	0.954799i	$-0.403929\pi$
$701$	416.749			0.594507			0.297253	+	0.954799i	$0.403929\pi$
$702$		0			0
$703$	−404.570	−	233.579i	−0.575491	−	0.332260i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−90.1692	+	155.303i	−0.127538	+	0.219665i
$708$		0			0
$709$	539.523	−	934.481i	0.760963	−	1.31803i	−0.181392	−	0.983411i	$-0.558060\pi$
$709$	539.523	−	934.481i	0.760963	−	1.31803i	0.942355	−	0.334616i	$-0.108606\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−148.539	+	85.7591i	−0.208330	+	0.120279i
$714$		0			0
$715$	−646.156	+	1119.17i	−0.903715	+	1.56528i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	493.175	−	284.735i	0.685918	−	0.396015i	−0.116163	−	0.993230i	$-0.537059\pi$
$719$	493.175	−	284.735i	0.685918	−	0.396015i	0.802081	+	0.597215i	$0.203726\pi$
$720$		0			0
$721$	−186.693	+	321.552i	−0.258937	+	0.445981i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	18.1554	+	31.4461i	0.0250419	+	0.0433739i
$726$		0			0
$727$	−190.853	+	110.189i	−0.262522	+	0.151567i	−0.625484	−	0.780237i	$-0.715099\pi$
$727$	−190.853	+	110.189i	−0.262522	+	0.151567i	0.362962	+	0.931804i	$0.381765\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			375.554i			0.513754i
$732$		0			0
$733$		−	148.974i		−	0.203239i	−0.994823	−	0.101619i	$-0.967598\pi$
$733$		−	148.974i		−	0.203239i	0.994823	−	0.101619i	$-0.0324024\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	652.141	+	1129.54i	0.884859	+	1.53262i
$738$		0			0
$739$	432.421	−	748.976i	0.585144	−	1.01350i	−0.409714	−	0.912214i	$-0.634371\pi$
$739$	432.421	−	748.976i	0.585144	−	1.01350i	0.994857	−	0.101285i	$-0.0322953\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	5.43335	−	9.41084i	0.00731272	−	0.0126660i	−0.862346	−	0.506320i	$-0.831006\pi$
$743$	5.43335	−	9.41084i	0.00731272	−	0.0126660i	0.869659	+	0.493654i	$0.164339\pi$
$744$		0			0
$745$			247.947i			0.332814i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	2.95920	−	1215.53i	0.00395087	−	1.62287i
$750$		0			0
$751$	1321.13			1.75916			0.879579	−	0.475754i	$-0.157825\pi$
$751$	1321.13			1.75916			0.879579	+	0.475754i	$0.157825\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	130.609i		−	0.172992i
$756$		0			0
$757$	1141.12			1.50743			0.753713	−	0.657203i	$-0.228261\pi$
$757$	1141.12			1.50743			0.753713	+	0.657203i	$0.228261\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	654.045i		−	0.859455i	−0.902959	−	0.429728i	$-0.858610\pi$
$761$		−	654.045i		−	0.859455i	0.902959	−	0.429728i	$-0.141390\pi$
$762$		0			0
$763$	1249.96	−	717.612i	1.63822	−	0.940514i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1096.52			−1.42962
$768$		0			0
$769$	−20.1935	−	11.6587i	−0.0262594	−	0.0151609i	0.486813	−	0.873506i	$-0.338159\pi$
$769$	−20.1935	−	11.6587i	−0.0262594	−	0.0151609i	−0.513072	+	0.858345i	$0.671493\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	342.130	+	197.529i	0.442600	+	0.255535i	0.704700	−	0.709506i	$-0.251082\pi$
$773$	342.130	+	197.529i	0.442600	+	0.255535i	−0.262100	+	0.965041i	$0.584415\pi$
$774$		0			0
$775$	−23.7943	+	13.7376i	−0.0307023	+	0.0177260i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−641.286			−0.823216
$780$		0			0
$781$	40.2339			0.0515159
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−510.908	−	884.918i	−0.650838	−	1.12728i
$786$		0			0
$787$	−221.886	+	128.106i	−0.281939	+	0.162778i	−0.634301	−	0.773086i	$-0.718712\pi$
$787$	−221.886	+	128.106i	−0.281939	+	0.162778i	0.352362	+	0.935864i	$0.385379\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−636.647	+	365.505i	−0.804863	+	0.462079i
$792$		0			0
$793$	−854.140	−	1479.41i	−1.07710	−	1.86559i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1249.36	−	721.316i	−1.56757	−	0.905039i	−0.996451	−	0.0841749i	$-0.973175\pi$
$797$	−1249.36	−	721.316i	−1.56757	−	0.905039i	−0.571123	−	0.820864i	$-0.693492\pi$
$798$		0			0
$799$	−623.957	−	1080.73i	−0.780922	−	1.35260i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−531.482	−	306.851i	−0.661870	−	0.382131i
$804$		0			0
$805$	−103.864	+	178.891i	−0.129024	+	0.222225i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	744.804	−	1290.04i	0.920648	−	1.59461i	0.122234	−	0.992501i	$-0.460994\pi$
$809$	744.804	−	1290.04i	0.920648	−	1.59461i	0.798414	−	0.602108i	$-0.205672\pi$
$810$		0			0
$811$			891.748i			1.09957i	0.835308	+	0.549783i	$0.185290\pi$
$811$			891.748i			1.09957i	−0.835308	+	0.549783i	$0.814710\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	187.665	−	108.349i	0.230264	−	0.132943i
$816$		0			0
$817$	338.395	+	195.372i	0.414192	+	0.239134i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	528.845	−	915.987i	0.644148	−	1.11570i	−0.340350	−	0.940299i	$-0.610546\pi$
$821$	528.845	−	915.987i	0.644148	−	1.11570i	0.984498	−	0.175398i	$-0.0561211\pi$
$822$		0			0
$823$	−342.747	−	593.655i	−0.416460	−	0.721330i	0.579120	−	0.815242i	$-0.303396\pi$
$823$	−342.747	−	593.655i	−0.416460	−	0.721330i	−0.995581	+	0.0939119i	$0.970063\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	1271.30			1.53724			0.768622	−	0.639703i	$-0.220943\pi$
$827$	1271.30			1.53724			0.768622	+	0.639703i	$0.220943\pi$
$828$		0			0
$829$	426.025	+	245.966i	0.513902	+	0.296702i	0.734436	−	0.678678i	$-0.237447\pi$
$829$	426.025	+	245.966i	0.513902	+	0.296702i	−0.220534	+	0.975379i	$0.570780\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	4.70733	−	966.791i	0.00565106	−	1.16061i
$834$		0			0
$835$	−765.358	+	1325.64i	−0.916596	+	1.58759i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−658.564	+	380.222i	−0.784939	+	0.453185i	−0.838178	−	0.545397i	$-0.816379\pi$
$839$	−658.564	+	380.222i	−0.784939	+	0.453185i	0.0532387	+	0.998582i	$0.483046\pi$
$840$		0			0
$841$	−286.580	+	496.371i	−0.340761	+	0.590215i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	931.965	−	538.071i	1.10292	−	0.636770i
$846$		0			0
$847$	−350.230	−	203.344i	−0.413495	−	0.240076i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	68.5845	+	118.792i	0.0805929	+	0.139591i
$852$		0			0
$853$	282.187	−	162.921i	0.330818	−	0.190998i	−0.325386	−	0.945581i	$-0.605494\pi$
$853$	282.187	−	162.921i	0.330818	−	0.190998i	0.656204	+	0.754583i	$0.272161\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			715.677i			0.835096i	0.908655	+	0.417548i	$0.137110\pi$
$857$			715.677i			0.835096i	−0.908655	+	0.417548i	$0.862890\pi$
$858$		0			0
$859$			562.338i			0.654643i	0.944913	+	0.327322i	$0.106146\pi$
$859$			562.338i			0.654643i	−0.944913	+	0.327322i	$0.893854\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−296.451	−	513.468i	−0.343512	−	0.594980i	0.641570	−	0.767064i	$-0.278283\pi$
$863$	−296.451	−	513.468i	−0.343512	−	0.594980i	−0.985082	+	0.172084i	$0.944950\pi$
$864$		0			0
$865$	−13.4444	+	23.2864i	−0.0155426	+	0.0269206i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−35.7027	+	61.8389i	−0.0410848	+	0.0711610i
$870$		0			0
$871$		−	1922.31i		−	2.20701i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−447.413	+	770.604i	−0.511330	+	0.880690i
$876$		0			0
$877$	−962.505			−1.09750			−0.548749	−	0.835987i	$-0.684896\pi$
$877$	−962.505			−1.09750			−0.548749	+	0.835987i	$0.684896\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	277.658i		−	0.315162i	−0.987506	−	0.157581i	$-0.949630\pi$
$881$		−	277.658i		−	0.315162i	0.987506	−	0.157581i	$-0.0503695\pi$
$882$		0			0
$883$	1470.15			1.66495			0.832475	−	0.554063i	$-0.186923\pi$
$883$	1470.15			1.66495			0.832475	+	0.554063i	$0.186923\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	753.784i		−	0.849813i	−0.905237	−	0.424906i	$-0.860307\pi$
$887$		−	753.784i		−	0.849813i	0.905237	−	0.424906i	$-0.139693\pi$
$888$		0			0
$889$	1108.22	+	643.435i	1.24659	+	0.723774i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1298.39			−1.45396
$894$		0			0
$895$	660.704	+	381.458i	0.738217	+	0.426210i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−926.692	−	535.026i	−1.03080	−	0.595134i
$900$		0			0
$901$	1025.01	−	591.793i	1.13764	−	0.656818i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−973.351			−1.07553
$906$		0			0
$907$	1317.87			1.45300			0.726500	−	0.687167i	$-0.241146\pi$
$907$	1317.87			1.45300			0.726500	+	0.687167i	$0.241146\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−287.491	−	497.950i	−0.315578	−	0.546597i	0.663982	−	0.747748i	$-0.268865\pi$
$911$	−287.491	−	497.950i	−0.315578	−	0.546597i	−0.979560	+	0.201151i	$0.935532\pi$
$912$		0			0
$913$	354.286	−	204.547i	0.388046	−	0.224038i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−515.549	+	887.958i	−0.562213	+	0.968330i
$918$		0			0
$919$	393.926	+	682.300i	0.428647	+	0.742438i	0.996753	−	0.0805171i	$-0.0256572\pi$
$919$	393.926	+	682.300i	0.428647	+	0.742438i	−0.568106	+	0.822955i	$0.692324\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−51.3539	−	29.6492i	−0.0556380	−	0.0321226i
$924$		0			0
$925$	10.9865	+	19.0291i	0.0118773	+	0.0205720i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−780.938	−	450.875i	−0.840622	−	0.485334i	0.0168533	−	0.999858i	$-0.494635\pi$
$929$	−780.938	−	450.875i	−0.840622	−	0.485334i	−0.857476	+	0.514524i	$0.827968\pi$
$930$		0			0
$931$	−868.682	−	507.189i	−0.933064	−	0.544779i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	646.813	−	1120.31i	0.691778	−	1.19819i
$936$		0			0
$937$		−	172.726i		−	0.184340i	−0.995743	−	0.0921699i	$-0.970620\pi$
$937$		−	172.726i		−	0.184340i	0.995743	−	0.0921699i	$-0.0293803\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−421.021	+	243.077i	−0.447419	+	0.258317i	−0.706739	−	0.707474i	$-0.749835\pi$
$941$	−421.021	+	243.077i	−0.447419	+	0.258317i	0.259321	+	0.965791i	$0.416501\pi$
$942$		0			0
$943$	163.070	+	94.1487i	0.172927	+	0.0998396i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	243.598	−	421.925i	0.257232	−	0.445538i	−0.708268	−	0.705944i	$-0.750523\pi$
$947$	243.598	−	421.925i	0.257232	−	0.445538i	0.965499	+	0.260406i	$0.0838563\pi$
$948$		0			0
$949$	452.250	+	783.320i	0.476554	+	0.825416i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	623.761			0.654524			0.327262	−	0.944934i	$-0.393874\pi$
$953$	623.761			0.654524			0.327262	+	0.944934i	$0.393874\pi$
$954$		0			0
$955$	1017.05	+	587.192i	1.06497	+	0.614861i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−186.504	−	108.284i	−0.194477	−	0.112914i
$960$		0			0
$961$	−75.6623	+	131.051i	−0.0787329	+	0.136369i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−582.160	+	336.110i	−0.603275	+	0.348301i
$966$		0			0
$967$	383.882	−	664.903i	0.396982	−	0.687593i	−0.596370	−	0.802710i	$-0.703391\pi$
$967$	383.882	−	664.903i	0.396982	−	0.687593i	0.993352	+	0.115116i	$0.0367241\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	909.315	−	524.993i	0.936473	−	0.540673i	0.0476198	−	0.998866i	$-0.484836\pi$
$971$	909.315	−	524.993i	0.936473	−	0.540673i	0.888853	+	0.458193i	$0.151503\pi$
$972$		0			0
$973$	−334.993	−	583.501i	−0.344289	−	0.599693i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	242.755	+	420.463i	0.248469	+	0.430362i	0.963101	−	0.269139i	$-0.0867391\pi$
$977$	242.755	+	420.463i	0.248469	+	0.430362i	−0.714632	+	0.699501i	$0.753406\pi$
$978$		0			0
$979$	1582.47	−	913.637i	1.61641	−	0.933235i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	1537.64i		−	1.56423i	−0.623131	−	0.782117i	$-0.714140\pi$
$983$		−	1537.64i		−	1.56423i	0.623131	−	0.782117i	$-0.285860\pi$
$984$		0			0
$985$		−	1127.33i		−	1.14450i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−57.3662	−	99.3612i	−0.0580042	−	0.100466i
$990$		0			0
$991$	179.412	−	310.750i	0.181041	−	0.313572i	−0.761194	−	0.648524i	$-0.775387\pi$
$991$	179.412	−	310.750i	0.181041	−	0.313572i	0.942235	+	0.334952i	$0.108720\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	161.628	−	279.948i	0.162441	−	0.281355i
$996$		0			0
$997$		−	6.81490i		−	0.00683540i	−0.999994	−	0.00341770i	$-0.998912\pi$
$997$		−	6.81490i		−	0.00683540i	0.999994	−	0.00341770i	$-0.00108789\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.3.p.a.577.4		32
3.2	odd	2		252.3.p.a.157.15	yes	32
7.5	odd	6		756.3.bd.a.145.13		32
9.2	odd	6		252.3.bd.a.241.10	yes	32
9.7	even	3		756.3.bd.a.73.13		32
21.5	even	6		252.3.bd.a.229.10	yes	32
63.47	even	6		252.3.p.a.61.15	✓	32
63.61	odd	6	inner	756.3.p.a.397.13		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.3.p.a.61.15	✓	32	63.47	even	6
252.3.p.a.157.15	yes	32	3.2	odd	2
252.3.bd.a.229.10	yes	32	21.5	even	6
252.3.bd.a.241.10	yes	32	9.2	odd	6
756.3.p.a.397.13		32	63.61	odd	6	inner
756.3.p.a.577.4		32	1.1	even	1	trivial
756.3.bd.a.73.13		32	9.7	even	3
756.3.bd.a.145.13		32	7.5	odd	6

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	756.p (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-4.90249i q^{5} +(-6.05364 - 3.51475i) q^{7} -13.3736 q^{11} +(17.0699 + 9.85532i) q^{13} +(-17.0873 - 9.86533i) q^{17} +(-17.7784 + 10.2644i) q^{19} +6.02775 q^{23} +0.965576 q^{25} +(18.8027 + 32.5672i) q^{29} +(-24.6426 + 14.2274i) q^{31} +(-17.2310 + 29.6779i) q^{35} +(11.3781 + 19.7075i) q^{37} +(27.0533 + 15.6192i) q^{41} +(-9.51702 - 16.4840i) q^{43} +(54.7739 + 31.6237i) q^{47} +(24.2931 + 42.5540i) q^{49} +(-29.9936 + 51.9504i) q^{53} +65.5642i q^{55} +(-48.1779 + 27.8155i) q^{59} +(-75.0566 - 43.3340i) q^{61} +(48.3156 - 83.6851i) q^{65} +(-48.7632 - 84.4603i) q^{67} -3.00845 q^{71} +(39.7410 + 22.9445i) q^{73} +(80.9592 + 47.0050i) q^{77} +(2.66963 - 4.62393i) q^{79} +(-26.4913 + 15.2948i) q^{83} +(-48.3647 + 83.7701i) q^{85} +(-118.327 + 68.3162i) q^{89} +(-68.6961 - 119.657i) q^{91} +(50.3209 + 87.1584i) q^{95} +(-152.175 + 87.8581i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + q^{7} + 12 q^{11} + 15 q^{13} + 27 q^{17} - 30 q^{23} - 160 q^{25} - 24 q^{29} - 24 q^{31} - 141 q^{35} + 11 q^{37} + 90 q^{41} - 16 q^{43} - 108 q^{47} - 61 q^{49} - 54 q^{53} - 45 q^{59} - 165 q^{61}+ \cdots - 57 q^{97}+O(q^{100}) \) 32 * q + q^7 + 12 * q^11 + 15 * q^13 + 27 * q^17 - 30 * q^23 - 160 * q^25 - 24 * q^29 - 24 * q^31 - 141 * q^35 + 11 * q^37 + 90 * q^41 - 16 * q^43 - 108 * q^47 - 61 * q^49 - 54 * q^53 - 45 * q^59 - 165 * q^61 + 33 * q^65 - 35 * q^67 - 138 * q^71 + 123 * q^77 + 37 * q^79 - 378 * q^83 + 30 * q^85 + 261 * q^89 + 45 * q^91 + 48 * q^95 - 57 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more