LMFDB - Embedded newform 756.4.t.c.269.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [756,4,Mod(269,756)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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

Self dual:	no
Analytic conductor:	$44.6054439643$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 43x^{10} + 1500x^{8} - 13455x^{6} + 88433x^{4} - 270824x^{2} + 602176 $ x^12 - 43x^10 + 1500x^8 - 13455x^6 + 88433x^4 - 270824*x^2 + 602176
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{9}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			269.5
Root			$1.74910 - 1.00984i$ of defining polynomial
Character	$\chi$	$=$	756.269
Dual form			756.4.t.c.593.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(6.90269 + 11.9558i) q^{5} +(8.82127 - 16.2845i) q^{7} +O(q^{10})$	$q+(6.90269 + 11.9558i) q^{5} +(8.82127 - 16.2845i) q^{7} +(-35.5912 - 20.5486i) q^{11} +57.5129i q^{13} +(56.4236 - 97.7285i) q^{17} +(77.1424 - 44.5382i) q^{19} +(-56.2992 + 32.5044i) q^{23} +(-32.7942 + 56.8013i) q^{25} -277.651i q^{29} +(228.040 + 131.659i) q^{31} +(255.585 - 6.94153i) q^{35} +(-144.939 - 251.042i) q^{37} -140.955 q^{41} +260.465 q^{43} +(238.298 + 412.744i) q^{47} +(-187.370 - 287.300i) q^{49} +(-86.0654 - 49.6899i) q^{53} -567.361i q^{55} +(-335.121 + 580.446i) q^{59} +(119.160 - 68.7970i) q^{61} +(-687.614 + 396.994i) q^{65} +(101.900 - 176.496i) q^{67} -601.833i q^{71} +(832.666 + 480.740i) q^{73} +(-648.582 + 398.320i) q^{77} +(320.201 + 554.604i) q^{79} +1372.20 q^{83} +1557.90 q^{85} +(181.813 + 314.909i) q^{89} +(936.570 + 507.337i) q^{91} +(1064.98 + 614.866i) q^{95} -828.102i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 42 q^{7}+O(q^{10}) $ 12 * q - 42 * q^7	$ 12 q - 42 q^{7} + 186 q^{19} - 246 q^{25} + 738 q^{31} - 336 q^{37} + 756 q^{43} - 1218 q^{49} - 426 q^{61} - 1440 q^{67} + 2730 q^{73} + 1992 q^{79} + 3288 q^{85} + 1764 q^{91}+O(q^{100}) $ 12 * q - 42 * q^7 + 186 * q^19 - 246 * q^25 + 738 * q^31 - 336 * q^37 + 756 * q^43 - 1218 * q^49 - 426 * q^61 - 1440 * q^67 + 2730 * q^73 + 1992 * q^79 + 3288 * q^85 + 1764 * q^91

Display 100 coefficients 10 coefficients

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
$5$	6.90269	+	11.9558i	0.617395	+	1.06936i	0.989959	+	0.141353i	$0.0451454\pi$
$5$	6.90269	+	11.9558i	0.617395	+	1.06936i	−0.372564	+	0.928007i	$0.621521\pi$
$6$		0			0
$7$	8.82127	−	16.2845i	0.476304	−	0.879281i
$7$	8.82127	−	16.2845i	0.476304	−	0.879281i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−35.5912	−	20.5486i	−0.975558	−	0.563239i	−0.0746318	−	0.997211i	$-0.523778\pi$
$11$	−35.5912	−	20.5486i	−0.975558	−	0.563239i	−0.900926	+	0.433972i	$0.857111\pi$
$12$		0			0
$13$			57.5129i			1.22702i	0.789688	+	0.613509i	$0.210242\pi$
$13$			57.5129i			1.22702i	−0.789688	+	0.613509i	$0.789758\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	56.4236	−	97.7285i	0.804984	−	1.39427i	−0.111317	−	0.993785i	$-0.535507\pi$
$17$	56.4236	−	97.7285i	0.804984	−	1.39427i	0.916302	−	0.400489i	$-0.131160\pi$
$18$		0			0
$19$	77.1424	−	44.5382i	0.931457	−	0.537777i	0.0441846	−	0.999023i	$-0.485931\pi$
$19$	77.1424	−	44.5382i	0.931457	−	0.537777i	0.887272	+	0.461247i	$0.152598\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−56.2992	+	32.5044i	−0.510400	+	0.294680i	−0.732998	−	0.680231i	$-0.761880\pi$
$23$	−56.2992	+	32.5044i	−0.510400	+	0.294680i	0.222598	+	0.974910i	$0.428546\pi$
$24$		0			0
$25$	−32.7942	+	56.8013i	−0.262354	+	0.454410i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	277.651i		−	1.77788i	−0.458022	−	0.888941i	$-0.651442\pi$
$29$		−	277.651i		−	1.77788i	0.458022	−	0.888941i	$-0.348558\pi$
$30$		0			0
$31$	228.040	+	131.659i	1.32120	+	0.762794i	0.983920	−	0.178610i	$-0.0571602\pi$
$31$	228.040	+	131.659i	1.32120	+	0.762794i	0.337279	+	0.941405i	$0.390493\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	255.585	−	6.94153i	1.23434	−	0.0335238i
$36$		0			0
$37$	−144.939	−	251.042i	−0.643997	−	1.11544i	−0.984532	−	0.175204i	$-0.943942\pi$
$37$	−144.939	−	251.042i	−0.643997	−	1.11544i	0.340535	−	0.940232i	$-0.389392\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−140.955			−0.536916			−0.268458	−	0.963291i	$-0.586514\pi$
$41$	−140.955			−0.536916			−0.268458	+	0.963291i	$0.586514\pi$
$42$		0			0
$43$	260.465			0.923732			0.461866	−	0.886950i	$-0.347180\pi$
$43$	260.465			0.923732			0.461866	+	0.886950i	$0.347180\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	238.298	+	412.744i	0.739559	+	1.28095i	0.952694	+	0.303932i	$0.0982994\pi$
$47$	238.298	+	412.744i	0.739559	+	1.28095i	−0.213135	+	0.977023i	$0.568367\pi$
$48$		0			0
$49$	−187.370	−	287.300i	−0.546270	−	0.837609i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−86.0654	−	49.6899i	−0.223056	−	0.128782i	0.384308	−	0.923205i	$-0.374440\pi$
$53$	−86.0654	−	49.6899i	−0.223056	−	0.128782i	−0.607365	+	0.794423i	$0.707773\pi$
$54$		0			0
$55$		−	567.361i		−	1.39096i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−335.121	+	580.446i	−0.739475	+	1.28081i	0.213257	+	0.976996i	$0.431593\pi$
$59$	−335.121	+	580.446i	−0.739475	+	1.28081i	−0.952732	+	0.303812i	$0.901741\pi$
$60$		0			0
$61$	119.160	−	68.7970i	0.250113	−	0.144403i	−0.369703	−	0.929150i	$-0.620541\pi$
$61$	119.160	−	68.7970i	0.250113	−	0.144403i	0.619816	+	0.784747i	$0.287207\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−687.614	+	396.994i	−1.31212	+	0.757555i
$66$		0			0
$67$	101.900	−	176.496i	0.185807	−	0.321827i	−0.758041	−	0.652207i	$-0.773843\pi$
$67$	101.900	−	176.496i	0.185807	−	0.321827i	0.943848	+	0.330380i	$0.107177\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	601.833i		−	1.00598i	−0.864293	−	0.502989i	$-0.832234\pi$
$71$		−	601.833i		−	1.00598i	0.864293	−	0.502989i	$-0.167766\pi$
$72$		0			0
$73$	832.666	+	480.740i	1.33502	+	0.770772i	0.986064	−	0.166369i	$-0.0532043\pi$
$73$	832.666	+	480.740i	1.33502	+	0.770772i	0.348952	+	0.937141i	$0.386538\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−648.582	+	398.320i	−0.959907	+	0.589517i
$78$		0			0
$79$	320.201	+	554.604i	0.456017	+	0.789845i	0.998746	−	0.0500628i	$-0.0159421\pi$
$79$	320.201	+	554.604i	0.456017	+	0.789845i	−0.542729	+	0.839908i	$0.682609\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1372.20			1.81468			0.907340	−	0.420398i	$-0.138109\pi$
$83$	1372.20			1.81468			0.907340	+	0.420398i	$0.138109\pi$
$84$		0			0
$85$	1557.90			1.98797
$86$		0			0
$87$		0			0
$88$		0			0
$89$	181.813	+	314.909i	0.216541	+	0.375060i	0.953748	−	0.300607i	$-0.0971892\pi$
$89$	181.813	+	314.909i	0.216541	+	0.375060i	−0.737207	+	0.675667i	$0.763856\pi$
$90$		0			0
$91$	936.570	+	507.337i	1.07889	+	0.584433i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1064.98	+	614.866i	1.15015	+	0.664042i
$96$		0			0
$97$		−	828.102i		−	0.866815i	−0.901198	−	0.433407i	$-0.857311\pi$
$97$		−	828.102i		−	0.866815i	0.901198	−	0.433407i	$-0.142689\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	709.753	−	1229.33i	0.699239	−	1.21112i	−0.269492	−	0.963003i	$-0.586856\pi$
$101$	709.753	−	1229.33i	0.699239	−	1.21112i	0.968731	−	0.248114i	$-0.0798108\pi$
$102$		0			0
$103$	1622.12	−	936.533i	1.55177	−	0.895916i	0.553774	−	0.832667i	$-0.313187\pi$
$103$	1622.12	−	936.533i	1.55177	−	0.895916i	0.997998	−	0.0632484i	$-0.0201460\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1337.23	+	772.053i	−1.20818	+	0.697543i	−0.962362	−	0.271772i	$-0.912390\pi$
$107$	−1337.23	+	772.053i	−1.20818	+	0.697543i	−0.245819	+	0.969316i	$0.579057\pi$
$108$		0			0
$109$	58.6482	−	101.582i	0.0515365	−	0.0892639i	−0.839106	−	0.543967i	$-0.816921\pi$
$109$	58.6482	−	101.582i	0.0515365	−	0.0892639i	0.890643	+	0.454704i	$0.150255\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	972.690i		−	0.809761i	−0.914370	−	0.404881i	$-0.867313\pi$
$113$		−	972.690i		−	0.809761i	0.914370	−	0.404881i	$-0.132687\pi$
$114$		0			0
$115$	−777.232	−	448.735i	−0.630237	−	0.363868i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1093.73	−	1780.92i	−0.842541	−	1.37190i
$120$		0			0
$121$	178.987	+	310.015i	0.134476	+	0.232919i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	820.199			0.586886
$126$		0			0
$127$	232.704			0.162592			0.0812958	−	0.996690i	$-0.474094\pi$
$127$	232.704			0.162592			0.0812958	+	0.996690i	$0.474094\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−305.933	−	529.891i	−0.204042	−	0.353410i	0.745785	−	0.666186i	$-0.232074\pi$
$131$	−305.933	−	529.891i	−0.204042	−	0.353410i	−0.949827	+	0.312776i	$0.898741\pi$
$132$		0			0
$133$	−44.7888	−	1649.11i	−0.0292006	−	1.07516i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1648.39	−	951.696i	−1.02796	−	0.593495i	−0.111563	−	0.993757i	$-0.535586\pi$
$137$	−1648.39	−	951.696i	−1.02796	−	0.593495i	−0.916401	+	0.400262i	$0.868919\pi$
$138$		0			0
$139$			597.088i			0.364348i	0.983266	+	0.182174i	$0.0583134\pi$
$139$			597.088i			0.364348i	−0.983266	+	0.182174i	$0.941687\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1181.81	−	2046.95i	0.691103	−	1.19703i
$144$		0			0
$145$	3319.55	−	1916.54i	1.90120	−	1.09766i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−543.947	+	314.048i	−0.299073	+	0.172670i	−0.642026	−	0.766683i	$-0.721906\pi$
$149$	−543.947	+	314.048i	−0.299073	+	0.172670i	0.342954	+	0.939352i	$0.388573\pi$
$150$		0			0
$151$	939.284	−	1626.89i	0.506211	−	0.876783i	−0.493763	−	0.869596i	$-0.664379\pi$
$151$	939.284	−	1626.89i	0.506211	−	0.876783i	0.999974	−	0.00718665i	$-0.00228760\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			3635.20i			1.88378i
$156$		0			0
$157$	160.799	+	92.8375i	0.0817400	+	0.0471926i	0.540313	−	0.841464i	$-0.318306\pi$
$157$	160.799	+	92.8375i	0.0817400	+	0.0471926i	−0.458573	+	0.888657i	$0.651639\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	32.6873	+	1203.54i	0.0160007	+	0.589142i
$162$		0			0
$163$	731.972	+	1267.81i	0.351733	+	0.609219i	0.986553	−	0.163441i	$-0.0522592\pi$
$163$	731.972	+	1267.81i	0.351733	+	0.609219i	−0.634820	+	0.772660i	$0.718926\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1283.16			0.594572			0.297286	−	0.954788i	$-0.403919\pi$
$167$	1283.16			0.594572			0.297286	+	0.954788i	$0.403919\pi$
$168$		0			0
$169$	−1110.74			−0.505571
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1539.71	+	2666.86i	0.676660	+	1.17201i	0.975981	+	0.217856i	$0.0699064\pi$
$173$	1539.71	+	2666.86i	0.676660	+	1.17201i	−0.299321	+	0.954152i	$0.596760\pi$
$174$		0			0
$175$	635.694	+	1035.10i	0.274594	+	0.447120i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1251.05	−	722.292i	−0.522389	−	0.301601i	0.215523	−	0.976499i	$-0.430854\pi$
$179$	−1251.05	−	722.292i	−0.522389	−	0.301601i	−0.737911	+	0.674898i	$0.764188\pi$
$180$		0			0
$181$			2205.51i			0.905714i	0.891583	+	0.452857i	$0.149595\pi$
$181$			2205.51i			0.905714i	−0.891583	+	0.452857i	$0.850405\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2000.94	−	3465.74i	0.795202	−	1.37733i
$186$		0			0
$187$	−4016.36	+	2318.85i	−1.57062	+	0.906797i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1349.66	−	779.228i	0.511299	−	0.295199i	−0.222068	−	0.975031i	$-0.571281\pi$
$191$	1349.66	−	779.228i	0.511299	−	0.295199i	0.733368	+	0.679832i	$0.237947\pi$
$192$		0			0
$193$	1926.18	−	3336.24i	0.718390	−	1.24429i	−0.243248	−	0.969964i	$-0.578213\pi$
$193$	1926.18	−	3336.24i	0.718390	−	1.24429i	0.961637	−	0.274324i	$-0.0884539\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			457.056i			0.165299i	0.996579	+	0.0826494i	$0.0263382\pi$
$197$			457.056i			0.165299i	−0.996579	+	0.0826494i	$0.973662\pi$
$198$		0			0
$199$	−2047.48	−	1182.11i	−0.729356	−	0.421094i	0.0888304	−	0.996047i	$-0.471687\pi$
$199$	−2047.48	−	1182.11i	−0.729356	−	0.421094i	−0.818187	+	0.574953i	$0.805020\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−4521.42	−	2449.24i	−1.56326	−	0.846811i
$204$		0			0
$205$	−972.972	−	1685.24i	−0.331489	−	0.574156i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−3660.78			−1.21159
$210$		0			0
$211$	5393.44			1.75971			0.879857	−	0.475239i	$-0.157638\pi$
$211$	5393.44			1.75971			0.879857	+	0.475239i	$0.157638\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1797.91	+	3114.06i	0.570308	+	0.987802i
$216$		0			0
$217$	4155.60	−	2552.12i	1.30000	−	0.798383i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5620.66	+	3245.09i	1.71080	+	0.987729i
$222$		0			0
$223$			882.724i			0.265074i	0.991178	+	0.132537i	$0.0423124\pi$
$223$			882.724i			0.265074i	−0.991178	+	0.132537i	$0.957688\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2766.99	+	4792.56i	−0.809037	+	1.40129i	0.104495	+	0.994525i	$0.466677\pi$
$227$	−2766.99	+	4792.56i	−0.809037	+	1.40129i	−0.913532	+	0.406768i	$0.866656\pi$
$228$		0			0
$229$	−2099.57	+	1212.19i	−0.605868	+	0.349798i	−0.771347	−	0.636415i	$-0.780416\pi$
$229$	−2099.57	+	1212.19i	−0.605868	+	0.349798i	0.165479	+	0.986213i	$0.447083\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5052.62	+	2917.13i	−1.42064	+	0.820204i	−0.996353	−	0.0853257i	$-0.972807\pi$
$233$	−5052.62	+	2917.13i	−1.42064	+	0.820204i	−0.424282	+	0.905530i	$0.639474\pi$
$234$		0			0
$235$	−3289.79	+	5698.08i	−0.913201	+	1.58171i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	2431.96i		−	0.658202i	−0.944295	−	0.329101i	$-0.893254\pi$
$239$		−	2431.96i		−	0.658202i	0.944295	−	0.329101i	$-0.106746\pi$
$240$		0			0
$241$	−3899.69	−	2251.49i	−1.04233	−	0.601788i	−0.121836	−	0.992550i	$-0.538878\pi$
$241$	−3899.69	−	2251.49i	−1.04233	−	0.601788i	−0.920492	+	0.390762i	$0.872212\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2141.54	−	4223.31i	0.558442	−	1.10130i
$246$		0			0
$247$	2561.52	+	4436.69i	0.659861	+	1.14291i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	623.293			0.156741			0.0783704	−	0.996924i	$-0.475028\pi$
$251$	623.293			0.156741			0.0783704	+	0.996924i	$0.475028\pi$
$252$		0			0
$253$	2671.67			0.663900
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2314.84	+	4009.42i	0.561852	+	0.973156i	0.997335	+	0.0729589i	$0.0232442\pi$
$257$	2314.84	+	4009.42i	0.561852	+	0.973156i	−0.435483	+	0.900197i	$0.643422\pi$
$258$		0			0
$259$	−5366.65	+	145.755i	−1.28752	+	0.0349682i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1073.98	+	620.062i	0.251804	+	0.145379i	0.620590	−	0.784135i	$-0.286893\pi$
$263$	1073.98	+	620.062i	0.251804	+	0.145379i	−0.368786	+	0.929514i	$0.620227\pi$
$264$		0			0
$265$		−	1371.98i		−	0.318037i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−194.109	+	336.207i	−0.0439965	+	0.0762041i	−0.887185	−	0.461414i	$-0.847342\pi$
$269$	−194.109	+	336.207i	−0.0439965	+	0.0762041i	0.843189	+	0.537618i	$0.180676\pi$
$270$		0			0
$271$	99.8873	−	57.6700i	0.0223901	−	0.0129269i	−0.488763	−	0.872417i	$-0.662552\pi$
$271$	99.8873	−	57.6700i	0.0223901	−	0.0129269i	0.511153	+	0.859490i	$0.329218\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2334.37	−	1347.75i	0.511883	−	0.295536i
$276$		0			0
$277$	−4188.29	+	7254.34i	−0.908484	+	1.57354i	−0.0923135	+	0.995730i	$0.529426\pi$
$277$	−4188.29	+	7254.34i	−0.908484	+	1.57354i	−0.816171	+	0.577811i	$0.803907\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	2525.53i		−	0.536157i	−0.963397	−	0.268079i	$-0.913611\pi$
$281$		−	2525.53i		−	0.536157i	0.963397	−	0.268079i	$-0.0863887\pi$
$282$		0			0
$283$	781.546	+	451.226i	0.164163	+	0.0947794i	0.579831	−	0.814737i	$-0.303119\pi$
$283$	781.546	+	451.226i	0.164163	+	0.0947794i	−0.415668	+	0.909517i	$0.636452\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1243.41	+	2295.39i	−0.255735	+	0.472100i
$288$		0			0
$289$	−3910.74	−	6773.61i	−0.795999	−	1.37871i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−9361.31			−1.86653			−0.933266	−	0.359187i	$-0.883054\pi$
$293$	−9361.31			−1.86653			−0.933266	+	0.359187i	$0.883054\pi$
$294$		0			0
$295$	−9252.94			−1.82619
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1869.42	−	3237.93i	−0.361577	−	0.626269i
$300$		0			0
$301$	2297.63	−	4241.54i	0.439977	−	0.812220i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1645.05	+	949.769i	0.308837	+	0.178307i
$306$		0			0
$307$			2174.87i			0.404321i	0.979352	+	0.202161i	$0.0647963\pi$
$307$			2174.87i			0.404321i	−0.979352	+	0.202161i	$0.935204\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1441.27	−	2496.35i	0.262787	−	0.455161i	−0.704194	−	0.710007i	$-0.748692\pi$
$311$	1441.27	−	2496.35i	0.262787	−	0.455161i	0.966982	+	0.254846i	$0.0820249\pi$
$312$		0			0
$313$	−4723.23	+	2726.96i	−0.852948	+	0.492450i	−0.861644	−	0.507513i	$-0.830565\pi$
$313$	−4723.23	+	2726.96i	−0.852948	+	0.492450i	0.00869669	+	0.999962i	$0.497232\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	6327.02	−	3652.91i	1.12101	−	0.647217i	0.179353	−	0.983785i	$-0.442599\pi$
$317$	6327.02	−	3652.91i	1.12101	−	0.647217i	0.941659	+	0.336568i	$0.109266\pi$
$318$		0			0
$319$	−5705.34	+	9881.93i	−1.00137	+	1.73443i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	10052.0i		−	1.73161i
$324$		0			0
$325$	−3266.81	−	1886.09i	−0.557569	−	0.321913i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	8823.42	−	239.638i	1.47857	−	0.0401571i
$330$		0			0
$331$	−751.986	−	1302.48i	−0.124873	−	0.216286i	0.796810	−	0.604229i	$-0.206519\pi$
$331$	−751.986	−	1302.48i	−0.124873	−	0.216286i	−0.921683	+	0.387943i	$0.873186\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2813.53			0.458865
$336$		0			0
$337$	−5276.54			−0.852913			−0.426456	−	0.904508i	$-0.640238\pi$
$337$	−5276.54			−0.852913			−0.426456	+	0.904508i	$0.640238\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−5410.80	−	9371.78i	−0.859271	−	1.48830i
$342$		0			0
$343$	−6331.38	+	516.886i	−0.996684	+	0.0813680i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−7507.70	−	4334.57i	−1.16148	−	0.670582i	−0.209823	−	0.977739i	$-0.567289\pi$
$347$	−7507.70	−	4334.57i	−1.16148	−	0.670582i	−0.951659	+	0.307157i	$0.900622\pi$
$348$		0			0
$349$		−	2809.46i		−	0.430908i	−0.976514	−	0.215454i	$-0.930877\pi$
$349$		−	2809.46i		−	0.430908i	0.976514	−	0.215454i	$-0.0691232\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	222.247	−	384.944i	0.0335100	−	0.0580411i	−0.848784	−	0.528740i	$-0.822665\pi$
$353$	222.247	−	384.944i	0.0335100	−	0.0580411i	0.882294	+	0.470699i	$0.155998\pi$
$354$		0			0
$355$	7195.40	−	4154.27i	1.07575	−	0.621086i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−219.665	+	126.824i	−0.0322938	+	0.0186448i	−0.516060	−	0.856552i	$-0.672602\pi$
$359$	−219.665	+	126.824i	−0.0322938	+	0.0186448i	0.483766	+	0.875197i	$0.339269\pi$
$360$		0			0
$361$	537.797	−	931.492i	0.0784075	−	0.135806i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			13273.6i			1.90348i
$366$		0			0
$367$	−11543.5	−	6664.65i	−1.64187	−	0.947934i	−0.980168	−	0.198168i	$-0.936501\pi$
$367$	−11543.5	−	6664.65i	−1.64187	−	0.947934i	−0.661702	−	0.749767i	$-0.730166\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1568.38	+	963.205i	−0.219478	+	0.134790i
$372$		0			0
$373$	4284.06	+	7420.21i	0.594692	+	1.03004i	0.993590	+	0.113042i	$0.0360594\pi$
$373$	4284.06	+	7420.21i	0.594692	+	1.03004i	−0.398898	+	0.916995i	$0.630607\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	15968.5			2.18149
$378$		0			0
$379$	5015.67			0.679783			0.339892	−	0.940465i	$-0.389610\pi$
$379$	5015.67			0.679783			0.339892	+	0.940465i	$0.389610\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−2374.05	−	4111.98i	−0.316732	−	0.548596i	0.663072	−	0.748556i	$-0.269252\pi$
$383$	−2374.05	−	4111.98i	−0.316732	−	0.548596i	−0.979804	+	0.199960i	$0.935919\pi$
$384$		0			0
$385$	−9239.20	−	5004.85i	−1.22305	−	0.662521i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−4157.27	−	2400.20i	−0.541856	−	0.312841i	0.203975	−	0.978976i	$-0.434614\pi$
$389$	−4157.27	−	2400.20i	−0.541856	−	0.312841i	−0.745831	+	0.666135i	$0.767947\pi$
$390$		0			0
$391$			7336.05i			0.948850i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4420.49	+	7656.51i	−0.563086	+	0.975294i
$396$		0			0
$397$	−2411.67	+	1392.38i	−0.304883	+	0.176024i	−0.644634	−	0.764491i	$-0.722990\pi$
$397$	−2411.67	+	1392.38i	−0.304883	+	0.176024i	0.339752	+	0.940515i	$0.389657\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−8763.27	+	5059.48i	−1.09131	+	0.630070i	−0.933926	−	0.357467i	$-0.883641\pi$
$401$	−8763.27	+	5059.48i	−1.09131	+	0.630070i	−0.157388	+	0.987537i	$0.550307\pi$
$402$		0			0
$403$	−7572.09	+	13115.2i	−0.935962	+	1.62113i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			11913.2i			1.45090i
$408$		0			0
$409$	−1342.22	−	774.928i	−0.162270	−	0.0936864i	0.416666	−	0.909060i	$-0.363198\pi$
$409$	−1342.22	−	774.928i	−0.162270	−	0.0936864i	−0.578936	+	0.815373i	$0.696532\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	6496.09	+	10577.5i	0.773975	+	1.26026i
$414$		0			0
$415$	9471.87	+	16405.8i	1.12037	+	1.94055i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−401.529			−0.0468161			−0.0234081	−	0.999726i	$-0.507452\pi$
$419$	−401.529			−0.0468161			−0.0234081	+	0.999726i	$0.507452\pi$
$420$		0			0
$421$	4507.73			0.521836			0.260918	−	0.965361i	$-0.415975\pi$
$421$	4507.73			0.521836			0.260918	+	0.965361i	$0.415975\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	3700.74	+	6409.87i	0.422382	+	0.731586i
$426$		0			0
$427$	−69.1841	−	2547.34i	−0.00784088	−	0.288699i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−1239.54	−	715.647i	−0.138530	−	0.0799803i	0.429133	−	0.903241i	$-0.358819\pi$
$431$	−1239.54	−	715.647i	−0.138530	−	0.0799803i	−0.567663	+	0.823261i	$0.692152\pi$
$432$		0			0
$433$		−	1115.14i		−	0.123765i	−0.998083	−	0.0618826i	$-0.980290\pi$
$433$		−	1115.14i		−	0.123765i	0.998083	−	0.0618826i	$-0.0197104\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2895.37	+	5014.93i	−0.316944	+	0.548962i
$438$		0			0
$439$	−4322.17	+	2495.41i	−0.469900	+	0.271297i	−0.716198	−	0.697897i	$-0.754119\pi$
$439$	−4322.17	+	2495.41i	−0.469900	+	0.271297i	0.246298	+	0.969194i	$0.420786\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2006.48	−	1158.44i	0.215193	−	0.124242i	−0.388529	−	0.921436i	$-0.627017\pi$
$443$	2006.48	−	1158.44i	0.215193	−	0.124242i	0.603723	+	0.797194i	$0.293683\pi$
$444$		0			0
$445$	−2510.00	+	4347.44i	−0.267383	+	0.463120i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	14186.0i		−	1.49104i	−0.666481	−	0.745522i	$-0.732200\pi$
$449$		−	14186.0i		−	1.49104i	0.666481	−	0.745522i	$-0.267800\pi$
$450$		0			0
$451$	5016.77	+	2896.43i	0.523792	+	0.302412i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	399.228	+	14699.4i	0.0411342	+	1.51455i
$456$		0			0
$457$	−6577.70	−	11392.9i	−0.673286	−	1.16617i	−0.976967	−	0.213392i	$-0.931549\pi$
$457$	−6577.70	−	11392.9i	−0.673286	−	1.16617i	0.303680	−	0.952774i	$-0.401785\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−14004.0			−1.41482			−0.707409	−	0.706804i	$-0.750136\pi$
$461$	−14004.0			−1.41482			−0.707409	+	0.706804i	$0.750136\pi$
$462$		0			0
$463$	7911.37			0.794109			0.397054	−	0.917795i	$-0.370032\pi$
$463$	7911.37			0.794109			0.397054	+	0.917795i	$0.370032\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−6052.64	−	10483.5i	−0.599749	−	1.03880i	−0.992858	−	0.119304i	$-0.961934\pi$
$467$	−6052.64	−	10483.5i	−0.599749	−	1.03880i	0.393108	−	0.919492i	$-0.371400\pi$
$468$		0			0
$469$	−1975.26	−	3216.31i	−0.194476	−	0.316664i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−9270.24	−	5352.17i	−0.901154	−	0.520282i
$474$		0			0
$475$			5842.38i			0.564351i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	4454.29	−	7715.05i	0.424889	−	0.735929i	−0.571521	−	0.820587i	$-0.693647\pi$
$479$	4454.29	−	7715.05i	0.424889	−	0.735929i	0.996410	+	0.0846585i	$0.0269799\pi$
$480$		0			0
$481$	14438.2	−	8335.89i	1.36866	−	0.790195i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	9900.63	−	5716.13i	0.926937	−	0.535167i
$486$		0			0
$487$	−3029.89	+	5247.93i	−0.281925	+	0.488309i	−0.971859	−	0.235564i	$-0.924306\pi$
$487$	−3029.89	+	5247.93i	−0.281925	+	0.488309i	0.689934	+	0.723873i	$0.257640\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			13210.0i			1.21418i	0.794634	+	0.607089i	$0.207663\pi$
$491$			13210.0i			1.21418i	−0.794634	+	0.607089i	$0.792337\pi$
$492$		0			0
$493$	−27134.5	−	15666.1i	−2.47885	−	1.43117i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−9800.56	−	5308.93i	−0.884538	−	0.479151i
$498$		0			0
$499$	1741.82	+	3016.91i	0.156261	+	0.270653i	0.933518	−	0.358532i	$-0.116722\pi$
$499$	1741.82	+	3016.91i	0.156261	+	0.270653i	−0.777256	+	0.629184i	$0.783389\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	16902.6			1.49831			0.749154	−	0.662396i	$-0.230461\pi$
$503$	16902.6			1.49831			0.749154	+	0.662396i	$0.230461\pi$
$504$		0			0
$505$	19596.8			1.72683
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−4357.87	−	7548.04i	−0.379487	−	0.657291i	0.611501	−	0.791244i	$-0.290566\pi$
$509$	−4357.87	−	7548.04i	−0.379487	−	0.657291i	−0.990988	+	0.133953i	$0.957233\pi$
$510$		0			0
$511$	15173.8	−	9318.82i	1.31360	−	0.806732i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	22394.0	+	12929.2i	1.91611	+	1.10627i
$516$		0			0
$517$		−	19586.7i		−	1.66619i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−11287.1	+	19549.9i	−0.949132	+	1.64394i	−0.201873	+	0.979412i	$0.564703\pi$
$521$	−11287.1	+	19549.9i	−0.949132	+	1.64394i	−0.747259	+	0.664533i	$0.768631\pi$
$522$		0			0
$523$	−12790.3	+	7384.48i	−1.06937	+	0.617401i	−0.928009	−	0.372557i	$-0.878481\pi$
$523$	−12790.3	+	7384.48i	−1.06937	+	0.617401i	−0.141361	+	0.989958i	$0.545148\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	25733.7	−	14857.3i	2.12709	−	1.22808i
$528$		0			0
$529$	−3970.43	+	6876.99i	−0.326328	+	0.565216i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	8106.76i		−	0.658805i
$534$		0			0
$535$	−18461.0	−	10658.5i	−1.49185	−	0.861320i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	765.130	+	14075.5i	0.0611437	+	1.12482i
$540$		0			0
$541$	−2903.12	−	5028.35i	−0.230711	−	0.399604i	0.727306	−	0.686313i	$-0.240772\pi$
$541$	−2903.12	−	5028.35i	−0.230711	−	0.399604i	−0.958018	+	0.286709i	$0.907439\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1619.32			0.127274
$546$		0			0
$547$	1466.05			0.114595			0.0572977	−	0.998357i	$-0.481752\pi$
$547$	1466.05			0.114595			0.0572977	+	0.998357i	$0.481752\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−12366.1	−	21418.7i	−0.956103	−	1.65602i
$552$		0			0
$553$	11856.0	−	322.002i	0.911699	−	0.0247612i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−16685.6	−	9633.41i	−1.26928	−	0.732820i	−0.294429	−	0.955673i	$-0.595130\pi$
$557$	−16685.6	−	9633.41i	−1.26928	−	0.732820i	−0.974852	+	0.222853i	$0.928463\pi$
$558$		0			0
$559$			14980.1i			1.13343i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1960.15	+	3395.08i	−0.146732	+	0.254148i	−0.930018	−	0.367514i	$-0.880209\pi$
$563$	−1960.15	+	3395.08i	−0.146732	+	0.254148i	0.783286	+	0.621662i	$0.213542\pi$
$564$		0			0
$565$	11629.3	−	6714.18i	0.865926	−	0.499943i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	8741.34	−	5046.82i	0.644035	−	0.371834i	−0.142132	−	0.989848i	$-0.545396\pi$
$569$	8741.34	−	5046.82i	0.644035	−	0.371834i	0.786167	+	0.618014i	$0.212062\pi$
$570$		0			0
$571$	−326.600	+	565.687i	−0.0239365	+	0.0414593i	−0.877746	−	0.479127i	$-0.840953\pi$
$571$	−326.600	+	565.687i	−0.0239365	+	0.0414593i	0.853809	+	0.520586i	$0.174287\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	4263.82i		−	0.309241i
$576$		0			0
$577$	−1603.38	−	925.713i	−0.115684	−	0.0667902i	0.441042	−	0.897487i	$-0.354609\pi$
$577$	−1603.38	−	925.713i	−0.115684	−	0.0667902i	−0.556725	+	0.830697i	$0.687942\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	12104.5	−	22345.6i	0.864339	−	1.59561i
$582$		0			0
$583$	2042.11	+	3537.04i	0.145070	+	0.251268i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−12502.1			−0.879075			−0.439537	−	0.898224i	$-0.644858\pi$
$587$	−12502.1			−0.879075			−0.439537	+	0.898224i	$0.644858\pi$
$588$		0			0
$589$	23455.4			1.64085
$590$		0			0
$591$		0			0
$592$		0			0
$593$	4196.37	+	7268.33i	0.290598	+	0.503330i	0.973951	−	0.226758i	$-0.0728126\pi$
$593$	4196.37	+	7268.33i	0.290598	+	0.503330i	−0.683354	+	0.730088i	$0.739479\pi$
$594$		0			0
$595$	13742.6	−	25369.6i	0.946879	−	1.74799i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1190.12	+	687.119i	0.0811806	+	0.0468696i	0.540041	−	0.841639i	$-0.318409\pi$
$599$	1190.12	+	687.119i	0.0811806	+	0.0468696i	−0.458860	+	0.888508i	$0.651742\pi$
$600$		0			0
$601$			18277.0i			1.24049i	0.784408	+	0.620245i	$0.212967\pi$
$601$			18277.0i			1.24049i	−0.784408	+	0.620245i	$0.787033\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2470.98	+	4279.87i	−0.166049	+	0.287606i
$606$		0			0
$607$	780.950	−	450.882i	0.0522204	−	0.0301495i	−0.473663	−	0.880706i	$-0.657068\pi$
$607$	780.950	−	450.882i	0.0522204	−	0.0301495i	0.525883	+	0.850557i	$0.323735\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−23738.1	+	13705.2i	−1.57175	+	0.907452i
$612$		0			0
$613$	8361.05	−	14481.8i	0.550896	−	0.954181i	−0.447314	−	0.894377i	$-0.647619\pi$
$613$	8361.05	−	14481.8i	0.550896	−	0.954181i	0.998210	−	0.0598035i	$-0.0190474\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	18190.2i		−	1.18689i	−0.804875	−	0.593444i	$-0.797768\pi$
$617$		−	18190.2i		−	1.18689i	0.804875	−	0.593444i	$-0.202232\pi$
$618$		0			0
$619$	22572.0	+	13031.9i	1.46566	+	0.846200i	0.999263	−	0.0383743i	$-0.0122179\pi$
$619$	22572.0	+	13031.9i	1.46566	+	0.846200i	0.466399	+	0.884575i	$0.345551\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	6731.97	−	182.836i	0.432922	−	0.0117579i
$624$		0			0
$625$	9760.86	+	16906.3i	0.624695	+	1.08200i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−32712.0			−2.07363
$630$		0			0
$631$	−371.934			−0.0234650			−0.0117325	−	0.999931i	$-0.503735\pi$
$631$	−371.934			−0.0234650			−0.0117325	+	0.999931i	$0.503735\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1606.28	+	2782.16i	0.100383	+	0.173869i
$636$		0			0
$637$	16523.5	−	10776.2i	1.02776	−	0.670282i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	10616.1	+	6129.20i	0.654150	+	0.377673i	0.790044	−	0.613050i	$-0.210058\pi$
$641$	10616.1	+	6129.20i	0.654150	+	0.377673i	−0.135895	+	0.990723i	$0.543391\pi$
$642$		0			0
$643$			23619.7i			1.44863i	0.689469	+	0.724315i	$0.257844\pi$
$643$			23619.7i			1.44863i	−0.689469	+	0.724315i	$0.742156\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	6707.48	−	11617.7i	0.407571	−	0.705933i	−0.587046	−	0.809554i	$-0.699709\pi$
$647$	6707.48	−	11617.7i	0.407571	−	0.705933i	0.994617	+	0.103620i	$0.0330426\pi$
$648$		0			0
$649$	23854.7	−	13772.5i	1.44280	−	0.833002i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	8873.40	−	5123.06i	0.531766	−	0.307015i	−0.209969	−	0.977708i	$-0.567336\pi$
$653$	8873.40	−	5123.06i	0.531766	−	0.307015i	0.741735	+	0.670693i	$0.234003\pi$
$654$		0			0
$655$	4223.52	−	7315.34i	0.251949	−	0.436388i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			27958.0i			1.65264i	0.563203	+	0.826318i	$0.309569\pi$
$659$			27958.0i			1.65264i	−0.563203	+	0.826318i	$0.690431\pi$
$660$		0			0
$661$	−6982.19	−	4031.17i	−0.410856	−	0.237208i	0.280301	−	0.959912i	$-0.409566\pi$
$661$	−6982.19	−	4031.17i	−0.410856	−	0.237208i	−0.691157	+	0.722704i	$0.742899\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	19407.3	−	11918.8i	1.13170	−	0.695023i
$666$		0			0
$667$	9024.88	+	15631.6i	0.523905	+	0.907431i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−5654.72			−0.325333
$672$		0			0
$673$	10696.6			0.612668			0.306334	−	0.951924i	$-0.400898\pi$
$673$	10696.6			0.612668			0.306334	+	0.951924i	$0.400898\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	14294.5	+	24758.8i	0.811495	+	1.40555i	0.911818	+	0.410596i	$0.134679\pi$
$677$	14294.5	+	24758.8i	0.811495	+	1.40555i	−0.100323	+	0.994955i	$0.531987\pi$
$678$		0			0
$679$	−13485.2	−	7304.91i	−0.762173	−	0.412867i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1675.51	+	967.357i	0.0938677	+	0.0541945i	0.546199	−	0.837655i	$-0.316074\pi$
$683$	1675.51	+	967.357i	0.0938677	+	0.0541945i	−0.452331	+	0.891850i	$0.649408\pi$
$684$		0			0
$685$		−	26277.0i		−	1.46568i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2857.81	−	4949.87i	0.158017	−	0.273694i
$690$		0			0
$691$	11556.5	−	6672.18i	0.636226	−	0.367325i	−0.146934	−	0.989146i	$-0.546940\pi$
$691$	11556.5	−	6672.18i	0.636226	−	0.367325i	0.783159	+	0.621821i	$0.213607\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−7138.67	+	4121.51i	−0.389619	+	0.224947i
$696$		0			0
$697$	−7953.21	+	13775.4i	−0.432209	+	0.748607i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			1011.59i			0.0545040i	0.999629	+	0.0272520i	$0.00867566\pi$
$701$			1011.59i			0.0545040i	−0.999629	+	0.0272520i	$0.991324\pi$
$702$		0			0
$703$	−22361.9	−	12910.7i	−1.19971	−	0.692653i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−13758.1	−	22402.2i	−0.731862	−	1.19169i
$708$		0			0
$709$	6697.92	+	11601.1i	0.354789	+	0.614513i	0.987082	−	0.160217i	$-0.0512193\pi$
$709$	6697.92	+	11601.1i	0.354789	+	0.614513i	−0.632293	+	0.774730i	$0.717886\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−17118.0			−0.899120
$714$		0			0
$715$	32630.6			1.70674
$716$		0			0
$717$		0			0
$718$		0			0
$719$	12849.9	+	22256.6i	0.666509	+	1.15443i	0.978874	+	0.204465i	$0.0655454\pi$
$719$	12849.9	+	22256.6i	0.666509	+	1.15443i	−0.312365	+	0.949962i	$0.601121\pi$
$720$		0			0
$721$	−941.802	−	34676.9i	−0.0486471	−	1.79117i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	15771.0	+	9105.36i	0.807888	+	0.466434i
$726$		0			0
$727$		−	19526.6i		−	0.996149i	−0.867134	−	0.498074i	$-0.834041\pi$
$727$		−	19526.6i		−	0.996149i	0.867134	−	0.498074i	$-0.165959\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	14696.3	−	25454.8i	0.743590	−	1.28794i
$732$		0			0
$733$	23713.8	−	13691.2i	1.19494	−	0.689897i	0.235514	−	0.971871i	$-0.424323\pi$
$733$	23713.8	−	13691.2i	1.19494	−	0.689897i	0.959422	+	0.281974i	$0.0909893\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−7253.47	+	4187.79i	−0.362531	+	0.209307i
$738$		0			0
$739$	−19235.8	+	33317.4i	−0.957512	+	1.65846i	−0.229001	+	0.973426i	$0.573546\pi$
$739$	−19235.8	+	33317.4i	−0.957512	+	1.65846i	−0.728511	+	0.685034i	$0.759787\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			14729.9i			0.727304i	0.931535	+	0.363652i	$0.118470\pi$
$743$			14729.9i			0.727304i	−0.931535	+	0.363652i	$0.881530\pi$
$744$		0			0
$745$	−7509.39	−	4335.55i	−0.369292	−	0.213211i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	776.397	+	28586.7i	0.0378757	+	1.39457i
$750$		0			0
$751$	−3886.66	−	6731.89i	−0.188850	−	0.327098i	0.756017	−	0.654552i	$-0.227143\pi$
$751$	−3886.66	−	6731.89i	−0.188850	−	0.327098i	−0.944867	+	0.327454i	$0.893809\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	25934.3			1.25013
$756$		0			0
$757$	33391.2			1.60320			0.801601	−	0.597860i	$-0.203982\pi$
$757$	33391.2			1.60320			0.801601	+	0.597860i	$0.203982\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	10098.8	+	17491.7i	0.481053	+	0.833209i	0.999764	−	0.0217412i	$-0.00692097\pi$
$761$	10098.8	+	17491.7i	0.481053	+	0.833209i	−0.518710	+	0.854950i	$0.673588\pi$
$762$		0			0
$763$	−1136.86	−	1851.14i	−0.0539410	−	0.0878318i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−33383.2	−	19273.8i	−1.57157	−	0.907348i
$768$		0			0
$769$		−	3862.66i		−	0.181133i	−0.995890	−	0.0905665i	$-0.971132\pi$
$769$		−	3862.66i		−	0.181133i	0.995890	−	0.0905665i	$-0.0288678\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−8478.71	+	14685.6i	−0.394512	+	0.683315i	−0.993039	−	0.117788i	$-0.962420\pi$
$773$	−8478.71	+	14685.6i	−0.394512	+	0.683315i	0.598527	+	0.801103i	$0.295753\pi$
$774$		0			0
$775$	−14956.8	+	8635.30i	−0.693243	+	0.400244i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−10873.6	+	6277.90i	−0.500114	+	0.288741i
$780$		0			0
$781$	−12366.8	+	21419.9i	−0.566606	+	0.981390i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			2563.31i			0.116546i
$786$		0			0
$787$	−24154.2	−	13945.4i	−1.09403	−	0.631640i	−0.159385	−	0.987216i	$-0.550951\pi$
$787$	−24154.2	−	13945.4i	−1.09403	−	0.631640i	−0.934647	+	0.355576i	$0.884285\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−15839.8	−	8580.36i	−0.712008	−	0.385692i
$792$		0			0
$793$	3956.72	+	6853.24i	0.177184	+	0.306893i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	6934.09			0.308178			0.154089	−	0.988057i	$-0.450756\pi$
$797$	6934.09			0.308178			0.154089	+	0.988057i	$0.450756\pi$
$798$		0			0
$799$	53782.5			2.38133
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−19757.0	−	34220.2i	−0.868257	−	1.50386i
$804$		0			0
$805$	−14163.6	+	8698.43i	−0.620126	+	0.380844i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	13288.6	+	7672.20i	0.577507	+	0.333424i	0.760142	−	0.649757i	$-0.225129\pi$
$809$	13288.6	+	7672.20i	0.577507	+	0.333424i	−0.182635	+	0.983181i	$0.558463\pi$
$810$		0			0
$811$			6315.36i			0.273443i	0.990610	+	0.136721i	$0.0436565\pi$
$811$			6315.36i			0.273443i	−0.990610	+	0.136721i	$0.956343\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−10105.2	+	17502.6i	−0.434317	+	0.752258i
$816$		0			0
$817$	20092.9	−	11600.6i	0.860416	−	0.496762i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	5211.61	−	3008.92i	0.221542	−	0.127908i	−0.385122	−	0.922866i	$-0.625841\pi$
$821$	5211.61	−	3008.92i	0.221542	−	0.127908i	0.606664	+	0.794958i	$0.292507\pi$
$822$		0			0
$823$	6811.63	−	11798.1i	0.288504	−	0.499703i	−0.684949	−	0.728591i	$-0.740176\pi$
$823$	6811.63	−	11798.1i	0.288504	−	0.499703i	0.973453	+	0.228888i	$0.0735089\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			14214.7i			0.597694i	0.954301	+	0.298847i	$0.0966020\pi$
$827$			14214.7i			0.597694i	−0.954301	+	0.298847i	$0.903398\pi$
$828$		0			0
$829$	−18508.4	−	10685.8i	−0.775418	−	0.447688i	0.0593859	−	0.998235i	$-0.481086\pi$
$829$	−18508.4	−	10685.8i	−0.775418	−	0.447688i	−0.834804	+	0.550547i	$0.814419\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−38649.5	+	2100.94i	−1.60760	+	0.0873870i
$834$		0			0
$835$	8857.22	+	15341.2i	0.367086	+	0.635812i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−14814.1			−0.609583			−0.304791	−	0.952419i	$-0.598587\pi$
$839$	−14814.1			−0.609583			−0.304791	+	0.952419i	$0.598587\pi$
$840$		0			0
$841$	−52701.3			−2.16086
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−7667.09	−	13279.8i	−0.312137	−	0.540637i
$846$		0			0
$847$	6627.33	−	179.994i	0.268852	−	0.00730185i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	16320.0	+	9422.33i	0.657392	+	0.379546i
$852$		0			0
$853$		−	72.2930i		−	0.00290183i	−0.999999	−	0.00145092i	$-0.999538\pi$
$853$		−	72.2930i		−	0.00290183i	0.999999	−	0.00145092i	$-0.000461841\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1544.70	−	2675.50i	0.0615705	−	0.106643i	−0.833597	−	0.552373i	$-0.813722\pi$
$857$	1544.70	−	2675.50i	0.0615705	−	0.106643i	0.895168	+	0.445730i	$0.147056\pi$
$858$		0			0
$859$	17965.2	−	10372.2i	0.713581	−	0.411986i	−0.0988044	−	0.995107i	$-0.531502\pi$
$859$	17965.2	−	10372.2i	0.713581	−	0.411986i	0.812386	+	0.583121i	$0.198168\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	3404.77	−	1965.75i	0.134299	−	0.0775374i	−0.431345	−	0.902187i	$-0.641961\pi$
$863$	3404.77	−	1965.75i	0.134299	−	0.0775374i	0.565644	+	0.824649i	$0.308628\pi$
$864$		0			0
$865$	−21256.3	+	36817.0i	−0.835533	+	1.44719i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	26318.6i		−	1.02739i
$870$		0			0
$871$	10150.8	+	5860.57i	0.394887	+	0.227988i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	7235.19	−	13356.5i	0.279536	−	0.516038i
$876$		0			0
$877$	−5314.79	−	9205.49i	−0.204638	−	0.354444i	0.745379	−	0.666641i	$-0.232268\pi$
$877$	−5314.79	−	9205.49i	−0.204638	−	0.354444i	−0.950017	+	0.312197i	$0.898935\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	41443.8			1.58488			0.792438	−	0.609952i	$-0.208811\pi$
$881$	41443.8			1.58488			0.792438	+	0.609952i	$0.208811\pi$
$882$		0			0
$883$	6698.61			0.255296			0.127648	−	0.991820i	$-0.459257\pi$
$883$	6698.61			0.255296			0.127648	+	0.991820i	$0.459257\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	8864.70	+	15354.1i	0.335567	+	0.581219i	0.983594	−	0.180399i	$-0.0577389\pi$
$887$	8864.70	+	15354.1i	0.335567	+	0.581219i	−0.648027	+	0.761618i	$0.724406\pi$
$888$		0			0
$889$	2052.74	−	3789.47i	0.0774429	−	0.142964i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	36765.7	+	21226.7i	1.37773	+	0.795436i
$894$		0			0
$895$		−	19943.0i		−	0.744829i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	36555.3	−	63315.6i	1.35616	−	2.34893i
$900$		0			0
$901$	−9712.24	+	5607.36i	−0.359114	+	0.207334i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−26368.6	+	15223.9i	−0.968534	+	0.559183i
$906$		0			0
$907$	−7415.53	+	12844.1i	−0.271476	+	0.470210i	−0.969240	−	0.246118i	$-0.920845\pi$
$907$	−7415.53	+	12844.1i	−0.271476	+	0.470210i	0.697764	+	0.716328i	$0.254178\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	23039.0i		−	0.837887i	−0.908012	−	0.418943i	$-0.862401\pi$
$911$		−	23039.0i		−	0.837887i	0.908012	−	0.418943i	$-0.137599\pi$
$912$		0			0
$913$	−48838.2	−	28196.7i	−1.77033	−	1.02210i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−11327.7	+	307.654i	−0.407933	+	0.0110792i
$918$		0			0
$919$	−20275.1	−	35117.5i	−0.727762	−	1.26052i	−0.957827	−	0.287345i	$-0.907227\pi$
$919$	−20275.1	−	35117.5i	−0.727762	−	1.26052i	0.230066	−	0.973175i	$-0.426106\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	34613.2			1.23435
$924$		0			0
$925$	19012.7			0.675821
$926$		0			0
$927$		0			0
$928$		0			0
$929$	11232.2	+	19454.7i	0.396680	+	0.687069i	0.993314	−	0.115444i	$-0.0368291\pi$
$929$	11232.2	+	19454.7i	0.396680	+	0.687069i	−0.596634	+	0.802513i	$0.703496\pi$
$930$		0			0
$931$	−27250.0	−	13817.9i	−0.959273	−	0.486426i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−55447.4	−	32012.6i	−1.93938	−	1.11970i
$936$		0			0
$937$			28667.9i			0.999509i	0.866167	+	0.499755i	$0.166576\pi$
$937$			28667.9i			0.999509i	−0.866167	+	0.499755i	$0.833424\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	3897.91	−	6751.38i	0.135035	−	0.233888i	−0.790576	−	0.612365i	$-0.790218\pi$
$941$	3897.91	−	6751.38i	0.135035	−	0.233888i	0.925611	+	0.378476i	$0.123552\pi$
$942$		0			0
$943$	7935.68	−	4581.67i	0.274042	−	0.158218i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	4880.63	−	2817.83i	0.167475	−	0.0966919i	−0.413920	−	0.910313i	$-0.635841\pi$
$947$	4880.63	−	2817.83i	0.167475	−	0.0966919i	0.581395	+	0.813622i	$0.302507\pi$
$948$		0			0
$949$	−27648.8	+	47889.0i	−0.945750	+	1.63809i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	28590.4i		−	0.971808i	−0.874012	−	0.485904i	$-0.838490\pi$
$953$		−	28590.4i		−	0.971808i	0.874012	−	0.485904i	$-0.161510\pi$
$954$		0			0
$955$	18632.6	+	10757.5i	0.631347	+	0.364509i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−30038.7	+	18448.0i	−1.01147	+	0.621185i
$960$		0			0
$961$	19772.6	+	34247.2i	0.663711	+	1.14958i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	53183.2			1.77412
$966$		0			0
$967$	6507.00			0.216392			0.108196	−	0.994130i	$-0.465493\pi$
$967$	6507.00			0.216392			0.108196	+	0.994130i	$0.465493\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	3949.43	+	6840.61i	0.130528	+	0.226082i	0.923880	−	0.382681i	$-0.124999\pi$
$971$	3949.43	+	6840.61i	0.130528	+	0.226082i	−0.793352	+	0.608763i	$0.791666\pi$
$972$		0			0
$973$	9723.28	+	5267.07i	0.320364	+	0.173540i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−23212.0	−	13401.5i	−0.760101	−	0.438845i	0.0692308	−	0.997601i	$-0.477946\pi$
$977$	−23212.0	−	13401.5i	−0.760101	−	0.438845i	−0.829332	+	0.558756i	$0.811279\pi$
$978$		0			0
$979$		−	14944.0i		−	0.487857i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−4616.33	+	7995.71i	−0.149784	+	0.259434i	−0.931148	−	0.364642i	$-0.881191\pi$
$983$	−4616.33	+	7995.71i	−0.149784	+	0.259434i	0.781363	+	0.624076i	$0.214525\pi$
$984$		0			0
$985$	−5464.47	+	3154.91i	−0.176764	+	0.102055i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−14664.0	+	8466.24i	−0.471473	+	0.272205i
$990$		0			0
$991$	−26268.9	+	45499.0i	−0.842037	+	1.45845i	0.0461326	+	0.998935i	$0.485310\pi$
$991$	−26268.9	+	45499.0i	−0.842037	+	1.45845i	−0.888170	+	0.459516i	$0.848023\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	32639.0i		−	1.03993i
$996$		0			0
$997$	20043.6	+	11572.2i	0.636696	+	0.367597i	0.783341	−	0.621593i	$-0.213514\pi$
$997$	20043.6	+	11572.2i	0.636696	+	0.367597i	−0.146645	+	0.989189i	$0.546847\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.4.t.c.269.5	yes	12
3.2	odd	2	inner	756.4.t.c.269.2	✓	12
7.5	odd	6	inner	756.4.t.c.593.2	yes	12
21.5	even	6	inner	756.4.t.c.593.5	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.4.t.c.269.2	✓	12	3.2	odd	2	inner
756.4.t.c.269.5	yes	12	1.1	even	1	trivial
756.4.t.c.593.2	yes	12	7.5	odd	6	inner
756.4.t.c.593.5	yes	12	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(6.90269 + 11.9558i) q^{5} +(8.82127 - 16.2845i) q^{7} +O(q^{10})\)	\(q+(6.90269 + 11.9558i) q^{5} +(8.82127 - 16.2845i) q^{7} +(-35.5912 - 20.5486i) q^{11} +57.5129i q^{13} +(56.4236 - 97.7285i) q^{17} +(77.1424 - 44.5382i) q^{19} +(-56.2992 + 32.5044i) q^{23} +(-32.7942 + 56.8013i) q^{25} -277.651i q^{29} +(228.040 + 131.659i) q^{31} +(255.585 - 6.94153i) q^{35} +(-144.939 - 251.042i) q^{37} -140.955 q^{41} +260.465 q^{43} +(238.298 + 412.744i) q^{47} +(-187.370 - 287.300i) q^{49} +(-86.0654 - 49.6899i) q^{53} -567.361i q^{55} +(-335.121 + 580.446i) q^{59} +(119.160 - 68.7970i) q^{61} +(-687.614 + 396.994i) q^{65} +(101.900 - 176.496i) q^{67} -601.833i q^{71} +(832.666 + 480.740i) q^{73} +(-648.582 + 398.320i) q^{77} +(320.201 + 554.604i) q^{79} +1372.20 q^{83} +1557.90 q^{85} +(181.813 + 314.909i) q^{89} +(936.570 + 507.337i) q^{91} +(1064.98 + 614.866i) q^{95} -828.102i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 42 q^{7}+O(q^{10}) \) 12 * q - 42 * q^7	\( 12 q - 42 q^{7} + 186 q^{19} - 246 q^{25} + 738 q^{31} - 336 q^{37} + 756 q^{43} - 1218 q^{49} - 426 q^{61} - 1440 q^{67} + 2730 q^{73} + 1992 q^{79} + 3288 q^{85} + 1764 q^{91}+O(q^{100}) \) 12 * q - 42 * q^7 + 186 * q^19 - 246 * q^25 + 738 * q^31 - 336 * q^37 + 756 * q^43 - 1218 * q^49 - 426 * q^61 - 1440 * q^67 + 2730 * q^73 + 1992 * q^79 + 3288 * q^85 + 1764 * q^91

Introduction

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