LMFDB - Embedded newform 756.4.t.e.269.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

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

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

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

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

Newform invariants

comment:select newform

sage:traces = [32,0,0,0,0,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$44.6054439643$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			269.10
Character	$\chi$	$=$	756.269
Dual form			756.4.t.e.593.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.13836 + 3.70374i) q^{5} +(-15.9269 + 9.45161i) q^{7} +(-35.6565 - 20.5863i) q^{11} -24.1846i q^{13} +(28.7633 - 49.8194i) q^{17} +(16.1346 - 9.31533i) q^{19} +(67.9773 - 39.2467i) q^{23} +(53.3549 - 92.4134i) q^{25} +202.514i q^{29} +(184.138 + 106.312i) q^{31} +(-69.0637 - 38.7783i) q^{35} +(116.462 + 201.719i) q^{37} +224.476 q^{41} -522.759 q^{43} +(190.401 + 329.785i) q^{47} +(164.334 - 301.070i) q^{49} +(569.999 + 329.089i) q^{53} -176.083i q^{55} +(-338.019 + 585.466i) q^{59} +(-126.822 + 73.2210i) q^{61} +(89.5734 - 51.7152i) q^{65} +(-221.248 + 383.212i) q^{67} -213.383i q^{71} +(-568.396 - 328.164i) q^{73} +(762.471 - 9.13483i) q^{77} +(247.191 + 428.148i) q^{79} +857.695 q^{83} +246.024 q^{85} +(339.378 + 587.821i) q^{89} +(228.583 + 385.186i) q^{91} +(69.0031 + 39.8390i) q^{95} -436.541i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 10 q^{7} + 72 q^{19} - 514 q^{25} - 714 q^{31} + 8 q^{37} + 368 q^{43} + 890 q^{49} + 1272 q^{61} + 1840 q^{67} + 324 q^{73} + 532 q^{79} - 1344 q^{85} - 4596 q^{91}+O(q^{100}) $ 32 * q + 10 * q^7 + 72 * q^19 - 514 * q^25 - 714 * q^31 + 8 * q^37 + 368 * q^43 + 890 * q^49 + 1272 * q^61 + 1840 * q^67 + 324 * q^73 + 532 * q^79 - 1344 * q^85 - 4596 * q^91

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	2.13836	+	3.70374i	0.191260	+	0.331273i	0.945668	−	0.325133i	$-0.105409\pi$
$5$	2.13836	+	3.70374i	0.191260	+	0.331273i	−0.754408	+	0.656406i	$0.772076\pi$
$6$		0			0
$7$	−15.9269	+	9.45161i	−0.859973	+	0.510339i
$7$	−15.9269	+	9.45161i	−0.859973	+	0.510339i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−35.6565	−	20.5863i	−0.977348	−	0.564272i	−0.0758796	−	0.997117i	$-0.524176\pi$
$11$	−35.6565	−	20.5863i	−0.977348	−	0.564272i	−0.901468	+	0.432845i	$0.857510\pi$
$12$		0			0
$13$		−	24.1846i		−	0.515969i	−0.966149	−	0.257984i	$-0.916942\pi$
$13$		−	24.1846i		−	0.515969i	0.966149	−	0.257984i	$-0.0830583\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	28.7633	−	49.8194i	0.410360	−	0.710764i	−0.584569	−	0.811344i	$-0.698736\pi$
$17$	28.7633	−	49.8194i	0.410360	−	0.710764i	0.994929	+	0.100580i	$0.0320698\pi$
$18$		0			0
$19$	16.1346	−	9.31533i	0.194818	−	0.112478i	−0.399418	−	0.916769i	$-0.630788\pi$
$19$	16.1346	−	9.31533i	0.194818	−	0.112478i	0.594236	+	0.804291i	$0.297455\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	67.9773	−	39.2467i	0.616272	−	0.355805i	−0.159144	−	0.987255i	$-0.550873\pi$
$23$	67.9773	−	39.2467i	0.616272	−	0.355805i	0.775416	+	0.631451i	$0.217540\pi$
$24$		0			0
$25$	53.3549	−	92.4134i	0.426839	−	0.739307i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			202.514i			1.29675i	0.761320	+	0.648377i	$0.224552\pi$
$29$			202.514i			1.29675i	−0.761320	+	0.648377i	$0.775448\pi$
$30$		0			0
$31$	184.138	+	106.312i	1.06684	+	0.615941i	0.927317	−	0.374277i	$-0.122109\pi$
$31$	184.138	+	106.312i	1.06684	+	0.615941i	0.139525	+	0.990219i	$0.455442\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−69.0637	−	38.7783i	−0.333540	−	0.187278i
$36$		0			0
$37$	116.462	+	201.719i	0.517467	+	0.896280i	0.999794	+	0.0202882i	$0.00645839\pi$
$37$	116.462	+	201.719i	0.517467	+	0.896280i	−0.482327	+	0.875991i	$0.660208\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	224.476			0.855056			0.427528	−	0.904002i	$-0.359385\pi$
$41$	224.476			0.855056			0.427528	+	0.904002i	$0.359385\pi$
$42$		0			0
$43$	−522.759			−1.85395			−0.926977	−	0.375117i	$-0.877602\pi$
$43$	−522.759			−1.85395			−0.926977	+	0.375117i	$0.877602\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	190.401	+	329.785i	0.590913	+	1.02349i	0.994110	+	0.108378i	$0.0345656\pi$
$47$	190.401	+	329.785i	0.590913	+	1.02349i	−0.403197	+	0.915113i	$0.632101\pi$
$48$		0			0
$49$	164.334	−	301.070i	0.479109	−	0.877756i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	569.999	+	329.089i	1.47727	+	0.852904i	0.999670	−	0.0256725i	$-0.00817270\pi$
$53$	569.999	+	329.089i	1.47727	+	0.852904i	0.477602	+	0.878576i	$0.341506\pi$
$54$		0			0
$55$		−	176.083i		−	0.431691i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−338.019	+	585.466i	−0.745870	+	1.29188i	0.203917	+	0.978988i	$0.434633\pi$
$59$	−338.019	+	585.466i	−0.745870	+	1.29188i	−0.949787	+	0.312896i	$0.898701\pi$
$60$		0			0
$61$	−126.822	+	73.2210i	−0.266196	+	0.153688i	−0.627158	−	0.778892i	$-0.715782\pi$
$61$	−126.822	+	73.2210i	−0.266196	+	0.153688i	0.360962	+	0.932581i	$0.382449\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	89.5734	−	51.7152i	0.170926	−	0.0986843i
$66$		0			0
$67$	−221.248	+	383.212i	−0.403429	+	0.698759i	−0.994137	−	0.108126i	$-0.965515\pi$
$67$	−221.248	+	383.212i	−0.403429	+	0.698759i	0.590709	+	0.806885i	$0.298848\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	213.383i		−	0.356674i	−0.983969	−	0.178337i	$-0.942928\pi$
$71$		−	213.383i		−	0.356674i	0.983969	−	0.178337i	$-0.0570717\pi$
$72$		0			0
$73$	−568.396	−	328.164i	−0.911312	−	0.526146i	−0.0304587	−	0.999536i	$-0.509697\pi$
$73$	−568.396	−	328.164i	−0.911312	−	0.526146i	−0.880853	+	0.473390i	$0.843030\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	762.471	−	9.13483i	1.12846	−	0.0135196i
$78$		0			0
$79$	247.191	+	428.148i	0.352040	+	0.609752i	0.986607	−	0.163116i	$-0.0521546\pi$
$79$	247.191	+	428.148i	0.352040	+	0.609752i	−0.634566	+	0.772868i	$0.718821\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	857.695			1.13427			0.567134	−	0.823625i	$-0.308052\pi$
$83$	857.695			1.13427			0.567134	+	0.823625i	$0.308052\pi$
$84$		0			0
$85$	246.024			0.313942
$86$		0			0
$87$		0			0
$88$		0			0
$89$	339.378	+	587.821i	0.404203	+	0.700100i	0.994228	−	0.107285i	$-0.0342157\pi$
$89$	339.378	+	587.821i	0.404203	+	0.700100i	−0.590026	+	0.807385i	$0.700882\pi$
$90$		0			0
$91$	228.583	+	385.186i	0.263319	+	0.443719i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	69.0031	+	39.8390i	0.0745218	+	0.0430252i
$96$		0			0
$97$		−	436.541i		−	0.456949i	−0.973550	−	0.228474i	$-0.926626\pi$
$97$		−	436.541i		−	0.456949i	0.973550	−	0.228474i	$-0.0733737\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	264.620	−	458.336i	0.260700	−	0.451546i	−0.705728	−	0.708483i	$-0.749380\pi$
$101$	264.620	−	458.336i	0.260700	−	0.451546i	0.966428	+	0.256937i	$0.0827133\pi$
$102$		0			0
$103$	1176.86	−	679.459i	1.12582	−	0.649991i	0.182938	−	0.983124i	$-0.441439\pi$
$103$	1176.86	−	679.459i	1.12582	−	0.649991i	0.942880	+	0.333133i	$0.108106\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	235.668	−	136.063i	0.212924	−	0.122932i	−0.389746	−	0.920923i	$-0.627437\pi$
$107$	235.668	−	136.063i	0.212924	−	0.122932i	0.602670	+	0.797991i	$0.294104\pi$
$108$		0			0
$109$	−220.323	+	381.611i	−0.193607	+	0.335337i	−0.946443	−	0.322871i	$-0.895352\pi$
$109$	−220.323	+	381.611i	−0.193607	+	0.335337i	0.752836	+	0.658208i	$0.228685\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			1096.06i			0.912465i	0.889861	+	0.456233i	$0.150801\pi$
$113$			1096.06i			0.912465i	−0.889861	+	0.456233i	$0.849199\pi$
$114$		0			0
$115$	290.719	+	167.847i	0.235737	+	0.136103i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	12.7632	+	1065.33i	0.00983197	+	0.820661i
$120$		0			0
$121$	182.089	+	315.387i	0.136806	+	0.236955i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	990.956			0.709070
$126$		0			0
$127$	−953.350			−0.666111			−0.333056	−	0.942907i	$-0.608080\pi$
$127$	−953.350			−0.666111			−0.333056	+	0.942907i	$0.608080\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	858.313	+	1486.64i	0.572452	+	0.991515i	0.996313	+	0.0857884i	$0.0273409\pi$
$131$	858.313	+	1486.64i	0.572452	+	0.991515i	−0.423862	+	0.905727i	$0.639326\pi$
$132$		0			0
$133$	−168.930	+	300.863i	−0.110136	+	0.196151i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1838.55	+	1061.48i	1.14655	+	0.661962i	0.948045	−	0.318138i	$-0.103057\pi$
$137$	1838.55	+	1061.48i	1.14655	+	0.661962i	0.198507	+	0.980099i	$0.436391\pi$
$138$		0			0
$139$		−	1768.95i		−	1.07943i	−0.841848	−	0.539715i	$-0.818532\pi$
$139$		−	1768.95i		−	1.07943i	0.841848	−	0.539715i	$-0.181468\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−497.870	+	862.336i	−0.291147	+	0.504281i
$144$		0			0
$145$	−750.058	+	433.046i	−0.429579	+	0.248018i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1922.68	−	1110.06i	1.05713	−	0.610333i	0.132491	−	0.991184i	$-0.457702\pi$
$149$	1922.68	−	1110.06i	1.05713	−	0.610333i	0.924636	+	0.380851i	$0.124369\pi$
$150$		0			0
$151$	298.447	−	516.925i	0.160843	−	0.278588i	−0.774328	−	0.632784i	$-0.781912\pi$
$151$	298.447	−	516.925i	0.160843	−	0.278588i	0.935171	+	0.354196i	$0.115245\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			909.330i			0.471221i
$156$		0			0
$157$	−78.6406	−	45.4032i	−0.0399758	−	0.0230801i	0.479879	−	0.877335i	$-0.340681\pi$
$157$	−78.6406	−	45.4032i	−0.0399758	−	0.0230801i	−0.519855	+	0.854255i	$0.674014\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−711.725	+	1267.57i	−0.348396	+	0.620490i
$162$		0			0
$163$	−584.493	−	1012.37i	−0.280865	−	0.486473i	0.690733	−	0.723110i	$-0.257288\pi$
$163$	−584.493	−	1012.37i	−0.280865	−	0.486473i	−0.971598	+	0.236637i	$0.923955\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	896.718			0.415510			0.207755	−	0.978181i	$-0.433384\pi$
$167$	896.718			0.415510			0.207755	+	0.978181i	$0.433384\pi$
$168$		0			0
$169$	1612.11			0.733776
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−540.506	−	936.184i	−0.237537	−	0.411426i	0.722470	−	0.691402i	$-0.243007\pi$
$173$	−540.506	−	936.184i	−0.237537	−	0.411426i	−0.960007	+	0.279976i	$0.909673\pi$
$174$		0			0
$175$	23.6754	+	1976.15i	0.0102268	+	0.853617i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3952.75	+	2282.12i	1.65052	+	0.952926i	0.976859	+	0.213882i	$0.0686109\pi$
$179$	3952.75	+	2282.12i	1.65052	+	0.952926i	0.673657	+	0.739044i	$0.264722\pi$
$180$		0			0
$181$			2695.13i			1.10678i	0.832922	+	0.553391i	$0.186666\pi$
$181$			2695.13i			1.10678i	−0.832922	+	0.553391i	$0.813334\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−498.076	+	862.692i	−0.197942	+	0.342845i
$186$		0			0
$187$	−2051.19	+	1184.26i	−0.802129	+	0.463109i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1241.02	+	716.501i	−0.470140	+	0.271436i	−0.716299	−	0.697794i	$-0.754165\pi$
$191$	−1241.02	+	716.501i	−0.470140	+	0.271436i	0.246158	+	0.969230i	$0.420832\pi$
$192$		0			0
$193$	1899.26	−	3289.61i	0.708350	−	1.22690i	−0.257119	−	0.966380i	$-0.582773\pi$
$193$	1899.26	−	3289.61i	0.708350	−	1.22690i	0.965469	−	0.260518i	$-0.0838934\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	1189.41i		−	0.430163i	−0.976596	−	0.215082i	$-0.930998\pi$
$197$		−	1189.41i		−	0.430163i	0.976596	−	0.215082i	$-0.0690017\pi$
$198$		0			0
$199$	−861.857	−	497.593i	−0.307012	−	0.177254i	0.338577	−	0.940939i	$-0.390055\pi$
$199$	−861.857	−	497.593i	−0.307012	−	0.177254i	−0.645589	+	0.763685i	$0.723388\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1914.08	−	3225.42i	−0.661784	−	1.11517i
$204$		0			0
$205$	480.010	+	831.401i	0.163538	+	0.283257i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−767.072			−0.253873
$210$		0			0
$211$	3662.87			1.19508			0.597542	−	0.801838i	$-0.296144\pi$
$211$	3662.87			1.19508			0.597542	+	0.801838i	$0.296144\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1117.85	−	1936.16i	−0.354588	−	0.614164i
$216$		0			0
$217$	−3937.57	+	47.1742i	−1.23179	+	0.0147576i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1204.86	−	695.627i	−0.366732	−	0.211733i
$222$		0			0
$223$			1643.63i			0.493568i	0.969071	+	0.246784i	$0.0793738\pi$
$223$			1643.63i			0.493568i	−0.969071	+	0.246784i	$0.920626\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−15.3747	+	26.6297i	−0.00449539	+	0.00778624i	−0.868264	−	0.496102i	$-0.834764\pi$
$227$	−15.3747	+	26.6297i	−0.00449539	+	0.00778624i	0.863769	+	0.503888i	$0.168098\pi$
$228$		0			0
$229$	5374.23	−	3102.81i	1.55083	−	0.895369i	0.552751	−	0.833347i	$-0.313578\pi$
$229$	5374.23	−	3102.81i	1.55083	−	0.895369i	0.998075	−	0.0620228i	$-0.0197552\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1147.38	−	662.439i	0.322606	−	0.186257i	−0.329947	−	0.943999i	$-0.607031\pi$
$233$	1147.38	−	662.439i	0.322606	−	0.186257i	0.652554	+	0.757742i	$0.273698\pi$
$234$		0			0
$235$	−814.292	+	1410.39i	−0.226036	+	0.391506i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	2248.29i		−	0.608492i	−0.952593	−	0.304246i	$-0.901595\pi$
$239$		−	2248.29i		−	0.608492i	0.952593	−	0.304246i	$-0.0984045\pi$
$240$		0			0
$241$	−4267.12	−	2463.62i	−1.14054	−	0.658489i	−0.193974	−	0.981007i	$-0.562138\pi$
$241$	−4267.12	−	2463.62i	−1.14054	−	0.658489i	−0.946564	+	0.322517i	$0.895471\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1466.49	−	35.1438i	0.382411	−	0.00916431i
$246$		0			0
$247$	−225.287	−	390.209i	−0.0580352	−	0.100520i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−6922.20			−1.74074			−0.870369	−	0.492400i	$-0.836120\pi$
$251$	−6922.20			−1.74074			−0.870369	+	0.492400i	$0.836120\pi$
$252$		0			0
$253$	−3231.77			−0.803083
$254$		0			0
$255$		0			0
$256$		0			0
$257$	310.575	+	537.932i	0.0753819	+	0.130565i	0.901252	−	0.433295i	$-0.142649\pi$
$257$	310.575	+	537.932i	0.0753819	+	0.130565i	−0.825870	+	0.563860i	$0.809316\pi$
$258$		0			0
$259$	−3761.45	−	2112.00i	−0.902414	−	0.506693i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2661.67	−	1536.72i	−0.624052	−	0.360297i	0.154393	−	0.988010i	$-0.450658\pi$
$263$	−2661.67	−	1536.72i	−0.624052	−	0.360297i	−0.778445	+	0.627713i	$0.783991\pi$
$264$		0			0
$265$			2814.84i			0.652507i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	593.308	−	1027.64i	0.134478	−	0.232923i	−0.790920	−	0.611920i	$-0.790398\pi$
$269$	593.308	−	1027.64i	0.134478	−	0.232923i	0.925398	+	0.378997i	$0.123731\pi$
$270$		0			0
$271$	−54.8423	+	31.6632i	−0.0122931	+	0.00709743i	−0.506134	−	0.862455i	$-0.668926\pi$
$271$	−54.8423	+	31.6632i	−0.0122931	+	0.00709743i	0.493841	+	0.869552i	$0.335593\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3804.89	+	2196.76i	−0.834341	+	0.481707i
$276$		0			0
$277$	−2953.02	+	5114.79i	−0.640542	+	1.10945i	0.344770	+	0.938687i	$0.387957\pi$
$277$	−2953.02	+	5114.79i	−0.640542	+	1.10945i	−0.985312	+	0.170764i	$0.945377\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	9070.73i		−	1.92567i	−0.270085	−	0.962836i	$-0.587052\pi$
$281$		−	9070.73i		−	1.92567i	0.270085	−	0.962836i	$-0.412948\pi$
$282$		0			0
$283$	5747.96	+	3318.58i	1.20735	+	0.697065i	0.962180	−	0.272415i	$-0.0878223\pi$
$283$	5747.96	+	3318.58i	1.20735	+	0.697065i	0.245172	+	0.969480i	$0.421156\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3575.22	+	2121.66i	−0.735325	+	0.436368i
$288$		0			0
$289$	801.849	+	1388.84i	0.163210	+	0.282688i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	5630.46			1.12265			0.561323	−	0.827597i	$-0.310293\pi$
$293$	5630.46			1.12265			0.561323	+	0.827597i	$0.310293\pi$
$294$		0			0
$295$	−2891.22			−0.570621
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−949.165	−	1644.00i	−0.183584	−	0.317977i
$300$		0			0
$301$	8325.95	−	4940.92i	1.59435	−	0.946145i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−542.383	−	313.145i	−0.101825	−	0.0587889i
$306$		0			0
$307$		−	2412.89i		−	0.448570i	−0.974524	−	0.224285i	$-0.927995\pi$
$307$		−	2412.89i		−	0.448570i	0.974524	−	0.224285i	$-0.0720047\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3830.84	+	6635.22i	−0.698480	+	1.20980i	0.270514	+	0.962716i	$0.412807\pi$
$311$	−3830.84	+	6635.22i	−0.698480	+	1.20980i	−0.968993	+	0.247087i	$0.920527\pi$
$312$		0			0
$313$	−142.607	+	82.3340i	−0.0257527	+	0.0148683i	−0.512821	−	0.858496i	$-0.671400\pi$
$313$	−142.607	+	82.3340i	−0.0257527	+	0.0148683i	0.487068	+	0.873364i	$0.338066\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3217.61	−	1857.69i	0.570091	−	0.329142i	−0.187095	−	0.982342i	$-0.559907\pi$
$317$	3217.61	−	1857.69i	0.570091	−	0.329142i	0.757186	+	0.653200i	$0.226574\pi$
$318$		0			0
$319$	4169.00	−	7220.92i	0.731722	−	1.26738i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	1071.76i		−	0.184626i
$324$		0			0
$325$	−2234.98	−	1290.36i	−0.381459	−	0.220236i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6149.51	−	3452.86i	−1.03050	−	0.578609i
$330$		0			0
$331$	−1993.16	−	3452.25i	−0.330979	−	0.573272i	0.651725	−	0.758455i	$-0.274045\pi$
$331$	−1993.16	−	3452.25i	−0.330979	−	0.573272i	−0.982704	+	0.185183i	$0.940712\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1892.43			−0.308640
$336$		0			0
$337$	7112.94			1.14975			0.574876	−	0.818240i	$-0.305050\pi$
$337$	7112.94			1.14975			0.574876	+	0.818240i	$0.305050\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−4377.13	−	7581.41i	−0.695117	−	1.20398i
$342$		0			0
$343$	228.258	+	6348.35i	0.0359322	+	0.999354i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3424.22	−	1976.97i	−0.529745	−	0.305848i	0.211168	−	0.977450i	$-0.432273\pi$
$347$	−3424.22	−	1976.97i	−0.529745	−	0.305848i	−0.740913	+	0.671601i	$0.765607\pi$
$348$		0			0
$349$			631.667i			0.0968836i	0.998826	+	0.0484418i	$0.0154255\pi$
$349$			631.667i			0.0968836i	−0.998826	+	0.0484418i	$0.984574\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4963.73	+	8597.43i	−0.748421	+	1.29630i	0.200159	+	0.979763i	$0.435854\pi$
$353$	−4963.73	+	8597.43i	−0.748421	+	1.29630i	−0.948579	+	0.316539i	$0.897479\pi$
$354$		0			0
$355$	790.314	−	456.288i	0.118156	−	0.0682176i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−678.529	+	391.749i	−0.0997532	+	0.0575925i	−0.549047	−	0.835792i	$-0.685009\pi$
$359$	−678.529	+	391.749i	−0.0997532	+	0.0575925i	0.449293	+	0.893384i	$0.351676\pi$
$360$		0			0
$361$	−3255.95	+	5639.47i	−0.474697	+	0.822200i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	2806.92i		−	0.402523i
$366$		0			0
$367$	4531.71	+	2616.39i	0.644560	+	0.372137i	0.786369	−	0.617757i	$-0.211959\pi$
$367$	4531.71	+	2616.39i	0.644560	+	0.372137i	−0.141809	+	0.989894i	$0.545292\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−12188.8	+	146.028i	−1.70569	+	0.0204351i
$372$		0			0
$373$	−1131.18	−	1959.27i	−0.157026	−	0.271976i	0.776769	−	0.629785i	$-0.216857\pi$
$373$	−1131.18	−	1959.27i	−0.157026	−	0.271976i	−0.933795	+	0.357809i	$0.883524\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4897.71			0.669084
$378$		0			0
$379$	−7342.89			−0.995196			−0.497598	−	0.867408i	$-0.665784\pi$
$379$	−7342.89			−0.995196			−0.497598	+	0.867408i	$0.665784\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−5824.04	−	10087.5i	−0.777009	−	1.34582i	−0.933658	−	0.358165i	$-0.883403\pi$
$383$	−5824.04	−	10087.5i	−0.777009	−	1.34582i	0.156649	−	0.987654i	$-0.449931\pi$
$384$		0			0
$385$	1664.27	+	2804.46i	0.220309	+	0.371243i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	7596.76	+	4385.99i	0.990157	+	0.571667i	0.905321	−	0.424728i	$-0.139630\pi$
$389$	7596.76	+	4385.99i	0.990157	+	0.571667i	0.0848357	+	0.996395i	$0.472963\pi$
$390$		0			0
$391$		−	4515.46i		−	0.584032i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1057.17	+	1831.06i	−0.134663	+	0.233243i
$396$		0			0
$397$	−7151.28	+	4128.79i	−0.904061	+	0.521960i	−0.878516	−	0.477714i	$-0.841466\pi$
$397$	−7151.28	+	4128.79i	−0.904061	+	0.521960i	−0.0255456	+	0.999674i	$0.508132\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	9520.48	−	5496.65i	1.18561	−	0.684513i	0.228305	−	0.973590i	$-0.426682\pi$
$401$	9520.48	−	5496.65i	1.18561	−	0.684513i	0.957306	+	0.289077i	$0.0933483\pi$
$402$		0			0
$403$	2571.11	−	4453.29i	0.317806	−	0.550457i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	9590.10i		−	1.16797i
$408$		0			0
$409$	2530.01	+	1460.70i	0.305871	+	0.176594i	0.645077	−	0.764117i	$-0.276825\pi$
$409$	2530.01	+	1460.70i	0.305871	+	0.176594i	−0.339207	+	0.940712i	$0.610159\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−149.991	−	12519.5i	−0.0178706	−	1.49163i
$414$		0			0
$415$	1834.06	+	3176.68i	0.216941	+	0.375752i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−14677.0			−1.71126			−0.855632	−	0.517585i	$-0.826831\pi$
$419$	−14677.0			−1.71126			−0.855632	+	0.517585i	$0.826831\pi$
$420$		0			0
$421$	−10512.8			−1.21701			−0.608507	−	0.793549i	$-0.708231\pi$
$421$	−10512.8			−1.21701			−0.608507	+	0.793549i	$0.708231\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3069.32	−	5316.22i	−0.350315	−	0.606764i
$426$		0			0
$427$	1327.84	−	2364.86i	0.150488	−	0.268018i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−7588.24	−	4381.07i	−0.848057	−	0.489626i	0.0119377	−	0.999929i	$-0.496200\pi$
$431$	−7588.24	−	4381.07i	−0.848057	−	0.489626i	−0.859995	+	0.510303i	$0.829533\pi$
$432$		0			0
$433$		−	5529.57i		−	0.613705i	−0.951757	−	0.306853i	$-0.900724\pi$
$433$		−	5529.57i		−	0.613705i	0.951757	−	0.306853i	$-0.0992758\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	731.193	−	1266.46i	0.0800405	−	0.138634i
$438$		0			0
$439$	−7000.68	+	4041.84i	−0.761103	+	0.439423i	−0.829692	−	0.558222i	$-0.811484\pi$
$439$	−7000.68	+	4041.84i	−0.761103	+	0.439423i	0.0685887	+	0.997645i	$0.478150\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−7862.68	+	4539.52i	−0.843266	+	0.486860i	−0.858373	−	0.513026i	$-0.828525\pi$
$443$	−7862.68	+	4539.52i	−0.843266	+	0.486860i	0.0151067	+	0.999886i	$0.495191\pi$
$444$		0			0
$445$	−1451.42	+	2513.94i	−0.154616	+	0.267803i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	5428.06i		−	0.570526i	−0.958449	−	0.285263i	$-0.907919\pi$
$449$		−	5428.06i		−	0.570526i	0.958449	−	0.285263i	$-0.0920809\pi$
$450$		0			0
$451$	−8004.03	−	4621.13i	−0.835687	−	0.482484i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−937.837	+	1670.28i	−0.0966296	+	0.172096i
$456$		0			0
$457$	1160.12	+	2009.39i	0.118749	+	0.205679i	0.919272	−	0.393623i	$-0.128778\pi$
$457$	1160.12	+	2009.39i	0.118749	+	0.205679i	−0.800523	+	0.599302i	$0.795445\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3952.56			0.399326			0.199663	−	0.979865i	$-0.436015\pi$
$461$	3952.56			0.399326			0.199663	+	0.979865i	$0.436015\pi$
$462$		0			0
$463$	6118.93			0.614192			0.307096	−	0.951679i	$-0.400643\pi$
$463$	6118.93			0.614192			0.307096	+	0.951679i	$0.400643\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	5237.11	+	9070.94i	0.518939	+	0.898829i	0.999758	+	0.0220087i	$0.00700616\pi$
$467$	5237.11	+	9070.94i	0.518939	+	0.898829i	−0.480819	+	0.876820i	$0.659661\pi$
$468$		0			0
$469$	−98.1752	−	8194.54i	−0.00966591	−	0.806799i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	18639.8	+	10761.7i	1.81196	+	1.04614i
$474$		0			0
$475$		−	1988.07i		−	0.192040i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5289.17	+	9161.11i	−0.504527	+	0.873866i	0.495460	+	0.868631i	$0.335000\pi$
$479$	−5289.17	+	9161.11i	−0.504527	+	0.873866i	−0.999986	+	0.00523496i	$0.998334\pi$
$480$		0			0
$481$	4878.48	−	2816.59i	0.462452	−	0.266997i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1616.84	−	933.480i	0.151375	−	0.0873962i
$486$		0			0
$487$	−4253.22	+	7366.79i	−0.395753	+	0.685464i	−0.993197	−	0.116447i	$-0.962850\pi$
$487$	−4253.22	+	7366.79i	−0.395753	+	0.685464i	0.597444	+	0.801910i	$0.296183\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			956.220i			0.0878892i	0.999034	+	0.0439446i	$0.0139925\pi$
$491$			956.220i			0.0878892i	−0.999034	+	0.0439446i	$0.986007\pi$
$492$		0			0
$493$	10089.1	+	5824.96i	0.921686	+	0.532136i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	2016.81	+	3398.53i	0.182025	+	0.306730i
$498$		0			0
$499$	5702.62	+	9877.23i	0.511592	+	0.886104i	0.999910	+	0.0134376i	$0.00427746\pi$
$499$	5702.62	+	9877.23i	0.511592	+	0.886104i	−0.488318	+	0.872666i	$0.662389\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	101.526			0.00899964			0.00449982	−	0.999990i	$-0.498568\pi$
$503$	101.526			0.00899964			0.00449982	+	0.999990i	$0.498568\pi$
$504$		0			0
$505$	2263.41			0.199446
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1164.58	+	2017.11i	0.101413	+	0.175652i	0.912267	−	0.409596i	$-0.134330\pi$
$509$	1164.58	+	2017.11i	0.101413	+	0.175652i	−0.810854	+	0.585248i	$0.800997\pi$
$510$		0			0
$511$	12154.5	−	145.617i	1.05222	−	0.0126061i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	5033.08	+	2905.85i	0.430648	+	0.248635i
$516$		0			0
$517$		−	15678.6i		−	1.33374i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−3994.60	+	6918.84i	−0.335905	+	0.581804i	−0.983658	−	0.180046i	$-0.942375\pi$
$521$	−3994.60	+	6918.84i	−0.335905	+	0.581804i	0.647753	+	0.761850i	$0.275709\pi$
$522$		0			0
$523$	−8064.07	+	4655.80i	−0.674220	+	0.389261i	−0.797674	−	0.603089i	$-0.793936\pi$
$523$	−8064.07	+	4655.80i	−0.674220	+	0.389261i	0.123453	+	0.992350i	$0.460603\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	10592.8	−	6115.75i	0.875578	−	0.505515i
$528$		0			0
$529$	−3002.89	+	5201.16i	−0.246806	+	0.427481i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	5428.86i		−	0.441182i
$534$		0			0
$535$	1007.88	+	581.901i	0.0814478	+	0.0470239i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−12057.5	+	7352.07i	−0.963549	+	0.587525i
$540$		0			0
$541$	−1273.38	−	2205.56i	−0.101196	−	0.175276i	0.810982	−	0.585071i	$-0.198934\pi$
$541$	−1273.38	−	2205.56i	−0.101196	−	0.175276i	−0.912178	+	0.409795i	$0.865600\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1884.52			−0.148117
$546$		0			0
$547$	16319.2			1.27561			0.637807	−	0.770196i	$-0.279842\pi$
$547$	16319.2			1.27561			0.637807	+	0.770196i	$0.279842\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1886.48	+	3267.48i	0.145856	+	0.252631i
$552$		0			0
$553$	−7983.68	−	4482.73i	−0.613925	−	0.344710i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	22168.6	+	12799.0i	1.68638	+	0.973632i	0.957252	+	0.289254i	$0.0934072\pi$
$557$	22168.6	+	12799.0i	1.68638	+	0.973632i	0.729127	+	0.684378i	$0.239926\pi$
$558$		0			0
$559$			12642.7i			0.956583i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−10293.7	+	17829.1i	−0.770561	+	1.33465i	0.166695	+	0.986009i	$0.446691\pi$
$563$	−10293.7	+	17829.1i	−0.770561	+	1.33465i	−0.937256	+	0.348642i	$0.886643\pi$
$564$		0			0
$565$	−4059.52	+	2343.76i	−0.302275	+	0.174518i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	10507.7	−	6066.64i	0.774178	−	0.446972i	−0.0601850	−	0.998187i	$-0.519169\pi$
$569$	10507.7	−	6066.64i	0.774178	−	0.446972i	0.834363	+	0.551215i	$0.185836\pi$
$570$		0			0
$571$	−917.059	+	1588.39i	−0.0672114	+	0.116414i	−0.897673	−	0.440663i	$-0.854744\pi$
$571$	−917.059	+	1588.39i	−0.0672114	+	0.116414i	0.830461	+	0.557076i	$0.188077\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	8376.02i		−	0.607485i
$576$		0			0
$577$	1014.52	+	585.733i	0.0731976	+	0.0422606i	0.536152	−	0.844121i	$-0.319877\pi$
$577$	1014.52	+	585.733i	0.0731976	+	0.0422606i	−0.462955	+	0.886382i	$0.653211\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−13660.5	+	8106.60i	−0.975441	+	0.578861i
$582$		0			0
$583$	−13549.4	−	23468.3i	−0.962540	−	1.66717i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	28344.8			1.99304			0.996520	−	0.0833542i	$-0.0265633\pi$
$587$	28344.8			1.99304			0.996520	+	0.0833542i	$0.0265633\pi$
$588$		0			0
$589$	3961.32			0.277120
$590$		0			0
$591$		0			0
$592$		0			0
$593$	3788.26	+	6561.47i	0.262336	+	0.454380i	0.966862	−	0.255298i	$-0.0821737\pi$
$593$	3788.26	+	6561.47i	0.262336	+	0.454380i	−0.704526	+	0.709678i	$0.748840\pi$
$594$		0			0
$595$	−3918.41	+	2325.33i	−0.269982	+	0.160217i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	25265.0	+	14586.7i	1.72337	+	0.994988i	0.911680	+	0.410900i	$0.134786\pi$
$599$	25265.0	+	14586.7i	1.72337	+	0.994988i	0.811690	+	0.584088i	$0.198548\pi$
$600$		0			0
$601$			20302.1i			1.37793i	0.724793	+	0.688967i	$0.241935\pi$
$601$			20302.1i			1.37793i	−0.724793	+	0.688967i	$0.758065\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−778.742	+	1348.82i	−0.0523312	+	0.0906403i
$606$		0			0
$607$	4664.27	−	2692.92i	0.311890	−	0.180069i	−0.335882	−	0.941904i	$-0.609034\pi$
$607$	4664.27	−	2692.92i	0.311890	−	0.180069i	0.647772	+	0.761835i	$0.275701\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	7975.71	−	4604.78i	0.528089	−	0.304892i
$612$		0			0
$613$	13571.5	−	23506.5i	0.894203	−	1.54881i	0.0594159	−	0.998233i	$-0.481076\pi$
$613$	13571.5	−	23506.5i	0.894203	−	1.54881i	0.834787	−	0.550572i	$-0.185590\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	22863.1i		−	1.49179i	−0.666063	−	0.745896i	$-0.732022\pi$
$617$		−	22863.1i		−	1.49179i	0.666063	−	0.745896i	$-0.267978\pi$
$618$		0			0
$619$	−18682.0	−	10786.0i	−1.21307	−	0.700367i	−0.249645	−	0.968338i	$-0.580314\pi$
$619$	−18682.0	−	10786.0i	−1.21307	−	0.700367i	−0.963427	+	0.267970i	$0.913647\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−10961.1	−	6154.51i	−0.704892	−	0.395787i
$624$		0			0
$625$	−4550.34	−	7881.43i	−0.291222	−	0.504411i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	13399.3			0.849391
$630$		0			0
$631$	−9149.66			−0.577246			−0.288623	−	0.957443i	$-0.593197\pi$
$631$	−9149.66			−0.577246			−0.288623	+	0.957443i	$0.593197\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2038.60	−	3530.96i	−0.127401	−	0.220664i
$636$		0			0
$637$	−7281.25	−	3974.35i	−0.452894	−	0.247205i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	9934.51	+	5735.69i	0.612152	+	0.353426i	0.773807	−	0.633421i	$-0.218350\pi$
$641$	9934.51	+	5735.69i	0.612152	+	0.353426i	−0.161655	+	0.986847i	$0.551683\pi$
$642$		0			0
$643$			21804.8i			1.33732i	0.743568	+	0.668660i	$0.233132\pi$
$643$			21804.8i			1.33732i	−0.743568	+	0.668660i	$0.766868\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−13667.6	+	23673.0i	−0.830495	+	1.43846i	0.0671520	+	0.997743i	$0.478609\pi$
$647$	−13667.6	+	23673.0i	−0.830495	+	1.43846i	−0.897647	+	0.440716i	$0.854725\pi$
$648$		0			0
$649$	24105.1	−	13917.1i	1.45795	−	0.841747i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−8234.37	+	4754.12i	−0.493470	+	0.284905i	−0.726013	−	0.687681i	$-0.758629\pi$
$653$	−8234.37	+	4754.12i	−0.493470	+	0.284905i	0.232543	+	0.972586i	$0.425295\pi$
$654$		0			0
$655$	−3670.76	+	6357.94i	−0.218975	+	0.379275i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			1178.95i			0.0696896i	0.999393	+	0.0348448i	$0.0110937\pi$
$659$			1178.95i			0.0696896i	−0.999393	+	0.0348448i	$0.988906\pi$
$660$		0			0
$661$	11846.9	+	6839.83i	0.697113	+	0.402479i	0.806271	−	0.591546i	$-0.201482\pi$
$661$	11846.9	+	6839.83i	0.697113	+	0.402479i	−0.109158	+	0.994024i	$0.534815\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1475.55	+	17.6779i	−0.0860442	+	0.00103086i
$666$		0			0
$667$	7948.00	+	13766.3i	0.461391	+	0.799153i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	6029.39			0.346888
$672$		0			0
$673$	−13877.3			−0.794845			−0.397422	−	0.917636i	$-0.630095\pi$
$673$	−13877.3			−0.794845			−0.397422	+	0.917636i	$0.630095\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−5249.25	−	9091.96i	−0.297998	−	0.516148i	0.677680	−	0.735357i	$-0.262986\pi$
$677$	−5249.25	−	9091.96i	−0.297998	−	0.516148i	−0.975678	+	0.219209i	$0.929652\pi$
$678$		0			0
$679$	4126.02	+	6952.76i	0.233199	+	0.392964i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	20028.5	+	11563.4i	1.12206	+	0.647823i	0.941927	−	0.335819i	$-0.109013\pi$
$683$	20028.5	+	11563.4i	1.12206	+	0.647823i	0.180136	+	0.983642i	$0.442346\pi$
$684$		0			0
$685$			9079.33i			0.506428i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	7958.88	−	13785.2i	0.440072	−	0.762226i
$690$		0			0
$691$	8736.41	−	5043.97i	0.480968	−	0.277687i	−0.239852	−	0.970809i	$-0.577099\pi$
$691$	8736.41	−	5043.97i	0.480968	−	0.277687i	0.720820	+	0.693123i	$0.243766\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	6551.75	−	3782.65i	0.357586	−	0.206452i
$696$		0			0
$697$	6456.67	−	11183.3i	0.350881	−	0.607743i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			11673.0i			0.628937i	0.949268	+	0.314468i	$0.101826\pi$
$701$			11673.0i			0.628937i	−0.949268	+	0.314468i	$0.898174\pi$
$702$		0			0
$703$	3758.15	+	2169.77i	0.201624	+	0.116407i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	117.421	+	9800.98i	0.00624622	+	0.521363i
$708$		0			0
$709$	3302.77	+	5720.57i	0.174948	+	0.303019i	0.940143	−	0.340779i	$-0.110691\pi$
$709$	3302.77	+	5720.57i	0.174948	+	0.303019i	−0.765195	+	0.643798i	$0.777358\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	16689.6			0.876619
$714$		0			0
$715$	−4258.49			−0.222739
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−6329.61	−	10963.2i	−0.328310	−	0.568649i	0.653867	−	0.756610i	$-0.273146\pi$
$719$	−6329.61	−	10963.2i	−0.328310	−	0.568649i	−0.982177	+	0.187961i	$0.939812\pi$
$720$		0			0
$721$	−12321.7	+	21944.9i	−0.636457	+	1.13352i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	18715.0	+	10805.1i	0.958699	+	0.553505i
$726$		0			0
$727$		−	20978.8i		−	1.07023i	−0.844778	−	0.535117i	$-0.820267\pi$
$727$		−	20978.8i		−	1.07023i	0.844778	−	0.535117i	$-0.179733\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−15036.3	+	26043.6i	−0.760788	+	1.31772i
$732$		0			0
$733$	−25434.5	+	14684.6i	−1.28164	+	0.739956i	−0.977148	−	0.212558i	$-0.931821\pi$
$733$	−25434.5	+	14684.6i	−1.28164	+	0.739956i	−0.304494	+	0.952514i	$0.598487\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	15777.8	−	9109.33i	0.788580	−	0.455287i
$738$		0			0
$739$	604.945	−	1047.80i	0.0301127	−	0.0521567i	−0.850576	−	0.525852i	$-0.823747\pi$
$739$	604.945	−	1047.80i	0.0301127	−	0.0521567i	0.880689	+	0.473695i	$0.157080\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			5344.74i			0.263902i	0.991256	+	0.131951i	$0.0421242\pi$
$743$			5344.74i			0.263902i	−0.991256	+	0.131951i	$0.957876\pi$
$744$		0			0
$745$	8222.74	+	4747.40i	0.404373	+	0.233465i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−2467.45	+	4394.50i	−0.120372	+	0.214381i
$750$		0			0
$751$	−12394.5	−	21468.0i	−0.602241	−	1.04311i	−0.992481	−	0.122398i	$-0.960941\pi$
$751$	−12394.5	−	21468.0i	−0.602241	−	1.04311i	0.390240	−	0.920713i	$-0.372392\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	2552.74			0.123051
$756$		0			0
$757$	7531.15			0.361591			0.180795	−	0.983521i	$-0.442133\pi$
$757$	7531.15			0.361591			0.180795	+	0.983521i	$0.442133\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	10714.8	+	18558.6i	0.510397	+	0.884034i	0.999927	+	0.0120477i	$0.00383500\pi$
$761$	10714.8	+	18558.6i	0.510397	+	0.884034i	−0.489530	+	0.871986i	$0.662832\pi$
$762$		0			0
$763$	−97.7650	−	8160.31i	−0.00463871	−	0.387186i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	14159.2	+	8174.84i	0.666572	+	0.384845i
$768$		0			0
$769$			5672.41i			0.265998i	0.991116	+	0.132999i	$0.0424607\pi$
$769$			5672.41i			0.265998i	−0.991116	+	0.132999i	$0.957539\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−15382.4	+	26643.1i	−0.715739	+	1.23970i	0.246935	+	0.969032i	$0.420577\pi$
$773$	−15382.4	+	26643.1i	−0.715739	+	1.23970i	−0.962674	+	0.270664i	$0.912757\pi$
$774$		0			0
$775$	19649.3	−	11344.5i	0.910739	−	0.525816i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	3621.84	−	2091.07i	0.166580	−	0.0961750i
$780$		0			0
$781$	−4392.75	+	7608.47i	−0.201261	+	0.348595i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	388.353i		−	0.0176572i
$786$		0			0
$787$	−2009.12	−	1159.96i	−0.0910003	−	0.0525390i	0.453809	−	0.891099i	$-0.350065\pi$
$787$	−2009.12	−	1159.96i	−0.0910003	−	0.0525390i	−0.544810	+	0.838560i	$0.683398\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−10359.5	−	17456.9i	−0.465666	−	0.784696i
$792$		0			0
$793$	1770.82	+	3067.15i	0.0792984	+	0.137349i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	8031.15			0.356936			0.178468	−	0.983946i	$-0.442886\pi$
$797$	8031.15			0.356936			0.178468	+	0.983946i	$0.442886\pi$
$798$		0			0
$799$	21906.3			0.969947
$800$		0			0
$801$		0			0
$802$		0			0
$803$	13511.3	+	23402.3i	0.593779	+	1.02846i
$804$		0			0
$805$	−6216.69	+	74.4794i	−0.272186	+	0.00326094i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	9791.93	+	5653.37i	0.425545	+	0.245688i	0.697447	−	0.716637i	$-0.254319\pi$
$809$	9791.93	+	5653.37i	0.425545	+	0.245688i	−0.271902	+	0.962325i	$0.587653\pi$
$810$		0			0
$811$		−	34258.3i		−	1.48332i	−0.670777	−	0.741659i	$-0.734039\pi$
$811$		−	34258.3i		−	1.48332i	0.670777	−	0.741659i	$-0.265961\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2499.71	−	4329.62i	0.107437	−	0.186086i
$816$		0			0
$817$	−8434.53	+	4869.68i	−0.361183	+	0.208529i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	15485.0	−	8940.25i	0.658257	−	0.380045i	−0.133355	−	0.991068i	$-0.542575\pi$
$821$	15485.0	−	8940.25i	0.658257	−	0.380045i	0.791613	+	0.611023i	$0.209242\pi$
$822$		0			0
$823$	−12259.4	+	21233.9i	−0.519241	+	0.899352i	0.480509	+	0.876990i	$0.340452\pi$
$823$	−12259.4	+	21233.9i	−0.519241	+	0.899352i	−0.999750	+	0.0223618i	$0.992881\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	25814.5i		−	1.08544i	−0.839914	−	0.542720i	$-0.817394\pi$
$827$		−	25814.5i		−	1.08544i	0.839914	−	0.542720i	$-0.182606\pi$
$828$		0			0
$829$	−24726.9	−	14276.1i	−1.03595	−	0.598104i	−0.117265	−	0.993101i	$-0.537413\pi$
$829$	−24726.9	−	14276.1i	−1.03595	−	0.598104i	−0.918683	+	0.394996i	$0.870746\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−10272.4	−	16846.8i	−0.427270	−	0.700729i
$834$		0			0
$835$	1917.50	+	3321.21i	0.0794705	+	0.137647i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−41794.9			−1.71981			−0.859904	−	0.510455i	$-0.829477\pi$
$839$	−41794.9			−1.71981			−0.859904	+	0.510455i	$0.829477\pi$
$840$		0			0
$841$	−16622.8			−0.681570
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3447.26	+	5970.82i	0.140342	+	0.243080i
$846$		0			0
$847$	−5881.04	−	3302.12i	−0.238577	−	0.133958i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	15833.6	+	9141.53i	0.637801	+	0.368235i
$852$		0			0
$853$			23442.1i			0.940962i	0.882410	+	0.470481i	$0.155920\pi$
$853$			23442.1i			0.940962i	−0.882410	+	0.470481i	$0.844080\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−17726.7	+	30703.5i	−0.706572	+	1.22382i	0.259550	+	0.965730i	$0.416426\pi$
$857$	−17726.7	+	30703.5i	−0.706572	+	1.22382i	−0.966121	+	0.258088i	$0.916907\pi$
$858$		0			0
$859$	−38342.8	+	22137.2i	−1.52298	+	0.879293i	−0.523350	+	0.852118i	$0.675318\pi$
$859$	−38342.8	+	22137.2i	−1.52298	+	0.879293i	−0.999631	+	0.0271751i	$0.991349\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	32865.7	−	18975.0i	1.29636	−	0.748455i	0.316588	−	0.948563i	$-0.397463\pi$
$863$	32865.7	−	18975.0i	1.29636	−	0.748455i	0.979774	+	0.200108i	$0.0641293\pi$
$864$		0			0
$865$	2311.59	−	4003.79i	0.0908628	−	0.157379i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	20355.0i		−	0.794586i
$870$		0			0
$871$	9267.83	+	5350.78i	0.360538	+	0.208157i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−15782.9	+	9366.12i	−0.609781	+	0.361866i
$876$		0			0
$877$	−622.088	−	1077.49i	−0.0239526	−	0.0414871i	0.853801	−	0.520600i	$-0.174292\pi$
$877$	−622.088	−	1077.49i	−0.0239526	−	0.0414871i	−0.877753	+	0.479113i	$0.840958\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	13355.7			0.510745			0.255373	−	0.966843i	$-0.417802\pi$
$881$	13355.7			0.510745			0.255373	+	0.966843i	$0.417802\pi$
$882$		0			0
$883$	36500.1			1.39108			0.695542	−	0.718485i	$-0.255164\pi$
$883$	36500.1			1.39108			0.695542	+	0.718485i	$0.255164\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−9794.39	−	16964.4i	−0.370759	−	0.642174i	0.618923	−	0.785451i	$-0.287569\pi$
$887$	−9794.39	−	16964.4i	−0.370759	−	0.642174i	−0.989683	+	0.143278i	$0.954236\pi$
$888$		0			0
$889$	15183.9	−	9010.69i	0.572838	−	0.339942i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	6144.11	+	3547.30i	0.230241	+	0.132929i
$894$		0			0
$895$			19519.9i			0.729028i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−21529.6	+	37290.4i	−0.798724	+	1.38343i
$900$		0			0
$901$	32790.1	−	18931.4i	1.21243	−	0.699995i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−9982.06	+	5763.15i	−0.366646	+	0.211683i
$906$		0			0
$907$	16841.8	−	29170.9i	0.616565	−	1.06792i	−0.373543	−	0.927613i	$-0.621857\pi$
$907$	16841.8	−	29170.9i	0.616565	−	1.06792i	0.990108	−	0.140309i	$-0.0448095\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	48956.2i		−	1.78045i	−0.455520	−	0.890226i	$-0.650547\pi$
$911$		−	48956.2i		−	1.78045i	0.455520	−	0.890226i	$-0.349453\pi$
$912$		0			0
$913$	−30582.4	−	17656.7i	−1.10857	−	0.640036i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−27721.4	−	15565.2i	−0.998302	−	0.560532i
$918$		0			0
$919$	23650.1	+	40963.2i	0.848906	+	1.47035i	0.882185	+	0.470902i	$0.156072\pi$
$919$	23650.1	+	40963.2i	0.848906	+	1.47035i	−0.0332794	+	0.999446i	$0.510595\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−5160.57			−0.184033
$924$		0			0
$925$	24855.3			0.883501
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−19097.8	−	33078.4i	−0.674467	−	1.16821i	−0.976624	−	0.214953i	$-0.931040\pi$
$929$	−19097.8	−	33078.4i	−0.674467	−	1.16821i	0.302158	−	0.953258i	$-0.402293\pi$
$930$		0			0
$931$	−153.097	−	6388.48i	−0.00538943	−	0.224892i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−8772.36	−	5064.72i	−0.306831	−	0.177149i
$936$		0			0
$937$		−	38926.2i		−	1.35716i	−0.734525	−	0.678582i	$-0.762595\pi$
$937$		−	38926.2i		−	1.35716i	0.734525	−	0.678582i	$-0.237405\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−23277.1	+	40317.1i	−0.806388	+	1.39670i	0.108962	+	0.994046i	$0.465247\pi$
$941$	−23277.1	+	40317.1i	−0.806388	+	1.39670i	−0.915350	+	0.402659i	$0.868086\pi$
$942$		0			0
$943$	15259.3	−	8809.95i	0.526947	−	0.304233i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	2755.38	−	1590.82i	0.0945490	−	0.0545879i	−0.451980	−	0.892028i	$-0.649282\pi$
$947$	2755.38	−	1590.82i	0.0945490	−	0.0545879i	0.546529	+	0.837440i	$0.315949\pi$
$948$		0			0
$949$	−7936.50	+	13746.4i	−0.271475	+	0.470208i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	37411.6i		−	1.27165i	−0.771834	−	0.635824i	$-0.780660\pi$
$953$		−	37411.6i		−	1.27165i	0.771834	−	0.635824i	$-0.219340\pi$
$954$		0			0
$955$	−5307.47	−	3064.27i	−0.179838	−	0.103830i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−39315.1	+	471.017i	−1.32383	+	0.0158602i
$960$		0			0
$961$	7708.94	+	13352.3i	0.258767	+	0.448198i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	16245.1			0.541917
$966$		0			0
$967$	−53691.7			−1.78553			−0.892766	−	0.450521i	$-0.851238\pi$
$967$	−53691.7			−1.78553			−0.892766	+	0.450521i	$0.851238\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−19762.8	−	34230.1i	−0.653159	−	1.13131i	−0.982352	−	0.187042i	$-0.940110\pi$
$971$	−19762.8	−	34230.1i	−0.653159	−	1.13131i	0.329193	−	0.944263i	$-0.393223\pi$
$972$		0			0
$973$	16719.5	+	28174.0i	0.550875	+	0.928281i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−721.916	−	416.798i	−0.0236398	−	0.0136485i	0.488134	−	0.872769i	$-0.337678\pi$
$977$	−721.916	−	416.798i	−0.0236398	−	0.0136485i	−0.511773	+	0.859121i	$0.671011\pi$
$978$		0			0
$979$		−	27946.1i		−	0.912321i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	23795.5	−	41215.0i	0.772084	−	1.33729i	−0.164336	−	0.986405i	$-0.552548\pi$
$983$	23795.5	−	41215.0i	0.772084	−	1.33729i	0.936419	−	0.350884i	$-0.114119\pi$
$984$		0			0
$985$	4405.28	−	2543.39i	0.142501	−	0.0822731i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−35535.8	+	20516.6i	−1.14254	+	0.659646i
$990$		0			0
$991$	−3373.33	+	5842.78i	−0.108131	+	0.187288i	−0.915013	−	0.403425i	$-0.867820\pi$
$991$	−3373.33	+	5842.78i	−0.108131	+	0.187288i	0.806882	+	0.590712i	$0.201153\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	4256.13i		−	0.135606i
$996$		0			0
$997$	45419.1	+	26222.7i	1.44277	+	0.832981i	0.998034	−	0.0626825i	$-0.0199656\pi$
$997$	45419.1	+	26222.7i	1.44277	+	0.832981i	0.444732	+	0.895664i	$0.353299\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.4.t.e.269.10	yes	32
3.2	odd	2	inner	756.4.t.e.269.7	✓	32
7.5	odd	6	inner	756.4.t.e.593.7	yes	32
21.5	even	6	inner	756.4.t.e.593.10	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.4.t.e.269.7	✓	32	3.2	odd	2	inner
756.4.t.e.269.10	yes	32	1.1	even	1	trivial
756.4.t.e.593.7	yes	32	7.5	odd	6	inner
756.4.t.e.593.10	yes	32	21.5	even	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+(2.13836 + 3.70374i) q^{5} +(-15.9269 + 9.45161i) q^{7} +(-35.6565 - 20.5863i) q^{11} -24.1846i q^{13} +(28.7633 - 49.8194i) q^{17} +(16.1346 - 9.31533i) q^{19} +(67.9773 - 39.2467i) q^{23} +(53.3549 - 92.4134i) q^{25} +202.514i q^{29} +(184.138 + 106.312i) q^{31} +(-69.0637 - 38.7783i) q^{35} +(116.462 + 201.719i) q^{37} +224.476 q^{41} -522.759 q^{43} +(190.401 + 329.785i) q^{47} +(164.334 - 301.070i) q^{49} +(569.999 + 329.089i) q^{53} -176.083i q^{55} +(-338.019 + 585.466i) q^{59} +(-126.822 + 73.2210i) q^{61} +(89.5734 - 51.7152i) q^{65} +(-221.248 + 383.212i) q^{67} -213.383i q^{71} +(-568.396 - 328.164i) q^{73} +(762.471 - 9.13483i) q^{77} +(247.191 + 428.148i) q^{79} +857.695 q^{83} +246.024 q^{85} +(339.378 + 587.821i) q^{89} +(228.583 + 385.186i) q^{91} +(69.0031 + 39.8390i) q^{95} -436.541i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 10 q^{7} + 72 q^{19} - 514 q^{25} - 714 q^{31} + 8 q^{37} + 368 q^{43} + 890 q^{49} + 1272 q^{61} + 1840 q^{67} + 324 q^{73} + 532 q^{79} - 1344 q^{85} - 4596 q^{91}+O(q^{100}) \) 32 * q + 10 * q^7 + 72 * q^19 - 514 * q^25 - 714 * q^31 + 8 * q^37 + 368 * q^43 + 890 * q^49 + 1272 * q^61 + 1840 * q^67 + 324 * q^73 + 532 * q^79 - 1344 * q^85 - 4596 * q^91

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