LMFDB - Embedded newform 756.3.p.a.397.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [756,3,Mod(397,756)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

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

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("756.397"); S:= CuspForms(chi, 3); N := Newforms(S);

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

Newform invariants

comment:select newform

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

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

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

Embedding invariants

Embedding label			397.9
Character	$\chi$	$=$	756.397
Dual form			756.3.p.a.577.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.963344i q^{5} +(6.86265 - 1.37988i) q^{7} -8.27537 q^{11} +(0.385122 - 0.222350i) q^{13} +(4.35489 - 2.51430i) q^{17} +(19.3354 + 11.1633i) q^{19} +15.0853 q^{23} +24.0720 q^{25} +(12.3863 - 21.4537i) q^{29} +(-21.0881 - 12.1752i) q^{31} +(1.32930 + 6.61109i) q^{35} +(-15.1201 + 26.1888i) q^{37} +(-21.0544 + 12.1558i) q^{41} +(11.5760 - 20.0503i) q^{43} +(-8.13670 + 4.69773i) q^{47} +(45.1918 - 18.9393i) q^{49} +(44.7372 + 77.4871i) q^{53} -7.97203i q^{55} +(23.7510 + 13.7127i) q^{59} +(68.9235 - 39.7930i) q^{61} +(0.214200 + 0.371005i) q^{65} +(7.09042 - 12.2810i) q^{67} +133.436 q^{71} +(51.2871 - 29.6106i) q^{73} +(-56.7910 + 11.4191i) q^{77} +(-9.98363 - 17.2921i) q^{79} +(101.021 + 58.3246i) q^{83} +(2.42213 + 4.19526i) q^{85} +(105.064 + 60.6588i) q^{89} +(2.33614 - 2.05734i) q^{91} +(-10.7541 + 18.6266i) q^{95} +(-53.2313 - 30.7331i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + q^{7} + 12 q^{11} + 15 q^{13} + 27 q^{17} - 30 q^{23} - 160 q^{25} - 24 q^{29} - 24 q^{31} - 141 q^{35} + 11 q^{37} + 90 q^{41} - 16 q^{43} - 108 q^{47} - 61 q^{49} - 54 q^{53} - 45 q^{59} - 165 q^{61}+ \cdots - 57 q^{97}+O(q^{100}) $ 32 * q + q^7 + 12 * q^11 + 15 * q^13 + 27 * q^17 - 30 * q^23 - 160 * q^25 - 24 * q^29 - 24 * q^31 - 141 * q^35 + 11 * q^37 + 90 * q^41 - 16 * q^43 - 108 * q^47 - 61 * q^49 - 54 * q^53 - 45 * q^59 - 165 * q^61 + 33 * q^65 - 35 * q^67 - 138 * q^71 + 123 * q^77 + 37 * q^79 - 378 * q^83 + 30 * q^85 + 261 * q^89 + 45 * q^91 + 48 * q^95 - 57 * q^97

Character values

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

$n$	$29$	$325$	$379$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{5}{6}\right)$	$1$

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
$5$			0.963344i			0.192669i	0.995349	+	0.0963344i	$0.0307118\pi$
$5$			0.963344i			0.192669i	−0.995349	+	0.0963344i	$0.969288\pi$
$6$		0			0
$7$	6.86265	−	1.37988i	0.980378	−	0.197126i
$7$	6.86265	−	1.37988i	0.980378	−	0.197126i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−8.27537			−0.752307			−0.376153	−	0.926557i	$-0.622753\pi$
$11$	−8.27537			−0.752307			−0.376153	+	0.926557i	$0.622753\pi$
$12$		0			0
$13$	0.385122	−	0.222350i	0.0296248	−	0.0171039i	−0.485114	−	0.874451i	$-0.661222\pi$
$13$	0.385122	−	0.222350i	0.0296248	−	0.0171039i	0.514739	+	0.857347i	$0.327889\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.35489	−	2.51430i	0.256170	−	0.147900i	−0.366416	−	0.930451i	$-0.619415\pi$
$17$	4.35489	−	2.51430i	0.256170	−	0.147900i	0.622586	+	0.782551i	$0.286082\pi$
$18$		0			0
$19$	19.3354	+	11.1633i	1.01765	+	0.587542i	0.913423	−	0.407011i	$-0.133429\pi$
$19$	19.3354	+	11.1633i	1.01765	+	0.587542i	0.104230	+	0.994553i	$0.466762\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	15.0853			0.655883			0.327942	−	0.944698i	$-0.393645\pi$
$23$	15.0853			0.655883			0.327942	+	0.944698i	$0.393645\pi$
$24$		0			0
$25$	24.0720			0.962879
$26$		0			0
$27$		0			0
$28$		0			0
$29$	12.3863	−	21.4537i	0.427113	−	0.739782i	−0.569502	−	0.821990i	$-0.692864\pi$
$29$	12.3863	−	21.4537i	0.427113	−	0.739782i	0.996615	+	0.0822080i	$0.0261972\pi$
$30$		0			0
$31$	−21.0881	−	12.1752i	−0.680263	−	0.392750i	0.119691	−	0.992811i	$-0.461810\pi$
$31$	−21.0881	−	12.1752i	−0.680263	−	0.392750i	−0.799954	+	0.600061i	$0.795143\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.32930	+	6.61109i	0.0379801	+	0.188888i
$36$		0			0
$37$	−15.1201	+	26.1888i	−0.408652	+	0.707806i	−0.994739	−	0.102442i	$-0.967334\pi$
$37$	−15.1201	+	26.1888i	−0.408652	+	0.707806i	0.586087	+	0.810248i	$0.300668\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−21.0544	+	12.1558i	−0.513523	+	0.296483i	−0.734281	−	0.678846i	$-0.762480\pi$
$41$	−21.0544	+	12.1558i	−0.513523	+	0.296483i	0.220758	+	0.975329i	$0.429147\pi$
$42$		0			0
$43$	11.5760	−	20.0503i	0.269210	−	0.466286i	−0.699448	−	0.714684i	$-0.746571\pi$
$43$	11.5760	−	20.0503i	0.269210	−	0.466286i	0.968658	+	0.248398i	$0.0799041\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−8.13670	+	4.69773i	−0.173121	+	0.0999517i	−0.584057	−	0.811713i	$-0.698535\pi$
$47$	−8.13670	+	4.69773i	−0.173121	+	0.0999517i	0.410935	+	0.911664i	$0.365202\pi$
$48$		0			0
$49$	45.1918	−	18.9393i	0.922282	−	0.386517i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	44.7372	+	77.4871i	0.844098	+	1.46202i	0.886402	+	0.462915i	$0.153197\pi$
$53$	44.7372	+	77.4871i	0.844098	+	1.46202i	−0.0423047	+	0.999105i	$0.513470\pi$
$54$		0			0
$55$		−	7.97203i		−	0.144946i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	23.7510	+	13.7127i	0.402560	+	0.232418i	0.687588	−	0.726101i	$-0.258670\pi$
$59$	23.7510	+	13.7127i	0.402560	+	0.232418i	−0.285028	+	0.958519i	$0.592003\pi$
$60$		0			0
$61$	68.9235	−	39.7930i	1.12989	−	0.652344i	0.185985	−	0.982553i	$-0.440452\pi$
$61$	68.9235	−	39.7930i	1.12989	−	0.652344i	0.943908	+	0.330208i	$0.107119\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.214200	+	0.371005i	0.00329538	+	0.00570777i
$66$		0			0
$67$	7.09042	−	12.2810i	0.105827	−	0.183298i	−0.808249	−	0.588841i	$-0.799584\pi$
$67$	7.09042	−	12.2810i	0.105827	−	0.183298i	0.914076	+	0.405543i	$0.132918\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	133.436			1.87938			0.939688	−	0.342032i	$-0.111115\pi$
$71$	133.436			1.87938			0.939688	+	0.342032i	$0.111115\pi$
$72$		0			0
$73$	51.2871	−	29.6106i	0.702563	−	0.405625i	−0.105738	−	0.994394i	$-0.533721\pi$
$73$	51.2871	−	29.6106i	0.702563	−	0.405625i	0.808301	+	0.588769i	$0.200387\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−56.7910	+	11.4191i	−0.737545	+	0.148299i
$78$		0			0
$79$	−9.98363	−	17.2921i	−0.126375	−	0.218888i	0.795895	−	0.605435i	$-0.207001\pi$
$79$	−9.98363	−	17.2921i	−0.126375	−	0.218888i	−0.922270	+	0.386547i	$0.873668\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	101.021	+	58.3246i	1.21712	+	0.702705i	0.964301	−	0.264808i	$-0.0853085\pi$
$83$	101.021	+	58.3246i	1.21712	+	0.702705i	0.252820	+	0.967513i	$0.418642\pi$
$84$		0			0
$85$	2.42213	+	4.19526i	0.0284957	+	0.0493559i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	105.064	+	60.6588i	1.18050	+	0.681560i	0.956129	−	0.292945i	$-0.0946352\pi$
$89$	105.064	+	60.6588i	1.18050	+	0.681560i	0.224367	+	0.974505i	$0.427969\pi$
$90$		0			0
$91$	2.33614	−	2.05734i	0.0256719	−	0.0226081i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−10.7541	+	18.6266i	−0.113201	+	0.196070i
$96$		0			0
$97$	−53.2313	−	30.7331i	−0.548776	−	0.316836i	0.199852	−	0.979826i	$-0.435954\pi$
$97$	−53.2313	−	30.7331i	−0.548776	−	0.316836i	−0.748628	+	0.662990i	$0.769287\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			119.021i			1.17842i	0.807979	+	0.589212i	$0.200562\pi$
$101$			119.021i			1.17842i	−0.807979	+	0.589212i	$0.799438\pi$
$102$		0			0
$103$		−	97.0278i		−	0.942018i	−0.882129	−	0.471009i	$-0.843890\pi$
$103$		−	97.0278i		−	0.942018i	0.882129	−	0.471009i	$-0.156110\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	38.1581	−	66.0918i	0.356618	−	0.617681i	−0.630775	−	0.775965i	$-0.717263\pi$
$107$	38.1581	−	66.0918i	0.356618	−	0.617681i	0.987394	+	0.158285i	$0.0505964\pi$
$108$		0			0
$109$	19.1865	+	33.2320i	0.176023	+	0.304880i	0.940515	−	0.339753i	$-0.110343\pi$
$109$	19.1865	+	33.2320i	0.176023	+	0.304880i	−0.764492	+	0.644633i	$0.777010\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−102.540	−	177.605i	−0.907437	−	1.57173i	−0.817611	−	0.575770i	$-0.804702\pi$
$113$	−102.540	−	177.605i	−0.907437	−	1.57173i	−0.0898260	−	0.995957i	$-0.528631\pi$
$114$		0			0
$115$			14.5323i			0.126368i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	26.4166	−	23.2640i	0.221988	−	0.195496i
$120$		0			0
$121$	−52.5182			−0.434035
$122$		0			0
$123$		0			0
$124$		0			0
$125$			47.2732i			0.378185i
$126$		0			0
$127$	−71.8116			−0.565446			−0.282723	−	0.959202i	$-0.591238\pi$
$127$	−71.8116			−0.565446			−0.282723	+	0.959202i	$0.591238\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	7.55009i		−	0.0576343i	−0.999585	−	0.0288172i	$-0.990826\pi$
$131$		−	7.55009i		−	0.0576343i	0.999585	−	0.0288172i	$-0.00917405\pi$
$132$		0			0
$133$	148.096	+	49.9292i	1.11350	+	0.375407i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−203.912			−1.48841			−0.744203	−	0.667954i	$-0.767170\pi$
$137$	−203.912			−1.48841			−0.744203	+	0.667954i	$0.767170\pi$
$138$		0			0
$139$	−77.3785	+	44.6745i	−0.556680	+	0.321399i	−0.751812	−	0.659378i	$-0.770820\pi$
$139$	−77.3785	+	44.6745i	−0.556680	+	0.321399i	0.195132	+	0.980777i	$0.437486\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.18703	+	1.84003i	−0.0222869	+	0.0128674i
$144$		0			0
$145$	20.6673	+	11.9323i	0.142533	+	0.0822914i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−126.472			−0.848808			−0.424404	−	0.905473i	$-0.639516\pi$
$149$	−126.472			−0.848808			−0.424404	+	0.905473i	$0.639516\pi$
$150$		0			0
$151$	33.9775			0.225017			0.112508	−	0.993651i	$-0.464111\pi$
$151$	33.9775			0.225017			0.112508	+	0.993651i	$0.464111\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	11.7289	−	20.3151i	0.0756706	−	0.131065i
$156$		0			0
$157$	39.5383	+	22.8274i	0.251836	+	0.145398i	0.620605	−	0.784124i	$-0.286887\pi$
$157$	39.5383	+	22.8274i	0.251836	+	0.145398i	−0.368769	+	0.929521i	$0.620220\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	103.525	−	20.8160i	0.643013	−	0.129292i
$162$		0			0
$163$	65.6487	−	113.707i	0.402753	−	0.697588i	−0.591304	−	0.806449i	$-0.701387\pi$
$163$	65.6487	−	113.707i	0.402753	−	0.697588i	0.994057	+	0.108860i	$0.0347201\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	205.982	−	118.924i	1.23343	−	0.712118i	0.265683	−	0.964060i	$-0.414403\pi$
$167$	205.982	−	118.924i	1.23343	−	0.712118i	0.967742	+	0.251942i	$0.0810692\pi$
$168$		0			0
$169$	−84.4011	+	146.187i	−0.499415	+	0.865012i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−268.118	+	154.798i	−1.54981	+	0.894785i	−0.551657	+	0.834071i	$0.686004\pi$
$173$	−268.118	+	154.798i	−1.54981	+	0.894785i	−0.998155	+	0.0607137i	$0.980662\pi$
$174$		0			0
$175$	165.197	−	33.2165i	0.943985	−	0.189809i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−151.419	−	262.266i	−0.845917	−	1.46517i	−0.884822	−	0.465928i	$-0.845720\pi$
$179$	−151.419	−	262.266i	−0.845917	−	1.46517i	0.0389054	−	0.999243i	$-0.487613\pi$
$180$		0			0
$181$			205.717i			1.13656i	0.822835	+	0.568280i	$0.192391\pi$
$181$			205.717i			1.13656i	−0.822835	+	0.568280i	$0.807609\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−25.2288	−	14.5659i	−0.136372	−	0.0787344i
$186$		0			0
$187$	−36.0383	+	20.8067i	−0.192718	+	0.111266i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	36.5455	+	63.2987i	0.191338	+	0.331407i	0.945694	−	0.325059i	$-0.105384\pi$
$191$	36.5455	+	63.2987i	0.191338	+	0.331407i	−0.754356	+	0.656465i	$0.772051\pi$
$192$		0			0
$193$	−177.413	+	307.288i	−0.919236	+	1.59216i	−0.118658	+	0.992935i	$0.537859\pi$
$193$	−177.413	+	307.288i	−0.919236	+	1.59216i	−0.800578	+	0.599229i	$0.795474\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.81361			0.0142823			0.00714113	−	0.999975i	$-0.497727\pi$
$197$	2.81361			0.0142823			0.00714113	+	0.999975i	$0.497727\pi$
$198$		0			0
$199$	92.6937	−	53.5167i	0.465797	−	0.268928i	−0.248681	−	0.968585i	$-0.579997\pi$
$199$	92.6937	−	53.5167i	0.465797	−	0.268928i	0.714479	+	0.699657i	$0.246664\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	55.3991	−	164.321i	0.272902	−	0.809461i
$204$		0			0
$205$	−11.7102	−	20.2827i	−0.0571230	−	0.0989399i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−160.008	−	92.3805i	−0.765587	−	0.442012i
$210$		0			0
$211$	0.310904	+	0.538502i	0.00147348	+	0.00255214i	0.866761	−	0.498723i	$-0.166198\pi$
$211$	0.310904	+	0.538502i	0.00147348	+	0.00255214i	−0.865288	+	0.501276i	$0.832864\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	19.3153	+	11.1517i	0.0898387	+	0.0518684i
$216$		0			0
$217$	−161.521	−	54.4552i	−0.744336	−	0.250946i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.11811	−	1.93662i	0.00505932	−	0.00876299i
$222$		0			0
$223$	−347.765	−	200.782i	−1.55948	−	0.900368i	−0.997306	−	0.0733508i	$-0.976631\pi$
$223$	−347.765	−	200.782i	−1.55948	−	0.900368i	−0.562177	−	0.827017i	$-0.690036\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			391.807i			1.72602i	0.505187	+	0.863010i	$0.331424\pi$
$227$			391.807i			1.72602i	−0.505187	+	0.863010i	$0.668576\pi$
$228$		0			0
$229$		−	128.808i		−	0.562480i	−0.959637	−	0.281240i	$-0.909254\pi$
$229$		−	128.808i		−	0.562480i	0.959637	−	0.281240i	$-0.0907458\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	17.8368	−	30.8942i	0.0765526	−	0.132593i	−0.825208	−	0.564829i	$-0.808942\pi$
$233$	17.8368	−	30.8942i	0.0765526	−	0.132593i	0.901760	+	0.432236i	$0.142275\pi$
$234$		0			0
$235$	−4.52553	−	7.83844i	−0.0192576	−	0.0333551i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−142.757	−	247.262i	−0.597308	−	1.03457i	−0.993217	−	0.116278i	$-0.962904\pi$
$239$	−142.757	−	247.262i	−0.597308	−	1.03457i	0.395909	−	0.918290i	$-0.370430\pi$
$240$		0			0
$241$			138.284i			0.573792i	0.957962	+	0.286896i	$0.0926234\pi$
$241$			138.284i			0.573792i	−0.957962	+	0.286896i	$0.907377\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	18.2451	+	43.5353i	0.0744696	+	0.177695i
$246$		0			0
$247$	9.92866			0.0401970
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	29.7450i		−	0.118506i	−0.998243	−	0.0592530i	$-0.981128\pi$
$251$		−	29.7450i		−	0.118506i	0.998243	−	0.0592530i	$-0.0188719\pi$
$252$		0			0
$253$	−124.837			−0.493425
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	289.004i		−	1.12453i	−0.826957	−	0.562265i	$-0.809930\pi$
$257$		−	289.004i		−	1.12453i	0.826957	−	0.562265i	$-0.190070\pi$
$258$		0			0
$259$	−67.6265	+	200.589i	−0.261106	+	0.774473i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−316.849			−1.20475			−0.602375	−	0.798213i	$-0.705779\pi$
$263$	−316.849			−1.20475			−0.602375	+	0.798213i	$0.705779\pi$
$264$		0			0
$265$	−74.6467	+	43.0973i	−0.281686	+	0.162631i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−154.361	+	89.1206i	−0.573834	+	0.331303i	−0.758679	−	0.651464i	$-0.774155\pi$
$269$	−154.361	+	89.1206i	−0.573834	+	0.331303i	0.184845	+	0.982768i	$0.440822\pi$
$270$		0			0
$271$	−299.052	−	172.658i	−1.10351	−	0.637114i	−0.166373	−	0.986063i	$-0.553205\pi$
$271$	−299.052	−	172.658i	−1.10351	−	0.637114i	−0.937142	+	0.348948i	$0.886539\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−199.204			−0.724380
$276$		0			0
$277$	−374.339			−1.35140			−0.675702	−	0.737175i	$-0.736159\pi$
$277$	−374.339			−1.35140			−0.675702	+	0.737175i	$0.736159\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	168.976	−	292.676i	0.601340	−	1.04155i	−0.391279	−	0.920272i	$-0.627967\pi$
$281$	168.976	−	292.676i	0.601340	−	1.04155i	0.992618	−	0.121279i	$-0.0386995\pi$
$282$		0			0
$283$	370.026	+	213.634i	1.30751	+	0.754892i	0.981680	−	0.190536i	$-0.0610225\pi$
$283$	370.026	+	213.634i	1.30751	+	0.754892i	0.325831	+	0.945428i	$0.394356\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−127.716	+	112.474i	−0.445002	+	0.391894i
$288$		0			0
$289$	−131.857	+	228.382i	−0.456251	+	0.790250i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	54.0168	−	31.1866i	0.184358	−	0.106439i	−0.404981	−	0.914325i	$-0.632722\pi$
$293$	54.0168	−	31.1866i	0.184358	−	0.106439i	0.589338	+	0.807886i	$0.299388\pi$
$294$		0			0
$295$	−13.2100	+	22.8804i	−0.0447797	+	0.0775607i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	5.80968	−	3.35422i	0.0194304	−	0.0112181i
$300$		0			0
$301$	51.7752	−	153.572i	0.172011	−	0.510205i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	38.3343	+	66.3970i	0.125686	+	0.217695i
$306$		0			0
$307$		−	473.374i		−	1.54194i	−0.636874	−	0.770968i	$-0.719773\pi$
$307$		−	473.374i		−	1.54194i	0.636874	−	0.770968i	$-0.280227\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	125.112	+	72.2333i	0.402288	+	0.232261i	0.687471	−	0.726212i	$-0.258721\pi$
$311$	125.112	+	72.2333i	0.402288	+	0.232261i	−0.285183	+	0.958473i	$0.592054\pi$
$312$		0			0
$313$	16.2589	−	9.38707i	0.0519453	−	0.0299906i	−0.473802	−	0.880631i	$-0.657119\pi$
$313$	16.2589	−	9.38707i	0.0519453	−	0.0299906i	0.525748	+	0.850641i	$0.323786\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	13.0645	+	22.6284i	0.0412131	+	0.0713831i	0.885896	−	0.463884i	$-0.153544\pi$
$317$	13.0645	+	22.6284i	0.0412131	+	0.0713831i	−0.844683	+	0.535267i	$0.820211\pi$
$318$		0			0
$319$	−102.501	+	177.537i	−0.321320	+	0.556543i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	112.271			0.347590
$324$		0			0
$325$	9.27064	−	5.35241i	0.0285251	−	0.0164690i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−49.3570	+	43.4666i	−0.150021	+	0.132117i
$330$		0			0
$331$	−140.657	−	243.624i	−0.424944	−	0.736025i	0.571471	−	0.820622i	$-0.306373\pi$
$331$	−140.657	−	243.624i	−0.424944	−	0.736025i	−0.996415	+	0.0845973i	$0.973040\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	11.8308	+	6.83051i	0.0353158	+	0.0203896i
$336$		0			0
$337$	−94.4669	−	163.621i	−0.280317	−	0.485523i	0.691146	−	0.722715i	$-0.257106\pi$
$337$	−94.4669	−	163.621i	−0.280317	−	0.485523i	−0.971463	+	0.237192i	$0.923773\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	174.512	+	100.755i	0.511766	+	0.295468i
$342$		0			0
$343$	284.002	−	192.333i	0.827993	−	0.560738i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−39.7467	+	68.8433i	−0.114544	+	0.198396i	−0.917597	−	0.397511i	$-0.869874\pi$
$347$	−39.7467	+	68.8433i	−0.114544	+	0.198396i	0.803054	+	0.595907i	$0.203207\pi$
$348$		0			0
$349$	256.280	+	147.963i	0.734327	+	0.423964i	0.820003	−	0.572359i	$-0.193972\pi$
$349$	256.280	+	147.963i	0.734327	+	0.423964i	−0.0856763	+	0.996323i	$0.527305\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			479.848i			1.35934i	0.733516	+	0.679672i	$0.237878\pi$
$353$			479.848i			1.35934i	−0.733516	+	0.679672i	$0.762122\pi$
$354$		0			0
$355$			128.544i			0.362097i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	267.870	−	463.964i	0.746156	−	1.29238i	−0.203497	−	0.979076i	$-0.565231\pi$
$359$	267.870	−	463.964i	0.746156	−	1.29238i	0.949653	−	0.313304i	$-0.101436\pi$
$360$		0			0
$361$	68.7387	+	119.059i	0.190412	+	0.329803i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	28.5252	+	49.4071i	0.0781512	+	0.135362i
$366$		0			0
$367$			272.723i			0.743115i	0.928410	+	0.371557i	$0.121176\pi$
$367$			272.723i			0.743115i	−0.928410	+	0.371557i	$0.878824\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	413.939	+	470.034i	1.11574	+	1.26694i
$372$		0			0
$373$	−671.704			−1.80082			−0.900408	−	0.435047i	$-0.856732\pi$
$373$	−671.704			−1.80082			−0.900408	+	0.435047i	$0.856732\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	11.0164i		−	0.0292212i
$378$		0			0
$379$	−140.974			−0.371962			−0.185981	−	0.982553i	$-0.559546\pi$
$379$	−140.974			−0.371962			−0.185981	+	0.982553i	$0.559546\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	397.648i		−	1.03825i	−0.854700	−	0.519123i	$-0.826259\pi$
$383$		−	397.648i		−	1.03825i	0.854700	−	0.519123i	$-0.173741\pi$
$384$		0			0
$385$	−11.0005	−	54.7092i	−0.0285726	−	0.142102i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−518.270			−1.33231			−0.666157	−	0.745812i	$-0.732062\pi$
$389$	−518.270			−1.33231			−0.666157	+	0.745812i	$0.732062\pi$
$390$		0			0
$391$	65.6949	−	37.9289i	0.168018	−	0.0970050i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	16.6583	−	9.61766i	0.0421729	−	0.0243485i
$396$		0			0
$397$	301.974	+	174.345i	0.760640	+	0.439156i	0.829525	−	0.558469i	$-0.188611\pi$
$397$	301.974	+	174.345i	0.760640	+	0.439156i	−0.0688856	+	0.997625i	$0.521944\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−210.523			−0.524996			−0.262498	−	0.964933i	$-0.584546\pi$
$401$	−210.523			−0.524996			−0.262498	+	0.964933i	$0.584546\pi$
$402$		0			0
$403$	−10.8287			−0.0268702
$404$		0			0
$405$		0			0
$406$		0			0
$407$	125.125	−	216.722i	0.307431	−	0.532487i
$408$		0			0
$409$	464.682	+	268.285i	1.13614	+	0.655952i	0.945473	−	0.325702i	$-0.105601\pi$
$409$	464.682	+	268.285i	1.13614	+	0.655952i	0.190670	+	0.981654i	$0.438934\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	181.917	+	61.3315i	0.440476	+	0.148502i
$414$		0			0
$415$	−56.1866	+	97.3180i	−0.135389	+	0.234501i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−368.723	+	212.882i	−0.880007	+	0.508072i	−0.870661	−	0.491884i	$-0.836308\pi$
$419$	−368.723	+	212.882i	−0.880007	+	0.508072i	−0.00934643	+	0.999956i	$0.502975\pi$
$420$		0			0
$421$	399.672	−	692.253i	0.949341	−	1.64431i	0.202523	−	0.979277i	$-0.435086\pi$
$421$	399.672	−	692.253i	0.949341	−	1.64431i	0.746818	−	0.665029i	$-0.231581\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	104.831	−	60.5241i	0.246661	−	0.142410i
$426$		0			0
$427$	418.088	−	368.192i	0.979129	−	0.862276i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−33.3100	−	57.6946i	−0.0772854	−	0.133862i	0.824792	−	0.565436i	$-0.191292\pi$
$431$	−33.3100	−	57.6946i	−0.0772854	−	0.133862i	−0.902078	+	0.431573i	$0.857959\pi$
$432$		0			0
$433$			532.074i			1.22881i	0.788991	+	0.614405i	$0.210604\pi$
$433$			532.074i			1.22881i	−0.788991	+	0.614405i	$0.789396\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	291.681	+	168.402i	0.667461	+	0.385359i
$438$		0			0
$439$	222.308	−	128.350i	0.506397	−	0.292368i	−0.224955	−	0.974369i	$-0.572223\pi$
$439$	222.308	−	128.350i	0.506397	−	0.292368i	0.731351	+	0.682001i	$0.238890\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	45.0204	+	77.9776i	0.101626	+	0.176022i	0.912355	−	0.409400i	$-0.134262\pi$
$443$	45.0204	+	77.9776i	0.101626	+	0.176022i	−0.810729	+	0.585422i	$0.800929\pi$
$444$		0			0
$445$	−58.4353	+	101.213i	−0.131315	+	0.227445i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	238.627			0.531464			0.265732	−	0.964047i	$-0.414386\pi$
$449$	238.627			0.531464			0.265732	+	0.964047i	$0.414386\pi$
$450$		0			0
$451$	174.233	−	100.594i	0.386327	−	0.223046i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1.98192	+	2.25050i	0.00435587	+	0.00494616i
$456$		0			0
$457$	312.577	+	541.400i	0.683977	+	1.18468i	0.973757	+	0.227589i	$0.0730843\pi$
$457$	312.577	+	541.400i	0.683977	+	1.18468i	−0.289781	+	0.957093i	$0.593582\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−612.475	−	353.613i	−1.32858	−	0.767056i	−0.343500	−	0.939153i	$-0.611612\pi$
$461$	−612.475	−	353.613i	−1.32858	−	0.767056i	−0.985080	+	0.172097i	$0.944946\pi$
$462$		0			0
$463$	−420.069	−	727.582i	−0.907277	−	1.57145i	−0.817831	−	0.575459i	$-0.804823\pi$
$463$	−420.069	−	727.582i	−0.907277	−	1.57145i	−0.0894469	−	0.995992i	$-0.528510\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	37.7190	+	21.7771i	0.0807688	+	0.0466319i	0.539840	−	0.841767i	$-0.318485\pi$
$467$	37.7190	+	21.7771i	0.0807688	+	0.0466319i	−0.459072	+	0.888399i	$0.651818\pi$
$468$		0			0
$469$	31.7127	−	94.0638i	0.0676177	−	0.200563i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−95.7960	+	165.924i	−0.202529	+	0.350790i
$474$		0			0
$475$	465.441	+	268.723i	0.979877	+	0.565732i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	670.529i		−	1.39985i	−0.714215	−	0.699926i	$-0.753216\pi$
$479$		−	670.529i		−	1.39985i	0.714215	−	0.699926i	$-0.246784\pi$
$480$		0			0
$481$			13.4478i			0.0279581i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	29.6065	−	51.2800i	0.0610444	−	0.105732i
$486$		0			0
$487$	365.228	+	632.594i	0.749955	+	1.29896i	0.947843	+	0.318736i	$0.103258\pi$
$487$	365.228	+	632.594i	0.749955	+	1.29896i	−0.197888	+	0.980225i	$0.563408\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	256.934	+	445.023i	0.523287	+	0.906360i	0.999633	+	0.0271018i	$0.00862782\pi$
$491$	256.934	+	445.023i	0.523287	+	0.906360i	−0.476346	+	0.879258i	$0.658039\pi$
$492$		0			0
$493$		−	124.571i		−	0.252680i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	915.722	−	184.126i	1.84250	−	0.370474i
$498$		0			0
$499$	153.056			0.306724			0.153362	−	0.988170i	$-0.450990\pi$
$499$	153.056			0.306724			0.153362	+	0.988170i	$0.450990\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			80.5356i			0.160110i	0.996790	+	0.0800552i	$0.0255097\pi$
$503$			80.5356i			0.160110i	−0.996790	+	0.0800552i	$0.974490\pi$
$504$		0			0
$505$	−114.658			−0.227045
$506$		0			0
$507$		0			0
$508$		0			0
$509$			66.7632i			0.131165i	0.997847	+	0.0655827i	$0.0208906\pi$
$509$			66.7632i			0.131165i	−0.997847	+	0.0655827i	$0.979109\pi$
$510$		0			0
$511$	311.106	−	273.977i	0.608818	−	0.536159i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	93.4711			0.181497
$516$		0			0
$517$	67.3342	−	38.8754i	0.130240	−	0.0751943i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−828.396	+	478.274i	−1.59001	+	0.917993i	−0.596707	+	0.802459i	$0.703525\pi$
$521$	−828.396	+	478.274i	−1.59001	+	0.917993i	−0.993303	+	0.115534i	$0.963142\pi$
$522$		0			0
$523$	−77.4777	−	44.7318i	−0.148141	−	0.0855292i	0.424097	−	0.905617i	$-0.360591\pi$
$523$	−77.4777	−	44.7318i	−0.148141	−	0.0855292i	−0.572238	+	0.820087i	$0.693925\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−122.449			−0.232350
$528$		0			0
$529$	−301.433			−0.569817
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−5.40569	+	9.36293i	−0.0101420	+	0.0175665i
$534$		0			0
$535$	63.6691	+	36.7594i	0.119008	+	0.0687092i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−373.979	+	156.730i	−0.693839	+	0.290779i
$540$		0			0
$541$	13.8403	−	23.9722i	0.0255829	−	0.0443109i	−0.852951	−	0.521992i	$-0.825189\pi$
$541$	13.8403	−	23.9722i	0.0255829	−	0.0443109i	0.878533	+	0.477681i	$0.158522\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−32.0138	+	18.4832i	−0.0587409	+	0.0339141i
$546$		0			0
$547$	339.847	−	588.632i	0.621292	−	1.07611i	−0.367953	−	0.929844i	$-0.619941\pi$
$547$	339.847	−	588.632i	0.621292	−	1.07611i	0.989245	−	0.146265i	$-0.0467253\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	478.988	−	276.544i	0.869307	−	0.501894i
$552$		0			0
$553$	−92.3753	−	104.894i	−0.167044	−	0.189681i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	172.676	+	299.083i	0.310010	+	0.536954i	0.978364	−	0.206890i	$-0.0663342\pi$
$557$	172.676	+	299.083i	0.310010	+	0.536954i	−0.668354	+	0.743843i	$0.733001\pi$
$558$		0			0
$559$		−	10.2957i		−	0.0184181i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	930.283	+	537.099i	1.65237	+	0.953995i	0.976095	+	0.217344i	$0.0697394\pi$
$563$	930.283	+	537.099i	1.65237	+	0.953995i	0.676273	+	0.736651i	$0.263594\pi$
$564$		0			0
$565$	171.095	−	98.7817i	0.302823	−	0.174835i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−307.619	−	532.811i	−0.540630	−	0.936399i	−0.998868	−	0.0475691i	$-0.984853\pi$
$569$	−307.619	−	532.811i	−0.540630	−	0.936399i	0.458238	−	0.888830i	$-0.348481\pi$
$570$		0			0
$571$	−248.251	+	429.983i	−0.434765	+	0.753035i	−0.997276	−	0.0737547i	$-0.976502\pi$
$571$	−248.251	+	429.983i	−0.434765	+	0.753035i	0.562512	+	0.826789i	$0.309835\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	363.133			0.631536
$576$		0			0
$577$	514.284	−	296.922i	0.891306	−	0.514596i	0.0169365	−	0.999857i	$-0.494609\pi$
$577$	514.284	−	296.922i	0.891306	−	0.514596i	0.874369	+	0.485261i	$0.161275\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	773.753	+	260.863i	1.33176	+	0.448990i
$582$		0			0
$583$	−370.217	−	641.234i	−0.635020	−	1.09989i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	322.786	+	186.360i	0.549890	+	0.317479i	0.749078	−	0.662482i	$-0.230497\pi$
$587$	322.786	+	186.360i	0.549890	+	0.317479i	−0.199187	+	0.979961i	$0.563830\pi$
$588$		0			0
$589$	−271.832	−	470.827i	−0.461514	−	0.799366i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	857.809	+	495.256i	1.44656	+	0.835171i	0.998274	−	0.0587202i	$-0.0187020\pi$
$593$	857.809	+	495.256i	1.44656	+	0.835171i	0.448284	+	0.893891i	$0.352035\pi$
$594$		0			0
$595$	22.4112	+	25.4483i	0.0376659	+	0.0427702i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−340.110	+	589.088i	−0.567796	+	0.983452i	0.428987	+	0.903311i	$0.358871\pi$
$599$	−340.110	+	589.088i	−0.567796	+	0.983452i	−0.996784	+	0.0801414i	$0.974463\pi$
$600$		0			0
$601$	132.268	+	76.3649i	0.220080	+	0.127063i	0.605987	−	0.795474i	$-0.292778\pi$
$601$	132.268	+	76.3649i	0.220080	+	0.127063i	−0.385908	+	0.922537i	$0.626112\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	50.5931i		−	0.0836250i
$606$		0			0
$607$		−	291.769i		−	0.480674i	−0.970690	−	0.240337i	$-0.922742\pi$
$607$		−	291.769i		−	0.480674i	0.970690	−	0.240337i	$-0.0772579\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.08908	+	3.61840i	−0.00341912	+	0.00592209i
$612$		0			0
$613$	−152.238	−	263.684i	−0.248349	−	0.430154i	0.714719	−	0.699412i	$-0.246555\pi$
$613$	−152.238	−	263.684i	−0.248349	−	0.430154i	−0.963068	+	0.269258i	$0.913221\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	42.3876	+	73.4174i	0.0686995	+	0.118991i	0.898329	−	0.439323i	$-0.144782\pi$
$617$	42.3876	+	73.4174i	0.0686995	+	0.118991i	−0.829630	+	0.558314i	$0.811448\pi$
$618$		0			0
$619$			649.483i			1.04925i	0.851335	+	0.524623i	$0.175794\pi$
$619$			649.483i			1.04925i	−0.851335	+	0.524623i	$0.824206\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	804.720	+	271.304i	1.29169	+	0.435480i
$624$		0			0
$625$	556.259			0.890014
$626$		0			0
$627$		0			0
$628$		0			0
$629$			152.066i			0.241758i
$630$		0			0
$631$	−46.9650			−0.0744295			−0.0372147	−	0.999307i	$-0.511849\pi$
$631$	−46.9650			−0.0744295			−0.0372147	+	0.999307i	$0.511849\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	69.1793i		−	0.108944i
$636$		0			0
$637$	13.1932	−	17.3424i	0.0207115	−	0.0272251i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−287.551			−0.448597			−0.224298	−	0.974520i	$-0.572009\pi$
$641$	−287.551			−0.448597			−0.224298	+	0.974520i	$0.572009\pi$
$642$		0			0
$643$	−7.84398	+	4.52872i	−0.0121990	+	0.00704312i	−0.506087	−	0.862482i	$-0.668909\pi$
$643$	−7.84398	+	4.52872i	−0.0121990	+	0.00704312i	0.493888	+	0.869525i	$0.335575\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−366.328	+	211.499i	−0.566194	+	0.326892i	−0.755628	−	0.655001i	$-0.772668\pi$
$647$	−366.328	+	211.499i	−0.566194	+	0.326892i	0.189434	+	0.981894i	$0.439335\pi$
$648$		0			0
$649$	−196.548	−	113.477i	−0.302848	−	0.174850i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−361.234			−0.553191			−0.276596	−	0.960986i	$-0.589206\pi$
$653$	−361.234			−0.553191			−0.276596	+	0.960986i	$0.589206\pi$
$654$		0			0
$655$	7.27333			0.0111043
$656$		0			0
$657$		0			0
$658$		0			0
$659$	271.135	−	469.619i	0.411433	−	0.712624i	−0.583613	−	0.812032i	$-0.698362\pi$
$659$	271.135	−	469.619i	0.411433	−	0.712624i	0.995047	+	0.0994081i	$0.0316949\pi$
$660$		0			0
$661$	772.501	+	446.003i	1.16868	+	0.674740i	0.953370	−	0.301804i	$-0.0975889\pi$
$661$	772.501	+	446.003i	1.16868	+	0.674740i	0.215315	+	0.976545i	$0.430922\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−48.0990	+	142.667i	−0.0723293	+	0.214538i
$666$		0			0
$667$	186.851	−	323.635i	0.280136	−	0.485211i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−570.368	+	329.302i	−0.850026	+	0.490763i
$672$		0			0
$673$	40.9317	−	70.8958i	0.0608197	−	0.105343i	−0.834012	−	0.551746i	$-0.813962\pi$
$673$	40.9317	−	70.8958i	0.0608197	−	0.105343i	0.894832	+	0.446403i	$0.147295\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	452.877	−	261.469i	0.668946	−	0.386216i	−0.126731	−	0.991937i	$-0.540448\pi$
$677$	452.877	−	261.469i	0.668946	−	0.386216i	0.795677	+	0.605721i	$0.207115\pi$
$678$		0			0
$679$	−407.715	−	137.457i	−0.600464	−	0.202441i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−466.207	−	807.494i	−0.682587	−	1.18228i	−0.974189	−	0.225736i	$-0.927521\pi$
$683$	−466.207	−	807.494i	−0.682587	−	1.18228i	0.291601	−	0.956540i	$-0.405812\pi$
$684$		0			0
$685$		−	196.437i		−	0.286769i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	34.4585	+	19.8947i	0.0500124	+	0.0288747i
$690$		0			0
$691$	−978.220	+	564.775i	−1.41566	+	0.817331i	−0.995914	−	0.0903117i	$-0.971214\pi$
$691$	−978.220	+	564.775i	−1.41566	+	0.817331i	−0.419745	+	0.907642i	$0.637880\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−43.0369	−	74.5421i	−0.0619236	−	0.107255i
$696$		0			0
$697$	−61.1265	+	105.874i	−0.0876995	+	0.151900i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−883.655			−1.26056			−0.630281	−	0.776367i	$-0.717061\pi$
$701$	−883.655			−1.26056			−0.630281	+	0.776367i	$0.717061\pi$
$702$		0			0
$703$	−584.707	+	337.581i	−0.831732	+	0.480200i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	164.235	+	816.797i	0.232298	+	1.15530i
$708$		0			0
$709$	118.989	+	206.096i	0.167827	+	0.290685i	0.937656	−	0.347566i	$-0.112992\pi$
$709$	118.989	+	206.096i	0.167827	+	0.290685i	−0.769829	+	0.638251i	$0.779658\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−318.121	−	183.667i	−0.446173	−	0.257598i
$714$		0			0
$715$	−1.77258	−	3.07020i	−0.00247914	−	0.00429399i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	54.4638	+	31.4447i	0.0757494	+	0.0437339i	0.537396	−	0.843330i	$-0.319408\pi$
$719$	54.4638	+	31.4447i	0.0757494	+	0.0437339i	−0.461647	+	0.887064i	$0.652741\pi$
$720$		0			0
$721$	−133.887	−	665.868i	−0.185696	−	0.923533i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	298.162	−	516.432i	0.411258	−	0.712321i
$726$		0			0
$727$	245.665	+	141.835i	0.337917	+	0.195096i	0.659350	−	0.751836i	$-0.270831\pi$
$727$	245.665	+	141.835i	0.337917	+	0.195096i	−0.321434	+	0.946932i	$0.604165\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	116.422i		−	0.159264i
$732$		0			0
$733$		−	486.378i		−	0.663544i	−0.943360	−	0.331772i	$-0.892353\pi$
$733$		−	486.378i		−	0.663544i	0.943360	−	0.331772i	$-0.107647\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−58.6758	+	101.630i	−0.0796144	+	0.137896i
$738$		0			0
$739$	378.929	+	656.325i	0.512759	+	0.888125i	0.999891	+	0.0147966i	$0.00471007\pi$
$739$	378.929	+	656.325i	0.512759	+	0.888125i	−0.487131	+	0.873329i	$0.661957\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	181.162	+	313.782i	0.243825	+	0.422317i	0.961801	−	0.273751i	$-0.0882644\pi$
$743$	181.162	+	313.782i	0.243825	+	0.422317i	−0.717976	+	0.696068i	$0.754931\pi$
$744$		0			0
$745$		−	121.836i		−	0.163539i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	170.667	−	506.219i	0.227860	−	0.675859i
$750$		0			0
$751$	−1353.58			−1.80236			−0.901182	−	0.433441i	$-0.857299\pi$
$751$	−1353.58			−1.80236			−0.901182	+	0.433441i	$0.857299\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			32.7320i			0.0433537i
$756$		0			0
$757$	−160.495			−0.212014			−0.106007	−	0.994365i	$-0.533807\pi$
$757$	−160.495			−0.212014			−0.106007	+	0.994365i	$0.533807\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	619.288i		−	0.813782i	−0.913477	−	0.406891i	$-0.866613\pi$
$761$		−	619.288i		−	0.813782i	0.913477	−	0.406891i	$-0.133387\pi$
$762$		0			0
$763$	177.526	+	201.584i	0.232669	+	0.264199i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	12.1961			0.0159010
$768$		0			0
$769$	133.828	−	77.2658i	0.174029	−	0.100476i	−0.410455	−	0.911881i	$-0.634630\pi$
$769$	133.828	−	77.2658i	0.174029	−	0.100476i	0.584484	+	0.811405i	$0.301297\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−554.122	+	319.923i	−0.716846	+	0.413871i	−0.813591	−	0.581438i	$-0.802490\pi$
$773$	−554.122	+	319.923i	−0.716846	+	0.413871i	0.0967447	+	0.995309i	$0.469157\pi$
$774$		0			0
$775$	−507.633	−	293.082i	−0.655011	−	0.378171i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−542.795			−0.696785
$780$		0			0
$781$	−1104.23			−1.41387
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−21.9907	+	38.0890i	−0.0280136	+	0.0485210i
$786$		0			0
$787$	428.370	+	247.320i	0.544308	+	0.314256i	0.746823	−	0.665023i	$-0.231578\pi$
$787$	428.370	+	247.320i	0.544308	+	0.314256i	−0.202515	+	0.979279i	$0.564911\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−948.773	−	1077.35i	−1.19946	−	1.36201i
$792$		0			0
$793$	17.6960	−	30.6503i	0.0223152	−	0.0386511i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	933.311	−	538.847i	1.17103	−	0.676095i	0.217108	−	0.976148i	$-0.430338\pi$
$797$	933.311	−	538.847i	1.17103	−	0.676095i	0.953923	+	0.300053i	$0.0970044\pi$
$798$		0			0
$799$	−23.6230	+	40.9162i	−0.0295657	+	0.0512092i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−424.420	+	245.039i	−0.528543	+	0.305154i
$804$		0			0
$805$	20.0529	+	99.7303i	0.0249105	+	0.123889i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−403.448	−	698.793i	−0.498700	−	0.863773i	0.501299	−	0.865274i	$-0.332856\pi$
$809$	−403.448	−	698.793i	−0.498700	−	0.863773i	−0.999999	+	0.00150078i	$0.999522\pi$
$810$		0			0
$811$			940.287i			1.15942i	0.814824	+	0.579708i	$0.196833\pi$
$811$			940.287i			1.15942i	−0.814824	+	0.579708i	$0.803167\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	109.539	+	63.2423i	0.134403	+	0.0775979i
$816$		0			0
$817$	447.655	−	258.454i	0.547925	−	0.316345i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	395.018	+	684.191i	0.481142	+	0.833363i	0.999766	−	0.0216400i	$-0.00688877\pi$
$821$	395.018	+	684.191i	0.481142	+	0.833363i	−0.518624	+	0.855003i	$0.673555\pi$
$822$		0			0
$823$	−494.196	+	855.972i	−0.600481	+	1.04006i	0.392267	+	0.919851i	$0.371691\pi$
$823$	−494.196	+	855.972i	−0.600481	+	1.04006i	−0.992748	+	0.120212i	$0.961642\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1264.20			−1.52865			−0.764327	−	0.644829i	$-0.776929\pi$
$827$	−1264.20			−1.52865			−0.764327	+	0.644829i	$0.776929\pi$
$828$		0			0
$829$	429.224	−	247.812i	0.517761	−	0.298929i	−0.218257	−	0.975891i	$-0.570037\pi$
$829$	429.224	−	247.812i	0.517761	−	0.298929i	0.736018	+	0.676962i	$0.236704\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	149.186	−	196.104i	0.179095	−	0.235419i
$834$		0			0
$835$	114.564	+	198.431i	0.137203	+	0.237642i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1056.00	−	609.684i	−1.25865	−	0.726680i	−0.285835	−	0.958279i	$-0.592271\pi$
$839$	−1056.00	−	609.684i	−1.25865	−	0.726680i	−0.972811	+	0.231599i	$0.925604\pi$
$840$		0			0
$841$	113.660	+	196.864i	0.135148	+	0.234084i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−140.828	−	81.3073i	−0.166661	−	0.0962216i
$846$		0			0
$847$	−360.414	+	72.4690i	−0.425518	+	0.0855597i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−228.092	+	395.066i	−0.268028	+	0.464238i
$852$		0			0
$853$	−774.869	−	447.371i	−0.908404	−	0.524467i	−0.0284869	−	0.999594i	$-0.509069\pi$
$853$	−774.869	−	447.371i	−0.908404	−	0.524467i	−0.879917	+	0.475127i	$0.842402\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			560.997i			0.654605i	0.944920	+	0.327303i	$0.106140\pi$
$857$			560.997i			0.654605i	−0.944920	+	0.327303i	$0.893860\pi$
$858$		0			0
$859$			1468.27i			1.70928i	0.519225	+	0.854638i	$0.326221\pi$
$859$			1468.27i			1.70928i	−0.519225	+	0.854638i	$0.673779\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	723.196	−	1252.61i	0.838003	−	1.45146i	−0.0535591	−	0.998565i	$-0.517057\pi$
$863$	723.196	−	1252.61i	0.838003	−	1.45146i	0.891562	−	0.452899i	$-0.149610\pi$
$864$		0			0
$865$	−149.123	−	258.289i	−0.172397	−	0.298600i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	82.6182	+	143.099i	0.0950727	+	0.164671i
$870$		0			0
$871$		−	6.30622i		−	0.00724021i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	65.2315	+	324.419i	0.0745503	+	0.370765i
$876$		0			0
$877$	1370.01			1.56215			0.781075	−	0.624438i	$-0.214672\pi$
$877$	1370.01			1.56215			0.781075	+	0.624438i	$0.214672\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1155.87i		−	1.31200i	−0.754763	−	0.655998i	$-0.772248\pi$
$881$		−	1155.87i		−	1.31200i	0.754763	−	0.655998i	$-0.227752\pi$
$882$		0			0
$883$	457.894			0.518567			0.259283	−	0.965801i	$-0.416514\pi$
$883$	457.894			0.518567			0.259283	+	0.965801i	$0.416514\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			64.4365i			0.0726455i	0.999340	+	0.0363227i	$0.0115644\pi$
$887$			64.4365i			0.0726455i	−0.999340	+	0.0363227i	$0.988436\pi$
$888$		0			0
$889$	−492.818	+	99.0917i	−0.554351	+	0.111464i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−209.769			−0.234903
$894$		0			0
$895$	252.652	−	145.869i	0.282293	−	0.162982i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−522.408	+	301.612i	−0.581099	+	0.335497i
$900$		0			0
$901$	389.651	+	224.965i	0.432465	+	0.249684i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−198.177			−0.218980
$906$		0			0
$907$	1027.59			1.13296			0.566480	−	0.824075i	$-0.308305\pi$
$907$	1027.59			1.13296			0.566480	+	0.824075i	$0.308305\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	838.411	−	1452.17i	0.920320	−	1.59404i	0.121399	−	0.992604i	$-0.461262\pi$
$911$	838.411	−	1452.17i	0.920320	−	1.59404i	0.798921	−	0.601437i	$-0.205405\pi$
$912$		0			0
$913$	−835.987	−	482.657i	−0.915648	−	0.528650i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−10.4183	−	51.8136i	−0.0113612	−	0.0565034i
$918$		0			0
$919$	−724.595	+	1255.03i	−0.788460	+	1.36565i	0.138450	+	0.990369i	$0.455788\pi$
$919$	−724.595	+	1255.03i	−0.788460	+	1.36565i	−0.926910	+	0.375283i	$0.877545\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	51.3890	−	29.6695i	0.0556761	−	0.0321446i
$924$		0			0
$925$	−363.971	+	630.416i	−0.393482	+	0.681531i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1429.67	+	825.418i	−1.53893	+	0.888501i	−0.540027	+	0.841647i	$0.681586\pi$
$929$	−1429.67	+	825.418i	−1.53893	+	0.888501i	−0.998902	+	0.0468537i	$0.985081\pi$
$930$		0			0
$931$	1085.23	+	138.291i	1.16566	+	0.148540i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−20.0440	−	34.7173i	−0.0214375	−	0.0371308i
$936$		0			0
$937$		−	1340.89i		−	1.43105i	−0.698588	−	0.715525i	$-0.746188\pi$
$937$		−	1340.89i		−	1.43105i	0.698588	−	0.715525i	$-0.253812\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	552.125	+	318.769i	0.586743	+	0.338756i	0.763808	−	0.645443i	$-0.223327\pi$
$941$	552.125	+	318.769i	0.586743	+	0.338756i	−0.177066	+	0.984199i	$0.556661\pi$
$942$		0			0
$943$	−317.613	+	183.374i	−0.336811	+	0.194458i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	506.734	+	877.689i	0.535094	+	0.926810i	0.999159	+	0.0410089i	$0.0130572\pi$
$947$	506.734	+	877.689i	0.535094	+	0.926810i	−0.464065	+	0.885801i	$0.653609\pi$
$948$		0			0
$949$	13.1679	−	22.8074i	0.0138755	−	0.0240331i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−81.0381			−0.0850347			−0.0425173	−	0.999096i	$-0.513538\pi$
$953$	−81.0381			−0.0850347			−0.0425173	+	0.999096i	$0.513538\pi$
$954$		0			0
$955$	−60.9784	+	35.2059i	−0.0638517	+	0.0368648i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1399.37	+	281.374i	−1.45920	+	0.293404i
$960$		0			0
$961$	−184.027	−	318.744i	−0.191495	−	0.331679i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−296.024	−	170.909i	−0.306760	−	0.177108i
$966$		0			0
$967$	−253.546	−	439.155i	−0.262199	−	0.454142i	0.704627	−	0.709578i	$-0.251114\pi$
$967$	−253.546	−	439.155i	−0.262199	−	0.454142i	−0.966826	+	0.255436i	$0.917781\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1301.46	−	751.397i	−1.34033	−	0.773839i	−0.353473	−	0.935445i	$-0.614999\pi$
$971$	−1301.46	−	751.397i	−1.34033	−	0.773839i	−0.986855	+	0.161606i	$0.948333\pi$
$972$		0			0
$973$	−469.376	+	413.358i	−0.482400	+	0.424829i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	310.361	−	537.562i	0.317668	−	0.550217i	−0.662333	−	0.749209i	$-0.730434\pi$
$977$	310.361	−	537.562i	0.317668	−	0.550217i	0.980001	+	0.198993i	$0.0637670\pi$
$978$		0			0
$979$	−869.445	−	501.974i	−0.888095	−	0.512742i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	1128.62i		−	1.14814i	−0.818808	−	0.574068i	$-0.805365\pi$
$983$		−	1128.62i		−	1.14814i	0.818808	−	0.574068i	$-0.194635\pi$
$984$		0			0
$985$			2.71047i			0.00275175i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	174.628	−	302.465i	0.176570	−	0.305829i
$990$		0			0
$991$	−501.599	−	868.795i	−0.506154	−	0.876685i	−0.999975	−	0.00712091i	$-0.997733\pi$
$991$	−501.599	−	868.795i	−0.506154	−	0.876685i	0.493820	−	0.869564i	$-0.335600\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	51.5550	+	89.2959i	0.0518141	+	0.0897446i
$996$		0			0
$997$			1095.10i			1.09839i	0.835694	+	0.549195i	$0.185066\pi$
$997$			1095.10i			1.09839i	−0.835694	+	0.549195i	$0.814934\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.3.p.a.397.9		32
3.2	odd	2		252.3.p.a.61.12	✓	32
7.3	odd	6		756.3.bd.a.73.9		32
9.4	even	3		756.3.bd.a.145.9		32
9.5	odd	6		252.3.bd.a.229.7	yes	32
21.17	even	6		252.3.bd.a.241.7	yes	32
63.31	odd	6	inner	756.3.p.a.577.8		32
63.59	even	6		252.3.p.a.157.12	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.3.p.a.61.12	✓	32	3.2	odd	2
252.3.p.a.157.12	yes	32	63.59	even	6
252.3.bd.a.229.7	yes	32	9.5	odd	6
252.3.bd.a.241.7	yes	32	21.17	even	6
756.3.p.a.397.9		32	1.1	even	1	trivial
756.3.p.a.577.8		32	63.31	odd	6	inner
756.3.bd.a.73.9		32	7.3	odd	6
756.3.bd.a.145.9		32	9.4	even	3

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+0.963344i q^{5} +(6.86265 - 1.37988i) q^{7} -8.27537 q^{11} +(0.385122 - 0.222350i) q^{13} +(4.35489 - 2.51430i) q^{17} +(19.3354 + 11.1633i) q^{19} +15.0853 q^{23} +24.0720 q^{25} +(12.3863 - 21.4537i) q^{29} +(-21.0881 - 12.1752i) q^{31} +(1.32930 + 6.61109i) q^{35} +(-15.1201 + 26.1888i) q^{37} +(-21.0544 + 12.1558i) q^{41} +(11.5760 - 20.0503i) q^{43} +(-8.13670 + 4.69773i) q^{47} +(45.1918 - 18.9393i) q^{49} +(44.7372 + 77.4871i) q^{53} -7.97203i q^{55} +(23.7510 + 13.7127i) q^{59} +(68.9235 - 39.7930i) q^{61} +(0.214200 + 0.371005i) q^{65} +(7.09042 - 12.2810i) q^{67} +133.436 q^{71} +(51.2871 - 29.6106i) q^{73} +(-56.7910 + 11.4191i) q^{77} +(-9.98363 - 17.2921i) q^{79} +(101.021 + 58.3246i) q^{83} +(2.42213 + 4.19526i) q^{85} +(105.064 + 60.6588i) q^{89} +(2.33614 - 2.05734i) q^{91} +(-10.7541 + 18.6266i) q^{95} +(-53.2313 - 30.7331i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + q^{7} + 12 q^{11} + 15 q^{13} + 27 q^{17} - 30 q^{23} - 160 q^{25} - 24 q^{29} - 24 q^{31} - 141 q^{35} + 11 q^{37} + 90 q^{41} - 16 q^{43} - 108 q^{47} - 61 q^{49} - 54 q^{53} - 45 q^{59} - 165 q^{61}+ \cdots - 57 q^{97}+O(q^{100}) \) 32 * q + q^7 + 12 * q^11 + 15 * q^13 + 27 * q^17 - 30 * q^23 - 160 * q^25 - 24 * q^29 - 24 * q^31 - 141 * q^35 + 11 * q^37 + 90 * q^41 - 16 * q^43 - 108 * q^47 - 61 * q^49 - 54 * q^53 - 45 * q^59 - 165 * q^61 + 33 * q^65 - 35 * q^67 - 138 * q^71 + 123 * q^77 + 37 * q^79 - 378 * q^83 + 30 * q^85 + 261 * q^89 + 45 * q^91 + 48 * q^95 - 57 * q^97

Introduction

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