LMFDB - Embedded newform 684.3.y.h.217.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,3,Mod(145,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("684.145");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	684.y (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$18.6376500822$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 56x^{6} - 154x^{5} + 917x^{4} - 1582x^{3} + 4294x^{2} - 3528x + 4971 $ x^8 - 4x^7 + 56x^6 - 154x^5 + 917x^4 - 1582x^3 + 4294x^2 - 3528*x + 4971
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 76)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			217.4
Root			$0.500000 - 1.77696i$ of defining polynomial
Character	$\chi$	$=$	684.217
Dual form			684.3.y.h.145.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.81674 - 8.34284i) q^{5} +7.59243 q^{7} +O(q^{10})$	$q+(4.81674 - 8.34284i) q^{5} +7.59243 q^{7} +4.61883 q^{11} +(19.8169 - 11.4413i) q^{13} +(-3.02053 + 5.23171i) q^{17} +(-2.21998 + 18.8699i) q^{19} +(11.8651 + 20.5510i) q^{23} +(-33.9020 - 58.7200i) q^{25} +(-10.1836 + 5.87948i) q^{29} +4.22078i q^{31} +(36.5708 - 63.3424i) q^{35} +11.8554i q^{37} +(7.64514 + 4.41392i) q^{41} +(-11.5058 + 19.9286i) q^{43} +(-33.0941 - 57.3207i) q^{47} +8.64494 q^{49} +(-40.7377 + 23.5199i) q^{53} +(22.2477 - 38.5342i) q^{55} +(7.52242 + 4.34307i) q^{59} +(-3.57907 - 6.19914i) q^{61} -220.439i q^{65} +(-77.3224 + 44.6421i) q^{67} +(67.9840 + 39.2506i) q^{71} +(43.1234 - 74.6919i) q^{73} +35.0682 q^{77} +(57.9637 + 33.4653i) q^{79} -5.30752 q^{83} +(29.0982 + 50.3996i) q^{85} +(99.6601 - 57.5388i) q^{89} +(150.458 - 86.8671i) q^{91} +(146.735 + 109.412i) q^{95} +(-49.2806 - 28.4521i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{5} - 12 q^{7}+O(q^{10}) $ 8 * q + q^5 - 12 * q^7	$ 8 q + q^{5} - 12 q^{7} + 10 q^{11} + 9 q^{13} - 23 q^{17} - 33 q^{19} + 31 q^{23} - 73 q^{25} + 105 q^{29} + 68 q^{35} - 18 q^{41} - 41 q^{43} - 107 q^{47} + 312 q^{49} - 39 q^{53} + 70 q^{55} - 348 q^{59} - 45 q^{61} - 432 q^{67} + 243 q^{71} + 16 q^{73} - 544 q^{77} + 75 q^{79} + 82 q^{83} + 109 q^{85} + 213 q^{89} + 222 q^{91} + 385 q^{95} + 144 q^{97}+O(q^{100}) $ 8 * q + q^5 - 12 * q^7 + 10 * q^11 + 9 * q^13 - 23 * q^17 - 33 * q^19 + 31 * q^23 - 73 * q^25 + 105 * q^29 + 68 * q^35 - 18 * q^41 - 41 * q^43 - 107 * q^47 + 312 * q^49 - 39 * q^53 + 70 * q^55 - 348 * q^59 - 45 * q^61 - 432 * q^67 + 243 * q^71 + 16 * q^73 - 544 * q^77 + 75 * q^79 + 82 * q^83 + 109 * q^85 + 213 * q^89 + 222 * q^91 + 385 * q^95 + 144 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$.

$n$	$325$	$343$	$533$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	4.81674	−	8.34284i	0.963348	−	1.66857i	0.249360	−	0.968411i	$-0.419780\pi$
$5$	4.81674	−	8.34284i	0.963348	−	1.66857i	0.713989	−	0.700157i	$-0.246887\pi$
$6$		0			0
$7$	7.59243			1.08463			0.542316	−	0.840174i	$-0.317547\pi$
$7$	7.59243			1.08463			0.542316	+	0.840174i	$0.317547\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.61883			0.419894			0.209947	−	0.977713i	$-0.432671\pi$
$11$	4.61883			0.419894			0.209947	+	0.977713i	$0.432671\pi$
$12$		0			0
$13$	19.8169	−	11.4413i	1.52438	−	0.880099i	0.524793	−	0.851230i	$-0.324143\pi$
$13$	19.8169	−	11.4413i	1.52438	−	0.880099i	0.999583	−	0.0288692i	$-0.00919064\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.02053	+	5.23171i	−0.177678	+	0.307748i	−0.941085	−	0.338171i	$-0.890192\pi$
$17$	−3.02053	+	5.23171i	−0.177678	+	0.307748i	0.763407	+	0.645918i	$0.223525\pi$
$18$		0			0
$19$	−2.21998	+	18.8699i	−0.116841	+	0.993151i
$19$	−2.21998	+	18.8699i	−0.116841	+	0.993151i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	11.8651	+	20.5510i	0.515875	+	0.893521i	0.999830	+	0.0184287i	$0.00586638\pi$
$23$	11.8651	+	20.5510i	0.515875	+	0.893521i	−0.483955	+	0.875093i	$0.660800\pi$
$24$		0			0
$25$	−33.9020	−	58.7200i	−1.35608	−	2.34880i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−10.1836	+	5.87948i	−0.351157	+	0.202741i	−0.665195	−	0.746670i	$-0.731652\pi$
$29$	−10.1836	+	5.87948i	−0.351157	+	0.202741i	0.314038	+	0.949411i	$0.398318\pi$
$30$		0			0
$31$			4.22078i			0.136154i	0.997680	+	0.0680771i	$0.0216864\pi$
$31$			4.22078i			0.136154i	−0.997680	+	0.0680771i	$0.978314\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	36.5708	−	63.3424i	1.04488	−	1.80978i
$36$		0			0
$37$			11.8554i			0.320417i	0.987083	+	0.160209i	$0.0512167\pi$
$37$			11.8554i			0.320417i	−0.987083	+	0.160209i	$0.948783\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	7.64514	+	4.41392i	0.186467	+	0.107657i	0.590327	−	0.807164i	$-0.298999\pi$
$41$	7.64514	+	4.41392i	0.186467	+	0.107657i	−0.403861	+	0.914821i	$0.632332\pi$
$42$		0			0
$43$	−11.5058	+	19.9286i	−0.267576	+	0.463456i	−0.968235	−	0.250041i	$-0.919556\pi$
$43$	−11.5058	+	19.9286i	−0.267576	+	0.463456i	0.700659	+	0.713496i	$0.252889\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−33.0941	−	57.3207i	−0.704130	−	1.21959i	−0.967004	−	0.254759i	$-0.918004\pi$
$47$	−33.0941	−	57.3207i	−0.704130	−	1.21959i	0.262874	−	0.964830i	$-0.415330\pi$
$48$		0			0
$49$	8.64494			0.176427
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−40.7377	+	23.5199i	−0.768635	+	0.443772i	−0.832388	−	0.554194i	$-0.813027\pi$
$53$	−40.7377	+	23.5199i	−0.768635	+	0.443772i	0.0637523	+	0.997966i	$0.479693\pi$
$54$		0			0
$55$	22.2477	−	38.5342i	0.404504	−	0.700622i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	7.52242	+	4.34307i	0.127499	+	0.0736114i	0.562393	−	0.826870i	$-0.309881\pi$
$59$	7.52242	+	4.34307i	0.127499	+	0.0736114i	−0.434894	+	0.900482i	$0.643214\pi$
$60$		0			0
$61$	−3.57907	−	6.19914i	−0.0586733	−	0.101625i	0.835197	−	0.549951i	$-0.185354\pi$
$61$	−3.57907	−	6.19914i	−0.0586733	−	0.101625i	−0.893870	+	0.448326i	$0.852020\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	220.439i		−	3.39137i
$66$		0			0
$67$	−77.3224	+	44.6421i	−1.15407	+	0.666300i	−0.949875	−	0.312631i	$-0.898790\pi$
$67$	−77.3224	+	44.6421i	−1.15407	+	0.666300i	−0.204191	+	0.978931i	$0.565456\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	67.9840	+	39.2506i	0.957521	+	0.552825i	0.895409	−	0.445244i	$-0.146883\pi$
$71$	67.9840	+	39.2506i	0.957521	+	0.552825i	0.0621119	+	0.998069i	$0.480216\pi$
$72$		0			0
$73$	43.1234	−	74.6919i	0.590731	−	1.02318i	−0.403403	−	0.915022i	$-0.632173\pi$
$73$	43.1234	−	74.6919i	0.590731	−	1.02318i	0.994134	−	0.108154i	$-0.0344939\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	35.0682			0.455431
$78$		0			0
$79$	57.9637	+	33.4653i	0.733717	+	0.423612i	0.819781	−	0.572678i	$-0.194095\pi$
$79$	57.9637	+	33.4653i	0.733717	+	0.423612i	−0.0860632	+	0.996290i	$0.527429\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5.30752			−0.0639460			−0.0319730	−	0.999489i	$-0.510179\pi$
$83$	−5.30752			−0.0639460			−0.0319730	+	0.999489i	$0.510179\pi$
$84$		0			0
$85$	29.0982	+	50.3996i	0.342332	+	0.592936i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	99.6601	−	57.5388i	1.11978	−	0.646503i	0.178432	−	0.983952i	$-0.442897\pi$
$89$	99.6601	−	57.5388i	1.11978	−	0.646503i	0.941344	+	0.337449i	$0.109564\pi$
$90$		0			0
$91$	150.458	−	86.8671i	1.65339	−	0.954584i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	146.735	+	109.412i	1.54458	+	1.15171i
$96$		0			0
$97$	−49.2806	−	28.4521i	−0.508047	−	0.293321i	0.223983	−	0.974593i	$-0.428094\pi$
$97$	−49.2806	−	28.4521i	−0.508047	−	0.293321i	−0.732031	+	0.681272i	$0.761427\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	17.2533	+	29.8836i	0.170825	+	0.295877i	0.938708	−	0.344712i	$-0.112023\pi$
$101$	17.2533	+	29.8836i	0.170825	+	0.295877i	−0.767884	+	0.640589i	$0.778690\pi$
$102$		0			0
$103$		−	50.3577i		−	0.488910i	−0.969661	−	0.244455i	$-0.921391\pi$
$103$		−	50.3577i		−	0.488910i	0.969661	−	0.244455i	$-0.0786090\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	184.188i		−	1.72138i	−0.509126	−	0.860692i	$-0.670031\pi$
$107$		−	184.188i		−	1.72138i	0.509126	−	0.860692i	$-0.329969\pi$
$108$		0			0
$109$	−17.7493	−	10.2476i	−0.162838	−	0.0940145i	0.416366	−	0.909197i	$-0.363303\pi$
$109$	−17.7493	−	10.2476i	−0.162838	−	0.0940145i	−0.579204	+	0.815182i	$0.696637\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			138.686i			1.22731i	0.789576	+	0.613653i	$0.210301\pi$
$113$			138.686i			1.22731i	−0.789576	+	0.613653i	$0.789699\pi$
$114$		0			0
$115$	228.605			1.98787
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−22.9331	+	39.7214i	−0.192715	+	0.333793i
$120$		0			0
$121$	−99.6664			−0.823689
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−412.351			−3.29881
$126$		0			0
$127$	−151.220	+	87.3070i	−1.19071	+	0.687457i	−0.958468	−	0.285201i	$-0.907940\pi$
$127$	−151.220	+	87.3070i	−1.19071	+	0.687457i	−0.232242	+	0.972658i	$0.574606\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−11.2446	+	19.4763i	−0.0858370	+	0.148674i	−0.905748	−	0.423818i	$-0.860690\pi$
$131$	−11.2446	+	19.4763i	−0.0858370	+	0.148674i	0.819911	+	0.572492i	$0.194023\pi$
$132$		0			0
$133$	−16.8550	+	143.268i	−0.126730	+	1.07720i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−38.3769	−	66.4708i	−0.280124	−	0.485188i	0.691291	−	0.722576i	$-0.257042\pi$
$137$	−38.3769	−	66.4708i	−0.280124	−	0.485188i	−0.971415	+	0.237388i	$0.923709\pi$
$138$		0			0
$139$	−84.7624	−	146.813i	−0.609801	−	1.05621i	−0.991273	−	0.131826i	$-0.957916\pi$
$139$	−84.7624	−	146.813i	−0.609801	−	1.05621i	0.381472	−	0.924381i	$-0.375417\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	91.5309	−	52.8454i	0.640077	−	0.369548i
$144$		0			0
$145$			113.280i			0.781240i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−97.6855	+	169.196i	−0.655607	+	1.13555i	0.326134	+	0.945324i	$0.394254\pi$
$149$	−97.6855	+	169.196i	−0.655607	+	1.13555i	−0.981741	+	0.190222i	$0.939079\pi$
$150$		0			0
$151$			162.149i			1.07383i	0.843635	+	0.536917i	$0.180411\pi$
$151$			162.149i			1.07383i	−0.843635	+	0.536917i	$0.819589\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	35.2133	+	20.3304i	0.227183	+	0.131164i
$156$		0			0
$157$	−65.6679	+	113.740i	−0.418267	+	0.724460i	−0.995765	−	0.0919322i	$-0.970696\pi$
$157$	−65.6679	+	113.740i	−0.418267	+	0.724460i	0.577498	+	0.816392i	$0.304029\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	90.0851	+	156.032i	0.559535	+	0.969142i
$162$		0			0
$163$	−99.1996			−0.608586			−0.304293	−	0.952578i	$-0.598420\pi$
$163$	−99.1996			−0.608586			−0.304293	+	0.952578i	$0.598420\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	183.454	−	105.917i	1.09853	−	0.634235i	0.162693	−	0.986677i	$-0.447982\pi$
$167$	183.454	−	105.917i	1.09853	−	0.634235i	0.935834	+	0.352442i	$0.114649\pi$
$168$		0			0
$169$	177.306	−	307.103i	1.04915	−	1.81718i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−147.823	−	85.3459i	−0.854471	−	0.493329i	0.00768594	−	0.999970i	$-0.497553\pi$
$173$	−147.823	−	85.3459i	−0.854471	−	0.493329i	−0.862157	+	0.506641i	$0.830887\pi$
$174$		0			0
$175$	−257.398	−	445.827i	−1.47085	−	2.54758i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			192.935i			1.07785i	0.842354	+	0.538925i	$0.181170\pi$
$179$			192.935i			1.07785i	−0.842354	+	0.538925i	$0.818830\pi$
$180$		0			0
$181$	−263.715	+	152.256i	−1.45699	+	0.841194i	−0.998862	−	0.0476914i	$-0.984814\pi$
$181$	−263.715	+	152.256i	−1.45699	+	0.841194i	−0.458129	+	0.888886i	$0.651480\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	98.9081	+	57.1046i	0.534638	+	0.308674i
$186$		0			0
$187$	−13.9513	+	24.1644i	−0.0746060	+	0.129221i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−201.179			−1.05329			−0.526647	−	0.850084i	$-0.676551\pi$
$191$	−201.179			−1.05329			−0.526647	+	0.850084i	$0.676551\pi$
$192$		0			0
$193$	33.0963	+	19.1081i	0.171483	+	0.0990059i	0.583285	−	0.812268i	$-0.301767\pi$
$193$	33.0963	+	19.1081i	0.171483	+	0.0990059i	−0.411802	+	0.911273i	$0.635100\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−100.500			−0.510151			−0.255076	−	0.966921i	$-0.582100\pi$
$197$	−100.500			−0.510151			−0.255076	+	0.966921i	$0.582100\pi$
$198$		0			0
$199$	49.2159	+	85.2444i	0.247316	+	0.428364i	0.962780	−	0.270285i	$-0.0871181\pi$
$199$	49.2159	+	85.2444i	0.247316	+	0.428364i	−0.715464	+	0.698649i	$0.753785\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−77.3179	+	44.6395i	−0.380876	+	0.219899i
$204$		0			0
$205$	73.6493	−	42.5215i	0.359265	−	0.207422i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−10.2537	+	87.1568i	−0.0490609	+	0.417018i
$210$		0			0
$211$	279.837	+	161.564i	1.32624	+	0.765707i	0.984716	−	0.174165i	$-0.0557226\pi$
$211$	279.837	+	161.564i	1.32624	+	0.765707i	0.341527	+	0.939872i	$0.389056\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	110.841	+	191.982i	0.515538	+	0.892938i
$216$		0			0
$217$			32.0460i			0.147677i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			138.235i			0.625497i
$222$		0			0
$223$	256.629	+	148.165i	1.15080	+	0.664415i	0.949082	−	0.315028i	$-0.102014\pi$
$223$	256.629	+	148.165i	1.15080	+	0.664415i	0.201718	+	0.979444i	$0.435347\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			214.461i			0.944762i	0.881394	+	0.472381i	$0.156605\pi$
$227$			214.461i			0.944762i	−0.881394	+	0.472381i	$0.843395\pi$
$228$		0			0
$229$	118.681			0.518256			0.259128	−	0.965843i	$-0.416565\pi$
$229$	118.681			0.518256			0.259128	+	0.965843i	$0.416565\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−91.9447	+	159.253i	−0.394612	+	0.683489i	−0.993052	−	0.117679i	$-0.962454\pi$
$233$	−91.9447	+	159.253i	−0.394612	+	0.683489i	0.598439	+	0.801168i	$0.295788\pi$
$234$		0			0
$235$	−637.623			−2.71329
$236$		0			0
$237$		0			0
$238$		0			0
$239$	21.2428			0.0888822			0.0444411	−	0.999012i	$-0.485849\pi$
$239$	21.2428			0.0888822			0.0444411	+	0.999012i	$0.485849\pi$
$240$		0			0
$241$	68.4562	−	39.5232i	0.284051	−	0.163997i	−0.351205	−	0.936299i	$-0.614228\pi$
$241$	68.4562	−	39.5232i	0.284051	−	0.163997i	0.635256	+	0.772302i	$0.280895\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	41.6404	−	72.1234i	0.169961	−	0.294381i
$246$		0			0
$247$	171.902	+	399.341i	0.695961	+	1.61677i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−180.962	−	313.436i	−0.720965	−	1.24875i	−0.960614	−	0.277888i	$-0.910366\pi$
$251$	−180.962	−	313.436i	−0.720965	−	1.24875i	0.239649	−	0.970860i	$-0.422968\pi$
$252$		0			0
$253$	54.8030	+	94.9216i	0.216613	+	0.375184i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−111.378	+	64.3044i	−0.433379	+	0.250212i	−0.700785	−	0.713372i	$-0.747167\pi$
$257$	−111.378	+	64.3044i	−0.433379	+	0.250212i	0.267406	+	0.963584i	$0.413834\pi$
$258$		0			0
$259$			90.0116i			0.347535i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	115.623	−	200.266i	0.439633	−	0.761466i	−0.558028	−	0.829822i	$-0.688442\pi$
$263$	115.623	−	200.266i	0.439633	−	0.761466i	0.997661	+	0.0683558i	$0.0217753\pi$
$264$		0			0
$265$			453.157i			1.71003i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	211.625	+	122.182i	0.786710	+	0.454207i	0.838803	−	0.544435i	$-0.183256\pi$
$269$	211.625	+	122.182i	0.786710	+	0.454207i	−0.0520931	+	0.998642i	$0.516589\pi$
$270$		0			0
$271$	221.354	−	383.397i	0.816805	−	1.41475i	−0.0912198	−	0.995831i	$-0.529077\pi$
$271$	221.354	−	383.397i	0.816805	−	1.41475i	0.908025	−	0.418917i	$-0.137590\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−156.588	−	271.218i	−0.569410	−	0.986247i
$276$		0			0
$277$	89.7022			0.323835			0.161917	−	0.986804i	$-0.448232\pi$
$277$	89.7022			0.323835			0.161917	+	0.986804i	$0.448232\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−368.458	+	212.729i	−1.31124	+	0.757044i	−0.982301	−	0.187308i	$-0.940024\pi$
$281$	−368.458	+	212.729i	−1.31124	+	0.757044i	−0.328937	+	0.944352i	$0.606690\pi$
$282$		0			0
$283$	74.4153	−	128.891i	0.262952	−	0.455446i	−0.704073	−	0.710127i	$-0.748637\pi$
$283$	74.4153	−	128.891i	0.262952	−	0.455446i	0.967025	+	0.254682i	$0.0819707\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	58.0452	+	33.5124i	0.202248	+	0.116768i
$288$		0			0
$289$	126.253	+	218.676i	0.436861	+	0.756665i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	42.5442i		−	0.145202i	−0.997361	−	0.0726011i	$-0.976870\pi$
$293$		−	42.5442i		−	0.145202i	0.997361	−	0.0726011i	$-0.0231300\pi$
$294$		0			0
$295$	72.4671	−	41.8389i	0.245651	−	0.141827i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	470.260	+	271.505i	1.57277	+	0.908042i
$300$		0			0
$301$	−87.3568	+	151.306i	−0.290222	+	0.502679i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−68.9579			−0.226091
$306$		0			0
$307$	249.511	+	144.055i	0.812740	+	0.469235i	0.847906	−	0.530146i	$-0.177863\pi$
$307$	249.511	+	144.055i	0.812740	+	0.469235i	−0.0351668	+	0.999381i	$0.511196\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−269.870			−0.867750			−0.433875	−	0.900973i	$-0.642854\pi$
$311$	−269.870			−0.867750			−0.433875	+	0.900973i	$0.642854\pi$
$312$		0			0
$313$	−88.8271	−	153.853i	−0.283793	−	0.491543i	0.688523	−	0.725214i	$-0.258259\pi$
$313$	−88.8271	−	153.853i	−0.283793	−	0.491543i	−0.972316	+	0.233671i	$0.924926\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	505.362	−	291.771i	1.59420	−	0.920413i	0.601627	−	0.798777i	$-0.294520\pi$
$317$	505.362	−	291.771i	1.59420	−	0.920413i	0.992575	−	0.121635i	$-0.0388138\pi$
$318$		0			0
$319$	−47.0362	+	27.1563i	−0.147449	+	0.0851296i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−92.0161	−	68.6112i	−0.284880	−	0.212419i
$324$		0			0
$325$	−1343.66	−	775.765i	−4.13435	−	2.38697i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−251.265	−	435.203i	−0.763722	−	1.32281i
$330$		0			0
$331$		−	32.6891i		−	0.0987586i	−0.998780	−	0.0493793i	$-0.984276\pi$
$331$		−	32.6891i		−	0.0987586i	0.998780	−	0.0493793i	$-0.0157243\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			860.118i			2.56752i
$336$		0			0
$337$	−255.755	−	147.660i	−0.758918	−	0.438161i	0.0699892	−	0.997548i	$-0.477704\pi$
$337$	−255.755	−	147.660i	−0.758918	−	0.438161i	−0.828907	+	0.559386i	$0.811037\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			19.4951i			0.0571704i
$342$		0			0
$343$	−306.393			−0.893274
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−85.7263	+	148.482i	−0.247050	+	0.427903i	−0.962706	−	0.270550i	$-0.912794\pi$
$347$	−85.7263	+	148.482i	−0.247050	+	0.427903i	0.715656	+	0.698453i	$0.246128\pi$
$348$		0			0
$349$	127.644			0.365743			0.182872	−	0.983137i	$-0.441461\pi$
$349$	127.644			0.365743			0.182872	+	0.983137i	$0.441461\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	184.494			0.522647			0.261323	−	0.965251i	$-0.415841\pi$
$353$	184.494			0.522647			0.261323	+	0.965251i	$0.415841\pi$
$354$		0			0
$355$	654.923	−	378.120i	1.84485	−	1.06513i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	131.720	−	228.145i	0.366907	−	0.635502i	−0.622173	−	0.782880i	$-0.713750\pi$
$359$	131.720	−	228.145i	0.366907	−	0.635502i	0.989080	+	0.147378i	$0.0470833\pi$
$360$		0			0
$361$	−351.143	−	83.7815i	−0.972696	−	0.232082i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−415.428	−	719.543i	−1.13816	−	1.97135i
$366$		0			0
$367$	60.5658	+	104.903i	0.165029	+	0.285839i	0.936666	−	0.350225i	$-0.113895\pi$
$367$	60.5658	+	104.903i	0.165029	+	0.285839i	−0.771636	+	0.636064i	$0.780561\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−309.298	+	178.573i	−0.833687	+	0.481329i
$372$		0			0
$373$			335.187i			0.898625i	0.893375	+	0.449312i	$0.148331\pi$
$373$			335.187i			0.898625i	−0.893375	+	0.449312i	$0.851669\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−134.538	+	233.026i	−0.356864	+	0.618106i
$378$		0			0
$379$			315.980i			0.833721i	0.908970	+	0.416861i	$0.136870\pi$
$379$			315.980i			0.833721i	−0.908970	+	0.416861i	$0.863130\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	408.291	+	235.727i	1.06603	+	0.615475i	0.927095	−	0.374825i	$-0.122297\pi$
$383$	408.291	+	235.727i	1.06603	+	0.615475i	0.138939	+	0.990301i	$0.455631\pi$
$384$		0			0
$385$	168.914	−	292.568i	0.438738	−	0.759917i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−235.508	−	407.912i	−0.605420	−	1.04862i	−0.991985	−	0.126355i	$-0.959672\pi$
$389$	−235.508	−	407.912i	−0.605420	−	1.04862i	0.386566	−	0.922262i	$-0.373661\pi$
$390$		0			0
$391$	−143.356			−0.366639
$392$		0			0
$393$		0			0
$394$		0			0
$395$	558.392	−	322.388i	1.41365	−	0.816172i
$396$		0			0
$397$	−110.236	+	190.934i	−0.277672	+	0.480942i	−0.970806	−	0.239867i	$-0.922896\pi$
$397$	−110.236	+	190.934i	−0.277672	+	0.480942i	0.693134	+	0.720809i	$0.256230\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−79.1787	−	45.7138i	−0.197453	−	0.114000i	0.398014	−	0.917379i	$-0.369700\pi$
$401$	−79.1787	−	45.7138i	−0.197453	−	0.114000i	−0.595467	+	0.803380i	$0.703033\pi$
$402$		0			0
$403$	48.2912	+	83.6428i	0.119829	+	0.207550i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			54.7583i			0.134541i
$408$		0			0
$409$	291.494	−	168.294i	0.712699	−	0.411477i	−0.0993606	−	0.995051i	$-0.531680\pi$
$409$	291.494	−	168.294i	0.712699	−	0.411477i	0.812060	+	0.583575i	$0.198346\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	57.1134	+	32.9745i	0.138289	+	0.0798413i
$414$		0			0
$415$	−25.5649	+	44.2798i	−0.0616023	+	0.106698i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	213.808			0.510281			0.255141	−	0.966904i	$-0.417878\pi$
$419$	213.808			0.510281			0.255141	+	0.966904i	$0.417878\pi$
$420$		0			0
$421$	524.575	+	302.863i	1.24602	+	0.719391i	0.970313	−	0.241851i	$-0.0777545\pi$
$421$	524.575	+	302.863i	1.24602	+	0.719391i	0.275708	+	0.961242i	$0.411088\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	409.608			0.963783
$426$		0			0
$427$	−27.1738	−	47.0665i	−0.0636390	−	0.110226i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−138.813	+	80.1437i	−0.322072	+	0.185948i	−0.652316	−	0.757947i	$-0.726202\pi$
$431$	−138.813	+	80.1437i	−0.322072	+	0.185948i	0.330244	+	0.943896i	$0.392869\pi$
$432$		0			0
$433$	377.175	−	217.762i	0.871073	−	0.502914i	0.00336832	−	0.999994i	$-0.498928\pi$
$433$	377.175	−	217.762i	0.871073	−	0.502914i	0.867705	+	0.497080i	$0.165594\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−414.135	+	178.270i	−0.947677	+	0.407941i
$438$		0			0
$439$	320.878	+	185.259i	0.730930	+	0.422003i	0.818762	−	0.574133i	$-0.194661\pi$
$439$	320.878	+	185.259i	0.730930	+	0.422003i	−0.0878324	+	0.996135i	$0.527994\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	144.384	+	250.081i	0.325924	+	0.564517i	0.981699	−	0.190439i	$-0.0609912\pi$
$443$	144.384	+	250.081i	0.325924	+	0.564517i	−0.655775	+	0.754957i	$0.727658\pi$
$444$		0			0
$445$		−	1108.60i		−	2.49123i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	76.8398i		−	0.171135i	−0.996332	−	0.0855677i	$-0.972730\pi$
$449$		−	76.8398i		−	0.171135i	0.996332	−	0.0855677i	$-0.0272704\pi$
$450$		0			0
$451$	35.3116	+	20.3872i	0.0782963	+	0.0452044i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	1673.67i		−	3.67839i
$456$		0			0
$457$	97.5752			0.213513			0.106756	−	0.994285i	$-0.465954\pi$
$457$	97.5752			0.213513			0.106756	+	0.994285i	$0.465954\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	156.003	−	270.204i	0.338401	−	0.586127i	−0.645732	−	0.763565i	$-0.723447\pi$
$461$	156.003	−	270.204i	0.338401	−	0.586127i	0.984132	+	0.177438i	$0.0567808\pi$
$462$		0			0
$463$	114.000			0.246220			0.123110	−	0.992393i	$-0.460713\pi$
$463$	114.000			0.246220			0.123110	+	0.992393i	$0.460713\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	651.304			1.39466			0.697328	−	0.716752i	$-0.254372\pi$
$467$	651.304			1.39466			0.697328	+	0.716752i	$0.254372\pi$
$468$		0			0
$469$	−587.065	+	338.942i	−1.25174	+	0.722691i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−53.1433	+	92.0469i	−0.112354	+	0.194602i
$474$		0			0
$475$	1183.30	−	509.369i	2.49116	−	1.07236i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	216.179	+	374.433i	0.451313	+	0.781697i	0.998468	−	0.0553341i	$-0.0176224\pi$
$479$	216.179	+	374.433i	0.451313	+	0.781697i	−0.547155	+	0.837031i	$0.684289\pi$
$480$		0			0
$481$	135.642	+	234.938i	0.281999	+	0.488437i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−474.743	+	274.093i	−0.978853	+	0.565141i
$486$		0			0
$487$		−	100.949i		−	0.207287i	−0.994615	−	0.103643i	$-0.966950\pi$
$487$		−	100.949i		−	0.207287i	0.994615	−	0.103643i	$-0.0330501\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	112.182	−	194.304i	0.228476	−	0.395732i	−0.728881	−	0.684641i	$-0.759959\pi$
$491$	112.182	−	194.304i	0.228476	−	0.395732i	0.957357	+	0.288909i	$0.0932924\pi$
$492$		0			0
$493$		−	71.0365i		−	0.144090i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	516.164	+	298.007i	1.03856	+	0.599612i
$498$		0			0
$499$	−332.278	+	575.523i	−0.665888	+	1.15335i	0.313155	+	0.949702i	$0.398614\pi$
$499$	−332.278	+	575.523i	−0.665888	+	1.15335i	−0.979044	+	0.203651i	$0.934719\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	180.202	+	312.119i	0.358255	+	0.620515i	0.987669	−	0.156554i	$-0.0500387\pi$
$503$	180.202	+	312.119i	0.358255	+	0.620515i	−0.629415	+	0.777070i	$0.716705\pi$
$504$		0			0
$505$	332.419			0.658255
$506$		0			0
$507$		0			0
$508$		0			0
$509$	670.407	−	387.060i	1.31711	−	0.760432i	0.333845	−	0.942628i	$-0.391654\pi$
$509$	670.407	−	387.060i	1.31711	−	0.760432i	0.983262	+	0.182196i	$0.0583206\pi$
$510$		0			0
$511$	327.411	−	567.092i	0.640726	−	1.10977i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−420.126	−	242.560i	−0.815780	−	0.470991i
$516$		0			0
$517$	−152.856	−	264.755i	−0.295660	−	0.512098i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			196.045i			0.376285i	0.982142	+	0.188143i	$0.0602468\pi$
$521$			196.045i			0.376285i	−0.982142	+	0.188143i	$0.939753\pi$
$522$		0			0
$523$	−285.968	+	165.104i	−0.546784	+	0.315686i	−0.747824	−	0.663897i	$-0.768901\pi$
$523$	−285.968	+	165.104i	−0.546784	+	0.315686i	0.201040	+	0.979583i	$0.435568\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−22.0819	−	12.7490i	−0.0419011	−	0.0241916i
$528$		0			0
$529$	−17.0622	+	29.5526i	−0.0322537	+	0.0558651i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	202.004			0.378994
$534$		0			0
$535$	−1536.65	−	887.186i	−2.87225	−	1.65829i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	39.9296			0.0740808
$540$		0			0
$541$	−328.577	−	569.111i	−0.607350	−	1.05196i	−0.991675	−	0.128764i	$-0.958899\pi$
$541$	−328.577	−	569.111i	−0.607350	−	1.05196i	0.384325	−	0.923198i	$-0.374434\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−170.988	+	98.7199i	−0.313739	+	0.181137i
$546$		0			0
$547$	51.3126	−	29.6253i	0.0938072	−	0.0541596i	−0.452363	−	0.891834i	$-0.649419\pi$
$547$	51.3126	−	29.6253i	0.0938072	−	0.0541596i	0.546170	+	0.837675i	$0.316085\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−88.3377	−	205.215i	−0.160322	−	0.372440i
$552$		0			0
$553$	440.085	+	254.083i	0.795814	+	0.459463i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−120.377	−	208.499i	−0.216117	−	0.374325i	0.737501	−	0.675346i	$-0.236006\pi$
$557$	−120.377	−	208.499i	−0.216117	−	0.374325i	−0.953617	+	0.301021i	$0.902673\pi$
$558$		0			0
$559$			526.564i			0.941974i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	919.141i		−	1.63258i	−0.577645	−	0.816288i	$-0.696028\pi$
$563$		−	919.141i		−	1.63258i	0.577645	−	0.816288i	$-0.303972\pi$
$564$		0			0
$565$	1157.03	+	668.012i	2.04784	+	1.18232i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	726.404i		−	1.27663i	−0.769774	−	0.638316i	$-0.779631\pi$
$569$		−	726.404i		−	1.27663i	0.769774	−	0.638316i	$-0.220369\pi$
$570$		0			0
$571$	−924.876			−1.61975			−0.809874	−	0.586604i	$-0.800464\pi$
$571$	−924.876			−1.61975			−0.809874	+	0.586604i	$0.800464\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	804.503	−	1393.44i	1.39913	−	2.42337i
$576$		0			0
$577$	161.154			0.279296			0.139648	−	0.990201i	$-0.455403\pi$
$577$	161.154			0.279296			0.139648	+	0.990201i	$0.455403\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−40.2969			−0.0693579
$582$		0			0
$583$	−188.161	+	108.635i	−0.322745	+	0.186337i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−294.997	+	510.950i	−0.502550	+	0.870442i	0.497445	+	0.867495i	$0.334271\pi$
$587$	−294.997	+	510.950i	−0.502550	+	0.870442i	−0.999996	+	0.00294722i	$0.999062\pi$
$588$		0			0
$589$	−79.6456	−	9.37006i	−0.135222	−	0.0159084i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	9.99383	+	17.3098i	0.0168530	+	0.0291903i	0.874329	−	0.485334i	$-0.161302\pi$
$593$	9.99383	+	17.3098i	0.0168530	+	0.0291903i	−0.857476	+	0.514524i	$0.827969\pi$
$594$		0			0
$595$	220.926	+	382.655i	0.371304	+	0.643118i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−931.550	+	537.831i	−1.55518	+	0.897881i	−0.557468	+	0.830198i	$0.688227\pi$
$599$	−931.550	+	537.831i	−1.55518	+	0.897881i	−0.997707	+	0.0676825i	$0.978440\pi$
$600$		0			0
$601$		−	14.5736i		−	0.0242489i	−0.999926	−	0.0121244i	$-0.996141\pi$
$601$		−	14.5736i		−	0.0242489i	0.999926	−	0.0121244i	$-0.00385942\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−480.067	+	831.501i	−0.793499	+	1.37438i
$606$		0			0
$607$		−	437.145i		−	0.720173i	−0.932919	−	0.360086i	$-0.882747\pi$
$607$		−	437.145i		−	0.720173i	0.932919	−	0.360086i	$-0.117253\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1311.65	−	757.279i	−2.14672	−	1.23941i
$612$		0			0
$613$	534.944	−	926.551i	0.872666	−	1.51150i	0.0134375	−	0.999910i	$-0.495723\pi$
$613$	534.944	−	926.551i	0.872666	−	1.51150i	0.859228	−	0.511592i	$-0.170944\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	99.8317	+	172.914i	0.161802	+	0.280249i	0.935515	−	0.353287i	$-0.114936\pi$
$617$	99.8317	+	172.914i	0.161802	+	0.280249i	−0.773713	+	0.633536i	$0.781603\pi$
$618$		0			0
$619$	−432.305			−0.698393			−0.349196	−	0.937050i	$-0.613545\pi$
$619$	−432.305			−0.698393			−0.349196	+	0.937050i	$0.613545\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	756.662	−	436.859i	1.21455	−	0.701218i
$624$		0			0
$625$	−1138.64	+	1972.18i	−1.82182	+	3.15549i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−62.0242	−	35.8097i	−0.0986077	−	0.0569312i
$630$		0			0
$631$	102.220	+	177.051i	0.161997	+	0.280587i	0.935585	−	0.353102i	$-0.114873\pi$
$631$	102.220	+	177.051i	0.161997	+	0.280587i	−0.773588	+	0.633689i	$0.781540\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			1682.14i			2.64904i
$636$		0			0
$637$	171.316	−	98.9093i	0.268942	−	0.155274i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	34.0248	+	19.6442i	0.0530808	+	0.0306462i	0.526306	−	0.850296i	$-0.323577\pi$
$641$	34.0248	+	19.6442i	0.0530808	+	0.0306462i	−0.473225	+	0.880942i	$0.656910\pi$
$642$		0			0
$643$	−383.822	+	664.799i	−0.596923	+	1.03390i	0.396349	+	0.918100i	$0.370277\pi$
$643$	−383.822	+	664.799i	−0.596923	+	1.03390i	−0.993272	+	0.115801i	$0.963056\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−963.932			−1.48985			−0.744924	−	0.667149i	$-0.767514\pi$
$647$	−963.932			−1.48985			−0.744924	+	0.667149i	$0.767514\pi$
$648$		0			0
$649$	34.7448	+	20.0599i	0.0535359	+	0.0309090i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	824.035			1.26192			0.630961	−	0.775815i	$-0.282661\pi$
$653$	824.035			1.26192			0.630961	+	0.775815i	$0.282661\pi$
$654$		0			0
$655$	108.325	+	187.625i	0.165382	+	0.286450i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	144.310	−	83.3177i	0.218984	−	0.126430i	−0.386496	−	0.922291i	$-0.626315\pi$
$659$	144.310	−	83.3177i	0.218984	−	0.126430i	0.605480	+	0.795861i	$0.292981\pi$
$660$		0			0
$661$	273.938	−	158.158i	0.414430	−	0.239271i	−0.278261	−	0.960505i	$-0.589758\pi$
$661$	273.938	−	158.158i	0.414430	−	0.239271i	0.692692	+	0.721234i	$0.256425\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1114.08	+	830.704i	1.67530	+	1.24918i
$666$		0			0
$667$	−241.658	−	139.522i	−0.362306	−	0.209178i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−16.5311	−	28.6328i	−0.0246366	−	0.0426718i
$672$		0			0
$673$			582.475i			0.865491i	0.901516	+	0.432745i	$0.142455\pi$
$673$			582.475i			0.865491i	−0.901516	+	0.432745i	$0.857545\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	956.276i		−	1.41252i	−0.707953	−	0.706260i	$-0.750381\pi$
$677$		−	956.276i		−	1.41252i	0.707953	−	0.706260i	$-0.249619\pi$
$678$		0			0
$679$	−374.159	−	216.021i	−0.551044	−	0.318146i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	240.373i		−	0.351937i	−0.984396	−	0.175969i	$-0.943694\pi$
$683$		−	240.373i		−	0.351937i	0.984396	−	0.175969i	$-0.0563057\pi$
$684$		0			0
$685$	−739.407			−1.07943
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−538.196	+	932.183i	−0.781126	+	1.35295i
$690$		0			0
$691$	868.890			1.25744			0.628719	−	0.777632i	$-0.283579\pi$
$691$	868.890			1.25744			0.628719	+	0.777632i	$0.283579\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1633.11			−2.34980
$696$		0			0
$697$	−46.1847	+	26.6648i	−0.0662622	+	0.0382565i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−467.106	+	809.052i	−0.666343	+	1.15414i	0.312576	+	0.949893i	$0.398808\pi$
$701$	−467.106	+	809.052i	−0.666343	+	1.15414i	−0.978919	+	0.204247i	$0.934525\pi$
$702$		0			0
$703$	−223.711	−	26.3189i	−0.318223	−	0.0374379i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	130.994	+	226.889i	0.185282	+	0.320918i
$708$		0			0
$709$	−514.968	−	891.950i	−0.726330	−	1.25804i	−0.958424	−	0.285347i	$-0.907891\pi$
$709$	−514.968	−	891.950i	−0.726330	−	1.25804i	0.232095	−	0.972693i	$-0.425442\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−86.7413	+	50.0801i	−0.121657	+	0.0702386i
$714$		0			0
$715$		−	1018.17i		−	1.42402i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−352.663	+	610.831i	−0.490491	+	0.849556i	−0.999940	−	0.0109450i	$-0.996516\pi$
$719$	−352.663	+	610.831i	−0.490491	+	0.849556i	0.509449	+	0.860501i	$0.329849\pi$
$720$		0			0
$721$		−	382.337i		−	0.530288i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	690.486	+	398.652i	0.952394	+	0.549865i
$726$		0			0
$727$	426.231	−	738.253i	0.586287	−	1.01548i	−0.408427	−	0.912791i	$-0.633922\pi$
$727$	426.231	−	738.253i	0.586287	−	1.01548i	0.994714	−	0.102688i	$-0.0327442\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−69.5070	−	120.390i	−0.0950849	−	0.164692i
$732$		0			0
$733$	1327.83			1.81150			0.905751	−	0.423809i	$-0.139307\pi$
$733$	1327.83			1.81150			0.905751	+	0.423809i	$0.139307\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−357.139	+	206.195i	−0.484585	+	0.279776i
$738$		0			0
$739$	−197.639	+	342.320i	−0.267441	+	0.463221i	−0.968200	−	0.250177i	$-0.919511\pi$
$739$	−197.639	+	342.320i	−0.267441	+	0.463221i	0.700760	+	0.713398i	$0.252845\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	916.709	+	529.262i	1.23379	+	0.712332i	0.967819	−	0.251648i	$-0.0809726\pi$
$743$	916.709	+	529.262i	1.23379	+	0.712332i	0.265976	+	0.963980i	$0.414306\pi$
$744$		0			0
$745$	941.052	+	1629.95i	1.26316	+	2.18785i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	1398.43i		−	1.86707i
$750$		0			0
$751$	207.032	−	119.530i	0.275674	−	0.159161i	−0.355789	−	0.934566i	$-0.615788\pi$
$751$	207.032	−	119.530i	0.275674	−	0.159161i	0.631464	+	0.775406i	$0.282455\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1352.78	+	781.030i	1.79177	+	1.03448i
$756$		0			0
$757$	92.7004	−	160.562i	0.122458	−	0.212103i	−0.798279	−	0.602288i	$-0.794256\pi$
$757$	92.7004	−	160.562i	0.122458	−	0.212103i	0.920736	+	0.390185i	$0.127589\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	712.211			0.935888			0.467944	−	0.883758i	$-0.344995\pi$
$761$	712.211			0.935888			0.467944	+	0.883758i	$0.344995\pi$
$762$		0			0
$763$	−134.760	−	77.8040i	−0.176619	−	0.101971i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	198.761			0.259141
$768$		0			0
$769$	−395.236	−	684.568i	−0.513961	−	0.890206i	−0.999869	−	0.0161960i	$-0.994844\pi$
$769$	−395.236	−	684.568i	−0.513961	−	0.890206i	0.485908	−	0.874010i	$-0.338489\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−334.502	+	193.125i	−0.432732	+	0.249838i	−0.700510	−	0.713643i	$-0.747044\pi$
$773$	−334.502	+	193.125i	−0.432732	+	0.249838i	0.267778	+	0.963481i	$0.413711\pi$
$774$		0			0
$775$	247.844	−	143.093i	0.319799	−	0.184636i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−100.262	+	134.464i	−0.128706	+	0.172611i
$780$		0			0
$781$	314.007	+	181.292i	0.402057	+	0.232128i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	632.611	+	1095.71i	0.805874	+	1.39581i
$786$		0			0
$787$			606.206i			0.770274i	0.922859	+	0.385137i	$0.125846\pi$
$787$			606.206i			0.770274i	−0.922859	+	0.385137i	$0.874154\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			1052.96i			1.33118i
$792$		0			0
$793$	−141.852	−	81.8984i	−0.178880	−	0.103277i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	332.769i		−	0.417526i	−0.977966	−	0.208763i	$-0.933056\pi$
$797$		−	332.769i		−	0.417526i	0.977966	−	0.208763i	$-0.0669438\pi$
$798$		0			0
$799$	399.847			0.500434
$800$		0			0
$801$		0			0
$802$		0			0
$803$	199.180	−	344.989i	0.248044	−	0.429626i
$804$		0			0
$805$	1735.67			2.15611
$806$		0			0
$807$		0			0
$808$		0			0
$809$	603.251			0.745675			0.372838	−	0.927897i	$-0.378385\pi$
$809$	603.251			0.745675			0.372838	+	0.927897i	$0.378385\pi$
$810$		0			0
$811$	−905.988	+	523.073i	−1.11713	+	0.644972i	−0.940665	−	0.339335i	$-0.889798\pi$
$811$	−905.988	+	523.073i	−1.11713	+	0.644972i	−0.176460	+	0.984308i	$0.556465\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−477.819	+	827.606i	−0.586281	+	1.01547i
$816$		0			0
$817$	−350.507	−	261.354i	−0.429017	−	0.319894i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	166.613	+	288.583i	0.202940	+	0.351502i	0.949474	−	0.313845i	$-0.101617\pi$
$821$	166.613	+	288.583i	0.202940	+	0.351502i	−0.746535	+	0.665346i	$0.768284\pi$
$822$		0			0
$823$	208.418	+	360.990i	0.253241	+	0.438627i	0.964416	−	0.264388i	$-0.0851700\pi$
$823$	208.418	+	360.990i	0.253241	+	0.438627i	−0.711175	+	0.703015i	$0.751837\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−734.915	+	424.303i	−0.888651	+	0.513063i	−0.873501	−	0.486822i	$-0.838156\pi$
$827$	−734.915	+	424.303i	−0.888651	+	0.513063i	−0.0151501	+	0.999885i	$0.504823\pi$
$828$		0			0
$829$			958.797i			1.15657i	0.815835	+	0.578285i	$0.196278\pi$
$829$			958.797i			1.15657i	−0.815835	+	0.578285i	$0.803722\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−26.1123	+	45.2278i	−0.0313473	+	0.0542951i
$834$		0			0
$835$		−	2040.70i		−	2.44396i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−931.117	−	537.581i	−1.10979	−	0.640740i	−0.171018	−	0.985268i	$-0.554706\pi$
$839$	−931.117	−	537.581i	−1.10979	−	0.640740i	−0.938776	+	0.344528i	$0.888039\pi$
$840$		0			0
$841$	−351.363	+	608.579i	−0.417792	+	0.723638i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1708.08	−	2958.47i	−2.02139	−	3.50115i
$846$		0			0
$847$	−756.710			−0.893400
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−243.641	+	140.666i	−0.286300	+	0.165295i
$852$		0			0
$853$	−83.4052	+	144.462i	−0.0977787	+	0.169358i	−0.910765	−	0.412925i	$-0.864507\pi$
$853$	−83.4052	+	144.462i	−0.0977787	+	0.169358i	0.812986	+	0.582283i	$0.197840\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−581.227	−	335.571i	−0.678211	−	0.391565i	0.120970	−	0.992656i	$-0.461400\pi$
$857$	−581.227	−	335.571i	−0.678211	−	0.391565i	−0.799181	+	0.601091i	$0.794733\pi$
$858$		0			0
$859$	−396.070	−	686.014i	−0.461083	−	0.798619i	0.537932	−	0.842988i	$-0.319206\pi$
$859$	−396.070	−	686.014i	−0.461083	−	0.798619i	−0.999015	+	0.0443690i	$0.985872\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	570.060i		−	0.660556i	−0.943884	−	0.330278i	$-0.892857\pi$
$863$		−	570.060i		−	0.660556i	0.943884	−	0.330278i	$-0.107143\pi$
$864$		0			0
$865$	−1424.05	+	822.178i	−1.64631	+	0.950495i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	267.725	+	154.571i	0.308084	+	0.177872i
$870$		0			0
$871$	−1021.53	+	1769.34i	−1.17282	+	2.03138i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−3130.75			−3.57800
$876$		0			0
$877$	255.489	+	147.506i	0.291321	+	0.168194i	0.638537	−	0.769591i	$-0.279540\pi$
$877$	255.489	+	147.506i	0.291321	+	0.168194i	−0.347216	+	0.937785i	$0.612873\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	976.298			1.10817			0.554085	−	0.832460i	$-0.313068\pi$
$881$	976.298			1.10817			0.554085	+	0.832460i	$0.313068\pi$
$882$		0			0
$883$	147.883	+	256.141i	0.167478	+	0.290081i	0.937533	−	0.347897i	$-0.113104\pi$
$883$	147.883	+	256.141i	0.167478	+	0.290081i	−0.770054	+	0.637978i	$0.779771\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−793.272	+	457.996i	−0.894331	+	0.516342i	−0.875357	−	0.483477i	$-0.839374\pi$
$887$	−793.272	+	457.996i	−0.894331	+	0.516342i	−0.0189746	+	0.999820i	$0.506040\pi$
$888$		0			0
$889$	−1148.13	+	662.872i	−1.29148	+	0.745638i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1155.10	−	497.231i	1.29351	−	0.556809i
$894$		0			0
$895$	1609.63	+	929.319i	1.79847	+	1.03835i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−24.8160	−	42.9826i	−0.0276040	−	0.0478116i
$900$		0			0
$901$		−	284.170i		−	0.315394i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			2933.51i			3.24145i
$906$		0			0
$907$	−407.014	−	234.989i	−0.448747	−	0.259084i	0.258554	−	0.965997i	$-0.416754\pi$
$907$	−407.014	−	234.989i	−0.448747	−	0.259084i	−0.707301	+	0.706913i	$0.750087\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			440.064i			0.483056i	0.970394	+	0.241528i	$0.0776486\pi$
$911$			440.064i			0.483056i	−0.970394	+	0.241528i	$0.922351\pi$
$912$		0			0
$913$	−24.5145			−0.0268505
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−85.3741	+	147.872i	−0.0931016	+	0.161257i
$918$		0			0
$919$	−609.475			−0.663194			−0.331597	−	0.943421i	$-0.607587\pi$
$919$	−609.475			−0.663194			−0.331597	+	0.943421i	$0.607587\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1796.31			1.94616
$924$		0			0
$925$	696.151	−	401.923i	0.752596	−	0.434512i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−663.351	+	1148.96i	−0.714048	+	1.23677i	0.249277	+	0.968432i	$0.419807\pi$
$929$	−663.351	+	1148.96i	−0.714048	+	1.23677i	−0.963326	+	0.268335i	$0.913526\pi$
$930$		0			0
$931$	−19.1916	+	163.129i	−0.0206140	+	0.175219i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	134.400	+	232.787i	0.143743	+	0.248970i
$936$		0			0
$937$	−911.033	−	1577.96i	−0.972288	−	1.68405i	−0.688610	−	0.725132i	$-0.741779\pi$
$937$	−911.033	−	1577.96i	−0.972288	−	1.68405i	−0.283678	−	0.958920i	$-0.591555\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−961.497	+	555.121i	−1.02178	+	0.589926i	−0.914620	−	0.404315i	$-0.867510\pi$
$941$	−961.497	+	555.121i	−1.02178	+	0.589926i	−0.107163	+	0.994242i	$0.534177\pi$
$942$		0			0
$943$			209.487i			0.222149i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−431.526	+	747.425i	−0.455677	+	0.789255i	−0.998727	−	0.0504450i	$-0.983936\pi$
$947$	−431.526	+	747.425i	−0.455677	+	0.789255i	0.543050	+	0.839700i	$0.317269\pi$
$948$		0			0
$949$		−	1973.55i		−	2.07961i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1494.57	+	862.890i	1.56828	+	0.905446i	0.996370	+	0.0851297i	$0.0271305\pi$
$953$	1494.57	+	862.890i	1.56828	+	0.905446i	0.571909	+	0.820317i	$0.306203\pi$
$954$		0			0
$955$	−969.029	+	1678.41i	−1.01469	+	1.75749i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−291.374	−	504.675i	−0.303831	−	0.526251i
$960$		0			0
$961$	943.185			0.981462
$962$		0			0
$963$		0			0
$964$		0			0
$965$	318.832	−	184.078i	0.330396	−	0.190754i
$966$		0			0
$967$	−767.279	+	1328.97i	−0.793463	+	1.37432i	0.130348	+	0.991468i	$0.458391\pi$
$967$	−767.279	+	1328.97i	−0.793463	+	1.37432i	−0.923811	+	0.382850i	$0.874943\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1048.94	−	605.609i	−1.08027	−	0.623696i	−0.149303	−	0.988792i	$-0.547703\pi$
$971$	−1048.94	−	605.609i	−1.08027	−	0.623696i	−0.930970	+	0.365096i	$0.881036\pi$
$972$		0			0
$973$	−643.552	−	1114.67i	−0.661410	−	1.14560i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	918.806i		−	0.940436i	−0.882550	−	0.470218i	$-0.844175\pi$
$977$		−	918.806i		−	0.940436i	0.882550	−	0.470218i	$-0.155825\pi$
$978$		0			0
$979$	460.313	−	265.762i	0.470187	−	0.271463i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	542.556	+	313.245i	0.551939	+	0.318662i	0.749904	−	0.661547i	$-0.230100\pi$
$983$	542.556	+	313.245i	0.551939	+	0.318662i	−0.197965	+	0.980209i	$0.563433\pi$
$984$		0			0
$985$	−484.081	+	838.453i	−0.491453	+	0.851222i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−546.070			−0.552143
$990$		0			0
$991$	−216.951	−	125.257i	−0.218921	−	0.126394i	0.386530	−	0.922277i	$-0.373674\pi$
$991$	−216.951	−	125.257i	−0.218921	−	0.126394i	−0.605451	+	0.795883i	$0.707007\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	948.241			0.953006
$996$		0			0
$997$	−643.120	−	1113.92i	−0.645055	−	1.11727i	−0.984289	−	0.176566i	$-0.943501\pi$
$997$	−643.120	−	1113.92i	−0.645055	−	1.11727i	0.339234	−	0.940702i	$-0.389832\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.3.y.h.217.4		8
3.2	odd	2		76.3.h.a.65.3	✓	8
12.11	even	2		304.3.r.c.65.2		8
19.12	odd	6	inner	684.3.y.h.145.4		8
57.8	even	6		1444.3.c.b.721.6		8
57.11	odd	6		1444.3.c.b.721.3		8
57.50	even	6		76.3.h.a.69.3	yes	8
228.107	odd	6		304.3.r.c.145.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.3.h.a.65.3	✓	8	3.2	odd	2
76.3.h.a.69.3	yes	8	57.50	even	6
304.3.r.c.65.2		8	12.11	even	2
304.3.r.c.145.2		8	228.107	odd	6
684.3.y.h.145.4		8	19.12	odd	6	inner
684.3.y.h.217.4		8	1.1	even	1	trivial
1444.3.c.b.721.3		8	57.11	odd	6
1444.3.c.b.721.6		8	57.8	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	684.y (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(4.81674 - 8.34284i) q^{5} +7.59243 q^{7} +O(q^{10})\)	\(q+(4.81674 - 8.34284i) q^{5} +7.59243 q^{7} +4.61883 q^{11} +(19.8169 - 11.4413i) q^{13} +(-3.02053 + 5.23171i) q^{17} +(-2.21998 + 18.8699i) q^{19} +(11.8651 + 20.5510i) q^{23} +(-33.9020 - 58.7200i) q^{25} +(-10.1836 + 5.87948i) q^{29} +4.22078i q^{31} +(36.5708 - 63.3424i) q^{35} +11.8554i q^{37} +(7.64514 + 4.41392i) q^{41} +(-11.5058 + 19.9286i) q^{43} +(-33.0941 - 57.3207i) q^{47} +8.64494 q^{49} +(-40.7377 + 23.5199i) q^{53} +(22.2477 - 38.5342i) q^{55} +(7.52242 + 4.34307i) q^{59} +(-3.57907 - 6.19914i) q^{61} -220.439i q^{65} +(-77.3224 + 44.6421i) q^{67} +(67.9840 + 39.2506i) q^{71} +(43.1234 - 74.6919i) q^{73} +35.0682 q^{77} +(57.9637 + 33.4653i) q^{79} -5.30752 q^{83} +(29.0982 + 50.3996i) q^{85} +(99.6601 - 57.5388i) q^{89} +(150.458 - 86.8671i) q^{91} +(146.735 + 109.412i) q^{95} +(-49.2806 - 28.4521i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{5} - 12 q^{7}+O(q^{10}) \) 8 * q + q^5 - 12 * q^7	\( 8 q + q^{5} - 12 q^{7} + 10 q^{11} + 9 q^{13} - 23 q^{17} - 33 q^{19} + 31 q^{23} - 73 q^{25} + 105 q^{29} + 68 q^{35} - 18 q^{41} - 41 q^{43} - 107 q^{47} + 312 q^{49} - 39 q^{53} + 70 q^{55} - 348 q^{59} - 45 q^{61} - 432 q^{67} + 243 q^{71} + 16 q^{73} - 544 q^{77} + 75 q^{79} + 82 q^{83} + 109 q^{85} + 213 q^{89} + 222 q^{91} + 385 q^{95} + 144 q^{97}+O(q^{100}) \) 8 * q + q^5 - 12 * q^7 + 10 * q^11 + 9 * q^13 - 23 * q^17 - 33 * q^19 + 31 * q^23 - 73 * q^25 + 105 * q^29 + 68 * q^35 - 18 * q^41 - 41 * q^43 - 107 * q^47 + 312 * q^49 - 39 * q^53 + 70 * q^55 - 348 * q^59 - 45 * q^61 - 432 * q^67 + 243 * q^71 + 16 * q^73 - 544 * q^77 + 75 * q^79 + 82 * q^83 + 109 * q^85 + 213 * q^89 + 222 * q^91 + 385 * q^95 + 144 * q^97

Introduction

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