LMFDB - Embedded newform 1728.3.q.l.449.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(449,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, 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("1728.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.q (of order $6$, degree $2$, not 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:	$47.0845896815$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 288)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			449.7
Character	$\chi$	$=$	1728.449
Dual form			1728.3.q.l.1601.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.15965 + 0.669525i) q^{5} +(-0.328661 - 0.569258i) q^{7} +O(q^{10})$	$q+(1.15965 + 0.669525i) q^{5} +(-0.328661 - 0.569258i) q^{7} +(-7.33786 + 4.23651i) q^{11} +(0.615997 - 1.06694i) q^{13} -5.78107i q^{17} +22.7103 q^{19} +(17.4792 + 10.0916i) q^{23} +(-11.6035 - 20.0978i) q^{25} +(-35.5451 + 20.5220i) q^{29} +(-7.57415 + 13.1188i) q^{31} -0.880187i q^{35} +51.2271 q^{37} +(-17.7563 - 10.2516i) q^{41} +(3.14476 + 5.44689i) q^{43} +(31.9185 - 18.4281i) q^{47} +(24.2840 - 42.0611i) q^{49} +85.9624i q^{53} -11.3458 q^{55} +(51.2682 + 29.5997i) q^{59} +(38.1516 + 66.0806i) q^{61} +(1.42868 - 0.824851i) q^{65} +(-48.2337 + 83.5433i) q^{67} +31.2026i q^{71} -24.9673 q^{73} +(4.82334 + 2.78475i) q^{77} +(67.4448 + 116.818i) q^{79} +(98.6013 - 56.9275i) q^{83} +(3.87057 - 6.70403i) q^{85} -135.098i q^{89} -0.809818 q^{91} +(26.3360 + 15.2051i) q^{95} +(-68.8218 - 119.203i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 60 q^{25} + 72 q^{29} + 36 q^{41} - 132 q^{49} - 96 q^{61} - 576 q^{65} + 24 q^{73} + 432 q^{77} + 96 q^{85} + 252 q^{97}+O(q^{100}) $ 24 * q + 60 * q^25 + 72 * q^29 + 36 * q^41 - 132 * q^49 - 96 * q^61 - 576 * q^65 + 24 * q^73 + 432 * q^77 + 96 * q^85 + 252 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

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
$5$	1.15965	+	0.669525i	0.231930	+	0.133905i	0.611462	−	0.791274i	$-0.290582\pi$
$5$	1.15965	+	0.669525i	0.231930	+	0.133905i	−0.379532	+	0.925179i	$0.623915\pi$
$6$		0			0
$7$	−0.328661	−	0.569258i	−0.0469516	−	0.0813226i	0.841595	−	0.540110i	$-0.181617\pi$
$7$	−0.328661	−	0.569258i	−0.0469516	−	0.0813226i	−0.888546	+	0.458787i	$0.848284\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−7.33786	+	4.23651i	−0.667078	+	0.385138i	−0.794968	−	0.606651i	$-0.792513\pi$
$11$	−7.33786	+	4.23651i	−0.667078	+	0.385138i	0.127891	+	0.991788i	$0.459179\pi$
$12$		0			0
$13$	0.615997	−	1.06694i	0.0473844	−	0.0820722i	−0.841360	−	0.540474i	$-0.818245\pi$
$13$	0.615997	−	1.06694i	0.0473844	−	0.0820722i	0.888745	+	0.458402i	$0.151578\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	5.78107i		−	0.340063i	−0.985439	−	0.170032i	$-0.945613\pi$
$17$		−	5.78107i		−	0.340063i	0.985439	−	0.170032i	$-0.0543870\pi$
$18$		0			0
$19$	22.7103			1.19528			0.597639	−	0.801765i	$-0.296106\pi$
$19$	22.7103			1.19528			0.597639	+	0.801765i	$0.296106\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	17.4792	+	10.0916i	0.759963	+	0.438765i	0.829283	−	0.558829i	$-0.188749\pi$
$23$	17.4792	+	10.0916i	0.759963	+	0.438765i	−0.0693192	+	0.997595i	$0.522083\pi$
$24$		0			0
$25$	−11.6035	−	20.0978i	−0.464139	−	0.803912i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−35.5451	+	20.5220i	−1.22569	+	0.707655i	−0.966126	−	0.258070i	$-0.916913\pi$
$29$	−35.5451	+	20.5220i	−1.22569	+	0.707655i	−0.259568	+	0.965725i	$0.583580\pi$
$30$		0			0
$31$	−7.57415	+	13.1188i	−0.244328	+	0.423188i	−0.961942	−	0.273253i	$-0.911901\pi$
$31$	−7.57415	+	13.1188i	−0.244328	+	0.423188i	0.717615	+	0.696440i	$0.245234\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	0.880187i		−	0.0251482i
$36$		0			0
$37$	51.2271			1.38452			0.692258	−	0.721651i	$-0.256616\pi$
$37$	51.2271			1.38452			0.692258	+	0.721651i	$0.256616\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−17.7563	−	10.2516i	−0.433079	−	0.250038i	0.267578	−	0.963536i	$-0.413777\pi$
$41$	−17.7563	−	10.2516i	−0.433079	−	0.250038i	−0.700658	+	0.713498i	$0.747110\pi$
$42$		0			0
$43$	3.14476	+	5.44689i	0.0731340	+	0.126672i	0.900273	−	0.435325i	$-0.143367\pi$
$43$	3.14476	+	5.44689i	0.0731340	+	0.126672i	−0.827139	+	0.561997i	$0.810033\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	31.9185	−	18.4281i	0.679117	−	0.392088i	−0.120406	−	0.992725i	$-0.538419\pi$
$47$	31.9185	−	18.4281i	0.679117	−	0.392088i	0.799522	+	0.600637i	$0.205086\pi$
$48$		0			0
$49$	24.2840	−	42.0611i	0.495591	−	0.858389i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			85.9624i			1.62193i	0.585094	+	0.810966i	$0.301058\pi$
$53$			85.9624i			1.62193i	−0.585094	+	0.810966i	$0.698942\pi$
$54$		0			0
$55$	−11.3458			−0.206287
$56$		0			0
$57$		0			0
$58$		0			0
$59$	51.2682	+	29.5997i	0.868953	+	0.501690i	0.867000	−	0.498308i	$-0.166045\pi$
$59$	51.2682	+	29.5997i	0.868953	+	0.501690i	0.00195303	+	0.999998i	$0.499378\pi$
$60$		0			0
$61$	38.1516	+	66.0806i	0.625437	+	1.08329i	0.988456	+	0.151506i	$0.0484124\pi$
$61$	38.1516	+	66.0806i	0.625437	+	1.08329i	−0.363020	+	0.931781i	$0.618254\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.42868	−	0.824851i	0.0219797	−	0.0126900i
$66$		0			0
$67$	−48.2337	+	83.5433i	−0.719907	+	1.24691i	0.241130	+	0.970493i	$0.422482\pi$
$67$	−48.2337	+	83.5433i	−0.719907	+	1.24691i	−0.961036	+	0.276422i	$0.910851\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			31.2026i			0.439473i	0.975559	+	0.219737i	$0.0705198\pi$
$71$			31.2026i			0.439473i	−0.975559	+	0.219737i	$0.929480\pi$
$72$		0			0
$73$	−24.9673			−0.342018			−0.171009	−	0.985269i	$-0.554703\pi$
$73$	−24.9673			−0.342018			−0.171009	+	0.985269i	$0.554703\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	4.82334	+	2.78475i	0.0626407	+	0.0361656i
$78$		0			0
$79$	67.4448	+	116.818i	0.853732	+	1.47871i	0.877817	+	0.478997i	$0.159000\pi$
$79$	67.4448	+	116.818i	0.853732	+	1.47871i	−0.0240847	+	0.999710i	$0.507667\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	98.6013	−	56.9275i	1.18797	−	0.685873i	0.230123	−	0.973162i	$-0.426087\pi$
$83$	98.6013	−	56.9275i	1.18797	−	0.685873i	0.957844	+	0.287288i	$0.0927538\pi$
$84$		0			0
$85$	3.87057	−	6.70403i	0.0455361	−	0.0788709i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	135.098i		−	1.51795i	−0.651118	−	0.758977i	$-0.725700\pi$
$89$		−	135.098i		−	1.51795i	0.651118	−	0.758977i	$-0.274300\pi$
$90$		0			0
$91$	−0.809818			−0.00889909
$92$		0			0
$93$		0			0
$94$		0			0
$95$	26.3360	+	15.2051i	0.277221	+	0.160054i
$96$		0			0
$97$	−68.8218	−	119.203i	−0.709503	−	1.22890i	−0.965042	−	0.262097i	$-0.915586\pi$
$97$	−68.8218	−	119.203i	−0.709503	−	1.22890i	0.255538	−	0.966799i	$-0.417747\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−102.305	+	59.0659i	−1.01292	+	0.584811i	−0.912046	−	0.410088i	$-0.865498\pi$
$101$	−102.305	+	59.0659i	−1.01292	+	0.584811i	−0.100876	+	0.994899i	$0.532165\pi$
$102$		0			0
$103$	17.9512	−	31.0923i	0.174283	−	0.301867i	−0.765630	−	0.643281i	$-0.777573\pi$
$103$	17.9512	−	31.0923i	0.174283	−	0.301867i	0.939913	+	0.341414i	$0.110906\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			51.3746i			0.480136i	0.970756	+	0.240068i	$0.0771698\pi$
$107$			51.3746i			0.480136i	−0.970756	+	0.240068i	$0.922830\pi$
$108$		0			0
$109$	65.4476			0.600436			0.300218	−	0.953871i	$-0.402940\pi$
$109$	65.4476			0.600436			0.300218	+	0.953871i	$0.402940\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	49.6877	+	28.6872i	0.439714	+	0.253869i	0.703476	−	0.710719i	$-0.251630\pi$
$113$	49.6877	+	28.6872i	0.439714	+	0.253869i	−0.263762	+	0.964588i	$0.584963\pi$
$114$		0			0
$115$	13.5131	+	23.4055i	0.117506	+	0.203526i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−3.29092	+	1.90001i	−0.0276548	+	0.0159665i
$120$		0			0
$121$	−24.6039	+	42.6152i	−0.203338	+	0.352192i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	64.5515i		−	0.516412i
$126$		0			0
$127$	−12.5134			−0.0985308			−0.0492654	−	0.998786i	$-0.515688\pi$
$127$	−12.5134			−0.0985308			−0.0492654	+	0.998786i	$0.515688\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	100.368	+	57.9476i	0.766169	+	0.442348i	0.831506	−	0.555515i	$-0.187479\pi$
$131$	100.368	+	57.9476i	0.766169	+	0.442348i	−0.0653374	+	0.997863i	$0.520812\pi$
$132$		0			0
$133$	−7.46399	−	12.9280i	−0.0561202	−	0.0972031i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	55.9545	−	32.3053i	0.408427	−	0.235805i	−0.281687	−	0.959506i	$-0.590894\pi$
$137$	55.9545	−	32.3053i	0.408427	−	0.235805i	0.690114	+	0.723701i	$0.257561\pi$
$138$		0			0
$139$	−86.2101	+	149.320i	−0.620216	+	1.07425i	0.369229	+	0.929339i	$0.379622\pi$
$139$	−86.2101	+	149.320i	−0.620216	+	1.07425i	−0.989445	+	0.144908i	$0.953711\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			10.4387i			0.0729980i
$144$		0			0
$145$	−54.9599			−0.379034
$146$		0			0
$147$		0			0
$148$		0			0
$149$	145.339	+	83.9114i	0.975428	+	0.563163i	0.900887	−	0.434055i	$-0.142917\pi$
$149$	145.339	+	83.9114i	0.975428	+	0.563163i	0.0745411	+	0.997218i	$0.476251\pi$
$150$		0			0
$151$	118.439	+	205.142i	0.784363	+	1.35856i	0.929379	+	0.369128i	$0.120343\pi$
$151$	118.439	+	205.142i	0.784363	+	1.35856i	−0.145015	+	0.989429i	$0.546323\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−17.5667	+	10.1422i	−0.113334	+	0.0654333i
$156$		0			0
$157$	−80.3922	+	139.243i	−0.512052	+	0.886900i	0.487850	+	0.872927i	$0.337781\pi$
$157$	−80.3922	+	139.243i	−0.512052	+	0.886900i	−0.999902	+	0.0139730i	$0.995552\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	13.2669i		−	0.0824029i
$162$		0			0
$163$	279.703			1.71597			0.857986	−	0.513673i	$-0.171716\pi$
$163$	279.703			1.71597			0.857986	+	0.513673i	$0.171716\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	199.901	+	115.413i	1.19701	+	0.691096i	0.959888	−	0.280383i	$-0.0904615\pi$
$167$	199.901	+	115.413i	1.19701	+	0.691096i	0.237125	+	0.971479i	$0.423795\pi$
$168$		0			0
$169$	83.7411	+	145.044i	0.495509	+	0.858248i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	91.9688	−	53.0982i	0.531612	−	0.306926i	−0.210061	−	0.977688i	$-0.567366\pi$
$173$	91.9688	−	53.0982i	0.531612	−	0.306926i	0.741673	+	0.670762i	$0.234033\pi$
$174$		0			0
$175$	−7.62722	+	13.2107i	−0.0435841	+	0.0754899i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			4.14055i			0.0231315i	0.999933	+	0.0115658i	$0.00368158\pi$
$179$			4.14055i			0.0231315i	−0.999933	+	0.0115658i	$0.996318\pi$
$180$		0			0
$181$	−186.430			−1.03000			−0.515001	−	0.857190i	$-0.672208\pi$
$181$	−186.430			−1.03000			−0.515001	+	0.857190i	$0.672208\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	59.4055	+	34.2978i	0.321111	+	0.185393i
$186$		0			0
$187$	24.4916	+	42.4207i	0.130971	+	0.226849i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	209.368	−	120.879i	1.09617	−	0.632872i	0.160955	−	0.986962i	$-0.448543\pi$
$191$	209.368	−	120.879i	1.09617	−	0.632872i	0.935211	+	0.354090i	$0.115209\pi$
$192$		0			0
$193$	−44.3377	+	76.7952i	−0.229729	+	0.397903i	−0.957728	−	0.287676i	$-0.907117\pi$
$193$	−44.3377	+	76.7952i	−0.229729	+	0.397903i	0.727999	+	0.685579i	$0.240451\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	224.293i		−	1.13854i	−0.822149	−	0.569272i	$-0.807225\pi$
$197$		−	224.293i		−	1.13854i	0.822149	−	0.569272i	$-0.192775\pi$
$198$		0			0
$199$	−371.640			−1.86754			−0.933770	−	0.357874i	$-0.883502\pi$
$199$	−371.640			−1.86754			−0.933770	+	0.357874i	$0.883502\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	23.3646	+	13.4896i	0.115097	+	0.0664510i
$204$		0			0
$205$	−13.7274	−	23.7765i	−0.0669628	−	0.115983i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−166.645	+	96.2124i	−0.797343	+	0.460346i
$210$		0			0
$211$	−69.1170	+	119.714i	−0.327569	+	0.567365i	−0.982029	−	0.188731i	$-0.939563\pi$
$211$	−69.1170	+	119.714i	−0.327569	+	0.567365i	0.654460	+	0.756096i	$0.272896\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			8.42198i			0.0391720i
$216$		0			0
$217$	9.95732			0.0458863
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−6.16805	−	3.56113i	−0.0279097	−	0.0161137i
$222$		0			0
$223$	112.891	+	195.533i	0.506238	+	0.876829i	0.999974	+	0.00721770i	$0.00229749\pi$
$223$	112.891	+	195.533i	0.506238	+	0.876829i	−0.493736	+	0.869612i	$0.664369\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	110.292	−	63.6769i	0.485866	−	0.280515i	−0.236992	−	0.971512i	$-0.576161\pi$
$227$	110.292	−	63.6769i	0.485866	−	0.280515i	0.722858	+	0.690997i	$0.242828\pi$
$228$		0			0
$229$	77.1871	−	133.692i	0.337061	−	0.583807i	−0.646817	−	0.762645i	$-0.723900\pi$
$229$	77.1871	−	133.692i	0.337061	−	0.583807i	0.983879	+	0.178838i	$0.0572337\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	20.3013i		−	0.0871300i	−0.999051	−	0.0435650i	$-0.986128\pi$
$233$		−	20.3013i		−	0.0871300i	0.999051	−	0.0435650i	$-0.0138716\pi$
$234$		0			0
$235$	49.3524			0.210010
$236$		0			0
$237$		0			0
$238$		0			0
$239$	223.391	+	128.975i	0.934691	+	0.539644i	0.888292	−	0.459279i	$-0.151892\pi$
$239$	223.391	+	128.975i	0.934691	+	0.539644i	0.0463986	+	0.998923i	$0.485226\pi$
$240$		0			0
$241$	38.0826	+	65.9610i	0.158019	+	0.273697i	0.934154	−	0.356869i	$-0.116156\pi$
$241$	38.0826	+	65.9610i	0.158019	+	0.273697i	−0.776135	+	0.630567i	$0.782823\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	56.3218	−	32.5174i	0.229885	−	0.132724i
$246$		0			0
$247$	13.9895	−	24.2305i	0.0566375	−	0.0980991i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			108.320i			0.431555i	0.976443	+	0.215778i	$0.0692286\pi$
$251$			108.320i			0.431555i	−0.976443	+	0.215778i	$0.930771\pi$
$252$		0			0
$253$	−171.013			−0.675940
$254$		0			0
$255$		0			0
$256$		0			0
$257$	289.661	+	167.236i	1.12709	+	0.650723i	0.943200	−	0.332225i	$-0.107799\pi$
$257$	289.661	+	167.236i	1.12709	+	0.650723i	0.183885	+	0.982948i	$0.441133\pi$
$258$		0			0
$259$	−16.8363	−	29.1614i	−0.0650052	−	0.112592i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−18.4612	+	10.6586i	−0.0701946	+	0.0405269i	−0.534687	−	0.845050i	$-0.679570\pi$
$263$	−18.4612	+	10.6586i	−0.0701946	+	0.0405269i	0.464492	+	0.885577i	$0.346237\pi$
$264$		0			0
$265$	−57.5539	+	99.6863i	−0.217185	+	0.376175i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	411.961i		−	1.53145i	−0.643167	−	0.765726i	$-0.722380\pi$
$269$		−	411.961i		−	1.53145i	0.643167	−	0.765726i	$-0.277620\pi$
$270$		0			0
$271$	169.050			0.623803			0.311901	−	0.950115i	$-0.399034\pi$
$271$	169.050			0.623803			0.311901	+	0.950115i	$0.399034\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	170.289	+	98.3165i	0.619234	+	0.357515i
$276$		0			0
$277$	−183.385	−	317.632i	−0.662040	−	1.14669i	−0.980079	−	0.198609i	$-0.936357\pi$
$277$	−183.385	−	317.632i	−0.662040	−	1.14669i	0.318039	−	0.948078i	$-0.396976\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−212.102	+	122.457i	−0.754812	+	0.435791i	−0.827430	−	0.561569i	$-0.810198\pi$
$281$	−212.102	+	122.457i	−0.754812	+	0.435791i	0.0726179	+	0.997360i	$0.476865\pi$
$282$		0			0
$283$	−134.798	+	233.478i	−0.476320	+	0.825010i	−0.999632	−	0.0271311i	$-0.991363\pi$
$283$	−134.798	+	233.478i	−0.476320	+	0.825010i	0.523312	+	0.852141i	$0.324696\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			13.4772i			0.0469588i
$288$		0			0
$289$	255.579			0.884357
$290$		0			0
$291$		0			0
$292$		0			0
$293$	109.202	+	63.0479i	0.372704	+	0.215181i	0.674639	−	0.738148i	$-0.264299\pi$
$293$	109.202	+	63.0479i	0.372704	+	0.215181i	−0.301935	+	0.953328i	$0.597633\pi$
$294$		0			0
$295$	39.6355	+	68.6507i	0.134358	+	0.232714i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	21.5342	−	12.4328i	0.0720208	−	0.0415812i
$300$		0			0
$301$	2.06712	−	3.58036i	0.00686752	−	0.0118949i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			102.174i			0.334996i
$306$		0			0
$307$	254.097			0.827679			0.413840	−	0.910350i	$-0.364187\pi$
$307$	254.097			0.827679			0.413840	+	0.910350i	$0.364187\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−450.132	−	259.884i	−1.44737	−	0.835640i	−0.449047	−	0.893508i	$-0.648236\pi$
$311$	−450.132	−	259.884i	−1.44737	−	0.835640i	−0.998324	+	0.0578683i	$0.981570\pi$
$312$		0			0
$313$	188.807	+	327.023i	0.603217	+	1.04480i	0.992331	+	0.123613i	$0.0394481\pi$
$313$	188.807	+	327.023i	0.603217	+	1.04480i	−0.389113	+	0.921190i	$0.627219\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	21.7374	−	12.5501i	0.0685722	−	0.0395902i	−0.465322	−	0.885142i	$-0.654061\pi$
$317$	21.7374	−	12.5501i	0.0685722	−	0.0395902i	0.533894	+	0.845551i	$0.320728\pi$
$318$		0			0
$319$	173.883	−	301.175i	0.545089	−	0.944121i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	131.290i		−	0.406470i
$324$		0			0
$325$	−28.5908			−0.0879718
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−20.9807	−	12.1132i	−0.0637712	−	0.0368183i
$330$		0			0
$331$	−267.865	−	463.956i	−0.809261	−	1.40168i	−0.913377	−	0.407116i	$-0.866535\pi$
$331$	−267.865	−	463.956i	−0.809261	−	1.40168i	0.104116	−	0.994565i	$-0.466799\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−111.869	+	64.5873i	−0.333936	+	0.192798i
$336$		0			0
$337$	108.211	−	187.427i	0.321101	−	0.556164i	−0.659614	−	0.751604i	$-0.729280\pi$
$337$	108.211	−	187.427i	0.321101	−	0.556164i	0.980716	+	0.195440i	$0.0626136\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	128.352i		−	0.376399i
$342$		0			0
$343$	−64.1336			−0.186978
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−205.090	−	118.409i	−0.591036	−	0.341235i	0.174471	−	0.984662i	$-0.444179\pi$
$347$	−205.090	−	118.409i	−0.591036	−	0.341235i	−0.765507	+	0.643427i	$0.777512\pi$
$348$		0			0
$349$	−69.5513	−	120.466i	−0.199287	−	0.345176i	0.749010	−	0.662558i	$-0.230529\pi$
$349$	−69.5513	−	120.466i	−0.199287	−	0.345176i	−0.948298	+	0.317383i	$0.897196\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−533.791	+	308.184i	−1.51216	+	0.873044i	−0.512257	+	0.858832i	$0.671190\pi$
$353$	−533.791	+	308.184i	−1.51216	+	0.873044i	−0.999899	+	0.0142112i	$0.995476\pi$
$354$		0			0
$355$	−20.8909	+	36.1841i	−0.0588476	+	0.101927i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			337.595i			0.940375i	0.882567	+	0.470187i	$0.155814\pi$
$359$			337.595i			0.940375i	−0.882567	+	0.470187i	$0.844186\pi$
$360$		0			0
$361$	154.757			0.428690
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−28.9534	−	16.7162i	−0.0793243	−	0.0457979i
$366$		0			0
$367$	60.3827	+	104.586i	0.164530	+	0.284975i	0.936488	−	0.350698i	$-0.114056\pi$
$367$	60.3827	+	104.586i	0.164530	+	0.284975i	−0.771958	+	0.635674i	$0.780722\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	48.9347	−	28.2525i	0.131900	−	0.0761523i
$372$		0			0
$373$	154.120	−	266.943i	0.413190	−	0.715666i	−0.582047	−	0.813155i	$-0.697748\pi$
$373$	154.120	−	266.943i	0.413190	−	0.715666i	0.995237	+	0.0974896i	$0.0310813\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			50.5659i			0.134127i
$378$		0			0
$379$	−231.359			−0.610446			−0.305223	−	0.952281i	$-0.598731\pi$
$379$	−231.359			−0.610446			−0.305223	+	0.952281i	$0.598731\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−167.447	−	96.6758i	−0.437200	−	0.252417i	0.265209	−	0.964191i	$-0.414559\pi$
$383$	−167.447	−	96.6758i	−0.437200	−	0.252417i	−0.702409	+	0.711773i	$0.747892\pi$
$384$		0			0
$385$	3.72892	+	6.45869i	0.00968552	+	0.0167758i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−590.703	+	341.042i	−1.51852	+	0.876715i	−0.518753	+	0.854924i	$0.673604\pi$
$389$	−590.703	+	341.042i	−1.51852	+	0.876715i	−0.999763	+	0.0217913i	$0.993063\pi$
$390$		0			0
$391$	58.3403	−	101.048i	0.149208	−	0.258436i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			180.624i			0.457276i
$396$		0			0
$397$	233.681			0.588618			0.294309	−	0.955710i	$-0.404911\pi$
$397$	233.681			0.588618			0.294309	+	0.955710i	$0.404911\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	235.711	+	136.088i	0.587809	+	0.339372i	0.764231	−	0.644943i	$-0.223119\pi$
$401$	235.711	+	136.088i	0.587809	+	0.339372i	−0.176422	+	0.984315i	$0.556452\pi$
$402$		0			0
$403$	9.33132	+	16.1623i	0.0231546	+	0.0401050i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−375.897	+	217.024i	−0.923579	+	0.533229i
$408$		0			0
$409$	92.1194	−	159.555i	0.225231	−	0.390111i	−0.731158	−	0.682208i	$-0.761020\pi$
$409$	92.1194	−	159.555i	0.225231	−	0.390111i	0.956389	+	0.292097i	$0.0943531\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	38.9131i		−	0.0942207i
$414$		0			0
$415$	152.457			0.367367
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−72.1233	−	41.6404i	−0.172132	−	0.0993805i	0.411459	−	0.911428i	$-0.365019\pi$
$419$	−72.1233	−	41.6404i	−0.172132	−	0.0993805i	−0.583591	+	0.812048i	$0.698353\pi$
$420$		0			0
$421$	213.853	+	370.405i	0.507965	+	0.879821i	0.999957	+	0.00922184i	$0.00293544\pi$
$421$	213.853	+	370.405i	0.507965	+	0.879821i	−0.491992	+	0.870599i	$0.663731\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−116.187	+	67.0805i	−0.273381	+	0.157837i
$426$		0			0
$427$	25.0779	−	43.4362i	0.0587305	−	0.101724i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	698.818i		−	1.62139i	−0.585470	−	0.810694i	$-0.699090\pi$
$431$		−	698.818i		−	1.62139i	0.585470	−	0.810694i	$-0.300910\pi$
$432$		0			0
$433$	−70.4515			−0.162705			−0.0813527	−	0.996685i	$-0.525924\pi$
$433$	−70.4515			−0.162705			−0.0813527	+	0.996685i	$0.525924\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	396.957	+	229.183i	0.908368	+	0.524446i
$438$		0			0
$439$	−365.285	−	632.692i	−0.832084	−	1.44121i	−0.896383	−	0.443280i	$-0.853815\pi$
$439$	−365.285	−	632.692i	−0.832084	−	1.44121i	0.0642994	−	0.997931i	$-0.479519\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−511.433	+	295.276i	−1.15448	+	0.666538i	−0.949974	−	0.312328i	$-0.898891\pi$
$443$	−511.433	+	295.276i	−1.15448	+	0.666538i	−0.204503	+	0.978866i	$0.565558\pi$
$444$		0			0
$445$	90.4514	−	156.666i	0.203261	−	0.352059i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	386.044i		−	0.859785i	−0.902880	−	0.429893i	$-0.858551\pi$
$449$		−	386.044i		−	0.859785i	0.902880	−	0.429893i	$-0.141449\pi$
$450$		0			0
$451$	173.724			0.385197
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−0.939105	−	0.542193i	−0.00206397	−	0.00119163i
$456$		0			0
$457$	−209.179	−	362.309i	−0.457722	−	0.792798i	0.541118	−	0.840947i	$-0.318001\pi$
$457$	−209.179	−	362.309i	−0.457722	−	0.792798i	−0.998840	+	0.0481488i	$0.984668\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−462.178	+	266.838i	−1.00255	+	0.578825i	−0.909003	−	0.416789i	$-0.863155\pi$
$461$	−462.178	+	266.838i	−1.00255	+	0.578825i	−0.0935513	+	0.995614i	$0.529822\pi$
$462$		0			0
$463$	−21.1080	+	36.5602i	−0.0455897	+	0.0789637i	−0.887920	−	0.459998i	$-0.847850\pi$
$463$	−21.1080	+	36.5602i	−0.0455897	+	0.0789637i	0.842330	+	0.538962i	$0.181183\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	734.531i		−	1.57287i	−0.617673	−	0.786435i	$-0.711924\pi$
$467$		−	734.531i		−	1.57287i	0.617673	−	0.786435i	$-0.288076\pi$
$468$		0			0
$469$	63.4102			0.135203
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−46.1516	−	26.6457i	−0.0975722	−	0.0563333i
$474$		0			0
$475$	−263.518	−	456.427i	−0.554775	−	0.960899i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	577.408	−	333.367i	1.20544	−	0.695964i	0.243684	−	0.969855i	$-0.421644\pi$
$479$	577.408	−	333.367i	1.20544	−	0.695964i	0.961761	+	0.273891i	$0.0883107\pi$
$480$		0			0
$481$	31.5557	−	54.6561i	0.0656044	−	0.113630i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	184.312i		−	0.380024i
$486$		0			0
$487$	−788.556			−1.61921			−0.809606	−	0.586974i	$-0.800319\pi$
$487$	−788.556			−1.61921			−0.809606	+	0.586974i	$0.800319\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−300.390	−	173.430i	−0.611792	−	0.353218i	0.161874	−	0.986811i	$-0.448246\pi$
$491$	−300.390	−	173.430i	−0.611792	−	0.353218i	−0.773667	+	0.633593i	$0.781579\pi$
$492$		0			0
$493$	118.639	+	205.489i	0.240647	+	0.416813i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	17.7623	−	10.2551i	0.0357391	−	0.0206340i
$498$		0			0
$499$	260.871	−	451.841i	0.522787	−	0.905493i	−0.476862	−	0.878978i	$-0.658226\pi$
$499$	260.871	−	451.841i	0.522787	−	0.905493i	0.999648	−	0.0265150i	$-0.00844096\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			244.735i			0.486550i	0.969957	+	0.243275i	$0.0782217\pi$
$503$			244.735i			0.486550i	−0.969957	+	0.243275i	$0.921778\pi$
$504$		0			0
$505$	−158.184			−0.313236
$506$		0			0
$507$		0			0
$508$		0			0
$509$	278.949	+	161.051i	0.548034	+	0.316407i	0.748329	−	0.663328i	$-0.230857\pi$
$509$	278.949	+	161.051i	0.548034	+	0.316407i	−0.200295	+	0.979736i	$0.564190\pi$
$510$		0			0
$511$	8.20579	+	14.2128i	0.0160583	+	0.0278138i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	41.6341	−	24.0375i	0.0808430	−	0.0466747i
$516$		0			0
$517$	−156.142	+	270.446i	−0.302016	+	0.523107i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	486.479i		−	0.933740i	−0.884326	−	0.466870i	$-0.845382\pi$
$521$		−	486.479i		−	0.933740i	0.884326	−	0.466870i	$-0.154618\pi$
$522$		0			0
$523$	−337.116			−0.644582			−0.322291	−	0.946641i	$-0.604453\pi$
$523$	−337.116			−0.644582			−0.322291	+	0.946641i	$0.604453\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	75.8409	+	43.7867i	0.143911	+	0.0830868i
$528$		0			0
$529$	−60.8194	−	105.342i	−0.114970	−	0.199135i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−21.8756	+	12.6299i	−0.0410424	+	0.0236958i
$534$		0			0
$535$	−34.3965	+	59.5766i	−0.0642926	+	0.111358i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			411.517i			0.763483i
$540$		0			0
$541$	−35.2083			−0.0650800			−0.0325400	−	0.999470i	$-0.510360\pi$
$541$	−35.2083			−0.0650800			−0.0325400	+	0.999470i	$0.510360\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	75.8963	+	43.8188i	0.139259	+	0.0804014i
$546$		0			0
$547$	321.377	+	556.641i	0.587526	+	1.01762i	0.994555	+	0.104210i	$0.0332313\pi$
$547$	321.377	+	556.641i	0.587526	+	1.01762i	−0.407030	+	0.913415i	$0.633435\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−807.240	+	466.060i	−1.46504	+	0.845844i
$552$		0			0
$553$	44.3330	−	76.7870i	0.0801681	−	0.138855i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			602.553i			1.08178i	0.841092	+	0.540892i	$0.181913\pi$
$557$			602.553i			1.08178i	−0.841092	+	0.540892i	$0.818087\pi$
$558$		0			0
$559$	7.74866			0.0138616
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−601.451	−	347.248i	−1.06830	−	0.616782i	−0.140581	−	0.990069i	$-0.544897\pi$
$563$	−601.451	−	347.248i	−1.06830	−	0.616782i	−0.927716	+	0.373288i	$0.878230\pi$
$564$		0			0
$565$	38.4136	+	66.5343i	0.0679886	+	0.117760i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	283.950	−	163.939i	0.499033	−	0.288117i	−0.229281	−	0.973360i	$-0.573637\pi$
$569$	283.950	−	163.939i	0.499033	−	0.288117i	0.728314	+	0.685243i	$0.240304\pi$
$570$		0			0
$571$	440.493	−	762.956i	0.771441	−	1.33617i	−0.165332	−	0.986238i	$-0.552870\pi$
$571$	440.493	−	762.956i	0.771441	−	1.33617i	0.936773	−	0.349937i	$-0.113797\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	468.390i		−	0.814592i
$576$		0			0
$577$	−394.926			−0.684447			−0.342224	−	0.939619i	$-0.611180\pi$
$577$	−394.926			−0.684447			−0.342224	+	0.939619i	$0.611180\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−64.8128	−	37.4197i	−0.111554	−	0.0644057i
$582$		0			0
$583$	−364.181	−	630.779i	−0.624667	−	1.08195i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−821.405	+	474.238i	−1.39933	+	0.807902i	−0.994322	−	0.106413i	$-0.966064\pi$
$587$	−821.405	+	474.238i	−1.39933	+	0.807902i	−0.405005	+	0.914314i	$0.632730\pi$
$588$		0			0
$589$	−172.011	+	297.932i	−0.292039	+	0.505827i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	1138.81i		−	1.92043i	−0.279269	−	0.960213i	$-0.590092\pi$
$593$		−	1138.81i		−	1.92043i	0.279269	−	0.960213i	$-0.409908\pi$
$594$		0			0
$595$	−5.08843			−0.00855198
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−286.895	−	165.639i	−0.478956	−	0.276525i	0.241025	−	0.970519i	$-0.422516\pi$
$599$	−286.895	−	165.639i	−0.478956	−	0.276525i	−0.719981	+	0.693993i	$0.755850\pi$
$600$		0			0
$601$	−416.186	−	720.856i	−0.692490	−	1.19943i	−0.971020	−	0.239000i	$-0.923180\pi$
$601$	−416.186	−	720.856i	−0.692490	−	1.19943i	0.278530	−	0.960428i	$-0.410153\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−57.0639	+	32.9459i	−0.0943205	+	0.0544560i
$606$		0			0
$607$	395.479	−	684.990i	0.651531	−	1.12848i	−0.331220	−	0.943553i	$-0.607460\pi$
$607$	395.479	−	684.990i	0.651531	−	1.12848i	0.982751	−	0.184931i	$-0.0592063\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	45.4067i		−	0.0743154i
$612$		0			0
$613$	795.300			1.29739			0.648695	−	0.761049i	$-0.275315\pi$
$613$	795.300			1.29739			0.648695	+	0.761049i	$0.275315\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	512.491	+	295.887i	0.830618	+	0.479558i	0.854064	−	0.520168i	$-0.174131\pi$
$617$	512.491	+	295.887i	0.830618	+	0.479558i	−0.0234462	+	0.999725i	$0.507464\pi$
$618$		0			0
$619$	−198.357	−	343.565i	−0.320448	−	0.555033i	0.660132	−	0.751149i	$-0.270500\pi$
$619$	−198.357	−	343.565i	−0.320448	−	0.555033i	−0.980581	+	0.196117i	$0.937167\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−76.9055	+	44.4014i	−0.123444	+	0.0712704i
$624$		0			0
$625$	−246.868	+	427.588i	−0.394989	+	0.684141i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	296.147i		−	0.470823i
$630$		0			0
$631$	451.164			0.714999			0.357499	−	0.933913i	$-0.383629\pi$
$631$	451.164			0.714999			0.357499	+	0.933913i	$0.383629\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−14.5112	−	8.37804i	−0.0228523	−	0.0131938i
$636$		0			0
$637$	−29.9177	−	51.8190i	−0.0469666	−	0.0813485i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	488.625	−	282.108i	0.762285	−	0.440105i	−0.0678306	−	0.997697i	$-0.521608\pi$
$641$	488.625	−	282.108i	0.762285	−	0.440105i	0.830116	+	0.557591i	$0.188274\pi$
$642$		0			0
$643$	209.098	−	362.169i	0.325192	−	0.563248i	−0.656360	−	0.754448i	$-0.727905\pi$
$643$	209.098	−	362.169i	0.325192	−	0.563248i	0.981551	+	0.191200i	$0.0612379\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			38.3697i			0.0593040i	0.999560	+	0.0296520i	$0.00943991\pi$
$647$			38.3697i			0.0593040i	−0.999560	+	0.0296520i	$0.990560\pi$
$648$		0			0
$649$	−501.599			−0.772879
$650$		0			0
$651$		0			0
$652$		0			0
$653$	89.3873	+	51.6078i	0.136887	+	0.0790318i	0.566880	−	0.823801i	$-0.308150\pi$
$653$	89.3873	+	51.6078i	0.136887	+	0.0790318i	−0.429993	+	0.902832i	$0.641484\pi$
$654$		0			0
$655$	77.5946	+	134.398i	0.118465	+	0.205188i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	680.807	−	393.064i	1.03309	−	0.596455i	0.115222	−	0.993340i	$-0.463242\pi$
$659$	680.807	−	393.064i	1.03309	−	0.596455i	0.917869	+	0.396885i	$0.129909\pi$
$660$		0			0
$661$	−431.654	+	747.646i	−0.653031	+	1.13108i	0.329353	+	0.944207i	$0.393170\pi$
$661$	−431.654	+	747.646i	−0.653031	+	1.13108i	−0.982384	+	0.186876i	$0.940164\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	19.9893i		−	0.0300591i
$666$		0			0
$667$	−828.398			−1.24198
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−559.902	−	323.260i	−0.834430	−	0.481758i
$672$		0			0
$673$	32.0001	+	55.4259i	0.0475485	+	0.0823564i	0.888820	−	0.458256i	$-0.151526\pi$
$673$	32.0001	+	55.4259i	0.0475485	+	0.0823564i	−0.841272	+	0.540613i	$0.818192\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−365.558	+	211.055i	−0.539967	+	0.311750i	−0.745066	−	0.666991i	$-0.767582\pi$
$677$	−365.558	+	211.055i	−0.539967	+	0.311750i	0.205098	+	0.978741i	$0.434249\pi$
$678$		0			0
$679$	−45.2381	+	78.3547i	−0.0666246	+	0.115397i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	59.7457i		−	0.0874754i	−0.999043	−	0.0437377i	$-0.986073\pi$
$683$		−	59.7457i		−	0.0874754i	0.999043	−	0.0437377i	$-0.0139266\pi$
$684$		0			0
$685$	86.5169			0.126302
$686$		0			0
$687$		0			0
$688$		0			0
$689$	91.7165	+	52.9526i	0.133115	+	0.0768542i
$690$		0			0
$691$	13.7385	+	23.7959i	0.0198821	+	0.0344368i	0.875795	−	0.482683i	$-0.160338\pi$
$691$	13.7385	+	23.7959i	0.0198821	+	0.0344368i	−0.855913	+	0.517120i	$0.827004\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−199.947	+	115.440i	−0.287694	+	0.166100i
$696$		0			0
$697$	−59.2651	+	102.650i	−0.0850289	+	0.147274i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			440.657i			0.628611i	0.949322	+	0.314306i	$0.101772\pi$
$701$			440.657i			0.628611i	−0.949322	+	0.314306i	$0.898228\pi$
$702$		0			0
$703$	1163.38			1.65488
$704$		0			0
$705$		0			0
$706$		0			0
$707$	67.2474	+	38.8253i	0.0951166	+	0.0549156i
$708$		0			0
$709$	562.466	+	974.219i	0.793323	+	1.37408i	0.923899	+	0.382637i	$0.124984\pi$
$709$	562.466	+	974.219i	0.793323	+	1.37408i	−0.130576	+	0.991438i	$0.541683\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−264.780	+	152.871i	−0.371360	+	0.214405i
$714$		0			0
$715$	−6.98898	+	12.1053i	−0.00977480	+	0.0169304i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			734.502i			1.02156i	0.859711	+	0.510780i	$0.170643\pi$
$719$			734.502i			1.02156i	−0.859711	+	0.510780i	$0.829357\pi$
$720$		0			0
$721$	−23.5994			−0.0327315
$722$		0			0
$723$		0			0
$724$		0			0
$725$	824.894	+	476.253i	1.13778	+	0.656900i
$726$		0			0
$727$	−605.328	−	1048.46i	−0.832639	−	1.44217i	−0.895938	−	0.444178i	$-0.853496\pi$
$727$	−605.328	−	1048.46i	−0.832639	−	1.44217i	0.0632997	−	0.997995i	$-0.479838\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	31.4889	−	18.1801i	0.0430764	−	0.0248702i
$732$		0			0
$733$	212.490	−	368.044i	0.289891	−	0.502106i	−0.683893	−	0.729583i	$-0.739714\pi$
$733$	212.490	−	368.044i	0.289891	−	0.502106i	0.973783	+	0.227477i	$0.0730476\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	817.371i		−	1.10905i
$738$		0			0
$739$	19.6676			0.0266138			0.0133069	−	0.999911i	$-0.495764\pi$
$739$	19.6676			0.0266138			0.0133069	+	0.999911i	$0.495764\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−673.114	−	388.622i	−0.905940	−	0.523045i	−0.0268177	−	0.999640i	$-0.508537\pi$
$743$	−673.114	−	388.622i	−0.905940	−	0.523045i	−0.879123	+	0.476595i	$0.841871\pi$
$744$		0			0
$745$	112.361	+	194.616i	0.150821	+	0.261229i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	29.2454	−	16.8848i	0.0390459	−	0.0225432i
$750$		0			0
$751$	−372.532	+	645.245i	−0.496048	+	0.859181i	−0.999990	−	0.00455689i	$-0.998549\pi$
$751$	−372.532	+	645.245i	−0.496048	+	0.859181i	0.503941	+	0.863738i	$0.331883\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			317.191i			0.420120i
$756$		0			0
$757$	−1278.50			−1.68890			−0.844449	−	0.535635i	$-0.820072\pi$
$757$	−1278.50			−1.68890			−0.844449	+	0.535635i	$0.820072\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	95.3728	+	55.0635i	0.125326	+	0.0723568i	0.561352	−	0.827577i	$-0.310281\pi$
$761$	95.3728	+	55.0635i	0.125326	+	0.0723568i	−0.436027	+	0.899934i	$0.643615\pi$
$762$		0			0
$763$	−21.5101	−	37.2565i	−0.0281915	−	0.0488290i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	63.1622	−	36.4667i	0.0823497	−	0.0475446i
$768$		0			0
$769$	−76.7696	+	132.969i	−0.0998304	+	0.172911i	−0.911614	−	0.411047i	$-0.865163\pi$
$769$	−76.7696	+	132.969i	−0.0998304	+	0.172911i	0.811784	+	0.583958i	$0.198497\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	1448.17i		−	1.87344i	−0.350073	−	0.936722i	$-0.613843\pi$
$773$		−	1448.17i		−	1.87344i	0.350073	−	0.936722i	$-0.386157\pi$
$774$		0			0
$775$	351.546			0.453608
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−403.249	−	232.816i	−0.517650	−	0.298865i
$780$		0			0
$781$	−132.190	−	228.960i	−0.169258	−	0.293163i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−186.454	+	107.649i	−0.237521	+	0.137133i
$786$		0			0
$787$	91.2986	−	158.134i	0.116008	−	0.200932i	−0.802174	−	0.597090i	$-0.796323\pi$
$787$	91.2986	−	158.134i	0.116008	−	0.200932i	0.918182	+	0.396158i	$0.129657\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	37.7135i		−	0.0476782i
$792$		0			0
$793$	94.0052			0.118544
$794$		0			0
$795$		0			0
$796$		0			0
$797$	622.667	+	359.497i	0.781263	+	0.451063i	0.836878	−	0.547390i	$-0.184378\pi$
$797$	622.667	+	359.497i	0.781263	+	0.451063i	−0.0556145	+	0.998452i	$0.517712\pi$
$798$		0			0
$799$	−106.534	−	184.523i	−0.133335	−	0.230943i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	183.207	−	105.774i	0.228153	−	0.131724i
$804$		0			0
$805$	8.88249	−	15.3849i	0.0110342	−	0.0191117i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			967.469i			1.19588i	0.801540	+	0.597942i	$0.204015\pi$
$809$			967.469i			1.19588i	−0.801540	+	0.597942i	$0.795985\pi$
$810$		0			0
$811$	−1014.11			−1.25044			−0.625221	−	0.780448i	$-0.714991\pi$
$811$	−1014.11			−1.25044			−0.625221	+	0.780448i	$0.714991\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	324.358	+	187.268i	0.397985	+	0.229777i
$816$		0			0
$817$	71.4185	+	123.700i	0.0874155	+	0.151408i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−71.8562	+	41.4862i	−0.0875228	+	0.0505313i	−0.543123	−	0.839653i	$-0.682758\pi$
$821$	−71.8562	+	41.4862i	−0.0875228	+	0.0505313i	0.455600	+	0.890185i	$0.349425\pi$
$822$		0			0
$823$	−491.669	+	851.595i	−0.597410	+	1.03474i	0.395792	+	0.918340i	$0.370470\pi$
$823$	−491.669	+	851.595i	−0.597410	+	1.03474i	−0.993202	+	0.116404i	$0.962863\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	352.196i		−	0.425871i	−0.977066	−	0.212936i	$-0.931697\pi$
$827$		−	352.196i		−	0.425871i	0.977066	−	0.212936i	$-0.0683025\pi$
$828$		0			0
$829$	63.3448			0.0764111			0.0382055	−	0.999270i	$-0.487836\pi$
$829$	63.3448			0.0764111			0.0382055	+	0.999270i	$0.487836\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−243.158	−	140.387i	−0.291907	−	0.168532i
$834$		0			0
$835$	154.544	+	267.678i	0.185082	+	0.320572i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−394.812	+	227.945i	−0.470575	+	0.271686i	−0.716480	−	0.697607i	$-0.754248\pi$
$839$	−394.812	+	227.945i	−0.470575	+	0.271686i	0.245906	+	0.969294i	$0.420915\pi$
$840$		0			0
$841$	421.804	−	730.586i	0.501550	−	0.868710i
$842$		0			0
$843$		0			0
$844$		0			0
$845$			224.267i			0.265405i
$846$		0			0
$847$	32.3454			0.0381882
$848$		0			0
$849$		0			0
$850$		0			0
$851$	895.406	+	516.963i	1.05218	+	0.607477i
$852$		0			0
$853$	−413.061	−	715.443i	−0.484245	−	0.838737i	0.515591	−	0.856835i	$-0.327572\pi$
$853$	−413.061	−	715.443i	−0.484245	−	0.838737i	−0.999836	+	0.0180976i	$0.994239\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	304.917	−	176.044i	0.355796	−	0.205419i	−0.311439	−	0.950266i	$-0.600811\pi$
$857$	304.917	−	176.044i	0.355796	−	0.205419i	0.667235	+	0.744847i	$0.267478\pi$
$858$		0			0
$859$	−346.426	+	600.028i	−0.403290	+	0.698519i	−0.994121	−	0.108276i	$-0.965467\pi$
$859$	−346.426	+	600.028i	−0.403290	+	0.698519i	0.590831	+	0.806796i	$0.298800\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1288.63i		−	1.49319i	−0.665277	−	0.746596i	$-0.731687\pi$
$863$		−	1288.63i		−	1.49319i	0.665277	−	0.746596i	$-0.268313\pi$
$864$		0			0
$865$	142.202			0.164396
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−989.801	−	571.462i	−1.13901	−	0.657608i
$870$		0			0
$871$	59.4237	+	102.925i	0.0682247	+	0.118169i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−36.7464	+	21.2156i	−0.0419959	+	0.0242464i
$876$		0			0
$877$	285.241	−	494.051i	0.325246	−	0.563342i	−0.656316	−	0.754486i	$-0.727886\pi$
$877$	285.241	−	494.051i	0.325246	−	0.563342i	0.981562	+	0.191144i	$0.0612196\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			42.1382i			0.0478300i	0.999714	+	0.0239150i	$0.00761310\pi$
$881$			42.1382i			0.0478300i	−0.999714	+	0.0239150i	$0.992387\pi$
$882$		0			0
$883$	99.1902			0.112333			0.0561666	−	0.998421i	$-0.482112\pi$
$883$	99.1902			0.112333			0.0561666	+	0.998421i	$0.482112\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1167.87	+	674.272i	1.31666	+	0.760172i	0.983189	−	0.182590i	$-0.0584482\pi$
$887$	1167.87	+	674.272i	1.31666	+	0.760172i	0.333467	+	0.942762i	$0.391782\pi$
$888$		0			0
$889$	4.11267	+	7.12336i	0.00462618	+	0.00801278i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	724.878	−	418.508i	0.811733	−	0.468654i
$894$		0			0
$895$	−2.77220	+	4.80159i	−0.00309743	+	0.00536490i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	621.747i		−	0.691598i
$900$		0			0
$901$	496.955			0.551559
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−216.194	−	124.820i	−0.238888	−	0.137922i
$906$		0			0
$907$	462.420	+	800.935i	0.509835	+	0.883059i	0.999935	+	0.0113935i	$0.00362674\pi$
$907$	462.420	+	800.935i	0.509835	+	0.883059i	−0.490100	+	0.871666i	$0.663040\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1134.47	−	654.987i	1.24530	−	0.718976i	0.275134	−	0.961406i	$-0.411278\pi$
$911$	1134.47	−	654.987i	1.24530	−	0.718976i	0.970169	+	0.242430i	$0.0779443\pi$
$912$		0			0
$913$	−482.348	+	835.451i	−0.528311	+	0.915062i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	76.1805i		−	0.0830757i
$918$		0			0
$919$	−591.406			−0.643532			−0.321766	−	0.946819i	$-0.604277\pi$
$919$	−591.406			−0.643532			−0.321766	+	0.946819i	$0.604277\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	33.2912	+	19.2207i	0.0360685	+	0.0208242i
$924$		0			0
$925$	−594.412	−	1029.55i	−0.642607	−	1.11303i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1063.25	+	613.870i	−1.14452	+	0.660786i	−0.947545	−	0.319623i	$-0.896444\pi$
$929$	−1063.25	+	613.870i	−1.14452	+	0.660786i	−0.196971	+	0.980409i	$0.563110\pi$
$930$		0			0
$931$	551.496	−	955.219i	0.592369	−	1.02601i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			65.5909i			0.0701507i
$936$		0			0
$937$	528.681			0.564228			0.282114	−	0.959381i	$-0.408964\pi$
$937$	528.681			0.564228			0.282114	+	0.959381i	$0.408964\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1554.27	−	897.360i	−1.65172	−	0.953624i	−0.976363	−	0.216137i	$-0.930654\pi$
$941$	−1554.27	−	897.360i	−1.65172	−	0.953624i	−0.675361	−	0.737487i	$-0.736012\pi$
$942$		0			0
$943$	−206.910	−	358.378i	−0.219416	−	0.380040i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−162.093	+	93.5844i	−0.171165	+	0.0988220i	−0.583135	−	0.812375i	$-0.698174\pi$
$947$	−162.093	+	93.5844i	−0.171165	+	0.0988220i	0.411970	+	0.911197i	$0.364841\pi$
$948$		0			0
$949$	−15.3798	+	26.6386i	−0.0162063	+	0.0280702i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1473.92i			1.54661i	0.634035	+	0.773304i	$0.281398\pi$
$953$			1473.92i			1.54661i	−0.634035	+	0.773304i	$0.718602\pi$
$954$		0			0
$955$	323.725			0.338979
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−36.7801	−	21.2350i	−0.0383526	−	0.0221429i
$960$		0			0
$961$	365.764	+	633.523i	0.380608	+	0.659233i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−102.833	+	59.3704i	−0.106562	+	0.0615237i
$966$		0			0
$967$	−526.580	+	912.063i	−0.544550	+	0.943189i	0.454085	+	0.890958i	$0.349966\pi$
$967$	−526.580	+	912.063i	−0.544550	+	0.943189i	−0.998635	+	0.0522302i	$0.983367\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			691.453i			0.712104i	0.934466	+	0.356052i	$0.115877\pi$
$971$			691.453i			0.712104i	−0.934466	+	0.356052i	$0.884123\pi$
$972$		0			0
$973$	113.336			0.116481
$974$		0			0
$975$		0			0
$976$		0			0
$977$	730.566	+	421.793i	0.747765	+	0.431722i	0.824886	−	0.565299i	$-0.191239\pi$
$977$	730.566	+	421.793i	0.747765	+	0.431722i	−0.0771208	+	0.997022i	$0.524573\pi$
$978$		0			0
$979$	572.344	+	991.329i	0.584621	+	1.01259i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	365.413	−	210.971i	0.371732	−	0.214620i	−0.302483	−	0.953155i	$-0.597815\pi$
$983$	365.413	−	210.971i	0.371732	−	0.214620i	0.674215	+	0.738535i	$0.264482\pi$
$984$		0			0
$985$	150.170	−	260.102i	0.152457	−	0.264063i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			126.943i			0.128355i
$990$		0			0
$991$	1063.60			1.07326			0.536630	−	0.843818i	$-0.319697\pi$
$991$	1063.60			1.07326			0.536630	+	0.843818i	$0.319697\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−430.973	−	248.822i	−0.433139	−	0.250073i
$996$		0			0
$997$	−276.703	−	479.264i	−0.277536	−	0.480706i	0.693236	−	0.720711i	$-0.256184\pi$
$997$	−276.703	−	479.264i	−0.277536	−	0.480706i	−0.970772	+	0.240005i	$0.922851\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.q.l.449.7		24
3.2	odd	2		576.3.q.k.257.12		24
4.3	odd	2	inner	1728.3.q.l.449.8		24
8.3	odd	2		864.3.q.b.449.6		24
8.5	even	2		864.3.q.b.449.5		24
9.2	odd	6	inner	1728.3.q.l.1601.7		24
9.7	even	3		576.3.q.k.65.12		24
12.11	even	2		576.3.q.k.257.1		24
24.5	odd	2		288.3.q.a.257.1	yes	24
24.11	even	2		288.3.q.a.257.12	yes	24
36.7	odd	6		576.3.q.k.65.1		24
36.11	even	6	inner	1728.3.q.l.1601.8		24
72.5	odd	6		2592.3.e.j.161.11		24
72.11	even	6		864.3.q.b.737.6		24
72.13	even	6		2592.3.e.j.161.12		24
72.29	odd	6		864.3.q.b.737.5		24
72.43	odd	6		288.3.q.a.65.12	yes	24
72.59	even	6		2592.3.e.j.161.13		24
72.61	even	6		288.3.q.a.65.1	✓	24
72.67	odd	6		2592.3.e.j.161.14		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.3.q.a.65.1	✓	24	72.61	even	6
288.3.q.a.65.12	yes	24	72.43	odd	6
288.3.q.a.257.1	yes	24	24.5	odd	2
288.3.q.a.257.12	yes	24	24.11	even	2
576.3.q.k.65.1		24	36.7	odd	6
576.3.q.k.65.12		24	9.7	even	3
576.3.q.k.257.1		24	12.11	even	2
576.3.q.k.257.12		24	3.2	odd	2
864.3.q.b.449.5		24	8.5	even	2
864.3.q.b.449.6		24	8.3	odd	2
864.3.q.b.737.5		24	72.29	odd	6
864.3.q.b.737.6		24	72.11	even	6
1728.3.q.l.449.7		24	1.1	even	1	trivial
1728.3.q.l.449.8		24	4.3	odd	2	inner
1728.3.q.l.1601.7		24	9.2	odd	6	inner
1728.3.q.l.1601.8		24	36.11	even	6	inner
2592.3.e.j.161.11		24	72.5	odd	6
2592.3.e.j.161.12		24	72.13	even	6
2592.3.e.j.161.13		24	72.59	even	6
2592.3.e.j.161.14		24	72.67	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.q (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.15965 + 0.669525i) q^{5} +(-0.328661 - 0.569258i) q^{7} +O(q^{10})\)	\(q+(1.15965 + 0.669525i) q^{5} +(-0.328661 - 0.569258i) q^{7} +(-7.33786 + 4.23651i) q^{11} +(0.615997 - 1.06694i) q^{13} -5.78107i q^{17} +22.7103 q^{19} +(17.4792 + 10.0916i) q^{23} +(-11.6035 - 20.0978i) q^{25} +(-35.5451 + 20.5220i) q^{29} +(-7.57415 + 13.1188i) q^{31} -0.880187i q^{35} +51.2271 q^{37} +(-17.7563 - 10.2516i) q^{41} +(3.14476 + 5.44689i) q^{43} +(31.9185 - 18.4281i) q^{47} +(24.2840 - 42.0611i) q^{49} +85.9624i q^{53} -11.3458 q^{55} +(51.2682 + 29.5997i) q^{59} +(38.1516 + 66.0806i) q^{61} +(1.42868 - 0.824851i) q^{65} +(-48.2337 + 83.5433i) q^{67} +31.2026i q^{71} -24.9673 q^{73} +(4.82334 + 2.78475i) q^{77} +(67.4448 + 116.818i) q^{79} +(98.6013 - 56.9275i) q^{83} +(3.87057 - 6.70403i) q^{85} -135.098i q^{89} -0.809818 q^{91} +(26.3360 + 15.2051i) q^{95} +(-68.8218 - 119.203i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 60 q^{25} + 72 q^{29} + 36 q^{41} - 132 q^{49} - 96 q^{61} - 576 q^{65} + 24 q^{73} + 432 q^{77} + 96 q^{85} + 252 q^{97}+O(q^{100}) \) 24 * q + 60 * q^25 + 72 * q^29 + 36 * q^41 - 132 * q^49 - 96 * q^61 - 576 * q^65 + 24 * q^73 + 432 * q^77 + 96 * q^85 + 252 * q^97

Introduction

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