LMFDB - Embedded newform 600.3.u.g.457.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,3,Mod(193,600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(600, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 0, 3]))

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

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

chi := DirichletCharacter("600.193");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	600.u (of order $4$, 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:	$16.3488158616$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			457.2
Root			$1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	600.457
Dual form			600.3.u.g.193.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 - 1.22474i) q^{3} +(7.67423 + 7.67423i) q^{7} -3.00000i q^{9} +O(q^{10})$	$q+(1.22474 - 1.22474i) q^{3} +(7.67423 + 7.67423i) q^{7} -3.00000i q^{9} +7.79796 q^{11} +(-3.67423 + 3.67423i) q^{13} +(-3.34847 - 3.34847i) q^{17} +28.3939i q^{19} +18.7980 q^{21} +(14.4495 - 14.4495i) q^{23} +(-3.67423 - 3.67423i) q^{27} +43.3939i q^{29} -40.7980 q^{31} +(9.55051 - 9.55051i) q^{33} +(12.8990 + 12.8990i) q^{37} +9.00000i q^{39} +37.7980 q^{41} +(36.5732 - 36.5732i) q^{43} +(52.2474 + 52.2474i) q^{47} +68.7878i q^{49} -8.20204 q^{51} +(47.1010 - 47.1010i) q^{53} +(34.7753 + 34.7753i) q^{57} -82.0000i q^{59} +55.4041 q^{61} +(23.0227 - 23.0227i) q^{63} +(-59.2702 - 59.2702i) q^{67} -35.3939i q^{69} -58.2020 q^{71} +(21.7071 - 21.7071i) q^{73} +(59.8434 + 59.8434i) q^{77} +91.9796i q^{79} -9.00000 q^{81} +(57.1464 - 57.1464i) q^{83} +(53.1464 + 53.1464i) q^{87} +120.404i q^{89} -56.3939 q^{91} +(-49.9671 + 49.9671i) q^{93} +(-103.270 - 103.270i) q^{97} -23.3939i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 16 q^{7}+O(q^{10}) $ 4 * q + 16 * q^7	$ 4 q + 16 q^{7} - 8 q^{11} + 16 q^{17} + 36 q^{21} + 48 q^{23} - 124 q^{31} + 48 q^{33} + 32 q^{37} + 112 q^{41} + 112 q^{43} + 160 q^{47} - 72 q^{51} + 208 q^{53} + 144 q^{57} + 300 q^{61} + 48 q^{63} - 144 q^{67} - 272 q^{71} + 224 q^{73} + 112 q^{77} - 36 q^{81} + 160 q^{83} + 144 q^{87} - 108 q^{91} - 48 q^{93} - 320 q^{97}+O(q^{100}) $ 4 * q + 16 * q^7 - 8 * q^11 + 16 * q^17 + 36 * q^21 + 48 * q^23 - 124 * q^31 + 48 * q^33 + 32 * q^37 + 112 * q^41 + 112 * q^43 + 160 * q^47 - 72 * q^51 + 208 * q^53 + 144 * q^57 + 300 * q^61 + 48 * q^63 - 144 * q^67 - 272 * q^71 + 224 * q^73 + 112 * q^77 - 36 * q^81 + 160 * q^83 + 144 * q^87 - 108 * q^91 - 48 * q^93 - 320 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$.

$n$	$151$	$301$	$401$	$577$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{1}{4}\right)$

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
$3$	1.22474	−	1.22474i	0.408248	−	0.408248i
$3$	1.22474	−	1.22474i	0.408248	−	0.408248i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	7.67423	+	7.67423i	1.09632	+	1.09632i	0.994837	+	0.101482i	$0.0323584\pi$
$7$	7.67423	+	7.67423i	1.09632	+	1.09632i	0.101482	+	0.994837i	$0.467642\pi$
$8$		0			0
$9$		−	3.00000i		−	0.333333i
$10$		0			0
$11$	7.79796			0.708905			0.354453	−	0.935074i	$-0.384667\pi$
$11$	7.79796			0.708905			0.354453	+	0.935074i	$0.384667\pi$
$12$		0			0
$13$	−3.67423	+	3.67423i	−0.282633	+	0.282633i	−0.834158	−	0.551525i	$-0.814046\pi$
$13$	−3.67423	+	3.67423i	−0.282633	+	0.282633i	0.551525	+	0.834158i	$0.314046\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.34847	−	3.34847i	−0.196969	−	0.196969i	0.601730	−	0.798699i	$-0.294478\pi$
$17$	−3.34847	−	3.34847i	−0.196969	−	0.196969i	−0.798699	+	0.601730i	$0.794478\pi$
$18$		0			0
$19$			28.3939i			1.49441i	0.664591	+	0.747207i	$0.268606\pi$
$19$			28.3939i			1.49441i	−0.664591	+	0.747207i	$0.731394\pi$
$20$		0			0
$21$	18.7980			0.895141
$22$		0			0
$23$	14.4495	−	14.4495i	0.628239	−	0.628239i	−0.319386	−	0.947625i	$-0.603477\pi$
$23$	14.4495	−	14.4495i	0.628239	−	0.628239i	0.947625	+	0.319386i	$0.103477\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−3.67423	−	3.67423i	−0.136083	−	0.136083i
$28$		0			0
$29$			43.3939i			1.49634i	0.663507	+	0.748170i	$0.269067\pi$
$29$			43.3939i			1.49634i	−0.663507	+	0.748170i	$0.730933\pi$
$30$		0			0
$31$	−40.7980			−1.31606			−0.658032	−	0.752990i	$-0.728611\pi$
$31$	−40.7980			−1.31606			−0.658032	+	0.752990i	$0.728611\pi$
$32$		0			0
$33$	9.55051	−	9.55051i	0.289409	−	0.289409i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	12.8990	+	12.8990i	0.348621	+	0.348621i	0.859596	−	0.510975i	$-0.170715\pi$
$37$	12.8990	+	12.8990i	0.348621	+	0.348621i	−0.510975	+	0.859596i	$0.670715\pi$
$38$		0			0
$39$			9.00000i			0.230769i
$40$		0			0
$41$	37.7980			0.921901			0.460951	−	0.887426i	$-0.347508\pi$
$41$	37.7980			0.921901			0.460951	+	0.887426i	$0.347508\pi$
$42$		0			0
$43$	36.5732	−	36.5732i	0.850540	−	0.850540i	−0.139660	−	0.990200i	$-0.544601\pi$
$43$	36.5732	−	36.5732i	0.850540	−	0.850540i	0.990200	+	0.139660i	$0.0446009\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	52.2474	+	52.2474i	1.11165	+	1.11165i	0.992928	+	0.118720i	$0.0378791\pi$
$47$	52.2474	+	52.2474i	1.11165	+	1.11165i	0.118720	+	0.992928i	$0.462121\pi$
$48$		0			0
$49$			68.7878i			1.40383i
$50$		0			0
$51$	−8.20204			−0.160824
$52$		0			0
$53$	47.1010	−	47.1010i	0.888699	−	0.888699i	−0.105700	−	0.994398i	$-0.533708\pi$
$53$	47.1010	−	47.1010i	0.888699	−	0.888699i	0.994398	+	0.105700i	$0.0337082\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	34.7753	+	34.7753i	0.610092	+	0.610092i
$58$		0			0
$59$		−	82.0000i		−	1.38983i	−0.719092	−	0.694915i	$-0.755442\pi$
$59$		−	82.0000i		−	1.38983i	0.719092	−	0.694915i	$-0.244558\pi$
$60$		0			0
$61$	55.4041			0.908264			0.454132	−	0.890935i	$-0.349950\pi$
$61$	55.4041			0.908264			0.454132	+	0.890935i	$0.349950\pi$
$62$		0			0
$63$	23.0227	−	23.0227i	0.365440	−	0.365440i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−59.2702	−	59.2702i	−0.884629	−	0.884629i	0.109372	−	0.994001i	$-0.465116\pi$
$67$	−59.2702	−	59.2702i	−0.884629	−	0.884629i	−0.994001	+	0.109372i	$0.965116\pi$
$68$		0			0
$69$		−	35.3939i		−	0.512955i
$70$		0			0
$71$	−58.2020			−0.819747			−0.409874	−	0.912142i	$-0.634427\pi$
$71$	−58.2020			−0.819747			−0.409874	+	0.912142i	$0.634427\pi$
$72$		0			0
$73$	21.7071	−	21.7071i	0.297358	−	0.297358i	−0.542620	−	0.839978i	$-0.682568\pi$
$73$	21.7071	−	21.7071i	0.297358	−	0.297358i	0.839978	+	0.542620i	$0.182568\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	59.8434	+	59.8434i	0.777187	+	0.777187i
$78$		0			0
$79$			91.9796i			1.16430i	0.813082	+	0.582149i	$0.197788\pi$
$79$			91.9796i			1.16430i	−0.813082	+	0.582149i	$0.802212\pi$
$80$		0			0
$81$	−9.00000			−0.111111
$82$		0			0
$83$	57.1464	−	57.1464i	0.688511	−	0.688511i	−0.273392	−	0.961903i	$-0.588145\pi$
$83$	57.1464	−	57.1464i	0.688511	−	0.688511i	0.961903	+	0.273392i	$0.0881455\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	53.1464	+	53.1464i	0.610878	+	0.610878i
$88$		0			0
$89$			120.404i			1.35285i	0.736509	+	0.676427i	$0.236473\pi$
$89$			120.404i			1.35285i	−0.736509	+	0.676427i	$0.763527\pi$
$90$		0			0
$91$	−56.3939			−0.619713
$92$		0			0
$93$	−49.9671	+	49.9671i	−0.537281	+	0.537281i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−103.270	−	103.270i	−1.06464	−	1.06464i	−0.997761	−	0.0668797i	$-0.978696\pi$
$97$	−103.270	−	103.270i	−1.06464	−	1.06464i	−0.0668797	−	0.997761i	$-0.521304\pi$
$98$		0			0
$99$		−	23.3939i		−	0.236302i
$100$		0			0
$101$	65.7980			0.651465			0.325732	−	0.945462i	$-0.394389\pi$
$101$	65.7980			0.651465			0.325732	+	0.945462i	$0.394389\pi$
$102$		0			0
$103$	93.5755	−	93.5755i	0.908500	−	0.908500i	−0.0876512	−	0.996151i	$-0.527936\pi$
$103$	93.5755	−	93.5755i	0.908500	−	0.908500i	0.996151	+	0.0876512i	$0.0279361\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−42.3837	−	42.3837i	−0.396109	−	0.396109i	0.480749	−	0.876858i	$-0.340365\pi$
$107$	−42.3837	−	42.3837i	−0.396109	−	0.396109i	−0.876858	+	0.480749i	$0.840365\pi$
$108$		0			0
$109$			114.171i			1.04744i	0.851889	+	0.523722i	$0.175457\pi$
$109$			114.171i			1.04744i	−0.851889	+	0.523722i	$0.824543\pi$
$110$		0			0
$111$	31.5959			0.284648
$112$		0			0
$113$	−138.788	+	138.788i	−1.22821	+	1.22821i	−0.263570	+	0.964640i	$0.584900\pi$
$113$	−138.788	+	138.788i	−1.22821	+	1.22821i	−0.964640	+	0.263570i	$0.915100\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	11.0227	+	11.0227i	0.0942111	+	0.0942111i
$118$		0			0
$119$		−	51.3939i		−	0.431881i
$120$		0			0
$121$	−60.1918			−0.497453
$122$		0			0
$123$	46.2929	−	46.2929i	0.376365	−	0.376365i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−63.5051	−	63.5051i	−0.500040	−	0.500040i	0.411410	−	0.911450i	$-0.365036\pi$
$127$	−63.5051	−	63.5051i	−0.500040	−	0.500040i	−0.911450	+	0.411410i	$0.865036\pi$
$128$		0			0
$129$		−	89.5857i		−	0.694463i
$130$		0			0
$131$	−7.41429			−0.0565976			−0.0282988	−	0.999600i	$-0.509009\pi$
$131$	−7.41429			−0.0565976			−0.0282988	+	0.999600i	$0.509009\pi$
$132$		0			0
$133$	−217.901	+	217.901i	−1.63836	+	1.63836i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	128.338	+	128.338i	0.936776	+	0.936776i	0.998117	−	0.0613412i	$-0.0195378\pi$
$137$	128.338	+	128.338i	0.936776	+	0.936776i	−0.0613412	+	0.998117i	$0.519538\pi$
$138$		0			0
$139$		−	18.8082i		−	0.135311i	−0.997709	−	0.0676553i	$-0.978448\pi$
$139$		−	18.8082i		−	0.135311i	0.997709	−	0.0676553i	$-0.0215518\pi$
$140$		0			0
$141$	127.980			0.907657
$142$		0			0
$143$	−28.6515	+	28.6515i	−0.200360	+	0.200360i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	84.2474	+	84.2474i	0.573112	+	0.573112i
$148$		0			0
$149$		−	293.151i		−	1.96746i	−0.179663	−	0.983728i	$-0.557501\pi$
$149$		−	293.151i		−	1.96746i	0.179663	−	0.983728i	$-0.442499\pi$
$150$		0			0
$151$	−102.778			−0.680646			−0.340323	−	0.940309i	$-0.610536\pi$
$151$	−102.778			−0.680646			−0.340323	+	0.940309i	$0.610536\pi$
$152$		0			0
$153$	−10.0454	+	10.0454i	−0.0656563	+	0.0656563i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−23.0885	−	23.0885i	−0.147061	−	0.147061i	0.629743	−	0.776804i	$-0.283160\pi$
$157$	−23.0885	−	23.0885i	−0.147061	−	0.147061i	−0.776804	+	0.629743i	$0.783160\pi$
$158$		0			0
$159$		−	115.373i		−	0.725619i
$160$		0			0
$161$	221.778			1.37750
$162$		0			0
$163$	−139.427	+	139.427i	−0.855379	+	0.855379i	−0.990790	−	0.135411i	$-0.956765\pi$
$163$	−139.427	+	139.427i	−0.855379	+	0.855379i	0.135411	+	0.990790i	$0.456765\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−180.182	−	180.182i	−1.07893	−	1.07893i	−0.996605	−	0.0823265i	$-0.973765\pi$
$167$	−180.182	−	180.182i	−1.07893	−	1.07893i	−0.0823265	−	0.996605i	$-0.526235\pi$
$168$		0			0
$169$			142.000i			0.840237i
$170$		0			0
$171$	85.1816			0.498138
$172$		0			0
$173$	181.015	−	181.015i	1.04633	−	1.04633i	0.0474549	−	0.998873i	$-0.484889\pi$
$173$	181.015	−	181.015i	1.04633	−	1.04633i	0.998873	−	0.0474549i	$-0.0151110\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−100.429	−	100.429i	−0.567396	−	0.567396i
$178$		0			0
$179$		−	326.767i		−	1.82552i	−0.408501	−	0.912758i	$-0.633948\pi$
$179$		−	326.767i		−	1.82552i	0.408501	−	0.912758i	$-0.366052\pi$
$180$		0			0
$181$	−154.959			−0.856128			−0.428064	−	0.903748i	$-0.640804\pi$
$181$	−154.959			−0.856128			−0.428064	+	0.903748i	$0.640804\pi$
$182$		0			0
$183$	67.8559	−	67.8559i	0.370797	−	0.370797i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−26.1112	−	26.1112i	−0.139632	−	0.139632i
$188$		0			0
$189$		−	56.3939i		−	0.298380i
$190$		0			0
$191$	−43.7980			−0.229309			−0.114654	−	0.993405i	$-0.536576\pi$
$191$	−43.7980			−0.229309			−0.114654	+	0.993405i	$0.536576\pi$
$192$		0			0
$193$	−208.260	+	208.260i	−1.07907	+	1.07907i	−0.0824739	+	0.996593i	$0.526282\pi$
$193$	−208.260	+	208.260i	−1.07907	+	1.07907i	−0.996593	+	0.0824739i	$0.973718\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	101.914	+	101.914i	0.517329	+	0.517329i	0.916762	−	0.399433i	$-0.130793\pi$
$197$	101.914	+	101.914i	0.517329	+	0.517329i	−0.399433	+	0.916762i	$0.630793\pi$
$198$		0			0
$199$			83.2020i			0.418101i	0.977905	+	0.209050i	$0.0670373\pi$
$199$			83.2020i			0.418101i	−0.977905	+	0.209050i	$0.932963\pi$
$200$		0			0
$201$	−145.182			−0.722297
$202$		0			0
$203$	−333.015	+	333.015i	−1.64047	+	1.64047i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−43.3485	−	43.3485i	−0.209413	−	0.209413i
$208$		0			0
$209$			221.414i			1.05940i
$210$		0			0
$211$	−339.565			−1.60931			−0.804657	−	0.593740i	$-0.797651\pi$
$211$	−339.565			−1.60931			−0.804657	+	0.593740i	$0.797651\pi$
$212$		0			0
$213$	−71.2827	+	71.2827i	−0.334660	+	0.334660i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−313.093	−	313.093i	−1.44283	−	1.44283i
$218$		0			0
$219$		−	53.1714i		−	0.242792i
$220$		0			0
$221$	24.6061			0.111340
$222$		0			0
$223$	57.3156	−	57.3156i	0.257020	−	0.257020i	−0.566821	−	0.823841i	$-0.691827\pi$
$223$	57.3156	−	57.3156i	0.257020	−	0.257020i	0.823841	+	0.566821i	$0.191827\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−64.5857	−	64.5857i	−0.284519	−	0.284519i	0.550389	−	0.834908i	$-0.314479\pi$
$227$	−64.5857	−	64.5857i	−0.284519	−	0.284519i	−0.834908	+	0.550389i	$0.814479\pi$
$228$		0			0
$229$		−	178.212i		−	0.778219i	−0.921191	−	0.389110i	$-0.872783\pi$
$229$		−	178.212i		−	0.778219i	0.921191	−	0.389110i	$-0.127217\pi$
$230$		0			0
$231$	146.586			0.634570
$232$		0			0
$233$	127.283	−	127.283i	0.546277	−	0.546277i	−0.379085	−	0.925362i	$-0.623761\pi$
$233$	127.283	−	127.283i	0.546277	−	0.546277i	0.925362	+	0.379085i	$0.123761\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	112.652	+	112.652i	0.475323	+	0.475323i
$238$		0			0
$239$		−	321.757i		−	1.34626i	−0.739522	−	0.673132i	$-0.764949\pi$
$239$		−	321.757i		−	1.34626i	0.739522	−	0.673132i	$-0.235051\pi$
$240$		0			0
$241$	−317.767			−1.31854			−0.659268	−	0.751908i	$-0.729134\pi$
$241$	−317.767			−1.31854			−0.659268	+	0.751908i	$0.729134\pi$
$242$		0			0
$243$	−11.0227	+	11.0227i	−0.0453609	+	0.0453609i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−104.326	−	104.326i	−0.422372	−	0.422372i
$248$		0			0
$249$		−	139.980i		−	0.562167i
$250$		0			0
$251$	−171.151			−0.681877			−0.340938	−	0.940086i	$-0.610745\pi$
$251$	−171.151			−0.681877			−0.340938	+	0.940086i	$0.610745\pi$
$252$		0			0
$253$	112.677	−	112.677i	0.445362	−	0.445362i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−133.262	−	133.262i	−0.518530	−	0.518530i	0.398596	−	0.917126i	$-0.369497\pi$
$257$	−133.262	−	133.262i	−0.518530	−	0.518530i	−0.917126	+	0.398596i	$0.869497\pi$
$258$		0			0
$259$			197.980i			0.764400i
$260$		0			0
$261$	130.182			0.498780
$262$		0			0
$263$	−135.464	+	135.464i	−0.515073	+	0.515073i	−0.916077	−	0.401003i	$-0.868662\pi$
$263$	−135.464	+	135.464i	−0.515073	+	0.515073i	0.401003	+	0.916077i	$0.368662\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	147.464	+	147.464i	0.552301	+	0.552301i
$268$		0			0
$269$		−	272.161i		−	1.01175i	−0.862606	−	0.505876i	$-0.831169\pi$
$269$		−	272.161i		−	1.01175i	0.862606	−	0.505876i	$-0.168831\pi$
$270$		0			0
$271$	403.576			1.48921			0.744604	−	0.667506i	$-0.232638\pi$
$271$	403.576			1.48921			0.744604	+	0.667506i	$0.232638\pi$
$272$		0			0
$273$	−69.0681	+	69.0681i	−0.252997	+	0.252997i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−22.4370	−	22.4370i	−0.0810000	−	0.0810000i	0.665446	−	0.746446i	$-0.268241\pi$
$277$	−22.4370	−	22.4370i	−0.0810000	−	0.0810000i	−0.746446	+	0.665446i	$0.768241\pi$
$278$		0			0
$279$			122.394i			0.438688i
$280$		0			0
$281$	−117.414			−0.417844			−0.208922	−	0.977932i	$-0.566996\pi$
$281$	−117.414			−0.417844			−0.208922	+	0.977932i	$0.566996\pi$
$282$		0			0
$283$	94.5278	−	94.5278i	0.334021	−	0.334021i	−0.520091	−	0.854111i	$-0.674102\pi$
$283$	94.5278	−	94.5278i	0.334021	−	0.334021i	0.854111	+	0.520091i	$0.174102\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	290.070	+	290.070i	1.01070	+	1.01070i
$288$		0			0
$289$		−	266.576i		−	0.922407i
$290$		0			0
$291$	−252.959			−0.869276
$292$		0			0
$293$	33.5097	−	33.5097i	0.114368	−	0.114368i	−0.647607	−	0.761975i	$-0.724230\pi$
$293$	33.5097	−	33.5097i	0.114368	−	0.114368i	0.761975	+	0.647607i	$0.224230\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−28.6515	−	28.6515i	−0.0964698	−	0.0964698i
$298$		0			0
$299$			106.182i			0.355123i
$300$		0			0
$301$	561.343			1.86493
$302$		0			0
$303$	80.5857	−	80.5857i	0.265959	−	0.265959i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	134.396	+	134.396i	0.437773	+	0.437773i	0.891262	−	0.453489i	$-0.149821\pi$
$307$	134.396	+	134.396i	0.437773	+	0.437773i	−0.453489	+	0.891262i	$0.649821\pi$
$308$		0			0
$309$		−	229.212i		−	0.741787i
$310$		0			0
$311$	147.535			0.474388			0.237194	−	0.971462i	$-0.423772\pi$
$311$	147.535			0.474388			0.237194	+	0.971462i	$0.423772\pi$
$312$		0			0
$313$	90.3712	−	90.3712i	0.288726	−	0.288726i	−0.547850	−	0.836576i	$-0.684554\pi$
$313$	90.3712	−	90.3712i	0.288726	−	0.288726i	0.836576	+	0.547850i	$0.184554\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	206.070	+	206.070i	0.650064	+	0.650064i	0.953008	−	0.302944i	$-0.0979695\pi$
$317$	206.070	+	206.070i	0.650064	+	0.650064i	−0.302944	+	0.953008i	$0.597970\pi$
$318$		0			0
$319$			338.384i			1.06076i
$320$		0			0
$321$	−103.818			−0.323422
$322$		0			0
$323$	95.0760	−	95.0760i	0.294353	−	0.294353i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	139.831	+	139.831i	0.427617	+	0.427617i
$328$		0			0
$329$			801.918i			2.43744i
$330$		0			0
$331$	355.576			1.07425			0.537123	−	0.843504i	$-0.319511\pi$
$331$	355.576			1.07425			0.537123	+	0.843504i	$0.319511\pi$
$332$		0			0
$333$	38.6969	−	38.6969i	0.116207	−	0.116207i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−132.062	−	132.062i	−0.391877	−	0.391877i	0.483479	−	0.875356i	$-0.339373\pi$
$337$	−132.062	−	132.062i	−0.391877	−	0.391877i	−0.875356	+	0.483479i	$0.839373\pi$
$338$		0			0
$339$			339.959i			1.00283i
$340$		0			0
$341$	−318.141			−0.932964
$342$		0			0
$343$	−151.856	+	151.856i	−0.442728	+	0.442728i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−114.318	−	114.318i	−0.329446	−	0.329446i	0.522930	−	0.852376i	$-0.324839\pi$
$347$	−114.318	−	114.318i	−0.329446	−	0.329446i	−0.852376	+	0.522930i	$0.824839\pi$
$348$		0			0
$349$		−	364.424i		−	1.04420i	−0.852885	−	0.522098i	$-0.825149\pi$
$349$		−	364.424i		−	1.04420i	0.852885	−	0.522098i	$-0.174851\pi$
$350$		0			0
$351$	27.0000			0.0769231
$352$		0			0
$353$	250.409	−	250.409i	0.709373	−	0.709373i	−0.257030	−	0.966403i	$-0.582744\pi$
$353$	250.409	−	250.409i	0.709373	−	0.709373i	0.966403	+	0.257030i	$0.0827440\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−62.9444	−	62.9444i	−0.176315	−	0.176315i
$358$		0			0
$359$			36.9490i			0.102922i	0.998675	+	0.0514610i	$0.0163878\pi$
$359$			36.9490i			0.102922i	−0.998675	+	0.0514610i	$0.983612\pi$
$360$		0			0
$361$	−445.212			−1.23327
$362$		0			0
$363$	−73.7196	+	73.7196i	−0.203084	+	0.203084i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−5.90127	−	5.90127i	−0.0160798	−	0.0160798i	0.699021	−	0.715101i	$-0.253619\pi$
$367$	−5.90127	−	5.90127i	−0.0160798	−	0.0160798i	−0.715101	+	0.699021i	$0.753619\pi$
$368$		0			0
$369$		−	113.394i		−	0.307300i
$370$		0			0
$371$	722.929			1.94859
$372$		0			0
$373$	172.482	−	172.482i	0.462419	−	0.462419i	−0.437028	−	0.899448i	$-0.643969\pi$
$373$	172.482	−	172.482i	0.462419	−	0.462419i	0.899448	+	0.437028i	$0.143969\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−159.439	−	159.439i	−0.422916	−	0.422916i
$378$		0			0
$379$			537.161i			1.41731i	0.705554	+	0.708656i	$0.250698\pi$
$379$			537.161i			1.41731i	−0.705554	+	0.708656i	$0.749302\pi$
$380$		0			0
$381$	−155.555			−0.408281
$382$		0			0
$383$	−407.110	+	407.110i	−1.06295	+	1.06295i	−0.0650702	+	0.997881i	$0.520727\pi$
$383$	−407.110	+	407.110i	−1.06295	+	1.06295i	−0.997881	+	0.0650702i	$0.979273\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−109.720	−	109.720i	−0.283513	−	0.283513i
$388$		0			0
$389$			379.696i			0.976082i	0.872821	+	0.488041i	$0.162288\pi$
$389$			379.696i			0.976082i	−0.872821	+	0.488041i	$0.837712\pi$
$390$		0			0
$391$	−96.7673			−0.247487
$392$		0			0
$393$	−9.08061	+	9.08061i	−0.0231059	+	0.0231059i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	371.048	+	371.048i	0.934629	+	0.934629i	0.997991	−	0.0633617i	$-0.0201822\pi$
$397$	371.048	+	371.048i	0.934629	+	0.934629i	−0.0633617	+	0.997991i	$0.520182\pi$
$398$		0			0
$399$			533.747i			1.33771i
$400$		0			0
$401$	−235.818			−0.588076			−0.294038	−	0.955794i	$-0.594999\pi$
$401$	−235.818			−0.588076			−0.294038	+	0.955794i	$0.594999\pi$
$402$		0			0
$403$	149.901	−	149.901i	0.371963	−	0.371963i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	100.586	+	100.586i	0.247139	+	0.247139i
$408$		0			0
$409$			672.131i			1.64335i	0.569956	+	0.821676i	$0.306960\pi$
$409$			672.131i			1.64335i	−0.569956	+	0.821676i	$0.693040\pi$
$410$		0			0
$411$	314.363			0.764874
$412$		0			0
$413$	629.287	−	629.287i	1.52370	−	1.52370i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−23.0352	−	23.0352i	−0.0552403	−	0.0552403i
$418$		0			0
$419$		−	106.243i		−	0.253563i	−0.991931	−	0.126781i	$-0.959535\pi$
$419$		−	106.243i		−	0.253563i	0.991931	−	0.126781i	$-0.0404647\pi$
$420$		0			0
$421$	−25.6776			−0.0609918			−0.0304959	−	0.999535i	$-0.509709\pi$
$421$	−25.6776			−0.0609918			−0.0304959	+	0.999535i	$0.509709\pi$
$422$		0			0
$423$	156.742	−	156.742i	0.370549	−	0.370549i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	425.184	+	425.184i	0.995747	+	0.995747i
$428$		0			0
$429$			70.1816i			0.163594i
$430$		0			0
$431$	−246.504			−0.571935			−0.285968	−	0.958239i	$-0.592315\pi$
$431$	−246.504			−0.571935			−0.285968	+	0.958239i	$0.592315\pi$
$432$		0			0
$433$	315.204	−	315.204i	0.727955	−	0.727955i	−0.242257	−	0.970212i	$-0.577888\pi$
$433$	315.204	−	315.204i	0.727955	−	0.727955i	0.970212	+	0.242257i	$0.0778879\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	410.277	+	410.277i	0.938849	+	0.938849i
$438$		0			0
$439$		−	435.141i		−	0.991209i	−0.868548	−	0.495605i	$-0.834947\pi$
$439$		−	435.141i		−	0.991209i	0.868548	−	0.495605i	$-0.165053\pi$
$440$		0			0
$441$	206.363			0.467944
$442$		0			0
$443$	70.6969	−	70.6969i	0.159587	−	0.159587i	−0.622797	−	0.782384i	$-0.714004\pi$
$443$	70.6969	−	70.6969i	0.159587	−	0.159587i	0.782384	+	0.622797i	$0.214004\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−359.035	−	359.035i	−0.803211	−	0.803211i
$448$		0			0
$449$			111.233i			0.247734i	0.992299	+	0.123867i	$0.0395296\pi$
$449$			111.233i			0.247734i	−0.992299	+	0.123867i	$0.960470\pi$
$450$		0			0
$451$	294.747			0.653541
$452$		0			0
$453$	−125.876	+	125.876i	−0.277873	+	0.277873i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	534.202	+	534.202i	1.16893	+	1.16893i	0.982460	+	0.186472i	$0.0597054\pi$
$457$	534.202	+	534.202i	1.16893	+	1.16893i	0.186472	+	0.982460i	$0.440295\pi$
$458$		0			0
$459$			24.6061i			0.0536081i
$460$		0			0
$461$	461.353			1.00077			0.500383	−	0.865804i	$-0.333193\pi$
$461$	461.353			1.00077			0.500383	+	0.865804i	$0.333193\pi$
$462$		0			0
$463$	92.5357	−	92.5357i	0.199861	−	0.199861i	−0.600079	−	0.799940i	$-0.704864\pi$
$463$	92.5357	−	92.5357i	0.199861	−	0.199861i	0.799940	+	0.600079i	$0.204864\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−384.974	−	384.974i	−0.824355	−	0.824355i	0.162374	−	0.986729i	$-0.448085\pi$
$467$	−384.974	−	384.974i	−0.824355	−	0.824355i	−0.986729	+	0.162374i	$0.948085\pi$
$468$		0			0
$469$		−	909.706i		−	1.93967i
$470$		0			0
$471$	−56.5551			−0.120075
$472$		0			0
$473$	285.196	−	285.196i	0.602952	−	0.602952i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−141.303	−	141.303i	−0.296233	−	0.296233i
$478$		0			0
$479$			81.5551i			0.170261i	0.996370	+	0.0851306i	$0.0271307\pi$
$479$			81.5551i			0.170261i	−0.996370	+	0.0851306i	$0.972869\pi$
$480$		0			0
$481$	−94.7878			−0.197064
$482$		0			0
$483$	271.621	−	271.621i	0.562362	−	0.562362i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	446.215	+	446.215i	0.916252	+	0.916252i	0.996754	−	0.0805028i	$-0.0256526\pi$
$487$	446.215	+	446.215i	0.916252	+	0.916252i	−0.0805028	+	0.996754i	$0.525653\pi$
$488$		0			0
$489$			341.524i			0.698414i
$490$		0			0
$491$	532.808			1.08515			0.542575	−	0.840008i	$-0.317450\pi$
$491$	532.808			1.08515			0.542575	+	0.840008i	$0.317450\pi$
$492$		0			0
$493$	145.303	−	145.303i	0.294732	−	0.294732i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−446.656	−	446.656i	−0.898704	−	0.898704i
$498$		0			0
$499$		−	322.292i		−	0.645875i	−0.946420	−	0.322938i	$-0.895330\pi$
$499$		−	322.292i		−	0.645875i	0.946420	−	0.322938i	$-0.104670\pi$
$500$		0			0
$501$	−441.353			−0.880944
$502$		0			0
$503$	−143.333	+	143.333i	−0.284956	+	0.284956i	−0.835082	−	0.550126i	$-0.814580\pi$
$503$	−143.333	+	143.333i	−0.284956	+	0.284956i	0.550126	+	0.835082i	$0.314580\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	173.914	+	173.914i	0.343025	+	0.343025i
$508$		0			0
$509$			247.090i			0.485442i	0.970096	+	0.242721i	$0.0780399\pi$
$509$			247.090i			0.485442i	−0.970096	+	0.242721i	$0.921960\pi$
$510$		0			0
$511$	333.171			0.651999
$512$		0			0
$513$	104.326	−	104.326i	0.203364	−	0.203364i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	407.423	+	407.423i	0.788053	+	0.788053i
$518$		0			0
$519$		−	443.394i		−	0.854323i
$520$		0			0
$521$	−1012.12			−1.94265			−0.971325	−	0.237757i	$-0.923588\pi$
$521$	−1012.12			−1.94265			−0.971325	+	0.237757i	$0.923588\pi$
$522$		0			0
$523$	25.4222	−	25.4222i	0.0486084	−	0.0486084i	−0.682385	−	0.730993i	$-0.739057\pi$
$523$	25.4222	−	25.4222i	0.0486084	−	0.0486084i	0.730993	+	0.682385i	$0.239057\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	136.611	+	136.611i	0.259223	+	0.259223i
$528$		0			0
$529$			111.424i			0.210632i
$530$		0			0
$531$	−246.000			−0.463277
$532$		0			0
$533$	−138.879	+	138.879i	−0.260560	+	0.260560i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−400.207	−	400.207i	−0.745264	−	0.745264i
$538$		0			0
$539$			536.404i			0.995184i
$540$		0			0
$541$	−340.110			−0.628669			−0.314335	−	0.949312i	$-0.601781\pi$
$541$	−340.110			−0.628669			−0.314335	+	0.949312i	$0.601781\pi$
$542$		0			0
$543$	−189.785	+	189.785i	−0.349513	+	0.349513i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−6.38367	−	6.38367i	−0.0116703	−	0.0116703i	0.701248	−	0.712918i	$-0.252627\pi$
$547$	−6.38367	−	6.38367i	−0.0116703	−	0.0116703i	−0.712918	+	0.701248i	$0.752627\pi$
$548$		0			0
$549$		−	166.212i		−	0.302755i
$550$		0			0
$551$	−1232.12			−2.23615
$552$		0			0
$553$	−705.873	+	705.873i	−1.27644	+	1.27644i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−6.85357	−	6.85357i	−0.0123044	−	0.0123044i	0.700928	−	0.713232i	$-0.252770\pi$
$557$	−6.85357	−	6.85357i	−0.0123044	−	0.0123044i	−0.713232	+	0.700928i	$0.752770\pi$
$558$		0			0
$559$			268.757i			0.480782i
$560$		0			0
$561$	−63.9592			−0.114009
$562$		0			0
$563$	290.954	−	290.954i	0.516791	−	0.516791i	−0.399808	−	0.916599i	$-0.630923\pi$
$563$	290.954	−	290.954i	0.516791	−	0.516791i	0.916599	+	0.399808i	$0.130923\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−69.0681	−	69.0681i	−0.121813	−	0.121813i
$568$		0			0
$569$		−	459.271i		−	0.807155i	−0.914945	−	0.403578i	$-0.867767\pi$
$569$		−	459.271i		−	0.807155i	0.914945	−	0.403578i	$-0.132233\pi$
$570$		0			0
$571$	533.929			0.935076			0.467538	−	0.883973i	$-0.345141\pi$
$571$	533.929			0.935076			0.467538	+	0.883973i	$0.345141\pi$
$572$		0			0
$573$	−53.6413	+	53.6413i	−0.0936149	+	0.0936149i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−654.875	−	654.875i	−1.13497	−	1.13497i	−0.989340	−	0.145626i	$-0.953480\pi$
$577$	−654.875	−	654.875i	−1.13497	−	1.13497i	−0.145626	−	0.989340i	$-0.546520\pi$
$578$		0			0
$579$			510.131i			0.881055i
$580$		0			0
$581$	877.110			1.50966
$582$		0			0
$583$	367.292	−	367.292i	0.630003	−	0.630003i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−817.626	−	817.626i	−1.39289	−	1.39289i	−0.818778	−	0.574111i	$-0.805348\pi$
$587$	−817.626	−	817.626i	−1.39289	−	1.39289i	−0.574111	−	0.818778i	$-0.694652\pi$
$588$		0			0
$589$		−	1158.41i		−	1.96674i
$590$		0			0
$591$	249.637			0.422397
$592$		0			0
$593$	−57.5755	+	57.5755i	−0.0970919	+	0.0970919i	−0.753984	−	0.656892i	$-0.771871\pi$
$593$	−57.5755	+	57.5755i	−0.0970919	+	0.0970919i	0.656892	+	0.753984i	$0.271871\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	101.901	+	101.901i	0.170689	+	0.170689i
$598$		0			0
$599$			888.282i			1.48294i	0.670986	+	0.741470i	$0.265871\pi$
$599$			888.282i			1.48294i	−0.670986	+	0.741470i	$0.734129\pi$
$600$		0			0
$601$	431.706			0.718313			0.359156	−	0.933277i	$-0.383064\pi$
$601$	431.706			0.718313			0.359156	+	0.933277i	$0.383064\pi$
$602$		0			0
$603$	−177.810	+	177.810i	−0.294876	+	0.294876i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	62.3337	+	62.3337i	0.102691	+	0.102691i	0.756586	−	0.653894i	$-0.226866\pi$
$607$	62.3337	+	62.3337i	0.102691	+	0.102691i	−0.653894	+	0.756586i	$0.726866\pi$
$608$		0			0
$609$			815.716i			1.33944i
$610$		0			0
$611$	−383.939			−0.628378
$612$		0			0
$613$	136.232	−	136.232i	0.222238	−	0.222238i	−0.587203	−	0.809440i	$-0.699771\pi$
$613$	136.232	−	136.232i	0.222238	−	0.222238i	0.809440	+	0.587203i	$0.199771\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	192.767	+	192.767i	0.312427	+	0.312427i	0.845849	−	0.533422i	$-0.179094\pi$
$617$	192.767	+	192.767i	0.312427	+	0.312427i	−0.533422	+	0.845849i	$0.679094\pi$
$618$		0			0
$619$		−	753.039i		−	1.21654i	−0.793730	−	0.608270i	$-0.791864\pi$
$619$		−	753.039i		−	1.21654i	0.793730	−	0.608270i	$-0.208136\pi$
$620$		0			0
$621$	−106.182			−0.170985
$622$		0			0
$623$	−924.009	+	924.009i	−1.48316	+	1.48316i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	271.176	+	271.176i	0.432498	+	0.432498i
$628$		0			0
$629$		−	86.3837i		−	0.137335i
$630$		0			0
$631$	400.271			0.634345			0.317172	−	0.948368i	$-0.397267\pi$
$631$	400.271			0.634345			0.317172	+	0.948368i	$0.397267\pi$
$632$		0			0
$633$	−415.881	+	415.881i	−0.657000	+	0.657000i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−252.742	−	252.742i	−0.396770	−	0.396770i
$638$		0			0
$639$			174.606i			0.273249i
$640$		0			0
$641$	−434.665			−0.678105			−0.339052	−	0.940767i	$-0.610106\pi$
$641$	−434.665			−0.678105			−0.339052	+	0.940767i	$0.610106\pi$
$642$		0			0
$643$	153.040	−	153.040i	0.238009	−	0.238009i	−0.578016	−	0.816025i	$-0.696173\pi$
$643$	153.040	−	153.040i	0.238009	−	0.238009i	0.816025	+	0.578016i	$0.196173\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−472.627	−	472.627i	−0.730489	−	0.730489i	0.240227	−	0.970717i	$-0.422778\pi$
$647$	−472.627	−	472.627i	−0.730489	−	0.730489i	−0.970717	+	0.240227i	$0.922778\pi$
$648$		0			0
$649$		−	639.433i		−	0.985258i
$650$		0			0
$651$	−766.918			−1.17806
$652$		0			0
$653$	−782.227	+	782.227i	−1.19790	+	1.19790i	−0.223102	+	0.974795i	$0.571618\pi$
$653$	−782.227	+	782.227i	−1.19790	+	1.19790i	−0.974795	+	0.223102i	$0.928382\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−65.1214	−	65.1214i	−0.0991194	−	0.0991194i
$658$		0			0
$659$		−	737.394i		−	1.11896i	−0.828844	−	0.559479i	$-0.811001\pi$
$659$		−	737.394i		−	1.11896i	0.828844	−	0.559479i	$-0.188999\pi$
$660$		0			0
$661$	1013.07			1.53263			0.766316	−	0.642464i	$-0.222088\pi$
$661$	1013.07			1.53263			0.766316	+	0.642464i	$0.222088\pi$
$662$		0			0
$663$	30.1362	−	30.1362i	0.0454543	−	0.0454543i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	627.019	+	627.019i	0.940059	+	0.940059i
$668$		0			0
$669$		−	140.394i		−	0.209856i
$670$		0			0
$671$	432.039			0.643873
$672$		0			0
$673$	−87.6867	+	87.6867i	−0.130292	+	0.130292i	−0.769246	−	0.638953i	$-0.779368\pi$
$673$	−87.6867	+	87.6867i	−0.130292	+	0.130292i	0.638953	+	0.769246i	$0.279368\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	453.873	+	453.873i	0.670418	+	0.670418i	0.957812	−	0.287394i	$-0.0927890\pi$
$677$	453.873	+	453.873i	0.670418	+	0.670418i	−0.287394	+	0.957812i	$0.592789\pi$
$678$		0			0
$679$		−	1585.04i		−	2.33437i
$680$		0			0
$681$	−158.202			−0.232308
$682$		0			0
$683$	132.767	−	132.767i	0.194388	−	0.194388i	−0.603201	−	0.797589i	$-0.706108\pi$
$683$	132.767	−	132.767i	0.194388	−	0.194388i	0.797589	+	0.603201i	$0.206108\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−218.265	−	218.265i	−0.317707	−	0.317707i
$688$		0			0
$689$			346.120i			0.502352i
$690$		0			0
$691$	−970.727			−1.40481			−0.702407	−	0.711775i	$-0.747891\pi$
$691$	−970.727			−1.40481			−0.702407	+	0.711775i	$0.747891\pi$
$692$		0			0
$693$	179.530	−	179.530i	0.259062	−	0.259062i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−126.565	−	126.565i	−0.181586	−	0.181586i
$698$		0			0
$699$		−	311.778i		−	0.446034i
$700$		0			0
$701$	136.729			0.195048			0.0975240	−	0.995233i	$-0.468908\pi$
$701$	136.729			0.195048			0.0975240	+	0.995233i	$0.468908\pi$
$702$		0			0
$703$	−366.252	+	366.252i	−0.520984	+	0.520984i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	504.949	+	504.949i	0.714214	+	0.714214i
$708$		0			0
$709$			488.171i			0.688535i	0.938872	+	0.344268i	$0.111873\pi$
$709$			488.171i			0.688535i	−0.938872	+	0.344268i	$0.888127\pi$
$710$		0			0
$711$	275.939			0.388100
$712$		0			0
$713$	−589.510	+	589.510i	−0.826802	+	0.826802i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−394.070	−	394.070i	−0.549610	−	0.549610i
$718$		0			0
$719$		−	1386.34i		−	1.92815i	−0.265628	−	0.964076i	$-0.585579\pi$
$719$		−	1386.34i		−	1.92815i	0.265628	−	0.964076i	$-0.414421\pi$
$720$		0			0
$721$	1436.24			1.99201
$722$		0			0
$723$	−389.184	+	389.184i	−0.538290	+	0.538290i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−671.593	−	671.593i	−0.923786	−	0.923786i	0.0735084	−	0.997295i	$-0.476580\pi$
$727$	−671.593	−	671.593i	−0.923786	−	0.923786i	−0.997295	+	0.0735084i	$0.976580\pi$
$728$		0			0
$729$			27.0000i			0.0370370i
$730$		0			0
$731$	−244.929			−0.335060
$732$		0			0
$733$	−177.303	+	177.303i	−0.241887	+	0.241887i	−0.817630	−	0.575744i	$-0.804713\pi$
$733$	−177.303	+	177.303i	−0.241887	+	0.241887i	0.575744	+	0.817630i	$0.304713\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−462.186	−	462.186i	−0.627118	−	0.627118i
$738$		0			0
$739$		−	487.131i		−	0.659175i	−0.944125	−	0.329588i	$-0.893090\pi$
$739$		−	487.131i		−	0.659175i	0.944125	−	0.329588i	$-0.106910\pi$
$740$		0			0
$741$	−255.545			−0.344865
$742$		0			0
$743$	200.777	−	200.777i	0.270224	−	0.270224i	−0.558966	−	0.829190i	$-0.688802\pi$
$743$	200.777	−	200.777i	0.270224	−	0.270224i	0.829190	+	0.558966i	$0.188802\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−171.439	−	171.439i	−0.229504	−	0.229504i
$748$		0			0
$749$		−	650.524i		−	0.868524i
$750$		0			0
$751$	108.424			0.144373			0.0721867	−	0.997391i	$-0.477002\pi$
$751$	108.424			0.144373			0.0721867	+	0.997391i	$0.477002\pi$
$752$		0			0
$753$	−209.616	+	209.616i	−0.278375	+	0.278375i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	230.412	+	230.412i	0.304375	+	0.304375i	0.842723	−	0.538348i	$-0.180951\pi$
$757$	230.412	+	230.412i	0.304375	+	0.304375i	−0.538348	+	0.842723i	$0.680951\pi$
$758$		0			0
$759$		−	276.000i		−	0.363636i
$760$		0			0
$761$	893.857			1.17458			0.587291	−	0.809376i	$-0.300194\pi$
$761$	893.857			1.17458			0.587291	+	0.809376i	$0.300194\pi$
$762$		0			0
$763$	−876.178	+	876.178i	−1.14833	+	1.14833i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	301.287	+	301.287i	0.392813	+	0.392813i
$768$		0			0
$769$			685.445i			0.891346i	0.895196	+	0.445673i	$0.147036\pi$
$769$			685.445i			0.891346i	−0.895196	+	0.445673i	$0.852964\pi$
$770$		0			0
$771$	−326.424			−0.423378
$772$		0			0
$773$	−95.1510	+	95.1510i	−0.123093	+	0.123093i	−0.765970	−	0.642877i	$-0.777741\pi$
$773$	−95.1510	+	95.1510i	−0.123093	+	0.123093i	0.642877	+	0.765970i	$0.277741\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	242.474	+	242.474i	0.312065	+	0.312065i
$778$		0			0
$779$			1073.23i			1.37770i
$780$		0			0
$781$	−453.857			−0.581123
$782$		0			0
$783$	159.439	−	159.439i	0.203626	−	0.203626i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−120.771	−	120.771i	−0.153457	−	0.153457i	0.626203	−	0.779660i	$-0.284608\pi$
$787$	−120.771	−	120.771i	−0.153457	−	0.153457i	−0.779660	+	0.626203i	$0.784608\pi$
$788$		0			0
$789$			331.818i			0.420556i
$790$		0			0
$791$	−2130.18			−2.69302
$792$		0			0
$793$	−203.568	+	203.568i	−0.256706	+	0.256706i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−810.161	−	810.161i	−1.01651	−	1.01651i	−0.999861	−	0.0166521i	$-0.994699\pi$
$797$	−810.161	−	810.161i	−1.01651	−	1.01651i	−0.0166521	−	0.999861i	$-0.505301\pi$
$798$		0			0
$799$		−	349.898i		−	0.437920i
$800$		0			0
$801$	361.212			0.450952
$802$		0			0
$803$	169.271	−	169.271i	0.210799	−	0.210799i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−333.328	−	333.328i	−0.413046	−	0.413046i
$808$		0			0
$809$		−	122.565i		−	0.151502i	−0.997127	−	0.0757511i	$-0.975865\pi$
$809$		−	122.565i		−	0.151502i	0.997127	−	0.0757511i	$-0.0241354\pi$
$810$		0			0
$811$	−1035.18			−1.27643			−0.638213	−	0.769860i	$-0.720326\pi$
$811$	−1035.18			−1.27643			−0.638213	+	0.769860i	$0.720326\pi$
$812$		0			0
$813$	494.277	−	494.277i	0.607967	−	0.607967i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	1038.46	+	1038.46i	1.27106	+	1.27106i
$818$		0			0
$819$			169.182i			0.206571i
$820$		0			0
$821$	537.878			0.655149			0.327575	−	0.944825i	$-0.393769\pi$
$821$	537.878			0.655149			0.327575	+	0.944825i	$0.393769\pi$
$822$		0			0
$823$	516.855	−	516.855i	0.628013	−	0.628013i	−0.319555	−	0.947568i	$-0.603533\pi$
$823$	516.855	−	516.855i	0.628013	−	0.628013i	0.947568	+	0.319555i	$0.103533\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−700.232	−	700.232i	−0.846713	−	0.846713i	0.143008	−	0.989721i	$-0.454322\pi$
$827$	−700.232	−	700.232i	−0.846713	−	0.846713i	−0.989721	+	0.143008i	$0.954322\pi$
$828$		0			0
$829$			537.110i			0.647901i	0.946074	+	0.323951i	$0.105011\pi$
$829$			537.110i			0.647901i	−0.946074	+	0.323951i	$0.894989\pi$
$830$		0			0
$831$	−54.9592			−0.0661362
$832$		0			0
$833$	230.334	−	230.334i	0.276511	−	0.276511i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	149.901	+	149.901i	0.179094	+	0.179094i
$838$		0			0
$839$			220.606i			0.262939i	0.991320	+	0.131470i	$0.0419696\pi$
$839$			220.606i			0.262939i	−0.991320	+	0.131470i	$0.958030\pi$
$840$		0			0
$841$	−1042.03			−1.23904
$842$		0			0
$843$	−143.803	+	143.803i	−0.170584	+	0.170584i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−461.926	−	461.926i	−0.545367	−	0.545367i
$848$		0			0
$849$		−	231.545i		−	0.272727i
$850$		0			0
$851$	372.767			0.438034
$852$		0			0
$853$	−308.269	+	308.269i	−0.361394	+	0.361394i	−0.864326	−	0.502932i	$-0.832255\pi$
$853$	−308.269	+	308.269i	−0.361394	+	0.361394i	0.502932	+	0.864326i	$0.332255\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−989.989	−	989.989i	−1.15518	−	1.15518i	−0.985499	−	0.169680i	$-0.945727\pi$
$857$	−989.989	−	989.989i	−1.15518	−	1.15518i	−0.169680	−	0.985499i	$-0.554273\pi$
$858$		0			0
$859$			629.837i			0.733221i	0.930375	+	0.366610i	$0.119482\pi$
$859$			629.837i			0.733221i	−0.930375	+	0.366610i	$0.880518\pi$
$860$		0			0
$861$	710.524			0.825232
$862$		0			0
$863$	1144.00	−	1144.00i	1.32561	−	1.32561i	0.416449	−	0.909159i	$-0.363274\pi$
$863$	1144.00	−	1144.00i	1.32561	−	1.32561i	0.909159	−	0.416449i	$-0.136726\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−326.487	−	326.487i	−0.376571	−	0.376571i
$868$		0			0
$869$			717.253i			0.825378i
$870$		0			0
$871$	435.545			0.500052
$872$		0			0
$873$	−309.810	+	309.810i	−0.354880	+	0.354880i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	336.260	+	336.260i	0.383421	+	0.383421i	0.872333	−	0.488912i	$-0.162606\pi$
$877$	336.260	+	336.260i	0.383421	+	0.383421i	−0.488912	+	0.872333i	$0.662606\pi$
$878$		0			0
$879$		−	82.0816i		−	0.0933807i
$880$		0			0
$881$	−932.967			−1.05899			−0.529493	−	0.848314i	$-0.677618\pi$
$881$	−932.967			−1.05899			−0.529493	+	0.848314i	$0.677618\pi$
$882$		0			0
$883$	180.623	−	180.623i	0.204556	−	0.204556i	−0.597393	−	0.801949i	$-0.703797\pi$
$883$	180.623	−	180.623i	0.204556	−	0.204556i	0.801949	+	0.597393i	$0.203797\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	853.106	+	853.106i	0.961788	+	0.961788i	0.999296	−	0.0375087i	$-0.0119422\pi$
$887$	853.106	+	853.106i	0.961788	+	0.961788i	−0.0375087	+	0.999296i	$0.511942\pi$
$888$		0			0
$889$		−	974.706i		−	1.09641i
$890$		0			0
$891$	−70.1816			−0.0787673
$892$		0			0
$893$	−1483.51	+	1483.51i	−1.66126	+	1.66126i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	130.045	+	130.045i	0.144978	+	0.144978i
$898$		0			0
$899$		−	1770.38i		−	1.96928i
$900$		0			0
$901$	−315.433			−0.350092
$902$		0			0
$903$	687.502	−	687.502i	0.761353	−	0.761353i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	464.313	+	464.313i	0.511922	+	0.511922i	0.915115	−	0.403193i	$-0.132100\pi$
$907$	464.313	+	464.313i	0.511922	+	0.511922i	−0.403193	+	0.915115i	$0.632100\pi$
$908$		0			0
$909$		−	197.394i		−	0.217155i
$910$		0			0
$911$	1456.95			1.59928			0.799642	−	0.600478i	$-0.205023\pi$
$911$	1456.95			1.59928			0.799642	+	0.600478i	$0.205023\pi$
$912$		0			0
$913$	445.626	−	445.626i	0.488089	−	0.488089i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−56.8990	−	56.8990i	−0.0620491	−	0.0620491i
$918$		0			0
$919$		−	519.969i		−	0.565799i	−0.959150	−	0.282900i	$-0.908704\pi$
$919$		−	519.969i		−	0.565799i	0.959150	−	0.282900i	$-0.0912963\pi$
$920$		0			0
$921$	329.202			0.357440
$922$		0			0
$923$	213.848	−	213.848i	0.231688	−	0.231688i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−280.727	−	280.727i	−0.302833	−	0.302833i
$928$		0			0
$929$			229.655i			0.247207i	0.992332	+	0.123603i	$0.0394451\pi$
$929$			229.655i			0.247207i	−0.992332	+	0.123603i	$0.960555\pi$
$930$		0			0
$931$	−1953.15			−2.09791
$932$		0			0
$933$	180.692	−	180.692i	0.193668	−	0.193668i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−176.003	−	176.003i	−0.187837	−	0.187837i	0.606923	−	0.794760i	$-0.292403\pi$
$937$	−176.003	−	176.003i	−0.187837	−	0.187837i	−0.794760	+	0.606923i	$0.792403\pi$
$938$		0			0
$939$		−	221.363i		−	0.235744i
$940$		0			0
$941$	331.857			0.352664			0.176332	−	0.984331i	$-0.443577\pi$
$941$	331.857			0.352664			0.176332	+	0.984331i	$0.443577\pi$
$942$		0			0
$943$	546.161	−	546.161i	0.579174	−	0.579174i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	404.733	+	404.733i	0.427385	+	0.427385i	0.887736	−	0.460352i	$-0.152277\pi$
$947$	404.733	+	404.733i	0.427385	+	0.427385i	−0.460352	+	0.887736i	$0.652277\pi$
$948$		0			0
$949$			159.514i			0.168087i
$950$		0			0
$951$	504.767			0.530775
$952$		0			0
$953$	581.344	−	581.344i	0.610015	−	0.610015i	−0.332935	−	0.942950i	$-0.608039\pi$
$953$	581.344	−	581.344i	0.610015	−	0.610015i	0.942950	+	0.332935i	$0.108039\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	414.434	+	414.434i	0.433055	+	0.433055i
$958$		0			0
$959$			1969.80i			2.05401i
$960$		0			0
$961$	703.473			0.732022
$962$		0			0
$963$	−127.151	+	127.151i	−0.132036	+	0.132036i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−254.434	−	254.434i	−0.263117	−	0.263117i	0.563202	−	0.826319i	$-0.309569\pi$
$967$	−254.434	−	254.434i	−0.263117	−	0.263117i	−0.826319	+	0.563202i	$0.809569\pi$
$968$		0			0
$969$		−	232.888i		−	0.240338i
$970$		0			0
$971$	1594.36			1.64198			0.820990	−	0.570942i	$-0.193422\pi$
$971$	1594.36			1.64198			0.820990	+	0.570942i	$0.193422\pi$
$972$		0			0
$973$	144.338	−	144.338i	0.148344	−	0.148344i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1052.56	+	1052.56i	1.07734	+	1.07734i	0.996747	+	0.0805924i	$0.0256812\pi$
$977$	1052.56	+	1052.56i	1.07734	+	1.07734i	0.0805924	+	0.996747i	$0.474319\pi$
$978$		0			0
$979$			938.906i			0.959046i
$980$		0			0
$981$	342.514			0.349148
$982$		0			0
$983$	838.352	−	838.352i	0.852850	−	0.852850i	−0.137633	−	0.990483i	$-0.543949\pi$
$983$	838.352	−	838.352i	0.852850	−	0.852850i	0.990483	+	0.137633i	$0.0439494\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	982.145	+	982.145i	0.995081	+	0.995081i
$988$		0			0
$989$		−	1056.93i		−	1.06868i
$990$		0			0
$991$	−764.937			−0.771884			−0.385942	−	0.922523i	$-0.626123\pi$
$991$	−764.937			−0.771884			−0.385942	+	0.922523i	$0.626123\pi$
$992$		0			0
$993$	435.489	−	435.489i	0.438559	−	0.438559i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	510.170	+	510.170i	0.511706	+	0.511706i	0.915049	−	0.403343i	$-0.132152\pi$
$997$	510.170	+	510.170i	0.511706	+	0.511706i	−0.403343	+	0.915049i	$0.632152\pi$
$998$		0			0
$999$		−	94.7878i		−	0.0948826i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.3.u.g.457.2	yes	4
3.2	odd	2		1800.3.v.p.1657.2		4
4.3	odd	2		1200.3.bg.c.1057.1		4
5.2	odd	4		600.3.u.b.193.1	✓	4
5.3	odd	4	inner	600.3.u.g.193.2	yes	4
5.4	even	2		600.3.u.b.457.1	yes	4
15.2	even	4		1800.3.v.i.793.1		4
15.8	even	4		1800.3.v.p.793.2		4
15.14	odd	2		1800.3.v.i.1657.1		4
20.3	even	4		1200.3.bg.c.193.1		4
20.7	even	4		1200.3.bg.n.193.2		4
20.19	odd	2		1200.3.bg.n.1057.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.3.u.b.193.1	✓	4	5.2	odd	4
600.3.u.b.457.1	yes	4	5.4	even	2
600.3.u.g.193.2	yes	4	5.3	odd	4	inner
600.3.u.g.457.2	yes	4	1.1	even	1	trivial
1200.3.bg.c.193.1		4	20.3	even	4
1200.3.bg.c.1057.1		4	4.3	odd	2
1200.3.bg.n.193.2		4	20.7	even	4
1200.3.bg.n.1057.2		4	20.19	odd	2
1800.3.v.i.793.1		4	15.2	even	4
1800.3.v.i.1657.1		4	15.14	odd	2
1800.3.v.p.793.2		4	15.8	even	4
1800.3.v.p.1657.2		4	3.2	odd	2

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:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	600.u (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 - 1.22474i) q^{3} +(7.67423 + 7.67423i) q^{7} -3.00000i q^{9} +O(q^{10})\)	\(q+(1.22474 - 1.22474i) q^{3} +(7.67423 + 7.67423i) q^{7} -3.00000i q^{9} +7.79796 q^{11} +(-3.67423 + 3.67423i) q^{13} +(-3.34847 - 3.34847i) q^{17} +28.3939i q^{19} +18.7980 q^{21} +(14.4495 - 14.4495i) q^{23} +(-3.67423 - 3.67423i) q^{27} +43.3939i q^{29} -40.7980 q^{31} +(9.55051 - 9.55051i) q^{33} +(12.8990 + 12.8990i) q^{37} +9.00000i q^{39} +37.7980 q^{41} +(36.5732 - 36.5732i) q^{43} +(52.2474 + 52.2474i) q^{47} +68.7878i q^{49} -8.20204 q^{51} +(47.1010 - 47.1010i) q^{53} +(34.7753 + 34.7753i) q^{57} -82.0000i q^{59} +55.4041 q^{61} +(23.0227 - 23.0227i) q^{63} +(-59.2702 - 59.2702i) q^{67} -35.3939i q^{69} -58.2020 q^{71} +(21.7071 - 21.7071i) q^{73} +(59.8434 + 59.8434i) q^{77} +91.9796i q^{79} -9.00000 q^{81} +(57.1464 - 57.1464i) q^{83} +(53.1464 + 53.1464i) q^{87} +120.404i q^{89} -56.3939 q^{91} +(-49.9671 + 49.9671i) q^{93} +(-103.270 - 103.270i) q^{97} -23.3939i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{7}+O(q^{10}) \) 4 * q + 16 * q^7	\( 4 q + 16 q^{7} - 8 q^{11} + 16 q^{17} + 36 q^{21} + 48 q^{23} - 124 q^{31} + 48 q^{33} + 32 q^{37} + 112 q^{41} + 112 q^{43} + 160 q^{47} - 72 q^{51} + 208 q^{53} + 144 q^{57} + 300 q^{61} + 48 q^{63} - 144 q^{67} - 272 q^{71} + 224 q^{73} + 112 q^{77} - 36 q^{81} + 160 q^{83} + 144 q^{87} - 108 q^{91} - 48 q^{93} - 320 q^{97}+O(q^{100}) \) 4 * q + 16 * q^7 - 8 * q^11 + 16 * q^17 + 36 * q^21 + 48 * q^23 - 124 * q^31 + 48 * q^33 + 32 * q^37 + 112 * q^41 + 112 * q^43 + 160 * q^47 - 72 * q^51 + 208 * q^53 + 144 * q^57 + 300 * q^61 + 48 * q^63 - 144 * q^67 - 272 * q^71 + 224 * q^73 + 112 * q^77 - 36 * q^81 + 160 * q^83 + 144 * q^87 - 108 * q^91 - 48 * q^93 - 320 * q^97

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