LMFDB - Embedded newform 1200.3.bg.c.1057.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,3,Mod(193,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, 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("1200.193");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1200.bg (of order $4$, 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:	$32.6976317232$
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:	no (minimal twist has level 600)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1057.1
Root			$1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1200.1057
Dual form			1200.3.bg.c.193.1

$q$-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 $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−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	$-0.967642\pi$
$7$	−7.67423	−	7.67423i	−1.09632	−	1.09632i	−0.101482	−	0.994837i	$-0.532358\pi$
$8$		0			0
$9$		−	3.00000i		−	0.333333i
$10$		0			0
$11$	−7.79796			−0.708905			−0.354453	−	0.935074i	$-0.615333\pi$
$11$	−7.79796			−0.708905			−0.354453	+	0.935074i	$0.615333\pi$
$12$		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	$-0.731394\pi$
$19$		−	28.3939i		−	1.49441i	0.664591	−	0.747207i	$-0.268606\pi$
$20$		0			0
$21$	18.7980			0.895141
$22$		0			0
$23$	−14.4495	+	14.4495i	−0.628239	+	0.628239i	−0.947625	−	0.319386i	$-0.896523\pi$
$23$	−14.4495	+	14.4495i	−0.628239	+	0.628239i	0.319386	+	0.947625i	$0.396523\pi$
$24$		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	$-0.271389\pi$
$31$	40.7980			1.31606			0.658032	+	0.752990i	$0.271389\pi$
$32$		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	−0.990200	−	0.139660i	$-0.955399\pi$
$43$	−36.5732	+	36.5732i	−0.850540	+	0.850540i	0.139660	+	0.990200i	$0.455399\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−52.2474	−	52.2474i	−1.11165	−	1.11165i	−0.992928	−	0.118720i	$-0.962121\pi$
$47$	−52.2474	−	52.2474i	−1.11165	−	1.11165i	−0.118720	−	0.992928i	$-0.537879\pi$
$48$		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	$0.244558\pi$
$59$			82.0000i			1.38983i	−0.719092	+	0.694915i	$0.755442\pi$
$60$		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	0.994001	−	0.109372i	$-0.0348839\pi$
$67$	59.2702	+	59.2702i	0.884629	+	0.884629i	−0.109372	+	0.994001i	$0.534884\pi$
$68$		0			0
$69$		−	35.3939i		−	0.512955i
$70$		0			0
$71$	58.2020			0.819747			0.409874	−	0.912142i	$-0.365573\pi$
$71$	58.2020			0.819747			0.409874	+	0.912142i	$0.365573\pi$
$72$		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	$-0.802212\pi$
$79$		−	91.9796i		−	1.16430i	0.813082	−	0.582149i	$-0.197788\pi$
$80$		0			0
$81$	−9.00000			−0.111111
$82$		0			0
$83$	−57.1464	+	57.1464i	−0.688511	+	0.688511i	−0.961903	−	0.273392i	$-0.911855\pi$
$83$	−57.1464	+	57.1464i	−0.688511	+	0.688511i	0.273392	+	0.961903i	$0.411855\pi$
$84$		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	−0.996151	−	0.0876512i	$-0.972064\pi$
$103$	−93.5755	+	93.5755i	−0.908500	+	0.908500i	0.0876512	+	0.996151i	$0.472064\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	42.3837	+	42.3837i	0.396109	+	0.396109i	0.876858	−	0.480749i	$-0.159635\pi$
$107$	42.3837	+	42.3837i	0.396109	+	0.396109i	−0.480749	+	0.876858i	$0.659635\pi$
$108$		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	0.911450	−	0.411410i	$-0.134964\pi$
$127$	63.5051	+	63.5051i	0.500040	+	0.500040i	−0.411410	+	0.911450i	$0.634964\pi$
$128$		0			0
$129$		−	89.5857i		−	0.694463i
$130$		0			0
$131$	7.41429			0.0565976			0.0282988	−	0.999600i	$-0.490991\pi$
$131$	7.41429			0.0565976			0.0282988	+	0.999600i	$0.490991\pi$
$132$		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	$0.0215518\pi$
$139$			18.8082i			0.135311i	−0.997709	+	0.0676553i	$0.978448\pi$
$140$		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	$-0.389464\pi$
$151$	102.778			0.680646			0.340323	+	0.940309i	$0.389464\pi$
$152$		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	−0.135411	−	0.990790i	$-0.543235\pi$
$163$	139.427	−	139.427i	0.855379	−	0.855379i	0.990790	+	0.135411i	$0.0432353\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	180.182	+	180.182i	1.07893	+	1.07893i	0.996605	+	0.0823265i	$0.0262350\pi$
$167$	180.182	+	180.182i	1.07893	+	1.07893i	0.0823265	+	0.996605i	$0.473765\pi$
$168$		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	$0.366052\pi$
$179$			326.767i			1.82552i	−0.408501	+	0.912758i	$0.633948\pi$
$180$		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	$-0.463424\pi$
$191$	43.7980			0.229309			0.114654	+	0.993405i	$0.463424\pi$
$192$		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	$-0.932963\pi$
$199$		−	83.2020i		−	0.418101i	0.977905	−	0.209050i	$-0.0670373\pi$
$200$		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	$-0.202349\pi$
$211$	339.565			1.60931			0.804657	+	0.593740i	$0.202349\pi$
$212$		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	−0.823841	−	0.566821i	$-0.808173\pi$
$223$	−57.3156	+	57.3156i	−0.257020	+	0.257020i	0.566821	+	0.823841i	$0.308173\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	64.5857	+	64.5857i	0.284519	+	0.284519i	0.834908	−	0.550389i	$-0.185521\pi$
$227$	64.5857	+	64.5857i	0.284519	+	0.284519i	−0.550389	+	0.834908i	$0.685521\pi$
$228$		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	$0.235051\pi$
$239$			321.757i			1.34626i	−0.739522	+	0.673132i	$0.764949\pi$
$240$		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	$-0.389255\pi$
$251$	171.151			0.681877			0.340938	+	0.940086i	$0.389255\pi$
$252$		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	−0.401003	−	0.916077i	$-0.631338\pi$
$263$	135.464	−	135.464i	0.515073	−	0.515073i	0.916077	+	0.401003i	$0.131338\pi$
$264$		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	$-0.767362\pi$
$271$	−403.576			−1.48921			−0.744604	+	0.667506i	$0.767362\pi$
$272$		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	−0.854111	−	0.520091i	$-0.825898\pi$
$283$	−94.5278	+	94.5278i	−0.334021	+	0.334021i	0.520091	+	0.854111i	$0.325898\pi$
$284$		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	0.453489	−	0.891262i	$-0.350179\pi$
$307$	−134.396	−	134.396i	−0.437773	−	0.437773i	−0.891262	+	0.453489i	$0.850179\pi$
$308$		0			0
$309$		−	229.212i		−	0.741787i
$310$		0			0
$311$	−147.535			−0.474388			−0.237194	−	0.971462i	$-0.576228\pi$
$311$	−147.535			−0.474388			−0.237194	+	0.971462i	$0.576228\pi$
$312$		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	$-0.680489\pi$
$331$	−355.576			−1.07425			−0.537123	+	0.843504i	$0.680489\pi$
$332$		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	0.852376	−	0.522930i	$-0.175161\pi$
$347$	114.318	+	114.318i	0.329446	+	0.329446i	−0.522930	+	0.852376i	$0.675161\pi$
$348$		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	$-0.983612\pi$
$359$		−	36.9490i		−	0.102922i	0.998675	−	0.0514610i	$-0.0163878\pi$
$360$		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	0.715101	−	0.699021i	$-0.246381\pi$
$367$	5.90127	+	5.90127i	0.0160798	+	0.0160798i	−0.699021	+	0.715101i	$0.746381\pi$
$368$		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	$-0.749302\pi$
$379$		−	537.161i		−	1.41731i	0.705554	−	0.708656i	$-0.250698\pi$
$380$		0			0
$381$	−155.555			−0.408281
$382$		0			0
$383$	407.110	−	407.110i	1.06295	−	1.06295i	0.0650702	−	0.997881i	$-0.479273\pi$
$383$	407.110	−	407.110i	1.06295	−	1.06295i	0.997881	−	0.0650702i	$-0.0207271\pi$
$384$		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	$0.0404647\pi$
$419$			106.243i			0.253563i	−0.991931	+	0.126781i	$0.959535\pi$
$420$		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	$-0.407685\pi$
$431$	246.504			0.571935			0.285968	+	0.958239i	$0.407685\pi$
$432$		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	$0.165053\pi$
$439$			435.141i			0.991209i	−0.868548	+	0.495605i	$0.834947\pi$
$440$		0			0
$441$	206.363			0.467944
$442$		0			0
$443$	−70.6969	+	70.6969i	−0.159587	+	0.159587i	−0.782384	−	0.622797i	$-0.785996\pi$
$443$	−70.6969	+	70.6969i	−0.159587	+	0.159587i	0.622797	+	0.782384i	$0.285996\pi$
$444$		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	−0.799940	−	0.600079i	$-0.795136\pi$
$463$	−92.5357	+	92.5357i	−0.199861	+	0.199861i	0.600079	+	0.799940i	$0.295136\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	384.974	+	384.974i	0.824355	+	0.824355i	0.986729	−	0.162374i	$-0.0519151\pi$
$467$	384.974	+	384.974i	0.824355	+	0.824355i	−0.162374	+	0.986729i	$0.551915\pi$
$468$		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	$-0.972869\pi$
$479$		−	81.5551i		−	0.170261i	0.996370	−	0.0851306i	$-0.0271307\pi$
$480$		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	0.0805028	−	0.996754i	$-0.474347\pi$
$487$	−446.215	−	446.215i	−0.916252	−	0.916252i	−0.996754	+	0.0805028i	$0.974347\pi$
$488$		0			0
$489$			341.524i			0.698414i
$490$		0			0
$491$	−532.808			−1.08515			−0.542575	−	0.840008i	$-0.682550\pi$
$491$	−532.808			−1.08515			−0.542575	+	0.840008i	$0.682550\pi$
$492$		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	$0.104670\pi$
$499$			322.292i			0.645875i	−0.946420	+	0.322938i	$0.895330\pi$
$500$		0			0
$501$	−441.353			−0.880944
$502$		0			0
$503$	143.333	−	143.333i	0.284956	−	0.284956i	−0.550126	−	0.835082i	$-0.685420\pi$
$503$	143.333	−	143.333i	0.284956	−	0.284956i	0.835082	+	0.550126i	$0.185420\pi$
$504$		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	−0.730993	−	0.682385i	$-0.760943\pi$
$523$	−25.4222	+	25.4222i	−0.0486084	+	0.0486084i	0.682385	+	0.730993i	$0.260943\pi$
$524$		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	0.712918	−	0.701248i	$-0.247373\pi$
$547$	6.38367	+	6.38367i	0.0116703	+	0.0116703i	−0.701248	+	0.712918i	$0.747373\pi$
$548$		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	−0.916599	−	0.399808i	$-0.869077\pi$
$563$	−290.954	+	290.954i	−0.516791	+	0.516791i	0.399808	+	0.916599i	$0.369077\pi$
$564$		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	$-0.654859\pi$
$571$	−533.929			−0.935076			−0.467538	+	0.883973i	$0.654859\pi$
$572$		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	$0.194652\pi$
$587$	817.626	+	817.626i	1.39289	+	1.39289i	0.574111	+	0.818778i	$0.305348\pi$
$588$		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	$-0.734129\pi$
$599$		−	888.282i		−	1.48294i	0.670986	−	0.741470i	$-0.265871\pi$
$600$		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	0.653894	−	0.756586i	$-0.273134\pi$
$607$	−62.3337	−	62.3337i	−0.102691	−	0.102691i	−0.756586	+	0.653894i	$0.773134\pi$
$608$		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	$0.208136\pi$
$619$			753.039i			1.21654i	−0.793730	+	0.608270i	$0.791864\pi$
$620$		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	$-0.602733\pi$
$631$	−400.271			−0.634345			−0.317172	+	0.948368i	$0.602733\pi$
$632$		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	−0.816025	−	0.578016i	$-0.803827\pi$
$643$	−153.040	+	153.040i	−0.238009	+	0.238009i	0.578016	+	0.816025i	$0.303827\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	472.627	+	472.627i	0.730489	+	0.730489i	0.970717	−	0.240227i	$-0.0772220\pi$
$647$	472.627	+	472.627i	0.730489	+	0.730489i	−0.240227	+	0.970717i	$0.577222\pi$
$648$		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	$0.188999\pi$
$659$			737.394i			1.11896i	−0.828844	+	0.559479i	$0.811001\pi$
$660$		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	−0.797589	−	0.603201i	$-0.793892\pi$
$683$	−132.767	+	132.767i	−0.194388	+	0.194388i	0.603201	+	0.797589i	$0.293892\pi$
$684$		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	$-0.252109\pi$
$691$	970.727			1.40481			0.702407	+	0.711775i	$0.252109\pi$
$692$		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	$0.414421\pi$
$719$			1386.34i			1.92815i	−0.265628	+	0.964076i	$0.585579\pi$
$720$		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	0.997295	−	0.0735084i	$-0.0234196\pi$
$727$	671.593	+	671.593i	0.923786	+	0.923786i	−0.0735084	+	0.997295i	$0.523420\pi$
$728$		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	$0.106910\pi$
$739$			487.131i			0.659175i	−0.944125	+	0.329588i	$0.893090\pi$
$740$		0			0
$741$	−255.545			−0.344865
$742$		0			0
$743$	−200.777	+	200.777i	−0.270224	+	0.270224i	−0.829190	−	0.558966i	$-0.811198\pi$
$743$	−200.777	+	200.777i	−0.270224	+	0.270224i	0.558966	+	0.829190i	$0.311198\pi$
$744$		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	$-0.522998\pi$
$751$	−108.424			−0.144373			−0.0721867	+	0.997391i	$0.522998\pi$
$752$		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	0.779660	−	0.626203i	$-0.215392\pi$
$787$	120.771	+	120.771i	0.153457	+	0.153457i	−0.626203	+	0.779660i	$0.715392\pi$
$788$		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	$-0.279674\pi$
$811$	1035.18			1.27643			0.638213	+	0.769860i	$0.279674\pi$
$812$		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	−0.947568	−	0.319555i	$-0.896467\pi$
$823$	−516.855	+	516.855i	−0.628013	+	0.628013i	0.319555	+	0.947568i	$0.396467\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	700.232	+	700.232i	0.846713	+	0.846713i	0.989721	−	0.143008i	$-0.0456776\pi$
$827$	700.232	+	700.232i	0.846713	+	0.846713i	−0.143008	+	0.989721i	$0.545678\pi$
$828$		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	$-0.958030\pi$
$839$		−	220.606i		−	0.262939i	0.991320	−	0.131470i	$-0.0419696\pi$
$840$		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	$-0.880518\pi$
$859$		−	629.837i		−	0.733221i	0.930375	−	0.366610i	$-0.119482\pi$
$860$		0			0
$861$	710.524			0.825232
$862$		0			0
$863$	−1144.00	+	1144.00i	−1.32561	+	1.32561i	−0.416449	+	0.909159i	$0.636726\pi$
$863$	−1144.00	+	1144.00i	−1.32561	+	1.32561i	−0.909159	+	0.416449i	$0.863274\pi$
$864$		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	−0.801949	−	0.597393i	$-0.796203\pi$
$883$	−180.623	+	180.623i	−0.204556	+	0.204556i	0.597393	+	0.801949i	$0.296203\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−853.106	−	853.106i	−0.961788	−	0.961788i	0.0375087	−	0.999296i	$-0.488058\pi$
$887$	−853.106	−	853.106i	−0.961788	−	0.961788i	−0.999296	+	0.0375087i	$0.988058\pi$
$888$		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	0.403193	−	0.915115i	$-0.367900\pi$
$907$	−464.313	−	464.313i	−0.511922	−	0.511922i	−0.915115	+	0.403193i	$0.867900\pi$
$908$		0			0
$909$		−	197.394i		−	0.217155i
$910$		0			0
$911$	−1456.95			−1.59928			−0.799642	−	0.600478i	$-0.794977\pi$
$911$	−1456.95			−1.59928			−0.799642	+	0.600478i	$0.794977\pi$
$912$		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	$0.0912963\pi$
$919$			519.969i			0.565799i	−0.959150	+	0.282900i	$0.908704\pi$
$920$		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	0.460352	−	0.887736i	$-0.347723\pi$
$947$	−404.733	−	404.733i	−0.427385	−	0.427385i	−0.887736	+	0.460352i	$0.847723\pi$
$948$		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	0.826319	−	0.563202i	$-0.190431\pi$
$967$	254.434	+	254.434i	0.263117	+	0.263117i	−0.563202	+	0.826319i	$0.690431\pi$
$968$		0			0
$969$		−	232.888i		−	0.240338i
$970$		0			0
$971$	−1594.36			−1.64198			−0.820990	−	0.570942i	$-0.806578\pi$
$971$	−1594.36			−1.64198			−0.820990	+	0.570942i	$0.806578\pi$
$972$		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	−0.990483	−	0.137633i	$-0.956051\pi$
$983$	−838.352	+	838.352i	−0.852850	+	0.852850i	0.137633	+	0.990483i	$0.456051\pi$
$984$		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	$-0.373877\pi$
$991$	764.937			0.771884			0.385942	+	0.922523i	$0.373877\pi$
$992$		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	1200.3.bg.c.1057.1		4
4.3	odd	2		600.3.u.g.457.2	yes	4
5.2	odd	4		1200.3.bg.n.193.2		4
5.3	odd	4	inner	1200.3.bg.c.193.1		4
5.4	even	2		1200.3.bg.n.1057.2		4
12.11	even	2		1800.3.v.p.1657.2		4
20.3	even	4		600.3.u.g.193.2	yes	4
20.7	even	4		600.3.u.b.193.1	✓	4
20.19	odd	2		600.3.u.b.457.1	yes	4
60.23	odd	4		1800.3.v.p.793.2		4
60.47	odd	4		1800.3.v.i.793.1		4
60.59	even	2		1800.3.v.i.1657.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.3.u.b.193.1	✓	4	20.7	even	4
600.3.u.b.457.1	yes	4	20.19	odd	2
600.3.u.g.193.2	yes	4	20.3	even	4
600.3.u.g.457.2	yes	4	4.3	odd	2
1200.3.bg.c.193.1		4	5.3	odd	4	inner
1200.3.bg.c.1057.1		4	1.1	even	1	trivial
1200.3.bg.n.193.2		4	5.2	odd	4
1200.3.bg.n.1057.2		4	5.4	even	2
1800.3.v.i.793.1		4	60.47	odd	4
1800.3.v.i.1657.1		4	60.59	even	2
1800.3.v.p.793.2		4	60.23	odd	4
1800.3.v.p.1657.2		4	12.11	even	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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1200.bg (of order \(4\), degree \(2\), not 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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