LMFDB - Embedded newform 1200.3.j.c.799.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1200,3,Mod(799,1200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1200.799"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1200.j (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,-120] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)

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

Self dual:	no
Analytic conductor:	$32.6976317232$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.1
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1200.799
Dual form			1200.3.j.c.799.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} -1.73205 q^{7} +3.00000 q^{9} -3.46410i q^{11} +11.0000i q^{13} +6.00000i q^{17} -19.0526i q^{19} +3.00000 q^{21} +17.3205 q^{23} -5.19615 q^{27} -30.0000 q^{29} -5.19615i q^{31} +6.00000i q^{33} +14.0000i q^{37} -19.0526i q^{39} +36.0000 q^{41} -5.19615 q^{43} -45.0333 q^{47} -46.0000 q^{49} -10.3923i q^{51} +72.0000i q^{53} +33.0000i q^{57} -38.1051i q^{59} +35.0000 q^{61} -5.19615 q^{63} -29.4449 q^{67} -30.0000 q^{69} +90.0666i q^{71} +62.0000i q^{73} +6.00000i q^{77} -76.2102i q^{79} +9.00000 q^{81} -72.7461 q^{83} +51.9615 q^{87} -144.000 q^{89} -19.0526i q^{91} +9.00000i q^{93} +181.000i q^{97} -10.3923i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{9} + 12 q^{21} - 120 q^{29} + 144 q^{41} - 184 q^{49} + 140 q^{61} - 120 q^{69} + 36 q^{81} - 576 q^{89}+O(q^{100}) $ 4 * q + 12 * q^9 + 12 * q^21 - 120 * q^29 + 144 * q^41 - 184 * q^49 + 140 * q^61 - 120 * q^69 + 36 * q^81 - 576 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$	−1.73205	−0.577350
$3$	−1.73205	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.73205	−0.247436	−0.123718	−	0.992317i	$-0.539482\pi$
$7$	−1.73205	−0.247436	−0.123718	+	0.992317i	$0.539482\pi$
$8$	0	0
$9$	3.00000	0.333333
$10$	0	0
$11$	− 3.46410i	− 0.314918i	−0.987525	−	0.157459i	$-0.949670\pi$
$11$	− 3.46410i	− 0.314918i	0.987525	−	0.157459i	$-0.0503303\pi$
$12$	0	0
$13$	11.0000i	0.846154i	0.906094	+	0.423077i	$0.139050\pi$
$13$	11.0000i	0.846154i	−0.906094	+	0.423077i	$0.860950\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.00000i	0.352941i	0.984306	+	0.176471i	$0.0564680\pi$
$17$	6.00000i	0.352941i	−0.984306	+	0.176471i	$0.943532\pi$
$18$	0	0
$19$	− 19.0526i	− 1.00277i	−0.865225	−	0.501383i	$-0.832825\pi$
$19$	− 19.0526i	− 1.00277i	0.865225	−	0.501383i	$-0.167175\pi$
$20$	0	0
$21$	3.00000	0.142857
$22$	0	0
$23$	17.3205	0.753066	0.376533	−	0.926403i	$-0.377116\pi$
$23$	17.3205	0.753066	0.376533	+	0.926403i	$0.377116\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.19615	−0.192450
$28$	0	0
$29$	−30.0000	−1.03448	−0.517241	−	0.855840i	$-0.673041\pi$
$29$	−30.0000	−1.03448	−0.517241	+	0.855840i	$0.673041\pi$
$30$	0	0
$31$	− 5.19615i	− 0.167618i	−0.996482	−	0.0838089i	$-0.973291\pi$
$31$	− 5.19615i	− 0.167618i	0.996482	−	0.0838089i	$-0.0267085\pi$
$32$	0	0
$33$	6.00000i	0.181818i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	14.0000i	0.378378i	0.981941	+	0.189189i	$0.0605859\pi$
$37$	14.0000i	0.378378i	−0.981941	+	0.189189i	$0.939414\pi$
$38$	0	0
$39$	− 19.0526i	− 0.488527i
$40$	0	0
$41$	36.0000	0.878049	0.439024	−	0.898475i	$-0.355324\pi$
$41$	36.0000	0.878049	0.439024	+	0.898475i	$0.355324\pi$
$42$	0	0
$43$	−5.19615	−0.120841	−0.0604204	−	0.998173i	$-0.519244\pi$
$43$	−5.19615	−0.120841	−0.0604204	+	0.998173i	$0.519244\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−45.0333	−0.958156	−0.479078	−	0.877772i	$-0.659029\pi$
$47$	−45.0333	−0.958156	−0.479078	+	0.877772i	$0.659029\pi$
$48$	0	0
$49$	−46.0000	−0.938776
$50$	0	0
$51$	− 10.3923i	− 0.203771i
$52$	0	0
$53$	72.0000i	1.35849i	0.733911	+	0.679245i	$0.237693\pi$
$53$	72.0000i	1.35849i	−0.733911	+	0.679245i	$0.762307\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	33.0000i	0.578947i
$58$	0	0
$59$	− 38.1051i	− 0.645849i	−0.946425	−	0.322925i	$-0.895334\pi$
$59$	− 38.1051i	− 0.645849i	0.946425	−	0.322925i	$-0.104666\pi$
$60$	0	0
$61$	35.0000	0.573770	0.286885	−	0.957965i	$-0.407380\pi$
$61$	35.0000	0.573770	0.286885	+	0.957965i	$0.407380\pi$
$62$	0	0
$63$	−5.19615	−0.0824786
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−29.4449	−0.439476	−0.219738	−	0.975559i	$-0.570520\pi$
$67$	−29.4449	−0.439476	−0.219738	+	0.975559i	$0.570520\pi$
$68$	0	0
$69$	−30.0000	−0.434783
$70$	0	0
$71$	90.0666i	1.26854i	0.773110	+	0.634272i	$0.218700\pi$
$71$	90.0666i	1.26854i	−0.773110	+	0.634272i	$0.781300\pi$
$72$	0	0
$73$	62.0000i	0.849315i	0.905354	+	0.424658i	$0.139605\pi$
$73$	62.0000i	0.849315i	−0.905354	+	0.424658i	$0.860395\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.00000i	0.0779221i
$78$	0	0
$79$	− 76.2102i	− 0.964687i	−0.875982	−	0.482343i	$-0.839786\pi$
$79$	− 76.2102i	− 0.964687i	0.875982	−	0.482343i	$-0.160214\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	−72.7461	−0.876459	−0.438230	−	0.898863i	$-0.644394\pi$
$83$	−72.7461	−0.876459	−0.438230	+	0.898863i	$0.644394\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	51.9615	0.597259
$88$	0	0
$89$	−144.000	−1.61798	−0.808989	−	0.587824i	$-0.799985\pi$
$89$	−144.000	−1.61798	−0.808989	+	0.587824i	$0.799985\pi$
$90$	0	0
$91$	− 19.0526i	− 0.209369i
$92$	0	0
$93$	9.00000i	0.0967742i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	181.000i	1.86598i	0.359903	+	0.932990i	$0.382810\pi$
$97$	181.000i	1.86598i	−0.359903	+	0.932990i	$0.617190\pi$
$98$	0	0
$99$	− 10.3923i	− 0.104973i
$100$	0	0
$101$	−48.0000	−0.475248	−0.237624	−	0.971357i	$-0.576369\pi$
$101$	−48.0000	−0.475248	−0.237624	+	0.971357i	$0.576369\pi$
$102$	0	0
$103$	−96.9948	−0.941698	−0.470849	−	0.882214i	$-0.656052\pi$
$103$	−96.9948	−0.941698	−0.470849	+	0.882214i	$0.656052\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−180.133	−1.68349	−0.841744	−	0.539876i	$-0.818471\pi$
$107$	−180.133	−1.68349	−0.841744	+	0.539876i	$0.818471\pi$
$108$	0	0
$109$	−169.000	−1.55046	−0.775229	−	0.631680i	$-0.782366\pi$
$109$	−169.000	−1.55046	−0.775229	+	0.631680i	$0.782366\pi$
$110$	0	0
$111$	− 24.2487i	− 0.218457i
$112$	0	0
$113$	12.0000i	0.106195i	0.998589	+	0.0530973i	$0.0169094\pi$
$113$	12.0000i	0.106195i	−0.998589	+	0.0530973i	$0.983091\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	33.0000i	0.282051i
$118$	0	0
$119$	− 10.3923i	− 0.0873303i
$120$	0	0
$121$	109.000	0.900826
$122$	0	0
$123$	−62.3538	−0.506942
$124$	0	0
$125$	0	0
$126$	0	0
$127$	103.923	0.818292	0.409146	−	0.912469i	$-0.365827\pi$
$127$	103.923	0.818292	0.409146	+	0.912469i	$0.365827\pi$
$128$	0	0
$129$	9.00000	0.0697674
$130$	0	0
$131$	69.2820i	0.528870i	0.964403	+	0.264435i	$0.0851855\pi$
$131$	69.2820i	0.528870i	−0.964403	+	0.264435i	$0.914814\pi$
$132$	0	0
$133$	33.0000i	0.248120i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	54.0000i	0.394161i	0.980387	+	0.197080i	$0.0631460\pi$
$137$	54.0000i	0.394161i	−0.980387	+	0.197080i	$0.936854\pi$
$138$	0	0
$139$	− 48.4974i	− 0.348902i	−0.984666	−	0.174451i	$-0.944185\pi$
$139$	− 48.4974i	− 0.348902i	0.984666	−	0.174451i	$-0.0558151\pi$
$140$	0	0
$141$	78.0000	0.553191
$142$	0	0
$143$	38.1051	0.266469
$144$	0	0
$145$	0	0
$146$	0	0
$147$	79.6743	0.542002
$148$	0	0
$149$	−234.000	−1.57047	−0.785235	−	0.619198i	$-0.787458\pi$
$149$	−234.000	−1.57047	−0.785235	+	0.619198i	$0.787458\pi$
$150$	0	0
$151$	164.545i	1.08970i	0.838533	+	0.544850i	$0.183414\pi$
$151$	164.545i	1.08970i	−0.838533	+	0.544850i	$0.816586\pi$
$152$	0	0
$153$	18.0000i	0.117647i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	35.0000i	0.222930i	0.993768	+	0.111465i	$0.0355543\pi$
$157$	35.0000i	0.222930i	−0.993768	+	0.111465i	$0.964446\pi$
$158$	0	0
$159$	− 124.708i	− 0.784325i
$160$	0	0
$161$	−30.0000	−0.186335
$162$	0	0
$163$	−174.937	−1.07323	−0.536617	−	0.843826i	$-0.680298\pi$
$163$	−174.937	−1.07323	−0.536617	+	0.843826i	$0.680298\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−138.564	−0.829725	−0.414862	−	0.909884i	$-0.636170\pi$
$167$	−138.564	−0.829725	−0.414862	+	0.909884i	$0.636170\pi$
$168$	0	0
$169$	48.0000	0.284024
$170$	0	0
$171$	− 57.1577i	− 0.334255i
$172$	0	0
$173$	198.000i	1.14451i	0.820076	+	0.572254i	$0.193931\pi$
$173$	198.000i	1.14451i	−0.820076	+	0.572254i	$0.806069\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	66.0000i	0.372881i
$178$	0	0
$179$	252.879i	1.41273i	0.707846	+	0.706367i	$0.249667\pi$
$179$	252.879i	1.41273i	−0.707846	+	0.706367i	$0.750333\pi$
$180$	0	0
$181$	−23.0000	−0.127072	−0.0635359	−	0.997980i	$-0.520238\pi$
$181$	−23.0000	−0.127072	−0.0635359	+	0.997980i	$0.520238\pi$
$182$	0	0
$183$	−60.6218	−0.331267
$184$	0	0
$185$	0	0
$186$	0	0
$187$	20.7846	0.111148
$188$	0	0
$189$	9.00000	0.0476190
$190$	0	0
$191$	239.023i	1.25143i	0.780052	+	0.625715i	$0.215193\pi$
$191$	239.023i	1.25143i	−0.780052	+	0.625715i	$0.784807\pi$
$192$	0	0
$193$	− 59.0000i	− 0.305699i	−0.988249	−	0.152850i	$-0.951155\pi$
$193$	− 59.0000i	− 0.305699i	0.988249	−	0.152850i	$-0.0488451\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 162.000i	− 0.822335i	−0.911560	−	0.411168i	$-0.865121\pi$
$197$	− 162.000i	− 0.822335i	0.911560	−	0.411168i	$-0.134879\pi$
$198$	0	0
$199$	− 219.970i	− 1.10538i	−0.833387	−	0.552690i	$-0.813602\pi$
$199$	− 219.970i	− 1.10538i	0.833387	−	0.552690i	$-0.186398\pi$
$200$	0	0
$201$	51.0000	0.253731
$202$	0	0
$203$	51.9615	0.255968
$204$	0	0
$205$	0	0
$206$	0	0
$207$	51.9615	0.251022
$208$	0	0
$209$	−66.0000	−0.315789
$210$	0	0
$211$	− 403.568i	− 1.91264i	−0.292316	−	0.956322i	$-0.594426\pi$
$211$	− 403.568i	− 1.91264i	0.292316	−	0.956322i	$-0.405574\pi$
$212$	0	0
$213$	− 156.000i	− 0.732394i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	9.00000i	0.0414747i
$218$	0	0
$219$	− 107.387i	− 0.490352i
$220$	0	0
$221$	−66.0000	−0.298643
$222$	0	0
$223$	223.435	1.00195	0.500974	−	0.865462i	$-0.332975\pi$
$223$	223.435	1.00195	0.500974	+	0.865462i	$0.332975\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−422.620	−1.86176	−0.930882	−	0.365320i	$-0.880960\pi$
$227$	−422.620	−1.86176	−0.930882	+	0.365320i	$0.880960\pi$
$228$	0	0
$229$	59.0000	0.257642	0.128821	−	0.991668i	$-0.458881\pi$
$229$	59.0000	0.257642	0.128821	+	0.991668i	$0.458881\pi$
$230$	0	0
$231$	− 10.3923i	− 0.0449883i
$232$	0	0
$233$	84.0000i	0.360515i	0.983619	+	0.180258i	$0.0576931\pi$
$233$	84.0000i	0.360515i	−0.983619	+	0.180258i	$0.942307\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	132.000i	0.556962i
$238$	0	0
$239$	193.990i	0.811672i	0.913946	+	0.405836i	$0.133020\pi$
$239$	193.990i	0.811672i	−0.913946	+	0.405836i	$0.866980\pi$
$240$	0	0
$241$	73.0000	0.302905	0.151452	−	0.988465i	$-0.451605\pi$
$241$	73.0000	0.302905	0.151452	+	0.988465i	$0.451605\pi$
$242$	0	0
$243$	−15.5885	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	209.578	0.848495
$248$	0	0
$249$	126.000	0.506024
$250$	0	0
$251$	− 34.6410i	− 0.138012i	−0.997616	−	0.0690060i	$-0.978017\pi$
$251$	− 34.6410i	− 0.138012i	0.997616	−	0.0690060i	$-0.0219828\pi$
$252$	0	0
$253$	− 60.0000i	− 0.237154i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 228.000i	− 0.887160i	−0.896235	−	0.443580i	$-0.853708\pi$
$257$	− 228.000i	− 0.887160i	0.896235	−	0.443580i	$-0.146292\pi$
$258$	0	0
$259$	− 24.2487i	− 0.0936244i
$260$	0	0
$261$	−90.0000	−0.344828
$262$	0	0
$263$	422.620	1.60692	0.803461	−	0.595358i	$-0.202990\pi$
$263$	422.620	1.60692	0.803461	+	0.595358i	$0.202990\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	249.415	0.934140
$268$	0	0
$269$	270.000	1.00372	0.501859	−	0.864950i	$-0.332650\pi$
$269$	270.000	1.00372	0.501859	+	0.864950i	$0.332650\pi$
$270$	0	0
$271$	34.6410i	0.127827i	0.997955	+	0.0639133i	$0.0203581\pi$
$271$	34.6410i	0.127827i	−0.997955	+	0.0639133i	$0.979642\pi$
$272$	0	0
$273$	33.0000i	0.120879i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	49.0000i	0.176895i	0.996081	+	0.0884477i	$0.0281906\pi$
$277$	49.0000i	0.176895i	−0.996081	+	0.0884477i	$0.971809\pi$
$278$	0	0
$279$	− 15.5885i	− 0.0558726i
$280$	0	0
$281$	−246.000	−0.875445	−0.437722	−	0.899110i	$-0.644215\pi$
$281$	−246.000	−0.875445	−0.437722	+	0.899110i	$0.644215\pi$
$282$	0	0
$283$	−140.296	−0.495746	−0.247873	−	0.968793i	$-0.579732\pi$
$283$	−140.296	−0.495746	−0.247873	+	0.968793i	$0.579732\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−62.3538	−0.217261
$288$	0	0
$289$	253.000	0.875433
$290$	0	0
$291$	− 313.501i	− 1.07732i
$292$	0	0
$293$	− 78.0000i	− 0.266212i	−0.991102	−	0.133106i	$-0.957505\pi$
$293$	− 78.0000i	− 0.266212i	0.991102	−	0.133106i	$-0.0424950\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	18.0000i	0.0606061i
$298$	0	0
$299$	190.526i	0.637209i
$300$	0	0
$301$	9.00000	0.0299003
$302$	0	0
$303$	83.1384	0.274384
$304$	0	0
$305$	0	0
$306$	0	0
$307$	202.650	0.660098	0.330049	−	0.943964i	$-0.392935\pi$
$307$	202.650	0.660098	0.330049	+	0.943964i	$0.392935\pi$
$308$	0	0
$309$	168.000	0.543689
$310$	0	0
$311$	370.659i	1.19183i	0.803048	+	0.595915i	$0.203210\pi$
$311$	370.659i	1.19183i	−0.803048	+	0.595915i	$0.796790\pi$
$312$	0	0
$313$	25.0000i	0.0798722i	0.999202	+	0.0399361i	$0.0127154\pi$
$313$	25.0000i	0.0798722i	−0.999202	+	0.0399361i	$0.987285\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 600.000i	− 1.89274i	−0.323078	−	0.946372i	$-0.604718\pi$
$317$	− 600.000i	− 1.89274i	0.323078	−	0.946372i	$-0.395282\pi$
$318$	0	0
$319$	103.923i	0.325778i
$320$	0	0
$321$	312.000	0.971963
$322$	0	0
$323$	114.315	0.353918
$324$	0	0
$325$	0	0
$326$	0	0
$327$	292.717	0.895158
$328$	0	0
$329$	78.0000	0.237082
$330$	0	0
$331$	− 408.764i	− 1.23494i	−0.786596	−	0.617468i	$-0.788158\pi$
$331$	− 408.764i	− 1.23494i	0.786596	−	0.617468i	$-0.211842\pi$
$332$	0	0
$333$	42.0000i	0.126126i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 203.000i	− 0.602374i	−0.953565	−	0.301187i	$-0.902617\pi$
$337$	− 203.000i	− 0.602374i	0.953565	−	0.301187i	$-0.0973828\pi$
$338$	0	0
$339$	− 20.7846i	− 0.0613115i
$340$	0	0
$341$	−18.0000	−0.0527859
$342$	0	0
$343$	164.545	0.479723
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−613.146	−1.76699	−0.883496	−	0.468439i	$-0.844816\pi$
$347$	−613.146	−1.76699	−0.883496	+	0.468439i	$0.844816\pi$
$348$	0	0
$349$	538.000	1.54155	0.770774	−	0.637109i	$-0.219870\pi$
$349$	538.000	1.54155	0.770774	+	0.637109i	$0.219870\pi$
$350$	0	0
$351$	− 57.1577i	− 0.162842i
$352$	0	0
$353$	− 498.000i	− 1.41076i	−0.708827	−	0.705382i	$-0.750775\pi$
$353$	− 498.000i	− 1.41076i	0.708827	−	0.705382i	$-0.249225\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	18.0000i	0.0504202i
$358$	0	0
$359$	90.0666i	0.250882i	0.992101	+	0.125441i	$0.0400346\pi$
$359$	90.0666i	0.250882i	−0.992101	+	0.125441i	$0.959965\pi$
$360$	0	0
$361$	−2.00000	−0.00554017
$362$	0	0
$363$	−188.794	−0.520092
$364$	0	0
$365$	0	0
$366$	0	0
$367$	569.845	1.55271	0.776355	−	0.630296i	$-0.217066\pi$
$367$	569.845	1.55271	0.776355	+	0.630296i	$0.217066\pi$
$368$	0	0
$369$	108.000	0.292683
$370$	0	0
$371$	− 124.708i	− 0.336139i
$372$	0	0
$373$	− 491.000i	− 1.31635i	−0.752863	−	0.658177i	$-0.771328\pi$
$373$	− 491.000i	− 1.31635i	0.752863	−	0.658177i	$-0.228672\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 330.000i	− 0.875332i
$378$	0	0
$379$	323.894i	0.854600i	0.904110	+	0.427300i	$0.140535\pi$
$379$	323.894i	0.854600i	−0.904110	+	0.427300i	$0.859465\pi$
$380$	0	0
$381$	−180.000	−0.472441
$382$	0	0
$383$	367.195	0.958733	0.479367	−	0.877615i	$-0.340866\pi$
$383$	367.195	0.958733	0.479367	+	0.877615i	$0.340866\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−15.5885	−0.0402803
$388$	0	0
$389$	132.000	0.339332	0.169666	−	0.985502i	$-0.445731\pi$
$389$	132.000	0.339332	0.169666	+	0.985502i	$0.445731\pi$
$390$	0	0
$391$	103.923i	0.265788i
$392$	0	0
$393$	− 120.000i	− 0.305344i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	251.000i	0.632242i	0.948719	+	0.316121i	$0.102381\pi$
$397$	251.000i	0.632242i	−0.948719	+	0.316121i	$0.897619\pi$
$398$	0	0
$399$	− 57.1577i	− 0.143252i
$400$	0	0
$401$	−720.000	−1.79551	−0.897756	−	0.440494i	$-0.854803\pi$
$401$	−720.000	−1.79551	−0.897756	+	0.440494i	$0.854803\pi$
$402$	0	0
$403$	57.1577	0.141830
$404$	0	0
$405$	0	0
$406$	0	0
$407$	48.4974	0.119158
$408$	0	0
$409$	515.000	1.25917	0.629584	−	0.776932i	$-0.283225\pi$
$409$	515.000	1.25917	0.629584	+	0.776932i	$0.283225\pi$
$410$	0	0
$411$	− 93.5307i	− 0.227569i
$412$	0	0
$413$	66.0000i	0.159806i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	84.0000i	0.201439i
$418$	0	0
$419$	789.815i	1.88500i	0.334206	+	0.942500i	$0.391532\pi$
$419$	789.815i	1.88500i	−0.334206	+	0.942500i	$0.608468\pi$
$420$	0	0
$421$	−278.000	−0.660333	−0.330166	−	0.943923i	$-0.607105\pi$
$421$	−278.000	−0.660333	−0.330166	+	0.943923i	$0.607105\pi$
$422$	0	0
$423$	−135.100	−0.319385
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−60.6218	−0.141971
$428$	0	0
$429$	−66.0000	−0.153846
$430$	0	0
$431$	627.002i	1.45476i	0.686234	+	0.727381i	$0.259263\pi$
$431$	627.002i	1.45476i	−0.686234	+	0.727381i	$0.740737\pi$
$432$	0	0
$433$	− 287.000i	− 0.662818i	−0.943487	−	0.331409i	$-0.892476\pi$
$433$	− 287.000i	− 0.662818i	0.943487	−	0.331409i	$-0.107524\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 330.000i	− 0.755149i
$438$	0	0
$439$	− 244.219i	− 0.556308i	−0.960537	−	0.278154i	$-0.910277\pi$
$439$	− 244.219i	− 0.556308i	0.960537	−	0.278154i	$-0.0897225\pi$
$440$	0	0
$441$	−138.000	−0.312925
$442$	0	0
$443$	−221.703	−0.500457	−0.250229	−	0.968187i	$-0.580506\pi$
$443$	−221.703	−0.500457	−0.250229	+	0.968187i	$0.580506\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	405.300	0.906711
$448$	0	0
$449$	−132.000	−0.293987	−0.146993	−	0.989137i	$-0.546960\pi$
$449$	−132.000	−0.293987	−0.146993	+	0.989137i	$0.546960\pi$
$450$	0	0
$451$	− 124.708i	− 0.276514i
$452$	0	0
$453$	− 285.000i	− 0.629139i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 434.000i	− 0.949672i	−0.880074	−	0.474836i	$-0.842507\pi$
$457$	− 434.000i	− 0.949672i	0.880074	−	0.474836i	$-0.157493\pi$
$458$	0	0
$459$	− 31.1769i	− 0.0679236i
$460$	0	0
$461$	312.000	0.676790	0.338395	−	0.941004i	$-0.390116\pi$
$461$	312.000	0.676790	0.338395	+	0.941004i	$0.390116\pi$
$462$	0	0
$463$	90.0666	0.194528	0.0972642	−	0.995259i	$-0.468991\pi$
$463$	90.0666	0.194528	0.0972642	+	0.995259i	$0.468991\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.46410	0.00741778	0.00370889	−	0.999993i	$-0.498819\pi$
$467$	3.46410	0.00741778	0.00370889	+	0.999993i	$0.498819\pi$
$468$	0	0
$469$	51.0000	0.108742
$470$	0	0
$471$	− 60.6218i	− 0.128709i
$472$	0	0
$473$	18.0000i	0.0380550i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	216.000i	0.452830i
$478$	0	0
$479$	737.854i	1.54040i	0.637800	+	0.770202i	$0.279845\pi$
$479$	737.854i	1.54040i	−0.637800	+	0.770202i	$0.720155\pi$
$480$	0	0
$481$	−154.000	−0.320166
$482$	0	0
$483$	51.9615	0.107581
$484$	0	0
$485$	0	0
$486$	0	0
$487$	258.076	0.529929	0.264965	−	0.964258i	$-0.414640\pi$
$487$	258.076	0.529929	0.264965	+	0.964258i	$0.414640\pi$
$488$	0	0
$489$	303.000	0.619632
$490$	0	0
$491$	− 332.554i	− 0.677299i	−0.940913	−	0.338649i	$-0.890030\pi$
$491$	− 332.554i	− 0.677299i	0.940913	−	0.338649i	$-0.109970\pi$
$492$	0	0
$493$	− 180.000i	− 0.365112i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 156.000i	− 0.313883i
$498$	0	0
$499$	306.573i	0.614375i	0.951649	+	0.307187i	$0.0993878\pi$
$499$	306.573i	0.614375i	−0.951649	+	0.307187i	$0.900612\pi$
$500$	0	0
$501$	240.000	0.479042
$502$	0	0
$503$	187.061	0.371892	0.185946	−	0.982560i	$-0.440465\pi$
$503$	187.061	0.371892	0.185946	+	0.982560i	$0.440465\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−83.1384	−0.163981
$508$	0	0
$509$	546.000	1.07269	0.536346	−	0.843998i	$-0.319804\pi$
$509$	546.000	1.07269	0.536346	+	0.843998i	$0.319804\pi$
$510$	0	0
$511$	− 107.387i	− 0.210151i
$512$	0	0
$513$	99.0000i	0.192982i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	156.000i	0.301741i
$518$	0	0
$519$	− 342.946i	− 0.660782i
$520$	0	0
$521$	−318.000	−0.610365	−0.305182	−	0.952294i	$-0.598717\pi$
$521$	−318.000	−0.610365	−0.305182	+	0.952294i	$0.598717\pi$
$522$	0	0
$523$	−895.470	−1.71218	−0.856090	−	0.516827i	$-0.827113\pi$
$523$	−895.470	−1.71218	−0.856090	+	0.516827i	$0.827113\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	31.1769	0.0591592
$528$	0	0
$529$	−229.000	−0.432892
$530$	0	0
$531$	− 114.315i	− 0.215283i
$532$	0	0
$533$	396.000i	0.742964i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 438.000i	− 0.815642i
$538$	0	0
$539$	159.349i	0.295638i
$540$	0	0
$541$	157.000	0.290203	0.145102	−	0.989417i	$-0.453649\pi$
$541$	157.000	0.290203	0.145102	+	0.989417i	$0.453649\pi$
$542$	0	0
$543$	39.8372	0.0733650
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−69.2820	−0.126658	−0.0633291	−	0.997993i	$-0.520172\pi$
$547$	−69.2820	−0.126658	−0.0633291	+	0.997993i	$0.520172\pi$
$548$	0	0
$549$	105.000	0.191257
$550$	0	0
$551$	571.577i	1.03734i
$552$	0	0
$553$	132.000i	0.238698i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	114.000i	0.204668i	0.994750	+	0.102334i	$0.0326310\pi$
$557$	114.000i	0.204668i	−0.994750	+	0.102334i	$0.967369\pi$
$558$	0	0
$559$	− 57.1577i	− 0.102250i
$560$	0	0
$561$	−36.0000	−0.0641711
$562$	0	0
$563$	128.172	0.227659	0.113829	−	0.993500i	$-0.463688\pi$
$563$	128.172	0.227659	0.113829	+	0.993500i	$0.463688\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−15.5885	−0.0274929
$568$	0	0
$569$	−450.000	−0.790861	−0.395431	−	0.918496i	$-0.629405\pi$
$569$	−450.000	−0.790861	−0.395431	+	0.918496i	$0.629405\pi$
$570$	0	0
$571$	− 22.5167i	− 0.0394337i	−0.999806	−	0.0197169i	$-0.993724\pi$
$571$	− 22.5167i	− 0.0394337i	0.999806	−	0.0197169i	$-0.00627648\pi$
$572$	0	0
$573$	− 414.000i	− 0.722513i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	311.000i	0.538995i	0.963001	+	0.269497i	$0.0868576\pi$
$577$	311.000i	0.538995i	−0.963001	+	0.269497i	$0.913142\pi$
$578$	0	0
$579$	102.191i	0.176496i
$580$	0	0
$581$	126.000	0.216867
$582$	0	0
$583$	249.415	0.427814
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−166.277	−0.283266	−0.141633	−	0.989919i	$-0.545235\pi$
$587$	−166.277	−0.283266	−0.141633	+	0.989919i	$0.545235\pi$
$588$	0	0
$589$	−99.0000	−0.168081
$590$	0	0
$591$	280.592i	0.474775i
$592$	0	0
$593$	− 636.000i	− 1.07251i	−0.844055	−	0.536256i	$-0.819838\pi$
$593$	− 636.000i	− 1.07251i	0.844055	−	0.536256i	$-0.180162\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	381.000i	0.638191i
$598$	0	0
$599$	− 769.031i	− 1.28386i	−0.766764	−	0.641929i	$-0.778135\pi$
$599$	− 769.031i	− 1.28386i	0.766764	−	0.641929i	$-0.221865\pi$
$600$	0	0
$601$	371.000	0.617304	0.308652	−	0.951175i	$-0.400122\pi$
$601$	371.000	0.617304	0.308652	+	0.951175i	$0.400122\pi$
$602$	0	0
$603$	−88.3346	−0.146492
$604$	0	0
$605$	0	0
$606$	0	0
$607$	533.472	0.878866	0.439433	−	0.898275i	$-0.355179\pi$
$607$	533.472	0.878866	0.439433	+	0.898275i	$0.355179\pi$
$608$	0	0
$609$	−90.0000	−0.147783
$610$	0	0
$611$	− 495.367i	− 0.810747i
$612$	0	0
$613$	− 374.000i	− 0.610114i	−0.952334	−	0.305057i	$-0.901324\pi$
$613$	− 374.000i	− 0.610114i	0.952334	−	0.305057i	$-0.0986755\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	240.000i	0.388979i	0.980905	+	0.194489i	$0.0623050\pi$
$617$	240.000i	0.388979i	−0.980905	+	0.194489i	$0.937695\pi$
$618$	0	0
$619$	− 621.806i	− 1.00453i	−0.864713	−	0.502267i	$-0.832500\pi$
$619$	− 621.806i	− 1.00453i	0.864713	−	0.502267i	$-0.167500\pi$
$620$	0	0
$621$	−90.0000	−0.144928
$622$	0	0
$623$	249.415	0.400346
$624$	0	0
$625$	0	0
$626$	0	0
$627$	114.315	0.182321
$628$	0	0
$629$	−84.0000	−0.133545
$630$	0	0
$631$	549.060i	0.870143i	0.900396	+	0.435071i	$0.143277\pi$
$631$	549.060i	0.870143i	−0.900396	+	0.435071i	$0.856723\pi$
$632$	0	0
$633$	699.000i	1.10427i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 506.000i	− 0.794349i
$638$	0	0
$639$	270.200i	0.422848i
$640$	0	0
$641$	−300.000	−0.468019	−0.234009	−	0.972234i	$-0.575185\pi$
$641$	−300.000	−0.468019	−0.234009	+	0.972234i	$0.575185\pi$
$642$	0	0
$643$	394.908	0.614164	0.307082	−	0.951683i	$-0.400647\pi$
$643$	394.908	0.614164	0.307082	+	0.951683i	$0.400647\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−699.749	−1.08153	−0.540764	−	0.841174i	$-0.681865\pi$
$647$	−699.749	−1.08153	−0.540764	+	0.841174i	$0.681865\pi$
$648$	0	0
$649$	−132.000	−0.203390
$650$	0	0
$651$	− 15.5885i	− 0.0239454i
$652$	0	0
$653$	534.000i	0.817764i	0.912587	+	0.408882i	$0.134081\pi$
$653$	534.000i	0.817764i	−0.912587	+	0.408882i	$0.865919\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	186.000i	0.283105i
$658$	0	0
$659$	− 436.477i	− 0.662332i	−0.943573	−	0.331166i	$-0.892558\pi$
$659$	− 436.477i	− 0.662332i	0.943573	−	0.331166i	$-0.107442\pi$
$660$	0	0
$661$	−34.0000	−0.0514372	−0.0257186	−	0.999669i	$-0.508187\pi$
$661$	−34.0000	−0.0514372	−0.0257186	+	0.999669i	$0.508187\pi$
$662$	0	0
$663$	114.315	0.172421
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−519.615	−0.779033
$668$	0	0
$669$	−387.000	−0.578475
$670$	0	0
$671$	− 121.244i	− 0.180691i
$672$	0	0
$673$	1234.00i	1.83358i	0.399368	+	0.916790i	$0.369229\pi$
$673$	1234.00i	1.83358i	−0.399368	+	0.916790i	$0.630771\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	534.000i	0.788774i	0.918944	+	0.394387i	$0.129043\pi$
$677$	534.000i	0.788774i	−0.918944	+	0.394387i	$0.870957\pi$
$678$	0	0
$679$	− 313.501i	− 0.461710i
$680$	0	0
$681$	732.000	1.07489
$682$	0	0
$683$	−526.543	−0.770927	−0.385464	−	0.922723i	$-0.625959\pi$
$683$	−526.543	−0.770927	−0.385464	+	0.922723i	$0.625959\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−102.191	−0.148750
$688$	0	0
$689$	−792.000	−1.14949
$690$	0	0
$691$	− 1122.37i	− 1.62427i	−0.583471	−	0.812134i	$-0.698306\pi$
$691$	− 1122.37i	− 1.62427i	0.583471	−	0.812134i	$-0.301694\pi$
$692$	0	0
$693$	18.0000i	0.0259740i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	216.000i	0.309900i
$698$	0	0
$699$	− 145.492i	− 0.208143i
$700$	0	0
$701$	306.000	0.436519	0.218260	−	0.975891i	$-0.429962\pi$
$701$	306.000	0.436519	0.218260	+	0.975891i	$0.429962\pi$
$702$	0	0
$703$	266.736	0.379425
$704$	0	0
$705$	0	0
$706$	0	0
$707$	83.1384	0.117593
$708$	0	0
$709$	−433.000	−0.610719	−0.305360	−	0.952237i	$-0.598777\pi$
$709$	−433.000	−0.610719	−0.305360	+	0.952237i	$0.598777\pi$
$710$	0	0
$711$	− 228.631i	− 0.321562i
$712$	0	0
$713$	− 90.0000i	− 0.126227i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 336.000i	− 0.468619i
$718$	0	0
$719$	− 585.433i	− 0.814233i	−0.913376	−	0.407116i	$-0.866534\pi$
$719$	− 585.433i	− 0.814233i	0.913376	−	0.407116i	$-0.133466\pi$
$720$	0	0
$721$	168.000	0.233010
$722$	0	0
$723$	−126.440	−0.174882
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−930.111	−1.27938	−0.639691	−	0.768632i	$-0.720938\pi$
$727$	−930.111	−1.27938	−0.639691	+	0.768632i	$0.720938\pi$
$728$	0	0
$729$	27.0000	0.0370370
$730$	0	0
$731$	− 31.1769i	− 0.0426497i
$732$	0	0
$733$	590.000i	0.804911i	0.915439	+	0.402456i	$0.131843\pi$
$733$	590.000i	0.804911i	−0.915439	+	0.402456i	$0.868157\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	102.000i	0.138399i
$738$	0	0
$739$	644.323i	0.871885i	0.899975	+	0.435942i	$0.143585\pi$
$739$	644.323i	0.871885i	−0.899975	+	0.435942i	$0.856415\pi$
$740$	0	0
$741$	−363.000	−0.489879
$742$	0	0
$743$	−741.318	−0.997736	−0.498868	−	0.866678i	$-0.666251\pi$
$743$	−741.318	−0.997736	−0.498868	+	0.866678i	$0.666251\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−218.238	−0.292153
$748$	0	0
$749$	312.000	0.416555
$750$	0	0
$751$	907.595i	1.20851i	0.796789	+	0.604257i	$0.206530\pi$
$751$	907.595i	1.20851i	−0.796789	+	0.604257i	$0.793470\pi$
$752$	0	0
$753$	60.0000i	0.0796813i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	841.000i	1.11096i	0.831528	+	0.555482i	$0.187466\pi$
$757$	841.000i	1.11096i	−0.831528	+	0.555482i	$0.812534\pi$
$758$	0	0
$759$	103.923i	0.136921i
$760$	0	0
$761$	924.000	1.21419	0.607096	−	0.794629i	$-0.292334\pi$
$761$	924.000	1.21419	0.607096	+	0.794629i	$0.292334\pi$
$762$	0	0
$763$	292.717	0.383639
$764$	0	0
$765$	0	0
$766$	0	0
$767$	419.156	0.546488
$768$	0	0
$769$	−347.000	−0.451235	−0.225618	−	0.974216i	$-0.572440\pi$
$769$	−347.000	−0.451235	−0.225618	+	0.974216i	$0.572440\pi$
$770$	0	0
$771$	394.908i	0.512202i
$772$	0	0
$773$	1404.00i	1.81630i	0.418645	+	0.908150i	$0.362505\pi$
$773$	1404.00i	1.81630i	−0.418645	+	0.908150i	$0.637495\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	42.0000i	0.0540541i
$778$	0	0
$779$	− 685.892i	− 0.880478i
$780$	0	0
$781$	312.000	0.399488
$782$	0	0
$783$	155.885	0.199086
$784$	0	0
$785$	0	0
$786$	0	0
$787$	43.3013	0.0550207	0.0275103	−	0.999622i	$-0.491242\pi$
$787$	43.3013	0.0550207	0.0275103	+	0.999622i	$0.491242\pi$
$788$	0	0
$789$	−732.000	−0.927757
$790$	0	0
$791$	− 20.7846i	− 0.0262764i
$792$	0	0
$793$	385.000i	0.485498i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	348.000i	0.436637i	0.975878	+	0.218319i	$0.0700572\pi$
$797$	348.000i	0.436637i	−0.975878	+	0.218319i	$0.929943\pi$
$798$	0	0
$799$	− 270.200i	− 0.338173i
$800$	0	0
$801$	−432.000	−0.539326
$802$	0	0
$803$	214.774	0.267465
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−467.654	−0.579497
$808$	0	0
$809$	−384.000	−0.474660	−0.237330	−	0.971429i	$-0.576272\pi$
$809$	−384.000	−0.474660	−0.237330	+	0.971429i	$0.576272\pi$
$810$	0	0
$811$	691.088i	0.852143i	0.904689	+	0.426072i	$0.140103\pi$
$811$	691.088i	0.852143i	−0.904689	+	0.426072i	$0.859897\pi$
$812$	0	0
$813$	− 60.0000i	− 0.0738007i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	99.0000i	0.121175i
$818$	0	0
$819$	− 57.1577i	− 0.0697896i
$820$	0	0
$821$	462.000	0.562728	0.281364	−	0.959601i	$-0.409213\pi$
$821$	462.000	0.562728	0.281364	+	0.959601i	$0.409213\pi$
$822$	0	0
$823$	−614.878	−0.747118	−0.373559	−	0.927606i	$-0.621863\pi$
$823$	−614.878	−0.747118	−0.373559	+	0.927606i	$0.621863\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−193.990	−0.234570	−0.117285	−	0.993098i	$-0.537419\pi$
$827$	−193.990	−0.234570	−0.117285	+	0.993098i	$0.537419\pi$
$828$	0	0
$829$	−1606.00	−1.93727	−0.968637	−	0.248480i	$-0.920069\pi$
$829$	−1606.00	−1.93727	−0.968637	+	0.248480i	$0.920069\pi$
$830$	0	0
$831$	− 84.8705i	− 0.102131i
$832$	0	0
$833$	− 276.000i	− 0.331333i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	27.0000i	0.0322581i
$838$	0	0
$839$	− 897.202i	− 1.06937i	−0.845051	−	0.534686i	$-0.820430\pi$
$839$	− 897.202i	− 1.06937i	0.845051	−	0.534686i	$-0.179570\pi$
$840$	0	0
$841$	59.0000	0.0701546
$842$	0	0
$843$	426.084	0.505438
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−188.794	−0.222897
$848$	0	0
$849$	243.000	0.286219
$850$	0	0
$851$	242.487i	0.284944i
$852$	0	0
$853$	− 575.000i	− 0.674091i	−0.941488	−	0.337046i	$-0.890572\pi$
$853$	− 575.000i	− 0.674091i	0.941488	−	0.337046i	$-0.109428\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1068.00i	1.24621i	0.782139	+	0.623104i	$0.214129\pi$
$857$	1068.00i	1.24621i	−0.782139	+	0.623104i	$0.785871\pi$
$858$	0	0
$859$	− 1233.22i	− 1.43565i	−0.696225	−	0.717823i	$-0.745139\pi$
$859$	− 1233.22i	− 1.43565i	0.696225	−	0.717823i	$-0.254861\pi$
$860$	0	0
$861$	108.000	0.125436
$862$	0	0
$863$	166.277	0.192673	0.0963365	−	0.995349i	$-0.469287\pi$
$863$	166.277	0.192673	0.0963365	+	0.995349i	$0.469287\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−438.209	−0.505431
$868$	0	0
$869$	−264.000	−0.303797
$870$	0	0
$871$	− 323.894i	− 0.371864i
$872$	0	0
$873$	543.000i	0.621993i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	731.000i	0.833523i	0.909016	+	0.416762i	$0.136835\pi$
$877$	731.000i	0.833523i	−0.909016	+	0.416762i	$0.863165\pi$
$878$	0	0
$879$	135.100i	0.153697i
$880$	0	0
$881$	48.0000	0.0544835	0.0272418	−	0.999629i	$-0.491328\pi$
$881$	48.0000	0.0544835	0.0272418	+	0.999629i	$0.491328\pi$
$882$	0	0
$883$	−1037.50	−1.17497	−0.587485	−	0.809235i	$-0.699882\pi$
$883$	−1037.50	−1.17497	−0.587485	+	0.809235i	$0.699882\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1375.25	1.55045	0.775225	−	0.631686i	$-0.217637\pi$
$887$	1375.25	1.55045	0.775225	+	0.631686i	$0.217637\pi$
$888$	0	0
$889$	−180.000	−0.202475
$890$	0	0
$891$	− 31.1769i	− 0.0349909i
$892$	0	0
$893$	858.000i	0.960806i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 330.000i	− 0.367893i
$898$	0	0
$899$	155.885i	0.173398i
$900$	0	0
$901$	−432.000	−0.479467
$902$	0	0
$903$	−15.5885	−0.0172630
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1531.13	−1.68813	−0.844064	−	0.536242i	$-0.819844\pi$
$907$	−1531.13	−1.68813	−0.844064	+	0.536242i	$0.819844\pi$
$908$	0	0
$909$	−144.000	−0.158416
$910$	0	0
$911$	− 1389.10i	− 1.52481i	−0.647098	−	0.762407i	$-0.724018\pi$
$911$	− 1389.10i	− 1.52481i	0.647098	−	0.762407i	$-0.275982\pi$
$912$	0	0
$913$	252.000i	0.276013i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 120.000i	− 0.130862i
$918$	0	0
$919$	140.296i	0.152662i	0.997083	+	0.0763309i	$0.0243205\pi$
$919$	140.296i	0.152662i	−0.997083	+	0.0763309i	$0.975679\pi$
$920$	0	0
$921$	−351.000	−0.381107
$922$	0	0
$923$	−990.733	−1.07338
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−290.985	−0.313899
$928$	0	0
$929$	−1002.00	−1.07858	−0.539290	−	0.842120i	$-0.681307\pi$
$929$	−1002.00	−1.07858	−0.539290	+	0.842120i	$0.681307\pi$
$930$	0	0
$931$	876.418i	0.941372i
$932$	0	0
$933$	− 642.000i	− 0.688103i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 661.000i	− 0.705443i	−0.935728	−	0.352721i	$-0.885256\pi$
$937$	− 661.000i	− 0.705443i	0.935728	−	0.352721i	$-0.114744\pi$
$938$	0	0
$939$	− 43.3013i	− 0.0461142i
$940$	0	0
$941$	1110.00	1.17960	0.589798	−	0.807551i	$-0.299207\pi$
$941$	1110.00	1.17960	0.589798	+	0.807551i	$0.299207\pi$
$942$	0	0
$943$	623.538	0.661228
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1008.05	1.06447	0.532235	−	0.846597i	$-0.321352\pi$
$947$	1008.05	1.06447	0.532235	+	0.846597i	$0.321352\pi$
$948$	0	0
$949$	−682.000	−0.718651
$950$	0	0
$951$	1039.23i	1.09278i
$952$	0	0
$953$	− 1680.00i	− 1.76285i	−0.472320	−	0.881427i	$-0.656583\pi$
$953$	− 1680.00i	− 1.76285i	0.472320	−	0.881427i	$-0.343417\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 180.000i	− 0.188088i
$958$	0	0
$959$	− 93.5307i	− 0.0975295i
$960$	0	0
$961$	934.000	0.971904
$962$	0	0
$963$	−540.400	−0.561163
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−616.610	−0.637653	−0.318826	−	0.947813i	$-0.603289\pi$
$967$	−616.610	−0.637653	−0.318826	+	0.947813i	$0.603289\pi$
$968$	0	0
$969$	−198.000	−0.204334
$970$	0	0
$971$	308.305i	0.317513i	0.987318	+	0.158756i	$0.0507485\pi$
$971$	308.305i	0.317513i	−0.987318	+	0.158756i	$0.949252\pi$
$972$	0	0
$973$	84.0000i	0.0863309i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1578.00i	1.61515i	0.589766	+	0.807574i	$0.299220\pi$
$977$	1578.00i	1.61515i	−0.589766	+	0.807574i	$0.700780\pi$
$978$	0	0
$979$	498.831i	0.509531i
$980$	0	0
$981$	−507.000	−0.516820
$982$	0	0
$983$	−1378.71	−1.40256	−0.701278	−	0.712888i	$-0.747387\pi$
$983$	−1378.71	−1.40256	−0.701278	+	0.712888i	$0.747387\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−135.100	−0.136879
$988$	0	0
$989$	−90.0000	−0.0910010
$990$	0	0
$991$	− 81.4064i	− 0.0821457i	−0.999156	−	0.0410728i	$-0.986922\pi$
$991$	− 81.4064i	− 0.0821457i	0.999156	−	0.0410728i	$-0.0130776\pi$
$992$	0	0
$993$	708.000i	0.712991i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 34.0000i	− 0.0341023i	−0.999855	−	0.0170512i	$-0.994572\pi$
$997$	− 34.0000i	− 0.0341023i	0.999855	−	0.0170512i	$-0.00542781\pi$
$998$	0	0
$999$	− 72.7461i	− 0.0728190i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.j.c.799.1		4
3.2	odd	2		3600.3.j.h.1999.2		4
4.3	odd	2	inner	1200.3.j.c.799.4		4
5.2	odd	4		1200.3.e.f.751.2	yes	2
5.3	odd	4		1200.3.e.c.751.1	✓	2
5.4	even	2	inner	1200.3.j.c.799.3		4
12.11	even	2		3600.3.j.h.1999.3		4
15.2	even	4		3600.3.e.r.3151.1		2
15.8	even	4		3600.3.e.l.3151.2		2
15.14	odd	2		3600.3.j.h.1999.4		4
20.3	even	4		1200.3.e.c.751.2	yes	2
20.7	even	4		1200.3.e.f.751.1	yes	2
20.19	odd	2	inner	1200.3.j.c.799.2		4
60.23	odd	4		3600.3.e.l.3151.1		2
60.47	odd	4		3600.3.e.r.3151.2		2
60.59	even	2		3600.3.j.h.1999.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1200.3.e.c.751.1	✓	2	5.3	odd	4
1200.3.e.c.751.2	yes	2	20.3	even	4
1200.3.e.f.751.1	yes	2	20.7	even	4
1200.3.e.f.751.2	yes	2	5.2	odd	4
1200.3.j.c.799.1		4	1.1	even	1	trivial
1200.3.j.c.799.2		4	20.19	odd	2	inner
1200.3.j.c.799.3		4	5.4	even	2	inner
1200.3.j.c.799.4		4	4.3	odd	2	inner
3600.3.e.l.3151.1		2	60.23	odd	4
3600.3.e.l.3151.2		2	15.8	even	4
3600.3.e.r.3151.1		2	15.2	even	4
3600.3.e.r.3151.2		2	60.47	odd	4
3600.3.j.h.1999.1		4	60.59	even	2
3600.3.j.h.1999.2		4	3.2	odd	2
3600.3.j.h.1999.3		4	12.11	even	2
3600.3.j.h.1999.4		4	15.14	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1200.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} -1.73205 q^{7} +3.00000 q^{9} -3.46410i q^{11} +11.0000i q^{13} +6.00000i q^{17} -19.0526i q^{19} +3.00000 q^{21} +17.3205 q^{23} -5.19615 q^{27} -30.0000 q^{29} -5.19615i q^{31} +6.00000i q^{33} +14.0000i q^{37} -19.0526i q^{39} +36.0000 q^{41} -5.19615 q^{43} -45.0333 q^{47} -46.0000 q^{49} -10.3923i q^{51} +72.0000i q^{53} +33.0000i q^{57} -38.1051i q^{59} +35.0000 q^{61} -5.19615 q^{63} -29.4449 q^{67} -30.0000 q^{69} +90.0666i q^{71} +62.0000i q^{73} +6.00000i q^{77} -76.2102i q^{79} +9.00000 q^{81} -72.7461 q^{83} +51.9615 q^{87} -144.000 q^{89} -19.0526i q^{91} +9.00000i q^{93} +181.000i q^{97} -10.3923i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{9} + 12 q^{21} - 120 q^{29} + 144 q^{41} - 184 q^{49} + 140 q^{61} - 120 q^{69} + 36 q^{81} - 576 q^{89}+O(q^{100}) \) 4 * q + 12 * q^9 + 12 * q^21 - 120 * q^29 + 144 * q^41 - 184 * q^49 + 140 * q^61 - 120 * q^69 + 36 * q^81 - 576 * q^89

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