LMFDB - Embedded newform 1200.3.e.f.751.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1200,3,Mod(751,1200)] 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("1200.751"); 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, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1200.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,-6,0,0,0,22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$32.6976317232$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			751.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1200.751
Dual form			1200.3.e.f.751.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} +1.73205i q^{7} -3.00000 q^{9} +3.46410i q^{11} +11.0000 q^{13} -6.00000 q^{17} -19.0526i q^{19} +3.00000 q^{21} +17.3205i q^{23} +5.19615i q^{27} +30.0000 q^{29} +5.19615i q^{31} +6.00000 q^{33} -14.0000 q^{37} -19.0526i q^{39} +36.0000 q^{41} -5.19615i q^{43} +45.0333i q^{47} +46.0000 q^{49} +10.3923i q^{51} +72.0000 q^{53} -33.0000 q^{57} -38.1051i q^{59} +35.0000 q^{61} -5.19615i q^{63} +29.4449i q^{67} +30.0000 q^{69} -90.0666i q^{71} +62.0000 q^{73} -6.00000 q^{77} -76.2102i q^{79} +9.00000 q^{81} -72.7461i q^{83} -51.9615i q^{87} +144.000 q^{89} +19.0526i q^{91} +9.00000 q^{93} -181.000 q^{97} -10.3923i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{9} + 22 q^{13} - 12 q^{17} + 6 q^{21} + 60 q^{29} + 12 q^{33} - 28 q^{37} + 72 q^{41} + 92 q^{49} + 144 q^{53} - 66 q^{57} + 70 q^{61} + 60 q^{69} + 124 q^{73} - 12 q^{77} + 18 q^{81} + 288 q^{89}+ \cdots - 362 q^{97}+O(q^{100}) $ 2 * q - 6 * q^9 + 22 * q^13 - 12 * q^17 + 6 * q^21 + 60 * q^29 + 12 * q^33 - 28 * q^37 + 72 * q^41 + 92 * q^49 + 144 * q^53 - 66 * q^57 + 70 * q^61 + 60 * q^69 + 124 * q^73 - 12 * q^77 + 18 * q^81 + 288 * q^89 + 18 * q^93 - 362 * q^97

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
$3$	− 1.73205i	− 0.577350i
$3$	− 1.73205i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.73205i	0.247436i	0.992317	+	0.123718i	$0.0394818\pi$
$7$	1.73205i	0.247436i	−0.992317	+	0.123718i	$0.960518\pi$
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	3.46410i	0.314918i	0.987525	+	0.157459i	$0.0503303\pi$
$11$	3.46410i	0.314918i	−0.987525	+	0.157459i	$0.949670\pi$
$12$	0	0
$13$	11.0000	0.846154	0.423077	−	0.906094i	$-0.360950\pi$
$13$	11.0000	0.846154	0.423077	+	0.906094i	$0.360950\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.00000	−0.352941	−0.176471	−	0.984306i	$-0.556468\pi$
$17$	−6.00000	−0.352941	−0.176471	+	0.984306i	$0.556468\pi$
$18$	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
$23$	17.3205i	0.753066i	0.926403	+	0.376533i	$0.122884\pi$
$23$	17.3205i	0.753066i	−0.926403	+	0.376533i	$0.877116\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.19615i	0.192450i
$28$	0	0
$29$	30.0000	1.03448	0.517241	−	0.855840i	$-0.326959\pi$
$29$	30.0000	1.03448	0.517241	+	0.855840i	$0.326959\pi$
$30$	0	0
$31$	5.19615i	0.167618i	0.996482	+	0.0838089i	$0.0267085\pi$
$31$	5.19615i	0.167618i	−0.996482	+	0.0838089i	$0.973291\pi$
$32$	0	0
$33$	6.00000	0.181818
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−14.0000	−0.378378	−0.189189	−	0.981941i	$-0.560586\pi$
$37$	−14.0000	−0.378378	−0.189189	+	0.981941i	$0.560586\pi$
$38$	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
$43$	− 5.19615i	− 0.120841i	−0.998173	−	0.0604204i	$-0.980756\pi$
$43$	− 5.19615i	− 0.120841i	0.998173	−	0.0604204i	$-0.0192441\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	45.0333i	0.958156i	0.877772	+	0.479078i	$0.159029\pi$
$47$	45.0333i	0.958156i	−0.877772	+	0.479078i	$0.840971\pi$
$48$	0	0
$49$	46.0000	0.938776
$50$	0	0
$51$	10.3923i	0.203771i
$52$	0	0
$53$	72.0000	1.35849	0.679245	−	0.733911i	$-0.262307\pi$
$53$	72.0000	1.35849	0.679245	+	0.733911i	$0.262307\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−33.0000	−0.578947
$58$	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
$63$	− 5.19615i	− 0.0824786i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	29.4449i	0.439476i	0.975559	+	0.219738i	$0.0705202\pi$
$67$	29.4449i	0.439476i	−0.975559	+	0.219738i	$0.929480\pi$
$68$	0	0
$69$	30.0000	0.434783
$70$	0	0
$71$	− 90.0666i	− 1.26854i	−0.773110	−	0.634272i	$-0.781300\pi$
$71$	− 90.0666i	− 1.26854i	0.773110	−	0.634272i	$-0.218700\pi$
$72$	0	0
$73$	62.0000	0.849315	0.424658	−	0.905354i	$-0.360395\pi$
$73$	62.0000	0.849315	0.424658	+	0.905354i	$0.360395\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.00000	−0.0779221
$78$	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
$83$	− 72.7461i	− 0.876459i	−0.898863	−	0.438230i	$-0.855606\pi$
$83$	− 72.7461i	− 0.876459i	0.898863	−	0.438230i	$-0.144394\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 51.9615i	− 0.597259i
$88$	0	0
$89$	144.000	1.61798	0.808989	−	0.587824i	$-0.200015\pi$
$89$	144.000	1.61798	0.808989	+	0.587824i	$0.200015\pi$
$90$	0	0
$91$	19.0526i	0.209369i
$92$	0	0
$93$	9.00000	0.0967742
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−181.000	−1.86598	−0.932990	−	0.359903i	$-0.882810\pi$
$97$	−181.000	−1.86598	−0.932990	+	0.359903i	$0.882810\pi$
$98$	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
$103$	− 96.9948i	− 0.941698i	−0.882214	−	0.470849i	$-0.843948\pi$
$103$	− 96.9948i	− 0.941698i	0.882214	−	0.470849i	$-0.156052\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	180.133i	1.68349i	0.539876	+	0.841744i	$0.318471\pi$
$107$	180.133i	1.68349i	−0.539876	+	0.841744i	$0.681529\pi$
$108$	0	0
$109$	169.000	1.55046	0.775229	−	0.631680i	$-0.217634\pi$
$109$	169.000	1.55046	0.775229	+	0.631680i	$0.217634\pi$
$110$	0	0
$111$	24.2487i	0.218457i
$112$	0	0
$113$	12.0000	0.106195	0.0530973	−	0.998589i	$-0.483091\pi$
$113$	12.0000	0.106195	0.0530973	+	0.998589i	$0.483091\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−33.0000	−0.282051
$118$	0	0
$119$	− 10.3923i	− 0.0873303i
$120$	0	0
$121$	109.000	0.900826
$122$	0	0
$123$	− 62.3538i	− 0.506942i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 103.923i	− 0.818292i	−0.912469	−	0.409146i	$-0.865827\pi$
$127$	− 103.923i	− 0.818292i	0.912469	−	0.409146i	$-0.134173\pi$
$128$	0	0
$129$	−9.00000	−0.0697674
$130$	0	0
$131$	− 69.2820i	− 0.528870i	−0.964403	−	0.264435i	$-0.914814\pi$
$131$	− 69.2820i	− 0.528870i	0.964403	−	0.264435i	$-0.0851855\pi$
$132$	0	0
$133$	33.0000	0.248120
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−54.0000	−0.394161	−0.197080	−	0.980387i	$-0.563146\pi$
$137$	−54.0000	−0.394161	−0.197080	+	0.980387i	$0.563146\pi$
$138$	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
$143$	38.1051i	0.266469i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 79.6743i	− 0.542002i
$148$	0	0
$149$	234.000	1.57047	0.785235	−	0.619198i	$-0.212542\pi$
$149$	234.000	1.57047	0.785235	+	0.619198i	$0.212542\pi$
$150$	0	0
$151$	− 164.545i	− 1.08970i	−0.838533	−	0.544850i	$-0.816586\pi$
$151$	− 164.545i	− 1.08970i	0.838533	−	0.544850i	$-0.183414\pi$
$152$	0	0
$153$	18.0000	0.117647
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−35.0000	−0.222930	−0.111465	−	0.993768i	$-0.535554\pi$
$157$	−35.0000	−0.222930	−0.111465	+	0.993768i	$0.535554\pi$
$158$	0	0
$159$	− 124.708i	− 0.784325i
$160$	0	0
$161$	−30.0000	−0.186335
$162$	0	0
$163$	− 174.937i	− 1.07323i	−0.843826	−	0.536617i	$-0.819702\pi$
$163$	− 174.937i	− 1.07323i	0.843826	−	0.536617i	$-0.180298\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	138.564i	0.829725i	0.909884	+	0.414862i	$0.136170\pi$
$167$	138.564i	0.829725i	−0.909884	+	0.414862i	$0.863830\pi$
$168$	0	0
$169$	−48.0000	−0.284024
$170$	0	0
$171$	57.1577i	0.334255i
$172$	0	0
$173$	198.000	1.14451	0.572254	−	0.820076i	$-0.306069\pi$
$173$	198.000	1.14451	0.572254	+	0.820076i	$0.306069\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−66.0000	−0.372881
$178$	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
$183$	− 60.6218i	− 0.331267i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 20.7846i	− 0.111148i
$188$	0	0
$189$	−9.00000	−0.0476190
$190$	0	0
$191$	− 239.023i	− 1.25143i	−0.780052	−	0.625715i	$-0.784807\pi$
$191$	− 239.023i	− 1.25143i	0.780052	−	0.625715i	$-0.215193\pi$
$192$	0	0
$193$	−59.0000	−0.305699	−0.152850	−	0.988249i	$-0.548845\pi$
$193$	−59.0000	−0.305699	−0.152850	+	0.988249i	$0.548845\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	162.000	0.822335	0.411168	−	0.911560i	$-0.365121\pi$
$197$	162.000	0.822335	0.411168	+	0.911560i	$0.365121\pi$
$198$	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
$203$	51.9615i	0.255968i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 51.9615i	− 0.251022i
$208$	0	0
$209$	66.0000	0.315789
$210$	0	0
$211$	403.568i	1.91264i	0.292316	+	0.956322i	$0.405574\pi$
$211$	403.568i	1.91264i	−0.292316	+	0.956322i	$0.594426\pi$
$212$	0	0
$213$	−156.000	−0.732394
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−9.00000	−0.0414747
$218$	0	0
$219$	− 107.387i	− 0.490352i
$220$	0	0
$221$	−66.0000	−0.298643
$222$	0	0
$223$	223.435i	1.00195i	0.865462	+	0.500974i	$0.167025\pi$
$223$	223.435i	1.00195i	−0.865462	+	0.500974i	$0.832975\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	422.620i	1.86176i	0.365320	+	0.930882i	$0.380960\pi$
$227$	422.620i	1.86176i	−0.365320	+	0.930882i	$0.619040\pi$
$228$	0	0
$229$	−59.0000	−0.257642	−0.128821	−	0.991668i	$-0.541119\pi$
$229$	−59.0000	−0.257642	−0.128821	+	0.991668i	$0.541119\pi$
$230$	0	0
$231$	10.3923i	0.0449883i
$232$	0	0
$233$	84.0000	0.360515	0.180258	−	0.983619i	$-0.442307\pi$
$233$	84.0000	0.360515	0.180258	+	0.983619i	$0.442307\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−132.000	−0.556962
$238$	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
$243$	− 15.5885i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 209.578i	− 0.848495i
$248$	0	0
$249$	−126.000	−0.506024
$250$	0	0
$251$	34.6410i	0.138012i	0.997616	+	0.0690060i	$0.0219828\pi$
$251$	34.6410i	0.138012i	−0.997616	+	0.0690060i	$0.978017\pi$
$252$	0	0
$253$	−60.0000	−0.237154
$254$	0	0
$255$	0	0
$256$	0	0
$257$	228.000	0.887160	0.443580	−	0.896235i	$-0.353708\pi$
$257$	228.000	0.887160	0.443580	+	0.896235i	$0.353708\pi$
$258$	0	0
$259$	− 24.2487i	− 0.0936244i
$260$	0	0
$261$	−90.0000	−0.344828
$262$	0	0
$263$	422.620i	1.60692i	0.595358	+	0.803461i	$0.297010\pi$
$263$	422.620i	1.60692i	−0.595358	+	0.803461i	$0.702990\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 249.415i	− 0.934140i
$268$	0	0
$269$	−270.000	−1.00372	−0.501859	−	0.864950i	$-0.667350\pi$
$269$	−270.000	−1.00372	−0.501859	+	0.864950i	$0.667350\pi$
$270$	0	0
$271$	− 34.6410i	− 0.127827i	−0.997955	−	0.0639133i	$-0.979642\pi$
$271$	− 34.6410i	− 0.127827i	0.997955	−	0.0639133i	$-0.0203581\pi$
$272$	0	0
$273$	33.0000	0.120879
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−49.0000	−0.176895	−0.0884477	−	0.996081i	$-0.528191\pi$
$277$	−49.0000	−0.176895	−0.0884477	+	0.996081i	$0.528191\pi$
$278$	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
$283$	− 140.296i	− 0.495746i	−0.968793	−	0.247873i	$-0.920268\pi$
$283$	− 140.296i	− 0.495746i	0.968793	−	0.247873i	$-0.0797316\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	62.3538i	0.217261i
$288$	0	0
$289$	−253.000	−0.875433
$290$	0	0
$291$	313.501i	1.07732i
$292$	0	0
$293$	−78.0000	−0.266212	−0.133106	−	0.991102i	$-0.542495\pi$
$293$	−78.0000	−0.266212	−0.133106	+	0.991102i	$0.542495\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−18.0000	−0.0606061
$298$	0	0
$299$	190.526i	0.637209i
$300$	0	0
$301$	9.00000	0.0299003
$302$	0	0
$303$	83.1384i	0.274384i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 202.650i	− 0.660098i	−0.943964	−	0.330049i	$-0.892935\pi$
$307$	− 202.650i	− 0.660098i	0.943964	−	0.330049i	$-0.107065\pi$
$308$	0	0
$309$	−168.000	−0.543689
$310$	0	0
$311$	− 370.659i	− 1.19183i	−0.803048	−	0.595915i	$-0.796790\pi$
$311$	− 370.659i	− 1.19183i	0.803048	−	0.595915i	$-0.203210\pi$
$312$	0	0
$313$	25.0000	0.0798722	0.0399361	−	0.999202i	$-0.487285\pi$
$313$	25.0000	0.0798722	0.0399361	+	0.999202i	$0.487285\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	600.000	1.89274	0.946372	−	0.323078i	$-0.104718\pi$
$317$	600.000	1.89274	0.946372	+	0.323078i	$0.104718\pi$
$318$	0	0
$319$	103.923i	0.325778i
$320$	0	0
$321$	312.000	0.971963
$322$	0	0
$323$	114.315i	0.353918i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 292.717i	− 0.895158i
$328$	0	0
$329$	−78.0000	−0.237082
$330$	0	0
$331$	408.764i	1.23494i	0.786596	+	0.617468i	$0.211842\pi$
$331$	408.764i	1.23494i	−0.786596	+	0.617468i	$0.788158\pi$
$332$	0	0
$333$	42.0000	0.126126
$334$	0	0
$335$	0	0
$336$	0	0
$337$	203.000	0.602374	0.301187	−	0.953565i	$-0.402617\pi$
$337$	203.000	0.602374	0.301187	+	0.953565i	$0.402617\pi$
$338$	0	0
$339$	− 20.7846i	− 0.0613115i
$340$	0	0
$341$	−18.0000	−0.0527859
$342$	0	0
$343$	164.545i	0.479723i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	613.146i	1.76699i	0.468439	+	0.883496i	$0.344816\pi$
$347$	613.146i	1.76699i	−0.468439	+	0.883496i	$0.655184\pi$
$348$	0	0
$349$	−538.000	−1.54155	−0.770774	−	0.637109i	$-0.780130\pi$
$349$	−538.000	−1.54155	−0.770774	+	0.637109i	$0.780130\pi$
$350$	0	0
$351$	57.1577i	0.162842i
$352$	0	0
$353$	−498.000	−1.41076	−0.705382	−	0.708827i	$-0.749225\pi$
$353$	−498.000	−1.41076	−0.705382	+	0.708827i	$0.749225\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−18.0000	−0.0504202
$358$	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
$363$	− 188.794i	− 0.520092i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 569.845i	− 1.55271i	−0.630296	−	0.776355i	$-0.717066\pi$
$367$	− 569.845i	− 1.55271i	0.630296	−	0.776355i	$-0.282934\pi$
$368$	0	0
$369$	−108.000	−0.292683
$370$	0	0
$371$	124.708i	0.336139i
$372$	0	0
$373$	−491.000	−1.31635	−0.658177	−	0.752863i	$-0.728672\pi$
$373$	−491.000	−1.31635	−0.658177	+	0.752863i	$0.728672\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	330.000	0.875332
$378$	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
$383$	367.195i	0.958733i	0.877615	+	0.479367i	$0.159134\pi$
$383$	367.195i	0.958733i	−0.877615	+	0.479367i	$0.840866\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	15.5885i	0.0402803i
$388$	0	0
$389$	−132.000	−0.339332	−0.169666	−	0.985502i	$-0.554269\pi$
$389$	−132.000	−0.339332	−0.169666	+	0.985502i	$0.554269\pi$
$390$	0	0
$391$	− 103.923i	− 0.265788i
$392$	0	0
$393$	−120.000	−0.305344
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−251.000	−0.632242	−0.316121	−	0.948719i	$-0.602381\pi$
$397$	−251.000	−0.632242	−0.316121	+	0.948719i	$0.602381\pi$
$398$	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
$403$	57.1577i	0.141830i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 48.4974i	− 0.119158i
$408$	0	0
$409$	−515.000	−1.25917	−0.629584	−	0.776932i	$-0.716775\pi$
$409$	−515.000	−1.25917	−0.629584	+	0.776932i	$0.716775\pi$
$410$	0	0
$411$	93.5307i	0.227569i
$412$	0	0
$413$	66.0000	0.159806
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−84.0000	−0.201439
$418$	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
$423$	− 135.100i	− 0.319385i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	60.6218i	0.141971i
$428$	0	0
$429$	66.0000	0.153846
$430$	0	0
$431$	− 627.002i	− 1.45476i	−0.686234	−	0.727381i	$-0.740737\pi$
$431$	− 627.002i	− 1.45476i	0.686234	−	0.727381i	$-0.259263\pi$
$432$	0	0
$433$	−287.000	−0.662818	−0.331409	−	0.943487i	$-0.607524\pi$
$433$	−287.000	−0.662818	−0.331409	+	0.943487i	$0.607524\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	330.000	0.755149
$438$	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
$443$	− 221.703i	− 0.500457i	−0.968187	−	0.250229i	$-0.919494\pi$
$443$	− 221.703i	− 0.500457i	0.968187	−	0.250229i	$-0.0805058\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 405.300i	− 0.906711i
$448$	0	0
$449$	132.000	0.293987	0.146993	−	0.989137i	$-0.453040\pi$
$449$	132.000	0.293987	0.146993	+	0.989137i	$0.453040\pi$
$450$	0	0
$451$	124.708i	0.276514i
$452$	0	0
$453$	−285.000	−0.629139
$454$	0	0
$455$	0	0
$456$	0	0
$457$	434.000	0.949672	0.474836	−	0.880074i	$-0.342507\pi$
$457$	434.000	0.949672	0.474836	+	0.880074i	$0.342507\pi$
$458$	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
$463$	90.0666i	0.194528i	0.995259	+	0.0972642i	$0.0310092\pi$
$463$	90.0666i	0.194528i	−0.995259	+	0.0972642i	$0.968991\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 3.46410i	− 0.00741778i	−0.999993	−	0.00370889i	$-0.998819\pi$
$467$	− 3.46410i	− 0.00741778i	0.999993	−	0.00370889i	$-0.00118058\pi$
$468$	0	0
$469$	−51.0000	−0.108742
$470$	0	0
$471$	60.6218i	0.128709i
$472$	0	0
$473$	18.0000	0.0380550
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−216.000	−0.452830
$478$	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
$483$	51.9615i	0.107581i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 258.076i	− 0.529929i	−0.964258	−	0.264965i	$-0.914640\pi$
$487$	− 258.076i	− 0.529929i	0.964258	−	0.264965i	$-0.0853603\pi$
$488$	0	0
$489$	−303.000	−0.619632
$490$	0	0
$491$	332.554i	0.677299i	0.940913	+	0.338649i	$0.109970\pi$
$491$	332.554i	0.677299i	−0.940913	+	0.338649i	$0.890030\pi$
$492$	0	0
$493$	−180.000	−0.365112
$494$	0	0
$495$	0	0
$496$	0	0
$497$	156.000	0.313883
$498$	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
$503$	187.061i	0.371892i	0.982560	+	0.185946i	$0.0595349\pi$
$503$	187.061i	0.371892i	−0.982560	+	0.185946i	$0.940465\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	83.1384i	0.163981i
$508$	0	0
$509$	−546.000	−1.07269	−0.536346	−	0.843998i	$-0.680196\pi$
$509$	−546.000	−1.07269	−0.536346	+	0.843998i	$0.680196\pi$
$510$	0	0
$511$	107.387i	0.210151i
$512$	0	0
$513$	99.0000	0.192982
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−156.000	−0.301741
$518$	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
$523$	− 895.470i	− 1.71218i	−0.516827	−	0.856090i	$-0.672887\pi$
$523$	− 895.470i	− 1.71218i	0.516827	−	0.856090i	$-0.327113\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 31.1769i	− 0.0591592i
$528$	0	0
$529$	229.000	0.432892
$530$	0	0
$531$	114.315i	0.215283i
$532$	0	0
$533$	396.000	0.742964
$534$	0	0
$535$	0	0
$536$	0	0
$537$	438.000	0.815642
$538$	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
$543$	39.8372i	0.0733650i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	69.2820i	0.126658i	0.997993	+	0.0633291i	$0.0201718\pi$
$547$	69.2820i	0.126658i	−0.997993	+	0.0633291i	$0.979828\pi$
$548$	0	0
$549$	−105.000	−0.191257
$550$	0	0
$551$	− 571.577i	− 1.03734i
$552$	0	0
$553$	132.000	0.238698
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−114.000	−0.204668	−0.102334	−	0.994750i	$-0.532631\pi$
$557$	−114.000	−0.204668	−0.102334	+	0.994750i	$0.532631\pi$
$558$	0	0
$559$	− 57.1577i	− 0.102250i
$560$	0	0
$561$	−36.0000	−0.0641711
$562$	0	0
$563$	128.172i	0.227659i	0.993500	+	0.113829i	$0.0363117\pi$
$563$	128.172i	0.227659i	−0.993500	+	0.113829i	$0.963688\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	15.5885i	0.0274929i
$568$	0	0
$569$	450.000	0.790861	0.395431	−	0.918496i	$-0.370595\pi$
$569$	450.000	0.790861	0.395431	+	0.918496i	$0.370595\pi$
$570$	0	0
$571$	22.5167i	0.0394337i	0.999806	+	0.0197169i	$0.00627648\pi$
$571$	22.5167i	0.0394337i	−0.999806	+	0.0197169i	$0.993724\pi$
$572$	0	0
$573$	−414.000	−0.722513
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−311.000	−0.538995	−0.269497	−	0.963001i	$-0.586858\pi$
$577$	−311.000	−0.538995	−0.269497	+	0.963001i	$0.586858\pi$
$578$	0	0
$579$	102.191i	0.176496i
$580$	0	0
$581$	126.000	0.216867
$582$	0	0
$583$	249.415i	0.427814i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	166.277i	0.283266i	0.989919	+	0.141633i	$0.0452352\pi$
$587$	166.277i	0.283266i	−0.989919	+	0.141633i	$0.954765\pi$
$588$	0	0
$589$	99.0000	0.168081
$590$	0	0
$591$	− 280.592i	− 0.474775i
$592$	0	0
$593$	−636.000	−1.07251	−0.536256	−	0.844055i	$-0.680162\pi$
$593$	−636.000	−1.07251	−0.536256	+	0.844055i	$0.680162\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−381.000	−0.638191
$598$	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
$603$	− 88.3346i	− 0.146492i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 533.472i	− 0.878866i	−0.898275	−	0.439433i	$-0.855179\pi$
$607$	− 533.472i	− 0.878866i	0.898275	−	0.439433i	$-0.144821\pi$
$608$	0	0
$609$	90.0000	0.147783
$610$	0	0
$611$	495.367i	0.810747i
$612$	0	0
$613$	−374.000	−0.610114	−0.305057	−	0.952334i	$-0.598676\pi$
$613$	−374.000	−0.610114	−0.305057	+	0.952334i	$0.598676\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−240.000	−0.388979	−0.194489	−	0.980905i	$-0.562305\pi$
$617$	−240.000	−0.388979	−0.194489	+	0.980905i	$0.562305\pi$
$618$	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
$623$	249.415i	0.400346i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 114.315i	− 0.182321i
$628$	0	0
$629$	84.0000	0.133545
$630$	0	0
$631$	− 549.060i	− 0.870143i	−0.900396	−	0.435071i	$-0.856723\pi$
$631$	− 549.060i	− 0.870143i	0.900396	−	0.435071i	$-0.143277\pi$
$632$	0	0
$633$	699.000	1.10427
$634$	0	0
$635$	0	0
$636$	0	0
$637$	506.000	0.794349
$638$	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
$643$	394.908i	0.614164i	0.951683	+	0.307082i	$0.0993526\pi$
$643$	394.908i	0.614164i	−0.951683	+	0.307082i	$0.900647\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	699.749i	1.08153i	0.841174	+	0.540764i	$0.181865\pi$
$647$	699.749i	1.08153i	−0.841174	+	0.540764i	$0.818135\pi$
$648$	0	0
$649$	132.000	0.203390
$650$	0	0
$651$	15.5885i	0.0239454i
$652$	0	0
$653$	534.000	0.817764	0.408882	−	0.912587i	$-0.365919\pi$
$653$	534.000	0.817764	0.408882	+	0.912587i	$0.365919\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−186.000	−0.283105
$658$	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
$663$	114.315i	0.172421i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	519.615i	0.779033i
$668$	0	0
$669$	387.000	0.578475
$670$	0	0
$671$	121.244i	0.180691i
$672$	0	0
$673$	1234.00	1.83358	0.916790	−	0.399368i	$-0.130771\pi$
$673$	1234.00	1.83358	0.916790	+	0.399368i	$0.130771\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−534.000	−0.788774	−0.394387	−	0.918944i	$-0.629043\pi$
$677$	−534.000	−0.788774	−0.394387	+	0.918944i	$0.629043\pi$
$678$	0	0
$679$	− 313.501i	− 0.461710i
$680$	0	0
$681$	732.000	1.07489
$682$	0	0
$683$	− 526.543i	− 0.770927i	−0.922723	−	0.385464i	$-0.874041\pi$
$683$	− 526.543i	− 0.770927i	0.922723	−	0.385464i	$-0.125959\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	102.191i	0.148750i
$688$	0	0
$689$	792.000	1.14949
$690$	0	0
$691$	1122.37i	1.62427i	0.583471	+	0.812134i	$0.301694\pi$
$691$	1122.37i	1.62427i	−0.583471	+	0.812134i	$0.698306\pi$
$692$	0	0
$693$	18.0000	0.0259740
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−216.000	−0.309900
$698$	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
$703$	266.736i	0.379425i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 83.1384i	− 0.117593i
$708$	0	0
$709$	433.000	0.610719	0.305360	−	0.952237i	$-0.401223\pi$
$709$	433.000	0.610719	0.305360	+	0.952237i	$0.401223\pi$
$710$	0	0
$711$	228.631i	0.321562i
$712$	0	0
$713$	−90.0000	−0.126227
$714$	0	0
$715$	0	0
$716$	0	0
$717$	336.000	0.468619
$718$	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
$723$	− 126.440i	− 0.174882i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	930.111i	1.27938i	0.768632	+	0.639691i	$0.220938\pi$
$727$	930.111i	1.27938i	−0.768632	+	0.639691i	$0.779062\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	31.1769i	0.0426497i
$732$	0	0
$733$	590.000	0.804911	0.402456	−	0.915439i	$-0.368157\pi$
$733$	590.000	0.804911	0.402456	+	0.915439i	$0.368157\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−102.000	−0.138399
$738$	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
$743$	− 741.318i	− 0.997736i	−0.866678	−	0.498868i	$-0.833749\pi$
$743$	− 741.318i	− 0.997736i	0.866678	−	0.498868i	$-0.166251\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	218.238i	0.292153i
$748$	0	0
$749$	−312.000	−0.416555
$750$	0	0
$751$	− 907.595i	− 1.20851i	−0.796789	−	0.604257i	$-0.793470\pi$
$751$	− 907.595i	− 1.20851i	0.796789	−	0.604257i	$-0.206530\pi$
$752$	0	0
$753$	60.0000	0.0796813
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−841.000	−1.11096	−0.555482	−	0.831528i	$-0.687466\pi$
$757$	−841.000	−1.11096	−0.555482	+	0.831528i	$0.687466\pi$
$758$	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
$763$	292.717i	0.383639i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 419.156i	− 0.546488i
$768$	0	0
$769$	347.000	0.451235	0.225618	−	0.974216i	$-0.427560\pi$
$769$	347.000	0.451235	0.225618	+	0.974216i	$0.427560\pi$
$770$	0	0
$771$	− 394.908i	− 0.512202i
$772$	0	0
$773$	1404.00	1.81630	0.908150	−	0.418645i	$-0.137495\pi$
$773$	1404.00	1.81630	0.908150	+	0.418645i	$0.137495\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−42.0000	−0.0540541
$778$	0	0
$779$	− 685.892i	− 0.880478i
$780$	0	0
$781$	312.000	0.399488
$782$	0	0
$783$	155.885i	0.199086i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 43.3013i	− 0.0550207i	−0.999622	−	0.0275103i	$-0.991242\pi$
$787$	− 43.3013i	− 0.0550207i	0.999622	−	0.0275103i	$-0.00875792\pi$
$788$	0	0
$789$	732.000	0.927757
$790$	0	0
$791$	20.7846i	0.0262764i
$792$	0	0
$793$	385.000	0.485498
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−348.000	−0.436637	−0.218319	−	0.975878i	$-0.570057\pi$
$797$	−348.000	−0.436637	−0.218319	+	0.975878i	$0.570057\pi$
$798$	0	0
$799$	− 270.200i	− 0.338173i
$800$	0	0
$801$	−432.000	−0.539326
$802$	0	0
$803$	214.774i	0.267465i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	467.654i	0.579497i
$808$	0	0
$809$	384.000	0.474660	0.237330	−	0.971429i	$-0.423728\pi$
$809$	384.000	0.474660	0.237330	+	0.971429i	$0.423728\pi$
$810$	0	0
$811$	− 691.088i	− 0.852143i	−0.904689	−	0.426072i	$-0.859897\pi$
$811$	− 691.088i	− 0.852143i	0.904689	−	0.426072i	$-0.140103\pi$
$812$	0	0
$813$	−60.0000	−0.0738007
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−99.0000	−0.121175
$818$	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
$823$	− 614.878i	− 0.747118i	−0.927606	−	0.373559i	$-0.878137\pi$
$823$	− 614.878i	− 0.747118i	0.927606	−	0.373559i	$-0.121863\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	193.990i	0.234570i	0.993098	+	0.117285i	$0.0374192\pi$
$827$	193.990i	0.234570i	−0.993098	+	0.117285i	$0.962581\pi$
$828$	0	0
$829$	1606.00	1.93727	0.968637	−	0.248480i	$-0.0799312\pi$
$829$	1606.00	1.93727	0.968637	+	0.248480i	$0.0799312\pi$
$830$	0	0
$831$	84.8705i	0.102131i
$832$	0	0
$833$	−276.000	−0.331333
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−27.0000	−0.0322581
$838$	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
$843$	426.084i	0.505438i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	188.794i	0.222897i
$848$	0	0
$849$	−243.000	−0.286219
$850$	0	0
$851$	− 242.487i	− 0.284944i
$852$	0	0
$853$	−575.000	−0.674091	−0.337046	−	0.941488i	$-0.609428\pi$
$853$	−575.000	−0.674091	−0.337046	+	0.941488i	$0.609428\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1068.00	−1.24621	−0.623104	−	0.782139i	$-0.714129\pi$
$857$	−1068.00	−1.24621	−0.623104	+	0.782139i	$0.714129\pi$
$858$	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
$863$	166.277i	0.192673i	0.995349	+	0.0963365i	$0.0307125\pi$
$863$	166.277i	0.192673i	−0.995349	+	0.0963365i	$0.969287\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	438.209i	0.505431i
$868$	0	0
$869$	264.000	0.303797
$870$	0	0
$871$	323.894i	0.371864i
$872$	0	0
$873$	543.000	0.621993
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−731.000	−0.833523	−0.416762	−	0.909016i	$-0.636835\pi$
$877$	−731.000	−0.833523	−0.416762	+	0.909016i	$0.636835\pi$
$878$	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
$883$	− 1037.50i	− 1.17497i	−0.809235	−	0.587485i	$-0.800118\pi$
$883$	− 1037.50i	− 1.17497i	0.809235	−	0.587485i	$-0.199882\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1375.25i	− 1.55045i	−0.631686	−	0.775225i	$-0.717637\pi$
$887$	− 1375.25i	− 1.55045i	0.631686	−	0.775225i	$-0.282363\pi$
$888$	0	0
$889$	180.000	0.202475
$890$	0	0
$891$	31.1769i	0.0349909i
$892$	0	0
$893$	858.000	0.960806
$894$	0	0
$895$	0	0
$896$	0	0
$897$	330.000	0.367893
$898$	0	0
$899$	155.885i	0.173398i
$900$	0	0
$901$	−432.000	−0.479467
$902$	0	0
$903$	− 15.5885i	− 0.0172630i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1531.13i	1.68813i	0.536242	+	0.844064i	$0.319844\pi$
$907$	1531.13i	1.68813i	−0.536242	+	0.844064i	$0.680156\pi$
$908$	0	0
$909$	144.000	0.158416
$910$	0	0
$911$	1389.10i	1.52481i	0.647098	+	0.762407i	$0.275982\pi$
$911$	1389.10i	1.52481i	−0.647098	+	0.762407i	$0.724018\pi$
$912$	0	0
$913$	252.000	0.276013
$914$	0	0
$915$	0	0
$916$	0	0
$917$	120.000	0.130862
$918$	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
$923$	− 990.733i	− 1.07338i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	290.985i	0.313899i
$928$	0	0
$929$	1002.00	1.07858	0.539290	−	0.842120i	$-0.318693\pi$
$929$	1002.00	1.07858	0.539290	+	0.842120i	$0.318693\pi$
$930$	0	0
$931$	− 876.418i	− 0.941372i
$932$	0	0
$933$	−642.000	−0.688103
$934$	0	0
$935$	0	0
$936$	0	0
$937$	661.000	0.705443	0.352721	−	0.935728i	$-0.385256\pi$
$937$	661.000	0.705443	0.352721	+	0.935728i	$0.385256\pi$
$938$	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
$943$	623.538i	0.661228i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1008.05i	− 1.06447i	−0.846597	−	0.532235i	$-0.821352\pi$
$947$	− 1008.05i	− 1.06447i	0.846597	−	0.532235i	$-0.178648\pi$
$948$	0	0
$949$	682.000	0.718651
$950$	0	0
$951$	− 1039.23i	− 1.09278i
$952$	0	0
$953$	−1680.00	−1.76285	−0.881427	−	0.472320i	$-0.843417\pi$
$953$	−1680.00	−1.76285	−0.881427	+	0.472320i	$0.843417\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	180.000	0.188088
$958$	0	0
$959$	− 93.5307i	− 0.0975295i
$960$	0	0
$961$	934.000	0.971904
$962$	0	0
$963$	− 540.400i	− 0.561163i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	616.610i	0.637653i	0.947813	+	0.318826i	$0.103289\pi$
$967$	616.610i	0.637653i	−0.947813	+	0.318826i	$0.896711\pi$
$968$	0	0
$969$	198.000	0.204334
$970$	0	0
$971$	− 308.305i	− 0.317513i	−0.987318	−	0.158756i	$-0.949252\pi$
$971$	− 308.305i	− 0.317513i	0.987318	−	0.158756i	$-0.0507485\pi$
$972$	0	0
$973$	84.0000	0.0863309
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1578.00	−1.61515	−0.807574	−	0.589766i	$-0.799220\pi$
$977$	−1578.00	−1.61515	−0.807574	+	0.589766i	$0.799220\pi$
$978$	0	0
$979$	498.831i	0.509531i
$980$	0	0
$981$	−507.000	−0.516820
$982$	0	0
$983$	− 1378.71i	− 1.40256i	−0.712888	−	0.701278i	$-0.752613\pi$
$983$	− 1378.71i	− 1.40256i	0.712888	−	0.701278i	$-0.247387\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	135.100i	0.136879i
$988$	0	0
$989$	90.0000	0.0910010
$990$	0	0
$991$	81.4064i	0.0821457i	0.999156	+	0.0410728i	$0.0130776\pi$
$991$	81.4064i	0.0821457i	−0.999156	+	0.0410728i	$0.986922\pi$
$992$	0	0
$993$	708.000	0.712991
$994$	0	0
$995$	0	0
$996$	0	0
$997$	34.0000	0.0341023	0.0170512	−	0.999855i	$-0.494572\pi$
$997$	34.0000	0.0341023	0.0170512	+	0.999855i	$0.494572\pi$
$998$	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.e.f.751.1	yes	2
3.2	odd	2		3600.3.e.r.3151.2		2
4.3	odd	2	inner	1200.3.e.f.751.2	yes	2
5.2	odd	4		1200.3.j.c.799.2		4
5.3	odd	4		1200.3.j.c.799.4		4
5.4	even	2		1200.3.e.c.751.2	yes	2
12.11	even	2		3600.3.e.r.3151.1		2
15.2	even	4		3600.3.j.h.1999.1		4
15.8	even	4		3600.3.j.h.1999.3		4
15.14	odd	2		3600.3.e.l.3151.1		2
20.3	even	4		1200.3.j.c.799.1		4
20.7	even	4		1200.3.j.c.799.3		4
20.19	odd	2		1200.3.e.c.751.1	✓	2
60.23	odd	4		3600.3.j.h.1999.2		4
60.47	odd	4		3600.3.j.h.1999.4		4
60.59	even	2		3600.3.e.l.3151.2		2

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

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.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} +1.73205i q^{7} -3.00000 q^{9} +3.46410i q^{11} +11.0000 q^{13} -6.00000 q^{17} -19.0526i q^{19} +3.00000 q^{21} +17.3205i q^{23} +5.19615i q^{27} +30.0000 q^{29} +5.19615i q^{31} +6.00000 q^{33} -14.0000 q^{37} -19.0526i q^{39} +36.0000 q^{41} -5.19615i q^{43} +45.0333i q^{47} +46.0000 q^{49} +10.3923i q^{51} +72.0000 q^{53} -33.0000 q^{57} -38.1051i q^{59} +35.0000 q^{61} -5.19615i q^{63} +29.4449i q^{67} +30.0000 q^{69} -90.0666i q^{71} +62.0000 q^{73} -6.00000 q^{77} -76.2102i q^{79} +9.00000 q^{81} -72.7461i q^{83} -51.9615i q^{87} +144.000 q^{89} +19.0526i q^{91} +9.00000 q^{93} -181.000 q^{97} -10.3923i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{9} + 22 q^{13} - 12 q^{17} + 6 q^{21} + 60 q^{29} + 12 q^{33} - 28 q^{37} + 72 q^{41} + 92 q^{49} + 144 q^{53} - 66 q^{57} + 70 q^{61} + 60 q^{69} + 124 q^{73} - 12 q^{77} + 18 q^{81} + 288 q^{89}+ \cdots - 362 q^{97}+O(q^{100}) \) 2 * q - 6 * q^9 + 22 * q^13 - 12 * q^17 + 6 * q^21 + 60 * q^29 + 12 * q^33 - 28 * q^37 + 72 * q^41 + 92 * q^49 + 144 * q^53 - 66 * q^57 + 70 * q^61 + 60 * q^69 + 124 * q^73 - 12 * q^77 + 18 * q^81 + 288 * q^89 + 18 * q^93 - 362 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more