LMFDB - Embedded newform 1200.3.e.i.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,-6,0,0,0,48] 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:	no (minimal twist has level 240)
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.i.751.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} -6.92820i q^{7} -3.00000 q^{9} +13.8564i q^{11} +24.0000 q^{13} -24.0000 q^{17} +27.7128i q^{19} -12.0000 q^{21} +34.6410i q^{23} +5.19615i q^{27} -10.0000 q^{29} -13.8564i q^{31} +24.0000 q^{33} +24.0000 q^{37} -41.5692i q^{39} -34.0000 q^{41} +20.7846i q^{43} -6.92820i q^{47} +1.00000 q^{49} +41.5692i q^{51} +48.0000 q^{53} +48.0000 q^{57} -13.8564i q^{59} +70.0000 q^{61} +20.7846i q^{63} +90.0666i q^{67} +60.0000 q^{69} +55.4256i q^{71} +48.0000 q^{73} +96.0000 q^{77} +41.5692i q^{79} +9.00000 q^{81} -90.0666i q^{83} +17.3205i q^{87} +14.0000 q^{89} -166.277i q^{91} -24.0000 q^{93} +96.0000 q^{97} -41.5692i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{9} + 48 q^{13} - 48 q^{17} - 24 q^{21} - 20 q^{29} + 48 q^{33} + 48 q^{37} - 68 q^{41} + 2 q^{49} + 96 q^{53} + 96 q^{57} + 140 q^{61} + 120 q^{69} + 96 q^{73} + 192 q^{77} + 18 q^{81} + 28 q^{89}+ \cdots + 192 q^{97}+O(q^{100}) $ 2 * q - 6 * q^9 + 48 * q^13 - 48 * q^17 - 24 * q^21 - 20 * q^29 + 48 * q^33 + 48 * q^37 - 68 * q^41 + 2 * q^49 + 96 * q^53 + 96 * q^57 + 140 * q^61 + 120 * q^69 + 96 * q^73 + 192 * q^77 + 18 * q^81 + 28 * q^89 - 48 * q^93 + 192 * 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
$7$	− 6.92820i	− 0.989743i	−0.868966	−	0.494872i	$-0.835215\pi$
$7$	− 6.92820i	− 0.989743i	0.868966	−	0.494872i	$-0.164785\pi$
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	13.8564i	1.25967i	0.776728	+	0.629837i	$0.216878\pi$
$11$	13.8564i	1.25967i	−0.776728	+	0.629837i	$0.783122\pi$
$12$	0	0
$13$	24.0000	1.84615	0.923077	−	0.384615i	$-0.125666\pi$
$13$	24.0000	1.84615	0.923077	+	0.384615i	$0.125666\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−24.0000	−1.41176	−0.705882	−	0.708329i	$-0.749449\pi$
$17$	−24.0000	−1.41176	−0.705882	+	0.708329i	$0.749449\pi$
$18$	0	0
$19$	27.7128i	1.45857i	0.684211	+	0.729285i	$0.260147\pi$
$19$	27.7128i	1.45857i	−0.684211	+	0.729285i	$0.739853\pi$
$20$	0	0
$21$	−12.0000	−0.571429
$22$	0	0
$23$	34.6410i	1.50613i	0.657945	+	0.753066i	$0.271426\pi$
$23$	34.6410i	1.50613i	−0.657945	+	0.753066i	$0.728574\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.19615i	0.192450i
$28$	0	0
$29$	−10.0000	−0.344828	−0.172414	−	0.985025i	$-0.555157\pi$
$29$	−10.0000	−0.344828	−0.172414	+	0.985025i	$0.555157\pi$
$30$	0	0
$31$	− 13.8564i	− 0.446981i	−0.974706	−	0.223490i	$-0.928255\pi$
$31$	− 13.8564i	− 0.446981i	0.974706	−	0.223490i	$-0.0717451\pi$
$32$	0	0
$33$	24.0000	0.727273
$34$	0	0
$35$	0	0
$36$	0	0
$37$	24.0000	0.648649	0.324324	−	0.945946i	$-0.394863\pi$
$37$	24.0000	0.648649	0.324324	+	0.945946i	$0.394863\pi$
$38$	0	0
$39$	− 41.5692i	− 1.06588i
$40$	0	0
$41$	−34.0000	−0.829268	−0.414634	−	0.909988i	$-0.636090\pi$
$41$	−34.0000	−0.829268	−0.414634	+	0.909988i	$0.636090\pi$
$42$	0	0
$43$	20.7846i	0.483363i	0.970356	+	0.241682i	$0.0776989\pi$
$43$	20.7846i	0.483363i	−0.970356	+	0.241682i	$0.922301\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 6.92820i	− 0.147409i	−0.997280	−	0.0737043i	$-0.976518\pi$
$47$	− 6.92820i	− 0.147409i	0.997280	−	0.0737043i	$-0.0234821\pi$
$48$	0	0
$49$	1.00000	0.0204082
$50$	0	0
$51$	41.5692i	0.815083i
$52$	0	0
$53$	48.0000	0.905660	0.452830	−	0.891597i	$-0.350414\pi$
$53$	48.0000	0.905660	0.452830	+	0.891597i	$0.350414\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	48.0000	0.842105
$58$	0	0
$59$	− 13.8564i	− 0.234854i	−0.993081	−	0.117427i	$-0.962535\pi$
$59$	− 13.8564i	− 0.234854i	0.993081	−	0.117427i	$-0.0374647\pi$
$60$	0	0
$61$	70.0000	1.14754	0.573770	−	0.819016i	$-0.305480\pi$
$61$	70.0000	1.14754	0.573770	+	0.819016i	$0.305480\pi$
$62$	0	0
$63$	20.7846i	0.329914i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	90.0666i	1.34428i	0.740425	+	0.672139i	$0.234624\pi$
$67$	90.0666i	1.34428i	−0.740425	+	0.672139i	$0.765376\pi$
$68$	0	0
$69$	60.0000	0.869565
$70$	0	0
$71$	55.4256i	0.780643i	0.920679	+	0.390321i	$0.127636\pi$
$71$	55.4256i	0.780643i	−0.920679	+	0.390321i	$0.872364\pi$
$72$	0	0
$73$	48.0000	0.657534	0.328767	−	0.944411i	$-0.393367\pi$
$73$	48.0000	0.657534	0.328767	+	0.944411i	$0.393367\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	96.0000	1.24675
$78$	0	0
$79$	41.5692i	0.526193i	0.964770	+	0.263096i	$0.0847437\pi$
$79$	41.5692i	0.526193i	−0.964770	+	0.263096i	$0.915256\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	− 90.0666i	− 1.08514i	−0.840011	−	0.542570i	$-0.817451\pi$
$83$	− 90.0666i	− 1.08514i	0.840011	−	0.542570i	$-0.182549\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	17.3205i	0.199086i
$88$	0	0
$89$	14.0000	0.157303	0.0786517	−	0.996902i	$-0.474939\pi$
$89$	14.0000	0.157303	0.0786517	+	0.996902i	$0.474939\pi$
$90$	0	0
$91$	− 166.277i	− 1.82722i
$92$	0	0
$93$	−24.0000	−0.258065
$94$	0	0
$95$	0	0
$96$	0	0
$97$	96.0000	0.989691	0.494845	−	0.868981i	$-0.335225\pi$
$97$	96.0000	0.989691	0.494845	+	0.868981i	$0.335225\pi$
$98$	0	0
$99$	− 41.5692i	− 0.419891i
$100$	0	0
$101$	−38.0000	−0.376238	−0.188119	−	0.982146i	$-0.560239\pi$
$101$	−38.0000	−0.376238	−0.188119	+	0.982146i	$0.560239\pi$
$102$	0	0
$103$	145.492i	1.41255i	0.707939	+	0.706273i	$0.249625\pi$
$103$	145.492i	1.41255i	−0.707939	+	0.706273i	$0.750375\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.92820i	0.0647496i	0.999476	+	0.0323748i	$0.0103070\pi$
$107$	6.92820i	0.0647496i	−0.999476	+	0.0323748i	$0.989693\pi$
$108$	0	0
$109$	−166.000	−1.52294	−0.761468	−	0.648203i	$-0.775521\pi$
$109$	−166.000	−1.52294	−0.761468	+	0.648203i	$0.775521\pi$
$110$	0	0
$111$	− 41.5692i	− 0.374497i
$112$	0	0
$113$	168.000	1.48673	0.743363	−	0.668888i	$-0.233230\pi$
$113$	168.000	1.48673	0.743363	+	0.668888i	$0.233230\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−72.0000	−0.615385
$118$	0	0
$119$	166.277i	1.39728i
$120$	0	0
$121$	−71.0000	−0.586777
$122$	0	0
$123$	58.8897i	0.478778i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 34.6410i	− 0.272764i	−0.990656	−	0.136382i	$-0.956453\pi$
$127$	− 34.6410i	− 0.272764i	0.990656	−	0.136382i	$-0.0435474\pi$
$128$	0	0
$129$	36.0000	0.279070
$130$	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
$133$	192.000	1.44361
$134$	0	0
$135$	0	0
$136$	0	0
$137$	24.0000	0.175182	0.0875912	−	0.996157i	$-0.472083\pi$
$137$	24.0000	0.175182	0.0875912	+	0.996157i	$0.472083\pi$
$138$	0	0
$139$	− 193.990i	− 1.39561i	−0.716288	−	0.697805i	$-0.754160\pi$
$139$	− 193.990i	− 1.39561i	0.716288	−	0.697805i	$-0.245840\pi$
$140$	0	0
$141$	−12.0000	−0.0851064
$142$	0	0
$143$	332.554i	2.32555i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1.73205i	− 0.0117827i
$148$	0	0
$149$	−86.0000	−0.577181	−0.288591	−	0.957453i	$-0.593187\pi$
$149$	−86.0000	−0.577181	−0.288591	+	0.957453i	$0.593187\pi$
$150$	0	0
$151$	69.2820i	0.458821i	0.973330	+	0.229411i	$0.0736799\pi$
$151$	69.2820i	0.458821i	−0.973330	+	0.229411i	$0.926320\pi$
$152$	0	0
$153$	72.0000	0.470588
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−120.000	−0.764331	−0.382166	−	0.924094i	$-0.624822\pi$
$157$	−120.000	−0.764331	−0.382166	+	0.924094i	$0.624822\pi$
$158$	0	0
$159$	− 83.1384i	− 0.522883i
$160$	0	0
$161$	240.000	1.49068
$162$	0	0
$163$	− 62.3538i	− 0.382539i	−0.981538	−	0.191269i	$-0.938740\pi$
$163$	− 62.3538i	− 0.382539i	0.981538	−	0.191269i	$-0.0612604\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	242.487i	1.45202i	0.687685	+	0.726009i	$0.258627\pi$
$167$	242.487i	1.45202i	−0.687685	+	0.726009i	$0.741373\pi$
$168$	0	0
$169$	407.000	2.40828
$170$	0	0
$171$	− 83.1384i	− 0.486190i
$172$	0	0
$173$	192.000	1.10983	0.554913	−	0.831908i	$-0.312751\pi$
$173$	192.000	1.10983	0.554913	+	0.831908i	$0.312751\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−24.0000	−0.135593
$178$	0	0
$179$	318.697i	1.78043i	0.455539	+	0.890216i	$0.349447\pi$
$179$	318.697i	1.78043i	−0.455539	+	0.890216i	$0.650553\pi$
$180$	0	0
$181$	122.000	0.674033	0.337017	−	0.941499i	$-0.390582\pi$
$181$	122.000	0.674033	0.337017	+	0.941499i	$0.390582\pi$
$182$	0	0
$183$	− 121.244i	− 0.662533i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 332.554i	− 1.77836i
$188$	0	0
$189$	36.0000	0.190476
$190$	0	0
$191$	− 193.990i	− 1.01565i	−0.861459	−	0.507826i	$-0.830449\pi$
$191$	− 193.990i	− 1.01565i	0.861459	−	0.507826i	$-0.169551\pi$
$192$	0	0
$193$	144.000	0.746114	0.373057	−	0.927808i	$-0.378310\pi$
$193$	144.000	0.746114	0.373057	+	0.927808i	$0.378310\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	48.0000	0.243655	0.121827	−	0.992551i	$-0.461125\pi$
$197$	48.0000	0.243655	0.121827	+	0.992551i	$0.461125\pi$
$198$	0	0
$199$	− 290.985i	− 1.46223i	−0.682252	−	0.731117i	$-0.738999\pi$
$199$	− 290.985i	− 1.46223i	0.682252	−	0.731117i	$-0.261001\pi$
$200$	0	0
$201$	156.000	0.776119
$202$	0	0
$203$	69.2820i	0.341291i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 103.923i	− 0.502044i
$208$	0	0
$209$	−384.000	−1.83732
$210$	0	0
$211$	193.990i	0.919382i	0.888079	+	0.459691i	$0.152040\pi$
$211$	193.990i	0.919382i	−0.888079	+	0.459691i	$0.847960\pi$
$212$	0	0
$213$	96.0000	0.450704
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−96.0000	−0.442396
$218$	0	0
$219$	− 83.1384i	− 0.379628i
$220$	0	0
$221$	−576.000	−2.60633
$222$	0	0
$223$	− 131.636i	− 0.590295i	−0.955452	−	0.295148i	$-0.904631\pi$
$223$	− 131.636i	− 0.590295i	0.955452	−	0.295148i	$-0.0953688\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 270.200i	− 1.19031i	−0.803612	−	0.595154i	$-0.797091\pi$
$227$	− 270.200i	− 1.19031i	0.803612	−	0.595154i	$-0.202909\pi$
$228$	0	0
$229$	166.000	0.724891	0.362445	−	0.932005i	$-0.381942\pi$
$229$	166.000	0.724891	0.362445	+	0.932005i	$0.381942\pi$
$230$	0	0
$231$	− 166.277i	− 0.719813i
$232$	0	0
$233$	−264.000	−1.13305	−0.566524	−	0.824046i	$-0.691712\pi$
$233$	−264.000	−1.13305	−0.566524	+	0.824046i	$0.691712\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	72.0000	0.303797
$238$	0	0
$239$	83.1384i	0.347860i	0.984758	+	0.173930i	$0.0556466\pi$
$239$	83.1384i	0.347860i	−0.984758	+	0.173930i	$0.944353\pi$
$240$	0	0
$241$	−242.000	−1.00415	−0.502075	−	0.864824i	$-0.667430\pi$
$241$	−242.000	−1.00415	−0.502075	+	0.864824i	$0.667430\pi$
$242$	0	0
$243$	− 15.5885i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	665.108i	2.69274i
$248$	0	0
$249$	−156.000	−0.626506
$250$	0	0
$251$	207.846i	0.828072i	0.910260	+	0.414036i	$0.135881\pi$
$251$	207.846i	0.828072i	−0.910260	+	0.414036i	$0.864119\pi$
$252$	0	0
$253$	−480.000	−1.89723
$254$	0	0
$255$	0	0
$256$	0	0
$257$	72.0000	0.280156	0.140078	−	0.990140i	$-0.455265\pi$
$257$	72.0000	0.280156	0.140078	+	0.990140i	$0.455265\pi$
$258$	0	0
$259$	− 166.277i	− 0.641996i
$260$	0	0
$261$	30.0000	0.114943
$262$	0	0
$263$	− 270.200i	− 1.02738i	−0.857977	−	0.513688i	$-0.828279\pi$
$263$	− 270.200i	− 1.02738i	0.857977	−	0.513688i	$-0.171721\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 24.2487i	− 0.0908191i
$268$	0	0
$269$	−230.000	−0.855019	−0.427509	−	0.904011i	$-0.640609\pi$
$269$	−230.000	−0.855019	−0.427509	+	0.904011i	$0.640609\pi$
$270$	0	0
$271$	207.846i	0.766960i	0.923549	+	0.383480i	$0.125274\pi$
$271$	207.846i	0.766960i	−0.923549	+	0.383480i	$0.874726\pi$
$272$	0	0
$273$	−288.000	−1.05495
$274$	0	0
$275$	0	0
$276$	0	0
$277$	264.000	0.953069	0.476534	−	0.879156i	$-0.341893\pi$
$277$	264.000	0.953069	0.476534	+	0.879156i	$0.341893\pi$
$278$	0	0
$279$	41.5692i	0.148994i
$280$	0	0
$281$	254.000	0.903915	0.451957	−	0.892040i	$-0.350726\pi$
$281$	254.000	0.903915	0.451957	+	0.892040i	$0.350726\pi$
$282$	0	0
$283$	214.774i	0.758920i	0.925208	+	0.379460i	$0.123890\pi$
$283$	214.774i	0.758920i	−0.925208	+	0.379460i	$0.876110\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	235.559i	0.820763i
$288$	0	0
$289$	287.000	0.993080
$290$	0	0
$291$	− 166.277i	− 0.571398i
$292$	0	0
$293$	−432.000	−1.47440	−0.737201	−	0.675673i	$-0.763853\pi$
$293$	−432.000	−1.47440	−0.737201	+	0.675673i	$0.763853\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−72.0000	−0.242424
$298$	0	0
$299$	831.384i	2.78055i
$300$	0	0
$301$	144.000	0.478405
$302$	0	0
$303$	65.8179i	0.217221i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 297.913i	− 0.970400i	−0.874403	−	0.485200i	$-0.838747\pi$
$307$	− 297.913i	− 0.970400i	0.874403	−	0.485200i	$-0.161253\pi$
$308$	0	0
$309$	252.000	0.815534
$310$	0	0
$311$	526.543i	1.69307i	0.532336	+	0.846533i	$0.321314\pi$
$311$	526.543i	1.69307i	−0.532336	+	0.846533i	$0.678686\pi$
$312$	0	0
$313$	−240.000	−0.766773	−0.383387	−	0.923588i	$-0.625242\pi$
$313$	−240.000	−0.766773	−0.383387	+	0.923588i	$0.625242\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−240.000	−0.757098	−0.378549	−	0.925581i	$-0.623577\pi$
$317$	−240.000	−0.757098	−0.378549	+	0.925581i	$0.623577\pi$
$318$	0	0
$319$	− 138.564i	− 0.434370i
$320$	0	0
$321$	12.0000	0.0373832
$322$	0	0
$323$	− 665.108i	− 2.05916i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	287.520i	0.879267i
$328$	0	0
$329$	−48.0000	−0.145897
$330$	0	0
$331$	110.851i	0.334898i	0.985881	+	0.167449i	$0.0535530\pi$
$331$	110.851i	0.334898i	−0.985881	+	0.167449i	$0.946447\pi$
$332$	0	0
$333$	−72.0000	−0.216216
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−48.0000	−0.142433	−0.0712166	−	0.997461i	$-0.522688\pi$
$337$	−48.0000	−0.142433	−0.0712166	+	0.997461i	$0.522688\pi$
$338$	0	0
$339$	− 290.985i	− 0.858361i
$340$	0	0
$341$	192.000	0.563050
$342$	0	0
$343$	− 346.410i	− 1.00994i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 685.892i	− 1.97663i	−0.152411	−	0.988317i	$-0.548704\pi$
$347$	− 685.892i	− 1.97663i	0.152411	−	0.988317i	$-0.451296\pi$
$348$	0	0
$349$	−458.000	−1.31232	−0.656160	−	0.754621i	$-0.727821\pi$
$349$	−458.000	−1.31232	−0.656160	+	0.754621i	$0.727821\pi$
$350$	0	0
$351$	124.708i	0.355292i
$352$	0	0
$353$	−72.0000	−0.203966	−0.101983	−	0.994786i	$-0.532519\pi$
$353$	−72.0000	−0.203966	−0.101983	+	0.994786i	$0.532519\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	288.000	0.806723
$358$	0	0
$359$	637.395i	1.77547i	0.460352	+	0.887736i	$0.347723\pi$
$359$	637.395i	1.77547i	−0.460352	+	0.887736i	$0.652277\pi$
$360$	0	0
$361$	−407.000	−1.12742
$362$	0	0
$363$	122.976i	0.338776i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 145.492i	− 0.396437i	−0.980158	−	0.198218i	$-0.936484\pi$
$367$	− 145.492i	− 0.396437i	0.980158	−	0.198218i	$-0.0635155\pi$
$368$	0	0
$369$	102.000	0.276423
$370$	0	0
$371$	− 332.554i	− 0.896371i
$372$	0	0
$373$	−24.0000	−0.0643432	−0.0321716	−	0.999482i	$-0.510242\pi$
$373$	−24.0000	−0.0643432	−0.0321716	+	0.999482i	$0.510242\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−240.000	−0.636605
$378$	0	0
$379$	− 332.554i	− 0.877451i	−0.898621	−	0.438725i	$-0.855430\pi$
$379$	− 332.554i	− 0.877451i	0.898621	−	0.438725i	$-0.144570\pi$
$380$	0	0
$381$	−60.0000	−0.157480
$382$	0	0
$383$	− 187.061i	− 0.488411i	−0.969723	−	0.244206i	$-0.921473\pi$
$383$	− 187.061i	− 0.488411i	0.969723	−	0.244206i	$-0.0785272\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 62.3538i	− 0.161121i
$388$	0	0
$389$	−182.000	−0.467866	−0.233933	−	0.972253i	$-0.575160\pi$
$389$	−182.000	−0.467866	−0.233933	+	0.972253i	$0.575160\pi$
$390$	0	0
$391$	− 831.384i	− 2.12630i
$392$	0	0
$393$	120.000	0.305344
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−264.000	−0.664987	−0.332494	−	0.943105i	$-0.607890\pi$
$397$	−264.000	−0.664987	−0.332494	+	0.943105i	$0.607890\pi$
$398$	0	0
$399$	− 332.554i	− 0.833468i
$400$	0	0
$401$	−350.000	−0.872818	−0.436409	−	0.899748i	$-0.643750\pi$
$401$	−350.000	−0.872818	−0.436409	+	0.899748i	$0.643750\pi$
$402$	0	0
$403$	− 332.554i	− 0.825195i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	332.554i	0.817085i
$408$	0	0
$409$	−190.000	−0.464548	−0.232274	−	0.972650i	$-0.574617\pi$
$409$	−190.000	−0.464548	−0.232274	+	0.972650i	$0.574617\pi$
$410$	0	0
$411$	− 41.5692i	− 0.101142i
$412$	0	0
$413$	−96.0000	−0.232446
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−336.000	−0.805755
$418$	0	0
$419$	− 789.815i	− 1.88500i	−0.334206	−	0.942500i	$-0.608468\pi$
$419$	− 789.815i	− 1.88500i	0.334206	−	0.942500i	$-0.391532\pi$
$420$	0	0
$421$	502.000	1.19240	0.596200	−	0.802836i	$-0.296677\pi$
$421$	502.000	1.19240	0.596200	+	0.802836i	$0.296677\pi$
$422$	0	0
$423$	20.7846i	0.0491362i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 484.974i	− 1.13577i
$428$	0	0
$429$	576.000	1.34266
$430$	0	0
$431$	193.990i	0.450092i	0.974348	+	0.225046i	$0.0722533\pi$
$431$	193.990i	0.450092i	−0.974348	+	0.225046i	$0.927747\pi$
$432$	0	0
$433$	−528.000	−1.21940	−0.609700	−	0.792632i	$-0.708710\pi$
$433$	−528.000	−1.21940	−0.609700	+	0.792632i	$0.708710\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−960.000	−2.19680
$438$	0	0
$439$	− 180.133i	− 0.410326i	−0.978728	−	0.205163i	$-0.934227\pi$
$439$	− 180.133i	− 0.410326i	0.978728	−	0.205163i	$-0.0657725\pi$
$440$	0	0
$441$	−3.00000	−0.00680272
$442$	0	0
$443$	713.605i	1.61085i	0.592700	+	0.805423i	$0.298062\pi$
$443$	713.605i	1.61085i	−0.592700	+	0.805423i	$0.701938\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	148.956i	0.333236i
$448$	0	0
$449$	82.0000	0.182628	0.0913140	−	0.995822i	$-0.470893\pi$
$449$	82.0000	0.182628	0.0913140	+	0.995822i	$0.470893\pi$
$450$	0	0
$451$	− 471.118i	− 1.04461i
$452$	0	0
$453$	120.000	0.264901
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−384.000	−0.840263	−0.420131	−	0.907463i	$-0.638016\pi$
$457$	−384.000	−0.840263	−0.420131	+	0.907463i	$0.638016\pi$
$458$	0	0
$459$	− 124.708i	− 0.271694i
$460$	0	0
$461$	−278.000	−0.603037	−0.301518	−	0.953460i	$-0.597493\pi$
$461$	−278.000	−0.603037	−0.301518	+	0.953460i	$0.597493\pi$
$462$	0	0
$463$	228.631i	0.493803i	0.969041	+	0.246901i	$0.0794124\pi$
$463$	228.631i	0.493803i	−0.969041	+	0.246901i	$0.920588\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 436.477i	− 0.934640i	−0.884088	−	0.467320i	$-0.845220\pi$
$467$	− 436.477i	− 0.934640i	0.884088	−	0.467320i	$-0.154780\pi$
$468$	0	0
$469$	624.000	1.33049
$470$	0	0
$471$	207.846i	0.441287i
$472$	0	0
$473$	−288.000	−0.608879
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−144.000	−0.301887
$478$	0	0
$479$	387.979i	0.809978i	0.914322	+	0.404989i	$0.132725\pi$
$479$	387.979i	0.809978i	−0.914322	+	0.404989i	$0.867275\pi$
$480$	0	0
$481$	576.000	1.19751
$482$	0	0
$483$	− 415.692i	− 0.860646i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	270.200i	0.554825i	0.960751	+	0.277413i	$0.0894769\pi$
$487$	270.200i	0.554825i	−0.960751	+	0.277413i	$0.910523\pi$
$488$	0	0
$489$	−108.000	−0.220859
$490$	0	0
$491$	568.113i	1.15705i	0.815664	+	0.578526i	$0.196372\pi$
$491$	568.113i	1.15705i	−0.815664	+	0.578526i	$0.803628\pi$
$492$	0	0
$493$	240.000	0.486815
$494$	0	0
$495$	0	0
$496$	0	0
$497$	384.000	0.772636
$498$	0	0
$499$	775.959i	1.55503i	0.628866	+	0.777514i	$0.283519\pi$
$499$	775.959i	1.55503i	−0.628866	+	0.777514i	$0.716481\pi$
$500$	0	0
$501$	420.000	0.838323
$502$	0	0
$503$	256.344i	0.509629i	0.966990	+	0.254815i	$0.0820145\pi$
$503$	256.344i	0.509629i	−0.966990	+	0.254815i	$0.917986\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 704.945i	− 1.39042i
$508$	0	0
$509$	374.000	0.734774	0.367387	−	0.930068i	$-0.380252\pi$
$509$	374.000	0.734774	0.367387	+	0.930068i	$0.380252\pi$
$510$	0	0
$511$	− 332.554i	− 0.650790i
$512$	0	0
$513$	−144.000	−0.280702
$514$	0	0
$515$	0	0
$516$	0	0
$517$	96.0000	0.185687
$518$	0	0
$519$	− 332.554i	− 0.640759i
$520$	0	0
$521$	82.0000	0.157390	0.0786948	−	0.996899i	$-0.474925\pi$
$521$	82.0000	0.157390	0.0786948	+	0.996899i	$0.474925\pi$
$522$	0	0
$523$	− 505.759i	− 0.967034i	−0.875335	−	0.483517i	$-0.839359\pi$
$523$	− 505.759i	− 0.967034i	0.875335	−	0.483517i	$-0.160641\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	332.554i	0.631032i
$528$	0	0
$529$	−671.000	−1.26843
$530$	0	0
$531$	41.5692i	0.0782848i
$532$	0	0
$533$	−816.000	−1.53096
$534$	0	0
$535$	0	0
$536$	0	0
$537$	552.000	1.02793
$538$	0	0
$539$	13.8564i	0.0257076i
$540$	0	0
$541$	842.000	1.55638	0.778189	−	0.628031i	$-0.216139\pi$
$541$	842.000	1.55638	0.778189	+	0.628031i	$0.216139\pi$
$542$	0	0
$543$	− 211.310i	− 0.389153i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	450.333i	0.823278i	0.911347	+	0.411639i	$0.135044\pi$
$547$	450.333i	0.823278i	−0.911347	+	0.411639i	$0.864956\pi$
$548$	0	0
$549$	−210.000	−0.382514
$550$	0	0
$551$	− 277.128i	− 0.502955i
$552$	0	0
$553$	288.000	0.520796
$554$	0	0
$555$	0	0
$556$	0	0
$557$	864.000	1.55117	0.775583	−	0.631245i	$-0.217456\pi$
$557$	864.000	1.55117	0.775583	+	0.631245i	$0.217456\pi$
$558$	0	0
$559$	498.831i	0.892362i
$560$	0	0
$561$	−576.000	−1.02674
$562$	0	0
$563$	− 616.610i	− 1.09522i	−0.836733	−	0.547611i	$-0.815537\pi$
$563$	− 616.610i	− 1.09522i	0.836733	−	0.547611i	$-0.184463\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 62.3538i	− 0.109971i
$568$	0	0
$569$	−830.000	−1.45870	−0.729350	−	0.684141i	$-0.760177\pi$
$569$	−830.000	−1.45870	−0.729350	+	0.684141i	$0.760177\pi$
$570$	0	0
$571$	55.4256i	0.0970676i	0.998822	+	0.0485338i	$0.0154549\pi$
$571$	55.4256i	0.0970676i	−0.998822	+	0.0485338i	$0.984545\pi$
$572$	0	0
$573$	−336.000	−0.586387
$574$	0	0
$575$	0	0
$576$	0	0
$577$	96.0000	0.166378	0.0831889	−	0.996534i	$-0.473490\pi$
$577$	96.0000	0.166378	0.0831889	+	0.996534i	$0.473490\pi$
$578$	0	0
$579$	− 249.415i	− 0.430769i
$580$	0	0
$581$	−624.000	−1.07401
$582$	0	0
$583$	665.108i	1.14084i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	755.174i	1.28650i	0.765657	+	0.643249i	$0.222414\pi$
$587$	755.174i	1.28650i	−0.765657	+	0.643249i	$0.777586\pi$
$588$	0	0
$589$	384.000	0.651952
$590$	0	0
$591$	− 83.1384i	− 0.140674i
$592$	0	0
$593$	456.000	0.768971	0.384486	−	0.923131i	$-0.374379\pi$
$593$	456.000	0.768971	0.384486	+	0.923131i	$0.374379\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−504.000	−0.844221
$598$	0	0
$599$	− 304.841i	− 0.508916i	−0.967084	−	0.254458i	$-0.918103\pi$
$599$	− 304.841i	− 0.508916i	0.967084	−	0.254458i	$-0.0818971\pi$
$600$	0	0
$601$	−1154.00	−1.92013	−0.960067	−	0.279772i	$-0.909741\pi$
$601$	−1154.00	−1.92013	−0.960067	+	0.279772i	$0.909741\pi$
$602$	0	0
$603$	− 270.200i	− 0.448093i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 117.779i	− 0.194035i	−0.995283	−	0.0970177i	$-0.969070\pi$
$607$	− 117.779i	− 0.194035i	0.995283	−	0.0970177i	$-0.0309303\pi$
$608$	0	0
$609$	120.000	0.197044
$610$	0	0
$611$	− 166.277i	− 0.272139i
$612$	0	0
$613$	−456.000	−0.743883	−0.371941	−	0.928256i	$-0.621308\pi$
$613$	−456.000	−0.743883	−0.371941	+	0.928256i	$0.621308\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	600.000	0.972447	0.486224	−	0.873834i	$-0.338374\pi$
$617$	600.000	0.972447	0.486224	+	0.873834i	$0.338374\pi$
$618$	0	0
$619$	249.415i	0.402933i	0.979495	+	0.201466i	$0.0645707\pi$
$619$	249.415i	0.402933i	−0.979495	+	0.201466i	$0.935429\pi$
$620$	0	0
$621$	−180.000	−0.289855
$622$	0	0
$623$	− 96.9948i	− 0.155690i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	665.108i	1.06078i
$628$	0	0
$629$	−576.000	−0.915739
$630$	0	0
$631$	− 13.8564i	− 0.0219594i	−0.999940	−	0.0109797i	$-0.996505\pi$
$631$	− 13.8564i	− 0.0219594i	0.999940	−	0.0109797i	$-0.00349502\pi$
$632$	0	0
$633$	336.000	0.530806
$634$	0	0
$635$	0	0
$636$	0	0
$637$	24.0000	0.0376766
$638$	0	0
$639$	− 166.277i	− 0.260214i
$640$	0	0
$641$	1070.00	1.66927	0.834633	−	0.550806i	$-0.185680\pi$
$641$	1070.00	1.66927	0.834633	+	0.550806i	$0.185680\pi$
$642$	0	0
$643$	− 921.451i	− 1.43305i	−0.697562	−	0.716525i	$-0.745732\pi$
$643$	− 921.451i	− 1.43305i	0.697562	−	0.716525i	$-0.254268\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 339.482i	− 0.524702i	−0.964973	−	0.262351i	$-0.915502\pi$
$647$	− 339.482i	− 0.524702i	0.964973	−	0.262351i	$-0.0844978\pi$
$648$	0	0
$649$	192.000	0.295840
$650$	0	0
$651$	166.277i	0.255418i
$652$	0	0
$653$	1056.00	1.61715	0.808576	−	0.588392i	$-0.200239\pi$
$653$	1056.00	1.61715	0.808576	+	0.588392i	$0.200239\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−144.000	−0.219178
$658$	0	0
$659$	956.092i	1.45082i	0.688316	+	0.725411i	$0.258350\pi$
$659$	956.092i	1.45082i	−0.688316	+	0.725411i	$0.741650\pi$
$660$	0	0
$661$	−554.000	−0.838124	−0.419062	−	0.907958i	$-0.637641\pi$
$661$	−554.000	−0.838124	−0.419062	+	0.907958i	$0.637641\pi$
$662$	0	0
$663$	997.661i	1.50477i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 346.410i	− 0.519356i
$668$	0	0
$669$	−228.000	−0.340807
$670$	0	0
$671$	969.948i	1.44553i
$672$	0	0
$673$	−624.000	−0.927192	−0.463596	−	0.886047i	$-0.653441\pi$
$673$	−624.000	−0.927192	−0.463596	+	0.886047i	$0.653441\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	864.000	1.27622	0.638109	−	0.769946i	$-0.279717\pi$
$677$	864.000	1.27622	0.638109	+	0.769946i	$0.279717\pi$
$678$	0	0
$679$	− 665.108i	− 0.979540i
$680$	0	0
$681$	−468.000	−0.687225
$682$	0	0
$683$	− 145.492i	− 0.213019i	−0.994312	−	0.106510i	$-0.966032\pi$
$683$	− 145.492i	− 0.213019i	0.994312	−	0.106510i	$-0.0339675\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 287.520i	− 0.418516i
$688$	0	0
$689$	1152.00	1.67199
$690$	0	0
$691$	748.246i	1.08285i	0.840751	+	0.541423i	$0.182114\pi$
$691$	748.246i	1.08285i	−0.840751	+	0.541423i	$0.817886\pi$
$692$	0	0
$693$	−288.000	−0.415584
$694$	0	0
$695$	0	0
$696$	0	0
$697$	816.000	1.17073
$698$	0	0
$699$	457.261i	0.654165i
$700$	0	0
$701$	−634.000	−0.904422	−0.452211	−	0.891911i	$-0.649365\pi$
$701$	−634.000	−0.904422	−0.452211	+	0.891911i	$0.649365\pi$
$702$	0	0
$703$	665.108i	0.946099i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	263.272i	0.372379i
$708$	0	0
$709$	358.000	0.504937	0.252468	−	0.967605i	$-0.418758\pi$
$709$	358.000	0.504937	0.252468	+	0.967605i	$0.418758\pi$
$710$	0	0
$711$	− 124.708i	− 0.175398i
$712$	0	0
$713$	480.000	0.673212
$714$	0	0
$715$	0	0
$716$	0	0
$717$	144.000	0.200837
$718$	0	0
$719$	− 471.118i	− 0.655240i	−0.944810	−	0.327620i	$-0.893753\pi$
$719$	− 471.118i	− 0.655240i	0.944810	−	0.327620i	$-0.106247\pi$
$720$	0	0
$721$	1008.00	1.39806
$722$	0	0
$723$	419.156i	0.579746i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	713.605i	0.981575i	0.871279	+	0.490787i	$0.163291\pi$
$727$	713.605i	0.981575i	−0.871279	+	0.490787i	$0.836709\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 498.831i	− 0.682395i
$732$	0	0
$733$	840.000	1.14598	0.572988	−	0.819564i	$-0.305784\pi$
$733$	840.000	1.14598	0.572988	+	0.819564i	$0.305784\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1248.00	−1.69335
$738$	0	0
$739$	− 609.682i	− 0.825009i	−0.910955	−	0.412505i	$-0.864654\pi$
$739$	− 609.682i	− 0.825009i	0.910955	−	0.412505i	$-0.135346\pi$
$740$	0	0
$741$	1152.00	1.55466
$742$	0	0
$743$	782.887i	1.05368i	0.849963	+	0.526842i	$0.176624\pi$
$743$	782.887i	1.05368i	−0.849963	+	0.526842i	$0.823376\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	270.200i	0.361713i
$748$	0	0
$749$	48.0000	0.0640854
$750$	0	0
$751$	872.954i	1.16239i	0.813765	+	0.581194i	$0.197414\pi$
$751$	872.954i	1.16239i	−0.813765	+	0.581194i	$0.802586\pi$
$752$	0	0
$753$	360.000	0.478088
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−504.000	−0.665786	−0.332893	−	0.942965i	$-0.608025\pi$
$757$	−504.000	−0.665786	−0.332893	+	0.942965i	$0.608025\pi$
$758$	0	0
$759$	831.384i	1.09537i
$760$	0	0
$761$	754.000	0.990802	0.495401	−	0.868665i	$-0.335021\pi$
$761$	754.000	0.990802	0.495401	+	0.868665i	$0.335021\pi$
$762$	0	0
$763$	1150.08i	1.50732i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 332.554i	− 0.433577i
$768$	0	0
$769$	−578.000	−0.751625	−0.375813	−	0.926696i	$-0.622636\pi$
$769$	−578.000	−0.751625	−0.375813	+	0.926696i	$0.622636\pi$
$770$	0	0
$771$	− 124.708i	− 0.161748i
$772$	0	0
$773$	1296.00	1.67658	0.838292	−	0.545221i	$-0.183554\pi$
$773$	1296.00	1.67658	0.838292	+	0.545221i	$0.183554\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−288.000	−0.370656
$778$	0	0
$779$	− 942.236i	− 1.20955i
$780$	0	0
$781$	−768.000	−0.983355
$782$	0	0
$783$	− 51.9615i	− 0.0663621i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 1351.00i	− 1.71665i	−0.513111	−	0.858323i	$-0.671507\pi$
$787$	− 1351.00i	− 1.71665i	0.513111	−	0.858323i	$-0.328493\pi$
$788$	0	0
$789$	−468.000	−0.593156
$790$	0	0
$791$	− 1163.94i	− 1.47148i
$792$	0	0
$793$	1680.00	2.11854
$794$	0	0
$795$	0	0
$796$	0	0
$797$	288.000	0.361355	0.180678	−	0.983542i	$-0.442171\pi$
$797$	288.000	0.361355	0.180678	+	0.983542i	$0.442171\pi$
$798$	0	0
$799$	166.277i	0.208106i
$800$	0	0
$801$	−42.0000	−0.0524345
$802$	0	0
$803$	665.108i	0.828278i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	398.372i	0.493645i
$808$	0	0
$809$	494.000	0.610630	0.305315	−	0.952251i	$-0.401238\pi$
$809$	494.000	0.610630	0.305315	+	0.952251i	$0.401238\pi$
$810$	0	0
$811$	− 720.533i	− 0.888450i	−0.895915	−	0.444225i	$-0.853479\pi$
$811$	− 720.533i	− 0.888450i	0.895915	−	0.444225i	$-0.146521\pi$
$812$	0	0
$813$	360.000	0.442804
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−576.000	−0.705018
$818$	0	0
$819$	498.831i	0.609073i
$820$	0	0
$821$	−298.000	−0.362972	−0.181486	−	0.983394i	$-0.558091\pi$
$821$	−298.000	−0.362972	−0.181486	+	0.983394i	$0.558091\pi$
$822$	0	0
$823$	866.025i	1.05228i	0.850398	+	0.526139i	$0.176361\pi$
$823$	866.025i	1.05228i	−0.850398	+	0.526139i	$0.823639\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	367.195i	0.444008i	0.975046	+	0.222004i	$0.0712598\pi$
$827$	367.195i	0.444008i	−0.975046	+	0.222004i	$0.928740\pi$
$828$	0	0
$829$	−454.000	−0.547648	−0.273824	−	0.961780i	$-0.588289\pi$
$829$	−454.000	−0.547648	−0.273824	+	0.961780i	$0.588289\pi$
$830$	0	0
$831$	− 457.261i	− 0.550254i
$832$	0	0
$833$	−24.0000	−0.0288115
$834$	0	0
$835$	0	0
$836$	0	0
$837$	72.0000	0.0860215
$838$	0	0
$839$	− 193.990i	− 0.231215i	−0.993295	−	0.115608i	$-0.963118\pi$
$839$	− 193.990i	− 0.231215i	0.993295	−	0.115608i	$-0.0368815\pi$
$840$	0	0
$841$	−741.000	−0.881094
$842$	0	0
$843$	− 439.941i	− 0.521875i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	491.902i	0.580758i
$848$	0	0
$849$	372.000	0.438163
$850$	0	0
$851$	831.384i	0.976950i
$852$	0	0
$853$	1560.00	1.82884	0.914420	−	0.404767i	$-0.132647\pi$
$853$	1560.00	1.82884	0.914420	+	0.404767i	$0.132647\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1272.00	−1.48425	−0.742124	−	0.670263i	$-0.766181\pi$
$857$	−1272.00	−1.48425	−0.742124	+	0.670263i	$0.766181\pi$
$858$	0	0
$859$	− 1053.09i	− 1.22595i	−0.790104	−	0.612973i	$-0.789974\pi$
$859$	− 1053.09i	− 1.22595i	0.790104	−	0.612973i	$-0.210026\pi$
$860$	0	0
$861$	408.000	0.473868
$862$	0	0
$863$	− 491.902i	− 0.569991i	−0.958529	−	0.284996i	$-0.908008\pi$
$863$	− 491.902i	− 0.569991i	0.958529	−	0.284996i	$-0.0919921\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 497.099i	− 0.573355i
$868$	0	0
$869$	−576.000	−0.662831
$870$	0	0
$871$	2161.60i	2.48174i
$872$	0	0
$873$	−288.000	−0.329897
$874$	0	0
$875$	0	0
$876$	0	0
$877$	936.000	1.06727	0.533637	−	0.845713i	$-0.320825\pi$
$877$	936.000	1.06727	0.533637	+	0.845713i	$0.320825\pi$
$878$	0	0
$879$	748.246i	0.851247i
$880$	0	0
$881$	−562.000	−0.637911	−0.318956	−	0.947770i	$-0.603332\pi$
$881$	−562.000	−0.637911	−0.318956	+	0.947770i	$0.603332\pi$
$882$	0	0
$883$	− 6.92820i	− 0.00784621i	−0.999992	−	0.00392310i	$-0.998751\pi$
$883$	− 6.92820i	− 0.00784621i	0.999992	−	0.00392310i	$-0.00124877\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 561.184i	− 0.632677i	−0.948646	−	0.316338i	$-0.897546\pi$
$887$	− 561.184i	− 0.632677i	0.948646	−	0.316338i	$-0.102454\pi$
$888$	0	0
$889$	−240.000	−0.269966
$890$	0	0
$891$	124.708i	0.139964i
$892$	0	0
$893$	192.000	0.215006
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1440.00	1.60535
$898$	0	0
$899$	138.564i	0.154131i
$900$	0	0
$901$	−1152.00	−1.27858
$902$	0	0
$903$	− 249.415i	− 0.276207i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1669.70i	1.84090i	0.390859	+	0.920450i	$0.372178\pi$
$907$	1669.70i	1.84090i	−0.390859	+	0.920450i	$0.627822\pi$
$908$	0	0
$909$	114.000	0.125413
$910$	0	0
$911$	− 1718.19i	− 1.88605i	−0.332717	−	0.943027i	$-0.607966\pi$
$911$	− 1718.19i	− 1.88605i	0.332717	−	0.943027i	$-0.392034\pi$
$912$	0	0
$913$	1248.00	1.36692
$914$	0	0
$915$	0	0
$916$	0	0
$917$	480.000	0.523446
$918$	0	0
$919$	595.825i	0.648341i	0.945999	+	0.324171i	$0.105085\pi$
$919$	595.825i	0.648341i	−0.945999	+	0.324171i	$0.894915\pi$
$920$	0	0
$921$	−516.000	−0.560261
$922$	0	0
$923$	1330.22i	1.44119i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 436.477i	− 0.470849i
$928$	0	0
$929$	1042.00	1.12164	0.560818	−	0.827939i	$-0.310487\pi$
$929$	1042.00	1.12164	0.560818	+	0.827939i	$0.310487\pi$
$930$	0	0
$931$	27.7128i	0.0297667i
$932$	0	0
$933$	912.000	0.977492
$934$	0	0
$935$	0	0
$936$	0	0
$937$	864.000	0.922092	0.461046	−	0.887376i	$-0.347474\pi$
$937$	864.000	0.922092	0.461046	+	0.887376i	$0.347474\pi$
$938$	0	0
$939$	415.692i	0.442697i
$940$	0	0
$941$	490.000	0.520723	0.260361	−	0.965511i	$-0.416158\pi$
$941$	490.000	0.520723	0.260361	+	0.965511i	$0.416158\pi$
$942$	0	0
$943$	− 1177.79i	− 1.24899i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1129.30i	− 1.19250i	−0.802799	−	0.596250i	$-0.796657\pi$
$947$	− 1129.30i	− 1.19250i	0.802799	−	0.596250i	$-0.203343\pi$
$948$	0	0
$949$	1152.00	1.21391
$950$	0	0
$951$	415.692i	0.437111i
$952$	0	0
$953$	−360.000	−0.377754	−0.188877	−	0.982001i	$-0.560485\pi$
$953$	−360.000	−0.377754	−0.188877	+	0.982001i	$0.560485\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−240.000	−0.250784
$958$	0	0
$959$	− 166.277i	− 0.173386i
$960$	0	0
$961$	769.000	0.800208
$962$	0	0
$963$	− 20.7846i	− 0.0215832i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 422.620i	− 0.437043i	−0.975832	−	0.218521i	$-0.929877\pi$
$967$	− 422.620i	− 0.437043i	0.975832	−	0.218521i	$-0.0701233\pi$
$968$	0	0
$969$	−1152.00	−1.18885
$970$	0	0
$971$	− 1233.22i	− 1.27005i	−0.772491	−	0.635026i	$-0.780989\pi$
$971$	− 1233.22i	− 1.27005i	0.772491	−	0.635026i	$-0.219011\pi$
$972$	0	0
$973$	−1344.00	−1.38129
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1128.00	1.15455	0.577277	−	0.816548i	$-0.304115\pi$
$977$	1128.00	1.15455	0.577277	+	0.816548i	$0.304115\pi$
$978$	0	0
$979$	193.990i	0.198151i
$980$	0	0
$981$	498.000	0.507645
$982$	0	0
$983$	284.056i	0.288969i	0.989507	+	0.144484i	$0.0461524\pi$
$983$	284.056i	0.288969i	−0.989507	+	0.144484i	$0.953848\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	83.1384i	0.0842335i
$988$	0	0
$989$	−720.000	−0.728008
$990$	0	0
$991$	− 1094.66i	− 1.10460i	−0.833646	−	0.552299i	$-0.813751\pi$
$991$	− 1094.66i	− 1.10460i	0.833646	−	0.552299i	$-0.186249\pi$
$992$	0	0
$993$	192.000	0.193353
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−24.0000	−0.0240722	−0.0120361	−	0.999928i	$-0.503831\pi$
$997$	−24.0000	−0.0240722	−0.0120361	+	0.999928i	$0.503831\pi$
$998$	0	0
$999$	124.708i	0.124832i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.e.i.751.1		2
3.2	odd	2		3600.3.e.x.3151.1		2
4.3	odd	2	inner	1200.3.e.i.751.2		2
5.2	odd	4		240.3.j.a.79.2	yes	4
5.3	odd	4		240.3.j.a.79.4	yes	4
5.4	even	2		1200.3.e.a.751.2		2
12.11	even	2		3600.3.e.x.3151.2		2
15.2	even	4		720.3.j.g.559.3		4
15.8	even	4		720.3.j.g.559.1		4
15.14	odd	2		3600.3.e.f.3151.2		2
20.3	even	4		240.3.j.a.79.1	✓	4
20.7	even	4		240.3.j.a.79.3	yes	4
20.19	odd	2		1200.3.e.a.751.1		2
40.3	even	4		960.3.j.d.319.4		4
40.13	odd	4		960.3.j.d.319.1		4
40.27	even	4		960.3.j.d.319.2		4
40.37	odd	4		960.3.j.d.319.3		4
60.23	odd	4		720.3.j.g.559.4		4
60.47	odd	4		720.3.j.g.559.2		4
60.59	even	2		3600.3.e.f.3151.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.3.j.a.79.1	✓	4	20.3	even	4
240.3.j.a.79.2	yes	4	5.2	odd	4
240.3.j.a.79.3	yes	4	20.7	even	4
240.3.j.a.79.4	yes	4	5.3	odd	4
720.3.j.g.559.1		4	15.8	even	4
720.3.j.g.559.2		4	60.47	odd	4
720.3.j.g.559.3		4	15.2	even	4
720.3.j.g.559.4		4	60.23	odd	4
960.3.j.d.319.1		4	40.13	odd	4
960.3.j.d.319.2		4	40.27	even	4
960.3.j.d.319.3		4	40.37	odd	4
960.3.j.d.319.4		4	40.3	even	4
1200.3.e.a.751.1		2	20.19	odd	2
1200.3.e.a.751.2		2	5.4	even	2
1200.3.e.i.751.1		2	1.1	even	1	trivial
1200.3.e.i.751.2		2	4.3	odd	2	inner
3600.3.e.f.3151.1		2	60.59	even	2
3600.3.e.f.3151.2		2	15.14	odd	2
3600.3.e.x.3151.1		2	3.2	odd	2
3600.3.e.x.3151.2		2	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 $\F_{q}$
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)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} -6.92820i q^{7} -3.00000 q^{9} +13.8564i q^{11} +24.0000 q^{13} -24.0000 q^{17} +27.7128i q^{19} -12.0000 q^{21} +34.6410i q^{23} +5.19615i q^{27} -10.0000 q^{29} -13.8564i q^{31} +24.0000 q^{33} +24.0000 q^{37} -41.5692i q^{39} -34.0000 q^{41} +20.7846i q^{43} -6.92820i q^{47} +1.00000 q^{49} +41.5692i q^{51} +48.0000 q^{53} +48.0000 q^{57} -13.8564i q^{59} +70.0000 q^{61} +20.7846i q^{63} +90.0666i q^{67} +60.0000 q^{69} +55.4256i q^{71} +48.0000 q^{73} +96.0000 q^{77} +41.5692i q^{79} +9.00000 q^{81} -90.0666i q^{83} +17.3205i q^{87} +14.0000 q^{89} -166.277i q^{91} -24.0000 q^{93} +96.0000 q^{97} -41.5692i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{9} + 48 q^{13} - 48 q^{17} - 24 q^{21} - 20 q^{29} + 48 q^{33} + 48 q^{37} - 68 q^{41} + 2 q^{49} + 96 q^{53} + 96 q^{57} + 140 q^{61} + 120 q^{69} + 96 q^{73} + 192 q^{77} + 18 q^{81} + 28 q^{89}+ \cdots + 192 q^{97}+O(q^{100}) \) 2 * q - 6 * q^9 + 48 * q^13 - 48 * q^17 - 24 * q^21 - 20 * q^29 + 48 * q^33 + 48 * q^37 - 68 * q^41 + 2 * q^49 + 96 * q^53 + 96 * q^57 + 140 * q^61 + 120 * q^69 + 96 * q^73 + 192 * q^77 + 18 * q^81 + 28 * q^89 - 48 * q^93 + 192 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more