LMFDB - Embedded newform 9300.2.g.p.3349.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9300,2,Mod(3349,9300)] 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("9300.3349"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9300.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,0,0,-6,0,-4,0,0,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$74.2608738798$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.5089536.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 16x^2 - 24*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1860)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3349.1
Root			$0.675970 - 0.675970i$ of defining polynomial
Character	$\chi$	$=$	9300.3349
Dual form			9300.2.g.p.3349.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -4.43807i q^{7} -1.00000 q^{9} +4.17226 q^{11} +4.43807i q^{13} +1.35194i q^{17} -2.70388 q^{19} -4.43807 q^{21} -1.35194i q^{23} +1.00000i q^{27} +7.25839 q^{29} -1.00000 q^{31} -4.17226i q^{33} +10.7826i q^{37} +4.43807 q^{39} +2.17226 q^{41} +9.46838i q^{43} +4.11644i q^{47} -12.6965 q^{49} +1.35194 q^{51} +12.8203i q^{53} +2.70388i q^{57} -5.25839 q^{59} +6.00000 q^{61} +4.43807i q^{63} +1.14195i q^{67} -1.35194 q^{69} -10.7268 q^{71} +10.4381i q^{73} -18.5168i q^{77} +2.05582 q^{79} +1.00000 q^{81} -6.99258i q^{83} -7.25839i q^{87} +2.55451 q^{89} +19.6965 q^{91} +1.00000i q^{93} +15.9245i q^{97} -4.17226 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{9} - 4 q^{11} - 8 q^{19} - 8 q^{21} - 6 q^{31} + 8 q^{39} - 16 q^{41} - 14 q^{49} + 4 q^{51} + 12 q^{59} + 36 q^{61} - 4 q^{69} + 6 q^{81} - 20 q^{89} + 56 q^{91} + 4 q^{99}+O(q^{100}) $ 6 * q - 6 * q^9 - 4 * q^11 - 8 * q^19 - 8 * q^21 - 6 * q^31 + 8 * q^39 - 16 * q^41 - 14 * q^49 + 4 * q^51 + 12 * q^59 + 36 * q^61 - 4 * q^69 + 6 * q^81 - 20 * q^89 + 56 * q^91 + 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/9300\mathbb{Z}\right)^\times$.

$n$	$1801$	$2977$	$3101$	$4651$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.43807i	− 1.67743i	−0.544569	−	0.838716i	$-0.683307\pi$
$7$	− 4.43807i	− 1.67743i	0.544569	−	0.838716i	$-0.316693\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.17226	1.25798	0.628992	−	0.777412i	$-0.283468\pi$
$11$	4.17226	1.25798	0.628992	+	0.777412i	$0.283468\pi$
$12$	0	0
$13$	4.43807i	1.23090i	0.788176	+	0.615449i	$0.211025\pi$
$13$	4.43807i	1.23090i	−0.788176	+	0.615449i	$0.788975\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.35194i	0.327893i	0.986469	+	0.163947i	$0.0524225\pi$
$17$	1.35194i	0.327893i	−0.986469	+	0.163947i	$0.947577\pi$
$18$	0	0
$19$	−2.70388	−0.620312	−0.310156	−	0.950686i	$-0.600381\pi$
$19$	−2.70388	−0.620312	−0.310156	+	0.950686i	$0.600381\pi$
$20$	0	0
$21$	−4.43807	−0.968466
$22$	0	0
$23$	− 1.35194i	− 0.281899i	−0.990017	−	0.140949i	$-0.954985\pi$
$23$	− 1.35194i	− 0.281899i	0.990017	−	0.140949i	$-0.0450155\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	7.25839	1.34785	0.673925	−	0.738800i	$-0.264607\pi$
$29$	7.25839	1.34785	0.673925	+	0.738800i	$0.264607\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	− 4.17226i	− 0.726297i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.7826i	1.77265i	0.463067	+	0.886323i	$0.346749\pi$
$37$	10.7826i	1.77265i	−0.463067	+	0.886323i	$0.653251\pi$
$38$	0	0
$39$	4.43807	0.710660
$40$	0	0
$41$	2.17226	0.339250	0.169625	−	0.985509i	$-0.445744\pi$
$41$	2.17226	0.339250	0.169625	+	0.985509i	$0.445744\pi$
$42$	0	0
$43$	9.46838i	1.44391i	0.691938	+	0.721957i	$0.256757\pi$
$43$	9.46838i	1.44391i	−0.691938	+	0.721957i	$0.743243\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.11644i	0.600445i	0.953869	+	0.300222i	$0.0970609\pi$
$47$	4.11644i	0.600445i	−0.953869	+	0.300222i	$0.902939\pi$
$48$	0	0
$49$	−12.6965	−1.81378
$50$	0	0
$51$	1.35194	0.189309
$52$	0	0
$53$	12.8203i	1.76101i	0.474040	+	0.880503i	$0.342795\pi$
$53$	12.8203i	1.76101i	−0.474040	+	0.880503i	$0.657205\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.70388i	0.358137i
$58$	0	0
$59$	−5.25839	−0.684584	−0.342292	−	0.939594i	$-0.611203\pi$
$59$	−5.25839	−0.684584	−0.342292	+	0.939594i	$0.611203\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	4.43807i	0.559144i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.14195i	0.139511i	0.997564	+	0.0697556i	$0.0222219\pi$
$67$	1.14195i	0.139511i	−0.997564	+	0.0697556i	$0.977778\pi$
$68$	0	0
$69$	−1.35194	−0.162754
$70$	0	0
$71$	−10.7268	−1.27303	−0.636517	−	0.771263i	$-0.719625\pi$
$71$	−10.7268	−1.27303	−0.636517	+	0.771263i	$0.719625\pi$
$72$	0	0
$73$	10.4381i	1.22168i	0.791753	+	0.610842i	$0.209169\pi$
$73$	10.4381i	1.22168i	−0.791753	+	0.610842i	$0.790831\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 18.5168i	− 2.11018i
$78$	0	0
$79$	2.05582	0.231298	0.115649	−	0.993290i	$-0.463105\pi$
$79$	2.05582	0.231298	0.115649	+	0.993290i	$0.463105\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 6.99258i	− 0.767536i	−0.923430	−	0.383768i	$-0.874626\pi$
$83$	− 6.99258i	− 0.767536i	0.923430	−	0.383768i	$-0.125374\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 7.25839i	− 0.778181i
$88$	0	0
$89$	2.55451	0.270778	0.135389	−	0.990793i	$-0.456772\pi$
$89$	2.55451	0.270778	0.135389	+	0.990793i	$0.456772\pi$
$90$	0	0
$91$	19.6965	2.06475
$92$	0	0
$93$	1.00000i	0.103695i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	15.9245i	1.61689i	0.588570	+	0.808446i	$0.299691\pi$
$97$	15.9245i	1.61689i	−0.588570	+	0.808446i	$0.700309\pi$
$98$	0	0
$99$	−4.17226	−0.419328
$100$	0	0
$101$	−0.876139	−0.0871791	−0.0435895	−	0.999050i	$-0.513879\pi$
$101$	−0.876139	−0.0871791	−0.0435895	+	0.999050i	$0.513879\pi$
$102$	0	0
$103$	6.43807i	0.634362i	0.948365	+	0.317181i	$0.102736\pi$
$103$	6.43807i	0.634362i	−0.948365	+	0.317181i	$0.897264\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 11.6965i	− 1.13074i	−0.824838	−	0.565370i	$-0.808733\pi$
$107$	− 11.6965i	− 1.13074i	0.824838	−	0.565370i	$-0.191267\pi$
$108$	0	0
$109$	−11.5242	−1.10382	−0.551909	−	0.833904i	$-0.686100\pi$
$109$	−11.5242	−1.10382	−0.551909	+	0.833904i	$0.686100\pi$
$110$	0	0
$111$	10.7826	1.02344
$112$	0	0
$113$	17.5800i	1.65379i	0.562357	+	0.826894i	$0.309895\pi$
$113$	17.5800i	1.65379i	−0.562357	+	0.826894i	$0.690105\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 4.43807i	− 0.410300i
$118$	0	0
$119$	6.00000	0.550019
$120$	0	0
$121$	6.40776	0.582523
$122$	0	0
$123$	− 2.17226i	− 0.195866i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 15.8129i	− 1.40317i	−0.712587	−	0.701584i	$-0.752476\pi$
$127$	− 15.8129i	− 1.40317i	0.712587	−	0.701584i	$-0.247524\pi$
$128$	0	0
$129$	9.46838	0.833645
$130$	0	0
$131$	−10.1345	−0.885458	−0.442729	−	0.896655i	$-0.645990\pi$
$131$	−10.1345	−0.885458	−0.442729	+	0.896655i	$0.645990\pi$
$132$	0	0
$133$	12.0000i	1.04053i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	15.1042i	1.29044i	0.763997	+	0.645220i	$0.223234\pi$
$137$	15.1042i	1.29044i	−0.763997	+	0.645220i	$0.776766\pi$
$138$	0	0
$139$	17.2207	1.46064	0.730319	−	0.683107i	$-0.239372\pi$
$139$	17.2207	1.46064	0.730319	+	0.683107i	$0.239372\pi$
$140$	0	0
$141$	4.11644	0.346667
$142$	0	0
$143$	18.5168i	1.54845i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	12.6965i	1.04719i
$148$	0	0
$149$	−1.82774	−0.149734	−0.0748671	−	0.997194i	$-0.523853\pi$
$149$	−1.82774	−0.149734	−0.0748671	+	0.997194i	$0.523853\pi$
$150$	0	0
$151$	−16.4610	−1.33957	−0.669787	−	0.742553i	$-0.733615\pi$
$151$	−16.4610	−1.33957	−0.669787	+	0.742553i	$0.733615\pi$
$152$	0	0
$153$	− 1.35194i	− 0.109298i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 4.53162i	− 0.361662i	−0.983514	−	0.180831i	$-0.942121\pi$
$157$	− 4.53162i	− 0.361662i	0.983514	−	0.180831i	$-0.0578788\pi$
$158$	0	0
$159$	12.8203	1.01672
$160$	0	0
$161$	−6.00000	−0.472866
$162$	0	0
$163$	− 2.78259i	− 0.217949i	−0.994045	−	0.108975i	$-0.965243\pi$
$163$	− 2.78259i	− 0.217949i	0.994045	−	0.108975i	$-0.0347567\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.7645i	0.987747i	0.869534	+	0.493873i	$0.164419\pi$
$167$	12.7645i	0.987747i	−0.869534	+	0.493873i	$0.835581\pi$
$168$	0	0
$169$	−6.69646	−0.515112
$170$	0	0
$171$	2.70388	0.206771
$172$	0	0
$173$	25.5800i	1.94481i	0.233294	+	0.972406i	$0.425050\pi$
$173$	25.5800i	1.94481i	−0.233294	+	0.972406i	$0.574950\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.25839i	0.395245i
$178$	0	0
$179$	−5.64064	−0.421601	−0.210801	−	0.977529i	$-0.567607\pi$
$179$	−5.64064	−0.421601	−0.210801	+	0.977529i	$0.567607\pi$
$180$	0	0
$181$	8.93676	0.664264	0.332132	−	0.943233i	$-0.392232\pi$
$181$	8.93676	0.664264	0.332132	+	0.943233i	$0.392232\pi$
$182$	0	0
$183$	− 6.00000i	− 0.443533i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5.64064i	0.412485i
$188$	0	0
$189$	4.43807	0.322822
$190$	0	0
$191$	−2.32163	−0.167987	−0.0839935	−	0.996466i	$-0.526767\pi$
$191$	−2.32163	−0.167987	−0.0839935	+	0.996466i	$0.526767\pi$
$192$	0	0
$193$	− 5.22066i	− 0.375791i	−0.982189	−	0.187896i	$-0.939833\pi$
$193$	− 5.22066i	− 0.375791i	0.982189	−	0.187896i	$-0.0601667\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.58482i	0.682890i	0.939902	+	0.341445i	$0.110916\pi$
$197$	9.58482i	0.682890i	−0.939902	+	0.341445i	$0.889084\pi$
$198$	0	0
$199$	−12.9171	−0.915670	−0.457835	−	0.889037i	$-0.651375\pi$
$199$	−12.9171	−0.915670	−0.457835	+	0.889037i	$0.651375\pi$
$200$	0	0
$201$	1.14195	0.0805468
$202$	0	0
$203$	− 32.2132i	− 2.26093i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.35194i	0.0939663i
$208$	0	0
$209$	−11.2813	−0.780343
$210$	0	0
$211$	5.82774	0.401198	0.200599	−	0.979673i	$-0.435711\pi$
$211$	5.82774	0.401198	0.200599	+	0.979673i	$0.435711\pi$
$212$	0	0
$213$	10.7268i	0.734986i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	4.43807i	0.301276i
$218$	0	0
$219$	10.4381	0.705339
$220$	0	0
$221$	−6.00000	−0.403604
$222$	0	0
$223$	− 2.00000i	− 0.133930i	−0.997755	−	0.0669650i	$-0.978668\pi$
$223$	− 2.00000i	− 0.133930i	0.997755	−	0.0669650i	$-0.0213316\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 25.7933i	− 1.71196i	−0.517010	−	0.855979i	$-0.672955\pi$
$227$	− 25.7933i	− 1.71196i	0.517010	−	0.855979i	$-0.327045\pi$
$228$	0	0
$229$	7.93937	0.524649	0.262324	−	0.964980i	$-0.415511\pi$
$229$	7.93937	0.524649	0.262324	+	0.964980i	$0.415511\pi$
$230$	0	0
$231$	−18.5168	−1.21831
$232$	0	0
$233$	24.4610i	1.60249i	0.598336	+	0.801246i	$0.295829\pi$
$233$	24.4610i	1.60249i	−0.598336	+	0.801246i	$0.704171\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 2.05582i	− 0.133540i
$238$	0	0
$239$	18.0000	1.16432	0.582162	−	0.813073i	$-0.302207\pi$
$239$	18.0000	1.16432	0.582162	+	0.813073i	$0.302207\pi$
$240$	0	0
$241$	−2.64325	−0.170267	−0.0851335	−	0.996370i	$-0.527132\pi$
$241$	−2.64325	−0.170267	−0.0851335	+	0.996370i	$0.527132\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 12.0000i	− 0.763542i
$248$	0	0
$249$	−6.99258	−0.443137
$250$	0	0
$251$	−16.4562	−1.03870	−0.519352	−	0.854561i	$-0.673827\pi$
$251$	−16.4562	−1.03870	−0.519352	+	0.854561i	$0.673827\pi$
$252$	0	0
$253$	− 5.64064i	− 0.354624i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 7.16484i	− 0.446931i	−0.974712	−	0.223465i	$-0.928263\pi$
$257$	− 7.16484i	− 0.446931i	0.974712	−	0.223465i	$-0.0717369\pi$
$258$	0	0
$259$	47.8539	2.97350
$260$	0	0
$261$	−7.25839	−0.449283
$262$	0	0
$263$	− 8.87614i	− 0.547326i	−0.961826	−	0.273663i	$-0.911765\pi$
$263$	− 8.87614i	− 0.547326i	0.961826	−	0.273663i	$-0.0882353\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 2.55451i	− 0.156334i
$268$	0	0
$269$	−3.90164	−0.237887	−0.118944	−	0.992901i	$-0.537951\pi$
$269$	−3.90164	−0.237887	−0.118944	+	0.992901i	$0.537951\pi$
$270$	0	0
$271$	−4.87614	−0.296204	−0.148102	−	0.988972i	$-0.547316\pi$
$271$	−4.87614	−0.296204	−0.148102	+	0.988972i	$0.547316\pi$
$272$	0	0
$273$	− 19.6965i	− 1.19208i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 29.5981i	− 1.77838i	−0.457540	−	0.889189i	$-0.651269\pi$
$277$	− 29.5981i	− 1.77838i	0.457540	−	0.889189i	$-0.348731\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	−15.2207	−0.907988	−0.453994	−	0.891005i	$-0.650001\pi$
$281$	−15.2207	−0.907988	−0.453994	+	0.891005i	$0.650001\pi$
$282$	0	0
$283$	− 20.0181i	− 1.18995i	−0.803744	−	0.594976i	$-0.797162\pi$
$283$	− 20.0181i	− 1.18995i	0.803744	−	0.594976i	$-0.202838\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 9.64064i	− 0.569069i
$288$	0	0
$289$	15.1723	0.892486
$290$	0	0
$291$	15.9245	0.933513
$292$	0	0
$293$	− 8.46096i	− 0.494295i	−0.968978	−	0.247147i	$-0.920507\pi$
$293$	− 8.46096i	− 0.494295i	0.968978	−	0.247147i	$-0.0794932\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.17226i	0.242099i
$298$	0	0
$299$	6.00000	0.346989
$300$	0	0
$301$	42.0213	2.42207
$302$	0	0
$303$	0.876139i	0.0503329i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	8.43807i	0.481586i	0.970576	+	0.240793i	$0.0774075\pi$
$307$	8.43807i	0.481586i	−0.970576	+	0.240793i	$0.922592\pi$
$308$	0	0
$309$	6.43807	0.366249
$310$	0	0
$311$	−20.6661	−1.17187	−0.585935	−	0.810358i	$-0.699273\pi$
$311$	−20.6661	−1.17187	−0.585935	+	0.810358i	$0.699273\pi$
$312$	0	0
$313$	21.3142i	1.20475i	0.798213	+	0.602375i	$0.205779\pi$
$313$	21.3142i	1.20475i	−0.798213	+	0.602375i	$0.794221\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 13.5242i	− 0.759595i	−0.925070	−	0.379797i	$-0.875994\pi$
$317$	− 13.5242i	− 0.759595i	0.925070	−	0.379797i	$-0.124006\pi$
$318$	0	0
$319$	30.2839	1.69557
$320$	0	0
$321$	−11.6965	−0.652833
$322$	0	0
$323$	− 3.65548i	− 0.203396i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	11.5242i	0.637290i
$328$	0	0
$329$	18.2691	1.00721
$330$	0	0
$331$	12.9926	0.714137	0.357068	−	0.934078i	$-0.383776\pi$
$331$	12.9926	0.714137	0.357068	+	0.934078i	$0.383776\pi$
$332$	0	0
$333$	− 10.7826i	− 0.590882i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 22.5955i	− 1.23085i	−0.788194	−	0.615427i	$-0.788983\pi$
$337$	− 22.5955i	− 1.23085i	0.788194	−	0.615427i	$-0.211017\pi$
$338$	0	0
$339$	17.5800	0.954815
$340$	0	0
$341$	−4.17226	−0.225941
$342$	0	0
$343$	25.2813i	1.36506i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0.419983i	0.0225459i	0.999936	+	0.0112729i	$0.00358836\pi$
$347$	0.419983i	0.0225459i	−0.999936	+	0.0112729i	$0.996412\pi$
$348$	0	0
$349$	11.8687	0.635318	0.317659	−	0.948205i	$-0.397103\pi$
$349$	11.8687	0.635318	0.317659	+	0.948205i	$0.397103\pi$
$350$	0	0
$351$	−4.43807	−0.236887
$352$	0	0
$353$	− 1.69646i	− 0.0902935i	−0.998980	−	0.0451467i	$-0.985624\pi$
$353$	− 1.69646i	− 0.0902935i	0.998980	−	0.0451467i	$-0.0143755\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 6.00000i	− 0.317554i
$358$	0	0
$359$	4.80223	0.253452	0.126726	−	0.991938i	$-0.459553\pi$
$359$	4.80223	0.253452	0.126726	+	0.991938i	$0.459553\pi$
$360$	0	0
$361$	−11.6890	−0.615213
$362$	0	0
$363$	− 6.40776i	− 0.336320i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	31.6258i	1.65085i	0.564509	+	0.825427i	$0.309065\pi$
$367$	31.6258i	1.65085i	−0.564509	+	0.825427i	$0.690935\pi$
$368$	0	0
$369$	−2.17226	−0.113083
$370$	0	0
$371$	56.8975	2.95397
$372$	0	0
$373$	17.9245i	0.928097i	0.885810	+	0.464049i	$0.153604\pi$
$373$	17.9245i	0.928097i	−0.885810	+	0.464049i	$0.846396\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	32.2132i	1.65907i
$378$	0	0
$379$	−32.8007	−1.68486	−0.842429	−	0.538808i	$-0.818875\pi$
$379$	−32.8007	−1.68486	−0.842429	+	0.538808i	$0.818875\pi$
$380$	0	0
$381$	−15.8129	−0.810120
$382$	0	0
$383$	− 29.8081i	− 1.52312i	−0.648093	−	0.761561i	$-0.724433\pi$
$383$	− 29.8081i	− 1.52312i	0.648093	−	0.761561i	$-0.275567\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 9.46838i	− 0.481305i
$388$	0	0
$389$	32.7629	1.66115	0.830574	−	0.556909i	$-0.188013\pi$
$389$	32.7629	1.66115	0.830574	+	0.556909i	$0.188013\pi$
$390$	0	0
$391$	1.82774	0.0924328
$392$	0	0
$393$	10.1345i	0.511219i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 34.4562i	− 1.72930i	−0.502371	−	0.864652i	$-0.667539\pi$
$397$	− 34.4562i	− 1.72930i	0.502371	−	0.864652i	$-0.332461\pi$
$398$	0	0
$399$	12.0000	0.600751
$400$	0	0
$401$	29.7145	1.48387	0.741937	−	0.670470i	$-0.233907\pi$
$401$	29.7145	1.48387	0.741937	+	0.670470i	$0.233907\pi$
$402$	0	0
$403$	− 4.43807i	− 0.221076i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	44.9878i	2.22996i
$408$	0	0
$409$	28.0968	1.38930	0.694649	−	0.719349i	$-0.255560\pi$
$409$	28.0968	1.38930	0.694649	+	0.719349i	$0.255560\pi$
$410$	0	0
$411$	15.1042	0.745036
$412$	0	0
$413$	23.3371i	1.14834i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 17.2207i	− 0.843299i
$418$	0	0
$419$	−6.83841	−0.334078	−0.167039	−	0.985950i	$-0.553421\pi$
$419$	−6.83841	−0.334078	−0.167039	+	0.985950i	$0.553421\pi$
$420$	0	0
$421$	−5.24030	−0.255397	−0.127698	−	0.991813i	$-0.540759\pi$
$421$	−5.24030	−0.255397	−0.127698	+	0.991813i	$0.540759\pi$
$422$	0	0
$423$	− 4.11644i	− 0.200148i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 26.6284i	− 1.28864i
$428$	0	0
$429$	18.5168	0.893999
$430$	0	0
$431$	6.83841	0.329395	0.164697	−	0.986344i	$-0.447335\pi$
$431$	6.83841	0.329395	0.164697	+	0.986344i	$0.447335\pi$
$432$	0	0
$433$	− 11.3142i	− 0.543726i	−0.962336	−	0.271863i	$-0.912360\pi$
$433$	− 11.3142i	− 0.543726i	0.962336	−	0.271863i	$-0.0876398\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.65548i	0.174865i
$438$	0	0
$439$	10.8155	0.516196	0.258098	−	0.966119i	$-0.416904\pi$
$439$	10.8155	0.516196	0.258098	+	0.966119i	$0.416904\pi$
$440$	0	0
$441$	12.6965	0.604593
$442$	0	0
$443$	− 17.4487i	− 0.829015i	−0.910046	−	0.414507i	$-0.863954\pi$
$443$	− 17.4487i	− 0.829015i	0.910046	−	0.414507i	$-0.136046\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1.82774i	0.0864491i
$448$	0	0
$449$	−3.96227	−0.186991	−0.0934955	−	0.995620i	$-0.529804\pi$
$449$	−3.96227	−0.186991	−0.0934955	+	0.995620i	$0.529804\pi$
$450$	0	0
$451$	9.06324	0.426771
$452$	0	0
$453$	16.4610i	0.773404i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 33.8310i	− 1.58255i	−0.611462	−	0.791273i	$-0.709418\pi$
$457$	− 33.8310i	− 1.58255i	0.611462	−	0.791273i	$-0.290582\pi$
$458$	0	0
$459$	−1.35194	−0.0631031
$460$	0	0
$461$	−7.55712	−0.351970	−0.175985	−	0.984393i	$-0.556311\pi$
$461$	−7.55712	−0.351970	−0.175985	+	0.984393i	$0.556311\pi$
$462$	0	0
$463$	28.7401i	1.33566i	0.744312	+	0.667832i	$0.232777\pi$
$463$	28.7401i	1.33566i	−0.744312	+	0.667832i	$0.767223\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.16484i	0.331549i	0.986164	+	0.165775i	$0.0530124\pi$
$467$	7.16484i	0.331549i	−0.986164	+	0.165775i	$0.946988\pi$
$468$	0	0
$469$	5.06804	0.234021
$470$	0	0
$471$	−4.53162	−0.208806
$472$	0	0
$473$	39.5046i	1.81642i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 12.8203i	− 0.587002i
$478$	0	0
$479$	30.8384	1.40904	0.704521	−	0.709683i	$-0.251162\pi$
$479$	30.8384	1.40904	0.704521	+	0.709683i	$0.251162\pi$
$480$	0	0
$481$	−47.8539	−2.18195
$482$	0	0
$483$	6.00000i	0.273009i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	6.17226i	0.279692i	0.990173	+	0.139846i	$0.0446607\pi$
$487$	6.17226i	0.279692i	−0.990173	+	0.139846i	$0.955339\pi$
$488$	0	0
$489$	−2.78259	−0.125833
$490$	0	0
$491$	1.10902	0.0500496	0.0250248	−	0.999687i	$-0.492034\pi$
$491$	1.10902	0.0500496	0.0250248	+	0.999687i	$0.492034\pi$
$492$	0	0
$493$	9.81290i	0.441951i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	47.6062i	2.13543i
$498$	0	0
$499$	30.2180	1.35275	0.676373	−	0.736559i	$-0.263551\pi$
$499$	30.2180	1.35275	0.676373	+	0.736559i	$0.263551\pi$
$500$	0	0
$501$	12.7645	0.570276
$502$	0	0
$503$	− 15.2765i	− 0.681144i	−0.940218	−	0.340572i	$-0.889379\pi$
$503$	− 15.2765i	− 0.681144i	0.940218	−	0.340572i	$-0.110621\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6.69646i	0.297400i
$508$	0	0
$509$	7.08613	0.314087	0.157044	−	0.987592i	$-0.449804\pi$
$509$	7.08613	0.314087	0.157044	+	0.987592i	$0.449804\pi$
$510$	0	0
$511$	46.3249	2.04929
$512$	0	0
$513$	− 2.70388i	− 0.119379i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	17.1749i	0.755350i
$518$	0	0
$519$	25.5800	1.12284
$520$	0	0
$521$	16.7645	0.734466	0.367233	−	0.930129i	$-0.380305\pi$
$521$	16.7645	0.734466	0.367233	+	0.930129i	$0.380305\pi$
$522$	0	0
$523$	− 20.9878i	− 0.917731i	−0.888506	−	0.458866i	$-0.848256\pi$
$523$	− 20.9878i	− 0.917731i	0.888506	−	0.458866i	$-0.151744\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1.35194i	− 0.0588914i
$528$	0	0
$529$	21.1723	0.920533
$530$	0	0
$531$	5.25839	0.228195
$532$	0	0
$533$	9.64064i	0.417583i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	5.64064i	0.243412i
$538$	0	0
$539$	−52.9729	−2.28171
$540$	0	0
$541$	−24.4003	−1.04905	−0.524526	−	0.851394i	$-0.675758\pi$
$541$	−24.4003	−1.04905	−0.524526	+	0.851394i	$0.675758\pi$
$542$	0	0
$543$	− 8.93676i	− 0.383513i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 23.4355i	− 1.00203i	−0.865439	−	0.501014i	$-0.832961\pi$
$547$	− 23.4355i	− 1.00203i	0.865439	−	0.501014i	$-0.167039\pi$
$548$	0	0
$549$	−6.00000	−0.256074
$550$	0	0
$551$	−19.6258	−0.836087
$552$	0	0
$553$	− 9.12386i	− 0.387986i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	26.1526i	1.10812i	0.832476	+	0.554061i	$0.186923\pi$
$557$	26.1526i	1.10812i	−0.832476	+	0.554061i	$0.813077\pi$
$558$	0	0
$559$	−42.0213	−1.77731
$560$	0	0
$561$	5.64064	0.238148
$562$	0	0
$563$	37.9049i	1.59750i	0.601663	+	0.798750i	$0.294505\pi$
$563$	37.9049i	1.59750i	−0.601663	+	0.798750i	$0.705495\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 4.43807i	− 0.186381i
$568$	0	0
$569$	9.19777	0.385590	0.192795	−	0.981239i	$-0.438245\pi$
$569$	9.19777	0.385590	0.192795	+	0.981239i	$0.438245\pi$
$570$	0	0
$571$	29.2207	1.22285	0.611423	−	0.791304i	$-0.290597\pi$
$571$	29.2207	1.22285	0.611423	+	0.791304i	$0.290597\pi$
$572$	0	0
$573$	2.32163i	0.0969873i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	18.2084i	0.758027i	0.925391	+	0.379014i	$0.123737\pi$
$577$	18.2084i	0.758027i	−0.925391	+	0.379014i	$0.876263\pi$
$578$	0	0
$579$	−5.22066	−0.216963
$580$	0	0
$581$	−31.0336	−1.28749
$582$	0	0
$583$	53.4897i	2.21532i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	40.3807i	1.66669i	0.552754	+	0.833345i	$0.313577\pi$
$587$	40.3807i	1.66669i	−0.552754	+	0.833345i	$0.686423\pi$
$588$	0	0
$589$	2.70388	0.111411
$590$	0	0
$591$	9.58482	0.394267
$592$	0	0
$593$	− 36.7497i	− 1.50913i	−0.656226	−	0.754564i	$-0.727848\pi$
$593$	− 36.7497i	− 1.50913i	0.656226	−	0.754564i	$-0.272152\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	12.9171i	0.528663i
$598$	0	0
$599$	12.5545	0.512963	0.256482	−	0.966549i	$-0.417437\pi$
$599$	12.5545	0.512963	0.256482	+	0.966549i	$0.417437\pi$
$600$	0	0
$601$	−30.0362	−1.22520	−0.612601	−	0.790393i	$-0.709877\pi$
$601$	−30.0362	−1.22520	−0.612601	+	0.790393i	$0.709877\pi$
$602$	0	0
$603$	− 1.14195i	− 0.0465037i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	10.1297i	0.411153i	0.978641	+	0.205576i	$0.0659069\pi$
$607$	10.1297i	0.411153i	−0.978641	+	0.205576i	$0.934093\pi$
$608$	0	0
$609$	−32.2132	−1.30535
$610$	0	0
$611$	−18.2691	−0.739087
$612$	0	0
$613$	− 38.8794i	− 1.57032i	−0.619291	−	0.785162i	$-0.712580\pi$
$613$	− 38.8794i	− 1.57032i	0.619291	−	0.785162i	$-0.287420\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	27.4078i	1.10339i	0.834044	+	0.551697i	$0.186020\pi$
$617$	27.4078i	1.10339i	−0.834044	+	0.551697i	$0.813980\pi$
$618$	0	0
$619$	−26.2887	−1.05663	−0.528316	−	0.849048i	$-0.677176\pi$
$619$	−26.2887	−1.05663	−0.528316	+	0.849048i	$0.677176\pi$
$620$	0	0
$621$	1.35194	0.0542515
$622$	0	0
$623$	− 11.3371i	− 0.454211i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	11.2813i	0.450531i
$628$	0	0
$629$	−14.5774	−0.581239
$630$	0	0
$631$	−3.87875	−0.154411	−0.0772053	−	0.997015i	$-0.524600\pi$
$631$	−3.87875	−0.154411	−0.0772053	+	0.997015i	$0.524600\pi$
$632$	0	0
$633$	− 5.82774i	− 0.231632i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 56.3478i	− 2.23258i
$638$	0	0
$639$	10.7268	0.424345
$640$	0	0
$641$	4.08874	0.161496	0.0807478	−	0.996735i	$-0.474269\pi$
$641$	4.08874	0.161496	0.0807478	+	0.996735i	$0.474269\pi$
$642$	0	0
$643$	15.2355i	0.600829i	0.953809	+	0.300415i	$0.0971251\pi$
$643$	15.2355i	0.600829i	−0.953809	+	0.300415i	$0.902875\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	21.6406i	0.850781i	0.905010	+	0.425391i	$0.139863\pi$
$647$	21.6406i	0.850781i	−0.905010	+	0.425391i	$0.860137\pi$
$648$	0	0
$649$	−21.9394	−0.861196
$650$	0	0
$651$	4.43807	0.173942
$652$	0	0
$653$	− 0.228078i	− 0.00892538i	−0.999990	−	0.00446269i	$-0.998579\pi$
$653$	− 0.228078i	− 0.00892538i	0.999990	−	0.00446269i	$-0.00142052\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 10.4381i	− 0.407228i
$658$	0	0
$659$	−50.9500	−1.98473	−0.992366	−	0.123328i	$-0.960643\pi$
$659$	−50.9500	−1.98473	−0.992366	+	0.123328i	$0.960643\pi$
$660$	0	0
$661$	−7.82774	−0.304464	−0.152232	−	0.988345i	$-0.548646\pi$
$661$	−7.82774	−0.304464	−0.152232	+	0.988345i	$0.548646\pi$
$662$	0	0
$663$	6.00000i	0.233021i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 9.81290i	− 0.379957i
$668$	0	0
$669$	−2.00000	−0.0773245
$670$	0	0
$671$	25.0336	0.966410
$672$	0	0
$673$	− 45.3962i	− 1.74989i	−0.484219	−	0.874947i	$-0.660896\pi$
$673$	− 45.3962i	− 1.74989i	0.484219	−	0.874947i	$-0.339104\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 51.9507i	− 1.99663i	−0.0580539	−	0.998313i	$-0.518490\pi$
$677$	− 51.9507i	− 1.99663i	0.0580539	−	0.998313i	$-0.481510\pi$
$678$	0	0
$679$	70.6742	2.71223
$680$	0	0
$681$	−25.7933	−0.988400
$682$	0	0
$683$	47.0894i	1.80183i	0.434001	+	0.900913i	$0.357102\pi$
$683$	47.0894i	1.80183i	−0.434001	+	0.900913i	$0.642898\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 7.93937i	− 0.302906i
$688$	0	0
$689$	−56.8975	−2.16762
$690$	0	0
$691$	−29.9097	−1.13782	−0.568909	−	0.822400i	$-0.692634\pi$
$691$	−29.9097	−1.13782	−0.568909	+	0.822400i	$0.692634\pi$
$692$	0	0
$693$	18.5168i	0.703394i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.93676i	0.111238i
$698$	0	0
$699$	24.4610	0.925199
$700$	0	0
$701$	40.4562	1.52801	0.764004	−	0.645212i	$-0.223231\pi$
$701$	40.4562	1.52801	0.764004	+	0.645212i	$0.223231\pi$
$702$	0	0
$703$	− 29.1548i	− 1.09959i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.88836i	0.146237i
$708$	0	0
$709$	52.7858	1.98241	0.991207	−	0.132321i	$-0.0422430\pi$
$709$	52.7858	1.98241	0.991207	+	0.132321i	$0.0422430\pi$
$710$	0	0
$711$	−2.05582	−0.0770992
$712$	0	0
$713$	1.35194i	0.0506305i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 18.0000i	− 0.672222i
$718$	0	0
$719$	−3.82774	−0.142751	−0.0713753	−	0.997450i	$-0.522739\pi$
$719$	−3.82774	−0.142751	−0.0713753	+	0.997450i	$0.522739\pi$
$720$	0	0
$721$	28.5726	1.06410
$722$	0	0
$723$	2.64325i	0.0983036i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	11.8554i	0.439694i	0.975534	+	0.219847i	$0.0705558\pi$
$727$	11.8554i	0.439694i	−0.975534	+	0.219847i	$0.929444\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−12.8007	−0.473450
$732$	0	0
$733$	− 5.16003i	− 0.190590i	−0.995449	−	0.0952951i	$-0.969621\pi$
$733$	− 5.16003i	− 0.190590i	0.995449	−	0.0952951i	$-0.0303795\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.76450i	0.175503i
$738$	0	0
$739$	9.52420	0.350353	0.175177	−	0.984537i	$-0.443950\pi$
$739$	9.52420	0.350353	0.175177	+	0.984537i	$0.443950\pi$
$740$	0	0
$741$	−12.0000	−0.440831
$742$	0	0
$743$	− 33.7885i	− 1.23958i	−0.784768	−	0.619789i	$-0.787218\pi$
$743$	− 33.7885i	− 1.23958i	0.784768	−	0.619789i	$-0.212782\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6.99258i	0.255845i
$748$	0	0
$749$	−51.9097	−1.89674
$750$	0	0
$751$	41.1090	1.50009	0.750045	−	0.661387i	$-0.230032\pi$
$751$	41.1090	1.50009	0.750045	+	0.661387i	$0.230032\pi$
$752$	0	0
$753$	16.4562i	0.599696i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	24.3626i	0.885474i	0.896652	+	0.442737i	$0.145992\pi$
$757$	24.3626i	0.885474i	−0.896652	+	0.442737i	$0.854008\pi$
$758$	0	0
$759$	−5.64064	−0.204742
$760$	0	0
$761$	26.4184	0.957667	0.478834	−	0.877906i	$-0.341060\pi$
$761$	26.4184	0.957667	0.478834	+	0.877906i	$0.341060\pi$
$762$	0	0
$763$	51.1452i	1.85158i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 23.3371i	− 0.842654i
$768$	0	0
$769$	−32.9023	−1.18649	−0.593243	−	0.805023i	$-0.702153\pi$
$769$	−32.9023	−1.18649	−0.593243	+	0.805023i	$0.702153\pi$
$770$	0	0
$771$	−7.16484	−0.258035
$772$	0	0
$773$	44.8661i	1.61372i	0.590741	+	0.806861i	$0.298835\pi$
$773$	44.8661i	1.61372i	−0.590741	+	0.806861i	$0.701165\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 47.8539i	− 1.71675i
$778$	0	0
$779$	−5.87353	−0.210441
$780$	0	0
$781$	−44.7549	−1.60146
$782$	0	0
$783$	7.25839i	0.259394i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	42.4265i	1.51234i	0.654375	+	0.756170i	$0.272932\pi$
$787$	42.4265i	1.51234i	−0.654375	+	0.756170i	$0.727068\pi$
$788$	0	0
$789$	−8.87614	−0.315999
$790$	0	0
$791$	78.0213	2.77412
$792$	0	0
$793$	26.6284i	0.945603i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 28.9681i	− 1.02610i	−0.858358	−	0.513052i	$-0.828515\pi$
$797$	− 28.9681i	− 1.02610i	0.858358	−	0.513052i	$-0.171485\pi$
$798$	0	0
$799$	−5.56518	−0.196882
$800$	0	0
$801$	−2.55451	−0.0902593
$802$	0	0
$803$	43.5503i	1.53686i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	3.90164i	0.137344i
$808$	0	0
$809$	−26.4429	−0.929682	−0.464841	−	0.885394i	$-0.653889\pi$
$809$	−26.4429	−0.929682	−0.464841	+	0.885394i	$0.653889\pi$
$810$	0	0
$811$	−26.2542	−0.921910	−0.460955	−	0.887423i	$-0.652493\pi$
$811$	−26.2542	−0.921910	−0.460955	+	0.887423i	$0.652493\pi$
$812$	0	0
$813$	4.87614i	0.171014i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 25.6014i	− 0.895678i
$818$	0	0
$819$	−19.6965	−0.688250
$820$	0	0
$821$	13.5571	0.473147	0.236573	−	0.971614i	$-0.423976\pi$
$821$	13.5571	0.473147	0.236573	+	0.971614i	$0.423976\pi$
$822$	0	0
$823$	44.8613i	1.56377i	0.623425	+	0.781883i	$0.285741\pi$
$823$	44.8613i	1.56377i	−0.623425	+	0.781883i	$0.714259\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	29.8081i	1.03653i	0.855220	+	0.518265i	$0.173422\pi$
$827$	29.8081i	1.03653i	−0.855220	+	0.518265i	$0.826578\pi$
$828$	0	0
$829$	43.4535	1.50920	0.754602	−	0.656183i	$-0.227830\pi$
$829$	43.4535	1.50920	0.754602	+	0.656183i	$0.227830\pi$
$830$	0	0
$831$	−29.5981	−1.02675
$832$	0	0
$833$	− 17.1648i	− 0.594727i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 1.00000i	− 0.0345651i
$838$	0	0
$839$	12.9958	0.448666	0.224333	−	0.974513i	$-0.427980\pi$
$839$	12.9958	0.448666	0.224333	+	0.974513i	$0.427980\pi$
$840$	0	0
$841$	23.6842	0.816698
$842$	0	0
$843$	15.2207i	0.524227i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 28.4381i	− 0.977144i
$848$	0	0
$849$	−20.0181	−0.687019
$850$	0	0
$851$	14.5774	0.499707
$852$	0	0
$853$	− 49.7523i	− 1.70349i	−0.523960	−	0.851743i	$-0.675546\pi$
$853$	− 49.7523i	− 1.70349i	0.523960	−	0.851743i	$-0.324454\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	21.2058i	0.724377i	0.932105	+	0.362189i	$0.117970\pi$
$857$	21.2058i	0.724377i	−0.932105	+	0.362189i	$0.882030\pi$
$858$	0	0
$859$	10.4710	0.357266	0.178633	−	0.983916i	$-0.442833\pi$
$859$	10.4710	0.357266	0.178633	+	0.983916i	$0.442833\pi$
$860$	0	0
$861$	−9.64064	−0.328552
$862$	0	0
$863$	− 4.83997i	− 0.164754i	−0.996601	−	0.0823772i	$-0.973749\pi$
$863$	− 4.83997i	− 0.164754i	0.996601	−	0.0823772i	$-0.0262512\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 15.1723i	− 0.515277i
$868$	0	0
$869$	8.57741	0.290969
$870$	0	0
$871$	−5.06804	−0.171724
$872$	0	0
$873$	− 15.9245i	− 0.538964i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	36.8007i	1.24267i	0.783545	+	0.621335i	$0.213409\pi$
$877$	36.8007i	1.24267i	−0.783545	+	0.621335i	$0.786591\pi$
$878$	0	0
$879$	−8.46096	−0.285381
$880$	0	0
$881$	−39.1319	−1.31839	−0.659194	−	0.751973i	$-0.729102\pi$
$881$	−39.1319	−1.31839	−0.659194	+	0.751973i	$0.729102\pi$
$882$	0	0
$883$	− 17.7671i	− 0.597911i	−0.954267	−	0.298956i	$-0.903362\pi$
$883$	− 17.7671i	− 0.597911i	0.954267	−	0.298956i	$-0.0966382\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	57.2520i	1.92233i	0.275966	+	0.961167i	$0.411002\pi$
$887$	57.2520i	1.92233i	−0.275966	+	0.961167i	$0.588998\pi$
$888$	0	0
$889$	−70.1788	−2.35372
$890$	0	0
$891$	4.17226	0.139776
$892$	0	0
$893$	− 11.1304i	− 0.372463i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 6.00000i	− 0.200334i
$898$	0	0
$899$	−7.25839	−0.242081
$900$	0	0
$901$	−17.3323	−0.577422
$902$	0	0
$903$	− 42.0213i	− 1.39838i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 16.8581i	− 0.559762i	−0.960035	−	0.279881i	$-0.909705\pi$
$907$	− 16.8581i	− 0.559762i	0.960035	−	0.279881i	$-0.0902951\pi$
$908$	0	0
$909$	0.876139	0.0290597
$910$	0	0
$911$	45.5800	1.51013	0.755067	−	0.655648i	$-0.227604\pi$
$911$	45.5800	1.51013	0.755067	+	0.655648i	$0.227604\pi$
$912$	0	0
$913$	− 29.1749i	− 0.965547i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	44.9777i	1.48530i
$918$	0	0
$919$	35.8491	1.18255	0.591276	−	0.806469i	$-0.298624\pi$
$919$	35.8491	1.18255	0.591276	+	0.806469i	$0.298624\pi$
$920$	0	0
$921$	8.43807	0.278044
$922$	0	0
$923$	− 47.6062i	− 1.56698i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 6.43807i	− 0.211454i
$928$	0	0
$929$	−45.7752	−1.50183	−0.750917	−	0.660396i	$-0.770388\pi$
$929$	−45.7752	−1.50183	−0.750917	+	0.660396i	$0.770388\pi$
$930$	0	0
$931$	34.3297	1.12511
$932$	0	0
$933$	20.6661i	0.676579i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 25.8277i	− 0.843755i	−0.906653	−	0.421878i	$-0.861371\pi$
$937$	− 25.8277i	− 0.843755i	0.906653	−	0.421878i	$-0.138629\pi$
$938$	0	0
$939$	21.3142	0.695563
$940$	0	0
$941$	−1.52095	−0.0495815	−0.0247908	−	0.999693i	$-0.507892\pi$
$941$	−1.52095	−0.0495815	−0.0247908	+	0.999693i	$0.507892\pi$
$942$	0	0
$943$	− 2.93676i	− 0.0956342i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 18.6481i	− 0.605981i	−0.952994	−	0.302990i	$-0.902015\pi$
$947$	− 18.6481i	− 0.605981i	0.952994	−	0.302990i	$-0.0979850\pi$
$948$	0	0
$949$	−46.3249	−1.50377
$950$	0	0
$951$	−13.5242	−0.438552
$952$	0	0
$953$	33.3519i	1.08038i	0.841545	+	0.540188i	$0.181647\pi$
$953$	33.3519i	1.08038i	−0.841545	+	0.540188i	$0.818353\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 30.2839i	− 0.978939i
$958$	0	0
$959$	67.0336	2.16463
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	11.6965i	0.376913i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2.00000i	0.0643157i	0.999483	+	0.0321578i	$0.0102379\pi$
$967$	2.00000i	0.0643157i	−0.999483	+	0.0321578i	$0.989762\pi$
$968$	0	0
$969$	−3.65548	−0.117431
$970$	0	0
$971$	21.6932	0.696168	0.348084	−	0.937463i	$-0.386832\pi$
$971$	21.6932	0.696168	0.348084	+	0.937463i	$0.386832\pi$
$972$	0	0
$973$	− 76.4265i	− 2.45012i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 17.2813i	− 0.552877i	−0.961032	−	0.276439i	$-0.910846\pi$
$977$	− 17.2813i	− 0.552877i	0.961032	−	0.276439i	$-0.0891543\pi$
$978$	0	0
$979$	10.6581	0.340634
$980$	0	0
$981$	11.5242	0.367939
$982$	0	0
$983$	29.1452i	0.929587i	0.885419	+	0.464794i	$0.153872\pi$
$983$	29.1452i	0.929587i	−0.885419	+	0.464794i	$0.846128\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 18.2691i	− 0.581511i
$988$	0	0
$989$	12.8007	0.407038
$990$	0	0
$991$	−5.34191	−0.169691	−0.0848457	−	0.996394i	$-0.527040\pi$
$991$	−5.34191	−0.169691	−0.0848457	+	0.996394i	$0.527040\pi$
$992$	0	0
$993$	− 12.9926i	− 0.412307i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 1.23550i	− 0.0391286i	−0.999809	−	0.0195643i	$-0.993772\pi$
$997$	− 1.23550i	− 0.0391286i	0.999809	−	0.0195643i	$-0.00622790\pi$
$998$	0	0
$999$	−10.7826	−0.341146

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9300.2.g.p.3349.1		6
5.2	odd	4		1860.2.a.f.1.3	✓	3
5.3	odd	4		9300.2.a.v.1.1		3
5.4	even	2	inner	9300.2.g.p.3349.6		6
15.2	even	4		5580.2.a.l.1.3		3
20.7	even	4		7440.2.a.bt.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1860.2.a.f.1.3	✓	3	5.2	odd	4
5580.2.a.l.1.3		3	15.2	even	4
7440.2.a.bt.1.1		3	20.7	even	4
9300.2.a.v.1.1		3	5.3	odd	4
9300.2.g.p.3349.1		6	1.1	even	1	trivial
9300.2.g.p.3349.6		6	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9300.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -4.43807i q^{7} -1.00000 q^{9} +4.17226 q^{11} +4.43807i q^{13} +1.35194i q^{17} -2.70388 q^{19} -4.43807 q^{21} -1.35194i q^{23} +1.00000i q^{27} +7.25839 q^{29} -1.00000 q^{31} -4.17226i q^{33} +10.7826i q^{37} +4.43807 q^{39} +2.17226 q^{41} +9.46838i q^{43} +4.11644i q^{47} -12.6965 q^{49} +1.35194 q^{51} +12.8203i q^{53} +2.70388i q^{57} -5.25839 q^{59} +6.00000 q^{61} +4.43807i q^{63} +1.14195i q^{67} -1.35194 q^{69} -10.7268 q^{71} +10.4381i q^{73} -18.5168i q^{77} +2.05582 q^{79} +1.00000 q^{81} -6.99258i q^{83} -7.25839i q^{87} +2.55451 q^{89} +19.6965 q^{91} +1.00000i q^{93} +15.9245i q^{97} -4.17226 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{9} - 4 q^{11} - 8 q^{19} - 8 q^{21} - 6 q^{31} + 8 q^{39} - 16 q^{41} - 14 q^{49} + 4 q^{51} + 12 q^{59} + 36 q^{61} - 4 q^{69} + 6 q^{81} - 20 q^{89} + 56 q^{91} + 4 q^{99}+O(q^{100}) \) 6 * q - 6 * q^9 - 4 * q^11 - 8 * q^19 - 8 * q^21 - 6 * q^31 + 8 * q^39 - 16 * q^41 - 14 * q^49 + 4 * q^51 + 12 * q^59 + 36 * q^61 - 4 * q^69 + 6 * q^81 - 20 * q^89 + 56 * q^91 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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