LMFDB - Embedded newform 3300.2.c.f.1849.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3300,2,Mod(1849,3300)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3300.1849"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3300.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,-2,0,-2,0,0,0,0,0,0,0,14,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)

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

Self dual:	no
Analytic conductor:	$26.3506326670$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	3300.1849
Dual form			3300.2.c.f.1849.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +3.00000i q^{7} -1.00000 q^{9} -1.00000 q^{11} -4.00000i q^{13} -1.00000i q^{17} +7.00000 q^{19} +3.00000 q^{21} -3.00000i q^{23} +1.00000i q^{27} -10.0000 q^{29} +1.00000i q^{33} +1.00000i q^{37} -4.00000 q^{39} +5.00000 q^{41} +5.00000i q^{47} -2.00000 q^{49} -1.00000 q^{51} -6.00000i q^{53} -7.00000i q^{57} -3.00000 q^{59} +6.00000 q^{61} -3.00000i q^{63} -14.0000i q^{67} -3.00000 q^{69} +15.0000 q^{71} +6.00000i q^{73} -3.00000i q^{77} +13.0000 q^{79} +1.00000 q^{81} -4.00000i q^{83} +10.0000i q^{87} +14.0000 q^{89} +12.0000 q^{91} -13.0000i q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9} - 2 q^{11} + 14 q^{19} + 6 q^{21} - 20 q^{29} - 8 q^{39} + 10 q^{41} - 4 q^{49} - 2 q^{51} - 6 q^{59} + 12 q^{61} - 6 q^{69} + 30 q^{71} + 26 q^{79} + 2 q^{81} + 28 q^{89} + 24 q^{91} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 - 2 * q^11 + 14 * q^19 + 6 * q^21 - 20 * q^29 - 8 * q^39 + 10 * q^41 - 4 * q^49 - 2 * q^51 - 6 * q^59 + 12 * q^61 - 6 * q^69 + 30 * q^71 + 26 * q^79 + 2 * q^81 + 28 * q^89 + 24 * q^91 + 2 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3300\mathbb{Z}\right)^\times$.

$n$	$1201$	$1651$	$2201$	$2377$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$7$	3.00000i	1.13389i	0.823754	+	0.566947i	$0.191875\pi$
$7$	3.00000i	1.13389i	−0.823754	+	0.566947i	$0.808125\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	− 4.00000i	− 1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	− 4.00000i	− 1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.00000i	− 0.242536i	−0.992620	−	0.121268i	$-0.961304\pi$
$17$	− 1.00000i	− 0.242536i	0.992620	−	0.121268i	$-0.0386960\pi$
$18$	0	0
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	0	0
$23$	− 3.00000i	− 0.625543i	−0.949828	−	0.312772i	$-0.898743\pi$
$23$	− 3.00000i	− 0.625543i	0.949828	−	0.312772i	$-0.101257\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	−10.0000	−1.85695	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000	−1.85695	−0.928477	+	0.371391i	$0.878881\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	1.00000i	0.174078i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.00000i	0.164399i	0.996616	+	0.0821995i	$0.0261945\pi$
$37$	1.00000i	0.164399i	−0.996616	+	0.0821995i	$0.973806\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	0	0
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.00000i	0.729325i	0.931140	+	0.364662i	$0.118816\pi$
$47$	5.00000i	0.729325i	−0.931140	+	0.364662i	$0.881184\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	0	0
$51$	−1.00000	−0.140028
$52$	0	0
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 7.00000i	− 0.927173i
$58$	0	0
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	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
$63$	− 3.00000i	− 0.377964i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 14.0000i	− 1.71037i	−0.518321	−	0.855186i	$-0.673443\pi$
$67$	− 14.0000i	− 1.71037i	0.518321	−	0.855186i	$-0.326557\pi$
$68$	0	0
$69$	−3.00000	−0.361158
$70$	0	0
$71$	15.0000	1.78017	0.890086	−	0.455792i	$-0.150644\pi$
$71$	15.0000	1.78017	0.890086	+	0.455792i	$0.150644\pi$
$72$	0	0
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 3.00000i	− 0.341882i
$78$	0	0
$79$	13.0000	1.46261	0.731307	−	0.682048i	$-0.238911\pi$
$79$	13.0000	1.46261	0.731307	+	0.682048i	$0.238911\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 4.00000i	− 0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$	− 4.00000i	− 0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	10.0000i	1.07211i
$88$	0	0
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	12.0000	1.25794
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 13.0000i	− 1.31995i	−0.751288	−	0.659975i	$-0.770567\pi$
$97$	− 13.0000i	− 1.31995i	0.751288	−	0.659975i	$-0.229433\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	0	0
$103$	− 12.0000i	− 1.18240i	−0.806527	−	0.591198i	$-0.798655\pi$
$103$	− 12.0000i	− 1.18240i	0.806527	−	0.591198i	$-0.201345\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 6.00000i	− 0.580042i	−0.957020	−	0.290021i	$-0.906338\pi$
$107$	− 6.00000i	− 0.580042i	0.957020	−	0.290021i	$-0.0936623\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	1.00000	0.0949158
$112$	0	0
$113$	− 10.0000i	− 0.940721i	−0.882474	−	0.470360i	$-0.844124\pi$
$113$	− 10.0000i	− 0.940721i	0.882474	−	0.470360i	$-0.155876\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.00000i	0.369800i
$118$	0	0
$119$	3.00000	0.275010
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	− 5.00000i	− 0.450835i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 19.0000i	− 1.68598i	−0.537931	−	0.842989i	$-0.680794\pi$
$127$	− 19.0000i	− 1.68598i	0.537931	−	0.842989i	$-0.319206\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	21.0000i	1.82093i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	5.00000	0.421076
$142$	0	0
$143$	4.00000i	0.334497i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.00000i	0.164957i
$148$	0	0
$149$	−21.0000	−1.72039	−0.860194	−	0.509968i	$-0.829657\pi$
$149$	−21.0000	−1.72039	−0.860194	+	0.509968i	$0.829657\pi$
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	0	0
$153$	1.00000i	0.0808452i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	10.0000i	0.798087i	0.916932	+	0.399043i	$0.130658\pi$
$157$	10.0000i	0.798087i	−0.916932	+	0.399043i	$0.869342\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	9.00000	0.709299
$162$	0	0
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.00000i	0.154765i	0.997001	+	0.0773823i	$0.0246562\pi$
$167$	2.00000i	0.154765i	−0.997001	+	0.0773823i	$0.975344\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	−7.00000	−0.535303
$172$	0	0
$173$	13.0000i	0.988372i	0.869356	+	0.494186i	$0.164534\pi$
$173$	13.0000i	0.988372i	−0.869356	+	0.494186i	$0.835466\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.00000i	0.225494i
$178$	0	0
$179$	−11.0000	−0.822179	−0.411089	−	0.911595i	$-0.634852\pi$
$179$	−11.0000	−0.822179	−0.411089	+	0.911595i	$0.634852\pi$
$180$	0	0
$181$	7.00000	0.520306	0.260153	−	0.965567i	$-0.416227\pi$
$181$	7.00000	0.520306	0.260153	+	0.965567i	$0.416227\pi$
$182$	0	0
$183$	− 6.00000i	− 0.443533i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.00000i	0.0731272i
$188$	0	0
$189$	−3.00000	−0.218218
$190$	0	0
$191$	−21.0000	−1.51951	−0.759753	−	0.650211i	$-0.774680\pi$
$191$	−21.0000	−1.51951	−0.759753	+	0.650211i	$0.774680\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 21.0000i	− 1.49619i	−0.663593	−	0.748094i	$-0.730969\pi$
$197$	− 21.0000i	− 1.49619i	0.663593	−	0.748094i	$-0.269031\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	−14.0000	−0.987484
$202$	0	0
$203$	− 30.0000i	− 2.10559i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3.00000i	0.208514i
$208$	0	0
$209$	−7.00000	−0.484200
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	0	0
$213$	− 15.0000i	− 1.02778i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	6.00000	0.405442
$220$	0	0
$221$	−4.00000	−0.269069
$222$	0	0
$223$	− 8.00000i	− 0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$	− 8.00000i	− 0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 6.00000i	− 0.398234i	−0.979976	−	0.199117i	$-0.936193\pi$
$227$	− 6.00000i	− 0.398234i	0.979976	−	0.199117i	$-0.0638074\pi$
$228$	0	0
$229$	−7.00000	−0.462573	−0.231287	−	0.972886i	$-0.574293\pi$
$229$	−7.00000	−0.462573	−0.231287	+	0.972886i	$0.574293\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	0	0
$233$	11.0000i	0.720634i	0.932830	+	0.360317i	$0.117331\pi$
$233$	11.0000i	0.720634i	−0.932830	+	0.360317i	$0.882669\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 13.0000i	− 0.844441i
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	28.0000	1.80364	0.901819	−	0.432113i	$-0.142232\pi$
$241$	28.0000	1.80364	0.901819	+	0.432113i	$0.142232\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 28.0000i	− 1.78160i
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	0	0
$253$	3.00000i	0.188608i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 22.0000i	− 1.37232i	−0.727450	−	0.686161i	$-0.759294\pi$
$257$	− 22.0000i	− 1.37232i	0.727450	−	0.686161i	$-0.240706\pi$
$258$	0	0
$259$	−3.00000	−0.186411
$260$	0	0
$261$	10.0000	0.618984
$262$	0	0
$263$	− 22.0000i	− 1.35658i	−0.734795	−	0.678289i	$-0.762722\pi$
$263$	− 22.0000i	− 1.35658i	0.734795	−	0.678289i	$-0.237278\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 14.0000i	− 0.856786i
$268$	0	0
$269$	16.0000	0.975537	0.487769	−	0.872973i	$-0.337811\pi$
$269$	16.0000	0.975537	0.487769	+	0.872973i	$0.337811\pi$
$270$	0	0
$271$	−25.0000	−1.51864	−0.759321	−	0.650716i	$-0.774469\pi$
$271$	−25.0000	−1.51864	−0.759321	+	0.650716i	$0.774469\pi$
$272$	0	0
$273$	− 12.0000i	− 0.726273i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.0000i	0.600842i	0.953807	+	0.300421i	$0.0971271\pi$
$277$	10.0000i	0.600842i	−0.953807	+	0.300421i	$0.902873\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	17.0000	1.01413	0.507067	−	0.861906i	$-0.330729\pi$
$281$	17.0000	1.01413	0.507067	+	0.861906i	$0.330729\pi$
$282$	0	0
$283$	21.0000i	1.24832i	0.781296	+	0.624160i	$0.214559\pi$
$283$	21.0000i	1.24832i	−0.781296	+	0.624160i	$0.785441\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	15.0000i	0.885422i
$288$	0	0
$289$	16.0000	0.941176
$290$	0	0
$291$	−13.0000	−0.762073
$292$	0	0
$293$	− 25.0000i	− 1.46052i	−0.683172	−	0.730258i	$-0.739400\pi$
$293$	− 25.0000i	− 1.46052i	0.683172	−	0.730258i	$-0.260600\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1.00000i	− 0.0580259i
$298$	0	0
$299$	−12.0000	−0.693978
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 3.00000i	− 0.172345i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 4.00000i	− 0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$	− 4.00000i	− 0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$	0	0
$309$	−12.0000	−0.682656
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	29.0000i	1.63918i	0.572953	+	0.819588i	$0.305798\pi$
$313$	29.0000i	1.63918i	−0.572953	+	0.819588i	$0.694202\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.0000i	0.673987i	0.941507	+	0.336994i	$0.109410\pi$
$317$	12.0000i	0.673987i	−0.941507	+	0.336994i	$0.890590\pi$
$318$	0	0
$319$	10.0000	0.559893
$320$	0	0
$321$	−6.00000	−0.334887
$322$	0	0
$323$	− 7.00000i	− 0.389490i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.00000i	0.110600i
$328$	0	0
$329$	−15.0000	−0.826977
$330$	0	0
$331$	−2.00000	−0.109930	−0.0549650	−	0.998488i	$-0.517505\pi$
$331$	−2.00000	−0.109930	−0.0549650	+	0.998488i	$0.517505\pi$
$332$	0	0
$333$	− 1.00000i	− 0.0547997i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	18.0000i	0.980522i	0.871576	+	0.490261i	$0.163099\pi$
$337$	18.0000i	0.980522i	−0.871576	+	0.490261i	$0.836901\pi$
$338$	0	0
$339$	−10.0000	−0.543125
$340$	0	0
$341$	0	0
$342$	0	0
$343$	15.0000i	0.809924i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	0	0
$353$	24.0000i	1.27739i	0.769460	+	0.638696i	$0.220526\pi$
$353$	24.0000i	1.27739i	−0.769460	+	0.638696i	$0.779474\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 3.00000i	− 0.158777i
$358$	0	0
$359$	36.0000	1.90001	0.950004	−	0.312239i	$-0.101079\pi$
$359$	36.0000	1.90001	0.950004	+	0.312239i	$0.101079\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	0	0
$363$	− 1.00000i	− 0.0524864i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 6.00000i	− 0.313197i	−0.987662	−	0.156599i	$-0.949947\pi$
$367$	− 6.00000i	− 0.313197i	0.987662	−	0.156599i	$-0.0500529\pi$
$368$	0	0
$369$	−5.00000	−0.260290
$370$	0	0
$371$	18.0000	0.934513
$372$	0	0
$373$	10.0000i	0.517780i	0.965907	+	0.258890i	$0.0833568\pi$
$373$	10.0000i	0.517780i	−0.965907	+	0.258890i	$0.916643\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	40.0000i	2.06010i
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	0	0
$381$	−19.0000	−0.973399
$382$	0	0
$383$	− 12.0000i	− 0.613171i	−0.951843	−	0.306586i	$-0.900813\pi$
$383$	− 12.0000i	− 0.613171i	0.951843	−	0.306586i	$-0.0991866\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−28.0000	−1.41966	−0.709828	−	0.704375i	$-0.751227\pi$
$389$	−28.0000	−1.41966	−0.709828	+	0.704375i	$0.751227\pi$
$390$	0	0
$391$	−3.00000	−0.151717
$392$	0	0
$393$	− 4.00000i	− 0.201773i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	14.0000i	0.702640i	0.936255	+	0.351320i	$0.114267\pi$
$397$	14.0000i	0.702640i	−0.936255	+	0.351320i	$0.885733\pi$
$398$	0	0
$399$	21.0000	1.05131
$400$	0	0
$401$	8.00000	0.399501	0.199750	−	0.979847i	$-0.435987\pi$
$401$	8.00000	0.399501	0.199750	+	0.979847i	$0.435987\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1.00000i	− 0.0495682i
$408$	0	0
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	0	0
$413$	− 9.00000i	− 0.442861i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.00000i	0.195881i
$418$	0	0
$419$	23.0000	1.12362	0.561812	−	0.827265i	$-0.310105\pi$
$419$	23.0000	1.12362	0.561812	+	0.827265i	$0.310105\pi$
$420$	0	0
$421$	−15.0000	−0.731055	−0.365528	−	0.930800i	$-0.619111\pi$
$421$	−15.0000	−0.731055	−0.365528	+	0.930800i	$0.619111\pi$
$422$	0	0
$423$	− 5.00000i	− 0.243108i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	18.0000i	0.871081i
$428$	0	0
$429$	4.00000	0.193122
$430$	0	0
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	0	0
$433$	34.0000i	1.63394i	0.576683	+	0.816968i	$0.304347\pi$
$433$	34.0000i	1.63394i	−0.576683	+	0.816968i	$0.695653\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 21.0000i	− 1.00457i
$438$	0	0
$439$	17.0000	0.811366	0.405683	−	0.914014i	$-0.367034\pi$
$439$	17.0000	0.811366	0.405683	+	0.914014i	$0.367034\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	9.00000i	0.427603i	0.976877	+	0.213801i	$0.0685846\pi$
$443$	9.00000i	0.427603i	−0.976877	+	0.213801i	$0.931415\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	21.0000i	0.993266i
$448$	0	0
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	0	0
$451$	−5.00000	−0.235441
$452$	0	0
$453$	− 20.0000i	− 0.939682i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 2.00000i	− 0.0935561i	−0.998905	−	0.0467780i	$-0.985105\pi$
$457$	− 2.00000i	− 0.0935561i	0.998905	−	0.0467780i	$-0.0148953\pi$
$458$	0	0
$459$	1.00000	0.0466760
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	38.0000i	1.76601i	0.469364	+	0.883005i	$0.344483\pi$
$463$	38.0000i	1.76601i	−0.469364	+	0.883005i	$0.655517\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 28.0000i	− 1.29569i	−0.761774	−	0.647843i	$-0.775671\pi$
$467$	− 28.0000i	− 1.29569i	0.761774	−	0.647843i	$-0.224329\pi$
$468$	0	0
$469$	42.0000	1.93938
$470$	0	0
$471$	10.0000	0.460776
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.00000i	0.274721i
$478$	0	0
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	− 9.00000i	− 0.409514i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 34.0000i	− 1.54069i	−0.637629	−	0.770344i	$-0.720085\pi$
$487$	− 34.0000i	− 1.54069i	0.637629	−	0.770344i	$-0.279915\pi$
$488$	0	0
$489$	−4.00000	−0.180886
$490$	0	0
$491$	−6.00000	−0.270776	−0.135388	−	0.990793i	$-0.543228\pi$
$491$	−6.00000	−0.270776	−0.135388	+	0.990793i	$0.543228\pi$
$492$	0	0
$493$	10.0000i	0.450377i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	45.0000i	2.01853i
$498$	0	0
$499$	−6.00000	−0.268597	−0.134298	−	0.990941i	$-0.542878\pi$
$499$	−6.00000	−0.268597	−0.134298	+	0.990941i	$0.542878\pi$
$500$	0	0
$501$	2.00000	0.0893534
$502$	0	0
$503$	− 10.0000i	− 0.445878i	−0.974832	−	0.222939i	$-0.928435\pi$
$503$	− 10.0000i	− 0.445878i	0.974832	−	0.222939i	$-0.0715651\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3.00000i	0.133235i
$508$	0	0
$509$	−34.0000	−1.50702	−0.753512	−	0.657434i	$-0.771642\pi$
$509$	−34.0000	−1.50702	−0.753512	+	0.657434i	$0.771642\pi$
$510$	0	0
$511$	−18.0000	−0.796273
$512$	0	0
$513$	7.00000i	0.309058i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 5.00000i	− 0.219900i
$518$	0	0
$519$	13.0000	0.570637
$520$	0	0
$521$	−22.0000	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	0	0
$523$	3.00000i	0.131181i	0.997847	+	0.0655904i	$0.0208931\pi$
$523$	3.00000i	0.131181i	−0.997847	+	0.0655904i	$0.979107\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	14.0000	0.608696
$530$	0	0
$531$	3.00000	0.130189
$532$	0	0
$533$	− 20.0000i	− 0.866296i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	11.0000i	0.474685i
$538$	0	0
$539$	2.00000	0.0861461
$540$	0	0
$541$	4.00000	0.171973	0.0859867	−	0.996296i	$-0.472596\pi$
$541$	4.00000	0.171973	0.0859867	+	0.996296i	$0.472596\pi$
$542$	0	0
$543$	− 7.00000i	− 0.300399i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 23.0000i	− 0.983409i	−0.870762	−	0.491704i	$-0.836374\pi$
$547$	− 23.0000i	− 0.983409i	0.870762	−	0.491704i	$-0.163626\pi$
$548$	0	0
$549$	−6.00000	−0.256074
$550$	0	0
$551$	−70.0000	−2.98210
$552$	0	0
$553$	39.0000i	1.65845i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	18.0000i	0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$	18.0000i	0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	1.00000	0.0422200
$562$	0	0
$563$	24.0000i	1.01148i	0.862686	+	0.505740i	$0.168780\pi$
$563$	24.0000i	1.01148i	−0.862686	+	0.505740i	$0.831220\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.00000i	0.125988i
$568$	0	0
$569$	5.00000	0.209611	0.104805	−	0.994493i	$-0.466578\pi$
$569$	5.00000	0.209611	0.104805	+	0.994493i	$0.466578\pi$
$570$	0	0
$571$	24.0000	1.00437	0.502184	−	0.864761i	$-0.332530\pi$
$571$	24.0000	1.00437	0.502184	+	0.864761i	$0.332530\pi$
$572$	0	0
$573$	21.0000i	0.877288i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	33.0000i	1.37381i	0.726748	+	0.686904i	$0.241031\pi$
$577$	33.0000i	1.37381i	−0.726748	+	0.686904i	$0.758969\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	6.00000i	0.248495i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.00000i	0.288921i	0.989511	+	0.144460i	$0.0461446\pi$
$587$	7.00000i	0.288921i	−0.989511	+	0.144460i	$0.953855\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−21.0000	−0.863825
$592$	0	0
$593$	− 26.0000i	− 1.06769i	−0.845582	−	0.533846i	$-0.820746\pi$
$593$	− 26.0000i	− 1.06769i	0.845582	−	0.533846i	$-0.179254\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	8.00000i	0.327418i
$598$	0	0
$599$	−5.00000	−0.204294	−0.102147	−	0.994769i	$-0.532571\pi$
$599$	−5.00000	−0.204294	−0.102147	+	0.994769i	$0.532571\pi$
$600$	0	0
$601$	−4.00000	−0.163163	−0.0815817	−	0.996667i	$-0.525997\pi$
$601$	−4.00000	−0.163163	−0.0815817	+	0.996667i	$0.525997\pi$
$602$	0	0
$603$	14.0000i	0.570124i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 24.0000i	− 0.974130i	−0.873366	−	0.487065i	$-0.838067\pi$
$607$	− 24.0000i	− 0.974130i	0.873366	−	0.487065i	$-0.161933\pi$
$608$	0	0
$609$	−30.0000	−1.21566
$610$	0	0
$611$	20.0000	0.809113
$612$	0	0
$613$	− 18.0000i	− 0.727013i	−0.931592	−	0.363507i	$-0.881579\pi$
$613$	− 18.0000i	− 0.727013i	0.931592	−	0.363507i	$-0.118421\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.0000i	0.483102i	0.970388	+	0.241551i	$0.0776561\pi$
$617$	12.0000i	0.483102i	−0.970388	+	0.241551i	$0.922344\pi$
$618$	0	0
$619$	8.00000	0.321547	0.160774	−	0.986991i	$-0.448601\pi$
$619$	8.00000	0.321547	0.160774	+	0.986991i	$0.448601\pi$
$620$	0	0
$621$	3.00000	0.120386
$622$	0	0
$623$	42.0000i	1.68269i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	7.00000i	0.279553i
$628$	0	0
$629$	1.00000	0.0398726
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	0	0
$633$	− 20.0000i	− 0.794929i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	8.00000i	0.316972i
$638$	0	0
$639$	−15.0000	−0.593391
$640$	0	0
$641$	−26.0000	−1.02694	−0.513469	−	0.858108i	$-0.671640\pi$
$641$	−26.0000	−1.02694	−0.513469	+	0.858108i	$0.671640\pi$
$642$	0	0
$643$	− 28.0000i	− 1.10421i	−0.833774	−	0.552106i	$-0.813824\pi$
$643$	− 28.0000i	− 1.10421i	0.833774	−	0.552106i	$-0.186176\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	37.0000i	1.45462i	0.686309	+	0.727310i	$0.259230\pi$
$647$	37.0000i	1.45462i	−0.686309	+	0.727310i	$0.740770\pi$
$648$	0	0
$649$	3.00000	0.117760
$650$	0	0
$651$	0	0
$652$	0	0
$653$	48.0000i	1.87839i	0.343391	+	0.939193i	$0.388424\pi$
$653$	48.0000i	1.87839i	−0.343391	+	0.939193i	$0.611576\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 6.00000i	− 0.234082i
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	−33.0000	−1.28355	−0.641776	−	0.766892i	$-0.721802\pi$
$661$	−33.0000	−1.28355	−0.641776	+	0.766892i	$0.721802\pi$
$662$	0	0
$663$	4.00000i	0.155347i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	30.0000i	1.16160i
$668$	0	0
$669$	−8.00000	−0.309298
$670$	0	0
$671$	−6.00000	−0.231627
$672$	0	0
$673$	− 34.0000i	− 1.31060i	−0.755367	−	0.655302i	$-0.772541\pi$
$673$	− 34.0000i	− 1.31060i	0.755367	−	0.655302i	$-0.227459\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	22.0000i	0.845529i	0.906240	+	0.422764i	$0.138940\pi$
$677$	22.0000i	0.845529i	−0.906240	+	0.422764i	$0.861060\pi$
$678$	0	0
$679$	39.0000	1.49668
$680$	0	0
$681$	−6.00000	−0.229920
$682$	0	0
$683$	− 37.0000i	− 1.41577i	−0.706330	−	0.707883i	$-0.749650\pi$
$683$	− 37.0000i	− 1.41577i	0.706330	−	0.707883i	$-0.250350\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	7.00000i	0.267067i
$688$	0	0
$689$	−24.0000	−0.914327
$690$	0	0
$691$	14.0000	0.532585	0.266293	−	0.963892i	$-0.414201\pi$
$691$	14.0000	0.532585	0.266293	+	0.963892i	$0.414201\pi$
$692$	0	0
$693$	3.00000i	0.113961i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 5.00000i	− 0.189389i
$698$	0	0
$699$	11.0000	0.416058
$700$	0	0
$701$	−27.0000	−1.01978	−0.509888	−	0.860241i	$-0.670313\pi$
$701$	−27.0000	−1.01978	−0.509888	+	0.860241i	$0.670313\pi$
$702$	0	0
$703$	7.00000i	0.264010i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	9.00000i	0.338480i
$708$	0	0
$709$	−3.00000	−0.112667	−0.0563337	−	0.998412i	$-0.517941\pi$
$709$	−3.00000	−0.112667	−0.0563337	+	0.998412i	$0.517941\pi$
$710$	0	0
$711$	−13.0000	−0.487538
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.00000i	0.224074i
$718$	0	0
$719$	20.0000	0.745874	0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000	0.745874	0.372937	+	0.927857i	$0.378351\pi$
$720$	0	0
$721$	36.0000	1.34071
$722$	0	0
$723$	− 28.0000i	− 1.04133i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	28.0000i	1.03846i	0.854634	+	0.519231i	$0.173782\pi$
$727$	28.0000i	1.03846i	−0.854634	+	0.519231i	$0.826218\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	26.0000i	0.960332i	0.877178	+	0.480166i	$0.159424\pi$
$733$	26.0000i	0.960332i	−0.877178	+	0.480166i	$0.840576\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	14.0000i	0.515697i
$738$	0	0
$739$	9.00000	0.331070	0.165535	−	0.986204i	$-0.447065\pi$
$739$	9.00000	0.331070	0.165535	+	0.986204i	$0.447065\pi$
$740$	0	0
$741$	−28.0000	−1.02861
$742$	0	0
$743$	− 8.00000i	− 0.293492i	−0.989174	−	0.146746i	$-0.953120\pi$
$743$	− 8.00000i	− 0.293492i	0.989174	−	0.146746i	$-0.0468799\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	4.00000i	0.146352i
$748$	0	0
$749$	18.0000	0.657706
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	0	0
$753$	24.0000i	0.874609i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	22.0000i	0.799604i	0.916602	+	0.399802i	$0.130921\pi$
$757$	22.0000i	0.799604i	−0.916602	+	0.399802i	$0.869079\pi$
$758$	0	0
$759$	3.00000	0.108893
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	0	0
$763$	− 6.00000i	− 0.217215i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	12.0000i	0.433295i
$768$	0	0
$769$	6.00000	0.216366	0.108183	−	0.994131i	$-0.465497\pi$
$769$	6.00000	0.216366	0.108183	+	0.994131i	$0.465497\pi$
$770$	0	0
$771$	−22.0000	−0.792311
$772$	0	0
$773$	− 18.0000i	− 0.647415i	−0.946157	−	0.323708i	$-0.895071\pi$
$773$	− 18.0000i	− 0.647415i	0.946157	−	0.323708i	$-0.104929\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3.00000i	0.107624i
$778$	0	0
$779$	35.0000	1.25401
$780$	0	0
$781$	−15.0000	−0.536742
$782$	0	0
$783$	− 10.0000i	− 0.357371i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 47.0000i	− 1.67537i	−0.546154	−	0.837685i	$-0.683909\pi$
$787$	− 47.0000i	− 1.67537i	0.546154	−	0.837685i	$-0.316091\pi$
$788$	0	0
$789$	−22.0000	−0.783221
$790$	0	0
$791$	30.0000	1.06668
$792$	0	0
$793$	− 24.0000i	− 0.852265i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 44.0000i	− 1.55856i	−0.626676	−	0.779280i	$-0.715585\pi$
$797$	− 44.0000i	− 1.55856i	0.626676	−	0.779280i	$-0.284415\pi$
$798$	0	0
$799$	5.00000	0.176887
$800$	0	0
$801$	−14.0000	−0.494666
$802$	0	0
$803$	− 6.00000i	− 0.211735i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 16.0000i	− 0.563227i
$808$	0	0
$809$	−51.0000	−1.79306	−0.896532	−	0.442978i	$-0.853922\pi$
$809$	−51.0000	−1.79306	−0.896532	+	0.442978i	$0.853922\pi$
$810$	0	0
$811$	−7.00000	−0.245803	−0.122902	−	0.992419i	$-0.539220\pi$
$811$	−7.00000	−0.245803	−0.122902	+	0.992419i	$0.539220\pi$
$812$	0	0
$813$	25.0000i	0.876788i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−12.0000	−0.419314
$820$	0	0
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	22.0000i	0.766872i	0.923567	+	0.383436i	$0.125259\pi$
$823$	22.0000i	0.766872i	−0.923567	+	0.383436i	$0.874741\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	56.0000i	1.94731i	0.228024	+	0.973655i	$0.426773\pi$
$827$	56.0000i	1.94731i	−0.228024	+	0.973655i	$0.573227\pi$
$828$	0	0
$829$	−6.00000	−0.208389	−0.104194	−	0.994557i	$-0.533226\pi$
$829$	−6.00000	−0.208389	−0.104194	+	0.994557i	$0.533226\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	0	0
$833$	2.00000i	0.0692959i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	4.00000	0.138095	0.0690477	−	0.997613i	$-0.478004\pi$
$839$	4.00000	0.138095	0.0690477	+	0.997613i	$0.478004\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	0	0
$843$	− 17.0000i	− 0.585511i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3.00000i	0.103081i
$848$	0	0
$849$	21.0000	0.720718
$850$	0	0
$851$	3.00000	0.102839
$852$	0	0
$853$	− 22.0000i	− 0.753266i	−0.926363	−	0.376633i	$-0.877082\pi$
$853$	− 22.0000i	− 0.753266i	0.926363	−	0.376633i	$-0.122918\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 9.00000i	− 0.307434i	−0.988115	−	0.153717i	$-0.950876\pi$
$857$	− 9.00000i	− 0.307434i	0.988115	−	0.153717i	$-0.0491244\pi$
$858$	0	0
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	0	0
$861$	15.0000	0.511199
$862$	0	0
$863$	48.0000i	1.63394i	0.576681	+	0.816970i	$0.304348\pi$
$863$	48.0000i	1.63394i	−0.576681	+	0.816970i	$0.695652\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 16.0000i	− 0.543388i
$868$	0	0
$869$	−13.0000	−0.440995
$870$	0	0
$871$	−56.0000	−1.89749
$872$	0	0
$873$	13.0000i	0.439983i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 8.00000i	− 0.270141i	−0.990836	−	0.135070i	$-0.956874\pi$
$877$	− 8.00000i	− 0.270141i	0.990836	−	0.135070i	$-0.0431261\pi$
$878$	0	0
$879$	−25.0000	−0.843229
$880$	0	0
$881$	−52.0000	−1.75192	−0.875962	−	0.482380i	$-0.839773\pi$
$881$	−52.0000	−1.75192	−0.875962	+	0.482380i	$0.839773\pi$
$882$	0	0
$883$	56.0000i	1.88455i	0.334840	+	0.942275i	$0.391318\pi$
$883$	56.0000i	1.88455i	−0.334840	+	0.942275i	$0.608682\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	20.0000i	0.671534i	0.941945	+	0.335767i	$0.108996\pi$
$887$	20.0000i	0.671534i	−0.941945	+	0.335767i	$0.891004\pi$
$888$	0	0
$889$	57.0000	1.91172
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	35.0000i	1.17123i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	12.0000i	0.400668i
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−6.00000	−0.199889
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 6.00000i	− 0.199227i	−0.995026	−	0.0996134i	$-0.968239\pi$
$907$	− 6.00000i	− 0.199227i	0.995026	−	0.0996134i	$-0.0317606\pi$
$908$	0	0
$909$	−3.00000	−0.0995037
$910$	0	0
$911$	−33.0000	−1.09334	−0.546669	−	0.837349i	$-0.684105\pi$
$911$	−33.0000	−1.09334	−0.546669	+	0.837349i	$0.684105\pi$
$912$	0	0
$913$	4.00000i	0.132381i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	12.0000i	0.396275i
$918$	0	0
$919$	9.00000	0.296883	0.148441	−	0.988921i	$-0.452574\pi$
$919$	9.00000	0.296883	0.148441	+	0.988921i	$0.452574\pi$
$920$	0	0
$921$	−4.00000	−0.131804
$922$	0	0
$923$	− 60.0000i	− 1.97492i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	12.0000i	0.394132i
$928$	0	0
$929$	−38.0000	−1.24674	−0.623370	−	0.781927i	$-0.714237\pi$
$929$	−38.0000	−1.24674	−0.623370	+	0.781927i	$0.714237\pi$
$930$	0	0
$931$	−14.0000	−0.458831
$932$	0	0
$933$	8.00000i	0.261908i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 12.0000i	− 0.392023i	−0.980602	−	0.196011i	$-0.937201\pi$
$937$	− 12.0000i	− 0.392023i	0.980602	−	0.196011i	$-0.0627990\pi$
$938$	0	0
$939$	29.0000	0.946379
$940$	0	0
$941$	15.0000	0.488986	0.244493	−	0.969651i	$-0.421378\pi$
$941$	15.0000	0.488986	0.244493	+	0.969651i	$0.421378\pi$
$942$	0	0
$943$	− 15.0000i	− 0.488467i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 57.0000i	− 1.85225i	−0.377215	−	0.926126i	$-0.623118\pi$
$947$	− 57.0000i	− 1.85225i	0.377215	−	0.926126i	$-0.376882\pi$
$948$	0	0
$949$	24.0000	0.779073
$950$	0	0
$951$	12.0000	0.389127
$952$	0	0
$953$	57.0000i	1.84641i	0.384307	+	0.923206i	$0.374441\pi$
$953$	57.0000i	1.84641i	−0.384307	+	0.923206i	$0.625559\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 10.0000i	− 0.323254i
$958$	0	0
$959$	−18.0000	−0.581250
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	6.00000i	0.193347i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	16.0000i	0.514525i	0.966342	+	0.257263i	$0.0828206\pi$
$967$	16.0000i	0.514525i	−0.966342	+	0.257263i	$0.917179\pi$
$968$	0	0
$969$	−7.00000	−0.224872
$970$	0	0
$971$	−15.0000	−0.481373	−0.240686	−	0.970603i	$-0.577373\pi$
$971$	−15.0000	−0.481373	−0.240686	+	0.970603i	$0.577373\pi$
$972$	0	0
$973$	− 12.0000i	− 0.384702i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 60.0000i	− 1.91957i	−0.280736	−	0.959785i	$-0.590579\pi$
$977$	− 60.0000i	− 1.91957i	0.280736	−	0.959785i	$-0.409421\pi$
$978$	0	0
$979$	−14.0000	−0.447442
$980$	0	0
$981$	2.00000	0.0638551
$982$	0	0
$983$	35.0000i	1.11633i	0.829731	+	0.558163i	$0.188494\pi$
$983$	35.0000i	1.11633i	−0.829731	+	0.558163i	$0.811506\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	15.0000i	0.477455i
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−46.0000	−1.46124	−0.730619	−	0.682785i	$-0.760768\pi$
$991$	−46.0000	−1.46124	−0.730619	+	0.682785i	$0.760768\pi$
$992$	0	0
$993$	2.00000i	0.0634681i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	10.0000i	0.316703i	0.987383	+	0.158352i	$0.0506179\pi$
$997$	10.0000i	0.316703i	−0.987383	+	0.158352i	$0.949382\pi$
$998$	0	0
$999$	−1.00000	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3300.2.c.f.1849.1		2
3.2	odd	2		9900.2.c.o.5149.2		2
5.2	odd	4		3300.2.a.a.1.1	✓	1
5.3	odd	4		3300.2.a.r.1.1	yes	1
5.4	even	2	inner	3300.2.c.f.1849.2		2
15.2	even	4		9900.2.a.c.1.1		1
15.8	even	4		9900.2.a.bb.1.1		1
15.14	odd	2		9900.2.c.o.5149.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3300.2.a.a.1.1	✓	1	5.2	odd	4
3300.2.a.r.1.1	yes	1	5.3	odd	4
3300.2.c.f.1849.1		2	1.1	even	1	trivial
3300.2.c.f.1849.2		2	5.4	even	2	inner
9900.2.a.c.1.1		1	15.2	even	4
9900.2.a.bb.1.1		1	15.8	even	4
9900.2.c.o.5149.1		2	15.14	odd	2
9900.2.c.o.5149.2		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3300.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +3.00000i q^{7} -1.00000 q^{9} -1.00000 q^{11} -4.00000i q^{13} -1.00000i q^{17} +7.00000 q^{19} +3.00000 q^{21} -3.00000i q^{23} +1.00000i q^{27} -10.0000 q^{29} +1.00000i q^{33} +1.00000i q^{37} -4.00000 q^{39} +5.00000 q^{41} +5.00000i q^{47} -2.00000 q^{49} -1.00000 q^{51} -6.00000i q^{53} -7.00000i q^{57} -3.00000 q^{59} +6.00000 q^{61} -3.00000i q^{63} -14.0000i q^{67} -3.00000 q^{69} +15.0000 q^{71} +6.00000i q^{73} -3.00000i q^{77} +13.0000 q^{79} +1.00000 q^{81} -4.00000i q^{83} +10.0000i q^{87} +14.0000 q^{89} +12.0000 q^{91} -13.0000i q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9} - 2 q^{11} + 14 q^{19} + 6 q^{21} - 20 q^{29} - 8 q^{39} + 10 q^{41} - 4 q^{49} - 2 q^{51} - 6 q^{59} + 12 q^{61} - 6 q^{69} + 30 q^{71} + 26 q^{79} + 2 q^{81} + 28 q^{89} + 24 q^{91} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 - 2 * q^11 + 14 * q^19 + 6 * q^21 - 20 * q^29 - 8 * q^39 + 10 * q^41 - 4 * q^49 - 2 * q^51 - 6 * q^59 + 12 * q^61 - 6 * q^69 + 30 * q^71 + 26 * q^79 + 2 * q^81 + 28 * q^89 + 24 * q^91 + 2 * q^99

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