LMFDB - Embedded newform 1900.2.c.b.1749.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,2,Mod(1749,1900)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1900.1749");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1900.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$15.1715763840$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 76)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1749.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1900.1749
Dual form			1900.2.c.b.1749.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{3} +3.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+2.00000i q^{3} +3.00000i q^{7} -1.00000 q^{9} +5.00000 q^{11} -4.00000i q^{13} +3.00000i q^{17} +1.00000 q^{19} -6.00000 q^{21} +8.00000i q^{23} +4.00000i q^{27} +2.00000 q^{29} +4.00000 q^{31} +10.0000i q^{33} -10.0000i q^{37} +8.00000 q^{39} +10.0000 q^{41} +1.00000i q^{43} +1.00000i q^{47} -2.00000 q^{49} -6.00000 q^{51} -4.00000i q^{53} +2.00000i q^{57} -6.00000 q^{59} -13.0000 q^{61} -3.00000i q^{63} +12.0000i q^{67} -16.0000 q^{69} +2.00000 q^{71} +9.00000i q^{73} +15.0000i q^{77} -8.00000 q^{79} -11.0000 q^{81} -12.0000i q^{83} +4.00000i q^{87} -12.0000 q^{89} +12.0000 q^{91} +8.00000i q^{93} +8.00000i q^{97} -5.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} + 10 q^{11} + 2 q^{19} - 12 q^{21} + 4 q^{29} + 8 q^{31} + 16 q^{39} + 20 q^{41} - 4 q^{49} - 12 q^{51} - 12 q^{59} - 26 q^{61} - 32 q^{69} + 4 q^{71} - 16 q^{79} - 22 q^{81} - 24 q^{89} + 24 q^{91} - 10 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 + 10 * q^11 + 2 * q^19 - 12 * q^21 + 4 * q^29 + 8 * q^31 + 16 * q^39 + 20 * q^41 - 4 * q^49 - 12 * q^51 - 12 * q^59 - 26 * q^61 - 32 * q^69 + 4 * q^71 - 16 * q^79 - 22 * q^81 - 24 * q^89 + 24 * q^91 - 10 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$.

$n$	$77$	$401$	$951$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	2.00000i	1.15470i	0.816497	+	0.577350i	$0.195913\pi$
$3$	2.00000i	1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$	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
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	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
$17$	3.00000i	0.727607i	0.931476	+	0.363803i	$0.118522\pi$
$17$	3.00000i	0.727607i	−0.931476	+	0.363803i	$0.881478\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−6.00000	−1.30931
$22$	0	0
$23$	8.00000i	1.66812i	0.551677	+	0.834058i	$0.313988\pi$
$23$	8.00000i	1.66812i	−0.551677	+	0.834058i	$0.686012\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.00000i	0.769800i
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	10.0000i	1.74078i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 10.0000i	− 1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	− 10.0000i	− 1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$	0	0
$39$	8.00000	1.28103
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	1.00000i	0.152499i	0.997089	+	0.0762493i	$0.0242945\pi$
$43$	1.00000i	0.152499i	−0.997089	+	0.0762493i	$0.975706\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.00000i	0.145865i	0.997337	+	0.0729325i	$0.0232358\pi$
$47$	1.00000i	0.145865i	−0.997337	+	0.0729325i	$0.976764\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	0	0
$51$	−6.00000	−0.840168
$52$	0	0
$53$	− 4.00000i	− 0.549442i	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	− 4.00000i	− 0.549442i	0.961524	−	0.274721i	$-0.0885855\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.00000i	0.264906i
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	−13.0000	−1.66448	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000	−1.66448	−0.832240	+	0.554416i	$0.812942\pi$
$62$	0	0
$63$	− 3.00000i	− 0.377964i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.0000i	1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$	12.0000i	1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$	0	0
$69$	−16.0000	−1.92617
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	0	0
$73$	9.00000i	1.05337i	0.850060	+	0.526685i	$0.176565\pi$
$73$	9.00000i	1.05337i	−0.850060	+	0.526685i	$0.823435\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	15.0000i	1.70941i
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.00000i	0.428845i
$88$	0	0
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	12.0000	1.25794
$92$	0	0
$93$	8.00000i	0.829561i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.00000i	0.812277i	0.913812	+	0.406138i	$0.133125\pi$
$97$	8.00000i	0.812277i	−0.913812	+	0.406138i	$0.866875\pi$
$98$	0	0
$99$	−5.00000	−0.502519
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	− 6.00000i	− 0.591198i	−0.955312	−	0.295599i	$-0.904481\pi$
$103$	− 6.00000i	− 0.591198i	0.955312	−	0.295599i	$-0.0955191\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 2.00000i	− 0.193347i	−0.995316	−	0.0966736i	$-0.969180\pi$
$107$	− 2.00000i	− 0.193347i	0.995316	−	0.0966736i	$-0.0308203\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	20.0000	1.89832
$112$	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
$119$	−9.00000	−0.825029
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	20.0000i	1.80334i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 6.00000i	− 0.532414i	−0.963916	−	0.266207i	$-0.914230\pi$
$127$	− 6.00000i	− 0.532414i	0.963916	−	0.266207i	$-0.0857705\pi$
$128$	0	0
$129$	−2.00000	−0.176090
$130$	0	0
$131$	−9.00000	−0.786334	−0.393167	−	0.919467i	$-0.628621\pi$
$131$	−9.00000	−0.786334	−0.393167	+	0.919467i	$0.628621\pi$
$132$	0	0
$133$	3.00000i	0.260133i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.0000i	0.939793i	0.882721	+	0.469897i	$0.155709\pi$
$137$	11.0000i	0.939793i	−0.882721	+	0.469897i	$0.844291\pi$
$138$	0	0
$139$	3.00000	0.254457	0.127228	−	0.991873i	$-0.459392\pi$
$139$	3.00000	0.254457	0.127228	+	0.991873i	$0.459392\pi$
$140$	0	0
$141$	−2.00000	−0.168430
$142$	0	0
$143$	− 20.0000i	− 1.67248i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 4.00000i	− 0.329914i
$148$	0	0
$149$	15.0000	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	0	0
$153$	− 3.00000i	− 0.242536i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.00000i	0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$	2.00000i	0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	0	0
$159$	8.00000	0.634441
$160$	0	0
$161$	−24.0000	−1.89146
$162$	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
$167$	6.00000i	0.464294i	0.972681	+	0.232147i	$0.0745750\pi$
$167$	6.00000i	0.464294i	−0.972681	+	0.232147i	$0.925425\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 12.0000i	− 0.901975i
$178$	0	0
$179$	−18.0000	−1.34538	−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000	−1.34538	−0.672692	+	0.739923i	$0.734862\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	− 26.0000i	− 1.92198i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	15.0000i	1.09691i
$188$	0	0
$189$	−12.0000	−0.872872
$190$	0	0
$191$	25.0000	1.80894	0.904468	−	0.426541i	$-0.140268\pi$
$191$	25.0000	1.80894	0.904468	+	0.426541i	$0.140268\pi$
$192$	0	0
$193$	12.0000i	0.863779i	0.901927	+	0.431889i	$0.142153\pi$
$193$	12.0000i	0.863779i	−0.901927	+	0.431889i	$0.857847\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2.00000i	− 0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$	− 2.00000i	− 0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$	0	0
$199$	7.00000	0.496217	0.248108	−	0.968732i	$-0.420191\pi$
$199$	7.00000	0.496217	0.248108	+	0.968732i	$0.420191\pi$
$200$	0	0
$201$	−24.0000	−1.69283
$202$	0	0
$203$	6.00000i	0.421117i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 8.00000i	− 0.556038i
$208$	0	0
$209$	5.00000	0.345857
$210$	0	0
$211$	18.0000	1.23917	0.619586	−	0.784929i	$-0.287301\pi$
$211$	18.0000	1.23917	0.619586	+	0.784929i	$0.287301\pi$
$212$	0	0
$213$	4.00000i	0.274075i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	12.0000i	0.814613i
$218$	0	0
$219$	−18.0000	−1.21633
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	2.00000i	0.133930i	0.997755	+	0.0669650i	$0.0213316\pi$
$223$	2.00000i	0.133930i	−0.997755	+	0.0669650i	$0.978668\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 4.00000i	− 0.265489i	−0.991150	−	0.132745i	$-0.957621\pi$
$227$	− 4.00000i	− 0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$	0	0
$229$	−17.0000	−1.12339	−0.561696	−	0.827344i	$-0.689851\pi$
$229$	−17.0000	−1.12339	−0.561696	+	0.827344i	$0.689851\pi$
$230$	0	0
$231$	−30.0000	−1.97386
$232$	0	0
$233$	3.00000i	0.196537i	0.995160	+	0.0982683i	$0.0313303\pi$
$233$	3.00000i	0.196537i	−0.995160	+	0.0982683i	$0.968670\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 16.0000i	− 1.03931i
$238$	0	0
$239$	−21.0000	−1.35838	−0.679189	−	0.733964i	$-0.737668\pi$
$239$	−21.0000	−1.35838	−0.679189	+	0.733964i	$0.737668\pi$
$240$	0	0
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	0	0
$243$	− 10.0000i	− 0.641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 4.00000i	− 0.254514i
$248$	0	0
$249$	24.0000	1.52094
$250$	0	0
$251$	11.0000	0.694314	0.347157	−	0.937807i	$-0.387147\pi$
$251$	11.0000	0.694314	0.347157	+	0.937807i	$0.387147\pi$
$252$	0	0
$253$	40.0000i	2.51478i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 32.0000i	− 1.99611i	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	− 32.0000i	− 1.99611i	0.0623783	−	0.998053i	$-0.480131\pi$
$258$	0	0
$259$	30.0000	1.86411
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	− 21.0000i	− 1.29492i	−0.762101	−	0.647458i	$-0.775832\pi$
$263$	− 21.0000i	− 1.29492i	0.762101	−	0.647458i	$-0.224168\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 24.0000i	− 1.46878i
$268$	0	0
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	0	0
$273$	24.0000i	1.45255i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 1.00000i	− 0.0600842i	−0.999549	−	0.0300421i	$-0.990436\pi$
$277$	− 1.00000i	− 0.0600842i	0.999549	−	0.0300421i	$-0.00956413\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	22.0000	1.31241	0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000	1.31241	0.656205	+	0.754583i	$0.272161\pi$
$282$	0	0
$283$	− 3.00000i	− 0.178331i	−0.996017	−	0.0891657i	$-0.971580\pi$
$283$	− 3.00000i	− 0.178331i	0.996017	−	0.0891657i	$-0.0284201\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	30.0000i	1.77084i
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	−16.0000	−0.937937
$292$	0	0
$293$	− 12.0000i	− 0.701047i	−0.936554	−	0.350524i	$-0.886004\pi$
$293$	− 12.0000i	− 0.701047i	0.936554	−	0.350524i	$-0.113996\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	20.0000i	1.16052i
$298$	0	0
$299$	32.0000	1.85061
$300$	0	0
$301$	−3.00000	−0.172917
$302$	0	0
$303$	− 20.0000i	− 1.14897i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 12.0000i	− 0.684876i	−0.939540	−	0.342438i	$-0.888747\pi$
$307$	− 12.0000i	− 0.684876i	0.939540	−	0.342438i	$-0.111253\pi$
$308$	0	0
$309$	12.0000	0.682656
$310$	0	0
$311$	7.00000	0.396934	0.198467	−	0.980108i	$-0.436404\pi$
$311$	7.00000	0.396934	0.198467	+	0.980108i	$0.436404\pi$
$312$	0	0
$313$	− 10.0000i	− 0.565233i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$	− 10.0000i	− 0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 30.0000i	− 1.68497i	−0.538721	−	0.842484i	$-0.681092\pi$
$317$	− 30.0000i	− 1.68497i	0.538721	−	0.842484i	$-0.318908\pi$
$318$	0	0
$319$	10.0000	0.559893
$320$	0	0
$321$	4.00000	0.223258
$322$	0	0
$323$	3.00000i	0.166924i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−3.00000	−0.165395
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	0	0
$333$	10.0000i	0.547997i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	32.0000i	1.74315i	0.490261	+	0.871576i	$0.336901\pi$
$337$	32.0000i	1.74315i	−0.490261	+	0.871576i	$0.663099\pi$
$338$	0	0
$339$	20.0000	1.08625
$340$	0	0
$341$	20.0000	1.08306
$342$	0	0
$343$	15.0000i	0.809924i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 19.0000i	− 1.01997i	−0.860182	−	0.509987i	$-0.829650\pi$
$347$	− 19.0000i	− 1.01997i	0.860182	−	0.509987i	$-0.170350\pi$
$348$	0	0
$349$	11.0000	0.588817	0.294408	−	0.955680i	$-0.404877\pi$
$349$	11.0000	0.588817	0.294408	+	0.955680i	$0.404877\pi$
$350$	0	0
$351$	16.0000	0.854017
$352$	0	0
$353$	− 6.00000i	− 0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$	− 6.00000i	− 0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 18.0000i	− 0.952661i
$358$	0	0
$359$	−21.0000	−1.10834	−0.554169	−	0.832404i	$-0.686964\pi$
$359$	−21.0000	−1.10834	−0.554169	+	0.832404i	$0.686964\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	28.0000i	1.46962i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 16.0000i	− 0.835193i	−0.908633	−	0.417597i	$-0.862873\pi$
$367$	− 16.0000i	− 0.835193i	0.908633	−	0.417597i	$-0.137127\pi$
$368$	0	0
$369$	−10.0000	−0.520579
$370$	0	0
$371$	12.0000	0.623009
$372$	0	0
$373$	− 4.00000i	− 0.207112i	−0.994624	−	0.103556i	$-0.966978\pi$
$373$	− 4.00000i	− 0.207112i	0.994624	−	0.103556i	$-0.0330221\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 8.00000i	− 0.412021i
$378$	0	0
$379$	30.0000	1.54100	0.770498	−	0.637442i	$-0.220007\pi$
$379$	30.0000	1.54100	0.770498	+	0.637442i	$0.220007\pi$
$380$	0	0
$381$	12.0000	0.614779
$382$	0	0
$383$	4.00000i	0.204390i	0.994764	+	0.102195i	$0.0325866\pi$
$383$	4.00000i	0.204390i	−0.994764	+	0.102195i	$0.967413\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1.00000i	− 0.0508329i
$388$	0	0
$389$	21.0000	1.06474	0.532371	−	0.846511i	$-0.321301\pi$
$389$	21.0000	1.06474	0.532371	+	0.846511i	$0.321301\pi$
$390$	0	0
$391$	−24.0000	−1.21373
$392$	0	0
$393$	− 18.0000i	− 0.907980i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 5.00000i	− 0.250943i	−0.992097	−	0.125471i	$-0.959956\pi$
$397$	− 5.00000i	− 0.250943i	0.992097	−	0.125471i	$-0.0400443\pi$
$398$	0	0
$399$	−6.00000	−0.300376
$400$	0	0
$401$	28.0000	1.39825	0.699127	−	0.714998i	$-0.253572\pi$
$401$	28.0000	1.39825	0.699127	+	0.714998i	$0.253572\pi$
$402$	0	0
$403$	− 16.0000i	− 0.797017i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 50.0000i	− 2.47841i
$408$	0	0
$409$	20.0000	0.988936	0.494468	−	0.869196i	$-0.335363\pi$
$409$	20.0000	0.988936	0.494468	+	0.869196i	$0.335363\pi$
$410$	0	0
$411$	−22.0000	−1.08518
$412$	0	0
$413$	− 18.0000i	− 0.885722i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	6.00000i	0.293821i
$418$	0	0
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	0	0
$421$	−40.0000	−1.94948	−0.974740	−	0.223341i	$-0.928304\pi$
$421$	−40.0000	−1.94948	−0.974740	+	0.223341i	$0.928304\pi$
$422$	0	0
$423$	− 1.00000i	− 0.0486217i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 39.0000i	− 1.88734i
$428$	0	0
$429$	40.0000	1.93122
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	2.00000i	0.0961139i	0.998845	+	0.0480569i	$0.0153029\pi$
$433$	2.00000i	0.0961139i	−0.998845	+	0.0480569i	$0.984697\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.00000i	0.382692i
$438$	0	0
$439$	−2.00000	−0.0954548	−0.0477274	−	0.998860i	$-0.515198\pi$
$439$	−2.00000	−0.0954548	−0.0477274	+	0.998860i	$0.515198\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	− 5.00000i	− 0.237557i	−0.992921	−	0.118779i	$-0.962102\pi$
$443$	− 5.00000i	− 0.237557i	0.992921	−	0.118779i	$-0.0378979\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	30.0000i	1.41895i
$448$	0	0
$449$	−16.0000	−0.755087	−0.377543	−	0.925992i	$-0.623231\pi$
$449$	−16.0000	−0.755087	−0.377543	+	0.925992i	$0.623231\pi$
$450$	0	0
$451$	50.0000	2.35441
$452$	0	0
$453$	4.00000i	0.187936i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	13.0000i	0.608114i	0.952654	+	0.304057i	$0.0983414\pi$
$457$	13.0000i	0.608114i	−0.952654	+	0.304057i	$0.901659\pi$
$458$	0	0
$459$	−12.0000	−0.560112
$460$	0	0
$461$	−19.0000	−0.884918	−0.442459	−	0.896789i	$-0.645894\pi$
$461$	−19.0000	−0.884918	−0.442459	+	0.896789i	$0.645894\pi$
$462$	0	0
$463$	19.0000i	0.883005i	0.897260	+	0.441502i	$0.145554\pi$
$463$	19.0000i	0.883005i	−0.897260	+	0.441502i	$0.854446\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5.00000i	0.231372i	0.993286	+	0.115686i	$0.0369067\pi$
$467$	5.00000i	0.231372i	−0.993286	+	0.115686i	$0.963093\pi$
$468$	0	0
$469$	−36.0000	−1.66233
$470$	0	0
$471$	−4.00000	−0.184310
$472$	0	0
$473$	5.00000i	0.229900i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	4.00000i	0.183147i
$478$	0	0
$479$	−4.00000	−0.182765	−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000	−0.182765	−0.0913823	+	0.995816i	$0.529129\pi$
$480$	0	0
$481$	−40.0000	−1.82384
$482$	0	0
$483$	− 48.0000i	− 2.18408i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	26.0000i	1.17817i	0.808070	+	0.589086i	$0.200512\pi$
$487$	26.0000i	1.17817i	−0.808070	+	0.589086i	$0.799488\pi$
$488$	0	0
$489$	8.00000	0.361773
$490$	0	0
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	0	0
$493$	6.00000i	0.270226i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.00000i	0.269137i
$498$	0	0
$499$	−19.0000	−0.850557	−0.425278	−	0.905063i	$-0.639824\pi$
$499$	−19.0000	−0.850557	−0.425278	+	0.905063i	$0.639824\pi$
$500$	0	0
$501$	−12.0000	−0.536120
$502$	0	0
$503$	− 36.0000i	− 1.60516i	−0.596544	−	0.802580i	$-0.703460\pi$
$503$	− 36.0000i	− 1.60516i	0.596544	−	0.802580i	$-0.296540\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 6.00000i	− 0.266469i
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	−27.0000	−1.19441
$512$	0	0
$513$	4.00000i	0.176604i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	5.00000i	0.219900i
$518$	0	0
$519$	−12.0000	−0.526742
$520$	0	0
$521$	32.0000	1.40195	0.700973	−	0.713188i	$-0.252749\pi$
$521$	32.0000	1.40195	0.700973	+	0.713188i	$0.252749\pi$
$522$	0	0
$523$	26.0000i	1.13690i	0.822718	+	0.568450i	$0.192457\pi$
$523$	26.0000i	1.13690i	−0.822718	+	0.568450i	$0.807543\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12.0000i	0.522728i
$528$	0	0
$529$	−41.0000	−1.78261
$530$	0	0
$531$	6.00000	0.260378
$532$	0	0
$533$	− 40.0000i	− 1.73259i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 36.0000i	− 1.55351i
$538$	0	0
$539$	−10.0000	−0.430730
$540$	0	0
$541$	11.0000	0.472927	0.236463	−	0.971640i	$-0.424012\pi$
$541$	11.0000	0.472927	0.236463	+	0.971640i	$0.424012\pi$
$542$	0	0
$543$	20.0000i	0.858282i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 28.0000i	− 1.19719i	−0.801050	−	0.598597i	$-0.795725\pi$
$547$	− 28.0000i	− 1.19719i	0.801050	−	0.598597i	$-0.204275\pi$
$548$	0	0
$549$	13.0000	0.554826
$550$	0	0
$551$	2.00000	0.0852029
$552$	0	0
$553$	− 24.0000i	− 1.02058i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 1.00000i	− 0.0423714i	−0.999776	−	0.0211857i	$-0.993256\pi$
$557$	− 1.00000i	− 0.0423714i	0.999776	−	0.0211857i	$-0.00674412\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	0	0
$561$	−30.0000	−1.26660
$562$	0	0
$563$	42.0000i	1.77009i	0.465506	+	0.885044i	$0.345872\pi$
$563$	42.0000i	1.77009i	−0.465506	+	0.885044i	$0.654128\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 33.0000i	− 1.38587i
$568$	0	0
$569$	8.00000	0.335377	0.167689	−	0.985840i	$-0.446370\pi$
$569$	8.00000	0.335377	0.167689	+	0.985840i	$0.446370\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	50.0000i	2.08878i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	13.0000i	0.541197i	0.962692	+	0.270599i	$0.0872216\pi$
$577$	13.0000i	0.541197i	−0.962692	+	0.270599i	$0.912778\pi$
$578$	0	0
$579$	−24.0000	−0.997406
$580$	0	0
$581$	36.0000	1.49353
$582$	0	0
$583$	− 20.0000i	− 0.828315i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 3.00000i	− 0.123823i	−0.998082	−	0.0619116i	$-0.980280\pi$
$587$	− 3.00000i	− 0.123823i	0.998082	−	0.0619116i	$-0.0197197\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	0	0
$591$	4.00000	0.164538
$592$	0	0
$593$	22.0000i	0.903432i	0.892162	+	0.451716i	$0.149188\pi$
$593$	22.0000i	0.903432i	−0.892162	+	0.451716i	$0.850812\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	14.0000i	0.572982i
$598$	0	0
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	− 12.0000i	− 0.488678i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 32.0000i	− 1.29884i	−0.760430	−	0.649420i	$-0.775012\pi$
$607$	− 32.0000i	− 1.29884i	0.760430	−	0.649420i	$-0.224988\pi$
$608$	0	0
$609$	−12.0000	−0.486265
$610$	0	0
$611$	4.00000	0.161823
$612$	0	0
$613$	− 47.0000i	− 1.89831i	−0.314806	−	0.949156i	$-0.601939\pi$
$613$	− 47.0000i	− 1.89831i	0.314806	−	0.949156i	$-0.398061\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 9.00000i	− 0.362326i	−0.983453	−	0.181163i	$-0.942014\pi$
$617$	− 9.00000i	− 0.362326i	0.983453	−	0.181163i	$-0.0579862\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	−32.0000	−1.28412
$622$	0	0
$623$	− 36.0000i	− 1.44231i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	10.0000i	0.399362i
$628$	0	0
$629$	30.0000	1.19618
$630$	0	0
$631$	−7.00000	−0.278666	−0.139333	−	0.990246i	$-0.544496\pi$
$631$	−7.00000	−0.278666	−0.139333	+	0.990246i	$0.544496\pi$
$632$	0	0
$633$	36.0000i	1.43087i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	8.00000i	0.316972i
$638$	0	0
$639$	−2.00000	−0.0791188
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	− 11.0000i	− 0.433798i	−0.976194	−	0.216899i	$-0.930406\pi$
$643$	− 11.0000i	− 0.433798i	0.976194	−	0.216899i	$-0.0695942\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7.00000i	0.275198i	0.990488	+	0.137599i	$0.0439386\pi$
$647$	7.00000i	0.275198i	−0.990488	+	0.137599i	$0.956061\pi$
$648$	0	0
$649$	−30.0000	−1.17760
$650$	0	0
$651$	−24.0000	−0.940634
$652$	0	0
$653$	− 27.0000i	− 1.05659i	−0.849060	−	0.528296i	$-0.822831\pi$
$653$	− 27.0000i	− 1.05659i	0.849060	−	0.528296i	$-0.177169\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 9.00000i	− 0.351123i
$658$	0	0
$659$	34.0000	1.32445	0.662226	−	0.749304i	$-0.269612\pi$
$659$	34.0000	1.32445	0.662226	+	0.749304i	$0.269612\pi$
$660$	0	0
$661$	16.0000	0.622328	0.311164	−	0.950356i	$-0.399281\pi$
$661$	16.0000	0.622328	0.311164	+	0.950356i	$0.399281\pi$
$662$	0	0
$663$	24.0000i	0.932083i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	16.0000i	0.619522i
$668$	0	0
$669$	−4.00000	−0.154649
$670$	0	0
$671$	−65.0000	−2.50930
$672$	0	0
$673$	− 10.0000i	− 0.385472i	−0.981251	−	0.192736i	$-0.938264\pi$
$673$	− 10.0000i	− 0.385472i	0.981251	−	0.192736i	$-0.0617360\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.00000i	0.0768662i	0.999261	+	0.0384331i	$0.0122367\pi$
$677$	2.00000i	0.0768662i	−0.999261	+	0.0384331i	$0.987763\pi$
$678$	0	0
$679$	−24.0000	−0.921035
$680$	0	0
$681$	8.00000	0.306561
$682$	0	0
$683$	− 20.0000i	− 0.765279i	−0.923898	−	0.382639i	$-0.875015\pi$
$683$	− 20.0000i	− 0.765279i	0.923898	−	0.382639i	$-0.124985\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 34.0000i	− 1.29718i
$688$	0	0
$689$	−16.0000	−0.609551
$690$	0	0
$691$	−33.0000	−1.25538	−0.627690	−	0.778464i	$-0.715999\pi$
$691$	−33.0000	−1.25538	−0.627690	+	0.778464i	$0.715999\pi$
$692$	0	0
$693$	− 15.0000i	− 0.569803i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	30.0000i	1.13633i
$698$	0	0
$699$	−6.00000	−0.226941
$700$	0	0
$701$	−10.0000	−0.377695	−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000	−0.377695	−0.188847	+	0.982006i	$0.560475\pi$
$702$	0	0
$703$	− 10.0000i	− 0.377157i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 30.0000i	− 1.12827i
$708$	0	0
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	0	0
$713$	32.0000i	1.19841i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 42.0000i	− 1.56852i
$718$	0	0
$719$	−13.0000	−0.484818	−0.242409	−	0.970174i	$-0.577938\pi$
$719$	−13.0000	−0.484818	−0.242409	+	0.970174i	$0.577938\pi$
$720$	0	0
$721$	18.0000	0.670355
$722$	0	0
$723$	− 52.0000i	− 1.93390i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 7.00000i	− 0.259616i	−0.991539	−	0.129808i	$-0.958564\pi$
$727$	− 7.00000i	− 0.259616i	0.991539	−	0.129808i	$-0.0414360\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−3.00000	−0.110959
$732$	0	0
$733$	− 6.00000i	− 0.221615i	−0.993842	−	0.110808i	$-0.964656\pi$
$733$	− 6.00000i	− 0.221615i	0.993842	−	0.110808i	$-0.0353437\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	60.0000i	2.21013i
$738$	0	0
$739$	43.0000	1.58178	0.790890	−	0.611958i	$-0.209618\pi$
$739$	43.0000	1.58178	0.790890	+	0.611958i	$0.209618\pi$
$740$	0	0
$741$	8.00000	0.293887
$742$	0	0
$743$	− 40.0000i	− 1.46746i	−0.679442	−	0.733729i	$-0.737778\pi$
$743$	− 40.0000i	− 1.46746i	0.679442	−	0.733729i	$-0.262222\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12.0000i	0.439057i
$748$	0	0
$749$	6.00000	0.219235
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	0	0
$753$	22.0000i	0.801725i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	5.00000i	0.181728i	0.995863	+	0.0908640i	$0.0289629\pi$
$757$	5.00000i	0.181728i	−0.995863	+	0.0908640i	$0.971037\pi$
$758$	0	0
$759$	−80.0000	−2.90382
$760$	0	0
$761$	9.00000	0.326250	0.163125	−	0.986605i	$-0.447843\pi$
$761$	9.00000	0.326250	0.163125	+	0.986605i	$0.447843\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	24.0000i	0.866590i
$768$	0	0
$769$	−31.0000	−1.11789	−0.558944	−	0.829205i	$-0.688793\pi$
$769$	−31.0000	−1.11789	−0.558944	+	0.829205i	$0.688793\pi$
$770$	0	0
$771$	64.0000	2.30490
$772$	0	0
$773$	18.0000i	0.647415i	0.946157	+	0.323708i	$0.104929\pi$
$773$	18.0000i	0.647415i	−0.946157	+	0.323708i	$0.895071\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	60.0000i	2.15249i
$778$	0	0
$779$	10.0000	0.358287
$780$	0	0
$781$	10.0000	0.357828
$782$	0	0
$783$	8.00000i	0.285897i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 4.00000i	− 0.142585i	−0.997455	−	0.0712923i	$-0.977288\pi$
$787$	− 4.00000i	− 0.142585i	0.997455	−	0.0712923i	$-0.0227123\pi$
$788$	0	0
$789$	42.0000	1.49524
$790$	0	0
$791$	30.0000	1.06668
$792$	0	0
$793$	52.0000i	1.84657i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	12.0000i	0.425062i	0.977154	+	0.212531i	$0.0681706\pi$
$797$	12.0000i	0.425062i	−0.977154	+	0.212531i	$0.931829\pi$
$798$	0	0
$799$	−3.00000	−0.106132
$800$	0	0
$801$	12.0000	0.423999
$802$	0	0
$803$	45.0000i	1.58802i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	48.0000i	1.68968i
$808$	0	0
$809$	−39.0000	−1.37117	−0.685583	−	0.727994i	$-0.740453\pi$
$809$	−39.0000	−1.37117	−0.685583	+	0.727994i	$0.740453\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	− 16.0000i	− 0.561144i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.00000i	0.0349856i
$818$	0	0
$819$	−12.0000	−0.419314
$820$	0	0
$821$	45.0000	1.57051	0.785255	−	0.619172i	$-0.212532\pi$
$821$	45.0000	1.57051	0.785255	+	0.619172i	$0.212532\pi$
$822$	0	0
$823$	53.0000i	1.84746i	0.383040	+	0.923732i	$0.374877\pi$
$823$	53.0000i	1.84746i	−0.383040	+	0.923732i	$0.625123\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	28.0000i	0.973655i	0.873498	+	0.486828i	$0.161846\pi$
$827$	28.0000i	0.973655i	−0.873498	+	0.486828i	$0.838154\pi$
$828$	0	0
$829$	48.0000	1.66711	0.833554	−	0.552437i	$-0.186302\pi$
$829$	48.0000	1.66711	0.833554	+	0.552437i	$0.186302\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	0	0
$833$	− 6.00000i	− 0.207888i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	16.0000i	0.553041i
$838$	0	0
$839$	10.0000	0.345238	0.172619	−	0.984989i	$-0.444777\pi$
$839$	10.0000	0.345238	0.172619	+	0.984989i	$0.444777\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	44.0000i	1.51544i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	42.0000i	1.44314i
$848$	0	0
$849$	6.00000	0.205919
$850$	0	0
$851$	80.0000	2.74236
$852$	0	0
$853$	42.0000i	1.43805i	0.694983	+	0.719026i	$0.255412\pi$
$853$	42.0000i	1.43805i	−0.694983	+	0.719026i	$0.744588\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 18.0000i	− 0.614868i	−0.951569	−	0.307434i	$-0.900530\pi$
$857$	− 18.0000i	− 0.614868i	0.951569	−	0.307434i	$-0.0994704\pi$
$858$	0	0
$859$	−1.00000	−0.0341196	−0.0170598	−	0.999854i	$-0.505431\pi$
$859$	−1.00000	−0.0341196	−0.0170598	+	0.999854i	$0.505431\pi$
$860$	0	0
$861$	−60.0000	−2.04479
$862$	0	0
$863$	6.00000i	0.204242i	0.994772	+	0.102121i	$0.0325630\pi$
$863$	6.00000i	0.204242i	−0.994772	+	0.102121i	$0.967437\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	16.0000i	0.543388i
$868$	0	0
$869$	−40.0000	−1.35691
$870$	0	0
$871$	48.0000	1.62642
$872$	0	0
$873$	− 8.00000i	− 0.270759i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 34.0000i	− 1.14810i	−0.818821	−	0.574049i	$-0.805372\pi$
$877$	− 34.0000i	− 1.14810i	0.818821	−	0.574049i	$-0.194628\pi$
$878$	0	0
$879$	24.0000	0.809500
$880$	0	0
$881$	−51.0000	−1.71823	−0.859117	−	0.511780i	$-0.828986\pi$
$881$	−51.0000	−1.71823	−0.859117	+	0.511780i	$0.828986\pi$
$882$	0	0
$883$	1.00000i	0.0336527i	0.999858	+	0.0168263i	$0.00535624\pi$
$883$	1.00000i	0.0336527i	−0.999858	+	0.0168263i	$0.994644\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 22.0000i	− 0.738688i	−0.929293	−	0.369344i	$-0.879582\pi$
$887$	− 22.0000i	− 0.738688i	0.929293	−	0.369344i	$-0.120418\pi$
$888$	0	0
$889$	18.0000	0.603701
$890$	0	0
$891$	−55.0000	−1.84257
$892$	0	0
$893$	1.00000i	0.0334637i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	64.0000i	2.13690i
$898$	0	0
$899$	8.00000	0.266815
$900$	0	0
$901$	12.0000	0.399778
$902$	0	0
$903$	− 6.00000i	− 0.199667i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	40.0000i	1.32818i	0.747653	+	0.664089i	$0.231180\pi$
$907$	40.0000i	1.32818i	−0.747653	+	0.664089i	$0.768820\pi$
$908$	0	0
$909$	10.0000	0.331679
$910$	0	0
$911$	−18.0000	−0.596367	−0.298183	−	0.954509i	$-0.596381\pi$
$911$	−18.0000	−0.596367	−0.298183	+	0.954509i	$0.596381\pi$
$912$	0	0
$913$	− 60.0000i	− 1.98571i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 27.0000i	− 0.891619i
$918$	0	0
$919$	28.0000	0.923635	0.461817	−	0.886975i	$-0.347198\pi$
$919$	28.0000	0.923635	0.461817	+	0.886975i	$0.347198\pi$
$920$	0	0
$921$	24.0000	0.790827
$922$	0	0
$923$	− 8.00000i	− 0.263323i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	6.00000i	0.197066i
$928$	0	0
$929$	−14.0000	−0.459325	−0.229663	−	0.973270i	$-0.573762\pi$
$929$	−14.0000	−0.459325	−0.229663	+	0.973270i	$0.573762\pi$
$930$	0	0
$931$	−2.00000	−0.0655474
$932$	0	0
$933$	14.0000i	0.458339i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	31.0000i	1.01273i	0.862320	+	0.506363i	$0.169010\pi$
$937$	31.0000i	1.01273i	−0.862320	+	0.506363i	$0.830990\pi$
$938$	0	0
$939$	20.0000	0.652675
$940$	0	0
$941$	−26.0000	−0.847576	−0.423788	−	0.905761i	$-0.639300\pi$
$941$	−26.0000	−0.847576	−0.423788	+	0.905761i	$0.639300\pi$
$942$	0	0
$943$	80.0000i	2.60516i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	12.0000i	0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$	12.0000i	0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$	0	0
$949$	36.0000	1.16861
$950$	0	0
$951$	60.0000	1.94563
$952$	0	0
$953$	− 16.0000i	− 0.518291i	−0.965838	−	0.259145i	$-0.916559\pi$
$953$	− 16.0000i	− 0.518291i	0.965838	−	0.259145i	$-0.0834409\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	20.0000i	0.646508i
$958$	0	0
$959$	−33.0000	−1.06563
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	2.00000i	0.0644491i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 48.0000i	− 1.54358i	−0.635880	−	0.771788i	$-0.719363\pi$
$967$	− 48.0000i	− 1.54358i	0.635880	−	0.771788i	$-0.280637\pi$
$968$	0	0
$969$	−6.00000	−0.192748
$970$	0	0
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	0	0
$973$	9.00000i	0.288527i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 56.0000i	− 1.79160i	−0.444459	−	0.895799i	$-0.646604\pi$
$977$	− 56.0000i	− 1.79160i	0.444459	−	0.895799i	$-0.353396\pi$
$978$	0	0
$979$	−60.0000	−1.91761
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 12.0000i	− 0.382741i	−0.981518	−	0.191370i	$-0.938707\pi$
$983$	− 12.0000i	− 0.382741i	0.981518	−	0.191370i	$-0.0612931\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 6.00000i	− 0.190982i
$988$	0	0
$989$	−8.00000	−0.254385
$990$	0	0
$991$	−38.0000	−1.20711	−0.603555	−	0.797321i	$-0.706250\pi$
$991$	−38.0000	−1.20711	−0.603555	+	0.797321i	$0.706250\pi$
$992$	0	0
$993$	− 8.00000i	− 0.253872i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 37.0000i	− 1.17180i	−0.810383	−	0.585901i	$-0.800741\pi$
$997$	− 37.0000i	− 1.17180i	0.810383	−	0.585901i	$-0.199259\pi$
$998$	0	0
$999$	40.0000	1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.2.c.b.1749.2		2
5.2	odd	4		76.2.a.a.1.1	✓	1
5.3	odd	4		1900.2.a.b.1.1		1
5.4	even	2	inner	1900.2.c.b.1749.1		2
15.2	even	4		684.2.a.b.1.1		1
20.3	even	4		7600.2.a.p.1.1		1
20.7	even	4		304.2.a.a.1.1		1
35.27	even	4		3724.2.a.a.1.1		1
40.27	even	4		1216.2.a.q.1.1		1
40.37	odd	4		1216.2.a.c.1.1		1
55.32	even	4		9196.2.a.f.1.1		1
60.47	odd	4		2736.2.a.q.1.1		1
95.7	odd	12		1444.2.e.a.429.1		2
95.12	even	12		1444.2.e.c.429.1		2
95.27	even	12		1444.2.e.c.653.1		2
95.37	even	4		1444.2.a.a.1.1		1
95.87	odd	12		1444.2.e.a.653.1		2
380.227	odd	4		5776.2.a.p.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.2.a.a.1.1	✓	1	5.2	odd	4
304.2.a.a.1.1		1	20.7	even	4
684.2.a.b.1.1		1	15.2	even	4
1216.2.a.c.1.1		1	40.37	odd	4
1216.2.a.q.1.1		1	40.27	even	4
1444.2.a.a.1.1		1	95.37	even	4
1444.2.e.a.429.1		2	95.7	odd	12
1444.2.e.a.653.1		2	95.87	odd	12
1444.2.e.c.429.1		2	95.12	even	12
1444.2.e.c.653.1		2	95.27	even	12
1900.2.a.b.1.1		1	5.3	odd	4
1900.2.c.b.1749.1		2	5.4	even	2	inner
1900.2.c.b.1749.2		2	1.1	even	1	trivial
2736.2.a.q.1.1		1	60.47	odd	4
3724.2.a.a.1.1		1	35.27	even	4
5776.2.a.p.1.1		1	380.227	odd	4
7600.2.a.p.1.1		1	20.3	even	4
9196.2.a.f.1.1		1	55.32	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1900.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{3} +3.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+2.00000i q^{3} +3.00000i q^{7} -1.00000 q^{9} +5.00000 q^{11} -4.00000i q^{13} +3.00000i q^{17} +1.00000 q^{19} -6.00000 q^{21} +8.00000i q^{23} +4.00000i q^{27} +2.00000 q^{29} +4.00000 q^{31} +10.0000i q^{33} -10.0000i q^{37} +8.00000 q^{39} +10.0000 q^{41} +1.00000i q^{43} +1.00000i q^{47} -2.00000 q^{49} -6.00000 q^{51} -4.00000i q^{53} +2.00000i q^{57} -6.00000 q^{59} -13.0000 q^{61} -3.00000i q^{63} +12.0000i q^{67} -16.0000 q^{69} +2.00000 q^{71} +9.00000i q^{73} +15.0000i q^{77} -8.00000 q^{79} -11.0000 q^{81} -12.0000i q^{83} +4.00000i q^{87} -12.0000 q^{89} +12.0000 q^{91} +8.00000i q^{93} +8.00000i q^{97} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} + 10 q^{11} + 2 q^{19} - 12 q^{21} + 4 q^{29} + 8 q^{31} + 16 q^{39} + 20 q^{41} - 4 q^{49} - 12 q^{51} - 12 q^{59} - 26 q^{61} - 32 q^{69} + 4 q^{71} - 16 q^{79} - 22 q^{81} - 24 q^{89} + 24 q^{91} - 10 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 + 10 * q^11 + 2 * q^19 - 12 * q^21 + 4 * q^29 + 8 * q^31 + 16 * q^39 + 20 * q^41 - 4 * q^49 - 12 * q^51 - 12 * q^59 - 26 * q^61 - 32 * q^69 + 4 * q^71 - 16 * q^79 - 22 * q^81 - 24 * q^89 + 24 * q^91 - 10 * q^99

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