LMFDB - Embedded newform 1925.2.b.e.1849.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1925,2,Mod(1849,1925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1925, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1925.1849");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1925 = 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1925.b (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:	$15.3712023891$
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 77)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1925.1849
Dual form			1925.2.b.e.1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} +2.00000 q^{4} -1.00000i q^{7} -6.00000 q^{9} -1.00000 q^{11} +6.00000i q^{12} +4.00000i q^{13} +4.00000 q^{16} +2.00000i q^{17} +6.00000 q^{19} +3.00000 q^{21} +5.00000i q^{23} -9.00000i q^{27} -2.00000i q^{28} -10.0000 q^{29} +1.00000 q^{31} -3.00000i q^{33} -12.0000 q^{36} -5.00000i q^{37} -12.0000 q^{39} -2.00000 q^{41} +8.00000i q^{43} -2.00000 q^{44} +8.00000i q^{47} +12.0000i q^{48} -1.00000 q^{49} -6.00000 q^{51} +8.00000i q^{52} +6.00000i q^{53} +18.0000i q^{57} -3.00000 q^{59} -2.00000 q^{61} +6.00000i q^{63} +8.00000 q^{64} -3.00000i q^{67} +4.00000i q^{68} -15.0000 q^{69} +1.00000 q^{71} -10.0000i q^{73} +12.0000 q^{76} +1.00000i q^{77} -6.00000 q^{79} +9.00000 q^{81} -12.0000i q^{83} +6.00000 q^{84} -30.0000i q^{87} +15.0000 q^{89} +4.00000 q^{91} +10.0000i q^{92} +3.00000i q^{93} -5.00000i q^{97} +6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{4} - 12 q^{9} - 2 q^{11} + 8 q^{16} + 12 q^{19} + 6 q^{21} - 20 q^{29} + 2 q^{31} - 24 q^{36} - 24 q^{39} - 4 q^{41} - 4 q^{44} - 2 q^{49} - 12 q^{51} - 6 q^{59} - 4 q^{61} + 16 q^{64} - 30 q^{69}+ \cdots + 12 q^{99}+O(q^{100}) $ 2 * q + 4 * q^4 - 12 * q^9 - 2 * q^11 + 8 * q^16 + 12 * q^19 + 6 * q^21 - 20 * q^29 + 2 * q^31 - 24 * q^36 - 24 * q^39 - 4 * q^41 - 4 * q^44 - 2 * q^49 - 12 * q^51 - 6 * q^59 - 4 * q^61 + 16 * q^64 - 30 * q^69 + 2 * q^71 + 24 * q^76 - 12 * q^79 + 18 * q^81 + 12 * q^84 + 30 * q^89 + 8 * q^91 + 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1925\mathbb{Z}\right)^\times$.

$n$	$276$	$1002$	$1751$
$\chi(n)$	$1$	$-1$	$1$

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	1.00000			$0$
$2$	0	0	−1.00000			$\pi$
$3$	3.00000i	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$3$	3.00000i	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$4$	2.00000	1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	−6.00000	−2.00000
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	6.00000i	1.73205i
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	2.00000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	2.00000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	0	0
$23$	5.00000i	1.04257i	0.853382	+	0.521286i	$0.174548\pi$
$23$	5.00000i	1.04257i	−0.853382	+	0.521286i	$0.825452\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 9.00000i	− 1.73205i
$28$	− 2.00000i	− 0.377964i
$29$	−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
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	0	0
$33$	− 3.00000i	− 0.522233i
$34$	0	0
$35$	0	0
$36$	−12.0000	−2.00000
$37$	− 5.00000i	− 0.821995i	−0.911636	−	0.410997i	$-0.865181\pi$
$37$	− 5.00000i	− 0.821995i	0.911636	−	0.410997i	$-0.134819\pi$
$38$	0	0
$39$	−12.0000	−1.92154
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	8.00000i	1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$	8.00000i	1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	0	0
$47$	8.00000i	1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$	8.00000i	1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$	12.0000i	1.73205i
$49$	−1.00000	−0.142857
$50$	0	0
$51$	−6.00000	−0.840168
$52$	8.00000i	1.10940i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	18.0000i	2.38416i
$58$	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
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	6.00000i	0.755929i
$64$	8.00000	1.00000
$65$	0	0
$66$	0	0
$67$	− 3.00000i	− 0.366508i	−0.983066	−	0.183254i	$-0.941337\pi$
$67$	− 3.00000i	− 0.366508i	0.983066	−	0.183254i	$-0.0586631\pi$
$68$	4.00000i	0.485071i
$69$	−15.0000	−1.80579
$70$	0	0
$71$	1.00000	0.118678	0.0593391	−	0.998238i	$-0.481101\pi$
$71$	1.00000	0.118678	0.0593391	+	0.998238i	$0.481101\pi$
$72$	0	0
$73$	− 10.0000i	− 1.17041i	−0.810885	−	0.585206i	$-0.801014\pi$
$73$	− 10.0000i	− 1.17041i	0.810885	−	0.585206i	$-0.198986\pi$
$74$	0	0
$75$	0	0
$76$	12.0000	1.37649
$77$	1.00000i	0.113961i
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	6.00000	0.654654
$85$	0	0
$86$	0	0
$87$	− 30.0000i	− 3.21634i
$88$	0	0
$89$	15.0000	1.59000	0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000	1.59000	0.794998	+	0.606612i	$0.207472\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	10.0000i	1.04257i
$93$	3.00000i	0.311086i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 5.00000i	− 0.507673i	−0.967247	−	0.253837i	$-0.918307\pi$
$97$	− 5.00000i	− 0.507673i	0.967247	−	0.253837i	$-0.0816925\pi$
$98$	0	0
$99$	6.00000	0.603023
$100$	0	0
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	12.0000i	1.18240i	0.806527	+	0.591198i	$0.201345\pi$
$103$	12.0000i	1.18240i	−0.806527	+	0.591198i	$0.798655\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 10.0000i	− 0.966736i	−0.875417	−	0.483368i	$-0.839413\pi$
$107$	− 10.0000i	− 0.966736i	0.875417	−	0.483368i	$-0.160587\pi$
$108$	− 18.0000i	− 1.73205i
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	15.0000	1.42374
$112$	− 4.00000i	− 0.377964i
$113$	19.0000i	1.78737i	0.448695	+	0.893685i	$0.351889\pi$
$113$	19.0000i	1.78737i	−0.448695	+	0.893685i	$0.648111\pi$
$114$	0	0
$115$	0	0
$116$	−20.0000	−1.85695
$117$	− 24.0000i	− 2.21880i
$118$	0	0
$119$	2.00000	0.183340
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	− 6.00000i	− 0.541002i
$124$	2.00000	0.179605
$125$	0	0
$126$	0	0
$127$	2.00000i	0.177471i	0.996055	+	0.0887357i	$0.0282826\pi$
$127$	2.00000i	0.177471i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$	0	0
$129$	−24.0000	−2.11308
$130$	0	0
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	− 6.00000i	− 0.522233i
$133$	− 6.00000i	− 0.520266i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 3.00000i	− 0.256307i	−0.991754	−	0.128154i	$-0.959095\pi$
$137$	− 3.00000i	− 0.256307i	0.991754	−	0.128154i	$-0.0409051\pi$
$138$	0	0
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	0	0
$141$	−24.0000	−2.02116
$142$	0	0
$143$	− 4.00000i	− 0.334497i
$144$	−24.0000	−2.00000
$145$	0	0
$146$	0	0
$147$	− 3.00000i	− 0.247436i
$148$	− 10.0000i	− 0.821995i
$149$	22.0000	1.80231	0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000	1.80231	0.901155	+	0.433497i	$0.142720\pi$
$150$	0	0
$151$	6.00000	0.488273	0.244137	−	0.969741i	$-0.421495\pi$
$151$	6.00000	0.488273	0.244137	+	0.969741i	$0.421495\pi$
$152$	0	0
$153$	− 12.0000i	− 0.970143i
$154$	0	0
$155$	0	0
$156$	−24.0000	−1.92154
$157$	7.00000i	0.558661i	0.960195	+	0.279330i	$0.0901125\pi$
$157$	7.00000i	0.558661i	−0.960195	+	0.279330i	$0.909888\pi$
$158$	0	0
$159$	−18.0000	−1.42749
$160$	0	0
$161$	5.00000	0.394055
$162$	0	0
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	−4.00000	−0.312348
$165$	0	0
$166$	0	0
$167$	− 2.00000i	− 0.154765i	−0.997001	−	0.0773823i	$-0.975344\pi$
$167$	− 2.00000i	− 0.154765i	0.997001	−	0.0773823i	$-0.0246562\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	−36.0000	−2.75299
$172$	16.0000i	1.21999i
$173$	− 16.0000i	− 1.21646i	−0.793762	−	0.608229i	$-0.791880\pi$
$173$	− 16.0000i	− 1.21646i	0.793762	−	0.608229i	$-0.208120\pi$
$174$	0	0
$175$	0	0
$176$	−4.00000	−0.301511
$177$	− 9.00000i	− 0.676481i
$178$	0	0
$179$	−1.00000	−0.0747435	−0.0373718	−	0.999301i	$-0.511899\pi$
$179$	−1.00000	−0.0747435	−0.0373718	+	0.999301i	$0.511899\pi$
$180$	0	0
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	0	0
$183$	− 6.00000i	− 0.443533i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 2.00000i	− 0.146254i
$188$	16.0000i	1.16692i
$189$	−9.00000	−0.654654
$190$	0	0
$191$	5.00000	0.361787	0.180894	−	0.983503i	$-0.442101\pi$
$191$	5.00000	0.361787	0.180894	+	0.983503i	$0.442101\pi$
$192$	24.0000i	1.73205i
$193$	− 14.0000i	− 1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	− 14.0000i	− 1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	0	0
$195$	0	0
$196$	−2.00000	−0.142857
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	9.00000	0.634811
$202$	0	0
$203$	10.0000i	0.701862i
$204$	−12.0000	−0.840168
$205$	0	0
$206$	0	0
$207$	− 30.0000i	− 2.08514i
$208$	16.0000i	1.10940i
$209$	−6.00000	−0.415029
$210$	0	0
$211$	−2.00000	−0.137686	−0.0688428	−	0.997628i	$-0.521931\pi$
$211$	−2.00000	−0.137686	−0.0688428	+	0.997628i	$0.521931\pi$
$212$	12.0000i	0.824163i
$213$	3.00000i	0.205557i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 1.00000i	− 0.0678844i
$218$	0	0
$219$	30.0000	2.02721
$220$	0	0
$221$	−8.00000	−0.538138
$222$	0	0
$223$	− 1.00000i	− 0.0669650i	−0.999439	−	0.0334825i	$-0.989340\pi$
$223$	− 1.00000i	− 0.0669650i	0.999439	−	0.0334825i	$-0.0106598\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.00000i	0.265489i	0.991150	+	0.132745i	$0.0423790\pi$
$227$	4.00000i	0.265489i	−0.991150	+	0.132745i	$0.957621\pi$
$228$	36.0000i	2.38416i
$229$	7.00000	0.462573	0.231287	−	0.972886i	$-0.425707\pi$
$229$	7.00000	0.462573	0.231287	+	0.972886i	$0.425707\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	0	0
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	0	0
$235$	0	0
$236$	−6.00000	−0.390567
$237$	− 18.0000i	− 1.16923i
$238$	0	0
$239$	−4.00000	−0.258738	−0.129369	−	0.991596i	$-0.541295\pi$
$239$	−4.00000	−0.258738	−0.129369	+	0.991596i	$0.541295\pi$
$240$	0	0
$241$	−12.0000	−0.772988	−0.386494	−	0.922292i	$-0.626314\pi$
$241$	−12.0000	−0.772988	−0.386494	+	0.922292i	$0.626314\pi$
$242$	0	0
$243$	0	0
$244$	−4.00000	−0.256074
$245$	0	0
$246$	0	0
$247$	24.0000i	1.52708i
$248$	0	0
$249$	36.0000	2.28141
$250$	0	0
$251$	−21.0000	−1.32551	−0.662754	−	0.748837i	$-0.730613\pi$
$251$	−21.0000	−1.32551	−0.662754	+	0.748837i	$0.730613\pi$
$252$	12.0000i	0.755929i
$253$	− 5.00000i	− 0.314347i
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	− 6.00000i	− 0.374270i	−0.982334	−	0.187135i	$-0.940080\pi$
$257$	− 6.00000i	− 0.374270i	0.982334	−	0.187135i	$-0.0599201\pi$
$258$	0	0
$259$	−5.00000	−0.310685
$260$	0	0
$261$	60.0000	3.71391
$262$	0	0
$263$	− 18.0000i	− 1.10993i	−0.831875	−	0.554964i	$-0.812732\pi$
$263$	− 18.0000i	− 1.10993i	0.831875	−	0.554964i	$-0.187268\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	45.0000i	2.75396i
$268$	− 6.00000i	− 0.366508i
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	8.00000i	0.485071i
$273$	12.0000i	0.726273i
$274$	0	0
$275$	0	0
$276$	−30.0000	−1.80579
$277$	24.0000i	1.44202i	0.692925	+	0.721010i	$0.256322\pi$
$277$	24.0000i	1.44202i	−0.692925	+	0.721010i	$0.743678\pi$
$278$	0	0
$279$	−6.00000	−0.359211
$280$	0	0
$281$	−4.00000	−0.238620	−0.119310	−	0.992857i	$-0.538068\pi$
$281$	−4.00000	−0.238620	−0.119310	+	0.992857i	$0.538068\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	2.00000	0.118678
$285$	0	0
$286$	0	0
$287$	2.00000i	0.118056i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	15.0000	0.879316
$292$	− 20.0000i	− 1.17041i
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	9.00000i	0.522233i
$298$	0	0
$299$	−20.0000	−1.15663
$300$	0	0
$301$	8.00000	0.461112
$302$	0	0
$303$	− 36.0000i	− 2.06815i
$304$	24.0000	1.37649
$305$	0	0
$306$	0	0
$307$	− 28.0000i	− 1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$	− 28.0000i	− 1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$	2.00000i	0.113961i
$309$	−36.0000	−2.04797
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	23.0000i	1.30004i	0.759918	+	0.650018i	$0.225239\pi$
$313$	23.0000i	1.30004i	−0.759918	+	0.650018i	$0.774761\pi$
$314$	0	0
$315$	0	0
$316$	−12.0000	−0.675053
$317$	9.00000i	0.505490i	0.967533	+	0.252745i	$0.0813334\pi$
$317$	9.00000i	0.505490i	−0.967533	+	0.252745i	$0.918667\pi$
$318$	0	0
$319$	10.0000	0.559893
$320$	0	0
$321$	30.0000	1.67444
$322$	0	0
$323$	12.0000i	0.667698i
$324$	18.0000	1.00000
$325$	0	0
$326$	0	0
$327$	− 12.0000i	− 0.663602i
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	−17.0000	−0.934405	−0.467202	−	0.884150i	$-0.654738\pi$
$331$	−17.0000	−0.934405	−0.467202	+	0.884150i	$0.654738\pi$
$332$	− 24.0000i	− 1.31717i
$333$	30.0000i	1.64399i
$334$	0	0
$335$	0	0
$336$	12.0000	0.654654
$337$	− 18.0000i	− 0.980522i	−0.871576	−	0.490261i	$-0.836901\pi$
$337$	− 18.0000i	− 0.980522i	0.871576	−	0.490261i	$-0.163099\pi$
$338$	0	0
$339$	−57.0000	−3.09582
$340$	0	0
$341$	−1.00000	−0.0541530
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	14.0000i	0.751559i	0.926709	+	0.375780i	$0.122625\pi$
$347$	14.0000i	0.751559i	−0.926709	+	0.375780i	$0.877375\pi$
$348$	− 60.0000i	− 3.21634i
$349$	34.0000	1.81998	0.909989	−	0.414632i	$-0.136090\pi$
$349$	34.0000	1.81998	0.909989	+	0.414632i	$0.136090\pi$
$350$	0	0
$351$	36.0000	1.92154
$352$	0	0
$353$	− 9.00000i	− 0.479022i	−0.970894	−	0.239511i	$-0.923013\pi$
$353$	− 9.00000i	− 0.479022i	0.970894	−	0.239511i	$-0.0769871\pi$
$354$	0	0
$355$	0	0
$356$	30.0000	1.59000
$357$	6.00000i	0.317554i
$358$	0	0
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	3.00000i	0.157459i
$364$	8.00000	0.419314
$365$	0	0
$366$	0	0
$367$	− 11.0000i	− 0.574195i	−0.957901	−	0.287098i	$-0.907310\pi$
$367$	− 11.0000i	− 0.574195i	0.957901	−	0.287098i	$-0.0926904\pi$
$368$	20.0000i	1.04257i
$369$	12.0000	0.624695
$370$	0	0
$371$	6.00000	0.311504
$372$	6.00000i	0.311086i
$373$	4.00000i	0.207112i	0.994624	+	0.103556i	$0.0330221\pi$
$373$	4.00000i	0.207112i	−0.994624	+	0.103556i	$0.966978\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 40.0000i	− 2.06010i
$378$	0	0
$379$	29.0000	1.48963	0.744815	−	0.667271i	$-0.232538\pi$
$379$	29.0000	1.48963	0.744815	+	0.667271i	$0.232538\pi$
$380$	0	0
$381$	−6.00000	−0.307389
$382$	0	0
$383$	− 17.0000i	− 0.868659i	−0.900754	−	0.434330i	$-0.856985\pi$
$383$	− 17.0000i	− 0.868659i	0.900754	−	0.434330i	$-0.143015\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 48.0000i	− 2.43998i
$388$	− 10.0000i	− 0.507673i
$389$	−9.00000	−0.456318	−0.228159	−	0.973624i	$-0.573271\pi$
$389$	−9.00000	−0.456318	−0.228159	+	0.973624i	$0.573271\pi$
$390$	0	0
$391$	−10.0000	−0.505722
$392$	0	0
$393$	54.0000i	2.72394i
$394$	0	0
$395$	0	0
$396$	12.0000	0.603023
$397$	18.0000i	0.903394i	0.892171	+	0.451697i	$0.149181\pi$
$397$	18.0000i	0.903394i	−0.892171	+	0.451697i	$0.850819\pi$
$398$	0	0
$399$	18.0000	0.901127
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	4.00000i	0.199254i
$404$	−24.0000	−1.19404
$405$	0	0
$406$	0	0
$407$	5.00000i	0.247841i
$408$	0	0
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	0	0
$411$	9.00000	0.443937
$412$	24.0000i	1.18240i
$413$	3.00000i	0.147620i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	30.0000i	1.46911i
$418$	0	0
$419$	−16.0000	−0.781651	−0.390826	−	0.920465i	$-0.627810\pi$
$419$	−16.0000	−0.781651	−0.390826	+	0.920465i	$0.627810\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	0	0
$423$	− 48.0000i	− 2.33384i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.00000i	0.0967868i
$428$	− 20.0000i	− 0.966736i
$429$	12.0000	0.579365
$430$	0	0
$431$	−20.0000	−0.963366	−0.481683	−	0.876346i	$-0.659974\pi$
$431$	−20.0000	−0.963366	−0.481683	+	0.876346i	$0.659974\pi$
$432$	− 36.0000i	− 1.73205i
$433$	25.0000i	1.20142i	0.799466	+	0.600712i	$0.205116\pi$
$433$	25.0000i	1.20142i	−0.799466	+	0.600712i	$0.794884\pi$
$434$	0	0
$435$	0	0
$436$	−8.00000	−0.383131
$437$	30.0000i	1.43509i
$438$	0	0
$439$	14.0000	0.668184	0.334092	−	0.942541i	$-0.391570\pi$
$439$	14.0000	0.668184	0.334092	+	0.942541i	$0.391570\pi$
$440$	0	0
$441$	6.00000	0.285714
$442$	0	0
$443$	39.0000i	1.85295i	0.376361	+	0.926473i	$0.377175\pi$
$443$	39.0000i	1.85295i	−0.376361	+	0.926473i	$0.622825\pi$
$444$	30.0000	1.42374
$445$	0	0
$446$	0	0
$447$	66.0000i	3.12169i
$448$	− 8.00000i	− 0.377964i
$449$	−15.0000	−0.707894	−0.353947	−	0.935266i	$-0.615161\pi$
$449$	−15.0000	−0.707894	−0.353947	+	0.935266i	$0.615161\pi$
$450$	0	0
$451$	2.00000	0.0941763
$452$	38.0000i	1.78737i
$453$	18.0000i	0.845714i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8.00000i	0.374224i	0.982339	+	0.187112i	$0.0599128\pi$
$457$	8.00000i	0.374224i	−0.982339	+	0.187112i	$0.940087\pi$
$458$	0	0
$459$	18.0000	0.840168
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	− 13.0000i	− 0.604161i	−0.953282	−	0.302081i	$-0.902319\pi$
$463$	− 13.0000i	− 0.604161i	0.953282	−	0.302081i	$-0.0976812\pi$
$464$	−40.0000	−1.85695
$465$	0	0
$466$	0	0
$467$	3.00000i	0.138823i	0.997588	+	0.0694117i	$0.0221122\pi$
$467$	3.00000i	0.138823i	−0.997588	+	0.0694117i	$0.977888\pi$
$468$	− 48.0000i	− 2.21880i
$469$	−3.00000	−0.138527
$470$	0	0
$471$	−21.0000	−0.967629
$472$	0	0
$473$	− 8.00000i	− 0.367840i
$474$	0	0
$475$	0	0
$476$	4.00000	0.183340
$477$	− 36.0000i	− 1.64833i
$478$	0	0
$479$	28.0000	1.27935	0.639676	−	0.768644i	$-0.279068\pi$
$479$	28.0000	1.27935	0.639676	+	0.768644i	$0.279068\pi$
$480$	0	0
$481$	20.0000	0.911922
$482$	0	0
$483$	15.0000i	0.682524i
$484$	2.00000	0.0909091
$485$	0	0
$486$	0	0
$487$	− 13.0000i	− 0.589086i	−0.955638	−	0.294543i	$-0.904833\pi$
$487$	− 13.0000i	− 0.589086i	0.955638	−	0.294543i	$-0.0951675\pi$
$488$	0	0
$489$	12.0000	0.542659
$490$	0	0
$491$	−30.0000	−1.35388	−0.676941	−	0.736038i	$-0.736695\pi$
$491$	−30.0000	−1.35388	−0.676941	+	0.736038i	$0.736695\pi$
$492$	− 12.0000i	− 0.541002i
$493$	− 20.0000i	− 0.900755i
$494$	0	0
$495$	0	0
$496$	4.00000	0.179605
$497$	− 1.00000i	− 0.0448561i
$498$	0	0
$499$	−44.0000	−1.96971	−0.984855	−	0.173379i	$-0.944532\pi$
$499$	−44.0000	−1.96971	−0.984855	+	0.173379i	$0.944532\pi$
$500$	0	0
$501$	6.00000	0.268060
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 9.00000i	− 0.399704i
$508$	4.00000i	0.177471i
$509$	31.0000	1.37405	0.687025	−	0.726633i	$-0.258916\pi$
$509$	31.0000	1.37405	0.687025	+	0.726633i	$0.258916\pi$
$510$	0	0
$511$	−10.0000	−0.442374
$512$	0	0
$513$	− 54.0000i	− 2.38416i
$514$	0	0
$515$	0	0
$516$	−48.0000	−2.11308
$517$	− 8.00000i	− 0.351840i
$518$	0	0
$519$	48.0000	2.10697
$520$	0	0
$521$	7.00000	0.306676	0.153338	−	0.988174i	$-0.450998\pi$
$521$	7.00000	0.306676	0.153338	+	0.988174i	$0.450998\pi$
$522$	0	0
$523$	− 32.0000i	− 1.39926i	−0.714504	−	0.699631i	$-0.753348\pi$
$523$	− 32.0000i	− 1.39926i	0.714504	−	0.699631i	$-0.246652\pi$
$524$	36.0000	1.57267
$525$	0	0
$526$	0	0
$527$	2.00000i	0.0871214i
$528$	− 12.0000i	− 0.522233i
$529$	−2.00000	−0.0869565
$530$	0	0
$531$	18.0000	0.781133
$532$	− 12.0000i	− 0.520266i
$533$	− 8.00000i	− 0.346518i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 3.00000i	− 0.129460i
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	32.0000	1.37579	0.687894	−	0.725811i	$-0.258536\pi$
$541$	32.0000	1.37579	0.687894	+	0.725811i	$0.258536\pi$
$542$	0	0
$543$	15.0000i	0.643712i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 24.0000i	− 1.02617i	−0.858339	−	0.513083i	$-0.828503\pi$
$547$	− 24.0000i	− 1.02617i	0.858339	−	0.513083i	$-0.171497\pi$
$548$	− 6.00000i	− 0.256307i
$549$	12.0000	0.512148
$550$	0	0
$551$	−60.0000	−2.55609
$552$	0	0
$553$	6.00000i	0.255146i
$554$	0	0
$555$	0	0
$556$	20.0000	0.848189
$557$	14.0000i	0.593199i	0.955002	+	0.296600i	$0.0958526\pi$
$557$	14.0000i	0.593199i	−0.955002	+	0.296600i	$0.904147\pi$
$558$	0	0
$559$	−32.0000	−1.35346
$560$	0	0
$561$	6.00000	0.253320
$562$	0	0
$563$	− 20.0000i	− 0.842900i	−0.906852	−	0.421450i	$-0.861521\pi$
$563$	− 20.0000i	− 0.842900i	0.906852	−	0.421450i	$-0.138479\pi$
$564$	−48.0000	−2.02116
$565$	0	0
$566$	0	0
$567$	− 9.00000i	− 0.377964i
$568$	0	0
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	− 8.00000i	− 0.334497i
$573$	15.0000i	0.626634i
$574$	0	0
$575$	0	0
$576$	−48.0000	−2.00000
$577$	− 25.0000i	− 1.04076i	−0.853934	−	0.520382i	$-0.825790\pi$
$577$	− 25.0000i	− 1.04076i	0.853934	−	0.520382i	$-0.174210\pi$
$578$	0	0
$579$	42.0000	1.74546
$580$	0	0
$581$	−12.0000	−0.497844
$582$	0	0
$583$	− 6.00000i	− 0.248495i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	36.0000i	1.48588i	0.669359	+	0.742940i	$0.266569\pi$
$587$	36.0000i	1.48588i	−0.669359	+	0.742940i	$0.733431\pi$
$588$	− 6.00000i	− 0.247436i
$589$	6.00000	0.247226
$590$	0	0
$591$	−54.0000	−2.22126
$592$	− 20.0000i	− 0.821995i
$593$	− 30.0000i	− 1.23195i	−0.787765	−	0.615976i	$-0.788762\pi$
$593$	− 30.0000i	− 1.23195i	0.787765	−	0.615976i	$-0.211238\pi$
$594$	0	0
$595$	0	0
$596$	44.0000	1.80231
$597$	24.0000i	0.982255i
$598$	0	0
$599$	48.0000	1.96123	0.980613	−	0.195952i	$-0.0627798\pi$
$599$	48.0000	1.96123	0.980613	+	0.195952i	$0.0627798\pi$
$600$	0	0
$601$	8.00000	0.326327	0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000	0.326327	0.163163	+	0.986599i	$0.447830\pi$
$602$	0	0
$603$	18.0000i	0.733017i
$604$	12.0000	0.488273
$605$	0	0
$606$	0	0
$607$	− 10.0000i	− 0.405887i	−0.979190	−	0.202944i	$-0.934949\pi$
$607$	− 10.0000i	− 0.405887i	0.979190	−	0.202944i	$-0.0650509\pi$
$608$	0	0
$609$	−30.0000	−1.21566
$610$	0	0
$611$	−32.0000	−1.29458
$612$	− 24.0000i	− 0.970143i
$613$	− 16.0000i	− 0.646234i	−0.946359	−	0.323117i	$-0.895269\pi$
$613$	− 16.0000i	− 0.646234i	0.946359	−	0.323117i	$-0.104731\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 30.0000i	− 1.20775i	−0.797077	−	0.603877i	$-0.793622\pi$
$617$	− 30.0000i	− 1.20775i	0.797077	−	0.603877i	$-0.206378\pi$
$618$	0	0
$619$	−17.0000	−0.683288	−0.341644	−	0.939829i	$-0.610984\pi$
$619$	−17.0000	−0.683288	−0.341644	+	0.939829i	$0.610984\pi$
$620$	0	0
$621$	45.0000	1.80579
$622$	0	0
$623$	− 15.0000i	− 0.600962i
$624$	−48.0000	−1.92154
$625$	0	0
$626$	0	0
$627$	− 18.0000i	− 0.718851i
$628$	14.0000i	0.558661i
$629$	10.0000	0.398726
$630$	0	0
$631$	27.0000	1.07485	0.537427	−	0.843311i	$-0.319397\pi$
$631$	27.0000	1.07485	0.537427	+	0.843311i	$0.319397\pi$
$632$	0	0
$633$	− 6.00000i	− 0.238479i
$634$	0	0
$635$	0	0
$636$	−36.0000	−1.42749
$637$	− 4.00000i	− 0.158486i
$638$	0	0
$639$	−6.00000	−0.237356
$640$	0	0
$641$	15.0000	0.592464	0.296232	−	0.955116i	$-0.404270\pi$
$641$	15.0000	0.592464	0.296232	+	0.955116i	$0.404270\pi$
$642$	0	0
$643$	29.0000i	1.14365i	0.820376	+	0.571824i	$0.193764\pi$
$643$	29.0000i	1.14365i	−0.820376	+	0.571824i	$0.806236\pi$
$644$	10.0000	0.394055
$645$	0	0
$646$	0	0
$647$	− 21.0000i	− 0.825595i	−0.910823	−	0.412798i	$-0.864552\pi$
$647$	− 21.0000i	− 0.825595i	0.910823	−	0.412798i	$-0.135448\pi$
$648$	0	0
$649$	3.00000	0.117760
$650$	0	0
$651$	3.00000	0.117579
$652$	− 8.00000i	− 0.313304i
$653$	17.0000i	0.665261i	0.943057	+	0.332631i	$0.107936\pi$
$653$	17.0000i	0.665261i	−0.943057	+	0.332631i	$0.892064\pi$
$654$	0	0
$655$	0	0
$656$	−8.00000	−0.312348
$657$	60.0000i	2.34082i
$658$	0	0
$659$	2.00000	0.0779089	0.0389545	−	0.999241i	$-0.487597\pi$
$659$	2.00000	0.0779089	0.0389545	+	0.999241i	$0.487597\pi$
$660$	0	0
$661$	35.0000	1.36134	0.680671	−	0.732589i	$-0.261688\pi$
$661$	35.0000	1.36134	0.680671	+	0.732589i	$0.261688\pi$
$662$	0	0
$663$	− 24.0000i	− 0.932083i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 50.0000i	− 1.93601i
$668$	− 4.00000i	− 0.154765i
$669$	3.00000	0.115987
$670$	0	0
$671$	2.00000	0.0772091
$672$	0	0
$673$	− 4.00000i	− 0.154189i	−0.997024	−	0.0770943i	$-0.975436\pi$
$673$	− 4.00000i	− 0.154189i	0.997024	−	0.0770943i	$-0.0245643\pi$
$674$	0	0
$675$	0	0
$676$	−6.00000	−0.230769
$677$	38.0000i	1.46046i	0.683202	+	0.730229i	$0.260587\pi$
$677$	38.0000i	1.46046i	−0.683202	+	0.730229i	$0.739413\pi$
$678$	0	0
$679$	−5.00000	−0.191882
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	− 12.0000i	− 0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$	− 12.0000i	− 0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	−72.0000	−2.75299
$685$	0	0
$686$	0	0
$687$	21.0000i	0.801200i
$688$	32.0000i	1.21999i
$689$	−24.0000	−0.914327
$690$	0	0
$691$	15.0000	0.570627	0.285313	−	0.958434i	$-0.407902\pi$
$691$	15.0000	0.570627	0.285313	+	0.958434i	$0.407902\pi$
$692$	− 32.0000i	− 1.21646i
$693$	− 6.00000i	− 0.227921i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 4.00000i	− 0.151511i
$698$	0	0
$699$	18.0000	0.680823
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	− 30.0000i	− 1.13147i
$704$	−8.00000	−0.301511
$705$	0	0
$706$	0	0
$707$	12.0000i	0.451306i
$708$	− 18.0000i	− 0.676481i
$709$	−39.0000	−1.46468	−0.732338	−	0.680941i	$-0.761571\pi$
$709$	−39.0000	−1.46468	−0.732338	+	0.680941i	$0.761571\pi$
$710$	0	0
$711$	36.0000	1.35011
$712$	0	0
$713$	5.00000i	0.187251i
$714$	0	0
$715$	0	0
$716$	−2.00000	−0.0747435
$717$	− 12.0000i	− 0.448148i
$718$	0	0
$719$	11.0000	0.410231	0.205115	−	0.978738i	$-0.434243\pi$
$719$	11.0000	0.410231	0.205115	+	0.978738i	$0.434243\pi$
$720$	0	0
$721$	12.0000	0.446903
$722$	0	0
$723$	− 36.0000i	− 1.33885i
$724$	10.0000	0.371647
$725$	0	0
$726$	0	0
$727$	− 19.0000i	− 0.704671i	−0.935874	−	0.352335i	$-0.885388\pi$
$727$	− 19.0000i	− 0.704671i	0.935874	−	0.352335i	$-0.114612\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	−16.0000	−0.591781
$732$	− 12.0000i	− 0.443533i
$733$	4.00000i	0.147743i	0.997268	+	0.0738717i	$0.0235355\pi$
$733$	4.00000i	0.147743i	−0.997268	+	0.0738717i	$0.976464\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.00000i	0.110506i
$738$	0	0
$739$	18.0000	0.662141	0.331070	−	0.943606i	$-0.392590\pi$
$739$	18.0000	0.662141	0.331070	+	0.943606i	$0.392590\pi$
$740$	0	0
$741$	−72.0000	−2.64499
$742$	0	0
$743$	24.0000i	0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$	24.0000i	0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	72.0000i	2.63434i
$748$	− 4.00000i	− 0.146254i
$749$	−10.0000	−0.365392
$750$	0	0
$751$	−23.0000	−0.839282	−0.419641	−	0.907690i	$-0.637844\pi$
$751$	−23.0000	−0.839282	−0.419641	+	0.907690i	$0.637844\pi$
$752$	32.0000i	1.16692i
$753$	− 63.0000i	− 2.29585i
$754$	0	0
$755$	0	0
$756$	−18.0000	−0.654654
$757$	38.0000i	1.38113i	0.723269	+	0.690567i	$0.242639\pi$
$757$	38.0000i	1.38113i	−0.723269	+	0.690567i	$0.757361\pi$
$758$	0	0
$759$	15.0000	0.544466
$760$	0	0
$761$	−48.0000	−1.74000	−0.869999	−	0.493053i	$-0.835881\pi$
$761$	−48.0000	−1.74000	−0.869999	+	0.493053i	$0.835881\pi$
$762$	0	0
$763$	4.00000i	0.144810i
$764$	10.0000	0.361787
$765$	0	0
$766$	0	0
$767$	− 12.0000i	− 0.433295i
$768$	48.0000i	1.73205i
$769$	−40.0000	−1.44244	−0.721218	−	0.692708i	$-0.756418\pi$
$769$	−40.0000	−1.44244	−0.721218	+	0.692708i	$0.756418\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	− 28.0000i	− 1.00774i
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 15.0000i	− 0.538122i
$778$	0	0
$779$	−12.0000	−0.429945
$780$	0	0
$781$	−1.00000	−0.0357828
$782$	0	0
$783$	90.0000i	3.21634i
$784$	−4.00000	−0.142857
$785$	0	0
$786$	0	0
$787$	− 22.0000i	− 0.784215i	−0.919919	−	0.392108i	$-0.871746\pi$
$787$	− 22.0000i	− 0.784215i	0.919919	−	0.392108i	$-0.128254\pi$
$788$	36.0000i	1.28245i
$789$	54.0000	1.92245
$790$	0	0
$791$	19.0000	0.675562
$792$	0	0
$793$	− 8.00000i	− 0.284088i
$794$	0	0
$795$	0	0
$796$	16.0000	0.567105
$797$	23.0000i	0.814702i	0.913272	+	0.407351i	$0.133547\pi$
$797$	23.0000i	0.814702i	−0.913272	+	0.407351i	$0.866453\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	0	0
$801$	−90.0000	−3.17999
$802$	0	0
$803$	10.0000i	0.352892i
$804$	18.0000	0.634811
$805$	0	0
$806$	0	0
$807$	54.0000i	1.90089i
$808$	0	0
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	−22.0000	−0.772524	−0.386262	−	0.922389i	$-0.626234\pi$
$811$	−22.0000	−0.772524	−0.386262	+	0.922389i	$0.626234\pi$
$812$	20.0000i	0.701862i
$813$	48.0000i	1.68343i
$814$	0	0
$815$	0	0
$816$	−24.0000	−0.840168
$817$	48.0000i	1.67931i
$818$	0	0
$819$	−24.0000	−0.838628
$820$	0	0
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	25.0000i	0.871445i	0.900081	+	0.435723i	$0.143507\pi$
$823$	25.0000i	0.871445i	−0.900081	+	0.435723i	$0.856493\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	20.0000i	0.695468i	0.937593	+	0.347734i	$0.113049\pi$
$827$	20.0000i	0.695468i	−0.937593	+	0.347734i	$0.886951\pi$
$828$	− 60.0000i	− 2.08514i
$829$	29.0000	1.00721	0.503606	−	0.863934i	$-0.332006\pi$
$829$	29.0000	1.00721	0.503606	+	0.863934i	$0.332006\pi$
$830$	0	0
$831$	−72.0000	−2.49765
$832$	32.0000i	1.10940i
$833$	− 2.00000i	− 0.0692959i
$834$	0	0
$835$	0	0
$836$	−12.0000	−0.415029
$837$	− 9.00000i	− 0.311086i
$838$	0	0
$839$	−45.0000	−1.55357	−0.776786	−	0.629764i	$-0.783151\pi$
$839$	−45.0000	−1.55357	−0.776786	+	0.629764i	$0.783151\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	0	0
$843$	− 12.0000i	− 0.413302i
$844$	−4.00000	−0.137686
$845$	0	0
$846$	0	0
$847$	− 1.00000i	− 0.0343604i
$848$	24.0000i	0.824163i
$849$	0	0
$850$	0	0
$851$	25.0000	0.856989
$852$	6.00000i	0.205557i
$853$	34.0000i	1.16414i	0.813139	+	0.582069i	$0.197757\pi$
$853$	34.0000i	1.16414i	−0.813139	+	0.582069i	$0.802243\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 28.0000i	− 0.956462i	−0.878234	−	0.478231i	$-0.841278\pi$
$857$	− 28.0000i	− 0.956462i	0.878234	−	0.478231i	$-0.158722\pi$
$858$	0	0
$859$	−55.0000	−1.87658	−0.938288	−	0.345855i	$-0.887589\pi$
$859$	−55.0000	−1.87658	−0.938288	+	0.345855i	$0.887589\pi$
$860$	0	0
$861$	−6.00000	−0.204479
$862$	0	0
$863$	− 52.0000i	− 1.77010i	−0.465495	−	0.885050i	$-0.654124\pi$
$863$	− 52.0000i	− 1.77010i	0.465495	−	0.885050i	$-0.345876\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	39.0000i	1.32451i
$868$	− 2.00000i	− 0.0678844i
$869$	6.00000	0.203536
$870$	0	0
$871$	12.0000	0.406604
$872$	0	0
$873$	30.0000i	1.01535i
$874$	0	0
$875$	0	0
$876$	60.0000	2.02721
$877$	− 38.0000i	− 1.28317i	−0.767052	−	0.641584i	$-0.778277\pi$
$877$	− 38.0000i	− 1.28317i	0.767052	−	0.641584i	$-0.221723\pi$
$878$	0	0
$879$	−18.0000	−0.607125
$880$	0	0
$881$	27.0000	0.909653	0.454827	−	0.890580i	$-0.349701\pi$
$881$	27.0000	0.909653	0.454827	+	0.890580i	$0.349701\pi$
$882$	0	0
$883$	44.0000i	1.48072i	0.672212	+	0.740359i	$0.265344\pi$
$883$	44.0000i	1.48072i	−0.672212	+	0.740359i	$0.734656\pi$
$884$	−16.0000	−0.538138
$885$	0	0
$886$	0	0
$887$	− 2.00000i	− 0.0671534i	−0.999436	−	0.0335767i	$-0.989310\pi$
$887$	− 2.00000i	− 0.0671534i	0.999436	−	0.0335767i	$-0.0106898\pi$
$888$	0	0
$889$	2.00000	0.0670778
$890$	0	0
$891$	−9.00000	−0.301511
$892$	− 2.00000i	− 0.0669650i
$893$	48.0000i	1.60626i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 60.0000i	− 2.00334i
$898$	0	0
$899$	−10.0000	−0.333519
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	24.0000i	0.798670i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 40.0000i	− 1.32818i	−0.747653	−	0.664089i	$-0.768820\pi$
$907$	− 40.0000i	− 1.32818i	0.747653	−	0.664089i	$-0.231180\pi$
$908$	8.00000i	0.265489i
$909$	72.0000	2.38809
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	72.0000i	2.38416i
$913$	12.0000i	0.397142i
$914$	0	0
$915$	0	0
$916$	14.0000	0.462573
$917$	− 18.0000i	− 0.594412i
$918$	0	0
$919$	−48.0000	−1.58337	−0.791687	−	0.610927i	$-0.790797\pi$
$919$	−48.0000	−1.58337	−0.791687	+	0.610927i	$0.790797\pi$
$920$	0	0
$921$	84.0000	2.76789
$922$	0	0
$923$	4.00000i	0.131662i
$924$	−6.00000	−0.197386
$925$	0	0
$926$	0	0
$927$	− 72.0000i	− 2.36479i
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	− 12.0000i	− 0.393073i
$933$	24.0000i	0.785725i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	36.0000i	1.17607i	0.808836	+	0.588034i	$0.200098\pi$
$937$	36.0000i	1.17607i	−0.808836	+	0.588034i	$0.799902\pi$
$938$	0	0
$939$	−69.0000	−2.25173
$940$	0	0
$941$	58.0000	1.89075	0.945373	−	0.325991i	$-0.105698\pi$
$941$	58.0000	1.89075	0.945373	+	0.325991i	$0.105698\pi$
$942$	0	0
$943$	− 10.0000i	− 0.325645i
$944$	−12.0000	−0.390567
$945$	0	0
$946$	0	0
$947$	5.00000i	0.162478i	0.996695	+	0.0812391i	$0.0258877\pi$
$947$	5.00000i	0.162478i	−0.996695	+	0.0812391i	$0.974112\pi$
$948$	− 36.0000i	− 1.16923i
$949$	40.0000	1.29845
$950$	0	0
$951$	−27.0000	−0.875535
$952$	0	0
$953$	− 44.0000i	− 1.42530i	−0.701520	−	0.712650i	$-0.747495\pi$
$953$	− 44.0000i	− 1.42530i	0.701520	−	0.712650i	$-0.252505\pi$
$954$	0	0
$955$	0	0
$956$	−8.00000	−0.258738
$957$	30.0000i	0.969762i
$958$	0	0
$959$	−3.00000	−0.0968751
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	60.0000i	1.93347i
$964$	−24.0000	−0.772988
$965$	0	0
$966$	0	0
$967$	− 34.0000i	− 1.09337i	−0.837340	−	0.546683i	$-0.815890\pi$
$967$	− 34.0000i	− 1.09337i	0.837340	−	0.546683i	$-0.184110\pi$
$968$	0	0
$969$	−36.0000	−1.15649
$970$	0	0
$971$	29.0000	0.930654	0.465327	−	0.885139i	$-0.345937\pi$
$971$	29.0000	0.930654	0.465327	+	0.885139i	$0.345937\pi$
$972$	0	0
$973$	− 10.0000i	− 0.320585i
$974$	0	0
$975$	0	0
$976$	−8.00000	−0.256074
$977$	− 31.0000i	− 0.991778i	−0.868386	−	0.495889i	$-0.834842\pi$
$977$	− 31.0000i	− 0.991778i	0.868386	−	0.495889i	$-0.165158\pi$
$978$	0	0
$979$	−15.0000	−0.479402
$980$	0	0
$981$	24.0000	0.766261
$982$	0	0
$983$	27.0000i	0.861166i	0.902551	+	0.430583i	$0.141692\pi$
$983$	27.0000i	0.861166i	−0.902551	+	0.430583i	$0.858308\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	24.0000i	0.763928i
$988$	48.0000i	1.52708i
$989$	−40.0000	−1.27193
$990$	0	0
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	0	0
$993$	− 51.0000i	− 1.61844i
$994$	0	0
$995$	0	0
$996$	72.0000	2.28141
$997$	− 12.0000i	− 0.380044i	−0.981780	−	0.190022i	$-0.939144\pi$
$997$	− 12.0000i	− 0.380044i	0.981780	−	0.190022i	$-0.0608559\pi$
$998$	0	0
$999$	−45.0000	−1.42374

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1925.2.b.e.1849.2		2
5.2	odd	4		1925.2.a.h.1.1		1
5.3	odd	4		77.2.a.a.1.1	✓	1
5.4	even	2	inner	1925.2.b.e.1849.1		2
15.8	even	4		693.2.a.c.1.1		1
20.3	even	4		1232.2.a.l.1.1		1
35.3	even	12		539.2.e.c.177.1		2
35.13	even	4		539.2.a.c.1.1		1
35.18	odd	12		539.2.e.f.177.1		2
35.23	odd	12		539.2.e.f.67.1		2
35.33	even	12		539.2.e.c.67.1		2
40.3	even	4		4928.2.a.a.1.1		1
40.13	odd	4		4928.2.a.bj.1.1		1
55.3	odd	20		847.2.f.i.372.1		4
55.8	even	20		847.2.f.h.372.1		4
55.13	even	20		847.2.f.h.323.1		4
55.18	even	20		847.2.f.h.148.1		4
55.28	even	20		847.2.f.h.729.1		4
55.38	odd	20		847.2.f.i.729.1		4
55.43	even	4		847.2.a.b.1.1		1
55.48	odd	20		847.2.f.i.148.1		4
55.53	odd	20		847.2.f.i.323.1		4
105.83	odd	4		4851.2.a.j.1.1		1
140.83	odd	4		8624.2.a.a.1.1		1
165.98	odd	4		7623.2.a.j.1.1		1
385.153	odd	4		5929.2.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.a.a.1.1	✓	1	5.3	odd	4
539.2.a.c.1.1		1	35.13	even	4
539.2.e.c.67.1		2	35.33	even	12
539.2.e.c.177.1		2	35.3	even	12
539.2.e.f.67.1		2	35.23	odd	12
539.2.e.f.177.1		2	35.18	odd	12
693.2.a.c.1.1		1	15.8	even	4
847.2.a.b.1.1		1	55.43	even	4
847.2.f.h.148.1		4	55.18	even	20
847.2.f.h.323.1		4	55.13	even	20
847.2.f.h.372.1		4	55.8	even	20
847.2.f.h.729.1		4	55.28	even	20
847.2.f.i.148.1		4	55.48	odd	20
847.2.f.i.323.1		4	55.53	odd	20
847.2.f.i.372.1		4	55.3	odd	20
847.2.f.i.729.1		4	55.38	odd	20
1232.2.a.l.1.1		1	20.3	even	4
1925.2.a.h.1.1		1	5.2	odd	4
1925.2.b.e.1849.1		2	5.4	even	2	inner
1925.2.b.e.1849.2		2	1.1	even	1	trivial
4851.2.a.j.1.1		1	105.83	odd	4
4928.2.a.a.1.1		1	40.3	even	4
4928.2.a.bj.1.1		1	40.13	odd	4
5929.2.a.f.1.1		1	385.153	odd	4
7623.2.a.j.1.1		1	165.98	odd	4
8624.2.a.a.1.1		1	140.83	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1925.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} +2.00000 q^{4} -1.00000i q^{7} -6.00000 q^{9} -1.00000 q^{11} +6.00000i q^{12} +4.00000i q^{13} +4.00000 q^{16} +2.00000i q^{17} +6.00000 q^{19} +3.00000 q^{21} +5.00000i q^{23} -9.00000i q^{27} -2.00000i q^{28} -10.0000 q^{29} +1.00000 q^{31} -3.00000i q^{33} -12.0000 q^{36} -5.00000i q^{37} -12.0000 q^{39} -2.00000 q^{41} +8.00000i q^{43} -2.00000 q^{44} +8.00000i q^{47} +12.0000i q^{48} -1.00000 q^{49} -6.00000 q^{51} +8.00000i q^{52} +6.00000i q^{53} +18.0000i q^{57} -3.00000 q^{59} -2.00000 q^{61} +6.00000i q^{63} +8.00000 q^{64} -3.00000i q^{67} +4.00000i q^{68} -15.0000 q^{69} +1.00000 q^{71} -10.0000i q^{73} +12.0000 q^{76} +1.00000i q^{77} -6.00000 q^{79} +9.00000 q^{81} -12.0000i q^{83} +6.00000 q^{84} -30.0000i q^{87} +15.0000 q^{89} +4.00000 q^{91} +10.0000i q^{92} +3.00000i q^{93} -5.00000i q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{4} - 12 q^{9} - 2 q^{11} + 8 q^{16} + 12 q^{19} + 6 q^{21} - 20 q^{29} + 2 q^{31} - 24 q^{36} - 24 q^{39} - 4 q^{41} - 4 q^{44} - 2 q^{49} - 12 q^{51} - 6 q^{59} - 4 q^{61} + 16 q^{64} - 30 q^{69}+ \cdots + 12 q^{99}+O(q^{100}) \) 2 * q + 4 * q^4 - 12 * q^9 - 2 * q^11 + 8 * q^16 + 12 * q^19 + 6 * q^21 - 20 * q^29 + 2 * q^31 - 24 * q^36 - 24 * q^39 - 4 * q^41 - 4 * q^44 - 2 * q^49 - 12 * q^51 - 6 * q^59 - 4 * q^61 + 16 * q^64 - 30 * q^69 + 2 * q^71 + 24 * q^76 - 12 * q^79 + 18 * q^81 + 12 * q^84 + 30 * q^89 + 8 * q^91 + 12 * q^99

Introduction

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