LMFDB - Embedded newform 1925.2.b.g.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:	$\Z[a_1, a_2, a_3]$
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.g.1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +2.00000 q^{4} -1.00000i q^{7} +2.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +2.00000 q^{4} -1.00000i q^{7} +2.00000 q^{9} -1.00000 q^{11} +2.00000i q^{12} -4.00000i q^{13} +4.00000 q^{16} +6.00000i q^{17} -2.00000 q^{19} +1.00000 q^{21} +3.00000i q^{23} +5.00000i q^{27} -2.00000i q^{28} +6.00000 q^{29} +5.00000 q^{31} -1.00000i q^{33} +4.00000 q^{36} -11.0000i q^{37} +4.00000 q^{39} +6.00000 q^{41} +8.00000i q^{43} -2.00000 q^{44} +4.00000i q^{48} -1.00000 q^{49} -6.00000 q^{51} -8.00000i q^{52} -6.00000i q^{53} -2.00000i q^{57} +9.00000 q^{59} -10.0000 q^{61} -2.00000i q^{63} +8.00000 q^{64} -5.00000i q^{67} +12.0000i q^{68} -3.00000 q^{69} +9.00000 q^{71} +2.00000i q^{73} -4.00000 q^{76} +1.00000i q^{77} +10.0000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +2.00000 q^{84} +6.00000i q^{87} +3.00000 q^{89} -4.00000 q^{91} +6.00000i q^{92} +5.00000i q^{93} +1.00000i q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{4} + 4 q^{9}+O(q^{10}) $ 2 * q + 4 * q^4 + 4 * q^9	$ 2 q + 4 q^{4} + 4 q^{9} - 2 q^{11} + 8 q^{16} - 4 q^{19} + 2 q^{21} + 12 q^{29} + 10 q^{31} + 8 q^{36} + 8 q^{39} + 12 q^{41} - 4 q^{44} - 2 q^{49} - 12 q^{51} + 18 q^{59} - 20 q^{61} + 16 q^{64} - 6 q^{69} + 18 q^{71} - 8 q^{76} + 20 q^{79} + 2 q^{81} + 4 q^{84} + 6 q^{89} - 8 q^{91} - 4 q^{99}+O(q^{100}) $ 2 * q + 4 * q^4 + 4 * q^9 - 2 * q^11 + 8 * q^16 - 4 * q^19 + 2 * q^21 + 12 * q^29 + 10 * q^31 + 8 * q^36 + 8 * q^39 + 12 * q^41 - 4 * q^44 - 2 * q^49 - 12 * q^51 + 18 * q^59 - 20 * q^61 + 16 * q^64 - 6 * q^69 + 18 * q^71 - 8 * q^76 + 20 * q^79 + 2 * q^81 + 4 * q^84 + 6 * q^89 - 8 * q^91 - 4 * q^99

Display 100 coefficients 10 coefficients

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			$\pi$
$3$	1.00000i	0.577350i	0.957427	+	0.288675i	$0.0932147\pi$
$3$	1.00000i	0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$	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
$9$	2.00000	0.666667
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	2.00000i	0.577350i
$13$	− 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
$16$	4.00000	1.00000
$17$	6.00000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	6.00000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	3.00000i	0.625543i	0.949828	+	0.312772i	$0.101257\pi$
$23$	3.00000i	0.625543i	−0.949828	+	0.312772i	$0.898743\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.00000i	0.962250i
$28$	− 2.00000i	− 0.377964i
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	0	0
$33$	− 1.00000i	− 0.174078i
$34$	0	0
$35$	0	0
$36$	4.00000	0.666667
$37$	− 11.0000i	− 1.80839i	−0.427121	−	0.904194i	$-0.640472\pi$
$37$	− 11.0000i	− 1.80839i	0.427121	−	0.904194i	$-0.359528\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	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
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	4.00000i	0.577350i
$49$	−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	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 2.00000i	− 0.264906i
$58$	0	0
$59$	9.00000	1.17170	0.585850	−	0.810419i	$-0.300761\pi$
$59$	9.00000	1.17170	0.585850	+	0.810419i	$0.300761\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	− 2.00000i	− 0.251976i
$64$	8.00000	1.00000
$65$	0	0
$66$	0	0
$67$	− 5.00000i	− 0.610847i	−0.952217	−	0.305424i	$-0.901202\pi$
$67$	− 5.00000i	− 0.610847i	0.952217	−	0.305424i	$-0.0987981\pi$
$68$	12.0000i	1.45521i
$69$	−3.00000	−0.361158
$70$	0	0
$71$	9.00000	1.06810	0.534052	−	0.845452i	$-0.320669\pi$
$71$	9.00000	1.06810	0.534052	+	0.845452i	$0.320669\pi$
$72$	0	0
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	0	0
$75$	0	0
$76$	−4.00000	−0.458831
$77$	1.00000i	0.113961i
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	0	0
$87$	6.00000i	0.643268i
$88$	0	0
$89$	3.00000	0.317999	0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000	0.317999	0.159000	+	0.987279i	$0.449173\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	6.00000i	0.625543i
$93$	5.00000i	0.518476i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1.00000i	0.101535i	0.998711	+	0.0507673i	$0.0161667\pi$
$97$	1.00000i	0.101535i	−0.998711	+	0.0507673i	$0.983833\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	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
$103$	− 4.00000i	− 0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$	− 4.00000i	− 0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$	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	$-0.0936623\pi$
$108$	10.0000i	0.962250i
$109$	−20.0000	−1.91565	−0.957826	−	0.287348i	$-0.907226\pi$
$109$	−20.0000	−1.91565	−0.957826	+	0.287348i	$0.907226\pi$
$110$	0	0
$111$	11.0000	1.04407
$112$	− 4.00000i	− 0.377964i
$113$	− 3.00000i	− 0.282216i	−0.989994	−	0.141108i	$-0.954933\pi$
$113$	− 3.00000i	− 0.282216i	0.989994	−	0.141108i	$-0.0450665\pi$
$114$	0	0
$115$	0	0
$116$	12.0000	1.11417
$117$	− 8.00000i	− 0.739600i
$118$	0	0
$119$	6.00000	0.550019
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	6.00000i	0.541002i
$124$	10.0000	0.898027
$125$	0	0
$126$	0	0
$127$	− 2.00000i	− 0.177471i	−0.996055	−	0.0887357i	$-0.971717\pi$
$127$	− 2.00000i	− 0.177471i	0.996055	−	0.0887357i	$-0.0282826\pi$
$128$	0	0
$129$	−8.00000	−0.704361
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	− 2.00000i	− 0.174078i
$133$	2.00000i	0.173422i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.00000i	0.256307i	0.991754	+	0.128154i	$0.0409051\pi$
$137$	3.00000i	0.256307i	−0.991754	+	0.128154i	$0.959095\pi$
$138$	0	0
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.00000i	0.334497i
$144$	8.00000	0.666667
$145$	0	0
$146$	0	0
$147$	− 1.00000i	− 0.0824786i
$148$	− 22.0000i	− 1.80839i
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	0	0
$153$	12.0000i	0.970143i
$154$	0	0
$155$	0	0
$156$	8.00000	0.640513
$157$	13.0000i	1.03751i	0.854922	+	0.518756i	$0.173605\pi$
$157$	13.0000i	1.03751i	−0.854922	+	0.518756i	$0.826395\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	3.00000	0.236433
$162$	0	0
$163$	20.0000i	1.56652i	0.621694	+	0.783260i	$0.286445\pi$
$163$	20.0000i	1.56652i	−0.621694	+	0.783260i	$0.713555\pi$
$164$	12.0000	0.937043
$165$	0	0
$166$	0	0
$167$	− 6.00000i	− 0.464294i	−0.972681	−	0.232147i	$-0.925425\pi$
$167$	− 6.00000i	− 0.464294i	0.972681	−	0.232147i	$-0.0745750\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	−4.00000	−0.305888
$172$	16.0000i	1.21999i
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	−4.00000	−0.301511
$177$	9.00000i	0.676481i
$178$	0	0
$179$	15.0000	1.12115	0.560576	−	0.828103i	$-0.310580\pi$
$179$	15.0000	1.12115	0.560576	+	0.828103i	$0.310580\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	0	0
$183$	− 10.0000i	− 0.739221i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 6.00000i	− 0.438763i
$188$	0	0
$189$	5.00000	0.363696
$190$	0	0
$191$	−27.0000	−1.95365	−0.976826	−	0.214036i	$-0.931339\pi$
$191$	−27.0000	−1.95365	−0.976826	+	0.214036i	$0.931339\pi$
$192$	8.00000i	0.577350i
$193$	14.0000i	1.00774i	0.863779	+	0.503871i	$0.168091\pi$
$193$	14.0000i	1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$	0	0
$195$	0	0
$196$	−2.00000	−0.142857
$197$	− 18.0000i	− 1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$	− 18.0000i	− 1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	5.00000	0.352673
$202$	0	0
$203$	− 6.00000i	− 0.421117i
$204$	−12.0000	−0.840168
$205$	0	0
$206$	0	0
$207$	6.00000i	0.417029i
$208$	− 16.0000i	− 1.10940i
$209$	2.00000	0.138343
$210$	0	0
$211$	14.0000	0.963800	0.481900	−	0.876226i	$-0.339947\pi$
$211$	14.0000	0.963800	0.481900	+	0.876226i	$0.339947\pi$
$212$	− 12.0000i	− 0.824163i
$213$	9.00000i	0.616670i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 5.00000i	− 0.339422i
$218$	0	0
$219$	−2.00000	−0.135147
$220$	0	0
$221$	24.0000	1.61441
$222$	0	0
$223$	− 19.0000i	− 1.27233i	−0.771551	−	0.636167i	$-0.780519\pi$
$223$	− 19.0000i	− 1.27233i	0.771551	−	0.636167i	$-0.219481\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	− 4.00000i	− 0.264906i
$229$	−5.00000	−0.330409	−0.165205	−	0.986259i	$-0.552828\pi$
$229$	−5.00000	−0.330409	−0.165205	+	0.986259i	$0.552828\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	6.00000i	0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$	6.00000i	0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	0	0
$235$	0	0
$236$	18.0000	1.17170
$237$	10.0000i	0.649570i
$238$	0	0
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	−28.0000	−1.80364	−0.901819	−	0.432113i	$-0.857768\pi$
$241$	−28.0000	−1.80364	−0.901819	+	0.432113i	$0.857768\pi$
$242$	0	0
$243$	16.0000i	1.02640i
$244$	−20.0000	−1.28037
$245$	0	0
$246$	0	0
$247$	8.00000i	0.509028i
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	−9.00000	−0.568075	−0.284037	−	0.958813i	$-0.591674\pi$
$251$	−9.00000	−0.568075	−0.284037	+	0.958813i	$0.591674\pi$
$252$	− 4.00000i	− 0.251976i
$253$	− 3.00000i	− 0.188608i
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	6.00000i	0.374270i	0.982334	+	0.187135i	$0.0599201\pi$
$257$	6.00000i	0.374270i	−0.982334	+	0.187135i	$0.940080\pi$
$258$	0	0
$259$	−11.0000	−0.683507
$260$	0	0
$261$	12.0000	0.742781
$262$	0	0
$263$	− 30.0000i	− 1.84988i	−0.380114	−	0.924940i	$-0.624115\pi$
$263$	− 30.0000i	− 1.84988i	0.380114	−	0.924940i	$-0.375885\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3.00000i	0.183597i
$268$	− 10.0000i	− 0.610847i
$269$	−30.0000	−1.82913	−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000	−1.82913	−0.914566	+	0.404436i	$0.867468\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	24.0000i	1.45521i
$273$	− 4.00000i	− 0.242091i
$274$	0	0
$275$	0	0
$276$	−6.00000	−0.361158
$277$	− 8.00000i	− 0.480673i	−0.970690	−	0.240337i	$-0.922742\pi$
$277$	− 8.00000i	− 0.480673i	0.970690	−	0.240337i	$-0.0772579\pi$
$278$	0	0
$279$	10.0000	0.598684
$280$	0	0
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	0	0
$283$	32.0000i	1.90220i	0.308879	+	0.951101i	$0.400046\pi$
$283$	32.0000i	1.90220i	−0.308879	+	0.951101i	$0.599954\pi$
$284$	18.0000	1.06810
$285$	0	0
$286$	0	0
$287$	− 6.00000i	− 0.354169i
$288$	0	0
$289$	−19.0000	−1.11765
$290$	0	0
$291$	−1.00000	−0.0586210
$292$	4.00000i	0.234082i
$293$	− 30.0000i	− 1.75262i	−0.481749	−	0.876309i	$-0.659998\pi$
$293$	− 30.0000i	− 1.75262i	0.481749	−	0.876309i	$-0.340002\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 5.00000i	− 0.290129i
$298$	0	0
$299$	12.0000	0.693978
$300$	0	0
$301$	8.00000	0.461112
$302$	0	0
$303$	− 12.0000i	− 0.689382i
$304$	−8.00000	−0.458831
$305$	0	0
$306$	0	0
$307$	− 20.0000i	− 1.14146i	−0.821138	−	0.570730i	$-0.806660\pi$
$307$	− 20.0000i	− 1.14146i	0.821138	−	0.570730i	$-0.193340\pi$
$308$	2.00000i	0.113961i
$309$	4.00000	0.227552
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	− 19.0000i	− 1.07394i	−0.843600	−	0.536972i	$-0.819568\pi$
$313$	− 19.0000i	− 1.07394i	0.843600	−	0.536972i	$-0.180432\pi$
$314$	0	0
$315$	0	0
$316$	20.0000	1.12509
$317$	− 9.00000i	− 0.505490i	−0.967533	−	0.252745i	$-0.918667\pi$
$317$	− 9.00000i	− 0.505490i	0.967533	−	0.252745i	$-0.0813334\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	0	0
$321$	6.00000	0.334887
$322$	0	0
$323$	− 12.0000i	− 0.667698i
$324$	2.00000	0.111111
$325$	0	0
$326$	0	0
$327$	− 20.0000i	− 1.10600i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−1.00000	−0.0549650	−0.0274825	−	0.999622i	$-0.508749\pi$
$331$	−1.00000	−0.0549650	−0.0274825	+	0.999622i	$0.508749\pi$
$332$	24.0000i	1.31717i
$333$	− 22.0000i	− 1.20559i
$334$	0	0
$335$	0	0
$336$	4.00000	0.218218
$337$	− 14.0000i	− 0.762629i	−0.924445	−	0.381314i	$-0.875472\pi$
$337$	− 14.0000i	− 0.762629i	0.924445	−	0.381314i	$-0.124528\pi$
$338$	0	0
$339$	3.00000	0.162938
$340$	0	0
$341$	−5.00000	−0.270765
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	18.0000i	0.966291i	0.875540	+	0.483145i	$0.160506\pi$
$347$	18.0000i	0.966291i	−0.875540	+	0.483145i	$0.839494\pi$
$348$	12.0000i	0.643268i
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	20.0000	1.06752
$352$	0	0
$353$	− 3.00000i	− 0.159674i	−0.996808	−	0.0798369i	$-0.974560\pi$
$353$	− 3.00000i	− 0.159674i	0.996808	−	0.0798369i	$-0.0254400\pi$
$354$	0	0
$355$	0	0
$356$	6.00000	0.317999
$357$	6.00000i	0.317554i
$358$	0	0
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	1.00000i	0.0524864i
$364$	−8.00000	−0.419314
$365$	0	0
$366$	0	0
$367$	− 17.0000i	− 0.887393i	−0.896177	−	0.443696i	$-0.853667\pi$
$367$	− 17.0000i	− 0.887393i	0.896177	−	0.443696i	$-0.146333\pi$
$368$	12.0000i	0.625543i
$369$	12.0000	0.624695
$370$	0	0
$371$	−6.00000	−0.311504
$372$	10.0000i	0.518476i
$373$	− 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
$377$	− 24.0000i	− 1.23606i
$378$	0	0
$379$	−11.0000	−0.565032	−0.282516	−	0.959263i	$-0.591169\pi$
$379$	−11.0000	−0.565032	−0.282516	+	0.959263i	$0.591169\pi$
$380$	0	0
$381$	2.00000	0.102463
$382$	0	0
$383$	21.0000i	1.07305i	0.843884	+	0.536525i	$0.180263\pi$
$383$	21.0000i	1.07305i	−0.843884	+	0.536525i	$0.819737\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	16.0000i	0.813326i
$388$	2.00000i	0.101535i
$389$	−33.0000	−1.67317	−0.836583	−	0.547840i	$-0.815450\pi$
$389$	−33.0000	−1.67317	−0.836583	+	0.547840i	$0.815450\pi$
$390$	0	0
$391$	−18.0000	−0.910299
$392$	0	0
$393$	− 6.00000i	− 0.302660i
$394$	0	0
$395$	0	0
$396$	−4.00000	−0.201008
$397$	− 2.00000i	− 0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	− 2.00000i	− 0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$	0	0
$399$	−2.00000	−0.100125
$400$	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
$403$	− 20.0000i	− 0.996271i
$404$	−24.0000	−1.19404
$405$	0	0
$406$	0	0
$407$	11.0000i	0.545250i
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	−3.00000	−0.147979
$412$	− 8.00000i	− 0.394132i
$413$	− 9.00000i	− 0.442861i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 14.0000i	− 0.685583i
$418$	0	0
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	10.0000i	0.483934i
$428$	− 12.0000i	− 0.580042i
$429$	−4.00000	−0.193122
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	20.0000i	0.962250i
$433$	11.0000i	0.528626i	0.964437	+	0.264313i	$0.0851452\pi$
$433$	11.0000i	0.528626i	−0.964437	+	0.264313i	$0.914855\pi$
$434$	0	0
$435$	0	0
$436$	−40.0000	−1.91565
$437$	− 6.00000i	− 0.287019i
$438$	0	0
$439$	−26.0000	−1.24091	−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000	−1.24091	−0.620456	+	0.784241i	$0.713053\pi$
$440$	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	$0.931415\pi$
$444$	22.0000	1.04407
$445$	0	0
$446$	0	0
$447$	6.00000i	0.283790i
$448$	− 8.00000i	− 0.377964i
$449$	9.00000	0.424736	0.212368	−	0.977190i	$-0.431882\pi$
$449$	9.00000	0.424736	0.212368	+	0.977190i	$0.431882\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	− 6.00000i	− 0.282216i
$453$	− 10.0000i	− 0.469841i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 8.00000i	− 0.374224i	−0.982339	−	0.187112i	$-0.940087\pi$
$457$	− 8.00000i	− 0.374224i	0.982339	−	0.187112i	$-0.0599128\pi$
$458$	0	0
$459$	−30.0000	−1.40028
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	5.00000i	0.232370i	0.993228	+	0.116185i	$0.0370665\pi$
$463$	5.00000i	0.232370i	−0.993228	+	0.116185i	$0.962933\pi$
$464$	24.0000	1.11417
$465$	0	0
$466$	0	0
$467$	− 15.0000i	− 0.694117i	−0.937843	−	0.347059i	$-0.887180\pi$
$467$	− 15.0000i	− 0.694117i	0.937843	−	0.347059i	$-0.112820\pi$
$468$	− 16.0000i	− 0.739600i
$469$	−5.00000	−0.230879
$470$	0	0
$471$	−13.0000	−0.599008
$472$	0	0
$473$	− 8.00000i	− 0.367840i
$474$	0	0
$475$	0	0
$476$	12.0000	0.550019
$477$	− 12.0000i	− 0.549442i
$478$	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
$481$	−44.0000	−2.00623
$482$	0	0
$483$	3.00000i	0.136505i
$484$	2.00000	0.0909091
$485$	0	0
$486$	0	0
$487$	− 11.0000i	− 0.498458i	−0.968445	−	0.249229i	$-0.919823\pi$
$487$	− 11.0000i	− 0.498458i	0.968445	−	0.249229i	$-0.0801771\pi$
$488$	0	0
$489$	−20.0000	−0.904431
$490$	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
$493$	36.0000i	1.62136i
$494$	0	0
$495$	0	0
$496$	20.0000	0.898027
$497$	− 9.00000i	− 0.403705i
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	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
$507$	− 3.00000i	− 0.133235i
$508$	− 4.00000i	− 0.177471i
$509$	−21.0000	−0.930809	−0.465404	−	0.885098i	$-0.654091\pi$
$509$	−21.0000	−0.930809	−0.465404	+	0.885098i	$0.654091\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	0	0
$513$	− 10.0000i	− 0.441511i
$514$	0	0
$515$	0	0
$516$	−16.0000	−0.704361
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	3.00000	0.131432	0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000	0.131432	0.0657162	+	0.997838i	$0.479067\pi$
$522$	0	0
$523$	− 16.0000i	− 0.699631i	−0.936819	−	0.349816i	$-0.886244\pi$
$523$	− 16.0000i	− 0.699631i	0.936819	−	0.349816i	$-0.113756\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	0	0
$527$	30.0000i	1.30682i
$528$	− 4.00000i	− 0.174078i
$529$	14.0000	0.608696
$530$	0	0
$531$	18.0000	0.781133
$532$	4.00000i	0.173422i
$533$	− 24.0000i	− 1.03956i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	15.0000i	0.647298i
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−16.0000	−0.687894	−0.343947	−	0.938989i	$-0.611764\pi$
$541$	−16.0000	−0.687894	−0.343947	+	0.938989i	$0.611764\pi$
$542$	0	0
$543$	− 7.00000i	− 0.300399i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 8.00000i	− 0.342055i	−0.985266	−	0.171028i	$-0.945291\pi$
$547$	− 8.00000i	− 0.342055i	0.985266	−	0.171028i	$-0.0547087\pi$
$548$	6.00000i	0.256307i
$549$	−20.0000	−0.853579
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	− 10.0000i	− 0.425243i
$554$	0	0
$555$	0	0
$556$	−28.0000	−1.18746
$557$	− 30.0000i	− 1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$	− 30.0000i	− 1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$	0	0
$559$	32.0000	1.35346
$560$	0	0
$561$	6.00000	0.253320
$562$	0	0
$563$	36.0000i	1.51722i	0.651546	+	0.758610i	$0.274121\pi$
$563$	36.0000i	1.51722i	−0.651546	+	0.758610i	$0.725879\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 1.00000i	− 0.0419961i
$568$	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
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	8.00000i	0.334497i
$573$	− 27.0000i	− 1.12794i
$574$	0	0
$575$	0	0
$576$	16.0000	0.666667
$577$	− 11.0000i	− 0.457936i	−0.973434	−	0.228968i	$-0.926465\pi$
$577$	− 11.0000i	− 0.457936i	0.973434	−	0.228968i	$-0.0735351\pi$
$578$	0	0
$579$	−14.0000	−0.581820
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	6.00000i	0.248495i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	− 2.00000i	− 0.0824786i
$589$	−10.0000	−0.412043
$590$	0	0
$591$	18.0000	0.740421
$592$	− 44.0000i	− 1.80839i
$593$	6.00000i	0.246390i	0.992382	+	0.123195i	$0.0393141\pi$
$593$	6.00000i	0.246390i	−0.992382	+	0.123195i	$0.960686\pi$
$594$	0	0
$595$	0	0
$596$	12.0000	0.491539
$597$	16.0000i	0.654836i
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	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
$603$	− 10.0000i	− 0.407231i
$604$	−20.0000	−0.813788
$605$	0	0
$606$	0	0
$607$	− 14.0000i	− 0.568242i	−0.958788	−	0.284121i	$-0.908298\pi$
$607$	− 14.0000i	− 0.568242i	0.958788	−	0.284121i	$-0.0917018\pi$
$608$	0	0
$609$	6.00000	0.243132
$610$	0	0
$611$	0	0
$612$	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	$0.206378\pi$
$617$	30.0000i	1.20775i	−0.797077	+	0.603877i	$0.793622\pi$
$618$	0	0
$619$	19.0000	0.763674	0.381837	−	0.924230i	$-0.375291\pi$
$619$	19.0000	0.763674	0.381837	+	0.924230i	$0.375291\pi$
$620$	0	0
$621$	−15.0000	−0.601929
$622$	0	0
$623$	− 3.00000i	− 0.120192i
$624$	16.0000	0.640513
$625$	0	0
$626$	0	0
$627$	2.00000i	0.0798723i
$628$	26.0000i	1.03751i
$629$	66.0000	2.63159
$630$	0	0
$631$	11.0000	0.437903	0.218952	−	0.975736i	$-0.429736\pi$
$631$	11.0000	0.437903	0.218952	+	0.975736i	$0.429736\pi$
$632$	0	0
$633$	14.0000i	0.556450i
$634$	0	0
$635$	0	0
$636$	12.0000	0.475831
$637$	4.00000i	0.158486i
$638$	0	0
$639$	18.0000	0.712069
$640$	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
$643$	− 49.0000i	− 1.93237i	−0.257847	−	0.966186i	$-0.583013\pi$
$643$	− 49.0000i	− 1.93237i	0.257847	−	0.966186i	$-0.416987\pi$
$644$	6.00000	0.236433
$645$	0	0
$646$	0	0
$647$	33.0000i	1.29736i	0.761060	+	0.648682i	$0.224679\pi$
$647$	33.0000i	1.29736i	−0.761060	+	0.648682i	$0.775321\pi$
$648$	0	0
$649$	−9.00000	−0.353281
$650$	0	0
$651$	5.00000	0.195965
$652$	40.0000i	1.56652i
$653$	39.0000i	1.52619i	0.646288	+	0.763094i	$0.276321\pi$
$653$	39.0000i	1.52619i	−0.646288	+	0.763094i	$0.723679\pi$
$654$	0	0
$655$	0	0
$656$	24.0000	0.937043
$657$	4.00000i	0.156055i
$658$	0	0
$659$	−30.0000	−1.16863	−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000	−1.16863	−0.584317	+	0.811525i	$0.698638\pi$
$660$	0	0
$661$	−49.0000	−1.90588	−0.952940	−	0.303160i	$-0.901958\pi$
$661$	−49.0000	−1.90588	−0.952940	+	0.303160i	$0.901958\pi$
$662$	0	0
$663$	24.0000i	0.932083i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	18.0000i	0.696963i
$668$	− 12.0000i	− 0.464294i
$669$	19.0000	0.734582
$670$	0	0
$671$	10.0000	0.386046
$672$	0	0
$673$	− 28.0000i	− 1.07932i	−0.841883	−	0.539660i	$-0.818553\pi$
$673$	− 28.0000i	− 1.07932i	0.841883	−	0.539660i	$-0.181447\pi$
$674$	0	0
$675$	0	0
$676$	−6.00000	−0.230769
$677$	18.0000i	0.691796i	0.938272	+	0.345898i	$0.112426\pi$
$677$	18.0000i	0.691796i	−0.938272	+	0.345898i	$0.887574\pi$
$678$	0	0
$679$	1.00000	0.0383765
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	−8.00000	−0.305888
$685$	0	0
$686$	0	0
$687$	− 5.00000i	− 0.190762i
$688$	32.0000i	1.21999i
$689$	−24.0000	−0.914327
$690$	0	0
$691$	35.0000	1.33146	0.665731	−	0.746191i	$-0.268120\pi$
$691$	35.0000	1.33146	0.665731	+	0.746191i	$0.268120\pi$
$692$	0	0
$693$	2.00000i	0.0759737i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	36.0000i	1.36360i
$698$	0	0
$699$	−6.00000	−0.226941
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	22.0000i	0.829746i
$704$	−8.00000	−0.301511
$705$	0	0
$706$	0	0
$707$	12.0000i	0.451306i
$708$	18.0000i	0.676481i
$709$	1.00000	0.0375558	0.0187779	−	0.999824i	$-0.494022\pi$
$709$	1.00000	0.0375558	0.0187779	+	0.999824i	$0.494022\pi$
$710$	0	0
$711$	20.0000	0.750059
$712$	0	0
$713$	15.0000i	0.561754i
$714$	0	0
$715$	0	0
$716$	30.0000	1.12115
$717$	12.0000i	0.448148i
$718$	0	0
$719$	39.0000	1.45445	0.727227	−	0.686397i	$-0.240809\pi$
$719$	39.0000	1.45445	0.727227	+	0.686397i	$0.240809\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	0	0
$723$	− 28.0000i	− 1.04133i
$724$	−14.0000	−0.520306
$725$	0	0
$726$	0	0
$727$	− 17.0000i	− 0.630495i	−0.949009	−	0.315248i	$-0.897912\pi$
$727$	− 17.0000i	− 0.630495i	0.949009	−	0.315248i	$-0.102088\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−48.0000	−1.77534
$732$	− 20.0000i	− 0.739221i
$733$	− 4.00000i	− 0.147743i	−0.997268	−	0.0738717i	$-0.976464\pi$
$733$	− 4.00000i	− 0.147743i	0.997268	−	0.0738717i	$-0.0235355\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.00000i	0.184177i
$738$	0	0
$739$	34.0000	1.25071	0.625355	−	0.780340i	$-0.284954\pi$
$739$	34.0000	1.25071	0.625355	+	0.780340i	$0.284954\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	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
$747$	24.0000i	0.878114i
$748$	− 12.0000i	− 0.438763i
$749$	−6.00000	−0.219235
$750$	0	0
$751$	−31.0000	−1.13121	−0.565603	−	0.824678i	$-0.691357\pi$
$751$	−31.0000	−1.13121	−0.565603	+	0.824678i	$0.691357\pi$
$752$	0	0
$753$	− 9.00000i	− 0.327978i
$754$	0	0
$755$	0	0
$756$	10.0000	0.363696
$757$	− 38.0000i	− 1.38113i	−0.723269	−	0.690567i	$-0.757361\pi$
$757$	− 38.0000i	− 1.38113i	0.723269	−	0.690567i	$-0.242639\pi$
$758$	0	0
$759$	3.00000	0.108893
$760$	0	0
$761$	48.0000	1.74000	0.869999	−	0.493053i	$-0.164119\pi$
$761$	48.0000	1.74000	0.869999	+	0.493053i	$0.164119\pi$
$762$	0	0
$763$	20.0000i	0.724049i
$764$	−54.0000	−1.95365
$765$	0	0
$766$	0	0
$767$	− 36.0000i	− 1.29988i
$768$	16.0000i	0.577350i
$769$	40.0000	1.44244	0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000	1.44244	0.721218	+	0.692708i	$0.243582\pi$
$770$	0	0
$771$	−6.00000	−0.216085
$772$	28.0000i	1.00774i
$773$	− 6.00000i	− 0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$	− 6.00000i	− 0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 11.0000i	− 0.394623i
$778$	0	0
$779$	−12.0000	−0.429945
$780$	0	0
$781$	−9.00000	−0.322045
$782$	0	0
$783$	30.0000i	1.07211i
$784$	−4.00000	−0.142857
$785$	0	0
$786$	0	0
$787$	− 50.0000i	− 1.78231i	−0.453701	−	0.891154i	$-0.649897\pi$
$787$	− 50.0000i	− 1.78231i	0.453701	−	0.891154i	$-0.350103\pi$
$788$	− 36.0000i	− 1.28245i
$789$	30.0000	1.06803
$790$	0	0
$791$	−3.00000	−0.106668
$792$	0	0
$793$	40.0000i	1.42044i
$794$	0	0
$795$	0	0
$796$	32.0000	1.13421
$797$	21.0000i	0.743858i	0.928261	+	0.371929i	$0.121304\pi$
$797$	21.0000i	0.743858i	−0.928261	+	0.371929i	$0.878696\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	− 2.00000i	− 0.0705785i
$804$	10.0000	0.352673
$805$	0	0
$806$	0	0
$807$	− 30.0000i	− 1.05605i
$808$	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
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	− 12.0000i	− 0.421117i
$813$	− 16.0000i	− 0.561144i
$814$	0	0
$815$	0	0
$816$	−24.0000	−0.840168
$817$	− 16.0000i	− 0.559769i
$818$	0	0
$819$	−8.00000	−0.279543
$820$	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
$823$	23.0000i	0.801730i	0.916137	+	0.400865i	$0.131290\pi$
$823$	23.0000i	0.801730i	−0.916137	+	0.400865i	$0.868710\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 36.0000i	− 1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$	− 36.0000i	− 1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$	12.0000i	0.417029i
$829$	25.0000	0.868286	0.434143	−	0.900844i	$-0.357051\pi$
$829$	25.0000	0.868286	0.434143	+	0.900844i	$0.357051\pi$
$830$	0	0
$831$	8.00000	0.277517
$832$	− 32.0000i	− 1.10940i
$833$	− 6.00000i	− 0.207888i
$834$	0	0
$835$	0	0
$836$	4.00000	0.138343
$837$	25.0000i	0.864126i
$838$	0	0
$839$	15.0000	0.517858	0.258929	−	0.965896i	$-0.416631\pi$
$839$	15.0000	0.517858	0.258929	+	0.965896i	$0.416631\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	12.0000i	0.413302i
$844$	28.0000	0.963800
$845$	0	0
$846$	0	0
$847$	− 1.00000i	− 0.0343604i
$848$	− 24.0000i	− 0.824163i
$849$	−32.0000	−1.09824
$850$	0	0
$851$	33.0000	1.13123
$852$	18.0000i	0.616670i
$853$	− 10.0000i	− 0.342393i	−0.985237	−	0.171197i	$-0.945237\pi$
$853$	− 10.0000i	− 0.342393i	0.985237	−	0.171197i	$-0.0547634\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	12.0000i	0.409912i	0.978771	+	0.204956i	$0.0657052\pi$
$857$	12.0000i	0.409912i	−0.978771	+	0.204956i	$0.934295\pi$
$858$	0	0
$859$	13.0000	0.443554	0.221777	−	0.975097i	$-0.428814\pi$
$859$	13.0000	0.443554	0.221777	+	0.975097i	$0.428814\pi$
$860$	0	0
$861$	6.00000	0.204479
$862$	0	0
$863$	− 12.0000i	− 0.408485i	−0.978920	−	0.204242i	$-0.934527\pi$
$863$	− 12.0000i	− 0.408485i	0.978920	−	0.204242i	$-0.0654731\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 19.0000i	− 0.645274i
$868$	− 10.0000i	− 0.339422i
$869$	−10.0000	−0.339227
$870$	0	0
$871$	−20.0000	−0.677674
$872$	0	0
$873$	2.00000i	0.0676897i
$874$	0	0
$875$	0	0
$876$	−4.00000	−0.135147
$877$	22.0000i	0.742887i	0.928456	+	0.371444i	$0.121137\pi$
$877$	22.0000i	0.742887i	−0.928456	+	0.371444i	$0.878863\pi$
$878$	0	0
$879$	30.0000	1.01187
$880$	0	0
$881$	−9.00000	−0.303218	−0.151609	−	0.988441i	$-0.548445\pi$
$881$	−9.00000	−0.303218	−0.151609	+	0.988441i	$0.548445\pi$
$882$	0	0
$883$	20.0000i	0.673054i	0.941674	+	0.336527i	$0.109252\pi$
$883$	20.0000i	0.673054i	−0.941674	+	0.336527i	$0.890748\pi$
$884$	48.0000	1.61441
$885$	0	0
$886$	0	0
$887$	42.0000i	1.41022i	0.709097	+	0.705111i	$0.249103\pi$
$887$	42.0000i	1.41022i	−0.709097	+	0.705111i	$0.750897\pi$
$888$	0	0
$889$	−2.00000	−0.0670778
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	− 38.0000i	− 1.27233i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	12.0000i	0.400668i
$898$	0	0
$899$	30.0000	1.00056
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	8.00000i	0.266223i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 8.00000i	− 0.265636i	−0.991140	−	0.132818i	$-0.957597\pi$
$907$	− 8.00000i	− 0.265636i	0.991140	−	0.132818i	$-0.0424025\pi$
$908$	24.0000i	0.796468i
$909$	−24.0000	−0.796030
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	− 8.00000i	− 0.264906i
$913$	− 12.0000i	− 0.397142i
$914$	0	0
$915$	0	0
$916$	−10.0000	−0.330409
$917$	6.00000i	0.198137i
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	0	0
$923$	− 36.0000i	− 1.18495i
$924$	−2.00000	−0.0657952
$925$	0	0
$926$	0	0
$927$	− 8.00000i	− 0.262754i
$928$	0	0
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	12.0000i	0.393073i
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 20.0000i	− 0.653372i	−0.945133	−	0.326686i	$-0.894068\pi$
$937$	− 20.0000i	− 0.653372i	0.945133	−	0.326686i	$-0.105932\pi$
$938$	0	0
$939$	19.0000	0.620042
$940$	0	0
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	0	0
$943$	18.0000i	0.586161i
$944$	36.0000	1.17170
$945$	0	0
$946$	0	0
$947$	27.0000i	0.877382i	0.898638	+	0.438691i	$0.144558\pi$
$947$	27.0000i	0.877382i	−0.898638	+	0.438691i	$0.855442\pi$
$948$	20.0000i	0.649570i
$949$	8.00000	0.259691
$950$	0	0
$951$	9.00000	0.291845
$952$	0	0
$953$	− 36.0000i	− 1.16615i	−0.812417	−	0.583077i	$-0.801849\pi$
$953$	− 36.0000i	− 1.16615i	0.812417	−	0.583077i	$-0.198151\pi$
$954$	0	0
$955$	0	0
$956$	24.0000	0.776215
$957$	− 6.00000i	− 0.193952i
$958$	0	0
$959$	3.00000	0.0968751
$960$	0	0
$961$	−6.00000	−0.193548
$962$	0	0
$963$	− 12.0000i	− 0.386695i
$964$	−56.0000	−1.80364
$965$	0	0
$966$	0	0
$967$	− 14.0000i	− 0.450210i	−0.974335	−	0.225105i	$-0.927728\pi$
$967$	− 14.0000i	− 0.450210i	0.974335	−	0.225105i	$-0.0722725\pi$
$968$	0	0
$969$	12.0000	0.385496
$970$	0	0
$971$	−39.0000	−1.25157	−0.625785	−	0.779996i	$-0.715221\pi$
$971$	−39.0000	−1.25157	−0.625785	+	0.779996i	$0.715221\pi$
$972$	32.0000i	1.02640i
$973$	14.0000i	0.448819i
$974$	0	0
$975$	0	0
$976$	−40.0000	−1.28037
$977$	− 9.00000i	− 0.287936i	−0.989582	−	0.143968i	$-0.954014\pi$
$977$	− 9.00000i	− 0.287936i	0.989582	−	0.143968i	$-0.0459862\pi$
$978$	0	0
$979$	−3.00000	−0.0958804
$980$	0	0
$981$	−40.0000	−1.27710
$982$	0	0
$983$	33.0000i	1.05254i	0.850319	+	0.526268i	$0.176409\pi$
$983$	33.0000i	1.05254i	−0.850319	+	0.526268i	$0.823591\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	16.0000i	0.509028i
$989$	−24.0000	−0.763156
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	0	0
$993$	− 1.00000i	− 0.0317340i
$994$	0	0
$995$	0	0
$996$	−24.0000	−0.760469
$997$	28.0000i	0.886769i	0.896332	+	0.443384i	$0.146222\pi$
$997$	28.0000i	0.886769i	−0.896332	+	0.443384i	$0.853778\pi$
$998$	0	0
$999$	55.0000	1.74012

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1925.2.b.g.1849.2		2
5.2	odd	4		77.2.a.b.1.1	✓	1
5.3	odd	4		1925.2.a.f.1.1		1
5.4	even	2	inner	1925.2.b.g.1849.1		2
15.2	even	4		693.2.a.b.1.1		1
20.7	even	4		1232.2.a.d.1.1		1
35.2	odd	12		539.2.e.d.67.1		2
35.12	even	12		539.2.e.e.67.1		2
35.17	even	12		539.2.e.e.177.1		2
35.27	even	4		539.2.a.b.1.1		1
35.32	odd	12		539.2.e.d.177.1		2
40.27	even	4		4928.2.a.x.1.1		1
40.37	odd	4		4928.2.a.i.1.1		1
55.2	even	20		847.2.f.g.323.1		4
55.7	even	20		847.2.f.g.148.1		4
55.17	even	20		847.2.f.g.729.1		4
55.27	odd	20		847.2.f.f.729.1		4
55.32	even	4		847.2.a.c.1.1		1
55.37	odd	20		847.2.f.f.148.1		4
55.42	odd	20		847.2.f.f.323.1		4
55.47	odd	20		847.2.f.f.372.1		4
55.52	even	20		847.2.f.g.372.1		4
105.62	odd	4		4851.2.a.k.1.1		1
140.27	odd	4		8624.2.a.s.1.1		1
165.32	odd	4		7623.2.a.i.1.1		1
385.307	odd	4		5929.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.a.b.1.1	✓	1	5.2	odd	4
539.2.a.b.1.1		1	35.27	even	4
539.2.e.d.67.1		2	35.2	odd	12
539.2.e.d.177.1		2	35.32	odd	12
539.2.e.e.67.1		2	35.12	even	12
539.2.e.e.177.1		2	35.17	even	12
693.2.a.b.1.1		1	15.2	even	4
847.2.a.c.1.1		1	55.32	even	4
847.2.f.f.148.1		4	55.37	odd	20
847.2.f.f.323.1		4	55.42	odd	20
847.2.f.f.372.1		4	55.47	odd	20
847.2.f.f.729.1		4	55.27	odd	20
847.2.f.g.148.1		4	55.7	even	20
847.2.f.g.323.1		4	55.2	even	20
847.2.f.g.372.1		4	55.52	even	20
847.2.f.g.729.1		4	55.17	even	20
1232.2.a.d.1.1		1	20.7	even	4
1925.2.a.f.1.1		1	5.3	odd	4
1925.2.b.g.1849.1		2	5.4	even	2	inner
1925.2.b.g.1849.2		2	1.1	even	1	trivial
4851.2.a.k.1.1		1	105.62	odd	4
4928.2.a.i.1.1		1	40.37	odd	4
4928.2.a.x.1.1		1	40.27	even	4
5929.2.a.d.1.1		1	385.307	odd	4
7623.2.a.i.1.1		1	165.32	odd	4
8624.2.a.s.1.1		1	140.27	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +2.00000 q^{4} -1.00000i q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +2.00000 q^{4} -1.00000i q^{7} +2.00000 q^{9} -1.00000 q^{11} +2.00000i q^{12} -4.00000i q^{13} +4.00000 q^{16} +6.00000i q^{17} -2.00000 q^{19} +1.00000 q^{21} +3.00000i q^{23} +5.00000i q^{27} -2.00000i q^{28} +6.00000 q^{29} +5.00000 q^{31} -1.00000i q^{33} +4.00000 q^{36} -11.0000i q^{37} +4.00000 q^{39} +6.00000 q^{41} +8.00000i q^{43} -2.00000 q^{44} +4.00000i q^{48} -1.00000 q^{49} -6.00000 q^{51} -8.00000i q^{52} -6.00000i q^{53} -2.00000i q^{57} +9.00000 q^{59} -10.0000 q^{61} -2.00000i q^{63} +8.00000 q^{64} -5.00000i q^{67} +12.0000i q^{68} -3.00000 q^{69} +9.00000 q^{71} +2.00000i q^{73} -4.00000 q^{76} +1.00000i q^{77} +10.0000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +2.00000 q^{84} +6.00000i q^{87} +3.00000 q^{89} -4.00000 q^{91} +6.00000i q^{92} +5.00000i q^{93} +1.00000i q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{4} + 4 q^{9}+O(q^{10}) \) 2 * q + 4 * q^4 + 4 * q^9	\( 2 q + 4 q^{4} + 4 q^{9} - 2 q^{11} + 8 q^{16} - 4 q^{19} + 2 q^{21} + 12 q^{29} + 10 q^{31} + 8 q^{36} + 8 q^{39} + 12 q^{41} - 4 q^{44} - 2 q^{49} - 12 q^{51} + 18 q^{59} - 20 q^{61} + 16 q^{64} - 6 q^{69} + 18 q^{71} - 8 q^{76} + 20 q^{79} + 2 q^{81} + 4 q^{84} + 6 q^{89} - 8 q^{91} - 4 q^{99}+O(q^{100}) \) 2 * q + 4 * q^4 + 4 * q^9 - 2 * q^11 + 8 * q^16 - 4 * q^19 + 2 * q^21 + 12 * q^29 + 10 * q^31 + 8 * q^36 + 8 * q^39 + 12 * q^41 - 4 * q^44 - 2 * q^49 - 12 * q^51 + 18 * q^59 - 20 * q^61 + 16 * q^64 - 6 * q^69 + 18 * q^71 - 8 * q^76 + 20 * q^79 + 2 * q^81 + 4 * q^84 + 6 * q^89 - 8 * q^91 - 4 * q^99

Introduction

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