LMFDB - Embedded newform 1925.2.b.a.1849.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1925,2,Mod(1849,1925)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [2,0,0,-4,0,0,0,0,6,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{2} -2.00000 q^{4} +1.00000i q^{7} +3.00000 q^{9} +1.00000 q^{11} -3.00000i q^{13} +2.00000 q^{14} -4.00000 q^{16} +3.00000i q^{17} -6.00000i q^{18} +7.00000 q^{19} -2.00000i q^{22} +4.00000i q^{23} -6.00000 q^{26} -2.00000i q^{28} +4.00000 q^{29} +2.00000 q^{31} +8.00000i q^{32} +6.00000 q^{34} -6.00000 q^{36} +3.00000i q^{37} -14.0000i q^{38} +5.00000 q^{41} +4.00000i q^{43} -2.00000 q^{44} +8.00000 q^{46} -8.00000i q^{47} -1.00000 q^{49} +6.00000i q^{52} -9.00000i q^{53} -8.00000i q^{58} +12.0000 q^{59} -9.00000 q^{61} -4.00000i q^{62} +3.00000i q^{63} +8.00000 q^{64} -13.0000i q^{67} -6.00000i q^{68} +3.00000 q^{71} +3.00000i q^{73} +6.00000 q^{74} -14.0000 q^{76} +1.00000i q^{77} -10.0000 q^{79} +9.00000 q^{81} -10.0000i q^{82} -12.0000i q^{83} +8.00000 q^{86} -10.0000 q^{89} +3.00000 q^{91} -8.00000i q^{92} -16.0000 q^{94} -4.00000i q^{97} +2.00000i q^{98} +3.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} + 6 q^{9} + 2 q^{11} + 4 q^{14} - 8 q^{16} + 14 q^{19} - 12 q^{26} + 8 q^{29} + 4 q^{31} + 12 q^{34} - 12 q^{36} + 10 q^{41} - 4 q^{44} + 16 q^{46} - 2 q^{49} + 24 q^{59} - 18 q^{61} + 16 q^{64}+ \cdots + 6 q^{99}+O(q^{100}) $ 2 * q - 4 * q^4 + 6 * q^9 + 2 * q^11 + 4 * q^14 - 8 * q^16 + 14 * q^19 - 12 * q^26 + 8 * q^29 + 4 * q^31 + 12 * q^34 - 12 * q^36 + 10 * q^41 - 4 * q^44 + 16 * q^46 - 2 * q^49 + 24 * q^59 - 18 * q^61 + 16 * q^64 + 6 * q^71 + 12 * q^74 - 28 * q^76 - 20 * q^79 + 18 * q^81 + 16 * q^86 - 20 * q^89 + 6 * q^91 - 32 * q^94 + 6 * 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 $

$2$	− 2.00000i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	− 2.00000i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	−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
$9$	3.00000	1.00000
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	− 3.00000i	− 0.832050i	−0.909353	−	0.416025i	$-0.863423\pi$
$13$	− 3.00000i	− 0.832050i	0.909353	−	0.416025i	$-0.136577\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	−4.00000	−1.00000
$17$	3.00000i	0.727607i	0.931476	+	0.363803i	$0.118522\pi$
$17$	3.00000i	0.727607i	−0.931476	+	0.363803i	$0.881478\pi$
$18$	− 6.00000i	− 1.41421i
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$20$	0	0
$21$	0	0
$22$	− 2.00000i	− 0.426401i
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	0	0
$25$	0	0
$26$	−6.00000	−1.17670
$27$	0	0
$28$	− 2.00000i	− 0.377964i
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	8.00000i	1.41421i
$33$	0	0
$34$	6.00000	1.02899
$35$	0	0
$36$	−6.00000	−1.00000
$37$	3.00000i	0.493197i	0.969118	+	0.246598i	$0.0793129\pi$
$37$	3.00000i	0.493197i	−0.969118	+	0.246598i	$0.920687\pi$
$38$	− 14.0000i	− 2.27110i
$39$	0	0
$40$	0	0
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	0	0
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	8.00000	1.17954
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	6.00000i	0.832050i
$53$	− 9.00000i	− 1.23625i	−0.786082	−	0.618123i	$-0.787894\pi$
$53$	− 9.00000i	− 1.23625i	0.786082	−	0.618123i	$-0.212106\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	− 8.00000i	− 1.05045i
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	−9.00000	−1.15233	−0.576166	−	0.817333i	$-0.695452\pi$
$61$	−9.00000	−1.15233	−0.576166	+	0.817333i	$0.695452\pi$
$62$	− 4.00000i	− 0.508001i
$63$	3.00000i	0.377964i
$64$	8.00000	1.00000
$65$	0	0
$66$	0	0
$67$	− 13.0000i	− 1.58820i	−0.607785	−	0.794101i	$-0.707942\pi$
$67$	− 13.0000i	− 1.58820i	0.607785	−	0.794101i	$-0.292058\pi$
$68$	− 6.00000i	− 0.727607i
$69$	0	0
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	0	0
$73$	3.00000i	0.351123i	0.984468	+	0.175562i	$0.0561742\pi$
$73$	3.00000i	0.351123i	−0.984468	+	0.175562i	$0.943826\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	−14.0000	−1.60591
$77$	1.00000i	0.113961i
$78$	0	0
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	− 10.0000i	− 1.10432i
$83$	− 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
$86$	8.00000	0.862662
$87$	0	0
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	3.00000	0.314485
$92$	− 8.00000i	− 0.834058i
$93$	0	0
$94$	−16.0000	−1.65027
$95$	0	0
$96$	0	0
$97$	− 4.00000i	− 0.406138i	−0.979164	−	0.203069i	$-0.934908\pi$
$97$	− 4.00000i	− 0.406138i	0.979164	−	0.203069i	$-0.0650917\pi$
$98$	2.00000i	0.202031i
$99$	3.00000	0.301511
$100$	0	0
$101$	−11.0000	−1.09454	−0.547270	−	0.836956i	$-0.684333\pi$
$101$	−11.0000	−1.09454	−0.547270	+	0.836956i	$0.684333\pi$
$102$	0	0
$103$	− 8.00000i	− 0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$	− 8.00000i	− 0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	0	0
$105$	0	0
$106$	−18.0000	−1.74831
$107$	14.0000i	1.35343i	0.736245	+	0.676716i	$0.236597\pi$
$107$	14.0000i	1.35343i	−0.736245	+	0.676716i	$0.763403\pi$
$108$	0	0
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	0	0
$112$	− 4.00000i	− 0.377964i
$113$	5.00000i	0.470360i	0.971952	+	0.235180i	$0.0755680\pi$
$113$	5.00000i	0.470360i	−0.971952	+	0.235180i	$0.924432\pi$
$114$	0	0
$115$	0	0
$116$	−8.00000	−0.742781
$117$	− 9.00000i	− 0.832050i
$118$	− 24.0000i	− 2.20938i
$119$	−3.00000	−0.275010
$120$	0	0
$121$	1.00000	0.0909091
$122$	18.0000i	1.62964i
$123$	0	0
$124$	−4.00000	−0.359211
$125$	0	0
$126$	6.00000	0.534522
$127$	− 18.0000i	− 1.59724i	−0.601834	−	0.798621i	$-0.705563\pi$
$127$	− 18.0000i	− 1.59724i	0.601834	−	0.798621i	$-0.294437\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−20.0000	−1.74741	−0.873704	−	0.486458i	$-0.838289\pi$
$131$	−20.0000	−1.74741	−0.873704	+	0.486458i	$0.838289\pi$
$132$	0	0
$133$	7.00000i	0.606977i
$134$	−26.0000	−2.24606
$135$	0	0
$136$	0	0
$137$	9.00000i	0.768922i	0.923141	+	0.384461i	$0.125613\pi$
$137$	9.00000i	0.768922i	−0.923141	+	0.384461i	$0.874387\pi$
$138$	0	0
$139$	17.0000	1.44192	0.720961	−	0.692976i	$-0.243701\pi$
$139$	17.0000	1.44192	0.720961	+	0.692976i	$0.243701\pi$
$140$	0	0
$141$	0	0
$142$	− 6.00000i	− 0.503509i
$143$	− 3.00000i	− 0.250873i
$144$	−12.0000	−1.00000
$145$	0	0
$146$	6.00000	0.496564
$147$	0	0
$148$	− 6.00000i	− 0.493197i
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	9.00000i	0.727607i
$154$	2.00000	0.161165
$155$	0	0
$156$	0	0
$157$	− 14.0000i	− 1.11732i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$	− 14.0000i	− 1.11732i	0.829396	−	0.558661i	$-0.188685\pi$
$158$	20.0000i	1.59111i
$159$	0	0
$160$	0	0
$161$	−4.00000	−0.315244
$162$	− 18.0000i	− 1.41421i
$163$	12.0000i	0.939913i	0.882690	+	0.469956i	$0.155730\pi$
$163$	12.0000i	0.939913i	−0.882690	+	0.469956i	$0.844270\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	−24.0000	−1.86276
$167$	− 3.00000i	− 0.232147i	−0.993241	−	0.116073i	$-0.962969\pi$
$167$	− 3.00000i	− 0.232147i	0.993241	−	0.116073i	$-0.0370308\pi$
$168$	0	0
$169$	4.00000	0.307692
$170$	0	0
$171$	21.0000	1.60591
$172$	− 8.00000i	− 0.609994i
$173$	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
$176$	−4.00000	−0.301511
$177$	0	0
$178$	20.0000i	1.49906i
$179$	−15.0000	−1.12115	−0.560576	−	0.828103i	$-0.689420\pi$
$179$	−15.0000	−1.12115	−0.560576	+	0.828103i	$0.689420\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	− 6.00000i	− 0.444750i
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.00000i	0.219382i
$188$	16.0000i	1.16692i
$189$	0	0
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	0	0
$193$	16.0000i	1.15171i	0.817554	+	0.575853i	$0.195330\pi$
$193$	16.0000i	1.15171i	−0.817554	+	0.575853i	$0.804670\pi$
$194$	−8.00000	−0.574367
$195$	0	0
$196$	2.00000	0.142857
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	− 6.00000i	− 0.426401i
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	0	0
$201$	0	0
$202$	22.0000i	1.54791i
$203$	4.00000i	0.280745i
$204$	0	0
$205$	0	0
$206$	−16.0000	−1.11477
$207$	12.0000i	0.834058i
$208$	12.0000i	0.832050i
$209$	7.00000	0.484200
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	18.0000i	1.23625i
$213$	0	0
$214$	28.0000	1.91404
$215$	0	0
$216$	0	0
$217$	2.00000i	0.135769i
$218$	− 32.0000i	− 2.16731i
$219$	0	0
$220$	0	0
$221$	9.00000	0.605406
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	−8.00000	−0.534522
$225$	0	0
$226$	10.0000	0.665190
$227$	3.00000i	0.199117i	0.995032	+	0.0995585i	$0.0317430\pi$
$227$	3.00000i	0.199117i	−0.995032	+	0.0995585i	$0.968257\pi$
$228$	0	0
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	24.0000i	1.57229i	0.618041	+	0.786146i	$0.287927\pi$
$233$	24.0000i	1.57229i	−0.618041	+	0.786146i	$0.712073\pi$
$234$	−18.0000	−1.17670
$235$	0	0
$236$	−24.0000	−1.56227
$237$	0	0
$238$	6.00000i	0.388922i
$239$	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
$241$	−11.0000	−0.708572	−0.354286	−	0.935137i	$-0.615276\pi$
$241$	−11.0000	−0.708572	−0.354286	+	0.935137i	$0.615276\pi$
$242$	− 2.00000i	− 0.128565i
$243$	0	0
$244$	18.0000	1.15233
$245$	0	0
$246$	0	0
$247$	− 21.0000i	− 1.33620i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	22.0000	1.38863	0.694314	−	0.719672i	$-0.255708\pi$
$251$	22.0000	1.38863	0.694314	+	0.719672i	$0.255708\pi$
$252$	− 6.00000i	− 0.377964i
$253$	4.00000i	0.251478i
$254$	−36.0000	−2.25884
$255$	0	0
$256$	16.0000	1.00000
$257$	8.00000i	0.499026i	0.968371	+	0.249513i	$0.0802706\pi$
$257$	8.00000i	0.499026i	−0.968371	+	0.249513i	$0.919729\pi$
$258$	0	0
$259$	−3.00000	−0.186411
$260$	0	0
$261$	12.0000	0.742781
$262$	40.0000i	2.47121i
$263$	30.0000i	1.84988i	0.380114	+	0.924940i	$0.375885\pi$
$263$	30.0000i	1.84988i	−0.380114	+	0.924940i	$0.624115\pi$
$264$	0	0
$265$	0	0
$266$	14.0000	0.858395
$267$	0	0
$268$	26.0000i	1.58820i
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	−27.0000	−1.64013	−0.820067	−	0.572268i	$-0.806064\pi$
$271$	−27.0000	−1.64013	−0.820067	+	0.572268i	$0.806064\pi$
$272$	− 12.0000i	− 0.727607i
$273$	0	0
$274$	18.0000	1.08742
$275$	0	0
$276$	0	0
$277$	− 10.0000i	− 0.600842i	−0.953807	−	0.300421i	$-0.902873\pi$
$277$	− 10.0000i	− 0.600842i	0.953807	−	0.300421i	$-0.0971271\pi$
$278$	− 34.0000i	− 2.03918i
$279$	6.00000	0.359211
$280$	0	0
$281$	−16.0000	−0.954480	−0.477240	−	0.878773i	$-0.658363\pi$
$281$	−16.0000	−0.954480	−0.477240	+	0.878773i	$0.658363\pi$
$282$	0	0
$283$	− 20.0000i	− 1.18888i	−0.804141	−	0.594438i	$-0.797374\pi$
$283$	− 20.0000i	− 1.18888i	0.804141	−	0.594438i	$-0.202626\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	−6.00000	−0.354787
$287$	5.00000i	0.295141i
$288$	24.0000i	1.41421i
$289$	8.00000	0.470588
$290$	0	0
$291$	0	0
$292$	− 6.00000i	− 0.351123i
$293$	− 10.0000i	− 0.584206i	−0.956387	−	0.292103i	$-0.905645\pi$
$293$	− 10.0000i	− 0.584206i	0.956387	−	0.292103i	$-0.0943550\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	12.0000	0.693978
$300$	0	0
$301$	−4.00000	−0.230556
$302$	− 16.0000i	− 0.920697i
$303$	0	0
$304$	−28.0000	−1.60591
$305$	0	0
$306$	18.0000	1.02899
$307$	− 19.0000i	− 1.08439i	−0.840254	−	0.542194i	$-0.817594\pi$
$307$	− 19.0000i	− 1.08439i	0.840254	−	0.542194i	$-0.182406\pi$
$308$	− 2.00000i	− 0.113961i
$309$	0	0
$310$	0	0
$311$	−14.0000	−0.793867	−0.396934	−	0.917847i	$-0.629926\pi$
$311$	−14.0000	−0.793867	−0.396934	+	0.917847i	$0.629926\pi$
$312$	0	0
$313$	− 28.0000i	− 1.58265i	−0.611393	−	0.791327i	$-0.709391\pi$
$313$	− 28.0000i	− 1.58265i	0.611393	−	0.791327i	$-0.290609\pi$
$314$	−28.0000	−1.58013
$315$	0	0
$316$	20.0000	1.12509
$317$	30.0000i	1.68497i	0.538721	+	0.842484i	$0.318908\pi$
$317$	30.0000i	1.68497i	−0.538721	+	0.842484i	$0.681092\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	0	0
$321$	0	0
$322$	8.00000i	0.445823i
$323$	21.0000i	1.16847i
$324$	−18.0000	−1.00000
$325$	0	0
$326$	24.0000	1.32924
$327$	0	0
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	1.00000	0.0549650	0.0274825	−	0.999622i	$-0.491251\pi$
$331$	1.00000	0.0549650	0.0274825	+	0.999622i	$0.491251\pi$
$332$	24.0000i	1.31717i
$333$	9.00000i	0.493197i
$334$	−6.00000	−0.328305
$335$	0	0
$336$	0	0
$337$	14.0000i	0.762629i	0.924445	+	0.381314i	$0.124528\pi$
$337$	14.0000i	0.762629i	−0.924445	+	0.381314i	$0.875472\pi$
$338$	− 8.00000i	− 0.435143i
$339$	0	0
$340$	0	0
$341$	2.00000	0.108306
$342$	− 42.0000i	− 2.27110i
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	0	0
$346$	12.0000	0.645124
$347$	− 10.0000i	− 0.536828i	−0.963304	−	0.268414i	$-0.913500\pi$
$347$	− 10.0000i	− 0.536828i	0.963304	−	0.268414i	$-0.0864995\pi$
$348$	0	0
$349$	1.00000	0.0535288	0.0267644	−	0.999642i	$-0.491480\pi$
$349$	1.00000	0.0535288	0.0267644	+	0.999642i	$0.491480\pi$
$350$	0	0
$351$	0	0
$352$	8.00000i	0.426401i
$353$	22.0000i	1.17094i	0.810693	+	0.585471i	$0.199090\pi$
$353$	22.0000i	1.17094i	−0.810693	+	0.585471i	$0.800910\pi$
$354$	0	0
$355$	0	0
$356$	20.0000	1.06000
$357$	0	0
$358$	30.0000i	1.58555i
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	− 4.00000i	− 0.210235i
$363$	0	0
$364$	−6.00000	−0.314485
$365$	0	0
$366$	0	0
$367$	22.0000i	1.14839i	0.818718	+	0.574195i	$0.194685\pi$
$367$	22.0000i	1.14839i	−0.818718	+	0.574195i	$0.805315\pi$
$368$	− 16.0000i	− 0.834058i
$369$	15.0000	0.780869
$370$	0	0
$371$	9.00000	0.467257
$372$	0	0
$373$	32.0000i	1.65690i	0.560065	+	0.828449i	$0.310776\pi$
$373$	32.0000i	1.65690i	−0.560065	+	0.828449i	$0.689224\pi$
$374$	6.00000	0.310253
$375$	0	0
$376$	0	0
$377$	− 12.0000i	− 0.618031i
$378$	0	0
$379$	12.0000	0.616399	0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000	0.616399	0.308199	+	0.951322i	$0.400274\pi$
$380$	0	0
$381$	0	0
$382$	32.0000i	1.63726i
$383$	− 16.0000i	− 0.817562i	−0.912633	−	0.408781i	$-0.865954\pi$
$383$	− 16.0000i	− 0.817562i	0.912633	−	0.408781i	$-0.134046\pi$
$384$	0	0
$385$	0	0
$386$	32.0000	1.62876
$387$	12.0000i	0.609994i
$388$	8.00000i	0.406138i
$389$	−19.0000	−0.963338	−0.481669	−	0.876353i	$-0.659969\pi$
$389$	−19.0000	−0.963338	−0.481669	+	0.876353i	$0.659969\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	−6.00000	−0.301511
$397$	6.00000i	0.301131i	0.988600	+	0.150566i	$0.0481095\pi$
$397$	6.00000i	0.301131i	−0.988600	+	0.150566i	$0.951890\pi$
$398$	− 20.0000i	− 1.00251i
$399$	0	0
$400$	0	0
$401$	15.0000	0.749064	0.374532	−	0.927214i	$-0.377803\pi$
$401$	15.0000	0.749064	0.374532	+	0.927214i	$0.377803\pi$
$402$	0	0
$403$	− 6.00000i	− 0.298881i
$404$	22.0000	1.09454
$405$	0	0
$406$	8.00000	0.397033
$407$	3.00000i	0.148704i
$408$	0	0
$409$	−23.0000	−1.13728	−0.568638	−	0.822588i	$-0.692530\pi$
$409$	−23.0000	−1.13728	−0.568638	+	0.822588i	$0.692530\pi$
$410$	0	0
$411$	0	0
$412$	16.0000i	0.788263i
$413$	12.0000i	0.590481i
$414$	24.0000	1.17954
$415$	0	0
$416$	24.0000	1.17670
$417$	0	0
$418$	− 14.0000i	− 0.684762i
$419$	−30.0000	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$419$	−30.0000	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$420$	0	0
$421$	13.0000	0.633581	0.316791	−	0.948495i	$-0.397395\pi$
$421$	13.0000	0.633581	0.316791	+	0.948495i	$0.397395\pi$
$422$	20.0000i	0.973585i
$423$	− 24.0000i	− 1.16692i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 9.00000i	− 0.435541i
$428$	− 28.0000i	− 1.35343i
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	14.0000i	0.672797i	0.941720	+	0.336399i	$0.109209\pi$
$433$	14.0000i	0.672797i	−0.941720	+	0.336399i	$0.890791\pi$
$434$	4.00000	0.192006
$435$	0	0
$436$	−32.0000	−1.53252
$437$	28.0000i	1.33942i
$438$	0	0
$439$	−7.00000	−0.334092	−0.167046	−	0.985949i	$-0.553423\pi$
$439$	−7.00000	−0.334092	−0.167046	+	0.985949i	$0.553423\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	− 18.0000i	− 0.856173i
$443$	− 15.0000i	− 0.712672i	−0.934358	−	0.356336i	$-0.884026\pi$
$443$	− 15.0000i	− 0.712672i	0.934358	−	0.356336i	$-0.115974\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	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
$451$	5.00000	0.235441
$452$	− 10.0000i	− 0.470360i
$453$	0	0
$454$	6.00000	0.281594
$455$	0	0
$456$	0	0
$457$	22.0000i	1.02912i	0.857455	+	0.514558i	$0.172044\pi$
$457$	22.0000i	1.02912i	−0.857455	+	0.514558i	$0.827956\pi$
$458$	− 44.0000i	− 2.05598i
$459$	0	0
$460$	0	0
$461$	−21.0000	−0.978068	−0.489034	−	0.872265i	$-0.662651\pi$
$461$	−21.0000	−0.978068	−0.489034	+	0.872265i	$0.662651\pi$
$462$	0	0
$463$	− 13.0000i	− 0.604161i	−0.953282	−	0.302081i	$-0.902319\pi$
$463$	− 13.0000i	− 0.604161i	0.953282	−	0.302081i	$-0.0976812\pi$
$464$	−16.0000	−0.742781
$465$	0	0
$466$	48.0000	2.22356
$467$	8.00000i	0.370196i	0.982720	+	0.185098i	$0.0592602\pi$
$467$	8.00000i	0.370196i	−0.982720	+	0.185098i	$0.940740\pi$
$468$	18.0000i	0.832050i
$469$	13.0000	0.600284
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4.00000i	0.183920i
$474$	0	0
$475$	0	0
$476$	6.00000	0.275010
$477$	− 27.0000i	− 1.23625i
$478$	− 24.0000i	− 1.09773i
$479$	3.00000	0.137073	0.0685367	−	0.997649i	$-0.478167\pi$
$479$	3.00000	0.137073	0.0685367	+	0.997649i	$0.478167\pi$
$480$	0	0
$481$	9.00000	0.410365
$482$	22.0000i	1.00207i
$483$	0	0
$484$	−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
$489$	0	0
$490$	0	0
$491$	−18.0000	−0.812329	−0.406164	−	0.913800i	$-0.633134\pi$
$491$	−18.0000	−0.812329	−0.406164	+	0.913800i	$0.633134\pi$
$492$	0	0
$493$	12.0000i	0.540453i
$494$	−42.0000	−1.88967
$495$	0	0
$496$	−8.00000	−0.359211
$497$	3.00000i	0.134568i
$498$	0	0
$499$	31.0000	1.38775	0.693875	−	0.720095i	$-0.255902\pi$
$499$	31.0000	1.38775	0.693875	+	0.720095i	$0.255902\pi$
$500$	0	0
$501$	0	0
$502$	− 44.0000i	− 1.96382i
$503$	− 31.0000i	− 1.38222i	−0.722749	−	0.691111i	$-0.757122\pi$
$503$	− 31.0000i	− 1.38222i	0.722749	−	0.691111i	$-0.242878\pi$
$504$	0	0
$505$	0	0
$506$	8.00000	0.355643
$507$	0	0
$508$	36.0000i	1.59724i
$509$	38.0000	1.68432	0.842160	−	0.539227i	$-0.181284\pi$
$509$	38.0000	1.68432	0.842160	+	0.539227i	$0.181284\pi$
$510$	0	0
$511$	−3.00000	−0.132712
$512$	− 32.0000i	− 1.41421i
$513$	0	0
$514$	16.0000	0.705730
$515$	0	0
$516$	0	0
$517$	− 8.00000i	− 0.351840i
$518$	6.00000i	0.263625i
$519$	0	0
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	− 24.0000i	− 1.05045i
$523$	− 13.0000i	− 0.568450i	−0.958758	−	0.284225i	$-0.908264\pi$
$523$	− 13.0000i	− 0.568450i	0.958758	−	0.284225i	$-0.0917363\pi$
$524$	40.0000	1.74741
$525$	0	0
$526$	60.0000	2.61612
$527$	6.00000i	0.261364i
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	36.0000	1.56227
$532$	− 14.0000i	− 0.606977i
$533$	− 15.0000i	− 0.649722i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	28.0000i	1.20717i
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	54.0000i	2.31950i
$543$	0	0
$544$	−24.0000	−1.02899
$545$	0	0
$546$	0	0
$547$	42.0000i	1.79579i	0.440209	+	0.897895i	$0.354904\pi$
$547$	42.0000i	1.79579i	−0.440209	+	0.897895i	$0.645096\pi$
$548$	− 18.0000i	− 0.768922i
$549$	−27.0000	−1.15233
$550$	0	0
$551$	28.0000	1.19284
$552$	0	0
$553$	− 10.0000i	− 0.425243i
$554$	−20.0000	−0.849719
$555$	0	0
$556$	−34.0000	−1.44192
$557$	16.0000i	0.677942i	0.940797	+	0.338971i	$0.110079\pi$
$557$	16.0000i	0.677942i	−0.940797	+	0.338971i	$0.889921\pi$
$558$	− 12.0000i	− 0.508001i
$559$	12.0000	0.507546
$560$	0	0
$561$	0	0
$562$	32.0000i	1.34984i
$563$	45.0000i	1.89652i	0.317489	+	0.948262i	$0.397160\pi$
$563$	45.0000i	1.89652i	−0.317489	+	0.948262i	$0.602840\pi$
$564$	0	0
$565$	0	0
$566$	−40.0000	−1.68133
$567$	9.00000i	0.377964i
$568$	0	0
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	2.00000	0.0836974	0.0418487	−	0.999124i	$-0.486675\pi$
$571$	2.00000	0.0836974	0.0418487	+	0.999124i	$0.486675\pi$
$572$	6.00000i	0.250873i
$573$	0	0
$574$	10.0000	0.417392
$575$	0	0
$576$	24.0000	1.00000
$577$	10.0000i	0.416305i	0.978096	+	0.208153i	$0.0667451\pi$
$577$	10.0000i	0.416305i	−0.978096	+	0.208153i	$0.933255\pi$
$578$	− 16.0000i	− 0.665512i
$579$	0	0
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	− 9.00000i	− 0.372742i
$584$	0	0
$585$	0	0
$586$	−20.0000	−0.826192
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	0	0
$589$	14.0000	0.576860
$590$	0	0
$591$	0	0
$592$	− 12.0000i	− 0.493197i
$593$	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
$596$	0	0
$597$	0	0
$598$	− 24.0000i	− 0.981433i
$599$	−3.00000	−0.122577	−0.0612883	−	0.998120i	$-0.519521\pi$
$599$	−3.00000	−0.122577	−0.0612883	+	0.998120i	$0.519521\pi$
$600$	0	0
$601$	34.0000	1.38689	0.693444	−	0.720510i	$-0.256092\pi$
$601$	34.0000	1.38689	0.693444	+	0.720510i	$0.256092\pi$
$602$	8.00000i	0.326056i
$603$	− 39.0000i	− 1.58820i
$604$	−16.0000	−0.651031
$605$	0	0
$606$	0	0
$607$	28.0000i	1.13648i	0.822861	+	0.568242i	$0.192376\pi$
$607$	28.0000i	1.13648i	−0.822861	+	0.568242i	$0.807624\pi$
$608$	56.0000i	2.27110i
$609$	0	0
$610$	0	0
$611$	−24.0000	−0.970936
$612$	− 18.0000i	− 0.727607i
$613$	− 16.0000i	− 0.646234i	−0.946359	−	0.323117i	$-0.895269\pi$
$613$	− 16.0000i	− 0.646234i	0.946359	−	0.323117i	$-0.104731\pi$
$614$	−38.0000	−1.53356
$615$	0	0
$616$	0	0
$617$	5.00000i	0.201292i	0.994922	+	0.100646i	$0.0320910\pi$
$617$	5.00000i	0.201292i	−0.994922	+	0.100646i	$0.967909\pi$
$618$	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
$621$	0	0
$622$	28.0000i	1.12270i
$623$	− 10.0000i	− 0.400642i
$624$	0	0
$625$	0	0
$626$	−56.0000	−2.23821
$627$	0	0
$628$	28.0000i	1.11732i
$629$	−9.00000	−0.358854
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	0	0
$633$	0	0
$634$	60.0000	2.38290
$635$	0	0
$636$	0	0
$637$	3.00000i	0.118864i
$638$	− 8.00000i	− 0.316723i
$639$	9.00000	0.356034
$640$	0	0
$641$	−27.0000	−1.06644	−0.533218	−	0.845978i	$-0.679017\pi$
$641$	−27.0000	−1.06644	−0.533218	+	0.845978i	$0.679017\pi$
$642$	0	0
$643$	− 6.00000i	− 0.236617i	−0.992977	−	0.118308i	$-0.962253\pi$
$643$	− 6.00000i	− 0.236617i	0.992977	−	0.118308i	$-0.0377472\pi$
$644$	8.00000	0.315244
$645$	0	0
$646$	42.0000	1.65247
$647$	2.00000i	0.0786281i	0.999227	+	0.0393141i	$0.0125173\pi$
$647$	2.00000i	0.0786281i	−0.999227	+	0.0393141i	$0.987483\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	0	0
$652$	− 24.0000i	− 0.939913i
$653$	− 39.0000i	− 1.52619i	−0.646288	−	0.763094i	$-0.723679\pi$
$653$	− 39.0000i	− 1.52619i	0.646288	−	0.763094i	$-0.276321\pi$
$654$	0	0
$655$	0	0
$656$	−20.0000	−0.780869
$657$	9.00000i	0.351123i
$658$	− 16.0000i	− 0.623745i
$659$	22.0000	0.856998	0.428499	−	0.903542i	$-0.359042\pi$
$659$	22.0000	0.856998	0.428499	+	0.903542i	$0.359042\pi$
$660$	0	0
$661$	12.0000	0.466746	0.233373	−	0.972387i	$-0.425024\pi$
$661$	12.0000	0.466746	0.233373	+	0.972387i	$0.425024\pi$
$662$	− 2.00000i	− 0.0777322i
$663$	0	0
$664$	0	0
$665$	0	0
$666$	18.0000	0.697486
$667$	16.0000i	0.619522i
$668$	6.00000i	0.232147i
$669$	0	0
$670$	0	0
$671$	−9.00000	−0.347441
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	28.0000	1.07852
$675$	0	0
$676$	−8.00000	−0.307692
$677$	− 5.00000i	− 0.192166i	−0.995373	−	0.0960828i	$-0.969369\pi$
$677$	− 5.00000i	− 0.192166i	0.995373	−	0.0960828i	$-0.0306314\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	0	0
$681$	0	0
$682$	− 4.00000i	− 0.153168i
$683$	36.0000i	1.37750i	0.724998	+	0.688751i	$0.241841\pi$
$683$	36.0000i	1.37750i	−0.724998	+	0.688751i	$0.758159\pi$
$684$	−42.0000	−1.60591
$685$	0	0
$686$	−2.00000	−0.0763604
$687$	0	0
$688$	− 16.0000i	− 0.609994i
$689$	−27.0000	−1.02862
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	− 12.0000i	− 0.456172i
$693$	3.00000i	0.113961i
$694$	−20.0000	−0.759190
$695$	0	0
$696$	0	0
$697$	15.0000i	0.568166i
$698$	− 2.00000i	− 0.0757011i
$699$	0	0
$700$	0	0
$701$	24.0000	0.906467	0.453234	−	0.891392i	$-0.350270\pi$
$701$	24.0000	0.906467	0.453234	+	0.891392i	$0.350270\pi$
$702$	0	0
$703$	21.0000i	0.792030i
$704$	8.00000	0.301511
$705$	0	0
$706$	44.0000	1.65596
$707$	− 11.0000i	− 0.413698i
$708$	0	0
$709$	7.00000	0.262891	0.131445	−	0.991323i	$-0.458038\pi$
$709$	7.00000	0.262891	0.131445	+	0.991323i	$0.458038\pi$
$710$	0	0
$711$	−30.0000	−1.12509
$712$	0	0
$713$	8.00000i	0.299602i
$714$	0	0
$715$	0	0
$716$	30.0000	1.12115
$717$	0	0
$718$	24.0000i	0.895672i
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	− 60.0000i	− 2.23297i
$723$	0	0
$724$	−4.00000	−0.148659
$725$	0	0
$726$	0	0
$727$	− 24.0000i	− 0.890111i	−0.895503	−	0.445055i	$-0.853184\pi$
$727$	− 24.0000i	− 0.890111i	0.895503	−	0.445055i	$-0.146816\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	−12.0000	−0.443836
$732$	0	0
$733$	− 14.0000i	− 0.517102i	−0.965998	−	0.258551i	$-0.916755\pi$
$733$	− 14.0000i	− 0.517102i	0.965998	−	0.258551i	$-0.0832450\pi$
$734$	44.0000	1.62407
$735$	0	0
$736$	−32.0000	−1.17954
$737$	− 13.0000i	− 0.478861i
$738$	− 30.0000i	− 1.10432i
$739$	10.0000	0.367856	0.183928	−	0.982940i	$-0.441119\pi$
$739$	10.0000	0.367856	0.183928	+	0.982940i	$0.441119\pi$
$740$	0	0
$741$	0	0
$742$	− 18.0000i	− 0.660801i
$743$	28.0000i	1.02722i	0.858024	+	0.513610i	$0.171692\pi$
$743$	28.0000i	1.02722i	−0.858024	+	0.513610i	$0.828308\pi$
$744$	0	0
$745$	0	0
$746$	64.0000	2.34321
$747$	− 36.0000i	− 1.31717i
$748$	− 6.00000i	− 0.219382i
$749$	−14.0000	−0.511549
$750$	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
$753$	0	0
$754$	−24.0000	−0.874028
$755$	0	0
$756$	0	0
$757$	22.0000i	0.799604i	0.916602	+	0.399802i	$0.130921\pi$
$757$	22.0000i	0.799604i	−0.916602	+	0.399802i	$0.869079\pi$
$758$	− 24.0000i	− 0.871719i
$759$	0	0
$760$	0	0
$761$	−25.0000	−0.906249	−0.453125	−	0.891447i	$-0.649691\pi$
$761$	−25.0000	−0.906249	−0.453125	+	0.891447i	$0.649691\pi$
$762$	0	0
$763$	16.0000i	0.579239i
$764$	32.0000	1.15772
$765$	0	0
$766$	−32.0000	−1.15621
$767$	− 36.0000i	− 1.29988i
$768$	0	0
$769$	−46.0000	−1.65880	−0.829401	−	0.558653i	$-0.811318\pi$
$769$	−46.0000	−1.65880	−0.829401	+	0.558653i	$0.811318\pi$
$770$	0	0
$771$	0	0
$772$	− 32.0000i	− 1.15171i
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	24.0000	0.862662
$775$	0	0
$776$	0	0
$777$	0	0
$778$	38.0000i	1.36237i
$779$	35.0000	1.25401
$780$	0	0
$781$	3.00000	0.107348
$782$	24.0000i	0.858238i
$783$	0	0
$784$	4.00000	0.142857
$785$	0	0
$786$	0	0
$787$	47.0000i	1.67537i	0.546154	+	0.837685i	$0.316091\pi$
$787$	47.0000i	1.67537i	−0.546154	+	0.837685i	$0.683909\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−5.00000	−0.177780
$792$	0	0
$793$	27.0000i	0.958798i
$794$	12.0000	0.425864
$795$	0	0
$796$	−20.0000	−0.708881
$797$	− 12.0000i	− 0.425062i	−0.977154	−	0.212531i	$-0.931829\pi$
$797$	− 12.0000i	− 0.425062i	0.977154	−	0.212531i	$-0.0681706\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	0	0
$801$	−30.0000	−1.06000
$802$	− 30.0000i	− 1.05934i
$803$	3.00000i	0.105868i
$804$	0	0
$805$	0	0
$806$	−12.0000	−0.422682
$807$	0	0
$808$	0	0
$809$	−54.0000	−1.89854	−0.949269	−	0.314464i	$-0.898175\pi$
$809$	−54.0000	−1.89854	−0.949269	+	0.314464i	$0.898175\pi$
$810$	0	0
$811$	−56.0000	−1.96643	−0.983213	−	0.182462i	$-0.941593\pi$
$811$	−56.0000	−1.96643	−0.983213	+	0.182462i	$0.941593\pi$
$812$	− 8.00000i	− 0.280745i
$813$	0	0
$814$	6.00000	0.210300
$815$	0	0
$816$	0	0
$817$	28.0000i	0.979596i
$818$	46.0000i	1.60835i
$819$	9.00000	0.314485
$820$	0	0
$821$	−50.0000	−1.74501	−0.872506	−	0.488603i	$-0.837507\pi$
$821$	−50.0000	−1.74501	−0.872506	+	0.488603i	$0.837507\pi$
$822$	0	0
$823$	16.0000i	0.557725i	0.960331	+	0.278862i	$0.0899574\pi$
$823$	16.0000i	0.557725i	−0.960331	+	0.278862i	$0.910043\pi$
$824$	0	0
$825$	0	0
$826$	24.0000	0.835067
$827$	36.0000i	1.25184i	0.779886	+	0.625921i	$0.215277\pi$
$827$	36.0000i	1.25184i	−0.779886	+	0.625921i	$0.784723\pi$
$828$	− 24.0000i	− 0.834058i
$829$	−44.0000	−1.52818	−0.764092	−	0.645108i	$-0.776812\pi$
$829$	−44.0000	−1.52818	−0.764092	+	0.645108i	$0.776812\pi$
$830$	0	0
$831$	0	0
$832$	− 24.0000i	− 0.832050i
$833$	− 3.00000i	− 0.103944i
$834$	0	0
$835$	0	0
$836$	−14.0000	−0.484200
$837$	0	0
$838$	60.0000i	2.07267i
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	− 26.0000i	− 0.896019i
$843$	0	0
$844$	20.0000	0.688428
$845$	0	0
$846$	−48.0000	−1.65027
$847$	1.00000i	0.0343604i
$848$	36.0000i	1.23625i
$849$	0	0
$850$	0	0
$851$	−12.0000	−0.411355
$852$	0	0
$853$	49.0000i	1.67773i	0.544341	+	0.838864i	$0.316780\pi$
$853$	49.0000i	1.67773i	−0.544341	+	0.838864i	$0.683220\pi$
$854$	−18.0000	−0.615947
$855$	0	0
$856$	0	0
$857$	− 7.00000i	− 0.239115i	−0.992827	−	0.119558i	$-0.961852\pi$
$857$	− 7.00000i	− 0.239115i	0.992827	−	0.119558i	$-0.0381477\pi$
$858$	0	0
$859$	8.00000	0.272956	0.136478	−	0.990643i	$-0.456422\pi$
$859$	8.00000	0.272956	0.136478	+	0.990643i	$0.456422\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 3.00000i	− 0.102121i	−0.998696	−	0.0510606i	$-0.983740\pi$
$863$	− 3.00000i	− 0.102121i	0.998696	−	0.0510606i	$-0.0162602\pi$
$864$	0	0
$865$	0	0
$866$	28.0000	0.951479
$867$	0	0
$868$	− 4.00000i	− 0.135769i
$869$	−10.0000	−0.339227
$870$	0	0
$871$	−39.0000	−1.32146
$872$	0	0
$873$	− 12.0000i	− 0.406138i
$874$	56.0000	1.89423
$875$	0	0
$876$	0	0
$877$	− 14.0000i	− 0.472746i	−0.971662	−	0.236373i	$-0.924041\pi$
$877$	− 14.0000i	− 0.472746i	0.971662	−	0.236373i	$-0.0759588\pi$
$878$	14.0000i	0.472477i
$879$	0	0
$880$	0	0
$881$	−34.0000	−1.14549	−0.572745	−	0.819734i	$-0.694121\pi$
$881$	−34.0000	−1.14549	−0.572745	+	0.819734i	$0.694121\pi$
$882$	6.00000i	0.202031i
$883$	− 23.0000i	− 0.774012i	−0.922077	−	0.387006i	$-0.873509\pi$
$883$	− 23.0000i	− 0.774012i	0.922077	−	0.387006i	$-0.126491\pi$
$884$	−18.0000	−0.605406
$885$	0	0
$886$	−30.0000	−1.00787
$887$	27.0000i	0.906571i	0.891365	+	0.453286i	$0.149748\pi$
$887$	27.0000i	0.906571i	−0.891365	+	0.453286i	$0.850252\pi$
$888$	0	0
$889$	18.0000	0.603701
$890$	0	0
$891$	9.00000	0.301511
$892$	0	0
$893$	− 56.0000i	− 1.87397i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	− 18.0000i	− 0.600668i
$899$	8.00000	0.266815
$900$	0	0
$901$	27.0000	0.899500
$902$	− 10.0000i	− 0.332964i
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 44.0000i	− 1.46100i	−0.682915	−	0.730498i	$-0.739288\pi$
$907$	− 44.0000i	− 1.46100i	0.682915	−	0.730498i	$-0.260712\pi$
$908$	− 6.00000i	− 0.199117i
$909$	−33.0000	−1.09454
$910$	0	0
$911$	13.0000	0.430709	0.215355	−	0.976536i	$-0.430909\pi$
$911$	13.0000	0.430709	0.215355	+	0.976536i	$0.430909\pi$
$912$	0	0
$913$	− 12.0000i	− 0.397142i
$914$	44.0000	1.45539
$915$	0	0
$916$	−44.0000	−1.45380
$917$	− 20.0000i	− 0.660458i
$918$	0	0
$919$	12.0000	0.395843	0.197922	−	0.980218i	$-0.436581\pi$
$919$	12.0000	0.395843	0.197922	+	0.980218i	$0.436581\pi$
$920$	0	0
$921$	0	0
$922$	42.0000i	1.38320i
$923$	− 9.00000i	− 0.296239i
$924$	0	0
$925$	0	0
$926$	−26.0000	−0.854413
$927$	− 24.0000i	− 0.788263i
$928$	32.0000i	1.05045i
$929$	16.0000	0.524943	0.262471	−	0.964940i	$-0.415462\pi$
$929$	16.0000	0.524943	0.262471	+	0.964940i	$0.415462\pi$
$930$	0	0
$931$	−7.00000	−0.229416
$932$	− 48.0000i	− 1.57229i
$933$	0	0
$934$	16.0000	0.523536
$935$	0	0
$936$	0	0
$937$	21.0000i	0.686040i	0.939328	+	0.343020i	$0.111450\pi$
$937$	21.0000i	0.686040i	−0.939328	+	0.343020i	$0.888550\pi$
$938$	− 26.0000i	− 0.848930i
$939$	0	0
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	0	0
$943$	20.0000i	0.651290i
$944$	−48.0000	−1.56227
$945$	0	0
$946$	8.00000	0.260102
$947$	43.0000i	1.39731i	0.715458	+	0.698656i	$0.246218\pi$
$947$	43.0000i	1.39731i	−0.715458	+	0.698656i	$0.753782\pi$
$948$	0	0
$949$	9.00000	0.292152
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 44.0000i	− 1.42530i	−0.701520	−	0.712650i	$-0.747495\pi$
$953$	− 44.0000i	− 1.42530i	0.701520	−	0.712650i	$-0.252505\pi$
$954$	−54.0000	−1.74831
$955$	0	0
$956$	−24.0000	−0.776215
$957$	0	0
$958$	− 6.00000i	− 0.193851i
$959$	−9.00000	−0.290625
$960$	0	0
$961$	−27.0000	−0.870968
$962$	− 18.0000i	− 0.580343i
$963$	42.0000i	1.35343i
$964$	22.0000	0.708572
$965$	0	0
$966$	0	0
$967$	− 42.0000i	− 1.35063i	−0.737530	−	0.675314i	$-0.764008\pi$
$967$	− 42.0000i	− 1.35063i	0.737530	−	0.675314i	$-0.235992\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	42.0000	1.34784	0.673922	−	0.738802i	$-0.264608\pi$
$971$	42.0000	1.34784	0.673922	+	0.738802i	$0.264608\pi$
$972$	0	0
$973$	17.0000i	0.544995i
$974$	−26.0000	−0.833094
$975$	0	0
$976$	36.0000	1.15233
$977$	21.0000i	0.671850i	0.941889	+	0.335925i	$0.109049\pi$
$977$	21.0000i	0.671850i	−0.941889	+	0.335925i	$0.890951\pi$
$978$	0	0
$979$	−10.0000	−0.319601
$980$	0	0
$981$	48.0000	1.53252
$982$	36.0000i	1.14881i
$983$	− 36.0000i	− 1.14822i	−0.818778	−	0.574111i	$-0.805348\pi$
$983$	− 36.0000i	− 1.14822i	0.818778	−	0.574111i	$-0.194652\pi$
$984$	0	0
$985$	0	0
$986$	24.0000	0.764316
$987$	0	0
$988$	42.0000i	1.33620i
$989$	−16.0000	−0.508770
$990$	0	0
$991$	−56.0000	−1.77890	−0.889449	−	0.457034i	$-0.848912\pi$
$991$	−56.0000	−1.77890	−0.889449	+	0.457034i	$0.848912\pi$
$992$	16.0000i	0.508001i
$993$	0	0
$994$	6.00000	0.190308
$995$	0	0
$996$	0	0
$997$	− 29.0000i	− 0.918439i	−0.888323	−	0.459220i	$-0.848129\pi$
$997$	− 29.0000i	− 0.918439i	0.888323	−	0.459220i	$-0.151871\pi$
$998$	− 62.0000i	− 1.96258i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1925.2.b.a.1849.1		2
5.2	odd	4		1925.2.a.m.1.1	yes	1
5.3	odd	4		1925.2.a.a.1.1	✓	1
5.4	even	2	inner	1925.2.b.a.1849.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1925.2.a.a.1.1	✓	1	5.3	odd	4
1925.2.a.m.1.1	yes	1	5.2	odd	4
1925.2.b.a.1849.1		2	1.1	even	1	trivial
1925.2.b.a.1849.2		2	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -2.00000 q^{4} +1.00000i q^{7} +3.00000 q^{9} +1.00000 q^{11} -3.00000i q^{13} +2.00000 q^{14} -4.00000 q^{16} +3.00000i q^{17} -6.00000i q^{18} +7.00000 q^{19} -2.00000i q^{22} +4.00000i q^{23} -6.00000 q^{26} -2.00000i q^{28} +4.00000 q^{29} +2.00000 q^{31} +8.00000i q^{32} +6.00000 q^{34} -6.00000 q^{36} +3.00000i q^{37} -14.0000i q^{38} +5.00000 q^{41} +4.00000i q^{43} -2.00000 q^{44} +8.00000 q^{46} -8.00000i q^{47} -1.00000 q^{49} +6.00000i q^{52} -9.00000i q^{53} -8.00000i q^{58} +12.0000 q^{59} -9.00000 q^{61} -4.00000i q^{62} +3.00000i q^{63} +8.00000 q^{64} -13.0000i q^{67} -6.00000i q^{68} +3.00000 q^{71} +3.00000i q^{73} +6.00000 q^{74} -14.0000 q^{76} +1.00000i q^{77} -10.0000 q^{79} +9.00000 q^{81} -10.0000i q^{82} -12.0000i q^{83} +8.00000 q^{86} -10.0000 q^{89} +3.00000 q^{91} -8.00000i q^{92} -16.0000 q^{94} -4.00000i q^{97} +2.00000i q^{98} +3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} + 6 q^{9} + 2 q^{11} + 4 q^{14} - 8 q^{16} + 14 q^{19} - 12 q^{26} + 8 q^{29} + 4 q^{31} + 12 q^{34} - 12 q^{36} + 10 q^{41} - 4 q^{44} + 16 q^{46} - 2 q^{49} + 24 q^{59} - 18 q^{61} + 16 q^{64}+ \cdots + 6 q^{99}+O(q^{100}) \) 2 * q - 4 * q^4 + 6 * q^9 + 2 * q^11 + 4 * q^14 - 8 * q^16 + 14 * q^19 - 12 * q^26 + 8 * q^29 + 4 * q^31 + 12 * q^34 - 12 * q^36 + 10 * q^41 - 4 * q^44 + 16 * q^46 - 2 * q^49 + 24 * q^59 - 18 * q^61 + 16 * q^64 + 6 * q^71 + 12 * q^74 - 28 * q^76 - 20 * q^79 + 18 * q^81 + 16 * q^86 - 20 * q^89 + 6 * q^91 - 32 * q^94 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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