LMFDB - Embedded newform 1925.2.b.d.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)

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);

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")

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 = [2,0,0,2,0,4,0,0,-2,0,2] 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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} +2.00000i q^{3} +1.00000 q^{4} +2.00000 q^{6} +1.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} +1.00000 q^{11} +2.00000i q^{12} +4.00000i q^{13} +1.00000 q^{14} -1.00000 q^{16} -4.00000i q^{17} +1.00000i q^{18} -2.00000 q^{21} -1.00000i q^{22} -4.00000i q^{23} +6.00000 q^{24} +4.00000 q^{26} +4.00000i q^{27} +1.00000i q^{28} +6.00000 q^{29} +10.0000 q^{31} -5.00000i q^{32} +2.00000i q^{33} -4.00000 q^{34} -1.00000 q^{36} +6.00000i q^{37} -8.00000 q^{39} +4.00000 q^{41} +2.00000i q^{42} +12.0000i q^{43} +1.00000 q^{44} -4.00000 q^{46} +10.0000i q^{47} -2.00000i q^{48} -1.00000 q^{49} +8.00000 q^{51} +4.00000i q^{52} -6.00000i q^{53} +4.00000 q^{54} +3.00000 q^{56} -6.00000i q^{58} -2.00000 q^{59} -10.0000i q^{62} -1.00000i q^{63} -7.00000 q^{64} +2.00000 q^{66} -8.00000i q^{67} -4.00000i q^{68} +8.00000 q^{69} -12.0000 q^{71} +3.00000i q^{72} -8.00000i q^{73} +6.00000 q^{74} +1.00000i q^{77} +8.00000i q^{78} -8.00000 q^{79} -11.0000 q^{81} -4.00000i q^{82} -2.00000 q^{84} +12.0000 q^{86} +12.0000i q^{87} -3.00000i q^{88} +6.00000 q^{89} -4.00000 q^{91} -4.00000i q^{92} +20.0000i q^{93} +10.0000 q^{94} +10.0000 q^{96} +10.0000i q^{97} +1.00000i q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + 4 q^{6} - 2 q^{9} + 2 q^{11} + 2 q^{14} - 2 q^{16} - 4 q^{21} + 12 q^{24} + 8 q^{26} + 12 q^{29} + 20 q^{31} - 8 q^{34} - 2 q^{36} - 16 q^{39} + 8 q^{41} + 2 q^{44} - 8 q^{46} - 2 q^{49}+ \cdots - 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^4 + 4 * q^6 - 2 * q^9 + 2 * q^11 + 2 * q^14 - 2 * q^16 - 4 * q^21 + 12 * q^24 + 8 * q^26 + 12 * q^29 + 20 * q^31 - 8 * q^34 - 2 * q^36 - 16 * q^39 + 8 * q^41 + 2 * q^44 - 8 * q^46 - 2 * q^49 + 16 * q^51 + 8 * q^54 + 6 * q^56 - 4 * q^59 - 14 * q^64 + 4 * q^66 + 16 * q^69 - 24 * q^71 + 12 * q^74 - 16 * q^79 - 22 * q^81 - 4 * q^84 + 24 * q^86 + 12 * q^89 - 8 * q^91 + 20 * q^94 + 20 * q^96 - 2 * 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 $

$2$	− 1.00000i	− 0.707107i	−0.935414	−	0.353553i	$-0.884973\pi$
$2$	− 1.00000i	− 0.707107i	0.935414	−	0.353553i	$-0.115027\pi$
$3$	2.00000i	1.15470i	0.816497	+	0.577350i	$0.195913\pi$
$3$	2.00000i	1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	2.00000	0.816497
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	− 3.00000i	− 1.06066i
$9$	−1.00000	−0.333333
$10$	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	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	−1.00000	−0.250000
$17$	− 4.00000i	− 0.970143i	−0.874475	−	0.485071i	$-0.838794\pi$
$17$	− 4.00000i	− 0.970143i	0.874475	−	0.485071i	$-0.161206\pi$
$18$	1.00000i	0.235702i
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	− 1.00000i	− 0.213201i
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	6.00000	1.22474
$25$	0	0
$26$	4.00000	0.784465
$27$	4.00000i	0.769800i
$28$	1.00000i	0.188982i
$29$	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
$31$	10.0000	1.79605	0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000	1.79605	0.898027	+	0.439941i	$0.145001\pi$
$32$	− 5.00000i	− 0.883883i
$33$	2.00000i	0.348155i
$34$	−4.00000	−0.685994
$35$	0	0
$36$	−1.00000	−0.166667
$37$	6.00000i	0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$	6.00000i	0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	0	0
$39$	−8.00000	−1.28103
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	2.00000i	0.308607i
$43$	12.0000i	1.82998i	0.403473	+	0.914991i	$0.367803\pi$
$43$	12.0000i	1.82998i	−0.403473	+	0.914991i	$0.632197\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−4.00000	−0.589768
$47$	10.0000i	1.45865i	0.684167	+	0.729325i	$0.260166\pi$
$47$	10.0000i	1.45865i	−0.684167	+	0.729325i	$0.739834\pi$
$48$	− 2.00000i	− 0.288675i
$49$	−1.00000	−0.142857
$50$	0	0
$51$	8.00000	1.12022
$52$	4.00000i	0.554700i
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	4.00000	0.544331
$55$	0	0
$56$	3.00000	0.400892
$57$	0	0
$58$	− 6.00000i	− 0.787839i
$59$	−2.00000	−0.260378	−0.130189	−	0.991489i	$-0.541558\pi$
$59$	−2.00000	−0.260378	−0.130189	+	0.991489i	$0.541558\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	− 10.0000i	− 1.27000i
$63$	− 1.00000i	− 0.125988i
$64$	−7.00000	−0.875000
$65$	0	0
$66$	2.00000	0.246183
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	− 4.00000i	− 0.485071i
$69$	8.00000	0.963087
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	3.00000i	0.353553i
$73$	− 8.00000i	− 0.936329i	−0.883641	−	0.468165i	$-0.844915\pi$
$73$	− 8.00000i	− 0.936329i	0.883641	−	0.468165i	$-0.155085\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	0	0
$77$	1.00000i	0.113961i
$78$	8.00000i	0.905822i
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	− 4.00000i	− 0.441726i
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	12.0000	1.29399
$87$	12.0000i	1.28654i
$88$	− 3.00000i	− 0.319801i
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	− 4.00000i	− 0.417029i
$93$	20.0000i	2.07390i
$94$	10.0000	1.03142
$95$	0	0
$96$	10.0000	1.02062
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	1.00000i	0.101015i
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−4.00000	−0.398015	−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000	−0.398015	−0.199007	+	0.979998i	$0.563772\pi$
$102$	− 8.00000i	− 0.792118i
$103$	14.0000i	1.37946i	0.724066	+	0.689730i	$0.242271\pi$
$103$	14.0000i	1.37946i	−0.724066	+	0.689730i	$0.757729\pi$
$104$	12.0000	1.17670
$105$	0	0
$106$	−6.00000	−0.582772
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	4.00000i	0.384900i
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	−12.0000	−1.13899
$112$	− 1.00000i	− 0.0944911i
$113$	18.0000i	1.69330i	0.532152	+	0.846649i	$0.321383\pi$
$113$	18.0000i	1.69330i	−0.532152	+	0.846649i	$0.678617\pi$
$114$	0	0
$115$	0	0
$116$	6.00000	0.557086
$117$	− 4.00000i	− 0.369800i
$118$	2.00000i	0.184115i
$119$	4.00000	0.366679
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	8.00000i	0.721336i
$124$	10.0000	0.898027
$125$	0	0
$126$	−1.00000	−0.0890871
$127$	− 8.00000i	− 0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$	− 8.00000i	− 0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	− 3.00000i	− 0.265165i
$129$	−24.0000	−2.11308
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	2.00000i	0.174078i
$133$	0	0
$134$	−8.00000	−0.691095
$135$	0	0
$136$	−12.0000	−1.02899
$137$	10.0000i	0.854358i	0.904167	+	0.427179i	$0.140493\pi$
$137$	10.0000i	0.854358i	−0.904167	+	0.427179i	$0.859507\pi$
$138$	− 8.00000i	− 0.681005i
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	0	0
$141$	−20.0000	−1.68430
$142$	12.0000i	1.00702i
$143$	4.00000i	0.334497i
$144$	1.00000	0.0833333
$145$	0	0
$146$	−8.00000	−0.662085
$147$	− 2.00000i	− 0.164957i
$148$	6.00000i	0.493197i
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	4.00000i	0.323381i
$154$	1.00000	0.0805823
$155$	0	0
$156$	−8.00000	−0.640513
$157$	− 14.0000i	− 1.11732i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$	− 14.0000i	− 1.11732i	0.829396	−	0.558661i	$-0.188685\pi$
$158$	8.00000i	0.636446i
$159$	12.0000	0.951662
$160$	0	0
$161$	4.00000	0.315244
$162$	11.0000i	0.864242i
$163$	− 8.00000i	− 0.626608i	−0.949653	−	0.313304i	$-0.898564\pi$
$163$	− 8.00000i	− 0.626608i	0.949653	−	0.313304i	$-0.101436\pi$
$164$	4.00000	0.312348
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	6.00000i	0.462910i
$169$	−3.00000	−0.230769
$170$	0	0
$171$	0	0
$172$	12.0000i	0.914991i
$173$	12.0000i	0.912343i	0.889892	+	0.456172i	$0.150780\pi$
$173$	12.0000i	0.912343i	−0.889892	+	0.456172i	$0.849220\pi$
$174$	12.0000	0.909718
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	− 4.00000i	− 0.300658i
$178$	− 6.00000i	− 0.449719i
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	4.00000i	0.296500i
$183$	0	0
$184$	−12.0000	−0.884652
$185$	0	0
$186$	20.0000	1.46647
$187$	− 4.00000i	− 0.292509i
$188$	10.0000i	0.729325i
$189$	−4.00000	−0.290957
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	− 14.0000i	− 1.01036i
$193$	− 14.0000i	− 1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	− 14.0000i	− 1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	−1.00000	−0.0714286
$197$	− 22.0000i	− 1.56744i	−0.621117	−	0.783718i	$-0.713321\pi$
$197$	− 22.0000i	− 1.56744i	0.621117	−	0.783718i	$-0.286679\pi$
$198$	1.00000i	0.0710669i
$199$	18.0000	1.27599	0.637993	−	0.770042i	$-0.279765\pi$
$199$	18.0000	1.27599	0.637993	+	0.770042i	$0.279765\pi$
$200$	0	0
$201$	16.0000	1.12855
$202$	4.00000i	0.281439i
$203$	6.00000i	0.421117i
$204$	8.00000	0.560112
$205$	0	0
$206$	14.0000	0.975426
$207$	4.00000i	0.278019i
$208$	− 4.00000i	− 0.277350i
$209$	0	0
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	− 6.00000i	− 0.412082i
$213$	− 24.0000i	− 1.64445i
$214$	−12.0000	−0.820303
$215$	0	0
$216$	12.0000	0.816497
$217$	10.0000i	0.678844i
$218$	− 14.0000i	− 0.948200i
$219$	16.0000	1.08118
$220$	0	0
$221$	16.0000	1.07628
$222$	12.0000i	0.805387i
$223$	22.0000i	1.47323i	0.676313	+	0.736614i	$0.263577\pi$
$223$	22.0000i	1.47323i	−0.676313	+	0.736614i	$0.736423\pi$
$224$	5.00000	0.334077
$225$	0	0
$226$	18.0000	1.19734
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	0	0
$229$	−18.0000	−1.18947	−0.594737	−	0.803921i	$-0.702744\pi$
$229$	−18.0000	−1.18947	−0.594737	+	0.803921i	$0.702744\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	− 18.0000i	− 1.18176i
$233$	− 18.0000i	− 1.17922i	−0.807688	−	0.589610i	$-0.799282\pi$
$233$	− 18.0000i	− 1.17922i	0.807688	−	0.589610i	$-0.200718\pi$
$234$	−4.00000	−0.261488
$235$	0	0
$236$	−2.00000	−0.130189
$237$	− 16.0000i	− 1.03931i
$238$	− 4.00000i	− 0.259281i
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−20.0000	−1.28831	−0.644157	−	0.764894i	$-0.722792\pi$
$241$	−20.0000	−1.28831	−0.644157	+	0.764894i	$0.722792\pi$
$242$	− 1.00000i	− 0.0642824i
$243$	− 10.0000i	− 0.641500i
$244$	0	0
$245$	0	0
$246$	8.00000	0.510061
$247$	0	0
$248$	− 30.0000i	− 1.90500i
$249$	0	0
$250$	0	0
$251$	−2.00000	−0.126239	−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000	−0.126239	−0.0631194	+	0.998006i	$0.520105\pi$
$252$	− 1.00000i	− 0.0629941i
$253$	− 4.00000i	− 0.251478i
$254$	−8.00000	−0.501965
$255$	0	0
$256$	−17.0000	−1.06250
$257$	14.0000i	0.873296i	0.899632	+	0.436648i	$0.143834\pi$
$257$	14.0000i	0.873296i	−0.899632	+	0.436648i	$0.856166\pi$
$258$	24.0000i	1.49417i
$259$	−6.00000	−0.372822
$260$	0	0
$261$	−6.00000	−0.371391
$262$	− 12.0000i	− 0.741362i
$263$	8.00000i	0.493301i	0.969104	+	0.246651i	$0.0793300\pi$
$263$	8.00000i	0.493301i	−0.969104	+	0.246651i	$0.920670\pi$
$264$	6.00000	0.369274
$265$	0	0
$266$	0	0
$267$	12.0000i	0.734388i
$268$	− 8.00000i	− 0.488678i
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	4.00000i	0.242536i
$273$	− 8.00000i	− 0.484182i
$274$	10.0000	0.604122
$275$	0	0
$276$	8.00000	0.481543
$277$	− 22.0000i	− 1.32185i	−0.750451	−	0.660926i	$-0.770164\pi$
$277$	− 22.0000i	− 1.32185i	0.750451	−	0.660926i	$-0.229836\pi$
$278$	− 8.00000i	− 0.479808i
$279$	−10.0000	−0.598684
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	20.0000i	1.19098i
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	4.00000	0.236525
$287$	4.00000i	0.236113i
$288$	5.00000i	0.294628i
$289$	1.00000	0.0588235
$290$	0	0
$291$	−20.0000	−1.17242
$292$	− 8.00000i	− 0.468165i
$293$	− 24.0000i	− 1.40209i	−0.713115	−	0.701047i	$-0.752716\pi$
$293$	− 24.0000i	− 1.40209i	0.713115	−	0.701047i	$-0.247284\pi$
$294$	−2.00000	−0.116642
$295$	0	0
$296$	18.0000	1.04623
$297$	4.00000i	0.232104i
$298$	− 10.0000i	− 0.579284i
$299$	16.0000	0.925304
$300$	0	0
$301$	−12.0000	−0.691669
$302$	16.0000i	0.920697i
$303$	− 8.00000i	− 0.459588i
$304$	0	0
$305$	0	0
$306$	4.00000	0.228665
$307$	− 20.0000i	− 1.14146i	−0.821138	−	0.570730i	$-0.806660\pi$
$307$	− 20.0000i	− 1.14146i	0.821138	−	0.570730i	$-0.193340\pi$
$308$	1.00000i	0.0569803i
$309$	−28.0000	−1.59286
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	24.0000i	1.35873i
$313$	2.00000i	0.113047i	0.998401	+	0.0565233i	$0.0180015\pi$
$313$	2.00000i	0.113047i	−0.998401	+	0.0565233i	$0.981998\pi$
$314$	−14.0000	−0.790066
$315$	0	0
$316$	−8.00000	−0.450035
$317$	2.00000i	0.112331i	0.998421	+	0.0561656i	$0.0178875\pi$
$317$	2.00000i	0.112331i	−0.998421	+	0.0561656i	$0.982113\pi$
$318$	− 12.0000i	− 0.672927i
$319$	6.00000	0.335936
$320$	0	0
$321$	24.0000	1.33955
$322$	− 4.00000i	− 0.222911i
$323$	0	0
$324$	−11.0000	−0.611111
$325$	0	0
$326$	−8.00000	−0.443079
$327$	28.0000i	1.54840i
$328$	− 12.0000i	− 0.662589i
$329$	−10.0000	−0.551318
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	0	0
$333$	− 6.00000i	− 0.328798i
$334$	0	0
$335$	0	0
$336$	2.00000	0.109109
$337$	− 14.0000i	− 0.762629i	−0.924445	−	0.381314i	$-0.875472\pi$
$337$	− 14.0000i	− 0.762629i	0.924445	−	0.381314i	$-0.124528\pi$
$338$	3.00000i	0.163178i
$339$	−36.0000	−1.95525
$340$	0	0
$341$	10.0000	0.541530
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	36.0000	1.94099
$345$	0	0
$346$	12.0000	0.645124
$347$	− 4.00000i	− 0.214731i	−0.994220	−	0.107366i	$-0.965758\pi$
$347$	− 4.00000i	− 0.214731i	0.994220	−	0.107366i	$-0.0342415\pi$
$348$	12.0000i	0.643268i
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	−16.0000	−0.854017
$352$	− 5.00000i	− 0.266501i
$353$	− 30.0000i	− 1.59674i	−0.602168	−	0.798369i	$-0.705696\pi$
$353$	− 30.0000i	− 1.59674i	0.602168	−	0.798369i	$-0.294304\pi$
$354$	−4.00000	−0.212598
$355$	0	0
$356$	6.00000	0.317999
$357$	8.00000i	0.423405i
$358$	12.0000i	0.634220i
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	− 10.0000i	− 0.525588i
$363$	2.00000i	0.104973i
$364$	−4.00000	−0.209657
$365$	0	0
$366$	0	0
$367$	− 22.0000i	− 1.14839i	−0.818718	−	0.574195i	$-0.805315\pi$
$367$	− 22.0000i	− 1.14839i	0.818718	−	0.574195i	$-0.194685\pi$
$368$	4.00000i	0.208514i
$369$	−4.00000	−0.208232
$370$	0	0
$371$	6.00000	0.311504
$372$	20.0000i	1.03695i
$373$	− 26.0000i	− 1.34623i	−0.739538	−	0.673114i	$-0.764956\pi$
$373$	− 26.0000i	− 1.34623i	0.739538	−	0.673114i	$-0.235044\pi$
$374$	−4.00000	−0.206835
$375$	0	0
$376$	30.0000	1.54713
$377$	24.0000i	1.23606i
$378$	4.00000i	0.205738i
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	− 8.00000i	− 0.409316i
$383$	2.00000i	0.102195i	0.998694	+	0.0510976i	$0.0162720\pi$
$383$	2.00000i	0.102195i	−0.998694	+	0.0510976i	$0.983728\pi$
$384$	6.00000	0.306186
$385$	0	0
$386$	−14.0000	−0.712581
$387$	− 12.0000i	− 0.609994i
$388$	10.0000i	0.507673i
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	3.00000i	0.151523i
$393$	24.0000i	1.21064i
$394$	−22.0000	−1.10834
$395$	0	0
$396$	−1.00000	−0.0502519
$397$	− 22.0000i	− 1.10415i	−0.833795	−	0.552074i	$-0.813837\pi$
$397$	− 22.0000i	− 1.10415i	0.833795	−	0.552074i	$-0.186163\pi$
$398$	− 18.0000i	− 0.902258i
$399$	0	0
$400$	0	0
$401$	−22.0000	−1.09863	−0.549314	−	0.835616i	$-0.685111\pi$
$401$	−22.0000	−1.09863	−0.549314	+	0.835616i	$0.685111\pi$
$402$	− 16.0000i	− 0.798007i
$403$	40.0000i	1.99254i
$404$	−4.00000	−0.199007
$405$	0	0
$406$	6.00000	0.297775
$407$	6.00000i	0.297409i
$408$	− 24.0000i	− 1.18818i
$409$	−24.0000	−1.18672	−0.593362	−	0.804936i	$-0.702200\pi$
$409$	−24.0000	−1.18672	−0.593362	+	0.804936i	$0.702200\pi$
$410$	0	0
$411$	−20.0000	−0.986527
$412$	14.0000i	0.689730i
$413$	− 2.00000i	− 0.0984136i
$414$	4.00000	0.196589
$415$	0	0
$416$	20.0000	0.980581
$417$	16.0000i	0.783523i
$418$	0	0
$419$	−2.00000	−0.0977064	−0.0488532	−	0.998806i	$-0.515557\pi$
$419$	−2.00000	−0.0977064	−0.0488532	+	0.998806i	$0.515557\pi$
$420$	0	0
$421$	−14.0000	−0.682318	−0.341159	−	0.940006i	$-0.610819\pi$
$421$	−14.0000	−0.682318	−0.341159	+	0.940006i	$0.610819\pi$
$422$	12.0000i	0.584151i
$423$	− 10.0000i	− 0.486217i
$424$	−18.0000	−0.874157
$425$	0	0
$426$	−24.0000	−1.16280
$427$	0	0
$428$	− 12.0000i	− 0.580042i
$429$	−8.00000	−0.386244
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	− 4.00000i	− 0.192450i
$433$	− 26.0000i	− 1.24948i	−0.780833	−	0.624740i	$-0.785205\pi$
$433$	− 26.0000i	− 1.24948i	0.780833	−	0.624740i	$-0.214795\pi$
$434$	10.0000	0.480015
$435$	0	0
$436$	14.0000	0.670478
$437$	0	0
$438$	− 16.0000i	− 0.764510i
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	− 16.0000i	− 0.761042i
$443$	4.00000i	0.190046i	0.995475	+	0.0950229i	$0.0302924\pi$
$443$	4.00000i	0.190046i	−0.995475	+	0.0950229i	$0.969708\pi$
$444$	−12.0000	−0.569495
$445$	0	0
$446$	22.0000	1.04173
$447$	20.0000i	0.945968i
$448$	− 7.00000i	− 0.330719i
$449$	10.0000	0.471929	0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000	0.471929	0.235965	+	0.971762i	$0.424175\pi$
$450$	0	0
$451$	4.00000	0.188353
$452$	18.0000i	0.846649i
$453$	− 32.0000i	− 1.50349i
$454$	−12.0000	−0.563188
$455$	0	0
$456$	0	0
$457$	− 18.0000i	− 0.842004i	−0.907060	−	0.421002i	$-0.861678\pi$
$457$	− 18.0000i	− 0.842004i	0.907060	−	0.421002i	$-0.138322\pi$
$458$	18.0000i	0.841085i
$459$	16.0000	0.746816
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	2.00000i	0.0930484i
$463$	4.00000i	0.185896i	0.995671	+	0.0929479i	$0.0296290\pi$
$463$	4.00000i	0.185896i	−0.995671	+	0.0929479i	$0.970371\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	−18.0000	−0.833834
$467$	− 30.0000i	− 1.38823i	−0.719862	−	0.694117i	$-0.755795\pi$
$467$	− 30.0000i	− 1.38823i	0.719862	−	0.694117i	$-0.244205\pi$
$468$	− 4.00000i	− 0.184900i
$469$	8.00000	0.369406
$470$	0	0
$471$	28.0000	1.29017
$472$	6.00000i	0.276172i
$473$	12.0000i	0.551761i
$474$	−16.0000	−0.734904
$475$	0	0
$476$	4.00000	0.183340
$477$	6.00000i	0.274721i
$478$	0	0
$479$	4.00000	0.182765	0.0913823	−	0.995816i	$-0.470871\pi$
$479$	4.00000	0.182765	0.0913823	+	0.995816i	$0.470871\pi$
$480$	0	0
$481$	−24.0000	−1.09431
$482$	20.0000i	0.910975i
$483$	8.00000i	0.364013i
$484$	1.00000	0.0454545
$485$	0	0
$486$	−10.0000	−0.453609
$487$	28.0000i	1.26880i	0.773004	+	0.634401i	$0.218753\pi$
$487$	28.0000i	1.26880i	−0.773004	+	0.634401i	$0.781247\pi$
$488$	0	0
$489$	16.0000	0.723545
$490$	0	0
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	8.00000i	0.360668i
$493$	− 24.0000i	− 1.08091i
$494$	0	0
$495$	0	0
$496$	−10.0000	−0.449013
$497$	− 12.0000i	− 0.538274i
$498$	0	0
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	0	0
$501$	0	0
$502$	2.00000i	0.0892644i
$503$	4.00000i	0.178351i	0.996016	+	0.0891756i	$0.0284232\pi$
$503$	4.00000i	0.178351i	−0.996016	+	0.0891756i	$0.971577\pi$
$504$	−3.00000	−0.133631
$505$	0	0
$506$	−4.00000	−0.177822
$507$	− 6.00000i	− 0.266469i
$508$	− 8.00000i	− 0.354943i
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	8.00000	0.353899
$512$	11.0000i	0.486136i
$513$	0	0
$514$	14.0000	0.617514
$515$	0	0
$516$	−24.0000	−1.05654
$517$	10.0000i	0.439799i
$518$	6.00000i	0.263625i
$519$	−24.0000	−1.05348
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	6.00000i	0.262613i
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	8.00000	0.348817
$527$	− 40.0000i	− 1.74243i
$528$	− 2.00000i	− 0.0870388i
$529$	7.00000	0.304348
$530$	0	0
$531$	2.00000	0.0867926
$532$	0	0
$533$	16.0000i	0.693037i
$534$	12.0000	0.519291
$535$	0	0
$536$	−24.0000	−1.03664
$537$	− 24.0000i	− 1.03568i
$538$	10.0000i	0.431131i
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−26.0000	−1.11783	−0.558914	−	0.829226i	$-0.688782\pi$
$541$	−26.0000	−1.11783	−0.558914	+	0.829226i	$0.688782\pi$
$542$	4.00000i	0.171815i
$543$	20.0000i	0.858282i
$544$	−20.0000	−0.857493
$545$	0	0
$546$	−8.00000	−0.342368
$547$	28.0000i	1.19719i	0.801050	+	0.598597i	$0.204275\pi$
$547$	28.0000i	1.19719i	−0.801050	+	0.598597i	$0.795725\pi$
$548$	10.0000i	0.427179i
$549$	0	0
$550$	0	0
$551$	0	0
$552$	− 24.0000i	− 1.02151i
$553$	− 8.00000i	− 0.340195i
$554$	−22.0000	−0.934690
$555$	0	0
$556$	8.00000	0.339276
$557$	− 22.0000i	− 0.932170i	−0.884740	−	0.466085i	$-0.845664\pi$
$557$	− 22.0000i	− 0.932170i	0.884740	−	0.466085i	$-0.154336\pi$
$558$	10.0000i	0.423334i
$559$	−48.0000	−2.03018
$560$	0	0
$561$	8.00000	0.337760
$562$	− 6.00000i	− 0.253095i
$563$	− 32.0000i	− 1.34864i	−0.738440	−	0.674320i	$-0.764437\pi$
$563$	− 32.0000i	− 1.34864i	0.738440	−	0.674320i	$-0.235563\pi$
$564$	−20.0000	−0.842152
$565$	0	0
$566$	4.00000	0.168133
$567$	− 11.0000i	− 0.461957i
$568$	36.0000i	1.51053i
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	4.00000i	0.167248i
$573$	16.0000i	0.668410i
$574$	4.00000	0.166957
$575$	0	0
$576$	7.00000	0.291667
$577$	18.0000i	0.749350i	0.927156	+	0.374675i	$0.122246\pi$
$577$	18.0000i	0.749350i	−0.927156	+	0.374675i	$0.877754\pi$
$578$	− 1.00000i	− 0.0415945i
$579$	28.0000	1.16364
$580$	0	0
$581$	0	0
$582$	20.0000i	0.829027i
$583$	− 6.00000i	− 0.248495i
$584$	−24.0000	−0.993127
$585$	0	0
$586$	−24.0000	−0.991431
$587$	2.00000i	0.0825488i	0.999148	+	0.0412744i	$0.0131418\pi$
$587$	2.00000i	0.0825488i	−0.999148	+	0.0412744i	$0.986858\pi$
$588$	− 2.00000i	− 0.0824786i
$589$	0	0
$590$	0	0
$591$	44.0000	1.80992
$592$	− 6.00000i	− 0.246598i
$593$	32.0000i	1.31408i	0.753855	+	0.657041i	$0.228192\pi$
$593$	32.0000i	1.31408i	−0.753855	+	0.657041i	$0.771808\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	10.0000	0.409616
$597$	36.0000i	1.47338i
$598$	− 16.0000i	− 0.654289i
$599$	20.0000	0.817178	0.408589	−	0.912719i	$-0.366021\pi$
$599$	20.0000	0.817178	0.408589	+	0.912719i	$0.366021\pi$
$600$	0	0
$601$	−28.0000	−1.14214	−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000	−1.14214	−0.571072	+	0.820900i	$0.693472\pi$
$602$	12.0000i	0.489083i
$603$	8.00000i	0.325785i
$604$	−16.0000	−0.651031
$605$	0	0
$606$	−8.00000	−0.324978
$607$	40.0000i	1.62355i	0.583970	+	0.811775i	$0.301498\pi$
$607$	40.0000i	1.62355i	−0.583970	+	0.811775i	$0.698502\pi$
$608$	0	0
$609$	−12.0000	−0.486265
$610$	0	0
$611$	−40.0000	−1.61823
$612$	4.00000i	0.161690i
$613$	26.0000i	1.05013i	0.851062	+	0.525065i	$0.175959\pi$
$613$	26.0000i	1.05013i	−0.851062	+	0.525065i	$0.824041\pi$
$614$	−20.0000	−0.807134
$615$	0	0
$616$	3.00000	0.120873
$617$	− 6.00000i	− 0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$	− 6.00000i	− 0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$	28.0000i	1.12633i
$619$	−14.0000	−0.562708	−0.281354	−	0.959604i	$-0.590783\pi$
$619$	−14.0000	−0.562708	−0.281354	+	0.959604i	$0.590783\pi$
$620$	0	0
$621$	16.0000	0.642058
$622$	18.0000i	0.721734i
$623$	6.00000i	0.240385i
$624$	8.00000	0.320256
$625$	0	0
$626$	2.00000	0.0799361
$627$	0	0
$628$	− 14.0000i	− 0.558661i
$629$	24.0000	0.956943
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	24.0000i	0.954669i
$633$	− 24.0000i	− 0.953914i
$634$	2.00000	0.0794301
$635$	0	0
$636$	12.0000	0.475831
$637$	− 4.00000i	− 0.158486i
$638$	− 6.00000i	− 0.237542i
$639$	12.0000	0.474713
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	− 24.0000i	− 0.947204i
$643$	14.0000i	0.552106i	0.961142	+	0.276053i	$0.0890266\pi$
$643$	14.0000i	0.552106i	−0.961142	+	0.276053i	$0.910973\pi$
$644$	4.00000	0.157622
$645$	0	0
$646$	0	0
$647$	22.0000i	0.864909i	0.901656	+	0.432455i	$0.142352\pi$
$647$	22.0000i	0.864909i	−0.901656	+	0.432455i	$0.857648\pi$
$648$	33.0000i	1.29636i
$649$	−2.00000	−0.0785069
$650$	0	0
$651$	−20.0000	−0.783862
$652$	− 8.00000i	− 0.313304i
$653$	− 26.0000i	− 1.01746i	−0.860927	−	0.508729i	$-0.830115\pi$
$653$	− 26.0000i	− 1.01746i	0.860927	−	0.508729i	$-0.169885\pi$
$654$	28.0000	1.09489
$655$	0	0
$656$	−4.00000	−0.156174
$657$	8.00000i	0.312110i
$658$	10.0000i	0.389841i
$659$	−4.00000	−0.155818	−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000	−0.155818	−0.0779089	+	0.996960i	$0.524824\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	20.0000i	0.777322i
$663$	32.0000i	1.24278i
$664$	0	0
$665$	0	0
$666$	−6.00000	−0.232495
$667$	− 24.0000i	− 0.929284i
$668$	0	0
$669$	−44.0000	−1.70114
$670$	0	0
$671$	0	0
$672$	10.0000i	0.385758i
$673$	− 34.0000i	− 1.31060i	−0.755367	−	0.655302i	$-0.772541\pi$
$673$	− 34.0000i	− 1.31060i	0.755367	−	0.655302i	$-0.227459\pi$
$674$	−14.0000	−0.539260
$675$	0	0
$676$	−3.00000	−0.115385
$677$	12.0000i	0.461197i	0.973049	+	0.230599i	$0.0740685\pi$
$677$	12.0000i	0.461197i	−0.973049	+	0.230599i	$0.925932\pi$
$678$	36.0000i	1.38257i
$679$	−10.0000	−0.383765
$680$	0	0
$681$	24.0000	0.919682
$682$	− 10.0000i	− 0.382920i
$683$	− 4.00000i	− 0.153056i	−0.997067	−	0.0765279i	$-0.975617\pi$
$683$	− 4.00000i	− 0.153056i	0.997067	−	0.0765279i	$-0.0243834\pi$
$684$	0	0
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	− 36.0000i	− 1.37349i
$688$	− 12.0000i	− 0.457496i
$689$	24.0000	0.914327
$690$	0	0
$691$	−46.0000	−1.74992	−0.874961	−	0.484193i	$-0.839113\pi$
$691$	−46.0000	−1.74992	−0.874961	+	0.484193i	$0.839113\pi$
$692$	12.0000i	0.456172i
$693$	− 1.00000i	− 0.0379869i
$694$	−4.00000	−0.151838
$695$	0	0
$696$	36.0000	1.36458
$697$	− 16.0000i	− 0.606043i
$698$	0	0
$699$	36.0000	1.36165
$700$	0	0
$701$	−22.0000	−0.830929	−0.415464	−	0.909610i	$-0.636381\pi$
$701$	−22.0000	−0.830929	−0.415464	+	0.909610i	$0.636381\pi$
$702$	16.0000i	0.603881i
$703$	0	0
$704$	−7.00000	−0.263822
$705$	0	0
$706$	−30.0000	−1.12906
$707$	− 4.00000i	− 0.150435i
$708$	− 4.00000i	− 0.150329i
$709$	34.0000	1.27690	0.638448	−	0.769665i	$-0.279577\pi$
$709$	34.0000	1.27690	0.638448	+	0.769665i	$0.279577\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	− 18.0000i	− 0.674579i
$713$	− 40.0000i	− 1.49801i
$714$	8.00000	0.299392
$715$	0	0
$716$	−12.0000	−0.448461
$717$	0	0
$718$	16.0000i	0.597115i
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	0	0
$721$	−14.0000	−0.521387
$722$	19.0000i	0.707107i
$723$	− 40.0000i	− 1.48762i
$724$	10.0000	0.371647
$725$	0	0
$726$	2.00000	0.0742270
$727$	− 18.0000i	− 0.667583i	−0.942647	−	0.333792i	$-0.891672\pi$
$727$	− 18.0000i	− 0.667583i	0.942647	−	0.333792i	$-0.108328\pi$
$728$	12.0000i	0.444750i
$729$	−13.0000	−0.481481
$730$	0	0
$731$	48.0000	1.77534
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	−22.0000	−0.812035
$735$	0	0
$736$	−20.0000	−0.737210
$737$	− 8.00000i	− 0.294684i
$738$	4.00000i	0.147242i
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	0	0
$741$	0	0
$742$	− 6.00000i	− 0.220267i
$743$	− 8.00000i	− 0.293492i	−0.989174	−	0.146746i	$-0.953120\pi$
$743$	− 8.00000i	− 0.293492i	0.989174	−	0.146746i	$-0.0468799\pi$
$744$	60.0000	2.19971
$745$	0	0
$746$	−26.0000	−0.951928
$747$	0	0
$748$	− 4.00000i	− 0.146254i
$749$	12.0000	0.438470
$750$	0	0
$751$	−20.0000	−0.729810	−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000	−0.729810	−0.364905	+	0.931045i	$0.618899\pi$
$752$	− 10.0000i	− 0.364662i
$753$	− 4.00000i	− 0.145768i
$754$	24.0000	0.874028
$755$	0	0
$756$	−4.00000	−0.145479
$757$	10.0000i	0.363456i	0.983349	+	0.181728i	$0.0581691\pi$
$757$	10.0000i	0.363456i	−0.983349	+	0.181728i	$0.941831\pi$
$758$	− 8.00000i	− 0.290573i
$759$	8.00000	0.290382
$760$	0	0
$761$	48.0000	1.74000	0.869999	−	0.493053i	$-0.164119\pi$
$761$	48.0000	1.74000	0.869999	+	0.493053i	$0.164119\pi$
$762$	− 16.0000i	− 0.579619i
$763$	14.0000i	0.506834i
$764$	8.00000	0.289430
$765$	0	0
$766$	2.00000	0.0722629
$767$	− 8.00000i	− 0.288863i
$768$	− 34.0000i	− 1.22687i
$769$	−32.0000	−1.15395	−0.576975	−	0.816762i	$-0.695767\pi$
$769$	−32.0000	−1.15395	−0.576975	+	0.816762i	$0.695767\pi$
$770$	0	0
$771$	−28.0000	−1.00840
$772$	− 14.0000i	− 0.503871i
$773$	− 30.0000i	− 1.07903i	−0.841978	−	0.539513i	$-0.818609\pi$
$773$	− 30.0000i	− 1.07903i	0.841978	−	0.539513i	$-0.181391\pi$
$774$	−12.0000	−0.431331
$775$	0	0
$776$	30.0000	1.07694
$777$	− 12.0000i	− 0.430498i
$778$	6.00000i	0.215110i
$779$	0	0
$780$	0	0
$781$	−12.0000	−0.429394
$782$	16.0000i	0.572159i
$783$	24.0000i	0.857690i
$784$	1.00000	0.0357143
$785$	0	0
$786$	24.0000	0.856052
$787$	− 16.0000i	− 0.570338i	−0.958477	−	0.285169i	$-0.907950\pi$
$787$	− 16.0000i	− 0.570338i	0.958477	−	0.285169i	$-0.0920498\pi$
$788$	− 22.0000i	− 0.783718i
$789$	−16.0000	−0.569615
$790$	0	0
$791$	−18.0000	−0.640006
$792$	3.00000i	0.106600i
$793$	0	0
$794$	−22.0000	−0.780751
$795$	0	0
$796$	18.0000	0.637993
$797$	− 14.0000i	− 0.495905i	−0.968772	−	0.247953i	$-0.920242\pi$
$797$	− 14.0000i	− 0.495905i	0.968772	−	0.247953i	$-0.0797578\pi$
$798$	0	0
$799$	40.0000	1.41510
$800$	0	0
$801$	−6.00000	−0.212000
$802$	22.0000i	0.776847i
$803$	− 8.00000i	− 0.282314i
$804$	16.0000	0.564276
$805$	0	0
$806$	40.0000	1.40894
$807$	− 20.0000i	− 0.704033i
$808$	12.0000i	0.422159i
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	6.00000i	0.210559i
$813$	− 8.00000i	− 0.280572i
$814$	6.00000	0.210300
$815$	0	0
$816$	−8.00000	−0.280056
$817$	0	0
$818$	24.0000i	0.839140i
$819$	4.00000	0.139771
$820$	0	0
$821$	46.0000	1.60541	0.802706	−	0.596376i	$-0.203393\pi$
$821$	46.0000	1.60541	0.802706	+	0.596376i	$0.203393\pi$
$822$	20.0000i	0.697580i
$823$	24.0000i	0.836587i	0.908312	+	0.418294i	$0.137372\pi$
$823$	24.0000i	0.836587i	−0.908312	+	0.418294i	$0.862628\pi$
$824$	42.0000	1.46314
$825$	0	0
$826$	−2.00000	−0.0695889
$827$	28.0000i	0.973655i	0.873498	+	0.486828i	$0.161846\pi$
$827$	28.0000i	0.973655i	−0.873498	+	0.486828i	$0.838154\pi$
$828$	4.00000i	0.139010i
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	44.0000	1.52634
$832$	− 28.0000i	− 0.970725i
$833$	4.00000i	0.138592i
$834$	16.0000	0.554035
$835$	0	0
$836$	0	0
$837$	40.0000i	1.38260i
$838$	2.00000i	0.0690889i
$839$	−34.0000	−1.17381	−0.586905	−	0.809656i	$-0.699654\pi$
$839$	−34.0000	−1.17381	−0.586905	+	0.809656i	$0.699654\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	14.0000i	0.482472i
$843$	12.0000i	0.413302i
$844$	−12.0000	−0.413057
$845$	0	0
$846$	−10.0000	−0.343807
$847$	1.00000i	0.0343604i
$848$	6.00000i	0.206041i
$849$	−8.00000	−0.274559
$850$	0	0
$851$	24.0000	0.822709
$852$	− 24.0000i	− 0.822226i
$853$	44.0000i	1.50653i	0.657716	+	0.753266i	$0.271523\pi$
$853$	44.0000i	1.50653i	−0.657716	+	0.753266i	$0.728477\pi$
$854$	0	0
$855$	0	0
$856$	−36.0000	−1.23045
$857$	− 56.0000i	− 1.91292i	−0.291858	−	0.956462i	$-0.594273\pi$
$857$	− 56.0000i	− 1.91292i	0.291858	−	0.956462i	$-0.405727\pi$
$858$	8.00000i	0.273115i
$859$	−6.00000	−0.204717	−0.102359	−	0.994748i	$-0.532639\pi$
$859$	−6.00000	−0.204717	−0.102359	+	0.994748i	$0.532639\pi$
$860$	0	0
$861$	−8.00000	−0.272639
$862$	0	0
$863$	24.0000i	0.816970i	0.912765	+	0.408485i	$0.133943\pi$
$863$	24.0000i	0.816970i	−0.912765	+	0.408485i	$0.866057\pi$
$864$	20.0000	0.680414
$865$	0	0
$866$	−26.0000	−0.883516
$867$	2.00000i	0.0679236i
$868$	10.0000i	0.339422i
$869$	−8.00000	−0.271381
$870$	0	0
$871$	32.0000	1.08428
$872$	− 42.0000i	− 1.42230i
$873$	− 10.0000i	− 0.338449i
$874$	0	0
$875$	0	0
$876$	16.0000	0.540590
$877$	− 42.0000i	− 1.41824i	−0.705088	−	0.709120i	$-0.749093\pi$
$877$	− 42.0000i	− 1.41824i	0.705088	−	0.709120i	$-0.250907\pi$
$878$	− 20.0000i	− 0.674967i
$879$	48.0000	1.61900
$880$	0	0
$881$	−34.0000	−1.14549	−0.572745	−	0.819734i	$-0.694121\pi$
$881$	−34.0000	−1.14549	−0.572745	+	0.819734i	$0.694121\pi$
$882$	− 1.00000i	− 0.0336718i
$883$	28.0000i	0.942275i	0.882060	+	0.471138i	$0.156156\pi$
$883$	28.0000i	0.942275i	−0.882060	+	0.471138i	$0.843844\pi$
$884$	16.0000	0.538138
$885$	0	0
$886$	4.00000	0.134383
$887$	28.0000i	0.940148i	0.882627	+	0.470074i	$0.155773\pi$
$887$	28.0000i	0.940148i	−0.882627	+	0.470074i	$0.844227\pi$
$888$	36.0000i	1.20808i
$889$	8.00000	0.268311
$890$	0	0
$891$	−11.0000	−0.368514
$892$	22.0000i	0.736614i
$893$	0	0
$894$	20.0000	0.668900
$895$	0	0
$896$	3.00000	0.100223
$897$	32.0000i	1.06845i
$898$	− 10.0000i	− 0.333704i
$899$	60.0000	2.00111
$900$	0	0
$901$	−24.0000	−0.799556
$902$	− 4.00000i	− 0.133185i
$903$	− 24.0000i	− 0.798670i
$904$	54.0000	1.79601
$905$	0	0
$906$	−32.0000	−1.06313
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	− 12.0000i	− 0.398234i
$909$	4.00000	0.132672
$910$	0	0
$911$	36.0000	1.19273	0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000	1.19273	0.596367	+	0.802712i	$0.296610\pi$
$912$	0	0
$913$	0	0
$914$	−18.0000	−0.595387
$915$	0	0
$916$	−18.0000	−0.594737
$917$	12.0000i	0.396275i
$918$	− 16.0000i	− 0.528079i
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	40.0000	1.31804
$922$	0	0
$923$	− 48.0000i	− 1.57994i
$924$	−2.00000	−0.0657952
$925$	0	0
$926$	4.00000	0.131448
$927$	− 14.0000i	− 0.459820i
$928$	− 30.0000i	− 0.984798i
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	0	0
$932$	− 18.0000i	− 0.589610i
$933$	− 36.0000i	− 1.17859i
$934$	−30.0000	−0.981630
$935$	0	0
$936$	−12.0000	−0.392232
$937$	16.0000i	0.522697i	0.965244	+	0.261349i	$0.0841672\pi$
$937$	16.0000i	0.522697i	−0.965244	+	0.261349i	$0.915833\pi$
$938$	− 8.00000i	− 0.261209i
$939$	−4.00000	−0.130535
$940$	0	0
$941$	−24.0000	−0.782378	−0.391189	−	0.920310i	$-0.627936\pi$
$941$	−24.0000	−0.782378	−0.391189	+	0.920310i	$0.627936\pi$
$942$	− 28.0000i	− 0.912289i
$943$	− 16.0000i	− 0.521032i
$944$	2.00000	0.0650945
$945$	0	0
$946$	12.0000	0.390154
$947$	36.0000i	1.16984i	0.811090	+	0.584921i	$0.198875\pi$
$947$	36.0000i	1.16984i	−0.811090	+	0.584921i	$0.801125\pi$
$948$	− 16.0000i	− 0.519656i
$949$	32.0000	1.03876
$950$	0	0
$951$	−4.00000	−0.129709
$952$	− 12.0000i	− 0.388922i
$953$	34.0000i	1.10137i	0.834714	+	0.550684i	$0.185633\pi$
$953$	34.0000i	1.10137i	−0.834714	+	0.550684i	$0.814367\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	0	0
$957$	12.0000i	0.387905i
$958$	− 4.00000i	− 0.129234i
$959$	−10.0000	−0.322917
$960$	0	0
$961$	69.0000	2.22581
$962$	24.0000i	0.773791i
$963$	12.0000i	0.386695i
$964$	−20.0000	−0.644157
$965$	0	0
$966$	8.00000	0.257396
$967$	− 40.0000i	− 1.28631i	−0.765735	−	0.643157i	$-0.777624\pi$
$967$	− 40.0000i	− 1.28631i	0.765735	−	0.643157i	$-0.222376\pi$
$968$	− 3.00000i	− 0.0964237i
$969$	0	0
$970$	0	0
$971$	14.0000	0.449281	0.224641	−	0.974442i	$-0.427879\pi$
$971$	14.0000	0.449281	0.224641	+	0.974442i	$0.427879\pi$
$972$	− 10.0000i	− 0.320750i
$973$	8.00000i	0.256468i
$974$	28.0000	0.897178
$975$	0	0
$976$	0	0
$977$	− 42.0000i	− 1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$	− 42.0000i	− 1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$	− 16.0000i	− 0.511624i
$979$	6.00000	0.191761
$980$	0	0
$981$	−14.0000	−0.446986
$982$	− 28.0000i	− 0.893516i
$983$	54.0000i	1.72233i	0.508323	+	0.861166i	$0.330265\pi$
$983$	54.0000i	1.72233i	−0.508323	+	0.861166i	$0.669735\pi$
$984$	24.0000	0.765092
$985$	0	0
$986$	−24.0000	−0.764316
$987$	− 20.0000i	− 0.636607i
$988$	0	0
$989$	48.0000	1.52631
$990$	0	0
$991$	52.0000	1.65183	0.825917	−	0.563791i	$-0.190658\pi$
$991$	52.0000	1.65183	0.825917	+	0.563791i	$0.190658\pi$
$992$	− 50.0000i	− 1.58750i
$993$	− 40.0000i	− 1.26936i
$994$	−12.0000	−0.380617
$995$	0	0
$996$	0	0
$997$	− 20.0000i	− 0.633406i	−0.948525	−	0.316703i	$-0.897424\pi$
$997$	− 20.0000i	− 0.633406i	0.948525	−	0.316703i	$-0.102576\pi$
$998$	− 16.0000i	− 0.506471i
$999$	−24.0000	−0.759326

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +2.00000i q^{3} +1.00000 q^{4} +2.00000 q^{6} +1.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} +1.00000 q^{11} +2.00000i q^{12} +4.00000i q^{13} +1.00000 q^{14} -1.00000 q^{16} -4.00000i q^{17} +1.00000i q^{18} -2.00000 q^{21} -1.00000i q^{22} -4.00000i q^{23} +6.00000 q^{24} +4.00000 q^{26} +4.00000i q^{27} +1.00000i q^{28} +6.00000 q^{29} +10.0000 q^{31} -5.00000i q^{32} +2.00000i q^{33} -4.00000 q^{34} -1.00000 q^{36} +6.00000i q^{37} -8.00000 q^{39} +4.00000 q^{41} +2.00000i q^{42} +12.0000i q^{43} +1.00000 q^{44} -4.00000 q^{46} +10.0000i q^{47} -2.00000i q^{48} -1.00000 q^{49} +8.00000 q^{51} +4.00000i q^{52} -6.00000i q^{53} +4.00000 q^{54} +3.00000 q^{56} -6.00000i q^{58} -2.00000 q^{59} -10.0000i q^{62} -1.00000i q^{63} -7.00000 q^{64} +2.00000 q^{66} -8.00000i q^{67} -4.00000i q^{68} +8.00000 q^{69} -12.0000 q^{71} +3.00000i q^{72} -8.00000i q^{73} +6.00000 q^{74} +1.00000i q^{77} +8.00000i q^{78} -8.00000 q^{79} -11.0000 q^{81} -4.00000i q^{82} -2.00000 q^{84} +12.0000 q^{86} +12.0000i q^{87} -3.00000i q^{88} +6.00000 q^{89} -4.00000 q^{91} -4.00000i q^{92} +20.0000i q^{93} +10.0000 q^{94} +10.0000 q^{96} +10.0000i q^{97} +1.00000i q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 4 q^{6} - 2 q^{9} + 2 q^{11} + 2 q^{14} - 2 q^{16} - 4 q^{21} + 12 q^{24} + 8 q^{26} + 12 q^{29} + 20 q^{31} - 8 q^{34} - 2 q^{36} - 16 q^{39} + 8 q^{41} + 2 q^{44} - 8 q^{46} - 2 q^{49}+ \cdots - 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^4 + 4 * q^6 - 2 * q^9 + 2 * q^11 + 2 * q^14 - 2 * q^16 - 4 * q^21 + 12 * q^24 + 8 * q^26 + 12 * q^29 + 20 * q^31 - 8 * q^34 - 2 * q^36 - 16 * q^39 + 8 * q^41 + 2 * q^44 - 8 * q^46 - 2 * q^49 + 16 * q^51 + 8 * q^54 + 6 * q^56 - 4 * q^59 - 14 * q^64 + 4 * q^66 + 16 * q^69 - 24 * q^71 + 12 * q^74 - 16 * q^79 - 22 * q^81 - 4 * q^84 + 24 * q^86 + 12 * q^89 - 8 * q^91 + 20 * q^94 + 20 * q^96 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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