LMFDB - Embedded newform 550.2.b.d.199.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [550,2,Mod(199,550)] 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("550.199"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(550, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 550 = 2 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	550.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,-2,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

Self dual:	no
Analytic conductor:	$4.39177211117$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	550.199
Dual form			550.2.b.d.199.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} +4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} +2.00000i q^{12} +5.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} -1.00000i q^{18} +7.00000 q^{19} +8.00000 q^{21} -1.00000i q^{22} +3.00000i q^{23} -2.00000 q^{24} -5.00000 q^{26} -4.00000i q^{27} -4.00000i q^{28} -3.00000 q^{29} +5.00000 q^{31} +1.00000i q^{32} +2.00000i q^{33} +1.00000 q^{36} +4.00000i q^{37} +7.00000i q^{38} +10.0000 q^{39} +12.0000 q^{41} +8.00000i q^{42} +5.00000i q^{43} +1.00000 q^{44} -3.00000 q^{46} -2.00000i q^{48} -9.00000 q^{49} -5.00000i q^{52} +6.00000i q^{53} +4.00000 q^{54} +4.00000 q^{56} -14.0000i q^{57} -3.00000i q^{58} -12.0000 q^{59} -10.0000 q^{61} +5.00000i q^{62} -4.00000i q^{63} -1.00000 q^{64} -2.00000 q^{66} -14.0000i q^{67} +6.00000 q^{69} +3.00000 q^{71} +1.00000i q^{72} +8.00000i q^{73} -4.00000 q^{74} -7.00000 q^{76} -4.00000i q^{77} +10.0000i q^{78} +4.00000 q^{79} -11.0000 q^{81} +12.0000i q^{82} -15.0000i q^{83} -8.00000 q^{84} -5.00000 q^{86} +6.00000i q^{87} +1.00000i q^{88} -3.00000 q^{89} -20.0000 q^{91} -3.00000i q^{92} -10.0000i q^{93} +2.00000 q^{96} +13.0000i q^{97} -9.00000i q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9} - 2 q^{11} - 8 q^{14} + 2 q^{16} + 14 q^{19} + 16 q^{21} - 4 q^{24} - 10 q^{26} - 6 q^{29} + 10 q^{31} + 2 q^{36} + 20 q^{39} + 24 q^{41} + 2 q^{44} - 6 q^{46} - 18 q^{49}+ \cdots + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9 - 2 * q^11 - 8 * q^14 + 2 * q^16 + 14 * q^19 + 16 * q^21 - 4 * q^24 - 10 * q^26 - 6 * q^29 + 10 * q^31 + 2 * q^36 + 20 * q^39 + 24 * q^41 + 2 * q^44 - 6 * q^46 - 18 * q^49 + 8 * q^54 + 8 * q^56 - 24 * q^59 - 20 * q^61 - 2 * q^64 - 4 * q^66 + 12 * q^69 + 6 * q^71 - 8 * q^74 - 14 * q^76 + 8 * q^79 - 22 * q^81 - 16 * q^84 - 10 * q^86 - 6 * q^89 - 40 * q^91 + 4 * q^96 + 2 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/550\mathbb{Z}\right)^\times$.

$n$	$101$	$177$
$\chi(n)$	$1$	$-1$

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
$2$	1.00000i	0.707107i
$3$	− 2.00000i	− 1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	− 2.00000i	− 1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	2.00000	0.816497
$7$	4.00000i	1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$	4.00000i	1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	2.00000i	0.577350i
$13$	5.00000i	1.38675i	0.720577	+	0.693375i	$0.243877\pi$
$13$	5.00000i	1.38675i	−0.720577	+	0.693375i	$0.756123\pi$
$14$	−4.00000	−1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	− 1.00000i	− 0.235702i
$19$	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
$21$	8.00000	1.74574
$22$	− 1.00000i	− 0.213201i
$23$	3.00000i	0.625543i	0.949828	+	0.312772i	$0.101257\pi$
$23$	3.00000i	0.625543i	−0.949828	+	0.312772i	$0.898743\pi$
$24$	−2.00000	−0.408248
$25$	0	0
$26$	−5.00000	−0.980581
$27$	− 4.00000i	− 0.769800i
$28$	− 4.00000i	− 0.755929i
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	1.00000i	0.176777i
$33$	2.00000i	0.348155i
$34$	0	0
$35$	0	0
$36$	1.00000	0.166667
$37$	4.00000i	0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$	4.00000i	0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$	7.00000i	1.13555i
$39$	10.0000	1.60128
$40$	0	0
$41$	12.0000	1.87409	0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000	1.87409	0.937043	+	0.349215i	$0.113552\pi$
$42$	8.00000i	1.23443i
$43$	5.00000i	0.762493i	0.924473	+	0.381246i	$0.124505\pi$
$43$	5.00000i	0.762493i	−0.924473	+	0.381246i	$0.875495\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−3.00000	−0.442326
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	− 2.00000i	− 0.288675i
$49$	−9.00000	−1.28571
$50$	0	0
$51$	0	0
$52$	− 5.00000i	− 0.693375i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	4.00000	0.544331
$55$	0	0
$56$	4.00000	0.534522
$57$	− 14.0000i	− 1.85435i
$58$	− 3.00000i	− 0.393919i
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	5.00000i	0.635001i
$63$	− 4.00000i	− 0.503953i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−2.00000	−0.246183
$67$	− 14.0000i	− 1.71037i	−0.518321	−	0.855186i	$-0.673443\pi$
$67$	− 14.0000i	− 1.71037i	0.518321	−	0.855186i	$-0.326557\pi$
$68$	0	0
$69$	6.00000	0.722315
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	1.00000i	0.117851i
$73$	8.00000i	0.936329i	0.883641	+	0.468165i	$0.155085\pi$
$73$	8.00000i	0.936329i	−0.883641	+	0.468165i	$0.844915\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	−7.00000	−0.802955
$77$	− 4.00000i	− 0.455842i
$78$	10.0000i	1.13228i
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	12.0000i	1.32518i
$83$	− 15.0000i	− 1.64646i	−0.567705	−	0.823232i	$-0.692169\pi$
$83$	− 15.0000i	− 1.64646i	0.567705	−	0.823232i	$-0.307831\pi$
$84$	−8.00000	−0.872872
$85$	0	0
$86$	−5.00000	−0.539164
$87$	6.00000i	0.643268i
$88$	1.00000i	0.106600i
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	0	0
$91$	−20.0000	−2.09657
$92$	− 3.00000i	− 0.312772i
$93$	− 10.0000i	− 1.03695i
$94$	0	0
$95$	0	0
$96$	2.00000	0.204124
$97$	13.0000i	1.31995i	0.751288	+	0.659975i	$0.229433\pi$
$97$	13.0000i	1.31995i	−0.751288	+	0.659975i	$0.770567\pi$
$98$	− 9.00000i	− 0.909137i
$99$	1.00000	0.100504
$100$	0	0
$101$	−9.00000	−0.895533	−0.447767	−	0.894150i	$-0.647781\pi$
$101$	−9.00000	−0.895533	−0.447767	+	0.894150i	$0.647781\pi$
$102$	0	0
$103$	− 7.00000i	− 0.689730i	−0.938652	−	0.344865i	$-0.887925\pi$
$103$	− 7.00000i	− 0.689730i	0.938652	−	0.344865i	$-0.112075\pi$
$104$	5.00000	0.490290
$105$	0	0
$106$	−6.00000	−0.582772
$107$	− 15.0000i	− 1.45010i	−0.688694	−	0.725052i	$-0.741816\pi$
$107$	− 15.0000i	− 1.45010i	0.688694	−	0.725052i	$-0.258184\pi$
$108$	4.00000i	0.384900i
$109$	−5.00000	−0.478913	−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	4.00000i	0.377964i
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	14.0000	1.31122
$115$	0	0
$116$	3.00000	0.278543
$117$	− 5.00000i	− 0.462250i
$118$	− 12.0000i	− 1.10469i
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	− 10.0000i	− 0.905357i
$123$	− 24.0000i	− 2.16401i
$124$	−5.00000	−0.449013
$125$	0	0
$126$	4.00000	0.356348
$127$	16.0000i	1.41977i	0.704317	+	0.709885i	$0.251253\pi$
$127$	16.0000i	1.41977i	−0.704317	+	0.709885i	$0.748747\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	10.0000	0.880451
$130$	0	0
$131$	21.0000	1.83478	0.917389	−	0.397991i	$-0.130293\pi$
$131$	21.0000	1.83478	0.917389	+	0.397991i	$0.130293\pi$
$132$	− 2.00000i	− 0.174078i
$133$	28.0000i	2.42791i
$134$	14.0000	1.20942
$135$	0	0
$136$	0	0
$137$	− 3.00000i	− 0.256307i	−0.991754	−	0.128154i	$-0.959095\pi$
$137$	− 3.00000i	− 0.256307i	0.991754	−	0.128154i	$-0.0409051\pi$
$138$	6.00000i	0.510754i
$139$	13.0000	1.10265	0.551323	−	0.834292i	$-0.314123\pi$
$139$	13.0000	1.10265	0.551323	+	0.834292i	$0.314123\pi$
$140$	0	0
$141$	0	0
$142$	3.00000i	0.251754i
$143$	− 5.00000i	− 0.418121i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−8.00000	−0.662085
$147$	18.0000i	1.48461i
$148$	− 4.00000i	− 0.328798i
$149$	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
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	− 7.00000i	− 0.567775i
$153$	0	0
$154$	4.00000	0.322329
$155$	0	0
$156$	−10.0000	−0.800641
$157$	− 14.0000i	− 1.11732i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$	− 14.0000i	− 1.11732i	0.829396	−	0.558661i	$-0.188685\pi$
$158$	4.00000i	0.318223i
$159$	12.0000	0.951662
$160$	0	0
$161$	−12.0000	−0.945732
$162$	− 11.0000i	− 0.864242i
$163$	− 16.0000i	− 1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$	− 16.0000i	− 1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$	−12.0000	−0.937043
$165$	0	0
$166$	15.0000	1.16423
$167$	− 24.0000i	− 1.85718i	−0.371113	−	0.928588i	$-0.621024\pi$
$167$	− 24.0000i	− 1.85718i	0.371113	−	0.928588i	$-0.378976\pi$
$168$	− 8.00000i	− 0.617213i
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−7.00000	−0.535303
$172$	− 5.00000i	− 0.381246i
$173$	21.0000i	1.59660i	0.602260	+	0.798300i	$0.294267\pi$
$173$	21.0000i	1.59660i	−0.602260	+	0.798300i	$0.705733\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	24.0000i	1.80395i
$178$	− 3.00000i	− 0.224860i
$179$	−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
$181$	20.0000	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$182$	− 20.0000i	− 1.48250i
$183$	20.0000i	1.47844i
$184$	3.00000	0.221163
$185$	0	0
$186$	10.0000	0.733236
$187$	0	0
$188$	0	0
$189$	16.0000	1.16383
$190$	0	0
$191$	15.0000	1.08536	0.542681	−	0.839939i	$-0.317409\pi$
$191$	15.0000	1.08536	0.542681	+	0.839939i	$0.317409\pi$
$192$	2.00000i	0.144338i
$193$	− 4.00000i	− 0.287926i	−0.989583	−	0.143963i	$-0.954015\pi$
$193$	− 4.00000i	− 0.287926i	0.989583	−	0.143963i	$-0.0459847\pi$
$194$	−13.0000	−0.933346
$195$	0	0
$196$	9.00000	0.642857
$197$	− 3.00000i	− 0.213741i	−0.994273	−	0.106871i	$-0.965917\pi$
$197$	− 3.00000i	− 0.213741i	0.994273	−	0.106871i	$-0.0340831\pi$
$198$	1.00000i	0.0710669i
$199$	−11.0000	−0.779769	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000	−0.779769	−0.389885	+	0.920864i	$0.627485\pi$
$200$	0	0
$201$	−28.0000	−1.97497
$202$	− 9.00000i	− 0.633238i
$203$	− 12.0000i	− 0.842235i
$204$	0	0
$205$	0	0
$206$	7.00000	0.487713
$207$	− 3.00000i	− 0.208514i
$208$	5.00000i	0.346688i
$209$	−7.00000	−0.484200
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	− 6.00000i	− 0.412082i
$213$	− 6.00000i	− 0.411113i
$214$	15.0000	1.02538
$215$	0	0
$216$	−4.00000	−0.272166
$217$	20.0000i	1.35769i
$218$	− 5.00000i	− 0.338643i
$219$	16.0000	1.08118
$220$	0	0
$221$	0	0
$222$	8.00000i	0.536925i
$223$	8.00000i	0.535720i	0.963458	+	0.267860i	$0.0863164\pi$
$223$	8.00000i	0.535720i	−0.963458	+	0.267860i	$0.913684\pi$
$224$	−4.00000	−0.267261
$225$	0	0
$226$	6.00000	0.399114
$227$	3.00000i	0.199117i	0.995032	+	0.0995585i	$0.0317430\pi$
$227$	3.00000i	0.199117i	−0.995032	+	0.0995585i	$0.968257\pi$
$228$	14.0000i	0.927173i
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	0	0
$231$	−8.00000	−0.526361
$232$	3.00000i	0.196960i
$233$	− 18.0000i	− 1.17922i	−0.807688	−	0.589610i	$-0.799282\pi$
$233$	− 18.0000i	− 1.17922i	0.807688	−	0.589610i	$-0.200718\pi$
$234$	5.00000	0.326860
$235$	0	0
$236$	12.0000	0.781133
$237$	− 8.00000i	− 0.519656i
$238$	0	0
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	1.00000i	0.0642824i
$243$	10.0000i	0.641500i
$244$	10.0000	0.640184
$245$	0	0
$246$	24.0000	1.53018
$247$	35.0000i	2.22700i
$248$	− 5.00000i	− 0.317500i
$249$	−30.0000	−1.90117
$250$	0	0
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	4.00000i	0.251976i
$253$	− 3.00000i	− 0.188608i
$254$	−16.0000	−1.00393
$255$	0	0
$256$	1.00000	0.0625000
$257$	9.00000i	0.561405i	0.959795	+	0.280702i	$0.0905674\pi$
$257$	9.00000i	0.561405i	−0.959795	+	0.280702i	$0.909433\pi$
$258$	10.0000i	0.622573i
$259$	−16.0000	−0.994192
$260$	0	0
$261$	3.00000	0.185695
$262$	21.0000i	1.29738i
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	2.00000	0.123091
$265$	0	0
$266$	−28.0000	−1.71679
$267$	6.00000i	0.367194i
$268$	14.0000i	0.855186i
$269$	−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
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	0	0
$273$	40.0000i	2.42091i
$274$	3.00000	0.181237
$275$	0	0
$276$	−6.00000	−0.361158
$277$	− 26.0000i	− 1.56219i	−0.624413	−	0.781094i	$-0.714662\pi$
$277$	− 26.0000i	− 1.56219i	0.624413	−	0.781094i	$-0.285338\pi$
$278$	13.0000i	0.779688i
$279$	−5.00000	−0.299342
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	−3.00000	−0.178017
$285$	0	0
$286$	5.00000	0.295656
$287$	48.0000i	2.83335i
$288$	− 1.00000i	− 0.0589256i
$289$	17.0000	1.00000
$290$	0	0
$291$	26.0000	1.52415
$292$	− 8.00000i	− 0.468165i
$293$	− 18.0000i	− 1.05157i	−0.850617	−	0.525786i	$-0.823771\pi$
$293$	− 18.0000i	− 1.05157i	0.850617	−	0.525786i	$-0.176229\pi$
$294$	−18.0000	−1.04978
$295$	0	0
$296$	4.00000	0.232495
$297$	4.00000i	0.232104i
$298$	6.00000i	0.347571i
$299$	−15.0000	−0.867472
$300$	0	0
$301$	−20.0000	−1.15278
$302$	− 16.0000i	− 0.920697i
$303$	18.0000i	1.03407i
$304$	7.00000	0.401478
$305$	0	0
$306$	0	0
$307$	16.0000i	0.913168i	0.889680	+	0.456584i	$0.150927\pi$
$307$	16.0000i	0.913168i	−0.889680	+	0.456584i	$0.849073\pi$
$308$	4.00000i	0.227921i
$309$	−14.0000	−0.796432
$310$	0	0
$311$	3.00000	0.170114	0.0850572	−	0.996376i	$-0.472893\pi$
$311$	3.00000	0.170114	0.0850572	+	0.996376i	$0.472893\pi$
$312$	− 10.0000i	− 0.566139i
$313$	26.0000i	1.46961i	0.678280	+	0.734803i	$0.262726\pi$
$313$	26.0000i	1.46961i	−0.678280	+	0.734803i	$0.737274\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	−4.00000	−0.225018
$317$	− 12.0000i	− 0.673987i	−0.941507	−	0.336994i	$-0.890590\pi$
$317$	− 12.0000i	− 0.673987i	0.941507	−	0.336994i	$-0.109410\pi$
$318$	12.0000i	0.672927i
$319$	3.00000	0.167968
$320$	0	0
$321$	−30.0000	−1.67444
$322$	− 12.0000i	− 0.668734i
$323$	0	0
$324$	11.0000	0.611111
$325$	0	0
$326$	16.0000	0.886158
$327$	10.0000i	0.553001i
$328$	− 12.0000i	− 0.662589i
$329$	0	0
$330$	0	0
$331$	32.0000	1.75888	0.879440	−	0.476011i	$-0.157918\pi$
$331$	32.0000	1.75888	0.879440	+	0.476011i	$0.157918\pi$
$332$	15.0000i	0.823232i
$333$	− 4.00000i	− 0.219199i
$334$	24.0000	1.31322
$335$	0	0
$336$	8.00000	0.436436
$337$	− 14.0000i	− 0.762629i	−0.924445	−	0.381314i	$-0.875472\pi$
$337$	− 14.0000i	− 0.762629i	0.924445	−	0.381314i	$-0.124528\pi$
$338$	− 12.0000i	− 0.652714i
$339$	−12.0000	−0.651751
$340$	0	0
$341$	−5.00000	−0.270765
$342$	− 7.00000i	− 0.378517i
$343$	− 8.00000i	− 0.431959i
$344$	5.00000	0.269582
$345$	0	0
$346$	−21.0000	−1.12897
$347$	12.0000i	0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	− 6.00000i	− 0.321634i
$349$	−11.0000	−0.588817	−0.294408	−	0.955680i	$-0.595123\pi$
$349$	−11.0000	−0.588817	−0.294408	+	0.955680i	$0.595123\pi$
$350$	0	0
$351$	20.0000	1.06752
$352$	− 1.00000i	− 0.0533002i
$353$	− 9.00000i	− 0.479022i	−0.970894	−	0.239511i	$-0.923013\pi$
$353$	− 9.00000i	− 0.479022i	0.970894	−	0.239511i	$-0.0769871\pi$
$354$	−24.0000	−1.27559
$355$	0	0
$356$	3.00000	0.159000
$357$	0	0
$358$	− 12.0000i	− 0.634220i
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	20.0000i	1.05118i
$363$	− 2.00000i	− 0.104973i
$364$	20.0000	1.04828
$365$	0	0
$366$	−20.0000	−1.04542
$367$	− 17.0000i	− 0.887393i	−0.896177	−	0.443696i	$-0.853667\pi$
$367$	− 17.0000i	− 0.887393i	0.896177	−	0.443696i	$-0.146333\pi$
$368$	3.00000i	0.156386i
$369$	−12.0000	−0.624695
$370$	0	0
$371$	−24.0000	−1.24602
$372$	10.0000i	0.518476i
$373$	2.00000i	0.103556i	0.998659	+	0.0517780i	$0.0164888\pi$
$373$	2.00000i	0.103556i	−0.998659	+	0.0517780i	$0.983511\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 15.0000i	− 0.772539i
$378$	16.0000i	0.822951i
$379$	−2.00000	−0.102733	−0.0513665	−	0.998680i	$-0.516358\pi$
$379$	−2.00000	−0.102733	−0.0513665	+	0.998680i	$0.516358\pi$
$380$	0	0
$381$	32.0000	1.63941
$382$	15.0000i	0.767467i
$383$	− 3.00000i	− 0.153293i	−0.997058	−	0.0766464i	$-0.975579\pi$
$383$	− 3.00000i	− 0.153293i	0.997058	−	0.0766464i	$-0.0244213\pi$
$384$	−2.00000	−0.102062
$385$	0	0
$386$	4.00000	0.203595
$387$	− 5.00000i	− 0.254164i
$388$	− 13.0000i	− 0.659975i
$389$	24.0000	1.21685	0.608424	−	0.793612i	$-0.291802\pi$
$389$	24.0000	1.21685	0.608424	+	0.793612i	$0.291802\pi$
$390$	0	0
$391$	0	0
$392$	9.00000i	0.454569i
$393$	− 42.0000i	− 2.11862i
$394$	3.00000	0.151138
$395$	0	0
$396$	−1.00000	−0.0502519
$397$	− 14.0000i	− 0.702640i	−0.936255	−	0.351320i	$-0.885733\pi$
$397$	− 14.0000i	− 0.702640i	0.936255	−	0.351320i	$-0.114267\pi$
$398$	− 11.0000i	− 0.551380i
$399$	56.0000	2.80351
$400$	0	0
$401$	−9.00000	−0.449439	−0.224719	−	0.974424i	$-0.572147\pi$
$401$	−9.00000	−0.449439	−0.224719	+	0.974424i	$0.572147\pi$
$402$	− 28.0000i	− 1.39651i
$403$	25.0000i	1.24534i
$404$	9.00000	0.447767
$405$	0	0
$406$	12.0000	0.595550
$407$	− 4.00000i	− 0.198273i
$408$	0	0
$409$	4.00000	0.197787	0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000	0.197787	0.0988936	+	0.995098i	$0.468470\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	7.00000i	0.344865i
$413$	− 48.0000i	− 2.36193i
$414$	3.00000	0.147442
$415$	0	0
$416$	−5.00000	−0.245145
$417$	− 26.0000i	− 1.27323i
$418$	− 7.00000i	− 0.342381i
$419$	−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
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	− 4.00000i	− 0.194717i
$423$	0	0
$424$	6.00000	0.291386
$425$	0	0
$426$	6.00000	0.290701
$427$	− 40.0000i	− 1.93574i
$428$	15.0000i	0.725052i
$429$	−10.0000	−0.482805
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	− 4.00000i	− 0.192450i
$433$	11.0000i	0.528626i	0.964437	+	0.264313i	$0.0851452\pi$
$433$	11.0000i	0.528626i	−0.964437	+	0.264313i	$0.914855\pi$
$434$	−20.0000	−0.960031
$435$	0	0
$436$	5.00000	0.239457
$437$	21.0000i	1.00457i
$438$	16.0000i	0.764510i
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	0	0
$443$	− 24.0000i	− 1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$	− 24.0000i	− 1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$	−8.00000	−0.379663
$445$	0	0
$446$	−8.00000	−0.378811
$447$	− 12.0000i	− 0.567581i
$448$	− 4.00000i	− 0.188982i
$449$	27.0000	1.27421	0.637104	−	0.770778i	$-0.280132\pi$
$449$	27.0000	1.27421	0.637104	+	0.770778i	$0.280132\pi$
$450$	0	0
$451$	−12.0000	−0.565058
$452$	6.00000i	0.282216i
$453$	32.0000i	1.50349i
$454$	−3.00000	−0.140797
$455$	0	0
$456$	−14.0000	−0.655610
$457$	28.0000i	1.30978i	0.755722	+	0.654892i	$0.227286\pi$
$457$	28.0000i	1.30978i	−0.755722	+	0.654892i	$0.772714\pi$
$458$	− 2.00000i	− 0.0934539i
$459$	0	0
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	− 8.00000i	− 0.372194i
$463$	− 13.0000i	− 0.604161i	−0.953282	−	0.302081i	$-0.902319\pi$
$463$	− 13.0000i	− 0.604161i	0.953282	−	0.302081i	$-0.0976812\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	18.0000	0.833834
$467$	− 18.0000i	− 0.832941i	−0.909149	−	0.416470i	$-0.863267\pi$
$467$	− 18.0000i	− 0.832941i	0.909149	−	0.416470i	$-0.136733\pi$
$468$	5.00000i	0.231125i
$469$	56.0000	2.58584
$470$	0	0
$471$	−28.0000	−1.29017
$472$	12.0000i	0.552345i
$473$	− 5.00000i	− 0.229900i
$474$	8.00000	0.367452
$475$	0	0
$476$	0	0
$477$	− 6.00000i	− 0.274721i
$478$	6.00000i	0.274434i
$479$	−6.00000	−0.274147	−0.137073	−	0.990561i	$-0.543770\pi$
$479$	−6.00000	−0.274147	−0.137073	+	0.990561i	$0.543770\pi$
$480$	0	0
$481$	−20.0000	−0.911922
$482$	− 10.0000i	− 0.455488i
$483$	24.0000i	1.09204i
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	−10.0000	−0.453609
$487$	− 29.0000i	− 1.31412i	−0.753840	−	0.657058i	$-0.771801\pi$
$487$	− 29.0000i	− 1.31412i	0.753840	−	0.657058i	$-0.228199\pi$
$488$	10.0000i	0.452679i
$489$	−32.0000	−1.44709
$490$	0	0
$491$	9.00000	0.406164	0.203082	−	0.979162i	$-0.434904\pi$
$491$	9.00000	0.406164	0.203082	+	0.979162i	$0.434904\pi$
$492$	24.0000i	1.08200i
$493$	0	0
$494$	−35.0000	−1.57472
$495$	0	0
$496$	5.00000	0.224507
$497$	12.0000i	0.538274i
$498$	− 30.0000i	− 1.34433i
$499$	10.0000	0.447661	0.223831	−	0.974628i	$-0.428144\pi$
$499$	10.0000	0.447661	0.223831	+	0.974628i	$0.428144\pi$
$500$	0	0
$501$	−48.0000	−2.14448
$502$	24.0000i	1.07117i
$503$	− 6.00000i	− 0.267527i	−0.991013	−	0.133763i	$-0.957294\pi$
$503$	− 6.00000i	− 0.267527i	0.991013	−	0.133763i	$-0.0427062\pi$
$504$	−4.00000	−0.178174
$505$	0	0
$506$	3.00000	0.133366
$507$	24.0000i	1.06588i
$508$	− 16.0000i	− 0.709885i
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	−32.0000	−1.41560
$512$	1.00000i	0.0441942i
$513$	− 28.0000i	− 1.23623i
$514$	−9.00000	−0.396973
$515$	0	0
$516$	−10.0000	−0.440225
$517$	0	0
$518$	− 16.0000i	− 0.703000i
$519$	42.0000	1.84360
$520$	0	0
$521$	−33.0000	−1.44576	−0.722878	−	0.690976i	$-0.757181\pi$
$521$	−33.0000	−1.44576	−0.722878	+	0.690976i	$0.757181\pi$
$522$	3.00000i	0.131306i
$523$	11.0000i	0.480996i	0.970650	+	0.240498i	$0.0773108\pi$
$523$	11.0000i	0.480996i	−0.970650	+	0.240498i	$0.922689\pi$
$524$	−21.0000	−0.917389
$525$	0	0
$526$	0	0
$527$	0	0
$528$	2.00000i	0.0870388i
$529$	14.0000	0.608696
$530$	0	0
$531$	12.0000	0.520756
$532$	− 28.0000i	− 1.21395i
$533$	60.0000i	2.59889i
$534$	−6.00000	−0.259645
$535$	0	0
$536$	−14.0000	−0.604708
$537$	24.0000i	1.03568i
$538$	− 30.0000i	− 1.29339i
$539$	9.00000	0.387657
$540$	0	0
$541$	−31.0000	−1.33279	−0.666397	−	0.745597i	$-0.732164\pi$
$541$	−31.0000	−1.33279	−0.666397	+	0.745597i	$0.732164\pi$
$542$	2.00000i	0.0859074i
$543$	− 40.0000i	− 1.71656i
$544$	0	0
$545$	0	0
$546$	−40.0000	−1.71184
$547$	1.00000i	0.0427569i	0.999771	+	0.0213785i	$0.00680549\pi$
$547$	1.00000i	0.0427569i	−0.999771	+	0.0213785i	$0.993195\pi$
$548$	3.00000i	0.128154i
$549$	10.0000	0.426790
$550$	0	0
$551$	−21.0000	−0.894630
$552$	− 6.00000i	− 0.255377i
$553$	16.0000i	0.680389i
$554$	26.0000	1.10463
$555$	0	0
$556$	−13.0000	−0.551323
$557$	− 27.0000i	− 1.14403i	−0.820244	−	0.572013i	$-0.806163\pi$
$557$	− 27.0000i	− 1.14403i	0.820244	−	0.572013i	$-0.193837\pi$
$558$	− 5.00000i	− 0.211667i
$559$	−25.0000	−1.05739
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12.0000i	0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$	12.0000i	0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$	0	0
$565$	0	0
$566$	4.00000	0.168133
$567$	− 44.0000i	− 1.84783i
$568$	− 3.00000i	− 0.125877i
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−7.00000	−0.292941	−0.146470	−	0.989215i	$-0.546791\pi$
$571$	−7.00000	−0.292941	−0.146470	+	0.989215i	$0.546791\pi$
$572$	5.00000i	0.209061i
$573$	− 30.0000i	− 1.25327i
$574$	−48.0000	−2.00348
$575$	0	0
$576$	1.00000	0.0416667
$577$	− 2.00000i	− 0.0832611i	−0.999133	−	0.0416305i	$-0.986745\pi$
$577$	− 2.00000i	− 0.0832611i	0.999133	−	0.0416305i	$-0.0132552\pi$
$578$	17.0000i	0.707107i
$579$	−8.00000	−0.332469
$580$	0	0
$581$	60.0000	2.48922
$582$	26.0000i	1.07773i
$583$	− 6.00000i	− 0.248495i
$584$	8.00000	0.331042
$585$	0	0
$586$	18.0000	0.743573
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	− 18.0000i	− 0.742307i
$589$	35.0000	1.44215
$590$	0	0
$591$	−6.00000	−0.246807
$592$	4.00000i	0.164399i
$593$	− 6.00000i	− 0.246390i	−0.992382	−	0.123195i	$-0.960686\pi$
$593$	− 6.00000i	− 0.246390i	0.992382	−	0.123195i	$-0.0393141\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	−6.00000	−0.245770
$597$	22.0000i	0.900400i
$598$	− 15.0000i	− 0.613396i
$599$	36.0000	1.47092	0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000	1.47092	0.735460	+	0.677568i	$0.236966\pi$
$600$	0	0
$601$	44.0000	1.79480	0.897399	−	0.441221i	$-0.145454\pi$
$601$	44.0000	1.79480	0.897399	+	0.441221i	$0.145454\pi$
$602$	− 20.0000i	− 0.815139i
$603$	14.0000i	0.570124i
$604$	16.0000	0.651031
$605$	0	0
$606$	−18.0000	−0.731200
$607$	− 14.0000i	− 0.568242i	−0.958788	−	0.284121i	$-0.908298\pi$
$607$	− 14.0000i	− 0.568242i	0.958788	−	0.284121i	$-0.0917018\pi$
$608$	7.00000i	0.283887i
$609$	−24.0000	−0.972529
$610$	0	0
$611$	0	0
$612$	0	0
$613$	26.0000i	1.05013i	0.851062	+	0.525065i	$0.175959\pi$
$613$	26.0000i	1.05013i	−0.851062	+	0.525065i	$0.824041\pi$
$614$	−16.0000	−0.645707
$615$	0	0
$616$	−4.00000	−0.161165
$617$	27.0000i	1.08698i	0.839416	+	0.543490i	$0.182897\pi$
$617$	27.0000i	1.08698i	−0.839416	+	0.543490i	$0.817103\pi$
$618$	− 14.0000i	− 0.563163i
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	12.0000	0.481543
$622$	3.00000i	0.120289i
$623$	− 12.0000i	− 0.480770i
$624$	10.0000	0.400320
$625$	0	0
$626$	−26.0000	−1.03917
$627$	14.0000i	0.559106i
$628$	14.0000i	0.558661i
$629$	0	0
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	− 4.00000i	− 0.159111i
$633$	8.00000i	0.317971i
$634$	12.0000	0.476581
$635$	0	0
$636$	−12.0000	−0.475831
$637$	− 45.0000i	− 1.78296i
$638$	3.00000i	0.118771i
$639$	−3.00000	−0.118678
$640$	0	0
$641$	−27.0000	−1.06644	−0.533218	−	0.845978i	$-0.679017\pi$
$641$	−27.0000	−1.06644	−0.533218	+	0.845978i	$0.679017\pi$
$642$	− 30.0000i	− 1.18401i
$643$	− 4.00000i	− 0.157745i	−0.996885	−	0.0788723i	$-0.974868\pi$
$643$	− 4.00000i	− 0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$	12.0000	0.472866
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	11.0000i	0.432121i
$649$	12.0000	0.471041
$650$	0	0
$651$	40.0000	1.56772
$652$	16.0000i	0.626608i
$653$	30.0000i	1.17399i	0.809590	+	0.586995i	$0.199689\pi$
$653$	30.0000i	1.17399i	−0.809590	+	0.586995i	$0.800311\pi$
$654$	−10.0000	−0.391031
$655$	0	0
$656$	12.0000	0.468521
$657$	− 8.00000i	− 0.312110i
$658$	0	0
$659$	−3.00000	−0.116863	−0.0584317	−	0.998291i	$-0.518610\pi$
$659$	−3.00000	−0.116863	−0.0584317	+	0.998291i	$0.518610\pi$
$660$	0	0
$661$	−16.0000	−0.622328	−0.311164	−	0.950356i	$-0.600719\pi$
$661$	−16.0000	−0.622328	−0.311164	+	0.950356i	$0.600719\pi$
$662$	32.0000i	1.24372i
$663$	0	0
$664$	−15.0000	−0.582113
$665$	0	0
$666$	4.00000	0.154997
$667$	− 9.00000i	− 0.348481i
$668$	24.0000i	0.928588i
$669$	16.0000	0.618596
$670$	0	0
$671$	10.0000	0.386046
$672$	8.00000i	0.308607i
$673$	− 4.00000i	− 0.154189i	−0.997024	−	0.0770943i	$-0.975436\pi$
$673$	− 4.00000i	− 0.154189i	0.997024	−	0.0770943i	$-0.0245643\pi$
$674$	14.0000	0.539260
$675$	0	0
$676$	12.0000	0.461538
$677$	− 9.00000i	− 0.345898i	−0.984931	−	0.172949i	$-0.944670\pi$
$677$	− 9.00000i	− 0.345898i	0.984931	−	0.172949i	$-0.0553296\pi$
$678$	− 12.0000i	− 0.460857i
$679$	−52.0000	−1.99558
$680$	0	0
$681$	6.00000	0.229920
$682$	− 5.00000i	− 0.191460i
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	7.00000	0.267652
$685$	0	0
$686$	8.00000	0.305441
$687$	4.00000i	0.152610i
$688$	5.00000i	0.190623i
$689$	−30.0000	−1.14291
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	− 21.0000i	− 0.798300i
$693$	4.00000i	0.151947i
$694$	−12.0000	−0.455514
$695$	0	0
$696$	6.00000	0.227429
$697$	0	0
$698$	− 11.0000i	− 0.416356i
$699$	−36.0000	−1.36165
$700$	0	0
$701$	45.0000	1.69963	0.849813	−	0.527084i	$-0.176715\pi$
$701$	45.0000	1.69963	0.849813	+	0.527084i	$0.176715\pi$
$702$	20.0000i	0.754851i
$703$	28.0000i	1.05604i
$704$	1.00000	0.0376889
$705$	0	0
$706$	9.00000	0.338719
$707$	− 36.0000i	− 1.35392i
$708$	− 24.0000i	− 0.901975i
$709$	16.0000	0.600893	0.300446	−	0.953799i	$-0.402864\pi$
$709$	16.0000	0.600893	0.300446	+	0.953799i	$0.402864\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	3.00000i	0.112430i
$713$	15.0000i	0.561754i
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	− 12.0000i	− 0.448148i
$718$	24.0000i	0.895672i
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	0	0
$721$	28.0000	1.04277
$722$	30.0000i	1.11648i
$723$	20.0000i	0.743808i
$724$	−20.0000	−0.743294
$725$	0	0
$726$	2.00000	0.0742270
$727$	− 5.00000i	− 0.185440i	−0.995692	−	0.0927199i	$-0.970444\pi$
$727$	− 5.00000i	− 0.185440i	0.995692	−	0.0927199i	$-0.0295561\pi$
$728$	20.0000i	0.741249i
$729$	−13.0000	−0.481481
$730$	0	0
$731$	0	0
$732$	− 20.0000i	− 0.739221i
$733$	5.00000i	0.184679i	0.995728	+	0.0923396i	$0.0294345\pi$
$733$	5.00000i	0.184679i	−0.995728	+	0.0923396i	$0.970565\pi$
$734$	17.0000	0.627481
$735$	0	0
$736$	−3.00000	−0.110581
$737$	14.0000i	0.515697i
$738$	− 12.0000i	− 0.441726i
$739$	16.0000	0.588570	0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000	0.588570	0.294285	+	0.955718i	$0.404919\pi$
$740$	0	0
$741$	70.0000	2.57151
$742$	− 24.0000i	− 0.881068i
$743$	54.0000i	1.98107i	0.137268	+	0.990534i	$0.456168\pi$
$743$	54.0000i	1.98107i	−0.137268	+	0.990534i	$0.543832\pi$
$744$	−10.0000	−0.366618
$745$	0	0
$746$	−2.00000	−0.0732252
$747$	15.0000i	0.548821i
$748$	0	0
$749$	60.0000	2.19235
$750$	0	0
$751$	−25.0000	−0.912263	−0.456131	−	0.889912i	$-0.650765\pi$
$751$	−25.0000	−0.912263	−0.456131	+	0.889912i	$0.650765\pi$
$752$	0	0
$753$	− 48.0000i	− 1.74922i
$754$	15.0000	0.546268
$755$	0	0
$756$	−16.0000	−0.581914
$757$	− 8.00000i	− 0.290765i	−0.989376	−	0.145382i	$-0.953559\pi$
$757$	− 8.00000i	− 0.290765i	0.989376	−	0.145382i	$-0.0464413\pi$
$758$	− 2.00000i	− 0.0726433i
$759$	−6.00000	−0.217786
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	32.0000i	1.15924i
$763$	− 20.0000i	− 0.724049i
$764$	−15.0000	−0.542681
$765$	0	0
$766$	3.00000	0.108394
$767$	− 60.0000i	− 2.16647i
$768$	− 2.00000i	− 0.0721688i
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	4.00000i	0.143963i
$773$	− 36.0000i	− 1.29483i	−0.762138	−	0.647415i	$-0.775850\pi$
$773$	− 36.0000i	− 1.29483i	0.762138	−	0.647415i	$-0.224150\pi$
$774$	5.00000	0.179721
$775$	0	0
$776$	13.0000	0.466673
$777$	32.0000i	1.14799i
$778$	24.0000i	0.860442i
$779$	84.0000	3.00961
$780$	0	0
$781$	−3.00000	−0.107348
$782$	0	0
$783$	12.0000i	0.428845i
$784$	−9.00000	−0.321429
$785$	0	0
$786$	42.0000	1.49809
$787$	28.0000i	0.998092i	0.866575	+	0.499046i	$0.166316\pi$
$787$	28.0000i	0.998092i	−0.866575	+	0.499046i	$0.833684\pi$
$788$	3.00000i	0.106871i
$789$	0	0
$790$	0	0
$791$	24.0000	0.853342
$792$	− 1.00000i	− 0.0355335i
$793$	− 50.0000i	− 1.77555i
$794$	14.0000	0.496841
$795$	0	0
$796$	11.0000	0.389885
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	56.0000i	1.98238i
$799$	0	0
$800$	0	0
$801$	3.00000	0.106000
$802$	− 9.00000i	− 0.317801i
$803$	− 8.00000i	− 0.282314i
$804$	28.0000	0.987484
$805$	0	0
$806$	−25.0000	−0.880587
$807$	60.0000i	2.11210i
$808$	9.00000i	0.316619i
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	0	0
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	12.0000i	0.421117i
$813$	− 4.00000i	− 0.140286i
$814$	4.00000	0.140200
$815$	0	0
$816$	0	0
$817$	35.0000i	1.22449i
$818$	4.00000i	0.139857i
$819$	20.0000	0.698857
$820$	0	0
$821$	−9.00000	−0.314102	−0.157051	−	0.987590i	$-0.550199\pi$
$821$	−9.00000	−0.314102	−0.157051	+	0.987590i	$0.550199\pi$
$822$	− 6.00000i	− 0.209274i
$823$	− 40.0000i	− 1.39431i	−0.716919	−	0.697156i	$-0.754448\pi$
$823$	− 40.0000i	− 1.39431i	0.716919	−	0.697156i	$-0.245552\pi$
$824$	−7.00000	−0.243857
$825$	0	0
$826$	48.0000	1.67013
$827$	27.0000i	0.938882i	0.882964	+	0.469441i	$0.155545\pi$
$827$	27.0000i	0.938882i	−0.882964	+	0.469441i	$0.844455\pi$
$828$	3.00000i	0.104257i
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	0	0
$831$	−52.0000	−1.80386
$832$	− 5.00000i	− 0.173344i
$833$	0	0
$834$	26.0000	0.900306
$835$	0	0
$836$	7.00000	0.242100
$837$	− 20.0000i	− 0.691301i
$838$	− 30.0000i	− 1.03633i
$839$	−48.0000	−1.65714	−0.828572	−	0.559883i	$-0.810846\pi$
$839$	−48.0000	−1.65714	−0.828572	+	0.559883i	$0.810846\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	26.0000i	0.896019i
$843$	0	0
$844$	4.00000	0.137686
$845$	0	0
$846$	0	0
$847$	4.00000i	0.137442i
$848$	6.00000i	0.206041i
$849$	−8.00000	−0.274559
$850$	0	0
$851$	−12.0000	−0.411355
$852$	6.00000i	0.205557i
$853$	− 10.0000i	− 0.342393i	−0.985237	−	0.171197i	$-0.945237\pi$
$853$	− 10.0000i	− 0.342393i	0.985237	−	0.171197i	$-0.0547634\pi$
$854$	40.0000	1.36877
$855$	0	0
$856$	−15.0000	−0.512689
$857$	48.0000i	1.63965i	0.572615	+	0.819824i	$0.305929\pi$
$857$	48.0000i	1.63965i	−0.572615	+	0.819824i	$0.694071\pi$
$858$	− 10.0000i	− 0.341394i
$859$	−26.0000	−0.887109	−0.443554	−	0.896248i	$-0.646283\pi$
$859$	−26.0000	−0.887109	−0.443554	+	0.896248i	$0.646283\pi$
$860$	0	0
$861$	96.0000	3.27167
$862$	12.0000i	0.408722i
$863$	− 39.0000i	− 1.32758i	−0.747921	−	0.663788i	$-0.768948\pi$
$863$	− 39.0000i	− 1.32758i	0.747921	−	0.663788i	$-0.231052\pi$
$864$	4.00000	0.136083
$865$	0	0
$866$	−11.0000	−0.373795
$867$	− 34.0000i	− 1.15470i
$868$	− 20.0000i	− 0.678844i
$869$	−4.00000	−0.135691
$870$	0	0
$871$	70.0000	2.37186
$872$	5.00000i	0.169321i
$873$	− 13.0000i	− 0.439983i
$874$	−21.0000	−0.710336
$875$	0	0
$876$	−16.0000	−0.540590
$877$	− 23.0000i	− 0.776655i	−0.921521	−	0.388327i	$-0.873053\pi$
$877$	− 23.0000i	− 0.776655i	0.921521	−	0.388327i	$-0.126947\pi$
$878$	− 20.0000i	− 0.674967i
$879$	−36.0000	−1.21425
$880$	0	0
$881$	−27.0000	−0.909653	−0.454827	−	0.890580i	$-0.650299\pi$
$881$	−27.0000	−0.909653	−0.454827	+	0.890580i	$0.650299\pi$
$882$	9.00000i	0.303046i
$883$	44.0000i	1.48072i	0.672212	+	0.740359i	$0.265344\pi$
$883$	44.0000i	1.48072i	−0.672212	+	0.740359i	$0.734656\pi$
$884$	0	0
$885$	0	0
$886$	24.0000	0.806296
$887$	− 6.00000i	− 0.201460i	−0.994914	−	0.100730i	$-0.967882\pi$
$887$	− 6.00000i	− 0.201460i	0.994914	−	0.100730i	$-0.0321179\pi$
$888$	− 8.00000i	− 0.268462i
$889$	−64.0000	−2.14649
$890$	0	0
$891$	11.0000	0.368514
$892$	− 8.00000i	− 0.267860i
$893$	0	0
$894$	12.0000	0.401340
$895$	0	0
$896$	4.00000	0.133631
$897$	30.0000i	1.00167i
$898$	27.0000i	0.901002i
$899$	−15.0000	−0.500278
$900$	0	0
$901$	0	0
$902$	− 12.0000i	− 0.399556i
$903$	40.0000i	1.33112i
$904$	−6.00000	−0.199557
$905$	0	0
$906$	−32.0000	−1.06313
$907$	34.0000i	1.12895i	0.825450	+	0.564476i	$0.190922\pi$
$907$	34.0000i	1.12895i	−0.825450	+	0.564476i	$0.809078\pi$
$908$	− 3.00000i	− 0.0995585i
$909$	9.00000	0.298511
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	− 14.0000i	− 0.463586i
$913$	15.0000i	0.496428i
$914$	−28.0000	−0.926158
$915$	0	0
$916$	2.00000	0.0660819
$917$	84.0000i	2.77392i
$918$	0	0
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	0	0
$921$	32.0000	1.05444
$922$	− 6.00000i	− 0.197599i
$923$	15.0000i	0.493731i
$924$	8.00000	0.263181
$925$	0	0
$926$	13.0000	0.427207
$927$	7.00000i	0.229910i
$928$	− 3.00000i	− 0.0984798i
$929$	33.0000	1.08269	0.541347	−	0.840799i	$-0.317914\pi$
$929$	33.0000	1.08269	0.541347	+	0.840799i	$0.317914\pi$
$930$	0	0
$931$	−63.0000	−2.06474
$932$	18.0000i	0.589610i
$933$	− 6.00000i	− 0.196431i
$934$	18.0000	0.588978
$935$	0	0
$936$	−5.00000	−0.163430
$937$	− 26.0000i	− 0.849383i	−0.905338	−	0.424691i	$-0.860383\pi$
$937$	− 26.0000i	− 0.849383i	0.905338	−	0.424691i	$-0.139617\pi$
$938$	56.0000i	1.82846i
$939$	52.0000	1.69696
$940$	0	0
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	− 28.0000i	− 0.912289i
$943$	36.0000i	1.17232i
$944$	−12.0000	−0.390567
$945$	0	0
$946$	5.00000	0.162564
$947$	− 30.0000i	− 0.974869i	−0.873160	−	0.487435i	$-0.837933\pi$
$947$	− 30.0000i	− 0.974869i	0.873160	−	0.487435i	$-0.162067\pi$
$948$	8.00000i	0.259828i
$949$	−40.0000	−1.29845
$950$	0	0
$951$	−24.0000	−0.778253
$952$	0	0
$953$	6.00000i	0.194359i	0.995267	+	0.0971795i	$0.0309821\pi$
$953$	6.00000i	0.194359i	−0.995267	+	0.0971795i	$0.969018\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	−6.00000	−0.194054
$957$	− 6.00000i	− 0.193952i
$958$	− 6.00000i	− 0.193851i
$959$	12.0000	0.387500
$960$	0	0
$961$	−6.00000	−0.193548
$962$	− 20.0000i	− 0.644826i
$963$	15.0000i	0.483368i
$964$	10.0000	0.322078
$965$	0	0
$966$	−24.0000	−0.772187
$967$	− 26.0000i	− 0.836104i	−0.908423	−	0.418052i	$-0.862713\pi$
$967$	− 26.0000i	− 0.836104i	0.908423	−	0.418052i	$-0.137287\pi$
$968$	− 1.00000i	− 0.0321412i
$969$	0	0
$970$	0	0
$971$	−6.00000	−0.192549	−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000	−0.192549	−0.0962746	+	0.995355i	$0.530693\pi$
$972$	− 10.0000i	− 0.320750i
$973$	52.0000i	1.66704i
$974$	29.0000	0.929220
$975$	0	0
$976$	−10.0000	−0.320092
$977$	42.0000i	1.34370i	0.740688	+	0.671850i	$0.234500\pi$
$977$	42.0000i	1.34370i	−0.740688	+	0.671850i	$0.765500\pi$
$978$	− 32.0000i	− 1.02325i
$979$	3.00000	0.0958804
$980$	0	0
$981$	5.00000	0.159638
$982$	9.00000i	0.287202i
$983$	39.0000i	1.24391i	0.783054	+	0.621953i	$0.213661\pi$
$983$	39.0000i	1.24391i	−0.783054	+	0.621953i	$0.786339\pi$
$984$	−24.0000	−0.765092
$985$	0	0
$986$	0	0
$987$	0	0
$988$	− 35.0000i	− 1.11350i
$989$	−15.0000	−0.476972
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	5.00000i	0.158750i
$993$	− 64.0000i	− 2.03098i
$994$	−12.0000	−0.380617
$995$	0	0
$996$	30.0000	0.950586
$997$	46.0000i	1.45683i	0.685134	+	0.728417i	$0.259744\pi$
$997$	46.0000i	1.45683i	−0.685134	+	0.728417i	$0.740256\pi$
$998$	10.0000i	0.316544i
$999$	16.0000	0.506218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	550.2.b.d.199.2		2
3.2	odd	2		4950.2.c.ba.199.1		2
4.3	odd	2		4400.2.b.e.4049.2		2
5.2	odd	4		550.2.a.a.1.1	✓	1
5.3	odd	4		550.2.a.m.1.1	yes	1
5.4	even	2	inner	550.2.b.d.199.1		2
15.2	even	4		4950.2.a.y.1.1		1
15.8	even	4		4950.2.a.u.1.1		1
15.14	odd	2		4950.2.c.ba.199.2		2
20.3	even	4		4400.2.a.d.1.1		1
20.7	even	4		4400.2.a.bc.1.1		1
20.19	odd	2		4400.2.b.e.4049.1		2
55.32	even	4		6050.2.a.bb.1.1		1
55.43	even	4		6050.2.a.n.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
550.2.a.a.1.1	✓	1	5.2	odd	4
550.2.a.m.1.1	yes	1	5.3	odd	4
550.2.b.d.199.1		2	5.4	even	2	inner
550.2.b.d.199.2		2	1.1	even	1	trivial
4400.2.a.d.1.1		1	20.3	even	4
4400.2.a.bc.1.1		1	20.7	even	4
4400.2.b.e.4049.1		2	20.19	odd	2
4400.2.b.e.4049.2		2	4.3	odd	2
4950.2.a.u.1.1		1	15.8	even	4
4950.2.a.y.1.1		1	15.2	even	4
4950.2.c.ba.199.1		2	3.2	odd	2
4950.2.c.ba.199.2		2	15.14	odd	2
6050.2.a.n.1.1		1	55.43	even	4
6050.2.a.bb.1.1		1	55.32	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	550.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} +4.00000i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.00000 q^{11} +2.00000i q^{12} +5.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} -1.00000i q^{18} +7.00000 q^{19} +8.00000 q^{21} -1.00000i q^{22} +3.00000i q^{23} -2.00000 q^{24} -5.00000 q^{26} -4.00000i q^{27} -4.00000i q^{28} -3.00000 q^{29} +5.00000 q^{31} +1.00000i q^{32} +2.00000i q^{33} +1.00000 q^{36} +4.00000i q^{37} +7.00000i q^{38} +10.0000 q^{39} +12.0000 q^{41} +8.00000i q^{42} +5.00000i q^{43} +1.00000 q^{44} -3.00000 q^{46} -2.00000i q^{48} -9.00000 q^{49} -5.00000i q^{52} +6.00000i q^{53} +4.00000 q^{54} +4.00000 q^{56} -14.0000i q^{57} -3.00000i q^{58} -12.0000 q^{59} -10.0000 q^{61} +5.00000i q^{62} -4.00000i q^{63} -1.00000 q^{64} -2.00000 q^{66} -14.0000i q^{67} +6.00000 q^{69} +3.00000 q^{71} +1.00000i q^{72} +8.00000i q^{73} -4.00000 q^{74} -7.00000 q^{76} -4.00000i q^{77} +10.0000i q^{78} +4.00000 q^{79} -11.0000 q^{81} +12.0000i q^{82} -15.0000i q^{83} -8.00000 q^{84} -5.00000 q^{86} +6.00000i q^{87} +1.00000i q^{88} -3.00000 q^{89} -20.0000 q^{91} -3.00000i q^{92} -10.0000i q^{93} +2.00000 q^{96} +13.0000i q^{97} -9.00000i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9} - 2 q^{11} - 8 q^{14} + 2 q^{16} + 14 q^{19} + 16 q^{21} - 4 q^{24} - 10 q^{26} - 6 q^{29} + 10 q^{31} + 2 q^{36} + 20 q^{39} + 24 q^{41} + 2 q^{44} - 6 q^{46} - 18 q^{49}+ \cdots + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9 - 2 * q^11 - 8 * q^14 + 2 * q^16 + 14 * q^19 + 16 * q^21 - 4 * q^24 - 10 * q^26 - 6 * q^29 + 10 * q^31 + 2 * q^36 + 20 * q^39 + 24 * q^41 + 2 * q^44 - 6 * q^46 - 18 * q^49 + 8 * q^54 + 8 * q^56 - 24 * q^59 - 20 * q^61 - 2 * q^64 - 4 * q^66 + 12 * q^69 + 6 * q^71 - 8 * q^74 - 14 * q^76 + 8 * q^79 - 22 * q^81 - 16 * q^84 - 10 * q^86 - 6 * q^89 - 40 * q^91 + 4 * q^96 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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