LMFDB - Embedded newform 4400.2.b.g.4049.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4400,2,Mod(4049,4400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4400, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4400.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4400 = 2^{4} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4400.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$35.1341768894$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 110)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	4400.4049
Dual form			4400.2.b.g.4049.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +5.00000i q^{7} +2.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +5.00000i q^{7} +2.00000 q^{9} -1.00000 q^{11} +2.00000i q^{13} -3.00000i q^{17} -7.00000 q^{19} +5.00000 q^{21} +6.00000i q^{23} -5.00000i q^{27} +3.00000 q^{29} +7.00000 q^{31} +1.00000i q^{33} +7.00000i q^{37} +2.00000 q^{39} +6.00000 q^{41} -8.00000i q^{43} +6.00000i q^{47} -18.0000 q^{49} -3.00000 q^{51} -3.00000i q^{53} +7.00000i q^{57} -6.00000 q^{59} -1.00000 q^{61} +10.0000i q^{63} +8.00000i q^{67} +6.00000 q^{69} -3.00000 q^{71} +2.00000i q^{73} -5.00000i q^{77} -10.0000 q^{79} +1.00000 q^{81} +6.00000i q^{83} -3.00000i q^{87} -9.00000 q^{89} -10.0000 q^{91} -7.00000i q^{93} +4.00000i q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{9}+O(q^{10}) $ 2 * q + 4 * q^9	$ 2 q + 4 q^{9} - 2 q^{11} - 14 q^{19} + 10 q^{21} + 6 q^{29} + 14 q^{31} + 4 q^{39} + 12 q^{41} - 36 q^{49} - 6 q^{51} - 12 q^{59} - 2 q^{61} + 12 q^{69} - 6 q^{71} - 20 q^{79} + 2 q^{81} - 18 q^{89} - 20 q^{91} - 4 q^{99}+O(q^{100}) $ 2 * q + 4 * q^9 - 2 * q^11 - 14 * q^19 + 10 * q^21 + 6 * q^29 + 14 * q^31 + 4 * q^39 + 12 * q^41 - 36 * q^49 - 6 * q^51 - 12 * q^59 - 2 * q^61 + 12 * q^69 - 6 * q^71 - 20 * q^79 + 2 * q^81 - 18 * q^89 - 20 * q^91 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times$.

$n$	$177$	$1201$	$2751$	$3301$
$\chi(n)$	$-1$	$1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 1.00000i	− 0.577350i	−0.957427	−	0.288675i	$-0.906785\pi$
$3$	− 1.00000i	− 0.577350i	0.957427	−	0.288675i	$-0.0932147\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	5.00000i	1.88982i	0.327327	+	0.944911i	$0.393852\pi$
$7$	5.00000i	1.88982i	−0.327327	+	0.944911i	$0.606148\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 3.00000i	− 0.727607i	−0.931476	−	0.363803i	$-0.881478\pi$
$17$	− 3.00000i	− 0.727607i	0.931476	−	0.363803i	$-0.118522\pi$
$18$	0	0
$19$	−7.00000	−1.60591	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000	−1.60591	−0.802955	+	0.596040i	$0.796740\pi$
$20$	0	0
$21$	5.00000	1.09109
$22$	0	0
$23$	6.00000i	1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$	6.00000i	1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 5.00000i	− 0.962250i
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	0	0
$33$	1.00000i	0.174078i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.00000i	1.15079i	0.817875	+	0.575396i	$0.195152\pi$
$37$	7.00000i	1.15079i	−0.817875	+	0.575396i	$0.804848\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.00000i	0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$	6.00000i	0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$	0	0
$49$	−18.0000	−2.57143
$50$	0	0
$51$	−3.00000	−0.420084
$52$	0	0
$53$	− 3.00000i	− 0.412082i	−0.978543	−	0.206041i	$-0.933942\pi$
$53$	− 3.00000i	− 0.412082i	0.978543	−	0.206041i	$-0.0660580\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	7.00000i	0.927173i
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	0	0
$63$	10.0000i	1.25988i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.00000i	0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$	8.00000i	0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$	0	0
$69$	6.00000	0.722315
$70$	0	0
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	0	0
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 5.00000i	− 0.569803i
$78$	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
$81$	1.00000	0.111111
$82$	0	0
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 3.00000i	− 0.321634i
$88$	0	0
$89$	−9.00000	−0.953998	−0.476999	−	0.878904i	$-0.658275\pi$
$89$	−9.00000	−0.953998	−0.476999	+	0.878904i	$0.658275\pi$
$90$	0	0
$91$	−10.0000	−1.04828
$92$	0	0
$93$	− 7.00000i	− 0.725866i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	4.00000i	0.406138i	0.979164	+	0.203069i	$0.0650917\pi$
$97$	4.00000i	0.406138i	−0.979164	+	0.203069i	$0.934908\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	4.00000i	0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$	4.00000i	0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	7.00000	0.664411
$112$	0	0
$113$	12.0000i	1.12887i	0.825479	+	0.564433i	$0.190905\pi$
$113$	12.0000i	1.12887i	−0.825479	+	0.564433i	$0.809095\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.00000i	0.369800i
$118$	0	0
$119$	15.0000	1.37505
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	− 6.00000i	− 0.541002i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 16.0000i	− 1.41977i	−0.704317	−	0.709885i	$-0.748747\pi$
$127$	− 16.0000i	− 1.41977i	0.704317	−	0.709885i	$-0.251253\pi$
$128$	0	0
$129$	−8.00000	−0.704361
$130$	0	0
$131$	3.00000	0.262111	0.131056	−	0.991375i	$-0.458163\pi$
$131$	3.00000	0.262111	0.131056	+	0.991375i	$0.458163\pi$
$132$	0	0
$133$	− 35.0000i	− 3.03488i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 12.0000i	− 1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$	− 12.0000i	− 1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	0	0
$143$	− 2.00000i	− 0.167248i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	18.0000i	1.48461i
$148$	0	0
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	0	0
$153$	− 6.00000i	− 0.485071i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	7.00000i	0.558661i	0.960195	+	0.279330i	$0.0901125\pi$
$157$	7.00000i	0.558661i	−0.960195	+	0.279330i	$0.909888\pi$
$158$	0	0
$159$	−3.00000	−0.237915
$160$	0	0
$161$	−30.0000	−2.36433
$162$	0	0
$163$	− 5.00000i	− 0.391630i	−0.980641	−	0.195815i	$-0.937265\pi$
$163$	− 5.00000i	− 0.391630i	0.980641	−	0.195815i	$-0.0627352\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	21.0000i	1.62503i	0.582941	+	0.812514i	$0.301902\pi$
$167$	21.0000i	1.62503i	−0.582941	+	0.812514i	$0.698098\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	−14.0000	−1.07061
$172$	0	0
$173$	18.0000i	1.36851i	0.729241	+	0.684257i	$0.239873\pi$
$173$	18.0000i	1.36851i	−0.729241	+	0.684257i	$0.760127\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.00000i	0.450988i
$178$	0	0
$179$	−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
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	1.00000i	0.0739221i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.00000i	0.219382i
$188$	0	0
$189$	25.0000	1.81848
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	23.0000i	1.65558i	0.561041	+	0.827788i	$0.310401\pi$
$193$	23.0000i	1.65558i	−0.561041	+	0.827788i	$0.689599\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	11.0000	0.779769	0.389885	−	0.920864i	$-0.372515\pi$
$199$	11.0000	0.779769	0.389885	+	0.920864i	$0.372515\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	0	0
$203$	15.0000i	1.05279i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	12.0000i	0.834058i
$208$	0	0
$209$	7.00000	0.484200
$210$	0	0
$211$	−5.00000	−0.344214	−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000	−0.344214	−0.172107	+	0.985078i	$0.555058\pi$
$212$	0	0
$213$	3.00000i	0.205557i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	35.0000i	2.37595i
$218$	0	0
$219$	2.00000	0.135147
$220$	0	0
$221$	6.00000	0.403604
$222$	0	0
$223$	− 14.0000i	− 0.937509i	−0.883328	−	0.468755i	$-0.844703\pi$
$223$	− 14.0000i	− 0.937509i	0.883328	−	0.468755i	$-0.155297\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 18.0000i	− 1.19470i	−0.801980	−	0.597351i	$-0.796220\pi$
$227$	− 18.0000i	− 1.19470i	0.801980	−	0.597351i	$-0.203780\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	−5.00000	−0.328976
$232$	0	0
$233$	27.0000i	1.76883i	0.466702	+	0.884414i	$0.345442\pi$
$233$	27.0000i	1.76883i	−0.466702	+	0.884414i	$0.654558\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	10.0000i	0.649570i
$238$	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
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	0	0
$243$	− 16.0000i	− 1.02640i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 14.0000i	− 0.890799i
$248$	0	0
$249$	6.00000	0.380235
$250$	0	0
$251$	30.0000	1.89358	0.946792	−	0.321847i	$-0.104304\pi$
$251$	30.0000	1.89358	0.946792	+	0.321847i	$0.104304\pi$
$252$	0	0
$253$	− 6.00000i	− 0.377217i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 6.00000i	− 0.374270i	−0.982334	−	0.187135i	$-0.940080\pi$
$257$	− 6.00000i	− 0.374270i	0.982334	−	0.187135i	$-0.0599201\pi$
$258$	0	0
$259$	−35.0000	−2.17479
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	9.00000i	0.554964i	0.960731	+	0.277482i	$0.0894999\pi$
$263$	9.00000i	0.554964i	−0.960731	+	0.277482i	$0.910500\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	9.00000i	0.550791i
$268$	0	0
$269$	12.0000	0.731653	0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000	0.731653	0.365826	+	0.930683i	$0.380786\pi$
$270$	0	0
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	0	0
$273$	10.0000i	0.605228i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	4.00000i	0.240337i	0.992754	+	0.120168i	$0.0383434\pi$
$277$	4.00000i	0.240337i	−0.992754	+	0.120168i	$0.961657\pi$
$278$	0	0
$279$	14.0000	0.838158
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	− 2.00000i	− 0.118888i	−0.998232	−	0.0594438i	$-0.981067\pi$
$283$	− 2.00000i	− 0.118888i	0.998232	−	0.0594438i	$-0.0189327\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	30.0000i	1.77084i
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	4.00000	0.234484
$292$	0	0
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.00000i	0.290129i
$298$	0	0
$299$	−12.0000	−0.693978
$300$	0	0
$301$	40.0000	2.30556
$302$	0	0
$303$	6.00000i	0.344691i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 10.0000i	− 0.570730i	−0.958419	−	0.285365i	$-0.907885\pi$
$307$	− 10.0000i	− 0.570730i	0.958419	−	0.285365i	$-0.0921148\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	15.0000	0.850572	0.425286	−	0.905059i	$-0.360174\pi$
$311$	15.0000	0.850572	0.425286	+	0.905059i	$0.360174\pi$
$312$	0	0
$313$	14.0000i	0.791327i	0.918396	+	0.395663i	$0.129485\pi$
$313$	14.0000i	0.791327i	−0.918396	+	0.395663i	$0.870515\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.0000i	1.17948i	0.807594	+	0.589739i	$0.200769\pi$
$317$	21.0000i	1.17948i	−0.807594	+	0.589739i	$0.799231\pi$
$318$	0	0
$319$	−3.00000	−0.167968
$320$	0	0
$321$	0	0
$322$	0	0
$323$	21.0000i	1.16847i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	14.0000i	0.774202i
$328$	0	0
$329$	−30.0000	−1.65395
$330$	0	0
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	0	0
$333$	14.0000i	0.767195i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 5.00000i	− 0.272367i	−0.990684	−	0.136184i	$-0.956516\pi$
$337$	− 5.00000i	− 0.272367i	0.990684	−	0.136184i	$-0.0434837\pi$
$338$	0	0
$339$	12.0000	0.651751
$340$	0	0
$341$	−7.00000	−0.379071
$342$	0	0
$343$	− 55.0000i	− 2.96972i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	30.0000i	1.61048i	0.592946	+	0.805242i	$0.297965\pi$
$347$	30.0000i	1.61048i	−0.592946	+	0.805242i	$0.702035\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	10.0000	0.533761
$352$	0	0
$353$	6.00000i	0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$	6.00000i	0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 15.0000i	− 0.793884i
$358$	0	0
$359$	−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
$362$	0	0
$363$	− 1.00000i	− 0.0524864i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 4.00000i	− 0.208798i	−0.994535	−	0.104399i	$-0.966708\pi$
$367$	− 4.00000i	− 0.208798i	0.994535	−	0.104399i	$-0.0332919\pi$
$368$	0	0
$369$	12.0000	0.624695
$370$	0	0
$371$	15.0000	0.778761
$372$	0	0
$373$	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
$377$	6.00000i	0.309016i
$378$	0	0
$379$	26.0000	1.33553	0.667765	−	0.744372i	$-0.267251\pi$
$379$	26.0000	1.33553	0.667765	+	0.744372i	$0.267251\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	0	0
$383$	18.0000i	0.919757i	0.887982	+	0.459879i	$0.152107\pi$
$383$	18.0000i	0.919757i	−0.887982	+	0.459879i	$0.847893\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 16.0000i	− 0.813326i
$388$	0	0
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	0	0
$393$	− 3.00000i	− 0.151330i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	22.0000i	1.10415i	0.833795	+	0.552074i	$0.186163\pi$
$397$	22.0000i	1.10415i	−0.833795	+	0.552074i	$0.813837\pi$
$398$	0	0
$399$	−35.0000	−1.75219
$400$	0	0
$401$	3.00000	0.149813	0.0749064	−	0.997191i	$-0.476134\pi$
$401$	3.00000	0.149813	0.0749064	+	0.997191i	$0.476134\pi$
$402$	0	0
$403$	14.0000i	0.697390i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 7.00000i	− 0.346977i
$408$	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
$411$	−12.0000	−0.591916
$412$	0	0
$413$	− 30.0000i	− 1.47620i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.00000i	0.195881i
$418$	0	0
$419$	24.0000	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$420$	0	0
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	0	0
$423$	12.0000i	0.583460i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 5.00000i	− 0.241967i
$428$	0	0
$429$	−2.00000	−0.0965609
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	− 28.0000i	− 1.34559i	−0.739827	−	0.672797i	$-0.765093\pi$
$433$	− 28.0000i	− 1.34559i	0.739827	−	0.672797i	$-0.234907\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 42.0000i	− 2.00913i
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	−36.0000	−1.71429
$442$	0	0
$443$	− 12.0000i	− 0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$	− 12.0000i	− 0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	15.0000i	0.709476i
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	0	0
$453$	2.00000i	0.0939682i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 17.0000i	− 0.795226i	−0.917553	−	0.397613i	$-0.869839\pi$
$457$	− 17.0000i	− 0.795226i	0.917553	−	0.397613i	$-0.130161\pi$
$458$	0	0
$459$	−15.0000	−0.700140
$460$	0	0
$461$	3.00000	0.139724	0.0698620	−	0.997557i	$-0.477744\pi$
$461$	3.00000	0.139724	0.0698620	+	0.997557i	$0.477744\pi$
$462$	0	0
$463$	22.0000i	1.02243i	0.859454	+	0.511213i	$0.170804\pi$
$463$	22.0000i	1.02243i	−0.859454	+	0.511213i	$0.829196\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 39.0000i	− 1.80470i	−0.430999	−	0.902352i	$-0.641839\pi$
$467$	− 39.0000i	− 1.80470i	0.430999	−	0.902352i	$-0.358161\pi$
$468$	0	0
$469$	−40.0000	−1.84703
$470$	0	0
$471$	7.00000	0.322543
$472$	0	0
$473$	8.00000i	0.367840i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 6.00000i	− 0.274721i
$478$	0	0
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	0	0
$481$	−14.0000	−0.638345
$482$	0	0
$483$	30.0000i	1.36505i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	8.00000i	0.362515i	0.983436	+	0.181257i	$0.0580167\pi$
$487$	8.00000i	0.362515i	−0.983436	+	0.181257i	$0.941983\pi$
$488$	0	0
$489$	−5.00000	−0.226108
$490$	0	0
$491$	−33.0000	−1.48927	−0.744635	−	0.667472i	$-0.767376\pi$
$491$	−33.0000	−1.48927	−0.744635	+	0.667472i	$0.767376\pi$
$492$	0	0
$493$	− 9.00000i	− 0.405340i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 15.0000i	− 0.672842i
$498$	0	0
$499$	44.0000	1.96971	0.984855	−	0.173379i	$-0.0554684\pi$
$499$	44.0000	1.96971	0.984855	+	0.173379i	$0.0554684\pi$
$500$	0	0
$501$	21.0000	0.938211
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 9.00000i	− 0.399704i
$508$	0	0
$509$	12.0000	0.531891	0.265945	−	0.963988i	$-0.414316\pi$
$509$	12.0000	0.531891	0.265945	+	0.963988i	$0.414316\pi$
$510$	0	0
$511$	−10.0000	−0.442374
$512$	0	0
$513$	35.0000i	1.54529i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 6.00000i	− 0.263880i
$518$	0	0
$519$	18.0000	0.790112
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	16.0000i	0.699631i	0.936819	+	0.349816i	$0.113756\pi$
$523$	16.0000i	0.699631i	−0.936819	+	0.349816i	$0.886244\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 21.0000i	− 0.914774i
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	12.0000i	0.519778i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	12.0000i	0.517838i
$538$	0	0
$539$	18.0000	0.775315
$540$	0	0
$541$	41.0000	1.76273	0.881364	−	0.472438i	$-0.156626\pi$
$541$	41.0000	1.76273	0.881364	+	0.472438i	$0.156626\pi$
$542$	0	0
$543$	22.0000i	0.944110i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	8.00000i	0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$	8.00000i	0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	−21.0000	−0.894630
$552$	0	0
$553$	− 50.0000i	− 2.12622i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.00000i	0.254228i	0.991888	+	0.127114i	$0.0405714\pi$
$557$	6.00000i	0.254228i	−0.991888	+	0.127114i	$0.959429\pi$
$558$	0	0
$559$	16.0000	0.676728
$560$	0	0
$561$	3.00000	0.126660
$562$	0	0
$563$	6.00000i	0.252870i	0.991975	+	0.126435i	$0.0403535\pi$
$563$	6.00000i	0.252870i	−0.991975	+	0.126435i	$0.959647\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	5.00000i	0.209980i
$568$	0	0
$569$	−24.0000	−1.00613	−0.503066	−	0.864248i	$-0.667795\pi$
$569$	−24.0000	−1.00613	−0.503066	+	0.864248i	$0.667795\pi$
$570$	0	0
$571$	25.0000	1.04622	0.523109	−	0.852266i	$-0.324772\pi$
$571$	25.0000	1.04622	0.523109	+	0.852266i	$0.324772\pi$
$572$	0	0
$573$	− 12.0000i	− 0.501307i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 38.0000i	− 1.58196i	−0.611842	−	0.790980i	$-0.709571\pi$
$577$	− 38.0000i	− 1.58196i	0.611842	−	0.790980i	$-0.290429\pi$
$578$	0	0
$579$	23.0000	0.955847
$580$	0	0
$581$	−30.0000	−1.24461
$582$	0	0
$583$	3.00000i	0.124247i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3.00000i	0.123823i	0.998082	+	0.0619116i	$0.0197197\pi$
$587$	3.00000i	0.123823i	−0.998082	+	0.0619116i	$0.980280\pi$
$588$	0	0
$589$	−49.0000	−2.01901
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 6.00000i	− 0.246390i	−0.992382	−	0.123195i	$-0.960686\pi$
$593$	− 6.00000i	− 0.246390i	0.992382	−	0.123195i	$-0.0393141\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 11.0000i	− 0.450200i
$598$	0	0
$599$	9.00000	0.367730	0.183865	−	0.982952i	$-0.441139\pi$
$599$	9.00000	0.367730	0.183865	+	0.982952i	$0.441139\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	16.0000i	0.651570i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 49.0000i	− 1.98885i	−0.105453	−	0.994424i	$-0.533629\pi$
$607$	− 49.0000i	− 1.98885i	0.105453	−	0.994424i	$-0.466371\pi$
$608$	0	0
$609$	15.0000	0.607831
$610$	0	0
$611$	−12.0000	−0.485468
$612$	0	0
$613$	2.00000i	0.0807792i	0.999184	+	0.0403896i	$0.0128599\pi$
$613$	2.00000i	0.0807792i	−0.999184	+	0.0403896i	$0.987140\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	48.0000i	1.93241i	0.257780	+	0.966204i	$0.417009\pi$
$617$	48.0000i	1.93241i	−0.257780	+	0.966204i	$0.582991\pi$
$618$	0	0
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	30.0000	1.20386
$622$	0	0
$623$	− 45.0000i	− 1.80289i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 7.00000i	− 0.279553i
$628$	0	0
$629$	21.0000	0.837325
$630$	0	0
$631$	−35.0000	−1.39333	−0.696664	−	0.717398i	$-0.745333\pi$
$631$	−35.0000	−1.39333	−0.696664	+	0.717398i	$0.745333\pi$
$632$	0	0
$633$	5.00000i	0.198732i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 36.0000i	− 1.42637i
$638$	0	0
$639$	−6.00000	−0.237356
$640$	0	0
$641$	−33.0000	−1.30342	−0.651711	−	0.758468i	$-0.725948\pi$
$641$	−33.0000	−1.30342	−0.651711	+	0.758468i	$0.725948\pi$
$642$	0	0
$643$	− 5.00000i	− 0.197181i	−0.995128	−	0.0985904i	$-0.968567\pi$
$643$	− 5.00000i	− 0.197181i	0.995128	−	0.0985904i	$-0.0314334\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.00000i	0.235884i	0.993020	+	0.117942i	$0.0376297\pi$
$647$	6.00000i	0.235884i	−0.993020	+	0.117942i	$0.962370\pi$
$648$	0	0
$649$	6.00000	0.235521
$650$	0	0
$651$	35.0000	1.37176
$652$	0	0
$653$	− 45.0000i	− 1.76099i	−0.474059	−	0.880493i	$-0.657212\pi$
$653$	− 45.0000i	− 1.76099i	0.474059	−	0.880493i	$-0.342788\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	4.00000i	0.156055i
$658$	0	0
$659$	−21.0000	−0.818044	−0.409022	−	0.912525i	$-0.634130\pi$
$659$	−21.0000	−0.818044	−0.409022	+	0.912525i	$0.634130\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	0	0
$663$	− 6.00000i	− 0.233021i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	18.0000i	0.696963i
$668$	0	0
$669$	−14.0000	−0.541271
$670$	0	0
$671$	1.00000	0.0386046
$672$	0	0
$673$	− 1.00000i	− 0.0385472i	−0.999814	−	0.0192736i	$-0.993865\pi$
$673$	− 1.00000i	− 0.0385472i	0.999814	−	0.0192736i	$-0.00613535\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 12.0000i	− 0.461197i	−0.973049	−	0.230599i	$-0.925932\pi$
$677$	− 12.0000i	− 0.461197i	0.973049	−	0.230599i	$-0.0740685\pi$
$678$	0	0
$679$	−20.0000	−0.767530
$680$	0	0
$681$	−18.0000	−0.689761
$682$	0	0
$683$	9.00000i	0.344375i	0.985064	+	0.172188i	$0.0550836\pi$
$683$	9.00000i	0.344375i	−0.985064	+	0.172188i	$0.944916\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	14.0000i	0.534133i
$688$	0	0
$689$	6.00000	0.228582
$690$	0	0
$691$	10.0000	0.380418	0.190209	−	0.981744i	$-0.439083\pi$
$691$	10.0000	0.380418	0.190209	+	0.981744i	$0.439083\pi$
$692$	0	0
$693$	− 10.0000i	− 0.379869i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 18.0000i	− 0.681799i
$698$	0	0
$699$	27.0000	1.02123
$700$	0	0
$701$	39.0000	1.47301	0.736505	−	0.676432i	$-0.236475\pi$
$701$	39.0000	1.47301	0.736505	+	0.676432i	$0.236475\pi$
$702$	0	0
$703$	− 49.0000i	− 1.84807i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 30.0000i	− 1.12827i
$708$	0	0
$709$	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
$711$	−20.0000	−0.750059
$712$	0	0
$713$	42.0000i	1.57291i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 6.00000i	− 0.224074i
$718$	0	0
$719$	−51.0000	−1.90198	−0.950990	−	0.309223i	$-0.899931\pi$
$719$	−51.0000	−1.90198	−0.950990	+	0.309223i	$0.899931\pi$
$720$	0	0
$721$	−20.0000	−0.744839
$722$	0	0
$723$	22.0000i	0.818189i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 10.0000i	− 0.370879i	−0.982656	−	0.185440i	$-0.940629\pi$
$727$	− 10.0000i	− 0.370879i	0.982656	−	0.185440i	$-0.0593710\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−24.0000	−0.887672
$732$	0	0
$733$	− 4.00000i	− 0.147743i	−0.997268	−	0.0738717i	$-0.976464\pi$
$733$	− 4.00000i	− 0.147743i	0.997268	−	0.0738717i	$-0.0235355\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 8.00000i	− 0.294684i
$738$	0	0
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	0	0
$741$	−14.0000	−0.514303
$742$	0	0
$743$	3.00000i	0.110059i	0.998485	+	0.0550297i	$0.0175253\pi$
$743$	3.00000i	0.110059i	−0.998485	+	0.0550297i	$0.982475\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12.0000i	0.439057i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−5.00000	−0.182453	−0.0912263	−	0.995830i	$-0.529079\pi$
$751$	−5.00000	−0.182453	−0.0912263	+	0.995830i	$0.529079\pi$
$752$	0	0
$753$	− 30.0000i	− 1.09326i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 26.0000i	− 0.944986i	−0.881334	−	0.472493i	$-0.843354\pi$
$757$	− 26.0000i	− 0.944986i	0.881334	−	0.472493i	$-0.156646\pi$
$758$	0	0
$759$	−6.00000	−0.217786
$760$	0	0
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	− 70.0000i	− 2.53417i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 12.0000i	− 0.433295i
$768$	0	0
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	0	0
$771$	−6.00000	−0.216085
$772$	0	0
$773$	9.00000i	0.323708i	0.986815	+	0.161854i	$0.0517473\pi$
$773$	9.00000i	0.323708i	−0.986815	+	0.161854i	$0.948253\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	35.0000i	1.25562i
$778$	0	0
$779$	−42.0000	−1.50481
$780$	0	0
$781$	3.00000	0.107348
$782$	0	0
$783$	− 15.0000i	− 0.536056i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	8.00000i	0.285169i	0.989783	+	0.142585i	$0.0455413\pi$
$787$	8.00000i	0.285169i	−0.989783	+	0.142585i	$0.954459\pi$
$788$	0	0
$789$	9.00000	0.320408
$790$	0	0
$791$	−60.0000	−2.13335
$792$	0	0
$793$	− 2.00000i	− 0.0710221i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 30.0000i	− 1.06265i	−0.847167	−	0.531327i	$-0.821693\pi$
$797$	− 30.0000i	− 1.06265i	0.847167	−	0.531327i	$-0.178307\pi$
$798$	0	0
$799$	18.0000	0.636794
$800$	0	0
$801$	−18.0000	−0.635999
$802$	0	0
$803$	− 2.00000i	− 0.0705785i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 12.0000i	− 0.422420i
$808$	0	0
$809$	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
$811$	13.0000	0.456492	0.228246	−	0.973604i	$-0.426701\pi$
$811$	13.0000	0.456492	0.228246	+	0.973604i	$0.426701\pi$
$812$	0	0
$813$	20.0000i	0.701431i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	56.0000i	1.95919i
$818$	0	0
$819$	−20.0000	−0.698857
$820$	0	0
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	0	0
$823$	4.00000i	0.139431i	0.997567	+	0.0697156i	$0.0222092\pi$
$823$	4.00000i	0.139431i	−0.997567	+	0.0697156i	$0.977791\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	0	0
$829$	52.0000	1.80603	0.903017	−	0.429604i	$-0.141347\pi$
$829$	52.0000	1.80603	0.903017	+	0.429604i	$0.141347\pi$
$830$	0	0
$831$	4.00000	0.138758
$832$	0	0
$833$	54.0000i	1.87099i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 35.0000i	− 1.20978i
$838$	0	0
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	− 30.0000i	− 1.03325i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	5.00000i	0.171802i
$848$	0	0
$849$	−2.00000	−0.0686398
$850$	0	0
$851$	−42.0000	−1.43974
$852$	0	0
$853$	− 34.0000i	− 1.16414i	−0.813139	−	0.582069i	$-0.802243\pi$
$853$	− 34.0000i	− 1.16414i	0.813139	−	0.582069i	$-0.197757\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 3.00000i	− 0.102478i	−0.998686	−	0.0512390i	$-0.983683\pi$
$857$	− 3.00000i	− 0.102478i	0.998686	−	0.0512390i	$-0.0163170\pi$
$858$	0	0
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	0	0
$861$	30.0000	1.02240
$862$	0	0
$863$	6.00000i	0.204242i	0.994772	+	0.102121i	$0.0325630\pi$
$863$	6.00000i	0.204242i	−0.994772	+	0.102121i	$0.967437\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 8.00000i	− 0.271694i
$868$	0	0
$869$	10.0000	0.339227
$870$	0	0
$871$	−16.0000	−0.542139
$872$	0	0
$873$	8.00000i	0.270759i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	22.0000i	0.742887i	0.928456	+	0.371444i	$0.121137\pi$
$877$	22.0000i	0.742887i	−0.928456	+	0.371444i	$0.878863\pi$
$878$	0	0
$879$	−6.00000	−0.202375
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	1.00000i	0.0336527i	0.999858	+	0.0168263i	$0.00535624\pi$
$883$	1.00000i	0.0336527i	−0.999858	+	0.0168263i	$0.994644\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	80.0000	2.68311
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	− 42.0000i	− 1.40548i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	12.0000i	0.400668i
$898$	0	0
$899$	21.0000	0.700389
$900$	0	0
$901$	−9.00000	−0.299833
$902$	0	0
$903$	− 40.0000i	− 1.33112i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	17.0000i	0.564476i	0.959344	+	0.282238i	$0.0910767\pi$
$907$	17.0000i	0.564476i	−0.959344	+	0.282238i	$0.908923\pi$
$908$	0	0
$909$	−12.0000	−0.398015
$910$	0	0
$911$	−15.0000	−0.496972	−0.248486	−	0.968635i	$-0.579933\pi$
$911$	−15.0000	−0.496972	−0.248486	+	0.968635i	$0.579933\pi$
$912$	0	0
$913$	− 6.00000i	− 0.198571i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	15.0000i	0.495344i
$918$	0	0
$919$	−4.00000	−0.131948	−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000	−0.131948	−0.0659739	+	0.997821i	$0.521015\pi$
$920$	0	0
$921$	−10.0000	−0.329511
$922$	0	0
$923$	− 6.00000i	− 0.197492i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	8.00000i	0.262754i
$928$	0	0
$929$	45.0000	1.47640	0.738201	−	0.674581i	$-0.235676\pi$
$929$	45.0000	1.47640	0.738201	+	0.674581i	$0.235676\pi$
$930$	0	0
$931$	126.000	4.12948
$932$	0	0
$933$	− 15.0000i	− 0.491078i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 26.0000i	− 0.849383i	−0.905338	−	0.424691i	$-0.860383\pi$
$937$	− 26.0000i	− 0.849383i	0.905338	−	0.424691i	$-0.139617\pi$
$938$	0	0
$939$	14.0000	0.456873
$940$	0	0
$941$	−15.0000	−0.488986	−0.244493	−	0.969651i	$-0.578622\pi$
$941$	−15.0000	−0.488986	−0.244493	+	0.969651i	$0.578622\pi$
$942$	0	0
$943$	36.0000i	1.17232i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.00000i	0.0974869i	0.998811	+	0.0487435i	$0.0155217\pi$
$947$	3.00000i	0.0974869i	−0.998811	+	0.0487435i	$0.984478\pi$
$948$	0	0
$949$	−4.00000	−0.129845
$950$	0	0
$951$	21.0000	0.680972
$952$	0	0
$953$	− 27.0000i	− 0.874616i	−0.899312	−	0.437308i	$-0.855932\pi$
$953$	− 27.0000i	− 0.874616i	0.899312	−	0.437308i	$-0.144068\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	3.00000i	0.0969762i
$958$	0	0
$959$	60.0000	1.93750
$960$	0	0
$961$	18.0000	0.580645
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 13.0000i	− 0.418052i	−0.977910	−	0.209026i	$-0.932971\pi$
$967$	− 13.0000i	− 0.418052i	0.977910	−	0.209026i	$-0.0670293\pi$
$968$	0	0
$969$	21.0000	0.674617
$970$	0	0
$971$	24.0000	0.770197	0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000	0.770197	0.385098	+	0.922876i	$0.374168\pi$
$972$	0	0
$973$	− 20.0000i	− 0.641171i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 12.0000i	− 0.383914i	−0.981403	−	0.191957i	$-0.938517\pi$
$977$	− 12.0000i	− 0.383914i	0.981403	−	0.191957i	$-0.0614834\pi$
$978$	0	0
$979$	9.00000	0.287641
$980$	0	0
$981$	−28.0000	−0.893971
$982$	0	0
$983$	6.00000i	0.191370i	0.995412	+	0.0956851i	$0.0305042\pi$
$983$	6.00000i	0.191370i	−0.995412	+	0.0956851i	$0.969496\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	30.0000i	0.954911i
$988$	0	0
$989$	48.0000	1.52631
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	0	0
$993$	8.00000i	0.253872i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	28.0000i	0.886769i	0.896332	+	0.443384i	$0.146222\pi$
$997$	28.0000i	0.886769i	−0.896332	+	0.443384i	$0.853778\pi$
$998$	0	0
$999$	35.0000	1.10735

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.b.g.4049.1		2
4.3	odd	2		550.2.b.b.199.2		2
5.2	odd	4		880.2.a.c.1.1		1
5.3	odd	4		4400.2.a.w.1.1		1
5.4	even	2	inner	4400.2.b.g.4049.2		2
12.11	even	2		4950.2.c.a.199.1		2
15.2	even	4		7920.2.a.s.1.1		1
20.3	even	4		550.2.a.i.1.1		1
20.7	even	4		110.2.a.a.1.1	✓	1
20.19	odd	2		550.2.b.b.199.1		2
40.27	even	4		3520.2.a.l.1.1		1
40.37	odd	4		3520.2.a.z.1.1		1
55.32	even	4		9680.2.a.j.1.1		1
60.23	odd	4		4950.2.a.a.1.1		1
60.47	odd	4		990.2.a.l.1.1		1
60.59	even	2		4950.2.c.a.199.2		2
140.27	odd	4		5390.2.a.h.1.1		1
220.43	odd	4		6050.2.a.i.1.1		1
220.87	odd	4		1210.2.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.2.a.a.1.1	✓	1	20.7	even	4
550.2.a.i.1.1		1	20.3	even	4
550.2.b.b.199.1		2	20.19	odd	2
550.2.b.b.199.2		2	4.3	odd	2
880.2.a.c.1.1		1	5.2	odd	4
990.2.a.l.1.1		1	60.47	odd	4
1210.2.a.k.1.1		1	220.87	odd	4
3520.2.a.l.1.1		1	40.27	even	4
3520.2.a.z.1.1		1	40.37	odd	4
4400.2.a.w.1.1		1	5.3	odd	4
4400.2.b.g.4049.1		2	1.1	even	1	trivial
4400.2.b.g.4049.2		2	5.4	even	2	inner
4950.2.a.a.1.1		1	60.23	odd	4
4950.2.c.a.199.1		2	12.11	even	2
4950.2.c.a.199.2		2	60.59	even	2
5390.2.a.h.1.1		1	140.27	odd	4
6050.2.a.i.1.1		1	220.43	odd	4
7920.2.a.s.1.1		1	15.2	even	4
9680.2.a.j.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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +5.00000i q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +5.00000i q^{7} +2.00000 q^{9} -1.00000 q^{11} +2.00000i q^{13} -3.00000i q^{17} -7.00000 q^{19} +5.00000 q^{21} +6.00000i q^{23} -5.00000i q^{27} +3.00000 q^{29} +7.00000 q^{31} +1.00000i q^{33} +7.00000i q^{37} +2.00000 q^{39} +6.00000 q^{41} -8.00000i q^{43} +6.00000i q^{47} -18.0000 q^{49} -3.00000 q^{51} -3.00000i q^{53} +7.00000i q^{57} -6.00000 q^{59} -1.00000 q^{61} +10.0000i q^{63} +8.00000i q^{67} +6.00000 q^{69} -3.00000 q^{71} +2.00000i q^{73} -5.00000i q^{77} -10.0000 q^{79} +1.00000 q^{81} +6.00000i q^{83} -3.00000i q^{87} -9.00000 q^{89} -10.0000 q^{91} -7.00000i q^{93} +4.00000i q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{9}+O(q^{10}) \) 2 * q + 4 * q^9	\( 2 q + 4 q^{9} - 2 q^{11} - 14 q^{19} + 10 q^{21} + 6 q^{29} + 14 q^{31} + 4 q^{39} + 12 q^{41} - 36 q^{49} - 6 q^{51} - 12 q^{59} - 2 q^{61} + 12 q^{69} - 6 q^{71} - 20 q^{79} + 2 q^{81} - 18 q^{89} - 20 q^{91} - 4 q^{99}+O(q^{100}) \) 2 * q + 4 * q^9 - 2 * q^11 - 14 * q^19 + 10 * q^21 + 6 * q^29 + 14 * q^31 + 4 * q^39 + 12 * q^41 - 36 * q^49 - 6 * q^51 - 12 * q^59 - 2 * q^61 + 12 * q^69 - 6 * q^71 - 20 * q^79 + 2 * q^81 - 18 * q^89 - 20 * q^91 - 4 * q^99

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