LMFDB - Embedded newform 2200.2.b.b.1849.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2200,2,Mod(1849,2200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2200, 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("2200.1849");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2200 = 2^{3} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2200.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:	$17.5670884447$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 440)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2200.1849
Dual form			2200.2.b.b.1849.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +1.00000i q^{7} -6.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} +1.00000i q^{7} -6.00000 q^{9} -1.00000 q^{11} +6.00000i q^{13} +3.00000i q^{17} +5.00000 q^{19} +3.00000 q^{21} +2.00000i q^{23} +9.00000i q^{27} +5.00000 q^{29} +5.00000 q^{31} +3.00000i q^{33} -1.00000i q^{37} +18.0000 q^{39} -2.00000 q^{41} -12.0000i q^{43} -2.00000i q^{47} +6.00000 q^{49} +9.00000 q^{51} +13.0000i q^{53} -15.0000i q^{57} -2.00000 q^{59} +1.00000 q^{61} -6.00000i q^{63} +16.0000i q^{67} +6.00000 q^{69} +15.0000 q^{71} -10.0000i q^{73} -1.00000i q^{77} -2.00000 q^{79} +9.00000 q^{81} +14.0000i q^{83} -15.0000i q^{87} -9.00000 q^{89} -6.00000 q^{91} -15.0000i q^{93} -16.0000i q^{97} +6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 12 q^{9}+O(q^{10}) $ 2 * q - 12 * q^9	$ 2 q - 12 q^{9} - 2 q^{11} + 10 q^{19} + 6 q^{21} + 10 q^{29} + 10 q^{31} + 36 q^{39} - 4 q^{41} + 12 q^{49} + 18 q^{51} - 4 q^{59} + 2 q^{61} + 12 q^{69} + 30 q^{71} - 4 q^{79} + 18 q^{81} - 18 q^{89} - 12 q^{91} + 12 q^{99}+O(q^{100}) $ 2 * q - 12 * q^9 - 2 * q^11 + 10 * q^19 + 6 * q^21 + 10 * q^29 + 10 * q^31 + 36 * q^39 - 4 * q^41 + 12 * q^49 + 18 * q^51 - 4 * q^59 + 2 * q^61 + 12 * q^69 + 30 * q^71 - 4 * q^79 + 18 * q^81 - 18 * q^89 - 12 * q^91 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2200\mathbb{Z}\right)^\times$.

$n$	$177$	$551$	$1101$	$1201$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	− 3.00000i	− 1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$3$	− 3.00000i	− 1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000i	0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$	1.00000i	0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$	0	0
$9$	−6.00000	−2.00000
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	6.00000i	1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$	6.00000i	1.66410i	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.00000i	0.727607i	0.931476	+	0.363803i	$0.118522\pi$
$17$	3.00000i	0.727607i	−0.931476	+	0.363803i	$0.881478\pi$
$18$	0	0
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	0	0
$23$	2.00000i	0.417029i	0.978019	+	0.208514i	$0.0668628\pi$
$23$	2.00000i	0.417029i	−0.978019	+	0.208514i	$0.933137\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	9.00000i	1.73205i
$28$	0	0
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	0	0
$33$	3.00000i	0.522233i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 1.00000i	− 0.164399i	−0.996616	−	0.0821995i	$-0.973806\pi$
$37$	− 1.00000i	− 0.164399i	0.996616	−	0.0821995i	$-0.0261945\pi$
$38$	0	0
$39$	18.0000	2.88231
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	− 12.0000i	− 1.82998i	−0.403473	−	0.914991i	$-0.632197\pi$
$43$	− 12.0000i	− 1.82998i	0.403473	−	0.914991i	$-0.367803\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 2.00000i	− 0.291730i	−0.989305	−	0.145865i	$-0.953403\pi$
$47$	− 2.00000i	− 0.291730i	0.989305	−	0.145865i	$-0.0465965\pi$
$48$	0	0
$49$	6.00000	0.857143
$50$	0	0
$51$	9.00000	1.26025
$52$	0	0
$53$	13.0000i	1.78569i	0.450367	+	0.892844i	$0.351293\pi$
$53$	13.0000i	1.78569i	−0.450367	+	0.892844i	$0.648707\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 15.0000i	− 1.98680i
$58$	0	0
$59$	−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
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	0	0
$63$	− 6.00000i	− 0.755929i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	16.0000i	1.95471i	0.211604	+	0.977356i	$0.432131\pi$
$67$	16.0000i	1.95471i	−0.211604	+	0.977356i	$0.567869\pi$
$68$	0	0
$69$	6.00000	0.722315
$70$	0	0
$71$	15.0000	1.78017	0.890086	−	0.455792i	$-0.150644\pi$
$71$	15.0000	1.78017	0.890086	+	0.455792i	$0.150644\pi$
$72$	0	0
$73$	− 10.0000i	− 1.17041i	−0.810885	−	0.585206i	$-0.801014\pi$
$73$	− 10.0000i	− 1.17041i	0.810885	−	0.585206i	$-0.198986\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1.00000i	− 0.113961i
$78$	0	0
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	14.0000i	1.53670i	0.640030	+	0.768350i	$0.278922\pi$
$83$	14.0000i	1.53670i	−0.640030	+	0.768350i	$0.721078\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 15.0000i	− 1.60817i
$88$	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
$91$	−6.00000	−0.628971
$92$	0	0
$93$	− 15.0000i	− 1.55543i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 16.0000i	− 1.62455i	−0.583272	−	0.812277i	$-0.698228\pi$
$97$	− 16.0000i	− 1.62455i	0.583272	−	0.812277i	$-0.301772\pi$
$98$	0	0
$99$	6.00000	0.603023
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	16.0000i	1.57653i	0.615338	+	0.788263i	$0.289020\pi$
$103$	16.0000i	1.57653i	−0.615338	+	0.788263i	$0.710980\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	−3.00000	−0.284747
$112$	0	0
$113$	16.0000i	1.50515i	0.658505	+	0.752577i	$0.271189\pi$
$113$	16.0000i	1.50515i	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 36.0000i	− 3.32820i
$118$	0	0
$119$	−3.00000	−0.275010
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	6.00000i	0.541002i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 8.00000i	− 0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$	− 8.00000i	− 0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	0	0
$129$	−36.0000	−3.16962
$130$	0	0
$131$	7.00000	0.611593	0.305796	−	0.952097i	$-0.401077\pi$
$131$	7.00000	0.611593	0.305796	+	0.952097i	$0.401077\pi$
$132$	0	0
$133$	5.00000i	0.433555i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.0000i	1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$	12.0000i	1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	0	0
$143$	− 6.00000i	− 0.501745i
$144$	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	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	0	0
$151$	18.0000	1.46482	0.732410	−	0.680864i	$-0.238396\pi$
$151$	18.0000	1.46482	0.732410	+	0.680864i	$0.238396\pi$
$152$	0	0
$153$	− 18.0000i	− 1.45521i
$154$	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
$159$	39.0000	3.09290
$160$	0	0
$161$	−2.00000	−0.157622
$162$	0	0
$163$	1.00000i	0.0783260i	0.999233	+	0.0391630i	$0.0124692\pi$
$163$	1.00000i	0.0783260i	−0.999233	+	0.0391630i	$0.987531\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.00000i	0.0773823i	0.999251	+	0.0386912i	$0.0123189\pi$
$167$	1.00000i	0.0773823i	−0.999251	+	0.0386912i	$0.987681\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	−30.0000	−2.29416
$172$	0	0
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.00000i	0.450988i
$178$	0	0
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	− 3.00000i	− 0.221766i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 3.00000i	− 0.219382i
$188$	0	0
$189$	−9.00000	−0.654654
$190$	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
$193$	− 7.00000i	− 0.503871i	−0.967744	−	0.251936i	$-0.918933\pi$
$193$	− 7.00000i	− 0.503871i	0.967744	−	0.251936i	$-0.0810671\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 24.0000i	− 1.70993i	−0.518686	−	0.854965i	$-0.673579\pi$
$197$	− 24.0000i	− 1.70993i	0.518686	−	0.854965i	$-0.326421\pi$
$198$	0	0
$199$	1.00000	0.0708881	0.0354441	−	0.999372i	$-0.488715\pi$
$199$	1.00000	0.0708881	0.0354441	+	0.999372i	$0.488715\pi$
$200$	0	0
$201$	48.0000	3.38566
$202$	0	0
$203$	5.00000i	0.350931i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 12.0000i	− 0.834058i
$208$	0	0
$209$	−5.00000	−0.345857
$210$	0	0
$211$	−1.00000	−0.0688428	−0.0344214	−	0.999407i	$-0.510959\pi$
$211$	−1.00000	−0.0688428	−0.0344214	+	0.999407i	$0.510959\pi$
$212$	0	0
$213$	− 45.0000i	− 3.08335i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	5.00000i	0.339422i
$218$	0	0
$219$	−30.0000	−2.02721
$220$	0	0
$221$	−18.0000	−1.21081
$222$	0	0
$223$	− 26.0000i	− 1.74109i	−0.492090	−	0.870544i	$-0.663767\pi$
$223$	− 26.0000i	− 1.74109i	0.492090	−	0.870544i	$-0.336233\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	18.0000i	1.19470i	0.801980	+	0.597351i	$0.203780\pi$
$227$	18.0000i	1.19470i	−0.801980	+	0.597351i	$0.796220\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	0	0
$233$	− 11.0000i	− 0.720634i	−0.932830	−	0.360317i	$-0.882669\pi$
$233$	− 11.0000i	− 0.720634i	0.932830	−	0.360317i	$-0.117331\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	6.00000i	0.389742i
$238$	0	0
$239$	2.00000	0.129369	0.0646846	−	0.997906i	$-0.479396\pi$
$239$	2.00000	0.129369	0.0646846	+	0.997906i	$0.479396\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	30.0000i	1.90885i
$248$	0	0
$249$	42.0000	2.66164
$250$	0	0
$251$	−10.0000	−0.631194	−0.315597	−	0.948893i	$-0.602205\pi$
$251$	−10.0000	−0.631194	−0.315597	+	0.948893i	$0.602205\pi$
$252$	0	0
$253$	− 2.00000i	− 0.125739i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 22.0000i	− 1.37232i	−0.727450	−	0.686161i	$-0.759294\pi$
$257$	− 22.0000i	− 1.37232i	0.727450	−	0.686161i	$-0.240706\pi$
$258$	0	0
$259$	1.00000	0.0621370
$260$	0	0
$261$	−30.0000	−1.85695
$262$	0	0
$263$	13.0000i	0.801614i	0.916162	+	0.400807i	$0.131270\pi$
$263$	13.0000i	0.801614i	−0.916162	+	0.400807i	$0.868730\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	27.0000i	1.65237i
$268$	0	0
$269$	−20.0000	−1.21942	−0.609711	−	0.792624i	$-0.708714\pi$
$269$	−20.0000	−1.21942	−0.609711	+	0.792624i	$0.708714\pi$
$270$	0	0
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	0	0
$273$	18.0000i	1.08941i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	16.0000i	0.961347i	0.876900	+	0.480673i	$0.159608\pi$
$277$	16.0000i	0.961347i	−0.876900	+	0.480673i	$0.840392\pi$
$278$	0	0
$279$	−30.0000	−1.79605
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	− 10.0000i	− 0.594438i	−0.954809	−	0.297219i	$-0.903941\pi$
$283$	− 10.0000i	− 0.594438i	0.954809	−	0.297219i	$-0.0960592\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 2.00000i	− 0.118056i
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	−48.0000	−2.81381
$292$	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
$297$	− 9.00000i	− 0.522233i
$298$	0	0
$299$	−12.0000	−0.693978
$300$	0	0
$301$	12.0000	0.691669
$302$	0	0
$303$	30.0000i	1.72345i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 22.0000i	− 1.25561i	−0.778372	−	0.627803i	$-0.783954\pi$
$307$	− 22.0000i	− 1.25561i	0.778372	−	0.627803i	$-0.216046\pi$
$308$	0	0
$309$	48.0000	2.73062
$310$	0	0
$311$	21.0000	1.19080	0.595400	−	0.803429i	$-0.296993\pi$
$311$	21.0000	1.19080	0.595400	+	0.803429i	$0.296993\pi$
$312$	0	0
$313$	30.0000i	1.69570i	0.530236	+	0.847850i	$0.322103\pi$
$313$	30.0000i	1.69570i	−0.530236	+	0.847850i	$0.677897\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 3.00000i	− 0.168497i	−0.996445	−	0.0842484i	$-0.973151\pi$
$317$	− 3.00000i	− 0.168497i	0.996445	−	0.0842484i	$-0.0268489\pi$
$318$	0	0
$319$	−5.00000	−0.279946
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	15.0000i	0.834622i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	30.0000i	1.65900i
$328$	0	0
$329$	2.00000	0.110264
$330$	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
$336$	0	0
$337$	29.0000i	1.57973i	0.613280	+	0.789865i	$0.289850\pi$
$337$	29.0000i	1.57973i	−0.613280	+	0.789865i	$0.710150\pi$
$338$	0	0
$339$	48.0000	2.60700
$340$	0	0
$341$	−5.00000	−0.270765
$342$	0	0
$343$	13.0000i	0.701934i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2.00000i	− 0.107366i	−0.998558	−	0.0536828i	$-0.982904\pi$
$347$	− 2.00000i	− 0.107366i	0.998558	−	0.0536828i	$-0.0170960\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	−54.0000	−2.88231
$352$	0	0
$353$	18.0000i	0.958043i	0.877803	+	0.479022i	$0.159008\pi$
$353$	18.0000i	0.958043i	−0.877803	+	0.479022i	$0.840992\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	9.00000i	0.476331i
$358$	0	0
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	0	0
$363$	− 3.00000i	− 0.157459i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 12.0000i	− 0.626395i	−0.949688	−	0.313197i	$-0.898600\pi$
$367$	− 12.0000i	− 0.626395i	0.949688	−	0.313197i	$-0.101400\pi$
$368$	0	0
$369$	12.0000	0.624695
$370$	0	0
$371$	−13.0000	−0.674926
$372$	0	0
$373$	− 14.0000i	− 0.724893i	−0.932005	−	0.362446i	$-0.881942\pi$
$373$	− 14.0000i	− 0.724893i	0.932005	−	0.362446i	$-0.118058\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	30.0000i	1.54508i
$378$	0	0
$379$	2.00000	0.102733	0.0513665	−	0.998680i	$-0.483642\pi$
$379$	2.00000	0.102733	0.0513665	+	0.998680i	$0.483642\pi$
$380$	0	0
$381$	−24.0000	−1.22956
$382$	0	0
$383$	14.0000i	0.715367i	0.933843	+	0.357683i	$0.116433\pi$
$383$	14.0000i	0.715367i	−0.933843	+	0.357683i	$0.883567\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	72.0000i	3.65997i
$388$	0	0
$389$	2.00000	0.101404	0.0507020	−	0.998714i	$-0.483854\pi$
$389$	2.00000	0.101404	0.0507020	+	0.998714i	$0.483854\pi$
$390$	0	0
$391$	−6.00000	−0.303433
$392$	0	0
$393$	− 21.0000i	− 1.05931i
$394$	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
$399$	15.0000	0.750939
$400$	0	0
$401$	−21.0000	−1.04869	−0.524345	−	0.851506i	$-0.675690\pi$
$401$	−21.0000	−1.04869	−0.524345	+	0.851506i	$0.675690\pi$
$402$	0	0
$403$	30.0000i	1.49441i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.00000i	0.0495682i
$408$	0	0
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	0	0
$411$	36.0000	1.77575
$412$	0	0
$413$	− 2.00000i	− 0.0984136i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 12.0000i	− 0.587643i
$418$	0	0
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	0	0
$421$	−36.0000	−1.75453	−0.877266	−	0.480004i	$-0.840635\pi$
$421$	−36.0000	−1.75453	−0.877266	+	0.480004i	$0.840635\pi$
$422$	0	0
$423$	12.0000i	0.583460i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.00000i	0.0483934i
$428$	0	0
$429$	−18.0000	−0.869048
$430$	0	0
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	0	0
$433$	− 4.00000i	− 0.192228i	−0.995370	−	0.0961139i	$-0.969359\pi$
$433$	− 4.00000i	− 0.192228i	0.995370	−	0.0961139i	$-0.0306413\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	10.0000i	0.478365i
$438$	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
$447$	− 45.0000i	− 2.12843i
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	2.00000	0.0941763
$452$	0	0
$453$	− 54.0000i	− 2.53714i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	33.0000i	1.54367i	0.635820	+	0.771837i	$0.280662\pi$
$457$	33.0000i	1.54367i	−0.635820	+	0.771837i	$0.719338\pi$
$458$	0	0
$459$	−27.0000	−1.26025
$460$	0	0
$461$	5.00000	0.232873	0.116437	−	0.993198i	$-0.462853\pi$
$461$	5.00000	0.232873	0.116437	+	0.993198i	$0.462853\pi$
$462$	0	0
$463$	− 6.00000i	− 0.278844i	−0.990233	−	0.139422i	$-0.955476\pi$
$463$	− 6.00000i	− 0.278844i	0.990233	−	0.139422i	$-0.0445244\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	27.0000i	1.24941i	0.780860	+	0.624705i	$0.214781\pi$
$467$	27.0000i	1.24941i	−0.780860	+	0.624705i	$0.785219\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	0	0
$471$	21.0000	0.967629
$472$	0	0
$473$	12.0000i	0.551761i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 78.0000i	− 3.57137i
$478$	0	0
$479$	−20.0000	−0.913823	−0.456912	−	0.889512i	$-0.651044\pi$
$479$	−20.0000	−0.913823	−0.456912	+	0.889512i	$0.651044\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	0	0
$483$	6.00000i	0.273009i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 12.0000i	− 0.543772i	−0.962329	−	0.271886i	$-0.912353\pi$
$487$	− 12.0000i	− 0.543772i	0.962329	−	0.271886i	$-0.0876473\pi$
$488$	0	0
$489$	3.00000	0.135665
$490$	0	0
$491$	−13.0000	−0.586682	−0.293341	−	0.956008i	$-0.594767\pi$
$491$	−13.0000	−0.586682	−0.293341	+	0.956008i	$0.594767\pi$
$492$	0	0
$493$	15.0000i	0.675566i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	15.0000i	0.672842i
$498$	0	0
$499$	8.00000	0.358129	0.179065	−	0.983837i	$-0.442693\pi$
$499$	8.00000	0.358129	0.179065	+	0.983837i	$0.442693\pi$
$500$	0	0
$501$	3.00000	0.134030
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	69.0000i	3.06440i
$508$	0	0
$509$	−8.00000	−0.354594	−0.177297	−	0.984157i	$-0.556735\pi$
$509$	−8.00000	−0.354594	−0.177297	+	0.984157i	$0.556735\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	0	0
$513$	45.0000i	1.98680i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2.00000i	0.0879599i
$518$	0	0
$519$	18.0000	0.790112
$520$	0	0
$521$	−26.0000	−1.13908	−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000	−1.13908	−0.569540	+	0.821963i	$0.692879\pi$
$522$	0	0
$523$	4.00000i	0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$	4.00000i	0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	15.0000i	0.653410i
$528$	0	0
$529$	19.0000	0.826087
$530$	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
$539$	−6.00000	−0.258438
$540$	0	0
$541$	15.0000	0.644900	0.322450	−	0.946586i	$-0.395494\pi$
$541$	15.0000	0.644900	0.322450	+	0.946586i	$0.395494\pi$
$542$	0	0
$543$	− 66.0000i	− 2.83233i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 36.0000i	− 1.53925i	−0.638497	−	0.769624i	$-0.720443\pi$
$547$	− 36.0000i	− 1.53925i	0.638497	−	0.769624i	$-0.279557\pi$
$548$	0	0
$549$	−6.00000	−0.256074
$550$	0	0
$551$	25.0000	1.06504
$552$	0	0
$553$	− 2.00000i	− 0.0850487i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 30.0000i	− 1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$	− 30.0000i	− 1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$	0	0
$559$	72.0000	3.04528
$560$	0	0
$561$	−9.00000	−0.379980
$562$	0	0
$563$	18.0000i	0.758610i	0.925272	+	0.379305i	$0.123837\pi$
$563$	18.0000i	0.758610i	−0.925272	+	0.379305i	$0.876163\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	9.00000i	0.377964i
$568$	0	0
$569$	−36.0000	−1.50920	−0.754599	−	0.656186i	$-0.772169\pi$
$569$	−36.0000	−1.50920	−0.754599	+	0.656186i	$0.772169\pi$
$570$	0	0
$571$	−19.0000	−0.795125	−0.397563	−	0.917575i	$-0.630144\pi$
$571$	−19.0000	−0.795125	−0.397563	+	0.917575i	$0.630144\pi$
$572$	0	0
$573$	− 36.0000i	− 1.50392i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 14.0000i	− 0.582828i	−0.956597	−	0.291414i	$-0.905874\pi$
$577$	− 14.0000i	− 0.582828i	0.956597	−	0.291414i	$-0.0941257\pi$
$578$	0	0
$579$	−21.0000	−0.872730
$580$	0	0
$581$	−14.0000	−0.580818
$582$	0	0
$583$	− 13.0000i	− 0.538405i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 39.0000i	− 1.60970i	−0.593477	−	0.804851i	$-0.702245\pi$
$587$	− 39.0000i	− 1.60970i	0.593477	−	0.804851i	$-0.297755\pi$
$588$	0	0
$589$	25.0000	1.03011
$590$	0	0
$591$	−72.0000	−2.96168
$592$	0	0
$593$	22.0000i	0.903432i	0.892162	+	0.451716i	$0.149188\pi$
$593$	22.0000i	0.903432i	−0.892162	+	0.451716i	$0.850812\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 3.00000i	− 0.122782i
$598$	0	0
$599$	27.0000	1.10319	0.551595	−	0.834112i	$-0.314019\pi$
$599$	27.0000	1.10319	0.551595	+	0.834112i	$0.314019\pi$
$600$	0	0
$601$	14.0000	0.571072	0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000	0.571072	0.285536	+	0.958368i	$0.407828\pi$
$602$	0	0
$603$	− 96.0000i	− 3.90942i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 21.0000i	− 0.852364i	−0.904638	−	0.426182i	$-0.859858\pi$
$607$	− 21.0000i	− 0.852364i	0.904638	−	0.426182i	$-0.140142\pi$
$608$	0	0
$609$	15.0000	0.607831
$610$	0	0
$611$	12.0000	0.485468
$612$	0	0
$613$	10.0000i	0.403896i	0.979396	+	0.201948i	$0.0647272\pi$
$613$	10.0000i	0.403896i	−0.979396	+	0.201948i	$0.935273\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.0000i	0.483102i	0.970388	+	0.241551i	$0.0776561\pi$
$617$	12.0000i	0.483102i	−0.970388	+	0.241551i	$0.922344\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	−18.0000	−0.722315
$622$	0	0
$623$	− 9.00000i	− 0.360577i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	15.0000i	0.599042i
$628$	0	0
$629$	3.00000	0.119618
$630$	0	0
$631$	15.0000	0.597141	0.298570	−	0.954388i	$-0.403490\pi$
$631$	15.0000	0.597141	0.298570	+	0.954388i	$0.403490\pi$
$632$	0	0
$633$	3.00000i	0.119239i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	36.0000i	1.42637i
$638$	0	0
$639$	−90.0000	−3.56034
$640$	0	0
$641$	−9.00000	−0.355479	−0.177739	−	0.984078i	$-0.556878\pi$
$641$	−9.00000	−0.355479	−0.177739	+	0.984078i	$0.556878\pi$
$642$	0	0
$643$	− 7.00000i	− 0.276053i	−0.990429	−	0.138027i	$-0.955924\pi$
$643$	− 7.00000i	− 0.276053i	0.990429	−	0.138027i	$-0.0440759\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 6.00000i	− 0.235884i	−0.993020	−	0.117942i	$-0.962370\pi$
$647$	− 6.00000i	− 0.235884i	0.993020	−	0.117942i	$-0.0376297\pi$
$648$	0	0
$649$	2.00000	0.0785069
$650$	0	0
$651$	15.0000	0.587896
$652$	0	0
$653$	19.0000i	0.743527i	0.928327	+	0.371764i	$0.121247\pi$
$653$	19.0000i	0.743527i	−0.928327	+	0.371764i	$0.878753\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	60.0000i	2.34082i
$658$	0	0
$659$	−49.0000	−1.90877	−0.954384	−	0.298580i	$-0.903487\pi$
$659$	−49.0000	−1.90877	−0.954384	+	0.298580i	$0.903487\pi$
$660$	0	0
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	0	0
$663$	54.0000i	2.09719i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	10.0000i	0.387202i
$668$	0	0
$669$	−78.0000	−3.01565
$670$	0	0
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	1.00000i	0.0385472i	0.999814	+	0.0192736i	$0.00613535\pi$
$673$	1.00000i	0.0385472i	−0.999814	+	0.0192736i	$0.993865\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	16.0000	0.614024
$680$	0	0
$681$	54.0000	2.06928
$682$	0	0
$683$	19.0000i	0.727015i	0.931591	+	0.363507i	$0.118421\pi$
$683$	19.0000i	0.727015i	−0.931591	+	0.363507i	$0.881579\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	18.0000i	0.686743i
$688$	0	0
$689$	−78.0000	−2.97156
$690$	0	0
$691$	34.0000	1.29342	0.646710	−	0.762736i	$-0.276144\pi$
$691$	34.0000	1.29342	0.646710	+	0.762736i	$0.276144\pi$
$692$	0	0
$693$	6.00000i	0.227921i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 6.00000i	− 0.227266i
$698$	0	0
$699$	−33.0000	−1.24817
$700$	0	0
$701$	−31.0000	−1.17085	−0.585427	−	0.810725i	$-0.699073\pi$
$701$	−31.0000	−1.17085	−0.585427	+	0.810725i	$0.699073\pi$
$702$	0	0
$703$	− 5.00000i	− 0.188579i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 10.0000i	− 0.376089i
$708$	0	0
$709$	−46.0000	−1.72757	−0.863783	−	0.503864i	$-0.831911\pi$
$709$	−46.0000	−1.72757	−0.863783	+	0.503864i	$0.831911\pi$
$710$	0	0
$711$	12.0000	0.450035
$712$	0	0
$713$	10.0000i	0.374503i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 6.00000i	− 0.224074i
$718$	0	0
$719$	−1.00000	−0.0372937	−0.0186469	−	0.999826i	$-0.505936\pi$
$719$	−1.00000	−0.0372937	−0.0186469	+	0.999826i	$0.505936\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	0	0
$723$	− 54.0000i	− 2.00828i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 26.0000i	− 0.964287i	−0.876092	−	0.482143i	$-0.839858\pi$
$727$	− 26.0000i	− 0.964287i	0.876092	−	0.482143i	$-0.160142\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	36.0000	1.33151
$732$	0	0
$733$	32.0000i	1.18195i	0.806691	+	0.590973i	$0.201256\pi$
$733$	32.0000i	1.18195i	−0.806691	+	0.590973i	$0.798744\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 16.0000i	− 0.589368i
$738$	0	0
$739$	8.00000	0.294285	0.147142	−	0.989115i	$-0.452992\pi$
$739$	8.00000	0.294285	0.147142	+	0.989115i	$0.452992\pi$
$740$	0	0
$741$	90.0000	3.30623
$742$	0	0
$743$	− 33.0000i	− 1.21065i	−0.795977	−	0.605326i	$-0.793043\pi$
$743$	− 33.0000i	− 1.21065i	0.795977	−	0.605326i	$-0.206957\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 84.0000i	− 3.07340i
$748$	0	0
$749$	4.00000	0.146157
$750$	0	0
$751$	−7.00000	−0.255434	−0.127717	−	0.991811i	$-0.540765\pi$
$751$	−7.00000	−0.255434	−0.127717	+	0.991811i	$0.540765\pi$
$752$	0	0
$753$	30.0000i	1.09326i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 2.00000i	− 0.0726912i	−0.999339	−	0.0363456i	$-0.988428\pi$
$757$	− 2.00000i	− 0.0726912i	0.999339	−	0.0363456i	$-0.0115717\pi$
$758$	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
$763$	− 10.0000i	− 0.362024i
$764$	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	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	−66.0000	−2.37693
$772$	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
$777$	− 3.00000i	− 0.107624i
$778$	0	0
$779$	−10.0000	−0.358287
$780$	0	0
$781$	−15.0000	−0.536742
$782$	0	0
$783$	45.0000i	1.60817i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 48.0000i	− 1.71102i	−0.517790	−	0.855508i	$-0.673245\pi$
$787$	− 48.0000i	− 1.71102i	0.517790	−	0.855508i	$-0.326755\pi$
$788$	0	0
$789$	39.0000	1.38844
$790$	0	0
$791$	−16.0000	−0.568895
$792$	0	0
$793$	6.00000i	0.213066i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	34.0000i	1.20434i	0.798367	+	0.602171i	$0.205697\pi$
$797$	34.0000i	1.20434i	−0.798367	+	0.602171i	$0.794303\pi$
$798$	0	0
$799$	6.00000	0.212265
$800$	0	0
$801$	54.0000	1.90800
$802$	0	0
$803$	10.0000i	0.352892i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	60.0000i	2.11210i
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	1.00000	0.0351147	0.0175574	−	0.999846i	$-0.494411\pi$
$811$	1.00000	0.0351147	0.0175574	+	0.999846i	$0.494411\pi$
$812$	0	0
$813$	− 72.0000i	− 2.52515i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 60.0000i	− 2.09913i
$818$	0	0
$819$	36.0000	1.25794
$820$	0	0
$821$	50.0000	1.74501	0.872506	−	0.488603i	$-0.162493\pi$
$821$	50.0000	1.74501	0.872506	+	0.488603i	$0.162493\pi$
$822$	0	0
$823$	− 44.0000i	− 1.53374i	−0.641800	−	0.766872i	$-0.721812\pi$
$823$	− 44.0000i	− 1.53374i	0.641800	−	0.766872i	$-0.278188\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 52.0000i	− 1.80822i	−0.427303	−	0.904109i	$-0.640536\pi$
$827$	− 52.0000i	− 1.80822i	0.427303	−	0.904109i	$-0.359464\pi$
$828$	0	0
$829$	−4.00000	−0.138926	−0.0694629	−	0.997585i	$-0.522129\pi$
$829$	−4.00000	−0.138926	−0.0694629	+	0.997585i	$0.522129\pi$
$830$	0	0
$831$	48.0000	1.66510
$832$	0	0
$833$	18.0000i	0.623663i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	45.0000i	1.55543i
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	− 54.0000i	− 1.85986i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.00000i	0.0343604i
$848$	0	0
$849$	−30.0000	−1.02960
$850$	0	0
$851$	2.00000	0.0685591
$852$	0	0
$853$	− 22.0000i	− 0.753266i	−0.926363	−	0.376633i	$-0.877082\pi$
$853$	− 22.0000i	− 0.753266i	0.926363	−	0.376633i	$-0.122918\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 21.0000i	− 0.717346i	−0.933463	−	0.358673i	$-0.883229\pi$
$857$	− 21.0000i	− 0.717346i	0.933463	−	0.358673i	$-0.116771\pi$
$858$	0	0
$859$	22.0000	0.750630	0.375315	−	0.926897i	$-0.377534\pi$
$859$	22.0000	0.750630	0.375315	+	0.926897i	$0.377534\pi$
$860$	0	0
$861$	−6.00000	−0.204479
$862$	0	0
$863$	46.0000i	1.56586i	0.622111	+	0.782929i	$0.286275\pi$
$863$	46.0000i	1.56586i	−0.622111	+	0.782929i	$0.713725\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 24.0000i	− 0.815083i
$868$	0	0
$869$	2.00000	0.0678454
$870$	0	0
$871$	−96.0000	−3.25284
$872$	0	0
$873$	96.0000i	3.24911i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.00000i	0.0675352i	0.999430	+	0.0337676i	$0.0107506\pi$
$877$	2.00000i	0.0675352i	−0.999430	+	0.0337676i	$0.989249\pi$
$878$	0	0
$879$	−18.0000	−0.607125
$880$	0	0
$881$	50.0000	1.68454	0.842271	−	0.539054i	$-0.181218\pi$
$881$	50.0000	1.68454	0.842271	+	0.539054i	$0.181218\pi$
$882$	0	0
$883$	− 5.00000i	− 0.168263i	−0.996455	−	0.0841317i	$-0.973188\pi$
$883$	− 5.00000i	− 0.168263i	0.996455	−	0.0841317i	$-0.0268116\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 40.0000i	− 1.34307i	−0.740973	−	0.671534i	$-0.765636\pi$
$887$	− 40.0000i	− 1.34307i	0.740973	−	0.671534i	$-0.234364\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	0	0
$891$	−9.00000	−0.301511
$892$	0	0
$893$	− 10.0000i	− 0.334637i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	36.0000i	1.20201i
$898$	0	0
$899$	25.0000	0.833797
$900$	0	0
$901$	−39.0000	−1.29928
$902$	0	0
$903$	− 36.0000i	− 1.19800i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 29.0000i	− 0.962929i	−0.876466	−	0.481465i	$-0.840105\pi$
$907$	− 29.0000i	− 0.962929i	0.876466	−	0.481465i	$-0.159895\pi$
$908$	0	0
$909$	60.0000	1.99007
$910$	0	0
$911$	3.00000	0.0993944	0.0496972	−	0.998764i	$-0.484174\pi$
$911$	3.00000	0.0993944	0.0496972	+	0.998764i	$0.484174\pi$
$912$	0	0
$913$	− 14.0000i	− 0.463332i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7.00000i	0.231160i
$918$	0	0
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	−66.0000	−2.17477
$922$	0	0
$923$	90.0000i	2.96239i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 96.0000i	− 3.15305i
$928$	0	0
$929$	−59.0000	−1.93573	−0.967864	−	0.251476i	$-0.919084\pi$
$929$	−59.0000	−1.93573	−0.967864	+	0.251476i	$0.919084\pi$
$930$	0	0
$931$	30.0000	0.983210
$932$	0	0
$933$	− 63.0000i	− 2.06253i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	10.0000i	0.326686i	0.986569	+	0.163343i	$0.0522277\pi$
$937$	10.0000i	0.326686i	−0.986569	+	0.163343i	$0.947772\pi$
$938$	0	0
$939$	90.0000	2.93704
$940$	0	0
$941$	−41.0000	−1.33656	−0.668281	−	0.743909i	$-0.732970\pi$
$941$	−41.0000	−1.33656	−0.668281	+	0.743909i	$0.732970\pi$
$942$	0	0
$943$	− 4.00000i	− 0.130258i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	41.0000i	1.33232i	0.745808	+	0.666160i	$0.232063\pi$
$947$	41.0000i	1.33232i	−0.745808	+	0.666160i	$0.767937\pi$
$948$	0	0
$949$	60.0000	1.94768
$950$	0	0
$951$	−9.00000	−0.291845
$952$	0	0
$953$	− 29.0000i	− 0.939402i	−0.882826	−	0.469701i	$-0.844362\pi$
$953$	− 29.0000i	− 0.939402i	0.882826	−	0.469701i	$-0.155638\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	15.0000i	0.484881i
$958$	0	0
$959$	−12.0000	−0.387500
$960$	0	0
$961$	−6.00000	−0.193548
$962$	0	0
$963$	24.0000i	0.773389i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	55.0000i	1.76868i	0.466843	+	0.884340i	$0.345391\pi$
$967$	55.0000i	1.76868i	−0.466843	+	0.884340i	$0.654609\pi$
$968$	0	0
$969$	45.0000	1.44561
$970$	0	0
$971$	44.0000	1.41203	0.706014	−	0.708198i	$-0.250492\pi$
$971$	44.0000	1.41203	0.706014	+	0.708198i	$0.250492\pi$
$972$	0	0
$973$	4.00000i	0.128234i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 8.00000i	− 0.255943i	−0.991778	−	0.127971i	$-0.959153\pi$
$977$	− 8.00000i	− 0.255943i	0.991778	−	0.127971i	$-0.0408466\pi$
$978$	0	0
$979$	9.00000	0.287641
$980$	0	0
$981$	60.0000	1.91565
$982$	0	0
$983$	14.0000i	0.446531i	0.974758	+	0.223265i	$0.0716716\pi$
$983$	14.0000i	0.446531i	−0.974758	+	0.223265i	$0.928328\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 6.00000i	− 0.190982i
$988$	0	0
$989$	24.0000	0.763156
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	60.0000i	1.90404i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	8.00000i	0.253363i	0.991943	+	0.126681i	$0.0404325\pi$
$997$	8.00000i	0.253363i	−0.991943	+	0.126681i	$0.959567\pi$
$998$	0	0
$999$	9.00000	0.284747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2200.2.b.b.1849.1		2
4.3	odd	2		4400.2.b.a.4049.2		2
5.2	odd	4		2200.2.a.a.1.1		1
5.3	odd	4		440.2.a.d.1.1	✓	1
5.4	even	2	inner	2200.2.b.b.1849.2		2
15.8	even	4		3960.2.a.f.1.1		1
20.3	even	4		880.2.a.a.1.1		1
20.7	even	4		4400.2.a.be.1.1		1
20.19	odd	2		4400.2.b.a.4049.1		2
40.3	even	4		3520.2.a.bh.1.1		1
40.13	odd	4		3520.2.a.a.1.1		1
55.43	even	4		4840.2.a.i.1.1		1
60.23	odd	4		7920.2.a.e.1.1		1
220.43	odd	4		9680.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.a.d.1.1	✓	1	5.3	odd	4
880.2.a.a.1.1		1	20.3	even	4
2200.2.a.a.1.1		1	5.2	odd	4
2200.2.b.b.1849.1		2	1.1	even	1	trivial
2200.2.b.b.1849.2		2	5.4	even	2	inner
3520.2.a.a.1.1		1	40.13	odd	4
3520.2.a.bh.1.1		1	40.3	even	4
3960.2.a.f.1.1		1	15.8	even	4
4400.2.a.be.1.1		1	20.7	even	4
4400.2.b.a.4049.1		2	20.19	odd	2
4400.2.b.a.4049.2		2	4.3	odd	2
4840.2.a.i.1.1		1	55.43	even	4
7920.2.a.e.1.1		1	60.23	odd	4
9680.2.a.a.1.1		1	220.43	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2200.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +1.00000i q^{7} -6.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} +1.00000i q^{7} -6.00000 q^{9} -1.00000 q^{11} +6.00000i q^{13} +3.00000i q^{17} +5.00000 q^{19} +3.00000 q^{21} +2.00000i q^{23} +9.00000i q^{27} +5.00000 q^{29} +5.00000 q^{31} +3.00000i q^{33} -1.00000i q^{37} +18.0000 q^{39} -2.00000 q^{41} -12.0000i q^{43} -2.00000i q^{47} +6.00000 q^{49} +9.00000 q^{51} +13.0000i q^{53} -15.0000i q^{57} -2.00000 q^{59} +1.00000 q^{61} -6.00000i q^{63} +16.0000i q^{67} +6.00000 q^{69} +15.0000 q^{71} -10.0000i q^{73} -1.00000i q^{77} -2.00000 q^{79} +9.00000 q^{81} +14.0000i q^{83} -15.0000i q^{87} -9.00000 q^{89} -6.00000 q^{91} -15.0000i q^{93} -16.0000i q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 12 q^{9}+O(q^{10}) \) 2 * q - 12 * q^9	\( 2 q - 12 q^{9} - 2 q^{11} + 10 q^{19} + 6 q^{21} + 10 q^{29} + 10 q^{31} + 36 q^{39} - 4 q^{41} + 12 q^{49} + 18 q^{51} - 4 q^{59} + 2 q^{61} + 12 q^{69} + 30 q^{71} - 4 q^{79} + 18 q^{81} - 18 q^{89} - 12 q^{91} + 12 q^{99}+O(q^{100}) \) 2 * q - 12 * q^9 - 2 * q^11 + 10 * q^19 + 6 * q^21 + 10 * q^29 + 10 * q^31 + 36 * q^39 - 4 * q^41 + 12 * q^49 + 18 * q^51 - 4 * q^59 + 2 * q^61 + 12 * q^69 + 30 * q^71 - 4 * q^79 + 18 * q^81 - 18 * q^89 - 12 * q^91 + 12 * q^99

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