LMFDB - Embedded newform 1800.2.m.a.899.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,2,Mod(899,1800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1800, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1800.899");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1800.m (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:	$14.3730723638$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			899.2
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1800.899
Dual form			1800.2.m.a.899.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} +4.24264 q^{7} -2.82843 q^{8} +O(q^{10})$	$q-1.41421 q^{2} +2.00000 q^{4} +4.24264 q^{7} -2.82843 q^{8} +1.41421i q^{11} -4.24264 q^{13} -6.00000 q^{14} +4.00000 q^{16} -2.82843 q^{17} +4.00000 q^{19} -2.00000i q^{22} +6.00000i q^{23} +6.00000 q^{26} +8.48528 q^{28} +6.00000 q^{29} +8.48528i q^{31} -5.65685 q^{32} +4.00000 q^{34} -4.24264 q^{37} -5.65685 q^{38} +9.89949i q^{41} +8.00000i q^{43} +2.82843i q^{44} -8.48528i q^{46} +11.0000 q^{49} -8.48528 q^{52} -12.0000i q^{53} -12.0000 q^{56} -8.48528 q^{58} -1.41421i q^{59} -8.48528i q^{61} -12.0000i q^{62} +8.00000 q^{64} -8.00000i q^{67} -5.65685 q^{68} +14.0000i q^{73} +6.00000 q^{74} +8.00000 q^{76} +6.00000i q^{77} -8.48528i q^{79} -14.0000i q^{82} +2.82843 q^{83} -11.3137i q^{86} -4.00000i q^{88} +7.07107i q^{89} -18.0000 q^{91} +12.0000i q^{92} +10.0000i q^{97} -15.5563 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4}+O(q^{10}) $ 4 * q + 8 * q^4	$ 4 q + 8 q^{4} - 24 q^{14} + 16 q^{16} + 16 q^{19} + 24 q^{26} + 24 q^{29} + 16 q^{34} + 44 q^{49} - 48 q^{56} + 32 q^{64} + 24 q^{74} + 32 q^{76} - 72 q^{91}+O(q^{100}) $ 4 * q + 8 * q^4 - 24 * q^14 + 16 * q^16 + 16 * q^19 + 24 * q^26 + 24 * q^29 + 16 * q^34 + 44 * q^49 - 48 * q^56 + 32 * q^64 + 24 * q^74 + 32 * q^76 - 72 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1001$	$1351$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

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 $

$2$	−1.41421	−1.00000
$2$	−1.41421	−1.00000
$3$	0	0
$3$	0	0
$4$	2.00000	1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.24264	1.60357	0.801784	−	0.597614i	$-0.203885\pi$
$7$	4.24264	1.60357	0.801784	+	0.597614i	$0.203885\pi$
$8$	−2.82843	−1.00000
$9$	0	0
$10$	0	0
$11$	1.41421i	0.426401i	0.977008	+	0.213201i	$0.0683888\pi$
$11$	1.41421i	0.426401i	−0.977008	+	0.213201i	$0.931611\pi$
$12$	0	0
$13$	−4.24264	−1.17670	−0.588348	−	0.808608i	$-0.700222\pi$
$13$	−4.24264	−1.17670	−0.588348	+	0.808608i	$0.700222\pi$
$14$	−6.00000	−1.60357
$15$	0	0
$16$	4.00000	1.00000
$17$	−2.82843	−0.685994	−0.342997	−	0.939336i	$-0.611442\pi$
$17$	−2.82843	−0.685994	−0.342997	+	0.939336i	$0.611442\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	− 2.00000i	− 0.426401i
$23$	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
$26$	6.00000	1.17670
$27$	0	0
$28$	8.48528	1.60357
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	8.48528i	1.52400i	0.647576	+	0.762001i	$0.275783\pi$
$31$	8.48528i	1.52400i	−0.647576	+	0.762001i	$0.724217\pi$
$32$	−5.65685	−1.00000
$33$	0	0
$34$	4.00000	0.685994
$35$	0	0
$36$	0	0
$37$	−4.24264	−0.697486	−0.348743	−	0.937218i	$-0.613391\pi$
$37$	−4.24264	−0.697486	−0.348743	+	0.937218i	$0.613391\pi$
$38$	−5.65685	−0.917663
$39$	0	0
$40$	0	0
$41$	9.89949i	1.54604i	0.634381	+	0.773021i	$0.281255\pi$
$41$	9.89949i	1.54604i	−0.634381	+	0.773021i	$0.718745\pi$
$42$	0	0
$43$	8.00000i	1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$	8.00000i	1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$	2.82843i	0.426401i
$45$	0	0
$46$	− 8.48528i	− 1.25109i
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	11.0000	1.57143
$50$	0	0
$51$	0	0
$52$	−8.48528	−1.17670
$53$	− 12.0000i	− 1.64833i	−0.566352	−	0.824163i	$-0.691646\pi$
$53$	− 12.0000i	− 1.64833i	0.566352	−	0.824163i	$-0.308354\pi$
$54$	0	0
$55$	0	0
$56$	−12.0000	−1.60357
$57$	0	0
$58$	−8.48528	−1.11417
$59$	− 1.41421i	− 0.184115i	−0.995754	−	0.0920575i	$-0.970656\pi$
$59$	− 1.41421i	− 0.184115i	0.995754	−	0.0920575i	$-0.0293443\pi$
$60$	0	0
$61$	− 8.48528i	− 1.08643i	−0.839594	−	0.543214i	$-0.817207\pi$
$61$	− 8.48528i	− 1.08643i	0.839594	−	0.543214i	$-0.182793\pi$
$62$	− 12.0000i	− 1.52400i
$63$	0	0
$64$	8.00000	1.00000
$65$	0	0
$66$	0	0
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	−5.65685	−0.685994
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	14.0000i	1.63858i	0.573382	+	0.819288i	$0.305631\pi$
$73$	14.0000i	1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	8.00000	0.917663
$77$	6.00000i	0.683763i
$78$	0	0
$79$	− 8.48528i	− 0.954669i	−0.878722	−	0.477334i	$-0.841603\pi$
$79$	− 8.48528i	− 0.954669i	0.878722	−	0.477334i	$-0.158397\pi$
$80$	0	0
$81$	0	0
$82$	− 14.0000i	− 1.54604i
$83$	2.82843	0.310460	0.155230	−	0.987878i	$-0.450388\pi$
$83$	2.82843	0.310460	0.155230	+	0.987878i	$0.450388\pi$
$84$	0	0
$85$	0	0
$86$	− 11.3137i	− 1.21999i
$87$	0	0
$88$	− 4.00000i	− 0.426401i
$89$	7.07107i	0.749532i	0.927119	+	0.374766i	$0.122277\pi$
$89$	7.07107i	0.749532i	−0.927119	+	0.374766i	$0.877723\pi$
$90$	0	0
$91$	−18.0000	−1.88691
$92$	12.0000i	1.25109i
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	−15.5563	−1.57143
$99$	0	0
$100$	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
$103$	4.24264	0.418040	0.209020	−	0.977911i	$-0.432973\pi$
$103$	4.24264	0.418040	0.209020	+	0.977911i	$0.432973\pi$
$104$	12.0000	1.17670
$105$	0	0
$106$	16.9706i	1.64833i
$107$	14.1421	1.36717	0.683586	−	0.729870i	$-0.260419\pi$
$107$	14.1421	1.36717	0.683586	+	0.729870i	$0.260419\pi$
$108$	0	0
$109$	8.48528i	0.812743i	0.913708	+	0.406371i	$0.133206\pi$
$109$	8.48528i	0.812743i	−0.913708	+	0.406371i	$0.866794\pi$
$110$	0	0
$111$	0	0
$112$	16.9706	1.60357
$113$	11.3137	1.06430	0.532152	−	0.846649i	$-0.321383\pi$
$113$	11.3137	1.06430	0.532152	+	0.846649i	$0.321383\pi$
$114$	0	0
$115$	0	0
$116$	12.0000	1.11417
$117$	0	0
$118$	2.00000i	0.184115i
$119$	−12.0000	−1.10004
$120$	0	0
$121$	9.00000	0.818182
$122$	12.0000i	1.08643i
$123$	0	0
$124$	16.9706i	1.52400i
$125$	0	0
$126$	0	0
$127$	−4.24264	−0.376473	−0.188237	−	0.982124i	$-0.560277\pi$
$127$	−4.24264	−0.376473	−0.188237	+	0.982124i	$0.560277\pi$
$128$	−11.3137	−1.00000
$129$	0	0
$130$	0	0
$131$	1.41421i	0.123560i	0.998090	+	0.0617802i	$0.0196778\pi$
$131$	1.41421i	0.123560i	−0.998090	+	0.0617802i	$0.980322\pi$
$132$	0	0
$133$	16.9706	1.47153
$134$	11.3137i	0.977356i
$135$	0	0
$136$	8.00000	0.685994
$137$	5.65685	0.483298	0.241649	−	0.970364i	$-0.422312\pi$
$137$	5.65685	0.483298	0.241649	+	0.970364i	$0.422312\pi$
$138$	0	0
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 6.00000i	− 0.501745i
$144$	0	0
$145$	0	0
$146$	− 19.7990i	− 1.63858i
$147$	0	0
$148$	−8.48528	−0.697486
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	− 16.9706i	− 1.38104i	−0.723311	−	0.690522i	$-0.757381\pi$
$151$	− 16.9706i	− 1.38104i	0.723311	−	0.690522i	$-0.242619\pi$
$152$	−11.3137	−0.917663
$153$	0	0
$154$	− 8.48528i	− 0.683763i
$155$	0	0
$156$	0	0
$157$	4.24264	0.338600	0.169300	−	0.985565i	$-0.445849\pi$
$157$	4.24264	0.338600	0.169300	+	0.985565i	$0.445849\pi$
$158$	12.0000i	0.954669i
$159$	0	0
$160$	0	0
$161$	25.4558i	2.00620i
$162$	0	0
$163$	20.0000i	1.56652i	0.621694	+	0.783260i	$0.286445\pi$
$163$	20.0000i	1.56652i	−0.621694	+	0.783260i	$0.713555\pi$
$164$	19.7990i	1.54604i
$165$	0	0
$166$	−4.00000	−0.310460
$167$	− 18.0000i	− 1.39288i	−0.717614	−	0.696441i	$-0.754766\pi$
$167$	− 18.0000i	− 1.39288i	0.717614	−	0.696441i	$-0.245234\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	0	0
$171$	0	0
$172$	16.0000i	1.21999i
$173$	24.0000i	1.82469i	0.409426	+	0.912343i	$0.365729\pi$
$173$	24.0000i	1.82469i	−0.409426	+	0.912343i	$0.634271\pi$
$174$	0	0
$175$	0	0
$176$	5.65685i	0.426401i
$177$	0	0
$178$	− 10.0000i	− 0.749532i
$179$	7.07107i	0.528516i	0.964452	+	0.264258i	$0.0851271\pi$
$179$	7.07107i	0.528516i	−0.964452	+	0.264258i	$0.914873\pi$
$180$	0	0
$181$	8.48528i	0.630706i	0.948974	+	0.315353i	$0.102123\pi$
$181$	8.48528i	0.630706i	−0.948974	+	0.315353i	$0.897877\pi$
$182$	25.4558	1.88691
$183$	0	0
$184$	− 16.9706i	− 1.25109i
$185$	0	0
$186$	0	0
$187$	− 4.00000i	− 0.292509i
$188$	0	0
$189$	0	0
$190$	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
$193$	2.00000i	0.143963i	0.997406	+	0.0719816i	$0.0229323\pi$
$193$	2.00000i	0.143963i	−0.997406	+	0.0719816i	$0.977068\pi$
$194$	− 14.1421i	− 1.01535i
$195$	0	0
$196$	22.0000	1.57143
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	0	0
$199$	16.9706i	1.20301i	0.798869	+	0.601506i	$0.205432\pi$
$199$	16.9706i	1.20301i	−0.798869	+	0.601506i	$0.794568\pi$
$200$	0	0
$201$	0	0
$202$	8.48528	0.597022
$203$	25.4558	1.78665
$204$	0	0
$205$	0	0
$206$	−6.00000	−0.418040
$207$	0	0
$208$	−16.9706	−1.17670
$209$	5.65685i	0.391293i
$210$	0	0
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	− 24.0000i	− 1.64833i
$213$	0	0
$214$	−20.0000	−1.36717
$215$	0	0
$216$	0	0
$217$	36.0000i	2.44384i
$218$	− 12.0000i	− 0.812743i
$219$	0	0
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	−21.2132	−1.42054	−0.710271	−	0.703929i	$-0.751427\pi$
$223$	−21.2132	−1.42054	−0.710271	+	0.703929i	$0.751427\pi$
$224$	−24.0000	−1.60357
$225$	0	0
$226$	−16.0000	−1.06430
$227$	22.6274	1.50183	0.750917	−	0.660396i	$-0.229612\pi$
$227$	22.6274	1.50183	0.750917	+	0.660396i	$0.229612\pi$
$228$	0	0
$229$	− 16.9706i	− 1.12145i	−0.828003	−	0.560723i	$-0.810523\pi$
$229$	− 16.9706i	− 1.12145i	0.828003	−	0.560723i	$-0.189477\pi$
$230$	0	0
$231$	0	0
$232$	−16.9706	−1.11417
$233$	19.7990	1.29707	0.648537	−	0.761183i	$-0.275381\pi$
$233$	19.7990	1.29707	0.648537	+	0.761183i	$0.275381\pi$
$234$	0	0
$235$	0	0
$236$	− 2.82843i	− 0.184115i
$237$	0	0
$238$	16.9706	1.10004
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	20.0000	1.28831	0.644157	−	0.764894i	$-0.277208\pi$
$241$	20.0000	1.28831	0.644157	+	0.764894i	$0.277208\pi$
$242$	−12.7279	−0.818182
$243$	0	0
$244$	− 16.9706i	− 1.08643i
$245$	0	0
$246$	0	0
$247$	−16.9706	−1.07981
$248$	− 24.0000i	− 1.52400i
$249$	0	0
$250$	0	0
$251$	1.41421i	0.0892644i	0.999003	+	0.0446322i	$0.0142116\pi$
$251$	1.41421i	0.0892644i	−0.999003	+	0.0446322i	$0.985788\pi$
$252$	0	0
$253$	−8.48528	−0.533465
$254$	6.00000	0.376473
$255$	0	0
$256$	16.0000	1.00000
$257$	−11.3137	−0.705730	−0.352865	−	0.935674i	$-0.614792\pi$
$257$	−11.3137	−0.705730	−0.352865	+	0.935674i	$0.614792\pi$
$258$	0	0
$259$	−18.0000	−1.11847
$260$	0	0
$261$	0	0
$262$	− 2.00000i	− 0.123560i
$263$	− 6.00000i	− 0.369976i	−0.982741	−	0.184988i	$-0.940775\pi$
$263$	− 6.00000i	− 0.369976i	0.982741	−	0.184988i	$-0.0592246\pi$
$264$	0	0
$265$	0	0
$266$	−24.0000	−1.47153
$267$	0	0
$268$	− 16.0000i	− 0.977356i
$269$	30.0000	1.82913	0.914566	−	0.404436i	$-0.132532\pi$
$269$	30.0000	1.82913	0.914566	+	0.404436i	$0.132532\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	−11.3137	−0.685994
$273$	0	0
$274$	−8.00000	−0.483298
$275$	0	0
$276$	0	0
$277$	4.24264	0.254916	0.127458	−	0.991844i	$-0.459318\pi$
$277$	4.24264	0.254916	0.127458	+	0.991844i	$0.459318\pi$
$278$	2.82843	0.169638
$279$	0	0
$280$	0	0
$281$	9.89949i	0.590554i	0.955412	+	0.295277i	$0.0954120\pi$
$281$	9.89949i	0.590554i	−0.955412	+	0.295277i	$0.904588\pi$
$282$	0	0
$283$	− 4.00000i	− 0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$	− 4.00000i	− 0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$	0	0
$285$	0	0
$286$	8.48528i	0.501745i
$287$	42.0000i	2.47918i
$288$	0	0
$289$	−9.00000	−0.529412
$290$	0	0
$291$	0	0
$292$	28.0000i	1.63858i
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	0	0
$295$	0	0
$296$	12.0000	0.697486
$297$	0	0
$298$	25.4558	1.47462
$299$	− 25.4558i	− 1.47215i
$300$	0	0
$301$	33.9411i	1.95633i
$302$	24.0000i	1.38104i
$303$	0	0
$304$	16.0000	0.917663
$305$	0	0
$306$	0	0
$307$	− 8.00000i	− 0.456584i	−0.973593	−	0.228292i	$-0.926686\pi$
$307$	− 8.00000i	− 0.456584i	0.973593	−	0.228292i	$-0.0733141\pi$
$308$	12.0000i	0.683763i
$309$	0	0
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	− 10.0000i	− 0.565233i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$	− 10.0000i	− 0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$	−6.00000	−0.338600
$315$	0	0
$316$	− 16.9706i	− 0.954669i
$317$	− 18.0000i	− 1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$	− 18.0000i	− 1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$	0	0
$319$	8.48528i	0.475085i
$320$	0	0
$321$	0	0
$322$	− 36.0000i	− 2.00620i
$323$	−11.3137	−0.629512
$324$	0	0
$325$	0	0
$326$	− 28.2843i	− 1.56652i
$327$	0	0
$328$	− 28.0000i	− 1.54604i
$329$	0	0
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	5.65685	0.310460
$333$	0	0
$334$	25.4558i	1.39288i
$335$	0	0
$336$	0	0
$337$	− 2.00000i	− 0.108947i	−0.998515	−	0.0544735i	$-0.982652\pi$
$337$	− 2.00000i	− 0.108947i	0.998515	−	0.0544735i	$-0.0173480\pi$
$338$	−7.07107	−0.384615
$339$	0	0
$340$	0	0
$341$	−12.0000	−0.649836
$342$	0	0
$343$	16.9706	0.916324
$344$	− 22.6274i	− 1.21999i
$345$	0	0
$346$	− 33.9411i	− 1.82469i
$347$	5.65685	0.303676	0.151838	−	0.988405i	$-0.451481\pi$
$347$	5.65685	0.303676	0.151838	+	0.988405i	$0.451481\pi$
$348$	0	0
$349$	8.48528i	0.454207i	0.973871	+	0.227103i	$0.0729255\pi$
$349$	8.48528i	0.454207i	−0.973871	+	0.227103i	$0.927074\pi$
$350$	0	0
$351$	0	0
$352$	− 8.00000i	− 0.426401i
$353$	2.82843	0.150542	0.0752710	−	0.997163i	$-0.476018\pi$
$353$	2.82843	0.150542	0.0752710	+	0.997163i	$0.476018\pi$
$354$	0	0
$355$	0	0
$356$	14.1421i	0.749532i
$357$	0	0
$358$	− 10.0000i	− 0.528516i
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 12.0000i	− 0.630706i
$363$	0	0
$364$	−36.0000	−1.88691
$365$	0	0
$366$	0	0
$367$	29.6985	1.55025	0.775124	−	0.631809i	$-0.217687\pi$
$367$	29.6985	1.55025	0.775124	+	0.631809i	$0.217687\pi$
$368$	24.0000i	1.25109i
$369$	0	0
$370$	0	0
$371$	− 50.9117i	− 2.64320i
$372$	0	0
$373$	−12.7279	−0.659027	−0.329513	−	0.944151i	$-0.606885\pi$
$373$	−12.7279	−0.659027	−0.329513	+	0.944151i	$0.606885\pi$
$374$	5.65685i	0.292509i
$375$	0	0
$376$	0	0
$377$	−25.4558	−1.31104
$378$	0	0
$379$	−26.0000	−1.33553	−0.667765	−	0.744372i	$-0.732749\pi$
$379$	−26.0000	−1.33553	−0.667765	+	0.744372i	$0.732749\pi$
$380$	0	0
$381$	0	0
$382$	−16.9706	−0.868290
$383$	− 24.0000i	− 1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$	− 24.0000i	− 1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$	0	0
$385$	0	0
$386$	− 2.82843i	− 0.143963i
$387$	0	0
$388$	20.0000i	1.01535i
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	− 16.9706i	− 0.858238i
$392$	−31.1127	−1.57143
$393$	0	0
$394$	8.48528i	0.427482i
$395$	0	0
$396$	0	0
$397$	12.7279	0.638796	0.319398	−	0.947621i	$-0.396519\pi$
$397$	12.7279	0.638796	0.319398	+	0.947621i	$0.396519\pi$
$398$	− 24.0000i	− 1.20301i
$399$	0	0
$400$	0	0
$401$	18.3848i	0.918092i	0.888413	+	0.459046i	$0.151809\pi$
$401$	18.3848i	0.918092i	−0.888413	+	0.459046i	$0.848191\pi$
$402$	0	0
$403$	− 36.0000i	− 1.79329i
$404$	−12.0000	−0.597022
$405$	0	0
$406$	−36.0000	−1.78665
$407$	− 6.00000i	− 0.297409i
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	0	0
$412$	8.48528	0.418040
$413$	− 6.00000i	− 0.295241i
$414$	0	0
$415$	0	0
$416$	24.0000	1.17670
$417$	0	0
$418$	− 8.00000i	− 0.391293i
$419$	− 26.8701i	− 1.31269i	−0.754462	−	0.656344i	$-0.772102\pi$
$419$	− 26.8701i	− 1.31269i	0.754462	−	0.656344i	$-0.227898\pi$
$420$	0	0
$421$	− 33.9411i	− 1.65419i	−0.562063	−	0.827095i	$-0.689992\pi$
$421$	− 33.9411i	− 1.65419i	0.562063	−	0.827095i	$-0.310008\pi$
$422$	31.1127	1.51454
$423$	0	0
$424$	33.9411i	1.64833i
$425$	0	0
$426$	0	0
$427$	− 36.0000i	− 1.74216i
$428$	28.2843	1.36717
$429$	0	0
$430$	0	0
$431$	−36.0000	−1.73406	−0.867029	−	0.498257i	$-0.833974\pi$
$431$	−36.0000	−1.73406	−0.867029	+	0.498257i	$0.833974\pi$
$432$	0	0
$433$	− 10.0000i	− 0.480569i	−0.970702	−	0.240285i	$-0.922759\pi$
$433$	− 10.0000i	− 0.480569i	0.970702	−	0.240285i	$-0.0772408\pi$
$434$	− 50.9117i	− 2.44384i
$435$	0	0
$436$	16.9706i	0.812743i
$437$	24.0000i	1.14808i
$438$	0	0
$439$	− 16.9706i	− 0.809961i	−0.914325	−	0.404980i	$-0.867278\pi$
$439$	− 16.9706i	− 0.809961i	0.914325	−	0.404980i	$-0.132722\pi$
$440$	0	0
$441$	0	0
$442$	−16.9706	−0.807207
$443$	11.3137	0.537531	0.268765	−	0.963206i	$-0.413384\pi$
$443$	11.3137	0.537531	0.268765	+	0.963206i	$0.413384\pi$
$444$	0	0
$445$	0	0
$446$	30.0000	1.42054
$447$	0	0
$448$	33.9411	1.60357
$449$	− 26.8701i	− 1.26808i	−0.773302	−	0.634038i	$-0.781396\pi$
$449$	− 26.8701i	− 1.26808i	0.773302	−	0.634038i	$-0.218604\pi$
$450$	0	0
$451$	−14.0000	−0.659234
$452$	22.6274	1.06430
$453$	0	0
$454$	−32.0000	−1.50183
$455$	0	0
$456$	0	0
$457$	− 2.00000i	− 0.0935561i	−0.998905	−	0.0467780i	$-0.985105\pi$
$457$	− 2.00000i	− 0.0935561i	0.998905	−	0.0467780i	$-0.0148953\pi$
$458$	24.0000i	1.12145i
$459$	0	0
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	12.7279	0.591517	0.295758	−	0.955263i	$-0.404428\pi$
$463$	12.7279	0.591517	0.295758	+	0.955263i	$0.404428\pi$
$464$	24.0000	1.11417
$465$	0	0
$466$	−28.0000	−1.29707
$467$	−19.7990	−0.916188	−0.458094	−	0.888904i	$-0.651468\pi$
$467$	−19.7990	−0.916188	−0.458094	+	0.888904i	$0.651468\pi$
$468$	0	0
$469$	− 33.9411i	− 1.56726i
$470$	0	0
$471$	0	0
$472$	4.00000i	0.184115i
$473$	−11.3137	−0.520205
$474$	0	0
$475$	0	0
$476$	−24.0000	−1.10004
$477$	0	0
$478$	−33.9411	−1.55243
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	18.0000	0.820729
$482$	−28.2843	−1.28831
$483$	0	0
$484$	18.0000	0.818182
$485$	0	0
$486$	0	0
$487$	−21.2132	−0.961262	−0.480631	−	0.876923i	$-0.659592\pi$
$487$	−21.2132	−0.961262	−0.480631	+	0.876923i	$0.659592\pi$
$488$	24.0000i	1.08643i
$489$	0	0
$490$	0	0
$491$	1.41421i	0.0638226i	0.999491	+	0.0319113i	$0.0101594\pi$
$491$	1.41421i	0.0638226i	−0.999491	+	0.0319113i	$0.989841\pi$
$492$	0	0
$493$	−16.9706	−0.764316
$494$	24.0000	1.07981
$495$	0	0
$496$	33.9411i	1.52400i
$497$	0	0
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	0	0
$502$	− 2.00000i	− 0.0892644i
$503$	24.0000i	1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$	24.0000i	1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$	0	0
$505$	0	0
$506$	12.0000	0.533465
$507$	0	0
$508$	−8.48528	−0.376473
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	59.3970i	2.62757i
$512$	−22.6274	−1.00000
$513$	0	0
$514$	16.0000	0.705730
$515$	0	0
$516$	0	0
$517$	0	0
$518$	25.4558	1.11847
$519$	0	0
$520$	0	0
$521$	− 15.5563i	− 0.681536i	−0.940147	−	0.340768i	$-0.889313\pi$
$521$	− 15.5563i	− 0.681536i	0.940147	−	0.340768i	$-0.110687\pi$
$522$	0	0
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	2.82843i	0.123560i
$525$	0	0
$526$	8.48528i	0.369976i
$527$	− 24.0000i	− 1.04546i
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	0	0
$532$	33.9411	1.47153
$533$	− 42.0000i	− 1.81922i
$534$	0	0
$535$	0	0
$536$	22.6274i	0.977356i
$537$	0	0
$538$	−42.4264	−1.82913
$539$	15.5563i	0.670059i
$540$	0	0
$541$	− 8.48528i	− 0.364811i	−0.983223	−	0.182405i	$-0.941612\pi$
$541$	− 8.48528i	− 0.364811i	0.983223	−	0.182405i	$-0.0583883\pi$
$542$	0	0
$543$	0	0
$544$	16.0000	0.685994
$545$	0	0
$546$	0	0
$547$	− 20.0000i	− 0.855138i	−0.903983	−	0.427569i	$-0.859370\pi$
$547$	− 20.0000i	− 0.855138i	0.903983	−	0.427569i	$-0.140630\pi$
$548$	11.3137	0.483298
$549$	0	0
$550$	0	0
$551$	24.0000	1.02243
$552$	0	0
$553$	− 36.0000i	− 1.53088i
$554$	−6.00000	−0.254916
$555$	0	0
$556$	−4.00000	−0.169638
$557$	− 12.0000i	− 0.508456i	−0.967144	−	0.254228i	$-0.918179\pi$
$557$	− 12.0000i	− 0.508456i	0.967144	−	0.254228i	$-0.0818214\pi$
$558$	0	0
$559$	− 33.9411i	− 1.43556i
$560$	0	0
$561$	0	0
$562$	− 14.0000i	− 0.590554i
$563$	−39.5980	−1.66886	−0.834428	−	0.551117i	$-0.814202\pi$
$563$	−39.5980	−1.66886	−0.834428	+	0.551117i	$0.814202\pi$
$564$	0	0
$565$	0	0
$566$	5.65685i	0.237775i
$567$	0	0
$568$	0	0
$569$	− 1.41421i	− 0.0592869i	−0.999561	−	0.0296435i	$-0.990563\pi$
$569$	− 1.41421i	− 0.0592869i	0.999561	−	0.0296435i	$-0.00943719\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	− 12.0000i	− 0.501745i
$573$	0	0
$574$	− 59.3970i	− 2.47918i
$575$	0	0
$576$	0	0
$577$	22.0000i	0.915872i	0.888985	+	0.457936i	$0.151411\pi$
$577$	22.0000i	0.915872i	−0.888985	+	0.457936i	$0.848589\pi$
$578$	12.7279	0.529412
$579$	0	0
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	16.9706	0.702849
$584$	− 39.5980i	− 1.63858i
$585$	0	0
$586$	− 8.48528i	− 0.350524i
$587$	14.1421	0.583708	0.291854	−	0.956463i	$-0.405728\pi$
$587$	14.1421	0.583708	0.291854	+	0.956463i	$0.405728\pi$
$588$	0	0
$589$	33.9411i	1.39852i
$590$	0	0
$591$	0	0
$592$	−16.9706	−0.697486
$593$	−5.65685	−0.232299	−0.116150	−	0.993232i	$-0.537055\pi$
$593$	−5.65685	−0.232299	−0.116150	+	0.993232i	$0.537055\pi$
$594$	0	0
$595$	0	0
$596$	−36.0000	−1.47462
$597$	0	0
$598$	36.0000i	1.47215i
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	32.0000	1.30531	0.652654	−	0.757656i	$-0.273656\pi$
$601$	32.0000	1.30531	0.652654	+	0.757656i	$0.273656\pi$
$602$	− 48.0000i	− 1.95633i
$603$	0	0
$604$	− 33.9411i	− 1.38104i
$605$	0	0
$606$	0	0
$607$	4.24264	0.172203	0.0861017	−	0.996286i	$-0.472559\pi$
$607$	4.24264	0.172203	0.0861017	+	0.996286i	$0.472559\pi$
$608$	−22.6274	−0.917663
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	12.7279	0.514076	0.257038	−	0.966401i	$-0.417253\pi$
$613$	12.7279	0.514076	0.257038	+	0.966401i	$0.417253\pi$
$614$	11.3137i	0.456584i
$615$	0	0
$616$	− 16.9706i	− 0.683763i
$617$	−36.7696	−1.48029	−0.740143	−	0.672449i	$-0.765242\pi$
$617$	−36.7696	−1.48029	−0.740143	+	0.672449i	$0.765242\pi$
$618$	0	0
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	0	0
$621$	0	0
$622$	16.9706	0.680458
$623$	30.0000i	1.20192i
$624$	0	0
$625$	0	0
$626$	14.1421i	0.565233i
$627$	0	0
$628$	8.48528	0.338600
$629$	12.0000	0.478471
$630$	0	0
$631$	− 8.48528i	− 0.337794i	−0.985634	−	0.168897i	$-0.945980\pi$
$631$	− 8.48528i	− 0.337794i	0.985634	−	0.168897i	$-0.0540205\pi$
$632$	24.0000i	0.954669i
$633$	0	0
$634$	25.4558i	1.01098i
$635$	0	0
$636$	0	0
$637$	−46.6690	−1.84909
$638$	− 12.0000i	− 0.475085i
$639$	0	0
$640$	0	0
$641$	− 41.0122i	− 1.61988i	−0.586510	−	0.809942i	$-0.699498\pi$
$641$	− 41.0122i	− 1.61988i	0.586510	−	0.809942i	$-0.300502\pi$
$642$	0	0
$643$	− 40.0000i	− 1.57745i	−0.614749	−	0.788723i	$-0.710743\pi$
$643$	− 40.0000i	− 1.57745i	0.614749	−	0.788723i	$-0.289257\pi$
$644$	50.9117i	2.00620i
$645$	0	0
$646$	16.0000	0.629512
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	2.00000	0.0785069
$650$	0	0
$651$	0	0
$652$	40.0000i	1.56652i
$653$	18.0000i	0.704394i	0.935926	+	0.352197i	$0.114565\pi$
$653$	18.0000i	0.704394i	−0.935926	+	0.352197i	$0.885435\pi$
$654$	0	0
$655$	0	0
$656$	39.5980i	1.54604i
$657$	0	0
$658$	0	0
$659$	− 9.89949i	− 0.385630i	−0.981235	−	0.192815i	$-0.938238\pi$
$659$	− 9.89949i	− 0.385630i	0.981235	−	0.192815i	$-0.0617617\pi$
$660$	0	0
$661$	− 25.4558i	− 0.990118i	−0.868859	−	0.495059i	$-0.835147\pi$
$661$	− 25.4558i	− 0.990118i	0.868859	−	0.495059i	$-0.164853\pi$
$662$	39.5980	1.53902
$663$	0	0
$664$	−8.00000	−0.310460
$665$	0	0
$666$	0	0
$667$	36.0000i	1.39393i
$668$	− 36.0000i	− 1.39288i
$669$	0	0
$670$	0	0
$671$	12.0000	0.463255
$672$	0	0
$673$	− 10.0000i	− 0.385472i	−0.981251	−	0.192736i	$-0.938264\pi$
$673$	− 10.0000i	− 0.385472i	0.981251	−	0.192736i	$-0.0617360\pi$
$674$	2.82843i	0.108947i
$675$	0	0
$676$	10.0000	0.384615
$677$	− 6.00000i	− 0.230599i	−0.993331	−	0.115299i	$-0.963217\pi$
$677$	− 6.00000i	− 0.230599i	0.993331	−	0.115299i	$-0.0367827\pi$
$678$	0	0
$679$	42.4264i	1.62818i
$680$	0	0
$681$	0	0
$682$	16.9706	0.649836
$683$	−14.1421	−0.541134	−0.270567	−	0.962701i	$-0.587211\pi$
$683$	−14.1421	−0.541134	−0.270567	+	0.962701i	$0.587211\pi$
$684$	0	0
$685$	0	0
$686$	−24.0000	−0.916324
$687$	0	0
$688$	32.0000i	1.21999i
$689$	50.9117i	1.93958i
$690$	0	0
$691$	44.0000	1.67384	0.836919	−	0.547326i	$-0.184354\pi$
$691$	44.0000	1.67384	0.836919	+	0.547326i	$0.184354\pi$
$692$	48.0000i	1.82469i
$693$	0	0
$694$	−8.00000	−0.303676
$695$	0	0
$696$	0	0
$697$	− 28.0000i	− 1.06058i
$698$	− 12.0000i	− 0.454207i
$699$	0	0
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	−16.9706	−0.640057
$704$	11.3137i	0.426401i
$705$	0	0
$706$	−4.00000	−0.150542
$707$	−25.4558	−0.957366
$708$	0	0
$709$	50.9117i	1.91203i	0.293320	+	0.956014i	$0.405240\pi$
$709$	50.9117i	1.91203i	−0.293320	+	0.956014i	$0.594760\pi$
$710$	0	0
$711$	0	0
$712$	− 20.0000i	− 0.749532i
$713$	−50.9117	−1.90666
$714$	0	0
$715$	0	0
$716$	14.1421i	0.528516i
$717$	0	0
$718$	0	0
$719$	−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
$721$	18.0000	0.670355
$722$	4.24264	0.157895
$723$	0	0
$724$	16.9706i	0.630706i
$725$	0	0
$726$	0	0
$727$	4.24264	0.157351	0.0786754	−	0.996900i	$-0.474931\pi$
$727$	4.24264	0.157351	0.0786754	+	0.996900i	$0.474931\pi$
$728$	50.9117	1.88691
$729$	0	0
$730$	0	0
$731$	− 22.6274i	− 0.836905i
$732$	0	0
$733$	12.7279	0.470117	0.235058	−	0.971981i	$-0.424472\pi$
$733$	12.7279	0.470117	0.235058	+	0.971981i	$0.424472\pi$
$734$	−42.0000	−1.55025
$735$	0	0
$736$	− 33.9411i	− 1.25109i
$737$	11.3137	0.416746
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	0	0
$742$	72.0000i	2.64320i
$743$	24.0000i	0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$	24.0000i	0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$	0	0
$745$	0	0
$746$	18.0000	0.659027
$747$	0	0
$748$	− 8.00000i	− 0.292509i
$749$	60.0000	2.19235
$750$	0	0
$751$	− 42.4264i	− 1.54816i	−0.633087	−	0.774081i	$-0.718212\pi$
$751$	− 42.4264i	− 1.54816i	0.633087	−	0.774081i	$-0.281788\pi$
$752$	0	0
$753$	0	0
$754$	36.0000	1.31104
$755$	0	0
$756$	0	0
$757$	4.24264	0.154201	0.0771007	−	0.997023i	$-0.475434\pi$
$757$	4.24264	0.154201	0.0771007	+	0.997023i	$0.475434\pi$
$758$	36.7696	1.33553
$759$	0	0
$760$	0	0
$761$	26.8701i	0.974039i	0.873391	+	0.487019i	$0.161916\pi$
$761$	26.8701i	0.974039i	−0.873391	+	0.487019i	$0.838084\pi$
$762$	0	0
$763$	36.0000i	1.30329i
$764$	24.0000	0.868290
$765$	0	0
$766$	33.9411i	1.22634i
$767$	6.00000i	0.216647i
$768$	0	0
$769$	40.0000	1.44244	0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000	1.44244	0.721218	+	0.692708i	$0.243582\pi$
$770$	0	0
$771$	0	0
$772$	4.00000i	0.143963i
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	0	0
$775$	0	0
$776$	− 28.2843i	− 1.01535i
$777$	0	0
$778$	42.4264	1.52106
$779$	39.5980i	1.41874i
$780$	0	0
$781$	0	0
$782$	24.0000i	0.858238i
$783$	0	0
$784$	44.0000	1.57143
$785$	0	0
$786$	0	0
$787$	− 32.0000i	− 1.14068i	−0.821410	−	0.570338i	$-0.806812\pi$
$787$	− 32.0000i	− 1.14068i	0.821410	−	0.570338i	$-0.193188\pi$
$788$	− 12.0000i	− 0.427482i
$789$	0	0
$790$	0	0
$791$	48.0000	1.70668
$792$	0	0
$793$	36.0000i	1.27840i
$794$	−18.0000	−0.638796
$795$	0	0
$796$	33.9411i	1.20301i
$797$	24.0000i	0.850124i	0.905164	+	0.425062i	$0.139748\pi$
$797$	24.0000i	0.850124i	−0.905164	+	0.425062i	$0.860252\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	− 26.0000i	− 0.918092i
$803$	−19.7990	−0.698691
$804$	0	0
$805$	0	0
$806$	50.9117i	1.79329i
$807$	0	0
$808$	16.9706	0.597022
$809$	− 18.3848i	− 0.646374i	−0.946335	−	0.323187i	$-0.895246\pi$
$809$	− 18.3848i	− 0.646374i	0.946335	−	0.323187i	$-0.104754\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	50.9117	1.78665
$813$	0	0
$814$	8.48528i	0.297409i
$815$	0	0
$816$	0	0
$817$	32.0000i	1.11954i
$818$	−14.1421	−0.494468
$819$	0	0
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	0	0
$823$	−4.24264	−0.147889	−0.0739446	−	0.997262i	$-0.523559\pi$
$823$	−4.24264	−0.147889	−0.0739446	+	0.997262i	$0.523559\pi$
$824$	−12.0000	−0.418040
$825$	0	0
$826$	8.48528i	0.295241i
$827$	5.65685	0.196708	0.0983540	−	0.995151i	$-0.468642\pi$
$827$	5.65685	0.196708	0.0983540	+	0.995151i	$0.468642\pi$
$828$	0	0
$829$	− 8.48528i	− 0.294706i	−0.989084	−	0.147353i	$-0.952925\pi$
$829$	− 8.48528i	− 0.294706i	0.989084	−	0.147353i	$-0.0470753\pi$
$830$	0	0
$831$	0	0
$832$	−33.9411	−1.17670
$833$	−31.1127	−1.07799
$834$	0	0
$835$	0	0
$836$	11.3137i	0.391293i
$837$	0	0
$838$	38.0000i	1.31269i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	48.0000i	1.65419i
$843$	0	0
$844$	−44.0000	−1.51454
$845$	0	0
$846$	0	0
$847$	38.1838	1.31201
$848$	− 48.0000i	− 1.64833i
$849$	0	0
$850$	0	0
$851$	− 25.4558i	− 0.872615i
$852$	0	0
$853$	29.6985	1.01686	0.508428	−	0.861104i	$-0.330227\pi$
$853$	29.6985	1.01686	0.508428	+	0.861104i	$0.330227\pi$
$854$	50.9117i	1.74216i
$855$	0	0
$856$	−40.0000	−1.36717
$857$	48.0833	1.64249	0.821246	−	0.570574i	$-0.193279\pi$
$857$	48.0833	1.64249	0.821246	+	0.570574i	$0.193279\pi$
$858$	0	0
$859$	46.0000	1.56950	0.784750	−	0.619813i	$-0.212791\pi$
$859$	46.0000	1.56950	0.784750	+	0.619813i	$0.212791\pi$
$860$	0	0
$861$	0	0
$862$	50.9117	1.73406
$863$	− 18.0000i	− 0.612727i	−0.951915	−	0.306364i	$-0.900888\pi$
$863$	− 18.0000i	− 0.612727i	0.951915	−	0.306364i	$-0.0991123\pi$
$864$	0	0
$865$	0	0
$866$	14.1421i	0.480569i
$867$	0	0
$868$	72.0000i	2.44384i
$869$	12.0000	0.407072
$870$	0	0
$871$	33.9411i	1.15005i
$872$	− 24.0000i	− 0.812743i
$873$	0	0
$874$	− 33.9411i	− 1.14808i
$875$	0	0
$876$	0	0
$877$	12.7279	0.429791	0.214896	−	0.976637i	$-0.431059\pi$
$877$	12.7279	0.429791	0.214896	+	0.976637i	$0.431059\pi$
$878$	24.0000i	0.809961i
$879$	0	0
$880$	0	0
$881$	− 32.5269i	− 1.09586i	−0.836524	−	0.547930i	$-0.815416\pi$
$881$	− 32.5269i	− 1.09586i	0.836524	−	0.547930i	$-0.184584\pi$
$882$	0	0
$883$	− 28.0000i	− 0.942275i	−0.882060	−	0.471138i	$-0.843844\pi$
$883$	− 28.0000i	− 0.942275i	0.882060	−	0.471138i	$-0.156156\pi$
$884$	24.0000	0.807207
$885$	0	0
$886$	−16.0000	−0.537531
$887$	− 18.0000i	− 0.604381i	−0.953248	−	0.302190i	$-0.902282\pi$
$887$	− 18.0000i	− 0.604381i	0.953248	−	0.302190i	$-0.0977178\pi$
$888$	0	0
$889$	−18.0000	−0.603701
$890$	0	0
$891$	0	0
$892$	−42.4264	−1.42054
$893$	0	0
$894$	0	0
$895$	0	0
$896$	−48.0000	−1.60357
$897$	0	0
$898$	38.0000i	1.26808i
$899$	50.9117i	1.69800i
$900$	0	0
$901$	33.9411i	1.13074i
$902$	19.7990	0.659234
$903$	0	0
$904$	−32.0000	−1.06430
$905$	0	0
$906$	0	0
$907$	28.0000i	0.929725i	0.885383	+	0.464862i	$0.153896\pi$
$907$	28.0000i	0.929725i	−0.885383	+	0.464862i	$0.846104\pi$
$908$	45.2548	1.50183
$909$	0	0
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	0	0
$913$	4.00000i	0.132381i
$914$	2.82843i	0.0935561i
$915$	0	0
$916$	− 33.9411i	− 1.12145i
$917$	6.00000i	0.198137i
$918$	0	0
$919$	25.4558i	0.839711i	0.907591	+	0.419855i	$0.137919\pi$
$919$	25.4558i	0.839711i	−0.907591	+	0.419855i	$0.862081\pi$
$920$	0	0
$921$	0	0
$922$	−25.4558	−0.838344
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−18.0000	−0.591517
$927$	0	0
$928$	−33.9411	−1.11417
$929$	41.0122i	1.34557i	0.739840	+	0.672783i	$0.234901\pi$
$929$	41.0122i	1.34557i	−0.739840	+	0.672783i	$0.765099\pi$
$930$	0	0
$931$	44.0000	1.44204
$932$	39.5980	1.29707
$933$	0	0
$934$	28.0000	0.916188
$935$	0	0
$936$	0	0
$937$	− 38.0000i	− 1.24141i	−0.784046	−	0.620703i	$-0.786847\pi$
$937$	− 38.0000i	− 1.24141i	0.784046	−	0.620703i	$-0.213153\pi$
$938$	48.0000i	1.56726i
$939$	0	0
$940$	0	0
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	0	0
$943$	−59.3970	−1.93423
$944$	− 5.65685i	− 0.184115i
$945$	0	0
$946$	16.0000	0.520205
$947$	22.6274	0.735292	0.367646	−	0.929966i	$-0.380164\pi$
$947$	22.6274	0.735292	0.367646	+	0.929966i	$0.380164\pi$
$948$	0	0
$949$	− 59.3970i	− 1.92811i
$950$	0	0
$951$	0	0
$952$	33.9411	1.10004
$953$	28.2843	0.916217	0.458109	−	0.888896i	$-0.348527\pi$
$953$	28.2843	0.916217	0.458109	+	0.888896i	$0.348527\pi$
$954$	0	0
$955$	0	0
$956$	48.0000	1.55243
$957$	0	0
$958$	0	0
$959$	24.0000	0.775000
$960$	0	0
$961$	−41.0000	−1.32258
$962$	−25.4558	−0.820729
$963$	0	0
$964$	40.0000	1.28831
$965$	0	0
$966$	0	0
$967$	−21.2132	−0.682171	−0.341085	−	0.940032i	$-0.610795\pi$
$967$	−21.2132	−0.682171	−0.341085	+	0.940032i	$0.610795\pi$
$968$	−25.4558	−0.818182
$969$	0	0
$970$	0	0
$971$	43.8406i	1.40691i	0.710739	+	0.703456i	$0.248361\pi$
$971$	43.8406i	1.40691i	−0.710739	+	0.703456i	$0.751639\pi$
$972$	0	0
$973$	−8.48528	−0.272026
$974$	30.0000	0.961262
$975$	0	0
$976$	− 33.9411i	− 1.08643i
$977$	−2.82843	−0.0904894	−0.0452447	−	0.998976i	$-0.514407\pi$
$977$	−2.82843	−0.0904894	−0.0452447	+	0.998976i	$0.514407\pi$
$978$	0	0
$979$	−10.0000	−0.319601
$980$	0	0
$981$	0	0
$982$	− 2.00000i	− 0.0638226i
$983$	− 24.0000i	− 0.765481i	−0.923856	−	0.382741i	$-0.874980\pi$
$983$	− 24.0000i	− 0.765481i	0.923856	−	0.382741i	$-0.125020\pi$
$984$	0	0
$985$	0	0
$986$	24.0000	0.764316
$987$	0	0
$988$	−33.9411	−1.07981
$989$	−48.0000	−1.52631
$990$	0	0
$991$	− 16.9706i	− 0.539088i	−0.962988	−	0.269544i	$-0.913127\pi$
$991$	− 16.9706i	− 0.539088i	0.962988	−	0.269544i	$-0.0868729\pi$
$992$	− 48.0000i	− 1.52400i
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−21.2132	−0.671829	−0.335914	−	0.941893i	$-0.609045\pi$
$997$	−21.2132	−0.671829	−0.335914	+	0.941893i	$0.609045\pi$
$998$	−5.65685	−0.179065
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.2.m.a.899.2		4
3.2	odd	2		1800.2.m.b.899.3		4
4.3	odd	2		7200.2.m.b.3599.1		4
5.2	odd	4		360.2.b.a.251.1	✓	2
5.3	odd	4		1800.2.b.b.251.2		2
5.4	even	2	inner	1800.2.m.a.899.4		4
8.3	odd	2		1800.2.m.b.899.2		4
8.5	even	2		7200.2.m.a.3599.3		4
12.11	even	2		7200.2.m.a.3599.2		4
15.2	even	4		360.2.b.b.251.2	yes	2
15.8	even	4		1800.2.b.a.251.1		2
15.14	odd	2		1800.2.m.b.899.1		4
20.3	even	4		7200.2.b.b.4751.2		2
20.7	even	4		1440.2.b.a.431.1		2
20.19	odd	2		7200.2.m.b.3599.3		4
24.5	odd	2		7200.2.m.b.3599.4		4
24.11	even	2	inner	1800.2.m.a.899.3		4
40.3	even	4		1800.2.b.a.251.2		2
40.13	odd	4		7200.2.b.a.4751.1		2
40.19	odd	2		1800.2.m.b.899.4		4
40.27	even	4		360.2.b.b.251.1	yes	2
40.29	even	2		7200.2.m.a.3599.1		4
40.37	odd	4		1440.2.b.b.431.2		2
60.23	odd	4		7200.2.b.a.4751.2		2
60.47	odd	4		1440.2.b.b.431.1		2
60.59	even	2		7200.2.m.a.3599.4		4
120.29	odd	2		7200.2.m.b.3599.2		4
120.53	even	4		7200.2.b.b.4751.1		2
120.59	even	2	inner	1800.2.m.a.899.1		4
120.77	even	4		1440.2.b.a.431.2		2
120.83	odd	4		1800.2.b.b.251.1		2
120.107	odd	4		360.2.b.a.251.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.b.a.251.1	✓	2	5.2	odd	4
360.2.b.a.251.2	yes	2	120.107	odd	4
360.2.b.b.251.1	yes	2	40.27	even	4
360.2.b.b.251.2	yes	2	15.2	even	4
1440.2.b.a.431.1		2	20.7	even	4
1440.2.b.a.431.2		2	120.77	even	4
1440.2.b.b.431.1		2	60.47	odd	4
1440.2.b.b.431.2		2	40.37	odd	4
1800.2.b.a.251.1		2	15.8	even	4
1800.2.b.a.251.2		2	40.3	even	4
1800.2.b.b.251.1		2	120.83	odd	4
1800.2.b.b.251.2		2	5.3	odd	4
1800.2.m.a.899.1		4	120.59	even	2	inner
1800.2.m.a.899.2		4	1.1	even	1	trivial
1800.2.m.a.899.3		4	24.11	even	2	inner
1800.2.m.a.899.4		4	5.4	even	2	inner
1800.2.m.b.899.1		4	15.14	odd	2
1800.2.m.b.899.2		4	8.3	odd	2
1800.2.m.b.899.3		4	3.2	odd	2
1800.2.m.b.899.4		4	40.19	odd	2
7200.2.b.a.4751.1		2	40.13	odd	4
7200.2.b.a.4751.2		2	60.23	odd	4
7200.2.b.b.4751.1		2	120.53	even	4
7200.2.b.b.4751.2		2	20.3	even	4
7200.2.m.a.3599.1		4	40.29	even	2
7200.2.m.a.3599.2		4	12.11	even	2
7200.2.m.a.3599.3		4	8.5	even	2
7200.2.m.a.3599.4		4	60.59	even	2
7200.2.m.b.3599.1		4	4.3	odd	2
7200.2.m.b.3599.2		4	120.29	odd	2
7200.2.m.b.3599.3		4	20.19	odd	2
7200.2.m.b.3599.4		4	24.5	odd	2

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:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1800.m (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} +4.24264 q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +2.00000 q^{4} +4.24264 q^{7} -2.82843 q^{8} +1.41421i q^{11} -4.24264 q^{13} -6.00000 q^{14} +4.00000 q^{16} -2.82843 q^{17} +4.00000 q^{19} -2.00000i q^{22} +6.00000i q^{23} +6.00000 q^{26} +8.48528 q^{28} +6.00000 q^{29} +8.48528i q^{31} -5.65685 q^{32} +4.00000 q^{34} -4.24264 q^{37} -5.65685 q^{38} +9.89949i q^{41} +8.00000i q^{43} +2.82843i q^{44} -8.48528i q^{46} +11.0000 q^{49} -8.48528 q^{52} -12.0000i q^{53} -12.0000 q^{56} -8.48528 q^{58} -1.41421i q^{59} -8.48528i q^{61} -12.0000i q^{62} +8.00000 q^{64} -8.00000i q^{67} -5.65685 q^{68} +14.0000i q^{73} +6.00000 q^{74} +8.00000 q^{76} +6.00000i q^{77} -8.48528i q^{79} -14.0000i q^{82} +2.82843 q^{83} -11.3137i q^{86} -4.00000i q^{88} +7.07107i q^{89} -18.0000 q^{91} +12.0000i q^{92} +10.0000i q^{97} -15.5563 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4}+O(q^{10}) \) 4 * q + 8 * q^4	\( 4 q + 8 q^{4} - 24 q^{14} + 16 q^{16} + 16 q^{19} + 24 q^{26} + 24 q^{29} + 16 q^{34} + 44 q^{49} - 48 q^{56} + 32 q^{64} + 24 q^{74} + 32 q^{76} - 72 q^{91}+O(q^{100}) \) 4 * q + 8 * q^4 - 24 * q^14 + 16 * q^16 + 16 * q^19 + 24 * q^26 + 24 * q^29 + 16 * q^34 + 44 * q^49 - 48 * q^56 + 32 * q^64 + 24 * q^74 + 32 * q^76 - 72 * q^91

Introduction

L-functions

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