LMFDB - Embedded newform 1800.2.b.b.251.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,2,Mod(251,1800)]

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, 0]))

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

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

chi := DirichletCharacter("1800.251");

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.b (of order $2$, degree $1$, 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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			251.1
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	1800.251
Dual form			1800.2.b.b.251.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.00000 q^{4} +4.24264i q^{7} +2.82843i q^{8} +O(q^{10})$	$q-1.41421i q^{2} -2.00000 q^{4} +4.24264i q^{7} +2.82843i q^{8} -1.41421i q^{11} +4.24264i q^{13} +6.00000 q^{14} +4.00000 q^{16} -2.82843i q^{17} -4.00000 q^{19} -2.00000 q^{22} -6.00000 q^{23} +6.00000 q^{26} -8.48528i q^{28} -6.00000 q^{29} -8.48528i q^{31} -5.65685i q^{32} -4.00000 q^{34} -4.24264i q^{37} +5.65685i q^{38} -9.89949i q^{41} -8.00000 q^{43} +2.82843i q^{44} +8.48528i q^{46} -11.0000 q^{49} -8.48528i q^{52} +12.0000 q^{53} -12.0000 q^{56} +8.48528i q^{58} -1.41421i q^{59} +8.48528i q^{61} -12.0000 q^{62} -8.00000 q^{64} -8.00000 q^{67} +5.65685i q^{68} -14.0000 q^{73} -6.00000 q^{74} +8.00000 q^{76} +6.00000 q^{77} -8.48528i q^{79} -14.0000 q^{82} -2.82843i q^{83} +11.3137i q^{86} +4.00000 q^{88} +7.07107i q^{89} -18.0000 q^{91} +12.0000 q^{92} +10.0000 q^{97} +15.5563i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4}+O(q^{10}) $ 2 * q - 4 * q^4	$ 2 q - 4 q^{4} + 12 q^{14} + 8 q^{16} - 8 q^{19} - 4 q^{22} - 12 q^{23} + 12 q^{26} - 12 q^{29} - 8 q^{34} - 16 q^{43} - 22 q^{49} + 24 q^{53} - 24 q^{56} - 24 q^{62} - 16 q^{64} - 16 q^{67} - 28 q^{73} - 12 q^{74} + 16 q^{76} + 12 q^{77} - 28 q^{82} + 8 q^{88} - 36 q^{91} + 24 q^{92} + 20 q^{97}+O(q^{100}) $ 2 * q - 4 * q^4 + 12 * q^14 + 8 * q^16 - 8 * q^19 - 4 * q^22 - 12 * q^23 + 12 * q^26 - 12 * q^29 - 8 * q^34 - 16 * q^43 - 22 * q^49 + 24 * q^53 - 24 * q^56 - 24 * q^62 - 16 * q^64 - 16 * q^67 - 28 * q^73 - 12 * q^74 + 16 * q^76 + 12 * q^77 - 28 * q^82 + 8 * q^88 - 36 * q^91 + 24 * q^92 + 20 * q^97

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 $

$2$	− 1.41421i	− 1.00000i
$2$	− 1.41421i	− 1.00000i
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.24264i	1.60357i	0.597614	+	0.801784i	$0.296115\pi$
$7$	4.24264i	1.60357i	−0.597614	+	0.801784i	$0.703885\pi$
$8$	2.82843i	1.00000i
$9$	0	0
$10$	0	0
$11$	− 1.41421i	− 0.426401i	−0.977008	−	0.213201i	$-0.931611\pi$
$11$	− 1.41421i	− 0.426401i	0.977008	−	0.213201i	$-0.0683888\pi$
$12$	0	0
$13$	4.24264i	1.17670i	0.808608	+	0.588348i	$0.200222\pi$
$13$	4.24264i	1.17670i	−0.808608	+	0.588348i	$0.799778\pi$
$14$	6.00000	1.60357
$15$	0	0
$16$	4.00000	1.00000
$17$	− 2.82843i	− 0.685994i	−0.939336	−	0.342997i	$-0.888558\pi$
$17$	− 2.82843i	− 0.685994i	0.939336	−	0.342997i	$-0.111442\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	0	0
$22$	−2.00000	−0.426401
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	0	0
$26$	6.00000	1.17670
$27$	0	0
$28$	− 8.48528i	− 1.60357i
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	− 8.48528i	− 1.52400i	−0.647576	−	0.762001i	$-0.724217\pi$
$31$	− 8.48528i	− 1.52400i	0.647576	−	0.762001i	$-0.275783\pi$
$32$	− 5.65685i	− 1.00000i
$33$	0	0
$34$	−4.00000	−0.685994
$35$	0	0
$36$	0	0
$37$	− 4.24264i	− 0.697486i	−0.937218	−	0.348743i	$-0.886609\pi$
$37$	− 4.24264i	− 0.697486i	0.937218	−	0.348743i	$-0.113391\pi$
$38$	5.65685i	0.917663i
$39$	0	0
$40$	0	0
$41$	− 9.89949i	− 1.54604i	−0.634381	−	0.773021i	$-0.718745\pi$
$41$	− 9.89949i	− 1.54604i	0.634381	−	0.773021i	$-0.281255\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	2.82843i	0.426401i
$45$	0	0
$46$	8.48528i	1.25109i
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−11.0000	−1.57143
$50$	0	0
$51$	0	0
$52$	− 8.48528i	− 1.17670i
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	0	0
$55$	0	0
$56$	−12.0000	−1.60357
$57$	0	0
$58$	8.48528i	1.11417i
$59$	− 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	$0.182793\pi$
$61$	8.48528i	1.08643i	−0.839594	+	0.543214i	$0.817207\pi$
$62$	−12.0000	−1.52400
$63$	0	0
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	5.65685i	0.685994i
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	8.00000	0.917663
$77$	6.00000	0.683763
$78$	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
$82$	−14.0000	−1.54604
$83$	− 2.82843i	− 0.310460i	−0.987878	−	0.155230i	$-0.950388\pi$
$83$	− 2.82843i	− 0.310460i	0.987878	−	0.155230i	$-0.0496119\pi$
$84$	0	0
$85$	0	0
$86$	11.3137i	1.21999i
$87$	0	0
$88$	4.00000	0.426401
$89$	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
$92$	12.0000	1.25109
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	15.5563i	1.57143i
$99$	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
$103$	− 4.24264i	− 0.418040i	−0.977911	−	0.209020i	$-0.932973\pi$
$103$	− 4.24264i	− 0.418040i	0.977911	−	0.209020i	$-0.0670273\pi$
$104$	−12.0000	−1.17670
$105$	0	0
$106$	− 16.9706i	− 1.64833i
$107$	14.1421i	1.36717i	0.729870	+	0.683586i	$0.239581\pi$
$107$	14.1421i	1.36717i	−0.729870	+	0.683586i	$0.760419\pi$
$108$	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
$112$	16.9706i	1.60357i
$113$	− 11.3137i	− 1.06430i	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	− 11.3137i	− 1.06430i	0.846649	−	0.532152i	$-0.178617\pi$
$114$	0	0
$115$	0	0
$116$	12.0000	1.11417
$117$	0	0
$118$	−2.00000	−0.184115
$119$	12.0000	1.10004
$120$	0	0
$121$	9.00000	0.818182
$122$	12.0000	1.08643
$123$	0	0
$124$	16.9706i	1.52400i
$125$	0	0
$126$	0	0
$127$	− 4.24264i	− 0.376473i	−0.982124	−	0.188237i	$-0.939723\pi$
$127$	− 4.24264i	− 0.376473i	0.982124	−	0.188237i	$-0.0602772\pi$
$128$	11.3137i	1.00000i
$129$	0	0
$130$	0	0
$131$	− 1.41421i	− 0.123560i	−0.998090	−	0.0617802i	$-0.980322\pi$
$131$	− 1.41421i	− 0.123560i	0.998090	−	0.0617802i	$-0.0196778\pi$
$132$	0	0
$133$	− 16.9706i	− 1.47153i
$134$	11.3137i	0.977356i
$135$	0	0
$136$	8.00000	0.685994
$137$	5.65685i	0.483298i	0.970364	+	0.241649i	$0.0776882\pi$
$137$	5.65685i	0.483298i	−0.970364	+	0.241649i	$0.922312\pi$
$138$	0	0
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	6.00000	0.501745
$144$	0	0
$145$	0	0
$146$	19.7990i	1.63858i
$147$	0	0
$148$	8.48528i	0.697486i
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	16.9706i	1.38104i	0.723311	+	0.690522i	$0.242619\pi$
$151$	16.9706i	1.38104i	−0.723311	+	0.690522i	$0.757381\pi$
$152$	− 11.3137i	− 0.917663i
$153$	0	0
$154$	− 8.48528i	− 0.683763i
$155$	0	0
$156$	0	0
$157$	4.24264i	0.338600i	0.985565	+	0.169300i	$0.0541506\pi$
$157$	4.24264i	0.338600i	−0.985565	+	0.169300i	$0.945849\pi$
$158$	−12.0000	−0.954669
$159$	0	0
$160$	0	0
$161$	− 25.4558i	− 2.00620i
$162$	0	0
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	19.7990i	1.54604i
$165$	0	0
$166$	−4.00000	−0.310460
$167$	−18.0000	−1.39288	−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000	−1.39288	−0.696441	+	0.717614i	$0.745234\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	0	0
$172$	16.0000	1.21999
$173$	−24.0000	−1.82469	−0.912343	−	0.409426i	$-0.865729\pi$
$173$	−24.0000	−1.82469	−0.912343	+	0.409426i	$0.865729\pi$
$174$	0	0
$175$	0	0
$176$	− 5.65685i	− 0.426401i
$177$	0	0
$178$	10.0000	0.749532
$179$	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	$-0.897877\pi$
$181$	− 8.48528i	− 0.630706i	0.948974	−	0.315353i	$-0.102123\pi$
$182$	25.4558i	1.88691i
$183$	0	0
$184$	− 16.9706i	− 1.25109i
$185$	0	0
$186$	0	0
$187$	−4.00000	−0.292509
$188$	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
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	− 14.1421i	− 1.01535i
$195$	0	0
$196$	22.0000	1.57143
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	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
$202$	8.48528i	0.597022i
$203$	− 25.4558i	− 1.78665i
$204$	0	0
$205$	0	0
$206$	−6.00000	−0.418040
$207$	0	0
$208$	16.9706i	1.17670i
$209$	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	$0.773468\pi$
$212$	−24.0000	−1.64833
$213$	0	0
$214$	20.0000	1.36717
$215$	0	0
$216$	0	0
$217$	36.0000	2.44384
$218$	12.0000	0.812743
$219$	0	0
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	21.2132i	1.42054i	0.703929	+	0.710271i	$0.251427\pi$
$223$	21.2132i	1.42054i	−0.703929	+	0.710271i	$0.748573\pi$
$224$	24.0000	1.60357
$225$	0	0
$226$	−16.0000	−1.06430
$227$	22.6274i	1.50183i	0.660396	+	0.750917i	$0.270388\pi$
$227$	22.6274i	1.50183i	−0.660396	+	0.750917i	$0.729612\pi$
$228$	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
$232$	− 16.9706i	− 1.11417i
$233$	− 19.7990i	− 1.29707i	−0.761183	−	0.648537i	$-0.775381\pi$
$233$	− 19.7990i	− 1.29707i	0.761183	−	0.648537i	$-0.224619\pi$
$234$	0	0
$235$	0	0
$236$	2.82843i	0.184115i
$237$	0	0
$238$	− 16.9706i	− 1.10004i
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	20.0000	1.28831	0.644157	−	0.764894i	$-0.277208\pi$
$241$	20.0000	1.28831	0.644157	+	0.764894i	$0.277208\pi$
$242$	− 12.7279i	− 0.818182i
$243$	0	0
$244$	− 16.9706i	− 1.08643i
$245$	0	0
$246$	0	0
$247$	− 16.9706i	− 1.07981i
$248$	24.0000	1.52400
$249$	0	0
$250$	0	0
$251$	− 1.41421i	− 0.0892644i	−0.999003	−	0.0446322i	$-0.985788\pi$
$251$	− 1.41421i	− 0.0892644i	0.999003	−	0.0446322i	$-0.0142116\pi$
$252$	0	0
$253$	8.48528i	0.533465i
$254$	−6.00000	−0.376473
$255$	0	0
$256$	16.0000	1.00000
$257$	− 11.3137i	− 0.705730i	−0.935674	−	0.352865i	$-0.885208\pi$
$257$	− 11.3137i	− 0.705730i	0.935674	−	0.352865i	$-0.114792\pi$
$258$	0	0
$259$	18.0000	1.11847
$260$	0	0
$261$	0	0
$262$	−2.00000	−0.123560
$263$	6.00000	0.369976	0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000	0.369976	0.184988	+	0.982741i	$0.440775\pi$
$264$	0	0
$265$	0	0
$266$	−24.0000	−1.47153
$267$	0	0
$268$	16.0000	0.977356
$269$	−30.0000	−1.82913	−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000	−1.82913	−0.914566	+	0.404436i	$0.867468\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	− 11.3137i	− 0.685994i
$273$	0	0
$274$	8.00000	0.483298
$275$	0	0
$276$	0	0
$277$	4.24264i	0.254916i	0.991844	+	0.127458i	$0.0406817\pi$
$277$	4.24264i	0.254916i	−0.991844	+	0.127458i	$0.959318\pi$
$278$	− 2.82843i	− 0.169638i
$279$	0	0
$280$	0	0
$281$	− 9.89949i	− 0.590554i	−0.955412	−	0.295277i	$-0.904588\pi$
$281$	− 9.89949i	− 0.590554i	0.955412	−	0.295277i	$-0.0954120\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	0	0
$286$	− 8.48528i	− 0.501745i
$287$	42.0000	2.47918
$288$	0	0
$289$	9.00000	0.529412
$290$	0	0
$291$	0	0
$292$	28.0000	1.63858
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	0	0
$296$	12.0000	0.697486
$297$	0	0
$298$	− 25.4558i	− 1.47462i
$299$	− 25.4558i	− 1.47215i
$300$	0	0
$301$	− 33.9411i	− 1.95633i
$302$	24.0000	1.38104
$303$	0	0
$304$	−16.0000	−0.917663
$305$	0	0
$306$	0	0
$307$	−8.00000	−0.456584	−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000	−0.456584	−0.228292	+	0.973593i	$0.573314\pi$
$308$	−12.0000	−0.683763
$309$	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
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	6.00000	0.338600
$315$	0	0
$316$	16.9706i	0.954669i
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	8.48528i	0.475085i
$320$	0	0
$321$	0	0
$322$	−36.0000	−2.00620
$323$	11.3137i	0.629512i
$324$	0	0
$325$	0	0
$326$	28.2843i	1.56652i
$327$	0	0
$328$	28.0000	1.54604
$329$	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	$0.779499\pi$
$332$	5.65685i	0.310460i
$333$	0	0
$334$	25.4558i	1.39288i
$335$	0	0
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	7.07107i	0.384615i
$339$	0	0
$340$	0	0
$341$	−12.0000	−0.649836
$342$	0	0
$343$	− 16.9706i	− 0.916324i
$344$	− 22.6274i	− 1.21999i
$345$	0	0
$346$	33.9411i	1.82469i
$347$	5.65685i	0.303676i	0.988405	+	0.151838i	$0.0485192\pi$
$347$	5.65685i	0.303676i	−0.988405	+	0.151838i	$0.951481\pi$
$348$	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
$352$	−8.00000	−0.426401
$353$	− 2.82843i	− 0.150542i	−0.997163	−	0.0752710i	$-0.976018\pi$
$353$	− 2.82843i	− 0.150542i	0.997163	−	0.0752710i	$-0.0239822\pi$
$354$	0	0
$355$	0	0
$356$	− 14.1421i	− 0.749532i
$357$	0	0
$358$	10.0000	0.528516
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	−12.0000	−0.630706
$363$	0	0
$364$	36.0000	1.88691
$365$	0	0
$366$	0	0
$367$	29.6985i	1.55025i	0.631809	+	0.775124i	$0.282313\pi$
$367$	29.6985i	1.55025i	−0.631809	+	0.775124i	$0.717687\pi$
$368$	−24.0000	−1.25109
$369$	0	0
$370$	0	0
$371$	50.9117i	2.64320i
$372$	0	0
$373$	12.7279i	0.659027i	0.944151	+	0.329513i	$0.106885\pi$
$373$	12.7279i	0.659027i	−0.944151	+	0.329513i	$0.893115\pi$
$374$	5.65685i	0.292509i
$375$	0	0
$376$	0	0
$377$	− 25.4558i	− 1.31104i
$378$	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
$381$	0	0
$382$	− 16.9706i	− 0.868290i
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	0	0
$385$	0	0
$386$	2.82843i	0.143963i
$387$	0	0
$388$	−20.0000	−1.01535
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	16.9706i	0.858238i
$392$	− 31.1127i	− 1.57143i
$393$	0	0
$394$	8.48528i	0.427482i
$395$	0	0
$396$	0	0
$397$	12.7279i	0.638796i	0.947621	+	0.319398i	$0.103481\pi$
$397$	12.7279i	0.638796i	−0.947621	+	0.319398i	$0.896519\pi$
$398$	24.0000	1.20301
$399$	0	0
$400$	0	0
$401$	− 18.3848i	− 0.918092i	−0.888413	−	0.459046i	$-0.848191\pi$
$401$	− 18.3848i	− 0.918092i	0.888413	−	0.459046i	$-0.151809\pi$
$402$	0	0
$403$	36.0000	1.79329
$404$	12.0000	0.597022
$405$	0	0
$406$	−36.0000	−1.78665
$407$	−6.00000	−0.297409
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	0	0
$412$	8.48528i	0.418040i
$413$	6.00000	0.295241
$414$	0	0
$415$	0	0
$416$	24.0000	1.17670
$417$	0	0
$418$	8.00000	0.391293
$419$	− 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	$0.310008\pi$
$421$	33.9411i	1.65419i	−0.562063	+	0.827095i	$0.689992\pi$
$422$	31.1127i	1.51454i
$423$	0	0
$424$	33.9411i	1.64833i
$425$	0	0
$426$	0	0
$427$	−36.0000	−1.74216
$428$	− 28.2843i	− 1.36717i
$429$	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
$433$	10.0000	0.480569	0.240285	−	0.970702i	$-0.422759\pi$
$433$	10.0000	0.480569	0.240285	+	0.970702i	$0.422759\pi$
$434$	− 50.9117i	− 2.44384i
$435$	0	0
$436$	− 16.9706i	− 0.812743i
$437$	24.0000	1.14808
$438$	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
$442$	− 16.9706i	− 0.807207i
$443$	− 11.3137i	− 0.537531i	−0.963206	−	0.268765i	$-0.913384\pi$
$443$	− 11.3137i	− 0.537531i	0.963206	−	0.268765i	$-0.0866156\pi$
$444$	0	0
$445$	0	0
$446$	30.0000	1.42054
$447$	0	0
$448$	− 33.9411i	− 1.60357i
$449$	− 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
$452$	22.6274i	1.06430i
$453$	0	0
$454$	32.0000	1.50183
$455$	0	0
$456$	0	0
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	−24.0000	−1.12145
$459$	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
$463$	− 12.7279i	− 0.591517i	−0.955263	−	0.295758i	$-0.904428\pi$
$463$	− 12.7279i	− 0.591517i	0.955263	−	0.295758i	$-0.0955723\pi$
$464$	−24.0000	−1.11417
$465$	0	0
$466$	−28.0000	−1.29707
$467$	− 19.7990i	− 0.916188i	−0.888904	−	0.458094i	$-0.848532\pi$
$467$	− 19.7990i	− 0.916188i	0.888904	−	0.458094i	$-0.151468\pi$
$468$	0	0
$469$	− 33.9411i	− 1.56726i
$470$	0	0
$471$	0	0
$472$	4.00000	0.184115
$473$	11.3137i	0.520205i
$474$	0	0
$475$	0	0
$476$	−24.0000	−1.10004
$477$	0	0
$478$	33.9411i	1.55243i
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	18.0000	0.820729
$482$	− 28.2843i	− 1.28831i
$483$	0	0
$484$	−18.0000	−0.818182
$485$	0	0
$486$	0	0
$487$	− 21.2132i	− 0.961262i	−0.876923	−	0.480631i	$-0.840408\pi$
$487$	− 21.2132i	− 0.961262i	0.876923	−	0.480631i	$-0.159592\pi$
$488$	−24.0000	−1.08643
$489$	0	0
$490$	0	0
$491$	− 1.41421i	− 0.0638226i	−0.999491	−	0.0319113i	$-0.989841\pi$
$491$	− 1.41421i	− 0.0638226i	0.999491	−	0.0319113i	$-0.0101594\pi$
$492$	0	0
$493$	16.9706i	0.764316i
$494$	−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	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	0	0
$502$	−2.00000	−0.0892644
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	0	0
$506$	12.0000	0.533465
$507$	0	0
$508$	8.48528i	0.376473i
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	− 59.3970i	− 2.62757i
$512$	− 22.6274i	− 1.00000i
$513$	0	0
$514$	−16.0000	−0.705730
$515$	0	0
$516$	0	0
$517$	0	0
$518$	− 25.4558i	− 1.11847i
$519$	0	0
$520$	0	0
$521$	15.5563i	0.681536i	0.940147	+	0.340768i	$0.110687\pi$
$521$	15.5563i	0.681536i	−0.940147	+	0.340768i	$0.889313\pi$
$522$	0	0
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	2.82843i	0.123560i
$525$	0	0
$526$	− 8.48528i	− 0.369976i
$527$	−24.0000	−1.04546
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	0	0
$532$	33.9411i	1.47153i
$533$	42.0000	1.81922
$534$	0	0
$535$	0	0
$536$	− 22.6274i	− 0.977356i
$537$	0	0
$538$	42.4264i	1.82913i
$539$	15.5563i	0.670059i
$540$	0	0
$541$	8.48528i	0.364811i	0.983223	+	0.182405i	$0.0583883\pi$
$541$	8.48528i	0.364811i	−0.983223	+	0.182405i	$0.941612\pi$
$542$	0	0
$543$	0	0
$544$	−16.0000	−0.685994
$545$	0	0
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	− 11.3137i	− 0.483298i
$549$	0	0
$550$	0	0
$551$	24.0000	1.02243
$552$	0	0
$553$	36.0000	1.53088
$554$	6.00000	0.254916
$555$	0	0
$556$	−4.00000	−0.169638
$557$	−12.0000	−0.508456	−0.254228	−	0.967144i	$-0.581821\pi$
$557$	−12.0000	−0.508456	−0.254228	+	0.967144i	$0.581821\pi$
$558$	0	0
$559$	− 33.9411i	− 1.43556i
$560$	0	0
$561$	0	0
$562$	−14.0000	−0.590554
$563$	39.5980i	1.66886i	0.551117	+	0.834428i	$0.314202\pi$
$563$	39.5980i	1.66886i	−0.551117	+	0.834428i	$0.685798\pi$
$564$	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	$0.526673\pi$
$572$	−12.0000	−0.501745
$573$	0	0
$574$	− 59.3970i	− 2.47918i
$575$	0	0
$576$	0	0
$577$	22.0000	0.915872	0.457936	−	0.888985i	$-0.348589\pi$
$577$	22.0000	0.915872	0.457936	+	0.888985i	$0.348589\pi$
$578$	− 12.7279i	− 0.529412i
$579$	0	0
$580$	0	0
$581$	12.0000	0.497844
$582$	0	0
$583$	− 16.9706i	− 0.702849i
$584$	− 39.5980i	− 1.63858i
$585$	0	0
$586$	8.48528i	0.350524i
$587$	14.1421i	0.583708i	0.956463	+	0.291854i	$0.0942722\pi$
$587$	14.1421i	0.583708i	−0.956463	+	0.291854i	$0.905728\pi$
$588$	0	0
$589$	33.9411i	1.39852i
$590$	0	0
$591$	0	0
$592$	− 16.9706i	− 0.697486i
$593$	5.65685i	0.232299i	0.993232	+	0.116150i	$0.0370552\pi$
$593$	5.65685i	0.232299i	−0.993232	+	0.116150i	$0.962945\pi$
$594$	0	0
$595$	0	0
$596$	−36.0000	−1.47462
$597$	0	0
$598$	−36.0000	−1.47215
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	32.0000	1.30531	0.652654	−	0.757656i	$-0.273656\pi$
$601$	32.0000	1.30531	0.652654	+	0.757656i	$0.273656\pi$
$602$	−48.0000	−1.95633
$603$	0	0
$604$	− 33.9411i	− 1.38104i
$605$	0	0
$606$	0	0
$607$	4.24264i	0.172203i	0.996286	+	0.0861017i	$0.0274410\pi$
$607$	4.24264i	0.172203i	−0.996286	+	0.0861017i	$0.972559\pi$
$608$	22.6274i	0.917663i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 12.7279i	− 0.514076i	−0.966401	−	0.257038i	$-0.917253\pi$
$613$	− 12.7279i	− 0.514076i	0.966401	−	0.257038i	$-0.0827465\pi$
$614$	11.3137i	0.456584i
$615$	0	0
$616$	16.9706i	0.683763i
$617$	− 36.7696i	− 1.48029i	−0.672449	−	0.740143i	$-0.734758\pi$
$617$	− 36.7696i	− 1.48029i	0.672449	−	0.740143i	$-0.265242\pi$
$618$	0	0
$619$	−10.0000	−0.401934	−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000	−0.401934	−0.200967	+	0.979598i	$0.564408\pi$
$620$	0	0
$621$	0	0
$622$	16.9706i	0.680458i
$623$	−30.0000	−1.20192
$624$	0	0
$625$	0	0
$626$	− 14.1421i	− 0.565233i
$627$	0	0
$628$	− 8.48528i	− 0.338600i
$629$	−12.0000	−0.478471
$630$	0	0
$631$	8.48528i	0.337794i	0.985634	+	0.168897i	$0.0540205\pi$
$631$	8.48528i	0.337794i	−0.985634	+	0.168897i	$0.945980\pi$
$632$	24.0000	0.954669
$633$	0	0
$634$	25.4558i	1.01098i
$635$	0	0
$636$	0	0
$637$	− 46.6690i	− 1.84909i
$638$	12.0000	0.475085
$639$	0	0
$640$	0	0
$641$	41.0122i	1.61988i	0.586510	+	0.809942i	$0.300502\pi$
$641$	41.0122i	1.61988i	−0.586510	+	0.809942i	$0.699498\pi$
$642$	0	0
$643$	40.0000	1.57745	0.788723	−	0.614749i	$-0.210743\pi$
$643$	40.0000	1.57745	0.788723	+	0.614749i	$0.210743\pi$
$644$	50.9117i	2.00620i
$645$	0	0
$646$	16.0000	0.629512
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	−2.00000	−0.0785069
$650$	0	0
$651$	0	0
$652$	40.0000	1.56652
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	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	$0.164853\pi$
$661$	25.4558i	0.990118i	−0.868859	+	0.495059i	$0.835147\pi$
$662$	39.5980i	1.53902i
$663$	0	0
$664$	8.00000	0.310460
$665$	0	0
$666$	0	0
$667$	36.0000	1.39393
$668$	36.0000	1.39288
$669$	0	0
$670$	0	0
$671$	12.0000	0.463255
$672$	0	0
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	2.82843i	0.108947i
$675$	0	0
$676$	10.0000	0.384615
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	0	0
$679$	42.4264i	1.62818i
$680$	0	0
$681$	0	0
$682$	16.9706i	0.649836i
$683$	14.1421i	0.541134i	0.962701	+	0.270567i	$0.0872111\pi$
$683$	14.1421i	0.541134i	−0.962701	+	0.270567i	$0.912789\pi$
$684$	0	0
$685$	0	0
$686$	−24.0000	−0.916324
$687$	0	0
$688$	−32.0000	−1.21999
$689$	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	$0.184354\pi$
$692$	48.0000	1.82469
$693$	0	0
$694$	8.00000	0.303676
$695$	0	0
$696$	0	0
$697$	−28.0000	−1.06058
$698$	12.0000	0.454207
$699$	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
$703$	16.9706i	0.640057i
$704$	11.3137i	0.426401i
$705$	0	0
$706$	−4.00000	−0.150542
$707$	− 25.4558i	− 0.957366i
$708$	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
$712$	−20.0000	−0.749532
$713$	50.9117i	1.90666i
$714$	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	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	18.0000	0.670355
$722$	4.24264i	0.157895i
$723$	0	0
$724$	16.9706i	0.630706i
$725$	0	0
$726$	0	0
$727$	4.24264i	0.157351i	0.996900	+	0.0786754i	$0.0250691\pi$
$727$	4.24264i	0.157351i	−0.996900	+	0.0786754i	$0.974931\pi$
$728$	− 50.9117i	− 1.88691i
$729$	0	0
$730$	0	0
$731$	22.6274i	0.836905i
$732$	0	0
$733$	− 12.7279i	− 0.470117i	−0.971981	−	0.235058i	$-0.924472\pi$
$733$	− 12.7279i	− 0.470117i	0.971981	−	0.235058i	$-0.0755281\pi$
$734$	42.0000	1.55025
$735$	0	0
$736$	33.9411i	1.25109i
$737$	11.3137i	0.416746i
$738$	0	0
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	0	0
$742$	72.0000	2.64320
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	0	0
$746$	18.0000	0.659027
$747$	0	0
$748$	8.00000	0.292509
$749$	−60.0000	−2.19235
$750$	0	0
$751$	42.4264i	1.54816i	0.633087	+	0.774081i	$0.281788\pi$
$751$	42.4264i	1.54816i	−0.633087	+	0.774081i	$0.718212\pi$
$752$	0	0
$753$	0	0
$754$	−36.0000	−1.31104
$755$	0	0
$756$	0	0
$757$	4.24264i	0.154201i	0.997023	+	0.0771007i	$0.0245663\pi$
$757$	4.24264i	0.154201i	−0.997023	+	0.0771007i	$0.975434\pi$
$758$	− 36.7696i	− 1.33553i
$759$	0	0
$760$	0	0
$761$	− 26.8701i	− 0.974039i	−0.873391	−	0.487019i	$-0.838084\pi$
$761$	− 26.8701i	− 0.974039i	0.873391	−	0.487019i	$-0.161916\pi$
$762$	0	0
$763$	−36.0000	−1.30329
$764$	−24.0000	−0.868290
$765$	0	0
$766$	− 33.9411i	− 1.22634i
$767$	6.00000	0.216647
$768$	0	0
$769$	−40.0000	−1.44244	−0.721218	−	0.692708i	$-0.756418\pi$
$769$	−40.0000	−1.44244	−0.721218	+	0.692708i	$0.756418\pi$
$770$	0	0
$771$	0	0
$772$	4.00000	0.143963
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	0	0
$775$	0	0
$776$	28.2843i	1.01535i
$777$	0	0
$778$	− 42.4264i	− 1.52106i
$779$	39.5980i	1.41874i
$780$	0	0
$781$	0	0
$782$	24.0000	0.858238
$783$	0	0
$784$	−44.0000	−1.57143
$785$	0	0
$786$	0	0
$787$	−32.0000	−1.14068	−0.570338	−	0.821410i	$-0.693188\pi$
$787$	−32.0000	−1.14068	−0.570338	+	0.821410i	$0.693188\pi$
$788$	12.0000	0.427482
$789$	0	0
$790$	0	0
$791$	48.0000	1.70668
$792$	0	0
$793$	−36.0000	−1.27840
$794$	18.0000	0.638796
$795$	0	0
$796$	− 33.9411i	− 1.20301i
$797$	24.0000	0.850124	0.425062	−	0.905164i	$-0.360252\pi$
$797$	24.0000	0.850124	0.425062	+	0.905164i	$0.360252\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	−26.0000	−0.918092
$803$	19.7990i	0.698691i
$804$	0	0
$805$	0	0
$806$	− 50.9117i	− 1.79329i
$807$	0	0
$808$	− 16.9706i	− 0.597022i
$809$	− 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	$0.488820\pi$
$812$	50.9117i	1.78665i
$813$	0	0
$814$	8.48528i	0.297409i
$815$	0	0
$816$	0	0
$817$	32.0000	1.11954
$818$	14.1421i	0.494468i
$819$	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
$823$	4.24264i	0.147889i	0.997262	+	0.0739446i	$0.0235588\pi$
$823$	4.24264i	0.147889i	−0.997262	+	0.0739446i	$0.976441\pi$
$824$	12.0000	0.418040
$825$	0	0
$826$	− 8.48528i	− 0.295241i
$827$	5.65685i	0.196708i	0.995151	+	0.0983540i	$0.0313578\pi$
$827$	5.65685i	0.196708i	−0.995151	+	0.0983540i	$0.968642\pi$
$828$	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
$832$	− 33.9411i	− 1.17670i
$833$	31.1127i	1.07799i
$834$	0	0
$835$	0	0
$836$	− 11.3137i	− 0.391293i
$837$	0	0
$838$	−38.0000	−1.31269
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	48.0000	1.65419
$843$	0	0
$844$	44.0000	1.51454
$845$	0	0
$846$	0	0
$847$	38.1838i	1.31201i
$848$	48.0000	1.64833
$849$	0	0
$850$	0	0
$851$	25.4558i	0.872615i
$852$	0	0
$853$	− 29.6985i	− 1.01686i	−0.861104	−	0.508428i	$-0.830227\pi$
$853$	− 29.6985i	− 1.01686i	0.861104	−	0.508428i	$-0.169773\pi$
$854$	50.9117i	1.74216i
$855$	0	0
$856$	−40.0000	−1.36717
$857$	48.0833i	1.64249i	0.570574	+	0.821246i	$0.306721\pi$
$857$	48.0833i	1.64249i	−0.570574	+	0.821246i	$0.693279\pi$
$858$	0	0
$859$	−46.0000	−1.56950	−0.784750	−	0.619813i	$-0.787209\pi$
$859$	−46.0000	−1.56950	−0.784750	+	0.619813i	$0.787209\pi$
$860$	0	0
$861$	0	0
$862$	50.9117i	1.73406i
$863$	18.0000	0.612727	0.306364	−	0.951915i	$-0.400888\pi$
$863$	18.0000	0.612727	0.306364	+	0.951915i	$0.400888\pi$
$864$	0	0
$865$	0	0
$866$	− 14.1421i	− 0.480569i
$867$	0	0
$868$	−72.0000	−2.44384
$869$	−12.0000	−0.407072
$870$	0	0
$871$	− 33.9411i	− 1.15005i
$872$	−24.0000	−0.812743
$873$	0	0
$874$	− 33.9411i	− 1.14808i
$875$	0	0
$876$	0	0
$877$	12.7279i	0.429791i	0.976637	+	0.214896i	$0.0689412\pi$
$877$	12.7279i	0.429791i	−0.976637	+	0.214896i	$0.931059\pi$
$878$	−24.0000	−0.809961
$879$	0	0
$880$	0	0
$881$	32.5269i	1.09586i	0.836524	+	0.547930i	$0.184584\pi$
$881$	32.5269i	1.09586i	−0.836524	+	0.547930i	$0.815416\pi$
$882$	0	0
$883$	28.0000	0.942275	0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000	0.942275	0.471138	+	0.882060i	$0.343844\pi$
$884$	−24.0000	−0.807207
$885$	0	0
$886$	−16.0000	−0.537531
$887$	−18.0000	−0.604381	−0.302190	−	0.953248i	$-0.597718\pi$
$887$	−18.0000	−0.604381	−0.302190	+	0.953248i	$0.597718\pi$
$888$	0	0
$889$	18.0000	0.603701
$890$	0	0
$891$	0	0
$892$	− 42.4264i	− 1.42054i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	−48.0000	−1.60357
$897$	0	0
$898$	−38.0000	−1.26808
$899$	50.9117i	1.69800i
$900$	0	0
$901$	− 33.9411i	− 1.13074i
$902$	19.7990i	0.659234i
$903$	0	0
$904$	32.0000	1.06430
$905$	0	0
$906$	0	0
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	− 45.2548i	− 1.50183i
$909$	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
$913$	−4.00000	−0.132381
$914$	2.82843i	0.0935561i
$915$	0	0
$916$	33.9411i	1.12145i
$917$	6.00000	0.198137
$918$	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
$922$	− 25.4558i	− 0.838344i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−18.0000	−0.591517
$927$	0	0
$928$	33.9411i	1.11417i
$929$	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
$932$	39.5980i	1.29707i
$933$	0	0
$934$	−28.0000	−0.916188
$935$	0	0
$936$	0	0
$937$	−38.0000	−1.24141	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000	−1.24141	−0.620703	+	0.784046i	$0.713153\pi$
$938$	−48.0000	−1.56726
$939$	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
$943$	59.3970i	1.93423i
$944$	− 5.65685i	− 0.184115i
$945$	0	0
$946$	16.0000	0.520205
$947$	22.6274i	0.735292i	0.929966	+	0.367646i	$0.119836\pi$
$947$	22.6274i	0.735292i	−0.929966	+	0.367646i	$0.880164\pi$
$948$	0	0
$949$	− 59.3970i	− 1.92811i
$950$	0	0
$951$	0	0
$952$	33.9411i	1.10004i
$953$	− 28.2843i	− 0.916217i	−0.888896	−	0.458109i	$-0.848527\pi$
$953$	− 28.2843i	− 0.916217i	0.888896	−	0.458109i	$-0.151473\pi$
$954$	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
$962$	− 25.4558i	− 0.820729i
$963$	0	0
$964$	−40.0000	−1.28831
$965$	0	0
$966$	0	0
$967$	− 21.2132i	− 0.682171i	−0.940032	−	0.341085i	$-0.889205\pi$
$967$	− 21.2132i	− 0.682171i	0.940032	−	0.341085i	$-0.110795\pi$
$968$	25.4558i	0.818182i
$969$	0	0
$970$	0	0
$971$	− 43.8406i	− 1.40691i	−0.710739	−	0.703456i	$-0.751639\pi$
$971$	− 43.8406i	− 1.40691i	0.710739	−	0.703456i	$-0.248361\pi$
$972$	0	0
$973$	8.48528i	0.272026i
$974$	−30.0000	−0.961262
$975$	0	0
$976$	33.9411i	1.08643i
$977$	− 2.82843i	− 0.0904894i	−0.998976	−	0.0452447i	$-0.985593\pi$
$977$	− 2.82843i	− 0.0904894i	0.998976	−	0.0452447i	$-0.0144068\pi$
$978$	0	0
$979$	10.0000	0.319601
$980$	0	0
$981$	0	0
$982$	−2.00000	−0.0638226
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	0	0
$985$	0	0
$986$	24.0000	0.764316
$987$	0	0
$988$	33.9411i	1.07981i
$989$	48.0000	1.52631
$990$	0	0
$991$	16.9706i	0.539088i	0.962988	+	0.269544i	$0.0868729\pi$
$991$	16.9706i	0.539088i	−0.962988	+	0.269544i	$0.913127\pi$
$992$	−48.0000	−1.52400
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 21.2132i	− 0.671829i	−0.941893	−	0.335914i	$-0.890955\pi$
$997$	− 21.2132i	− 0.671829i	0.941893	−	0.335914i	$-0.109045\pi$
$998$	5.65685i	0.179065i
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.b.a.251.1	✓	2	120.59	even	2
360.2.b.a.251.2	yes	2	5.4	even	2
360.2.b.b.251.1	yes	2	15.14	odd	2
360.2.b.b.251.2	yes	2	40.19	odd	2
1440.2.b.a.431.1		2	120.29	odd	2
1440.2.b.a.431.2		2	20.19	odd	2
1440.2.b.b.431.1		2	40.29	even	2
1440.2.b.b.431.2		2	60.59	even	2
1800.2.b.a.251.1		2	8.3	odd	2
1800.2.b.a.251.2		2	3.2	odd	2
1800.2.b.b.251.1		2	1.1	even	1	trivial
1800.2.b.b.251.2		2	24.11	even	2	inner
1800.2.m.a.899.1		4	5.3	odd	4
1800.2.m.a.899.2		4	120.107	odd	4
1800.2.m.a.899.3		4	5.2	odd	4
1800.2.m.a.899.4		4	120.83	odd	4
1800.2.m.b.899.1		4	40.3	even	4
1800.2.m.b.899.2		4	15.2	even	4
1800.2.m.b.899.3		4	40.27	even	4
1800.2.m.b.899.4		4	15.8	even	4
7200.2.b.a.4751.1		2	12.11	even	2
7200.2.b.a.4751.2		2	8.5	even	2
7200.2.b.b.4751.1		2	4.3	odd	2
7200.2.b.b.4751.2		2	24.5	odd	2
7200.2.m.a.3599.1		4	60.23	odd	4
7200.2.m.a.3599.2		4	40.37	odd	4
7200.2.m.a.3599.3		4	60.47	odd	4
7200.2.m.a.3599.4		4	40.13	odd	4
7200.2.m.b.3599.1		4	120.77	even	4
7200.2.m.b.3599.2		4	20.3	even	4
7200.2.m.b.3599.3		4	120.53	even	4
7200.2.m.b.3599.4		4	20.7	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:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1800.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} +4.24264i q^{7} +2.82843i q^{8} +O(q^{10})\)	\(q-1.41421i q^{2} -2.00000 q^{4} +4.24264i q^{7} +2.82843i q^{8} -1.41421i q^{11} +4.24264i q^{13} +6.00000 q^{14} +4.00000 q^{16} -2.82843i q^{17} -4.00000 q^{19} -2.00000 q^{22} -6.00000 q^{23} +6.00000 q^{26} -8.48528i q^{28} -6.00000 q^{29} -8.48528i q^{31} -5.65685i q^{32} -4.00000 q^{34} -4.24264i q^{37} +5.65685i q^{38} -9.89949i q^{41} -8.00000 q^{43} +2.82843i q^{44} +8.48528i q^{46} -11.0000 q^{49} -8.48528i q^{52} +12.0000 q^{53} -12.0000 q^{56} +8.48528i q^{58} -1.41421i q^{59} +8.48528i q^{61} -12.0000 q^{62} -8.00000 q^{64} -8.00000 q^{67} +5.65685i q^{68} -14.0000 q^{73} -6.00000 q^{74} +8.00000 q^{76} +6.00000 q^{77} -8.48528i q^{79} -14.0000 q^{82} -2.82843i q^{83} +11.3137i q^{86} +4.00000 q^{88} +7.07107i q^{89} -18.0000 q^{91} +12.0000 q^{92} +10.0000 q^{97} +15.5563i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4}+O(q^{10}) \) 2 * q - 4 * q^4	\( 2 q - 4 q^{4} + 12 q^{14} + 8 q^{16} - 8 q^{19} - 4 q^{22} - 12 q^{23} + 12 q^{26} - 12 q^{29} - 8 q^{34} - 16 q^{43} - 22 q^{49} + 24 q^{53} - 24 q^{56} - 24 q^{62} - 16 q^{64} - 16 q^{67} - 28 q^{73} - 12 q^{74} + 16 q^{76} + 12 q^{77} - 28 q^{82} + 8 q^{88} - 36 q^{91} + 24 q^{92} + 20 q^{97}+O(q^{100}) \) 2 * q - 4 * q^4 + 12 * q^14 + 8 * q^16 - 8 * q^19 - 4 * q^22 - 12 * q^23 + 12 * q^26 - 12 * q^29 - 8 * q^34 - 16 * q^43 - 22 * q^49 + 24 * q^53 - 24 * q^56 - 24 * q^62 - 16 * q^64 - 16 * q^67 - 28 * q^73 - 12 * q^74 + 16 * q^76 + 12 * q^77 - 28 * q^82 + 8 * q^88 - 36 * q^91 + 24 * q^92 + 20 * q^97

Introduction

L-functions

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Database

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