LMFDB - Embedded newform 180.10.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [180,10,Mod(1,180)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(180, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("180.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 180 = 2^{2} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	180.a (trivial)

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:	yes
Analytic conductor:	$92.7064505095$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{79}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 79 $ x^2 - 79
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	no (minimal twist has level 20)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-8.88819$ of defining polynomial
Character	$\chi$	$=$	180.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+625.000 q^{5} -10002.6 q^{7} +O(q^{10})$	$q+625.000 q^{5} -10002.6 q^{7} -47093.7 q^{11} +9362.93 q^{13} -108521. q^{17} -665173. q^{19} -576426. q^{23} +390625. q^{25} +2.61067e6 q^{29} +3.87896e6 q^{31} -6.25160e6 q^{35} +1.41599e7 q^{37} -4.62193e6 q^{41} +8.31227e6 q^{43} -2.51923e7 q^{47} +5.96977e7 q^{49} -3.49333e7 q^{53} -2.94335e7 q^{55} -6.71868e6 q^{59} -4.75982e6 q^{61} +5.85183e6 q^{65} -1.38772e8 q^{67} -3.54359e8 q^{71} +2.41596e8 q^{73} +4.71058e8 q^{77} +2.61173e8 q^{79} +6.55898e8 q^{83} -6.78253e7 q^{85} +1.00221e9 q^{89} -9.36534e7 q^{91} -4.15733e8 q^{95} +1.24869e9 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 1250 q^{5} - 380 q^{7}+O(q^{10}) $ 2 * q + 1250 * q^5 - 380 * q^7	$ 2 q + 1250 q^{5} - 380 q^{7} - 102720 q^{11} + 179140 q^{13} - 316020 q^{17} + 137272 q^{19} + 665460 q^{23} + 781250 q^{25} + 6893748 q^{29} + 291832 q^{31} - 237500 q^{35} + 11261380 q^{37} - 29773452 q^{41} - 11708180 q^{43} - 62493300 q^{47} + 111937914 q^{49} - 9417780 q^{53} - 64200000 q^{55} + 92930856 q^{59} + 195673924 q^{61} + 111962500 q^{65} - 219767420 q^{67} - 311207016 q^{71} - 99224060 q^{73} - 64210560 q^{77} + 542261776 q^{79} + 1256915700 q^{83} - 197512500 q^{85} + 462291852 q^{89} + 1540037768 q^{91} + 85795000 q^{95} + 1671716740 q^{97}+O(q^{100}) $ 2 * q + 1250 * q^5 - 380 * q^7 - 102720 * q^11 + 179140 * q^13 - 316020 * q^17 + 137272 * q^19 + 665460 * q^23 + 781250 * q^25 + 6893748 * q^29 + 291832 * q^31 - 237500 * q^35 + 11261380 * q^37 - 29773452 * q^41 - 11708180 * q^43 - 62493300 * q^47 + 111937914 * q^49 - 9417780 * q^53 - 64200000 * q^55 + 92930856 * q^59 + 195673924 * q^61 + 111962500 * q^65 - 219767420 * q^67 - 311207016 * q^71 - 99224060 * q^73 - 64210560 * q^77 + 542261776 * q^79 + 1256915700 * q^83 - 197512500 * q^85 + 462291852 * q^89 + 1540037768 * q^91 + 85795000 * q^95 + 1671716740 * q^97

Display 100 coefficients 10 coefficients

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	625.000	0.447214
$5$	625.000	0.447214
$6$	0	0
$7$	−10002.6	−1.57460	−0.787300	−	0.616570i	$-0.788522\pi$
$7$	−10002.6	−1.57460	−0.787300	+	0.616570i	$0.788522\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−47093.7	−0.969830	−0.484915	−	0.874561i	$-0.661149\pi$
$11$	−47093.7	−0.969830	−0.484915	+	0.874561i	$0.661149\pi$
$12$	0	0
$13$	9362.93	0.0909216	0.0454608	−	0.998966i	$-0.485524\pi$
$13$	9362.93	0.0909216	0.0454608	+	0.998966i	$0.485524\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−108521.	−0.315131	−0.157566	−	0.987509i	$-0.550365\pi$
$17$	−108521.	−0.315131	−0.157566	+	0.987509i	$0.550365\pi$
$18$	0	0
$19$	−665173.	−1.17096	−0.585482	−	0.810685i	$-0.699095\pi$
$19$	−665173.	−1.17096	−0.585482	+	0.810685i	$0.699095\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−576426.	−0.429505	−0.214752	−	0.976669i	$-0.568894\pi$
$23$	−576426.	−0.429505	−0.214752	+	0.976669i	$0.568894\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.61067e6	0.685427	0.342714	−	0.939440i	$-0.388654\pi$
$29$	2.61067e6	0.685427	0.342714	+	0.939440i	$0.388654\pi$
$30$	0	0
$31$	3.87896e6	0.754375	0.377188	−	0.926137i	$-0.376891\pi$
$31$	3.87896e6	0.754375	0.377188	+	0.926137i	$0.376891\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−6.25160e6	−0.704183
$36$	0	0
$37$	1.41599e7	1.24209	0.621046	−	0.783774i	$-0.286708\pi$
$37$	1.41599e7	1.24209	0.621046	+	0.783774i	$0.286708\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.62193e6	−0.255444	−0.127722	−	0.991810i	$-0.540767\pi$
$41$	−4.62193e6	−0.255444	−0.127722	+	0.991810i	$0.540767\pi$
$42$	0	0
$43$	8.31227e6	0.370776	0.185388	−	0.982665i	$-0.440646\pi$
$43$	8.31227e6	0.370776	0.185388	+	0.982665i	$0.440646\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.51923e7	−0.753056	−0.376528	−	0.926405i	$-0.622882\pi$
$47$	−2.51923e7	−0.753056	−0.376528	+	0.926405i	$0.622882\pi$
$48$	0	0
$49$	5.96977e7	1.47937
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−3.49333e7	−0.608133	−0.304066	−	0.952651i	$-0.598344\pi$
$53$	−3.49333e7	−0.608133	−0.304066	+	0.952651i	$0.598344\pi$
$54$	0	0
$55$	−2.94335e7	−0.433721
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6.71868e6	−0.0721855	−0.0360928	−	0.999348i	$-0.511491\pi$
$59$	−6.71868e6	−0.0721855	−0.0360928	+	0.999348i	$0.511491\pi$
$60$	0	0
$61$	−4.75982e6	−0.0440156	−0.0220078	−	0.999758i	$-0.507006\pi$
$61$	−4.75982e6	−0.0440156	−0.0220078	+	0.999758i	$0.507006\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.85183e6	0.0406614
$66$	0	0
$67$	−1.38772e8	−0.841330	−0.420665	−	0.907216i	$-0.638203\pi$
$67$	−1.38772e8	−0.841330	−0.420665	+	0.907216i	$0.638203\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−3.54359e8	−1.65494	−0.827468	−	0.561513i	$-0.810219\pi$
$71$	−3.54359e8	−1.65494	−0.827468	+	0.561513i	$0.810219\pi$
$72$	0	0
$73$	2.41596e8	0.995719	0.497859	−	0.867258i	$-0.334120\pi$
$73$	2.41596e8	0.995719	0.497859	+	0.867258i	$0.334120\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.71058e8	1.52709
$78$	0	0
$79$	2.61173e8	0.754409	0.377204	−	0.926130i	$-0.376885\pi$
$79$	2.61173e8	0.754409	0.377204	+	0.926130i	$0.376885\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.55898e8	1.51700	0.758499	−	0.651674i	$-0.225933\pi$
$83$	6.55898e8	1.51700	0.758499	+	0.651674i	$0.225933\pi$
$84$	0	0
$85$	−6.78253e7	−0.140931
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.00221e9	1.69318	0.846590	−	0.532245i	$-0.178652\pi$
$89$	1.00221e9	1.69318	0.846590	+	0.532245i	$0.178652\pi$
$90$	0	0
$91$	−9.36534e7	−0.143165
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.15733e8	−0.523671
$96$	0	0
$97$	1.24869e9	1.43213	0.716065	−	0.698034i	$-0.245942\pi$
$97$	1.24869e9	1.43213	0.716065	+	0.698034i	$0.245942\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	9.39590e7	0.0898446	0.0449223	−	0.998990i	$-0.485696\pi$
$101$	9.39590e7	0.0898446	0.0449223	+	0.998990i	$0.485696\pi$
$102$	0	0
$103$	−3.58300e8	−0.313675	−0.156837	−	0.987624i	$-0.550130\pi$
$103$	−3.58300e8	−0.313675	−0.156837	+	0.987624i	$0.550130\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.02197e9	0.753719	0.376860	−	0.926270i	$-0.377004\pi$
$107$	1.02197e9	0.753719	0.376860	+	0.926270i	$0.377004\pi$
$108$	0	0
$109$	−8.94205e8	−0.606761	−0.303380	−	0.952869i	$-0.598115\pi$
$109$	−8.94205e8	−0.606761	−0.303380	+	0.952869i	$0.598115\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.11113e9	−0.641077	−0.320539	−	0.947235i	$-0.603864\pi$
$113$	−1.11113e9	−0.641077	−0.320539	+	0.947235i	$0.603864\pi$
$114$	0	0
$115$	−3.60266e8	−0.192080
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.08548e9	0.496206
$120$	0	0
$121$	−1.40134e8	−0.0594306
$122$	0	0
$123$	0	0
$124$	0	0
$125$	2.44141e8	0.0894427
$126$	0	0
$127$	5.51824e9	1.88228	0.941139	−	0.338020i	$-0.109757\pi$
$127$	5.51824e9	1.88228	0.941139	+	0.338020i	$0.109757\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−4.66527e9	−1.38406	−0.692031	−	0.721867i	$-0.743284\pi$
$131$	−4.66527e9	−1.38406	−0.692031	+	0.721867i	$0.743284\pi$
$132$	0	0
$133$	6.65344e9	1.84380
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.97325e9	1.69119	0.845595	−	0.533825i	$-0.179246\pi$
$137$	6.97325e9	1.69119	0.845595	+	0.533825i	$0.179246\pi$
$138$	0	0
$139$	6.27736e9	1.42630	0.713149	−	0.701013i	$-0.247268\pi$
$139$	6.27736e9	1.42630	0.713149	+	0.701013i	$0.247268\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−4.40935e8	−0.0881784
$144$	0	0
$145$	1.63167e9	0.306532
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.62156e9	0.269523	0.134761	−	0.990878i	$-0.456973\pi$
$149$	1.62156e9	0.269523	0.134761	+	0.990878i	$0.456973\pi$
$150$	0	0
$151$	1.17677e10	1.84202	0.921011	−	0.389537i	$-0.127365\pi$
$151$	1.17677e10	1.84202	0.921011	+	0.389537i	$0.127365\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.42435e9	0.337367
$156$	0	0
$157$	−1.71288e9	−0.224998	−0.112499	−	0.993652i	$-0.535886\pi$
$157$	−1.71288e9	−0.224998	−0.112499	+	0.993652i	$0.535886\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.76574e9	0.676298
$162$	0	0
$163$	9.98636e9	1.10806	0.554030	−	0.832497i	$-0.313089\pi$
$163$	9.98636e9	1.10806	0.554030	+	0.832497i	$0.313089\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.56153e10	−1.55355	−0.776776	−	0.629777i	$-0.783146\pi$
$167$	−1.56153e10	−1.55355	−0.776776	+	0.629777i	$0.783146\pi$
$168$	0	0
$169$	−1.05168e10	−0.991733
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−4.61727e8	−0.0391902	−0.0195951	−	0.999808i	$-0.506238\pi$
$173$	−4.61727e8	−0.0391902	−0.0195951	+	0.999808i	$0.506238\pi$
$174$	0	0
$175$	−3.90725e9	−0.314920
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.56860e10	1.14202	0.571011	−	0.820943i	$-0.306551\pi$
$179$	1.56860e10	1.14202	0.571011	+	0.820943i	$0.306551\pi$
$180$	0	0
$181$	5.26075e8	0.0364329	0.0182165	−	0.999834i	$-0.494201\pi$
$181$	5.26075e8	0.0364329	0.0182165	+	0.999834i	$0.494201\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	8.84996e9	0.555480
$186$	0	0
$187$	5.11063e9	0.305624
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.60641e10	1.41708	0.708538	−	0.705673i	$-0.249355\pi$
$191$	2.60641e10	1.41708	0.708538	+	0.705673i	$0.249355\pi$
$192$	0	0
$193$	4.83514e9	0.250843	0.125421	−	0.992104i	$-0.459972\pi$
$193$	4.83514e9	0.250843	0.125421	+	0.992104i	$0.459972\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.09275e10	−0.516919	−0.258460	−	0.966022i	$-0.583215\pi$
$197$	−1.09275e10	−0.516919	−0.258460	+	0.966022i	$0.583215\pi$
$198$	0	0
$199$	−2.24042e10	−1.01272	−0.506361	−	0.862322i	$-0.669010\pi$
$199$	−2.24042e10	−1.01272	−0.506361	+	0.862322i	$0.669010\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.61134e10	−1.07927
$204$	0	0
$205$	−2.88871e9	−0.114238
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3.13255e10	1.13564
$210$	0	0
$211$	3.17129e10	1.10145	0.550725	−	0.834687i	$-0.314351\pi$
$211$	3.17129e10	1.10145	0.550725	+	0.834687i	$0.314351\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.19517e9	0.165816
$216$	0	0
$217$	−3.87995e10	−1.18784
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.01607e9	−0.0286522
$222$	0	0
$223$	1.52662e10	0.413388	0.206694	−	0.978406i	$-0.433730\pi$
$223$	1.52662e10	0.413388	0.206694	+	0.978406i	$0.433730\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	7.53481e9	0.188346	0.0941729	−	0.995556i	$-0.469979\pi$
$227$	7.53481e9	0.188346	0.0941729	+	0.995556i	$0.469979\pi$
$228$	0	0
$229$	−1.28244e10	−0.308160	−0.154080	−	0.988058i	$-0.549241\pi$
$229$	−1.28244e10	−0.308160	−0.154080	+	0.988058i	$0.549241\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.06272e10	−0.458500	−0.229250	−	0.973368i	$-0.573627\pi$
$233$	−2.06272e10	−0.458500	−0.229250	+	0.973368i	$0.573627\pi$
$234$	0	0
$235$	−1.57452e10	−0.336777
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−3.96664e10	−0.786380	−0.393190	−	0.919457i	$-0.628628\pi$
$239$	−3.96664e10	−0.786380	−0.393190	+	0.919457i	$0.628628\pi$
$240$	0	0
$241$	−3.75642e9	−0.0717294	−0.0358647	−	0.999357i	$-0.511419\pi$
$241$	−3.75642e9	−0.0717294	−0.0358647	+	0.999357i	$0.511419\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.73111e10	0.661592
$246$	0	0
$247$	−6.22797e9	−0.106466
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−7.74656e10	−1.23191	−0.615953	−	0.787783i	$-0.711229\pi$
$251$	−7.74656e10	−1.23191	−0.615953	+	0.787783i	$0.711229\pi$
$252$	0	0
$253$	2.71460e10	0.416546
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.31990e10	−0.474707	−0.237354	−	0.971423i	$-0.576280\pi$
$257$	−3.31990e10	−0.474707	−0.237354	+	0.971423i	$0.576280\pi$
$258$	0	0
$259$	−1.41636e11	−1.95580
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1.00754e11	−1.29856	−0.649281	−	0.760549i	$-0.724930\pi$
$263$	−1.00754e11	−1.29856	−0.649281	+	0.760549i	$0.724930\pi$
$264$	0	0
$265$	−2.18333e10	−0.271965
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7.49946e10	0.873262	0.436631	−	0.899641i	$-0.356172\pi$
$269$	7.49946e10	0.873262	0.436631	+	0.899641i	$0.356172\pi$
$270$	0	0
$271$	−5.80768e10	−0.654095	−0.327048	−	0.945008i	$-0.606054\pi$
$271$	−5.80768e10	−0.654095	−0.327048	+	0.945008i	$0.606054\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.83960e10	−0.193966
$276$	0	0
$277$	1.44538e11	1.47511	0.737554	−	0.675289i	$-0.235981\pi$
$277$	1.44538e11	1.47511	0.737554	+	0.675289i	$0.235981\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1.73877e11	−1.66366	−0.831828	−	0.555034i	$-0.812705\pi$
$281$	−1.73877e11	−1.66366	−0.831828	+	0.555034i	$0.812705\pi$
$282$	0	0
$283$	−3.86637e8	−0.00358315	−0.00179157	−	0.999998i	$-0.500570\pi$
$283$	−3.86637e8	−0.00358315	−0.00179157	+	0.999998i	$0.500570\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.62311e10	0.402222
$288$	0	0
$289$	−1.06811e11	−0.900692
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−6.25206e10	−0.495586	−0.247793	−	0.968813i	$-0.579705\pi$
$293$	−6.25206e10	−0.495586	−0.247793	+	0.968813i	$0.579705\pi$
$294$	0	0
$295$	−4.19918e9	−0.0322823
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−5.39703e9	−0.0390512
$300$	0	0
$301$	−8.31440e10	−0.583824
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.97489e9	−0.0196844
$306$	0	0
$307$	−1.57636e10	−0.101282	−0.0506410	−	0.998717i	$-0.516126\pi$
$307$	−1.57636e10	−0.101282	−0.0506410	+	0.998717i	$0.516126\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2.01301e10	0.122018	0.0610091	−	0.998137i	$-0.480568\pi$
$311$	2.01301e10	0.122018	0.0610091	+	0.998137i	$0.480568\pi$
$312$	0	0
$313$	−6.03790e10	−0.355579	−0.177790	−	0.984069i	$-0.556895\pi$
$313$	−6.03790e10	−0.355579	−0.177790	+	0.984069i	$0.556895\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.24292e11	0.691313	0.345657	−	0.938361i	$-0.387656\pi$
$317$	1.24292e11	0.691313	0.345657	+	0.938361i	$0.387656\pi$
$318$	0	0
$319$	−1.22946e11	−0.664748
$320$	0	0
$321$	0	0
$322$	0	0
$323$	7.21850e10	0.369008
$324$	0	0
$325$	3.65740e9	0.0181843
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.51988e11	1.18576
$330$	0	0
$331$	−1.14638e11	−0.524930	−0.262465	−	0.964941i	$-0.584535\pi$
$331$	−1.14638e11	−0.524930	−0.262465	+	0.964941i	$0.584535\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−8.67327e10	−0.376254
$336$	0	0
$337$	−3.40785e11	−1.43928	−0.719642	−	0.694345i	$-0.755694\pi$
$337$	−3.40785e11	−1.43928	−0.719642	+	0.694345i	$0.755694\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.82674e11	−0.731615
$342$	0	0
$343$	−1.93491e11	−0.754809
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.38240e11	1.25240	0.626198	−	0.779664i	$-0.284610\pi$
$347$	3.38240e11	1.25240	0.626198	+	0.779664i	$0.284610\pi$
$348$	0	0
$349$	2.34359e11	0.845603	0.422802	−	0.906222i	$-0.361047\pi$
$349$	2.34359e11	0.845603	0.422802	+	0.906222i	$0.361047\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−3.56202e11	−1.22098	−0.610492	−	0.792022i	$-0.709028\pi$
$353$	−3.56202e11	−1.22098	−0.610492	+	0.792022i	$0.709028\pi$
$354$	0	0
$355$	−2.21474e11	−0.740110
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.37815e11	−1.07338	−0.536691	−	0.843779i	$-0.680326\pi$
$359$	−3.37815e11	−1.07338	−0.536691	+	0.843779i	$0.680326\pi$
$360$	0	0
$361$	1.19768e11	0.371157
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.50997e11	0.445299
$366$	0	0
$367$	1.36383e11	0.392431	0.196215	−	0.980561i	$-0.437135\pi$
$367$	1.36383e11	0.392431	0.196215	+	0.980561i	$0.437135\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.49423e11	0.957566
$372$	0	0
$373$	5.08929e11	1.36134	0.680672	−	0.732589i	$-0.261688\pi$
$373$	5.08929e11	1.36134	0.680672	+	0.732589i	$0.261688\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.44436e10	0.0623201
$378$	0	0
$379$	9.70726e10	0.241669	0.120834	−	0.992673i	$-0.461443\pi$
$379$	9.70726e10	0.241669	0.120834	+	0.992673i	$0.461443\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7.13792e11	1.69503	0.847514	−	0.530772i	$-0.178098\pi$
$383$	7.13792e11	1.69503	0.847514	+	0.530772i	$0.178098\pi$
$384$	0	0
$385$	2.94411e11	0.682937
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2.48149e11	−0.549465	−0.274732	−	0.961521i	$-0.588589\pi$
$389$	−2.48149e11	−0.549465	−0.274732	+	0.961521i	$0.588589\pi$
$390$	0	0
$391$	6.25540e10	0.135350
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.63233e11	0.337382
$396$	0	0
$397$	7.04445e11	1.42328	0.711640	−	0.702545i	$-0.247953\pi$
$397$	7.04445e11	1.42328	0.711640	+	0.702545i	$0.247953\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.82127e11	0.544873	0.272437	−	0.962174i	$-0.412170\pi$
$401$	2.82127e11	0.544873	0.272437	+	0.962174i	$0.412170\pi$
$402$	0	0
$403$	3.63184e10	0.0685890
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6.66844e11	−1.20462
$408$	0	0
$409$	9.15011e11	1.61686	0.808428	−	0.588595i	$-0.200319\pi$
$409$	9.15011e11	1.61686	0.808428	+	0.588595i	$0.200319\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	6.72041e10	0.113663
$414$	0	0
$415$	4.09937e11	0.678422
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.19010e12	−1.88635	−0.943174	−	0.332299i	$-0.892176\pi$
$419$	−1.19010e12	−1.88635	−0.943174	+	0.332299i	$0.892176\pi$
$420$	0	0
$421$	7.44534e11	1.15509	0.577544	−	0.816360i	$-0.304011\pi$
$421$	7.44534e11	1.15509	0.577544	+	0.816360i	$0.304011\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−4.23908e10	−0.0630263
$426$	0	0
$427$	4.76104e10	0.0693069
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.47758e11	0.485433	0.242716	−	0.970097i	$-0.421962\pi$
$431$	3.47758e11	0.485433	0.242716	+	0.970097i	$0.421962\pi$
$432$	0	0
$433$	−8.13744e11	−1.11248	−0.556240	−	0.831022i	$-0.687756\pi$
$433$	−8.13744e11	−1.11248	−0.556240	+	0.831022i	$0.687756\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.83423e11	0.502935
$438$	0	0
$439$	−1.09730e12	−1.41005	−0.705023	−	0.709184i	$-0.749063\pi$
$439$	−1.09730e12	−1.41005	−0.705023	+	0.709184i	$0.749063\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−6.59236e10	−0.0813251	−0.0406625	−	0.999173i	$-0.512947\pi$
$443$	−6.59236e10	−0.0813251	−0.0406625	+	0.999173i	$0.512947\pi$
$444$	0	0
$445$	6.26381e11	0.757213
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2.15220e11	−0.249905	−0.124952	−	0.992163i	$-0.539878\pi$
$449$	−2.15220e11	−0.249905	−0.124952	+	0.992163i	$0.539878\pi$
$450$	0	0
$451$	2.17664e11	0.247737
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5.85334e10	−0.0640254
$456$	0	0
$457$	7.36259e11	0.789601	0.394801	−	0.918767i	$-0.370814\pi$
$457$	7.36259e11	0.789601	0.394801	+	0.918767i	$0.370814\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.23730e12	1.27591	0.637957	−	0.770072i	$-0.279780\pi$
$461$	1.23730e12	1.27591	0.637957	+	0.770072i	$0.279780\pi$
$462$	0	0
$463$	−7.36969e11	−0.745306	−0.372653	−	0.927971i	$-0.621552\pi$
$463$	−7.36969e11	−0.745306	−0.372653	+	0.927971i	$0.621552\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.22439e12	1.19123	0.595614	−	0.803271i	$-0.296909\pi$
$467$	1.22439e12	1.19123	0.595614	+	0.803271i	$0.296909\pi$
$468$	0	0
$469$	1.38808e12	1.32476
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−3.91455e11	−0.359590
$474$	0	0
$475$	−2.59833e11	−0.234193
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1.11970e12	−0.971833	−0.485916	−	0.874005i	$-0.661514\pi$
$479$	−1.11970e12	−0.971833	−0.485916	+	0.874005i	$0.661514\pi$
$480$	0	0
$481$	1.32579e11	0.112933
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7.80432e11	0.640468
$486$	0	0
$487$	−2.25337e11	−0.181532	−0.0907659	−	0.995872i	$-0.528932\pi$
$487$	−2.25337e11	−0.181532	−0.0907659	+	0.995872i	$0.528932\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.60716e11	0.124794	0.0623970	−	0.998051i	$-0.480126\pi$
$491$	1.60716e11	0.124794	0.0623970	+	0.998051i	$0.480126\pi$
$492$	0	0
$493$	−2.83312e11	−0.216000
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.54450e12	2.60586
$498$	0	0
$499$	8.05187e11	0.581359	0.290680	−	0.956820i	$-0.406119\pi$
$499$	8.05187e11	0.581359	0.290680	+	0.956820i	$0.406119\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.68493e12	1.17361	0.586807	−	0.809727i	$-0.300385\pi$
$503$	1.68493e12	1.17361	0.586807	+	0.809727i	$0.300385\pi$
$504$	0	0
$505$	5.87244e10	0.0401797
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1.50208e12	−0.991890	−0.495945	−	0.868354i	$-0.665178\pi$
$509$	−1.50208e12	−0.991890	−0.495945	+	0.868354i	$0.665178\pi$
$510$	0	0
$511$	−2.41658e12	−1.56786
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2.23938e11	−0.140280
$516$	0	0
$517$	1.18640e12	0.730336
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−9.24724e11	−0.549848	−0.274924	−	0.961466i	$-0.588653\pi$
$521$	−9.24724e11	−0.549848	−0.274924	+	0.961466i	$0.588653\pi$
$522$	0	0
$523$	−3.81782e10	−0.0223130	−0.0111565	−	0.999938i	$-0.503551\pi$
$523$	−3.81782e10	−0.0223130	−0.0111565	+	0.999938i	$0.503551\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.20947e11	−0.237727
$528$	0	0
$529$	−1.46889e12	−0.815526
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−4.32748e10	−0.0232254
$534$	0	0
$535$	6.38729e11	0.337073
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.81139e12	−1.43473
$540$	0	0
$541$	−1.05812e12	−0.531064	−0.265532	−	0.964102i	$-0.585548\pi$
$541$	−1.05812e12	−0.531064	−0.265532	+	0.964102i	$0.585548\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−5.58878e11	−0.271352
$546$	0	0
$547$	−1.05156e12	−0.502215	−0.251108	−	0.967959i	$-0.580795\pi$
$547$	−1.05156e12	−0.502215	−0.251108	+	0.967959i	$0.580795\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.73655e12	−0.802611
$552$	0	0
$553$	−2.61240e12	−1.18789
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3.92684e11	0.172860	0.0864301	−	0.996258i	$-0.472454\pi$
$557$	3.92684e11	0.172860	0.0864301	+	0.996258i	$0.472454\pi$
$558$	0	0
$559$	7.78272e10	0.0337115
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−2.36287e12	−0.991180	−0.495590	−	0.868557i	$-0.665048\pi$
$563$	−2.36287e12	−0.991180	−0.495590	+	0.868557i	$0.665048\pi$
$564$	0	0
$565$	−6.94454e11	−0.286699
$566$	0	0
$567$	0	0
$568$	0	0
$569$	2.50603e12	1.00226	0.501130	−	0.865372i	$-0.332918\pi$
$569$	2.50603e12	1.00226	0.501130	+	0.865372i	$0.332918\pi$
$570$	0	0
$571$	−2.99195e12	−1.17785	−0.588927	−	0.808187i	$-0.700449\pi$
$571$	−2.99195e12	−1.17785	−0.588927	+	0.808187i	$0.700449\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.25166e11	−0.0859010
$576$	0	0
$577$	−3.06622e11	−0.115163	−0.0575815	−	0.998341i	$-0.518339\pi$
$577$	−3.06622e11	−0.115163	−0.0575815	+	0.998341i	$0.518339\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−6.56067e12	−2.38867
$582$	0	0
$583$	1.64514e12	0.589785
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.80823e12	0.976249	0.488125	−	0.872774i	$-0.337681\pi$
$587$	2.80823e12	0.976249	0.488125	+	0.872774i	$0.337681\pi$
$588$	0	0
$589$	−2.58018e12	−0.883346
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.34954e12	0.448166	0.224083	−	0.974570i	$-0.428061\pi$
$593$	1.34954e12	0.448166	0.224083	+	0.974570i	$0.428061\pi$
$594$	0	0
$595$	6.78427e11	0.221910
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−3.38598e12	−1.07464	−0.537321	−	0.843378i	$-0.680564\pi$
$599$	−3.38598e12	−1.07464	−0.537321	+	0.843378i	$0.680564\pi$
$600$	0	0
$601$	4.62837e12	1.44708	0.723540	−	0.690282i	$-0.242514\pi$
$601$	4.62837e12	1.44708	0.723540	+	0.690282i	$0.242514\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−8.75839e10	−0.0265782
$606$	0	0
$607$	4.78828e12	1.43163	0.715814	−	0.698291i	$-0.246056\pi$
$607$	4.78828e12	1.43163	0.715814	+	0.698291i	$0.246056\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2.35874e11	−0.0684690
$612$	0	0
$613$	−1.02749e12	−0.293905	−0.146953	−	0.989144i	$-0.546946\pi$
$613$	−1.02749e12	−0.293905	−0.146953	+	0.989144i	$0.546946\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4.53079e12	1.25861	0.629304	−	0.777159i	$-0.283340\pi$
$617$	4.53079e12	1.25861	0.629304	+	0.777159i	$0.283340\pi$
$618$	0	0
$619$	−3.42902e12	−0.938777	−0.469388	−	0.882992i	$-0.655526\pi$
$619$	−3.42902e12	−0.938777	−0.469388	+	0.882992i	$0.655526\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.00247e13	−2.66608
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.53664e12	−0.391422
$630$	0	0
$631$	−3.52587e12	−0.885389	−0.442695	−	0.896672i	$-0.645977\pi$
$631$	−3.52587e12	−0.885389	−0.442695	+	0.896672i	$0.645977\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	3.44890e12	0.841780
$636$	0	0
$637$	5.58946e11	0.134506
$638$	0	0
$639$	0	0
$640$	0	0
$641$	7.00458e12	1.63878	0.819390	−	0.573236i	$-0.194312\pi$
$641$	7.00458e12	1.63878	0.819390	+	0.573236i	$0.194312\pi$
$642$	0	0
$643$	−4.31047e12	−0.994432	−0.497216	−	0.867627i	$-0.665644\pi$
$643$	−4.31047e12	−0.994432	−0.497216	+	0.867627i	$0.665644\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.84949e12	0.639290	0.319645	−	0.947537i	$-0.396436\pi$
$647$	2.84949e12	0.639290	0.319645	+	0.947537i	$0.396436\pi$
$648$	0	0
$649$	3.16407e11	0.0700077
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.93213e12	−0.415841	−0.207920	−	0.978146i	$-0.566669\pi$
$653$	−1.93213e12	−0.415841	−0.207920	+	0.978146i	$0.566669\pi$
$654$	0	0
$655$	−2.91579e12	−0.618972
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−8.58675e12	−1.77355	−0.886777	−	0.462198i	$-0.847061\pi$
$659$	−8.58675e12	−1.77355	−0.886777	+	0.462198i	$0.847061\pi$
$660$	0	0
$661$	1.11478e12	0.227133	0.113567	−	0.993530i	$-0.463772\pi$
$661$	1.11478e12	0.227133	0.113567	+	0.993530i	$0.463772\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.15840e12	0.824573
$666$	0	0
$667$	−1.50486e12	−0.294394
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.24157e11	0.0426876
$672$	0	0
$673$	7.57716e12	1.42377	0.711883	−	0.702298i	$-0.247843\pi$
$673$	7.57716e12	1.42377	0.711883	+	0.702298i	$0.247843\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5.08558e12	0.930447	0.465223	−	0.885193i	$-0.345974\pi$
$677$	5.08558e12	0.930447	0.465223	+	0.885193i	$0.345974\pi$
$678$	0	0
$679$	−1.24901e13	−2.25503
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.78278e11	0.0313476	0.0156738	−	0.999877i	$-0.495011\pi$
$683$	1.78278e11	0.0313476	0.0156738	+	0.999877i	$0.495011\pi$
$684$	0	0
$685$	4.35828e12	0.756323
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−3.27078e11	−0.0552924
$690$	0	0
$691$	−8.66612e11	−0.144602	−0.0723008	−	0.997383i	$-0.523034\pi$
$691$	−8.66612e11	−0.144602	−0.0723008	+	0.997383i	$0.523034\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.92335e12	0.637860
$696$	0	0
$697$	5.01574e11	0.0804985
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−5.18099e12	−0.810367	−0.405183	−	0.914235i	$-0.632792\pi$
$701$	−5.18099e12	−0.810367	−0.405183	+	0.914235i	$0.632792\pi$
$702$	0	0
$703$	−9.41882e12	−1.45445
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−9.39831e11	−0.141469
$708$	0	0
$709$	−1.22351e13	−1.81844	−0.909218	−	0.416320i	$-0.863319\pi$
$709$	−1.22351e13	−1.81844	−0.909218	+	0.416320i	$0.863319\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2.23593e12	−0.324008
$714$	0	0
$715$	−2.75584e11	−0.0394346
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.11881e13	1.56127	0.780634	−	0.624989i	$-0.214896\pi$
$719$	1.11881e13	1.56127	0.780634	+	0.624989i	$0.214896\pi$
$720$	0	0
$721$	3.58392e12	0.493913
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.01979e12	0.137085
$726$	0	0
$727$	7.94265e12	1.05453	0.527267	−	0.849699i	$-0.323217\pi$
$727$	7.94265e12	1.05453	0.527267	+	0.849699i	$0.323217\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−9.02052e11	−0.116843
$732$	0	0
$733$	1.09012e13	1.39478	0.697392	−	0.716690i	$-0.254344\pi$
$733$	1.09012e13	1.39478	0.697392	+	0.716690i	$0.254344\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.53530e12	0.815947
$738$	0	0
$739$	1.54854e12	0.190996	0.0954978	−	0.995430i	$-0.469556\pi$
$739$	1.54854e12	0.190996	0.0954978	+	0.995430i	$0.469556\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−5.96254e12	−0.717764	−0.358882	−	0.933383i	$-0.616842\pi$
$743$	−5.96254e12	−0.717764	−0.358882	+	0.933383i	$0.616842\pi$
$744$	0	0
$745$	1.01348e12	0.120534
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1.02223e13	−1.18681
$750$	0	0
$751$	9.92051e12	1.13803	0.569016	−	0.822326i	$-0.307324\pi$
$751$	9.92051e12	1.13803	0.569016	+	0.822326i	$0.307324\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	7.35480e12	0.823777
$756$	0	0
$757$	1.23733e13	1.36947	0.684737	−	0.728791i	$-0.259917\pi$
$757$	1.23733e13	1.36947	0.684737	+	0.728791i	$0.259917\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	8.52217e12	0.921127	0.460563	−	0.887627i	$-0.347647\pi$
$761$	8.52217e12	0.921127	0.460563	+	0.887627i	$0.347647\pi$
$762$	0	0
$763$	8.94434e12	0.955406
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6.29066e10	−0.00656322
$768$	0	0
$769$	2.99595e12	0.308934	0.154467	−	0.987998i	$-0.450634\pi$
$769$	2.99595e12	0.308934	0.154467	+	0.987998i	$0.450634\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−6.44491e11	−0.0649245	−0.0324623	−	0.999473i	$-0.510335\pi$
$773$	−6.44491e11	−0.0649245	−0.0324623	+	0.999473i	$0.510335\pi$
$774$	0	0
$775$	1.51522e12	0.150875
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.07438e12	0.299116
$780$	0	0
$781$	1.66881e13	1.60501
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1.07055e12	−0.100622
$786$	0	0
$787$	8.05096e12	0.748103	0.374052	−	0.927408i	$-0.377968\pi$
$787$	8.05096e12	0.748103	0.374052	+	0.927408i	$0.377968\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.11141e13	1.00944
$792$	0	0
$793$	−4.45659e10	−0.00400196
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.11381e13	0.977799	0.488899	−	0.872340i	$-0.337399\pi$
$797$	1.11381e13	0.977799	0.488899	+	0.872340i	$0.337399\pi$
$798$	0	0
$799$	2.73388e12	0.237312
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1.13776e13	−0.965678
$804$	0	0
$805$	3.60358e12	0.302450
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.69716e13	−1.39301	−0.696506	−	0.717550i	$-0.745263\pi$
$809$	−1.69716e13	−1.39301	−0.696506	+	0.717550i	$0.745263\pi$
$810$	0	0
$811$	−1.17583e13	−0.954444	−0.477222	−	0.878783i	$-0.658356\pi$
$811$	−1.17583e13	−0.954444	−0.477222	+	0.878783i	$0.658356\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	6.24148e12	0.495539
$816$	0	0
$817$	−5.52910e12	−0.434165
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.99657e13	−1.53370	−0.766851	−	0.641825i	$-0.778177\pi$
$821$	−1.99657e13	−1.53370	−0.766851	+	0.641825i	$0.778177\pi$
$822$	0	0
$823$	5.52813e11	0.0420029	0.0210014	−	0.999779i	$-0.493315\pi$
$823$	5.52813e11	0.0420029	0.0210014	+	0.999779i	$0.493315\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.57036e13	1.16741	0.583707	−	0.811965i	$-0.301602\pi$
$827$	1.57036e13	1.16741	0.583707	+	0.811965i	$0.301602\pi$
$828$	0	0
$829$	1.05928e13	0.778958	0.389479	−	0.921035i	$-0.372655\pi$
$829$	1.05928e13	0.778958	0.389479	+	0.921035i	$0.372655\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−6.47843e12	−0.466195
$834$	0	0
$835$	−9.75955e12	−0.694770
$836$	0	0
$837$	0	0
$838$	0	0
$839$	6.55506e11	0.0456718	0.0228359	−	0.999739i	$-0.492730\pi$
$839$	6.55506e11	0.0456718	0.0228359	+	0.999739i	$0.492730\pi$
$840$	0	0
$841$	−7.69153e12	−0.530189
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−6.57302e12	−0.443517
$846$	0	0
$847$	1.40170e12	0.0935794
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−8.16215e12	−0.533484
$852$	0	0
$853$	2.08805e13	1.35043	0.675213	−	0.737622i	$-0.264051\pi$
$853$	2.08805e13	1.35043	0.675213	+	0.737622i	$0.264051\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.19074e13	1.38732	0.693660	−	0.720303i	$-0.255997\pi$
$857$	2.19074e13	1.38732	0.693660	+	0.720303i	$0.255997\pi$
$858$	0	0
$859$	1.47238e13	0.922680	0.461340	−	0.887223i	$-0.347369\pi$
$859$	1.47238e13	0.922680	0.461340	+	0.887223i	$0.347369\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.74805e12	0.168646	0.0843230	−	0.996438i	$-0.473127\pi$
$863$	2.74805e12	0.168646	0.0843230	+	0.996438i	$0.473127\pi$
$864$	0	0
$865$	−2.88579e11	−0.0175264
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.22996e13	−0.731648
$870$	0	0
$871$	−1.29932e12	−0.0764950
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2.44203e12	−0.140837
$876$	0	0
$877$	6.68274e12	0.381466	0.190733	−	0.981642i	$-0.438913\pi$
$877$	6.68274e12	0.381466	0.190733	+	0.981642i	$0.438913\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.42384e13	−0.796289	−0.398144	−	0.917323i	$-0.630346\pi$
$881$	−1.42384e13	−0.796289	−0.398144	+	0.917323i	$0.630346\pi$
$882$	0	0
$883$	−2.46098e13	−1.36234	−0.681170	−	0.732125i	$-0.738529\pi$
$883$	−2.46098e13	−1.36234	−0.681170	+	0.732125i	$0.738529\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	9.12625e12	0.495035	0.247518	−	0.968883i	$-0.420385\pi$
$887$	9.12625e12	0.495035	0.247518	+	0.968883i	$0.420385\pi$
$888$	0	0
$889$	−5.51966e13	−2.96384
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.67572e13	0.881802
$894$	0	0
$895$	9.80376e12	0.510727
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.01267e13	0.517069
$900$	0	0
$901$	3.79098e12	0.191642
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3.28797e11	0.0162933
$906$	0	0
$907$	3.32446e13	1.63113	0.815564	−	0.578668i	$-0.196427\pi$
$907$	3.32446e13	1.63113	0.815564	+	0.578668i	$0.196427\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.42660e13	0.686232	0.343116	−	0.939293i	$-0.388518\pi$
$911$	1.42660e13	0.686232	0.343116	+	0.939293i	$0.388518\pi$
$912$	0	0
$913$	−3.08887e13	−1.47123
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.66647e13	2.17935
$918$	0	0
$919$	3.23171e13	1.49456	0.747278	−	0.664512i	$-0.231360\pi$
$919$	3.23171e13	1.49456	0.747278	+	0.664512i	$0.231360\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−3.31784e12	−0.150469
$924$	0	0
$925$	5.53123e12	0.248418
$926$	0	0
$927$	0	0
$928$	0	0
$929$	3.49040e12	0.153746	0.0768732	−	0.997041i	$-0.475506\pi$
$929$	3.49040e12	0.153746	0.0768732	+	0.997041i	$0.475506\pi$
$930$	0	0
$931$	−3.97093e13	−1.73228
$932$	0	0
$933$	0	0
$934$	0	0
$935$	3.19414e12	0.136679
$936$	0	0
$937$	−1.48063e13	−0.627507	−0.313754	−	0.949504i	$-0.601587\pi$
$937$	−1.48063e13	−0.627507	−0.313754	+	0.949504i	$0.601587\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−3.90935e13	−1.62537	−0.812684	−	0.582705i	$-0.801994\pi$
$941$	−3.90935e13	−1.62537	−0.812684	+	0.582705i	$0.801994\pi$
$942$	0	0
$943$	2.66420e12	0.109714
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.30013e13	−0.929346	−0.464673	−	0.885482i	$-0.653828\pi$
$947$	−2.30013e13	−0.929346	−0.464673	+	0.885482i	$0.653828\pi$
$948$	0	0
$949$	2.26205e12	0.0905323
$950$	0	0
$951$	0	0
$952$	0	0
$953$	3.32845e13	1.30714	0.653572	−	0.756864i	$-0.273270\pi$
$953$	3.32845e13	1.30714	0.653572	+	0.756864i	$0.273270\pi$
$954$	0	0
$955$	1.62901e13	0.633735
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6.97504e13	−2.66295
$960$	0	0
$961$	−1.13933e13	−0.430918
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.02196e12	0.112180
$966$	0	0
$967$	−5.77848e12	−0.212518	−0.106259	−	0.994339i	$-0.533887\pi$
$967$	−5.77848e12	−0.212518	−0.106259	+	0.994339i	$0.533887\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	4.46200e13	1.61080	0.805402	−	0.592728i	$-0.201949\pi$
$971$	4.46200e13	1.61080	0.805402	+	0.592728i	$0.201949\pi$
$972$	0	0
$973$	−6.27897e13	−2.24585
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−3.90697e13	−1.37188	−0.685938	−	0.727660i	$-0.740608\pi$
$977$	−3.90697e13	−1.37188	−0.685938	+	0.727660i	$0.740608\pi$
$978$	0	0
$979$	−4.71977e13	−1.64210
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−1.96137e12	−0.0669990	−0.0334995	−	0.999439i	$-0.510665\pi$
$983$	−1.96137e12	−0.0669990	−0.0334995	+	0.999439i	$0.510665\pi$
$984$	0	0
$985$	−6.82969e12	−0.231173
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−4.79141e12	−0.159250
$990$	0	0
$991$	−4.33785e13	−1.42871	−0.714354	−	0.699785i	$-0.753279\pi$
$991$	−4.33785e13	−1.42871	−0.714354	+	0.699785i	$0.753279\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1.40026e13	−0.452903
$996$	0	0
$997$	−2.79360e12	−0.0895440	−0.0447720	−	0.998997i	$-0.514256\pi$
$997$	−2.79360e12	−0.0895440	−0.0447720	+	0.998997i	$0.514256\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	180.10.a.e.1.1		2
3.2	odd	2		20.10.a.b.1.1	✓	2
12.11	even	2		80.10.a.j.1.2		2
15.2	even	4		100.10.c.c.49.4		4
15.8	even	4		100.10.c.c.49.1		4
15.14	odd	2		100.10.a.c.1.2		2
24.5	odd	2		320.10.a.t.1.2		2
24.11	even	2		320.10.a.l.1.1		2
60.23	odd	4		400.10.c.l.49.4		4
60.47	odd	4		400.10.c.l.49.1		4
60.59	even	2		400.10.a.l.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.10.a.b.1.1	✓	2	3.2	odd	2
80.10.a.j.1.2		2	12.11	even	2
100.10.a.c.1.2		2	15.14	odd	2
100.10.c.c.49.1		4	15.8	even	4
100.10.c.c.49.4		4	15.2	even	4
180.10.a.e.1.1		2	1.1	even	1	trivial
320.10.a.l.1.1		2	24.11	even	2
320.10.a.t.1.2		2	24.5	odd	2
400.10.a.l.1.1		2	60.59	even	2
400.10.c.l.49.1		4	60.47	odd	4
400.10.c.l.49.4		4	60.23	odd	4

Overview	Random
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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:	\( N \)	\(=\)	\( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	180.a (trivial)

\(f(q)\)	\(=\)	\(q+625.000 q^{5} -10002.6 q^{7} +O(q^{10})\)	\(q+625.000 q^{5} -10002.6 q^{7} -47093.7 q^{11} +9362.93 q^{13} -108521. q^{17} -665173. q^{19} -576426. q^{23} +390625. q^{25} +2.61067e6 q^{29} +3.87896e6 q^{31} -6.25160e6 q^{35} +1.41599e7 q^{37} -4.62193e6 q^{41} +8.31227e6 q^{43} -2.51923e7 q^{47} +5.96977e7 q^{49} -3.49333e7 q^{53} -2.94335e7 q^{55} -6.71868e6 q^{59} -4.75982e6 q^{61} +5.85183e6 q^{65} -1.38772e8 q^{67} -3.54359e8 q^{71} +2.41596e8 q^{73} +4.71058e8 q^{77} +2.61173e8 q^{79} +6.55898e8 q^{83} -6.78253e7 q^{85} +1.00221e9 q^{89} -9.36534e7 q^{91} -4.15733e8 q^{95} +1.24869e9 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 1250 q^{5} - 380 q^{7}+O(q^{10}) \) 2 * q + 1250 * q^5 - 380 * q^7	\( 2 q + 1250 q^{5} - 380 q^{7} - 102720 q^{11} + 179140 q^{13} - 316020 q^{17} + 137272 q^{19} + 665460 q^{23} + 781250 q^{25} + 6893748 q^{29} + 291832 q^{31} - 237500 q^{35} + 11261380 q^{37} - 29773452 q^{41} - 11708180 q^{43} - 62493300 q^{47} + 111937914 q^{49} - 9417780 q^{53} - 64200000 q^{55} + 92930856 q^{59} + 195673924 q^{61} + 111962500 q^{65} - 219767420 q^{67} - 311207016 q^{71} - 99224060 q^{73} - 64210560 q^{77} + 542261776 q^{79} + 1256915700 q^{83} - 197512500 q^{85} + 462291852 q^{89} + 1540037768 q^{91} + 85795000 q^{95} + 1671716740 q^{97}+O(q^{100}) \) 2 * q + 1250 * q^5 - 380 * q^7 - 102720 * q^11 + 179140 * q^13 - 316020 * q^17 + 137272 * q^19 + 665460 * q^23 + 781250 * q^25 + 6893748 * q^29 + 291832 * q^31 - 237500 * q^35 + 11261380 * q^37 - 29773452 * q^41 - 11708180 * q^43 - 62493300 * q^47 + 111937914 * q^49 - 9417780 * q^53 - 64200000 * q^55 + 92930856 * q^59 + 195673924 * q^61 + 111962500 * q^65 - 219767420 * q^67 - 311207016 * q^71 - 99224060 * q^73 - 64210560 * q^77 + 542261776 * q^79 + 1256915700 * q^83 - 197512500 * q^85 + 462291852 * q^89 + 1540037768 * q^91 + 85795000 * q^95 + 1671716740 * q^97

Introduction

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