LMFDB - Embedded newform 72.8.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,8,Mod(1,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("72.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	72.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:	$22.4917218349$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 8)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	72.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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
$5$	82.0000	0.293372	0.146686	−	0.989183i	$-0.453139\pi$
$5$	82.0000	0.293372	0.146686	+	0.989183i	$0.453139\pi$
$6$	0	0
$7$	−456.000	−0.502483	−0.251242	−	0.967924i	$-0.580839\pi$
$7$	−456.000	−0.502483	−0.251242	+	0.967924i	$0.580839\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2524.00	0.571762	0.285881	−	0.958265i	$-0.407714\pi$
$11$	2524.00	0.571762	0.285881	+	0.958265i	$0.407714\pi$
$12$	0	0
$13$	−10778.0	−1.36062	−0.680309	−	0.732925i	$-0.738155\pi$
$13$	−10778.0	−1.36062	−0.680309	+	0.732925i	$0.738155\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	11150.0	0.550432	0.275216	−	0.961382i	$-0.411251\pi$
$17$	11150.0	0.550432	0.275216	+	0.961382i	$0.411251\pi$
$18$	0	0
$19$	4124.00	0.137937	0.0689685	−	0.997619i	$-0.478029\pi$
$19$	4124.00	0.137937	0.0689685	+	0.997619i	$0.478029\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−81704.0	−1.40022	−0.700109	−	0.714036i	$-0.746865\pi$
$23$	−81704.0	−1.40022	−0.700109	+	0.714036i	$0.746865\pi$
$24$	0	0
$25$	−71401.0	−0.913933
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−99798.0	−0.759852	−0.379926	−	0.925017i	$-0.624051\pi$
$29$	−99798.0	−0.759852	−0.379926	+	0.925017i	$0.624051\pi$
$30$	0	0
$31$	−40480.0	−0.244048	−0.122024	−	0.992527i	$-0.538938\pi$
$31$	−40480.0	−0.244048	−0.122024	+	0.992527i	$0.538938\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−37392.0	−0.147415
$36$	0	0
$37$	−419442.	−1.36134	−0.680669	−	0.732591i	$-0.738311\pi$
$37$	−419442.	−1.36134	−0.680669	+	0.732591i	$0.738311\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−141402.	−0.320414	−0.160207	−	0.987083i	$-0.551216\pi$
$41$	−141402.	−0.320414	−0.160207	+	0.987083i	$0.551216\pi$
$42$	0	0
$43$	−690428.	−1.32428	−0.662138	−	0.749382i	$-0.730351\pi$
$43$	−690428.	−1.32428	−0.662138	+	0.749382i	$0.730351\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	682032.	0.958213	0.479107	−	0.877757i	$-0.340961\pi$
$47$	682032.	0.958213	0.479107	+	0.877757i	$0.340961\pi$
$48$	0	0
$49$	−615607.	−0.747510
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.81312e6	−1.67286	−0.836432	−	0.548071i	$-0.815362\pi$
$53$	−1.81312e6	−1.67286	−0.836432	+	0.548071i	$0.815362\pi$
$54$	0	0
$55$	206968.	0.167739
$56$	0	0
$57$	0	0
$58$	0	0
$59$	966028.	0.612361	0.306181	−	0.951973i	$-0.400949\pi$
$59$	966028.	0.612361	0.306181	+	0.951973i	$0.400949\pi$
$60$	0	0
$61$	1.88767e6	1.06481	0.532404	−	0.846490i	$-0.321289\pi$
$61$	1.88767e6	1.06481	0.532404	+	0.846490i	$0.321289\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−883796.	−0.399168
$66$	0	0
$67$	2.96587e6	1.20473	0.602365	−	0.798220i	$-0.294225\pi$
$67$	2.96587e6	1.20473	0.602365	+	0.798220i	$0.294225\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.54823e6	0.844957	0.422479	−	0.906373i	$-0.361160\pi$
$71$	2.54823e6	0.844957	0.422479	+	0.906373i	$0.361160\pi$
$72$	0	0
$73$	−1.68033e6	−0.505549	−0.252775	−	0.967525i	$-0.581343\pi$
$73$	−1.68033e6	−0.505549	−0.252775	+	0.967525i	$0.581343\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.15094e6	−0.287301
$78$	0	0
$79$	4.03806e6	0.921464	0.460732	−	0.887539i	$-0.347587\pi$
$79$	4.03806e6	0.921464	0.460732	+	0.887539i	$0.347587\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	5.38576e6	1.03389	0.516945	−	0.856019i	$-0.327069\pi$
$83$	5.38576e6	1.03389	0.516945	+	0.856019i	$0.327069\pi$
$84$	0	0
$85$	914300.	0.161481
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.47305e6	0.973293	0.486647	−	0.873599i	$-0.338220\pi$
$89$	6.47305e6	0.973293	0.486647	+	0.873599i	$0.338220\pi$
$90$	0	0
$91$	4.91477e6	0.683688
$92$	0	0
$93$	0	0
$94$	0	0
$95$	338168.	0.0404669
$96$	0	0
$97$	−6.06576e6	−0.674814	−0.337407	−	0.941359i	$-0.609550\pi$
$97$	−6.06576e6	−0.674814	−0.337407	+	0.941359i	$0.609550\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−9.70069e6	−0.936866	−0.468433	−	0.883499i	$-0.655181\pi$
$101$	−9.70069e6	−0.936866	−0.468433	+	0.883499i	$0.655181\pi$
$102$	0	0
$103$	4.10159e6	0.369847	0.184924	−	0.982753i	$-0.440796\pi$
$103$	4.10159e6	0.369847	0.184924	+	0.982753i	$0.440796\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−72900.0	−0.00575287	−0.00287643	−	0.999996i	$-0.500916\pi$
$107$	−72900.0	−0.00575287	−0.00287643	+	0.999996i	$0.500916\pi$
$108$	0	0
$109$	9.55841e6	0.706957	0.353478	−	0.935443i	$-0.384999\pi$
$109$	9.55841e6	0.706957	0.353478	+	0.935443i	$0.384999\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−9.33890e6	−0.608865	−0.304433	−	0.952534i	$-0.598467\pi$
$113$	−9.33890e6	−0.608865	−0.304433	+	0.952534i	$0.598467\pi$
$114$	0	0
$115$	−6.69973e6	−0.410785
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.08440e6	−0.276583
$120$	0	0
$121$	−1.31166e7	−0.673089
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.22611e7	−0.561495
$126$	0	0
$127$	−3.59794e7	−1.55862	−0.779311	−	0.626637i	$-0.784431\pi$
$127$	−3.59794e7	−1.55862	−0.779311	+	0.626637i	$0.784431\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	676052.	0.0262743	0.0131371	−	0.999914i	$-0.495818\pi$
$131$	676052.	0.0262743	0.0131371	+	0.999914i	$0.495818\pi$
$132$	0	0
$133$	−1.88054e6	−0.0693111
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.95841e7	0.982962	0.491481	−	0.870888i	$-0.336456\pi$
$137$	2.95841e7	0.982962	0.491481	+	0.870888i	$0.336456\pi$
$138$	0	0
$139$	−3.19084e7	−1.00775	−0.503876	−	0.863776i	$-0.668093\pi$
$139$	−3.19084e7	−1.00775	−0.503876	+	0.863776i	$0.668093\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.72037e7	−0.777949
$144$	0	0
$145$	−8.18344e6	−0.222919
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.16603e7	0.288773	0.144386	−	0.989521i	$-0.453879\pi$
$149$	1.16603e7	0.288773	0.144386	+	0.989521i	$0.453879\pi$
$150$	0	0
$151$	−1.76295e7	−0.416698	−0.208349	−	0.978055i	$-0.566809\pi$
$151$	−1.76295e7	−0.416698	−0.208349	+	0.978055i	$0.566809\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3.31936e6	−0.0715968
$156$	0	0
$157$	6.34658e6	0.130885	0.0654427	−	0.997856i	$-0.479154\pi$
$157$	6.34658e6	0.130885	0.0654427	+	0.997856i	$0.479154\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.72570e7	0.703587
$162$	0	0
$163$	8.04234e7	1.45454	0.727271	−	0.686351i	$-0.240789\pi$
$163$	8.04234e7	1.45454	0.727271	+	0.686351i	$0.240789\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.14767e8	−1.90682	−0.953411	−	0.301673i	$-0.902455\pi$
$167$	−1.14767e8	−1.90682	−0.953411	+	0.301673i	$0.902455\pi$
$168$	0	0
$169$	5.34168e7	0.851283
$170$	0	0
$171$	0	0
$172$	0	0
$173$	6.33755e7	0.930594	0.465297	−	0.885155i	$-0.345947\pi$
$173$	6.33755e7	0.930594	0.465297	+	0.885155i	$0.345947\pi$
$174$	0	0
$175$	3.25589e7	0.459236
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.13228e7	0.147559	0.0737796	−	0.997275i	$-0.476494\pi$
$179$	1.13228e7	0.147559	0.0737796	+	0.997275i	$0.476494\pi$
$180$	0	0
$181$	−5.22650e6	−0.0655143	−0.0327571	−	0.999463i	$-0.510429\pi$
$181$	−5.22650e6	−0.0655143	−0.0327571	+	0.999463i	$0.510429\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−3.43942e7	−0.399379
$186$	0	0
$187$	2.81426e7	0.314716
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.50301e7	0.882990	0.441495	−	0.897264i	$-0.354448\pi$
$191$	8.50301e7	0.882990	0.441495	+	0.897264i	$0.354448\pi$
$192$	0	0
$193$	1.15092e8	1.15237	0.576186	−	0.817319i	$-0.304540\pi$
$193$	1.15092e8	1.15237	0.576186	+	0.817319i	$0.304540\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.38522e8	1.29088	0.645441	−	0.763810i	$-0.276674\pi$
$197$	1.38522e8	1.29088	0.645441	+	0.763810i	$0.276674\pi$
$198$	0	0
$199$	−2.19614e7	−0.197548	−0.0987742	−	0.995110i	$-0.531492\pi$
$199$	−2.19614e7	−0.197548	−0.0987742	+	0.995110i	$0.531492\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4.55079e7	0.381813
$204$	0	0
$205$	−1.15950e7	−0.0940007
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.04090e7	0.0788671
$210$	0	0
$211$	−6.10208e7	−0.447187	−0.223594	−	0.974682i	$-0.571779\pi$
$211$	−6.10208e7	−0.447187	−0.223594	+	0.974682i	$0.571779\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5.66151e7	−0.388506
$216$	0	0
$217$	1.84589e7	0.122630
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.20175e8	−0.748928
$222$	0	0
$223$	−4.22448e7	−0.255098	−0.127549	−	0.991832i	$-0.540711\pi$
$223$	−4.22448e7	−0.255098	−0.127549	+	0.991832i	$0.540711\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.39102e8	1.35673	0.678364	−	0.734726i	$-0.262689\pi$
$227$	2.39102e8	1.35673	0.678364	+	0.734726i	$0.262689\pi$
$228$	0	0
$229$	−4.67889e7	−0.257465	−0.128733	−	0.991679i	$-0.541091\pi$
$229$	−4.67889e7	−0.257465	−0.128733	+	0.991679i	$0.541091\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.45225e8	−1.78795	−0.893977	−	0.448113i	$-0.852096\pi$
$233$	−3.45225e8	−1.78795	−0.893977	+	0.448113i	$0.852096\pi$
$234$	0	0
$235$	5.59266e7	0.281113
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2.34413e8	−1.11068	−0.555340	−	0.831624i	$-0.687412\pi$
$239$	−2.34413e8	−1.11068	−0.555340	+	0.831624i	$0.687412\pi$
$240$	0	0
$241$	−1.09557e8	−0.504175	−0.252087	−	0.967705i	$-0.581117\pi$
$241$	−1.09557e8	−0.504175	−0.252087	+	0.967705i	$0.581117\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−5.04798e7	−0.219299
$246$	0	0
$247$	−4.44485e7	−0.187680
$248$	0	0
$249$	0	0
$250$	0	0
$251$	3.94031e8	1.57280	0.786398	−	0.617720i	$-0.211943\pi$
$251$	3.94031e8	1.57280	0.786398	+	0.617720i	$0.211943\pi$
$252$	0	0
$253$	−2.06221e8	−0.800591
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.19064e8	−1.17250	−0.586248	−	0.810131i	$-0.699396\pi$
$257$	−3.19064e8	−1.17250	−0.586248	+	0.810131i	$0.699396\pi$
$258$	0	0
$259$	1.91266e8	0.684050
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2.19359e8	−0.743549	−0.371774	−	0.928323i	$-0.621250\pi$
$263$	−2.19359e8	−0.743549	−0.371774	+	0.928323i	$0.621250\pi$
$264$	0	0
$265$	−1.48676e8	−0.490772
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.48033e8	0.463687	0.231844	−	0.972753i	$-0.425524\pi$
$269$	1.48033e8	0.463687	0.231844	+	0.972753i	$0.425524\pi$
$270$	0	0
$271$	−3.69934e8	−1.12910	−0.564549	−	0.825399i	$-0.690950\pi$
$271$	−3.69934e8	−1.12910	−0.564549	+	0.825399i	$0.690950\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.80216e8	−0.522552
$276$	0	0
$277$	−3.95860e8	−1.11908	−0.559541	−	0.828803i	$-0.689023\pi$
$277$	−3.95860e8	−1.11908	−0.559541	+	0.828803i	$0.689023\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5.97760e8	1.60714	0.803572	−	0.595208i	$-0.202930\pi$
$281$	5.97760e8	1.60714	0.803572	+	0.595208i	$0.202930\pi$
$282$	0	0
$283$	8.05797e7	0.211336	0.105668	−	0.994401i	$-0.466302\pi$
$283$	8.05797e7	0.211336	0.105668	+	0.994401i	$0.466302\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.44793e7	0.161003
$288$	0	0
$289$	−2.86016e8	−0.697025
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−7.54530e8	−1.75243	−0.876213	−	0.481924i	$-0.839938\pi$
$293$	−7.54530e8	−1.75243	−0.876213	+	0.481924i	$0.839938\pi$
$294$	0	0
$295$	7.92143e7	0.179650
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8.80606e8	1.90516
$300$	0	0
$301$	3.14835e8	0.665427
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.54789e8	0.312385
$306$	0	0
$307$	8.20472e8	1.61838	0.809188	−	0.587549i	$-0.199907\pi$
$307$	8.20472e8	1.61838	0.809188	+	0.587549i	$0.199907\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6.53503e8	−1.23193	−0.615965	−	0.787773i	$-0.711234\pi$
$311$	−6.53503e8	−1.23193	−0.615965	+	0.787773i	$0.711234\pi$
$312$	0	0
$313$	6.63587e8	1.22319	0.611594	−	0.791172i	$-0.290529\pi$
$313$	6.63587e8	1.22319	0.611594	+	0.791172i	$0.290529\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.54718e8	0.625426	0.312713	−	0.949848i	$-0.398762\pi$
$317$	3.54718e8	0.625426	0.312713	+	0.949848i	$0.398762\pi$
$318$	0	0
$319$	−2.51890e8	−0.434454
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.59826e7	0.0759250
$324$	0	0
$325$	7.69560e8	1.24351
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−3.11007e8	−0.481486
$330$	0	0
$331$	−3.05543e8	−0.463100	−0.231550	−	0.972823i	$-0.574380\pi$
$331$	−3.05543e8	−0.463100	−0.231550	+	0.972823i	$0.574380\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.43201e8	0.353434
$336$	0	0
$337$	3.54965e7	0.0505220	0.0252610	−	0.999681i	$-0.491958\pi$
$337$	3.54965e7	0.0505220	0.0252610	+	0.999681i	$0.491958\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.02172e8	−0.139537
$342$	0	0
$343$	6.56252e8	0.878095
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.90594e8	0.244882	0.122441	−	0.992476i	$-0.460928\pi$
$347$	1.90594e8	0.244882	0.122441	+	0.992476i	$0.460928\pi$
$348$	0	0
$349$	8.60864e8	1.08404	0.542020	−	0.840366i	$-0.317660\pi$
$349$	8.60864e8	1.08404	0.542020	+	0.840366i	$0.317660\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.04544e9	1.26500	0.632498	−	0.774562i	$-0.282030\pi$
$353$	1.04544e9	1.26500	0.632498	+	0.774562i	$0.282030\pi$
$354$	0	0
$355$	2.08955e8	0.247887
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7.63303e8	−0.870696	−0.435348	−	0.900262i	$-0.643375\pi$
$359$	−7.63303e8	−0.870696	−0.435348	+	0.900262i	$0.643375\pi$
$360$	0	0
$361$	−8.76864e8	−0.980973
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1.37787e8	−0.148314
$366$	0	0
$367$	−1.38692e9	−1.46460	−0.732302	−	0.680980i	$-0.761554\pi$
$367$	−1.38692e9	−1.46460	−0.732302	+	0.680980i	$0.761554\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	8.26782e8	0.840586
$372$	0	0
$373$	4.77105e8	0.476029	0.238015	−	0.971262i	$-0.423503\pi$
$373$	4.77105e8	0.476029	0.238015	+	0.971262i	$0.423503\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.07562e9	1.03387
$378$	0	0
$379$	−3.92468e8	−0.370311	−0.185156	−	0.982709i	$-0.559279\pi$
$379$	−3.92468e8	−0.370311	−0.185156	+	0.982709i	$0.559279\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2.10409e9	−1.91368	−0.956839	−	0.290617i	$-0.906139\pi$
$383$	−2.10409e9	−1.91368	−0.956839	+	0.290617i	$0.906139\pi$
$384$	0	0
$385$	−9.43774e7	−0.0842860
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.26019e9	1.08546	0.542730	−	0.839907i	$-0.317391\pi$
$389$	1.26019e9	1.08546	0.542730	+	0.839907i	$0.317391\pi$
$390$	0	0
$391$	−9.11000e8	−0.770725
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3.31121e8	0.270332
$396$	0	0
$397$	−9.81298e8	−0.787107	−0.393554	−	0.919302i	$-0.628754\pi$
$397$	−9.81298e8	−0.787107	−0.393554	+	0.919302i	$0.628754\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−9.09981e8	−0.704737	−0.352369	−	0.935861i	$-0.614624\pi$
$401$	−9.09981e8	−0.704737	−0.352369	+	0.935861i	$0.614624\pi$
$402$	0	0
$403$	4.36293e8	0.332056
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.05867e9	−0.778361
$408$	0	0
$409$	−3.55609e7	−0.0257004	−0.0128502	−	0.999917i	$-0.504090\pi$
$409$	−3.55609e7	−0.0257004	−0.0128502	+	0.999917i	$0.504090\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−4.40509e8	−0.307701
$414$	0	0
$415$	4.41633e8	0.303314
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−2.65360e9	−1.76233	−0.881163	−	0.472813i	$-0.843239\pi$
$419$	−2.65360e9	−1.76233	−0.881163	+	0.472813i	$0.843239\pi$
$420$	0	0
$421$	−1.12113e9	−0.732264	−0.366132	−	0.930563i	$-0.619318\pi$
$421$	−1.12113e9	−0.732264	−0.366132	+	0.930563i	$0.619318\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−7.96121e8	−0.503058
$426$	0	0
$427$	−8.60778e8	−0.535049
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.06344e9	0.639799	0.319900	−	0.947451i	$-0.396351\pi$
$431$	1.06344e9	0.639799	0.319900	+	0.947451i	$0.396351\pi$
$432$	0	0
$433$	−7.05962e8	−0.417901	−0.208951	−	0.977926i	$-0.567005\pi$
$433$	−7.05962e8	−0.417901	−0.208951	+	0.977926i	$0.567005\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.36947e8	−0.193142
$438$	0	0
$439$	1.48506e9	0.837760	0.418880	−	0.908042i	$-0.362423\pi$
$439$	1.48506e9	0.837760	0.418880	+	0.908042i	$0.362423\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.22153e8	0.394654	0.197327	−	0.980338i	$-0.436774\pi$
$443$	7.22153e8	0.394654	0.197327	+	0.980338i	$0.436774\pi$
$444$	0	0
$445$	5.30790e8	0.285537
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.22968e9	0.641109	0.320554	−	0.947230i	$-0.396131\pi$
$449$	1.22968e9	0.641109	0.320554	+	0.947230i	$0.396131\pi$
$450$	0	0
$451$	−3.56899e8	−0.183201
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4.03011e8	0.200575
$456$	0	0
$457$	−8.85551e7	−0.0434017	−0.0217009	−	0.999765i	$-0.506908\pi$
$457$	−8.85551e7	−0.0434017	−0.0217009	+	0.999765i	$0.506908\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−2.10937e8	−0.100277	−0.0501384	−	0.998742i	$-0.515966\pi$
$461$	−2.10937e8	−0.100277	−0.0501384	+	0.998742i	$0.515966\pi$
$462$	0	0
$463$	−3.29775e9	−1.54413	−0.772066	−	0.635543i	$-0.780776\pi$
$463$	−3.29775e9	−1.54413	−0.772066	+	0.635543i	$0.780776\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−8.82873e7	−0.0401134	−0.0200567	−	0.999799i	$-0.506385\pi$
$467$	−8.82873e7	−0.0401134	−0.0200567	+	0.999799i	$0.506385\pi$
$468$	0	0
$469$	−1.35244e9	−0.605357
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.74264e9	−0.757171
$474$	0	0
$475$	−2.94458e8	−0.126065
$476$	0	0
$477$	0	0
$478$	0	0
$479$	4.51507e9	1.87711	0.938557	−	0.345125i	$-0.112164\pi$
$479$	4.51507e9	1.87711	0.938557	+	0.345125i	$0.112164\pi$
$480$	0	0
$481$	4.52075e9	1.85226
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.97392e8	−0.197972
$486$	0	0
$487$	3.31338e9	1.29993	0.649964	−	0.759965i	$-0.274784\pi$
$487$	3.31338e9	1.29993	0.649964	+	0.759965i	$0.274784\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	4.01694e9	1.53147	0.765737	−	0.643154i	$-0.222374\pi$
$491$	4.01694e9	1.53147	0.765737	+	0.643154i	$0.222374\pi$
$492$	0	0
$493$	−1.11275e9	−0.418247
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.16199e9	−0.424577
$498$	0	0
$499$	2.70976e9	0.976290	0.488145	−	0.872763i	$-0.337674\pi$
$499$	2.70976e9	0.976290	0.488145	+	0.872763i	$0.337674\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−3.04579e8	−0.106712	−0.0533558	−	0.998576i	$-0.516992\pi$
$503$	−3.04579e8	−0.106712	−0.0533558	+	0.998576i	$0.516992\pi$
$504$	0	0
$505$	−7.95456e8	−0.274850
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.88202e8	0.0632575	0.0316287	−	0.999500i	$-0.489931\pi$
$509$	1.88202e8	0.0632575	0.0316287	+	0.999500i	$0.489931\pi$
$510$	0	0
$511$	7.66229e8	0.254030
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.36331e8	0.108503
$516$	0	0
$517$	1.72145e9	0.547870
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−4.14963e9	−1.28552	−0.642758	−	0.766069i	$-0.722210\pi$
$521$	−4.14963e9	−1.28552	−0.642758	+	0.766069i	$0.722210\pi$
$522$	0	0
$523$	2.51360e9	0.768318	0.384159	−	0.923267i	$-0.374491\pi$
$523$	2.51360e9	0.768318	0.384159	+	0.923267i	$0.374491\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.51352e8	−0.134332
$528$	0	0
$529$	3.27072e9	0.960613
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.52403e9	0.435962
$534$	0	0
$535$	−5.97780e6	−0.00168773
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.55379e9	−0.427398
$540$	0	0
$541$	−1.32416e9	−0.359543	−0.179772	−	0.983708i	$-0.557536\pi$
$541$	−1.32416e9	−0.359543	−0.179772	+	0.983708i	$0.557536\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7.83789e8	0.207401
$546$	0	0
$547$	5.58047e8	0.145786	0.0728929	−	0.997340i	$-0.476777\pi$
$547$	5.58047e8	0.145786	0.0728929	+	0.997340i	$0.476777\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−4.11567e8	−0.104812
$552$	0	0
$553$	−1.84136e9	−0.463020
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3.30331e9	0.809946	0.404973	−	0.914329i	$-0.367281\pi$
$557$	3.30331e9	0.809946	0.404973	+	0.914329i	$0.367281\pi$
$558$	0	0
$559$	7.44143e9	1.80184
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.22011e8	0.0288152	0.0144076	−	0.999896i	$-0.495414\pi$
$563$	1.22011e8	0.0288152	0.0144076	+	0.999896i	$0.495414\pi$
$564$	0	0
$565$	−7.65790e8	−0.178624
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5.00925e8	−0.113993	−0.0569967	−	0.998374i	$-0.518152\pi$
$569$	−5.00925e8	−0.113993	−0.0569967	+	0.998374i	$0.518152\pi$
$570$	0	0
$571$	−6.98702e9	−1.57060	−0.785300	−	0.619116i	$-0.787491\pi$
$571$	−6.98702e9	−1.57060	−0.785300	+	0.619116i	$0.787491\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	5.83375e9	1.27971
$576$	0	0
$577$	−8.16573e9	−1.76962	−0.884809	−	0.465954i	$-0.845711\pi$
$577$	−8.16573e9	−1.76962	−0.884809	+	0.465954i	$0.845711\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.45591e9	−0.519512
$582$	0	0
$583$	−4.57631e9	−0.956479
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−8.53182e9	−1.74104	−0.870519	−	0.492135i	$-0.836217\pi$
$587$	−8.53182e9	−1.74104	−0.870519	+	0.492135i	$0.836217\pi$
$588$	0	0
$589$	−1.66940e8	−0.0336632
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.71175e9	0.337092	0.168546	−	0.985694i	$-0.446093\pi$
$593$	1.71175e9	0.337092	0.168546	+	0.985694i	$0.446093\pi$
$594$	0	0
$595$	−4.16921e8	−0.0811417
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4.77362e9	0.907516	0.453758	−	0.891125i	$-0.350083\pi$
$599$	4.77362e9	0.907516	0.453758	+	0.891125i	$0.350083\pi$
$600$	0	0
$601$	7.89998e8	0.148445	0.0742224	−	0.997242i	$-0.476353\pi$
$601$	7.89998e8	0.148445	0.0742224	+	0.997242i	$0.476353\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.07556e9	−0.197465
$606$	0	0
$607$	−1.82652e9	−0.331485	−0.165743	−	0.986169i	$-0.553002\pi$
$607$	−1.82652e9	−0.331485	−0.165743	+	0.986169i	$0.553002\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−7.35094e9	−1.30376
$612$	0	0
$613$	−6.90339e9	−1.21046	−0.605231	−	0.796050i	$-0.706919\pi$
$613$	−6.90339e9	−1.21046	−0.605231	+	0.796050i	$0.706919\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5.69235e9	0.975649	0.487825	−	0.872942i	$-0.337791\pi$
$617$	5.69235e9	0.975649	0.487825	+	0.872942i	$0.337791\pi$
$618$	0	0
$619$	−4.28594e9	−0.726321	−0.363161	−	0.931727i	$-0.618302\pi$
$619$	−4.28594e9	−0.726321	−0.363161	+	0.931727i	$0.618302\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2.95171e9	−0.489064
$624$	0	0
$625$	4.57279e9	0.749206
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4.67678e9	−0.749324
$630$	0	0
$631$	−5.61602e8	−0.0889869	−0.0444935	−	0.999010i	$-0.514167\pi$
$631$	−5.61602e8	−0.0889869	−0.0444935	+	0.999010i	$0.514167\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−2.95031e9	−0.457256
$636$	0	0
$637$	6.63501e9	1.01708
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−5.17445e9	−0.775998	−0.387999	−	0.921660i	$-0.626834\pi$
$641$	−5.17445e9	−0.775998	−0.387999	+	0.921660i	$0.626834\pi$
$642$	0	0
$643$	−1.04374e10	−1.54830	−0.774148	−	0.633004i	$-0.781822\pi$
$643$	−1.04374e10	−1.54830	−0.774148	+	0.633004i	$0.781822\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	9.71623e8	0.141037	0.0705185	−	0.997510i	$-0.477535\pi$
$647$	9.71623e8	0.141037	0.0705185	+	0.997510i	$0.477535\pi$
$648$	0	0
$649$	2.43825e9	0.350125
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7.25223e9	1.01924	0.509619	−	0.860400i	$-0.329786\pi$
$653$	7.25223e9	1.01924	0.509619	+	0.860400i	$0.329786\pi$
$654$	0	0
$655$	5.54363e7	0.00770814
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−3.81924e9	−0.519851	−0.259925	−	0.965629i	$-0.583698\pi$
$659$	−3.81924e9	−0.519851	−0.259925	+	0.965629i	$0.583698\pi$
$660$	0	0
$661$	1.07881e10	1.45292	0.726459	−	0.687210i	$-0.241165\pi$
$661$	1.07881e10	1.45292	0.726459	+	0.687210i	$0.241165\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.54205e8	−0.0203339
$666$	0	0
$667$	8.15390e9	1.06396
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4.76448e9	0.608817
$672$	0	0
$673$	−6.34833e9	−0.802798	−0.401399	−	0.915903i	$-0.631476\pi$
$673$	−6.34833e9	−0.802798	−0.401399	+	0.915903i	$0.631476\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8.82566e9	−1.09317	−0.546584	−	0.837404i	$-0.684072\pi$
$677$	−8.82566e9	−1.09317	−0.546584	+	0.837404i	$0.684072\pi$
$678$	0	0
$679$	2.76599e9	0.339083
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−4.92331e9	−0.591268	−0.295634	−	0.955301i	$-0.595531\pi$
$683$	−4.92331e9	−0.591268	−0.295634	+	0.955301i	$0.595531\pi$
$684$	0	0
$685$	2.42590e9	0.288374
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.95418e10	2.27613
$690$	0	0
$691$	−5.68449e9	−0.655418	−0.327709	−	0.944779i	$-0.606277\pi$
$691$	−5.68449e9	−0.655418	−0.327709	+	0.944779i	$0.606277\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.61649e9	−0.295646
$696$	0	0
$697$	−1.57663e9	−0.176366
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.70567e9	0.187017	0.0935085	−	0.995618i	$-0.470192\pi$
$701$	1.70567e9	0.187017	0.0935085	+	0.995618i	$0.470192\pi$
$702$	0	0
$703$	−1.72978e9	−0.187779
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.42351e9	0.470760
$708$	0	0
$709$	4.52189e9	0.476495	0.238248	−	0.971204i	$-0.423427\pi$
$709$	4.52189e9	0.476495	0.238248	+	0.971204i	$0.423427\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3.30738e9	0.341720
$714$	0	0
$715$	−2.23070e9	−0.228229
$716$	0	0
$717$	0	0
$718$	0	0
$719$	3.09206e9	0.310239	0.155120	−	0.987896i	$-0.450424\pi$
$719$	3.09206e9	0.310239	0.155120	+	0.987896i	$0.450424\pi$
$720$	0	0
$721$	−1.87033e9	−0.185842
$722$	0	0
$723$	0	0
$724$	0	0
$725$	7.12568e9	0.694453
$726$	0	0
$727$	1.44622e10	1.39593	0.697965	−	0.716132i	$-0.254089\pi$
$727$	1.44622e10	1.39593	0.697965	+	0.716132i	$0.254089\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−7.69827e9	−0.728924
$732$	0	0
$733$	3.15415e9	0.295814	0.147907	−	0.989001i	$-0.452746\pi$
$733$	3.15415e9	0.295814	0.147907	+	0.989001i	$0.452746\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.48585e9	0.688819
$738$	0	0
$739$	1.54236e10	1.40582	0.702912	−	0.711277i	$-0.251883\pi$
$739$	1.54236e10	1.40582	0.702912	+	0.711277i	$0.251883\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.59520e10	1.42677	0.713385	−	0.700772i	$-0.247161\pi$
$743$	1.59520e10	1.42677	0.713385	+	0.700772i	$0.247161\pi$
$744$	0	0
$745$	9.56141e8	0.0847179
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.32424e7	0.00289072
$750$	0	0
$751$	6.13964e9	0.528936	0.264468	−	0.964395i	$-0.414804\pi$
$751$	6.13964e9	0.528936	0.264468	+	0.964395i	$0.414804\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1.44562e9	−0.122248
$756$	0	0
$757$	−1.42818e10	−1.19660	−0.598299	−	0.801273i	$-0.704157\pi$
$757$	−1.42818e10	−1.19660	−0.598299	+	0.801273i	$0.704157\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.47536e10	−1.21353	−0.606767	−	0.794880i	$-0.707534\pi$
$761$	−1.47536e10	−1.21353	−0.606767	+	0.794880i	$0.707534\pi$
$762$	0	0
$763$	−4.35863e9	−0.355234
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.04118e10	−0.833190
$768$	0	0
$769$	1.97592e10	1.56685	0.783424	−	0.621487i	$-0.213471\pi$
$769$	1.97592e10	1.56685	0.783424	+	0.621487i	$0.213471\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1.01370e10	−0.789374	−0.394687	−	0.918816i	$-0.629147\pi$
$773$	−1.01370e10	−0.789374	−0.394687	+	0.918816i	$0.629147\pi$
$774$	0	0
$775$	2.89031e9	0.223043
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5.83142e8	−0.0441970
$780$	0	0
$781$	6.43174e9	0.483114
$782$	0	0
$783$	0	0
$784$	0	0
$785$	5.20420e8	0.0383981
$786$	0	0
$787$	−1.27882e10	−0.935188	−0.467594	−	0.883943i	$-0.654879\pi$
$787$	−1.27882e10	−0.935188	−0.467594	+	0.883943i	$0.654879\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	4.25854e9	0.305945
$792$	0	0
$793$	−2.03453e10	−1.44880
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7.38617e9	0.516791	0.258396	−	0.966039i	$-0.416806\pi$
$797$	7.38617e9	0.516791	0.258396	+	0.966039i	$0.416806\pi$
$798$	0	0
$799$	7.60466e9	0.527431
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−4.24114e9	−0.289054
$804$	0	0
$805$	3.05508e9	0.206413
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.53742e10	−1.02087	−0.510437	−	0.859915i	$-0.670516\pi$
$809$	−1.53742e10	−1.02087	−0.510437	+	0.859915i	$0.670516\pi$
$810$	0	0
$811$	9.77882e9	0.643744	0.321872	−	0.946783i	$-0.395688\pi$
$811$	9.77882e9	0.643744	0.321872	+	0.946783i	$0.395688\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	6.59472e9	0.426722
$816$	0	0
$817$	−2.84733e9	−0.182667
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.83470e10	−1.15708	−0.578540	−	0.815654i	$-0.696377\pi$
$821$	−1.83470e10	−1.15708	−0.578540	+	0.815654i	$0.696377\pi$
$822$	0	0
$823$	−3.16960e10	−1.98201	−0.991004	−	0.133829i	$-0.957273\pi$
$823$	−3.16960e10	−1.98201	−0.991004	+	0.133829i	$0.957273\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.12845e9	0.376774	0.188387	−	0.982095i	$-0.439674\pi$
$827$	6.12845e9	0.376774	0.188387	+	0.982095i	$0.439674\pi$
$828$	0	0
$829$	−1.24652e10	−0.759904	−0.379952	−	0.925006i	$-0.624060\pi$
$829$	−1.24652e10	−0.759904	−0.379952	+	0.925006i	$0.624060\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−6.86402e9	−0.411454
$834$	0	0
$835$	−9.41091e9	−0.559409
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.82237e10	1.06530	0.532648	−	0.846337i	$-0.321197\pi$
$839$	1.82237e10	1.06530	0.532648	+	0.846337i	$0.321197\pi$
$840$	0	0
$841$	−7.29024e9	−0.422625
$842$	0	0
$843$	0	0
$844$	0	0
$845$	4.38017e9	0.249743
$846$	0	0
$847$	5.98117e9	0.338216
$848$	0	0
$849$	0	0
$850$	0	0
$851$	3.42701e10	1.90617
$852$	0	0
$853$	−2.48619e10	−1.37155	−0.685777	−	0.727812i	$-0.740537\pi$
$853$	−2.48619e10	−1.37155	−0.685777	+	0.727812i	$0.740537\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.96761e10	−1.61055	−0.805275	−	0.592902i	$-0.797982\pi$
$857$	−2.96761e10	−1.61055	−0.805275	+	0.592902i	$0.797982\pi$
$858$	0	0
$859$	1.14772e10	0.617819	0.308910	−	0.951091i	$-0.400036\pi$
$859$	1.14772e10	0.617819	0.308910	+	0.951091i	$0.400036\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−2.13485e10	−1.13066	−0.565328	−	0.824866i	$-0.691250\pi$
$863$	−2.13485e10	−1.13066	−0.565328	+	0.824866i	$0.691250\pi$
$864$	0	0
$865$	5.19679e9	0.273010
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.01921e10	0.526858
$870$	0	0
$871$	−3.19661e10	−1.63918
$872$	0	0
$873$	0	0
$874$	0	0
$875$	5.59108e9	0.282142
$876$	0	0
$877$	−7.92753e9	−0.396862	−0.198431	−	0.980115i	$-0.563585\pi$
$877$	−7.92753e9	−0.396862	−0.198431	+	0.980115i	$0.563585\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−7.32045e9	−0.360680	−0.180340	−	0.983604i	$-0.557720\pi$
$881$	−7.32045e9	−0.360680	−0.180340	+	0.983604i	$0.557720\pi$
$882$	0	0
$883$	−3.54988e9	−0.173521	−0.0867604	−	0.996229i	$-0.527651\pi$
$883$	−3.54988e9	−0.173521	−0.0867604	+	0.996229i	$0.527651\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−5.80634e9	−0.279364	−0.139682	−	0.990196i	$-0.544608\pi$
$887$	−5.80634e9	−0.279364	−0.139682	+	0.990196i	$0.544608\pi$
$888$	0	0
$889$	1.64066e10	0.783182
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2.81270e9	0.132173
$894$	0	0
$895$	9.28466e8	0.0432898
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.03982e9	0.185440
$900$	0	0
$901$	−2.02163e10	−0.920798
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4.28573e8	−0.0192201
$906$	0	0
$907$	1.78240e10	0.793196	0.396598	−	0.917992i	$-0.370191\pi$
$907$	1.78240e10	0.793196	0.396598	+	0.917992i	$0.370191\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−1.87703e10	−0.822538	−0.411269	−	0.911514i	$-0.634914\pi$
$911$	−1.87703e10	−0.822538	−0.411269	+	0.911514i	$0.634914\pi$
$912$	0	0
$913$	1.35937e10	0.591138
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3.08280e8	−0.0132024
$918$	0	0
$919$	3.75844e10	1.59736	0.798681	−	0.601754i	$-0.205531\pi$
$919$	3.75844e10	1.59736	0.798681	+	0.601754i	$0.205531\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−2.74648e10	−1.14966
$924$	0	0
$925$	2.99486e10	1.24417
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.92372e10	0.787205	0.393602	−	0.919281i	$-0.371229\pi$
$929$	1.92372e10	0.787205	0.393602	+	0.919281i	$0.371229\pi$
$930$	0	0
$931$	−2.53876e9	−0.103109
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.30769e9	0.0923289
$936$	0	0
$937$	1.04732e9	0.0415900	0.0207950	−	0.999784i	$-0.493380\pi$
$937$	1.04732e9	0.0415900	0.0207950	+	0.999784i	$0.493380\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	7.97861e9	0.312150	0.156075	−	0.987745i	$-0.450116\pi$
$941$	7.97861e9	0.312150	0.156075	+	0.987745i	$0.450116\pi$
$942$	0	0
$943$	1.15531e10	0.448650
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.26943e9	0.163360	0.0816799	−	0.996659i	$-0.473971\pi$
$947$	4.26943e9	0.163360	0.0816799	+	0.996659i	$0.473971\pi$
$948$	0	0
$949$	1.81106e10	0.687860
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.06048e10	0.396897	0.198449	−	0.980111i	$-0.436410\pi$
$953$	1.06048e10	0.396897	0.198449	+	0.980111i	$0.436410\pi$
$954$	0	0
$955$	6.97247e9	0.259045
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−1.34904e10	−0.493922
$960$	0	0
$961$	−2.58740e10	−0.940441
$962$	0	0
$963$	0	0
$964$	0	0
$965$	9.43750e9	0.338074
$966$	0	0
$967$	−1.65090e10	−0.587123	−0.293562	−	0.955940i	$-0.594841\pi$
$967$	−1.65090e10	−0.587123	−0.293562	+	0.955940i	$0.594841\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	2.46094e10	0.862649	0.431324	−	0.902197i	$-0.358046\pi$
$971$	2.46094e10	0.862649	0.431324	+	0.902197i	$0.358046\pi$
$972$	0	0
$973$	1.45503e10	0.506379
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.53886e10	0.870979	0.435489	−	0.900194i	$-0.356575\pi$
$977$	2.53886e10	0.870979	0.435489	+	0.900194i	$0.356575\pi$
$978$	0	0
$979$	1.63380e10	0.556492
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−1.87585e10	−0.629884	−0.314942	−	0.949111i	$-0.601985\pi$
$983$	−1.87585e10	−0.629884	−0.314942	+	0.949111i	$0.601985\pi$
$984$	0	0
$985$	1.13588e10	0.378709
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5.64107e10	1.85428
$990$	0	0
$991$	−3.59792e9	−0.117434	−0.0587170	−	0.998275i	$-0.518701\pi$
$991$	−3.59792e9	−0.117434	−0.0587170	+	0.998275i	$0.518701\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1.80083e9	−0.0579552
$996$	0	0
$997$	−1.34287e10	−0.429143	−0.214571	−	0.976708i	$-0.568835\pi$
$997$	−1.34287e10	−0.429143	−0.214571	+	0.976708i	$0.568835\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.8.a.d.1.1		1
3.2	odd	2		8.8.a.a.1.1	✓	1
4.3	odd	2		144.8.a.g.1.1		1
8.3	odd	2		576.8.a.k.1.1		1
8.5	even	2		576.8.a.j.1.1		1
12.11	even	2		16.8.a.c.1.1		1
15.2	even	4		200.8.c.a.49.2		2
15.8	even	4		200.8.c.a.49.1		2
15.14	odd	2		200.8.a.i.1.1		1
21.20	even	2		392.8.a.d.1.1		1
24.5	odd	2		64.8.a.g.1.1		1
24.11	even	2		64.8.a.a.1.1		1
48.5	odd	4		256.8.b.e.129.2		2
48.11	even	4		256.8.b.c.129.1		2
48.29	odd	4		256.8.b.e.129.1		2
48.35	even	4		256.8.b.c.129.2		2
60.23	odd	4		400.8.c.b.49.2		2
60.47	odd	4		400.8.c.b.49.1		2
60.59	even	2		400.8.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.8.a.a.1.1	✓	1	3.2	odd	2
16.8.a.c.1.1		1	12.11	even	2
64.8.a.a.1.1		1	24.11	even	2
64.8.a.g.1.1		1	24.5	odd	2
72.8.a.d.1.1		1	1.1	even	1	trivial
144.8.a.g.1.1		1	4.3	odd	2
200.8.a.i.1.1		1	15.14	odd	2
200.8.c.a.49.1		2	15.8	even	4
200.8.c.a.49.2		2	15.2	even	4
256.8.b.c.129.1		2	48.11	even	4
256.8.b.c.129.2		2	48.35	even	4
256.8.b.e.129.1		2	48.29	odd	4
256.8.b.e.129.2		2	48.5	odd	4
392.8.a.d.1.1		1	21.20	even	2
400.8.a.b.1.1		1	60.59	even	2
400.8.c.b.49.1		2	60.47	odd	4
400.8.c.b.49.2		2	60.23	odd	4
576.8.a.j.1.1		1	8.5	even	2
576.8.a.k.1.1		1	8.3	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:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	72.a (trivial)

Introduction

L-functions

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