LMFDB - Embedded newform 45.10.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,10,Mod(1,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("45.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	45.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:	$23.1766126274$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	45.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} -496.000 q^{4} -625.000 q^{5} -7680.00 q^{7} -4032.00 q^{8} +O(q^{10})$

\(q+4.00000 q^{2} -496.000 q^{4} -625.000 q^{5} -7680.00 q^{7} -4032.00 q^{8} -2500.00 q^{10} +86404.0 q^{11} -149978. q^{13} -30720.0 q^{14} +237824. q^{16} +207622. q^{17} +716284. q^{19} +310000. q^{20} +345616. q^{22} -1.36992e6 q^{23} +390625. q^{25} -599912. q^{26} +3.80928e6 q^{28} +3.19440e6 q^{29} -2.34900e6 q^{31} +3.01568e6 q^{32} +830488. q^{34} +4.80000e6 q^{35} +1.87357e7 q^{37} +2.86514e6 q^{38} +2.52000e6 q^{40} +2.92826e7 q^{41} -1.51672e6 q^{43} -4.28564e7 q^{44} -5.47968e6 q^{46} -615752. q^{47} +1.86288e7 q^{49} +1.56250e6 q^{50} +7.43891e7 q^{52} -4.74743e6 q^{53} -5.40025e7 q^{55} +3.09658e7 q^{56} +1.27776e7 q^{58} -6.06161e7 q^{59} -1.26746e8 q^{61} -9.39600e6 q^{62} -1.09703e8 q^{64} +9.37362e7 q^{65} -1.11183e8 q^{67} -1.02981e8 q^{68} +1.92000e7 q^{70} +1.75552e8 q^{71} -6.12334e7 q^{73} +7.49428e7 q^{74} -3.55277e8 q^{76} -6.63583e8 q^{77} +2.34431e8 q^{79} -1.48640e8 q^{80} +1.17131e8 q^{82} -1.18910e8 q^{83} -1.29764e8 q^{85} -6.06690e6 q^{86} -3.48381e8 q^{88} +3.16534e8 q^{89} +1.15183e9 q^{91} +6.79480e8 q^{92} -2.46301e6 q^{94} -4.47678e8 q^{95} +2.42912e8 q^{97} +7.45152e7 q^{98} +O(q^{100})\)

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 $

$2$	4.00000	0.176777	0.0883883	−	0.996086i	$-0.471828\pi$
$2$	4.00000	0.176777	0.0883883	+	0.996086i	$0.471828\pi$
$3$	0	0
$3$	0	0
$4$	−496.000	−0.968750
$5$	−625.000	−0.447214
$5$	−625.000	−0.447214
$6$	0	0
$7$	−7680.00	−1.20898	−0.604491	−	0.796612i	$-0.706624\pi$
$7$	−7680.00	−1.20898	−0.604491	+	0.796612i	$0.706624\pi$
$8$	−4032.00	−0.348029
$9$	0	0
$10$	−2500.00	−0.0790569
$11$	86404.0	1.77937	0.889686	−	0.456573i	$-0.150923\pi$
$11$	86404.0	1.77937	0.889686	+	0.456573i	$0.150923\pi$
$12$	0	0
$13$	−149978.	−1.45641	−0.728203	−	0.685361i	$-0.759644\pi$
$13$	−149978.	−1.45641	−0.728203	+	0.685361i	$0.759644\pi$
$14$	−30720.0	−0.213720
$15$	0	0
$16$	237824.	0.907227
$17$	207622.	0.602911	0.301456	−	0.953480i	$-0.402528\pi$
$17$	207622.	0.602911	0.301456	+	0.953480i	$0.402528\pi$
$18$	0	0
$19$	716284.	1.26094	0.630469	−	0.776214i	$-0.282862\pi$
$19$	716284.	1.26094	0.630469	+	0.776214i	$0.282862\pi$
$20$	310000.	0.433238
$21$	0	0
$22$	345616.	0.314552
$23$	−1.36992e6	−1.02075	−0.510376	−	0.859952i	$-0.670494\pi$
$23$	−1.36992e6	−1.02075	−0.510376	+	0.859952i	$0.670494\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	−599912.	−0.257459
$27$	0	0
$28$	3.80928e6	1.17120
$29$	3.19440e6	0.838684	0.419342	−	0.907828i	$-0.362261\pi$
$29$	3.19440e6	0.838684	0.419342	+	0.907828i	$0.362261\pi$
$30$	0	0
$31$	−2.34900e6	−0.456831	−0.228415	−	0.973564i	$-0.573354\pi$
$31$	−2.34900e6	−0.456831	−0.228415	+	0.973564i	$0.573354\pi$
$32$	3.01568e6	0.508406
$33$	0	0
$34$	830488.	0.106581
$35$	4.80000e6	0.540673
$36$	0	0
$37$	1.87357e7	1.64347	0.821736	−	0.569868i	$-0.193006\pi$
$37$	1.87357e7	1.64347	0.821736	+	0.569868i	$0.193006\pi$
$38$	2.86514e6	0.222905
$39$	0	0
$40$	2.52000e6	0.155643
$41$	2.92826e7	1.61839	0.809194	−	0.587541i	$-0.199904\pi$
$41$	2.92826e7	1.61839	0.809194	+	0.587541i	$0.199904\pi$
$42$	0	0
$43$	−1.51672e6	−0.0676548	−0.0338274	−	0.999428i	$-0.510770\pi$
$43$	−1.51672e6	−0.0676548	−0.0338274	+	0.999428i	$0.510770\pi$
$44$	−4.28564e7	−1.72377
$45$	0	0
$46$	−5.47968e6	−0.180445
$47$	−615752.	−0.0184063	−0.00920313	−	0.999958i	$-0.502929\pi$
$47$	−615752.	−0.0184063	−0.00920313	+	0.999958i	$0.502929\pi$
$48$	0	0
$49$	1.86288e7	0.461639
$50$	1.56250e6	0.0353553
$51$	0	0
$52$	7.43891e7	1.41089
$53$	−4.74743e6	−0.0826451	−0.0413226	−	0.999146i	$-0.513157\pi$
$53$	−4.74743e6	−0.0826451	−0.0413226	+	0.999146i	$0.513157\pi$
$54$	0	0
$55$	−5.40025e7	−0.795759
$56$	3.09658e7	0.420761
$57$	0	0
$58$	1.27776e7	0.148260
$59$	−6.06161e7	−0.651259	−0.325630	−	0.945497i	$-0.605576\pi$
$59$	−6.06161e7	−0.651259	−0.325630	+	0.945497i	$0.605576\pi$
$60$	0	0
$61$	−1.26746e8	−1.17206	−0.586029	−	0.810290i	$-0.699309\pi$
$61$	−1.26746e8	−1.17206	−0.586029	+	0.810290i	$0.699309\pi$
$62$	−9.39600e6	−0.0807570
$63$	0	0
$64$	−1.09703e8	−0.817352
$65$	9.37362e7	0.651325
$66$	0	0
$67$	−1.11183e8	−0.674063	−0.337031	−	0.941493i	$-0.609423\pi$
$67$	−1.11183e8	−0.674063	−0.337031	+	0.941493i	$0.609423\pi$
$68$	−1.02981e8	−0.584070
$69$	0	0
$70$	1.92000e7	0.0955785
$71$	1.75552e8	0.819865	0.409932	−	0.912116i	$-0.365552\pi$
$71$	1.75552e8	0.819865	0.409932	+	0.912116i	$0.365552\pi$
$72$	0	0
$73$	−6.12334e7	−0.252369	−0.126184	−	0.992007i	$-0.540273\pi$
$73$	−6.12334e7	−0.252369	−0.126184	+	0.992007i	$0.540273\pi$
$74$	7.49428e7	0.290528
$75$	0	0
$76$	−3.55277e8	−1.22153
$77$	−6.63583e8	−2.15123
$78$	0	0
$79$	2.34431e8	0.677163	0.338582	−	0.940937i	$-0.390053\pi$
$79$	2.34431e8	0.677163	0.338582	+	0.940937i	$0.390053\pi$
$80$	−1.48640e8	−0.405724
$81$	0	0
$82$	1.17131e8	0.286093
$83$	−1.18910e8	−0.275023	−0.137511	−	0.990500i	$-0.543910\pi$
$83$	−1.18910e8	−0.275023	−0.137511	+	0.990500i	$0.543910\pi$
$84$	0	0
$85$	−1.29764e8	−0.269630
$86$	−6.06690e6	−0.0119598
$87$	0	0
$88$	−3.48381e8	−0.619273
$89$	3.16534e8	0.534768	0.267384	−	0.963590i	$-0.413841\pi$
$89$	3.16534e8	0.534768	0.267384	+	0.963590i	$0.413841\pi$
$90$	0	0
$91$	1.15183e9	1.76077
$92$	6.79480e8	0.988853
$93$	0	0
$94$	−2.46301e6	−0.00325380
$95$	−4.47678e8	−0.563909
$96$	0	0
$97$	2.42912e8	0.278597	0.139299	−	0.990250i	$-0.455515\pi$
$97$	2.42912e8	0.278597	0.139299	+	0.990250i	$0.455515\pi$
$98$	7.45152e7	0.0816070
$99$	0	0
$100$	−1.93750e8	−0.193750
$101$	6.53803e8	0.625173	0.312587	−	0.949889i	$-0.398805\pi$
$101$	6.53803e8	0.625173	0.312587	+	0.949889i	$0.398805\pi$
$102$	0	0
$103$	1.40420e9	1.22931	0.614656	−	0.788795i	$-0.289295\pi$
$103$	1.40420e9	1.22931	0.614656	+	0.788795i	$0.289295\pi$
$104$	6.04711e8	0.506872
$105$	0	0
$106$	−1.89897e7	−0.0146097
$107$	1.83854e9	1.35595	0.677977	−	0.735083i	$-0.262857\pi$
$107$	1.83854e9	1.35595	0.677977	+	0.735083i	$0.262857\pi$
$108$	0	0
$109$	−9.33452e8	−0.633392	−0.316696	−	0.948527i	$-0.602574\pi$
$109$	−9.33452e8	−0.633392	−0.316696	+	0.948527i	$0.602574\pi$
$110$	−2.16010e8	−0.140672
$111$	0	0
$112$	−1.82649e9	−1.09682
$113$	9.28534e7	0.0535728	0.0267864	−	0.999641i	$-0.491473\pi$
$113$	9.28534e7	0.0535728	0.0267864	+	0.999641i	$0.491473\pi$
$114$	0	0
$115$	8.56200e8	0.456494
$116$	−1.58442e9	−0.812476
$117$	0	0
$118$	−2.42464e8	−0.115127
$119$	−1.59454e9	−0.728909
$120$	0	0
$121$	5.10770e9	2.16616
$122$	−5.06983e8	−0.207192
$123$	0	0
$124$	1.16510e9	0.442555
$125$	−2.44141e8	−0.0894427
$126$	0	0
$127$	1.73819e9	0.592900	0.296450	−	0.955048i	$-0.404197\pi$
$127$	1.73819e9	0.592900	0.296450	+	0.955048i	$0.404197\pi$
$128$	−1.98284e9	−0.652894
$129$	0	0
$130$	3.74945e8	0.115139
$131$	2.49730e9	0.740882	0.370441	−	0.928856i	$-0.379207\pi$
$131$	2.49730e9	0.740882	0.370441	+	0.928856i	$0.379207\pi$
$132$	0	0
$133$	−5.50106e9	−1.52445
$134$	−4.44731e8	−0.119159
$135$	0	0
$136$	−8.37132e8	−0.209831
$137$	7.96226e9	1.93105	0.965526	−	0.260306i	$-0.0838234\pi$
$137$	7.96226e9	1.93105	0.965526	+	0.260306i	$0.0838234\pi$
$138$	0	0
$139$	−2.85565e9	−0.648842	−0.324421	−	0.945913i	$-0.605169\pi$
$139$	−2.85565e9	−0.648842	−0.324421	+	0.945913i	$0.605169\pi$
$140$	−2.38080e9	−0.523777
$141$	0	0
$142$	7.02206e8	0.144933
$143$	−1.29587e10	−2.59149
$144$	0	0
$145$	−1.99650e9	−0.375071
$146$	−2.44933e8	−0.0446129
$147$	0	0
$148$	−9.29291e9	−1.59211
$149$	9.63383e9	1.60126	0.800628	−	0.599161i	$-0.204499\pi$
$149$	9.63383e9	1.60126	0.800628	+	0.599161i	$0.204499\pi$
$150$	0	0
$151$	−5.38292e9	−0.842601	−0.421300	−	0.906921i	$-0.638426\pi$
$151$	−5.38292e9	−0.842601	−0.421300	+	0.906921i	$0.638426\pi$
$152$	−2.88806e9	−0.438843
$153$	0	0
$154$	−2.65433e9	−0.380287
$155$	1.46812e9	0.204301
$156$	0	0
$157$	5.19434e8	0.0682310	0.0341155	−	0.999418i	$-0.489139\pi$
$157$	5.19434e8	0.0682310	0.0341155	+	0.999418i	$0.489139\pi$
$158$	9.37725e8	0.119707
$159$	0	0
$160$	−1.88480e9	−0.227366
$161$	1.05210e10	1.23407
$162$	0	0
$163$	9.41239e9	1.04437	0.522187	−	0.852831i	$-0.325116\pi$
$163$	9.41239e9	1.04437	0.522187	+	0.852831i	$0.325116\pi$
$164$	−1.45242e10	−1.56781
$165$	0	0
$166$	−4.75642e8	−0.0486176
$167$	−9.37241e9	−0.932453	−0.466227	−	0.884665i	$-0.654387\pi$
$167$	−9.37241e9	−0.932453	−0.466227	+	0.884665i	$0.654387\pi$
$168$	0	0
$169$	1.18889e10	1.12112
$170$	−5.19055e8	−0.0476643
$171$	0	0
$172$	7.52295e8	0.0655406
$173$	−1.23573e10	−1.04886	−0.524428	−	0.851455i	$-0.675721\pi$
$173$	−1.23573e10	−1.04886	−0.524428	+	0.851455i	$0.675721\pi$
$174$	0	0
$175$	−3.00000e9	−0.241797
$176$	2.05489e10	1.61429
$177$	0	0
$178$	1.26614e9	0.0945346
$179$	6.66040e8	0.0484910	0.0242455	−	0.999706i	$-0.492282\pi$
$179$	6.66040e8	0.0484910	0.0242455	+	0.999706i	$0.492282\pi$
$180$	0	0
$181$	5.27207e9	0.365113	0.182557	−	0.983195i	$-0.441563\pi$
$181$	5.27207e9	0.365113	0.182557	+	0.983195i	$0.441563\pi$
$182$	4.60732e9	0.311263
$183$	0	0
$184$	5.52352e9	0.355251
$185$	−1.17098e10	−0.734983
$186$	0	0
$187$	1.79394e10	1.07280
$188$	3.05413e8	0.0178311
$189$	0	0
$190$	−1.79071e9	−0.0996860
$191$	2.93896e10	1.59788	0.798939	−	0.601412i	$-0.205395\pi$
$191$	2.93896e10	1.59788	0.798939	+	0.601412i	$0.205395\pi$
$192$	0	0
$193$	−1.48746e10	−0.771681	−0.385841	−	0.922565i	$-0.626089\pi$
$193$	−1.48746e10	−0.771681	−0.385841	+	0.922565i	$0.626089\pi$
$194$	9.71649e8	0.0492495
$195$	0	0
$196$	−9.23988e9	−0.447213
$197$	−4.98675e9	−0.235895	−0.117948	−	0.993020i	$-0.537632\pi$
$197$	−4.98675e9	−0.235895	−0.117948	+	0.993020i	$0.537632\pi$
$198$	0	0
$199$	1.45527e10	0.657816	0.328908	−	0.944362i	$-0.393319\pi$
$199$	1.45527e10	0.657816	0.328908	+	0.944362i	$0.393319\pi$
$200$	−1.57500e9	−0.0696058
$201$	0	0
$202$	2.61521e9	0.110516
$203$	−2.45330e10	−1.01395
$204$	0	0
$205$	−1.83016e10	−0.723765
$206$	5.61681e9	0.217314
$207$	0	0
$208$	−3.56684e10	−1.32129
$209$	6.18898e10	2.24368
$210$	0	0
$211$	5.15407e10	1.79011	0.895054	−	0.445959i	$-0.147137\pi$
$211$	5.15407e10	1.79011	0.895054	+	0.445959i	$0.147137\pi$
$212$	2.35473e9	0.0800624
$213$	0	0
$214$	7.35414e9	0.239701
$215$	9.47952e8	0.0302561
$216$	0	0
$217$	1.80403e10	0.552300
$218$	−3.73381e9	−0.111969
$219$	0	0
$220$	2.67852e10	0.770892
$221$	−3.11387e10	−0.878083
$222$	0	0
$223$	4.61272e10	1.24907	0.624533	−	0.780998i	$-0.285289\pi$
$223$	4.61272e10	1.24907	0.624533	+	0.780998i	$0.285289\pi$
$224$	−2.31604e10	−0.614654
$225$	0	0
$226$	3.71414e8	0.00947043
$227$	3.75833e10	0.939460	0.469730	−	0.882810i	$-0.344351\pi$
$227$	3.75833e10	0.939460	0.469730	+	0.882810i	$0.344351\pi$
$228$	0	0
$229$	−6.41082e10	−1.54047	−0.770236	−	0.637759i	$-0.779862\pi$
$229$	−6.41082e10	−1.54047	−0.770236	+	0.637759i	$0.779862\pi$
$230$	3.42480e9	0.0806975
$231$	0	0
$232$	−1.28798e10	−0.291887
$233$	−6.96578e10	−1.54835	−0.774174	−	0.632973i	$-0.781834\pi$
$233$	−6.96578e10	−1.54835	−0.774174	+	0.632973i	$0.781834\pi$
$234$	0	0
$235$	3.84845e8	0.00823153
$236$	3.00656e10	0.630907
$237$	0	0
$238$	−6.37815e9	−0.128854
$239$	−6.65825e10	−1.31999	−0.659993	−	0.751272i	$-0.729441\pi$
$239$	−6.65825e10	−1.31999	−0.659993	+	0.751272i	$0.729441\pi$
$240$	0	0
$241$	−4.41659e10	−0.843354	−0.421677	−	0.906746i	$-0.638558\pi$
$241$	−4.41659e10	−0.843354	−0.421677	+	0.906746i	$0.638558\pi$
$242$	2.04308e10	0.382927
$243$	0	0
$244$	6.28659e10	1.13543
$245$	−1.16430e10	−0.206451
$246$	0	0
$247$	−1.07427e11	−1.83644
$248$	9.47117e9	0.158990
$249$	0	0
$250$	−9.76562e8	−0.0158114
$251$	−8.36236e10	−1.32983	−0.664916	−	0.746918i	$-0.731533\pi$
$251$	−8.36236e10	−1.32983	−0.664916	+	0.746918i	$0.731533\pi$
$252$	0	0
$253$	−1.18367e11	−1.81630
$254$	6.95277e9	0.104811
$255$	0	0
$256$	4.82367e10	0.701936
$257$	8.65274e10	1.23724	0.618621	−	0.785690i	$-0.287692\pi$
$257$	8.65274e10	1.23724	0.618621	+	0.785690i	$0.287692\pi$
$258$	0	0
$259$	−1.43890e11	−1.98693
$260$	−4.64932e10	−0.630971
$261$	0	0
$262$	9.98919e9	0.130971
$263$	9.61535e10	1.23927	0.619633	−	0.784892i	$-0.287282\pi$
$263$	9.61535e10	1.23927	0.619633	+	0.784892i	$0.287282\pi$
$264$	0	0
$265$	2.96714e9	0.0369600
$266$	−2.20042e10	−0.269488
$267$	0	0
$268$	5.51466e10	0.652998
$269$	1.09505e10	0.127511	0.0637557	−	0.997966i	$-0.479692\pi$
$269$	1.09505e10	0.127511	0.0637557	+	0.997966i	$0.479692\pi$
$270$	0	0
$271$	7.80287e10	0.878805	0.439403	−	0.898290i	$-0.355190\pi$
$271$	7.80287e10	0.878805	0.439403	+	0.898290i	$0.355190\pi$
$272$	4.93775e10	0.546977
$273$	0	0
$274$	3.18491e10	0.341365
$275$	3.37516e10	0.355874
$276$	0	0
$277$	6.56840e10	0.670349	0.335174	−	0.942156i	$-0.391205\pi$
$277$	6.56840e10	0.670349	0.335174	+	0.942156i	$0.391205\pi$
$278$	−1.14226e10	−0.114700
$279$	0	0
$280$	−1.93536e10	−0.188170
$281$	6.44906e10	0.617046	0.308523	−	0.951217i	$-0.400165\pi$
$281$	6.44906e10	0.617046	0.308523	+	0.951217i	$0.400165\pi$
$282$	0	0
$283$	9.63133e10	0.892580	0.446290	−	0.894888i	$-0.352745\pi$
$283$	9.63133e10	0.892580	0.446290	+	0.894888i	$0.352745\pi$
$284$	−8.70736e10	−0.794244
$285$	0	0
$286$	−5.18348e10	−0.458115
$287$	−2.24891e11	−1.95660
$288$	0	0
$289$	−7.54810e10	−0.636498
$290$	−7.98600e9	−0.0663038
$291$	0	0
$292$	3.03717e10	0.244482
$293$	−8.16308e10	−0.647068	−0.323534	−	0.946217i	$-0.604871\pi$
$293$	−8.16308e10	−0.647068	−0.323534	+	0.946217i	$0.604871\pi$
$294$	0	0
$295$	3.78850e10	0.291252
$296$	−7.55424e10	−0.571976
$297$	0	0
$298$	3.85353e10	0.283065
$299$	2.05458e11	1.48663
$300$	0	0
$301$	1.16484e10	0.0817935
$302$	−2.15317e10	−0.148952
$303$	0	0
$304$	1.70350e11	1.14396
$305$	7.92161e10	0.524160
$306$	0	0
$307$	−2.95582e10	−0.189914	−0.0949568	−	0.995481i	$-0.530271\pi$
$307$	−2.95582e10	−0.189914	−0.0949568	+	0.995481i	$0.530271\pi$
$308$	3.29137e11	2.08400
$309$	0	0
$310$	5.87250e9	0.0361156
$311$	3.99071e10	0.241896	0.120948	−	0.992659i	$-0.461407\pi$
$311$	3.99071e10	0.241896	0.120948	+	0.992659i	$0.461407\pi$
$312$	0	0
$313$	1.85371e11	1.09167	0.545836	−	0.837892i	$-0.316212\pi$
$313$	1.85371e11	1.09167	0.545836	+	0.837892i	$0.316212\pi$
$314$	2.07774e9	0.0120617
$315$	0	0
$316$	−1.16278e11	−0.656002
$317$	−2.68895e11	−1.49560	−0.747800	−	0.663924i	$-0.768890\pi$
$317$	−2.68895e11	−1.49560	−0.747800	+	0.663924i	$0.768890\pi$
$318$	0	0
$319$	2.76009e11	1.49233
$320$	6.85645e10	0.365531
$321$	0	0
$322$	4.20839e10	0.218155
$323$	1.48716e11	0.760234
$324$	0	0
$325$	−5.85852e10	−0.291281
$326$	3.76496e10	0.184621
$327$	0	0
$328$	−1.18068e11	−0.563246
$329$	4.72898e9	0.0222528
$330$	0	0
$331$	−4.29099e11	−1.96486	−0.982430	−	0.186629i	$-0.940244\pi$
$331$	−4.29099e11	−1.96486	−0.982430	+	0.186629i	$0.940244\pi$
$332$	5.89796e10	0.266428
$333$	0	0
$334$	−3.74896e10	−0.164836
$335$	6.94892e10	0.301450
$336$	0	0
$337$	−2.02598e10	−0.0855657	−0.0427828	−	0.999084i	$-0.513622\pi$
$337$	−2.02598e10	−0.0855657	−0.0427828	+	0.999084i	$0.513622\pi$
$338$	4.75556e10	0.198188
$339$	0	0
$340$	6.43628e10	0.261204
$341$	−2.02963e11	−0.812872
$342$	0	0
$343$	1.66847e11	0.650869
$344$	6.11543e9	0.0235458
$345$	0	0
$346$	−4.94292e10	−0.185413
$347$	−2.92783e10	−0.108409	−0.0542043	−	0.998530i	$-0.517262\pi$
$347$	−2.92783e10	−0.108409	−0.0542043	+	0.998530i	$0.517262\pi$
$348$	0	0
$349$	7.05132e10	0.254423	0.127211	−	0.991876i	$-0.459397\pi$
$349$	7.05132e10	0.254423	0.127211	+	0.991876i	$0.459397\pi$
$350$	−1.20000e10	−0.0427440
$351$	0	0
$352$	2.60567e11	0.904643
$353$	6.57350e10	0.225325	0.112663	−	0.993633i	$-0.464062\pi$
$353$	6.57350e10	0.225325	0.112663	+	0.993633i	$0.464062\pi$
$354$	0	0
$355$	−1.09720e11	−0.366655
$356$	−1.57001e11	−0.518057
$357$	0	0
$358$	2.66416e9	0.00857209
$359$	−5.81702e11	−1.84831	−0.924157	−	0.382013i	$-0.875231\pi$
$359$	−5.81702e11	−1.84831	−0.924157	+	0.382013i	$0.875231\pi$
$360$	0	0
$361$	1.90375e11	0.589967
$362$	2.10883e10	0.0645435
$363$	0	0
$364$	−5.71308e11	−1.70575
$365$	3.82708e10	0.112863
$366$	0	0
$367$	−4.17070e11	−1.20008	−0.600042	−	0.799969i	$-0.704849\pi$
$367$	−4.17070e11	−1.20008	−0.600042	+	0.799969i	$0.704849\pi$
$368$	−3.25800e11	−0.926053
$369$	0	0
$370$	−4.68393e10	−0.129928
$371$	3.64603e10	0.0999165
$372$	0	0
$373$	−7.60417e10	−0.203405	−0.101703	−	0.994815i	$-0.532429\pi$
$373$	−7.60417e10	−0.203405	−0.101703	+	0.994815i	$0.532429\pi$
$374$	7.17575e10	0.189647
$375$	0	0
$376$	2.48271e9	0.00640591
$377$	−4.79090e11	−1.22147
$378$	0	0
$379$	−1.79180e11	−0.446080	−0.223040	−	0.974809i	$-0.571598\pi$
$379$	−1.79180e11	−0.446080	−0.223040	+	0.974809i	$0.571598\pi$
$380$	2.22048e11	0.546287
$381$	0	0
$382$	1.17558e11	0.282467
$383$	7.95018e11	1.88792	0.943958	−	0.330066i	$-0.107071\pi$
$383$	7.95018e11	1.88792	0.943958	+	0.330066i	$0.107071\pi$
$384$	0	0
$385$	4.14739e11	0.962059
$386$	−5.94985e10	−0.136415
$387$	0	0
$388$	−1.20484e11	−0.269891
$389$	1.79533e11	0.397532	0.198766	−	0.980047i	$-0.436307\pi$
$389$	1.79533e11	0.397532	0.198766	+	0.980047i	$0.436307\pi$
$390$	0	0
$391$	−2.84426e11	−0.615422
$392$	−7.51113e10	−0.160664
$393$	0	0
$394$	−1.99470e10	−0.0417008
$395$	−1.46519e11	−0.302837
$396$	0	0
$397$	−3.43730e11	−0.694480	−0.347240	−	0.937776i	$-0.612881\pi$
$397$	−3.43730e11	−0.694480	−0.347240	+	0.937776i	$0.612881\pi$
$398$	5.82108e10	0.116287
$399$	0	0
$400$	9.29000e10	0.181445
$401$	−7.72080e11	−1.49112	−0.745560	−	0.666438i	$-0.767818\pi$
$401$	−7.72080e11	−1.49112	−0.745560	+	0.666438i	$0.767818\pi$
$402$	0	0
$403$	3.52298e11	0.665331
$404$	−3.24286e11	−0.605637
$405$	0	0
$406$	−9.81320e10	−0.179244
$407$	1.61884e12	2.92435
$408$	0	0
$409$	2.60632e11	0.460546	0.230273	−	0.973126i	$-0.426038\pi$
$409$	2.60632e11	0.460546	0.230273	+	0.973126i	$0.426038\pi$
$410$	−7.32066e10	−0.127945
$411$	0	0
$412$	−6.96484e11	−1.19090
$413$	4.65531e11	0.787361
$414$	0	0
$415$	7.43190e10	0.122994
$416$	−4.52286e11	−0.740445
$417$	0	0
$418$	2.47559e11	0.396630
$419$	5.60166e11	0.887879	0.443939	−	0.896057i	$-0.353581\pi$
$419$	5.60166e11	0.887879	0.443939	+	0.896057i	$0.353581\pi$
$420$	0	0
$421$	1.68321e11	0.261137	0.130569	−	0.991439i	$-0.458320\pi$
$421$	1.68321e11	0.261137	0.130569	+	0.991439i	$0.458320\pi$
$422$	2.06163e11	0.316449
$423$	0	0
$424$	1.91416e10	0.0287629
$425$	8.11023e10	0.120582
$426$	0	0
$427$	9.73407e11	1.41700
$428$	−9.11913e11	−1.31358
$429$	0	0
$430$	3.79181e9	0.00534858
$431$	−4.48383e11	−0.625895	−0.312948	−	0.949770i	$-0.601316\pi$
$431$	−4.48383e11	−0.625895	−0.312948	+	0.949770i	$0.601316\pi$
$432$	0	0
$433$	−1.08485e12	−1.48311	−0.741556	−	0.670891i	$-0.765912\pi$
$433$	−1.08485e12	−1.48311	−0.741556	+	0.670891i	$0.765912\pi$
$434$	7.21613e10	0.0976339
$435$	0	0
$436$	4.62992e11	0.613599
$437$	−9.81252e11	−1.28711
$438$	0	0
$439$	4.60548e11	0.591814	0.295907	−	0.955217i	$-0.404378\pi$
$439$	4.60548e11	0.591814	0.295907	+	0.955217i	$0.404378\pi$
$440$	2.17738e11	0.276947
$441$	0	0
$442$	−1.24555e11	−0.155225
$443$	1.32095e10	0.0162956	0.00814779	−	0.999967i	$-0.497406\pi$
$443$	1.32095e10	0.0162956	0.00814779	+	0.999967i	$0.497406\pi$
$444$	0	0
$445$	−1.97834e11	−0.239156
$446$	1.84509e11	0.220806
$447$	0	0
$448$	8.42520e11	0.988165
$449$	6.91889e11	0.803393	0.401696	−	0.915773i	$-0.368421\pi$
$449$	6.91889e11	0.803393	0.401696	+	0.915773i	$0.368421\pi$
$450$	0	0
$451$	2.53014e12	2.87971
$452$	−4.60553e10	−0.0518987
$453$	0	0
$454$	1.50333e11	0.166075
$455$	−7.19894e11	−0.787440
$456$	0	0
$457$	−3.73135e11	−0.400168	−0.200084	−	0.979779i	$-0.564122\pi$
$457$	−3.73135e11	−0.400168	−0.200084	+	0.979779i	$0.564122\pi$
$458$	−2.56433e11	−0.272320
$459$	0	0
$460$	−4.24675e11	−0.442228
$461$	1.45940e12	1.50494	0.752470	−	0.658627i	$-0.228862\pi$
$461$	1.45940e12	1.50494	0.752470	+	0.658627i	$0.228862\pi$
$462$	0	0
$463$	1.34213e11	0.135732	0.0678658	−	0.997694i	$-0.478381\pi$
$463$	1.34213e11	0.135732	0.0678658	+	0.997694i	$0.478381\pi$
$464$	7.59705e11	0.760877
$465$	0	0
$466$	−2.78631e11	−0.273712
$467$	3.64531e10	0.0354657	0.0177329	−	0.999843i	$-0.494355\pi$
$467$	3.64531e10	0.0354657	0.0177329	+	0.999843i	$0.494355\pi$
$468$	0	0
$469$	8.53883e11	0.814930
$470$	1.53938e9	0.00145514
$471$	0	0
$472$	2.44404e11	0.226657
$473$	−1.31051e11	−0.120383
$474$	0	0
$475$	2.79798e11	0.252188
$476$	7.90890e11	0.706131
$477$	0	0
$478$	−2.66330e11	−0.233343
$479$	−8.82280e11	−0.765767	−0.382883	−	0.923797i	$-0.625069\pi$
$479$	−8.82280e11	−0.765767	−0.382883	+	0.923797i	$0.625069\pi$
$480$	0	0
$481$	−2.80994e12	−2.39356
$482$	−1.76663e11	−0.149085
$483$	0	0
$484$	−2.53342e12	−2.09847
$485$	−1.51820e11	−0.124592
$486$	0	0
$487$	5.09840e11	0.410727	0.205364	−	0.978686i	$-0.434162\pi$
$487$	5.09840e11	0.410727	0.205364	+	0.978686i	$0.434162\pi$
$488$	5.11039e11	0.407910
$489$	0	0
$490$	−4.65720e10	−0.0364958
$491$	1.52131e12	1.18128	0.590638	−	0.806937i	$-0.298876\pi$
$491$	1.52131e12	1.18128	0.590638	+	0.806937i	$0.298876\pi$
$492$	0	0
$493$	6.63228e11	0.505652
$494$	−4.29707e11	−0.324640
$495$	0	0
$496$	−5.58649e11	−0.414449
$497$	−1.34824e12	−0.991202
$498$	0	0
$499$	7.03413e11	0.507876	0.253938	−	0.967220i	$-0.418274\pi$
$499$	7.03413e11	0.507876	0.253938	+	0.967220i	$0.418274\pi$
$500$	1.21094e11	0.0866476
$501$	0	0
$502$	−3.34494e11	−0.235083
$503$	−3.78018e8	−0.000263304 0	−0.000131652	−	1.00000i	$-0.500042\pi$
$503$	−3.78018e8	−0.000263304 0	−0.000131652		1.00000i	$0.500042\pi$
$504$	0	0
$505$	−4.08627e11	−0.279586
$506$	−4.73466e11	−0.321079
$507$	0	0
$508$	−8.62144e11	−0.574372
$509$	−1.32057e12	−0.872027	−0.436013	−	0.899940i	$-0.643610\pi$
$509$	−1.32057e12	−0.872027	−0.436013	+	0.899940i	$0.643610\pi$
$510$	0	0
$511$	4.70272e11	0.305109
$512$	1.20816e12	0.776980
$513$	0	0
$514$	3.46110e11	0.218715
$515$	−8.77627e11	−0.549765
$516$	0	0
$517$	−5.32034e10	−0.0327516
$518$	−5.75561e11	−0.351243
$519$	0	0
$520$	−3.77945e11	−0.226680
$521$	1.31853e12	0.784009	0.392005	−	0.919963i	$-0.371782\pi$
$521$	1.31853e12	0.784009	0.392005	+	0.919963i	$0.371782\pi$
$522$	0	0
$523$	−1.69211e12	−0.988945	−0.494472	−	0.869193i	$-0.664639\pi$
$523$	−1.69211e12	−0.988945	−0.494472	+	0.869193i	$0.664639\pi$
$524$	−1.23866e12	−0.717730
$525$	0	0
$526$	3.84614e11	0.219073
$527$	−4.87704e11	−0.275428
$528$	0	0
$529$	7.55281e10	0.0419332
$530$	1.18686e10	0.00653367
$531$	0	0
$532$	2.72853e12	1.47681
$533$	−4.39175e12	−2.35703
$534$	0	0
$535$	−1.14908e12	−0.606401
$536$	4.48288e11	0.234594
$537$	0	0
$538$	4.38021e10	0.0225411
$539$	1.60960e12	0.821427
$540$	0	0
$541$	−1.86369e12	−0.935373	−0.467687	−	0.883894i	$-0.654912\pi$
$541$	−1.86369e12	−0.935373	−0.467687	+	0.883894i	$0.654912\pi$
$542$	3.12115e11	0.155352
$543$	0	0
$544$	6.26122e11	0.306523
$545$	5.83408e11	0.283262
$546$	0	0
$547$	4.37242e11	0.208823	0.104412	−	0.994534i	$-0.466704\pi$
$547$	4.37242e11	0.208823	0.104412	+	0.994534i	$0.466704\pi$
$548$	−3.94928e12	−1.87071
$549$	0	0
$550$	1.35006e11	0.0629103
$551$	2.28810e12	1.05753
$552$	0	0
$553$	−1.80043e12	−0.818679
$554$	2.62736e11	0.118502
$555$	0	0
$556$	1.41640e12	0.628565
$557$	7.09146e11	0.312167	0.156084	−	0.987744i	$-0.450113\pi$
$557$	7.09146e11	0.312167	0.156084	+	0.987744i	$0.450113\pi$
$558$	0	0
$559$	2.27475e11	0.0985328
$560$	1.14156e12	0.490513
$561$	0	0
$562$	2.57962e11	0.109079
$563$	−3.77472e10	−0.0158342	−0.00791711	−	0.999969i	$-0.502520\pi$
$563$	−3.77472e10	−0.0158342	−0.00791711	+	0.999969i	$0.502520\pi$
$564$	0	0
$565$	−5.80334e10	−0.0239585
$566$	3.85253e11	0.157787
$567$	0	0
$568$	−7.07824e11	−0.285337
$569$	3.56270e11	0.142487	0.0712433	−	0.997459i	$-0.477303\pi$
$569$	3.56270e11	0.142487	0.0712433	+	0.997459i	$0.477303\pi$
$570$	0	0
$571$	3.87932e12	1.52719	0.763596	−	0.645694i	$-0.223432\pi$
$571$	3.87932e12	1.52719	0.763596	+	0.645694i	$0.223432\pi$
$572$	6.42751e12	2.51050
$573$	0	0
$574$	−8.99562e11	−0.345882
$575$	−5.35125e11	−0.204150
$576$	0	0
$577$	−4.28876e12	−1.61080	−0.805399	−	0.592734i	$-0.798049\pi$
$577$	−4.28876e12	−1.61080	−0.805399	+	0.592734i	$0.798049\pi$
$578$	−3.01924e11	−0.112518
$579$	0	0
$580$	9.90265e11	0.363350
$581$	9.13232e11	0.332498
$582$	0	0
$583$	−4.10197e11	−0.147056
$584$	2.46893e11	0.0878316
$585$	0	0
$586$	−3.26523e11	−0.114387
$587$	−4.43245e12	−1.54089	−0.770447	−	0.637504i	$-0.779967\pi$
$587$	−4.43245e12	−1.54089	−0.770447	+	0.637504i	$0.779967\pi$
$588$	0	0
$589$	−1.68255e12	−0.576036
$590$	1.51540e11	0.0514865
$591$	0	0
$592$	4.45580e12	1.49100
$593$	5.10104e12	1.69400	0.846998	−	0.531596i	$-0.178407\pi$
$593$	5.10104e12	1.69400	0.846998	+	0.531596i	$0.178407\pi$
$594$	0	0
$595$	9.96586e11	0.325978
$596$	−4.77838e12	−1.55122
$597$	0	0
$598$	8.21831e11	0.262801
$599$	7.04599e11	0.223626	0.111813	−	0.993729i	$-0.464334\pi$
$599$	7.04599e11	0.223626	0.111813	+	0.993729i	$0.464334\pi$
$600$	0	0
$601$	−1.73879e12	−0.543641	−0.271821	−	0.962348i	$-0.587626\pi$
$601$	−1.73879e12	−0.543641	−0.271821	+	0.962348i	$0.587626\pi$
$602$	4.65938e10	0.0144592
$603$	0	0
$604$	2.66993e12	0.816269
$605$	−3.19231e12	−0.968738
$606$	0	0
$607$	−5.78292e11	−0.172901	−0.0864507	−	0.996256i	$-0.527553\pi$
$607$	−5.78292e11	−0.172901	−0.0864507	+	0.996256i	$0.527553\pi$
$608$	2.16008e12	0.641068
$609$	0	0
$610$	3.16864e11	0.0926593
$611$	9.23493e10	0.0268070
$612$	0	0
$613$	3.74595e12	1.07150	0.535748	−	0.844378i	$-0.320030\pi$
$613$	3.74595e12	1.07150	0.535748	+	0.844378i	$0.320030\pi$
$614$	−1.18233e11	−0.0335723
$615$	0	0
$616$	2.67557e12	0.748691
$617$	3.94875e12	1.09692	0.548461	−	0.836176i	$-0.315214\pi$
$617$	3.94875e12	1.09692	0.548461	+	0.836176i	$0.315214\pi$
$618$	0	0
$619$	3.42253e12	0.937000	0.468500	−	0.883463i	$-0.344795\pi$
$619$	3.42253e12	0.937000	0.468500	+	0.883463i	$0.344795\pi$
$620$	−7.28190e11	−0.197917
$621$	0	0
$622$	1.59628e11	0.0427615
$623$	−2.43098e12	−0.646526
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	7.41484e11	0.192982
$627$	0	0
$628$	−2.57639e11	−0.0660988
$629$	3.88995e12	0.990868
$630$	0	0
$631$	5.84755e12	1.46839	0.734196	−	0.678938i	$-0.237560\pi$
$631$	5.84755e12	1.46839	0.734196	+	0.678938i	$0.237560\pi$
$632$	−9.45226e11	−0.235673
$633$	0	0
$634$	−1.07558e12	−0.264387
$635$	−1.08637e12	−0.265153
$636$	0	0
$637$	−2.79391e12	−0.672334
$638$	1.10404e12	0.263809
$639$	0	0
$640$	1.23928e12	0.291983
$641$	−2.66671e12	−0.623899	−0.311950	−	0.950099i	$-0.600982\pi$
$641$	−2.66671e12	−0.623899	−0.311950	+	0.950099i	$0.600982\pi$
$642$	0	0
$643$	9.68716e10	0.0223484	0.0111742	−	0.999938i	$-0.496443\pi$
$643$	9.68716e10	0.0223484	0.0111742	+	0.999938i	$0.496443\pi$
$644$	−5.21841e12	−1.19551
$645$	0	0
$646$	5.94865e11	0.134392
$647$	−4.47368e10	−0.0100368	−0.00501840	−	0.999987i	$-0.501597\pi$
$647$	−4.47368e10	−0.0100368	−0.00501840	+	0.999987i	$0.501597\pi$
$648$	0	0
$649$	−5.23747e12	−1.15883
$650$	−2.34341e11	−0.0514917
$651$	0	0
$652$	−4.66855e12	−1.01174
$653$	−4.95385e12	−1.06619	−0.533094	−	0.846056i	$-0.678971\pi$
$653$	−4.95385e12	−1.06619	−0.533094	+	0.846056i	$0.678971\pi$
$654$	0	0
$655$	−1.56081e12	−0.331333
$656$	6.96411e12	1.46824
$657$	0	0
$658$	1.89159e10	0.00393378
$659$	5.85077e12	1.20845	0.604225	−	0.796814i	$-0.293483\pi$
$659$	5.85077e12	1.20845	0.604225	+	0.796814i	$0.293483\pi$
$660$	0	0
$661$	−8.81007e11	−0.179503	−0.0897517	−	0.995964i	$-0.528607\pi$
$661$	−8.81007e11	−0.179503	−0.0897517	+	0.995964i	$0.528607\pi$
$662$	−1.71640e12	−0.347342
$663$	0	0
$664$	4.79447e11	0.0957159
$665$	3.43816e12	0.681756
$666$	0	0
$667$	−4.37608e12	−0.856088
$668$	4.64872e12	0.903314
$669$	0	0
$670$	2.77957e11	0.0532894
$671$	−1.09513e13	−2.08553
$672$	0	0
$673$	−8.66521e12	−1.62821	−0.814107	−	0.580715i	$-0.802773\pi$
$673$	−8.66521e12	−1.62821	−0.814107	+	0.580715i	$0.802773\pi$
$674$	−8.10390e10	−0.0151260
$675$	0	0
$676$	−5.89689e12	−1.08608
$677$	−8.98549e12	−1.64397	−0.821983	−	0.569512i	$-0.807132\pi$
$677$	−8.98549e12	−1.64397	−0.821983	+	0.569512i	$0.807132\pi$
$678$	0	0
$679$	−1.86557e12	−0.336819
$680$	5.23207e11	0.0938391
$681$	0	0
$682$	−8.11852e11	−0.143697
$683$	3.86477e12	0.679564	0.339782	−	0.940504i	$-0.389647\pi$
$683$	3.86477e12	0.679564	0.339782	+	0.940504i	$0.389647\pi$
$684$	0	0
$685$	−4.97642e12	−0.863593
$686$	6.67386e11	0.115059
$687$	0	0
$688$	−3.60713e11	−0.0613782
$689$	7.12010e11	0.120365
$690$	0	0
$691$	−7.08564e12	−1.18230	−0.591150	−	0.806561i	$-0.701326\pi$
$691$	−7.08564e12	−1.18230	−0.591150	+	0.806561i	$0.701326\pi$
$692$	6.12922e12	1.01608
$693$	0	0
$694$	−1.17113e11	−0.0191641
$695$	1.78478e12	0.290171
$696$	0	0
$697$	6.07972e12	0.975744
$698$	2.82053e11	0.0449760
$699$	0	0
$700$	1.48800e12	0.234240
$701$	−4.69380e12	−0.734165	−0.367083	−	0.930188i	$-0.619643\pi$
$701$	−4.69380e12	−0.734165	−0.367083	+	0.930188i	$0.619643\pi$
$702$	0	0
$703$	1.34201e13	2.07232
$704$	−9.47879e12	−1.45437
$705$	0	0
$706$	2.62940e11	0.0398323
$707$	−5.02120e12	−0.755824
$708$	0	0
$709$	1.06645e13	1.58501	0.792503	−	0.609868i	$-0.208777\pi$
$709$	1.06645e13	1.58501	0.792503	+	0.609868i	$0.208777\pi$
$710$	−4.38879e11	−0.0648160
$711$	0	0
$712$	−1.27627e12	−0.186115
$713$	3.21794e12	0.466311
$714$	0	0
$715$	8.09919e12	1.15895
$716$	−3.30356e11	−0.0469757
$717$	0	0
$718$	−2.32681e12	−0.326739
$719$	8.15663e12	1.13823	0.569116	−	0.822257i	$-0.307285\pi$
$719$	8.15663e12	1.13823	0.569116	+	0.822257i	$0.307285\pi$
$720$	0	0
$721$	−1.07843e13	−1.48622
$722$	7.61500e11	0.104292
$723$	0	0
$724$	−2.61495e12	−0.353703
$725$	1.24781e12	0.167737
$726$	0	0
$727$	6.64771e12	0.882606	0.441303	−	0.897358i	$-0.354516\pi$
$727$	6.64771e12	0.882606	0.441303	+	0.897358i	$0.354516\pi$
$728$	−4.64418e12	−0.612799
$729$	0	0
$730$	1.53083e11	0.0199515
$731$	−3.14905e11	−0.0407898
$732$	0	0
$733$	7.07821e12	0.905640	0.452820	−	0.891602i	$-0.350418\pi$
$733$	7.07821e12	0.905640	0.452820	+	0.891602i	$0.350418\pi$
$734$	−1.66828e12	−0.212147
$735$	0	0
$736$	−4.13124e12	−0.518956
$737$	−9.60663e12	−1.19941
$738$	0	0
$739$	−2.61052e12	−0.321979	−0.160989	−	0.986956i	$-0.551468\pi$
$739$	−2.61052e12	−0.321979	−0.160989	+	0.986956i	$0.551468\pi$
$740$	5.80807e12	0.712015
$741$	0	0
$742$	1.45841e11	0.0176629
$743$	1.41841e13	1.70747	0.853734	−	0.520709i	$-0.174333\pi$
$743$	1.41841e13	1.70747	0.853734	+	0.520709i	$0.174333\pi$
$744$	0	0
$745$	−6.02115e12	−0.716104
$746$	−3.04167e11	−0.0359573
$747$	0	0
$748$	−8.89793e12	−1.03928
$749$	−1.41199e13	−1.63932
$750$	0	0
$751$	8.44355e11	0.0968603	0.0484301	−	0.998827i	$-0.484578\pi$
$751$	8.44355e11	0.0968603	0.0484301	+	0.998827i	$0.484578\pi$
$752$	−1.46441e11	−0.0166986
$753$	0	0
$754$	−1.91636e12	−0.215927
$755$	3.36433e12	0.376822
$756$	0	0
$757$	9.05305e12	1.00199	0.500995	−	0.865450i	$-0.332968\pi$
$757$	9.05305e12	1.00199	0.500995	+	0.865450i	$0.332968\pi$
$758$	−7.16719e11	−0.0788565
$759$	0	0
$760$	1.80504e12	0.196257
$761$	6.97701e12	0.754116	0.377058	−	0.926190i	$-0.376936\pi$
$761$	6.97701e12	0.754116	0.377058	+	0.926190i	$0.376936\pi$
$762$	0	0
$763$	7.16891e12	0.765760
$764$	−1.45772e13	−1.54794
$765$	0	0
$766$	3.18007e12	0.333739
$767$	9.09108e12	0.948498
$768$	0	0
$769$	−1.07233e13	−1.10576	−0.552879	−	0.833261i	$-0.686471\pi$
$769$	−1.07233e13	−1.10576	−0.552879	+	0.833261i	$0.686471\pi$
$770$	1.65896e12	0.170070
$771$	0	0
$772$	7.37781e12	0.747566
$773$	1.37568e13	1.38583	0.692916	−	0.721019i	$-0.256326\pi$
$773$	1.37568e13	1.38583	0.692916	+	0.721019i	$0.256326\pi$
$774$	0	0
$775$	−9.17578e11	−0.0913662
$776$	−9.79422e11	−0.0969599
$777$	0	0
$778$	7.18134e11	0.0702744
$779$	2.09747e13	2.04069
$780$	0	0
$781$	1.51684e13	1.45884
$782$	−1.13770e12	−0.108792
$783$	0	0
$784$	4.43037e12	0.418811
$785$	−3.24646e11	−0.0305138
$786$	0	0
$787$	−1.27539e13	−1.18510	−0.592551	−	0.805533i	$-0.701879\pi$
$787$	−1.27539e13	−1.18510	−0.592551	+	0.805533i	$0.701879\pi$
$788$	2.47343e12	0.228524
$789$	0	0
$790$	−5.86078e11	−0.0535345
$791$	−7.13114e11	−0.0647686
$792$	0	0
$793$	1.90091e13	1.70699
$794$	−1.37492e12	−0.122768
$795$	0	0
$796$	−7.21814e12	−0.637260
$797$	−1.27845e13	−1.12233	−0.561164	−	0.827704i	$-0.689646\pi$
$797$	−1.27845e13	−1.12233	−0.561164	+	0.827704i	$0.689646\pi$
$798$	0	0
$799$	−1.27844e11	−0.0110973
$800$	1.17800e12	0.101681
$801$	0	0
$802$	−3.08832e12	−0.263595
$803$	−5.29081e12	−0.449057
$804$	0	0
$805$	−6.57562e12	−0.551893
$806$	1.40919e12	0.117615
$807$	0	0
$808$	−2.63613e12	−0.217579
$809$	−6.03746e12	−0.495548	−0.247774	−	0.968818i	$-0.579699\pi$
$809$	−6.03746e12	−0.495548	−0.247774	+	0.968818i	$0.579699\pi$
$810$	0	0
$811$	−2.50043e12	−0.202965	−0.101482	−	0.994837i	$-0.532359\pi$
$811$	−2.50043e12	−0.202965	−0.101482	+	0.994837i	$0.532359\pi$
$812$	1.21684e13	0.982269
$813$	0	0
$814$	6.47536e12	0.516957
$815$	−5.88275e12	−0.467058
$816$	0	0
$817$	−1.08641e12	−0.0853085
$818$	1.04253e12	0.0814138
$819$	0	0
$820$	9.07762e12	0.701148
$821$	4.21082e12	0.323461	0.161731	−	0.986835i	$-0.448292\pi$
$821$	4.21082e12	0.323461	0.161731	+	0.986835i	$0.448292\pi$
$822$	0	0
$823$	2.08206e11	0.0158196	0.00790978	−	0.999969i	$-0.497482\pi$
$823$	2.08206e11	0.0158196	0.00790978	+	0.999969i	$0.497482\pi$
$824$	−5.66174e12	−0.427836
$825$	0	0
$826$	1.86213e12	0.139187
$827$	9.26106e12	0.688472	0.344236	−	0.938883i	$-0.388138\pi$
$827$	9.26106e12	0.688472	0.344236	+	0.938883i	$0.388138\pi$
$828$	0	0
$829$	2.42762e13	1.78519	0.892597	−	0.450856i	$-0.148881\pi$
$829$	2.42762e13	1.78519	0.892597	+	0.450856i	$0.148881\pi$
$830$	2.97276e11	0.0217424
$831$	0	0
$832$	1.64531e13	1.19040
$833$	3.86775e12	0.278327
$834$	0	0
$835$	5.85776e12	0.417006
$836$	−3.06973e13	−2.17356
$837$	0	0
$838$	2.24066e12	0.156956
$839$	1.25546e13	0.874728	0.437364	−	0.899285i	$-0.355912\pi$
$839$	1.25546e13	0.874728	0.437364	+	0.899285i	$0.355912\pi$
$840$	0	0
$841$	−4.30294e12	−0.296608
$842$	6.73284e11	0.0461630
$843$	0	0
$844$	−2.55642e13	−1.73417
$845$	−7.43056e12	−0.501379
$846$	0	0
$847$	−3.92272e13	−2.61886
$848$	−1.12905e12	−0.0749778
$849$	0	0
$850$	3.24409e11	0.0213161
$851$	−2.56664e13	−1.67758
$852$	0	0
$853$	−1.36320e13	−0.881632	−0.440816	−	0.897597i	$-0.645311\pi$
$853$	−1.36320e13	−0.881632	−0.440816	+	0.897597i	$0.645311\pi$
$854$	3.89363e12	0.250492
$855$	0	0
$856$	−7.41297e12	−0.471911
$857$	−1.73987e13	−1.10180	−0.550901	−	0.834570i	$-0.685716\pi$
$857$	−1.73987e13	−1.10180	−0.550901	+	0.834570i	$0.685716\pi$
$858$	0	0
$859$	−6.43355e10	−0.00403163	−0.00201582	−	0.999998i	$-0.500642\pi$
$859$	−6.43355e10	−0.00403163	−0.00201582	+	0.999998i	$0.500642\pi$
$860$	−4.70184e11	−0.0293106
$861$	0	0
$862$	−1.79353e12	−0.110644
$863$	−3.32120e12	−0.203820	−0.101910	−	0.994794i	$-0.532495\pi$
$863$	−3.32120e12	−0.203820	−0.101910	+	0.994794i	$0.532495\pi$
$864$	0	0
$865$	7.72331e12	0.469063
$866$	−4.33940e12	−0.262180
$867$	0	0
$868$	−8.94800e12	−0.535041
$869$	2.02558e13	1.20493
$870$	0	0
$871$	1.66750e13	0.981709
$872$	3.76368e12	0.220439
$873$	0	0
$874$	−3.92501e12	−0.227530
$875$	1.87500e12	0.108135
$876$	0	0
$877$	1.95832e12	0.111786	0.0558928	−	0.998437i	$-0.482200\pi$
$877$	1.95832e12	0.111786	0.0558928	+	0.998437i	$0.482200\pi$
$878$	1.84219e12	0.104619
$879$	0	0
$880$	−1.28431e13	−0.721934
$881$	3.02253e13	1.69036	0.845179	−	0.534483i	$-0.179494\pi$
$881$	3.02253e13	1.69036	0.845179	+	0.534483i	$0.179494\pi$
$882$	0	0
$883$	−9.30097e12	−0.514879	−0.257439	−	0.966294i	$-0.582879\pi$
$883$	−9.30097e12	−0.514879	−0.257439	+	0.966294i	$0.582879\pi$
$884$	1.54448e13	0.850643
$885$	0	0
$886$	5.28380e10	0.00288068
$887$	−9.27171e12	−0.502925	−0.251463	−	0.967867i	$-0.580912\pi$
$887$	−9.27171e12	−0.502925	−0.251463	+	0.967867i	$0.580912\pi$
$888$	0	0
$889$	−1.33493e13	−0.716805
$890$	−7.91336e11	−0.0422772
$891$	0	0
$892$	−2.28791e13	−1.21003
$893$	−4.41053e11	−0.0232092
$894$	0	0
$895$	−4.16275e11	−0.0216859
$896$	1.52282e13	0.789338
$897$	0	0
$898$	2.76756e12	0.142021
$899$	−7.50365e12	−0.383137
$900$	0	0
$901$	−9.85671e11	−0.0498276
$902$	1.01205e13	0.509066
$903$	0	0
$904$	−3.74385e11	−0.0186449
$905$	−3.29504e12	−0.163284
$906$	0	0
$907$	1.32868e12	0.0651908	0.0325954	−	0.999469i	$-0.489623\pi$
$907$	1.32868e12	0.0651908	0.0325954	+	0.999469i	$0.489623\pi$
$908$	−1.86413e13	−0.910102
$909$	0	0
$910$	−2.87958e12	−0.139201
$911$	−2.71297e12	−0.130501	−0.0652503	−	0.997869i	$-0.520785\pi$
$911$	−2.71297e12	−0.130501	−0.0652503	+	0.997869i	$0.520785\pi$
$912$	0	0
$913$	−1.02743e13	−0.489368
$914$	−1.49254e12	−0.0707404
$915$	0	0
$916$	3.17977e13	1.49233
$917$	−1.91792e13	−0.895714
$918$	0	0
$919$	1.32139e12	0.0611100	0.0305550	−	0.999533i	$-0.490273\pi$
$919$	1.32139e12	0.0611100	0.0305550	+	0.999533i	$0.490273\pi$
$920$	−3.45220e12	−0.158873
$921$	0	0
$922$	5.83758e12	0.266038
$923$	−2.63289e13	−1.19406
$924$	0	0
$925$	7.31864e12	0.328694
$926$	5.36853e11	0.0239942
$927$	0	0
$928$	9.63329e12	0.426392
$929$	−1.16124e12	−0.0511507	−0.0255754	−	0.999673i	$-0.508142\pi$
$929$	−1.16124e12	−0.0511507	−0.0255754	+	0.999673i	$0.508142\pi$
$930$	0	0
$931$	1.33435e13	0.582098
$932$	3.45503e13	1.49996
$933$	0	0
$934$	1.45813e11	0.00626952
$935$	−1.12121e13	−0.479772
$936$	0	0
$937$	3.40914e13	1.44483	0.722415	−	0.691460i	$-0.243032\pi$
$937$	3.40914e13	1.44483	0.722415	+	0.691460i	$0.243032\pi$
$938$	3.41553e12	0.144061
$939$	0	0
$940$	−1.90883e11	−0.00797429
$941$	1.60215e12	0.0666114	0.0333057	−	0.999445i	$-0.489397\pi$
$941$	1.60215e12	0.0666114	0.0333057	+	0.999445i	$0.489397\pi$
$942$	0	0
$943$	−4.01149e13	−1.65197
$944$	−1.44160e13	−0.590840
$945$	0	0
$946$	−5.24204e11	−0.0212809
$947$	3.38850e13	1.36909	0.684546	−	0.728969i	$-0.260000\pi$
$947$	3.38850e13	1.36909	0.684546	+	0.728969i	$0.260000\pi$
$948$	0	0
$949$	9.18366e12	0.367551
$950$	1.11919e12	0.0445809
$951$	0	0
$952$	6.42917e12	0.253682
$953$	−2.15757e13	−0.847320	−0.423660	−	0.905821i	$-0.639255\pi$
$953$	−2.15757e13	−0.847320	−0.423660	+	0.905821i	$0.639255\pi$
$954$	0	0
$955$	−1.83685e13	−0.714592
$956$	3.30249e13	1.27874
$957$	0	0
$958$	−3.52912e12	−0.135370
$959$	−6.11502e13	−2.33461
$960$	0	0
$961$	−2.09218e13	−0.791306
$962$	−1.12398e13	−0.423126
$963$	0	0
$964$	2.19063e13	0.816999
$965$	9.29664e12	0.345106
$966$	0	0
$967$	3.06249e13	1.12630	0.563152	−	0.826353i	$-0.309589\pi$
$967$	3.06249e13	1.12630	0.563152	+	0.826353i	$0.309589\pi$
$968$	−2.05943e13	−0.753888
$969$	0	0
$970$	−6.07281e11	−0.0220250
$971$	−1.92365e12	−0.0694447	−0.0347224	−	0.999397i	$-0.511055\pi$
$971$	−1.92365e12	−0.0694447	−0.0347224	+	0.999397i	$0.511055\pi$
$972$	0	0
$973$	2.19314e13	0.784438
$974$	2.03936e12	0.0726070
$975$	0	0
$976$	−3.01432e13	−1.06332
$977$	1.83893e13	0.645714	0.322857	−	0.946448i	$-0.395357\pi$
$977$	1.83893e13	0.645714	0.322857	+	0.946448i	$0.395357\pi$
$978$	0	0
$979$	2.73498e13	0.951552
$980$	5.77493e12	0.200000
$981$	0	0
$982$	6.08524e12	0.208822
$983$	4.63273e13	1.58251	0.791254	−	0.611488i	$-0.209429\pi$
$983$	4.63273e13	1.58251	0.791254	+	0.611488i	$0.209429\pi$
$984$	0	0
$985$	3.11672e12	0.105496
$986$	2.65291e12	0.0893875
$987$	0	0
$988$	5.32837e13	1.77905
$989$	2.07779e12	0.0690587
$990$	0	0
$991$	2.76252e12	0.0909857	0.0454929	−	0.998965i	$-0.485514\pi$
$991$	2.76252e12	0.0909857	0.0454929	+	0.998965i	$0.485514\pi$
$992$	−7.08383e12	−0.232255
$993$	0	0
$994$	−5.39295e12	−0.175221
$995$	−9.09544e12	−0.294184
$996$	0	0
$997$	−1.74502e13	−0.559337	−0.279668	−	0.960097i	$-0.590224\pi$
$997$	−1.74502e13	−0.559337	−0.279668	+	0.960097i	$0.590224\pi$
$998$	2.81365e12	0.0897807
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.10.a.b.1.1		1
3.2	odd	2		15.10.a.a.1.1	✓	1
5.2	odd	4		225.10.b.e.199.2		2
5.3	odd	4		225.10.b.e.199.1		2
5.4	even	2		225.10.a.c.1.1		1
12.11	even	2		240.10.a.c.1.1		1
15.2	even	4		75.10.b.d.49.1		2
15.8	even	4		75.10.b.d.49.2		2
15.14	odd	2		75.10.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.10.a.a.1.1	✓	1	3.2	odd	2
45.10.a.b.1.1		1	1.1	even	1	trivial
75.10.a.c.1.1		1	15.14	odd	2
75.10.b.d.49.1		2	15.2	even	4
75.10.b.d.49.2		2	15.8	even	4
225.10.a.c.1.1		1	5.4	even	2
225.10.b.e.199.1		2	5.3	odd	4
225.10.b.e.199.2		2	5.2	odd	4
240.10.a.c.1.1		1	12.11	even	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:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	45.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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