LMFDB - Embedded newform 28.12.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [28,12,Mod(1,28)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(28, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("28.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 28 = 2^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	28.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:	$21.5136090557$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 11269x - 111300 $ x^3 - 11269*x - 111300
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4}\cdot 3\cdot 7 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$110.786$ of defining polynomial
Character	$\chi$	$=$	28.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-158.956 q^{3} +5794.30 q^{5} +16807.0 q^{7} -151880. q^{9} +O(q^{10})$	$q-158.956 q^{3} +5794.30 q^{5} +16807.0 q^{7} -151880. q^{9} -182351. q^{11} +499142. q^{13} -921041. q^{15} +5.59277e6 q^{17} +9.67414e6 q^{19} -2.67158e6 q^{21} +3.12545e7 q^{23} -1.52542e7 q^{25} +5.23009e7 q^{27} +9.02844e7 q^{29} +2.58342e8 q^{31} +2.89858e7 q^{33} +9.73848e7 q^{35} +9.45474e7 q^{37} -7.93417e7 q^{39} +8.68434e8 q^{41} +6.33820e8 q^{43} -8.80038e8 q^{45} -1.40146e9 q^{47} +2.82475e8 q^{49} -8.89007e8 q^{51} -4.90516e9 q^{53} -1.05660e9 q^{55} -1.53777e9 q^{57} +2.31273e8 q^{59} -2.27817e9 q^{61} -2.55264e9 q^{63} +2.89218e9 q^{65} +4.67122e9 q^{67} -4.96811e9 q^{69} -4.69060e9 q^{71} -8.30010e9 q^{73} +2.42475e9 q^{75} -3.06477e9 q^{77} +4.37744e10 q^{79} +1.85915e10 q^{81} +2.21339e10 q^{83} +3.24062e10 q^{85} -1.43513e10 q^{87} +4.27452e10 q^{89} +8.38907e9 q^{91} -4.10651e10 q^{93} +5.60549e10 q^{95} -5.30341e10 q^{97} +2.76954e10 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 1132 q^{3} + 4986 q^{5} + 50421 q^{7} + 370759 q^{9}+O(q^{10}) $ 3 * q + 1132 * q^3 + 4986 * q^5 + 50421 * q^7 + 370759 * q^9	$ 3 q + 1132 q^{3} + 4986 q^{5} + 50421 q^{7} + 370759 q^{9} + 863796 q^{11} + 1006962 q^{13} + 1990704 q^{15} - 2561418 q^{17} - 4380708 q^{19} + 19025524 q^{21} + 29203392 q^{23} + 157500381 q^{25} + 217312408 q^{27} + 382205778 q^{29} + 302574048 q^{31} + 587964816 q^{33} + 83799702 q^{35} + 110922954 q^{37} - 364261312 q^{39} + 60153150 q^{41} - 858558972 q^{43} + 3341217234 q^{45} + 416749152 q^{47} + 847425747 q^{49} - 7453626648 q^{51} - 1976465382 q^{53} - 10626661512 q^{55} - 11041098904 q^{57} - 4571596908 q^{59} - 8042615598 q^{61} + 6231346513 q^{63} - 45509674572 q^{65} + 15101535372 q^{67} - 18210771120 q^{69} + 21355612152 q^{71} + 29008383630 q^{73} + 111158791924 q^{75} + 14517819372 q^{77} - 7958497584 q^{79} + 72564681139 q^{81} - 7461285060 q^{83} - 66658496220 q^{85} + 170953812216 q^{87} - 10300974450 q^{89} + 16924010334 q^{91} - 6942785584 q^{93} + 27813662208 q^{95} - 181271913834 q^{97} + 150953802948 q^{99}+O(q^{100}) $ 3 * q + 1132 * q^3 + 4986 * q^5 + 50421 * q^7 + 370759 * q^9 + 863796 * q^11 + 1006962 * q^13 + 1990704 * q^15 - 2561418 * q^17 - 4380708 * q^19 + 19025524 * q^21 + 29203392 * q^23 + 157500381 * q^25 + 217312408 * q^27 + 382205778 * q^29 + 302574048 * q^31 + 587964816 * q^33 + 83799702 * q^35 + 110922954 * q^37 - 364261312 * q^39 + 60153150 * q^41 - 858558972 * q^43 + 3341217234 * q^45 + 416749152 * q^47 + 847425747 * q^49 - 7453626648 * q^51 - 1976465382 * q^53 - 10626661512 * q^55 - 11041098904 * q^57 - 4571596908 * q^59 - 8042615598 * q^61 + 6231346513 * q^63 - 45509674572 * q^65 + 15101535372 * q^67 - 18210771120 * q^69 + 21355612152 * q^71 + 29008383630 * q^73 + 111158791924 * q^75 + 14517819372 * q^77 - 7958497584 * q^79 + 72564681139 * q^81 - 7461285060 * q^83 - 66658496220 * q^85 + 170953812216 * q^87 - 10300974450 * q^89 + 16924010334 * q^91 - 6942785584 * q^93 + 27813662208 * q^95 - 181271913834 * q^97 + 150953802948 * q^99

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
$3$	−158.956	−0.377669	−0.188834	−	0.982009i	$-0.560471\pi$
$3$	−158.956	−0.377669	−0.188834	+	0.982009i	$0.560471\pi$
$4$	0	0
$5$	5794.30	0.829213	0.414606	−	0.910001i	$-0.363919\pi$
$5$	5794.30	0.829213	0.414606	+	0.910001i	$0.363919\pi$
$6$	0	0
$7$	16807.0	0.377964
$7$	16807.0	0.377964
$8$	0	0
$9$	−151880.	−0.857366
$10$	0	0
$11$	−182351.	−0.341388	−0.170694	−	0.985324i	$-0.554601\pi$
$11$	−182351.	−0.341388	−0.170694	+	0.985324i	$0.554601\pi$
$12$	0	0
$13$	499142.	0.372851	0.186425	−	0.982469i	$-0.440310\pi$
$13$	499142.	0.372851	0.186425	+	0.982469i	$0.440310\pi$
$14$	0	0
$15$	−921041.	−0.313168
$16$	0	0
$17$	5.59277e6	0.955340	0.477670	−	0.878539i	$-0.341481\pi$
$17$	5.59277e6	0.955340	0.477670	+	0.878539i	$0.341481\pi$
$18$	0	0
$19$	9.67414e6	0.896330	0.448165	−	0.893951i	$-0.352078\pi$
$19$	9.67414e6	0.896330	0.448165	+	0.893951i	$0.352078\pi$
$20$	0	0
$21$	−2.67158e6	−0.142745
$22$	0	0
$23$	3.12545e7	1.01253	0.506267	−	0.862377i	$-0.331025\pi$
$23$	3.12545e7	1.01253	0.506267	+	0.862377i	$0.331025\pi$
$24$	0	0
$25$	−1.52542e7	−0.312406
$26$	0	0
$27$	5.23009e7	0.701469
$28$	0	0
$29$	9.02844e7	0.817379	0.408690	−	0.912673i	$-0.365986\pi$
$29$	9.02844e7	0.817379	0.408690	+	0.912673i	$0.365986\pi$
$30$	0	0
$31$	2.58342e8	1.62071	0.810356	−	0.585938i	$-0.199274\pi$
$31$	2.58342e8	1.62071	0.810356	+	0.585938i	$0.199274\pi$
$32$	0	0
$33$	2.89858e7	0.128931
$34$	0	0
$35$	9.73848e7	0.313413
$36$	0	0
$37$	9.45474e7	0.224151	0.112075	−	0.993700i	$-0.464250\pi$
$37$	9.45474e7	0.224151	0.112075	+	0.993700i	$0.464250\pi$
$38$	0	0
$39$	−7.93417e7	−0.140814
$40$	0	0
$41$	8.68434e8	1.17065	0.585323	−	0.810800i	$-0.300968\pi$
$41$	8.68434e8	1.17065	0.585323	+	0.810800i	$0.300968\pi$
$42$	0	0
$43$	6.33820e8	0.657491	0.328745	−	0.944419i	$-0.393374\pi$
$43$	6.33820e8	0.657491	0.328745	+	0.944419i	$0.393374\pi$
$44$	0	0
$45$	−8.80038e8	−0.710939
$46$	0	0
$47$	−1.40146e9	−0.891338	−0.445669	−	0.895198i	$-0.647034\pi$
$47$	−1.40146e9	−0.891338	−0.445669	+	0.895198i	$0.647034\pi$
$48$	0	0
$49$	2.82475e8	0.142857
$50$	0	0
$51$	−8.89007e8	−0.360802
$52$	0	0
$53$	−4.90516e9	−1.61115	−0.805574	−	0.592495i	$-0.798143\pi$
$53$	−4.90516e9	−1.61115	−0.805574	+	0.592495i	$0.798143\pi$
$54$	0	0
$55$	−1.05660e9	−0.283083
$56$	0	0
$57$	−1.53777e9	−0.338516
$58$	0	0
$59$	2.31273e8	0.0421152	0.0210576	−	0.999778i	$-0.493297\pi$
$59$	2.31273e8	0.0421152	0.0210576	+	0.999778i	$0.493297\pi$
$60$	0	0
$61$	−2.27817e9	−0.345360	−0.172680	−	0.984978i	$-0.555243\pi$
$61$	−2.27817e9	−0.345360	−0.172680	+	0.984978i	$0.555243\pi$
$62$	0	0
$63$	−2.55264e9	−0.324054
$64$	0	0
$65$	2.89218e9	0.309173
$66$	0	0
$67$	4.67122e9	0.422687	0.211343	−	0.977412i	$-0.432216\pi$
$67$	4.67122e9	0.422687	0.211343	+	0.977412i	$0.432216\pi$
$68$	0	0
$69$	−4.96811e9	−0.382403
$70$	0	0
$71$	−4.69060e9	−0.308537	−0.154269	−	0.988029i	$-0.549302\pi$
$71$	−4.69060e9	−0.308537	−0.154269	+	0.988029i	$0.549302\pi$
$72$	0	0
$73$	−8.30010e9	−0.468606	−0.234303	−	0.972164i	$-0.575281\pi$
$73$	−8.30010e9	−0.468606	−0.234303	+	0.972164i	$0.575281\pi$
$74$	0	0
$75$	2.42475e9	0.117986
$76$	0	0
$77$	−3.06477e9	−0.129032
$78$	0	0
$79$	4.37744e10	1.60056	0.800278	−	0.599628i	$-0.204685\pi$
$79$	4.37744e10	1.60056	0.800278	+	0.599628i	$0.204685\pi$
$80$	0	0
$81$	1.85915e10	0.592443
$82$	0	0
$83$	2.21339e10	0.616776	0.308388	−	0.951261i	$-0.400210\pi$
$83$	2.21339e10	0.616776	0.308388	+	0.951261i	$0.400210\pi$
$84$	0	0
$85$	3.24062e10	0.792180
$86$	0	0
$87$	−1.43513e10	−0.308699
$88$	0	0
$89$	4.27452e10	0.811413	0.405707	−	0.914003i	$-0.367025\pi$
$89$	4.27452e10	0.811413	0.405707	+	0.914003i	$0.367025\pi$
$90$	0	0
$91$	8.38907e9	0.140924
$92$	0	0
$93$	−4.10651e10	−0.612092
$94$	0	0
$95$	5.60549e10	0.743248
$96$	0	0
$97$	−5.30341e10	−0.627062	−0.313531	−	0.949578i	$-0.601512\pi$
$97$	−5.30341e10	−0.627062	−0.313531	+	0.949578i	$0.601512\pi$
$98$	0	0
$99$	2.76954e10	0.292694
$100$	0	0
$101$	−3.67739e10	−0.348154	−0.174077	−	0.984732i	$-0.555694\pi$
$101$	−3.67739e10	−0.348154	−0.174077	+	0.984732i	$0.555694\pi$
$102$	0	0
$103$	−7.35288e10	−0.624961	−0.312480	−	0.949924i	$-0.601160\pi$
$103$	−7.35288e10	−0.624961	−0.312480	+	0.949924i	$0.601160\pi$
$104$	0	0
$105$	−1.54799e10	−0.118366
$106$	0	0
$107$	8.56549e10	0.590394	0.295197	−	0.955436i	$-0.404615\pi$
$107$	8.56549e10	0.590394	0.295197	+	0.955436i	$0.404615\pi$
$108$	0	0
$109$	−3.17304e11	−1.97529	−0.987643	−	0.156721i	$-0.949908\pi$
$109$	−3.17304e11	−1.97529	−0.987643	+	0.156721i	$0.949908\pi$
$110$	0	0
$111$	−1.50289e10	−0.0846547
$112$	0	0
$113$	−1.60720e11	−0.820613	−0.410307	−	0.911948i	$-0.634578\pi$
$113$	−1.60720e11	−0.820613	−0.410307	+	0.911948i	$0.634578\pi$
$114$	0	0
$115$	1.81098e11	0.839606
$116$	0	0
$117$	−7.58096e10	−0.319670
$118$	0	0
$119$	9.39977e10	0.361085
$120$	0	0
$121$	−2.52060e11	−0.883454
$122$	0	0
$123$	−1.38043e11	−0.442116
$124$	0	0
$125$	−3.71312e11	−1.08826
$126$	0	0
$127$	2.99574e11	0.804608	0.402304	−	0.915506i	$-0.368210\pi$
$127$	2.99574e11	0.804608	0.402304	+	0.915506i	$0.368210\pi$
$128$	0	0
$129$	−1.00750e11	−0.248314
$130$	0	0
$131$	3.12938e11	0.708705	0.354353	−	0.935112i	$-0.384701\pi$
$131$	3.12938e11	0.708705	0.354353	+	0.935112i	$0.384701\pi$
$132$	0	0
$133$	1.62593e11	0.338781
$134$	0	0
$135$	3.03047e11	0.581667
$136$	0	0
$137$	3.08870e11	0.546780	0.273390	−	0.961903i	$-0.411855\pi$
$137$	3.08870e11	0.546780	0.273390	+	0.961903i	$0.411855\pi$
$138$	0	0
$139$	−6.80809e10	−0.111287	−0.0556434	−	0.998451i	$-0.517721\pi$
$139$	−6.80809e10	−0.111287	−0.0556434	+	0.998451i	$0.517721\pi$
$140$	0	0
$141$	2.22771e11	0.336630
$142$	0	0
$143$	−9.10189e10	−0.127287
$144$	0	0
$145$	5.23135e11	0.677781
$146$	0	0
$147$	−4.49012e10	−0.0539527
$148$	0	0
$149$	1.41000e11	0.157288	0.0786440	−	0.996903i	$-0.474941\pi$
$149$	1.41000e11	0.157288	0.0786440	+	0.996903i	$0.474941\pi$
$150$	0	0
$151$	1.72220e11	0.178530	0.0892648	−	0.996008i	$-0.471548\pi$
$151$	1.72220e11	0.178530	0.0892648	+	0.996008i	$0.471548\pi$
$152$	0	0
$153$	−8.49429e11	−0.819076
$154$	0	0
$155$	1.49691e12	1.34391
$156$	0	0
$157$	−1.35620e11	−0.113469	−0.0567344	−	0.998389i	$-0.518069\pi$
$157$	−1.35620e11	−0.113469	−0.0567344	+	0.998389i	$0.518069\pi$
$158$	0	0
$159$	7.79706e11	0.608481
$160$	0	0
$161$	5.25295e11	0.382702
$162$	0	0
$163$	2.33121e12	1.58690	0.793452	−	0.608633i	$-0.208282\pi$
$163$	2.33121e12	1.58690	0.793452	+	0.608633i	$0.208282\pi$
$164$	0	0
$165$	1.67953e11	0.106912
$166$	0	0
$167$	−2.48250e12	−1.47893	−0.739466	−	0.673194i	$-0.764922\pi$
$167$	−2.48250e12	−1.47893	−0.739466	+	0.673194i	$0.764922\pi$
$168$	0	0
$169$	−1.54302e12	−0.860982
$170$	0	0
$171$	−1.46931e12	−0.768483
$172$	0	0
$173$	7.99340e11	0.392173	0.196087	−	0.980587i	$-0.437177\pi$
$173$	7.99340e11	0.392173	0.196087	+	0.980587i	$0.437177\pi$
$174$	0	0
$175$	−2.56378e11	−0.118078
$176$	0	0
$177$	−3.67623e10	−0.0159056
$178$	0	0
$179$	−3.06300e12	−1.24582	−0.622910	−	0.782294i	$-0.714050\pi$
$179$	−3.06300e12	−1.24582	−0.622910	+	0.782294i	$0.714050\pi$
$180$	0	0
$181$	−5.10164e12	−1.95199	−0.975996	−	0.217789i	$-0.930116\pi$
$181$	−5.10164e12	−1.95199	−0.975996	+	0.217789i	$0.930116\pi$
$182$	0	0
$183$	3.62129e11	0.130432
$184$	0	0
$185$	5.47836e11	0.185869
$186$	0	0
$187$	−1.01985e12	−0.326141
$188$	0	0
$189$	8.79022e11	0.265130
$190$	0	0
$191$	3.21291e12	0.914565	0.457282	−	0.889322i	$-0.348823\pi$
$191$	3.21291e12	0.914565	0.457282	+	0.889322i	$0.348823\pi$
$192$	0	0
$193$	−4.86872e12	−1.30873	−0.654365	−	0.756179i	$-0.727064\pi$
$193$	−4.86872e12	−1.30873	−0.654365	+	0.756179i	$0.727064\pi$
$194$	0	0
$195$	−4.59730e11	−0.116765
$196$	0	0
$197$	7.88978e12	1.89453	0.947263	−	0.320457i	$-0.103836\pi$
$197$	7.88978e12	1.89453	0.947263	+	0.320457i	$0.103836\pi$
$198$	0	0
$199$	−5.87885e12	−1.33537	−0.667684	−	0.744445i	$-0.732714\pi$
$199$	−5.87885e12	−1.33537	−0.667684	+	0.744445i	$0.732714\pi$
$200$	0	0
$201$	−7.42520e11	−0.159636
$202$	0	0
$203$	1.51741e12	0.308940
$204$	0	0
$205$	5.03197e12	0.970715
$206$	0	0
$207$	−4.74693e12	−0.868113
$208$	0	0
$209$	−1.76409e12	−0.305996
$210$	0	0
$211$	5.37728e12	0.885134	0.442567	−	0.896735i	$-0.354068\pi$
$211$	5.37728e12	0.885134	0.442567	+	0.896735i	$0.354068\pi$
$212$	0	0
$213$	7.45602e11	0.116525
$214$	0	0
$215$	3.67254e12	0.545199
$216$	0	0
$217$	4.34195e12	0.612571
$218$	0	0
$219$	1.31935e12	0.176978
$220$	0	0
$221$	2.79158e12	0.356199
$222$	0	0
$223$	−1.39082e13	−1.68886	−0.844431	−	0.535665i	$-0.820061\pi$
$223$	−1.39082e13	−1.68886	−0.844431	+	0.535665i	$0.820061\pi$
$224$	0	0
$225$	2.31681e12	0.267847
$226$	0	0
$227$	4.65668e12	0.512784	0.256392	−	0.966573i	$-0.417466\pi$
$227$	4.65668e12	0.512784	0.256392	+	0.966573i	$0.417466\pi$
$228$	0	0
$229$	1.57614e13	1.65387	0.826934	−	0.562300i	$-0.190083\pi$
$229$	1.57614e13	1.65387	0.826934	+	0.562300i	$0.190083\pi$
$230$	0	0
$231$	4.87165e11	0.0487315
$232$	0	0
$233$	1.94075e13	1.85145	0.925726	−	0.378194i	$-0.123455\pi$
$233$	1.94075e13	1.85145	0.925726	+	0.378194i	$0.123455\pi$
$234$	0	0
$235$	−8.12048e12	−0.739109
$236$	0	0
$237$	−6.95822e12	−0.604480
$238$	0	0
$239$	1.08340e13	0.898672	0.449336	−	0.893363i	$-0.351661\pi$
$239$	1.08340e13	0.898672	0.449336	+	0.893363i	$0.351661\pi$
$240$	0	0
$241$	−4.43982e12	−0.351780	−0.175890	−	0.984410i	$-0.556280\pi$
$241$	−4.43982e12	−0.351780	−0.175890	+	0.984410i	$0.556280\pi$
$242$	0	0
$243$	−1.22202e13	−0.925217
$244$	0	0
$245$	1.63675e12	0.118459
$246$	0	0
$247$	4.82877e12	0.334197
$248$	0	0
$249$	−3.51832e12	−0.232937
$250$	0	0
$251$	−2.26617e13	−1.43577	−0.717887	−	0.696159i	$-0.754891\pi$
$251$	−2.26617e13	−1.43577	−0.717887	+	0.696159i	$0.754891\pi$
$252$	0	0
$253$	−5.69929e12	−0.345667
$254$	0	0
$255$	−5.15117e12	−0.299182
$256$	0	0
$257$	2.30037e13	1.27987	0.639935	−	0.768429i	$-0.278961\pi$
$257$	2.30037e13	1.27987	0.639935	+	0.768429i	$0.278961\pi$
$258$	0	0
$259$	1.58906e12	0.0847210
$260$	0	0
$261$	−1.37124e13	−0.700794
$262$	0	0
$263$	6.20470e12	0.304064	0.152032	−	0.988376i	$-0.451418\pi$
$263$	6.20470e12	0.304064	0.152032	+	0.988376i	$0.451418\pi$
$264$	0	0
$265$	−2.84219e13	−1.33598
$266$	0	0
$267$	−6.79462e12	−0.306445
$268$	0	0
$269$	−1.38147e13	−0.598003	−0.299002	−	0.954253i	$-0.596654\pi$
$269$	−1.38147e13	−0.598003	−0.299002	+	0.954253i	$0.596654\pi$
$270$	0	0
$271$	4.43875e12	0.184471	0.0922357	−	0.995737i	$-0.470599\pi$
$271$	4.43875e12	0.184471	0.0922357	+	0.995737i	$0.470599\pi$
$272$	0	0
$273$	−1.33350e12	−0.0532228
$274$	0	0
$275$	2.78162e12	0.106652
$276$	0	0
$277$	−1.58721e13	−0.584783	−0.292391	−	0.956299i	$-0.594451\pi$
$277$	−1.58721e13	−0.584783	−0.292391	+	0.956299i	$0.594451\pi$
$278$	0	0
$279$	−3.92370e13	−1.38954
$280$	0	0
$281$	2.17431e13	0.740348	0.370174	−	0.928962i	$-0.379298\pi$
$281$	2.17431e13	0.740348	0.370174	+	0.928962i	$0.379298\pi$
$282$	0	0
$283$	−1.28231e13	−0.419922	−0.209961	−	0.977710i	$-0.567334\pi$
$283$	−1.28231e13	−0.419922	−0.209961	+	0.977710i	$0.567334\pi$
$284$	0	0
$285$	−8.91028e12	−0.280701
$286$	0	0
$287$	1.45958e13	0.442463
$288$	0	0
$289$	−2.99281e12	−0.0873256
$290$	0	0
$291$	8.43011e12	0.236822
$292$	0	0
$293$	4.28150e13	1.15831	0.579154	−	0.815218i	$-0.303383\pi$
$293$	4.28150e13	1.15831	0.579154	+	0.815218i	$0.303383\pi$
$294$	0	0
$295$	1.34007e12	0.0349225
$296$	0	0
$297$	−9.53712e12	−0.239473
$298$	0	0
$299$	1.56004e13	0.377524
$300$	0	0
$301$	1.06526e13	0.248508
$302$	0	0
$303$	5.84544e12	0.131487
$304$	0	0
$305$	−1.32004e13	−0.286377
$306$	0	0
$307$	5.49480e13	1.14998	0.574991	−	0.818160i	$-0.305005\pi$
$307$	5.49480e13	1.14998	0.574991	+	0.818160i	$0.305005\pi$
$308$	0	0
$309$	1.16879e13	0.236028
$310$	0	0
$311$	9.40405e13	1.83287	0.916437	−	0.400178i	$-0.131052\pi$
$311$	9.40405e13	1.83287	0.916437	+	0.400178i	$0.131052\pi$
$312$	0	0
$313$	−4.90137e13	−0.922197	−0.461099	−	0.887349i	$-0.652545\pi$
$313$	−4.90137e13	−0.922197	−0.461099	+	0.887349i	$0.652545\pi$
$314$	0	0
$315$	−1.47908e13	−0.268710
$316$	0	0
$317$	6.47329e13	1.13579	0.567896	−	0.823100i	$-0.307758\pi$
$317$	6.47329e13	1.13579	0.567896	+	0.823100i	$0.307758\pi$
$318$	0	0
$319$	−1.64634e13	−0.279043
$320$	0	0
$321$	−1.36154e13	−0.222973
$322$	0	0
$323$	5.41053e13	0.856299
$324$	0	0
$325$	−7.61401e12	−0.116481
$326$	0	0
$327$	5.04375e13	0.746004
$328$	0	0
$329$	−2.35543e13	−0.336894
$330$	0	0
$331$	−5.69893e13	−0.788386	−0.394193	−	0.919028i	$-0.628976\pi$
$331$	−5.69893e13	−0.788386	−0.394193	+	0.919028i	$0.628976\pi$
$332$	0	0
$333$	−1.43598e13	−0.192179
$334$	0	0
$335$	2.70664e13	0.350497
$336$	0	0
$337$	−1.34664e14	−1.68767	−0.843834	−	0.536604i	$-0.819707\pi$
$337$	−1.34664e14	−1.68767	−0.843834	+	0.536604i	$0.819707\pi$
$338$	0	0
$339$	2.55475e13	0.309920
$340$	0	0
$341$	−4.71089e13	−0.553291
$342$	0	0
$343$	4.74756e12	0.0539949
$344$	0	0
$345$	−2.87867e13	−0.317093
$346$	0	0
$347$	8.90396e13	0.950104	0.475052	−	0.879958i	$-0.342429\pi$
$347$	8.90396e13	0.950104	0.475052	+	0.879958i	$0.342429\pi$
$348$	0	0
$349$	1.16684e14	1.20635	0.603174	−	0.797610i	$-0.293902\pi$
$349$	1.16684e14	1.20635	0.603174	+	0.797610i	$0.293902\pi$
$350$	0	0
$351$	2.61056e13	0.261543
$352$	0	0
$353$	−6.64827e13	−0.645576	−0.322788	−	0.946471i	$-0.604620\pi$
$353$	−6.64827e13	−0.645576	−0.322788	+	0.946471i	$0.604620\pi$
$354$	0	0
$355$	−2.71788e13	−0.255843
$356$	0	0
$357$	−1.49415e13	−0.136370
$358$	0	0
$359$	−7.15976e13	−0.633693	−0.316847	−	0.948477i	$-0.602624\pi$
$359$	−7.15976e13	−0.633693	−0.316847	+	0.948477i	$0.602624\pi$
$360$	0	0
$361$	−2.29012e13	−0.196593
$362$	0	0
$363$	4.00665e13	0.333653
$364$	0	0
$365$	−4.80933e13	−0.388574
$366$	0	0
$367$	−1.32908e14	−1.04205	−0.521026	−	0.853541i	$-0.674450\pi$
$367$	−1.32908e14	−1.04205	−0.521026	+	0.853541i	$0.674450\pi$
$368$	0	0
$369$	−1.31898e14	−1.00367
$370$	0	0
$371$	−8.24410e13	−0.608957
$372$	0	0
$373$	−1.03081e13	−0.0739231	−0.0369616	−	0.999317i	$-0.511768\pi$
$373$	−1.03081e13	−0.0739231	−0.0369616	+	0.999317i	$0.511768\pi$
$374$	0	0
$375$	5.90225e13	0.411003
$376$	0	0
$377$	4.50647e13	0.304761
$378$	0	0
$379$	−2.16958e14	−1.42515	−0.712575	−	0.701595i	$-0.752471\pi$
$379$	−2.16958e14	−1.42515	−0.712575	+	0.701595i	$0.752471\pi$
$380$	0	0
$381$	−4.76193e13	−0.303875
$382$	0	0
$383$	−1.06519e14	−0.660439	−0.330219	−	0.943904i	$-0.607123\pi$
$383$	−1.06519e14	−0.660439	−0.330219	+	0.943904i	$0.607123\pi$
$384$	0	0
$385$	−1.77582e13	−0.106995
$386$	0	0
$387$	−9.62645e13	−0.563710
$388$	0	0
$389$	2.94992e14	1.67914	0.839570	−	0.543251i	$-0.182807\pi$
$389$	2.94992e14	1.67914	0.839570	+	0.543251i	$0.182807\pi$
$390$	0	0
$391$	1.74799e14	0.967314
$392$	0	0
$393$	−4.97434e13	−0.267656
$394$	0	0
$395$	2.53642e14	1.32720
$396$	0	0
$397$	1.96658e14	1.00084	0.500419	−	0.865783i	$-0.333179\pi$
$397$	1.96658e14	1.00084	0.500419	+	0.865783i	$0.333179\pi$
$398$	0	0
$399$	−2.58452e13	−0.127947
$400$	0	0
$401$	−3.24807e14	−1.56434	−0.782169	−	0.623066i	$-0.785887\pi$
$401$	−3.24807e14	−1.56434	−0.782169	+	0.623066i	$0.785887\pi$
$402$	0	0
$403$	1.28949e14	0.604284
$404$	0	0
$405$	1.07725e14	0.491262
$406$	0	0
$407$	−1.72408e13	−0.0765223
$408$	0	0
$409$	1.53170e14	0.661752	0.330876	−	0.943674i	$-0.392656\pi$
$409$	1.53170e14	0.661752	0.330876	+	0.943674i	$0.392656\pi$
$410$	0	0
$411$	−4.90969e13	−0.206502
$412$	0	0
$413$	3.88701e12	0.0159180
$414$	0	0
$415$	1.28250e14	0.511439
$416$	0	0
$417$	1.08219e13	0.0420296
$418$	0	0
$419$	3.14926e14	1.19133	0.595665	−	0.803233i	$-0.296889\pi$
$419$	3.14926e14	1.19133	0.595665	+	0.803233i	$0.296889\pi$
$420$	0	0
$421$	2.23239e13	0.0822656	0.0411328	−	0.999154i	$-0.486903\pi$
$421$	2.23239e13	0.0822656	0.0411328	+	0.999154i	$0.486903\pi$
$422$	0	0
$423$	2.12853e14	0.764203
$424$	0	0
$425$	−8.53133e13	−0.298454
$426$	0	0
$427$	−3.82892e13	−0.130534
$428$	0	0
$429$	1.44680e13	0.0480722
$430$	0	0
$431$	−3.95137e14	−1.27974	−0.639871	−	0.768483i	$-0.721012\pi$
$431$	−3.95137e14	−1.27974	−0.639871	+	0.768483i	$0.721012\pi$
$432$	0	0
$433$	4.45401e14	1.40627	0.703133	−	0.711058i	$-0.251784\pi$
$433$	4.45401e14	1.40627	0.703133	+	0.711058i	$0.251784\pi$
$434$	0	0
$435$	−8.31556e13	−0.255977
$436$	0	0
$437$	3.02361e14	0.907564
$438$	0	0
$439$	3.29533e13	0.0964594	0.0482297	−	0.998836i	$-0.484642\pi$
$439$	3.29533e13	0.0964594	0.0482297	+	0.998836i	$0.484642\pi$
$440$	0	0
$441$	−4.29023e13	−0.122481
$442$	0	0
$443$	−3.41910e14	−0.952120	−0.476060	−	0.879413i	$-0.657936\pi$
$443$	−3.41910e14	−0.952120	−0.476060	+	0.879413i	$0.657936\pi$
$444$	0	0
$445$	2.47678e14	0.672834
$446$	0	0
$447$	−2.24129e13	−0.0594027
$448$	0	0
$449$	−8.04902e13	−0.208155	−0.104078	−	0.994569i	$-0.533189\pi$
$449$	−8.04902e13	−0.208155	−0.104078	+	0.994569i	$0.533189\pi$
$450$	0	0
$451$	−1.58360e14	−0.399644
$452$	0	0
$453$	−2.73755e13	−0.0674250
$454$	0	0
$455$	4.86088e13	0.116856
$456$	0	0
$457$	−4.29694e14	−1.00837	−0.504185	−	0.863595i	$-0.668207\pi$
$457$	−4.29694e14	−1.00837	−0.504185	+	0.863595i	$0.668207\pi$
$458$	0	0
$459$	2.92507e14	0.670142
$460$	0	0
$461$	1.66103e14	0.371555	0.185778	−	0.982592i	$-0.440520\pi$
$461$	1.66103e14	0.371555	0.185778	+	0.982592i	$0.440520\pi$
$462$	0	0
$463$	−5.81638e14	−1.27045	−0.635225	−	0.772327i	$-0.719093\pi$
$463$	−5.81638e14	−1.27045	−0.635225	+	0.772327i	$0.719093\pi$
$464$	0	0
$465$	−2.37944e14	−0.507555
$466$	0	0
$467$	−6.43429e14	−1.34047	−0.670236	−	0.742148i	$-0.733807\pi$
$467$	−6.43429e14	−1.34047	−0.670236	+	0.742148i	$0.733807\pi$
$468$	0	0
$469$	7.85092e13	0.159761
$470$	0	0
$471$	2.15577e13	0.0428536
$472$	0	0
$473$	−1.15578e14	−0.224459
$474$	0	0
$475$	−1.47571e14	−0.280019
$476$	0	0
$477$	7.44995e14	1.38134
$478$	0	0
$479$	−6.39195e14	−1.15821	−0.579106	−	0.815252i	$-0.696598\pi$
$479$	−6.39195e14	−1.15821	−0.579106	+	0.815252i	$0.696598\pi$
$480$	0	0
$481$	4.71925e13	0.0835748
$482$	0	0
$483$	−8.34989e13	−0.144535
$484$	0	0
$485$	−3.07295e14	−0.519968
$486$	0	0
$487$	−3.41289e14	−0.564564	−0.282282	−	0.959331i	$-0.591091\pi$
$487$	−3.41289e14	−0.564564	−0.282282	+	0.959331i	$0.591091\pi$
$488$	0	0
$489$	−3.70561e14	−0.599324
$490$	0	0
$491$	−2.10683e14	−0.333182	−0.166591	−	0.986026i	$-0.553276\pi$
$491$	−2.10683e14	−0.333182	−0.166591	+	0.986026i	$0.553276\pi$
$492$	0	0
$493$	5.04940e14	0.780875
$494$	0	0
$495$	1.60476e14	0.242706
$496$	0	0
$497$	−7.88350e13	−0.116616
$498$	0	0
$499$	7.09436e14	1.02650	0.513251	−	0.858238i	$-0.328441\pi$
$499$	7.09436e14	1.02650	0.513251	+	0.858238i	$0.328441\pi$
$500$	0	0
$501$	3.94609e14	0.558546
$502$	0	0
$503$	−6.26362e14	−0.867364	−0.433682	−	0.901066i	$-0.642786\pi$
$503$	−6.26362e14	−0.867364	−0.433682	+	0.901066i	$0.642786\pi$
$504$	0	0
$505$	−2.13079e14	−0.288694
$506$	0	0
$507$	2.45273e14	0.325166
$508$	0	0
$509$	3.57529e14	0.463835	0.231917	−	0.972735i	$-0.425500\pi$
$509$	3.57529e14	0.463835	0.231917	+	0.972735i	$0.425500\pi$
$510$	0	0
$511$	−1.39500e14	−0.177116
$512$	0	0
$513$	5.05967e14	0.628748
$514$	0	0
$515$	−4.26048e14	−0.518226
$516$	0	0
$517$	2.55557e14	0.304292
$518$	0	0
$519$	−1.27060e14	−0.148112
$520$	0	0
$521$	−1.39109e15	−1.58763	−0.793813	−	0.608162i	$-0.791907\pi$
$521$	−1.39109e15	−1.58763	−0.793813	+	0.608162i	$0.791907\pi$
$522$	0	0
$523$	4.47096e14	0.499622	0.249811	−	0.968295i	$-0.419632\pi$
$523$	4.47096e14	0.499622	0.249811	+	0.968295i	$0.419632\pi$
$524$	0	0
$525$	4.07528e13	0.0445946
$526$	0	0
$527$	1.44485e15	1.54833
$528$	0	0
$529$	2.40353e13	0.0252257
$530$	0	0
$531$	−3.51257e13	−0.0361082
$532$	0	0
$533$	4.33472e14	0.436477
$534$	0	0
$535$	4.96310e14	0.489562
$536$	0	0
$537$	4.86883e14	0.470507
$538$	0	0
$539$	−5.15096e13	−0.0487697
$540$	0	0
$541$	5.31236e14	0.492836	0.246418	−	0.969164i	$-0.420746\pi$
$541$	5.31236e14	0.492836	0.246418	+	0.969164i	$0.420746\pi$
$542$	0	0
$543$	8.10939e14	0.737206
$544$	0	0
$545$	−1.83856e15	−1.63793
$546$	0	0
$547$	1.99720e15	1.74378	0.871888	−	0.489705i	$-0.162896\pi$
$547$	1.99720e15	1.74378	0.871888	+	0.489705i	$0.162896\pi$
$548$	0	0
$549$	3.46008e14	0.296100
$550$	0	0
$551$	8.73424e14	0.732641
$552$	0	0
$553$	7.35716e14	0.604954
$554$	0	0
$555$	−8.70820e13	−0.0701967
$556$	0	0
$557$	−4.91708e14	−0.388601	−0.194301	−	0.980942i	$-0.562244\pi$
$557$	−4.91708e14	−0.388601	−0.194301	+	0.980942i	$0.562244\pi$
$558$	0	0
$559$	3.16366e14	0.245146
$560$	0	0
$561$	1.62111e14	0.123173
$562$	0	0
$563$	9.47224e14	0.705759	0.352880	−	0.935669i	$-0.385203\pi$
$563$	9.47224e14	0.705759	0.352880	+	0.935669i	$0.385203\pi$
$564$	0	0
$565$	−9.31260e14	−0.680463
$566$	0	0
$567$	3.12467e14	0.223923
$568$	0	0
$569$	−2.95934e14	−0.208007	−0.104003	−	0.994577i	$-0.533165\pi$
$569$	−2.95934e14	−0.208007	−0.104003	+	0.994577i	$0.533165\pi$
$570$	0	0
$571$	−2.54748e15	−1.75636	−0.878178	−	0.478333i	$-0.841241\pi$
$571$	−2.54748e15	−1.75636	−0.878178	+	0.478333i	$0.841241\pi$
$572$	0	0
$573$	−5.10712e14	−0.345403
$574$	0	0
$575$	−4.76763e14	−0.316322
$576$	0	0
$577$	−1.69968e15	−1.10637	−0.553186	−	0.833058i	$-0.686588\pi$
$577$	−1.69968e15	−1.10637	−0.553186	+	0.833058i	$0.686588\pi$
$578$	0	0
$579$	7.73915e14	0.494266
$580$	0	0
$581$	3.72004e14	0.233119
$582$	0	0
$583$	8.94459e14	0.550026
$584$	0	0
$585$	−4.39263e14	−0.265074
$586$	0	0
$587$	−1.72206e15	−1.01986	−0.509928	−	0.860217i	$-0.670328\pi$
$587$	−1.72206e15	−1.01986	−0.509928	+	0.860217i	$0.670328\pi$
$588$	0	0
$589$	2.49924e15	1.45269
$590$	0	0
$591$	−1.25413e15	−0.715503
$592$	0	0
$593$	−3.28431e15	−1.83926	−0.919629	−	0.392787i	$-0.871511\pi$
$593$	−3.28431e15	−1.83926	−0.919629	+	0.392787i	$0.871511\pi$
$594$	0	0
$595$	5.44651e14	0.299416
$596$	0	0
$597$	9.34481e14	0.504327
$598$	0	0
$599$	−1.78426e15	−0.945391	−0.472695	−	0.881226i	$-0.656719\pi$
$599$	−1.78426e15	−0.945391	−0.472695	+	0.881226i	$0.656719\pi$
$600$	0	0
$601$	−2.88375e15	−1.50019	−0.750097	−	0.661328i	$-0.769993\pi$
$601$	−2.88375e15	−1.50019	−0.750097	+	0.661328i	$0.769993\pi$
$602$	0	0
$603$	−7.09464e14	−0.362398
$604$	0	0
$605$	−1.46051e15	−0.732572
$606$	0	0
$607$	2.90437e15	1.43059	0.715293	−	0.698824i	$-0.246293\pi$
$607$	2.90437e15	1.43059	0.715293	+	0.698824i	$0.246293\pi$
$608$	0	0
$609$	−2.41202e14	−0.116677
$610$	0	0
$611$	−6.99527e14	−0.332336
$612$	0	0
$613$	−1.67767e14	−0.0782841	−0.0391420	−	0.999234i	$-0.512462\pi$
$613$	−1.67767e14	−0.0782841	−0.0391420	+	0.999234i	$0.512462\pi$
$614$	0	0
$615$	−7.99863e14	−0.366609
$616$	0	0
$617$	1.09704e15	0.493916	0.246958	−	0.969026i	$-0.420569\pi$
$617$	1.09704e15	0.493916	0.246958	+	0.969026i	$0.420569\pi$
$618$	0	0
$619$	2.74678e15	1.21486	0.607429	−	0.794374i	$-0.292201\pi$
$619$	2.74678e15	1.21486	0.607429	+	0.794374i	$0.292201\pi$
$620$	0	0
$621$	1.63464e15	0.710262
$622$	0	0
$623$	7.18418e14	0.306685
$624$	0	0
$625$	−1.40666e15	−0.589996
$626$	0	0
$627$	2.80413e14	0.115565
$628$	0	0
$629$	5.28782e14	0.214140
$630$	0	0
$631$	−2.45153e15	−0.975608	−0.487804	−	0.872953i	$-0.662202\pi$
$631$	−2.45153e15	−0.975608	−0.487804	+	0.872953i	$0.662202\pi$
$632$	0	0
$633$	−8.54753e14	−0.334287
$634$	0	0
$635$	1.73582e15	0.667191
$636$	0	0
$637$	1.40995e14	0.0532644
$638$	0	0
$639$	7.12408e14	0.264530
$640$	0	0
$641$	−1.78421e15	−0.651220	−0.325610	−	0.945504i	$-0.605570\pi$
$641$	−1.78421e15	−0.651220	−0.325610	+	0.945504i	$0.605570\pi$
$642$	0	0
$643$	−5.11541e15	−1.83536	−0.917678	−	0.397324i	$-0.869939\pi$
$643$	−5.11541e15	−1.83536	−0.917678	+	0.397324i	$0.869939\pi$
$644$	0	0
$645$	−5.83774e14	−0.205905
$646$	0	0
$647$	4.33839e15	1.50437	0.752185	−	0.658952i	$-0.229000\pi$
$647$	4.33839e15	1.50437	0.752185	+	0.658952i	$0.229000\pi$
$648$	0	0
$649$	−4.21728e13	−0.0143776
$650$	0	0
$651$	−6.90181e14	−0.231349
$652$	0	0
$653$	3.13516e15	1.03332	0.516662	−	0.856189i	$-0.327174\pi$
$653$	3.13516e15	1.03332	0.516662	+	0.856189i	$0.327174\pi$
$654$	0	0
$655$	1.81325e15	0.587668
$656$	0	0
$657$	1.26062e15	0.401767
$658$	0	0
$659$	2.94468e15	0.922930	0.461465	−	0.887159i	$-0.347324\pi$
$659$	2.94468e15	0.922930	0.461465	+	0.887159i	$0.347324\pi$
$660$	0	0
$661$	−3.48122e15	−1.07306	−0.536529	−	0.843882i	$-0.680265\pi$
$661$	−3.48122e15	−1.07306	−0.536529	+	0.843882i	$0.680265\pi$
$662$	0	0
$663$	−4.43740e14	−0.134525
$664$	0	0
$665$	9.42115e14	0.280921
$666$	0	0
$667$	2.82180e15	0.827625
$668$	0	0
$669$	2.21080e15	0.637830
$670$	0	0
$671$	4.15426e14	0.117902
$672$	0	0
$673$	4.02445e15	1.12363	0.561815	−	0.827263i	$-0.310103\pi$
$673$	4.02445e15	1.12363	0.561815	+	0.827263i	$0.310103\pi$
$674$	0	0
$675$	−7.97809e14	−0.219143
$676$	0	0
$677$	4.11451e15	1.11194	0.555969	−	0.831203i	$-0.312347\pi$
$677$	4.11451e15	1.11194	0.555969	+	0.831203i	$0.312347\pi$
$678$	0	0
$679$	−8.91344e14	−0.237007
$680$	0	0
$681$	−7.40209e14	−0.193662
$682$	0	0
$683$	3.89066e15	1.00163	0.500817	−	0.865553i	$-0.333033\pi$
$683$	3.89066e15	1.00163	0.500817	+	0.865553i	$0.333033\pi$
$684$	0	0
$685$	1.78969e15	0.453397
$686$	0	0
$687$	−2.50538e15	−0.624614
$688$	0	0
$689$	−2.44837e15	−0.600718
$690$	0	0
$691$	−5.00642e15	−1.20892	−0.604461	−	0.796635i	$-0.706611\pi$
$691$	−5.00642e15	−1.20892	−0.604461	+	0.796635i	$0.706611\pi$
$692$	0	0
$693$	4.65477e14	0.110628
$694$	0	0
$695$	−3.94481e14	−0.0922805
$696$	0	0
$697$	4.85695e15	1.11836
$698$	0	0
$699$	−3.08495e15	−0.699236
$700$	0	0
$701$	−1.31859e15	−0.294213	−0.147106	−	0.989121i	$-0.546996\pi$
$701$	−1.31859e15	−0.294213	−0.147106	+	0.989121i	$0.546996\pi$
$702$	0	0
$703$	9.14665e14	0.200913
$704$	0	0
$705$	1.29080e15	0.279138
$706$	0	0
$707$	−6.18058e14	−0.131590
$708$	0	0
$709$	−3.96963e15	−0.832138	−0.416069	−	0.909333i	$-0.636593\pi$
$709$	−3.96963e15	−0.832138	−0.416069	+	0.909333i	$0.636593\pi$
$710$	0	0
$711$	−6.64845e15	−1.37226
$712$	0	0
$713$	8.07436e15	1.64103
$714$	0	0
$715$	−5.27391e14	−0.105548
$716$	0	0
$717$	−1.72214e15	−0.339400
$718$	0	0
$719$	9.02246e14	0.175112	0.0875561	−	0.996160i	$-0.472094\pi$
$719$	9.02246e14	0.175112	0.0875561	+	0.996160i	$0.472094\pi$
$720$	0	0
$721$	−1.23580e15	−0.236213
$722$	0	0
$723$	7.05738e14	0.132856
$724$	0	0
$725$	−1.37722e15	−0.255354
$726$	0	0
$727$	5.00190e15	0.913474	0.456737	−	0.889602i	$-0.349018\pi$
$727$	5.00190e15	0.913474	0.456737	+	0.889602i	$0.349018\pi$
$728$	0	0
$729$	−1.35095e15	−0.243018
$730$	0	0
$731$	3.54481e15	0.628127
$732$	0	0
$733$	−8.06085e15	−1.40705	−0.703524	−	0.710671i	$-0.748391\pi$
$733$	−8.06085e15	−1.40705	−0.703524	+	0.710671i	$0.748391\pi$
$734$	0	0
$735$	−2.60171e14	−0.0447382
$736$	0	0
$737$	−8.51801e14	−0.144300
$738$	0	0
$739$	−7.90434e15	−1.31923	−0.659616	−	0.751602i	$-0.729281\pi$
$739$	−7.90434e15	−1.31923	−0.659616	+	0.751602i	$0.729281\pi$
$740$	0	0
$741$	−7.67563e14	−0.126216
$742$	0	0
$743$	−9.43316e15	−1.52834	−0.764168	−	0.645017i	$-0.776850\pi$
$743$	−9.43316e15	−1.52834	−0.764168	+	0.645017i	$0.776850\pi$
$744$	0	0
$745$	8.16998e14	0.130425
$746$	0	0
$747$	−3.36169e15	−0.528803
$748$	0	0
$749$	1.43960e15	0.223148
$750$	0	0
$751$	−8.66699e15	−1.32388	−0.661940	−	0.749556i	$-0.730267\pi$
$751$	−8.66699e15	−1.32388	−0.661940	+	0.749556i	$0.730267\pi$
$752$	0	0
$753$	3.60222e15	0.542247
$754$	0	0
$755$	9.97894e14	0.148039
$756$	0	0
$757$	5.38943e15	0.787980	0.393990	−	0.919115i	$-0.371094\pi$
$757$	5.38943e15	0.787980	0.393990	+	0.919115i	$0.371094\pi$
$758$	0	0
$759$	9.05938e14	0.130548
$760$	0	0
$761$	−1.98058e15	−0.281305	−0.140652	−	0.990059i	$-0.544920\pi$
$761$	−1.98058e15	−0.281305	−0.140652	+	0.990059i	$0.544920\pi$
$762$	0	0
$763$	−5.33293e15	−0.746588
$764$	0	0
$765$	−4.92185e15	−0.679188
$766$	0	0
$767$	1.15438e14	0.0157027
$768$	0	0
$769$	9.02399e15	1.21005	0.605025	−	0.796206i	$-0.293163\pi$
$769$	9.02399e15	1.21005	0.605025	+	0.796206i	$0.293163\pi$
$770$	0	0
$771$	−3.65659e15	−0.483367
$772$	0	0
$773$	−2.06414e15	−0.269000	−0.134500	−	0.990914i	$-0.542943\pi$
$773$	−2.06414e15	−0.269000	−0.134500	+	0.990914i	$0.542943\pi$
$774$	0	0
$775$	−3.94080e15	−0.506320
$776$	0	0
$777$	−2.52591e14	−0.0319965
$778$	0	0
$779$	8.40136e15	1.04928
$780$	0	0
$781$	8.55336e14	0.105331
$782$	0	0
$783$	4.72196e15	0.573366
$784$	0	0
$785$	−7.85825e14	−0.0940898
$786$	0	0
$787$	1.24521e16	1.47021	0.735107	−	0.677951i	$-0.237132\pi$
$787$	1.24521e16	1.47021	0.735107	+	0.677951i	$0.237132\pi$
$788$	0	0
$789$	−9.86277e14	−0.114835
$790$	0	0
$791$	−2.70122e15	−0.310163
$792$	0	0
$793$	−1.13713e15	−0.128768
$794$	0	0
$795$	4.51785e15	0.504560
$796$	0	0
$797$	1.48282e16	1.63330	0.816652	−	0.577130i	$-0.195828\pi$
$797$	1.48282e16	1.63330	0.816652	+	0.577130i	$0.195828\pi$
$798$	0	0
$799$	−7.83804e15	−0.851530
$800$	0	0
$801$	−6.49213e15	−0.695678
$802$	0	0
$803$	1.51353e15	0.159976
$804$	0	0
$805$	3.04372e15	0.317341
$806$	0	0
$807$	2.19593e15	0.225847
$808$	0	0
$809$	1.90159e16	1.92930	0.964652	−	0.263527i	$-0.0848857\pi$
$809$	1.90159e16	1.92930	0.964652	+	0.263527i	$0.0848857\pi$
$810$	0	0
$811$	−1.58504e14	−0.0158645	−0.00793226	−	0.999969i	$-0.502525\pi$
$811$	−1.58504e14	−0.0158645	−0.00793226	+	0.999969i	$0.502525\pi$
$812$	0	0
$813$	−7.05567e14	−0.0696691
$814$	0	0
$815$	1.35078e16	1.31588
$816$	0	0
$817$	6.13167e15	0.589328
$818$	0	0
$819$	−1.27413e15	−0.120824
$820$	0	0
$821$	−2.03365e16	−1.90278	−0.951390	−	0.307990i	$-0.900344\pi$
$821$	−2.03365e16	−1.90278	−0.951390	+	0.307990i	$0.900344\pi$
$822$	0	0
$823$	−6.61731e13	−0.00610917	−0.00305459	−	0.999995i	$-0.500972\pi$
$823$	−6.61731e13	−0.00610917	−0.00305459	+	0.999995i	$0.500972\pi$
$824$	0	0
$825$	−4.42156e14	−0.0402790
$826$	0	0
$827$	5.46949e15	0.491662	0.245831	−	0.969313i	$-0.420939\pi$
$827$	5.46949e15	0.491662	0.245831	+	0.969313i	$0.420939\pi$
$828$	0	0
$829$	−6.24572e15	−0.554029	−0.277015	−	0.960866i	$-0.589345\pi$
$829$	−6.24572e15	−0.554029	−0.277015	+	0.960866i	$0.589345\pi$
$830$	0	0
$831$	2.52297e15	0.220854
$832$	0	0
$833$	1.57982e15	0.136477
$834$	0	0
$835$	−1.43843e16	−1.22635
$836$	0	0
$837$	1.35115e16	1.13688
$838$	0	0
$839$	1.56154e16	1.29677	0.648386	−	0.761312i	$-0.275444\pi$
$839$	1.56154e16	1.29677	0.648386	+	0.761312i	$0.275444\pi$
$840$	0	0
$841$	−4.04924e15	−0.331891
$842$	0	0
$843$	−3.45620e15	−0.279606
$844$	0	0
$845$	−8.94071e15	−0.713937
$846$	0	0
$847$	−4.23637e15	−0.333914
$848$	0	0
$849$	2.03832e15	0.158592
$850$	0	0
$851$	2.95503e15	0.226960
$852$	0	0
$853$	8.40801e14	0.0637490	0.0318745	−	0.999492i	$-0.489852\pi$
$853$	8.40801e14	0.0637490	0.0318745	+	0.999492i	$0.489852\pi$
$854$	0	0
$855$	−8.51361e15	−0.637236
$856$	0	0
$857$	−7.66011e15	−0.566032	−0.283016	−	0.959115i	$-0.591335\pi$
$857$	−7.66011e15	−0.566032	−0.283016	+	0.959115i	$0.591335\pi$
$858$	0	0
$859$	5.41003e15	0.394673	0.197336	−	0.980336i	$-0.436771\pi$
$859$	5.41003e15	0.394673	0.197336	+	0.980336i	$0.436771\pi$
$860$	0	0
$861$	−2.32009e15	−0.167104
$862$	0	0
$863$	1.81454e16	1.29035	0.645176	−	0.764034i	$-0.276784\pi$
$863$	1.81454e16	1.29035	0.645176	+	0.764034i	$0.276784\pi$
$864$	0	0
$865$	4.63162e15	0.325195
$866$	0	0
$867$	4.75727e14	0.0329801
$868$	0	0
$869$	−7.98230e15	−0.546411
$870$	0	0
$871$	2.33160e15	0.157599
$872$	0	0
$873$	8.05481e15	0.537622
$874$	0	0
$875$	−6.24065e15	−0.411325
$876$	0	0
$877$	−6.04433e15	−0.393415	−0.196707	−	0.980462i	$-0.563025\pi$
$877$	−6.04433e15	−0.393415	−0.196707	+	0.980462i	$0.563025\pi$
$878$	0	0
$879$	−6.80571e15	−0.437457
$880$	0	0
$881$	−1.43664e16	−0.911973	−0.455986	−	0.889987i	$-0.650713\pi$
$881$	−1.43664e16	−0.911973	−0.455986	+	0.889987i	$0.650713\pi$
$882$	0	0
$883$	−1.49995e15	−0.0940360	−0.0470180	−	0.998894i	$-0.514972\pi$
$883$	−1.49995e15	−0.0940360	−0.0470180	+	0.998894i	$0.514972\pi$
$884$	0	0
$885$	−2.13012e14	−0.0131891
$886$	0	0
$887$	−7.70581e14	−0.0471236	−0.0235618	−	0.999722i	$-0.507501\pi$
$887$	−7.70581e14	−0.0471236	−0.0235618	+	0.999722i	$0.507501\pi$
$888$	0	0
$889$	5.03495e15	0.304113
$890$	0	0
$891$	−3.39018e15	−0.202253
$892$	0	0
$893$	−1.35579e16	−0.798932
$894$	0	0
$895$	−1.77479e16	−1.03305
$896$	0	0
$897$	−2.47979e15	−0.142579
$898$	0	0
$899$	2.33243e16	1.32474
$900$	0	0
$901$	−2.74334e16	−1.53919
$902$	0	0
$903$	−1.69330e15	−0.0938537
$904$	0	0
$905$	−2.95605e16	−1.61862
$906$	0	0
$907$	−8.03321e15	−0.434559	−0.217280	−	0.976109i	$-0.569718\pi$
$907$	−8.03321e15	−0.434559	−0.217280	+	0.976109i	$0.569718\pi$
$908$	0	0
$909$	5.58521e15	0.298496
$910$	0	0
$911$	−2.36909e15	−0.125092	−0.0625460	−	0.998042i	$-0.519922\pi$
$911$	−2.36909e15	−0.125092	−0.0625460	+	0.998042i	$0.519922\pi$
$912$	0	0
$913$	−4.03613e15	−0.210560
$914$	0	0
$915$	2.09829e15	0.108155
$916$	0	0
$917$	5.25954e15	0.267865
$918$	0	0
$919$	1.96422e16	0.988448	0.494224	−	0.869335i	$-0.335452\pi$
$919$	1.96422e16	0.988448	0.494224	+	0.869335i	$0.335452\pi$
$920$	0	0
$921$	−8.73434e15	−0.434312
$922$	0	0
$923$	−2.34128e15	−0.115039
$924$	0	0
$925$	−1.44225e15	−0.0700261
$926$	0	0
$927$	1.11675e16	0.535820
$928$	0	0
$929$	−1.06153e16	−0.503323	−0.251661	−	0.967815i	$-0.580977\pi$
$929$	−1.06153e16	−0.503323	−0.251661	+	0.967815i	$0.580977\pi$
$930$	0	0
$931$	2.73271e15	0.128047
$932$	0	0
$933$	−1.49483e16	−0.692220
$934$	0	0
$935$	−5.90930e15	−0.270441
$936$	0	0
$937$	3.27953e16	1.48335	0.741676	−	0.670758i	$-0.234031\pi$
$937$	3.27953e16	1.48335	0.741676	+	0.670758i	$0.234031\pi$
$938$	0	0
$939$	7.79104e15	0.348285
$940$	0	0
$941$	−4.48095e16	−1.97983	−0.989914	−	0.141671i	$-0.954752\pi$
$941$	−4.48095e16	−1.97983	−0.989914	+	0.141671i	$0.954752\pi$
$942$	0	0
$943$	2.71425e16	1.18532
$944$	0	0
$945$	5.09331e15	0.219850
$946$	0	0
$947$	2.87511e16	1.22668	0.613338	−	0.789820i	$-0.289826\pi$
$947$	2.87511e16	1.22668	0.613338	+	0.789820i	$0.289826\pi$
$948$	0	0
$949$	−4.14292e15	−0.174720
$950$	0	0
$951$	−1.02897e16	−0.428953
$952$	0	0
$953$	3.88967e16	1.60288	0.801441	−	0.598073i	$-0.204067\pi$
$953$	3.88967e16	1.60288	0.801441	+	0.598073i	$0.204067\pi$
$954$	0	0
$955$	1.86165e16	0.758369
$956$	0	0
$957$	2.61697e15	0.105386
$958$	0	0
$959$	5.19118e15	0.206664
$960$	0	0
$961$	4.13321e16	1.62671
$962$	0	0
$963$	−1.30093e16	−0.506184
$964$	0	0
$965$	−2.82109e16	−1.08522
$966$	0	0
$967$	−1.85205e15	−0.0704380	−0.0352190	−	0.999380i	$-0.511213\pi$
$967$	−1.85205e15	−0.0704380	−0.0352190	+	0.999380i	$0.511213\pi$
$968$	0	0
$969$	−8.60038e15	−0.323398
$970$	0	0
$971$	−1.87407e16	−0.696754	−0.348377	−	0.937355i	$-0.613267\pi$
$971$	−1.87407e16	−0.696754	−0.348377	+	0.937355i	$0.613267\pi$
$972$	0	0
$973$	−1.14424e15	−0.0420625
$974$	0	0
$975$	1.21030e15	0.0439912
$976$	0	0
$977$	1.32671e16	0.476822	0.238411	−	0.971164i	$-0.423374\pi$
$977$	1.32671e16	0.476822	0.238411	+	0.971164i	$0.423374\pi$
$978$	0	0
$979$	−7.79462e15	−0.277007
$980$	0	0
$981$	4.81921e16	1.69354
$982$	0	0
$983$	7.51642e15	0.261196	0.130598	−	0.991435i	$-0.458310\pi$
$983$	7.51642e15	0.261196	0.130598	+	0.991435i	$0.458310\pi$
$984$	0	0
$985$	4.57157e16	1.57097
$986$	0	0
$987$	3.74411e15	0.127234
$988$	0	0
$989$	1.98097e16	0.665732
$990$	0	0
$991$	−6.20718e14	−0.0206295	−0.0103148	−	0.999947i	$-0.503283\pi$
$991$	−6.20718e14	−0.0206295	−0.0103148	+	0.999947i	$0.503283\pi$
$992$	0	0
$993$	9.05881e15	0.297749
$994$	0	0
$995$	−3.40638e16	−1.10730
$996$	0	0
$997$	−3.42702e16	−1.10178	−0.550888	−	0.834579i	$-0.685711\pi$
$997$	−3.42702e16	−1.10178	−0.550888	+	0.834579i	$0.685711\pi$
$998$	0	0
$999$	4.94491e15	0.157235

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	28.12.a.b.1.1	✓	3
3.2	odd	2		252.12.a.e.1.2		3
4.3	odd	2		112.12.a.f.1.3		3
7.2	even	3		196.12.e.c.165.3		6
7.3	odd	6		196.12.e.f.177.1		6
7.4	even	3		196.12.e.c.177.3		6
7.5	odd	6		196.12.e.f.165.1		6
7.6	odd	2		196.12.a.b.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.12.a.b.1.1	✓	3	1.1	even	1	trivial
112.12.a.f.1.3		3	4.3	odd	2
196.12.a.b.1.3		3	7.6	odd	2
196.12.e.c.165.3		6	7.2	even	3
196.12.e.c.177.3		6	7.4	even	3
196.12.e.f.165.1		6	7.5	odd	6
196.12.e.f.177.1		6	7.3	odd	6
252.12.a.e.1.2		3	3.2	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:	\( N \)	\(=\)	\( 28 = 2^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	28.a (trivial)

\(f(q)\)	\(=\)	\(q-158.956 q^{3} +5794.30 q^{5} +16807.0 q^{7} -151880. q^{9} +O(q^{10})\)	\(q-158.956 q^{3} +5794.30 q^{5} +16807.0 q^{7} -151880. q^{9} -182351. q^{11} +499142. q^{13} -921041. q^{15} +5.59277e6 q^{17} +9.67414e6 q^{19} -2.67158e6 q^{21} +3.12545e7 q^{23} -1.52542e7 q^{25} +5.23009e7 q^{27} +9.02844e7 q^{29} +2.58342e8 q^{31} +2.89858e7 q^{33} +9.73848e7 q^{35} +9.45474e7 q^{37} -7.93417e7 q^{39} +8.68434e8 q^{41} +6.33820e8 q^{43} -8.80038e8 q^{45} -1.40146e9 q^{47} +2.82475e8 q^{49} -8.89007e8 q^{51} -4.90516e9 q^{53} -1.05660e9 q^{55} -1.53777e9 q^{57} +2.31273e8 q^{59} -2.27817e9 q^{61} -2.55264e9 q^{63} +2.89218e9 q^{65} +4.67122e9 q^{67} -4.96811e9 q^{69} -4.69060e9 q^{71} -8.30010e9 q^{73} +2.42475e9 q^{75} -3.06477e9 q^{77} +4.37744e10 q^{79} +1.85915e10 q^{81} +2.21339e10 q^{83} +3.24062e10 q^{85} -1.43513e10 q^{87} +4.27452e10 q^{89} +8.38907e9 q^{91} -4.10651e10 q^{93} +5.60549e10 q^{95} -5.30341e10 q^{97} +2.76954e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 1132 q^{3} + 4986 q^{5} + 50421 q^{7} + 370759 q^{9}+O(q^{10}) \) 3 * q + 1132 * q^3 + 4986 * q^5 + 50421 * q^7 + 370759 * q^9	\( 3 q + 1132 q^{3} + 4986 q^{5} + 50421 q^{7} + 370759 q^{9} + 863796 q^{11} + 1006962 q^{13} + 1990704 q^{15} - 2561418 q^{17} - 4380708 q^{19} + 19025524 q^{21} + 29203392 q^{23} + 157500381 q^{25} + 217312408 q^{27} + 382205778 q^{29} + 302574048 q^{31} + 587964816 q^{33} + 83799702 q^{35} + 110922954 q^{37} - 364261312 q^{39} + 60153150 q^{41} - 858558972 q^{43} + 3341217234 q^{45} + 416749152 q^{47} + 847425747 q^{49} - 7453626648 q^{51} - 1976465382 q^{53} - 10626661512 q^{55} - 11041098904 q^{57} - 4571596908 q^{59} - 8042615598 q^{61} + 6231346513 q^{63} - 45509674572 q^{65} + 15101535372 q^{67} - 18210771120 q^{69} + 21355612152 q^{71} + 29008383630 q^{73} + 111158791924 q^{75} + 14517819372 q^{77} - 7958497584 q^{79} + 72564681139 q^{81} - 7461285060 q^{83} - 66658496220 q^{85} + 170953812216 q^{87} - 10300974450 q^{89} + 16924010334 q^{91} - 6942785584 q^{93} + 27813662208 q^{95} - 181271913834 q^{97} + 150953802948 q^{99}+O(q^{100}) \) 3 * q + 1132 * q^3 + 4986 * q^5 + 50421 * q^7 + 370759 * q^9 + 863796 * q^11 + 1006962 * q^13 + 1990704 * q^15 - 2561418 * q^17 - 4380708 * q^19 + 19025524 * q^21 + 29203392 * q^23 + 157500381 * q^25 + 217312408 * q^27 + 382205778 * q^29 + 302574048 * q^31 + 587964816 * q^33 + 83799702 * q^35 + 110922954 * q^37 - 364261312 * q^39 + 60153150 * q^41 - 858558972 * q^43 + 3341217234 * q^45 + 416749152 * q^47 + 847425747 * q^49 - 7453626648 * q^51 - 1976465382 * q^53 - 10626661512 * q^55 - 11041098904 * q^57 - 4571596908 * q^59 - 8042615598 * q^61 + 6231346513 * q^63 - 45509674572 * q^65 + 15101535372 * q^67 - 18210771120 * q^69 + 21355612152 * q^71 + 29008383630 * q^73 + 111158791924 * q^75 + 14517819372 * q^77 - 7958497584 * q^79 + 72564681139 * q^81 - 7461285060 * q^83 - 66658496220 * q^85 + 170953812216 * q^87 - 10300974450 * q^89 + 16924010334 * q^91 - 6942785584 * q^93 + 27813662208 * q^95 - 181271913834 * q^97 + 150953802948 * q^99

Introduction

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