LMFDB - Embedded newform 140.15.h.b.69.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [140,15,Mod(69,140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(140, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 15, names="a")

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

chi := DirichletCharacter("140.69");

S:= CuspForms(chi, 15);

N := Newforms(S);

Level:	$ N $	$=$	$ 140 = 2^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 15 $
Character orbit:	$[\chi]$	$=$	140.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$174.060555413$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{105}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 26 $ x^2 - x - 26
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 7 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			69.1
Root			$5.62348$ of defining polynomial
Character	$\chi$	$=$	140.69

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+545.063 q^{3} -78125.0 q^{5} +823543. q^{7} -4.48588e6 q^{9} +O(q^{10})$	$q+545.063 q^{3} -78125.0 q^{5} +823543. q^{7} -4.48588e6 q^{9} +2.82472e7 q^{11} +1.25463e8 q^{13} -4.25830e7 q^{15} +7.72493e8 q^{17} +4.48882e8 q^{21} +6.10352e9 q^{25} -5.05210e9 q^{27} -2.37653e10 q^{29} +1.53965e10 q^{33} -6.43393e10 q^{35} +6.83849e10 q^{39} +3.50459e11 q^{45} -9.77758e11 q^{47} +6.78223e11 q^{49} +4.21057e11 q^{51} -2.20681e12 q^{55} -3.69431e12 q^{63} -9.80176e12 q^{65} -1.79056e12 q^{71} +2.20336e13 q^{73} +3.32680e12 q^{75} +2.32628e13 q^{77} +3.71246e13 q^{79} +1.87021e13 q^{81} +3.37268e13 q^{83} -6.03510e13 q^{85} -1.29536e13 q^{87} +1.03324e14 q^{91} -1.50611e14 q^{97} -1.26713e14 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4031 q^{3} - 156250 q^{5} + 1647086 q^{7} + 2882915 q^{9}+O(q^{10}) $ 2 * q + 4031 * q^3 - 156250 * q^5 + 1647086 * q^7 + 2882915 * q^9	$ 2 q + 4031 q^{3} - 156250 q^{5} + 1647086 q^{7} + 2882915 q^{9} + 37379173 q^{11} + 65279611 q^{13} - 314921875 q^{15} + 626193259 q^{17} + 3319701833 q^{21} + 12207031250 q^{25} + 3961912691 q^{27} + 9775649497 q^{29} + 47230040239 q^{33} - 128678593750 q^{35} - 141408966047 q^{39} - 225227734375 q^{45} - 719081600801 q^{47} + 1356446145698 q^{49} - 88934587703 q^{51} - 2920247890625 q^{55} + 2374204467845 q^{63} - 5099969609375 q^{65} - 3581117991356 q^{71} + 44067195256828 q^{73} + 24603271484375 q^{75} + 30783356269939 q^{77} + 27088287440917 q^{79} + 14879681566028 q^{81} + 67453508527948 q^{83} - 48921348359375 q^{85} + 103968109323331 q^{87} + 53760566681773 q^{91} - 24587561871581 q^{97} - 59421580676570 q^{99}+O(q^{100}) $ 2 * q + 4031 * q^3 - 156250 * q^5 + 1647086 * q^7 + 2882915 * q^9 + 37379173 * q^11 + 65279611 * q^13 - 314921875 * q^15 + 626193259 * q^17 + 3319701833 * q^21 + 12207031250 * q^25 + 3961912691 * q^27 + 9775649497 * q^29 + 47230040239 * q^33 - 128678593750 * q^35 - 141408966047 * q^39 - 225227734375 * q^45 - 719081600801 * q^47 + 1356446145698 * q^49 - 88934587703 * q^51 - 2920247890625 * q^55 + 2374204467845 * q^63 - 5099969609375 * q^65 - 3581117991356 * q^71 + 44067195256828 * q^73 + 24603271484375 * q^75 + 30783356269939 * q^77 + 27088287440917 * q^79 + 14879681566028 * q^81 + 67453508527948 * q^83 - 48921348359375 * q^85 + 103968109323331 * q^87 + 53760566681773 * q^91 - 24587561871581 * q^97 - 59421580676570 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/140\mathbb{Z}\right)^\times$.

$n$	$57$	$71$	$101$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	545.063	0.249228	0.124614	−	0.992205i	$-0.460231\pi$
$3$	545.063	0.249228	0.124614	+	0.992205i	$0.460231\pi$
$4$	0	0
$5$	−78125.0	−1.00000
$5$	−78125.0	−1.00000
$6$	0	0
$7$	823543.	1.00000
$7$	823543.	1.00000
$8$	0	0
$9$	−4.48588e6	−0.937885
$10$	0	0
$11$	2.82472e7	1.44953	0.724763	−	0.688998i	$-0.241949\pi$
$11$	2.82472e7	1.44953	0.724763	+	0.688998i	$0.241949\pi$
$12$	0	0
$13$	1.25463e8	1.99945	0.999725	−	0.0234474i	$-0.00746421\pi$
$13$	1.25463e8	1.99945	0.999725	+	0.0234474i	$0.00746421\pi$
$14$	0	0
$15$	−4.25830e7	−0.249228
$16$	0	0
$17$	7.72493e8	1.88257	0.941287	−	0.337607i	$-0.109618\pi$
$17$	7.72493e8	1.88257	0.941287	+	0.337607i	$0.109618\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	4.48882e8	0.249228
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	6.10352e9	1.00000
$26$	0	0
$27$	−5.05210e9	−0.482976
$28$	0	0
$29$	−2.37653e10	−1.37771	−0.688855	−	0.724900i	$-0.741886\pi$
$29$	−2.37653e10	−1.37771	−0.688855	+	0.724900i	$0.741886\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	1.53965e10	0.361263
$34$	0	0
$35$	−6.43393e10	−1.00000
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	6.83849e10	0.498320
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	3.50459e11	0.937885
$46$	0	0
$47$	−9.77758e11	−1.92995	−0.964975	−	0.262340i	$-0.915506\pi$
$47$	−9.77758e11	−1.92995	−0.964975	+	0.262340i	$0.915506\pi$
$48$	0	0
$49$	6.78223e11	1.00000
$50$	0	0
$51$	4.21057e11	0.469191
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	−2.20681e12	−1.44953
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	−3.69431e12	−0.937885
$64$	0	0
$65$	−9.80176e12	−1.99945
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.79056e12	−0.196870	−0.0984351	−	0.995143i	$-0.531384\pi$
$71$	−1.79056e12	−0.196870	−0.0984351	+	0.995143i	$0.531384\pi$
$72$	0	0
$73$	2.20336e13	1.99446	0.997230	−	0.0743777i	$-0.0236970\pi$
$73$	2.20336e13	1.99446	0.997230	+	0.0743777i	$0.0236970\pi$
$74$	0	0
$75$	3.32680e12	0.249228
$76$	0	0
$77$	2.32628e13	1.44953
$78$	0	0
$79$	3.71246e13	1.93318	0.966590	−	0.256327i	$-0.0825123\pi$
$79$	3.71246e13	1.93318	0.966590	+	0.256327i	$0.0825123\pi$
$80$	0	0
$81$	1.87021e13	0.817514
$82$	0	0
$83$	3.37268e13	1.24288	0.621438	−	0.783463i	$-0.286549\pi$
$83$	3.37268e13	1.24288	0.621438	+	0.783463i	$0.286549\pi$
$84$	0	0
$85$	−6.03510e13	−1.88257
$86$	0	0
$87$	−1.29536e13	−0.343364
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	1.03324e14	1.99945
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1.50611e14	−1.86404	−0.932019	−	0.362409i	$-0.881954\pi$
$97$	−1.50611e14	−1.86404	−0.932019	+	0.362409i	$0.881954\pi$
$98$	0	0
$99$	−1.26713e14	−1.35949
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	1.74370e14	1.41779	0.708896	−	0.705313i	$-0.249194\pi$
$103$	1.74370e14	1.41779	0.708896	+	0.705313i	$0.249194\pi$
$104$	0	0
$105$	−3.50689e13	−0.249228
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	−2.99697e14	−1.63945	−0.819724	−	0.572759i	$-0.805873\pi$
$109$	−2.99697e14	−1.63945	−0.819724	+	0.572759i	$0.805873\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−5.62809e14	−1.87525
$118$	0	0
$119$	6.36181e14	1.88257
$120$	0	0
$121$	4.18153e14	1.10113
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−4.76837e14	−1.00000
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	3.94695e14	0.482976
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	−5.32939e14	−0.480999
$142$	0	0
$143$	3.54396e15	2.89826
$144$	0	0
$145$	1.85666e15	1.37771
$146$	0	0
$147$	3.69674e14	0.249228
$148$	0	0
$149$	3.07982e15	1.88895	0.944477	−	0.328577i	$-0.106569\pi$
$149$	3.07982e15	1.88895	0.944477	+	0.328577i	$0.106569\pi$
$150$	0	0
$151$	−3.49188e15	−1.95084	−0.975419	−	0.220358i	$-0.929278\pi$
$151$	−3.49188e15	−1.95084	−0.975419	+	0.220358i	$0.929278\pi$
$152$	0	0
$153$	−3.46531e15	−1.76564
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−2.59952e14	−0.110560	−0.0552798	−	0.998471i	$-0.517605\pi$
$157$	−2.59952e14	−0.110560	−0.0552798	+	0.998471i	$0.517605\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	−1.20285e15	−0.361263
$166$	0	0
$167$	−7.12643e15	−1.96724	−0.983619	−	0.180262i	$-0.942305\pi$
$167$	−7.12643e15	−1.96724	−0.983619	+	0.180262i	$0.942305\pi$
$168$	0	0
$169$	1.18035e16	2.99780
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−5.69537e15	−1.22800	−0.614002	−	0.789305i	$-0.710441\pi$
$173$	−5.69537e15	−1.22800	−0.614002	+	0.789305i	$0.710441\pi$
$174$	0	0
$175$	5.02651e15	1.00000
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.16777e16	1.98330	0.991648	−	0.128976i	$-0.0411692\pi$
$179$	1.16777e16	1.98330	0.991648	+	0.128976i	$0.0411692\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.18207e16	2.72884
$188$	0	0
$189$	−4.16062e15	−0.482976
$190$	0	0
$191$	1.11549e16	1.20291	0.601454	−	0.798907i	$-0.294588\pi$
$191$	1.11549e16	1.20291	0.601454	+	0.798907i	$0.294588\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	−5.34257e15	−0.498320
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.95718e16	−1.37771
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	1.10024e16	0.590894	0.295447	−	0.955359i	$-0.404531\pi$
$211$	1.10024e16	0.590894	0.295447	+	0.955359i	$0.404531\pi$
$212$	0	0
$213$	−9.75967e14	−0.0490657
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1.20097e16	0.497076
$220$	0	0
$221$	9.69189e16	3.76411
$222$	0	0
$223$	2.59887e16	0.947654	0.473827	−	0.880618i	$-0.342872\pi$
$223$	2.59887e16	0.947654	0.473827	+	0.880618i	$0.342872\pi$
$224$	0	0
$225$	−2.73796e16	−0.937885
$226$	0	0
$227$	4.13023e15	0.132982	0.0664911	−	0.997787i	$-0.478820\pi$
$227$	4.13023e15	0.132982	0.0664911	+	0.997787i	$0.478820\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	1.26797e16	0.361263
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	7.63873e16	1.92995
$236$	0	0
$237$	2.02352e16	0.481804
$238$	0	0
$239$	4.03303e16	0.905412	0.452706	−	0.891660i	$-0.350459\pi$
$239$	4.03303e16	0.905412	0.452706	+	0.891660i	$0.350459\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	3.43579e16	0.686724
$244$	0	0
$245$	−5.29862e16	−1.00000
$246$	0	0
$247$	0	0
$248$	0	0
$249$	1.83832e16	0.309760
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−3.28951e16	−0.469191
$256$	0	0
$257$	−5.60794e16	−0.757307	−0.378653	−	0.925539i	$-0.623613\pi$
$257$	−5.60794e16	−0.757307	−0.378653	+	0.925539i	$0.623613\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1.06608e17	1.29213
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	5.63179e16	0.498320
$274$	0	0
$275$	1.72407e17	1.44953
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−3.71075e16	−0.268236	−0.134118	−	0.990965i	$-0.542820\pi$
$281$	−3.71075e16	−0.268236	−0.134118	+	0.990965i	$0.542820\pi$
$282$	0	0
$283$	−1.58967e17	−1.09346	−0.546730	−	0.837309i	$-0.684128\pi$
$283$	−1.58967e17	−1.09346	−0.546730	+	0.837309i	$0.684128\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	4.28368e17	2.54409
$290$	0	0
$291$	−8.20925e16	−0.464571
$292$	0	0
$293$	2.71136e17	1.46256	0.731279	−	0.682078i	$-0.238924\pi$
$293$	2.71136e17	1.46256	0.731279	+	0.682078i	$0.238924\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.42708e17	−0.700087
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4.03888e16	0.157142	0.0785710	−	0.996909i	$-0.474964\pi$
$307$	4.03888e16	0.157142	0.0785710	+	0.996909i	$0.474964\pi$
$308$	0	0
$309$	9.50428e16	0.353354
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	−2.46075e17	−0.836099	−0.418049	−	0.908424i	$-0.637286\pi$
$313$	−2.46075e17	−0.836099	−0.418049	+	0.908424i	$0.637286\pi$
$314$	0	0
$315$	2.88618e17	0.937885
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	−6.71303e17	−1.99703
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	7.65763e17	1.99945
$326$	0	0
$327$	−1.63354e17	−0.408597
$328$	0	0
$329$	−8.05226e17	−1.92995
$330$	0	0
$331$	8.69979e17	1.99854	0.999270	−	0.0381916i	$-0.0121597\pi$
$331$	8.69979e17	1.99854	0.999270	+	0.0381916i	$0.0121597\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	5.58546e17	1.00000
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	−6.33849e17	−0.965687
$352$	0	0
$353$	−8.87449e17	−1.29933	−0.649666	−	0.760220i	$-0.725091\pi$
$353$	−8.87449e17	−1.29933	−0.649666	+	0.760220i	$0.725091\pi$
$354$	0	0
$355$	1.39887e17	0.196870
$356$	0	0
$357$	3.46759e17	0.469191
$358$	0	0
$359$	−4.05103e17	−0.527113	−0.263557	−	0.964644i	$-0.584896\pi$
$359$	−4.05103e17	−0.527113	−0.263557	+	0.964644i	$0.584896\pi$
$360$	0	0
$361$	7.99007e17	1.00000
$362$	0	0
$363$	2.27920e17	0.274432
$364$	0	0
$365$	−1.72137e18	−1.99446
$366$	0	0
$367$	1.72977e18	1.92897	0.964487	−	0.264131i	$-0.0850850\pi$
$367$	1.72977e18	1.92897	0.964487	+	0.264131i	$0.0850850\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	−2.59906e17	−0.249228
$376$	0	0
$377$	−2.98166e18	−2.75466
$378$	0	0
$379$	−2.10679e18	−1.87563	−0.937816	−	0.347134i	$-0.887155\pi$
$379$	−2.10679e18	−1.87563	−0.937816	+	0.347134i	$0.887155\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2.19092e18	−1.81232	−0.906161	−	0.422934i	$-0.861000\pi$
$383$	−2.19092e18	−1.81232	−0.906161	+	0.422934i	$0.861000\pi$
$384$	0	0
$385$	−1.81740e18	−1.44953
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.69550e18	1.99982	0.999912	−	0.0132570i	$-0.00421995\pi$
$389$	2.69550e18	1.99982	0.999912	+	0.0132570i	$0.00421995\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.90036e18	−1.93318
$396$	0	0
$397$	1.76746e18	1.13714	0.568572	−	0.822634i	$-0.307496\pi$
$397$	1.76746e18	1.13714	0.568572	+	0.822634i	$0.307496\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3.31276e18	−1.98692	−0.993459	−	0.114192i	$-0.963572\pi$
$401$	−3.31276e18	−1.98692	−0.993459	+	0.114192i	$0.963572\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−1.46110e18	−0.817514
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−2.63490e18	−1.24288
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	−2.43093e18	−1.03704	−0.518522	−	0.855064i	$-0.673518\pi$
$421$	−2.43093e18	−1.03704	−0.518522	+	0.855064i	$0.673518\pi$
$422$	0	0
$423$	4.38610e18	1.81007
$424$	0	0
$425$	4.71492e18	1.88257
$426$	0	0
$427$	0	0
$428$	0	0
$429$	1.93168e18	0.722328
$430$	0	0
$431$	−2.59467e18	−0.939165	−0.469582	−	0.882889i	$-0.655596\pi$
$431$	−2.59467e18	−0.939165	−0.469582	+	0.882889i	$0.655596\pi$
$432$	0	0
$433$	5.11915e18	1.79384	0.896919	−	0.442196i	$-0.145800\pi$
$433$	5.11915e18	1.79384	0.896919	+	0.442196i	$0.145800\pi$
$434$	0	0
$435$	1.01200e18	0.343364
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−3.04242e18	−0.937885
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1.67869e18	0.470781
$448$	0	0
$449$	−2.71535e18	−0.738077	−0.369039	−	0.929414i	$-0.620313\pi$
$449$	−2.71535e18	−0.738077	−0.369039	+	0.929414i	$0.620313\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−1.90330e18	−0.486204
$454$	0	0
$455$	−8.07217e18	−1.99945
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	−3.90271e18	−0.909238
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.12608e17	0.0645329	0.0322665	−	0.999479i	$-0.489727\pi$
$467$	3.12608e17	0.0645329	0.0322665	+	0.999479i	$0.489727\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−1.41690e17	−0.0275546
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.17665e19	1.86404
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.33629e19	1.94237	0.971184	−	0.238332i	$-0.0766006\pi$
$491$	1.33629e19	1.94237	0.971184	+	0.238332i	$0.0766006\pi$
$492$	0	0
$493$	−1.83585e19	−2.59364
$494$	0	0
$495$	9.89948e18	1.35949
$496$	0	0
$497$	−1.47460e18	−0.196870
$498$	0	0
$499$	−1.36420e19	−1.77082	−0.885409	−	0.464813i	$-0.846122\pi$
$499$	−1.36420e19	−1.77082	−0.885409	+	0.464813i	$0.846122\pi$
$500$	0	0
$501$	−3.88435e18	−0.490291
$502$	0	0
$503$	1.42659e19	1.75114	0.875572	−	0.483088i	$-0.160485\pi$
$503$	1.42659e19	1.75114	0.875572	+	0.483088i	$0.160485\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6.43363e18	0.747137
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	1.81456e19	1.99446
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1.36227e19	−1.41779
$516$	0	0
$517$	−2.76189e19	−2.79752
$518$	0	0
$519$	−3.10434e18	−0.306053
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	−1.70696e19	−1.59482	−0.797410	−	0.603438i	$-0.793797\pi$
$523$	−1.70696e19	−1.59482	−0.797410	+	0.603438i	$0.793797\pi$
$524$	0	0
$525$	2.73976e18	0.249228
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1.15928e19	1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	6.36510e18	0.494294
$538$	0	0
$539$	1.91579e19	1.44953
$540$	0	0
$541$	7.47862e18	0.551367	0.275683	−	0.961248i	$-0.411096\pi$
$541$	7.47862e18	0.551367	0.275683	+	0.961248i	$0.411096\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.34139e19	1.63945
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	3.05737e19	1.93318
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	1.18937e19	0.680105
$562$	0	0
$563$	−2.25342e18	−0.125685	−0.0628426	−	0.998023i	$-0.520017\pi$
$563$	−2.25342e18	−0.125685	−0.0628426	+	0.998023i	$0.520017\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.54020e19	0.817514
$568$	0	0
$569$	7.46856e18	0.386768	0.193384	−	0.981123i	$-0.438054\pi$
$569$	7.46856e18	0.386768	0.193384	+	0.981123i	$0.438054\pi$
$570$	0	0
$571$	2.62350e19	1.32565	0.662825	−	0.748774i	$-0.269357\pi$
$571$	2.62350e19	1.32565	0.662825	+	0.748774i	$0.269357\pi$
$572$	0	0
$573$	6.08012e18	0.299799
$574$	0	0
$575$	0	0
$576$	0	0
$577$	4.00375e19	1.88034	0.940169	−	0.340708i	$-0.110667\pi$
$577$	4.00375e19	1.88034	0.940169	+	0.340708i	$0.110667\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2.77754e19	1.24288
$582$	0	0
$583$	0	0
$584$	0	0
$585$	4.39695e19	1.87525
$586$	0	0
$587$	−4.11324e19	−1.71284	−0.856420	−	0.516280i	$-0.827317\pi$
$587$	−4.11324e19	−1.71284	−0.856420	+	0.516280i	$0.827317\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1.74447e19	−0.676522	−0.338261	−	0.941052i	$-0.609839\pi$
$593$	−1.74447e19	−0.676522	−0.338261	+	0.941052i	$0.609839\pi$
$594$	0	0
$595$	−4.97017e19	−1.88257
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4.20725e19	1.52058	0.760292	−	0.649582i	$-0.225056\pi$
$599$	4.20725e19	1.52058	0.760292	+	0.649582i	$0.225056\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.26682e19	−1.10113
$606$	0	0
$607$	−4.81903e19	−1.58723	−0.793613	−	0.608422i	$-0.791803\pi$
$607$	−4.81903e19	−1.58723	−0.793613	+	0.608422i	$0.791803\pi$
$608$	0	0
$609$	−1.06678e19	−0.343364
$610$	0	0
$611$	−1.22672e20	−3.85884
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	3.72529e19	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	5.16000e19	1.29552	0.647761	−	0.761844i	$-0.275706\pi$
$631$	5.16000e19	1.29552	0.647761	+	0.761844i	$0.275706\pi$
$632$	0	0
$633$	5.99699e18	0.147268
$634$	0	0
$635$	0	0
$636$	0	0
$637$	8.50916e19	1.99945
$638$	0	0
$639$	8.03223e18	0.184642
$640$	0	0
$641$	−3.26194e19	−0.733617	−0.366809	−	0.930296i	$-0.619550\pi$
$641$	−3.26194e19	−0.733617	−0.366809	+	0.930296i	$0.619550\pi$
$642$	0	0
$643$	8.83446e19	1.94403	0.972016	−	0.234916i	$-0.0754815\pi$
$643$	8.83446e19	1.94403	0.972016	+	0.234916i	$0.0754815\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.96839e19	0.625447	0.312724	−	0.949844i	$-0.398759\pi$
$647$	2.96839e19	0.625447	0.312724	+	0.949844i	$0.398759\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−9.88400e19	−1.87057
$658$	0	0
$659$	9.58717e19	1.77620	0.888101	−	0.459649i	$-0.152025\pi$
$659$	9.58717e19	1.77620	0.888101	+	0.459649i	$0.152025\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	5.28269e19	0.938124
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	1.41654e19	0.236182
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	−3.08356e19	−0.482976
$676$	0	0
$677$	5.95328e19	0.913346	0.456673	−	0.889635i	$-0.349041\pi$
$677$	5.95328e19	0.913346	0.456673	+	0.889635i	$0.349041\pi$
$678$	0	0
$679$	−1.24035e20	−1.86404
$680$	0	0
$681$	2.25123e18	0.0331429
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	−1.04354e20	−1.35949
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−8.24234e19	−0.990888	−0.495444	−	0.868640i	$-0.664995\pi$
$701$	−8.24234e19	−0.990888	−0.495444	+	0.868640i	$0.664995\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	4.16359e19	0.480999
$706$	0	0
$707$	0	0
$708$	0	0
$709$	4.64250e19	0.515500	0.257750	−	0.966212i	$-0.417019\pi$
$709$	4.64250e19	0.515500	0.257750	+	0.966212i	$0.417019\pi$
$710$	0	0
$711$	−1.66536e20	−1.81310
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−2.76872e20	−2.89826
$716$	0	0
$717$	2.19825e19	0.225654
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	1.43602e20	1.41779
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.45052e20	−1.37771
$726$	0	0
$727$	−4.15590e19	−0.387190	−0.193595	−	0.981081i	$-0.562015\pi$
$727$	−4.15590e19	−0.387190	−0.193595	+	0.981081i	$0.562015\pi$
$728$	0	0
$729$	−7.07244e19	−0.646363
$730$	0	0
$731$	0	0
$732$	0	0
$733$	9.44667e19	0.830904	0.415452	−	0.909615i	$-0.363623\pi$
$733$	9.44667e19	0.830904	0.415452	+	0.909615i	$0.363623\pi$
$734$	0	0
$735$	−2.88808e19	−0.249228
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−5.58631e19	−0.464102	−0.232051	−	0.972704i	$-0.574544\pi$
$739$	−5.58631e19	−0.464102	−0.232051	+	0.972704i	$0.574544\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	−2.40611e20	−1.88895
$746$	0	0
$747$	−1.51294e20	−1.16568
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−8.28717e19	−0.615073	−0.307537	−	0.951536i	$-0.599505\pi$
$751$	−8.28717e19	−0.615073	−0.307537	+	0.951536i	$0.599505\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.72803e20	1.95084
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	−2.46814e20	−1.63945
$764$	0	0
$765$	2.70727e20	1.76564
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	−3.05668e19	−0.188742
$772$	0	0
$773$	1.30358e19	0.0790464	0.0395232	−	0.999219i	$-0.487416\pi$
$773$	1.30358e19	0.0790464	0.0395232	+	0.999219i	$0.487416\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−5.05782e19	−0.285369
$782$	0	0
$783$	1.20065e20	0.665401
$784$	0	0
$785$	2.03088e19	0.110560
$786$	0	0
$787$	2.59515e19	0.138784	0.0693919	−	0.997589i	$-0.477894\pi$
$787$	2.59515e19	0.138784	0.0693919	+	0.997589i	$0.477894\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−4.05616e20	−1.98567	−0.992834	−	0.119505i	$-0.961869\pi$
$797$	−4.05616e20	−1.98567	−0.992834	+	0.119505i	$0.961869\pi$
$798$	0	0
$799$	−7.55311e20	−3.63328
$800$	0	0
$801$	0	0
$802$	0	0
$803$	6.22387e20	2.89102
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2.66240e20	1.17391	0.586954	−	0.809620i	$-0.300327\pi$
$809$	2.66240e20	1.17391	0.586954	+	0.809620i	$0.300327\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−4.63498e20	−1.87525
$820$	0	0
$821$	−4.63013e20	−1.84158	−0.920792	−	0.390054i	$-0.872456\pi$
$821$	−4.63013e20	−1.84158	−0.920792	+	0.390054i	$0.872456\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	9.39727e19	0.361263
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	5.23923e20	1.88257
$834$	0	0
$835$	5.56752e20	1.96724
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	2.67232e20	0.898082
$842$	0	0
$843$	−2.02259e19	−0.0668521
$844$	0	0
$845$	−9.22146e20	−2.99780
$846$	0	0
$847$	3.44367e20	1.10113
$848$	0	0
$849$	−8.66472e19	−0.272521
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−2.27204e20	−0.691470	−0.345735	−	0.938332i	$-0.612370\pi$
$853$	−2.27204e20	−0.691470	−0.345735	+	0.938332i	$0.612370\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.14755e20	0.337991	0.168995	−	0.985617i	$-0.445948\pi$
$857$	1.14755e20	0.337991	0.168995	+	0.985617i	$0.445948\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	4.44951e20	1.22800
$866$	0	0
$867$	2.33487e20	0.634058
$868$	0	0
$869$	1.04867e21	2.80220
$870$	0	0
$871$	0	0
$872$	0	0
$873$	6.75623e20	1.74825
$874$	0	0
$875$	−3.92696e20	−1.00000
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	1.47786e20	0.364511
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−7.94212e20	−1.83853	−0.919266	−	0.393637i	$-0.871217\pi$
$887$	−7.94212e20	−1.83853	−0.919266	+	0.393637i	$0.871217\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	5.28281e20	1.18501
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−9.12323e20	−1.98330
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	9.81917e20	1.88559	0.942793	−	0.333379i	$-0.108189\pi$
$911$	9.81917e20	1.88559	0.942793	+	0.333379i	$0.108189\pi$
$912$	0	0
$913$	9.52686e20	1.80158
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−1.93258e20	−0.349082	−0.174541	−	0.984650i	$-0.555844\pi$
$919$	−1.93258e20	−0.349082	−0.174541	+	0.984650i	$0.555844\pi$
$920$	0	0
$921$	2.20144e19	0.0391643
$922$	0	0
$923$	−2.24648e20	−0.393632
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−7.82204e20	−1.32973
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1.70475e21	−2.72884
$936$	0	0
$937$	1.07637e21	1.69740	0.848699	−	0.528877i	$-0.177387\pi$
$937$	1.07637e21	1.69740	0.848699	+	0.528877i	$0.177387\pi$
$938$	0	0
$939$	−1.34126e20	−0.208380
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	3.25049e20	0.482976
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	2.76439e21	3.98782
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	−8.71477e20	−1.20291
$956$	0	0
$957$	−3.65902e20	−0.497716
$958$	0	0
$959$	0	0
$960$	0	0
$961$	7.56944e20	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	4.17388e20	0.498320
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	1.34441e21	1.53761
$982$	0	0
$983$	−1.03527e20	−0.116729	−0.0583644	−	0.998295i	$-0.518589\pi$
$983$	−1.03527e20	−0.116729	−0.0583644	+	0.998295i	$0.518589\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−4.38898e20	−0.480999
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−9.51633e20	−1.01380	−0.506902	−	0.862004i	$-0.669209\pi$
$991$	−9.51633e20	−1.01380	−0.506902	+	0.862004i	$0.669209\pi$
$992$	0	0
$993$	4.74193e20	0.498093
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.95791e21	−1.99953	−0.999764	−	0.0217340i	$-0.993081\pi$
$997$	−1.95791e21	−1.99953	−0.999764	+	0.0217340i	$0.993081\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.15.h.b.69.1	yes	2
5.4	even	2		140.15.h.a.69.2	✓	2
7.6	odd	2		140.15.h.a.69.2	✓	2
35.34	odd	2	CM	140.15.h.b.69.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.15.h.a.69.2	✓	2	5.4	even	2
140.15.h.a.69.2	✓	2	7.6	odd	2
140.15.h.b.69.1	yes	2	1.1	even	1	trivial
140.15.h.b.69.1	yes	2	35.34	odd	2	CM

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:	\( N \)	\(=\)	\( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	140.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+545.063 q^{3} -78125.0 q^{5} +823543. q^{7} -4.48588e6 q^{9} +O(q^{10})\)	\(q+545.063 q^{3} -78125.0 q^{5} +823543. q^{7} -4.48588e6 q^{9} +2.82472e7 q^{11} +1.25463e8 q^{13} -4.25830e7 q^{15} +7.72493e8 q^{17} +4.48882e8 q^{21} +6.10352e9 q^{25} -5.05210e9 q^{27} -2.37653e10 q^{29} +1.53965e10 q^{33} -6.43393e10 q^{35} +6.83849e10 q^{39} +3.50459e11 q^{45} -9.77758e11 q^{47} +6.78223e11 q^{49} +4.21057e11 q^{51} -2.20681e12 q^{55} -3.69431e12 q^{63} -9.80176e12 q^{65} -1.79056e12 q^{71} +2.20336e13 q^{73} +3.32680e12 q^{75} +2.32628e13 q^{77} +3.71246e13 q^{79} +1.87021e13 q^{81} +3.37268e13 q^{83} -6.03510e13 q^{85} -1.29536e13 q^{87} +1.03324e14 q^{91} -1.50611e14 q^{97} -1.26713e14 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4031 q^{3} - 156250 q^{5} + 1647086 q^{7} + 2882915 q^{9}+O(q^{10}) \) 2 * q + 4031 * q^3 - 156250 * q^5 + 1647086 * q^7 + 2882915 * q^9	\( 2 q + 4031 q^{3} - 156250 q^{5} + 1647086 q^{7} + 2882915 q^{9} + 37379173 q^{11} + 65279611 q^{13} - 314921875 q^{15} + 626193259 q^{17} + 3319701833 q^{21} + 12207031250 q^{25} + 3961912691 q^{27} + 9775649497 q^{29} + 47230040239 q^{33} - 128678593750 q^{35} - 141408966047 q^{39} - 225227734375 q^{45} - 719081600801 q^{47} + 1356446145698 q^{49} - 88934587703 q^{51} - 2920247890625 q^{55} + 2374204467845 q^{63} - 5099969609375 q^{65} - 3581117991356 q^{71} + 44067195256828 q^{73} + 24603271484375 q^{75} + 30783356269939 q^{77} + 27088287440917 q^{79} + 14879681566028 q^{81} + 67453508527948 q^{83} - 48921348359375 q^{85} + 103968109323331 q^{87} + 53760566681773 q^{91} - 24587561871581 q^{97} - 59421580676570 q^{99}+O(q^{100}) \) 2 * q + 4031 * q^3 - 156250 * q^5 + 1647086 * q^7 + 2882915 * q^9 + 37379173 * q^11 + 65279611 * q^13 - 314921875 * q^15 + 626193259 * q^17 + 3319701833 * q^21 + 12207031250 * q^25 + 3961912691 * q^27 + 9775649497 * q^29 + 47230040239 * q^33 - 128678593750 * q^35 - 141408966047 * q^39 - 225227734375 * q^45 - 719081600801 * q^47 + 1356446145698 * q^49 - 88934587703 * q^51 - 2920247890625 * q^55 + 2374204467845 * q^63 - 5099969609375 * q^65 - 3581117991356 * q^71 + 44067195256828 * q^73 + 24603271484375 * q^75 + 30783356269939 * q^77 + 27088287440917 * q^79 + 14879681566028 * q^81 + 67453508527948 * q^83 - 48921348359375 * q^85 + 103968109323331 * q^87 + 53760566681773 * q^91 - 24587561871581 * q^97 - 59421580676570 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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