LMFDB - Embedded newform 140.6.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [140,6,Mod(1,140)]

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, 0, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("140.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 140 = 2^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	140.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$22.4537347738$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1009}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 252 $ x^2 - x - 252
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-15.3824$ of defining polynomial
Character	$\chi$	$=$	140.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.38238 q^{3} -25.0000 q^{5} +49.0000 q^{7} -223.795 q^{9} +579.441 q^{11} -426.030 q^{13} -109.560 q^{15} -2010.09 q^{17} -402.294 q^{19} +214.737 q^{21} +1516.77 q^{23} +625.000 q^{25} -2045.67 q^{27} -7789.51 q^{29} -4700.59 q^{31} +2539.33 q^{33} -1225.00 q^{35} +1773.42 q^{37} -1867.03 q^{39} -7354.41 q^{41} -8562.31 q^{43} +5594.87 q^{45} +3748.17 q^{47} +2401.00 q^{49} -8808.98 q^{51} -33514.8 q^{53} -14486.0 q^{55} -1763.01 q^{57} -11302.6 q^{59} -24964.2 q^{61} -10965.9 q^{63} +10650.7 q^{65} -35315.7 q^{67} +6647.06 q^{69} +70973.7 q^{71} +17070.3 q^{73} +2738.99 q^{75} +28392.6 q^{77} +10103.6 q^{79} +45417.2 q^{81} +46105.8 q^{83} +50252.2 q^{85} -34136.6 q^{87} -21494.0 q^{89} -20875.5 q^{91} -20599.8 q^{93} +10057.4 q^{95} -65031.4 q^{97} -129676. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 23 q^{3} - 50 q^{5} + 98 q^{7} + 283 q^{9} + 873 q^{11} - 185 q^{13} + 575 q^{15} - 2019 q^{17} - 614 q^{19} - 1127 q^{21} - 1350 q^{23} + 1250 q^{25} - 9269 q^{27} - 999 q^{29} - 9020 q^{31} - 5499 q^{33}+ \cdots + 19098 q^{99}+O(q^{100}) $ 2 * q - 23 * q^3 - 50 * q^5 + 98 * q^7 + 283 * q^9 + 873 * q^11 - 185 * q^13 + 575 * q^15 - 2019 * q^17 - 614 * q^19 - 1127 * q^21 - 1350 * q^23 + 1250 * q^25 - 9269 * q^27 - 999 * q^29 - 9020 * q^31 - 5499 * q^33 - 2450 * q^35 - 9032 * q^37 - 8467 * q^39 - 18330 * q^41 - 3974 * q^43 - 7075 * q^45 - 21759 * q^47 + 4802 * q^49 - 8565 * q^51 + 2154 * q^53 - 21825 * q^55 + 4034 * q^57 - 28704 * q^59 + 12394 * q^61 + 13867 * q^63 + 4625 * q^65 - 101888 * q^67 + 85146 * q^69 + 91632 * q^71 - 35996 * q^73 - 14375 * q^75 + 42777 * q^77 - 10001 * q^79 + 120058 * q^81 + 78108 * q^83 + 50475 * q^85 - 220077 * q^87 + 33438 * q^89 - 9065 * q^91 + 97676 * q^93 + 15350 * q^95 - 85751 * q^97 + 19098 * q^99

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
$3$	4.38238	0.281130	0.140565	−	0.990071i	$-0.455108\pi$
$3$	4.38238	0.281130	0.140565	+	0.990071i	$0.455108\pi$
$4$	0	0
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	−223.795	−0.920966
$10$	0	0
$11$	579.441	1.44387	0.721935	−	0.691961i	$-0.243253\pi$
$11$	579.441	1.44387	0.721935	+	0.691961i	$0.243253\pi$
$12$	0	0
$13$	−426.030	−0.699168	−0.349584	−	0.936905i	$-0.613677\pi$
$13$	−426.030	−0.699168	−0.349584	+	0.936905i	$0.613677\pi$
$14$	0	0
$15$	−109.560	−0.125725
$16$	0	0
$17$	−2010.09	−1.68691	−0.843457	−	0.537196i	$-0.819484\pi$
$17$	−2010.09	−1.68691	−0.843457	+	0.537196i	$0.819484\pi$
$18$	0	0
$19$	−402.294	−0.255658	−0.127829	−	0.991796i	$-0.540801\pi$
$19$	−402.294	−0.255658	−0.127829	+	0.991796i	$0.540801\pi$
$20$	0	0
$21$	214.737	0.106257
$22$	0	0
$23$	1516.77	0.597860	0.298930	−	0.954275i	$-0.403370\pi$
$23$	1516.77	0.597860	0.298930	+	0.954275i	$0.403370\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−2045.67	−0.540041
$28$	0	0
$29$	−7789.51	−1.71995	−0.859974	−	0.510338i	$-0.829520\pi$
$29$	−7789.51	−1.71995	−0.859974	+	0.510338i	$0.829520\pi$
$30$	0	0
$31$	−4700.59	−0.878513	−0.439256	−	0.898362i	$-0.644758\pi$
$31$	−4700.59	−0.878513	−0.439256	+	0.898362i	$0.644758\pi$
$32$	0	0
$33$	2539.33	0.405915
$34$	0	0
$35$	−1225.00	−0.169031
$36$	0	0
$37$	1773.42	0.212965	0.106482	−	0.994315i	$-0.466041\pi$
$37$	1773.42	0.212965	0.106482	+	0.994315i	$0.466041\pi$
$38$	0	0
$39$	−1867.03	−0.196557
$40$	0	0
$41$	−7354.41	−0.683263	−0.341632	−	0.939834i	$-0.610980\pi$
$41$	−7354.41	−0.683263	−0.341632	+	0.939834i	$0.610980\pi$
$42$	0	0
$43$	−8562.31	−0.706187	−0.353093	−	0.935588i	$-0.614870\pi$
$43$	−8562.31	−0.706187	−0.353093	+	0.935588i	$0.614870\pi$
$44$	0	0
$45$	5594.87	0.411869
$46$	0	0
$47$	3748.17	0.247500	0.123750	−	0.992313i	$-0.460508\pi$
$47$	3748.17	0.247500	0.123750	+	0.992313i	$0.460508\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−8808.98	−0.474242
$52$	0	0
$53$	−33514.8	−1.63888	−0.819440	−	0.573165i	$-0.805716\pi$
$53$	−33514.8	−1.63888	−0.819440	+	0.573165i	$0.805716\pi$
$54$	0	0
$55$	−14486.0	−0.645718
$56$	0	0
$57$	−1763.01	−0.0718732
$58$	0	0
$59$	−11302.6	−0.422715	−0.211358	−	0.977409i	$-0.567788\pi$
$59$	−11302.6	−0.422715	−0.211358	+	0.977409i	$0.567788\pi$
$60$	0	0
$61$	−24964.2	−0.859001	−0.429500	−	0.903067i	$-0.641310\pi$
$61$	−24964.2	−0.859001	−0.429500	+	0.903067i	$0.641310\pi$
$62$	0	0
$63$	−10965.9	−0.348092
$64$	0	0
$65$	10650.7	0.312678
$66$	0	0
$67$	−35315.7	−0.961128	−0.480564	−	0.876960i	$-0.659568\pi$
$67$	−35315.7	−0.961128	−0.480564	+	0.876960i	$0.659568\pi$
$68$	0	0
$69$	6647.06	0.168076
$70$	0	0
$71$	70973.7	1.67090	0.835452	−	0.549563i	$-0.185206\pi$
$71$	70973.7	1.67090	0.835452	+	0.549563i	$0.185206\pi$
$72$	0	0
$73$	17070.3	0.374916	0.187458	−	0.982273i	$-0.439975\pi$
$73$	17070.3	0.374916	0.187458	+	0.982273i	$0.439975\pi$
$74$	0	0
$75$	2738.99	0.0562260
$76$	0	0
$77$	28392.6	0.545731
$78$	0	0
$79$	10103.6	0.182142	0.0910710	−	0.995844i	$-0.470971\pi$
$79$	10103.6	0.182142	0.0910710	+	0.995844i	$0.470971\pi$
$80$	0	0
$81$	45417.2	0.769144
$82$	0	0
$83$	46105.8	0.734616	0.367308	−	0.930099i	$-0.380280\pi$
$83$	46105.8	0.734616	0.367308	+	0.930099i	$0.380280\pi$
$84$	0	0
$85$	50252.2	0.754411
$86$	0	0
$87$	−34136.6	−0.483529
$88$	0	0
$89$	−21494.0	−0.287635	−0.143818	−	0.989604i	$-0.545938\pi$
$89$	−21494.0	−0.287635	−0.143818	+	0.989604i	$0.545938\pi$
$90$	0	0
$91$	−20875.5	−0.264261
$92$	0	0
$93$	−20599.8	−0.246976
$94$	0	0
$95$	10057.4	0.114334
$96$	0	0
$97$	−65031.4	−0.701769	−0.350884	−	0.936419i	$-0.614119\pi$
$97$	−65031.4	−0.701769	−0.350884	+	0.936419i	$0.614119\pi$
$98$	0	0
$99$	−129676.	−1.32975
$100$	0	0
$101$	200097.	1.95181	0.975903	−	0.218205i	$-0.0700201\pi$
$101$	200097.	1.95181	0.975903	+	0.218205i	$0.0700201\pi$
$102$	0	0
$103$	38990.8	0.362134	0.181067	−	0.983471i	$-0.442045\pi$
$103$	38990.8	0.362134	0.181067	+	0.983471i	$0.442045\pi$
$104$	0	0
$105$	−5368.42	−0.0475196
$106$	0	0
$107$	208797.	1.76305	0.881526	−	0.472135i	$-0.156516\pi$
$107$	208797.	1.76305	0.881526	+	0.472135i	$0.156516\pi$
$108$	0	0
$109$	152328.	1.22804	0.614020	−	0.789291i	$-0.289551\pi$
$109$	152328.	1.22804	0.614020	+	0.789291i	$0.289551\pi$
$110$	0	0
$111$	7771.81	0.0598708
$112$	0	0
$113$	107692.	0.793389	0.396694	−	0.917951i	$-0.370157\pi$
$113$	107692.	0.793389	0.396694	+	0.917951i	$0.370157\pi$
$114$	0	0
$115$	−37919.2	−0.267371
$116$	0	0
$117$	95343.3	0.643910
$118$	0	0
$119$	−98494.4	−0.637594
$120$	0	0
$121$	174701.	1.08476
$122$	0	0
$123$	−32229.8	−0.192086
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	−199498.	−1.09756	−0.548781	−	0.835966i	$-0.684908\pi$
$127$	−199498.	−1.09756	−0.548781	+	0.835966i	$0.684908\pi$
$128$	0	0
$129$	−37523.3	−0.198530
$130$	0	0
$131$	−137776.	−0.701448	−0.350724	−	0.936479i	$-0.614064\pi$
$131$	−137776.	−0.701448	−0.350724	+	0.936479i	$0.614064\pi$
$132$	0	0
$133$	−19712.4	−0.0966297
$134$	0	0
$135$	51141.8	0.241514
$136$	0	0
$137$	−236889.	−1.07831	−0.539154	−	0.842207i	$-0.681256\pi$
$137$	−236889.	−1.07831	−0.539154	+	0.842207i	$0.681256\pi$
$138$	0	0
$139$	297901.	1.30778	0.653891	−	0.756588i	$-0.273135\pi$
$139$	297901.	1.30778	0.653891	+	0.756588i	$0.273135\pi$
$140$	0	0
$141$	16425.9	0.0695796
$142$	0	0
$143$	−246859.	−1.00951
$144$	0	0
$145$	194738.	0.769184
$146$	0	0
$147$	10522.1	0.0401614
$148$	0	0
$149$	−48444.4	−0.178763	−0.0893816	−	0.995997i	$-0.528489\pi$
$149$	−48444.4	−0.178763	−0.0893816	+	0.995997i	$0.528489\pi$
$150$	0	0
$151$	57169.5	0.204043	0.102022	−	0.994782i	$-0.467469\pi$
$151$	57169.5	0.204043	0.102022	+	0.994782i	$0.467469\pi$
$152$	0	0
$153$	449848.	1.55359
$154$	0	0
$155$	117515.	0.392883
$156$	0	0
$157$	−370032.	−1.19809	−0.599046	−	0.800715i	$-0.704453\pi$
$157$	−370032.	−1.19809	−0.599046	+	0.800715i	$0.704453\pi$
$158$	0	0
$159$	−146875.	−0.460738
$160$	0	0
$161$	74321.7	0.225970
$162$	0	0
$163$	−264040.	−0.778397	−0.389198	−	0.921154i	$-0.627248\pi$
$163$	−264040.	−0.778397	−0.389198	+	0.921154i	$0.627248\pi$
$164$	0	0
$165$	−63483.3	−0.181531
$166$	0	0
$167$	−676427.	−1.87685	−0.938426	−	0.345480i	$-0.887716\pi$
$167$	−676427.	−1.87685	−0.938426	+	0.345480i	$0.887716\pi$
$168$	0	0
$169$	−189791.	−0.511164
$170$	0	0
$171$	90031.3	0.235453
$172$	0	0
$173$	−198499.	−0.504246	−0.252123	−	0.967695i	$-0.581129\pi$
$173$	−198499.	−0.504246	−0.252123	+	0.967695i	$0.581129\pi$
$174$	0	0
$175$	30625.0	0.0755929
$176$	0	0
$177$	−49532.2	−0.118838
$178$	0	0
$179$	−389818.	−0.909346	−0.454673	−	0.890659i	$-0.650244\pi$
$179$	−389818.	−0.909346	−0.454673	+	0.890659i	$0.650244\pi$
$180$	0	0
$181$	803751.	1.82358	0.911790	−	0.410656i	$-0.134700\pi$
$181$	803751.	1.82358	0.911790	+	0.410656i	$0.134700\pi$
$182$	0	0
$183$	−109403.	−0.241491
$184$	0	0
$185$	−44335.6	−0.0952408
$186$	0	0
$187$	−1.16473e6	−2.43568
$188$	0	0
$189$	−100238.	−0.204116
$190$	0	0
$191$	931541.	1.84765	0.923823	−	0.382820i	$-0.125047\pi$
$191$	931541.	1.84765	0.923823	+	0.382820i	$0.125047\pi$
$192$	0	0
$193$	−592332.	−1.14465	−0.572324	−	0.820028i	$-0.693958\pi$
$193$	−592332.	−1.14465	−0.572324	+	0.820028i	$0.693958\pi$
$194$	0	0
$195$	46675.6	0.0879030
$196$	0	0
$197$	782068.	1.43575	0.717875	−	0.696172i	$-0.245115\pi$
$197$	782068.	1.43575	0.717875	+	0.696172i	$0.245115\pi$
$198$	0	0
$199$	−383375.	−0.686264	−0.343132	−	0.939287i	$-0.611488\pi$
$199$	−383375.	−0.686264	−0.343132	+	0.939287i	$0.611488\pi$
$200$	0	0
$201$	−154767.	−0.270202
$202$	0	0
$203$	−381686.	−0.650079
$204$	0	0
$205$	183860.	0.305565
$206$	0	0
$207$	−339445.	−0.550609
$208$	0	0
$209$	−233106.	−0.369137
$210$	0	0
$211$	−145997.	−0.225756	−0.112878	−	0.993609i	$-0.536007\pi$
$211$	−145997.	−0.225756	−0.112878	+	0.993609i	$0.536007\pi$
$212$	0	0
$213$	311034.	0.469741
$214$	0	0
$215$	214058.	0.315816
$216$	0	0
$217$	−230329.	−0.332047
$218$	0	0
$219$	74808.5	0.105400
$220$	0	0
$221$	856359.	1.17944
$222$	0	0
$223$	−634335.	−0.854194	−0.427097	−	0.904206i	$-0.640464\pi$
$223$	−634335.	−0.854194	−0.427097	+	0.904206i	$0.640464\pi$
$224$	0	0
$225$	−139872.	−0.184193
$226$	0	0
$227$	−439350.	−0.565909	−0.282954	−	0.959133i	$-0.591314\pi$
$227$	−439350.	−0.565909	−0.282954	+	0.959133i	$0.591314\pi$
$228$	0	0
$229$	−858587.	−1.08192	−0.540961	−	0.841048i	$-0.681939\pi$
$229$	−858587.	−1.08192	−0.540961	+	0.841048i	$0.681939\pi$
$230$	0	0
$231$	124427.	0.153421
$232$	0	0
$233$	1.13183e6	1.36581	0.682904	−	0.730508i	$-0.260717\pi$
$233$	1.13183e6	1.36581	0.682904	+	0.730508i	$0.260717\pi$
$234$	0	0
$235$	−93704.3	−0.110685
$236$	0	0
$237$	44278.0	0.0512056
$238$	0	0
$239$	502366.	0.568887	0.284443	−	0.958693i	$-0.408191\pi$
$239$	502366.	0.568887	0.284443	+	0.958693i	$0.408191\pi$
$240$	0	0
$241$	−1.36123e6	−1.50969	−0.754846	−	0.655902i	$-0.772289\pi$
$241$	−1.36123e6	−1.50969	−0.754846	+	0.655902i	$0.772289\pi$
$242$	0	0
$243$	696134.	0.756270
$244$	0	0
$245$	−60025.0	−0.0638877
$246$	0	0
$247$	171389.	0.178748
$248$	0	0
$249$	202053.	0.206522
$250$	0	0
$251$	378383.	0.379095	0.189547	−	0.981872i	$-0.439298\pi$
$251$	378383.	0.379095	0.189547	+	0.981872i	$0.439298\pi$
$252$	0	0
$253$	878878.	0.863232
$254$	0	0
$255$	220224.	0.212088
$256$	0	0
$257$	−909210.	−0.858680	−0.429340	−	0.903143i	$-0.641254\pi$
$257$	−909210.	−0.858680	−0.429340	+	0.903143i	$0.641254\pi$
$258$	0	0
$259$	86897.7	0.0804931
$260$	0	0
$261$	1.74325e6	1.58401
$262$	0	0
$263$	333491.	0.297300	0.148650	−	0.988890i	$-0.452507\pi$
$263$	333491.	0.297300	0.148650	+	0.988890i	$0.452507\pi$
$264$	0	0
$265$	837871.	0.732930
$266$	0	0
$267$	−94194.9	−0.0808629
$268$	0	0
$269$	−830167.	−0.699495	−0.349747	−	0.936844i	$-0.613733\pi$
$269$	−830167.	−0.699495	−0.349747	+	0.936844i	$0.613733\pi$
$270$	0	0
$271$	1.58079e6	1.30752	0.653762	−	0.756700i	$-0.273190\pi$
$271$	1.58079e6	1.30752	0.653762	+	0.756700i	$0.273190\pi$
$272$	0	0
$273$	−91484.2	−0.0742916
$274$	0	0
$275$	362151.	0.288774
$276$	0	0
$277$	60252.2	0.0471817	0.0235908	−	0.999722i	$-0.492490\pi$
$277$	60252.2	0.0471817	0.0235908	+	0.999722i	$0.492490\pi$
$278$	0	0
$279$	1.05197e6	0.809080
$280$	0	0
$281$	1.53242e6	1.15774	0.578872	−	0.815418i	$-0.303493\pi$
$281$	1.53242e6	1.15774	0.578872	+	0.815418i	$0.303493\pi$
$282$	0	0
$283$	−2.39082e6	−1.77452	−0.887259	−	0.461272i	$-0.847393\pi$
$283$	−2.39082e6	−1.77452	−0.887259	+	0.461272i	$0.847393\pi$
$284$	0	0
$285$	44075.2	0.0321427
$286$	0	0
$287$	−360366.	−0.258249
$288$	0	0
$289$	2.62060e6	1.84568
$290$	0	0
$291$	−284992.	−0.197288
$292$	0	0
$293$	274332.	0.186684	0.0933419	−	0.995634i	$-0.470245\pi$
$293$	274332.	0.186684	0.0933419	+	0.995634i	$0.470245\pi$
$294$	0	0
$295$	282565.	0.189044
$296$	0	0
$297$	−1.18535e6	−0.779748
$298$	0	0
$299$	−646189.	−0.418005
$300$	0	0
$301$	−419553.	−0.266914
$302$	0	0
$303$	876900.	0.548711
$304$	0	0
$305$	624106.	0.384157
$306$	0	0
$307$	2.79580e6	1.69301	0.846507	−	0.532378i	$-0.178702\pi$
$307$	2.79580e6	1.69301	0.846507	+	0.532378i	$0.178702\pi$
$308$	0	0
$309$	170873.	0.101807
$310$	0	0
$311$	2.05010e6	1.20192	0.600959	−	0.799280i	$-0.294785\pi$
$311$	2.05010e6	1.20192	0.600959	+	0.799280i	$0.294785\pi$
$312$	0	0
$313$	−1.67383e6	−0.965717	−0.482858	−	0.875698i	$-0.660401\pi$
$313$	−1.67383e6	−0.965717	−0.482858	+	0.875698i	$0.660401\pi$
$314$	0	0
$315$	274149.	0.155672
$316$	0	0
$317$	198590.	0.110996	0.0554982	−	0.998459i	$-0.482325\pi$
$317$	198590.	0.110996	0.0554982	+	0.998459i	$0.482325\pi$
$318$	0	0
$319$	−4.51357e6	−2.48338
$320$	0	0
$321$	915029.	0.495647
$322$	0	0
$323$	808648.	0.431274
$324$	0	0
$325$	−266269.	−0.139834
$326$	0	0
$327$	667558.	0.345239
$328$	0	0
$329$	183660.	0.0935461
$330$	0	0
$331$	367129.	0.184182	0.0920912	−	0.995751i	$-0.470645\pi$
$331$	367129.	0.184182	0.0920912	+	0.995751i	$0.470645\pi$
$332$	0	0
$333$	−396883.	−0.196133
$334$	0	0
$335$	882893.	0.429830
$336$	0	0
$337$	−3.40086e6	−1.63122	−0.815612	−	0.578599i	$-0.803600\pi$
$337$	−3.40086e6	−1.63122	−0.815612	+	0.578599i	$0.803600\pi$
$338$	0	0
$339$	471946.	0.223045
$340$	0	0
$341$	−2.72372e6	−1.26846
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	−166176.	−0.0751660
$346$	0	0
$347$	−130304.	−0.0580945	−0.0290472	−	0.999578i	$-0.509247\pi$
$347$	−130304.	−0.0580945	−0.0290472	+	0.999578i	$0.509247\pi$
$348$	0	0
$349$	−3.48504e6	−1.53159	−0.765797	−	0.643082i	$-0.777655\pi$
$349$	−3.48504e6	−1.53159	−0.765797	+	0.643082i	$0.777655\pi$
$350$	0	0
$351$	871518.	0.377579
$352$	0	0
$353$	501512.	0.214212	0.107106	−	0.994248i	$-0.465842\pi$
$353$	501512.	0.214212	0.107106	+	0.994248i	$0.465842\pi$
$354$	0	0
$355$	−1.77434e6	−0.747251
$356$	0	0
$357$	−431640.	−0.179247
$358$	0	0
$359$	2.20994e6	0.904992	0.452496	−	0.891766i	$-0.350534\pi$
$359$	2.20994e6	0.904992	0.452496	+	0.891766i	$0.350534\pi$
$360$	0	0
$361$	−2.31426e6	−0.934639
$362$	0	0
$363$	765608.	0.304958
$364$	0	0
$365$	−426757.	−0.167667
$366$	0	0
$367$	−1.20727e6	−0.467885	−0.233943	−	0.972250i	$-0.575163\pi$
$367$	−1.20727e6	−0.467885	−0.233943	+	0.972250i	$0.575163\pi$
$368$	0	0
$369$	1.64588e6	0.629262
$370$	0	0
$371$	−1.64223e6	−0.619439
$372$	0	0
$373$	2.21175e6	0.823120	0.411560	−	0.911383i	$-0.364984\pi$
$373$	2.21175e6	0.823120	0.411560	+	0.911383i	$0.364984\pi$
$374$	0	0
$375$	−68474.7	−0.0251450
$376$	0	0
$377$	3.31857e6	1.20253
$378$	0	0
$379$	−725319.	−0.259377	−0.129688	−	0.991555i	$-0.541398\pi$
$379$	−725319.	−0.259377	−0.129688	+	0.991555i	$0.541398\pi$
$380$	0	0
$381$	−874276.	−0.308557
$382$	0	0
$383$	−4.53778e6	−1.58069	−0.790344	−	0.612663i	$-0.790098\pi$
$383$	−4.53778e6	−1.58069	−0.790344	+	0.612663i	$0.790098\pi$
$384$	0	0
$385$	−709816.	−0.244058
$386$	0	0
$387$	1.91620e6	0.650374
$388$	0	0
$389$	629591.	0.210952	0.105476	−	0.994422i	$-0.466363\pi$
$389$	629591.	0.210952	0.105476	+	0.994422i	$0.466363\pi$
$390$	0	0
$391$	−3.04884e6	−1.00854
$392$	0	0
$393$	−603787.	−0.197198
$394$	0	0
$395$	−252591.	−0.0814564
$396$	0	0
$397$	−1.66148e6	−0.529076	−0.264538	−	0.964375i	$-0.585220\pi$
$397$	−1.66148e6	−0.529076	−0.264538	+	0.964375i	$0.585220\pi$
$398$	0	0
$399$	−86387.3	−0.0271655
$400$	0	0
$401$	−1.66267e6	−0.516350	−0.258175	−	0.966098i	$-0.583121\pi$
$401$	−1.66267e6	−0.516350	−0.258175	+	0.966098i	$0.583121\pi$
$402$	0	0
$403$	2.00259e6	0.614228
$404$	0	0
$405$	−1.13543e6	−0.343972
$406$	0	0
$407$	1.02759e6	0.307493
$408$	0	0
$409$	−5.39697e6	−1.59530	−0.797649	−	0.603123i	$-0.793923\pi$
$409$	−5.39697e6	−1.59530	−0.797649	+	0.603123i	$0.793923\pi$
$410$	0	0
$411$	−1.03814e6	−0.303145
$412$	0	0
$413$	−553827.	−0.159771
$414$	0	0
$415$	−1.15264e6	−0.328530
$416$	0	0
$417$	1.30552e6	0.367657
$418$	0	0
$419$	5.80750e6	1.61605	0.808024	−	0.589150i	$-0.200537\pi$
$419$	5.80750e6	1.61605	0.808024	+	0.589150i	$0.200537\pi$
$420$	0	0
$421$	4.45344e6	1.22459	0.612295	−	0.790629i	$-0.290247\pi$
$421$	4.45344e6	1.22459	0.612295	+	0.790629i	$0.290247\pi$
$422$	0	0
$423$	−838821.	−0.227939
$424$	0	0
$425$	−1.25631e6	−0.337383
$426$	0	0
$427$	−1.22325e6	−0.324672
$428$	0	0
$429$	−1.08183e6	−0.283803
$430$	0	0
$431$	−906343.	−0.235017	−0.117509	−	0.993072i	$-0.537491\pi$
$431$	−906343.	−0.235017	−0.117509	+	0.993072i	$0.537491\pi$
$432$	0	0
$433$	−5.02034e6	−1.28681	−0.643403	−	0.765528i	$-0.722478\pi$
$433$	−5.02034e6	−1.28681	−0.643403	+	0.765528i	$0.722478\pi$
$434$	0	0
$435$	853415.	0.216241
$436$	0	0
$437$	−610187.	−0.152848
$438$	0	0
$439$	3.24699e6	0.804118	0.402059	−	0.915614i	$-0.368295\pi$
$439$	3.24699e6	0.804118	0.402059	+	0.915614i	$0.368295\pi$
$440$	0	0
$441$	−537331.	−0.131567
$442$	0	0
$443$	4.31230e6	1.04400	0.521999	−	0.852946i	$-0.325186\pi$
$443$	4.31230e6	1.04400	0.521999	+	0.852946i	$0.325186\pi$
$444$	0	0
$445$	537350.	0.128634
$446$	0	0
$447$	−212302.	−0.0502556
$448$	0	0
$449$	−3.53475e6	−0.827453	−0.413726	−	0.910401i	$-0.635773\pi$
$449$	−3.53475e6	−0.827453	−0.413726	+	0.910401i	$0.635773\pi$
$450$	0	0
$451$	−4.26145e6	−0.986543
$452$	0	0
$453$	250538.	0.0573626
$454$	0	0
$455$	521887.	0.118181
$456$	0	0
$457$	−3.80009e6	−0.851144	−0.425572	−	0.904925i	$-0.639927\pi$
$457$	−3.80009e6	−0.851144	−0.425572	+	0.904925i	$0.639927\pi$
$458$	0	0
$459$	4.11198e6	0.911003
$460$	0	0
$461$	219847.	0.0481801	0.0240900	−	0.999710i	$-0.492331\pi$
$461$	219847.	0.0481801	0.0240900	+	0.999710i	$0.492331\pi$
$462$	0	0
$463$	−2.69126e6	−0.583449	−0.291725	−	0.956502i	$-0.594229\pi$
$463$	−2.69126e6	−0.583449	−0.291725	+	0.956502i	$0.594229\pi$
$464$	0	0
$465$	514994.	0.110451
$466$	0	0
$467$	3.68553e6	0.782003	0.391001	−	0.920390i	$-0.372129\pi$
$467$	3.68553e6	0.782003	0.391001	+	0.920390i	$0.372129\pi$
$468$	0	0
$469$	−1.73047e6	−0.363272
$470$	0	0
$471$	−1.62162e6	−0.336819
$472$	0	0
$473$	−4.96135e6	−1.01964
$474$	0	0
$475$	−251434.	−0.0511317
$476$	0	0
$477$	7.50044e6	1.50935
$478$	0	0
$479$	6.44698e6	1.28386	0.641930	−	0.766763i	$-0.278134\pi$
$479$	6.44698e6	1.28386	0.641930	+	0.766763i	$0.278134\pi$
$480$	0	0
$481$	−755531.	−0.148898
$482$	0	0
$483$	325706.	0.0635269
$484$	0	0
$485$	1.62579e6	0.313840
$486$	0	0
$487$	−2.38314e6	−0.455331	−0.227665	−	0.973739i	$-0.573109\pi$
$487$	−2.38314e6	−0.455331	−0.227665	+	0.973739i	$0.573109\pi$
$488$	0	0
$489$	−1.15712e6	−0.218831
$490$	0	0
$491$	−6.68620e6	−1.25163	−0.625815	−	0.779972i	$-0.715233\pi$
$491$	−6.68620e6	−1.25163	−0.625815	+	0.779972i	$0.715233\pi$
$492$	0	0
$493$	1.56576e7	2.90141
$494$	0	0
$495$	3.24190e6	0.594684
$496$	0	0
$497$	3.47771e6	0.631542
$498$	0	0
$499$	8.76182e6	1.57523	0.787613	−	0.616170i	$-0.211317\pi$
$499$	8.76182e6	1.57523	0.787613	+	0.616170i	$0.211317\pi$
$500$	0	0
$501$	−2.96436e6	−0.527639
$502$	0	0
$503$	5.39464e6	0.950698	0.475349	−	0.879797i	$-0.342322\pi$
$503$	5.39464e6	0.950698	0.475349	+	0.879797i	$0.342322\pi$
$504$	0	0
$505$	−5.00242e6	−0.872874
$506$	0	0
$507$	−831738.	−0.143703
$508$	0	0
$509$	2.97015e6	0.508141	0.254070	−	0.967186i	$-0.418231\pi$
$509$	2.97015e6	0.508141	0.254070	+	0.967186i	$0.418231\pi$
$510$	0	0
$511$	836444.	0.141705
$512$	0	0
$513$	822962.	0.138066
$514$	0	0
$515$	−974771.	−0.161951
$516$	0	0
$517$	2.17185e6	0.357357
$518$	0	0
$519$	−869897.	−0.141759
$520$	0	0
$521$	−4.08245e6	−0.658910	−0.329455	−	0.944171i	$-0.606865\pi$
$521$	−4.08245e6	−0.658910	−0.329455	+	0.944171i	$0.606865\pi$
$522$	0	0
$523$	−1.01220e6	−0.161812	−0.0809061	−	0.996722i	$-0.525781\pi$
$523$	−1.01220e6	−0.161812	−0.0809061	+	0.996722i	$0.525781\pi$
$524$	0	0
$525$	134210.	0.0212514
$526$	0	0
$527$	9.44861e6	1.48198
$528$	0	0
$529$	−4.13576e6	−0.642563
$530$	0	0
$531$	2.52946e6	0.389306
$532$	0	0
$533$	3.13320e6	0.477716
$534$	0	0
$535$	−5.21993e6	−0.788461
$536$	0	0
$537$	−1.70833e6	−0.255644
$538$	0	0
$539$	1.39124e6	0.206267
$540$	0	0
$541$	−3.59552e6	−0.528164	−0.264082	−	0.964500i	$-0.585069\pi$
$541$	−3.59552e6	−0.528164	−0.264082	+	0.964500i	$0.585069\pi$
$542$	0	0
$543$	3.52234e6	0.512663
$544$	0	0
$545$	−3.80819e6	−0.549196
$546$	0	0
$547$	1.08289e7	1.54745	0.773726	−	0.633520i	$-0.218391\pi$
$547$	1.08289e7	1.54745	0.773726	+	0.633520i	$0.218391\pi$
$548$	0	0
$549$	5.58686e6	0.791111
$550$	0	0
$551$	3.13368e6	0.439719
$552$	0	0
$553$	495079.	0.0688432
$554$	0	0
$555$	−194295.	−0.0267750
$556$	0	0
$557$	9.30164e6	1.27034	0.635172	−	0.772371i	$-0.280929\pi$
$557$	9.30164e6	1.27034	0.635172	+	0.772371i	$0.280929\pi$
$558$	0	0
$559$	3.64780e6	0.493743
$560$	0	0
$561$	−5.10429e6	−0.684744
$562$	0	0
$563$	−7.94494e6	−1.05638	−0.528189	−	0.849127i	$-0.677129\pi$
$563$	−7.94494e6	−1.05638	−0.528189	+	0.849127i	$0.677129\pi$
$564$	0	0
$565$	−2.69229e6	−0.354814
$566$	0	0
$567$	2.22544e6	0.290709
$568$	0	0
$569$	−5.70178e6	−0.738295	−0.369148	−	0.929371i	$-0.620350\pi$
$569$	−5.70178e6	−0.738295	−0.369148	+	0.929371i	$0.620350\pi$
$570$	0	0
$571$	−4.62136e6	−0.593171	−0.296585	−	0.955006i	$-0.595848\pi$
$571$	−4.62136e6	−0.593171	−0.296585	+	0.955006i	$0.595848\pi$
$572$	0	0
$573$	4.08237e6	0.519428
$574$	0	0
$575$	947980.	0.119572
$576$	0	0
$577$	6.35670e6	0.794863	0.397431	−	0.917632i	$-0.369902\pi$
$577$	6.35670e6	0.794863	0.397431	+	0.917632i	$0.369902\pi$
$578$	0	0
$579$	−2.59582e6	−0.321795
$580$	0	0
$581$	2.25918e6	0.277659
$582$	0	0
$583$	−1.94199e7	−2.36633
$584$	0	0
$585$	−2.38358e6	−0.287965
$586$	0	0
$587$	−1.35002e7	−1.61713	−0.808565	−	0.588407i	$-0.799755\pi$
$587$	−1.35002e7	−1.61713	−0.808565	+	0.588407i	$0.799755\pi$
$588$	0	0
$589$	1.89102e6	0.224599
$590$	0	0
$591$	3.42732e6	0.403632
$592$	0	0
$593$	4.01400e6	0.468750	0.234375	−	0.972146i	$-0.424696\pi$
$593$	4.01400e6	0.468750	0.234375	+	0.972146i	$0.424696\pi$
$594$	0	0
$595$	2.46236e6	0.285141
$596$	0	0
$597$	−1.68010e6	−0.192929
$598$	0	0
$599$	1.18393e7	1.34821	0.674104	−	0.738636i	$-0.264530\pi$
$599$	1.18393e7	1.34821	0.674104	+	0.738636i	$0.264530\pi$
$600$	0	0
$601$	−5.06735e6	−0.572262	−0.286131	−	0.958190i	$-0.592369\pi$
$601$	−5.06735e6	−0.572262	−0.286131	+	0.958190i	$0.592369\pi$
$602$	0	0
$603$	7.90348e6	0.885166
$604$	0	0
$605$	−4.36753e6	−0.485119
$606$	0	0
$607$	1.03970e7	1.14534	0.572671	−	0.819786i	$-0.305907\pi$
$607$	1.03970e7	1.14534	0.572671	+	0.819786i	$0.305907\pi$
$608$	0	0
$609$	−1.67269e6	−0.182757
$610$	0	0
$611$	−1.59683e6	−0.173044
$612$	0	0
$613$	6.54981e6	0.704008	0.352004	−	0.935998i	$-0.385500\pi$
$613$	6.54981e6	0.704008	0.352004	+	0.935998i	$0.385500\pi$
$614$	0	0
$615$	805745.	0.0859033
$616$	0	0
$617$	5.70921e6	0.603758	0.301879	−	0.953346i	$-0.402386\pi$
$617$	5.70921e6	0.603758	0.301879	+	0.953346i	$0.402386\pi$
$618$	0	0
$619$	4.25418e6	0.446261	0.223131	−	0.974789i	$-0.428372\pi$
$619$	4.25418e6	0.446261	0.223131	+	0.974789i	$0.428372\pi$
$620$	0	0
$621$	−3.10281e6	−0.322869
$622$	0	0
$623$	−1.05321e6	−0.108716
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−1.02156e6	−0.103775
$628$	0	0
$629$	−3.56474e6	−0.359254
$630$	0	0
$631$	−1.60836e7	−1.60809	−0.804043	−	0.594570i	$-0.797322\pi$
$631$	−1.60836e7	−1.60809	−0.804043	+	0.594570i	$0.797322\pi$
$632$	0	0
$633$	−639816.	−0.0634666
$634$	0	0
$635$	4.98745e6	0.490845
$636$	0	0
$637$	−1.02290e6	−0.0998812
$638$	0	0
$639$	−1.58835e7	−1.53885
$640$	0	0
$641$	4.17421e6	0.401263	0.200631	−	0.979667i	$-0.435701\pi$
$641$	4.17421e6	0.401263	0.200631	+	0.979667i	$0.435701\pi$
$642$	0	0
$643$	−1.12908e7	−1.07695	−0.538477	−	0.842640i	$-0.681000\pi$
$643$	−1.12908e7	−1.07695	−0.538477	+	0.842640i	$0.681000\pi$
$644$	0	0
$645$	938082.	0.0887854
$646$	0	0
$647$	−1.08901e7	−1.02275	−0.511377	−	0.859357i	$-0.670864\pi$
$647$	−1.08901e7	−1.02275	−0.511377	+	0.859357i	$0.670864\pi$
$648$	0	0
$649$	−6.54918e6	−0.610345
$650$	0	0
$651$	−1.00939e6	−0.0933482
$652$	0	0
$653$	−2.96997e6	−0.272565	−0.136282	−	0.990670i	$-0.543515\pi$
$653$	−2.96997e6	−0.272565	−0.136282	+	0.990670i	$0.543515\pi$
$654$	0	0
$655$	3.44440e6	0.313697
$656$	0	0
$657$	−3.82024e6	−0.345285
$658$	0	0
$659$	−1.77817e7	−1.59500	−0.797500	−	0.603319i	$-0.793845\pi$
$659$	−1.77817e7	−1.59500	−0.797500	+	0.603319i	$0.793845\pi$
$660$	0	0
$661$	4.10248e6	0.365210	0.182605	−	0.983186i	$-0.441547\pi$
$661$	4.10248e6	0.365210	0.182605	+	0.983186i	$0.441547\pi$
$662$	0	0
$663$	3.75289e6	0.331575
$664$	0	0
$665$	492810.	0.0432141
$666$	0	0
$667$	−1.18149e7	−1.02829
$668$	0	0
$669$	−2.77990e6	−0.240139
$670$	0	0
$671$	−1.44653e7	−1.24028
$672$	0	0
$673$	−1.92507e6	−0.163836	−0.0819181	−	0.996639i	$-0.526105\pi$
$673$	−1.92507e6	−0.163836	−0.0819181	+	0.996639i	$0.526105\pi$
$674$	0	0
$675$	−1.27855e6	−0.108008
$676$	0	0
$677$	−3.54800e6	−0.297517	−0.148759	−	0.988874i	$-0.547528\pi$
$677$	−3.54800e6	−0.297517	−0.148759	+	0.988874i	$0.547528\pi$
$678$	0	0
$679$	−3.18654e6	−0.265244
$680$	0	0
$681$	−1.92540e6	−0.159094
$682$	0	0
$683$	−8.58361e6	−0.704074	−0.352037	−	0.935986i	$-0.614511\pi$
$683$	−8.58361e6	−0.704074	−0.352037	+	0.935986i	$0.614511\pi$
$684$	0	0
$685$	5.92222e6	0.482234
$686$	0	0
$687$	−3.76266e6	−0.304160
$688$	0	0
$689$	1.42783e7	1.14585
$690$	0	0
$691$	−1.70540e6	−0.135872	−0.0679361	−	0.997690i	$-0.521641\pi$
$691$	−1.70540e6	−0.135872	−0.0679361	+	0.997690i	$0.521641\pi$
$692$	0	0
$693$	−6.35412e6	−0.502600
$694$	0	0
$695$	−7.44754e6	−0.584858
$696$	0	0
$697$	1.47830e7	1.15261
$698$	0	0
$699$	4.96009e6	0.383969
$700$	0	0
$701$	1.10522e7	0.849481	0.424741	−	0.905315i	$-0.360365\pi$
$701$	1.10522e7	0.849481	0.424741	+	0.905315i	$0.360365\pi$
$702$	0	0
$703$	−713438.	−0.0544462
$704$	0	0
$705$	−410648.	−0.0311169
$706$	0	0
$707$	9.80474e6	0.737713
$708$	0	0
$709$	3.28816e6	0.245662	0.122831	−	0.992428i	$-0.460803\pi$
$709$	3.28816e6	0.245662	0.122831	+	0.992428i	$0.460803\pi$
$710$	0	0
$711$	−2.26114e6	−0.167747
$712$	0	0
$713$	−7.12970e6	−0.525228
$714$	0	0
$715$	6.17149e6	0.451466
$716$	0	0
$717$	2.20156e6	0.159931
$718$	0	0
$719$	2.07563e7	1.49736	0.748681	−	0.662930i	$-0.230687\pi$
$719$	2.07563e7	1.49736	0.748681	+	0.662930i	$0.230687\pi$
$720$	0	0
$721$	1.91055e6	0.136874
$722$	0	0
$723$	−5.96542e6	−0.424420
$724$	0	0
$725$	−4.86845e6	−0.343990
$726$	0	0
$727$	2.10774e6	0.147904	0.0739522	−	0.997262i	$-0.476439\pi$
$727$	2.10774e6	0.147904	0.0739522	+	0.997262i	$0.476439\pi$
$728$	0	0
$729$	−7.98566e6	−0.556534
$730$	0	0
$731$	1.72110e7	1.19128
$732$	0	0
$733$	1.68040e7	1.15519	0.577595	−	0.816323i	$-0.303991\pi$
$733$	1.68040e7	1.15519	0.577595	+	0.816323i	$0.303991\pi$
$734$	0	0
$735$	−263052.	−0.0179607
$736$	0	0
$737$	−2.04634e7	−1.38774
$738$	0	0
$739$	−7.40020e6	−0.498462	−0.249231	−	0.968444i	$-0.580178\pi$
$739$	−7.40020e6	−0.498462	−0.249231	+	0.968444i	$0.580178\pi$
$740$	0	0
$741$	751094.	0.0502514
$742$	0	0
$743$	−1.80394e7	−1.19881	−0.599405	−	0.800446i	$-0.704596\pi$
$743$	−1.80394e7	−1.19881	−0.599405	+	0.800446i	$0.704596\pi$
$744$	0	0
$745$	1.21111e6	0.0799453
$746$	0	0
$747$	−1.03182e7	−0.676556
$748$	0	0
$749$	1.02311e7	0.666371
$750$	0	0
$751$	−2.49611e7	−1.61497	−0.807484	−	0.589889i	$-0.799171\pi$
$751$	−2.49611e7	−1.61497	−0.807484	+	0.589889i	$0.799171\pi$
$752$	0	0
$753$	1.65822e6	0.106575
$754$	0	0
$755$	−1.42924e6	−0.0912508
$756$	0	0
$757$	6.25759e6	0.396887	0.198444	−	0.980112i	$-0.436411\pi$
$757$	6.25759e6	0.396887	0.198444	+	0.980112i	$0.436411\pi$
$758$	0	0
$759$	3.85158e6	0.242680
$760$	0	0
$761$	−1.04767e7	−0.655785	−0.327893	−	0.944715i	$-0.606338\pi$
$761$	−1.04767e7	−0.655785	−0.327893	+	0.944715i	$0.606338\pi$
$762$	0	0
$763$	7.46405e6	0.464155
$764$	0	0
$765$	−1.12462e7	−0.694787
$766$	0	0
$767$	4.81524e6	0.295549
$768$	0	0
$769$	−1.15960e7	−0.707117	−0.353558	−	0.935412i	$-0.615028\pi$
$769$	−1.15960e7	−0.707117	−0.353558	+	0.935412i	$0.615028\pi$
$770$	0	0
$771$	−3.98450e6	−0.241401
$772$	0	0
$773$	2.20438e7	1.32690	0.663451	−	0.748220i	$-0.269091\pi$
$773$	2.20438e7	1.32690	0.663451	+	0.748220i	$0.269091\pi$
$774$	0	0
$775$	−2.93787e6	−0.175703
$776$	0	0
$777$	380819.	0.0226290
$778$	0	0
$779$	2.95864e6	0.174682
$780$	0	0
$781$	4.11251e7	2.41257
$782$	0	0
$783$	1.59348e7	0.928842
$784$	0	0
$785$	9.25079e6	0.535803
$786$	0	0
$787$	−1.32085e7	−0.760179	−0.380090	−	0.924950i	$-0.624107\pi$
$787$	−1.32085e7	−0.760179	−0.380090	+	0.924950i	$0.624107\pi$
$788$	0	0
$789$	1.46149e6	0.0835800
$790$	0	0
$791$	5.27689e6	0.299873
$792$	0	0
$793$	1.06355e7	0.600586
$794$	0	0
$795$	3.67187e6	0.206048
$796$	0	0
$797$	2.64400e7	1.47440	0.737200	−	0.675675i	$-0.236148\pi$
$797$	2.64400e7	1.47440	0.737200	+	0.675675i	$0.236148\pi$
$798$	0	0
$799$	−7.53416e6	−0.417511
$800$	0	0
$801$	4.81025e6	0.264902
$802$	0	0
$803$	9.89124e6	0.541330
$804$	0	0
$805$	−1.85804e6	−0.101057
$806$	0	0
$807$	−3.63811e6	−0.196649
$808$	0	0
$809$	−3.50288e7	−1.88171	−0.940856	−	0.338805i	$-0.889977\pi$
$809$	−3.50288e7	−1.88171	−0.940856	+	0.338805i	$0.889977\pi$
$810$	0	0
$811$	1.36252e7	0.727432	0.363716	−	0.931510i	$-0.381508\pi$
$811$	1.36252e7	0.727432	0.363716	+	0.931510i	$0.381508\pi$
$812$	0	0
$813$	6.92760e6	0.367584
$814$	0	0
$815$	6.60101e6	0.348110
$816$	0	0
$817$	3.44457e6	0.180542
$818$	0	0
$819$	4.67182e6	0.243375
$820$	0	0
$821$	1.13414e7	0.587229	0.293615	−	0.955924i	$-0.405142\pi$
$821$	1.13414e7	0.587229	0.293615	+	0.955924i	$0.405142\pi$
$822$	0	0
$823$	1.11368e6	0.0573138	0.0286569	−	0.999589i	$-0.490877\pi$
$823$	1.11368e6	0.0573138	0.0286569	+	0.999589i	$0.490877\pi$
$824$	0	0
$825$	1.58708e6	0.0811829
$826$	0	0
$827$	−3.71049e7	−1.88655	−0.943273	−	0.332017i	$-0.892271\pi$
$827$	−3.71049e7	−1.88655	−0.943273	+	0.332017i	$0.892271\pi$
$828$	0	0
$829$	9.73384e6	0.491924	0.245962	−	0.969279i	$-0.420896\pi$
$829$	9.73384e6	0.491924	0.245962	+	0.969279i	$0.420896\pi$
$830$	0	0
$831$	264048.	0.0132642
$832$	0	0
$833$	−4.82623e6	−0.240988
$834$	0	0
$835$	1.69107e7	0.839354
$836$	0	0
$837$	9.61586e6	0.474433
$838$	0	0
$839$	−7.76027e6	−0.380603	−0.190301	−	0.981726i	$-0.560946\pi$
$839$	−7.76027e6	−0.380603	−0.190301	+	0.981726i	$0.560946\pi$
$840$	0	0
$841$	4.01654e7	1.95822
$842$	0	0
$843$	6.71566e6	0.325477
$844$	0	0
$845$	4.74479e6	0.228599
$846$	0	0
$847$	8.56037e6	0.410000
$848$	0	0
$849$	−1.04775e7	−0.498870
$850$	0	0
$851$	2.68987e6	0.127323
$852$	0	0
$853$	−3.05251e7	−1.43643	−0.718215	−	0.695821i	$-0.755041\pi$
$853$	−3.05251e7	−1.43643	−0.718215	+	0.695821i	$0.755041\pi$
$854$	0	0
$855$	−2.25078e6	−0.105298
$856$	0	0
$857$	−3.01825e7	−1.40379	−0.701896	−	0.712279i	$-0.747663\pi$
$857$	−3.01825e7	−1.40379	−0.701896	+	0.712279i	$0.747663\pi$
$858$	0	0
$859$	2.85929e7	1.32213	0.661066	−	0.750328i	$-0.270104\pi$
$859$	2.85929e7	1.32213	0.661066	+	0.750328i	$0.270104\pi$
$860$	0	0
$861$	−1.57926e6	−0.0726016
$862$	0	0
$863$	−1.44726e7	−0.661484	−0.330742	−	0.943721i	$-0.607299\pi$
$863$	−1.44726e7	−0.661484	−0.330742	+	0.943721i	$0.607299\pi$
$864$	0	0
$865$	4.96247e6	0.225506
$866$	0	0
$867$	1.14845e7	0.518876
$868$	0	0
$869$	5.85447e6	0.262989
$870$	0	0
$871$	1.50456e7	0.671990
$872$	0	0
$873$	1.45537e7	0.646305
$874$	0	0
$875$	−765625.	−0.0338062
$876$	0	0
$877$	−3.61028e7	−1.58505	−0.792524	−	0.609841i	$-0.791233\pi$
$877$	−3.61028e7	−1.58505	−0.792524	+	0.609841i	$0.791233\pi$
$878$	0	0
$879$	1.20223e6	0.0524824
$880$	0	0
$881$	−2.45112e7	−1.06396	−0.531980	−	0.846757i	$-0.678552\pi$
$881$	−2.45112e7	−1.06396	−0.531980	+	0.846757i	$0.678552\pi$
$882$	0	0
$883$	−2.55558e7	−1.10303	−0.551516	−	0.834164i	$-0.685950\pi$
$883$	−2.55558e7	−1.10303	−0.551516	+	0.834164i	$0.685950\pi$
$884$	0	0
$885$	1.23831e6	0.0531459
$886$	0	0
$887$	−9.89572e6	−0.422317	−0.211158	−	0.977452i	$-0.567724\pi$
$887$	−9.89572e6	−0.422317	−0.211158	+	0.977452i	$0.567724\pi$
$888$	0	0
$889$	−9.77540e6	−0.414840
$890$	0	0
$891$	2.63166e7	1.11054
$892$	0	0
$893$	−1.50787e6	−0.0632754
$894$	0	0
$895$	9.74544e6	0.406672
$896$	0	0
$897$	−2.83185e6	−0.117514
$898$	0	0
$899$	3.66153e7	1.51100
$900$	0	0
$901$	6.73678e7	2.76465
$902$	0	0
$903$	−1.83864e6	−0.0750373
$904$	0	0
$905$	−2.00938e7	−0.815530
$906$	0	0
$907$	1.39247e7	0.562040	0.281020	−	0.959702i	$-0.409327\pi$
$907$	1.39247e7	0.562040	0.281020	+	0.959702i	$0.409327\pi$
$908$	0	0
$909$	−4.47806e7	−1.79755
$910$	0	0
$911$	8.43300e6	0.336656	0.168328	−	0.985731i	$-0.446163\pi$
$911$	8.43300e6	0.336656	0.168328	+	0.985731i	$0.446163\pi$
$912$	0	0
$913$	2.67156e7	1.06069
$914$	0	0
$915$	2.73507e6	0.107998
$916$	0	0
$917$	−6.75102e6	−0.265122
$918$	0	0
$919$	4.62946e7	1.80818	0.904091	−	0.427341i	$-0.140550\pi$
$919$	4.62946e7	1.80818	0.904091	+	0.427341i	$0.140550\pi$
$920$	0	0
$921$	1.22523e7	0.475956
$922$	0	0
$923$	−3.02369e7	−1.16824
$924$	0	0
$925$	1.10839e6	0.0425930
$926$	0	0
$927$	−8.72594e6	−0.333513
$928$	0	0
$929$	−1.45748e7	−0.554067	−0.277034	−	0.960860i	$-0.589351\pi$
$929$	−1.45748e7	−0.554067	−0.277034	+	0.960860i	$0.589351\pi$
$930$	0	0
$931$	−965909.	−0.0365226
$932$	0	0
$933$	8.98434e6	0.337895
$934$	0	0
$935$	2.91182e7	1.08927
$936$	0	0
$937$	−1.25424e7	−0.466693	−0.233346	−	0.972394i	$-0.574968\pi$
$937$	−1.25424e7	−0.466693	−0.233346	+	0.972394i	$0.574968\pi$
$938$	0	0
$939$	−7.33535e6	−0.271492
$940$	0	0
$941$	2.62772e7	0.967396	0.483698	−	0.875235i	$-0.339293\pi$
$941$	2.62772e7	0.967396	0.483698	+	0.875235i	$0.339293\pi$
$942$	0	0
$943$	−1.11549e7	−0.408496
$944$	0	0
$945$	2.50595e6	0.0912836
$946$	0	0
$947$	−3.31403e7	−1.20083	−0.600415	−	0.799688i	$-0.704998\pi$
$947$	−3.31403e7	−1.20083	−0.600415	+	0.799688i	$0.704998\pi$
$948$	0	0
$949$	−7.27246e6	−0.262129
$950$	0	0
$951$	870296.	0.0312044
$952$	0	0
$953$	2.77205e7	0.988709	0.494355	−	0.869260i	$-0.335404\pi$
$953$	2.77205e7	0.988709	0.494355	+	0.869260i	$0.335404\pi$
$954$	0	0
$955$	−2.32885e7	−0.826292
$956$	0	0
$957$	−1.97802e7	−0.698152
$958$	0	0
$959$	−1.16075e7	−0.407562
$960$	0	0
$961$	−6.53362e6	−0.228216
$962$	0	0
$963$	−4.67277e7	−1.62371
$964$	0	0
$965$	1.48083e7	0.511902
$966$	0	0
$967$	1.77930e7	0.611905	0.305953	−	0.952047i	$-0.401025\pi$
$967$	1.77930e7	0.611905	0.305953	+	0.952047i	$0.401025\pi$
$968$	0	0
$969$	3.54380e6	0.121244
$970$	0	0
$971$	−2.95500e7	−1.00579	−0.502897	−	0.864346i	$-0.667733\pi$
$971$	−2.95500e7	−1.00579	−0.502897	+	0.864346i	$0.667733\pi$
$972$	0	0
$973$	1.45972e7	0.494295
$974$	0	0
$975$	−1.16689e6	−0.0393114
$976$	0	0
$977$	−5.26510e6	−0.176470	−0.0882349	−	0.996100i	$-0.528123\pi$
$977$	−5.26510e6	−0.176470	−0.0882349	+	0.996100i	$0.528123\pi$
$978$	0	0
$979$	−1.24545e7	−0.415308
$980$	0	0
$981$	−3.40901e7	−1.13098
$982$	0	0
$983$	−2.14800e7	−0.709006	−0.354503	−	0.935055i	$-0.615350\pi$
$983$	−2.14800e7	−0.709006	−0.354503	+	0.935055i	$0.615350\pi$
$984$	0	0
$985$	−1.95517e7	−0.642087
$986$	0	0
$987$	804870.	0.0262986
$988$	0	0
$989$	−1.29870e7	−0.422201
$990$	0	0
$991$	−4.73852e7	−1.53271	−0.766353	−	0.642420i	$-0.777930\pi$
$991$	−4.73852e7	−1.53271	−0.766353	+	0.642420i	$0.777930\pi$
$992$	0	0
$993$	1.60890e6	0.0517792
$994$	0	0
$995$	9.58438e6	0.306907
$996$	0	0
$997$	−4.20284e7	−1.33908	−0.669538	−	0.742778i	$-0.733508\pi$
$997$	−4.20284e7	−1.33908	−0.669538	+	0.742778i	$0.733508\pi$
$998$	0	0
$999$	−3.62784e6	−0.115010

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.6.a.a.1.2	✓	2
4.3	odd	2		560.6.a.p.1.1		2
5.2	odd	4		700.6.e.e.449.2		4
5.3	odd	4		700.6.e.e.449.3		4
5.4	even	2		700.6.a.h.1.1		2
7.6	odd	2		980.6.a.g.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.6.a.a.1.2	✓	2	1.1	even	1	trivial
560.6.a.p.1.1		2	4.3	odd	2
700.6.a.h.1.1		2	5.4	even	2
700.6.e.e.449.2		4	5.2	odd	4
700.6.e.e.449.3		4	5.3	odd	4
980.6.a.g.1.1		2	7.6	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:	\( N \)	\(=\)	\( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	140.a (trivial)

\(f(q)\)	\(=\)	\(q+4.38238 q^{3} -25.0000 q^{5} +49.0000 q^{7} -223.795 q^{9} +579.441 q^{11} -426.030 q^{13} -109.560 q^{15} -2010.09 q^{17} -402.294 q^{19} +214.737 q^{21} +1516.77 q^{23} +625.000 q^{25} -2045.67 q^{27} -7789.51 q^{29} -4700.59 q^{31} +2539.33 q^{33} -1225.00 q^{35} +1773.42 q^{37} -1867.03 q^{39} -7354.41 q^{41} -8562.31 q^{43} +5594.87 q^{45} +3748.17 q^{47} +2401.00 q^{49} -8808.98 q^{51} -33514.8 q^{53} -14486.0 q^{55} -1763.01 q^{57} -11302.6 q^{59} -24964.2 q^{61} -10965.9 q^{63} +10650.7 q^{65} -35315.7 q^{67} +6647.06 q^{69} +70973.7 q^{71} +17070.3 q^{73} +2738.99 q^{75} +28392.6 q^{77} +10103.6 q^{79} +45417.2 q^{81} +46105.8 q^{83} +50252.2 q^{85} -34136.6 q^{87} -21494.0 q^{89} -20875.5 q^{91} -20599.8 q^{93} +10057.4 q^{95} -65031.4 q^{97} -129676. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 23 q^{3} - 50 q^{5} + 98 q^{7} + 283 q^{9} + 873 q^{11} - 185 q^{13} + 575 q^{15} - 2019 q^{17} - 614 q^{19} - 1127 q^{21} - 1350 q^{23} + 1250 q^{25} - 9269 q^{27} - 999 q^{29} - 9020 q^{31} - 5499 q^{33}+ \cdots + 19098 q^{99}+O(q^{100}) \) 2 * q - 23 * q^3 - 50 * q^5 + 98 * q^7 + 283 * q^9 + 873 * q^11 - 185 * q^13 + 575 * q^15 - 2019 * q^17 - 614 * q^19 - 1127 * q^21 - 1350 * q^23 + 1250 * q^25 - 9269 * q^27 - 999 * q^29 - 9020 * q^31 - 5499 * q^33 - 2450 * q^35 - 9032 * q^37 - 8467 * q^39 - 18330 * q^41 - 3974 * q^43 - 7075 * q^45 - 21759 * q^47 + 4802 * q^49 - 8565 * q^51 + 2154 * q^53 - 21825 * q^55 + 4034 * q^57 - 28704 * q^59 + 12394 * q^61 + 13867 * q^63 + 4625 * q^65 - 101888 * q^67 + 85146 * q^69 + 91632 * q^71 - 35996 * q^73 - 14375 * q^75 + 42777 * q^77 - 10001 * q^79 + 120058 * q^81 + 78108 * q^83 + 50475 * q^85 - 220077 * q^87 + 33438 * q^89 - 9065 * q^91 + 97676 * q^93 + 15350 * q^95 - 85751 * q^97 + 19098 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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