LMFDB - Embedded newform 1040.6.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1040,6,Mod(1,1040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1040.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1040 = 2^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1040.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:	$166.799172605$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1458804.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 361x - 1139 $ x^3 - x^2 - 361*x - 1139
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 130)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.28300$ of defining polynomial
Character	$\chi$	$=$	1040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.71700 q^{3} +25.0000 q^{5} +10.2576 q^{7} -229.184 q^{9} -197.274 q^{11} +169.000 q^{13} +92.9250 q^{15} -949.722 q^{17} -2233.30 q^{19} +38.1276 q^{21} +367.911 q^{23} +625.000 q^{25} -1755.11 q^{27} -6602.16 q^{29} +9911.27 q^{31} -733.267 q^{33} +256.441 q^{35} -9397.70 q^{37} +628.173 q^{39} +20588.3 q^{41} +16341.8 q^{43} -5729.60 q^{45} +13538.5 q^{47} -16701.8 q^{49} -3530.12 q^{51} -35049.6 q^{53} -4931.84 q^{55} -8301.20 q^{57} +6170.98 q^{59} +13746.1 q^{61} -2350.88 q^{63} +4225.00 q^{65} +62136.4 q^{67} +1367.52 q^{69} +38355.6 q^{71} +59967.3 q^{73} +2323.13 q^{75} -2023.56 q^{77} -89670.2 q^{79} +49167.9 q^{81} +38134.8 q^{83} -23743.1 q^{85} -24540.2 q^{87} -78278.2 q^{89} +1733.54 q^{91} +36840.2 q^{93} -55832.6 q^{95} +80567.1 q^{97} +45212.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 22 q^{3} + 75 q^{5} + 234 q^{7} + 155 q^{9} - 48 q^{11} + 507 q^{13} + 550 q^{15} + 1506 q^{17} + 360 q^{19} + 6904 q^{21} + 2370 q^{23} + 1875 q^{25} + 10168 q^{27} - 3078 q^{29} + 5388 q^{31} - 22572 q^{33}+ \cdots - 350340 q^{99}+O(q^{100}) $ 3 * q + 22 * q^3 + 75 * q^5 + 234 * q^7 + 155 * q^9 - 48 * q^11 + 507 * q^13 + 550 * q^15 + 1506 * q^17 + 360 * q^19 + 6904 * q^21 + 2370 * q^23 + 1875 * q^25 + 10168 * q^27 - 3078 * q^29 + 5388 * q^31 - 22572 * q^33 + 5850 * q^35 - 25362 * q^37 + 3718 * q^39 - 15906 * q^41 + 39306 * q^43 + 3875 * q^45 + 17778 * q^47 + 7767 * q^49 + 33972 * q^51 + 9246 * q^53 - 1200 * q^55 - 2996 * q^57 + 77760 * q^59 + 17982 * q^61 + 128762 * q^63 + 12675 * q^65 + 2922 * q^67 - 31860 * q^69 + 4944 * q^71 + 43278 * q^73 + 13750 * q^75 - 144420 * q^77 - 42120 * q^79 + 146519 * q^81 - 58098 * q^83 + 37650 * q^85 - 2016 * q^87 + 19614 * q^89 + 39546 * q^91 + 162796 * q^93 + 9000 * q^95 - 87078 * q^97 - 350340 * 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
$3$	3.71700	0.238446	0.119223	−	0.992868i	$-0.461960\pi$
$3$	3.71700	0.238446	0.119223	+	0.992868i	$0.461960\pi$
$4$	0	0
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	10.2576	0.0791228	0.0395614	−	0.999217i	$-0.487404\pi$
$7$	10.2576	0.0791228	0.0395614	+	0.999217i	$0.487404\pi$
$8$	0	0
$9$	−229.184	−0.943144
$10$	0	0
$11$	−197.274	−0.491572	−0.245786	−	0.969324i	$-0.579046\pi$
$11$	−197.274	−0.491572	−0.245786	+	0.969324i	$0.579046\pi$
$12$	0	0
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	0	0
$15$	92.9250	0.106636
$16$	0	0
$17$	−949.722	−0.797029	−0.398515	−	0.917162i	$-0.630474\pi$
$17$	−949.722	−0.797029	−0.398515	+	0.917162i	$0.630474\pi$
$18$	0	0
$19$	−2233.30	−1.41927	−0.709633	−	0.704571i	$-0.751139\pi$
$19$	−2233.30	−1.41927	−0.709633	+	0.704571i	$0.751139\pi$
$20$	0	0
$21$	38.1276	0.0188665
$22$	0	0
$23$	367.911	0.145018	0.0725092	−	0.997368i	$-0.476899\pi$
$23$	367.911	0.145018	0.0725092	+	0.997368i	$0.476899\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−1755.11	−0.463334
$28$	0	0
$29$	−6602.16	−1.45778	−0.728888	−	0.684633i	$-0.759963\pi$
$29$	−6602.16	−1.45778	−0.728888	+	0.684633i	$0.759963\pi$
$30$	0	0
$31$	9911.27	1.85236	0.926179	−	0.377084i	$-0.123073\pi$
$31$	9911.27	1.85236	0.926179	+	0.377084i	$0.123073\pi$
$32$	0	0
$33$	−733.267	−0.117213
$34$	0	0
$35$	256.441	0.0353848
$36$	0	0
$37$	−9397.70	−1.12854	−0.564271	−	0.825590i	$-0.690842\pi$
$37$	−9397.70	−1.12854	−0.564271	+	0.825590i	$0.690842\pi$
$38$	0	0
$39$	628.173	0.0661330
$40$	0	0
$41$	20588.3	1.91276	0.956380	−	0.292126i	$-0.0943626\pi$
$41$	20588.3	1.91276	0.956380	+	0.292126i	$0.0943626\pi$
$42$	0	0
$43$	16341.8	1.34781	0.673904	−	0.738819i	$-0.264616\pi$
$43$	16341.8	1.34781	0.673904	+	0.738819i	$0.264616\pi$
$44$	0	0
$45$	−5729.60	−0.421787
$46$	0	0
$47$	13538.5	0.893977	0.446989	−	0.894540i	$-0.352496\pi$
$47$	13538.5	0.893977	0.446989	+	0.894540i	$0.352496\pi$
$48$	0	0
$49$	−16701.8	−0.993740
$50$	0	0
$51$	−3530.12	−0.190048
$52$	0	0
$53$	−35049.6	−1.71393	−0.856965	−	0.515374i	$-0.827653\pi$
$53$	−35049.6	−1.71393	−0.856965	+	0.515374i	$0.827653\pi$
$54$	0	0
$55$	−4931.84	−0.219838
$56$	0	0
$57$	−8301.20	−0.338418
$58$	0	0
$59$	6170.98	0.230794	0.115397	−	0.993319i	$-0.463186\pi$
$59$	6170.98	0.230794	0.115397	+	0.993319i	$0.463186\pi$
$60$	0	0
$61$	13746.1	0.472993	0.236496	−	0.971632i	$-0.424001\pi$
$61$	13746.1	0.472993	0.236496	+	0.971632i	$0.424001\pi$
$62$	0	0
$63$	−2350.88	−0.0746242
$64$	0	0
$65$	4225.00	0.124035
$66$	0	0
$67$	62136.4	1.69106	0.845530	−	0.533927i	$-0.179284\pi$
$67$	62136.4	1.69106	0.845530	+	0.533927i	$0.179284\pi$
$68$	0	0
$69$	1367.52	0.0345790
$70$	0	0
$71$	38355.6	0.902989	0.451495	−	0.892274i	$-0.350891\pi$
$71$	38355.6	0.902989	0.451495	+	0.892274i	$0.350891\pi$
$72$	0	0
$73$	59967.3	1.31707	0.658533	−	0.752552i	$-0.271177\pi$
$73$	59967.3	1.31707	0.658533	+	0.752552i	$0.271177\pi$
$74$	0	0
$75$	2323.13	0.0476891
$76$	0	0
$77$	−2023.56	−0.0388946
$78$	0	0
$79$	−89670.2	−1.61652	−0.808259	−	0.588827i	$-0.799590\pi$
$79$	−89670.2	−1.61652	−0.808259	+	0.588827i	$0.799590\pi$
$80$	0	0
$81$	49167.9	0.832664
$82$	0	0
$83$	38134.8	0.607611	0.303806	−	0.952734i	$-0.401743\pi$
$83$	38134.8	0.607611	0.303806	+	0.952734i	$0.401743\pi$
$84$	0	0
$85$	−23743.1	−0.356442
$86$	0	0
$87$	−24540.2	−0.347601
$88$	0	0
$89$	−78278.2	−1.04753	−0.523764	−	0.851863i	$-0.675473\pi$
$89$	−78278.2	−1.04753	−0.523764	+	0.851863i	$0.675473\pi$
$90$	0	0
$91$	1733.54	0.0219447
$92$	0	0
$93$	36840.2	0.441687
$94$	0	0
$95$	−55832.6	−0.634715
$96$	0	0
$97$	80567.1	0.869417	0.434709	−	0.900571i	$-0.356851\pi$
$97$	80567.1	0.869417	0.434709	+	0.900571i	$0.356851\pi$
$98$	0	0
$99$	45212.0	0.463623
$100$	0	0
$101$	−10770.8	−0.105062	−0.0525309	−	0.998619i	$-0.516729\pi$
$101$	−10770.8	−0.105062	−0.0525309	+	0.998619i	$0.516729\pi$
$102$	0	0
$103$	−44993.4	−0.417884	−0.208942	−	0.977928i	$-0.567002\pi$
$103$	−44993.4	−0.417884	−0.208942	+	0.977928i	$0.567002\pi$
$104$	0	0
$105$	953.190	0.00843735
$106$	0	0
$107$	194227.	1.64003	0.820013	−	0.572345i	$-0.193966\pi$
$107$	194227.	1.64003	0.820013	+	0.572345i	$0.193966\pi$
$108$	0	0
$109$	216396.	1.74455	0.872275	−	0.489016i	$-0.162644\pi$
$109$	216396.	1.74455	0.872275	+	0.489016i	$0.162644\pi$
$110$	0	0
$111$	−34931.3	−0.269096
$112$	0	0
$113$	141318.	1.04112	0.520562	−	0.853824i	$-0.325722\pi$
$113$	141318.	1.04112	0.520562	+	0.853824i	$0.325722\pi$
$114$	0	0
$115$	9197.77	0.0648542
$116$	0	0
$117$	−38732.1	−0.261581
$118$	0	0
$119$	−9741.89	−0.0630632
$120$	0	0
$121$	−122134.	−0.758356
$122$	0	0
$123$	76526.7	0.456089
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	188285.	1.03587	0.517935	−	0.855420i	$-0.326701\pi$
$127$	188285.	1.03587	0.517935	+	0.855420i	$0.326701\pi$
$128$	0	0
$129$	60742.4	0.321379
$130$	0	0
$131$	−263916.	−1.34365	−0.671826	−	0.740709i	$-0.734490\pi$
$131$	−263916.	−1.34365	−0.671826	+	0.740709i	$0.734490\pi$
$132$	0	0
$133$	−22908.4	−0.112296
$134$	0	0
$135$	−43877.7	−0.207209
$136$	0	0
$137$	−337417.	−1.53591	−0.767955	−	0.640504i	$-0.778725\pi$
$137$	−337417.	−1.53591	−0.767955	+	0.640504i	$0.778725\pi$
$138$	0	0
$139$	86203.3	0.378431	0.189216	−	0.981936i	$-0.439405\pi$
$139$	86203.3	0.378431	0.189216	+	0.981936i	$0.439405\pi$
$140$	0	0
$141$	50322.7	0.213165
$142$	0	0
$143$	−33339.3	−0.136338
$144$	0	0
$145$	−165054.	−0.651937
$146$	0	0
$147$	−62080.5	−0.236953
$148$	0	0
$149$	−290191.	−1.07082	−0.535412	−	0.844591i	$-0.679844\pi$
$149$	−290191.	−1.07082	−0.535412	+	0.844591i	$0.679844\pi$
$150$	0	0
$151$	−129889.	−0.463585	−0.231793	−	0.972765i	$-0.574459\pi$
$151$	−129889.	−0.463585	−0.231793	+	0.972765i	$0.574459\pi$
$152$	0	0
$153$	217661.	0.751713
$154$	0	0
$155$	247782.	0.828400
$156$	0	0
$157$	−46206.2	−0.149607	−0.0748033	−	0.997198i	$-0.523833\pi$
$157$	−46206.2	−0.149607	−0.0748033	+	0.997198i	$0.523833\pi$
$158$	0	0
$159$	−130279.	−0.408679
$160$	0	0
$161$	3773.89	0.0114743
$162$	0	0
$163$	−379076.	−1.11753	−0.558763	−	0.829328i	$-0.688724\pi$
$163$	−379076.	−1.11753	−0.558763	+	0.829328i	$0.688724\pi$
$164$	0	0
$165$	−18331.7	−0.0524194
$166$	0	0
$167$	569176.	1.57927	0.789633	−	0.613580i	$-0.210271\pi$
$167$	569176.	1.57927	0.789633	+	0.613580i	$0.210271\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	0	0
$171$	511838.	1.33857
$172$	0	0
$173$	−113968.	−0.289513	−0.144757	−	0.989467i	$-0.546240\pi$
$173$	−113968.	−0.289513	−0.144757	+	0.989467i	$0.546240\pi$
$174$	0	0
$175$	6411.02	0.0158246
$176$	0	0
$177$	22937.5	0.0550318
$178$	0	0
$179$	−320294.	−0.747165	−0.373582	−	0.927597i	$-0.621871\pi$
$179$	−320294.	−0.747165	−0.373582	+	0.927597i	$0.621871\pi$
$180$	0	0
$181$	451978.	1.02547	0.512733	−	0.858548i	$-0.328633\pi$
$181$	451978.	1.02547	0.512733	+	0.858548i	$0.328633\pi$
$182$	0	0
$183$	51094.2	0.112783
$184$	0	0
$185$	−234943.	−0.504699
$186$	0	0
$187$	187355.	0.391798
$188$	0	0
$189$	−18003.2	−0.0366603
$190$	0	0
$191$	−182918.	−0.362805	−0.181402	−	0.983409i	$-0.558064\pi$
$191$	−182918.	−0.362805	−0.181402	+	0.983409i	$0.558064\pi$
$192$	0	0
$193$	508696.	0.983026	0.491513	−	0.870870i	$-0.336444\pi$
$193$	508696.	0.983026	0.491513	+	0.870870i	$0.336444\pi$
$194$	0	0
$195$	15704.3	0.0295756
$196$	0	0
$197$	111666.	0.205000	0.102500	−	0.994733i	$-0.467316\pi$
$197$	111666.	0.205000	0.102500	+	0.994733i	$0.467316\pi$
$198$	0	0
$199$	−521218.	−0.933011	−0.466505	−	0.884518i	$-0.654487\pi$
$199$	−521218.	−0.933011	−0.466505	+	0.884518i	$0.654487\pi$
$200$	0	0
$201$	230961.	0.403226
$202$	0	0
$203$	−67722.4	−0.115343
$204$	0	0
$205$	514707.	0.855412
$206$	0	0
$207$	−84319.2	−0.136773
$208$	0	0
$209$	440572.	0.697672
$210$	0	0
$211$	543969.	0.841140	0.420570	−	0.907260i	$-0.361830\pi$
$211$	543969.	0.841140	0.420570	+	0.907260i	$0.361830\pi$
$212$	0	0
$213$	142568.	0.215314
$214$	0	0
$215$	408544.	0.602758
$216$	0	0
$217$	101666.	0.146564
$218$	0	0
$219$	222899.	0.314049
$220$	0	0
$221$	−160503.	−0.221056
$222$	0	0
$223$	−114372.	−0.154013	−0.0770063	−	0.997031i	$-0.524536\pi$
$223$	−114372.	−0.154013	−0.0770063	+	0.997031i	$0.524536\pi$
$224$	0	0
$225$	−143240.	−0.188629
$226$	0	0
$227$	213250.	0.274678	0.137339	−	0.990524i	$-0.456145\pi$
$227$	213250.	0.274678	0.137339	+	0.990524i	$0.456145\pi$
$228$	0	0
$229$	1.18502e6	1.49326	0.746631	−	0.665239i	$-0.231670\pi$
$229$	1.18502e6	1.49326	0.746631	+	0.665239i	$0.231670\pi$
$230$	0	0
$231$	−7521.58	−0.00927425
$232$	0	0
$233$	1.37128e6	1.65476	0.827381	−	0.561641i	$-0.189830\pi$
$233$	1.37128e6	1.65476	0.827381	+	0.561641i	$0.189830\pi$
$234$	0	0
$235$	338463.	0.399799
$236$	0	0
$237$	−333304.	−0.385452
$238$	0	0
$239$	1.09024e6	1.23460	0.617301	−	0.786727i	$-0.288226\pi$
$239$	1.09024e6	1.23460	0.617301	+	0.786727i	$0.288226\pi$
$240$	0	0
$241$	84897.3	0.0941567	0.0470784	−	0.998891i	$-0.485009\pi$
$241$	84897.3	0.0941567	0.0470784	+	0.998891i	$0.485009\pi$
$242$	0	0
$243$	609249.	0.661879
$244$	0	0
$245$	−417545.	−0.444414
$246$	0	0
$247$	−377429.	−0.393634
$248$	0	0
$249$	141747.	0.144882
$250$	0	0
$251$	−425949.	−0.426750	−0.213375	−	0.976970i	$-0.568446\pi$
$251$	−425949.	−0.426750	−0.213375	+	0.976970i	$0.568446\pi$
$252$	0	0
$253$	−72579.1	−0.0712870
$254$	0	0
$255$	−88253.0	−0.0849922
$256$	0	0
$257$	250941.	0.236995	0.118497	−	0.992954i	$-0.462192\pi$
$257$	250941.	0.236995	0.118497	+	0.992954i	$0.462192\pi$
$258$	0	0
$259$	−96398.1	−0.0892933
$260$	0	0
$261$	1.51311e6	1.37489
$262$	0	0
$263$	1.48299e6	1.32205	0.661026	−	0.750363i	$-0.270121\pi$
$263$	1.48299e6	1.32205	0.661026	+	0.750363i	$0.270121\pi$
$264$	0	0
$265$	−876239.	−0.766493
$266$	0	0
$267$	−290960.	−0.249779
$268$	0	0
$269$	398773.	0.336004	0.168002	−	0.985787i	$-0.446268\pi$
$269$	398773.	0.336004	0.168002	+	0.985787i	$0.446268\pi$
$270$	0	0
$271$	−182496.	−0.150949	−0.0754743	−	0.997148i	$-0.524047\pi$
$271$	−182496.	−0.150949	−0.0754743	+	0.997148i	$0.524047\pi$
$272$	0	0
$273$	6443.57	0.00523262
$274$	0	0
$275$	−123296.	−0.0983145
$276$	0	0
$277$	−79202.9	−0.0620215	−0.0310107	−	0.999519i	$-0.509873\pi$
$277$	−79202.9	−0.0620215	−0.0310107	+	0.999519i	$0.509873\pi$
$278$	0	0
$279$	−2.27150e6	−1.74704
$280$	0	0
$281$	−1.51199e6	−1.14231	−0.571154	−	0.820843i	$-0.693504\pi$
$281$	−1.51199e6	−1.14231	−0.571154	+	0.820843i	$0.693504\pi$
$282$	0	0
$283$	−2.50781e6	−1.86135	−0.930675	−	0.365848i	$-0.880779\pi$
$283$	−2.50781e6	−1.86135	−0.930675	+	0.365848i	$0.880779\pi$
$284$	0	0
$285$	−207530.	−0.151345
$286$	0	0
$287$	211187.	0.151343
$288$	0	0
$289$	−517885.	−0.364744
$290$	0	0
$291$	299468.	0.207309
$292$	0	0
$293$	1.75384e6	1.19349	0.596747	−	0.802429i	$-0.296459\pi$
$293$	1.75384e6	1.19349	0.596747	+	0.802429i	$0.296459\pi$
$294$	0	0
$295$	154274.	0.103214
$296$	0	0
$297$	346237.	0.227762
$298$	0	0
$299$	62176.9	0.0402208
$300$	0	0
$301$	167628.	0.106642
$302$	0	0
$303$	−40035.1	−0.0250515
$304$	0	0
$305$	343652.	0.211529
$306$	0	0
$307$	1.55298e6	0.940414	0.470207	−	0.882556i	$-0.344179\pi$
$307$	1.55298e6	0.940414	0.470207	+	0.882556i	$0.344179\pi$
$308$	0	0
$309$	−167240.	−0.0996426
$310$	0	0
$311$	188125.	0.110292	0.0551462	−	0.998478i	$-0.482438\pi$
$311$	188125.	0.110292	0.0551462	+	0.998478i	$0.482438\pi$
$312$	0	0
$313$	846464.	0.488369	0.244184	−	0.969729i	$-0.421480\pi$
$313$	846464.	0.488369	0.244184	+	0.969729i	$0.421480\pi$
$314$	0	0
$315$	−58772.1	−0.0333729
$316$	0	0
$317$	−303332.	−0.169539	−0.0847696	−	0.996401i	$-0.527015\pi$
$317$	−303332.	−0.169539	−0.0847696	+	0.996401i	$0.527015\pi$
$318$	0	0
$319$	1.30243e6	0.716603
$320$	0	0
$321$	721943.	0.391057
$322$	0	0
$323$	2.12102e6	1.13120
$324$	0	0
$325$	105625.	0.0554700
$326$	0	0
$327$	804345.	0.415980
$328$	0	0
$329$	138873.	0.0707340
$330$	0	0
$331$	3.36224e6	1.68678	0.843390	−	0.537301i	$-0.180556\pi$
$331$	3.36224e6	1.68678	0.843390	+	0.537301i	$0.180556\pi$
$332$	0	0
$333$	2.15380e6	1.06438
$334$	0	0
$335$	1.55341e6	0.756265
$336$	0	0
$337$	2.42001e6	1.16076	0.580380	−	0.814346i	$-0.302904\pi$
$337$	2.42001e6	1.16076	0.580380	+	0.814346i	$0.302904\pi$
$338$	0	0
$339$	525281.	0.248252
$340$	0	0
$341$	−1.95523e6	−0.910568
$342$	0	0
$343$	−343720.	−0.157750
$344$	0	0
$345$	34188.1	0.0154642
$346$	0	0
$347$	208658.	0.0930274	0.0465137	−	0.998918i	$-0.485189\pi$
$347$	208658.	0.0930274	0.0465137	+	0.998918i	$0.485189\pi$
$348$	0	0
$349$	−2.56139e6	−1.12567	−0.562837	−	0.826568i	$-0.690290\pi$
$349$	−2.56139e6	−1.12567	−0.562837	+	0.826568i	$0.690290\pi$
$350$	0	0
$351$	−296613.	−0.128506
$352$	0	0
$353$	−2.22758e6	−0.951472	−0.475736	−	0.879588i	$-0.657818\pi$
$353$	−2.22758e6	−0.951472	−0.475736	+	0.879588i	$0.657818\pi$
$354$	0	0
$355$	958889.	0.403829
$356$	0	0
$357$	−36210.6	−0.0150372
$358$	0	0
$359$	4.69711e6	1.92351	0.961756	−	0.273908i	$-0.0883164\pi$
$359$	4.69711e6	1.92351	0.961756	+	0.273908i	$0.0883164\pi$
$360$	0	0
$361$	2.51155e6	1.01432
$362$	0	0
$363$	−453973.	−0.180827
$364$	0	0
$365$	1.49918e6	0.589010
$366$	0	0
$367$	3.22496e6	1.24985	0.624926	−	0.780684i	$-0.285129\pi$
$367$	3.22496e6	1.24985	0.624926	+	0.780684i	$0.285129\pi$
$368$	0	0
$369$	−4.71850e6	−1.80401
$370$	0	0
$371$	−359525.	−0.135611
$372$	0	0
$373$	−3.09204e6	−1.15073	−0.575364	−	0.817898i	$-0.695140\pi$
$373$	−3.09204e6	−1.15073	−0.575364	+	0.817898i	$0.695140\pi$
$374$	0	0
$375$	58078.1	0.0213272
$376$	0	0
$377$	−1.11576e6	−0.404314
$378$	0	0
$379$	3.63851e6	1.30115	0.650573	−	0.759444i	$-0.274529\pi$
$379$	3.63851e6	1.30115	0.650573	+	0.759444i	$0.274529\pi$
$380$	0	0
$381$	699854.	0.246999
$382$	0	0
$383$	−1.89242e6	−0.659204	−0.329602	−	0.944120i	$-0.606915\pi$
$383$	−1.89242e6	−0.659204	−0.329602	+	0.944120i	$0.606915\pi$
$384$	0	0
$385$	−50589.0	−0.0173942
$386$	0	0
$387$	−3.74527e6	−1.27118
$388$	0	0
$389$	79129.5	0.0265133	0.0132567	−	0.999912i	$-0.495780\pi$
$389$	79129.5	0.0265133	0.0132567	+	0.999912i	$0.495780\pi$
$390$	0	0
$391$	−349413.	−0.115584
$392$	0	0
$393$	−980975.	−0.320388
$394$	0	0
$395$	−2.24176e6	−0.722929
$396$	0	0
$397$	−1.82235e6	−0.580305	−0.290153	−	0.956980i	$-0.593706\pi$
$397$	−1.82235e6	−0.580305	−0.290153	+	0.956980i	$0.593706\pi$
$398$	0	0
$399$	−85150.6	−0.0267766
$400$	0	0
$401$	−117774.	−0.0365754	−0.0182877	−	0.999833i	$-0.505821\pi$
$401$	−117774.	−0.0365754	−0.0182877	+	0.999833i	$0.505821\pi$
$402$	0	0
$403$	1.67500e6	0.513752
$404$	0	0
$405$	1.22920e6	0.372378
$406$	0	0
$407$	1.85392e6	0.554760
$408$	0	0
$409$	3.73101e6	1.10285	0.551427	−	0.834223i	$-0.314083\pi$
$409$	3.73101e6	1.10285	0.551427	+	0.834223i	$0.314083\pi$
$410$	0	0
$411$	−1.25418e6	−0.366231
$412$	0	0
$413$	63299.6	0.0182610
$414$	0	0
$415$	953369.	0.271732
$416$	0	0
$417$	320418.	0.0902353
$418$	0	0
$419$	2.69321e6	0.749438	0.374719	−	0.927138i	$-0.377739\pi$
$419$	2.69321e6	0.749438	0.374719	+	0.927138i	$0.377739\pi$
$420$	0	0
$421$	5.56789e6	1.53103	0.765517	−	0.643415i	$-0.222483\pi$
$421$	5.56789e6	1.53103	0.765517	+	0.643415i	$0.222483\pi$
$422$	0	0
$423$	−3.10281e6	−0.843149
$424$	0	0
$425$	−593576.	−0.159406
$426$	0	0
$427$	141002.	0.0374245
$428$	0	0
$429$	−123922.	−0.0325091
$430$	0	0
$431$	3.75659e6	0.974093	0.487047	−	0.873376i	$-0.338074\pi$
$431$	3.75659e6	0.974093	0.487047	+	0.873376i	$0.338074\pi$
$432$	0	0
$433$	1.60978e6	0.412615	0.206308	−	0.978487i	$-0.433855\pi$
$433$	1.60978e6	0.412615	0.206308	+	0.978487i	$0.433855\pi$
$434$	0	0
$435$	−613506.	−0.155452
$436$	0	0
$437$	−821657.	−0.205820
$438$	0	0
$439$	1.92307e6	0.476248	0.238124	−	0.971235i	$-0.423468\pi$
$439$	1.92307e6	0.476248	0.238124	+	0.971235i	$0.423468\pi$
$440$	0	0
$441$	3.82778e6	0.937239
$442$	0	0
$443$	1.43060e6	0.346346	0.173173	−	0.984891i	$-0.444598\pi$
$443$	1.43060e6	0.346346	0.173173	+	0.984891i	$0.444598\pi$
$444$	0	0
$445$	−1.95695e6	−0.468469
$446$	0	0
$447$	−1.07864e6	−0.255333
$448$	0	0
$449$	−3.30270e6	−0.773131	−0.386565	−	0.922262i	$-0.626339\pi$
$449$	−3.30270e6	−0.773131	−0.386565	+	0.922262i	$0.626339\pi$
$450$	0	0
$451$	−4.06153e6	−0.940260
$452$	0	0
$453$	−482797.	−0.110540
$454$	0	0
$455$	43338.5	0.00981398
$456$	0	0
$457$	6.85658e6	1.53574	0.767869	−	0.640606i	$-0.221317\pi$
$457$	6.85658e6	1.53574	0.767869	+	0.640606i	$0.221317\pi$
$458$	0	0
$459$	1.66687e6	0.369291
$460$	0	0
$461$	8.33435e6	1.82650	0.913250	−	0.407400i	$-0.133565\pi$
$461$	8.33435e6	1.82650	0.913250	+	0.407400i	$0.133565\pi$
$462$	0	0
$463$	2.87024e6	0.622250	0.311125	−	0.950369i	$-0.399294\pi$
$463$	2.87024e6	0.622250	0.311125	+	0.950369i	$0.399294\pi$
$464$	0	0
$465$	921005.	0.197528
$466$	0	0
$467$	−6.61814e6	−1.40425	−0.702124	−	0.712055i	$-0.747765\pi$
$467$	−6.61814e6	−1.40425	−0.702124	+	0.712055i	$0.747765\pi$
$468$	0	0
$469$	637372.	0.133801
$470$	0	0
$471$	−171748.	−0.0356731
$472$	0	0
$473$	−3.22380e6	−0.662545
$474$	0	0
$475$	−1.39582e6	−0.283853
$476$	0	0
$477$	8.03280e6	1.61648
$478$	0	0
$479$	1.47435e6	0.293603	0.146802	−	0.989166i	$-0.453102\pi$
$479$	1.47435e6	0.293603	0.146802	+	0.989166i	$0.453102\pi$
$480$	0	0
$481$	−1.58821e6	−0.313001
$482$	0	0
$483$	14027.6	0.00273599
$484$	0	0
$485$	2.01418e6	0.388815
$486$	0	0
$487$	−2.63151e6	−0.502786	−0.251393	−	0.967885i	$-0.580889\pi$
$487$	−2.63151e6	−0.502786	−0.251393	+	0.967885i	$0.580889\pi$
$488$	0	0
$489$	−1.40903e6	−0.266469
$490$	0	0
$491$	−4.70461e6	−0.880684	−0.440342	−	0.897830i	$-0.645143\pi$
$491$	−4.70461e6	−0.880684	−0.440342	+	0.897830i	$0.645143\pi$
$492$	0	0
$493$	6.27022e6	1.16189
$494$	0	0
$495$	1.13030e6	0.207339
$496$	0	0
$497$	393437.	0.0714471
$498$	0	0
$499$	−409148.	−0.0735578	−0.0367789	−	0.999323i	$-0.511710\pi$
$499$	−409148.	−0.0735578	−0.0367789	+	0.999323i	$0.511710\pi$
$500$	0	0
$501$	2.11563e6	0.376569
$502$	0	0
$503$	−9.62836e6	−1.69681	−0.848404	−	0.529350i	$-0.822436\pi$
$503$	−9.62836e6	−1.69681	−0.848404	+	0.529350i	$0.822436\pi$
$504$	0	0
$505$	−269270.	−0.0469851
$506$	0	0
$507$	106161.	0.0183420
$508$	0	0
$509$	−1.24824e6	−0.213552	−0.106776	−	0.994283i	$-0.534053\pi$
$509$	−1.24824e6	−0.213552	−0.106776	+	0.994283i	$0.534053\pi$
$510$	0	0
$511$	615122.	0.104210
$512$	0	0
$513$	3.91969e6	0.657595
$514$	0	0
$515$	−1.12483e6	−0.186883
$516$	0	0
$517$	−2.67079e6	−0.439455
$518$	0	0
$519$	−423620.	−0.0690332
$520$	0	0
$521$	−7.13050e6	−1.15087	−0.575434	−	0.817848i	$-0.695167\pi$
$521$	−7.13050e6	−1.15087	−0.575434	+	0.817848i	$0.695167\pi$
$522$	0	0
$523$	−4.85487e6	−0.776110	−0.388055	−	0.921636i	$-0.626853\pi$
$523$	−4.85487e6	−0.776110	−0.388055	+	0.921636i	$0.626853\pi$
$524$	0	0
$525$	23829.8	0.00377330
$526$	0	0
$527$	−9.41295e6	−1.47638
$528$	0	0
$529$	−6.30098e6	−0.978970
$530$	0	0
$531$	−1.41429e6	−0.217672
$532$	0	0
$533$	3.47942e6	0.530504
$534$	0	0
$535$	4.85568e6	0.733442
$536$	0	0
$537$	−1.19053e6	−0.178158
$538$	0	0
$539$	3.29482e6	0.488495
$540$	0	0
$541$	−1.13451e7	−1.66653	−0.833266	−	0.552872i	$-0.813532\pi$
$541$	−1.13451e7	−1.66653	−0.833266	+	0.552872i	$0.813532\pi$
$542$	0	0
$543$	1.68000e6	0.244518
$544$	0	0
$545$	5.40990e6	0.780186
$546$	0	0
$547$	−401928.	−0.0574355	−0.0287177	−	0.999588i	$-0.509142\pi$
$547$	−401928.	−0.0574355	−0.0287177	+	0.999588i	$0.509142\pi$
$548$	0	0
$549$	−3.15038e6	−0.446100
$550$	0	0
$551$	1.47446e7	2.06897
$552$	0	0
$553$	−919803.	−0.127903
$554$	0	0
$555$	−873282.	−0.120343
$556$	0	0
$557$	7.58501e6	1.03590	0.517950	−	0.855411i	$-0.326695\pi$
$557$	7.58501e6	1.03590	0.517950	+	0.855411i	$0.326695\pi$
$558$	0	0
$559$	2.76176e6	0.373814
$560$	0	0
$561$	696400.	0.0934225
$562$	0	0
$563$	9.36555e6	1.24527	0.622633	−	0.782514i	$-0.286063\pi$
$563$	9.36555e6	1.24527	0.622633	+	0.782514i	$0.286063\pi$
$564$	0	0
$565$	3.53296e6	0.465605
$566$	0	0
$567$	504346.	0.0658827
$568$	0	0
$569$	1.36255e7	1.76430	0.882150	−	0.470969i	$-0.156096\pi$
$569$	1.36255e7	1.76430	0.882150	+	0.470969i	$0.156096\pi$
$570$	0	0
$571$	−1.29016e7	−1.65598	−0.827988	−	0.560746i	$-0.810515\pi$
$571$	−1.29016e7	−1.65598	−0.827988	+	0.560746i	$0.810515\pi$
$572$	0	0
$573$	−679907.	−0.0865093
$574$	0	0
$575$	229944.	0.0290037
$576$	0	0
$577$	3.12690e6	0.390998	0.195499	−	0.980704i	$-0.437367\pi$
$577$	3.12690e6	0.390998	0.195499	+	0.980704i	$0.437367\pi$
$578$	0	0
$579$	1.89082e6	0.234398
$580$	0	0
$581$	391172.	0.0480759
$582$	0	0
$583$	6.91436e6	0.842521
$584$	0	0
$585$	−968302.	−0.116983
$586$	0	0
$587$	1.61598e7	1.93571	0.967854	−	0.251511i	$-0.0809275\pi$
$587$	1.61598e7	1.93571	0.967854	+	0.251511i	$0.0809275\pi$
$588$	0	0
$589$	−2.21349e7	−2.62899
$590$	0	0
$591$	415062.	0.0488814
$592$	0	0
$593$	−5.02852e6	−0.587223	−0.293612	−	0.955925i	$-0.594857\pi$
$593$	−5.02852e6	−0.587223	−0.293612	+	0.955925i	$0.594857\pi$
$594$	0	0
$595$	−243547.	−0.0282027
$596$	0	0
$597$	−1.93737e6	−0.222472
$598$	0	0
$599$	−6.83761e6	−0.778642	−0.389321	−	0.921102i	$-0.627290\pi$
$599$	−6.83761e6	−0.778642	−0.389321	+	0.921102i	$0.627290\pi$
$600$	0	0
$601$	−8.01912e6	−0.905608	−0.452804	−	0.891610i	$-0.649576\pi$
$601$	−8.01912e6	−0.905608	−0.452804	+	0.891610i	$0.649576\pi$
$602$	0	0
$603$	−1.42407e7	−1.59491
$604$	0	0
$605$	−3.05335e6	−0.339147
$606$	0	0
$607$	−7.83186e6	−0.862766	−0.431383	−	0.902169i	$-0.641974\pi$
$607$	−7.83186e6	−0.862766	−0.431383	+	0.902169i	$0.641974\pi$
$608$	0	0
$609$	−251724.	−0.0275031
$610$	0	0
$611$	2.28801e6	0.247945
$612$	0	0
$613$	−1.57922e7	−1.69743	−0.848716	−	0.528849i	$-0.822624\pi$
$613$	−1.57922e7	−1.69743	−0.848716	+	0.528849i	$0.822624\pi$
$614$	0	0
$615$	1.91317e6	0.203969
$616$	0	0
$617$	8.38747e6	0.886989	0.443494	−	0.896277i	$-0.353739\pi$
$617$	8.38747e6	0.886989	0.443494	+	0.896277i	$0.353739\pi$
$618$	0	0
$619$	−1.35269e6	−0.141896	−0.0709481	−	0.997480i	$-0.522602\pi$
$619$	−1.35269e6	−0.141896	−0.0709481	+	0.997480i	$0.522602\pi$
$620$	0	0
$621$	−645723.	−0.0671920
$622$	0	0
$623$	−802948.	−0.0828834
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	1.63761e6	0.166357
$628$	0	0
$629$	8.92521e6	0.899480
$630$	0	0
$631$	7.20270e6	0.720149	0.360074	−	0.932924i	$-0.382751\pi$
$631$	7.20270e6	0.720149	0.360074	+	0.932924i	$0.382751\pi$
$632$	0	0
$633$	2.02193e6	0.200566
$634$	0	0
$635$	4.70712e6	0.463255
$636$	0	0
$637$	−2.82260e6	−0.275614
$638$	0	0
$639$	−8.79048e6	−0.851649
$640$	0	0
$641$	2.63844e6	0.253631	0.126815	−	0.991926i	$-0.459524\pi$
$641$	2.63844e6	0.253631	0.126815	+	0.991926i	$0.459524\pi$
$642$	0	0
$643$	−6.76792e6	−0.645547	−0.322774	−	0.946476i	$-0.604615\pi$
$643$	−6.76792e6	−0.645547	−0.322774	+	0.946476i	$0.604615\pi$
$644$	0	0
$645$	1.51856e6	0.143725
$646$	0	0
$647$	−1.01251e7	−0.950907	−0.475453	−	0.879741i	$-0.657716\pi$
$647$	−1.01251e7	−0.950907	−0.475453	+	0.879741i	$0.657716\pi$
$648$	0	0
$649$	−1.21737e6	−0.113452
$650$	0	0
$651$	377893.	0.0349475
$652$	0	0
$653$	−8.17558e6	−0.750301	−0.375150	−	0.926964i	$-0.622409\pi$
$653$	−8.17558e6	−0.750301	−0.375150	+	0.926964i	$0.622409\pi$
$654$	0	0
$655$	−6.59789e6	−0.600900
$656$	0	0
$657$	−1.37435e7	−1.24218
$658$	0	0
$659$	1.24909e7	1.12042	0.560211	−	0.828350i	$-0.310720\pi$
$659$	1.24909e7	1.12042	0.560211	+	0.828350i	$0.310720\pi$
$660$	0	0
$661$	5.43688e6	0.484001	0.242001	−	0.970276i	$-0.422196\pi$
$661$	5.43688e6	0.484001	0.242001	+	0.970276i	$0.422196\pi$
$662$	0	0
$663$	−596590.	−0.0527099
$664$	0	0
$665$	−572710.	−0.0502205
$666$	0	0
$667$	−2.42900e6	−0.211404
$668$	0	0
$669$	−425120.	−0.0367236
$670$	0	0
$671$	−2.71174e6	−0.232510
$672$	0	0
$673$	9.60528e6	0.817471	0.408736	−	0.912653i	$-0.365970\pi$
$673$	9.60528e6	0.817471	0.408736	+	0.912653i	$0.365970\pi$
$674$	0	0
$675$	−1.09694e6	−0.0926669
$676$	0	0
$677$	5.99107e6	0.502380	0.251190	−	0.967938i	$-0.419178\pi$
$677$	5.99107e6	0.502380	0.251190	+	0.967938i	$0.419178\pi$
$678$	0	0
$679$	826427.	0.0687907
$680$	0	0
$681$	792650.	0.0654958
$682$	0	0
$683$	−1.49302e7	−1.22465	−0.612327	−	0.790605i	$-0.709766\pi$
$683$	−1.49302e7	−1.22465	−0.612327	+	0.790605i	$0.709766\pi$
$684$	0	0
$685$	−8.43543e6	−0.686880
$686$	0	0
$687$	4.40471e6	0.356062
$688$	0	0
$689$	−5.92338e6	−0.475359
$690$	0	0
$691$	4.19350e6	0.334104	0.167052	−	0.985948i	$-0.446575\pi$
$691$	4.19350e6	0.334104	0.167052	+	0.985948i	$0.446575\pi$
$692$	0	0
$693$	463767.	0.0366832
$694$	0	0
$695$	2.15508e6	0.169240
$696$	0	0
$697$	−1.95531e7	−1.52453
$698$	0	0
$699$	5.09704e6	0.394571
$700$	0	0
$701$	−1.88086e7	−1.44564	−0.722820	−	0.691036i	$-0.757155\pi$
$701$	−1.88086e7	−1.44564	−0.722820	+	0.691036i	$0.757155\pi$
$702$	0	0
$703$	2.09879e7	1.60170
$704$	0	0
$705$	1.25807e6	0.0953303
$706$	0	0
$707$	−110483.	−0.00831278
$708$	0	0
$709$	1.59830e7	1.19411	0.597054	−	0.802201i	$-0.296338\pi$
$709$	1.59830e7	1.19411	0.597054	+	0.802201i	$0.296338\pi$
$710$	0	0
$711$	2.05510e7	1.52461
$712$	0	0
$713$	3.64646e6	0.268626
$714$	0	0
$715$	−833482.	−0.0609721
$716$	0	0
$717$	4.05242e6	0.294386
$718$	0	0
$719$	−1.01853e7	−0.734771	−0.367386	−	0.930069i	$-0.619747\pi$
$719$	−1.01853e7	−0.734771	−0.367386	+	0.930069i	$0.619747\pi$
$720$	0	0
$721$	−461525.	−0.0330641
$722$	0	0
$723$	315564.	0.0224513
$724$	0	0
$725$	−4.12635e6	−0.291555
$726$	0	0
$727$	4.80166e6	0.336942	0.168471	−	0.985707i	$-0.446117\pi$
$727$	4.80166e6	0.336942	0.168471	+	0.985707i	$0.446117\pi$
$728$	0	0
$729$	−9.68323e6	−0.674841
$730$	0	0
$731$	−1.55201e7	−1.07424
$732$	0	0
$733$	4.57724e6	0.314662	0.157331	−	0.987546i	$-0.449711\pi$
$733$	4.57724e6	0.314662	0.157331	+	0.987546i	$0.449711\pi$
$734$	0	0
$735$	−1.55201e6	−0.105969
$736$	0	0
$737$	−1.22579e7	−0.831279
$738$	0	0
$739$	−2.54398e7	−1.71358	−0.856788	−	0.515669i	$-0.827543\pi$
$739$	−2.54398e7	−1.71358	−0.856788	+	0.515669i	$0.827543\pi$
$740$	0	0
$741$	−1.40290e6	−0.0938603
$742$	0	0
$743$	1.06776e7	0.709583	0.354791	−	0.934946i	$-0.384552\pi$
$743$	1.06776e7	0.709583	0.354791	+	0.934946i	$0.384552\pi$
$744$	0	0
$745$	−7.25477e6	−0.478887
$746$	0	0
$747$	−8.73987e6	−0.573065
$748$	0	0
$749$	1.99231e6	0.129763
$750$	0	0
$751$	−1.24759e7	−0.807184	−0.403592	−	0.914939i	$-0.632238\pi$
$751$	−1.24759e7	−0.807184	−0.403592	+	0.914939i	$0.632238\pi$
$752$	0	0
$753$	−1.58325e6	−0.101757
$754$	0	0
$755$	−3.24722e6	−0.207322
$756$	0	0
$757$	1.89987e6	0.120499	0.0602495	−	0.998183i	$-0.480810\pi$
$757$	1.89987e6	0.120499	0.0602495	+	0.998183i	$0.480810\pi$
$758$	0	0
$759$	−269777.	−0.0169981
$760$	0	0
$761$	1.16295e7	0.727949	0.363975	−	0.931409i	$-0.381420\pi$
$761$	1.16295e7	0.727949	0.363975	+	0.931409i	$0.381420\pi$
$762$	0	0
$763$	2.21971e6	0.138034
$764$	0	0
$765$	5.44153e6	0.336176
$766$	0	0
$767$	1.04290e6	0.0640107
$768$	0	0
$769$	−2.05493e7	−1.25309	−0.626544	−	0.779386i	$-0.715531\pi$
$769$	−2.05493e7	−1.25309	−0.626544	+	0.779386i	$0.715531\pi$
$770$	0	0
$771$	932748.	0.0565104
$772$	0	0
$773$	−5.50647e6	−0.331455	−0.165727	−	0.986172i	$-0.552997\pi$
$773$	−5.50647e6	−0.331455	−0.165727	+	0.986172i	$0.552997\pi$
$774$	0	0
$775$	6.19454e6	0.370472
$776$	0	0
$777$	−358312.	−0.0212916
$778$	0	0
$779$	−4.59799e7	−2.71472
$780$	0	0
$781$	−7.56655e6	−0.443885
$782$	0	0
$783$	1.15875e7	0.675438
$784$	0	0
$785$	−1.15515e6	−0.0669061
$786$	0	0
$787$	−1.66836e7	−0.960182	−0.480091	−	0.877219i	$-0.659396\pi$
$787$	−1.66836e7	−0.960182	−0.480091	+	0.877219i	$0.659396\pi$
$788$	0	0
$789$	5.51228e6	0.315238
$790$	0	0
$791$	1.44959e6	0.0823767
$792$	0	0
$793$	2.32309e6	0.131185
$794$	0	0
$795$	−3.25698e6	−0.182767
$796$	0	0
$797$	−1.53978e7	−0.858643	−0.429321	−	0.903152i	$-0.641247\pi$
$797$	−1.53978e7	−0.858643	−0.429321	+	0.903152i	$0.641247\pi$
$798$	0	0
$799$	−1.28578e7	−0.712526
$800$	0	0
$801$	1.79401e7	0.987969
$802$	0	0
$803$	−1.18300e7	−0.647434
$804$	0	0
$805$	94347.3	0.00513144
$806$	0	0
$807$	1.48224e6	0.0801188
$808$	0	0
$809$	−2.01482e7	−1.08234	−0.541171	−	0.840912i	$-0.682019\pi$
$809$	−2.01482e7	−1.08234	−0.541171	+	0.840912i	$0.682019\pi$
$810$	0	0
$811$	3.88568e6	0.207451	0.103725	−	0.994606i	$-0.466924\pi$
$811$	3.88568e6	0.207451	0.103725	+	0.994606i	$0.466924\pi$
$812$	0	0
$813$	−678337.	−0.0359931
$814$	0	0
$815$	−9.47690e6	−0.499773
$816$	0	0
$817$	−3.64961e7	−1.91290
$818$	0	0
$819$	−397299.	−0.0206970
$820$	0	0
$821$	1.69007e7	0.875079	0.437539	−	0.899199i	$-0.355850\pi$
$821$	1.69007e7	0.875079	0.437539	+	0.899199i	$0.355850\pi$
$822$	0	0
$823$	−166131.	−0.00854971	−0.00427485	−	0.999991i	$-0.501361\pi$
$823$	−166131.	−0.00854971	−0.00427485	+	0.999991i	$0.501361\pi$
$824$	0	0
$825$	−458292.	−0.0234427
$826$	0	0
$827$	1.64293e7	0.835324	0.417662	−	0.908602i	$-0.362850\pi$
$827$	1.64293e7	0.835324	0.417662	+	0.908602i	$0.362850\pi$
$828$	0	0
$829$	811068.	0.0409894	0.0204947	−	0.999790i	$-0.493476\pi$
$829$	811068.	0.0409894	0.0204947	+	0.999790i	$0.493476\pi$
$830$	0	0
$831$	−294397.	−0.0147888
$832$	0	0
$833$	1.58621e7	0.792040
$834$	0	0
$835$	1.42294e7	0.706269
$836$	0	0
$837$	−1.73953e7	−0.858261
$838$	0	0
$839$	−1.09462e7	−0.536855	−0.268427	−	0.963300i	$-0.586504\pi$
$839$	−1.09462e7	−0.536855	−0.268427	+	0.963300i	$0.586504\pi$
$840$	0	0
$841$	2.30773e7	1.12511
$842$	0	0
$843$	−5.62007e6	−0.272378
$844$	0	0
$845$	714025.	0.0344010
$846$	0	0
$847$	−1.25281e6	−0.0600033
$848$	0	0
$849$	−9.32152e6	−0.443831
$850$	0	0
$851$	−3.45752e6	−0.163659
$852$	0	0
$853$	1.24011e7	0.583564	0.291782	−	0.956485i	$-0.405752\pi$
$853$	1.24011e7	0.583564	0.291782	+	0.956485i	$0.405752\pi$
$854$	0	0
$855$	1.27959e7	0.598628
$856$	0	0
$857$	1.78406e7	0.829768	0.414884	−	0.909874i	$-0.363822\pi$
$857$	1.78406e7	0.829768	0.414884	+	0.909874i	$0.363822\pi$
$858$	0	0
$859$	−1.43038e7	−0.661408	−0.330704	−	0.943735i	$-0.607286\pi$
$859$	−1.43038e7	−0.661408	−0.330704	+	0.943735i	$0.607286\pi$
$860$	0	0
$861$	784982.	0.0360871
$862$	0	0
$863$	−2.13905e7	−0.977673	−0.488836	−	0.872376i	$-0.662578\pi$
$863$	−2.13905e7	−0.977673	−0.488836	+	0.872376i	$0.662578\pi$
$864$	0	0
$865$	−2.84920e6	−0.129474
$866$	0	0
$867$	−1.92498e6	−0.0869717
$868$	0	0
$869$	1.76896e7	0.794636
$870$	0	0
$871$	1.05011e7	0.469016
$872$	0	0
$873$	−1.84647e7	−0.819985
$874$	0	0
$875$	160275.	0.00707696
$876$	0	0
$877$	−3.28786e7	−1.44349	−0.721745	−	0.692159i	$-0.756660\pi$
$877$	−3.28786e7	−1.44349	−0.721745	+	0.692159i	$0.756660\pi$
$878$	0	0
$879$	6.51902e6	0.284584
$880$	0	0
$881$	−2.63729e7	−1.14477	−0.572384	−	0.819986i	$-0.693981\pi$
$881$	−2.63729e7	−1.14477	−0.572384	+	0.819986i	$0.693981\pi$
$882$	0	0
$883$	1.89435e7	0.817634	0.408817	−	0.912616i	$-0.365941\pi$
$883$	1.89435e7	0.817634	0.408817	+	0.912616i	$0.365941\pi$
$884$	0	0
$885$	573438.	0.0246110
$886$	0	0
$887$	−1.00174e7	−0.427509	−0.213755	−	0.976887i	$-0.568569\pi$
$887$	−1.00174e7	−0.427509	−0.213755	+	0.976887i	$0.568569\pi$
$888$	0	0
$889$	1.93135e6	0.0819610
$890$	0	0
$891$	−9.69955e6	−0.409314
$892$	0	0
$893$	−3.02356e7	−1.26879
$894$	0	0
$895$	−8.00735e6	−0.334142
$896$	0	0
$897$	231112.	0.00959049
$898$	0	0
$899$	−6.54357e7	−2.70032
$900$	0	0
$901$	3.32874e7	1.36605
$902$	0	0
$903$	623072.	0.0254284
$904$	0	0
$905$	1.12995e7	0.458602
$906$	0	0
$907$	2.20644e7	0.890581	0.445290	−	0.895386i	$-0.353100\pi$
$907$	2.20644e7	0.890581	0.445290	+	0.895386i	$0.353100\pi$
$908$	0	0
$909$	2.46850e6	0.0990884
$910$	0	0
$911$	2.02691e7	0.809167	0.404583	−	0.914501i	$-0.367417\pi$
$911$	2.02691e7	0.809167	0.404583	+	0.914501i	$0.367417\pi$
$912$	0	0
$913$	−7.52299e6	−0.298685
$914$	0	0
$915$	1.27736e6	0.0504381
$916$	0	0
$917$	−2.70715e6	−0.106314
$918$	0	0
$919$	1.79004e7	0.699154	0.349577	−	0.936908i	$-0.386325\pi$
$919$	1.79004e7	0.699154	0.349577	+	0.936908i	$0.386325\pi$
$920$	0	0
$921$	5.77242e6	0.224238
$922$	0	0
$923$	6.48209e6	0.250444
$924$	0	0
$925$	−5.87356e6	−0.225708
$926$	0	0
$927$	1.03118e7	0.394124
$928$	0	0
$929$	9.25686e6	0.351904	0.175952	−	0.984399i	$-0.443700\pi$
$929$	9.25686e6	0.351904	0.175952	+	0.984399i	$0.443700\pi$
$930$	0	0
$931$	3.73002e7	1.41038
$932$	0	0
$933$	699260.	0.0262987
$934$	0	0
$935$	4.68388e6	0.175217
$936$	0	0
$937$	−8.47287e6	−0.315269	−0.157635	−	0.987498i	$-0.550387\pi$
$937$	−8.47287e6	−0.315269	−0.157635	+	0.987498i	$0.550387\pi$
$938$	0	0
$939$	3.14631e6	0.116449
$940$	0	0
$941$	−1.22548e7	−0.451161	−0.225580	−	0.974225i	$-0.572428\pi$
$941$	−1.22548e7	−0.451161	−0.225580	+	0.974225i	$0.572428\pi$
$942$	0	0
$943$	7.57465e6	0.277385
$944$	0	0
$945$	−450081.	−0.0163950
$946$	0	0
$947$	−2.12327e7	−0.769363	−0.384681	−	0.923049i	$-0.625689\pi$
$947$	−2.12327e7	−0.769363	−0.384681	+	0.923049i	$0.625689\pi$
$948$	0	0
$949$	1.01345e7	0.365288
$950$	0	0
$951$	−1.12749e6	−0.0404259
$952$	0	0
$953$	−2.31217e7	−0.824684	−0.412342	−	0.911029i	$-0.635289\pi$
$953$	−2.31217e7	−0.824684	−0.412342	+	0.911029i	$0.635289\pi$
$954$	0	0
$955$	−4.57295e6	−0.162251
$956$	0	0
$957$	4.84114e6	0.170871
$958$	0	0
$959$	−3.46110e6	−0.121525
$960$	0	0
$961$	6.96040e7	2.43123
$962$	0	0
$963$	−4.45137e7	−1.54678
$964$	0	0
$965$	1.27174e7	0.439622
$966$	0	0
$967$	−1.92947e7	−0.663548	−0.331774	−	0.943359i	$-0.607647\pi$
$967$	−1.92947e7	−0.663548	−0.331774	+	0.943359i	$0.607647\pi$
$968$	0	0
$969$	7.88383e6	0.269729
$970$	0	0
$971$	−1.63863e7	−0.557743	−0.278871	−	0.960328i	$-0.589960\pi$
$971$	−1.63863e7	−0.557743	−0.278871	+	0.960328i	$0.589960\pi$
$972$	0	0
$973$	884241.	0.0299425
$974$	0	0
$975$	392608.	0.0132266
$976$	0	0
$977$	3.83317e6	0.128476	0.0642380	−	0.997935i	$-0.479538\pi$
$977$	3.83317e6	0.128476	0.0642380	+	0.997935i	$0.479538\pi$
$978$	0	0
$979$	1.54422e7	0.514936
$980$	0	0
$981$	−4.95945e7	−1.64536
$982$	0	0
$983$	−2.80937e7	−0.927310	−0.463655	−	0.886016i	$-0.653462\pi$
$983$	−2.80937e7	−0.927310	−0.463655	+	0.886016i	$0.653462\pi$
$984$	0	0
$985$	2.79164e6	0.0916789
$986$	0	0
$987$	516191.	0.0168662
$988$	0	0
$989$	6.01231e6	0.195457
$990$	0	0
$991$	4.83597e7	1.56423	0.782113	−	0.623137i	$-0.214142\pi$
$991$	4.83597e7	1.56423	0.782113	+	0.623137i	$0.214142\pi$
$992$	0	0
$993$	1.24974e7	0.402206
$994$	0	0
$995$	−1.30305e7	−0.417255
$996$	0	0
$997$	−4.50212e7	−1.43443	−0.717215	−	0.696852i	$-0.754584\pi$
$997$	−4.50212e7	−1.43443	−0.717215	+	0.696852i	$0.754584\pi$
$998$	0	0
$999$	1.64940e7	0.522892

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1040.6.a.l.1.2		3
4.3	odd	2		130.6.a.f.1.2	✓	3
20.3	even	4		650.6.b.j.599.5		6
20.7	even	4		650.6.b.j.599.2		6
20.19	odd	2		650.6.a.j.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.6.a.f.1.2	✓	3	4.3	odd	2
650.6.a.j.1.2		3	20.19	odd	2
650.6.b.j.599.2		6	20.7	even	4
650.6.b.j.599.5		6	20.3	even	4
1040.6.a.l.1.2		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1040.a (trivial)

\(f(q)\)	\(=\)	\(q+3.71700 q^{3} +25.0000 q^{5} +10.2576 q^{7} -229.184 q^{9} -197.274 q^{11} +169.000 q^{13} +92.9250 q^{15} -949.722 q^{17} -2233.30 q^{19} +38.1276 q^{21} +367.911 q^{23} +625.000 q^{25} -1755.11 q^{27} -6602.16 q^{29} +9911.27 q^{31} -733.267 q^{33} +256.441 q^{35} -9397.70 q^{37} +628.173 q^{39} +20588.3 q^{41} +16341.8 q^{43} -5729.60 q^{45} +13538.5 q^{47} -16701.8 q^{49} -3530.12 q^{51} -35049.6 q^{53} -4931.84 q^{55} -8301.20 q^{57} +6170.98 q^{59} +13746.1 q^{61} -2350.88 q^{63} +4225.00 q^{65} +62136.4 q^{67} +1367.52 q^{69} +38355.6 q^{71} +59967.3 q^{73} +2323.13 q^{75} -2023.56 q^{77} -89670.2 q^{79} +49167.9 q^{81} +38134.8 q^{83} -23743.1 q^{85} -24540.2 q^{87} -78278.2 q^{89} +1733.54 q^{91} +36840.2 q^{93} -55832.6 q^{95} +80567.1 q^{97} +45212.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 22 q^{3} + 75 q^{5} + 234 q^{7} + 155 q^{9} - 48 q^{11} + 507 q^{13} + 550 q^{15} + 1506 q^{17} + 360 q^{19} + 6904 q^{21} + 2370 q^{23} + 1875 q^{25} + 10168 q^{27} - 3078 q^{29} + 5388 q^{31} - 22572 q^{33}+ \cdots - 350340 q^{99}+O(q^{100}) \) 3 * q + 22 * q^3 + 75 * q^5 + 234 * q^7 + 155 * q^9 - 48 * q^11 + 507 * q^13 + 550 * q^15 + 1506 * q^17 + 360 * q^19 + 6904 * q^21 + 2370 * q^23 + 1875 * q^25 + 10168 * q^27 - 3078 * q^29 + 5388 * q^31 - 22572 * q^33 + 5850 * q^35 - 25362 * q^37 + 3718 * q^39 - 15906 * q^41 + 39306 * q^43 + 3875 * q^45 + 17778 * q^47 + 7767 * q^49 + 33972 * q^51 + 9246 * q^53 - 1200 * q^55 - 2996 * q^57 + 77760 * q^59 + 17982 * q^61 + 128762 * q^63 + 12675 * q^65 + 2922 * q^67 - 31860 * q^69 + 4944 * q^71 + 43278 * q^73 + 13750 * q^75 - 144420 * q^77 - 42120 * q^79 + 146519 * q^81 - 58098 * q^83 + 37650 * q^85 - 2016 * q^87 + 19614 * q^89 + 39546 * q^91 + 162796 * q^93 + 9000 * q^95 - 87078 * q^97 - 350340 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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