LMFDB - Embedded newform 369.6.a.e.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [369,6,Mod(1,369)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(369, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("369.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 369 = 3^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	369.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:	$59.1816295110$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 3 x^{9} - 259 x^{8} + 639 x^{7} + 22422 x^{6} - 38356 x^{5} - 735592 x^{4} + 422608 x^{3} + \cdots - 24923264 $ x^10 - 3x^9 - 259x^8 + 639x^7 + 22422x^6 - 38356x^5 - 735592x^4 + 422608x^3 + 8253728x^2 - 28352*x - 24923264
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 41)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.00833$ of defining polynomial
Character	$\chi$	$=$	369.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00833 q^{2} -27.9666 q^{4} -74.6642 q^{5} +126.297 q^{7} +120.433 q^{8} +O(q^{10})$	\(q-2.00833 q^{2} -27.9666 q^{4} -74.6642 q^{5} +126.297 q^{7} +120.433 q^{8} +149.950 q^{10} -732.971 q^{11} -948.864 q^{13} -253.647 q^{14} +653.063 q^{16} -447.346 q^{17} +506.762 q^{19} +2088.11 q^{20} +1472.05 q^{22} -3649.99 q^{23} +2449.75 q^{25} +1905.63 q^{26} -3532.11 q^{28} -6037.46 q^{29} -528.129 q^{31} -5165.41 q^{32} +898.419 q^{34} -9429.91 q^{35} -10277.5 q^{37} -1017.75 q^{38} -8992.02 q^{40} -1681.00 q^{41} -3661.14 q^{43} +20498.7 q^{44} +7330.38 q^{46} -16793.7 q^{47} -855.947 q^{49} -4919.90 q^{50} +26536.5 q^{52} +6959.90 q^{53} +54726.7 q^{55} +15210.3 q^{56} +12125.2 q^{58} -5214.02 q^{59} +18393.2 q^{61} +1060.66 q^{62} -10524.2 q^{64} +70846.2 q^{65} +21799.0 q^{67} +12510.8 q^{68} +18938.4 q^{70} +45250.6 q^{71} -31348.1 q^{73} +20640.5 q^{74} -14172.4 q^{76} -92572.3 q^{77} +71611.3 q^{79} -48760.5 q^{80} +3376.00 q^{82} +8237.92 q^{83} +33400.8 q^{85} +7352.77 q^{86} -88273.6 q^{88} -444.314 q^{89} -119839. q^{91} +102078. q^{92} +33727.3 q^{94} -37837.0 q^{95} +45300.6 q^{97} +1719.02 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 3 q^{2} + 207 q^{4} - 32 q^{5} + 342 q^{7} - 249 q^{8}+O(q^{10}) $ 10 * q - 3 * q^2 + 207 * q^4 - 32 * q^5 + 342 * q^7 - 249 * q^8	$ 10 q - 3 q^{2} + 207 q^{4} - 32 q^{5} + 342 q^{7} - 249 q^{8} + 102 q^{10} - 846 q^{11} + 1504 q^{13} - 3468 q^{14} + 5859 q^{16} - 560 q^{17} + 4240 q^{19} + 6182 q^{20} - 2628 q^{22} + 1508 q^{23} + 11734 q^{25} + 22014 q^{26} - 8662 q^{28} + 124 q^{29} + 10384 q^{31} + 6619 q^{32} + 802 q^{34} + 17890 q^{35} + 5524 q^{37} + 46098 q^{38} - 61738 q^{40} - 16810 q^{41} + 24160 q^{43} + 21594 q^{44} + 42404 q^{46} - 58984 q^{47} + 70326 q^{49} - 6817 q^{50} + 64374 q^{52} - 23456 q^{53} + 96426 q^{55} - 80184 q^{56} - 13378 q^{58} - 52428 q^{59} + 113540 q^{61} + 113008 q^{62} + 37363 q^{64} + 22340 q^{65} + 85506 q^{67} + 71406 q^{68} - 71946 q^{70} - 75236 q^{71} - 85148 q^{73} + 23462 q^{74} + 113376 q^{76} + 172896 q^{77} + 178200 q^{79} + 401850 q^{80} + 5043 q^{82} + 125412 q^{83} - 245912 q^{85} + 18848 q^{86} - 135952 q^{88} + 62696 q^{89} + 30056 q^{91} + 236372 q^{92} + 419014 q^{94} - 28002 q^{95} + 154548 q^{97} - 288367 q^{98}+O(q^{100}) $ 10 * q - 3 * q^2 + 207 * q^4 - 32 * q^5 + 342 * q^7 - 249 * q^8 + 102 * q^10 - 846 * q^11 + 1504 * q^13 - 3468 * q^14 + 5859 * q^16 - 560 * q^17 + 4240 * q^19 + 6182 * q^20 - 2628 * q^22 + 1508 * q^23 + 11734 * q^25 + 22014 * q^26 - 8662 * q^28 + 124 * q^29 + 10384 * q^31 + 6619 * q^32 + 802 * q^34 + 17890 * q^35 + 5524 * q^37 + 46098 * q^38 - 61738 * q^40 - 16810 * q^41 + 24160 * q^43 + 21594 * q^44 + 42404 * q^46 - 58984 * q^47 + 70326 * q^49 - 6817 * q^50 + 64374 * q^52 - 23456 * q^53 + 96426 * q^55 - 80184 * q^56 - 13378 * q^58 - 52428 * q^59 + 113540 * q^61 + 113008 * q^62 + 37363 * q^64 + 22340 * q^65 + 85506 * q^67 + 71406 * q^68 - 71946 * q^70 - 75236 * q^71 - 85148 * q^73 + 23462 * q^74 + 113376 * q^76 + 172896 * q^77 + 178200 * q^79 + 401850 * q^80 + 5043 * q^82 + 125412 * q^83 - 245912 * q^85 + 18848 * q^86 - 135952 * q^88 + 62696 * q^89 + 30056 * q^91 + 236372 * q^92 + 419014 * q^94 - 28002 * q^95 + 154548 * q^97 - 288367 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.00833	−0.355026	−0.177513	−	0.984118i	$-0.556805\pi$
$2$	−2.00833	−0.355026	−0.177513	+	0.984118i	$0.556805\pi$
$3$	0	0
$3$	0	0
$4$	−27.9666	−0.873957
$5$	−74.6642	−1.33563	−0.667817	−	0.744325i	$-0.732771\pi$
$5$	−74.6642	−1.33563	−0.667817	+	0.744325i	$0.732771\pi$
$6$	0	0
$7$	126.297	0.974203	0.487102	−	0.873345i	$-0.338054\pi$
$7$	126.297	0.974203	0.487102	+	0.873345i	$0.338054\pi$
$8$	120.433	0.665303
$9$	0	0
$10$	149.950	0.474185
$11$	−732.971	−1.82644	−0.913219	−	0.407469i	$-0.866411\pi$
$11$	−732.971	−1.82644	−0.913219	+	0.407469i	$0.866411\pi$
$12$	0	0
$13$	−948.864	−1.55720	−0.778602	−	0.627518i	$-0.784071\pi$
$13$	−948.864	−1.55720	−0.778602	+	0.627518i	$0.784071\pi$
$14$	−253.647	−0.345867
$15$	0	0
$16$	653.063	0.637757
$17$	−447.346	−0.375424	−0.187712	−	0.982224i	$-0.560107\pi$
$17$	−447.346	−0.375424	−0.187712	+	0.982224i	$0.560107\pi$
$18$	0	0
$19$	506.762	0.322048	0.161024	−	0.986951i	$-0.448520\pi$
$19$	506.762	0.322048	0.161024	+	0.986951i	$0.448520\pi$
$20$	2088.11	1.16729
$21$	0	0
$22$	1472.05	0.648433
$23$	−3649.99	−1.43871	−0.719353	−	0.694645i	$-0.755561\pi$
$23$	−3649.99	−1.43871	−0.719353	+	0.694645i	$0.755561\pi$
$24$	0	0
$25$	2449.75	0.783920
$26$	1905.63	0.552848
$27$	0	0
$28$	−3532.11	−0.851411
$29$	−6037.46	−1.33309	−0.666545	−	0.745465i	$-0.732227\pi$
$29$	−6037.46	−1.33309	−0.666545	+	0.745465i	$0.732227\pi$
$30$	0	0
$31$	−528.129	−0.0987042	−0.0493521	−	0.998781i	$-0.515716\pi$
$31$	−528.129	−0.0987042	−0.0493521	+	0.998781i	$0.515716\pi$
$32$	−5165.41	−0.891723
$33$	0	0
$34$	898.419	0.133285
$35$	−9429.91	−1.30118
$36$	0	0
$37$	−10277.5	−1.23419	−0.617095	−	0.786889i	$-0.711691\pi$
$37$	−10277.5	−1.23419	−0.617095	+	0.786889i	$0.711691\pi$
$38$	−1017.75	−0.114335
$39$	0	0
$40$	−8992.02	−0.888602
$41$	−1681.00	−0.156174
$41$	−1681.00	−0.156174
$42$	0	0
$43$	−3661.14	−0.301957	−0.150978	−	0.988537i	$-0.548242\pi$
$43$	−3661.14	−0.301957	−0.150978	+	0.988537i	$0.548242\pi$
$44$	20498.7	1.59623
$45$	0	0
$46$	7330.38	0.510777
$47$	−16793.7	−1.10892	−0.554462	−	0.832209i	$-0.687076\pi$
$47$	−16793.7	−1.10892	−0.554462	+	0.832209i	$0.687076\pi$
$48$	0	0
$49$	−855.947	−0.0509280
$50$	−4919.90	−0.278312
$51$	0	0
$52$	26536.5	1.36093
$53$	6959.90	0.340340	0.170170	−	0.985415i	$-0.445568\pi$
$53$	6959.90	0.340340	0.170170	+	0.985415i	$0.445568\pi$
$54$	0	0
$55$	54726.7	2.43945
$56$	15210.3	0.648140
$57$	0	0
$58$	12125.2	0.473281
$59$	−5214.02	−0.195004	−0.0975018	−	0.995235i	$-0.531085\pi$
$59$	−5214.02	−0.195004	−0.0975018	+	0.995235i	$0.531085\pi$
$60$	0	0
$61$	18393.2	0.632898	0.316449	−	0.948610i	$-0.397509\pi$
$61$	18393.2	0.632898	0.316449	+	0.948610i	$0.397509\pi$
$62$	1060.66	0.0350425
$63$	0	0
$64$	−10524.2	−0.321172
$65$	70846.2	2.07986
$66$	0	0
$67$	21799.0	0.593266	0.296633	−	0.954992i	$-0.404136\pi$
$67$	21799.0	0.593266	0.296633	+	0.954992i	$0.404136\pi$
$68$	12510.8	0.328104
$69$	0	0
$70$	18938.4	0.461952
$71$	45250.6	1.06532	0.532658	−	0.846330i	$-0.321193\pi$
$71$	45250.6	1.06532	0.532658	+	0.846330i	$0.321193\pi$
$72$	0	0
$73$	−31348.1	−0.688501	−0.344250	−	0.938878i	$-0.611867\pi$
$73$	−31348.1	−0.688501	−0.344250	+	0.938878i	$0.611867\pi$
$74$	20640.5	0.438169
$75$	0	0
$76$	−14172.4	−0.281456
$77$	−92572.3	−1.77932
$78$	0	0
$79$	71611.3	1.29096	0.645482	−	0.763776i	$-0.276657\pi$
$79$	71611.3	1.29096	0.645482	+	0.763776i	$0.276657\pi$
$80$	−48760.5	−0.851810
$81$	0	0
$82$	3376.00	0.0554457
$83$	8237.92	0.131257	0.0656285	−	0.997844i	$-0.479095\pi$
$83$	8237.92	0.131257	0.0656285	+	0.997844i	$0.479095\pi$
$84$	0	0
$85$	33400.8	0.501429
$86$	7352.77	0.107202
$87$	0	0
$88$	−88273.6	−1.21513
$89$	−444.314	−0.00594586	−0.00297293	−	0.999996i	$-0.500946\pi$
$89$	−444.314	−0.00594586	−0.00297293	+	0.999996i	$0.500946\pi$
$90$	0	0
$91$	−119839.	−1.51703
$92$	102078.	1.25737
$93$	0	0
$94$	33727.3	0.393696
$95$	−37837.0	−0.430138
$96$	0	0
$97$	45300.6	0.488849	0.244424	−	0.969668i	$-0.421401\pi$
$97$	45300.6	0.488849	0.244424	+	0.969668i	$0.421401\pi$
$98$	1719.02	0.0180808
$99$	0	0
$100$	−68511.2	−0.685112
$101$	145004.	1.41441	0.707207	−	0.707007i	$-0.249955\pi$
$101$	145004.	1.41441	0.707207	+	0.707007i	$0.249955\pi$
$102$	0	0
$103$	−42152.0	−0.391494	−0.195747	−	0.980654i	$-0.562713\pi$
$103$	−42152.0	−0.391494	−0.195747	+	0.980654i	$0.562713\pi$
$104$	−114274.	−1.03601
$105$	0	0
$106$	−13977.8	−0.120830
$107$	−2295.91	−0.0193863	−0.00969316	−	0.999953i	$-0.503085\pi$
$107$	−2295.91	−0.0193863	−0.00969316	+	0.999953i	$0.503085\pi$
$108$	0	0
$109$	18226.1	0.146936	0.0734678	−	0.997298i	$-0.476593\pi$
$109$	18226.1	0.146936	0.0734678	+	0.997298i	$0.476593\pi$
$110$	−109909.	−0.866069
$111$	0	0
$112$	82480.2	0.621305
$113$	−42216.8	−0.311021	−0.155510	−	0.987834i	$-0.549702\pi$
$113$	−42216.8	−0.311021	−0.155510	+	0.987834i	$0.549702\pi$
$114$	0	0
$115$	272524.	1.92158
$116$	168847.	1.16506
$117$	0	0
$118$	10471.5	0.0692313
$119$	−56498.7	−0.365739
$120$	0	0
$121$	376195.	2.33587
$122$	−36939.7	−0.224695
$123$	0	0
$124$	14770.0	0.0862632
$125$	50417.1	0.288604
$126$	0	0
$127$	−247425.	−1.36124	−0.680618	−	0.732638i	$-0.738289\pi$
$127$	−247425.	−1.36124	−0.680618	+	0.732638i	$0.738289\pi$
$128$	186429.	1.00575
$129$	0	0
$130$	−142283.	−0.738403
$131$	109910.	0.559574	0.279787	−	0.960062i	$-0.409736\pi$
$131$	109910.	0.559574	0.279787	+	0.960062i	$0.409736\pi$
$132$	0	0
$133$	64002.8	0.313740
$134$	−43779.5	−0.210625
$135$	0	0
$136$	−53875.1	−0.249770
$137$	−105426.	−0.479895	−0.239948	−	0.970786i	$-0.577130\pi$
$137$	−105426.	−0.479895	−0.239948	+	0.970786i	$0.577130\pi$
$138$	0	0
$139$	−244289.	−1.07243	−0.536213	−	0.844082i	$-0.680146\pi$
$139$	−244289.	−1.07243	−0.536213	+	0.844082i	$0.680146\pi$
$140$	263723.	1.13717
$141$	0	0
$142$	−90878.1	−0.378215
$143$	695490.	2.84414
$144$	0	0
$145$	450782.	1.78052
$146$	62957.4	0.244436
$147$	0	0
$148$	287426.	1.07863
$149$	−226376.	−0.835344	−0.417672	−	0.908598i	$-0.637154\pi$
$149$	−226376.	−0.835344	−0.417672	+	0.908598i	$0.637154\pi$
$150$	0	0
$151$	388030.	1.38492	0.692458	−	0.721458i	$-0.256528\pi$
$151$	388030.	1.38492	0.692458	+	0.721458i	$0.256528\pi$
$152$	61030.8	0.214259
$153$	0	0
$154$	185916.	0.631705
$155$	39432.3	0.131833
$156$	0	0
$157$	−305827.	−0.990209	−0.495104	−	0.868833i	$-0.664870\pi$
$157$	−305827.	−0.990209	−0.495104	+	0.868833i	$0.664870\pi$
$158$	−143819.	−0.458325
$159$	0	0
$160$	385672.	1.19102
$161$	−460984.	−1.40159
$162$	0	0
$163$	−436627.	−1.28719	−0.643593	−	0.765368i	$-0.722557\pi$
$163$	−436627.	−1.28719	−0.643593	+	0.765368i	$0.722557\pi$
$164$	47011.9	0.136489
$165$	0	0
$166$	−16544.5	−0.0465996
$167$	29287.0	0.0812613	0.0406307	−	0.999174i	$-0.487063\pi$
$167$	29287.0	0.0812613	0.0406307	+	0.999174i	$0.487063\pi$
$168$	0	0
$169$	529051.	1.42489
$170$	−67079.7	−0.178020
$171$	0	0
$172$	102390.	0.263897
$173$	−171998.	−0.436925	−0.218463	−	0.975845i	$-0.570104\pi$
$173$	−171998.	−0.436925	−0.218463	+	0.975845i	$0.570104\pi$
$174$	0	0
$175$	309397.	0.763697
$176$	−478676.	−1.16482
$177$	0	0
$178$	892.329	0.00211094
$179$	−581307.	−1.35604	−0.678021	−	0.735043i	$-0.737162\pi$
$179$	−581307.	−1.35604	−0.678021	+	0.735043i	$0.737162\pi$
$180$	0	0
$181$	645486.	1.46450	0.732252	−	0.681034i	$-0.238469\pi$
$181$	645486.	1.46450	0.732252	+	0.681034i	$0.238469\pi$
$182$	240677.	0.538586
$183$	0	0
$184$	−439578.	−0.957175
$185$	767360.	1.64843
$186$	0	0
$187$	327892.	0.685688
$188$	469663.	0.969151
$189$	0	0
$190$	75989.2	0.152710
$191$	−525364.	−1.04202	−0.521011	−	0.853550i	$-0.674445\pi$
$191$	−525364.	−1.04202	−0.521011	+	0.853550i	$0.674445\pi$
$192$	0	0
$193$	−165595.	−0.320004	−0.160002	−	0.987117i	$-0.551150\pi$
$193$	−165595.	−0.320004	−0.160002	+	0.987117i	$0.551150\pi$
$194$	−90978.5	−0.173554
$195$	0	0
$196$	23937.9	0.0445089
$197$	129599.	0.237923	0.118961	−	0.992899i	$-0.462044\pi$
$197$	129599.	0.237923	0.118961	+	0.992899i	$0.462044\pi$
$198$	0	0
$199$	−208362.	−0.372981	−0.186490	−	0.982457i	$-0.559711\pi$
$199$	−208362.	−0.372981	−0.186490	+	0.982457i	$0.559711\pi$
$200$	295030.	0.521544
$201$	0	0
$202$	−291216.	−0.502153
$203$	−762516.	−1.29870
$204$	0	0
$205$	125511.	0.208591
$206$	84655.1	0.138990
$207$	0	0
$208$	−619668.	−0.993118
$209$	−371442.	−0.588200
$210$	0	0
$211$	−982645.	−1.51946	−0.759732	−	0.650237i	$-0.774670\pi$
$211$	−982645.	−1.51946	−0.759732	+	0.650237i	$0.774670\pi$
$212$	−194645.	−0.297443
$213$	0	0
$214$	4610.94	0.00688264
$215$	273356.	0.403304
$216$	0	0
$217$	−66701.3	−0.0961579
$218$	−36604.0	−0.0521659
$219$	0	0
$220$	−1.53052e6	−2.13198
$221$	424471.	0.584611
$222$	0	0
$223$	178185.	0.239943	0.119971	−	0.992777i	$-0.461720\pi$
$223$	178185.	0.239943	0.119971	+	0.992777i	$0.461720\pi$
$224$	−652379.	−0.868720
$225$	0	0
$226$	84785.3	0.110420
$227$	97981.4	0.126206	0.0631029	−	0.998007i	$-0.479900\pi$
$227$	97981.4	0.126206	0.0631029	+	0.998007i	$0.479900\pi$
$228$	0	0
$229$	−1.12371e6	−1.41601	−0.708004	−	0.706208i	$-0.750404\pi$
$229$	−1.12371e6	−1.41601	−0.708004	+	0.706208i	$0.750404\pi$
$230$	−547317.	−0.682212
$231$	0	0
$232$	−727107.	−0.886908
$233$	948911.	1.14508	0.572540	−	0.819877i	$-0.305958\pi$
$233$	948911.	1.14508	0.572540	+	0.819877i	$0.305958\pi$
$234$	0	0
$235$	1.25389e6	1.48112
$236$	145818.	0.170425
$237$	0	0
$238$	113468.	0.129847
$239$	−143412.	−0.162402	−0.0812010	−	0.996698i	$-0.525876\pi$
$239$	−143412.	−0.162402	−0.0812010	+	0.996698i	$0.525876\pi$
$240$	0	0
$241$	−758845.	−0.841609	−0.420805	−	0.907151i	$-0.638252\pi$
$241$	−758845.	−0.841609	−0.420805	+	0.907151i	$0.638252\pi$
$242$	−755523.	−0.829296
$243$	0	0
$244$	−514397.	−0.553125
$245$	63908.7	0.0680212
$246$	0	0
$247$	−480849.	−0.501495
$248$	−63604.0	−0.0656682
$249$	0	0
$250$	−101254.	−0.102462
$251$	384559.	0.385282	0.192641	−	0.981269i	$-0.438295\pi$
$251$	384559.	0.385282	0.192641	+	0.981269i	$0.438295\pi$
$252$	0	0
$253$	2.67533e6	2.62771
$254$	496910.	0.483274
$255$	0	0
$256$	−37637.8	−0.0358942
$257$	1.13441e6	1.07136	0.535682	−	0.844420i	$-0.320055\pi$
$257$	1.13441e6	1.07136	0.535682	+	0.844420i	$0.320055\pi$
$258$	0	0
$259$	−1.29802e6	−1.20235
$260$	−1.98133e6	−1.81770
$261$	0	0
$262$	−220735.	−0.198663
$263$	951705.	0.848424	0.424212	−	0.905563i	$-0.360551\pi$
$263$	951705.	0.848424	0.424212	+	0.905563i	$0.360551\pi$
$264$	0	0
$265$	−519656.	−0.454570
$266$	−128539.	−0.111386
$267$	0	0
$268$	−609644.	−0.518488
$269$	2.13808e6	1.80154	0.900768	−	0.434300i	$-0.143004\pi$
$269$	2.13808e6	1.80154	0.900768	+	0.434300i	$0.143004\pi$
$270$	0	0
$271$	1.43926e6	1.19046	0.595230	−	0.803555i	$-0.297061\pi$
$271$	1.43926e6	1.19046	0.595230	+	0.803555i	$0.297061\pi$
$272$	−292145.	−0.239429
$273$	0	0
$274$	211730.	0.170375
$275$	−1.79559e6	−1.43178
$276$	0	0
$277$	−1.14694e6	−0.898131	−0.449066	−	0.893499i	$-0.648243\pi$
$277$	−1.14694e6	−0.898131	−0.449066	+	0.893499i	$0.648243\pi$
$278$	490614.	0.380739
$279$	0	0
$280$	−1.13567e6	−0.865679
$281$	−1.94822e6	−1.47188	−0.735940	−	0.677046i	$-0.763260\pi$
$281$	−1.94822e6	−1.47188	−0.735940	+	0.677046i	$0.763260\pi$
$282$	0	0
$283$	−1.41049e6	−1.04690	−0.523450	−	0.852056i	$-0.675355\pi$
$283$	−1.41049e6	−1.04690	−0.523450	+	0.852056i	$0.675355\pi$
$284$	−1.26551e6	−0.931040
$285$	0	0
$286$	−1.39677e6	−1.00974
$287$	−212306.	−0.152145
$288$	0	0
$289$	−1.21974e6	−0.859057
$290$	−905319.	−0.632131
$291$	0	0
$292$	876701.	0.601720
$293$	976872.	0.664766	0.332383	−	0.943144i	$-0.392147\pi$
$293$	976872.	0.664766	0.332383	+	0.943144i	$0.392147\pi$
$294$	0	0
$295$	389301.	0.260453
$296$	−1.23774e6	−0.821110
$297$	0	0
$298$	454638.	0.296569
$299$	3.46334e6	2.24036
$300$	0	0
$301$	−462392.	−0.294167
$302$	−779293.	−0.491681
$303$	0	0
$304$	330948.	0.205388
$305$	−1.37332e6	−0.845321
$306$	0	0
$307$	−1.96092e6	−1.18745	−0.593723	−	0.804670i	$-0.702342\pi$
$307$	−1.96092e6	−1.18745	−0.593723	+	0.804670i	$0.702342\pi$
$308$	2.58893e6	1.55505
$309$	0	0
$310$	−79193.1	−0.0468040
$311$	−2.66867e6	−1.56457	−0.782284	−	0.622922i	$-0.785945\pi$
$311$	−2.66867e6	−1.56457	−0.782284	+	0.622922i	$0.785945\pi$
$312$	0	0
$313$	2.29241e6	1.32261	0.661303	−	0.750118i	$-0.270004\pi$
$313$	2.29241e6	1.32261	0.661303	+	0.750118i	$0.270004\pi$
$314$	614201.	0.351550
$315$	0	0
$316$	−2.00273e6	−1.12825
$317$	1.71583e6	0.959015	0.479508	−	0.877538i	$-0.340815\pi$
$317$	1.71583e6	0.959015	0.479508	+	0.877538i	$0.340815\pi$
$318$	0	0
$319$	4.42528e6	2.43480
$320$	785779.	0.428969
$321$	0	0
$322$	925808.	0.497601
$323$	−226698.	−0.120904
$324$	0	0
$325$	−2.32448e6	−1.22072
$326$	876891.	0.456984
$327$	0	0
$328$	−202447.	−0.103903
$329$	−2.12100e6	−1.08032
$330$	0	0
$331$	2.03801e6	1.02244	0.511218	−	0.859451i	$-0.329194\pi$
$331$	2.03801e6	1.02244	0.511218	+	0.859451i	$0.329194\pi$
$332$	−230387.	−0.114713
$333$	0	0
$334$	−58817.9	−0.0288499
$335$	−1.62760e6	−0.792386
$336$	0	0
$337$	1.56295e6	0.749670	0.374835	−	0.927092i	$-0.377699\pi$
$337$	1.56295e6	0.749670	0.374835	+	0.927092i	$0.377699\pi$
$338$	−1.06251e6	−0.505872
$339$	0	0
$340$	−934106.	−0.438227
$341$	387103.	0.180277
$342$	0	0
$343$	−2.23079e6	−1.02382
$344$	−440921.	−0.200893
$345$	0	0
$346$	345428.	0.155120
$347$	−1.46841e6	−0.654671	−0.327336	−	0.944908i	$-0.606151\pi$
$347$	−1.46841e6	−0.654671	−0.327336	+	0.944908i	$0.606151\pi$
$348$	0	0
$349$	−182768.	−0.0803224	−0.0401612	−	0.999193i	$-0.512787\pi$
$349$	−182768.	−0.0803224	−0.0401612	+	0.999193i	$0.512787\pi$
$350$	−621371.	−0.271132
$351$	0	0
$352$	3.78610e6	1.62868
$353$	−1.89626e6	−0.809956	−0.404978	−	0.914326i	$-0.632721\pi$
$353$	−1.89626e6	−0.809956	−0.404978	+	0.914326i	$0.632721\pi$
$354$	0	0
$355$	−3.37860e6	−1.42287
$356$	12426.0	0.00519643
$357$	0	0
$358$	1.16746e6	0.481430
$359$	−2.06782e6	−0.846793	−0.423397	−	0.905944i	$-0.639162\pi$
$359$	−2.06782e6	−0.846793	−0.423397	+	0.905944i	$0.639162\pi$
$360$	0	0
$361$	−2.21929e6	−0.896285
$362$	−1.29635e6	−0.519937
$363$	0	0
$364$	3.35150e6	1.32582
$365$	2.34058e6	0.919586
$366$	0	0
$367$	−1.37545e6	−0.533063	−0.266531	−	0.963826i	$-0.585878\pi$
$367$	−1.37545e6	−0.533063	−0.266531	+	0.963826i	$0.585878\pi$
$368$	−2.38367e6	−0.917544
$369$	0	0
$370$	−1.54111e6	−0.585234
$371$	879018.	0.331561
$372$	0	0
$373$	1.67587e6	0.623688	0.311844	−	0.950133i	$-0.399053\pi$
$373$	1.67587e6	0.623688	0.311844	+	0.950133i	$0.399053\pi$
$374$	−658514.	−0.243437
$375$	0	0
$376$	−2.02251e6	−0.737770
$377$	5.72873e6	2.07589
$378$	0	0
$379$	−1.66000e6	−0.593624	−0.296812	−	0.954936i	$-0.595923\pi$
$379$	−1.66000e6	−0.593624	−0.296812	+	0.954936i	$0.595923\pi$
$380$	1.05817e6	0.375922
$381$	0	0
$382$	1.05510e6	0.369944
$383$	3.79497e6	1.32194	0.660969	−	0.750413i	$-0.270145\pi$
$383$	3.79497e6	1.32194	0.660969	+	0.750413i	$0.270145\pi$
$384$	0	0
$385$	6.91184e6	2.37652
$386$	332570.	0.113610
$387$	0	0
$388$	−1.26690e6	−0.427232
$389$	271386.	0.0909315	0.0454657	−	0.998966i	$-0.485523\pi$
$389$	271386.	0.0909315	0.0454657	+	0.998966i	$0.485523\pi$
$390$	0	0
$391$	1.63281e6	0.540124
$392$	−103084.	−0.0338826
$393$	0	0
$394$	−260277.	−0.0844687
$395$	−5.34681e6	−1.72426
$396$	0	0
$397$	−1.08718e6	−0.346200	−0.173100	−	0.984904i	$-0.555378\pi$
$397$	−1.08718e6	−0.346200	−0.173100	+	0.984904i	$0.555378\pi$
$398$	418460.	0.132418
$399$	0	0
$400$	1.59984e6	0.499950
$401$	4.08800e6	1.26955	0.634775	−	0.772697i	$-0.281093\pi$
$401$	4.08800e6	1.26955	0.634775	+	0.772697i	$0.281093\pi$
$402$	0	0
$403$	501122.	0.153703
$404$	−4.05527e6	−1.23614
$405$	0	0
$406$	1.53138e6	0.461072
$407$	7.53308e6	2.25417
$408$	0	0
$409$	114795.	0.0339324	0.0169662	−	0.999856i	$-0.494599\pi$
$409$	114795.	0.0339324	0.0169662	+	0.999856i	$0.494599\pi$
$410$	−252067.	−0.0740552
$411$	0	0
$412$	1.17885e6	0.342149
$413$	−658517.	−0.189973
$414$	0	0
$415$	−615078.	−0.175311
$416$	4.90128e6	1.38860
$417$	0	0
$418$	745978.	0.208826
$419$	−2.33238e6	−0.649030	−0.324515	−	0.945880i	$-0.605201\pi$
$419$	−2.33238e6	−0.649030	−0.324515	+	0.945880i	$0.605201\pi$
$420$	0	0
$421$	−6.50089e6	−1.78759	−0.893794	−	0.448477i	$-0.851967\pi$
$421$	−6.50089e6	−1.78759	−0.893794	+	0.448477i	$0.851967\pi$
$422$	1.97347e6	0.539449
$423$	0	0
$424$	838200.	0.226429
$425$	−1.09589e6	−0.294302
$426$	0	0
$427$	2.32302e6	0.616571
$428$	64208.8	0.0169428
$429$	0	0
$430$	−548989.	−0.143183
$431$	−2.19292e6	−0.568629	−0.284314	−	0.958731i	$-0.591766\pi$
$431$	−2.19292e6	−0.568629	−0.284314	+	0.958731i	$0.591766\pi$
$432$	0	0
$433$	6.89217e6	1.76659	0.883296	−	0.468816i	$-0.155319\pi$
$433$	6.89217e6	1.76659	0.883296	+	0.468816i	$0.155319\pi$
$434$	133958.	0.0341385
$435$	0	0
$436$	−509721.	−0.128415
$437$	−1.84968e6	−0.463332
$438$	0	0
$439$	−5.27181e6	−1.30557	−0.652783	−	0.757545i	$-0.726398\pi$
$439$	−5.27181e6	−1.30557	−0.652783	+	0.757545i	$0.726398\pi$
$440$	6.59088e6	1.62298
$441$	0	0
$442$	−852477.	−0.207552
$443$	−165140.	−0.0399801	−0.0199901	−	0.999800i	$-0.506363\pi$
$443$	−165140.	−0.0399801	−0.0199901	+	0.999800i	$0.506363\pi$
$444$	0	0
$445$	33174.4	0.00794150
$446$	−357853.	−0.0851859
$447$	0	0
$448$	−1.32918e6	−0.312887
$449$	−6.57754e6	−1.53974	−0.769870	−	0.638201i	$-0.779679\pi$
$449$	−6.57754e6	−1.53974	−0.769870	+	0.638201i	$0.779679\pi$
$450$	0	0
$451$	1.23212e6	0.285242
$452$	1.18066e6	0.271819
$453$	0	0
$454$	−196779.	−0.0448063
$455$	8.94770e6	2.02620
$456$	0	0
$457$	6.54286e6	1.46547	0.732736	−	0.680513i	$-0.238243\pi$
$457$	6.54286e6	1.46547	0.732736	+	0.680513i	$0.238243\pi$
$458$	2.25678e6	0.502720
$459$	0	0
$460$	−7.62156e6	−1.67938
$461$	−2.70103e6	−0.591940	−0.295970	−	0.955197i	$-0.595643\pi$
$461$	−2.70103e6	−0.591940	−0.295970	+	0.955197i	$0.595643\pi$
$462$	0	0
$463$	−3.42507e6	−0.742534	−0.371267	−	0.928526i	$-0.621077\pi$
$463$	−3.42507e6	−0.742534	−0.371267	+	0.928526i	$0.621077\pi$
$464$	−3.94284e6	−0.850187
$465$	0	0
$466$	−1.90573e6	−0.406533
$467$	−2.25373e6	−0.478201	−0.239100	−	0.970995i	$-0.576853\pi$
$467$	−2.25373e6	−0.478201	−0.239100	+	0.970995i	$0.576853\pi$
$468$	0	0
$469$	2.75316e6	0.577961
$470$	−2.51822e6	−0.525834
$471$	0	0
$472$	−627938.	−0.129736
$473$	2.68351e6	0.551505
$474$	0	0
$475$	1.24144e6	0.252460
$476$	1.58008e6	0.319640
$477$	0	0
$478$	288019.	0.0576569
$479$	−637109.	−0.126875	−0.0634373	−	0.997986i	$-0.520206\pi$
$479$	−637109.	−0.126875	−0.0634373	+	0.997986i	$0.520206\pi$
$480$	0	0
$481$	9.75193e6	1.92189
$482$	1.52401e6	0.298793
$483$	0	0
$484$	−1.05209e7	−2.04145
$485$	−3.38233e6	−0.652923
$486$	0	0
$487$	9.24018e6	1.76546	0.882730	−	0.469880i	$-0.155703\pi$
$487$	9.24018e6	1.76546	0.882730	+	0.469880i	$0.155703\pi$
$488$	2.21515e6	0.421069
$489$	0	0
$490$	−128350.	−0.0241493
$491$	2.97142e6	0.556238	0.278119	−	0.960547i	$-0.410289\pi$
$491$	2.97142e6	0.556238	0.278119	+	0.960547i	$0.410289\pi$
$492$	0	0
$493$	2.70083e6	0.500473
$494$	965703.	0.178044
$495$	0	0
$496$	−344901.	−0.0629493
$497$	5.71504e6	1.03783
$498$	0	0
$499$	−2.16754e6	−0.389688	−0.194844	−	0.980834i	$-0.562420\pi$
$499$	−2.16754e6	−0.389688	−0.194844	+	0.980834i	$0.562420\pi$
$500$	−1.40999e6	−0.252227
$501$	0	0
$502$	−772322.	−0.136785
$503$	444093.	0.0782626	0.0391313	−	0.999234i	$-0.487541\pi$
$503$	444093.	0.0782626	0.0391313	+	0.999234i	$0.487541\pi$
$504$	0	0
$505$	−1.08266e7	−1.88914
$506$	−5.37295e6	−0.932903
$507$	0	0
$508$	6.91963e6	1.18966
$509$	1.16617e7	1.99511	0.997555	−	0.0698795i	$-0.0222615\pi$
$509$	1.16617e7	1.99511	0.997555	+	0.0698795i	$0.0222615\pi$
$510$	0	0
$511$	−3.95919e6	−0.670740
$512$	−5.89015e6	−0.993004
$513$	0	0
$514$	−2.27827e6	−0.380362
$515$	3.14725e6	0.522893
$516$	0	0
$517$	1.23093e7	2.02538
$518$	2.60685e6	0.426866
$519$	0	0
$520$	8.53220e6	1.38373
$521$	−3.33390e6	−0.538094	−0.269047	−	0.963127i	$-0.586709\pi$
$521$	−3.33390e6	−0.538094	−0.269047	+	0.963127i	$0.586709\pi$
$522$	0	0
$523$	−1.13005e7	−1.80652	−0.903259	−	0.429095i	$-0.858833\pi$
$523$	−1.13005e7	−1.80652	−0.903259	+	0.429095i	$0.858833\pi$
$524$	−3.07380e6	−0.489043
$525$	0	0
$526$	−1.91134e6	−0.301212
$527$	236256.	0.0370559
$528$	0	0
$529$	6.88607e6	1.06987
$530$	1.04364e6	0.161384
$531$	0	0
$532$	−1.78994e6	−0.274195
$533$	1.59504e6	0.243195
$534$	0	0
$535$	171422.	0.0258930
$536$	2.62531e6	0.394701
$537$	0	0
$538$	−4.29397e6	−0.639592
$539$	627384.	0.0930169
$540$	0	0
$541$	9.27640e6	1.36266	0.681328	−	0.731978i	$-0.261403\pi$
$541$	9.27640e6	1.36266	0.681328	+	0.731978i	$0.261403\pi$
$542$	−2.89050e6	−0.422644
$543$	0	0
$544$	2.31073e6	0.334774
$545$	−1.36084e6	−0.196252
$546$	0	0
$547$	−4.38460e6	−0.626558	−0.313279	−	0.949661i	$-0.601428\pi$
$547$	−4.38460e6	−0.626558	−0.313279	+	0.949661i	$0.601428\pi$
$548$	2.94841e6	0.419408
$549$	0	0
$550$	3.60614e6	0.508319
$551$	−3.05956e6	−0.429319
$552$	0	0
$553$	9.04433e6	1.25766
$554$	2.30342e6	0.318860
$555$	0	0
$556$	6.83195e6	0.937255
$557$	414479.	0.0566063	0.0283031	−	0.999599i	$-0.490990\pi$
$557$	414479.	0.0566063	0.0283031	+	0.999599i	$0.490990\pi$
$558$	0	0
$559$	3.47392e6	0.470209
$560$	−6.15832e6	−0.829836
$561$	0	0
$562$	3.91267e6	0.522556
$563$	−1.15148e7	−1.53103	−0.765516	−	0.643417i	$-0.777516\pi$
$563$	−1.15148e7	−1.53103	−0.765516	+	0.643417i	$0.777516\pi$
$564$	0	0
$565$	3.15209e6	0.415410
$566$	2.83274e6	0.371677
$567$	0	0
$568$	5.44965e6	0.708758
$569$	8.36373e6	1.08298	0.541489	−	0.840708i	$-0.317861\pi$
$569$	8.36373e6	1.08298	0.541489	+	0.840708i	$0.317861\pi$
$570$	0	0
$571$	1.17323e7	1.50588	0.752942	−	0.658087i	$-0.228634\pi$
$571$	1.17323e7	1.50588	0.752942	+	0.658087i	$0.228634\pi$
$572$	−1.94505e7	−2.48565
$573$	0	0
$574$	426380.	0.0540154
$575$	−8.94156e6	−1.12783
$576$	0	0
$577$	934469.	0.116849	0.0584245	−	0.998292i	$-0.481392\pi$
$577$	934469.	0.116849	0.0584245	+	0.998292i	$0.481392\pi$
$578$	2.44964e6	0.304987
$579$	0	0
$580$	−1.26069e7	−1.55610
$581$	1.04043e6	0.127871
$582$	0	0
$583$	−5.10140e6	−0.621610
$584$	−3.77534e6	−0.458062
$585$	0	0
$586$	−1.96188e6	−0.236009
$587$	3.06040e6	0.366592	0.183296	−	0.983058i	$-0.441323\pi$
$587$	3.06040e6	0.366592	0.183296	+	0.983058i	$0.441323\pi$
$588$	0	0
$589$	−267636.	−0.0317875
$590$	−781844.	−0.0924677
$591$	0	0
$592$	−6.71184e6	−0.787113
$593$	−1.82527e6	−0.213152	−0.106576	−	0.994305i	$-0.533989\pi$
$593$	−1.82527e6	−0.213152	−0.106576	+	0.994305i	$0.533989\pi$
$594$	0	0
$595$	4.21843e6	0.488493
$596$	6.33098e6	0.730055
$597$	0	0
$598$	−6.95553e6	−0.795385
$599$	9.09331e6	1.03551	0.517756	−	0.855528i	$-0.326768\pi$
$599$	9.09331e6	1.03551	0.517756	+	0.855528i	$0.326768\pi$
$600$	0	0
$601$	2.93415e6	0.331357	0.165678	−	0.986180i	$-0.447019\pi$
$601$	2.93415e6	0.331357	0.165678	+	0.986180i	$0.447019\pi$
$602$	928636.	0.104437
$603$	0	0
$604$	−1.08519e7	−1.21036
$605$	−2.80883e7	−3.11987
$606$	0	0
$607$	5.23197e6	0.576360	0.288180	−	0.957576i	$-0.406950\pi$
$607$	5.23197e6	0.576360	0.288180	+	0.957576i	$0.406950\pi$
$608$	−2.61764e6	−0.287178
$609$	0	0
$610$	2.75807e6	0.300111
$611$	1.59349e7	1.72682
$612$	0	0
$613$	−1.32549e7	−1.42470	−0.712352	−	0.701823i	$-0.752370\pi$
$613$	−1.32549e7	−1.42470	−0.712352	+	0.701823i	$0.752370\pi$
$614$	3.93817e6	0.421574
$615$	0	0
$616$	−1.11487e7	−1.18379
$617$	−8.83110e6	−0.933903	−0.466951	−	0.884283i	$-0.654648\pi$
$617$	−8.83110e6	−0.933903	−0.466951	+	0.884283i	$0.654648\pi$
$618$	0	0
$619$	3.76389e6	0.394830	0.197415	−	0.980320i	$-0.436745\pi$
$619$	3.76389e6	0.394830	0.197415	+	0.980320i	$0.436745\pi$
$620$	−1.10279e6	−0.115216
$621$	0	0
$622$	5.35957e6	0.555462
$623$	−56115.7	−0.00579248
$624$	0	0
$625$	−1.14198e7	−1.16939
$626$	−4.60391e6	−0.469560
$627$	0	0
$628$	8.55295e6	0.865400
$629$	4.59759e6	0.463344
$630$	0	0
$631$	6.67735e6	0.667622	0.333811	−	0.942640i	$-0.391665\pi$
$631$	6.67735e6	0.667622	0.333811	+	0.942640i	$0.391665\pi$
$632$	8.62435e6	0.858882
$633$	0	0
$634$	−3.44595e6	−0.340475
$635$	1.84738e7	1.81811
$636$	0	0
$637$	812178.	0.0793054
$638$	−8.88742e6	−0.864418
$639$	0	0
$640$	−1.39196e7	−1.34331
$641$	1.27863e7	1.22913	0.614566	−	0.788865i	$-0.289331\pi$
$641$	1.27863e7	1.22913	0.614566	+	0.788865i	$0.289331\pi$
$642$	0	0
$643$	1.12041e7	1.06869	0.534344	−	0.845267i	$-0.320559\pi$
$643$	1.12041e7	1.06869	0.534344	+	0.845267i	$0.320559\pi$
$644$	1.28922e7	1.22493
$645$	0	0
$646$	455285.	0.0429242
$647$	1.78490e7	1.67631	0.838154	−	0.545434i	$-0.183635\pi$
$647$	1.78490e7	1.67631	0.838154	+	0.545434i	$0.183635\pi$
$648$	0	0
$649$	3.82172e6	0.356162
$650$	4.66832e6	0.433388
$651$	0	0
$652$	1.22110e7	1.12495
$653$	−1.72317e7	−1.58141	−0.790706	−	0.612195i	$-0.790287\pi$
$653$	−1.72317e7	−1.58141	−0.790706	+	0.612195i	$0.790287\pi$
$654$	0	0
$655$	−8.20632e6	−0.747386
$656$	−1.09780e6	−0.0996009
$657$	0	0
$658$	4.25967e6	0.383540
$659$	−8.55241e6	−0.767140	−0.383570	−	0.923512i	$-0.625306\pi$
$659$	−8.55241e6	−0.767140	−0.383570	+	0.923512i	$0.625306\pi$
$660$	0	0
$661$	−1.53106e7	−1.36297	−0.681487	−	0.731831i	$-0.738666\pi$
$661$	−1.53106e7	−1.36297	−0.681487	+	0.731831i	$0.738666\pi$
$662$	−4.09300e6	−0.362991
$663$	0	0
$664$	992115.	0.0873257
$665$	−4.77872e6	−0.419042
$666$	0	0
$667$	2.20366e7	1.91792
$668$	−819058.	−0.0710189
$669$	0	0
$670$	3.26877e6	0.281317
$671$	−1.34817e7	−1.15595
$672$	0	0
$673$	−6.91094e6	−0.588166	−0.294083	−	0.955780i	$-0.595014\pi$
$673$	−6.91094e6	−0.588166	−0.294083	+	0.955780i	$0.595014\pi$
$674$	−3.13892e6	−0.266152
$675$	0	0
$676$	−1.47958e7	−1.24529
$677$	−1.32490e7	−1.11099	−0.555497	−	0.831519i	$-0.687472\pi$
$677$	−1.32490e7	−1.11099	−0.555497	+	0.831519i	$0.687472\pi$
$678$	0	0
$679$	5.72135e6	0.476238
$680$	4.02255e6	0.333602
$681$	0	0
$682$	−777430.	−0.0640030
$683$	9.66117e6	0.792461	0.396230	−	0.918151i	$-0.370318\pi$
$683$	9.66117e6	0.792461	0.396230	+	0.918151i	$0.370318\pi$
$684$	0	0
$685$	7.87156e6	0.640965
$686$	4.48015e6	0.363482
$687$	0	0
$688$	−2.39095e6	−0.192575
$689$	−6.60400e6	−0.529980
$690$	0	0
$691$	−1.96634e7	−1.56662	−0.783311	−	0.621631i	$-0.786471\pi$
$691$	−1.96634e7	−1.56662	−0.783311	+	0.621631i	$0.786471\pi$
$692$	4.81019e6	0.381854
$693$	0	0
$694$	2.94905e6	0.232425
$695$	1.82397e7	1.43237
$696$	0	0
$697$	751989.	0.0586313
$698$	367058.	0.0285165
$699$	0	0
$700$	−8.65279e6	−0.667438
$701$	−9.82975e6	−0.755523	−0.377761	−	0.925903i	$-0.623306\pi$
$701$	−9.82975e6	−0.755523	−0.377761	+	0.925903i	$0.623306\pi$
$702$	0	0
$703$	−5.20824e6	−0.397468
$704$	7.71391e6	0.586601
$705$	0	0
$706$	3.80832e6	0.287555
$707$	1.83136e7	1.37793
$708$	0	0
$709$	−1.94356e7	−1.45205	−0.726026	−	0.687667i	$-0.758635\pi$
$709$	−1.94356e7	−1.45205	−0.726026	+	0.687667i	$0.758635\pi$
$710$	6.78535e6	0.505157
$711$	0	0
$712$	−53509.9	−0.00395580
$713$	1.92766e6	0.142006
$714$	0	0
$715$	−5.19282e7	−3.79873
$716$	1.62572e7	1.18512
$717$	0	0
$718$	4.15287e6	0.300633
$719$	−8.16016e6	−0.588676	−0.294338	−	0.955701i	$-0.595099\pi$
$719$	−8.16016e6	−0.588676	−0.294338	+	0.955701i	$0.595099\pi$
$720$	0	0
$721$	−5.32369e6	−0.381395
$722$	4.45707e6	0.318204
$723$	0	0
$724$	−1.80521e7	−1.27991
$725$	−1.47903e7	−1.04503
$726$	0	0
$727$	2.08807e7	1.46524	0.732620	−	0.680638i	$-0.238298\pi$
$727$	2.08807e7	1.46524	0.732620	+	0.680638i	$0.238298\pi$
$728$	−1.44326e7	−1.00929
$729$	0	0
$730$	−4.70066e6	−0.326477
$731$	1.63780e6	0.113362
$732$	0	0
$733$	−1.86849e7	−1.28449	−0.642247	−	0.766498i	$-0.721998\pi$
$733$	−1.86849e7	−1.28449	−0.642247	+	0.766498i	$0.721998\pi$
$734$	2.76235e6	0.189251
$735$	0	0
$736$	1.88537e7	1.28293
$737$	−1.59780e7	−1.08356
$738$	0	0
$739$	1.89041e7	1.27334	0.636669	−	0.771137i	$-0.280312\pi$
$739$	1.89041e7	1.27334	0.636669	+	0.771137i	$0.280312\pi$
$740$	−2.14605e7	−1.44065
$741$	0	0
$742$	−1.76536e6	−0.117713
$743$	−1.64341e7	−1.09213	−0.546063	−	0.837744i	$-0.683874\pi$
$743$	−1.64341e7	−1.09213	−0.546063	+	0.837744i	$0.683874\pi$
$744$	0	0
$745$	1.69022e7	1.11571
$746$	−3.36569e6	−0.221425
$747$	0	0
$748$	−9.17002e6	−0.599261
$749$	−289968.	−0.0188862
$750$	0	0
$751$	2.30723e7	1.49276	0.746382	−	0.665518i	$-0.231789\pi$
$751$	2.30723e7	1.49276	0.746382	+	0.665518i	$0.231789\pi$
$752$	−1.09673e7	−0.707223
$753$	0	0
$754$	−1.15052e7	−0.736996
$755$	−2.89720e7	−1.84974
$756$	0	0
$757$	−1.60940e7	−1.02076	−0.510381	−	0.859949i	$-0.670495\pi$
$757$	−1.60940e7	−1.02076	−0.510381	+	0.859949i	$0.670495\pi$
$758$	3.33384e6	0.210752
$759$	0	0
$760$	−4.55682e6	−0.286172
$761$	2.87571e7	1.80005	0.900023	−	0.435842i	$-0.143549\pi$
$761$	2.87571e7	1.80005	0.900023	+	0.435842i	$0.143549\pi$
$762$	0	0
$763$	2.30191e6	0.143145
$764$	1.46926e7	0.910681
$765$	0	0
$766$	−7.62154e6	−0.469322
$767$	4.94740e6	0.303660
$768$	0	0
$769$	−9.25896e6	−0.564607	−0.282304	−	0.959325i	$-0.591099\pi$
$769$	−9.25896e6	−0.564607	−0.282304	+	0.959325i	$0.591099\pi$
$770$	−1.38813e7	−0.843727
$771$	0	0
$772$	4.63114e6	0.279669
$773$	−2.04713e7	−1.23224	−0.616122	−	0.787651i	$-0.711297\pi$
$773$	−2.04713e7	−1.23224	−0.616122	+	0.787651i	$0.711297\pi$
$774$	0	0
$775$	−1.29378e6	−0.0773762
$776$	5.45567e6	0.325232
$777$	0	0
$778$	−545033.	−0.0322830
$779$	−851868.	−0.0502954
$780$	0	0
$781$	−3.31674e7	−1.94573
$782$	−3.27922e6	−0.191758
$783$	0	0
$784$	−558988.	−0.0324797
$785$	2.28343e7	1.32256
$786$	0	0
$787$	7.74235e6	0.445591	0.222795	−	0.974865i	$-0.428482\pi$
$787$	7.74235e6	0.445591	0.222795	+	0.974865i	$0.428482\pi$
$788$	−3.62444e6	−0.207934
$789$	0	0
$790$	1.07381e7	0.612155
$791$	−5.33188e6	−0.302998
$792$	0	0
$793$	−1.74527e7	−0.985552
$794$	2.18343e6	0.122910
$795$	0	0
$796$	5.82718e6	0.325969
$797$	−2.40220e7	−1.33956	−0.669782	−	0.742558i	$-0.733613\pi$
$797$	−2.40220e7	−1.33956	−0.669782	+	0.742558i	$0.733613\pi$
$798$	0	0
$799$	7.51259e6	0.416316
$800$	−1.26540e7	−0.699040
$801$	0	0
$802$	−8.21005e6	−0.450723
$803$	2.29773e7	1.25750
$804$	0	0
$805$	3.44190e7	1.87201
$806$	−1.00642e6	−0.0545684
$807$	0	0
$808$	1.74632e7	0.941013
$809$	1.71095e7	0.919105	0.459552	−	0.888151i	$-0.348010\pi$
$809$	1.71095e7	0.919105	0.459552	+	0.888151i	$0.348010\pi$
$810$	0	0
$811$	−3.10663e7	−1.65859	−0.829293	−	0.558814i	$-0.811256\pi$
$811$	−3.10663e7	−1.65859	−0.829293	+	0.558814i	$0.811256\pi$
$812$	2.13250e7	1.13501
$813$	0	0
$814$	−1.51289e7	−0.800289
$815$	3.26004e7	1.71921
$816$	0	0
$817$	−1.85533e6	−0.0972446
$818$	−230546.	−0.0120469
$819$	0	0
$820$	−3.51011e6	−0.182300
$821$	−2.44442e6	−0.126566	−0.0632831	−	0.997996i	$-0.520157\pi$
$821$	−2.44442e6	−0.126566	−0.0632831	+	0.997996i	$0.520157\pi$
$822$	0	0
$823$	−2.53877e7	−1.30654	−0.653271	−	0.757124i	$-0.726604\pi$
$823$	−2.53877e7	−1.30654	−0.653271	+	0.757124i	$0.726604\pi$
$824$	−5.07648e6	−0.260462
$825$	0	0
$826$	1.32252e6	0.0674454
$827$	−2.63726e7	−1.34088	−0.670440	−	0.741964i	$-0.733894\pi$
$827$	−2.63726e7	−1.34088	−0.670440	+	0.741964i	$0.733894\pi$
$828$	0	0
$829$	9.19562e6	0.464724	0.232362	−	0.972629i	$-0.425355\pi$
$829$	9.19562e6	0.464724	0.232362	+	0.972629i	$0.425355\pi$
$830$	1.23528e6	0.0622401
$831$	0	0
$832$	9.98601e6	0.500131
$833$	382905.	0.0191196
$834$	0	0
$835$	−2.18669e6	−0.108535
$836$	1.03880e7	0.514062
$837$	0	0
$838$	4.68420e6	0.230423
$839$	2.29518e7	1.12567	0.562836	−	0.826568i	$-0.309710\pi$
$839$	2.29518e7	1.12567	0.562836	+	0.826568i	$0.309710\pi$
$840$	0	0
$841$	1.59398e7	0.777126
$842$	1.30559e7	0.634640
$843$	0	0
$844$	2.74812e7	1.32795
$845$	−3.95012e7	−1.90313
$846$	0	0
$847$	4.75125e7	2.27562
$848$	4.54526e6	0.217054
$849$	0	0
$850$	2.20090e6	0.104485
$851$	3.75126e7	1.77564
$852$	0	0
$853$	−2.83018e7	−1.33181	−0.665904	−	0.746038i	$-0.731954\pi$
$853$	−2.83018e7	−1.33181	−0.665904	+	0.746038i	$0.731954\pi$
$854$	−4.66539e6	−0.218899
$855$	0	0
$856$	−276503.	−0.0128978
$857$	1.91532e7	0.890819	0.445409	−	0.895327i	$-0.353058\pi$
$857$	1.91532e7	0.890819	0.445409	+	0.895327i	$0.353058\pi$
$858$	0	0
$859$	1.24842e7	0.577267	0.288633	−	0.957440i	$-0.406799\pi$
$859$	1.24842e7	0.577267	0.288633	+	0.957440i	$0.406799\pi$
$860$	−7.64484e6	−0.352470
$861$	0	0
$862$	4.40410e6	0.201878
$863$	−3.33719e7	−1.52530	−0.762648	−	0.646813i	$-0.776101\pi$
$863$	−3.33719e7	−1.52530	−0.762648	+	0.646813i	$0.776101\pi$
$864$	0	0
$865$	1.28421e7	0.583572
$866$	−1.38417e7	−0.627185
$867$	0	0
$868$	1.86541e6	0.0840379
$869$	−5.24890e7	−2.35786
$870$	0	0
$871$	−2.06843e7	−0.923836
$872$	2.19501e6	0.0977566
$873$	0	0
$874$	3.71476e6	0.164495
$875$	6.36755e6	0.281159
$876$	0	0
$877$	−2.00140e6	−0.0878688	−0.0439344	−	0.999034i	$-0.513989\pi$
$877$	−2.00140e6	−0.0878688	−0.0439344	+	0.999034i	$0.513989\pi$
$878$	1.05875e7	0.463509
$879$	0	0
$880$	3.57400e7	1.55578
$881$	7.28531e6	0.316234	0.158117	−	0.987420i	$-0.449458\pi$
$881$	7.28531e6	0.316234	0.158117	+	0.987420i	$0.449458\pi$
$882$	0	0
$883$	−4.26084e7	−1.83905	−0.919525	−	0.393032i	$-0.871426\pi$
$883$	−4.26084e7	−1.83905	−0.919525	+	0.393032i	$0.871426\pi$
$884$	−1.18710e7	−0.510925
$885$	0	0
$886$	331656.	0.0141940
$887$	3.71824e7	1.58682	0.793412	−	0.608685i	$-0.208303\pi$
$887$	3.71824e7	1.58682	0.793412	+	0.608685i	$0.208303\pi$
$888$	0	0
$889$	−3.12491e7	−1.32612
$890$	−66625.1	−0.00281944
$891$	0	0
$892$	−4.98322e6	−0.209700
$893$	−8.51041e6	−0.357126
$894$	0	0
$895$	4.34029e7	1.81118
$896$	2.35455e7	0.979803
$897$	0	0
$898$	1.32099e7	0.546647
$899$	3.18855e6	0.131581
$900$	0	0
$901$	−3.11349e6	−0.127772
$902$	−2.47451e6	−0.101268
$903$	0	0
$904$	−5.08429e6	−0.206923
$905$	−4.81947e7	−1.95604
$906$	0	0
$907$	−1.55191e7	−0.626396	−0.313198	−	0.949688i	$-0.601400\pi$
$907$	−1.55191e7	−0.626396	−0.313198	+	0.949688i	$0.601400\pi$
$908$	−2.74021e6	−0.110298
$909$	0	0
$910$	−1.79699e7	−0.719354
$911$	−2.03355e7	−0.811818	−0.405909	−	0.913913i	$-0.633045\pi$
$911$	−2.03355e7	−0.811818	−0.405909	+	0.913913i	$0.633045\pi$
$912$	0	0
$913$	−6.03816e6	−0.239733
$914$	−1.31402e7	−0.520280
$915$	0	0
$916$	3.14264e7	1.23753
$917$	1.38813e7	0.545139
$918$	0	0
$919$	3.27109e7	1.27762	0.638812	−	0.769363i	$-0.279426\pi$
$919$	3.27109e7	1.27762	0.638812	+	0.769363i	$0.279426\pi$
$920$	3.28208e7	1.27844
$921$	0	0
$922$	5.42456e6	0.210154
$923$	−4.29367e7	−1.65892
$924$	0	0
$925$	−2.51772e7	−0.967506
$926$	6.87866e6	0.263619
$927$	0	0
$928$	3.11860e7	1.18875
$929$	−9.88499e6	−0.375783	−0.187891	−	0.982190i	$-0.560165\pi$
$929$	−9.88499e6	−0.375783	−0.187891	+	0.982190i	$0.560165\pi$
$930$	0	0
$931$	−433762.	−0.0164013
$932$	−2.65378e7	−1.00075
$933$	0	0
$934$	4.52624e6	0.169774
$935$	−2.44818e7	−0.915828
$936$	0	0
$937$	−1.51794e7	−0.564816	−0.282408	−	0.959294i	$-0.591133\pi$
$937$	−1.51794e7	−0.564816	−0.282408	+	0.959294i	$0.591133\pi$
$938$	−5.52924e6	−0.205191
$939$	0	0
$940$	−3.50670e7	−1.29443
$941$	1.74843e7	0.643686	0.321843	−	0.946793i	$-0.395698\pi$
$941$	1.74843e7	0.643686	0.321843	+	0.946793i	$0.395698\pi$
$942$	0	0
$943$	6.13563e6	0.224688
$944$	−3.40508e6	−0.124365
$945$	0	0
$946$	−5.38936e6	−0.195799
$947$	−1.22387e7	−0.443468	−0.221734	−	0.975107i	$-0.571172\pi$
$947$	−1.22387e7	−0.443468	−0.221734	+	0.975107i	$0.571172\pi$
$948$	0	0
$949$	2.97451e7	1.07214
$950$	−2.49322e6	−0.0896297
$951$	0	0
$952$	−6.80429e6	−0.243327
$953$	1.83349e7	0.653953	0.326977	−	0.945032i	$-0.393970\pi$
$953$	1.83349e7	0.653953	0.326977	+	0.945032i	$0.393970\pi$
$954$	0	0
$955$	3.92259e7	1.39176
$956$	4.01076e6	0.141932
$957$	0	0
$958$	1.27952e6	0.0450438
$959$	−1.33150e7	−0.467516
$960$	0	0
$961$	−2.83502e7	−0.990257
$962$	−1.95851e7	−0.682319
$963$	0	0
$964$	2.12223e7	0.735530
$965$	1.23641e7	0.427408
$966$	0	0
$967$	−1.52514e7	−0.524498	−0.262249	−	0.965000i	$-0.584464\pi$
$967$	−1.52514e7	−0.524498	−0.262249	+	0.965000i	$0.584464\pi$
$968$	4.53062e7	1.55406
$969$	0	0
$970$	6.79284e6	0.231805
$971$	−1.47157e7	−0.500878	−0.250439	−	0.968132i	$-0.580575\pi$
$971$	−1.47157e7	−0.500878	−0.250439	+	0.968132i	$0.580575\pi$
$972$	0	0
$973$	−3.08531e7	−1.04476
$974$	−1.85573e7	−0.626784
$975$	0	0
$976$	1.20119e7	0.403635
$977$	−2.39180e6	−0.0801655	−0.0400828	−	0.999196i	$-0.512762\pi$
$977$	−2.39180e6	−0.0801655	−0.0400828	+	0.999196i	$0.512762\pi$
$978$	0	0
$979$	325669.	0.0108597
$980$	−1.78731e6	−0.0594476
$981$	0	0
$982$	−5.96760e6	−0.197479
$983$	3.20155e7	1.05676	0.528380	−	0.849008i	$-0.322800\pi$
$983$	3.20155e7	1.05676	0.528380	+	0.849008i	$0.322800\pi$
$984$	0	0
$985$	−9.67641e6	−0.317778
$986$	−5.42416e6	−0.177681
$987$	0	0
$988$	1.34477e7	0.438284
$989$	1.33631e7	0.434427
$990$	0	0
$991$	2.54263e7	0.822432	0.411216	−	0.911538i	$-0.365104\pi$
$991$	2.54263e7	0.822432	0.411216	+	0.911538i	$0.365104\pi$
$992$	2.72800e6	0.0880168
$993$	0	0
$994$	−1.14777e7	−0.368458
$995$	1.55572e7	0.498166
$996$	0	0
$997$	4.79018e7	1.52621	0.763104	−	0.646276i	$-0.223674\pi$
$997$	4.79018e7	1.52621	0.763104	+	0.646276i	$0.223674\pi$
$998$	4.35314e6	0.138349
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	369.6.a.e.1.5		10
3.2	odd	2		41.6.a.b.1.6	✓	10
12.11	even	2		656.6.a.g.1.10		10
15.14	odd	2		1025.6.a.b.1.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
41.6.a.b.1.6	✓	10	3.2	odd	2
369.6.a.e.1.5		10	1.1	even	1	trivial
656.6.a.g.1.10		10	12.11	even	2
1025.6.a.b.1.5		10	15.14	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:	\( N \)	\(=\)	\( 369 = 3^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	369.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00833 q^{2} -27.9666 q^{4} -74.6642 q^{5} +126.297 q^{7} +120.433 q^{8} +O(q^{10})\)	\(q-2.00833 q^{2} -27.9666 q^{4} -74.6642 q^{5} +126.297 q^{7} +120.433 q^{8} +149.950 q^{10} -732.971 q^{11} -948.864 q^{13} -253.647 q^{14} +653.063 q^{16} -447.346 q^{17} +506.762 q^{19} +2088.11 q^{20} +1472.05 q^{22} -3649.99 q^{23} +2449.75 q^{25} +1905.63 q^{26} -3532.11 q^{28} -6037.46 q^{29} -528.129 q^{31} -5165.41 q^{32} +898.419 q^{34} -9429.91 q^{35} -10277.5 q^{37} -1017.75 q^{38} -8992.02 q^{40} -1681.00 q^{41} -3661.14 q^{43} +20498.7 q^{44} +7330.38 q^{46} -16793.7 q^{47} -855.947 q^{49} -4919.90 q^{50} +26536.5 q^{52} +6959.90 q^{53} +54726.7 q^{55} +15210.3 q^{56} +12125.2 q^{58} -5214.02 q^{59} +18393.2 q^{61} +1060.66 q^{62} -10524.2 q^{64} +70846.2 q^{65} +21799.0 q^{67} +12510.8 q^{68} +18938.4 q^{70} +45250.6 q^{71} -31348.1 q^{73} +20640.5 q^{74} -14172.4 q^{76} -92572.3 q^{77} +71611.3 q^{79} -48760.5 q^{80} +3376.00 q^{82} +8237.92 q^{83} +33400.8 q^{85} +7352.77 q^{86} -88273.6 q^{88} -444.314 q^{89} -119839. q^{91} +102078. q^{92} +33727.3 q^{94} -37837.0 q^{95} +45300.6 q^{97} +1719.02 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 3 q^{2} + 207 q^{4} - 32 q^{5} + 342 q^{7} - 249 q^{8}+O(q^{10}) \) 10 * q - 3 * q^2 + 207 * q^4 - 32 * q^5 + 342 * q^7 - 249 * q^8	\( 10 q - 3 q^{2} + 207 q^{4} - 32 q^{5} + 342 q^{7} - 249 q^{8} + 102 q^{10} - 846 q^{11} + 1504 q^{13} - 3468 q^{14} + 5859 q^{16} - 560 q^{17} + 4240 q^{19} + 6182 q^{20} - 2628 q^{22} + 1508 q^{23} + 11734 q^{25} + 22014 q^{26} - 8662 q^{28} + 124 q^{29} + 10384 q^{31} + 6619 q^{32} + 802 q^{34} + 17890 q^{35} + 5524 q^{37} + 46098 q^{38} - 61738 q^{40} - 16810 q^{41} + 24160 q^{43} + 21594 q^{44} + 42404 q^{46} - 58984 q^{47} + 70326 q^{49} - 6817 q^{50} + 64374 q^{52} - 23456 q^{53} + 96426 q^{55} - 80184 q^{56} - 13378 q^{58} - 52428 q^{59} + 113540 q^{61} + 113008 q^{62} + 37363 q^{64} + 22340 q^{65} + 85506 q^{67} + 71406 q^{68} - 71946 q^{70} - 75236 q^{71} - 85148 q^{73} + 23462 q^{74} + 113376 q^{76} + 172896 q^{77} + 178200 q^{79} + 401850 q^{80} + 5043 q^{82} + 125412 q^{83} - 245912 q^{85} + 18848 q^{86} - 135952 q^{88} + 62696 q^{89} + 30056 q^{91} + 236372 q^{92} + 419014 q^{94} - 28002 q^{95} + 154548 q^{97} - 288367 q^{98}+O(q^{100}) \) 10 * q - 3 * q^2 + 207 * q^4 - 32 * q^5 + 342 * q^7 - 249 * q^8 + 102 * q^10 - 846 * q^11 + 1504 * q^13 - 3468 * q^14 + 5859 * q^16 - 560 * q^17 + 4240 * q^19 + 6182 * q^20 - 2628 * q^22 + 1508 * q^23 + 11734 * q^25 + 22014 * q^26 - 8662 * q^28 + 124 * q^29 + 10384 * q^31 + 6619 * q^32 + 802 * q^34 + 17890 * q^35 + 5524 * q^37 + 46098 * q^38 - 61738 * q^40 - 16810 * q^41 + 24160 * q^43 + 21594 * q^44 + 42404 * q^46 - 58984 * q^47 + 70326 * q^49 - 6817 * q^50 + 64374 * q^52 - 23456 * q^53 + 96426 * q^55 - 80184 * q^56 - 13378 * q^58 - 52428 * q^59 + 113540 * q^61 + 113008 * q^62 + 37363 * q^64 + 22340 * q^65 + 85506 * q^67 + 71406 * q^68 - 71946 * q^70 - 75236 * q^71 - 85148 * q^73 + 23462 * q^74 + 113376 * q^76 + 172896 * q^77 + 178200 * q^79 + 401850 * q^80 + 5043 * q^82 + 125412 * q^83 - 245912 * q^85 + 18848 * q^86 - 135952 * q^88 + 62696 * q^89 + 30056 * q^91 + 236372 * q^92 + 419014 * q^94 - 28002 * q^95 + 154548 * q^97 - 288367 * q^98

Introduction

L-functions

Modular forms

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Database

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