LMFDB - Embedded newform 135.4.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [135,4,Mod(1,135)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(135, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("135.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 135 = 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	135.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:	$7.96525785077$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.5637.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 23x + 6 $ x^3 - x^2 - 23*x + 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-4.45938$ of defining polynomial
Character	$\chi$	$=$	135.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.45938 q^{2} +11.8861 q^{4} +5.00000 q^{5} +5.08123 q^{7} +17.3296 q^{8} +O(q^{10})$	\(q+4.45938 q^{2} +11.8861 q^{4} +5.00000 q^{5} +5.08123 q^{7} +17.3296 q^{8} +22.2969 q^{10} +58.3007 q^{11} +21.2119 q^{13} +22.6592 q^{14} -17.8095 q^{16} -68.8451 q^{17} -40.8133 q^{19} +59.4305 q^{20} +259.985 q^{22} -144.318 q^{23} +25.0000 q^{25} +94.5921 q^{26} +60.3960 q^{28} -220.058 q^{29} +291.545 q^{31} -218.056 q^{32} -307.006 q^{34} +25.4062 q^{35} +260.637 q^{37} -182.002 q^{38} +86.6479 q^{40} -169.766 q^{41} -438.596 q^{43} +692.967 q^{44} -643.571 q^{46} -255.481 q^{47} -317.181 q^{49} +111.485 q^{50} +252.127 q^{52} +214.714 q^{53} +291.503 q^{55} +88.0557 q^{56} -981.322 q^{58} +331.524 q^{59} +54.9647 q^{61} +1300.11 q^{62} -829.920 q^{64} +106.060 q^{65} +758.179 q^{67} -818.299 q^{68} +113.296 q^{70} -904.348 q^{71} +866.622 q^{73} +1162.28 q^{74} -485.110 q^{76} +296.239 q^{77} +206.961 q^{79} -89.0475 q^{80} -757.054 q^{82} -463.397 q^{83} -344.225 q^{85} -1955.87 q^{86} +1010.33 q^{88} +601.736 q^{89} +107.783 q^{91} -1715.38 q^{92} -1139.29 q^{94} -204.066 q^{95} +229.363 q^{97} -1414.43 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 23 q^{4} + 15 q^{5} + 44 q^{7} - 36 q^{8}+O(q^{10}) $ 3 * q - q^2 + 23 * q^4 + 15 * q^5 + 44 * q^7 - 36 * q^8	$ 3 q - q^{2} + 23 q^{4} + 15 q^{5} + 44 q^{7} - 36 q^{8} - 5 q^{10} + 38 q^{11} + 28 q^{13} - 108 q^{14} + 191 q^{16} - 19 q^{17} + 187 q^{19} + 115 q^{20} + 122 q^{22} - 81 q^{23} + 75 q^{25} + 416 q^{26} + 410 q^{28} + 160 q^{29} + 227 q^{31} - 569 q^{32} + 17 q^{34} + 220 q^{35} + 78 q^{37} - 757 q^{38} - 180 q^{40} - 338 q^{41} + 22 q^{43} + 1636 q^{44} - 1425 q^{46} - 472 q^{47} - 197 q^{49} - 25 q^{50} - 1566 q^{52} + 521 q^{53} + 190 q^{55} - 1254 q^{56} - 2096 q^{58} + 140 q^{59} + 595 q^{61} + 1407 q^{62} - 918 q^{64} + 140 q^{65} + 878 q^{67} - 3053 q^{68} - 540 q^{70} - 602 q^{71} + 1294 q^{73} + 2878 q^{74} + 525 q^{76} + 288 q^{77} + 629 q^{79} + 955 q^{80} - 1682 q^{82} - 1287 q^{83} - 95 q^{85} - 3730 q^{86} - 858 q^{88} + 2154 q^{89} - 440 q^{91} + 1959 q^{92} - 1108 q^{94} + 935 q^{95} + 1392 q^{97} - 2693 q^{98}+O(q^{100}) $ 3 * q - q^2 + 23 * q^4 + 15 * q^5 + 44 * q^7 - 36 * q^8 - 5 * q^10 + 38 * q^11 + 28 * q^13 - 108 * q^14 + 191 * q^16 - 19 * q^17 + 187 * q^19 + 115 * q^20 + 122 * q^22 - 81 * q^23 + 75 * q^25 + 416 * q^26 + 410 * q^28 + 160 * q^29 + 227 * q^31 - 569 * q^32 + 17 * q^34 + 220 * q^35 + 78 * q^37 - 757 * q^38 - 180 * q^40 - 338 * q^41 + 22 * q^43 + 1636 * q^44 - 1425 * q^46 - 472 * q^47 - 197 * q^49 - 25 * q^50 - 1566 * q^52 + 521 * q^53 + 190 * q^55 - 1254 * q^56 - 2096 * q^58 + 140 * q^59 + 595 * q^61 + 1407 * q^62 - 918 * q^64 + 140 * q^65 + 878 * q^67 - 3053 * q^68 - 540 * q^70 - 602 * q^71 + 1294 * q^73 + 2878 * q^74 + 525 * q^76 + 288 * q^77 + 629 * q^79 + 955 * q^80 - 1682 * q^82 - 1287 * q^83 - 95 * q^85 - 3730 * q^86 - 858 * q^88 + 2154 * q^89 - 440 * q^91 + 1959 * q^92 - 1108 * q^94 + 935 * q^95 + 1392 * q^97 - 2693 * 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 $

$2$	4.45938	1.57663	0.788315	−	0.615272i	$-0.210954\pi$
$2$	4.45938	1.57663	0.788315	+	0.615272i	$0.210954\pi$
$3$	0	0
$3$	0	0
$4$	11.8861	1.48576
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	5.08123	0.274361	0.137180	−	0.990546i	$-0.456196\pi$
$7$	5.08123	0.274361	0.137180	+	0.990546i	$0.456196\pi$
$8$	17.3296	0.765867
$9$	0	0
$10$	22.2969	0.705090
$11$	58.3007	1.59803	0.799014	−	0.601312i	$-0.205355\pi$
$11$	58.3007	1.59803	0.799014	+	0.601312i	$0.205355\pi$
$12$	0	0
$13$	21.2119	0.452548	0.226274	−	0.974064i	$-0.427345\pi$
$13$	21.2119	0.452548	0.226274	+	0.974064i	$0.427345\pi$
$14$	22.6592	0.432566
$15$	0	0
$16$	−17.8095	−0.278274
$17$	−68.8451	−0.982199	−0.491099	−	0.871104i	$-0.663405\pi$
$17$	−68.8451	−0.982199	−0.491099	+	0.871104i	$0.663405\pi$
$18$	0	0
$19$	−40.8133	−0.492800	−0.246400	−	0.969168i	$-0.579248\pi$
$19$	−40.8133	−0.492800	−0.246400	+	0.969168i	$0.579248\pi$
$20$	59.4305	0.664453
$21$	0	0
$22$	259.985	2.51950
$23$	−144.318	−1.30837	−0.654184	−	0.756336i	$-0.726988\pi$
$23$	−144.318	−1.30837	−0.654184	+	0.756336i	$0.726988\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	94.5921	0.713501
$27$	0	0
$28$	60.3960	0.407635
$29$	−220.058	−1.40909	−0.704547	−	0.709657i	$-0.748850\pi$
$29$	−220.058	−1.40909	−0.704547	+	0.709657i	$0.748850\pi$
$30$	0	0
$31$	291.545	1.68913	0.844566	−	0.535452i	$-0.179859\pi$
$31$	291.545	1.68913	0.844566	+	0.535452i	$0.179859\pi$
$32$	−218.056	−1.20460
$33$	0	0
$34$	−307.006	−1.54856
$35$	25.4062	0.122698
$36$	0	0
$37$	260.637	1.15807	0.579033	−	0.815304i	$-0.303430\pi$
$37$	260.637	1.15807	0.579033	+	0.815304i	$0.303430\pi$
$38$	−182.002	−0.776964
$39$	0	0
$40$	86.6479	0.342506
$41$	−169.766	−0.646660	−0.323330	−	0.946286i	$-0.604802\pi$
$41$	−169.766	−0.646660	−0.323330	+	0.946286i	$0.604802\pi$
$42$	0	0
$43$	−438.596	−1.55547	−0.777735	−	0.628592i	$-0.783631\pi$
$43$	−438.596	−1.55547	−0.777735	+	0.628592i	$0.783631\pi$
$44$	692.967	2.37429
$45$	0	0
$46$	−643.571	−2.06281
$47$	−255.481	−0.792887	−0.396444	−	0.918059i	$-0.629756\pi$
$47$	−255.481	−0.792887	−0.396444	+	0.918059i	$0.629756\pi$
$48$	0	0
$49$	−317.181	−0.924726
$50$	111.485	0.315326
$51$	0	0
$52$	252.127	0.672379
$53$	214.714	0.556477	0.278239	−	0.960512i	$-0.410249\pi$
$53$	214.714	0.556477	0.278239	+	0.960512i	$0.410249\pi$
$54$	0	0
$55$	291.503	0.714660
$56$	88.0557	0.210124
$57$	0	0
$58$	−981.322	−2.22162
$59$	331.524	0.731537	0.365769	−	0.930706i	$-0.380806\pi$
$59$	331.524	0.731537	0.365769	+	0.930706i	$0.380806\pi$
$60$	0	0
$61$	54.9647	0.115369	0.0576845	−	0.998335i	$-0.481628\pi$
$61$	54.9647	0.115369	0.0576845	+	0.998335i	$0.481628\pi$
$62$	1300.11	2.66314
$63$	0	0
$64$	−829.920	−1.62094
$65$	106.060	0.202386
$66$	0	0
$67$	758.179	1.38248	0.691241	−	0.722624i	$-0.257064\pi$
$67$	758.179	1.38248	0.691241	+	0.722624i	$0.257064\pi$
$68$	−818.299	−1.45931
$69$	0	0
$70$	113.296	0.193449
$71$	−904.348	−1.51164	−0.755819	−	0.654780i	$-0.772761\pi$
$71$	−904.348	−1.51164	−0.755819	+	0.654780i	$0.772761\pi$
$72$	0	0
$73$	866.622	1.38946	0.694729	−	0.719271i	$-0.255524\pi$
$73$	866.622	1.38946	0.694729	+	0.719271i	$0.255524\pi$
$74$	1162.28	1.82584
$75$	0	0
$76$	−485.110	−0.732184
$77$	296.239	0.438437
$78$	0	0
$79$	206.961	0.294746	0.147373	−	0.989081i	$-0.452918\pi$
$79$	206.961	0.294746	0.147373	+	0.989081i	$0.452918\pi$
$80$	−89.0475	−0.124448
$81$	0	0
$82$	−757.054	−1.01954
$83$	−463.397	−0.612825	−0.306412	−	0.951899i	$-0.599129\pi$
$83$	−463.397	−0.612825	−0.306412	+	0.951899i	$0.599129\pi$
$84$	0	0
$85$	−344.225	−0.439253
$86$	−1955.87	−2.45240
$87$	0	0
$88$	1010.33	1.22388
$89$	601.736	0.716673	0.358337	−	0.933592i	$-0.383344\pi$
$89$	601.736	0.716673	0.358337	+	0.933592i	$0.383344\pi$
$90$	0	0
$91$	107.783	0.124162
$92$	−1715.38	−1.94392
$93$	0	0
$94$	−1139.29	−1.25009
$95$	−204.066	−0.220387
$96$	0	0
$97$	229.363	0.240086	0.120043	−	0.992769i	$-0.461697\pi$
$97$	229.363	0.240086	0.120043	+	0.992769i	$0.461697\pi$
$98$	−1414.43	−1.45795
$99$	0	0
$100$	297.152	0.297152
$101$	1345.66	1.32573	0.662863	−	0.748740i	$-0.269341\pi$
$101$	1345.66	1.32573	0.662863	+	0.748740i	$0.269341\pi$
$102$	0	0
$103$	−1596.30	−1.52707	−0.763534	−	0.645768i	$-0.776537\pi$
$103$	−1596.30	−1.52707	−0.763534	+	0.645768i	$0.776537\pi$
$104$	367.594	0.346592
$105$	0	0
$106$	957.494	0.877358
$107$	958.786	0.866256	0.433128	−	0.901333i	$-0.357410\pi$
$107$	958.786	0.866256	0.433128	+	0.901333i	$0.357410\pi$
$108$	0	0
$109$	1690.23	1.48527	0.742635	−	0.669696i	$-0.233576\pi$
$109$	1690.23	1.48527	0.742635	+	0.669696i	$0.233576\pi$
$110$	1299.93	1.12675
$111$	0	0
$112$	−90.4943	−0.0763474
$113$	−11.6211	−0.00967456	−0.00483728	−	0.999988i	$-0.501540\pi$
$113$	−11.6211	−0.00967456	−0.00483728	+	0.999988i	$0.501540\pi$
$114$	0	0
$115$	−721.592	−0.585120
$116$	−2615.63	−2.09358
$117$	0	0
$118$	1478.39	1.15336
$119$	−349.818	−0.269477
$120$	0	0
$121$	2067.97	1.55370
$122$	245.109	0.181894
$123$	0	0
$124$	3465.34	2.50965
$125$	125.000	0.0894427
$126$	0	0
$127$	309.141	0.215999	0.107999	−	0.994151i	$-0.465556\pi$
$127$	309.141	0.215999	0.107999	+	0.994151i	$0.465556\pi$
$128$	−1956.48	−1.35102
$129$	0	0
$130$	472.960	0.319087
$131$	2785.03	1.85747	0.928736	−	0.370742i	$-0.120897\pi$
$131$	2785.03	1.85747	0.928736	+	0.370742i	$0.120897\pi$
$132$	0	0
$133$	−207.382	−0.135205
$134$	3381.01	2.17966
$135$	0	0
$136$	−1193.06	−0.752233
$137$	−2489.41	−1.55244	−0.776222	−	0.630460i	$-0.782866\pi$
$137$	−2489.41	−1.55244	−0.776222	+	0.630460i	$0.782866\pi$
$138$	0	0
$139$	1786.05	1.08986	0.544931	−	0.838481i	$-0.316556\pi$
$139$	1786.05	1.08986	0.544931	+	0.838481i	$0.316556\pi$
$140$	301.980	0.182300
$141$	0	0
$142$	−4032.83	−2.38329
$143$	1236.67	0.723185
$144$	0	0
$145$	−1100.29	−0.630166
$146$	3864.60	2.19066
$147$	0	0
$148$	3097.95	1.72061
$149$	1568.83	0.862575	0.431288	−	0.902214i	$-0.358059\pi$
$149$	1568.83	0.862575	0.431288	+	0.902214i	$0.358059\pi$
$150$	0	0
$151$	−438.327	−0.236229	−0.118114	−	0.993000i	$-0.537685\pi$
$151$	−438.327	−0.236229	−0.118114	+	0.993000i	$0.537685\pi$
$152$	−707.277	−0.377419
$153$	0	0
$154$	1321.04	0.691252
$155$	1457.73	0.755403
$156$	0	0
$157$	−44.7479	−0.0227469	−0.0113735	−	0.999935i	$-0.503620\pi$
$157$	−44.7479	−0.0227469	−0.0113735	+	0.999935i	$0.503620\pi$
$158$	922.917	0.464705
$159$	0	0
$160$	−1090.28	−0.538714
$161$	−733.315	−0.358965
$162$	0	0
$163$	−2611.84	−1.25506	−0.627531	−	0.778591i	$-0.715935\pi$
$163$	−2611.84	−1.25506	−0.627531	+	0.778591i	$0.715935\pi$
$164$	−2017.86	−0.960783
$165$	0	0
$166$	−2066.47	−0.966198
$167$	−188.947	−0.0875516	−0.0437758	−	0.999041i	$-0.513939\pi$
$167$	−188.947	−0.0875516	−0.0437758	+	0.999041i	$0.513939\pi$
$168$	0	0
$169$	−1747.05	−0.795200
$170$	−1535.03	−0.692539
$171$	0	0
$172$	−5213.19	−2.31106
$173$	−1505.02	−0.661413	−0.330707	−	0.943734i	$-0.607287\pi$
$173$	−1505.02	−0.661413	−0.330707	+	0.943734i	$0.607287\pi$
$174$	0	0
$175$	127.031	0.0548722
$176$	−1038.31	−0.444689
$177$	0	0
$178$	2683.37	1.12993
$179$	−3136.62	−1.30973	−0.654865	−	0.755746i	$-0.727275\pi$
$179$	−3136.62	−1.30973	−0.654865	+	0.755746i	$0.727275\pi$
$180$	0	0
$181$	4512.67	1.85317	0.926586	−	0.376084i	$-0.122730\pi$
$181$	4512.67	1.85317	0.926586	+	0.376084i	$0.122730\pi$
$182$	480.644	0.195757
$183$	0	0
$184$	−2500.98	−1.00204
$185$	1303.18	0.517902
$186$	0	0
$187$	−4013.71	−1.56958
$188$	−3036.67	−1.17804
$189$	0	0
$190$	−910.010	−0.347469
$191$	1207.43	0.457418	0.228709	−	0.973495i	$-0.426550\pi$
$191$	1207.43	0.457418	0.228709	+	0.973495i	$0.426550\pi$
$192$	0	0
$193$	923.164	0.344305	0.172152	−	0.985070i	$-0.444928\pi$
$193$	923.164	0.344305	0.172152	+	0.985070i	$0.444928\pi$
$194$	1022.82	0.378526
$195$	0	0
$196$	−3770.04	−1.37392
$197$	−1180.87	−0.427075	−0.213537	−	0.976935i	$-0.568499\pi$
$197$	−1180.87	−0.427075	−0.213537	+	0.976935i	$0.568499\pi$
$198$	0	0
$199$	−839.805	−0.299157	−0.149578	−	0.988750i	$-0.547792\pi$
$199$	−839.805	−0.299157	−0.149578	+	0.988750i	$0.547792\pi$
$200$	433.240	0.153173
$201$	0	0
$202$	6000.82	2.09018
$203$	−1118.17	−0.386600
$204$	0	0
$205$	−848.832	−0.289195
$206$	−7118.51	−2.40762
$207$	0	0
$208$	−377.774	−0.125932
$209$	−2379.44	−0.787509
$210$	0	0
$211$	−2589.65	−0.844923	−0.422461	−	0.906381i	$-0.638834\pi$
$211$	−2589.65	−0.844923	−0.422461	+	0.906381i	$0.638834\pi$
$212$	2552.12	0.826792
$213$	0	0
$214$	4275.59	1.36576
$215$	−2192.98	−0.695627
$216$	0	0
$217$	1481.41	0.463432
$218$	7537.38	2.34172
$219$	0	0
$220$	3464.84	1.06181
$221$	−1460.34	−0.444492
$222$	0	0
$223$	−4180.76	−1.25544	−0.627722	−	0.778437i	$-0.716013\pi$
$223$	−4180.76	−1.25544	−0.627722	+	0.778437i	$0.716013\pi$
$224$	−1107.99	−0.330495
$225$	0	0
$226$	−51.8232	−0.0152532
$227$	−2602.67	−0.760993	−0.380497	−	0.924782i	$-0.624247\pi$
$227$	−2602.67	−0.760993	−0.380497	+	0.924782i	$0.624247\pi$
$228$	0	0
$229$	−1845.35	−0.532508	−0.266254	−	0.963903i	$-0.585786\pi$
$229$	−1845.35	−0.532508	−0.266254	+	0.963903i	$0.585786\pi$
$230$	−3217.85	−0.922517
$231$	0	0
$232$	−3813.51	−1.07918
$233$	240.637	0.0676594	0.0338297	−	0.999428i	$-0.489230\pi$
$233$	240.637	0.0676594	0.0338297	+	0.999428i	$0.489230\pi$
$234$	0	0
$235$	−1277.40	−0.354590
$236$	3940.52	1.08689
$237$	0	0
$238$	−1559.97	−0.424865
$239$	2567.64	0.694924	0.347462	−	0.937694i	$-0.387044\pi$
$239$	2567.64	0.694924	0.347462	+	0.937694i	$0.387044\pi$
$240$	0	0
$241$	3987.99	1.06593	0.532965	−	0.846137i	$-0.321078\pi$
$241$	3987.99	1.06593	0.532965	+	0.846137i	$0.321078\pi$
$242$	9221.86	2.44960
$243$	0	0
$244$	653.315	0.171411
$245$	−1585.91	−0.413550
$246$	0	0
$247$	−865.728	−0.223016
$248$	5052.36	1.29365
$249$	0	0
$250$	557.423	0.141018
$251$	967.393	0.243272	0.121636	−	0.992575i	$-0.461186\pi$
$251$	967.393	0.243272	0.121636	+	0.992575i	$0.461186\pi$
$252$	0	0
$253$	−8413.86	−2.09081
$254$	1378.58	0.340550
$255$	0	0
$256$	−2085.34	−0.509116
$257$	3743.03	0.908498	0.454249	−	0.890875i	$-0.349908\pi$
$257$	3743.03	0.908498	0.454249	+	0.890875i	$0.349908\pi$
$258$	0	0
$259$	1324.36	0.317728
$260$	1260.63	0.300697
$261$	0	0
$262$	12419.5	2.92855
$263$	−2497.47	−0.585554	−0.292777	−	0.956181i	$-0.594579\pi$
$263$	−2497.47	−0.585554	−0.292777	+	0.956181i	$0.594579\pi$
$264$	0	0
$265$	1073.57	0.248864
$266$	−924.795	−0.213168
$267$	0	0
$268$	9011.79	2.05404
$269$	−2142.91	−0.485709	−0.242855	−	0.970063i	$-0.578084\pi$
$269$	−2142.91	−0.485709	−0.242855	+	0.970063i	$0.578084\pi$
$270$	0	0
$271$	1540.64	0.345341	0.172671	−	0.984980i	$-0.444760\pi$
$271$	1540.64	0.345341	0.172671	+	0.984980i	$0.444760\pi$
$272$	1226.10	0.273320
$273$	0	0
$274$	−11101.2	−2.44763
$275$	1457.52	0.319606
$276$	0	0
$277$	6777.80	1.47018	0.735088	−	0.677972i	$-0.237141\pi$
$277$	6777.80	1.47018	0.735088	+	0.677972i	$0.237141\pi$
$278$	7964.69	1.71831
$279$	0	0
$280$	440.278	0.0939702
$281$	827.653	0.175707	0.0878535	−	0.996133i	$-0.471999\pi$
$281$	827.653	0.175707	0.0878535	+	0.996133i	$0.471999\pi$
$282$	0	0
$283$	−3171.98	−0.666270	−0.333135	−	0.942879i	$-0.608106\pi$
$283$	−3171.98	−0.666270	−0.333135	+	0.942879i	$0.608106\pi$
$284$	−10749.2	−2.24593
$285$	0	0
$286$	5514.78	1.14020
$287$	−862.623	−0.177418
$288$	0	0
$289$	−173.358	−0.0352856
$290$	−4906.61	−0.993539
$291$	0	0
$292$	10300.8	2.06440
$293$	1376.02	0.274362	0.137181	−	0.990546i	$-0.456196\pi$
$293$	1376.02	0.274362	0.137181	+	0.990546i	$0.456196\pi$
$294$	0	0
$295$	1657.62	0.327153
$296$	4516.73	0.886923
$297$	0	0
$298$	6996.02	1.35996
$299$	−3061.27	−0.592099
$300$	0	0
$301$	−2228.61	−0.426760
$302$	−1954.67	−0.372445
$303$	0	0
$304$	726.864	0.137133
$305$	274.823	0.0515946
$306$	0	0
$307$	−119.504	−0.0222165	−0.0111083	−	0.999938i	$-0.503536\pi$
$307$	−119.504	−0.0222165	−0.0111083	+	0.999938i	$0.503536\pi$
$308$	3521.13	0.651412
$309$	0	0
$310$	6500.56	1.19099
$311$	2139.35	0.390069	0.195035	−	0.980796i	$-0.437518\pi$
$311$	2139.35	0.390069	0.195035	+	0.980796i	$0.437518\pi$
$312$	0	0
$313$	5163.50	0.932455	0.466227	−	0.884665i	$-0.345613\pi$
$313$	5163.50	0.932455	0.466227	+	0.884665i	$0.345613\pi$
$314$	−199.548	−0.0358635
$315$	0	0
$316$	2459.95	0.437922
$317$	−8631.69	−1.52935	−0.764675	−	0.644416i	$-0.777101\pi$
$317$	−8631.69	−1.52935	−0.764675	+	0.644416i	$0.777101\pi$
$318$	0	0
$319$	−12829.5	−2.25177
$320$	−4149.60	−0.724905
$321$	0	0
$322$	−3270.13	−0.565955
$323$	2809.79	0.484028
$324$	0	0
$325$	530.298	0.0905097
$326$	−11647.2	−1.97877
$327$	0	0
$328$	−2941.98	−0.495255
$329$	−1298.16	−0.217537
$330$	0	0
$331$	−2942.34	−0.488597	−0.244298	−	0.969700i	$-0.578558\pi$
$331$	−2942.34	−0.488597	−0.244298	+	0.969700i	$0.578558\pi$
$332$	−5507.98	−0.910512
$333$	0	0
$334$	−842.585	−0.138037
$335$	3790.90	0.618265
$336$	0	0
$337$	9897.46	1.59985	0.799924	−	0.600101i	$-0.204873\pi$
$337$	9897.46	1.59985	0.799924	+	0.600101i	$0.204873\pi$
$338$	−7790.79	−1.25374
$339$	0	0
$340$	−4091.49	−0.652625
$341$	16997.3	2.69928
$342$	0	0
$343$	−3354.53	−0.528070
$344$	−7600.68	−1.19128
$345$	0	0
$346$	−6711.46	−1.04280
$347$	−12015.7	−1.85890	−0.929451	−	0.368947i	$-0.879718\pi$
$347$	−12015.7	−1.85890	−0.929451	+	0.368947i	$0.879718\pi$
$348$	0	0
$349$	7894.62	1.21086	0.605428	−	0.795900i	$-0.293002\pi$
$349$	7894.62	1.21086	0.605428	+	0.795900i	$0.293002\pi$
$350$	566.479	0.0865131
$351$	0	0
$352$	−12712.8	−1.92499
$353$	741.154	0.111750	0.0558748	−	0.998438i	$-0.482205\pi$
$353$	741.154	0.111750	0.0558748	+	0.998438i	$0.482205\pi$
$354$	0	0
$355$	−4521.74	−0.676025
$356$	7152.30	1.06481
$357$	0	0
$358$	−13987.4	−2.06496
$359$	−11564.4	−1.70012	−0.850060	−	0.526685i	$-0.823435\pi$
$359$	−11564.4	−1.70012	−0.850060	+	0.526685i	$0.823435\pi$
$360$	0	0
$361$	−5193.28	−0.757148
$362$	20123.7	2.92177
$363$	0	0
$364$	1281.12	0.184474
$365$	4333.11	0.621385
$366$	0	0
$367$	−1148.57	−0.163365	−0.0816823	−	0.996658i	$-0.526029\pi$
$367$	−1148.57	−0.163365	−0.0816823	+	0.996658i	$0.526029\pi$
$368$	2570.24	0.364084
$369$	0	0
$370$	5811.39	0.816540
$371$	1091.01	0.152676
$372$	0	0
$373$	−4602.98	−0.638963	−0.319481	−	0.947593i	$-0.603509\pi$
$373$	−4602.98	−0.638963	−0.319481	+	0.947593i	$0.603509\pi$
$374$	−17898.7	−2.47465
$375$	0	0
$376$	−4427.38	−0.607246
$377$	−4667.85	−0.637683
$378$	0	0
$379$	−3988.46	−0.540563	−0.270282	−	0.962781i	$-0.587117\pi$
$379$	−3988.46	−0.540563	−0.270282	+	0.962781i	$0.587117\pi$
$380$	−2425.55	−0.327443
$381$	0	0
$382$	5384.41	0.721179
$383$	7788.20	1.03906	0.519528	−	0.854454i	$-0.326108\pi$
$383$	7788.20	1.03906	0.519528	+	0.854454i	$0.326108\pi$
$384$	0	0
$385$	1481.20	0.196075
$386$	4116.74	0.542841
$387$	0	0
$388$	2726.23	0.356710
$389$	8524.87	1.11113	0.555563	−	0.831474i	$-0.312503\pi$
$389$	8524.87	1.11113	0.555563	+	0.831474i	$0.312503\pi$
$390$	0	0
$391$	9935.60	1.28508
$392$	−5496.62	−0.708217
$393$	0	0
$394$	−5265.97	−0.673339
$395$	1034.80	0.131814
$396$	0	0
$397$	−155.729	−0.0196872	−0.00984361	−	0.999952i	$-0.503133\pi$
$397$	−155.729	−0.0196872	−0.00984361	+	0.999952i	$0.503133\pi$
$398$	−3745.01	−0.471659
$399$	0	0
$400$	−445.238	−0.0556547
$401$	5933.96	0.738972	0.369486	−	0.929236i	$-0.379534\pi$
$401$	5933.96	0.738972	0.369486	+	0.929236i	$0.379534\pi$
$402$	0	0
$403$	6184.23	0.764414
$404$	15994.7	1.96971
$405$	0	0
$406$	−4986.33	−0.609526
$407$	15195.3	1.85062
$408$	0	0
$409$	14161.4	1.71207	0.856035	−	0.516917i	$-0.172921\pi$
$409$	14161.4	1.71207	0.856035	+	0.516917i	$0.172921\pi$
$410$	−3785.27	−0.455954
$411$	0	0
$412$	−18973.8	−2.26886
$413$	1684.55	0.200705
$414$	0	0
$415$	−2316.99	−0.274064
$416$	−4625.39	−0.545140
$417$	0	0
$418$	−10610.8	−1.24161
$419$	9624.24	1.12214	0.561068	−	0.827770i	$-0.310391\pi$
$419$	9624.24	1.12214	0.561068	+	0.827770i	$0.310391\pi$
$420$	0	0
$421$	−1536.26	−0.177845	−0.0889223	−	0.996039i	$-0.528342\pi$
$421$	−1536.26	−0.177845	−0.0889223	+	0.996039i	$0.528342\pi$
$422$	−11548.2	−1.33213
$423$	0	0
$424$	3720.91	0.426187
$425$	−1721.13	−0.196440
$426$	0	0
$427$	279.288	0.0316527
$428$	11396.2	1.28705
$429$	0	0
$430$	−9779.33	−1.09675
$431$	11582.2	1.29441	0.647207	−	0.762314i	$-0.275937\pi$
$431$	11582.2	1.29441	0.647207	+	0.762314i	$0.275937\pi$
$432$	0	0
$433$	14892.6	1.65287	0.826437	−	0.563029i	$-0.190364\pi$
$433$	14892.6	1.65287	0.826437	+	0.563029i	$0.190364\pi$
$434$	6606.17	0.730660
$435$	0	0
$436$	20090.2	2.20676
$437$	5890.10	0.644764
$438$	0	0
$439$	−1642.51	−0.178571	−0.0892853	−	0.996006i	$-0.528458\pi$
$439$	−1642.51	−0.178571	−0.0892853	+	0.996006i	$0.528458\pi$
$440$	5051.63	0.547334
$441$	0	0
$442$	−6512.20	−0.700800
$443$	3916.31	0.420021	0.210011	−	0.977699i	$-0.432650\pi$
$443$	3916.31	0.420021	0.210011	+	0.977699i	$0.432650\pi$
$444$	0	0
$445$	3008.68	0.320506
$446$	−18643.6	−1.97937
$447$	0	0
$448$	−4217.02	−0.444722
$449$	−3985.25	−0.418876	−0.209438	−	0.977822i	$-0.567163\pi$
$449$	−3985.25	−0.418876	−0.209438	+	0.977822i	$0.567163\pi$
$450$	0	0
$451$	−9897.50	−1.03338
$452$	−138.130	−0.0143741
$453$	0	0
$454$	−11606.3	−1.19980
$455$	538.914	0.0555267
$456$	0	0
$457$	−14177.8	−1.45122	−0.725611	−	0.688105i	$-0.758443\pi$
$457$	−14177.8	−1.45122	−0.725611	+	0.688105i	$0.758443\pi$
$458$	−8229.13	−0.839567
$459$	0	0
$460$	−8576.91	−0.869349
$461$	−16394.3	−1.65631	−0.828154	−	0.560500i	$-0.810609\pi$
$461$	−16394.3	−1.65631	−0.828154	+	0.560500i	$0.810609\pi$
$462$	0	0
$463$	3319.60	0.333207	0.166603	−	0.986024i	$-0.446720\pi$
$463$	3319.60	0.333207	0.166603	+	0.986024i	$0.446720\pi$
$464$	3919.12	0.392114
$465$	0	0
$466$	1073.09	0.106674
$467$	−2529.79	−0.250674	−0.125337	−	0.992114i	$-0.540001\pi$
$467$	−2529.79	−0.250674	−0.125337	+	0.992114i	$0.540001\pi$
$468$	0	0
$469$	3852.49	0.379299
$470$	−5696.43	−0.559057
$471$	0	0
$472$	5745.17	0.560260
$473$	−25570.4	−2.48569
$474$	0	0
$475$	−1020.33	−0.0985601
$476$	−4157.97	−0.400379
$477$	0	0
$478$	11450.1	1.09564
$479$	−8646.75	−0.824802	−0.412401	−	0.911002i	$-0.635310\pi$
$479$	−8646.75	−0.824802	−0.412401	+	0.911002i	$0.635310\pi$
$480$	0	0
$481$	5528.60	0.524080
$482$	17784.0	1.68058
$483$	0	0
$484$	24580.1	2.30842
$485$	1146.82	0.107370
$486$	0	0
$487$	−15251.7	−1.41914	−0.709569	−	0.704636i	$-0.751110\pi$
$487$	−15251.7	−1.41914	−0.709569	+	0.704636i	$0.751110\pi$
$488$	952.515	0.0883572
$489$	0	0
$490$	−7072.16	−0.652015
$491$	−1766.87	−0.162398	−0.0811992	−	0.996698i	$-0.525875\pi$
$491$	−1766.87	−0.162398	−0.0811992	+	0.996698i	$0.525875\pi$
$492$	0	0
$493$	15149.9	1.38401
$494$	−3860.61	−0.351614
$495$	0	0
$496$	−5192.28	−0.470041
$497$	−4595.20	−0.414734
$498$	0	0
$499$	11733.8	1.05266	0.526332	−	0.850279i	$-0.323567\pi$
$499$	11733.8	1.05266	0.526332	+	0.850279i	$0.323567\pi$
$500$	1485.76	0.132891
$501$	0	0
$502$	4313.98	0.383550
$503$	−977.608	−0.0866589	−0.0433294	−	0.999061i	$-0.513797\pi$
$503$	−977.608	−0.0866589	−0.0433294	+	0.999061i	$0.513797\pi$
$504$	0	0
$505$	6728.31	0.592883
$506$	−37520.6	−3.29643
$507$	0	0
$508$	3674.48	0.320923
$509$	9674.72	0.842484	0.421242	−	0.906948i	$-0.361594\pi$
$509$	9674.72	0.842484	0.421242	+	0.906948i	$0.361594\pi$
$510$	0	0
$511$	4403.51	0.381213
$512$	6352.52	0.548329
$513$	0	0
$514$	16691.6	1.43237
$515$	−7981.49	−0.682926
$516$	0	0
$517$	−14894.7	−1.26706
$518$	5905.81	0.500939
$519$	0	0
$520$	1837.97	0.155000
$521$	5178.95	0.435497	0.217748	−	0.976005i	$-0.430129\pi$
$521$	5178.95	0.435497	0.217748	+	0.976005i	$0.430129\pi$
$522$	0	0
$523$	14280.7	1.19398	0.596992	−	0.802248i	$-0.296363\pi$
$523$	14280.7	1.19398	0.596992	+	0.802248i	$0.296363\pi$
$524$	33103.1	2.75976
$525$	0	0
$526$	−11137.2	−0.923203
$527$	−20071.5	−1.65906
$528$	0	0
$529$	8660.78	0.711826
$530$	4787.47	0.392367
$531$	0	0
$532$	−2464.96	−0.200883
$533$	−3601.07	−0.292645
$534$	0	0
$535$	4793.93	0.387401
$536$	13138.9	1.05880
$537$	0	0
$538$	−9556.08	−0.765784
$539$	−18491.9	−1.47774
$540$	0	0
$541$	12923.8	1.02706	0.513529	−	0.858072i	$-0.328338\pi$
$541$	12923.8	1.02706	0.513529	+	0.858072i	$0.328338\pi$
$542$	6870.32	0.544475
$543$	0	0
$544$	15012.1	1.18316
$545$	8451.14	0.664233
$546$	0	0
$547$	13653.3	1.06723	0.533614	−	0.845728i	$-0.320834\pi$
$547$	13653.3	1.06723	0.533614	+	0.845728i	$0.320834\pi$
$548$	−29589.4	−2.30656
$549$	0	0
$550$	6499.63	0.503900
$551$	8981.28	0.694402
$552$	0	0
$553$	1051.62	0.0808666
$554$	30224.8	2.31792
$555$	0	0
$556$	21229.2	1.61928
$557$	−9313.63	−0.708494	−0.354247	−	0.935152i	$-0.615263\pi$
$557$	−9313.63	−0.708494	−0.354247	+	0.935152i	$0.615263\pi$
$558$	0	0
$559$	−9303.46	−0.703925
$560$	−452.471	−0.0341436
$561$	0	0
$562$	3690.82	0.277025
$563$	−10625.4	−0.795394	−0.397697	−	0.917517i	$-0.630190\pi$
$563$	−10625.4	−0.795394	−0.397697	+	0.917517i	$0.630190\pi$
$564$	0	0
$565$	−58.1057	−0.00432660
$566$	−14145.1	−1.05046
$567$	0	0
$568$	−15672.0	−1.15771
$569$	9060.50	0.667550	0.333775	−	0.942653i	$-0.391677\pi$
$569$	9060.50	0.667550	0.333775	+	0.942653i	$0.391677\pi$
$570$	0	0
$571$	−21379.1	−1.56688	−0.783440	−	0.621467i	$-0.786537\pi$
$571$	−21379.1	−1.56688	−0.783440	+	0.621467i	$0.786537\pi$
$572$	14699.2	1.07448
$573$	0	0
$574$	−3846.77	−0.279723
$575$	−3607.96	−0.261674
$576$	0	0
$577$	6347.76	0.457991	0.228996	−	0.973427i	$-0.426456\pi$
$577$	6347.76	0.457991	0.228996	+	0.973427i	$0.426456\pi$
$578$	−773.071	−0.0556324
$579$	0	0
$580$	−13078.1	−0.936277
$581$	−2354.63	−0.168135
$582$	0	0
$583$	12518.0	0.889266
$584$	15018.2	1.06414
$585$	0	0
$586$	6136.22	0.432568
$587$	14773.7	1.03880	0.519401	−	0.854531i	$-0.326155\pi$
$587$	14773.7	1.03880	0.519401	+	0.854531i	$0.326155\pi$
$588$	0	0
$589$	−11898.9	−0.832405
$590$	7391.95	0.515800
$591$	0	0
$592$	−4641.81	−0.322259
$593$	26868.1	1.86061	0.930304	−	0.366790i	$-0.119543\pi$
$593$	26868.1	1.86061	0.930304	+	0.366790i	$0.119543\pi$
$594$	0	0
$595$	−1749.09	−0.120514
$596$	18647.3	1.28158
$597$	0	0
$598$	−13651.4	−0.933522
$599$	−1749.83	−0.119359	−0.0596794	−	0.998218i	$-0.519008\pi$
$599$	−1749.83	−0.119359	−0.0596794	+	0.998218i	$0.519008\pi$
$600$	0	0
$601$	−17964.0	−1.21924	−0.609622	−	0.792692i	$-0.708679\pi$
$601$	−17964.0	−1.21924	−0.609622	+	0.792692i	$0.708679\pi$
$602$	−9938.21	−0.672843
$603$	0	0
$604$	−5209.99	−0.350979
$605$	10339.8	0.694834
$606$	0	0
$607$	−4418.22	−0.295437	−0.147718	−	0.989029i	$-0.547193\pi$
$607$	−4418.22	−0.295437	−0.147718	+	0.989029i	$0.547193\pi$
$608$	8899.58	0.593628
$609$	0	0
$610$	1225.54	0.0813455
$611$	−5419.24	−0.358820
$612$	0	0
$613$	−20179.7	−1.32961	−0.664804	−	0.747018i	$-0.731485\pi$
$613$	−20179.7	−1.32961	−0.664804	+	0.747018i	$0.731485\pi$
$614$	−532.915	−0.0350272
$615$	0	0
$616$	5133.71	0.335784
$617$	−4673.01	−0.304908	−0.152454	−	0.988311i	$-0.548718\pi$
$617$	−4673.01	−0.304908	−0.152454	+	0.988311i	$0.548718\pi$
$618$	0	0
$619$	−19976.8	−1.29715	−0.648574	−	0.761151i	$-0.724634\pi$
$619$	−19976.8	−1.29715	−0.648574	+	0.761151i	$0.724634\pi$
$620$	17326.7	1.12235
$621$	0	0
$622$	9540.19	0.614994
$623$	3057.56	0.196627
$624$	0	0
$625$	625.000	0.0400000
$626$	23026.0	1.47014
$627$	0	0
$628$	−531.877	−0.0337965
$629$	−17943.5	−1.13745
$630$	0	0
$631$	10457.5	0.659757	0.329879	−	0.944023i	$-0.392992\pi$
$631$	10457.5	0.659757	0.329879	+	0.944023i	$0.392992\pi$
$632$	3586.54	0.225736
$633$	0	0
$634$	−38492.0	−2.41122
$635$	1545.70	0.0965975
$636$	0	0
$637$	−6728.02	−0.418483
$638$	−57211.8	−3.55021
$639$	0	0
$640$	−9782.40	−0.604193
$641$	4395.86	0.270867	0.135434	−	0.990786i	$-0.456757\pi$
$641$	4395.86	0.270867	0.135434	+	0.990786i	$0.456757\pi$
$642$	0	0
$643$	5786.36	0.354886	0.177443	−	0.984131i	$-0.443217\pi$
$643$	5786.36	0.354886	0.177443	+	0.984131i	$0.443217\pi$
$644$	−8716.26	−0.533336
$645$	0	0
$646$	12529.9	0.763133
$647$	−25367.7	−1.54143	−0.770717	−	0.637178i	$-0.780102\pi$
$647$	−25367.7	−1.54143	−0.770717	+	0.637178i	$0.780102\pi$
$648$	0	0
$649$	19328.1	1.16902
$650$	2364.80	0.142700
$651$	0	0
$652$	−31044.6	−1.86472
$653$	−21633.2	−1.29644	−0.648218	−	0.761454i	$-0.724486\pi$
$653$	−21633.2	−1.29644	−0.648218	+	0.761454i	$0.724486\pi$
$654$	0	0
$655$	13925.1	0.830687
$656$	3023.46	0.179948
$657$	0	0
$658$	−5788.98	−0.342976
$659$	18312.4	1.08248	0.541238	−	0.840870i	$-0.317956\pi$
$659$	18312.4	1.08248	0.541238	+	0.840870i	$0.317956\pi$
$660$	0	0
$661$	−5526.08	−0.325174	−0.162587	−	0.986694i	$-0.551984\pi$
$661$	−5526.08	−0.325174	−0.162587	+	0.986694i	$0.551984\pi$
$662$	−13121.0	−0.770337
$663$	0	0
$664$	−8030.48	−0.469342
$665$	−1036.91	−0.0604656
$666$	0	0
$667$	31758.4	1.84361
$668$	−2245.84	−0.130081
$669$	0	0
$670$	16905.1	0.974775
$671$	3204.48	0.184363
$672$	0	0
$673$	1437.24	0.0823204	0.0411602	−	0.999153i	$-0.486895\pi$
$673$	1437.24	0.0823204	0.0411602	+	0.999153i	$0.486895\pi$
$674$	44136.6	2.52237
$675$	0	0
$676$	−20765.7	−1.18148
$677$	−23405.9	−1.32875	−0.664373	−	0.747401i	$-0.731301\pi$
$677$	−23405.9	−1.32875	−0.664373	+	0.747401i	$0.731301\pi$
$678$	0	0
$679$	1165.45	0.0658701
$680$	−5965.28	−0.336409
$681$	0	0
$682$	75797.4	4.25577
$683$	19227.4	1.07719	0.538593	−	0.842566i	$-0.318956\pi$
$683$	19227.4	1.07719	0.538593	+	0.842566i	$0.318956\pi$
$684$	0	0
$685$	−12447.1	−0.694274
$686$	−14959.2	−0.832570
$687$	0	0
$688$	7811.17	0.432846
$689$	4554.50	0.251833
$690$	0	0
$691$	−35284.8	−1.94254	−0.971271	−	0.237975i	$-0.923516\pi$
$691$	−35284.8	−1.94254	−0.971271	+	0.237975i	$0.923516\pi$
$692$	−17888.8	−0.982703
$693$	0	0
$694$	−53582.8	−2.93080
$695$	8930.26	0.487401
$696$	0	0
$697$	11687.6	0.635149
$698$	35205.1	1.90907
$699$	0	0
$700$	1509.90	0.0815270
$701$	9173.00	0.494236	0.247118	−	0.968985i	$-0.420516\pi$
$701$	9173.00	0.494236	0.247118	+	0.968985i	$0.420516\pi$
$702$	0	0
$703$	−10637.4	−0.570695
$704$	−48384.9	−2.59030
$705$	0	0
$706$	3305.09	0.176188
$707$	6837.63	0.363728
$708$	0	0
$709$	−33951.6	−1.79842	−0.899210	−	0.437517i	$-0.855858\pi$
$709$	−33951.6	−1.79842	−0.899210	+	0.437517i	$0.855858\pi$
$710$	−20164.2	−1.06584
$711$	0	0
$712$	10427.8	0.548876
$713$	−42075.3	−2.21001
$714$	0	0
$715$	6183.35	0.323418
$716$	−37282.1	−1.94595
$717$	0	0
$718$	−51569.9	−2.68046
$719$	−6727.90	−0.348968	−0.174484	−	0.984660i	$-0.555826\pi$
$719$	−6727.90	−0.348968	−0.174484	+	0.984660i	$0.555826\pi$
$720$	0	0
$721$	−8111.17	−0.418968
$722$	−23158.8	−1.19374
$723$	0	0
$724$	53638.0	2.75337
$725$	−5501.45	−0.281819
$726$	0	0
$727$	−36726.1	−1.87359	−0.936793	−	0.349885i	$-0.886221\pi$
$727$	−36726.1	−1.87359	−0.936793	+	0.349885i	$0.886221\pi$
$728$	1867.83	0.0950912
$729$	0	0
$730$	19323.0	0.979694
$731$	30195.1	1.52778
$732$	0	0
$733$	26691.4	1.34498	0.672489	−	0.740108i	$-0.265225\pi$
$733$	26691.4	1.34498	0.672489	+	0.740108i	$0.265225\pi$
$734$	−5121.91	−0.257566
$735$	0	0
$736$	31469.5	1.57606
$737$	44202.4	2.20925
$738$	0	0
$739$	12207.0	0.607634	0.303817	−	0.952730i	$-0.401739\pi$
$739$	12207.0	0.607634	0.303817	+	0.952730i	$0.401739\pi$
$740$	15489.8	0.769480
$741$	0	0
$742$	4865.25	0.240713
$743$	−12473.3	−0.615882	−0.307941	−	0.951405i	$-0.599640\pi$
$743$	−12473.3	−0.615882	−0.307941	+	0.951405i	$0.599640\pi$
$744$	0	0
$745$	7844.16	0.385755
$746$	−20526.4	−1.00741
$747$	0	0
$748$	−47707.4	−2.33202
$749$	4871.82	0.237667
$750$	0	0
$751$	−15102.6	−0.733825	−0.366913	−	0.930255i	$-0.619585\pi$
$751$	−15102.6	−0.733825	−0.366913	+	0.930255i	$0.619585\pi$
$752$	4549.99	0.220640
$753$	0	0
$754$	−20815.7	−1.00539
$755$	−2191.63	−0.105645
$756$	0	0
$757$	−3418.34	−0.164124	−0.0820618	−	0.996627i	$-0.526150\pi$
$757$	−3418.34	−0.164124	−0.0820618	+	0.996627i	$0.526150\pi$
$758$	−17786.1	−0.852268
$759$	0	0
$760$	−3536.39	−0.168787
$761$	−12684.5	−0.604224	−0.302112	−	0.953272i	$-0.597692\pi$
$761$	−12684.5	−0.604224	−0.302112	+	0.953272i	$0.597692\pi$
$762$	0	0
$763$	8588.45	0.407500
$764$	14351.7	0.679614
$765$	0	0
$766$	34730.6	1.63821
$767$	7032.25	0.331056
$768$	0	0
$769$	−27580.2	−1.29333	−0.646663	−	0.762776i	$-0.723836\pi$
$769$	−27580.2	−1.29333	−0.646663	+	0.762776i	$0.723836\pi$
$770$	6605.22	0.309137
$771$	0	0
$772$	10972.8	0.511555
$773$	17386.0	0.808966	0.404483	−	0.914546i	$-0.367452\pi$
$773$	17386.0	0.808966	0.404483	+	0.914546i	$0.367452\pi$
$774$	0	0
$775$	7288.63	0.337826
$776$	3974.77	0.183874
$777$	0	0
$778$	38015.7	1.75183
$779$	6928.72	0.318674
$780$	0	0
$781$	−52724.1	−2.41564
$782$	44306.7	2.02609
$783$	0	0
$784$	5648.84	0.257327
$785$	−223.739	−0.0101727
$786$	0	0
$787$	4680.29	0.211988	0.105994	−	0.994367i	$-0.466198\pi$
$787$	4680.29	0.211988	0.105994	+	0.994367i	$0.466198\pi$
$788$	−14036.0	−0.634531
$789$	0	0
$790$	4614.58	0.207822
$791$	−59.0498	−0.00265432
$792$	0	0
$793$	1165.91	0.0522100
$794$	−694.456	−0.0310395
$795$	0	0
$796$	−9982.00	−0.444476
$797$	7278.62	0.323490	0.161745	−	0.986833i	$-0.448288\pi$
$797$	7278.62	0.323490	0.161745	+	0.986833i	$0.448288\pi$
$798$	0	0
$799$	17588.6	0.778773
$800$	−5451.40	−0.240920
$801$	0	0
$802$	26461.8	1.16508
$803$	50524.7	2.22039
$804$	0	0
$805$	−3666.58	−0.160534
$806$	27577.9	1.20520
$807$	0	0
$808$	23319.8	1.01533
$809$	29212.5	1.26954	0.634770	−	0.772701i	$-0.281095\pi$
$809$	29212.5	1.26954	0.634770	+	0.772701i	$0.281095\pi$
$810$	0	0
$811$	41992.4	1.81819	0.909094	−	0.416590i	$-0.136775\pi$
$811$	41992.4	1.81819	0.909094	+	0.416590i	$0.136775\pi$
$812$	−13290.6	−0.574396
$813$	0	0
$814$	67761.6	2.91774
$815$	−13059.2	−0.561281
$816$	0	0
$817$	17900.5	0.766536
$818$	63151.2	2.69930
$819$	0	0
$820$	−10089.3	−0.429675
$821$	−8722.99	−0.370809	−0.185405	−	0.982662i	$-0.559360\pi$
$821$	−8722.99	−0.370809	−0.185405	+	0.982662i	$0.559360\pi$
$822$	0	0
$823$	13584.8	0.575379	0.287690	−	0.957724i	$-0.407113\pi$
$823$	13584.8	0.575379	0.287690	+	0.957724i	$0.407113\pi$
$824$	−27663.2	−1.16953
$825$	0	0
$826$	7512.05	0.316438
$827$	26573.6	1.11736	0.558679	−	0.829384i	$-0.311308\pi$
$827$	26573.6	1.11736	0.558679	+	0.829384i	$0.311308\pi$
$828$	0	0
$829$	−43238.4	−1.81150	−0.905748	−	0.423816i	$-0.860690\pi$
$829$	−43238.4	−1.81150	−0.905748	+	0.423816i	$0.860690\pi$
$830$	−10332.3	−0.432097
$831$	0	0
$832$	−17604.2	−0.733552
$833$	21836.3	0.908265
$834$	0	0
$835$	−944.733	−0.0391543
$836$	−28282.3	−1.17005
$837$	0	0
$838$	42918.2	1.76919
$839$	−4093.21	−0.168431	−0.0842154	−	0.996448i	$-0.526838\pi$
$839$	−4093.21	−0.168431	−0.0842154	+	0.996448i	$0.526838\pi$
$840$	0	0
$841$	24036.5	0.985546
$842$	−6850.75	−0.280395
$843$	0	0
$844$	−30780.8	−1.25535
$845$	−8735.27	−0.355624
$846$	0	0
$847$	10507.8	0.426273
$848$	−3823.96	−0.154853
$849$	0	0
$850$	−7675.16	−0.309713
$851$	−37614.7	−1.51517
$852$	0	0
$853$	−20201.5	−0.810886	−0.405443	−	0.914120i	$-0.632883\pi$
$853$	−20201.5	−0.810886	−0.405443	+	0.914120i	$0.632883\pi$
$854$	1245.45	0.0499046
$855$	0	0
$856$	16615.4	0.663436
$857$	4551.65	0.181425	0.0907126	−	0.995877i	$-0.471086\pi$
$857$	4551.65	0.181425	0.0907126	+	0.995877i	$0.471086\pi$
$858$	0	0
$859$	11962.6	0.475154	0.237577	−	0.971369i	$-0.423647\pi$
$859$	11962.6	0.475154	0.237577	+	0.971369i	$0.423647\pi$
$860$	−26065.9	−1.03354
$861$	0	0
$862$	51649.3	2.04081
$863$	−7164.61	−0.282603	−0.141301	−	0.989967i	$-0.545129\pi$
$863$	−7164.61	−0.282603	−0.141301	+	0.989967i	$0.545129\pi$
$864$	0	0
$865$	−7525.10	−0.295793
$866$	66412.0	2.60597
$867$	0	0
$868$	17608.2	0.688549
$869$	12065.9	0.471012
$870$	0	0
$871$	16082.4	0.625640
$872$	29291.0	1.13752
$873$	0	0
$874$	26266.2	1.01655
$875$	635.154	0.0245396
$876$	0	0
$877$	−19218.7	−0.739987	−0.369994	−	0.929034i	$-0.620640\pi$
$877$	−19218.7	−0.739987	−0.369994	+	0.929034i	$0.620640\pi$
$878$	−7324.56	−0.281540
$879$	0	0
$880$	−5191.53	−0.198871
$881$	−45914.5	−1.75585	−0.877923	−	0.478802i	$-0.841071\pi$
$881$	−45914.5	−1.75585	−0.877923	+	0.478802i	$0.841071\pi$
$882$	0	0
$883$	−44656.7	−1.70194	−0.850972	−	0.525211i	$-0.823986\pi$
$883$	−44656.7	−1.70194	−0.850972	+	0.525211i	$0.823986\pi$
$884$	−17357.7	−0.660410
$885$	0	0
$886$	17464.3	0.662218
$887$	14975.2	0.566873	0.283437	−	0.958991i	$-0.408525\pi$
$887$	14975.2	0.566873	0.283437	+	0.958991i	$0.408525\pi$
$888$	0	0
$889$	1570.82	0.0592616
$890$	13416.9	0.505319
$891$	0	0
$892$	−49692.9	−1.86529
$893$	10427.0	0.390735
$894$	0	0
$895$	−15683.1	−0.585729
$896$	−9941.33	−0.370666
$897$	0	0
$898$	−17771.7	−0.660413
$899$	−64156.8	−2.38015
$900$	0	0
$901$	−14782.0	−0.546571
$902$	−44136.7	−1.62926
$903$	0	0
$904$	−201.390	−0.00740943
$905$	22563.3	0.828763
$906$	0	0
$907$	−14818.1	−0.542479	−0.271240	−	0.962512i	$-0.587434\pi$
$907$	−14818.1	−0.542479	−0.271240	+	0.962512i	$0.587434\pi$
$908$	−30935.6	−1.13065
$909$	0	0
$910$	2403.22	0.0875451
$911$	21846.5	0.794518	0.397259	−	0.917707i	$-0.369961\pi$
$911$	21846.5	0.794518	0.397259	+	0.917707i	$0.369961\pi$
$912$	0	0
$913$	−27016.4	−0.979312
$914$	−63224.2	−2.28804
$915$	0	0
$916$	−21934.0	−0.791179
$917$	14151.4	0.509618
$918$	0	0
$919$	28878.9	1.03659	0.518296	−	0.855201i	$-0.326566\pi$
$919$	28878.9	1.03659	0.518296	+	0.855201i	$0.326566\pi$
$920$	−12504.9	−0.448124
$921$	0	0
$922$	−73108.4	−2.61139
$923$	−19182.9	−0.684089
$924$	0	0
$925$	6515.92	0.231613
$926$	14803.4	0.525344
$927$	0	0
$928$	47985.0	1.69740
$929$	4608.79	0.162766	0.0813829	−	0.996683i	$-0.474066\pi$
$929$	4608.79	0.162766	0.0813829	+	0.996683i	$0.474066\pi$
$930$	0	0
$931$	12945.2	0.455705
$932$	2860.23	0.100526
$933$	0	0
$934$	−11281.3	−0.395220
$935$	−20068.6	−0.701938
$936$	0	0
$937$	−21063.8	−0.734391	−0.367195	−	0.930144i	$-0.619682\pi$
$937$	−21063.8	−0.734391	−0.367195	+	0.930144i	$0.619682\pi$
$938$	17179.7	0.598014
$939$	0	0
$940$	−15183.3	−0.526836
$941$	−20244.8	−0.701339	−0.350670	−	0.936499i	$-0.614046\pi$
$941$	−20244.8	−0.701339	−0.350670	+	0.936499i	$0.614046\pi$
$942$	0	0
$943$	24500.4	0.846069
$944$	−5904.27	−0.203567
$945$	0	0
$946$	−114028.	−3.91901
$947$	−29978.3	−1.02868	−0.514342	−	0.857585i	$-0.671964\pi$
$947$	−29978.3	−1.02868	−0.514342	+	0.857585i	$0.671964\pi$
$948$	0	0
$949$	18382.7	0.628797
$950$	−4550.05	−0.155393
$951$	0	0
$952$	−6062.20	−0.206383
$953$	5782.65	0.196556	0.0982782	−	0.995159i	$-0.468666\pi$
$953$	5782.65	0.196556	0.0982782	+	0.995159i	$0.468666\pi$
$954$	0	0
$955$	6037.17	0.204564
$956$	30519.2	1.03249
$957$	0	0
$958$	−38559.2	−1.30041
$959$	−12649.3	−0.425930
$960$	0	0
$961$	55207.7	1.85317
$962$	24654.2	0.826281
$963$	0	0
$964$	47401.6	1.58372
$965$	4615.82	0.153978
$966$	0	0
$967$	−26119.2	−0.868600	−0.434300	−	0.900768i	$-0.643004\pi$
$967$	−26119.2	−0.868600	−0.434300	+	0.900768i	$0.643004\pi$
$968$	35837.0	1.18992
$969$	0	0
$970$	5114.09	0.169282
$971$	5101.93	0.168619	0.0843093	−	0.996440i	$-0.473132\pi$
$971$	5101.93	0.168619	0.0843093	+	0.996440i	$0.473132\pi$
$972$	0	0
$973$	9075.34	0.299016
$974$	−68013.1	−2.23746
$975$	0	0
$976$	−978.894	−0.0321041
$977$	−45902.4	−1.50312	−0.751559	−	0.659666i	$-0.770698\pi$
$977$	−45902.4	−1.50312	−0.751559	+	0.659666i	$0.770698\pi$
$978$	0	0
$979$	35081.6	1.14526
$980$	−18850.2	−0.614437
$981$	0	0
$982$	−7879.14	−0.256042
$983$	13416.5	0.435321	0.217661	−	0.976025i	$-0.430157\pi$
$983$	13416.5	0.435321	0.217661	+	0.976025i	$0.430157\pi$
$984$	0	0
$985$	−5904.37	−0.190994
$986$	67559.2	2.18207
$987$	0	0
$988$	−10290.1	−0.331349
$989$	63297.4	2.03513
$990$	0	0
$991$	3806.66	0.122021	0.0610104	−	0.998137i	$-0.480568\pi$
$991$	3806.66	0.122021	0.0610104	+	0.998137i	$0.480568\pi$
$992$	−63573.2	−2.03473
$993$	0	0
$994$	−20491.8	−0.653883
$995$	−4199.02	−0.133787
$996$	0	0
$997$	−22523.8	−0.715483	−0.357742	−	0.933821i	$-0.616453\pi$
$997$	−22523.8	−0.715483	−0.357742	+	0.933821i	$0.616453\pi$
$998$	52325.7	1.65966
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.4.a.f.1.3	✓	3
3.2	odd	2		135.4.a.g.1.1	yes	3
4.3	odd	2		2160.4.a.bm.1.3		3
5.2	odd	4		675.4.b.l.649.5		6
5.3	odd	4		675.4.b.l.649.2		6
5.4	even	2		675.4.a.r.1.1		3
9.2	odd	6		405.4.e.r.271.3		6
9.4	even	3		405.4.e.t.136.1		6
9.5	odd	6		405.4.e.r.136.3		6
9.7	even	3		405.4.e.t.271.1		6
12.11	even	2		2160.4.a.be.1.3		3
15.2	even	4		675.4.b.k.649.2		6
15.8	even	4		675.4.b.k.649.5		6
15.14	odd	2		675.4.a.q.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.a.f.1.3	✓	3	1.1	even	1	trivial
135.4.a.g.1.1	yes	3	3.2	odd	2
405.4.e.r.136.3		6	9.5	odd	6
405.4.e.r.271.3		6	9.2	odd	6
405.4.e.t.136.1		6	9.4	even	3
405.4.e.t.271.1		6	9.7	even	3
675.4.a.q.1.3		3	15.14	odd	2
675.4.a.r.1.1		3	5.4	even	2
675.4.b.k.649.2		6	15.2	even	4
675.4.b.k.649.5		6	15.8	even	4
675.4.b.l.649.2		6	5.3	odd	4
675.4.b.l.649.5		6	5.2	odd	4
2160.4.a.be.1.3		3	12.11	even	2
2160.4.a.bm.1.3		3	4.3	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:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	135.a (trivial)

\(f(q)\)	\(=\)	\(q+4.45938 q^{2} +11.8861 q^{4} +5.00000 q^{5} +5.08123 q^{7} +17.3296 q^{8} +O(q^{10})\)	\(q+4.45938 q^{2} +11.8861 q^{4} +5.00000 q^{5} +5.08123 q^{7} +17.3296 q^{8} +22.2969 q^{10} +58.3007 q^{11} +21.2119 q^{13} +22.6592 q^{14} -17.8095 q^{16} -68.8451 q^{17} -40.8133 q^{19} +59.4305 q^{20} +259.985 q^{22} -144.318 q^{23} +25.0000 q^{25} +94.5921 q^{26} +60.3960 q^{28} -220.058 q^{29} +291.545 q^{31} -218.056 q^{32} -307.006 q^{34} +25.4062 q^{35} +260.637 q^{37} -182.002 q^{38} +86.6479 q^{40} -169.766 q^{41} -438.596 q^{43} +692.967 q^{44} -643.571 q^{46} -255.481 q^{47} -317.181 q^{49} +111.485 q^{50} +252.127 q^{52} +214.714 q^{53} +291.503 q^{55} +88.0557 q^{56} -981.322 q^{58} +331.524 q^{59} +54.9647 q^{61} +1300.11 q^{62} -829.920 q^{64} +106.060 q^{65} +758.179 q^{67} -818.299 q^{68} +113.296 q^{70} -904.348 q^{71} +866.622 q^{73} +1162.28 q^{74} -485.110 q^{76} +296.239 q^{77} +206.961 q^{79} -89.0475 q^{80} -757.054 q^{82} -463.397 q^{83} -344.225 q^{85} -1955.87 q^{86} +1010.33 q^{88} +601.736 q^{89} +107.783 q^{91} -1715.38 q^{92} -1139.29 q^{94} -204.066 q^{95} +229.363 q^{97} -1414.43 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 23 q^{4} + 15 q^{5} + 44 q^{7} - 36 q^{8}+O(q^{10}) \) 3 * q - q^2 + 23 * q^4 + 15 * q^5 + 44 * q^7 - 36 * q^8	\( 3 q - q^{2} + 23 q^{4} + 15 q^{5} + 44 q^{7} - 36 q^{8} - 5 q^{10} + 38 q^{11} + 28 q^{13} - 108 q^{14} + 191 q^{16} - 19 q^{17} + 187 q^{19} + 115 q^{20} + 122 q^{22} - 81 q^{23} + 75 q^{25} + 416 q^{26} + 410 q^{28} + 160 q^{29} + 227 q^{31} - 569 q^{32} + 17 q^{34} + 220 q^{35} + 78 q^{37} - 757 q^{38} - 180 q^{40} - 338 q^{41} + 22 q^{43} + 1636 q^{44} - 1425 q^{46} - 472 q^{47} - 197 q^{49} - 25 q^{50} - 1566 q^{52} + 521 q^{53} + 190 q^{55} - 1254 q^{56} - 2096 q^{58} + 140 q^{59} + 595 q^{61} + 1407 q^{62} - 918 q^{64} + 140 q^{65} + 878 q^{67} - 3053 q^{68} - 540 q^{70} - 602 q^{71} + 1294 q^{73} + 2878 q^{74} + 525 q^{76} + 288 q^{77} + 629 q^{79} + 955 q^{80} - 1682 q^{82} - 1287 q^{83} - 95 q^{85} - 3730 q^{86} - 858 q^{88} + 2154 q^{89} - 440 q^{91} + 1959 q^{92} - 1108 q^{94} + 935 q^{95} + 1392 q^{97} - 2693 q^{98}+O(q^{100}) \) 3 * q - q^2 + 23 * q^4 + 15 * q^5 + 44 * q^7 - 36 * q^8 - 5 * q^10 + 38 * q^11 + 28 * q^13 - 108 * q^14 + 191 * q^16 - 19 * q^17 + 187 * q^19 + 115 * q^20 + 122 * q^22 - 81 * q^23 + 75 * q^25 + 416 * q^26 + 410 * q^28 + 160 * q^29 + 227 * q^31 - 569 * q^32 + 17 * q^34 + 220 * q^35 + 78 * q^37 - 757 * q^38 - 180 * q^40 - 338 * q^41 + 22 * q^43 + 1636 * q^44 - 1425 * q^46 - 472 * q^47 - 197 * q^49 - 25 * q^50 - 1566 * q^52 + 521 * q^53 + 190 * q^55 - 1254 * q^56 - 2096 * q^58 + 140 * q^59 + 595 * q^61 + 1407 * q^62 - 918 * q^64 + 140 * q^65 + 878 * q^67 - 3053 * q^68 - 540 * q^70 - 602 * q^71 + 1294 * q^73 + 2878 * q^74 + 525 * q^76 + 288 * q^77 + 629 * q^79 + 955 * q^80 - 1682 * q^82 - 1287 * q^83 - 95 * q^85 - 3730 * q^86 - 858 * q^88 + 2154 * q^89 - 440 * q^91 + 1959 * q^92 - 1108 * q^94 + 935 * q^95 + 1392 * q^97 - 2693 * q^98

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