LMFDB - Embedded newform 567.4.a.i.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [567,4,Mod(1,567)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(567, 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("567.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 567 = 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	567.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:	$33.4540829733$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 49x^{6} + 138x^{5} + 708x^{4} - 1941x^{3} - 2506x^{2} + 8592x - 4616 $ x^8 - 3x^7 - 49x^6 + 138x^5 + 708x^4 - 1941x^3 - 2506x^2 + 8592*x - 4616
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 63)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.93948$ of defining polynomial
Character	$\chi$	$=$	567.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.93948 q^{2} +0.640533 q^{4} -2.56885 q^{5} -7.00000 q^{7} +21.6330 q^{8} +O(q^{10})$	\(q-2.93948 q^{2} +0.640533 q^{4} -2.56885 q^{5} -7.00000 q^{7} +21.6330 q^{8} +7.55109 q^{10} -0.514272 q^{11} +65.4434 q^{13} +20.5763 q^{14} -68.7140 q^{16} -3.37738 q^{17} -123.871 q^{19} -1.64544 q^{20} +1.51169 q^{22} -25.1184 q^{23} -118.401 q^{25} -192.369 q^{26} -4.48373 q^{28} +237.463 q^{29} -89.9139 q^{31} +28.9193 q^{32} +9.92772 q^{34} +17.9820 q^{35} -67.4721 q^{37} +364.115 q^{38} -55.5720 q^{40} +287.316 q^{41} -435.500 q^{43} -0.329409 q^{44} +73.8349 q^{46} +54.0547 q^{47} +49.0000 q^{49} +348.037 q^{50} +41.9186 q^{52} +272.147 q^{53} +1.32109 q^{55} -151.431 q^{56} -698.017 q^{58} -516.496 q^{59} +251.711 q^{61} +264.300 q^{62} +464.704 q^{64} -168.114 q^{65} +461.750 q^{67} -2.16332 q^{68} -52.8576 q^{70} +532.933 q^{71} -360.888 q^{73} +198.333 q^{74} -79.3432 q^{76} +3.59991 q^{77} +762.674 q^{79} +176.516 q^{80} -844.559 q^{82} -944.843 q^{83} +8.67598 q^{85} +1280.14 q^{86} -11.1253 q^{88} +1494.37 q^{89} -458.104 q^{91} -16.0891 q^{92} -158.893 q^{94} +318.205 q^{95} +1345.58 q^{97} -144.034 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 3 q^{2} + 43 q^{4} + 30 q^{5} - 56 q^{7} + 6 q^{8}+O(q^{10}) $ 8 * q + 3 * q^2 + 43 * q^4 + 30 * q^5 - 56 * q^7 + 6 * q^8	$ 8 q + 3 q^{2} + 43 q^{4} + 30 q^{5} - 56 q^{7} + 6 q^{8} - 14 q^{10} + 24 q^{11} + 68 q^{13} - 21 q^{14} + 103 q^{16} + 168 q^{17} + 176 q^{19} + 330 q^{20} + 151 q^{22} + 228 q^{23} + 244 q^{25} + 795 q^{26} - 301 q^{28} + 618 q^{29} + 72 q^{31} + 786 q^{32} - 261 q^{34} - 210 q^{35} + 210 q^{37} + 1032 q^{38} - 375 q^{40} + 420 q^{41} - 2 q^{43} + 387 q^{44} + 402 q^{46} + 570 q^{47} + 392 q^{49} + 1110 q^{50} - 431 q^{52} + 528 q^{53} - 838 q^{55} - 42 q^{56} + 37 q^{58} - 150 q^{59} + 578 q^{61} + 1170 q^{62} - 112 q^{64} - 366 q^{65} - 898 q^{67} + 2526 q^{68} + 98 q^{70} + 882 q^{71} + 972 q^{73} - 222 q^{74} + 1423 q^{76} - 168 q^{77} - 158 q^{79} + 2475 q^{80} - 211 q^{82} + 2958 q^{83} - 774 q^{85} - 114 q^{86} + 1317 q^{88} + 4380 q^{89} - 476 q^{91} + 4629 q^{92} - 3234 q^{94} + 930 q^{95} - 60 q^{97} + 147 q^{98}+O(q^{100}) $ 8 * q + 3 * q^2 + 43 * q^4 + 30 * q^5 - 56 * q^7 + 6 * q^8 - 14 * q^10 + 24 * q^11 + 68 * q^13 - 21 * q^14 + 103 * q^16 + 168 * q^17 + 176 * q^19 + 330 * q^20 + 151 * q^22 + 228 * q^23 + 244 * q^25 + 795 * q^26 - 301 * q^28 + 618 * q^29 + 72 * q^31 + 786 * q^32 - 261 * q^34 - 210 * q^35 + 210 * q^37 + 1032 * q^38 - 375 * q^40 + 420 * q^41 - 2 * q^43 + 387 * q^44 + 402 * q^46 + 570 * q^47 + 392 * q^49 + 1110 * q^50 - 431 * q^52 + 528 * q^53 - 838 * q^55 - 42 * q^56 + 37 * q^58 - 150 * q^59 + 578 * q^61 + 1170 * q^62 - 112 * q^64 - 366 * q^65 - 898 * q^67 + 2526 * q^68 + 98 * q^70 + 882 * q^71 + 972 * q^73 - 222 * q^74 + 1423 * q^76 - 168 * q^77 - 158 * q^79 + 2475 * q^80 - 211 * q^82 + 2958 * q^83 - 774 * q^85 - 114 * q^86 + 1317 * q^88 + 4380 * q^89 - 476 * q^91 + 4629 * q^92 - 3234 * q^94 + 930 * q^95 - 60 * q^97 + 147 * 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 $

$2$	−2.93948	−1.03926	−0.519631	−	0.854391i	$-0.673931\pi$
$2$	−2.93948	−1.03926	−0.519631	+	0.854391i	$0.673931\pi$
$3$	0	0
$3$	0	0
$4$	0.640533	0.0800666
$5$	−2.56885	−0.229765	−0.114883	−	0.993379i	$-0.536649\pi$
$5$	−2.56885	−0.229765	−0.114883	+	0.993379i	$0.536649\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	21.6330	0.956052
$9$	0	0
$10$	7.55109	0.238786
$11$	−0.514272	−0.0140963	−0.00704814	−	0.999975i	$-0.502244\pi$
$11$	−0.514272	−0.0140963	−0.00704814	+	0.999975i	$0.502244\pi$
$12$	0	0
$13$	65.4434	1.39621	0.698105	−	0.715996i	$-0.254027\pi$
$13$	65.4434	1.39621	0.698105	+	0.715996i	$0.254027\pi$
$14$	20.5763	0.392804
$15$	0	0
$16$	−68.7140	−1.07366
$17$	−3.37738	−0.0481843	−0.0240922	−	0.999710i	$-0.507670\pi$
$17$	−3.37738	−0.0481843	−0.0240922	+	0.999710i	$0.507670\pi$
$18$	0	0
$19$	−123.871	−1.49568	−0.747838	−	0.663881i	$-0.768908\pi$
$19$	−123.871	−1.49568	−0.747838	+	0.663881i	$0.768908\pi$
$20$	−1.64544	−0.0183965
$21$	0	0
$22$	1.51169	0.0146497
$23$	−25.1184	−0.227719	−0.113860	−	0.993497i	$-0.536321\pi$
$23$	−25.1184	−0.227719	−0.113860	+	0.993497i	$0.536321\pi$
$24$	0	0
$25$	−118.401	−0.947208
$26$	−192.369	−1.45103
$27$	0	0
$28$	−4.48373	−0.0302623
$29$	237.463	1.52054	0.760272	−	0.649605i	$-0.225066\pi$
$29$	237.463	1.52054	0.760272	+	0.649605i	$0.225066\pi$
$30$	0	0
$31$	−89.9139	−0.520936	−0.260468	−	0.965482i	$-0.583877\pi$
$31$	−89.9139	−0.520936	−0.260468	+	0.965482i	$0.583877\pi$
$32$	28.9193	0.159758
$33$	0	0
$34$	9.92772	0.0500762
$35$	17.9820	0.0868431
$36$	0	0
$37$	−67.4721	−0.299793	−0.149897	−	0.988702i	$-0.547894\pi$
$37$	−67.4721	−0.299793	−0.149897	+	0.988702i	$0.547894\pi$
$38$	364.115	1.55440
$39$	0	0
$40$	−55.5720	−0.219667
$41$	287.316	1.09442	0.547210	−	0.836995i	$-0.315690\pi$
$41$	287.316	1.09442	0.547210	+	0.836995i	$0.315690\pi$
$42$	0	0
$43$	−435.500	−1.54449	−0.772245	−	0.635325i	$-0.780866\pi$
$43$	−435.500	−1.54449	−0.772245	+	0.635325i	$0.780866\pi$
$44$	−0.329409	−0.00112864
$45$	0	0
$46$	73.8349	0.236660
$47$	54.0547	0.167759	0.0838797	−	0.996476i	$-0.473269\pi$
$47$	54.0547	0.167759	0.0838797	+	0.996476i	$0.473269\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	348.037	0.984398
$51$	0	0
$52$	41.9186	0.111790
$53$	272.147	0.705325	0.352662	−	0.935751i	$-0.385276\pi$
$53$	272.147	0.705325	0.352662	+	0.935751i	$0.385276\pi$
$54$	0	0
$55$	1.32109	0.00323883
$56$	−151.431	−0.361354
$57$	0	0
$58$	−698.017	−1.58024
$59$	−516.496	−1.13970	−0.569848	−	0.821750i	$-0.692998\pi$
$59$	−516.496	−1.13970	−0.569848	+	0.821750i	$0.692998\pi$
$60$	0	0
$61$	251.711	0.528332	0.264166	−	0.964477i	$-0.414903\pi$
$61$	251.711	0.528332	0.264166	+	0.964477i	$0.414903\pi$
$62$	264.300	0.541389
$63$	0	0
$64$	464.704	0.907625
$65$	−168.114	−0.320800
$66$	0	0
$67$	461.750	0.841967	0.420984	−	0.907068i	$-0.361685\pi$
$67$	461.750	0.841967	0.420984	+	0.907068i	$0.361685\pi$
$68$	−2.16332	−0.00385796
$69$	0	0
$70$	−52.8576	−0.0902527
$71$	532.933	0.890809	0.445405	−	0.895329i	$-0.353060\pi$
$71$	532.933	0.890809	0.445405	+	0.895329i	$0.353060\pi$
$72$	0	0
$73$	−360.888	−0.578613	−0.289306	−	0.957237i	$-0.593425\pi$
$73$	−360.888	−0.578613	−0.289306	+	0.957237i	$0.593425\pi$
$74$	198.333	0.311564
$75$	0	0
$76$	−79.3432	−0.119754
$77$	3.59991	0.00532789
$78$	0	0
$79$	762.674	1.08617	0.543086	−	0.839677i	$-0.317256\pi$
$79$	762.674	1.08617	0.543086	+	0.839677i	$0.317256\pi$
$80$	176.516	0.246689
$81$	0	0
$82$	−844.559	−1.13739
$83$	−944.843	−1.24952	−0.624759	−	0.780818i	$-0.714803\pi$
$83$	−944.843	−1.24952	−0.624759	+	0.780818i	$0.714803\pi$
$84$	0	0
$85$	8.67598	0.0110711
$86$	1280.14	1.60513
$87$	0	0
$88$	−11.1253	−0.0134768
$89$	1494.37	1.77981	0.889906	−	0.456145i	$-0.150770\pi$
$89$	1494.37	1.77981	0.889906	+	0.456145i	$0.150770\pi$
$90$	0	0
$91$	−458.104	−0.527718
$92$	−16.0891	−0.0182327
$93$	0	0
$94$	−158.893	−0.174346
$95$	318.205	0.343654
$96$	0	0
$97$	1345.58	1.40849	0.704243	−	0.709959i	$-0.251287\pi$
$97$	1345.58	1.40849	0.704243	+	0.709959i	$0.251287\pi$
$98$	−144.034	−0.148466
$99$	0	0
$100$	−75.8398	−0.0758398
$101$	1779.71	1.75334	0.876670	−	0.481092i	$-0.159760\pi$
$101$	1779.71	1.75334	0.876670	+	0.481092i	$0.159760\pi$
$102$	0	0
$103$	−37.3017	−0.0356839	−0.0178420	−	0.999841i	$-0.505680\pi$
$103$	−37.3017	−0.0356839	−0.0178420	+	0.999841i	$0.505680\pi$
$104$	1415.74	1.33485
$105$	0	0
$106$	−799.969	−0.733018
$107$	389.494	0.351905	0.175953	−	0.984399i	$-0.443699\pi$
$107$	389.494	0.351905	0.175953	+	0.984399i	$0.443699\pi$
$108$	0	0
$109$	115.830	0.101784	0.0508922	−	0.998704i	$-0.483794\pi$
$109$	115.830	0.101784	0.0508922	+	0.998704i	$0.483794\pi$
$110$	−3.88332	−0.00336600
$111$	0	0
$112$	480.998	0.405804
$113$	−1920.18	−1.59854	−0.799270	−	0.600971i	$-0.794781\pi$
$113$	−1920.18	−1.59854	−0.799270	+	0.600971i	$0.794781\pi$
$114$	0	0
$115$	64.5253	0.0523219
$116$	152.103	0.121745
$117$	0	0
$118$	1518.23	1.18444
$119$	23.6416	0.0182120
$120$	0	0
$121$	−1330.74	−0.999801
$122$	−739.898	−0.549076
$123$	0	0
$124$	−57.5928	−0.0417096
$125$	625.261	0.447401
$126$	0	0
$127$	2674.36	1.86859	0.934295	−	0.356502i	$-0.116031\pi$
$127$	2674.36	1.86859	0.934295	+	0.356502i	$0.116031\pi$
$128$	−1597.34	−1.10302
$129$	0	0
$130$	494.169	0.333396
$131$	1237.50	0.825348	0.412674	−	0.910879i	$-0.364595\pi$
$131$	1237.50	0.825348	0.412674	+	0.910879i	$0.364595\pi$
$132$	0	0
$133$	867.094	0.565313
$134$	−1357.31	−0.875025
$135$	0	0
$136$	−73.0627	−0.0460668
$137$	2378.33	1.48317	0.741586	−	0.670858i	$-0.234074\pi$
$137$	2378.33	1.48317	0.741586	+	0.670858i	$0.234074\pi$
$138$	0	0
$139$	1164.47	0.710569	0.355284	−	0.934758i	$-0.384384\pi$
$139$	1164.47	0.710569	0.355284	+	0.934758i	$0.384384\pi$
$140$	11.5180	0.00695323
$141$	0	0
$142$	−1566.54	−0.925785
$143$	−33.6557	−0.0196814
$144$	0	0
$145$	−610.007	−0.349368
$146$	1060.82	0.601331
$147$	0	0
$148$	−43.2181	−0.0240034
$149$	1679.39	0.923362	0.461681	−	0.887046i	$-0.347246\pi$
$149$	1679.39	0.923362	0.461681	+	0.887046i	$0.347246\pi$
$150$	0	0
$151$	864.499	0.465907	0.232953	−	0.972488i	$-0.425161\pi$
$151$	864.499	0.465907	0.232953	+	0.972488i	$0.425161\pi$
$152$	−2679.69	−1.42995
$153$	0	0
$154$	−10.5819	−0.00553708
$155$	230.976	0.119693
$156$	0	0
$157$	1723.42	0.876074	0.438037	−	0.898957i	$-0.355674\pi$
$157$	1723.42	0.876074	0.438037	+	0.898957i	$0.355674\pi$
$158$	−2241.86	−1.12882
$159$	0	0
$160$	−74.2895	−0.0367069
$161$	175.828	0.0860697
$162$	0	0
$163$	2035.48	0.978105	0.489052	−	0.872254i	$-0.337343\pi$
$163$	2035.48	0.978105	0.489052	+	0.872254i	$0.337343\pi$
$164$	184.035	0.0876266
$165$	0	0
$166$	2777.35	1.29858
$167$	1610.71	0.746351	0.373175	−	0.927761i	$-0.378269\pi$
$167$	1610.71	0.746351	0.373175	+	0.927761i	$0.378269\pi$
$168$	0	0
$169$	2085.83	0.949401
$170$	−25.5029	−0.0115058
$171$	0	0
$172$	−278.952	−0.123662
$173$	2011.71	0.884089	0.442045	−	0.896993i	$-0.354253\pi$
$173$	2011.71	0.884089	0.442045	+	0.896993i	$0.354253\pi$
$174$	0	0
$175$	828.807	0.358011
$176$	35.3377	0.0151345
$177$	0	0
$178$	−4392.68	−1.84969
$179$	−1778.67	−0.742704	−0.371352	−	0.928492i	$-0.621106\pi$
$179$	−1778.67	−0.742704	−0.371352	+	0.928492i	$0.621106\pi$
$180$	0	0
$181$	−608.294	−0.249802	−0.124901	−	0.992169i	$-0.539861\pi$
$181$	−608.294	−0.249802	−0.124901	+	0.992169i	$0.539861\pi$
$182$	1346.59	0.548437
$183$	0	0
$184$	−543.385	−0.217711
$185$	173.326	0.0688820
$186$	0	0
$187$	1.73689	0.000679220 0
$188$	34.6239	0.0134319
$189$	0	0
$190$	−935.357	−0.357147
$191$	−4394.84	−1.66492	−0.832459	−	0.554086i	$-0.813068\pi$
$191$	−4394.84	−1.66492	−0.832459	+	0.554086i	$0.813068\pi$
$192$	0	0
$193$	1186.92	0.442675	0.221337	−	0.975197i	$-0.428958\pi$
$193$	1186.92	0.442675	0.221337	+	0.975197i	$0.428958\pi$
$194$	−3955.31	−1.46379
$195$	0	0
$196$	31.3861	0.0114381
$197$	−816.364	−0.295246	−0.147623	−	0.989044i	$-0.547162\pi$
$197$	−816.364	−0.295246	−0.147623	+	0.989044i	$0.547162\pi$
$198$	0	0
$199$	−5392.46	−1.92091	−0.960455	−	0.278434i	$-0.910185\pi$
$199$	−5392.46	−1.92091	−0.960455	+	0.278434i	$0.910185\pi$
$200$	−2561.37	−0.905580
$201$	0	0
$202$	−5231.41	−1.82218
$203$	−1662.24	−0.574711
$204$	0	0
$205$	−738.073	−0.251460
$206$	109.648	0.0370850
$207$	0	0
$208$	−4496.87	−1.49905
$209$	63.7032	0.0210835
$210$	0	0
$211$	2616.37	0.853643	0.426822	−	0.904336i	$-0.359633\pi$
$211$	2616.37	0.853643	0.426822	+	0.904336i	$0.359633\pi$
$212$	174.319	0.0564730
$213$	0	0
$214$	−1144.91	−0.365722
$215$	1118.73	0.354870
$216$	0	0
$217$	629.397	0.196895
$218$	−340.479	−0.105781
$219$	0	0
$220$	0.846202	0.000259322 0
$221$	−221.027	−0.0672754
$222$	0	0
$223$	−843.898	−0.253415	−0.126708	−	0.991940i	$-0.540441\pi$
$223$	−843.898	−0.253415	−0.126708	+	0.991940i	$0.540441\pi$
$224$	−202.435	−0.0603829
$225$	0	0
$226$	5644.32	1.66130
$227$	−3349.28	−0.979292	−0.489646	−	0.871921i	$-0.662874\pi$
$227$	−3349.28	−0.979292	−0.489646	+	0.871921i	$0.662874\pi$
$228$	0	0
$229$	−690.623	−0.199291	−0.0996455	−	0.995023i	$-0.531771\pi$
$229$	−690.623	−0.199291	−0.0996455	+	0.995023i	$0.531771\pi$
$230$	−189.671	−0.0543762
$231$	0	0
$232$	5137.03	1.45372
$233$	−4203.93	−1.18201	−0.591005	−	0.806668i	$-0.701269\pi$
$233$	−4203.93	−1.18201	−0.591005	+	0.806668i	$0.701269\pi$
$234$	0	0
$235$	−138.859	−0.0385453
$236$	−330.833	−0.0912516
$237$	0	0
$238$	−69.4941	−0.0189270
$239$	4307.31	1.16576	0.582881	−	0.812558i	$-0.301926\pi$
$239$	4307.31	1.16576	0.582881	+	0.812558i	$0.301926\pi$
$240$	0	0
$241$	−5338.50	−1.42690	−0.713450	−	0.700707i	$-0.752868\pi$
$241$	−5338.50	−1.42690	−0.713450	+	0.700707i	$0.752868\pi$
$242$	3911.67	1.03906
$243$	0	0
$244$	161.229	0.0423018
$245$	−125.874	−0.0328236
$246$	0	0
$247$	−8106.51	−2.08828
$248$	−1945.11	−0.498042
$249$	0	0
$250$	−1837.94	−0.464967
$251$	2972.68	0.747546	0.373773	−	0.927520i	$-0.378064\pi$
$251$	2972.68	0.747546	0.373773	+	0.927520i	$0.378064\pi$
$252$	0	0
$253$	12.9177	0.00320999
$254$	−7861.22	−1.94195
$255$	0	0
$256$	977.720	0.238701
$257$	−5115.82	−1.24170	−0.620848	−	0.783931i	$-0.713212\pi$
$257$	−5115.82	−1.24170	−0.620848	+	0.783931i	$0.713212\pi$
$258$	0	0
$259$	472.305	0.113311
$260$	−107.683	−0.0256854
$261$	0	0
$262$	−3637.60	−0.857753
$263$	7270.03	1.70452	0.852261	−	0.523116i	$-0.175231\pi$
$263$	7270.03	1.70452	0.852261	+	0.523116i	$0.175231\pi$
$264$	0	0
$265$	−699.105	−0.162059
$266$	−2548.80	−0.587508
$267$	0	0
$268$	295.766	0.0674135
$269$	793.367	0.179823	0.0899116	−	0.995950i	$-0.471342\pi$
$269$	793.367	0.179823	0.0899116	+	0.995950i	$0.471342\pi$
$270$	0	0
$271$	7063.92	1.58340	0.791702	−	0.610907i	$-0.209195\pi$
$271$	7063.92	1.58340	0.791702	+	0.610907i	$0.209195\pi$
$272$	232.073	0.0517334
$273$	0	0
$274$	−6991.05	−1.54140
$275$	60.8904	0.0133521
$276$	0	0
$277$	7158.56	1.55277	0.776383	−	0.630261i	$-0.217052\pi$
$277$	7158.56	1.55277	0.776383	+	0.630261i	$0.217052\pi$
$278$	−3422.94	−0.738468
$279$	0	0
$280$	389.004	0.0830265
$281$	−7559.85	−1.60492	−0.802461	−	0.596705i	$-0.796476\pi$
$281$	−7559.85	−1.60492	−0.802461	+	0.596705i	$0.796476\pi$
$282$	0	0
$283$	−3796.47	−0.797443	−0.398722	−	0.917072i	$-0.630546\pi$
$283$	−3796.47	−0.797443	−0.398722	+	0.917072i	$0.630546\pi$
$284$	341.361	0.0713241
$285$	0	0
$286$	98.9303	0.0204541
$287$	−2011.21	−0.413652
$288$	0	0
$289$	−4901.59	−0.997678
$290$	1793.10	0.363085
$291$	0	0
$292$	−231.161	−0.0463276
$293$	602.803	0.120192	0.0600958	−	0.998193i	$-0.480859\pi$
$293$	602.803	0.120192	0.0600958	+	0.998193i	$0.480859\pi$
$294$	0	0
$295$	1326.80	0.261862
$296$	−1459.62	−0.286618
$297$	0	0
$298$	−4936.53	−0.959616
$299$	−1643.83	−0.317944
$300$	0	0
$301$	3048.50	0.583763
$302$	−2541.18	−0.484200
$303$	0	0
$304$	8511.64	1.60584
$305$	−646.608	−0.121392
$306$	0	0
$307$	9215.96	1.71330	0.856649	−	0.515900i	$-0.172542\pi$
$307$	9215.96	1.71330	0.856649	+	0.515900i	$0.172542\pi$
$308$	2.30586	0.000426586 0
$309$	0	0
$310$	−678.948	−0.124392
$311$	2193.57	0.399955	0.199977	−	0.979801i	$-0.435913\pi$
$311$	2193.57	0.399955	0.199977	+	0.979801i	$0.435913\pi$
$312$	0	0
$313$	−6208.20	−1.12111	−0.560556	−	0.828116i	$-0.689413\pi$
$313$	−6208.20	−1.12111	−0.560556	+	0.828116i	$0.689413\pi$
$314$	−5065.95	−0.910471
$315$	0	0
$316$	488.518	0.0869661
$317$	6756.19	1.19705	0.598526	−	0.801103i	$-0.295753\pi$
$317$	6756.19	1.19705	0.598526	+	0.801103i	$0.295753\pi$
$318$	0	0
$319$	−122.121	−0.0214340
$320$	−1193.76	−0.208541
$321$	0	0
$322$	−516.844	−0.0894490
$323$	418.357	0.0720682
$324$	0	0
$325$	−7748.56	−1.32250
$326$	−5983.25	−1.01651
$327$	0	0
$328$	6215.51	1.04632
$329$	−378.383	−0.0634071
$330$	0	0
$331$	−5692.19	−0.945229	−0.472615	−	0.881269i	$-0.656690\pi$
$331$	−5692.19	−0.945229	−0.472615	+	0.881269i	$0.656690\pi$
$332$	−605.203	−0.100045
$333$	0	0
$334$	−4734.65	−0.775655
$335$	−1186.17	−0.193455
$336$	0	0
$337$	9528.00	1.54013	0.770064	−	0.637967i	$-0.220224\pi$
$337$	9528.00	1.54013	0.770064	+	0.637967i	$0.220224\pi$
$338$	−6131.27	−0.986677
$339$	0	0
$340$	5.55725	0.000886424 0
$341$	46.2403	0.00734326
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−9421.16	−1.47661
$345$	0	0
$346$	−5913.38	−0.918801
$347$	6723.08	1.04010	0.520049	−	0.854136i	$-0.325914\pi$
$347$	6723.08	1.04010	0.520049	+	0.854136i	$0.325914\pi$
$348$	0	0
$349$	−3224.96	−0.494637	−0.247319	−	0.968934i	$-0.579549\pi$
$349$	−3224.96	−0.494637	−0.247319	+	0.968934i	$0.579549\pi$
$350$	−2436.26	−0.372067
$351$	0	0
$352$	−14.8724	−0.00225199
$353$	7200.11	1.08562	0.542809	−	0.839856i	$-0.317361\pi$
$353$	7200.11	1.08562	0.542809	+	0.839856i	$0.317361\pi$
$354$	0	0
$355$	−1369.03	−0.204677
$356$	957.195	0.142504
$357$	0	0
$358$	5228.36	0.771864
$359$	3642.82	0.535545	0.267772	−	0.963482i	$-0.413712\pi$
$359$	3642.82	0.535545	0.267772	+	0.963482i	$0.413712\pi$
$360$	0	0
$361$	8484.92	1.23705
$362$	1788.07	0.259610
$363$	0	0
$364$	−293.431	−0.0422526
$365$	927.068	0.132945
$366$	0	0
$367$	−2832.77	−0.402914	−0.201457	−	0.979497i	$-0.564568\pi$
$367$	−2832.77	−0.402914	−0.201457	+	0.979497i	$0.564568\pi$
$368$	1725.98	0.244492
$369$	0	0
$370$	−509.488	−0.0715865
$371$	−1905.03	−0.266588
$372$	0	0
$373$	5166.34	0.717166	0.358583	−	0.933498i	$-0.383260\pi$
$373$	5166.34	0.717166	0.358583	+	0.933498i	$0.383260\pi$
$374$	−5.10555	−0.000705888 0
$375$	0	0
$376$	1169.37	0.160387
$377$	15540.4	2.12300
$378$	0	0
$379$	56.3096	0.00763174	0.00381587	−	0.999993i	$-0.498785\pi$
$379$	56.3096	0.00763174	0.00381587	+	0.999993i	$0.498785\pi$
$380$	203.821	0.0275152
$381$	0	0
$382$	12918.5	1.73029
$383$	−5055.39	−0.674461	−0.337230	−	0.941422i	$-0.609490\pi$
$383$	−5055.39	−0.674461	−0.337230	+	0.941422i	$0.609490\pi$
$384$	0	0
$385$	−9.24763	−0.00122416
$386$	−3488.92	−0.460055
$387$	0	0
$388$	861.889	0.112773
$389$	2535.98	0.330539	0.165269	−	0.986248i	$-0.447151\pi$
$389$	2535.98	0.330539	0.165269	+	0.986248i	$0.447151\pi$
$390$	0	0
$391$	84.8341	0.0109725
$392$	1060.02	0.136579
$393$	0	0
$394$	2399.68	0.306838
$395$	−1959.20	−0.249564
$396$	0	0
$397$	3204.52	0.405114	0.202557	−	0.979270i	$-0.435075\pi$
$397$	3204.52	0.405114	0.202557	+	0.979270i	$0.435075\pi$
$398$	15851.0	1.99633
$399$	0	0
$400$	8135.80	1.01698
$401$	1551.92	0.193265	0.0966323	−	0.995320i	$-0.469193\pi$
$401$	1551.92	0.193265	0.0966323	+	0.995320i	$0.469193\pi$
$402$	0	0
$403$	−5884.27	−0.727336
$404$	1139.96	0.140384
$405$	0	0
$406$	4886.12	0.597276
$407$	34.6991	0.00422597
$408$	0	0
$409$	566.211	0.0684531	0.0342265	−	0.999414i	$-0.489103\pi$
$409$	566.211	0.0684531	0.0342265	+	0.999414i	$0.489103\pi$
$410$	2169.55	0.261333
$411$	0	0
$412$	−23.8930	−0.00285709
$413$	3615.47	0.430764
$414$	0	0
$415$	2427.16	0.287096
$416$	1892.58	0.223056
$417$	0	0
$418$	−187.254	−0.0219113
$419$	10645.8	1.24125	0.620624	−	0.784108i	$-0.286879\pi$
$419$	10645.8	1.24125	0.620624	+	0.784108i	$0.286879\pi$
$420$	0	0
$421$	−9283.60	−1.07471	−0.537357	−	0.843355i	$-0.680577\pi$
$421$	−9283.60	−1.07471	−0.537357	+	0.843355i	$0.680577\pi$
$422$	−7690.78	−0.887159
$423$	0	0
$424$	5887.35	0.674328
$425$	399.885	0.0456406
$426$	0	0
$427$	−1761.97	−0.199691
$428$	249.484	0.0281759
$429$	0	0
$430$	−3288.50	−0.368803
$431$	−7494.55	−0.837587	−0.418793	−	0.908082i	$-0.637547\pi$
$431$	−7494.55	−0.837587	−0.418793	+	0.908082i	$0.637547\pi$
$432$	0	0
$433$	−1564.51	−0.173639	−0.0868195	−	0.996224i	$-0.527670\pi$
$433$	−1564.51	−0.173639	−0.0868195	+	0.996224i	$0.527670\pi$
$434$	−1850.10	−0.204626
$435$	0	0
$436$	74.1929	0.00814953
$437$	3111.42	0.340594
$438$	0	0
$439$	11397.1	1.23907	0.619537	−	0.784967i	$-0.287320\pi$
$439$	11397.1	1.23907	0.619537	+	0.784967i	$0.287320\pi$
$440$	28.5791	0.00309649
$441$	0	0
$442$	649.704	0.0699169
$443$	4335.52	0.464982	0.232491	−	0.972599i	$-0.425312\pi$
$443$	4335.52	0.464982	0.232491	+	0.972599i	$0.425312\pi$
$444$	0	0
$445$	−3838.82	−0.408939
$446$	2480.62	0.263365
$447$	0	0
$448$	−3252.93	−0.343050
$449$	−14016.9	−1.47327	−0.736634	−	0.676291i	$-0.763586\pi$
$449$	−14016.9	−1.47327	−0.736634	+	0.676291i	$0.763586\pi$
$450$	0	0
$451$	−147.759	−0.0154272
$452$	−1229.94	−0.127990
$453$	0	0
$454$	9845.13	1.01774
$455$	1176.80	0.121251
$456$	0	0
$457$	2506.06	0.256518	0.128259	−	0.991741i	$-0.459061\pi$
$457$	2506.06	0.256518	0.128259	+	0.991741i	$0.459061\pi$
$458$	2030.07	0.207116
$459$	0	0
$460$	41.3306	0.00418924
$461$	13664.0	1.38047	0.690235	−	0.723585i	$-0.257507\pi$
$461$	13664.0	1.38047	0.690235	+	0.723585i	$0.257507\pi$
$462$	0	0
$463$	1590.36	0.159634	0.0798168	−	0.996810i	$-0.474566\pi$
$463$	1590.36	0.159634	0.0798168	+	0.996810i	$0.474566\pi$
$464$	−16317.0	−1.63254
$465$	0	0
$466$	12357.4	1.22842
$467$	−8980.42	−0.889859	−0.444930	−	0.895566i	$-0.646771\pi$
$467$	−8980.42	−0.889859	−0.444930	+	0.895566i	$0.646771\pi$
$468$	0	0
$469$	−3232.25	−0.318234
$470$	408.172	0.0400587
$471$	0	0
$472$	−11173.4	−1.08961
$473$	223.966	0.0217716
$474$	0	0
$475$	14666.4	1.41672
$476$	15.1432	0.00145817
$477$	0	0
$478$	−12661.3	−1.21153
$479$	4391.79	0.418927	0.209464	−	0.977816i	$-0.432828\pi$
$479$	4391.79	0.418927	0.209464	+	0.977816i	$0.432828\pi$
$480$	0	0
$481$	−4415.60	−0.418574
$482$	15692.4	1.48292
$483$	0	0
$484$	−852.380	−0.0800507
$485$	−3456.60	−0.323621
$486$	0	0
$487$	9440.93	0.878458	0.439229	−	0.898375i	$-0.355252\pi$
$487$	9440.93	0.878458	0.439229	+	0.898375i	$0.355252\pi$
$488$	5445.26	0.505113
$489$	0	0
$490$	370.003	0.0341123
$491$	−7309.23	−0.671814	−0.335907	−	0.941895i	$-0.609043\pi$
$491$	−7309.23	−0.671814	−0.335907	+	0.941895i	$0.609043\pi$
$492$	0	0
$493$	−802.001	−0.0732664
$494$	23828.9	2.17027
$495$	0	0
$496$	6178.34	0.559306
$497$	−3730.53	−0.336694
$498$	0	0
$499$	−8594.72	−0.771047	−0.385524	−	0.922698i	$-0.625979\pi$
$499$	−8594.72	−0.771047	−0.385524	+	0.922698i	$0.625979\pi$
$500$	400.501	0.0358219
$501$	0	0
$502$	−8738.14	−0.776897
$503$	13124.1	1.16337	0.581684	−	0.813415i	$-0.302394\pi$
$503$	13124.1	1.16337	0.581684	+	0.813415i	$0.302394\pi$
$504$	0	0
$505$	−4571.80	−0.402856
$506$	−37.9712	−0.00333602
$507$	0	0
$508$	1713.01	0.149612
$509$	21261.2	1.85145	0.925725	−	0.378198i	$-0.123456\pi$
$509$	21261.2	1.85145	0.925725	+	0.378198i	$0.123456\pi$
$510$	0	0
$511$	2526.21	0.218695
$512$	9904.75	0.854946
$513$	0	0
$514$	15037.8	1.29045
$515$	95.8226	0.00819893
$516$	0	0
$517$	−27.7989	−0.00236478
$518$	−1388.33	−0.117760
$519$	0	0
$520$	−3636.82	−0.306702
$521$	18918.3	1.59084	0.795419	−	0.606059i	$-0.207251\pi$
$521$	18918.3	1.59084	0.795419	+	0.606059i	$0.207251\pi$
$522$	0	0
$523$	−3728.42	−0.311725	−0.155863	−	0.987779i	$-0.549816\pi$
$523$	−3728.42	−0.311725	−0.155863	+	0.987779i	$0.549816\pi$
$524$	792.658	0.0660829
$525$	0	0
$526$	−21370.1	−1.77145
$527$	303.673	0.0251010
$528$	0	0
$529$	−11536.1	−0.948144
$530$	2055.00	0.168422
$531$	0	0
$532$	555.402	0.0452627
$533$	18802.9	1.52804
$534$	0	0
$535$	−1000.55	−0.0808555
$536$	9989.04	0.804965
$537$	0	0
$538$	−2332.09	−0.186884
$539$	−25.1994	−0.00201375
$540$	0	0
$541$	−20984.0	−1.66761	−0.833803	−	0.552062i	$-0.813841\pi$
$541$	−20984.0	−1.66761	−0.833803	+	0.552062i	$0.813841\pi$
$542$	−20764.2	−1.64557
$543$	0	0
$544$	−97.6714	−0.00769784
$545$	−297.550	−0.0233865
$546$	0	0
$547$	−2840.49	−0.222030	−0.111015	−	0.993819i	$-0.535410\pi$
$547$	−2840.49	−0.222030	−0.111015	+	0.993819i	$0.535410\pi$
$548$	1523.40	0.118753
$549$	0	0
$550$	−178.986	−0.0138763
$551$	−29414.7	−2.27424
$552$	0	0
$553$	−5338.72	−0.410534
$554$	−21042.4	−1.61373
$555$	0	0
$556$	745.882	0.0568929
$557$	5824.36	0.443063	0.221532	−	0.975153i	$-0.428894\pi$
$557$	5824.36	0.443063	0.221532	+	0.975153i	$0.428894\pi$
$558$	0	0
$559$	−28500.6	−2.15643
$560$	−1235.61	−0.0932396
$561$	0	0
$562$	22222.0	1.66793
$563$	7380.87	0.552516	0.276258	−	0.961083i	$-0.410906\pi$
$563$	7380.87	0.552516	0.276258	+	0.961083i	$0.410906\pi$
$564$	0	0
$565$	4932.65	0.367289
$566$	11159.6	0.828753
$567$	0	0
$568$	11528.9	0.851660
$569$	−12051.2	−0.887892	−0.443946	−	0.896053i	$-0.646422\pi$
$569$	−12051.2	−0.887892	−0.443946	+	0.896053i	$0.646422\pi$
$570$	0	0
$571$	−18270.9	−1.33908	−0.669538	−	0.742778i	$-0.733508\pi$
$571$	−18270.9	−1.33908	−0.669538	+	0.742778i	$0.733508\pi$
$572$	−21.5576	−0.00157582
$573$	0	0
$574$	5911.92	0.429893
$575$	2974.04	0.215697
$576$	0	0
$577$	10165.1	0.733413	0.366707	−	0.930337i	$-0.380485\pi$
$577$	10165.1	0.733413	0.366707	+	0.930337i	$0.380485\pi$
$578$	14408.1	1.03685
$579$	0	0
$580$	−390.730	−0.0279727
$581$	6613.90	0.472273
$582$	0	0
$583$	−139.958	−0.00994245
$584$	−7807.08	−0.553184
$585$	0	0
$586$	−1771.93	−0.124911
$587$	−4635.34	−0.325930	−0.162965	−	0.986632i	$-0.552106\pi$
$587$	−4635.34	−0.325930	−0.162965	+	0.986632i	$0.552106\pi$
$588$	0	0
$589$	11137.7	0.779152
$590$	−3900.11	−0.272144
$591$	0	0
$592$	4636.28	0.321875
$593$	16282.3	1.12755	0.563773	−	0.825930i	$-0.309349\pi$
$593$	16282.3	1.12755	0.563773	+	0.825930i	$0.309349\pi$
$594$	0	0
$595$	−60.7319	−0.00418448
$596$	1075.70	0.0739305
$597$	0	0
$598$	4832.00	0.330427
$599$	9759.63	0.665722	0.332861	−	0.942976i	$-0.391986\pi$
$599$	9759.63	0.665722	0.332861	+	0.942976i	$0.391986\pi$
$600$	0	0
$601$	11844.3	0.803894	0.401947	−	0.915663i	$-0.368334\pi$
$601$	11844.3	0.803894	0.401947	+	0.915663i	$0.368334\pi$
$602$	−8960.99	−0.606683
$603$	0	0
$604$	553.740	0.0373036
$605$	3418.46	0.229719
$606$	0	0
$607$	−22206.1	−1.48488	−0.742438	−	0.669915i	$-0.766331\pi$
$607$	−22206.1	−1.48488	−0.742438	+	0.669915i	$0.766331\pi$
$608$	−3582.25	−0.238947
$609$	0	0
$610$	1900.69	0.126158
$611$	3537.52	0.234227
$612$	0	0
$613$	−19461.8	−1.28231	−0.641154	−	0.767412i	$-0.721544\pi$
$613$	−19461.8	−1.28231	−0.641154	+	0.767412i	$0.721544\pi$
$614$	−27090.1	−1.78057
$615$	0	0
$616$	77.8768	0.00509374
$617$	−15820.2	−1.03225	−0.516124	−	0.856514i	$-0.672625\pi$
$617$	−15820.2	−1.03225	−0.516124	+	0.856514i	$0.672625\pi$
$618$	0	0
$619$	−15398.0	−0.999838	−0.499919	−	0.866072i	$-0.666637\pi$
$619$	−15398.0	−0.999838	−0.499919	+	0.866072i	$0.666637\pi$
$620$	147.948	0.00958341
$621$	0	0
$622$	−6447.95	−0.415658
$623$	−10460.6	−0.672705
$624$	0	0
$625$	13193.9	0.844411
$626$	18248.9	1.16513
$627$	0	0
$628$	1103.91	0.0701443
$629$	227.879	0.0144453
$630$	0	0
$631$	15072.8	0.950930	0.475465	−	0.879735i	$-0.342280\pi$
$631$	15072.8	0.950930	0.475465	+	0.879735i	$0.342280\pi$
$632$	16498.9	1.03844
$633$	0	0
$634$	−19859.7	−1.24405
$635$	−6870.03	−0.429337
$636$	0	0
$637$	3206.73	0.199459
$638$	358.971	0.0222755
$639$	0	0
$640$	4103.34	0.253435
$641$	20448.2	1.25999	0.629996	−	0.776598i	$-0.283056\pi$
$641$	20448.2	1.25999	0.629996	+	0.776598i	$0.283056\pi$
$642$	0	0
$643$	8343.12	0.511696	0.255848	−	0.966717i	$-0.417645\pi$
$643$	8343.12	0.511696	0.255848	+	0.966717i	$0.417645\pi$
$644$	112.624	0.00689131
$645$	0	0
$646$	−1229.75	−0.0748978
$647$	−2134.15	−0.129679	−0.0648394	−	0.997896i	$-0.520653\pi$
$647$	−2134.15	−0.129679	−0.0648394	+	0.997896i	$0.520653\pi$
$648$	0	0
$649$	265.620	0.0160655
$650$	22776.7	1.37443
$651$	0	0
$652$	1303.79	0.0783136
$653$	−25095.3	−1.50391	−0.751956	−	0.659213i	$-0.770890\pi$
$653$	−25095.3	−1.50391	−0.751956	+	0.659213i	$0.770890\pi$
$654$	0	0
$655$	−3178.95	−0.189636
$656$	−19742.6	−1.17503
$657$	0	0
$658$	1112.25	0.0658966
$659$	21027.4	1.24296	0.621481	−	0.783429i	$-0.286531\pi$
$659$	21027.4	1.24296	0.621481	+	0.783429i	$0.286531\pi$
$660$	0	0
$661$	−2656.34	−0.156308	−0.0781541	−	0.996941i	$-0.524903\pi$
$661$	−2656.34	−0.156308	−0.0781541	+	0.996941i	$0.524903\pi$
$662$	16732.1	0.982341
$663$	0	0
$664$	−20439.8	−1.19460
$665$	−2227.44	−0.129889
$666$	0	0
$667$	−5964.68	−0.346257
$668$	1031.71	0.0597578
$669$	0	0
$670$	3486.72	0.201050
$671$	−129.448	−0.00744751
$672$	0	0
$673$	−15791.3	−0.904471	−0.452235	−	0.891899i	$-0.649373\pi$
$673$	−15791.3	−0.904471	−0.452235	+	0.891899i	$0.649373\pi$
$674$	−28007.3	−1.60060
$675$	0	0
$676$	1336.05	0.0760154
$677$	4012.08	0.227765	0.113882	−	0.993494i	$-0.463671\pi$
$677$	4012.08	0.227765	0.113882	+	0.993494i	$0.463671\pi$
$678$	0	0
$679$	−9419.07	−0.532357
$680$	187.687	0.0105845
$681$	0	0
$682$	−135.922	−0.00763157
$683$	−7438.45	−0.416727	−0.208363	−	0.978051i	$-0.566814\pi$
$683$	−7438.45	−0.416727	−0.208363	+	0.978051i	$0.566814\pi$
$684$	0	0
$685$	−6109.58	−0.340781
$686$	1008.24	0.0561149
$687$	0	0
$688$	29924.9	1.65825
$689$	17810.2	0.984781
$690$	0	0
$691$	16071.1	0.884764	0.442382	−	0.896827i	$-0.354133\pi$
$691$	16071.1	0.884764	0.442382	+	0.896827i	$0.354133\pi$
$692$	1288.57	0.0707860
$693$	0	0
$694$	−19762.4	−1.08093
$695$	−2991.35	−0.163264
$696$	0	0
$697$	−970.374	−0.0527339
$698$	9479.71	0.514058
$699$	0	0
$700$	530.878	0.0286647
$701$	−27269.4	−1.46926	−0.734630	−	0.678467i	$-0.762644\pi$
$701$	−27269.4	−1.46926	−0.734630	+	0.678467i	$0.762644\pi$
$702$	0	0
$703$	8357.81	0.448394
$704$	−238.985	−0.0127941
$705$	0	0
$706$	−21164.6	−1.12824
$707$	−12457.9	−0.662700
$708$	0	0
$709$	12825.7	0.679377	0.339689	−	0.940538i	$-0.389678\pi$
$709$	12825.7	0.679377	0.339689	+	0.940538i	$0.389678\pi$
$710$	4024.22	0.212713
$711$	0	0
$712$	32327.8	1.70159
$713$	2258.49	0.118627
$714$	0	0
$715$	86.4566	0.00452209
$716$	−1139.30	−0.0594658
$717$	0	0
$718$	−10708.0	−0.556572
$719$	−9140.58	−0.474111	−0.237056	−	0.971496i	$-0.576182\pi$
$719$	−9140.58	−0.474111	−0.237056	+	0.971496i	$0.576182\pi$
$720$	0	0
$721$	261.112	0.0134873
$722$	−24941.2	−1.28562
$723$	0	0
$724$	−389.632	−0.0200008
$725$	−28115.8	−1.44027
$726$	0	0
$727$	−11651.7	−0.594412	−0.297206	−	0.954813i	$-0.596055\pi$
$727$	−11651.7	−0.594412	−0.297206	+	0.954813i	$0.596055\pi$
$728$	−9910.15	−0.504526
$729$	0	0
$730$	−2725.10	−0.138165
$731$	1470.85	0.0744203
$732$	0	0
$733$	18831.0	0.948894	0.474447	−	0.880284i	$-0.342648\pi$
$733$	18831.0	0.948894	0.474447	+	0.880284i	$0.342648\pi$
$734$	8326.86	0.418733
$735$	0	0
$736$	−726.406	−0.0363800
$737$	−237.466	−0.0118686
$738$	0	0
$739$	7467.51	0.371714	0.185857	−	0.982577i	$-0.440494\pi$
$739$	7467.51	0.371714	0.185857	+	0.982577i	$0.440494\pi$
$740$	111.021	0.00551515
$741$	0	0
$742$	5599.79	0.277055
$743$	−12467.1	−0.615576	−0.307788	−	0.951455i	$-0.599589\pi$
$743$	−12467.1	−0.615576	−0.307788	+	0.951455i	$0.599589\pi$
$744$	0	0
$745$	−4314.10	−0.212156
$746$	−15186.3	−0.745324
$747$	0	0
$748$	1.11254	5.43828e−5 0
$749$	−2726.46	−0.133008
$750$	0	0
$751$	−2456.69	−0.119368	−0.0596842	−	0.998217i	$-0.519009\pi$
$751$	−2456.69	−0.119368	−0.0596842	+	0.998217i	$0.519009\pi$
$752$	−3714.32	−0.180116
$753$	0	0
$754$	−45680.6	−2.20635
$755$	−2220.77	−0.107049
$756$	0	0
$757$	−7246.72	−0.347935	−0.173967	−	0.984751i	$-0.555659\pi$
$757$	−7246.72	−0.347935	−0.173967	+	0.984751i	$0.555659\pi$
$758$	−165.521	−0.00793138
$759$	0	0
$760$	6883.73	0.328552
$761$	−12683.0	−0.604149	−0.302075	−	0.953284i	$-0.597679\pi$
$761$	−12683.0	−0.604149	−0.302075	+	0.953284i	$0.597679\pi$
$762$	0	0
$763$	−810.809	−0.0384709
$764$	−2815.04	−0.133304
$765$	0	0
$766$	14860.2	0.700942
$767$	−33801.2	−1.59125
$768$	0	0
$769$	−14297.4	−0.670454	−0.335227	−	0.942137i	$-0.608813\pi$
$769$	−14297.4	−0.670454	−0.335227	+	0.942137i	$0.608813\pi$
$770$	27.1832	0.00127223
$771$	0	0
$772$	760.260	0.0354435
$773$	6703.00	0.311889	0.155944	−	0.987766i	$-0.450158\pi$
$773$	6703.00	0.311889	0.155944	+	0.987766i	$0.450158\pi$
$774$	0	0
$775$	10645.9	0.493435
$776$	29109.0	1.34659
$777$	0	0
$778$	−7454.47	−0.343516
$779$	−35590.0	−1.63690
$780$	0	0
$781$	−274.073	−0.0125571
$782$	−249.368	−0.0114033
$783$	0	0
$784$	−3366.99	−0.153379
$785$	−4427.20	−0.201291
$786$	0	0
$787$	34935.0	1.58233	0.791167	−	0.611600i	$-0.209474\pi$
$787$	34935.0	1.58233	0.791167	+	0.611600i	$0.209474\pi$
$788$	−522.908	−0.0236394
$789$	0	0
$790$	5759.02	0.259363
$791$	13441.2	0.604192
$792$	0	0
$793$	16472.8	0.737662
$794$	−9419.62	−0.421020
$795$	0	0
$796$	−3454.05	−0.153801
$797$	2614.16	0.116184	0.0580918	−	0.998311i	$-0.481498\pi$
$797$	2614.16	0.116184	0.0580918	+	0.998311i	$0.481498\pi$
$798$	0	0
$799$	−182.563	−0.00808338
$800$	−3424.08	−0.151324
$801$	0	0
$802$	−4561.83	−0.200853
$803$	185.595	0.00815628
$804$	0	0
$805$	−451.677	−0.0197758
$806$	17296.7	0.755893
$807$	0	0
$808$	38500.4	1.67628
$809$	14079.7	0.611885	0.305942	−	0.952050i	$-0.401029\pi$
$809$	14079.7	0.611885	0.305942	+	0.952050i	$0.401029\pi$
$810$	0	0
$811$	360.517	0.0156097	0.00780485	−	0.999970i	$-0.497516\pi$
$811$	360.517	0.0156097	0.00780485	+	0.999970i	$0.497516\pi$
$812$	−1064.72	−0.0460152
$813$	0	0
$814$	−101.997	−0.00439189
$815$	−5228.85	−0.224734
$816$	0	0
$817$	53945.6	2.31006
$818$	−1664.36	−0.0711407
$819$	0	0
$820$	−472.760	−0.0201335
$821$	−20005.9	−0.850439	−0.425220	−	0.905090i	$-0.639803\pi$
$821$	−20005.9	−0.850439	−0.425220	+	0.905090i	$0.639803\pi$
$822$	0	0
$823$	11187.2	0.473828	0.236914	−	0.971531i	$-0.423864\pi$
$823$	11187.2	0.473828	0.236914	+	0.971531i	$0.423864\pi$
$824$	−806.948	−0.0341157
$825$	0	0
$826$	−10627.6	−0.447677
$827$	−10206.8	−0.429172	−0.214586	−	0.976705i	$-0.568840\pi$
$827$	−10206.8	−0.429172	−0.214586	+	0.976705i	$0.568840\pi$
$828$	0	0
$829$	12084.0	0.506267	0.253134	−	0.967431i	$-0.418539\pi$
$829$	12084.0	0.506267	0.253134	+	0.967431i	$0.418539\pi$
$830$	−7134.59	−0.298368
$831$	0	0
$832$	30411.8	1.26724
$833$	−165.491	−0.00688348
$834$	0	0
$835$	−4137.68	−0.171485
$836$	40.8040	0.00168808
$837$	0	0
$838$	−31293.2	−1.28998
$839$	−5470.25	−0.225094	−0.112547	−	0.993646i	$-0.535901\pi$
$839$	−5470.25	−0.225094	−0.112547	+	0.993646i	$0.535901\pi$
$840$	0	0
$841$	31999.6	1.31205
$842$	27288.9	1.11691
$843$	0	0
$844$	1675.87	0.0683483
$845$	−5358.20	−0.218139
$846$	0	0
$847$	9315.15	0.377889
$848$	−18700.3	−0.757276
$849$	0	0
$850$	−1175.45	−0.0474326
$851$	1694.79	0.0682686
$852$	0	0
$853$	−2405.08	−0.0965398	−0.0482699	−	0.998834i	$-0.515371\pi$
$853$	−2405.08	−0.0965398	−0.0482699	+	0.998834i	$0.515371\pi$
$854$	5179.29	0.207531
$855$	0	0
$856$	8425.93	0.336440
$857$	−66.0880	−0.00263422	−0.00131711	−	0.999999i	$-0.500419\pi$
$857$	−66.0880	−0.00263422	−0.00131711	+	0.999999i	$0.500419\pi$
$858$	0	0
$859$	30565.7	1.21407	0.607037	−	0.794673i	$-0.292358\pi$
$859$	30565.7	1.21407	0.607037	+	0.794673i	$0.292358\pi$
$860$	716.587	0.0284133
$861$	0	0
$862$	22030.1	0.870473
$863$	−3384.92	−0.133516	−0.0667579	−	0.997769i	$-0.521266\pi$
$863$	−3384.92	−0.133516	−0.0667579	+	0.997769i	$0.521266\pi$
$864$	0	0
$865$	−5167.78	−0.203133
$866$	4598.85	0.180456
$867$	0	0
$868$	403.150	0.0157647
$869$	−392.222	−0.0153110
$870$	0	0
$871$	30218.5	1.17556
$872$	2505.75	0.0973111
$873$	0	0
$874$	−9145.97	−0.353967
$875$	−4376.83	−0.169102
$876$	0	0
$877$	−21340.0	−0.821664	−0.410832	−	0.911711i	$-0.634762\pi$
$877$	−21340.0	−0.821664	−0.410832	+	0.911711i	$0.634762\pi$
$878$	−33501.5	−1.28772
$879$	0	0
$880$	−90.7774	−0.00347739
$881$	24424.6	0.934037	0.467019	−	0.884247i	$-0.345328\pi$
$881$	24424.6	0.934037	0.467019	+	0.884247i	$0.345328\pi$
$882$	0	0
$883$	−39237.6	−1.49541	−0.747707	−	0.664029i	$-0.768845\pi$
$883$	−39237.6	−1.49541	−0.747707	+	0.664029i	$0.768845\pi$
$884$	−141.575	−0.00538652
$885$	0	0
$886$	−12744.2	−0.483238
$887$	−16580.6	−0.627647	−0.313823	−	0.949481i	$-0.601610\pi$
$887$	−16580.6	−0.627647	−0.313823	+	0.949481i	$0.601610\pi$
$888$	0	0
$889$	−18720.5	−0.706260
$890$	11284.1	0.424995
$891$	0	0
$892$	−540.545	−0.0202901
$893$	−6695.79	−0.250914
$894$	0	0
$895$	4569.14	0.170647
$896$	11181.4	0.416902
$897$	0	0
$898$	41202.3	1.53111
$899$	−21351.2	−0.792106
$900$	0	0
$901$	−919.142	−0.0339856
$902$	434.334	0.0160330
$903$	0	0
$904$	−41539.2	−1.52829
$905$	1562.62	0.0573958
$906$	0	0
$907$	−31392.2	−1.14924	−0.574619	−	0.818421i	$-0.694850\pi$
$907$	−31392.2	−1.14924	−0.574619	+	0.818421i	$0.694850\pi$
$908$	−2145.32	−0.0784087
$909$	0	0
$910$	−3459.18	−0.126012
$911$	−51535.7	−1.87426	−0.937131	−	0.348977i	$-0.886529\pi$
$911$	−51535.7	−1.87426	−0.937131	+	0.348977i	$0.886529\pi$
$912$	0	0
$913$	485.907	0.0176135
$914$	−7366.52	−0.266589
$915$	0	0
$916$	−442.367	−0.0159566
$917$	−8662.48	−0.311952
$918$	0	0
$919$	27365.1	0.982253	0.491127	−	0.871088i	$-0.336585\pi$
$919$	27365.1	0.982253	0.491127	+	0.871088i	$0.336585\pi$
$920$	1395.88	0.0500225
$921$	0	0
$922$	−40165.1	−1.43467
$923$	34876.9	1.24376
$924$	0	0
$925$	7988.77	0.283967
$926$	−4674.83	−0.165901
$927$	0	0
$928$	6867.26	0.242919
$929$	33324.4	1.17690	0.588450	−	0.808534i	$-0.299739\pi$
$929$	33324.4	1.17690	0.588450	+	0.808534i	$0.299739\pi$
$930$	0	0
$931$	−6069.66	−0.213668
$932$	−2692.76	−0.0946396
$933$	0	0
$934$	26397.8	0.924797
$935$	−4.46182	−0.000156061 0
$936$	0	0
$937$	49600.6	1.72933	0.864664	−	0.502351i	$-0.167531\pi$
$937$	49600.6	1.72933	0.864664	+	0.502351i	$0.167531\pi$
$938$	9501.14	0.330728
$939$	0	0
$940$	−88.9436	−0.00308619
$941$	−12149.0	−0.420878	−0.210439	−	0.977607i	$-0.567489\pi$
$941$	−12149.0	−0.420878	−0.210439	+	0.977607i	$0.567489\pi$
$942$	0	0
$943$	−7216.91	−0.249220
$944$	35490.5	1.22364
$945$	0	0
$946$	−658.342	−0.0226264
$947$	−3637.85	−0.124830	−0.0624151	−	0.998050i	$-0.519880\pi$
$947$	−3637.85	−0.124830	−0.0624151	+	0.998050i	$0.519880\pi$
$948$	0	0
$949$	−23617.7	−0.807865
$950$	−43111.6	−1.47234
$951$	0	0
$952$	511.439	0.0174116
$953$	−26019.5	−0.884421	−0.442210	−	0.896911i	$-0.645806\pi$
$953$	−26019.5	−0.884421	−0.442210	+	0.896911i	$0.645806\pi$
$954$	0	0
$955$	11289.7	0.382540
$956$	2758.98	0.0933386
$957$	0	0
$958$	−12909.6	−0.435375
$959$	−16648.3	−0.560586
$960$	0	0
$961$	−21706.5	−0.728626
$962$	12979.6	0.435009
$963$	0	0
$964$	−3419.48	−0.114247
$965$	−3049.02	−0.101711
$966$	0	0
$967$	−21914.7	−0.728779	−0.364390	−	0.931247i	$-0.618722\pi$
$967$	−21914.7	−0.728779	−0.364390	+	0.931247i	$0.618722\pi$
$968$	−28787.8	−0.955862
$969$	0	0
$970$	10160.6	0.336327
$971$	−11091.9	−0.366587	−0.183293	−	0.983058i	$-0.558676\pi$
$971$	−11091.9	−0.366587	−0.183293	+	0.983058i	$0.558676\pi$
$972$	0	0
$973$	−8151.29	−0.268570
$974$	−27751.4	−0.912949
$975$	0	0
$976$	−17296.0	−0.567247
$977$	56320.9	1.84428	0.922142	−	0.386853i	$-0.126438\pi$
$977$	56320.9	1.84428	0.922142	+	0.386853i	$0.126438\pi$
$978$	0	0
$979$	−768.515	−0.0250887
$980$	−80.6263	−0.00262807
$981$	0	0
$982$	21485.3	0.698192
$983$	−141.191	−0.00458117	−0.00229059	−	0.999997i	$-0.500729\pi$
$983$	−141.191	−0.00458117	−0.00229059	+	0.999997i	$0.500729\pi$
$984$	0	0
$985$	2097.12	0.0678373
$986$	2357.47	0.0761430
$987$	0	0
$988$	−5192.49	−0.167201
$989$	10939.0	0.351710
$990$	0	0
$991$	−43398.1	−1.39111	−0.695553	−	0.718475i	$-0.744840\pi$
$991$	−43398.1	−1.39111	−0.695553	+	0.718475i	$0.744840\pi$
$992$	−2600.25	−0.0832238
$993$	0	0
$994$	10965.8	0.349914
$995$	13852.4	0.441358
$996$	0	0
$997$	39006.1	1.23905	0.619527	−	0.784976i	$-0.287325\pi$
$997$	39006.1	1.23905	0.619527	+	0.784976i	$0.287325\pi$
$998$	25264.0	0.801320
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	567.4.a.i.1.3		8
3.2	odd	2		567.4.a.g.1.6		8
9.2	odd	6		189.4.f.b.64.3		16
9.4	even	3		63.4.f.b.43.6	yes	16
9.5	odd	6		189.4.f.b.127.3		16
9.7	even	3		63.4.f.b.22.6	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.f.b.22.6	✓	16	9.7	even	3
63.4.f.b.43.6	yes	16	9.4	even	3
189.4.f.b.64.3		16	9.2	odd	6
189.4.f.b.127.3		16	9.5	odd	6
567.4.a.g.1.6		8	3.2	odd	2
567.4.a.i.1.3		8	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:	\( N \)	\(=\)	\( 567 = 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	567.a (trivial)

\(f(q)\)	\(=\)	\(q-2.93948 q^{2} +0.640533 q^{4} -2.56885 q^{5} -7.00000 q^{7} +21.6330 q^{8} +O(q^{10})\)	\(q-2.93948 q^{2} +0.640533 q^{4} -2.56885 q^{5} -7.00000 q^{7} +21.6330 q^{8} +7.55109 q^{10} -0.514272 q^{11} +65.4434 q^{13} +20.5763 q^{14} -68.7140 q^{16} -3.37738 q^{17} -123.871 q^{19} -1.64544 q^{20} +1.51169 q^{22} -25.1184 q^{23} -118.401 q^{25} -192.369 q^{26} -4.48373 q^{28} +237.463 q^{29} -89.9139 q^{31} +28.9193 q^{32} +9.92772 q^{34} +17.9820 q^{35} -67.4721 q^{37} +364.115 q^{38} -55.5720 q^{40} +287.316 q^{41} -435.500 q^{43} -0.329409 q^{44} +73.8349 q^{46} +54.0547 q^{47} +49.0000 q^{49} +348.037 q^{50} +41.9186 q^{52} +272.147 q^{53} +1.32109 q^{55} -151.431 q^{56} -698.017 q^{58} -516.496 q^{59} +251.711 q^{61} +264.300 q^{62} +464.704 q^{64} -168.114 q^{65} +461.750 q^{67} -2.16332 q^{68} -52.8576 q^{70} +532.933 q^{71} -360.888 q^{73} +198.333 q^{74} -79.3432 q^{76} +3.59991 q^{77} +762.674 q^{79} +176.516 q^{80} -844.559 q^{82} -944.843 q^{83} +8.67598 q^{85} +1280.14 q^{86} -11.1253 q^{88} +1494.37 q^{89} -458.104 q^{91} -16.0891 q^{92} -158.893 q^{94} +318.205 q^{95} +1345.58 q^{97} -144.034 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{2} + 43 q^{4} + 30 q^{5} - 56 q^{7} + 6 q^{8}+O(q^{10}) \) 8 * q + 3 * q^2 + 43 * q^4 + 30 * q^5 - 56 * q^7 + 6 * q^8	\( 8 q + 3 q^{2} + 43 q^{4} + 30 q^{5} - 56 q^{7} + 6 q^{8} - 14 q^{10} + 24 q^{11} + 68 q^{13} - 21 q^{14} + 103 q^{16} + 168 q^{17} + 176 q^{19} + 330 q^{20} + 151 q^{22} + 228 q^{23} + 244 q^{25} + 795 q^{26} - 301 q^{28} + 618 q^{29} + 72 q^{31} + 786 q^{32} - 261 q^{34} - 210 q^{35} + 210 q^{37} + 1032 q^{38} - 375 q^{40} + 420 q^{41} - 2 q^{43} + 387 q^{44} + 402 q^{46} + 570 q^{47} + 392 q^{49} + 1110 q^{50} - 431 q^{52} + 528 q^{53} - 838 q^{55} - 42 q^{56} + 37 q^{58} - 150 q^{59} + 578 q^{61} + 1170 q^{62} - 112 q^{64} - 366 q^{65} - 898 q^{67} + 2526 q^{68} + 98 q^{70} + 882 q^{71} + 972 q^{73} - 222 q^{74} + 1423 q^{76} - 168 q^{77} - 158 q^{79} + 2475 q^{80} - 211 q^{82} + 2958 q^{83} - 774 q^{85} - 114 q^{86} + 1317 q^{88} + 4380 q^{89} - 476 q^{91} + 4629 q^{92} - 3234 q^{94} + 930 q^{95} - 60 q^{97} + 147 q^{98}+O(q^{100}) \) 8 * q + 3 * q^2 + 43 * q^4 + 30 * q^5 - 56 * q^7 + 6 * q^8 - 14 * q^10 + 24 * q^11 + 68 * q^13 - 21 * q^14 + 103 * q^16 + 168 * q^17 + 176 * q^19 + 330 * q^20 + 151 * q^22 + 228 * q^23 + 244 * q^25 + 795 * q^26 - 301 * q^28 + 618 * q^29 + 72 * q^31 + 786 * q^32 - 261 * q^34 - 210 * q^35 + 210 * q^37 + 1032 * q^38 - 375 * q^40 + 420 * q^41 - 2 * q^43 + 387 * q^44 + 402 * q^46 + 570 * q^47 + 392 * q^49 + 1110 * q^50 - 431 * q^52 + 528 * q^53 - 838 * q^55 - 42 * q^56 + 37 * q^58 - 150 * q^59 + 578 * q^61 + 1170 * q^62 - 112 * q^64 - 366 * q^65 - 898 * q^67 + 2526 * q^68 + 98 * q^70 + 882 * q^71 + 972 * q^73 - 222 * q^74 + 1423 * q^76 - 168 * q^77 - 158 * q^79 + 2475 * q^80 - 211 * q^82 + 2958 * q^83 - 774 * q^85 - 114 * q^86 + 1317 * q^88 + 4380 * q^89 - 476 * q^91 + 4629 * q^92 - 3234 * q^94 + 930 * q^95 - 60 * q^97 + 147 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more