LMFDB - Embedded newform 531.6.a.c.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,6,Mod(1,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, 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("531.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	531.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:	$85.1638083207$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} - 269 x^{10} + 143 x^{9} + 25384 x^{8} + 8539 x^{7} - 1009736 x^{6} - 720516 x^{5} + \cdots + 49172480 $ x^12 - 2x^11 - 269x^10 + 143x^9 + 25384x^8 + 8539x^7 - 1009736x^6 - 720516x^5 + 15565376x^4 + 6775664x^3 - 75006848x^2 + 21512960*x + 49172480
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$8.10567$ of defining polynomial
Character	$\chi$	$=$	531.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.10567 q^{2} +5.27923 q^{4} +41.8018 q^{5} +208.248 q^{7} -163.148 q^{8} +O(q^{10})$	\(q+6.10567 q^{2} +5.27923 q^{4} +41.8018 q^{5} +208.248 q^{7} -163.148 q^{8} +255.228 q^{10} -149.105 q^{11} -952.113 q^{13} +1271.49 q^{14} -1165.07 q^{16} -1073.60 q^{17} -486.976 q^{19} +220.681 q^{20} -910.386 q^{22} -2198.23 q^{23} -1377.61 q^{25} -5813.29 q^{26} +1099.39 q^{28} +2548.98 q^{29} +1027.11 q^{31} -1892.76 q^{32} -6555.08 q^{34} +8705.13 q^{35} -3164.40 q^{37} -2973.31 q^{38} -6819.88 q^{40} -14579.4 q^{41} +13357.1 q^{43} -787.159 q^{44} -13421.7 q^{46} -15809.5 q^{47} +26560.2 q^{49} -8411.26 q^{50} -5026.42 q^{52} -746.932 q^{53} -6232.85 q^{55} -33975.3 q^{56} +15563.2 q^{58} +3481.00 q^{59} +31646.6 q^{61} +6271.22 q^{62} +25725.5 q^{64} -39800.0 q^{65} -36947.6 q^{67} -5667.81 q^{68} +53150.6 q^{70} -67026.7 q^{71} +15479.2 q^{73} -19320.8 q^{74} -2570.86 q^{76} -31050.8 q^{77} +45796.7 q^{79} -48701.8 q^{80} -89017.1 q^{82} -16456.1 q^{83} -44878.6 q^{85} +81554.1 q^{86} +24326.2 q^{88} +28480.0 q^{89} -198276. q^{91} -11604.9 q^{92} -96527.9 q^{94} -20356.4 q^{95} +79923.6 q^{97} +162168. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 22 q^{2} + 198 q^{4} - 158 q^{5} + 413 q^{7} - 723 q^{8}+O(q^{10}) $ 12 * q - 22 * q^2 + 198 * q^4 - 158 * q^5 + 413 * q^7 - 723 * q^8	$ 12 q - 22 q^{2} + 198 q^{4} - 158 q^{5} + 413 q^{7} - 723 q^{8} + 601 q^{10} - 1480 q^{11} + 472 q^{13} - 1065 q^{14} + 6370 q^{16} - 1565 q^{17} + 3939 q^{19} - 8033 q^{20} - 1738 q^{22} - 7245 q^{23} + 9690 q^{25} - 3764 q^{26} + 12154 q^{28} - 10003 q^{29} + 7295 q^{31} - 11628 q^{32} - 16344 q^{34} - 11015 q^{35} + 6741 q^{37} - 3035 q^{38} + 5572 q^{40} - 34025 q^{41} - 6336 q^{43} - 41168 q^{44} + 2345 q^{46} - 66167 q^{47} + 28319 q^{49} - 31173 q^{50} + 16440 q^{52} - 62290 q^{53} + 55764 q^{55} - 107306 q^{56} + 37952 q^{58} + 41772 q^{59} + 68469 q^{61} - 99190 q^{62} + 68525 q^{64} - 80156 q^{65} + 113310 q^{67} - 33887 q^{68} + 32034 q^{70} - 84520 q^{71} + 135895 q^{73} + 31962 q^{74} - 61848 q^{76} + 3799 q^{77} + 14122 q^{79} - 77609 q^{80} - 1501 q^{82} - 114463 q^{83} - 101097 q^{85} + 203536 q^{86} - 244967 q^{88} - 189109 q^{89} - 168249 q^{91} + 71946 q^{92} - 472284 q^{94} - 21923 q^{95} - 76192 q^{97} + 17544 q^{98}+O(q^{100}) $ 12 * q - 22 * q^2 + 198 * q^4 - 158 * q^5 + 413 * q^7 - 723 * q^8 + 601 * q^10 - 1480 * q^11 + 472 * q^13 - 1065 * q^14 + 6370 * q^16 - 1565 * q^17 + 3939 * q^19 - 8033 * q^20 - 1738 * q^22 - 7245 * q^23 + 9690 * q^25 - 3764 * q^26 + 12154 * q^28 - 10003 * q^29 + 7295 * q^31 - 11628 * q^32 - 16344 * q^34 - 11015 * q^35 + 6741 * q^37 - 3035 * q^38 + 5572 * q^40 - 34025 * q^41 - 6336 * q^43 - 41168 * q^44 + 2345 * q^46 - 66167 * q^47 + 28319 * q^49 - 31173 * q^50 + 16440 * q^52 - 62290 * q^53 + 55764 * q^55 - 107306 * q^56 + 37952 * q^58 + 41772 * q^59 + 68469 * q^61 - 99190 * q^62 + 68525 * q^64 - 80156 * q^65 + 113310 * q^67 - 33887 * q^68 + 32034 * q^70 - 84520 * q^71 + 135895 * q^73 + 31962 * q^74 - 61848 * q^76 + 3799 * q^77 + 14122 * q^79 - 77609 * q^80 - 1501 * q^82 - 114463 * q^83 - 101097 * q^85 + 203536 * q^86 - 244967 * q^88 - 189109 * q^89 - 168249 * q^91 + 71946 * q^92 - 472284 * q^94 - 21923 * q^95 - 76192 * q^97 + 17544 * 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 $

$2$	6.10567	1.07934	0.539670	−	0.841876i	$-0.318549\pi$
$2$	6.10567	1.07934	0.539670	+	0.841876i	$0.318549\pi$
$3$	0	0
$3$	0	0
$4$	5.27923	0.164976
$5$	41.8018	0.747772	0.373886	−	0.927475i	$-0.378025\pi$
$5$	41.8018	0.747772	0.373886	+	0.927475i	$0.378025\pi$
$6$	0	0
$7$	208.248	1.60633	0.803166	−	0.595755i	$-0.203147\pi$
$7$	208.248	1.60633	0.803166	+	0.595755i	$0.203147\pi$
$8$	−163.148	−0.901275
$9$	0	0
$10$	255.228	0.807101
$11$	−149.105	−0.371544	−0.185772	−	0.982593i	$-0.559479\pi$
$11$	−149.105	−0.371544	−0.185772	+	0.982593i	$0.559479\pi$
$12$	0	0
$13$	−952.113	−1.56254	−0.781268	−	0.624196i	$-0.785427\pi$
$13$	−952.113	−1.56254	−0.781268	+	0.624196i	$0.785427\pi$
$14$	1271.49	1.73378
$15$	0	0
$16$	−1165.07	−1.13776
$17$	−1073.60	−0.900994	−0.450497	−	0.892778i	$-0.648753\pi$
$17$	−1073.60	−0.900994	−0.450497	+	0.892778i	$0.648753\pi$
$18$	0	0
$19$	−486.976	−0.309473	−0.154737	−	0.987956i	$-0.549453\pi$
$19$	−486.976	−0.309473	−0.154737	+	0.987956i	$0.549453\pi$
$20$	220.681	0.123364
$21$	0	0
$22$	−910.386	−0.401022
$23$	−2198.23	−0.866469	−0.433235	−	0.901281i	$-0.642628\pi$
$23$	−2198.23	−0.866469	−0.433235	+	0.901281i	$0.642628\pi$
$24$	0	0
$25$	−1377.61	−0.440836
$26$	−5813.29	−1.68651
$27$	0	0
$28$	1099.39	0.265006
$29$	2548.98	0.562823	0.281411	−	0.959587i	$-0.409197\pi$
$29$	2548.98	0.562823	0.281411	+	0.959587i	$0.409197\pi$
$30$	0	0
$31$	1027.11	0.191961	0.0959807	−	0.995383i	$-0.469401\pi$
$31$	1027.11	0.191961	0.0959807	+	0.995383i	$0.469401\pi$
$32$	−1892.76	−0.326754
$33$	0	0
$34$	−6555.08	−0.972480
$35$	8705.13	1.20117
$36$	0	0
$37$	−3164.40	−0.380003	−0.190001	−	0.981784i	$-0.560849\pi$
$37$	−3164.40	−0.380003	−0.190001	+	0.981784i	$0.560849\pi$
$38$	−2973.31	−0.334027
$39$	0	0
$40$	−6819.88	−0.673949
$41$	−14579.4	−1.35450	−0.677252	−	0.735751i	$-0.736829\pi$
$41$	−14579.4	−1.35450	−0.677252	+	0.735751i	$0.736829\pi$
$42$	0	0
$43$	13357.1	1.10164	0.550822	−	0.834623i	$-0.314314\pi$
$43$	13357.1	1.10164	0.550822	+	0.834623i	$0.314314\pi$
$44$	−787.159	−0.0612958
$45$	0	0
$46$	−13421.7	−0.935215
$47$	−15809.5	−1.04394	−0.521969	−	0.852965i	$-0.674802\pi$
$47$	−15809.5	−1.04394	−0.521969	+	0.852965i	$0.674802\pi$
$48$	0	0
$49$	26560.2	1.58030
$50$	−8411.26	−0.475813
$51$	0	0
$52$	−5026.42	−0.257781
$53$	−746.932	−0.0365251	−0.0182625	−	0.999833i	$-0.505813\pi$
$53$	−746.932	−0.0365251	−0.0182625	+	0.999833i	$0.505813\pi$
$54$	0	0
$55$	−6232.85	−0.277830
$56$	−33975.3	−1.44775
$57$	0	0
$58$	15563.2	0.607478
$59$	3481.00	0.130189
$59$	3481.00	0.130189
$60$	0	0
$61$	31646.6	1.08894	0.544468	−	0.838782i	$-0.316732\pi$
$61$	31646.6	1.08894	0.544468	+	0.838782i	$0.316732\pi$
$62$	6271.22	0.207192
$63$	0	0
$64$	25725.5	0.785080
$65$	−39800.0	−1.16842
$66$	0	0
$67$	−36947.6	−1.00554	−0.502770	−	0.864420i	$-0.667686\pi$
$67$	−36947.6	−1.00554	−0.502770	+	0.864420i	$0.667686\pi$
$68$	−5667.81	−0.148642
$69$	0	0
$70$	53150.6	1.29647
$71$	−67026.7	−1.57798	−0.788991	−	0.614405i	$-0.789396\pi$
$71$	−67026.7	−1.57798	−0.788991	+	0.614405i	$0.789396\pi$
$72$	0	0
$73$	15479.2	0.339971	0.169986	−	0.985447i	$-0.445628\pi$
$73$	15479.2	0.339971	0.169986	+	0.985447i	$0.445628\pi$
$74$	−19320.8	−0.410153
$75$	0	0
$76$	−2570.86	−0.0510557
$77$	−31050.8	−0.596823
$78$	0	0
$79$	45796.7	0.825594	0.412797	−	0.910823i	$-0.364552\pi$
$79$	45796.7	0.825594	0.412797	+	0.910823i	$0.364552\pi$
$80$	−48701.8	−0.850785
$81$	0	0
$82$	−89017.1	−1.46197
$83$	−16456.1	−0.262199	−0.131100	−	0.991369i	$-0.541851\pi$
$83$	−16456.1	−0.262199	−0.131100	+	0.991369i	$0.541851\pi$
$84$	0	0
$85$	−44878.6	−0.673739
$86$	81554.1	1.18905
$87$	0	0
$88$	24326.2	0.334863
$89$	28480.0	0.381123	0.190562	−	0.981675i	$-0.438969\pi$
$89$	28480.0	0.381123	0.190562	+	0.981675i	$0.438969\pi$
$90$	0	0
$91$	−198276.	−2.50995
$92$	−11604.9	−0.142947
$93$	0	0
$94$	−96527.9	−1.12676
$95$	−20356.4	−0.231416
$96$	0	0
$97$	79923.6	0.862473	0.431237	−	0.902239i	$-0.358077\pi$
$97$	79923.6	0.862473	0.431237	+	0.902239i	$0.358077\pi$
$98$	162168.	1.70569
$99$	0	0
$100$	−7272.74	−0.0727274
$101$	−134102.	−1.30808	−0.654038	−	0.756462i	$-0.726926\pi$
$101$	−134102.	−1.30808	−0.654038	+	0.756462i	$0.726926\pi$
$102$	0	0
$103$	−119371.	−1.10868	−0.554339	−	0.832291i	$-0.687029\pi$
$103$	−119371.	−1.10868	−0.554339	+	0.832291i	$0.687029\pi$
$104$	155336.	1.40828
$105$	0	0
$106$	−4560.52	−0.0394230
$107$	−61222.4	−0.516953	−0.258476	−	0.966018i	$-0.583220\pi$
$107$	−61222.4	−0.516953	−0.258476	+	0.966018i	$0.583220\pi$
$108$	0	0
$109$	118858.	0.958211	0.479106	−	0.877757i	$-0.340961\pi$
$109$	118858.	0.958211	0.479106	+	0.877757i	$0.340961\pi$
$110$	−38055.7	−0.299874
$111$	0	0
$112$	−242622.	−1.82762
$113$	−248492.	−1.83069	−0.915347	−	0.402665i	$-0.868084\pi$
$113$	−248492.	−1.83069	−0.915347	+	0.402665i	$0.868084\pi$
$114$	0	0
$115$	−91889.8	−0.647922
$116$	13456.7	0.0928522
$117$	0	0
$118$	21253.8	0.140518
$119$	−223576.	−1.44730
$120$	0	0
$121$	−138819.	−0.861955
$122$	193224.	1.17533
$123$	0	0
$124$	5422.37	0.0316690
$125$	−188217.	−1.07742
$126$	0	0
$127$	−184071.	−1.01269	−0.506343	−	0.862332i	$-0.669003\pi$
$127$	−184071.	−1.01269	−0.506343	+	0.862332i	$0.669003\pi$
$128$	217640.	1.17412
$129$	0	0
$130$	−243006.	−1.26112
$131$	256895.	1.30791	0.653953	−	0.756535i	$-0.273109\pi$
$131$	256895.	1.30791	0.653953	+	0.756535i	$0.273109\pi$
$132$	0	0
$133$	−101412.	−0.497117
$134$	−225590.	−1.08532
$135$	0	0
$136$	175157.	0.812044
$137$	118936.	0.541394	0.270697	−	0.962665i	$-0.412746\pi$
$137$	118936.	0.541394	0.270697	+	0.962665i	$0.412746\pi$
$138$	0	0
$139$	−147451.	−0.647306	−0.323653	−	0.946176i	$-0.604911\pi$
$139$	−147451.	−0.647306	−0.323653	+	0.946176i	$0.604911\pi$
$140$	45956.4	0.198164
$141$	0	0
$142$	−409243.	−1.70318
$143$	141965.	0.580551
$144$	0	0
$145$	106552.	0.420863
$146$	94511.0	0.366945
$147$	0	0
$148$	−16705.6	−0.0626913
$149$	−125781.	−0.464139	−0.232070	−	0.972699i	$-0.574550\pi$
$149$	−125781.	−0.464139	−0.232070	+	0.972699i	$0.574550\pi$
$150$	0	0
$151$	181920.	0.649288	0.324644	−	0.945836i	$-0.394755\pi$
$151$	181920.	0.649288	0.324644	+	0.945836i	$0.394755\pi$
$152$	79449.2	0.278921
$153$	0	0
$154$	−189586.	−0.644176
$155$	42935.1	0.143543
$156$	0	0
$157$	252024.	0.816004	0.408002	−	0.912981i	$-0.366226\pi$
$157$	252024.	0.816004	0.408002	+	0.912981i	$0.366226\pi$
$158$	279620.	0.891097
$159$	0	0
$160$	−79120.7	−0.244338
$161$	−457776.	−1.39184
$162$	0	0
$163$	−284724.	−0.839373	−0.419687	−	0.907669i	$-0.637860\pi$
$163$	−284724.	−0.839373	−0.419687	+	0.907669i	$0.637860\pi$
$164$	−76968.1	−0.223461
$165$	0	0
$166$	−100475.	−0.283002
$167$	71081.6	0.197227	0.0986134	−	0.995126i	$-0.468559\pi$
$167$	71081.6	0.197227	0.0986134	+	0.995126i	$0.468559\pi$
$168$	0	0
$169$	535226.	1.44152
$170$	−274014.	−0.727194
$171$	0	0
$172$	70515.2	0.181745
$173$	420683.	1.06866	0.534331	−	0.845276i	$-0.320564\pi$
$173$	420683.	1.06866	0.534331	+	0.845276i	$0.320564\pi$
$174$	0	0
$175$	−286885.	−0.708130
$176$	173717.	0.422727
$177$	0	0
$178$	173890.	0.411362
$179$	49547.5	0.115582	0.0577909	−	0.998329i	$-0.481594\pi$
$179$	49547.5	0.115582	0.0577909	+	0.998329i	$0.481594\pi$
$180$	0	0
$181$	−5910.71	−0.0134104	−0.00670522	−	0.999978i	$-0.502134\pi$
$181$	−5910.71	−0.0134104	−0.00670522	+	0.999978i	$0.502134\pi$
$182$	−1.21061e6	−2.70909
$183$	0	0
$184$	358637.	0.780927
$185$	−132277.	−0.284156
$186$	0	0
$187$	160080.	0.334759
$188$	−83462.2	−0.172225
$189$	0	0
$190$	−124290.	−0.249776
$191$	−745180.	−1.47801	−0.739006	−	0.673699i	$-0.764704\pi$
$191$	−745180.	−1.47801	−0.739006	+	0.673699i	$0.764704\pi$
$192$	0	0
$193$	−29909.0	−0.0577975	−0.0288987	−	0.999582i	$-0.509200\pi$
$193$	−29909.0	−0.0577975	−0.0288987	+	0.999582i	$0.509200\pi$
$194$	487987.	0.930902
$195$	0	0
$196$	140217.	0.260712
$197$	500243.	0.918365	0.459183	−	0.888342i	$-0.348142\pi$
$197$	500243.	0.918365	0.459183	+	0.888342i	$0.348142\pi$
$198$	0	0
$199$	−517673.	−0.926664	−0.463332	−	0.886185i	$-0.653346\pi$
$199$	−517673.	−0.926664	−0.463332	+	0.886185i	$0.653346\pi$
$200$	224755.	0.397315
$201$	0	0
$202$	−818785.	−1.41186
$203$	530820.	0.904081
$204$	0	0
$205$	−609445.	−1.01286
$206$	−728839.	−1.19664
$207$	0	0
$208$	1.10927e6	1.77779
$209$	72610.5	0.114983
$210$	0	0
$211$	1.25719e6	1.94400	0.971999	−	0.234984i	$-0.0755037\pi$
$211$	1.25719e6	1.94400	0.971999	+	0.234984i	$0.0755037\pi$
$212$	−3943.23	−0.00602576
$213$	0	0
$214$	−373804.	−0.557968
$215$	558350.	0.823779
$216$	0	0
$217$	213894.	0.308354
$218$	725707.	1.03424
$219$	0	0
$220$	−32904.6	−0.0458353
$221$	1.02219e6	1.40784
$222$	0	0
$223$	967522.	1.30286	0.651431	−	0.758708i	$-0.274169\pi$
$223$	967522.	1.30286	0.651431	+	0.758708i	$0.274169\pi$
$224$	−394163.	−0.524876
$225$	0	0
$226$	−1.51721e6	−1.97594
$227$	374162.	0.481943	0.240971	−	0.970532i	$-0.422534\pi$
$227$	374162.	0.481943	0.240971	+	0.970532i	$0.422534\pi$
$228$	0	0
$229$	65812.0	0.0829309	0.0414654	−	0.999140i	$-0.486797\pi$
$229$	65812.0	0.0829309	0.0414654	+	0.999140i	$0.486797\pi$
$230$	−561049.	−0.699328
$231$	0	0
$232$	−415862.	−0.507258
$233$	633048.	0.763918	0.381959	−	0.924179i	$-0.375249\pi$
$233$	633048.	0.763918	0.381959	+	0.924179i	$0.375249\pi$
$234$	0	0
$235$	−660866.	−0.780628
$236$	18377.0	0.0214780
$237$	0	0
$238$	−1.36508e6	−1.56213
$239$	−728479.	−0.824940	−0.412470	−	0.910971i	$-0.635334\pi$
$239$	−728479.	−0.824940	−0.412470	+	0.910971i	$0.635334\pi$
$240$	0	0
$241$	1.22438e6	1.35792	0.678960	−	0.734175i	$-0.262431\pi$
$241$	1.22438e6	1.35792	0.678960	+	0.734175i	$0.262431\pi$
$242$	−847582.	−0.930343
$243$	0	0
$244$	167070.	0.179648
$245$	1.11026e6	1.18171
$246$	0	0
$247$	463656.	0.483563
$248$	−167572.	−0.173010
$249$	0	0
$250$	−1.14919e6	−1.16290
$251$	388944.	0.389675	0.194838	−	0.980835i	$-0.437582\pi$
$251$	388944.	0.389675	0.194838	+	0.980835i	$0.437582\pi$
$252$	0	0
$253$	327767.	0.321931
$254$	−1.12387e6	−1.09303
$255$	0	0
$256$	505621.	0.482198
$257$	−1.35199e6	−1.27685	−0.638427	−	0.769683i	$-0.720415\pi$
$257$	−1.35199e6	−1.27685	−0.638427	+	0.769683i	$0.720415\pi$
$258$	0	0
$259$	−658980.	−0.610411
$260$	−210113.	−0.192761
$261$	0	0
$262$	1.56851e6	1.41168
$263$	685770.	0.611348	0.305674	−	0.952136i	$-0.401118\pi$
$263$	685770.	0.611348	0.305674	+	0.952136i	$0.401118\pi$
$264$	0	0
$265$	−31223.1	−0.0273125
$266$	−619186.	−0.536559
$267$	0	0
$268$	−195055.	−0.165890
$269$	−2.15145e6	−1.81281	−0.906403	−	0.422413i	$-0.861183\pi$
$269$	−2.15145e6	−1.81281	−0.906403	+	0.422413i	$0.861183\pi$
$270$	0	0
$271$	−2.36646e6	−1.95738	−0.978692	−	0.205336i	$-0.934171\pi$
$271$	−2.36646e6	−1.95738	−0.978692	+	0.205336i	$0.934171\pi$
$272$	1.25082e6	1.02511
$273$	0	0
$274$	726186.	0.584348
$275$	205409.	0.163790
$276$	0	0
$277$	2.37254e6	1.85786	0.928930	−	0.370254i	$-0.120729\pi$
$277$	2.37254e6	1.85786	0.928930	+	0.370254i	$0.120729\pi$
$278$	−900285.	−0.698663
$279$	0	0
$280$	−1.42023e6	−1.08259
$281$	386861.	0.292273	0.146137	−	0.989264i	$-0.453316\pi$
$281$	386861.	0.292273	0.146137	+	0.989264i	$0.453316\pi$
$282$	0	0
$283$	19175.2	0.0142323	0.00711613	−	0.999975i	$-0.497735\pi$
$283$	19175.2	0.0142323	0.00711613	+	0.999975i	$0.497735\pi$
$284$	−353849.	−0.260329
$285$	0	0
$286$	866790.	0.626612
$287$	−3.03613e6	−2.17578
$288$	0	0
$289$	−267230.	−0.188209
$290$	650571.	0.454255
$291$	0	0
$292$	81718.4	0.0560870
$293$	1.93969e6	1.31997	0.659985	−	0.751279i	$-0.270563\pi$
$293$	1.93969e6	1.31997	0.659985	+	0.751279i	$0.270563\pi$
$294$	0	0
$295$	145512.	0.0973517
$296$	516266.	0.342487
$297$	0	0
$298$	−767976.	−0.500965
$299$	2.09296e6	1.35389
$300$	0	0
$301$	2.78159e6	1.76961
$302$	1.11074e6	0.700803
$303$	0	0
$304$	567358.	0.352106
$305$	1.32288e6	0.814276
$306$	0	0
$307$	−110640.	−0.0669988	−0.0334994	−	0.999439i	$-0.510665\pi$
$307$	−110640.	−0.0669988	−0.0334994	+	0.999439i	$0.510665\pi$
$308$	−163924.	−0.0984615
$309$	0	0
$310$	262148.	0.154932
$311$	−2.13892e6	−1.25399	−0.626994	−	0.779024i	$-0.715715\pi$
$311$	−2.13892e6	−1.25399	−0.626994	+	0.779024i	$0.715715\pi$
$312$	0	0
$313$	2.94572e6	1.69953	0.849767	−	0.527158i	$-0.176742\pi$
$313$	2.94572e6	1.69953	0.849767	+	0.527158i	$0.176742\pi$
$314$	1.53877e6	0.880747
$315$	0	0
$316$	241771.	0.136203
$317$	−1.02771e6	−0.574412	−0.287206	−	0.957869i	$-0.592726\pi$
$317$	−1.02771e6	−0.574412	−0.287206	+	0.957869i	$0.592726\pi$
$318$	0	0
$319$	−380066.	−0.209113
$320$	1.07537e6	0.587061
$321$	0	0
$322$	−2.79503e6	−1.50227
$323$	522819.	0.278834
$324$	0	0
$325$	1.31164e6	0.688823
$326$	−1.73843e6	−0.905970
$327$	0	0
$328$	2.37861e6	1.22078
$329$	−3.29230e6	−1.67691
$330$	0	0
$331$	−1.99485e6	−1.00078	−0.500391	−	0.865799i	$-0.666811\pi$
$331$	−1.99485e6	−1.00078	−0.500391	+	0.865799i	$0.666811\pi$
$332$	−86875.4	−0.0432565
$333$	0	0
$334$	434001.	0.212875
$335$	−1.54447e6	−0.751915
$336$	0	0
$337$	−2.56238e6	−1.22905	−0.614523	−	0.788899i	$-0.710652\pi$
$337$	−2.56238e6	−1.22905	−0.614523	+	0.788899i	$0.710652\pi$
$338$	3.26791e6	1.55589
$339$	0	0
$340$	−236924.	−0.111151
$341$	−153148.	−0.0713221
$342$	0	0
$343$	2.03108e6	0.932162
$344$	−2.17919e6	−0.992884
$345$	0	0
$346$	2.56855e6	1.15345
$347$	1.22248e6	0.545026	0.272513	−	0.962152i	$-0.412145\pi$
$347$	1.22248e6	0.545026	0.272513	+	0.962152i	$0.412145\pi$
$348$	0	0
$349$	−3.21103e6	−1.41117	−0.705587	−	0.708623i	$-0.749317\pi$
$349$	−3.21103e6	−1.41117	−0.705587	+	0.708623i	$0.749317\pi$
$350$	−1.75163e6	−0.764313
$351$	0	0
$352$	282220.	0.121403
$353$	4.36299e6	1.86358	0.931789	−	0.363001i	$-0.118248\pi$
$353$	4.36299e6	1.86358	0.931789	+	0.363001i	$0.118248\pi$
$354$	0	0
$355$	−2.80183e6	−1.17997
$356$	150353.	0.0628761
$357$	0	0
$358$	302521.	0.124752
$359$	−38987.2	−0.0159656	−0.00798281	−	0.999968i	$-0.502541\pi$
$359$	−38987.2	−0.0159656	−0.00798281	+	0.999968i	$0.502541\pi$
$360$	0	0
$361$	−2.23895e6	−0.904226
$362$	−36088.9	−0.0144744
$363$	0	0
$364$	−1.04674e6	−0.414082
$365$	647059.	0.254221
$366$	0	0
$367$	−615070.	−0.238374	−0.119187	−	0.992872i	$-0.538029\pi$
$367$	−615070.	−0.238374	−0.119187	+	0.992872i	$0.538029\pi$
$368$	2.56108e6	0.985833
$369$	0	0
$370$	−807643.	−0.306701
$371$	−155547.	−0.0586715
$372$	0	0
$373$	−2.59722e6	−0.966579	−0.483289	−	0.875461i	$-0.660558\pi$
$373$	−2.59722e6	−0.966579	−0.483289	+	0.875461i	$0.660558\pi$
$374$	977394.	0.361319
$375$	0	0
$376$	2.57930e6	0.940875
$377$	−2.42692e6	−0.879431
$378$	0	0
$379$	−294437.	−0.105292	−0.0526459	−	0.998613i	$-0.516765\pi$
$379$	−294437.	−0.105292	−0.0526459	+	0.998613i	$0.516765\pi$
$380$	−107466.	−0.0381780
$381$	0	0
$382$	−4.54983e6	−1.59528
$383$	−2.51121e6	−0.874753	−0.437377	−	0.899278i	$-0.644092\pi$
$383$	−2.51121e6	−0.874753	−0.437377	+	0.899278i	$0.644092\pi$
$384$	0	0
$385$	−1.29798e6	−0.446288
$386$	−182615.	−0.0623831
$387$	0	0
$388$	421935.	0.142287
$389$	3.42294e6	1.14690	0.573449	−	0.819241i	$-0.305605\pi$
$389$	3.42294e6	1.14690	0.573449	+	0.819241i	$0.305605\pi$
$390$	0	0
$391$	2.36003e6	0.780684
$392$	−4.33325e6	−1.42429
$393$	0	0
$394$	3.05432e6	0.991229
$395$	1.91438e6	0.617356
$396$	0	0
$397$	−4.77198e6	−1.51957	−0.759787	−	0.650172i	$-0.774697\pi$
$397$	−4.77198e6	−1.51957	−0.759787	+	0.650172i	$0.774697\pi$
$398$	−3.16074e6	−1.00019
$399$	0	0
$400$	1.60501e6	0.501565
$401$	−1.57375e6	−0.488737	−0.244369	−	0.969682i	$-0.578581\pi$
$401$	−1.57375e6	−0.488737	−0.244369	+	0.969682i	$0.578581\pi$
$402$	0	0
$403$	−977928.	−0.299947
$404$	−707957.	−0.215801
$405$	0	0
$406$	3.24101e6	0.975811
$407$	471828.	0.141188
$408$	0	0
$409$	−4.72041e6	−1.39531	−0.697657	−	0.716432i	$-0.745774\pi$
$409$	−4.72041e6	−1.39531	−0.697657	+	0.716432i	$0.745774\pi$
$410$	−3.72107e6	−1.09322
$411$	0	0
$412$	−630186.	−0.182905
$413$	724911.	0.209127
$414$	0	0
$415$	−687893.	−0.196065
$416$	1.80212e6	0.510565
$417$	0	0
$418$	443336.	0.124106
$419$	1.33538e6	0.371595	0.185798	−	0.982588i	$-0.440513\pi$
$419$	1.33538e6	0.371595	0.185798	+	0.982588i	$0.440513\pi$
$420$	0	0
$421$	−338764.	−0.0931521	−0.0465760	−	0.998915i	$-0.514831\pi$
$421$	−338764.	−0.0931521	−0.0465760	+	0.998915i	$0.514831\pi$
$422$	7.67601e6	2.09824
$423$	0	0
$424$	121861.	0.0329192
$425$	1.47901e6	0.397191
$426$	0	0
$427$	6.59033e6	1.74919
$428$	−323207.	−0.0852848
$429$	0	0
$430$	3.40910e6	0.889138
$431$	7.24422e6	1.87844	0.939222	−	0.343311i	$-0.111548\pi$
$431$	7.24422e6	1.87844	0.939222	+	0.343311i	$0.111548\pi$
$432$	0	0
$433$	6.16246e6	1.57955	0.789777	−	0.613394i	$-0.210196\pi$
$433$	6.16246e6	1.57955	0.789777	+	0.613394i	$0.210196\pi$
$434$	1.30597e6	0.332819
$435$	0	0
$436$	627478.	0.158082
$437$	1.07048e6	0.268149
$438$	0	0
$439$	4.65424e6	1.15262	0.576311	−	0.817230i	$-0.304492\pi$
$439$	4.65424e6	1.15262	0.576311	+	0.817230i	$0.304492\pi$
$440$	1.01688e6	0.250402
$441$	0	0
$442$	6.24117e6	1.51953
$443$	1.25610e6	0.304098	0.152049	−	0.988373i	$-0.451413\pi$
$443$	1.25610e6	0.304098	0.152049	+	0.988373i	$0.451413\pi$
$444$	0	0
$445$	1.19051e6	0.284993
$446$	5.90737e6	1.40623
$447$	0	0
$448$	5.35728e6	1.26110
$449$	4.46355e6	1.04488	0.522438	−	0.852677i	$-0.325023\pi$
$449$	4.46355e6	1.04488	0.522438	+	0.852677i	$0.325023\pi$
$450$	0	0
$451$	2.17386e6	0.503258
$452$	−1.31185e6	−0.302021
$453$	0	0
$454$	2.28451e6	0.520180
$455$	−8.28826e6	−1.87687
$456$	0	0
$457$	4.01815e6	0.899986	0.449993	−	0.893032i	$-0.351427\pi$
$457$	4.01815e6	0.899986	0.449993	+	0.893032i	$0.351427\pi$
$458$	401826.	0.0895106
$459$	0	0
$460$	−485107.	−0.106891
$461$	−1.98402e6	−0.434804	−0.217402	−	0.976082i	$-0.569758\pi$
$461$	−1.98402e6	−0.434804	−0.217402	+	0.976082i	$0.569758\pi$
$462$	0	0
$463$	2.31083e6	0.500975	0.250488	−	0.968120i	$-0.419409\pi$
$463$	2.31083e6	0.500975	0.250488	+	0.968120i	$0.419409\pi$
$464$	−2.96973e6	−0.640357
$465$	0	0
$466$	3.86518e6	0.824528
$467$	−2.65831e6	−0.564044	−0.282022	−	0.959408i	$-0.591005\pi$
$467$	−2.65831e6	−0.564044	−0.282022	+	0.959408i	$0.591005\pi$
$468$	0	0
$469$	−7.69426e6	−1.61523
$470$	−4.03503e6	−0.842563
$471$	0	0
$472$	−567919.	−0.117336
$473$	−1.99161e6	−0.409309
$474$	0	0
$475$	670864.	0.136427
$476$	−1.18031e6	−0.238769
$477$	0	0
$478$	−4.44785e6	−0.890391
$479$	6.22698e6	1.24005	0.620024	−	0.784582i	$-0.287123\pi$
$479$	6.22698e6	1.24005	0.620024	+	0.784582i	$0.287123\pi$
$480$	0	0
$481$	3.01287e6	0.593768
$482$	7.47567e6	1.46566
$483$	0	0
$484$	−732856.	−0.142202
$485$	3.34095e6	0.644934
$486$	0	0
$487$	8.29735e6	1.58532	0.792660	−	0.609664i	$-0.208696\pi$
$487$	8.29735e6	1.58532	0.792660	+	0.609664i	$0.208696\pi$
$488$	−5.16308e6	−0.981431
$489$	0	0
$490$	6.77890e6	1.27547
$491$	6.71368e6	1.25677	0.628386	−	0.777901i	$-0.283716\pi$
$491$	6.71368e6	1.25677	0.628386	+	0.777901i	$0.283716\pi$
$492$	0	0
$493$	−2.73660e6	−0.507100
$494$	2.83093e6	0.521930
$495$	0	0
$496$	−1.19665e6	−0.218406
$497$	−1.39582e7	−2.53476
$498$	0	0
$499$	−1.07848e7	−1.93892	−0.969458	−	0.245258i	$-0.921127\pi$
$499$	−1.07848e7	−1.93892	−0.969458	+	0.245258i	$0.921127\pi$
$500$	−993641.	−0.177748
$501$	0	0
$502$	2.37477e6	0.420592
$503$	6.41855e6	1.13114	0.565570	−	0.824700i	$-0.308656\pi$
$503$	6.41855e6	1.13114	0.565570	+	0.824700i	$0.308656\pi$
$504$	0	0
$505$	−5.60571e6	−0.978143
$506$	2.00124e6	0.347474
$507$	0	0
$508$	−971751.	−0.167069
$509$	−2.78903e6	−0.477153	−0.238577	−	0.971124i	$-0.576681\pi$
$509$	−2.78903e6	−0.477153	−0.238577	+	0.971124i	$0.576681\pi$
$510$	0	0
$511$	3.22351e6	0.546107
$512$	−3.87732e6	−0.653667
$513$	0	0
$514$	−8.25482e6	−1.37816
$515$	−4.98991e6	−0.829038
$516$	0	0
$517$	2.35728e6	0.387869
$518$	−4.02351e6	−0.658842
$519$	0	0
$520$	6.49330e6	1.05307
$521$	−6.38091e6	−1.02988	−0.514942	−	0.857225i	$-0.672187\pi$
$521$	−6.38091e6	−1.02988	−0.514942	+	0.857225i	$0.672187\pi$
$522$	0	0
$523$	−2.42857e6	−0.388237	−0.194119	−	0.980978i	$-0.562185\pi$
$523$	−2.42857e6	−0.388237	−0.194119	+	0.980978i	$0.562185\pi$
$524$	1.35621e6	0.215773
$525$	0	0
$526$	4.18708e6	0.659853
$527$	−1.10271e6	−0.172956
$528$	0	0
$529$	−1.60414e6	−0.249231
$530$	−190638.	−0.0294794
$531$	0	0
$532$	−535375.	−0.0820124
$533$	1.38812e7	2.11646
$534$	0	0
$535$	−2.55920e6	−0.386563
$536$	6.02793e6	0.906268
$537$	0	0
$538$	−1.31361e7	−1.95664
$539$	−3.96025e6	−0.587153
$540$	0	0
$541$	9.78542e6	1.43743	0.718714	−	0.695305i	$-0.244731\pi$
$541$	9.78542e6	1.43743	0.718714	+	0.695305i	$0.244731\pi$
$542$	−1.44488e7	−2.11268
$543$	0	0
$544$	2.03208e6	0.294403
$545$	4.96846e6	0.716524
$546$	0	0
$547$	1.26732e7	1.81100	0.905501	−	0.424345i	$-0.139496\pi$
$547$	1.26732e7	1.81100	0.905501	+	0.424345i	$0.139496\pi$
$548$	627892.	0.0893169
$549$	0	0
$550$	1.25416e6	0.176785
$551$	−1.24129e6	−0.174179
$552$	0	0
$553$	9.53707e6	1.32618
$554$	1.44859e7	2.00526
$555$	0	0
$556$	−778426.	−0.106790
$557$	−9.24073e6	−1.26203	−0.631013	−	0.775773i	$-0.717360\pi$
$557$	−9.24073e6	−1.26203	−0.631013	+	0.775773i	$0.717360\pi$
$558$	0	0
$559$	−1.27175e7	−1.72136
$560$	−1.01420e7	−1.36664
$561$	0	0
$562$	2.36205e6	0.315462
$563$	−1.33652e7	−1.77707	−0.888535	−	0.458809i	$-0.848276\pi$
$563$	−1.33652e7	−1.77707	−0.888535	+	0.458809i	$0.848276\pi$
$564$	0	0
$565$	−1.03874e7	−1.36894
$566$	117078.	0.0153615
$567$	0	0
$568$	1.09353e7	1.42220
$569$	8.45895e6	1.09531	0.547653	−	0.836705i	$-0.315521\pi$
$569$	8.45895e6	1.09531	0.547653	+	0.836705i	$0.315521\pi$
$570$	0	0
$571$	1.49334e6	0.191677	0.0958385	−	0.995397i	$-0.469447\pi$
$571$	1.49334e6	0.191677	0.0958385	+	0.995397i	$0.469447\pi$
$572$	749464.	0.0957769
$573$	0	0
$574$	−1.85376e7	−2.34841
$575$	3.02831e6	0.381971
$576$	0	0
$577$	2.62239e6	0.327912	0.163956	−	0.986468i	$-0.447575\pi$
$577$	2.62239e6	0.327912	0.163956	+	0.986468i	$0.447575\pi$
$578$	−1.63162e6	−0.203142
$579$	0	0
$580$	562512.	0.0694323
$581$	−3.42694e6	−0.421179
$582$	0	0
$583$	111371.	0.0135707
$584$	−2.52541e6	−0.306408
$585$	0	0
$586$	1.18431e7	1.42470
$587$	−9.30156e6	−1.11419	−0.557097	−	0.830448i	$-0.688085\pi$
$587$	−9.30156e6	−1.11419	−0.557097	+	0.830448i	$0.688085\pi$
$588$	0	0
$589$	−500179.	−0.0594070
$590$	888448.	0.105076
$591$	0	0
$592$	3.68673e6	0.432352
$593$	1.04915e7	1.22518	0.612592	−	0.790399i	$-0.290127\pi$
$593$	1.04915e7	1.22518	0.612592	+	0.790399i	$0.290127\pi$
$594$	0	0
$595$	−9.34586e6	−1.08225
$596$	−664025.	−0.0765718
$597$	0	0
$598$	1.27789e7	1.46131
$599$	3.77401e6	0.429770	0.214885	−	0.976639i	$-0.431062\pi$
$599$	3.77401e6	0.429770	0.214885	+	0.976639i	$0.431062\pi$
$600$	0	0
$601$	−7.42586e6	−0.838611	−0.419305	−	0.907845i	$-0.637726\pi$
$601$	−7.42586e6	−0.838611	−0.419305	+	0.907845i	$0.637726\pi$
$602$	1.69835e7	1.91001
$603$	0	0
$604$	960396.	0.107117
$605$	−5.80287e6	−0.644546
$606$	0	0
$607$	9.37397e6	1.03265	0.516323	−	0.856394i	$-0.327300\pi$
$607$	9.37397e6	1.03265	0.516323	+	0.856394i	$0.327300\pi$
$608$	921729.	0.101122
$609$	0	0
$610$	8.07709e6	0.878881
$611$	1.50525e7	1.63119
$612$	0	0
$613$	−1.51042e7	−1.62348	−0.811740	−	0.584019i	$-0.801479\pi$
$613$	−1.51042e7	−1.62348	−0.811740	+	0.584019i	$0.801479\pi$
$614$	−675533.	−0.0723145
$615$	0	0
$616$	5.06588e6	0.537902
$617$	−1.17572e7	−1.24335	−0.621674	−	0.783276i	$-0.713547\pi$
$617$	−1.17572e7	−1.24335	−0.621674	+	0.783276i	$0.713547\pi$
$618$	0	0
$619$	−1.34123e7	−1.40695	−0.703473	−	0.710722i	$-0.748368\pi$
$619$	−1.34123e7	−1.40695	−0.703473	+	0.710722i	$0.748368\pi$
$620$	226664.	0.0236812
$621$	0	0
$622$	−1.30595e7	−1.35348
$623$	5.93090e6	0.612210
$624$	0	0
$625$	−3.56276e6	−0.364827
$626$	1.79856e7	1.83438
$627$	0	0
$628$	1.33049e6	0.134621
$629$	3.39731e6	0.342381
$630$	0	0
$631$	−1.63331e7	−1.63303	−0.816516	−	0.577323i	$-0.804097\pi$
$631$	−1.63331e7	−1.63303	−0.816516	+	0.577323i	$0.804097\pi$
$632$	−7.47165e6	−0.744087
$633$	0	0
$634$	−6.27488e6	−0.619986
$635$	−7.69447e6	−0.757259
$636$	0	0
$637$	−2.52883e7	−2.46928
$638$	−2.32056e6	−0.225705
$639$	0	0
$640$	9.09773e6	0.877977
$641$	−3.02654e6	−0.290938	−0.145469	−	0.989363i	$-0.546469\pi$
$641$	−3.02654e6	−0.290938	−0.145469	+	0.989363i	$0.546469\pi$
$642$	0	0
$643$	−1.43363e7	−1.36744	−0.683722	−	0.729743i	$-0.739640\pi$
$643$	−1.43363e7	−1.36744	−0.683722	+	0.729743i	$0.739640\pi$
$644$	−2.41671e6	−0.229620
$645$	0	0
$646$	3.19216e6	0.300957
$647$	8.34526e6	0.783753	0.391876	−	0.920018i	$-0.371826\pi$
$647$	8.34526e6	0.783753	0.391876	+	0.920018i	$0.371826\pi$
$648$	0	0
$649$	−519034.	−0.0483709
$650$	8.00847e6	0.743474
$651$	0	0
$652$	−1.50312e6	−0.138476
$653$	1.91948e7	1.76157	0.880787	−	0.473513i	$-0.157014\pi$
$653$	1.91948e7	1.76157	0.880787	+	0.473513i	$0.157014\pi$
$654$	0	0
$655$	1.07386e7	0.978016
$656$	1.69860e7	1.54110
$657$	0	0
$658$	−2.01017e7	−1.80996
$659$	1.01224e7	0.907967	0.453984	−	0.891010i	$-0.350002\pi$
$659$	1.01224e7	0.907967	0.453984	+	0.891010i	$0.350002\pi$
$660$	0	0
$661$	1.37317e7	1.22242	0.611211	−	0.791467i	$-0.290683\pi$
$661$	1.37317e7	1.22242	0.611211	+	0.791467i	$0.290683\pi$
$662$	−1.21799e7	−1.08019
$663$	0	0
$664$	2.68478e6	0.236314
$665$	−4.23919e6	−0.371731
$666$	0	0
$667$	−5.60324e6	−0.487669
$668$	375256.	0.0325377
$669$	0	0
$670$	−9.43005e6	−0.811572
$671$	−4.71866e6	−0.404587
$672$	0	0
$673$	−1.99795e7	−1.70038	−0.850191	−	0.526475i	$-0.823513\pi$
$673$	−1.99795e7	−1.70038	−0.850191	+	0.526475i	$0.823513\pi$
$674$	−1.56450e7	−1.32656
$675$	0	0
$676$	2.82558e6	0.237816
$677$	1.45995e7	1.22424	0.612118	−	0.790767i	$-0.290318\pi$
$677$	1.45995e7	1.22424	0.612118	+	0.790767i	$0.290318\pi$
$678$	0	0
$679$	1.66439e7	1.38542
$680$	7.32186e6	0.607224
$681$	0	0
$682$	−935069.	−0.0769809
$683$	−1.29809e7	−1.06477	−0.532383	−	0.846504i	$-0.678703\pi$
$683$	−1.29809e7	−1.06477	−0.532383	+	0.846504i	$0.678703\pi$
$684$	0	0
$685$	4.97175e6	0.404839
$686$	1.24011e7	1.00612
$687$	0	0
$688$	−1.55619e7	−1.25340
$689$	711164.	0.0570718
$690$	0	0
$691$	−1.24354e7	−0.990753	−0.495376	−	0.868678i	$-0.664970\pi$
$691$	−1.24354e7	−0.990753	−0.495376	+	0.868678i	$0.664970\pi$
$692$	2.22088e6	0.176303
$693$	0	0
$694$	7.46404e6	0.588268
$695$	−6.16369e6	−0.484038
$696$	0	0
$697$	1.56525e7	1.22040
$698$	−1.96055e7	−1.52314
$699$	0	0
$700$	−1.51453e6	−0.116824
$701$	3.72318e6	0.286166	0.143083	−	0.989711i	$-0.454298\pi$
$701$	3.72318e6	0.286166	0.143083	+	0.989711i	$0.454298\pi$
$702$	0	0
$703$	1.54099e6	0.117601
$704$	−3.83580e6	−0.291692
$705$	0	0
$706$	2.66390e7	2.01144
$707$	−2.79265e7	−2.10120
$708$	0	0
$709$	−9.20964e6	−0.688061	−0.344030	−	0.938958i	$-0.611792\pi$
$709$	−9.20964e6	−0.688061	−0.344030	+	0.938958i	$0.611792\pi$
$710$	−1.71071e7	−1.27359
$711$	0	0
$712$	−4.64646e6	−0.343497
$713$	−2.25783e6	−0.166329
$714$	0	0
$715$	5.93437e6	0.434120
$716$	261573.	0.0190682
$717$	0	0
$718$	−238043.	−0.0172324
$719$	4.15670e6	0.299865	0.149933	−	0.988696i	$-0.452094\pi$
$719$	4.15670e6	0.299865	0.149933	+	0.988696i	$0.452094\pi$
$720$	0	0
$721$	−2.48587e7	−1.78090
$722$	−1.36703e7	−0.975968
$723$	0	0
$724$	−31204.0	−0.00221240
$725$	−3.51151e6	−0.248113
$726$	0	0
$727$	−1.87342e6	−0.131462	−0.0657309	−	0.997837i	$-0.520938\pi$
$727$	−1.87342e6	−0.131462	−0.0657309	+	0.997837i	$0.520938\pi$
$728$	3.23483e7	2.26216
$729$	0	0
$730$	3.95073e6	0.274391
$731$	−1.43402e7	−0.992575
$732$	0	0
$733$	−1.70190e7	−1.16997	−0.584986	−	0.811044i	$-0.698900\pi$
$733$	−1.70190e7	−1.16997	−0.584986	+	0.811044i	$0.698900\pi$
$734$	−3.75542e6	−0.257287
$735$	0	0
$736$	4.16072e6	0.283122
$737$	5.50907e6	0.373602
$738$	0	0
$739$	−1.51158e7	−1.01817	−0.509084	−	0.860717i	$-0.670016\pi$
$739$	−1.51158e7	−1.01817	−0.509084	+	0.860717i	$0.670016\pi$
$740$	−698323.	−0.0468789
$741$	0	0
$742$	−949719.	−0.0633265
$743$	−1.88824e7	−1.25483	−0.627416	−	0.778684i	$-0.715888\pi$
$743$	−1.88824e7	−1.25483	−0.627416	+	0.778684i	$0.715888\pi$
$744$	0	0
$745$	−5.25786e6	−0.347071
$746$	−1.58578e7	−1.04327
$747$	0	0
$748$	845098.	0.0552272
$749$	−1.27494e7	−0.830398
$750$	0	0
$751$	−1.00270e7	−0.648739	−0.324369	−	0.945931i	$-0.605152\pi$
$751$	−1.00270e7	−0.648739	−0.324369	+	0.945931i	$0.605152\pi$
$752$	1.84191e7	1.18775
$753$	0	0
$754$	−1.48180e7	−0.949206
$755$	7.60456e6	0.485520
$756$	0	0
$757$	−2.49293e7	−1.58114	−0.790570	−	0.612372i	$-0.790215\pi$
$757$	−2.49293e7	−1.58114	−0.790570	+	0.612372i	$0.790215\pi$
$758$	−1.79774e6	−0.113646
$759$	0	0
$760$	3.32112e6	0.208569
$761$	1.02441e7	0.641229	0.320614	−	0.947210i	$-0.396111\pi$
$761$	1.02441e7	0.641229	0.320614	+	0.947210i	$0.396111\pi$
$762$	0	0
$763$	2.47519e7	1.53921
$764$	−3.93398e6	−0.243836
$765$	0	0
$766$	−1.53326e7	−0.944157
$767$	−3.31431e6	−0.203425
$768$	0	0
$769$	2.17564e7	1.32670	0.663348	−	0.748311i	$-0.269135\pi$
$769$	2.17564e7	1.32670	0.663348	+	0.748311i	$0.269135\pi$
$770$	−7.92502e6	−0.481697
$771$	0	0
$772$	−157897.	−0.00953519
$773$	−2.30398e7	−1.38685	−0.693425	−	0.720529i	$-0.743899\pi$
$773$	−2.30398e7	−1.38685	−0.693425	+	0.720529i	$0.743899\pi$
$774$	0	0
$775$	−1.41497e6	−0.0846236
$776$	−1.30394e7	−0.777326
$777$	0	0
$778$	2.08993e7	1.23789
$779$	7.09982e6	0.419183
$780$	0	0
$781$	9.99401e6	0.586289
$782$	1.44096e7	0.842624
$783$	0	0
$784$	−3.09443e7	−1.79801
$785$	1.05350e7	0.610186
$786$	0	0
$787$	3.85957e6	0.222127	0.111064	−	0.993813i	$-0.464574\pi$
$787$	3.85957e6	0.222127	0.111064	+	0.993813i	$0.464574\pi$
$788$	2.64090e6	0.151508
$789$	0	0
$790$	1.16886e7	0.666338
$791$	−5.17479e7	−2.94071
$792$	0	0
$793$	−3.01311e7	−1.70150
$794$	−2.91361e7	−1.64014
$795$	0	0
$796$	−2.73291e6	−0.152877
$797$	−1.58425e7	−0.883444	−0.441722	−	0.897152i	$-0.645632\pi$
$797$	−1.58425e7	−0.883444	−0.441722	+	0.897152i	$0.645632\pi$
$798$	0	0
$799$	1.69732e7	0.940582
$800$	2.60749e6	0.144045
$801$	0	0
$802$	−9.60881e6	−0.527514
$803$	−2.30803e6	−0.126314
$804$	0	0
$805$	−1.91359e7	−1.04078
$806$	−5.97091e6	−0.323745
$807$	0	0
$808$	2.18786e7	1.17894
$809$	−1.45751e7	−0.782961	−0.391481	−	0.920186i	$-0.628037\pi$
$809$	−1.45751e7	−0.782961	−0.391481	+	0.920186i	$0.628037\pi$
$810$	0	0
$811$	3.03425e7	1.61994	0.809970	−	0.586472i	$-0.199484\pi$
$811$	3.03425e7	1.61994	0.809970	+	0.586472i	$0.199484\pi$
$812$	2.80232e6	0.149152
$813$	0	0
$814$	2.88082e6	0.152390
$815$	−1.19020e7	−0.627660
$816$	0	0
$817$	−6.50458e6	−0.340929
$818$	−2.88213e7	−1.50602
$819$	0	0
$820$	−3.21740e6	−0.167098
$821$	−3.22288e6	−0.166873	−0.0834366	−	0.996513i	$-0.526590\pi$
$821$	−3.22288e6	−0.166873	−0.0834366	+	0.996513i	$0.526590\pi$
$822$	0	0
$823$	−4.75007e6	−0.244456	−0.122228	−	0.992502i	$-0.539004\pi$
$823$	−4.75007e6	−0.244456	−0.122228	+	0.992502i	$0.539004\pi$
$824$	1.94751e7	0.999223
$825$	0	0
$826$	4.42607e6	0.225719
$827$	−1.44496e7	−0.734671	−0.367336	−	0.930088i	$-0.619730\pi$
$827$	−1.44496e7	−0.734671	−0.367336	+	0.930088i	$0.619730\pi$
$828$	0	0
$829$	−2.10004e7	−1.06131	−0.530653	−	0.847589i	$-0.678053\pi$
$829$	−2.10004e7	−1.06131	−0.530653	+	0.847589i	$0.678053\pi$
$830$	−4.20005e6	−0.211621
$831$	0	0
$832$	−2.44936e7	−1.22672
$833$	−2.85151e7	−1.42385
$834$	0	0
$835$	2.97133e6	0.147481
$836$	383327.	0.0189694
$837$	0	0
$838$	8.15340e6	0.401078
$839$	−1.45792e6	−0.0715040	−0.0357520	−	0.999361i	$-0.511383\pi$
$839$	−1.45792e6	−0.0715040	−0.0357520	+	0.999361i	$0.511383\pi$
$840$	0	0
$841$	−1.40138e7	−0.683230
$842$	−2.06838e6	−0.100543
$843$	0	0
$844$	6.63701e6	0.320713
$845$	2.23734e7	1.07793
$846$	0	0
$847$	−2.89087e7	−1.38459
$848$	870224.	0.0415568
$849$	0	0
$850$	9.03036e6	0.428704
$851$	6.95607e6	0.329261
$852$	0	0
$853$	−1.47110e7	−0.692262	−0.346131	−	0.938186i	$-0.612505\pi$
$853$	−1.47110e7	−0.692262	−0.346131	+	0.938186i	$0.612505\pi$
$854$	4.02384e7	1.88797
$855$	0	0
$856$	9.98833e6	0.465917
$857$	−2.25591e7	−1.04923	−0.524615	−	0.851340i	$-0.675791\pi$
$857$	−2.25591e7	−1.04923	−0.524615	+	0.851340i	$0.675791\pi$
$858$	0	0
$859$	3.74722e7	1.73271	0.866356	−	0.499427i	$-0.166456\pi$
$859$	3.74722e7	1.73271	0.866356	+	0.499427i	$0.166456\pi$
$860$	2.94766e6	0.135904
$861$	0	0
$862$	4.42308e7	2.02748
$863$	−7.18311e6	−0.328311	−0.164156	−	0.986434i	$-0.552490\pi$
$863$	−7.18311e6	−0.328311	−0.164156	+	0.986434i	$0.552490\pi$
$864$	0	0
$865$	1.75853e7	0.799116
$866$	3.76260e7	1.70488
$867$	0	0
$868$	1.12920e6	0.0508710
$869$	−6.82851e6	−0.306744
$870$	0	0
$871$	3.51783e7	1.57119
$872$	−1.93914e7	−0.863612
$873$	0	0
$874$	6.53602e6	0.289424
$875$	−3.91958e7	−1.73069
$876$	0	0
$877$	4.17457e7	1.83279	0.916394	−	0.400277i	$-0.131086\pi$
$877$	4.17457e7	1.83279	0.916394	+	0.400277i	$0.131086\pi$
$878$	2.84172e7	1.24407
$879$	0	0
$880$	7.26167e6	0.316104
$881$	8.72480e6	0.378718	0.189359	−	0.981908i	$-0.439359\pi$
$881$	8.72480e6	0.378718	0.189359	+	0.981908i	$0.439359\pi$
$882$	0	0
$883$	2.98453e6	0.128817	0.0644087	−	0.997924i	$-0.479484\pi$
$883$	2.98453e6	0.128817	0.0644087	+	0.997924i	$0.479484\pi$
$884$	5.39639e6	0.232259
$885$	0	0
$886$	7.66932e6	0.328226
$887$	2.69623e7	1.15066	0.575332	−	0.817920i	$-0.304873\pi$
$887$	2.69623e7	1.15066	0.575332	+	0.817920i	$0.304873\pi$
$888$	0	0
$889$	−3.83323e7	−1.62671
$890$	7.26889e6	0.307605
$891$	0	0
$892$	5.10777e6	0.214941
$893$	7.69886e6	0.323071
$894$	0	0
$895$	2.07117e6	0.0864288
$896$	4.53230e7	1.88603
$897$	0	0
$898$	2.72530e7	1.12778
$899$	2.61809e6	0.108040
$900$	0	0
$901$	801910.	0.0329089
$902$	1.32729e7	0.543187
$903$	0	0
$904$	4.05410e7	1.64996
$905$	−247078.	−0.0100280
$906$	0	0
$907$	−1.11838e7	−0.451409	−0.225704	−	0.974196i	$-0.572468\pi$
$907$	−1.11838e7	−0.451409	−0.225704	+	0.974196i	$0.572468\pi$
$908$	1.97529e6	0.0795090
$909$	0	0
$910$	−5.06054e7	−2.02579
$911$	1.36404e7	0.544542	0.272271	−	0.962221i	$-0.412225\pi$
$911$	1.36404e7	0.544542	0.272271	+	0.962221i	$0.412225\pi$
$912$	0	0
$913$	2.45368e6	0.0974185
$914$	2.45335e7	0.971391
$915$	0	0
$916$	347437.	0.0136816
$917$	5.34977e7	2.10093
$918$	0	0
$919$	3.56348e7	1.39183	0.695914	−	0.718125i	$-0.254999\pi$
$919$	3.56348e7	1.39183	0.695914	+	0.718125i	$0.254999\pi$
$920$	1.49917e7	0.583956
$921$	0	0
$922$	−1.21138e7	−0.469302
$923$	6.38170e7	2.46565
$924$	0	0
$925$	4.35932e6	0.167519
$926$	1.41092e7	0.540723
$927$	0	0
$928$	−4.82461e6	−0.183905
$929$	2.06763e7	0.786021	0.393011	−	0.919534i	$-0.371434\pi$
$929$	2.06763e7	0.786021	0.393011	+	0.919534i	$0.371434\pi$
$930$	0	0
$931$	−1.29342e7	−0.489062
$932$	3.34201e6	0.126028
$933$	0	0
$934$	−1.62307e7	−0.608795
$935$	6.69161e6	0.250324
$936$	0	0
$937$	964568.	0.0358909	0.0179454	−	0.999839i	$-0.494287\pi$
$937$	964568.	0.0358909	0.0179454	+	0.999839i	$0.494287\pi$
$938$	−4.69786e7	−1.74338
$939$	0	0
$940$	−3.48887e6	−0.128785
$941$	−3.97475e7	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$941$	−3.97475e7	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$942$	0	0
$943$	3.20489e7	1.17364
$944$	−4.05559e6	−0.148124
$945$	0	0
$946$	−1.21601e7	−0.441784
$947$	−4.51166e6	−0.163479	−0.0817395	−	0.996654i	$-0.526048\pi$
$947$	−4.51166e6	−0.163479	−0.0817395	+	0.996654i	$0.526048\pi$
$948$	0	0
$949$	−1.47380e7	−0.531217
$950$	4.09608e6	0.147251
$951$	0	0
$952$	3.64760e7	1.30441
$953$	2.78914e7	0.994807	0.497403	−	0.867519i	$-0.334287\pi$
$953$	2.78914e7	0.994807	0.497403	+	0.867519i	$0.334287\pi$
$954$	0	0
$955$	−3.11498e7	−1.10522
$956$	−3.84581e6	−0.136095
$957$	0	0
$958$	3.80199e7	1.33843
$959$	2.47682e7	0.869658
$960$	0	0
$961$	−2.75742e7	−0.963151
$962$	1.83956e7	0.640878
$963$	0	0
$964$	6.46379e6	0.224024
$965$	−1.25025e6	−0.0432193
$966$	0	0
$967$	8.43317e6	0.290018	0.145009	−	0.989430i	$-0.453679\pi$
$967$	8.43317e6	0.290018	0.145009	+	0.989430i	$0.453679\pi$
$968$	2.26480e7	0.776859
$969$	0	0
$970$	2.03987e7	0.696103
$971$	7.74992e6	0.263784	0.131892	−	0.991264i	$-0.457895\pi$
$971$	7.74992e6	0.263784	0.131892	+	0.991264i	$0.457895\pi$
$972$	0	0
$973$	−3.07063e7	−1.03979
$974$	5.06609e7	1.71110
$975$	0	0
$976$	−3.68703e7	−1.23895
$977$	2.03500e6	0.0682069	0.0341035	−	0.999418i	$-0.489142\pi$
$977$	2.03500e6	0.0682069	0.0341035	+	0.999418i	$0.489142\pi$
$978$	0	0
$979$	−4.24651e6	−0.141604
$980$	5.86133e6	0.194953
$981$	0	0
$982$	4.09915e7	1.35649
$983$	−4.87946e7	−1.61060	−0.805301	−	0.592866i	$-0.797996\pi$
$983$	−4.87946e7	−1.61060	−0.805301	+	0.592866i	$0.797996\pi$
$984$	0	0
$985$	2.09110e7	0.686728
$986$	−1.67088e7	−0.547334
$987$	0	0
$988$	2.44775e6	0.0797763
$989$	−2.93620e7	−0.954540
$990$	0	0
$991$	4.77463e7	1.54439	0.772193	−	0.635388i	$-0.219160\pi$
$991$	4.77463e7	1.54439	0.772193	+	0.635388i	$0.219160\pi$
$992$	−1.94408e6	−0.0627242
$993$	0	0
$994$	−8.52240e7	−2.73587
$995$	−2.16396e7	−0.692934
$996$	0	0
$997$	−4.32326e7	−1.37744	−0.688721	−	0.725026i	$-0.741828\pi$
$997$	−4.32326e7	−1.37744	−0.688721	+	0.725026i	$0.741828\pi$
$998$	−6.58482e7	−2.09275
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.6.a.c.1.11		12
3.2	odd	2		177.6.a.c.1.2	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.6.a.c.1.2	✓	12	3.2	odd	2
531.6.a.c.1.11		12	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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)

\(f(q)\)	\(=\)	\(q+6.10567 q^{2} +5.27923 q^{4} +41.8018 q^{5} +208.248 q^{7} -163.148 q^{8} +O(q^{10})\)	\(q+6.10567 q^{2} +5.27923 q^{4} +41.8018 q^{5} +208.248 q^{7} -163.148 q^{8} +255.228 q^{10} -149.105 q^{11} -952.113 q^{13} +1271.49 q^{14} -1165.07 q^{16} -1073.60 q^{17} -486.976 q^{19} +220.681 q^{20} -910.386 q^{22} -2198.23 q^{23} -1377.61 q^{25} -5813.29 q^{26} +1099.39 q^{28} +2548.98 q^{29} +1027.11 q^{31} -1892.76 q^{32} -6555.08 q^{34} +8705.13 q^{35} -3164.40 q^{37} -2973.31 q^{38} -6819.88 q^{40} -14579.4 q^{41} +13357.1 q^{43} -787.159 q^{44} -13421.7 q^{46} -15809.5 q^{47} +26560.2 q^{49} -8411.26 q^{50} -5026.42 q^{52} -746.932 q^{53} -6232.85 q^{55} -33975.3 q^{56} +15563.2 q^{58} +3481.00 q^{59} +31646.6 q^{61} +6271.22 q^{62} +25725.5 q^{64} -39800.0 q^{65} -36947.6 q^{67} -5667.81 q^{68} +53150.6 q^{70} -67026.7 q^{71} +15479.2 q^{73} -19320.8 q^{74} -2570.86 q^{76} -31050.8 q^{77} +45796.7 q^{79} -48701.8 q^{80} -89017.1 q^{82} -16456.1 q^{83} -44878.6 q^{85} +81554.1 q^{86} +24326.2 q^{88} +28480.0 q^{89} -198276. q^{91} -11604.9 q^{92} -96527.9 q^{94} -20356.4 q^{95} +79923.6 q^{97} +162168. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 22 q^{2} + 198 q^{4} - 158 q^{5} + 413 q^{7} - 723 q^{8}+O(q^{10}) \) 12 * q - 22 * q^2 + 198 * q^4 - 158 * q^5 + 413 * q^7 - 723 * q^8	\( 12 q - 22 q^{2} + 198 q^{4} - 158 q^{5} + 413 q^{7} - 723 q^{8} + 601 q^{10} - 1480 q^{11} + 472 q^{13} - 1065 q^{14} + 6370 q^{16} - 1565 q^{17} + 3939 q^{19} - 8033 q^{20} - 1738 q^{22} - 7245 q^{23} + 9690 q^{25} - 3764 q^{26} + 12154 q^{28} - 10003 q^{29} + 7295 q^{31} - 11628 q^{32} - 16344 q^{34} - 11015 q^{35} + 6741 q^{37} - 3035 q^{38} + 5572 q^{40} - 34025 q^{41} - 6336 q^{43} - 41168 q^{44} + 2345 q^{46} - 66167 q^{47} + 28319 q^{49} - 31173 q^{50} + 16440 q^{52} - 62290 q^{53} + 55764 q^{55} - 107306 q^{56} + 37952 q^{58} + 41772 q^{59} + 68469 q^{61} - 99190 q^{62} + 68525 q^{64} - 80156 q^{65} + 113310 q^{67} - 33887 q^{68} + 32034 q^{70} - 84520 q^{71} + 135895 q^{73} + 31962 q^{74} - 61848 q^{76} + 3799 q^{77} + 14122 q^{79} - 77609 q^{80} - 1501 q^{82} - 114463 q^{83} - 101097 q^{85} + 203536 q^{86} - 244967 q^{88} - 189109 q^{89} - 168249 q^{91} + 71946 q^{92} - 472284 q^{94} - 21923 q^{95} - 76192 q^{97} + 17544 q^{98}+O(q^{100}) \) 12 * q - 22 * q^2 + 198 * q^4 - 158 * q^5 + 413 * q^7 - 723 * q^8 + 601 * q^10 - 1480 * q^11 + 472 * q^13 - 1065 * q^14 + 6370 * q^16 - 1565 * q^17 + 3939 * q^19 - 8033 * q^20 - 1738 * q^22 - 7245 * q^23 + 9690 * q^25 - 3764 * q^26 + 12154 * q^28 - 10003 * q^29 + 7295 * q^31 - 11628 * q^32 - 16344 * q^34 - 11015 * q^35 + 6741 * q^37 - 3035 * q^38 + 5572 * q^40 - 34025 * q^41 - 6336 * q^43 - 41168 * q^44 + 2345 * q^46 - 66167 * q^47 + 28319 * q^49 - 31173 * q^50 + 16440 * q^52 - 62290 * q^53 + 55764 * q^55 - 107306 * q^56 + 37952 * q^58 + 41772 * q^59 + 68469 * q^61 - 99190 * q^62 + 68525 * q^64 - 80156 * q^65 + 113310 * q^67 - 33887 * q^68 + 32034 * q^70 - 84520 * q^71 + 135895 * q^73 + 31962 * q^74 - 61848 * q^76 + 3799 * q^77 + 14122 * q^79 - 77609 * q^80 - 1501 * q^82 - 114463 * q^83 - 101097 * q^85 + 203536 * q^86 - 244967 * q^88 - 189109 * q^89 - 168249 * q^91 + 71946 * q^92 - 472284 * q^94 - 21923 * q^95 - 76192 * q^97 + 17544 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more