LMFDB - Embedded newform 531.6.a.d.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} - 4 x^{11} - 283 x^{10} + 1045 x^{9} + 27968 x^{8} - 94393 x^{7} - 1130486 x^{6} + \cdots - 50564480 $ x^12 - 4x^11 - 283x^10 + 1045x^9 + 27968x^8 - 94393x^7 - 1130486x^6 + 3566264x^5 + 15496192x^4 - 53008480x^3 - 16576192x^2 + 120303168*x - 50564480
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			$8.94324$ 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)

$f(q)$	$=$	$q+8.94324 q^{2} +47.9816 q^{4} -48.3356 q^{5} -32.5863 q^{7} +142.927 q^{8} +O(q^{10})$	\(q+8.94324 q^{2} +47.9816 q^{4} -48.3356 q^{5} -32.5863 q^{7} +142.927 q^{8} -432.277 q^{10} +607.250 q^{11} -114.450 q^{13} -291.427 q^{14} -257.180 q^{16} -1477.59 q^{17} +1197.95 q^{19} -2319.22 q^{20} +5430.79 q^{22} +815.109 q^{23} -788.665 q^{25} -1023.55 q^{26} -1563.54 q^{28} -7054.74 q^{29} -2013.45 q^{31} -6873.69 q^{32} -13214.4 q^{34} +1575.08 q^{35} +2888.36 q^{37} +10713.6 q^{38} -6908.46 q^{40} -7977.13 q^{41} -11709.8 q^{43} +29136.8 q^{44} +7289.72 q^{46} -2217.46 q^{47} -15745.1 q^{49} -7053.22 q^{50} -5491.47 q^{52} +16868.6 q^{53} -29351.8 q^{55} -4657.46 q^{56} -63092.2 q^{58} +3481.00 q^{59} -11356.2 q^{61} -18006.8 q^{62} -53243.2 q^{64} +5532.00 q^{65} -29351.9 q^{67} -70896.9 q^{68} +14086.3 q^{70} -61841.8 q^{71} -6951.65 q^{73} +25831.3 q^{74} +57479.7 q^{76} -19788.1 q^{77} -74447.1 q^{79} +12431.0 q^{80} -71341.4 q^{82} +64837.6 q^{83} +71420.1 q^{85} -104724. q^{86} +86792.4 q^{88} +6479.60 q^{89} +3729.49 q^{91} +39110.2 q^{92} -19831.3 q^{94} -57903.9 q^{95} +34765.8 q^{97} -140812. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{2} + 198 q^{4} - 36 q^{5} - 411 q^{7} + 69 q^{8}+O(q^{10}) $ 12 * q + 4 * q^2 + 198 * q^4 - 36 * q^5 - 411 * q^7 + 69 * q^8	$ 12 q + 4 q^{2} + 198 q^{4} - 36 q^{5} - 411 q^{7} + 69 q^{8} - 863 q^{10} - 492 q^{11} - 974 q^{13} + 967 q^{14} + 6370 q^{16} + 1463 q^{17} - 3189 q^{19} + 835 q^{20} - 2726 q^{22} + 2617 q^{23} + 8642 q^{25} - 2414 q^{26} - 20458 q^{28} + 1963 q^{29} - 11929 q^{31} + 14382 q^{32} - 20744 q^{34} - 1829 q^{35} - 28105 q^{37} + 23475 q^{38} - 100576 q^{40} + 7585 q^{41} - 33146 q^{43} - 26014 q^{44} - 142851 q^{46} + 79215 q^{47} - 32569 q^{49} + 136019 q^{50} - 248218 q^{52} + 12220 q^{53} - 117770 q^{55} + 186728 q^{56} - 188072 q^{58} + 41772 q^{59} - 54195 q^{61} - 36230 q^{62} + 45197 q^{64} - 42368 q^{65} + 24224 q^{67} + 209639 q^{68} - 35684 q^{70} - 60254 q^{71} - 15385 q^{73} - 214638 q^{74} - 167504 q^{76} + 17169 q^{77} - 27054 q^{79} - 216899 q^{80} + 37917 q^{82} + 117595 q^{83} - 121585 q^{85} - 306756 q^{86} - 105799 q^{88} + 36033 q^{89} - 32217 q^{91} + 30906 q^{92} + 128392 q^{94} + 50721 q^{95} - 196914 q^{97} - 574100 q^{98}+O(q^{100}) $ 12 * q + 4 * q^2 + 198 * q^4 - 36 * q^5 - 411 * q^7 + 69 * q^8 - 863 * q^10 - 492 * q^11 - 974 * q^13 + 967 * q^14 + 6370 * q^16 + 1463 * q^17 - 3189 * q^19 + 835 * q^20 - 2726 * q^22 + 2617 * q^23 + 8642 * q^25 - 2414 * q^26 - 20458 * q^28 + 1963 * q^29 - 11929 * q^31 + 14382 * q^32 - 20744 * q^34 - 1829 * q^35 - 28105 * q^37 + 23475 * q^38 - 100576 * q^40 + 7585 * q^41 - 33146 * q^43 - 26014 * q^44 - 142851 * q^46 + 79215 * q^47 - 32569 * q^49 + 136019 * q^50 - 248218 * q^52 + 12220 * q^53 - 117770 * q^55 + 186728 * q^56 - 188072 * q^58 + 41772 * q^59 - 54195 * q^61 - 36230 * q^62 + 45197 * q^64 - 42368 * q^65 + 24224 * q^67 + 209639 * q^68 - 35684 * q^70 - 60254 * q^71 - 15385 * q^73 - 214638 * q^74 - 167504 * q^76 + 17169 * q^77 - 27054 * q^79 - 216899 * q^80 + 37917 * q^82 + 117595 * q^83 - 121585 * q^85 - 306756 * q^86 - 105799 * q^88 + 36033 * q^89 - 32217 * q^91 + 30906 * q^92 + 128392 * q^94 + 50721 * q^95 - 196914 * q^97 - 574100 * 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 $

$2$	8.94324	1.58096	0.790478	−	0.612490i	$-0.209832\pi$
$2$	8.94324	1.58096	0.790478	+	0.612490i	$0.209832\pi$
$3$	0	0
$3$	0	0
$4$	47.9816	1.49942
$5$	−48.3356	−0.864654	−0.432327	−	0.901717i	$-0.642307\pi$
$5$	−48.3356	−0.864654	−0.432327	+	0.901717i	$0.642307\pi$
$6$	0	0
$7$	−32.5863	−0.251357	−0.125678	−	0.992071i	$-0.540111\pi$
$7$	−32.5863	−0.251357	−0.125678	+	0.992071i	$0.540111\pi$
$8$	142.927	0.789567
$9$	0	0
$10$	−432.277	−1.36698
$11$	607.250	1.51316	0.756582	−	0.653899i	$-0.226868\pi$
$11$	607.250	1.51316	0.756582	+	0.653899i	$0.226868\pi$
$12$	0	0
$13$	−114.450	−0.187826	−0.0939131	−	0.995580i	$-0.529938\pi$
$13$	−114.450	−0.187826	−0.0939131	+	0.995580i	$0.529938\pi$
$14$	−291.427	−0.397384
$15$	0	0
$16$	−257.180	−0.251153
$17$	−1477.59	−1.24003	−0.620013	−	0.784592i	$-0.712873\pi$
$17$	−1477.59	−1.24003	−0.620013	+	0.784592i	$0.712873\pi$
$18$	0	0
$19$	1197.95	0.761300	0.380650	−	0.924719i	$-0.375700\pi$
$19$	1197.95	0.761300	0.380650	+	0.924719i	$0.375700\pi$
$20$	−2319.22	−1.29648
$21$	0	0
$22$	5430.79	2.39225
$23$	815.109	0.321289	0.160645	−	0.987012i	$-0.448643\pi$
$23$	815.109	0.321289	0.160645	+	0.987012i	$0.448643\pi$
$24$	0	0
$25$	−788.665	−0.252373
$26$	−1023.55	−0.296945
$27$	0	0
$28$	−1563.54	−0.376890
$29$	−7054.74	−1.55771	−0.778854	−	0.627205i	$-0.784199\pi$
$29$	−7054.74	−1.55771	−0.778854	+	0.627205i	$0.784199\pi$
$30$	0	0
$31$	−2013.45	−0.376302	−0.188151	−	0.982140i	$-0.560249\pi$
$31$	−2013.45	−0.376302	−0.188151	+	0.982140i	$0.560249\pi$
$32$	−6873.69	−1.18663
$33$	0	0
$34$	−13214.4	−1.96043
$35$	1575.08	0.217337
$36$	0	0
$37$	2888.36	0.346855	0.173427	−	0.984847i	$-0.444516\pi$
$37$	2888.36	0.346855	0.173427	+	0.984847i	$0.444516\pi$
$38$	10713.6	1.20358
$39$	0	0
$40$	−6908.46	−0.682702
$41$	−7977.13	−0.741117	−0.370559	−	0.928809i	$-0.620834\pi$
$41$	−7977.13	−0.741117	−0.370559	+	0.928809i	$0.620834\pi$
$42$	0	0
$43$	−11709.8	−0.965784	−0.482892	−	0.875680i	$-0.660414\pi$
$43$	−11709.8	−0.965784	−0.482892	+	0.875680i	$0.660414\pi$
$44$	29136.8	2.26887
$45$	0	0
$46$	7289.72	0.507944
$47$	−2217.46	−0.146424	−0.0732119	−	0.997316i	$-0.523325\pi$
$47$	−2217.46	−0.146424	−0.0732119	+	0.997316i	$0.523325\pi$
$48$	0	0
$49$	−15745.1	−0.936820
$50$	−7053.22	−0.398990
$51$	0	0
$52$	−5491.47	−0.281631
$53$	16868.6	0.824878	0.412439	−	0.910985i	$-0.364677\pi$
$53$	16868.6	0.824878	0.412439	+	0.910985i	$0.364677\pi$
$54$	0	0
$55$	−29351.8	−1.30836
$56$	−4657.46	−0.198463
$57$	0	0
$58$	−63092.2	−2.46267
$59$	3481.00	0.130189
$59$	3481.00	0.130189
$60$	0	0
$61$	−11356.2	−0.390758	−0.195379	−	0.980728i	$-0.562594\pi$
$61$	−11356.2	−0.390758	−0.195379	+	0.980728i	$0.562594\pi$
$62$	−18006.8	−0.594917
$63$	0	0
$64$	−53243.2	−1.62486
$65$	5532.00	0.162405
$66$	0	0
$67$	−29351.9	−0.798821	−0.399410	−	0.916772i	$-0.630785\pi$
$67$	−29351.9	−0.798821	−0.399410	+	0.916772i	$0.630785\pi$
$68$	−70896.9	−1.85932
$69$	0	0
$70$	14086.3	0.343600
$71$	−61841.8	−1.45592	−0.727958	−	0.685622i	$-0.759530\pi$
$71$	−61841.8	−1.45592	−0.727958	+	0.685622i	$0.759530\pi$
$72$	0	0
$73$	−6951.65	−0.152680	−0.0763398	−	0.997082i	$-0.524323\pi$
$73$	−6951.65	−0.152680	−0.0763398	+	0.997082i	$0.524323\pi$
$74$	25831.3	0.548362
$75$	0	0
$76$	57479.7	1.14151
$77$	−19788.1	−0.380344
$78$	0	0
$79$	−74447.1	−1.34208	−0.671042	−	0.741419i	$-0.734153\pi$
$79$	−74447.1	−1.34208	−0.671042	+	0.741419i	$0.734153\pi$
$80$	12431.0	0.217160
$81$	0	0
$82$	−71341.4	−1.17167
$83$	64837.6	1.03307	0.516537	−	0.856265i	$-0.327221\pi$
$83$	64837.6	1.03307	0.516537	+	0.856265i	$0.327221\pi$
$84$	0	0
$85$	71420.1	1.07219
$86$	−104724.	−1.52686
$87$	0	0
$88$	86792.4	1.19474
$89$	6479.60	0.0867108	0.0433554	−	0.999060i	$-0.486195\pi$
$89$	6479.60	0.0867108	0.0433554	+	0.999060i	$0.486195\pi$
$90$	0	0
$91$	3729.49	0.0472114
$92$	39110.2	0.481749
$93$	0	0
$94$	−19831.3	−0.231490
$95$	−57903.9	−0.658262
$96$	0	0
$97$	34765.8	0.375166	0.187583	−	0.982249i	$-0.439935\pi$
$97$	34765.8	0.375166	0.187583	+	0.982249i	$0.439935\pi$
$98$	−140812.	−1.48107
$99$	0	0
$100$	−37841.4	−0.378414
$101$	10413.0	0.101571	0.0507857	−	0.998710i	$-0.483827\pi$
$101$	10413.0	0.101571	0.0507857	+	0.998710i	$0.483827\pi$
$102$	0	0
$103$	−187352.	−1.74007	−0.870033	−	0.492993i	$-0.835903\pi$
$103$	−187352.	−1.74007	−0.870033	+	0.492993i	$0.835903\pi$
$104$	−16357.9	−0.148301
$105$	0	0
$106$	150860.	1.30410
$107$	−18946.3	−0.159979	−0.0799897	−	0.996796i	$-0.525489\pi$
$107$	−18946.3	−0.159979	−0.0799897	+	0.996796i	$0.525489\pi$
$108$	0	0
$109$	12701.4	0.102396	0.0511981	−	0.998689i	$-0.483696\pi$
$109$	12701.4	0.102396	0.0511981	+	0.998689i	$0.483696\pi$
$110$	−262501.	−2.06847
$111$	0	0
$112$	8380.57	0.0631289
$113$	90034.0	0.663301	0.331650	−	0.943402i	$-0.392395\pi$
$113$	90034.0	0.663301	0.331650	+	0.943402i	$0.392395\pi$
$114$	0	0
$115$	−39398.8	−0.277804
$116$	−338497.	−2.33566
$117$	0	0
$118$	31131.4	0.205823
$119$	48149.1	0.311689
$120$	0	0
$121$	207702.	1.28967
$122$	−101561.	−0.617772
$123$	0	0
$124$	−96608.5	−0.564236
$125$	189170.	1.08287
$126$	0	0
$127$	−75773.2	−0.416875	−0.208438	−	0.978036i	$-0.566838\pi$
$127$	−75773.2	−0.416875	−0.208438	+	0.978036i	$0.566838\pi$
$128$	−256209.	−1.38220
$129$	0	0
$130$	49474.0	0.256755
$131$	279838.	1.42472	0.712358	−	0.701816i	$-0.247627\pi$
$131$	279838.	1.42472	0.712358	+	0.701816i	$0.247627\pi$
$132$	0	0
$133$	−39036.9	−0.191358
$134$	−262501.	−1.26290
$135$	0	0
$136$	−211187.	−0.979083
$137$	309329.	1.40805	0.704026	−	0.710174i	$-0.251384\pi$
$137$	309329.	1.40805	0.704026	+	0.710174i	$0.251384\pi$
$138$	0	0
$139$	440414.	1.93341	0.966706	−	0.255889i	$-0.0823682\pi$
$139$	440414.	1.93341	0.966706	+	0.255889i	$0.0823682\pi$
$140$	75574.8	0.325880
$141$	0	0
$142$	−553066.	−2.30174
$143$	−69499.6	−0.284212
$144$	0	0
$145$	340995.	1.34688
$146$	−62170.3	−0.241380
$147$	0	0
$148$	138588.	0.520082
$149$	35329.4	0.130368	0.0651840	−	0.997873i	$-0.479237\pi$
$149$	35329.4	0.130368	0.0651840	+	0.997873i	$0.479237\pi$
$150$	0	0
$151$	−277734.	−0.991258	−0.495629	−	0.868534i	$-0.665062\pi$
$151$	−277734.	−0.991258	−0.495629	+	0.868534i	$0.665062\pi$
$152$	171220.	0.601097
$153$	0	0
$154$	−176969.	−0.601307
$155$	97321.4	0.325371
$156$	0	0
$157$	123296.	0.399208	0.199604	−	0.979877i	$-0.436034\pi$
$157$	123296.	0.399208	0.199604	+	0.979877i	$0.436034\pi$
$158$	−665798.	−2.12178
$159$	0	0
$160$	332244.	1.02602
$161$	−26561.4	−0.0807582
$162$	0	0
$163$	−516622.	−1.52301	−0.761507	−	0.648157i	$-0.775540\pi$
$163$	−516622.	−1.52301	−0.761507	+	0.648157i	$0.775540\pi$
$164$	−382755.	−1.11125
$165$	0	0
$166$	579858.	1.63325
$167$	−265123.	−0.735625	−0.367812	−	0.929900i	$-0.619893\pi$
$167$	−265123.	−0.735625	−0.367812	+	0.929900i	$0.619893\pi$
$168$	0	0
$169$	−358194.	−0.964721
$170$	638727.	1.69509
$171$	0	0
$172$	−561856.	−1.44812
$173$	571531.	1.45186	0.725930	−	0.687769i	$-0.241410\pi$
$173$	571531.	1.45186	0.725930	+	0.687769i	$0.241410\pi$
$174$	0	0
$175$	25699.7	0.0634356
$176$	−156173.	−0.380035
$177$	0	0
$178$	57948.6	0.137086
$179$	−70392.7	−0.164208	−0.0821041	−	0.996624i	$-0.526164\pi$
$179$	−70392.7	−0.164208	−0.0821041	+	0.996624i	$0.526164\pi$
$180$	0	0
$181$	−69490.3	−0.157662	−0.0788312	−	0.996888i	$-0.525119\pi$
$181$	−69490.3	−0.157662	−0.0788312	+	0.996888i	$0.525119\pi$
$182$	33353.8	0.0746391
$183$	0	0
$184$	116501.	0.253679
$185$	−139611.	−0.299909
$186$	0	0
$187$	−897265.	−1.87636
$188$	−106397.	−0.219551
$189$	0	0
$190$	−517848.	−1.04068
$191$	790909.	1.56871	0.784356	−	0.620311i	$-0.212993\pi$
$191$	790909.	1.56871	0.784356	+	0.620311i	$0.212993\pi$
$192$	0	0
$193$	−970964.	−1.87633	−0.938166	−	0.346185i	$-0.887477\pi$
$193$	−970964.	−1.87633	−0.938166	+	0.346185i	$0.887477\pi$
$194$	310919.	0.593121
$195$	0	0
$196$	−755476.	−1.40469
$197$	619957.	1.13814	0.569070	−	0.822289i	$-0.307303\pi$
$197$	619957.	1.13814	0.569070	+	0.822289i	$0.307303\pi$
$198$	0	0
$199$	−175483.	−0.314124	−0.157062	−	0.987589i	$-0.550202\pi$
$199$	−175483.	−0.314124	−0.157062	+	0.987589i	$0.550202\pi$
$200$	−112721.	−0.199265
$201$	0	0
$202$	93125.7	0.160580
$203$	229888.	0.391540
$204$	0	0
$205$	385580.	0.640810
$206$	−1.67554e6	−2.75097
$207$	0	0
$208$	29434.2	0.0471731
$209$	727458.	1.15197
$210$	0	0
$211$	561184.	0.867759	0.433880	−	0.900971i	$-0.357144\pi$
$211$	561184.	0.867759	0.433880	+	0.900971i	$0.357144\pi$
$212$	809383.	1.23684
$213$	0	0
$214$	−169441.	−0.252921
$215$	566003.	0.835069
$216$	0	0
$217$	65611.0	0.0945861
$218$	113591.	0.161884
$219$	0	0
$220$	−1.40835e6	−1.96179
$221$	169109.	0.232909
$222$	0	0
$223$	−346056.	−0.465998	−0.232999	−	0.972477i	$-0.574854\pi$
$223$	−346056.	−0.465998	−0.232999	+	0.972477i	$0.574854\pi$
$224$	223988.	0.298267
$225$	0	0
$226$	805196.	1.04865
$227$	69353.3	0.0893310	0.0446655	−	0.999002i	$-0.485778\pi$
$227$	69353.3	0.0893310	0.0446655	+	0.999002i	$0.485778\pi$
$228$	0	0
$229$	1.03395e6	1.30289	0.651447	−	0.758694i	$-0.274162\pi$
$229$	1.03395e6	1.30289	0.651447	+	0.758694i	$0.274162\pi$
$230$	−352353.	−0.439196
$231$	0	0
$232$	−1.00831e6	−1.22991
$233$	−1.08238e6	−1.30614	−0.653069	−	0.757298i	$-0.726519\pi$
$233$	−1.08238e6	−1.30614	−0.653069	+	0.757298i	$0.726519\pi$
$234$	0	0
$235$	107182.	0.126606
$236$	167024.	0.195208
$237$	0	0
$238$	430609.	0.492766
$239$	−526608.	−0.596338	−0.298169	−	0.954513i	$-0.596376\pi$
$239$	−526608.	−0.596338	−0.298169	+	0.954513i	$0.596376\pi$
$240$	0	0
$241$	1.09725e6	1.21692	0.608461	−	0.793584i	$-0.291787\pi$
$241$	1.09725e6	1.21692	0.608461	+	0.793584i	$0.291787\pi$
$242$	1.85753e6	2.03891
$243$	0	0
$244$	−544888.	−0.585912
$245$	761051.	0.810025
$246$	0	0
$247$	−137105.	−0.142992
$248$	−287776.	−0.297116
$249$	0	0
$250$	1.69179e6	1.71197
$251$	−729854.	−0.731226	−0.365613	−	0.930767i	$-0.619141\pi$
$251$	−729854.	−0.731226	−0.365613	+	0.930767i	$0.619141\pi$
$252$	0	0
$253$	494975.	0.486163
$254$	−677658.	−0.659062
$255$	0	0
$256$	−587557.	−0.560338
$257$	−722806.	−0.682636	−0.341318	−	0.939948i	$-0.610873\pi$
$257$	−722806.	−0.682636	−0.341318	+	0.939948i	$0.610873\pi$
$258$	0	0
$259$	−94121.1	−0.0871842
$260$	265434.	0.243513
$261$	0	0
$262$	2.50266e6	2.25241
$263$	1.30671e6	1.16490	0.582452	−	0.812865i	$-0.302094\pi$
$263$	1.30671e6	1.16490	0.582452	+	0.812865i	$0.302094\pi$
$264$	0	0
$265$	−815356.	−0.713235
$266$	−349116.	−0.302528
$267$	0	0
$268$	−1.40835e6	−1.19777
$269$	−2.09858e6	−1.76825	−0.884127	−	0.467246i	$-0.845246\pi$
$269$	−2.09858e6	−1.76825	−0.884127	+	0.467246i	$0.845246\pi$
$270$	0	0
$271$	1.04873e6	0.867440	0.433720	−	0.901048i	$-0.357201\pi$
$271$	1.04873e6	0.867440	0.433720	+	0.901048i	$0.357201\pi$
$272$	380006.	0.311436
$273$	0	0
$274$	2.76640e6	2.22607
$275$	−478917.	−0.381881
$276$	0	0
$277$	−191493.	−0.149953	−0.0749763	−	0.997185i	$-0.523888\pi$
$277$	−191493.	−0.149953	−0.0749763	+	0.997185i	$0.523888\pi$
$278$	3.93873e6	3.05664
$279$	0	0
$280$	225121.	0.171602
$281$	1.21812e6	0.920292	0.460146	−	0.887843i	$-0.347797\pi$
$281$	1.21812e6	0.920292	0.460146	+	0.887843i	$0.347797\pi$
$282$	0	0
$283$	798367.	0.592566	0.296283	−	0.955100i	$-0.404253\pi$
$283$	798367.	0.592566	0.296283	+	0.955100i	$0.404253\pi$
$284$	−2.96727e6	−2.18303
$285$	0	0
$286$	−621552.	−0.449327
$287$	259945.	0.186285
$288$	0	0
$289$	763404.	0.537662
$290$	3.04960e6	2.12936
$291$	0	0
$292$	−333551.	−0.228931
$293$	1.05915e6	0.720754	0.360377	−	0.932807i	$-0.382648\pi$
$293$	1.05915e6	0.720754	0.360377	+	0.932807i	$0.382648\pi$
$294$	0	0
$295$	−168256.	−0.112568
$296$	412825.	0.273865
$297$	0	0
$298$	315960.	0.206106
$299$	−93288.9	−0.0603465
$300$	0	0
$301$	381581.	0.242756
$302$	−2.48384e6	−1.56714
$303$	0	0
$304$	−308090.	−0.191203
$305$	548909.	0.337871
$306$	0	0
$307$	80118.4	0.0485161	0.0242581	−	0.999706i	$-0.492278\pi$
$307$	80118.4	0.0485161	0.0242581	+	0.999706i	$0.492278\pi$
$308$	−949462.	−0.570296
$309$	0	0
$310$	870369.	0.514398
$311$	−2.98406e6	−1.74947	−0.874736	−	0.484600i	$-0.838965\pi$
$311$	−2.98406e6	−1.74947	−0.874736	+	0.484600i	$0.838965\pi$
$312$	0	0
$313$	678550.	0.391491	0.195745	−	0.980655i	$-0.437287\pi$
$313$	678550.	0.391491	0.195745	+	0.980655i	$0.437287\pi$
$314$	1.10266e6	0.631130
$315$	0	0
$316$	−3.57209e6	−2.01235
$317$	−1.81380e6	−1.01377	−0.506887	−	0.862013i	$-0.669204\pi$
$317$	−1.81380e6	−1.01377	−0.506887	+	0.862013i	$0.669204\pi$
$318$	0	0
$319$	−4.28399e6	−2.35707
$320$	2.57355e6	1.40494
$321$	0	0
$322$	−237545.	−0.127675
$323$	−1.77008e6	−0.944031
$324$	0	0
$325$	90262.4	0.0474022
$326$	−4.62027e6	−2.40782
$327$	0	0
$328$	−1.14015e6	−0.585161
$329$	72258.9	0.0368046
$330$	0	0
$331$	−2.91425e6	−1.46203	−0.731016	−	0.682360i	$-0.760954\pi$
$331$	−2.91425e6	−1.46203	−0.731016	+	0.682360i	$0.760954\pi$
$332$	3.11101e6	1.54902
$333$	0	0
$334$	−2.37106e6	−1.16299
$335$	1.41874e6	0.690704
$336$	0	0
$337$	−1.92879e6	−0.925145	−0.462572	−	0.886582i	$-0.653073\pi$
$337$	−1.92879e6	−0.925145	−0.462572	+	0.886582i	$0.653073\pi$
$338$	−3.20342e6	−1.52518
$339$	0	0
$340$	3.42685e6	1.60767
$341$	−1.22267e6	−0.569407
$342$	0	0
$343$	1.06075e6	0.486833
$344$	−1.67365e6	−0.762551
$345$	0	0
$346$	5.11134e6	2.29533
$347$	2.63458e6	1.17459	0.587297	−	0.809372i	$-0.300192\pi$
$347$	2.63458e6	1.17459	0.587297	+	0.809372i	$0.300192\pi$
$348$	0	0
$349$	−338492.	−0.148759	−0.0743797	−	0.997230i	$-0.523698\pi$
$349$	−338492.	−0.148759	−0.0743797	+	0.997230i	$0.523698\pi$
$350$	229839.	0.100289
$351$	0	0
$352$	−4.17405e6	−1.79556
$353$	1.24130e6	0.530200	0.265100	−	0.964221i	$-0.414595\pi$
$353$	1.24130e6	0.530200	0.265100	+	0.964221i	$0.414595\pi$
$354$	0	0
$355$	2.98916e6	1.25886
$356$	310901.	0.130016
$357$	0	0
$358$	−629539.	−0.259606
$359$	1.85276e6	0.758723	0.379361	−	0.925249i	$-0.376144\pi$
$359$	1.85276e6	0.758723	0.379361	+	0.925249i	$0.376144\pi$
$360$	0	0
$361$	−1.04101e6	−0.420422
$362$	−621469.	−0.249257
$363$	0	0
$364$	178947.	0.0707898
$365$	336013.	0.132015
$366$	0	0
$367$	2.21751e6	0.859410	0.429705	−	0.902969i	$-0.358618\pi$
$367$	2.21751e6	0.859410	0.429705	+	0.902969i	$0.358618\pi$
$368$	−209630.	−0.0806927
$369$	0	0
$370$	−1.24857e6	−0.474144
$371$	−549687.	−0.207339
$372$	0	0
$373$	−1.99276e6	−0.741623	−0.370811	−	0.928708i	$-0.620920\pi$
$373$	−1.99276e6	−0.741623	−0.370811	+	0.928708i	$0.620920\pi$
$374$	−8.02445e6	−2.96645
$375$	0	0
$376$	−316935.	−0.115611
$377$	807413.	0.292578
$378$	0	0
$379$	1.04393e6	0.373312	0.186656	−	0.982425i	$-0.440235\pi$
$379$	1.04393e6	0.373312	0.186656	+	0.982425i	$0.440235\pi$
$380$	−2.77832e6	−0.987013
$381$	0	0
$382$	7.07329e6	2.48007
$383$	−1.07017e6	−0.372783	−0.186391	−	0.982476i	$-0.559679\pi$
$383$	−1.07017e6	−0.372783	−0.186391	+	0.982476i	$0.559679\pi$
$384$	0	0
$385$	956469.	0.328866
$386$	−8.68356e6	−2.96640
$387$	0	0
$388$	1.66812e6	0.562533
$389$	−1.68363e6	−0.564122	−0.282061	−	0.959396i	$-0.591018\pi$
$389$	−1.68363e6	−0.564122	−0.282061	+	0.959396i	$0.591018\pi$
$390$	0	0
$391$	−1.20439e6	−0.398407
$392$	−2.25040e6	−0.739682
$393$	0	0
$394$	5.54442e6	1.79935
$395$	3.59845e6	1.16044
$396$	0	0
$397$	−812204.	−0.258636	−0.129318	−	0.991603i	$-0.541279\pi$
$397$	−812204.	−0.258636	−0.129318	+	0.991603i	$0.541279\pi$
$398$	−1.56938e6	−0.496617
$399$	0	0
$400$	202829.	0.0633841
$401$	2.25492e6	0.700279	0.350139	−	0.936698i	$-0.386134\pi$
$401$	2.25492e6	0.700279	0.350139	+	0.936698i	$0.386134\pi$
$402$	0	0
$403$	230439.	0.0706794
$404$	499631.	0.152299
$405$	0	0
$406$	2.05594e6	0.619008
$407$	1.75396e6	0.524848
$408$	0	0
$409$	949748.	0.280737	0.140369	−	0.990099i	$-0.455171\pi$
$409$	949748.	0.280737	0.140369	+	0.990099i	$0.455171\pi$
$410$	3.44833e6	1.01309
$411$	0	0
$412$	−8.98945e6	−2.60910
$413$	−113433.	−0.0327238
$414$	0	0
$415$	−3.13397e6	−0.893252
$416$	786691.	0.222880
$417$	0	0
$418$	6.50583e6	1.82122
$419$	328196.	0.0913267	0.0456634	−	0.998957i	$-0.485460\pi$
$419$	328196.	0.0913267	0.0456634	+	0.998957i	$0.485460\pi$
$420$	0	0
$421$	6.14766e6	1.69046	0.845230	−	0.534403i	$-0.179463\pi$
$421$	6.14766e6	1.69046	0.845230	+	0.534403i	$0.179463\pi$
$422$	5.01880e6	1.37189
$423$	0	0
$424$	2.41098e6	0.651297
$425$	1.16532e6	0.312949
$426$	0	0
$427$	370057.	0.0982197
$428$	−909071.	−0.239877
$429$	0	0
$430$	5.06190e6	1.32021
$431$	−3.28062e6	−0.850674	−0.425337	−	0.905035i	$-0.639844\pi$
$431$	−3.28062e6	−0.850674	−0.425337	+	0.905035i	$0.639844\pi$
$432$	0	0
$433$	−1.79870e6	−0.461040	−0.230520	−	0.973068i	$-0.574043\pi$
$433$	−1.79870e6	−0.461040	−0.230520	+	0.973068i	$0.574043\pi$
$434$	586775.	0.149536
$435$	0	0
$436$	609431.	0.153535
$437$	976463.	0.244598
$438$	0	0
$439$	1.62646e6	0.402794	0.201397	−	0.979510i	$-0.435452\pi$
$439$	1.62646e6	0.402794	0.201397	+	0.979510i	$0.435452\pi$
$440$	−4.19517e6	−1.03304
$441$	0	0
$442$	1.51238e6	0.368219
$443$	−3.82966e6	−0.927151	−0.463576	−	0.886057i	$-0.653434\pi$
$443$	−3.82966e6	−0.927151	−0.463576	+	0.886057i	$0.653434\pi$
$444$	0	0
$445$	−313196.	−0.0749748
$446$	−3.09486e6	−0.736723
$447$	0	0
$448$	1.73500e6	0.408418
$449$	4.45166e6	1.04209	0.521046	−	0.853529i	$-0.325542\pi$
$449$	4.45166e6	1.04209	0.521046	+	0.853529i	$0.325542\pi$
$450$	0	0
$451$	−4.84411e6	−1.12143
$452$	4.31997e6	0.994569
$453$	0	0
$454$	620243.	0.141228
$455$	−180268.	−0.0408215
$456$	0	0
$457$	−1.17886e6	−0.264042	−0.132021	−	0.991247i	$-0.542147\pi$
$457$	−1.17886e6	−0.264042	−0.132021	+	0.991247i	$0.542147\pi$
$458$	9.24683e6	2.05982
$459$	0	0
$460$	−1.89042e6	−0.416546
$461$	−1.73221e6	−0.379620	−0.189810	−	0.981821i	$-0.560787\pi$
$461$	−1.73221e6	−0.379620	−0.189810	+	0.981821i	$0.560787\pi$
$462$	0	0
$463$	1.35619e6	0.294015	0.147007	−	0.989135i	$-0.453036\pi$
$463$	1.35619e6	0.294015	0.147007	+	0.989135i	$0.453036\pi$
$464$	1.81434e6	0.391223
$465$	0	0
$466$	−9.67997e6	−2.06495
$467$	3.59227e6	0.762214	0.381107	−	0.924531i	$-0.375543\pi$
$467$	3.59227e6	0.762214	0.381107	+	0.924531i	$0.375543\pi$
$468$	0	0
$469$	956471.	0.200789
$470$	958558.	0.200158
$471$	0	0
$472$	497528.	0.102793
$473$	−7.11080e6	−1.46139
$474$	0	0
$475$	−944784.	−0.192131
$476$	2.31027e6	0.467353
$477$	0	0
$478$	−4.70958e6	−0.942784
$479$	5.70119e6	1.13534	0.567671	−	0.823255i	$-0.307844\pi$
$479$	5.70119e6	1.13534	0.567671	+	0.823255i	$0.307844\pi$
$480$	0	0
$481$	−330572.	−0.0651484
$482$	9.81296e6	1.92390
$483$	0	0
$484$	9.96586e6	1.93375
$485$	−1.68043e6	−0.324389
$486$	0	0
$487$	−2.32159e6	−0.443571	−0.221786	−	0.975095i	$-0.571189\pi$
$487$	−2.32159e6	−0.443571	−0.221786	+	0.975095i	$0.571189\pi$
$488$	−1.62310e6	−0.308530
$489$	0	0
$490$	6.80626e6	1.28061
$491$	7.27492e6	1.36183	0.680917	−	0.732360i	$-0.261581\pi$
$491$	7.27492e6	1.36183	0.680917	+	0.732360i	$0.261581\pi$
$492$	0	0
$493$	1.04240e7	1.93160
$494$	−1.22617e6	−0.226064
$495$	0	0
$496$	517820.	0.0945093
$497$	2.01520e6	0.365954
$498$	0	0
$499$	2.13472e6	0.383786	0.191893	−	0.981416i	$-0.438537\pi$
$499$	2.13472e6	0.383786	0.191893	+	0.981416i	$0.438537\pi$
$500$	9.07665e6	1.62368
$501$	0	0
$502$	−6.52726e6	−1.15604
$503$	−6.73329e6	−1.18661	−0.593304	−	0.804979i	$-0.702177\pi$
$503$	−6.73329e6	−1.18661	−0.593304	+	0.804979i	$0.702177\pi$
$504$	0	0
$505$	−503318.	−0.0878242
$506$	4.42668e6	0.768603
$507$	0	0
$508$	−3.63571e6	−0.625073
$509$	1.65266e6	0.282742	0.141371	−	0.989957i	$-0.454849\pi$
$509$	1.65266e6	0.282742	0.141371	+	0.989957i	$0.454849\pi$
$510$	0	0
$511$	226529.	0.0383770
$512$	2.94403e6	0.496327
$513$	0	0
$514$	−6.46423e6	−1.07922
$515$	9.05579e6	1.50456
$516$	0	0
$517$	−1.34655e6	−0.221563
$518$	−841748.	−0.137834
$519$	0	0
$520$	790671.	0.128229
$521$	4.60489e6	0.743233	0.371617	−	0.928386i	$-0.378804\pi$
$521$	4.60489e6	0.743233	0.371617	+	0.928386i	$0.378804\pi$
$522$	0	0
$523$	−797422.	−0.127478	−0.0637388	−	0.997967i	$-0.520302\pi$
$523$	−797422.	−0.127478	−0.0637388	+	0.997967i	$0.520302\pi$
$524$	1.34271e7	2.13625
$525$	0	0
$526$	1.16862e7	1.84166
$527$	2.97505e6	0.466624
$528$	0	0
$529$	−5.77194e6	−0.896773
$530$	−7.29192e6	−1.12759
$531$	0	0
$532$	−1.87305e6	−0.286926
$533$	912979.	0.139201
$534$	0	0
$535$	915780.	0.138327
$536$	−4.19518e6	−0.630722
$537$	0	0
$538$	−1.87681e7	−2.79553
$539$	−9.56124e6	−1.41756
$540$	0	0
$541$	−7.75717e6	−1.13949	−0.569745	−	0.821822i	$-0.692958\pi$
$541$	−7.75717e6	−1.13949	−0.569745	+	0.821822i	$0.692958\pi$
$542$	9.37902e6	1.37138
$543$	0	0
$544$	1.01565e7	1.47145
$545$	−613929.	−0.0885374
$546$	0	0
$547$	6.87407e6	0.982304	0.491152	−	0.871074i	$-0.336576\pi$
$547$	6.87407e6	0.982304	0.491152	+	0.871074i	$0.336576\pi$
$548$	1.48421e7	2.11127
$549$	0	0
$550$	−4.28307e6	−0.603738
$551$	−8.45125e6	−1.18588
$552$	0	0
$553$	2.42596e6	0.337342
$554$	−1.71257e6	−0.237068
$555$	0	0
$556$	2.11318e7	2.89900
$557$	1.03785e7	1.41741	0.708707	−	0.705503i	$-0.249279\pi$
$557$	1.03785e7	1.41741	0.708707	+	0.705503i	$0.249279\pi$
$558$	0	0
$559$	1.34019e6	0.181399
$560$	−405080.	−0.0545847
$561$	0	0
$562$	1.08940e7	1.45494
$563$	1.17737e7	1.56546	0.782730	−	0.622361i	$-0.213827\pi$
$563$	1.17737e7	1.56546	0.782730	+	0.622361i	$0.213827\pi$
$564$	0	0
$565$	−4.35185e6	−0.573526
$566$	7.13999e6	0.936821
$567$	0	0
$568$	−8.83885e6	−1.14954
$569$	1.71606e6	0.222204	0.111102	−	0.993809i	$-0.464562\pi$
$569$	1.71606e6	0.222204	0.111102	+	0.993809i	$0.464562\pi$
$570$	0	0
$571$	−1.38827e7	−1.78190	−0.890950	−	0.454102i	$-0.849960\pi$
$571$	−1.38827e7	−1.78190	−0.890950	+	0.454102i	$0.849960\pi$
$572$	−3.33470e6	−0.426154
$573$	0	0
$574$	2.32475e6	0.294508
$575$	−642848.	−0.0810846
$576$	0	0
$577$	4.75324e6	0.594360	0.297180	−	0.954821i	$-0.403954\pi$
$577$	4.75324e6	0.594360	0.297180	+	0.954821i	$0.403954\pi$
$578$	6.82730e6	0.850021
$579$	0	0
$580$	1.63615e7	2.01954
$581$	−2.11282e6	−0.259670
$582$	0	0
$583$	1.02435e7	1.24818
$584$	−993578.	−0.120551
$585$	0	0
$586$	9.47221e6	1.13948
$587$	1.92855e6	0.231013	0.115506	−	0.993307i	$-0.463151\pi$
$587$	1.92855e6	0.231013	0.115506	+	0.993307i	$0.463151\pi$
$588$	0	0
$589$	−2.41202e6	−0.286479
$590$	−1.50476e6	−0.177966
$591$	0	0
$592$	−742830.	−0.0871135
$593$	−1.17565e7	−1.37291	−0.686453	−	0.727174i	$-0.740833\pi$
$593$	−1.17565e7	−1.37291	−0.686453	+	0.727174i	$0.740833\pi$
$594$	0	0
$595$	−2.32732e6	−0.269503
$596$	1.69516e6	0.195477
$597$	0	0
$598$	−834305.	−0.0954052
$599$	−1.36231e7	−1.55135	−0.775673	−	0.631136i	$-0.782589\pi$
$599$	−1.36231e7	−1.55135	−0.775673	+	0.631136i	$0.782589\pi$
$600$	0	0
$601$	5.50257e6	0.621411	0.310706	−	0.950506i	$-0.399435\pi$
$601$	5.50257e6	0.621411	0.310706	+	0.950506i	$0.399435\pi$
$602$	3.41257e6	0.383787
$603$	0	0
$604$	−1.33261e7	−1.48632
$605$	−1.00394e7	−1.11512
$606$	0	0
$607$	529691.	0.0583513	0.0291757	−	0.999574i	$-0.490712\pi$
$607$	529691.	0.0583513	0.0291757	+	0.999574i	$0.490712\pi$
$608$	−8.23436e6	−0.903380
$609$	0	0
$610$	4.90902e6	0.534159
$611$	253788.	0.0275022
$612$	0	0
$613$	−6.94496e6	−0.746480	−0.373240	−	0.927735i	$-0.621753\pi$
$613$	−6.94496e6	−0.746480	−0.373240	+	0.927735i	$0.621753\pi$
$614$	716518.	0.0767019
$615$	0	0
$616$	−2.82824e6	−0.300307
$617$	−1.72574e7	−1.82500	−0.912501	−	0.409075i	$-0.865852\pi$
$617$	−1.72574e7	−1.82500	−0.912501	+	0.409075i	$0.865852\pi$
$618$	0	0
$619$	−1.09446e7	−1.14809	−0.574043	−	0.818825i	$-0.694626\pi$
$619$	−1.09446e7	−1.14809	−0.574043	+	0.818825i	$0.694626\pi$
$620$	4.66963e6	0.487869
$621$	0	0
$622$	−2.66872e7	−2.76584
$623$	−211146.	−0.0217953
$624$	0	0
$625$	−6.67905e6	−0.683935
$626$	6.06844e6	0.618930
$627$	0	0
$628$	5.91592e6	0.598581
$629$	−4.26780e6	−0.430108
$630$	0	0
$631$	−1.74968e7	−1.74939	−0.874694	−	0.484676i	$-0.838937\pi$
$631$	−1.74968e7	−1.74939	−0.874694	+	0.484676i	$0.838937\pi$
$632$	−1.06405e7	−1.05967
$633$	0	0
$634$	−1.62212e7	−1.60273
$635$	3.66255e6	0.360453
$636$	0	0
$637$	1.80202e6	0.175959
$638$	−3.83128e7	−3.72642
$639$	0	0
$640$	1.23840e7	1.19512
$641$	1.02308e6	0.0983478	0.0491739	−	0.998790i	$-0.484341\pi$
$641$	1.02308e6	0.0983478	0.0491739	+	0.998790i	$0.484341\pi$
$642$	0	0
$643$	1.59762e7	1.52387	0.761933	−	0.647655i	$-0.224250\pi$
$643$	1.59762e7	1.52387	0.761933	+	0.647655i	$0.224250\pi$
$644$	−1.27446e6	−0.121091
$645$	0	0
$646$	−1.58302e7	−1.49247
$647$	−1.48605e7	−1.39564	−0.697820	−	0.716273i	$-0.745847\pi$
$647$	−1.48605e7	−1.39564	−0.697820	+	0.716273i	$0.745847\pi$
$648$	0	0
$649$	2.11384e6	0.196997
$650$	807239.	0.0749408
$651$	0	0
$652$	−2.47883e7	−2.28364
$653$	−1.12385e7	−1.03140	−0.515699	−	0.856770i	$-0.672468\pi$
$653$	−1.12385e7	−1.03140	−0.515699	+	0.856770i	$0.672468\pi$
$654$	0	0
$655$	−1.35261e7	−1.23189
$656$	2.05156e6	0.186134
$657$	0	0
$658$	646229.	0.0581864
$659$	−1.66209e7	−1.49088	−0.745439	−	0.666574i	$-0.767760\pi$
$659$	−1.66209e7	−1.49088	−0.745439	+	0.666574i	$0.767760\pi$
$660$	0	0
$661$	5.12621e6	0.456344	0.228172	−	0.973621i	$-0.426725\pi$
$661$	5.12621e6	0.456344	0.228172	+	0.973621i	$0.426725\pi$
$662$	−2.60628e7	−2.31141
$663$	0	0
$664$	9.26703e6	0.815681
$665$	1.88687e6	0.165458
$666$	0	0
$667$	−5.75038e6	−0.500475
$668$	−1.27210e7	−1.10301
$669$	0	0
$670$	1.26882e7	1.09197
$671$	−6.89605e6	−0.591281
$672$	0	0
$673$	2.11025e6	0.179596	0.0897978	−	0.995960i	$-0.471378\pi$
$673$	2.11025e6	0.179596	0.0897978	+	0.995960i	$0.471378\pi$
$674$	−1.72496e7	−1.46261
$675$	0	0
$676$	−1.71867e7	−1.44653
$677$	−4.85784e6	−0.407354	−0.203677	−	0.979038i	$-0.565289\pi$
$677$	−4.85784e6	−0.407354	−0.203677	+	0.979038i	$0.565289\pi$
$678$	0	0
$679$	−1.13289e6	−0.0943005
$680$	1.02078e7	0.846568
$681$	0	0
$682$	−1.09346e7	−0.900208
$683$	1.60898e7	1.31977	0.659887	−	0.751365i	$-0.270604\pi$
$683$	1.60898e7	1.31977	0.659887	+	0.751365i	$0.270604\pi$
$684$	0	0
$685$	−1.49516e7	−1.21748
$686$	9.48658e6	0.769661
$687$	0	0
$688$	3.01154e6	0.242559
$689$	−1.93061e6	−0.154934
$690$	0	0
$691$	5.42126e6	0.431922	0.215961	−	0.976402i	$-0.430712\pi$
$691$	5.42126e6	0.431922	0.215961	+	0.976402i	$0.430712\pi$
$692$	2.74229e7	2.17695
$693$	0	0
$694$	2.35617e7	1.85698
$695$	−2.12877e7	−1.67173
$696$	0	0
$697$	1.17869e7	0.919004
$698$	−3.02721e6	−0.235182
$699$	0	0
$700$	1.23311e6	0.0951168
$701$	−1.47704e7	−1.13527	−0.567633	−	0.823282i	$-0.692141\pi$
$701$	−1.47704e7	−1.13527	−0.567633	+	0.823282i	$0.692141\pi$
$702$	0	0
$703$	3.46012e6	0.264060
$704$	−3.23320e7	−2.45867
$705$	0	0
$706$	1.11012e7	0.838223
$707$	−339321.	−0.0255306
$708$	0	0
$709$	2.41386e7	1.80341	0.901707	−	0.432347i	$-0.142314\pi$
$709$	2.41386e7	1.80341	0.901707	+	0.432347i	$0.142314\pi$
$710$	2.67328e7	1.99021
$711$	0	0
$712$	926108.	0.0684639
$713$	−1.64118e6	−0.120902
$714$	0	0
$715$	3.35931e6	0.245745
$716$	−3.37755e6	−0.246218
$717$	0	0
$718$	1.65697e7	1.19951
$719$	−1.09977e7	−0.793374	−0.396687	−	0.917954i	$-0.629840\pi$
$719$	−1.09977e7	−0.793374	−0.396687	+	0.917954i	$0.629840\pi$
$720$	0	0
$721$	6.10512e6	0.437377
$722$	−9.30997e6	−0.664669
$723$	0	0
$724$	−3.33425e6	−0.236403
$725$	5.56383e6	0.393123
$726$	0	0
$727$	−1.70462e7	−1.19616	−0.598082	−	0.801435i	$-0.704070\pi$
$727$	−1.70462e7	−1.19616	−0.598082	+	0.801435i	$0.704070\pi$
$728$	533045.	0.0372765
$729$	0	0
$730$	3.00504e6	0.208710
$731$	1.73023e7	1.19760
$732$	0	0
$733$	−337753.	−0.0232188	−0.0116094	−	0.999933i	$-0.503695\pi$
$733$	−337753.	−0.0232188	−0.0116094	+	0.999933i	$0.503695\pi$
$734$	1.98317e7	1.35869
$735$	0	0
$736$	−5.60280e6	−0.381251
$737$	−1.78240e7	−1.20875
$738$	0	0
$739$	−805741.	−0.0542731	−0.0271365	−	0.999632i	$-0.508639\pi$
$739$	−805741.	−0.0542731	−0.0271365	+	0.999632i	$0.508639\pi$
$740$	−6.69875e6	−0.449691
$741$	0	0
$742$	−4.91598e6	−0.327793
$743$	2.67091e7	1.77495	0.887477	−	0.460851i	$-0.152456\pi$
$743$	2.67091e7	1.77495	0.887477	+	0.460851i	$0.152456\pi$
$744$	0	0
$745$	−1.70767e6	−0.112723
$746$	−1.78217e7	−1.17247
$747$	0	0
$748$	−4.30521e7	−2.81346
$749$	617389.	0.0402119
$750$	0	0
$751$	−3.82078e6	−0.247202	−0.123601	−	0.992332i	$-0.539444\pi$
$751$	−3.82078e6	−0.247202	−0.123601	+	0.992332i	$0.539444\pi$
$752$	570288.	0.0367747
$753$	0	0
$754$	7.22089e6	0.462554
$755$	1.34245e7	0.857096
$756$	0	0
$757$	8.80899e6	0.558710	0.279355	−	0.960188i	$-0.409879\pi$
$757$	8.80899e6	0.558710	0.279355	+	0.960188i	$0.409879\pi$
$758$	9.33609e6	0.590190
$759$	0	0
$760$	−8.27602e6	−0.519741
$761$	−2.10564e6	−0.131802	−0.0659012	−	0.997826i	$-0.520992\pi$
$761$	−2.10564e6	−0.131802	−0.0659012	+	0.997826i	$0.520992\pi$
$762$	0	0
$763$	−413891.	−0.0257380
$764$	3.79491e7	2.35216
$765$	0	0
$766$	−9.57079e6	−0.589353
$767$	−398399.	−0.0244529
$768$	0	0
$769$	−6.98302e6	−0.425821	−0.212911	−	0.977072i	$-0.568294\pi$
$769$	−6.98302e6	−0.425821	−0.212911	+	0.977072i	$0.568294\pi$
$770$	8.55393e6	0.519923
$771$	0	0
$772$	−4.65884e7	−2.81342
$773$	1.43123e7	0.861511	0.430756	−	0.902469i	$-0.358247\pi$
$773$	1.43123e7	0.861511	0.430756	+	0.902469i	$0.358247\pi$
$774$	0	0
$775$	1.58794e6	0.0949684
$776$	4.96897e6	0.296219
$777$	0	0
$778$	−1.50571e7	−0.891852
$779$	−9.55623e6	−0.564213
$780$	0	0
$781$	−3.75534e7	−2.20304
$782$	−1.07712e7	−0.629864
$783$	0	0
$784$	4.04934e6	0.235285
$785$	−5.95958e6	−0.345177
$786$	0	0
$787$	−1.83449e7	−1.05579	−0.527897	−	0.849308i	$-0.677019\pi$
$787$	−1.83449e7	−1.05579	−0.527897	+	0.849308i	$0.677019\pi$
$788$	2.97465e7	1.70655
$789$	0	0
$790$	3.21818e7	1.83460
$791$	−2.93388e6	−0.166725
$792$	0	0
$793$	1.29971e6	0.0733946
$794$	−7.26373e6	−0.408892
$795$	0	0
$796$	−8.41993e6	−0.471005
$797$	1.80668e7	1.00748	0.503740	−	0.863855i	$-0.331957\pi$
$797$	1.80668e7	1.00748	0.503740	+	0.863855i	$0.331957\pi$
$798$	0	0
$799$	3.27649e6	0.181569
$800$	5.42104e6	0.299473
$801$	0	0
$802$	2.01663e7	1.10711
$803$	−4.22139e6	−0.231029
$804$	0	0
$805$	1.28386e6	0.0698279
$806$	2.06087e6	0.111741
$807$	0	0
$808$	1.48829e6	0.0801974
$809$	−1.71475e7	−0.921150	−0.460575	−	0.887621i	$-0.652357\pi$
$809$	−1.71475e7	−0.921150	−0.460575	+	0.887621i	$0.652357\pi$
$810$	0	0
$811$	−2.71687e7	−1.45050	−0.725249	−	0.688487i	$-0.758275\pi$
$811$	−2.71687e7	−1.45050	−0.725249	+	0.688487i	$0.758275\pi$
$812$	1.10304e7	0.587085
$813$	0	0
$814$	1.56861e7	0.829762
$815$	2.49712e7	1.31688
$816$	0	0
$817$	−1.40278e7	−0.735251
$818$	8.49383e6	0.443834
$819$	0	0
$820$	1.85007e7	0.960846
$821$	−2.36514e7	−1.22462	−0.612308	−	0.790619i	$-0.709759\pi$
$821$	−2.36514e7	−1.22462	−0.612308	+	0.790619i	$0.709759\pi$
$822$	0	0
$823$	−3.65582e7	−1.88142	−0.940708	−	0.339216i	$-0.889838\pi$
$823$	−3.65582e7	−1.88142	−0.940708	+	0.339216i	$0.889838\pi$
$824$	−2.67777e7	−1.37390
$825$	0	0
$826$	−1.01446e6	−0.0517350
$827$	−3.43682e7	−1.74740	−0.873702	−	0.486461i	$-0.838288\pi$
$827$	−3.43682e7	−1.74740	−0.873702	+	0.486461i	$0.838288\pi$
$828$	0	0
$829$	−6.66101e6	−0.336631	−0.168315	−	0.985733i	$-0.553833\pi$
$829$	−6.66101e6	−0.336631	−0.168315	+	0.985733i	$0.553833\pi$
$830$	−2.80278e7	−1.41219
$831$	0	0
$832$	6.09367e6	0.305190
$833$	2.32648e7	1.16168
$834$	0	0
$835$	1.28149e7	0.636061
$836$	3.49045e7	1.72729
$837$	0	0
$838$	2.93513e6	0.144384
$839$	−2.74300e7	−1.34531	−0.672654	−	0.739957i	$-0.734846\pi$
$839$	−2.74300e7	−1.34531	−0.672654	+	0.739957i	$0.734846\pi$
$840$	0	0
$841$	2.92582e7	1.42645
$842$	5.49800e7	2.67254
$843$	0	0
$844$	2.69265e7	1.30114
$845$	1.73136e7	0.834151
$846$	0	0
$847$	−6.76824e6	−0.324166
$848$	−4.33828e6	−0.207171
$849$	0	0
$850$	1.04217e7	0.494758
$851$	2.35433e6	0.111441
$852$	0	0
$853$	3.05075e7	1.43560	0.717800	−	0.696249i	$-0.245149\pi$
$853$	3.05075e7	1.43560	0.717800	+	0.696249i	$0.245149\pi$
$854$	3.30950e6	0.155281
$855$	0	0
$856$	−2.70793e6	−0.126314
$857$	−3.59631e7	−1.67265	−0.836324	−	0.548235i	$-0.815300\pi$
$857$	−3.59631e7	−1.67265	−0.836324	+	0.548235i	$0.815300\pi$
$858$	0	0
$859$	2.39847e7	1.10905	0.554525	−	0.832167i	$-0.312900\pi$
$859$	2.39847e7	1.10905	0.554525	+	0.832167i	$0.312900\pi$
$860$	2.71577e7	1.25212
$861$	0	0
$862$	−2.93394e7	−1.34488
$863$	7.98871e6	0.365132	0.182566	−	0.983194i	$-0.441560\pi$
$863$	7.98871e6	0.365132	0.182566	+	0.983194i	$0.441560\pi$
$864$	0	0
$865$	−2.76253e7	−1.25536
$866$	−1.60862e7	−0.728885
$867$	0	0
$868$	3.14812e6	0.141825
$869$	−4.52080e7	−2.03079
$870$	0	0
$871$	3.35932e6	0.150039
$872$	1.81537e6	0.0808487
$873$	0	0
$874$	8.73274e6	0.386698
$875$	−6.16434e6	−0.272186
$876$	0	0
$877$	1.96183e7	0.861316	0.430658	−	0.902515i	$-0.358281\pi$
$877$	1.96183e7	0.861316	0.430658	+	0.902515i	$0.358281\pi$
$878$	1.45459e7	0.636800
$879$	0	0
$880$	7.54872e6	0.328599
$881$	−2.55652e7	−1.10971	−0.554854	−	0.831947i	$-0.687226\pi$
$881$	−2.55652e7	−1.10971	−0.554854	+	0.831947i	$0.687226\pi$
$882$	0	0
$883$	1.76039e7	0.759812	0.379906	−	0.925025i	$-0.375956\pi$
$883$	1.76039e7	0.759812	0.379906	+	0.925025i	$0.375956\pi$
$884$	8.11412e6	0.349229
$885$	0	0
$886$	−3.42495e7	−1.46579
$887$	−3.35376e7	−1.43127	−0.715637	−	0.698473i	$-0.753863\pi$
$887$	−3.35376e7	−1.43127	−0.715637	+	0.698473i	$0.753863\pi$
$888$	0	0
$889$	2.46917e6	0.104784
$890$	−2.80098e6	−0.118532
$891$	0	0
$892$	−1.66043e7	−0.698729
$893$	−2.65642e6	−0.111472
$894$	0	0
$895$	3.40248e6	0.141983
$896$	8.34892e6	0.347424
$897$	0	0
$898$	3.98122e7	1.64750
$899$	1.42044e7	0.586169
$900$	0	0
$901$	−2.49248e7	−1.02287
$902$	−4.33221e7	−1.77293
$903$	0	0
$904$	1.28683e7	0.523720
$905$	3.35886e6	0.136323
$906$	0	0
$907$	−4.58984e7	−1.85259	−0.926296	−	0.376798i	$-0.877025\pi$
$907$	−4.58984e7	−1.85259	−0.926296	+	0.376798i	$0.877025\pi$
$908$	3.32768e6	0.133945
$909$	0	0
$910$	−1.61218e6	−0.0645370
$911$	3.40938e7	1.36107	0.680534	−	0.732716i	$-0.261748\pi$
$911$	3.40938e7	1.36107	0.680534	+	0.732716i	$0.261748\pi$
$912$	0	0
$913$	3.93727e7	1.56321
$914$	−1.05429e7	−0.417439
$915$	0	0
$916$	4.96103e7	1.95359
$917$	−9.11889e6	−0.358112
$918$	0	0
$919$	−4.71301e7	−1.84081	−0.920406	−	0.390965i	$-0.872141\pi$
$919$	−4.71301e7	−1.84081	−0.920406	+	0.390965i	$0.872141\pi$
$920$	−5.63115e6	−0.219345
$921$	0	0
$922$	−1.54916e7	−0.600162
$923$	7.07777e6	0.273459
$924$	0	0
$925$	−2.27795e6	−0.0875367
$926$	1.21288e7	0.464825
$927$	0	0
$928$	4.84921e7	1.84842
$929$	−3.41936e7	−1.29989	−0.649944	−	0.759982i	$-0.725208\pi$
$929$	−3.41936e7	−1.29989	−0.649944	+	0.759982i	$0.725208\pi$
$930$	0	0
$931$	−1.88619e7	−0.713201
$932$	−5.19342e7	−1.95845
$933$	0	0
$934$	3.21265e7	1.20503
$935$	4.33699e7	1.62240
$936$	0	0
$937$	−4.54494e7	−1.69114	−0.845569	−	0.533866i	$-0.820739\pi$
$937$	−4.54494e7	−1.69114	−0.845569	+	0.533866i	$0.820739\pi$
$938$	8.55395e6	0.317438
$939$	0	0
$940$	5.14278e6	0.189836
$941$	5.33355e7	1.96355	0.981776	−	0.190043i	$-0.0608629\pi$
$941$	5.33355e7	1.96355	0.981776	+	0.190043i	$0.0608629\pi$
$942$	0	0
$943$	−6.50223e6	−0.238113
$944$	−895245.	−0.0326973
$945$	0	0
$946$	−6.35936e7	−2.31039
$947$	−2.21988e7	−0.804367	−0.402183	−	0.915559i	$-0.631749\pi$
$947$	−2.21988e7	−0.804367	−0.402183	+	0.915559i	$0.631749\pi$
$948$	0	0
$949$	795614.	0.0286772
$950$	−8.44943e6	−0.303752
$951$	0	0
$952$	6.88180e6	0.246099
$953$	43754.2	0.00156059	0.000780293	−	1.00000i	$-0.499752\pi$
$953$	43754.2	0.00156059	0.000780293		1.00000i	$0.499752\pi$
$954$	0	0
$955$	−3.82291e7	−1.35639
$956$	−2.52675e7	−0.894163
$957$	0	0
$958$	5.09871e7	1.79493
$959$	−1.00799e7	−0.353923
$960$	0	0
$961$	−2.45752e7	−0.858397
$962$	−2.95639e6	−0.102997
$963$	0	0
$964$	5.26477e7	1.82468
$965$	4.69322e7	1.62238
$966$	0	0
$967$	4.71228e7	1.62056	0.810280	−	0.586043i	$-0.199315\pi$
$967$	4.71228e7	1.62056	0.810280	+	0.586043i	$0.199315\pi$
$968$	2.96862e7	1.01828
$969$	0	0
$970$	−1.50285e7	−0.512845
$971$	3.47681e7	1.18340	0.591701	−	0.806157i	$-0.298457\pi$
$971$	3.47681e7	1.18340	0.591701	+	0.806157i	$0.298457\pi$
$972$	0	0
$973$	−1.43515e7	−0.485976
$974$	−2.07626e7	−0.701267
$975$	0	0
$976$	2.92059e6	0.0981400
$977$	4.21133e7	1.41151	0.705753	−	0.708458i	$-0.250609\pi$
$977$	4.21133e7	1.41151	0.705753	+	0.708458i	$0.250609\pi$
$978$	0	0
$979$	3.93474e6	0.131208
$980$	3.65164e7	1.21457
$981$	0	0
$982$	6.50613e7	2.15300
$983$	1.65545e7	0.546427	0.273214	−	0.961953i	$-0.411913\pi$
$983$	1.65545e7	0.546427	0.273214	+	0.961953i	$0.411913\pi$
$984$	0	0
$985$	−2.99660e7	−0.984098
$986$	9.32242e7	3.05377
$987$	0	0
$988$	−6.57853e6	−0.214406
$989$	−9.54480e6	−0.310296
$990$	0	0
$991$	5.06855e6	0.163946	0.0819728	−	0.996635i	$-0.473878\pi$
$991$	5.06855e6	0.163946	0.0819728	+	0.996635i	$0.473878\pi$
$992$	1.38398e7	0.446531
$993$	0	0
$994$	1.80224e7	0.578557
$995$	8.48207e6	0.271609
$996$	0	0
$997$	3.70847e6	0.118156	0.0590782	−	0.998253i	$-0.481184\pi$
$997$	3.70847e6	0.118156	0.0590782	+	0.998253i	$0.481184\pi$
$998$	1.90913e7	0.606749
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.6.a.b.1.2	✓	12	3.2	odd	2
531.6.a.d.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)

\(f(q)\)	\(=\)	\(q+8.94324 q^{2} +47.9816 q^{4} -48.3356 q^{5} -32.5863 q^{7} +142.927 q^{8} +O(q^{10})\)	\(q+8.94324 q^{2} +47.9816 q^{4} -48.3356 q^{5} -32.5863 q^{7} +142.927 q^{8} -432.277 q^{10} +607.250 q^{11} -114.450 q^{13} -291.427 q^{14} -257.180 q^{16} -1477.59 q^{17} +1197.95 q^{19} -2319.22 q^{20} +5430.79 q^{22} +815.109 q^{23} -788.665 q^{25} -1023.55 q^{26} -1563.54 q^{28} -7054.74 q^{29} -2013.45 q^{31} -6873.69 q^{32} -13214.4 q^{34} +1575.08 q^{35} +2888.36 q^{37} +10713.6 q^{38} -6908.46 q^{40} -7977.13 q^{41} -11709.8 q^{43} +29136.8 q^{44} +7289.72 q^{46} -2217.46 q^{47} -15745.1 q^{49} -7053.22 q^{50} -5491.47 q^{52} +16868.6 q^{53} -29351.8 q^{55} -4657.46 q^{56} -63092.2 q^{58} +3481.00 q^{59} -11356.2 q^{61} -18006.8 q^{62} -53243.2 q^{64} +5532.00 q^{65} -29351.9 q^{67} -70896.9 q^{68} +14086.3 q^{70} -61841.8 q^{71} -6951.65 q^{73} +25831.3 q^{74} +57479.7 q^{76} -19788.1 q^{77} -74447.1 q^{79} +12431.0 q^{80} -71341.4 q^{82} +64837.6 q^{83} +71420.1 q^{85} -104724. q^{86} +86792.4 q^{88} +6479.60 q^{89} +3729.49 q^{91} +39110.2 q^{92} -19831.3 q^{94} -57903.9 q^{95} +34765.8 q^{97} -140812. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{2} + 198 q^{4} - 36 q^{5} - 411 q^{7} + 69 q^{8}+O(q^{10}) \) 12 * q + 4 * q^2 + 198 * q^4 - 36 * q^5 - 411 * q^7 + 69 * q^8	\( 12 q + 4 q^{2} + 198 q^{4} - 36 q^{5} - 411 q^{7} + 69 q^{8} - 863 q^{10} - 492 q^{11} - 974 q^{13} + 967 q^{14} + 6370 q^{16} + 1463 q^{17} - 3189 q^{19} + 835 q^{20} - 2726 q^{22} + 2617 q^{23} + 8642 q^{25} - 2414 q^{26} - 20458 q^{28} + 1963 q^{29} - 11929 q^{31} + 14382 q^{32} - 20744 q^{34} - 1829 q^{35} - 28105 q^{37} + 23475 q^{38} - 100576 q^{40} + 7585 q^{41} - 33146 q^{43} - 26014 q^{44} - 142851 q^{46} + 79215 q^{47} - 32569 q^{49} + 136019 q^{50} - 248218 q^{52} + 12220 q^{53} - 117770 q^{55} + 186728 q^{56} - 188072 q^{58} + 41772 q^{59} - 54195 q^{61} - 36230 q^{62} + 45197 q^{64} - 42368 q^{65} + 24224 q^{67} + 209639 q^{68} - 35684 q^{70} - 60254 q^{71} - 15385 q^{73} - 214638 q^{74} - 167504 q^{76} + 17169 q^{77} - 27054 q^{79} - 216899 q^{80} + 37917 q^{82} + 117595 q^{83} - 121585 q^{85} - 306756 q^{86} - 105799 q^{88} + 36033 q^{89} - 32217 q^{91} + 30906 q^{92} + 128392 q^{94} + 50721 q^{95} - 196914 q^{97} - 574100 q^{98}+O(q^{100}) \) 12 * q + 4 * q^2 + 198 * q^4 - 36 * q^5 - 411 * q^7 + 69 * q^8 - 863 * q^10 - 492 * q^11 - 974 * q^13 + 967 * q^14 + 6370 * q^16 + 1463 * q^17 - 3189 * q^19 + 835 * q^20 - 2726 * q^22 + 2617 * q^23 + 8642 * q^25 - 2414 * q^26 - 20458 * q^28 + 1963 * q^29 - 11929 * q^31 + 14382 * q^32 - 20744 * q^34 - 1829 * q^35 - 28105 * q^37 + 23475 * q^38 - 100576 * q^40 + 7585 * q^41 - 33146 * q^43 - 26014 * q^44 - 142851 * q^46 + 79215 * q^47 - 32569 * q^49 + 136019 * q^50 - 248218 * q^52 + 12220 * q^53 - 117770 * q^55 + 186728 * q^56 - 188072 * q^58 + 41772 * q^59 - 54195 * q^61 - 36230 * q^62 + 45197 * q^64 - 42368 * q^65 + 24224 * q^67 + 209639 * q^68 - 35684 * q^70 - 60254 * q^71 - 15385 * q^73 - 214638 * q^74 - 167504 * q^76 + 17169 * q^77 - 27054 * q^79 - 216899 * q^80 + 37917 * q^82 + 117595 * q^83 - 121585 * q^85 - 306756 * q^86 - 105799 * q^88 + 36033 * q^89 - 32217 * q^91 + 30906 * q^92 + 128392 * q^94 + 50721 * q^95 - 196914 * q^97 - 574100 * q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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