LMFDB - Embedded newform 531.6.a.b.1.3

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:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 5 x^{10} - 238 x^{9} + 1067 x^{8} + 20782 x^{7} - 79077 x^{6} - 813818 x^{5} + 2364885 x^{4} + \cdots - 14846072 $ x^11 - 5x^10 - 238x^9 + 1067x^8 + 20782x^7 - 79077x^6 - 813818x^5 + 2364885x^4 + 13849341x^3 - 23890558x^2 - 74443300x - 14846072
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$7.91273$ 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)

$f(q)$	$=$	$q-6.91273 q^{2} +15.7858 q^{4} +11.1343 q^{5} -193.283 q^{7} +112.084 q^{8} +O(q^{10})$	\(q-6.91273 q^{2} +15.7858 q^{4} +11.1343 q^{5} -193.283 q^{7} +112.084 q^{8} -76.9686 q^{10} +125.065 q^{11} +1116.22 q^{13} +1336.12 q^{14} -1279.95 q^{16} +1948.69 q^{17} -2116.73 q^{19} +175.764 q^{20} -864.538 q^{22} +2537.35 q^{23} -3001.03 q^{25} -7716.11 q^{26} -3051.13 q^{28} -4531.40 q^{29} -902.692 q^{31} +5261.27 q^{32} -13470.8 q^{34} -2152.08 q^{35} +15206.0 q^{37} +14632.4 q^{38} +1247.98 q^{40} +8454.58 q^{41} -9993.30 q^{43} +1974.25 q^{44} -17540.0 q^{46} -2702.02 q^{47} +20551.5 q^{49} +20745.3 q^{50} +17620.4 q^{52} -23313.3 q^{53} +1392.51 q^{55} -21664.0 q^{56} +31324.3 q^{58} -3481.00 q^{59} -29163.2 q^{61} +6240.06 q^{62} +4588.78 q^{64} +12428.3 q^{65} -9692.20 q^{67} +30761.7 q^{68} +14876.7 q^{70} +6707.84 q^{71} +51879.5 q^{73} -105115. q^{74} -33414.2 q^{76} -24172.9 q^{77} -70812.0 q^{79} -14251.4 q^{80} -58444.2 q^{82} -31987.2 q^{83} +21697.4 q^{85} +69081.0 q^{86} +14017.8 q^{88} -78629.4 q^{89} -215746. q^{91} +40054.2 q^{92} +18678.3 q^{94} -23568.3 q^{95} +21462.8 q^{97} -142067. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 6 q^{2} + 150 q^{4} + 192 q^{5} - 371 q^{7} + 621 q^{8}+O(q^{10}) $ 11 * q + 6 * q^2 + 150 * q^4 + 192 * q^5 - 371 * q^7 + 621 * q^8	$ 11 q + 6 q^{2} + 150 q^{4} + 192 q^{5} - 371 q^{7} + 621 q^{8} - 399 q^{10} + 698 q^{11} - 1556 q^{13} + 1679 q^{14} - 2662 q^{16} + 4793 q^{17} - 3753 q^{19} + 11023 q^{20} - 9534 q^{22} + 7323 q^{23} + 7867 q^{25} + 4844 q^{26} + 3650 q^{28} + 15467 q^{29} - 5151 q^{31} + 15368 q^{32} + 8452 q^{34} + 23285 q^{35} + 8623 q^{37} - 15205 q^{38} + 41530 q^{40} + 6369 q^{41} - 20506 q^{43} + 55632 q^{44} - 45191 q^{46} + 47899 q^{47} - 10322 q^{49} + 102147 q^{50} - 292 q^{52} + 80048 q^{53} - 2114 q^{55} + 108126 q^{56} - 58294 q^{58} - 38291 q^{59} - 82527 q^{61} + 67438 q^{62} - 51411 q^{64} + 167646 q^{65} - 166976 q^{67} + 136533 q^{68} + 76140 q^{70} + 183560 q^{71} - 36809 q^{73} + 116686 q^{74} + 55580 q^{76} + 164885 q^{77} - 281518 q^{79} + 32683 q^{80} + 178815 q^{82} + 254691 q^{83} + 4763 q^{85} - 349324 q^{86} + 251285 q^{88} + 89687 q^{89} + 34897 q^{91} + 20240 q^{92} + 96548 q^{94} + 155113 q^{95} - 45828 q^{97} - 465864 q^{98}+O(q^{100}) $ 11 * q + 6 * q^2 + 150 * q^4 + 192 * q^5 - 371 * q^7 + 621 * q^8 - 399 * q^10 + 698 * q^11 - 1556 * q^13 + 1679 * q^14 - 2662 * q^16 + 4793 * q^17 - 3753 * q^19 + 11023 * q^20 - 9534 * q^22 + 7323 * q^23 + 7867 * q^25 + 4844 * q^26 + 3650 * q^28 + 15467 * q^29 - 5151 * q^31 + 15368 * q^32 + 8452 * q^34 + 23285 * q^35 + 8623 * q^37 - 15205 * q^38 + 41530 * q^40 + 6369 * q^41 - 20506 * q^43 + 55632 * q^44 - 45191 * q^46 + 47899 * q^47 - 10322 * q^49 + 102147 * q^50 - 292 * q^52 + 80048 * q^53 - 2114 * q^55 + 108126 * q^56 - 58294 * q^58 - 38291 * q^59 - 82527 * q^61 + 67438 * q^62 - 51411 * q^64 + 167646 * q^65 - 166976 * q^67 + 136533 * q^68 + 76140 * q^70 + 183560 * q^71 - 36809 * q^73 + 116686 * q^74 + 55580 * q^76 + 164885 * q^77 - 281518 * q^79 + 32683 * q^80 + 178815 * q^82 + 254691 * q^83 + 4763 * q^85 - 349324 * q^86 + 251285 * q^88 + 89687 * q^89 + 34897 * q^91 + 20240 * q^92 + 96548 * q^94 + 155113 * q^95 - 45828 * q^97 - 465864 * 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 $

$2$	−6.91273	−1.22201	−0.611005	−	0.791627i	$-0.709234\pi$
$2$	−6.91273	−1.22201	−0.611005	+	0.791627i	$0.709234\pi$
$3$	0	0
$3$	0	0
$4$	15.7858	0.493306
$5$	11.1343	0.199177	0.0995885	−	0.995029i	$-0.468247\pi$
$5$	11.1343	0.199177	0.0995885	+	0.995029i	$0.468247\pi$
$6$	0	0
$7$	−193.283	−1.49090	−0.745452	−	0.666560i	$-0.767766\pi$
$7$	−193.283	−1.49090	−0.745452	+	0.666560i	$0.767766\pi$
$8$	112.084	0.619185
$9$	0	0
$10$	−76.9686	−0.243396
$11$	125.065	0.311640	0.155820	−	0.987785i	$-0.450198\pi$
$11$	125.065	0.311640	0.155820	+	0.987785i	$0.450198\pi$
$12$	0	0
$13$	1116.22	1.83185	0.915927	−	0.401346i	$-0.131457\pi$
$13$	1116.22	1.83185	0.915927	+	0.401346i	$0.131457\pi$
$14$	1336.12	1.82190
$15$	0	0
$16$	−1279.95	−1.24996
$17$	1948.69	1.63539	0.817695	−	0.575652i	$-0.195252\pi$
$17$	1948.69	1.63539	0.817695	+	0.575652i	$0.195252\pi$
$18$	0	0
$19$	−2116.73	−1.34518	−0.672591	−	0.740014i	$-0.734819\pi$
$19$	−2116.73	−1.34518	−0.672591	+	0.740014i	$0.734819\pi$
$20$	175.764	0.0982552
$21$	0	0
$22$	−864.538	−0.380827
$23$	2537.35	1.00014	0.500071	−	0.865985i	$-0.333307\pi$
$23$	2537.35	1.00014	0.500071	+	0.865985i	$0.333307\pi$
$24$	0	0
$25$	−3001.03	−0.960329
$26$	−7716.11	−2.23854
$27$	0	0
$28$	−3051.13	−0.735472
$29$	−4531.40	−1.00055	−0.500273	−	0.865867i	$-0.666767\pi$
$29$	−4531.40	−1.00055	−0.500273	+	0.865867i	$0.666767\pi$
$30$	0	0
$31$	−902.692	−0.168708	−0.0843539	−	0.996436i	$-0.526883\pi$
$31$	−902.692	−0.168708	−0.0843539	+	0.996436i	$0.526883\pi$
$32$	5261.27	0.908272
$33$	0	0
$34$	−13470.8	−1.99846
$35$	−2152.08	−0.296953
$36$	0	0
$37$	15206.0	1.82604	0.913020	−	0.407916i	$-0.133744\pi$
$37$	15206.0	1.82604	0.913020	+	0.407916i	$0.133744\pi$
$38$	14632.4	1.64382
$39$	0	0
$40$	1247.98	0.123327
$41$	8454.58	0.785475	0.392737	−	0.919651i	$-0.371528\pi$
$41$	8454.58	0.785475	0.392737	+	0.919651i	$0.371528\pi$
$42$	0	0
$43$	−9993.30	−0.824210	−0.412105	−	0.911136i	$-0.635206\pi$
$43$	−9993.30	−0.824210	−0.412105	+	0.911136i	$0.635206\pi$
$44$	1974.25	0.153734
$45$	0	0
$46$	−17540.0	−1.22218
$47$	−2702.02	−0.178420	−0.0892100	−	0.996013i	$-0.528434\pi$
$47$	−2702.02	−0.178420	−0.0892100	+	0.996013i	$0.528434\pi$
$48$	0	0
$49$	20551.5	1.22279
$50$	20745.3	1.17353
$51$	0	0
$52$	17620.4	0.903664
$53$	−23313.3	−1.14003	−0.570013	−	0.821636i	$-0.693062\pi$
$53$	−23313.3	−1.14003	−0.570013	+	0.821636i	$0.693062\pi$
$54$	0	0
$55$	1392.51	0.0620715
$56$	−21664.0	−0.923144
$57$	0	0
$58$	31324.3	1.22268
$59$	−3481.00	−0.130189
$59$	−3481.00	−0.130189
$60$	0	0
$61$	−29163.2	−1.00348	−0.501742	−	0.865017i	$-0.667307\pi$
$61$	−29163.2	−1.00348	−0.501742	+	0.865017i	$0.667307\pi$
$62$	6240.06	0.206162
$63$	0	0
$64$	4588.78	0.140039
$65$	12428.3	0.364863
$66$	0	0
$67$	−9692.20	−0.263776	−0.131888	−	0.991265i	$-0.542104\pi$
$67$	−9692.20	−0.263776	−0.131888	+	0.991265i	$0.542104\pi$
$68$	30761.7	0.806748
$69$	0	0
$70$	14876.7	0.362880
$71$	6707.84	0.157920	0.0789599	−	0.996878i	$-0.474840\pi$
$71$	6707.84	0.157920	0.0789599	+	0.996878i	$0.474840\pi$
$72$	0	0
$73$	51879.5	1.13943	0.569716	−	0.821841i	$-0.307053\pi$
$73$	51879.5	1.13943	0.569716	+	0.821841i	$0.307053\pi$
$74$	−105115.	−2.23144
$75$	0	0
$76$	−33414.2	−0.663586
$77$	−24172.9	−0.464625
$78$	0	0
$79$	−70812.0	−1.27655	−0.638277	−	0.769807i	$-0.720352\pi$
$79$	−70812.0	−1.27655	−0.638277	+	0.769807i	$0.720352\pi$
$80$	−14251.4	−0.248962
$81$	0	0
$82$	−58444.2	−0.959857
$83$	−31987.2	−0.509660	−0.254830	−	0.966986i	$-0.582020\pi$
$83$	−31987.2	−0.509660	−0.254830	+	0.966986i	$0.582020\pi$
$84$	0	0
$85$	21697.4	0.325732
$86$	69081.0	1.00719
$87$	0	0
$88$	14017.8	0.192963
$89$	−78629.4	−1.05223	−0.526114	−	0.850414i	$-0.676351\pi$
$89$	−78629.4	−1.05223	−0.526114	+	0.850414i	$0.676351\pi$
$90$	0	0
$91$	−215746.	−2.73112
$92$	40054.2	0.493376
$93$	0	0
$94$	18678.3	0.218031
$95$	−23568.3	−0.267929
$96$	0	0
$97$	21462.8	0.231610	0.115805	−	0.993272i	$-0.463055\pi$
$97$	21462.8	0.231610	0.115805	+	0.993272i	$0.463055\pi$
$98$	−142067.	−1.49426
$99$	0	0
$100$	−47373.6	−0.473736
$101$	−4675.59	−0.0456072	−0.0228036	−	0.999740i	$-0.507259\pi$
$101$	−4675.59	−0.0456072	−0.0228036	+	0.999740i	$0.507259\pi$
$102$	0	0
$103$	71375.2	0.662909	0.331455	−	0.943471i	$-0.392461\pi$
$103$	71375.2	0.662909	0.331455	+	0.943471i	$0.392461\pi$
$104$	125111.	1.13426
$105$	0	0
$106$	161159.	1.39312
$107$	227244.	1.91882	0.959409	−	0.282019i	$-0.0910040\pi$
$107$	227244.	1.91882	0.959409	+	0.282019i	$0.0910040\pi$
$108$	0	0
$109$	19845.6	0.159992	0.0799961	−	0.996795i	$-0.474509\pi$
$109$	19845.6	0.159992	0.0799961	+	0.996795i	$0.474509\pi$
$110$	−9626.05	−0.0758519
$111$	0	0
$112$	247394.	1.86356
$113$	−69900.0	−0.514969	−0.257484	−	0.966282i	$-0.582894\pi$
$113$	−69900.0	−0.514969	−0.257484	+	0.966282i	$0.582894\pi$
$114$	0	0
$115$	28251.7	0.199205
$116$	−71531.8	−0.493576
$117$	0	0
$118$	24063.2	0.159092
$119$	−376650.	−2.43821
$120$	0	0
$121$	−145410.	−0.902881
$122$	201597.	1.22627
$123$	0	0
$124$	−14249.7	−0.0832246
$125$	−68209.2	−0.390452
$126$	0	0
$127$	99756.3	0.548821	0.274411	−	0.961613i	$-0.411517\pi$
$127$	99756.3	0.548821	0.274411	+	0.961613i	$0.411517\pi$
$128$	−200082.	−1.07940
$129$	0	0
$130$	−85913.7	−0.445866
$131$	197499.	1.00551	0.502756	−	0.864428i	$-0.332319\pi$
$131$	197499.	1.00551	0.502756	+	0.864428i	$0.332319\pi$
$132$	0	0
$133$	409128.	2.00554
$134$	66999.6	0.322337
$135$	0	0
$136$	218418.	1.01261
$137$	244667.	1.11371	0.556857	−	0.830608i	$-0.312007\pi$
$137$	244667.	1.11371	0.556857	+	0.830608i	$0.312007\pi$
$138$	0	0
$139$	−204234.	−0.896584	−0.448292	−	0.893887i	$-0.647968\pi$
$139$	−204234.	−0.896584	−0.448292	+	0.893887i	$0.647968\pi$
$140$	−33972.3	−0.146489
$141$	0	0
$142$	−46369.4	−0.192979
$143$	139600.	0.570879
$144$	0	0
$145$	−50454.1	−0.199286
$146$	−358629.	−1.39240
$147$	0	0
$148$	240038.	0.900796
$149$	410416.	1.51446	0.757232	−	0.653146i	$-0.226551\pi$
$149$	410416.	1.51446	0.757232	+	0.653146i	$0.226551\pi$
$150$	0	0
$151$	−388786.	−1.38761	−0.693807	−	0.720161i	$-0.744068\pi$
$151$	−388786.	−1.38761	−0.693807	+	0.720161i	$0.744068\pi$
$152$	−237252.	−0.832916
$153$	0	0
$154$	167101.	0.567776
$155$	−10050.9	−0.0336027
$156$	0	0
$157$	336248.	1.08871	0.544353	−	0.838856i	$-0.316775\pi$
$157$	336248.	1.08871	0.544353	+	0.838856i	$0.316775\pi$
$158$	489504.	1.55996
$159$	0	0
$160$	58580.7	0.180907
$161$	−490428.	−1.49111
$162$	0	0
$163$	299758.	0.883694	0.441847	−	0.897091i	$-0.354323\pi$
$163$	299758.	0.883694	0.441847	+	0.897091i	$0.354323\pi$
$164$	133462.	0.387479
$165$	0	0
$166$	221119.	0.622810
$167$	−263356.	−0.730722	−0.365361	−	0.930866i	$-0.619054\pi$
$167$	−263356.	−0.730722	−0.365361	+	0.930866i	$0.619054\pi$
$168$	0	0
$169$	874650.	2.35569
$170$	−149988.	−0.398047
$171$	0	0
$172$	−157752.	−0.406588
$173$	695281.	1.76622	0.883111	−	0.469164i	$-0.155445\pi$
$173$	695281.	1.76622	0.883111	+	0.469164i	$0.155445\pi$
$174$	0	0
$175$	580049.	1.43176
$176$	−160077.	−0.389536
$177$	0	0
$178$	543543.	1.28583
$179$	−450239.	−1.05029	−0.525147	−	0.851012i	$-0.675990\pi$
$179$	−450239.	−1.05029	−0.525147	+	0.851012i	$0.675990\pi$
$180$	0	0
$181$	289973.	0.657901	0.328951	−	0.944347i	$-0.393305\pi$
$181$	289973.	0.657901	0.328951	+	0.944347i	$0.393305\pi$
$182$	1.49140e6	3.33745
$183$	0	0
$184$	284398.	0.619272
$185$	169308.	0.363705
$186$	0	0
$187$	243713.	0.509653
$188$	−42653.5	−0.0880157
$189$	0	0
$190$	162922.	0.327412
$191$	272091.	0.539673	0.269836	−	0.962906i	$-0.413030\pi$
$191$	272091.	0.539673	0.269836	+	0.962906i	$0.413030\pi$
$192$	0	0
$193$	−800409.	−1.54675	−0.773373	−	0.633951i	$-0.781432\pi$
$193$	−800409.	−1.54675	−0.773373	+	0.633951i	$0.781432\pi$
$194$	−148367.	−0.283030
$195$	0	0
$196$	324421.	0.603211
$197$	639502.	1.17402	0.587012	−	0.809579i	$-0.300304\pi$
$197$	639502.	1.17402	0.587012	+	0.809579i	$0.300304\pi$
$198$	0	0
$199$	812910.	1.45516	0.727579	−	0.686024i	$-0.240646\pi$
$199$	812910.	1.45516	0.727579	+	0.686024i	$0.240646\pi$
$200$	−336368.	−0.594621
$201$	0	0
$202$	32321.1	0.0557324
$203$	875845.	1.49172
$204$	0	0
$205$	94136.0	0.156448
$206$	−493397.	−0.810081
$207$	0	0
$208$	−1.42871e6	−2.28973
$209$	−264728.	−0.419212
$210$	0	0
$211$	134678.	0.208253	0.104126	−	0.994564i	$-0.466795\pi$
$211$	134678.	0.208253	0.104126	+	0.994564i	$0.466795\pi$
$212$	−368019.	−0.562382
$213$	0	0
$214$	−1.57088e6	−2.34481
$215$	−111269.	−0.164164
$216$	0	0
$217$	174475.	0.251527
$218$	−137187.	−0.195512
$219$	0	0
$220$	21981.9	0.0306202
$221$	2.17517e6	2.99579
$222$	0	0
$223$	−922919.	−1.24280	−0.621401	−	0.783493i	$-0.713436\pi$
$223$	−922919.	−1.24280	−0.621401	+	0.783493i	$0.713436\pi$
$224$	−1.01692e6	−1.35415
$225$	0	0
$226$	483200.	0.629297
$227$	936298.	1.20601	0.603003	−	0.797739i	$-0.293971\pi$
$227$	936298.	1.20601	0.603003	+	0.797739i	$0.293971\pi$
$228$	0	0
$229$	−1.13593e6	−1.43141	−0.715706	−	0.698402i	$-0.753895\pi$
$229$	−1.13593e6	−1.43141	−0.715706	+	0.698402i	$0.753895\pi$
$230$	−195297.	−0.243430
$231$	0	0
$232$	−507899.	−0.619523
$233$	1.39667e6	1.68540	0.842700	−	0.538383i	$-0.180965\pi$
$233$	1.39667e6	1.68540	0.842700	+	0.538383i	$0.180965\pi$
$234$	0	0
$235$	−30085.2	−0.0355372
$236$	−54950.4	−0.0642230
$237$	0	0
$238$	2.60368e6	2.97951
$239$	−833356.	−0.943704	−0.471852	−	0.881678i	$-0.656414\pi$
$239$	−833356.	−0.943704	−0.471852	+	0.881678i	$0.656414\pi$
$240$	0	0
$241$	1.27013e6	1.40866	0.704331	−	0.709871i	$-0.251247\pi$
$241$	1.27013e6	1.40866	0.704331	+	0.709871i	$0.251247\pi$
$242$	1.00518e6	1.10333
$243$	0	0
$244$	−460364.	−0.495025
$245$	228827.	0.243552
$246$	0	0
$247$	−2.36273e6	−2.46418
$248$	−101178.	−0.104461
$249$	0	0
$250$	471512.	0.477136
$251$	791023.	0.792510	0.396255	−	0.918141i	$-0.370310\pi$
$251$	791023.	0.792510	0.396255	+	0.918141i	$0.370310\pi$
$252$	0	0
$253$	317334.	0.311684
$254$	−689588.	−0.670665
$255$	0	0
$256$	1.23627e6	1.17900
$257$	1.83299e6	1.73112	0.865559	−	0.500807i	$-0.166963\pi$
$257$	1.83299e6	1.73112	0.865559	+	0.500807i	$0.166963\pi$
$258$	0	0
$259$	−2.93906e6	−2.72245
$260$	196191.	0.179989
$261$	0	0
$262$	−1.36526e6	−1.22874
$263$	12651.0	0.0112781	0.00563905	−	0.999984i	$-0.498205\pi$
$263$	12651.0	0.0112781	0.00563905	+	0.999984i	$0.498205\pi$
$264$	0	0
$265$	−259578.	−0.227067
$266$	−2.82819e6	−2.45078
$267$	0	0
$268$	−152999.	−0.130122
$269$	148582.	0.125195	0.0625973	−	0.998039i	$-0.480062\pi$
$269$	148582.	0.125195	0.0625973	+	0.998039i	$0.480062\pi$
$270$	0	0
$271$	941611.	0.778840	0.389420	−	0.921060i	$-0.372675\pi$
$271$	941611.	0.778840	0.389420	+	0.921060i	$0.372675\pi$
$272$	−2.49424e6	−2.04416
$273$	0	0
$274$	−1.69131e6	−1.36097
$275$	−375323.	−0.299277
$276$	0	0
$277$	2.13299e6	1.67028	0.835140	−	0.550038i	$-0.185387\pi$
$277$	2.13299e6	1.67028	0.835140	+	0.550038i	$0.185387\pi$
$278$	1.41181e6	1.09563
$279$	0	0
$280$	−241215.	−0.183869
$281$	890716.	0.672935	0.336468	−	0.941695i	$-0.390768\pi$
$281$	890716.	0.672935	0.336468	+	0.941695i	$0.390768\pi$
$282$	0	0
$283$	817581.	0.606826	0.303413	−	0.952859i	$-0.401874\pi$
$283$	817581.	0.606826	0.303413	+	0.952859i	$0.401874\pi$
$284$	105889.	0.0779028
$285$	0	0
$286$	−965013.	−0.697619
$287$	−1.63413e6	−1.17107
$288$	0	0
$289$	2.37755e6	1.67450
$290$	348775.	0.243529
$291$	0	0
$292$	818959.	0.562089
$293$	−610411.	−0.415388	−0.207694	−	0.978194i	$-0.566596\pi$
$293$	−610411.	−0.415388	−0.207694	+	0.978194i	$0.566596\pi$
$294$	0	0
$295$	−38758.6	−0.0259306
$296$	1.70435e6	1.13066
$297$	0	0
$298$	−2.83710e6	−1.85069
$299$	2.83224e6	1.83211
$300$	0	0
$301$	1.93154e6	1.22882
$302$	2.68757e6	1.69568
$303$	0	0
$304$	2.70931e6	1.68142
$305$	−324713.	−0.199871
$306$	0	0
$307$	−1.00422e6	−0.608112	−0.304056	−	0.952654i	$-0.598341\pi$
$307$	−1.00422e6	−0.608112	−0.304056	+	0.952654i	$0.598341\pi$
$308$	−381589.	−0.229202
$309$	0	0
$310$	69478.9	0.0410628
$311$	35455.9	0.0207868	0.0103934	−	0.999946i	$-0.496692\pi$
$311$	35455.9	0.0207868	0.0103934	+	0.999946i	$0.496692\pi$
$312$	0	0
$313$	−1.63361e6	−0.942511	−0.471255	−	0.881997i	$-0.656199\pi$
$313$	−1.63361e6	−0.942511	−0.471255	+	0.881997i	$0.656199\pi$
$314$	−2.32439e6	−1.33041
$315$	0	0
$316$	−1.11782e6	−0.629732
$317$	620992.	0.347086	0.173543	−	0.984826i	$-0.444478\pi$
$317$	620992.	0.347086	0.173543	+	0.984826i	$0.444478\pi$
$318$	0	0
$319$	−566718.	−0.311810
$320$	51093.0	0.0278925
$321$	0	0
$322$	3.39020e6	1.82216
$323$	−4.12485e6	−2.19990
$324$	0	0
$325$	−3.34980e6	−1.75918
$326$	−2.07215e6	−1.07988
$327$	0	0
$328$	947626.	0.486354
$329$	522255.	0.266007
$330$	0	0
$331$	2.75397e6	1.38162	0.690812	−	0.723034i	$-0.257253\pi$
$331$	2.75397e6	1.38162	0.690812	+	0.723034i	$0.257253\pi$
$332$	−504943.	−0.251419
$333$	0	0
$334$	1.82051e6	0.892949
$335$	−107916.	−0.0525381
$336$	0	0
$337$	467039.	0.224016	0.112008	−	0.993707i	$-0.464272\pi$
$337$	467039.	0.224016	0.112008	+	0.993707i	$0.464272\pi$
$338$	−6.04622e6	−2.87867
$339$	0	0
$340$	342511.	0.160686
$341$	−112895.	−0.0525761
$342$	0	0
$343$	−723744.	−0.332162
$344$	−1.12009e6	−0.510338
$345$	0	0
$346$	−4.80629e6	−2.15834
$347$	−3.27711e6	−1.46106	−0.730530	−	0.682881i	$-0.760727\pi$
$347$	−3.27711e6	−1.46106	−0.730530	+	0.682881i	$0.760727\pi$
$348$	0	0
$349$	−3.92940e6	−1.72688	−0.863441	−	0.504450i	$-0.831695\pi$
$349$	−3.92940e6	−1.72688	−0.863441	+	0.504450i	$0.831695\pi$
$350$	−4.00972e6	−1.74962
$351$	0	0
$352$	658000.	0.283054
$353$	−3.34115e6	−1.42712	−0.713558	−	0.700597i	$-0.752917\pi$
$353$	−3.34115e6	−1.42712	−0.713558	+	0.700597i	$0.752917\pi$
$354$	0	0
$355$	74687.3	0.0314540
$356$	−1.24123e6	−0.519070
$357$	0	0
$358$	3.11238e6	1.28347
$359$	772755.	0.316450	0.158225	−	0.987403i	$-0.449423\pi$
$359$	772755.	0.316450	0.158225	+	0.987403i	$0.449423\pi$
$360$	0	0
$361$	2.00444e6	0.809514
$362$	−2.00450e6	−0.803961
$363$	0	0
$364$	−3.40573e6	−1.34728
$365$	577643.	0.226949
$366$	0	0
$367$	−640958.	−0.248407	−0.124204	−	0.992257i	$-0.539638\pi$
$367$	−640958.	−0.248407	−0.124204	+	0.992257i	$0.539638\pi$
$368$	−3.24770e6	−1.25013
$369$	0	0
$370$	−1.17038e6	−0.444451
$371$	4.50608e6	1.69967
$372$	0	0
$373$	−2.37093e6	−0.882360	−0.441180	−	0.897419i	$-0.645440\pi$
$373$	−2.37093e6	−0.882360	−0.441180	+	0.897419i	$0.645440\pi$
$374$	−1.68472e6	−0.622800
$375$	0	0
$376$	−302854.	−0.110475
$377$	−5.05803e6	−1.83286
$378$	0	0
$379$	−2.59822e6	−0.929134	−0.464567	−	0.885538i	$-0.653790\pi$
$379$	−2.59822e6	−0.929134	−0.464567	+	0.885538i	$0.653790\pi$
$380$	−372045.	−0.132171
$381$	0	0
$382$	−1.88089e6	−0.659485
$383$	2.73909e6	0.954135	0.477067	−	0.878867i	$-0.341700\pi$
$383$	2.73909e6	0.954135	0.477067	+	0.878867i	$0.341700\pi$
$384$	0	0
$385$	−269149.	−0.0925426
$386$	5.53301e6	1.89014
$387$	0	0
$388$	338808.	0.114255
$389$	−1.26060e6	−0.422381	−0.211190	−	0.977445i	$-0.567734\pi$
$389$	−1.26060e6	−0.422381	−0.211190	+	0.977445i	$0.567734\pi$
$390$	0	0
$391$	4.94453e6	1.63562
$392$	2.30350e6	0.757134
$393$	0	0
$394$	−4.42071e6	−1.43467
$395$	−788444.	−0.254260
$396$	0	0
$397$	146031.	0.0465017	0.0232508	−	0.999730i	$-0.492598\pi$
$397$	146031.	0.0465017	0.0232508	+	0.999730i	$0.492598\pi$
$398$	−5.61943e6	−1.77822
$399$	0	0
$400$	3.84118e6	1.20037
$401$	2.94941e6	0.915956	0.457978	−	0.888964i	$-0.348574\pi$
$401$	2.94941e6	0.915956	0.457978	+	0.888964i	$0.348574\pi$
$402$	0	0
$403$	−1.00760e6	−0.309048
$404$	−73808.0	−0.0224983
$405$	0	0
$406$	−6.05447e6	−1.82289
$407$	1.90173e6	0.569067
$408$	0	0
$409$	3.43709e6	1.01597	0.507987	−	0.861365i	$-0.330390\pi$
$409$	3.43709e6	1.01597	0.507987	+	0.861365i	$0.330390\pi$
$410$	−650737.	−0.191181
$411$	0	0
$412$	1.12671e6	0.327017
$413$	672819.	0.194099
$414$	0	0
$415$	−356156.	−0.101513
$416$	5.87273e6	1.66382
$417$	0	0
$418$	1.82999e6	0.512281
$419$	6.48770e6	1.80533	0.902664	−	0.430347i	$-0.141609\pi$
$419$	6.48770e6	1.80533	0.902664	+	0.430347i	$0.141609\pi$
$420$	0	0
$421$	−3.88945e6	−1.06951	−0.534753	−	0.845008i	$-0.679595\pi$
$421$	−3.88945e6	−1.06951	−0.534753	+	0.845008i	$0.679595\pi$
$422$	−930992.	−0.254487
$423$	0	0
$424$	−2.61306e6	−0.705886
$425$	−5.84808e6	−1.57051
$426$	0	0
$427$	5.63676e6	1.49610
$428$	3.58723e6	0.946565
$429$	0	0
$430$	769170.	0.200609
$431$	−268690.	−0.0696721	−0.0348361	−	0.999393i	$-0.511091\pi$
$431$	−268690.	−0.0696721	−0.0348361	+	0.999393i	$0.511091\pi$
$432$	0	0
$433$	−1.71353e6	−0.439210	−0.219605	−	0.975589i	$-0.570477\pi$
$433$	−1.71353e6	−0.439210	−0.219605	+	0.975589i	$0.570477\pi$
$434$	−1.20610e6	−0.307368
$435$	0	0
$436$	313279.	0.0789251
$437$	−5.37089e6	−1.34537
$438$	0	0
$439$	663061.	0.164207	0.0821035	−	0.996624i	$-0.473836\pi$
$439$	663061.	0.164207	0.0821035	+	0.996624i	$0.473836\pi$
$440$	156079.	0.0384337
$441$	0	0
$442$	−1.50363e7	−3.66089
$443$	1.92044e6	0.464934	0.232467	−	0.972604i	$-0.425320\pi$
$443$	1.92044e6	0.464934	0.232467	+	0.972604i	$0.425320\pi$
$444$	0	0
$445$	−875485.	−0.209579
$446$	6.37989e6	1.51871
$447$	0	0
$448$	−886936.	−0.208784
$449$	641166.	0.150091	0.0750454	−	0.997180i	$-0.476090\pi$
$449$	641166.	0.150091	0.0750454	+	0.997180i	$0.476090\pi$
$450$	0	0
$451$	1.05737e6	0.244785
$452$	−1.10343e6	−0.254037
$453$	0	0
$454$	−6.47237e6	−1.47375
$455$	−2.40219e6	−0.543975
$456$	0	0
$457$	2.82002e6	0.631629	0.315814	−	0.948821i	$-0.397722\pi$
$457$	2.82002e6	0.631629	0.315814	+	0.948821i	$0.397722\pi$
$458$	7.85241e6	1.74920
$459$	0	0
$460$	445976.	0.0982691
$461$	4.36961e6	0.957613	0.478806	−	0.877920i	$-0.341070\pi$
$461$	4.36961e6	0.957613	0.478806	+	0.877920i	$0.341070\pi$
$462$	0	0
$463$	712732.	0.154516	0.0772580	−	0.997011i	$-0.475383\pi$
$463$	712732.	0.154516	0.0772580	+	0.997011i	$0.475383\pi$
$464$	5.79999e6	1.25064
$465$	0	0
$466$	−9.65478e6	−2.05957
$467$	−2.02176e6	−0.428981	−0.214491	−	0.976726i	$-0.568809\pi$
$467$	−2.02176e6	−0.428981	−0.214491	+	0.976726i	$0.568809\pi$
$468$	0	0
$469$	1.87334e6	0.393265
$470$	207970.	0.0434267
$471$	0	0
$472$	−390166.	−0.0806110
$473$	−1.24981e6	−0.256857
$474$	0	0
$475$	6.35236e6	1.29182
$476$	−5.94572e6	−1.20278
$477$	0	0
$478$	5.76076e6	1.15322
$479$	−5.57705e6	−1.11062	−0.555310	−	0.831643i	$-0.687400\pi$
$479$	−5.57705e6	−1.11062	−0.555310	+	0.831643i	$0.687400\pi$
$480$	0	0
$481$	1.69732e7	3.34504
$482$	−8.78009e6	−1.72140
$483$	0	0
$484$	−2.29541e6	−0.445396
$485$	238974.	0.0461314
$486$	0	0
$487$	−1.31991e6	−0.252186	−0.126093	−	0.992018i	$-0.540244\pi$
$487$	−1.31991e6	−0.252186	−0.126093	+	0.992018i	$0.540244\pi$
$488$	−3.26874e6	−0.621342
$489$	0	0
$490$	−1.58182e6	−0.297623
$491$	−5.36809e6	−1.00488	−0.502442	−	0.864611i	$-0.667565\pi$
$491$	−5.36809e6	−1.00488	−0.502442	+	0.864611i	$0.667565\pi$
$492$	0	0
$493$	−8.83031e6	−1.63628
$494$	1.63329e7	3.01125
$495$	0	0
$496$	1.15540e6	0.210877
$497$	−1.29651e6	−0.235443
$498$	0	0
$499$	−1.33480e6	−0.239974	−0.119987	−	0.992775i	$-0.538285\pi$
$499$	−1.33480e6	−0.239974	−0.119987	+	0.992775i	$0.538285\pi$
$500$	−1.07674e6	−0.192612
$501$	0	0
$502$	−5.46812e6	−0.968454
$503$	7.34574e6	1.29454	0.647270	−	0.762260i	$-0.275910\pi$
$503$	7.34574e6	1.29454	0.647270	+	0.762260i	$0.275910\pi$
$504$	0	0
$505$	−52059.6	−0.00908390
$506$	−2.19364e6	−0.380881
$507$	0	0
$508$	1.57473e6	0.270737
$509$	−1.37057e6	−0.234480	−0.117240	−	0.993104i	$-0.537405\pi$
$509$	−1.37057e6	−0.234480	−0.117240	+	0.993104i	$0.537405\pi$
$510$	0	0
$511$	−1.00274e7	−1.69878
$512$	−2.14338e6	−0.361346
$513$	0	0
$514$	−1.26709e7	−2.11544
$515$	794714.	0.132036
$516$	0	0
$517$	−337927.	−0.0556028
$518$	2.03169e7	3.32686
$519$	0	0
$520$	1.39302e6	0.225917
$521$	1.52552e6	0.246220	0.123110	−	0.992393i	$-0.460713\pi$
$521$	1.52552e6	0.246220	0.123110	+	0.992393i	$0.460713\pi$
$522$	0	0
$523$	4.78382e6	0.764751	0.382376	−	0.924007i	$-0.375106\pi$
$523$	4.78382e6	0.764751	0.382376	+	0.924007i	$0.375106\pi$
$524$	3.11768e6	0.496025
$525$	0	0
$526$	−87452.9	−0.0137819
$527$	−1.75907e6	−0.275903
$528$	0	0
$529$	1823.55	0.000283320 0
$530$	1.79439e6	0.277478
$531$	0	0
$532$	6.45842e6	0.989343
$533$	9.43715e6	1.43887
$534$	0	0
$535$	2.53021e6	0.382184
$536$	−1.08634e6	−0.163326
$537$	0	0
$538$	−1.02711e6	−0.152989
$539$	2.57026e6	0.381071
$540$	0	0
$541$	1.22324e7	1.79688	0.898440	−	0.439096i	$-0.144701\pi$
$541$	1.22324e7	1.79688	0.898440	+	0.439096i	$0.144701\pi$
$542$	−6.50910e6	−0.951749
$543$	0	0
$544$	1.02526e7	1.48538
$545$	220968.	0.0318667
$546$	0	0
$547$	1.06742e7	1.52534	0.762669	−	0.646789i	$-0.223889\pi$
$547$	1.06742e7	1.52534	0.762669	+	0.646789i	$0.223889\pi$
$548$	3.86226e6	0.549402
$549$	0	0
$550$	2.59450e6	0.365719
$551$	9.59174e6	1.34592
$552$	0	0
$553$	1.36868e7	1.90322
$554$	−1.47448e7	−2.04110
$555$	0	0
$556$	−3.22400e6	−0.442291
$557$	−5.16477e6	−0.705363	−0.352682	−	0.935743i	$-0.614730\pi$
$557$	−5.16477e6	−0.705363	−0.352682	+	0.935743i	$0.614730\pi$
$558$	0	0
$559$	−1.11547e7	−1.50983
$560$	2.75456e6	0.371179
$561$	0	0
$562$	−6.15727e6	−0.822333
$563$	4.48550e6	0.596403	0.298201	−	0.954503i	$-0.403613\pi$
$563$	4.48550e6	0.596403	0.298201	+	0.954503i	$0.403613\pi$
$564$	0	0
$565$	−778289.	−0.102570
$566$	−5.65171e6	−0.741547
$567$	0	0
$568$	751844.	0.0977815
$569$	−1.02493e7	−1.32713	−0.663566	−	0.748118i	$-0.730958\pi$
$569$	−1.02493e7	−1.32713	−0.663566	+	0.748118i	$0.730958\pi$
$570$	0	0
$571$	−7.69920e6	−0.988224	−0.494112	−	0.869398i	$-0.664507\pi$
$571$	−7.69920e6	−0.988224	−0.494112	+	0.869398i	$0.664507\pi$
$572$	2.20369e6	0.281618
$573$	0	0
$574$	1.12963e7	1.43105
$575$	−7.61467e6	−0.960465
$576$	0	0
$577$	1.53456e7	1.91887	0.959435	−	0.281929i	$-0.0909743\pi$
$577$	1.53456e7	1.91887	0.959435	+	0.281929i	$0.0909743\pi$
$578$	−1.64354e7	−2.04625
$579$	0	0
$580$	−796458.	−0.0983089
$581$	6.18259e6	0.759854
$582$	0	0
$583$	−2.91568e6	−0.355278
$584$	5.81488e6	0.705519
$585$	0	0
$586$	4.21961e6	0.507607
$587$	1.11377e7	1.33413	0.667066	−	0.744999i	$-0.267550\pi$
$587$	1.11377e7	1.33413	0.667066	+	0.744999i	$0.267550\pi$
$588$	0	0
$589$	1.91075e6	0.226943
$590$	267928.	0.0316875
$591$	0	0
$592$	−1.94630e7	−2.28247
$593$	−2.86489e6	−0.334558	−0.167279	−	0.985910i	$-0.553498\pi$
$593$	−2.86489e6	−0.334558	−0.167279	+	0.985910i	$0.553498\pi$
$594$	0	0
$595$	−4.19375e6	−0.485635
$596$	6.47875e6	0.747094
$597$	0	0
$598$	−1.95785e7	−2.23886
$599$	−5.10093e6	−0.580874	−0.290437	−	0.956894i	$-0.593801\pi$
$599$	−5.10093e6	−0.580874	−0.290437	+	0.956894i	$0.593801\pi$
$600$	0	0
$601$	6.57438e6	0.742452	0.371226	−	0.928543i	$-0.378937\pi$
$601$	6.57438e6	0.742452	0.371226	+	0.928543i	$0.378937\pi$
$602$	−1.33522e7	−1.50163
$603$	0	0
$604$	−6.13730e6	−0.684518
$605$	−1.61904e6	−0.179833
$606$	0	0
$607$	−2.53033e6	−0.278744	−0.139372	−	0.990240i	$-0.544508\pi$
$607$	−2.53033e6	−0.278744	−0.139372	+	0.990240i	$0.544508\pi$
$608$	−1.11367e7	−1.22179
$609$	0	0
$610$	2.24465e6	0.244244
$611$	−3.01604e6	−0.326839
$612$	0	0
$613$	−6.47911e6	−0.696409	−0.348204	−	0.937419i	$-0.613208\pi$
$613$	−6.47911e6	−0.696409	−0.348204	+	0.937419i	$0.613208\pi$
$614$	6.94191e6	0.743118
$615$	0	0
$616$	−2.70941e6	−0.287689
$617$	−499984.	−0.0528741	−0.0264371	−	0.999650i	$-0.508416\pi$
$617$	−499984.	−0.0528741	−0.0264371	+	0.999650i	$0.508416\pi$
$618$	0	0
$619$	−2.37362e6	−0.248991	−0.124496	−	0.992220i	$-0.539731\pi$
$619$	−2.37362e6	−0.248991	−0.124496	+	0.992220i	$0.539731\pi$
$620$	−158661.	−0.0165764
$621$	0	0
$622$	−245097.	−0.0254016
$623$	1.51977e7	1.56877
$624$	0	0
$625$	8.61875e6	0.882560
$626$	1.12927e7	1.15176
$627$	0	0
$628$	5.30794e6	0.537065
$629$	2.96318e7	2.98629
$630$	0	0
$631$	968968.	0.0968804	0.0484402	−	0.998826i	$-0.484575\pi$
$631$	968968.	0.0968804	0.0484402	+	0.998826i	$0.484575\pi$
$632$	−7.93692e6	−0.790422
$633$	0	0
$634$	−4.29275e6	−0.424143
$635$	1.11072e6	0.109313
$636$	0	0
$637$	2.29399e7	2.23998
$638$	3.91757e6	0.381035
$639$	0	0
$640$	−2.22778e6	−0.214992
$641$	1.30259e7	1.25216	0.626082	−	0.779757i	$-0.284657\pi$
$641$	1.30259e7	1.25216	0.626082	+	0.779757i	$0.284657\pi$
$642$	0	0
$643$	8.48030e6	0.808879	0.404440	−	0.914565i	$-0.367467\pi$
$643$	8.48030e6	0.808879	0.404440	+	0.914565i	$0.367467\pi$
$644$	−7.74180e6	−0.735576
$645$	0	0
$646$	2.85140e7	2.68829
$647$	1.14711e7	1.07732	0.538660	−	0.842523i	$-0.318931\pi$
$647$	1.14711e7	1.07732	0.538660	+	0.842523i	$0.318931\pi$
$648$	0	0
$649$	−435350.	−0.0405721
$650$	2.31563e7	2.14974
$651$	0	0
$652$	4.73192e6	0.435931
$653$	−1.41233e7	−1.29614	−0.648072	−	0.761579i	$-0.724425\pi$
$653$	−1.41233e7	−1.29614	−0.648072	+	0.761579i	$0.724425\pi$
$654$	0	0
$655$	2.19902e6	0.200275
$656$	−1.08215e7	−0.981808
$657$	0	0
$658$	−3.61021e6	−0.325063
$659$	−4.29463e6	−0.385223	−0.192611	−	0.981275i	$-0.561696\pi$
$659$	−4.29463e6	−0.385223	−0.192611	+	0.981275i	$0.561696\pi$
$660$	0	0
$661$	1.57272e7	1.40006	0.700032	−	0.714111i	$-0.253169\pi$
$661$	1.57272e7	1.40006	0.700032	+	0.714111i	$0.253169\pi$
$662$	−1.90375e7	−1.68836
$663$	0	0
$664$	−3.58526e6	−0.315574
$665$	4.55537e6	0.399456
$666$	0	0
$667$	−1.14978e7	−1.00069
$668$	−4.15729e6	−0.360470
$669$	0	0
$670$	745995.	0.0642020
$671$	−3.64729e6	−0.312726
$672$	0	0
$673$	−2.25572e6	−0.191976	−0.0959881	−	0.995382i	$-0.530601\pi$
$673$	−2.25572e6	−0.191976	−0.0959881	+	0.995382i	$0.530601\pi$
$674$	−3.22851e6	−0.273749
$675$	0	0
$676$	1.38070e7	1.16207
$677$	1.19113e7	0.998818	0.499409	−	0.866366i	$-0.333551\pi$
$677$	1.19113e7	0.998818	0.499409	+	0.866366i	$0.333551\pi$
$678$	0	0
$679$	−4.14841e6	−0.345308
$680$	2.43194e6	0.201688
$681$	0	0
$682$	780412.	0.0642485
$683$	−2.37444e7	−1.94765	−0.973823	−	0.227309i	$-0.927007\pi$
$683$	−2.37444e7	−1.94765	−0.973823	+	0.227309i	$0.927007\pi$
$684$	0	0
$685$	2.72420e6	0.221826
$686$	5.00304e6	0.405905
$687$	0	0
$688$	1.27910e7	1.03023
$689$	−2.60228e7	−2.08836
$690$	0	0
$691$	−671726.	−0.0535176	−0.0267588	−	0.999642i	$-0.508519\pi$
$691$	−671726.	−0.0535176	−0.0267588	+	0.999642i	$0.508519\pi$
$692$	1.09756e7	0.871288
$693$	0	0
$694$	2.26538e7	1.78543
$695$	−2.27401e6	−0.178579
$696$	0	0
$697$	1.64754e7	1.28456
$698$	2.71629e7	2.11027
$699$	0	0
$700$	9.15653e6	0.706294
$701$	−7.09210e6	−0.545105	−0.272552	−	0.962141i	$-0.587868\pi$
$701$	−7.09210e6	−0.545105	−0.272552	+	0.962141i	$0.587868\pi$
$702$	0	0
$703$	−3.21869e7	−2.45635
$704$	573895.	0.0436416
$705$	0	0
$706$	2.30964e7	1.74395
$707$	903715.	0.0679959
$708$	0	0
$709$	1.22100e7	0.912222	0.456111	−	0.889923i	$-0.349242\pi$
$709$	1.22100e7	0.912222	0.456111	+	0.889923i	$0.349242\pi$
$710$	−516293.	−0.0384371
$711$	0	0
$712$	−8.81312e6	−0.651523
$713$	−2.29045e6	−0.168732
$714$	0	0
$715$	1.55435e6	0.113706
$716$	−7.10738e6	−0.518116
$717$	0	0
$718$	−5.34184e6	−0.386705
$719$	1.62202e7	1.17013	0.585066	−	0.810986i	$-0.301069\pi$
$719$	1.62202e7	1.17013	0.585066	+	0.810986i	$0.301069\pi$
$720$	0	0
$721$	−1.37956e7	−0.988333
$722$	−1.38561e7	−0.989234
$723$	0	0
$724$	4.57745e6	0.324547
$725$	1.35989e7	0.960854
$726$	0	0
$727$	2.20651e7	1.54835	0.774175	−	0.632972i	$-0.218165\pi$
$727$	2.20651e7	1.54835	0.774175	+	0.632972i	$0.218165\pi$
$728$	−2.41818e7	−1.69106
$729$	0	0
$730$	−3.99309e6	−0.277333
$731$	−1.94739e7	−1.34790
$732$	0	0
$733$	−1.13061e7	−0.777233	−0.388617	−	0.921400i	$-0.627047\pi$
$733$	−1.13061e7	−0.777233	−0.388617	+	0.921400i	$0.627047\pi$
$734$	4.43077e6	0.303556
$735$	0	0
$736$	1.33497e7	0.908401
$737$	−1.21215e6	−0.0822032
$738$	0	0
$739$	9.23997e6	0.622386	0.311193	−	0.950347i	$-0.399272\pi$
$739$	9.23997e6	0.622386	0.311193	+	0.950347i	$0.399272\pi$
$740$	2.67267e6	0.179418
$741$	0	0
$742$	−3.11493e7	−2.07701
$743$	−1.10636e7	−0.735233	−0.367617	−	0.929977i	$-0.619826\pi$
$743$	−1.10636e7	−0.735233	−0.367617	+	0.929977i	$0.619826\pi$
$744$	0	0
$745$	4.56971e6	0.301646
$746$	1.63896e7	1.07825
$747$	0	0
$748$	3.84720e6	0.251415
$749$	−4.39226e7	−2.86077
$750$	0	0
$751$	−1.63941e6	−0.106069	−0.0530343	−	0.998593i	$-0.516889\pi$
$751$	−1.63941e6	−0.106069	−0.0530343	+	0.998593i	$0.516889\pi$
$752$	3.45846e6	0.223017
$753$	0	0
$754$	3.49648e7	2.23977
$755$	−4.32887e6	−0.276381
$756$	0	0
$757$	−3.02306e7	−1.91738	−0.958688	−	0.284461i	$-0.908185\pi$
$757$	−3.02306e7	−1.91738	−0.958688	+	0.284461i	$0.908185\pi$
$758$	1.79608e7	1.13541
$759$	0	0
$760$	−2.64164e6	−0.165898
$761$	−2.90933e7	−1.82109	−0.910546	−	0.413408i	$-0.864338\pi$
$761$	−2.90933e7	−1.82109	−0.910546	+	0.413408i	$0.864338\pi$
$762$	0	0
$763$	−3.83583e6	−0.238533
$764$	4.29517e6	0.266224
$765$	0	0
$766$	−1.89346e7	−1.16596
$767$	−3.88556e6	−0.238487
$768$	0	0
$769$	−6.83882e6	−0.417028	−0.208514	−	0.978019i	$-0.566863\pi$
$769$	−6.83882e6	−0.417028	−0.208514	+	0.978019i	$0.566863\pi$
$770$	1.86056e6	0.113088
$771$	0	0
$772$	−1.26351e7	−0.763019
$773$	2.33000e7	1.40252	0.701258	−	0.712907i	$-0.252622\pi$
$773$	2.33000e7	1.40252	0.701258	+	0.712907i	$0.252622\pi$
$774$	0	0
$775$	2.70900e6	0.162015
$776$	2.40565e6	0.143409
$777$	0	0
$778$	8.71420e6	0.516153
$779$	−1.78960e7	−1.05661
$780$	0	0
$781$	838914.	0.0492141
$782$	−3.41802e7	−1.99874
$783$	0	0
$784$	−2.63049e7	−1.52844
$785$	3.74389e6	0.216845
$786$	0	0
$787$	−2.73298e7	−1.57290	−0.786448	−	0.617657i	$-0.788082\pi$
$787$	−2.73298e7	−1.57290	−0.786448	+	0.617657i	$0.788082\pi$
$788$	1.00951e7	0.579153
$789$	0	0
$790$	5.45030e6	0.310708
$791$	1.35105e7	0.767769
$792$	0	0
$793$	−3.25525e7	−1.83824
$794$	−1.00947e6	−0.0568255
$795$	0	0
$796$	1.28324e7	0.717838
$797$	−2.61123e7	−1.45613	−0.728064	−	0.685509i	$-0.759580\pi$
$797$	−2.61123e7	−1.45613	−0.728064	+	0.685509i	$0.759580\pi$
$798$	0	0
$799$	−5.26541e6	−0.291786
$800$	−1.57892e7	−0.872240
$801$	0	0
$802$	−2.03885e7	−1.11931
$803$	6.48830e6	0.355093
$804$	0	0
$805$	−5.46059e6	−0.296996
$806$	6.96527e6	0.377659
$807$	0	0
$808$	−524061.	−0.0282393
$809$	−1.70399e7	−0.915369	−0.457684	−	0.889115i	$-0.651321\pi$
$809$	−1.70399e7	−0.915369	−0.457684	+	0.889115i	$0.651321\pi$
$810$	0	0
$811$	−1.19527e7	−0.638136	−0.319068	−	0.947732i	$-0.603370\pi$
$811$	−1.19527e7	−0.638136	−0.319068	+	0.947732i	$0.603370\pi$
$812$	1.38259e7	0.735874
$813$	0	0
$814$	−1.31462e7	−0.695405
$815$	3.33760e6	0.176011
$816$	0	0
$817$	2.11531e7	1.10871
$818$	−2.37597e7	−1.24153
$819$	0	0
$820$	1.48601e6	0.0771770
$821$	2.76018e7	1.42915	0.714577	−	0.699556i	$-0.246619\pi$
$821$	2.76018e7	1.42915	0.714577	+	0.699556i	$0.246619\pi$
$822$	0	0
$823$	1.60010e7	0.823472	0.411736	−	0.911303i	$-0.364923\pi$
$823$	1.60010e7	0.823472	0.411736	+	0.911303i	$0.364923\pi$
$824$	8.00004e6	0.410463
$825$	0	0
$826$	−4.65102e6	−0.237191
$827$	−2.61565e7	−1.32989	−0.664945	−	0.746892i	$-0.731545\pi$
$827$	−2.61565e7	−1.32989	−0.664945	+	0.746892i	$0.731545\pi$
$828$	0	0
$829$	1.29226e7	0.653076	0.326538	−	0.945184i	$-0.394118\pi$
$829$	1.29226e7	0.653076	0.326538	+	0.945184i	$0.394118\pi$
$830$	2.46201e6	0.124049
$831$	0	0
$832$	5.12209e6	0.256530
$833$	4.00485e7	1.99974
$834$	0	0
$835$	−2.93229e6	−0.145543
$836$	−4.17894e6	−0.206800
$837$	0	0
$838$	−4.48477e7	−2.20613
$839$	2.17442e7	1.06644	0.533222	−	0.845975i	$-0.320981\pi$
$839$	2.17442e7	1.06644	0.533222	+	0.845975i	$0.320981\pi$
$840$	0	0
$841$	22444.5	0.00109426
$842$	2.68867e7	1.30695
$843$	0	0
$844$	2.12600e6	0.102732
$845$	9.73864e6	0.469198
$846$	0	0
$847$	2.81053e7	1.34611
$848$	2.98400e7	1.42498
$849$	0	0
$850$	4.04262e7	1.91918
$851$	3.85830e7	1.82630
$852$	0	0
$853$	−2.35835e7	−1.10978	−0.554888	−	0.831925i	$-0.687239\pi$
$853$	−2.35835e7	−1.10978	−0.554888	+	0.831925i	$0.687239\pi$
$854$	−3.89654e7	−1.82824
$855$	0	0
$856$	2.54705e7	1.18810
$857$	1.24991e7	0.581337	0.290668	−	0.956824i	$-0.406122\pi$
$857$	1.24991e7	0.581337	0.290668	+	0.956824i	$0.406122\pi$
$858$	0	0
$859$	3.62400e7	1.67573	0.837867	−	0.545875i	$-0.183803\pi$
$859$	3.62400e7	1.67573	0.837867	+	0.545875i	$0.183803\pi$
$860$	−1.75646e6	−0.0809829
$861$	0	0
$862$	1.85738e6	0.0851400
$863$	1.92175e7	0.878353	0.439177	−	0.898401i	$-0.355270\pi$
$863$	1.92175e7	0.878353	0.439177	+	0.898401i	$0.355270\pi$
$864$	0	0
$865$	7.74149e6	0.351791
$866$	1.18452e7	0.536719
$867$	0	0
$868$	2.75423e6	0.124080
$869$	−8.85608e6	−0.397825
$870$	0	0
$871$	−1.08186e7	−0.483199
$872$	2.22438e6	0.0990646
$873$	0	0
$874$	3.71275e7	1.64406
$875$	1.31837e7	0.582126
$876$	0	0
$877$	2.97880e7	1.30780	0.653901	−	0.756580i	$-0.273131\pi$
$877$	2.97880e7	1.30780	0.653901	+	0.756580i	$0.273131\pi$
$878$	−4.58356e6	−0.200663
$879$	0	0
$880$	−1.78235e6	−0.0775866
$881$	−1.64782e7	−0.715272	−0.357636	−	0.933861i	$-0.616417\pi$
$881$	−1.64782e7	−0.715272	−0.357636	+	0.933861i	$0.616417\pi$
$882$	0	0
$883$	−1.95035e7	−0.841802	−0.420901	−	0.907106i	$-0.638286\pi$
$883$	−1.95035e7	−0.841802	−0.420901	+	0.907106i	$0.638286\pi$
$884$	3.43367e7	1.47784
$885$	0	0
$886$	−1.32755e7	−0.568153
$887$	−2.38782e7	−1.01904	−0.509522	−	0.860457i	$-0.670178\pi$
$887$	−2.38782e7	−1.01904	−0.509522	+	0.860457i	$0.670178\pi$
$888$	0	0
$889$	−1.92812e7	−0.818240
$890$	6.05199e6	0.256108
$891$	0	0
$892$	−1.45690e7	−0.613081
$893$	5.71944e6	0.240007
$894$	0	0
$895$	−5.01311e6	−0.209194
$896$	3.86725e7	1.60928
$897$	0	0
$898$	−4.43220e6	−0.183412
$899$	4.09046e6	0.168800
$900$	0	0
$901$	−4.54305e7	−1.86439
$902$	−7.30931e6	−0.299130
$903$	0	0
$904$	−7.83470e6	−0.318861
$905$	3.22865e6	0.131039
$906$	0	0
$907$	3.15297e7	1.27263	0.636315	−	0.771429i	$-0.280458\pi$
$907$	3.15297e7	1.27263	0.636315	+	0.771429i	$0.280458\pi$
$908$	1.47802e7	0.594930
$909$	0	0
$910$	1.66057e7	0.664743
$911$	1.40130e7	0.559418	0.279709	−	0.960085i	$-0.409762\pi$
$911$	1.40130e7	0.559418	0.279709	+	0.960085i	$0.409762\pi$
$912$	0	0
$913$	−4.00047e6	−0.158831
$914$	−1.94940e7	−0.771856
$915$	0	0
$916$	−1.79316e7	−0.706124
$917$	−3.81733e7	−1.49912
$918$	0	0
$919$	−3.77111e7	−1.47293	−0.736463	−	0.676478i	$-0.763505\pi$
$919$	−3.77111e7	−1.47293	−0.736463	+	0.676478i	$0.763505\pi$
$920$	3.16658e6	0.123345
$921$	0	0
$922$	−3.02059e7	−1.17021
$923$	7.48741e6	0.289286
$924$	0	0
$925$	−4.56336e7	−1.75360
$926$	−4.92692e6	−0.188820
$927$	0	0
$928$	−2.38409e7	−0.908769
$929$	2.22014e7	0.843997	0.421998	−	0.906597i	$-0.361329\pi$
$929$	2.22014e7	0.843997	0.421998	+	0.906597i	$0.361329\pi$
$930$	0	0
$931$	−4.35019e7	−1.64488
$932$	2.20475e7	0.831418
$933$	0	0
$934$	1.39759e7	0.524219
$935$	2.71358e6	0.101511
$936$	0	0
$937$	3.15581e7	1.17425	0.587127	−	0.809495i	$-0.300259\pi$
$937$	3.15581e7	1.17425	0.587127	+	0.809495i	$0.300259\pi$
$938$	−1.29499e7	−0.480573
$939$	0	0
$940$	−474918.	−0.0175307
$941$	1.61367e7	0.594075	0.297037	−	0.954866i	$-0.404001\pi$
$941$	1.61367e7	0.594075	0.297037	+	0.954866i	$0.404001\pi$
$942$	0	0
$943$	2.14523e7	0.785586
$944$	4.45552e6	0.162730
$945$	0	0
$946$	8.63959e6	0.313881
$947$	5.18477e7	1.87869	0.939343	−	0.342978i	$-0.111436\pi$
$947$	5.18477e7	1.87869	0.939343	+	0.342978i	$0.111436\pi$
$948$	0	0
$949$	5.79088e7	2.08727
$950$	−4.39121e7	−1.57861
$951$	0	0
$952$	−4.22166e7	−1.50970
$953$	−3.02972e7	−1.08061	−0.540307	−	0.841468i	$-0.681692\pi$
$953$	−3.02972e7	−1.08061	−0.540307	+	0.841468i	$0.681692\pi$
$954$	0	0
$955$	3.02955e6	0.107490
$956$	−1.31552e7	−0.465535
$957$	0	0
$958$	3.85526e7	1.35719
$959$	−4.72900e7	−1.66044
$960$	0	0
$961$	−2.78143e7	−0.971538
$962$	−1.17331e8	−4.08766
$963$	0	0
$964$	2.00501e7	0.694902
$965$	−8.91202e6	−0.308076
$966$	0	0
$967$	1.15030e7	0.395591	0.197795	−	0.980243i	$-0.436622\pi$
$967$	1.15030e7	0.395591	0.197795	+	0.980243i	$0.436622\pi$
$968$	−1.62982e7	−0.559050
$969$	0	0
$970$	−1.65196e6	−0.0563730
$971$	1.92341e7	0.654673	0.327336	−	0.944908i	$-0.393849\pi$
$971$	1.92341e7	0.654673	0.327336	+	0.944908i	$0.393849\pi$
$972$	0	0
$973$	3.94751e7	1.33672
$974$	9.12417e6	0.308174
$975$	0	0
$976$	3.73275e7	1.25431
$977$	7.41320e6	0.248467	0.124234	−	0.992253i	$-0.460353\pi$
$977$	7.41320e6	0.248467	0.124234	+	0.992253i	$0.460353\pi$
$978$	0	0
$979$	−9.83376e6	−0.327916
$980$	3.61221e6	0.120146
$981$	0	0
$982$	3.71081e7	1.22798
$983$	4.98279e7	1.64471	0.822354	−	0.568976i	$-0.192660\pi$
$983$	4.98279e7	1.64471	0.822354	+	0.568976i	$0.192660\pi$
$984$	0	0
$985$	7.12043e6	0.233838
$986$	6.10415e7	1.99955
$987$	0	0
$988$	−3.72976e7	−1.21559
$989$	−2.53565e7	−0.824327
$990$	0	0
$991$	3.87158e7	1.25229	0.626144	−	0.779708i	$-0.284632\pi$
$991$	3.87158e7	1.25229	0.626144	+	0.779708i	$0.284632\pi$
$992$	−4.74931e6	−0.153233
$993$	0	0
$994$	8.96244e6	0.287714
$995$	9.05121e6	0.289834
$996$	0	0
$997$	1.92043e7	0.611873	0.305937	−	0.952052i	$-0.401030\pi$
$997$	1.92043e7	0.611873	0.305937	+	0.952052i	$0.401030\pi$
$998$	9.22708e6	0.293250
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.6.a.b.1.3		11
3.2	odd	2		177.6.a.a.1.9	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.6.a.a.1.9	✓	11	3.2	odd	2
531.6.a.b.1.3		11	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)

\(f(q)\)	\(=\)	\(q-6.91273 q^{2} +15.7858 q^{4} +11.1343 q^{5} -193.283 q^{7} +112.084 q^{8} +O(q^{10})\)	\(q-6.91273 q^{2} +15.7858 q^{4} +11.1343 q^{5} -193.283 q^{7} +112.084 q^{8} -76.9686 q^{10} +125.065 q^{11} +1116.22 q^{13} +1336.12 q^{14} -1279.95 q^{16} +1948.69 q^{17} -2116.73 q^{19} +175.764 q^{20} -864.538 q^{22} +2537.35 q^{23} -3001.03 q^{25} -7716.11 q^{26} -3051.13 q^{28} -4531.40 q^{29} -902.692 q^{31} +5261.27 q^{32} -13470.8 q^{34} -2152.08 q^{35} +15206.0 q^{37} +14632.4 q^{38} +1247.98 q^{40} +8454.58 q^{41} -9993.30 q^{43} +1974.25 q^{44} -17540.0 q^{46} -2702.02 q^{47} +20551.5 q^{49} +20745.3 q^{50} +17620.4 q^{52} -23313.3 q^{53} +1392.51 q^{55} -21664.0 q^{56} +31324.3 q^{58} -3481.00 q^{59} -29163.2 q^{61} +6240.06 q^{62} +4588.78 q^{64} +12428.3 q^{65} -9692.20 q^{67} +30761.7 q^{68} +14876.7 q^{70} +6707.84 q^{71} +51879.5 q^{73} -105115. q^{74} -33414.2 q^{76} -24172.9 q^{77} -70812.0 q^{79} -14251.4 q^{80} -58444.2 q^{82} -31987.2 q^{83} +21697.4 q^{85} +69081.0 q^{86} +14017.8 q^{88} -78629.4 q^{89} -215746. q^{91} +40054.2 q^{92} +18678.3 q^{94} -23568.3 q^{95} +21462.8 q^{97} -142067. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 6 q^{2} + 150 q^{4} + 192 q^{5} - 371 q^{7} + 621 q^{8}+O(q^{10}) \) 11 * q + 6 * q^2 + 150 * q^4 + 192 * q^5 - 371 * q^7 + 621 * q^8	\( 11 q + 6 q^{2} + 150 q^{4} + 192 q^{5} - 371 q^{7} + 621 q^{8} - 399 q^{10} + 698 q^{11} - 1556 q^{13} + 1679 q^{14} - 2662 q^{16} + 4793 q^{17} - 3753 q^{19} + 11023 q^{20} - 9534 q^{22} + 7323 q^{23} + 7867 q^{25} + 4844 q^{26} + 3650 q^{28} + 15467 q^{29} - 5151 q^{31} + 15368 q^{32} + 8452 q^{34} + 23285 q^{35} + 8623 q^{37} - 15205 q^{38} + 41530 q^{40} + 6369 q^{41} - 20506 q^{43} + 55632 q^{44} - 45191 q^{46} + 47899 q^{47} - 10322 q^{49} + 102147 q^{50} - 292 q^{52} + 80048 q^{53} - 2114 q^{55} + 108126 q^{56} - 58294 q^{58} - 38291 q^{59} - 82527 q^{61} + 67438 q^{62} - 51411 q^{64} + 167646 q^{65} - 166976 q^{67} + 136533 q^{68} + 76140 q^{70} + 183560 q^{71} - 36809 q^{73} + 116686 q^{74} + 55580 q^{76} + 164885 q^{77} - 281518 q^{79} + 32683 q^{80} + 178815 q^{82} + 254691 q^{83} + 4763 q^{85} - 349324 q^{86} + 251285 q^{88} + 89687 q^{89} + 34897 q^{91} + 20240 q^{92} + 96548 q^{94} + 155113 q^{95} - 45828 q^{97} - 465864 q^{98}+O(q^{100}) \) 11 * q + 6 * q^2 + 150 * q^4 + 192 * q^5 - 371 * q^7 + 621 * q^8 - 399 * q^10 + 698 * q^11 - 1556 * q^13 + 1679 * q^14 - 2662 * q^16 + 4793 * q^17 - 3753 * q^19 + 11023 * q^20 - 9534 * q^22 + 7323 * q^23 + 7867 * q^25 + 4844 * q^26 + 3650 * q^28 + 15467 * q^29 - 5151 * q^31 + 15368 * q^32 + 8452 * q^34 + 23285 * q^35 + 8623 * q^37 - 15205 * q^38 + 41530 * q^40 + 6369 * q^41 - 20506 * q^43 + 55632 * q^44 - 45191 * q^46 + 47899 * q^47 - 10322 * q^49 + 102147 * q^50 - 292 * q^52 + 80048 * q^53 - 2114 * q^55 + 108126 * q^56 - 58294 * q^58 - 38291 * q^59 - 82527 * q^61 + 67438 * q^62 - 51411 * q^64 + 167646 * q^65 - 166976 * q^67 + 136533 * q^68 + 76140 * q^70 + 183560 * q^71 - 36809 * q^73 + 116686 * q^74 + 55580 * q^76 + 164885 * q^77 - 281518 * q^79 + 32683 * q^80 + 178815 * q^82 + 254691 * q^83 + 4763 * q^85 - 349324 * q^86 + 251285 * q^88 + 89687 * q^89 + 34897 * q^91 + 20240 * q^92 + 96548 * q^94 + 155113 * q^95 - 45828 * q^97 - 465864 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more