LMFDB - Embedded newform 531.6.a.b.1.9

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			1.9
Root			$-5.70379$ 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)

$f(q)$	$=$	$q+6.70379 q^{2} +12.9408 q^{4} -105.016 q^{5} -129.262 q^{7} -127.769 q^{8} +O(q^{10})$	\(q+6.70379 q^{2} +12.9408 q^{4} -105.016 q^{5} -129.262 q^{7} -127.769 q^{8} -704.003 q^{10} +188.535 q^{11} -958.805 q^{13} -866.546 q^{14} -1270.64 q^{16} -82.6984 q^{17} -2066.89 q^{19} -1358.99 q^{20} +1263.90 q^{22} +589.875 q^{23} +7903.29 q^{25} -6427.63 q^{26} -1672.76 q^{28} +1121.65 q^{29} +940.683 q^{31} -4429.51 q^{32} -554.393 q^{34} +13574.5 q^{35} +575.784 q^{37} -13856.0 q^{38} +13417.7 q^{40} +5521.52 q^{41} -16502.9 q^{43} +2439.80 q^{44} +3954.40 q^{46} -23876.5 q^{47} -98.3079 q^{49} +52982.0 q^{50} -12407.7 q^{52} +39649.3 q^{53} -19799.2 q^{55} +16515.7 q^{56} +7519.30 q^{58} -3481.00 q^{59} +1264.98 q^{61} +6306.14 q^{62} +10966.0 q^{64} +100690. q^{65} -49177.7 q^{67} -1070.19 q^{68} +91000.9 q^{70} +53621.6 q^{71} +30870.6 q^{73} +3859.93 q^{74} -26747.3 q^{76} -24370.5 q^{77} -26724.2 q^{79} +133437. q^{80} +37015.1 q^{82} -27418.4 q^{83} +8684.63 q^{85} -110632. q^{86} -24088.9 q^{88} +30163.0 q^{89} +123937. q^{91} +7633.46 q^{92} -160063. q^{94} +217056. q^{95} -80237.8 q^{97} -659.035 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	6.70379	1.18507	0.592537	−	0.805543i	$-0.298126\pi$
$2$	6.70379	1.18507	0.592537	+	0.805543i	$0.298126\pi$
$3$	0	0
$3$	0	0
$4$	12.9408	0.404401
$5$	−105.016	−1.87858	−0.939289	−	0.343128i	$-0.888513\pi$
$5$	−105.016	−1.87858	−0.939289	+	0.343128i	$0.888513\pi$
$6$	0	0
$7$	−129.262	−0.997071	−0.498536	−	0.866869i	$-0.666129\pi$
$7$	−129.262	−0.997071	−0.498536	+	0.866869i	$0.666129\pi$
$8$	−127.769	−0.705829
$9$	0	0
$10$	−704.003	−2.22625
$11$	188.535	0.469798	0.234899	−	0.972020i	$-0.424524\pi$
$11$	188.535	0.469798	0.234899	+	0.972020i	$0.424524\pi$
$12$	0	0
$13$	−958.805	−1.57352	−0.786760	−	0.617260i	$-0.788243\pi$
$13$	−958.805	−1.57352	−0.786760	+	0.617260i	$0.788243\pi$
$14$	−866.546	−1.18160
$15$	0	0
$16$	−1270.64	−1.24086
$17$	−82.6984	−0.0694025	−0.0347012	−	0.999398i	$-0.511048\pi$
$17$	−82.6984	−0.0694025	−0.0347012	+	0.999398i	$0.511048\pi$
$18$	0	0
$19$	−2066.89	−1.31351	−0.656755	−	0.754104i	$-0.728072\pi$
$19$	−2066.89	−1.31351	−0.656755	+	0.754104i	$0.728072\pi$
$20$	−1358.99	−0.759698
$21$	0	0
$22$	1263.90	0.556745
$23$	589.875	0.232509	0.116255	−	0.993219i	$-0.462911\pi$
$23$	589.875	0.232509	0.116255	+	0.993219i	$0.462911\pi$
$24$	0	0
$25$	7903.29	2.52905
$26$	−6427.63	−1.86474
$27$	0	0
$28$	−1672.76	−0.403216
$29$	1121.65	0.247664	0.123832	−	0.992303i	$-0.460482\pi$
$29$	1121.65	0.247664	0.123832	+	0.992303i	$0.460482\pi$
$30$	0	0
$31$	940.683	0.175808	0.0879041	−	0.996129i	$-0.471983\pi$
$31$	940.683	0.175808	0.0879041	+	0.996129i	$0.471983\pi$
$32$	−4429.51	−0.764682
$33$	0	0
$34$	−554.393	−0.0822471
$35$	13574.5	1.87307
$36$	0	0
$37$	575.784	0.0691441	0.0345720	−	0.999402i	$-0.488993\pi$
$37$	575.784	0.0691441	0.0345720	+	0.999402i	$0.488993\pi$
$38$	−13856.0	−1.55661
$39$	0	0
$40$	13417.7	1.32596
$41$	5521.52	0.512979	0.256489	−	0.966547i	$-0.417434\pi$
$41$	5521.52	0.512979	0.256489	+	0.966547i	$0.417434\pi$
$42$	0	0
$43$	−16502.9	−1.36110	−0.680551	−	0.732701i	$-0.738259\pi$
$43$	−16502.9	−1.36110	−0.680551	+	0.732701i	$0.738259\pi$
$44$	2439.80	0.189987
$45$	0	0
$46$	3954.40	0.275541
$47$	−23876.5	−1.57662	−0.788309	−	0.615279i	$-0.789043\pi$
$47$	−23876.5	−1.57662	−0.788309	+	0.615279i	$0.789043\pi$
$48$	0	0
$49$	−98.3079	−0.00584922
$50$	52982.0	2.99711
$51$	0	0
$52$	−12407.7	−0.636332
$53$	39649.3	1.93886	0.969428	−	0.245376i	$-0.0789113\pi$
$53$	39649.3	1.93886	0.969428	+	0.245376i	$0.0789113\pi$
$54$	0	0
$55$	−19799.2	−0.882551
$56$	16515.7	0.703762
$57$	0	0
$58$	7519.30	0.293500
$59$	−3481.00	−0.130189
$59$	−3481.00	−0.130189
$60$	0	0
$61$	1264.98	0.0435270	0.0217635	−	0.999763i	$-0.493072\pi$
$61$	1264.98	0.0435270	0.0217635	+	0.999763i	$0.493072\pi$
$62$	6306.14	0.208346
$63$	0	0
$64$	10966.0	0.334655
$65$	100690.	2.95598
$66$	0	0
$67$	−49177.7	−1.33838	−0.669192	−	0.743089i	$-0.733360\pi$
$67$	−49177.7	−1.33838	−0.669192	+	0.743089i	$0.733360\pi$
$68$	−1070.19	−0.0280664
$69$	0	0
$70$	91000.9	2.21973
$71$	53621.6	1.26239	0.631195	−	0.775624i	$-0.282565\pi$
$71$	53621.6	1.26239	0.631195	+	0.775624i	$0.282565\pi$
$72$	0	0
$73$	30870.6	0.678012	0.339006	−	0.940784i	$-0.389909\pi$
$73$	30870.6	0.678012	0.339006	+	0.940784i	$0.389909\pi$
$74$	3859.93	0.0819409
$75$	0	0
$76$	−26747.3	−0.531184
$77$	−24370.5	−0.468422
$78$	0	0
$79$	−26724.2	−0.481768	−0.240884	−	0.970554i	$-0.577437\pi$
$79$	−26724.2	−0.481768	−0.240884	+	0.970554i	$0.577437\pi$
$80$	133437.	2.33105
$81$	0	0
$82$	37015.1	0.607918
$83$	−27418.4	−0.436864	−0.218432	−	0.975852i	$-0.570094\pi$
$83$	−27418.4	−0.436864	−0.218432	+	0.975852i	$0.570094\pi$
$84$	0	0
$85$	8684.63	0.130378
$86$	−110632.	−1.61301
$87$	0	0
$88$	−24088.9	−0.331597
$89$	30163.0	0.403645	0.201822	−	0.979422i	$-0.435314\pi$
$89$	30163.0	0.403645	0.201822	+	0.979422i	$0.435314\pi$
$90$	0	0
$91$	123937.	1.56891
$92$	7633.46	0.0940269
$93$	0	0
$94$	−160063.	−1.86841
$95$	217056.	2.46753
$96$	0	0
$97$	−80237.8	−0.865864	−0.432932	−	0.901427i	$-0.642521\pi$
$97$	−80237.8	−0.865864	−0.432932	+	0.901427i	$0.642521\pi$
$98$	−659.035	−0.00693176
$99$	0	0
$100$	102275.	1.02275
$101$	−94103.2	−0.917912	−0.458956	−	0.888459i	$-0.651776\pi$
$101$	−94103.2	−0.917912	−0.458956	+	0.888459i	$0.651776\pi$
$102$	0	0
$103$	−35090.7	−0.325911	−0.162955	−	0.986633i	$-0.552103\pi$
$103$	−35090.7	−0.325911	−0.162955	+	0.986633i	$0.552103\pi$
$104$	122505.	1.11064
$105$	0	0
$106$	265800.	2.29769
$107$	160286.	1.35343	0.676715	−	0.736245i	$-0.263403\pi$
$107$	160286.	1.35343	0.676715	+	0.736245i	$0.263403\pi$
$108$	0	0
$109$	240043.	1.93518	0.967592	−	0.252518i	$-0.0812588\pi$
$109$	240043.	1.93518	0.967592	+	0.252518i	$0.0812588\pi$
$110$	−132729.	−1.04589
$111$	0	0
$112$	164246.	1.23723
$113$	−185172.	−1.36421	−0.682103	−	0.731257i	$-0.738934\pi$
$113$	−185172.	−1.36421	−0.682103	+	0.731257i	$0.738934\pi$
$114$	0	0
$115$	−61946.1	−0.436787
$116$	14515.1	0.100155
$117$	0	0
$118$	−23335.9	−0.154284
$119$	10689.8	0.0691992
$120$	0	0
$121$	−125505.	−0.779290
$122$	8480.15	0.0515827
$123$	0	0
$124$	12173.2	0.0710969
$125$	−501795.	−2.87244
$126$	0	0
$127$	−102539.	−0.564132	−0.282066	−	0.959395i	$-0.591020\pi$
$127$	−102539.	−0.564132	−0.282066	+	0.959395i	$0.591020\pi$
$128$	215258.	1.16127
$129$	0	0
$130$	675002.	3.50305
$131$	−166552.	−0.847952	−0.423976	−	0.905673i	$-0.639366\pi$
$131$	−166552.	−0.847952	−0.423976	+	0.905673i	$0.639366\pi$
$132$	0	0
$133$	267171.	1.30966
$134$	−329677.	−1.58609
$135$	0	0
$136$	10566.3	0.0489863
$137$	−373498.	−1.70015	−0.850073	−	0.526664i	$-0.823442\pi$
$137$	−373498.	−1.70015	−0.850073	+	0.526664i	$0.823442\pi$
$138$	0	0
$139$	−2560.95	−0.0112425	−0.00562126	−	0.999984i	$-0.501789\pi$
$139$	−2560.95	−0.0112425	−0.00562126	+	0.999984i	$0.501789\pi$
$140$	175666.	0.757473
$141$	0	0
$142$	359468.	1.49603
$143$	−180769.	−0.739236
$144$	0	0
$145$	−117791.	−0.465255
$146$	206950.	0.803495
$147$	0	0
$148$	7451.11	0.0279619
$149$	529574.	1.95416	0.977082	−	0.212863i	$-0.0682789\pi$
$149$	529574.	1.95416	0.977082	+	0.212863i	$0.0682789\pi$
$150$	0	0
$151$	−477364.	−1.70376	−0.851878	−	0.523740i	$-0.824536\pi$
$151$	−477364.	−1.70376	−0.851878	+	0.523740i	$0.824536\pi$
$152$	264084.	0.927115
$153$	0	0
$154$	−163375.	−0.555115
$155$	−98786.4	−0.330269
$156$	0	0
$157$	−210317.	−0.680965	−0.340483	−	0.940251i	$-0.610590\pi$
$157$	−210317.	−0.680965	−0.340483	+	0.940251i	$0.610590\pi$
$158$	−179154.	−0.570930
$159$	0	0
$160$	465168.	1.43651
$161$	−76248.5	−0.231828
$162$	0	0
$163$	82201.3	0.242331	0.121166	−	0.992632i	$-0.461337\pi$
$163$	82201.3	0.242331	0.121166	+	0.992632i	$0.461337\pi$
$164$	71453.0	0.207449
$165$	0	0
$166$	−183807.	−0.517716
$167$	713691.	1.98024	0.990122	−	0.140206i	$-0.0447766\pi$
$167$	713691.	1.98024	0.990122	+	0.140206i	$0.0447766\pi$
$168$	0	0
$169$	548014.	1.47596
$170$	58219.9	0.154507
$171$	0	0
$172$	−213562.	−0.550430
$173$	−465524.	−1.18257	−0.591286	−	0.806462i	$-0.701379\pi$
$173$	−465524.	−1.18257	−0.591286	+	0.806462i	$0.701379\pi$
$174$	0	0
$175$	−1.02160e6	−2.52164
$176$	−239561.	−0.582954
$177$	0	0
$178$	202206.	0.478349
$179$	40052.8	0.0934331	0.0467165	−	0.998908i	$-0.485124\pi$
$179$	40052.8	0.0934331	0.0467165	+	0.998908i	$0.485124\pi$
$180$	0	0
$181$	346928.	0.787123	0.393562	−	0.919298i	$-0.371243\pi$
$181$	346928.	0.787123	0.393562	+	0.919298i	$0.371243\pi$
$182$	830849.	1.85927
$183$	0	0
$184$	−75367.6	−0.164112
$185$	−60466.3	−0.129893
$186$	0	0
$187$	−15591.6	−0.0326051
$188$	−308982.	−0.637585
$189$	0	0
$190$	1.45510e6	2.92421
$191$	−176540.	−0.350155	−0.175078	−	0.984555i	$-0.556018\pi$
$191$	−176540.	−0.350155	−0.175078	+	0.984555i	$0.556018\pi$
$192$	0	0
$193$	−438623.	−0.847614	−0.423807	−	0.905753i	$-0.639307\pi$
$193$	−438623.	−0.847614	−0.423807	+	0.905753i	$0.639307\pi$
$194$	−537897.	−1.02611
$195$	0	0
$196$	−1272.18	−0.00236543
$197$	670092.	1.23018	0.615091	−	0.788456i	$-0.289119\pi$
$197$	670092.	1.23018	0.615091	+	0.788456i	$0.289119\pi$
$198$	0	0
$199$	−236725.	−0.423752	−0.211876	−	0.977297i	$-0.567957\pi$
$199$	−236725.	−0.423752	−0.211876	+	0.977297i	$0.567957\pi$
$200$	−1.00979e6	−1.78508
$201$	0	0
$202$	−630848.	−1.08779
$203$	−144987.	−0.246938
$204$	0	0
$205$	−579847.	−0.963670
$206$	−235241.	−0.386228
$207$	0	0
$208$	1.21830e6	1.95252
$209$	−389682.	−0.617084
$210$	0	0
$211$	−739722.	−1.14383	−0.571916	−	0.820312i	$-0.693800\pi$
$211$	−739722.	−1.14383	−0.571916	+	0.820312i	$0.693800\pi$
$212$	513094.	0.784075
$213$	0	0
$214$	1.07452e6	1.60391
$215$	1.73307e6	2.55693
$216$	0	0
$217$	−121595.	−0.175293
$218$	1.60920e6	2.29334
$219$	0	0
$220$	−256217.	−0.356904
$221$	79291.7	0.109206
$222$	0	0
$223$	126063.	0.169757	0.0848783	−	0.996391i	$-0.472950\pi$
$223$	126063.	0.169757	0.0848783	+	0.996391i	$0.472950\pi$
$224$	572568.	0.762443
$225$	0	0
$226$	−1.24136e6	−1.61668
$227$	−411990.	−0.530667	−0.265333	−	0.964157i	$-0.585482\pi$
$227$	−411990.	−0.530667	−0.265333	+	0.964157i	$0.585482\pi$
$228$	0	0
$229$	750528.	0.945754	0.472877	−	0.881128i	$-0.343215\pi$
$229$	750528.	0.945754	0.472877	+	0.881128i	$0.343215\pi$
$230$	−415274.	−0.517624
$231$	0	0
$232$	−143312.	−0.174808
$233$	149621.	0.180553	0.0902764	−	0.995917i	$-0.471225\pi$
$233$	149621.	0.180553	0.0902764	+	0.995917i	$0.471225\pi$
$234$	0	0
$235$	2.50741e6	2.96180
$236$	−45047.0	−0.0526485
$237$	0	0
$238$	71662.0	0.0820062
$239$	−171621.	−0.194346	−0.0971730	−	0.995268i	$-0.530980\pi$
$239$	−171621.	−0.194346	−0.0971730	+	0.995268i	$0.530980\pi$
$240$	0	0
$241$	−267062.	−0.296189	−0.148094	−	0.988973i	$-0.547314\pi$
$241$	−267062.	−0.296189	−0.148094	+	0.988973i	$0.547314\pi$
$242$	−841362.	−0.923516
$243$	0	0
$244$	16369.8	0.0176023
$245$	10323.9	0.0109882
$246$	0	0
$247$	1.98175e6	2.06683
$248$	−120190.	−0.124091
$249$	0	0
$250$	−3.36393e6	−3.40406
$251$	−1.10039e6	−1.10246	−0.551231	−	0.834352i	$-0.685842\pi$
$251$	−1.10039e6	−1.10246	−0.551231	+	0.834352i	$0.685842\pi$
$252$	0	0
$253$	111212.	0.109232
$254$	−687402.	−0.668538
$255$	0	0
$256$	1.09213e6	1.04154
$257$	567643.	0.536096	0.268048	−	0.963406i	$-0.413621\pi$
$257$	567643.	0.536096	0.268048	+	0.963406i	$0.413621\pi$
$258$	0	0
$259$	−74427.0	−0.0689416
$260$	1.30301e6	1.19540
$261$	0	0
$262$	−1.11653e6	−1.00489
$263$	−1.09213e6	−0.973613	−0.486807	−	0.873510i	$-0.661838\pi$
$263$	−1.09213e6	−0.973613	−0.486807	+	0.873510i	$0.661838\pi$
$264$	0	0
$265$	−4.16379e6	−3.64229
$266$	1.79106e6	1.55205
$267$	0	0
$268$	−636399.	−0.541244
$269$	−1.75427e6	−1.47814	−0.739069	−	0.673630i	$-0.764734\pi$
$269$	−1.75427e6	−1.47814	−0.739069	+	0.673630i	$0.764734\pi$
$270$	0	0
$271$	1.21698e6	1.00661	0.503306	−	0.864108i	$-0.332117\pi$
$271$	1.21698e6	1.00661	0.503306	+	0.864108i	$0.332117\pi$
$272$	105080.	0.0861188
$273$	0	0
$274$	−2.50385e6	−2.01480
$275$	1.49005e6	1.18814
$276$	0	0
$277$	2.13269e6	1.67004	0.835022	−	0.550217i	$-0.185455\pi$
$277$	2.13269e6	1.67004	0.835022	+	0.550217i	$0.185455\pi$
$278$	−17168.1	−0.0133232
$279$	0	0
$280$	−1.73440e6	−1.32207
$281$	−948331.	−0.716464	−0.358232	−	0.933633i	$-0.616620\pi$
$281$	−948331.	−0.716464	−0.358232	+	0.933633i	$0.616620\pi$
$282$	0	0
$283$	1.97106e6	1.46297	0.731484	−	0.681859i	$-0.238828\pi$
$283$	1.97106e6	1.46297	0.731484	+	0.681859i	$0.238828\pi$
$284$	693907.	0.510511
$285$	0	0
$286$	−1.21184e6	−0.876049
$287$	−713724.	−0.511476
$288$	0	0
$289$	−1.41302e6	−0.995183
$290$	−789644.	−0.551362
$291$	0	0
$292$	399490.	0.274189
$293$	−588573.	−0.400526	−0.200263	−	0.979742i	$-0.564180\pi$
$293$	−588573.	−0.400526	−0.200263	+	0.979742i	$0.564180\pi$
$294$	0	0
$295$	365559.	0.244570
$296$	−73567.2	−0.0488039
$297$	0	0
$298$	3.55015e6	2.31583
$299$	−565575.	−0.365858
$300$	0	0
$301$	2.13321e6	1.35711
$302$	−3.20015e6	−2.01908
$303$	0	0
$304$	2.62628e6	1.62988
$305$	−132842.	−0.0817687
$306$	0	0
$307$	−1.76408e6	−1.06825	−0.534125	−	0.845405i	$-0.679359\pi$
$307$	−1.76408e6	−1.06825	−0.534125	+	0.845405i	$0.679359\pi$
$308$	−315374.	−0.189430
$309$	0	0
$310$	−662244.	−0.391393
$311$	1.05327e6	0.617501	0.308750	−	0.951143i	$-0.400089\pi$
$311$	1.05327e6	0.617501	0.308750	+	0.951143i	$0.400089\pi$
$312$	0	0
$313$	1.11290e6	0.642089	0.321045	−	0.947064i	$-0.395966\pi$
$313$	1.11290e6	0.642089	0.321045	+	0.947064i	$0.395966\pi$
$314$	−1.40992e6	−0.806994
$315$	0	0
$316$	−345833.	−0.194827
$317$	878252.	0.490875	0.245438	−	0.969412i	$-0.421068\pi$
$317$	878252.	0.490875	0.245438	+	0.969412i	$0.421068\pi$
$318$	0	0
$319$	211470.	0.116352
$320$	−1.15160e6	−0.628676
$321$	0	0
$322$	−511154.	−0.274734
$323$	170929.	0.0911609
$324$	0	0
$325$	−7.57771e6	−3.97951
$326$	551060.	0.287181
$327$	0	0
$328$	−705478.	−0.362076
$329$	3.08633e6	1.57200
$330$	0	0
$331$	2.13208e6	1.06963	0.534815	−	0.844969i	$-0.320381\pi$
$331$	2.13208e6	1.06963	0.534815	+	0.844969i	$0.320381\pi$
$332$	−354816.	−0.176668
$333$	0	0
$334$	4.78443e6	2.34674
$335$	5.16443e6	2.51426
$336$	0	0
$337$	1.12736e6	0.540739	0.270370	−	0.962757i	$-0.412854\pi$
$337$	1.12736e6	0.540739	0.270370	+	0.962757i	$0.412854\pi$
$338$	3.67377e6	1.74912
$339$	0	0
$340$	112386.	0.0527249
$341$	177352.	0.0825943
$342$	0	0
$343$	2.18522e6	1.00290
$344$	2.10856e6	0.960705
$345$	0	0
$346$	−3.12078e6	−1.40143
$347$	2.16376e6	0.964683	0.482342	−	0.875983i	$-0.339786\pi$
$347$	2.16376e6	0.964683	0.482342	+	0.875983i	$0.339786\pi$
$348$	0	0
$349$	1.74582e6	0.767250	0.383625	−	0.923489i	$-0.374676\pi$
$349$	1.74582e6	0.767250	0.383625	+	0.923489i	$0.374676\pi$
$350$	−6.84856e6	−2.98834
$351$	0	0
$352$	−835120.	−0.359246
$353$	−3.74324e6	−1.59886	−0.799431	−	0.600758i	$-0.794866\pi$
$353$	−3.74324e6	−1.59886	−0.799431	+	0.600758i	$0.794866\pi$
$354$	0	0
$355$	−5.63110e6	−2.37150
$356$	390334.	0.163234
$357$	0	0
$358$	268506.	0.110725
$359$	2.31593e6	0.948395	0.474197	−	0.880419i	$-0.342738\pi$
$359$	2.31593e6	0.948395	0.474197	+	0.880419i	$0.342738\pi$
$360$	0	0
$361$	1.79594e6	0.725310
$362$	2.32573e6	0.932800
$363$	0	0
$364$	1.60385e6	0.634468
$365$	−3.24189e6	−1.27370
$366$	0	0
$367$	−3.24429e6	−1.25735	−0.628673	−	0.777670i	$-0.716402\pi$
$367$	−3.24429e6	−1.25735	−0.628673	+	0.777670i	$0.716402\pi$
$368$	−749519.	−0.288512
$369$	0	0
$370$	−405353.	−0.153932
$371$	−5.12515e6	−1.93318
$372$	0	0
$373$	−2.30146e6	−0.856508	−0.428254	−	0.903658i	$-0.640871\pi$
$373$	−2.30146e6	−0.856508	−0.428254	+	0.903658i	$0.640871\pi$
$374$	−104523.	−0.0386395
$375$	0	0
$376$	3.05068e6	1.11282
$377$	−1.07544e6	−0.389703
$378$	0	0
$379$	2.81081e6	1.00515	0.502577	−	0.864532i	$-0.332385\pi$
$379$	2.81081e6	1.00515	0.502577	+	0.864532i	$0.332385\pi$
$380$	2.80888e6	0.997871
$381$	0	0
$382$	−1.18349e6	−0.414960
$383$	−580596.	−0.202245	−0.101122	−	0.994874i	$-0.532243\pi$
$383$	−580596.	−0.202245	−0.101122	+	0.994874i	$0.532243\pi$
$384$	0	0
$385$	2.55928e6	0.879967
$386$	−2.94044e6	−1.00449
$387$	0	0
$388$	−1.03834e6	−0.350156
$389$	4.35788e6	1.46016	0.730082	−	0.683360i	$-0.239482\pi$
$389$	4.35788e6	1.46016	0.730082	+	0.683360i	$0.239482\pi$
$390$	0	0
$391$	−48781.7	−0.0161367
$392$	12560.7	0.00412855
$393$	0	0
$394$	4.49216e6	1.45786
$395$	2.80646e6	0.905037
$396$	0	0
$397$	1.87737e6	0.597826	0.298913	−	0.954280i	$-0.403376\pi$
$397$	1.87737e6	0.597826	0.298913	+	0.954280i	$0.403376\pi$
$398$	−1.58696e6	−0.502178
$399$	0	0
$400$	−1.00422e7	−3.13820
$401$	4.27006e6	1.32609	0.663046	−	0.748579i	$-0.269263\pi$
$401$	4.27006e6	1.32609	0.663046	+	0.748579i	$0.269263\pi$
$402$	0	0
$403$	−901932.	−0.276637
$404$	−1.21777e6	−0.371204
$405$	0	0
$406$	−971961.	−0.292640
$407$	108556.	0.0324837
$408$	0	0
$409$	−265237.	−0.0784017	−0.0392008	−	0.999231i	$-0.512481\pi$
$409$	−265237.	−0.0784017	−0.0392008	+	0.999231i	$0.512481\pi$
$410$	−3.88717e6	−1.14202
$411$	0	0
$412$	−454102.	−0.131799
$413$	449961.	0.129808
$414$	0	0
$415$	2.87936e6	0.820683
$416$	4.24704e6	1.20324
$417$	0	0
$418$	−2.61235e6	−0.731291
$419$	5.27516e6	1.46791	0.733957	−	0.679196i	$-0.237672\pi$
$419$	5.27516e6	1.46791	0.733957	+	0.679196i	$0.237672\pi$
$420$	0	0
$421$	5.84793e6	1.60804	0.804020	−	0.594602i	$-0.202690\pi$
$421$	5.84793e6	1.60804	0.804020	+	0.594602i	$0.202690\pi$
$422$	−4.95894e6	−1.35553
$423$	0	0
$424$	−5.06594e6	−1.36850
$425$	−653589.	−0.175522
$426$	0	0
$427$	−163514.	−0.0433995
$428$	2.07423e6	0.547328
$429$	0	0
$430$	1.16181e7	3.03016
$431$	−7.13139e6	−1.84919	−0.924594	−	0.380955i	$-0.875595\pi$
$431$	−7.13139e6	−1.84919	−0.924594	+	0.380955i	$0.875595\pi$
$432$	0	0
$433$	−3.38539e6	−0.867739	−0.433869	−	0.900976i	$-0.642852\pi$
$433$	−3.38539e6	−0.867739	−0.433869	+	0.900976i	$0.642852\pi$
$434$	−815145.	−0.207735
$435$	0	0
$436$	3.10635e6	0.782590
$437$	−1.21921e6	−0.305403
$438$	0	0
$439$	2.34305e6	0.580257	0.290129	−	0.956988i	$-0.406302\pi$
$439$	2.34305e6	0.580257	0.290129	+	0.956988i	$0.406302\pi$
$440$	2.52971e6	0.622931
$441$	0	0
$442$	531555.	0.129417
$443$	999893.	0.242072	0.121036	−	0.992648i	$-0.461378\pi$
$443$	999893.	0.242072	0.121036	+	0.992648i	$0.461378\pi$
$444$	0	0
$445$	−3.16759e6	−0.758278
$446$	845102.	0.201174
$447$	0	0
$448$	−1.41749e6	−0.333675
$449$	−7.27002e6	−1.70184	−0.850921	−	0.525293i	$-0.823956\pi$
$449$	−7.27002e6	−1.70184	−0.850921	+	0.525293i	$0.823956\pi$
$450$	0	0
$451$	1.04100e6	0.240996
$452$	−2.39628e6	−0.551685
$453$	0	0
$454$	−2.76189e6	−0.628879
$455$	−1.30153e7	−2.94732
$456$	0	0
$457$	−6.00303e6	−1.34456	−0.672279	−	0.740297i	$-0.734685\pi$
$457$	−6.00303e6	−1.34456	−0.672279	+	0.740297i	$0.734685\pi$
$458$	5.03138e6	1.12079
$459$	0	0
$460$	−801633.	−0.176637
$461$	2.14478e6	0.470035	0.235018	−	0.971991i	$-0.424485\pi$
$461$	2.14478e6	0.470035	0.235018	+	0.971991i	$0.424485\pi$
$462$	0	0
$463$	−4.81062e6	−1.04291	−0.521457	−	0.853278i	$-0.674611\pi$
$463$	−4.81062e6	−1.04291	−0.521457	+	0.853278i	$0.674611\pi$
$464$	−1.42521e6	−0.307316
$465$	0	0
$466$	1.00303e6	0.213968
$467$	6.00558e6	1.27427	0.637137	−	0.770750i	$-0.280118\pi$
$467$	6.00558e6	1.27427	0.637137	+	0.770750i	$0.280118\pi$
$468$	0	0
$469$	6.35681e6	1.33446
$470$	1.68092e7	3.50995
$471$	0	0
$472$	444763.	0.0918912
$473$	−3.11139e6	−0.639442
$474$	0	0
$475$	−1.63352e7	−3.32194
$476$	138334.	0.0279842
$477$	0	0
$478$	−1.15051e6	−0.230314
$479$	−4.87817e6	−0.971445	−0.485723	−	0.874113i	$-0.661444\pi$
$479$	−4.87817e6	−0.971445	−0.485723	+	0.874113i	$0.661444\pi$
$480$	0	0
$481$	−552064.	−0.108800
$482$	−1.79032e6	−0.351006
$483$	0	0
$484$	−1.62414e6	−0.315145
$485$	8.42622e6	1.62659
$486$	0	0
$487$	−3.07733e6	−0.587966	−0.293983	−	0.955811i	$-0.594981\pi$
$487$	−3.07733e6	−0.587966	−0.293983	+	0.955811i	$0.594981\pi$
$488$	−161625.	−0.0307226
$489$	0	0
$490$	69209.0	0.0130218
$491$	−8.06992e6	−1.51066	−0.755328	−	0.655347i	$-0.772522\pi$
$491$	−8.06992e6	−1.51066	−0.755328	+	0.655347i	$0.772522\pi$
$492$	0	0
$493$	−92758.6	−0.0171885
$494$	1.32852e7	2.44935
$495$	0	0
$496$	−1.19527e6	−0.218153
$497$	−6.93124e6	−1.25869
$498$	0	0
$499$	4.28183e6	0.769800	0.384900	−	0.922958i	$-0.374236\pi$
$499$	4.28183e6	0.769800	0.384900	+	0.922958i	$0.374236\pi$
$500$	−6.49364e6	−1.16162
$501$	0	0
$502$	−7.37681e6	−1.30650
$503$	2.87943e6	0.507442	0.253721	−	0.967277i	$-0.418345\pi$
$503$	2.87943e6	0.507442	0.253721	+	0.967277i	$0.418345\pi$
$504$	0	0
$505$	9.88231e6	1.72437
$506$	745544.	0.129448
$507$	0	0
$508$	−1.32694e6	−0.228135
$509$	−3.51526e6	−0.601400	−0.300700	−	0.953719i	$-0.597220\pi$
$509$	−3.51526e6	−0.601400	−0.300700	+	0.953719i	$0.597220\pi$
$510$	0	0
$511$	−3.99040e6	−0.676026
$512$	433178.	0.0730284
$513$	0	0
$514$	3.80536e6	0.635313
$515$	3.68507e6	0.612249
$516$	0	0
$517$	−4.50157e6	−0.740692
$518$	−498943.	−0.0817009
$519$	0	0
$520$	−1.28650e7	−2.08642
$521$	−3.02436e6	−0.488134	−0.244067	−	0.969758i	$-0.578482\pi$
$521$	−3.02436e6	−0.488134	−0.244067	+	0.969758i	$0.578482\pi$
$522$	0	0
$523$	−4.60919e6	−0.736836	−0.368418	−	0.929660i	$-0.620100\pi$
$523$	−4.60919e6	−0.736836	−0.368418	+	0.929660i	$0.620100\pi$
$524$	−2.15532e6	−0.342912
$525$	0	0
$526$	−7.32144e6	−1.15380
$527$	−77793.0	−0.0122015
$528$	0	0
$529$	−6.08839e6	−0.945939
$530$	−2.79132e7	−4.31638
$531$	0	0
$532$	3.45741e6	0.529629
$533$	−5.29407e6	−0.807182
$534$	0	0
$535$	−1.68325e7	−2.54252
$536$	6.28337e6	0.944671
$537$	0	0
$538$	−1.17602e7	−1.75170
$539$	−18534.5	−0.00274795
$540$	0	0
$541$	3.48957e6	0.512600	0.256300	−	0.966597i	$-0.417496\pi$
$541$	3.48957e6	0.512600	0.256300	+	0.966597i	$0.417496\pi$
$542$	8.15841e6	1.19291
$543$	0	0
$544$	366314.	0.0530708
$545$	−2.52082e7	−3.63539
$546$	0	0
$547$	−5.80736e6	−0.829871	−0.414935	−	0.909851i	$-0.636196\pi$
$547$	−5.80736e6	−0.829871	−0.414935	+	0.909851i	$0.636196\pi$
$548$	−4.83336e6	−0.687540
$549$	0	0
$550$	9.98898e6	1.40804
$551$	−2.31833e6	−0.325309
$552$	0	0
$553$	3.45443e6	0.480356
$554$	1.42971e7	1.97913
$555$	0	0
$556$	−33140.8	−0.00454648
$557$	1.35811e7	1.85480	0.927399	−	0.374075i	$-0.122040\pi$
$557$	1.35811e7	1.85480	0.927399	+	0.374075i	$0.122040\pi$
$558$	0	0
$559$	1.58231e7	2.14172
$560$	−1.72484e7	−2.32423
$561$	0	0
$562$	−6.35742e6	−0.849063
$563$	−4.82039e6	−0.640930	−0.320465	−	0.947260i	$-0.603839\pi$
$563$	−4.82039e6	−0.640930	−0.320465	+	0.947260i	$0.603839\pi$
$564$	0	0
$565$	1.94460e7	2.56276
$566$	1.32136e7	1.73372
$567$	0	0
$568$	−6.85116e6	−0.891032
$569$	897706.	0.116239	0.0581197	−	0.998310i	$-0.481489\pi$
$569$	897706.	0.116239	0.0581197	+	0.998310i	$0.481489\pi$
$570$	0	0
$571$	6.48530e6	0.832415	0.416207	−	0.909270i	$-0.363359\pi$
$571$	6.48530e6	0.832415	0.416207	+	0.909270i	$0.363359\pi$
$572$	−2.33929e6	−0.298947
$573$	0	0
$574$	−4.78466e6	−0.606137
$575$	4.66195e6	0.588028
$576$	0	0
$577$	−3.92319e6	−0.490569	−0.245285	−	0.969451i	$-0.578881\pi$
$577$	−3.92319e6	−0.490569	−0.245285	+	0.969451i	$0.578881\pi$
$578$	−9.47258e6	−1.17937
$579$	0	0
$580$	−1.52431e6	−0.188149
$581$	3.54416e6	0.435585
$582$	0	0
$583$	7.47529e6	0.910870
$584$	−3.94430e6	−0.478561
$585$	0	0
$586$	−3.94567e6	−0.474653
$587$	−1.83842e6	−0.220217	−0.110108	−	0.993920i	$-0.535120\pi$
$587$	−1.83842e6	−0.220217	−0.110108	+	0.993920i	$0.535120\pi$
$588$	0	0
$589$	−1.94429e6	−0.230926
$590$	2.45063e6	0.289833
$591$	0	0
$592$	−731615.	−0.0857982
$593$	−2.60549e6	−0.304265	−0.152133	−	0.988360i	$-0.548614\pi$
$593$	−2.60549e6	−0.304265	−0.152133	+	0.988360i	$0.548614\pi$
$594$	0	0
$595$	−1.12259e6	−0.129996
$596$	6.85312e6	0.790265
$597$	0	0
$598$	−3.79150e6	−0.433569
$599$	790975.	0.0900732	0.0450366	−	0.998985i	$-0.485660\pi$
$599$	790975.	0.0900732	0.0450366	+	0.998985i	$0.485660\pi$
$600$	0	0
$601$	1.28795e7	1.45449	0.727246	−	0.686377i	$-0.240800\pi$
$601$	1.28795e7	1.45449	0.727246	+	0.686377i	$0.240800\pi$
$602$	1.43006e7	1.60828
$603$	0	0
$604$	−6.17749e6	−0.689000
$605$	1.31800e7	1.46396
$606$	0	0
$607$	5.79335e6	0.638202	0.319101	−	0.947721i	$-0.396619\pi$
$607$	5.79335e6	0.638202	0.319101	+	0.947721i	$0.396619\pi$
$608$	9.15532e6	1.00442
$609$	0	0
$610$	−890548.	−0.0969020
$611$	2.28929e7	2.48084
$612$	0	0
$613$	6.59875e6	0.709268	0.354634	−	0.935005i	$-0.384605\pi$
$613$	6.59875e6	0.709268	0.354634	+	0.935005i	$0.384605\pi$
$614$	−1.18260e7	−1.26596
$615$	0	0
$616$	3.11379e6	0.330626
$617$	2.10218e6	0.222309	0.111155	−	0.993803i	$-0.464545\pi$
$617$	2.10218e6	0.222309	0.111155	+	0.993803i	$0.464545\pi$
$618$	0	0
$619$	8.05878e6	0.845362	0.422681	−	0.906278i	$-0.361089\pi$
$619$	8.05878e6	0.845362	0.422681	+	0.906278i	$0.361089\pi$
$620$	−1.27838e6	−0.133561
$621$	0	0
$622$	7.06088e6	0.731784
$623$	−3.89893e6	−0.402463
$624$	0	0
$625$	2.79986e7	2.86705
$626$	7.46065e6	0.760923
$627$	0	0
$628$	−2.72167e6	−0.275383
$629$	−47616.4	−0.00479877
$630$	0	0
$631$	2.29082e6	0.229043	0.114521	−	0.993421i	$-0.463467\pi$
$631$	2.29082e6	0.229043	0.114521	+	0.993421i	$0.463467\pi$
$632$	3.41452e6	0.340046
$633$	0	0
$634$	5.88762e6	0.581723
$635$	1.07682e7	1.05977
$636$	0	0
$637$	94258.1	0.00920386
$638$	1.41765e6	0.137885
$639$	0	0
$640$	−2.26055e7	−2.18154
$641$	1.22229e7	1.17498	0.587489	−	0.809232i	$-0.300116\pi$
$641$	1.22229e7	1.17498	0.587489	+	0.809232i	$0.300116\pi$
$642$	0	0
$643$	1.94577e7	1.85594	0.927969	−	0.372656i	$-0.121553\pi$
$643$	1.94577e7	1.85594	0.927969	+	0.372656i	$0.121553\pi$
$644$	−986718.	−0.0937515
$645$	0	0
$646$	1.14587e6	0.108032
$647$	1.97484e7	1.85469	0.927346	−	0.374205i	$-0.122084\pi$
$647$	1.97484e7	1.85469	0.927346	+	0.374205i	$0.122084\pi$
$648$	0	0
$649$	−656291.	−0.0611625
$650$	−5.07994e7	−4.71602
$651$	0	0
$652$	1.06375e6	0.0979989
$653$	−4.80387e6	−0.440868	−0.220434	−	0.975402i	$-0.570747\pi$
$653$	−4.80387e6	−0.440868	−0.220434	+	0.975402i	$0.570747\pi$
$654$	0	0
$655$	1.74905e7	1.59294
$656$	−7.01588e6	−0.636535
$657$	0	0
$658$	2.06901e7	1.86294
$659$	6.39074e6	0.573242	0.286621	−	0.958044i	$-0.407468\pi$
$659$	6.39074e6	0.573242	0.286621	+	0.958044i	$0.407468\pi$
$660$	0	0
$661$	−2.04223e6	−0.181803	−0.0909017	−	0.995860i	$-0.528975\pi$
$661$	−2.04223e6	−0.181803	−0.0909017	+	0.995860i	$0.528975\pi$
$662$	1.42930e7	1.26759
$663$	0	0
$664$	3.50321e6	0.308352
$665$	−2.80571e7	−2.46030
$666$	0	0
$667$	661633.	0.0575841
$668$	9.23574e6	0.800812
$669$	0	0
$670$	3.46212e7	2.97958
$671$	238493.	0.0204489
$672$	0	0
$673$	−7.12679e6	−0.606535	−0.303268	−	0.952905i	$-0.598078\pi$
$673$	−7.12679e6	−0.606535	−0.303268	+	0.952905i	$0.598078\pi$
$674$	7.55758e6	0.640816
$675$	0	0
$676$	7.09175e6	0.596880
$677$	−9.46228e6	−0.793458	−0.396729	−	0.917936i	$-0.629855\pi$
$677$	−9.46228e6	−0.793458	−0.396729	+	0.917936i	$0.629855\pi$
$678$	0	0
$679$	1.03717e7	0.863328
$680$	−1.10962e6	−0.0920245
$681$	0	0
$682$	1.18893e6	0.0978803
$683$	2.26865e7	1.86087	0.930433	−	0.366462i	$-0.119431\pi$
$683$	2.26865e7	1.86087	0.930433	+	0.366462i	$0.119431\pi$
$684$	0	0
$685$	3.92231e7	3.19386
$686$	1.46492e7	1.18851
$687$	0	0
$688$	2.09693e7	1.68894
$689$	−3.80159e7	−3.05083
$690$	0	0
$691$	8.35889e6	0.665969	0.332984	−	0.942932i	$-0.391944\pi$
$691$	8.35889e6	0.665969	0.332984	+	0.942932i	$0.391944\pi$
$692$	−6.02427e6	−0.478232
$693$	0	0
$694$	1.45054e7	1.14322
$695$	268940.	0.0211199
$696$	0	0
$697$	−456621.	−0.0356020
$698$	1.17036e7	0.909248
$699$	0	0
$700$	−1.32203e7	−1.01975
$701$	4.42447e6	0.340068	0.170034	−	0.985438i	$-0.445612\pi$
$701$	4.42447e6	0.340068	0.170034	+	0.985438i	$0.445612\pi$
$702$	0	0
$703$	−1.19008e6	−0.0908215
$704$	2.06748e6	0.157220
$705$	0	0
$706$	−2.50939e7	−1.89477
$707$	1.21640e7	0.915223
$708$	0	0
$709$	−5.65728e6	−0.422661	−0.211330	−	0.977415i	$-0.567780\pi$
$709$	−5.65728e6	−0.422661	−0.211330	+	0.977415i	$0.567780\pi$
$710$	−3.77497e7	−2.81040
$711$	0	0
$712$	−3.85389e6	−0.284904
$713$	554885.	0.0408770
$714$	0	0
$715$	1.89835e7	1.38871
$716$	518316.	0.0377844
$717$	0	0
$718$	1.55255e7	1.12392
$719$	797010.	0.0574965	0.0287483	−	0.999587i	$-0.490848\pi$
$719$	797010.	0.0574965	0.0287483	+	0.999587i	$0.490848\pi$
$720$	0	0
$721$	4.53590e6	0.324956
$722$	1.20396e7	0.859546
$723$	0	0
$724$	4.48953e6	0.318313
$725$	8.86472e6	0.626354
$726$	0	0
$727$	−2.01191e7	−1.41180	−0.705900	−	0.708312i	$-0.749457\pi$
$727$	−2.01191e7	−1.41180	−0.705900	+	0.708312i	$0.749457\pi$
$728$	−1.58353e7	−1.10738
$729$	0	0
$730$	−2.17330e7	−1.50943
$731$	1.36477e6	0.0944638
$732$	0	0
$733$	−2.35216e7	−1.61699	−0.808493	−	0.588505i	$-0.799717\pi$
$733$	−2.35216e7	−1.61699	−0.808493	+	0.588505i	$0.799717\pi$
$734$	−2.17491e7	−1.49005
$735$	0	0
$736$	−2.61286e6	−0.177796
$737$	−9.27173e6	−0.628770
$738$	0	0
$739$	−2.16190e7	−1.45621	−0.728107	−	0.685464i	$-0.759600\pi$
$739$	−2.16190e7	−1.45621	−0.728107	+	0.685464i	$0.759600\pi$
$740$	−782483.	−0.0525286
$741$	0	0
$742$	−3.43579e7	−2.29096
$743$	−2.43927e6	−0.162102	−0.0810510	−	0.996710i	$-0.525828\pi$
$743$	−2.43927e6	−0.162102	−0.0810510	+	0.996710i	$0.525828\pi$
$744$	0	0
$745$	−5.56136e7	−3.67105
$746$	−1.54285e7	−1.01503
$747$	0	0
$748$	−201768.	−0.0131855
$749$	−2.07189e7	−1.34947
$750$	0	0
$751$	−7.79043e6	−0.504036	−0.252018	−	0.967723i	$-0.581094\pi$
$751$	−7.79043e6	−0.504036	−0.252018	+	0.967723i	$0.581094\pi$
$752$	3.03385e7	1.95636
$753$	0	0
$754$	−7.20955e6	−0.461827
$755$	5.01307e7	3.20064
$756$	0	0
$757$	2.03779e7	1.29247	0.646235	−	0.763138i	$-0.276343\pi$
$757$	2.03779e7	1.29247	0.646235	+	0.763138i	$0.276343\pi$
$758$	1.88431e7	1.19118
$759$	0	0
$760$	−2.77330e7	−1.74166
$761$	2.59966e7	1.62725	0.813626	−	0.581389i	$-0.197490\pi$
$761$	2.59966e7	1.62725	0.813626	+	0.581389i	$0.197490\pi$
$762$	0	0
$763$	−3.10284e7	−1.92952
$764$	−2.28458e6	−0.141603
$765$	0	0
$766$	−3.89219e6	−0.239675
$767$	3.33760e6	0.204855
$768$	0	0
$769$	2.44602e7	1.49157	0.745785	−	0.666187i	$-0.232075\pi$
$769$	2.44602e7	1.49157	0.745785	+	0.666187i	$0.232075\pi$
$770$	1.71569e7	1.04283
$771$	0	0
$772$	−5.67614e6	−0.342776
$773$	−1.92647e7	−1.15962	−0.579808	−	0.814753i	$-0.696872\pi$
$773$	−1.92647e7	−1.15962	−0.579808	+	0.814753i	$0.696872\pi$
$774$	0	0
$775$	7.43449e6	0.444628
$776$	1.02519e7	0.611152
$777$	0	0
$778$	2.92143e7	1.73040
$779$	−1.14124e7	−0.673803
$780$	0	0
$781$	1.01096e7	0.593068
$782$	−327022.	−0.0191232
$783$	0	0
$784$	124914.	0.00725807
$785$	2.20866e7	1.27925
$786$	0	0
$787$	−2.18742e7	−1.25891	−0.629455	−	0.777037i	$-0.716722\pi$
$787$	−2.18742e7	−1.25891	−0.629455	+	0.777037i	$0.716722\pi$
$788$	8.67154e6	0.497486
$789$	0	0
$790$	1.88139e7	1.07254
$791$	2.39357e7	1.36021
$792$	0	0
$793$	−1.21287e6	−0.0684905
$794$	1.25855e7	0.708468
$795$	0	0
$796$	−3.06342e6	−0.171366
$797$	2.81755e7	1.57118	0.785591	−	0.618746i	$-0.212359\pi$
$797$	2.81755e7	1.57118	0.785591	+	0.618746i	$0.212359\pi$
$798$	0	0
$799$	1.97455e6	0.109421
$800$	−3.50077e7	−1.93392
$801$	0	0
$802$	2.86256e7	1.57152
$803$	5.82019e6	0.318529
$804$	0	0
$805$	8.00728e6	0.435507
$806$	−6.04636e6	−0.327836
$807$	0	0
$808$	1.20234e7	0.647889
$809$	−1.83693e7	−0.986785	−0.493392	−	0.869807i	$-0.664243\pi$
$809$	−1.83693e7	−0.986785	−0.493392	+	0.869807i	$0.664243\pi$
$810$	0	0
$811$	2.97018e7	1.58574	0.792868	−	0.609393i	$-0.208587\pi$
$811$	2.97018e7	1.58574	0.792868	+	0.609393i	$0.208587\pi$
$812$	−1.87625e6	−0.0998619
$813$	0	0
$814$	727734.	0.0384956
$815$	−8.63242e6	−0.455238
$816$	0	0
$817$	3.41098e7	1.78782
$818$	−1.77809e6	−0.0929118
$819$	0	0
$820$	−7.50369e6	−0.389709
$821$	2.76967e6	0.143407	0.0717034	−	0.997426i	$-0.477156\pi$
$821$	2.76967e6	0.143407	0.0717034	+	0.997426i	$0.477156\pi$
$822$	0	0
$823$	1.00360e7	0.516489	0.258244	−	0.966080i	$-0.416856\pi$
$823$	1.00360e7	0.516489	0.258244	+	0.966080i	$0.416856\pi$
$824$	4.48349e6	0.230037
$825$	0	0
$826$	3.01645e6	0.153832
$827$	3.34891e7	1.70271	0.851354	−	0.524591i	$-0.175782\pi$
$827$	3.34891e7	1.70271	0.851354	+	0.524591i	$0.175782\pi$
$828$	0	0
$829$	1.64842e7	0.833072	0.416536	−	0.909119i	$-0.363244\pi$
$829$	1.64842e7	0.833072	0.416536	+	0.909119i	$0.363244\pi$
$830$	1.93026e7	0.972570
$831$	0	0
$832$	−1.05142e7	−0.526587
$833$	8129.90	0.000405950 0
$834$	0	0
$835$	−7.49487e7	−3.72004
$836$	−5.04280e6	−0.249549
$837$	0	0
$838$	3.53636e7	1.73959
$839$	2.49583e7	1.22408	0.612042	−	0.790826i	$-0.290349\pi$
$839$	2.49583e7	1.22408	0.612042	+	0.790826i	$0.290349\pi$
$840$	0	0
$841$	−1.92531e7	−0.938663
$842$	3.92033e7	1.90565
$843$	0	0
$844$	−9.57260e6	−0.462566
$845$	−5.75501e7	−2.77271
$846$	0	0
$847$	1.62231e7	0.777008
$848$	−5.03800e7	−2.40585
$849$	0	0
$850$	−4.38153e6	−0.208007
$851$	339640.	0.0160766
$852$	0	0
$853$	−2.83557e7	−1.33434	−0.667171	−	0.744904i	$-0.732495\pi$
$853$	−2.83557e7	−1.33434	−0.667171	+	0.744904i	$0.732495\pi$
$854$	−1.09616e6	−0.0514316
$855$	0	0
$856$	−2.04795e7	−0.955291
$857$	1.22086e7	0.567822	0.283911	−	0.958851i	$-0.408368\pi$
$857$	1.22086e7	0.567822	0.283911	+	0.958851i	$0.408368\pi$
$858$	0	0
$859$	1.60844e7	0.743740	0.371870	−	0.928285i	$-0.378717\pi$
$859$	1.60844e7	0.743740	0.371870	+	0.928285i	$0.378717\pi$
$860$	2.24273e7	1.03403
$861$	0	0
$862$	−4.78073e7	−2.19142
$863$	−656036.	−0.0299848	−0.0149924	−	0.999888i	$-0.504772\pi$
$863$	−656036.	−0.0299848	−0.0149924	+	0.999888i	$0.504772\pi$
$864$	0	0
$865$	4.88874e7	2.22155
$866$	−2.26949e7	−1.02833
$867$	0	0
$868$	−1.57353e6	−0.0708887
$869$	−5.03846e6	−0.226333
$870$	0	0
$871$	4.71518e7	2.10597
$872$	−3.06700e7	−1.36591
$873$	0	0
$874$	−8.17331e6	−0.361926
$875$	6.48631e7	2.86403
$876$	0	0
$877$	−9.37944e6	−0.411792	−0.205896	−	0.978574i	$-0.566011\pi$
$877$	−9.37944e6	−0.411792	−0.205896	+	0.978574i	$0.566011\pi$
$878$	1.57073e7	0.687648
$879$	0	0
$880$	2.51576e7	1.09512
$881$	−950816.	−0.0412721	−0.0206361	−	0.999787i	$-0.506569\pi$
$881$	−950816.	−0.0412721	−0.0206361	+	0.999787i	$0.506569\pi$
$882$	0	0
$883$	−1.58113e7	−0.682442	−0.341221	−	0.939983i	$-0.610840\pi$
$883$	−1.58113e7	−0.682442	−0.341221	+	0.939983i	$0.610840\pi$
$884$	1.02610e6	0.0441630
$885$	0	0
$886$	6.70308e6	0.286873
$887$	−2.23299e7	−0.952969	−0.476484	−	0.879183i	$-0.658089\pi$
$887$	−2.23299e7	−0.952969	−0.476484	+	0.879183i	$0.658089\pi$
$888$	0	0
$889$	1.32544e7	0.562480
$890$	−2.12348e7	−0.898616
$891$	0	0
$892$	1.63136e6	0.0686497
$893$	4.93502e7	2.07091
$894$	0	0
$895$	−4.20617e6	−0.175521
$896$	−2.78247e7	−1.15787
$897$	0	0
$898$	−4.87367e7	−2.01681
$899$	1.05512e6	0.0435413
$900$	0	0
$901$	−3.27893e6	−0.134561
$902$	6.97866e6	0.285598
$903$	0	0
$904$	2.36592e7	0.962896
$905$	−3.64329e7	−1.47867
$906$	0	0
$907$	−4.23650e7	−1.70997	−0.854986	−	0.518651i	$-0.826434\pi$
$907$	−4.23650e7	−1.70997	−0.854986	+	0.518651i	$0.826434\pi$
$908$	−5.33148e6	−0.214602
$909$	0	0
$910$	−8.72521e7	−3.49279
$911$	−6.22591e6	−0.248546	−0.124273	−	0.992248i	$-0.539660\pi$
$911$	−6.22591e6	−0.248546	−0.124273	+	0.992248i	$0.539660\pi$
$912$	0	0
$913$	−5.16933e6	−0.205238
$914$	−4.02430e7	−1.59340
$915$	0	0
$916$	9.71245e6	0.382463
$917$	2.15288e7	0.845468
$918$	0	0
$919$	5.05205e7	1.97324	0.986618	−	0.163051i	$-0.0521336\pi$
$919$	5.05205e7	1.97324	0.986618	+	0.163051i	$0.0521336\pi$
$920$	7.91478e6	0.308297
$921$	0	0
$922$	1.43782e7	0.557026
$923$	−5.14126e7	−1.98639
$924$	0	0
$925$	4.55058e6	0.174869
$926$	−3.22494e7	−1.23593
$927$	0	0
$928$	−4.96836e6	−0.189384
$929$	3.87776e7	1.47415	0.737075	−	0.675811i	$-0.236206\pi$
$929$	3.87776e7	1.47415	0.737075	+	0.675811i	$0.236206\pi$
$930$	0	0
$931$	203192.	0.00768301
$932$	1.93622e6	0.0730156
$933$	0	0
$934$	4.02602e7	1.51011
$935$	1.63736e6	0.0612512
$936$	0	0
$937$	2.07364e7	0.771586	0.385793	−	0.922585i	$-0.373928\pi$
$937$	2.07364e7	0.771586	0.385793	+	0.922585i	$0.373928\pi$
$938$	4.26147e7	1.58144
$939$	0	0
$940$	3.24479e7	1.19775
$941$	−1.54725e7	−0.569623	−0.284811	−	0.958584i	$-0.591931\pi$
$941$	−1.54725e7	−0.569623	−0.284811	+	0.958584i	$0.591931\pi$
$942$	0	0
$943$	3.25701e6	0.119272
$944$	4.42310e6	0.161546
$945$	0	0
$946$	−2.08581e7	−0.757787
$947$	−3.92696e7	−1.42292	−0.711462	−	0.702724i	$-0.751967\pi$
$947$	−3.92696e7	−1.42292	−0.711462	+	0.702724i	$0.751967\pi$
$948$	0	0
$949$	−2.95989e7	−1.06687
$950$	−1.09508e8	−3.93674
$951$	0	0
$952$	−1.36582e6	−0.0488428
$953$	−1.72996e7	−0.617027	−0.308513	−	0.951220i	$-0.599831\pi$
$953$	−1.72996e7	−0.617027	−0.308513	+	0.951220i	$0.599831\pi$
$954$	0	0
$955$	1.85395e7	0.657794
$956$	−2.22092e6	−0.0785936
$957$	0	0
$958$	−3.27023e7	−1.15123
$959$	4.82791e7	1.69517
$960$	0	0
$961$	−2.77443e7	−0.969092
$962$	−3.70092e6	−0.128936
$963$	0	0
$964$	−3.45599e6	−0.119779
$965$	4.60623e7	1.59231
$966$	0	0
$967$	−9.19283e6	−0.316143	−0.158071	−	0.987428i	$-0.550528\pi$
$967$	−9.19283e6	−0.316143	−0.158071	+	0.987428i	$0.550528\pi$
$968$	1.60357e7	0.550046
$969$	0	0
$970$	5.64876e7	1.92763
$971$	−2.71756e7	−0.924979	−0.462489	−	0.886625i	$-0.653044\pi$
$971$	−2.71756e7	−0.924979	−0.462489	+	0.886625i	$0.653044\pi$
$972$	0	0
$973$	331033.	0.0112096
$974$	−2.06298e7	−0.696783
$975$	0	0
$976$	−1.60733e6	−0.0540109
$977$	2.95289e7	0.989715	0.494858	−	0.868974i	$-0.335220\pi$
$977$	2.95289e7	0.989715	0.494858	+	0.868974i	$0.335220\pi$
$978$	0	0
$979$	5.68679e6	0.189631
$980$	133599.	0.00444364
$981$	0	0
$982$	−5.40991e7	−1.79024
$983$	3.03956e7	1.00329	0.501645	−	0.865074i	$-0.332728\pi$
$983$	3.03956e7	1.00329	0.501645	+	0.865074i	$0.332728\pi$
$984$	0	0
$985$	−7.03702e7	−2.31099
$986$	−621834.	−0.0203696
$987$	0	0
$988$	2.56454e7	0.835829
$989$	−9.73467e6	−0.316469
$990$	0	0
$991$	−1.19636e7	−0.386969	−0.193485	−	0.981103i	$-0.561979\pi$
$991$	−1.19636e7	−0.386969	−0.193485	+	0.981103i	$0.561979\pi$
$992$	−4.16677e6	−0.134437
$993$	0	0
$994$	−4.64656e7	−1.49164
$995$	2.48599e7	0.796051
$996$	0	0
$997$	6.46829e6	0.206088	0.103044	−	0.994677i	$-0.467142\pi$
$997$	6.46829e6	0.206088	0.103044	+	0.994677i	$0.467142\pi$
$998$	2.87045e7	0.912271
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q+6.70379 q^{2} +12.9408 q^{4} -105.016 q^{5} -129.262 q^{7} -127.769 q^{8} +O(q^{10})\)	\(q+6.70379 q^{2} +12.9408 q^{4} -105.016 q^{5} -129.262 q^{7} -127.769 q^{8} -704.003 q^{10} +188.535 q^{11} -958.805 q^{13} -866.546 q^{14} -1270.64 q^{16} -82.6984 q^{17} -2066.89 q^{19} -1358.99 q^{20} +1263.90 q^{22} +589.875 q^{23} +7903.29 q^{25} -6427.63 q^{26} -1672.76 q^{28} +1121.65 q^{29} +940.683 q^{31} -4429.51 q^{32} -554.393 q^{34} +13574.5 q^{35} +575.784 q^{37} -13856.0 q^{38} +13417.7 q^{40} +5521.52 q^{41} -16502.9 q^{43} +2439.80 q^{44} +3954.40 q^{46} -23876.5 q^{47} -98.3079 q^{49} +52982.0 q^{50} -12407.7 q^{52} +39649.3 q^{53} -19799.2 q^{55} +16515.7 q^{56} +7519.30 q^{58} -3481.00 q^{59} +1264.98 q^{61} +6306.14 q^{62} +10966.0 q^{64} +100690. q^{65} -49177.7 q^{67} -1070.19 q^{68} +91000.9 q^{70} +53621.6 q^{71} +30870.6 q^{73} +3859.93 q^{74} -26747.3 q^{76} -24370.5 q^{77} -26724.2 q^{79} +133437. q^{80} +37015.1 q^{82} -27418.4 q^{83} +8684.63 q^{85} -110632. q^{86} -24088.9 q^{88} +30163.0 q^{89} +123937. q^{91} +7633.46 q^{92} -160063. q^{94} +217056. q^{95} -80237.8 q^{97} -659.035 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 6 q^{2} + 150 q^{4} + 192 q^{5} - 371 q^{7} + 621 q^{8}+O(q^{10}) \) 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

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