LMFDB - Embedded newform 124.6.a.b.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [124,6,Mod(1,124)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(124, 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("124.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 124 = 2^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	124.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:	$19.8875936568$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 847x^{4} + 1184x^{3} + 199815x^{2} - 13326x - 12452553 $ x^6 - 2x^5 - 847x^4 + 1184x^3 + 199815x^2 - 13326*x - 12452553
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$9.40536$ of defining polynomial
Character	$\chi$	$=$	124.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+12.4054 q^{3} -65.1867 q^{5} +116.704 q^{7} -89.1069 q^{9} +O(q^{10})$	$q+12.4054 q^{3} -65.1867 q^{5} +116.704 q^{7} -89.1069 q^{9} +115.836 q^{11} +564.654 q^{13} -808.664 q^{15} +2224.15 q^{17} +1085.48 q^{19} +1447.76 q^{21} -16.5836 q^{23} +1124.30 q^{25} -4119.91 q^{27} +8009.90 q^{29} -961.000 q^{31} +1436.99 q^{33} -7607.56 q^{35} -8025.55 q^{37} +7004.74 q^{39} +8791.37 q^{41} +7781.35 q^{43} +5808.58 q^{45} +28792.3 q^{47} -3187.11 q^{49} +27591.4 q^{51} +35549.8 q^{53} -7550.97 q^{55} +13465.8 q^{57} -27594.5 q^{59} -26772.6 q^{61} -10399.2 q^{63} -36807.9 q^{65} -5610.06 q^{67} -205.726 q^{69} -7453.03 q^{71} -11148.6 q^{73} +13947.4 q^{75} +13518.6 q^{77} -104147. q^{79} -29456.0 q^{81} +51115.3 q^{83} -144985. q^{85} +99365.8 q^{87} -92711.2 q^{89} +65897.5 q^{91} -11921.6 q^{93} -70758.9 q^{95} +95474.3 q^{97} -10321.8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 20 q^{3} + 25 q^{5} + 39 q^{7} + 306 q^{9}+O(q^{10}) $ 6 * q + 20 * q^3 + 25 * q^5 + 39 * q^7 + 306 * q^9	$ 6 q + 20 q^{3} + 25 q^{5} + 39 q^{7} + 306 q^{9} + 280 q^{11} + 1214 q^{13} + 1914 q^{15} + 796 q^{17} + 3147 q^{19} + 1082 q^{21} + 9122 q^{23} + 6481 q^{25} + 7316 q^{27} + 13020 q^{29} - 5766 q^{31} + 24804 q^{33} + 20059 q^{35} + 21678 q^{37} + 30680 q^{39} + 3227 q^{41} + 37882 q^{43} + 26169 q^{45} + 29708 q^{47} + 34849 q^{49} + 24432 q^{51} - 9976 q^{53} + 23758 q^{55} + 17318 q^{57} + 20573 q^{59} + 21610 q^{61} - 17697 q^{63} + 3894 q^{65} - 17024 q^{67} - 83692 q^{69} + 44509 q^{71} - 161864 q^{73} - 49430 q^{75} - 144202 q^{77} - 24420 q^{79} - 181158 q^{81} - 114160 q^{83} - 228882 q^{85} - 56180 q^{87} - 199742 q^{89} - 186774 q^{91} - 19220 q^{93} - 12793 q^{95} - 282951 q^{97} + 34060 q^{99}+O(q^{100}) $ 6 * q + 20 * q^3 + 25 * q^5 + 39 * q^7 + 306 * q^9 + 280 * q^11 + 1214 * q^13 + 1914 * q^15 + 796 * q^17 + 3147 * q^19 + 1082 * q^21 + 9122 * q^23 + 6481 * q^25 + 7316 * q^27 + 13020 * q^29 - 5766 * q^31 + 24804 * q^33 + 20059 * q^35 + 21678 * q^37 + 30680 * q^39 + 3227 * q^41 + 37882 * q^43 + 26169 * q^45 + 29708 * q^47 + 34849 * q^49 + 24432 * q^51 - 9976 * q^53 + 23758 * q^55 + 17318 * q^57 + 20573 * q^59 + 21610 * q^61 - 17697 * q^63 + 3894 * q^65 - 17024 * q^67 - 83692 * q^69 + 44509 * q^71 - 161864 * q^73 - 49430 * q^75 - 144202 * q^77 - 24420 * q^79 - 181158 * q^81 - 114160 * q^83 - 228882 * q^85 - 56180 * q^87 - 199742 * q^89 - 186774 * q^91 - 19220 * q^93 - 12793 * q^95 - 282951 * q^97 + 34060 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	12.4054	0.795804	0.397902	−	0.917428i	$-0.369738\pi$
$3$	12.4054	0.795804	0.397902	+	0.917428i	$0.369738\pi$
$4$	0	0
$5$	−65.1867	−1.16609	−0.583047	−	0.812438i	$-0.698140\pi$
$5$	−65.1867	−1.16609	−0.583047	+	0.812438i	$0.698140\pi$
$6$	0	0
$7$	116.704	0.900206	0.450103	−	0.892977i	$-0.351387\pi$
$7$	116.704	0.900206	0.450103	+	0.892977i	$0.351387\pi$
$8$	0	0
$9$	−89.1069	−0.366695
$10$	0	0
$11$	115.836	0.288644	0.144322	−	0.989531i	$-0.453900\pi$
$11$	115.836	0.288644	0.144322	+	0.989531i	$0.453900\pi$
$12$	0	0
$13$	564.654	0.926668	0.463334	−	0.886184i	$-0.346653\pi$
$13$	564.654	0.926668	0.463334	+	0.886184i	$0.346653\pi$
$14$	0	0
$15$	−808.664	−0.927983
$16$	0	0
$17$	2224.15	1.86656	0.933281	−	0.359147i	$-0.116932\pi$
$17$	2224.15	1.86656	0.933281	+	0.359147i	$0.116932\pi$
$18$	0	0
$19$	1085.48	0.689824	0.344912	−	0.938635i	$-0.387909\pi$
$19$	1085.48	0.689824	0.344912	+	0.938635i	$0.387909\pi$
$20$	0	0
$21$	1447.76	0.716388
$22$	0	0
$23$	−16.5836	−0.00653672	−0.00326836	−	0.999995i	$-0.501040\pi$
$23$	−16.5836	−0.00653672	−0.00326836	+	0.999995i	$0.501040\pi$
$24$	0	0
$25$	1124.30	0.359776
$26$	0	0
$27$	−4119.91	−1.08762
$28$	0	0
$29$	8009.90	1.76861	0.884305	−	0.466909i	$-0.154632\pi$
$29$	8009.90	1.76861	0.884305	+	0.466909i	$0.154632\pi$
$30$	0	0
$31$	−961.000	−0.179605
$31$	−961.000	−0.179605
$32$	0	0
$33$	1436.99	0.229704
$34$	0	0
$35$	−7607.56	−1.04972
$36$	0	0
$37$	−8025.55	−0.963763	−0.481882	−	0.876236i	$-0.660047\pi$
$37$	−8025.55	−0.963763	−0.481882	+	0.876236i	$0.660047\pi$
$38$	0	0
$39$	7004.74	0.737446
$40$	0	0
$41$	8791.37	0.816765	0.408382	−	0.912811i	$-0.366093\pi$
$41$	8791.37	0.816765	0.408382	+	0.912811i	$0.366093\pi$
$42$	0	0
$43$	7781.35	0.641777	0.320888	−	0.947117i	$-0.396019\pi$
$43$	7781.35	0.641777	0.320888	+	0.947117i	$0.396019\pi$
$44$	0	0
$45$	5808.58	0.427601
$46$	0	0
$47$	28792.3	1.90122	0.950609	−	0.310391i	$-0.100460\pi$
$47$	28792.3	1.90122	0.950609	+	0.310391i	$0.100460\pi$
$48$	0	0
$49$	−3187.11	−0.189630
$50$	0	0
$51$	27591.4	1.48542
$52$	0	0
$53$	35549.8	1.73839	0.869195	−	0.494470i	$-0.164638\pi$
$53$	35549.8	1.73839	0.869195	+	0.494470i	$0.164638\pi$
$54$	0	0
$55$	−7550.97	−0.336586
$56$	0	0
$57$	13465.8	0.548965
$58$	0	0
$59$	−27594.5	−1.03203	−0.516016	−	0.856579i	$-0.672585\pi$
$59$	−27594.5	−1.03203	−0.516016	+	0.856579i	$0.672585\pi$
$60$	0	0
$61$	−26772.6	−0.921227	−0.460613	−	0.887601i	$-0.652371\pi$
$61$	−26772.6	−0.921227	−0.460613	+	0.887601i	$0.652371\pi$
$62$	0	0
$63$	−10399.2	−0.330101
$64$	0	0
$65$	−36807.9	−1.08058
$66$	0	0
$67$	−5610.06	−0.152679	−0.0763397	−	0.997082i	$-0.524323\pi$
$67$	−5610.06	−0.152679	−0.0763397	+	0.997082i	$0.524323\pi$
$68$	0	0
$69$	−205.726	−0.00520195
$70$	0	0
$71$	−7453.03	−0.175464	−0.0877318	−	0.996144i	$-0.527962\pi$
$71$	−7453.03	−0.175464	−0.0877318	+	0.996144i	$0.527962\pi$
$72$	0	0
$73$	−11148.6	−0.244857	−0.122428	−	0.992477i	$-0.539068\pi$
$73$	−11148.6	−0.244857	−0.122428	+	0.992477i	$0.539068\pi$
$74$	0	0
$75$	13947.4	0.286312
$76$	0	0
$77$	13518.6	0.259839
$78$	0	0
$79$	−104147.	−1.87751	−0.938753	−	0.344592i	$-0.888017\pi$
$79$	−104147.	−1.87751	−0.938753	+	0.344592i	$0.888017\pi$
$80$	0	0
$81$	−29456.0	−0.498839
$82$	0	0
$83$	51115.3	0.814433	0.407216	−	0.913332i	$-0.366499\pi$
$83$	51115.3	0.814433	0.407216	+	0.913332i	$0.366499\pi$
$84$	0	0
$85$	−144985.	−2.17659
$86$	0	0
$87$	99365.8	1.40747
$88$	0	0
$89$	−92711.2	−1.24067	−0.620337	−	0.784336i	$-0.713004\pi$
$89$	−92711.2	−1.24067	−0.620337	+	0.784336i	$0.713004\pi$
$90$	0	0
$91$	65897.5	0.834191
$92$	0	0
$93$	−11921.6	−0.142931
$94$	0	0
$95$	−70758.9	−0.804400
$96$	0	0
$97$	95474.3	1.03028	0.515142	−	0.857105i	$-0.327739\pi$
$97$	95474.3	1.03028	0.515142	+	0.857105i	$0.327739\pi$
$98$	0	0
$99$	−10321.8	−0.105844
$100$	0	0
$101$	−9908.83	−0.0966538	−0.0483269	−	0.998832i	$-0.515389\pi$
$101$	−9908.83	−0.0966538	−0.0483269	+	0.998832i	$0.515389\pi$
$102$	0	0
$103$	−10619.4	−0.0986299	−0.0493150	−	0.998783i	$-0.515704\pi$
$103$	−10619.4	−0.0986299	−0.0493150	+	0.998783i	$0.515704\pi$
$104$	0	0
$105$	−94374.6	−0.835376
$106$	0	0
$107$	−48844.1	−0.412432	−0.206216	−	0.978506i	$-0.566115\pi$
$107$	−48844.1	−0.412432	−0.206216	+	0.978506i	$0.566115\pi$
$108$	0	0
$109$	−4155.82	−0.0335036	−0.0167518	−	0.999860i	$-0.505333\pi$
$109$	−4155.82	−0.0335036	−0.0167518	+	0.999860i	$0.505333\pi$
$110$	0	0
$111$	−99559.9	−0.766967
$112$	0	0
$113$	−131686.	−0.970157	−0.485078	−	0.874471i	$-0.661209\pi$
$113$	−131686.	−0.970157	−0.485078	+	0.874471i	$0.661209\pi$
$114$	0	0
$115$	1081.03	0.00762243
$116$	0	0
$117$	−50314.6	−0.339805
$118$	0	0
$119$	259568.	1.68029
$120$	0	0
$121$	−147633.	−0.916685
$122$	0	0
$123$	109060.	0.649985
$124$	0	0
$125$	130419.	0.746561
$126$	0	0
$127$	−22758.4	−0.125208	−0.0626040	−	0.998038i	$-0.519940\pi$
$127$	−22758.4	−0.125208	−0.0626040	+	0.998038i	$0.519940\pi$
$128$	0	0
$129$	96530.5	0.510729
$130$	0	0
$131$	−22620.5	−0.115166	−0.0575830	−	0.998341i	$-0.518339\pi$
$131$	−22620.5	−0.115166	−0.0575830	+	0.998341i	$0.518339\pi$
$132$	0	0
$133$	126680.	0.620984
$134$	0	0
$135$	268563.	1.26827
$136$	0	0
$137$	−208512.	−0.949139	−0.474570	−	0.880218i	$-0.657396\pi$
$137$	−208512.	−0.949139	−0.474570	+	0.880218i	$0.657396\pi$
$138$	0	0
$139$	107528.	0.472047	0.236023	−	0.971747i	$-0.424156\pi$
$139$	107528.	0.472047	0.236023	+	0.971747i	$0.424156\pi$
$140$	0	0
$141$	357179.	1.51300
$142$	0	0
$143$	65407.4	0.267477
$144$	0	0
$145$	−522139.	−2.06237
$146$	0	0
$147$	−39537.3	−0.150908
$148$	0	0
$149$	−184072.	−0.679239	−0.339620	−	0.940563i	$-0.610298\pi$
$149$	−184072.	−0.679239	−0.339620	+	0.940563i	$0.610298\pi$
$150$	0	0
$151$	−69122.2	−0.246703	−0.123352	−	0.992363i	$-0.539364\pi$
$151$	−69122.2	−0.246703	−0.123352	+	0.992363i	$0.539364\pi$
$152$	0	0
$153$	−198188.	−0.684459
$154$	0	0
$155$	62644.4	0.209437
$156$	0	0
$157$	539322.	1.74622	0.873110	−	0.487523i	$-0.162099\pi$
$157$	539322.	1.74622	0.873110	+	0.487523i	$0.162099\pi$
$158$	0	0
$159$	441008.	1.38342
$160$	0	0
$161$	−1935.38	−0.00588439
$162$	0	0
$163$	320741.	0.945553	0.472776	−	0.881182i	$-0.343252\pi$
$163$	320741.	0.945553	0.472776	+	0.881182i	$0.343252\pi$
$164$	0	0
$165$	−93672.6	−0.267857
$166$	0	0
$167$	595873.	1.65334	0.826671	−	0.562686i	$-0.190232\pi$
$167$	595873.	1.65334	0.826671	+	0.562686i	$0.190232\pi$
$168$	0	0
$169$	−52458.9	−0.141287
$170$	0	0
$171$	−96723.9	−0.252955
$172$	0	0
$173$	−192255.	−0.488386	−0.244193	−	0.969727i	$-0.578523\pi$
$173$	−192255.	−0.488386	−0.244193	+	0.969727i	$0.578523\pi$
$174$	0	0
$175$	131211.	0.323873
$176$	0	0
$177$	−342320.	−0.821295
$178$	0	0
$179$	−670437.	−1.56396	−0.781980	−	0.623304i	$-0.785790\pi$
$179$	−670437.	−1.56396	−0.781980	+	0.623304i	$0.785790\pi$
$180$	0	0
$181$	245546.	0.557104	0.278552	−	0.960421i	$-0.410145\pi$
$181$	245546.	0.557104	0.278552	+	0.960421i	$0.410145\pi$
$182$	0	0
$183$	−332124.	−0.733116
$184$	0	0
$185$	523159.	1.12384
$186$	0	0
$187$	257637.	0.538772
$188$	0	0
$189$	−480811.	−0.979084
$190$	0	0
$191$	720955.	1.42996	0.714982	−	0.699143i	$-0.246435\pi$
$191$	720955.	1.42996	0.714982	+	0.699143i	$0.246435\pi$
$192$	0	0
$193$	662692.	1.28061	0.640307	−	0.768119i	$-0.278807\pi$
$193$	662692.	1.28061	0.640307	+	0.768119i	$0.278807\pi$
$194$	0	0
$195$	−456615.	−0.859932
$196$	0	0
$197$	−245771.	−0.451196	−0.225598	−	0.974220i	$-0.572434\pi$
$197$	−245771.	−0.451196	−0.225598	+	0.974220i	$0.572434\pi$
$198$	0	0
$199$	828616.	1.48327	0.741636	−	0.670803i	$-0.234050\pi$
$199$	828616.	1.48327	0.741636	+	0.670803i	$0.234050\pi$
$200$	0	0
$201$	−69594.8	−0.121503
$202$	0	0
$203$	934790.	1.59211
$204$	0	0
$205$	−573080.	−0.952425
$206$	0	0
$207$	1477.72	0.00239698
$208$	0	0
$209$	125738.	0.199114
$210$	0	0
$211$	−817270.	−1.26375	−0.631873	−	0.775072i	$-0.717713\pi$
$211$	−817270.	−1.26375	−0.631873	+	0.775072i	$0.717713\pi$
$212$	0	0
$213$	−92457.6	−0.139635
$214$	0	0
$215$	−507240.	−0.748372
$216$	0	0
$217$	−112153.	−0.161682
$218$	0	0
$219$	−138302.	−0.194858
$220$	0	0
$221$	1.25588e6	1.72968
$222$	0	0
$223$	−689552.	−0.928549	−0.464275	−	0.885691i	$-0.653685\pi$
$223$	−689552.	−0.928549	−0.464275	+	0.885691i	$0.653685\pi$
$224$	0	0
$225$	−100183.	−0.131928
$226$	0	0
$227$	771501.	0.993737	0.496869	−	0.867826i	$-0.334483\pi$
$227$	771501.	0.993737	0.496869	+	0.867826i	$0.334483\pi$
$228$	0	0
$229$	−670857.	−0.845359	−0.422679	−	0.906279i	$-0.638910\pi$
$229$	−670857.	−0.845359	−0.422679	+	0.906279i	$0.638910\pi$
$230$	0	0
$231$	167703.	0.206781
$232$	0	0
$233$	−614524.	−0.741565	−0.370783	−	0.928720i	$-0.620910\pi$
$233$	−614524.	−0.741565	−0.370783	+	0.928720i	$0.620910\pi$
$234$	0	0
$235$	−1.87687e6	−2.21700
$236$	0	0
$237$	−1.29199e6	−1.49413
$238$	0	0
$239$	11314.7	0.0128130	0.00640648	−	0.999979i	$-0.497961\pi$
$239$	11314.7	0.0128130	0.00640648	+	0.999979i	$0.497961\pi$
$240$	0	0
$241$	−838091.	−0.929498	−0.464749	−	0.885442i	$-0.653856\pi$
$241$	−838091.	−0.929498	−0.464749	+	0.885442i	$0.653856\pi$
$242$	0	0
$243$	635726.	0.690644
$244$	0	0
$245$	207757.	0.221126
$246$	0	0
$247$	612921.	0.639238
$248$	0	0
$249$	634103.	0.648129
$250$	0	0
$251$	−940294.	−0.942061	−0.471031	−	0.882117i	$-0.656118\pi$
$251$	−940294.	−0.942061	−0.471031	+	0.882117i	$0.656118\pi$
$252$	0	0
$253$	−1920.98	−0.00188678
$254$	0	0
$255$	−1.79859e6	−1.73214
$256$	0	0
$257$	723683.	0.683464	0.341732	−	0.939797i	$-0.388986\pi$
$257$	723683.	0.683464	0.341732	+	0.939797i	$0.388986\pi$
$258$	0	0
$259$	−936616.	−0.867585
$260$	0	0
$261$	−713738.	−0.648541
$262$	0	0
$263$	1.66414e6	1.48354	0.741771	−	0.670654i	$-0.233986\pi$
$263$	1.66414e6	1.48354	0.741771	+	0.670654i	$0.233986\pi$
$264$	0	0
$265$	−2.31737e6	−2.02713
$266$	0	0
$267$	−1.15012e6	−0.987333
$268$	0	0
$269$	95037.3	0.0800780	0.0400390	−	0.999198i	$-0.487252\pi$
$269$	95037.3	0.0800780	0.0400390	+	0.999198i	$0.487252\pi$
$270$	0	0
$271$	−1.78955e6	−1.48020	−0.740100	−	0.672497i	$-0.765222\pi$
$271$	−1.78955e6	−1.48020	−0.740100	+	0.672497i	$0.765222\pi$
$272$	0	0
$273$	817483.	0.663853
$274$	0	0
$275$	130235.	0.103847
$276$	0	0
$277$	1.71170e6	1.34038	0.670189	−	0.742191i	$-0.266213\pi$
$277$	1.71170e6	1.34038	0.670189	+	0.742191i	$0.266213\pi$
$278$	0	0
$279$	85631.8	0.0658604
$280$	0	0
$281$	−114473.	−0.0864841	−0.0432421	−	0.999065i	$-0.513769\pi$
$281$	−114473.	−0.0864841	−0.0432421	+	0.999065i	$0.513769\pi$
$282$	0	0
$283$	−1.09533e6	−0.812979	−0.406489	−	0.913655i	$-0.633247\pi$
$283$	−1.09533e6	−0.812979	−0.406489	+	0.913655i	$0.633247\pi$
$284$	0	0
$285$	−877790.	−0.640145
$286$	0	0
$287$	1.02599e6	0.735256
$288$	0	0
$289$	3.52700e6	2.48405
$290$	0	0
$291$	1.18439e6	0.819905
$292$	0	0
$293$	−586668.	−0.399230	−0.199615	−	0.979874i	$-0.563969\pi$
$293$	−586668.	−0.399230	−0.199615	+	0.979874i	$0.563969\pi$
$294$	0	0
$295$	1.79880e6	1.20345
$296$	0	0
$297$	−477234.	−0.313936
$298$	0	0
$299$	−9364.00	−0.00605736
$300$	0	0
$301$	908117.	0.577731
$302$	0	0
$303$	−122923.	−0.0769175
$304$	0	0
$305$	1.74522e6	1.07424
$306$	0	0
$307$	1.68034e6	1.01754	0.508770	−	0.860902i	$-0.330100\pi$
$307$	1.68034e6	1.01754	0.508770	+	0.860902i	$0.330100\pi$
$308$	0	0
$309$	−131738.	−0.0784902
$310$	0	0
$311$	−1.39159e6	−0.815853	−0.407926	−	0.913015i	$-0.633748\pi$
$311$	−1.39159e6	−0.815853	−0.407926	+	0.913015i	$0.633748\pi$
$312$	0	0
$313$	−2.81842e6	−1.62609	−0.813045	−	0.582201i	$-0.802192\pi$
$313$	−2.81842e6	−1.62609	−0.813045	+	0.582201i	$0.802192\pi$
$314$	0	0
$315$	677887.	0.384929
$316$	0	0
$317$	1.67040e6	0.933626	0.466813	−	0.884356i	$-0.345402\pi$
$317$	1.67040e6	0.933626	0.466813	+	0.884356i	$0.345402\pi$
$318$	0	0
$319$	927837.	0.510499
$320$	0	0
$321$	−605929.	−0.328216
$322$	0	0
$323$	2.41428e6	1.28760
$324$	0	0
$325$	634841.	0.333393
$326$	0	0
$327$	−51554.5	−0.0266623
$328$	0	0
$329$	3.36019e6	1.71149
$330$	0	0
$331$	−829261.	−0.416027	−0.208013	−	0.978126i	$-0.566700\pi$
$331$	−829261.	−0.416027	−0.208013	+	0.978126i	$0.566700\pi$
$332$	0	0
$333$	715132.	0.353407
$334$	0	0
$335$	365701.	0.178039
$336$	0	0
$337$	−2.75999e6	−1.32383	−0.661916	−	0.749578i	$-0.730257\pi$
$337$	−2.75999e6	−1.32383	−0.661916	+	0.749578i	$0.730257\pi$
$338$	0	0
$339$	−1.63361e6	−0.772055
$340$	0	0
$341$	−111319.	−0.0518420
$342$	0	0
$343$	−2.33340e6	−1.07091
$344$	0	0
$345$	13410.6	0.00606596
$346$	0	0
$347$	−2.26494e6	−1.00979	−0.504897	−	0.863180i	$-0.668469\pi$
$347$	−2.26494e6	−1.00979	−0.504897	+	0.863180i	$0.668469\pi$
$348$	0	0
$349$	3.28272e6	1.44268	0.721341	−	0.692580i	$-0.243526\pi$
$349$	3.28272e6	1.44268	0.721341	+	0.692580i	$0.243526\pi$
$350$	0	0
$351$	−2.32632e6	−1.00786
$352$	0	0
$353$	−865114.	−0.369519	−0.184759	−	0.982784i	$-0.559151\pi$
$353$	−865114.	−0.369519	−0.184759	+	0.982784i	$0.559151\pi$
$354$	0	0
$355$	485838.	0.204607
$356$	0	0
$357$	3.22004e6	1.33718
$358$	0	0
$359$	224545.	0.0919534	0.0459767	−	0.998943i	$-0.485360\pi$
$359$	224545.	0.0919534	0.0459767	+	0.998943i	$0.485360\pi$
$360$	0	0
$361$	−1.29783e6	−0.524143
$362$	0	0
$363$	−1.83144e6	−0.729502
$364$	0	0
$365$	726738.	0.285526
$366$	0	0
$367$	1.58368e6	0.613765	0.306882	−	0.951747i	$-0.400714\pi$
$367$	1.58368e6	0.613765	0.306882	+	0.951747i	$0.400714\pi$
$368$	0	0
$369$	−783372.	−0.299504
$370$	0	0
$371$	4.14881e6	1.56491
$372$	0	0
$373$	−3.15625e6	−1.17462	−0.587312	−	0.809361i	$-0.699814\pi$
$373$	−3.15625e6	−1.17462	−0.587312	+	0.809361i	$0.699814\pi$
$374$	0	0
$375$	1.61789e6	0.594117
$376$	0	0
$377$	4.52282e6	1.63891
$378$	0	0
$379$	1.92070e6	0.686849	0.343424	−	0.939180i	$-0.388413\pi$
$379$	1.92070e6	0.686849	0.343424	+	0.939180i	$0.388413\pi$
$380$	0	0
$381$	−282326.	−0.0996410
$382$	0	0
$383$	−2.64123e6	−0.920046	−0.460023	−	0.887907i	$-0.652159\pi$
$383$	−2.64123e6	−0.920046	−0.460023	+	0.887907i	$0.652159\pi$
$384$	0	0
$385$	−881231.	−0.302997
$386$	0	0
$387$	−693373.	−0.235337
$388$	0	0
$389$	748955.	0.250947	0.125473	−	0.992097i	$-0.459955\pi$
$389$	748955.	0.250947	0.125473	+	0.992097i	$0.459955\pi$
$390$	0	0
$391$	−36884.5	−0.0122012
$392$	0	0
$393$	−280616.	−0.0916496
$394$	0	0
$395$	6.78903e6	2.18935
$396$	0	0
$397$	3.81150e6	1.21372	0.606862	−	0.794807i	$-0.292428\pi$
$397$	3.81150e6	1.21372	0.606862	+	0.794807i	$0.292428\pi$
$398$	0	0
$399$	1.57152e6	0.494182
$400$	0	0
$401$	−3.07392e6	−0.954624	−0.477312	−	0.878734i	$-0.658389\pi$
$401$	−3.07392e6	−0.954624	−0.477312	+	0.878734i	$0.658389\pi$
$402$	0	0
$403$	−542632.	−0.166434
$404$	0	0
$405$	1.92014e6	0.581694
$406$	0	0
$407$	−929649.	−0.278185
$408$	0	0
$409$	−1.51680e6	−0.448352	−0.224176	−	0.974549i	$-0.571969\pi$
$409$	−1.51680e6	−0.448352	−0.224176	+	0.974549i	$0.571969\pi$
$410$	0	0
$411$	−2.58667e6	−0.755329
$412$	0	0
$413$	−3.22040e6	−0.929041
$414$	0	0
$415$	−3.33203e6	−0.949706
$416$	0	0
$417$	1.33392e6	0.375657
$418$	0	0
$419$	2.74255e6	0.763166	0.381583	−	0.924335i	$-0.375379\pi$
$419$	2.74255e6	0.763166	0.381583	+	0.924335i	$0.375379\pi$
$420$	0	0
$421$	−1.24441e6	−0.342183	−0.171091	−	0.985255i	$-0.554729\pi$
$421$	−1.24441e6	−0.342183	−0.171091	+	0.985255i	$0.554729\pi$
$422$	0	0
$423$	−2.56559e6	−0.697168
$424$	0	0
$425$	2.50062e6	0.671545
$426$	0	0
$427$	−3.12448e6	−0.829293
$428$	0	0
$429$	811402.	0.212859
$430$	0	0
$431$	2.70694e6	0.701917	0.350958	−	0.936391i	$-0.385856\pi$
$431$	2.70694e6	0.701917	0.350958	+	0.936391i	$0.385856\pi$
$432$	0	0
$433$	−795131.	−0.203807	−0.101903	−	0.994794i	$-0.532493\pi$
$433$	−795131.	−0.203807	−0.101903	+	0.994794i	$0.532493\pi$
$434$	0	0
$435$	−6.47732e6	−1.64124
$436$	0	0
$437$	−18001.2	−0.00450919
$438$	0	0
$439$	−6.58546e6	−1.63089	−0.815445	−	0.578834i	$-0.803508\pi$
$439$	−6.58546e6	−1.63089	−0.815445	+	0.578834i	$0.803508\pi$
$440$	0	0
$441$	283994.	0.0695364
$442$	0	0
$443$	2.94797e6	0.713696	0.356848	−	0.934162i	$-0.383851\pi$
$443$	2.94797e6	0.713696	0.356848	+	0.934162i	$0.383851\pi$
$444$	0	0
$445$	6.04354e6	1.44674
$446$	0	0
$447$	−2.28348e6	−0.540542
$448$	0	0
$449$	−4.15636e6	−0.972964	−0.486482	−	0.873691i	$-0.661720\pi$
$449$	−4.15636e6	−0.972964	−0.486482	+	0.873691i	$0.661720\pi$
$450$	0	0
$451$	1.01836e6	0.235754
$452$	0	0
$453$	−857486.	−0.196328
$454$	0	0
$455$	−4.29564e6	−0.972746
$456$	0	0
$457$	374186.	0.0838102	0.0419051	−	0.999122i	$-0.486657\pi$
$457$	374186.	0.0838102	0.0419051	+	0.999122i	$0.486657\pi$
$458$	0	0
$459$	−9.16331e6	−2.03011
$460$	0	0
$461$	2.55441e6	0.559806	0.279903	−	0.960028i	$-0.409698\pi$
$461$	2.55441e6	0.559806	0.279903	+	0.960028i	$0.409698\pi$
$462$	0	0
$463$	−8.73527e6	−1.89376	−0.946878	−	0.321594i	$-0.895781\pi$
$463$	−8.73527e6	−1.89376	−0.946878	+	0.321594i	$0.895781\pi$
$464$	0	0
$465$	777126.	0.166671
$466$	0	0
$467$	−2.94803e6	−0.625518	−0.312759	−	0.949832i	$-0.601253\pi$
$467$	−2.94803e6	−0.625518	−0.312759	+	0.949832i	$0.601253\pi$
$468$	0	0
$469$	−654718.	−0.137443
$470$	0	0
$471$	6.69049e6	1.38965
$472$	0	0
$473$	901362.	0.185245
$474$	0	0
$475$	1.22041e6	0.248182
$476$	0	0
$477$	−3.16773e6	−0.637459
$478$	0	0
$479$	1.49781e6	0.298276	0.149138	−	0.988816i	$-0.452350\pi$
$479$	1.49781e6	0.298276	0.149138	+	0.988816i	$0.452350\pi$
$480$	0	0
$481$	−4.53166e6	−0.893088
$482$	0	0
$483$	−24009.1	−0.00468282
$484$	0	0
$485$	−6.22365e6	−1.20141
$486$	0	0
$487$	−1.57909e6	−0.301707	−0.150853	−	0.988556i	$-0.548202\pi$
$487$	−1.57909e6	−0.301707	−0.150853	+	0.988556i	$0.548202\pi$
$488$	0	0
$489$	3.97891e6	0.752475
$490$	0	0
$491$	1.16112e6	0.217356	0.108678	−	0.994077i	$-0.465338\pi$
$491$	1.16112e6	0.217356	0.108678	+	0.994077i	$0.465338\pi$
$492$	0	0
$493$	1.78153e7	3.30122
$494$	0	0
$495$	672844.	0.123425
$496$	0	0
$497$	−869801.	−0.157953
$498$	0	0
$499$	−655236.	−0.117800	−0.0589001	−	0.998264i	$-0.518759\pi$
$499$	−655236.	−0.117800	−0.0589001	+	0.998264i	$0.518759\pi$
$500$	0	0
$501$	7.39202e6	1.31574
$502$	0	0
$503$	−663180.	−0.116872	−0.0584362	−	0.998291i	$-0.518611\pi$
$503$	−663180.	−0.116872	−0.0584362	+	0.998291i	$0.518611\pi$
$504$	0	0
$505$	645923.	0.112707
$506$	0	0
$507$	−650772.	−0.112437
$508$	0	0
$509$	−1.02416e7	−1.75215	−0.876075	−	0.482175i	$-0.839847\pi$
$509$	−1.02416e7	−1.75215	−0.876075	+	0.482175i	$0.839847\pi$
$510$	0	0
$511$	−1.30109e6	−0.220421
$512$	0	0
$513$	−4.47208e6	−0.750268
$514$	0	0
$515$	692246.	0.115012
$516$	0	0
$517$	3.33519e6	0.548775
$518$	0	0
$519$	−2.38500e6	−0.388659
$520$	0	0
$521$	−1.00963e7	−1.62954	−0.814772	−	0.579782i	$-0.803138\pi$
$521$	−1.00963e7	−1.62954	−0.814772	+	0.579782i	$0.803138\pi$
$522$	0	0
$523$	−2.46620e6	−0.394252	−0.197126	−	0.980378i	$-0.563161\pi$
$523$	−2.46620e6	−0.394252	−0.197126	+	0.980378i	$0.563161\pi$
$524$	0	0
$525$	1.62772e6	0.257739
$526$	0	0
$527$	−2.13741e6	−0.335244
$528$	0	0
$529$	−6.43607e6	−0.999957
$530$	0	0
$531$	2.45886e6	0.378441
$532$	0	0
$533$	4.96408e6	0.756869
$534$	0	0
$535$	3.18398e6	0.480935
$536$	0	0
$537$	−8.31702e6	−1.24461
$538$	0	0
$539$	−369183.	−0.0547355
$540$	0	0
$541$	−1.40825e6	−0.206864	−0.103432	−	0.994637i	$-0.532982\pi$
$541$	−1.40825e6	−0.206864	−0.103432	+	0.994637i	$0.532982\pi$
$542$	0	0
$543$	3.04609e6	0.443346
$544$	0	0
$545$	270904.	0.0390683
$546$	0	0
$547$	5.96148e6	0.851895	0.425947	−	0.904748i	$-0.359941\pi$
$547$	5.96148e6	0.851895	0.425947	+	0.904748i	$0.359941\pi$
$548$	0	0
$549$	2.38563e6	0.337809
$550$	0	0
$551$	8.69460e6	1.22003
$552$	0	0
$553$	−1.21545e7	−1.69014
$554$	0	0
$555$	6.48997e6	0.894356
$556$	0	0
$557$	1.04484e7	1.42696	0.713482	−	0.700673i	$-0.247117\pi$
$557$	1.04484e7	1.42696	0.713482	+	0.700673i	$0.247117\pi$
$558$	0	0
$559$	4.39377e6	0.594714
$560$	0	0
$561$	3.19609e6	0.428757
$562$	0	0
$563$	819230.	0.108927	0.0544634	−	0.998516i	$-0.482655\pi$
$563$	819230.	0.108927	0.0544634	+	0.998516i	$0.482655\pi$
$564$	0	0
$565$	8.58414e6	1.13129
$566$	0	0
$567$	−3.43764e6	−0.449058
$568$	0	0
$569$	501105.	0.0648856	0.0324428	−	0.999474i	$-0.489671\pi$
$569$	501105.	0.0648856	0.0324428	+	0.999474i	$0.489671\pi$
$570$	0	0
$571$	4.43886e6	0.569746	0.284873	−	0.958565i	$-0.408049\pi$
$571$	4.43886e6	0.569746	0.284873	+	0.958565i	$0.408049\pi$
$572$	0	0
$573$	8.94371e6	1.13797
$574$	0	0
$575$	−18645.0	−0.00235176
$576$	0	0
$577$	−1.34606e7	−1.68315	−0.841577	−	0.540138i	$-0.818372\pi$
$577$	−1.34606e7	−1.68315	−0.841577	+	0.540138i	$0.818372\pi$
$578$	0	0
$579$	8.22094e6	1.01912
$580$	0	0
$581$	5.96537e6	0.733157
$582$	0	0
$583$	4.11795e6	0.501776
$584$	0	0
$585$	3.27984e6	0.396244
$586$	0	0
$587$	1.62638e7	1.94817	0.974086	−	0.226178i	$-0.0726230\pi$
$587$	1.62638e7	1.94817	0.974086	+	0.226178i	$0.0726230\pi$
$588$	0	0
$589$	−1.04315e6	−0.123896
$590$	0	0
$591$	−3.04888e6	−0.359064
$592$	0	0
$593$	−1.19589e7	−1.39655	−0.698274	−	0.715831i	$-0.746048\pi$
$593$	−1.19589e7	−1.39655	−0.698274	+	0.715831i	$0.746048\pi$
$594$	0	0
$595$	−1.69204e7	−1.95938
$596$	0	0
$597$	1.02793e7	1.18039
$598$	0	0
$599$	1.07320e7	1.22212	0.611058	−	0.791586i	$-0.290744\pi$
$599$	1.07320e7	1.22212	0.611058	+	0.791586i	$0.290744\pi$
$600$	0	0
$601$	−3.05716e6	−0.345248	−0.172624	−	0.984988i	$-0.555225\pi$
$601$	−3.05716e6	−0.345248	−0.172624	+	0.984988i	$0.555225\pi$
$602$	0	0
$603$	499895.	0.0559868
$604$	0	0
$605$	9.62370e6	1.06894
$606$	0	0
$607$	1.45849e7	1.60669	0.803343	−	0.595517i	$-0.203053\pi$
$607$	1.45849e7	1.60669	0.803343	+	0.595517i	$0.203053\pi$
$608$	0	0
$609$	1.15964e7	1.26701
$610$	0	0
$611$	1.62577e7	1.76180
$612$	0	0
$613$	−6.06610e6	−0.652016	−0.326008	−	0.945367i	$-0.605704\pi$
$613$	−6.06610e6	−0.652016	−0.326008	+	0.945367i	$0.605704\pi$
$614$	0	0
$615$	−7.10927e6	−0.757944
$616$	0	0
$617$	−9.82154e6	−1.03864	−0.519322	−	0.854579i	$-0.673815\pi$
$617$	−9.82154e6	−1.03864	−0.519322	+	0.854579i	$0.673815\pi$
$618$	0	0
$619$	5.27410e6	0.553250	0.276625	−	0.960978i	$-0.410784\pi$
$619$	5.27410e6	0.553250	0.276625	+	0.960978i	$0.410784\pi$
$620$	0	0
$621$	68323.0	0.00710948
$622$	0	0
$623$	−1.08198e7	−1.11686
$624$	0	0
$625$	−1.20150e7	−1.23034
$626$	0	0
$627$	1.55983e6	0.158456
$628$	0	0
$629$	−1.78501e7	−1.79892
$630$	0	0
$631$	1.69803e7	1.69774	0.848872	−	0.528599i	$-0.177282\pi$
$631$	1.69803e7	1.69774	0.848872	+	0.528599i	$0.177282\pi$
$632$	0	0
$633$	−1.01385e7	−1.00569
$634$	0	0
$635$	1.48354e6	0.146004
$636$	0	0
$637$	−1.79961e6	−0.175724
$638$	0	0
$639$	664117.	0.0643417
$640$	0	0
$641$	1.02050e6	0.0980999	0.0490499	−	0.998796i	$-0.484381\pi$
$641$	1.02050e6	0.0980999	0.0490499	+	0.998796i	$0.484381\pi$
$642$	0	0
$643$	3.73712e6	0.356459	0.178229	−	0.983989i	$-0.442963\pi$
$643$	3.73712e6	0.356459	0.178229	+	0.983989i	$0.442963\pi$
$644$	0	0
$645$	−6.29250e6	−0.595558
$646$	0	0
$647$	1.12582e7	1.05732	0.528662	−	0.848833i	$-0.322694\pi$
$647$	1.12582e7	1.05732	0.528662	+	0.848833i	$0.322694\pi$
$648$	0	0
$649$	−3.19645e6	−0.297890
$650$	0	0
$651$	−1.39130e6	−0.128667
$652$	0	0
$653$	−4.54202e6	−0.416837	−0.208418	−	0.978040i	$-0.566832\pi$
$653$	−4.54202e6	−0.416837	−0.208418	+	0.978040i	$0.566832\pi$
$654$	0	0
$655$	1.47456e6	0.134294
$656$	0	0
$657$	993415.	0.0897878
$658$	0	0
$659$	−1.71201e7	−1.53565	−0.767827	−	0.640657i	$-0.778662\pi$
$659$	−1.71201e7	−1.53565	−0.767827	+	0.640657i	$0.778662\pi$
$660$	0	0
$661$	−1.77627e7	−1.58127	−0.790635	−	0.612288i	$-0.790249\pi$
$661$	−1.77627e7	−1.58127	−0.790635	+	0.612288i	$0.790249\pi$
$662$	0	0
$663$	1.55796e7	1.37649
$664$	0	0
$665$	−8.25787e6	−0.724126
$666$	0	0
$667$	−132833.	−0.0115609
$668$	0	0
$669$	−8.55414e6	−0.738944
$670$	0	0
$671$	−3.10124e6	−0.265907
$672$	0	0
$673$	6.11592e6	0.520504	0.260252	−	0.965541i	$-0.416194\pi$
$673$	6.11592e6	0.520504	0.260252	+	0.965541i	$0.416194\pi$
$674$	0	0
$675$	−4.63201e6	−0.391301
$676$	0	0
$677$	−6.17788e6	−0.518045	−0.259023	−	0.965871i	$-0.583400\pi$
$677$	−6.17788e6	−0.518045	−0.259023	+	0.965871i	$0.583400\pi$
$678$	0	0
$679$	1.11423e7	0.927467
$680$	0	0
$681$	9.57075e6	0.790821
$682$	0	0
$683$	−2.22684e7	−1.82657	−0.913286	−	0.407320i	$-0.866463\pi$
$683$	−2.22684e7	−1.82657	−0.913286	+	0.407320i	$0.866463\pi$
$684$	0	0
$685$	1.35922e7	1.10679
$686$	0	0
$687$	−8.32222e6	−0.672740
$688$	0	0
$689$	2.00733e7	1.61091
$690$	0	0
$691$	1.14776e7	0.914440	0.457220	−	0.889354i	$-0.348845\pi$
$691$	1.14776e7	0.914440	0.457220	+	0.889354i	$0.348845\pi$
$692$	0	0
$693$	−1.20460e6	−0.0952817
$694$	0	0
$695$	−7.00940e6	−0.550451
$696$	0	0
$697$	1.95534e7	1.52454
$698$	0	0
$699$	−7.62340e6	−0.590141
$700$	0	0
$701$	−4.14523e6	−0.318605	−0.159303	−	0.987230i	$-0.550925\pi$
$701$	−4.14523e6	−0.318605	−0.159303	+	0.987230i	$0.550925\pi$
$702$	0	0
$703$	−8.71159e6	−0.664827
$704$	0	0
$705$	−2.32833e7	−1.76430
$706$	0	0
$707$	−1.15640e6	−0.0870083
$708$	0	0
$709$	−1.75067e7	−1.30794	−0.653972	−	0.756519i	$-0.726899\pi$
$709$	−1.75067e7	−1.30794	−0.653972	+	0.756519i	$0.726899\pi$
$710$	0	0
$711$	9.28027e6	0.688472
$712$	0	0
$713$	15936.9	0.00117403
$714$	0	0
$715$	−4.26369e6	−0.311904
$716$	0	0
$717$	140363.	0.0101966
$718$	0	0
$719$	−1.91068e7	−1.37837	−0.689185	−	0.724585i	$-0.742031\pi$
$719$	−1.91068e7	−1.37837	−0.689185	+	0.724585i	$0.742031\pi$
$720$	0	0
$721$	−1.23933e6	−0.0887872
$722$	0	0
$723$	−1.03968e7	−0.739699
$724$	0	0
$725$	9.00554e6	0.636304
$726$	0	0
$727$	−2.59411e7	−1.82034	−0.910169	−	0.414238i	$-0.864048\pi$
$727$	−2.59411e7	−1.82034	−0.910169	+	0.414238i	$0.864048\pi$
$728$	0	0
$729$	1.50442e7	1.04846
$730$	0	0
$731$	1.73069e7	1.19792
$732$	0	0
$733$	4.18318e6	0.287572	0.143786	−	0.989609i	$-0.454072\pi$
$733$	4.18318e6	0.287572	0.143786	+	0.989609i	$0.454072\pi$
$734$	0	0
$735$	2.57730e6	0.175973
$736$	0	0
$737$	−649848.	−0.0440700
$738$	0	0
$739$	2.85901e7	1.92577	0.962886	−	0.269910i	$-0.0869938\pi$
$739$	2.85901e7	1.92577	0.962886	+	0.269910i	$0.0869938\pi$
$740$	0	0
$741$	7.60351e6	0.508708
$742$	0	0
$743$	−9.21266e6	−0.612228	−0.306114	−	0.951995i	$-0.599029\pi$
$743$	−9.21266e6	−0.612228	−0.306114	+	0.951995i	$0.599029\pi$
$744$	0	0
$745$	1.19991e7	0.792057
$746$	0	0
$747$	−4.55472e6	−0.298649
$748$	0	0
$749$	−5.70032e6	−0.371274
$750$	0	0
$751$	−6.64136e6	−0.429692	−0.214846	−	0.976648i	$-0.568925\pi$
$751$	−6.64136e6	−0.429692	−0.214846	+	0.976648i	$0.568925\pi$
$752$	0	0
$753$	−1.16647e7	−0.749697
$754$	0	0
$755$	4.50585e6	0.287680
$756$	0	0
$757$	6101.18	0.000386967 0	0.000193484	−	1.00000i	$-0.499938\pi$
$757$	6101.18	0.000386967 0	0.000193484		1.00000i	$0.499938\pi$
$758$	0	0
$759$	−23830.5	−0.00150151
$760$	0	0
$761$	1.90776e7	1.19416	0.597079	−	0.802183i	$-0.296328\pi$
$761$	1.90776e7	1.19416	0.597079	+	0.802183i	$0.296328\pi$
$762$	0	0
$763$	−485002.	−0.0301601
$764$	0	0
$765$	1.29192e7	0.798144
$766$	0	0
$767$	−1.55814e7	−0.956350
$768$	0	0
$769$	−2.18700e7	−1.33362	−0.666811	−	0.745227i	$-0.732341\pi$
$769$	−2.18700e7	−1.33362	−0.666811	+	0.745227i	$0.732341\pi$
$770$	0	0
$771$	8.97756e6	0.543904
$772$	0	0
$773$	−4.64857e6	−0.279815	−0.139907	−	0.990165i	$-0.544681\pi$
$773$	−4.64857e6	−0.279815	−0.139907	+	0.990165i	$0.544681\pi$
$774$	0	0
$775$	−1.08045e6	−0.0646177
$776$	0	0
$777$	−1.16191e7	−0.690428
$778$	0	0
$779$	9.54287e6	0.563424
$780$	0	0
$781$	−863331.	−0.0506465
$782$	0	0
$783$	−3.30001e7	−1.92358
$784$	0	0
$785$	−3.51566e7	−2.03626
$786$	0	0
$787$	−4.64640e6	−0.267411	−0.133706	−	0.991021i	$-0.542688\pi$
$787$	−4.64640e6	−0.267411	−0.133706	+	0.991021i	$0.542688\pi$
$788$	0	0
$789$	2.06442e7	1.18061
$790$	0	0
$791$	−1.53683e7	−0.873341
$792$	0	0
$793$	−1.51173e7	−0.853671
$794$	0	0
$795$	−2.87478e7	−1.61320
$796$	0	0
$797$	1.42378e7	0.793957	0.396979	−	0.917828i	$-0.370059\pi$
$797$	1.42378e7	0.793957	0.396979	+	0.917828i	$0.370059\pi$
$798$	0	0
$799$	6.40385e7	3.54874
$800$	0	0
$801$	8.26122e6	0.454949
$802$	0	0
$803$	−1.29141e6	−0.0706764
$804$	0	0
$805$	126161.	0.00686175
$806$	0	0
$807$	1.17897e6	0.0637264
$808$	0	0
$809$	−1.37620e7	−0.739283	−0.369642	−	0.929174i	$-0.620520\pi$
$809$	−1.37620e7	−0.739283	−0.369642	+	0.929174i	$0.620520\pi$
$810$	0	0
$811$	−5.90814e6	−0.315427	−0.157713	−	0.987485i	$-0.550412\pi$
$811$	−5.90814e6	−0.315427	−0.157713	+	0.987485i	$0.550412\pi$
$812$	0	0
$813$	−2.22000e7	−1.17795
$814$	0	0
$815$	−2.09081e7	−1.10260
$816$	0	0
$817$	8.44652e6	0.442713
$818$	0	0
$819$	−5.87193e6	−0.305894
$820$	0	0
$821$	2.11066e7	1.09285	0.546426	−	0.837508i	$-0.315988\pi$
$821$	2.11066e7	1.09285	0.546426	+	0.837508i	$0.315988\pi$
$822$	0	0
$823$	3.04766e7	1.56844	0.784219	−	0.620484i	$-0.213064\pi$
$823$	3.04766e7	1.56844	0.784219	+	0.620484i	$0.213064\pi$
$824$	0	0
$825$	1.61561e6	0.0826421
$826$	0	0
$827$	−1.92606e7	−0.979277	−0.489639	−	0.871925i	$-0.662871\pi$
$827$	−1.92606e7	−0.979277	−0.489639	+	0.871925i	$0.662871\pi$
$828$	0	0
$829$	−1.98613e7	−1.00374	−0.501870	−	0.864943i	$-0.667355\pi$
$829$	−1.98613e7	−1.00374	−0.501870	+	0.864943i	$0.667355\pi$
$830$	0	0
$831$	2.12342e7	1.06668
$832$	0	0
$833$	−7.08862e6	−0.353956
$834$	0	0
$835$	−3.88430e7	−1.92795
$836$	0	0
$837$	3.95923e6	0.195343
$838$	0	0
$839$	3.24146e7	1.58977	0.794887	−	0.606758i	$-0.207530\pi$
$839$	3.24146e7	1.58977	0.794887	+	0.606758i	$0.207530\pi$
$840$	0	0
$841$	4.36474e7	2.12798
$842$	0	0
$843$	−1.42008e6	−0.0688245
$844$	0	0
$845$	3.41962e6	0.164754
$846$	0	0
$847$	−1.72294e7	−0.825205
$848$	0	0
$849$	−1.35880e7	−0.646972
$850$	0	0
$851$	133093.	0.00629985
$852$	0	0
$853$	−1.48598e7	−0.699265	−0.349632	−	0.936887i	$-0.613694\pi$
$853$	−1.48598e7	−0.699265	−0.349632	+	0.936887i	$0.613694\pi$
$854$	0	0
$855$	6.30511e6	0.294970
$856$	0	0
$857$	−1.88696e7	−0.877628	−0.438814	−	0.898578i	$-0.644601\pi$
$857$	−1.88696e7	−0.877628	−0.438814	+	0.898578i	$0.644601\pi$
$858$	0	0
$859$	1.41175e7	0.652793	0.326396	−	0.945233i	$-0.394166\pi$
$859$	1.41175e7	0.652793	0.326396	+	0.945233i	$0.394166\pi$
$860$	0	0
$861$	1.27278e7	0.585120
$862$	0	0
$863$	−1.32859e7	−0.607245	−0.303623	−	0.952792i	$-0.598196\pi$
$863$	−1.32859e7	−0.607245	−0.303623	+	0.952792i	$0.598196\pi$
$864$	0	0
$865$	1.25325e7	0.569504
$866$	0	0
$867$	4.37537e7	1.97682
$868$	0	0
$869$	−1.20640e7	−0.541931
$870$	0	0
$871$	−3.16774e6	−0.141483
$872$	0	0
$873$	−8.50742e6	−0.377800
$874$	0	0
$875$	1.52204e7	0.672059
$876$	0	0
$877$	3.02279e7	1.32712	0.663558	−	0.748125i	$-0.269046\pi$
$877$	3.02279e7	1.32712	0.663558	+	0.748125i	$0.269046\pi$
$878$	0	0
$879$	−7.27782e6	−0.317709
$880$	0	0
$881$	−1.70236e7	−0.738944	−0.369472	−	0.929242i	$-0.620461\pi$
$881$	−1.70236e7	−0.738944	−0.369472	+	0.929242i	$0.620461\pi$
$882$	0	0
$883$	3.76106e6	0.162334	0.0811669	−	0.996701i	$-0.474135\pi$
$883$	3.76106e6	0.162334	0.0811669	+	0.996701i	$0.474135\pi$
$884$	0	0
$885$	2.23147e7	0.957708
$886$	0	0
$887$	−1.02878e7	−0.439050	−0.219525	−	0.975607i	$-0.570451\pi$
$887$	−1.02878e7	−0.439050	−0.219525	+	0.975607i	$0.570451\pi$
$888$	0	0
$889$	−2.65600e6	−0.112713
$890$	0	0
$891$	−3.41207e6	−0.143987
$892$	0	0
$893$	3.12535e7	1.31151
$894$	0	0
$895$	4.37036e7	1.82372
$896$	0	0
$897$	−116164.	−0.00482048
$898$	0	0
$899$	−7.69752e6	−0.317652
$900$	0	0
$901$	7.90682e7	3.24481
$902$	0	0
$903$	1.12655e7	0.459761
$904$	0	0
$905$	−1.60063e7	−0.649636
$906$	0	0
$907$	1.37046e7	0.553155	0.276578	−	0.960992i	$-0.410800\pi$
$907$	1.37046e7	0.553155	0.276578	+	0.960992i	$0.410800\pi$
$908$	0	0
$909$	882945.	0.0354425
$910$	0	0
$911$	4.91159e7	1.96077	0.980385	−	0.197093i	$-0.0631502\pi$
$911$	4.91159e7	1.96077	0.980385	+	0.197093i	$0.0631502\pi$
$912$	0	0
$913$	5.92100e6	0.235081
$914$	0	0
$915$	2.16501e7	0.854883
$916$	0	0
$917$	−2.63991e6	−0.103673
$918$	0	0
$919$	−1.73927e7	−0.679325	−0.339662	−	0.940547i	$-0.610313\pi$
$919$	−1.73927e7	−0.679325	−0.339662	+	0.940547i	$0.610313\pi$
$920$	0	0
$921$	2.08452e7	0.809763
$922$	0	0
$923$	−4.20839e6	−0.162597
$924$	0	0
$925$	−9.02313e6	−0.346739
$926$	0	0
$927$	946266.	0.0361671
$928$	0	0
$929$	−2.41687e7	−0.918787	−0.459393	−	0.888233i	$-0.651933\pi$
$929$	−2.41687e7	−0.918787	−0.459393	+	0.888233i	$0.651933\pi$
$930$	0	0
$931$	−3.45955e6	−0.130811
$932$	0	0
$933$	−1.72632e7	−0.649259
$934$	0	0
$935$	−1.67945e7	−0.628259
$936$	0	0
$937$	4.87981e7	1.81574	0.907870	−	0.419251i	$-0.137707\pi$
$937$	4.87981e7	1.81574	0.907870	+	0.419251i	$0.137707\pi$
$938$	0	0
$939$	−3.49635e7	−1.29405
$940$	0	0
$941$	−3.54488e7	−1.30505	−0.652525	−	0.757767i	$-0.726290\pi$
$941$	−3.54488e7	−1.30505	−0.652525	+	0.757767i	$0.726290\pi$
$942$	0	0
$943$	−145793.	−0.00533896
$944$	0	0
$945$	3.13425e7	1.14170
$946$	0	0
$947$	2.12301e7	0.769268	0.384634	−	0.923069i	$-0.374328\pi$
$947$	2.12301e7	0.769268	0.384634	+	0.923069i	$0.374328\pi$
$948$	0	0
$949$	−6.29508e6	−0.226901
$950$	0	0
$951$	2.07219e7	0.742984
$952$	0	0
$953$	−2.33430e7	−0.832575	−0.416288	−	0.909233i	$-0.636669\pi$
$953$	−2.33430e7	−0.832575	−0.416288	+	0.909233i	$0.636669\pi$
$954$	0	0
$955$	−4.69967e7	−1.66747
$956$	0	0
$957$	1.15102e7	0.406257
$958$	0	0
$959$	−2.43343e7	−0.854420
$960$	0	0
$961$	923521.	0.0322581
$962$	0	0
$963$	4.35235e6	0.151237
$964$	0	0
$965$	−4.31987e7	−1.49332
$966$	0	0
$967$	−4.62806e7	−1.59159	−0.795797	−	0.605563i	$-0.792948\pi$
$967$	−4.62806e7	−1.59159	−0.795797	+	0.605563i	$0.792948\pi$
$968$	0	0
$969$	2.99500e7	1.02468
$970$	0	0
$971$	−7.86754e6	−0.267788	−0.133894	−	0.990996i	$-0.542748\pi$
$971$	−7.86754e6	−0.267788	−0.133894	+	0.990996i	$0.542748\pi$
$972$	0	0
$973$	1.25490e7	0.424939
$974$	0	0
$975$	7.87543e6	0.265316
$976$	0	0
$977$	3.19249e7	1.07002	0.535012	−	0.844845i	$-0.320307\pi$
$977$	3.19249e7	1.07002	0.535012	+	0.844845i	$0.320307\pi$
$978$	0	0
$979$	−1.07393e7	−0.358113
$980$	0	0
$981$	370313.	0.0122856
$982$	0	0
$983$	−2.40204e7	−0.792859	−0.396429	−	0.918065i	$-0.629751\pi$
$983$	−2.40204e7	−0.792859	−0.396429	+	0.918065i	$0.629751\pi$
$984$	0	0
$985$	1.60210e7	0.526138
$986$	0	0
$987$	4.16843e7	1.36201
$988$	0	0
$989$	−129043.	−0.00419511
$990$	0	0
$991$	−1.90707e7	−0.616853	−0.308426	−	0.951248i	$-0.599802\pi$
$991$	−1.90707e7	−0.616853	−0.308426	+	0.951248i	$0.599802\pi$
$992$	0	0
$993$	−1.02873e7	−0.331076
$994$	0	0
$995$	−5.40147e7	−1.72963
$996$	0	0
$997$	−2.60568e7	−0.830201	−0.415100	−	0.909776i	$-0.636254\pi$
$997$	−2.60568e7	−0.830201	−0.415100	+	0.909776i	$0.636254\pi$
$998$	0	0
$999$	3.30645e7	1.04821

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.6.a.b.1.4	✓	6
4.3	odd	2		496.6.a.f.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.6.a.b.1.4	✓	6	1.1	even	1	trivial
496.6.a.f.1.3		6	4.3	odd	2

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:	\( N \)	\(=\)	\( 124 = 2^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	124.a (trivial)

\(f(q)\)	\(=\)	\(q+12.4054 q^{3} -65.1867 q^{5} +116.704 q^{7} -89.1069 q^{9} +O(q^{10})\)	\(q+12.4054 q^{3} -65.1867 q^{5} +116.704 q^{7} -89.1069 q^{9} +115.836 q^{11} +564.654 q^{13} -808.664 q^{15} +2224.15 q^{17} +1085.48 q^{19} +1447.76 q^{21} -16.5836 q^{23} +1124.30 q^{25} -4119.91 q^{27} +8009.90 q^{29} -961.000 q^{31} +1436.99 q^{33} -7607.56 q^{35} -8025.55 q^{37} +7004.74 q^{39} +8791.37 q^{41} +7781.35 q^{43} +5808.58 q^{45} +28792.3 q^{47} -3187.11 q^{49} +27591.4 q^{51} +35549.8 q^{53} -7550.97 q^{55} +13465.8 q^{57} -27594.5 q^{59} -26772.6 q^{61} -10399.2 q^{63} -36807.9 q^{65} -5610.06 q^{67} -205.726 q^{69} -7453.03 q^{71} -11148.6 q^{73} +13947.4 q^{75} +13518.6 q^{77} -104147. q^{79} -29456.0 q^{81} +51115.3 q^{83} -144985. q^{85} +99365.8 q^{87} -92711.2 q^{89} +65897.5 q^{91} -11921.6 q^{93} -70758.9 q^{95} +95474.3 q^{97} -10321.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 20 q^{3} + 25 q^{5} + 39 q^{7} + 306 q^{9}+O(q^{10}) \) 6 * q + 20 * q^3 + 25 * q^5 + 39 * q^7 + 306 * q^9	\( 6 q + 20 q^{3} + 25 q^{5} + 39 q^{7} + 306 q^{9} + 280 q^{11} + 1214 q^{13} + 1914 q^{15} + 796 q^{17} + 3147 q^{19} + 1082 q^{21} + 9122 q^{23} + 6481 q^{25} + 7316 q^{27} + 13020 q^{29} - 5766 q^{31} + 24804 q^{33} + 20059 q^{35} + 21678 q^{37} + 30680 q^{39} + 3227 q^{41} + 37882 q^{43} + 26169 q^{45} + 29708 q^{47} + 34849 q^{49} + 24432 q^{51} - 9976 q^{53} + 23758 q^{55} + 17318 q^{57} + 20573 q^{59} + 21610 q^{61} - 17697 q^{63} + 3894 q^{65} - 17024 q^{67} - 83692 q^{69} + 44509 q^{71} - 161864 q^{73} - 49430 q^{75} - 144202 q^{77} - 24420 q^{79} - 181158 q^{81} - 114160 q^{83} - 228882 q^{85} - 56180 q^{87} - 199742 q^{89} - 186774 q^{91} - 19220 q^{93} - 12793 q^{95} - 282951 q^{97} + 34060 q^{99}+O(q^{100}) \) 6 * q + 20 * q^3 + 25 * q^5 + 39 * q^7 + 306 * q^9 + 280 * q^11 + 1214 * q^13 + 1914 * q^15 + 796 * q^17 + 3147 * q^19 + 1082 * q^21 + 9122 * q^23 + 6481 * q^25 + 7316 * q^27 + 13020 * q^29 - 5766 * q^31 + 24804 * q^33 + 20059 * q^35 + 21678 * q^37 + 30680 * q^39 + 3227 * q^41 + 37882 * q^43 + 26169 * q^45 + 29708 * q^47 + 34849 * q^49 + 24432 * q^51 - 9976 * q^53 + 23758 * q^55 + 17318 * q^57 + 20573 * q^59 + 21610 * q^61 - 17697 * q^63 + 3894 * q^65 - 17024 * q^67 - 83692 * q^69 + 44509 * q^71 - 161864 * q^73 - 49430 * q^75 - 144202 * q^77 - 24420 * q^79 - 181158 * q^81 - 114160 * q^83 - 228882 * q^85 - 56180 * q^87 - 199742 * q^89 - 186774 * q^91 - 19220 * q^93 - 12793 * q^95 - 282951 * q^97 + 34060 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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