LMFDB - Embedded newform 124.6.a.b.1.5

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			1.5
Root			$18.2423$ 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)

$f(q)$	$=$	$q+21.2423 q^{3} +45.9749 q^{5} -213.890 q^{7} +208.235 q^{9} +O(q^{10})$	$q+21.2423 q^{3} +45.9749 q^{5} -213.890 q^{7} +208.235 q^{9} +697.283 q^{11} +1030.53 q^{13} +976.612 q^{15} -967.895 q^{17} +1982.30 q^{19} -4543.51 q^{21} +1865.28 q^{23} -1011.31 q^{25} -738.490 q^{27} +4575.83 q^{29} -961.000 q^{31} +14811.9 q^{33} -9833.56 q^{35} +6411.15 q^{37} +21890.8 q^{39} -17512.0 q^{41} +2024.92 q^{43} +9573.58 q^{45} -4403.85 q^{47} +28941.8 q^{49} -20560.3 q^{51} -23788.9 q^{53} +32057.5 q^{55} +42108.6 q^{57} -24320.6 q^{59} +37484.8 q^{61} -44539.3 q^{63} +47378.4 q^{65} -1132.04 q^{67} +39622.8 q^{69} +60714.8 q^{71} -59542.7 q^{73} -21482.5 q^{75} -149142. q^{77} -50857.8 q^{79} -66288.3 q^{81} -73658.0 q^{83} -44498.9 q^{85} +97201.1 q^{87} +17839.3 q^{89} -220419. q^{91} -20413.8 q^{93} +91136.0 q^{95} -175952. q^{97} +145199. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	21.2423	1.36269	0.681347	−	0.731961i	$-0.261395\pi$
$3$	21.2423	1.36269	0.681347	+	0.731961i	$0.261395\pi$
$4$	0	0
$5$	45.9749	0.822424	0.411212	−	0.911540i	$-0.365106\pi$
$5$	45.9749	0.822424	0.411212	+	0.911540i	$0.365106\pi$
$6$	0	0
$7$	−213.890	−1.64985	−0.824926	−	0.565241i	$-0.808783\pi$
$7$	−213.890	−1.64985	−0.824926	+	0.565241i	$0.808783\pi$
$8$	0	0
$9$	208.235	0.856934
$10$	0	0
$11$	697.283	1.73751	0.868755	−	0.495242i	$-0.164921\pi$
$11$	697.283	1.73751	0.868755	+	0.495242i	$0.164921\pi$
$12$	0	0
$13$	1030.53	1.69123	0.845613	−	0.533796i	$-0.179235\pi$
$13$	1030.53	1.69123	0.845613	+	0.533796i	$0.179235\pi$
$14$	0	0
$15$	976.612	1.12071
$16$	0	0
$17$	−967.895	−0.812281	−0.406140	−	0.913811i	$-0.633126\pi$
$17$	−967.895	−0.812281	−0.406140	+	0.913811i	$0.633126\pi$
$18$	0	0
$19$	1982.30	1.25975	0.629876	−	0.776695i	$-0.283106\pi$
$19$	1982.30	1.25975	0.629876	+	0.776695i	$0.283106\pi$
$20$	0	0
$21$	−4543.51	−2.24824
$22$	0	0
$23$	1865.28	0.735232	0.367616	−	0.929978i	$-0.380174\pi$
$23$	1865.28	0.735232	0.367616	+	0.929978i	$0.380174\pi$
$24$	0	0
$25$	−1011.31	−0.323619
$26$	0	0
$27$	−738.490	−0.194955
$28$	0	0
$29$	4575.83	1.01036	0.505178	−	0.863015i	$-0.331427\pi$
$29$	4575.83	1.01036	0.505178	+	0.863015i	$0.331427\pi$
$30$	0	0
$31$	−961.000	−0.179605
$31$	−961.000	−0.179605
$32$	0	0
$33$	14811.9	2.36769
$34$	0	0
$35$	−9833.56	−1.35688
$36$	0	0
$37$	6411.15	0.769896	0.384948	−	0.922938i	$-0.374219\pi$
$37$	6411.15	0.769896	0.384948	+	0.922938i	$0.374219\pi$
$38$	0	0
$39$	21890.8	2.30462
$40$	0	0
$41$	−17512.0	−1.62696	−0.813481	−	0.581592i	$-0.802430\pi$
$41$	−17512.0	−1.62696	−0.813481	+	0.581592i	$0.802430\pi$
$42$	0	0
$43$	2024.92	0.167008	0.0835040	−	0.996507i	$-0.473389\pi$
$43$	2024.92	0.167008	0.0835040	+	0.996507i	$0.473389\pi$
$44$	0	0
$45$	9573.58	0.704763
$46$	0	0
$47$	−4403.85	−0.290795	−0.145398	−	0.989373i	$-0.546446\pi$
$47$	−4403.85	−0.290795	−0.145398	+	0.989373i	$0.546446\pi$
$48$	0	0
$49$	28941.8	1.72201
$50$	0	0
$51$	−20560.3	−1.10689
$52$	0	0
$53$	−23788.9	−1.16328	−0.581640	−	0.813446i	$-0.697589\pi$
$53$	−23788.9	−1.16328	−0.581640	+	0.813446i	$0.697589\pi$
$54$	0	0
$55$	32057.5	1.42897
$56$	0	0
$57$	42108.6	1.71666
$58$	0	0
$59$	−24320.6	−0.909585	−0.454793	−	0.890597i	$-0.650287\pi$
$59$	−24320.6	−0.909585	−0.454793	+	0.890597i	$0.650287\pi$
$60$	0	0
$61$	37484.8	1.28982	0.644912	−	0.764257i	$-0.276894\pi$
$61$	37484.8	1.28982	0.644912	+	0.764257i	$0.276894\pi$
$62$	0	0
$63$	−44539.3	−1.41381
$64$	0	0
$65$	47378.4	1.39090
$66$	0	0
$67$	−1132.04	−0.0308087	−0.0154043	−	0.999881i	$-0.504904\pi$
$67$	−1132.04	−0.0308087	−0.0154043	+	0.999881i	$0.504904\pi$
$68$	0	0
$69$	39622.8	1.00190
$70$	0	0
$71$	60714.8	1.42938	0.714692	−	0.699440i	$-0.246567\pi$
$71$	60714.8	1.42938	0.714692	+	0.699440i	$0.246567\pi$
$72$	0	0
$73$	−59542.7	−1.30774	−0.653870	−	0.756607i	$-0.726856\pi$
$73$	−59542.7	−1.30774	−0.653870	+	0.756607i	$0.726856\pi$
$74$	0	0
$75$	−21482.5	−0.440993
$76$	0	0
$77$	−149142.	−2.86663
$78$	0	0
$79$	−50857.8	−0.916831	−0.458416	−	0.888738i	$-0.651583\pi$
$79$	−50857.8	−0.916831	−0.458416	+	0.888738i	$0.651583\pi$
$80$	0	0
$81$	−66288.3	−1.12260
$82$	0	0
$83$	−73658.0	−1.17361	−0.586806	−	0.809728i	$-0.699615\pi$
$83$	−73658.0	−1.17361	−0.586806	+	0.809728i	$0.699615\pi$
$84$	0	0
$85$	−44498.9	−0.668039
$86$	0	0
$87$	97201.1	1.37681
$88$	0	0
$89$	17839.3	0.238728	0.119364	−	0.992851i	$-0.461914\pi$
$89$	17839.3	0.238728	0.119364	+	0.992851i	$0.461914\pi$
$90$	0	0
$91$	−220419.	−2.79027
$92$	0	0
$93$	−20413.8	−0.244747
$94$	0	0
$95$	91136.0	1.03605
$96$	0	0
$97$	−175952.	−1.89873	−0.949367	−	0.314169i	$-0.898274\pi$
$97$	−175952.	−1.89873	−0.949367	+	0.314169i	$0.898274\pi$
$98$	0	0
$99$	145199.	1.48893
$100$	0	0
$101$	−62986.8	−0.614392	−0.307196	−	0.951646i	$-0.599391\pi$
$101$	−62986.8	−0.614392	−0.307196	+	0.951646i	$0.599391\pi$
$102$	0	0
$103$	59220.3	0.550019	0.275010	−	0.961441i	$-0.411319\pi$
$103$	59220.3	0.550019	0.275010	+	0.961441i	$0.411319\pi$
$104$	0	0
$105$	−208887.	−1.84901
$106$	0	0
$107$	211429.	1.78527	0.892636	−	0.450779i	$-0.148854\pi$
$107$	211429.	1.78527	0.892636	+	0.450779i	$0.148854\pi$
$108$	0	0
$109$	112763.	0.909078	0.454539	−	0.890727i	$-0.349804\pi$
$109$	112763.	0.909078	0.454539	+	0.890727i	$0.349804\pi$
$110$	0	0
$111$	136188.	1.04913
$112$	0	0
$113$	−197668.	−1.45626	−0.728132	−	0.685437i	$-0.759611\pi$
$113$	−197668.	−1.45626	−0.728132	+	0.685437i	$0.759611\pi$
$114$	0	0
$115$	85756.1	0.604673
$116$	0	0
$117$	214592.	1.44927
$118$	0	0
$119$	207023.	1.34014
$120$	0	0
$121$	325152.	2.01894
$122$	0	0
$123$	−371996.	−2.21705
$124$	0	0
$125$	−190166.	−1.08858
$126$	0	0
$127$	−105344.	−0.579564	−0.289782	−	0.957093i	$-0.593583\pi$
$127$	−105344.	−0.579564	−0.289782	+	0.957093i	$0.593583\pi$
$128$	0	0
$129$	43014.0	0.227581
$130$	0	0
$131$	−8959.36	−0.0456141	−0.0228070	−	0.999740i	$-0.507260\pi$
$131$	−8959.36	−0.0456141	−0.0228070	+	0.999740i	$0.507260\pi$
$132$	0	0
$133$	−423994.	−2.07840
$134$	0	0
$135$	−33952.0	−0.160336
$136$	0	0
$137$	−44039.3	−0.200465	−0.100233	−	0.994964i	$-0.531959\pi$
$137$	−44039.3	−0.200465	−0.100233	+	0.994964i	$0.531959\pi$
$138$	0	0
$139$	−68824.9	−0.302140	−0.151070	−	0.988523i	$-0.548272\pi$
$139$	−68824.9	−0.302140	−0.151070	+	0.988523i	$0.548272\pi$
$140$	0	0
$141$	−93547.8	−0.396265
$142$	0	0
$143$	718570.	2.93852
$144$	0	0
$145$	210373.	0.830941
$146$	0	0
$147$	614790.	2.34657
$148$	0	0
$149$	84588.2	0.312136	0.156068	−	0.987746i	$-0.450118\pi$
$149$	84588.2	0.312136	0.156068	+	0.987746i	$0.450118\pi$
$150$	0	0
$151$	−187069.	−0.667668	−0.333834	−	0.942632i	$-0.608342\pi$
$151$	−187069.	−0.667668	−0.333834	+	0.942632i	$0.608342\pi$
$152$	0	0
$153$	−201550.	−0.696071
$154$	0	0
$155$	−44181.9	−0.147712
$156$	0	0
$157$	−436580.	−1.41356	−0.706781	−	0.707432i	$-0.749854\pi$
$157$	−436580.	−1.41356	−0.706781	+	0.707432i	$0.749854\pi$
$158$	0	0
$159$	−505330.	−1.58519
$160$	0	0
$161$	−398964.	−1.21302
$162$	0	0
$163$	159297.	0.469610	0.234805	−	0.972042i	$-0.424555\pi$
$163$	159297.	0.469610	0.234805	+	0.972042i	$0.424555\pi$
$164$	0	0
$165$	680975.	1.94725
$166$	0	0
$167$	115211.	0.319671	0.159836	−	0.987144i	$-0.448904\pi$
$167$	115211.	0.319671	0.159836	+	0.987144i	$0.448904\pi$
$168$	0	0
$169$	690696.	1.86025
$170$	0	0
$171$	412784.	1.07952
$172$	0	0
$173$	−105083.	−0.266942	−0.133471	−	0.991053i	$-0.542612\pi$
$173$	−105083.	−0.266942	−0.133471	+	0.991053i	$0.542612\pi$
$174$	0	0
$175$	216309.	0.533923
$176$	0	0
$177$	−516624.	−1.23949
$178$	0	0
$179$	268771.	0.626974	0.313487	−	0.949592i	$-0.398503\pi$
$179$	268771.	0.626974	0.313487	+	0.949592i	$0.398503\pi$
$180$	0	0
$181$	−792331.	−1.79767	−0.898835	−	0.438287i	$-0.855585\pi$
$181$	−792331.	−1.79767	−0.898835	+	0.438287i	$0.855585\pi$
$182$	0	0
$183$	796263.	1.75764
$184$	0	0
$185$	294752.	0.633181
$186$	0	0
$187$	−674897.	−1.41135
$188$	0	0
$189$	157955.	0.321647
$190$	0	0
$191$	226162.	0.448576	0.224288	−	0.974523i	$-0.427994\pi$
$191$	226162.	0.448576	0.224288	+	0.974523i	$0.427994\pi$
$192$	0	0
$193$	380954.	0.736171	0.368086	−	0.929792i	$-0.380013\pi$
$193$	380954.	0.736171	0.368086	+	0.929792i	$0.380013\pi$
$194$	0	0
$195$	1.00643e6	1.89538
$196$	0	0
$197$	61464.5	0.112839	0.0564195	−	0.998407i	$-0.482032\pi$
$197$	61464.5	0.112839	0.0564195	+	0.998407i	$0.482032\pi$
$198$	0	0
$199$	44165.0	0.0790579	0.0395289	−	0.999218i	$-0.487414\pi$
$199$	44165.0	0.0790579	0.0395289	+	0.999218i	$0.487414\pi$
$200$	0	0
$201$	−24047.0	−0.0419828
$202$	0	0
$203$	−978723.	−1.66694
$204$	0	0
$205$	−805115.	−1.33805
$206$	0	0
$207$	388417.	0.630046
$208$	0	0
$209$	1.38222e6	2.18883
$210$	0	0
$211$	215052.	0.332536	0.166268	−	0.986081i	$-0.446828\pi$
$211$	215052.	0.332536	0.166268	+	0.986081i	$0.446828\pi$
$212$	0	0
$213$	1.28972e6	1.94781
$214$	0	0
$215$	93095.6	0.137351
$216$	0	0
$217$	205548.	0.296322
$218$	0	0
$219$	−1.26482e6	−1.78205
$220$	0	0
$221$	−997444.	−1.37375
$222$	0	0
$223$	−502559.	−0.676744	−0.338372	−	0.941012i	$-0.609876\pi$
$223$	−502559.	−0.676744	−0.338372	+	0.941012i	$0.609876\pi$
$224$	0	0
$225$	−210590.	−0.277320
$226$	0	0
$227$	−787569.	−1.01443	−0.507217	−	0.861818i	$-0.669326\pi$
$227$	−787569.	−1.01443	−0.507217	+	0.861818i	$0.669326\pi$
$228$	0	0
$229$	−195784.	−0.246710	−0.123355	−	0.992363i	$-0.539365\pi$
$229$	−195784.	−0.246710	−0.123355	+	0.992363i	$0.539365\pi$
$230$	0	0
$231$	−3.16811e6	−3.90634
$232$	0	0
$233$	310597.	0.374807	0.187404	−	0.982283i	$-0.439993\pi$
$233$	310597.	0.374807	0.187404	+	0.982283i	$0.439993\pi$
$234$	0	0
$235$	−202466.	−0.239157
$236$	0	0
$237$	−1.08034e6	−1.24936
$238$	0	0
$239$	388152.	0.439548	0.219774	−	0.975551i	$-0.429468\pi$
$239$	388152.	0.439548	0.219774	+	0.975551i	$0.429468\pi$
$240$	0	0
$241$	−1.33488e6	−1.48047	−0.740236	−	0.672347i	$-0.765286\pi$
$241$	−1.33488e6	−1.48047	−0.740236	+	0.672347i	$0.765286\pi$
$242$	0	0
$243$	−1.22866e6	−1.33480
$244$	0	0
$245$	1.33060e6	1.41622
$246$	0	0
$247$	2.04282e6	2.13053
$248$	0	0
$249$	−1.56466e6	−1.59927
$250$	0	0
$251$	1.18318e6	1.18540	0.592700	−	0.805423i	$-0.298062\pi$
$251$	1.18318e6	1.18540	0.592700	+	0.805423i	$0.298062\pi$
$252$	0	0
$253$	1.30063e6	1.27747
$254$	0	0
$255$	−945258.	−0.910332
$256$	0	0
$257$	153386.	0.144862	0.0724309	−	0.997373i	$-0.476924\pi$
$257$	153386.	0.144862	0.0724309	+	0.997373i	$0.476924\pi$
$258$	0	0
$259$	−1.37128e6	−1.27021
$260$	0	0
$261$	952847.	0.865809
$262$	0	0
$263$	497222.	0.443262	0.221631	−	0.975131i	$-0.428862\pi$
$263$	497222.	0.443262	0.221631	+	0.975131i	$0.428862\pi$
$264$	0	0
$265$	−1.09369e6	−0.956709
$266$	0	0
$267$	378949.	0.325314
$268$	0	0
$269$	−604703.	−0.509520	−0.254760	−	0.967004i	$-0.581996\pi$
$269$	−604703.	−0.509520	−0.254760	+	0.967004i	$0.581996\pi$
$270$	0	0
$271$	921615.	0.762301	0.381150	−	0.924513i	$-0.375528\pi$
$271$	921615.	0.762301	0.381150	+	0.924513i	$0.375528\pi$
$272$	0	0
$273$	−4.68221e6	−3.80229
$274$	0	0
$275$	−705168.	−0.562291
$276$	0	0
$277$	1.76901e6	1.38526	0.692630	−	0.721293i	$-0.256452\pi$
$277$	1.76901e6	1.38526	0.692630	+	0.721293i	$0.256452\pi$
$278$	0	0
$279$	−200114.	−0.153910
$280$	0	0
$281$	−1.31928e6	−0.996717	−0.498359	−	0.866971i	$-0.666064\pi$
$281$	−1.31928e6	−0.996717	−0.498359	+	0.866971i	$0.666064\pi$
$282$	0	0
$283$	1.97349e6	1.46476	0.732382	−	0.680893i	$-0.238408\pi$
$283$	1.97349e6	1.46476	0.732382	+	0.680893i	$0.238408\pi$
$284$	0	0
$285$	1.93594e6	1.41182
$286$	0	0
$287$	3.74565e6	2.68424
$288$	0	0
$289$	−483036.	−0.340200
$290$	0	0
$291$	−3.73762e6	−2.58739
$292$	0	0
$293$	2.19163e6	1.49142	0.745709	−	0.666272i	$-0.232111\pi$
$293$	2.19163e6	1.49142	0.745709	+	0.666272i	$0.232111\pi$
$294$	0	0
$295$	−1.11813e6	−0.748065
$296$	0	0
$297$	−514936.	−0.338737
$298$	0	0
$299$	1.92223e6	1.24344
$300$	0	0
$301$	−433110.	−0.275538
$302$	0	0
$303$	−1.33798e6	−0.837229
$304$	0	0
$305$	1.72336e6	1.06078
$306$	0	0
$307$	2.61576e6	1.58399	0.791993	−	0.610530i	$-0.209044\pi$
$307$	2.61576e6	1.58399	0.791993	+	0.610530i	$0.209044\pi$
$308$	0	0
$309$	1.25798e6	0.749508
$310$	0	0
$311$	1.07374e6	0.629501	0.314751	−	0.949174i	$-0.398079\pi$
$311$	1.07374e6	0.629501	0.314751	+	0.949174i	$0.398079\pi$
$312$	0	0
$313$	−2.37060e6	−1.36772	−0.683861	−	0.729612i	$-0.739700\pi$
$313$	−2.37060e6	−1.36772	−0.683861	+	0.729612i	$0.739700\pi$
$314$	0	0
$315$	−2.04769e6	−1.16275
$316$	0	0
$317$	308279.	0.172304	0.0861520	−	0.996282i	$-0.472543\pi$
$317$	308279.	0.172304	0.0861520	+	0.996282i	$0.472543\pi$
$318$	0	0
$319$	3.19065e6	1.75550
$320$	0	0
$321$	4.49123e6	2.43278
$322$	0	0
$323$	−1.91866e6	−1.02327
$324$	0	0
$325$	−1.04218e6	−0.547313
$326$	0	0
$327$	2.39535e6	1.23879
$328$	0	0
$329$	941937.	0.479769
$330$	0	0
$331$	−1.74749e6	−0.876688	−0.438344	−	0.898807i	$-0.644435\pi$
$331$	−1.74749e6	−0.876688	−0.438344	+	0.898807i	$0.644435\pi$
$332$	0	0
$333$	1.33503e6	0.659750
$334$	0	0
$335$	−52045.2	−0.0253378
$336$	0	0
$337$	−2.01088e6	−0.964519	−0.482260	−	0.876028i	$-0.660184\pi$
$337$	−2.01088e6	−0.964519	−0.482260	+	0.876028i	$0.660184\pi$
$338$	0	0
$339$	−4.19892e6	−1.98444
$340$	0	0
$341$	−670089.	−0.312066
$342$	0	0
$343$	−2.59551e6	−1.19121
$344$	0	0
$345$	1.82166e6	0.823984
$346$	0	0
$347$	202823.	0.0904259	0.0452130	−	0.998977i	$-0.485603\pi$
$347$	202823.	0.0904259	0.0452130	+	0.998977i	$0.485603\pi$
$348$	0	0
$349$	−308259.	−0.135473	−0.0677364	−	0.997703i	$-0.521578\pi$
$349$	−308259.	−0.135473	−0.0677364	+	0.997703i	$0.521578\pi$
$350$	0	0
$351$	−761035.	−0.329713
$352$	0	0
$353$	3.99838e6	1.70784	0.853921	−	0.520403i	$-0.174218\pi$
$353$	3.99838e6	1.70784	0.853921	+	0.520403i	$0.174218\pi$
$354$	0	0
$355$	2.79136e6	1.17556
$356$	0	0
$357$	4.39764e6	1.82620
$358$	0	0
$359$	−2.32769e6	−0.953209	−0.476604	−	0.879118i	$-0.658133\pi$
$359$	−2.32769e6	−0.953209	−0.476604	+	0.879118i	$0.658133\pi$
$360$	0	0
$361$	1.45341e6	0.586977
$362$	0	0
$363$	6.90698e6	2.75120
$364$	0	0
$365$	−2.73747e6	−1.07552
$366$	0	0
$367$	−289191.	−0.112078	−0.0560388	−	0.998429i	$-0.517847\pi$
$367$	−289191.	−0.112078	−0.0560388	+	0.998429i	$0.517847\pi$
$368$	0	0
$369$	−3.64662e6	−1.39420
$370$	0	0
$371$	5.08819e6	1.91924
$372$	0	0
$373$	−2.04782e6	−0.762112	−0.381056	−	0.924552i	$-0.624440\pi$
$373$	−2.04782e6	−0.762112	−0.381056	+	0.924552i	$0.624440\pi$
$374$	0	0
$375$	−4.03957e6	−1.48340
$376$	0	0
$377$	4.71552e6	1.70874
$378$	0	0
$379$	2.20637e6	0.789007	0.394503	−	0.918894i	$-0.370917\pi$
$379$	2.20637e6	0.789007	0.394503	+	0.918894i	$0.370917\pi$
$380$	0	0
$381$	−2.23775e6	−0.789768
$382$	0	0
$383$	518047.	0.180457	0.0902283	−	0.995921i	$-0.471240\pi$
$383$	518047.	0.180457	0.0902283	+	0.995921i	$0.471240\pi$
$384$	0	0
$385$	−6.85677e6	−2.35759
$386$	0	0
$387$	421659.	0.143115
$388$	0	0
$389$	1.26381e6	0.423457	0.211728	−	0.977329i	$-0.432091\pi$
$389$	1.26381e6	0.423457	0.211728	+	0.977329i	$0.432091\pi$
$390$	0	0
$391$	−1.80540e6	−0.597215
$392$	0	0
$393$	−190317.	−0.0621580
$394$	0	0
$395$	−2.33818e6	−0.754024
$396$	0	0
$397$	−10199.8	−0.00324799	−0.00162400	−	0.999999i	$-0.500517\pi$
$397$	−10199.8	−0.00324799	−0.00162400	+	0.999999i	$0.500517\pi$
$398$	0	0
$399$	−9.00659e6	−2.83223
$400$	0	0
$401$	−2.53591e6	−0.787540	−0.393770	−	0.919209i	$-0.628829\pi$
$401$	−2.53591e6	−0.787540	−0.393770	+	0.919209i	$0.628829\pi$
$402$	0	0
$403$	−990338.	−0.303753
$404$	0	0
$405$	−3.04760e6	−0.923252
$406$	0	0
$407$	4.47039e6	1.33770
$408$	0	0
$409$	3.40783e6	1.00733	0.503663	−	0.863900i	$-0.331985\pi$
$409$	3.40783e6	1.00733	0.503663	+	0.863900i	$0.331985\pi$
$410$	0	0
$411$	−935496.	−0.273173
$412$	0	0
$413$	5.20192e6	1.50068
$414$	0	0
$415$	−3.38642e6	−0.965206
$416$	0	0
$417$	−1.46200e6	−0.411725
$418$	0	0
$419$	2.23196e6	0.621086	0.310543	−	0.950559i	$-0.399489\pi$
$419$	2.23196e6	0.621086	0.310543	+	0.950559i	$0.399489\pi$
$420$	0	0
$421$	3.61151e6	0.993079	0.496540	−	0.868014i	$-0.334604\pi$
$421$	3.61151e6	0.993079	0.496540	+	0.868014i	$0.334604\pi$
$422$	0	0
$423$	−917035.	−0.249192
$424$	0	0
$425$	978841.	0.262869
$426$	0	0
$427$	−8.01761e6	−2.12802
$428$	0	0
$429$	1.52641e7	4.00431
$430$	0	0
$431$	5.25369e6	1.36230	0.681148	−	0.732146i	$-0.261481\pi$
$431$	5.25369e6	1.36230	0.681148	+	0.732146i	$0.261481\pi$
$432$	0	0
$433$	2.51678e6	0.645098	0.322549	−	0.946553i	$-0.395460\pi$
$433$	2.51678e6	0.645098	0.322549	+	0.946553i	$0.395460\pi$
$434$	0	0
$435$	4.46881e6	1.13232
$436$	0	0
$437$	3.69755e6	0.926211
$438$	0	0
$439$	5.78321e6	1.43221	0.716107	−	0.697991i	$-0.245922\pi$
$439$	5.78321e6	1.43221	0.716107	+	0.697991i	$0.245922\pi$
$440$	0	0
$441$	6.02670e6	1.47565
$442$	0	0
$443$	−5.99581e6	−1.45157	−0.725786	−	0.687920i	$-0.758524\pi$
$443$	−5.99581e6	−1.45157	−0.725786	+	0.687920i	$0.758524\pi$
$444$	0	0
$445$	820162.	0.196336
$446$	0	0
$447$	1.79685e6	0.425346
$448$	0	0
$449$	−5.03538e6	−1.17873	−0.589367	−	0.807865i	$-0.700623\pi$
$449$	−5.03538e6	−1.17873	−0.589367	+	0.807865i	$0.700623\pi$
$450$	0	0
$451$	−1.22109e7	−2.82686
$452$	0	0
$453$	−3.97378e6	−0.909827
$454$	0	0
$455$	−1.01338e7	−2.29479
$456$	0	0
$457$	−7.40557e6	−1.65870	−0.829350	−	0.558729i	$-0.811289\pi$
$457$	−7.40557e6	−1.65870	−0.829350	+	0.558729i	$0.811289\pi$
$458$	0	0
$459$	714780.	0.158358
$460$	0	0
$461$	−876502.	−0.192088	−0.0960440	−	0.995377i	$-0.530619\pi$
$461$	−876502.	−0.192088	−0.0960440	+	0.995377i	$0.530619\pi$
$462$	0	0
$463$	2.90675e6	0.630167	0.315084	−	0.949064i	$-0.397967\pi$
$463$	2.90675e6	0.630167	0.315084	+	0.949064i	$0.397967\pi$
$464$	0	0
$465$	−938524.	−0.201286
$466$	0	0
$467$	2.21723e6	0.470455	0.235228	−	0.971940i	$-0.424416\pi$
$467$	2.21723e6	0.470455	0.235228	+	0.971940i	$0.424416\pi$
$468$	0	0
$469$	242131.	0.0508297
$470$	0	0
$471$	−9.27397e6	−1.92625
$472$	0	0
$473$	1.41194e6	0.290178
$474$	0	0
$475$	−2.00472e6	−0.407680
$476$	0	0
$477$	−4.95367e6	−0.996854
$478$	0	0
$479$	−7.67280e6	−1.52797	−0.763985	−	0.645234i	$-0.776760\pi$
$479$	−7.67280e6	−1.52797	−0.763985	+	0.645234i	$0.776760\pi$
$480$	0	0
$481$	6.60688e6	1.30207
$482$	0	0
$483$	−8.47492e6	−1.65298
$484$	0	0
$485$	−8.08936e6	−1.56156
$486$	0	0
$487$	−7.95009e6	−1.51897	−0.759486	−	0.650524i	$-0.774549\pi$
$487$	−7.95009e6	−1.51897	−0.759486	+	0.650524i	$0.774549\pi$
$488$	0	0
$489$	3.38383e6	0.639935
$490$	0	0
$491$	12422.2	0.00232538	0.00116269	−	0.999999i	$-0.499630\pi$
$491$	12422.2	0.00232538	0.00116269	+	0.999999i	$0.499630\pi$
$492$	0	0
$493$	−4.42892e6	−0.820693
$494$	0	0
$495$	6.67549e6	1.22453
$496$	0	0
$497$	−1.29863e7	−2.35827
$498$	0	0
$499$	−841551.	−0.151297	−0.0756483	−	0.997135i	$-0.524103\pi$
$499$	−841551.	−0.151297	−0.0756483	+	0.997135i	$0.524103\pi$
$500$	0	0
$501$	2.44735e6	0.435614
$502$	0	0
$503$	517545.	0.0912070	0.0456035	−	0.998960i	$-0.485479\pi$
$503$	517545.	0.0912070	0.0456035	+	0.998960i	$0.485479\pi$
$504$	0	0
$505$	−2.89581e6	−0.505291
$506$	0	0
$507$	1.46720e7	2.53495
$508$	0	0
$509$	1.74010e6	0.297701	0.148850	−	0.988860i	$-0.452443\pi$
$509$	1.74010e6	0.297701	0.148850	+	0.988860i	$0.452443\pi$
$510$	0	0
$511$	1.27356e7	2.15758
$512$	0	0
$513$	−1.46391e6	−0.245595
$514$	0	0
$515$	2.72265e6	0.452349
$516$	0	0
$517$	−3.07073e6	−0.505260
$518$	0	0
$519$	−2.23221e6	−0.363761
$520$	0	0
$521$	−2.90754e6	−0.469280	−0.234640	−	0.972082i	$-0.575391\pi$
$521$	−2.90754e6	−0.469280	−0.234640	+	0.972082i	$0.575391\pi$
$522$	0	0
$523$	4.55970e6	0.728924	0.364462	−	0.931218i	$-0.381253\pi$
$523$	4.55970e6	0.728924	0.364462	+	0.931218i	$0.381253\pi$
$524$	0	0
$525$	4.59489e6	0.727573
$526$	0	0
$527$	930147.	0.145890
$528$	0	0
$529$	−2.95707e6	−0.459433
$530$	0	0
$531$	−5.06439e6	−0.779454
$532$	0	0
$533$	−1.80467e7	−2.75156
$534$	0	0
$535$	9.72041e6	1.46825
$536$	0	0
$537$	5.70931e6	0.854374
$538$	0	0
$539$	2.01806e7	2.99201
$540$	0	0
$541$	8.26252e6	1.21372	0.606861	−	0.794808i	$-0.292429\pi$
$541$	8.26252e6	1.21372	0.606861	+	0.794808i	$0.292429\pi$
$542$	0	0
$543$	−1.68309e7	−2.44967
$544$	0	0
$545$	5.18428e6	0.747648
$546$	0	0
$547$	−6.79180e6	−0.970546	−0.485273	−	0.874363i	$-0.661280\pi$
$547$	−6.79180e6	−0.970546	−0.485273	+	0.874363i	$0.661280\pi$
$548$	0	0
$549$	7.80565e6	1.10529
$550$	0	0
$551$	9.07066e6	1.27280
$552$	0	0
$553$	1.08780e7	1.51264
$554$	0	0
$555$	6.26121e6	0.862831
$556$	0	0
$557$	8.56598e6	1.16987	0.584937	−	0.811079i	$-0.301119\pi$
$557$	8.56598e6	1.16987	0.584937	+	0.811079i	$0.301119\pi$
$558$	0	0
$559$	2.08674e6	0.282448
$560$	0	0
$561$	−1.43364e7	−1.92323
$562$	0	0
$563$	7.21769e6	0.959681	0.479841	−	0.877356i	$-0.340694\pi$
$563$	7.21769e6	0.959681	0.479841	+	0.877356i	$0.340694\pi$
$564$	0	0
$565$	−9.08776e6	−1.19767
$566$	0	0
$567$	1.41784e7	1.85212
$568$	0	0
$569$	−7.96941e6	−1.03192	−0.515960	−	0.856613i	$-0.672565\pi$
$569$	−7.96941e6	−1.03192	−0.515960	+	0.856613i	$0.672565\pi$
$570$	0	0
$571$	7.25439e6	0.931131	0.465565	−	0.885013i	$-0.345851\pi$
$571$	7.25439e6	0.931131	0.465565	+	0.885013i	$0.345851\pi$
$572$	0	0
$573$	4.80419e6	0.611271
$574$	0	0
$575$	−1.88638e6	−0.237935
$576$	0	0
$577$	−1.36975e7	−1.71278	−0.856389	−	0.516330i	$-0.827298\pi$
$577$	−1.36975e7	−1.71278	−0.856389	+	0.516330i	$0.827298\pi$
$578$	0	0
$579$	8.09233e6	1.00318
$580$	0	0
$581$	1.57547e7	1.93628
$582$	0	0
$583$	−1.65876e7	−2.02121
$584$	0	0
$585$	9.86585e6	1.19191
$586$	0	0
$587$	−5.71998e6	−0.685172	−0.342586	−	0.939486i	$-0.611303\pi$
$587$	−5.71998e6	−0.685172	−0.342586	+	0.939486i	$0.611303\pi$
$588$	0	0
$589$	−1.90499e6	−0.226258
$590$	0	0
$591$	1.30565e6	0.153765
$592$	0	0
$593$	5.70095e6	0.665749	0.332875	−	0.942971i	$-0.391981\pi$
$593$	5.70095e6	0.665749	0.332875	+	0.942971i	$0.391981\pi$
$594$	0	0
$595$	9.51785e6	1.10216
$596$	0	0
$597$	938165.	0.107732
$598$	0	0
$599$	9.98531e6	1.13709	0.568544	−	0.822653i	$-0.307507\pi$
$599$	9.98531e6	1.13709	0.568544	+	0.822653i	$0.307507\pi$
$600$	0	0
$601$	5.42406e6	0.612546	0.306273	−	0.951944i	$-0.400918\pi$
$601$	5.42406e6	0.612546	0.306273	+	0.951944i	$0.400918\pi$
$602$	0	0
$603$	−235729.	−0.0264010
$604$	0	0
$605$	1.49488e7	1.66043
$606$	0	0
$607$	−1.76417e7	−1.94343	−0.971715	−	0.236159i	$-0.924111\pi$
$607$	−1.76417e7	−1.94343	−0.971715	+	0.236159i	$0.924111\pi$
$608$	0	0
$609$	−2.07903e7	−2.27153
$610$	0	0
$611$	−4.53829e6	−0.491801
$612$	0	0
$613$	1.57020e6	0.168774	0.0843868	−	0.996433i	$-0.473107\pi$
$613$	1.57020e6	0.168774	0.0843868	+	0.996433i	$0.473107\pi$
$614$	0	0
$615$	−1.71025e7	−1.82336
$616$	0	0
$617$	−438619.	−0.0463847	−0.0231923	−	0.999731i	$-0.507383\pi$
$617$	−438619.	−0.0463847	−0.0231923	+	0.999731i	$0.507383\pi$
$618$	0	0
$619$	7.43471e6	0.779898	0.389949	−	0.920837i	$-0.372493\pi$
$619$	7.43471e6	0.779898	0.389949	+	0.920837i	$0.372493\pi$
$620$	0	0
$621$	−1.37749e6	−0.143337
$622$	0	0
$623$	−3.81565e6	−0.393866
$624$	0	0
$625$	−5.58254e6	−0.571652
$626$	0	0
$627$	2.93616e7	2.98271
$628$	0	0
$629$	−6.20532e6	−0.625371
$630$	0	0
$631$	−1.91820e7	−1.91788	−0.958940	−	0.283608i	$-0.908468\pi$
$631$	−1.91820e7	−1.91788	−0.958940	+	0.283608i	$0.908468\pi$
$632$	0	0
$633$	4.56821e6	0.453144
$634$	0	0
$635$	−4.84319e6	−0.476647
$636$	0	0
$637$	2.98254e7	2.91231
$638$	0	0
$639$	1.26429e7	1.22489
$640$	0	0
$641$	4.53331e6	0.435783	0.217891	−	0.975973i	$-0.430082\pi$
$641$	4.53331e6	0.435783	0.217891	+	0.975973i	$0.430082\pi$
$642$	0	0
$643$	3.58220e6	0.341682	0.170841	−	0.985299i	$-0.445352\pi$
$643$	3.58220e6	0.341682	0.170841	+	0.985299i	$0.445352\pi$
$644$	0	0
$645$	1.97756e6	0.187168
$646$	0	0
$647$	1.04781e6	0.0984064	0.0492032	−	0.998789i	$-0.484332\pi$
$647$	1.04781e6	0.0984064	0.0492032	+	0.998789i	$0.484332\pi$
$648$	0	0
$649$	−1.69583e7	−1.58041
$650$	0	0
$651$	4.36631e6	0.403796
$652$	0	0
$653$	944156.	0.0866485	0.0433242	−	0.999061i	$-0.486205\pi$
$653$	944156.	0.0866485	0.0433242	+	0.999061i	$0.486205\pi$
$654$	0	0
$655$	−411906.	−0.0375141
$656$	0	0
$657$	−1.23989e7	−1.12065
$658$	0	0
$659$	−5.36498e6	−0.481232	−0.240616	−	0.970620i	$-0.577349\pi$
$659$	−5.36498e6	−0.481232	−0.240616	+	0.970620i	$0.577349\pi$
$660$	0	0
$661$	−5.94506e6	−0.529240	−0.264620	−	0.964353i	$-0.585247\pi$
$661$	−5.94506e6	−0.529240	−0.264620	+	0.964353i	$0.585247\pi$
$662$	0	0
$663$	−2.11880e7	−1.87200
$664$	0	0
$665$	−1.94931e7	−1.70933
$666$	0	0
$667$	8.53520e6	0.742847
$668$	0	0
$669$	−1.06755e7	−0.922195
$670$	0	0
$671$	2.61375e7	2.24108
$672$	0	0
$673$	441299.	0.0375574	0.0187787	−	0.999824i	$-0.494022\pi$
$673$	441299.	0.0375574	0.0187787	+	0.999824i	$0.494022\pi$
$674$	0	0
$675$	746841.	0.0630912
$676$	0	0
$677$	−1.33010e7	−1.11536	−0.557679	−	0.830057i	$-0.688308\pi$
$677$	−1.33010e7	−1.11536	−0.557679	+	0.830057i	$0.688308\pi$
$678$	0	0
$679$	3.76343e7	3.13263
$680$	0	0
$681$	−1.67298e7	−1.38236
$682$	0	0
$683$	−8.85105e6	−0.726011	−0.363005	−	0.931787i	$-0.618249\pi$
$683$	−8.85105e6	−0.726011	−0.363005	+	0.931787i	$0.618249\pi$
$684$	0	0
$685$	−2.02470e6	−0.164867
$686$	0	0
$687$	−4.15889e6	−0.336191
$688$	0	0
$689$	−2.45151e7	−1.96737
$690$	0	0
$691$	−1.59156e7	−1.26803	−0.634013	−	0.773323i	$-0.718593\pi$
$691$	−1.59156e7	−1.26803	−0.634013	+	0.773323i	$0.718593\pi$
$692$	0	0
$693$	−3.10565e7	−2.45651
$694$	0	0
$695$	−3.16422e6	−0.248487
$696$	0	0
$697$	1.69498e7	1.32155
$698$	0	0
$699$	6.59780e6	0.510748
$700$	0	0
$701$	1.28734e7	0.989460	0.494730	−	0.869047i	$-0.335267\pi$
$701$	1.28734e7	0.989460	0.494730	+	0.869047i	$0.335267\pi$
$702$	0	0
$703$	1.27088e7	0.969878
$704$	0	0
$705$	−4.30085e6	−0.325898
$706$	0	0
$707$	1.34722e7	1.01366
$708$	0	0
$709$	2.19038e7	1.63645	0.818226	−	0.574897i	$-0.194958\pi$
$709$	2.19038e7	1.63645	0.818226	+	0.574897i	$0.194958\pi$
$710$	0	0
$711$	−1.05904e7	−0.785664
$712$	0	0
$713$	−1.79253e6	−0.132052
$714$	0	0
$715$	3.30362e7	2.41671
$716$	0	0
$717$	8.24523e6	0.598970
$718$	0	0
$719$	−2.57175e7	−1.85527	−0.927635	−	0.373488i	$-0.878162\pi$
$719$	−2.57175e7	−1.85527	−0.927635	+	0.373488i	$0.878162\pi$
$720$	0	0
$721$	−1.26666e7	−0.907450
$722$	0	0
$723$	−2.83560e7	−2.01743
$724$	0	0
$725$	−4.62758e6	−0.326970
$726$	0	0
$727$	1.82322e7	1.27939	0.639693	−	0.768630i	$-0.279061\pi$
$727$	1.82322e7	1.27939	0.639693	+	0.768630i	$0.279061\pi$
$728$	0	0
$729$	−9.99155e6	−0.696328
$730$	0	0
$731$	−1.95991e6	−0.135657
$732$	0	0
$733$	−1.65047e7	−1.13461	−0.567307	−	0.823506i	$-0.692015\pi$
$733$	−1.65047e7	−1.13461	−0.567307	+	0.823506i	$0.692015\pi$
$734$	0	0
$735$	2.82649e7	1.92988
$736$	0	0
$737$	−789349.	−0.0535304
$738$	0	0
$739$	1.74204e6	0.117340	0.0586700	−	0.998277i	$-0.481314\pi$
$739$	1.74204e6	0.117340	0.0586700	+	0.998277i	$0.481314\pi$
$740$	0	0
$741$	4.33941e7	2.90326
$742$	0	0
$743$	5.29258e6	0.351719	0.175859	−	0.984415i	$-0.443730\pi$
$743$	5.29258e6	0.351719	0.175859	+	0.984415i	$0.443730\pi$
$744$	0	0
$745$	3.88893e6	0.256708
$746$	0	0
$747$	−1.53382e7	−1.00571
$748$	0	0
$749$	−4.52224e7	−2.94543
$750$	0	0
$751$	−9.30819e6	−0.602234	−0.301117	−	0.953587i	$-0.597359\pi$
$751$	−9.30819e6	−0.602234	−0.301117	+	0.953587i	$0.597359\pi$
$752$	0	0
$753$	2.51334e7	1.61534
$754$	0	0
$755$	−8.60050e6	−0.549106
$756$	0	0
$757$	−1.13574e7	−0.720341	−0.360171	−	0.932886i	$-0.617282\pi$
$757$	−1.13574e7	−0.720341	−0.360171	+	0.932886i	$0.617282\pi$
$758$	0	0
$759$	2.76283e7	1.74081
$760$	0	0
$761$	1.96551e7	1.23031	0.615153	−	0.788408i	$-0.289094\pi$
$761$	1.96551e7	1.23031	0.615153	+	0.788408i	$0.289094\pi$
$762$	0	0
$763$	−2.41189e7	−1.49984
$764$	0	0
$765$	−9.26622e6	−0.572465
$766$	0	0
$767$	−2.50630e7	−1.53831
$768$	0	0
$769$	1.93050e6	0.117721	0.0588605	−	0.998266i	$-0.481253\pi$
$769$	1.93050e6	0.117721	0.0588605	+	0.998266i	$0.481253\pi$
$770$	0	0
$771$	3.25828e6	0.197402
$772$	0	0
$773$	4.12587e6	0.248351	0.124176	−	0.992260i	$-0.460371\pi$
$773$	4.12587e6	0.248351	0.124176	+	0.992260i	$0.460371\pi$
$774$	0	0
$775$	971868.	0.0581237
$776$	0	0
$777$	−2.91291e7	−1.73091
$778$	0	0
$779$	−3.47141e7	−2.04957
$780$	0	0
$781$	4.23354e7	2.48357
$782$	0	0
$783$	−3.37920e6	−0.196974
$784$	0	0
$785$	−2.00717e7	−1.16255
$786$	0	0
$787$	−2.18000e7	−1.25464	−0.627322	−	0.778760i	$-0.715849\pi$
$787$	−2.18000e7	−1.25464	−0.627322	+	0.778760i	$0.715849\pi$
$788$	0	0
$789$	1.05621e7	0.604031
$790$	0	0
$791$	4.22791e7	2.40262
$792$	0	0
$793$	3.86292e7	2.18139
$794$	0	0
$795$	−2.32325e7	−1.30370
$796$	0	0
$797$	−3.35815e7	−1.87264	−0.936321	−	0.351144i	$-0.885793\pi$
$797$	−3.35815e7	−1.87264	−0.936321	+	0.351144i	$0.885793\pi$
$798$	0	0
$799$	4.26246e6	0.236207
$800$	0	0
$801$	3.71477e6	0.204574
$802$	0	0
$803$	−4.15181e7	−2.27221
$804$	0	0
$805$	−1.83423e7	−0.997620
$806$	0	0
$807$	−1.28453e7	−0.694320
$808$	0	0
$809$	−4.58991e6	−0.246566	−0.123283	−	0.992372i	$-0.539342\pi$
$809$	−4.58991e6	−0.246566	−0.123283	+	0.992372i	$0.539342\pi$
$810$	0	0
$811$	5.43841e6	0.290349	0.145174	−	0.989406i	$-0.453626\pi$
$811$	5.43841e6	0.290349	0.145174	+	0.989406i	$0.453626\pi$
$812$	0	0
$813$	1.95772e7	1.03878
$814$	0	0
$815$	7.32365e6	0.386219
$816$	0	0
$817$	4.01400e6	0.210389
$818$	0	0
$819$	−4.58990e7	−2.39108
$820$	0	0
$821$	2.70788e7	1.40208	0.701039	−	0.713123i	$-0.252720\pi$
$821$	2.70788e7	1.40208	0.701039	+	0.713123i	$0.252720\pi$
$822$	0	0
$823$	2.20562e6	0.113509	0.0567547	−	0.998388i	$-0.481925\pi$
$823$	2.20562e6	0.113509	0.0567547	+	0.998388i	$0.481925\pi$
$824$	0	0
$825$	−1.49794e7	−0.766230
$826$	0	0
$827$	2.46462e7	1.25310	0.626551	−	0.779380i	$-0.284466\pi$
$827$	2.46462e7	1.25310	0.626551	+	0.779380i	$0.284466\pi$
$828$	0	0
$829$	1.43578e7	0.725606	0.362803	−	0.931866i	$-0.381820\pi$
$829$	1.43578e7	0.725606	0.362803	+	0.931866i	$0.381820\pi$
$830$	0	0
$831$	3.75779e7	1.88768
$832$	0	0
$833$	−2.80126e7	−1.39875
$834$	0	0
$835$	5.29682e6	0.262905
$836$	0	0
$837$	709688.	0.0350150
$838$	0	0
$839$	1.72991e7	0.848434	0.424217	−	0.905561i	$-0.360549\pi$
$839$	1.72991e7	0.848434	0.424217	+	0.905561i	$0.360549\pi$
$840$	0	0
$841$	427052.	0.0208205
$842$	0	0
$843$	−2.80246e7	−1.35822
$844$	0	0
$845$	3.17547e7	1.52991
$846$	0	0
$847$	−6.95468e7	−3.33095
$848$	0	0
$849$	4.19214e7	1.99603
$850$	0	0
$851$	1.19586e7	0.566052
$852$	0	0
$853$	−1.28723e7	−0.605736	−0.302868	−	0.953033i	$-0.597944\pi$
$853$	−1.28723e7	−0.605736	−0.302868	+	0.953033i	$0.597944\pi$
$854$	0	0
$855$	1.89777e7	0.887827
$856$	0	0
$857$	1.06968e7	0.497508	0.248754	−	0.968567i	$-0.419979\pi$
$857$	1.06968e7	0.497508	0.248754	+	0.968567i	$0.419979\pi$
$858$	0	0
$859$	−9.50379e6	−0.439455	−0.219727	−	0.975561i	$-0.570517\pi$
$859$	−9.50379e6	−0.439455	−0.219727	+	0.975561i	$0.570517\pi$
$860$	0	0
$861$	7.95661e7	3.65780
$862$	0	0
$863$	−4.21540e6	−0.192669	−0.0963345	−	0.995349i	$-0.530712\pi$
$863$	−4.21540e6	−0.192669	−0.0963345	+	0.995349i	$0.530712\pi$
$864$	0	0
$865$	−4.83119e6	−0.219540
$866$	0	0
$867$	−1.02608e7	−0.463589
$868$	0	0
$869$	−3.54622e7	−1.59300
$870$	0	0
$871$	−1.16660e6	−0.0521044
$872$	0	0
$873$	−3.66393e7	−1.62709
$874$	0	0
$875$	4.06746e7	1.79599
$876$	0	0
$877$	2.28925e7	1.00507	0.502533	−	0.864558i	$-0.332402\pi$
$877$	2.28925e7	1.00507	0.502533	+	0.864558i	$0.332402\pi$
$878$	0	0
$879$	4.65553e7	2.03234
$880$	0	0
$881$	3.30374e7	1.43406	0.717028	−	0.697045i	$-0.245502\pi$
$881$	3.30374e7	1.43406	0.717028	+	0.697045i	$0.245502\pi$
$882$	0	0
$883$	−1.45560e7	−0.628259	−0.314130	−	0.949380i	$-0.601713\pi$
$883$	−1.45560e7	−0.628259	−0.314130	+	0.949380i	$0.601713\pi$
$884$	0	0
$885$	−2.37517e7	−1.01938
$886$	0	0
$887$	2.27263e7	0.969882	0.484941	−	0.874547i	$-0.338841\pi$
$887$	2.27263e7	0.969882	0.484941	+	0.874547i	$0.338841\pi$
$888$	0	0
$889$	2.25320e7	0.956194
$890$	0	0
$891$	−4.62217e7	−1.95053
$892$	0	0
$893$	−8.72974e6	−0.366330
$894$	0	0
$895$	1.23567e7	0.515639
$896$	0	0
$897$	4.08325e7	1.69443
$898$	0	0
$899$	−4.39737e6	−0.181465
$900$	0	0
$901$	2.30251e7	0.944909
$902$	0	0
$903$	−9.20025e6	−0.375474
$904$	0	0
$905$	−3.64273e7	−1.47845
$906$	0	0
$907$	−1.86351e7	−0.752166	−0.376083	−	0.926586i	$-0.622729\pi$
$907$	−1.86351e7	−0.752166	−0.376083	+	0.926586i	$0.622729\pi$
$908$	0	0
$909$	−1.31160e7	−0.526494
$910$	0	0
$911$	−8.29370e6	−0.331095	−0.165547	−	0.986202i	$-0.552939\pi$
$911$	−8.29370e6	−0.331095	−0.165547	+	0.986202i	$0.552939\pi$
$912$	0	0
$913$	−5.13604e7	−2.03916
$914$	0	0
$915$	3.66081e7	1.44552
$916$	0	0
$917$	1.91632e6	0.0752564
$918$	0	0
$919$	−6.17971e6	−0.241368	−0.120684	−	0.992691i	$-0.538509\pi$
$919$	−6.17971e6	−0.241368	−0.120684	+	0.992691i	$0.538509\pi$
$920$	0	0
$921$	5.55647e7	2.15849
$922$	0	0
$923$	6.25683e7	2.41741
$924$	0	0
$925$	−6.48366e6	−0.249153
$926$	0	0
$927$	1.23317e7	0.471330
$928$	0	0
$929$	473623.	0.0180050	0.00900252	−	0.999959i	$-0.497134\pi$
$929$	473623.	0.0180050	0.00900252	+	0.999959i	$0.497134\pi$
$930$	0	0
$931$	5.73713e7	2.16931
$932$	0	0
$933$	2.28086e7	0.857817
$934$	0	0
$935$	−3.10283e7	−1.16072
$936$	0	0
$937$	4.48153e7	1.66754	0.833771	−	0.552110i	$-0.186177\pi$
$937$	4.48153e7	1.66754	0.833771	+	0.552110i	$0.186177\pi$
$938$	0	0
$939$	−5.03570e7	−1.86379
$940$	0	0
$941$	2.20110e7	0.810335	0.405168	−	0.914242i	$-0.367213\pi$
$941$	2.20110e7	0.810335	0.405168	+	0.914242i	$0.367213\pi$
$942$	0	0
$943$	−3.26649e7	−1.19620
$944$	0	0
$945$	7.26198e6	0.264530
$946$	0	0
$947$	9.13022e6	0.330831	0.165416	−	0.986224i	$-0.447103\pi$
$947$	9.13022e6	0.330831	0.165416	+	0.986224i	$0.447103\pi$
$948$	0	0
$949$	−6.13604e7	−2.21168
$950$	0	0
$951$	6.54855e6	0.234798
$952$	0	0
$953$	−3.75914e7	−1.34078	−0.670389	−	0.742010i	$-0.733873\pi$
$953$	−3.75914e7	−1.34078	−0.670389	+	0.742010i	$0.733873\pi$
$954$	0	0
$955$	1.03978e7	0.368919
$956$	0	0
$957$	6.77766e7	2.39221
$958$	0	0
$959$	9.41955e6	0.330738
$960$	0	0
$961$	923521.	0.0322581
$962$	0	0
$963$	4.40268e7	1.52986
$964$	0	0
$965$	1.75143e7	0.605445
$966$	0	0
$967$	2.85896e7	0.983201	0.491600	−	0.870821i	$-0.336412\pi$
$967$	2.85896e7	0.983201	0.491600	+	0.870821i	$0.336412\pi$
$968$	0	0
$969$	−4.07567e7	−1.39441
$970$	0	0
$971$	−5.57864e7	−1.89880	−0.949402	−	0.314062i	$-0.898310\pi$
$971$	−5.57864e7	−1.89880	−0.949402	+	0.314062i	$0.898310\pi$
$972$	0	0
$973$	1.47209e7	0.498486
$974$	0	0
$975$	−2.21383e7	−0.745819
$976$	0	0
$977$	5.43851e6	0.182282	0.0911409	−	0.995838i	$-0.470949\pi$
$977$	5.43851e6	0.182282	0.0911409	+	0.995838i	$0.470949\pi$
$978$	0	0
$979$	1.24391e7	0.414793
$980$	0	0
$981$	2.34812e7	0.779020
$982$	0	0
$983$	5.03425e7	1.66169	0.830847	−	0.556502i	$-0.187857\pi$
$983$	5.03425e7	1.66169	0.830847	+	0.556502i	$0.187857\pi$
$984$	0	0
$985$	2.82583e6	0.0928015
$986$	0	0
$987$	2.00089e7	0.653778
$988$	0	0
$989$	3.77705e6	0.122790
$990$	0	0
$991$	3.53436e7	1.14321	0.571606	−	0.820528i	$-0.306320\pi$
$991$	3.53436e7	1.14321	0.571606	+	0.820528i	$0.306320\pi$
$992$	0	0
$993$	−3.71207e7	−1.19466
$994$	0	0
$995$	2.03048e6	0.0650191
$996$	0	0
$997$	−1.50172e7	−0.478466	−0.239233	−	0.970962i	$-0.576896\pi$
$997$	−1.50172e7	−0.478466	−0.239233	+	0.970962i	$0.576896\pi$
$998$	0	0
$999$	−4.73457e6	−0.150095

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	124.6.a.b.1.5	✓	6
4.3	odd	2		496.6.a.f.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
124.6.a.b.1.5	✓	6	1.1	even	1	trivial
496.6.a.f.1.2		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)

\(f(q)\)	\(=\)	\(q+21.2423 q^{3} +45.9749 q^{5} -213.890 q^{7} +208.235 q^{9} +O(q^{10})\)	\(q+21.2423 q^{3} +45.9749 q^{5} -213.890 q^{7} +208.235 q^{9} +697.283 q^{11} +1030.53 q^{13} +976.612 q^{15} -967.895 q^{17} +1982.30 q^{19} -4543.51 q^{21} +1865.28 q^{23} -1011.31 q^{25} -738.490 q^{27} +4575.83 q^{29} -961.000 q^{31} +14811.9 q^{33} -9833.56 q^{35} +6411.15 q^{37} +21890.8 q^{39} -17512.0 q^{41} +2024.92 q^{43} +9573.58 q^{45} -4403.85 q^{47} +28941.8 q^{49} -20560.3 q^{51} -23788.9 q^{53} +32057.5 q^{55} +42108.6 q^{57} -24320.6 q^{59} +37484.8 q^{61} -44539.3 q^{63} +47378.4 q^{65} -1132.04 q^{67} +39622.8 q^{69} +60714.8 q^{71} -59542.7 q^{73} -21482.5 q^{75} -149142. q^{77} -50857.8 q^{79} -66288.3 q^{81} -73658.0 q^{83} -44498.9 q^{85} +97201.1 q^{87} +17839.3 q^{89} -220419. q^{91} -20413.8 q^{93} +91136.0 q^{95} -175952. q^{97} +145199. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 20 q^{3} + 25 q^{5} + 39 q^{7} + 306 q^{9}+O(q^{10}) \) 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

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