LMFDB - Embedded newform 126.8.a.k.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [126,8,Mod(1,126)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("126.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(126, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 126 = 2 \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	126.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-16,0,128,168] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$39.3605132110$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{3691}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3691 $ x^2 - 3691
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$60.7536$ of defining polynomial
Character	$\chi$	$=$	126.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-8.00000 q^{2} +64.0000 q^{4} +448.522 q^{5} +343.000 q^{7} -512.000 q^{8} -3588.17 q^{10} -5299.65 q^{11} -11506.5 q^{13} -2744.00 q^{14} +4096.00 q^{16} -6946.61 q^{17} -5416.52 q^{19} +28705.4 q^{20} +42397.2 q^{22} -50952.2 q^{23} +123047. q^{25} +92052.1 q^{26} +21952.0 q^{28} -237212. q^{29} +171756. q^{31} -32768.0 q^{32} +55572.9 q^{34} +153843. q^{35} +260898. q^{37} +43332.1 q^{38} -229643. q^{40} -406515. q^{41} -566697. q^{43} -339178. q^{44} +407617. q^{46} +109487. q^{47} +117649. q^{49} -984373. q^{50} -736417. q^{52} -592466. q^{53} -2.37701e6 q^{55} -175616. q^{56} +1.89770e6 q^{58} -1.62378e6 q^{59} +1.18797e6 q^{61} -1.37405e6 q^{62} +262144. q^{64} -5.16092e6 q^{65} +4.22954e6 q^{67} -444583. q^{68} -1.23074e6 q^{70} +3.02794e6 q^{71} -5.86494e6 q^{73} -2.08718e6 q^{74} -346657. q^{76} -1.81778e6 q^{77} -6.06964e6 q^{79} +1.83714e6 q^{80} +3.25212e6 q^{82} -1.11643e6 q^{83} -3.11570e6 q^{85} +4.53358e6 q^{86} +2.71342e6 q^{88} -8.36015e6 q^{89} -3.94674e6 q^{91} -3.26094e6 q^{92} -875894. q^{94} -2.42943e6 q^{95} -1.28495e7 q^{97} -941192. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 16 q^{2} + 128 q^{4} + 168 q^{5} + 686 q^{7} - 1024 q^{8} - 1344 q^{10} - 5496 q^{11} - 5516 q^{13} - 5488 q^{14} + 8192 q^{16} - 10248 q^{17} + 6664 q^{19} + 10752 q^{20} + 43968 q^{22} - 45768 q^{23}+ \cdots - 1882384 q^{98}+O(q^{100}) $ 2 * q - 16 * q^2 + 128 * q^4 + 168 * q^5 + 686 * q^7 - 1024 * q^8 - 1344 * q^10 - 5496 * q^11 - 5516 * q^13 - 5488 * q^14 + 8192 * q^16 - 10248 * q^17 + 6664 * q^19 + 10752 * q^20 + 43968 * q^22 - 45768 * q^23 + 123614 * q^25 + 44128 * q^26 + 43904 * q^28 - 188640 * q^29 + 221032 * q^31 - 65536 * q^32 + 81984 * q^34 + 57624 * q^35 + 399316 * q^37 - 53312 * q^38 - 86016 * q^40 - 562968 * q^41 - 153560 * q^43 - 351744 * q^44 + 366144 * q^46 - 651504 * q^47 + 235298 * q^49 - 988912 * q^50 - 353024 * q^52 - 2032080 * q^53 - 2321928 * q^55 - 351232 * q^56 + 1509120 * q^58 - 3864336 * q^59 - 721028 * q^61 - 1768256 * q^62 + 524288 * q^64 - 6841392 * q^65 - 604376 * q^67 - 655872 * q^68 - 460992 * q^70 - 389592 * q^71 - 6935684 * q^73 - 3194528 * q^74 + 426496 * q^76 - 1885128 * q^77 - 7974992 * q^79 + 688128 * q^80 + 4503744 * q^82 - 2209536 * q^83 - 2189592 * q^85 + 1228480 * q^86 + 2813952 * q^88 - 1659000 * q^89 - 1891988 * q^91 - 2929152 * q^92 + 5212032 * q^94 - 5818272 * q^95 - 4807460 * q^97 - 1882384 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−8.00000	−0.707107
$2$	−8.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	64.0000	0.500000
$5$	448.522	1.60468	0.802340	−	0.596867i	$-0.203588\pi$
$5$	448.522	1.60468	0.802340	+	0.596867i	$0.203588\pi$
$6$	0	0
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	−512.000	−0.353553
$9$	0	0
$10$	−3588.17	−1.13468
$11$	−5299.65	−1.20053	−0.600265	−	0.799801i	$-0.704938\pi$
$11$	−5299.65	−1.20053	−0.600265	+	0.799801i	$0.704938\pi$
$12$	0	0
$13$	−11506.5	−1.45259	−0.726294	−	0.687385i	$-0.758759\pi$
$13$	−11506.5	−1.45259	−0.726294	+	0.687385i	$0.758759\pi$
$14$	−2744.00	−0.267261
$15$	0	0
$16$	4096.00	0.250000
$17$	−6946.61	−0.342927	−0.171463	−	0.985190i	$-0.554850\pi$
$17$	−6946.61	−0.342927	−0.171463	+	0.985190i	$0.554850\pi$
$18$	0	0
$19$	−5416.52	−0.181168	−0.0905842	−	0.995889i	$-0.528873\pi$
$19$	−5416.52	−0.181168	−0.0905842	+	0.995889i	$0.528873\pi$
$20$	28705.4	0.802340
$21$	0	0
$22$	42397.2	0.848903
$23$	−50952.2	−0.873203	−0.436601	−	0.899655i	$-0.643818\pi$
$23$	−50952.2	−0.873203	−0.436601	+	0.899655i	$0.643818\pi$
$24$	0	0
$25$	123047.	1.57500
$26$	92052.1	1.02713
$27$	0	0
$28$	21952.0	0.188982
$29$	−237212.	−1.80611	−0.903056	−	0.429524i	$-0.858682\pi$
$29$	−237212.	−1.80611	−0.903056	+	0.429524i	$0.858682\pi$
$30$	0	0
$31$	171756.	1.03549	0.517744	−	0.855536i	$-0.326772\pi$
$31$	171756.	1.03549	0.517744	+	0.855536i	$0.326772\pi$
$32$	−32768.0	−0.176777
$33$	0	0
$34$	55572.9	0.242486
$35$	153843.	0.606512
$36$	0	0
$37$	260898.	0.846767	0.423384	−	0.905950i	$-0.360842\pi$
$37$	260898.	0.846767	0.423384	+	0.905950i	$0.360842\pi$
$38$	43332.1	0.128105
$39$	0	0
$40$	−229643.	−0.567340
$41$	−406515.	−0.921156	−0.460578	−	0.887619i	$-0.652358\pi$
$41$	−406515.	−0.921156	−0.460578	+	0.887619i	$0.652358\pi$
$42$	0	0
$43$	−566697.	−1.08695	−0.543477	−	0.839424i	$-0.682893\pi$
$43$	−566697.	−1.08695	−0.543477	+	0.839424i	$0.682893\pi$
$44$	−339178.	−0.600265
$45$	0	0
$46$	407617.	0.617448
$47$	109487.	0.153822	0.0769111	−	0.997038i	$-0.475494\pi$
$47$	109487.	0.153822	0.0769111	+	0.997038i	$0.475494\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	−984373.	−1.11369
$51$	0	0
$52$	−736417.	−0.726294
$53$	−592466.	−0.546635	−0.273318	−	0.961924i	$-0.588121\pi$
$53$	−592466.	−0.546635	−0.273318	+	0.961924i	$0.588121\pi$
$54$	0	0
$55$	−2.37701e6	−1.92647
$56$	−175616.	−0.133631
$57$	0	0
$58$	1.89770e6	1.27711
$59$	−1.62378e6	−1.02931	−0.514655	−	0.857398i	$-0.672080\pi$
$59$	−1.62378e6	−1.02931	−0.514655	+	0.857398i	$0.672080\pi$
$60$	0	0
$61$	1.18797e6	0.670120	0.335060	−	0.942197i	$-0.391243\pi$
$61$	1.18797e6	0.670120	0.335060	+	0.942197i	$0.391243\pi$
$62$	−1.37405e6	−0.732201
$63$	0	0
$64$	262144.	0.125000
$65$	−5.16092e6	−2.33094
$66$	0	0
$67$	4.22954e6	1.71803	0.859017	−	0.511947i	$-0.171076\pi$
$67$	4.22954e6	1.71803	0.859017	+	0.511947i	$0.171076\pi$
$68$	−444583.	−0.171463
$69$	0	0
$70$	−1.23074e6	−0.428869
$71$	3.02794e6	1.00402	0.502011	−	0.864861i	$-0.332594\pi$
$71$	3.02794e6	1.00402	0.502011	+	0.864861i	$0.332594\pi$
$72$	0	0
$73$	−5.86494e6	−1.76455	−0.882274	−	0.470737i	$-0.843988\pi$
$73$	−5.86494e6	−1.76455	−0.882274	+	0.470737i	$0.843988\pi$
$74$	−2.08718e6	−0.598755
$75$	0	0
$76$	−346657.	−0.0905842
$77$	−1.81778e6	−0.453758
$78$	0	0
$79$	−6.06964e6	−1.38506	−0.692530	−	0.721389i	$-0.743504\pi$
$79$	−6.06964e6	−1.38506	−0.692530	+	0.721389i	$0.743504\pi$
$80$	1.83714e6	0.401170
$81$	0	0
$82$	3.25212e6	0.651355
$83$	−1.11643e6	−0.214318	−0.107159	−	0.994242i	$-0.534175\pi$
$83$	−1.11643e6	−0.214318	−0.107159	+	0.994242i	$0.534175\pi$
$84$	0	0
$85$	−3.11570e6	−0.550288
$86$	4.53358e6	0.768593
$87$	0	0
$88$	2.71342e6	0.424451
$89$	−8.36015e6	−1.25704	−0.628520	−	0.777793i	$-0.716339\pi$
$89$	−8.36015e6	−1.25704	−0.628520	+	0.777793i	$0.716339\pi$
$90$	0	0
$91$	−3.94674e6	−0.549026
$92$	−3.26094e6	−0.436601
$93$	0	0
$94$	−875894.	−0.108769
$95$	−2.42943e6	−0.290717
$96$	0	0
$97$	−1.28495e7	−1.42950	−0.714749	−	0.699381i	$-0.753459\pi$
$97$	−1.28495e7	−1.42950	−0.714749	+	0.699381i	$0.753459\pi$
$98$	−941192.	−0.101015
$99$	0	0
$100$	7.87498e6	0.787498
$101$	−1.14659e7	−1.10734	−0.553671	−	0.832735i	$-0.686774\pi$
$101$	−1.14659e7	−1.10734	−0.553671	+	0.832735i	$0.686774\pi$
$102$	0	0
$103$	1.66835e7	1.50438	0.752190	−	0.658946i	$-0.228997\pi$
$103$	1.66835e7	1.50438	0.752190	+	0.658946i	$0.228997\pi$
$104$	5.89134e6	0.513567
$105$	0	0
$106$	4.73973e6	0.386530
$107$	7.97687e6	0.629491	0.314745	−	0.949176i	$-0.398081\pi$
$107$	7.97687e6	0.629491	0.314745	+	0.949176i	$0.398081\pi$
$108$	0	0
$109$	2.28499e7	1.69002	0.845011	−	0.534749i	$-0.179594\pi$
$109$	2.28499e7	1.69002	0.845011	+	0.534749i	$0.179594\pi$
$110$	1.90161e7	1.36222
$111$	0	0
$112$	1.40493e6	0.0944911
$113$	1.37201e7	0.894505	0.447252	−	0.894408i	$-0.352403\pi$
$113$	1.37201e7	0.894505	0.447252	+	0.894408i	$0.352403\pi$
$114$	0	0
$115$	−2.28531e7	−1.40121
$116$	−1.51816e7	−0.903056
$117$	0	0
$118$	1.29903e7	0.727832
$119$	−2.38269e6	−0.129614
$120$	0	0
$121$	8.59913e6	0.441271
$122$	−9.50379e6	−0.473846
$123$	0	0
$124$	1.09924e7	0.517744
$125$	2.01483e7	0.922686
$126$	0	0
$127$	−1.82759e7	−0.791710	−0.395855	−	0.918313i	$-0.629552\pi$
$127$	−1.82759e7	−0.791710	−0.395855	+	0.918313i	$0.629552\pi$
$128$	−2.09715e6	−0.0883883
$129$	0	0
$130$	4.12874e7	1.64822
$131$	4.51203e7	1.75357	0.876784	−	0.480885i	$-0.159685\pi$
$131$	4.51203e7	1.75357	0.876784	+	0.480885i	$0.159685\pi$
$132$	0	0
$133$	−1.85787e6	−0.0684752
$134$	−3.38364e7	−1.21483
$135$	0	0
$136$	3.55666e6	0.121243
$137$	2.78865e7	0.926557	0.463279	−	0.886213i	$-0.346673\pi$
$137$	2.78865e7	0.926557	0.463279	+	0.886213i	$0.346673\pi$
$138$	0	0
$139$	1.17880e7	0.372295	0.186148	−	0.982522i	$-0.440400\pi$
$139$	1.17880e7	0.372295	0.186148	+	0.982522i	$0.440400\pi$
$140$	9.84595e6	0.303256
$141$	0	0
$142$	−2.42235e7	−0.709950
$143$	6.09805e7	1.74387
$144$	0	0
$145$	−1.06395e8	−2.89823
$146$	4.69195e7	1.24772
$147$	0	0
$148$	1.66974e7	0.423384
$149$	−3.84472e7	−0.952166	−0.476083	−	0.879400i	$-0.657944\pi$
$149$	−3.84472e7	−0.952166	−0.476083	+	0.879400i	$0.657944\pi$
$150$	0	0
$151$	−3.91861e7	−0.926217	−0.463108	−	0.886302i	$-0.653266\pi$
$151$	−3.91861e7	−0.926217	−0.463108	+	0.886302i	$0.653266\pi$
$152$	2.77326e6	0.0640527
$153$	0	0
$154$	1.45422e7	0.320855
$155$	7.70361e7	1.66163
$156$	0	0
$157$	−3.56409e7	−0.735021	−0.367510	−	0.930019i	$-0.619790\pi$
$157$	−3.56409e7	−0.735021	−0.367510	+	0.930019i	$0.619790\pi$
$158$	4.85571e7	0.979385
$159$	0	0
$160$	−1.46972e7	−0.283670
$161$	−1.74766e7	−0.330040
$162$	0	0
$163$	−3.32960e7	−0.602192	−0.301096	−	0.953594i	$-0.597353\pi$
$163$	−3.32960e7	−0.602192	−0.301096	+	0.953594i	$0.597353\pi$
$164$	−2.60170e7	−0.460578
$165$	0	0
$166$	8.93146e6	0.151546
$167$	−7.23060e7	−1.20134	−0.600671	−	0.799497i	$-0.705100\pi$
$167$	−7.23060e7	−1.20134	−0.600671	+	0.799497i	$0.705100\pi$
$168$	0	0
$169$	6.96515e7	1.11001
$170$	2.49256e7	0.389112
$171$	0	0
$172$	−3.62686e7	−0.543477
$173$	−8.34486e7	−1.22534	−0.612672	−	0.790337i	$-0.709905\pi$
$173$	−8.34486e7	−1.22534	−0.612672	+	0.790337i	$0.709905\pi$
$174$	0	0
$175$	4.22050e7	0.595293
$176$	−2.17074e7	−0.300132
$177$	0	0
$178$	6.68812e7	0.888862
$179$	−1.15464e7	−0.150474	−0.0752371	−	0.997166i	$-0.523971\pi$
$179$	−1.15464e7	−0.150474	−0.0752371	+	0.997166i	$0.523971\pi$
$180$	0	0
$181$	6.08217e7	0.762401	0.381200	−	0.924492i	$-0.375511\pi$
$181$	6.08217e7	0.762401	0.381200	+	0.924492i	$0.375511\pi$
$182$	3.15739e7	0.388220
$183$	0	0
$184$	2.60875e7	0.308724
$185$	1.17018e8	1.35879
$186$	0	0
$187$	3.68146e7	0.411694
$188$	7.00715e6	0.0769111
$189$	0	0
$190$	1.94354e7	0.205568
$191$	−1.06179e8	−1.10261	−0.551307	−	0.834302i	$-0.685871\pi$
$191$	−1.06179e8	−1.10261	−0.551307	+	0.834302i	$0.685871\pi$
$192$	0	0
$193$	−1.16101e8	−1.16248	−0.581241	−	0.813731i	$-0.697433\pi$
$193$	−1.16101e8	−1.16248	−0.581241	+	0.813731i	$0.697433\pi$
$194$	1.02796e8	1.01081
$195$	0	0
$196$	7.52954e6	0.0714286
$197$	6.91557e6	0.0644460	0.0322230	−	0.999481i	$-0.489741\pi$
$197$	6.91557e6	0.0644460	0.0322230	+	0.999481i	$0.489741\pi$
$198$	0	0
$199$	−1.35406e8	−1.21801	−0.609006	−	0.793166i	$-0.708431\pi$
$199$	−1.35406e8	−1.21801	−0.609006	+	0.793166i	$0.708431\pi$
$200$	−6.29999e7	−0.556845
$201$	0	0
$202$	9.17269e7	0.783009
$203$	−8.13639e7	−0.682646
$204$	0	0
$205$	−1.82331e8	−1.47816
$206$	−1.33468e8	−1.06376
$207$	0	0
$208$	−4.71307e7	−0.363147
$209$	2.87057e7	0.217498
$210$	0	0
$211$	1.90693e8	1.39748	0.698742	−	0.715373i	$-0.253743\pi$
$211$	1.90693e8	1.39748	0.698742	+	0.715373i	$0.253743\pi$
$212$	−3.79178e7	−0.273318
$213$	0	0
$214$	−6.38150e7	−0.445117
$215$	−2.54176e8	−1.74421
$216$	0	0
$217$	5.89122e7	0.391378
$218$	−1.82800e8	−1.19503
$219$	0	0
$220$	−1.52129e8	−0.963233
$221$	7.99313e7	0.498131
$222$	0	0
$223$	2.56628e8	1.54966	0.774830	−	0.632170i	$-0.217836\pi$
$223$	2.56628e8	1.54966	0.774830	+	0.632170i	$0.217836\pi$
$224$	−1.12394e7	−0.0668153
$225$	0	0
$226$	−1.09761e8	−0.632510
$227$	−1.03716e8	−0.588514	−0.294257	−	0.955726i	$-0.595072\pi$
$227$	−1.03716e8	−0.588514	−0.294257	+	0.955726i	$0.595072\pi$
$228$	0	0
$229$	−2.19819e6	−0.0120960	−0.00604798	−	0.999982i	$-0.501925\pi$
$229$	−2.19819e6	−0.0120960	−0.00604798	+	0.999982i	$0.501925\pi$
$230$	1.82825e8	0.990806
$231$	0	0
$232$	1.21453e8	0.638557
$233$	1.81985e8	0.942517	0.471259	−	0.881995i	$-0.343800\pi$
$233$	1.81985e8	0.942517	0.471259	+	0.881995i	$0.343800\pi$
$234$	0	0
$235$	4.91072e7	0.246835
$236$	−1.03922e8	−0.514655
$237$	0	0
$238$	1.90615e7	0.0916511
$239$	−2.69191e8	−1.27546	−0.637732	−	0.770258i	$-0.720127\pi$
$239$	−2.69191e8	−1.27546	−0.637732	+	0.770258i	$0.720127\pi$
$240$	0	0
$241$	1.97228e8	0.907629	0.453815	−	0.891096i	$-0.350063\pi$
$241$	1.97228e8	0.907629	0.453815	+	0.891096i	$0.350063\pi$
$242$	−6.87931e7	−0.312026
$243$	0	0
$244$	7.60303e7	0.335060
$245$	5.27681e7	0.229240
$246$	0	0
$247$	6.23253e7	0.263163
$248$	−8.79389e7	−0.366100
$249$	0	0
$250$	−1.61187e8	−0.652437
$251$	2.40008e8	0.958004	0.479002	−	0.877814i	$-0.340999\pi$
$251$	2.40008e8	0.958004	0.479002	+	0.877814i	$0.340999\pi$
$252$	0	0
$253$	2.70029e8	1.04831
$254$	1.46207e8	0.559823
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	−3.67989e8	−1.35229	−0.676144	−	0.736769i	$-0.736350\pi$
$257$	−3.67989e8	−1.35229	−0.676144	+	0.736769i	$0.736350\pi$
$258$	0	0
$259$	8.94879e7	0.320048
$260$	−3.30299e8	−1.16547
$261$	0	0
$262$	−3.60962e8	−1.23996
$263$	1.34513e8	0.455953	0.227977	−	0.973667i	$-0.426789\pi$
$263$	1.34513e8	0.455953	0.227977	+	0.973667i	$0.426789\pi$
$264$	0	0
$265$	−2.65734e8	−0.877175
$266$	1.48629e7	0.0484193
$267$	0	0
$268$	2.70691e8	0.859017
$269$	−6.15309e7	−0.192735	−0.0963675	−	0.995346i	$-0.530722\pi$
$269$	−6.15309e7	−0.192735	−0.0963675	+	0.995346i	$0.530722\pi$
$270$	0	0
$271$	4.17868e7	0.127540	0.0637700	−	0.997965i	$-0.479688\pi$
$271$	4.17868e7	0.127540	0.0637700	+	0.997965i	$0.479688\pi$
$272$	−2.84533e7	−0.0857317
$273$	0	0
$274$	−2.23092e8	−0.655175
$275$	−6.52104e8	−1.89083
$276$	0	0
$277$	6.59838e8	1.86534	0.932671	−	0.360729i	$-0.117472\pi$
$277$	6.59838e8	1.86534	0.932671	+	0.360729i	$0.117472\pi$
$278$	−9.43038e7	−0.263253
$279$	0	0
$280$	−7.87676e7	−0.214434
$281$	1.45549e8	0.391325	0.195662	−	0.980671i	$-0.437314\pi$
$281$	1.45549e8	0.391325	0.195662	+	0.980671i	$0.437314\pi$
$282$	0	0
$283$	−3.13808e8	−0.823021	−0.411511	−	0.911405i	$-0.634999\pi$
$283$	−3.13808e8	−0.823021	−0.411511	+	0.911405i	$0.634999\pi$
$284$	1.93788e8	0.502011
$285$	0	0
$286$	−4.87844e8	−1.23311
$287$	−1.39435e8	−0.348164
$288$	0	0
$289$	−3.62083e8	−0.882401
$290$	8.51159e8	2.04936
$291$	0	0
$292$	−3.75356e8	−0.882274
$293$	9.29302e7	0.215834	0.107917	−	0.994160i	$-0.465582\pi$
$293$	9.29302e7	0.215834	0.107917	+	0.994160i	$0.465582\pi$
$294$	0	0
$295$	−7.28302e8	−1.65171
$296$	−1.33580e8	−0.299377
$297$	0	0
$298$	3.07578e8	0.673283
$299$	5.86282e8	1.26840
$300$	0	0
$301$	−1.94377e8	−0.410830
$302$	3.13489e8	0.654934
$303$	0	0
$304$	−2.21861e7	−0.0452921
$305$	5.32832e8	1.07533
$306$	0	0
$307$	3.50287e8	0.690939	0.345469	−	0.938430i	$-0.387720\pi$
$307$	3.50287e8	0.690939	0.345469	+	0.938430i	$0.387720\pi$
$308$	−1.16338e8	−0.226879
$309$	0	0
$310$	−6.16289e8	−1.17495
$311$	3.03239e8	0.571641	0.285820	−	0.958283i	$-0.407734\pi$
$311$	3.03239e8	0.571641	0.285820	+	0.958283i	$0.407734\pi$
$312$	0	0
$313$	1.08029e8	0.199130	0.0995649	−	0.995031i	$-0.468255\pi$
$313$	1.08029e8	0.199130	0.0995649	+	0.995031i	$0.468255\pi$
$314$	2.85127e8	0.519738
$315$	0	0
$316$	−3.88457e8	−0.692530
$317$	−3.21944e8	−0.567640	−0.283820	−	0.958878i	$-0.591602\pi$
$317$	−3.21944e8	−0.567640	−0.283820	+	0.958878i	$0.591602\pi$
$318$	0	0
$319$	1.25714e9	2.16829
$320$	1.17577e8	0.200585
$321$	0	0
$322$	1.39813e8	0.233373
$323$	3.76264e7	0.0621275
$324$	0	0
$325$	−1.41584e9	−2.28782
$326$	2.66368e8	0.425814
$327$	0	0
$328$	2.08136e8	0.325678
$329$	3.75540e7	0.0581394
$330$	0	0
$331$	1.12397e9	1.70355	0.851776	−	0.523907i	$-0.175526\pi$
$331$	1.12397e9	1.70355	0.851776	+	0.523907i	$0.175526\pi$
$332$	−7.14517e7	−0.107159
$333$	0	0
$334$	5.78448e8	0.849477
$335$	1.89704e9	2.75689
$336$	0	0
$337$	1.17136e9	1.66720	0.833598	−	0.552372i	$-0.186277\pi$
$337$	1.17136e9	1.66720	0.833598	+	0.552372i	$0.186277\pi$
$338$	−5.57212e8	−0.784895
$339$	0	0
$340$	−1.99405e8	−0.275144
$341$	−9.10245e8	−1.24313
$342$	0	0
$343$	4.03536e7	0.0539949
$344$	2.90149e8	0.384296
$345$	0	0
$346$	6.67589e8	0.866449
$347$	1.38737e9	1.78254	0.891271	−	0.453470i	$-0.149814\pi$
$347$	1.38737e9	1.78254	0.891271	+	0.453470i	$0.149814\pi$
$348$	0	0
$349$	−5.90839e8	−0.744013	−0.372006	−	0.928230i	$-0.621330\pi$
$349$	−5.90839e8	−0.744013	−0.372006	+	0.928230i	$0.621330\pi$
$350$	−3.37640e8	−0.420936
$351$	0	0
$352$	1.73659e8	0.212226
$353$	1.37642e9	1.66548	0.832742	−	0.553661i	$-0.186770\pi$
$353$	1.37642e9	1.66548	0.832742	+	0.553661i	$0.186770\pi$
$354$	0	0
$355$	1.35810e9	1.61113
$356$	−5.35050e8	−0.628520
$357$	0	0
$358$	9.23714e7	0.106401
$359$	−3.97273e7	−0.0453167	−0.0226583	−	0.999743i	$-0.507213\pi$
$359$	−3.97273e7	−0.0453167	−0.0226583	+	0.999743i	$0.507213\pi$
$360$	0	0
$361$	−8.64533e8	−0.967178
$362$	−4.86573e8	−0.539099
$363$	0	0
$364$	−2.52591e8	−0.274513
$365$	−2.63055e9	−2.83153
$366$	0	0
$367$	6.15437e8	0.649908	0.324954	−	0.945730i	$-0.394651\pi$
$367$	6.15437e8	0.649908	0.324954	+	0.945730i	$0.394651\pi$
$368$	−2.08700e8	−0.218301
$369$	0	0
$370$	−9.36146e8	−0.960810
$371$	−2.03216e8	−0.206609
$372$	0	0
$373$	−2.96660e8	−0.295991	−0.147995	−	0.988988i	$-0.547282\pi$
$373$	−2.96660e8	−0.295991	−0.147995	+	0.988988i	$0.547282\pi$
$374$	−2.94517e8	−0.291112
$375$	0	0
$376$	−5.60572e7	−0.0543844
$377$	2.72949e9	2.62353
$378$	0	0
$379$	1.12605e8	0.106248	0.0531239	−	0.998588i	$-0.483082\pi$
$379$	1.12605e8	0.106248	0.0531239	+	0.998588i	$0.483082\pi$
$380$	−1.55483e8	−0.145359
$381$	0	0
$382$	8.49436e8	0.779666
$383$	1.93832e9	1.76291	0.881455	−	0.472269i	$-0.156565\pi$
$383$	1.93832e9	1.76291	0.881455	+	0.472269i	$0.156565\pi$
$384$	0	0
$385$	−8.15314e8	−0.728136
$386$	9.28810e8	0.821999
$387$	0	0
$388$	−8.22366e8	−0.714749
$389$	−1.63944e9	−1.41212	−0.706062	−	0.708150i	$-0.749530\pi$
$389$	−1.63944e9	−1.41212	−0.706062	+	0.708150i	$0.749530\pi$
$390$	0	0
$391$	3.53945e8	0.299445
$392$	−6.02363e7	−0.0505076
$393$	0	0
$394$	−5.53245e7	−0.0455702
$395$	−2.72237e9	−2.22258
$396$	0	0
$397$	−5.62631e8	−0.451291	−0.225645	−	0.974209i	$-0.572449\pi$
$397$	−5.62631e8	−0.451291	−0.225645	+	0.974209i	$0.572449\pi$
$398$	1.08325e9	0.861264
$399$	0	0
$400$	5.03999e8	0.393749
$401$	−5.91452e8	−0.458051	−0.229026	−	0.973420i	$-0.573554\pi$
$401$	−5.91452e8	−0.458051	−0.229026	+	0.973420i	$0.573554\pi$
$402$	0	0
$403$	−1.97631e9	−1.50414
$404$	−7.33815e8	−0.553671
$405$	0	0
$406$	6.50911e8	0.482704
$407$	−1.38267e9	−1.01657
$408$	0	0
$409$	1.02432e9	0.740290	0.370145	−	0.928974i	$-0.379308\pi$
$409$	1.02432e9	0.740290	0.370145	+	0.928974i	$0.379308\pi$
$410$	1.45865e9	1.04522
$411$	0	0
$412$	1.06775e9	0.752190
$413$	−5.56957e8	−0.389042
$414$	0	0
$415$	−5.00744e8	−0.343912
$416$	3.77046e8	0.256784
$417$	0	0
$418$	−2.29645e8	−0.153794
$419$	8.89500e8	0.590741	0.295370	−	0.955383i	$-0.404557\pi$
$419$	8.89500e8	0.590741	0.295370	+	0.955383i	$0.404557\pi$
$420$	0	0
$421$	−2.43115e9	−1.58791	−0.793954	−	0.607978i	$-0.791981\pi$
$421$	−2.43115e9	−1.58791	−0.793954	+	0.607978i	$0.791981\pi$
$422$	−1.52555e9	−0.988171
$423$	0	0
$424$	3.03343e8	0.193265
$425$	−8.54757e8	−0.540109
$426$	0	0
$427$	4.07475e8	0.253281
$428$	5.10520e8	0.314745
$429$	0	0
$430$	2.03341e9	1.23335
$431$	−1.65199e9	−0.993886	−0.496943	−	0.867783i	$-0.665544\pi$
$431$	−1.65199e9	−0.993886	−0.496943	+	0.867783i	$0.665544\pi$
$432$	0	0
$433$	−1.07947e8	−0.0639001	−0.0319500	−	0.999489i	$-0.510172\pi$
$433$	−1.07947e8	−0.0639001	−0.0319500	+	0.999489i	$0.510172\pi$
$434$	−4.71297e8	−0.276746
$435$	0	0
$436$	1.46240e9	0.845011
$437$	2.75983e8	0.158197
$438$	0	0
$439$	−5.49941e8	−0.310234	−0.155117	−	0.987896i	$-0.549576\pi$
$439$	−5.49941e8	−0.310234	−0.155117	+	0.987896i	$0.549576\pi$
$440$	1.21703e9	0.681108
$441$	0	0
$442$	−6.39450e8	−0.352232
$443$	−1.16826e9	−0.638452	−0.319226	−	0.947679i	$-0.603423\pi$
$443$	−1.16826e9	−0.638452	−0.319226	+	0.947679i	$0.603423\pi$
$444$	0	0
$445$	−3.74971e9	−2.01715
$446$	−2.05302e9	−1.09577
$447$	0	0
$448$	8.99154e7	0.0472456
$449$	−2.80689e9	−1.46340	−0.731701	−	0.681626i	$-0.761273\pi$
$449$	−2.80689e9	−1.46340	−0.731701	+	0.681626i	$0.761273\pi$
$450$	0	0
$451$	2.15439e9	1.10587
$452$	8.78086e8	0.447252
$453$	0	0
$454$	8.29731e8	0.416142
$455$	−1.77020e9	−0.881011
$456$	0	0
$457$	−1.30612e9	−0.640143	−0.320071	−	0.947393i	$-0.603707\pi$
$457$	−1.30612e9	−0.640143	−0.320071	+	0.947393i	$0.603707\pi$
$458$	1.75855e7	0.00855314
$459$	0	0
$460$	−1.46260e9	−0.700606
$461$	3.55976e9	1.69226	0.846130	−	0.532977i	$-0.178927\pi$
$461$	3.55976e9	1.69226	0.846130	+	0.532977i	$0.178927\pi$
$462$	0	0
$463$	2.25381e9	1.05532	0.527659	−	0.849456i	$-0.323070\pi$
$463$	2.25381e9	1.05532	0.527659	+	0.849456i	$0.323070\pi$
$464$	−9.71622e8	−0.451528
$465$	0	0
$466$	−1.45588e9	−0.666460
$467$	−2.27560e9	−1.03392	−0.516960	−	0.856009i	$-0.672937\pi$
$467$	−2.27560e9	−1.03392	−0.516960	+	0.856009i	$0.672937\pi$
$468$	0	0
$469$	1.45073e9	0.649356
$470$	−3.92858e8	−0.174539
$471$	0	0
$472$	8.31377e8	0.363916
$473$	3.00330e9	1.30492
$474$	0	0
$475$	−6.66484e8	−0.285340
$476$	−1.52492e8	−0.0648071
$477$	0	0
$478$	2.15353e9	0.901890
$479$	2.87757e9	1.19633	0.598166	−	0.801372i	$-0.295896\pi$
$479$	2.87757e9	1.19633	0.598166	+	0.801372i	$0.295896\pi$
$480$	0	0
$481$	−3.00202e9	−1.23000
$482$	−1.57782e9	−0.641791
$483$	0	0
$484$	5.50344e8	0.220636
$485$	−5.76326e9	−2.29389
$486$	0	0
$487$	−1.54199e9	−0.604966	−0.302483	−	0.953155i	$-0.597816\pi$
$487$	−1.54199e9	−0.604966	−0.302483	+	0.953155i	$0.597816\pi$
$488$	−6.08243e8	−0.236923
$489$	0	0
$490$	−4.22145e8	−0.162097
$491$	−3.22726e9	−1.23041	−0.615204	−	0.788368i	$-0.710926\pi$
$491$	−3.22726e9	−1.23041	−0.615204	+	0.788368i	$0.710926\pi$
$492$	0	0
$493$	1.64782e9	0.619364
$494$	−4.98602e8	−0.186084
$495$	0	0
$496$	7.03511e8	0.258872
$497$	1.03858e9	0.379484
$498$	0	0
$499$	−2.30284e9	−0.829682	−0.414841	−	0.909894i	$-0.636163\pi$
$499$	−2.30284e9	−0.829682	−0.414841	+	0.909894i	$0.636163\pi$
$500$	1.28949e9	0.461343
$501$	0	0
$502$	−1.92006e9	−0.677411
$503$	−1.57135e9	−0.550536	−0.275268	−	0.961368i	$-0.588767\pi$
$503$	−1.57135e9	−0.550536	−0.275268	+	0.961368i	$0.588767\pi$
$504$	0	0
$505$	−5.14269e9	−1.77693
$506$	−2.16023e9	−0.741264
$507$	0	0
$508$	−1.16966e9	−0.395855
$509$	−2.38208e9	−0.800654	−0.400327	−	0.916372i	$-0.631103\pi$
$509$	−2.38208e9	−0.800654	−0.400327	+	0.916372i	$0.631103\pi$
$510$	0	0
$511$	−2.01167e9	−0.666936
$512$	−1.34218e8	−0.0441942
$513$	0	0
$514$	2.94391e9	0.956212
$515$	7.48293e9	2.41405
$516$	0	0
$517$	−5.80242e8	−0.184668
$518$	−7.15903e8	−0.226308
$519$	0	0
$520$	2.64239e9	0.824111
$521$	3.03375e9	0.939825	0.469913	−	0.882713i	$-0.344285\pi$
$521$	3.03375e9	0.939825	0.469913	+	0.882713i	$0.344285\pi$
$522$	0	0
$523$	6.04017e9	1.84626	0.923130	−	0.384488i	$-0.125622\pi$
$523$	6.04017e9	1.84626	0.923130	+	0.384488i	$0.125622\pi$
$524$	2.88770e9	0.876784
$525$	0	0
$526$	−1.07611e9	−0.322407
$527$	−1.19312e9	−0.355097
$528$	0	0
$529$	−8.08702e8	−0.237517
$530$	2.12587e9	0.620256
$531$	0	0
$532$	−1.18903e8	−0.0342376
$533$	4.67757e9	1.33806
$534$	0	0
$535$	3.57780e9	1.01013
$536$	−2.16553e9	−0.607417
$537$	0	0
$538$	4.92247e8	0.136284
$539$	−6.23499e8	−0.171504
$540$	0	0
$541$	7.70631e8	0.209245	0.104623	−	0.994512i	$-0.466637\pi$
$541$	7.70631e8	0.209245	0.104623	+	0.994512i	$0.466637\pi$
$542$	−3.34294e8	−0.0901845
$543$	0	0
$544$	2.27626e8	0.0606215
$545$	1.02487e10	2.71194
$546$	0	0
$547$	5.56573e9	1.45401	0.727004	−	0.686634i	$-0.240912\pi$
$547$	5.56573e9	1.45401	0.727004	+	0.686634i	$0.240912\pi$
$548$	1.78474e9	0.463279
$549$	0	0
$550$	5.21683e9	1.33702
$551$	1.28487e9	0.327210
$552$	0	0
$553$	−2.08189e9	−0.523503
$554$	−5.27871e9	−1.31900
$555$	0	0
$556$	7.54431e8	0.186148
$557$	−1.80145e9	−0.441702	−0.220851	−	0.975308i	$-0.570884\pi$
$557$	−1.80145e9	−0.441702	−0.220851	+	0.975308i	$0.570884\pi$
$558$	0	0
$559$	6.52071e9	1.57890
$560$	6.30141e8	0.151628
$561$	0	0
$562$	−1.16439e9	−0.276708
$563$	−2.58295e9	−0.610011	−0.305005	−	0.952351i	$-0.598658\pi$
$563$	−2.58295e9	−0.610011	−0.305005	+	0.952351i	$0.598658\pi$
$564$	0	0
$565$	6.15376e9	1.43539
$566$	2.51046e9	0.581964
$567$	0	0
$568$	−1.55031e9	−0.354975
$569$	1.40190e9	0.319025	0.159512	−	0.987196i	$-0.449008\pi$
$569$	1.40190e9	0.319025	0.159512	+	0.987196i	$0.449008\pi$
$570$	0	0
$571$	−7.25650e8	−0.163118	−0.0815588	−	0.996669i	$-0.525990\pi$
$571$	−7.25650e8	−0.163118	−0.0815588	+	0.996669i	$0.525990\pi$
$572$	3.90275e9	0.871937
$573$	0	0
$574$	1.11548e9	0.246189
$575$	−6.26949e9	−1.37529
$576$	0	0
$577$	−6.05247e9	−1.31165	−0.655824	−	0.754914i	$-0.727679\pi$
$577$	−6.05247e9	−1.31165	−0.655824	+	0.754914i	$0.727679\pi$
$578$	2.89667e9	0.623952
$579$	0	0
$580$	−6.80927e9	−1.44912
$581$	−3.82936e8	−0.0810047
$582$	0	0
$583$	3.13986e9	0.656252
$584$	3.00285e9	0.623862
$585$	0	0
$586$	−7.43441e8	−0.152618
$587$	6.10151e9	1.24510	0.622550	−	0.782580i	$-0.286097\pi$
$587$	6.10151e9	1.24510	0.622550	+	0.782580i	$0.286097\pi$
$588$	0	0
$589$	−9.30318e8	−0.187598
$590$	5.82641e9	1.16794
$591$	0	0
$592$	1.06864e9	0.211692
$593$	−6.57021e9	−1.29386	−0.646931	−	0.762548i	$-0.723948\pi$
$593$	−6.57021e9	−1.29386	−0.646931	+	0.762548i	$0.723948\pi$
$594$	0	0
$595$	−1.06869e9	−0.207989
$596$	−2.46062e9	−0.476083
$597$	0	0
$598$	−4.69026e9	−0.896897
$599$	9.50755e9	1.80749	0.903743	−	0.428076i	$-0.140809\pi$
$599$	9.50755e9	1.80749	0.903743	+	0.428076i	$0.140809\pi$
$600$	0	0
$601$	−8.37769e9	−1.57421	−0.787106	−	0.616817i	$-0.788422\pi$
$601$	−8.37769e9	−1.57421	−0.787106	+	0.616817i	$0.788422\pi$
$602$	1.55502e9	0.290501
$603$	0	0
$604$	−2.50791e9	−0.463108
$605$	3.85690e9	0.708099
$606$	0	0
$607$	4.19736e9	0.761756	0.380878	−	0.924625i	$-0.375622\pi$
$607$	4.19736e9	0.761756	0.380878	+	0.924625i	$0.375622\pi$
$608$	1.77488e8	0.0320264
$609$	0	0
$610$	−4.26266e9	−0.760371
$611$	−1.25981e9	−0.223440
$612$	0	0
$613$	−4.10106e9	−0.719093	−0.359546	−	0.933127i	$-0.617069\pi$
$613$	−4.10106e9	−0.719093	−0.359546	+	0.933127i	$0.617069\pi$
$614$	−2.80229e9	−0.488567
$615$	0	0
$616$	9.30704e8	0.160428
$617$	7.10665e9	1.21806	0.609028	−	0.793149i	$-0.291560\pi$
$617$	7.10665e9	1.21806	0.609028	+	0.793149i	$0.291560\pi$
$618$	0	0
$619$	−2.58428e9	−0.437948	−0.218974	−	0.975731i	$-0.570271\pi$
$619$	−2.58428e9	−0.437948	−0.218974	+	0.975731i	$0.570271\pi$
$620$	4.93031e9	0.830813
$621$	0	0
$622$	−2.42591e9	−0.404211
$623$	−2.86753e9	−0.475117
$624$	0	0
$625$	−5.76061e8	−0.0943818
$626$	−8.64234e8	−0.140806
$627$	0	0
$628$	−2.28102e9	−0.367510
$629$	−1.81235e9	−0.290379
$630$	0	0
$631$	−3.29857e9	−0.522665	−0.261332	−	0.965249i	$-0.584162\pi$
$631$	−3.29857e9	−0.522665	−0.261332	+	0.965249i	$0.584162\pi$
$632$	3.10766e9	0.489692
$633$	0	0
$634$	2.57555e9	0.401382
$635$	−8.19714e9	−1.27044
$636$	0	0
$637$	−1.35373e9	−0.207512
$638$	−1.00571e10	−1.53321
$639$	0	0
$640$	−9.40618e8	−0.141835
$641$	−7.23119e9	−1.08444	−0.542221	−	0.840236i	$-0.682416\pi$
$641$	−7.23119e9	−1.08444	−0.542221	+	0.840236i	$0.682416\pi$
$642$	0	0
$643$	−7.61434e9	−1.12952	−0.564760	−	0.825255i	$-0.691031\pi$
$643$	−7.61434e9	−1.12952	−0.564760	+	0.825255i	$0.691031\pi$
$644$	−1.11850e9	−0.165020
$645$	0	0
$646$	−3.01011e8	−0.0439308
$647$	4.63934e8	0.0673428	0.0336714	−	0.999433i	$-0.489280\pi$
$647$	4.63934e8	0.0673428	0.0336714	+	0.999433i	$0.489280\pi$
$648$	0	0
$649$	8.60548e9	1.23572
$650$	1.13267e10	1.61773
$651$	0	0
$652$	−2.13094e9	−0.301096
$653$	−3.44460e9	−0.484108	−0.242054	−	0.970263i	$-0.577821\pi$
$653$	−3.44460e9	−0.484108	−0.242054	+	0.970263i	$0.577821\pi$
$654$	0	0
$655$	2.02374e10	2.81391
$656$	−1.66509e9	−0.230289
$657$	0	0
$658$	−3.00432e8	−0.0411107
$659$	6.21207e9	0.845546	0.422773	−	0.906236i	$-0.361057\pi$
$659$	6.21207e9	0.845546	0.422773	+	0.906236i	$0.361057\pi$
$660$	0	0
$661$	−5.15390e9	−0.694114	−0.347057	−	0.937844i	$-0.612819\pi$
$661$	−5.15390e9	−0.694114	−0.347057	+	0.937844i	$0.612819\pi$
$662$	−8.99173e9	−1.20459
$663$	0	0
$664$	5.71614e8	0.0757730
$665$	−8.33293e8	−0.109881
$666$	0	0
$667$	1.20865e10	1.57710
$668$	−4.62758e9	−0.600671
$669$	0	0
$670$	−1.51763e10	−1.94942
$671$	−6.29585e9	−0.804499
$672$	0	0
$673$	1.51567e9	0.191669	0.0958346	−	0.995397i	$-0.469448\pi$
$673$	1.51567e9	0.191669	0.0958346	+	0.995397i	$0.469448\pi$
$674$	−9.37090e9	−1.17889
$675$	0	0
$676$	4.45769e9	0.555005
$677$	2.55494e9	0.316461	0.158231	−	0.987402i	$-0.449421\pi$
$677$	2.55494e9	0.316461	0.158231	+	0.987402i	$0.449421\pi$
$678$	0	0
$679$	−4.40737e9	−0.540300
$680$	1.59524e9	0.194556
$681$	0	0
$682$	7.28196e9	0.879028
$683$	1.40038e10	1.68179	0.840896	−	0.541197i	$-0.182029\pi$
$683$	1.40038e10	1.68179	0.840896	+	0.541197i	$0.182029\pi$
$684$	0	0
$685$	1.25077e10	1.48683
$686$	−3.22829e8	−0.0381802
$687$	0	0
$688$	−2.32119e9	−0.271739
$689$	6.81722e9	0.794036
$690$	0	0
$691$	−5.80692e9	−0.669534	−0.334767	−	0.942301i	$-0.608658\pi$
$691$	−5.80692e9	−0.669534	−0.334767	+	0.942301i	$0.608658\pi$
$692$	−5.34071e9	−0.612672
$693$	0	0
$694$	−1.10990e10	−1.26045
$695$	5.28716e9	0.597415
$696$	0	0
$697$	2.82390e9	0.315889
$698$	4.72672e9	0.526097
$699$	0	0
$700$	2.70112e9	0.297646
$701$	9.32076e9	1.02197	0.510985	−	0.859589i	$-0.329281\pi$
$701$	9.32076e9	1.02197	0.510985	+	0.859589i	$0.329281\pi$
$702$	0	0
$703$	−1.41316e9	−0.153408
$704$	−1.38927e9	−0.150066
$705$	0	0
$706$	−1.10114e10	−1.17768
$707$	−3.93279e9	−0.418536
$708$	0	0
$709$	−1.15982e10	−1.22216	−0.611080	−	0.791569i	$-0.709265\pi$
$709$	−1.15982e10	−1.22216	−0.611080	+	0.791569i	$0.709265\pi$
$710$	−1.08648e10	−1.13924
$711$	0	0
$712$	4.28040e9	0.444431
$713$	−8.75132e9	−0.904191
$714$	0	0
$715$	2.73511e10	2.79836
$716$	−7.38971e8	−0.0752371
$717$	0	0
$718$	3.17818e8	0.0320437
$719$	−1.50137e9	−0.150638	−0.0753191	−	0.997159i	$-0.523998\pi$
$719$	−1.50137e9	−0.150638	−0.0753191	+	0.997159i	$0.523998\pi$
$720$	0	0
$721$	5.72245e9	0.568602
$722$	6.91626e9	0.683898
$723$	0	0
$724$	3.89259e9	0.381200
$725$	−2.91882e10	−2.84462
$726$	0	0
$727$	5.04679e9	0.487130	0.243565	−	0.969885i	$-0.421683\pi$
$727$	5.04679e9	0.487130	0.243565	+	0.969885i	$0.421683\pi$
$728$	2.02073e9	0.194110
$729$	0	0
$730$	2.10444e10	2.00220
$731$	3.93662e9	0.372746
$732$	0	0
$733$	5.83993e9	0.547701	0.273850	−	0.961772i	$-0.411703\pi$
$733$	5.83993e9	0.547701	0.273850	+	0.961772i	$0.411703\pi$
$734$	−4.92349e9	−0.459555
$735$	0	0
$736$	1.66960e9	0.154362
$737$	−2.24151e10	−2.06255
$738$	0	0
$739$	−1.77537e10	−1.61820	−0.809101	−	0.587669i	$-0.800046\pi$
$739$	−1.77537e10	−1.61820	−0.809101	+	0.587669i	$0.800046\pi$
$740$	7.48917e9	0.679395
$741$	0	0
$742$	1.62573e9	0.146094
$743$	−1.81295e10	−1.62153	−0.810765	−	0.585371i	$-0.800949\pi$
$743$	−1.81295e10	−1.62153	−0.810765	+	0.585371i	$0.800949\pi$
$744$	0	0
$745$	−1.72444e10	−1.52792
$746$	2.37328e9	0.209297
$747$	0	0
$748$	2.35613e9	0.205847
$749$	2.73607e9	0.237925
$750$	0	0
$751$	−2.20680e10	−1.90118	−0.950590	−	0.310450i	$-0.899520\pi$
$751$	−2.20680e10	−1.90118	−0.950590	+	0.310450i	$0.899520\pi$
$752$	4.48458e8	0.0384556
$753$	0	0
$754$	−2.18359e10	−1.85512
$755$	−1.75758e10	−1.48628
$756$	0	0
$757$	−3.25204e9	−0.272471	−0.136236	−	0.990676i	$-0.543500\pi$
$757$	−3.25204e9	−0.272471	−0.136236	+	0.990676i	$0.543500\pi$
$758$	−9.00838e8	−0.0751285
$759$	0	0
$760$	1.24387e9	0.102784
$761$	−5.79031e9	−0.476272	−0.238136	−	0.971232i	$-0.576536\pi$
$761$	−5.79031e9	−0.476272	−0.238136	+	0.971232i	$0.576536\pi$
$762$	0	0
$763$	7.83753e9	0.638768
$764$	−6.79549e9	−0.551307
$765$	0	0
$766$	−1.55066e10	−1.24656
$767$	1.86841e10	1.49516
$768$	0	0
$769$	1.86517e10	1.47903	0.739515	−	0.673140i	$-0.235055\pi$
$769$	1.86517e10	1.47903	0.739515	+	0.673140i	$0.235055\pi$
$770$	6.52251e9	0.514870
$771$	0	0
$772$	−7.43048e9	−0.581241
$773$	−1.15337e10	−0.898133	−0.449066	−	0.893498i	$-0.648243\pi$
$773$	−1.15337e10	−0.898133	−0.449066	+	0.893498i	$0.648243\pi$
$774$	0	0
$775$	2.11340e10	1.63089
$776$	6.57892e9	0.505404
$777$	0	0
$778$	1.31155e10	0.998522
$779$	2.20190e9	0.166884
$780$	0	0
$781$	−1.60470e10	−1.20536
$782$	−2.83156e9	−0.211739
$783$	0	0
$784$	4.81890e8	0.0357143
$785$	−1.59857e10	−1.17947
$786$	0	0
$787$	1.70616e10	1.24769	0.623847	−	0.781547i	$-0.285569\pi$
$787$	1.70616e10	1.24769	0.623847	+	0.781547i	$0.285569\pi$
$788$	4.42596e8	0.0322230
$789$	0	0
$790$	2.17789e10	1.57160
$791$	4.70599e9	0.338091
$792$	0	0
$793$	−1.36694e10	−0.973407
$794$	4.50104e9	0.319111
$795$	0	0
$796$	−8.66597e9	−0.609006
$797$	2.24165e10	1.56843	0.784213	−	0.620491i	$-0.213067\pi$
$797$	2.24165e10	1.56843	0.784213	+	0.620491i	$0.213067\pi$
$798$	0	0
$799$	−7.60562e8	−0.0527498
$800$	−4.03199e9	−0.278423
$801$	0	0
$802$	4.73161e9	0.323891
$803$	3.10821e10	2.11839
$804$	0	0
$805$	−7.83863e9	−0.529608
$806$	1.58105e10	1.06359
$807$	0	0
$808$	5.87052e9	0.391505
$809$	−2.53527e10	−1.68347	−0.841733	−	0.539894i	$-0.818464\pi$
$809$	−2.53527e10	−1.68347	−0.841733	+	0.539894i	$0.818464\pi$
$810$	0	0
$811$	1.47598e10	0.971647	0.485823	−	0.874057i	$-0.338520\pi$
$811$	1.47598e10	0.971647	0.485823	+	0.874057i	$0.338520\pi$
$812$	−5.20729e9	−0.341323
$813$	0	0
$814$	1.10613e10	0.718823
$815$	−1.49340e10	−0.966326
$816$	0	0
$817$	3.06952e9	0.196922
$818$	−8.19452e9	−0.523464
$819$	0	0
$820$	−1.16692e10	−0.739080
$821$	1.90667e10	1.20247	0.601236	−	0.799072i	$-0.294675\pi$
$821$	1.90667e10	1.20247	0.601236	+	0.799072i	$0.294675\pi$
$822$	0	0
$823$	−7.79947e9	−0.487714	−0.243857	−	0.969811i	$-0.578413\pi$
$823$	−7.79947e9	−0.487714	−0.243857	+	0.969811i	$0.578413\pi$
$824$	−8.54197e9	−0.531879
$825$	0	0
$826$	4.45566e9	0.275094
$827$	1.51860e10	0.933628	0.466814	−	0.884356i	$-0.345402\pi$
$827$	1.51860e10	0.933628	0.466814	+	0.884356i	$0.345402\pi$
$828$	0	0
$829$	2.96321e8	0.0180643	0.00903217	−	0.999959i	$-0.497125\pi$
$829$	2.96321e8	0.0180643	0.00903217	+	0.999959i	$0.497125\pi$
$830$	4.00595e9	0.243183
$831$	0	0
$832$	−3.01636e9	−0.181573
$833$	−8.17261e8	−0.0489896
$834$	0	0
$835$	−3.24308e10	−1.92777
$836$	1.83716e9	0.108749
$837$	0	0
$838$	−7.11600e9	−0.417717
$839$	−2.96140e10	−1.73113	−0.865566	−	0.500795i	$-0.833041\pi$
$839$	−2.96140e10	−1.73113	−0.865566	+	0.500795i	$0.833041\pi$
$840$	0	0
$841$	3.90199e10	2.26204
$842$	1.94492e10	1.12282
$843$	0	0
$844$	1.22044e10	0.698742
$845$	3.12402e10	1.78121
$846$	0	0
$847$	2.94950e9	0.166785
$848$	−2.42674e9	−0.136659
$849$	0	0
$850$	6.83805e9	0.381915
$851$	−1.32933e10	−0.739400
$852$	0	0
$853$	−1.19384e10	−0.658606	−0.329303	−	0.944224i	$-0.606814\pi$
$853$	−1.19384e10	−0.658606	−0.329303	+	0.944224i	$0.606814\pi$
$854$	−3.25980e9	−0.179097
$855$	0	0
$856$	−4.08416e9	−0.222559
$857$	1.12597e10	0.611072	0.305536	−	0.952181i	$-0.401164\pi$
$857$	1.12597e10	0.611072	0.305536	+	0.952181i	$0.401164\pi$
$858$	0	0
$859$	−1.27863e10	−0.688284	−0.344142	−	0.938918i	$-0.611830\pi$
$859$	−1.27863e10	−0.688284	−0.344142	+	0.938918i	$0.611830\pi$
$860$	−1.62673e10	−0.872107
$861$	0	0
$862$	1.32159e10	0.702783
$863$	3.27370e10	1.73381	0.866904	−	0.498474i	$-0.166106\pi$
$863$	3.27370e10	1.73381	0.866904	+	0.498474i	$0.166106\pi$
$864$	0	0
$865$	−3.74285e10	−1.96628
$866$	8.63573e8	0.0451842
$867$	0	0
$868$	3.77038e9	0.195689
$869$	3.21670e10	1.66280
$870$	0	0
$871$	−4.86673e10	−2.49559
$872$	−1.16992e10	−0.597513
$873$	0	0
$874$	−2.20787e9	−0.111862
$875$	6.91087e9	0.348742
$876$	0	0
$877$	−7.47496e8	−0.0374205	−0.0187103	−	0.999825i	$-0.505956\pi$
$877$	−7.47496e8	−0.0374205	−0.0187103	+	0.999825i	$0.505956\pi$
$878$	4.39952e9	0.219369
$879$	0	0
$880$	−9.73623e9	−0.481616
$881$	2.71804e10	1.33918	0.669591	−	0.742730i	$-0.266469\pi$
$881$	2.71804e10	1.33918	0.669591	+	0.742730i	$0.266469\pi$
$882$	0	0
$883$	−3.01672e10	−1.47460	−0.737298	−	0.675567i	$-0.763899\pi$
$883$	−3.01672e10	−1.47460	−0.737298	+	0.675567i	$0.763899\pi$
$884$	5.11560e9	0.249066
$885$	0	0
$886$	9.34612e9	0.451454
$887$	4.76420e9	0.229223	0.114611	−	0.993410i	$-0.463438\pi$
$887$	4.76420e9	0.229223	0.114611	+	0.993410i	$0.463438\pi$
$888$	0	0
$889$	−6.26864e9	−0.299238
$890$	2.99977e10	1.42634
$891$	0	0
$892$	1.64242e10	0.774830
$893$	−5.93037e8	−0.0278677
$894$	0	0
$895$	−5.17882e9	−0.241463
$896$	−7.19323e8	−0.0334077
$897$	0	0
$898$	2.24551e10	1.03478
$899$	−4.07426e10	−1.87021
$900$	0	0
$901$	4.11563e9	0.187456
$902$	−1.72351e10	−0.781971
$903$	0	0
$904$	−7.02469e9	−0.316255
$905$	2.72798e10	1.22341
$906$	0	0
$907$	−1.97305e10	−0.878036	−0.439018	−	0.898478i	$-0.644674\pi$
$907$	−1.97305e10	−0.878036	−0.439018	+	0.898478i	$0.644674\pi$
$908$	−6.63785e9	−0.294257
$909$	0	0
$910$	1.41616e10	0.622969
$911$	2.00829e10	0.880059	0.440029	−	0.897983i	$-0.354968\pi$
$911$	2.00829e10	0.880059	0.440029	+	0.897983i	$0.354968\pi$
$912$	0	0
$913$	5.91670e9	0.257295
$914$	1.04490e10	0.452649
$915$	0	0
$916$	−1.40684e8	−0.00604798
$917$	1.54763e10	0.662786
$918$	0	0
$919$	1.19622e10	0.508402	0.254201	−	0.967151i	$-0.418188\pi$
$919$	1.19622e10	0.508402	0.254201	+	0.967151i	$0.418188\pi$
$920$	1.17008e10	0.495403
$921$	0	0
$922$	−2.84781e10	−1.19661
$923$	−3.48410e10	−1.45843
$924$	0	0
$925$	3.21026e10	1.33366
$926$	−1.80305e10	−0.746222
$927$	0	0
$928$	7.77298e9	0.319278
$929$	−2.41229e10	−0.987130	−0.493565	−	0.869709i	$-0.664307\pi$
$929$	−2.41229e10	−0.987130	−0.493565	+	0.869709i	$0.664307\pi$
$930$	0	0
$931$	−6.37248e8	−0.0258812
$932$	1.16470e10	0.471259
$933$	0	0
$934$	1.82048e10	0.731092
$935$	1.65121e10	0.660637
$936$	0	0
$937$	−1.06927e10	−0.424617	−0.212308	−	0.977203i	$-0.568098\pi$
$937$	−1.06927e10	−0.424617	−0.212308	+	0.977203i	$0.568098\pi$
$938$	−1.16059e10	−0.459164
$939$	0	0
$940$	3.14286e9	0.123418
$941$	3.74453e10	1.46499	0.732493	−	0.680775i	$-0.238357\pi$
$941$	3.74453e10	1.46499	0.732493	+	0.680775i	$0.238357\pi$
$942$	0	0
$943$	2.07128e10	0.804356
$944$	−6.65101e9	−0.257327
$945$	0	0
$946$	−2.40264e10	−0.922719
$947$	2.02422e10	0.774522	0.387261	−	0.921970i	$-0.373421\pi$
$947$	2.02422e10	0.774522	0.387261	+	0.921970i	$0.373421\pi$
$948$	0	0
$949$	6.74850e10	2.56316
$950$	5.33187e9	0.201766
$951$	0	0
$952$	1.21994e9	0.0458255
$953$	1.03125e10	0.385955	0.192978	−	0.981203i	$-0.438186\pi$
$953$	1.03125e10	0.385955	0.192978	+	0.981203i	$0.438186\pi$
$954$	0	0
$955$	−4.76238e10	−1.76934
$956$	−1.72283e10	−0.637732
$957$	0	0
$958$	−2.30206e10	−0.845935
$959$	9.56507e9	0.350206
$960$	0	0
$961$	1.98738e9	0.0722353
$962$	2.40162e10	0.869744
$963$	0	0
$964$	1.26226e10	0.453815
$965$	−5.20739e10	−1.86541
$966$	0	0
$967$	5.65967e9	0.201279	0.100640	−	0.994923i	$-0.467911\pi$
$967$	5.65967e9	0.201279	0.100640	+	0.994923i	$0.467911\pi$
$968$	−4.40276e9	−0.156013
$969$	0	0
$970$	4.61061e10	1.62202
$971$	−5.37776e9	−0.188510	−0.0942549	−	0.995548i	$-0.530047\pi$
$971$	−5.37776e9	−0.188510	−0.0942549	+	0.995548i	$0.530047\pi$
$972$	0	0
$973$	4.04328e9	0.140714
$974$	1.23359e10	0.427776
$975$	0	0
$976$	4.86594e9	0.167530
$977$	−5.70574e9	−0.195741	−0.0978703	−	0.995199i	$-0.531203\pi$
$977$	−5.70574e9	−0.195741	−0.0978703	+	0.995199i	$0.531203\pi$
$978$	0	0
$979$	4.43059e10	1.50911
$980$	3.37716e9	0.114620
$981$	0	0
$982$	2.58181e10	0.870030
$983$	−4.81415e10	−1.61653	−0.808263	−	0.588822i	$-0.799592\pi$
$983$	−4.81415e10	−1.61653	−0.808263	+	0.588822i	$0.799592\pi$
$984$	0	0
$985$	3.10178e9	0.103415
$986$	−1.31826e10	−0.437957
$987$	0	0
$988$	3.98882e9	0.131582
$989$	2.88744e10	0.949132
$990$	0	0
$991$	−7.12168e7	−0.00232447	−0.00116224	−	0.999999i	$-0.500370\pi$
$991$	−7.12168e7	−0.00232447	−0.00116224	+	0.999999i	$0.500370\pi$
$992$	−5.62809e9	−0.183050
$993$	0	0
$994$	−8.30867e9	−0.268336
$995$	−6.07324e10	−1.95452
$996$	0	0
$997$	1.21125e10	0.387080	0.193540	−	0.981092i	$-0.438003\pi$
$997$	1.21125e10	0.387080	0.193540	+	0.981092i	$0.438003\pi$
$998$	1.84227e10	0.586674
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	126.8.a.k.1.2	✓	2
3.2	odd	2		126.8.a.l.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.8.a.k.1.2	✓	2	1.1	even	1	trivial
126.8.a.l.1.1	yes	2	3.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	126.a (trivial)

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} +64.0000 q^{4} +448.522 q^{5} +343.000 q^{7} -512.000 q^{8} -3588.17 q^{10} -5299.65 q^{11} -11506.5 q^{13} -2744.00 q^{14} +4096.00 q^{16} -6946.61 q^{17} -5416.52 q^{19} +28705.4 q^{20} +42397.2 q^{22} -50952.2 q^{23} +123047. q^{25} +92052.1 q^{26} +21952.0 q^{28} -237212. q^{29} +171756. q^{31} -32768.0 q^{32} +55572.9 q^{34} +153843. q^{35} +260898. q^{37} +43332.1 q^{38} -229643. q^{40} -406515. q^{41} -566697. q^{43} -339178. q^{44} +407617. q^{46} +109487. q^{47} +117649. q^{49} -984373. q^{50} -736417. q^{52} -592466. q^{53} -2.37701e6 q^{55} -175616. q^{56} +1.89770e6 q^{58} -1.62378e6 q^{59} +1.18797e6 q^{61} -1.37405e6 q^{62} +262144. q^{64} -5.16092e6 q^{65} +4.22954e6 q^{67} -444583. q^{68} -1.23074e6 q^{70} +3.02794e6 q^{71} -5.86494e6 q^{73} -2.08718e6 q^{74} -346657. q^{76} -1.81778e6 q^{77} -6.06964e6 q^{79} +1.83714e6 q^{80} +3.25212e6 q^{82} -1.11643e6 q^{83} -3.11570e6 q^{85} +4.53358e6 q^{86} +2.71342e6 q^{88} -8.36015e6 q^{89} -3.94674e6 q^{91} -3.26094e6 q^{92} -875894. q^{94} -2.42943e6 q^{95} -1.28495e7 q^{97} -941192. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 16 q^{2} + 128 q^{4} + 168 q^{5} + 686 q^{7} - 1024 q^{8} - 1344 q^{10} - 5496 q^{11} - 5516 q^{13} - 5488 q^{14} + 8192 q^{16} - 10248 q^{17} + 6664 q^{19} + 10752 q^{20} + 43968 q^{22} - 45768 q^{23}+ \cdots - 1882384 q^{98}+O(q^{100}) \) 2 * q - 16 * q^2 + 128 * q^4 + 168 * q^5 + 686 * q^7 - 1024 * q^8 - 1344 * q^10 - 5496 * q^11 - 5516 * q^13 - 5488 * q^14 + 8192 * q^16 - 10248 * q^17 + 6664 * q^19 + 10752 * q^20 + 43968 * q^22 - 45768 * q^23 + 123614 * q^25 + 44128 * q^26 + 43904 * q^28 - 188640 * q^29 + 221032 * q^31 - 65536 * q^32 + 81984 * q^34 + 57624 * q^35 + 399316 * q^37 - 53312 * q^38 - 86016 * q^40 - 562968 * q^41 - 153560 * q^43 - 351744 * q^44 + 366144 * q^46 - 651504 * q^47 + 235298 * q^49 - 988912 * q^50 - 353024 * q^52 - 2032080 * q^53 - 2321928 * q^55 - 351232 * q^56 + 1509120 * q^58 - 3864336 * q^59 - 721028 * q^61 - 1768256 * q^62 + 524288 * q^64 - 6841392 * q^65 - 604376 * q^67 - 655872 * q^68 - 460992 * q^70 - 389592 * q^71 - 6935684 * q^73 - 3194528 * q^74 + 426496 * q^76 - 1885128 * q^77 - 7974992 * q^79 + 688128 * q^80 + 4503744 * q^82 - 2209536 * q^83 - 2189592 * q^85 + 1228480 * q^86 + 2813952 * q^88 - 1659000 * q^89 - 1891988 * q^91 - 2929152 * q^92 + 5212032 * q^94 - 5818272 * q^95 - 4807460 * q^97 - 1882384 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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