LMFDB - Embedded newform 114.8.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [114,8,Mod(1,114)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(114, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("114.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 114 = 2 \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	114.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:	$35.6118929052$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	114.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -140.000 q^{5} -216.000 q^{6} -60.0000 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})$

\(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -140.000 q^{5} -216.000 q^{6} -60.0000 q^{7} +512.000 q^{8} +729.000 q^{9} -1120.00 q^{10} +3976.00 q^{11} -1728.00 q^{12} +1592.00 q^{13} -480.000 q^{14} +3780.00 q^{15} +4096.00 q^{16} -23682.0 q^{17} +5832.00 q^{18} -6859.00 q^{19} -8960.00 q^{20} +1620.00 q^{21} +31808.0 q^{22} -70370.0 q^{23} -13824.0 q^{24} -58525.0 q^{25} +12736.0 q^{26} -19683.0 q^{27} -3840.00 q^{28} +97502.0 q^{29} +30240.0 q^{30} -234890. q^{31} +32768.0 q^{32} -107352. q^{33} -189456. q^{34} +8400.00 q^{35} +46656.0 q^{36} +146968. q^{37} -54872.0 q^{38} -42984.0 q^{39} -71680.0 q^{40} -232282. q^{41} +12960.0 q^{42} -409028. q^{43} +254464. q^{44} -102060. q^{45} -562960. q^{46} -891710. q^{47} -110592. q^{48} -819943. q^{49} -468200. q^{50} +639414. q^{51} +101888. q^{52} +30862.0 q^{53} -157464. q^{54} -556640. q^{55} -30720.0 q^{56} +185193. q^{57} +780016. q^{58} -374820. q^{59} +241920. q^{60} -1.54811e6 q^{61} -1.87912e6 q^{62} -43740.0 q^{63} +262144. q^{64} -222880. q^{65} -858816. q^{66} +3.94493e6 q^{67} -1.51565e6 q^{68} +1.89999e6 q^{69} +67200.0 q^{70} +477232. q^{71} +373248. q^{72} +514206. q^{73} +1.17574e6 q^{74} +1.58018e6 q^{75} -438976. q^{76} -238560. q^{77} -343872. q^{78} -4.86638e6 q^{79} -573440. q^{80} +531441. q^{81} -1.85826e6 q^{82} +2.04320e6 q^{83} +103680. q^{84} +3.31548e6 q^{85} -3.27222e6 q^{86} -2.63255e6 q^{87} +2.03571e6 q^{88} -7.82242e6 q^{89} -816480. q^{90} -95520.0 q^{91} -4.50368e6 q^{92} +6.34203e6 q^{93} -7.13368e6 q^{94} +960260. q^{95} -884736. q^{96} -2.74709e6 q^{97} -6.55954e6 q^{98} +2.89850e6 q^{99} +O(q^{100})\)

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 $

$2$	8.00000	0.707107
$2$	8.00000	0.707107
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	64.0000	0.500000
$5$	−140.000	−0.500879	−0.250440	−	0.968132i	$-0.580575\pi$
$5$	−140.000	−0.500879	−0.250440	+	0.968132i	$0.580575\pi$
$6$	−216.000	−0.408248
$7$	−60.0000	−0.0661162	−0.0330581	−	0.999453i	$-0.510525\pi$
$7$	−60.0000	−0.0661162	−0.0330581	+	0.999453i	$0.510525\pi$
$8$	512.000	0.353553
$9$	729.000	0.333333
$10$	−1120.00	−0.354175
$11$	3976.00	0.900683	0.450342	−	0.892856i	$-0.351302\pi$
$11$	3976.00	0.900683	0.450342	+	0.892856i	$0.351302\pi$
$12$	−1728.00	−0.288675
$13$	1592.00	0.200975	0.100487	−	0.994938i	$-0.467960\pi$
$13$	1592.00	0.200975	0.100487	+	0.994938i	$0.467960\pi$
$14$	−480.000	−0.0467512
$15$	3780.00	0.289183
$16$	4096.00	0.250000
$17$	−23682.0	−1.16909	−0.584544	−	0.811362i	$-0.698727\pi$
$17$	−23682.0	−1.16909	−0.584544	+	0.811362i	$0.698727\pi$
$18$	5832.00	0.235702
$19$	−6859.00	−0.229416
$19$	−6859.00	−0.229416
$20$	−8960.00	−0.250440
$21$	1620.00	0.0381722
$22$	31808.0	0.636879
$23$	−70370.0	−1.20598	−0.602990	−	0.797749i	$-0.706024\pi$
$23$	−70370.0	−1.20598	−0.602990	+	0.797749i	$0.706024\pi$
$24$	−13824.0	−0.204124
$25$	−58525.0	−0.749120
$26$	12736.0	0.142111
$27$	−19683.0	−0.192450
$28$	−3840.00	−0.0330581
$29$	97502.0	0.742370	0.371185	−	0.928559i	$-0.378952\pi$
$29$	97502.0	0.742370	0.371185	+	0.928559i	$0.378952\pi$
$30$	30240.0	0.204483
$31$	−234890.	−1.41612	−0.708058	−	0.706155i	$-0.750428\pi$
$31$	−234890.	−1.41612	−0.708058	+	0.706155i	$0.750428\pi$
$32$	32768.0	0.176777
$33$	−107352.	−0.520010
$34$	−189456.	−0.826670
$35$	8400.00	0.0331162
$36$	46656.0	0.166667
$37$	146968.	0.476998	0.238499	−	0.971143i	$-0.423345\pi$
$37$	146968.	0.476998	0.238499	+	0.971143i	$0.423345\pi$
$38$	−54872.0	−0.162221
$39$	−42984.0	−0.116033
$40$	−71680.0	−0.177088
$41$	−232282.	−0.526347	−0.263173	−	0.964749i	$-0.584769\pi$
$41$	−232282.	−0.526347	−0.263173	+	0.964749i	$0.584769\pi$
$42$	12960.0	0.0269918
$43$	−409028.	−0.784537	−0.392269	−	0.919851i	$-0.628310\pi$
$43$	−409028.	−0.784537	−0.392269	+	0.919851i	$0.628310\pi$
$44$	254464.	0.450342
$45$	−102060.	−0.166960
$46$	−562960.	−0.852757
$47$	−891710.	−1.25280	−0.626399	−	0.779502i	$-0.715472\pi$
$47$	−891710.	−1.25280	−0.626399	+	0.779502i	$0.715472\pi$
$48$	−110592.	−0.144338
$49$	−819943.	−0.995629
$50$	−468200.	−0.529708
$51$	639414.	0.674973
$52$	101888.	0.100487
$53$	30862.0	0.0284747	0.0142373	−	0.999899i	$-0.495468\pi$
$53$	30862.0	0.0284747	0.0142373	+	0.999899i	$0.495468\pi$
$54$	−157464.	−0.136083
$55$	−556640.	−0.451133
$56$	−30720.0	−0.0233756
$57$	185193.	0.132453
$58$	780016.	0.524935
$59$	−374820.	−0.237597	−0.118798	−	0.992918i	$-0.537904\pi$
$59$	−374820.	−0.237597	−0.118798	+	0.992918i	$0.537904\pi$
$60$	241920.	0.144591
$61$	−1.54811e6	−0.873265	−0.436633	−	0.899640i	$-0.643829\pi$
$61$	−1.54811e6	−0.873265	−0.436633	+	0.899640i	$0.643829\pi$
$62$	−1.87912e6	−1.00134
$63$	−43740.0	−0.0220387
$64$	262144.	0.125000
$65$	−222880.	−0.100664
$66$	−858816.	−0.367702
$67$	3.94493e6	1.60242	0.801212	−	0.598381i	$-0.204189\pi$
$67$	3.94493e6	1.60242	0.801212	+	0.598381i	$0.204189\pi$
$68$	−1.51565e6	−0.584544
$69$	1.89999e6	0.696273
$70$	67200.0	0.0234167
$71$	477232.	0.158243	0.0791217	−	0.996865i	$-0.474788\pi$
$71$	477232.	0.158243	0.0791217	+	0.996865i	$0.474788\pi$
$72$	373248.	0.117851
$73$	514206.	0.154706	0.0773530	−	0.997004i	$-0.475353\pi$
$73$	514206.	0.154706	0.0773530	+	0.997004i	$0.475353\pi$
$74$	1.17574e6	0.337289
$75$	1.58018e6	0.432505
$76$	−438976.	−0.114708
$77$	−238560.	−0.0595498
$78$	−343872.	−0.0820476
$79$	−4.86638e6	−1.11048	−0.555241	−	0.831690i	$-0.687374\pi$
$79$	−4.86638e6	−1.11048	−0.555241	+	0.831690i	$0.687374\pi$
$80$	−573440.	−0.125220
$81$	531441.	0.111111
$82$	−1.85826e6	−0.372183
$83$	2.04320e6	0.392228	0.196114	−	0.980581i	$-0.437168\pi$
$83$	2.04320e6	0.392228	0.196114	+	0.980581i	$0.437168\pi$
$84$	103680.	0.0190861
$85$	3.31548e6	0.585572
$86$	−3.27222e6	−0.554751
$87$	−2.63255e6	−0.428608
$88$	2.03571e6	0.318440
$89$	−7.82242e6	−1.17619	−0.588093	−	0.808793i	$-0.700121\pi$
$89$	−7.82242e6	−1.17619	−0.588093	+	0.808793i	$0.700121\pi$
$90$	−816480.	−0.118058
$91$	−95520.0	−0.0132877
$92$	−4.50368e6	−0.602990
$93$	6.34203e6	0.817595
$94$	−7.13368e6	−0.885862
$95$	960260.	0.114910
$96$	−884736.	−0.102062
$97$	−2.74709e6	−0.305613	−0.152806	−	0.988256i	$-0.548831\pi$
$97$	−2.74709e6	−0.305613	−0.152806	+	0.988256i	$0.548831\pi$
$98$	−6.55954e6	−0.704016
$99$	2.89850e6	0.300228
$100$	−3.74560e6	−0.374560
$101$	5.01197e6	0.484043	0.242021	−	0.970271i	$-0.422190\pi$
$101$	5.01197e6	0.484043	0.242021	+	0.970271i	$0.422190\pi$
$102$	5.11531e6	0.477278
$103$	1.19905e7	1.08120	0.540602	−	0.841279i	$-0.318197\pi$
$103$	1.19905e7	1.08120	0.540602	+	0.841279i	$0.318197\pi$
$104$	815104.	0.0710553
$105$	−226800.	−0.0191197
$106$	246896.	0.0201346
$107$	−1.58310e7	−1.24930	−0.624648	−	0.780906i	$-0.714758\pi$
$107$	−1.58310e7	−1.24930	−0.624648	+	0.780906i	$0.714758\pi$
$108$	−1.25971e6	−0.0962250
$109$	−1.16711e7	−0.863216	−0.431608	−	0.902061i	$-0.642054\pi$
$109$	−1.16711e7	−0.863216	−0.431608	+	0.902061i	$0.642054\pi$
$110$	−4.45312e6	−0.318999
$111$	−3.96814e6	−0.275395
$112$	−245760.	−0.0165291
$113$	2.50559e7	1.63356	0.816782	−	0.576946i	$-0.195756\pi$
$113$	2.50559e7	1.63356	0.816782	+	0.576946i	$0.195756\pi$
$114$	1.48154e6	0.0936586
$115$	9.85180e6	0.604050
$116$	6.24013e6	0.371185
$117$	1.16057e6	0.0669916
$118$	−2.99856e6	−0.168006
$119$	1.42092e6	0.0772957
$120$	1.93536e6	0.102242
$121$	−3.67859e6	−0.188770
$122$	−1.23848e7	−0.617492
$123$	6.27161e6	0.303887
$124$	−1.50330e7	−0.708058
$125$	1.91310e7	0.876098
$126$	−349920.	−0.0155837
$127$	3.48760e7	1.51082	0.755411	−	0.655251i	$-0.227437\pi$
$127$	3.48760e7	1.51082	0.755411	+	0.655251i	$0.227437\pi$
$128$	2.09715e6	0.0883883
$129$	1.10438e7	0.452953
$130$	−1.78304e6	−0.0711802
$131$	4.10350e7	1.59480	0.797398	−	0.603454i	$-0.206209\pi$
$131$	4.10350e7	1.59480	0.797398	+	0.603454i	$0.206209\pi$
$132$	−6.87053e6	−0.260005
$133$	411540.	0.0151681
$134$	3.15594e7	1.13308
$135$	2.75562e6	0.0963943
$136$	−1.21252e7	−0.413335
$137$	3.26488e7	1.08479	0.542395	−	0.840124i	$-0.317518\pi$
$137$	3.26488e7	1.08479	0.542395	+	0.840124i	$0.317518\pi$
$138$	1.51999e7	0.492339
$139$	1.46257e6	0.0461919	0.0230959	−	0.999733i	$-0.492648\pi$
$139$	1.46257e6	0.0461919	0.0230959	+	0.999733i	$0.492648\pi$
$140$	537600.	0.0165581
$141$	2.40762e7	0.723303
$142$	3.81786e6	0.111895
$143$	6.32979e6	0.181014
$144$	2.98598e6	0.0833333
$145$	−1.36503e7	−0.371838
$146$	4.11365e6	0.109394
$147$	2.21385e7	0.574826
$148$	9.40595e6	0.238499
$149$	7.26156e6	0.179837	0.0899183	−	0.995949i	$-0.471339\pi$
$149$	7.26156e6	0.179837	0.0899183	+	0.995949i	$0.471339\pi$
$150$	1.26414e7	0.305827
$151$	−1.89085e7	−0.446928	−0.223464	−	0.974712i	$-0.571737\pi$
$151$	−1.89085e7	−0.446928	−0.223464	+	0.974712i	$0.571737\pi$
$152$	−3.51181e6	−0.0811107
$153$	−1.72642e7	−0.389696
$154$	−1.90848e6	−0.0421080
$155$	3.28846e7	0.709303
$156$	−2.75098e6	−0.0580164
$157$	−3.83686e7	−0.791275	−0.395637	−	0.918407i	$-0.629476\pi$
$157$	−3.83686e7	−0.791275	−0.395637	+	0.918407i	$0.629476\pi$
$158$	−3.89311e7	−0.785229
$159$	−833274.	−0.0164399
$160$	−4.58752e6	−0.0885438
$161$	4.22220e6	0.0797349
$162$	4.25153e6	0.0785674
$163$	−3.82161e7	−0.691178	−0.345589	−	0.938386i	$-0.612321\pi$
$163$	−3.82161e7	−0.691178	−0.345589	+	0.938386i	$0.612321\pi$
$164$	−1.48660e7	−0.263173
$165$	1.50293e7	0.260462
$166$	1.63456e7	0.277347
$167$	3.85662e7	0.640766	0.320383	−	0.947288i	$-0.396188\pi$
$167$	3.85662e7	0.640766	0.320383	+	0.947288i	$0.396188\pi$
$168$	829440.	0.0134959
$169$	−6.02141e7	−0.959609
$170$	2.65238e7	0.414062
$171$	−5.00021e6	−0.0764719
$172$	−2.61778e7	−0.392269
$173$	−1.92761e6	−0.0283047	−0.0141524	−	0.999900i	$-0.504505\pi$
$173$	−1.92761e6	−0.0283047	−0.0141524	+	0.999900i	$0.504505\pi$
$174$	−2.10604e7	−0.303071
$175$	3.51150e6	0.0495290
$176$	1.62857e7	0.225171
$177$	1.01201e7	0.137177
$178$	−6.25794e7	−0.831690
$179$	2.03169e7	0.264772	0.132386	−	0.991198i	$-0.457736\pi$
$179$	2.03169e7	0.264772	0.132386	+	0.991198i	$0.457736\pi$
$180$	−6.53184e6	−0.0834799
$181$	−9.01240e7	−1.12971	−0.564853	−	0.825192i	$-0.691067\pi$
$181$	−9.01240e7	−1.12971	−0.564853	+	0.825192i	$0.691067\pi$
$182$	−764160.	−0.00939581
$183$	4.17989e7	0.504180
$184$	−3.60294e7	−0.426378
$185$	−2.05755e7	−0.238918
$186$	5.07362e7	0.578127
$187$	−9.41596e7	−1.05298
$188$	−5.70694e7	−0.626399
$189$	1.18098e6	0.0127241
$190$	7.68208e6	0.0812533
$191$	1.71867e7	0.178475	0.0892373	−	0.996010i	$-0.471557\pi$
$191$	1.71867e7	0.178475	0.0892373	+	0.996010i	$0.471557\pi$
$192$	−7.07789e6	−0.0721688
$193$	−2.27116e7	−0.227404	−0.113702	−	0.993515i	$-0.536271\pi$
$193$	−2.27116e7	−0.227404	−0.113702	+	0.993515i	$0.536271\pi$
$194$	−2.19767e7	−0.216101
$195$	6.01776e6	0.0581184
$196$	−5.24764e7	−0.497814
$197$	−3.89071e7	−0.362574	−0.181287	−	0.983430i	$-0.558026\pi$
$197$	−3.89071e7	−0.362574	−0.181287	+	0.983430i	$0.558026\pi$
$198$	2.31880e7	0.212293
$199$	3.15005e7	0.283356	0.141678	−	0.989913i	$-0.454750\pi$
$199$	3.15005e7	0.283356	0.141678	+	0.989913i	$0.454750\pi$
$200$	−2.99648e7	−0.264854
$201$	−1.06513e8	−0.925160
$202$	4.00958e7	0.342270
$203$	−5.85012e6	−0.0490827
$204$	4.09225e7	0.337487
$205$	3.25195e7	0.263636
$206$	9.59241e7	0.764526
$207$	−5.12997e7	−0.401993
$208$	6.52083e6	0.0502437
$209$	−2.72714e7	−0.206631
$210$	−1.81440e6	−0.0135197
$211$	−2.13254e8	−1.56282	−0.781411	−	0.624017i	$-0.785500\pi$
$211$	−2.13254e8	−1.56282	−0.781411	+	0.624017i	$0.785500\pi$
$212$	1.97517e6	0.0142373
$213$	−1.28853e7	−0.0913618
$214$	−1.26648e8	−0.883386
$215$	5.72639e7	0.392958
$216$	−1.00777e7	−0.0680414
$217$	1.40934e7	0.0936282
$218$	−9.33689e7	−0.610386
$219$	−1.38836e7	−0.0893195
$220$	−3.56250e7	−0.225567
$221$	−3.77017e7	−0.234957
$222$	−3.17451e7	−0.194734
$223$	−1.54287e8	−0.931669	−0.465835	−	0.884872i	$-0.654246\pi$
$223$	−1.54287e8	−0.931669	−0.465835	+	0.884872i	$0.654246\pi$
$224$	−1.96608e6	−0.0116878
$225$	−4.26647e7	−0.249707
$226$	2.00448e8	1.15510
$227$	2.57804e8	1.46285	0.731424	−	0.681923i	$-0.238856\pi$
$227$	2.57804e8	1.46285	0.731424	+	0.681923i	$0.238856\pi$
$228$	1.18524e7	0.0662266
$229$	−2.20286e8	−1.21217	−0.606083	−	0.795401i	$-0.707260\pi$
$229$	−2.20286e8	−1.21217	−0.606083	+	0.795401i	$0.707260\pi$
$230$	7.88144e7	0.427128
$231$	6.44112e6	0.0343811
$232$	4.99210e7	0.262468
$233$	−8.40759e7	−0.435438	−0.217719	−	0.976012i	$-0.569862\pi$
$233$	−8.40759e7	−0.435438	−0.217719	+	0.976012i	$0.569862\pi$
$234$	9.28454e6	0.0473702
$235$	1.24839e8	0.627501
$236$	−2.39885e7	−0.118798
$237$	1.31392e8	0.641137
$238$	1.13674e7	0.0546563
$239$	3.44606e8	1.63279	0.816394	−	0.577496i	$-0.195970\pi$
$239$	3.44606e8	1.63279	0.816394	+	0.577496i	$0.195970\pi$
$240$	1.54829e7	0.0722957
$241$	−3.55672e8	−1.63678	−0.818390	−	0.574663i	$-0.805133\pi$
$241$	−3.55672e8	−1.63678	−0.818390	+	0.574663i	$0.805133\pi$
$242$	−2.94288e7	−0.133481
$243$	−1.43489e7	−0.0641500
$244$	−9.90788e7	−0.436633
$245$	1.14792e8	0.498690
$246$	5.01729e7	0.214880
$247$	−1.09195e7	−0.0461068
$248$	−1.20264e8	−0.500672
$249$	−5.51665e7	−0.226453
$250$	1.53048e8	0.619495
$251$	−3.56122e8	−1.42148	−0.710741	−	0.703454i	$-0.751640\pi$
$251$	−3.56122e8	−1.42148	−0.710741	+	0.703454i	$0.751640\pi$
$252$	−2.79936e6	−0.0110194
$253$	−2.79791e8	−1.08621
$254$	2.79008e8	1.06831
$255$	−8.95180e7	−0.338080
$256$	1.67772e7	0.0625000
$257$	−3.42186e8	−1.25747	−0.628733	−	0.777621i	$-0.716426\pi$
$257$	−3.42186e8	−1.25747	−0.628733	+	0.777621i	$0.716426\pi$
$258$	8.83500e7	0.320286
$259$	−8.81808e6	−0.0315373
$260$	−1.42643e7	−0.0503320
$261$	7.10790e7	0.247457
$262$	3.28280e8	1.12769
$263$	−4.42164e7	−0.149878	−0.0749390	−	0.997188i	$-0.523876\pi$
$263$	−4.42164e7	−0.149878	−0.0749390	+	0.997188i	$0.523876\pi$
$264$	−5.49642e7	−0.183851
$265$	−4.32068e6	−0.0142624
$266$	3.29232e6	0.0107255
$267$	2.11205e8	0.679072
$268$	2.52475e8	0.801212
$269$	1.69227e8	0.530075	0.265038	−	0.964238i	$-0.414616\pi$
$269$	1.69227e8	0.530075	0.265038	+	0.964238i	$0.414616\pi$
$270$	2.20450e7	0.0681610
$271$	5.16570e8	1.57666	0.788328	−	0.615256i	$-0.210947\pi$
$271$	5.16570e8	1.57666	0.788328	+	0.615256i	$0.210947\pi$
$272$	−9.70015e7	−0.292272
$273$	2.57904e6	0.00767165
$274$	2.61191e8	0.767062
$275$	−2.32695e8	−0.674720
$276$	1.21599e8	0.348136
$277$	1.22133e8	0.345267	0.172634	−	0.984986i	$-0.444772\pi$
$277$	1.22133e8	0.345267	0.172634	+	0.984986i	$0.444772\pi$
$278$	1.17006e7	0.0326626
$279$	−1.71235e8	−0.472038
$280$	4.30080e6	0.0117084
$281$	5.90978e8	1.58891	0.794454	−	0.607324i	$-0.207757\pi$
$281$	5.90978e8	1.58891	0.794454	+	0.607324i	$0.207757\pi$
$282$	1.92609e8	0.511453
$283$	−7.94587e7	−0.208396	−0.104198	−	0.994557i	$-0.533228\pi$
$283$	−7.94587e7	−0.208396	−0.104198	+	0.994557i	$0.533228\pi$
$284$	3.05428e7	0.0791217
$285$	−2.59270e7	−0.0663431
$286$	5.06383e7	0.127997
$287$	1.39369e7	0.0348001
$288$	2.38879e7	0.0589256
$289$	1.50498e8	0.366766
$290$	−1.09202e8	−0.262929
$291$	7.41714e7	0.176446
$292$	3.29092e7	0.0773530
$293$	2.60799e8	0.605717	0.302859	−	0.953035i	$-0.402059\pi$
$293$	2.60799e8	0.605717	0.302859	+	0.953035i	$0.402059\pi$
$294$	1.77108e8	0.406464
$295$	5.24748e7	0.119007
$296$	7.52476e7	0.168644
$297$	−7.82596e7	−0.173337
$298$	5.80925e7	0.127164
$299$	−1.12029e8	−0.242371
$300$	1.01131e8	0.216252
$301$	2.45417e7	0.0518706
$302$	−1.51268e8	−0.316026
$303$	−1.35323e8	−0.279462
$304$	−2.80945e7	−0.0573539
$305$	2.16735e8	0.437401
$306$	−1.38113e8	−0.275557
$307$	−5.34239e8	−1.05378	−0.526892	−	0.849932i	$-0.676643\pi$
$307$	−5.34239e8	−1.05378	−0.526892	+	0.849932i	$0.676643\pi$
$308$	−1.52678e7	−0.0297749
$309$	−3.23744e8	−0.624233
$310$	2.63077e8	0.501553
$311$	4.80186e7	0.0905208	0.0452604	−	0.998975i	$-0.485588\pi$
$311$	4.80186e7	0.0905208	0.0452604	+	0.998975i	$0.485588\pi$
$312$	−2.20078e7	−0.0410238
$313$	1.09816e8	0.202423	0.101212	−	0.994865i	$-0.467728\pi$
$313$	1.09816e8	0.202423	0.101212	+	0.994865i	$0.467728\pi$
$314$	−3.06949e8	−0.559516
$315$	6.12360e6	0.0110387
$316$	−3.11448e8	−0.555241
$317$	1.34389e8	0.236949	0.118475	−	0.992957i	$-0.462200\pi$
$317$	1.34389e8	0.236949	0.118475	+	0.992957i	$0.462200\pi$
$318$	−6.66619e6	−0.0116247
$319$	3.87668e8	0.668640
$320$	−3.67002e7	−0.0626099
$321$	4.27438e8	0.721282
$322$	3.37776e7	0.0563811
$323$	1.62435e8	0.268207
$324$	3.40122e7	0.0555556
$325$	−9.31718e7	−0.150554
$326$	−3.05729e8	−0.488737
$327$	3.15120e8	0.498378
$328$	−1.18928e8	−0.186092
$329$	5.35026e7	0.0828303
$330$	1.20234e8	0.184174
$331$	7.58407e8	1.14949	0.574744	−	0.818334i	$-0.305102\pi$
$331$	7.58407e8	1.14949	0.574744	+	0.818334i	$0.305102\pi$
$332$	1.30765e8	0.196114
$333$	1.07140e8	0.158999
$334$	3.08530e8	0.453090
$335$	−5.52290e8	−0.802621
$336$	6.63552e6	0.00954306
$337$	8.07942e8	1.14994	0.574970	−	0.818174i	$-0.305014\pi$
$337$	8.07942e8	1.14994	0.574970	+	0.818174i	$0.305014\pi$
$338$	−4.81712e8	−0.678546
$339$	−6.76511e8	−0.943139
$340$	2.12191e8	0.292786
$341$	−9.33923e8	−1.27547
$342$	−4.00017e7	−0.0540738
$343$	9.86092e7	0.131943
$344$	−2.09422e8	−0.277376
$345$	−2.65999e8	−0.348749
$346$	−1.54209e7	−0.0200145
$347$	6.66016e8	0.855720	0.427860	−	0.903845i	$-0.359268\pi$
$347$	6.66016e8	0.855720	0.427860	+	0.903845i	$0.359268\pi$
$348$	−1.68483e8	−0.214304
$349$	−4.33016e8	−0.545275	−0.272637	−	0.962117i	$-0.587896\pi$
$349$	−4.33016e8	−0.545275	−0.272637	+	0.962117i	$0.587896\pi$
$350$	2.80920e7	0.0350223
$351$	−3.13353e7	−0.0386776
$352$	1.30286e8	0.159220
$353$	−4.89896e8	−0.592779	−0.296389	−	0.955067i	$-0.595783\pi$
$353$	−4.89896e8	−0.592779	−0.296389	+	0.955067i	$0.595783\pi$
$354$	8.09611e7	0.0969985
$355$	−6.68125e7	−0.0792608
$356$	−5.00635e8	−0.588093
$357$	−3.83648e7	−0.0446267
$358$	1.62535e8	0.187222
$359$	2.29170e8	0.261413	0.130706	−	0.991421i	$-0.458275\pi$
$359$	2.29170e8	0.261413	0.130706	+	0.991421i	$0.458275\pi$
$360$	−5.22547e7	−0.0590292
$361$	4.70459e7	0.0526316
$362$	−7.20992e8	−0.798823
$363$	9.93221e7	0.108986
$364$	−6.11328e6	−0.00664384
$365$	−7.19888e7	−0.0774890
$366$	3.34391e8	0.356509
$367$	8.21190e8	0.867186	0.433593	−	0.901109i	$-0.357246\pi$
$367$	8.21190e8	0.867186	0.433593	+	0.901109i	$0.357246\pi$
$368$	−2.88236e8	−0.301495
$369$	−1.69334e8	−0.175449
$370$	−1.64604e8	−0.168941
$371$	−1.85172e6	−0.00188264
$372$	4.05890e8	0.408797
$373$	−4.08014e8	−0.407094	−0.203547	−	0.979065i	$-0.565247\pi$
$373$	−4.08014e8	−0.407094	−0.203547	+	0.979065i	$0.565247\pi$
$374$	−7.53277e8	−0.744568
$375$	−5.16537e8	−0.505815
$376$	−4.56556e8	−0.442931
$377$	1.55223e8	0.149198
$378$	9.44784e6	0.00899728
$379$	4.46874e8	0.421645	0.210823	−	0.977524i	$-0.432386\pi$
$379$	4.46874e8	0.421645	0.210823	+	0.977524i	$0.432386\pi$
$380$	6.14566e7	0.0574548
$381$	−9.41652e8	−0.872274
$382$	1.37494e8	0.126201
$383$	6.45580e8	0.587157	0.293579	−	0.955935i	$-0.405154\pi$
$383$	6.45580e8	0.587157	0.293579	+	0.955935i	$0.405154\pi$
$384$	−5.66231e7	−0.0510310
$385$	3.33984e7	0.0298272
$386$	−1.81693e8	−0.160799
$387$	−2.98181e8	−0.261512
$388$	−1.75814e8	−0.152806
$389$	−5.57343e8	−0.480064	−0.240032	−	0.970765i	$-0.577158\pi$
$389$	−5.57343e8	−0.480064	−0.240032	+	0.970765i	$0.577158\pi$
$390$	4.81421e7	0.0410959
$391$	1.66650e9	1.40990
$392$	−4.19811e8	−0.352008
$393$	−1.10795e9	−0.920756
$394$	−3.11257e8	−0.256379
$395$	6.81293e8	0.556217
$396$	1.85504e8	0.150114
$397$	8.63277e8	0.692442	0.346221	−	0.938153i	$-0.387465\pi$
$397$	8.63277e8	0.692442	0.346221	+	0.938153i	$0.387465\pi$
$398$	2.52004e8	0.200363
$399$	−1.11116e7	−0.00875731
$400$	−2.39718e8	−0.187280
$401$	−7.40467e8	−0.573456	−0.286728	−	0.958012i	$-0.592568\pi$
$401$	−7.40467e8	−0.573456	−0.286728	+	0.958012i	$0.592568\pi$
$402$	−8.52104e8	−0.654187
$403$	−3.73945e8	−0.284603
$404$	3.20766e8	0.242021
$405$	−7.44017e7	−0.0556532
$406$	−4.68010e7	−0.0347067
$407$	5.84345e8	0.429624
$408$	3.27380e8	0.238639
$409$	5.71726e8	0.413196	0.206598	−	0.978426i	$-0.433761\pi$
$409$	5.71726e8	0.413196	0.206598	+	0.978426i	$0.433761\pi$
$410$	2.60156e8	0.186419
$411$	−8.81518e8	−0.626304
$412$	7.67393e8	0.540602
$413$	2.24892e7	0.0157090
$414$	−4.10398e8	−0.284252
$415$	−2.86049e8	−0.196459
$416$	5.21667e7	0.0355276
$417$	−3.94894e7	−0.0266689
$418$	−2.18171e8	−0.146110
$419$	7.81791e8	0.519208	0.259604	−	0.965715i	$-0.416408\pi$
$419$	7.81791e8	0.519208	0.259604	+	0.965715i	$0.416408\pi$
$420$	−1.45152e7	−0.00955984
$421$	−2.77148e9	−1.81019	−0.905096	−	0.425208i	$-0.860201\pi$
$421$	−2.77148e9	−1.81019	−0.905096	+	0.425208i	$0.860201\pi$
$422$	−1.70604e9	−1.10508
$423$	−6.50057e8	−0.417599
$424$	1.58013e7	0.0100673
$425$	1.38599e9	0.875787
$426$	−1.03082e8	−0.0646026
$427$	9.28864e7	0.0577370
$428$	−1.01319e9	−0.624648
$429$	−1.70904e8	−0.104509
$430$	4.58111e8	0.277863
$431$	−2.72165e9	−1.63743	−0.818715	−	0.574201i	$-0.805313\pi$
$431$	−2.72165e9	−1.63743	−0.818715	+	0.574201i	$0.805313\pi$
$432$	−8.06216e7	−0.0481125
$433$	6.43051e8	0.380660	0.190330	−	0.981720i	$-0.439044\pi$
$433$	6.43051e8	0.380660	0.190330	+	0.981720i	$0.439044\pi$
$434$	1.12747e8	0.0662051
$435$	3.68558e8	0.214681
$436$	−7.46951e8	−0.431608
$437$	4.82668e8	0.276671
$438$	−1.11068e8	−0.0631585
$439$	5.49202e8	0.309818	0.154909	−	0.987929i	$-0.450492\pi$
$439$	5.49202e8	0.309818	0.154909	+	0.987929i	$0.450492\pi$
$440$	−2.85000e8	−0.159500
$441$	−5.97738e8	−0.331876
$442$	−3.01614e8	−0.166140
$443$	3.45473e9	1.88799	0.943997	−	0.329955i	$-0.107033\pi$
$443$	3.45473e9	1.88799	0.943997	+	0.329955i	$0.107033\pi$
$444$	−2.53961e8	−0.137698
$445$	1.09514e9	0.589127
$446$	−1.23429e9	−0.658790
$447$	−1.96062e8	−0.103829
$448$	−1.57286e7	−0.00826453
$449$	1.97261e9	1.02844	0.514219	−	0.857659i	$-0.328082\pi$
$449$	1.97261e9	1.02844	0.514219	+	0.857659i	$0.328082\pi$
$450$	−3.41318e8	−0.176569
$451$	−9.23553e8	−0.474072
$452$	1.60358e9	0.816782
$453$	5.10529e8	0.258034
$454$	2.06243e9	1.03439
$455$	1.33728e7	0.00665553
$456$	9.48188e7	0.0468293
$457$	−2.66401e9	−1.30566	−0.652828	−	0.757506i	$-0.726418\pi$
$457$	−2.66401e9	−1.30566	−0.652828	+	0.757506i	$0.726418\pi$
$458$	−1.76229e9	−0.857131
$459$	4.66133e8	0.224991
$460$	6.30515e8	0.302025
$461$	−2.59194e9	−1.23217	−0.616086	−	0.787679i	$-0.711282\pi$
$461$	−2.59194e9	−1.23217	−0.616086	+	0.787679i	$0.711282\pi$
$462$	5.15290e7	0.0243111
$463$	4.25872e8	0.199409	0.0997046	−	0.995017i	$-0.468210\pi$
$463$	4.25872e8	0.199409	0.0997046	+	0.995017i	$0.468210\pi$
$464$	3.99368e8	0.185593
$465$	−8.87884e8	−0.409516
$466$	−6.72608e8	−0.307901
$467$	2.19915e9	0.999186	0.499593	−	0.866260i	$-0.333483\pi$
$467$	2.19915e9	0.999186	0.499593	+	0.866260i	$0.333483\pi$
$468$	7.42764e7	0.0334958
$469$	−2.36696e8	−0.105946
$470$	9.98715e8	0.443710
$471$	1.03595e9	0.456843
$472$	−1.91908e8	−0.0840032
$473$	−1.62630e9	−0.706619
$474$	1.05114e9	0.453352
$475$	4.01423e8	0.171860
$476$	9.09389e7	0.0386478
$477$	2.24984e7	0.00949155
$478$	2.75684e9	1.15455
$479$	2.33462e9	0.970604	0.485302	−	0.874347i	$-0.338710\pi$
$479$	2.33462e9	0.970604	0.485302	+	0.874347i	$0.338710\pi$
$480$	1.23863e8	0.0511208
$481$	2.33973e8	0.0958646
$482$	−2.84538e9	−1.15738
$483$	−1.13999e8	−0.0460349
$484$	−2.35430e8	−0.0943850
$485$	3.84593e8	0.153075
$486$	−1.14791e8	−0.0453609
$487$	2.84905e9	1.11776	0.558881	−	0.829248i	$-0.311231\pi$
$487$	2.84905e9	1.11776	0.558881	+	0.829248i	$0.311231\pi$
$488$	−7.92630e8	−0.308746
$489$	1.03184e9	0.399052
$490$	9.18336e8	0.352627
$491$	2.07570e8	0.0791370	0.0395685	−	0.999217i	$-0.487402\pi$
$491$	2.07570e8	0.0791370	0.0395685	+	0.999217i	$0.487402\pi$
$492$	4.01383e8	0.151943
$493$	−2.30904e9	−0.867896
$494$	−8.73562e7	−0.0326024
$495$	−4.05791e8	−0.150378
$496$	−9.62109e8	−0.354029
$497$	−2.86339e7	−0.0104625
$498$	−4.41332e8	−0.160126
$499$	−1.60656e9	−0.578821	−0.289411	−	0.957205i	$-0.593459\pi$
$499$	−1.60656e9	−0.578821	−0.289411	+	0.957205i	$0.593459\pi$
$500$	1.22438e9	0.438049
$501$	−1.04129e9	−0.369947
$502$	−2.84898e9	−1.00514
$503$	−7.45357e8	−0.261142	−0.130571	−	0.991439i	$-0.541681\pi$
$503$	−7.45357e8	−0.261142	−0.130571	+	0.991439i	$0.541681\pi$
$504$	−2.23949e7	−0.00779187
$505$	−7.01676e8	−0.242447
$506$	−2.23833e9	−0.768063
$507$	1.62578e9	0.554031
$508$	2.23206e9	0.755411
$509$	6.48997e8	0.218137	0.109069	−	0.994034i	$-0.465213\pi$
$509$	6.48997e8	0.218137	0.109069	+	0.994034i	$0.465213\pi$
$510$	−7.16144e8	−0.239059
$511$	−3.08524e7	−0.0102286
$512$	1.34218e8	0.0441942
$513$	1.35006e8	0.0441511
$514$	−2.73749e9	−0.889163
$515$	−1.67867e9	−0.541552
$516$	7.06800e8	0.226476
$517$	−3.54544e9	−1.12837
$518$	−7.05446e7	−0.0223003
$519$	5.20456e7	0.0163417
$520$	−1.14115e8	−0.0355901
$521$	1.87790e9	0.581756	0.290878	−	0.956760i	$-0.406053\pi$
$521$	1.87790e9	0.581756	0.290878	+	0.956760i	$0.406053\pi$
$522$	5.68632e8	0.174978
$523$	7.95942e8	0.243291	0.121645	−	0.992574i	$-0.461183\pi$
$523$	7.95942e8	0.243291	0.121645	+	0.992574i	$0.461183\pi$
$524$	2.62624e9	0.797398
$525$	−9.48105e7	−0.0285956
$526$	−3.53731e8	−0.105980
$527$	5.56266e9	1.65556
$528$	−4.39714e8	−0.130002
$529$	1.54711e9	0.454388
$530$	−3.45654e7	−0.0100850
$531$	−2.73244e8	−0.0791990
$532$	2.63386e7	0.00758405
$533$	−3.69793e8	−0.105782
$534$	1.68964e9	0.480176
$535$	2.21634e9	0.625747
$536$	2.01980e9	0.566542
$537$	−5.48557e8	−0.152866
$538$	1.35382e9	0.374820
$539$	−3.26009e9	−0.896746
$540$	1.76360e8	0.0481971
$541$	1.69561e9	0.460401	0.230200	−	0.973143i	$-0.426062\pi$
$541$	1.69561e9	0.460401	0.230200	+	0.973143i	$0.426062\pi$
$542$	4.13256e9	1.11486
$543$	2.43335e9	0.652236
$544$	−7.76012e8	−0.206667
$545$	1.63396e9	0.432367
$546$	2.06323e7	0.00542468
$547$	4.74479e9	1.23954	0.619772	−	0.784782i	$-0.287225\pi$
$547$	4.74479e9	1.23954	0.619772	+	0.784782i	$0.287225\pi$
$548$	2.08952e9	0.542395
$549$	−1.12857e9	−0.291088
$550$	−1.86156e9	−0.477099
$551$	−6.68766e8	−0.170311
$552$	9.72795e8	0.246170
$553$	2.91983e8	0.0734209
$554$	9.77067e8	0.244141
$555$	5.55539e8	0.137940
$556$	9.36046e7	0.0230959
$557$	−3.48853e8	−0.0855359	−0.0427680	−	0.999085i	$-0.513618\pi$
$557$	−3.48853e8	−0.0855359	−0.0427680	+	0.999085i	$0.513618\pi$
$558$	−1.36988e9	−0.333782
$559$	−6.51173e8	−0.157672
$560$	3.44064e7	0.00827906
$561$	2.54231e9	0.607937
$562$	4.72782e9	1.12353
$563$	−3.43999e9	−0.812414	−0.406207	−	0.913781i	$-0.633149\pi$
$563$	−3.43999e9	−0.812414	−0.406207	+	0.913781i	$0.633149\pi$
$564$	1.54087e9	0.361652
$565$	−3.50783e9	−0.818219
$566$	−6.35669e8	−0.147358
$567$	−3.18865e7	−0.00734625
$568$	2.44343e8	0.0559475
$569$	−3.97632e9	−0.904874	−0.452437	−	0.891796i	$-0.649445\pi$
$569$	−3.97632e9	−0.904874	−0.452437	+	0.891796i	$0.649445\pi$
$570$	−2.07416e8	−0.0469116
$571$	3.57221e9	0.802992	0.401496	−	0.915861i	$-0.368490\pi$
$571$	3.57221e9	0.802992	0.401496	+	0.915861i	$0.368490\pi$
$572$	4.05107e8	0.0905072
$573$	−4.64042e8	−0.103042
$574$	1.11495e8	0.0246074
$575$	4.11840e9	0.903424
$576$	1.91103e8	0.0416667
$577$	−8.72951e9	−1.89180	−0.945898	−	0.324464i	$-0.894816\pi$
$577$	−8.72951e9	−1.89180	−0.945898	+	0.324464i	$0.894816\pi$
$578$	1.20399e9	0.259343
$579$	6.13215e8	0.131292
$580$	−8.73618e8	−0.185919
$581$	−1.22592e8	−0.0259326
$582$	5.93371e8	0.124766
$583$	1.22707e8	0.0256466
$584$	2.63273e8	0.0546968
$585$	−1.62480e8	−0.0335547
$586$	2.08640e9	0.428307
$587$	5.04360e9	1.02922	0.514609	−	0.857425i	$-0.327937\pi$
$587$	5.04360e9	1.02922	0.514609	+	0.857425i	$0.327937\pi$
$588$	1.41686e9	0.287413
$589$	1.61111e9	0.324879
$590$	4.19798e8	0.0841509
$591$	1.05049e9	0.209332
$592$	6.01981e8	0.119250
$593$	9.81995e8	0.193383	0.0966914	−	0.995314i	$-0.469174\pi$
$593$	9.81995e8	0.193383	0.0966914	+	0.995314i	$0.469174\pi$
$594$	−6.26077e8	−0.122567
$595$	−1.98929e8	−0.0387158
$596$	4.64740e8	0.0899183
$597$	−8.50514e8	−0.163596
$598$	−8.96232e8	−0.171382
$599$	4.42128e9	0.840532	0.420266	−	0.907401i	$-0.361937\pi$
$599$	4.42128e9	0.840532	0.420266	+	0.907401i	$0.361937\pi$
$600$	8.09050e8	0.152913
$601$	−6.88288e9	−1.29333	−0.646665	−	0.762774i	$-0.723837\pi$
$601$	−6.88288e9	−1.29333	−0.646665	+	0.762774i	$0.723837\pi$
$602$	1.96333e8	0.0366781
$603$	2.87585e9	0.534141
$604$	−1.21014e9	−0.223464
$605$	5.15003e8	0.0945510
$606$	−1.08259e9	−0.197610
$607$	4.74995e7	0.00862043	0.00431021	−	0.999991i	$-0.498628\pi$
$607$	4.74995e7	0.00862043	0.00431021	+	0.999991i	$0.498628\pi$
$608$	−2.24756e8	−0.0405554
$609$	1.57953e8	0.0283379
$610$	1.73388e9	0.309289
$611$	−1.41960e9	−0.251781
$612$	−1.10491e9	−0.194848
$613$	−1.58379e9	−0.277706	−0.138853	−	0.990313i	$-0.544342\pi$
$613$	−1.58379e9	−0.277706	−0.138853	+	0.990313i	$0.544342\pi$
$614$	−4.27392e9	−0.745138
$615$	−8.78026e8	−0.152210
$616$	−1.22143e8	−0.0210540
$617$	−8.24623e9	−1.41338	−0.706688	−	0.707526i	$-0.749811\pi$
$617$	−8.24623e9	−1.41338	−0.706688	+	0.707526i	$0.749811\pi$
$618$	−2.58995e9	−0.441400
$619$	1.55899e9	0.264195	0.132098	−	0.991237i	$-0.457829\pi$
$619$	1.55899e9	0.264195	0.132098	+	0.991237i	$0.457829\pi$
$620$	2.10461e9	0.354651
$621$	1.38509e9	0.232091
$622$	3.84149e8	0.0640078
$623$	4.69345e8	0.0777650
$624$	−1.76062e8	−0.0290082
$625$	1.89393e9	0.310301
$626$	8.78527e8	0.143135
$627$	7.36327e8	0.119298
$628$	−2.45559e9	−0.395637
$629$	−3.48050e9	−0.557653
$630$	4.89888e7	0.00780557
$631$	−9.99568e9	−1.58383	−0.791917	−	0.610629i	$-0.790917\pi$
$631$	−9.99568e9	−1.58383	−0.791917	+	0.610629i	$0.790917\pi$
$632$	−2.49159e9	−0.392615
$633$	5.75787e9	0.902295
$634$	1.07511e9	0.167548
$635$	−4.88264e9	−0.756740
$636$	−5.33295e7	−0.00821993
$637$	−1.30535e9	−0.200096
$638$	3.10134e9	0.472800
$639$	3.47902e8	0.0527478
$640$	−2.93601e8	−0.0442719
$641$	−1.10804e10	−1.66169	−0.830847	−	0.556501i	$-0.812144\pi$
$641$	−1.10804e10	−1.66169	−0.830847	+	0.556501i	$0.812144\pi$
$642$	3.41950e9	0.510023
$643$	−7.79012e9	−1.15559	−0.577797	−	0.816180i	$-0.696088\pi$
$643$	−7.79012e9	−1.15559	−0.577797	+	0.816180i	$0.696088\pi$
$644$	2.70221e8	0.0398674
$645$	−1.54613e9	−0.226875
$646$	1.29948e9	0.189651
$647$	−2.42734e9	−0.352342	−0.176171	−	0.984360i	$-0.556371\pi$
$647$	−2.42734e9	−0.352342	−0.176171	+	0.984360i	$0.556371\pi$
$648$	2.72098e8	0.0392837
$649$	−1.49028e9	−0.213999
$650$	−7.45374e8	−0.106458
$651$	−3.80522e8	−0.0540563
$652$	−2.44583e9	−0.345589
$653$	−8.71403e7	−0.0122468	−0.00612340	−	0.999981i	$-0.501949\pi$
$653$	−8.71403e7	−0.0122468	−0.00612340	+	0.999981i	$0.501949\pi$
$654$	2.52096e9	0.352407
$655$	−5.74490e9	−0.798800
$656$	−9.51427e8	−0.131587
$657$	3.74856e8	0.0515687
$658$	4.28021e8	0.0585699
$659$	−1.47455e9	−0.200706	−0.100353	−	0.994952i	$-0.531997\pi$
$659$	−1.47455e9	−0.200706	−0.100353	+	0.994952i	$0.531997\pi$
$660$	9.61874e8	0.130231
$661$	8.03606e9	1.08228	0.541138	−	0.840934i	$-0.317994\pi$
$661$	8.03606e9	1.08228	0.541138	+	0.840934i	$0.317994\pi$
$662$	6.06725e9	0.812810
$663$	1.01795e9	0.135653
$664$	1.04612e9	0.138673
$665$	−5.76156e7	−0.00759739
$666$	8.57117e8	0.112430
$667$	−6.86122e9	−0.895284
$668$	2.46824e9	0.320383
$669$	4.16574e9	0.537899
$670$	−4.41832e9	−0.567538
$671$	−6.15527e9	−0.786535
$672$	5.30842e7	0.00674796
$673$	3.29390e9	0.416541	0.208271	−	0.978071i	$-0.433217\pi$
$673$	3.29390e9	0.416541	0.208271	+	0.978071i	$0.433217\pi$
$674$	6.46353e9	0.813131
$675$	1.15195e9	0.144168
$676$	−3.85370e9	−0.479805
$677$	−1.46019e10	−1.80862	−0.904312	−	0.426872i	$-0.859615\pi$
$677$	−1.46019e10	−1.80862	−0.904312	+	0.426872i	$0.859615\pi$
$678$	−5.41209e9	−0.666900
$679$	1.64825e8	0.0202060
$680$	1.69753e9	0.207031
$681$	−6.96071e9	−0.844575
$682$	−7.47138e9	−0.901894
$683$	−1.38424e10	−1.66241	−0.831205	−	0.555966i	$-0.812348\pi$
$683$	−1.38424e10	−1.66241	−0.831205	+	0.555966i	$0.812348\pi$
$684$	−3.20014e8	−0.0382360
$685$	−4.57084e9	−0.543349
$686$	7.88873e8	0.0932981
$687$	5.94772e9	0.699845
$688$	−1.67538e9	−0.196134
$689$	4.91323e7	0.00572269
$690$	−2.12799e9	−0.246603
$691$	−9.73399e9	−1.12232	−0.561161	−	0.827706i	$-0.689645\pi$
$691$	−9.73399e9	−1.12232	−0.561161	+	0.827706i	$0.689645\pi$
$692$	−1.23367e8	−0.0141524
$693$	−1.73910e8	−0.0198499
$694$	5.32813e9	0.605085
$695$	−2.04760e8	−0.0231365
$696$	−1.34787e9	−0.151536
$697$	5.50090e9	0.615346
$698$	−3.46413e9	−0.385567
$699$	2.27005e9	0.251400
$700$	2.24736e8	0.0247645
$701$	6.41266e9	0.703114	0.351557	−	0.936167i	$-0.385652\pi$
$701$	6.41266e9	0.703114	0.351557	+	0.936167i	$0.385652\pi$
$702$	−2.50683e8	−0.0273492
$703$	−1.00805e9	−0.109431
$704$	1.04228e9	0.112585
$705$	−3.37066e9	−0.362288
$706$	−3.91917e9	−0.419158
$707$	−3.00718e8	−0.0320031
$708$	6.47689e8	0.0685883
$709$	3.13580e9	0.330435	0.165218	−	0.986257i	$-0.447167\pi$
$709$	3.13580e9	0.330435	0.165218	+	0.986257i	$0.447167\pi$
$710$	−5.34500e8	−0.0560458
$711$	−3.54759e9	−0.370161
$712$	−4.00508e9	−0.415845
$713$	1.65292e10	1.70781
$714$	−3.06919e8	−0.0315558
$715$	−8.86171e8	−0.0906664
$716$	1.30028e9	0.132386
$717$	−9.30435e9	−0.942690
$718$	1.83336e9	0.184847
$719$	9.58263e9	0.961465	0.480732	−	0.876867i	$-0.340371\pi$
$719$	9.58263e9	0.961465	0.480732	+	0.876867i	$0.340371\pi$
$720$	−4.18038e8	−0.0417399
$721$	−7.19431e8	−0.0714851
$722$	3.76367e8	0.0372161
$723$	9.60315e9	0.944996
$724$	−5.76793e9	−0.564853
$725$	−5.70630e9	−0.556124
$726$	7.94577e8	0.0770651
$727$	−1.44451e9	−0.139428	−0.0697140	−	0.997567i	$-0.522209\pi$
$727$	−1.44451e9	−0.139428	−0.0697140	+	0.997567i	$0.522209\pi$
$728$	−4.89062e7	−0.00469791
$729$	3.87420e8	0.0370370
$730$	−5.75911e8	−0.0547930
$731$	9.68660e9	0.917193
$732$	2.67513e9	0.252090
$733$	−1.06170e10	−0.995720	−0.497860	−	0.867257i	$-0.665881\pi$
$733$	−1.06170e10	−0.995720	−0.497860	+	0.867257i	$0.665881\pi$
$734$	6.56952e9	0.613193
$735$	−3.09938e9	−0.287919
$736$	−2.30588e9	−0.213189
$737$	1.56850e10	1.44328
$738$	−1.35467e9	−0.124061
$739$	6.26841e9	0.571349	0.285675	−	0.958327i	$-0.407782\pi$
$739$	6.26841e9	0.571349	0.285675	+	0.958327i	$0.407782\pi$
$740$	−1.31683e9	−0.119459
$741$	2.94827e8	0.0266197
$742$	−1.48138e7	−0.00133123
$743$	−1.63416e10	−1.46161	−0.730807	−	0.682585i	$-0.760856\pi$
$743$	−1.63416e10	−1.46161	−0.730807	+	0.682585i	$0.760856\pi$
$744$	3.24712e9	0.289063
$745$	−1.01662e9	−0.0900764
$746$	−3.26411e9	−0.287859
$747$	1.48950e9	0.130743
$748$	−6.02622e9	−0.526489
$749$	9.49861e8	0.0825988
$750$	−4.13230e9	−0.357665
$751$	1.25375e10	1.08012	0.540059	−	0.841627i	$-0.318402\pi$
$751$	1.25375e10	1.08012	0.540059	+	0.841627i	$0.318402\pi$
$752$	−3.65244e9	−0.313200
$753$	9.61530e9	0.820692
$754$	1.24179e9	0.105499
$755$	2.64719e9	0.223857
$756$	7.55827e7	0.00636204
$757$	1.63686e10	1.37143	0.685717	−	0.727868i	$-0.259489\pi$
$757$	1.63686e10	1.37143	0.685717	+	0.727868i	$0.259489\pi$
$758$	3.57499e9	0.298148
$759$	7.55436e9	0.627121
$760$	4.91653e8	0.0406267
$761$	9.04308e9	0.743824	0.371912	−	0.928268i	$-0.378702\pi$
$761$	9.04308e9	0.743824	0.371912	+	0.928268i	$0.378702\pi$
$762$	−7.53321e9	−0.616791
$763$	7.00267e8	0.0570726
$764$	1.09995e9	0.0892373
$765$	2.41698e9	0.195191
$766$	5.16464e9	0.415183
$767$	−5.96713e8	−0.0477510
$768$	−4.52985e8	−0.0360844
$769$	1.30077e10	1.03148	0.515738	−	0.856746i	$-0.327518\pi$
$769$	1.30077e10	1.03148	0.515738	+	0.856746i	$0.327518\pi$
$770$	2.67187e8	0.0210910
$771$	9.23902e9	0.725998
$772$	−1.45355e9	−0.113702
$773$	6.47257e9	0.504021	0.252010	−	0.967725i	$-0.418908\pi$
$773$	6.47257e9	0.504021	0.252010	+	0.967725i	$0.418908\pi$
$774$	−2.38545e9	−0.184917
$775$	1.37469e10	1.06084
$776$	−1.40651e9	−0.108051
$777$	2.38088e8	0.0182081
$778$	−4.45874e9	−0.339456
$779$	1.59322e9	0.120752
$780$	3.85137e8	0.0290592
$781$	1.89747e9	0.142527
$782$	1.33320e10	0.996947
$783$	−1.91913e9	−0.142869
$784$	−3.35849e9	−0.248907
$785$	5.37161e9	0.396333
$786$	−8.86356e9	−0.651073
$787$	3.96773e9	0.290155	0.145077	−	0.989420i	$-0.453657\pi$
$787$	3.96773e9	0.290155	0.145077	+	0.989420i	$0.453657\pi$
$788$	−2.49005e9	−0.181287
$789$	1.19384e9	0.0865321
$790$	5.45035e9	0.393305
$791$	−1.50336e9	−0.108005
$792$	1.48403e9	0.106147
$793$	−2.46458e9	−0.175504
$794$	6.90622e9	0.489631
$795$	1.16658e8	0.00823438
$796$	2.01603e9	0.141678
$797$	4.11589e9	0.287979	0.143989	−	0.989579i	$-0.454007\pi$
$797$	4.11589e9	0.287979	0.143989	+	0.989579i	$0.454007\pi$
$798$	−8.88926e7	−0.00619235
$799$	2.11175e10	1.46463
$800$	−1.91775e9	−0.132427
$801$	−5.70255e9	−0.392062
$802$	−5.92373e9	−0.405495
$803$	2.04448e9	0.139341
$804$	−6.81684e9	−0.462580
$805$	−5.91108e8	−0.0399375
$806$	−2.99156e9	−0.201245
$807$	−4.56914e9	−0.306039
$808$	2.56613e9	0.171135
$809$	2.30240e10	1.52884	0.764419	−	0.644720i	$-0.223026\pi$
$809$	2.30240e10	1.52884	0.764419	+	0.644720i	$0.223026\pi$
$810$	−5.95214e8	−0.0393528
$811$	1.34840e10	0.887655	0.443827	−	0.896112i	$-0.353620\pi$
$811$	1.34840e10	0.887655	0.443827	+	0.896112i	$0.353620\pi$
$812$	−3.74408e8	−0.0245414
$813$	−1.39474e10	−0.910282
$814$	4.67476e9	0.303790
$815$	5.35026e9	0.346197
$816$	2.61904e9	0.168743
$817$	2.80552e9	0.179985
$818$	4.57380e9	0.292174
$819$	−6.96341e7	−0.00442923
$820$	2.08125e9	0.131818
$821$	−1.16988e10	−0.737800	−0.368900	−	0.929469i	$-0.620266\pi$
$821$	−1.16988e10	−0.737800	−0.368900	+	0.929469i	$0.620266\pi$
$822$	−7.05215e9	−0.442864
$823$	−1.24855e10	−0.780742	−0.390371	−	0.920658i	$-0.627653\pi$
$823$	−1.24855e10	−0.780742	−0.390371	+	0.920658i	$0.627653\pi$
$824$	6.13914e9	0.382263
$825$	6.28278e9	0.389550
$826$	1.79914e8	0.0111079
$827$	7.27524e9	0.447278	0.223639	−	0.974672i	$-0.428206\pi$
$827$	7.27524e9	0.447278	0.223639	+	0.974672i	$0.428206\pi$
$828$	−3.28318e9	−0.200997
$829$	1.31781e9	0.0803362	0.0401681	−	0.999193i	$-0.487211\pi$
$829$	1.31781e9	0.0803362	0.0401681	+	0.999193i	$0.487211\pi$
$830$	−2.28839e9	−0.138917
$831$	−3.29760e9	−0.199340
$832$	4.17333e8	0.0251218
$833$	1.94179e10	1.16398
$834$	−3.15916e8	−0.0188577
$835$	−5.39927e9	−0.320946
$836$	−1.74537e9	−0.103315
$837$	4.62334e9	0.272532
$838$	6.25433e9	0.367136
$839$	−1.60143e10	−0.936144	−0.468072	−	0.883690i	$-0.655051\pi$
$839$	−1.60143e10	−0.936144	−0.468072	+	0.883690i	$0.655051\pi$
$840$	−1.16122e8	−0.00675983
$841$	−7.74324e9	−0.448886
$842$	−2.21718e10	−1.28000
$843$	−1.59564e10	−0.917357
$844$	−1.36483e10	−0.781411
$845$	8.42997e9	0.480648
$846$	−5.20045e9	−0.295287
$847$	2.20716e8	0.0124808
$848$	1.26411e8	0.00711866
$849$	2.14538e9	0.120317
$850$	1.10879e10	0.619275
$851$	−1.03421e10	−0.575250
$852$	−8.24657e8	−0.0456809
$853$	4.05900e9	0.223922	0.111961	−	0.993713i	$-0.464287\pi$
$853$	4.05900e9	0.223922	0.111961	+	0.993713i	$0.464287\pi$
$854$	7.43091e8	0.0408262
$855$	7.00030e8	0.0383032
$856$	−8.10548e9	−0.441693
$857$	−8.21878e9	−0.446041	−0.223020	−	0.974814i	$-0.571592\pi$
$857$	−8.21878e9	−0.446041	−0.223020	+	0.974814i	$0.571592\pi$
$858$	−1.36724e9	−0.0738989
$859$	1.40044e10	0.753856	0.376928	−	0.926242i	$-0.376980\pi$
$859$	1.40044e10	0.753856	0.376928	+	0.926242i	$0.376980\pi$
$860$	3.66489e9	0.196479
$861$	−3.76297e8	−0.0200918
$862$	−2.17732e10	−1.15784
$863$	−9.59295e9	−0.508059	−0.254030	−	0.967196i	$-0.581756\pi$
$863$	−9.59295e9	−0.508059	−0.254030	+	0.967196i	$0.581756\pi$
$864$	−6.44973e8	−0.0340207
$865$	2.69866e8	0.0141772
$866$	5.14441e9	0.269167
$867$	−4.06346e9	−0.211753
$868$	9.01978e8	0.0468141
$869$	−1.93487e10	−1.00019
$870$	2.94846e9	0.151802
$871$	6.28033e9	0.322047
$872$	−5.97561e9	−0.305193
$873$	−2.00263e9	−0.101871
$874$	3.86134e9	0.195636
$875$	−1.14786e9	−0.0579243
$876$	−8.88548e8	−0.0446598
$877$	3.02321e10	1.51346	0.756728	−	0.653730i	$-0.226797\pi$
$877$	3.02321e10	1.51346	0.756728	+	0.653730i	$0.226797\pi$
$878$	4.39361e9	0.219074
$879$	−7.04159e9	−0.349711
$880$	−2.28000e9	−0.112783
$881$	−2.72724e10	−1.34371	−0.671857	−	0.740681i	$-0.734503\pi$
$881$	−2.72724e10	−1.34371	−0.671857	+	0.740681i	$0.734503\pi$
$882$	−4.78191e9	−0.234672
$883$	2.81269e10	1.37486	0.687430	−	0.726250i	$-0.258739\pi$
$883$	2.81269e10	1.37486	0.687430	+	0.726250i	$0.258739\pi$
$884$	−2.41291e9	−0.117479
$885$	−1.41682e9	−0.0687089
$886$	2.76378e10	1.33501
$887$	−2.13080e10	−1.02520	−0.512601	−	0.858627i	$-0.671318\pi$
$887$	−2.13080e10	−1.02520	−0.512601	+	0.858627i	$0.671318\pi$
$888$	−2.03169e9	−0.0973669
$889$	−2.09256e9	−0.0998899
$890$	8.76111e9	0.416576
$891$	2.11301e9	0.100076
$892$	−9.87435e9	−0.465835
$893$	6.11624e9	0.287412
$894$	−1.56850e9	−0.0734180
$895$	−2.84437e9	−0.132619
$896$	−1.25829e8	−0.00584390
$897$	3.02478e9	0.139933
$898$	1.57808e10	0.727215
$899$	−2.29022e10	−1.05128
$900$	−2.73054e9	−0.124853
$901$	−7.30874e8	−0.0332894
$902$	−7.38843e9	−0.335219
$903$	−6.62625e8	−0.0299475
$904$	1.28286e10	0.577552
$905$	1.26174e10	0.565846
$906$	4.08423e9	0.182458
$907$	−3.80170e10	−1.69181	−0.845907	−	0.533331i	$-0.820940\pi$
$907$	−3.80170e10	−1.69181	−0.845907	+	0.533331i	$0.820940\pi$
$908$	1.64995e10	0.731424
$909$	3.65373e9	0.161348
$910$	1.06982e8	0.00470617
$911$	−2.26348e10	−0.991886	−0.495943	−	0.868355i	$-0.665178\pi$
$911$	−2.26348e10	−0.991886	−0.495943	+	0.868355i	$0.665178\pi$
$912$	7.58551e8	0.0331133
$913$	8.12378e9	0.353273
$914$	−2.13121e10	−0.923239
$915$	−5.85184e9	−0.252533
$916$	−1.40983e10	−0.606083
$917$	−2.46210e9	−0.105442
$918$	3.72906e9	0.159093
$919$	2.62191e10	1.11433	0.557165	−	0.830402i	$-0.311889\pi$
$919$	2.62191e10	1.11433	0.557165	+	0.830402i	$0.311889\pi$
$920$	5.04412e9	0.213564
$921$	1.44245e10	0.608402
$922$	−2.07355e10	−0.871277
$923$	7.59753e8	0.0318029
$924$	4.12232e8	0.0171905
$925$	−8.60130e9	−0.357329
$926$	3.40697e9	0.141004
$927$	8.74108e9	0.360401
$928$	3.19495e9	0.131234
$929$	−3.87695e10	−1.58648	−0.793241	−	0.608908i	$-0.791608\pi$
$929$	−3.87695e10	−1.58648	−0.793241	+	0.608908i	$0.791608\pi$
$930$	−7.10307e9	−0.289572
$931$	5.62399e9	0.228413
$932$	−5.38086e9	−0.217719
$933$	−1.29650e9	−0.0522622
$934$	1.75932e10	0.706531
$935$	1.31823e10	0.527415
$936$	5.94211e8	0.0236851
$937$	−2.23699e10	−0.888334	−0.444167	−	0.895944i	$-0.646500\pi$
$937$	−2.23699e10	−0.888334	−0.444167	+	0.895944i	$0.646500\pi$
$938$	−1.89357e9	−0.0749153
$939$	−2.96503e9	−0.116869
$940$	7.98972e9	0.313750
$941$	6.11474e9	0.239229	0.119615	−	0.992820i	$-0.461834\pi$
$941$	6.11474e9	0.239229	0.119615	+	0.992820i	$0.461834\pi$
$942$	8.28762e9	0.323037
$943$	1.63457e10	0.634764
$944$	−1.53526e9	−0.0593992
$945$	−1.65337e8	−0.00637322
$946$	−1.30104e10	−0.499655
$947$	−4.02823e10	−1.54131	−0.770655	−	0.637253i	$-0.780071\pi$
$947$	−4.02823e10	−1.54131	−0.770655	+	0.637253i	$0.780071\pi$
$948$	8.40911e9	0.320568
$949$	8.18616e8	0.0310920
$950$	3.21138e9	0.121523
$951$	−3.62849e9	−0.136803
$952$	7.27511e8	0.0273282
$953$	4.96543e10	1.85837	0.929184	−	0.369619i	$-0.120512\pi$
$953$	4.96543e10	1.85837	0.929184	+	0.369619i	$0.120512\pi$
$954$	1.79987e8	0.00671154
$955$	−2.40614e9	−0.0893942
$956$	2.20548e10	0.816394
$957$	−1.04670e10	−0.386040
$958$	1.86770e10	0.686321
$959$	−1.95893e9	−0.0717222
$960$	9.90904e8	0.0361478
$961$	2.76607e10	1.00538
$962$	1.87178e9	0.0677865
$963$	−1.15408e10	−0.416432
$964$	−2.27630e10	−0.818390
$965$	3.17963e9	0.113902
$966$	−9.11995e8	−0.0325516
$967$	5.05766e10	1.79869	0.899346	−	0.437237i	$-0.144043\pi$
$967$	5.05766e10	1.79869	0.899346	+	0.437237i	$0.144043\pi$
$968$	−1.88344e9	−0.0667403
$969$	−4.38574e9	−0.154849
$970$	3.07674e9	0.108241
$971$	−3.07016e10	−1.07620	−0.538101	−	0.842881i	$-0.680858\pi$
$971$	−3.07016e10	−1.07620	−0.538101	+	0.842881i	$0.680858\pi$
$972$	−9.18330e8	−0.0320750
$973$	−8.77543e7	−0.00305403
$974$	2.27924e10	0.790377
$975$	2.51564e9	0.0869225
$976$	−6.34104e9	−0.218316
$977$	−2.26055e10	−0.775501	−0.387750	−	0.921764i	$-0.626748\pi$
$977$	−2.26055e10	−0.775501	−0.387750	+	0.921764i	$0.626748\pi$
$978$	8.25469e9	0.282172
$979$	−3.11019e10	−1.05937
$980$	7.34669e9	0.249345
$981$	−8.50824e9	−0.287739
$982$	1.66056e9	0.0559583
$983$	−5.01849e10	−1.68514	−0.842569	−	0.538588i	$-0.818958\pi$
$983$	−5.01849e10	−1.68514	−0.842569	+	0.538588i	$0.818958\pi$
$984$	3.21107e9	0.107440
$985$	5.44699e9	0.181606
$986$	−1.84723e10	−0.613695
$987$	−1.44457e9	−0.0478221
$988$	−6.98850e8	−0.0230534
$989$	2.87833e10	0.946136
$990$	−3.24632e9	−0.106333
$991$	−4.83054e10	−1.57666	−0.788330	−	0.615252i	$-0.789054\pi$
$991$	−4.83054e10	−1.57666	−0.788330	+	0.615252i	$0.789054\pi$
$992$	−7.69688e9	−0.250336
$993$	−2.04770e10	−0.663657
$994$	−2.29071e8	−0.00739807
$995$	−4.41007e9	−0.141927
$996$	−3.53066e9	−0.113226
$997$	1.18580e10	0.378948	0.189474	−	0.981886i	$-0.439322\pi$
$997$	1.18580e10	0.378948	0.189474	+	0.981886i	$0.439322\pi$
$998$	−1.28525e10	−0.409288
$999$	−2.89277e9	−0.0917983

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	114.8.a.c.1.1	✓	1
3.2	odd	2		342.8.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.8.a.c.1.1	✓	1	1.1	even	1	trivial
342.8.a.c.1.1		1	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 114 = 2 \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	114.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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