LMFDB - Embedded newform 336.8.a.t.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [336,8,Mod(1,336)] 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("336.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	336.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-81,0,414] 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:	$104.961368563$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4400x - 26848 $ x^3 - x^2 - 4400*x - 26848
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 168)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-6.16367$ of defining polynomial
Character	$\chi$	$=$	336.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-27.0000 q^{3} +99.0180 q^{5} -343.000 q^{7} +729.000 q^{9} +3695.45 q^{11} +7236.44 q^{13} -2673.49 q^{15} +18523.3 q^{17} +9508.41 q^{19} +9261.00 q^{21} -70977.4 q^{23} -68320.4 q^{25} -19683.0 q^{27} +43155.0 q^{29} +15305.4 q^{31} -99777.3 q^{33} -33963.2 q^{35} +369151. q^{37} -195384. q^{39} -26475.6 q^{41} -275186. q^{43} +72184.1 q^{45} +114034. q^{47} +117649. q^{49} -500130. q^{51} -347795. q^{53} +365916. q^{55} -256727. q^{57} +238451. q^{59} +1.52156e6 q^{61} -250047. q^{63} +716537. q^{65} -688978. q^{67} +1.91639e6 q^{69} -4.42654e6 q^{71} -894592. q^{73} +1.84465e6 q^{75} -1.26754e6 q^{77} -2.74442e6 q^{79} +531441. q^{81} -5.51853e6 q^{83} +1.83414e6 q^{85} -1.16518e6 q^{87} +4.73609e6 q^{89} -2.48210e6 q^{91} -413245. q^{93} +941504. q^{95} +5.65023e6 q^{97} +2.69399e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 81 q^{3} + 414 q^{5} - 1029 q^{7} + 2187 q^{9} - 9552 q^{11} + 954 q^{13} - 11178 q^{15} - 30 q^{17} - 39588 q^{19} + 27783 q^{21} + 31164 q^{23} + 139581 q^{25} - 59049 q^{27} + 38898 q^{29} - 147888 q^{31}+ \cdots - 6963408 q^{99}+O(q^{100}) $ 3 * q - 81 * q^3 + 414 * q^5 - 1029 * q^7 + 2187 * q^9 - 9552 * q^11 + 954 * q^13 - 11178 * q^15 - 30 * q^17 - 39588 * q^19 + 27783 * q^21 + 31164 * q^23 + 139581 * q^25 - 59049 * q^27 + 38898 * q^29 - 147888 * q^31 + 257904 * q^33 - 142002 * q^35 - 217662 * q^37 - 25758 * q^39 + 174594 * q^41 - 193644 * q^43 + 301806 * q^45 - 675864 * q^47 + 352947 * q^49 + 810 * q^51 + 476898 * q^53 - 2055576 * q^55 + 1068876 * q^57 - 2107092 * q^59 + 190962 * q^61 - 750141 * q^63 - 922572 * q^65 - 3016380 * q^67 - 841428 * q^69 - 1285404 * q^71 + 6735462 * q^73 - 3768687 * q^75 + 3276336 * q^77 - 2985240 * q^79 + 1594323 * q^81 - 6168276 * q^83 + 14892156 * q^85 - 1050246 * q^87 + 3937938 * q^89 - 327222 * q^91 + 3992976 * q^93 - 24467016 * q^95 + 22816878 * q^97 - 6963408 * q^99

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
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	0	0
$5$	99.0180	0.354257	0.177129	−	0.984188i	$-0.443319\pi$
$5$	99.0180	0.354257	0.177129	+	0.984188i	$0.443319\pi$
$6$	0	0
$7$	−343.000	−0.377964
$7$	−343.000	−0.377964
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	3695.45	0.837131	0.418566	−	0.908187i	$-0.362533\pi$
$11$	3695.45	0.837131	0.418566	+	0.908187i	$0.362533\pi$
$12$	0	0
$13$	7236.44	0.913530	0.456765	−	0.889587i	$-0.349008\pi$
$13$	7236.44	0.913530	0.456765	+	0.889587i	$0.349008\pi$
$14$	0	0
$15$	−2673.49	−0.204531
$16$	0	0
$17$	18523.3	0.914425	0.457212	−	0.889358i	$-0.348848\pi$
$17$	18523.3	0.914425	0.457212	+	0.889358i	$0.348848\pi$
$18$	0	0
$19$	9508.41	0.318032	0.159016	−	0.987276i	$-0.449168\pi$
$19$	9508.41	0.318032	0.159016	+	0.987276i	$0.449168\pi$
$20$	0	0
$21$	9261.00	0.218218
$22$	0	0
$23$	−70977.4	−1.21639	−0.608195	−	0.793788i	$-0.708106\pi$
$23$	−70977.4	−1.21639	−0.608195	+	0.793788i	$0.708106\pi$
$24$	0	0
$25$	−68320.4	−0.874502
$26$	0	0
$27$	−19683.0	−0.192450
$28$	0	0
$29$	43155.0	0.328577	0.164289	−	0.986412i	$-0.447467\pi$
$29$	43155.0	0.328577	0.164289	+	0.986412i	$0.447467\pi$
$30$	0	0
$31$	15305.4	0.0922736	0.0461368	−	0.998935i	$-0.485309\pi$
$31$	15305.4	0.0922736	0.0461368	+	0.998935i	$0.485309\pi$
$32$	0	0
$33$	−99777.3	−0.483318
$34$	0	0
$35$	−33963.2	−0.133897
$36$	0	0
$37$	369151.	1.19811	0.599057	−	0.800706i	$-0.295542\pi$
$37$	369151.	1.19811	0.599057	+	0.800706i	$0.295542\pi$
$38$	0	0
$39$	−195384.	−0.527427
$40$	0	0
$41$	−26475.6	−0.0599932	−0.0299966	−	0.999550i	$-0.509550\pi$
$41$	−26475.6	−0.0599932	−0.0299966	+	0.999550i	$0.509550\pi$
$42$	0	0
$43$	−275186.	−0.527821	−0.263910	−	0.964547i	$-0.585012\pi$
$43$	−275186.	−0.527821	−0.263910	+	0.964547i	$0.585012\pi$
$44$	0	0
$45$	72184.1	0.118086
$46$	0	0
$47$	114034.	0.160210	0.0801051	−	0.996786i	$-0.474474\pi$
$47$	114034.	0.160210	0.0801051	+	0.996786i	$0.474474\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	0	0
$51$	−500130.	−0.527943
$52$	0	0
$53$	−347795.	−0.320891	−0.160446	−	0.987045i	$-0.551293\pi$
$53$	−347795.	−0.320891	−0.160446	+	0.987045i	$0.551293\pi$
$54$	0	0
$55$	365916.	0.296560
$56$	0	0
$57$	−256727.	−0.183616
$58$	0	0
$59$	238451.	0.151153	0.0755767	−	0.997140i	$-0.475920\pi$
$59$	238451.	0.151153	0.0755767	+	0.997140i	$0.475920\pi$
$60$	0	0
$61$	1.52156e6	0.858289	0.429144	−	0.903236i	$-0.358815\pi$
$61$	1.52156e6	0.858289	0.429144	+	0.903236i	$0.358815\pi$
$62$	0	0
$63$	−250047.	−0.125988
$64$	0	0
$65$	716537.	0.323625
$66$	0	0
$67$	−688978.	−0.279862	−0.139931	−	0.990161i	$-0.544688\pi$
$67$	−688978.	−0.279862	−0.139931	+	0.990161i	$0.544688\pi$
$68$	0	0
$69$	1.91639e6	0.702283
$70$	0	0
$71$	−4.42654e6	−1.46778	−0.733889	−	0.679269i	$-0.762297\pi$
$71$	−4.42654e6	−1.46778	−0.733889	+	0.679269i	$0.762297\pi$
$72$	0	0
$73$	−894592.	−0.269150	−0.134575	−	0.990903i	$-0.542967\pi$
$73$	−894592.	−0.269150	−0.134575	+	0.990903i	$0.542967\pi$
$74$	0	0
$75$	1.84465e6	0.504894
$76$	0	0
$77$	−1.26754e6	−0.316406
$78$	0	0
$79$	−2.74442e6	−0.626260	−0.313130	−	0.949710i	$-0.601378\pi$
$79$	−2.74442e6	−0.626260	−0.313130	+	0.949710i	$0.601378\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	−5.51853e6	−1.05938	−0.529688	−	0.848193i	$-0.677691\pi$
$83$	−5.51853e6	−1.05938	−0.529688	+	0.848193i	$0.677691\pi$
$84$	0	0
$85$	1.83414e6	0.323942
$86$	0	0
$87$	−1.16518e6	−0.189704
$88$	0	0
$89$	4.73609e6	0.712122	0.356061	−	0.934463i	$-0.384119\pi$
$89$	4.73609e6	0.712122	0.356061	+	0.934463i	$0.384119\pi$
$90$	0	0
$91$	−2.48210e6	−0.345282
$92$	0	0
$93$	−413245.	−0.0532742
$94$	0	0
$95$	941504.	0.112665
$96$	0	0
$97$	5.65023e6	0.628586	0.314293	−	0.949326i	$-0.398233\pi$
$97$	5.65023e6	0.628586	0.314293	+	0.949326i	$0.398233\pi$
$98$	0	0
$99$	2.69399e6	0.279044
$100$	0	0
$101$	1.13470e7	1.09587	0.547933	−	0.836522i	$-0.315415\pi$
$101$	1.13470e7	1.09587	0.547933	+	0.836522i	$0.315415\pi$
$102$	0	0
$103$	5.62575e6	0.507283	0.253642	−	0.967298i	$-0.418372\pi$
$103$	5.62575e6	0.507283	0.253642	+	0.967298i	$0.418372\pi$
$104$	0	0
$105$	917005.	0.0773053
$106$	0	0
$107$	4.43444e6	0.349941	0.174971	−	0.984574i	$-0.444017\pi$
$107$	4.43444e6	0.349941	0.174971	+	0.984574i	$0.444017\pi$
$108$	0	0
$109$	2.58869e7	1.91464	0.957319	−	0.289035i	$-0.0933343\pi$
$109$	2.58869e7	1.91464	0.957319	+	0.289035i	$0.0933343\pi$
$110$	0	0
$111$	−9.96709e6	−0.691732
$112$	0	0
$113$	1.92136e7	1.25266	0.626332	−	0.779556i	$-0.284555\pi$
$113$	1.92136e7	1.25266	0.626332	+	0.779556i	$0.284555\pi$
$114$	0	0
$115$	−7.02804e6	−0.430915
$116$	0	0
$117$	5.27536e6	0.304510
$118$	0	0
$119$	−6.35350e6	−0.345620
$120$	0	0
$121$	−5.83079e6	−0.299212
$122$	0	0
$123$	714841.	0.0346371
$124$	0	0
$125$	−1.45007e7	−0.664056
$126$	0	0
$127$	4.22100e7	1.82853	0.914266	−	0.405114i	$-0.132768\pi$
$127$	4.22100e7	1.82853	0.914266	+	0.405114i	$0.132768\pi$
$128$	0	0
$129$	7.43002e6	0.304738
$130$	0	0
$131$	1.21641e7	0.472749	0.236374	−	0.971662i	$-0.424041\pi$
$131$	1.21641e7	0.472749	0.236374	+	0.971662i	$0.424041\pi$
$132$	0	0
$133$	−3.26138e6	−0.120205
$134$	0	0
$135$	−1.94897e6	−0.0681769
$136$	0	0
$137$	3.12537e7	1.03843	0.519217	−	0.854642i	$-0.326224\pi$
$137$	3.12537e7	1.03843	0.519217	+	0.854642i	$0.326224\pi$
$138$	0	0
$139$	3.56212e7	1.12501	0.562505	−	0.826794i	$-0.309838\pi$
$139$	3.56212e7	1.12501	0.562505	+	0.826794i	$0.309838\pi$
$140$	0	0
$141$	−3.07891e6	−0.0924975
$142$	0	0
$143$	2.67419e7	0.764745
$144$	0	0
$145$	4.27312e6	0.116401
$146$	0	0
$147$	−3.17652e6	−0.0824786
$148$	0	0
$149$	4.44845e7	1.10168	0.550842	−	0.834610i	$-0.314307\pi$
$149$	4.44845e7	1.10168	0.550842	+	0.834610i	$0.314307\pi$
$150$	0	0
$151$	1.58932e6	0.0375657	0.0187828	−	0.999824i	$-0.494021\pi$
$151$	1.58932e6	0.0375657	0.0187828	+	0.999824i	$0.494021\pi$
$152$	0	0
$153$	1.35035e7	0.304808
$154$	0	0
$155$	1.51551e6	0.0326886
$156$	0	0
$157$	−7.40717e7	−1.52758	−0.763790	−	0.645465i	$-0.776664\pi$
$157$	−7.40717e7	−1.52758	−0.763790	+	0.645465i	$0.776664\pi$
$158$	0	0
$159$	9.39047e6	0.185267
$160$	0	0
$161$	2.43452e7	0.459752
$162$	0	0
$163$	4.14405e7	0.749495	0.374747	−	0.927127i	$-0.377729\pi$
$163$	4.14405e7	0.749495	0.374747	+	0.927127i	$0.377729\pi$
$164$	0	0
$165$	−9.87974e6	−0.171219
$166$	0	0
$167$	6.33911e7	1.05322	0.526612	−	0.850106i	$-0.323462\pi$
$167$	6.33911e7	1.05322	0.526612	+	0.850106i	$0.323462\pi$
$168$	0	0
$169$	−1.03825e7	−0.165462
$170$	0	0
$171$	6.93163e6	0.106011
$172$	0	0
$173$	−8.20895e7	−1.20539	−0.602693	−	0.797973i	$-0.705906\pi$
$173$	−8.20895e7	−1.20539	−0.602693	+	0.797973i	$0.705906\pi$
$174$	0	0
$175$	2.34339e7	0.330531
$176$	0	0
$177$	−6.43819e6	−0.0872684
$178$	0	0
$179$	1.08169e8	1.40967	0.704833	−	0.709373i	$-0.251022\pi$
$179$	1.08169e8	1.40967	0.704833	+	0.709373i	$0.251022\pi$
$180$	0	0
$181$	4.64906e7	0.582760	0.291380	−	0.956607i	$-0.405886\pi$
$181$	4.64906e7	0.582760	0.291380	+	0.956607i	$0.405886\pi$
$182$	0	0
$183$	−4.10820e7	−0.495533
$184$	0	0
$185$	3.65526e7	0.424441
$186$	0	0
$187$	6.84521e7	0.765493
$188$	0	0
$189$	6.75127e6	0.0727393
$190$	0	0
$191$	−8.40245e7	−0.872548	−0.436274	−	0.899814i	$-0.643702\pi$
$191$	−8.40245e7	−0.872548	−0.436274	+	0.899814i	$0.643702\pi$
$192$	0	0
$193$	−4.46018e7	−0.446582	−0.223291	−	0.974752i	$-0.571680\pi$
$193$	−4.46018e7	−0.446582	−0.223291	+	0.974752i	$0.571680\pi$
$194$	0	0
$195$	−1.93465e7	−0.186845
$196$	0	0
$197$	−3.04765e7	−0.284010	−0.142005	−	0.989866i	$-0.545355\pi$
$197$	−3.04765e7	−0.284010	−0.142005	+	0.989866i	$0.545355\pi$
$198$	0	0
$199$	3.70526e7	0.333298	0.166649	−	0.986016i	$-0.446705\pi$
$199$	3.70526e7	0.333298	0.166649	+	0.986016i	$0.446705\pi$
$200$	0	0
$201$	1.86024e7	0.161578
$202$	0	0
$203$	−1.48022e7	−0.124191
$204$	0	0
$205$	−2.62156e6	−0.0212531
$206$	0	0
$207$	−5.17425e7	−0.405463
$208$	0	0
$209$	3.51379e7	0.266234
$210$	0	0
$211$	1.38109e8	1.01212	0.506060	−	0.862498i	$-0.331101\pi$
$211$	1.38109e8	1.01212	0.506060	+	0.862498i	$0.331101\pi$
$212$	0	0
$213$	1.19517e8	0.847422
$214$	0	0
$215$	−2.72484e7	−0.186985
$216$	0	0
$217$	−5.24974e6	−0.0348761
$218$	0	0
$219$	2.41540e7	0.155394
$220$	0	0
$221$	1.34043e8	0.835355
$222$	0	0
$223$	2.07529e8	1.25317	0.626586	−	0.779352i	$-0.284452\pi$
$223$	2.07529e8	1.25317	0.626586	+	0.779352i	$0.284452\pi$
$224$	0	0
$225$	−4.98056e7	−0.291501
$226$	0	0
$227$	7.24650e7	0.411185	0.205593	−	0.978638i	$-0.434088\pi$
$227$	7.24650e7	0.411185	0.205593	+	0.978638i	$0.434088\pi$
$228$	0	0
$229$	−2.87710e8	−1.58318	−0.791590	−	0.611053i	$-0.790746\pi$
$229$	−2.87710e8	−1.58318	−0.791590	+	0.611053i	$0.790746\pi$
$230$	0	0
$231$	3.42236e7	0.182677
$232$	0	0
$233$	9.01798e7	0.467050	0.233525	−	0.972351i	$-0.424974\pi$
$233$	9.01798e7	0.467050	0.233525	+	0.972351i	$0.424974\pi$
$234$	0	0
$235$	1.12914e7	0.0567557
$236$	0	0
$237$	7.40992e7	0.361572
$238$	0	0
$239$	2.72154e8	1.28950	0.644752	−	0.764392i	$-0.276961\pi$
$239$	2.72154e8	1.28950	0.644752	+	0.764392i	$0.276961\pi$
$240$	0	0
$241$	−1.36758e7	−0.0629349	−0.0314675	−	0.999505i	$-0.510018\pi$
$241$	−1.36758e7	−0.0629349	−0.0314675	+	0.999505i	$0.510018\pi$
$242$	0	0
$243$	−1.43489e7	−0.0641500
$244$	0	0
$245$	1.16494e7	0.0506082
$246$	0	0
$247$	6.88070e7	0.290532
$248$	0	0
$249$	1.49000e8	0.611630
$250$	0	0
$251$	−1.79647e8	−0.717072	−0.358536	−	0.933516i	$-0.616724\pi$
$251$	−1.79647e8	−0.717072	−0.358536	+	0.933516i	$0.616724\pi$
$252$	0	0
$253$	−2.62294e8	−1.01828
$254$	0	0
$255$	−4.95219e7	−0.187028
$256$	0	0
$257$	2.08839e8	0.767442	0.383721	−	0.923449i	$-0.374642\pi$
$257$	2.08839e8	0.767442	0.383721	+	0.923449i	$0.374642\pi$
$258$	0	0
$259$	−1.26619e8	−0.452845
$260$	0	0
$261$	3.14600e7	0.109526
$262$	0	0
$263$	419999.	0.00142365	0.000711825	−	1.00000i	$-0.499773\pi$
$263$	419999.	0.00142365	0.000711825		1.00000i	$0.499773\pi$
$264$	0	0
$265$	−3.44380e7	−0.113678
$266$	0	0
$267$	−1.27874e8	−0.411144
$268$	0	0
$269$	2.26486e8	0.709429	0.354715	−	0.934975i	$-0.384578\pi$
$269$	2.26486e8	0.709429	0.354715	+	0.934975i	$0.384578\pi$
$270$	0	0
$271$	2.36234e8	0.721024	0.360512	−	0.932755i	$-0.382602\pi$
$271$	2.36234e8	0.721024	0.360512	+	0.932755i	$0.382602\pi$
$272$	0	0
$273$	6.70166e7	0.199349
$274$	0	0
$275$	−2.52475e8	−0.732072
$276$	0	0
$277$	−3.22906e8	−0.912846	−0.456423	−	0.889763i	$-0.650870\pi$
$277$	−3.22906e8	−0.912846	−0.456423	+	0.889763i	$0.650870\pi$
$278$	0	0
$279$	1.11576e7	0.0307579
$280$	0	0
$281$	3.12684e8	0.840684	0.420342	−	0.907366i	$-0.361910\pi$
$281$	3.12684e8	0.840684	0.420342	+	0.907366i	$0.361910\pi$
$282$	0	0
$283$	−3.61865e8	−0.949062	−0.474531	−	0.880239i	$-0.657382\pi$
$283$	−3.61865e8	−0.949062	−0.474531	+	0.880239i	$0.657382\pi$
$284$	0	0
$285$	−2.54206e7	−0.0650472
$286$	0	0
$287$	9.08113e6	0.0226753
$288$	0	0
$289$	−6.72248e7	−0.163828
$290$	0	0
$291$	−1.52556e8	−0.362914
$292$	0	0
$293$	5.13942e8	1.19365	0.596826	−	0.802371i	$-0.296428\pi$
$293$	5.13942e8	1.19365	0.596826	+	0.802371i	$0.296428\pi$
$294$	0	0
$295$	2.36110e7	0.0535472
$296$	0	0
$297$	−7.27376e7	−0.161106
$298$	0	0
$299$	−5.13623e8	−1.11121
$300$	0	0
$301$	9.43888e7	0.199498
$302$	0	0
$303$	−3.06370e8	−0.632699
$304$	0	0
$305$	1.50661e8	0.304055
$306$	0	0
$307$	1.35306e8	0.266890	0.133445	−	0.991056i	$-0.457396\pi$
$307$	1.35306e8	0.266890	0.133445	+	0.991056i	$0.457396\pi$
$308$	0	0
$309$	−1.51895e8	−0.292880
$310$	0	0
$311$	−4.15771e7	−0.0783778	−0.0391889	−	0.999232i	$-0.512477\pi$
$311$	−4.15771e7	−0.0783778	−0.0391889	+	0.999232i	$0.512477\pi$
$312$	0	0
$313$	7.71847e8	1.42274	0.711370	−	0.702817i	$-0.248075\pi$
$313$	7.71847e8	1.42274	0.711370	+	0.702817i	$0.248075\pi$
$314$	0	0
$315$	−2.47591e7	−0.0446322
$316$	0	0
$317$	502888.	0.000886674 0	0.000443337	−	1.00000i	$-0.499859\pi$
$317$	502888.	0.000886674 0	0.000443337		1.00000i	$0.499859\pi$
$318$	0	0
$319$	1.59477e8	0.275062
$320$	0	0
$321$	−1.19730e8	−0.202039
$322$	0	0
$323$	1.76127e8	0.290816
$324$	0	0
$325$	−4.94397e8	−0.798884
$326$	0	0
$327$	−6.98945e8	−1.10542
$328$	0	0
$329$	−3.91135e7	−0.0605538
$330$	0	0
$331$	−2.31667e8	−0.351129	−0.175564	−	0.984468i	$-0.556175\pi$
$331$	−2.31667e8	−0.351129	−0.175564	+	0.984468i	$0.556175\pi$
$332$	0	0
$333$	2.69111e8	0.399372
$334$	0	0
$335$	−6.82212e7	−0.0991432
$336$	0	0
$337$	−2.52157e8	−0.358895	−0.179447	−	0.983768i	$-0.557431\pi$
$337$	−2.52157e8	−0.358895	−0.179447	+	0.983768i	$0.557431\pi$
$338$	0	0
$339$	−5.18768e8	−0.723226
$340$	0	0
$341$	5.65602e7	0.0772451
$342$	0	0
$343$	−4.03536e7	−0.0539949
$344$	0	0
$345$	1.89757e8	0.248789
$346$	0	0
$347$	−3.88142e8	−0.498698	−0.249349	−	0.968414i	$-0.580217\pi$
$347$	−3.88142e8	−0.498698	−0.249349	+	0.968414i	$0.580217\pi$
$348$	0	0
$349$	2.15716e8	0.271640	0.135820	−	0.990734i	$-0.456633\pi$
$349$	2.15716e8	0.271640	0.135820	+	0.990734i	$0.456633\pi$
$350$	0	0
$351$	−1.42435e8	−0.175809
$352$	0	0
$353$	−1.01922e9	−1.23327	−0.616635	−	0.787250i	$-0.711504\pi$
$353$	−1.01922e9	−1.23327	−0.616635	+	0.787250i	$0.711504\pi$
$354$	0	0
$355$	−4.38307e8	−0.519972
$356$	0	0
$357$	1.71545e8	0.199544
$358$	0	0
$359$	−7.53375e8	−0.859371	−0.429686	−	0.902979i	$-0.641376\pi$
$359$	−7.53375e8	−0.859371	−0.429686	+	0.902979i	$0.641376\pi$
$360$	0	0
$361$	−8.03462e8	−0.898856
$362$	0	0
$363$	1.57431e8	0.172750
$364$	0	0
$365$	−8.85807e7	−0.0953485
$366$	0	0
$367$	−1.07509e8	−0.113531	−0.0567654	−	0.998388i	$-0.518079\pi$
$367$	−1.07509e8	−0.113531	−0.0567654	+	0.998388i	$0.518079\pi$
$368$	0	0
$369$	−1.93007e7	−0.0199977
$370$	0	0
$371$	1.19294e8	0.121285
$372$	0	0
$373$	−8.64265e8	−0.862315	−0.431157	−	0.902277i	$-0.641895\pi$
$373$	−8.64265e8	−0.862315	−0.431157	+	0.902277i	$0.641895\pi$
$374$	0	0
$375$	3.91520e8	0.383393
$376$	0	0
$377$	3.12288e8	0.300165
$378$	0	0
$379$	5.55521e8	0.524159	0.262080	−	0.965046i	$-0.415592\pi$
$379$	5.55521e8	0.524159	0.262080	+	0.965046i	$0.415592\pi$
$380$	0	0
$381$	−1.13967e9	−1.05570
$382$	0	0
$383$	8.64373e8	0.786150	0.393075	−	0.919506i	$-0.371411\pi$
$383$	8.64373e8	0.786150	0.393075	+	0.919506i	$0.371411\pi$
$384$	0	0
$385$	−1.25509e8	−0.112089
$386$	0	0
$387$	−2.00611e8	−0.175940
$388$	0	0
$389$	5.29805e8	0.456344	0.228172	−	0.973621i	$-0.426725\pi$
$389$	5.29805e8	0.456344	0.228172	+	0.973621i	$0.426725\pi$
$390$	0	0
$391$	−1.31474e9	−1.11230
$392$	0	0
$393$	−3.28431e8	−0.272942
$394$	0	0
$395$	−2.71746e8	−0.221857
$396$	0	0
$397$	1.08934e7	0.00873768	0.00436884	−	0.999990i	$-0.498609\pi$
$397$	1.08934e7	0.00873768	0.00436884	+	0.999990i	$0.498609\pi$
$398$	0	0
$399$	8.80574e7	0.0694002
$400$	0	0
$401$	7.35030e8	0.569245	0.284623	−	0.958640i	$-0.408132\pi$
$401$	7.35030e8	0.569245	0.284623	+	0.958640i	$0.408132\pi$
$402$	0	0
$403$	1.10756e8	0.0842947
$404$	0	0
$405$	5.26222e7	0.0393619
$406$	0	0
$407$	1.36418e9	1.00298
$408$	0	0
$409$	−2.64878e9	−1.91432	−0.957160	−	0.289558i	$-0.906492\pi$
$409$	−2.64878e9	−1.91432	−0.957160	+	0.289558i	$0.906492\pi$
$410$	0	0
$411$	−8.43849e8	−0.599540
$412$	0	0
$413$	−8.17888e7	−0.0571306
$414$	0	0
$415$	−5.46433e8	−0.375292
$416$	0	0
$417$	−9.61771e8	−0.649525
$418$	0	0
$419$	−1.02562e9	−0.681140	−0.340570	−	0.940219i	$-0.610620\pi$
$419$	−1.02562e9	−0.681140	−0.340570	+	0.940219i	$0.610620\pi$
$420$	0	0
$421$	−2.18207e9	−1.42522	−0.712608	−	0.701563i	$-0.752486\pi$
$421$	−2.18207e9	−1.42522	−0.712608	+	0.701563i	$0.752486\pi$
$422$	0	0
$423$	8.31305e7	0.0534034
$424$	0	0
$425$	−1.26552e9	−0.799666
$426$	0	0
$427$	−5.21894e8	−0.324403
$428$	0	0
$429$	−7.22032e8	−0.441526
$430$	0	0
$431$	2.96388e9	1.78316	0.891578	−	0.452866i	$-0.149599\pi$
$431$	2.96388e9	1.78316	0.891578	+	0.452866i	$0.149599\pi$
$432$	0	0
$433$	−6.91187e8	−0.409155	−0.204578	−	0.978850i	$-0.565582\pi$
$433$	−6.91187e8	−0.409155	−0.204578	+	0.978850i	$0.565582\pi$
$434$	0	0
$435$	−1.15374e8	−0.0672042
$436$	0	0
$437$	−6.74882e8	−0.386850
$438$	0	0
$439$	−9.87163e8	−0.556882	−0.278441	−	0.960453i	$-0.589818\pi$
$439$	−9.87163e8	−0.556882	−0.278441	+	0.960453i	$0.589818\pi$
$440$	0	0
$441$	8.57661e7	0.0476190
$442$	0	0
$443$	2.74519e9	1.50024	0.750118	−	0.661304i	$-0.229997\pi$
$443$	2.74519e9	1.50024	0.750118	+	0.661304i	$0.229997\pi$
$444$	0	0
$445$	4.68958e8	0.252275
$446$	0	0
$447$	−1.20108e9	−0.636057
$448$	0	0
$449$	−4.80638e8	−0.250585	−0.125293	−	0.992120i	$-0.539987\pi$
$449$	−4.80638e8	−0.250585	−0.125293	+	0.992120i	$0.539987\pi$
$450$	0	0
$451$	−9.78394e7	−0.0502222
$452$	0	0
$453$	−4.29115e7	−0.0216885
$454$	0	0
$455$	−2.45772e8	−0.122319
$456$	0	0
$457$	−8.43345e8	−0.413332	−0.206666	−	0.978412i	$-0.566261\pi$
$457$	−8.43345e8	−0.413332	−0.206666	+	0.978412i	$0.566261\pi$
$458$	0	0
$459$	−3.64595e8	−0.175981
$460$	0	0
$461$	2.31800e9	1.10195	0.550973	−	0.834523i	$-0.314257\pi$
$461$	2.31800e9	1.10195	0.550973	+	0.834523i	$0.314257\pi$
$462$	0	0
$463$	8.07564e8	0.378132	0.189066	−	0.981964i	$-0.439454\pi$
$463$	8.07564e8	0.378132	0.189066	+	0.981964i	$0.439454\pi$
$464$	0	0
$465$	−4.09186e7	−0.0188728
$466$	0	0
$467$	−3.95717e9	−1.79794	−0.898970	−	0.438011i	$-0.855683\pi$
$467$	−3.95717e9	−1.79794	−0.898970	+	0.438011i	$0.855683\pi$
$468$	0	0
$469$	2.36320e8	0.105778
$470$	0	0
$471$	1.99994e9	0.881948
$472$	0	0
$473$	−1.01694e9	−0.441855
$474$	0	0
$475$	−6.49619e8	−0.278119
$476$	0	0
$477$	−2.53543e8	−0.106964
$478$	0	0
$479$	3.24574e9	1.34940	0.674698	−	0.738094i	$-0.264274\pi$
$479$	3.24574e9	1.34940	0.674698	+	0.738094i	$0.264274\pi$
$480$	0	0
$481$	2.67134e9	1.09451
$482$	0	0
$483$	−6.57321e8	−0.265438
$484$	0	0
$485$	5.59474e8	0.222681
$486$	0	0
$487$	1.11073e9	0.435769	0.217885	−	0.975975i	$-0.430084\pi$
$487$	1.11073e9	0.435769	0.217885	+	0.975975i	$0.430084\pi$
$488$	0	0
$489$	−1.11889e9	−0.432721
$490$	0	0
$491$	3.46672e9	1.32170	0.660851	−	0.750517i	$-0.270195\pi$
$491$	3.46672e9	1.32170	0.660851	+	0.750517i	$0.270195\pi$
$492$	0	0
$493$	7.99374e8	0.300459
$494$	0	0
$495$	2.66753e8	0.0988533
$496$	0	0
$497$	1.51830e9	0.554768
$498$	0	0
$499$	−2.69715e9	−0.971746	−0.485873	−	0.874029i	$-0.661498\pi$
$499$	−2.69715e9	−0.971746	−0.485873	+	0.874029i	$0.661498\pi$
$500$	0	0
$501$	−1.71156e9	−0.608079
$502$	0	0
$503$	1.75257e9	0.614027	0.307014	−	0.951705i	$-0.400670\pi$
$503$	1.75257e9	0.614027	0.307014	+	0.951705i	$0.400670\pi$
$504$	0	0
$505$	1.12356e9	0.388219
$506$	0	0
$507$	2.80328e8	0.0955296
$508$	0	0
$509$	3.25027e9	1.09247	0.546233	−	0.837634i	$-0.316061\pi$
$509$	3.25027e9	1.09247	0.546233	+	0.837634i	$0.316061\pi$
$510$	0	0
$511$	3.06845e8	0.101729
$512$	0	0
$513$	−1.87154e8	−0.0612052
$514$	0	0
$515$	5.57051e8	0.179709
$516$	0	0
$517$	4.21406e8	0.134117
$518$	0	0
$519$	2.21642e9	0.695930
$520$	0	0
$521$	−1.74813e9	−0.541554	−0.270777	−	0.962642i	$-0.587281\pi$
$521$	−1.74813e9	−0.541554	−0.270777	+	0.962642i	$0.587281\pi$
$522$	0	0
$523$	−5.96418e9	−1.82304	−0.911518	−	0.411261i	$-0.865088\pi$
$523$	−5.96418e9	−1.82304	−0.911518	+	0.411261i	$0.865088\pi$
$524$	0	0
$525$	−6.32716e8	−0.190832
$526$	0	0
$527$	2.83506e8	0.0843773
$528$	0	0
$529$	1.63296e9	0.479602
$530$	0	0
$531$	1.73831e8	0.0503845
$532$	0	0
$533$	−1.91589e8	−0.0548057
$534$	0	0
$535$	4.39089e8	0.123969
$536$	0	0
$537$	−2.92056e9	−0.813871
$538$	0	0
$539$	4.34767e8	0.119590
$540$	0	0
$541$	4.42149e9	1.20054	0.600272	−	0.799796i	$-0.295059\pi$
$541$	4.42149e9	1.20054	0.600272	+	0.799796i	$0.295059\pi$
$542$	0	0
$543$	−1.25525e9	−0.336457
$544$	0	0
$545$	2.56326e9	0.678275
$546$	0	0
$547$	3.66088e9	0.956380	0.478190	−	0.878256i	$-0.341293\pi$
$547$	3.66088e9	0.956380	0.478190	+	0.878256i	$0.341293\pi$
$548$	0	0
$549$	1.10921e9	0.286096
$550$	0	0
$551$	4.10335e8	0.104498
$552$	0	0
$553$	9.41334e8	0.236704
$554$	0	0
$555$	−9.86921e8	−0.245051
$556$	0	0
$557$	−4.16407e9	−1.02100	−0.510499	−	0.859878i	$-0.670539\pi$
$557$	−4.16407e9	−1.02100	−0.510499	+	0.859878i	$0.670539\pi$
$558$	0	0
$559$	−1.99137e9	−0.482180
$560$	0	0
$561$	−1.84821e9	−0.441958
$562$	0	0
$563$	1.88169e9	0.444394	0.222197	−	0.975002i	$-0.428677\pi$
$563$	1.88169e9	0.444394	0.222197	+	0.975002i	$0.428677\pi$
$564$	0	0
$565$	1.90249e9	0.443766
$566$	0	0
$567$	−1.82284e8	−0.0419961
$568$	0	0
$569$	7.84632e9	1.78556	0.892778	−	0.450497i	$-0.148753\pi$
$569$	7.84632e9	1.78556	0.892778	+	0.450497i	$0.148753\pi$
$570$	0	0
$571$	−3.53026e9	−0.793562	−0.396781	−	0.917913i	$-0.629873\pi$
$571$	−3.53026e9	−0.793562	−0.396781	+	0.917913i	$0.629873\pi$
$572$	0	0
$573$	2.26866e9	0.503766
$574$	0	0
$575$	4.84921e9	1.06373
$576$	0	0
$577$	−1.30432e9	−0.282663	−0.141331	−	0.989962i	$-0.545138\pi$
$577$	−1.30432e9	−0.282663	−0.141331	+	0.989962i	$0.545138\pi$
$578$	0	0
$579$	1.20425e9	0.257834
$580$	0	0
$581$	1.89285e9	0.400406
$582$	0	0
$583$	−1.28526e9	−0.268628
$584$	0	0
$585$	5.22356e8	0.107875
$586$	0	0
$587$	1.96984e9	0.401973	0.200986	−	0.979594i	$-0.435585\pi$
$587$	1.96984e9	0.401973	0.200986	+	0.979594i	$0.435585\pi$
$588$	0	0
$589$	1.45530e8	0.0293459
$590$	0	0
$591$	8.22867e8	0.163973
$592$	0	0
$593$	6.32425e9	1.24542	0.622712	−	0.782451i	$-0.286031\pi$
$593$	6.32425e9	1.24542	0.622712	+	0.782451i	$0.286031\pi$
$594$	0	0
$595$	−6.29111e8	−0.122438
$596$	0	0
$597$	−1.00042e9	−0.192430
$598$	0	0
$599$	3.45689e9	0.657191	0.328596	−	0.944471i	$-0.393425\pi$
$599$	3.45689e9	0.657191	0.328596	+	0.944471i	$0.393425\pi$
$600$	0	0
$601$	7.76561e9	1.45920	0.729600	−	0.683874i	$-0.239706\pi$
$601$	7.76561e9	1.45920	0.729600	+	0.683874i	$0.239706\pi$
$602$	0	0
$603$	−5.02265e8	−0.0932873
$604$	0	0
$605$	−5.77353e8	−0.105998
$606$	0	0
$607$	1.59452e9	0.289381	0.144691	−	0.989477i	$-0.453781\pi$
$607$	1.59452e9	0.289381	0.144691	+	0.989477i	$0.453781\pi$
$608$	0	0
$609$	3.99658e8	0.0717015
$610$	0	0
$611$	8.25197e8	0.146357
$612$	0	0
$613$	9.11892e8	0.159894	0.0799469	−	0.996799i	$-0.474525\pi$
$613$	9.11892e8	0.159894	0.0799469	+	0.996799i	$0.474525\pi$
$614$	0	0
$615$	7.07821e7	0.0122705
$616$	0	0
$617$	−5.96004e9	−1.02153	−0.510765	−	0.859720i	$-0.670638\pi$
$617$	−5.96004e9	−1.02153	−0.510765	+	0.859720i	$0.670638\pi$
$618$	0	0
$619$	−1.13437e10	−1.92237	−0.961185	−	0.275905i	$-0.911023\pi$
$619$	−1.13437e10	−1.92237	−0.961185	+	0.275905i	$0.911023\pi$
$620$	0	0
$621$	1.39705e9	0.234094
$622$	0	0
$623$	−1.62448e9	−0.269157
$624$	0	0
$625$	3.90170e9	0.639255
$626$	0	0
$627$	−9.48723e8	−0.153710
$628$	0	0
$629$	6.83791e9	1.09559
$630$	0	0
$631$	7.83067e9	1.24078	0.620392	−	0.784292i	$-0.286974\pi$
$631$	7.83067e9	1.24078	0.620392	+	0.784292i	$0.286974\pi$
$632$	0	0
$633$	−3.72893e9	−0.584348
$634$	0	0
$635$	4.17955e9	0.647771
$636$	0	0
$637$	8.51360e8	0.130504
$638$	0	0
$639$	−3.22695e9	−0.489260
$640$	0	0
$641$	5.57392e9	0.835906	0.417953	−	0.908469i	$-0.362748\pi$
$641$	5.57392e9	0.835906	0.417953	+	0.908469i	$0.362748\pi$
$642$	0	0
$643$	−1.15645e10	−1.71549	−0.857744	−	0.514077i	$-0.828134\pi$
$643$	−1.15645e10	−1.71549	−0.857744	+	0.514077i	$0.828134\pi$
$644$	0	0
$645$	7.35705e8	0.107956
$646$	0	0
$647$	3.85801e9	0.560013	0.280007	−	0.959998i	$-0.409663\pi$
$647$	3.85801e9	0.560013	0.280007	+	0.959998i	$0.409663\pi$
$648$	0	0
$649$	8.81186e8	0.126535
$650$	0	0
$651$	1.41743e8	0.0201358
$652$	0	0
$653$	6.73377e9	0.946373	0.473187	−	0.880962i	$-0.343104\pi$
$653$	6.73377e9	0.946373	0.473187	+	0.880962i	$0.343104\pi$
$654$	0	0
$655$	1.20446e9	0.167475
$656$	0	0
$657$	−6.52158e8	−0.0897168
$658$	0	0
$659$	2.72188e9	0.370485	0.185242	−	0.982693i	$-0.440693\pi$
$659$	2.72188e9	0.370485	0.185242	+	0.982693i	$0.440693\pi$
$660$	0	0
$661$	−3.23681e9	−0.435925	−0.217962	−	0.975957i	$-0.569941\pi$
$661$	−3.23681e9	−0.435925	−0.217962	+	0.975957i	$0.569941\pi$
$662$	0	0
$663$	−3.61916e9	−0.482292
$664$	0	0
$665$	−3.22936e8	−0.0425834
$666$	0	0
$667$	−3.06303e9	−0.399678
$668$	0	0
$669$	−5.60327e9	−0.723519
$670$	0	0
$671$	5.62284e9	0.718500
$672$	0	0
$673$	−1.04198e10	−1.31768	−0.658838	−	0.752285i	$-0.728952\pi$
$673$	−1.04198e10	−1.31768	−0.658838	+	0.752285i	$0.728952\pi$
$674$	0	0
$675$	1.34475e9	0.168298
$676$	0	0
$677$	8.01041e9	0.992189	0.496094	−	0.868269i	$-0.334767\pi$
$677$	8.01041e9	0.992189	0.496094	+	0.868269i	$0.334767\pi$
$678$	0	0
$679$	−1.93803e9	−0.237583
$680$	0	0
$681$	−1.95655e9	−0.237398
$682$	0	0
$683$	−7.15349e9	−0.859104	−0.429552	−	0.903042i	$-0.641328\pi$
$683$	−7.15349e9	−0.859104	−0.429552	+	0.903042i	$0.641328\pi$
$684$	0	0
$685$	3.09467e9	0.367873
$686$	0	0
$687$	7.76816e9	0.914049
$688$	0	0
$689$	−2.51680e9	−0.293144
$690$	0	0
$691$	−1.25797e10	−1.45043	−0.725216	−	0.688521i	$-0.758260\pi$
$691$	−1.25797e10	−1.45043	−0.725216	+	0.688521i	$0.758260\pi$
$692$	0	0
$693$	−9.24037e8	−0.105469
$694$	0	0
$695$	3.52713e9	0.398543
$696$	0	0
$697$	−4.90416e8	−0.0548593
$698$	0	0
$699$	−2.43485e9	−0.269651
$700$	0	0
$701$	−2.83761e9	−0.311128	−0.155564	−	0.987826i	$-0.549719\pi$
$701$	−2.83761e9	−0.311128	−0.155564	+	0.987826i	$0.549719\pi$
$702$	0	0
$703$	3.51004e9	0.381038
$704$	0	0
$705$	−3.04867e8	−0.0327679
$706$	0	0
$707$	−3.89203e9	−0.414198
$708$	0	0
$709$	1.32544e10	1.39668	0.698341	−	0.715765i	$-0.253922\pi$
$709$	1.32544e10	1.39668	0.698341	+	0.715765i	$0.253922\pi$
$710$	0	0
$711$	−2.00068e9	−0.208753
$712$	0	0
$713$	−1.08633e9	−0.112241
$714$	0	0
$715$	2.64793e9	0.270917
$716$	0	0
$717$	−7.34816e9	−0.744495
$718$	0	0
$719$	5.30334e9	0.532107	0.266053	−	0.963958i	$-0.414280\pi$
$719$	5.30334e9	0.532107	0.266053	+	0.963958i	$0.414280\pi$
$720$	0	0
$721$	−1.92963e9	−0.191735
$722$	0	0
$723$	3.69245e8	0.0363355
$724$	0	0
$725$	−2.94837e9	−0.287342
$726$	0	0
$727$	1.14181e10	1.10210	0.551052	−	0.834471i	$-0.314226\pi$
$727$	1.14181e10	1.10210	0.551052	+	0.834471i	$0.314226\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−5.09736e9	−0.482652
$732$	0	0
$733$	−1.12247e10	−1.05271	−0.526355	−	0.850265i	$-0.676442\pi$
$733$	−1.12247e10	−1.05271	−0.526355	+	0.850265i	$0.676442\pi$
$734$	0	0
$735$	−3.14533e8	−0.0292187
$736$	0	0
$737$	−2.54609e9	−0.234281
$738$	0	0
$739$	1.91114e10	1.74196	0.870980	−	0.491319i	$-0.163485\pi$
$739$	1.91114e10	1.74196	0.870980	+	0.491319i	$0.163485\pi$
$740$	0	0
$741$	−1.85779e9	−0.167738
$742$	0	0
$743$	1.48601e10	1.32911	0.664554	−	0.747241i	$-0.268622\pi$
$743$	1.48601e10	1.32911	0.664554	+	0.747241i	$0.268622\pi$
$744$	0	0
$745$	4.40477e9	0.390280
$746$	0	0
$747$	−4.02301e9	−0.353125
$748$	0	0
$749$	−1.52101e9	−0.132265
$750$	0	0
$751$	2.11232e10	1.81978	0.909892	−	0.414846i	$-0.136165\pi$
$751$	2.11232e10	1.81978	0.909892	+	0.414846i	$0.136165\pi$
$752$	0	0
$753$	4.85048e9	0.414002
$754$	0	0
$755$	1.57371e8	0.0133079
$756$	0	0
$757$	−1.24838e10	−1.04595	−0.522977	−	0.852347i	$-0.675179\pi$
$757$	−1.24838e10	−1.04595	−0.522977	+	0.852347i	$0.675179\pi$
$758$	0	0
$759$	7.08193e9	0.587903
$760$	0	0
$761$	−1.10735e10	−0.910836	−0.455418	−	0.890278i	$-0.650510\pi$
$761$	−1.10735e10	−0.910836	−0.455418	+	0.890278i	$0.650510\pi$
$762$	0	0
$763$	−8.87919e9	−0.723665
$764$	0	0
$765$	1.33709e9	0.107981
$766$	0	0
$767$	1.72554e9	0.138083
$768$	0	0
$769$	−1.71862e9	−0.136282	−0.0681409	−	0.997676i	$-0.521707\pi$
$769$	−1.71862e9	−0.136282	−0.0681409	+	0.997676i	$0.521707\pi$
$770$	0	0
$771$	−5.63865e9	−0.443083
$772$	0	0
$773$	−6.13658e8	−0.0477857	−0.0238929	−	0.999715i	$-0.507606\pi$
$773$	−6.13658e8	−0.0477857	−0.0238929	+	0.999715i	$0.507606\pi$
$774$	0	0
$775$	−1.04567e9	−0.0806934
$776$	0	0
$777$	3.41871e9	0.261450
$778$	0	0
$779$	−2.51741e8	−0.0190798
$780$	0	0
$781$	−1.63581e10	−1.22872
$782$	0	0
$783$	−8.49419e8	−0.0632348
$784$	0	0
$785$	−7.33443e9	−0.541156
$786$	0	0
$787$	1.38237e10	1.01091	0.505455	−	0.862853i	$-0.331325\pi$
$787$	1.38237e10	1.01091	0.505455	+	0.862853i	$0.331325\pi$
$788$	0	0
$789$	−1.13400e7	−0.000821944 0
$790$	0	0
$791$	−6.59028e9	−0.473463
$792$	0	0
$793$	1.10106e10	0.784073
$794$	0	0
$795$	9.29825e8	0.0656321
$796$	0	0
$797$	2.37930e9	0.166474	0.0832368	−	0.996530i	$-0.473474\pi$
$797$	2.37930e9	0.166474	0.0832368	+	0.996530i	$0.473474\pi$
$798$	0	0
$799$	2.11228e9	0.146500
$800$	0	0
$801$	3.45261e9	0.237374
$802$	0	0
$803$	−3.30592e9	−0.225314
$804$	0	0
$805$	2.41062e9	0.162871
$806$	0	0
$807$	−6.11513e9	−0.409589
$808$	0	0
$809$	−4.24121e8	−0.0281624	−0.0140812	−	0.999901i	$-0.504482\pi$
$809$	−4.24121e8	−0.0281624	−0.0140812	+	0.999901i	$0.504482\pi$
$810$	0	0
$811$	−7.61503e9	−0.501301	−0.250651	−	0.968078i	$-0.580645\pi$
$811$	−7.61503e9	−0.501301	−0.250651	+	0.968078i	$0.580645\pi$
$812$	0	0
$813$	−6.37831e9	−0.416283
$814$	0	0
$815$	4.10336e9	0.265514
$816$	0	0
$817$	−2.61658e9	−0.167864
$818$	0	0
$819$	−1.80945e9	−0.115094
$820$	0	0
$821$	−8.22147e9	−0.518500	−0.259250	−	0.965810i	$-0.583475\pi$
$821$	−8.22147e9	−0.518500	−0.259250	+	0.965810i	$0.583475\pi$
$822$	0	0
$823$	−5.71466e9	−0.357348	−0.178674	−	0.983908i	$-0.557181\pi$
$823$	−5.71466e9	−0.357348	−0.178674	+	0.983908i	$0.557181\pi$
$824$	0	0
$825$	6.81683e9	0.422662
$826$	0	0
$827$	−9.12989e9	−0.561302	−0.280651	−	0.959810i	$-0.590550\pi$
$827$	−9.12989e9	−0.561302	−0.280651	+	0.959810i	$0.590550\pi$
$828$	0	0
$829$	−6.93786e9	−0.422946	−0.211473	−	0.977384i	$-0.567826\pi$
$829$	−6.93786e9	−0.422946	−0.211473	+	0.977384i	$0.567826\pi$
$830$	0	0
$831$	8.71848e9	0.527032
$832$	0	0
$833$	2.17925e9	0.130632
$834$	0	0
$835$	6.27686e9	0.373113
$836$	0	0
$837$	−3.01255e8	−0.0177581
$838$	0	0
$839$	−5.69445e9	−0.332878	−0.166439	−	0.986052i	$-0.553227\pi$
$839$	−5.69445e9	−0.332878	−0.166439	+	0.986052i	$0.553227\pi$
$840$	0	0
$841$	−1.53875e10	−0.892037
$842$	0	0
$843$	−8.44246e9	−0.485369
$844$	0	0
$845$	−1.02805e9	−0.0586162
$846$	0	0
$847$	1.99996e9	0.113091
$848$	0	0
$849$	9.77036e9	0.547941
$850$	0	0
$851$	−2.62014e10	−1.45737
$852$	0	0
$853$	−2.87194e10	−1.58436	−0.792180	−	0.610288i	$-0.791054\pi$
$853$	−2.87194e10	−1.58436	−0.792180	+	0.610288i	$0.791054\pi$
$854$	0	0
$855$	6.86356e8	0.0375550
$856$	0	0
$857$	−1.23544e10	−0.670483	−0.335241	−	0.942132i	$-0.608818\pi$
$857$	−1.23544e10	−0.670483	−0.335241	+	0.942132i	$0.608818\pi$
$858$	0	0
$859$	−5.46709e9	−0.294293	−0.147146	−	0.989115i	$-0.547009\pi$
$859$	−5.46709e9	−0.294293	−0.147146	+	0.989115i	$0.547009\pi$
$860$	0	0
$861$	−2.45191e8	−0.0130916
$862$	0	0
$863$	2.03580e10	1.07820	0.539098	−	0.842243i	$-0.318765\pi$
$863$	2.03580e10	1.07820	0.539098	+	0.842243i	$0.318765\pi$
$864$	0	0
$865$	−8.12834e9	−0.427017
$866$	0	0
$867$	1.81507e9	0.0945859
$868$	0	0
$869$	−1.01419e10	−0.524262
$870$	0	0
$871$	−4.98575e9	−0.255662
$872$	0	0
$873$	4.11902e9	0.209529
$874$	0	0
$875$	4.97375e9	0.250990
$876$	0	0
$877$	3.28998e10	1.64700	0.823502	−	0.567313i	$-0.192017\pi$
$877$	3.28998e10	1.64700	0.823502	+	0.567313i	$0.192017\pi$
$878$	0	0
$879$	−1.38764e10	−0.689155
$880$	0	0
$881$	3.29513e10	1.62352	0.811759	−	0.583992i	$-0.198510\pi$
$881$	3.29513e10	1.62352	0.811759	+	0.583992i	$0.198510\pi$
$882$	0	0
$883$	−9.84034e9	−0.481003	−0.240501	−	0.970649i	$-0.577312\pi$
$883$	−9.84034e9	−0.481003	−0.240501	+	0.970649i	$0.577312\pi$
$884$	0	0
$885$	−6.37496e8	−0.0309155
$886$	0	0
$887$	−3.83050e10	−1.84299	−0.921494	−	0.388392i	$-0.873031\pi$
$887$	−3.83050e10	−1.84299	−0.921494	+	0.388392i	$0.873031\pi$
$888$	0	0
$889$	−1.44780e10	−0.691120
$890$	0	0
$891$	1.96392e9	0.0930146
$892$	0	0
$893$	1.08428e9	0.0509519
$894$	0	0
$895$	1.07106e10	0.499385
$896$	0	0
$897$	1.38678e10	0.641556
$898$	0	0
$899$	6.60502e8	0.0303190
$900$	0	0
$901$	−6.44232e9	−0.293431
$902$	0	0
$903$	−2.54850e9	−0.115180
$904$	0	0
$905$	4.60340e9	0.206447
$906$	0	0
$907$	−6.85297e9	−0.304967	−0.152484	−	0.988306i	$-0.548727\pi$
$907$	−6.85297e9	−0.304967	−0.152484	+	0.988306i	$0.548727\pi$
$908$	0	0
$909$	8.27199e9	0.365289
$910$	0	0
$911$	−1.68852e9	−0.0739933	−0.0369966	−	0.999315i	$-0.511779\pi$
$911$	−1.68852e9	−0.0739933	−0.0369966	+	0.999315i	$0.511779\pi$
$912$	0	0
$913$	−2.03935e10	−0.886836
$914$	0	0
$915$	−4.06786e9	−0.175546
$916$	0	0
$917$	−4.17229e9	−0.178682
$918$	0	0
$919$	−8.49261e9	−0.360942	−0.180471	−	0.983580i	$-0.557762\pi$
$919$	−8.49261e9	−0.360942	−0.180471	+	0.983580i	$0.557762\pi$
$920$	0	0
$921$	−3.65326e9	−0.154089
$922$	0	0
$923$	−3.20324e10	−1.34086
$924$	0	0
$925$	−2.52206e10	−1.04775
$926$	0	0
$927$	4.10117e9	0.169094
$928$	0	0
$929$	−1.01380e10	−0.414854	−0.207427	−	0.978251i	$-0.566509\pi$
$929$	−1.01380e10	−0.414854	−0.207427	+	0.978251i	$0.566509\pi$
$930$	0	0
$931$	1.11865e9	0.0454331
$932$	0	0
$933$	1.12258e9	0.0452514
$934$	0	0
$935$	6.77799e9	0.271182
$936$	0	0
$937$	2.73102e10	1.08452	0.542258	−	0.840212i	$-0.317569\pi$
$937$	2.73102e10	1.08452	0.542258	+	0.840212i	$0.317569\pi$
$938$	0	0
$939$	−2.08399e10	−0.821420
$940$	0	0
$941$	−1.11756e10	−0.437229	−0.218614	−	0.975811i	$-0.570154\pi$
$941$	−1.11756e10	−0.437229	−0.218614	+	0.975811i	$0.570154\pi$
$942$	0	0
$943$	1.87917e9	0.0729751
$944$	0	0
$945$	6.68497e8	0.0257684
$946$	0	0
$947$	−1.14842e10	−0.439416	−0.219708	−	0.975566i	$-0.570510\pi$
$947$	−1.14842e10	−0.439416	−0.219708	+	0.975566i	$0.570510\pi$
$948$	0	0
$949$	−6.47366e9	−0.245877
$950$	0	0
$951$	−1.35780e7	−0.000511922 0
$952$	0	0
$953$	3.24527e10	1.21458	0.607289	−	0.794481i	$-0.292257\pi$
$953$	3.24527e10	1.21458	0.607289	+	0.794481i	$0.292257\pi$
$954$	0	0
$955$	−8.31994e9	−0.309107
$956$	0	0
$957$	−4.30588e9	−0.158807
$958$	0	0
$959$	−1.07200e10	−0.392491
$960$	0	0
$961$	−2.72784e10	−0.991486
$962$	0	0
$963$	3.23270e9	0.116647
$964$	0	0
$965$	−4.41638e9	−0.158205
$966$	0	0
$967$	1.48412e10	0.527807	0.263903	−	0.964549i	$-0.414990\pi$
$967$	1.48412e10	0.527807	0.263903	+	0.964549i	$0.414990\pi$
$968$	0	0
$969$	−4.75544e9	−0.167903
$970$	0	0
$971$	−5.11293e10	−1.79227	−0.896134	−	0.443784i	$-0.853636\pi$
$971$	−5.11293e10	−1.79227	−0.896134	+	0.443784i	$0.853636\pi$
$972$	0	0
$973$	−1.22181e10	−0.425214
$974$	0	0
$975$	1.33487e10	0.461236
$976$	0	0
$977$	−3.72350e10	−1.27738	−0.638691	−	0.769464i	$-0.720524\pi$
$977$	−3.72350e10	−1.27738	−0.638691	+	0.769464i	$0.720524\pi$
$978$	0	0
$979$	1.75020e10	0.596140
$980$	0	0
$981$	1.88715e10	0.638212
$982$	0	0
$983$	−2.51039e10	−0.842953	−0.421476	−	0.906839i	$-0.638488\pi$
$983$	−2.51039e10	−0.842953	−0.421476	+	0.906839i	$0.638488\pi$
$984$	0	0
$985$	−3.01773e9	−0.100613
$986$	0	0
$987$	1.05607e9	0.0349608
$988$	0	0
$989$	1.95320e10	0.642036
$990$	0	0
$991$	−9.08794e9	−0.296625	−0.148312	−	0.988941i	$-0.547384\pi$
$991$	−9.08794e9	−0.296625	−0.148312	+	0.988941i	$0.547384\pi$
$992$	0	0
$993$	6.25501e9	0.202724
$994$	0	0
$995$	3.66887e9	0.118073
$996$	0	0
$997$	3.04492e10	0.973068	0.486534	−	0.873662i	$-0.338261\pi$
$997$	3.04492e10	0.973068	0.486534	+	0.873662i	$0.338261\pi$
$998$	0	0
$999$	−7.26601e9	−0.230577

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.8.a.t.1.2		3
4.3	odd	2		168.8.a.i.1.2	✓	3
12.11	even	2		504.8.a.h.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.8.a.i.1.2	✓	3	4.3	odd	2
336.8.a.t.1.2		3	1.1	even	1	trivial
504.8.a.h.1.2		3	12.11	even	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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	336.a (trivial)

\(f(q)\)	\(=\)	\(q-27.0000 q^{3} +99.0180 q^{5} -343.000 q^{7} +729.000 q^{9} +3695.45 q^{11} +7236.44 q^{13} -2673.49 q^{15} +18523.3 q^{17} +9508.41 q^{19} +9261.00 q^{21} -70977.4 q^{23} -68320.4 q^{25} -19683.0 q^{27} +43155.0 q^{29} +15305.4 q^{31} -99777.3 q^{33} -33963.2 q^{35} +369151. q^{37} -195384. q^{39} -26475.6 q^{41} -275186. q^{43} +72184.1 q^{45} +114034. q^{47} +117649. q^{49} -500130. q^{51} -347795. q^{53} +365916. q^{55} -256727. q^{57} +238451. q^{59} +1.52156e6 q^{61} -250047. q^{63} +716537. q^{65} -688978. q^{67} +1.91639e6 q^{69} -4.42654e6 q^{71} -894592. q^{73} +1.84465e6 q^{75} -1.26754e6 q^{77} -2.74442e6 q^{79} +531441. q^{81} -5.51853e6 q^{83} +1.83414e6 q^{85} -1.16518e6 q^{87} +4.73609e6 q^{89} -2.48210e6 q^{91} -413245. q^{93} +941504. q^{95} +5.65023e6 q^{97} +2.69399e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 81 q^{3} + 414 q^{5} - 1029 q^{7} + 2187 q^{9} - 9552 q^{11} + 954 q^{13} - 11178 q^{15} - 30 q^{17} - 39588 q^{19} + 27783 q^{21} + 31164 q^{23} + 139581 q^{25} - 59049 q^{27} + 38898 q^{29} - 147888 q^{31}+ \cdots - 6963408 q^{99}+O(q^{100}) \) 3 * q - 81 * q^3 + 414 * q^5 - 1029 * q^7 + 2187 * q^9 - 9552 * q^11 + 954 * q^13 - 11178 * q^15 - 30 * q^17 - 39588 * q^19 + 27783 * q^21 + 31164 * q^23 + 139581 * q^25 - 59049 * q^27 + 38898 * q^29 - 147888 * q^31 + 257904 * q^33 - 142002 * q^35 - 217662 * q^37 - 25758 * q^39 + 174594 * q^41 - 193644 * q^43 + 301806 * q^45 - 675864 * q^47 + 352947 * q^49 + 810 * q^51 + 476898 * q^53 - 2055576 * q^55 + 1068876 * q^57 - 2107092 * q^59 + 190962 * q^61 - 750141 * q^63 - 922572 * q^65 - 3016380 * q^67 - 841428 * q^69 - 1285404 * q^71 + 6735462 * q^73 - 3768687 * q^75 + 3276336 * q^77 - 2985240 * q^79 + 1594323 * q^81 - 6168276 * q^83 + 14892156 * q^85 - 1050246 * q^87 + 3937938 * q^89 - 327222 * q^91 + 3992976 * q^93 - 24467016 * q^95 + 22816878 * q^97 - 6963408 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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