LMFDB - Embedded newform 63.18.a.d.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [63,18,Mod(1,63)] 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("63.1"); S:= CuspForms(chi, 18); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(63, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 18, names="a")

Level:	$ N $	$=$	$ 63 = 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	63.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,375] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$115.429915027$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 401750x^{2} - 42202572x + 5013099432 $ x^4 - x^3 - 401750x^2 - 42202572x + 5013099432
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{3}\cdot 7 $
Twist minimal:	no (minimal twist has level 21)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$71.1690$ of defining polynomial
Character	$\chi$	$=$	63.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+22.8310 q^{2} -130551. q^{4} -632611. q^{5} -5.76480e6 q^{7} -5.97310e6 q^{8} -1.44431e7 q^{10} +2.15066e8 q^{11} +3.03827e9 q^{13} -1.31616e8 q^{14} +1.69752e10 q^{16} -5.45123e10 q^{17} -1.24821e11 q^{19} +8.25879e10 q^{20} +4.91016e9 q^{22} -1.14102e11 q^{23} -3.62743e11 q^{25} +6.93665e10 q^{26} +7.52599e11 q^{28} -3.39139e12 q^{29} -7.84404e12 q^{31} +1.17047e12 q^{32} -1.24457e12 q^{34} +3.64688e12 q^{35} -1.41235e13 q^{37} -2.84979e12 q^{38} +3.77865e12 q^{40} -2.48174e13 q^{41} -9.60227e13 q^{43} -2.80770e13 q^{44} -2.60506e12 q^{46} +2.94778e14 q^{47} +3.32329e13 q^{49} -8.28176e12 q^{50} -3.96648e14 q^{52} +4.16239e14 q^{53} -1.36053e14 q^{55} +3.44337e13 q^{56} -7.74287e13 q^{58} +1.54226e15 q^{59} +1.78601e15 q^{61} -1.79087e14 q^{62} -2.19825e15 q^{64} -1.92204e15 q^{65} +8.00135e14 q^{67} +7.11662e15 q^{68} +8.32617e13 q^{70} -8.31181e15 q^{71} -1.03998e16 q^{73} -3.22453e14 q^{74} +1.62955e16 q^{76} -1.23981e15 q^{77} +9.94360e15 q^{79} -1.07387e16 q^{80} -5.66605e14 q^{82} -1.61457e16 q^{83} +3.44851e16 q^{85} -2.19229e15 q^{86} -1.28461e15 q^{88} -2.80887e16 q^{89} -1.75150e16 q^{91} +1.48961e16 q^{92} +6.73005e15 q^{94} +7.89633e16 q^{95} -2.34017e16 q^{97} +7.58740e14 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 375 q^{2} + 314369 q^{4} + 154140 q^{5} - 23059204 q^{7} + 3766143 q^{8} + 23352830 q^{10} + 1452022884 q^{11} + 1793559040 q^{13} - 2161800375 q^{14} + 36719981057 q^{16} - 68581449948 q^{17} - 2578000592 q^{19}+ \cdots + 12\!\cdots\!75 q^{98}+O(q^{100}) $ 4 * q + 375 * q^2 + 314369 * q^4 + 154140 * q^5 - 23059204 * q^7 + 3766143 * q^8 + 23352830 * q^10 + 1452022884 * q^11 + 1793559040 * q^13 - 2161800375 * q^14 + 36719981057 * q^16 - 68581449948 * q^17 - 2578000592 * q^19 - 132998517630 * q^20 + 270530423852 * q^22 - 649239533556 * q^23 - 708242296300 * q^25 + 2508729280746 * q^26 - 1812274725569 * q^28 + 3258162962760 * q^29 + 2414795507136 * q^31 + 18040133068671 * q^32 + 8860338750210 * q^34 - 888586426140 * q^35 + 23164217924208 * q^37 + 67011195804708 * q^38 - 115859765446530 * q^40 + 117516076237164 * q^41 - 175144919320720 * q^43 + 379640394802500 * q^44 - 441875935197000 * q^46 + 392013343869480 * q^47 + 132931722278404 * q^49 + 366102837059625 * q^50 - 647834999852794 * q^52 + 86395731614976 * q^53 - 987784569502880 * q^55 - 21711064932543 * q^56 - 2149209668160466 * q^58 + 23145424510920 * q^59 + 3424273979460952 * q^61 + 4542097297046784 * q^62 - 3840648962754815 * q^64 - 2015724107462520 * q^65 + 2234127752098744 * q^67 + 1956571325658750 * q^68 - 134624417736830 * q^70 - 2815776345321804 * q^71 + 11330038422344784 * q^73 - 2782049125778694 * q^74 + 33041781126411260 * q^76 - 8370622973706084 * q^77 - 12177038540666920 * q^79 - 35816456249313150 * q^80 + 79244490955525210 * q^82 + 4389820874951568 * q^83 + 46318093724780760 * q^85 + 79314655634282340 * q^86 + 93140307906096828 * q^88 - 100131520699583508 * q^89 - 10339510947351040 * q^91 + 14621524874654712 * q^92 + 120993183153224352 * q^94 + 115775864256900240 * q^95 - 44947325255962464 * q^97 + 12462348963600375 * 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 $

$2$	22.8310	0.0630622	0.0315311	−	0.999503i	$-0.489962\pi$
$2$	22.8310	0.0630622	0.0315311	+	0.999503i	$0.489962\pi$
$3$	0	0
$3$	0	0
$4$	−130551.	−0.996023
$5$	−632611.	−0.724256	−0.362128	−	0.932128i	$-0.617950\pi$
$5$	−632611.	−0.724256	−0.362128	+	0.932128i	$0.617950\pi$
$6$	0	0
$7$	−5.76480e6	−0.377964
$7$	−5.76480e6	−0.377964
$8$	−5.97310e6	−0.125874
$9$	0	0
$10$	−1.44431e7	−0.0456732
$11$	2.15066e8	0.302506	0.151253	−	0.988495i	$-0.451669\pi$
$11$	2.15066e8	0.302506	0.151253	+	0.988495i	$0.451669\pi$
$12$	0	0
$13$	3.03827e9	1.03302	0.516508	−	0.856282i	$-0.327231\pi$
$13$	3.03827e9	1.03302	0.516508	+	0.856282i	$0.327231\pi$
$14$	−1.31616e8	−0.0238353
$15$	0	0
$16$	1.69752e10	0.988085
$17$	−5.45123e10	−1.89530	−0.947652	−	0.319305i	$-0.896551\pi$
$17$	−5.45123e10	−1.89530	−0.947652	+	0.319305i	$0.896551\pi$
$18$	0	0
$19$	−1.24821e11	−1.68610	−0.843048	−	0.537838i	$-0.819241\pi$
$19$	−1.24821e11	−1.68610	−0.843048	+	0.537838i	$0.819241\pi$
$20$	8.25879e10	0.721375
$21$	0	0
$22$	4.91016e9	0.0190767
$23$	−1.14102e11	−0.303813	−0.151907	−	0.988395i	$-0.548541\pi$
$23$	−1.14102e11	−0.303813	−0.151907	+	0.988395i	$0.548541\pi$
$24$	0	0
$25$	−3.62743e11	−0.475454
$26$	6.93665e10	0.0651443
$27$	0	0
$28$	7.52599e11	0.376461
$29$	−3.39139e12	−1.25891	−0.629455	−	0.777037i	$-0.716722\pi$
$29$	−3.39139e12	−1.25891	−0.629455	+	0.777037i	$0.716722\pi$
$30$	0	0
$31$	−7.84404e12	−1.65183	−0.825915	−	0.563795i	$-0.809341\pi$
$31$	−7.84404e12	−1.65183	−0.825915	+	0.563795i	$0.809341\pi$
$32$	1.17047e12	0.188184
$33$	0	0
$34$	−1.24457e12	−0.119522
$35$	3.64688e12	0.273743
$36$	0	0
$37$	−1.41235e13	−0.661039	−0.330519	−	0.943799i	$-0.607224\pi$
$37$	−1.41235e13	−0.661039	−0.330519	+	0.943799i	$0.607224\pi$
$38$	−2.84979e12	−0.106329
$39$	0	0
$40$	3.77865e12	0.0911647
$41$	−2.48174e13	−0.485393	−0.242696	−	0.970102i	$-0.578032\pi$
$41$	−2.48174e13	−0.485393	−0.242696	+	0.970102i	$0.578032\pi$
$42$	0	0
$43$	−9.60227e13	−1.25283	−0.626415	−	0.779490i	$-0.715478\pi$
$43$	−9.60227e13	−1.25283	−0.626415	+	0.779490i	$0.715478\pi$
$44$	−2.80770e13	−0.301303
$45$	0	0
$46$	−2.60506e12	−0.0191591
$47$	2.94778e14	1.80577	0.902885	−	0.429881i	$-0.141445\pi$
$47$	2.94778e14	1.80577	0.902885	+	0.429881i	$0.141445\pi$
$48$	0	0
$49$	3.32329e13	0.142857
$50$	−8.28176e12	−0.0299832
$51$	0	0
$52$	−3.96648e14	−1.02891
$53$	4.16239e14	0.918328	0.459164	−	0.888352i	$-0.348149\pi$
$53$	4.16239e14	0.918328	0.459164	+	0.888352i	$0.348149\pi$
$54$	0	0
$55$	−1.36053e14	−0.219092
$56$	3.44337e13	0.0475758
$57$	0	0
$58$	−7.74287e13	−0.0793897
$59$	1.54226e15	1.36746	0.683729	−	0.729736i	$-0.260357\pi$
$59$	1.54226e15	1.36746	0.683729	+	0.729736i	$0.260357\pi$
$60$	0	0
$61$	1.78601e15	1.19284	0.596418	−	0.802674i	$-0.296590\pi$
$61$	1.78601e15	1.19284	0.596418	+	0.802674i	$0.296590\pi$
$62$	−1.79087e14	−0.104168
$63$	0	0
$64$	−2.19825e15	−0.976218
$65$	−1.92204e15	−0.748168
$66$	0	0
$67$	8.00135e14	0.240728	0.120364	−	0.992730i	$-0.461594\pi$
$67$	8.00135e14	0.240728	0.120364	+	0.992730i	$0.461594\pi$
$68$	7.11662e15	1.88777
$69$	0	0
$70$	8.32617e13	0.0172628
$71$	−8.31181e15	−1.52756	−0.763782	−	0.645474i	$-0.776660\pi$
$71$	−8.31181e15	−1.52756	−0.763782	+	0.645474i	$0.776660\pi$
$72$	0	0
$73$	−1.03998e16	−1.50932	−0.754659	−	0.656117i	$-0.772198\pi$
$73$	−1.03998e16	−1.50932	−0.754659	+	0.656117i	$0.772198\pi$
$74$	−3.22453e14	−0.0416866
$75$	0	0
$76$	1.62955e16	1.67939
$77$	−1.23981e15	−0.114336
$78$	0	0
$79$	9.94360e15	0.737418	0.368709	−	0.929545i	$-0.379800\pi$
$79$	9.94360e15	0.737418	0.368709	+	0.929545i	$0.379800\pi$
$80$	−1.07387e16	−0.715626
$81$	0	0
$82$	−5.66605e14	−0.0306099
$83$	−1.61457e16	−0.786854	−0.393427	−	0.919356i	$-0.628711\pi$
$83$	−1.61457e16	−0.786854	−0.393427	+	0.919356i	$0.628711\pi$
$84$	0	0
$85$	3.44851e16	1.37268
$86$	−2.19229e15	−0.0790062
$87$	0	0
$88$	−1.28461e15	−0.0380775
$89$	−2.80887e16	−0.756340	−0.378170	−	0.925736i	$-0.623447\pi$
$89$	−2.80887e16	−0.756340	−0.378170	+	0.925736i	$0.623447\pi$
$90$	0	0
$91$	−1.75150e16	−0.390444
$92$	1.48961e16	0.302605
$93$	0	0
$94$	6.73005e15	0.113876
$95$	7.89633e16	1.22117
$96$	0	0
$97$	−2.34017e16	−0.303171	−0.151586	−	0.988444i	$-0.548438\pi$
$97$	−2.34017e16	−0.303171	−0.151586	+	0.988444i	$0.548438\pi$
$98$	7.58740e14	0.00900889
$99$	0	0
$100$	4.73563e16	0.473563
$101$	1.47193e17	1.35255	0.676276	−	0.736648i	$-0.263592\pi$
$101$	1.47193e17	1.35255	0.676276	+	0.736648i	$0.263592\pi$
$102$	0	0
$103$	−9.97084e16	−0.775560	−0.387780	−	0.921752i	$-0.626758\pi$
$103$	−9.97084e16	−0.775560	−0.387780	+	0.921752i	$0.626758\pi$
$104$	−1.81479e16	−0.130030
$105$	0	0
$106$	9.50313e15	0.0579118
$107$	8.56145e16	0.481710	0.240855	−	0.970561i	$-0.422572\pi$
$107$	8.56145e16	0.481710	0.240855	+	0.970561i	$0.422572\pi$
$108$	0	0
$109$	−4.92382e15	−0.0236688	−0.0118344	−	0.999930i	$-0.503767\pi$
$109$	−4.92382e15	−0.0236688	−0.0118344	+	0.999930i	$0.503767\pi$
$110$	−3.10622e15	−0.0138164
$111$	0	0
$112$	−9.78585e16	−0.373461
$113$	8.04913e16	0.284828	0.142414	−	0.989807i	$-0.454514\pi$
$113$	8.04913e16	0.284828	0.142414	+	0.989807i	$0.454514\pi$
$114$	0	0
$115$	7.21822e16	0.220039
$116$	4.42749e17	1.25390
$117$	0	0
$118$	3.52112e16	0.0862350
$119$	3.14253e17	0.716358
$120$	0	0
$121$	−4.59194e17	−0.908490
$122$	4.07763e16	0.0752228
$123$	0	0
$124$	1.02404e18	1.64526
$125$	7.12119e17	1.06861
$126$	0	0
$127$	−1.81476e17	−0.237951	−0.118975	−	0.992897i	$-0.537961\pi$
$127$	−1.81476e17	−0.237951	−0.118975	+	0.992897i	$0.537961\pi$
$128$	−2.03603e17	−0.249747
$129$	0	0
$130$	−4.38821e16	−0.0471811
$131$	−5.17043e16	−0.0520860	−0.0260430	−	0.999661i	$-0.508291\pi$
$131$	−5.17043e16	−0.0520860	−0.0260430	+	0.999661i	$0.508291\pi$
$132$	0	0
$133$	7.19569e17	0.637285
$134$	1.82678e16	0.0151809
$135$	0	0
$136$	3.25607e17	0.238569
$137$	−1.39158e18	−0.958038	−0.479019	−	0.877804i	$-0.659008\pi$
$137$	−1.39158e18	−0.958038	−0.479019	+	0.877804i	$0.659008\pi$
$138$	0	0
$139$	9.31425e17	0.566920	0.283460	−	0.958984i	$-0.408518\pi$
$139$	9.31425e17	0.566920	0.283460	+	0.958984i	$0.408518\pi$
$140$	−4.76103e17	−0.272654
$141$	0	0
$142$	−1.89766e17	−0.0963316
$143$	6.53428e17	0.312494
$144$	0	0
$145$	2.14543e18	0.911773
$146$	−2.37437e17	−0.0951809
$147$	0	0
$148$	1.84383e18	0.658410
$149$	5.44442e18	1.83598	0.917991	−	0.396601i	$-0.129810\pi$
$149$	5.44442e18	1.83598	0.917991	+	0.396601i	$0.129810\pi$
$150$	0	0
$151$	8.94329e16	0.0269273	0.0134637	−	0.999909i	$-0.495714\pi$
$151$	8.94329e16	0.0269273	0.0134637	+	0.999909i	$0.495714\pi$
$152$	7.45569e17	0.212235
$153$	0	0
$154$	−2.83061e16	−0.00721031
$155$	4.96223e18	1.19635
$156$	0	0
$157$	1.44620e18	0.312665	0.156333	−	0.987704i	$-0.450033\pi$
$157$	1.44620e18	0.312665	0.156333	+	0.987704i	$0.450033\pi$
$158$	2.27022e17	0.0465032
$159$	0	0
$160$	−7.40449e17	−0.136294
$161$	6.57776e17	0.114831
$162$	0	0
$163$	−4.82679e17	−0.0758690	−0.0379345	−	0.999280i	$-0.512078\pi$
$163$	−4.82679e17	−0.0758690	−0.0379345	+	0.999280i	$0.512078\pi$
$164$	3.23993e18	0.483463
$165$	0	0
$166$	−3.68623e17	−0.0496207
$167$	6.19189e18	0.792015	0.396007	−	0.918247i	$-0.370395\pi$
$167$	6.19189e18	0.792015	0.396007	+	0.918247i	$0.370395\pi$
$168$	0	0
$169$	5.80650e17	0.0671240
$170$	7.87328e17	0.0865645
$171$	0	0
$172$	1.25358e19	1.24785
$173$	1.36734e19	1.29564	0.647822	−	0.761792i	$-0.275680\pi$
$173$	1.36734e19	1.29564	0.647822	+	0.761792i	$0.275680\pi$
$174$	0	0
$175$	2.09114e18	0.179705
$176$	3.65078e18	0.298902
$177$	0	0
$178$	−6.41293e17	−0.0476965
$179$	1.95585e19	1.38703	0.693513	−	0.720444i	$-0.256062\pi$
$179$	1.95585e19	1.38703	0.693513	+	0.720444i	$0.256062\pi$
$180$	0	0
$181$	−1.87533e19	−1.21007	−0.605034	−	0.796200i	$-0.706840\pi$
$181$	−1.87533e19	−1.21007	−0.605034	+	0.796200i	$0.706840\pi$
$182$	−3.99884e17	−0.0246222
$183$	0	0
$184$	6.81543e17	0.0382421
$185$	8.93467e18	0.478761
$186$	0	0
$187$	−1.17237e19	−0.573341
$188$	−3.84834e19	−1.79859
$189$	0	0
$190$	1.80281e18	0.0770094
$191$	−2.54663e19	−1.04036	−0.520178	−	0.854058i	$-0.674134\pi$
$191$	−2.54663e19	−1.04036	−0.520178	+	0.854058i	$0.674134\pi$
$192$	0	0
$193$	−5.35805e18	−0.200341	−0.100170	−	0.994970i	$-0.531939\pi$
$193$	−5.35805e18	−0.200341	−0.100170	+	0.994970i	$0.531939\pi$
$194$	−5.34284e17	−0.0191186
$195$	0	0
$196$	−4.33858e18	−0.142289
$197$	1.09188e19	0.342934	0.171467	−	0.985190i	$-0.445149\pi$
$197$	1.09188e19	0.342934	0.171467	+	0.985190i	$0.445149\pi$
$198$	0	0
$199$	2.24150e19	0.646081	0.323040	−	0.946385i	$-0.395295\pi$
$199$	2.24150e19	0.646081	0.323040	+	0.946385i	$0.395295\pi$
$200$	2.16670e18	0.0598471
$201$	0	0
$202$	3.36055e18	0.0852950
$203$	1.95507e19	0.475823
$204$	0	0
$205$	1.56998e19	0.351548
$206$	−2.27644e18	−0.0489085
$207$	0	0
$208$	5.15751e19	1.02071
$209$	−2.68448e19	−0.510054
$210$	0	0
$211$	−8.37629e19	−1.46775	−0.733873	−	0.679287i	$-0.762289\pi$
$211$	−8.37629e19	−1.46775	−0.733873	+	0.679287i	$0.762289\pi$
$212$	−5.43403e19	−0.914675
$213$	0	0
$214$	1.95466e18	0.0303777
$215$	6.07450e19	0.907369
$216$	0	0
$217$	4.52193e19	0.624333
$218$	−1.12416e17	−0.00149261
$219$	0	0
$220$	1.77618e19	0.218220
$221$	−1.65623e20	−1.95788
$222$	0	0
$223$	4.25519e19	0.465938	0.232969	−	0.972484i	$-0.425156\pi$
$223$	4.25519e19	0.465938	0.232969	+	0.972484i	$0.425156\pi$
$224$	−6.74750e18	−0.0711270
$225$	0	0
$226$	1.83769e18	0.0179619
$227$	−5.72079e19	−0.538563	−0.269282	−	0.963062i	$-0.586786\pi$
$227$	−5.72079e19	−0.538563	−0.269282	+	0.963062i	$0.586786\pi$
$228$	0	0
$229$	6.95149e19	0.607402	0.303701	−	0.952767i	$-0.401778\pi$
$229$	6.95149e19	0.607402	0.303701	+	0.952767i	$0.401778\pi$
$230$	1.64799e18	0.0138761
$231$	0	0
$232$	2.02571e19	0.158464
$233$	1.65993e20	1.25189	0.625943	−	0.779869i	$-0.284714\pi$
$233$	1.65993e20	1.25189	0.625943	+	0.779869i	$0.284714\pi$
$234$	0	0
$235$	−1.86480e20	−1.30784
$236$	−2.01343e20	−1.36202
$237$	0	0
$238$	7.17469e18	0.0451751
$239$	9.53139e19	0.579127	0.289564	−	0.957159i	$-0.406490\pi$
$239$	9.53139e19	0.579127	0.289564	+	0.957159i	$0.406490\pi$
$240$	0	0
$241$	−1.85793e20	−1.05168	−0.525842	−	0.850582i	$-0.676250\pi$
$241$	−1.85793e20	−1.05168	−0.525842	+	0.850582i	$0.676250\pi$
$242$	−1.04838e19	−0.0572914
$243$	0	0
$244$	−2.33165e20	−1.18809
$245$	−2.10235e19	−0.103465
$246$	0	0
$247$	−3.79240e20	−1.74177
$248$	4.68532e19	0.207922
$249$	0	0
$250$	1.62584e19	0.0673886
$251$	−4.61913e20	−1.85069	−0.925345	−	0.379126i	$-0.876225\pi$
$251$	−4.61913e20	−1.85069	−0.925345	+	0.379126i	$0.876225\pi$
$252$	0	0
$253$	−2.45395e19	−0.0919053
$254$	−4.14327e18	−0.0150057
$255$	0	0
$256$	2.83480e20	0.960468
$257$	−1.22729e20	−0.402267	−0.201134	−	0.979564i	$-0.564463\pi$
$257$	−1.22729e20	−0.402267	−0.201134	+	0.979564i	$0.564463\pi$
$258$	0	0
$259$	8.14190e19	0.249849
$260$	2.50924e20	0.745193
$261$	0	0
$262$	−1.18046e18	−0.00328466
$263$	−1.46641e20	−0.395030	−0.197515	−	0.980300i	$-0.563287\pi$
$263$	−1.46641e20	−0.395030	−0.197515	+	0.980300i	$0.563287\pi$
$264$	0	0
$265$	−2.63317e20	−0.665104
$266$	1.64285e19	0.0401886
$267$	0	0
$268$	−1.04458e20	−0.239771
$269$	2.05884e19	0.0457855	0.0228927	−	0.999738i	$-0.492712\pi$
$269$	2.05884e19	0.0457855	0.0228927	+	0.999738i	$0.492712\pi$
$270$	0	0
$271$	2.90773e20	0.607177	0.303588	−	0.952803i	$-0.401815\pi$
$271$	2.90773e20	0.607177	0.303588	+	0.952803i	$0.401815\pi$
$272$	−9.25356e20	−1.87272
$273$	0	0
$274$	−3.17711e19	−0.0604160
$275$	−7.80136e19	−0.143828
$276$	0	0
$277$	−6.72880e20	−1.16643	−0.583216	−	0.812317i	$-0.698206\pi$
$277$	−6.72880e20	−1.16643	−0.583216	+	0.812317i	$0.698206\pi$
$278$	2.12653e19	0.0357512
$279$	0	0
$280$	−2.17832e19	−0.0344570
$281$	4.87548e20	0.748193	0.374096	−	0.927390i	$-0.377953\pi$
$281$	4.87548e20	0.748193	0.374096	+	0.927390i	$0.377953\pi$
$282$	0	0
$283$	3.20008e20	0.462355	0.231178	−	0.972912i	$-0.425742\pi$
$283$	3.20008e20	0.462355	0.231178	+	0.972912i	$0.425742\pi$
$284$	1.08511e21	1.52149
$285$	0	0
$286$	1.49184e19	0.0197065
$287$	1.43067e20	0.183461
$288$	0	0
$289$	2.14435e21	2.59218
$290$	4.89823e19	0.0574984
$291$	0	0
$292$	1.35770e21	1.50332
$293$	−5.49606e20	−0.591121	−0.295561	−	0.955324i	$-0.595506\pi$
$293$	−5.49606e20	−0.591121	−0.295561	+	0.955324i	$0.595506\pi$
$294$	0	0
$295$	−9.75648e20	−0.990389
$296$	8.43609e19	0.0832074
$297$	0	0
$298$	1.24301e20	0.115781
$299$	−3.46673e20	−0.313844
$300$	0	0
$301$	5.53552e20	0.473525
$302$	2.04184e18	0.00169810
$303$	0	0
$304$	−2.11886e21	−1.66601
$305$	−1.12985e21	−0.863918
$306$	0	0
$307$	2.37264e21	1.71615	0.858075	−	0.513524i	$-0.171660\pi$
$307$	2.37264e21	1.71615	0.858075	+	0.513524i	$0.171660\pi$
$308$	1.61858e20	0.113882
$309$	0	0
$310$	1.13292e20	0.0754443
$311$	−3.09533e20	−0.200559	−0.100280	−	0.994959i	$-0.531974\pi$
$311$	−3.09533e20	−0.200559	−0.100280	+	0.994959i	$0.531974\pi$
$312$	0	0
$313$	−1.45959e21	−0.895582	−0.447791	−	0.894138i	$-0.647789\pi$
$313$	−1.45959e21	−0.895582	−0.447791	+	0.894138i	$0.647789\pi$
$314$	3.30180e19	0.0197174
$315$	0	0
$316$	−1.29814e21	−0.734485
$317$	6.93529e19	0.0381998	0.0190999	−	0.999818i	$-0.493920\pi$
$317$	6.93529e19	0.0381998	0.0190999	+	0.999818i	$0.493920\pi$
$318$	0	0
$319$	−7.29372e20	−0.380828
$320$	1.39064e21	0.707031
$321$	0	0
$322$	1.50176e19	0.00724148
$323$	6.80429e21	3.19567
$324$	0	0
$325$	−1.10211e21	−0.491152
$326$	−1.10200e19	−0.00478446
$327$	0	0
$328$	1.48237e20	0.0610982
$329$	−1.69933e21	−0.682517
$330$	0	0
$331$	3.45866e21	1.31938	0.659690	−	0.751538i	$-0.270688\pi$
$331$	3.45866e21	1.31938	0.659690	+	0.751538i	$0.270688\pi$
$332$	2.10784e21	0.783724
$333$	0	0
$334$	1.41367e20	0.0499462
$335$	−5.06174e20	−0.174349
$336$	0	0
$337$	−1.53324e21	−0.502060	−0.251030	−	0.967979i	$-0.580769\pi$
$337$	−1.53324e21	−0.502060	−0.251030	+	0.967979i	$0.580769\pi$
$338$	1.32568e19	0.00423299
$339$	0	0
$340$	−4.50206e21	−1.36723
$341$	−1.68699e21	−0.499688
$342$	0	0
$343$	−1.91581e20	−0.0539949
$344$	5.73553e20	0.157698
$345$	0	0
$346$	3.12178e20	0.0817062
$347$	1.30135e21	0.332349	0.166174	−	0.986096i	$-0.446859\pi$
$347$	1.30135e21	0.332349	0.166174	+	0.986096i	$0.446859\pi$
$348$	0	0
$349$	3.06887e21	0.746385	0.373193	−	0.927754i	$-0.378263\pi$
$349$	3.06887e21	0.746385	0.373193	+	0.927754i	$0.378263\pi$
$350$	4.77427e19	0.0113326
$351$	0	0
$352$	2.51727e20	0.0569269
$353$	6.07765e21	1.34169	0.670843	−	0.741599i	$-0.265932\pi$
$353$	6.07765e21	1.34169	0.670843	+	0.741599i	$0.265932\pi$
$354$	0	0
$355$	5.25814e21	1.10635
$356$	3.66701e21	0.753332
$357$	0	0
$358$	4.46539e20	0.0874689
$359$	4.15229e21	0.794301	0.397150	−	0.917754i	$-0.369999\pi$
$359$	4.15229e21	0.794301	0.397150	+	0.917754i	$0.369999\pi$
$360$	0	0
$361$	1.00999e22	1.84292
$362$	−4.28156e20	−0.0763095
$363$	0	0
$364$	2.28660e21	0.388891
$365$	6.57903e21	1.09313
$366$	0	0
$367$	−1.61653e21	−0.256403	−0.128202	−	0.991748i	$-0.540920\pi$
$367$	−1.61653e21	−0.256403	−0.128202	+	0.991748i	$0.540920\pi$
$368$	−1.93690e21	−0.300194
$369$	0	0
$370$	2.03987e20	0.0301917
$371$	−2.39953e21	−0.347095
$372$	0	0
$373$	−1.36639e21	−0.188820	−0.0944101	−	0.995533i	$-0.530097\pi$
$373$	−1.36639e21	−0.188820	−0.0944101	+	0.995533i	$0.530097\pi$
$374$	−2.67664e20	−0.0361561
$375$	0	0
$376$	−1.76074e21	−0.227299
$377$	−1.03039e22	−1.30048
$378$	0	0
$379$	−1.13977e22	−1.37525	−0.687627	−	0.726064i	$-0.741348\pi$
$379$	−1.13977e22	−1.37525	−0.687627	+	0.726064i	$0.741348\pi$
$380$	−1.03087e22	−1.21631
$381$	0	0
$382$	−5.81420e20	−0.0656071
$383$	−1.65483e22	−1.82627	−0.913134	−	0.407659i	$-0.866345\pi$
$383$	−1.65483e22	−1.82627	−0.913134	+	0.407659i	$0.866345\pi$
$384$	0	0
$385$	7.84319e20	0.0828088
$386$	−1.22329e20	−0.0126339
$387$	0	0
$388$	3.05511e21	0.301966
$389$	1.18214e22	1.14313	0.571566	−	0.820556i	$-0.306336\pi$
$389$	1.18214e22	1.14313	0.571566	+	0.820556i	$0.306336\pi$
$390$	0	0
$391$	6.21997e21	0.575819
$392$	−1.98504e20	−0.0179819
$393$	0	0
$394$	2.49286e20	0.0216262
$395$	−6.29043e21	−0.534079
$396$	0	0
$397$	1.51153e22	1.22941	0.614706	−	0.788757i	$-0.289275\pi$
$397$	1.51153e22	1.22941	0.614706	+	0.788757i	$0.289275\pi$
$398$	5.11755e20	0.0407433
$399$	0	0
$400$	−6.15762e21	−0.469789
$401$	−5.07171e21	−0.378815	−0.189408	−	0.981899i	$-0.560657\pi$
$401$	−5.07171e21	−0.378815	−0.189408	+	0.981899i	$0.560657\pi$
$402$	0	0
$403$	−2.38323e22	−1.70637
$404$	−1.92161e22	−1.34717
$405$	0	0
$406$	4.46361e20	0.0300065
$407$	−3.03748e21	−0.199968
$408$	0	0
$409$	−2.16376e22	−1.36635	−0.683175	−	0.730255i	$-0.739401\pi$
$409$	−2.16376e22	−1.36635	−0.683175	+	0.730255i	$0.739401\pi$
$410$	3.58441e20	0.0221694
$411$	0	0
$412$	1.30170e22	0.772475
$413$	−8.89080e21	−0.516851
$414$	0	0
$415$	1.02140e22	0.569883
$416$	3.55619e21	0.194398
$417$	0	0
$418$	−6.12892e20	−0.0321651
$419$	1.30411e22	0.670650	0.335325	−	0.942103i	$-0.391154\pi$
$419$	1.30411e22	0.670650	0.335325	+	0.942103i	$0.391154\pi$
$420$	0	0
$421$	−3.30097e22	−1.63021	−0.815107	−	0.579311i	$-0.803322\pi$
$421$	−3.30097e22	−1.63021	−0.815107	+	0.579311i	$0.803322\pi$
$422$	−1.91239e21	−0.0925593
$423$	0	0
$424$	−2.48623e21	−0.115593
$425$	1.97739e22	0.901130
$426$	0	0
$427$	−1.02960e22	−0.450849
$428$	−1.11770e22	−0.479794
$429$	0	0
$430$	1.38687e21	0.0572207
$431$	1.16277e22	0.470367	0.235184	−	0.971951i	$-0.424431\pi$
$431$	1.16277e22	0.470367	0.235184	+	0.971951i	$0.424431\pi$
$432$	0	0
$433$	−3.00470e21	−0.116857	−0.0584283	−	0.998292i	$-0.518609\pi$
$433$	−3.00470e21	−0.116857	−0.0584283	+	0.998292i	$0.518609\pi$
$434$	1.03240e21	0.0393718
$435$	0	0
$436$	6.42809e20	0.0235747
$437$	1.42424e22	0.512259
$438$	0	0
$439$	7.56531e21	0.261745	0.130872	−	0.991399i	$-0.458222\pi$
$439$	7.56531e21	0.261745	0.130872	+	0.991399i	$0.458222\pi$
$440$	8.12658e20	0.0275778
$441$	0	0
$442$	−3.78133e21	−0.123468
$443$	−3.71990e22	−1.19151	−0.595757	−	0.803165i	$-0.703148\pi$
$443$	−3.71990e22	−1.19151	−0.595757	+	0.803165i	$0.703148\pi$
$444$	0	0
$445$	1.77692e22	0.547784
$446$	9.71501e20	0.0293831
$447$	0	0
$448$	1.26725e22	0.368976
$449$	8.54040e21	0.243997	0.121998	−	0.992530i	$-0.461070\pi$
$449$	8.54040e21	0.243997	0.121998	+	0.992530i	$0.461070\pi$
$450$	0	0
$451$	−5.33738e21	−0.146834
$452$	−1.05082e22	−0.283695
$453$	0	0
$454$	−1.30611e21	−0.0339630
$455$	1.10802e22	0.282781
$456$	0	0
$457$	−7.46991e22	−1.83666	−0.918328	−	0.395821i	$-0.870460\pi$
$457$	−7.46991e22	−1.83666	−0.918328	+	0.395821i	$0.870460\pi$
$458$	1.58709e21	0.0383041
$459$	0	0
$460$	−9.42345e21	−0.219163
$461$	−2.06905e22	−0.472404	−0.236202	−	0.971704i	$-0.575903\pi$
$461$	−2.06905e22	−0.472404	−0.236202	+	0.971704i	$0.575903\pi$
$462$	0	0
$463$	4.05941e22	0.893357	0.446678	−	0.894695i	$-0.352607\pi$
$463$	4.05941e22	0.893357	0.446678	+	0.894695i	$0.352607\pi$
$464$	−5.75694e22	−1.24391
$465$	0	0
$466$	3.78978e21	0.0789466
$467$	−2.89680e22	−0.592550	−0.296275	−	0.955103i	$-0.595744\pi$
$467$	−2.89680e22	−0.592550	−0.296275	+	0.955103i	$0.595744\pi$
$468$	0	0
$469$	−4.61262e21	−0.0909868
$470$	−4.25751e21	−0.0824752
$471$	0	0
$472$	−9.21204e21	−0.172127
$473$	−2.06512e22	−0.378988
$474$	0	0
$475$	4.52779e22	0.801661
$476$	−4.10259e22	−0.713509
$477$	0	0
$478$	2.17611e21	0.0365211
$479$	8.74483e22	1.44178	0.720891	−	0.693048i	$-0.243733\pi$
$479$	8.74483e22	1.44178	0.720891	+	0.693048i	$0.243733\pi$
$480$	0	0
$481$	−4.29109e22	−0.682864
$482$	−4.24184e21	−0.0663215
$483$	0	0
$484$	5.99481e22	0.904877
$485$	1.48042e22	0.219573
$486$	0	0
$487$	6.80544e22	0.974675	0.487337	−	0.873214i	$-0.337968\pi$
$487$	6.80544e22	0.974675	0.487337	+	0.873214i	$0.337968\pi$
$488$	−1.06680e22	−0.150146
$489$	0	0
$490$	−4.79987e20	−0.00652474
$491$	−5.62422e22	−0.751398	−0.375699	−	0.926742i	$-0.622597\pi$
$491$	−5.62422e22	−0.751398	−0.375699	+	0.926742i	$0.622597\pi$
$492$	0	0
$493$	1.84873e23	2.38602
$494$	−8.65841e21	−0.109840
$495$	0	0
$496$	−1.33154e23	−1.63215
$497$	4.79159e22	0.577365
$498$	0	0
$499$	−2.44206e22	−0.284382	−0.142191	−	0.989839i	$-0.545415\pi$
$499$	−2.44206e22	−0.284382	−0.142191	+	0.989839i	$0.545415\pi$
$500$	−9.29677e22	−1.06436
$501$	0	0
$502$	−1.05459e22	−0.116709
$503$	1.47102e23	1.60063	0.800316	−	0.599579i	$-0.204665\pi$
$503$	1.47102e23	1.60063	0.800316	+	0.599579i	$0.204665\pi$
$504$	0	0
$505$	−9.31157e22	−0.979594
$506$	−5.60259e20	−0.00579575
$507$	0	0
$508$	2.36918e22	0.237005
$509$	1.49402e23	1.46979	0.734895	−	0.678181i	$-0.237231\pi$
$509$	1.49402e23	1.46979	0.734895	+	0.678181i	$0.237231\pi$
$510$	0	0
$511$	5.99528e22	0.570469
$512$	3.31588e22	0.310316
$513$	0	0
$514$	−2.80201e21	−0.0253679
$515$	6.30766e22	0.561703
$516$	0	0
$517$	6.33966e22	0.546256
$518$	1.85887e21	0.0157560
$519$	0	0
$520$	1.14805e22	0.0941746
$521$	−9.28095e22	−0.748983	−0.374491	−	0.927230i	$-0.622183\pi$
$521$	−9.28095e22	−0.748983	−0.374491	+	0.927230i	$0.622183\pi$
$522$	0	0
$523$	1.12343e23	0.877567	0.438784	−	0.898593i	$-0.355409\pi$
$523$	1.12343e23	0.877567	0.438784	+	0.898593i	$0.355409\pi$
$524$	6.75004e21	0.0518788
$525$	0	0
$526$	−3.34794e21	−0.0249115
$527$	4.27597e23	3.13072
$528$	0	0
$529$	−1.28031e23	−0.907697
$530$	−6.01179e21	−0.0419429
$531$	0	0
$532$	−9.39403e22	−0.634750
$533$	−7.54019e22	−0.501419
$534$	0	0
$535$	−5.41607e22	−0.348881
$536$	−4.77928e21	−0.0303014
$537$	0	0
$538$	4.70053e20	0.00288733
$539$	7.14727e21	0.0432151
$540$	0	0
$541$	9.57428e22	0.560957	0.280478	−	0.959860i	$-0.409507\pi$
$541$	9.57428e22	0.560957	0.280478	+	0.959860i	$0.409507\pi$
$542$	6.63863e21	0.0382899
$543$	0	0
$544$	−6.38048e22	−0.356667
$545$	3.11487e21	0.0171423
$546$	0	0
$547$	2.89093e23	1.54222	0.771109	−	0.636703i	$-0.219702\pi$
$547$	2.89093e23	1.54222	0.771109	+	0.636703i	$0.219702\pi$
$548$	1.81672e23	0.954228
$549$	0	0
$550$	−1.78112e21	−0.00907008
$551$	4.23317e23	2.12264
$552$	0	0
$553$	−5.73229e22	−0.278718
$554$	−1.53625e22	−0.0735578
$555$	0	0
$556$	−1.21598e23	−0.564666
$557$	−1.50045e23	−0.686202	−0.343101	−	0.939299i	$-0.611477\pi$
$557$	−1.50045e23	−0.686202	−0.343101	+	0.939299i	$0.611477\pi$
$558$	0	0
$559$	−2.91743e23	−1.29419
$560$	6.19064e22	0.270481
$561$	0	0
$562$	1.11312e22	0.0471827
$563$	−1.09521e23	−0.457274	−0.228637	−	0.973512i	$-0.573427\pi$
$563$	−1.09521e23	−0.457274	−0.228637	+	0.973512i	$0.573427\pi$
$564$	0	0
$565$	−5.09197e22	−0.206288
$566$	7.30609e21	0.0291572
$567$	0	0
$568$	4.96472e22	0.192280
$569$	−5.09552e23	−1.94417	−0.972085	−	0.234629i	$-0.924612\pi$
$569$	−5.09552e23	−1.94417	−0.972085	+	0.234629i	$0.924612\pi$
$570$	0	0
$571$	1.87057e23	0.692736	0.346368	−	0.938099i	$-0.387415\pi$
$571$	1.87057e23	0.692736	0.346368	+	0.938099i	$0.387415\pi$
$572$	−8.53055e22	−0.311251
$573$	0	0
$574$	3.26636e21	0.0115695
$575$	4.13897e22	0.144449
$576$	0	0
$577$	2.15129e23	0.728963	0.364481	−	0.931211i	$-0.381246\pi$
$577$	2.15129e23	0.728963	0.364481	+	0.931211i	$0.381246\pi$
$578$	4.89576e22	0.163468
$579$	0	0
$580$	−2.80088e23	−0.908147
$581$	9.30769e22	0.297403
$582$	0	0
$583$	8.95188e22	0.277799
$584$	6.21190e22	0.189983
$585$	0	0
$586$	−1.25480e22	−0.0372774
$587$	−6.14319e23	−1.79875	−0.899374	−	0.437180i	$-0.855977\pi$
$587$	−6.14319e23	−1.79875	−0.899374	+	0.437180i	$0.855977\pi$
$588$	0	0
$589$	9.79102e23	2.78514
$590$	−2.22750e22	−0.0624561
$591$	0	0
$592$	−2.39749e23	−0.653163
$593$	−1.42058e23	−0.381505	−0.190753	−	0.981638i	$-0.561093\pi$
$593$	−1.42058e23	−0.381505	−0.190753	+	0.981638i	$0.561093\pi$
$594$	0	0
$595$	−1.98800e23	−0.518826
$596$	−7.10773e23	−1.82868
$597$	0	0
$598$	−7.91487e21	−0.0197917
$599$	−3.29790e23	−0.813036	−0.406518	−	0.913643i	$-0.633257\pi$
$599$	−3.29790e23	−0.813036	−0.406518	+	0.913643i	$0.633257\pi$
$600$	0	0
$601$	4.90531e23	1.17553	0.587765	−	0.809032i	$-0.300008\pi$
$601$	4.90531e23	1.17553	0.587765	+	0.809032i	$0.300008\pi$
$602$	1.26381e22	0.0298615
$603$	0	0
$604$	−1.16755e22	−0.0268202
$605$	2.90491e23	0.657979
$606$	0	0
$607$	−6.11981e23	−1.34783	−0.673914	−	0.738810i	$-0.735388\pi$
$607$	−6.11981e23	−1.34783	−0.673914	+	0.738810i	$0.735388\pi$
$608$	−1.46099e23	−0.317297
$609$	0	0
$610$	−2.57956e22	−0.0544805
$611$	8.95613e23	1.86539
$612$	0	0
$613$	−2.98524e23	−0.604735	−0.302368	−	0.953191i	$-0.597777\pi$
$613$	−2.98524e23	−0.604735	−0.302368	+	0.953191i	$0.597777\pi$
$614$	5.41696e22	0.108224
$615$	0	0
$616$	7.40552e21	0.0143919
$617$	−1.98952e23	−0.381351	−0.190675	−	0.981653i	$-0.561068\pi$
$617$	−1.98952e23	−0.381351	−0.190675	+	0.981653i	$0.561068\pi$
$618$	0	0
$619$	−9.02032e22	−0.168210	−0.0841049	−	0.996457i	$-0.526803\pi$
$619$	−9.02032e22	−0.168210	−0.0841049	+	0.996457i	$0.526803\pi$
$620$	−6.47822e23	−1.19159
$621$	0	0
$622$	−7.06692e21	−0.0126477
$623$	1.61926e23	0.285870
$624$	0	0
$625$	−1.73744e23	−0.298490
$626$	−3.33239e22	−0.0564774
$627$	0	0
$628$	−1.88802e23	−0.311422
$629$	7.69904e23	1.25287
$630$	0	0
$631$	−5.31284e23	−0.841545	−0.420772	−	0.907166i	$-0.638241\pi$
$631$	−5.31284e23	−0.841545	−0.420772	+	0.907166i	$0.638241\pi$
$632$	−5.93941e22	−0.0928214
$633$	0	0
$634$	1.58339e21	0.00240896
$635$	1.14804e23	0.172337
$636$	0	0
$637$	1.00971e23	0.147574
$638$	−1.66523e22	−0.0240158
$639$	0	0
$640$	1.28802e23	0.180881
$641$	−3.16452e23	−0.438545	−0.219273	−	0.975664i	$-0.570368\pi$
$641$	−3.16452e23	−0.438545	−0.219273	+	0.975664i	$0.570368\pi$
$642$	0	0
$643$	−1.13986e24	−1.53836	−0.769178	−	0.639034i	$-0.779334\pi$
$643$	−1.13986e24	−1.53836	−0.769178	+	0.639034i	$0.779334\pi$
$644$	−8.58731e22	−0.114374
$645$	0	0
$646$	1.55348e23	0.201526
$647$	−8.52628e23	−1.09162	−0.545812	−	0.837908i	$-0.683779\pi$
$647$	−8.52628e23	−1.09162	−0.545812	+	0.837908i	$0.683779\pi$
$648$	0	0
$649$	3.31687e23	0.413664
$650$	−2.51622e22	−0.0309731
$651$	0	0
$652$	6.30142e22	0.0755673
$653$	3.70205e23	0.438208	0.219104	−	0.975701i	$-0.429687\pi$
$653$	3.70205e23	0.438208	0.219104	+	0.975701i	$0.429687\pi$
$654$	0	0
$655$	3.27087e22	0.0377235
$656$	−4.21280e23	−0.479610
$657$	0	0
$658$	−3.87974e22	−0.0430410
$659$	4.91109e23	0.537839	0.268919	−	0.963163i	$-0.413333\pi$
$659$	4.91109e23	0.537839	0.268919	+	0.963163i	$0.413333\pi$
$660$	0	0
$661$	−2.79560e23	−0.298375	−0.149188	−	0.988809i	$-0.547666\pi$
$661$	−2.79560e23	−0.298375	−0.149188	+	0.988809i	$0.547666\pi$
$662$	7.89645e22	0.0832030
$663$	0	0
$664$	9.64400e22	0.0990441
$665$	−4.55207e23	−0.461557
$666$	0	0
$667$	3.86965e23	0.382474
$668$	−8.08356e23	−0.788865
$669$	0	0
$670$	−1.15564e22	−0.0109948
$671$	3.84110e23	0.360840
$672$	0	0
$673$	−1.66622e24	−1.52618	−0.763089	−	0.646293i	$-0.776318\pi$
$673$	−1.66622e24	−1.52618	−0.763089	+	0.646293i	$0.776318\pi$
$674$	−3.50053e22	−0.0316610
$675$	0	0
$676$	−7.58043e22	−0.0668570
$677$	2.01718e23	0.175687	0.0878437	−	0.996134i	$-0.472002\pi$
$677$	2.01718e23	0.175687	0.0878437	+	0.996134i	$0.472002\pi$
$678$	0	0
$679$	1.34906e23	0.114588
$680$	−2.05983e23	−0.172785
$681$	0	0
$682$	−3.85155e22	−0.0315114
$683$	7.42953e23	0.600323	0.300162	−	0.953888i	$-0.402959\pi$
$683$	7.42953e23	0.600323	0.300162	+	0.953888i	$0.402959\pi$
$684$	0	0
$685$	8.80328e23	0.693864
$686$	−4.37398e21	−0.00340504
$687$	0	0
$688$	−1.63000e24	−1.23790
$689$	1.26464e24	0.948648
$690$	0	0
$691$	9.47125e23	0.693177	0.346588	−	0.938017i	$-0.387340\pi$
$691$	9.47125e23	0.693177	0.346588	+	0.938017i	$0.387340\pi$
$692$	−1.78508e24	−1.29049
$693$	0	0
$694$	2.97111e22	0.0209586
$695$	−5.89230e23	−0.410595
$696$	0	0
$697$	1.35285e24	0.919967
$698$	7.00653e22	0.0470687
$699$	0	0
$700$	−2.73000e23	−0.178990
$701$	−1.00735e24	−0.652498	−0.326249	−	0.945284i	$-0.605785\pi$
$701$	−1.00735e24	−0.652498	−0.326249	+	0.945284i	$0.605785\pi$
$702$	0	0
$703$	1.76291e24	1.11458
$704$	−4.72768e23	−0.295312
$705$	0	0
$706$	1.38759e23	0.0846097
$707$	−8.48536e23	−0.511217
$708$	0	0
$709$	1.54511e24	0.908793	0.454397	−	0.890799i	$-0.349855\pi$
$709$	1.54511e24	0.908793	0.454397	+	0.890799i	$0.349855\pi$
$710$	1.20048e23	0.0697687
$711$	0	0
$712$	1.67777e23	0.0952033
$713$	8.95021e23	0.501848
$714$	0	0
$715$	−4.13366e23	−0.226325
$716$	−2.55338e24	−1.38151
$717$	0	0
$718$	9.48008e22	0.0500903
$719$	2.63001e24	1.37329	0.686644	−	0.726994i	$-0.259083\pi$
$719$	2.63001e24	1.37329	0.686644	+	0.726994i	$0.259083\pi$
$720$	0	0
$721$	5.74799e23	0.293134
$722$	2.30591e23	0.116219
$723$	0	0
$724$	2.44826e24	1.20526
$725$	1.23020e24	0.598554
$726$	0	0
$727$	−8.10454e23	−0.385200	−0.192600	−	0.981277i	$-0.561692\pi$
$727$	−8.10454e23	−0.385200	−0.192600	+	0.981277i	$0.561692\pi$
$728$	1.04619e23	0.0491466
$729$	0	0
$730$	1.50206e23	0.0689353
$731$	5.23442e24	2.37449
$732$	0	0
$733$	1.02146e24	0.452730	0.226365	−	0.974043i	$-0.427316\pi$
$733$	1.02146e24	0.452730	0.226365	+	0.974043i	$0.427316\pi$
$734$	−3.69070e22	−0.0161694
$735$	0	0
$736$	−1.33553e23	−0.0571730
$737$	1.72082e23	0.0728218
$738$	0	0
$739$	−1.46607e24	−0.606286	−0.303143	−	0.952945i	$-0.598036\pi$
$739$	−1.46607e24	−0.606286	−0.303143	+	0.952945i	$0.598036\pi$
$740$	−1.16643e24	−0.476857
$741$	0	0
$742$	−5.47836e22	−0.0218886
$743$	7.21717e23	0.285077	0.142538	−	0.989789i	$-0.454474\pi$
$743$	7.21717e23	0.285077	0.142538	+	0.989789i	$0.454474\pi$
$744$	0	0
$745$	−3.44420e24	−1.32972
$746$	−3.11959e22	−0.0119074
$747$	0	0
$748$	1.53054e24	0.571060
$749$	−4.93551e23	−0.182069
$750$	0	0
$751$	−1.69381e24	−0.610835	−0.305418	−	0.952219i	$-0.598796\pi$
$751$	−1.69381e24	−0.610835	−0.305418	+	0.952219i	$0.598796\pi$
$752$	5.00390e24	1.78426
$753$	0	0
$754$	−2.35249e23	−0.0820109
$755$	−5.65762e22	−0.0195023
$756$	0	0
$757$	−4.95241e23	−0.166917	−0.0834587	−	0.996511i	$-0.526597\pi$
$757$	−4.95241e23	−0.166917	−0.0834587	+	0.996511i	$0.526597\pi$
$758$	−2.60220e23	−0.0867266
$759$	0	0
$760$	−4.71655e23	−0.153712
$761$	−4.97441e24	−1.60314	−0.801571	−	0.597900i	$-0.796002\pi$
$761$	−4.97441e24	−1.60314	−0.801571	+	0.597900i	$0.796002\pi$
$762$	0	0
$763$	2.83849e22	0.00894598
$764$	3.32464e24	1.03622
$765$	0	0
$766$	−3.77814e23	−0.115169
$767$	4.68578e24	1.41261
$768$	0	0
$769$	4.76740e24	1.40575	0.702875	−	0.711313i	$-0.251899\pi$
$769$	4.76740e24	1.40575	0.702875	+	0.711313i	$0.251899\pi$
$770$	1.79068e22	0.00522211
$771$	0	0
$772$	6.99497e23	0.199544
$773$	2.99309e24	0.844489	0.422244	−	0.906482i	$-0.361242\pi$
$773$	2.99309e24	0.844489	0.422244	+	0.906482i	$0.361242\pi$
$774$	0	0
$775$	2.84537e24	0.785369
$776$	1.39781e23	0.0381613
$777$	0	0
$778$	2.69893e23	0.0720884
$779$	3.09774e24	0.818419
$780$	0	0
$781$	−1.78759e24	−0.462097
$782$	1.42008e23	0.0363124
$783$	0	0
$784$	5.64135e23	0.141155
$785$	−9.14879e23	−0.226450
$786$	0	0
$787$	7.61086e23	0.184352	0.0921762	−	0.995743i	$-0.470618\pi$
$787$	7.61086e23	0.184352	0.0921762	+	0.995743i	$0.470618\pi$
$788$	−1.42545e24	−0.341570
$789$	0	0
$790$	−1.43617e23	−0.0336802
$791$	−4.64016e23	−0.107655
$792$	0	0
$793$	5.42638e24	1.23222
$794$	3.45096e23	0.0775294
$795$	0	0
$796$	−2.92629e24	−0.643511
$797$	7.11328e23	0.154766	0.0773828	−	0.997001i	$-0.475344\pi$
$797$	7.11328e23	0.154766	0.0773828	+	0.997001i	$0.475344\pi$
$798$	0	0
$799$	−1.60690e25	−3.42248
$800$	−4.24578e23	−0.0894730
$801$	0	0
$802$	−1.15792e23	−0.0238889
$803$	−2.23664e24	−0.456578
$804$	0	0
$805$	−4.16116e23	−0.0831667
$806$	−5.44114e23	−0.107607
$807$	0	0
$808$	−8.79196e23	−0.170251
$809$	−7.09657e24	−1.35984	−0.679918	−	0.733288i	$-0.737984\pi$
$809$	−7.09657e24	−1.35984	−0.679918	+	0.733288i	$0.737984\pi$
$810$	0	0
$811$	3.32935e24	0.624715	0.312357	−	0.949965i	$-0.398881\pi$
$811$	3.32935e24	0.624715	0.312357	+	0.949965i	$0.398881\pi$
$812$	−2.55236e24	−0.473931
$813$	0	0
$814$	−6.93486e22	−0.0126104
$815$	3.05348e23	0.0549485
$816$	0	0
$817$	1.19857e25	2.11239
$818$	−4.94008e23	−0.0861650
$819$	0	0
$820$	−2.04962e24	−0.350150
$821$	3.86094e24	0.652794	0.326397	−	0.945233i	$-0.394165\pi$
$821$	3.86094e24	0.652794	0.326397	+	0.945233i	$0.394165\pi$
$822$	0	0
$823$	−6.16918e24	−1.02171	−0.510856	−	0.859666i	$-0.670672\pi$
$823$	−6.16918e24	−1.02171	−0.510856	+	0.859666i	$0.670672\pi$
$824$	5.95568e23	0.0976225
$825$	0	0
$826$	−2.02985e23	−0.0325937
$827$	−9.38348e24	−1.49131	−0.745654	−	0.666334i	$-0.767863\pi$
$827$	−9.38348e24	−1.49131	−0.745654	+	0.666334i	$0.767863\pi$
$828$	0	0
$829$	7.02298e24	1.09347	0.546736	−	0.837305i	$-0.315870\pi$
$829$	7.02298e24	1.09347	0.546736	+	0.837305i	$0.315870\pi$
$830$	2.33195e23	0.0359381
$831$	0	0
$832$	−6.67886e24	−1.00845
$833$	−1.81160e24	−0.270758
$834$	0	0
$835$	−3.91706e24	−0.573621
$836$	3.50461e24	0.508026
$837$	0	0
$838$	2.97741e23	0.0422927
$839$	−6.14770e24	−0.864442	−0.432221	−	0.901768i	$-0.642270\pi$
$839$	−6.14770e24	−0.864442	−0.432221	+	0.901768i	$0.642270\pi$
$840$	0	0
$841$	4.24438e24	0.584855
$842$	−7.53644e23	−0.102805
$843$	0	0
$844$	1.09353e25	1.46191
$845$	−3.67326e23	−0.0486149
$846$	0	0
$847$	2.64716e24	0.343377
$848$	7.06573e24	0.907386
$849$	0	0
$850$	4.51458e23	0.0568272
$851$	1.61152e24	0.200832
$852$	0	0
$853$	−1.87599e24	−0.229174	−0.114587	−	0.993413i	$-0.536554\pi$
$853$	−1.87599e24	−0.229174	−0.114587	+	0.993413i	$0.536554\pi$
$854$	−2.35067e23	−0.0284316
$855$	0	0
$856$	−5.11384e23	−0.0606345
$857$	−5.85280e24	−0.687110	−0.343555	−	0.939132i	$-0.611631\pi$
$857$	−5.85280e24	−0.687110	−0.343555	+	0.939132i	$0.611631\pi$
$858$	0	0
$859$	−5.70291e24	−0.656379	−0.328189	−	0.944612i	$-0.606438\pi$
$859$	−5.70291e24	−0.656379	−0.328189	+	0.944612i	$0.606438\pi$
$860$	−7.93031e24	−0.903760
$861$	0	0
$862$	2.65471e23	0.0296624
$863$	−1.44974e25	−1.60398	−0.801991	−	0.597336i	$-0.796226\pi$
$863$	−1.44974e25	−1.60398	−0.801991	+	0.597336i	$0.796226\pi$
$864$	0	0
$865$	−8.64997e24	−0.938378
$866$	−6.86001e22	−0.00736924
$867$	0	0
$868$	−5.90342e24	−0.621850
$869$	2.13853e24	0.223073
$870$	0	0
$871$	2.43102e24	0.248677
$872$	2.94105e22	0.00297928
$873$	0	0
$874$	3.25166e23	0.0323042
$875$	−4.10522e24	−0.403895
$876$	0	0
$877$	1.53001e25	1.47638	0.738191	−	0.674592i	$-0.235680\pi$
$877$	1.53001e25	1.47638	0.738191	+	0.674592i	$0.235680\pi$
$878$	1.72723e23	0.0165062
$879$	0	0
$880$	−2.30953e24	−0.216481
$881$	−9.68780e24	−0.899352	−0.449676	−	0.893192i	$-0.648461\pi$
$881$	−9.68780e24	−0.899352	−0.449676	+	0.893192i	$0.648461\pi$
$882$	0	0
$883$	5.45095e24	0.496371	0.248185	−	0.968713i	$-0.420166\pi$
$883$	5.45095e24	0.496371	0.248185	+	0.968713i	$0.420166\pi$
$884$	2.16222e25	1.95009
$885$	0	0
$886$	−8.49288e23	−0.0751395
$887$	8.37406e24	0.733813	0.366907	−	0.930258i	$-0.380417\pi$
$887$	8.37406e24	0.733813	0.366907	+	0.930258i	$0.380417\pi$
$888$	0	0
$889$	1.04617e24	0.0899370
$890$	4.05689e23	0.0345444
$891$	0	0
$892$	−5.55518e24	−0.464085
$893$	−3.67945e25	−3.04470
$894$	0	0
$895$	−1.23729e25	−1.00456
$896$	1.17373e24	0.0943955
$897$	0	0
$898$	1.94985e23	0.0153870
$899$	2.66022e25	2.07951
$900$	0	0
$901$	−2.26901e25	−1.74051
$902$	−1.21857e23	−0.00925969
$903$	0	0
$904$	−4.80782e23	−0.0358523
$905$	1.18635e25	0.876398
$906$	0	0
$907$	−2.07998e24	−0.150798	−0.0753992	−	0.997153i	$-0.524023\pi$
$907$	−2.07998e24	−0.150798	−0.0753992	+	0.997153i	$0.524023\pi$
$908$	7.46854e24	0.536421
$909$	0	0
$910$	2.52971e23	0.0178328
$911$	−1.50752e25	−1.05283	−0.526413	−	0.850229i	$-0.676463\pi$
$911$	−1.50752e25	−1.05283	−0.526413	+	0.850229i	$0.676463\pi$
$912$	0	0
$913$	−3.47240e24	−0.238028
$914$	−1.70545e24	−0.115824
$915$	0	0
$916$	−9.07523e24	−0.604987
$917$	2.98065e23	0.0196866
$918$	0	0
$919$	−8.40599e24	−0.545013	−0.272506	−	0.962154i	$-0.587853\pi$
$919$	−8.40599e24	−0.545013	−0.272506	+	0.962154i	$0.587853\pi$
$920$	−4.31152e23	−0.0276970
$921$	0	0
$922$	−4.72384e23	−0.0297908
$923$	−2.52535e25	−1.57800
$924$	0	0
$925$	5.12319e24	0.314294
$926$	9.26803e23	0.0563370
$927$	0	0
$928$	−3.96951e24	−0.236907
$929$	−2.49512e23	−0.0147557	−0.00737783	−	0.999973i	$-0.502348\pi$
$929$	−2.49512e23	−0.0147557	−0.00737783	+	0.999973i	$0.502348\pi$
$930$	0	0
$931$	−4.14817e24	−0.240871
$932$	−2.16705e25	−1.24691
$933$	0	0
$934$	−6.61367e23	−0.0373675
$935$	7.41657e24	0.415245
$936$	0	0
$937$	4.52595e24	0.248842	0.124421	−	0.992230i	$-0.460293\pi$
$937$	4.52595e24	0.248842	0.124421	+	0.992230i	$0.460293\pi$
$938$	−1.05311e23	−0.00573783
$939$	0	0
$940$	2.43450e25	1.30264
$941$	8.41772e24	0.446357	0.223179	−	0.974778i	$-0.428357\pi$
$941$	8.41772e24	0.446357	0.223179	+	0.974778i	$0.428357\pi$
$942$	0	0
$943$	2.83172e24	0.147469
$944$	2.61801e25	1.35117
$945$	0	0
$946$	−4.71487e23	−0.0238998
$947$	3.41158e25	1.71388	0.856941	−	0.515414i	$-0.172362\pi$
$947$	3.41158e25	1.71388	0.856941	+	0.515414i	$0.172362\pi$
$948$	0	0
$949$	−3.15974e25	−1.55915
$950$	1.03374e24	0.0505545
$951$	0	0
$952$	−1.87706e24	−0.0901705
$953$	−2.42458e25	−1.15437	−0.577187	−	0.816612i	$-0.695850\pi$
$953$	−2.42458e25	−1.15437	−0.577187	+	0.816612i	$0.695850\pi$
$954$	0	0
$955$	1.61103e25	0.753484
$956$	−1.24433e25	−0.576824
$957$	0	0
$958$	1.99653e24	0.0909220
$959$	8.02217e24	0.362104
$960$	0	0
$961$	3.89788e25	1.72854
$962$	−9.79697e23	−0.0430629
$963$	0	0
$964$	2.42555e25	1.04750
$965$	3.38956e24	0.145098
$966$	0	0
$967$	−2.29517e25	−0.965363	−0.482681	−	0.875796i	$-0.660337\pi$
$967$	−2.29517e25	−0.965363	−0.482681	+	0.875796i	$0.660337\pi$
$968$	2.74281e24	0.114355
$969$	0	0
$970$	3.37994e23	0.0138468
$971$	4.54882e25	1.84729	0.923646	−	0.383246i	$-0.125194\pi$
$971$	4.54882e25	1.84729	0.923646	+	0.383246i	$0.125194\pi$
$972$	0	0
$973$	−5.36948e24	−0.214276
$974$	1.55375e24	0.0614651
$975$	0	0
$976$	3.03178e25	1.17862
$977$	−2.56796e25	−0.989655	−0.494827	−	0.868991i	$-0.664769\pi$
$977$	−2.56796e25	−0.989655	−0.494827	+	0.868991i	$0.664769\pi$
$978$	0	0
$979$	−6.04093e24	−0.228797
$980$	2.74464e24	0.103054
$981$	0	0
$982$	−1.28406e24	−0.0473848
$983$	1.97584e25	0.722848	0.361424	−	0.932402i	$-0.382291\pi$
$983$	1.97584e25	0.722848	0.361424	+	0.932402i	$0.382291\pi$
$984$	0	0
$985$	−6.90733e24	−0.248372
$986$	4.22082e24	0.150468
$987$	0	0
$988$	4.95101e25	1.73484
$989$	1.09564e25	0.380626
$990$	0	0
$991$	5.45422e25	1.86254	0.931271	−	0.364326i	$-0.118701\pi$
$991$	5.45422e25	1.86254	0.931271	+	0.364326i	$0.118701\pi$
$992$	−9.18117e24	−0.310849
$993$	0	0
$994$	1.09397e24	0.0364099
$995$	−1.41800e25	−0.467928
$996$	0	0
$997$	5.43418e25	1.76289	0.881445	−	0.472287i	$-0.156571\pi$
$997$	5.43418e25	1.76289	0.881445	+	0.472287i	$0.156571\pi$
$998$	−5.57546e23	−0.0179338
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.18.a.d.1.2		4
3.2	odd	2		21.18.a.b.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.18.a.b.1.3	✓	4	3.2	odd	2
63.18.a.d.1.2		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	63.a (trivial)

\(f(q)\)	\(=\)	\(q+22.8310 q^{2} -130551. q^{4} -632611. q^{5} -5.76480e6 q^{7} -5.97310e6 q^{8} -1.44431e7 q^{10} +2.15066e8 q^{11} +3.03827e9 q^{13} -1.31616e8 q^{14} +1.69752e10 q^{16} -5.45123e10 q^{17} -1.24821e11 q^{19} +8.25879e10 q^{20} +4.91016e9 q^{22} -1.14102e11 q^{23} -3.62743e11 q^{25} +6.93665e10 q^{26} +7.52599e11 q^{28} -3.39139e12 q^{29} -7.84404e12 q^{31} +1.17047e12 q^{32} -1.24457e12 q^{34} +3.64688e12 q^{35} -1.41235e13 q^{37} -2.84979e12 q^{38} +3.77865e12 q^{40} -2.48174e13 q^{41} -9.60227e13 q^{43} -2.80770e13 q^{44} -2.60506e12 q^{46} +2.94778e14 q^{47} +3.32329e13 q^{49} -8.28176e12 q^{50} -3.96648e14 q^{52} +4.16239e14 q^{53} -1.36053e14 q^{55} +3.44337e13 q^{56} -7.74287e13 q^{58} +1.54226e15 q^{59} +1.78601e15 q^{61} -1.79087e14 q^{62} -2.19825e15 q^{64} -1.92204e15 q^{65} +8.00135e14 q^{67} +7.11662e15 q^{68} +8.32617e13 q^{70} -8.31181e15 q^{71} -1.03998e16 q^{73} -3.22453e14 q^{74} +1.62955e16 q^{76} -1.23981e15 q^{77} +9.94360e15 q^{79} -1.07387e16 q^{80} -5.66605e14 q^{82} -1.61457e16 q^{83} +3.44851e16 q^{85} -2.19229e15 q^{86} -1.28461e15 q^{88} -2.80887e16 q^{89} -1.75150e16 q^{91} +1.48961e16 q^{92} +6.73005e15 q^{94} +7.89633e16 q^{95} -2.34017e16 q^{97} +7.58740e14 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 375 q^{2} + 314369 q^{4} + 154140 q^{5} - 23059204 q^{7} + 3766143 q^{8} + 23352830 q^{10} + 1452022884 q^{11} + 1793559040 q^{13} - 2161800375 q^{14} + 36719981057 q^{16} - 68581449948 q^{17} - 2578000592 q^{19}+ \cdots + 12\!\cdots\!75 q^{98}+O(q^{100}) \) 4 * q + 375 * q^2 + 314369 * q^4 + 154140 * q^5 - 23059204 * q^7 + 3766143 * q^8 + 23352830 * q^10 + 1452022884 * q^11 + 1793559040 * q^13 - 2161800375 * q^14 + 36719981057 * q^16 - 68581449948 * q^17 - 2578000592 * q^19 - 132998517630 * q^20 + 270530423852 * q^22 - 649239533556 * q^23 - 708242296300 * q^25 + 2508729280746 * q^26 - 1812274725569 * q^28 + 3258162962760 * q^29 + 2414795507136 * q^31 + 18040133068671 * q^32 + 8860338750210 * q^34 - 888586426140 * q^35 + 23164217924208 * q^37 + 67011195804708 * q^38 - 115859765446530 * q^40 + 117516076237164 * q^41 - 175144919320720 * q^43 + 379640394802500 * q^44 - 441875935197000 * q^46 + 392013343869480 * q^47 + 132931722278404 * q^49 + 366102837059625 * q^50 - 647834999852794 * q^52 + 86395731614976 * q^53 - 987784569502880 * q^55 - 21711064932543 * q^56 - 2149209668160466 * q^58 + 23145424510920 * q^59 + 3424273979460952 * q^61 + 4542097297046784 * q^62 - 3840648962754815 * q^64 - 2015724107462520 * q^65 + 2234127752098744 * q^67 + 1956571325658750 * q^68 - 134624417736830 * q^70 - 2815776345321804 * q^71 + 11330038422344784 * q^73 - 2782049125778694 * q^74 + 33041781126411260 * q^76 - 8370622973706084 * q^77 - 12177038540666920 * q^79 - 35816456249313150 * q^80 + 79244490955525210 * q^82 + 4389820874951568 * q^83 + 46318093724780760 * q^85 + 79314655634282340 * q^86 + 93140307906096828 * q^88 - 100131520699583508 * q^89 - 10339510947351040 * q^91 + 14621524874654712 * q^92 + 120993183153224352 * q^94 + 115775864256900240 * q^95 - 44947325255962464 * q^97 + 12462348963600375 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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