LMFDB - Embedded newform 169.8.a.g.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [169,8,Mod(1,169)] 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("169.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(169, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	169.a (trivial)

Newform invariants

comment:select newform

sage:traces = [14,0] 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:	$52.7930693068$
Analytic rank:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 1279 x^{12} + 629380 x^{10} - 148562016 x^{8} + 16872573312 x^{6} - 790180980480 x^{4} + \cdots - 4669637050368 $ x^14 - 1279x^12 + 629380x^10 - 148562016x^8 + 16872573312x^6 - 790180980480x^4 + 10484997239808x^2 - 4669637050368
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{13}\cdot 3^{3}\cdot 13^{6} $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.679146$ of defining polynomial
Character	$\chi$	$=$	169.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.679146 q^{2} +62.4534 q^{3} -127.539 q^{4} -439.155 q^{5} -42.4150 q^{6} +1302.66 q^{7} +173.548 q^{8} +1713.42 q^{9} +298.251 q^{10} +574.818 q^{11} -7965.23 q^{12} -884.695 q^{14} -27426.7 q^{15} +16207.1 q^{16} -11268.5 q^{17} -1163.66 q^{18} +41049.5 q^{19} +56009.3 q^{20} +81355.4 q^{21} -390.385 q^{22} -44265.7 q^{23} +10838.7 q^{24} +114732. q^{25} -29576.4 q^{27} -166139. q^{28} -145512. q^{29} +18626.8 q^{30} -122045. q^{31} -33221.1 q^{32} +35899.3 q^{33} +7652.99 q^{34} -572069. q^{35} -218528. q^{36} +42002.0 q^{37} -27878.6 q^{38} -76214.6 q^{40} +87690.6 q^{41} -55252.2 q^{42} -749752. q^{43} -73311.5 q^{44} -752459. q^{45} +30062.9 q^{46} -940126. q^{47} +1.01219e6 q^{48} +873376. q^{49} -77920.0 q^{50} -703759. q^{51} -924214. q^{53} +20086.7 q^{54} -252434. q^{55} +226074. q^{56} +2.56368e6 q^{57} +98824.0 q^{58} -632344. q^{59} +3.49797e6 q^{60} -64406.5 q^{61} +82886.6 q^{62} +2.23201e6 q^{63} -2.05195e6 q^{64} -24380.9 q^{66} +1.94403e6 q^{67} +1.43718e6 q^{68} -2.76454e6 q^{69} +388519. q^{70} -4.72403e6 q^{71} +297361. q^{72} +1.92731e6 q^{73} -28525.5 q^{74} +7.16542e6 q^{75} -5.23540e6 q^{76} +748791. q^{77} -1.50287e6 q^{79} -7.11743e6 q^{80} -5.59441e6 q^{81} -59554.7 q^{82} +1.87974e6 q^{83} -1.03760e7 q^{84} +4.94864e6 q^{85} +509191. q^{86} -9.08772e6 q^{87} +99758.5 q^{88} +4.37470e6 q^{89} +511030. q^{90} +5.64559e6 q^{92} -7.62215e6 q^{93} +638483. q^{94} -1.80271e7 q^{95} -2.07477e6 q^{96} -1.77364e6 q^{97} -593150. q^{98} +984906. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 52 q^{3} + 766 q^{4} + 6982 q^{9} - 1018 q^{10} - 38380 q^{12} - 47916 q^{14} + 1266 q^{16} - 76806 q^{17} + 251764 q^{22} - 137100 q^{23} - 39380 q^{25} - 432400 q^{27} - 443166 q^{29} - 315780 q^{30}+ \cdots - 86840772 q^{95}+O(q^{100}) $ 14 * q - 52 * q^3 + 766 * q^4 + 6982 * q^9 - 1018 * q^10 - 38380 * q^12 - 47916 * q^14 + 1266 * q^16 - 76806 * q^17 + 251764 * q^22 - 137100 * q^23 - 39380 * q^25 - 432400 * q^27 - 443166 * q^29 - 315780 * q^30 - 1319232 * q^35 + 1271350 * q^36 - 953640 * q^38 - 77182 * q^40 - 412812 * q^42 - 1765384 * q^43 - 2267260 * q^48 + 5307958 * q^49 + 38964 * q^51 - 1791210 * q^53 - 4907072 * q^55 - 14915904 * q^56 - 7717050 * q^61 + 3808284 * q^62 - 14570922 * q^64 - 31956396 * q^66 - 19139238 * q^68 - 27487680 * q^69 - 17280210 * q^74 + 20632304 * q^75 - 19599336 * q^77 - 30688440 * q^79 - 27772442 * q^81 - 74719438 * q^82 - 25539372 * q^87 + 54091072 * q^88 - 41193546 * q^90 - 47685588 * q^92 - 38277620 * q^94 - 86840772 * q^95

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 $

$2$	−0.679146	−0.0600286	−0.0300143	−	0.999549i	$-0.509555\pi$
$2$	−0.679146	−0.0600286	−0.0300143	+	0.999549i	$0.509555\pi$
$3$	62.4534	1.33546	0.667731	−	0.744403i	$-0.267266\pi$
$3$	62.4534	1.33546	0.667731	+	0.744403i	$0.267266\pi$
$4$	−127.539	−0.996397
$5$	−439.155	−1.57117	−0.785585	−	0.618754i	$-0.787638\pi$
$5$	−439.155	−1.57117	−0.785585	+	0.618754i	$0.787638\pi$
$6$	−42.4150	−0.0801659
$7$	1302.66	1.43545	0.717724	−	0.696328i	$-0.245184\pi$
$7$	1302.66	1.43545	0.717724	+	0.696328i	$0.245184\pi$
$8$	173.548	0.119841
$9$	1713.42	0.783459
$10$	298.251	0.0943151
$11$	574.818	0.130213	0.0651067	−	0.997878i	$-0.479261\pi$
$11$	574.818	0.130213	0.0651067	+	0.997878i	$0.479261\pi$
$12$	−7965.23	−1.33065
$13$	0	0
$13$	0	0
$14$	−884.695	−0.0861679
$15$	−27426.7	−2.09824
$16$	16207.1	0.989203
$17$	−11268.5	−0.556284	−0.278142	−	0.960540i	$-0.589719\pi$
$17$	−11268.5	−0.556284	−0.278142	+	0.960540i	$0.589719\pi$
$18$	−1163.66	−0.0470299
$19$	41049.5	1.37300	0.686499	−	0.727131i	$-0.259147\pi$
$19$	41049.5	1.37300	0.686499	+	0.727131i	$0.259147\pi$
$20$	56009.3	1.56551
$21$	81355.4	1.91699
$22$	−390.385	−0.00781653
$23$	−44265.7	−0.758612	−0.379306	−	0.925271i	$-0.623837\pi$
$23$	−44265.7	−0.758612	−0.379306	+	0.925271i	$0.623837\pi$
$24$	10838.7	0.160043
$25$	114732.	1.46857
$26$	0	0
$27$	−29576.4	−0.289183
$28$	−166139.	−1.43028
$29$	−145512.	−1.10791	−0.553957	−	0.832545i	$-0.686883\pi$
$29$	−145512.	−1.10791	−0.553957	+	0.832545i	$0.686883\pi$
$30$	18626.8	0.125954
$31$	−122045.	−0.735793	−0.367896	−	0.929867i	$-0.619922\pi$
$31$	−122045.	−0.735793	−0.367896	+	0.929867i	$0.619922\pi$
$32$	−33221.1	−0.179221
$33$	35899.3	0.173895
$34$	7652.99	0.0333930
$35$	−572069.	−2.25533
$36$	−218528.	−0.780635
$37$	42002.0	0.136322	0.0681608	−	0.997674i	$-0.478287\pi$
$37$	42002.0	0.136322	0.0681608	+	0.997674i	$0.478287\pi$
$38$	−27878.6	−0.0824191
$39$	0	0
$40$	−76214.6	−0.188290
$41$	87690.6	0.198705	0.0993526	−	0.995052i	$-0.468323\pi$
$41$	87690.6	0.198705	0.0993526	+	0.995052i	$0.468323\pi$
$42$	−55252.2	−0.115074
$43$	−749752.	−1.43806	−0.719032	−	0.694977i	$-0.755415\pi$
$43$	−749752.	−1.43806	−0.719032	+	0.694977i	$0.755415\pi$
$44$	−73311.5	−0.129744
$45$	−752459.	−1.23095
$46$	30062.9	0.0455384
$47$	−940126.	−1.32082	−0.660410	−	0.750905i	$-0.729617\pi$
$47$	−940126.	−1.32082	−0.660410	+	0.750905i	$0.729617\pi$
$48$	1.01219e6	1.32104
$49$	873376.	1.06051
$50$	−77920.0	−0.0881564
$51$	−703759.	−0.742896
$52$	0	0
$53$	−924214.	−0.852721	−0.426360	−	0.904553i	$-0.640204\pi$
$53$	−924214.	−0.852721	−0.426360	+	0.904553i	$0.640204\pi$
$54$	20086.7	0.0173592
$55$	−252434.	−0.204587
$56$	226074.	0.172025
$57$	2.56368e6	1.83359
$58$	98824.0	0.0665066
$59$	−632344.	−0.400840	−0.200420	−	0.979710i	$-0.564231\pi$
$59$	−632344.	−0.400840	−0.200420	+	0.979710i	$0.564231\pi$
$60$	3.49797e6	2.09068
$61$	−64406.5	−0.0363308	−0.0181654	−	0.999835i	$-0.505783\pi$
$61$	−64406.5	−0.0363308	−0.0181654	+	0.999835i	$0.505783\pi$
$62$	82886.6	0.0441686
$63$	2.23201e6	1.12461
$64$	−2.05195e6	−0.978444
$65$	0	0
$66$	−24380.9	−0.0104387
$67$	1.94403e6	0.789660	0.394830	−	0.918754i	$-0.370803\pi$
$67$	1.94403e6	0.789660	0.394830	+	0.918754i	$0.370803\pi$
$68$	1.43718e6	0.554280
$69$	−2.76454e6	−1.01310
$70$	388519.	0.135384
$71$	−4.72403e6	−1.56642	−0.783211	−	0.621756i	$-0.786419\pi$
$71$	−4.72403e6	−1.56642	−0.783211	+	0.621756i	$0.786419\pi$
$72$	297361.	0.0938904
$73$	1.92731e6	0.579857	0.289928	−	0.957048i	$-0.406368\pi$
$73$	1.92731e6	0.579857	0.289928	+	0.957048i	$0.406368\pi$
$74$	−28525.5	−0.00818319
$75$	7.16542e6	1.96122
$76$	−5.23540e6	−1.36805
$77$	748791.	0.186915
$78$	0	0
$79$	−1.50287e6	−0.342947	−0.171474	−	0.985189i	$-0.554853\pi$
$79$	−1.50287e6	−0.342947	−0.171474	+	0.985189i	$0.554853\pi$
$80$	−7.11743e6	−1.55421
$81$	−5.59441e6	−1.16965
$82$	−59554.7	−0.0119280
$83$	1.87974e6	0.360848	0.180424	−	0.983589i	$-0.442253\pi$
$83$	1.87974e6	0.360848	0.180424	+	0.983589i	$0.442253\pi$
$84$	−1.03760e7	−1.91008
$85$	4.94864e6	0.874017
$86$	509191.	0.0863250
$87$	−9.08772e6	−1.47958
$88$	99758.5	0.0156049
$89$	4.37470e6	0.657784	0.328892	−	0.944368i	$-0.393325\pi$
$89$	4.37470e6	0.657784	0.328892	+	0.944368i	$0.393325\pi$
$90$	511030.	0.0738920
$91$	0	0
$92$	5.64559e6	0.755879
$93$	−7.62215e6	−0.982623
$94$	638483.	0.0792869
$95$	−1.80271e7	−2.15721
$96$	−2.07477e6	−0.239343
$97$	−1.77364e6	−0.197317	−0.0986583	−	0.995121i	$-0.531455\pi$
$97$	−1.77364e6	−0.197317	−0.0986583	+	0.995121i	$0.531455\pi$
$98$	−593150.	−0.0636610
$99$	984906.	0.102017
$100$	−1.46328e7	−1.46328
$101$	−9.63946e6	−0.930953	−0.465477	−	0.885060i	$-0.654117\pi$
$101$	−9.63946e6	−0.930953	−0.465477	+	0.885060i	$0.654117\pi$
$102$	477955.	0.0445950
$103$	−7.14668e6	−0.644427	−0.322214	−	0.946667i	$-0.604427\pi$
$103$	−7.14668e6	−0.644427	−0.322214	+	0.946667i	$0.604427\pi$
$104$	0	0
$105$	−3.57277e7	−3.01191
$106$	627676.	0.0511876
$107$	−2.06005e7	−1.62568	−0.812839	−	0.582488i	$-0.802079\pi$
$107$	−2.06005e7	−1.62568	−0.812839	+	0.582488i	$0.802079\pi$
$108$	3.77214e6	0.288141
$109$	1.05713e7	0.781872	0.390936	−	0.920418i	$-0.372151\pi$
$109$	1.05713e7	0.781872	0.390936	+	0.920418i	$0.372151\pi$
$110$	171440.	0.0122811
$111$	2.62317e6	0.182052
$112$	2.11123e7	1.41995
$113$	2.68071e7	1.74774	0.873868	−	0.486163i	$-0.161604\pi$
$113$	2.68071e7	1.74774	0.873868	+	0.486163i	$0.161604\pi$
$114$	−1.74111e6	−0.110068
$115$	1.94395e7	1.19191
$116$	1.85584e7	1.10392
$117$	0	0
$118$	429454.	0.0240619
$119$	−1.46791e7	−0.798517
$120$	−4.75986e6	−0.251455
$121$	−1.91568e7	−0.983044
$122$	43741.4	0.00218089
$123$	5.47657e6	0.265363
$124$	1.55655e7	0.733141
$125$	−1.60763e7	−0.736209
$126$	−1.51586e6	−0.0675090
$127$	−8.07064e6	−0.349619	−0.174810	−	0.984602i	$-0.555931\pi$
$127$	−8.07064e6	−0.349619	−0.174810	+	0.984602i	$0.555931\pi$
$128$	5.64588e6	0.237956
$129$	−4.68246e7	−1.92048
$130$	0	0
$131$	1.26390e7	0.491206	0.245603	−	0.969370i	$-0.421014\pi$
$131$	1.26390e7	0.491206	0.245603	+	0.969370i	$0.421014\pi$
$132$	−4.57855e6	−0.173268
$133$	5.34734e7	1.97087
$134$	−1.32028e6	−0.0474022
$135$	1.29886e7	0.454355
$136$	−1.95564e6	−0.0666656
$137$	−2.83228e7	−0.941052	−0.470526	−	0.882386i	$-0.655936\pi$
$137$	−2.83228e7	−0.941052	−0.470526	+	0.882386i	$0.655936\pi$
$138$	1.87753e6	0.0608148
$139$	−8.88148e6	−0.280500	−0.140250	−	0.990116i	$-0.544791\pi$
$139$	−8.88148e6	−0.280500	−0.140250	+	0.990116i	$0.544791\pi$
$140$	7.29610e7	2.24721
$141$	−5.87140e7	−1.76390
$142$	3.20831e6	0.0940301
$143$	0	0
$144$	2.77696e7	0.774999
$145$	6.39024e7	1.74072
$146$	−1.30892e6	−0.0348080
$147$	5.45453e7	1.41627
$148$	−5.35689e6	−0.135830
$149$	−5.08520e7	−1.25938	−0.629689	−	0.776848i	$-0.716817\pi$
$149$	−5.08520e7	−1.25938	−0.629689	+	0.776848i	$0.716817\pi$
$150$	−4.86637e6	−0.117730
$151$	−9.76974e6	−0.230921	−0.115461	−	0.993312i	$-0.536834\pi$
$151$	−9.76974e6	−0.230921	−0.115461	+	0.993312i	$0.536834\pi$
$152$	7.12406e6	0.164541
$153$	−1.93078e7	−0.435826
$154$	−508538.	−0.0112202
$155$	5.35969e7	1.15606
$156$	0	0
$157$	6.53474e7	1.34766	0.673828	−	0.738888i	$-0.264649\pi$
$157$	6.53474e7	1.34766	0.673828	+	0.738888i	$0.264649\pi$
$158$	1.02067e6	0.0205867
$159$	−5.77203e7	−1.13878
$160$	1.45892e7	0.281587
$161$	−5.76631e7	−1.08895
$162$	3.79942e6	0.0702125
$163$	−2.01980e7	−0.365302	−0.182651	−	0.983178i	$-0.558468\pi$
$163$	−2.01980e7	−0.365302	−0.182651	+	0.983178i	$0.558468\pi$
$164$	−1.11839e7	−0.197989
$165$	−1.57654e7	−0.273219
$166$	−1.27662e6	−0.0216612
$167$	−6.77083e6	−0.112495	−0.0562477	−	0.998417i	$-0.517914\pi$
$167$	−6.77083e6	−0.112495	−0.0562477	+	0.998417i	$0.517914\pi$
$168$	1.41191e7	0.229733
$169$	0	0
$170$	−3.36085e6	−0.0524660
$171$	7.03351e7	1.07569
$172$	9.56225e7	1.43288
$173$	8.67045e7	1.27315	0.636576	−	0.771214i	$-0.280350\pi$
$173$	8.67045e7	1.27315	0.636576	+	0.771214i	$0.280350\pi$
$174$	6.17189e6	0.0888170
$175$	1.49457e8	2.10806
$176$	9.31612e6	0.128807
$177$	−3.94920e7	−0.535307
$178$	−2.97106e6	−0.0394858
$179$	−8.94442e7	−1.16565	−0.582823	−	0.812599i	$-0.698052\pi$
$179$	−8.94442e7	−1.16565	−0.582823	+	0.812599i	$0.698052\pi$
$180$	9.59677e7	1.22651
$181$	2.55748e7	0.320581	0.160290	−	0.987070i	$-0.448757\pi$
$181$	2.55748e7	0.320581	0.160290	+	0.987070i	$0.448757\pi$
$182$	0	0
$183$	−4.02240e6	−0.0485184
$184$	−7.68223e6	−0.0909128
$185$	−1.84454e7	−0.214184
$186$	5.17655e6	0.0589855
$187$	−6.47736e6	−0.0724356
$188$	1.19902e8	1.31606
$189$	−3.85280e7	−0.415107
$190$	1.22430e7	0.129494
$191$	1.45282e8	1.50868	0.754339	−	0.656486i	$-0.227958\pi$
$191$	1.45282e8	1.50868	0.754339	+	0.656486i	$0.227958\pi$
$192$	−1.28151e8	−1.30668
$193$	−9.12233e7	−0.913388	−0.456694	−	0.889624i	$-0.650967\pi$
$193$	−9.12233e7	−0.913388	−0.456694	+	0.889624i	$0.650967\pi$
$194$	1.20456e6	0.0118446
$195$	0	0
$196$	−1.11389e8	−1.05669
$197$	−1.70257e8	−1.58662	−0.793311	−	0.608817i	$-0.791644\pi$
$197$	−1.70257e8	−1.58662	−0.793311	+	0.608817i	$0.791644\pi$
$198$	−668895.	−0.00612392
$199$	−1.39523e8	−1.25504	−0.627521	−	0.778599i	$-0.715931\pi$
$199$	−1.39523e8	−1.25504	−0.627521	+	0.778599i	$0.715931\pi$
$200$	1.99116e7	0.175995
$201$	1.21411e8	1.05456
$202$	6.54660e6	0.0558838
$203$	−1.89553e8	−1.59035
$204$	8.97565e7	0.740219
$205$	−3.85098e7	−0.312200
$206$	4.85364e6	0.0386841
$207$	−7.58459e7	−0.594341
$208$	0	0
$209$	2.35960e7	0.178783
$210$	2.42643e7	0.180801
$211$	8.37079e7	0.613448	0.306724	−	0.951799i	$-0.400767\pi$
$211$	8.37079e7	0.613448	0.306724	+	0.951799i	$0.400767\pi$
$212$	1.17873e8	0.849648
$213$	−2.95032e8	−2.09190
$214$	1.39908e7	0.0975872
$215$	3.29258e8	2.25944
$216$	−5.13293e6	−0.0346559
$217$	−1.58983e8	−1.05619
$218$	−7.17946e6	−0.0469347
$219$	1.20367e8	0.774377
$220$	3.21951e7	0.203850
$221$	0	0
$222$	−1.78152e6	−0.0109283
$223$	9.51011e7	0.574273	0.287137	−	0.957890i	$-0.407297\pi$
$223$	9.51011e7	0.574273	0.287137	+	0.957890i	$0.407297\pi$
$224$	−4.32758e7	−0.257263
$225$	1.96585e8	1.15057
$226$	−1.82060e7	−0.104914
$227$	9.88410e7	0.560850	0.280425	−	0.959876i	$-0.409525\pi$
$227$	9.88410e7	0.560850	0.280425	+	0.959876i	$0.409525\pi$
$228$	−3.26968e8	−1.82698
$229$	8.72426e7	0.480070	0.240035	−	0.970764i	$-0.422841\pi$
$229$	8.72426e7	0.480070	0.240035	+	0.970764i	$0.422841\pi$
$230$	−1.32023e7	−0.0715486
$231$	4.67645e7	0.249617
$232$	−2.52534e7	−0.132773
$233$	2.42490e8	1.25588	0.627940	−	0.778262i	$-0.283898\pi$
$233$	2.42490e8	1.25588	0.627940	+	0.778262i	$0.283898\pi$
$234$	0	0
$235$	4.12861e8	2.07523
$236$	8.06484e7	0.399396
$237$	−9.38595e7	−0.457993
$238$	9.96923e6	0.0479338
$239$	4.08512e8	1.93558	0.967792	−	0.251750i	$-0.0810060\pi$
$239$	4.08512e8	1.93558	0.967792	+	0.251750i	$0.0810060\pi$
$240$	−4.44508e8	−2.07558
$241$	−2.19249e8	−1.00897	−0.504484	−	0.863421i	$-0.668317\pi$
$241$	−2.19249e8	−1.00897	−0.504484	+	0.863421i	$0.668317\pi$
$242$	1.30102e7	0.0590108
$243$	−2.84706e8	−1.27284
$244$	8.21432e6	0.0361999
$245$	−3.83548e8	−1.66624
$246$	−3.71939e6	−0.0159294
$247$	0	0
$248$	−2.11807e7	−0.0881780
$249$	1.17396e8	0.481899
$250$	1.09182e7	0.0441936
$251$	1.48564e7	0.0593003	0.0296501	−	0.999560i	$-0.490561\pi$
$251$	1.48564e7	0.0593003	0.0296501	+	0.999560i	$0.490561\pi$
$252$	−2.84667e8	−1.12056
$253$	−2.54447e7	−0.0987815
$254$	5.48114e6	0.0209871
$255$	3.09059e8	1.16722
$256$	2.58815e8	0.964160
$257$	4.99625e8	1.83602	0.918011	−	0.396555i	$-0.129794\pi$
$257$	4.99625e8	1.83602	0.918011	+	0.396555i	$0.129794\pi$
$258$	3.18007e7	0.115284
$259$	5.47143e7	0.195682
$260$	0	0
$261$	−2.49324e8	−0.868005
$262$	−8.58374e6	−0.0294864
$263$	3.03450e7	0.102859	0.0514295	−	0.998677i	$-0.483622\pi$
$263$	3.03450e7	0.102859	0.0514295	+	0.998677i	$0.483622\pi$
$264$	6.23026e6	0.0208397
$265$	4.05873e8	1.33977
$266$	−3.63163e7	−0.118308
$267$	2.73215e8	0.878445
$268$	−2.47939e8	−0.786815
$269$	1.10526e8	0.346204	0.173102	−	0.984904i	$-0.444621\pi$
$269$	1.10526e8	0.346204	0.173102	+	0.984904i	$0.444621\pi$
$270$	−8.82119e6	−0.0272743
$271$	5.04061e7	0.153848	0.0769238	−	0.997037i	$-0.475490\pi$
$271$	5.04061e7	0.153848	0.0769238	+	0.997037i	$0.475490\pi$
$272$	−1.82630e8	−0.550278
$273$	0	0
$274$	1.92353e7	0.0564900
$275$	6.59502e7	0.191228
$276$	3.52586e8	1.00945
$277$	−5.38594e8	−1.52259	−0.761294	−	0.648406i	$-0.775436\pi$
$277$	−5.38594e8	−1.52259	−0.761294	+	0.648406i	$0.775436\pi$
$278$	6.03182e6	0.0168380
$279$	−2.09115e8	−0.576463
$280$	−9.92816e7	−0.270281
$281$	1.43482e8	0.385766	0.192883	−	0.981222i	$-0.438216\pi$
$281$	1.43482e8	0.385766	0.192883	+	0.981222i	$0.438216\pi$
$282$	3.98754e7	0.105885
$283$	1.01247e8	0.265540	0.132770	−	0.991147i	$-0.457613\pi$
$283$	1.01247e8	0.265540	0.132770	+	0.991147i	$0.457613\pi$
$284$	6.02497e8	1.56078
$285$	−1.12585e9	−2.88088
$286$	0	0
$287$	1.14231e8	0.285231
$288$	−5.69219e7	−0.140412
$289$	−2.83359e8	−0.690548
$290$	−4.33991e7	−0.104493
$291$	−1.10770e8	−0.263509
$292$	−2.45806e8	−0.577768
$293$	3.06304e8	0.711405	0.355702	−	0.934599i	$-0.384242\pi$
$293$	3.06304e8	0.711405	0.355702	+	0.934599i	$0.384242\pi$
$294$	−3.70442e7	−0.0850168
$295$	2.77697e8	0.629788
$296$	7.28938e6	0.0163369
$297$	−1.70010e7	−0.0376555
$298$	3.45359e7	0.0755986
$299$	0	0
$300$	−9.13869e8	−1.95416
$301$	−9.76671e8	−2.06427
$302$	6.63508e6	0.0138619
$303$	−6.02017e8	−1.24325
$304$	6.65293e8	1.35817
$305$	2.82845e7	0.0570819
$306$	1.31128e7	0.0261620
$307$	8.86416e6	0.0174845	0.00874225	−	0.999962i	$-0.497217\pi$
$307$	8.86416e6	0.0174845	0.00874225	+	0.999962i	$0.497217\pi$
$308$	−9.54999e7	−0.186241
$309$	−4.46334e8	−0.860608
$310$	−3.64001e7	−0.0693964
$311$	3.95364e8	0.745309	0.372654	−	0.927970i	$-0.378448\pi$
$311$	3.95364e8	0.745309	0.372654	+	0.927970i	$0.378448\pi$
$312$	0	0
$313$	−8.88408e8	−1.63760	−0.818799	−	0.574080i	$-0.805360\pi$
$313$	−8.88408e8	−1.63760	−0.818799	+	0.574080i	$0.805360\pi$
$314$	−4.43804e7	−0.0808979
$315$	−9.80197e8	−1.76696
$316$	1.91675e8	0.341712
$317$	−3.40256e8	−0.599928	−0.299964	−	0.953951i	$-0.596975\pi$
$317$	−3.40256e8	−0.599928	−0.299964	+	0.953951i	$0.596975\pi$
$318$	3.92005e7	0.0683591
$319$	−8.36429e7	−0.144265
$320$	9.01123e8	1.53730
$321$	−1.28657e9	−2.17103
$322$	3.91617e7	0.0653680
$323$	−4.62568e8	−0.763777
$324$	7.13504e8	1.16544
$325$	0	0
$326$	1.37174e7	0.0219286
$327$	6.60213e8	1.04416
$328$	1.52185e7	0.0238130
$329$	−1.22466e9	−1.89597
$330$	1.07070e7	0.0164009
$331$	−5.49409e8	−0.832717	−0.416359	−	0.909200i	$-0.636694\pi$
$331$	−5.49409e8	−0.832717	−0.416359	+	0.909200i	$0.636694\pi$
$332$	−2.39740e8	−0.359548
$333$	7.19673e7	0.106802
$334$	4.59838e6	0.00675294
$335$	−8.53729e8	−1.24069
$336$	1.31854e9	1.89629
$337$	6.88538e8	0.979993	0.489997	−	0.871724i	$-0.336998\pi$
$337$	6.88538e8	0.979993	0.489997	+	0.871724i	$0.336998\pi$
$338$	0	0
$339$	1.67420e9	2.33404
$340$	−6.31144e8	−0.870867
$341$	−7.01538e7	−0.0958100
$342$	−4.77678e7	−0.0645720
$343$	6.49158e7	0.0868602
$344$	−1.30118e8	−0.172339
$345$	1.21406e9	1.59175
$346$	−5.88850e7	−0.0764255
$347$	−6.87155e8	−0.882879	−0.441440	−	0.897291i	$-0.645532\pi$
$347$	−6.87155e8	−0.882879	−0.441440	+	0.897291i	$0.645532\pi$
$348$	1.15904e9	1.47425
$349$	−1.03428e9	−1.30241	−0.651206	−	0.758901i	$-0.725737\pi$
$349$	−1.03428e9	−1.30241	−0.651206	+	0.758901i	$0.725737\pi$
$350$	−1.01503e8	−0.126544
$351$	0	0
$352$	−1.90961e7	−0.0233370
$353$	−1.07291e9	−1.29823	−0.649116	−	0.760690i	$-0.724861\pi$
$353$	−1.07291e9	−1.29823	−0.649116	+	0.760690i	$0.724861\pi$
$354$	2.68209e7	0.0321337
$355$	2.07458e9	2.46111
$356$	−5.57943e8	−0.655413
$357$	−9.16757e8	−1.06639
$358$	6.07457e7	0.0699721
$359$	−6.57582e8	−0.750100	−0.375050	−	0.927005i	$-0.622374\pi$
$359$	−6.57582e8	−0.750100	−0.375050	+	0.927005i	$0.622374\pi$
$360$	−1.30588e8	−0.147518
$361$	7.91187e8	0.885123
$362$	−1.73690e7	−0.0192440
$363$	−1.19640e9	−1.31282
$364$	0	0
$365$	−8.46387e8	−0.911054
$366$	2.73180e6	0.00291249
$367$	−8.94686e8	−0.944799	−0.472400	−	0.881384i	$-0.656612\pi$
$367$	−8.94686e8	−0.944799	−0.472400	+	0.881384i	$0.656612\pi$
$368$	−7.17418e8	−0.750421
$369$	1.50251e8	0.155677
$370$	1.25271e7	0.0128572
$371$	−1.20393e9	−1.22404
$372$	9.72119e8	0.979082
$373$	6.84177e8	0.682633	0.341317	−	0.939948i	$-0.389127\pi$
$373$	6.84177e8	0.682633	0.341317	+	0.939948i	$0.389127\pi$
$374$	4.39907e6	0.00434821
$375$	−1.00402e9	−0.983180
$376$	−1.63157e8	−0.158288
$377$	0	0
$378$	2.61661e7	0.0249183
$379$	1.13972e9	1.07537	0.537687	−	0.843144i	$-0.319298\pi$
$379$	1.13972e9	1.07537	0.537687	+	0.843144i	$0.319298\pi$
$380$	2.29915e9	2.14944
$381$	−5.04039e8	−0.466903
$382$	−9.86680e7	−0.0905638
$383$	1.34437e9	1.22271	0.611355	−	0.791356i	$-0.290625\pi$
$383$	1.34437e9	1.22271	0.611355	+	0.791356i	$0.290625\pi$
$384$	3.52604e8	0.317781
$385$	−3.28835e8	−0.293674
$386$	6.19540e7	0.0548294
$387$	−1.28464e9	−1.12666
$388$	2.26208e8	0.196606
$389$	−4.06758e8	−0.350359	−0.175179	−	0.984537i	$-0.556051\pi$
$389$	−4.06758e8	−0.350359	−0.175179	+	0.984537i	$0.556051\pi$
$390$	0	0
$391$	4.98810e8	0.422004
$392$	1.51573e8	0.127093
$393$	7.89350e8	0.655987
$394$	1.15629e8	0.0952427
$395$	6.59995e8	0.538829
$396$	−1.25614e8	−0.101649
$397$	−1.96883e8	−0.157922	−0.0789609	−	0.996878i	$-0.525160\pi$
$397$	−1.96883e8	−0.157922	−0.0789609	+	0.996878i	$0.525160\pi$
$398$	9.47562e7	0.0753385
$399$	3.33960e9	2.63202
$400$	1.85948e9	1.45272
$401$	1.00018e9	0.774588	0.387294	−	0.921956i	$-0.373410\pi$
$401$	1.00018e9	0.774588	0.387294	+	0.921956i	$0.373410\pi$
$402$	−8.24558e7	−0.0633038
$403$	0	0
$404$	1.22940e9	0.927599
$405$	2.45681e9	1.83772
$406$	1.28734e8	0.0954667
$407$	2.41435e7	0.0177509
$408$	−1.22136e8	−0.0890293
$409$	1.62401e9	1.17370	0.586850	−	0.809696i	$-0.300368\pi$
$409$	1.62401e9	1.17370	0.586850	+	0.809696i	$0.300368\pi$
$410$	2.61538e7	0.0187409
$411$	−1.76885e9	−1.25674
$412$	9.11478e8	0.642105
$413$	−8.23729e8	−0.575386
$414$	5.15104e7	0.0356775
$415$	−8.25497e8	−0.566953
$416$	0	0
$417$	−5.54678e8	−0.374598
$418$	−1.60251e7	−0.0107321
$419$	−1.91920e8	−0.127459	−0.0637297	−	0.997967i	$-0.520300\pi$
$419$	−1.91920e8	−0.127459	−0.0637297	+	0.997967i	$0.520300\pi$
$420$	4.55666e9	3.00106
$421$	1.17016e9	0.764292	0.382146	−	0.924102i	$-0.375185\pi$
$421$	1.17016e9	0.764292	0.382146	+	0.924102i	$0.375185\pi$
$422$	−5.68499e7	−0.0368244
$423$	−1.61083e9	−1.03481
$424$	−1.60396e8	−0.102191
$425$	−1.29287e9	−0.816944
$426$	2.00370e8	0.125574
$427$	−8.38997e7	−0.0521510
$428$	2.62736e9	1.61982
$429$	0	0
$430$	−2.23614e8	−0.135631
$431$	2.11589e8	0.127298	0.0636491	−	0.997972i	$-0.479726\pi$
$431$	2.11589e8	0.127298	0.0636491	+	0.997972i	$0.479726\pi$
$432$	−4.79348e8	−0.286060
$433$	−1.17596e9	−0.696119	−0.348060	−	0.937472i	$-0.613159\pi$
$433$	−1.17596e9	−0.696119	−0.348060	+	0.937472i	$0.613159\pi$
$434$	1.07973e8	0.0634017
$435$	3.99092e9	2.32467
$436$	−1.34825e9	−0.779055
$437$	−1.81708e9	−1.04157
$438$	−8.17466e7	−0.0464848
$439$	−2.42430e9	−1.36760	−0.683802	−	0.729668i	$-0.739675\pi$
$439$	−2.42430e9	−1.36760	−0.683802	+	0.729668i	$0.739675\pi$
$440$	−4.38095e7	−0.0245179
$441$	1.49646e9	0.830866
$442$	0	0
$443$	8.11271e7	0.0443356	0.0221678	−	0.999754i	$-0.492943\pi$
$443$	8.11271e7	0.0443356	0.0221678	+	0.999754i	$0.492943\pi$
$444$	−3.34556e8	−0.181396
$445$	−1.92117e9	−1.03349
$446$	−6.45876e7	−0.0344728
$447$	−3.17588e9	−1.68185
$448$	−2.67299e9	−1.40451
$449$	2.64894e7	0.0138105	0.00690525	−	0.999976i	$-0.497802\pi$
$449$	2.64894e7	0.0138105	0.00690525	+	0.999976i	$0.497802\pi$
$450$	−1.33510e8	−0.0690669
$451$	5.04061e7	0.0258741
$452$	−3.41895e9	−1.74144
$453$	−6.10153e8	−0.308386
$454$	−6.71275e7	−0.0336670
$455$	0	0
$456$	4.44921e8	0.219739
$457$	2.64093e9	1.29434	0.647172	−	0.762344i	$-0.275951\pi$
$457$	2.64093e9	1.29434	0.647172	+	0.762344i	$0.275951\pi$
$458$	−5.92505e7	−0.0288179
$459$	3.33283e8	0.160868
$460$	−2.47929e9	−1.18761
$461$	−2.20686e8	−0.104911	−0.0524556	−	0.998623i	$-0.516705\pi$
$461$	−2.20686e8	−0.104911	−0.0524556	+	0.998623i	$0.516705\pi$
$462$	−3.17599e7	−0.0149842
$463$	−1.40461e9	−0.657691	−0.328845	−	0.944384i	$-0.606659\pi$
$463$	−1.40461e9	−0.657691	−0.328845	+	0.944384i	$0.606659\pi$
$464$	−2.35833e9	−1.09595
$465$	3.34731e9	1.54387
$466$	−1.64686e8	−0.0753887
$467$	−1.70943e9	−0.776681	−0.388341	−	0.921516i	$-0.626952\pi$
$467$	−1.70943e9	−0.776681	−0.388341	+	0.921516i	$0.626952\pi$
$468$	0	0
$469$	2.53240e9	1.13352
$470$	−2.80393e8	−0.124573
$471$	4.08116e9	1.79974
$472$	−1.09742e8	−0.0480371
$473$	−4.30971e8	−0.187255
$474$	6.37443e7	0.0274927
$475$	4.70970e9	2.01635
$476$	1.87215e9	0.795640
$477$	−1.58357e9	−0.668071
$478$	−2.77439e8	−0.116190
$479$	−2.16475e9	−0.899980	−0.449990	−	0.893034i	$-0.648572\pi$
$479$	−2.16475e9	−0.899980	−0.449990	+	0.893034i	$0.648572\pi$
$480$	9.11147e8	0.376049
$481$	0	0
$482$	1.48902e8	0.0605669
$483$	−3.60125e9	−1.45425
$484$	2.44323e9	0.979502
$485$	7.78902e8	0.310018
$486$	1.93357e8	0.0764069
$487$	3.61830e9	1.41956	0.709779	−	0.704425i	$-0.248795\pi$
$487$	3.61830e9	1.41956	0.709779	+	0.704425i	$0.248795\pi$
$488$	−1.11776e7	−0.00435392
$489$	−1.26143e9	−0.487847
$490$	2.60485e8	0.100022
$491$	−1.79288e9	−0.683544	−0.341772	−	0.939783i	$-0.611027\pi$
$491$	−1.79288e9	−0.683544	−0.341772	+	0.939783i	$0.611027\pi$
$492$	−6.98475e8	−0.264407
$493$	1.63971e9	0.616315
$494$	0	0
$495$	−4.32527e8	−0.160286
$496$	−1.97800e9	−0.727848
$497$	−6.15380e9	−2.24852
$498$	−7.97291e7	−0.0289277
$499$	−1.87794e9	−0.676596	−0.338298	−	0.941039i	$-0.609851\pi$
$499$	−1.87794e9	−0.676596	−0.338298	+	0.941039i	$0.609851\pi$
$500$	2.05035e9	0.733556
$501$	−4.22861e8	−0.150233
$502$	−1.00897e7	−0.00355971
$503$	−1.13195e8	−0.0396589	−0.0198294	−	0.999803i	$-0.506312\pi$
$503$	−1.13195e8	−0.0396589	−0.0198294	+	0.999803i	$0.506312\pi$
$504$	3.87360e8	0.134775
$505$	4.23322e9	1.46269
$506$	1.72807e7	0.00592971
$507$	0	0
$508$	1.02932e9	0.348359
$509$	−5.42611e9	−1.82380	−0.911898	−	0.410418i	$-0.865383\pi$
$509$	−5.42611e9	−1.82380	−0.911898	+	0.410418i	$0.865383\pi$
$510$	−2.09896e8	−0.0700663
$511$	2.51062e9	0.832355
$512$	−8.98445e8	−0.295833
$513$	−1.21410e9	−0.397047
$514$	−3.39318e8	−0.110214
$515$	3.13850e9	1.01250
$516$	5.97195e9	1.91356
$517$	−5.40401e8	−0.171988
$518$	−3.71590e7	−0.0117465
$519$	5.41499e9	1.70025
$520$	0	0
$521$	−2.65903e9	−0.823741	−0.411871	−	0.911242i	$-0.635124\pi$
$521$	−2.65903e9	−0.823741	−0.411871	+	0.911242i	$0.635124\pi$
$522$	1.69327e8	0.0521051
$523$	4.04445e9	1.23624	0.618122	−	0.786082i	$-0.287894\pi$
$523$	4.04445e9	1.23624	0.618122	+	0.786082i	$0.287894\pi$
$524$	−1.61197e9	−0.489436
$525$	9.33410e9	2.81524
$526$	−2.06087e7	−0.00617448
$527$	1.37527e9	0.409310
$528$	5.81823e8	0.172017
$529$	−1.44537e9	−0.424507
$530$	−2.75647e8	−0.0804244
$531$	−1.08347e9	−0.314042
$532$	−6.81994e9	−1.96377
$533$	0	0
$534$	−1.85553e8	−0.0527318
$535$	9.04682e9	2.55422
$536$	3.37382e8	0.0946336
$537$	−5.58609e9	−1.55668
$538$	−7.50633e7	−0.0207821
$539$	5.02032e8	0.138093
$540$	−1.65656e9	−0.452718
$541$	6.97904e9	1.89498	0.947491	−	0.319782i	$-0.103610\pi$
$541$	6.97904e9	1.89498	0.947491	+	0.319782i	$0.103610\pi$
$542$	−3.42331e7	−0.00923525
$543$	1.59723e9	0.428124
$544$	3.74354e8	0.0996980
$545$	−4.64244e9	−1.22845
$546$	0	0
$547$	5.47434e7	0.0143013	0.00715065	−	0.999974i	$-0.497724\pi$
$547$	5.47434e7	0.0143013	0.00715065	+	0.999974i	$0.497724\pi$
$548$	3.61225e9	0.937661
$549$	−1.10356e8	−0.0284637
$550$	−4.47898e7	−0.0114791
$551$	−5.97320e9	−1.52116
$552$	−4.79781e8	−0.121411
$553$	−1.95773e9	−0.492283
$554$	3.65784e8	0.0913989
$555$	−1.15198e9	−0.286035
$556$	1.13273e9	0.279490
$557$	2.17057e8	0.0532207	0.0266103	−	0.999646i	$-0.491529\pi$
$557$	2.17057e8	0.0532207	0.0266103	+	0.999646i	$0.491529\pi$
$558$	1.42020e8	0.0346043
$559$	0	0
$560$	−9.27158e9	−2.23098
$561$	−4.04533e8	−0.0967350
$562$	−9.74450e7	−0.0231570
$563$	4.79943e9	1.13347	0.566735	−	0.823900i	$-0.308206\pi$
$563$	4.79943e9	1.13347	0.566735	+	0.823900i	$0.308206\pi$
$564$	7.48831e9	1.75755
$565$	−1.17725e10	−2.74599
$566$	−6.87616e7	−0.0159400
$567$	−7.28760e9	−1.67897
$568$	−8.19847e8	−0.187721
$569$	5.22411e9	1.18883	0.594415	−	0.804158i	$-0.297384\pi$
$569$	5.22411e9	1.18883	0.594415	+	0.804158i	$0.297384\pi$
$570$	7.64618e8	0.172935
$571$	9.53468e8	0.214328	0.107164	−	0.994241i	$-0.465823\pi$
$571$	9.53468e8	0.214328	0.107164	+	0.994241i	$0.465823\pi$
$572$	0	0
$573$	9.07338e9	2.01478
$574$	−7.75794e7	−0.0171220
$575$	−5.07871e9	−1.11408
$576$	−3.51585e9	−0.766571
$577$	−7.41729e6	−0.00160742	−0.000803711	−	1.00000i	$-0.500256\pi$
$577$	−7.41729e6	−0.00160742	−0.000803711		1.00000i	$0.500256\pi$
$578$	1.92442e8	0.0414526
$579$	−5.69720e9	−1.21980
$580$	−8.15004e9	−1.73445
$581$	2.44866e9	0.517979
$582$	7.52288e7	0.0158181
$583$	−5.31254e8	−0.111036
$584$	3.34481e8	0.0694906
$585$	0	0
$586$	−2.08025e8	−0.0427046
$587$	−6.00981e9	−1.22639	−0.613193	−	0.789933i	$-0.710115\pi$
$587$	−6.00981e9	−1.22639	−0.613193	+	0.789933i	$0.710115\pi$
$588$	−6.95664e9	−1.41117
$589$	−5.00990e9	−1.01024
$590$	−1.88597e8	−0.0378053
$591$	−1.06331e10	−2.11887
$592$	6.80731e8	0.134850
$593$	1.00461e9	0.197836	0.0989178	−	0.995096i	$-0.468462\pi$
$593$	1.00461e9	0.197836	0.0989178	+	0.995096i	$0.468462\pi$
$594$	1.15462e7	0.00226041
$595$	6.44639e9	1.25461
$596$	6.48560e9	1.25484
$597$	−8.71365e9	−1.67606
$598$	0	0
$599$	−4.70768e9	−0.894979	−0.447489	−	0.894289i	$-0.647682\pi$
$599$	−4.70768e9	−0.894979	−0.447489	+	0.894289i	$0.647682\pi$
$600$	1.24355e9	0.235035
$601$	6.05279e9	1.13735	0.568676	−	0.822562i	$-0.307456\pi$
$601$	6.05279e9	1.13735	0.568676	+	0.822562i	$0.307456\pi$
$602$	6.63302e8	0.123915
$603$	3.33094e9	0.618666
$604$	1.24602e9	0.230089
$605$	8.41279e9	1.54453
$606$	4.08857e8	0.0746307
$607$	−7.12726e9	−1.29349	−0.646744	−	0.762707i	$-0.723870\pi$
$607$	−7.12726e9	−1.29349	−0.646744	+	0.762707i	$0.723870\pi$
$608$	−1.36371e9	−0.246071
$609$	−1.18382e10	−2.12386
$610$	−1.92093e7	−0.00342655
$611$	0	0
$612$	2.46249e9	0.434255
$613$	1.00062e9	0.175452	0.0877258	−	0.996145i	$-0.472040\pi$
$613$	1.00062e9	0.175452	0.0877258	+	0.996145i	$0.472040\pi$
$614$	−6.02006e6	−0.00104957
$615$	−2.40507e9	−0.416931
$616$	1.29951e8	0.0224000
$617$	−3.71765e9	−0.637192	−0.318596	−	0.947891i	$-0.603211\pi$
$617$	−3.71765e9	−0.637192	−0.318596	+	0.947891i	$0.603211\pi$
$618$	3.03126e8	0.0516611
$619$	8.30664e9	1.40769	0.703847	−	0.710352i	$-0.251464\pi$
$619$	8.30664e9	1.40769	0.703847	+	0.710352i	$0.251464\pi$
$620$	−6.83568e9	−1.15189
$621$	1.30922e9	0.219378
$622$	−2.68510e8	−0.0447398
$623$	5.69874e9	0.944214
$624$	0	0
$625$	−1.90347e9	−0.311864
$626$	6.03359e8	0.0983027
$627$	1.47365e9	0.238757
$628$	−8.33432e9	−1.34280
$629$	−4.73302e8	−0.0758335
$630$	6.65697e8	0.106068
$631$	−7.83901e9	−1.24211	−0.621053	−	0.783769i	$-0.713295\pi$
$631$	−7.83901e9	−1.24211	−0.621053	+	0.783769i	$0.713295\pi$
$632$	−2.60821e8	−0.0410991
$633$	5.22784e9	0.819236
$634$	2.31084e8	0.0360128
$635$	3.54427e9	0.549311
$636$	7.36157e9	1.13467
$637$	0	0
$638$	5.68058e7	0.00866004
$639$	−8.09427e9	−1.22723
$640$	−2.47942e9	−0.373869
$641$	1.27214e9	0.190779	0.0953895	−	0.995440i	$-0.469590\pi$
$641$	1.27214e9	0.190779	0.0953895	+	0.995440i	$0.469590\pi$
$642$	8.73770e8	0.130324
$643$	5.67912e9	0.842447	0.421224	−	0.906957i	$-0.361601\pi$
$643$	5.67912e9	0.842447	0.421224	+	0.906957i	$0.361601\pi$
$644$	7.35428e9	1.08502
$645$	2.05633e10	3.01740
$646$	3.14151e8	0.0458485
$647$	−9.83025e9	−1.42692	−0.713460	−	0.700696i	$-0.752873\pi$
$647$	−9.83025e9	−1.42692	−0.713460	+	0.700696i	$0.752873\pi$
$648$	−9.70899e8	−0.140172
$649$	−3.63483e8	−0.0521948
$650$	0	0
$651$	−9.92905e9	−1.41050
$652$	2.57603e9	0.363986
$653$	−3.85413e9	−0.541665	−0.270832	−	0.962626i	$-0.587299\pi$
$653$	−3.85413e9	−0.541665	−0.270832	+	0.962626i	$0.587299\pi$
$654$	−4.48381e8	−0.0626795
$655$	−5.55049e9	−0.771769
$656$	1.42121e9	0.196560
$657$	3.30229e9	0.454294
$658$	8.31725e8	0.113812
$659$	−4.91498e9	−0.668995	−0.334498	−	0.942397i	$-0.608567\pi$
$659$	−4.91498e9	−0.668995	−0.334498	+	0.942397i	$0.608567\pi$
$660$	2.01069e9	0.272234
$661$	1.27915e10	1.72272	0.861360	−	0.507995i	$-0.169613\pi$
$661$	1.27915e10	1.72272	0.861360	+	0.507995i	$0.169613\pi$
$662$	3.73129e8	0.0499869
$663$	0	0
$664$	3.26225e8	0.0432443
$665$	−2.34831e10	−3.09657
$666$	−4.88763e7	−0.00641119
$667$	6.44120e9	0.840478
$668$	8.63544e8	0.112090
$669$	5.93939e9	0.766920
$670$	5.79807e8	0.0744769
$671$	−3.70220e7	−0.00473076
$672$	−2.70272e9	−0.343565
$673$	−1.09851e10	−1.38916	−0.694581	−	0.719414i	$-0.744410\pi$
$673$	−1.09851e10	−1.38916	−0.694581	+	0.719414i	$0.744410\pi$
$674$	−4.67618e8	−0.0588276
$675$	−3.39337e9	−0.424686
$676$	0	0
$677$	3.14306e9	0.389307	0.194653	−	0.980872i	$-0.437642\pi$
$677$	3.14306e9	0.389307	0.194653	+	0.980872i	$0.437642\pi$
$678$	−1.13702e9	−0.140109
$679$	−2.31044e9	−0.283238
$680$	8.58827e8	0.104743
$681$	6.17296e9	0.748994
$682$	4.76447e7	0.00575134
$683$	1.54717e10	1.85808	0.929042	−	0.369975i	$-0.120634\pi$
$683$	1.54717e10	1.85808	0.929042	+	0.369975i	$0.120634\pi$
$684$	−8.97046e9	−1.07181
$685$	1.24381e10	1.47855
$686$	−4.40873e7	−0.00521409
$687$	5.44859e9	0.641115
$688$	−1.21513e10	−1.42254
$689$	0	0
$690$	−8.24526e8	−0.0955504
$691$	4.62295e9	0.533023	0.266512	−	0.963832i	$-0.414129\pi$
$691$	4.62295e9	0.533023	0.266512	+	0.963832i	$0.414129\pi$
$692$	−1.10582e10	−1.26856
$693$	1.28300e9	0.146440
$694$	4.66678e8	0.0529980
$695$	3.90035e9	0.440714
$696$	−1.57716e9	−0.177314
$697$	−9.88145e8	−0.110537
$698$	7.02426e8	0.0781820
$699$	1.51443e10	1.67718
$700$	−1.90616e10	−2.10047
$701$	1.41767e10	1.55440	0.777200	−	0.629253i	$-0.216639\pi$
$701$	1.41767e10	1.55440	0.777200	+	0.629253i	$0.216639\pi$
$702$	0	0
$703$	1.72416e9	0.187169
$704$	−1.17949e9	−0.127407
$705$	2.57846e10	2.77139
$706$	7.28663e8	0.0779310
$707$	−1.25569e10	−1.33634
$708$	5.03676e9	0.533378
$709$	−8.17094e9	−0.861014	−0.430507	−	0.902587i	$-0.641665\pi$
$709$	−8.17094e9	−0.861014	−0.430507	+	0.902587i	$0.641665\pi$
$710$	−1.40895e9	−0.147737
$711$	−2.57506e9	−0.268685
$712$	7.59221e8	0.0788294
$713$	5.40242e9	0.558181
$714$	6.22612e8	0.0640138
$715$	0	0
$716$	1.14076e10	1.16145
$717$	2.55130e10	2.58490
$718$	4.46594e8	0.0450274
$719$	−1.54234e10	−1.54749	−0.773745	−	0.633497i	$-0.781619\pi$
$719$	−1.54234e10	−1.54749	−0.773745	+	0.633497i	$0.781619\pi$
$720$	−1.21952e10	−1.21766
$721$	−9.30968e9	−0.925042
$722$	−5.37331e8	−0.0531327
$723$	−1.36928e10	−1.34744
$724$	−3.26178e9	−0.319426
$725$	−1.66950e10	−1.62705
$726$	8.12533e8	0.0788066
$727$	−5.70391e9	−0.550557	−0.275279	−	0.961365i	$-0.588770\pi$
$727$	−5.70391e9	−0.550557	−0.275279	+	0.961365i	$0.588770\pi$
$728$	0	0
$729$	−5.54588e9	−0.530181
$730$	5.74820e8	0.0546893
$731$	8.44862e9	0.799972
$732$	5.13012e8	0.0483436
$733$	1.66495e10	1.56148	0.780742	−	0.624854i	$-0.214841\pi$
$733$	1.66495e10	1.56148	0.780742	+	0.624854i	$0.214841\pi$
$734$	6.07623e8	0.0567150
$735$	−2.39539e10	−2.22520
$736$	1.47056e9	0.135959
$737$	1.11746e9	0.102824
$738$	−1.02042e8	−0.00934509
$739$	−4.92732e9	−0.449112	−0.224556	−	0.974461i	$-0.572093\pi$
$739$	−4.92732e9	−0.449112	−0.224556	+	0.974461i	$0.572093\pi$
$740$	2.35251e9	0.213412
$741$	0	0
$742$	8.17647e8	0.0734772
$743$	9.91603e9	0.886904	0.443452	−	0.896298i	$-0.353754\pi$
$743$	9.91603e9	0.886904	0.443452	+	0.896298i	$0.353754\pi$
$744$	−1.32281e9	−0.117758
$745$	2.23319e10	1.97870
$746$	−4.64656e8	−0.0409775
$747$	3.22079e9	0.282709
$748$	8.26114e8	0.0721746
$749$	−2.68354e10	−2.33358
$750$	6.81876e8	0.0590189
$751$	2.25292e10	1.94091	0.970456	−	0.241277i	$-0.0775663\pi$
$751$	2.25292e10	1.94091	0.970456	+	0.241277i	$0.0775663\pi$
$752$	−1.52367e10	−1.30656
$753$	9.27835e8	0.0791933
$754$	0	0
$755$	4.29043e9	0.362816
$756$	4.91381e9	0.413611
$757$	−2.21638e10	−1.85699	−0.928493	−	0.371351i	$-0.878895\pi$
$757$	−2.21638e10	−1.85699	−0.928493	+	0.371351i	$0.878895\pi$
$758$	−7.74034e8	−0.0645532
$759$	−1.58911e9	−0.131919
$760$	−3.12857e9	−0.258522
$761$	−7.67281e9	−0.631115	−0.315557	−	0.948907i	$-0.602191\pi$
$761$	−7.67281e9	−0.631115	−0.315557	+	0.948907i	$0.602191\pi$
$762$	3.42316e8	0.0280275
$763$	1.37708e10	1.12234
$764$	−1.85291e10	−1.50324
$765$	8.47912e9	0.684756
$766$	−9.13025e8	−0.0733976
$767$	0	0
$768$	1.61639e10	1.28760
$769$	8.97812e9	0.711940	0.355970	−	0.934497i	$-0.384151\pi$
$769$	8.97812e9	0.711940	0.355970	+	0.934497i	$0.384151\pi$
$770$	2.23327e8	0.0176289
$771$	3.12032e10	2.45194
$772$	1.16345e10	0.910097
$773$	1.12009e10	0.872218	0.436109	−	0.899894i	$-0.356356\pi$
$773$	1.12009e10	0.872218	0.436109	+	0.899894i	$0.356356\pi$
$774$	8.72460e8	0.0676320
$775$	−1.40026e10	−1.08057
$776$	−3.07811e8	−0.0236466
$777$	3.41709e9	0.261326
$778$	2.76248e8	0.0210315
$779$	3.59965e9	0.272822
$780$	0	0
$781$	−2.71546e9	−0.203969
$782$	−3.38765e8	−0.0253323
$783$	4.30373e9	0.320390
$784$	1.41549e10	1.04906
$785$	−2.86976e10	−2.11740
$786$	−5.36084e8	−0.0393780
$787$	1.10539e9	0.0808360	0.0404180	−	0.999183i	$-0.487131\pi$
$787$	1.10539e9	0.0808360	0.0404180	+	0.999183i	$0.487131\pi$
$788$	2.17144e10	1.58090
$789$	1.89515e9	0.137364
$790$	−4.48233e8	−0.0323451
$791$	3.49206e10	2.50878
$792$	1.70929e8	0.0122258
$793$	0	0
$794$	1.33712e8	0.00947982
$795$	2.53482e10	1.78921
$796$	1.77945e10	1.25052
$797$	−1.49938e10	−1.04908	−0.524540	−	0.851386i	$-0.675763\pi$
$797$	−1.49938e10	−1.04908	−0.524540	+	0.851386i	$0.675763\pi$
$798$	−2.26807e9	−0.157996
$799$	1.05939e10	0.734751
$800$	−3.81154e9	−0.263200
$801$	7.49571e9	0.515346
$802$	−6.79265e8	−0.0464974
$803$	1.10785e9	0.0755051
$804$	−1.54846e10	−1.05076
$805$	2.53230e10	1.71092
$806$	0	0
$807$	6.90272e9	0.462342
$808$	−1.67291e9	−0.111566
$809$	1.53212e10	1.01736	0.508679	−	0.860956i	$-0.330134\pi$
$809$	1.53212e10	1.01736	0.508679	+	0.860956i	$0.330134\pi$
$810$	−1.66853e9	−0.110316
$811$	1.14391e10	0.753041	0.376521	−	0.926408i	$-0.377120\pi$
$811$	1.14391e10	0.753041	0.376521	+	0.926408i	$0.377120\pi$
$812$	2.41753e10	1.58462
$813$	3.14803e9	0.205458
$814$	−1.63970e7	−0.00106556
$815$	8.87006e9	0.573951
$816$	−1.14059e10	−0.734875
$817$	−3.07769e10	−1.97446
$818$	−1.10294e9	−0.0704555
$819$	0	0
$820$	4.91149e9	0.311075
$821$	−2.17796e10	−1.37356	−0.686782	−	0.726864i	$-0.740977\pi$
$821$	−2.17796e10	−1.37356	−0.686782	+	0.726864i	$0.740977\pi$
$822$	1.20131e9	0.0754403
$823$	−2.16314e9	−0.135265	−0.0676324	−	0.997710i	$-0.521544\pi$
$823$	−2.16314e9	−0.135265	−0.0676324	+	0.997710i	$0.521544\pi$
$824$	−1.24029e9	−0.0772287
$825$	4.11881e9	0.255378
$826$	5.59432e8	0.0345396
$827$	2.27847e10	1.40079	0.700395	−	0.713755i	$-0.253007\pi$
$827$	2.27847e10	1.40079	0.700395	+	0.713755i	$0.253007\pi$
$828$	9.67329e9	0.592200
$829$	9.83839e9	0.599768	0.299884	−	0.953976i	$-0.403052\pi$
$829$	9.83839e9	0.599768	0.299884	+	0.953976i	$0.403052\pi$
$830$	5.60633e8	0.0340334
$831$	−3.36370e10	−2.03336
$832$	0	0
$833$	−9.84168e9	−0.589945
$834$	3.76708e8	0.0224866
$835$	2.97345e9	0.176749
$836$	−3.00940e9	−0.178138
$837$	3.60967e9	0.212779
$838$	1.30342e8	0.00765121
$839$	2.09272e10	1.22333	0.611666	−	0.791116i	$-0.290500\pi$
$839$	2.09272e10	1.22333	0.611666	+	0.791116i	$0.290500\pi$
$840$	−6.20047e9	−0.360950
$841$	3.92391e9	0.227475
$842$	−7.94712e8	−0.0458793
$843$	8.96092e9	0.515176
$844$	−1.06760e10	−0.611237
$845$	0	0
$846$	1.09399e9	0.0621180
$847$	−2.49547e10	−1.41111
$848$	−1.49788e10	−0.843514
$849$	6.32322e9	0.354619
$850$	8.78045e8	0.0490400
$851$	−1.85925e9	−0.103415
$852$	3.76280e10	2.08436
$853$	−2.52741e10	−1.39429	−0.697147	−	0.716928i	$-0.745548\pi$
$853$	−2.52741e10	−1.39429	−0.697147	+	0.716928i	$0.745548\pi$
$854$	5.69801e7	0.00313055
$855$	−3.08880e10	−1.69009
$856$	−3.57518e9	−0.194823
$857$	2.67167e10	1.44994	0.724969	−	0.688782i	$-0.241854\pi$
$857$	2.67167e10	1.44994	0.724969	+	0.688782i	$0.241854\pi$
$858$	0	0
$859$	−8.51182e9	−0.458191	−0.229095	−	0.973404i	$-0.573577\pi$
$859$	−8.51182e9	−0.458191	−0.229095	+	0.973404i	$0.573577\pi$
$860$	−4.19931e10	−2.25130
$861$	7.13410e9	0.380915
$862$	−1.43700e8	−0.00764153
$863$	−5.94525e9	−0.314871	−0.157435	−	0.987529i	$-0.550323\pi$
$863$	−5.94525e9	−0.314871	−0.157435	+	0.987529i	$0.550323\pi$
$864$	9.82563e8	0.0518277
$865$	−3.80767e10	−2.00034
$866$	7.98646e8	0.0417871
$867$	−1.76967e10	−0.922201
$868$	2.02766e10	1.05239
$869$	−8.63878e8	−0.0446564
$870$	−2.71042e9	−0.139547
$871$	0	0
$872$	1.83463e9	0.0937002
$873$	−3.03899e9	−0.154589
$874$	1.23406e9	0.0625242
$875$	−2.09419e10	−1.05679
$876$	−1.53514e10	−0.771587
$877$	1.06461e10	0.532955	0.266477	−	0.963841i	$-0.414140\pi$
$877$	1.06461e10	0.532955	0.266477	+	0.963841i	$0.414140\pi$
$878$	1.64645e9	0.0820953
$879$	1.91297e10	0.950054
$880$	−4.09122e9	−0.202378
$881$	−3.83199e9	−0.188803	−0.0944014	−	0.995534i	$-0.530094\pi$
$881$	−3.83199e9	−0.188803	−0.0944014	+	0.995534i	$0.530094\pi$
$882$	−1.01632e9	−0.0498757
$883$	1.75487e10	0.857792	0.428896	−	0.903354i	$-0.358903\pi$
$883$	1.75487e10	0.857792	0.428896	+	0.903354i	$0.358903\pi$
$884$	0	0
$885$	1.73431e10	0.841058
$886$	−5.50972e7	−0.00266141
$887$	−1.07458e10	−0.517020	−0.258510	−	0.966009i	$-0.583231\pi$
$887$	−1.07458e10	−0.517020	−0.258510	+	0.966009i	$0.583231\pi$
$888$	4.55246e8	0.0218173
$889$	−1.05133e10	−0.501860
$890$	1.30476e9	0.0620389
$891$	−3.21576e9	−0.152304
$892$	−1.21291e10	−0.572204
$893$	−3.85917e10	−1.81348
$894$	2.15688e9	0.100959
$895$	3.92799e10	1.83143
$896$	7.35465e9	0.341573
$897$	0	0
$898$	−1.79902e7	−0.000829025 0
$899$	1.77591e10	0.815195
$900$	−2.50722e10	−1.14642
$901$	1.04145e10	0.474355
$902$	−3.42331e7	−0.00155319
$903$	−6.09964e10	−2.75675
$904$	4.65233e9	0.209450
$905$	−1.12313e10	−0.503687
$906$	4.14383e8	0.0185120
$907$	−2.07199e10	−0.922067	−0.461034	−	0.887383i	$-0.652521\pi$
$907$	−2.07199e10	−0.922067	−0.461034	+	0.887383i	$0.652521\pi$
$908$	−1.26061e10	−0.558829
$909$	−1.65165e10	−0.729363
$910$	0	0
$911$	−3.76677e10	−1.65065	−0.825325	−	0.564658i	$-0.809008\pi$
$911$	−3.76677e10	−1.65065	−0.825325	+	0.564658i	$0.809008\pi$
$912$	4.15498e10	1.81379
$913$	1.08051e9	0.0469872
$914$	−1.79358e9	−0.0776977
$915$	1.76646e9	0.0762307
$916$	−1.11268e10	−0.478340
$917$	1.64643e10	0.705101
$918$	−2.26348e8	−0.00965667
$919$	1.56150e10	0.663646	0.331823	−	0.943342i	$-0.392336\pi$
$919$	1.56150e10	0.663646	0.331823	+	0.943342i	$0.392336\pi$
$920$	3.37369e9	0.142839
$921$	5.53597e8	0.0233499
$922$	1.49878e8	0.00629768
$923$	0	0
$924$	−5.96429e9	−0.248718
$925$	4.81899e9	0.200198
$926$	9.53934e8	0.0394802
$927$	−1.22453e10	−0.504882
$928$	4.83408e9	0.198562
$929$	2.75954e10	1.12923	0.564615	−	0.825355i	$-0.309025\pi$
$929$	2.75954e10	1.12923	0.564615	+	0.825355i	$0.309025\pi$
$930$	−2.27331e9	−0.0926762
$931$	3.58516e10	1.45608
$932$	−3.09269e10	−1.25135
$933$	2.46918e10	0.995332
$934$	1.16095e9	0.0466231
$935$	2.84457e9	0.113809
$936$	0	0
$937$	−3.75571e9	−0.149143	−0.0745715	−	0.997216i	$-0.523759\pi$
$937$	−3.75571e9	−0.149143	−0.0745715	+	0.997216i	$0.523759\pi$
$938$	−1.71987e9	−0.0680434
$939$	−5.54841e10	−2.18695
$940$	−5.26558e10	−2.06775
$941$	3.77084e10	1.47528	0.737641	−	0.675193i	$-0.235940\pi$
$941$	3.77084e10	1.47528	0.737641	+	0.675193i	$0.235940\pi$
$942$	−2.77171e9	−0.108036
$943$	−3.88168e9	−0.150740
$944$	−1.02485e10	−0.396512
$945$	1.69198e10	0.652203
$946$	2.92692e8	0.0112407
$947$	−4.58140e10	−1.75297	−0.876483	−	0.481433i	$-0.840117\pi$
$947$	−4.58140e10	−1.75297	−0.876483	+	0.481433i	$0.840117\pi$
$948$	1.19707e10	0.456343
$949$	0	0
$950$	−3.19857e9	−0.121039
$951$	−2.12502e10	−0.801181
$952$	−2.54752e9	−0.0956950
$953$	1.83031e10	0.685013	0.342507	−	0.939515i	$-0.388724\pi$
$953$	1.83031e10	0.685013	0.342507	+	0.939515i	$0.388724\pi$
$954$	1.07547e9	0.0401034
$955$	−6.38015e10	−2.37039
$956$	−5.21011e10	−1.92861
$957$	−5.22378e9	−0.192661
$958$	1.47018e9	0.0540245
$959$	−3.68949e10	−1.35083
$960$	5.62782e10	2.05301
$961$	−1.26175e10	−0.458609
$962$	0	0
$963$	−3.52974e10	−1.27365
$964$	2.79627e10	1.00533
$965$	4.00612e10	1.43509
$966$	2.44578e9	0.0872965
$967$	2.62013e10	0.931816	0.465908	−	0.884833i	$-0.345728\pi$
$967$	2.62013e10	0.931816	0.465908	+	0.884833i	$0.345728\pi$
$968$	−3.32462e9	−0.117809
$969$	−2.88889e10	−1.02000
$970$	−5.28988e8	−0.0186099
$971$	3.04960e9	0.106899	0.0534497	−	0.998571i	$-0.482978\pi$
$971$	3.04960e9	0.106899	0.0534497	+	0.998571i	$0.482978\pi$
$972$	3.63110e10	1.26826
$973$	−1.15695e10	−0.402644
$974$	−2.45735e9	−0.0852141
$975$	0	0
$976$	−1.04384e9	−0.0359385
$977$	−4.40597e10	−1.51151	−0.755754	−	0.654856i	$-0.772729\pi$
$977$	−4.40597e10	−1.51151	−0.755754	+	0.654856i	$0.772729\pi$
$978$	8.56698e8	0.0292848
$979$	2.51465e9	0.0856522
$980$	4.89172e10	1.66024
$981$	1.81131e10	0.612564
$982$	1.21763e9	0.0410322
$983$	3.19323e10	1.07224	0.536122	−	0.844141i	$-0.319889\pi$
$983$	3.19323e10	1.07224	0.536122	+	0.844141i	$0.319889\pi$
$984$	9.50449e8	0.0318014
$985$	7.47693e10	2.49285
$986$	−1.11360e9	−0.0369965
$987$	−7.64843e10	−2.53199
$988$	0	0
$989$	3.31883e10	1.09093
$990$	2.93749e8	0.00962172
$991$	3.66598e10	1.19655	0.598277	−	0.801289i	$-0.295852\pi$
$991$	3.66598e10	1.19655	0.598277	+	0.801289i	$0.295852\pi$
$992$	4.05449e9	0.131870
$993$	−3.43124e10	−1.11206
$994$	4.17933e9	0.134975
$995$	6.12721e10	1.97189
$996$	−1.49725e10	−0.480162
$997$	5.86084e10	1.87295	0.936476	−	0.350732i	$-0.114067\pi$
$997$	5.86084e10	1.87295	0.936476	+	0.350732i	$0.114067\pi$
$998$	1.27539e9	0.0406151
$999$	−1.24227e9	−0.0394218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.8.a.g.1.7		14
13.5	odd	4		169.8.b.d.168.8		14
13.6	odd	12		13.8.e.a.10.4	yes	14
13.8	odd	4		169.8.b.d.168.7		14
13.11	odd	12		13.8.e.a.4.4	✓	14
13.12	even	2	inner	169.8.a.g.1.8		14
39.11	even	12		117.8.q.b.82.4		14
39.32	even	12		117.8.q.b.10.4		14
52.11	even	12		208.8.w.a.17.7		14
52.19	even	12		208.8.w.a.49.7		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.8.e.a.4.4	✓	14	13.11	odd	12
13.8.e.a.10.4	yes	14	13.6	odd	12
117.8.q.b.10.4		14	39.32	even	12
117.8.q.b.82.4		14	39.11	even	12
169.8.a.g.1.7		14	1.1	even	1	trivial
169.8.a.g.1.8		14	13.12	even	2	inner
169.8.b.d.168.7		14	13.8	odd	4
169.8.b.d.168.8		14	13.5	odd	4
208.8.w.a.17.7		14	52.11	even	12
208.8.w.a.49.7		14	52.19	even	12

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:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	169.a (trivial)

\(f(q)\)	\(=\)	\(q-0.679146 q^{2} +62.4534 q^{3} -127.539 q^{4} -439.155 q^{5} -42.4150 q^{6} +1302.66 q^{7} +173.548 q^{8} +1713.42 q^{9} +298.251 q^{10} +574.818 q^{11} -7965.23 q^{12} -884.695 q^{14} -27426.7 q^{15} +16207.1 q^{16} -11268.5 q^{17} -1163.66 q^{18} +41049.5 q^{19} +56009.3 q^{20} +81355.4 q^{21} -390.385 q^{22} -44265.7 q^{23} +10838.7 q^{24} +114732. q^{25} -29576.4 q^{27} -166139. q^{28} -145512. q^{29} +18626.8 q^{30} -122045. q^{31} -33221.1 q^{32} +35899.3 q^{33} +7652.99 q^{34} -572069. q^{35} -218528. q^{36} +42002.0 q^{37} -27878.6 q^{38} -76214.6 q^{40} +87690.6 q^{41} -55252.2 q^{42} -749752. q^{43} -73311.5 q^{44} -752459. q^{45} +30062.9 q^{46} -940126. q^{47} +1.01219e6 q^{48} +873376. q^{49} -77920.0 q^{50} -703759. q^{51} -924214. q^{53} +20086.7 q^{54} -252434. q^{55} +226074. q^{56} +2.56368e6 q^{57} +98824.0 q^{58} -632344. q^{59} +3.49797e6 q^{60} -64406.5 q^{61} +82886.6 q^{62} +2.23201e6 q^{63} -2.05195e6 q^{64} -24380.9 q^{66} +1.94403e6 q^{67} +1.43718e6 q^{68} -2.76454e6 q^{69} +388519. q^{70} -4.72403e6 q^{71} +297361. q^{72} +1.92731e6 q^{73} -28525.5 q^{74} +7.16542e6 q^{75} -5.23540e6 q^{76} +748791. q^{77} -1.50287e6 q^{79} -7.11743e6 q^{80} -5.59441e6 q^{81} -59554.7 q^{82} +1.87974e6 q^{83} -1.03760e7 q^{84} +4.94864e6 q^{85} +509191. q^{86} -9.08772e6 q^{87} +99758.5 q^{88} +4.37470e6 q^{89} +511030. q^{90} +5.64559e6 q^{92} -7.62215e6 q^{93} +638483. q^{94} -1.80271e7 q^{95} -2.07477e6 q^{96} -1.77364e6 q^{97} -593150. q^{98} +984906. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 52 q^{3} + 766 q^{4} + 6982 q^{9} - 1018 q^{10} - 38380 q^{12} - 47916 q^{14} + 1266 q^{16} - 76806 q^{17} + 251764 q^{22} - 137100 q^{23} - 39380 q^{25} - 432400 q^{27} - 443166 q^{29} - 315780 q^{30}+ \cdots - 86840772 q^{95}+O(q^{100}) \) 14 * q - 52 * q^3 + 766 * q^4 + 6982 * q^9 - 1018 * q^10 - 38380 * q^12 - 47916 * q^14 + 1266 * q^16 - 76806 * q^17 + 251764 * q^22 - 137100 * q^23 - 39380 * q^25 - 432400 * q^27 - 443166 * q^29 - 315780 * q^30 - 1319232 * q^35 + 1271350 * q^36 - 953640 * q^38 - 77182 * q^40 - 412812 * q^42 - 1765384 * q^43 - 2267260 * q^48 + 5307958 * q^49 + 38964 * q^51 - 1791210 * q^53 - 4907072 * q^55 - 14915904 * q^56 - 7717050 * q^61 + 3808284 * q^62 - 14570922 * q^64 - 31956396 * q^66 - 19139238 * q^68 - 27487680 * q^69 - 17280210 * q^74 + 20632304 * q^75 - 19599336 * q^77 - 30688440 * q^79 - 27772442 * q^81 - 74719438 * q^82 - 25539372 * q^87 + 54091072 * q^88 - 41193546 * q^90 - 47685588 * q^92 - 38277620 * q^94 - 86840772 * q^95

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