LMFDB - Embedded newform 350.8.c.k.99.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [350,8,Mod(99,350)] 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("350.99"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(350, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	350.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-256,0,1120] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	no
Analytic conductor:	$109.334758919$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{1969})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 985x^{2} + 242064 $ x^4 + 985*x^2 + 242064
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.4
Root			$21.6867i$ of defining polynomial
Character	$\chi$	$=$	350.99
Dual form			350.8.c.k.99.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+8.00000i q^{2} +9.37342i q^{3} -64.0000 q^{4} -74.9873 q^{6} +343.000i q^{7} -512.000i q^{8} +2099.14 q^{9} +3881.05 q^{11} -599.899i q^{12} +11585.6i q^{13} -2744.00 q^{14} +4096.00 q^{16} -15242.4i q^{17} +16793.1i q^{18} +32461.7 q^{19} -3215.08 q^{21} +31048.4i q^{22} -56146.1i q^{23} +4799.19 q^{24} -92684.6 q^{26} +40175.8i q^{27} -21952.0i q^{28} +26442.0 q^{29} -45448.2 q^{31} +32768.0i q^{32} +36378.7i q^{33} +121939. q^{34} -134345. q^{36} -555245. i q^{37} +259694. i q^{38} -108596. q^{39} +306385. q^{41} -25720.7i q^{42} -780755. i q^{43} -248387. q^{44} +449169. q^{46} -531243. i q^{47} +38393.5i q^{48} -117649. q^{49} +142873. q^{51} -741477. i q^{52} +363593. i q^{53} -321406. q^{54} +175616. q^{56} +304277. i q^{57} +211536. i q^{58} +2.14095e6 q^{59} +888824. q^{61} -363586. i q^{62} +720005. i q^{63} -262144. q^{64} -291030. q^{66} +4.34541e6i q^{67} +975513. i q^{68} +526281. q^{69} +663207. q^{71} -1.07476e6i q^{72} -1.34202e6i q^{73} +4.44196e6 q^{74} -2.07755e6 q^{76} +1.33120e6i q^{77} -868771. i q^{78} -7.05834e6 q^{79} +4.21423e6 q^{81} +2.45108e6i q^{82} -6.60874e6i q^{83} +205765. q^{84} +6.24604e6 q^{86} +247851. i q^{87} -1.98710e6i q^{88} +5.32998e6 q^{89} -3.97385e6 q^{91} +3.59335e6i q^{92} -426005. i q^{93} +4.24994e6 q^{94} -307148. q^{96} -2.09810e6i q^{97} -941192. i q^{98} +8.14686e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 256 q^{4} + 1120 q^{6} - 4028 q^{9} - 6840 q^{11} - 10976 q^{14} + 16384 q^{16} + 86716 q^{19} + 48020 q^{21} - 71680 q^{24} - 102368 q^{26} - 319152 q^{29} - 287224 q^{31} + 615552 q^{34} + 257792 q^{36}+ \cdots + 76354200 q^{99}+O(q^{100}) $ 4 * q - 256 * q^4 + 1120 * q^6 - 4028 * q^9 - 6840 * q^11 - 10976 * q^14 + 16384 * q^16 + 86716 * q^19 + 48020 * q^21 - 71680 * q^24 - 102368 * q^26 - 319152 * q^29 - 287224 * q^31 + 615552 * q^34 + 257792 * q^36 - 1040704 * q^39 + 129696 * q^41 + 437760 * q^44 + 1438848 * q^46 - 470596 * q^49 - 3401880 * q^51 - 3088960 * q^54 + 702464 * q^56 + 8367324 * q^59 - 561316 * q^61 - 1048576 * q^64 - 9854208 * q^66 - 4310208 * q^69 - 1238544 * q^71 + 4349312 * q^74 - 5549824 * q^76 - 9313232 * q^79 + 14707484 * q^81 - 3073280 * q^84 + 24447424 * q^86 + 34482840 * q^89 - 4389028 * q^91 - 7766976 * q^94 + 4587520 * q^96 + 76354200 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/350\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$	8.00000i	0.707107i
$2$	8.00000i	0.707107i
$3$	9.37342i	0.200435i	0.994966	+	0.100217i	$0.0319539\pi$
$3$	9.37342i	0.200435i	−0.994966	+	0.100217i	$0.968046\pi$
$4$	−64.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	−74.9873	−0.141729
$7$	343.000i	0.377964i
$7$	343.000i	0.377964i
$8$	− 512.000i	− 0.353553i
$9$	2099.14	0.959826
$10$	0	0
$11$	3881.05	0.879174	0.439587	−	0.898200i	$-0.355125\pi$
$11$	3881.05	0.879174	0.439587	+	0.898200i	$0.355125\pi$
$12$	− 599.899i	− 0.100217i
$13$	11585.6i	1.46257i	0.682073	+	0.731284i	$0.261078\pi$
$13$	11585.6i	1.46257i	−0.682073	+	0.731284i	$0.738922\pi$
$14$	−2744.00	−0.267261
$15$	0	0
$16$	4096.00	0.250000
$17$	− 15242.4i	− 0.752457i	−0.926527	−	0.376229i	$-0.877221\pi$
$17$	− 15242.4i	− 0.752457i	0.926527	−	0.376229i	$-0.122779\pi$
$18$	16793.1i	0.678699i
$19$	32461.7	1.08576	0.542880	−	0.839810i	$-0.317334\pi$
$19$	32461.7	1.08576	0.542880	+	0.839810i	$0.317334\pi$
$20$	0	0
$21$	−3215.08	−0.0757573
$22$	31048.4i	0.621670i
$23$	− 56146.1i	− 0.962215i	−0.876662	−	0.481108i	$-0.840235\pi$
$23$	− 56146.1i	− 0.962215i	0.876662	−	0.481108i	$-0.159765\pi$
$24$	4799.19	0.0708645
$25$	0	0
$26$	−92684.6	−1.03419
$27$	40175.8i	0.392818i
$28$	− 21952.0i	− 0.188982i
$29$	26442.0	0.201326	0.100663	−	0.994921i	$-0.467904\pi$
$29$	26442.0	0.201326	0.100663	+	0.994921i	$0.467904\pi$
$30$	0	0
$31$	−45448.2	−0.274000	−0.137000	−	0.990571i	$-0.543746\pi$
$31$	−45448.2	−0.274000	−0.137000	+	0.990571i	$0.543746\pi$
$32$	32768.0i	0.176777i
$33$	36378.7i	0.176217i
$34$	121939.	0.532068
$35$	0	0
$36$	−134345.	−0.479913
$37$	− 555245.i	− 1.80210i	−0.433717	−	0.901049i	$-0.642798\pi$
$37$	− 555245.i	− 1.80210i	0.433717	−	0.901049i	$-0.357202\pi$
$38$	259694.i	0.767749i
$39$	−108596.	−0.293150
$40$	0	0
$41$	306385.	0.694264	0.347132	−	0.937816i	$-0.387156\pi$
$41$	306385.	0.694264	0.347132	+	0.937816i	$0.387156\pi$
$42$	− 25720.7i	− 0.0535685i
$43$	− 780755.i	− 1.49753i	−0.662836	−	0.748765i	$-0.730647\pi$
$43$	− 780755.i	− 1.49753i	0.662836	−	0.748765i	$-0.269353\pi$
$44$	−248387.	−0.439587
$45$	0	0
$46$	449169.	0.680389
$47$	− 531243.i	− 0.746364i	−0.927758	−	0.373182i	$-0.878267\pi$
$47$	− 531243.i	− 0.746364i	0.927758	−	0.373182i	$-0.121733\pi$
$48$	38393.5i	0.0501087i
$49$	−117649.	−0.142857
$50$	0	0
$51$	142873.	0.150819
$52$	− 741477.i	− 0.731284i
$53$	363593.i	0.335467i	0.985832	+	0.167733i	$0.0536448\pi$
$53$	363593.i	0.335467i	−0.985832	+	0.167733i	$0.946355\pi$
$54$	−321406.	−0.277764
$55$	0	0
$56$	175616.	0.133631
$57$	304277.i	0.217624i
$58$	211536.i	0.142359i
$59$	2.14095e6	1.35714	0.678570	−	0.734535i	$-0.262600\pi$
$59$	2.14095e6	1.35714	0.678570	+	0.734535i	$0.262600\pi$
$60$	0	0
$61$	888824.	0.501373	0.250687	−	0.968068i	$-0.419344\pi$
$61$	888824.	0.501373	0.250687	+	0.968068i	$0.419344\pi$
$62$	− 363586.i	− 0.193747i
$63$	720005.i	0.362780i
$64$	−262144.	−0.125000
$65$	0	0
$66$	−291030.	−0.124604
$67$	4.34541e6i	1.76510i	0.470221	+	0.882549i	$0.344174\pi$
$67$	4.34541e6i	1.76510i	−0.470221	+	0.882549i	$0.655826\pi$
$68$	975513.i	0.376229i
$69$	526281.	0.192862
$70$	0	0
$71$	663207.	0.219910	0.109955	−	0.993937i	$-0.464929\pi$
$71$	663207.	0.219910	0.109955	+	0.993937i	$0.464929\pi$
$72$	− 1.07476e6i	− 0.339350i
$73$	− 1.34202e6i	− 0.403765i	−0.979410	−	0.201882i	$-0.935294\pi$
$73$	− 1.34202e6i	− 0.403765i	0.979410	−	0.201882i	$-0.0647059\pi$
$74$	4.44196e6	1.27428
$75$	0	0
$76$	−2.07755e6	−0.542880
$77$	1.33120e6i	0.332297i
$78$	− 868771.i	− 0.207288i
$79$	−7.05834e6	−1.61067	−0.805337	−	0.592818i	$-0.798016\pi$
$79$	−7.05834e6	−1.61067	−0.805337	+	0.592818i	$0.798016\pi$
$80$	0	0
$81$	4.21423e6	0.881091
$82$	2.45108e6i	0.490919i
$83$	− 6.60874e6i	− 1.26866i	−0.773063	−	0.634330i	$-0.781276\pi$
$83$	− 6.60874e6i	− 1.26866i	0.773063	−	0.634330i	$-0.218724\pi$
$84$	205765.	0.0378786
$85$	0	0
$86$	6.24604e6	1.05891
$87$	247851.i	0.0403528i
$88$	− 1.98710e6i	− 0.310835i
$89$	5.32998e6	0.801420	0.400710	−	0.916205i	$-0.368763\pi$
$89$	5.32998e6	0.801420	0.400710	+	0.916205i	$0.368763\pi$
$90$	0	0
$91$	−3.97385e6	−0.552799
$92$	3.59335e6i	0.481108i
$93$	− 426005.i	− 0.0549192i
$94$	4.24994e6	0.527759
$95$	0	0
$96$	−307148.	−0.0354322
$97$	− 2.09810e6i	− 0.233413i	−0.993166	−	0.116707i	$-0.962766\pi$
$97$	− 2.09810e6i	− 0.233413i	0.993166	−	0.116707i	$-0.0372337\pi$
$98$	− 941192.i	− 0.101015i
$99$	8.14686e6	0.843854
$100$	0	0
$101$	1.95538e7	1.88845	0.944227	−	0.329294i	$-0.106811\pi$
$101$	1.95538e7	1.88845	0.944227	+	0.329294i	$0.106811\pi$
$102$	1.14299e6i	0.106645i
$103$	8.95093e6i	0.807119i	0.914953	+	0.403560i	$0.132227\pi$
$103$	8.95093e6i	0.807119i	−0.914953	+	0.403560i	$0.867773\pi$
$104$	5.93181e6	0.517096
$105$	0	0
$106$	−2.90874e6	−0.237211
$107$	− 1.39509e7i	− 1.10093i	−0.834858	−	0.550466i	$-0.814450\pi$
$107$	− 1.39509e7i	− 1.10093i	0.834858	−	0.550466i	$-0.185550\pi$
$108$	− 2.57125e6i	− 0.196409i
$109$	−5.84295e6	−0.432155	−0.216077	−	0.976376i	$-0.569326\pi$
$109$	−5.84295e6	−0.432155	−0.216077	+	0.976376i	$0.569326\pi$
$110$	0	0
$111$	5.20454e6	0.361203
$112$	1.40493e6i	0.0944911i
$113$	1.20951e7i	0.788562i	0.918990	+	0.394281i	$0.129006\pi$
$113$	1.20951e7i	0.788562i	−0.918990	+	0.394281i	$0.870994\pi$
$114$	−2.43422e6	−0.153884
$115$	0	0
$116$	−1.69229e6	−0.100663
$117$	2.43197e7i	1.40381i
$118$	1.71276e7i	0.959644i
$119$	5.22814e6	0.284402
$120$	0	0
$121$	−4.42462e6	−0.227053
$122$	7.11059e6i	0.354524i
$123$	2.87188e6i	0.139155i
$124$	2.90868e6	0.137000
$125$	0	0
$126$	−5.76004e6	−0.256524
$127$	− 325760.i	− 0.0141119i	−0.999975	−	0.00705594i	$-0.997754\pi$
$127$	− 325760.i	− 0.0141119i	0.999975	−	0.00705594i	$-0.00224599\pi$
$128$	− 2.09715e6i	− 0.0883883i
$129$	7.31834e6	0.300157
$130$	0	0
$131$	6.38049e6	0.247973	0.123986	−	0.992284i	$-0.460432\pi$
$131$	6.38049e6	0.247973	0.123986	+	0.992284i	$0.460432\pi$
$132$	− 2.32824e6i	− 0.0881086i
$133$	1.11344e7i	0.410379i
$134$	−3.47633e7	−1.24811
$135$	0	0
$136$	−7.80411e6	−0.266034
$137$	4.62324e6i	0.153612i	0.997046	+	0.0768059i	$0.0244722\pi$
$137$	4.62324e6i	0.153612i	−0.997046	+	0.0768059i	$0.975528\pi$
$138$	4.21025e6i	0.136374i
$139$	3.55394e7	1.12243	0.561214	−	0.827671i	$-0.310335\pi$
$139$	3.55394e7	1.12243	0.561214	+	0.827671i	$0.310335\pi$
$140$	0	0
$141$	4.97956e6	0.149597
$142$	5.30565e6i	0.155500i
$143$	4.49642e7i	1.28585i
$144$	8.59807e6	0.239956
$145$	0	0
$146$	1.07362e7	0.285505
$147$	− 1.10277e6i	− 0.0286336i
$148$	3.55357e7i	0.901049i
$149$	−1.01708e7	−0.251885	−0.125943	−	0.992038i	$-0.540196\pi$
$149$	−1.01708e7	−0.251885	−0.125943	+	0.992038i	$0.540196\pi$
$150$	0	0
$151$	2.45272e7	0.579735	0.289867	−	0.957067i	$-0.406389\pi$
$151$	2.45272e7	0.579735	0.289867	+	0.957067i	$0.406389\pi$
$152$	− 1.66204e7i	− 0.383874i
$153$	− 3.19959e7i	− 0.722228i
$154$	−1.06496e7	−0.234969
$155$	0	0
$156$	6.95017e6	0.146575
$157$	− 1.27379e7i	− 0.262693i	−0.991337	−	0.131346i	$-0.958070\pi$
$157$	− 1.27379e7i	− 0.262693i	0.991337	−	0.131346i	$-0.0419300\pi$
$158$	− 5.64667e7i	− 1.13892i
$159$	−3.40810e6	−0.0672393
$160$	0	0
$161$	1.92581e7	0.363683
$162$	3.37139e7i	0.623026i
$163$	8.51531e7i	1.54008i	0.637995	+	0.770041i	$0.279764\pi$
$163$	8.51531e7i	1.54008i	−0.637995	+	0.770041i	$0.720236\pi$
$164$	−1.96087e7	−0.347132
$165$	0	0
$166$	5.28699e7	0.897078
$167$	− 3.74260e6i	− 0.0621822i	−0.999517	−	0.0310911i	$-0.990102\pi$
$167$	− 3.74260e6i	− 0.0621822i	0.999517	−	0.0310911i	$-0.00989820\pi$
$168$	1.64612e6i	0.0267842i
$169$	−7.14770e7	−1.13910
$170$	0	0
$171$	6.81417e7	1.04214
$172$	4.99683e7i	0.748765i
$173$	7.26865e7i	1.06731i	0.845701	+	0.533657i	$0.179183\pi$
$173$	7.26865e7i	1.06731i	−0.845701	+	0.533657i	$0.820817\pi$
$174$	−1.98281e6	−0.0285338
$175$	0	0
$176$	1.58968e7	0.219794
$177$	2.00680e7i	0.272018i
$178$	4.26398e7i	0.566690i
$179$	−1.02445e8	−1.33507	−0.667536	−	0.744578i	$-0.732651\pi$
$179$	−1.02445e8	−1.33507	−0.667536	+	0.744578i	$0.732651\pi$
$180$	0	0
$181$	4.79161e6	0.0600630	0.0300315	−	0.999549i	$-0.490439\pi$
$181$	4.79161e6	0.0600630	0.0300315	+	0.999549i	$0.490439\pi$
$182$	− 3.17908e7i	− 0.390888i
$183$	8.33131e6i	0.100493i
$184$	−2.87468e7	−0.340194
$185$	0	0
$186$	3.40804e6	0.0388337
$187$	− 5.91565e7i	− 0.661541i
$188$	3.39996e7i	0.373182i
$189$	−1.37803e7	−0.148471
$190$	0	0
$191$	2.60173e7	0.270175	0.135087	−	0.990834i	$-0.456868\pi$
$191$	2.60173e7	0.270175	0.135087	+	0.990834i	$0.456868\pi$
$192$	− 2.45718e6i	− 0.0250544i
$193$	7.58326e7i	0.759286i	0.925133	+	0.379643i	$0.123953\pi$
$193$	7.58326e7i	0.759286i	−0.925133	+	0.379643i	$0.876047\pi$
$194$	1.67848e7	0.165048
$195$	0	0
$196$	7.52954e6	0.0714286
$197$	1.19192e8i	1.11075i	0.831601	+	0.555374i	$0.187425\pi$
$197$	1.19192e8i	1.11075i	−0.831601	+	0.555374i	$0.812575\pi$
$198$	6.51749e7i	0.596695i
$199$	−6.72662e7	−0.605078	−0.302539	−	0.953137i	$-0.597834\pi$
$199$	−6.72662e7	−0.605078	−0.302539	+	0.953137i	$0.597834\pi$
$200$	0	0
$201$	−4.07313e7	−0.353787
$202$	1.56430e8i	1.33534i
$203$	9.06959e6i	0.0760942i
$204$	−9.14389e6	−0.0754094
$205$	0	0
$206$	−7.16074e7	−0.570720
$207$	− 1.17858e8i	− 0.923559i
$208$	4.74545e7i	0.365642i
$209$	1.25986e8	0.954573
$210$	0	0
$211$	2.32421e8	1.70328	0.851641	−	0.524126i	$-0.175608\pi$
$211$	2.32421e8	1.70328	0.851641	+	0.524126i	$0.175608\pi$
$212$	− 2.32699e7i	− 0.167733i
$213$	6.21651e6i	0.0440776i
$214$	1.11608e8	0.778476
$215$	0	0
$216$	2.05700e7	0.138882
$217$	− 1.55887e7i	− 0.103562i
$218$	− 4.67436e7i	− 0.305580i
$219$	1.25793e7	0.0809286
$220$	0	0
$221$	1.76592e8	1.10052
$222$	4.16363e7i	0.255409i
$223$	2.53319e8i	1.52968i	0.644221	+	0.764840i	$0.277182\pi$
$223$	2.53319e8i	1.52968i	−0.644221	+	0.764840i	$0.722818\pi$
$224$	−1.12394e7	−0.0668153
$225$	0	0
$226$	−9.67610e7	−0.557598
$227$	4.50124e7i	0.255412i	0.991812	+	0.127706i	$0.0407615\pi$
$227$	4.50124e7i	0.255412i	−0.991812	+	0.127706i	$0.959239\pi$
$228$	− 1.94738e7i	− 0.108812i
$229$	−1.86973e8	−1.02886	−0.514429	−	0.857533i	$-0.671996\pi$
$229$	−1.86973e8	−1.02886	−0.514429	+	0.857533i	$0.671996\pi$
$230$	0	0
$231$	−1.24779e7	−0.0666039
$232$	− 1.35383e7i	− 0.0711796i
$233$	2.57588e8i	1.33408i	0.745024	+	0.667038i	$0.232438\pi$
$233$	2.57588e8i	1.33408i	−0.745024	+	0.667038i	$0.767562\pi$
$234$	−1.94558e8	−0.992644
$235$	0	0
$236$	−1.37021e8	−0.678570
$237$	− 6.61607e7i	− 0.322835i
$238$	4.18251e7i	0.201103i
$239$	3.15071e8	1.49285	0.746425	−	0.665469i	$-0.231769\pi$
$239$	3.15071e8	1.49285	0.746425	+	0.665469i	$0.231769\pi$
$240$	0	0
$241$	8.39900e7	0.386516	0.193258	−	0.981148i	$-0.438095\pi$
$241$	8.39900e7	0.386516	0.193258	+	0.981148i	$0.438095\pi$
$242$	− 3.53970e7i	− 0.160551i
$243$	1.27366e8i	0.569419i
$244$	−5.68847e7	−0.250687
$245$	0	0
$246$	−2.29750e7	−0.0983973
$247$	3.76088e8i	1.58800i
$248$	2.32695e7i	0.0968737i
$249$	6.19464e7	0.254284
$250$	0	0
$251$	−5.33130e6	−0.0212802	−0.0106401	−	0.999943i	$-0.503387\pi$
$251$	−5.33130e6	−0.0212802	−0.0106401	+	0.999943i	$0.503387\pi$
$252$	− 4.60803e7i	− 0.181390i
$253$	− 2.17906e8i	− 0.845955i
$254$	2.60608e6	0.00997861
$255$	0	0
$256$	1.67772e7	0.0625000
$257$	− 2.09942e8i	− 0.771495i	−0.922604	−	0.385747i	$-0.873944\pi$
$257$	− 2.09942e8i	− 0.771495i	0.922604	−	0.385747i	$-0.126056\pi$
$258$	5.85467e7i	0.212243i
$259$	1.90449e8	0.681129
$260$	0	0
$261$	5.55053e7	0.193238
$262$	5.10439e7i	0.175343i
$263$	2.50805e8i	0.850143i	0.905160	+	0.425071i	$0.139751\pi$
$263$	2.50805e8i	0.850143i	−0.905160	+	0.425071i	$0.860249\pi$
$264$	1.86259e7	0.0623022
$265$	0	0
$266$	−8.90750e7	−0.290182
$267$	4.99601e7i	0.160633i
$268$	− 2.78106e8i	− 0.882549i
$269$	−3.39197e8	−1.06248	−0.531238	−	0.847223i	$-0.678273\pi$
$269$	−3.39197e8	−1.06248	−0.531238	+	0.847223i	$0.678273\pi$
$270$	0	0
$271$	7.59574e7	0.231834	0.115917	−	0.993259i	$-0.463019\pi$
$271$	7.59574e7	0.231834	0.115917	+	0.993259i	$0.463019\pi$
$272$	− 6.24328e7i	− 0.188114i
$273$	− 3.72486e7i	− 0.110800i
$274$	−3.69859e7	−0.108620
$275$	0	0
$276$	−3.36820e7	−0.0964308
$277$	− 5.00883e7i	− 0.141598i	−0.997491	−	0.0707990i	$-0.977445\pi$
$277$	− 5.00883e7i	− 0.141598i	0.997491	−	0.0707990i	$-0.0225549\pi$
$278$	2.84315e8i	0.793676i
$279$	−9.54021e7	−0.262992
$280$	0	0
$281$	3.41917e8	0.919283	0.459641	−	0.888105i	$-0.347978\pi$
$281$	3.41917e8	0.919283	0.459641	+	0.888105i	$0.347978\pi$
$282$	3.98365e7i	0.105781i
$283$	− 3.69077e8i	− 0.967977i	−0.875074	−	0.483988i	$-0.839188\pi$
$283$	− 3.69077e8i	− 0.967977i	0.875074	−	0.483988i	$-0.160812\pi$
$284$	−4.24452e7	−0.109955
$285$	0	0
$286$	−3.59714e8	−0.909234
$287$	1.05090e8i	0.262407i
$288$	6.87846e7i	0.169675i
$289$	1.78008e8	0.433808
$290$	0	0
$291$	1.96664e7	0.0467842
$292$	8.58892e7i	0.201882i
$293$	− 5.92780e8i	− 1.37676i	−0.725352	−	0.688378i	$-0.758323\pi$
$293$	− 5.92780e8i	− 1.37676i	0.725352	−	0.688378i	$-0.241677\pi$
$294$	8.82218e6	0.0202470
$295$	0	0
$296$	−2.84285e8	−0.637138
$297$	1.55924e8i	0.345355i
$298$	− 8.13663e7i	− 0.178110i
$299$	6.50485e8	1.40730
$300$	0	0
$301$	2.67799e8	0.566013
$302$	1.96218e8i	0.409934i
$303$	1.83286e8i	0.378512i
$304$	1.32963e8	0.271440
$305$	0	0
$306$	2.55967e8	0.510692
$307$	− 9.43846e8i	− 1.86173i	−0.365363	−	0.930865i	$-0.619055\pi$
$307$	− 9.43846e8i	− 1.86173i	0.365363	−	0.930865i	$-0.380945\pi$
$308$	− 8.51968e7i	− 0.166148i
$309$	−8.39008e7	−0.161775
$310$	0	0
$311$	−1.64065e8	−0.309282	−0.154641	−	0.987971i	$-0.549422\pi$
$311$	−1.64065e8	−0.309282	−0.154641	+	0.987971i	$0.549422\pi$
$312$	5.56014e7i	0.103644i
$313$	− 3.46012e8i	− 0.637801i	−0.947788	−	0.318901i	$-0.896686\pi$
$313$	− 3.46012e8i	− 0.637801i	0.947788	−	0.318901i	$-0.103314\pi$
$314$	1.01903e8	0.185752
$315$	0	0
$316$	4.51734e8	0.805337
$317$	− 9.53240e8i	− 1.68072i	−0.542030	−	0.840359i	$-0.682344\pi$
$317$	− 9.53240e8i	− 1.68072i	0.542030	−	0.840359i	$-0.317656\pi$
$318$	− 2.72648e7i	− 0.0475453i
$319$	1.02623e8	0.177001
$320$	0	0
$321$	1.30768e8	0.220665
$322$	1.54065e8i	0.257163i
$323$	− 4.94795e8i	− 0.816989i
$324$	−2.69711e8	−0.440546
$325$	0	0
$326$	−6.81225e8	−1.08900
$327$	− 5.47684e7i	− 0.0866189i
$328$	− 1.56869e8i	− 0.245459i
$329$	1.82216e8	0.282099
$330$	0	0
$331$	2.24263e8	0.339906	0.169953	−	0.985452i	$-0.445638\pi$
$331$	2.24263e8	0.339906	0.169953	+	0.985452i	$0.445638\pi$
$332$	4.22959e8i	0.634330i
$333$	− 1.16554e9i	− 1.72970i
$334$	2.99408e7	0.0439694
$335$	0	0
$336$	−1.31690e7	−0.0189393
$337$	3.56750e8i	0.507761i	0.967236	+	0.253880i	$0.0817069\pi$
$337$	3.56750e8i	0.507761i	−0.967236	+	0.253880i	$0.918293\pi$
$338$	− 5.71816e8i	− 0.805468i
$339$	−1.13373e8	−0.158055
$340$	0	0
$341$	−1.76387e8	−0.240894
$342$	5.45134e8i	0.736905i
$343$	− 4.03536e7i	− 0.0539949i
$344$	−3.99747e8	−0.529456
$345$	0	0
$346$	−5.81492e8	−0.754705
$347$	1.61839e7i	0.0207936i	0.999946	+	0.0103968i	$0.00330946\pi$
$347$	1.61839e7i	0.0207936i	−0.999946	+	0.0103968i	$0.996691\pi$
$348$	− 1.58625e7i	− 0.0201764i
$349$	−2.07780e8	−0.261647	−0.130823	−	0.991406i	$-0.541762\pi$
$349$	−2.07780e8	−0.261647	−0.130823	+	0.991406i	$0.541762\pi$
$350$	0	0
$351$	−4.65459e8	−0.574522
$352$	1.27174e8i	0.155417i
$353$	1.31624e9i	1.59266i	0.604865	+	0.796328i	$0.293227\pi$
$353$	1.31624e9i	1.59266i	−0.604865	+	0.796328i	$0.706773\pi$
$354$	−1.60544e8	−0.192346
$355$	0	0
$356$	−3.41119e8	−0.400710
$357$	4.90055e7i	0.0570041i
$358$	− 8.19559e8i	− 0.944038i
$359$	6.19271e8	0.706400	0.353200	−	0.935548i	$-0.385094\pi$
$359$	6.19271e8	0.706400	0.353200	+	0.935548i	$0.385094\pi$
$360$	0	0
$361$	1.59893e8	0.178877
$362$	3.83329e7i	0.0424709i
$363$	− 4.14738e7i	− 0.0455093i
$364$	2.54327e8	0.276399
$365$	0	0
$366$	−6.66505e7	−0.0710591
$367$	2.78628e8i	0.294235i	0.989119	+	0.147117i	$0.0469995\pi$
$367$	2.78628e8i	0.294235i	−0.989119	+	0.147117i	$0.953000\pi$
$368$	− 2.29974e8i	− 0.240554i
$369$	6.43146e8	0.666373
$370$	0	0
$371$	−1.24712e8	−0.126795
$372$	2.72643e7i	0.0274596i
$373$	1.65056e8i	0.164684i	0.996604	+	0.0823420i	$0.0262400\pi$
$373$	1.65056e8i	0.164684i	−0.996604	+	0.0823420i	$0.973760\pi$
$374$	4.73252e8	0.467780
$375$	0	0
$376$	−2.71996e8	−0.263880
$377$	3.06345e8i	0.294453i
$378$	− 1.10242e8i	− 0.104985i
$379$	−4.29433e8	−0.405190	−0.202595	−	0.979263i	$-0.564937\pi$
$379$	−4.29433e8	−0.405190	−0.202595	+	0.979263i	$0.564937\pi$
$380$	0	0
$381$	3.05348e6	0.00282851
$382$	2.08138e8i	0.191043i
$383$	− 1.34378e9i	− 1.22217i	−0.791566	−	0.611084i	$-0.790734\pi$
$383$	− 1.34378e9i	− 1.22217i	0.791566	−	0.611084i	$-0.209266\pi$
$384$	1.96575e7	0.0177161
$385$	0	0
$386$	−6.06661e8	−0.536897
$387$	− 1.63891e9i	− 1.43737i
$388$	1.34279e8i	0.116707i
$389$	1.07871e8	0.0929139	0.0464569	−	0.998920i	$-0.485207\pi$
$389$	1.07871e8	0.0929139	0.0464569	+	0.998920i	$0.485207\pi$
$390$	0	0
$391$	−8.55801e8	−0.724026
$392$	6.02363e7i	0.0505076i
$393$	5.98070e7i	0.0497024i
$394$	−9.53537e8	−0.785418
$395$	0	0
$396$	−5.21399e8	−0.421927
$397$	− 1.86525e9i	− 1.49613i	−0.663625	−	0.748066i	$-0.730983\pi$
$397$	− 1.86525e9i	− 1.49613i	0.663625	−	0.748066i	$-0.269017\pi$
$398$	− 5.38130e8i	− 0.427855i
$399$	−1.04367e8	−0.0822543
$400$	0	0
$401$	9.94724e6	0.00770367	0.00385183	−	0.999993i	$-0.498774\pi$
$401$	9.94724e6	0.00770367	0.00385183	+	0.999993i	$0.498774\pi$
$402$	− 3.25850e8i	− 0.250165i
$403$	− 5.26543e8i	− 0.400744i
$404$	−1.25144e9	−0.944227
$405$	0	0
$406$	−7.25567e7	−0.0538067
$407$	− 2.15493e9i	− 1.58436i
$408$	− 7.31511e7i	− 0.0533225i
$409$	8.76376e8	0.633372	0.316686	−	0.948530i	$-0.397430\pi$
$409$	8.76376e8	0.633372	0.316686	+	0.948530i	$0.397430\pi$
$410$	0	0
$411$	−4.33356e7	−0.0307892
$412$	− 5.72859e8i	− 0.403560i
$413$	7.34347e8i	0.512951i
$414$	9.42868e8	0.653055
$415$	0	0
$416$	−3.79636e8	−0.258548
$417$	3.33126e8i	0.224974i
$418$	1.00789e9i	0.674985i
$419$	2.01605e9	1.33891	0.669457	−	0.742851i	$-0.266527\pi$
$419$	2.01605e9	1.33891	0.669457	+	0.742851i	$0.266527\pi$
$420$	0	0
$421$	5.69546e8	0.371999	0.186000	−	0.982550i	$-0.440448\pi$
$421$	5.69546e8	0.371999	0.186000	+	0.982550i	$0.440448\pi$
$422$	1.85937e9i	1.20440i
$423$	− 1.11515e9i	− 0.716380i
$424$	1.86159e8	0.118605
$425$	0	0
$426$	−4.97321e7	−0.0311676
$427$	3.04867e8i	0.189501i
$428$	8.92860e8i	0.550466i
$429$	−4.21468e8	−0.257730
$430$	0	0
$431$	−7.71940e8	−0.464422	−0.232211	−	0.972665i	$-0.574596\pi$
$431$	−7.71940e8	−0.464422	−0.232211	+	0.972665i	$0.574596\pi$
$432$	1.64560e8i	0.0982044i
$433$	− 1.24002e9i	− 0.734043i	−0.930213	−	0.367021i	$-0.880378\pi$
$433$	− 1.24002e9i	− 0.734043i	0.930213	−	0.367021i	$-0.119622\pi$
$434$	1.24710e8	0.0732296
$435$	0	0
$436$	3.73949e8	0.216077
$437$	− 1.82260e9i	− 1.04474i
$438$	1.00634e8i	0.0572252i
$439$	−1.90621e9	−1.07534	−0.537669	−	0.843156i	$-0.680695\pi$
$439$	−1.90621e9	−1.07534	−0.537669	+	0.843156i	$0.680695\pi$
$440$	0	0
$441$	−2.46962e8	−0.137118
$442$	1.41274e9i	0.778185i
$443$	5.04738e8i	0.275837i	0.990444	+	0.137919i	$0.0440412\pi$
$443$	5.04738e8i	0.275837i	−0.990444	+	0.137919i	$0.955959\pi$
$444$	−3.33091e8	−0.180602
$445$	0	0
$446$	−2.02655e9	−1.08165
$447$	− 9.53351e7i	− 0.0504866i
$448$	− 8.99154e7i	− 0.0472456i
$449$	2.02721e9	1.05691	0.528453	−	0.848962i	$-0.322772\pi$
$449$	2.02721e9	1.05691	0.528453	+	0.848962i	$0.322772\pi$
$450$	0	0
$451$	1.18910e9	0.610379
$452$	− 7.74088e8i	− 0.394281i
$453$	2.29904e8i	0.116199i
$454$	−3.60100e8	−0.180604
$455$	0	0
$456$	1.55790e8	0.0769419
$457$	2.03969e9i	0.999672i	0.866120	+	0.499836i	$0.166606\pi$
$457$	2.03969e9i	0.999672i	−0.866120	+	0.499836i	$0.833394\pi$
$458$	− 1.49579e9i	− 0.727512i
$459$	6.12375e8	0.295579
$460$	0	0
$461$	−3.41390e9	−1.62292	−0.811461	−	0.584406i	$-0.801327\pi$
$461$	−3.41390e9	−1.62292	−0.811461	+	0.584406i	$0.801327\pi$
$462$	− 9.98231e7i	− 0.0470960i
$463$	1.97893e9i	0.926608i	0.886199	+	0.463304i	$0.153336\pi$
$463$	1.97893e9i	0.926608i	−0.886199	+	0.463304i	$0.846664\pi$
$464$	1.08306e8	0.0503316
$465$	0	0
$466$	−2.06071e9	−0.943334
$467$	1.86422e9i	0.847007i	0.905894	+	0.423504i	$0.139200\pi$
$467$	1.86422e9i	0.847007i	−0.905894	+	0.423504i	$0.860800\pi$
$468$	− 1.55646e9i	− 0.701905i
$469$	−1.49047e9	−0.667144
$470$	0	0
$471$	1.19397e8	0.0526528
$472$	− 1.09617e9i	− 0.479822i
$473$	− 3.03015e9i	− 1.31659i
$474$	5.29286e8	0.228279
$475$	0	0
$476$	−3.34601e8	−0.142201
$477$	7.63231e8i	0.321990i
$478$	2.52057e9i	1.05560i
$479$	−3.71828e9	−1.54585	−0.772926	−	0.634496i	$-0.781208\pi$
$479$	−3.71828e9	−1.54585	−0.772926	+	0.634496i	$0.781208\pi$
$480$	0	0
$481$	6.43283e9	2.63569
$482$	6.71920e8i	0.273308i
$483$	1.80514e8i	0.0728948i
$484$	2.83176e8	0.113526
$485$	0	0
$486$	−1.01893e9	−0.402640
$487$	− 2.16341e9i	− 0.848767i	−0.905483	−	0.424383i	$-0.860491\pi$
$487$	− 2.16341e9i	− 0.848767i	0.905483	−	0.424383i	$-0.139509\pi$
$488$	− 4.55078e8i	− 0.177262i
$489$	−7.98175e8	−0.308686
$490$	0	0
$491$	2.37608e8	0.0905890	0.0452945	−	0.998974i	$-0.485577\pi$
$491$	2.37608e8	0.0905890	0.0452945	+	0.998974i	$0.485577\pi$
$492$	− 1.83800e8i	− 0.0695774i
$493$	− 4.03039e8i	− 0.151489i
$494$	−3.00870e9	−1.12288
$495$	0	0
$496$	−1.86156e8	−0.0685000
$497$	2.27480e8i	0.0831181i
$498$	4.95571e8i	0.179806i
$499$	−2.65166e9	−0.955358	−0.477679	−	0.878535i	$-0.658522\pi$
$499$	−2.65166e9	−0.955358	−0.477679	+	0.878535i	$0.658522\pi$
$500$	0	0
$501$	3.50810e7	0.0124635
$502$	− 4.26504e7i	− 0.0150474i
$503$	− 2.15865e9i	− 0.756300i	−0.925744	−	0.378150i	$-0.876560\pi$
$503$	− 2.15865e9i	− 0.756300i	0.925744	−	0.378150i	$-0.123440\pi$
$504$	3.68642e8	0.128262
$505$	0	0
$506$	1.74325e9	0.598180
$507$	− 6.69984e8i	− 0.228316i
$508$	2.08486e7i	0.00705594i
$509$	−2.47831e9	−0.832998	−0.416499	−	0.909136i	$-0.636743\pi$
$509$	−2.47831e9	−0.832998	−0.416499	+	0.909136i	$0.636743\pi$
$510$	0	0
$511$	4.60312e8	0.152609
$512$	1.34218e8i	0.0441942i
$513$	1.30418e9i	0.426506i
$514$	1.67953e9	0.545529
$515$	0	0
$516$	−4.68374e8	−0.150079
$517$	− 2.06178e9i	− 0.656184i
$518$	1.52359e9i	0.481631i
$519$	−6.81321e8	−0.213927
$520$	0	0
$521$	−3.99735e9	−1.23834	−0.619170	−	0.785257i	$-0.712531\pi$
$521$	−3.99735e9	−1.23834	−0.619170	+	0.785257i	$0.712531\pi$
$522$	4.44043e8i	0.136640i
$523$	3.93132e9i	1.20166i	0.799376	+	0.600831i	$0.205163\pi$
$523$	3.93132e9i	1.20166i	−0.799376	+	0.600831i	$0.794837\pi$
$524$	−4.08351e8	−0.123986
$525$	0	0
$526$	−2.00644e9	−0.601142
$527$	6.92739e8i	0.206173i
$528$	1.49007e8i	0.0440543i
$529$	2.52441e8	0.0741421
$530$	0	0
$531$	4.49416e9	1.30262
$532$	− 7.12600e8i	− 0.205190i
$533$	3.54965e9i	1.01541i
$534$	−3.99681e8	−0.113584
$535$	0	0
$536$	2.22485e9	0.624056
$537$	− 9.60258e8i	− 0.267595i
$538$	− 2.71358e9i	− 0.751284i
$539$	−4.56602e8	−0.125596
$540$	0	0
$541$	−2.74766e9	−0.746057	−0.373028	−	0.927820i	$-0.621681\pi$
$541$	−2.74766e9	−0.746057	−0.373028	+	0.927820i	$0.621681\pi$
$542$	6.07659e8i	0.163932i
$543$	4.49138e7i	0.0120387i
$544$	4.99463e8	0.133017
$545$	0	0
$546$	2.97989e8	0.0783475
$547$	− 1.40581e9i	− 0.367258i	−0.982996	−	0.183629i	$-0.941216\pi$
$547$	− 1.40581e9i	− 0.367258i	0.982996	−	0.183629i	$-0.0587845\pi$
$548$	− 2.95887e8i	− 0.0768059i
$549$	1.86576e9	0.481231
$550$	0	0
$551$	8.58352e8	0.218592
$552$	− 2.69456e8i	− 0.0681869i
$553$	− 2.42101e9i	− 0.608777i
$554$	4.00707e8	0.100125
$555$	0	0
$556$	−2.27452e9	−0.561214
$557$	9.87805e8i	0.242202i	0.992640	+	0.121101i	$0.0386425\pi$
$557$	9.87805e8i	0.242202i	−0.992640	+	0.121101i	$0.961357\pi$
$558$	− 7.63217e8i	− 0.185964i
$559$	9.04550e9	2.19024
$560$	0	0
$561$	5.54498e8	0.132596
$562$	2.73534e9i	0.650031i
$563$	− 1.88115e9i	− 0.444268i	−0.975016	−	0.222134i	$-0.928698\pi$
$563$	− 1.88115e9i	− 0.444268i	0.975016	−	0.222134i	$-0.0713022\pi$
$564$	−3.18692e8	−0.0747987
$565$	0	0
$566$	2.95262e9	0.684463
$567$	1.44548e9i	0.333021i
$568$	− 3.39562e8i	− 0.0777499i
$569$	3.88531e9	0.884164	0.442082	−	0.896975i	$-0.354240\pi$
$569$	3.88531e9	0.884164	0.442082	+	0.896975i	$0.354240\pi$
$570$	0	0
$571$	2.44900e9	0.550505	0.275253	−	0.961372i	$-0.411238\pi$
$571$	2.44900e9	0.550505	0.275253	+	0.961372i	$0.411238\pi$
$572$	− 2.87771e9i	− 0.642926i
$573$	2.43871e8i	0.0541525i
$574$	−8.40722e8	−0.185550
$575$	0	0
$576$	−5.50277e8	−0.119978
$577$	2.88218e9i	0.624605i	0.949983	+	0.312303i	$0.101100\pi$
$577$	2.88218e9i	0.624605i	−0.949983	+	0.312303i	$0.898900\pi$
$578$	1.42407e9i	0.306748i
$579$	−7.10811e8	−0.152188
$580$	0	0
$581$	2.26680e9	0.479508
$582$	1.57331e8i	0.0330814i
$583$	1.41112e9i	0.294934i
$584$	−6.87114e8	−0.142752
$585$	0	0
$586$	4.74224e9	0.973513
$587$	4.72250e9i	0.963692i	0.876256	+	0.481846i	$0.160034\pi$
$587$	4.72250e9i	0.963692i	−0.876256	+	0.481846i	$0.839966\pi$
$588$	7.05775e7i	0.0143168i
$589$	−1.47533e9	−0.297499
$590$	0	0
$591$	−1.11724e9	−0.222633
$592$	− 2.27428e9i	− 0.450525i
$593$	− 1.23167e8i	− 0.0242550i	−0.999926	−	0.0121275i	$-0.996140\pi$
$593$	− 1.23167e8i	− 0.0242550i	0.999926	−	0.0121275i	$-0.00386040\pi$
$594$	−1.24739e9	−0.244203
$595$	0	0
$596$	6.50931e8	0.125943
$597$	− 6.30514e8i	− 0.121279i
$598$	5.20388e9i	0.995115i
$599$	−1.35497e9	−0.257594	−0.128797	−	0.991671i	$-0.541112\pi$
$599$	−1.35497e9	−0.257594	−0.128797	+	0.991671i	$0.541112\pi$
$600$	0	0
$601$	−5.25107e9	−0.986704	−0.493352	−	0.869830i	$-0.664228\pi$
$601$	−5.25107e9	−0.986704	−0.493352	+	0.869830i	$0.664228\pi$
$602$	2.14239e9i	0.400231i
$603$	9.12161e9i	1.69419i
$604$	−1.56974e9	−0.289867
$605$	0	0
$606$	−1.46629e9	−0.267649
$607$	4.99012e9i	0.905630i	0.891605	+	0.452815i	$0.149580\pi$
$607$	4.99012e9i	0.905630i	−0.891605	+	0.452815i	$0.850420\pi$
$608$	1.06371e9i	0.191937i
$609$	−8.50130e7	−0.0152519
$610$	0	0
$611$	6.15476e9	1.09161
$612$	2.04774e9i	0.361114i
$613$	2.46028e9i	0.431393i	0.976460	+	0.215696i	$0.0692022\pi$
$613$	2.46028e9i	0.431393i	−0.976460	+	0.215696i	$0.930798\pi$
$614$	7.55077e9	1.31644
$615$	0	0
$616$	6.81575e8	0.117485
$617$	− 1.93223e9i	− 0.331178i	−0.986195	−	0.165589i	$-0.947047\pi$
$617$	− 1.93223e9i	− 0.331178i	0.986195	−	0.165589i	$-0.0529525\pi$
$618$	− 6.71206e8i	− 0.114392i
$619$	8.13699e9	1.37894	0.689472	−	0.724313i	$-0.257843\pi$
$619$	8.13699e9	1.37894	0.689472	+	0.724313i	$0.257843\pi$
$620$	0	0
$621$	2.25571e9	0.377975
$622$	− 1.31252e9i	− 0.218695i
$623$	1.82818e9i	0.302908i
$624$	−4.44811e8	−0.0732874
$625$	0	0
$626$	2.76809e9	0.450994
$627$	1.18092e9i	0.191330i
$628$	8.15223e8i	0.131346i
$629$	−8.46326e9	−1.35600
$630$	0	0
$631$	8.83737e9	1.40030	0.700149	−	0.713997i	$-0.253117\pi$
$631$	8.83737e9	1.40030	0.700149	+	0.713997i	$0.253117\pi$
$632$	3.61387e9i	0.569459i
$633$	2.17858e9i	0.341397i
$634$	7.62592e9	1.18845
$635$	0	0
$636$	2.18119e8	0.0336196
$637$	− 1.36303e9i	− 0.208938i
$638$	8.20981e8i	0.125159i
$639$	1.39216e9	0.211075
$640$	0	0
$641$	−1.25905e10	−1.88817	−0.944086	−	0.329700i	$-0.893052\pi$
$641$	−1.25905e10	−1.88817	−0.944086	+	0.329700i	$0.893052\pi$
$642$	1.04614e9i	0.156034i
$643$	− 6.47578e9i	− 0.960625i	−0.877098	−	0.480312i	$-0.840523\pi$
$643$	− 6.47578e9i	− 0.960625i	0.877098	−	0.480312i	$-0.159477\pi$
$644$	−1.23252e9	−0.181842
$645$	0	0
$646$	3.95836e9	0.577698
$647$	8.69520e9i	1.26216i	0.775717	+	0.631081i	$0.217388\pi$
$647$	8.69520e9i	1.26216i	−0.775717	+	0.631081i	$0.782612\pi$
$648$	− 2.15769e9i	− 0.311513i
$649$	8.30914e9	1.19316
$650$	0	0
$651$	1.46120e8	0.0207575
$652$	− 5.44980e9i	− 0.770041i
$653$	3.87898e9i	0.545156i	0.962134	+	0.272578i	$0.0878763\pi$
$653$	3.87898e9i	0.545156i	−0.962134	+	0.272578i	$0.912124\pi$
$654$	4.38147e8	0.0612488
$655$	0	0
$656$	1.25495e9	0.173566
$657$	− 2.81708e9i	− 0.387544i
$658$	1.45773e9i	0.199474i
$659$	1.30333e9	0.177400	0.0887000	−	0.996058i	$-0.471729\pi$
$659$	1.30333e9	0.177400	0.0887000	+	0.996058i	$0.471729\pi$
$660$	0	0
$661$	6.85888e9	0.923736	0.461868	−	0.886949i	$-0.347179\pi$
$661$	6.85888e9	0.923736	0.461868	+	0.886949i	$0.347179\pi$
$662$	1.79410e9i	0.240350i
$663$	1.65527e9i	0.220583i
$664$	−3.38367e9	−0.448539
$665$	0	0
$666$	9.32429e9	1.22308
$667$	− 1.48461e9i	− 0.193719i
$668$	2.39527e8i	0.0310911i
$669$	−2.37446e9	−0.306601
$670$	0	0
$671$	3.44957e9	0.440794
$672$	− 1.05352e8i	− 0.0133921i
$673$	6.44953e9i	0.815596i	0.913072	+	0.407798i	$0.133703\pi$
$673$	6.44953e9i	0.815596i	−0.913072	+	0.407798i	$0.866297\pi$
$674$	−2.85400e9	−0.359041
$675$	0	0
$676$	4.57453e9	0.569552
$677$	3.83027e9i	0.474426i	0.971458	+	0.237213i	$0.0762340\pi$
$677$	3.83027e9i	0.474426i	−0.971458	+	0.237213i	$0.923766\pi$
$678$	− 9.06981e8i	− 0.111762i
$679$	7.19649e8	0.0882219
$680$	0	0
$681$	−4.21920e8	−0.0511936
$682$	− 1.41109e9i	− 0.170338i
$683$	1.97836e9i	0.237592i	0.992919	+	0.118796i	$0.0379035\pi$
$683$	1.97836e9i	0.237592i	−0.992919	+	0.118796i	$0.962097\pi$
$684$	−4.36107e9	−0.521071
$685$	0	0
$686$	3.22829e8	0.0381802
$687$	− 1.75258e9i	− 0.206219i
$688$	− 3.19797e9i	− 0.374382i
$689$	−4.21243e9	−0.490643
$690$	0	0
$691$	1.00255e10	1.15593	0.577967	−	0.816060i	$-0.303846\pi$
$691$	1.00255e10	1.15593	0.577967	+	0.816060i	$0.303846\pi$
$692$	− 4.65193e9i	− 0.533657i
$693$	2.79437e9i	0.318947i
$694$	−1.29471e8	−0.0147033
$695$	0	0
$696$	1.26900e8	0.0142669
$697$	− 4.67005e9i	− 0.522404i
$698$	− 1.66224e9i	− 0.185012i
$699$	−2.41448e9	−0.267395
$700$	0	0
$701$	1.21616e9	0.133345	0.0666727	−	0.997775i	$-0.478762\pi$
$701$	1.21616e9	0.133345	0.0666727	+	0.997775i	$0.478762\pi$
$702$	− 3.72367e9i	− 0.406249i
$703$	− 1.80242e10i	− 1.95665i
$704$	−1.01739e9	−0.109897
$705$	0	0
$706$	−1.05299e10	−1.12618
$707$	6.70696e9i	0.713769i
$708$	− 1.28435e9i	− 0.136009i
$709$	−1.04706e10	−1.10334	−0.551671	−	0.834062i	$-0.686010\pi$
$709$	−1.04706e10	−1.10334	−0.551671	+	0.834062i	$0.686010\pi$
$710$	0	0
$711$	−1.48164e10	−1.54597
$712$	− 2.72895e9i	− 0.283345i
$713$	2.55174e9i	0.263647i
$714$	−3.92044e8	−0.0403080
$715$	0	0
$716$	6.55647e9	0.667536
$717$	2.95330e9i	0.299219i
$718$	4.95417e9i	0.499500i
$719$	2.99586e9	0.300587	0.150294	−	0.988641i	$-0.451978\pi$
$719$	2.99586e9	0.300587	0.150294	+	0.988641i	$0.451978\pi$
$720$	0	0
$721$	−3.07017e9	−0.305062
$722$	1.27914e9i	0.126485i
$723$	7.87273e8i	0.0774714i
$724$	−3.06663e8	−0.0300315
$725$	0	0
$726$	3.31790e8	0.0321800
$727$	− 6.52469e9i	− 0.629781i	−0.949128	−	0.314890i	$-0.898032\pi$
$727$	− 6.52469e9i	− 0.629781i	0.949128	−	0.314890i	$-0.101968\pi$
$728$	2.03461e9i	0.195444i
$729$	8.02267e9	0.766960
$730$	0	0
$731$	−1.19006e10	−1.12683
$732$	− 5.33204e8i	− 0.0502464i
$733$	1.19856e10i	1.12408i	0.827110	+	0.562040i	$0.189983\pi$
$733$	1.19856e10i	1.12408i	−0.827110	+	0.562040i	$0.810017\pi$
$734$	−2.22903e9	−0.208056
$735$	0	0
$736$	1.83980e9	0.170097
$737$	1.68647e10i	1.55183i
$738$	5.14517e9i	0.471197i
$739$	−1.81843e10	−1.65746	−0.828728	−	0.559651i	$-0.810935\pi$
$739$	−1.81843e10	−1.65746	−0.828728	+	0.559651i	$0.810935\pi$
$740$	0	0
$741$	−3.52523e9	−0.318290
$742$	− 9.97698e8i	− 0.0896573i
$743$	− 6.10111e9i	− 0.545692i	−0.962058	−	0.272846i	$-0.912035\pi$
$743$	− 6.10111e9i	− 0.545692i	0.962058	−	0.272846i	$-0.0879650\pi$
$744$	−2.18114e8	−0.0194169
$745$	0	0
$746$	−1.32045e9	−0.116449
$747$	− 1.38727e10i	− 1.21769i
$748$	3.78602e9i	0.330771i
$749$	4.78517e9	0.416113
$750$	0	0
$751$	−3.69384e9	−0.318228	−0.159114	−	0.987260i	$-0.550864\pi$
$751$	−3.69384e9	−0.318228	−0.159114	+	0.987260i	$0.550864\pi$
$752$	− 2.17597e9i	− 0.186591i
$753$	− 4.99725e7i	− 0.00426529i
$754$	−2.45076e9	−0.208210
$755$	0	0
$756$	8.81938e8	0.0742356
$757$	1.35894e10i	1.13858i	0.822135	+	0.569292i	$0.192783\pi$
$757$	1.35894e10i	1.13858i	−0.822135	+	0.569292i	$0.807217\pi$
$758$	− 3.43546e9i	− 0.286512i
$759$	2.04252e9	0.169559
$760$	0	0
$761$	1.79994e10	1.48051	0.740255	−	0.672326i	$-0.234705\pi$
$761$	1.79994e10	1.48051	0.740255	+	0.672326i	$0.234705\pi$
$762$	2.44279e7i	0.00200006i
$763$	− 2.00413e9i	− 0.163339i
$764$	−1.66511e9	−0.135087
$765$	0	0
$766$	1.07502e10	0.864204
$767$	2.48042e10i	1.98491i
$768$	1.57260e8i	0.0125272i
$769$	−1.41340e10	−1.12079	−0.560394	−	0.828226i	$-0.689350\pi$
$769$	−1.41340e10	−1.12079	−0.560394	+	0.828226i	$0.689350\pi$
$770$	0	0
$771$	1.96787e9	0.154634
$772$	− 4.85329e9i	− 0.379643i
$773$	7.39484e9i	0.575838i	0.957655	+	0.287919i	$0.0929634\pi$
$773$	7.39484e9i	0.575838i	−0.957655	+	0.287919i	$0.907037\pi$
$774$	1.31113e10	1.01637
$775$	0	0
$776$	−1.07423e9	−0.0825241
$777$	1.78516e9i	0.136522i
$778$	8.62967e8i	0.0657000i
$779$	9.94581e9	0.753805
$780$	0	0
$781$	2.57394e9	0.193339
$782$	− 6.84641e9i	− 0.511964i
$783$	1.06233e9i	0.0790845i
$784$	−4.81890e8	−0.0357143
$785$	0	0
$786$	−4.78456e8	−0.0351449
$787$	− 9.49146e9i	− 0.694099i	−0.937847	−	0.347050i	$-0.887184\pi$
$787$	− 9.49146e9i	− 0.694099i	0.937847	−	0.347050i	$-0.112816\pi$
$788$	− 7.62829e9i	− 0.555374i
$789$	−2.35090e9	−0.170398
$790$	0	0
$791$	−4.14863e9	−0.298049
$792$	− 4.17119e9i	− 0.298347i
$793$	1.02975e10i	0.733292i
$794$	1.49220e10	1.05792
$795$	0	0
$796$	4.30504e9	0.302539
$797$	− 8.83350e9i	− 0.618057i	−0.951053	−	0.309029i	$-0.899996\pi$
$797$	− 8.83350e9i	− 0.618057i	0.951053	−	0.309029i	$-0.100004\pi$
$798$	− 8.34937e8i	− 0.0581626i
$799$	−8.09742e9	−0.561607
$800$	0	0
$801$	1.11884e10	0.769224
$802$	7.95779e7i	0.00544731i
$803$	− 5.20844e9i	− 0.354980i
$804$	2.60680e9	0.176894
$805$	0	0
$806$	4.21235e9	0.283368
$807$	− 3.17943e9i	− 0.212957i
$808$	− 1.00116e10i	− 0.667670i
$809$	−6.15622e9	−0.408784	−0.204392	−	0.978889i	$-0.565522\pi$
$809$	−6.15622e9	−0.408784	−0.204392	+	0.978889i	$0.565522\pi$
$810$	0	0
$811$	−1.53945e10	−1.01343	−0.506715	−	0.862114i	$-0.669140\pi$
$811$	−1.53945e10	−1.01343	−0.506715	+	0.862114i	$0.669140\pi$
$812$	− 5.80454e8i	− 0.0380471i
$813$	7.11980e8i	0.0464677i
$814$	1.72395e10	1.12031
$815$	0	0
$816$	5.85209e8	0.0377047
$817$	− 2.53447e10i	− 1.62596i
$818$	7.01101e9i	0.447862i
$819$	−8.34167e9	−0.530590
$820$	0	0
$821$	−2.74907e10	−1.73374	−0.866871	−	0.498532i	$-0.833872\pi$
$821$	−2.74907e10	−1.73374	−0.866871	+	0.498532i	$0.833872\pi$
$822$	− 3.46684e8i	− 0.0217712i
$823$	− 1.61300e9i	− 0.100864i	−0.998728	−	0.0504318i	$-0.983940\pi$
$823$	− 1.61300e9i	− 0.100864i	0.998728	−	0.0504318i	$-0.0160597\pi$
$824$	4.58288e9	0.285360
$825$	0	0
$826$	−5.87477e9	−0.362711
$827$	9.03950e9i	0.555745i	0.960618	+	0.277872i	$0.0896292\pi$
$827$	9.03950e9i	0.555745i	−0.960618	+	0.277872i	$0.910371\pi$
$828$	7.54294e9i	0.461779i
$829$	−2.46326e10	−1.50165	−0.750826	−	0.660501i	$-0.770344\pi$
$829$	−2.46326e10	−1.50165	−0.750826	+	0.660501i	$0.770344\pi$
$830$	0	0
$831$	4.69499e8	0.0283812
$832$	− 3.03709e9i	− 0.182821i
$833$	1.79325e9i	0.107494i
$834$	−2.66500e9	−0.159080
$835$	0	0
$836$	−8.06308e9	−0.477286
$837$	− 1.82592e9i	− 0.107632i
$838$	1.61284e10i	0.946755i
$839$	6.84347e9	0.400046	0.200023	−	0.979791i	$-0.435898\pi$
$839$	6.84347e9	0.400046	0.200023	+	0.979791i	$0.435898\pi$
$840$	0	0
$841$	−1.65507e10	−0.959468
$842$	4.55637e9i	0.263043i
$843$	3.20493e9i	0.184256i
$844$	−1.48749e10	−0.851641
$845$	0	0
$846$	8.92123e9	0.506557
$847$	− 1.51764e9i	− 0.0858179i
$848$	1.48928e9i	0.0838667i
$849$	3.45952e9	0.194016
$850$	0	0
$851$	−3.11748e10	−1.73401
$852$	− 3.97857e8i	− 0.0220388i
$853$	− 3.54072e10i	− 1.95330i	−0.214826	−	0.976652i	$-0.568919\pi$
$853$	− 3.54072e10i	− 1.95330i	0.214826	−	0.976652i	$-0.431081\pi$
$854$	−2.43893e9	−0.133998
$855$	0	0
$856$	−7.14288e9	−0.389238
$857$	− 1.87299e10i	− 1.01649i	−0.861213	−	0.508245i	$-0.830294\pi$
$857$	− 1.87299e10i	− 1.01649i	0.861213	−	0.508245i	$-0.169706\pi$
$858$	− 3.37175e9i	− 0.182242i
$859$	5.38272e9	0.289752	0.144876	−	0.989450i	$-0.453722\pi$
$859$	5.38272e9	0.289752	0.144876	+	0.989450i	$0.453722\pi$
$860$	0	0
$861$	−9.85054e8	−0.0525956
$862$	− 6.17552e9i	− 0.328396i
$863$	− 7.29444e9i	− 0.386326i	−0.981167	−	0.193163i	$-0.938125\pi$
$863$	− 7.29444e9i	− 0.386326i	0.981167	−	0.193163i	$-0.0618747\pi$
$864$	−1.31648e9	−0.0694410
$865$	0	0
$866$	9.92016e9	0.519046
$867$	1.66854e9i	0.0869503i
$868$	9.97679e8i	0.0517811i
$869$	−2.73938e10	−1.41606
$870$	0	0
$871$	−5.03440e10	−2.58157
$872$	2.99159e9i	0.152790i
$873$	− 4.40421e9i	− 0.224036i
$874$	1.45808e10	0.738740
$875$	0	0
$876$	−8.05075e8	−0.0404643
$877$	3.05623e10i	1.52998i	0.644039	+	0.764992i	$0.277257\pi$
$877$	3.05623e10i	1.52998i	−0.644039	+	0.764992i	$0.722743\pi$
$878$	− 1.52497e10i	− 0.760379i
$879$	5.55637e9	0.275950
$880$	0	0
$881$	−3.10736e10	−1.53100	−0.765500	−	0.643436i	$-0.777508\pi$
$881$	−3.10736e10	−1.53100	−0.765500	+	0.643436i	$0.777508\pi$
$882$	− 1.97569e9i	− 0.0969571i
$883$	− 9.10432e9i	− 0.445026i	−0.974930	−	0.222513i	$-0.928574\pi$
$883$	− 9.10432e9i	− 0.445026i	0.974930	−	0.222513i	$-0.0714259\pi$
$884$	−1.13019e10	−0.550260
$885$	0	0
$886$	−4.03790e9	−0.195046
$887$	− 5.03417e8i	− 0.0242212i	−0.999927	−	0.0121106i	$-0.996145\pi$
$887$	− 5.03417e8i	− 0.0242212i	0.999927	−	0.0121106i	$-0.00385501\pi$
$888$	− 2.66472e9i	− 0.127705i
$889$	1.11736e8	0.00533379
$890$	0	0
$891$	1.63557e10	0.774633
$892$	− 1.62124e10i	− 0.764840i
$893$	− 1.72451e10i	− 0.810373i
$894$	7.62680e8	0.0356994
$895$	0	0
$896$	7.19323e8	0.0334077
$897$	6.09727e9i	0.282073i
$898$	1.62177e10i	0.747346i
$899$	−1.20174e9	−0.0551634
$900$	0	0
$901$	5.54202e9	0.252424
$902$	9.51278e9i	0.431603i
$903$	2.51019e9i	0.113449i
$904$	6.19271e9	0.278799
$905$	0	0
$906$	−1.83923e9	−0.0821652
$907$	3.55392e10i	1.58155i	0.612108	+	0.790774i	$0.290322\pi$
$907$	3.55392e10i	1.58155i	−0.612108	+	0.790774i	$0.709678\pi$
$908$	− 2.88080e9i	− 0.127706i
$909$	4.10462e10	1.81259
$910$	0	0
$911$	3.60841e10	1.58125	0.790627	−	0.612299i	$-0.209755\pi$
$911$	3.60841e10	1.58125	0.790627	+	0.612299i	$0.209755\pi$
$912$	1.24632e9i	0.0544061i
$913$	− 2.56488e10i	− 1.11537i
$914$	−1.63175e10	−0.706875
$915$	0	0
$916$	1.19663e10	0.514429
$917$	2.18851e9i	0.0937250i
$918$	4.89900e9i	0.209006i
$919$	7.06975e9	0.300469	0.150235	−	0.988650i	$-0.451997\pi$
$919$	7.06975e9	0.300469	0.150235	+	0.988650i	$0.451997\pi$
$920$	0	0
$921$	8.84706e9	0.373156
$922$	− 2.73112e10i	− 1.14758i
$923$	7.68363e9i	0.321633i
$924$	7.98585e8	0.0333019
$925$	0	0
$926$	−1.58314e10	−0.655211
$927$	1.87892e10i	0.774694i
$928$	8.66450e8i	0.0355898i
$929$	1.55102e10	0.634691	0.317345	−	0.948310i	$-0.397209\pi$
$929$	1.55102e10	0.634691	0.317345	+	0.948310i	$0.397209\pi$
$930$	0	0
$931$	−3.81909e9	−0.155109
$932$	− 1.64856e10i	− 0.667038i
$933$	− 1.53785e9i	− 0.0619910i
$934$	−1.49137e10	−0.598925
$935$	0	0
$936$	1.24517e10	0.496322
$937$	− 3.61814e10i	− 1.43680i	−0.695630	−	0.718400i	$-0.744875\pi$
$937$	− 3.61814e10i	− 1.43680i	0.695630	−	0.718400i	$-0.255125\pi$
$938$	− 1.19238e10i	− 0.471742i
$939$	3.24331e9	0.127838
$940$	0	0
$941$	1.86518e10	0.729720	0.364860	−	0.931062i	$-0.381117\pi$
$941$	1.86518e10	0.729720	0.364860	+	0.931062i	$0.381117\pi$
$942$	9.55178e8i	0.0372311i
$943$	− 1.72023e10i	− 0.668031i
$944$	8.76934e9	0.339285
$945$	0	0
$946$	2.42412e10	0.930969
$947$	8.55630e9i	0.327387i	0.986511	+	0.163693i	$0.0523407\pi$
$947$	8.55630e9i	0.327387i	−0.986511	+	0.163693i	$0.947659\pi$
$948$	4.23429e9i	0.161418i
$949$	1.55481e10	0.590533
$950$	0	0
$951$	8.93512e9	0.336875
$952$	− 2.67681e9i	− 0.100551i
$953$	5.92820e8i	0.0221870i	0.999938	+	0.0110935i	$0.00353124\pi$
$953$	5.92820e8i	0.0221870i	−0.999938	+	0.0110935i	$0.996469\pi$
$954$	−6.10585e9	−0.227681
$955$	0	0
$956$	−2.01646e10	−0.746425
$957$	9.61924e8i	0.0354772i
$958$	− 2.97462e10i	− 1.09308i
$959$	−1.58577e9	−0.0580598
$960$	0	0
$961$	−2.54471e10	−0.924924
$962$	5.14626e10i	1.86371i
$963$	− 2.92850e10i	− 1.05670i
$964$	−5.37536e9	−0.193258
$965$	0	0
$966$	−1.44411e9	−0.0515444
$967$	− 5.25393e10i	− 1.86849i	−0.356626	−	0.934247i	$-0.616073\pi$
$967$	− 5.25393e10i	− 1.86849i	0.356626	−	0.934247i	$-0.383927\pi$
$968$	2.26541e9i	0.0802753i
$969$	4.63792e9	0.163753
$970$	0	0
$971$	−9.16576e9	−0.321293	−0.160647	−	0.987012i	$-0.551358\pi$
$971$	−9.16576e9	−0.321293	−0.160647	+	0.987012i	$0.551358\pi$
$972$	− 8.15143e9i	− 0.284710i
$973$	1.21900e10i	0.424238i
$974$	1.73073e10	0.600169
$975$	0	0
$976$	3.64062e9	0.125343
$977$	− 2.01603e10i	− 0.691618i	−0.938305	−	0.345809i	$-0.887604\pi$
$977$	− 2.01603e10i	− 0.691618i	0.938305	−	0.345809i	$-0.112396\pi$
$978$	− 6.38540e9i	− 0.218274i
$979$	2.06859e10	0.704588
$980$	0	0
$981$	−1.22652e10	−0.414793
$982$	1.90086e9i	0.0640561i
$983$	1.18989e9i	0.0399549i	0.999800	+	0.0199775i	$0.00635945\pi$
$983$	1.18989e9i	0.0399549i	−0.999800	+	0.0199775i	$0.993641\pi$
$984$	1.47040e9	0.0491986
$985$	0	0
$986$	3.22431e9	0.107119
$987$	1.70799e9i	0.0565425i
$988$	− 2.40696e10i	− 0.793999i
$989$	−4.38364e10	−1.44095
$990$	0	0
$991$	2.09986e10	0.685382	0.342691	−	0.939448i	$-0.388662\pi$
$991$	2.09986e10	0.685382	0.342691	+	0.939448i	$0.388662\pi$
$992$	− 1.48925e9i	− 0.0484368i
$993$	2.10211e9i	0.0681291i
$994$	−1.81984e9	−0.0587734
$995$	0	0
$996$	−3.96457e9	−0.127142
$997$	3.21657e10i	1.02792i	0.857814	+	0.513961i	$0.171822\pi$
$997$	3.21657e10i	1.02792i	−0.857814	+	0.513961i	$0.828178\pi$
$998$	− 2.12133e10i	− 0.675540i
$999$	2.23074e10	0.707896

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.8.c.k.99.4		4
5.2	odd	4		350.8.a.j.1.2		2
5.3	odd	4		14.8.a.c.1.1	✓	2
5.4	even	2	inner	350.8.c.k.99.1		4
15.8	even	4		126.8.a.i.1.1		2
20.3	even	4		112.8.a.g.1.2		2
35.3	even	12		98.8.c.k.79.1		4
35.13	even	4		98.8.a.g.1.2		2
35.18	odd	12		98.8.c.g.79.2		4
35.23	odd	12		98.8.c.g.67.2		4
35.33	even	12		98.8.c.k.67.1		4
40.3	even	4		448.8.a.s.1.1		2
40.13	odd	4		448.8.a.l.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.8.a.c.1.1	✓	2	5.3	odd	4
98.8.a.g.1.2		2	35.13	even	4
98.8.c.g.67.2		4	35.23	odd	12
98.8.c.g.79.2		4	35.18	odd	12
98.8.c.k.67.1		4	35.33	even	12
98.8.c.k.79.1		4	35.3	even	12
112.8.a.g.1.2		2	20.3	even	4
126.8.a.i.1.1		2	15.8	even	4
350.8.a.j.1.2		2	5.2	odd	4
350.8.c.k.99.1		4	5.4	even	2	inner
350.8.c.k.99.4		4	1.1	even	1	trivial
448.8.a.l.1.2		2	40.13	odd	4
448.8.a.s.1.1		2	40.3	even	4

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:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	350.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+8.00000i q^{2} +9.37342i q^{3} -64.0000 q^{4} -74.9873 q^{6} +343.000i q^{7} -512.000i q^{8} +2099.14 q^{9} +3881.05 q^{11} -599.899i q^{12} +11585.6i q^{13} -2744.00 q^{14} +4096.00 q^{16} -15242.4i q^{17} +16793.1i q^{18} +32461.7 q^{19} -3215.08 q^{21} +31048.4i q^{22} -56146.1i q^{23} +4799.19 q^{24} -92684.6 q^{26} +40175.8i q^{27} -21952.0i q^{28} +26442.0 q^{29} -45448.2 q^{31} +32768.0i q^{32} +36378.7i q^{33} +121939. q^{34} -134345. q^{36} -555245. i q^{37} +259694. i q^{38} -108596. q^{39} +306385. q^{41} -25720.7i q^{42} -780755. i q^{43} -248387. q^{44} +449169. q^{46} -531243. i q^{47} +38393.5i q^{48} -117649. q^{49} +142873. q^{51} -741477. i q^{52} +363593. i q^{53} -321406. q^{54} +175616. q^{56} +304277. i q^{57} +211536. i q^{58} +2.14095e6 q^{59} +888824. q^{61} -363586. i q^{62} +720005. i q^{63} -262144. q^{64} -291030. q^{66} +4.34541e6i q^{67} +975513. i q^{68} +526281. q^{69} +663207. q^{71} -1.07476e6i q^{72} -1.34202e6i q^{73} +4.44196e6 q^{74} -2.07755e6 q^{76} +1.33120e6i q^{77} -868771. i q^{78} -7.05834e6 q^{79} +4.21423e6 q^{81} +2.45108e6i q^{82} -6.60874e6i q^{83} +205765. q^{84} +6.24604e6 q^{86} +247851. i q^{87} -1.98710e6i q^{88} +5.32998e6 q^{89} -3.97385e6 q^{91} +3.59335e6i q^{92} -426005. i q^{93} +4.24994e6 q^{94} -307148. q^{96} -2.09810e6i q^{97} -941192. i q^{98} +8.14686e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 256 q^{4} + 1120 q^{6} - 4028 q^{9} - 6840 q^{11} - 10976 q^{14} + 16384 q^{16} + 86716 q^{19} + 48020 q^{21} - 71680 q^{24} - 102368 q^{26} - 319152 q^{29} - 287224 q^{31} + 615552 q^{34} + 257792 q^{36}+ \cdots + 76354200 q^{99}+O(q^{100}) \) 4 * q - 256 * q^4 + 1120 * q^6 - 4028 * q^9 - 6840 * q^11 - 10976 * q^14 + 16384 * q^16 + 86716 * q^19 + 48020 * q^21 - 71680 * q^24 - 102368 * q^26 - 319152 * q^29 - 287224 * q^31 + 615552 * q^34 + 257792 * q^36 - 1040704 * q^39 + 129696 * q^41 + 437760 * q^44 + 1438848 * q^46 - 470596 * q^49 - 3401880 * q^51 - 3088960 * q^54 + 702464 * q^56 + 8367324 * q^59 - 561316 * q^61 - 1048576 * q^64 - 9854208 * q^66 - 4310208 * q^69 - 1238544 * q^71 + 4349312 * q^74 - 5549824 * q^76 - 9313232 * q^79 + 14707484 * q^81 - 3073280 * q^84 + 24447424 * q^86 + 34482840 * q^89 - 4389028 * q^91 - 7766976 * q^94 + 4587520 * q^96 + 76354200 * q^99

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