LMFDB - Embedded newform 363.8.a.r.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [363,8,Mod(1,363)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(363, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("363.1"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 363 = 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	363.a (trivial)

Newform invariants

comment:select newform

sage:traces = [14,23] 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:	$113.395764251$
Analytic rank:	$0$
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} - 2 x^{13} - 1289 x^{12} + 2366 x^{11} + 623758 x^{10} - 908404 x^{9} - 141535137 x^{8} + \cdots - 20874968128476 $ x^14 - 2x^13 - 1289x^12 + 2366x^11 + 623758x^10 - 908404x^9 - 141535137x^8 + 122490614x^7 + 15472711238x^6 - 831247148x^5 - 774565663469x^4 - 859523952378x^3 + 14942261922049x^2 + 32570153376222*x - 20874968128476
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{4}\cdot 11^{8} $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-18.6486$ of defining polynomial
Character	$\chi$	$=$	363.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-16.0306 q^{2} +27.0000 q^{3} +128.979 q^{4} +322.621 q^{5} -432.825 q^{6} +992.956 q^{7} -15.6903 q^{8} +729.000 q^{9} -5171.79 q^{10} +3482.43 q^{12} +14438.4 q^{13} -15917.6 q^{14} +8710.76 q^{15} -16257.8 q^{16} -21255.8 q^{17} -11686.3 q^{18} +43770.9 q^{19} +41611.2 q^{20} +26809.8 q^{21} +22525.3 q^{23} -423.637 q^{24} +25959.1 q^{25} -231455. q^{26} +19683.0 q^{27} +128070. q^{28} +180217. q^{29} -139638. q^{30} -133133. q^{31} +262629. q^{32} +340743. q^{34} +320348. q^{35} +94025.5 q^{36} +368583. q^{37} -701672. q^{38} +389836. q^{39} -5062.01 q^{40} -20045.0 q^{41} -429776. q^{42} +366072. q^{43} +235190. q^{45} -361093. q^{46} +582130. q^{47} -438959. q^{48} +162419. q^{49} -416139. q^{50} -573908. q^{51} +1.86224e6 q^{52} +12091.0 q^{53} -315529. q^{54} -15579.8 q^{56} +1.18181e6 q^{57} -2.88898e6 q^{58} -635842. q^{59} +1.12350e6 q^{60} +3.07463e6 q^{61} +2.13420e6 q^{62} +723865. q^{63} -2.12910e6 q^{64} +4.65811e6 q^{65} +442702. q^{67} -2.74155e6 q^{68} +608184. q^{69} -5.13536e6 q^{70} -1.43752e6 q^{71} -11438.2 q^{72} -3.49069e6 q^{73} -5.90859e6 q^{74} +700896. q^{75} +5.64552e6 q^{76} -6.24928e6 q^{78} -8.08855e6 q^{79} -5.24509e6 q^{80} +531441. q^{81} +321332. q^{82} -2.67768e6 q^{83} +3.45790e6 q^{84} -6.85757e6 q^{85} -5.86834e6 q^{86} +4.86587e6 q^{87} +2.91852e6 q^{89} -3.77023e6 q^{90} +1.43367e7 q^{91} +2.90529e6 q^{92} -3.59459e6 q^{93} -9.33186e6 q^{94} +1.41214e7 q^{95} +7.09099e6 q^{96} -9.41306e6 q^{97} -2.60367e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 23 q^{2} + 378 q^{3} + 845 q^{4} + 69 q^{5} + 621 q^{6} + 3278 q^{7} + 4602 q^{8} + 10206 q^{9} + 5320 q^{10} + 22815 q^{12} + 32188 q^{13} - 7794 q^{14} + 1863 q^{15} + 86137 q^{16} + 42917 q^{17}+ \cdots - 70266089 q^{98}+O(q^{100}) $ 14 * q + 23 * q^2 + 378 * q^3 + 845 * q^4 + 69 * q^5 + 621 * q^6 + 3278 * q^7 + 4602 * q^8 + 10206 * q^9 + 5320 * q^10 + 22815 * q^12 + 32188 * q^13 - 7794 * q^14 + 1863 * q^15 + 86137 * q^16 + 42917 * q^17 + 16767 * q^18 + 50419 * q^19 + 54475 * q^20 + 88506 * q^21 - 83492 * q^23 + 124254 * q^24 + 336991 * q^25 + 234897 * q^26 + 275562 * q^27 + 697184 * q^28 + 227028 * q^29 + 143640 * q^30 - 36481 * q^31 + 151749 * q^32 - 614562 * q^34 - 258783 * q^35 + 616005 * q^36 + 634524 * q^37 - 662891 * q^38 + 869076 * q^39 + 1585604 * q^40 + 1286544 * q^41 - 210438 * q^42 + 2632644 * q^43 + 50301 * q^45 + 1290627 * q^46 - 1692245 * q^47 + 2325699 * q^48 + 842030 * q^49 + 7870914 * q^50 + 1158759 * q^51 + 6419781 * q^52 + 1211317 * q^53 + 452709 * q^54 - 9100953 * q^56 + 1361313 * q^57 - 6265711 * q^58 + 2551943 * q^59 + 1470825 * q^60 + 10221163 * q^61 + 5723657 * q^62 + 2389662 * q^63 + 6349332 * q^64 + 7684087 * q^65 - 3570631 * q^67 + 13112925 * q^68 - 2254284 * q^69 + 3559265 * q^70 + 4528437 * q^71 + 3354858 * q^72 + 13052276 * q^73 + 4896205 * q^74 + 9098757 * q^75 + 9845214 * q^76 + 6342219 * q^78 + 14911090 * q^79 - 9395382 * q^80 + 7440174 * q^81 + 17478001 * q^82 + 7068900 * q^83 + 18823968 * q^84 + 24423120 * q^85 + 12792479 * q^86 + 6129756 * q^87 + 4800832 * q^89 + 3878280 * q^90 - 15295476 * q^91 - 3195753 * q^92 - 984987 * q^93 - 43320002 * q^94 + 7614177 * q^95 + 4097223 * q^96 - 11578773 * q^97 - 70266089 * q^98

Coefficient data

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

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

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

$2$	−16.0306	−1.41691	−0.708457	−	0.705754i	$-0.750609\pi$
$2$	−16.0306	−1.41691	−0.708457	+	0.705754i	$0.750609\pi$
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	128.979	1.00765
$5$	322.621	1.15424	0.577121	−	0.816658i	$-0.304176\pi$
$5$	322.621	1.15424	0.577121	+	0.816658i	$0.304176\pi$
$6$	−432.825	−0.818056
$7$	992.956	1.09418	0.547088	−	0.837075i	$-0.315736\pi$
$7$	992.956	1.09418	0.547088	+	0.837075i	$0.315736\pi$
$8$	−15.6903	−0.0108347
$9$	729.000	0.333333
$10$	−5171.79	−1.63546
$11$	0	0
$11$	0	0
$12$	3482.43	0.581765
$13$	14438.4	1.82270	0.911352	−	0.411628i	$-0.135040\pi$
$13$	14438.4	1.82270	0.911352	+	0.411628i	$0.135040\pi$
$14$	−15917.6	−1.55035
$15$	8710.76	0.666402
$16$	−16257.8	−0.992295
$17$	−21255.8	−1.04932	−0.524659	−	0.851313i	$-0.675807\pi$
$17$	−21255.8	−1.04932	−0.524659	+	0.851313i	$0.675807\pi$
$18$	−11686.3	−0.472305
$19$	43770.9	1.46402	0.732012	−	0.681292i	$-0.238582\pi$
$19$	43770.9	1.46402	0.732012	+	0.681292i	$0.238582\pi$
$20$	41611.2	1.16307
$21$	26809.8	0.631723
$22$	0	0
$23$	22525.3	0.386032	0.193016	−	0.981196i	$-0.438173\pi$
$23$	22525.3	0.386032	0.193016	+	0.981196i	$0.438173\pi$
$24$	−423.637	−0.00625540
$25$	25959.1	0.332277
$26$	−231455.	−2.58262
$27$	19683.0	0.192450
$28$	128070.	1.10254
$29$	180217.	1.37216	0.686078	−	0.727528i	$-0.259331\pi$
$29$	180217.	1.37216	0.686078	+	0.727528i	$0.259331\pi$
$30$	−139638.	−0.944235
$31$	−133133.	−0.802639	−0.401319	−	0.915938i	$-0.631448\pi$
$31$	−133133.	−0.802639	−0.401319	+	0.915938i	$0.631448\pi$
$32$	262629.	1.41683
$33$	0	0
$34$	340743.	1.48679
$35$	320348.	1.26294
$36$	94025.5	0.335882
$37$	368583.	1.19627	0.598135	−	0.801396i	$-0.295909\pi$
$37$	368583.	1.19627	0.598135	+	0.801396i	$0.295909\pi$
$38$	−701672.	−2.07440
$39$	389836.	1.05234
$40$	−5062.01	−0.0125058
$41$	−20045.0	−0.0454216	−0.0227108	−	0.999742i	$-0.507230\pi$
$41$	−20045.0	−0.0454216	−0.0227108	+	0.999742i	$0.507230\pi$
$42$	−429776.	−0.895097
$43$	366072.	0.702146	0.351073	−	0.936348i	$-0.385817\pi$
$43$	366072.	0.702146	0.351073	+	0.936348i	$0.385817\pi$
$44$	0	0
$45$	235190.	0.384748
$46$	−361093.	−0.546975
$47$	582130.	0.817857	0.408928	−	0.912567i	$-0.365903\pi$
$47$	582130.	0.817857	0.408928	+	0.912567i	$0.365903\pi$
$48$	−438959.	−0.572902
$49$	162419.	0.197220
$50$	−416139.	−0.470808
$51$	−573908.	−0.605824
$52$	1.86224e6	1.83664
$53$	12091.0	0.0111557	0.00557785	−	0.999984i	$-0.498225\pi$
$53$	12091.0	0.0111557	0.00557785	+	0.999984i	$0.498225\pi$
$54$	−315529.	−0.272685
$55$	0	0
$56$	−15579.8	−0.0118550
$57$	1.18181e6	0.845254
$58$	−2.88898e6	−1.94423
$59$	−635842.	−0.403058	−0.201529	−	0.979483i	$-0.564591\pi$
$59$	−635842.	−0.403058	−0.201529	+	0.979483i	$0.564591\pi$
$60$	1.12350e6	0.671498
$61$	3.07463e6	1.73436	0.867179	−	0.497996i	$-0.165931\pi$
$61$	3.07463e6	1.73436	0.867179	+	0.497996i	$0.165931\pi$
$62$	2.13420e6	1.13727
$63$	723865.	0.364725
$64$	−2.12910e6	−1.01523
$65$	4.65811e6	2.10384
$66$	0	0
$67$	442702.	0.179825	0.0899124	−	0.995950i	$-0.471341\pi$
$67$	442702.	0.179825	0.0899124	+	0.995950i	$0.471341\pi$
$68$	−2.74155e6	−1.05734
$69$	608184.	0.222876
$70$	−5.13536e6	−1.78948
$71$	−1.43752e6	−0.476662	−0.238331	−	0.971184i	$-0.576600\pi$
$71$	−1.43752e6	−0.476662	−0.238331	+	0.971184i	$0.576600\pi$
$72$	−11438.2	−0.00361156
$73$	−3.49069e6	−1.05022	−0.525111	−	0.851034i	$-0.675976\pi$
$73$	−3.49069e6	−1.05022	−0.525111	+	0.851034i	$0.675976\pi$
$74$	−5.90859e6	−1.69501
$75$	700896.	0.191840
$76$	5.64552e6	1.47522
$77$	0	0
$78$	−6.24928e6	−1.49107
$79$	−8.08855e6	−1.84576	−0.922882	−	0.385083i	$-0.874173\pi$
$79$	−8.08855e6	−1.84576	−0.922882	+	0.385083i	$0.874173\pi$
$80$	−5.24509e6	−1.14535
$81$	531441.	0.111111
$82$	321332.	0.0643585
$83$	−2.67768e6	−0.514027	−0.257013	−	0.966408i	$-0.582738\pi$
$83$	−2.67768e6	−0.514027	−0.257013	+	0.966408i	$0.582738\pi$
$84$	3.45790e6	0.636553
$85$	−6.85757e6	−1.21117
$86$	−5.86834e6	−0.994880
$87$	4.86587e6	0.792215
$88$	0	0
$89$	2.91852e6	0.438831	0.219415	−	0.975632i	$-0.429585\pi$
$89$	2.91852e6	0.438831	0.219415	+	0.975632i	$0.429585\pi$
$90$	−3.77023e6	−0.545154
$91$	1.43367e7	1.99436
$92$	2.90529e6	0.388984
$93$	−3.59459e6	−0.463404
$94$	−9.33186e6	−1.15883
$95$	1.41214e7	1.68984
$96$	7.09099e6	0.818008
$97$	−9.41306e6	−1.04720	−0.523600	−	0.851964i	$-0.675411\pi$
$97$	−9.41306e6	−1.04720	−0.523600	+	0.851964i	$0.675411\pi$
$98$	−2.60367e6	−0.279444
$99$	0	0
$100$	3.34817e6	0.334817
$101$	−9.84888e6	−0.951178	−0.475589	−	0.879668i	$-0.657765\pi$
$101$	−9.84888e6	−0.951178	−0.475589	+	0.879668i	$0.657765\pi$
$102$	9.20006e6	0.858401
$103$	−2.61799e6	−0.236069	−0.118034	−	0.993010i	$-0.537659\pi$
$103$	−2.61799e6	−0.236069	−0.118034	+	0.993010i	$0.537659\pi$
$104$	−226542.	−0.0197484
$105$	8.64940e6	0.729161
$106$	−193826.	−0.0158067
$107$	−9.50506e6	−0.750087	−0.375043	−	0.927007i	$-0.622372\pi$
$107$	−9.50506e6	−0.750087	−0.375043	+	0.927007i	$0.622372\pi$
$108$	2.53869e6	0.193922
$109$	6.17065e6	0.456392	0.228196	−	0.973615i	$-0.426717\pi$
$109$	6.17065e6	0.456392	0.228196	+	0.973615i	$0.426717\pi$
$110$	0	0
$111$	9.95173e6	0.690666
$112$	−1.61432e7	−1.08574
$113$	−1.78915e7	−1.16646	−0.583232	−	0.812306i	$-0.698212\pi$
$113$	−1.78915e7	−1.16646	−0.583232	+	0.812306i	$0.698212\pi$
$114$	−1.89451e7	−1.19765
$115$	7.26714e6	0.445575
$116$	2.32442e7	1.38265
$117$	1.05256e7	0.607568
$118$	1.01929e7	0.571098
$119$	−2.11061e7	−1.14814
$120$	−136674.	−0.00722025
$121$	0	0
$122$	−4.92881e7	−2.45744
$123$	−541214.	−0.0262241
$124$	−1.71713e7	−0.808776
$125$	−1.68298e7	−0.770715
$126$	−1.16040e7	−0.516784
$127$	1.74127e6	0.0754314	0.0377157	−	0.999289i	$-0.487992\pi$
$127$	1.74127e6	0.0754314	0.0377157	+	0.999289i	$0.487992\pi$
$128$	514124.	0.0216687
$129$	9.88395e6	0.405384
$130$	−7.46722e7	−2.98097
$131$	1.17590e7	0.457003	0.228502	−	0.973544i	$-0.426617\pi$
$131$	1.17590e7	0.457003	0.228502	+	0.973544i	$0.426617\pi$
$132$	0	0
$133$	4.34626e7	1.60190
$134$	−7.09676e6	−0.254796
$135$	6.35014e6	0.222134
$136$	333510.	0.0113690
$137$	−4.46391e7	−1.48318	−0.741589	−	0.670855i	$-0.765927\pi$
$137$	−4.46391e7	−1.48318	−0.741589	+	0.670855i	$0.765927\pi$
$138$	−9.74952e6	−0.315796
$139$	−6.87526e6	−0.217139	−0.108569	−	0.994089i	$-0.534627\pi$
$139$	−6.87526e6	−0.217139	−0.108569	+	0.994089i	$0.534627\pi$
$140$	4.13181e7	1.27260
$141$	1.57175e7	0.472190
$142$	2.30443e7	0.675389
$143$	0	0
$144$	−1.18519e7	−0.330765
$145$	5.81418e7	1.58380
$146$	5.59577e7	1.48808
$147$	4.38532e6	0.113865
$148$	4.75393e7	1.20542
$149$	−6.92543e7	−1.71512	−0.857561	−	0.514382i	$-0.828021\pi$
$149$	−6.92543e7	−1.71512	−0.857561	+	0.514382i	$0.828021\pi$
$150$	−1.12358e7	−0.271821
$151$	6.48231e7	1.53218	0.766092	−	0.642731i	$-0.222199\pi$
$151$	6.48231e7	1.53218	0.766092	+	0.642731i	$0.222199\pi$
$152$	−686777.	−0.0158622
$153$	−1.54955e7	−0.349773
$154$	0	0
$155$	−4.29515e7	−0.926440
$156$	5.02805e7	1.06039
$157$	5.23787e7	1.08020	0.540102	−	0.841600i	$-0.318386\pi$
$157$	5.23787e7	1.08020	0.540102	+	0.841600i	$0.318386\pi$
$158$	1.29664e8	2.61529
$159$	326457.	0.00644075
$160$	8.47296e7	1.63537
$161$	2.23667e7	0.422387
$162$	−8.51930e6	−0.157435
$163$	2.00487e7	0.362602	0.181301	−	0.983428i	$-0.441969\pi$
$163$	2.00487e7	0.362602	0.181301	+	0.983428i	$0.441969\pi$
$164$	−2.58538e6	−0.0457689
$165$	0	0
$166$	4.29247e7	0.728332
$167$	−2.68917e7	−0.446798	−0.223399	−	0.974727i	$-0.571715\pi$
$167$	−2.68917e7	−0.446798	−0.223399	+	0.974727i	$0.571715\pi$
$168$	−420653.	−0.00684450
$169$	1.45718e8	2.32225
$170$	1.09931e8	1.71612
$171$	3.19090e7	0.488008
$172$	4.72156e7	0.707515
$173$	1.03395e8	1.51824	0.759118	−	0.650953i	$-0.225631\pi$
$173$	1.03395e8	1.51824	0.759118	+	0.650953i	$0.225631\pi$
$174$	−7.80026e7	−1.12250
$175$	2.57763e7	0.363569
$176$	0	0
$177$	−1.71677e7	−0.232705
$178$	−4.67854e7	−0.621786
$179$	−9.15502e7	−1.19309	−0.596546	−	0.802579i	$-0.703461\pi$
$179$	−9.15502e7	−1.19309	−0.596546	+	0.802579i	$0.703461\pi$
$180$	3.03346e7	0.387690
$181$	−1.95434e7	−0.244977	−0.122489	−	0.992470i	$-0.539087\pi$
$181$	−1.95434e7	−0.244977	−0.122489	+	0.992470i	$0.539087\pi$
$182$	−2.29825e8	−2.82584
$183$	8.30151e7	1.00133
$184$	−353428.	−0.00418253
$185$	1.18912e8	1.38079
$186$	5.76234e7	0.656604
$187$	0	0
$188$	7.50824e7	0.824110
$189$	1.95444e7	0.210574
$190$	−2.26374e8	−2.39436
$191$	−5.06038e6	−0.0525492	−0.0262746	−	0.999655i	$-0.508364\pi$
$191$	−5.06038e6	−0.0525492	−0.0262746	+	0.999655i	$0.508364\pi$
$192$	−5.74857e7	−0.586146
$193$	−6.66308e7	−0.667151	−0.333576	−	0.942723i	$-0.608255\pi$
$193$	−6.66308e7	−0.667151	−0.333576	+	0.942723i	$0.608255\pi$
$194$	1.50897e8	1.48379
$195$	1.25769e8	1.21465
$196$	2.09487e7	0.198728
$197$	1.88947e8	1.76080	0.880398	−	0.474236i	$-0.157276\pi$
$197$	1.88947e8	1.76080	0.880398	+	0.474236i	$0.157276\pi$
$198$	0	0
$199$	1.11378e6	0.0100187	0.00500935	−	0.999987i	$-0.498405\pi$
$199$	1.11378e6	0.0100187	0.00500935	+	0.999987i	$0.498405\pi$
$200$	−407306.	−0.00360011
$201$	1.19529e7	0.103822
$202$	1.57883e8	1.34774
$203$	1.78948e8	1.50138
$204$	−7.40219e7	−0.610456
$205$	−6.46692e6	−0.0524275
$206$	4.19679e7	0.334489
$207$	1.64210e7	0.128677
$208$	−2.34735e8	−1.80866
$209$	0	0
$210$	−1.38655e8	−1.03316
$211$	2.44732e8	1.79350	0.896751	−	0.442536i	$-0.145921\pi$
$211$	2.44732e8	1.79350	0.896751	+	0.442536i	$0.145921\pi$
$212$	1.55948e6	0.0112410
$213$	−3.88131e7	−0.275201
$214$	1.52371e8	1.06281
$215$	1.18102e8	0.810447
$216$	−308832.	−0.00208513
$217$	−1.32195e8	−0.878228
$218$	−9.89189e7	−0.646669
$219$	−9.42486e7	−0.606346
$220$	0	0
$221$	−3.06899e8	−1.91260
$222$	−1.59532e8	−0.978615
$223$	8.32265e7	0.502568	0.251284	−	0.967913i	$-0.419147\pi$
$223$	8.32265e7	0.502568	0.251284	+	0.967913i	$0.419147\pi$
$224$	2.60779e8	1.55026
$225$	1.89242e7	0.110759
$226$	2.86810e8	1.65278
$227$	−1.87772e8	−1.06547	−0.532734	−	0.846283i	$-0.678835\pi$
$227$	−1.87772e8	−1.06547	−0.532734	+	0.846283i	$0.678835\pi$
$228$	1.52429e8	0.851717
$229$	5.33918e7	0.293799	0.146899	−	0.989151i	$-0.453071\pi$
$229$	5.33918e7	0.293799	0.146899	+	0.989151i	$0.453071\pi$
$230$	−1.16496e8	−0.631342
$231$	0	0
$232$	−2.82766e6	−0.0148669
$233$	6.92698e7	0.358755	0.179378	−	0.983780i	$-0.442592\pi$
$233$	6.92698e7	0.358755	0.179378	+	0.983780i	$0.442592\pi$
$234$	−1.68731e8	−0.860872
$235$	1.87807e8	0.944005
$236$	−8.20101e7	−0.406140
$237$	−2.18391e8	−1.06565
$238$	3.38343e8	1.62681
$239$	6.76984e7	0.320764	0.160382	−	0.987055i	$-0.448727\pi$
$239$	6.76984e7	0.320764	0.160382	+	0.987055i	$0.448727\pi$
$240$	−1.41617e8	−0.661268
$241$	5.69151e7	0.261919	0.130960	−	0.991388i	$-0.458194\pi$
$241$	5.69151e7	0.261919	0.130960	+	0.991388i	$0.458194\pi$
$242$	0	0
$243$	1.43489e7	0.0641500
$244$	3.96562e8	1.74762
$245$	5.23998e7	0.227640
$246$	8.67597e6	0.0371574
$247$	6.31980e8	2.66848
$248$	2.08890e6	0.00869632
$249$	−7.22974e7	−0.296773
$250$	2.69791e8	1.09204
$251$	−1.58681e8	−0.633385	−0.316693	−	0.948528i	$-0.602572\pi$
$251$	−1.58681e8	−0.633385	−0.316693	+	0.948528i	$0.602572\pi$
$252$	9.33632e7	0.367514
$253$	0	0
$254$	−2.79135e7	−0.106880
$255$	−1.85154e8	−0.699268
$256$	2.64283e8	0.984532
$257$	−2.00591e8	−0.737134	−0.368567	−	0.929601i	$-0.620151\pi$
$257$	−2.00591e8	−0.737134	−0.368567	+	0.929601i	$0.620151\pi$
$258$	−1.58445e8	−0.574394
$259$	3.65987e8	1.30893
$260$	6.00798e8	2.11993
$261$	1.31378e8	0.457385
$262$	−1.88503e8	−0.647534
$263$	−5.37363e8	−1.82147	−0.910737	−	0.412987i	$-0.864486\pi$
$263$	−5.37363e8	−1.82147	−0.910737	+	0.412987i	$0.864486\pi$
$264$	0	0
$265$	3.90081e6	0.0128764
$266$	−6.96730e8	−2.26975
$267$	7.87999e7	0.253359
$268$	5.70991e7	0.181200
$269$	−3.96719e7	−0.124265	−0.0621327	−	0.998068i	$-0.519790\pi$
$269$	−3.96719e7	−0.124265	−0.0621327	+	0.998068i	$0.519790\pi$
$270$	−1.01796e8	−0.314745
$271$	−6.31573e7	−0.192766	−0.0963832	−	0.995344i	$-0.530727\pi$
$271$	−6.31573e7	−0.192766	−0.0963832	+	0.995344i	$0.530727\pi$
$272$	3.45572e8	1.04123
$273$	3.87090e8	1.15144
$274$	7.15589e8	2.10154
$275$	0	0
$276$	7.84428e7	0.224580
$277$	−3.69188e8	−1.04368	−0.521841	−	0.853043i	$-0.674755\pi$
$277$	−3.69188e8	−1.04368	−0.521841	+	0.853043i	$0.674755\pi$
$278$	1.10214e8	0.307667
$279$	−9.70541e7	−0.267546
$280$	−5.02635e6	−0.0136836
$281$	4.31318e8	1.15964	0.579822	−	0.814743i	$-0.303122\pi$
$281$	4.31318e8	1.15964	0.579822	+	0.814743i	$0.303122\pi$
$282$	−2.51960e8	−0.669052
$283$	−6.91736e8	−1.81421	−0.907105	−	0.420904i	$-0.861713\pi$
$283$	−6.91736e8	−1.81421	−0.907105	+	0.420904i	$0.861713\pi$
$284$	−1.85410e8	−0.480307
$285$	3.81278e8	0.975629
$286$	0	0
$287$	−1.99038e7	−0.0496992
$288$	1.91457e8	0.472277
$289$	4.14719e7	0.101068
$290$	−9.32046e8	−2.24411
$291$	−2.54152e8	−0.604601
$292$	−4.50225e8	−1.05825
$293$	−3.99794e8	−0.928539	−0.464269	−	0.885694i	$-0.653683\pi$
$293$	−3.99794e8	−0.928539	−0.464269	+	0.885694i	$0.653683\pi$
$294$	−7.02992e7	−0.161337
$295$	−2.05136e8	−0.465227
$296$	−5.78316e6	−0.0129612
$297$	0	0
$298$	1.11019e9	2.43018
$299$	3.25229e8	0.703623
$300$	9.04007e7	0.193307
$301$	3.63494e8	0.768271
$302$	−1.03915e9	−2.17097
$303$	−2.65920e8	−0.549163
$304$	−7.11617e8	−1.45274
$305$	9.91940e8	2.00187
$306$	2.48402e8	0.495598
$307$	2.11615e8	0.417410	0.208705	−	0.977979i	$-0.433075\pi$
$307$	2.11615e8	0.417410	0.208705	+	0.977979i	$0.433075\pi$
$308$	0	0
$309$	−7.06858e7	−0.136294
$310$	6.88537e8	1.31269
$311$	−2.81020e7	−0.0529755	−0.0264878	−	0.999649i	$-0.508432\pi$
$311$	−2.81020e7	−0.0529755	−0.0264878	+	0.999649i	$0.508432\pi$
$312$	−6.11663e6	−0.0114017
$313$	3.26533e6	0.00601897	0.00300948	−	0.999995i	$-0.499042\pi$
$313$	3.26533e6	0.00601897	0.00300948	+	0.999995i	$0.499042\pi$
$314$	−8.39659e8	−1.53056
$315$	2.33534e8	0.420981
$316$	−1.04325e9	−1.85988
$317$	−4.32150e8	−0.761951	−0.380976	−	0.924585i	$-0.624412\pi$
$317$	−4.32150e8	−0.761951	−0.380976	+	0.924585i	$0.624412\pi$
$318$	−5.23329e6	−0.00912599
$319$	0	0
$320$	−6.86892e8	−1.17183
$321$	−2.56637e8	−0.433063
$322$	−3.58550e8	−0.598486
$323$	−9.30387e8	−1.53623
$324$	6.85446e7	0.111961
$325$	3.74807e8	0.605642
$326$	−3.21392e8	−0.513776
$327$	1.66608e8	0.263498
$328$	314511.	0.000492127 0
$329$	5.78029e8	0.894879
$330$	0	0
$331$	3.76606e8	0.570807	0.285404	−	0.958407i	$-0.407872\pi$
$331$	3.76606e8	0.570807	0.285404	+	0.958407i	$0.407872\pi$
$332$	−3.45364e8	−0.517957
$333$	2.68697e8	0.398756
$334$	4.31090e8	0.633075
$335$	1.42825e8	0.207561
$336$	−4.35868e8	−0.626855
$337$	3.98784e8	0.567588	0.283794	−	0.958885i	$-0.408407\pi$
$337$	3.98784e8	0.567588	0.283794	+	0.958885i	$0.408407\pi$
$338$	−2.33594e9	−3.29043
$339$	−4.83070e8	−0.673458
$340$	−8.84481e8	−1.22043
$341$	0	0
$342$	−5.11519e8	−0.691465
$343$	−6.56467e8	−0.878382
$344$	−5.74377e6	−0.00760751
$345$	1.96213e8	0.257253
$346$	−1.65748e9	−2.15121
$347$	−2.24353e7	−0.0288256	−0.0144128	−	0.999896i	$-0.504588\pi$
$347$	−2.24353e7	−0.0288256	−0.0144128	+	0.999896i	$0.504588\pi$
$348$	6.27594e8	0.798273
$349$	−2.76106e8	−0.347685	−0.173843	−	0.984773i	$-0.555618\pi$
$349$	−2.76106e8	−0.347685	−0.173843	+	0.984773i	$0.555618\pi$
$350$	−4.13208e8	−0.515146
$351$	2.84190e8	0.350780
$352$	0	0
$353$	−8.63323e8	−1.04463	−0.522314	−	0.852753i	$-0.674931\pi$
$353$	−8.63323e8	−1.04463	−0.522314	+	0.852753i	$0.674931\pi$
$354$	2.75208e8	0.329724
$355$	−4.63775e8	−0.550184
$356$	3.76427e8	0.442186
$357$	−5.69865e8	−0.662878
$358$	1.46760e9	1.69051
$359$	5.95095e8	0.678821	0.339411	−	0.940638i	$-0.389772\pi$
$359$	5.95095e8	0.678821	0.339411	+	0.940638i	$0.389772\pi$
$360$	−3.69020e6	−0.00416861
$361$	1.02202e9	1.14336
$362$	3.13292e8	0.347112
$363$	0	0
$364$	1.84913e9	2.00961
$365$	−1.12617e9	−1.21221
$366$	−1.33078e9	−1.41880
$367$	−1.48566e9	−1.56888	−0.784438	−	0.620207i	$-0.787048\pi$
$367$	−1.48566e9	−1.56888	−0.784438	+	0.620207i	$0.787048\pi$
$368$	−3.66211e8	−0.383058
$369$	−1.46128e7	−0.0151405
$370$	−1.90623e9	−1.95645
$371$	1.20058e7	0.0122063
$372$	−4.63626e8	−0.466947
$373$	9.31856e8	0.929753	0.464877	−	0.885375i	$-0.346099\pi$
$373$	9.31856e8	0.929753	0.464877	+	0.885375i	$0.346099\pi$
$374$	0	0
$375$	−4.54404e8	−0.444972
$376$	−9.13377e6	−0.00886120
$377$	2.60204e9	2.50104
$378$	−3.13307e8	−0.298366
$379$	1.47733e9	1.39393	0.696966	−	0.717104i	$-0.254533\pi$
$379$	1.47733e9	1.39393	0.696966	+	0.717104i	$0.254533\pi$
$380$	1.82136e9	1.70276
$381$	4.70142e7	0.0435503
$382$	8.11207e7	0.0744578
$383$	1.64065e9	1.49218	0.746090	−	0.665846i	$-0.231929\pi$
$383$	1.64065e9	1.49218	0.746090	+	0.665846i	$0.231929\pi$
$384$	1.38813e7	0.0125104
$385$	0	0
$386$	1.06813e9	0.945296
$387$	2.66867e8	0.234049
$388$	−1.21408e9	−1.05521
$389$	−1.20892e9	−1.04130	−0.520648	−	0.853772i	$-0.674309\pi$
$389$	−1.20892e9	−1.04130	−0.520648	+	0.853772i	$0.674309\pi$
$390$	−2.01615e9	−1.72106
$391$	−4.78795e8	−0.405070
$392$	−2.54840e6	−0.00213682
$393$	3.17492e8	0.263851
$394$	−3.02893e9	−2.49490
$395$	−2.60953e9	−2.13046
$396$	0	0
$397$	3.60149e8	0.288879	0.144439	−	0.989514i	$-0.453862\pi$
$397$	3.60149e8	0.288879	0.144439	+	0.989514i	$0.453862\pi$
$398$	−1.78544e7	−0.0141957
$399$	1.17349e9	0.924856
$400$	−4.22037e8	−0.329716
$401$	2.22451e9	1.72278	0.861390	−	0.507945i	$-0.169595\pi$
$401$	2.22451e9	1.72278	0.861390	+	0.507945i	$0.169595\pi$
$402$	−1.91612e8	−0.147107
$403$	−1.92222e9	−1.46297
$404$	−1.27030e9	−0.958452
$405$	1.71454e8	0.128249
$406$	−2.86864e9	−2.12733
$407$	0	0
$408$	9.00477e6	0.00656390
$409$	−1.26528e9	−0.914439	−0.457219	−	0.889354i	$-0.651155\pi$
$409$	−1.26528e9	−0.914439	−0.457219	+	0.889354i	$0.651155\pi$
$410$	1.03668e8	0.0742853
$411$	−1.20525e9	−0.856313
$412$	−3.37665e8	−0.237874
$413$	−6.31363e8	−0.441016
$414$	−2.63237e8	−0.182325
$415$	−8.63876e8	−0.593312
$416$	3.79194e9	2.58247
$417$	−1.85632e8	−0.125365
$418$	0	0
$419$	−5.76631e8	−0.382956	−0.191478	−	0.981497i	$-0.561328\pi$
$419$	−5.76631e8	−0.382956	−0.191478	+	0.981497i	$0.561328\pi$
$420$	1.11559e9	0.734737
$421$	−6.27680e8	−0.409969	−0.204984	−	0.978765i	$-0.565714\pi$
$421$	−6.27680e8	−0.409969	−0.204984	+	0.978765i	$0.565714\pi$
$422$	−3.92319e9	−2.54124
$423$	4.24372e8	0.272619
$424$	−189711.	−0.000120868 0
$425$	−5.51783e8	−0.348664
$426$	6.22196e8	0.389936
$427$	3.05298e9	1.89769
$428$	−1.22595e9	−0.755822
$429$	0	0
$430$	−1.89325e9	−1.14833
$431$	−4.12434e8	−0.248133	−0.124066	−	0.992274i	$-0.539594\pi$
$431$	−4.12434e8	−0.248133	−0.124066	+	0.992274i	$0.539594\pi$
$432$	−3.20001e8	−0.190967
$433$	1.92048e9	1.13684	0.568422	−	0.822737i	$-0.307554\pi$
$433$	1.92048e9	1.13684	0.568422	+	0.822737i	$0.307554\pi$
$434$	2.11917e9	1.24437
$435$	1.56983e9	0.914408
$436$	7.95883e8	0.459882
$437$	9.85954e8	0.565160
$438$	1.51086e9	0.859141
$439$	3.46529e9	1.95485	0.977427	−	0.211275i	$-0.0677614\pi$
$439$	3.46529e9	1.95485	0.977427	+	0.211275i	$0.0677614\pi$
$440$	0	0
$441$	1.18404e8	0.0657401
$442$	4.91977e9	2.70998
$443$	2.28463e8	0.124854	0.0624271	−	0.998050i	$-0.480116\pi$
$443$	2.28463e8	0.124854	0.0624271	+	0.998050i	$0.480116\pi$
$444$	1.28356e9	0.695948
$445$	9.41574e8	0.506517
$446$	−1.33417e9	−0.712096
$447$	−1.86987e9	−0.990226
$448$	−2.11410e9	−1.11084
$449$	1.93253e9	1.00755	0.503773	−	0.863836i	$-0.331945\pi$
$449$	1.93253e9	1.00755	0.503773	+	0.863836i	$0.331945\pi$
$450$	−3.03365e8	−0.156936
$451$	0	0
$452$	−2.30762e9	−1.17538
$453$	1.75022e9	0.884606
$454$	3.01009e9	1.50968
$455$	4.62530e9	2.30197
$456$	−1.85430e7	−0.00915805
$457$	−9.40231e8	−0.460816	−0.230408	−	0.973094i	$-0.574006\pi$
$457$	−9.40231e8	−0.460816	−0.230408	+	0.973094i	$0.574006\pi$
$458$	−8.55900e8	−0.416288
$459$	−4.18379e8	−0.201941
$460$	9.37306e8	0.448982
$461$	1.27339e9	0.605352	0.302676	−	0.953094i	$-0.402120\pi$
$461$	1.27339e9	0.605352	0.302676	+	0.953094i	$0.402120\pi$
$462$	0	0
$463$	3.08457e9	1.44431	0.722156	−	0.691730i	$-0.243151\pi$
$463$	3.08457e9	1.44431	0.722156	+	0.691730i	$0.243151\pi$
$464$	−2.92993e9	−1.36158
$465$	−1.15969e9	−0.534881
$466$	−1.11043e9	−0.508325
$467$	1.91563e9	0.870365	0.435183	−	0.900342i	$-0.356684\pi$
$467$	1.91563e9	0.870365	0.435183	+	0.900342i	$0.356684\pi$
$468$	1.35757e9	0.612214
$469$	4.39583e8	0.196760
$470$	−3.01065e9	−1.33757
$471$	1.41422e9	0.623656
$472$	9.97654e6	0.00436700
$473$	0	0
$474$	3.50093e9	1.50994
$475$	1.13625e9	0.486461
$476$	−2.72224e9	−1.15692
$477$	8.81435e6	0.00371857
$478$	−1.08524e9	−0.454495
$479$	1.45734e9	0.605880	0.302940	−	0.953010i	$-0.402032\pi$
$479$	1.45734e9	0.605880	0.302940	+	0.953010i	$0.402032\pi$
$480$	2.28770e9	0.944180
$481$	5.32173e9	2.18044
$482$	−9.12381e8	−0.371117
$483$	6.03900e8	0.243865
$484$	0	0
$485$	−3.03685e9	−1.20872
$486$	−2.30021e8	−0.0908951
$487$	1.04638e9	0.410523	0.205261	−	0.978707i	$-0.434196\pi$
$487$	1.04638e9	0.410523	0.205261	+	0.978707i	$0.434196\pi$
$488$	−4.82418e7	−0.0187912
$489$	5.41315e8	0.209348
$490$	−8.39999e8	−0.322547
$491$	−2.27978e9	−0.869177	−0.434588	−	0.900629i	$-0.643106\pi$
$491$	−2.27978e9	−0.869177	−0.434588	+	0.900629i	$0.643106\pi$
$492$	−6.98052e7	−0.0264247
$493$	−3.83067e9	−1.43983
$494$	−1.01310e10	−3.78101
$495$	0	0
$496$	2.16445e9	0.796454
$497$	−1.42740e9	−0.521552
$498$	1.15897e9	0.420503
$499$	−4.78157e8	−0.172274	−0.0861368	−	0.996283i	$-0.527452\pi$
$499$	−4.78157e8	−0.172274	−0.0861368	+	0.996283i	$0.527452\pi$
$500$	−2.17069e9	−0.776608
$501$	−7.26077e8	−0.257959
$502$	2.54375e9	0.897453
$503$	−3.65939e9	−1.28210	−0.641048	−	0.767501i	$-0.721500\pi$
$503$	−3.65939e9	−1.28210	−0.641048	+	0.767501i	$0.721500\pi$
$504$	−1.13576e7	−0.00395168
$505$	−3.17745e9	−1.09789
$506$	0	0
$507$	3.93438e9	1.34075
$508$	2.24586e8	0.0760082
$509$	−4.79542e9	−1.61181	−0.805906	−	0.592044i	$-0.798321\pi$
$509$	−4.79542e9	−1.61181	−0.805906	+	0.592044i	$0.798321\pi$
$510$	2.96813e9	0.990803
$511$	−3.46610e9	−1.14913
$512$	−4.30242e9	−1.41667
$513$	8.61543e8	0.281751
$514$	3.21559e9	1.04446
$515$	−8.44619e8	−0.272480
$516$	1.27482e9	0.408484
$517$	0	0
$518$	−5.86697e9	−1.85464
$519$	2.79167e9	0.876554
$520$	−7.30871e7	−0.0227944
$521$	2.12623e9	0.658687	0.329344	−	0.944210i	$-0.393173\pi$
$521$	2.12623e9	0.658687	0.329344	+	0.944210i	$0.393173\pi$
$522$	−2.10607e9	−0.648076
$523$	−5.50855e8	−0.168376	−0.0841882	−	0.996450i	$-0.526830\pi$
$523$	−5.50855e8	−0.168376	−0.0841882	+	0.996450i	$0.526830\pi$
$524$	1.51666e9	0.460498
$525$	6.95959e8	0.209907
$526$	8.61423e9	2.58087
$527$	2.82986e9	0.842223
$528$	0	0
$529$	−2.89744e9	−0.850979
$530$	−6.25322e7	−0.0182447
$531$	−4.63529e8	−0.134353
$532$	5.60575e9	1.61415
$533$	−2.89417e8	−0.0827901
$534$	−1.26321e9	−0.358988
$535$	−3.06653e9	−0.865782
$536$	−6.94611e6	−0.00194834
$537$	−2.47186e9	−0.688832
$538$	6.35963e8	0.176074
$539$	0	0
$540$	8.19034e8	0.223833
$541$	4.39966e9	1.19462	0.597309	−	0.802011i	$-0.296237\pi$
$541$	4.39966e9	1.19462	0.597309	+	0.802011i	$0.296237\pi$
$542$	1.01245e9	0.273133
$543$	−5.27672e8	−0.141438
$544$	−5.58241e9	−1.48671
$545$	1.99078e9	0.526787
$546$	−6.20527e9	−1.63150
$547$	−1.38269e9	−0.361218	−0.180609	−	0.983555i	$-0.557807\pi$
$547$	−1.38269e9	−0.361218	−0.180609	+	0.983555i	$0.557807\pi$
$548$	−5.75749e9	−1.49452
$549$	2.24141e9	0.578119
$550$	0	0
$551$	7.88828e9	2.00887
$552$	−9.54257e6	−0.00241479
$553$	−8.03158e9	−2.01959
$554$	5.91829e9	1.47881
$555$	3.21063e9	0.797197
$556$	−8.86763e8	−0.218799
$557$	2.68689e9	0.658804	0.329402	−	0.944190i	$-0.393153\pi$
$557$	2.68689e9	0.658804	0.329402	+	0.944190i	$0.393153\pi$
$558$	1.55583e9	0.379090
$559$	5.28548e9	1.27980
$560$	−5.20815e9	−1.25321
$561$	0	0
$562$	−6.91426e9	−1.64312
$563$	1.74583e9	0.412309	0.206155	−	0.978519i	$-0.433905\pi$
$563$	1.74583e9	0.412309	0.206155	+	0.978519i	$0.433905\pi$
$564$	2.02722e9	0.475800
$565$	−5.77216e9	−1.34638
$566$	1.10889e10	2.57058
$567$	5.27698e8	0.121575
$568$	2.25551e7	0.00516447
$569$	−3.49925e9	−0.796311	−0.398155	−	0.917318i	$-0.630349\pi$
$569$	−3.49925e9	−0.796311	−0.398155	+	0.917318i	$0.630349\pi$
$570$	−6.11210e9	−1.38238
$571$	3.99974e9	0.899094	0.449547	−	0.893257i	$-0.351585\pi$
$571$	3.99974e9	0.899094	0.449547	+	0.893257i	$0.351585\pi$
$572$	0	0
$573$	−1.36630e8	−0.0303393
$574$	3.19069e8	0.0704195
$575$	5.84737e8	0.128269
$576$	−1.55211e9	−0.338411
$577$	2.02940e9	0.439797	0.219898	−	0.975523i	$-0.429427\pi$
$577$	2.02940e9	0.439797	0.219898	+	0.975523i	$0.429427\pi$
$578$	−6.64818e8	−0.143204
$579$	−1.79903e9	−0.385180
$580$	7.49906e9	1.59591
$581$	−2.65882e9	−0.562435
$582$	4.07421e9	0.856668
$583$	0	0
$584$	5.47699e7	0.0113788
$585$	3.39577e9	0.701281
$586$	6.40893e9	1.31566
$587$	−5.08799e9	−1.03828	−0.519139	−	0.854690i	$-0.673747\pi$
$587$	−5.08799e9	−1.03828	−0.519139	+	0.854690i	$0.673747\pi$
$588$	5.65614e8	0.114736
$589$	−5.82736e9	−1.17508
$590$	3.28844e9	0.659186
$591$	5.10158e9	1.01660
$592$	−5.99233e9	−1.18705
$593$	6.28144e9	1.23700	0.618498	−	0.785787i	$-0.287742\pi$
$593$	6.28144e9	1.23700	0.618498	+	0.785787i	$0.287742\pi$
$594$	0	0
$595$	−6.80927e9	−1.32523
$596$	−8.93234e9	−1.72824
$597$	3.00719e7	0.00578430
$598$	−5.21360e9	−0.996973
$599$	3.27746e9	0.623080	0.311540	−	0.950233i	$-0.399155\pi$
$599$	3.27746e9	0.623080	0.311540	+	0.950233i	$0.399155\pi$
$600$	−1.09972e7	−0.00207852
$601$	7.23030e9	1.35861	0.679306	−	0.733855i	$-0.262281\pi$
$601$	7.23030e9	1.35861	0.679306	+	0.733855i	$0.262281\pi$
$602$	−5.82701e9	−1.08857
$603$	3.22730e8	0.0599416
$604$	8.36081e9	1.54390
$605$	0	0
$606$	4.26284e9	0.778117
$607$	−9.44913e9	−1.71487	−0.857435	−	0.514591i	$-0.827944\pi$
$607$	−9.44913e9	−1.71487	−0.857435	+	0.514591i	$0.827944\pi$
$608$	1.14955e10	2.07427
$609$	4.83160e9	0.866822
$610$	−1.59014e10	−2.83648
$611$	8.40500e9	1.49071
$612$	−1.99859e9	−0.352447
$613$	9.37577e9	1.64397	0.821987	−	0.569506i	$-0.192865\pi$
$613$	9.37577e9	1.64397	0.821987	+	0.569506i	$0.192865\pi$
$614$	−3.39231e9	−0.591434
$615$	−1.74607e8	−0.0302690
$616$	0	0
$617$	5.19986e9	0.891239	0.445619	−	0.895223i	$-0.352984\pi$
$617$	5.19986e9	0.891239	0.445619	+	0.895223i	$0.352984\pi$
$618$	1.13313e9	0.193117
$619$	7.78909e9	1.31999	0.659993	−	0.751272i	$-0.270559\pi$
$619$	7.78909e9	1.31999	0.659993	+	0.751272i	$0.270559\pi$
$620$	−5.53983e9	−0.933524
$621$	4.43366e8	0.0742919
$622$	4.50490e8	0.0750618
$623$	2.89796e9	0.480158
$624$	−6.33786e9	−1.04423
$625$	−7.45770e9	−1.22187
$626$	−5.23451e7	−0.00852836
$627$	0	0
$628$	6.75573e9	1.08846
$629$	−7.83453e9	−1.25527
$630$	−3.74368e9	−0.596495
$631$	−1.35874e9	−0.215294	−0.107647	−	0.994189i	$-0.534332\pi$
$631$	−1.35874e9	−0.215294	−0.107647	+	0.994189i	$0.534332\pi$
$632$	1.26912e8	0.0199982
$633$	6.60776e9	1.03548
$634$	6.92761e9	1.07962
$635$	5.61768e8	0.0870661
$636$	4.21061e7	0.00649000
$637$	2.34507e9	0.359474
$638$	0	0
$639$	−1.04795e9	−0.158887
$640$	1.65867e8	0.0250109
$641$	6.32968e9	0.949246	0.474623	−	0.880189i	$-0.342584\pi$
$641$	6.32968e9	0.949246	0.474623	+	0.880189i	$0.342584\pi$
$642$	4.11403e9	0.613613
$643$	−5.55981e9	−0.824748	−0.412374	−	0.911015i	$-0.635300\pi$
$643$	−5.55981e9	−0.824748	−0.412374	+	0.911015i	$0.635300\pi$
$644$	2.88482e9	0.425617
$645$	3.18877e9	0.467912
$646$	1.49146e10	2.17670
$647$	2.76364e9	0.401160	0.200580	−	0.979677i	$-0.435717\pi$
$647$	2.76364e9	0.401160	0.200580	+	0.979677i	$0.435717\pi$
$648$	−8.33845e6	−0.00120385
$649$	0	0
$650$	−6.00837e9	−0.858143
$651$	−3.56928e9	−0.507045
$652$	2.58586e9	0.365374
$653$	−8.64892e9	−1.21553	−0.607765	−	0.794117i	$-0.707934\pi$
$653$	−8.64892e9	−1.21553	−0.607765	+	0.794117i	$0.707934\pi$
$654$	−2.67081e9	−0.373354
$655$	3.79368e9	0.527493
$656$	3.25886e8	0.0450716
$657$	−2.54471e9	−0.350074
$658$	−9.26613e9	−1.26797
$659$	−6.77833e9	−0.922623	−0.461311	−	0.887238i	$-0.652621\pi$
$659$	−6.77833e9	−0.922623	−0.461311	+	0.887238i	$0.652621\pi$
$660$	0	0
$661$	−1.43126e10	−1.92759	−0.963795	−	0.266643i	$-0.914086\pi$
$661$	−1.43126e10	−1.92759	−0.963795	+	0.266643i	$0.914086\pi$
$662$	−6.03721e9	−0.808785
$663$	−8.28629e9	−1.10424
$664$	4.20136e7	0.00556931
$665$	1.40219e10	1.84898
$666$	−4.30736e9	−0.565004
$667$	4.05945e9	0.529697
$668$	−3.46846e9	−0.450214
$669$	2.24712e9	0.290158
$670$	−2.28956e9	−0.294097
$671$	0	0
$672$	7.04104e9	0.895045
$673$	−1.02245e10	−1.29297	−0.646487	−	0.762925i	$-0.723762\pi$
$673$	−1.02245e10	−1.29297	−0.646487	+	0.762925i	$0.723762\pi$
$674$	−6.39273e9	−0.804223
$675$	5.10953e8	0.0639467
$676$	1.87945e10	2.34001
$677$	−3.53176e9	−0.437452	−0.218726	−	0.975786i	$-0.570190\pi$
$677$	−3.53176e9	−0.437452	−0.218726	+	0.975786i	$0.570190\pi$
$678$	7.74387e9	0.954233
$679$	−9.34675e9	−1.14582
$680$	1.07597e8	0.0131226
$681$	−5.06984e9	−0.615148
$682$	0	0
$683$	−3.60355e9	−0.432770	−0.216385	−	0.976308i	$-0.569427\pi$
$683$	−3.60355e9	−0.432770	−0.216385	+	0.976308i	$0.569427\pi$
$684$	4.11558e9	0.491739
$685$	−1.44015e10	−1.71195
$686$	1.05235e10	1.24459
$687$	1.44158e9	0.169625
$688$	−5.95151e9	−0.696736
$689$	1.74574e8	0.0203336
$690$	−3.14540e9	−0.364505
$691$	−1.01407e10	−1.16921	−0.584605	−	0.811318i	$-0.698751\pi$
$691$	−1.01407e10	−1.16921	−0.584605	+	0.811318i	$0.698751\pi$
$692$	1.33358e10	1.52985
$693$	0	0
$694$	3.59650e8	0.0408434
$695$	−2.21810e9	−0.250631
$696$	−7.63468e7	−0.00858338
$697$	4.26073e8	0.0476616
$698$	4.42613e9	0.492640
$699$	1.87028e9	0.207127
$700$	3.32459e9	0.366349
$701$	7.60003e9	0.833301	0.416651	−	0.909067i	$-0.363204\pi$
$701$	7.60003e9	0.833301	0.416651	+	0.909067i	$0.363204\pi$
$702$	−4.55573e9	−0.497025
$703$	1.61332e10	1.75137
$704$	0	0
$705$	5.07079e9	0.545022
$706$	1.38395e10	1.48015
$707$	−9.77950e9	−1.04076
$708$	−2.21427e9	−0.234485
$709$	−1.21320e10	−1.27841	−0.639206	−	0.769036i	$-0.720737\pi$
$709$	−1.21320e10	−1.27841	−0.639206	+	0.769036i	$0.720737\pi$
$710$	7.43457e9	0.779563
$711$	−5.89656e9	−0.615255
$712$	−4.57923e7	−0.00475459
$713$	−2.99887e9	−0.309845
$714$	9.13526e9	0.939241
$715$	0	0
$716$	−1.18080e10	−1.20221
$717$	1.82786e9	0.185193
$718$	−9.53970e9	−0.961832
$719$	−1.08273e10	−1.08635	−0.543176	−	0.839619i	$-0.682778\pi$
$719$	−1.08273e10	−1.08635	−0.543176	+	0.839619i	$0.682778\pi$
$720$	−3.82367e9	−0.381783
$721$	−2.59955e9	−0.258300
$722$	−1.63836e10	−1.62005
$723$	1.53671e9	0.151219
$724$	−2.52069e9	−0.246850
$725$	4.67828e9	0.455936
$726$	0	0
$727$	−2.33605e9	−0.225482	−0.112741	−	0.993624i	$-0.535963\pi$
$727$	−2.33605e9	−0.225482	−0.112741	+	0.993624i	$0.535963\pi$
$728$	−2.24946e8	−0.0216082
$729$	3.87420e8	0.0370370
$730$	1.80531e10	1.71760
$731$	−7.78117e9	−0.736774
$732$	1.07072e10	1.00899
$733$	1.46677e10	1.37562	0.687808	−	0.725893i	$-0.258573\pi$
$733$	1.46677e10	1.37562	0.687808	+	0.725893i	$0.258573\pi$
$734$	2.38160e10	2.22296
$735$	1.41480e9	0.131428
$736$	5.91581e9	0.546943
$737$	0	0
$738$	2.34251e8	0.0214528
$739$	8.70105e8	0.0793078	0.0396539	−	0.999213i	$-0.487374\pi$
$739$	8.70105e8	0.0793078	0.0396539	+	0.999213i	$0.487374\pi$
$740$	1.53372e10	1.39134
$741$	1.70635e10	1.54065
$742$	−1.92460e8	−0.0172953
$743$	1.03941e10	0.929663	0.464831	−	0.885399i	$-0.346115\pi$
$743$	1.03941e10	0.929663	0.464831	+	0.885399i	$0.346115\pi$
$744$	5.64002e7	0.00502083
$745$	−2.23429e10	−1.97967
$746$	−1.49382e10	−1.31738
$747$	−1.95203e9	−0.171342
$748$	0	0
$749$	−9.43811e9	−0.820727
$750$	7.28436e9	0.630488
$751$	−7.76502e9	−0.668964	−0.334482	−	0.942402i	$-0.608561\pi$
$751$	−7.76502e9	−0.668964	−0.334482	+	0.942402i	$0.608561\pi$
$752$	−9.46412e9	−0.811555
$753$	−4.28440e9	−0.365685
$754$	−4.17122e10	−3.54375
$755$	2.09133e10	1.76851
$756$	2.52081e9	0.212184
$757$	−1.35473e9	−0.113506	−0.0567528	−	0.998388i	$-0.518075\pi$
$757$	−1.35473e9	−0.113506	−0.0567528	+	0.998388i	$0.518075\pi$
$758$	−2.36825e10	−1.97508
$759$	0	0
$760$	−2.21569e8	−0.0183088
$761$	3.16501e9	0.260332	0.130166	−	0.991492i	$-0.458449\pi$
$761$	3.16501e9	0.260332	0.130166	+	0.991492i	$0.458449\pi$
$762$	−7.53664e8	−0.0617071
$763$	6.12719e9	0.499373
$764$	−6.52682e8	−0.0529511
$765$	−4.99917e9	−0.403722
$766$	−2.63006e10	−2.11429
$767$	−9.18052e9	−0.734655
$768$	7.13565e9	0.568420
$769$	1.56443e10	1.24055	0.620273	−	0.784386i	$-0.287022\pi$
$769$	1.56443e10	1.24055	0.620273	+	0.784386i	$0.287022\pi$
$770$	0	0
$771$	−5.41597e9	−0.425584
$772$	−8.59396e9	−0.672253
$773$	−1.21091e9	−0.0942936	−0.0471468	−	0.998888i	$-0.515013\pi$
$773$	−1.21091e9	−0.0942936	−0.0471468	+	0.998888i	$0.515013\pi$
$774$	−4.27802e9	−0.331627
$775$	−3.45602e9	−0.266698
$776$	1.47693e8	0.0113461
$777$	9.88164e9	0.755710
$778$	1.93797e10	1.47543
$779$	−8.77387e8	−0.0664982
$780$	1.62215e10	1.22394
$781$	0	0
$782$	7.67534e9	0.573950
$783$	3.54722e9	0.264072
$784$	−2.64057e9	−0.195701
$785$	1.68984e10	1.24682
$786$	−5.08957e9	−0.373854
$787$	2.18541e10	1.59817	0.799084	−	0.601220i	$-0.205318\pi$
$787$	2.18541e10	1.59817	0.799084	+	0.601220i	$0.205318\pi$
$788$	2.43702e10	1.77426
$789$	−1.45088e10	−1.05163
$790$	4.18323e10	3.01868
$791$	−1.77654e10	−1.27632
$792$	0	0
$793$	4.43927e10	3.16122
$794$	−5.77339e9	−0.409316
$795$	1.05322e8	0.00743419
$796$	1.43653e8	0.0100953
$797$	2.09736e10	1.46747	0.733734	−	0.679437i	$-0.237776\pi$
$797$	2.09736e10	1.46747	0.733734	+	0.679437i	$0.237776\pi$
$798$	−1.88117e10	−1.31044
$799$	−1.23736e10	−0.858191
$800$	6.81762e9	0.470780
$801$	2.12760e9	0.146277
$802$	−3.56602e10	−2.44103
$803$	0	0
$804$	1.54168e9	0.104616
$805$	7.21595e9	0.487537
$806$	3.08143e10	2.07291
$807$	−1.07114e9	−0.0717447
$808$	1.54532e8	0.0103057
$809$	−2.28362e9	−0.151637	−0.0758184	−	0.997122i	$-0.524157\pi$
$809$	−2.28362e9	−0.151637	−0.0758184	+	0.997122i	$0.524157\pi$
$810$	−2.74850e9	−0.181718
$811$	6.19706e9	0.407956	0.203978	−	0.978976i	$-0.434613\pi$
$811$	6.19706e9	0.407956	0.203978	+	0.978976i	$0.434613\pi$
$812$	2.30805e10	1.51286
$813$	−1.70525e9	−0.111294
$814$	0	0
$815$	6.46813e9	0.418530
$816$	9.33045e9	0.601156
$817$	1.60233e10	1.02796
$818$	2.02831e10	1.29568
$819$	1.04514e10	0.664786
$820$	−8.34096e8	−0.0528284
$821$	−5.98432e9	−0.377410	−0.188705	−	0.982034i	$-0.560429\pi$
$821$	−5.98432e9	−0.377410	−0.188705	+	0.982034i	$0.560429\pi$
$822$	1.93209e10	1.21332
$823$	−7.55206e9	−0.472243	−0.236122	−	0.971724i	$-0.575876\pi$
$823$	−7.55206e9	−0.472243	−0.236122	+	0.971724i	$0.575876\pi$
$824$	4.10770e7	0.00255772
$825$	0	0
$826$	1.01211e10	0.624882
$827$	2.89939e10	1.78253	0.891265	−	0.453482i	$-0.149818\pi$
$827$	2.89939e10	1.78253	0.891265	+	0.453482i	$0.149818\pi$
$828$	2.11796e9	0.129661
$829$	−3.92715e9	−0.239407	−0.119703	−	0.992810i	$-0.538194\pi$
$829$	−3.92715e9	−0.239407	−0.119703	+	0.992810i	$0.538194\pi$
$830$	1.38484e10	0.840672
$831$	−9.96808e9	−0.602571
$832$	−3.07407e10	−1.85047
$833$	−3.45236e9	−0.206947
$834$	2.97578e9	0.177632
$835$	−8.67583e9	−0.515713
$836$	0	0
$837$	−2.62046e9	−0.154468
$838$	9.24371e9	0.542616
$839$	−3.17273e10	−1.85467	−0.927334	−	0.374235i	$-0.877905\pi$
$839$	−3.17273e10	−1.85467	−0.927334	+	0.374235i	$0.877905\pi$
$840$	−1.35711e8	−0.00790022
$841$	1.52284e10	0.882813
$842$	1.00621e10	0.580891
$843$	1.16456e10	0.669521
$844$	3.15652e10	1.80722
$845$	4.70116e10	2.68044
$846$	−6.80293e9	−0.386278
$847$	0	0
$848$	−1.96573e8	−0.0110697
$849$	−1.86769e10	−1.04743
$850$	8.84538e9	0.494027
$851$	8.30244e9	0.461798
$852$	−5.00607e9	−0.277305
$853$	2.55424e10	1.40909	0.704546	−	0.709658i	$-0.251151\pi$
$853$	2.55424e10	1.40909	0.704546	+	0.709658i	$0.251151\pi$
$854$	−4.89409e10	−2.68887
$855$	1.02945e10	0.563279
$856$	1.49137e8	0.00812694
$857$	7.63993e9	0.414626	0.207313	−	0.978275i	$-0.433528\pi$
$857$	7.63993e9	0.414626	0.207313	+	0.978275i	$0.433528\pi$
$858$	0	0
$859$	−2.29262e10	−1.23412	−0.617058	−	0.786917i	$-0.711676\pi$
$859$	−2.29262e10	−1.23412	−0.617058	+	0.786917i	$0.711676\pi$
$860$	1.52327e10	0.816644
$861$	−5.37402e8	−0.0286938
$862$	6.61155e9	0.351583
$863$	−3.32382e10	−1.76035	−0.880177	−	0.474646i	$-0.842576\pi$
$863$	−3.32382e10	−1.76035	−0.880177	+	0.474646i	$0.842576\pi$
$864$	5.16933e9	0.272669
$865$	3.33574e10	1.75241
$866$	−3.07863e10	−1.61081
$867$	1.11974e9	0.0583514
$868$	−1.70504e10	−0.884943
$869$	0	0
$870$	−2.51652e10	−1.29564
$871$	6.39189e9	0.327767
$872$	−9.68192e7	−0.00494486
$873$	−6.86212e9	−0.349067
$874$	−1.58054e10	−0.800784
$875$	−1.67113e10	−0.843297
$876$	−1.21561e10	−0.610983
$877$	−3.34265e9	−0.167337	−0.0836687	−	0.996494i	$-0.526664\pi$
$877$	−3.34265e9	−0.167337	−0.0836687	+	0.996494i	$0.526664\pi$
$878$	−5.55506e10	−2.76986
$879$	−1.07944e10	−0.536092
$880$	0	0
$881$	−3.41347e10	−1.68183	−0.840913	−	0.541171i	$-0.817981\pi$
$881$	−3.41347e10	−1.68183	−0.840913	+	0.541171i	$0.817981\pi$
$882$	−1.89808e9	−0.0931481
$883$	−3.78866e8	−0.0185192	−0.00925962	−	0.999957i	$-0.502947\pi$
$883$	−3.78866e8	−0.0185192	−0.00925962	+	0.999957i	$0.502947\pi$
$884$	−3.95835e10	−1.92722
$885$	−5.53867e9	−0.268599
$886$	−3.66239e9	−0.176908
$887$	9.48340e9	0.456280	0.228140	−	0.973628i	$-0.426736\pi$
$887$	9.48340e9	0.456280	0.228140	+	0.973628i	$0.426736\pi$
$888$	−1.56145e8	−0.00748314
$889$	1.72900e9	0.0825352
$890$	−1.50940e10	−0.717692
$891$	0	0
$892$	1.07345e10	0.506411
$893$	2.54803e10	1.19736
$894$	2.99750e10	1.40307
$895$	−2.95360e10	−1.37712
$896$	5.10503e8	0.0237094
$897$	8.78118e9	0.406237
$898$	−3.09796e10	−1.42761
$899$	−2.39929e10	−1.10135
$900$	2.44082e9	0.111606
$901$	−2.57005e8	−0.0117059
$902$	0	0
$903$	9.81433e9	0.443561
$904$	2.80722e8	0.0126382
$905$	−6.30511e9	−0.282763
$906$	−2.80571e10	−1.25341
$907$	3.10341e10	1.38107	0.690533	−	0.723301i	$-0.257376\pi$
$907$	3.10341e10	1.38107	0.690533	+	0.723301i	$0.257376\pi$
$908$	−2.42186e10	−1.07361
$909$	−7.17983e9	−0.317059
$910$	−7.41462e10	−3.26170
$911$	2.61852e10	1.14747	0.573735	−	0.819041i	$-0.305494\pi$
$911$	2.61852e10	1.14747	0.573735	+	0.819041i	$0.305494\pi$
$912$	−1.92137e10	−0.838741
$913$	0	0
$914$	1.50724e10	0.652937
$915$	2.67824e10	1.15578
$916$	6.88641e9	0.296046
$917$	1.16761e10	0.500042
$918$	6.70684e9	0.286134
$919$	2.39284e10	1.01697	0.508487	−	0.861070i	$-0.330205\pi$
$919$	2.39284e10	1.01697	0.508487	+	0.861070i	$0.330205\pi$
$920$	−1.14023e8	−0.00482766
$921$	5.71361e9	0.240992
$922$	−2.04131e10	−0.857732
$923$	−2.07555e10	−0.868814
$924$	0	0
$925$	9.56808e9	0.397492
$926$	−4.94473e10	−2.04647
$927$	−1.90852e9	−0.0786895
$928$	4.73304e10	1.94411
$929$	−3.19314e8	−0.0130666	−0.00653330	−	0.999979i	$-0.502080\pi$
$929$	−3.19314e8	−0.0130666	−0.00653330	+	0.999979i	$0.502080\pi$
$930$	1.85905e10	0.757880
$931$	7.10924e9	0.288735
$932$	8.93433e9	0.361498
$933$	−7.58753e8	−0.0305854
$934$	−3.07085e10	−1.23323
$935$	0	0
$936$	−1.65149e8	−0.00658280
$937$	−2.24942e10	−0.893270	−0.446635	−	0.894716i	$-0.647378\pi$
$937$	−2.24942e10	−0.893270	−0.446635	+	0.894716i	$0.647378\pi$
$938$	−7.04677e9	−0.278792
$939$	8.81639e7	0.00347505
$940$	2.42231e10	0.951224
$941$	−3.22210e10	−1.26059	−0.630296	−	0.776355i	$-0.717067\pi$
$941$	−3.22210e10	−1.26059	−0.630296	+	0.776355i	$0.717067\pi$
$942$	−2.26708e10	−0.883667
$943$	−4.51520e8	−0.0175342
$944$	1.03374e10	0.399952
$945$	6.30542e9	0.243054
$946$	0	0
$947$	2.26003e10	0.864747	0.432373	−	0.901695i	$-0.357676\pi$
$947$	2.26003e10	0.864747	0.432373	+	0.901695i	$0.357676\pi$
$948$	−2.81678e10	−1.07380
$949$	−5.03999e10	−1.91424
$950$	−1.82148e10	−0.689273
$951$	−1.16680e10	−0.439913
$952$	3.31161e8	0.0124397
$953$	−2.14078e10	−0.801211	−0.400606	−	0.916251i	$-0.631200\pi$
$953$	−2.14078e10	−0.801211	−0.400606	+	0.916251i	$0.631200\pi$
$954$	−1.41299e8	−0.00526889
$955$	−1.63258e9	−0.0606546
$956$	8.73166e9	0.323217
$957$	0	0
$958$	−2.33620e10	−0.858480
$959$	−4.43247e10	−1.62286
$960$	−1.85461e10	−0.676555
$961$	−9.78818e9	−0.355771
$962$	−8.53103e10	−3.08950
$963$	−6.92919e9	−0.250029
$964$	7.34084e9	0.263922
$965$	−2.14965e10	−0.770055
$966$	−9.68085e9	−0.345536
$967$	1.75493e10	0.624118	0.312059	−	0.950063i	$-0.398981\pi$
$967$	1.75493e10	0.624118	0.312059	+	0.950063i	$0.398981\pi$
$968$	0	0
$969$	−2.51205e10	−0.886940
$970$	4.86823e10	1.71266
$971$	1.27674e9	0.0447545	0.0223772	−	0.999750i	$-0.492877\pi$
$971$	1.27674e9	0.0447545	0.0223772	+	0.999750i	$0.492877\pi$
$972$	1.85070e9	0.0646406
$973$	−6.82683e9	−0.237588
$974$	−1.67740e10	−0.581676
$975$	1.01198e10	0.349668
$976$	−4.99866e10	−1.72099
$977$	1.99064e10	0.682908	0.341454	−	0.939898i	$-0.389081\pi$
$977$	1.99064e10	0.682908	0.341454	+	0.939898i	$0.389081\pi$
$978$	−8.67759e9	−0.296629
$979$	0	0
$980$	6.75847e9	0.229381
$981$	4.49840e9	0.152131
$982$	3.65462e10	1.23155
$983$	−3.29569e10	−1.10665	−0.553324	−	0.832966i	$-0.686641\pi$
$983$	−3.29569e10	−1.10665	−0.553324	+	0.832966i	$0.686641\pi$
$984$	8.49180e6	0.000284130 0
$985$	6.09583e10	2.03239
$986$	6.14078e10	2.04011
$987$	1.56068e10	0.516658
$988$	8.15120e10	2.68889
$989$	8.24590e9	0.271051
$990$	0	0
$991$	1.80350e10	0.588652	0.294326	−	0.955705i	$-0.404905\pi$
$991$	1.80350e10	0.588652	0.294326	+	0.955705i	$0.404905\pi$
$992$	−3.49647e10	−1.13720
$993$	1.01684e10	0.329556
$994$	2.28820e10	0.738994
$995$	3.59327e8	0.0115640
$996$	−9.32483e9	−0.299043
$997$	−9.32859e9	−0.298115	−0.149057	−	0.988829i	$-0.547624\pi$
$997$	−9.32859e9	−0.298115	−0.149057	+	0.988829i	$0.547624\pi$
$998$	7.66512e9	0.244097
$999$	7.25481e9	0.230222

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.8.a.r.1.2		14
11.7	odd	10		33.8.e.b.16.7	✓	28
11.8	odd	10		33.8.e.b.31.7	yes	28
11.10	odd	2		363.8.a.o.1.13		14
33.8	even	10		99.8.f.b.64.1		28
33.29	even	10		99.8.f.b.82.1		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.e.b.16.7	✓	28	11.7	odd	10
33.8.e.b.31.7	yes	28	11.8	odd	10
99.8.f.b.64.1		28	33.8	even	10
99.8.f.b.82.1		28	33.29	even	10
363.8.a.o.1.13		14	11.10	odd	2
363.8.a.r.1.2		14	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	363.a (trivial)

\(f(q)\)	\(=\)	\(q-16.0306 q^{2} +27.0000 q^{3} +128.979 q^{4} +322.621 q^{5} -432.825 q^{6} +992.956 q^{7} -15.6903 q^{8} +729.000 q^{9} -5171.79 q^{10} +3482.43 q^{12} +14438.4 q^{13} -15917.6 q^{14} +8710.76 q^{15} -16257.8 q^{16} -21255.8 q^{17} -11686.3 q^{18} +43770.9 q^{19} +41611.2 q^{20} +26809.8 q^{21} +22525.3 q^{23} -423.637 q^{24} +25959.1 q^{25} -231455. q^{26} +19683.0 q^{27} +128070. q^{28} +180217. q^{29} -139638. q^{30} -133133. q^{31} +262629. q^{32} +340743. q^{34} +320348. q^{35} +94025.5 q^{36} +368583. q^{37} -701672. q^{38} +389836. q^{39} -5062.01 q^{40} -20045.0 q^{41} -429776. q^{42} +366072. q^{43} +235190. q^{45} -361093. q^{46} +582130. q^{47} -438959. q^{48} +162419. q^{49} -416139. q^{50} -573908. q^{51} +1.86224e6 q^{52} +12091.0 q^{53} -315529. q^{54} -15579.8 q^{56} +1.18181e6 q^{57} -2.88898e6 q^{58} -635842. q^{59} +1.12350e6 q^{60} +3.07463e6 q^{61} +2.13420e6 q^{62} +723865. q^{63} -2.12910e6 q^{64} +4.65811e6 q^{65} +442702. q^{67} -2.74155e6 q^{68} +608184. q^{69} -5.13536e6 q^{70} -1.43752e6 q^{71} -11438.2 q^{72} -3.49069e6 q^{73} -5.90859e6 q^{74} +700896. q^{75} +5.64552e6 q^{76} -6.24928e6 q^{78} -8.08855e6 q^{79} -5.24509e6 q^{80} +531441. q^{81} +321332. q^{82} -2.67768e6 q^{83} +3.45790e6 q^{84} -6.85757e6 q^{85} -5.86834e6 q^{86} +4.86587e6 q^{87} +2.91852e6 q^{89} -3.77023e6 q^{90} +1.43367e7 q^{91} +2.90529e6 q^{92} -3.59459e6 q^{93} -9.33186e6 q^{94} +1.41214e7 q^{95} +7.09099e6 q^{96} -9.41306e6 q^{97} -2.60367e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 23 q^{2} + 378 q^{3} + 845 q^{4} + 69 q^{5} + 621 q^{6} + 3278 q^{7} + 4602 q^{8} + 10206 q^{9} + 5320 q^{10} + 22815 q^{12} + 32188 q^{13} - 7794 q^{14} + 1863 q^{15} + 86137 q^{16} + 42917 q^{17}+ \cdots - 70266089 q^{98}+O(q^{100}) \) 14 * q + 23 * q^2 + 378 * q^3 + 845 * q^4 + 69 * q^5 + 621 * q^6 + 3278 * q^7 + 4602 * q^8 + 10206 * q^9 + 5320 * q^10 + 22815 * q^12 + 32188 * q^13 - 7794 * q^14 + 1863 * q^15 + 86137 * q^16 + 42917 * q^17 + 16767 * q^18 + 50419 * q^19 + 54475 * q^20 + 88506 * q^21 - 83492 * q^23 + 124254 * q^24 + 336991 * q^25 + 234897 * q^26 + 275562 * q^27 + 697184 * q^28 + 227028 * q^29 + 143640 * q^30 - 36481 * q^31 + 151749 * q^32 - 614562 * q^34 - 258783 * q^35 + 616005 * q^36 + 634524 * q^37 - 662891 * q^38 + 869076 * q^39 + 1585604 * q^40 + 1286544 * q^41 - 210438 * q^42 + 2632644 * q^43 + 50301 * q^45 + 1290627 * q^46 - 1692245 * q^47 + 2325699 * q^48 + 842030 * q^49 + 7870914 * q^50 + 1158759 * q^51 + 6419781 * q^52 + 1211317 * q^53 + 452709 * q^54 - 9100953 * q^56 + 1361313 * q^57 - 6265711 * q^58 + 2551943 * q^59 + 1470825 * q^60 + 10221163 * q^61 + 5723657 * q^62 + 2389662 * q^63 + 6349332 * q^64 + 7684087 * q^65 - 3570631 * q^67 + 13112925 * q^68 - 2254284 * q^69 + 3559265 * q^70 + 4528437 * q^71 + 3354858 * q^72 + 13052276 * q^73 + 4896205 * q^74 + 9098757 * q^75 + 9845214 * q^76 + 6342219 * q^78 + 14911090 * q^79 - 9395382 * q^80 + 7440174 * q^81 + 17478001 * q^82 + 7068900 * q^83 + 18823968 * q^84 + 24423120 * q^85 + 12792479 * q^86 + 6129756 * q^87 + 4800832 * q^89 + 3878280 * q^90 - 15295476 * q^91 - 3195753 * q^92 - 984987 * q^93 - 43320002 * q^94 + 7614177 * q^95 + 4097223 * q^96 - 11578773 * q^97 - 70266089 * q^98

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