LMFDB - Embedded newform 363.8.a.r.1.9

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			1.9
Root			$9.29307$ 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)

$f(q)$	$=$	\(q+9.67503 q^{2} +27.0000 q^{3} -34.3938 q^{4} +124.761 q^{5} +261.226 q^{6} +573.194 q^{7} -1571.16 q^{8} +729.000 q^{9} +1207.07 q^{10} -928.631 q^{12} -3163.99 q^{13} +5545.67 q^{14} +3368.55 q^{15} -10798.7 q^{16} +18563.9 q^{17} +7053.10 q^{18} -2446.74 q^{19} -4291.00 q^{20} +15476.2 q^{21} +53581.5 q^{23} -42421.4 q^{24} -62559.7 q^{25} -30611.7 q^{26} +19683.0 q^{27} -19714.3 q^{28} +49944.1 q^{29} +32590.8 q^{30} +4084.45 q^{31} +96631.6 q^{32} +179606. q^{34} +71512.3 q^{35} -25073.0 q^{36} +410718. q^{37} -23672.3 q^{38} -85427.8 q^{39} -196020. q^{40} +476351. q^{41} +149733. q^{42} -384285. q^{43} +90950.8 q^{45} +518403. q^{46} +275873. q^{47} -291564. q^{48} -494991. q^{49} -605267. q^{50} +501224. q^{51} +108822. q^{52} +1.25843e6 q^{53} +190434. q^{54} -900583. q^{56} -66062.0 q^{57} +483211. q^{58} -2.25283e6 q^{59} -115857. q^{60} +2.95340e6 q^{61} +39517.2 q^{62} +417859. q^{63} +2.31714e6 q^{64} -394743. q^{65} +3.06864e6 q^{67} -638481. q^{68} +1.44670e6 q^{69} +691884. q^{70} -888980. q^{71} -1.14538e6 q^{72} -4.33847e6 q^{73} +3.97371e6 q^{74} -1.68911e6 q^{75} +84152.6 q^{76} -826516. q^{78} -1.05430e6 q^{79} -1.34725e6 q^{80} +531441. q^{81} +4.60871e6 q^{82} +7.07334e6 q^{83} -532286. q^{84} +2.31605e6 q^{85} -3.71797e6 q^{86} +1.34849e6 q^{87} +5.68686e6 q^{89} +879952. q^{90} -1.81358e6 q^{91} -1.84287e6 q^{92} +110280. q^{93} +2.66908e6 q^{94} -305258. q^{95} +2.60905e6 q^{96} +9.89847e6 q^{97} -4.78906e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	9.67503	0.855160	0.427580	−	0.903977i	$-0.359366\pi$
$2$	9.67503	0.855160	0.427580	+	0.903977i	$0.359366\pi$
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	−34.3938	−0.268701
$5$	124.761	0.446359	0.223179	−	0.974777i	$-0.428356\pi$
$5$	124.761	0.446359	0.223179	+	0.974777i	$0.428356\pi$
$6$	261.226	0.493727
$7$	573.194	0.631624	0.315812	−	0.948822i	$-0.397723\pi$
$7$	573.194	0.631624	0.315812	+	0.948822i	$0.397723\pi$
$8$	−1571.16	−1.08494
$9$	729.000	0.333333
$10$	1207.07	0.381708
$11$	0	0
$11$	0	0
$12$	−928.631	−0.155135
$13$	−3163.99	−0.399423	−0.199712	−	0.979855i	$-0.564001\pi$
$13$	−3163.99	−0.399423	−0.199712	+	0.979855i	$0.564001\pi$
$14$	5545.67	0.540140
$15$	3368.55	0.257705
$16$	−10798.7	−0.659098
$17$	18563.9	0.916426	0.458213	−	0.888843i	$-0.348490\pi$
$17$	18563.9	0.916426	0.458213	+	0.888843i	$0.348490\pi$
$18$	7053.10	0.285053
$19$	−2446.74	−0.0818371	−0.0409186	−	0.999162i	$-0.513028\pi$
$19$	−2446.74	−0.0818371	−0.0409186	+	0.999162i	$0.513028\pi$
$20$	−4291.00	−0.119937
$21$	15476.2	0.364668
$22$	0	0
$23$	53581.5	0.918264	0.459132	−	0.888368i	$-0.348160\pi$
$23$	53581.5	0.918264	0.459132	+	0.888368i	$0.348160\pi$
$24$	−42421.4	−0.626392
$25$	−62559.7	−0.800764
$26$	−30611.7	−0.341571
$27$	19683.0	0.192450
$28$	−19714.3	−0.169718
$29$	49944.1	0.380269	0.190135	−	0.981758i	$-0.439108\pi$
$29$	49944.1	0.380269	0.190135	+	0.981758i	$0.439108\pi$
$30$	32590.8	0.220379
$31$	4084.45	0.0246245	0.0123123	−	0.999924i	$-0.496081\pi$
$31$	4084.45	0.0246245	0.0123123	+	0.999924i	$0.496081\pi$
$32$	96631.6	0.521308
$33$	0	0
$34$	179606.	0.783691
$35$	71512.3	0.281931
$36$	−25073.0	−0.0895671
$37$	410718.	1.33302	0.666511	−	0.745495i	$-0.267787\pi$
$37$	410718.	1.33302	0.666511	+	0.745495i	$0.267787\pi$
$38$	−23672.3	−0.0699839
$39$	−85427.8	−0.230607
$40$	−196020.	−0.484273
$41$	476351.	1.07940	0.539702	−	0.841856i	$-0.318537\pi$
$41$	476351.	1.07940	0.539702	+	0.841856i	$0.318537\pi$
$42$	149733.	0.311850
$43$	−384285.	−0.737078	−0.368539	−	0.929612i	$-0.620142\pi$
$43$	−384285.	−0.737078	−0.368539	+	0.929612i	$0.620142\pi$
$44$	0	0
$45$	90950.8	0.148786
$46$	518403.	0.785263
$47$	275873.	0.387585	0.193793	−	0.981043i	$-0.437921\pi$
$47$	275873.	0.387585	0.193793	+	0.981043i	$0.437921\pi$
$48$	−291564.	−0.380531
$49$	−494991.	−0.601051
$50$	−605267.	−0.684781
$51$	501224.	0.529099
$52$	108822.	0.107326
$53$	1.25843e6	1.16109	0.580543	−	0.814230i	$-0.302840\pi$
$53$	1.25843e6	1.16109	0.580543	+	0.814230i	$0.302840\pi$
$54$	190434.	0.164576
$55$	0	0
$56$	−900583.	−0.685276
$57$	−66062.0	−0.0472487
$58$	483211.	0.325191
$59$	−2.25283e6	−1.42806	−0.714031	−	0.700114i	$-0.753133\pi$
$59$	−2.25283e6	−1.42806	−0.714031	+	0.700114i	$0.753133\pi$
$60$	−115857.	−0.0692457
$61$	2.95340e6	1.66597	0.832987	−	0.553292i	$-0.186629\pi$
$61$	2.95340e6	1.66597	0.832987	+	0.553292i	$0.186629\pi$
$62$	39517.2	0.0210579
$63$	417859.	0.210541
$64$	2.31714e6	1.10490
$65$	−394743.	−0.178286
$66$	0	0
$67$	3.06864e6	1.24647	0.623237	−	0.782033i	$-0.285817\pi$
$67$	3.06864e6	1.24647	0.623237	+	0.782033i	$0.285817\pi$
$68$	−638481.	−0.246245
$69$	1.44670e6	0.530160
$70$	691884.	0.241096
$71$	−888980.	−0.294773	−0.147387	−	0.989079i	$-0.547086\pi$
$71$	−888980.	−0.294773	−0.147387	+	0.989079i	$0.547086\pi$
$72$	−1.14538e6	−0.361648
$73$	−4.33847e6	−1.30529	−0.652645	−	0.757664i	$-0.726340\pi$
$73$	−4.33847e6	−1.30529	−0.652645	+	0.757664i	$0.726340\pi$
$74$	3.97371e6	1.13995
$75$	−1.68911e6	−0.462321
$76$	84152.6	0.0219897
$77$	0	0
$78$	−826516.	−0.197206
$79$	−1.05430e6	−0.240586	−0.120293	−	0.992738i	$-0.538383\pi$
$79$	−1.05430e6	−0.240586	−0.120293	+	0.992738i	$0.538383\pi$
$80$	−1.34725e6	−0.294194
$81$	531441.	0.111111
$82$	4.60871e6	0.923063
$83$	7.07334e6	1.35785	0.678925	−	0.734208i	$-0.262446\pi$
$83$	7.07334e6	1.35785	0.678925	+	0.734208i	$0.262446\pi$
$84$	−532286.	−0.0979868
$85$	2.31605e6	0.409054
$86$	−3.71797e6	−0.630320
$87$	1.34849e6	0.219549
$88$	0	0
$89$	5.68686e6	0.855081	0.427541	−	0.903996i	$-0.359380\pi$
$89$	5.68686e6	0.855081	0.427541	+	0.903996i	$0.359380\pi$
$90$	879952.	0.127236
$91$	−1.81358e6	−0.252286
$92$	−1.84287e6	−0.246739
$93$	110280.	0.0142170
$94$	2.66908e6	0.331448
$95$	−305258.	−0.0365287
$96$	2.60905e6	0.300977
$97$	9.89847e6	1.10120	0.550601	−	0.834769i	$-0.314399\pi$
$97$	9.89847e6	1.10120	0.550601	+	0.834769i	$0.314399\pi$
$98$	−4.78906e6	−0.513995
$99$	0	0
$100$	2.15166e6	0.215166
$101$	−565721.	−0.0546358	−0.0273179	−	0.999627i	$-0.508697\pi$
$101$	−565721.	−0.0546358	−0.0273179	+	0.999627i	$0.508697\pi$
$102$	4.84936e6	0.452464
$103$	−2.32693e6	−0.209823	−0.104911	−	0.994482i	$-0.533456\pi$
$103$	−2.32693e6	−0.209823	−0.104911	+	0.994482i	$0.533456\pi$
$104$	4.97115e6	0.433352
$105$	1.93083e6	0.162773
$106$	1.21754e7	0.992914
$107$	5.37424e6	0.424105	0.212053	−	0.977258i	$-0.431985\pi$
$107$	5.37424e6	0.424105	0.212053	+	0.977258i	$0.431985\pi$
$108$	−676972.	−0.0517116
$109$	7.78058e6	0.575466	0.287733	−	0.957711i	$-0.407099\pi$
$109$	7.78058e6	0.575466	0.287733	+	0.957711i	$0.407099\pi$
$110$	0	0
$111$	1.10894e7	0.769621
$112$	−6.18974e6	−0.416302
$113$	1.18509e7	0.772642	0.386321	−	0.922364i	$-0.373746\pi$
$113$	1.18509e7	0.772642	0.386321	+	0.922364i	$0.373746\pi$
$114$	−639152.	−0.0404052
$115$	6.68489e6	0.409875
$116$	−1.71777e6	−0.102179
$117$	−2.30655e6	−0.133141
$118$	−2.17962e7	−1.22122
$119$	1.06407e7	0.578837
$120$	−5.29254e6	−0.279595
$121$	0	0
$122$	2.85743e7	1.42468
$123$	1.28615e7	0.623194
$124$	−140480.	−0.00661663
$125$	−1.75520e7	−0.803786
$126$	4.04280e6	0.180047
$127$	5.81235e6	0.251790	0.125895	−	0.992044i	$-0.459820\pi$
$127$	5.81235e6	0.251790	0.125895	+	0.992044i	$0.459820\pi$
$128$	1.00496e7	0.423559
$129$	−1.03757e7	−0.425552
$130$	−3.81915e6	−0.152463
$131$	3.03352e7	1.17896	0.589478	−	0.807784i	$-0.299333\pi$
$131$	3.03352e7	1.17896	0.589478	+	0.807784i	$0.299333\pi$
$132$	0	0
$133$	−1.40246e6	−0.0516903
$134$	2.96891e7	1.06594
$135$	2.45567e6	0.0859018
$136$	−2.91669e7	−0.994269
$137$	4.03402e7	1.34034	0.670172	−	0.742205i	$-0.266220\pi$
$137$	4.03402e7	1.34034	0.670172	+	0.742205i	$0.266220\pi$
$138$	1.39969e7	0.453372
$139$	2.96923e6	0.0937761	0.0468881	−	0.998900i	$-0.485070\pi$
$139$	2.96923e6	0.0937761	0.0468881	+	0.998900i	$0.485070\pi$
$140$	−2.45958e6	−0.0757552
$141$	7.44858e6	0.223773
$142$	−8.60091e6	−0.252078
$143$	0	0
$144$	−7.87223e6	−0.219699
$145$	6.23108e6	0.169736
$146$	−4.19748e7	−1.11623
$147$	−1.33648e7	−0.347017
$148$	−1.41261e7	−0.358185
$149$	1.68066e7	0.416225	0.208112	−	0.978105i	$-0.433268\pi$
$149$	1.68066e7	0.416225	0.208112	+	0.978105i	$0.433268\pi$
$150$	−1.63422e7	−0.395359
$151$	3.24440e7	0.766858	0.383429	−	0.923570i	$-0.374743\pi$
$151$	3.24440e7	0.766858	0.383429	+	0.923570i	$0.374743\pi$
$152$	3.84423e6	0.0887886
$153$	1.35331e7	0.305475
$154$	0	0
$155$	509580.	0.0109914
$156$	2.93818e6	0.0619645
$157$	−8.61870e7	−1.77743	−0.888716	−	0.458458i	$-0.848402\pi$
$157$	−8.61870e7	−1.77743	−0.888716	+	0.458458i	$0.848402\pi$
$158$	−1.02004e7	−0.205740
$159$	3.39777e7	0.670353
$160$	1.20559e7	0.232690
$161$	3.07126e7	0.579998
$162$	5.14171e6	0.0950178
$163$	−9.85445e7	−1.78228	−0.891139	−	0.453729i	$-0.850093\pi$
$163$	−9.85445e7	−1.78228	−0.891139	+	0.453729i	$0.850093\pi$
$164$	−1.63835e7	−0.290037
$165$	0	0
$166$	6.84348e7	1.16118
$167$	7.93480e7	1.31834	0.659172	−	0.751992i	$-0.270907\pi$
$167$	7.93480e7	1.31834	0.659172	+	0.751992i	$0.270907\pi$
$168$	−2.43157e7	−0.395644
$169$	−5.27377e7	−0.840461
$170$	2.24078e7	0.349807
$171$	−1.78367e6	−0.0272790
$172$	1.32170e7	0.198054
$173$	−9.93434e7	−1.45874	−0.729369	−	0.684120i	$-0.760186\pi$
$173$	−9.93434e7	−1.45874	−0.729369	+	0.684120i	$0.760186\pi$
$174$	1.30467e7	0.187749
$175$	−3.58589e7	−0.505782
$176$	0	0
$177$	−6.08265e7	−0.824492
$178$	5.50205e7	0.731231
$179$	−4.54952e6	−0.0592898	−0.0296449	−	0.999560i	$-0.509438\pi$
$179$	−4.54952e6	−0.0592898	−0.0296449	+	0.999560i	$0.509438\pi$
$180$	−3.12814e6	−0.0399790
$181$	5.07274e7	0.635869	0.317934	−	0.948113i	$-0.397011\pi$
$181$	5.07274e7	0.635869	0.317934	+	0.948113i	$0.397011\pi$
$182$	−1.75465e7	−0.215744
$183$	7.97419e7	0.961851
$184$	−8.41854e7	−0.996264
$185$	5.12416e7	0.595006
$186$	1.06696e6	0.0121578
$187$	0	0
$188$	−9.48832e6	−0.104145
$189$	1.12822e7	0.121556
$190$	−2.95338e6	−0.0312379
$191$	−4.83473e7	−0.502060	−0.251030	−	0.967979i	$-0.580769\pi$
$191$	−4.83473e7	−0.502060	−0.251030	+	0.967979i	$0.580769\pi$
$192$	6.25629e7	0.637914
$193$	1.74942e8	1.75163	0.875817	−	0.482643i	$-0.160323\pi$
$193$	1.74942e8	1.75163	0.875817	+	0.482643i	$0.160323\pi$
$194$	9.57680e7	0.941704
$195$	−1.06581e7	−0.102934
$196$	1.70246e7	0.161503
$197$	−2.11899e7	−0.197468	−0.0987341	−	0.995114i	$-0.531479\pi$
$197$	−2.11899e7	−0.197468	−0.0987341	+	0.995114i	$0.531479\pi$
$198$	0	0
$199$	−7.73307e7	−0.695611	−0.347805	−	0.937567i	$-0.613073\pi$
$199$	−7.73307e7	−0.695611	−0.347805	+	0.937567i	$0.613073\pi$
$200$	9.82916e7	0.868783
$201$	8.28531e7	0.719652
$202$	−5.47337e6	−0.0467223
$203$	2.86277e7	0.240187
$204$	−1.72390e7	−0.142169
$205$	5.94301e7	0.481801
$206$	−2.25131e7	−0.179432
$207$	3.90609e7	0.306088
$208$	3.41669e7	0.263259
$209$	0	0
$210$	1.86809e7	0.139197
$211$	2.63850e8	1.93361	0.966803	−	0.255522i	$-0.0822474\pi$
$211$	2.63850e8	1.93361	0.966803	+	0.255522i	$0.0822474\pi$
$212$	−4.32822e7	−0.311985
$213$	−2.40025e7	−0.170187
$214$	5.19959e7	0.362678
$215$	−4.79437e7	−0.329001
$216$	−3.09252e7	−0.208797
$217$	2.34118e6	0.0155534
$218$	7.52774e7	0.492115
$219$	−1.17139e8	−0.753609
$220$	0	0
$221$	−5.87359e7	−0.366042
$222$	1.07290e8	0.658149
$223$	−2.57211e8	−1.55318	−0.776591	−	0.630005i	$-0.783053\pi$
$223$	−2.57211e8	−1.55318	−0.776591	+	0.630005i	$0.783053\pi$
$224$	5.53887e7	0.329271
$225$	−4.56060e7	−0.266921
$226$	1.14658e8	0.660733
$227$	8.87259e6	0.0503454	0.0251727	−	0.999683i	$-0.491986\pi$
$227$	8.87259e6	0.0503454	0.0251727	+	0.999683i	$0.491986\pi$
$228$	2.27212e6	0.0126958
$229$	−1.46442e8	−0.805825	−0.402912	−	0.915239i	$-0.632002\pi$
$229$	−1.46442e8	−0.805825	−0.402912	+	0.915239i	$0.632002\pi$
$230$	6.46765e7	0.350509
$231$	0	0
$232$	−7.84704e7	−0.412570
$233$	3.33331e8	1.72635	0.863176	−	0.504903i	$-0.168472\pi$
$233$	3.33331e8	1.72635	0.863176	+	0.504903i	$0.168472\pi$
$234$	−2.23159e7	−0.113857
$235$	3.44183e7	0.173002
$236$	7.74835e7	0.383722
$237$	−2.84662e7	−0.138903
$238$	1.02949e8	0.494998
$239$	1.44380e8	0.684090	0.342045	−	0.939684i	$-0.388881\pi$
$239$	1.44380e8	0.684090	0.342045	+	0.939684i	$0.388881\pi$
$240$	−3.63758e7	−0.169853
$241$	3.50011e8	1.61073	0.805363	−	0.592782i	$-0.201970\pi$
$241$	3.50011e8	1.61073	0.805363	+	0.592782i	$0.201970\pi$
$242$	0	0
$243$	1.43489e7	0.0641500
$244$	−1.01579e8	−0.447649
$245$	−6.17556e7	−0.268284
$246$	1.24435e8	0.532930
$247$	7.74147e6	0.0326877
$248$	−6.41734e6	−0.0267162
$249$	1.90980e8	0.783954
$250$	−1.69816e8	−0.687366
$251$	−1.51800e8	−0.605917	−0.302959	−	0.953004i	$-0.597974\pi$
$251$	−1.51800e8	−0.605917	−0.302959	+	0.953004i	$0.597974\pi$
$252$	−1.43717e7	−0.0565727
$253$	0	0
$254$	5.62347e7	0.215321
$255$	6.25333e7	0.236168
$256$	−1.99364e8	−0.742690
$257$	−4.66807e8	−1.71542	−0.857712	−	0.514130i	$-0.828115\pi$
$257$	−4.66807e8	−1.71542	−0.857712	+	0.514130i	$0.828115\pi$
$258$	−1.00385e8	−0.363915
$259$	2.35421e8	0.841969
$260$	1.35767e7	0.0479057
$261$	3.64093e7	0.126756
$262$	2.93494e8	1.00820
$263$	−3.64927e8	−1.23698	−0.618488	−	0.785794i	$-0.712254\pi$
$263$	−3.64927e8	−1.23698	−0.618488	+	0.785794i	$0.712254\pi$
$264$	0	0
$265$	1.57003e8	0.518261
$266$	−1.35688e7	−0.0442035
$267$	1.53545e8	0.493681
$268$	−1.05542e8	−0.334929
$269$	−5.74730e6	−0.0180024	−0.00900121	−	0.999959i	$-0.502865\pi$
$269$	−5.74730e6	−0.0180024	−0.00900121	+	0.999959i	$0.502865\pi$
$270$	2.37587e7	0.0734597
$271$	−2.57543e8	−0.786062	−0.393031	−	0.919525i	$-0.628574\pi$
$271$	−2.57543e8	−0.786062	−0.393031	+	0.919525i	$0.628574\pi$
$272$	−2.00465e8	−0.604015
$273$	−4.89667e7	−0.145657
$274$	3.90293e8	1.14621
$275$	0	0
$276$	−4.97575e7	−0.142455
$277$	−6.73103e8	−1.90284	−0.951420	−	0.307895i	$-0.900375\pi$
$277$	−6.73103e8	−1.90284	−0.951420	+	0.307895i	$0.900375\pi$
$278$	2.87274e7	0.0801936
$279$	2.97756e6	0.00820817
$280$	−1.12358e8	−0.305879
$281$	1.34371e8	0.361271	0.180636	−	0.983550i	$-0.442185\pi$
$281$	1.34371e8	0.361271	0.180636	+	0.983550i	$0.442185\pi$
$282$	7.20653e7	0.191361
$283$	2.37735e8	0.623507	0.311754	−	0.950163i	$-0.399084\pi$
$283$	2.37735e8	0.623507	0.311754	+	0.950163i	$0.399084\pi$
$284$	3.05754e7	0.0792059
$285$	−8.24197e6	−0.0210899
$286$	0	0
$287$	2.73042e8	0.681777
$288$	7.04445e7	0.173769
$289$	−6.57215e7	−0.160164
$290$	6.02859e7	0.145152
$291$	2.67259e8	0.635779
$292$	1.49216e8	0.350733
$293$	1.79274e8	0.416370	0.208185	−	0.978089i	$-0.433244\pi$
$293$	1.79274e8	0.416370	0.208185	+	0.978089i	$0.433244\pi$
$294$	−1.29305e8	−0.296755
$295$	−2.81066e8	−0.637428
$296$	−6.45305e8	−1.44625
$297$	0	0
$298$	1.62604e8	0.355939
$299$	−1.69532e8	−0.366776
$300$	5.80949e7	0.124226
$301$	−2.20270e8	−0.465556
$302$	3.13897e8	0.655786
$303$	−1.52745e7	−0.0315440
$304$	2.64216e7	0.0539387
$305$	3.68470e8	0.743622
$306$	1.30933e8	0.261230
$307$	−5.67323e7	−0.111904	−0.0559521	−	0.998433i	$-0.517819\pi$
$307$	−5.67323e7	−0.111904	−0.0559521	+	0.998433i	$0.517819\pi$
$308$	0	0
$309$	−6.28270e7	−0.121141
$310$	4.93020e6	0.00939937
$311$	3.92348e8	0.739622	0.369811	−	0.929107i	$-0.379422\pi$
$311$	3.92348e8	0.739622	0.369811	+	0.929107i	$0.379422\pi$
$312$	1.34221e8	0.250196
$313$	1.21098e8	0.223220	0.111610	−	0.993752i	$-0.464399\pi$
$313$	1.21098e8	0.223220	0.111610	+	0.993752i	$0.464399\pi$
$314$	−8.33862e8	−1.51999
$315$	5.21325e7	0.0939770
$316$	3.62615e7	0.0646458
$317$	1.00505e9	1.77206	0.886030	−	0.463628i	$-0.153453\pi$
$317$	1.00505e9	1.77206	0.886030	+	0.463628i	$0.153453\pi$
$318$	3.28735e8	0.573259
$319$	0	0
$320$	2.89089e8	0.493182
$321$	1.45104e8	0.244857
$322$	2.97146e8	0.495991
$323$	−4.54210e7	−0.0749977
$324$	−1.82783e7	−0.0298557
$325$	1.97938e8	0.319844
$326$	−9.53421e8	−1.52413
$327$	2.10076e8	0.332245
$328$	−7.48426e8	−1.17109
$329$	1.58129e8	0.244808
$330$	0	0
$331$	5.92014e8	0.897292	0.448646	−	0.893710i	$-0.351906\pi$
$331$	5.92014e8	0.897292	0.448646	+	0.893710i	$0.351906\pi$
$332$	−2.43279e8	−0.364856
$333$	2.99413e8	0.444341
$334$	7.67695e8	1.12739
$335$	3.82846e8	0.556375
$336$	−1.67123e8	−0.240352
$337$	−5.64648e8	−0.803662	−0.401831	−	0.915714i	$-0.631626\pi$
$337$	−5.64648e8	−0.803662	−0.401831	+	0.915714i	$0.631626\pi$
$338$	−5.10239e8	−0.718729
$339$	3.19976e8	0.446085
$340$	−7.96576e7	−0.109913
$341$	0	0
$342$	−1.72571e7	−0.0233280
$343$	−7.55776e8	−1.01126
$344$	6.03774e8	0.799687
$345$	1.80492e8	0.236642
$346$	−9.61150e8	−1.24746
$347$	−1.09791e9	−1.41063	−0.705316	−	0.708893i	$-0.749195\pi$
$347$	−1.09791e9	−1.41063	−0.705316	+	0.708893i	$0.749195\pi$
$348$	−4.63797e7	−0.0589930
$349$	3.96232e8	0.498954	0.249477	−	0.968381i	$-0.419741\pi$
$349$	3.96232e8	0.498954	0.249477	+	0.968381i	$0.419741\pi$
$350$	−3.46936e8	−0.432524
$351$	−6.22768e7	−0.0768691
$352$	0	0
$353$	−5.76998e8	−0.698173	−0.349087	−	0.937091i	$-0.613508\pi$
$353$	−5.76998e8	−0.698173	−0.349087	+	0.937091i	$0.613508\pi$
$354$	−5.88499e8	−0.705073
$355$	−1.10910e8	−0.131575
$356$	−1.95592e8	−0.229761
$357$	2.87299e8	0.334191
$358$	−4.40167e7	−0.0507022
$359$	−1.52749e9	−1.74240	−0.871198	−	0.490931i	$-0.836657\pi$
$359$	−1.52749e9	−1.74240	−0.871198	+	0.490931i	$0.836657\pi$
$360$	−1.42899e8	−0.161424
$361$	−8.87885e8	−0.993303
$362$	4.90789e8	0.543770
$363$	0	0
$364$	6.23759e7	0.0677894
$365$	−5.41272e8	−0.582627
$366$	7.71505e8	0.822537
$367$	2.00435e8	0.211662	0.105831	−	0.994384i	$-0.466250\pi$
$367$	2.00435e8	0.211662	0.105831	+	0.994384i	$0.466250\pi$
$368$	−5.78609e8	−0.605226
$369$	3.47260e8	0.359801
$370$	4.95764e8	0.508825
$371$	7.21326e8	0.733370
$372$	−3.79295e6	−0.00382012
$373$	−3.56277e8	−0.355473	−0.177736	−	0.984078i	$-0.556877\pi$
$373$	−3.56277e8	−0.355473	−0.177736	+	0.984078i	$0.556877\pi$
$374$	0	0
$375$	−4.73903e8	−0.464066
$376$	−4.33443e8	−0.420508
$377$	−1.58023e8	−0.151888
$378$	1.09155e8	0.103950
$379$	−1.59685e9	−1.50670	−0.753350	−	0.657620i	$-0.771563\pi$
$379$	−1.59685e9	−1.50670	−0.753350	+	0.657620i	$0.771563\pi$
$380$	1.04990e7	0.00981531
$381$	1.56933e8	0.145371
$382$	−4.67762e8	−0.429342
$383$	1.48648e9	1.35196	0.675978	−	0.736922i	$-0.263721\pi$
$383$	1.48648e9	1.35196	0.675978	+	0.736922i	$0.263721\pi$
$384$	2.71339e8	0.244542
$385$	0	0
$386$	1.69257e9	1.49793
$387$	−2.80143e8	−0.245693
$388$	−3.40446e8	−0.295894
$389$	−9.27712e8	−0.799078	−0.399539	−	0.916716i	$-0.630830\pi$
$389$	−9.27712e8	−0.799078	−0.399539	+	0.916716i	$0.630830\pi$
$390$	−1.03117e8	−0.0880246
$391$	9.94680e8	0.841521
$392$	7.77713e8	0.652106
$393$	8.19051e8	0.680671
$394$	−2.05013e8	−0.168867
$395$	−1.31536e8	−0.107388
$396$	0	0
$397$	−1.77582e9	−1.42440	−0.712199	−	0.701977i	$-0.752301\pi$
$397$	−1.77582e9	−1.42440	−0.712199	+	0.701977i	$0.752301\pi$
$398$	−7.48177e8	−0.594859
$399$	−3.78664e7	−0.0298434
$400$	6.75561e8	0.527782
$401$	−2.05053e9	−1.58804	−0.794019	−	0.607893i	$-0.792015\pi$
$401$	−2.05053e9	−1.58804	−0.794019	+	0.607893i	$0.792015\pi$
$402$	8.01607e8	0.615418
$403$	−1.29232e7	−0.00983560
$404$	1.94573e7	0.0146807
$405$	6.63031e7	0.0495954
$406$	2.76974e8	0.205399
$407$	0	0
$408$	−7.87506e8	−0.574042
$409$	−2.08910e9	−1.50983	−0.754915	−	0.655823i	$-0.772322\pi$
$409$	−2.08910e9	−1.50983	−0.754915	+	0.655823i	$0.772322\pi$
$410$	5.74988e8	0.412017
$411$	1.08919e9	0.773849
$412$	8.00318e7	0.0563796
$413$	−1.29131e9	−0.901999
$414$	3.77916e8	0.261754
$415$	8.82478e8	0.606088
$416$	−3.05742e8	−0.208223
$417$	8.01693e7	0.0541417
$418$	0	0
$419$	−4.22741e8	−0.280754	−0.140377	−	0.990098i	$-0.544831\pi$
$419$	−4.22741e8	−0.280754	−0.140377	+	0.990098i	$0.544831\pi$
$420$	−6.64086e7	−0.0437373
$421$	2.06494e8	0.134872	0.0674358	−	0.997724i	$-0.478518\pi$
$421$	2.06494e8	0.134872	0.0674358	+	0.997724i	$0.478518\pi$
$422$	2.55275e9	1.65354
$423$	2.01112e8	0.129195
$424$	−1.97720e9	−1.25971
$425$	−1.16135e9	−0.733841
$426$	−2.32225e8	−0.145537
$427$	1.69287e9	1.05227
$428$	−1.84840e8	−0.113958
$429$	0	0
$430$	−4.63857e8	−0.281349
$431$	2.43670e9	1.46599	0.732997	−	0.680232i	$-0.238121\pi$
$431$	2.43670e9	1.46599	0.732997	+	0.680232i	$0.238121\pi$
$432$	−2.12550e8	−0.126844
$433$	8.58378e8	0.508126	0.254063	−	0.967188i	$-0.418233\pi$
$433$	8.58378e8	0.508126	0.254063	+	0.967188i	$0.418233\pi$
$434$	2.26510e7	0.0133007
$435$	1.68239e8	0.0979974
$436$	−2.67603e8	−0.154628
$437$	−1.31100e8	−0.0751481
$438$	−1.13332e9	−0.644456
$439$	−1.06533e9	−0.600977	−0.300488	−	0.953785i	$-0.597150\pi$
$439$	−1.06533e9	−0.600977	−0.300488	+	0.953785i	$0.597150\pi$
$440$	0	0
$441$	−3.60849e8	−0.200350
$442$	−5.68272e8	−0.313024
$443$	−8.62280e8	−0.471233	−0.235616	−	0.971846i	$-0.575711\pi$
$443$	−8.62280e8	−0.471233	−0.235616	+	0.971846i	$0.575711\pi$
$444$	−3.81405e8	−0.206798
$445$	7.09498e8	0.381673
$446$	−2.48852e9	−1.32822
$447$	4.53778e8	0.240307
$448$	1.32817e9	0.697882
$449$	−1.99009e9	−1.03755	−0.518776	−	0.854910i	$-0.673612\pi$
$449$	−1.99009e9	−1.03755	−0.518776	+	0.854910i	$0.673612\pi$
$450$	−4.41240e8	−0.228260
$451$	0	0
$452$	−4.07599e8	−0.207610
$453$	8.75988e8	0.442746
$454$	8.58426e7	0.0430534
$455$	−2.26264e8	−0.112610
$456$	1.03794e8	0.0512621
$457$	2.35723e9	1.15530	0.577651	−	0.816284i	$-0.303969\pi$
$457$	2.35723e9	1.15530	0.577651	+	0.816284i	$0.303969\pi$
$458$	−1.41683e9	−0.689109
$459$	3.65393e8	0.176366
$460$	−2.29918e8	−0.110134
$461$	−3.94154e9	−1.87375	−0.936877	−	0.349659i	$-0.886298\pi$
$461$	−3.94154e9	−1.87375	−0.936877	+	0.349659i	$0.886298\pi$
$462$	0	0
$463$	−4.20652e9	−1.96965	−0.984827	−	0.173540i	$-0.944480\pi$
$463$	−4.20652e9	−1.96965	−0.984827	+	0.173540i	$0.944480\pi$
$464$	−5.39330e8	−0.250635
$465$	1.37587e7	0.00634586
$466$	3.22498e9	1.47631
$467$	−1.39296e9	−0.632891	−0.316446	−	0.948611i	$-0.602490\pi$
$467$	−1.39296e9	−0.632891	−0.316446	+	0.948611i	$0.602490\pi$
$468$	7.93309e7	0.0357752
$469$	1.75892e9	0.787303
$470$	3.32998e8	0.147944
$471$	−2.32705e9	−1.02620
$472$	3.53957e9	1.54937
$473$	0	0
$474$	−2.75411e8	−0.118784
$475$	1.53067e8	0.0655322
$476$	−3.65974e8	−0.155534
$477$	9.17397e8	0.387029
$478$	1.39688e9	0.585006
$479$	−3.17355e9	−1.31938	−0.659692	−	0.751536i	$-0.729313\pi$
$479$	−3.17355e9	−1.31938	−0.659692	+	0.751536i	$0.729313\pi$
$480$	3.25508e8	0.134344
$481$	−1.29951e9	−0.532440
$482$	3.38636e9	1.37743
$483$	8.29241e8	0.334862
$484$	0	0
$485$	1.23494e9	0.491531
$486$	1.38826e8	0.0548585
$487$	3.12380e9	1.22555	0.612777	−	0.790256i	$-0.290052\pi$
$487$	3.12380e9	1.22555	0.612777	+	0.790256i	$0.290052\pi$
$488$	−4.64028e9	−1.80749
$489$	−2.66070e9	−1.02900
$490$	−5.97488e8	−0.229426
$491$	1.40305e9	0.534919	0.267460	−	0.963569i	$-0.413816\pi$
$491$	1.40305e9	0.534919	0.267460	+	0.963569i	$0.413816\pi$
$492$	−4.42355e8	−0.167453
$493$	9.27156e8	0.348488
$494$	7.48990e7	0.0279532
$495$	0	0
$496$	−4.41066e7	−0.0162300
$497$	−5.09558e8	−0.186186
$498$	1.84774e9	0.670407
$499$	−5.13505e8	−0.185009	−0.0925045	−	0.995712i	$-0.529487\pi$
$499$	−5.13505e8	−0.185009	−0.0925045	+	0.995712i	$0.529487\pi$
$500$	6.03678e8	0.215978
$501$	2.14240e9	0.761146
$502$	−1.46867e9	−0.518156
$503$	−1.89517e8	−0.0663988	−0.0331994	−	0.999449i	$-0.510570\pi$
$503$	−1.89517e8	−0.0663988	−0.0331994	+	0.999449i	$0.510570\pi$
$504$	−6.56525e8	−0.228425
$505$	−7.05799e7	−0.0243872
$506$	0	0
$507$	−1.42392e9	−0.485240
$508$	−1.99909e8	−0.0676563
$509$	−3.61263e9	−1.21426	−0.607129	−	0.794603i	$-0.707679\pi$
$509$	−3.61263e9	−1.21426	−0.607129	+	0.794603i	$0.707679\pi$
$510$	6.05011e8	0.201961
$511$	−2.48679e9	−0.824452
$512$	−3.21520e9	−1.05868
$513$	−4.81592e7	−0.0157496
$514$	−4.51638e9	−1.46696
$515$	−2.90310e8	−0.0936562
$516$	3.56859e8	0.114346
$517$	0	0
$518$	2.27771e9	0.720018
$519$	−2.68227e9	−0.842203
$520$	6.20206e8	0.193430
$521$	1.47376e9	0.456556	0.228278	−	0.973596i	$-0.426690\pi$
$521$	1.47376e9	0.456556	0.228278	+	0.973596i	$0.426690\pi$
$522$	3.52261e8	0.108397
$523$	1.08792e9	0.332539	0.166269	−	0.986080i	$-0.446828\pi$
$523$	1.08792e9	0.332539	0.166269	+	0.986080i	$0.446828\pi$
$524$	−1.04334e9	−0.316787
$525$	−9.68189e8	−0.292013
$526$	−3.53068e9	−1.05781
$527$	7.58232e7	0.0225665
$528$	0	0
$529$	−5.33845e8	−0.156791
$530$	1.51901e9	0.443196
$531$	−1.64232e9	−0.476021
$532$	4.82358e7	0.0138893
$533$	−1.50717e9	−0.431139
$534$	1.48555e9	0.422177
$535$	6.70495e8	0.189303
$536$	−4.82133e9	−1.35235
$537$	−1.22837e8	−0.0342310
$538$	−5.56053e7	−0.0153949
$539$	0	0
$540$	−8.44598e7	−0.0230819
$541$	6.53456e9	1.77429	0.887147	−	0.461487i	$-0.152684\pi$
$541$	6.53456e9	1.77429	0.887147	+	0.461487i	$0.152684\pi$
$542$	−2.49174e9	−0.672209
$543$	1.36964e9	0.367119
$544$	1.79386e9	0.477740
$545$	9.70713e8	0.256864
$546$	−4.73754e8	−0.124560
$547$	4.90695e8	0.128191	0.0640953	−	0.997944i	$-0.479584\pi$
$547$	4.90695e8	0.128191	0.0640953	+	0.997944i	$0.479584\pi$
$548$	−1.38745e9	−0.360152
$549$	2.15303e9	0.555325
$550$	0	0
$551$	−1.22200e8	−0.0311202
$552$	−2.27301e9	−0.575193
$553$	−6.04321e8	−0.151960
$554$	−6.51229e9	−1.62723
$555$	1.38352e9	0.343527
$556$	−1.02123e8	−0.0251978
$557$	−4.90546e9	−1.20278	−0.601390	−	0.798955i	$-0.705386\pi$
$557$	−4.90546e9	−1.20278	−0.601390	+	0.798955i	$0.705386\pi$
$558$	2.88080e7	0.00701930
$559$	1.21587e9	0.294406
$560$	−7.72238e8	−0.185820
$561$	0	0
$562$	1.30004e9	0.308945
$563$	−3.18017e9	−0.751054	−0.375527	−	0.926811i	$-0.622538\pi$
$563$	−3.18017e9	−0.751054	−0.375527	+	0.926811i	$0.622538\pi$
$564$	−2.56185e8	−0.0601280
$565$	1.47854e9	0.344876
$566$	2.30010e9	0.533198
$567$	3.04619e8	0.0701805
$568$	1.39673e9	0.319812
$569$	5.05558e9	1.15048	0.575239	−	0.817986i	$-0.304909\pi$
$569$	5.05558e9	1.15048	0.575239	+	0.817986i	$0.304909\pi$
$570$	−7.97413e7	−0.0180352
$571$	−2.03114e9	−0.456577	−0.228288	−	0.973594i	$-0.573313\pi$
$571$	−2.03114e9	−0.456577	−0.228288	+	0.973594i	$0.573313\pi$
$572$	0	0
$573$	−1.30538e9	−0.289865
$574$	2.64169e9	0.583029
$575$	−3.35204e9	−0.735313
$576$	1.68920e9	0.368300
$577$	7.40240e9	1.60420	0.802098	−	0.597193i	$-0.203717\pi$
$577$	7.40240e9	1.60420	0.802098	+	0.597193i	$0.203717\pi$
$578$	−6.35858e8	−0.136966
$579$	4.72343e9	1.01131
$580$	−2.14310e8	−0.0456084
$581$	4.05440e9	0.857650
$582$	2.58574e9	0.543693
$583$	0	0
$584$	6.81645e9	1.41616
$585$	−2.87768e8	−0.0594287
$586$	1.73448e9	0.356063
$587$	−6.41318e9	−1.30870	−0.654350	−	0.756192i	$-0.727058\pi$
$587$	−6.41318e9	−1.30870	−0.654350	+	0.756192i	$0.727058\pi$
$588$	4.59664e8	0.0932439
$589$	−9.99359e6	−0.00201520
$590$	−2.71932e9	−0.545103
$591$	−5.72127e8	−0.114008
$592$	−4.43520e9	−0.878593
$593$	−2.56715e9	−0.505546	−0.252773	−	0.967526i	$-0.581343\pi$
$593$	−2.56715e9	−0.505546	−0.252773	+	0.967526i	$0.581343\pi$
$594$	0	0
$595$	1.32754e9	0.258369
$596$	−5.78042e8	−0.111840
$597$	−2.08793e9	−0.401611
$598$	−1.64022e9	−0.313652
$599$	1.80533e9	0.343213	0.171606	−	0.985166i	$-0.445104\pi$
$599$	1.80533e9	0.343213	0.171606	+	0.985166i	$0.445104\pi$
$600$	2.65387e9	0.501592
$601$	4.73077e9	0.888937	0.444469	−	0.895794i	$-0.353393\pi$
$601$	4.73077e9	0.888937	0.444469	+	0.895794i	$0.353393\pi$
$602$	−2.13112e9	−0.398125
$603$	2.23704e9	0.415492
$604$	−1.11587e9	−0.206056
$605$	0	0
$606$	−1.47781e8	−0.0269752
$607$	1.82088e9	0.330462	0.165231	−	0.986255i	$-0.447163\pi$
$607$	1.82088e9	0.330462	0.165231	+	0.986255i	$0.447163\pi$
$608$	−2.36433e8	−0.0426624
$609$	7.72947e8	0.138672
$610$	3.56496e9	0.635916
$611$	−8.72861e8	−0.154811
$612$	−4.65453e8	−0.0820816
$613$	9.87628e9	1.73174	0.865868	−	0.500272i	$-0.166767\pi$
$613$	9.87628e9	1.73174	0.865868	+	0.500272i	$0.166767\pi$
$614$	−5.48887e8	−0.0956960
$615$	1.60461e9	0.278168
$616$	0	0
$617$	5.97195e9	1.02357	0.511785	−	0.859113i	$-0.328984\pi$
$617$	5.97195e9	1.02357	0.511785	+	0.859113i	$0.328984\pi$
$618$	−6.07854e8	−0.103595
$619$	1.03283e10	1.75030	0.875148	−	0.483855i	$-0.160764\pi$
$619$	1.03283e10	1.75030	0.875148	+	0.483855i	$0.160764\pi$
$620$	−1.75264e7	−0.00295339
$621$	1.05465e9	0.176720
$622$	3.79598e9	0.632495
$623$	3.25967e9	0.540090
$624$	9.22506e8	0.151993
$625$	2.69767e9	0.441987
$626$	1.17163e9	0.190889
$627$	0	0
$628$	2.96430e9	0.477598
$629$	7.62451e9	1.22162
$630$	5.04383e8	0.0803653
$631$	1.06783e9	0.169200	0.0846001	−	0.996415i	$-0.473039\pi$
$631$	1.06783e9	0.169200	0.0846001	+	0.996415i	$0.473039\pi$
$632$	1.65649e9	0.261022
$633$	7.12394e9	1.11637
$634$	9.72385e9	1.51539
$635$	7.25155e8	0.112389
$636$	−1.16862e9	−0.180125
$637$	1.56615e9	0.240074
$638$	0	0
$639$	−6.48067e8	−0.0982577
$640$	1.25380e9	0.189059
$641$	−4.99491e9	−0.749073	−0.374537	−	0.927212i	$-0.622198\pi$
$641$	−4.99491e9	−0.749073	−0.374537	+	0.927212i	$0.622198\pi$
$642$	1.40389e9	0.209392
$643$	−3.95028e9	−0.585988	−0.292994	−	0.956114i	$-0.594652\pi$
$643$	−3.95028e9	−0.585988	−0.292994	+	0.956114i	$0.594652\pi$
$644$	−1.05632e9	−0.155846
$645$	−1.29448e9	−0.189949
$646$	−4.39449e8	−0.0641350
$647$	1.36380e10	1.97964	0.989818	−	0.142342i	$-0.0454632\pi$
$647$	1.36380e10	1.97964	0.989818	+	0.142342i	$0.0454632\pi$
$648$	−8.34981e8	−0.120549
$649$	0	0
$650$	1.91506e9	0.273518
$651$	6.32119e7	0.00897978
$652$	3.38931e9	0.478901
$653$	−9.41459e9	−1.32314	−0.661569	−	0.749884i	$-0.730109\pi$
$653$	−9.41459e9	−1.32314	−0.661569	+	0.749884i	$0.730109\pi$
$654$	2.03249e9	0.284123
$655$	3.78466e9	0.526237
$656$	−5.14396e9	−0.711433
$657$	−3.16275e9	−0.435096
$658$	1.52990e9	0.209350
$659$	−2.82516e9	−0.384542	−0.192271	−	0.981342i	$-0.561585\pi$
$659$	−2.82516e9	−0.384542	−0.192271	+	0.981342i	$0.561585\pi$
$660$	0	0
$661$	3.49818e8	0.0471125	0.0235563	−	0.999723i	$-0.492501\pi$
$661$	3.49818e8	0.0471125	0.0235563	+	0.999723i	$0.492501\pi$
$662$	5.72775e9	0.767328
$663$	−1.58587e9	−0.211334
$664$	−1.11134e10	−1.47319
$665$	−1.74972e8	−0.0230724
$666$	2.89683e9	0.379982
$667$	2.67608e9	0.349188
$668$	−2.72908e9	−0.354241
$669$	−6.94470e9	−0.896730
$670$	3.70405e9	0.475789
$671$	0	0
$672$	1.49549e9	0.190105
$673$	5.38548e8	0.0681038	0.0340519	−	0.999420i	$-0.489159\pi$
$673$	5.38548e8	0.0681038	0.0340519	+	0.999420i	$0.489159\pi$
$674$	−5.46299e9	−0.687260
$675$	−1.23136e9	−0.154107
$676$	1.81385e9	0.225833
$677$	3.90200e9	0.483311	0.241656	−	0.970362i	$-0.422310\pi$
$677$	3.90200e9	0.483311	0.241656	+	0.970362i	$0.422310\pi$
$678$	3.09577e9	0.381474
$679$	5.67375e9	0.695546
$680$	−3.63889e9	−0.443801
$681$	2.39560e8	0.0290669
$682$	0	0
$683$	9.62524e9	1.15595	0.577975	−	0.816055i	$-0.303843\pi$
$683$	9.62524e9	1.15595	0.577975	+	0.816055i	$0.303843\pi$
$684$	6.13473e7	0.00732991
$685$	5.03289e9	0.598275
$686$	−7.31216e9	−0.864791
$687$	−3.95393e9	−0.465243
$688$	4.14976e9	0.485807
$689$	−3.98167e9	−0.463765
$690$	1.74627e9	0.202366
$691$	7.86926e9	0.907321	0.453661	−	0.891175i	$-0.350118\pi$
$691$	7.86926e9	0.907321	0.453661	+	0.891175i	$0.350118\pi$
$692$	3.41679e9	0.391965
$693$	0	0
$694$	−1.06223e10	−1.20632
$695$	3.70444e8	0.0418578
$696$	−2.11870e9	−0.238198
$697$	8.84292e9	0.989193
$698$	3.83356e9	0.426686
$699$	8.99993e9	0.996710
$700$	1.23332e9	0.135904
$701$	−5.25279e9	−0.575940	−0.287970	−	0.957639i	$-0.592980\pi$
$701$	−5.25279e9	−0.575940	−0.287970	+	0.957639i	$0.592980\pi$
$702$	−6.02530e8	−0.0657354
$703$	−1.00492e9	−0.109091
$704$	0	0
$705$	9.29293e8	0.0998828
$706$	−5.58248e9	−0.597050
$707$	−3.24268e8	−0.0345093
$708$	2.09205e9	0.221542
$709$	6.44495e9	0.679138	0.339569	−	0.940581i	$-0.389719\pi$
$709$	6.44495e9	0.679138	0.339569	+	0.940581i	$0.389719\pi$
$710$	−1.07306e9	−0.112517
$711$	−7.68587e8	−0.0801954
$712$	−8.93499e9	−0.927714
$713$	2.18851e8	0.0226118
$714$	2.77963e9	0.285787
$715$	0	0
$716$	1.56475e8	0.0159312
$717$	3.89825e9	0.394959
$718$	−1.47785e10	−1.49003
$719$	1.32048e10	1.32490	0.662448	−	0.749108i	$-0.269518\pi$
$719$	1.32048e10	1.32490	0.662448	+	0.749108i	$0.269518\pi$
$720$	−9.82147e8	−0.0980648
$721$	−1.33378e9	−0.132529
$722$	−8.59032e9	−0.849433
$723$	9.45029e9	0.929953
$724$	−1.74471e9	−0.170859
$725$	−3.12449e9	−0.304506
$726$	0	0
$727$	4.12522e8	0.0398177	0.0199089	−	0.999802i	$-0.493662\pi$
$727$	4.12522e8	0.0398177	0.0199089	+	0.999802i	$0.493662\pi$
$728$	2.84944e9	0.273715
$729$	3.87420e8	0.0370370
$730$	−5.23682e9	−0.498239
$731$	−7.13381e9	−0.675477
$732$	−2.74262e9	−0.258451
$733$	−9.85636e9	−0.924384	−0.462192	−	0.886780i	$-0.652937\pi$
$733$	−9.85636e9	−0.924384	−0.462192	+	0.886780i	$0.652937\pi$
$734$	1.93922e9	0.181005
$735$	−1.66740e9	−0.154894
$736$	5.17767e9	0.478698
$737$	0	0
$738$	3.35975e9	0.307688
$739$	1.87922e10	1.71287	0.856433	−	0.516259i	$-0.172676\pi$
$739$	1.87922e10	1.71287	0.856433	+	0.516259i	$0.172676\pi$
$740$	−1.76239e9	−0.159879
$741$	2.09020e8	0.0188722
$742$	6.97885e9	0.627149
$743$	−1.03291e9	−0.0923849	−0.0461925	−	0.998933i	$-0.514709\pi$
$743$	−1.03291e9	−0.0923849	−0.0461925	+	0.998933i	$0.514709\pi$
$744$	−1.73268e8	−0.0154246
$745$	2.09681e9	0.185786
$746$	−3.44699e9	−0.303986
$747$	5.15647e9	0.452616
$748$	0	0
$749$	3.08048e9	0.267875
$750$	−4.58503e9	−0.396851
$751$	−7.76631e9	−0.669075	−0.334538	−	0.942382i	$-0.608580\pi$
$751$	−7.76631e9	−0.669075	−0.334538	+	0.942382i	$0.608580\pi$
$752$	−2.97907e9	−0.255457
$753$	−4.09860e9	−0.349827
$754$	−1.52887e9	−0.129889
$755$	4.04774e9	0.342294
$756$	−3.88037e8	−0.0326623
$757$	1.28550e10	1.07705	0.538527	−	0.842608i	$-0.318981\pi$
$757$	1.28550e10	1.07705	0.538527	+	0.842608i	$0.318981\pi$
$758$	−1.54496e10	−1.28847
$759$	0	0
$760$	4.79611e8	0.0396316
$761$	2.01529e10	1.65764	0.828822	−	0.559512i	$-0.189012\pi$
$761$	2.01529e10	1.65764	0.828822	+	0.559512i	$0.189012\pi$
$762$	1.51834e9	0.124316
$763$	4.45979e9	0.363478
$764$	1.66285e9	0.134904
$765$	1.68840e9	0.136351
$766$	1.43817e10	1.15614
$767$	7.12795e9	0.570402
$768$	−5.38284e9	−0.428792
$769$	−8.91607e8	−0.0707020	−0.0353510	−	0.999375i	$-0.511255\pi$
$769$	−8.91607e8	−0.0707020	−0.0353510	+	0.999375i	$0.511255\pi$
$770$	0	0
$771$	−1.26038e10	−0.990401
$772$	−6.01691e9	−0.470666
$773$	1.76079e10	1.37113	0.685567	−	0.728009i	$-0.259554\pi$
$773$	1.76079e10	1.37113	0.685567	+	0.728009i	$0.259554\pi$
$774$	−2.71040e9	−0.210107
$775$	−2.55522e8	−0.0197184
$776$	−1.55521e10	−1.19474
$777$	6.35637e9	0.486111
$778$	−8.97564e9	−0.683340
$779$	−1.16551e9	−0.0883353
$780$	3.66571e8	0.0276584
$781$	0	0
$782$	9.62356e9	0.719635
$783$	9.83050e8	0.0731829
$784$	5.34525e9	0.396152
$785$	−1.07528e10	−0.793372
$786$	7.92435e9	0.582083
$787$	1.31004e10	0.958020	0.479010	−	0.877810i	$-0.340996\pi$
$787$	1.31004e10	0.958020	0.479010	+	0.877810i	$0.340996\pi$
$788$	7.28800e8	0.0530599
$789$	−9.85303e9	−0.714168
$790$	−1.27262e9	−0.0918337
$791$	6.79290e9	0.488020
$792$	0	0
$793$	−9.34455e9	−0.665429
$794$	−1.71811e10	−1.21809
$795$	4.23909e9	0.299218
$796$	2.65969e9	0.186911
$797$	1.29295e9	0.0904645	0.0452322	−	0.998976i	$-0.485597\pi$
$797$	1.29295e9	0.0904645	0.0452322	+	0.998976i	$0.485597\pi$
$798$	−3.66358e8	−0.0255209
$799$	5.12128e9	0.355193
$800$	−6.04524e9	−0.417445
$801$	4.14572e9	0.285027
$802$	−1.98390e10	−1.35803
$803$	0	0
$804$	−2.84963e9	−0.193371
$805$	3.83174e9	0.258887
$806$	−1.25032e8	−0.00841102
$807$	−1.55177e8	−0.0103937
$808$	8.88840e8	0.0592767
$809$	−1.26438e10	−0.839575	−0.419788	−	0.907622i	$-0.637895\pi$
$809$	−1.26438e10	−0.839575	−0.419788	+	0.907622i	$0.637895\pi$
$810$	6.41485e8	0.0424120
$811$	−8.39807e9	−0.552849	−0.276424	−	0.961036i	$-0.589150\pi$
$811$	−8.39807e9	−0.552849	−0.276424	+	0.961036i	$0.589150\pi$
$812$	−9.84613e8	−0.0645386
$813$	−6.95366e9	−0.453833
$814$	0	0
$815$	−1.22945e10	−0.795536
$816$	−5.41256e9	−0.348728
$817$	9.40245e8	0.0603203
$818$	−2.02121e10	−1.29115
$819$	−1.32210e9	−0.0840952
$820$	−2.04402e9	−0.129461
$821$	6.50997e9	0.410561	0.205280	−	0.978703i	$-0.434189\pi$
$821$	6.50997e9	0.410561	0.205280	+	0.978703i	$0.434189\pi$
$822$	1.05379e10	0.661764
$823$	−1.84146e10	−1.15150	−0.575749	−	0.817626i	$-0.695290\pi$
$823$	−1.84146e10	−1.15150	−0.575749	+	0.817626i	$0.695290\pi$
$824$	3.65599e9	0.227646
$825$	0	0
$826$	−1.24935e10	−0.771353
$827$	−6.19442e9	−0.380830	−0.190415	−	0.981704i	$-0.560983\pi$
$827$	−6.19442e9	−0.380830	−0.190415	+	0.981704i	$0.560983\pi$
$828$	−1.34345e9	−0.0822462
$829$	7.07900e9	0.431550	0.215775	−	0.976443i	$-0.430772\pi$
$829$	7.07900e9	0.431550	0.215775	+	0.976443i	$0.430772\pi$
$830$	8.53800e9	0.518302
$831$	−1.81738e10	−1.09861
$832$	−7.33142e9	−0.441323
$833$	−9.18895e9	−0.550818
$834$	7.75640e8	0.0462998
$835$	9.89954e9	0.588454
$836$	0	0
$837$	8.03942e7	0.00473899
$838$	−4.09003e9	−0.240089
$839$	−2.03331e10	−1.18860	−0.594301	−	0.804243i	$-0.702571\pi$
$839$	−2.03331e10	−1.18860	−0.594301	+	0.804243i	$0.702571\pi$
$840$	−3.03366e9	−0.176599
$841$	−1.47555e10	−0.855395
$842$	1.99784e9	0.115337
$843$	3.62801e9	0.208580
$844$	−9.07478e9	−0.519562
$845$	−6.57961e9	−0.375147
$846$	1.94576e9	0.110483
$847$	0	0
$848$	−1.35894e10	−0.765270
$849$	6.41886e9	0.359982
$850$	−1.12361e10	−0.627551
$851$	2.20069e10	1.22407
$852$	8.25535e8	0.0457296
$853$	3.15354e9	0.173971	0.0869855	−	0.996210i	$-0.472277\pi$
$853$	3.15354e9	0.173971	0.0869855	+	0.996210i	$0.472277\pi$
$854$	1.63786e10	0.899859
$855$	−2.22533e8	−0.0121762
$856$	−8.44381e9	−0.460130
$857$	−2.67053e9	−0.144932	−0.0724660	−	0.997371i	$-0.523087\pi$
$857$	−2.67053e9	−0.144932	−0.0724660	+	0.997371i	$0.523087\pi$
$858$	0	0
$859$	3.38071e10	1.81984	0.909918	−	0.414787i	$-0.136144\pi$
$859$	3.38071e10	1.81984	0.909918	+	0.414787i	$0.136144\pi$
$860$	1.64897e9	0.0884030
$861$	7.37213e9	0.393624
$862$	2.35752e10	1.25366
$863$	−7.11596e9	−0.376873	−0.188437	−	0.982085i	$-0.560342\pi$
$863$	−7.11596e9	−0.376873	−0.188437	+	0.982085i	$0.560342\pi$
$864$	1.90200e9	0.100326
$865$	−1.23942e10	−0.651121
$866$	8.30484e9	0.434529
$867$	−1.77448e9	−0.0924708
$868$	−8.05221e7	−0.00417923
$869$	0	0
$870$	1.62772e9	0.0838035
$871$	−9.70914e9	−0.497871
$872$	−1.22246e10	−0.624347
$873$	7.21598e9	0.367067
$874$	−1.26840e9	−0.0642637
$875$	−1.00607e10	−0.507691
$876$	4.02884e9	0.202496
$877$	1.06498e10	0.533143	0.266571	−	0.963815i	$-0.414109\pi$
$877$	1.06498e10	0.533143	0.266571	+	0.963815i	$0.414109\pi$
$878$	−1.03071e10	−0.513931
$879$	4.84039e9	0.240391
$880$	0	0
$881$	−3.41012e10	−1.68017	−0.840087	−	0.542452i	$-0.817496\pi$
$881$	−3.41012e10	−1.68017	−0.840087	+	0.542452i	$0.817496\pi$
$882$	−3.49122e9	−0.171332
$883$	−1.05079e10	−0.513636	−0.256818	−	0.966460i	$-0.582674\pi$
$883$	−1.05079e10	−0.513636	−0.256818	+	0.966460i	$0.582674\pi$
$884$	2.02015e9	0.0983559
$885$	−7.58878e9	−0.368019
$886$	−8.34259e9	−0.402979
$887$	−2.28910e10	−1.10137	−0.550683	−	0.834714i	$-0.685633\pi$
$887$	−2.28910e10	−1.10137	−0.550683	+	0.834714i	$0.685633\pi$
$888$	−1.74232e10	−0.834994
$889$	3.33161e9	0.159037
$890$	6.86442e9	0.326391
$891$	0	0
$892$	8.84645e9	0.417342
$893$	−6.74991e8	−0.0317189
$894$	4.39032e9	0.205501
$895$	−5.67602e8	−0.0264645
$896$	5.76037e9	0.267530
$897$	−4.57735e9	−0.211758
$898$	−1.92541e10	−0.887273
$899$	2.03994e8	0.00936394
$900$	1.56856e9	0.0717221
$901$	2.33614e10	1.06405
$902$	0	0
$903$	−5.94728e9	−0.268789
$904$	−1.86198e10	−0.838273
$905$	6.32880e9	0.283826
$906$	8.47521e9	0.378618
$907$	−4.23584e10	−1.88501	−0.942507	−	0.334187i	$-0.891538\pi$
$907$	−4.23584e10	−1.88501	−0.942507	+	0.334187i	$0.891538\pi$
$908$	−3.05162e8	−0.0135279
$909$	−4.12410e8	−0.0182119
$910$	−2.18911e9	−0.0962994
$911$	1.51565e10	0.664179	0.332090	−	0.943248i	$-0.392246\pi$
$911$	1.51565e10	0.664179	0.332090	+	0.943248i	$0.392246\pi$
$912$	7.13382e8	0.0311415
$913$	0	0
$914$	2.28063e10	0.987969
$915$	9.94868e9	0.429330
$916$	5.03668e9	0.216526
$917$	1.73880e10	0.744658
$918$	3.53518e9	0.150821
$919$	−9.49608e9	−0.403590	−0.201795	−	0.979428i	$-0.564677\pi$
$919$	−9.49608e9	−0.403590	−0.201795	+	0.979428i	$0.564677\pi$
$920$	−1.05031e10	−0.444691
$921$	−1.53177e9	−0.0646079
$922$	−3.81345e10	−1.60236
$923$	2.81273e9	0.117739
$924$	0	0
$925$	−2.56944e10	−1.06744
$926$	−4.06983e10	−1.68437
$927$	−1.69633e9	−0.0699409
$928$	4.82618e9	0.198237
$929$	−2.78782e10	−1.14080	−0.570400	−	0.821367i	$-0.693212\pi$
$929$	−2.78782e10	−1.14080	−0.570400	+	0.821367i	$0.693212\pi$
$930$	1.33115e8	0.00542673
$931$	1.21112e9	0.0491883
$932$	−1.14645e10	−0.463873
$933$	1.05934e10	0.427021
$934$	−1.34769e10	−0.541223
$935$	0	0
$936$	3.62397e9	0.144451
$937$	−4.12788e10	−1.63923	−0.819613	−	0.572918i	$-0.805811\pi$
$937$	−4.12788e10	−1.63923	−0.819613	+	0.572918i	$0.805811\pi$
$938$	1.70176e10	0.673270
$939$	3.26966e9	0.128876
$940$	−1.18377e9	−0.0464859
$941$	−6.09047e9	−0.238280	−0.119140	−	0.992877i	$-0.538014\pi$
$941$	−6.09047e9	−0.238280	−0.119140	+	0.992877i	$0.538014\pi$
$942$	−2.25143e10	−0.877566
$943$	2.55236e10	0.991177
$944$	2.43276e10	0.941234
$945$	1.40758e9	0.0542576
$946$	0	0
$947$	−1.28487e10	−0.491627	−0.245813	−	0.969317i	$-0.579055\pi$
$947$	−1.28487e10	−0.491627	−0.245813	+	0.969317i	$0.579055\pi$
$948$	9.79060e8	0.0373233
$949$	1.37269e10	0.521363
$950$	1.48093e9	0.0560406
$951$	2.71362e10	1.02310
$952$	−1.67183e10	−0.628004
$953$	1.24365e10	0.465451	0.232725	−	0.972542i	$-0.425236\pi$
$953$	1.24365e10	0.465451	0.232725	+	0.972542i	$0.425236\pi$
$954$	8.87585e9	0.330971
$955$	−6.03186e9	−0.224099
$956$	−4.96575e9	−0.183816
$957$	0	0
$958$	−3.07042e10	−1.12828
$959$	2.31228e10	0.846594
$960$	7.80541e9	0.284739
$961$	−2.74959e10	−0.999394
$962$	−1.25728e10	−0.455322
$963$	3.91782e9	0.141368
$964$	−1.20382e10	−0.432804
$965$	2.18259e10	0.781857
$966$	8.02293e9	0.286361
$967$	−6.53237e8	−0.0232316	−0.0116158	−	0.999933i	$-0.503698\pi$
$967$	−6.53237e8	−0.0232316	−0.0116158	+	0.999933i	$0.503698\pi$
$968$	0	0
$969$	−1.22637e9	−0.0432999
$970$	1.19481e10	0.420338
$971$	4.53217e10	1.58869	0.794345	−	0.607467i	$-0.207814\pi$
$971$	4.53217e10	1.58869	0.794345	+	0.607467i	$0.207814\pi$
$972$	−4.93513e8	−0.0172372
$973$	1.70195e9	0.0592313
$974$	3.02229e10	1.04804
$975$	5.34433e9	0.184662
$976$	−3.18928e10	−1.09804
$977$	−1.24884e10	−0.428427	−0.214213	−	0.976787i	$-0.568719\pi$
$977$	−1.24884e10	−0.428427	−0.214213	+	0.976787i	$0.568719\pi$
$978$	−2.57424e10	−0.879959
$979$	0	0
$980$	2.12401e9	0.0720883
$981$	5.67204e9	0.191822
$982$	1.35746e10	0.457441
$983$	4.58286e10	1.53886	0.769430	−	0.638731i	$-0.220540\pi$
$983$	4.58286e10	1.53886	0.769430	+	0.638731i	$0.220540\pi$
$984$	−2.02075e10	−0.676130
$985$	−2.64367e9	−0.0881416
$986$	8.97026e9	0.298013
$987$	4.26949e9	0.141340
$988$	−2.66258e8	−0.00878322
$989$	−2.05906e10	−0.676832
$990$	0	0
$991$	2.77986e10	0.907329	0.453665	−	0.891172i	$-0.350116\pi$
$991$	2.77986e10	0.907329	0.453665	+	0.891172i	$0.350116\pi$
$992$	3.94687e8	0.0128369
$993$	1.59844e10	0.518052
$994$	−4.92999e9	−0.159219
$995$	−9.64786e9	−0.310492
$996$	−6.56853e9	−0.210650
$997$	1.15787e10	0.370023	0.185011	−	0.982736i	$-0.440768\pi$
$997$	1.15787e10	0.370023	0.185011	+	0.982736i	$0.440768\pi$
$998$	−4.96818e9	−0.158212
$999$	8.08416e9	0.256540

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.8.a.r.1.9		14
11.2	odd	10		33.8.e.b.4.5	✓	28
11.6	odd	10		33.8.e.b.25.5	yes	28
11.10	odd	2		363.8.a.o.1.6		14
33.2	even	10		99.8.f.b.37.3		28
33.17	even	10		99.8.f.b.91.3		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.8.e.b.4.5	✓	28	11.2	odd	10
33.8.e.b.25.5	yes	28	11.6	odd	10
99.8.f.b.37.3		28	33.2	even	10
99.8.f.b.91.3		28	33.17	even	10
363.8.a.o.1.6		14	11.10	odd	2
363.8.a.r.1.9		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)

\(f(q)\)	\(=\)	\(q+9.67503 q^{2} +27.0000 q^{3} -34.3938 q^{4} +124.761 q^{5} +261.226 q^{6} +573.194 q^{7} -1571.16 q^{8} +729.000 q^{9} +1207.07 q^{10} -928.631 q^{12} -3163.99 q^{13} +5545.67 q^{14} +3368.55 q^{15} -10798.7 q^{16} +18563.9 q^{17} +7053.10 q^{18} -2446.74 q^{19} -4291.00 q^{20} +15476.2 q^{21} +53581.5 q^{23} -42421.4 q^{24} -62559.7 q^{25} -30611.7 q^{26} +19683.0 q^{27} -19714.3 q^{28} +49944.1 q^{29} +32590.8 q^{30} +4084.45 q^{31} +96631.6 q^{32} +179606. q^{34} +71512.3 q^{35} -25073.0 q^{36} +410718. q^{37} -23672.3 q^{38} -85427.8 q^{39} -196020. q^{40} +476351. q^{41} +149733. q^{42} -384285. q^{43} +90950.8 q^{45} +518403. q^{46} +275873. q^{47} -291564. q^{48} -494991. q^{49} -605267. q^{50} +501224. q^{51} +108822. q^{52} +1.25843e6 q^{53} +190434. q^{54} -900583. q^{56} -66062.0 q^{57} +483211. q^{58} -2.25283e6 q^{59} -115857. q^{60} +2.95340e6 q^{61} +39517.2 q^{62} +417859. q^{63} +2.31714e6 q^{64} -394743. q^{65} +3.06864e6 q^{67} -638481. q^{68} +1.44670e6 q^{69} +691884. q^{70} -888980. q^{71} -1.14538e6 q^{72} -4.33847e6 q^{73} +3.97371e6 q^{74} -1.68911e6 q^{75} +84152.6 q^{76} -826516. q^{78} -1.05430e6 q^{79} -1.34725e6 q^{80} +531441. q^{81} +4.60871e6 q^{82} +7.07334e6 q^{83} -532286. q^{84} +2.31605e6 q^{85} -3.71797e6 q^{86} +1.34849e6 q^{87} +5.68686e6 q^{89} +879952. q^{90} -1.81358e6 q^{91} -1.84287e6 q^{92} +110280. q^{93} +2.66908e6 q^{94} -305258. q^{95} +2.60905e6 q^{96} +9.89847e6 q^{97} -4.78906e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 23 q^{2} + 378 q^{3} + 845 q^{4} + 69 q^{5} + 621 q^{6} + 3278 q^{7} + 4602 q^{8} + 10206 q^{9} + 5320 q^{10} + 22815 q^{12} + 32188 q^{13} - 7794 q^{14} + 1863 q^{15} + 86137 q^{16} + 42917 q^{17}+ \cdots - 70266089 q^{98}+O(q^{100}) \) 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

Introduction

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