LMFDB - Embedded newform 98.8.a.l.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [98,8,Mod(1,98)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("98.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(98, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 98 = 2 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	98.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-32,0,256,0,0,0,-2048,7228] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$30.6137324974$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{1801})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 903x^{2} + 904x + 200702 $ x^4 - 2x^3 - 903x^2 + 904*x + 200702
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3}\cdot 7^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$20.3049$ of defining polynomial
Character	$\chi$	$=$	98.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-8.00000 q^{2} -40.2177 q^{3} +64.0000 q^{4} +98.9949 q^{5} +321.741 q^{6} -512.000 q^{8} -569.539 q^{9} -791.960 q^{10} +3779.21 q^{11} -2573.93 q^{12} +4944.73 q^{13} -3981.35 q^{15} +4096.00 q^{16} -25954.4 q^{17} +4556.31 q^{18} -43331.3 q^{19} +6335.68 q^{20} -30233.7 q^{22} -35219.3 q^{23} +20591.4 q^{24} -68325.0 q^{25} -39557.8 q^{26} +110862. q^{27} +152177. q^{29} +31850.8 q^{30} +33696.2 q^{31} -32768.0 q^{32} -151991. q^{33} +207635. q^{34} -36450.5 q^{36} +394229. q^{37} +346651. q^{38} -198865. q^{39} -50685.4 q^{40} +148840. q^{41} +871204. q^{43} +241870. q^{44} -56381.5 q^{45} +281754. q^{46} -725308. q^{47} -164732. q^{48} +546600. q^{50} +1.04383e6 q^{51} +316463. q^{52} +1.81855e6 q^{53} -886893. q^{54} +374123. q^{55} +1.74269e6 q^{57} -1.21741e6 q^{58} +1.14304e6 q^{59} -254806. q^{60} -2.06761e6 q^{61} -269570. q^{62} +262144. q^{64} +489503. q^{65} +1.21593e6 q^{66} +2.31943e6 q^{67} -1.66108e6 q^{68} +1.41644e6 q^{69} -1.15064e6 q^{71} +291604. q^{72} -919421. q^{73} -3.15384e6 q^{74} +2.74787e6 q^{75} -2.77321e6 q^{76} +1.59092e6 q^{78} +6.22035e6 q^{79} +405483. q^{80} -3.21301e6 q^{81} -1.19072e6 q^{82} +6.86089e6 q^{83} -2.56935e6 q^{85} -6.96963e6 q^{86} -6.12020e6 q^{87} -1.93496e6 q^{88} +6.98286e6 q^{89} +451052. q^{90} -2.25403e6 q^{92} -1.35518e6 q^{93} +5.80246e6 q^{94} -4.28958e6 q^{95} +1.31785e6 q^{96} -2.35300e6 q^{97} -2.15241e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 32 q^{2} + 256 q^{4} - 2048 q^{8} + 7228 q^{9} - 6272 q^{11} + 7840 q^{15} + 16384 q^{16} - 57824 q^{18} + 50176 q^{22} - 12544 q^{23} - 273300 q^{25} - 75736 q^{29} - 62720 q^{30} - 131072 q^{32}+ \cdots - 62164928 q^{99}+O(q^{100}) $ 4 * q - 32 * q^2 + 256 * q^4 - 2048 * q^8 + 7228 * q^9 - 6272 * q^11 + 7840 * q^15 + 16384 * q^16 - 57824 * q^18 + 50176 * q^22 - 12544 * q^23 - 273300 * q^25 - 75736 * q^29 - 62720 * q^30 - 131072 * q^32 + 462592 * q^36 + 1234696 * q^37 + 1771200 * q^39 + 2821760 * q^43 - 401408 * q^44 + 100352 * q^46 + 2186400 * q^50 + 5532304 * q^51 + 1285336 * q^53 + 9157160 * q^57 + 605888 * q^58 + 501760 * q^60 + 1048576 * q^64 + 3669120 * q^65 + 5342160 * q^67 + 2883552 * q^71 - 3700736 * q^72 - 9877568 * q^74 - 14169600 * q^78 + 6016448 * q^79 + 713228 * q^81 - 866320 * q^85 - 22574080 * q^86 + 3211264 * q^88 - 802816 * q^92 + 11291088 * q^93 - 1544480 * q^95 - 62164928 * q^99

Coefficient data

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

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

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

$2$	−8.00000	−0.707107
$2$	−8.00000	−0.707107
$3$	−40.2177	−0.859988	−0.429994	−	0.902832i	$-0.641484\pi$
$3$	−40.2177	−0.859988	−0.429994	+	0.902832i	$0.641484\pi$
$4$	64.0000	0.500000
$5$	98.9949	0.354175	0.177088	−	0.984195i	$-0.443332\pi$
$5$	98.9949	0.354175	0.177088	+	0.984195i	$0.443332\pi$
$6$	321.741	0.608104
$7$	0	0
$7$	0	0
$8$	−512.000	−0.353553
$9$	−569.539	−0.260420
$10$	−791.960	−0.250440
$11$	3779.21	0.856105	0.428052	−	0.903754i	$-0.359200\pi$
$11$	3779.21	0.856105	0.428052	+	0.903754i	$0.359200\pi$
$12$	−2573.93	−0.429994
$13$	4944.73	0.624224	0.312112	−	0.950045i	$-0.398964\pi$
$13$	4944.73	0.624224	0.312112	+	0.950045i	$0.398964\pi$
$14$	0	0
$15$	−3981.35	−0.304586
$16$	4096.00	0.250000
$17$	−25954.4	−1.28127	−0.640633	−	0.767847i	$-0.721328\pi$
$17$	−25954.4	−1.28127	−0.640633	+	0.767847i	$0.721328\pi$
$18$	4556.31	0.184145
$19$	−43331.3	−1.44932	−0.724661	−	0.689106i	$-0.758003\pi$
$19$	−43331.3	−1.44932	−0.724661	+	0.689106i	$0.758003\pi$
$20$	6335.68	0.177088
$21$	0	0
$22$	−30233.7	−0.605357
$23$	−35219.3	−0.603577	−0.301789	−	0.953375i	$-0.597584\pi$
$23$	−35219.3	−0.603577	−0.301789	+	0.953375i	$0.597584\pi$
$24$	20591.4	0.304052
$25$	−68325.0	−0.874560
$26$	−39557.8	−0.441393
$27$	110862.	1.08395
$28$	0	0
$29$	152177.	1.15866	0.579329	−	0.815094i	$-0.303315\pi$
$29$	152177.	1.15866	0.579329	+	0.815094i	$0.303315\pi$
$30$	31850.8	0.215375
$31$	33696.2	0.203149	0.101575	−	0.994828i	$-0.467612\pi$
$31$	33696.2	0.203149	0.101575	+	0.994828i	$0.467612\pi$
$32$	−32768.0	−0.176777
$33$	−151991.	−0.736240
$34$	207635.	0.905992
$35$	0	0
$36$	−36450.5	−0.130210
$37$	394229.	1.27951	0.639754	−	0.768580i	$-0.279036\pi$
$37$	394229.	1.27951	0.639754	+	0.768580i	$0.279036\pi$
$38$	346651.	1.02482
$39$	−198865.	−0.536826
$40$	−50685.4	−0.125220
$41$	148840.	0.337269	0.168635	−	0.985679i	$-0.446064\pi$
$41$	148840.	0.337269	0.168635	+	0.985679i	$0.446064\pi$
$42$	0	0
$43$	871204.	1.67101	0.835507	−	0.549480i	$-0.185174\pi$
$43$	871204.	1.67101	0.835507	+	0.549480i	$0.185174\pi$
$44$	241870.	0.428052
$45$	−56381.5	−0.0922343
$46$	281754.	0.426794
$47$	−725308.	−1.01901	−0.509507	−	0.860467i	$-0.670172\pi$
$47$	−725308.	−1.01901	−0.509507	+	0.860467i	$0.670172\pi$
$48$	−164732.	−0.214997
$49$	0	0
$50$	546600.	0.618407
$51$	1.04383e6	1.10187
$52$	316463.	0.312112
$53$	1.81855e6	1.67788	0.838939	−	0.544225i	$-0.183176\pi$
$53$	1.81855e6	1.67788	0.838939	+	0.544225i	$0.183176\pi$
$54$	−886893.	−0.766466
$55$	374123.	0.303211
$56$	0	0
$57$	1.74269e6	1.24640
$58$	−1.21741e6	−0.819295
$59$	1.14304e6	0.724569	0.362284	−	0.932068i	$-0.381997\pi$
$59$	1.14304e6	0.724569	0.362284	+	0.932068i	$0.381997\pi$
$60$	−254806.	−0.152293
$61$	−2.06761e6	−1.16631	−0.583155	−	0.812361i	$-0.698182\pi$
$61$	−2.06761e6	−1.16631	−0.583155	+	0.812361i	$0.698182\pi$
$62$	−269570.	−0.143648
$63$	0	0
$64$	262144.	0.125000
$65$	489503.	0.221085
$66$	1.21593e6	0.520600
$67$	2.31943e6	0.942147	0.471074	−	0.882094i	$-0.343867\pi$
$67$	2.31943e6	0.942147	0.471074	+	0.882094i	$0.343867\pi$
$68$	−1.66108e6	−0.640633
$69$	1.41644e6	0.519069
$70$	0	0
$71$	−1.15064e6	−0.381534	−0.190767	−	0.981635i	$-0.561098\pi$
$71$	−1.15064e6	−0.381534	−0.190767	+	0.981635i	$0.561098\pi$
$72$	291604.	0.0920724
$73$	−919421.	−0.276620	−0.138310	−	0.990389i	$-0.544167\pi$
$73$	−919421.	−0.276620	−0.138310	+	0.990389i	$0.544167\pi$
$74$	−3.15384e6	−0.904749
$75$	2.74787e6	0.752111
$76$	−2.77321e6	−0.724661
$77$	0	0
$78$	1.59092e6	0.379593
$79$	6.22035e6	1.41945	0.709725	−	0.704479i	$-0.248819\pi$
$79$	6.22035e6	1.41945	0.709725	+	0.704479i	$0.248819\pi$
$80$	405483.	0.0885438
$81$	−3.21301e6	−0.671761
$82$	−1.19072e6	−0.238485
$83$	6.86089e6	1.31706	0.658532	−	0.752552i	$-0.271178\pi$
$83$	6.86089e6	1.31706	0.658532	+	0.752552i	$0.271178\pi$
$84$	0	0
$85$	−2.56935e6	−0.453793
$86$	−6.96963e6	−1.18159
$87$	−6.12020e6	−0.996433
$88$	−1.93496e6	−0.302679
$89$	6.98286e6	1.04995	0.524974	−	0.851118i	$-0.324075\pi$
$89$	6.98286e6	1.04995	0.524974	+	0.851118i	$0.324075\pi$
$90$	451052.	0.0652195
$91$	0	0
$92$	−2.25403e6	−0.301789
$93$	−1.35518e6	−0.174706
$94$	5.80246e6	0.720551
$95$	−4.28958e6	−0.513313
$96$	1.31785e6	0.152026
$97$	−2.35300e6	−0.261770	−0.130885	−	0.991398i	$-0.541782\pi$
$97$	−2.35300e6	−0.261770	−0.130885	+	0.991398i	$0.541782\pi$
$98$	0	0
$99$	−2.15241e6	−0.222947
$100$	−4.37280e6	−0.437280
$101$	−1.80656e7	−1.74473	−0.872364	−	0.488856i	$-0.837414\pi$
$101$	−1.80656e7	−1.74473	−0.872364	+	0.488856i	$0.837414\pi$
$102$	−8.35060e6	−0.779143
$103$	1.47032e7	1.32581	0.662906	−	0.748703i	$-0.269323\pi$
$103$	1.47032e7	1.32581	0.662906	+	0.748703i	$0.269323\pi$
$104$	−2.53170e6	−0.220697
$105$	0	0
$106$	−1.45484e7	−1.18644
$107$	−2.99447e6	−0.236307	−0.118154	−	0.992995i	$-0.537698\pi$
$107$	−2.99447e6	−0.236307	−0.118154	+	0.992995i	$0.537698\pi$
$108$	7.09514e6	0.541973
$109$	1.78392e7	1.31942	0.659710	−	0.751520i	$-0.270679\pi$
$109$	1.78392e7	1.31942	0.659710	+	0.751520i	$0.270679\pi$
$110$	−2.99298e6	−0.214403
$111$	−1.58550e7	−1.10036
$112$	0	0
$113$	−2.07812e7	−1.35486	−0.677431	−	0.735586i	$-0.736907\pi$
$113$	−2.07812e7	−1.35486	−0.677431	+	0.735586i	$0.736907\pi$
$114$	−1.39415e7	−0.881337
$115$	−3.48653e6	−0.213772
$116$	9.73931e6	0.579329
$117$	−2.81621e6	−0.162561
$118$	−9.14432e6	−0.512347
$119$	0	0
$120$	2.03845e6	0.107688
$121$	−5.20473e6	−0.267085
$122$	1.65409e7	0.824706
$123$	−5.98601e6	−0.290048
$124$	2.15656e6	0.101575
$125$	−1.44978e7	−0.663922
$126$	0	0
$127$	1.51943e7	0.658215	0.329107	−	0.944293i	$-0.393252\pi$
$127$	1.51943e7	0.658215	0.329107	+	0.944293i	$0.393252\pi$
$128$	−2.09715e6	−0.0883883
$129$	−3.50378e7	−1.43705
$130$	−3.91602e6	−0.156330
$131$	3.92464e7	1.52528	0.762642	−	0.646821i	$-0.223902\pi$
$131$	3.92464e7	1.52528	0.762642	+	0.646821i	$0.223902\pi$
$132$	−9.72743e6	−0.368120
$133$	0	0
$134$	−1.85554e7	−0.666199
$135$	1.09747e7	0.383907
$136$	1.32886e7	0.452996
$137$	2.73797e7	0.909717	0.454858	−	0.890564i	$-0.349690\pi$
$137$	2.73797e7	0.909717	0.454858	+	0.890564i	$0.349690\pi$
$138$	−1.13315e7	−0.367038
$139$	1.67008e7	0.527456	0.263728	−	0.964597i	$-0.415048\pi$
$139$	1.67008e7	0.527456	0.263728	+	0.964597i	$0.415048\pi$
$140$	0	0
$141$	2.91702e7	0.876340
$142$	9.20509e6	0.269786
$143$	1.86872e7	0.534401
$144$	−2.33283e6	−0.0651050
$145$	1.50647e7	0.410368
$146$	7.35537e6	0.195600
$147$	0	0
$148$	2.52307e7	0.639754
$149$	1.01132e7	0.250459	0.125230	−	0.992128i	$-0.460033\pi$
$149$	1.01132e7	0.250459	0.125230	+	0.992128i	$0.460033\pi$
$150$	−2.19830e7	−0.531823
$151$	−2.23141e7	−0.527424	−0.263712	−	0.964601i	$-0.584947\pi$
$151$	−2.23141e7	−0.527424	−0.263712	+	0.964601i	$0.584947\pi$
$152$	2.21857e7	0.512412
$153$	1.47820e7	0.333668
$154$	0	0
$155$	3.33575e6	0.0719504
$156$	−1.27274e7	−0.268413
$157$	4.77487e7	0.984719	0.492360	−	0.870392i	$-0.336134\pi$
$157$	4.77487e7	0.984719	0.492360	+	0.870392i	$0.336134\pi$
$158$	−4.97628e7	−1.00370
$159$	−7.31380e7	−1.44296
$160$	−3.24387e6	−0.0626099
$161$	0	0
$162$	2.57041e7	0.475007
$163$	7.55940e7	1.36720	0.683598	−	0.729859i	$-0.260414\pi$
$163$	7.55940e7	1.36720	0.683598	+	0.729859i	$0.260414\pi$
$164$	9.52578e6	0.168635
$165$	−1.50464e7	−0.260758
$166$	−5.48871e7	−0.931305
$167$	5.56387e7	0.924420	0.462210	−	0.886771i	$-0.347057\pi$
$167$	5.56387e7	0.924420	0.462210	+	0.886771i	$0.347057\pi$
$168$	0	0
$169$	−3.82982e7	−0.610344
$170$	2.05548e7	0.320880
$171$	2.46789e7	0.377432
$172$	5.57570e7	0.835507
$173$	−8.43239e7	−1.23820	−0.619098	−	0.785314i	$-0.712502\pi$
$173$	−8.43239e7	−1.23820	−0.619098	+	0.785314i	$0.712502\pi$
$174$	4.89616e7	0.704584
$175$	0	0
$176$	1.54797e7	0.214026
$177$	−4.59704e7	−0.623121
$178$	−5.58628e7	−0.742426
$179$	−2.62488e7	−0.342077	−0.171039	−	0.985264i	$-0.554712\pi$
$179$	−2.62488e7	−0.342077	−0.171039	+	0.985264i	$0.554712\pi$
$180$	−3.60841e6	−0.0461171
$181$	7.16289e7	0.897869	0.448935	−	0.893565i	$-0.351804\pi$
$181$	7.16289e7	0.897869	0.448935	+	0.893565i	$0.351804\pi$
$182$	0	0
$183$	8.31545e7	1.00301
$184$	1.80323e7	0.213397
$185$	3.90267e7	0.453170
$186$	1.08415e7	0.123536
$187$	−9.80871e7	−1.09690
$188$	−4.64197e7	−0.509507
$189$	0	0
$190$	3.43167e7	0.362967
$191$	−6.53616e7	−0.678743	−0.339372	−	0.940652i	$-0.610214\pi$
$191$	−6.53616e7	−0.678743	−0.339372	+	0.940652i	$0.610214\pi$
$192$	−1.05428e7	−0.107499
$193$	6.33769e7	0.634572	0.317286	−	0.948330i	$-0.397229\pi$
$193$	6.33769e7	0.634572	0.317286	+	0.948330i	$0.397229\pi$
$194$	1.88240e7	0.185100
$195$	−1.96867e7	−0.190130
$196$	0	0
$197$	−8.21193e7	−0.765268	−0.382634	−	0.923900i	$-0.624983\pi$
$197$	−8.21193e7	−0.765268	−0.382634	+	0.923900i	$0.624983\pi$
$198$	1.72193e7	0.157647
$199$	2.80666e7	0.252467	0.126233	−	0.992001i	$-0.459711\pi$
$199$	2.80666e7	0.252467	0.126233	+	0.992001i	$0.459711\pi$
$200$	3.49824e7	0.309204
$201$	−9.32820e7	−0.810236
$202$	1.44525e8	1.23371
$203$	0	0
$204$	6.68048e7	0.550937
$205$	1.47344e7	0.119452
$206$	−1.17626e8	−0.937491
$207$	2.00587e7	0.157184
$208$	2.02536e7	0.156056
$209$	−1.63758e8	−1.24077
$210$	0	0
$211$	1.62082e8	1.18781	0.593905	−	0.804535i	$-0.297586\pi$
$211$	1.62082e8	1.18781	0.593905	+	0.804535i	$0.297586\pi$
$212$	1.16387e8	0.838939
$213$	4.62759e7	0.328115
$214$	2.39558e7	0.167095
$215$	8.62448e7	0.591831
$216$	−5.67611e7	−0.383233
$217$	0	0
$218$	−1.42714e8	−0.932971
$219$	3.69770e7	0.237890
$220$	2.39439e7	0.151605
$221$	−1.28337e8	−0.799798
$222$	1.26840e8	0.778073
$223$	2.91229e6	0.0175860	0.00879300	−	0.999961i	$-0.497201\pi$
$223$	2.91229e6	0.0175860	0.00879300	+	0.999961i	$0.497201\pi$
$224$	0	0
$225$	3.89137e7	0.227753
$226$	1.66249e8	0.958033
$227$	1.70208e8	0.965806	0.482903	−	0.875674i	$-0.339582\pi$
$227$	1.70208e8	0.965806	0.482903	+	0.875674i	$0.339582\pi$
$228$	1.11532e8	0.623200
$229$	2.61310e8	1.43791	0.718954	−	0.695057i	$-0.244621\pi$
$229$	2.61310e8	1.43791	0.718954	+	0.695057i	$0.244621\pi$
$230$	2.78922e7	0.151160
$231$	0	0
$232$	−7.79145e7	−0.409648
$233$	−2.75249e8	−1.42554	−0.712771	−	0.701396i	$-0.752560\pi$
$233$	−2.75249e8	−1.42554	−0.712771	+	0.701396i	$0.752560\pi$
$234$	2.25297e7	0.114948
$235$	−7.18018e7	−0.360909
$236$	7.31546e7	0.362284
$237$	−2.50168e8	−1.22071
$238$	0	0
$239$	−1.83277e8	−0.868392	−0.434196	−	0.900819i	$-0.642967\pi$
$239$	−1.83277e8	−0.868392	−0.434196	+	0.900819i	$0.642967\pi$
$240$	−1.63076e7	−0.0761466
$241$	−8.49864e7	−0.391102	−0.195551	−	0.980694i	$-0.562650\pi$
$241$	−8.49864e7	−0.391102	−0.195551	+	0.980694i	$0.562650\pi$
$242$	4.16378e7	0.188857
$243$	−1.13234e8	−0.506240
$244$	−1.32327e8	−0.583155
$245$	0	0
$246$	4.78881e7	0.205095
$247$	−2.14262e8	−0.904701
$248$	−1.72525e7	−0.0718241
$249$	−2.75929e8	−1.13266
$250$	1.15982e8	0.469464
$251$	−2.02873e8	−0.809777	−0.404888	−	0.914366i	$-0.632690\pi$
$251$	−2.02873e8	−0.809777	−0.404888	+	0.914366i	$0.632690\pi$
$252$	0	0
$253$	−1.33101e8	−0.516725
$254$	−1.21554e8	−0.465428
$255$	1.03333e8	0.390256
$256$	1.67772e7	0.0625000
$257$	4.82854e8	1.77439	0.887196	−	0.461393i	$-0.152650\pi$
$257$	4.82854e8	1.77439	0.887196	+	0.461393i	$0.152650\pi$
$258$	2.80302e8	1.01615
$259$	0	0
$260$	3.13282e7	0.110542
$261$	−8.66706e7	−0.301738
$262$	−3.13971e8	−1.07854
$263$	2.11160e8	0.715758	0.357879	−	0.933768i	$-0.383500\pi$
$263$	2.11160e8	0.715758	0.357879	+	0.933768i	$0.383500\pi$
$264$	7.78195e7	0.260300
$265$	1.80028e8	0.594263
$266$	0	0
$267$	−2.80834e8	−0.902943
$268$	1.48443e8	0.471074
$269$	−3.12650e8	−0.979321	−0.489660	−	0.871913i	$-0.662879\pi$
$269$	−3.12650e8	−0.979321	−0.489660	+	0.871913i	$0.662879\pi$
$270$	−8.77979e7	−0.271463
$271$	−4.54515e8	−1.38725	−0.693627	−	0.720335i	$-0.743988\pi$
$271$	−4.54515e8	−1.38725	−0.693627	+	0.720335i	$0.743988\pi$
$272$	−1.06309e8	−0.320317
$273$	0	0
$274$	−2.19037e8	−0.643267
$275$	−2.58215e8	−0.748715
$276$	9.06520e7	0.259535
$277$	−1.47766e7	−0.0417729	−0.0208864	−	0.999782i	$-0.506649\pi$
$277$	−1.47766e7	−0.0417729	−0.0208864	+	0.999782i	$0.506649\pi$
$278$	−1.33607e8	−0.372968
$279$	−1.91913e7	−0.0529041
$280$	0	0
$281$	−3.26508e8	−0.877852	−0.438926	−	0.898523i	$-0.644641\pi$
$281$	−3.26508e8	−0.877852	−0.438926	+	0.898523i	$0.644641\pi$
$282$	−2.33362e8	−0.619666
$283$	−5.20692e8	−1.36562	−0.682808	−	0.730598i	$-0.739242\pi$
$283$	−5.20692e8	−1.36562	−0.682808	+	0.730598i	$0.739242\pi$
$284$	−7.36407e7	−0.190767
$285$	1.72517e8	0.441444
$286$	−1.49497e8	−0.377879
$287$	0	0
$288$	1.86626e7	0.0460362
$289$	2.63292e8	0.641645
$290$	−1.20518e8	−0.290174
$291$	9.46321e7	0.225120
$292$	−5.88429e7	−0.138310
$293$	−1.89711e8	−0.440611	−0.220306	−	0.975431i	$-0.570705\pi$
$293$	−1.89711e8	−0.440611	−0.220306	+	0.975431i	$0.570705\pi$
$294$	0	0
$295$	1.13155e8	0.256624
$296$	−2.01845e8	−0.452374
$297$	4.18969e8	0.927972
$298$	−8.09057e7	−0.177102
$299$	−1.74150e8	−0.376768
$300$	1.75864e8	0.376056
$301$	0	0
$302$	1.78513e8	0.372945
$303$	7.26557e8	1.50045
$304$	−1.77485e8	−0.362330
$305$	−2.04683e8	−0.413078
$306$	−1.18256e8	−0.235939
$307$	1.00531e9	1.98297	0.991486	−	0.130214i	$-0.0415663\pi$
$307$	1.00531e9	1.98297	0.991486	+	0.130214i	$0.0415663\pi$
$308$	0	0
$309$	−5.91329e8	−1.14018
$310$	−2.66860e7	−0.0508766
$311$	2.92979e8	0.552300	0.276150	−	0.961115i	$-0.410941\pi$
$311$	2.92979e8	0.552300	0.276150	+	0.961115i	$0.410941\pi$
$312$	1.01819e8	0.189796
$313$	5.90505e8	1.08847	0.544237	−	0.838931i	$-0.316819\pi$
$313$	5.90505e8	1.08847	0.544237	+	0.838931i	$0.316819\pi$
$314$	−3.81989e8	−0.696302
$315$	0	0
$316$	3.98103e8	0.709725
$317$	−3.93367e8	−0.693569	−0.346785	−	0.937945i	$-0.612727\pi$
$317$	−3.93367e8	−0.693569	−0.346785	+	0.937945i	$0.612727\pi$
$318$	5.85104e8	1.02032
$319$	5.75108e8	0.991933
$320$	2.59509e7	0.0442719
$321$	1.20431e8	0.203222
$322$	0	0
$323$	1.12464e9	1.85697
$324$	−2.05633e8	−0.335881
$325$	−3.37849e8	−0.545922
$326$	−6.04752e8	−0.966753
$327$	−7.17452e8	−1.13469
$328$	−7.62062e7	−0.119243
$329$	0	0
$330$	1.20371e8	0.184384
$331$	8.93074e8	1.35360	0.676798	−	0.736168i	$-0.263367\pi$
$331$	8.93074e8	1.35360	0.676798	+	0.736168i	$0.263367\pi$
$332$	4.39097e8	0.658532
$333$	−2.24529e8	−0.333210
$334$	−4.45109e8	−0.653663
$335$	2.29612e8	0.333685
$336$	0	0
$337$	1.22512e9	1.74371	0.871855	−	0.489765i	$-0.162917\pi$
$337$	1.22512e9	1.74371	0.871855	+	0.489765i	$0.162917\pi$
$338$	3.06386e8	0.431578
$339$	8.35770e8	1.16517
$340$	−1.64439e8	−0.226896
$341$	1.27345e8	0.173917
$342$	−1.97431e8	−0.266885
$343$	0	0
$344$	−4.46056e8	−0.590793
$345$	1.40220e8	0.183841
$346$	6.74592e8	0.875537
$347$	3.03828e8	0.390368	0.195184	−	0.980767i	$-0.437470\pi$
$347$	3.03828e8	0.390368	0.195184	+	0.980767i	$0.437470\pi$
$348$	−3.91693e8	−0.498216
$349$	−8.97387e8	−1.13003	−0.565016	−	0.825080i	$-0.691130\pi$
$349$	−8.97387e8	−1.13003	−0.565016	+	0.825080i	$0.691130\pi$
$350$	0	0
$351$	5.48180e8	0.676626
$352$	−1.23837e8	−0.151339
$353$	−1.92936e8	−0.233454	−0.116727	−	0.993164i	$-0.537240\pi$
$353$	−1.92936e8	−0.233454	−0.116727	+	0.993164i	$0.537240\pi$
$354$	3.67763e8	0.440613
$355$	−1.13907e8	−0.135130
$356$	4.46903e8	0.524974
$357$	0	0
$358$	2.09991e8	0.241885
$359$	5.45101e8	0.621793	0.310897	−	0.950444i	$-0.399371\pi$
$359$	5.45101e8	0.621793	0.310897	+	0.950444i	$0.399371\pi$
$360$	2.88673e7	0.0326097
$361$	9.83734e8	1.10053
$362$	−5.73031e8	−0.634889
$363$	2.09322e8	0.229690
$364$	0	0
$365$	−9.10180e7	−0.0979721
$366$	−6.65236e8	−0.709238
$367$	−1.38009e9	−1.45739	−0.728694	−	0.684839i	$-0.759873\pi$
$367$	−1.38009e9	−1.45739	−0.728694	+	0.684839i	$0.759873\pi$
$368$	−1.44258e8	−0.150894
$369$	−8.47703e7	−0.0878317
$370$	−3.12214e8	−0.320439
$371$	0	0
$372$	−8.67317e7	−0.0873530
$373$	1.58943e9	1.58584	0.792921	−	0.609324i	$-0.208559\pi$
$373$	1.58943e9	1.58584	0.792921	+	0.609324i	$0.208559\pi$
$374$	7.84697e8	0.775624
$375$	5.83068e8	0.570966
$376$	3.71358e8	0.360276
$377$	7.52473e8	0.723263
$378$	0	0
$379$	8.75070e8	0.825669	0.412834	−	0.910806i	$-0.364539\pi$
$379$	8.75070e8	0.825669	0.412834	+	0.910806i	$0.364539\pi$
$380$	−2.74533e8	−0.256657
$381$	−6.11079e8	−0.566057
$382$	5.22892e8	0.479944
$383$	−1.72010e9	−1.56443	−0.782217	−	0.623006i	$-0.785911\pi$
$383$	−1.72010e9	−1.56443	−0.782217	+	0.623006i	$0.785911\pi$
$384$	8.43426e7	0.0760129
$385$	0	0
$386$	−5.07015e8	−0.448710
$387$	−4.96184e8	−0.435166
$388$	−1.50592e8	−0.130885
$389$	−5.90498e8	−0.508621	−0.254311	−	0.967123i	$-0.581849\pi$
$389$	−5.90498e8	−0.508621	−0.254311	+	0.967123i	$0.581849\pi$
$390$	1.57493e8	0.134442
$391$	9.14095e8	0.773344
$392$	0	0
$393$	−1.57840e9	−1.31173
$394$	6.56955e8	0.541126
$395$	6.15784e8	0.502734
$396$	−1.37754e8	−0.111473
$397$	1.60244e9	1.28533	0.642664	−	0.766148i	$-0.277829\pi$
$397$	1.60244e9	1.28533	0.642664	+	0.766148i	$0.277829\pi$
$398$	−2.24533e8	−0.178521
$399$	0	0
$400$	−2.79859e8	−0.218640
$401$	−2.09943e9	−1.62591	−0.812953	−	0.582329i	$-0.802141\pi$
$401$	−2.09943e9	−1.62591	−0.812953	+	0.582329i	$0.802141\pi$
$402$	7.46256e8	0.572923
$403$	1.66619e8	0.126811
$404$	−1.15620e9	−0.872364
$405$	−3.18072e8	−0.237921
$406$	0	0
$407$	1.48988e9	1.09539
$408$	−5.34438e8	−0.389571
$409$	−2.46721e9	−1.78310	−0.891548	−	0.452926i	$-0.850380\pi$
$409$	−2.46721e9	−1.78310	−0.891548	+	0.452926i	$0.850380\pi$
$410$	−1.17875e8	−0.0844656
$411$	−1.10115e9	−0.782346
$412$	9.41006e8	0.662906
$413$	0	0
$414$	−1.60470e8	−0.111146
$415$	6.79193e8	0.466471
$416$	−1.62029e8	−0.110348
$417$	−6.71669e8	−0.453606
$418$	1.31007e9	0.877357
$419$	−7.76135e8	−0.515452	−0.257726	−	0.966218i	$-0.582973\pi$
$419$	−7.76135e8	−0.515452	−0.257726	+	0.966218i	$0.582973\pi$
$420$	0	0
$421$	−2.83817e9	−1.85375	−0.926876	−	0.375369i	$-0.877516\pi$
$421$	−2.83817e9	−1.85375	−0.926876	+	0.375369i	$0.877516\pi$
$422$	−1.29666e9	−0.839908
$423$	4.13091e8	0.265372
$424$	−9.31099e8	−0.593220
$425$	1.77333e9	1.12054
$426$	−3.70207e8	−0.232012
$427$	0	0
$428$	−1.91646e8	−0.118154
$429$	−7.51555e8	−0.459579
$430$	−6.89958e8	−0.418488
$431$	−2.40721e9	−1.44825	−0.724126	−	0.689667i	$-0.757757\pi$
$431$	−2.40721e9	−1.44825	−0.724126	+	0.689667i	$0.757757\pi$
$432$	4.54089e8	0.270987
$433$	−3.89959e7	−0.0230840	−0.0115420	−	0.999933i	$-0.503674\pi$
$433$	−3.89959e7	−0.0230840	−0.0115420	+	0.999933i	$0.503674\pi$
$434$	0	0
$435$	−6.05869e8	−0.352912
$436$	1.14171e9	0.659710
$437$	1.52610e9	0.874777
$438$	−2.95816e8	−0.168214
$439$	2.57665e9	1.45355	0.726775	−	0.686875i	$-0.241018\pi$
$439$	2.57665e9	1.45355	0.726775	+	0.686875i	$0.241018\pi$
$440$	−1.91551e8	−0.107201
$441$	0	0
$442$	1.02670e9	0.565542
$443$	1.74303e9	0.952558	0.476279	−	0.879294i	$-0.341985\pi$
$443$	1.74303e9	0.952558	0.476279	+	0.879294i	$0.341985\pi$
$444$	−1.01472e9	−0.550181
$445$	6.91267e8	0.371866
$446$	−2.32983e7	−0.0124352
$447$	−4.06730e8	−0.215392
$448$	0	0
$449$	−1.79538e9	−0.936037	−0.468019	−	0.883719i	$-0.655032\pi$
$449$	−1.79538e9	−0.936037	−0.468019	+	0.883719i	$0.655032\pi$
$450$	−3.11310e8	−0.161046
$451$	5.62499e8	0.288738
$452$	−1.32999e9	−0.677431
$453$	8.97421e8	0.453579
$454$	−1.36166e9	−0.682928
$455$	0	0
$456$	−8.92255e8	−0.440669
$457$	−3.06780e9	−1.50356	−0.751780	−	0.659414i	$-0.770804\pi$
$457$	−3.06780e9	−1.50356	−0.751780	+	0.659414i	$0.770804\pi$
$458$	−2.09048e9	−1.01675
$459$	−2.87734e9	−1.38882
$460$	−2.23138e8	−0.106886
$461$	−2.33753e9	−1.11123	−0.555616	−	0.831439i	$-0.687518\pi$
$461$	−2.33753e9	−1.11123	−0.555616	+	0.831439i	$0.687518\pi$
$462$	0	0
$463$	1.03447e8	0.0484377	0.0242188	−	0.999707i	$-0.492290\pi$
$463$	1.03447e8	0.0484377	0.0242188	+	0.999707i	$0.492290\pi$
$464$	6.23316e8	0.289665
$465$	−1.34156e8	−0.0618765
$466$	2.20199e9	1.00801
$467$	−4.79172e8	−0.217712	−0.108856	−	0.994058i	$-0.534719\pi$
$467$	−4.79172e8	−0.217712	−0.108856	+	0.994058i	$0.534719\pi$
$468$	−1.80238e8	−0.0812803
$469$	0	0
$470$	5.74415e8	0.255201
$471$	−1.92034e9	−0.846847
$472$	−5.85237e8	−0.256174
$473$	3.29246e9	1.43056
$474$	2.00135e9	0.863173
$475$	2.96061e9	1.26752
$476$	0	0
$477$	−1.03574e9	−0.436953
$478$	1.46622e9	0.614046
$479$	3.61346e9	1.50227	0.751136	−	0.660147i	$-0.229506\pi$
$479$	3.61346e9	1.50227	0.751136	+	0.660147i	$0.229506\pi$
$480$	1.30461e8	0.0538438
$481$	1.94936e9	0.798700
$482$	6.79891e8	0.276551
$483$	0	0
$484$	−3.33103e8	−0.133542
$485$	−2.32935e8	−0.0927126
$486$	9.05875e8	0.357965
$487$	3.62741e9	1.42313	0.711565	−	0.702620i	$-0.247987\pi$
$487$	3.62741e9	1.42313	0.711565	+	0.702620i	$0.247987\pi$
$488$	1.05862e9	0.412353
$489$	−3.04021e9	−1.17577
$490$	0	0
$491$	7.23318e7	0.0275768	0.0137884	−	0.999905i	$-0.495611\pi$
$491$	7.23318e7	0.0275768	0.0137884	+	0.999905i	$0.495611\pi$
$492$	−3.83105e8	−0.145024
$493$	−3.94966e9	−1.48455
$494$	1.71409e9	0.639720
$495$	−2.13077e8	−0.0789622
$496$	1.38020e8	0.0507873
$497$	0	0
$498$	2.20743e9	0.800912
$499$	2.73543e9	0.985541	0.492770	−	0.870159i	$-0.335984\pi$
$499$	2.73543e9	0.985541	0.492770	+	0.870159i	$0.335984\pi$
$500$	−9.27860e8	−0.331961
$501$	−2.23766e9	−0.794990
$502$	1.62298e9	0.572599
$503$	−9.77342e8	−0.342420	−0.171210	−	0.985235i	$-0.554768\pi$
$503$	−9.77342e8	−0.342420	−0.171210	+	0.985235i	$0.554768\pi$
$504$	0	0
$505$	−1.78840e9	−0.617940
$506$	1.06481e9	0.365380
$507$	1.54026e9	0.524889
$508$	9.72435e8	0.329107
$509$	−5.12379e9	−1.72218	−0.861091	−	0.508451i	$-0.830218\pi$
$509$	−5.12379e9	−1.72218	−0.861091	+	0.508451i	$0.830218\pi$
$510$	−8.26667e8	−0.275953
$511$	0	0
$512$	−1.34218e8	−0.0441942
$513$	−4.80378e9	−1.57099
$514$	−3.86283e9	−1.25468
$515$	1.45554e9	0.469570
$516$	−2.24242e9	−0.718526
$517$	−2.74109e9	−0.872382
$518$	0	0
$519$	3.39131e9	1.06483
$520$	−2.50626e8	−0.0781652
$521$	5.06397e9	1.56877	0.784384	−	0.620275i	$-0.212979\pi$
$521$	5.06397e9	1.56877	0.784384	+	0.620275i	$0.212979\pi$
$522$	6.93364e8	0.213361
$523$	−2.17853e9	−0.665898	−0.332949	−	0.942945i	$-0.608044\pi$
$523$	−2.17853e9	−0.665898	−0.332949	+	0.942945i	$0.608044\pi$
$524$	2.51177e9	0.762642
$525$	0	0
$526$	−1.68928e9	−0.506118
$527$	−8.74564e8	−0.260288
$528$	−6.22556e8	−0.184060
$529$	−2.16443e9	−0.635694
$530$	−1.44022e9	−0.420207
$531$	−6.51006e8	−0.188692
$532$	0	0
$533$	7.35975e8	0.210532
$534$	2.24667e9	0.638477
$535$	−2.96438e8	−0.0836942
$536$	−1.18755e9	−0.333099
$537$	1.05567e9	0.294183
$538$	2.50120e9	0.692484
$539$	0	0
$540$	7.02383e8	0.191953
$541$	3.32200e9	0.902005	0.451002	−	0.892523i	$-0.351067\pi$
$541$	3.32200e9	0.902005	0.451002	+	0.892523i	$0.351067\pi$
$542$	3.63612e9	0.980936
$543$	−2.88075e9	−0.772157
$544$	8.50473e8	0.226498
$545$	1.76599e9	0.467306
$546$	0	0
$547$	2.97826e9	0.778049	0.389025	−	0.921227i	$-0.372812\pi$
$547$	2.97826e9	0.778049	0.389025	+	0.921227i	$0.372812\pi$
$548$	1.75230e9	0.454858
$549$	1.17758e9	0.303731
$550$	2.06572e9	0.529421
$551$	−6.59403e9	−1.67927
$552$	−7.25216e8	−0.183519
$553$	0	0
$554$	1.18213e8	0.0295379
$555$	−1.56956e9	−0.389721
$556$	1.06885e9	0.263728
$557$	−1.63407e9	−0.400662	−0.200331	−	0.979728i	$-0.564202\pi$
$557$	−1.63407e9	−0.400662	−0.200331	+	0.979728i	$0.564202\pi$
$558$	1.53530e8	0.0374089
$559$	4.30786e9	1.04309
$560$	0	0
$561$	3.94484e9	0.943320
$562$	2.61206e9	0.620735
$563$	4.04502e8	0.0955303	0.0477652	−	0.998859i	$-0.484790\pi$
$563$	4.04502e8	0.0955303	0.0477652	+	0.998859i	$0.484790\pi$
$564$	1.86689e9	0.438170
$565$	−2.05723e9	−0.479859
$566$	4.16554e9	0.965637
$567$	0	0
$568$	5.89126e8	0.134893
$569$	−8.81934e7	−0.0200698	−0.0100349	−	0.999950i	$-0.503194\pi$
$569$	−8.81934e7	−0.0200698	−0.0100349	+	0.999950i	$0.503194\pi$
$570$	−1.38014e9	−0.312148
$571$	−4.25590e9	−0.956676	−0.478338	−	0.878176i	$-0.658761\pi$
$571$	−4.25590e9	−0.956676	−0.478338	+	0.878176i	$0.658761\pi$
$572$	1.19598e9	0.267201
$573$	2.62869e9	0.583711
$574$	0	0
$575$	2.40636e9	0.527865
$576$	−1.49301e8	−0.0325525
$577$	−1.80650e9	−0.391493	−0.195746	−	0.980655i	$-0.562713\pi$
$577$	−1.80650e9	−0.391493	−0.195746	+	0.980655i	$0.562713\pi$
$578$	−2.10633e9	−0.453711
$579$	−2.54887e9	−0.545724
$580$	9.64143e8	0.205184
$581$	0	0
$582$	−7.57057e8	−0.159184
$583$	6.87270e9	1.43644
$584$	4.70743e8	0.0978001
$585$	−2.78791e8	−0.0575749
$586$	1.51769e9	0.311559
$587$	1.17257e9	0.239279	0.119640	−	0.992817i	$-0.461826\pi$
$587$	1.17257e9	0.239279	0.119640	+	0.992817i	$0.461826\pi$
$588$	0	0
$589$	−1.46010e9	−0.294428
$590$	−9.05242e8	−0.181461
$591$	3.30265e9	0.658122
$592$	1.61476e9	0.319877
$593$	4.44047e9	0.874455	0.437227	−	0.899351i	$-0.355961\pi$
$593$	4.44047e9	0.874455	0.437227	+	0.899351i	$0.355961\pi$
$594$	−3.35176e9	−0.656175
$595$	0	0
$596$	6.47246e8	0.125230
$597$	−1.12877e9	−0.217118
$598$	1.39320e9	0.266415
$599$	1.54770e9	0.294235	0.147117	−	0.989119i	$-0.453000\pi$
$599$	1.54770e9	0.294235	0.147117	+	0.989119i	$0.453000\pi$
$600$	−1.40691e9	−0.265912
$601$	8.34134e9	1.56738	0.783691	−	0.621151i	$-0.213335\pi$
$601$	8.34134e9	1.56738	0.783691	+	0.621151i	$0.213335\pi$
$602$	0	0
$603$	−1.32100e9	−0.245354
$604$	−1.42810e9	−0.263712
$605$	−5.15242e8	−0.0945948
$606$	−5.81246e9	−1.06098
$607$	−5.27449e7	−0.00957238	−0.00478619	−	0.999989i	$-0.501523\pi$
$607$	−5.27449e7	−0.00957238	−0.00478619	+	0.999989i	$0.501523\pi$
$608$	1.41988e9	0.256206
$609$	0	0
$610$	1.63746e9	0.292090
$611$	−3.58645e9	−0.636093
$612$	9.46050e8	0.166834
$613$	2.51149e9	0.440373	0.220186	−	0.975458i	$-0.429333\pi$
$613$	2.51149e9	0.440373	0.220186	+	0.975458i	$0.429333\pi$
$614$	−8.04250e9	−1.40217
$615$	−5.92585e8	−0.102728
$616$	0	0
$617$	4.91512e9	0.842434	0.421217	−	0.906960i	$-0.361603\pi$
$617$	4.91512e9	0.842434	0.421217	+	0.906960i	$0.361603\pi$
$618$	4.73063e9	0.806231
$619$	−2.21335e9	−0.375087	−0.187543	−	0.982256i	$-0.560053\pi$
$619$	−2.21335e9	−0.375087	−0.187543	+	0.982256i	$0.560053\pi$
$620$	2.13488e8	0.0359752
$621$	−3.90446e9	−0.654246
$622$	−2.34383e9	−0.390535
$623$	0	0
$624$	−8.14553e8	−0.134206
$625$	3.90268e9	0.639415
$626$	−4.72404e9	−0.769668
$627$	6.58598e9	1.06705
$628$	3.05591e9	0.492360
$629$	−1.02320e10	−1.63939
$630$	0	0
$631$	−4.58134e9	−0.725922	−0.362961	−	0.931804i	$-0.618234\pi$
$631$	−4.58134e9	−0.725922	−0.362961	+	0.931804i	$0.618234\pi$
$632$	−3.18482e9	−0.501851
$633$	−6.51857e9	−1.02150
$634$	3.14693e9	0.490428
$635$	1.50416e9	0.233123
$636$	−4.68083e9	−0.721478
$637$	0	0
$638$	−4.60087e9	−0.701402
$639$	6.55332e8	0.0993592
$640$	−2.07607e8	−0.0313050
$641$	1.45075e9	0.217565	0.108783	−	0.994066i	$-0.465305\pi$
$641$	1.45075e9	0.217565	0.108783	+	0.994066i	$0.465305\pi$
$642$	−9.63446e8	−0.143699
$643$	−2.87720e9	−0.426807	−0.213404	−	0.976964i	$-0.568455\pi$
$643$	−2.87720e9	−0.426807	−0.213404	+	0.976964i	$0.568455\pi$
$644$	0	0
$645$	−3.46856e9	−0.508968
$646$	−8.99711e9	−1.31307
$647$	4.04829e9	0.587634	0.293817	−	0.955862i	$-0.405074\pi$
$647$	4.04829e9	0.587634	0.293817	+	0.955862i	$0.405074\pi$
$648$	1.64506e9	0.237503
$649$	4.31979e9	0.620307
$650$	2.70279e9	0.386025
$651$	0	0
$652$	4.83802e9	0.683598
$653$	3.89699e9	0.547688	0.273844	−	0.961774i	$-0.411705\pi$
$653$	3.89699e9	0.547688	0.273844	+	0.961774i	$0.411705\pi$
$654$	5.73961e9	0.802344
$655$	3.88520e9	0.540217
$656$	6.09650e8	0.0843174
$657$	5.23646e8	0.0720375
$658$	0	0
$659$	−1.61283e9	−0.219528	−0.109764	−	0.993958i	$-0.535010\pi$
$659$	−1.61283e9	−0.219528	−0.109764	+	0.993958i	$0.535010\pi$
$660$	−9.62967e8	−0.130379
$661$	−8.68218e8	−0.116929	−0.0584647	−	0.998289i	$-0.518620\pi$
$661$	−8.68218e8	−0.116929	−0.0584647	+	0.998289i	$0.518620\pi$
$662$	−7.14459e9	−0.957138
$663$	5.16143e9	0.687817
$664$	−3.51278e9	−0.465653
$665$	0	0
$666$	1.79623e9	0.235615
$667$	−5.35956e9	−0.699340
$668$	3.56087e9	0.462210
$669$	−1.17125e8	−0.0151238
$670$	−1.83689e9	−0.235951
$671$	−7.81394e9	−0.998484
$672$	0	0
$673$	−3.31380e9	−0.419058	−0.209529	−	0.977802i	$-0.567193\pi$
$673$	−3.31380e9	−0.419058	−0.209529	+	0.977802i	$0.567193\pi$
$674$	−9.80096e9	−1.23299
$675$	−7.57462e9	−0.947976
$676$	−2.45108e9	−0.305172
$677$	1.19606e10	1.48146	0.740732	−	0.671801i	$-0.234479\pi$
$677$	1.19606e10	1.48146	0.740732	+	0.671801i	$0.234479\pi$
$678$	−6.68616e9	−0.823897
$679$	0	0
$680$	1.31551e9	0.160440
$681$	−6.84537e9	−0.830582
$682$	−1.01876e9	−0.122978
$683$	9.98538e9	1.19920	0.599601	−	0.800299i	$-0.295326\pi$
$683$	9.98538e9	1.19920	0.599601	+	0.800299i	$0.295326\pi$
$684$	1.57945e9	0.188716
$685$	2.71045e9	0.322199
$686$	0	0
$687$	−1.05093e10	−1.23658
$688$	3.56845e9	0.417753
$689$	8.99225e9	1.04737
$690$	−1.12176e9	−0.129996
$691$	4.47288e9	0.515720	0.257860	−	0.966182i	$-0.416983\pi$
$691$	4.47288e9	0.515720	0.257860	+	0.966182i	$0.416983\pi$
$692$	−5.39673e9	−0.619098
$693$	0	0
$694$	−2.43062e9	−0.276032
$695$	1.65330e9	0.186812
$696$	3.13354e9	0.352292
$697$	−3.86306e9	−0.432132
$698$	7.17910e9	0.799053
$699$	1.10699e10	1.22595
$700$	0	0
$701$	−5.17153e9	−0.567029	−0.283515	−	0.958968i	$-0.591500\pi$
$701$	−5.17153e9	−0.567029	−0.283515	+	0.958968i	$0.591500\pi$
$702$	−4.38544e9	−0.478447
$703$	−1.70825e10	−1.85442
$704$	9.90698e8	0.107013
$705$	2.88770e9	0.310378
$706$	1.54349e9	0.165077
$707$	0	0
$708$	−2.94211e9	−0.311560
$709$	1.46593e10	1.54473	0.772364	−	0.635180i	$-0.219074\pi$
$709$	1.46593e10	1.54473	0.772364	+	0.635180i	$0.219074\pi$
$710$	9.11257e8	0.0955514
$711$	−3.54273e9	−0.369653
$712$	−3.57522e9	−0.371213
$713$	−1.18676e9	−0.122616
$714$	0	0
$715$	1.84994e9	0.189272
$716$	−1.67993e9	−0.171039
$717$	7.37098e9	0.746807
$718$	−4.36080e9	−0.439674
$719$	1.53624e10	1.54137	0.770686	−	0.637215i	$-0.219914\pi$
$719$	1.53624e10	1.54137	0.770686	+	0.637215i	$0.219914\pi$
$720$	−2.30938e8	−0.0230586
$721$	0	0
$722$	−7.86987e9	−0.778193
$723$	3.41796e9	0.336343
$724$	4.58425e9	0.448935
$725$	−1.03975e10	−1.01332
$726$	−1.67458e9	−0.162415
$727$	−1.61641e10	−1.56021	−0.780104	−	0.625650i	$-0.784834\pi$
$727$	−1.61641e10	−1.56021	−0.780104	+	0.625650i	$0.784834\pi$
$728$	0	0
$729$	1.15809e10	1.10712
$730$	7.28144e8	0.0692767
$731$	−2.26116e10	−2.14101
$732$	5.32189e9	0.501507
$733$	3.39753e9	0.318639	0.159320	−	0.987227i	$-0.449070\pi$
$733$	3.39753e9	0.318639	0.159320	+	0.987227i	$0.449070\pi$
$734$	1.10407e10	1.03053
$735$	0	0
$736$	1.15407e9	0.106698
$737$	8.76561e9	0.806577
$738$	6.78162e8	0.0621064
$739$	−1.00733e10	−0.918157	−0.459079	−	0.888396i	$-0.651820\pi$
$739$	−1.00733e10	−0.918157	−0.459079	+	0.888396i	$0.651820\pi$
$740$	2.49771e9	0.226585
$741$	8.61711e9	0.778033
$742$	0	0
$743$	1.17736e9	0.105305	0.0526523	−	0.998613i	$-0.483233\pi$
$743$	1.17736e9	0.105305	0.0526523	+	0.998613i	$0.483233\pi$
$744$	6.93854e8	0.0617679
$745$	1.00116e9	0.0887065
$746$	−1.27154e10	−1.12136
$747$	−3.90754e9	−0.342990
$748$	−6.27758e9	−0.548449
$749$	0	0
$750$	−4.66455e9	−0.403734
$751$	−1.84155e10	−1.58651	−0.793256	−	0.608888i	$-0.791616\pi$
$751$	−1.84155e10	−1.58651	−0.793256	+	0.608888i	$0.791616\pi$
$752$	−2.97086e9	−0.254753
$753$	8.15906e9	0.696398
$754$	−6.01978e9	−0.511424
$755$	−2.20898e9	−0.186800
$756$	0	0
$757$	4.60910e9	0.386171	0.193086	−	0.981182i	$-0.438150\pi$
$757$	4.60910e9	0.386171	0.193086	+	0.981182i	$0.438150\pi$
$758$	−7.00056e9	−0.583836
$759$	5.35302e9	0.444378
$760$	2.19627e9	0.181484
$761$	−1.32369e10	−1.08878	−0.544389	−	0.838833i	$-0.683238\pi$
$761$	−1.32369e10	−1.08878	−0.544389	+	0.838833i	$0.683238\pi$
$762$	4.88863e9	0.400263
$763$	0	0
$764$	−4.18314e9	−0.339372
$765$	1.46335e9	0.118177
$766$	1.37608e10	1.10622
$767$	5.65202e9	0.452293
$768$	−6.74741e8	−0.0537493
$769$	1.16678e10	0.925225	0.462613	−	0.886561i	$-0.346912\pi$
$769$	1.16678e10	0.925225	0.462613	+	0.886561i	$0.346912\pi$
$770$	0	0
$771$	−1.94192e10	−1.52596
$772$	4.05612e9	0.317286
$773$	−1.55330e10	−1.20956	−0.604781	−	0.796392i	$-0.706739\pi$
$773$	−1.55330e10	−1.20956	−0.604781	+	0.796392i	$0.706739\pi$
$774$	3.96947e9	0.307708
$775$	−2.30229e9	−0.177666
$776$	1.20474e9	0.0925498
$777$	0	0
$778$	4.72398e9	0.359650
$779$	−6.44945e9	−0.488812
$780$	−1.25995e9	−0.0950651
$781$	−4.34850e9	−0.326633
$782$	−7.31276e9	−0.546837
$783$	1.68706e10	1.25592
$784$	0	0
$785$	4.72688e9	0.348763
$786$	1.26272e10	0.927530
$787$	1.04461e10	0.763907	0.381954	−	0.924181i	$-0.375251\pi$
$787$	1.04461e10	0.763907	0.381954	+	0.924181i	$0.375251\pi$
$788$	−5.25564e9	−0.382634
$789$	−8.49236e9	−0.615544
$790$	−4.92627e9	−0.355487
$791$	0	0
$792$	1.10203e9	0.0788236
$793$	−1.02238e10	−0.728040
$794$	−1.28195e10	−0.908864
$795$	−7.24029e9	−0.511059
$796$	1.79626e9	0.126233
$797$	−1.06705e10	−0.746584	−0.373292	−	0.927714i	$-0.621771\pi$
$797$	−1.06705e10	−0.746584	−0.373292	+	0.927714i	$0.621771\pi$
$798$	0	0
$799$	1.88249e10	1.30563
$800$	2.23887e9	0.154602
$801$	−3.97701e9	−0.273428
$802$	1.67954e10	1.14969
$803$	−3.47469e9	−0.236816
$804$	−5.97005e9	−0.405118
$805$	0	0
$806$	−1.33295e9	−0.0896687
$807$	1.25740e10	0.842205
$808$	9.24960e9	0.616855
$809$	1.89825e10	1.26048	0.630238	−	0.776402i	$-0.282957\pi$
$809$	1.89825e10	1.26048	0.630238	+	0.776402i	$0.282957\pi$
$810$	2.54458e9	0.168236
$811$	1.30672e9	0.0860222	0.0430111	−	0.999075i	$-0.486305\pi$
$811$	1.30672e9	0.0860222	0.0430111	+	0.999075i	$0.486305\pi$
$812$	0	0
$813$	1.82795e10	1.19302
$814$	−1.19190e10	−0.774560
$815$	7.48342e9	0.484227
$816$	4.27551e9	0.275469
$817$	−3.77504e10	−2.42184
$818$	1.97377e10	1.26084
$819$	0	0
$820$	9.43004e8	0.0597262
$821$	−1.89023e9	−0.119210	−0.0596052	−	0.998222i	$-0.518984\pi$
$821$	−1.89023e9	−0.119210	−0.0596052	+	0.998222i	$0.518984\pi$
$822$	8.80917e9	0.553202
$823$	4.63014e9	0.289531	0.144765	−	0.989466i	$-0.453757\pi$
$823$	4.63014e9	0.289531	0.144765	+	0.989466i	$0.453757\pi$
$824$	−7.52805e9	−0.468745
$825$	1.03848e10	0.643886
$826$	0	0
$827$	−9.32056e9	−0.573024	−0.286512	−	0.958077i	$-0.592496\pi$
$827$	−9.32056e9	−0.573024	−0.286512	+	0.958077i	$0.592496\pi$
$828$	1.28376e9	0.0785918
$829$	2.28125e10	1.39070	0.695349	−	0.718672i	$-0.255250\pi$
$829$	2.28125e10	1.39070	0.695349	+	0.718672i	$0.255250\pi$
$830$	−5.43355e9	−0.329845
$831$	5.94279e8	0.0359242
$832$	1.29623e9	0.0780280
$833$	0	0
$834$	5.37335e9	0.320748
$835$	5.50795e9	0.327406
$836$	−1.04805e10	−0.620385
$837$	3.73561e9	0.220203
$838$	6.20908e9	0.364480
$839$	−2.53946e10	−1.48448	−0.742241	−	0.670133i	$-0.766237\pi$
$839$	−2.53946e10	−1.48448	−0.742241	+	0.670133i	$0.766237\pi$
$840$	0	0
$841$	5.90790e9	0.342489
$842$	2.27054e10	1.31080
$843$	1.31314e10	0.754943
$844$	1.03733e10	0.593905
$845$	−3.79133e9	−0.216169
$846$	−3.30473e9	−0.187646
$847$	0	0
$848$	7.44879e9	0.419470
$849$	2.09410e10	1.17441
$850$	−1.41867e10	−0.792345
$851$	−1.38845e10	−0.772282
$852$	2.96166e9	0.164058
$853$	5.55257e9	0.306318	0.153159	−	0.988202i	$-0.451055\pi$
$853$	5.55257e9	0.306318	0.153159	+	0.988202i	$0.451055\pi$
$854$	0	0
$855$	2.44308e9	0.133677
$856$	1.53317e9	0.0835473
$857$	1.72828e10	0.937953	0.468977	−	0.883211i	$-0.344623\pi$
$857$	1.72828e10	0.937953	0.468977	+	0.883211i	$0.344623\pi$
$858$	6.01244e9	0.324971
$859$	2.95579e9	0.159110	0.0795550	−	0.996830i	$-0.474650\pi$
$859$	2.95579e9	0.159110	0.0795550	+	0.996830i	$0.474650\pi$
$860$	5.51966e9	0.295916
$861$	0	0
$862$	1.92577e10	1.02407
$863$	5.09978e9	0.270093	0.135047	−	0.990839i	$-0.456882\pi$
$863$	5.09978e9	0.270093	0.135047	+	0.990839i	$0.456882\pi$
$864$	−3.63271e9	−0.191616
$865$	−8.34764e9	−0.438538
$866$	3.11967e8	0.0163229
$867$	−1.05890e10	−0.551807
$868$	0	0
$869$	2.35080e10	1.21520
$870$	4.84695e9	0.249546
$871$	1.14689e10	0.588111
$872$	−9.13368e9	−0.466485
$873$	1.34012e9	0.0681703
$874$	−1.22088e10	−0.618561
$875$	0	0
$876$	2.36653e9	0.118945
$877$	2.38261e9	0.119276	0.0596382	−	0.998220i	$-0.481005\pi$
$877$	2.38261e9	0.119276	0.0596382	+	0.998220i	$0.481005\pi$
$878$	−2.06132e10	−1.02782
$879$	7.62973e9	0.378920
$880$	1.53241e9	0.0758027
$881$	1.38819e10	0.683964	0.341982	−	0.939707i	$-0.388902\pi$
$881$	1.38819e10	0.683964	0.341982	+	0.939707i	$0.388902\pi$
$882$	0	0
$883$	−1.59763e10	−0.780932	−0.390466	−	0.920617i	$-0.627686\pi$
$883$	−1.59763e10	−0.780932	−0.390466	+	0.920617i	$0.627686\pi$
$884$	−8.21359e9	−0.399899
$885$	−4.55084e9	−0.220694
$886$	−1.39442e10	−0.673560
$887$	3.80171e10	1.82914	0.914568	−	0.404431i	$-0.132530\pi$
$887$	3.80171e10	1.82914	0.914568	+	0.404431i	$0.132530\pi$
$888$	8.11775e9	0.389037
$889$	0	0
$890$	−5.53014e9	−0.262949
$891$	−1.21427e10	−0.575098
$892$	1.86386e8	0.00879300
$893$	3.14286e10	1.47688
$894$	3.25384e9	0.152305
$895$	−2.59850e9	−0.121155
$896$	0	0
$897$	7.00390e9	0.324016
$898$	1.43630e10	0.661878
$899$	5.12778e9	0.235380
$900$	2.49048e9	0.113876
$901$	−4.71994e10	−2.14981
$902$	−4.49999e9	−0.204169
$903$	0	0
$904$	1.06400e10	0.479016
$905$	7.09090e9	0.318003
$906$	−7.17937e9	−0.320728
$907$	−2.10728e10	−0.937772	−0.468886	−	0.883259i	$-0.655345\pi$
$907$	−2.10728e10	−0.937772	−0.468886	+	0.883259i	$0.655345\pi$
$908$	1.08933e10	0.482903
$909$	1.02891e10	0.454362
$910$	0	0
$911$	−2.32696e9	−0.101970	−0.0509852	−	0.998699i	$-0.516236\pi$
$911$	−2.32696e9	−0.101970	−0.0509852	+	0.998699i	$0.516236\pi$
$912$	7.13804e9	0.311600
$913$	2.59288e10	1.12755
$914$	2.45424e10	1.06318
$915$	8.23187e9	0.355243
$916$	1.67238e10	0.718954
$917$	0	0
$918$	2.30188e10	0.982047
$919$	−4.56675e10	−1.94090	−0.970450	−	0.241301i	$-0.922426\pi$
$919$	−4.56675e10	−1.94090	−0.970450	+	0.241301i	$0.922426\pi$
$920$	1.78510e9	0.0755798
$921$	−4.04313e10	−1.70533
$922$	1.87003e10	0.785760
$923$	−5.68958e9	−0.238163
$924$	0	0
$925$	−2.69357e10	−1.11901
$926$	−8.27574e8	−0.0342506
$927$	−8.37405e9	−0.345268
$928$	−4.98653e9	−0.204824
$929$	1.60424e10	0.656470	0.328235	−	0.944596i	$-0.393546\pi$
$929$	1.60424e10	0.656470	0.328235	+	0.944596i	$0.393546\pi$
$930$	1.07325e9	0.0437533
$931$	0	0
$932$	−1.76159e10	−0.712771
$933$	−1.17829e10	−0.474972
$934$	3.83337e9	0.153945
$935$	−9.71013e9	−0.388494
$936$	1.44190e9	0.0574738
$937$	−3.60968e10	−1.43344	−0.716722	−	0.697359i	$-0.754358\pi$
$937$	−3.60968e10	−1.43344	−0.716722	+	0.697359i	$0.754358\pi$
$938$	0	0
$939$	−2.37487e10	−0.936076
$940$	−4.59532e9	−0.180455
$941$	8.61873e9	0.337194	0.168597	−	0.985685i	$-0.446076\pi$
$941$	8.61873e9	0.337194	0.168597	+	0.985685i	$0.446076\pi$
$942$	1.53627e10	0.598811
$943$	−5.24205e9	−0.203568
$944$	4.68189e9	0.181142
$945$	0	0
$946$	−2.63397e10	−1.01156
$947$	−3.74410e9	−0.143259	−0.0716295	−	0.997431i	$-0.522820\pi$
$947$	−3.74410e9	−0.143259	−0.0716295	+	0.997431i	$0.522820\pi$
$948$	−1.60108e10	−0.610355
$949$	−4.54628e9	−0.172673
$950$	−2.36849e10	−0.896271
$951$	1.58203e10	0.596462
$952$	0	0
$953$	−9.19089e9	−0.343979	−0.171990	−	0.985099i	$-0.555020\pi$
$953$	−9.19089e9	−0.343979	−0.171990	+	0.985099i	$0.555020\pi$
$954$	8.28589e9	0.308973
$955$	−6.47046e9	−0.240394
$956$	−1.17297e10	−0.434196
$957$	−2.31295e10	−0.853051
$958$	−2.89077e10	−1.06227
$959$	0	0
$960$	−1.04369e9	−0.0380733
$961$	−2.63772e10	−0.958730
$962$	−1.55949e10	−0.564766
$963$	1.70547e9	0.0615392
$964$	−5.43913e9	−0.195551
$965$	6.27400e9	0.224749
$966$	0	0
$967$	−2.94926e10	−1.04887	−0.524434	−	0.851451i	$-0.675723\pi$
$967$	−2.94926e10	−1.04887	−0.524434	+	0.851451i	$0.675723\pi$
$968$	2.66482e9	0.0944287
$969$	−4.52304e10	−1.59697
$970$	1.86348e9	0.0655577
$971$	2.71931e10	0.953215	0.476608	−	0.879116i	$-0.341866\pi$
$971$	2.71931e10	0.953215	0.476608	+	0.879116i	$0.341866\pi$
$972$	−7.24700e9	−0.253120
$973$	0	0
$974$	−2.90193e10	−1.00631
$975$	1.35875e10	0.469486
$976$	−8.46893e9	−0.291578
$977$	4.00042e10	1.37238	0.686191	−	0.727422i	$-0.259282\pi$
$977$	4.00042e10	1.37238	0.686191	+	0.727422i	$0.259282\pi$
$978$	2.43217e10	0.831397
$979$	2.63897e10	0.898866
$980$	0	0
$981$	−1.01601e10	−0.343603
$982$	−5.78655e8	−0.0194998
$983$	−8.39258e9	−0.281811	−0.140905	−	0.990023i	$-0.545001\pi$
$983$	−8.39258e9	−0.281811	−0.140905	+	0.990023i	$0.545001\pi$
$984$	3.06484e9	0.102547
$985$	−8.12940e9	−0.271039
$986$	3.15972e10	1.04974
$987$	0	0
$988$	−1.37127e10	−0.452351
$989$	−3.06832e10	−1.00859
$990$	1.70462e9	0.0558347
$991$	−2.36957e10	−0.773415	−0.386707	−	0.922203i	$-0.626388\pi$
$991$	−2.36957e10	−0.773415	−0.386707	+	0.922203i	$0.626388\pi$
$992$	−1.10416e9	−0.0359120
$993$	−3.59173e10	−1.16408
$994$	0	0
$995$	2.77845e9	0.0894174
$996$	−1.76595e10	−0.566330
$997$	1.02812e9	0.0328558	0.0164279	−	0.999865i	$-0.494771\pi$
$997$	1.02812e9	0.0328558	0.0164279	+	0.999865i	$0.494771\pi$
$998$	−2.18835e10	−0.696882
$999$	4.37049e10	1.38692

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	98.8.a.l.1.2	✓	4
7.2	even	3		98.8.c.n.67.3		8
7.3	odd	6		98.8.c.n.79.2		8
7.4	even	3		98.8.c.n.79.3		8
7.5	odd	6		98.8.c.n.67.2		8
7.6	odd	2	inner	98.8.a.l.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
98.8.a.l.1.2	✓	4	1.1	even	1	trivial
98.8.a.l.1.3	yes	4	7.6	odd	2	inner
98.8.c.n.67.2		8	7.5	odd	6
98.8.c.n.67.3		8	7.2	even	3
98.8.c.n.79.2		8	7.3	odd	6
98.8.c.n.79.3		8	7.4	even	3

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:	\( N \)	\(=\)	\( 98 = 2 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	98.a (trivial)

\(f(q)\)	\(=\)	\(q-8.00000 q^{2} -40.2177 q^{3} +64.0000 q^{4} +98.9949 q^{5} +321.741 q^{6} -512.000 q^{8} -569.539 q^{9} -791.960 q^{10} +3779.21 q^{11} -2573.93 q^{12} +4944.73 q^{13} -3981.35 q^{15} +4096.00 q^{16} -25954.4 q^{17} +4556.31 q^{18} -43331.3 q^{19} +6335.68 q^{20} -30233.7 q^{22} -35219.3 q^{23} +20591.4 q^{24} -68325.0 q^{25} -39557.8 q^{26} +110862. q^{27} +152177. q^{29} +31850.8 q^{30} +33696.2 q^{31} -32768.0 q^{32} -151991. q^{33} +207635. q^{34} -36450.5 q^{36} +394229. q^{37} +346651. q^{38} -198865. q^{39} -50685.4 q^{40} +148840. q^{41} +871204. q^{43} +241870. q^{44} -56381.5 q^{45} +281754. q^{46} -725308. q^{47} -164732. q^{48} +546600. q^{50} +1.04383e6 q^{51} +316463. q^{52} +1.81855e6 q^{53} -886893. q^{54} +374123. q^{55} +1.74269e6 q^{57} -1.21741e6 q^{58} +1.14304e6 q^{59} -254806. q^{60} -2.06761e6 q^{61} -269570. q^{62} +262144. q^{64} +489503. q^{65} +1.21593e6 q^{66} +2.31943e6 q^{67} -1.66108e6 q^{68} +1.41644e6 q^{69} -1.15064e6 q^{71} +291604. q^{72} -919421. q^{73} -3.15384e6 q^{74} +2.74787e6 q^{75} -2.77321e6 q^{76} +1.59092e6 q^{78} +6.22035e6 q^{79} +405483. q^{80} -3.21301e6 q^{81} -1.19072e6 q^{82} +6.86089e6 q^{83} -2.56935e6 q^{85} -6.96963e6 q^{86} -6.12020e6 q^{87} -1.93496e6 q^{88} +6.98286e6 q^{89} +451052. q^{90} -2.25403e6 q^{92} -1.35518e6 q^{93} +5.80246e6 q^{94} -4.28958e6 q^{95} +1.31785e6 q^{96} -2.35300e6 q^{97} -2.15241e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 32 q^{2} + 256 q^{4} - 2048 q^{8} + 7228 q^{9} - 6272 q^{11} + 7840 q^{15} + 16384 q^{16} - 57824 q^{18} + 50176 q^{22} - 12544 q^{23} - 273300 q^{25} - 75736 q^{29} - 62720 q^{30} - 131072 q^{32}+ \cdots - 62164928 q^{99}+O(q^{100}) \) 4 * q - 32 * q^2 + 256 * q^4 - 2048 * q^8 + 7228 * q^9 - 6272 * q^11 + 7840 * q^15 + 16384 * q^16 - 57824 * q^18 + 50176 * q^22 - 12544 * q^23 - 273300 * q^25 - 75736 * q^29 - 62720 * q^30 - 131072 * q^32 + 462592 * q^36 + 1234696 * q^37 + 1771200 * q^39 + 2821760 * q^43 - 401408 * q^44 + 100352 * q^46 + 2186400 * q^50 + 5532304 * q^51 + 1285336 * q^53 + 9157160 * q^57 + 605888 * q^58 + 501760 * q^60 + 1048576 * q^64 + 3669120 * q^65 + 5342160 * q^67 + 2883552 * q^71 - 3700736 * q^72 - 9877568 * q^74 - 14169600 * q^78 + 6016448 * q^79 + 713228 * q^81 - 866320 * q^85 - 22574080 * q^86 + 3211264 * q^88 - 802816 * q^92 + 11291088 * q^93 - 1544480 * q^95 - 62164928 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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