LMFDB - Embedded newform 294.5.c.b.97.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [294,5,Mod(97,294)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(294, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("294.97");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 294 = 2 \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	294.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$30.3907691467$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.339738624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 $ x^8 - 4x^6 + 14x^4 - 8*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.3
Root			$-0.662827 - 0.382683i$ of defining polynomial
Character	$\chi$	$=$	294.97
Dual form			294.5.c.b.97.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843 q^{2} +5.19615i q^{3} +8.00000 q^{4} +10.0519i q^{5} -14.6969i q^{6} -22.6274 q^{8} -27.0000 q^{9} +O(q^{10})$	\(q-2.82843 q^{2} +5.19615i q^{3} +8.00000 q^{4} +10.0519i q^{5} -14.6969i q^{6} -22.6274 q^{8} -27.0000 q^{9} -28.4311i q^{10} +49.7072 q^{11} +41.5692i q^{12} -116.591i q^{13} -52.2313 q^{15} +64.0000 q^{16} -273.727i q^{17} +76.3675 q^{18} +12.1464i q^{19} +80.4154i q^{20} -140.593 q^{22} -13.6305 q^{23} -117.576i q^{24} +523.959 q^{25} +329.770i q^{26} -140.296i q^{27} -485.811 q^{29} +147.732 q^{30} -1797.96i q^{31} -181.019 q^{32} +258.286i q^{33} +774.217i q^{34} -216.000 q^{36} -533.706 q^{37} -34.3552i q^{38} +605.827 q^{39} -227.449i q^{40} +2680.49i q^{41} -378.695 q^{43} +397.657 q^{44} -271.402i q^{45} +38.5527 q^{46} -522.709i q^{47} +332.554i q^{48} -1481.98 q^{50} +1422.33 q^{51} -932.732i q^{52} +5575.84 q^{53} +396.817i q^{54} +499.653i q^{55} -63.1146 q^{57} +1374.08 q^{58} +27.3196i q^{59} -417.851 q^{60} -3123.41i q^{61} +5085.40i q^{62} +512.000 q^{64} +1171.97 q^{65} -730.543i q^{66} +7758.73 q^{67} -2189.82i q^{68} -70.8259i q^{69} +7556.02 q^{71} +610.940 q^{72} -4181.97i q^{73} +1509.55 q^{74} +2722.57i q^{75} +97.1713i q^{76} -1713.54 q^{78} +5598.55 q^{79} +643.323i q^{80} +729.000 q^{81} -7581.57i q^{82} +3074.79i q^{83} +2751.48 q^{85} +1071.11 q^{86} -2524.35i q^{87} -1124.74 q^{88} +3360.59i q^{89} +767.641i q^{90} -109.044 q^{92} +9342.47 q^{93} +1478.44i q^{94} -122.095 q^{95} -940.604i q^{96} -9460.26i q^{97} -1342.09 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 64 q^{4} - 216 q^{9}+O(q^{10}) $ 8 * q + 64 * q^4 - 216 * q^9	$ 8 q + 64 q^{4} - 216 q^{9} + 512 q^{16} - 1312 q^{22} - 272 q^{23} - 2808 q^{25} + 400 q^{29} + 3168 q^{30} - 1728 q^{36} - 3328 q^{37} + 656 q^{43} - 2400 q^{46} + 800 q^{50} + 7200 q^{51} + 9264 q^{53} + 1152 q^{57} - 11488 q^{58} + 4096 q^{64} - 15696 q^{65} + 26816 q^{67} + 28192 q^{71} - 4512 q^{74} + 9216 q^{78} + 19728 q^{79} + 5832 q^{81} - 49632 q^{85} + 5888 q^{86} - 10496 q^{88} - 2176 q^{92} + 15264 q^{93} - 92752 q^{95}+O(q^{100}) $ 8 * q + 64 * q^4 - 216 * q^9 + 512 * q^16 - 1312 * q^22 - 272 * q^23 - 2808 * q^25 + 400 * q^29 + 3168 * q^30 - 1728 * q^36 - 3328 * q^37 + 656 * q^43 - 2400 * q^46 + 800 * q^50 + 7200 * q^51 + 9264 * q^53 + 1152 * q^57 - 11488 * q^58 + 4096 * q^64 - 15696 * q^65 + 26816 * q^67 + 28192 * q^71 - 4512 * q^74 + 9216 * q^78 + 19728 * q^79 + 5832 * q^81 - 49632 * q^85 + 5888 * q^86 - 10496 * q^88 - 2176 * q^92 + 15264 * q^93 - 92752 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/294\mathbb{Z}\right)^\times$.

$n$	$197$	$199$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$	−2.82843	−0.707107
$2$	−2.82843	−0.707107
$3$	5.19615i	0.577350i
$3$	5.19615i	0.577350i
$4$	8.00000	0.500000
$5$	10.0519i	0.402077i	0.979583	+	0.201038i	$0.0644316\pi$
$5$	10.0519i	0.402077i	−0.979583	+	0.201038i	$0.935568\pi$
$6$	− 14.6969i	− 0.408248i
$7$	0	0
$7$	0	0
$8$	−22.6274	−0.353553
$9$	−27.0000	−0.333333
$10$	− 28.4311i	− 0.284311i
$11$	49.7072	0.410803	0.205402	−	0.978678i	$-0.434150\pi$
$11$	49.7072	0.410803	0.205402	+	0.978678i	$0.434150\pi$
$12$	41.5692i	0.288675i
$13$	− 116.591i	− 0.689890i	−0.938623	−	0.344945i	$-0.887898\pi$
$13$	− 116.591i	− 0.689890i	0.938623	−	0.344945i	$-0.112102\pi$
$14$	0	0
$15$	−52.2313	−0.232139
$16$	64.0000	0.250000
$17$	− 273.727i	− 0.947152i	−0.880753	−	0.473576i	$-0.842963\pi$
$17$	− 273.727i	− 0.947152i	0.880753	−	0.473576i	$-0.157037\pi$
$18$	76.3675	0.235702
$19$	12.1464i	0.0336466i	0.999858	+	0.0168233i	$0.00535527\pi$
$19$	12.1464i	0.0336466i	−0.999858	+	0.0168233i	$0.994645\pi$
$20$	80.4154i	0.201038i
$21$	0	0
$22$	−140.593	−0.290482
$23$	−13.6305	−0.0257665	−0.0128832	−	0.999917i	$-0.504101\pi$
$23$	−13.6305	−0.0257665	−0.0128832	+	0.999917i	$0.504101\pi$
$24$	− 117.576i	− 0.204124i
$25$	523.959	0.838334
$26$	329.770i	0.487826i
$27$	− 140.296i	− 0.192450i
$28$	0	0
$29$	−485.811	−0.577658	−0.288829	−	0.957381i	$-0.593266\pi$
$29$	−485.811	−0.577658	−0.288829	+	0.957381i	$0.593266\pi$
$30$	147.732	0.164147
$31$	− 1797.96i	− 1.87092i	−0.353426	−	0.935462i	$-0.614983\pi$
$31$	− 1797.96i	− 1.87092i	0.353426	−	0.935462i	$-0.385017\pi$
$32$	−181.019	−0.176777
$33$	258.286i	0.237177i
$34$	774.217i	0.669738i
$35$	0	0
$36$	−216.000	−0.166667
$37$	−533.706	−0.389851	−0.194925	−	0.980818i	$-0.562447\pi$
$37$	−533.706	−0.389851	−0.194925	+	0.980818i	$0.562447\pi$
$38$	− 34.3552i	− 0.0237917i
$39$	605.827	0.398308
$40$	− 227.449i	− 0.142156i
$41$	2680.49i	1.59458i	0.603596	+	0.797290i	$0.293734\pi$
$41$	2680.49i	1.59458i	−0.603596	+	0.797290i	$0.706266\pi$
$42$	0	0
$43$	−378.695	−0.204810	−0.102405	−	0.994743i	$-0.532654\pi$
$43$	−378.695	−0.204810	−0.102405	+	0.994743i	$0.532654\pi$
$44$	397.657	0.205402
$45$	− 271.402i	− 0.134026i
$46$	38.5527	0.0182196
$47$	− 522.709i	− 0.236627i	−0.992976	−	0.118313i	$-0.962251\pi$
$47$	− 522.709i	− 0.236627i	0.992976	−	0.118313i	$-0.0377487\pi$
$48$	332.554i	0.144338i
$49$	0	0
$50$	−1481.98	−0.592792
$51$	1422.33	0.546839
$52$	− 932.732i	− 0.344945i
$53$	5575.84	1.98499	0.992496	−	0.122274i	$-0.0390188\pi$
$53$	5575.84	1.98499	0.992496	+	0.122274i	$0.0390188\pi$
$54$	396.817i	0.136083i
$55$	499.653i	0.165174i
$56$	0	0
$57$	−63.1146	−0.0194259
$58$	1374.08	0.408466
$59$	27.3196i	0.00784819i	0.999992	+	0.00392410i	$0.00124908\pi$
$59$	27.3196i	0.00784819i	−0.999992	+	0.00392410i	$0.998751\pi$
$60$	−417.851	−0.116070
$61$	− 3123.41i	− 0.839400i	−0.907663	−	0.419700i	$-0.862135\pi$
$61$	− 3123.41i	− 0.839400i	0.907663	−	0.419700i	$-0.137865\pi$
$62$	5085.40i	1.32294i
$63$	0	0
$64$	512.000	0.125000
$65$	1171.97	0.277389
$66$	− 730.543i	− 0.167710i
$67$	7758.73	1.72839	0.864193	−	0.503160i	$-0.167829\pi$
$67$	7758.73	1.72839	0.864193	+	0.503160i	$0.167829\pi$
$68$	− 2189.82i	− 0.473576i
$69$	− 70.8259i	− 0.0148763i
$70$	0	0
$71$	7556.02	1.49891	0.749456	−	0.662054i	$-0.230315\pi$
$71$	7556.02	1.49891	0.749456	+	0.662054i	$0.230315\pi$
$72$	610.940	0.117851
$73$	− 4181.97i	− 0.784758i	−0.919804	−	0.392379i	$-0.871652\pi$
$73$	− 4181.97i	− 0.784758i	0.919804	−	0.392379i	$-0.128348\pi$
$74$	1509.55	0.275666
$75$	2722.57i	0.484012i
$76$	97.1713i	0.0168233i
$77$	0	0
$78$	−1713.54	−0.281646
$79$	5598.55	0.897060	0.448530	−	0.893768i	$-0.351948\pi$
$79$	5598.55	0.897060	0.448530	+	0.893768i	$0.351948\pi$
$80$	643.323i	0.100519i
$81$	729.000	0.111111
$82$	− 7581.57i	− 1.12754i
$83$	3074.79i	0.446334i	0.974780	+	0.223167i	$0.0716395\pi$
$83$	3074.79i	0.446334i	−0.974780	+	0.223167i	$0.928360\pi$
$84$	0	0
$85$	2751.48	0.380828
$86$	1071.11	0.144823
$87$	− 2524.35i	− 0.333511i
$88$	−1124.74	−0.145241
$89$	3360.59i	0.424263i	0.977241	+	0.212132i	$0.0680406\pi$
$89$	3360.59i	0.424263i	−0.977241	+	0.212132i	$0.931959\pi$
$90$	767.641i	0.0947704i
$91$	0	0
$92$	−109.044	−0.0128832
$93$	9342.47	1.08018
$94$	1478.44i	0.167320i
$95$	−122.095	−0.0135285
$96$	− 940.604i	− 0.102062i
$97$	− 9460.26i	− 1.00545i	−0.864447	−	0.502724i	$-0.832331\pi$
$97$	− 9460.26i	− 1.00545i	0.864447	−	0.502724i	$-0.167669\pi$
$98$	0	0
$99$	−1342.09	−0.136934
$100$	4191.67	0.419167
$101$	− 1420.92i	− 0.139293i	−0.997572	−	0.0696463i	$-0.977813\pi$
$101$	− 1420.92i	− 0.139293i	0.997572	−	0.0696463i	$-0.0221871\pi$
$102$	−4022.95	−0.386673
$103$	− 298.486i	− 0.0281351i	−0.999901	−	0.0140676i	$-0.995522\pi$
$103$	− 298.486i	− 0.0281351i	0.999901	−	0.0140676i	$-0.00447799\pi$
$104$	2638.16i	0.243913i
$105$	0	0
$106$	−15770.9	−1.40360
$107$	7853.23	0.685932	0.342966	−	0.939348i	$-0.388568\pi$
$107$	7853.23	0.685932	0.342966	+	0.939348i	$0.388568\pi$
$108$	− 1122.37i	− 0.0962250i
$109$	14158.4	1.19169	0.595843	−	0.803101i	$-0.296818\pi$
$109$	14158.4	1.19169	0.595843	+	0.803101i	$0.296818\pi$
$110$	− 1413.23i	− 0.116796i
$111$	− 2773.22i	− 0.225081i
$112$	0	0
$113$	20465.0	1.60271	0.801354	−	0.598190i	$-0.204113\pi$
$113$	20465.0	1.60271	0.801354	+	0.598190i	$0.204113\pi$
$114$	178.515	0.0137362
$115$	− 137.012i	− 0.0103601i
$116$	−3886.49	−0.288829
$117$	3147.97i	0.229963i
$118$	− 77.2714i	− 0.00554951i
$119$	0	0
$120$	1181.86	0.0820736
$121$	−12170.2	−0.831241
$122$	8834.33i	0.593545i
$123$	−13928.2	−0.920631
$124$	− 14383.7i	− 0.935462i
$125$	11549.2i	0.739152i
$126$	0	0
$127$	−12273.9	−0.760985	−0.380492	−	0.924784i	$-0.624246\pi$
$127$	−12273.9	−0.760985	−0.380492	+	0.924784i	$0.624246\pi$
$128$	−1448.15	−0.0883883
$129$	− 1967.75i	− 0.118247i
$130$	−3314.83	−0.196144
$131$	19903.1i	1.15979i	0.814693	+	0.579893i	$0.196906\pi$
$131$	19903.1i	1.15979i	−0.814693	+	0.579893i	$0.803094\pi$
$132$	2066.29i	0.118589i
$133$	0	0
$134$	−21945.0	−1.22215
$135$	1410.25	0.0773797
$136$	6193.74i	0.334869i
$137$	−10347.0	−0.551279	−0.275639	−	0.961261i	$-0.588890\pi$
$137$	−10347.0	−0.551279	−0.275639	+	0.961261i	$0.588890\pi$
$138$	200.326i	0.0105191i
$139$	− 37392.1i	− 1.93531i	−0.252280	−	0.967654i	$-0.581181\pi$
$139$	− 37392.1i	− 1.93531i	0.252280	−	0.967654i	$-0.418819\pi$
$140$	0	0
$141$	2716.07	0.136617
$142$	−21371.6	−1.05989
$143$	− 5795.43i	− 0.283409i
$144$	−1728.00	−0.0833333
$145$	− 4883.33i	− 0.232263i
$146$	11828.4i	0.554907i
$147$	0	0
$148$	−4269.65	−0.194925
$149$	−28642.5	−1.29014	−0.645072	−	0.764122i	$-0.723172\pi$
$149$	−28642.5	−1.29014	−0.645072	+	0.764122i	$0.723172\pi$
$150$	− 7700.59i	− 0.342248i
$151$	301.080	0.0132047	0.00660234	−	0.999978i	$-0.497898\pi$
$151$	301.080	0.0132047	0.00660234	+	0.999978i	$0.497898\pi$
$152$	− 274.842i	− 0.0118959i
$153$	7390.63i	0.315717i
$154$	0	0
$155$	18072.9	0.752256
$156$	4846.62	0.199154
$157$	23803.5i	0.965700i	0.875703	+	0.482850i	$0.160398\pi$
$157$	23803.5i	0.965700i	−0.875703	+	0.482850i	$0.839602\pi$
$158$	−15835.1	−0.634317
$159$	28972.9i	1.14604i
$160$	− 1819.59i	− 0.0710778i
$161$	0	0
$162$	−2061.92	−0.0785674
$163$	−49899.0	−1.87809	−0.939046	−	0.343791i	$-0.888289\pi$
$163$	−49899.0	−1.87809	−0.939046	+	0.343791i	$0.888289\pi$
$164$	21443.9i	0.797290i
$165$	−2596.27	−0.0953635
$166$	− 8696.83i	− 0.315606i
$167$	− 19448.8i	− 0.697364i	−0.937241	−	0.348682i	$-0.886629\pi$
$167$	− 19448.8i	− 0.697364i	0.937241	−	0.348682i	$-0.113371\pi$
$168$	0	0
$169$	14967.4	0.524052
$170$	−7782.37	−0.269286
$171$	− 327.953i	− 0.0112155i
$172$	−3029.56	−0.102405
$173$	23288.7i	0.778131i	0.921210	+	0.389066i	$0.127202\pi$
$173$	23288.7i	0.778131i	−0.921210	+	0.389066i	$0.872798\pi$
$174$	7139.93i	0.235828i
$175$	0	0
$176$	3181.26	0.102701
$177$	−141.957	−0.00453116
$178$	− 9505.18i	− 0.299999i
$179$	−45172.5	−1.40984	−0.704918	−	0.709289i	$-0.749016\pi$
$179$	−45172.5	−1.40984	−0.704918	+	0.709289i	$0.749016\pi$
$180$	− 2171.22i	− 0.0670128i
$181$	− 31334.1i	− 0.956445i	−0.878239	−	0.478222i	$-0.841281\pi$
$181$	− 31334.1i	− 0.956445i	0.878239	−	0.478222i	$-0.158719\pi$
$182$	0	0
$183$	16229.7	0.484628
$184$	308.422	0.00910982
$185$	− 5364.77i	− 0.156750i
$186$	−26424.5	−0.763802
$187$	− 13606.2i	− 0.389093i
$188$	− 4181.67i	− 0.118313i
$189$	0	0
$190$	345.336	0.00956610
$191$	−14201.6	−0.389288	−0.194644	−	0.980874i	$-0.562355\pi$
$191$	−14201.6	−0.389288	−0.194644	+	0.980874i	$0.562355\pi$
$192$	2660.43i	0.0721688i
$193$	42567.2	1.14277	0.571387	−	0.820680i	$-0.306405\pi$
$193$	42567.2	1.14277	0.571387	+	0.820680i	$0.306405\pi$
$194$	26757.7i	0.710959i
$195$	6089.73i	0.160151i
$196$	0	0
$197$	−1304.76	−0.0336200	−0.0168100	−	0.999859i	$-0.505351\pi$
$197$	−1304.76	−0.0336200	−0.0168100	+	0.999859i	$0.505351\pi$
$198$	3796.01	0.0968272
$199$	30884.8i	0.779899i	0.920836	+	0.389950i	$0.127508\pi$
$199$	30884.8i	0.779899i	−0.920836	+	0.389950i	$0.872492\pi$
$200$	−11855.8	−0.296396
$201$	40315.5i	0.997885i
$202$	4018.98i	0.0984948i
$203$	0	0
$204$	11378.6	0.273419
$205$	−26944.1	−0.641144
$206$	844.245i	0.0198945i
$207$	368.022	0.00858882
$208$	− 7461.85i	− 0.172473i
$209$	603.764i	0.0138221i
$210$	0	0
$211$	−16332.8	−0.366856	−0.183428	−	0.983033i	$-0.558719\pi$
$211$	−16332.8	−0.366856	−0.183428	+	0.983033i	$0.558719\pi$
$212$	44606.8	0.992496
$213$	39262.2i	0.865397i
$214$	−22212.3	−0.485027
$215$	− 3806.61i	− 0.0823496i
$216$	3174.54i	0.0680414i
$217$	0	0
$218$	−40046.0	−0.842649
$219$	21730.2	0.453080
$220$	3997.22i	0.0825872i
$221$	−31914.2	−0.653431
$222$	7843.84i	0.159156i
$223$	− 58757.4i	− 1.18155i	−0.806836	−	0.590776i	$-0.798822\pi$
$223$	− 58757.4i	− 1.18155i	0.806836	−	0.590776i	$-0.201178\pi$
$224$	0	0
$225$	−14146.9	−0.279445
$226$	−57883.7	−1.13329
$227$	− 5312.77i	− 0.103103i	−0.998670	−	0.0515513i	$-0.983583\pi$
$227$	− 5312.77i	− 0.103103i	0.998670	−	0.0515513i	$-0.0164166\pi$
$228$	−504.917	−0.00971293
$229$	− 83209.3i	− 1.58672i	−0.608751	−	0.793361i	$-0.708329\pi$
$229$	− 83209.3i	− 1.58672i	0.608751	−	0.793361i	$-0.291671\pi$
$230$	387.529i	0.00732570i
$231$	0	0
$232$	10992.6	0.204233
$233$	87512.7	1.61198	0.805989	−	0.591930i	$-0.201634\pi$
$233$	87512.7	1.61198	0.805989	+	0.591930i	$0.201634\pi$
$234$	− 8903.80i	− 0.162609i
$235$	5254.23	0.0951422
$236$	218.556i	0.00392410i
$237$	29090.9i	0.517918i
$238$	0	0
$239$	−7550.84	−0.132190	−0.0660951	−	0.997813i	$-0.521054\pi$
$239$	−7550.84	−0.132190	−0.0660951	+	0.997813i	$0.521054\pi$
$240$	−3342.80	−0.0580348
$241$	− 19956.7i	− 0.343601i	−0.985132	−	0.171800i	$-0.945042\pi$
$241$	− 19956.7i	− 0.343601i	0.985132	−	0.171800i	$-0.0549584\pi$
$242$	34422.5	0.587776
$243$	3788.00i	0.0641500i
$244$	− 24987.2i	− 0.419700i
$245$	0	0
$246$	39395.0	0.650985
$247$	1416.17	0.0232124
$248$	40683.2i	0.661472i
$249$	−15977.1	−0.257691
$250$	− 32666.2i	− 0.522659i
$251$	− 66288.3i	− 1.05218i	−0.850429	−	0.526089i	$-0.823658\pi$
$251$	− 66288.3i	− 1.05218i	0.850429	−	0.526089i	$-0.176342\pi$
$252$	0	0
$253$	−677.531	−0.0105849
$254$	34715.9	0.538098
$255$	14297.1i	0.219871i
$256$	4096.00	0.0625000
$257$	− 31453.3i	− 0.476212i	−0.971239	−	0.238106i	$-0.923473\pi$
$257$	− 31453.3i	− 0.476212i	0.971239	−	0.238106i	$-0.0765265\pi$
$258$	5565.65i	0.0836135i
$259$	0	0
$260$	9375.75	0.138694
$261$	13116.9	0.192553
$262$	− 56294.4i	− 0.820093i
$263$	−57022.9	−0.824400	−0.412200	−	0.911093i	$-0.635239\pi$
$263$	−57022.9	−0.824400	−0.412200	+	0.911093i	$0.635239\pi$
$264$	− 5844.35i	− 0.0838548i
$265$	56048.0i	0.798120i
$266$	0	0
$267$	−17462.1	−0.244949
$268$	62069.8	0.864193
$269$	− 92213.6i	− 1.27435i	−0.770717	−	0.637177i	$-0.780102\pi$
$269$	− 92213.6i	− 1.27435i	0.770717	−	0.637177i	$-0.219898\pi$
$270$	−3988.78	−0.0547157
$271$	− 101538.i	− 1.38258i	−0.722578	−	0.691289i	$-0.757043\pi$
$271$	− 101538.i	− 1.38258i	0.722578	−	0.691289i	$-0.242957\pi$
$272$	− 17518.5i	− 0.236788i
$273$	0	0
$274$	29265.6	0.389813
$275$	26044.5	0.344390
$276$	− 566.607i	− 0.00743814i
$277$	108108.	1.40896	0.704479	−	0.709724i	$-0.251181\pi$
$277$	108108.	1.40896	0.704479	+	0.709724i	$0.251181\pi$
$278$	105761.i	1.36847i
$279$	48544.9i	0.623642i
$280$	0	0
$281$	−1656.90	−0.0209838	−0.0104919	−	0.999945i	$-0.503340\pi$
$281$	−1656.90	−0.0209838	−0.0104919	+	0.999945i	$0.503340\pi$
$282$	−7682.22	−0.0966025
$283$	82652.1i	1.03200i	0.856588	+	0.516001i	$0.172580\pi$
$283$	82652.1i	1.03200i	−0.856588	+	0.516001i	$0.827420\pi$
$284$	60448.1	0.749456
$285$	− 634.423i	− 0.00781069i
$286$	16392.0i	0.200400i
$287$	0	0
$288$	4887.52	0.0589256
$289$	8594.52	0.102903
$290$	13812.2i	0.164235i
$291$	49157.0	0.580496
$292$	− 33455.8i	− 0.392379i
$293$	129210.i	1.50508i	0.658546	+	0.752540i	$0.271172\pi$
$293$	129210.i	1.50508i	−0.658546	+	0.752540i	$0.728828\pi$
$294$	0	0
$295$	−274.614	−0.00315558
$296$	12076.4	0.137833
$297$	− 6973.72i	− 0.0790591i
$298$	81013.1	0.912269
$299$	1589.19i	0.0177760i
$300$	21780.6i	0.242006i
$301$	0	0
$302$	−851.582	−0.00933711
$303$	7383.34	0.0804207
$304$	777.371i	0.00841164i
$305$	31396.2	0.337503
$306$	− 20903.9i	− 0.223246i
$307$	83560.9i	0.886597i	0.896374	+	0.443299i	$0.146192\pi$
$307$	83560.9i	0.886597i	−0.896374	+	0.443299i	$0.853808\pi$
$308$	0	0
$309$	1550.98	0.0162438
$310$	−51118.0	−0.531925
$311$	− 133785.i	− 1.38320i	−0.722280	−	0.691601i	$-0.756906\pi$
$311$	− 133785.i	− 1.38320i	0.722280	−	0.691601i	$-0.243094\pi$
$312$	−13708.3	−0.140823
$313$	56426.9i	0.575967i	0.957635	+	0.287984i	$0.0929849\pi$
$313$	56426.9i	0.575967i	−0.957635	+	0.287984i	$0.907015\pi$
$314$	− 67326.6i	− 0.682853i
$315$	0	0
$316$	44788.4	0.448530
$317$	105288.	1.04775	0.523877	−	0.851794i	$-0.324485\pi$
$317$	105288.	1.04775	0.523877	+	0.851794i	$0.324485\pi$
$318$	− 81947.8i	− 0.810370i
$319$	−24148.3	−0.237304
$320$	5146.58i	0.0502596i
$321$	40806.6i	0.396023i
$322$	0	0
$323$	3324.80	0.0318684
$324$	5832.00	0.0555556
$325$	− 61089.1i	− 0.578358i
$326$	141136.	1.32801
$327$	73569.3i	0.688020i
$328$	− 60652.6i	− 0.563769i
$329$	0	0
$330$	7343.36	0.0674322
$331$	−16322.4	−0.148980	−0.0744902	−	0.997222i	$-0.523733\pi$
$331$	−16322.4	−0.148980	−0.0744902	+	0.997222i	$0.523733\pi$
$332$	24598.4i	0.223167i
$333$	14410.1	0.129950
$334$	55009.5i	0.493111i
$335$	77990.1i	0.694944i
$336$	0	0
$337$	−139852.	−1.23143	−0.615715	−	0.787969i	$-0.711133\pi$
$337$	−139852.	−1.23143	−0.615715	+	0.787969i	$0.711133\pi$
$338$	−42334.3	−0.370560
$339$	106339.i	0.925324i
$340$	22011.9	0.190414
$341$	− 89371.4i	− 0.768582i
$342$	927.592i	0.00793057i
$343$	0	0
$344$	8568.88	0.0724114
$345$	711.937	0.00598141
$346$	− 65870.4i	− 0.550222i
$347$	6815.35	0.0566017	0.0283008	−	0.999599i	$-0.490990\pi$
$347$	6815.35	0.0566017	0.0283008	+	0.999599i	$0.490990\pi$
$348$	− 20194.8i	− 0.166756i
$349$	174743.i	1.43466i	0.696733	+	0.717331i	$0.254636\pi$
$349$	174743.i	1.43466i	−0.696733	+	0.717331i	$0.745364\pi$
$350$	0	0
$351$	−16357.3	−0.132769
$352$	−8997.96	−0.0726204
$353$	− 138634.i	− 1.11255i	−0.830998	−	0.556276i	$-0.812230\pi$
$353$	− 138634.i	− 1.11255i	0.830998	−	0.556276i	$-0.187770\pi$
$354$	401.514	0.00320401
$355$	75952.5i	0.602678i
$356$	26884.7i	0.212132i
$357$	0	0
$358$	127767.	0.996904
$359$	−124042.	−0.962453	−0.481226	−	0.876596i	$-0.659809\pi$
$359$	−124042.	−0.962453	−0.481226	+	0.876596i	$0.659809\pi$
$360$	6141.12i	0.0473852i
$361$	130173.	0.998868
$362$	88626.2i	0.676309i
$363$	− 63238.2i	− 0.479917i
$364$	0	0
$365$	42036.9	0.315533
$366$	−45904.5	−0.342683
$367$	41154.5i	0.305552i	0.988261	+	0.152776i	$0.0488213\pi$
$367$	41154.5i	0.305552i	−0.988261	+	0.152776i	$0.951179\pi$
$368$	−872.349	−0.00644161
$369$	− 72373.2i	− 0.531527i
$370$	15173.9i	0.110839i
$371$	0	0
$372$	74739.7	0.540089
$373$	147495.	1.06013	0.530066	−	0.847956i	$-0.322167\pi$
$373$	147495.	1.06013	0.530066	+	0.847956i	$0.322167\pi$
$374$	38484.1i	0.275130i
$375$	−60011.6	−0.426749
$376$	11827.5i	0.0836602i
$377$	56641.4i	0.398521i
$378$	0	0
$379$	−257420.	−1.79211	−0.896054	−	0.443944i	$-0.853579\pi$
$379$	−257420.	−1.79211	−0.896054	+	0.443944i	$0.853579\pi$
$380$	−976.759	−0.00676426
$381$	− 63777.2i	− 0.439355i
$382$	40168.3	0.275268
$383$	− 189027.i	− 1.28863i	−0.764761	−	0.644314i	$-0.777143\pi$
$383$	− 189027.i	− 1.28863i	0.764761	−	0.644314i	$-0.222857\pi$
$384$	− 7524.83i	− 0.0510310i
$385$	0	0
$386$	−120398.	−0.808064
$387$	10224.8	0.0682702
$388$	− 75682.1i	− 0.502724i
$389$	−38858.4	−0.256794	−0.128397	−	0.991723i	$-0.540983\pi$
$389$	−38858.4	−0.256794	−0.128397	+	0.991723i	$0.540983\pi$
$390$	− 17224.3i	− 0.113244i
$391$	3731.02i	0.0244048i
$392$	0	0
$393$	−103419.	−0.669603
$394$	3690.42	0.0237730
$395$	56276.2i	0.360687i
$396$	−10736.7	−0.0684672
$397$	91013.0i	0.577460i	0.957411	+	0.288730i	$0.0932331\pi$
$397$	91013.0i	0.577460i	−0.957411	+	0.288730i	$0.906767\pi$
$398$	− 87355.4i	− 0.551472i
$399$	0	0
$400$	33533.4	0.209584
$401$	−69038.4	−0.429341	−0.214670	−	0.976687i	$-0.568868\pi$
$401$	−69038.4	−0.429341	−0.214670	+	0.976687i	$0.568868\pi$
$402$	− 114030.i	− 0.705611i
$403$	−209627.	−1.29073
$404$	− 11367.4i	− 0.0696463i
$405$	7327.85i	0.0446752i
$406$	0	0
$407$	−26529.0	−0.160152
$408$	−32183.6	−0.193337
$409$	− 16577.6i	− 0.0991006i	−0.998772	−	0.0495503i	$-0.984221\pi$
$409$	− 16577.6i	− 0.0991006i	0.998772	−	0.0495503i	$-0.0157788\pi$
$410$	76209.4	0.453357
$411$	− 53764.3i	− 0.318281i
$412$	− 2387.89i	− 0.0140676i
$413$	0	0
$414$	−1040.92	−0.00607321
$415$	−30907.6	−0.179461
$416$	21105.3i	0.121957i
$417$	194295.	1.11735
$418$	− 1707.70i	− 0.00977371i
$419$	− 17162.1i	− 0.0977560i	−0.998805	−	0.0488780i	$-0.984435\pi$
$419$	− 17162.1i	− 0.0977560i	0.998805	−	0.0488780i	$-0.0155646\pi$
$420$	0	0
$421$	−15784.7	−0.0890576	−0.0445288	−	0.999008i	$-0.514179\pi$
$421$	−15784.7	−0.0890576	−0.0445288	+	0.999008i	$0.514179\pi$
$422$	46196.1	0.259406
$423$	14113.1i	0.0788756i
$424$	−126167.	−0.701801
$425$	− 143422.i	− 0.794030i
$426$	− 111050.i	− 0.611928i
$427$	0	0
$428$	62825.9	0.342966
$429$	30113.9	0.163626
$430$	10766.7i	0.0582299i
$431$	−282924.	−1.52305	−0.761526	−	0.648134i	$-0.775549\pi$
$431$	−282924.	−1.52305	−0.761526	+	0.648134i	$0.775549\pi$
$432$	− 8978.95i	− 0.0481125i
$433$	− 187998.i	− 1.00272i	−0.865240	−	0.501359i	$-0.832834\pi$
$433$	− 187998.i	− 1.00272i	0.865240	−	0.501359i	$-0.167166\pi$
$434$	0	0
$435$	25374.5	0.134097
$436$	113267.	0.595843
$437$	− 165.561i	0 0.000866953i
$438$	−61462.2	−0.320376
$439$	226125.i	1.17333i	0.809830	+	0.586664i	$0.199559\pi$
$439$	226125.i	1.17333i	−0.809830	+	0.586664i	$0.800441\pi$
$440$	− 11305.8i	− 0.0583980i
$441$	0	0
$442$	90267.1	0.462046
$443$	−264235.	−1.34643	−0.673213	−	0.739449i	$-0.735086\pi$
$443$	−264235.	−1.34643	−0.673213	+	0.739449i	$0.735086\pi$
$444$	− 22185.7i	− 0.112540i
$445$	−33780.4	−0.170586
$446$	166191.i	0.835483i
$447$	− 148831.i	− 0.744865i
$448$	0	0
$449$	−91015.4	−0.451463	−0.225732	−	0.974190i	$-0.572477\pi$
$449$	−91015.4	−0.451463	−0.225732	+	0.974190i	$0.572477\pi$
$450$	40013.4	0.197597
$451$	133240.i	0.655058i
$452$	163720.	0.801354
$453$	1564.46i	0.00762372i
$454$	15026.8i	0.0729045i
$455$	0	0
$456$	1428.12	0.00686808
$457$	281635.	1.34851	0.674255	−	0.738498i	$-0.264465\pi$
$457$	281635.	1.34851	0.674255	+	0.738498i	$0.264465\pi$
$458$	235351.i	1.12198i
$459$	−38402.8	−0.182280
$460$	− 1096.10i	− 0.00518005i
$461$	− 6432.69i	− 0.0302685i	−0.999885	−	0.0151343i	$-0.995182\pi$
$461$	− 6432.69i	− 0.0302685i	0.999885	−	0.0151343i	$-0.00481757\pi$
$462$	0	0
$463$	339986.	1.58599	0.792993	−	0.609231i	$-0.208522\pi$
$463$	339986.	1.58599	0.792993	+	0.609231i	$0.208522\pi$
$464$	−31091.9	−0.144415
$465$	93909.8i	0.434315i
$466$	−247523.	−1.13984
$467$	− 37823.3i	− 0.173431i	−0.996233	−	0.0867153i	$-0.972363\pi$
$467$	− 37823.3i	− 0.173431i	0.996233	−	0.0867153i	$-0.0276371\pi$
$468$	25183.8i	0.114982i
$469$	0	0
$470$	−14861.2	−0.0672757
$471$	−123687.	−0.557547
$472$	− 618.171i	− 0.00277476i
$473$	−18823.8	−0.0841368
$474$	− 82281.6i	− 0.366223i
$475$	6364.22i	0.0282071i
$476$	0	0
$477$	−150548.	−0.661664
$478$	21357.0	0.0934726
$479$	415442.i	1.81067i	0.424697	+	0.905336i	$0.360381\pi$
$479$	415442.i	1.81067i	−0.424697	+	0.905336i	$0.639619\pi$
$480$	9454.88	0.0410368
$481$	62225.5i	0.268954i
$482$	56446.0i	0.242963i
$483$	0	0
$484$	−97361.6	−0.415620
$485$	95093.8	0.404267
$486$	− 10714.1i	− 0.0453609i
$487$	−313237.	−1.32073	−0.660367	−	0.750943i	$-0.729599\pi$
$487$	−313237.	−1.32073	−0.660367	+	0.750943i	$0.729599\pi$
$488$	70674.6i	0.296773i
$489$	− 259283.i	− 1.08432i
$490$	0	0
$491$	433585.	1.79850	0.899252	−	0.437430i	$-0.144111\pi$
$491$	433585.	1.79850	0.899252	+	0.437430i	$0.144111\pi$
$492$	−111426.	−0.460316
$493$	132980.i	0.547131i
$494$	−4005.53	−0.0164137
$495$	− 13490.6i	− 0.0550581i
$496$	− 115069.i	− 0.467731i
$497$	0	0
$498$	45190.1	0.182215
$499$	−36576.4	−0.146893	−0.0734463	−	0.997299i	$-0.523400\pi$
$499$	−36576.4	−0.146893	−0.0734463	+	0.997299i	$0.523400\pi$
$500$	92394.0i	0.369576i
$501$	101059.	0.402624
$502$	187492.i	0.744003i
$503$	− 293860.i	− 1.16146i	−0.814096	−	0.580730i	$-0.802767\pi$
$503$	− 293860.i	− 1.16146i	0.814096	−	0.580730i	$-0.197233\pi$
$504$	0	0
$505$	14283.0	0.0560064
$506$	1916.35	0.00748468
$507$	77773.1i	0.302561i
$508$	−98191.4	−0.380492
$509$	− 277991.i	− 1.07299i	−0.843904	−	0.536494i	$-0.819748\pi$
$509$	− 277991.i	− 1.07299i	0.843904	−	0.536494i	$-0.180252\pi$
$510$	− 40438.4i	− 0.155472i
$511$	0	0
$512$	−11585.2	−0.0441942
$513$	1704.09	0.00647529
$514$	88963.4i	0.336733i
$515$	3000.36	0.0113125
$516$	− 15742.0i	− 0.0591237i
$517$	− 25982.4i	− 0.0972070i
$518$	0	0
$519$	−121012.	−0.449254
$520$	−26518.6	−0.0980718
$521$	− 323148.i	− 1.19049i	−0.803543	−	0.595246i	$-0.797055\pi$
$521$	− 323148.i	− 1.19049i	0.803543	−	0.595246i	$-0.202945\pi$
$522$	−37100.2	−0.136155
$523$	223912.i	0.818603i	0.912399	+	0.409301i	$0.134228\pi$
$523$	223912.i	0.818603i	−0.912399	+	0.409301i	$0.865772\pi$
$524$	159225.i	0.579893i
$525$	0	0
$526$	161285.	0.582939
$527$	−492150.	−1.77205
$528$	16530.3i	0.0592943i
$529$	−279655.	−0.999336
$530$	− 158528.i	− 0.564356i
$531$	− 737.628i	− 0.00261606i
$532$	0	0
$533$	312522.	1.10009
$534$	49390.4	0.173205
$535$	78940.1i	0.275797i
$536$	−175560.	−0.611077
$537$	− 234723.i	− 0.813969i
$538$	260819.i	0.901105i
$539$	0	0
$540$	11282.0	0.0386899
$541$	−80581.1	−0.275321	−0.137660	−	0.990480i	$-0.543958\pi$
$541$	−80581.1	−0.275321	−0.137660	+	0.990480i	$0.543958\pi$
$542$	287193.i	0.977630i
$543$	162817.	0.552204
$544$	49549.9i	0.167434i
$545$	142319.i	0.479149i
$546$	0	0
$547$	71665.0	0.239515	0.119757	−	0.992803i	$-0.461788\pi$
$547$	71665.0	0.239515	0.119757	+	0.992803i	$0.461788\pi$
$548$	−82775.6	−0.275639
$549$	84332.0i	0.279800i
$550$	−73665.0	−0.243521
$551$	− 5900.86i	− 0.0194362i
$552$	1602.61i	0.00525956i
$553$	0	0
$554$	−305776.	−0.996284
$555$	27876.2	0.0904997
$556$	− 299137.i	− 0.967654i
$557$	363705.	1.17230	0.586151	−	0.810202i	$-0.300643\pi$
$557$	363705.	1.17230	0.586151	+	0.810202i	$0.300643\pi$
$558$	− 137306.i	− 0.440981i
$559$	44152.5i	0.141297i
$560$	0	0
$561$	70699.9	0.224643
$562$	4686.42	0.0148378
$563$	191316.i	0.603580i	0.953374	+	0.301790i	$0.0975842\pi$
$563$	191316.i	0.603580i	−0.953374	+	0.301790i	$0.902416\pi$
$564$	21728.6	0.0683083
$565$	205712.i	0.644412i
$566$	− 233775.i	− 0.729736i
$567$	0	0
$568$	−170973.	−0.529945
$569$	−216919.	−0.669997	−0.334998	−	0.942219i	$-0.608736\pi$
$569$	−216919.	−0.669997	−0.334998	+	0.942219i	$0.608736\pi$
$570$	1794.42i	0.00552299i
$571$	−293440.	−0.900010	−0.450005	−	0.893026i	$-0.648578\pi$
$571$	−293440.	−0.900010	−0.450005	+	0.893026i	$0.648578\pi$
$572$	− 46363.4i	− 0.141705i
$573$	− 73793.8i	− 0.224756i
$574$	0	0
$575$	−7141.80	−0.0216009
$576$	−13824.0	−0.0416667
$577$	− 282246.i	− 0.847765i	−0.905717	−	0.423883i	$-0.860667\pi$
$577$	− 282246.i	− 0.847765i	0.905717	−	0.423883i	$-0.139333\pi$
$578$	−24309.0	−0.0727631
$579$	221186.i	0.659781i
$580$	− 39066.7i	− 0.116132i
$581$	0	0
$582$	−139037.	−0.410472
$583$	277159.	0.815441
$584$	94627.2i	0.277454i
$585$	−31643.1	−0.0924630
$586$	− 365460.i	− 1.06425i
$587$	− 434682.i	− 1.26152i	−0.775976	−	0.630762i	$-0.782742\pi$
$587$	− 434682.i	− 1.26152i	0.775976	−	0.630762i	$-0.217258\pi$
$588$	0	0
$589$	21838.8	0.0629502
$590$	776.726	0.00223133
$591$	− 6779.73i	− 0.0194105i
$592$	−34157.2	−0.0974627
$593$	310462.i	0.882873i	0.897292	+	0.441437i	$0.145531\pi$
$593$	310462.i	0.882873i	−0.897292	+	0.441437i	$0.854469\pi$
$594$	19724.7i	0.0559032i
$595$	0	0
$596$	−229140.	−0.645072
$597$	−160482.	−0.450275
$598$	− 4494.92i	− 0.0125695i
$599$	−216469.	−0.603313	−0.301656	−	0.953417i	$-0.597540\pi$
$599$	−216469.	−0.603313	−0.301656	+	0.953417i	$0.597540\pi$
$600$	− 61604.7i	− 0.171124i
$601$	486920.i	1.34806i	0.738705	+	0.674029i	$0.235438\pi$
$601$	486920.i	1.34806i	−0.738705	+	0.674029i	$0.764562\pi$
$602$	0	0
$603$	−209486.	−0.576129
$604$	2408.64	0.00660234
$605$	− 122334.i	− 0.334223i
$606$	−20883.2	−0.0568660
$607$	416967.i	1.13168i	0.824514	+	0.565841i	$0.191448\pi$
$607$	416967.i	1.13168i	−0.824514	+	0.565841i	$0.808552\pi$
$608$	− 2198.74i	− 0.00594793i
$609$	0	0
$610$	−88802.0	−0.238651
$611$	−60943.3	−0.163247
$612$	59125.0i	0.157859i
$613$	472526.	1.25749	0.628745	−	0.777611i	$-0.283569\pi$
$613$	472526.	1.25749	0.628745	+	0.777611i	$0.283569\pi$
$614$	− 236346.i	− 0.626919i
$615$	− 140006.i	− 0.370165i
$616$	0	0
$617$	−133106.	−0.349646	−0.174823	−	0.984600i	$-0.555935\pi$
$617$	−133106.	−0.349646	−0.174823	+	0.984600i	$0.555935\pi$
$618$	−4386.83	−0.0114861
$619$	191417.i	0.499573i	0.968301	+	0.249787i	$0.0803605\pi$
$619$	191417.i	0.499573i	−0.968301	+	0.249787i	$0.919639\pi$
$620$	144584.	0.376128
$621$	1912.30i	0.00495876i
$622$	378400.i	0.978071i
$623$	0	0
$624$	38772.9	0.0995771
$625$	211382.	0.541138
$626$	− 159599.i	− 0.407270i
$627$	−3137.25	−0.00798020
$628$	190428.i	0.482850i
$629$	146090.i	0.369248i
$630$	0	0
$631$	136069.	0.341743	0.170871	−	0.985293i	$-0.445342\pi$
$631$	136069.	0.341743	0.170871	+	0.985293i	$0.445342\pi$
$632$	−126681.	−0.317159
$633$	− 84867.7i	− 0.211804i
$634$	−297799.	−0.740873
$635$	− 123377.i	− 0.305974i
$636$	231784.i	0.573018i
$637$	0	0
$638$	68301.7	0.167799
$639$	−204012.	−0.499637
$640$	− 14556.7i	− 0.0355389i
$641$	420854.	1.02427	0.512136	−	0.858905i	$-0.328855\pi$
$641$	420854.	1.02427	0.512136	+	0.858905i	$0.328855\pi$
$642$	− 115418.i	− 0.280030i
$643$	− 216803.i	− 0.524376i	−0.965017	−	0.262188i	$-0.915556\pi$
$643$	− 216803.i	− 0.524376i	0.965017	−	0.262188i	$-0.0844441\pi$
$644$	0	0
$645$	19779.7	0.0475445
$646$	−9403.96	−0.0225344
$647$	− 544116.i	− 1.29982i	−0.760012	−	0.649910i	$-0.774807\pi$
$647$	− 544116.i	− 1.29982i	0.760012	−	0.649910i	$-0.225193\pi$
$648$	−16495.4	−0.0392837
$649$	1357.98i	0.00322406i
$650$	172786.i	0.408961i
$651$	0	0
$652$	−399192.	−0.939046
$653$	429311.	1.00681	0.503403	−	0.864052i	$-0.332081\pi$
$653$	429311.	1.00681	0.503403	+	0.864052i	$0.332081\pi$
$654$	− 208085.i	− 0.486504i
$655$	−200064.	−0.466323
$656$	171551.i	0.398645i
$657$	112913.i	0.261586i
$658$	0	0
$659$	−392889.	−0.904689	−0.452344	−	0.891843i	$-0.649412\pi$
$659$	−392889.	−0.904689	−0.452344	+	0.891843i	$0.649412\pi$
$660$	−20770.2	−0.0476818
$661$	− 442689.i	− 1.01320i	−0.862180	−	0.506601i	$-0.830902\pi$
$661$	− 442689.i	− 1.01320i	0.862180	−	0.506601i	$-0.169098\pi$
$662$	46166.8	0.105345
$663$	− 165831.i	− 0.377259i
$664$	− 69574.6i	− 0.157803i
$665$	0	0
$666$	−40757.8	−0.0918888
$667$	6621.82	0.0148842
$668$	− 155590.i	− 0.348682i
$669$	305312.	0.682169
$670$	− 220589.i	− 0.491400i
$671$	− 155256.i	− 0.344828i
$672$	0	0
$673$	250367.	0.552772	0.276386	−	0.961047i	$-0.410863\pi$
$673$	250367.	0.552772	0.276386	+	0.961047i	$0.410863\pi$
$674$	395562.	0.870753
$675$	− 73509.4i	− 0.161337i
$676$	119739.	0.262026
$677$	590260.i	1.28785i	0.765088	+	0.643926i	$0.222695\pi$
$677$	590260.i	1.28785i	−0.765088	+	0.643926i	$0.777305\pi$
$678$	− 300773.i	− 0.654303i
$679$	0	0
$680$	−62259.0	−0.134643
$681$	27606.0	0.0595263
$682$	252781.i	0.543469i
$683$	547349.	1.17334	0.586669	−	0.809827i	$-0.300439\pi$
$683$	547349.	1.17334	0.586669	+	0.809827i	$0.300439\pi$
$684$	− 2623.63i	− 0.00560776i
$685$	− 104007.i	− 0.221656i
$686$	0	0
$687$	432368.	0.916095
$688$	−24236.5	−0.0512026
$689$	− 650096.i	− 1.36943i
$690$	−2013.66	−0.00422949
$691$	864887.i	1.81135i	0.423969	+	0.905677i	$0.360637\pi$
$691$	864887.i	1.81135i	−0.423969	+	0.905677i	$0.639363\pi$
$692$	186310.i	0.389066i
$693$	0	0
$694$	−19276.7	−0.0400234
$695$	375862.	0.778143
$696$	57119.4i	0.117914i
$697$	733722.	1.51031
$698$	− 494249.i	− 1.01446i
$699$	454729.i	0.930676i
$700$	0	0
$701$	84934.5	0.172842	0.0864208	−	0.996259i	$-0.472457\pi$
$701$	84934.5	0.172842	0.0864208	+	0.996259i	$0.472457\pi$
$702$	46265.5	0.0938822
$703$	− 6482.61i	− 0.0131172i
$704$	25450.1	0.0513504
$705$	27301.8i	0.0549304i
$706$	392116.i	0.786693i
$707$	0	0
$708$	−1135.65	−0.00226558
$709$	591425.	1.17654	0.588271	−	0.808664i	$-0.299809\pi$
$709$	591425.	1.17654	0.588271	+	0.808664i	$0.299809\pi$
$710$	− 214826.i	− 0.426158i
$711$	−151161.	−0.299020
$712$	− 76041.5i	− 0.150000i
$713$	24507.0i	0.0482071i
$714$	0	0
$715$	58255.2	0.113952
$716$	−361380.	−0.704918
$717$	− 39235.3i	− 0.0763200i
$718$	350843.	0.680557
$719$	− 535114.i	− 1.03512i	−0.855648	−	0.517558i	$-0.826841\pi$
$719$	− 535114.i	− 1.03512i	0.855648	−	0.517558i	$-0.173159\pi$
$720$	− 17369.7i	− 0.0335064i
$721$	0	0
$722$	−368186.	−0.706306
$723$	103698.	0.198378
$724$	− 250673.i	− 0.478222i
$725$	−254545.	−0.484271
$726$	178865.i	0.339353i
$727$	151226.i	0.286126i	0.989714	+	0.143063i	$0.0456952\pi$
$727$	151226.i	0.286126i	−0.989714	+	0.143063i	$0.954305\pi$
$728$	0	0
$729$	−19683.0	−0.0370370
$730$	−118898.	−0.223115
$731$	103659.i	0.193987i
$732$	129838.	0.242314
$733$	− 745396.i	− 1.38733i	−0.720298	−	0.693664i	$-0.755995\pi$
$733$	− 745396.i	− 1.38733i	0.720298	−	0.693664i	$-0.244005\pi$
$734$	− 116403.i	− 0.216058i
$735$	0	0
$736$	2467.38	0.00455491
$737$	385664.	0.710027
$738$	204702.i	0.375846i
$739$	635215.	1.16314	0.581570	−	0.813496i	$-0.302439\pi$
$739$	635215.	1.16314	0.581570	+	0.813496i	$0.302439\pi$
$740$	− 42918.2i	− 0.0783750i
$741$	7358.62i	0.0134017i
$742$	0	0
$743$	635983.	1.15204	0.576021	−	0.817435i	$-0.304605\pi$
$743$	635983.	1.15204	0.576021	+	0.817435i	$0.304605\pi$
$744$	−211396.	−0.381901
$745$	− 287912.i	− 0.518737i
$746$	−417179.	−0.749627
$747$	− 83019.4i	− 0.148778i
$748$	− 108850.i	− 0.194547i
$749$	0	0
$750$	169739.	0.301757
$751$	−640955.	−1.13644	−0.568221	−	0.822876i	$-0.692368\pi$
$751$	−640955.	−1.13644	−0.568221	+	0.822876i	$0.692368\pi$
$752$	− 33453.4i	− 0.0591567i
$753$	344444.	0.607476
$754$	− 160206.i	− 0.281797i
$755$	3026.43i	0.00530929i
$756$	0	0
$757$	235571.	0.411084	0.205542	−	0.978648i	$-0.434104\pi$
$757$	235571.	0.411084	0.205542	+	0.978648i	$0.434104\pi$
$758$	728095.	1.26721
$759$	− 3520.56i	− 0.00611122i
$760$	2762.69	0.00478305
$761$	112704.i	0.194612i	0.995255	+	0.0973060i	$0.0310225\pi$
$761$	112704.i	0.194612i	−0.995255	+	0.0973060i	$0.968977\pi$
$762$	180389.i	0.310671i
$763$	0	0
$764$	−113613.	−0.194644
$765$	−74290.0	−0.126943
$766$	534650.i	0.911197i
$767$	3185.23	0.00541439
$768$	21283.4i	0.0360844i
$769$	783953.i	1.32568i	0.748763	+	0.662838i	$0.230648\pi$
$769$	783953.i	1.32568i	−0.748763	+	0.662838i	$0.769352\pi$
$770$	0	0
$771$	163436.	0.274941
$772$	340538.	0.571387
$773$	551618.i	0.923165i	0.887097	+	0.461583i	$0.152718\pi$
$773$	551618.i	0.923165i	−0.887097	+	0.461583i	$0.847282\pi$
$774$	−28920.0	−0.0482743
$775$	− 942056.i	− 1.56846i
$776$	214061.i	0.355480i
$777$	0	0
$778$	109908.	0.181581
$779$	−32558.3	−0.0536522
$780$	48717.8i	0.0800753i
$781$	375588.	0.615758
$782$	− 10552.9i	− 0.0172568i
$783$	68157.4i	0.111170i
$784$	0	0
$785$	−239271.	−0.388286
$786$	292514.	0.473481
$787$	430357.i	0.694831i	0.937711	+	0.347415i	$0.112941\pi$
$787$	430357.i	0.694831i	−0.937711	+	0.347415i	$0.887059\pi$
$788$	−10438.1	−0.0168100
$789$	− 296300.i	− 0.475968i
$790$	− 159173.i	− 0.255044i
$791$	0	0
$792$	30368.1	0.0484136
$793$	−364162.	−0.579094
$794$	− 257424.i	− 0.408326i
$795$	−291234.	−0.460795
$796$	247078.i	0.389950i
$797$	195264.i	0.307401i	0.988117	+	0.153700i	$0.0491191\pi$
$797$	195264.i	0.307401i	−0.988117	+	0.153700i	$0.950881\pi$
$798$	0	0
$799$	−143079.	−0.224122
$800$	−94846.7	−0.148198
$801$	− 90735.9i	− 0.141421i
$802$	195270.	0.303590
$803$	− 207874.i	− 0.322381i
$804$	322524.i	0.498942i
$805$	0	0
$806$	592914.	0.912686
$807$	479156.	0.735749
$808$	32151.9i	0.0492474i
$809$	−216365.	−0.330591	−0.165295	−	0.986244i	$-0.552858\pi$
$809$	−216365.	−0.330591	−0.165295	+	0.986244i	$0.552858\pi$
$810$	− 20726.3i	− 0.0315901i
$811$	− 710891.i	− 1.08084i	−0.841395	−	0.540420i	$-0.818265\pi$
$811$	− 710891.i	− 1.08084i	0.841395	−	0.540420i	$-0.181735\pi$
$812$	0	0
$813$	527607.	0.798232
$814$	75035.4	0.113245
$815$	− 501581.i	− 0.755138i
$816$	91028.9	0.136710
$817$	− 4599.78i	− 0.00689117i
$818$	46888.7i	0.0700747i
$819$	0	0
$820$	−215553.	−0.320572
$821$	726479.	1.07780	0.538898	−	0.842371i	$-0.318841\pi$
$821$	726479.	1.07780	0.538898	+	0.842371i	$0.318841\pi$
$822$	152069.i	0.225059i
$823$	78522.8	0.115930	0.0579650	−	0.998319i	$-0.481539\pi$
$823$	78522.8	0.115930	0.0579650	+	0.998319i	$0.481539\pi$
$824$	6753.96i	0.00994727i
$825$	135331.i	0.198834i
$826$	0	0
$827$	−1.02365e6	−1.49672	−0.748358	−	0.663295i	$-0.769158\pi$
$827$	−1.02365e6	−1.49672	−0.748358	+	0.663295i	$0.769158\pi$
$828$	2944.18	0.00429441
$829$	684766.i	0.996398i	0.867063	+	0.498199i	$0.166005\pi$
$829$	684766.i	0.996398i	−0.867063	+	0.498199i	$0.833995\pi$
$830$	87419.9	0.126898
$831$	561746.i	0.813463i
$832$	− 59694.8i	− 0.0862363i
$833$	0	0
$834$	−549549.	−0.790086
$835$	195498.	0.280394
$836$	4830.11i	0.00691106i
$837$	−252247.	−0.360060
$838$	48541.9i	0.0691239i
$839$	− 324207.i	− 0.460573i	−0.973123	−	0.230287i	$-0.926034\pi$
$839$	− 324207.i	− 0.460573i	0.973123	−	0.230287i	$-0.0739663\pi$
$840$	0	0
$841$	−471269.	−0.666311
$842$	44645.8	0.0629732
$843$	− 8609.51i	− 0.0121150i
$844$	−130662.	−0.183428
$845$	150452.i	0.210709i
$846$	− 39918.0i	− 0.0557735i
$847$	0	0
$848$	356854.	0.496248
$849$	−429473.	−0.595827
$850$	405658.i	0.561464i
$851$	7274.66	0.0100451
$852$	314098.i	0.432699i
$853$	85715.2i	0.117804i	0.998264	+	0.0589020i	$0.0187599\pi$
$853$	85715.2i	0.117804i	−0.998264	+	0.0589020i	$0.981240\pi$
$854$	0	0
$855$	3296.56	0.00450950
$856$	−177698.	−0.242514
$857$	929894.i	1.26611i	0.774106	+	0.633056i	$0.218200\pi$
$857$	929894.i	1.26611i	−0.774106	+	0.633056i	$0.781800\pi$
$858$	−85175.1	−0.115701
$859$	− 620659.i	− 0.841137i	−0.907261	−	0.420568i	$-0.861831\pi$
$859$	− 620659.i	− 0.841137i	0.907261	−	0.420568i	$-0.138169\pi$
$860$	− 30452.9i	− 0.0411748i
$861$	0	0
$862$	800229.	1.07696
$863$	89390.7	0.120025	0.0600124	−	0.998198i	$-0.480886\pi$
$863$	89390.7	0.120025	0.0600124	+	0.998198i	$0.480886\pi$
$864$	25396.3i	0.0340207i
$865$	−234096.	−0.312869
$866$	531740.i	0.709028i
$867$	44658.4i	0.0594108i
$868$	0	0
$869$	278288.	0.368515
$870$	−71770.0	−0.0948210
$871$	− 904601.i	− 1.19240i
$872$	−320368.	−0.421324
$873$	255427.i	0.335149i
$874$	468.278i	0 0.000613028i
$875$	0	0
$876$	173841.	0.226540
$877$	−902464.	−1.17336	−0.586679	−	0.809819i	$-0.699565\pi$
$877$	−902464.	−1.17336	−0.586679	+	0.809819i	$0.699565\pi$
$878$	− 639578.i	− 0.829669i
$879$	−671393.	−0.868959
$880$	31977.8i	0.0412936i
$881$	472588.i	0.608878i	0.952532	+	0.304439i	$0.0984690\pi$
$881$	472588.i	0.608878i	−0.952532	+	0.304439i	$0.901531\pi$
$882$	0	0
$883$	416715.	0.534463	0.267232	−	0.963632i	$-0.413891\pi$
$883$	416715.	0.534463	0.267232	+	0.963632i	$0.413891\pi$
$884$	−255314.	−0.326716
$885$	− 1426.94i	− 0.00182187i
$886$	747369.	0.952067
$887$	340060.i	0.432223i	0.976369	+	0.216112i	$0.0693375\pi$
$887$	340060.i	0.432223i	−0.976369	+	0.216112i	$0.930662\pi$
$888$	62750.8i	0.0795780i
$889$	0	0
$890$	95545.4	0.120623
$891$	36236.5	0.0456448
$892$	− 470059.i	− 0.590776i
$893$	6349.04	0.00796168
$894$	420957.i	0.526699i
$895$	− 454071.i	− 0.566862i
$896$	0	0
$897$	−8257.70	−0.0102630
$898$	257430.	0.319233
$899$	873468.i	1.08076i
$900$	−113175.	−0.139722
$901$	− 1.52626e6i	− 1.88009i
$902$	− 376858.i	− 0.463196i
$903$	0	0
$904$	−463070.	−0.566643
$905$	314968.	0.384564
$906$	− 4424.95i	− 0.00539078i
$907$	−1.19203e6	−1.44901	−0.724506	−	0.689268i	$-0.757932\pi$
$907$	−1.19203e6	−1.44901	−0.724506	+	0.689268i	$0.757932\pi$
$908$	− 42502.2i	− 0.0515513i
$909$	38365.0i	0.0464309i
$910$	0	0
$911$	−1.29802e6	−1.56403	−0.782016	−	0.623259i	$-0.785808\pi$
$911$	−1.29802e6	−1.56403	−0.782016	+	0.623259i	$0.785808\pi$
$912$	−4039.34	−0.00485647
$913$	152839.i	0.183355i
$914$	−796584.	−0.953541
$915$	163140.i	0.194858i
$916$	− 665674.i	− 0.793361i
$917$	0	0
$918$	108620.	0.128891
$919$	−370370.	−0.438535	−0.219268	−	0.975665i	$-0.570367\pi$
$919$	−370370.	−0.438535	−0.219268	+	0.975665i	$0.570367\pi$
$920$	3100.23i	0.00366285i
$921$	−434195.	−0.511877
$922$	18194.4i	0.0214031i
$923$	− 880967.i	− 1.03408i
$924$	0	0
$925$	−279640.	−0.326825
$926$	−961626.	−1.12146
$927$	8059.11i	0.00937838i
$928$	87941.1	0.102117
$929$	1.34493e6i	1.55836i	0.626801	+	0.779180i	$0.284364\pi$
$929$	1.34493e6i	1.55836i	−0.626801	+	0.779180i	$0.715636\pi$
$930$	− 265617.i	− 0.307107i
$931$	0	0
$932$	700101.	0.805989
$933$	695165.	0.798592
$934$	106981.i	0.122634i
$935$	136768.	0.156445
$936$	− 71230.4i	− 0.0813043i
$937$	− 1.40472e6i	− 1.59996i	−0.600023	−	0.799982i	$-0.704842\pi$
$937$	− 1.40472e6i	− 1.59996i	0.600023	−	0.799982i	$-0.295158\pi$
$938$	0	0
$939$	−293203.	−0.332535
$940$	42033.8	0.0475711
$941$	− 797821.i	− 0.901003i	−0.892776	−	0.450502i	$-0.851245\pi$
$941$	− 797821.i	− 0.901003i	0.892776	−	0.450502i	$-0.148755\pi$
$942$	349839.	0.394245
$943$	− 36536.3i	− 0.0410867i
$944$	1748.45i	0.00196205i
$945$	0	0
$946$	53241.9	0.0594937
$947$	−926759.	−1.03340	−0.516698	−	0.856168i	$-0.672839\pi$
$947$	−926759.	−1.03340	−0.516698	+	0.856168i	$0.672839\pi$
$948$	232727.i	0.258959i
$949$	−487582.	−0.541397
$950$	− 18000.7i	− 0.0199454i
$951$	547091.i	0.604921i
$952$	0	0
$953$	793467.	0.873662	0.436831	−	0.899544i	$-0.356101\pi$
$953$	793467.	0.873662	0.436831	+	0.899544i	$0.356101\pi$
$954$	425813.	0.467867
$955$	− 142754.i	− 0.156524i
$956$	−60406.7	−0.0660951
$957$	− 125478.i	− 0.137007i
$958$	− 1.17505e6i	− 1.28034i
$959$	0	0
$960$	−26742.4	−0.0290174
$961$	−2.30913e6	−2.50036
$962$	− 176000.i	− 0.190179i
$963$	−212037.	−0.228644
$964$	− 159653.i	− 0.171800i
$965$	427882.i	0.459483i
$966$	0	0
$967$	−581924.	−0.622320	−0.311160	−	0.950358i	$-0.600717\pi$
$967$	−581924.	−0.622320	−0.311160	+	0.950358i	$0.600717\pi$
$968$	275380.	0.293888
$969$	17276.2i	0.0183992i
$970$	−268966.	−0.285860
$971$	1.00548e6i	1.06643i	0.845979	+	0.533216i	$0.179017\pi$
$971$	1.00548e6i	1.06643i	−0.845979	+	0.533216i	$0.820983\pi$
$972$	30304.0i	0.0320750i
$973$	0	0
$974$	885968.	0.933900
$975$	317428.	0.333915
$976$	− 199898.i	− 0.209850i
$977$	−1.29706e6	−1.35885	−0.679426	−	0.733744i	$-0.737771\pi$
$977$	−1.29706e6	−1.35885	−0.679426	+	0.733744i	$0.737771\pi$
$978$	733363.i	0.766728i
$979$	167045.i	0.174289i
$980$	0	0
$981$	−382277.	−0.397229
$982$	−1.22636e6	−1.27173
$983$	1.87942e6i	1.94498i	0.232936	+	0.972492i	$0.425167\pi$
$983$	1.87942e6i	1.94498i	−0.232936	+	0.972492i	$0.574833\pi$
$984$	315160.	0.325492
$985$	− 13115.3i	− 0.0135178i
$986$	− 376123.i	− 0.386880i
$987$	0	0
$988$	11329.3	0.0116062
$989$	5161.78	0.00527724
$990$	38157.2i	0.0389320i
$991$	852133.	0.867681	0.433841	−	0.900990i	$-0.357158\pi$
$991$	852133.	0.867681	0.433841	+	0.900990i	$0.357158\pi$
$992$	325465.i	0.330736i
$993$	− 84813.9i	− 0.0860139i
$994$	0	0
$995$	−310452.	−0.313579
$996$	−127817.	−0.128845
$997$	150934.i	0.151844i	0.997114	+	0.0759218i	$0.0241899\pi$
$997$	150934.i	0.151844i	−0.997114	+	0.0759218i	$0.975810\pi$
$998$	103454.	0.103869
$999$	74876.9i	0.0750269i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	294.5.c.b.97.3	yes	8
3.2	odd	2		882.5.c.f.685.6		8
7.2	even	3		294.5.g.e.31.3		8
7.3	odd	6		294.5.g.e.19.3		8
7.4	even	3		294.5.g.g.19.4		8
7.5	odd	6		294.5.g.g.31.4		8
7.6	odd	2	inner	294.5.c.b.97.2	✓	8
21.20	even	2		882.5.c.f.685.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
294.5.c.b.97.2	✓	8	7.6	odd	2	inner
294.5.c.b.97.3	yes	8	1.1	even	1	trivial
294.5.g.e.19.3		8	7.3	odd	6
294.5.g.e.31.3		8	7.2	even	3
294.5.g.g.19.4		8	7.4	even	3
294.5.g.g.31.4		8	7.5	odd	6
882.5.c.f.685.6		8	3.2	odd	2
882.5.c.f.685.7		8	21.20	even	2

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	294.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.82843 q^{2} +5.19615i q^{3} +8.00000 q^{4} +10.0519i q^{5} -14.6969i q^{6} -22.6274 q^{8} -27.0000 q^{9} +O(q^{10})\)	\(q-2.82843 q^{2} +5.19615i q^{3} +8.00000 q^{4} +10.0519i q^{5} -14.6969i q^{6} -22.6274 q^{8} -27.0000 q^{9} -28.4311i q^{10} +49.7072 q^{11} +41.5692i q^{12} -116.591i q^{13} -52.2313 q^{15} +64.0000 q^{16} -273.727i q^{17} +76.3675 q^{18} +12.1464i q^{19} +80.4154i q^{20} -140.593 q^{22} -13.6305 q^{23} -117.576i q^{24} +523.959 q^{25} +329.770i q^{26} -140.296i q^{27} -485.811 q^{29} +147.732 q^{30} -1797.96i q^{31} -181.019 q^{32} +258.286i q^{33} +774.217i q^{34} -216.000 q^{36} -533.706 q^{37} -34.3552i q^{38} +605.827 q^{39} -227.449i q^{40} +2680.49i q^{41} -378.695 q^{43} +397.657 q^{44} -271.402i q^{45} +38.5527 q^{46} -522.709i q^{47} +332.554i q^{48} -1481.98 q^{50} +1422.33 q^{51} -932.732i q^{52} +5575.84 q^{53} +396.817i q^{54} +499.653i q^{55} -63.1146 q^{57} +1374.08 q^{58} +27.3196i q^{59} -417.851 q^{60} -3123.41i q^{61} +5085.40i q^{62} +512.000 q^{64} +1171.97 q^{65} -730.543i q^{66} +7758.73 q^{67} -2189.82i q^{68} -70.8259i q^{69} +7556.02 q^{71} +610.940 q^{72} -4181.97i q^{73} +1509.55 q^{74} +2722.57i q^{75} +97.1713i q^{76} -1713.54 q^{78} +5598.55 q^{79} +643.323i q^{80} +729.000 q^{81} -7581.57i q^{82} +3074.79i q^{83} +2751.48 q^{85} +1071.11 q^{86} -2524.35i q^{87} -1124.74 q^{88} +3360.59i q^{89} +767.641i q^{90} -109.044 q^{92} +9342.47 q^{93} +1478.44i q^{94} -122.095 q^{95} -940.604i q^{96} -9460.26i q^{97} -1342.09 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 64 q^{4} - 216 q^{9}+O(q^{10}) \) 8 * q + 64 * q^4 - 216 * q^9	\( 8 q + 64 q^{4} - 216 q^{9} + 512 q^{16} - 1312 q^{22} - 272 q^{23} - 2808 q^{25} + 400 q^{29} + 3168 q^{30} - 1728 q^{36} - 3328 q^{37} + 656 q^{43} - 2400 q^{46} + 800 q^{50} + 7200 q^{51} + 9264 q^{53} + 1152 q^{57} - 11488 q^{58} + 4096 q^{64} - 15696 q^{65} + 26816 q^{67} + 28192 q^{71} - 4512 q^{74} + 9216 q^{78} + 19728 q^{79} + 5832 q^{81} - 49632 q^{85} + 5888 q^{86} - 10496 q^{88} - 2176 q^{92} + 15264 q^{93} - 92752 q^{95}+O(q^{100}) \) 8 * q + 64 * q^4 - 216 * q^9 + 512 * q^16 - 1312 * q^22 - 272 * q^23 - 2808 * q^25 + 400 * q^29 + 3168 * q^30 - 1728 * q^36 - 3328 * q^37 + 656 * q^43 - 2400 * q^46 + 800 * q^50 + 7200 * q^51 + 9264 * q^53 + 1152 * q^57 - 11488 * q^58 + 4096 * q^64 - 15696 * q^65 + 26816 * q^67 + 28192 * q^71 - 4512 * q^74 + 9216 * q^78 + 19728 * q^79 + 5832 * q^81 - 49632 * q^85 + 5888 * q^86 - 10496 * q^88 - 2176 * q^92 + 15264 * q^93 - 92752 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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