LMFDB - Embedded newform 387.8.a.e.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [387,8,Mod(1,387)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(387, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("387.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 387 = 3^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	387.a (trivial)

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:	yes
Analytic conductor:	$120.893004862$
Analytic rank:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 1295 x^{11} + 4518 x^{10} + 633722 x^{9} - 1900560 x^{8} - 148352102 x^{7} + \cdots - 31740122163456 $ x^13 - 4x^12 - 1295x^11 + 4518x^10 + 633722x^9 - 1900560x^8 - 148352102x^7 + 362252852x^6 + 17377933853x^5 - 32944267708x^4 - 960497335723x^3 + 1496754347766x^2 + 19389495114120x - 31740122163456
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 129)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.65100$ of defining polynomial
Character	$\chi$	$=$	387.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.651000 q^{2} -127.576 q^{4} -131.920 q^{5} -1640.88 q^{7} -166.380 q^{8} +O(q^{10})$	\(q+0.651000 q^{2} -127.576 q^{4} -131.920 q^{5} -1640.88 q^{7} -166.380 q^{8} -85.8799 q^{10} +4662.73 q^{11} -8706.11 q^{13} -1068.21 q^{14} +16221.4 q^{16} +25534.1 q^{17} -9246.29 q^{19} +16829.8 q^{20} +3035.44 q^{22} +8437.14 q^{23} -60722.1 q^{25} -5667.68 q^{26} +209337. q^{28} +238673. q^{29} +88996.3 q^{31} +31856.8 q^{32} +16622.7 q^{34} +216465. q^{35} +297827. q^{37} -6019.34 q^{38} +21948.9 q^{40} -135127. q^{41} -79507.0 q^{43} -594853. q^{44} +5492.58 q^{46} -701078. q^{47} +1.86895e6 q^{49} -39530.1 q^{50} +1.11069e6 q^{52} +48047.2 q^{53} -615106. q^{55} +273010. q^{56} +155376. q^{58} +673642. q^{59} -674173. q^{61} +57936.6 q^{62} -2.05561e6 q^{64} +1.14851e6 q^{65} +1.92668e6 q^{67} -3.25754e6 q^{68} +140919. q^{70} -1.19165e6 q^{71} -6.03113e6 q^{73} +193885. q^{74} +1.17961e6 q^{76} -7.65098e6 q^{77} +6.47171e6 q^{79} -2.13993e6 q^{80} -87967.6 q^{82} +845995. q^{83} -3.36845e6 q^{85} -51759.1 q^{86} -775785. q^{88} -1.30527e7 q^{89} +1.42857e7 q^{91} -1.07638e6 q^{92} -456402. q^{94} +1.21977e6 q^{95} +9590.25 q^{97} +1.21669e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 9 q^{2} + 947 q^{4} + 266 q^{5} + 6 q^{7} - 3465 q^{8}+O(q^{10}) $ 13 * q - 9 * q^2 + 947 * q^4 + 266 * q^5 + 6 * q^7 - 3465 * q^8	\( 13 q - 9 q^{2} + 947 q^{4} + 266 q^{5} + 6 q^{7} - 3465 q^{8} - 104 q^{10} - 14218 q^{11} - 7644 q^{13} + 8970 q^{14} + 47811 q^{16} + 48156 q^{17} - 64464 q^{19} + 57914 q^{20} + 26850 q^{22} - 150260 q^{23} + 260915 q^{25} - 74694 q^{26} - 45608 q^{28} - 229626 q^{29} + 511518 q^{31} - 475449 q^{32} - 927790 q^{34} - 204172 q^{35} - 589576 q^{37} - 33540 q^{38} + 55316 q^{40} - 1656956 q^{41} - 1033591 q^{43} - 6317740 q^{44} + 5371478 q^{46} - 3197146 q^{47} + 3545993 q^{49} - 12234087 q^{50} + 5476920 q^{52} - 2301906 q^{53} + 1638708 q^{55} - 9569574 q^{56} + 6308936 q^{58} - 594912 q^{59} + 641576 q^{61} - 16711512 q^{62} + 15392259 q^{64} - 3394176 q^{65} + 4046258 q^{67} - 8031714 q^{68} + 11589246 q^{70} - 5506996 q^{71} + 763246 q^{73} - 11316924 q^{74} - 14274670 q^{76} - 3672676 q^{77} + 12051704 q^{79} - 785638 q^{80} + 5618894 q^{82} + 4951914 q^{83} - 11071848 q^{85} + 715563 q^{86} + 13455806 q^{88} - 20662594 q^{89} - 13410744 q^{91} - 36021878 q^{92} - 11411628 q^{94} + 5494840 q^{95} + 6095056 q^{97} + 388467 q^{98}+O(q^{100}) \) 13 * q - 9 * q^2 + 947 * q^4 + 266 * q^5 + 6 * q^7 - 3465 * q^8 - 104 * q^10 - 14218 * q^11 - 7644 * q^13 + 8970 * q^14 + 47811 * q^16 + 48156 * q^17 - 64464 * q^19 + 57914 * q^20 + 26850 * q^22 - 150260 * q^23 + 260915 * q^25 - 74694 * q^26 - 45608 * q^28 - 229626 * q^29 + 511518 * q^31 - 475449 * q^32 - 927790 * q^34 - 204172 * q^35 - 589576 * q^37 - 33540 * q^38 + 55316 * q^40 - 1656956 * q^41 - 1033591 * q^43 - 6317740 * q^44 + 5371478 * q^46 - 3197146 * q^47 + 3545993 * q^49 - 12234087 * q^50 + 5476920 * q^52 - 2301906 * q^53 + 1638708 * q^55 - 9569574 * q^56 + 6308936 * q^58 - 594912 * q^59 + 641576 * q^61 - 16711512 * q^62 + 15392259 * q^64 - 3394176 * q^65 + 4046258 * q^67 - 8031714 * q^68 + 11589246 * q^70 - 5506996 * q^71 + 763246 * q^73 - 11316924 * q^74 - 14274670 * q^76 - 3672676 * q^77 + 12051704 * q^79 - 785638 * q^80 + 5618894 * q^82 + 4951914 * q^83 - 11071848 * q^85 + 715563 * q^86 + 13455806 * q^88 - 20662594 * q^89 - 13410744 * q^91 - 36021878 * q^92 - 11411628 * q^94 + 5494840 * q^95 + 6095056 * q^97 + 388467 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	0.651000	0.0575409	0.0287704	−	0.999586i	$-0.490841\pi$
$2$	0.651000	0.0575409	0.0287704	+	0.999586i	$0.490841\pi$
$3$	0	0
$3$	0	0
$4$	−127.576	−0.996689
$5$	−131.920	−0.471971	−0.235986	−	0.971757i	$-0.575832\pi$
$5$	−131.920	−0.471971	−0.235986	+	0.971757i	$0.575832\pi$
$6$	0	0
$7$	−1640.88	−1.80815	−0.904074	−	0.427376i	$-0.859438\pi$
$7$	−1640.88	−1.80815	−0.904074	+	0.427376i	$0.859438\pi$
$8$	−166.380	−0.114891
$9$	0	0
$10$	−85.8799	−0.0271576
$11$	4662.73	1.05625	0.528123	−	0.849168i	$-0.322896\pi$
$11$	4662.73	1.05625	0.528123	+	0.849168i	$0.322896\pi$
$12$	0	0
$13$	−8706.11	−1.09906	−0.549532	−	0.835473i	$-0.685194\pi$
$13$	−8706.11	−1.09906	−0.549532	+	0.835473i	$0.685194\pi$
$14$	−1068.21	−0.104042
$15$	0	0
$16$	16221.4	0.990078
$17$	25534.1	1.26052	0.630259	−	0.776385i	$-0.282949\pi$
$17$	25534.1	1.26052	0.630259	+	0.776385i	$0.282949\pi$
$18$	0	0
$19$	−9246.29	−0.309264	−0.154632	−	0.987972i	$-0.549419\pi$
$19$	−9246.29	−0.309264	−0.154632	+	0.987972i	$0.549419\pi$
$20$	16829.8	0.470408
$21$	0	0
$22$	3035.44	0.0607774
$23$	8437.14	0.144593	0.0722966	−	0.997383i	$-0.476967\pi$
$23$	8437.14	0.144593	0.0722966	+	0.997383i	$0.476967\pi$
$24$	0	0
$25$	−60722.1	−0.777243
$26$	−5667.68	−0.0632410
$27$	0	0
$28$	209337.	1.80216
$29$	238673.	1.81723	0.908617	−	0.417630i	$-0.137139\pi$
$29$	238673.	1.81723	0.908617	+	0.417630i	$0.137139\pi$
$30$	0	0
$31$	88996.3	0.536545	0.268272	−	0.963343i	$-0.413547\pi$
$31$	88996.3	0.536545	0.268272	+	0.963343i	$0.413547\pi$
$32$	31856.8	0.171861
$33$	0	0
$34$	16622.7	0.0725313
$35$	216465.	0.853394
$36$	0	0
$37$	297827.	0.966624	0.483312	−	0.875448i	$-0.339434\pi$
$37$	297827.	0.966624	0.483312	+	0.875448i	$0.339434\pi$
$38$	−6019.34	−0.0177953
$39$	0	0
$40$	21948.9	0.0542253
$41$	−135127.	−0.306195	−0.153097	−	0.988211i	$-0.548925\pi$
$41$	−135127.	−0.306195	−0.153097	+	0.988211i	$0.548925\pi$
$42$	0	0
$43$	−79507.0	−0.152499
$43$	−79507.0	−0.152499
$44$	−594853.	−1.05275
$45$	0	0
$46$	5492.58	0.00832002
$47$	−701078.	−0.984972	−0.492486	−	0.870320i	$-0.663912\pi$
$47$	−701078.	−0.984972	−0.492486	+	0.870320i	$0.663912\pi$
$48$	0	0
$49$	1.86895e6	2.26940
$50$	−39530.1	−0.0447232
$51$	0	0
$52$	1.11069e6	1.09542
$53$	48047.2	0.0443305	0.0221653	−	0.999754i	$-0.492944\pi$
$53$	48047.2	0.0443305	0.0221653	+	0.999754i	$0.492944\pi$
$54$	0	0
$55$	−615106.	−0.498518
$56$	273010.	0.207740
$57$	0	0
$58$	155376.	0.104565
$59$	673642.	0.427019	0.213509	−	0.976941i	$-0.431511\pi$
$59$	673642.	0.427019	0.213509	+	0.976941i	$0.431511\pi$
$60$	0	0
$61$	−674173.	−0.380292	−0.190146	−	0.981756i	$-0.560896\pi$
$61$	−674173.	−0.380292	−0.190146	+	0.981756i	$0.560896\pi$
$62$	57936.6	0.0308732
$63$	0	0
$64$	−2.05561e6	−0.980189
$65$	1.14851e6	0.518726
$66$	0	0
$67$	1.92668e6	0.782613	0.391307	−	0.920260i	$-0.372023\pi$
$67$	1.92668e6	0.782613	0.391307	+	0.920260i	$0.372023\pi$
$68$	−3.25754e6	−1.25634
$69$	0	0
$70$	140919.	0.0491050
$71$	−1.19165e6	−0.395135	−0.197567	−	0.980289i	$-0.563304\pi$
$71$	−1.19165e6	−0.395135	−0.197567	+	0.980289i	$0.563304\pi$
$72$	0	0
$73$	−6.03113e6	−1.81455	−0.907274	−	0.420540i	$-0.861841\pi$
$73$	−6.03113e6	−1.81455	−0.907274	+	0.420540i	$0.861841\pi$
$74$	193885.	0.0556204
$75$	0	0
$76$	1.17961e6	0.308240
$77$	−7.65098e6	−1.90985
$78$	0	0
$79$	6.47171e6	1.47681	0.738405	−	0.674358i	$-0.235579\pi$
$79$	6.47171e6	1.47681	0.738405	+	0.674358i	$0.235579\pi$
$80$	−2.13993e6	−0.467288
$81$	0	0
$82$	−87967.6	−0.0176187
$83$	845995.	0.162403	0.0812015	−	0.996698i	$-0.474124\pi$
$83$	845995.	0.162403	0.0812015	+	0.996698i	$0.474124\pi$
$84$	0	0
$85$	−3.36845e6	−0.594928
$86$	−51759.1	−0.00877490
$87$	0	0
$88$	−775785.	−0.121353
$89$	−1.30527e7	−1.96262	−0.981308	−	0.192443i	$-0.938359\pi$
$89$	−1.30527e7	−1.96262	−0.981308	+	0.192443i	$0.938359\pi$
$90$	0	0
$91$	1.42857e7	1.98727
$92$	−1.07638e6	−0.144115
$93$	0	0
$94$	−456402.	−0.0566761
$95$	1.21977e6	0.145964
$96$	0	0
$97$	9590.25	0.00106691	0.000533456	−	1.00000i	$-0.499830\pi$
$97$	9590.25	0.00106691	0.000533456		1.00000i	$0.499830\pi$
$98$	1.21669e6	0.130583
$99$	0	0
$100$	7.74670e6	0.774670
$101$	1.70802e7	1.64956	0.824780	−	0.565454i	$-0.191299\pi$
$101$	1.70802e7	1.64956	0.824780	+	0.565454i	$0.191299\pi$
$102$	0	0
$103$	−1.98888e6	−0.179340	−0.0896702	−	0.995972i	$-0.528581\pi$
$103$	−1.98888e6	−0.179340	−0.0896702	+	0.995972i	$0.528581\pi$
$104$	1.44853e6	0.126273
$105$	0	0
$106$	31278.8	0.00255082
$107$	1.70759e7	1.34753	0.673767	−	0.738943i	$-0.264675\pi$
$107$	1.70759e7	1.34753	0.673767	+	0.738943i	$0.264675\pi$
$108$	0	0
$109$	−1.38909e7	−1.02740	−0.513698	−	0.857971i	$-0.671725\pi$
$109$	−1.38909e7	−1.02740	−0.513698	+	0.857971i	$0.671725\pi$
$110$	−400435.	−0.0286851
$111$	0	0
$112$	−2.66175e7	−1.79021
$113$	1.47471e7	0.961460	0.480730	−	0.876869i	$-0.340372\pi$
$113$	1.47471e7	0.961460	0.480730	+	0.876869i	$0.340372\pi$
$114$	0	0
$115$	−1.11303e6	−0.0682438
$116$	−3.04490e7	−1.81122
$117$	0	0
$118$	438541.	0.0245710
$119$	−4.18984e7	−2.27920
$120$	0	0
$121$	2.25384e6	0.115658
$122$	−438887.	−0.0218823
$123$	0	0
$124$	−1.13538e7	−0.534768
$125$	1.83167e7	0.838807
$126$	0	0
$127$	−2.65344e7	−1.14947	−0.574733	−	0.818341i	$-0.694894\pi$
$127$	−2.65344e7	−1.14947	−0.574733	+	0.818341i	$0.694894\pi$
$128$	−5.41587e6	−0.228262
$129$	0	0
$130$	747680.	0.0298479
$131$	−3.06318e7	−1.19048	−0.595241	−	0.803547i	$-0.702943\pi$
$131$	−3.06318e7	−1.19048	−0.595241	+	0.803547i	$0.702943\pi$
$132$	0	0
$133$	1.51721e7	0.559196
$134$	1.25427e6	0.0450322
$135$	0	0
$136$	−4.24837e6	−0.144822
$137$	2.73180e7	0.907669	0.453835	−	0.891086i	$-0.350056\pi$
$137$	2.73180e7	0.907669	0.453835	+	0.891086i	$0.350056\pi$
$138$	0	0
$139$	−4.17640e6	−0.131902	−0.0659508	−	0.997823i	$-0.521008\pi$
$139$	−4.17640e6	−0.131902	−0.0659508	+	0.997823i	$0.521008\pi$
$140$	−2.76158e7	−0.850568
$141$	0	0
$142$	−775766.	−0.0227364
$143$	−4.05942e7	−1.16088
$144$	0	0
$145$	−3.14858e7	−0.857682
$146$	−3.92627e6	−0.104411
$147$	0	0
$148$	−3.79956e7	−0.963424
$149$	2.65095e7	0.656523	0.328262	−	0.944587i	$-0.393537\pi$
$149$	2.65095e7	0.656523	0.328262	+	0.944587i	$0.393537\pi$
$150$	0	0
$151$	−1.13944e7	−0.269323	−0.134661	−	0.990892i	$-0.542995\pi$
$151$	−1.13944e7	−0.269323	−0.134661	+	0.990892i	$0.542995\pi$
$152$	1.53840e6	0.0355318
$153$	0	0
$154$	−4.98079e6	−0.109894
$155$	−1.17404e7	−0.253233
$156$	0	0
$157$	−4.02050e7	−0.829146	−0.414573	−	0.910016i	$-0.636069\pi$
$157$	−4.02050e7	−0.829146	−0.414573	+	0.910016i	$0.636069\pi$
$158$	4.21309e6	0.0849769
$159$	0	0
$160$	−4.20255e6	−0.0811135
$161$	−1.38444e7	−0.261446
$162$	0	0
$163$	5.16096e7	0.933412	0.466706	−	0.884412i	$-0.345441\pi$
$163$	5.16096e7	0.933412	0.466706	+	0.884412i	$0.345441\pi$
$164$	1.72390e7	0.305181
$165$	0	0
$166$	550743.	0.00934481
$167$	−1.92808e7	−0.320344	−0.160172	−	0.987089i	$-0.551205\pi$
$167$	−1.92808e7	−0.320344	−0.160172	+	0.987089i	$0.551205\pi$
$168$	0	0
$169$	1.30479e7	0.207940
$170$	−2.19287e6	−0.0342327
$171$	0	0
$172$	1.01432e7	0.151994
$173$	9.04396e7	1.32800	0.663999	−	0.747733i	$-0.268858\pi$
$173$	9.04396e7	1.32800	0.663999	+	0.747733i	$0.268858\pi$
$174$	0	0
$175$	9.96378e7	1.40537
$176$	7.56361e7	1.04577
$177$	0	0
$178$	−8.49732e6	−0.112931
$179$	1.19956e7	0.156327	0.0781636	−	0.996941i	$-0.475094\pi$
$179$	1.19956e7	0.156327	0.0781636	+	0.996941i	$0.475094\pi$
$180$	0	0
$181$	−1.43022e8	−1.79279	−0.896393	−	0.443260i	$-0.853822\pi$
$181$	−1.43022e8	−1.79279	−0.896393	+	0.443260i	$0.853822\pi$
$182$	9.30000e6	0.114349
$183$	0	0
$184$	−1.40377e6	−0.0166125
$185$	−3.92893e7	−0.456218
$186$	0	0
$187$	1.19058e8	1.33142
$188$	8.94409e7	0.981711
$189$	0	0
$190$	794071.	0.00839888
$191$	−2.03183e6	−0.0210995	−0.0105497	−	0.999944i	$-0.503358\pi$
$191$	−2.03183e6	−0.0210995	−0.0105497	+	0.999944i	$0.503358\pi$
$192$	0	0
$193$	4.99407e7	0.500039	0.250019	−	0.968241i	$-0.419563\pi$
$193$	4.99407e7	0.500039	0.250019	+	0.968241i	$0.419563\pi$
$194$	6243.26	6.13911e−5 0
$195$	0	0
$196$	−2.38433e8	−2.26189
$197$	−1.69902e8	−1.58331	−0.791656	−	0.610967i	$-0.790781\pi$
$197$	−1.69902e8	−1.58331	−0.791656	+	0.610967i	$0.790781\pi$
$198$	0	0
$199$	−3.53364e7	−0.317861	−0.158930	−	0.987290i	$-0.550805\pi$
$199$	−3.53364e7	−0.317861	−0.158930	+	0.987290i	$0.550805\pi$
$200$	1.01030e7	0.0892984
$201$	0	0
$202$	1.11192e7	0.0949170
$203$	−3.91635e8	−3.28583
$204$	0	0
$205$	1.78259e7	0.144515
$206$	−1.29476e6	−0.0103194
$207$	0	0
$208$	−1.41226e8	−1.08816
$209$	−4.31129e7	−0.326660
$210$	0	0
$211$	2.16309e8	1.58521	0.792604	−	0.609737i	$-0.208725\pi$
$211$	2.16309e8	1.58521	0.792604	+	0.609737i	$0.208725\pi$
$212$	−6.12968e6	−0.0441837
$213$	0	0
$214$	1.11164e7	0.0775383
$215$	1.04886e7	0.0719749
$216$	0	0
$217$	−1.46032e8	−0.970152
$218$	−9.04298e6	−0.0591172
$219$	0	0
$220$	7.84729e7	0.496867
$221$	−2.22303e8	−1.38539
$222$	0	0
$223$	−1.93132e8	−1.16624	−0.583120	−	0.812386i	$-0.698168\pi$
$223$	−1.93132e8	−1.16624	−0.583120	+	0.812386i	$0.698168\pi$
$224$	−5.22733e7	−0.310750
$225$	0	0
$226$	9.60035e6	0.0553232
$227$	−8.93697e7	−0.507107	−0.253554	−	0.967321i	$-0.581599\pi$
$227$	−8.93697e7	−0.507107	−0.253554	+	0.967321i	$0.581599\pi$
$228$	0	0
$229$	3.06554e8	1.68687	0.843437	−	0.537229i	$-0.180529\pi$
$229$	3.06554e8	1.68687	0.843437	+	0.537229i	$0.180529\pi$
$230$	−724581.	−0.00392681
$231$	0	0
$232$	−3.97105e7	−0.208784
$233$	−6.70594e6	−0.0347307	−0.0173654	−	0.999849i	$-0.505528\pi$
$233$	−6.70594e6	−0.0347307	−0.0173654	+	0.999849i	$0.505528\pi$
$234$	0	0
$235$	9.24862e7	0.464878
$236$	−8.59406e7	−0.425605
$237$	0	0
$238$	−2.72759e7	−0.131147
$239$	−5.22905e7	−0.247759	−0.123880	−	0.992297i	$-0.539534\pi$
$239$	−5.22905e7	−0.247759	−0.123880	+	0.992297i	$0.539534\pi$
$240$	0	0
$241$	9.01747e7	0.414978	0.207489	−	0.978237i	$-0.433471\pi$
$241$	9.01747e7	0.414978	0.207489	+	0.978237i	$0.433471\pi$
$242$	1.46725e6	0.00665504
$243$	0	0
$244$	8.60084e7	0.379032
$245$	−2.46552e8	−1.07109
$246$	0	0
$247$	8.04993e7	0.339901
$248$	−1.48072e7	−0.0616442
$249$	0	0
$250$	1.19242e7	0.0482657
$251$	2.64322e8	1.05506	0.527528	−	0.849537i	$-0.323119\pi$
$251$	2.64322e8	1.05506	0.527528	+	0.849537i	$0.323119\pi$
$252$	0	0
$253$	3.93401e7	0.152726
$254$	−1.72739e7	−0.0661412
$255$	0	0
$256$	2.59592e8	0.967055
$257$	−3.55090e8	−1.30489	−0.652443	−	0.757838i	$-0.726256\pi$
$257$	−3.55090e8	−1.30489	−0.652443	+	0.757838i	$0.726256\pi$
$258$	0	0
$259$	−4.88698e8	−1.74780
$260$	−1.46523e8	−0.517008
$261$	0	0
$262$	−1.99413e7	−0.0685014
$263$	−3.45490e8	−1.17109	−0.585544	−	0.810640i	$-0.699119\pi$
$263$	−3.45490e8	−1.17109	−0.585544	+	0.810640i	$0.699119\pi$
$264$	0	0
$265$	−6.33839e6	−0.0209227
$266$	9.87702e6	0.0321766
$267$	0	0
$268$	−2.45798e8	−0.780022
$269$	−5.48063e8	−1.71671	−0.858357	−	0.513053i	$-0.828514\pi$
$269$	−5.48063e8	−1.71671	−0.858357	+	0.513053i	$0.828514\pi$
$270$	0	0
$271$	−3.56164e8	−1.08707	−0.543535	−	0.839386i	$-0.682915\pi$
$271$	−3.56164e8	−1.08707	−0.543535	+	0.839386i	$0.682915\pi$
$272$	4.14200e8	1.24801
$273$	0	0
$274$	1.77841e7	0.0522281
$275$	−2.83131e8	−0.820961
$276$	0	0
$277$	−2.33147e8	−0.659099	−0.329549	−	0.944138i	$-0.606897\pi$
$277$	−2.33147e8	−0.659099	−0.329549	+	0.944138i	$0.606897\pi$
$278$	−2.71884e6	−0.00758973
$279$	0	0
$280$	−3.60155e7	−0.0980474
$281$	−1.30535e8	−0.350957	−0.175479	−	0.984483i	$-0.556147\pi$
$281$	−1.30535e8	−0.350957	−0.175479	+	0.984483i	$0.556147\pi$
$282$	0	0
$283$	−6.91521e8	−1.81365	−0.906824	−	0.421510i	$-0.861500\pi$
$283$	−6.91521e8	−1.81365	−0.906824	+	0.421510i	$0.861500\pi$
$284$	1.52026e8	0.393827
$285$	0	0
$286$	−2.64269e7	−0.0667981
$287$	2.21727e8	0.553646
$288$	0	0
$289$	2.41651e8	0.588907
$290$	−2.04972e7	−0.0493518
$291$	0	0
$292$	7.69428e8	1.80854
$293$	6.09230e8	1.41496	0.707481	−	0.706732i	$-0.249831\pi$
$293$	6.09230e8	1.41496	0.707481	+	0.706732i	$0.249831\pi$
$294$	0	0
$295$	−8.88667e7	−0.201540
$296$	−4.95525e7	−0.111057
$297$	0	0
$298$	1.72577e7	0.0377769
$299$	−7.34547e7	−0.158917
$300$	0	0
$301$	1.30462e8	0.275740
$302$	−7.41777e6	−0.0154971
$303$	0	0
$304$	−1.49988e8	−0.306196
$305$	8.89368e7	0.179487
$306$	0	0
$307$	6.45657e8	1.27355	0.636777	−	0.771048i	$-0.280267\pi$
$307$	6.45657e8	1.27355	0.636777	+	0.771048i	$0.280267\pi$
$308$	9.76083e8	1.90353
$309$	0	0
$310$	−7.64299e6	−0.0145713
$311$	−5.23168e8	−0.986234	−0.493117	−	0.869963i	$-0.664142\pi$
$311$	−5.23168e8	−0.986234	−0.493117	+	0.869963i	$0.664142\pi$
$312$	0	0
$313$	−2.19926e8	−0.405389	−0.202694	−	0.979242i	$-0.564970\pi$
$313$	−2.19926e8	−0.405389	−0.202694	+	0.979242i	$0.564970\pi$
$314$	−2.61735e7	−0.0477098
$315$	0	0
$316$	−8.25637e8	−1.47192
$317$	−2.96922e8	−0.523521	−0.261761	−	0.965133i	$-0.584303\pi$
$317$	−2.96922e8	−0.523521	−0.261761	+	0.965133i	$0.584303\pi$
$318$	0	0
$319$	1.11287e9	1.91945
$320$	2.71175e8	0.462621
$321$	0	0
$322$	−9.01268e6	−0.0150438
$323$	−2.36096e8	−0.389833
$324$	0	0
$325$	5.28654e8	0.854240
$326$	3.35978e7	0.0537093
$327$	0	0
$328$	2.24824e7	0.0351791
$329$	1.15039e9	1.78098
$330$	0	0
$331$	1.83197e8	0.277664	0.138832	−	0.990316i	$-0.455665\pi$
$331$	1.83197e8	0.277664	0.138832	+	0.990316i	$0.455665\pi$
$332$	−1.07929e8	−0.161865
$333$	0	0
$334$	−1.25518e7	−0.0184329
$335$	−2.54167e8	−0.369371
$336$	0	0
$337$	−2.81140e8	−0.400145	−0.200073	−	0.979781i	$-0.564118\pi$
$337$	−2.81140e8	−0.400145	−0.200073	+	0.979781i	$0.564118\pi$
$338$	8.49420e6	0.0119650
$339$	0	0
$340$	4.29735e8	0.592958
$341$	4.14965e8	0.566724
$342$	0	0
$343$	−1.71539e9	−2.29526
$344$	1.32284e7	0.0175207
$345$	0	0
$346$	5.88762e7	0.0764141
$347$	1.17181e9	1.50558	0.752790	−	0.658261i	$-0.228708\pi$
$347$	1.17181e9	1.50558	0.752790	+	0.658261i	$0.228708\pi$
$348$	0	0
$349$	5.32505e8	0.670556	0.335278	−	0.942119i	$-0.391170\pi$
$349$	5.32505e8	0.670556	0.335278	+	0.942119i	$0.391170\pi$
$350$	6.48643e7	0.0808663
$351$	0	0
$352$	1.48540e8	0.181528
$353$	5.99342e8	0.725209	0.362605	−	0.931943i	$-0.381888\pi$
$353$	5.99342e8	0.725209	0.362605	+	0.931943i	$0.381888\pi$
$354$	0	0
$355$	1.57203e8	0.186492
$356$	1.66521e9	1.95612
$357$	0	0
$358$	7.80911e6	0.00899520
$359$	1.49256e9	1.70256	0.851279	−	0.524714i	$-0.175828\pi$
$359$	1.49256e9	1.70256	0.851279	+	0.524714i	$0.175828\pi$
$360$	0	0
$361$	−8.08378e8	−0.904356
$362$	−9.31075e7	−0.103158
$363$	0	0
$364$	−1.82252e9	−1.98069
$365$	7.95626e8	0.856414
$366$	0	0
$367$	2.77043e8	0.292560	0.146280	−	0.989243i	$-0.453270\pi$
$367$	2.77043e8	0.292560	0.146280	+	0.989243i	$0.453270\pi$
$368$	1.36863e8	0.143159
$369$	0	0
$370$	−2.55773e7	−0.0262512
$371$	−7.88398e7	−0.0801562
$372$	0	0
$373$	−1.69390e9	−1.69008	−0.845041	−	0.534701i	$-0.820424\pi$
$373$	−1.69390e9	−1.69008	−0.845041	+	0.534701i	$0.820424\pi$
$374$	7.75071e7	0.0766110
$375$	0	0
$376$	1.16646e8	0.113165
$377$	−2.07792e9	−1.99726
$378$	0	0
$379$	1.30235e9	1.22882	0.614412	−	0.788985i	$-0.289393\pi$
$379$	1.30235e9	1.22882	0.614412	+	0.788985i	$0.289393\pi$
$380$	−1.55614e8	−0.145481
$381$	0	0
$382$	−1.32272e6	−0.00121408
$383$	−1.56161e9	−1.42029	−0.710147	−	0.704054i	$-0.751371\pi$
$383$	−1.56161e9	−1.42029	−0.710147	+	0.704054i	$0.751371\pi$
$384$	0	0
$385$	1.00932e9	0.901394
$386$	3.25114e7	0.0287727
$387$	0	0
$388$	−1.22349e6	−0.00106338
$389$	2.07252e9	1.78515	0.892575	−	0.450898i	$-0.148896\pi$
$389$	2.07252e9	1.78515	0.892575	+	0.450898i	$0.148896\pi$
$390$	0	0
$391$	2.15435e8	0.182262
$392$	−3.10956e8	−0.260734
$393$	0	0
$394$	−1.10606e8	−0.0911051
$395$	−8.53748e8	−0.697011
$396$	0	0
$397$	1.16600e9	0.935258	0.467629	−	0.883925i	$-0.345108\pi$
$397$	1.16600e9	0.935258	0.467629	+	0.883925i	$0.345108\pi$
$398$	−2.30040e7	−0.0182900
$399$	0	0
$400$	−9.85000e8	−0.769532
$401$	−5.00400e8	−0.387536	−0.193768	−	0.981047i	$-0.562071\pi$
$401$	−5.00400e8	−0.387536	−0.193768	+	0.981047i	$0.562071\pi$
$402$	0	0
$403$	−7.74812e8	−0.589696
$404$	−2.17903e9	−1.64410
$405$	0	0
$406$	−2.54954e8	−0.189069
$407$	1.38868e9	1.02099
$408$	0	0
$409$	6.28780e8	0.454431	0.227215	−	0.973845i	$-0.427038\pi$
$409$	6.28780e8	0.454431	0.227215	+	0.973845i	$0.427038\pi$
$410$	1.16047e7	0.00831552
$411$	0	0
$412$	2.53734e8	0.178747
$413$	−1.10537e9	−0.772113
$414$	0	0
$415$	−1.11604e8	−0.0766496
$416$	−2.77349e8	−0.188886
$417$	0	0
$418$	−2.80665e7	−0.0187963
$419$	2.19914e9	1.46051	0.730253	−	0.683177i	$-0.239402\pi$
$419$	2.19914e9	1.46051	0.730253	+	0.683177i	$0.239402\pi$
$420$	0	0
$421$	1.68678e9	1.10172	0.550858	−	0.834599i	$-0.314300\pi$
$421$	1.68678e9	1.10172	0.550858	+	0.834599i	$0.314300\pi$
$422$	1.40817e8	0.0912142
$423$	0	0
$424$	−7.99411e6	−0.00509319
$425$	−1.55048e9	−0.979730
$426$	0	0
$427$	1.10624e9	0.687624
$428$	−2.17848e9	−1.34307
$429$	0	0
$430$	6.82805e6	0.00414150
$431$	1.08128e9	0.650532	0.325266	−	0.945623i	$-0.394546\pi$
$431$	1.08128e9	0.650532	0.325266	+	0.945623i	$0.394546\pi$
$432$	0	0
$433$	−1.29545e9	−0.766854	−0.383427	−	0.923571i	$-0.625256\pi$
$433$	−1.29545e9	−0.766854	−0.383427	+	0.923571i	$0.625256\pi$
$434$	−9.50671e7	−0.0558234
$435$	0	0
$436$	1.77215e9	1.02399
$437$	−7.80123e7	−0.0447175
$438$	0	0
$439$	1.63847e9	0.924297	0.462148	−	0.886803i	$-0.347079\pi$
$439$	1.63847e9	0.924297	0.462148	+	0.886803i	$0.347079\pi$
$440$	1.02342e8	0.0572753
$441$	0	0
$442$	−1.44719e8	−0.0797165
$443$	2.06571e9	1.12890	0.564450	−	0.825467i	$-0.309088\pi$
$443$	2.06571e9	1.12890	0.564450	+	0.825467i	$0.309088\pi$
$444$	0	0
$445$	1.72191e9	0.926298
$446$	−1.25729e8	−0.0671064
$447$	0	0
$448$	3.37300e9	1.77233
$449$	2.46381e9	1.28453	0.642266	−	0.766482i	$-0.277994\pi$
$449$	2.46381e9	1.28453	0.642266	+	0.766482i	$0.277994\pi$
$450$	0	0
$451$	−6.30059e8	−0.323417
$452$	−1.88138e9	−0.958277
$453$	0	0
$454$	−5.81797e7	−0.0291794
$455$	−1.88457e9	−0.937933
$456$	0	0
$457$	8.80150e8	0.431370	0.215685	−	0.976463i	$-0.430802\pi$
$457$	8.80150e8	0.431370	0.215685	+	0.976463i	$0.430802\pi$
$458$	1.99567e8	0.0970641
$459$	0	0
$460$	1.41996e8	0.0680179
$461$	−2.11439e9	−1.00515	−0.502575	−	0.864533i	$-0.667614\pi$
$461$	−2.11439e9	−1.00515	−0.502575	+	0.864533i	$0.667614\pi$
$462$	0	0
$463$	1.57049e9	0.735361	0.367680	−	0.929952i	$-0.380152\pi$
$463$	1.57049e9	0.735361	0.367680	+	0.929952i	$0.380152\pi$
$464$	3.87163e9	1.79920
$465$	0	0
$466$	−4.36557e6	−0.00199844
$467$	1.01523e9	0.461270	0.230635	−	0.973040i	$-0.425920\pi$
$467$	1.01523e9	0.461270	0.230635	+	0.973040i	$0.425920\pi$
$468$	0	0
$469$	−3.16145e9	−1.41508
$470$	6.02085e7	0.0267495
$471$	0	0
$472$	−1.12081e8	−0.0490607
$473$	−3.70719e8	−0.161076
$474$	0	0
$475$	5.61455e8	0.240374
$476$	5.34524e9	2.27166
$477$	0	0
$478$	−3.40411e7	−0.0142563
$479$	2.70045e9	1.12269	0.561347	−	0.827581i	$-0.310283\pi$
$479$	2.70045e9	1.12269	0.561347	+	0.827581i	$0.310283\pi$
$480$	0	0
$481$	−2.59291e9	−1.06238
$482$	5.87038e7	0.0238782
$483$	0	0
$484$	−2.87537e8	−0.115275
$485$	−1.26515e6	−0.000503552 0
$486$	0	0
$487$	−5.31924e8	−0.208688	−0.104344	−	0.994541i	$-0.533274\pi$
$487$	−5.31924e8	−0.208688	−0.104344	+	0.994541i	$0.533274\pi$
$488$	1.12169e8	0.0436921
$489$	0	0
$490$	−1.60505e8	−0.0616315
$491$	−3.40336e9	−1.29755	−0.648773	−	0.760982i	$-0.724718\pi$
$491$	−3.40336e9	−1.29755	−0.648773	+	0.760982i	$0.724718\pi$
$492$	0	0
$493$	6.09431e9	2.29066
$494$	5.24051e7	0.0195582
$495$	0	0
$496$	1.44365e9	0.531221
$497$	1.95536e9	0.714463
$498$	0	0
$499$	−1.05372e9	−0.379641	−0.189821	−	0.981819i	$-0.560791\pi$
$499$	−1.05372e9	−0.379641	−0.189821	+	0.981819i	$0.560791\pi$
$500$	−2.33678e9	−0.836030
$501$	0	0
$502$	1.72074e8	0.0607089
$503$	−4.62954e9	−1.62200	−0.810999	−	0.585048i	$-0.801076\pi$
$503$	−4.62954e9	−1.62200	−0.810999	+	0.585048i	$0.801076\pi$
$504$	0	0
$505$	−2.25322e9	−0.778544
$506$	2.56104e7	0.00878800
$507$	0	0
$508$	3.38516e9	1.14566
$509$	1.26324e9	0.424592	0.212296	−	0.977205i	$-0.431906\pi$
$509$	1.26324e9	0.424592	0.212296	+	0.977205i	$0.431906\pi$
$510$	0	0
$511$	9.89636e9	3.28097
$512$	8.62226e8	0.283907
$513$	0	0
$514$	−2.31164e8	−0.0750842
$515$	2.62373e8	0.0846435
$516$	0	0
$517$	−3.26894e9	−1.04037
$518$	−3.18143e8	−0.100570
$519$	0	0
$520$	−1.91089e8	−0.0595970
$521$	−4.00375e9	−1.24032	−0.620162	−	0.784474i	$-0.712933\pi$
$521$	−4.00375e9	−1.24032	−0.620162	+	0.784474i	$0.712933\pi$
$522$	0	0
$523$	−4.27665e9	−1.30722	−0.653608	−	0.756833i	$-0.726746\pi$
$523$	−4.27665e9	−1.30722	−0.653608	+	0.756833i	$0.726746\pi$
$524$	3.90789e9	1.18654
$525$	0	0
$526$	−2.24914e8	−0.0673854
$527$	2.27244e9	0.676324
$528$	0	0
$529$	−3.33364e9	−0.979093
$530$	−4.12629e6	−0.00120391
$531$	0	0
$532$	−1.93559e9	−0.557344
$533$	1.17643e9	0.336528
$534$	0	0
$535$	−2.25265e9	−0.635997
$536$	−3.20561e8	−0.0899154
$537$	0	0
$538$	−3.56790e8	−0.0987812
$539$	8.71440e9	2.39705
$540$	0	0
$541$	1.00182e9	0.272019	0.136010	−	0.990708i	$-0.456572\pi$
$541$	1.00182e9	0.272019	0.136010	+	0.990708i	$0.456572\pi$
$542$	−2.31863e8	−0.0625510
$543$	0	0
$544$	8.13435e8	0.216634
$545$	1.83249e9	0.484901
$546$	0	0
$547$	−1.64658e9	−0.430157	−0.215079	−	0.976597i	$-0.569001\pi$
$547$	−1.64658e9	−0.430157	−0.215079	+	0.976597i	$0.569001\pi$
$548$	−3.48513e9	−0.904664
$549$	0	0
$550$	−1.84318e8	−0.0472388
$551$	−2.20684e9	−0.562006
$552$	0	0
$553$	−1.06193e10	−2.67029
$554$	−1.51779e8	−0.0379251
$555$	0	0
$556$	5.32809e8	0.131465
$557$	−4.09548e9	−1.00418	−0.502090	−	0.864815i	$-0.667435\pi$
$557$	−4.09548e9	−1.00418	−0.502090	+	0.864815i	$0.667435\pi$
$558$	0	0
$559$	6.92197e8	0.167606
$560$	3.51137e9	0.844926
$561$	0	0
$562$	−8.49782e7	−0.0201944
$563$	−2.73116e9	−0.645013	−0.322507	−	0.946567i	$-0.604525\pi$
$563$	−2.73116e9	−0.645013	−0.322507	+	0.946567i	$0.604525\pi$
$564$	0	0
$565$	−1.94543e9	−0.453781
$566$	−4.50180e8	−0.104359
$567$	0	0
$568$	1.98267e8	0.0453975
$569$	−3.76201e9	−0.856105	−0.428052	−	0.903754i	$-0.640800\pi$
$569$	−3.76201e9	−0.856105	−0.428052	+	0.903754i	$0.640800\pi$
$570$	0	0
$571$	−4.05314e9	−0.911098	−0.455549	−	0.890211i	$-0.650557\pi$
$571$	−4.05314e9	−0.911098	−0.455549	+	0.890211i	$0.650557\pi$
$572$	5.17886e9	1.15704
$573$	0	0
$574$	1.44344e8	0.0318573
$575$	−5.12321e8	−0.112384
$576$	0	0
$577$	−5.96852e8	−0.129346	−0.0646728	−	0.997907i	$-0.520600\pi$
$577$	−5.96852e8	−0.129346	−0.0646728	+	0.997907i	$0.520600\pi$
$578$	1.57315e8	0.0338862
$579$	0	0
$580$	4.01683e9	0.854842
$581$	−1.38818e9	−0.293649
$582$	0	0
$583$	2.24031e8	0.0468240
$584$	1.00346e9	0.208476
$585$	0	0
$586$	3.96609e8	0.0814181
$587$	−8.39512e9	−1.71314	−0.856571	−	0.516029i	$-0.827410\pi$
$587$	−8.39512e9	−1.71314	−0.856571	+	0.516029i	$0.827410\pi$
$588$	0	0
$589$	−8.22885e8	−0.165934
$590$	−5.78523e7	−0.0115968
$591$	0	0
$592$	4.83118e9	0.957033
$593$	4.12835e9	0.812989	0.406495	−	0.913653i	$-0.366751\pi$
$593$	4.12835e9	0.812989	0.406495	+	0.913653i	$0.366751\pi$
$594$	0	0
$595$	5.52724e9	1.07572
$596$	−3.38198e9	−0.654349
$597$	0	0
$598$	−4.78191e7	−0.00914423
$599$	−2.16463e9	−0.411518	−0.205759	−	0.978603i	$-0.565966\pi$
$599$	−2.16463e9	−0.411518	−0.205759	+	0.978603i	$0.565966\pi$
$600$	0	0
$601$	1.68118e9	0.315902	0.157951	−	0.987447i	$-0.449511\pi$
$601$	1.68118e9	0.315902	0.157951	+	0.987447i	$0.449511\pi$
$602$	8.49305e7	0.0158663
$603$	0	0
$604$	1.45366e9	0.268431
$605$	−2.97327e8	−0.0545871
$606$	0	0
$607$	−3.74202e9	−0.679118	−0.339559	−	0.940585i	$-0.610278\pi$
$607$	−3.74202e9	−0.679118	−0.339559	+	0.940585i	$0.610278\pi$
$608$	−2.94558e8	−0.0531505
$609$	0	0
$610$	5.78979e7	0.0103278
$611$	6.10367e9	1.08255
$612$	0	0
$613$	3.61510e9	0.633882	0.316941	−	0.948445i	$-0.397344\pi$
$613$	3.61510e9	0.633882	0.316941	+	0.948445i	$0.397344\pi$
$614$	4.20323e8	0.0732814
$615$	0	0
$616$	1.27297e9	0.219425
$617$	−7.33591e9	−1.25735	−0.628675	−	0.777668i	$-0.716402\pi$
$617$	−7.33591e9	−1.25735	−0.628675	+	0.777668i	$0.716402\pi$
$618$	0	0
$619$	−4.29644e8	−0.0728101	−0.0364050	−	0.999337i	$-0.511591\pi$
$619$	−4.29644e8	−0.0728101	−0.0364050	+	0.999337i	$0.511591\pi$
$620$	1.49779e9	0.252395
$621$	0	0
$622$	−3.40583e8	−0.0567487
$623$	2.14179e10	3.54870
$624$	0	0
$625$	2.32758e9	0.381351
$626$	−1.43172e8	−0.0233264
$627$	0	0
$628$	5.12920e9	0.826401
$629$	7.60473e9	1.21845
$630$	0	0
$631$	−1.13341e10	−1.79591	−0.897955	−	0.440088i	$-0.854947\pi$
$631$	−1.13341e10	−1.79591	−0.897955	+	0.440088i	$0.854947\pi$
$632$	−1.07677e9	−0.169672
$633$	0	0
$634$	−1.93296e8	−0.0301239
$635$	3.50041e9	0.542514
$636$	0	0
$637$	−1.62713e10	−2.49421
$638$	7.24478e8	0.110447
$639$	0	0
$640$	7.14462e8	0.107733
$641$	3.51134e9	0.526586	0.263293	−	0.964716i	$-0.415191\pi$
$641$	3.51134e9	0.526586	0.263293	+	0.964716i	$0.415191\pi$
$642$	0	0
$643$	6.39119e8	0.0948076	0.0474038	−	0.998876i	$-0.484905\pi$
$643$	6.39119e8	0.0948076	0.0474038	+	0.998876i	$0.484905\pi$
$644$	1.76621e9	0.260580
$645$	0	0
$646$	−1.53698e8	−0.0224314
$647$	−1.23706e10	−1.79567	−0.897836	−	0.440329i	$-0.854862\pi$
$647$	−1.23706e10	−1.79567	−0.897836	+	0.440329i	$0.854862\pi$
$648$	0	0
$649$	3.14101e9	0.451037
$650$	3.44154e8	0.0491537
$651$	0	0
$652$	−6.58415e9	−0.930322
$653$	−1.24728e9	−0.175295	−0.0876474	−	0.996152i	$-0.527935\pi$
$653$	−1.24728e9	−0.175295	−0.0876474	+	0.996152i	$0.527935\pi$
$654$	0	0
$655$	4.04094e9	0.561873
$656$	−2.19195e9	−0.303157
$657$	0	0
$658$	7.48902e8	0.102479
$659$	−1.05379e10	−1.43435	−0.717176	−	0.696892i	$-0.754566\pi$
$659$	−1.05379e10	−1.43435	−0.717176	+	0.696892i	$0.754566\pi$
$660$	0	0
$661$	−9.46132e9	−1.27423	−0.637113	−	0.770770i	$-0.719872\pi$
$661$	−9.46132e9	−1.27423	−0.637113	+	0.770770i	$0.719872\pi$
$662$	1.19261e8	0.0159770
$663$	0	0
$664$	−1.40757e8	−0.0186587
$665$	−2.00150e9	−0.263924
$666$	0	0
$667$	2.01372e9	0.262760
$668$	2.45977e9	0.319284
$669$	0	0
$670$	−1.65463e8	−0.0212539
$671$	−3.14348e9	−0.401682
$672$	0	0
$673$	−4.96969e9	−0.628458	−0.314229	−	0.949347i	$-0.601746\pi$
$673$	−4.96969e9	−0.628458	−0.314229	+	0.949347i	$0.601746\pi$
$674$	−1.83022e8	−0.0230247
$675$	0	0
$676$	−1.66460e9	−0.207251
$677$	−8.69252e9	−1.07668	−0.538338	−	0.842729i	$-0.680948\pi$
$677$	−8.69252e9	−1.07668	−0.538338	+	0.842729i	$0.680948\pi$
$678$	0	0
$679$	−1.57365e7	−0.00192914
$680$	5.60444e8	0.0683520
$681$	0	0
$682$	2.70142e8	0.0326098
$683$	4.17502e9	0.501402	0.250701	−	0.968065i	$-0.419339\pi$
$683$	4.17502e9	0.501402	0.250701	+	0.968065i	$0.419339\pi$
$684$	0	0
$685$	−3.60379e9	−0.428394
$686$	−1.11672e9	−0.132072
$687$	0	0
$688$	−1.28972e9	−0.150985
$689$	−4.18305e8	−0.0487220
$690$	0	0
$691$	1.05393e10	1.21517	0.607587	−	0.794253i	$-0.292137\pi$
$691$	1.05393e10	1.21517	0.607587	+	0.794253i	$0.292137\pi$
$692$	−1.15379e10	−1.32360
$693$	0	0
$694$	7.62848e8	0.0866323
$695$	5.50950e8	0.0622537
$696$	0	0
$697$	−3.45034e9	−0.385964
$698$	3.46661e8	0.0385843
$699$	0	0
$700$	−1.27114e10	−1.40072
$701$	−1.00619e10	−1.10324	−0.551618	−	0.834097i	$-0.685989\pi$
$701$	−1.00619e10	−1.10324	−0.551618	+	0.834097i	$0.685989\pi$
$702$	0	0
$703$	−2.75379e9	−0.298942
$704$	−9.58472e9	−1.03532
$705$	0	0
$706$	3.90172e8	0.0417292
$707$	−2.80266e10	−2.98265
$708$	0	0
$709$	1.41819e10	1.49442	0.747211	−	0.664587i	$-0.231392\pi$
$709$	1.41819e10	1.49442	0.747211	+	0.664587i	$0.231392\pi$
$710$	1.02339e8	0.0107309
$711$	0	0
$712$	2.17171e9	0.225487
$713$	7.50874e8	0.0775807
$714$	0	0
$715$	5.35519e9	0.547903
$716$	−1.53035e9	−0.155810
$717$	0	0
$718$	9.71659e8	0.0979666
$719$	1.49402e9	0.149902	0.0749508	−	0.997187i	$-0.476120\pi$
$719$	1.49402e9	0.149902	0.0749508	+	0.997187i	$0.476120\pi$
$720$	0	0
$721$	3.26352e9	0.324274
$722$	−5.26254e8	−0.0520374
$723$	0	0
$724$	1.82462e10	1.78685
$725$	−1.44928e10	−1.41243
$726$	0	0
$727$	−9.88594e9	−0.954218	−0.477109	−	0.878844i	$-0.658315\pi$
$727$	−9.88594e9	−0.954218	−0.477109	+	0.878844i	$0.658315\pi$
$728$	−2.37686e9	−0.228320
$729$	0	0
$730$	5.17953e8	0.0492788
$731$	−2.03014e9	−0.192227
$732$	0	0
$733$	2.13118e9	0.199874	0.0999371	−	0.994994i	$-0.468136\pi$
$733$	2.13118e9	0.199874	0.0999371	+	0.994994i	$0.468136\pi$
$734$	1.80355e8	0.0168342
$735$	0	0
$736$	2.68781e8	0.0248500
$737$	8.98357e9	0.826633
$738$	0	0
$739$	1.38201e9	0.125967	0.0629835	−	0.998015i	$-0.479938\pi$
$739$	1.38201e9	0.125967	0.0629835	+	0.998015i	$0.479938\pi$
$740$	5.01238e9	0.454708
$741$	0	0
$742$	−5.13248e7	−0.00461225
$743$	1.54223e10	1.37939	0.689697	−	0.724098i	$-0.257744\pi$
$743$	1.54223e10	1.37939	0.689697	+	0.724098i	$0.257744\pi$
$744$	0	0
$745$	−3.49713e9	−0.309860
$746$	−1.10273e9	−0.0972488
$747$	0	0
$748$	−1.51890e10	−1.32701
$749$	−2.80195e10	−2.43654
$750$	0	0
$751$	1.55035e10	1.33564	0.667821	−	0.744322i	$-0.267227\pi$
$751$	1.55035e10	1.33564	0.667821	+	0.744322i	$0.267227\pi$
$752$	−1.13725e10	−0.975200
$753$	0	0
$754$	−1.35273e9	−0.114924
$755$	1.50315e9	0.127113
$756$	0	0
$757$	−2.23258e10	−1.87056	−0.935278	−	0.353915i	$-0.884850\pi$
$757$	−2.23258e10	−1.87056	−0.935278	+	0.353915i	$0.884850\pi$
$758$	8.47830e8	0.0707076
$759$	0	0
$760$	−2.02946e8	−0.0167700
$761$	5.14709e9	0.423366	0.211683	−	0.977338i	$-0.432106\pi$
$761$	5.14709e9	0.423366	0.211683	+	0.977338i	$0.432106\pi$
$762$	0	0
$763$	2.27933e10	1.85768
$764$	2.59214e8	0.0210296
$765$	0	0
$766$	−1.01661e9	−0.0817249
$767$	−5.86480e9	−0.469320
$768$	0	0
$769$	−1.08946e10	−0.863910	−0.431955	−	0.901895i	$-0.642176\pi$
$769$	−1.08946e10	−0.863910	−0.431955	+	0.901895i	$0.642176\pi$
$770$	6.57066e8	0.0518670
$771$	0	0
$772$	−6.37124e9	−0.498383
$773$	−1.95101e10	−1.51925	−0.759627	−	0.650359i	$-0.774618\pi$
$773$	−1.95101e10	−1.51925	−0.759627	+	0.650359i	$0.774618\pi$
$774$	0	0
$775$	−5.40404e9	−0.417026
$776$	−1.59563e6	−0.000122579 0
$777$	0	0
$778$	1.34921e9	0.102719
$779$	1.24942e9	0.0946952
$780$	0	0
$781$	−5.55635e9	−0.417360
$782$	1.40248e8	0.0104875
$783$	0	0
$784$	3.03170e10	2.24688
$785$	5.30384e9	0.391333
$786$	0	0
$787$	1.20289e9	0.0879658	0.0439829	−	0.999032i	$-0.485995\pi$
$787$	1.20289e9	0.0879658	0.0439829	+	0.999032i	$0.485995\pi$
$788$	2.16754e10	1.57807
$789$	0	0
$790$	−5.55790e8	−0.0401066
$791$	−2.41982e10	−1.73846
$792$	0	0
$793$	5.86942e9	0.417964
$794$	7.59066e8	0.0538156
$795$	0	0
$796$	4.50809e9	0.316808
$797$	6.93230e9	0.485035	0.242518	−	0.970147i	$-0.422027\pi$
$797$	6.93230e9	0.485035	0.242518	+	0.970147i	$0.422027\pi$
$798$	0	0
$799$	−1.79014e10	−1.24158
$800$	−1.93442e9	−0.133578
$801$	0	0
$802$	−3.25761e8	−0.0222992
$803$	−2.81215e10	−1.91661
$804$	0	0
$805$	1.82635e9	0.123395
$806$	−5.04403e8	−0.0339316
$807$	0	0
$808$	−2.84181e9	−0.189520
$809$	6.12187e9	0.406504	0.203252	−	0.979126i	$-0.434849\pi$
$809$	6.12187e9	0.406504	0.203252	+	0.979126i	$0.434849\pi$
$810$	0	0
$811$	−1.04739e10	−0.689503	−0.344752	−	0.938694i	$-0.612037\pi$
$811$	−1.04739e10	−0.689503	−0.344752	+	0.938694i	$0.612037\pi$
$812$	4.99633e10	3.27495
$813$	0	0
$814$	9.04034e8	0.0587488
$815$	−6.80833e9	−0.440544
$816$	0	0
$817$	7.35145e8	0.0471624
$818$	4.09336e8	0.0261483
$819$	0	0
$820$	−2.27416e9	−0.144037
$821$	−1.95563e10	−1.23335	−0.616675	−	0.787218i	$-0.711521\pi$
$821$	−1.95563e10	−1.23335	−0.616675	+	0.787218i	$0.711521\pi$
$822$	0	0
$823$	−2.65822e10	−1.66223	−0.831117	−	0.556097i	$-0.812298\pi$
$823$	−2.65822e10	−1.66223	−0.831117	+	0.556097i	$0.812298\pi$
$824$	3.30910e8	0.0206046
$825$	0	0
$826$	−7.19594e8	−0.0444280
$827$	1.30868e10	0.804568	0.402284	−	0.915515i	$-0.368216\pi$
$827$	1.30868e10	0.804568	0.402284	+	0.915515i	$0.368216\pi$
$828$	0	0
$829$	1.93101e10	1.17718	0.588592	−	0.808430i	$-0.299682\pi$
$829$	1.93101e10	1.17718	0.588592	+	0.808430i	$0.299682\pi$
$830$	−7.26540e7	−0.00441048
$831$	0	0
$832$	1.78963e10	1.07729
$833$	4.77219e10	2.86062
$834$	0	0
$835$	2.54352e9	0.151193
$836$	5.50018e9	0.325578
$837$	0	0
$838$	1.43164e9	0.0840387
$839$	−2.78532e10	−1.62820	−0.814101	−	0.580724i	$-0.802770\pi$
$839$	−2.78532e10	−1.62820	−0.814101	+	0.580724i	$0.802770\pi$
$840$	0	0
$841$	3.97151e10	2.30234
$842$	1.09809e9	0.0633937
$843$	0	0
$844$	−2.75959e10	−1.57996
$845$	−1.72128e9	−0.0981415
$846$	0	0
$847$	−3.69829e9	−0.209126
$848$	7.79395e8	0.0438907
$849$	0	0
$850$	−1.00937e9	−0.0563745
$851$	2.51281e9	0.139767
$852$	0	0
$853$	−3.15528e10	−1.74067	−0.870334	−	0.492463i	$-0.836097\pi$
$853$	−3.15528e10	−1.74067	−0.870334	+	0.492463i	$0.836097\pi$
$854$	7.20161e8	0.0395664
$855$	0	0
$856$	−2.84109e9	−0.154820
$857$	1.80328e10	0.978658	0.489329	−	0.872099i	$-0.337242\pi$
$857$	1.80328e10	0.978658	0.489329	+	0.872099i	$0.337242\pi$
$858$	0	0
$859$	1.44022e10	0.775271	0.387636	−	0.921813i	$-0.373292\pi$
$859$	1.44022e10	0.775271	0.387636	+	0.921813i	$0.373292\pi$
$860$	−1.33809e9	−0.0717366
$861$	0	0
$862$	7.03916e8	0.0374322
$863$	3.20522e10	1.69754	0.848770	−	0.528762i	$-0.177344\pi$
$863$	3.20522e10	1.69754	0.848770	+	0.528762i	$0.177344\pi$
$864$	0	0
$865$	−1.19308e10	−0.626777
$866$	−8.43337e8	−0.0441254
$867$	0	0
$868$	1.86302e10	0.966940
$869$	3.01758e10	1.55988
$870$	0	0
$871$	−1.67739e10	−0.860142
$872$	2.31117e9	0.118039
$873$	0	0
$874$	−5.07860e7	−0.00257309
$875$	−3.00555e10	−1.51669
$876$	0	0
$877$	7.00824e8	0.0350841	0.0175420	−	0.999846i	$-0.494416\pi$
$877$	7.00824e8	0.0350841	0.0175420	+	0.999846i	$0.494416\pi$
$878$	1.06664e9	0.0531848
$879$	0	0
$880$	−9.97791e9	−0.493572
$881$	−1.64677e10	−0.811366	−0.405683	−	0.914014i	$-0.632966\pi$
$881$	−1.64677e10	−0.811366	−0.405683	+	0.914014i	$0.632966\pi$
$882$	0	0
$883$	1.63683e9	0.0800093	0.0400047	−	0.999199i	$-0.487263\pi$
$883$	1.63683e9	0.0800093	0.0400047	+	0.999199i	$0.487263\pi$
$884$	2.83605e10	1.38080
$885$	0	0
$886$	1.34478e9	0.0649579
$887$	−1.57468e10	−0.757632	−0.378816	−	0.925472i	$-0.623669\pi$
$887$	−1.57468e10	−0.757632	−0.378816	+	0.925472i	$0.623669\pi$
$888$	0	0
$889$	4.35398e10	2.07840
$890$	1.12097e9	0.0533000
$891$	0	0
$892$	2.46391e10	1.16238
$893$	6.48237e9	0.304617
$894$	0	0
$895$	−1.58245e9	−0.0737819
$896$	8.88681e9	0.412732
$897$	0	0
$898$	1.60394e9	0.0739131
$899$	2.12410e10	0.975027
$900$	0	0
$901$	1.22684e9	0.0558794
$902$	−4.10169e8	−0.0186097
$903$	0	0
$904$	−2.45362e9	−0.110463
$905$	1.88675e10	0.846143
$906$	0	0
$907$	2.93771e10	1.30733	0.653663	−	0.756786i	$-0.273231\pi$
$907$	2.93771e10	1.30733	0.653663	+	0.756786i	$0.273231\pi$
$908$	1.14014e10	0.505428
$909$	0	0
$910$	−1.22685e9	−0.0539695
$911$	−2.70404e10	−1.18495	−0.592474	−	0.805590i	$-0.701849\pi$
$911$	−2.70404e10	−1.18495	−0.592474	+	0.805590i	$0.701849\pi$
$912$	0	0
$913$	3.94464e9	0.171538
$914$	5.72978e8	0.0248214
$915$	0	0
$916$	−3.91090e10	−1.68129
$917$	5.02632e10	2.15257
$918$	0	0
$919$	−4.08926e10	−1.73796	−0.868981	−	0.494845i	$-0.835225\pi$
$919$	−4.08926e10	−1.73796	−0.868981	+	0.494845i	$0.835225\pi$
$920$	1.85186e8	0.00784061
$921$	0	0
$922$	−1.37647e9	−0.0578372
$923$	1.03747e10	0.434278
$924$	0	0
$925$	−1.80847e10	−0.751302
$926$	1.02239e9	0.0423133
$927$	0	0
$928$	7.60338e9	0.312312
$929$	3.00278e10	1.22876	0.614381	−	0.789009i	$-0.289406\pi$
$929$	3.00278e10	1.22876	0.614381	+	0.789009i	$0.289406\pi$
$930$	0	0
$931$	−1.72808e10	−0.701845
$932$	8.55518e8	0.0346157
$933$	0	0
$934$	6.60915e8	0.0265419
$935$	−1.57062e10	−0.628391
$936$	0	0
$937$	4.95682e9	0.196841	0.0984203	−	0.995145i	$-0.468621\pi$
$937$	4.95682e9	0.196841	0.0984203	+	0.995145i	$0.468621\pi$
$938$	−2.05811e9	−0.0814250
$939$	0	0
$940$	−1.17990e10	−0.463339
$941$	−4.50316e10	−1.76179	−0.880893	−	0.473315i	$-0.843057\pi$
$941$	−4.50316e10	−1.76179	−0.880893	+	0.473315i	$0.843057\pi$
$942$	0	0
$943$	−1.14008e9	−0.0442737
$944$	1.09274e10	0.422782
$945$	0	0
$946$	−2.41338e8	−0.00926846
$947$	−2.59153e10	−0.991587	−0.495794	−	0.868440i	$-0.665123\pi$
$947$	−2.59153e10	−0.991587	−0.495794	+	0.868440i	$0.665123\pi$
$948$	0	0
$949$	5.25077e10	1.99430
$950$	3.65507e8	0.0138313
$951$	0	0
$952$	6.97107e9	0.261861
$953$	−9.05322e9	−0.338827	−0.169413	−	0.985545i	$-0.554187\pi$
$953$	−9.05322e9	−0.338827	−0.169413	+	0.985545i	$0.554187\pi$
$954$	0	0
$955$	2.68039e8	0.00995833
$956$	6.67102e9	0.246939
$957$	0	0
$958$	1.75799e9	0.0646008
$959$	−4.48257e10	−1.64120
$960$	0	0
$961$	−1.95923e10	−0.712120
$962$	−1.68799e9	−0.0611303
$963$	0	0
$964$	−1.15041e10	−0.413604
$965$	−6.58817e9	−0.236004
$966$	0	0
$967$	2.45025e10	0.871401	0.435700	−	0.900092i	$-0.356501\pi$
$967$	2.45025e10	0.871401	0.435700	+	0.900092i	$0.356501\pi$
$968$	−3.74995e8	−0.0132881
$969$	0	0
$970$	−823610.	−2.89748e−5 0
$971$	−1.14727e10	−0.402161	−0.201080	−	0.979575i	$-0.564445\pi$
$971$	−1.14727e10	−0.402161	−0.201080	+	0.979575i	$0.564445\pi$
$972$	0	0
$973$	6.85297e9	0.238498
$974$	−3.46283e8	−0.0120081
$975$	0	0
$976$	−1.09360e10	−0.376518
$977$	−8.81945e8	−0.0302559	−0.0151280	−	0.999886i	$-0.504816\pi$
$977$	−8.81945e8	−0.0302559	−0.0151280	+	0.999886i	$0.504816\pi$
$978$	0	0
$979$	−6.08612e10	−2.07301
$980$	3.14541e10	1.06755
$981$	0	0
$982$	−2.21559e9	−0.0746619
$983$	1.78172e9	0.0598277	0.0299138	−	0.999552i	$-0.490477\pi$
$983$	1.78172e9	0.0598277	0.0299138	+	0.999552i	$0.490477\pi$
$984$	0	0
$985$	2.24134e10	0.747277
$986$	3.96740e9	0.131806
$987$	0	0
$988$	−1.02698e10	−0.338776
$989$	−6.70812e8	−0.0220503
$990$	0	0
$991$	−1.00559e10	−0.328220	−0.164110	−	0.986442i	$-0.552475\pi$
$991$	−1.00559e10	−0.328220	−0.164110	+	0.986442i	$0.552475\pi$
$992$	2.83514e9	0.0922112
$993$	0	0
$994$	1.27294e9	0.0411108
$995$	4.66158e9	0.150021
$996$	0	0
$997$	−3.03522e8	−0.00969968	−0.00484984	−	0.999988i	$-0.501544\pi$
$997$	−3.03522e8	−0.00969968	−0.00484984	+	0.999988i	$0.501544\pi$
$998$	−6.85972e8	−0.0218449
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	387.8.a.e.1.7		13
3.2	odd	2		129.8.a.c.1.7	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
129.8.a.c.1.7	✓	13	3.2	odd	2
387.8.a.e.1.7		13	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	387.a (trivial)

\(f(q)\)	\(=\)	\(q+0.651000 q^{2} -127.576 q^{4} -131.920 q^{5} -1640.88 q^{7} -166.380 q^{8} +O(q^{10})\)	\(q+0.651000 q^{2} -127.576 q^{4} -131.920 q^{5} -1640.88 q^{7} -166.380 q^{8} -85.8799 q^{10} +4662.73 q^{11} -8706.11 q^{13} -1068.21 q^{14} +16221.4 q^{16} +25534.1 q^{17} -9246.29 q^{19} +16829.8 q^{20} +3035.44 q^{22} +8437.14 q^{23} -60722.1 q^{25} -5667.68 q^{26} +209337. q^{28} +238673. q^{29} +88996.3 q^{31} +31856.8 q^{32} +16622.7 q^{34} +216465. q^{35} +297827. q^{37} -6019.34 q^{38} +21948.9 q^{40} -135127. q^{41} -79507.0 q^{43} -594853. q^{44} +5492.58 q^{46} -701078. q^{47} +1.86895e6 q^{49} -39530.1 q^{50} +1.11069e6 q^{52} +48047.2 q^{53} -615106. q^{55} +273010. q^{56} +155376. q^{58} +673642. q^{59} -674173. q^{61} +57936.6 q^{62} -2.05561e6 q^{64} +1.14851e6 q^{65} +1.92668e6 q^{67} -3.25754e6 q^{68} +140919. q^{70} -1.19165e6 q^{71} -6.03113e6 q^{73} +193885. q^{74} +1.17961e6 q^{76} -7.65098e6 q^{77} +6.47171e6 q^{79} -2.13993e6 q^{80} -87967.6 q^{82} +845995. q^{83} -3.36845e6 q^{85} -51759.1 q^{86} -775785. q^{88} -1.30527e7 q^{89} +1.42857e7 q^{91} -1.07638e6 q^{92} -456402. q^{94} +1.21977e6 q^{95} +9590.25 q^{97} +1.21669e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 9 q^{2} + 947 q^{4} + 266 q^{5} + 6 q^{7} - 3465 q^{8}+O(q^{10}) \) 13 * q - 9 * q^2 + 947 * q^4 + 266 * q^5 + 6 * q^7 - 3465 * q^8	\( 13 q - 9 q^{2} + 947 q^{4} + 266 q^{5} + 6 q^{7} - 3465 q^{8} - 104 q^{10} - 14218 q^{11} - 7644 q^{13} + 8970 q^{14} + 47811 q^{16} + 48156 q^{17} - 64464 q^{19} + 57914 q^{20} + 26850 q^{22} - 150260 q^{23} + 260915 q^{25} - 74694 q^{26} - 45608 q^{28} - 229626 q^{29} + 511518 q^{31} - 475449 q^{32} - 927790 q^{34} - 204172 q^{35} - 589576 q^{37} - 33540 q^{38} + 55316 q^{40} - 1656956 q^{41} - 1033591 q^{43} - 6317740 q^{44} + 5371478 q^{46} - 3197146 q^{47} + 3545993 q^{49} - 12234087 q^{50} + 5476920 q^{52} - 2301906 q^{53} + 1638708 q^{55} - 9569574 q^{56} + 6308936 q^{58} - 594912 q^{59} + 641576 q^{61} - 16711512 q^{62} + 15392259 q^{64} - 3394176 q^{65} + 4046258 q^{67} - 8031714 q^{68} + 11589246 q^{70} - 5506996 q^{71} + 763246 q^{73} - 11316924 q^{74} - 14274670 q^{76} - 3672676 q^{77} + 12051704 q^{79} - 785638 q^{80} + 5618894 q^{82} + 4951914 q^{83} - 11071848 q^{85} + 715563 q^{86} + 13455806 q^{88} - 20662594 q^{89} - 13410744 q^{91} - 36021878 q^{92} - 11411628 q^{94} + 5494840 q^{95} + 6095056 q^{97} + 388467 q^{98}+O(q^{100}) \) 13 * q - 9 * q^2 + 947 * q^4 + 266 * q^5 + 6 * q^7 - 3465 * q^8 - 104 * q^10 - 14218 * q^11 - 7644 * q^13 + 8970 * q^14 + 47811 * q^16 + 48156 * q^17 - 64464 * q^19 + 57914 * q^20 + 26850 * q^22 - 150260 * q^23 + 260915 * q^25 - 74694 * q^26 - 45608 * q^28 - 229626 * q^29 + 511518 * q^31 - 475449 * q^32 - 927790 * q^34 - 204172 * q^35 - 589576 * q^37 - 33540 * q^38 + 55316 * q^40 - 1656956 * q^41 - 1033591 * q^43 - 6317740 * q^44 + 5371478 * q^46 - 3197146 * q^47 + 3545993 * q^49 - 12234087 * q^50 + 5476920 * q^52 - 2301906 * q^53 + 1638708 * q^55 - 9569574 * q^56 + 6308936 * q^58 - 594912 * q^59 + 641576 * q^61 - 16711512 * q^62 + 15392259 * q^64 - 3394176 * q^65 + 4046258 * q^67 - 8031714 * q^68 + 11589246 * q^70 - 5506996 * q^71 + 763246 * q^73 - 11316924 * q^74 - 14274670 * q^76 - 3672676 * q^77 + 12051704 * q^79 - 785638 * q^80 + 5618894 * q^82 + 4951914 * q^83 - 11071848 * q^85 + 715563 * q^86 + 13455806 * q^88 - 20662594 * q^89 - 13410744 * q^91 - 36021878 * q^92 - 11411628 * q^94 + 5494840 * q^95 + 6095056 * q^97 + 388467 * q^98

Introduction

L-functions

Modular forms

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Database

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