LMFDB - Embedded newform 755.2.a.j.1.15

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [755,2,Mod(1,755)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(755, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("755.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 755 = 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	755.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,-2,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$6.02870535261$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 2 x^{14} - 22 x^{13} + 48 x^{12} + 171 x^{11} - 423 x^{10} - 527 x^{9} + 1641 x^{8} + 400 x^{7} + \cdots + 8 $ x^15 - 2x^14 - 22x^13 + 48x^12 + 171x^11 - 423x^10 - 527x^9 + 1641x^8 + 400x^7 - 2651x^6 + 421x^5 + 1591x^4 - 396x^3 - 288x^2 + 60x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$-2.68739$ of defining polynomial
Character	$\chi$	$=$	755.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.68739 q^{2} -2.24250 q^{3} +5.22206 q^{4} +1.00000 q^{5} -6.02646 q^{6} +4.36035 q^{7} +8.65893 q^{8} +2.02879 q^{9} +2.68739 q^{10} -3.66815 q^{11} -11.7105 q^{12} -1.55634 q^{13} +11.7180 q^{14} -2.24250 q^{15} +12.8258 q^{16} -4.96026 q^{17} +5.45216 q^{18} +4.96247 q^{19} +5.22206 q^{20} -9.77808 q^{21} -9.85776 q^{22} +0.669094 q^{23} -19.4176 q^{24} +1.00000 q^{25} -4.18250 q^{26} +2.17793 q^{27} +22.7700 q^{28} +1.54861 q^{29} -6.02646 q^{30} +0.257397 q^{31} +17.1500 q^{32} +8.22583 q^{33} -13.3301 q^{34} +4.36035 q^{35} +10.5945 q^{36} +7.73344 q^{37} +13.3361 q^{38} +3.49009 q^{39} +8.65893 q^{40} -6.42744 q^{41} -26.2775 q^{42} -10.0061 q^{43} -19.1553 q^{44} +2.02879 q^{45} +1.79812 q^{46} -12.7698 q^{47} -28.7618 q^{48} +12.0127 q^{49} +2.68739 q^{50} +11.1234 q^{51} -8.12731 q^{52} +8.41253 q^{53} +5.85293 q^{54} -3.66815 q^{55} +37.7560 q^{56} -11.1283 q^{57} +4.16171 q^{58} -12.5321 q^{59} -11.7105 q^{60} +2.04436 q^{61} +0.691725 q^{62} +8.84626 q^{63} +20.4372 q^{64} -1.55634 q^{65} +22.1060 q^{66} +5.84404 q^{67} -25.9028 q^{68} -1.50044 q^{69} +11.7180 q^{70} +2.52657 q^{71} +17.5672 q^{72} -9.55113 q^{73} +20.7828 q^{74} -2.24250 q^{75} +25.9143 q^{76} -15.9944 q^{77} +9.37924 q^{78} -14.1668 q^{79} +12.8258 q^{80} -10.9704 q^{81} -17.2730 q^{82} +8.19796 q^{83} -51.0617 q^{84} -4.96026 q^{85} -26.8904 q^{86} -3.47275 q^{87} -31.7623 q^{88} -10.6656 q^{89} +5.45216 q^{90} -6.78620 q^{91} +3.49405 q^{92} -0.577211 q^{93} -34.3173 q^{94} +4.96247 q^{95} -38.4589 q^{96} +8.31172 q^{97} +32.2827 q^{98} -7.44193 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 2 q^{2} + 5 q^{3} + 18 q^{4} + 15 q^{5} - 4 q^{6} + 11 q^{7} + 12 q^{8} + 10 q^{9} - 2 q^{10} + 2 q^{11} + 4 q^{12} + 11 q^{13} - 9 q^{14} + 5 q^{15} + 40 q^{16} + 25 q^{17} - 15 q^{18} - 3 q^{19}+ \cdots - 11 q^{99}+O(q^{100}) $ 15 * q - 2 * q^2 + 5 * q^3 + 18 * q^4 + 15 * q^5 - 4 * q^6 + 11 * q^7 + 12 * q^8 + 10 * q^9 - 2 * q^10 + 2 * q^11 + 4 * q^12 + 11 * q^13 - 9 * q^14 + 5 * q^15 + 40 * q^16 + 25 * q^17 - 15 * q^18 - 3 * q^19 + 18 * q^20 - 11 * q^21 + 13 * q^22 + 8 * q^23 - 21 * q^24 + 15 * q^25 - 45 * q^26 + 20 * q^27 + 29 * q^28 + 24 * q^29 - 4 * q^30 - 11 * q^31 + 27 * q^32 + 26 * q^33 - 11 * q^34 + 11 * q^35 - 22 * q^36 + 35 * q^37 + 2 * q^38 + 11 * q^39 + 12 * q^40 - 20 * q^41 - 17 * q^42 - 13 * q^43 - 32 * q^44 + 10 * q^45 - 5 * q^46 + 6 * q^47 - 11 * q^48 + 40 * q^49 - 2 * q^50 + 23 * q^51 + 27 * q^52 + 38 * q^53 - 30 * q^54 + 2 * q^55 - 36 * q^56 + 4 * q^57 + 6 * q^58 - 15 * q^59 + 4 * q^60 + 33 * q^61 - 13 * q^62 + 2 * q^63 + 74 * q^64 + 11 * q^65 + 8 * q^66 + 24 * q^67 + 8 * q^68 + 13 * q^69 - 9 * q^70 + 19 * q^71 + 56 * q^72 + 4 * q^73 - 28 * q^74 + 5 * q^75 + 19 * q^76 + 14 * q^77 - 15 * q^78 - 4 * q^79 + 40 * q^80 - 13 * q^81 + 33 * q^82 + 19 * q^83 - 71 * q^84 + 25 * q^85 - 59 * q^86 + 49 * q^87 - 48 * q^88 - 35 * q^89 - 15 * q^90 - q^91 + 18 * q^92 + 2 * q^93 - 70 * q^94 - 3 * q^95 - 35 * q^96 + 55 * q^97 - 64 * q^98 - 11 * q^99

Coefficient data

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

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

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

$2$	2.68739	1.90027	0.950136	−	0.311837i	$-0.100944\pi$
$2$	2.68739	1.90027	0.950136	+	0.311837i	$0.100944\pi$
$3$	−2.24250	−1.29471	−0.647353	−	0.762190i	$-0.724124\pi$
$3$	−2.24250	−1.29471	−0.647353	+	0.762190i	$0.724124\pi$
$4$	5.22206	2.61103
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−6.02646	−2.46029
$7$	4.36035	1.64806	0.824029	−	0.566547i	$-0.191721\pi$
$7$	4.36035	1.64806	0.824029	+	0.566547i	$0.191721\pi$
$8$	8.65893	3.06139
$9$	2.02879	0.676265
$10$	2.68739	0.849827
$11$	−3.66815	−1.10599	−0.552995	−	0.833185i	$-0.686515\pi$
$11$	−3.66815	−1.10599	−0.552995	+	0.833185i	$0.686515\pi$
$12$	−11.7105	−3.38052
$13$	−1.55634	−0.431652	−0.215826	−	0.976432i	$-0.569244\pi$
$13$	−1.55634	−0.431652	−0.215826	+	0.976432i	$0.569244\pi$
$14$	11.7180	3.13176
$15$	−2.24250	−0.579010
$16$	12.8258	3.20645
$17$	−4.96026	−1.20304	−0.601520	−	0.798858i	$-0.705438\pi$
$17$	−4.96026	−1.20304	−0.601520	+	0.798858i	$0.705438\pi$
$18$	5.45216	1.28509
$19$	4.96247	1.13847	0.569234	−	0.822175i	$-0.307240\pi$
$19$	4.96247	1.13847	0.569234	+	0.822175i	$0.307240\pi$
$20$	5.22206	1.16769
$21$	−9.77808	−2.13375
$22$	−9.85776	−2.10168
$23$	0.669094	0.139516	0.0697579	−	0.997564i	$-0.477777\pi$
$23$	0.669094	0.139516	0.0697579	+	0.997564i	$0.477777\pi$
$24$	−19.4176	−3.96361
$25$	1.00000	0.200000
$26$	−4.18250	−0.820255
$27$	2.17793	0.419142
$28$	22.7700	4.30313
$29$	1.54861	0.287569	0.143785	−	0.989609i	$-0.454073\pi$
$29$	1.54861	0.287569	0.143785	+	0.989609i	$0.454073\pi$
$30$	−6.02646	−1.10028
$31$	0.257397	0.0462298	0.0231149	−	0.999733i	$-0.492642\pi$
$31$	0.257397	0.0462298	0.0231149	+	0.999733i	$0.492642\pi$
$32$	17.1500	3.03173
$33$	8.22583	1.43193
$34$	−13.3301	−2.28610
$35$	4.36035	0.737034
$36$	10.5945	1.76575
$37$	7.73344	1.27137	0.635685	−	0.771949i	$-0.280718\pi$
$37$	7.73344	1.27137	0.635685	+	0.771949i	$0.280718\pi$
$38$	13.3361	2.16340
$39$	3.49009	0.558862
$40$	8.65893	1.36910
$41$	−6.42744	−1.00380	−0.501899	−	0.864926i	$-0.667365\pi$
$41$	−6.42744	−1.00380	−0.501899	+	0.864926i	$0.667365\pi$
$42$	−26.2775	−4.05471
$43$	−10.0061	−1.52592	−0.762961	−	0.646444i	$-0.776255\pi$
$43$	−10.0061	−1.52592	−0.762961	+	0.646444i	$0.776255\pi$
$44$	−19.1553	−2.88777
$45$	2.02879	0.302435
$46$	1.79812	0.265118
$47$	−12.7698	−1.86266	−0.931330	−	0.364175i	$-0.881351\pi$
$47$	−12.7698	−1.86266	−0.931330	+	0.364175i	$0.881351\pi$
$48$	−28.7618	−4.15141
$49$	12.0127	1.71610
$50$	2.68739	0.380054
$51$	11.1234	1.55758
$52$	−8.12731	−1.12706
$53$	8.41253	1.15555	0.577775	−	0.816196i	$-0.303921\pi$
$53$	8.41253	1.15555	0.577775	+	0.816196i	$0.303921\pi$
$54$	5.85293	0.796483
$55$	−3.66815	−0.494614
$56$	37.7560	5.04536
$57$	−11.1283	−1.47398
$58$	4.16171	0.546459
$59$	−12.5321	−1.63154	−0.815769	−	0.578377i	$-0.803686\pi$
$59$	−12.5321	−1.63154	−0.815769	+	0.578377i	$0.803686\pi$
$60$	−11.7105	−1.51181
$61$	2.04436	0.261754	0.130877	−	0.991399i	$-0.458221\pi$
$61$	2.04436	0.261754	0.130877	+	0.991399i	$0.458221\pi$
$62$	0.691725	0.0878491
$63$	8.84626	1.11452
$64$	20.4372	2.55465
$65$	−1.55634	−0.193040
$66$	22.1060	2.72106
$67$	5.84404	0.713963	0.356981	−	0.934111i	$-0.383806\pi$
$67$	5.84404	0.713963	0.356981	+	0.934111i	$0.383806\pi$
$68$	−25.9028	−3.14117
$69$	−1.50044	−0.180632
$70$	11.7180	1.40056
$71$	2.52657	0.299849	0.149924	−	0.988697i	$-0.452097\pi$
$71$	2.52657	0.299849	0.149924	+	0.988697i	$0.452097\pi$
$72$	17.5672	2.07031
$73$	−9.55113	−1.11788	−0.558938	−	0.829210i	$-0.688791\pi$
$73$	−9.55113	−1.11788	−0.558938	+	0.829210i	$0.688791\pi$
$74$	20.7828	2.41595
$75$	−2.24250	−0.258941
$76$	25.9143	2.97258
$77$	−15.9944	−1.82274
$78$	9.37924	1.06199
$79$	−14.1668	−1.59389	−0.796947	−	0.604049i	$-0.793553\pi$
$79$	−14.1668	−1.59389	−0.796947	+	0.604049i	$0.793553\pi$
$80$	12.8258	1.43397
$81$	−10.9704	−1.21893
$82$	−17.2730	−1.90749
$83$	8.19796	0.899843	0.449921	−	0.893068i	$-0.351452\pi$
$83$	8.19796	0.899843	0.449921	+	0.893068i	$0.351452\pi$
$84$	−51.0617	−5.57129
$85$	−4.96026	−0.538016
$86$	−26.8904	−2.89967
$87$	−3.47275	−0.372318
$88$	−31.7623	−3.38587
$89$	−10.6656	−1.13055	−0.565276	−	0.824902i	$-0.691230\pi$
$89$	−10.6656	−1.13055	−0.565276	+	0.824902i	$0.691230\pi$
$90$	5.45216	0.574708
$91$	−6.78620	−0.711387
$92$	3.49405	0.364280
$93$	−0.577211	−0.0598540
$94$	−34.3173	−3.53956
$95$	4.96247	0.509139
$96$	−38.4589	−3.92520
$97$	8.31172	0.843927	0.421964	−	0.906613i	$-0.361341\pi$
$97$	8.31172	0.843927	0.421964	+	0.906613i	$0.361341\pi$
$98$	32.2827	3.26105
$99$	−7.44193	−0.747942
$100$	5.22206	0.522206
$101$	−5.48623	−0.545900	−0.272950	−	0.962028i	$-0.587999\pi$
$101$	−5.48623	−0.545900	−0.272950	+	0.962028i	$0.587999\pi$
$102$	29.8928	2.95983
$103$	−5.86789	−0.578180	−0.289090	−	0.957302i	$-0.593353\pi$
$103$	−5.86789	−0.578180	−0.289090	+	0.957302i	$0.593353\pi$
$104$	−13.4763	−1.32146
$105$	−9.77808	−0.954243
$106$	22.6078	2.19586
$107$	−11.3157	−1.09393	−0.546965	−	0.837155i	$-0.684217\pi$
$107$	−11.3157	−1.09393	−0.546965	+	0.837155i	$0.684217\pi$
$108$	11.3733	1.09439
$109$	20.5560	1.96891	0.984453	−	0.175647i	$-0.0562017\pi$
$109$	20.5560	1.96891	0.984453	+	0.175647i	$0.0562017\pi$
$110$	−9.85776	−0.939900
$111$	−17.3422	−1.64605
$112$	55.9250	5.28441
$113$	7.52896	0.708265	0.354132	−	0.935195i	$-0.384776\pi$
$113$	7.52896	0.708265	0.354132	+	0.935195i	$0.384776\pi$
$114$	−29.9061	−2.80097
$115$	0.669094	0.0623933
$116$	8.08692	0.750852
$117$	−3.15750	−0.291911
$118$	−33.6786	−3.10037
$119$	−21.6285	−1.98268
$120$	−19.4176	−1.77258
$121$	2.45535	0.223214
$122$	5.49400	0.497404
$123$	14.4135	1.29962
$124$	1.34414	0.120707
$125$	1.00000	0.0894427
$126$	23.7733	2.11790
$127$	−6.42621	−0.570234	−0.285117	−	0.958493i	$-0.592032\pi$
$127$	−6.42621	−0.570234	−0.285117	+	0.958493i	$0.592032\pi$
$128$	20.6227	1.82281
$129$	22.4388	1.97562
$130$	−4.18250	−0.366829
$131$	−15.4958	−1.35387	−0.676937	−	0.736041i	$-0.736693\pi$
$131$	−15.4958	−1.35387	−0.676937	+	0.736041i	$0.736693\pi$
$132$	42.9558	3.73882
$133$	21.6381	1.87626
$134$	15.7052	1.35672
$135$	2.17793	0.187446
$136$	−42.9505	−3.68298
$137$	17.7263	1.51446	0.757230	−	0.653148i	$-0.226552\pi$
$137$	17.7263	1.51446	0.757230	+	0.653148i	$0.226552\pi$
$138$	−4.03227	−0.343250
$139$	−13.9341	−1.18188	−0.590938	−	0.806717i	$-0.701242\pi$
$139$	−13.9341	−1.18188	−0.590938	+	0.806717i	$0.701242\pi$
$140$	22.7700	1.92442
$141$	28.6362	2.41160
$142$	6.78988	0.569794
$143$	5.70890	0.477402
$144$	26.0209	2.16841
$145$	1.54861	0.128605
$146$	−25.6676	−2.12427
$147$	−26.9384	−2.22184
$148$	40.3845	3.31958
$149$	1.32148	0.108260	0.0541300	−	0.998534i	$-0.482761\pi$
$149$	1.32148	0.108260	0.0541300	+	0.998534i	$0.482761\pi$
$150$	−6.02646	−0.492059
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	42.9697	3.48530
$153$	−10.0633	−0.813574
$154$	−42.9833	−3.46369
$155$	0.257397	0.0206746
$156$	18.2255	1.45921
$157$	15.0978	1.20494	0.602469	−	0.798142i	$-0.294184\pi$
$157$	15.0978	1.20494	0.602469	+	0.798142i	$0.294184\pi$
$158$	−38.0718	−3.02883
$159$	−18.8651	−1.49610
$160$	17.1500	1.35583
$161$	2.91749	0.229930
$162$	−29.4817	−2.31630
$163$	19.4034	1.51979	0.759896	−	0.650044i	$-0.225250\pi$
$163$	19.4034	1.51979	0.759896	+	0.650044i	$0.225250\pi$
$164$	−33.5645	−2.62094
$165$	8.22583	0.640380
$166$	22.0311	1.70995
$167$	−1.65789	−0.128292	−0.0641458	−	0.997941i	$-0.520432\pi$
$167$	−1.65789	−0.128292	−0.0641458	+	0.997941i	$0.520432\pi$
$168$	−84.6677	−6.53226
$169$	−10.5778	−0.813677
$170$	−13.3301	−1.02238
$171$	10.0678	0.769907
$172$	−52.2527	−3.98423
$173$	7.82396	0.594845	0.297422	−	0.954746i	$-0.403873\pi$
$173$	7.82396	0.594845	0.297422	+	0.954746i	$0.403873\pi$
$174$	−9.33263	−0.707505
$175$	4.36035	0.329612
$176$	−47.0470	−3.54630
$177$	28.1032	2.11236
$178$	−28.6627	−2.14836
$179$	11.9486	0.893077	0.446539	−	0.894764i	$-0.352657\pi$
$179$	11.9486	0.893077	0.446539	+	0.894764i	$0.352657\pi$
$180$	10.5945	0.789667
$181$	−8.73869	−0.649542	−0.324771	−	0.945793i	$-0.605287\pi$
$181$	−8.73869	−0.649542	−0.324771	+	0.945793i	$0.605287\pi$
$182$	−18.2372	−1.35183
$183$	−4.58448	−0.338895
$184$	5.79364	0.427113
$185$	7.73344	0.568574
$186$	−1.55119	−0.113739
$187$	18.1950	1.33055
$188$	−66.6844	−4.86346
$189$	9.49652	0.690770
$190$	13.3361	0.967502
$191$	−16.6183	−1.20246	−0.601230	−	0.799076i	$-0.705323\pi$
$191$	−16.6183	−1.20246	−0.601230	+	0.799076i	$0.705323\pi$
$192$	−45.8304	−3.30753
$193$	1.10778	0.0797396	0.0398698	−	0.999205i	$-0.487306\pi$
$193$	1.10778	0.0797396	0.0398698	+	0.999205i	$0.487306\pi$
$194$	22.3368	1.60369
$195$	3.49009	0.249931
$196$	62.7309	4.48078
$197$	1.15265	0.0821226	0.0410613	−	0.999157i	$-0.486926\pi$
$197$	1.15265	0.0821226	0.0410613	+	0.999157i	$0.486926\pi$
$198$	−19.9994	−1.42129
$199$	27.6806	1.96222	0.981111	−	0.193444i	$-0.0619657\pi$
$199$	27.6806	1.96222	0.981111	+	0.193444i	$0.0619657\pi$
$200$	8.65893	0.612279
$201$	−13.1052	−0.924372
$202$	−14.7436	−1.03736
$203$	6.75248	0.473931
$204$	58.0869	4.06690
$205$	−6.42744	−0.448912
$206$	−15.7693	−1.09870
$207$	1.35745	0.0943496
$208$	−19.9613	−1.38407
$209$	−18.2031	−1.25914
$210$	−26.2775	−1.81332
$211$	−0.168136	−0.0115750	−0.00578748	−	0.999983i	$-0.501842\pi$
$211$	−0.168136	−0.0115750	−0.00578748	+	0.999983i	$0.501842\pi$
$212$	43.9308	3.01718
$213$	−5.66583	−0.388216
$214$	−30.4097	−2.07876
$215$	−10.0061	−0.682413
$216$	18.8585	1.28316
$217$	1.12234	0.0761894
$218$	55.2419	3.74146
$219$	21.4184	1.44732
$220$	−19.1553	−1.29145
$221$	7.71986	0.519294
$222$	−46.6053	−3.12794
$223$	−17.3112	−1.15924	−0.579622	−	0.814885i	$-0.696800\pi$
$223$	−17.3112	−1.15924	−0.579622	+	0.814885i	$0.696800\pi$
$224$	74.7802	4.99646
$225$	2.02879	0.135253
$226$	20.2332	1.34589
$227$	21.3695	1.41834	0.709172	−	0.705035i	$-0.249069\pi$
$227$	21.3695	1.41834	0.709172	+	0.705035i	$0.249069\pi$
$228$	−58.1128	−3.84861
$229$	−4.63441	−0.306250	−0.153125	−	0.988207i	$-0.548934\pi$
$229$	−4.63441	−0.306250	−0.153125	+	0.988207i	$0.548934\pi$
$230$	1.79812	0.118564
$231$	35.8675	2.35991
$232$	13.4093	0.880363
$233$	20.1793	1.32199	0.660996	−	0.750389i	$-0.270134\pi$
$233$	20.1793	1.32199	0.660996	+	0.750389i	$0.270134\pi$
$234$	−8.48543	−0.554710
$235$	−12.7698	−0.833007
$236$	−65.4433	−4.26000
$237$	31.7691	2.06363
$238$	−58.1242	−3.76763
$239$	−9.21562	−0.596109	−0.298054	−	0.954549i	$-0.596338\pi$
$239$	−9.21562	−0.596109	−0.298054	+	0.954549i	$0.596338\pi$
$240$	−28.7618	−1.85657
$241$	−5.96336	−0.384134	−0.192067	−	0.981382i	$-0.561519\pi$
$241$	−5.96336	−0.384134	−0.192067	+	0.981382i	$0.561519\pi$
$242$	6.59849	0.424167
$243$	18.0673	1.15902
$244$	10.6758	0.683448
$245$	12.0127	0.767462
$246$	38.7347	2.46964
$247$	−7.72330	−0.491422
$248$	2.22878	0.141528
$249$	−18.3839	−1.16503
$250$	2.68739	0.169965
$251$	1.43579	0.0906263	0.0453132	−	0.998973i	$-0.485571\pi$
$251$	1.43579	0.0906263	0.0453132	+	0.998973i	$0.485571\pi$
$252$	46.1957	2.91006
$253$	−2.45434	−0.154303
$254$	−17.2697	−1.08360
$255$	11.1234	0.696572
$256$	14.5468	0.909176
$257$	31.5332	1.96698	0.983492	−	0.180951i	$-0.0579175\pi$
$257$	31.5332	1.96698	0.983492	+	0.180951i	$0.0579175\pi$
$258$	60.3017	3.75422
$259$	33.7205	2.09529
$260$	−8.12731	−0.504035
$261$	3.14181	0.194473
$262$	−41.6432	−2.57273
$263$	−3.66593	−0.226051	−0.113025	−	0.993592i	$-0.536054\pi$
$263$	−3.66593	−0.226051	−0.113025	+	0.993592i	$0.536054\pi$
$264$	71.2268	4.38371
$265$	8.41253	0.516778
$266$	58.1501	3.56541
$267$	23.9176	1.46373
$268$	30.5179	1.86418
$269$	26.4543	1.61295	0.806473	−	0.591271i	$-0.201374\pi$
$269$	26.4543	1.61295	0.806473	+	0.591271i	$0.201374\pi$
$270$	5.85293	0.356198
$271$	19.6830	1.19566	0.597829	−	0.801624i	$-0.296030\pi$
$271$	19.6830	1.19566	0.597829	+	0.801624i	$0.296030\pi$
$272$	−63.6193	−3.85748
$273$	15.2180	0.921038
$274$	47.6375	2.87788
$275$	−3.66815	−0.221198
$276$	−7.83539	−0.471635
$277$	−5.98163	−0.359401	−0.179701	−	0.983721i	$-0.557513\pi$
$277$	−5.98163	−0.359401	−0.179701	+	0.983721i	$0.557513\pi$
$278$	−37.4464	−2.24588
$279$	0.522205	0.0312636
$280$	37.7560	2.25635
$281$	−12.0703	−0.720055	−0.360028	−	0.932942i	$-0.617233\pi$
$281$	−12.0703	−0.720055	−0.360028	+	0.932942i	$0.617233\pi$
$282$	76.9565	4.58269
$283$	18.1059	1.07629	0.538143	−	0.842854i	$-0.319126\pi$
$283$	18.1059	1.07629	0.538143	+	0.842854i	$0.319126\pi$
$284$	13.1939	0.782914
$285$	−11.1283	−0.659185
$286$	15.3420	0.907194
$287$	−28.0259	−1.65432
$288$	34.7939	2.05025
$289$	7.60418	0.447304
$290$	4.16171	0.244384
$291$	−18.6390	−1.09264
$292$	−49.8766	−2.91881
$293$	−6.10149	−0.356453	−0.178227	−	0.983989i	$-0.557036\pi$
$293$	−6.10149	−0.356453	−0.178227	+	0.983989i	$0.557036\pi$
$294$	−72.3940	−4.22210
$295$	−12.5321	−0.729646
$296$	66.9633	3.89216
$297$	−7.98897	−0.463567
$298$	3.55134	0.205723
$299$	−1.04134	−0.0602222
$300$	−11.7105	−0.676104
$301$	−43.6303	−2.51481
$302$	−2.68739	−0.154642
$303$	12.3029	0.706781
$304$	63.6476	3.65044
$305$	2.04436	0.117060
$306$	−27.0441	−1.54601
$307$	−12.7488	−0.727610	−0.363805	−	0.931475i	$-0.618523\pi$
$307$	−12.7488	−0.727610	−0.363805	+	0.931475i	$0.618523\pi$
$308$	−83.5240	−4.75922
$309$	13.1587	0.748574
$310$	0.691725	0.0392873
$311$	16.9795	0.962820	0.481410	−	0.876495i	$-0.340125\pi$
$311$	16.9795	0.962820	0.481410	+	0.876495i	$0.340125\pi$
$312$	30.2205	1.71090
$313$	19.1732	1.08373	0.541866	−	0.840465i	$-0.317718\pi$
$313$	19.1732	1.08373	0.541866	+	0.840465i	$0.317718\pi$
$314$	40.5738	2.28971
$315$	8.84626	0.498430
$316$	−73.9801	−4.16171
$317$	13.5363	0.760273	0.380137	−	0.924930i	$-0.375877\pi$
$317$	13.5363	0.760273	0.380137	+	0.924930i	$0.375877\pi$
$318$	−50.6978	−2.84299
$319$	−5.68053	−0.318049
$320$	20.4372	1.14248
$321$	25.3754	1.41632
$322$	7.84042	0.436929
$323$	−24.6151	−1.36962
$324$	−57.2880	−3.18266
$325$	−1.55634	−0.0863303
$326$	52.1445	2.88802
$327$	−46.0967	−2.54916
$328$	−55.6547	−3.07302
$329$	−55.6807	−3.06977
$330$	22.1060	1.21689
$331$	16.2226	0.891676	0.445838	−	0.895114i	$-0.352906\pi$
$331$	16.2226	0.891676	0.445838	+	0.895114i	$0.352906\pi$
$332$	42.8102	2.34952
$333$	15.6896	0.859782
$334$	−4.45540	−0.243789
$335$	5.84404	0.319294
$336$	−125.412	−6.84176
$337$	−30.6411	−1.66913	−0.834563	−	0.550912i	$-0.814280\pi$
$337$	−30.6411	−1.66913	−0.834563	+	0.550912i	$0.814280\pi$
$338$	−28.4267	−1.54621
$339$	−16.8837	−0.916995
$340$	−25.9028	−1.40478
$341$	−0.944170	−0.0511297
$342$	27.0562	1.46303
$343$	21.8571	1.18017
$344$	−86.6425	−4.67145
$345$	−1.50044	−0.0807810
$346$	21.0260	1.13037
$347$	9.53223	0.511717	0.255858	−	0.966714i	$-0.417642\pi$
$347$	9.53223	0.511717	0.255858	+	0.966714i	$0.417642\pi$
$348$	−18.1349	−0.972133
$349$	3.24465	0.173682	0.0868410	−	0.996222i	$-0.472323\pi$
$349$	3.24465	0.173682	0.0868410	+	0.996222i	$0.472323\pi$
$350$	11.7180	0.626352
$351$	−3.38960	−0.180923
$352$	−62.9090	−3.35306
$353$	22.9666	1.22239	0.611194	−	0.791481i	$-0.290690\pi$
$353$	22.9666	1.22239	0.611194	+	0.791481i	$0.290690\pi$
$354$	75.5242	4.01406
$355$	2.52657	0.134096
$356$	−55.6965	−2.95191
$357$	48.5018	2.56699
$358$	32.1104	1.69709
$359$	−23.1190	−1.22018	−0.610088	−	0.792334i	$-0.708866\pi$
$359$	−23.1190	−1.22018	−0.610088	+	0.792334i	$0.708866\pi$
$360$	17.5672	0.925872
$361$	5.62612	0.296112
$362$	−23.4843	−1.23431
$363$	−5.50613	−0.288997
$364$	−35.4380	−1.85745
$365$	−9.55113	−0.499929
$366$	−12.3203	−0.643992
$367$	−15.2229	−0.794630	−0.397315	−	0.917682i	$-0.630058\pi$
$367$	−15.2229	−0.794630	−0.397315	+	0.917682i	$0.630058\pi$
$368$	8.58166	0.447350
$369$	−13.0400	−0.678833
$370$	20.7828	1.08044
$371$	36.6816	1.90441
$372$	−3.01423	−0.156281
$373$	−11.9013	−0.616228	−0.308114	−	0.951349i	$-0.599698\pi$
$373$	−11.9013	−0.616228	−0.308114	+	0.951349i	$0.599698\pi$
$374$	48.8970	2.52841
$375$	−2.24250	−0.115802
$376$	−110.572	−5.70234
$377$	−2.41016	−0.124130
$378$	25.5209	1.31265
$379$	−4.38825	−0.225409	−0.112705	−	0.993629i	$-0.535951\pi$
$379$	−4.38825	−0.225409	−0.112705	+	0.993629i	$0.535951\pi$
$380$	25.9143	1.32938
$381$	14.4108	0.738286
$382$	−44.6599	−2.28500
$383$	7.23777	0.369833	0.184916	−	0.982754i	$-0.440799\pi$
$383$	7.23777	0.369833	0.184916	+	0.982754i	$0.440799\pi$
$384$	−46.2464	−2.36000
$385$	−15.9944	−0.815152
$386$	2.97703	0.151527
$387$	−20.3004	−1.03193
$388$	43.4043	2.20352
$389$	7.85847	0.398440	0.199220	−	0.979955i	$-0.436159\pi$
$389$	7.85847	0.398440	0.199220	+	0.979955i	$0.436159\pi$
$390$	9.37924	0.474936
$391$	−3.31888	−0.167843
$392$	104.017	5.25365
$393$	34.7493	1.75287
$394$	3.09761	0.156055
$395$	−14.1668	−0.712811
$396$	−38.8622	−1.95290
$397$	−2.13209	−0.107006	−0.0535032	−	0.998568i	$-0.517039\pi$
$397$	−2.13209	−0.107006	−0.0535032	+	0.998568i	$0.517039\pi$
$398$	74.3884	3.72876
$399$	−48.5234	−2.42921
$400$	12.8258	0.641290
$401$	−12.4391	−0.621177	−0.310588	−	0.950545i	$-0.600526\pi$
$401$	−12.4391	−0.621177	−0.310588	+	0.950545i	$0.600526\pi$
$402$	−35.2189	−1.75656
$403$	−0.400597	−0.0199552
$404$	−28.6494	−1.42536
$405$	−10.9704	−0.545122
$406$	18.1465	0.900597
$407$	−28.3674	−1.40612
$408$	96.3165	4.76838
$409$	−15.0838	−0.745846	−0.372923	−	0.927862i	$-0.621644\pi$
$409$	−15.0838	−0.745846	−0.372923	+	0.927862i	$0.621644\pi$
$410$	−17.2730	−0.853054
$411$	−39.7512	−1.96078
$412$	−30.6425	−1.50965
$413$	−54.6443	−2.68887
$414$	3.64801	0.179290
$415$	8.19796	0.402422
$416$	−26.6913	−1.30865
$417$	31.2472	1.53018
$418$	−48.9188	−2.39270
$419$	24.5966	1.20162	0.600812	−	0.799390i	$-0.294844\pi$
$419$	24.5966	1.20162	0.600812	+	0.799390i	$0.294844\pi$
$420$	−51.0617	−2.49156
$421$	−31.2859	−1.52478	−0.762390	−	0.647118i	$-0.775974\pi$
$421$	−31.2859	−1.52478	−0.762390	+	0.647118i	$0.775974\pi$
$422$	−0.451847	−0.0219956
$423$	−25.9072	−1.25965
$424$	72.8435	3.53759
$425$	−4.96026	−0.240608
$426$	−15.2263	−0.737716
$427$	8.91415	0.431386
$428$	−59.0913	−2.85629
$429$	−12.8022	−0.618096
$430$	−26.8904	−1.29677
$431$	11.2799	0.543333	0.271666	−	0.962392i	$-0.412425\pi$
$431$	11.2799	0.543333	0.271666	+	0.962392i	$0.412425\pi$
$432$	27.9336	1.34396
$433$	28.3466	1.36225	0.681126	−	0.732166i	$-0.261490\pi$
$433$	28.3466	1.36225	0.681126	+	0.732166i	$0.261490\pi$
$434$	3.01616	0.144781
$435$	−3.47275	−0.166506
$436$	107.345	5.14087
$437$	3.32036	0.158834
$438$	57.5595	2.75030
$439$	−30.6946	−1.46497	−0.732487	−	0.680781i	$-0.761641\pi$
$439$	−30.6946	−1.46497	−0.732487	+	0.680781i	$0.761641\pi$
$440$	−31.7623	−1.51421
$441$	24.3713	1.16054
$442$	20.7463	0.986800
$443$	18.8874	0.897367	0.448683	−	0.893691i	$-0.351893\pi$
$443$	18.8874	0.897367	0.448683	+	0.893691i	$0.351893\pi$
$444$	−90.5621	−4.29789
$445$	−10.6656	−0.505599
$446$	−46.5220	−2.20288
$447$	−2.96342	−0.140165
$448$	89.1135	4.21022
$449$	32.8560	1.55057	0.775286	−	0.631610i	$-0.217606\pi$
$449$	32.8560	1.55057	0.775286	+	0.631610i	$0.217606\pi$
$450$	5.45216	0.257017
$451$	23.5768	1.11019
$452$	39.3167	1.84930
$453$	2.24250	0.105362
$454$	57.4282	2.69524
$455$	−6.78620	−0.318142
$456$	−96.3594	−4.51244
$457$	0.971427	0.0454414	0.0227207	−	0.999742i	$-0.492767\pi$
$457$	0.971427	0.0454414	0.0227207	+	0.999742i	$0.492767\pi$
$458$	−12.4545	−0.581959
$459$	−10.8031	−0.504244
$460$	3.49405	0.162911
$461$	−39.7472	−1.85121	−0.925607	−	0.378487i	$-0.876445\pi$
$461$	−39.7472	−1.85121	−0.925607	+	0.378487i	$0.876445\pi$
$462$	96.3899	4.48447
$463$	24.9119	1.15775	0.578876	−	0.815416i	$-0.303491\pi$
$463$	24.9119	1.15775	0.578876	+	0.815416i	$0.303491\pi$
$464$	19.8621	0.922076
$465$	−0.577211	−0.0267675
$466$	54.2298	2.51214
$467$	−1.40035	−0.0648004	−0.0324002	−	0.999475i	$-0.510315\pi$
$467$	−1.40035	−0.0648004	−0.0324002	+	0.999475i	$0.510315\pi$
$468$	−16.4886	−0.762188
$469$	25.4821	1.17665
$470$	−34.3173	−1.58294
$471$	−33.8569	−1.56004
$472$	−108.514	−4.99478
$473$	36.7041	1.68766
$474$	85.3760	3.92145
$475$	4.96247	0.227694
$476$	−112.945	−5.17684
$477$	17.0673	0.781458
$478$	−24.7659	−1.13277
$479$	−0.236189	−0.0107917	−0.00539587	−	0.999985i	$-0.501718\pi$
$479$	−0.236189	−0.0107917	−0.00539587	+	0.999985i	$0.501718\pi$
$480$	−38.4589	−1.75540
$481$	−12.0359	−0.548789
$482$	−16.0259	−0.729958
$483$	−6.54245	−0.297692
$484$	12.8220	0.582819
$485$	8.31172	0.377416
$486$	48.5538	2.20244
$487$	−6.91742	−0.313458	−0.156729	−	0.987642i	$-0.550095\pi$
$487$	−6.91742	−0.313458	−0.156729	+	0.987642i	$0.550095\pi$
$488$	17.7020	0.801332
$489$	−43.5121	−1.96769
$490$	32.2827	1.45839
$491$	11.8160	0.533248	0.266624	−	0.963801i	$-0.414092\pi$
$491$	11.8160	0.533248	0.266624	+	0.963801i	$0.414092\pi$
$492$	75.2682	3.39335
$493$	−7.68150	−0.345957
$494$	−20.7555	−0.933835
$495$	−7.44193	−0.334490
$496$	3.30131	0.148233
$497$	11.0167	0.494168
$498$	−49.4047	−2.21388
$499$	28.7161	1.28551	0.642755	−	0.766072i	$-0.277791\pi$
$499$	28.7161	1.28551	0.642755	+	0.766072i	$0.277791\pi$
$500$	5.22206	0.233538
$501$	3.71782	0.166100
$502$	3.85853	0.172215
$503$	−4.41453	−0.196834	−0.0984170	−	0.995145i	$-0.531378\pi$
$503$	−4.41453	−0.196834	−0.0984170	+	0.995145i	$0.531378\pi$
$504$	76.5992	3.41200
$505$	−5.48623	−0.244134
$506$	−6.59576	−0.293217
$507$	23.7207	1.05347
$508$	−33.5581	−1.48890
$509$	11.3256	0.501999	0.251000	−	0.967987i	$-0.419241\pi$
$509$	11.3256	0.501999	0.251000	+	0.967987i	$0.419241\pi$
$510$	29.8928	1.32368
$511$	−41.6463	−1.84232
$512$	−2.15248	−0.0951272
$513$	10.8079	0.477180
$514$	84.7419	3.73780
$515$	−5.86789	−0.258570
$516$	117.177	5.15841
$517$	46.8414	2.06008
$518$	90.6202	3.98162
$519$	−17.5452	−0.770149
$520$	−13.4763	−0.590973
$521$	−17.7315	−0.776830	−0.388415	−	0.921485i	$-0.626977\pi$
$521$	−17.7315	−0.776830	−0.388415	+	0.921485i	$0.626977\pi$
$522$	8.44326	0.369551
$523$	−0.595231	−0.0260276	−0.0130138	−	0.999915i	$-0.504143\pi$
$523$	−0.595231	−0.0260276	−0.0130138	+	0.999915i	$0.504143\pi$
$524$	−80.9200	−3.53500
$525$	−9.77808	−0.426750
$526$	−9.85178	−0.429558
$527$	−1.27675	−0.0556163
$528$	105.503	4.59142
$529$	−22.5523	−0.980535
$530$	22.6078	0.982018
$531$	−25.4250	−1.10335
$532$	112.996	4.89898
$533$	10.0033	0.433291
$534$	64.2759	2.78149
$535$	−11.3157	−0.489221
$536$	50.6031	2.18572
$537$	−26.7946	−1.15627
$538$	71.0929	3.06503
$539$	−44.0644	−1.89799
$540$	11.3733	0.489427
$541$	−5.05218	−0.217210	−0.108605	−	0.994085i	$-0.534638\pi$
$541$	−5.05218	−0.217210	−0.108605	+	0.994085i	$0.534638\pi$
$542$	52.8959	2.27207
$543$	19.5965	0.840966
$544$	−85.0686	−3.64729
$545$	20.5560	0.880522
$546$	40.8968	1.75022
$547$	−41.3122	−1.76638	−0.883190	−	0.469015i	$-0.844609\pi$
$547$	−41.3122	−1.76638	−0.883190	+	0.469015i	$0.844609\pi$
$548$	92.5678	3.95430
$549$	4.14760	0.177015
$550$	−9.85776	−0.420336
$551$	7.68492	0.327389
$552$	−12.9922	−0.552985
$553$	−61.7724	−2.62683
$554$	−16.0750	−0.682960
$555$	−17.3422	−0.736136
$556$	−72.7648	−3.08591
$557$	−26.5034	−1.12298	−0.561492	−	0.827482i	$-0.689772\pi$
$557$	−26.5034	−1.12298	−0.561492	+	0.827482i	$0.689772\pi$
$558$	1.40337	0.0594093
$559$	15.5730	0.658667
$560$	55.9250	2.36326
$561$	−40.8022	−1.72267
$562$	−32.4377	−1.36830
$563$	−1.07691	−0.0453863	−0.0226931	−	0.999742i	$-0.507224\pi$
$563$	−1.07691	−0.0453863	−0.0226931	+	0.999742i	$0.507224\pi$
$564$	149.540	6.29676
$565$	7.52896	0.316746
$566$	48.6577	2.04524
$567$	−47.8347	−2.00887
$568$	21.8774	0.917955
$569$	28.6547	1.20127	0.600633	−	0.799525i	$-0.294915\pi$
$569$	28.6547	1.20127	0.600633	+	0.799525i	$0.294915\pi$
$570$	−29.9061	−1.25263
$571$	34.0036	1.42301	0.711504	−	0.702682i	$-0.248014\pi$
$571$	34.0036	1.42301	0.711504	+	0.702682i	$0.248014\pi$
$572$	29.8122	1.24651
$573$	37.2666	1.55683
$574$	−75.3165	−3.14365
$575$	0.669094	0.0279031
$576$	41.4629	1.72762
$577$	12.0297	0.500803	0.250401	−	0.968142i	$-0.419437\pi$
$577$	12.0297	0.500803	0.250401	+	0.968142i	$0.419437\pi$
$578$	20.4354	0.850000
$579$	−2.48419	−0.103239
$580$	8.08692	0.335791
$581$	35.7460	1.48299
$582$	−50.0903	−2.07631
$583$	−30.8585	−1.27803
$584$	−82.7026	−3.42226
$585$	−3.15750	−0.130547
$586$	−16.3971	−0.677357
$587$	−0.350778	−0.0144782	−0.00723908	−	0.999974i	$-0.502304\pi$
$587$	−0.350778	−0.0144782	−0.00723908	+	0.999974i	$0.502304\pi$
$588$	−140.674	−5.80130
$589$	1.27732	0.0526312
$590$	−33.6786	−1.38653
$591$	−2.58481	−0.106325
$592$	99.1875	4.07658
$593$	10.9339	0.449000	0.224500	−	0.974474i	$-0.427925\pi$
$593$	10.9339	0.449000	0.224500	+	0.974474i	$0.427925\pi$
$594$	−21.4695	−0.880903
$595$	−21.6285	−0.886681
$596$	6.90086	0.282670
$597$	−62.0736	−2.54050
$598$	−2.79848	−0.114438
$599$	2.20514	0.0900997	0.0450499	−	0.998985i	$-0.485655\pi$
$599$	2.20514	0.0900997	0.0450499	+	0.998985i	$0.485655\pi$
$600$	−19.4176	−0.792721
$601$	44.4388	1.81270	0.906349	−	0.422530i	$-0.138858\pi$
$601$	44.4388	1.81270	0.906349	+	0.422530i	$0.138858\pi$
$602$	−117.252	−4.77882
$603$	11.8564	0.482828
$604$	−5.22206	−0.212483
$605$	2.45535	0.0998243
$606$	33.0626	1.34308
$607$	18.3226	0.743693	0.371847	−	0.928294i	$-0.378725\pi$
$607$	18.3226	0.743693	0.371847	+	0.928294i	$0.378725\pi$
$608$	85.1065	3.45153
$609$	−15.1424	−0.613601
$610$	5.49400	0.222446
$611$	19.8741	0.804021
$612$	−52.5514	−2.12426
$613$	10.1657	0.410591	0.205295	−	0.978700i	$-0.434185\pi$
$613$	10.1657	0.410591	0.205295	+	0.978700i	$0.434185\pi$
$614$	−34.2609	−1.38266
$615$	14.4135	0.581209
$616$	−138.495	−5.58011
$617$	34.3116	1.38133	0.690666	−	0.723174i	$-0.257318\pi$
$617$	34.3116	1.38133	0.690666	+	0.723174i	$0.257318\pi$
$618$	35.3626	1.42249
$619$	14.1975	0.570645	0.285322	−	0.958432i	$-0.407899\pi$
$619$	14.1975	0.570645	0.285322	+	0.958432i	$0.407899\pi$
$620$	1.34414	0.0539820
$621$	1.45724	0.0584769
$622$	45.6306	1.82962
$623$	−46.5058	−1.86322
$624$	44.7632	1.79196
$625$	1.00000	0.0400000
$626$	51.5258	2.05938
$627$	40.8204	1.63021
$628$	78.8418	3.14613
$629$	−38.3599	−1.52951
$630$	23.7733	0.947153
$631$	8.66504	0.344950	0.172475	−	0.985014i	$-0.444824\pi$
$631$	8.66504	0.344950	0.172475	+	0.985014i	$0.444824\pi$
$632$	−122.670	−4.87954
$633$	0.377045	0.0149862
$634$	36.3773	1.44473
$635$	−6.42621	−0.255016
$636$	−98.5146	−3.90636
$637$	−18.6958	−0.740756
$638$	−15.2658	−0.604379
$639$	5.12589	0.202777
$640$	20.6227	0.815185
$641$	23.7598	0.938455	0.469228	−	0.883077i	$-0.344532\pi$
$641$	23.7598	0.938455	0.469228	+	0.883077i	$0.344532\pi$
$642$	68.1937	2.69139
$643$	−24.4982	−0.966114	−0.483057	−	0.875589i	$-0.660474\pi$
$643$	−24.4982	−0.966114	−0.483057	+	0.875589i	$0.660474\pi$
$644$	15.2353	0.600354
$645$	22.4388	0.883525
$646$	−66.1505	−2.60266
$647$	−19.4956	−0.766453	−0.383227	−	0.923654i	$-0.625187\pi$
$647$	−19.4956	−0.766453	−0.383227	+	0.923654i	$0.625187\pi$
$648$	−94.9917	−3.73163
$649$	45.9696	1.80447
$650$	−4.18250	−0.164051
$651$	−2.51684	−0.0986429
$652$	101.326	3.96822
$653$	−22.3201	−0.873451	−0.436726	−	0.899595i	$-0.643862\pi$
$653$	−22.3201	−0.873451	−0.436726	+	0.899595i	$0.643862\pi$
$654$	−123.880	−4.84409
$655$	−15.4958	−0.605471
$656$	−82.4370	−3.21862
$657$	−19.3773	−0.755980
$658$	−149.636	−5.83340
$659$	−9.16557	−0.357040	−0.178520	−	0.983936i	$-0.557131\pi$
$659$	−9.16557	−0.357040	−0.178520	+	0.983936i	$0.557131\pi$
$660$	42.9558	1.67205
$661$	33.4441	1.30082	0.650411	−	0.759582i	$-0.274597\pi$
$661$	33.4441	1.30082	0.650411	+	0.759582i	$0.274597\pi$
$662$	43.5965	1.69443
$663$	−17.3118	−0.672333
$664$	70.9855	2.75477
$665$	21.6381	0.839091
$666$	42.1639	1.63382
$667$	1.03616	0.0401204
$668$	−8.65761	−0.334973
$669$	38.8204	1.50088
$670$	15.7052	0.606745
$671$	−7.49904	−0.289497
$672$	−167.694	−6.46895
$673$	−19.3135	−0.744482	−0.372241	−	0.928136i	$-0.621411\pi$
$673$	−19.3135	−0.744482	−0.372241	+	0.928136i	$0.621411\pi$
$674$	−82.3445	−3.17179
$675$	2.17793	0.0838284
$676$	−55.2379	−2.12453
$677$	−12.6226	−0.485127	−0.242564	−	0.970136i	$-0.577988\pi$
$677$	−12.6226	−0.485127	−0.242564	+	0.970136i	$0.577988\pi$
$678$	−45.3730	−1.74254
$679$	36.2420	1.39084
$680$	−42.9505	−1.64708
$681$	−47.9211	−1.83634
$682$	−2.53735	−0.0971603
$683$	−32.8466	−1.25684	−0.628420	−	0.777874i	$-0.716298\pi$
$683$	−32.8466	−1.25684	−0.628420	+	0.777874i	$0.716298\pi$
$684$	52.5748	2.01025
$685$	17.7263	0.677287
$686$	58.7384	2.24264
$687$	10.3926	0.396504
$688$	−128.337	−4.89279
$689$	−13.0928	−0.498795
$690$	−4.03227	−0.153506
$691$	34.9545	1.32973	0.664865	−	0.746963i	$-0.268489\pi$
$691$	34.9545	1.32973	0.664865	+	0.746963i	$0.268489\pi$
$692$	40.8572	1.55316
$693$	−32.4494	−1.23265
$694$	25.6168	0.972401
$695$	−13.9341	−0.528551
$696$	−30.0703	−1.13981
$697$	31.8818	1.20761
$698$	8.71963	0.330043
$699$	−45.2521	−1.71159
$700$	22.7700	0.860626
$701$	−33.2005	−1.25396	−0.626982	−	0.779034i	$-0.715710\pi$
$701$	−33.2005	−1.25396	−0.626982	+	0.779034i	$0.715710\pi$
$702$	−9.10917	−0.343803
$703$	38.3770	1.44741
$704$	−74.9669	−2.82542
$705$	28.6362	1.07850
$706$	61.7201	2.32287
$707$	−23.9219	−0.899676
$708$	146.756	5.51545
$709$	9.14187	0.343330	0.171665	−	0.985155i	$-0.445085\pi$
$709$	9.14187	0.343330	0.171665	+	0.985155i	$0.445085\pi$
$710$	6.78988	0.254820
$711$	−28.7416	−1.07790
$712$	−92.3528	−3.46107
$713$	0.172222	0.00644978
$714$	130.343	4.87797
$715$	5.70890	0.213501
$716$	62.3961	2.33185
$717$	20.6660	0.771786
$718$	−62.1299	−2.31867
$719$	−19.6337	−0.732212	−0.366106	−	0.930573i	$-0.619309\pi$
$719$	−19.6337	−0.732212	−0.366106	+	0.930573i	$0.619309\pi$
$720$	26.0209	0.969742
$721$	−25.5861	−0.952875
$722$	15.1196	0.562692
$723$	13.3728	0.497340
$724$	−45.6340	−1.69597
$725$	1.54861	0.0575138
$726$	−14.7971	−0.549172
$727$	17.1123	0.634659	0.317329	−	0.948315i	$-0.397214\pi$
$727$	17.1123	0.634659	0.317329	+	0.948315i	$0.397214\pi$
$728$	−58.7612	−2.17784
$729$	−7.60467	−0.281654
$730$	−25.6676	−0.950001
$731$	49.6331	1.83575
$732$	−23.9404	−0.884864
$733$	−7.31140	−0.270053	−0.135026	−	0.990842i	$-0.543112\pi$
$733$	−7.31140	−0.270053	−0.135026	+	0.990842i	$0.543112\pi$
$734$	−40.9099	−1.51001
$735$	−26.9384	−0.993638
$736$	11.4750	0.422973
$737$	−21.4368	−0.789636
$738$	−35.0434	−1.28997
$739$	−36.7491	−1.35184	−0.675919	−	0.736976i	$-0.736253\pi$
$739$	−36.7491	−1.35184	−0.675919	+	0.736976i	$0.736253\pi$
$740$	40.3845	1.48456
$741$	17.3195	0.636247
$742$	98.5778	3.61890
$743$	5.89493	0.216264	0.108132	−	0.994137i	$-0.465513\pi$
$743$	5.89493	0.216264	0.108132	+	0.994137i	$0.465513\pi$
$744$	−4.99803	−0.183237
$745$	1.32148	0.0484154
$746$	−31.9835	−1.17100
$747$	16.6320	0.608532
$748$	95.0154	3.47411
$749$	−49.3405	−1.80286
$750$	−6.02646	−0.220055
$751$	21.6455	0.789855	0.394927	−	0.918712i	$-0.370770\pi$
$751$	21.6455	0.789855	0.394927	+	0.918712i	$0.370770\pi$
$752$	−163.782	−5.97252
$753$	−3.21976	−0.117334
$754$	−6.47705	−0.235880
$755$	−1.00000	−0.0363937
$756$	49.5914	1.80362
$757$	48.6556	1.76842	0.884209	−	0.467092i	$-0.154698\pi$
$757$	48.6556	1.76842	0.884209	+	0.467092i	$0.154698\pi$
$758$	−11.7929	−0.428338
$759$	5.50385	0.199777
$760$	42.9697	1.55867
$761$	29.7632	1.07891	0.539457	−	0.842013i	$-0.318630\pi$
$761$	29.7632	1.07891	0.539457	+	0.842013i	$0.318630\pi$
$762$	38.7273	1.40294
$763$	89.6314	3.24487
$764$	−86.7820	−3.13966
$765$	−10.0633	−0.363841
$766$	19.4507	0.702783
$767$	19.5042	0.704256
$768$	−32.6212	−1.17712
$769$	−9.26564	−0.334127	−0.167064	−	0.985946i	$-0.553429\pi$
$769$	−9.26564	−0.334127	−0.167064	+	0.985946i	$0.553429\pi$
$770$	−42.9833	−1.54901
$771$	−70.7130	−2.54667
$772$	5.78489	0.208203
$773$	−2.81463	−0.101235	−0.0506176	−	0.998718i	$-0.516119\pi$
$773$	−2.81463	−0.101235	−0.0506176	+	0.998718i	$0.516119\pi$
$774$	−54.5551	−1.96094
$775$	0.257397	0.00924596
$776$	71.9706	2.58359
$777$	−75.6182	−2.71279
$778$	21.1188	0.757144
$779$	−31.8960	−1.14279
$780$	18.2255	0.652577
$781$	−9.26785	−0.331630
$782$	−8.91912	−0.318947
$783$	3.37275	0.120532
$784$	154.072	5.50258
$785$	15.0978	0.538865
$786$	93.3848	3.33093
$787$	52.0752	1.85628	0.928141	−	0.372229i	$-0.121406\pi$
$787$	52.0752	1.85628	0.928141	+	0.372229i	$0.121406\pi$
$788$	6.01919	0.214425
$789$	8.22084	0.292670
$790$	−38.0718	−1.35453
$791$	32.8289	1.16726
$792$	−64.4392	−2.28975
$793$	−3.18173	−0.112987
$794$	−5.72975	−0.203341
$795$	−18.8651	−0.669076
$796$	144.550	5.12342
$797$	−4.80658	−0.170258	−0.0851289	−	0.996370i	$-0.527130\pi$
$797$	−4.80658	−0.170258	−0.0851289	+	0.996370i	$0.527130\pi$
$798$	−130.401	−4.61616
$799$	63.3413	2.24085
$800$	17.1500	0.606345
$801$	−21.6383	−0.764553
$802$	−33.4286	−1.18040
$803$	35.0350	1.23636
$804$	−68.4363	−2.41356
$805$	2.91749	0.102828
$806$	−1.07656	−0.0379202
$807$	−59.3236	−2.08829
$808$	−47.5049	−1.67122
$809$	−15.3607	−0.540053	−0.270026	−	0.962853i	$-0.587032\pi$
$809$	−15.3607	−0.540053	−0.270026	+	0.962853i	$0.587032\pi$
$810$	−29.4817	−1.03588
$811$	−8.60622	−0.302205	−0.151103	−	0.988518i	$-0.548282\pi$
$811$	−8.60622	−0.302205	−0.151103	+	0.988518i	$0.548282\pi$
$812$	35.2618	1.23745
$813$	−44.1391	−1.54803
$814$	−76.2343	−2.67201
$815$	19.4034	0.679672
$816$	142.666	4.99431
$817$	−49.6552	−1.73722
$818$	−40.5360	−1.41731
$819$	−13.7678	−0.481086
$820$	−33.5645	−1.17212
$821$	5.49446	0.191758	0.0958790	−	0.995393i	$-0.469434\pi$
$821$	5.49446	0.191758	0.0958790	+	0.995393i	$0.469434\pi$
$822$	−106.827	−3.72602
$823$	−20.4903	−0.714246	−0.357123	−	0.934057i	$-0.616242\pi$
$823$	−20.4903	−0.714246	−0.357123	+	0.934057i	$0.616242\pi$
$824$	−50.8096	−1.77004
$825$	8.22583	0.286387
$826$	−146.851	−5.10958
$827$	−15.4004	−0.535525	−0.267762	−	0.963485i	$-0.586284\pi$
$827$	−15.4004	−0.535525	−0.267762	+	0.963485i	$0.586284\pi$
$828$	7.08871	0.246350
$829$	38.4859	1.33667	0.668335	−	0.743860i	$-0.267007\pi$
$829$	38.4859	1.33667	0.668335	+	0.743860i	$0.267007\pi$
$830$	22.0311	0.764711
$831$	13.4138	0.465319
$832$	−31.8073	−1.10272
$833$	−59.5860	−2.06453
$834$	83.9734	2.90776
$835$	−1.65789	−0.0573737
$836$	−95.0577	−3.28764
$837$	0.560590	0.0193768
$838$	66.1007	2.28341
$839$	−10.7933	−0.372628	−0.186314	−	0.982490i	$-0.559654\pi$
$839$	−10.7933	−0.372628	−0.186314	+	0.982490i	$0.559654\pi$
$840$	−84.6677	−2.92131
$841$	−26.6018	−0.917304
$842$	−84.0773	−2.89750
$843$	27.0677	0.932260
$844$	−0.878017	−0.0302226
$845$	−10.5778	−0.363887
$846$	−69.6228	−2.39368
$847$	10.7062	0.367870
$848$	107.897	3.70521
$849$	−40.6025	−1.39347
$850$	−13.3301	−0.457220
$851$	5.17440	0.177376
$852$	−29.5873	−1.01364
$853$	43.9591	1.50513	0.752565	−	0.658518i	$-0.228816\pi$
$853$	43.9591	1.50513	0.752565	+	0.658518i	$0.228816\pi$
$854$	23.9558	0.819750
$855$	10.0678	0.344313
$856$	−97.9819	−3.34895
$857$	−22.0819	−0.754302	−0.377151	−	0.926152i	$-0.623096\pi$
$857$	−22.0819	−0.754302	−0.377151	+	0.926152i	$0.623096\pi$
$858$	−34.4045	−1.17455
$859$	−27.9821	−0.954737	−0.477368	−	0.878703i	$-0.658409\pi$
$859$	−27.9821	−0.954737	−0.477368	+	0.878703i	$0.658409\pi$
$860$	−52.2527	−1.78180
$861$	62.8480	2.14185
$862$	30.3134	1.03248
$863$	−19.3392	−0.658312	−0.329156	−	0.944275i	$-0.606764\pi$
$863$	−19.3392	−0.658312	−0.329156	+	0.944275i	$0.606764\pi$
$864$	37.3515	1.27072
$865$	7.82396	0.266023
$866$	76.1784	2.58865
$867$	−17.0523	−0.579128
$868$	5.86093	0.198933
$869$	51.9662	1.76283
$870$	−9.33263	−0.316406
$871$	−9.09532	−0.308183
$872$	177.993	6.02760
$873$	16.8628	0.570718
$874$	8.92310	0.301828
$875$	4.36035	0.147407
$876$	111.848	3.77900
$877$	6.01523	0.203120	0.101560	−	0.994829i	$-0.467617\pi$
$877$	6.01523	0.203120	0.101560	+	0.994829i	$0.467617\pi$
$878$	−82.4884	−2.78385
$879$	13.6826	0.461502
$880$	−47.0470	−1.58595
$881$	1.17425	0.0395614	0.0197807	−	0.999804i	$-0.493703\pi$
$881$	1.17425	0.0395614	0.0197807	+	0.999804i	$0.493703\pi$
$882$	65.4951	2.20533
$883$	8.64906	0.291064	0.145532	−	0.989354i	$-0.453511\pi$
$883$	8.64906	0.291064	0.145532	+	0.989354i	$0.453511\pi$
$884$	40.3136	1.35589
$885$	28.1032	0.944678
$886$	50.7578	1.70524
$887$	−15.9927	−0.536981	−0.268490	−	0.963282i	$-0.586525\pi$
$887$	−15.9927	−0.536981	−0.268490	+	0.963282i	$0.586525\pi$
$888$	−150.165	−5.03921
$889$	−28.0206	−0.939779
$890$	−28.6627	−0.960774
$891$	40.2410	1.34813
$892$	−90.4002	−3.02682
$893$	−63.3696	−2.12058
$894$	−7.96387	−0.266352
$895$	11.9486	0.399396
$896$	89.9224	3.00409
$897$	2.33520	0.0779701
$898$	88.2970	2.94651
$899$	0.398606	0.0132943
$900$	10.5945	0.353150
$901$	−41.7284	−1.39017
$902$	63.3601	2.10966
$903$	97.8409	3.25594
$904$	65.1927	2.16828
$905$	−8.73869	−0.290484
$906$	6.02646	0.200216
$907$	33.4382	1.11030	0.555149	−	0.831751i	$-0.312661\pi$
$907$	33.4382	1.11030	0.555149	+	0.831751i	$0.312661\pi$
$908$	111.593	3.70334
$909$	−11.1304	−0.369173
$910$	−18.2372	−0.604556
$911$	−48.0514	−1.59201	−0.796007	−	0.605288i	$-0.793058\pi$
$911$	−48.0514	−1.59201	−0.796007	+	0.605288i	$0.793058\pi$
$912$	−142.730	−4.72625
$913$	−30.0714	−0.995217
$914$	2.61060	0.0863510
$915$	−4.58448	−0.151558
$916$	−24.2012	−0.799629
$917$	−67.5671	−2.23126
$918$	−29.0321	−0.958201
$919$	−37.0111	−1.22088	−0.610441	−	0.792062i	$-0.709008\pi$
$919$	−37.0111	−1.22088	−0.610441	+	0.792062i	$0.709008\pi$
$920$	5.79364	0.191011
$921$	28.5890	0.942041
$922$	−106.816	−3.51781
$923$	−3.93221	−0.129430
$924$	187.302	6.16179
$925$	7.73344	0.254274
$926$	66.9479	2.20004
$927$	−11.9047	−0.391003
$928$	26.5587	0.871831
$929$	11.2820	0.370151	0.185075	−	0.982724i	$-0.440747\pi$
$929$	11.2820	0.370151	0.185075	+	0.982724i	$0.440747\pi$
$930$	−1.55119	−0.0508656
$931$	59.6126	1.95372
$932$	105.378	3.45176
$933$	−38.0765	−1.24657
$934$	−3.76328	−0.123138
$935$	18.1950	0.595040
$936$	−27.3406	−0.893654
$937$	−21.1801	−0.691923	−0.345962	−	0.938249i	$-0.612447\pi$
$937$	−21.1801	−0.691923	−0.345962	+	0.938249i	$0.612447\pi$
$938$	68.4802	2.23596
$939$	−42.9958	−1.40311
$940$	−66.6844	−2.17501
$941$	−27.4870	−0.896051	−0.448025	−	0.894021i	$-0.647873\pi$
$941$	−27.4870	−0.896051	−0.448025	+	0.894021i	$0.647873\pi$
$942$	−90.9866	−2.96450
$943$	−4.30056	−0.140045
$944$	−160.734	−5.23144
$945$	9.49652	0.308922
$946$	98.6382	3.20700
$947$	−55.1778	−1.79304	−0.896519	−	0.443006i	$-0.853912\pi$
$947$	−55.1778	−1.79304	−0.896519	+	0.443006i	$0.853912\pi$
$948$	165.900	5.38819
$949$	14.8648	0.482533
$950$	13.3361	0.432680
$951$	−30.3551	−0.984331
$952$	−187.280	−6.06976
$953$	14.8717	0.481743	0.240872	−	0.970557i	$-0.422567\pi$
$953$	14.8717	0.481743	0.240872	+	0.970557i	$0.422567\pi$
$954$	45.8665	1.48498
$955$	−16.6183	−0.537757
$956$	−48.1245	−1.55646
$957$	12.7386	0.411780
$958$	−0.634731	−0.0205072
$959$	77.2929	2.49592
$960$	−45.8304	−1.47917
$961$	−30.9337	−0.997863
$962$	−32.3451	−1.04285
$963$	−22.9572	−0.739787
$964$	−31.1410	−1.00298
$965$	1.10778	0.0356606
$966$	−17.5821	−0.565695
$967$	−11.4253	−0.367413	−0.183707	−	0.982981i	$-0.558810\pi$
$967$	−11.4253	−0.367413	−0.183707	+	0.982981i	$0.558810\pi$
$968$	21.2607	0.683346
$969$	55.1994	1.77326
$970$	22.3368	0.717192
$971$	−52.4182	−1.68218	−0.841090	−	0.540895i	$-0.818086\pi$
$971$	−52.4182	−1.68218	−0.841090	+	0.540895i	$0.818086\pi$
$972$	94.3483	3.02622
$973$	−60.7576	−1.94780
$974$	−18.5898	−0.595655
$975$	3.49009	0.111772
$976$	26.2206	0.839301
$977$	21.1523	0.676723	0.338362	−	0.941016i	$-0.390127\pi$
$977$	21.1523	0.676723	0.338362	+	0.941016i	$0.390127\pi$
$978$	−116.934	−3.73914
$979$	39.1231	1.25038
$980$	62.7309	2.00387
$981$	41.7039	1.33150
$982$	31.7542	1.01332
$983$	−12.9076	−0.411690	−0.205845	−	0.978585i	$-0.565994\pi$
$983$	−12.9076	−0.411690	−0.205845	+	0.978585i	$0.565994\pi$
$984$	124.806	3.97866
$985$	1.15265	0.0367264
$986$	−20.6432	−0.657412
$987$	124.864	3.97446
$988$	−40.3316	−1.28312
$989$	−6.69505	−0.212890
$990$	−19.9994	−0.635622
$991$	−18.6955	−0.593881	−0.296941	−	0.954896i	$-0.595966\pi$
$991$	−18.6955	−0.593881	−0.296941	+	0.954896i	$0.595966\pi$
$992$	4.41436	0.140156
$993$	−36.3792	−1.15446
$994$	29.6063	0.939054
$995$	27.6806	0.877533
$996$	−96.0018	−3.04193
$997$	23.0522	0.730070	0.365035	−	0.930994i	$-0.381057\pi$
$997$	23.0522	0.730070	0.365035	+	0.930994i	$0.381057\pi$
$998$	77.1714	2.44282
$999$	16.8428	0.532884

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	755.2.a.j.1.15	✓	15
3.2	odd	2		6795.2.a.bh.1.1		15
5.4	even	2		3775.2.a.q.1.1		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
755.2.a.j.1.15	✓	15	1.1	even	1	trivial
3775.2.a.q.1.1		15	5.4	even	2
6795.2.a.bh.1.1		15	3.2	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 755 = 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	755.a (trivial)

\(f(q)\)	\(=\)	\(q+2.68739 q^{2} -2.24250 q^{3} +5.22206 q^{4} +1.00000 q^{5} -6.02646 q^{6} +4.36035 q^{7} +8.65893 q^{8} +2.02879 q^{9} +2.68739 q^{10} -3.66815 q^{11} -11.7105 q^{12} -1.55634 q^{13} +11.7180 q^{14} -2.24250 q^{15} +12.8258 q^{16} -4.96026 q^{17} +5.45216 q^{18} +4.96247 q^{19} +5.22206 q^{20} -9.77808 q^{21} -9.85776 q^{22} +0.669094 q^{23} -19.4176 q^{24} +1.00000 q^{25} -4.18250 q^{26} +2.17793 q^{27} +22.7700 q^{28} +1.54861 q^{29} -6.02646 q^{30} +0.257397 q^{31} +17.1500 q^{32} +8.22583 q^{33} -13.3301 q^{34} +4.36035 q^{35} +10.5945 q^{36} +7.73344 q^{37} +13.3361 q^{38} +3.49009 q^{39} +8.65893 q^{40} -6.42744 q^{41} -26.2775 q^{42} -10.0061 q^{43} -19.1553 q^{44} +2.02879 q^{45} +1.79812 q^{46} -12.7698 q^{47} -28.7618 q^{48} +12.0127 q^{49} +2.68739 q^{50} +11.1234 q^{51} -8.12731 q^{52} +8.41253 q^{53} +5.85293 q^{54} -3.66815 q^{55} +37.7560 q^{56} -11.1283 q^{57} +4.16171 q^{58} -12.5321 q^{59} -11.7105 q^{60} +2.04436 q^{61} +0.691725 q^{62} +8.84626 q^{63} +20.4372 q^{64} -1.55634 q^{65} +22.1060 q^{66} +5.84404 q^{67} -25.9028 q^{68} -1.50044 q^{69} +11.7180 q^{70} +2.52657 q^{71} +17.5672 q^{72} -9.55113 q^{73} +20.7828 q^{74} -2.24250 q^{75} +25.9143 q^{76} -15.9944 q^{77} +9.37924 q^{78} -14.1668 q^{79} +12.8258 q^{80} -10.9704 q^{81} -17.2730 q^{82} +8.19796 q^{83} -51.0617 q^{84} -4.96026 q^{85} -26.8904 q^{86} -3.47275 q^{87} -31.7623 q^{88} -10.6656 q^{89} +5.45216 q^{90} -6.78620 q^{91} +3.49405 q^{92} -0.577211 q^{93} -34.3173 q^{94} +4.96247 q^{95} -38.4589 q^{96} +8.31172 q^{97} +32.2827 q^{98} -7.44193 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 2 q^{2} + 5 q^{3} + 18 q^{4} + 15 q^{5} - 4 q^{6} + 11 q^{7} + 12 q^{8} + 10 q^{9} - 2 q^{10} + 2 q^{11} + 4 q^{12} + 11 q^{13} - 9 q^{14} + 5 q^{15} + 40 q^{16} + 25 q^{17} - 15 q^{18} - 3 q^{19}+ \cdots - 11 q^{99}+O(q^{100}) \) 15 * q - 2 * q^2 + 5 * q^3 + 18 * q^4 + 15 * q^5 - 4 * q^6 + 11 * q^7 + 12 * q^8 + 10 * q^9 - 2 * q^10 + 2 * q^11 + 4 * q^12 + 11 * q^13 - 9 * q^14 + 5 * q^15 + 40 * q^16 + 25 * q^17 - 15 * q^18 - 3 * q^19 + 18 * q^20 - 11 * q^21 + 13 * q^22 + 8 * q^23 - 21 * q^24 + 15 * q^25 - 45 * q^26 + 20 * q^27 + 29 * q^28 + 24 * q^29 - 4 * q^30 - 11 * q^31 + 27 * q^32 + 26 * q^33 - 11 * q^34 + 11 * q^35 - 22 * q^36 + 35 * q^37 + 2 * q^38 + 11 * q^39 + 12 * q^40 - 20 * q^41 - 17 * q^42 - 13 * q^43 - 32 * q^44 + 10 * q^45 - 5 * q^46 + 6 * q^47 - 11 * q^48 + 40 * q^49 - 2 * q^50 + 23 * q^51 + 27 * q^52 + 38 * q^53 - 30 * q^54 + 2 * q^55 - 36 * q^56 + 4 * q^57 + 6 * q^58 - 15 * q^59 + 4 * q^60 + 33 * q^61 - 13 * q^62 + 2 * q^63 + 74 * q^64 + 11 * q^65 + 8 * q^66 + 24 * q^67 + 8 * q^68 + 13 * q^69 - 9 * q^70 + 19 * q^71 + 56 * q^72 + 4 * q^73 - 28 * q^74 + 5 * q^75 + 19 * q^76 + 14 * q^77 - 15 * q^78 - 4 * q^79 + 40 * q^80 - 13 * q^81 + 33 * q^82 + 19 * q^83 - 71 * q^84 + 25 * q^85 - 59 * q^86 + 49 * q^87 - 48 * q^88 - 35 * q^89 - 15 * q^90 - q^91 + 18 * q^92 + 2 * q^93 - 70 * q^94 - 3 * q^95 - 35 * q^96 + 55 * q^97 - 64 * q^98 - 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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