LMFDB - Embedded newform 4033.2.a.f.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4033,2,Mod(1,4033)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4033, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4033.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4033 = 37 \cdot 109 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4033.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:	$32.2036671352$
Analytic rank:	$0$
Dimension:	$85$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	4033.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.27621 q^{2} -2.60153 q^{3} +3.18111 q^{4} +2.30412 q^{5} +5.92161 q^{6} +1.78008 q^{7} -2.68845 q^{8} +3.76795 q^{9} +O(q^{10})$	\(q-2.27621 q^{2} -2.60153 q^{3} +3.18111 q^{4} +2.30412 q^{5} +5.92161 q^{6} +1.78008 q^{7} -2.68845 q^{8} +3.76795 q^{9} -5.24464 q^{10} +3.06113 q^{11} -8.27574 q^{12} +2.16510 q^{13} -4.05183 q^{14} -5.99422 q^{15} -0.242762 q^{16} -2.52261 q^{17} -8.57662 q^{18} -3.18685 q^{19} +7.32965 q^{20} -4.63093 q^{21} -6.96777 q^{22} -4.94642 q^{23} +6.99407 q^{24} +0.308954 q^{25} -4.92822 q^{26} -1.99784 q^{27} +5.66263 q^{28} +4.73582 q^{29} +13.6441 q^{30} -4.47090 q^{31} +5.92947 q^{32} -7.96362 q^{33} +5.74198 q^{34} +4.10152 q^{35} +11.9863 q^{36} +1.00000 q^{37} +7.25393 q^{38} -5.63258 q^{39} -6.19450 q^{40} -7.23262 q^{41} +10.5409 q^{42} +11.0794 q^{43} +9.73780 q^{44} +8.68179 q^{45} +11.2591 q^{46} +7.47598 q^{47} +0.631553 q^{48} -3.83131 q^{49} -0.703243 q^{50} +6.56264 q^{51} +6.88743 q^{52} -4.10255 q^{53} +4.54749 q^{54} +7.05321 q^{55} -4.78565 q^{56} +8.29068 q^{57} -10.7797 q^{58} -2.91589 q^{59} -19.0683 q^{60} +5.95510 q^{61} +10.1767 q^{62} +6.70725 q^{63} -13.0112 q^{64} +4.98865 q^{65} +18.1268 q^{66} +14.6837 q^{67} -8.02470 q^{68} +12.8683 q^{69} -9.33589 q^{70} -2.66884 q^{71} -10.1299 q^{72} -6.69619 q^{73} -2.27621 q^{74} -0.803752 q^{75} -10.1377 q^{76} +5.44907 q^{77} +12.8209 q^{78} +5.77436 q^{79} -0.559353 q^{80} -6.10641 q^{81} +16.4629 q^{82} +10.0472 q^{83} -14.7315 q^{84} -5.81239 q^{85} -25.2189 q^{86} -12.3204 q^{87} -8.22969 q^{88} +11.7392 q^{89} -19.7615 q^{90} +3.85406 q^{91} -15.7351 q^{92} +11.6312 q^{93} -17.0169 q^{94} -7.34288 q^{95} -15.4257 q^{96} +0.218286 q^{97} +8.72085 q^{98} +11.5342 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 85 q + 11 q^{2} + 21 q^{3} + 93 q^{4} + 12 q^{5} + 4 q^{6} + 17 q^{7} + 30 q^{8} + 98 q^{9}+O(q^{10}) $ 85 * q + 11 * q^2 + 21 * q^3 + 93 * q^4 + 12 * q^5 + 4 * q^6 + 17 * q^7 + 30 * q^8 + 98 * q^9	\( 85 q + 11 q^{2} + 21 q^{3} + 93 q^{4} + 12 q^{5} + 4 q^{6} + 17 q^{7} + 30 q^{8} + 98 q^{9} + 9 q^{10} + 37 q^{11} + 44 q^{12} + 14 q^{13} + 26 q^{14} + 27 q^{15} + 85 q^{16} + 34 q^{17} + 3 q^{18} + 15 q^{19} + 15 q^{20} + 17 q^{21} + q^{22} + 72 q^{23} + 15 q^{24} + 85 q^{25} + 33 q^{26} + 69 q^{27} + 7 q^{28} + 19 q^{29} - 9 q^{30} + 23 q^{31} + 51 q^{32} + 32 q^{33} + 49 q^{34} + 40 q^{35} + 121 q^{36} + 85 q^{37} + 84 q^{38} + 39 q^{39} + 22 q^{40} + 55 q^{41} - 28 q^{42} + 78 q^{44} + 28 q^{45} + 17 q^{46} + 184 q^{47} + 97 q^{48} + 88 q^{49} + 26 q^{50} + 27 q^{51} + 73 q^{52} + 64 q^{53} + 31 q^{54} + 39 q^{55} + 68 q^{56} - 33 q^{57} + 28 q^{58} + 60 q^{59} - 22 q^{60} + 7 q^{61} + 70 q^{62} + 28 q^{63} + 102 q^{64} + 17 q^{65} - 15 q^{66} + 82 q^{67} + 92 q^{68} + 22 q^{69} - 41 q^{70} + 113 q^{71} - 19 q^{73} + 11 q^{74} + 45 q^{75} + 34 q^{76} + 64 q^{77} + 29 q^{78} + 23 q^{79} + 54 q^{80} + 149 q^{81} + 4 q^{82} + 100 q^{83} - 49 q^{84} - 5 q^{85} - 24 q^{86} + 65 q^{87} + 14 q^{88} + 84 q^{89} - 21 q^{90} + 32 q^{91} + 95 q^{92} + 19 q^{93} - 47 q^{94} + 102 q^{95} + 29 q^{96} + 7 q^{97} + 26 q^{98} + 107 q^{99}+O(q^{100}) \) 85 * q + 11 * q^2 + 21 * q^3 + 93 * q^4 + 12 * q^5 + 4 * q^6 + 17 * q^7 + 30 * q^8 + 98 * q^9 + 9 * q^10 + 37 * q^11 + 44 * q^12 + 14 * q^13 + 26 * q^14 + 27 * q^15 + 85 * q^16 + 34 * q^17 + 3 * q^18 + 15 * q^19 + 15 * q^20 + 17 * q^21 + q^22 + 72 * q^23 + 15 * q^24 + 85 * q^25 + 33 * q^26 + 69 * q^27 + 7 * q^28 + 19 * q^29 - 9 * q^30 + 23 * q^31 + 51 * q^32 + 32 * q^33 + 49 * q^34 + 40 * q^35 + 121 * q^36 + 85 * q^37 + 84 * q^38 + 39 * q^39 + 22 * q^40 + 55 * q^41 - 28 * q^42 + 78 * q^44 + 28 * q^45 + 17 * q^46 + 184 * q^47 + 97 * q^48 + 88 * q^49 + 26 * q^50 + 27 * q^51 + 73 * q^52 + 64 * q^53 + 31 * q^54 + 39 * q^55 + 68 * q^56 - 33 * q^57 + 28 * q^58 + 60 * q^59 - 22 * q^60 + 7 * q^61 + 70 * q^62 + 28 * q^63 + 102 * q^64 + 17 * q^65 - 15 * q^66 + 82 * q^67 + 92 * q^68 + 22 * q^69 - 41 * q^70 + 113 * q^71 - 19 * q^73 + 11 * q^74 + 45 * q^75 + 34 * q^76 + 64 * q^77 + 29 * q^78 + 23 * q^79 + 54 * q^80 + 149 * q^81 + 4 * q^82 + 100 * q^83 - 49 * q^84 - 5 * q^85 - 24 * q^86 + 65 * q^87 + 14 * q^88 + 84 * q^89 - 21 * q^90 + 32 * q^91 + 95 * q^92 + 19 * q^93 - 47 * q^94 + 102 * q^95 + 29 * q^96 + 7 * q^97 + 26 * q^98 + 107 * q^99

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 $

$2$	−2.27621	−1.60952	−0.804760	−	0.593600i	$-0.797706\pi$
$2$	−2.27621	−1.60952	−0.804760	+	0.593600i	$0.797706\pi$
$3$	−2.60153	−1.50199	−0.750996	−	0.660306i	$-0.770426\pi$
$3$	−2.60153	−1.50199	−0.750996	+	0.660306i	$0.770426\pi$
$4$	3.18111	1.59055
$5$	2.30412	1.03043	0.515216	−	0.857060i	$-0.327712\pi$
$5$	2.30412	1.03043	0.515216	+	0.857060i	$0.327712\pi$
$6$	5.92161	2.41749
$7$	1.78008	0.672808	0.336404	−	0.941718i	$-0.390789\pi$
$7$	1.78008	0.672808	0.336404	+	0.941718i	$0.390789\pi$
$8$	−2.68845	−0.950510
$9$	3.76795	1.25598
$10$	−5.24464	−1.65850
$11$	3.06113	0.922966	0.461483	−	0.887149i	$-0.347317\pi$
$11$	3.06113	0.922966	0.461483	+	0.887149i	$0.347317\pi$
$12$	−8.27574	−2.38900
$13$	2.16510	0.600492	0.300246	−	0.953862i	$-0.402931\pi$
$13$	2.16510	0.600492	0.300246	+	0.953862i	$0.402931\pi$
$14$	−4.05183	−1.08290
$15$	−5.99422	−1.54770
$16$	−0.242762	−0.0606906
$17$	−2.52261	−0.611823	−0.305912	−	0.952060i	$-0.598961\pi$
$17$	−2.52261	−0.611823	−0.305912	+	0.952060i	$0.598961\pi$
$18$	−8.57662	−2.02153
$19$	−3.18685	−0.731114	−0.365557	−	0.930789i	$-0.619121\pi$
$19$	−3.18685	−0.731114	−0.365557	+	0.930789i	$0.619121\pi$
$20$	7.32965	1.63896
$21$	−4.63093	−1.01055
$22$	−6.96777	−1.48553
$23$	−4.94642	−1.03140	−0.515700	−	0.856769i	$-0.672468\pi$
$23$	−4.94642	−1.03140	−0.515700	+	0.856769i	$0.672468\pi$
$24$	6.99407	1.42766
$25$	0.308954	0.0617908
$26$	−4.92822	−0.966503
$27$	−1.99784	−0.384484
$28$	5.66263	1.07014
$29$	4.73582	0.879419	0.439710	−	0.898140i	$-0.355081\pi$
$29$	4.73582	0.879419	0.439710	+	0.898140i	$0.355081\pi$
$30$	13.6441	2.49106
$31$	−4.47090	−0.802998	−0.401499	−	0.915860i	$-0.631511\pi$
$31$	−4.47090	−0.802998	−0.401499	+	0.915860i	$0.631511\pi$
$32$	5.92947	1.04819
$33$	−7.96362	−1.38629
$34$	5.74198	0.984742
$35$	4.10152	0.693283
$36$	11.9863	1.99771
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	7.25393	1.17674
$39$	−5.63258	−0.901934
$40$	−6.19450	−0.979436
$41$	−7.23262	−1.12955	−0.564773	−	0.825246i	$-0.691036\pi$
$41$	−7.23262	−1.12955	−0.564773	+	0.825246i	$0.691036\pi$
$42$	10.5409	1.62650
$43$	11.0794	1.68959	0.844793	−	0.535093i	$-0.179723\pi$
$43$	11.0794	1.68959	0.844793	+	0.535093i	$0.179723\pi$
$44$	9.73780	1.46803
$45$	8.68179	1.29421
$46$	11.2591	1.66006
$47$	7.47598	1.09048	0.545242	−	0.838279i	$-0.316438\pi$
$47$	7.47598	1.09048	0.545242	+	0.838279i	$0.316438\pi$
$48$	0.631553	0.0911568
$49$	−3.83131	−0.547330
$50$	−0.703243	−0.0994535
$51$	6.56264	0.918954
$52$	6.88743	0.955115
$53$	−4.10255	−0.563528	−0.281764	−	0.959484i	$-0.590920\pi$
$53$	−4.10255	−0.563528	−0.281764	+	0.959484i	$0.590920\pi$
$54$	4.54749	0.618835
$55$	7.05321	0.951054
$56$	−4.78565	−0.639510
$57$	8.29068	1.09813
$58$	−10.7797	−1.41544
$59$	−2.91589	−0.379616	−0.189808	−	0.981821i	$-0.560787\pi$
$59$	−2.91589	−0.379616	−0.189808	+	0.981821i	$0.560787\pi$
$60$	−19.0683	−2.46170
$61$	5.95510	0.762473	0.381236	−	0.924478i	$-0.375498\pi$
$61$	5.95510	0.762473	0.381236	+	0.924478i	$0.375498\pi$
$62$	10.1767	1.29244
$63$	6.70725	0.845035
$64$	−13.0112	−1.62640
$65$	4.98865	0.618766
$66$	18.1268	2.23126
$67$	14.6837	1.79390	0.896952	−	0.442127i	$-0.145776\pi$
$67$	14.6837	1.79390	0.896952	+	0.442127i	$0.145776\pi$
$68$	−8.02470	−0.973138
$69$	12.8683	1.54916
$70$	−9.33589	−1.11585
$71$	−2.66884	−0.316733	−0.158366	−	0.987380i	$-0.550623\pi$
$71$	−2.66884	−0.316733	−0.158366	+	0.987380i	$0.550623\pi$
$72$	−10.1299	−1.19382
$73$	−6.69619	−0.783729	−0.391865	−	0.920023i	$-0.628170\pi$
$73$	−6.69619	−0.783729	−0.391865	+	0.920023i	$0.628170\pi$
$74$	−2.27621	−0.264603
$75$	−0.803752	−0.0928093
$76$	−10.1377	−1.16288
$77$	5.44907	0.620979
$78$	12.8209	1.45168
$79$	5.77436	0.649666	0.324833	−	0.945771i	$-0.394692\pi$
$79$	5.77436	0.649666	0.324833	+	0.945771i	$0.394692\pi$
$80$	−0.559353	−0.0625375
$81$	−6.10641	−0.678490
$82$	16.4629	1.81803
$83$	10.0472	1.10282	0.551411	−	0.834234i	$-0.314090\pi$
$83$	10.0472	1.10282	0.551411	+	0.834234i	$0.314090\pi$
$84$	−14.7315	−1.60734
$85$	−5.81239	−0.630442
$86$	−25.2189	−2.71942
$87$	−12.3204	−1.32088
$88$	−8.22969	−0.877288
$89$	11.7392	1.24436	0.622178	−	0.782876i	$-0.286248\pi$
$89$	11.7392	1.24436	0.622178	+	0.782876i	$0.286248\pi$
$90$	−19.7615	−2.08305
$91$	3.85406	0.404015
$92$	−15.7351	−1.64050
$93$	11.6312	1.20610
$94$	−17.0169	−1.75516
$95$	−7.34288	−0.753363
$96$	−15.4257	−1.57438
$97$	0.218286	0.0221635	0.0110818	−	0.999939i	$-0.496472\pi$
$97$	0.218286	0.0221635	0.0110818	+	0.999939i	$0.496472\pi$
$98$	8.72085	0.880939
$99$	11.5342	1.15923
$100$	0.982816	0.0982816
$101$	1.67934	0.167100	0.0835502	−	0.996504i	$-0.473374\pi$
$101$	1.67934	0.167100	0.0835502	+	0.996504i	$0.473374\pi$
$102$	−14.9379	−1.47907
$103$	0.793145	0.0781509	0.0390754	−	0.999236i	$-0.487559\pi$
$103$	0.793145	0.0781509	0.0390754	+	0.999236i	$0.487559\pi$
$104$	−5.82077	−0.570773
$105$	−10.6702	−1.04131
$106$	9.33824	0.907010
$107$	12.3414	1.19309	0.596545	−	0.802579i	$-0.296540\pi$
$107$	12.3414	1.19309	0.596545	+	0.802579i	$0.296540\pi$
$108$	−6.35534	−0.611543
$109$	−1.00000	−0.0957826
$109$	−1.00000	−0.0957826
$110$	−16.0545	−1.53074
$111$	−2.60153	−0.246926
$112$	−0.432137	−0.0408331
$113$	−7.35071	−0.691497	−0.345749	−	0.938327i	$-0.612375\pi$
$113$	−7.35071	−0.691497	−0.345749	+	0.938327i	$0.612375\pi$
$114$	−18.8713	−1.76746
$115$	−11.3971	−1.06279
$116$	15.0652	1.39876
$117$	8.15800	0.754207
$118$	6.63715	0.611000
$119$	−4.49045	−0.411639
$120$	16.1152	1.47111
$121$	−1.62947	−0.148133
$122$	−13.5550	−1.22722
$123$	18.8159	1.69657
$124$	−14.2224	−1.27721
$125$	−10.8087	−0.966761
$126$	−15.2671	−1.36010
$127$	9.40888	0.834904	0.417452	−	0.908699i	$-0.362923\pi$
$127$	9.40888	0.834904	0.417452	+	0.908699i	$0.362923\pi$
$128$	17.7571	1.56952
$129$	−28.8233	−2.53775
$130$	−11.3552	−0.995916
$131$	13.4113	1.17175	0.585874	−	0.810402i	$-0.300751\pi$
$131$	13.4113	1.17175	0.585874	+	0.810402i	$0.300751\pi$
$132$	−25.3332	−2.20497
$133$	−5.67285	−0.491899
$134$	−33.4232	−2.88733
$135$	−4.60325	−0.396185
$136$	6.78191	0.581544
$137$	0.961966	0.0821863	0.0410931	−	0.999155i	$-0.486916\pi$
$137$	0.961966	0.0821863	0.0410931	+	0.999155i	$0.486916\pi$
$138$	−29.2908	−2.49340
$139$	−17.1814	−1.45731	−0.728653	−	0.684883i	$-0.759853\pi$
$139$	−17.1814	−1.45731	−0.728653	+	0.684883i	$0.759853\pi$
$140$	13.0474	1.10270
$141$	−19.4490	−1.63790
$142$	6.07482	0.509787
$143$	6.62767	0.554233
$144$	−0.914716	−0.0762263
$145$	10.9119	0.906182
$146$	15.2419	1.26143
$147$	9.96726	0.822086
$148$	3.18111	0.261486
$149$	6.61781	0.542152	0.271076	−	0.962558i	$-0.412621\pi$
$149$	6.61781	0.542152	0.271076	+	0.962558i	$0.412621\pi$
$150$	1.82951	0.149378
$151$	−0.0342275	−0.00278540	−0.00139270	−	0.999999i	$-0.500443\pi$
$151$	−0.0342275	−0.00278540	−0.00139270	+	0.999999i	$0.500443\pi$
$152$	8.56768	0.694931
$153$	−9.50507	−0.768439
$154$	−12.4032	−0.999477
$155$	−10.3015	−0.827435
$156$	−17.9178	−1.43458
$157$	0.603937	0.0481994	0.0240997	−	0.999710i	$-0.492328\pi$
$157$	0.603937	0.0481994	0.0240997	+	0.999710i	$0.492328\pi$
$158$	−13.1436	−1.04565
$159$	10.6729	0.846416
$160$	13.6622	1.08009
$161$	−8.80504	−0.693934
$162$	13.8994	1.09204
$163$	−19.7366	−1.54589	−0.772943	−	0.634475i	$-0.781216\pi$
$163$	−19.7366	−1.54589	−0.772943	+	0.634475i	$0.781216\pi$
$164$	−23.0078	−1.79660
$165$	−18.3491	−1.42848
$166$	−22.8695	−1.77501
$167$	9.92336	0.767893	0.383946	−	0.923355i	$-0.374565\pi$
$167$	9.92336	0.767893	0.383946	+	0.923355i	$0.374565\pi$
$168$	12.4500	0.960539
$169$	−8.31233	−0.639410
$170$	13.2302	1.01471
$171$	−12.0079	−0.918266
$172$	35.2447	2.68738
$173$	22.0909	1.67954	0.839769	−	0.542943i	$-0.182690\pi$
$173$	22.0909	1.67954	0.839769	+	0.542943i	$0.182690\pi$
$174$	28.0437	2.12599
$175$	0.549963	0.0415733
$176$	−0.743128	−0.0560154
$177$	7.58576	0.570181
$178$	−26.7209	−2.00282
$179$	24.1558	1.80549	0.902746	−	0.430175i	$-0.141548\pi$
$179$	24.1558	1.80549	0.902746	+	0.430175i	$0.141548\pi$
$180$	27.6177	2.05850
$181$	12.9350	0.961452	0.480726	−	0.876871i	$-0.340373\pi$
$181$	12.9350	0.961452	0.480726	+	0.876871i	$0.340373\pi$
$182$	−8.77263	−0.650271
$183$	−15.4924	−1.14523
$184$	13.2982	0.980356
$185$	2.30412	0.169402
$186$	−26.4749	−1.94124
$187$	−7.72205	−0.564692
$188$	23.7819	1.73447
$189$	−3.55631	−0.258684
$190$	16.7139	1.21255
$191$	−12.0463	−0.871642	−0.435821	−	0.900033i	$-0.643542\pi$
$191$	−12.0463	−0.871642	−0.435821	+	0.900033i	$0.643542\pi$
$192$	33.8489	2.44283
$193$	−6.94115	−0.499635	−0.249817	−	0.968293i	$-0.580371\pi$
$193$	−6.94115	−0.499635	−0.249817	+	0.968293i	$0.580371\pi$
$194$	−0.496863	−0.0356727
$195$	−12.9781	−0.929382
$196$	−12.1878	−0.870558
$197$	−9.13803	−0.651058	−0.325529	−	0.945532i	$-0.605542\pi$
$197$	−9.13803	−0.651058	−0.325529	+	0.945532i	$0.605542\pi$
$198$	−26.2542	−1.86580
$199$	1.17005	0.0829430	0.0414715	−	0.999140i	$-0.486795\pi$
$199$	1.17005	0.0829430	0.0414715	+	0.999140i	$0.486795\pi$
$200$	−0.830606	−0.0587327
$201$	−38.2002	−2.69443
$202$	−3.82252	−0.268952
$203$	8.43014	0.591680
$204$	20.8765	1.46165
$205$	−16.6648	−1.16392
$206$	−1.80536	−0.125785
$207$	−18.6379	−1.29542
$208$	−0.525606	−0.0364442
$209$	−9.75538	−0.674793
$210$	24.2876	1.67600
$211$	−9.49782	−0.653857	−0.326928	−	0.945049i	$-0.606014\pi$
$211$	−9.49782	−0.653857	−0.326928	+	0.945049i	$0.606014\pi$
$212$	−13.0507	−0.896323
$213$	6.94305	0.475730
$214$	−28.0916	−1.92030
$215$	25.5281	1.74100
$216$	5.37108	0.365456
$217$	−7.95857	−0.540263
$218$	2.27621	0.154164
$219$	17.4203	1.17716
$220$	22.4370	1.51270
$221$	−5.46171	−0.367395
$222$	5.92161	0.397433
$223$	6.74230	0.451498	0.225749	−	0.974186i	$-0.427517\pi$
$223$	6.74230	0.451498	0.225749	+	0.974186i	$0.427517\pi$
$224$	10.5549	0.705232
$225$	1.16412	0.0776082
$226$	16.7317	1.11298
$227$	6.33521	0.420483	0.210241	−	0.977650i	$-0.432575\pi$
$227$	6.33521	0.420483	0.210241	+	0.977650i	$0.432575\pi$
$228$	26.3736	1.74663
$229$	−11.9262	−0.788103	−0.394051	−	0.919088i	$-0.628927\pi$
$229$	−11.9262	−0.788103	−0.394051	+	0.919088i	$0.628927\pi$
$230$	25.9422	1.71058
$231$	−14.1759	−0.932705
$232$	−12.7320	−0.835896
$233$	22.3073	1.46140	0.730700	−	0.682698i	$-0.239194\pi$
$233$	22.3073	1.46140	0.730700	+	0.682698i	$0.239194\pi$
$234$	−18.5693	−1.21391
$235$	17.2255	1.12367
$236$	−9.27575	−0.603800
$237$	−15.0222	−0.975794
$238$	10.2212	0.662541
$239$	−26.4152	−1.70866	−0.854329	−	0.519732i	$-0.826032\pi$
$239$	−26.4152	−1.70866	−0.854329	+	0.519732i	$0.826032\pi$
$240$	1.45517	0.0939309
$241$	5.64714	0.363764	0.181882	−	0.983320i	$-0.441781\pi$
$241$	5.64714	0.363764	0.181882	+	0.983320i	$0.441781\pi$
$242$	3.70900	0.238424
$243$	21.8795	1.40357
$244$	18.9438	1.21275
$245$	−8.82779	−0.563987
$246$	−42.8288	−2.73066
$247$	−6.89986	−0.439028
$248$	12.0198	0.763257
$249$	−26.1380	−1.65643
$250$	24.6029	1.55602
$251$	13.8354	0.873285	0.436642	−	0.899635i	$-0.356168\pi$
$251$	13.8354	0.873285	0.436642	+	0.899635i	$0.356168\pi$
$252$	21.3365	1.34407
$253$	−15.1417	−0.951948
$254$	−21.4166	−1.34379
$255$	15.1211	0.946920
$256$	−14.3966	−0.899785
$257$	23.9869	1.49626	0.748132	−	0.663550i	$-0.230951\pi$
$257$	23.9869	1.49626	0.748132	+	0.663550i	$0.230951\pi$
$258$	65.6077	4.08455
$259$	1.78008	0.110609
$260$	15.8694	0.984181
$261$	17.8443	1.10454
$262$	−30.5268	−1.88595
$263$	7.12389	0.439278	0.219639	−	0.975581i	$-0.429512\pi$
$263$	7.12389	0.439278	0.219639	+	0.975581i	$0.429512\pi$
$264$	21.4098	1.31768
$265$	−9.45275	−0.580678
$266$	12.9126	0.791721
$267$	−30.5399	−1.86901
$268$	46.7106	2.85330
$269$	−3.57039	−0.217691	−0.108845	−	0.994059i	$-0.534715\pi$
$269$	−3.57039	−0.217691	−0.108845	+	0.994059i	$0.534715\pi$
$270$	10.4779	0.637668
$271$	14.5470	0.883669	0.441835	−	0.897097i	$-0.354328\pi$
$271$	14.5470	0.883669	0.441835	+	0.897097i	$0.354328\pi$
$272$	0.612395	0.0371319
$273$	−10.0264	−0.606828
$274$	−2.18963	−0.132280
$275$	0.945749	0.0570308
$276$	40.9353	2.46402
$277$	−6.75810	−0.406055	−0.203027	−	0.979173i	$-0.565078\pi$
$277$	−6.75810	−0.406055	−0.203027	+	0.979173i	$0.565078\pi$
$278$	39.1083	2.34556
$279$	−16.8461	−1.00855
$280$	−11.0267	−0.658972
$281$	15.7938	0.942180	0.471090	−	0.882085i	$-0.343861\pi$
$281$	15.7938	0.942180	0.471090	+	0.882085i	$0.343861\pi$
$282$	44.2699	2.63623
$283$	3.42122	0.203371	0.101685	−	0.994817i	$-0.467577\pi$
$283$	3.42122	0.203371	0.101685	+	0.994817i	$0.467577\pi$
$284$	−8.48986	−0.503780
$285$	19.1027	1.13155
$286$	−15.0859	−0.892050
$287$	−12.8747	−0.759967
$288$	22.3419	1.31651
$289$	−10.6364	−0.625672
$290$	−24.8377	−1.45852
$291$	−0.567876	−0.0332895
$292$	−21.3013	−1.24656
$293$	21.3139	1.24517	0.622587	−	0.782551i	$-0.286082\pi$
$293$	21.3139	1.24517	0.622587	+	0.782551i	$0.286082\pi$
$294$	−22.6875	−1.32316
$295$	−6.71854	−0.391169
$296$	−2.68845	−0.156263
$297$	−6.11565	−0.354866
$298$	−15.0635	−0.872604
$299$	−10.7095	−0.619347
$300$	−2.55682	−0.147618
$301$	19.7222	1.13677
$302$	0.0779089	0.00448315
$303$	−4.36885	−0.250984
$304$	0.773647	0.0443717
$305$	13.7213	0.785677
$306$	21.6355	1.23682
$307$	13.2408	0.755691	0.377845	−	0.925869i	$-0.376665\pi$
$307$	13.2408	0.755691	0.377845	+	0.925869i	$0.376665\pi$
$308$	17.3341	0.987700
$309$	−2.06339	−0.117382
$310$	23.4483	1.33177
$311$	−14.1311	−0.801299	−0.400649	−	0.916232i	$-0.631215\pi$
$311$	−14.1311	−0.801299	−0.400649	+	0.916232i	$0.631215\pi$
$312$	15.1429	0.857297
$313$	−15.4827	−0.875132	−0.437566	−	0.899186i	$-0.644159\pi$
$313$	−15.4827	−0.875132	−0.437566	+	0.899186i	$0.644159\pi$
$314$	−1.37468	−0.0775779
$315$	15.4543	0.870751
$316$	18.3689	1.03333
$317$	−21.1989	−1.19065	−0.595325	−	0.803485i	$-0.702977\pi$
$317$	−21.1989	−1.19065	−0.595325	+	0.803485i	$0.702977\pi$
$318$	−24.2937	−1.36232
$319$	14.4970	0.811674
$320$	−29.9792	−1.67589
$321$	−32.1066	−1.79201
$322$	20.0421	1.11690
$323$	8.03919	0.447312
$324$	−19.4252	−1.07918
$325$	0.668917	0.0371049
$326$	44.9245	2.48814
$327$	2.60153	0.143865
$328$	19.4445	1.07364
$329$	13.3079	0.733686
$330$	41.7663	2.29916
$331$	8.52828	0.468757	0.234378	−	0.972145i	$-0.424695\pi$
$331$	8.52828	0.468757	0.234378	+	0.972145i	$0.424695\pi$
$332$	31.9612	1.75410
$333$	3.76795	0.206482
$334$	−22.5876	−1.23594
$335$	33.8331	1.84850
$336$	1.12422	0.0613310
$337$	−26.3753	−1.43675	−0.718377	−	0.695654i	$-0.755115\pi$
$337$	−26.3753	−1.43675	−0.718377	+	0.695654i	$0.755115\pi$
$338$	18.9206	1.02914
$339$	19.1231	1.03862
$340$	−18.4899	−1.00275
$341$	−13.6860	−0.741140
$342$	27.3324	1.47797
$343$	−19.2806	−1.04106
$344$	−29.7863	−1.60597
$345$	29.6500	1.59630
$346$	−50.2834	−2.70325
$347$	−1.09619	−0.0588468	−0.0294234	−	0.999567i	$-0.509367\pi$
$347$	−1.09619	−0.0588468	−0.0294234	+	0.999567i	$0.509367\pi$
$348$	−39.1924	−2.10093
$349$	33.8074	1.80967	0.904835	−	0.425761i	$-0.139994\pi$
$349$	33.8074	1.80967	0.904835	+	0.425761i	$0.139994\pi$
$350$	−1.25183	−0.0669131
$351$	−4.32553	−0.230880
$352$	18.1509	0.967446
$353$	−8.07857	−0.429979	−0.214990	−	0.976616i	$-0.568972\pi$
$353$	−8.07857	−0.429979	−0.214990	+	0.976616i	$0.568972\pi$
$354$	−17.2667	−0.917717
$355$	−6.14931	−0.326371
$356$	37.3438	1.97922
$357$	11.6820	0.618279
$358$	−54.9836	−2.90597
$359$	21.8228	1.15177	0.575883	−	0.817532i	$-0.304659\pi$
$359$	21.8228	1.15177	0.575883	+	0.817532i	$0.304659\pi$
$360$	−23.3405	−1.23015
$361$	−8.84398	−0.465473
$362$	−29.4427	−1.54748
$363$	4.23910	0.222495
$364$	12.2602	0.642608
$365$	−15.4288	−0.807580
$366$	35.2638	1.84327
$367$	27.7672	1.44944	0.724719	−	0.689044i	$-0.241969\pi$
$367$	27.7672	1.44944	0.724719	+	0.689044i	$0.241969\pi$
$368$	1.20081	0.0625963
$369$	−27.2521	−1.41869
$370$	−5.24464	−0.272656
$371$	−7.30287	−0.379146
$372$	37.0000	1.91836
$373$	16.1304	0.835200	0.417600	−	0.908631i	$-0.362871\pi$
$373$	16.1304	0.835200	0.417600	+	0.908631i	$0.362871\pi$
$374$	17.5770	0.908883
$375$	28.1192	1.45207
$376$	−20.0988	−1.03652
$377$	10.2535	0.528084
$378$	8.09490	0.416357
$379$	27.0434	1.38913	0.694563	−	0.719432i	$-0.255598\pi$
$379$	27.0434	1.38913	0.694563	+	0.719432i	$0.255598\pi$
$380$	−23.3585	−1.19827
$381$	−24.4775	−1.25402
$382$	27.4199	1.40293
$383$	28.9379	1.47866	0.739328	−	0.673345i	$-0.235143\pi$
$383$	28.9379	1.47866	0.739328	+	0.673345i	$0.235143\pi$
$384$	−46.1957	−2.35741
$385$	12.5553	0.639876
$386$	15.7995	0.804172
$387$	41.7464	2.12209
$388$	0.694390	0.0352523
$389$	26.0718	1.32189	0.660946	−	0.750433i	$-0.270155\pi$
$389$	26.0718	1.32189	0.660946	+	0.750433i	$0.270155\pi$
$390$	29.5409	1.49586
$391$	12.4779	0.631035
$392$	10.3003	0.520242
$393$	−34.8898	−1.75996
$394$	20.8000	1.04789
$395$	13.3048	0.669437
$396$	36.6915	1.84382
$397$	−5.57264	−0.279683	−0.139841	−	0.990174i	$-0.544659\pi$
$397$	−5.57264	−0.279683	−0.139841	+	0.990174i	$0.544659\pi$
$398$	−2.66328	−0.133498
$399$	14.7581	0.738829
$400$	−0.0750024	−0.00375012
$401$	−7.45875	−0.372472	−0.186236	−	0.982505i	$-0.559629\pi$
$401$	−7.45875	−0.372472	−0.186236	+	0.982505i	$0.559629\pi$
$402$	86.9514	4.33674
$403$	−9.67997	−0.482193
$404$	5.34216	0.265782
$405$	−14.0699	−0.699138
$406$	−19.1887	−0.952321
$407$	3.06113	0.151735
$408$	−17.6433	−0.873474
$409$	−26.7456	−1.32248	−0.661242	−	0.750173i	$-0.729970\pi$
$409$	−26.7456	−1.32248	−0.661242	+	0.750173i	$0.729970\pi$
$410$	37.9325	1.87335
$411$	−2.50258	−0.123443
$412$	2.52308	0.124303
$413$	−5.19051	−0.255408
$414$	42.4236	2.08501
$415$	23.1499	1.13638
$416$	12.8379	0.629431
$417$	44.6978	2.18886
$418$	22.2052	1.08609
$419$	−20.2407	−0.988824	−0.494412	−	0.869228i	$-0.664617\pi$
$419$	−20.2407	−0.988824	−0.494412	+	0.869228i	$0.664617\pi$
$420$	−33.9431	−1.65625
$421$	12.8898	0.628210	0.314105	−	0.949388i	$-0.398296\pi$
$421$	12.8898	0.628210	0.314105	+	0.949388i	$0.398296\pi$
$422$	21.6190	1.05240
$423$	28.1691	1.36963
$424$	11.0295	0.535639
$425$	−0.779371	−0.0378050
$426$	−15.8038	−0.765697
$427$	10.6006	0.512997
$428$	39.2594	1.89768
$429$	−17.2421	−0.832455
$430$	−58.1073	−2.80218
$431$	−19.9228	−0.959647	−0.479824	−	0.877365i	$-0.659299\pi$
$431$	−19.9228	−0.959647	−0.479824	+	0.877365i	$0.659299\pi$
$432$	0.485000	0.0233346
$433$	32.8998	1.58106	0.790532	−	0.612420i	$-0.209804\pi$
$433$	32.8998	1.58106	0.790532	+	0.612420i	$0.209804\pi$
$434$	18.1153	0.869564
$435$	−28.3876	−1.36108
$436$	−3.18111	−0.152348
$437$	15.7635	0.754071
$438$	−39.6522	−1.89466
$439$	−0.709157	−0.0338462	−0.0169231	−	0.999857i	$-0.505387\pi$
$439$	−0.709157	−0.0338462	−0.0169231	+	0.999857i	$0.505387\pi$
$440$	−18.9622	−0.903986
$441$	−14.4362	−0.687437
$442$	12.4320	0.591329
$443$	13.9415	0.662379	0.331189	−	0.943564i	$-0.392550\pi$
$443$	13.9415	0.662379	0.331189	+	0.943564i	$0.392550\pi$
$444$	−8.27574	−0.392749
$445$	27.0485	1.28222
$446$	−15.3469	−0.726695
$447$	−17.2164	−0.814308
$448$	−23.1609	−1.09425
$449$	7.39625	0.349051	0.174525	−	0.984653i	$-0.444161\pi$
$449$	7.39625	0.349051	0.174525	+	0.984653i	$0.444161\pi$
$450$	−2.64978	−0.124912
$451$	−22.1400	−1.04253
$452$	−23.3834	−1.09986
$453$	0.0890439	0.00418365
$454$	−14.4202	−0.676776
$455$	8.88020	0.416310
$456$	−22.2891	−1.04378
$457$	−21.2702	−0.994976	−0.497488	−	0.867471i	$-0.665744\pi$
$457$	−21.2702	−0.994976	−0.497488	+	0.867471i	$0.665744\pi$
$458$	27.1464	1.26847
$459$	5.03977	0.235236
$460$	−36.2555	−1.69042
$461$	−4.36175	−0.203147	−0.101573	−	0.994828i	$-0.532388\pi$
$461$	−4.36175	−0.203147	−0.101573	+	0.994828i	$0.532388\pi$
$462$	32.2672	1.50121
$463$	−23.2399	−1.08005	−0.540026	−	0.841649i	$-0.681585\pi$
$463$	−23.2399	−1.08005	−0.540026	+	0.841649i	$0.681585\pi$
$464$	−1.14968	−0.0533725
$465$	26.7996	1.24280
$466$	−50.7760	−2.35215
$467$	37.6605	1.74272	0.871360	−	0.490645i	$-0.163239\pi$
$467$	37.6605	1.74272	0.871360	+	0.490645i	$0.163239\pi$
$468$	25.9515	1.19961
$469$	26.1383	1.20695
$470$	−39.2089	−1.80857
$471$	−1.57116	−0.0723951
$472$	7.83920	0.360829
$473$	33.9154	1.55943
$474$	34.1935	1.57056
$475$	−0.984590	−0.0451761
$476$	−14.2846	−0.654735
$477$	−15.4582	−0.707782
$478$	60.1265	2.75012
$479$	18.8384	0.860749	0.430374	−	0.902651i	$-0.358382\pi$
$479$	18.8384	0.860749	0.430374	+	0.902651i	$0.358382\pi$
$480$	−35.5426	−1.62229
$481$	2.16510	0.0987202
$482$	−12.8540	−0.585485
$483$	22.9065	1.04228
$484$	−5.18351	−0.235614
$485$	0.502956	0.0228380
$486$	−49.8023	−2.25908
$487$	−10.3555	−0.469251	−0.234625	−	0.972086i	$-0.575386\pi$
$487$	−10.3555	−0.469251	−0.234625	+	0.972086i	$0.575386\pi$
$488$	−16.0100	−0.724738
$489$	51.3452	2.32191
$490$	20.0939	0.907748
$491$	41.7718	1.88514	0.942568	−	0.334013i	$-0.108403\pi$
$491$	41.7718	1.88514	0.942568	+	0.334013i	$0.108403\pi$
$492$	59.8553	2.69849
$493$	−11.9466	−0.538049
$494$	15.7055	0.706624
$495$	26.5761	1.19451
$496$	1.08537	0.0487344
$497$	−4.75074	−0.213100
$498$	59.4955	2.66606
$499$	11.4619	0.513106	0.256553	−	0.966530i	$-0.417413\pi$
$499$	11.4619	0.513106	0.256553	+	0.966530i	$0.417413\pi$
$500$	−34.3837	−1.53769
$501$	−25.8159	−1.15337
$502$	−31.4923	−1.40557
$503$	0.661678	0.0295028	0.0147514	−	0.999891i	$-0.495304\pi$
$503$	0.661678	0.0295028	0.0147514	+	0.999891i	$0.495304\pi$
$504$	−18.0321	−0.803213
$505$	3.86939	0.172186
$506$	34.4655	1.53218
$507$	21.6248	0.960389
$508$	29.9307	1.32796
$509$	4.60859	0.204272	0.102136	−	0.994770i	$-0.467432\pi$
$509$	4.60859	0.204272	0.102136	+	0.994770i	$0.467432\pi$
$510$	−34.4187	−1.52409
$511$	−11.9198	−0.527299
$512$	−2.74476	−0.121302
$513$	6.36681	0.281102
$514$	−54.5992	−2.40827
$515$	1.82750	0.0805292
$516$	−91.6899	−4.03642
$517$	22.8850	1.00648
$518$	−4.05183	−0.178027
$519$	−57.4701	−2.52266
$520$	−13.4117	−0.588143
$521$	−27.7391	−1.21527	−0.607636	−	0.794216i	$-0.707882\pi$
$521$	−27.7391	−1.21527	−0.607636	+	0.794216i	$0.707882\pi$
$522$	−40.6173	−1.77777
$523$	19.4634	0.851074	0.425537	−	0.904941i	$-0.360085\pi$
$523$	19.4634	0.851074	0.425537	+	0.904941i	$0.360085\pi$
$524$	42.6627	1.86373
$525$	−1.43074	−0.0624428
$526$	−16.2154	−0.707027
$527$	11.2783	0.491293
$528$	1.93327	0.0841347
$529$	1.46710	0.0637870
$530$	21.5164	0.934613
$531$	−10.9869	−0.476791
$532$	−18.0460	−0.782392
$533$	−15.6594	−0.678283
$534$	69.5151	3.00821
$535$	28.4361	1.22940
$536$	−39.4765	−1.70512
$537$	−62.8421	−2.71184
$538$	8.12695	0.350378
$539$	−11.7281	−0.505167
$540$	−14.6434	−0.630154
$541$	−42.4407	−1.82467	−0.912333	−	0.409448i	$-0.865721\pi$
$541$	−42.4407	−1.82467	−0.912333	+	0.409448i	$0.865721\pi$
$542$	−33.1120	−1.42228
$543$	−33.6508	−1.44409
$544$	−14.9577	−0.641308
$545$	−2.30412	−0.0986975
$546$	22.8222	0.976702
$547$	−2.03806	−0.0871410	−0.0435705	−	0.999050i	$-0.513873\pi$
$547$	−2.03806	−0.0871410	−0.0435705	+	0.999050i	$0.513873\pi$
$548$	3.06012	0.130722
$549$	22.4385	0.957653
$550$	−2.15272	−0.0917922
$551$	−15.0923	−0.642956
$552$	−34.5956	−1.47249
$553$	10.2788	0.437100
$554$	15.3828	0.653553
$555$	−5.99422	−0.254441
$556$	−54.6559	−2.31792
$557$	−28.6139	−1.21241	−0.606204	−	0.795309i	$-0.707309\pi$
$557$	−28.6139	−1.21241	−0.606204	+	0.795309i	$0.707309\pi$
$558$	38.3452	1.62328
$559$	23.9880	1.01458
$560$	−0.995693	−0.0420757
$561$	20.0891	0.848163
$562$	−35.9500	−1.51646
$563$	−9.54571	−0.402304	−0.201152	−	0.979560i	$-0.564468\pi$
$563$	−9.54571	−0.402304	−0.201152	+	0.979560i	$0.564468\pi$
$564$	−61.8693	−2.60517
$565$	−16.9369	−0.712541
$566$	−7.78741	−0.327329
$567$	−10.8699	−0.456493
$568$	7.17502	0.301057
$569$	39.3428	1.64933	0.824667	−	0.565618i	$-0.191362\pi$
$569$	39.3428	1.64933	0.824667	+	0.565618i	$0.191362\pi$
$570$	−43.4817	−1.82125
$571$	−4.95288	−0.207272	−0.103636	−	0.994615i	$-0.533048\pi$
$571$	−4.95288	−0.207272	−0.103636	+	0.994615i	$0.533048\pi$
$572$	21.0833	0.881539
$573$	31.3389	1.30920
$574$	29.3054	1.22318
$575$	−1.52822	−0.0637311
$576$	−49.0254	−2.04272
$577$	−7.41897	−0.308856	−0.154428	−	0.988004i	$-0.549353\pi$
$577$	−7.41897	−0.308856	−0.154428	+	0.988004i	$0.549353\pi$
$578$	24.2107	1.00703
$579$	18.0576	0.750448
$580$	34.7119	1.44133
$581$	17.8848	0.741987
$582$	1.29260	0.0535801
$583$	−12.5584	−0.520118
$584$	18.0023	0.744942
$585$	18.7970	0.777159
$586$	−48.5149	−2.00413
$587$	−25.6191	−1.05741	−0.528707	−	0.848805i	$-0.677323\pi$
$587$	−25.6191	−1.05741	−0.528707	+	0.848805i	$0.677323\pi$
$588$	31.7069	1.30757
$589$	14.2481	0.587083
$590$	15.2928	0.629594
$591$	23.7728	0.977884
$592$	−0.242762	−0.00997747
$593$	7.74015	0.317850	0.158925	−	0.987291i	$-0.449197\pi$
$593$	7.74015	0.317850	0.158925	+	0.987291i	$0.449197\pi$
$594$	13.9205	0.571164
$595$	−10.3465	−0.424166
$596$	21.0520	0.862322
$597$	−3.04393	−0.124580
$598$	24.3771	0.996852
$599$	−19.4907	−0.796367	−0.398184	−	0.917306i	$-0.630359\pi$
$599$	−19.4907	−0.796367	−0.398184	+	0.917306i	$0.630359\pi$
$600$	2.16085	0.0882162
$601$	21.3851	0.872318	0.436159	−	0.899870i	$-0.356339\pi$
$601$	21.3851	0.872318	0.436159	+	0.899870i	$0.356339\pi$
$602$	−44.8917	−1.82965
$603$	55.3276	2.25311
$604$	−0.108882	−0.00443033
$605$	−3.75448	−0.152641
$606$	9.94439	0.403963
$607$	−31.6412	−1.28428	−0.642138	−	0.766589i	$-0.721953\pi$
$607$	−31.6412	−1.28428	−0.642138	+	0.766589i	$0.721953\pi$
$608$	−18.8963	−0.766348
$609$	−21.9312	−0.888699
$610$	−31.2324	−1.26456
$611$	16.1863	0.654827
$612$	−30.2367	−1.22224
$613$	12.4421	0.502531	0.251265	−	0.967918i	$-0.419153\pi$
$613$	12.4421	0.502531	0.251265	+	0.967918i	$0.419153\pi$
$614$	−30.1387	−1.21630
$615$	43.3540	1.74820
$616$	−14.6495	−0.590246
$617$	−1.84147	−0.0741346	−0.0370673	−	0.999313i	$-0.511802\pi$
$617$	−1.84147	−0.0741346	−0.0370673	+	0.999313i	$0.511802\pi$
$618$	4.69669	0.188929
$619$	−33.5204	−1.34730	−0.673650	−	0.739051i	$-0.735274\pi$
$619$	−33.5204	−1.34730	−0.673650	+	0.739051i	$0.735274\pi$
$620$	−32.7701	−1.31608
$621$	9.88215	0.396557
$622$	32.1652	1.28971
$623$	20.8968	0.837212
$624$	1.36738	0.0547389
$625$	−26.4493	−1.05797
$626$	35.2417	1.40854
$627$	25.3789	1.01353
$628$	1.92119	0.0766637
$629$	−2.52261	−0.100583
$630$	−35.1771	−1.40149
$631$	31.6029	1.25809	0.629045	−	0.777369i	$-0.283446\pi$
$631$	31.6029	1.25809	0.629045	+	0.777369i	$0.283446\pi$
$632$	−15.5241	−0.617514
$633$	24.7088	0.982088
$634$	48.2531	1.91638
$635$	21.6792	0.860312
$636$	33.9516	1.34627
$637$	−8.29518	−0.328667
$638$	−32.9981	−1.30641
$639$	−10.0560	−0.397811
$640$	40.9145	1.61729
$641$	−43.4450	−1.71597	−0.857987	−	0.513672i	$-0.828285\pi$
$641$	−43.4450	−1.71597	−0.857987	+	0.513672i	$0.828285\pi$
$642$	73.0811	2.88428
$643$	−29.3200	−1.15627	−0.578134	−	0.815942i	$-0.696219\pi$
$643$	−29.3200	−1.15627	−0.578134	+	0.815942i	$0.696219\pi$
$644$	−28.0098	−1.10374
$645$	−66.4122	−2.61498
$646$	−18.2988	−0.719958
$647$	32.3450	1.27161	0.635807	−	0.771848i	$-0.280668\pi$
$647$	32.3450	1.27161	0.635807	+	0.771848i	$0.280668\pi$
$648$	16.4168	0.644911
$649$	−8.92591	−0.350373
$650$	−1.52259	−0.0597210
$651$	20.7044	0.811471
$652$	−62.7842	−2.45882
$653$	13.4291	0.525523	0.262761	−	0.964861i	$-0.415367\pi$
$653$	13.4291	0.525523	0.262761	+	0.964861i	$0.415367\pi$
$654$	−5.92161	−0.231553
$655$	30.9012	1.20741
$656$	1.75581	0.0685528
$657$	−25.2309	−0.984350
$658$	−30.2914	−1.18088
$659$	−25.3452	−0.987310	−0.493655	−	0.869658i	$-0.664339\pi$
$659$	−25.3452	−0.987310	−0.493655	+	0.869658i	$0.664339\pi$
$660$	−58.3705	−2.27207
$661$	−11.0144	−0.428411	−0.214205	−	0.976789i	$-0.568716\pi$
$661$	−11.0144	−0.428411	−0.214205	+	0.976789i	$0.568716\pi$
$662$	−19.4121	−0.754473
$663$	14.2088	0.551824
$664$	−27.0113	−1.04824
$665$	−13.0709	−0.506869
$666$	−8.57662	−0.332337
$667$	−23.4254	−0.907033
$668$	31.5673	1.22138
$669$	−17.5403	−0.678147
$670$	−77.0110	−2.97519
$671$	18.2294	0.703737
$672$	−27.4590	−1.05925
$673$	29.2102	1.12597	0.562985	−	0.826467i	$-0.309653\pi$
$673$	29.2102	1.12597	0.562985	+	0.826467i	$0.309653\pi$
$674$	60.0356	2.31248
$675$	−0.617240	−0.0237576
$676$	−26.4424	−1.01702
$677$	19.6309	0.754478	0.377239	−	0.926116i	$-0.376874\pi$
$677$	19.6309	0.754478	0.377239	+	0.926116i	$0.376874\pi$
$678$	−43.5281	−1.67169
$679$	0.388566	0.0149118
$680$	15.6263	0.599241
$681$	−16.4812	−0.631562
$682$	31.1522	1.19288
$683$	−41.0700	−1.57150	−0.785749	−	0.618545i	$-0.787722\pi$
$683$	−41.0700	−1.57150	−0.785749	+	0.618545i	$0.787722\pi$
$684$	−38.1984	−1.46055
$685$	2.21648	0.0846874
$686$	43.8866	1.67560
$687$	31.0262	1.18372
$688$	−2.68965	−0.102542
$689$	−8.88244	−0.338394
$690$	−67.4894	−2.56928
$691$	−3.77631	−0.143658	−0.0718288	−	0.997417i	$-0.522884\pi$
$691$	−3.77631	−0.143658	−0.0718288	+	0.997417i	$0.522884\pi$
$692$	70.2735	2.67140
$693$	20.5318	0.779938
$694$	2.49516	0.0947151
$695$	−39.5879	−1.50165
$696$	33.1226	1.25551
$697$	18.2451	0.691082
$698$	−76.9527	−2.91270
$699$	−58.0331	−2.19501
$700$	1.74949	0.0661246
$701$	−36.7183	−1.38683	−0.693416	−	0.720538i	$-0.743895\pi$
$701$	−36.7183	−1.38683	−0.693416	+	0.720538i	$0.743895\pi$
$702$	9.84578	0.371605
$703$	−3.18685	−0.120194
$704$	−39.8289	−1.50111
$705$	−44.8127	−1.68774
$706$	18.3885	0.692060
$707$	2.98936	0.112426
$708$	24.1311	0.906903
$709$	24.8667	0.933889	0.466945	−	0.884287i	$-0.345355\pi$
$709$	24.8667	0.933889	0.466945	+	0.884287i	$0.345355\pi$
$710$	13.9971	0.525301
$711$	21.7575	0.815969
$712$	−31.5603	−1.18277
$713$	22.1150	0.828212
$714$	−26.5907	−0.995133
$715$	15.2709	0.571100
$716$	76.8423	2.87173
$717$	68.7199	2.56639
$718$	−49.6732	−1.85379
$719$	23.2438	0.866846	0.433423	−	0.901191i	$-0.357306\pi$
$719$	23.2438	0.866846	0.433423	+	0.901191i	$0.357306\pi$
$720$	−2.10761	−0.0785461
$721$	1.41186	0.0525805
$722$	20.1307	0.749187
$723$	−14.6912	−0.546371
$724$	41.1477	1.52924
$725$	1.46315	0.0543400
$726$	−9.64907	−0.358111
$727$	−2.05092	−0.0760643	−0.0380321	−	0.999277i	$-0.512109\pi$
$727$	−2.05092	−0.0760643	−0.0380321	+	0.999277i	$0.512109\pi$
$728$	−10.3614	−0.384020
$729$	−38.6009	−1.42966
$730$	35.1191	1.29982
$731$	−27.9489	−1.03373
$732$	−49.2829	−1.82155
$733$	−13.0524	−0.482102	−0.241051	−	0.970512i	$-0.577492\pi$
$733$	−13.0524	−0.482102	−0.241051	+	0.970512i	$0.577492\pi$
$734$	−63.2039	−2.33290
$735$	22.9657	0.847104
$736$	−29.3297	−1.08111
$737$	44.9489	1.65571
$738$	62.0315	2.28341
$739$	−7.45642	−0.274289	−0.137144	−	0.990551i	$-0.543792\pi$
$739$	−7.45642	−0.274289	−0.137144	+	0.990551i	$0.543792\pi$
$740$	7.32965	0.269443
$741$	17.9502	0.659417
$742$	16.6228	0.610243
$743$	−23.4636	−0.860796	−0.430398	−	0.902639i	$-0.641627\pi$
$743$	−23.4636	−0.860796	−0.430398	+	0.902639i	$0.641627\pi$
$744$	−31.2698	−1.14641
$745$	15.2482	0.558651
$746$	−36.7161	−1.34427
$747$	37.8573	1.38513
$748$	−24.5647	−0.898174
$749$	21.9687	0.802720
$750$	−64.0050	−2.33713
$751$	−32.2120	−1.17543	−0.587716	−	0.809067i	$-0.699973\pi$
$751$	−32.2120	−1.17543	−0.587716	+	0.809067i	$0.699973\pi$
$752$	−1.81489	−0.0661821
$753$	−35.9933	−1.31167
$754$	−23.3391	−0.849962
$755$	−0.0788643	−0.00287016
$756$	−11.3130	−0.411451
$757$	48.7545	1.77201	0.886006	−	0.463673i	$-0.153469\pi$
$757$	48.7545	1.77201	0.886006	+	0.463673i	$0.153469\pi$
$758$	−61.5563	−2.23583
$759$	39.3914	1.42982
$760$	19.7409	0.716079
$761$	22.9274	0.831119	0.415560	−	0.909566i	$-0.363586\pi$
$761$	22.9274	0.831119	0.415560	+	0.909566i	$0.363586\pi$
$762$	55.7158	2.01837
$763$	−1.78008	−0.0644433
$764$	−38.3207	−1.38639
$765$	−21.9008	−0.791825
$766$	−65.8686	−2.37993
$767$	−6.31319	−0.227956
$768$	37.4531	1.35147
$769$	13.6321	0.491588	0.245794	−	0.969322i	$-0.420951\pi$
$769$	13.6321	0.491588	0.245794	+	0.969322i	$0.420951\pi$
$770$	−28.5784	−1.02989
$771$	−62.4027	−2.24738
$772$	−22.0806	−0.794697
$773$	−1.06631	−0.0383525	−0.0191763	−	0.999816i	$-0.506104\pi$
$773$	−1.06631	−0.0383525	−0.0191763	+	0.999816i	$0.506104\pi$
$774$	−95.0235	−3.41555
$775$	−1.38130	−0.0496179
$776$	−0.586849	−0.0210667
$777$	−4.63093	−0.166134
$778$	−59.3447	−2.12761
$779$	23.0493	0.825827
$780$	−41.2848	−1.47823
$781$	−8.16966	−0.292333
$782$	−28.4023	−1.01566
$783$	−9.46140	−0.338123
$784$	0.930098	0.0332178
$785$	1.39154	0.0496662
$786$	79.4164	2.83269
$787$	40.9392	1.45932	0.729662	−	0.683808i	$-0.239678\pi$
$787$	40.9392	1.45932	0.729662	+	0.683808i	$0.239678\pi$
$788$	−29.0691	−1.03554
$789$	−18.5330	−0.659793
$790$	−30.2844	−1.07747
$791$	−13.0849	−0.465244
$792$	−31.0091	−1.10186
$793$	12.8934	0.457859
$794$	12.6845	0.450155
$795$	24.5916	0.872174
$796$	3.72207	0.131925
$797$	−20.2227	−0.716323	−0.358162	−	0.933660i	$-0.616596\pi$
$797$	−20.2227	−0.716323	−0.358162	+	0.933660i	$0.616596\pi$
$798$	−33.5924	−1.18916
$799$	−18.8590	−0.667184
$800$	1.83193	0.0647686
$801$	44.2328	1.56289
$802$	16.9776	0.599502
$803$	−20.4979	−0.723356
$804$	−121.519	−4.28564
$805$	−20.2878	−0.715052
$806$	22.0336	0.776100
$807$	9.28848	0.326970
$808$	−4.51481	−0.158831
$809$	39.8473	1.40096	0.700479	−	0.713673i	$-0.252970\pi$
$809$	39.8473	1.40096	0.700479	+	0.713673i	$0.252970\pi$
$810$	32.0259	1.12528
$811$	−40.2351	−1.41284	−0.706422	−	0.707791i	$-0.749692\pi$
$811$	−40.2351	−1.41284	−0.706422	+	0.707791i	$0.749692\pi$
$812$	26.8172	0.941099
$813$	−37.8445	−1.32726
$814$	−6.96777	−0.244220
$815$	−45.4753	−1.59293
$816$	−1.59316	−0.0557719
$817$	−35.3083	−1.23528
$818$	60.8784	2.12856
$819$	14.5219	0.507436
$820$	−53.0126	−1.85128
$821$	10.2981	0.359406	0.179703	−	0.983721i	$-0.442486\pi$
$821$	10.2981	0.359406	0.179703	+	0.983721i	$0.442486\pi$
$822$	5.69639	0.198684
$823$	−0.385341	−0.0134322	−0.00671608	−	0.999977i	$-0.502138\pi$
$823$	−0.385341	−0.0134322	−0.00671608	+	0.999977i	$0.502138\pi$
$824$	−2.13233	−0.0742831
$825$	−2.46039	−0.0856599
$826$	11.8147	0.411085
$827$	25.4124	0.883674	0.441837	−	0.897095i	$-0.354327\pi$
$827$	25.4124	0.883674	0.441837	+	0.897095i	$0.354327\pi$
$828$	−59.2891	−2.06044
$829$	−24.1983	−0.840441	−0.420220	−	0.907422i	$-0.638047\pi$
$829$	−24.1983	−0.840441	−0.420220	+	0.907422i	$0.638047\pi$
$830$	−52.6939	−1.82903
$831$	17.5814	0.609891
$832$	−28.1705	−0.976637
$833$	9.66491	0.334869
$834$	−101.741	−3.52302
$835$	22.8646	0.791261
$836$	−31.0329	−1.07330
$837$	8.93214	0.308740
$838$	46.0720	1.59153
$839$	21.5310	0.743334	0.371667	−	0.928366i	$-0.378786\pi$
$839$	21.5310	0.743334	0.371667	+	0.928366i	$0.378786\pi$
$840$	28.6863	0.989771
$841$	−6.57203	−0.226622
$842$	−29.3398	−1.01112
$843$	−41.0881	−1.41515
$844$	−30.2136	−1.03999
$845$	−19.1526	−0.658869
$846$	−64.1187	−2.20445
$847$	−2.90058	−0.0996652
$848$	0.995944	0.0342009
$849$	−8.90041	−0.305461
$850$	1.77401	0.0608480
$851$	−4.94642	−0.169561
$852$	22.0866	0.756675
$853$	23.7664	0.813745	0.406872	−	0.913485i	$-0.366619\pi$
$853$	23.7664	0.813745	0.406872	+	0.913485i	$0.366619\pi$
$854$	−24.1291	−0.825680
$855$	−27.6676	−0.946211
$856$	−33.1793	−1.13404
$857$	13.5643	0.463347	0.231673	−	0.972794i	$-0.425580\pi$
$857$	13.5643	0.463347	0.231673	+	0.972794i	$0.425580\pi$
$858$	39.2465	1.33985
$859$	2.32578	0.0793546	0.0396773	−	0.999213i	$-0.487367\pi$
$859$	2.32578	0.0793546	0.0396773	+	0.999213i	$0.487367\pi$
$860$	81.2078	2.76916
$861$	33.4938	1.14146
$862$	45.3484	1.54457
$863$	45.2954	1.54187	0.770936	−	0.636912i	$-0.219789\pi$
$863$	45.2954	1.54187	0.770936	+	0.636912i	$0.219789\pi$
$864$	−11.8461	−0.403013
$865$	50.9000	1.73065
$866$	−74.8867	−2.54475
$867$	27.6710	0.939756
$868$	−25.3171	−0.859318
$869$	17.6761	0.599620
$870$	64.6159	2.19068
$871$	31.7918	1.07722
$872$	2.68845	0.0910423
$873$	0.822489	0.0278370
$874$	−35.8810	−1.21369
$875$	−19.2404	−0.650444
$876$	55.4159	1.87233
$877$	−10.3254	−0.348664	−0.174332	−	0.984687i	$-0.555777\pi$
$877$	−10.3254	−0.348664	−0.174332	+	0.984687i	$0.555777\pi$
$878$	1.61419	0.0544762
$879$	−55.4488	−1.87024
$880$	−1.71225	−0.0577200
$881$	39.2242	1.32150	0.660748	−	0.750608i	$-0.270239\pi$
$881$	39.2242	1.32150	0.660748	+	0.750608i	$0.270239\pi$
$882$	32.8597	1.10644
$883$	6.47276	0.217826	0.108913	−	0.994051i	$-0.465263\pi$
$883$	6.47276	0.217826	0.108913	+	0.994051i	$0.465263\pi$
$884$	−17.3743	−0.584361
$885$	17.4785	0.587532
$886$	−31.7336	−1.06611
$887$	−15.6828	−0.526576	−0.263288	−	0.964717i	$-0.584807\pi$
$887$	−15.6828	−0.526576	−0.263288	+	0.964717i	$0.584807\pi$
$888$	6.99407	0.234706
$889$	16.7486	0.561729
$890$	−61.5680	−2.06377
$891$	−18.6925	−0.626224
$892$	21.4480	0.718132
$893$	−23.8249	−0.797268
$894$	39.1881	1.31065
$895$	55.6579	1.86044
$896$	31.6092	1.05599
$897$	27.8611	0.930255
$898$	−16.8354	−0.561804
$899$	−21.1734	−0.706172
$900$	3.70320	0.123440
$901$	10.3491	0.344780
$902$	50.3952	1.67798
$903$	−51.3078	−1.70742
$904$	19.7620	0.657275
$905$	29.8038	0.990711
$906$	−0.202682	−0.00673367
$907$	39.0872	1.29787	0.648935	−	0.760844i	$-0.275215\pi$
$907$	39.0872	1.29787	0.648935	+	0.760844i	$0.275215\pi$
$908$	20.1530	0.668801
$909$	6.32766	0.209875
$910$	−20.2132	−0.670060
$911$	28.8625	0.956256	0.478128	−	0.878290i	$-0.341315\pi$
$911$	28.8625	0.956256	0.478128	+	0.878290i	$0.341315\pi$
$912$	−2.01267	−0.0666460
$913$	30.7558	1.01787
$914$	48.4152	1.60143
$915$	−35.6962	−1.18008
$916$	−37.9384	−1.25352
$917$	23.8732	0.788361
$918$	−11.4715	−0.378617
$919$	−9.09122	−0.299892	−0.149946	−	0.988694i	$-0.547910\pi$
$919$	−9.09122	−0.299892	−0.149946	+	0.988694i	$0.547910\pi$
$920$	30.6406	1.01019
$921$	−34.4462	−1.13504
$922$	9.92824	0.326969
$923$	−5.77830	−0.190195
$924$	−45.0951	−1.48352
$925$	0.308954	0.0101583
$926$	52.8989	1.73836
$927$	2.98853	0.0981561
$928$	28.0809	0.921800
$929$	14.8996	0.488839	0.244420	−	0.969670i	$-0.421403\pi$
$929$	14.8996	0.488839	0.244420	+	0.969670i	$0.421403\pi$
$930$	−61.0014	−2.00031
$931$	12.2098	0.400161
$932$	70.9620	2.32444
$933$	36.7623	1.20354
$934$	−85.7230	−2.80494
$935$	−17.7925	−0.581877
$936$	−21.9323	−0.716881
$937$	−21.3836	−0.698570	−0.349285	−	0.937016i	$-0.613576\pi$
$937$	−21.3836	−0.698570	−0.349285	+	0.937016i	$0.613576\pi$
$938$	−59.4960	−1.94261
$939$	40.2786	1.31444
$940$	54.7963	1.78726
$941$	57.8756	1.88669	0.943346	−	0.331811i	$-0.107660\pi$
$941$	57.8756	1.88669	0.943346	+	0.331811i	$0.107660\pi$
$942$	3.57628	0.116521
$943$	35.7756	1.16501
$944$	0.707867	0.0230391
$945$	−8.19416	−0.266556
$946$	−77.1984	−2.50994
$947$	−19.8508	−0.645064	−0.322532	−	0.946559i	$-0.604534\pi$
$947$	−19.8508	−0.645064	−0.322532	+	0.946559i	$0.604534\pi$
$948$	−47.7871	−1.55205
$949$	−14.4979	−0.470623
$950$	2.24113	0.0727118
$951$	55.1496	1.78835
$952$	12.0723	0.391267
$953$	−48.2282	−1.56226	−0.781132	−	0.624366i	$-0.785357\pi$
$953$	−48.2282	−1.56226	−0.781132	+	0.624366i	$0.785357\pi$
$954$	35.1860	1.13919
$955$	−27.7562	−0.898168
$956$	−84.0297	−2.71771
$957$	−37.7143	−1.21913
$958$	−42.8801	−1.38539
$959$	1.71238	0.0552955
$960$	77.9918	2.51718
$961$	−11.0110	−0.355195
$962$	−4.92822	−0.158892
$963$	46.5018	1.49850
$964$	17.9642	0.578586
$965$	−15.9932	−0.514840
$966$	−52.1400	−1.67758
$967$	32.9861	1.06076	0.530380	−	0.847760i	$-0.322049\pi$
$967$	32.9861	1.06076	0.530380	+	0.847760i	$0.322049\pi$
$968$	4.38074	0.140802
$969$	−20.9142	−0.671860
$970$	−1.14483	−0.0367583
$971$	−16.3461	−0.524571	−0.262285	−	0.964990i	$-0.584476\pi$
$971$	−16.3461	−0.524571	−0.262285	+	0.964990i	$0.584476\pi$
$972$	69.6011	2.23246
$973$	−30.5843	−0.980486
$974$	23.5711	0.755268
$975$	−1.74021	−0.0557312
$976$	−1.44567	−0.0462749
$977$	35.7362	1.14330	0.571651	−	0.820497i	$-0.306303\pi$
$977$	35.7362	1.14330	0.571651	+	0.820497i	$0.306303\pi$
$978$	−116.872	−3.73716
$979$	35.9353	1.14850
$980$	−28.0822	−0.897051
$981$	−3.76795	−0.120301
$982$	−95.0813	−3.03417
$983$	27.5874	0.879902	0.439951	−	0.898022i	$-0.354996\pi$
$983$	27.5874	0.879902	0.439951	+	0.898022i	$0.354996\pi$
$984$	−50.5855	−1.61261
$985$	−21.0551	−0.670871
$986$	27.1930	0.866001
$987$	−34.6208	−1.10199
$988$	−21.9492	−0.698298
$989$	−54.8032	−1.74264
$990$	−60.4927	−1.92258
$991$	−47.8239	−1.51918	−0.759589	−	0.650404i	$-0.774600\pi$
$991$	−47.8239	−1.51918	−0.759589	+	0.650404i	$0.774600\pi$
$992$	−26.5101	−0.841696
$993$	−22.1866	−0.704069
$994$	10.8137	0.342989
$995$	2.69594	0.0854671
$996$	−83.1479	−2.63464
$997$	9.60225	0.304106	0.152053	−	0.988372i	$-0.451411\pi$
$997$	9.60225	0.304106	0.152053	+	0.988372i	$0.451411\pi$
$998$	−26.0897	−0.825854
$999$	−1.99784	−0.0632088

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4033.2.a.f.1.9	✓	85

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4033.2.a.f.1.9	✓	85	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:	\( N \)	\(=\)	\( 4033 = 37 \cdot 109 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4033.a (trivial)

\(f(q)\)	\(=\)	\(q-2.27621 q^{2} -2.60153 q^{3} +3.18111 q^{4} +2.30412 q^{5} +5.92161 q^{6} +1.78008 q^{7} -2.68845 q^{8} +3.76795 q^{9} +O(q^{10})\)	\(q-2.27621 q^{2} -2.60153 q^{3} +3.18111 q^{4} +2.30412 q^{5} +5.92161 q^{6} +1.78008 q^{7} -2.68845 q^{8} +3.76795 q^{9} -5.24464 q^{10} +3.06113 q^{11} -8.27574 q^{12} +2.16510 q^{13} -4.05183 q^{14} -5.99422 q^{15} -0.242762 q^{16} -2.52261 q^{17} -8.57662 q^{18} -3.18685 q^{19} +7.32965 q^{20} -4.63093 q^{21} -6.96777 q^{22} -4.94642 q^{23} +6.99407 q^{24} +0.308954 q^{25} -4.92822 q^{26} -1.99784 q^{27} +5.66263 q^{28} +4.73582 q^{29} +13.6441 q^{30} -4.47090 q^{31} +5.92947 q^{32} -7.96362 q^{33} +5.74198 q^{34} +4.10152 q^{35} +11.9863 q^{36} +1.00000 q^{37} +7.25393 q^{38} -5.63258 q^{39} -6.19450 q^{40} -7.23262 q^{41} +10.5409 q^{42} +11.0794 q^{43} +9.73780 q^{44} +8.68179 q^{45} +11.2591 q^{46} +7.47598 q^{47} +0.631553 q^{48} -3.83131 q^{49} -0.703243 q^{50} +6.56264 q^{51} +6.88743 q^{52} -4.10255 q^{53} +4.54749 q^{54} +7.05321 q^{55} -4.78565 q^{56} +8.29068 q^{57} -10.7797 q^{58} -2.91589 q^{59} -19.0683 q^{60} +5.95510 q^{61} +10.1767 q^{62} +6.70725 q^{63} -13.0112 q^{64} +4.98865 q^{65} +18.1268 q^{66} +14.6837 q^{67} -8.02470 q^{68} +12.8683 q^{69} -9.33589 q^{70} -2.66884 q^{71} -10.1299 q^{72} -6.69619 q^{73} -2.27621 q^{74} -0.803752 q^{75} -10.1377 q^{76} +5.44907 q^{77} +12.8209 q^{78} +5.77436 q^{79} -0.559353 q^{80} -6.10641 q^{81} +16.4629 q^{82} +10.0472 q^{83} -14.7315 q^{84} -5.81239 q^{85} -25.2189 q^{86} -12.3204 q^{87} -8.22969 q^{88} +11.7392 q^{89} -19.7615 q^{90} +3.85406 q^{91} -15.7351 q^{92} +11.6312 q^{93} -17.0169 q^{94} -7.34288 q^{95} -15.4257 q^{96} +0.218286 q^{97} +8.72085 q^{98} +11.5342 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 85 q + 11 q^{2} + 21 q^{3} + 93 q^{4} + 12 q^{5} + 4 q^{6} + 17 q^{7} + 30 q^{8} + 98 q^{9}+O(q^{10}) \) 85 * q + 11 * q^2 + 21 * q^3 + 93 * q^4 + 12 * q^5 + 4 * q^6 + 17 * q^7 + 30 * q^8 + 98 * q^9	\( 85 q + 11 q^{2} + 21 q^{3} + 93 q^{4} + 12 q^{5} + 4 q^{6} + 17 q^{7} + 30 q^{8} + 98 q^{9} + 9 q^{10} + 37 q^{11} + 44 q^{12} + 14 q^{13} + 26 q^{14} + 27 q^{15} + 85 q^{16} + 34 q^{17} + 3 q^{18} + 15 q^{19} + 15 q^{20} + 17 q^{21} + q^{22} + 72 q^{23} + 15 q^{24} + 85 q^{25} + 33 q^{26} + 69 q^{27} + 7 q^{28} + 19 q^{29} - 9 q^{30} + 23 q^{31} + 51 q^{32} + 32 q^{33} + 49 q^{34} + 40 q^{35} + 121 q^{36} + 85 q^{37} + 84 q^{38} + 39 q^{39} + 22 q^{40} + 55 q^{41} - 28 q^{42} + 78 q^{44} + 28 q^{45} + 17 q^{46} + 184 q^{47} + 97 q^{48} + 88 q^{49} + 26 q^{50} + 27 q^{51} + 73 q^{52} + 64 q^{53} + 31 q^{54} + 39 q^{55} + 68 q^{56} - 33 q^{57} + 28 q^{58} + 60 q^{59} - 22 q^{60} + 7 q^{61} + 70 q^{62} + 28 q^{63} + 102 q^{64} + 17 q^{65} - 15 q^{66} + 82 q^{67} + 92 q^{68} + 22 q^{69} - 41 q^{70} + 113 q^{71} - 19 q^{73} + 11 q^{74} + 45 q^{75} + 34 q^{76} + 64 q^{77} + 29 q^{78} + 23 q^{79} + 54 q^{80} + 149 q^{81} + 4 q^{82} + 100 q^{83} - 49 q^{84} - 5 q^{85} - 24 q^{86} + 65 q^{87} + 14 q^{88} + 84 q^{89} - 21 q^{90} + 32 q^{91} + 95 q^{92} + 19 q^{93} - 47 q^{94} + 102 q^{95} + 29 q^{96} + 7 q^{97} + 26 q^{98} + 107 q^{99}+O(q^{100}) \) 85 * q + 11 * q^2 + 21 * q^3 + 93 * q^4 + 12 * q^5 + 4 * q^6 + 17 * q^7 + 30 * q^8 + 98 * q^9 + 9 * q^10 + 37 * q^11 + 44 * q^12 + 14 * q^13 + 26 * q^14 + 27 * q^15 + 85 * q^16 + 34 * q^17 + 3 * q^18 + 15 * q^19 + 15 * q^20 + 17 * q^21 + q^22 + 72 * q^23 + 15 * q^24 + 85 * q^25 + 33 * q^26 + 69 * q^27 + 7 * q^28 + 19 * q^29 - 9 * q^30 + 23 * q^31 + 51 * q^32 + 32 * q^33 + 49 * q^34 + 40 * q^35 + 121 * q^36 + 85 * q^37 + 84 * q^38 + 39 * q^39 + 22 * q^40 + 55 * q^41 - 28 * q^42 + 78 * q^44 + 28 * q^45 + 17 * q^46 + 184 * q^47 + 97 * q^48 + 88 * q^49 + 26 * q^50 + 27 * q^51 + 73 * q^52 + 64 * q^53 + 31 * q^54 + 39 * q^55 + 68 * q^56 - 33 * q^57 + 28 * q^58 + 60 * q^59 - 22 * q^60 + 7 * q^61 + 70 * q^62 + 28 * q^63 + 102 * q^64 + 17 * q^65 - 15 * q^66 + 82 * q^67 + 92 * q^68 + 22 * q^69 - 41 * q^70 + 113 * q^71 - 19 * q^73 + 11 * q^74 + 45 * q^75 + 34 * q^76 + 64 * q^77 + 29 * q^78 + 23 * q^79 + 54 * q^80 + 149 * q^81 + 4 * q^82 + 100 * q^83 - 49 * q^84 - 5 * q^85 - 24 * q^86 + 65 * q^87 + 14 * q^88 + 84 * q^89 - 21 * q^90 + 32 * q^91 + 95 * q^92 + 19 * q^93 - 47 * q^94 + 102 * q^95 + 29 * q^96 + 7 * q^97 + 26 * q^98 + 107 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more