LMFDB - Embedded newform 8045.2.a.d.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8045,2,Mod(1,8045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8045, 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("8045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8045 = 5 \cdot 1609 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8045.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:	$64.2396484261$
Analytic rank:	$0$
Dimension:	$141$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	8045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.59840 q^{2} +0.965890 q^{3} +4.75167 q^{4} -1.00000 q^{5} -2.50977 q^{6} -4.65256 q^{7} -7.14995 q^{8} -2.06706 q^{9} +O(q^{10})$	\(q-2.59840 q^{2} +0.965890 q^{3} +4.75167 q^{4} -1.00000 q^{5} -2.50977 q^{6} -4.65256 q^{7} -7.14995 q^{8} -2.06706 q^{9} +2.59840 q^{10} +0.588989 q^{11} +4.58959 q^{12} -1.68304 q^{13} +12.0892 q^{14} -0.965890 q^{15} +9.07506 q^{16} +5.37113 q^{17} +5.37104 q^{18} +4.46235 q^{19} -4.75167 q^{20} -4.49386 q^{21} -1.53043 q^{22} -3.73737 q^{23} -6.90606 q^{24} +1.00000 q^{25} +4.37320 q^{26} -4.89422 q^{27} -22.1075 q^{28} -8.08041 q^{29} +2.50977 q^{30} -3.88729 q^{31} -9.28074 q^{32} +0.568899 q^{33} -13.9563 q^{34} +4.65256 q^{35} -9.82198 q^{36} -5.98658 q^{37} -11.5950 q^{38} -1.62563 q^{39} +7.14995 q^{40} +7.11079 q^{41} +11.6768 q^{42} -6.88086 q^{43} +2.79869 q^{44} +2.06706 q^{45} +9.71117 q^{46} -4.46008 q^{47} +8.76551 q^{48} +14.6463 q^{49} -2.59840 q^{50} +5.18792 q^{51} -7.99725 q^{52} -2.36567 q^{53} +12.7171 q^{54} -0.588989 q^{55} +33.2656 q^{56} +4.31014 q^{57} +20.9961 q^{58} -5.97416 q^{59} -4.58959 q^{60} -4.95979 q^{61} +10.1007 q^{62} +9.61711 q^{63} +5.96493 q^{64} +1.68304 q^{65} -1.47823 q^{66} +3.93113 q^{67} +25.5219 q^{68} -3.60988 q^{69} -12.0892 q^{70} +6.50757 q^{71} +14.7794 q^{72} -3.35748 q^{73} +15.5555 q^{74} +0.965890 q^{75} +21.2036 q^{76} -2.74031 q^{77} +4.22403 q^{78} -4.00680 q^{79} -9.07506 q^{80} +1.47390 q^{81} -18.4767 q^{82} -8.75675 q^{83} -21.3534 q^{84} -5.37113 q^{85} +17.8792 q^{86} -7.80478 q^{87} -4.21124 q^{88} -0.109338 q^{89} -5.37104 q^{90} +7.83044 q^{91} -17.7588 q^{92} -3.75469 q^{93} +11.5891 q^{94} -4.46235 q^{95} -8.96417 q^{96} +3.96360 q^{97} -38.0570 q^{98} -1.21747 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 141 q - 8 q^{2} + 11 q^{3} + 158 q^{4} - 141 q^{5} + 23 q^{6} + 29 q^{7} - 21 q^{8} + 160 q^{9}+O(q^{10}) $ 141 * q - 8 * q^2 + 11 * q^3 + 158 * q^4 - 141 * q^5 + 23 * q^6 + 29 * q^7 - 21 * q^8 + 160 * q^9	\( 141 q - 8 q^{2} + 11 q^{3} + 158 q^{4} - 141 q^{5} + 23 q^{6} + 29 q^{7} - 21 q^{8} + 160 q^{9} + 8 q^{10} + 32 q^{11} + 17 q^{12} + 35 q^{13} + 18 q^{14} - 11 q^{15} + 188 q^{16} - 13 q^{17} - 16 q^{18} + 152 q^{19} - 158 q^{20} + 40 q^{21} + 14 q^{22} - 77 q^{23} + 69 q^{24} + 141 q^{25} + 27 q^{26} + 38 q^{27} + 67 q^{28} + 22 q^{29} - 23 q^{30} + 86 q^{31} - 65 q^{32} + 51 q^{33} + 79 q^{34} - 29 q^{35} + 191 q^{36} + 45 q^{37} - 9 q^{38} + 55 q^{39} + 21 q^{40} + 36 q^{41} + 6 q^{42} + 132 q^{43} + 74 q^{44} - 160 q^{45} + 72 q^{46} - 16 q^{47} + 22 q^{48} + 212 q^{49} - 8 q^{50} + 82 q^{51} + 106 q^{52} - 28 q^{53} + 86 q^{54} - 32 q^{55} + 27 q^{56} + 10 q^{57} + 23 q^{58} + 71 q^{59} - 17 q^{60} + 116 q^{61} - 36 q^{62} + 45 q^{63} + 237 q^{64} - 35 q^{65} + 69 q^{66} + 99 q^{67} - 7 q^{68} + 45 q^{69} - 18 q^{70} + 34 q^{71} - 53 q^{72} + 125 q^{73} + 50 q^{74} + 11 q^{75} + 271 q^{76} - 31 q^{77} + 2 q^{78} + 101 q^{79} - 188 q^{80} + 221 q^{81} + 67 q^{82} + 67 q^{83} + 141 q^{84} + 13 q^{85} + 48 q^{86} - 21 q^{87} + 71 q^{88} + 79 q^{89} + 16 q^{90} + 228 q^{91} - 198 q^{92} - 12 q^{93} + 114 q^{94} - 152 q^{95} + 129 q^{96} + 98 q^{97} - 31 q^{98} + 195 q^{99}+O(q^{100}) \) 141 * q - 8 * q^2 + 11 * q^3 + 158 * q^4 - 141 * q^5 + 23 * q^6 + 29 * q^7 - 21 * q^8 + 160 * q^9 + 8 * q^10 + 32 * q^11 + 17 * q^12 + 35 * q^13 + 18 * q^14 - 11 * q^15 + 188 * q^16 - 13 * q^17 - 16 * q^18 + 152 * q^19 - 158 * q^20 + 40 * q^21 + 14 * q^22 - 77 * q^23 + 69 * q^24 + 141 * q^25 + 27 * q^26 + 38 * q^27 + 67 * q^28 + 22 * q^29 - 23 * q^30 + 86 * q^31 - 65 * q^32 + 51 * q^33 + 79 * q^34 - 29 * q^35 + 191 * q^36 + 45 * q^37 - 9 * q^38 + 55 * q^39 + 21 * q^40 + 36 * q^41 + 6 * q^42 + 132 * q^43 + 74 * q^44 - 160 * q^45 + 72 * q^46 - 16 * q^47 + 22 * q^48 + 212 * q^49 - 8 * q^50 + 82 * q^51 + 106 * q^52 - 28 * q^53 + 86 * q^54 - 32 * q^55 + 27 * q^56 + 10 * q^57 + 23 * q^58 + 71 * q^59 - 17 * q^60 + 116 * q^61 - 36 * q^62 + 45 * q^63 + 237 * q^64 - 35 * q^65 + 69 * q^66 + 99 * q^67 - 7 * q^68 + 45 * q^69 - 18 * q^70 + 34 * q^71 - 53 * q^72 + 125 * q^73 + 50 * q^74 + 11 * q^75 + 271 * q^76 - 31 * q^77 + 2 * q^78 + 101 * q^79 - 188 * q^80 + 221 * q^81 + 67 * q^82 + 67 * q^83 + 141 * q^84 + 13 * q^85 + 48 * q^86 - 21 * q^87 + 71 * q^88 + 79 * q^89 + 16 * q^90 + 228 * q^91 - 198 * q^92 - 12 * q^93 + 114 * q^94 - 152 * q^95 + 129 * q^96 + 98 * q^97 - 31 * q^98 + 195 * 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 $

$2$	−2.59840	−1.83735	−0.918673	−	0.395020i	$-0.870738\pi$
$2$	−2.59840	−1.83735	−0.918673	+	0.395020i	$0.870738\pi$
$3$	0.965890	0.557657	0.278828	−	0.960341i	$-0.410054\pi$
$3$	0.965890	0.557657	0.278828	+	0.960341i	$0.410054\pi$
$4$	4.75167	2.37584
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−2.50977	−1.02461
$7$	−4.65256	−1.75850	−0.879251	−	0.476358i	$-0.841957\pi$
$7$	−4.65256	−1.75850	−0.879251	+	0.476358i	$0.841957\pi$
$8$	−7.14995	−2.52789
$9$	−2.06706	−0.689019
$10$	2.59840	0.821686
$11$	0.588989	0.177587	0.0887935	−	0.996050i	$-0.471699\pi$
$11$	0.588989	0.177587	0.0887935	+	0.996050i	$0.471699\pi$
$12$	4.58959	1.32490
$13$	−1.68304	−0.466791	−0.233395	−	0.972382i	$-0.574984\pi$
$13$	−1.68304	−0.466791	−0.233395	+	0.972382i	$0.574984\pi$
$14$	12.0892	3.23098
$15$	−0.965890	−0.249392
$16$	9.07506	2.26877
$17$	5.37113	1.30269	0.651346	−	0.758781i	$-0.274205\pi$
$17$	5.37113	1.30269	0.651346	+	0.758781i	$0.274205\pi$
$18$	5.37104	1.26597
$19$	4.46235	1.02373	0.511867	−	0.859065i	$-0.328954\pi$
$19$	4.46235	1.02373	0.511867	+	0.859065i	$0.328954\pi$
$20$	−4.75167	−1.06251
$21$	−4.49386	−0.980641
$22$	−1.53043	−0.326289
$23$	−3.73737	−0.779295	−0.389647	−	0.920964i	$-0.627403\pi$
$23$	−3.73737	−0.779295	−0.389647	+	0.920964i	$0.627403\pi$
$24$	−6.90606	−1.40969
$25$	1.00000	0.200000
$26$	4.37320	0.857656
$27$	−4.89422	−0.941893
$28$	−22.1075	−4.17792
$29$	−8.08041	−1.50049	−0.750247	−	0.661158i	$-0.770065\pi$
$29$	−8.08041	−1.50049	−0.750247	+	0.661158i	$0.770065\pi$
$30$	2.50977	0.458219
$31$	−3.88729	−0.698177	−0.349089	−	0.937090i	$-0.613509\pi$
$31$	−3.88729	−0.698177	−0.349089	+	0.937090i	$0.613509\pi$
$32$	−9.28074	−1.64062
$33$	0.568899	0.0990326
$34$	−13.9563	−2.39349
$35$	4.65256	0.786426
$36$	−9.82198	−1.63700
$37$	−5.98658	−0.984188	−0.492094	−	0.870542i	$-0.663768\pi$
$37$	−5.98658	−0.984188	−0.492094	+	0.870542i	$0.663768\pi$
$38$	−11.5950	−1.88095
$39$	−1.62563	−0.260309
$40$	7.14995	1.13051
$41$	7.11079	1.11052	0.555259	−	0.831677i	$-0.312619\pi$
$41$	7.11079	1.11052	0.555259	+	0.831677i	$0.312619\pi$
$42$	11.6768	1.80178
$43$	−6.88086	−1.04932	−0.524661	−	0.851311i	$-0.675808\pi$
$43$	−6.88086	−1.04932	−0.524661	+	0.851311i	$0.675808\pi$
$44$	2.79869	0.421918
$45$	2.06706	0.308139
$46$	9.71117	1.43183
$47$	−4.46008	−0.650570	−0.325285	−	0.945616i	$-0.605460\pi$
$47$	−4.46008	−0.650570	−0.325285	+	0.945616i	$0.605460\pi$
$48$	8.76551	1.26519
$49$	14.6463	2.09233
$50$	−2.59840	−0.367469
$51$	5.18792	0.726454
$52$	−7.99725	−1.10902
$53$	−2.36567	−0.324949	−0.162475	−	0.986713i	$-0.551948\pi$
$53$	−2.36567	−0.324949	−0.162475	+	0.986713i	$0.551948\pi$
$54$	12.7171	1.73058
$55$	−0.588989	−0.0794193
$56$	33.2656	4.44530
$57$	4.31014	0.570892
$58$	20.9961	2.75693
$59$	−5.97416	−0.777770	−0.388885	−	0.921286i	$-0.627140\pi$
$59$	−5.97416	−0.777770	−0.388885	+	0.921286i	$0.627140\pi$
$60$	−4.58959	−0.592514
$61$	−4.95979	−0.635036	−0.317518	−	0.948252i	$-0.602849\pi$
$61$	−4.95979	−0.635036	−0.317518	+	0.948252i	$0.602849\pi$
$62$	10.1007	1.28279
$63$	9.61711	1.21164
$64$	5.96493	0.745616
$65$	1.68304	0.208755
$66$	−1.47823	−0.181957
$67$	3.93113	0.480264	0.240132	−	0.970740i	$-0.422809\pi$
$67$	3.93113	0.480264	0.240132	+	0.970740i	$0.422809\pi$
$68$	25.5219	3.09498
$69$	−3.60988	−0.434579
$70$	−12.0892	−1.44494
$71$	6.50757	0.772306	0.386153	−	0.922435i	$-0.373804\pi$
$71$	6.50757	0.772306	0.386153	+	0.922435i	$0.373804\pi$
$72$	14.7794	1.74176
$73$	−3.35748	−0.392963	−0.196482	−	0.980507i	$-0.562952\pi$
$73$	−3.35748	−0.392963	−0.196482	+	0.980507i	$0.562952\pi$
$74$	15.5555	1.80829
$75$	0.965890	0.111531
$76$	21.2036	2.43222
$77$	−2.74031	−0.312287
$78$	4.22403	0.478277
$79$	−4.00680	−0.450800	−0.225400	−	0.974266i	$-0.572369\pi$
$79$	−4.00680	−0.450800	−0.225400	+	0.974266i	$0.572369\pi$
$80$	−9.07506	−1.01462
$81$	1.47390	0.163766
$82$	−18.4767	−2.04041
$83$	−8.75675	−0.961178	−0.480589	−	0.876946i	$-0.659577\pi$
$83$	−8.75675	−0.961178	−0.480589	+	0.876946i	$0.659577\pi$
$84$	−21.3534	−2.32984
$85$	−5.37113	−0.582581
$86$	17.8792	1.92797
$87$	−7.80478	−0.836761
$88$	−4.21124	−0.448920
$89$	−0.109338	−0.0115898	−0.00579491	−	0.999983i	$-0.501845\pi$
$89$	−0.109338	−0.0115898	−0.00579491	+	0.999983i	$0.501845\pi$
$90$	−5.37104	−0.566157
$91$	7.83044	0.820853
$92$	−17.7588	−1.85148
$93$	−3.75469	−0.389343
$94$	11.5891	1.19532
$95$	−4.46235	−0.457828
$96$	−8.96417	−0.914902
$97$	3.96360	0.402443	0.201221	−	0.979546i	$-0.435509\pi$
$97$	3.96360	0.402443	0.201221	+	0.979546i	$0.435509\pi$
$98$	−38.0570	−3.84434
$99$	−1.21747	−0.122361
$100$	4.75167	0.475167
$101$	−15.1693	−1.50940	−0.754701	−	0.656069i	$-0.772218\pi$
$101$	−15.1693	−1.50940	−0.754701	+	0.656069i	$0.772218\pi$
$102$	−13.4803	−1.33475
$103$	12.9859	1.27954	0.639768	−	0.768568i	$-0.279030\pi$
$103$	12.9859	1.27954	0.639768	+	0.768568i	$0.279030\pi$
$104$	12.0336	1.17999
$105$	4.49386	0.438556
$106$	6.14694	0.597044
$107$	−20.3845	−1.97064	−0.985322	−	0.170707i	$-0.945395\pi$
$107$	−20.3845	−1.97064	−0.985322	+	0.170707i	$0.945395\pi$
$108$	−23.2557	−2.23778
$109$	11.2150	1.07420	0.537102	−	0.843517i	$-0.319519\pi$
$109$	11.2150	1.07420	0.537102	+	0.843517i	$0.319519\pi$
$110$	1.53043	0.145921
$111$	−5.78238	−0.548839
$112$	−42.2223	−3.98963
$113$	3.63107	0.341582	0.170791	−	0.985307i	$-0.445368\pi$
$113$	3.63107	0.341582	0.170791	+	0.985307i	$0.445368\pi$
$114$	−11.1995	−1.04893
$115$	3.73737	0.348511
$116$	−38.3955	−3.56493
$117$	3.47894	0.321628
$118$	15.5233	1.42903
$119$	−24.9895	−2.29079
$120$	6.90606	0.630434
$121$	−10.6531	−0.968463
$122$	12.8875	1.16678
$123$	6.86824	0.619288
$124$	−18.4711	−1.65876
$125$	−1.00000	−0.0894427
$126$	−24.9891	−2.22620
$127$	−5.29870	−0.470184	−0.235092	−	0.971973i	$-0.575539\pi$
$127$	−5.29870	−0.470184	−0.235092	+	0.971973i	$0.575539\pi$
$128$	3.06222	0.270665
$129$	−6.64615	−0.585161
$130$	−4.37320	−0.383555
$131$	20.8012	1.81741	0.908705	−	0.417439i	$-0.137072\pi$
$131$	20.8012	1.81741	0.908705	+	0.417439i	$0.137072\pi$
$132$	2.70322	0.235285
$133$	−20.7614	−1.80024
$134$	−10.2146	−0.882411
$135$	4.89422	0.421227
$136$	−38.4033	−3.29306
$137$	−20.1405	−1.72072	−0.860358	−	0.509691i	$-0.829760\pi$
$137$	−20.1405	−1.72072	−0.860358	+	0.509691i	$0.829760\pi$
$138$	9.37992	0.798471
$139$	8.44415	0.716224	0.358112	−	0.933679i	$-0.383421\pi$
$139$	8.44415	0.716224	0.358112	+	0.933679i	$0.383421\pi$
$140$	22.1075	1.86842
$141$	−4.30795	−0.362795
$142$	−16.9093	−1.41899
$143$	−0.991292	−0.0828960
$144$	−18.7587	−1.56322
$145$	8.08041	0.671041
$146$	8.72407	0.722009
$147$	14.1467	1.16680
$148$	−28.4463	−2.33827
$149$	−8.66675	−0.710008	−0.355004	−	0.934865i	$-0.615520\pi$
$149$	−8.66675	−0.710008	−0.355004	+	0.934865i	$0.615520\pi$
$150$	−2.50977	−0.204922
$151$	−8.58582	−0.698704	−0.349352	−	0.936991i	$-0.613598\pi$
$151$	−8.58582	−0.698704	−0.349352	+	0.936991i	$0.613598\pi$
$152$	−31.9056	−2.58788
$153$	−11.1024	−0.897579
$154$	7.12042	0.573779
$155$	3.88729	0.312234
$156$	−7.72446	−0.618452
$157$	−2.80790	−0.224094	−0.112047	−	0.993703i	$-0.535741\pi$
$157$	−2.80790	−0.224094	−0.112047	+	0.993703i	$0.535741\pi$
$158$	10.4113	0.828275
$159$	−2.28497	−0.181210
$160$	9.28074	0.733707
$161$	17.3883	1.37039
$162$	−3.82977	−0.300895
$163$	−1.45654	−0.114085	−0.0570424	−	0.998372i	$-0.518167\pi$
$163$	−1.45654	−0.114085	−0.0570424	+	0.998372i	$0.518167\pi$
$164$	33.7882	2.63841
$165$	−0.568899	−0.0442887
$166$	22.7535	1.76602
$167$	−7.85927	−0.608169	−0.304084	−	0.952645i	$-0.598350\pi$
$167$	−7.85927	−0.608169	−0.304084	+	0.952645i	$0.598350\pi$
$168$	32.1309	2.47895
$169$	−10.1674	−0.782106
$170$	13.9563	1.07040
$171$	−9.22394	−0.705372
$172$	−32.6956	−2.49302
$173$	−4.15721	−0.316067	−0.158033	−	0.987434i	$-0.550515\pi$
$173$	−4.15721	−0.316067	−0.158033	+	0.987434i	$0.550515\pi$
$174$	20.2799	1.53742
$175$	−4.65256	−0.351701
$176$	5.34512	0.402903
$177$	−5.77038	−0.433728
$178$	0.284104	0.0212945
$179$	11.0966	0.829396	0.414698	−	0.909959i	$-0.363887\pi$
$179$	11.0966	0.829396	0.414698	+	0.909959i	$0.363887\pi$
$180$	9.82198	0.732087
$181$	−12.2167	−0.908057	−0.454029	−	0.890987i	$-0.650014\pi$
$181$	−12.2167	−0.908057	−0.454029	+	0.890987i	$0.650014\pi$
$182$	−20.3466	−1.50819
$183$	−4.79061	−0.354132
$184$	26.7220	1.96997
$185$	5.98658	0.440142
$186$	9.75618	0.715358
$187$	3.16354	0.231341
$188$	−21.1929	−1.54565
$189$	22.7707	1.65632
$190$	11.5950	0.841187
$191$	8.94626	0.647329	0.323664	−	0.946172i	$-0.395085\pi$
$191$	8.94626	0.647329	0.323664	+	0.946172i	$0.395085\pi$
$192$	5.76146	0.415798
$193$	5.30957	0.382191	0.191095	−	0.981571i	$-0.438796\pi$
$193$	5.30957	0.382191	0.191095	+	0.981571i	$0.438796\pi$
$194$	−10.2990	−0.739426
$195$	1.62563	0.116414
$196$	69.5946	4.97104
$197$	−11.6051	−0.826828	−0.413414	−	0.910543i	$-0.635664\pi$
$197$	−11.6051	−0.826828	−0.413414	+	0.910543i	$0.635664\pi$
$198$	3.16348	0.224819
$199$	−5.41252	−0.383684	−0.191842	−	0.981426i	$-0.561446\pi$
$199$	−5.41252	−0.383684	−0.191842	+	0.981426i	$0.561446\pi$
$200$	−7.14995	−0.505578
$201$	3.79704	0.267823
$202$	39.4159	2.77329
$203$	37.5946	2.63862
$204$	24.6513	1.72594
$205$	−7.11079	−0.496639
$206$	−33.7425	−2.35095
$207$	7.72535	0.536949
$208$	−15.2737	−1.05904
$209$	2.62828	0.181802
$210$	−11.6768	−0.805779
$211$	12.7294	0.876331	0.438166	−	0.898894i	$-0.355628\pi$
$211$	12.7294	0.876331	0.438166	+	0.898894i	$0.355628\pi$
$212$	−11.2409	−0.772026
$213$	6.28559	0.430682
$214$	52.9670	3.62075
$215$	6.88086	0.469271
$216$	34.9934	2.38100
$217$	18.0858	1.22775
$218$	−29.1411	−1.97368
$219$	−3.24296	−0.219139
$220$	−2.79869	−0.188687
$221$	−9.03982	−0.608084
$222$	15.0249	1.00841
$223$	19.2128	1.28659	0.643293	−	0.765620i	$-0.277568\pi$
$223$	19.2128	1.28659	0.643293	+	0.765620i	$0.277568\pi$
$224$	43.1792	2.88503
$225$	−2.06706	−0.137804
$226$	−9.43496	−0.627605
$227$	6.09258	0.404379	0.202189	−	0.979346i	$-0.435194\pi$
$227$	6.09258	0.404379	0.202189	+	0.979346i	$0.435194\pi$
$228$	20.4804	1.35635
$229$	20.9847	1.38671	0.693355	−	0.720596i	$-0.256132\pi$
$229$	20.9847	1.38671	0.693355	+	0.720596i	$0.256132\pi$
$230$	−9.71117	−0.640335
$231$	−2.64684	−0.174149
$232$	57.7745	3.79308
$233$	−2.96916	−0.194516	−0.0972580	−	0.995259i	$-0.531007\pi$
$233$	−2.96916	−0.194516	−0.0972580	+	0.995259i	$0.531007\pi$
$234$	−9.03966	−0.590941
$235$	4.46008	0.290944
$236$	−28.3873	−1.84785
$237$	−3.87012	−0.251392
$238$	64.9328	4.20897
$239$	0.866940	0.0560777	0.0280388	−	0.999607i	$-0.491074\pi$
$239$	0.866940	0.0560777	0.0280388	+	0.999607i	$0.491074\pi$
$240$	−8.76551	−0.565811
$241$	−1.08474	−0.0698743	−0.0349371	−	0.999390i	$-0.511123\pi$
$241$	−1.08474	−0.0698743	−0.0349371	+	0.999390i	$0.511123\pi$
$242$	27.6810	1.77940
$243$	16.1063	1.03322
$244$	−23.5673	−1.50874
$245$	−14.6463	−0.935720
$246$	−17.8464	−1.13785
$247$	−7.51031	−0.477869
$248$	27.7939	1.76491
$249$	−8.45805	−0.536007
$250$	2.59840	0.164337
$251$	5.64339	0.356208	0.178104	−	0.984012i	$-0.443004\pi$
$251$	5.64339	0.356208	0.178104	+	0.984012i	$0.443004\pi$
$252$	45.6974	2.87866
$253$	−2.20127	−0.138393
$254$	13.7681	0.863890
$255$	−5.18792	−0.324880
$256$	−19.8867	−1.24292
$257$	−15.4796	−0.965588	−0.482794	−	0.875734i	$-0.660378\pi$
$257$	−15.4796	−0.965588	−0.482794	+	0.875734i	$0.660378\pi$
$258$	17.2694	1.07514
$259$	27.8529	1.73070
$260$	7.99725	0.495968
$261$	16.7027	1.03387
$262$	−54.0498	−3.33921
$263$	−10.0896	−0.622150	−0.311075	−	0.950385i	$-0.600689\pi$
$263$	−10.0896	−0.622150	−0.311075	+	0.950385i	$0.600689\pi$
$264$	−4.06760	−0.250343
$265$	2.36567	0.145322
$266$	53.9463	3.30766
$267$	−0.105609	−0.00646314
$268$	18.6795	1.14103
$269$	−30.8029	−1.87809	−0.939044	−	0.343797i	$-0.888287\pi$
$269$	−30.8029	−1.87809	−0.939044	+	0.343797i	$0.888287\pi$
$270$	−12.7171	−0.773940
$271$	5.52308	0.335503	0.167752	−	0.985829i	$-0.446349\pi$
$271$	5.52308	0.335503	0.167752	+	0.985829i	$0.446349\pi$
$272$	48.7434	2.95550
$273$	7.56334	0.457754
$274$	52.3329	3.16155
$275$	0.588989	0.0355174
$276$	−17.1530	−1.03249
$277$	9.45628	0.568173	0.284086	−	0.958799i	$-0.408310\pi$
$277$	9.45628	0.568173	0.284086	+	0.958799i	$0.408310\pi$
$278$	−21.9413	−1.31595
$279$	8.03524	0.481057
$280$	−33.2656	−1.98800
$281$	25.2092	1.50385	0.751926	−	0.659247i	$-0.229125\pi$
$281$	25.2092	1.50385	0.751926	+	0.659247i	$0.229125\pi$
$282$	11.1938	0.666579
$283$	1.00084	0.0594939	0.0297469	−	0.999557i	$-0.490530\pi$
$283$	1.00084	0.0594939	0.0297469	+	0.999557i	$0.490530\pi$
$284$	30.9219	1.83487
$285$	−4.31014	−0.255311
$286$	2.57577	0.152308
$287$	−33.0834	−1.95285
$288$	19.1838	1.13042
$289$	11.8491	0.697004
$290$	−20.9961	−1.23293
$291$	3.82840	0.224425
$292$	−15.9537	−0.933617
$293$	−11.0640	−0.646367	−0.323184	−	0.946336i	$-0.604753\pi$
$293$	−11.0640	−0.646367	−0.323184	+	0.946336i	$0.604753\pi$
$294$	−36.7589	−2.14382
$295$	5.97416	0.347829
$296$	42.8037	2.48792
$297$	−2.88264	−0.167268
$298$	22.5197	1.30453
$299$	6.29013	0.363768
$300$	4.58959	0.264980
$301$	32.0136	1.84523
$302$	22.3094	1.28376
$303$	−14.6519	−0.841728
$304$	40.4961	2.32261
$305$	4.95979	0.283997
$306$	28.8486	1.64916
$307$	−5.85718	−0.334287	−0.167144	−	0.985933i	$-0.553454\pi$
$307$	−5.85718	−0.334287	−0.167144	+	0.985933i	$0.553454\pi$
$308$	−13.0211	−0.741944
$309$	12.5429	0.713542
$310$	−10.1007	−0.573682
$311$	−34.1170	−1.93460	−0.967299	−	0.253639i	$-0.918373\pi$
$311$	−34.1170	−1.93460	−0.967299	+	0.253639i	$0.918373\pi$
$312$	11.6232	0.658032
$313$	−30.1762	−1.70566	−0.852830	−	0.522189i	$-0.825115\pi$
$313$	−30.1762	−1.70566	−0.852830	+	0.522189i	$0.825115\pi$
$314$	7.29603	0.411739
$315$	−9.61711	−0.541863
$316$	−19.0390	−1.07103
$317$	7.30606	0.410349	0.205175	−	0.978725i	$-0.434224\pi$
$317$	7.30606	0.410349	0.205175	+	0.978725i	$0.434224\pi$
$318$	5.93727	0.332945
$319$	−4.75927	−0.266468
$320$	−5.96493	−0.333449
$321$	−19.6892	−1.09894
$322$	−45.1818	−2.51788
$323$	23.9679	1.33361
$324$	7.00348	0.389082
$325$	−1.68304	−0.0933581
$326$	3.78467	0.209613
$327$	10.8325	0.599037
$328$	−50.8418	−2.80727
$329$	20.7508	1.14403
$330$	1.47823	0.0813737
$331$	15.7409	0.865199	0.432600	−	0.901586i	$-0.357596\pi$
$331$	15.7409	0.865199	0.432600	+	0.901586i	$0.357596\pi$
$332$	−41.6092	−2.28360
$333$	12.3746	0.678124
$334$	20.4215	1.11742
$335$	−3.93113	−0.214781
$336$	−40.7821	−2.22484
$337$	34.5083	1.87979	0.939894	−	0.341468i	$-0.110924\pi$
$337$	34.5083	1.87979	0.939894	+	0.341468i	$0.110924\pi$
$338$	26.4189	1.43700
$339$	3.50721	0.190486
$340$	−25.5219	−1.38412
$341$	−2.28957	−0.123987
$342$	23.9675	1.29601
$343$	−35.5750	−1.92087
$344$	49.1978	2.65257
$345$	3.60988	0.194350
$346$	10.8021	0.580723
$347$	−27.1067	−1.45517	−0.727583	−	0.686020i	$-0.759356\pi$
$347$	−27.1067	−1.45517	−0.727583	+	0.686020i	$0.759356\pi$
$348$	−37.0858	−1.98801
$349$	21.6552	1.15917	0.579587	−	0.814910i	$-0.303214\pi$
$349$	21.6552	1.15917	0.579587	+	0.814910i	$0.303214\pi$
$350$	12.0892	0.646195
$351$	8.23715	0.439667
$352$	−5.46626	−0.291352
$353$	5.15871	0.274571	0.137285	−	0.990532i	$-0.456162\pi$
$353$	5.15871	0.274571	0.137285	+	0.990532i	$0.456162\pi$
$354$	14.9937	0.796909
$355$	−6.50757	−0.345386
$356$	−0.519539	−0.0275355
$357$	−24.1371	−1.27747
$358$	−28.8333	−1.52389
$359$	1.30196	0.0687149	0.0343574	−	0.999410i	$-0.489062\pi$
$359$	1.30196	0.0687149	0.0343574	+	0.999410i	$0.489062\pi$
$360$	−14.7794	−0.778940
$361$	0.912583	0.0480307
$362$	31.7438	1.66841
$363$	−10.2897	−0.540070
$364$	37.2077	1.95021
$365$	3.35748	0.175739
$366$	12.4479	0.650663
$367$	−18.0522	−0.942318	−0.471159	−	0.882048i	$-0.656164\pi$
$367$	−18.0522	−0.942318	−0.471159	+	0.882048i	$0.656164\pi$
$368$	−33.9168	−1.76804
$369$	−14.6984	−0.765168
$370$	−15.5555	−0.808693
$371$	11.0064	0.571424
$372$	−17.8411	−0.925016
$373$	17.0002	0.880238	0.440119	−	0.897940i	$-0.354936\pi$
$373$	17.0002	0.880238	0.440119	+	0.897940i	$0.354936\pi$
$374$	−8.22014	−0.425053
$375$	−0.965890	−0.0498783
$376$	31.8894	1.64457
$377$	13.5996	0.700417
$378$	−59.1672	−3.04323
$379$	13.5900	0.698071	0.349036	−	0.937109i	$-0.386509\pi$
$379$	13.5900	0.698071	0.349036	+	0.937109i	$0.386509\pi$
$380$	−21.2036	−1.08772
$381$	−5.11796	−0.262201
$382$	−23.2460	−1.18937
$383$	8.48021	0.433319	0.216659	−	0.976247i	$-0.430484\pi$
$383$	8.48021	0.433319	0.216659	+	0.976247i	$0.430484\pi$
$384$	2.95777	0.150938
$385$	2.74031	0.139659
$386$	−13.7964	−0.702217
$387$	14.2231	0.723002
$388$	18.8338	0.956139
$389$	29.3209	1.48663	0.743314	−	0.668942i	$-0.233253\pi$
$389$	29.3209	1.48663	0.743314	+	0.668942i	$0.233253\pi$
$390$	−4.22403	−0.213892
$391$	−20.0739	−1.01518
$392$	−104.720	−5.28918
$393$	20.0917	1.01349
$394$	30.1546	1.51917
$395$	4.00680	0.201604
$396$	−5.78504	−0.290709
$397$	33.7044	1.69158	0.845788	−	0.533520i	$-0.179131\pi$
$397$	33.7044	1.69158	0.845788	+	0.533520i	$0.179131\pi$
$398$	14.0639	0.704959
$399$	−20.0532	−1.00392
$400$	9.07506	0.453753
$401$	−5.35555	−0.267443	−0.133722	−	0.991019i	$-0.542693\pi$
$401$	−5.35555	−0.267443	−0.133722	+	0.991019i	$0.542693\pi$
$402$	−9.86622	−0.492082
$403$	6.54245	0.325903
$404$	−72.0796	−3.58609
$405$	−1.47390	−0.0732385
$406$	−97.6857	−4.84806
$407$	−3.52603	−0.174779
$408$	−37.0934	−1.83640
$409$	6.77109	0.334809	0.167404	−	0.985888i	$-0.446461\pi$
$409$	6.77109	0.334809	0.167404	+	0.985888i	$0.446461\pi$
$410$	18.4767	0.912497
$411$	−19.4535	−0.959568
$412$	61.7047	3.03997
$413$	27.7952	1.36771
$414$	−20.0735	−0.986561
$415$	8.75675	0.429852
$416$	15.6198	0.765825
$417$	8.15612	0.399407
$418$	−6.82931	−0.334033
$419$	−20.1559	−0.984680	−0.492340	−	0.870403i	$-0.663858\pi$
$419$	−20.1559	−0.984680	−0.492340	+	0.870403i	$0.663858\pi$
$420$	21.3534	1.04194
$421$	−10.8344	−0.528034	−0.264017	−	0.964518i	$-0.585048\pi$
$421$	−10.8344	−0.528034	−0.264017	+	0.964518i	$0.585048\pi$
$422$	−33.0762	−1.61012
$423$	9.21924	0.448255
$424$	16.9144	0.821435
$425$	5.37113	0.260538
$426$	−16.3325	−0.791311
$427$	23.0757	1.11671
$428$	−96.8605	−4.68193
$429$	−0.957478	−0.0462275
$430$	−17.8792	−0.862212
$431$	−18.7680	−0.904021	−0.452011	−	0.892012i	$-0.649293\pi$
$431$	−18.7680	−0.904021	−0.452011	+	0.892012i	$0.649293\pi$
$432$	−44.4153	−2.13693
$433$	38.1472	1.83324	0.916619	−	0.399762i	$-0.130907\pi$
$433$	38.1472	1.83324	0.916619	+	0.399762i	$0.130907\pi$
$434$	−46.9942	−2.25579
$435$	7.80478	0.374211
$436$	53.2901	2.55213
$437$	−16.6774	−0.797790
$438$	8.42649	0.402633
$439$	28.7956	1.37434	0.687169	−	0.726498i	$-0.258853\pi$
$439$	28.7956	1.37434	0.687169	+	0.726498i	$0.258853\pi$
$440$	4.21124	0.200763
$441$	−30.2748	−1.44166
$442$	23.4891	1.11726
$443$	0.954164	0.0453337	0.0226668	−	0.999743i	$-0.492784\pi$
$443$	0.954164	0.0453337	0.0226668	+	0.999743i	$0.492784\pi$
$444$	−27.4760	−1.30395
$445$	0.109338	0.00518313
$446$	−49.9225	−2.36390
$447$	−8.37112	−0.395941
$448$	−27.7522	−1.31117
$449$	−4.28001	−0.201986	−0.100993	−	0.994887i	$-0.532202\pi$
$449$	−4.28001	−0.201986	−0.100993	+	0.994887i	$0.532202\pi$
$450$	5.37104	0.253193
$451$	4.18818	0.197214
$452$	17.2537	0.811544
$453$	−8.29296	−0.389637
$454$	−15.8309	−0.742983
$455$	−7.83044	−0.367097
$456$	−30.8173	−1.44315
$457$	−1.12426	−0.0525905	−0.0262953	−	0.999654i	$-0.508371\pi$
$457$	−1.12426	−0.0525905	−0.0262953	+	0.999654i	$0.508371\pi$
$458$	−54.5267	−2.54786
$459$	−26.2875	−1.22700
$460$	17.7588	0.828006
$461$	−4.57582	−0.213117	−0.106558	−	0.994306i	$-0.533983\pi$
$461$	−4.57582	−0.213117	−0.106558	+	0.994306i	$0.533983\pi$
$462$	6.87754	0.319972
$463$	30.3602	1.41096	0.705480	−	0.708730i	$-0.250732\pi$
$463$	30.3602	1.41096	0.705480	+	0.708730i	$0.250732\pi$
$464$	−73.3302	−3.40427
$465$	3.75469	0.174120
$466$	7.71505	0.357393
$467$	−19.9743	−0.924301	−0.462150	−	0.886802i	$-0.652922\pi$
$467$	−19.9743	−0.924301	−0.462150	+	0.886802i	$0.652922\pi$
$468$	16.5308	0.764135
$469$	−18.2898	−0.844546
$470$	−11.5891	−0.534564
$471$	−2.71212	−0.124968
$472$	42.7149	1.96611
$473$	−4.05275	−0.186346
$474$	10.0561	0.461893
$475$	4.46235	0.204747
$476$	−118.742	−5.44254
$477$	4.88997	0.223896
$478$	−2.25266	−0.103034
$479$	40.7743	1.86302	0.931512	−	0.363711i	$-0.118490\pi$
$479$	40.7743	1.86302	0.931512	+	0.363711i	$0.118490\pi$
$480$	8.96417	0.409156
$481$	10.0756	0.459410
$482$	2.81859	0.128383
$483$	16.7952	0.764208
$484$	−50.6200	−2.30091
$485$	−3.96360	−0.179978
$486$	−41.8505	−1.89838
$487$	−5.15089	−0.233409	−0.116705	−	0.993167i	$-0.537233\pi$
$487$	−5.15089	−0.233409	−0.116705	+	0.993167i	$0.537233\pi$
$488$	35.4622	1.60530
$489$	−1.40686	−0.0636202
$490$	38.0570	1.71924
$491$	36.3407	1.64003	0.820017	−	0.572339i	$-0.193964\pi$
$491$	36.3407	1.64003	0.820017	+	0.572339i	$0.193964\pi$
$492$	32.6356	1.47133
$493$	−43.4009	−1.95468
$494$	19.5148	0.878011
$495$	1.21747	0.0547214
$496$	−35.2774	−1.58400
$497$	−30.2769	−1.35810
$498$	21.9774	0.984830
$499$	−3.00558	−0.134548	−0.0672741	−	0.997735i	$-0.521430\pi$
$499$	−3.00558	−0.134548	−0.0672741	+	0.997735i	$0.521430\pi$
$500$	−4.75167	−0.212501
$501$	−7.59119	−0.339149
$502$	−14.6638	−0.654477
$503$	33.1690	1.47893	0.739465	−	0.673195i	$-0.235078\pi$
$503$	33.1690	1.47893	0.739465	+	0.673195i	$0.235078\pi$
$504$	−68.7618	−3.06290
$505$	15.1693	0.675025
$506$	5.71977	0.254275
$507$	−9.82057	−0.436147
$508$	−25.1777	−1.11708
$509$	39.3344	1.74347	0.871733	−	0.489981i	$-0.162996\pi$
$509$	39.3344	1.74347	0.871733	+	0.489981i	$0.162996\pi$
$510$	13.4803	0.596917
$511$	15.6209	0.691027
$512$	45.5492	2.01301
$513$	−21.8397	−0.964247
$514$	40.2221	1.77412
$515$	−12.9859	−0.572226
$516$	−31.5804	−1.39025
$517$	−2.62694	−0.115533
$518$	−72.3730	−3.17989
$519$	−4.01540	−0.176257
$520$	−12.0336	−0.527710
$521$	−5.68102	−0.248890	−0.124445	−	0.992227i	$-0.539715\pi$
$521$	−5.68102	−0.248890	−0.124445	+	0.992227i	$0.539715\pi$
$522$	−43.4002	−1.89957
$523$	44.4626	1.94422	0.972108	−	0.234535i	$-0.0753567\pi$
$523$	44.4626	1.94422	0.972108	+	0.234535i	$0.0753567\pi$
$524$	98.8406	4.31787
$525$	−4.49386	−0.196128
$526$	26.2168	1.14310
$527$	−20.8791	−0.909509
$528$	5.16279	0.224682
$529$	−9.03209	−0.392700
$530$	−6.14694	−0.267006
$531$	12.3489	0.535898
$532$	−98.6513	−4.27707
$533$	−11.9677	−0.518380
$534$	0.274413	0.0118750
$535$	20.3845	0.881299
$536$	−28.1074	−1.21405
$537$	10.7181	0.462518
$538$	80.0383	3.45070
$539$	8.62653	0.371571
$540$	23.2557	1.00077
$541$	14.1718	0.609294	0.304647	−	0.952465i	$-0.401462\pi$
$541$	14.1718	0.609294	0.304647	+	0.952465i	$0.401462\pi$
$542$	−14.3512	−0.616435
$543$	−11.7999	−0.506384
$544$	−49.8481	−2.13722
$545$	−11.2150	−0.480399
$546$	−19.6526	−0.841052
$547$	13.8168	0.590765	0.295382	−	0.955379i	$-0.404553\pi$
$547$	13.8168	0.590765	0.295382	+	0.955379i	$0.404553\pi$
$548$	−95.7009	−4.08814
$549$	10.2522	0.437552
$550$	−1.53043	−0.0652577
$551$	−36.0576	−1.53611
$552$	25.8105	1.09857
$553$	18.6419	0.792733
$554$	−24.5712	−1.04393
$555$	5.78238	0.245448
$556$	40.1239	1.70163
$557$	−13.2746	−0.562461	−0.281231	−	0.959640i	$-0.590743\pi$
$557$	−13.2746	−0.562461	−0.281231	+	0.959640i	$0.590743\pi$
$558$	−20.8788	−0.883869
$559$	11.5807	0.489813
$560$	42.2223	1.78422
$561$	3.05563	0.129009
$562$	−65.5034	−2.76310
$563$	19.2828	0.812671	0.406336	−	0.913724i	$-0.366806\pi$
$563$	19.2828	0.812671	0.406336	+	0.913724i	$0.366806\pi$
$564$	−20.4700	−0.861941
$565$	−3.63107	−0.152760
$566$	−2.60059	−0.109311
$567$	−6.85740	−0.287984
$568$	−46.5288	−1.95230
$569$	−1.85339	−0.0776980	−0.0388490	−	0.999245i	$-0.512369\pi$
$569$	−1.85339	−0.0776980	−0.0388490	+	0.999245i	$0.512369\pi$
$570$	11.1995	0.469094
$571$	−3.87420	−0.162130	−0.0810652	−	0.996709i	$-0.525832\pi$
$571$	−3.87420	−0.162130	−0.0810652	+	0.996709i	$0.525832\pi$
$572$	−4.71029	−0.196947
$573$	8.64110	0.360987
$574$	85.9638	3.58806
$575$	−3.73737	−0.155859
$576$	−12.3298	−0.513743
$577$	−23.6289	−0.983684	−0.491842	−	0.870684i	$-0.663676\pi$
$577$	−23.6289	−0.983684	−0.491842	+	0.870684i	$0.663676\pi$
$578$	−30.7886	−1.28064
$579$	5.12846	0.213131
$580$	38.3955	1.59429
$581$	40.7413	1.69023
$582$	−9.94772	−0.412346
$583$	−1.39335	−0.0577067
$584$	24.0058	0.993367
$585$	−3.47894	−0.143836
$586$	28.7487	1.18760
$587$	−0.0546991	−0.00225767	−0.00112884	−	0.999999i	$-0.500359\pi$
$587$	−0.0546991	−0.00225767	−0.00112884	+	0.999999i	$0.500359\pi$
$588$	67.2207	2.77213
$589$	−17.3464	−0.714748
$590$	−15.5233	−0.639082
$591$	−11.2092	−0.461086
$592$	−54.3286	−2.23289
$593$	−21.9799	−0.902607	−0.451303	−	0.892371i	$-0.649041\pi$
$593$	−21.9799	−0.902607	−0.451303	+	0.892371i	$0.649041\pi$
$594$	7.49025	0.307329
$595$	24.9895	1.02447
$596$	−41.1816	−1.68686
$597$	−5.22790	−0.213964
$598$	−16.3443	−0.668367
$599$	−4.60262	−0.188058	−0.0940290	−	0.995569i	$-0.529975\pi$
$599$	−4.60262	−0.188058	−0.0940290	+	0.995569i	$0.529975\pi$
$600$	−6.90606	−0.281939
$601$	23.4953	0.958393	0.479197	−	0.877708i	$-0.340928\pi$
$601$	23.4953	0.958393	0.479197	+	0.877708i	$0.340928\pi$
$602$	−83.1842	−3.39033
$603$	−8.12588	−0.330911
$604$	−40.7970	−1.66001
$605$	10.6531	0.433110
$606$	38.0714	1.54655
$607$	2.34582	0.0952140	0.0476070	−	0.998866i	$-0.484840\pi$
$607$	2.34582	0.0952140	0.0476070	+	0.998866i	$0.484840\pi$
$608$	−41.4139	−1.67956
$609$	36.3122	1.47145
$610$	−12.8875	−0.521800
$611$	7.50649	0.303680
$612$	−52.7552	−2.13250
$613$	7.75402	0.313182	0.156591	−	0.987664i	$-0.449950\pi$
$613$	7.75402	0.313182	0.156591	+	0.987664i	$0.449950\pi$
$614$	15.2193	0.614201
$615$	−6.86824	−0.276954
$616$	19.5931	0.789427
$617$	−8.78985	−0.353866	−0.176933	−	0.984223i	$-0.556618\pi$
$617$	−8.78985	−0.353866	−0.176933	+	0.984223i	$0.556618\pi$
$618$	−32.5915	−1.31102
$619$	29.6318	1.19100	0.595502	−	0.803354i	$-0.296953\pi$
$619$	29.6318	1.19100	0.595502	+	0.803354i	$0.296953\pi$
$620$	18.4711	0.741818
$621$	18.2915	0.734012
$622$	88.6496	3.55452
$623$	0.508702	0.0203807
$624$	−14.7527	−0.590580
$625$	1.00000	0.0400000
$626$	78.4098	3.13389
$627$	2.53863	0.101383
$628$	−13.3422	−0.532412
$629$	−32.1547	−1.28209
$630$	24.9891	0.995589
$631$	2.71914	0.108247	0.0541235	−	0.998534i	$-0.482764\pi$
$631$	2.71914	0.108247	0.0541235	+	0.998534i	$0.482764\pi$
$632$	28.6484	1.13957
$633$	12.2952	0.488692
$634$	−18.9841	−0.753953
$635$	5.29870	0.210273
$636$	−10.8574	−0.430526
$637$	−24.6503	−0.976681
$638$	12.3665	0.489594
$639$	−13.4515	−0.532134
$640$	−3.06222	−0.121045
$641$	−40.5526	−1.60173	−0.800866	−	0.598844i	$-0.795627\pi$
$641$	−40.5526	−1.60173	−0.800866	+	0.598844i	$0.795627\pi$
$642$	51.1603	2.01914
$643$	−15.4894	−0.610843	−0.305422	−	0.952217i	$-0.598797\pi$
$643$	−15.4894	−0.610843	−0.305422	+	0.952217i	$0.598797\pi$
$644$	82.6237	3.25583
$645$	6.64615	0.261692
$646$	−62.2781	−2.45030
$647$	6.53812	0.257040	0.128520	−	0.991707i	$-0.458977\pi$
$647$	6.53812	0.257040	0.128520	+	0.991707i	$0.458977\pi$
$648$	−10.5383	−0.413983
$649$	−3.51872	−0.138122
$650$	4.37320	0.171531
$651$	17.4689	0.684661
$652$	−6.92100	−0.271047
$653$	38.2358	1.49628	0.748141	−	0.663539i	$-0.230947\pi$
$653$	38.2358	1.49628	0.748141	+	0.663539i	$0.230947\pi$
$654$	−28.1471	−1.10064
$655$	−20.8012	−0.812770
$656$	64.5309	2.51951
$657$	6.94010	0.270759
$658$	−53.9189	−2.10198
$659$	2.40323	0.0936166	0.0468083	−	0.998904i	$-0.485095\pi$
$659$	2.40323	0.0936166	0.0468083	+	0.998904i	$0.485095\pi$
$660$	−2.70322	−0.105223
$661$	−17.8134	−0.692861	−0.346430	−	0.938076i	$-0.612606\pi$
$661$	−17.8134	−0.692861	−0.346430	+	0.938076i	$0.612606\pi$
$662$	−40.9012	−1.58967
$663$	−8.73147	−0.339102
$664$	62.6103	2.42975
$665$	20.7614	0.805091
$666$	−32.1541	−1.24595
$667$	30.1994	1.16933
$668$	−37.3447	−1.44491
$669$	18.5575	0.717473
$670$	10.2146	0.394626
$671$	−2.92126	−0.112774
$672$	41.7063	1.60886
$673$	−43.1693	−1.66406	−0.832028	−	0.554734i	$-0.812820\pi$
$673$	−43.1693	−1.66406	−0.832028	+	0.554734i	$0.812820\pi$
$674$	−89.6663	−3.45382
$675$	−4.89422	−0.188379
$676$	−48.3121	−1.85816
$677$	2.53198	0.0973120	0.0486560	−	0.998816i	$-0.484506\pi$
$677$	2.53198	0.0973120	0.0486560	+	0.998816i	$0.484506\pi$
$678$	−9.11313	−0.349988
$679$	−18.4409	−0.707697
$680$	38.4033	1.47270
$681$	5.88476	0.225504
$682$	5.94922	0.227807
$683$	19.7894	0.757221	0.378610	−	0.925556i	$-0.376402\pi$
$683$	19.7894	0.757221	0.378610	+	0.925556i	$0.376402\pi$
$684$	−43.8291	−1.67585
$685$	20.1405	0.769527
$686$	92.4381	3.52930
$687$	20.2689	0.773308
$688$	−62.4442	−2.38066
$689$	3.98150	0.151683
$690$	−9.37992	−0.357087
$691$	−36.2034	−1.37724	−0.688621	−	0.725122i	$-0.741784\pi$
$691$	−36.2034	−1.37724	−0.688621	+	0.725122i	$0.741784\pi$
$692$	−19.7537	−0.750923
$693$	5.66438	0.215172
$694$	70.4341	2.67364
$695$	−8.44415	−0.320305
$696$	55.8038	2.11524
$697$	38.1930	1.44666
$698$	−56.2688	−2.12980
$699$	−2.86788	−0.108473
$700$	−22.1075	−0.835583
$701$	−1.76835	−0.0667898	−0.0333949	−	0.999442i	$-0.510632\pi$
$701$	−1.76835	−0.0667898	−0.0333949	+	0.999442i	$0.510632\pi$
$702$	−21.4034	−0.807820
$703$	−26.7142	−1.00755
$704$	3.51328	0.132412
$705$	4.30795	0.162247
$706$	−13.4044	−0.504481
$707$	70.5761	2.65429
$708$	−27.4190	−1.03047
$709$	−6.03774	−0.226752	−0.113376	−	0.993552i	$-0.536166\pi$
$709$	−6.03774	−0.226752	−0.113376	+	0.993552i	$0.536166\pi$
$710$	16.9093	0.634593
$711$	8.28228	0.310610
$712$	0.781762	0.0292978
$713$	14.5282	0.544086
$714$	62.7179	2.34716
$715$	0.991292	0.0370722
$716$	52.7273	1.97051
$717$	0.837368	0.0312721
$718$	−3.38301	−0.126253
$719$	−11.2125	−0.418157	−0.209078	−	0.977899i	$-0.567046\pi$
$719$	−11.2125	−0.418157	−0.209078	+	0.977899i	$0.567046\pi$
$720$	18.7587	0.699095
$721$	−60.4176	−2.25007
$722$	−2.37125	−0.0882489
$723$	−1.04774	−0.0389658
$724$	−58.0496	−2.15740
$725$	−8.08041	−0.300099
$726$	26.7368	0.992295
$727$	−34.3638	−1.27448	−0.637242	−	0.770664i	$-0.719925\pi$
$727$	−34.3638	−1.27448	−0.637242	+	0.770664i	$0.719925\pi$
$728$	−55.9872	−2.07502
$729$	11.1352	0.412415
$730$	−8.72407	−0.322892
$731$	−36.9580	−1.36694
$732$	−22.7634	−0.841360
$733$	5.48428	0.202566	0.101283	−	0.994858i	$-0.467705\pi$
$733$	5.48428	0.202566	0.101283	+	0.994858i	$0.467705\pi$
$734$	46.9068	1.73136
$735$	−14.1467	−0.521810
$736$	34.6855	1.27853
$737$	2.31540	0.0852887
$738$	38.1923	1.40588
$739$	7.22711	0.265854	0.132927	−	0.991126i	$-0.457562\pi$
$739$	7.22711	0.265854	0.132927	+	0.991126i	$0.457562\pi$
$740$	28.4463	1.04571
$741$	−7.25413	−0.266487
$742$	−28.5990	−1.04990
$743$	49.3271	1.80964	0.904818	−	0.425800i	$-0.140007\pi$
$743$	49.3271	1.80964	0.904818	+	0.425800i	$0.140007\pi$
$744$	26.8458	0.984216
$745$	8.66675	0.317525
$746$	−44.1733	−1.61730
$747$	18.1007	0.662270
$748$	15.0321	0.549629
$749$	94.8401	3.46538
$750$	2.50977	0.0916437
$751$	8.68468	0.316909	0.158454	−	0.987366i	$-0.449349\pi$
$751$	8.68468	0.316909	0.158454	+	0.987366i	$0.449349\pi$
$752$	−40.4755	−1.47599
$753$	5.45089	0.198642
$754$	−35.3373	−1.28691
$755$	8.58582	0.312470
$756$	108.199	3.93515
$757$	3.88960	0.141370	0.0706850	−	0.997499i	$-0.477481\pi$
$757$	3.88960	0.141370	0.0706850	+	0.997499i	$0.477481\pi$
$758$	−35.3122	−1.28260
$759$	−2.12618	−0.0771756
$760$	31.9056	1.15734
$761$	6.66644	0.241658	0.120829	−	0.992673i	$-0.461445\pi$
$761$	6.66644	0.241658	0.120829	+	0.992673i	$0.461445\pi$
$762$	13.2985	0.481754
$763$	−52.1786	−1.88899
$764$	42.5097	1.53795
$765$	11.1024	0.401410
$766$	−22.0350	−0.796156
$767$	10.0547	0.363056
$768$	−19.2084	−0.693123
$769$	36.7597	1.32559	0.662794	−	0.748802i	$-0.269370\pi$
$769$	36.7597	1.32559	0.662794	+	0.748802i	$0.269370\pi$
$770$	−7.12042	−0.256602
$771$	−14.9515	−0.538467
$772$	25.2293	0.908024
$773$	−20.2934	−0.729904	−0.364952	−	0.931026i	$-0.618915\pi$
$773$	−20.2934	−0.729904	−0.364952	+	0.931026i	$0.618915\pi$
$774$	−36.9574	−1.32841
$775$	−3.88729	−0.139635
$776$	−28.3396	−1.01733
$777$	26.9029	0.965135
$778$	−76.1874	−2.73145
$779$	31.7308	1.13688
$780$	7.72446	0.276580
$781$	3.83289	0.137152
$782$	52.1600	1.86524
$783$	39.5473	1.41330
$784$	132.916	4.74701
$785$	2.80790	0.100218
$786$	−52.2062	−1.86213
$787$	19.0184	0.677931	0.338966	−	0.940799i	$-0.389923\pi$
$787$	19.0184	0.677931	0.338966	+	0.940799i	$0.389923\pi$
$788$	−55.1436	−1.96441
$789$	−9.74542	−0.346946
$790$	−10.4113	−0.370416
$791$	−16.8938	−0.600673
$792$	8.70488	0.309314
$793$	8.34751	0.296429
$794$	−87.5775	−3.10801
$795$	2.28497	0.0810396
$796$	−25.7186	−0.911570
$797$	−23.5320	−0.833547	−0.416774	−	0.909010i	$-0.636839\pi$
$797$	−23.5320	−0.833547	−0.416774	+	0.909010i	$0.636839\pi$
$798$	52.1062	1.84454
$799$	−23.9557	−0.847492
$800$	−9.28074	−0.328124
$801$	0.226008	0.00798561
$802$	13.9158	0.491386
$803$	−1.97752	−0.0697852
$804$	18.0423	0.636303
$805$	−17.3883	−0.612858
$806$	−16.9999	−0.598796
$807$	−29.7522	−1.04733
$808$	108.460	3.81560
$809$	7.55566	0.265643	0.132821	−	0.991140i	$-0.457596\pi$
$809$	7.55566	0.265643	0.132821	+	0.991140i	$0.457596\pi$
$810$	3.82977	0.134564
$811$	−15.5849	−0.547261	−0.273631	−	0.961835i	$-0.588225\pi$
$811$	−15.5849	−0.547261	−0.273631	+	0.961835i	$0.588225\pi$
$812$	178.637	6.26894
$813$	5.33468	0.187095
$814$	9.16204	0.321129
$815$	1.45654	0.0510203
$816$	47.0807	1.64816
$817$	−30.7048	−1.07423
$818$	−17.5940	−0.615159
$819$	−16.1860	−0.565583
$820$	−33.7882	−1.17993
$821$	51.8498	1.80957	0.904784	−	0.425870i	$-0.140032\pi$
$821$	51.8498	1.80957	0.904784	+	0.425870i	$0.140032\pi$
$822$	50.5478	1.76306
$823$	−2.61546	−0.0911692	−0.0455846	−	0.998960i	$-0.514515\pi$
$823$	−2.61546	−0.0911692	−0.0455846	+	0.998960i	$0.514515\pi$
$824$	−92.8483	−3.23453
$825$	0.568899	0.0198065
$826$	−72.2229	−2.51296
$827$	39.9813	1.39029	0.695143	−	0.718872i	$-0.255341\pi$
$827$	39.9813	1.39029	0.695143	+	0.718872i	$0.255341\pi$
$828$	36.7084	1.27570
$829$	14.0184	0.486879	0.243439	−	0.969916i	$-0.421724\pi$
$829$	14.0184	0.486879	0.243439	+	0.969916i	$0.421724\pi$
$830$	−22.7535	−0.789786
$831$	9.13372	0.316845
$832$	−10.0392	−0.348046
$833$	78.6674	2.72566
$834$	−21.1929	−0.733848
$835$	7.85927	0.271981
$836$	12.4887	0.431931
$837$	19.0252	0.657608
$838$	52.3731	1.80920
$839$	−13.4531	−0.464454	−0.232227	−	0.972662i	$-0.574601\pi$
$839$	−13.4531	−0.464454	−0.232227	+	0.972662i	$0.574601\pi$
$840$	−32.1309	−1.10862
$841$	36.2930	1.25148
$842$	28.1520	0.970181
$843$	24.3493	0.838633
$844$	60.4862	2.08202
$845$	10.1674	0.349769
$846$	−23.9553	−0.823599
$847$	49.5642	1.70304
$848$	−21.4686	−0.737234
$849$	0.966703	0.0331772
$850$	−13.9563	−0.478699
$851$	22.3740	0.766972
$852$	29.8671	1.02323
$853$	22.1193	0.757350	0.378675	−	0.925530i	$-0.376380\pi$
$853$	22.1193	0.757350	0.378675	+	0.925530i	$0.376380\pi$
$854$	−59.9599	−2.05179
$855$	9.22394	0.315452
$856$	145.748	4.98157
$857$	−3.87219	−0.132272	−0.0661358	−	0.997811i	$-0.521067\pi$
$857$	−3.87219	−0.132272	−0.0661358	+	0.997811i	$0.521067\pi$
$858$	2.48791	0.0849358
$859$	−40.1600	−1.37024	−0.685122	−	0.728429i	$-0.740251\pi$
$859$	−40.1600	−1.37024	−0.685122	+	0.728429i	$0.740251\pi$
$860$	32.6956	1.11491
$861$	−31.9549	−1.08902
$862$	48.7667	1.66100
$863$	35.9275	1.22299	0.611493	−	0.791250i	$-0.290569\pi$
$863$	35.9275	1.22299	0.611493	+	0.791250i	$0.290569\pi$
$864$	45.4220	1.54529
$865$	4.15721	0.141349
$866$	−99.1216	−3.36829
$867$	11.4449	0.388689
$868$	85.9380	2.91693
$869$	−2.35996	−0.0800562
$870$	−20.2799	−0.687554
$871$	−6.61624	−0.224183
$872$	−80.1868	−2.71547
$873$	−8.19299	−0.277291
$874$	43.3346	1.46582
$875$	4.65256	0.157285
$876$	−15.4095	−0.520638
$877$	18.9375	0.639475	0.319738	−	0.947506i	$-0.396405\pi$
$877$	18.9375	0.639475	0.319738	+	0.947506i	$0.396405\pi$
$878$	−74.8224	−2.52513
$879$	−10.6866	−0.360451
$880$	−5.34512	−0.180184
$881$	14.6391	0.493204	0.246602	−	0.969117i	$-0.420686\pi$
$881$	14.6391	0.493204	0.246602	+	0.969117i	$0.420686\pi$
$882$	78.6660	2.64882
$883$	48.8451	1.64377	0.821884	−	0.569655i	$-0.192923\pi$
$883$	48.8451	1.64377	0.821884	+	0.569655i	$0.192923\pi$
$884$	−42.9543	−1.44471
$885$	5.77038	0.193969
$886$	−2.47930	−0.0832937
$887$	−34.5640	−1.16055	−0.580274	−	0.814422i	$-0.697054\pi$
$887$	−34.5640	−1.16055	−0.580274	+	0.814422i	$0.697054\pi$
$888$	41.3437	1.38740
$889$	24.6525	0.826819
$890$	−0.284104	−0.00952319
$891$	0.868110	0.0290828
$892$	91.2930	3.05672
$893$	−19.9025	−0.666010
$894$	21.7515	0.727480
$895$	−11.0966	−0.370917
$896$	−14.2472	−0.475965
$897$	6.07557	0.202857
$898$	11.1212	0.371118
$899$	31.4109	1.04761
$900$	−9.82198	−0.327399
$901$	−12.7063	−0.423308
$902$	−10.8826	−0.362350
$903$	30.9216	1.02901
$904$	−25.9620	−0.863482
$905$	12.2167	0.406096
$906$	21.5484	0.715898
$907$	19.6423	0.652213	0.326107	−	0.945333i	$-0.394263\pi$
$907$	19.6423	0.652213	0.326107	+	0.945333i	$0.394263\pi$
$908$	28.9500	0.960738
$909$	31.3558	1.04001
$910$	20.3466	0.674483
$911$	−39.7131	−1.31575	−0.657877	−	0.753126i	$-0.728545\pi$
$911$	−39.7131	−1.31575	−0.657877	+	0.753126i	$0.728545\pi$
$912$	39.1148	1.29522
$913$	−5.15763	−0.170693
$914$	2.92127	0.0966269
$915$	4.79061	0.158373
$916$	99.7126	3.29460
$917$	−96.7789	−3.19592
$918$	68.3054	2.25441
$919$	−10.0250	−0.330695	−0.165347	−	0.986235i	$-0.552875\pi$
$919$	−10.0250	−0.330695	−0.165347	+	0.986235i	$0.552875\pi$
$920$	−26.7220	−0.880997
$921$	−5.65739	−0.186417
$922$	11.8898	0.391569
$923$	−10.9525	−0.360505
$924$	−12.5769	−0.413750
$925$	−5.98658	−0.196838
$926$	−78.8880	−2.59242
$927$	−26.8425	−0.881625
$928$	74.9921	2.46174
$929$	11.1042	0.364316	0.182158	−	0.983269i	$-0.441692\pi$
$929$	11.1042	0.364316	0.182158	+	0.983269i	$0.441692\pi$
$930$	−9.75618	−0.319918
$931$	65.3571	2.14199
$932$	−14.1085	−0.462138
$933$	−32.9533	−1.07884
$934$	51.9012	1.69826
$935$	−3.16354	−0.103459
$936$	−24.8742	−0.813039
$937$	0.796719	0.0260277	0.0130138	−	0.999915i	$-0.495857\pi$
$937$	0.796719	0.0260277	0.0130138	+	0.999915i	$0.495857\pi$
$938$	47.5243	1.55172
$939$	−29.1469	−0.951172
$940$	21.1929	0.691235
$941$	−13.4767	−0.439327	−0.219664	−	0.975576i	$-0.570496\pi$
$941$	−13.4767	−0.439327	−0.219664	+	0.975576i	$0.570496\pi$
$942$	7.04716	0.229609
$943$	−26.5756	−0.865421
$944$	−54.2159	−1.76458
$945$	−22.7707	−0.740729
$946$	10.5307	0.342382
$947$	28.0274	0.910769	0.455385	−	0.890295i	$-0.349502\pi$
$947$	28.0274	0.910769	0.455385	+	0.890295i	$0.349502\pi$
$948$	−18.3896	−0.597266
$949$	5.65077	0.183432
$950$	−11.5950	−0.376190
$951$	7.05685	0.228834
$952$	178.674	5.79085
$953$	−30.8842	−1.00044	−0.500219	−	0.865899i	$-0.666747\pi$
$953$	−30.8842	−1.00044	−0.500219	+	0.865899i	$0.666747\pi$
$954$	−12.7061	−0.411375
$955$	−8.94626	−0.289494
$956$	4.11942	0.133231
$957$	−4.59693	−0.148598
$958$	−105.948	−3.42302
$959$	93.7047	3.02588
$960$	−5.76146	−0.185950
$961$	−15.8890	−0.512548
$962$	−26.1805	−0.844094
$963$	42.1359	1.35781
$964$	−5.15433	−0.166010
$965$	−5.30957	−0.170921
$966$	−43.6406	−1.40411
$967$	−13.4540	−0.432652	−0.216326	−	0.976321i	$-0.569407\pi$
$967$	−13.4540	−0.432652	−0.216326	+	0.976321i	$0.569407\pi$
$968$	76.1690	2.44817
$969$	23.1503	0.743696
$970$	10.2990	0.330682
$971$	−40.0680	−1.28584	−0.642922	−	0.765932i	$-0.722278\pi$
$971$	−40.0680	−1.28584	−0.642922	+	0.765932i	$0.722278\pi$
$972$	76.5318	2.45476
$973$	−39.2869	−1.25948
$974$	13.3841	0.428853
$975$	−1.62563	−0.0520618
$976$	−45.0104	−1.44075
$977$	−23.2927	−0.745199	−0.372600	−	0.927992i	$-0.621534\pi$
$977$	−23.2927	−0.745199	−0.372600	+	0.927992i	$0.621534\pi$
$978$	3.65557	0.116892
$979$	−0.0643990	−0.00205820
$980$	−69.5946	−2.22312
$981$	−23.1821	−0.740147
$982$	−94.4277	−3.01331
$983$	−3.78303	−0.120660	−0.0603299	−	0.998178i	$-0.519215\pi$
$983$	−3.78303	−0.120660	−0.0603299	+	0.998178i	$0.519215\pi$
$984$	−49.1075	−1.56549
$985$	11.6051	0.369769
$986$	112.773	3.59142
$987$	20.0430	0.637975
$988$	−35.6865	−1.13534
$989$	25.7163	0.817731
$990$	−3.16348	−0.100542
$991$	9.35844	0.297281	0.148640	−	0.988891i	$-0.452510\pi$
$991$	9.35844	0.297281	0.148640	+	0.988891i	$0.452510\pi$
$992$	36.0769	1.14544
$993$	15.2040	0.482484
$994$	78.6714	2.49530
$995$	5.41252	0.171589
$996$	−40.1899	−1.27347
$997$	26.8713	0.851023	0.425511	−	0.904953i	$-0.360094\pi$
$997$	26.8713	0.851023	0.425511	+	0.904953i	$0.360094\pi$
$998$	7.80969	0.247211
$999$	29.2996	0.926999

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8045.2.a.d.1.10	✓	141

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8045.2.a.d.1.10	✓	141	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:	\( N \)	\(=\)	\( 8045 = 5 \cdot 1609 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8045.a (trivial)

\(f(q)\)	\(=\)	\(q-2.59840 q^{2} +0.965890 q^{3} +4.75167 q^{4} -1.00000 q^{5} -2.50977 q^{6} -4.65256 q^{7} -7.14995 q^{8} -2.06706 q^{9} +O(q^{10})\)	\(q-2.59840 q^{2} +0.965890 q^{3} +4.75167 q^{4} -1.00000 q^{5} -2.50977 q^{6} -4.65256 q^{7} -7.14995 q^{8} -2.06706 q^{9} +2.59840 q^{10} +0.588989 q^{11} +4.58959 q^{12} -1.68304 q^{13} +12.0892 q^{14} -0.965890 q^{15} +9.07506 q^{16} +5.37113 q^{17} +5.37104 q^{18} +4.46235 q^{19} -4.75167 q^{20} -4.49386 q^{21} -1.53043 q^{22} -3.73737 q^{23} -6.90606 q^{24} +1.00000 q^{25} +4.37320 q^{26} -4.89422 q^{27} -22.1075 q^{28} -8.08041 q^{29} +2.50977 q^{30} -3.88729 q^{31} -9.28074 q^{32} +0.568899 q^{33} -13.9563 q^{34} +4.65256 q^{35} -9.82198 q^{36} -5.98658 q^{37} -11.5950 q^{38} -1.62563 q^{39} +7.14995 q^{40} +7.11079 q^{41} +11.6768 q^{42} -6.88086 q^{43} +2.79869 q^{44} +2.06706 q^{45} +9.71117 q^{46} -4.46008 q^{47} +8.76551 q^{48} +14.6463 q^{49} -2.59840 q^{50} +5.18792 q^{51} -7.99725 q^{52} -2.36567 q^{53} +12.7171 q^{54} -0.588989 q^{55} +33.2656 q^{56} +4.31014 q^{57} +20.9961 q^{58} -5.97416 q^{59} -4.58959 q^{60} -4.95979 q^{61} +10.1007 q^{62} +9.61711 q^{63} +5.96493 q^{64} +1.68304 q^{65} -1.47823 q^{66} +3.93113 q^{67} +25.5219 q^{68} -3.60988 q^{69} -12.0892 q^{70} +6.50757 q^{71} +14.7794 q^{72} -3.35748 q^{73} +15.5555 q^{74} +0.965890 q^{75} +21.2036 q^{76} -2.74031 q^{77} +4.22403 q^{78} -4.00680 q^{79} -9.07506 q^{80} +1.47390 q^{81} -18.4767 q^{82} -8.75675 q^{83} -21.3534 q^{84} -5.37113 q^{85} +17.8792 q^{86} -7.80478 q^{87} -4.21124 q^{88} -0.109338 q^{89} -5.37104 q^{90} +7.83044 q^{91} -17.7588 q^{92} -3.75469 q^{93} +11.5891 q^{94} -4.46235 q^{95} -8.96417 q^{96} +3.96360 q^{97} -38.0570 q^{98} -1.21747 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 141 q - 8 q^{2} + 11 q^{3} + 158 q^{4} - 141 q^{5} + 23 q^{6} + 29 q^{7} - 21 q^{8} + 160 q^{9}+O(q^{10}) \) 141 * q - 8 * q^2 + 11 * q^3 + 158 * q^4 - 141 * q^5 + 23 * q^6 + 29 * q^7 - 21 * q^8 + 160 * q^9	\( 141 q - 8 q^{2} + 11 q^{3} + 158 q^{4} - 141 q^{5} + 23 q^{6} + 29 q^{7} - 21 q^{8} + 160 q^{9} + 8 q^{10} + 32 q^{11} + 17 q^{12} + 35 q^{13} + 18 q^{14} - 11 q^{15} + 188 q^{16} - 13 q^{17} - 16 q^{18} + 152 q^{19} - 158 q^{20} + 40 q^{21} + 14 q^{22} - 77 q^{23} + 69 q^{24} + 141 q^{25} + 27 q^{26} + 38 q^{27} + 67 q^{28} + 22 q^{29} - 23 q^{30} + 86 q^{31} - 65 q^{32} + 51 q^{33} + 79 q^{34} - 29 q^{35} + 191 q^{36} + 45 q^{37} - 9 q^{38} + 55 q^{39} + 21 q^{40} + 36 q^{41} + 6 q^{42} + 132 q^{43} + 74 q^{44} - 160 q^{45} + 72 q^{46} - 16 q^{47} + 22 q^{48} + 212 q^{49} - 8 q^{50} + 82 q^{51} + 106 q^{52} - 28 q^{53} + 86 q^{54} - 32 q^{55} + 27 q^{56} + 10 q^{57} + 23 q^{58} + 71 q^{59} - 17 q^{60} + 116 q^{61} - 36 q^{62} + 45 q^{63} + 237 q^{64} - 35 q^{65} + 69 q^{66} + 99 q^{67} - 7 q^{68} + 45 q^{69} - 18 q^{70} + 34 q^{71} - 53 q^{72} + 125 q^{73} + 50 q^{74} + 11 q^{75} + 271 q^{76} - 31 q^{77} + 2 q^{78} + 101 q^{79} - 188 q^{80} + 221 q^{81} + 67 q^{82} + 67 q^{83} + 141 q^{84} + 13 q^{85} + 48 q^{86} - 21 q^{87} + 71 q^{88} + 79 q^{89} + 16 q^{90} + 228 q^{91} - 198 q^{92} - 12 q^{93} + 114 q^{94} - 152 q^{95} + 129 q^{96} + 98 q^{97} - 31 q^{98} + 195 q^{99}+O(q^{100}) \) 141 * q - 8 * q^2 + 11 * q^3 + 158 * q^4 - 141 * q^5 + 23 * q^6 + 29 * q^7 - 21 * q^8 + 160 * q^9 + 8 * q^10 + 32 * q^11 + 17 * q^12 + 35 * q^13 + 18 * q^14 - 11 * q^15 + 188 * q^16 - 13 * q^17 - 16 * q^18 + 152 * q^19 - 158 * q^20 + 40 * q^21 + 14 * q^22 - 77 * q^23 + 69 * q^24 + 141 * q^25 + 27 * q^26 + 38 * q^27 + 67 * q^28 + 22 * q^29 - 23 * q^30 + 86 * q^31 - 65 * q^32 + 51 * q^33 + 79 * q^34 - 29 * q^35 + 191 * q^36 + 45 * q^37 - 9 * q^38 + 55 * q^39 + 21 * q^40 + 36 * q^41 + 6 * q^42 + 132 * q^43 + 74 * q^44 - 160 * q^45 + 72 * q^46 - 16 * q^47 + 22 * q^48 + 212 * q^49 - 8 * q^50 + 82 * q^51 + 106 * q^52 - 28 * q^53 + 86 * q^54 - 32 * q^55 + 27 * q^56 + 10 * q^57 + 23 * q^58 + 71 * q^59 - 17 * q^60 + 116 * q^61 - 36 * q^62 + 45 * q^63 + 237 * q^64 - 35 * q^65 + 69 * q^66 + 99 * q^67 - 7 * q^68 + 45 * q^69 - 18 * q^70 + 34 * q^71 - 53 * q^72 + 125 * q^73 + 50 * q^74 + 11 * q^75 + 271 * q^76 - 31 * q^77 + 2 * q^78 + 101 * q^79 - 188 * q^80 + 221 * q^81 + 67 * q^82 + 67 * q^83 + 141 * q^84 + 13 * q^85 + 48 * q^86 - 21 * q^87 + 71 * q^88 + 79 * q^89 + 16 * q^90 + 228 * q^91 - 198 * q^92 - 12 * q^93 + 114 * q^94 - 152 * q^95 + 129 * q^96 + 98 * q^97 - 31 * q^98 + 195 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more