LMFDB - Embedded newform 4334.2.a.e.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4334,2,Mod(1,4334)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4334, 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("4334.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4334 = 2 \cdot 11 \cdot 197 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4334.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:	$34.6071642360$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	4334.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.02498 q^{3} +1.00000 q^{4} -1.91029 q^{5} +1.02498 q^{6} +0.691966 q^{7} -1.00000 q^{8} -1.94941 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.02498 q^{3} +1.00000 q^{4} -1.91029 q^{5} +1.02498 q^{6} +0.691966 q^{7} -1.00000 q^{8} -1.94941 q^{9} +1.91029 q^{10} -1.00000 q^{11} -1.02498 q^{12} -0.362562 q^{13} -0.691966 q^{14} +1.95801 q^{15} +1.00000 q^{16} -6.99685 q^{17} +1.94941 q^{18} +2.28021 q^{19} -1.91029 q^{20} -0.709254 q^{21} +1.00000 q^{22} +6.46590 q^{23} +1.02498 q^{24} -1.35080 q^{25} +0.362562 q^{26} +5.07306 q^{27} +0.691966 q^{28} +1.00614 q^{29} -1.95801 q^{30} -8.75557 q^{31} -1.00000 q^{32} +1.02498 q^{33} +6.99685 q^{34} -1.32185 q^{35} -1.94941 q^{36} -2.02334 q^{37} -2.28021 q^{38} +0.371620 q^{39} +1.91029 q^{40} -3.00686 q^{41} +0.709254 q^{42} -5.96518 q^{43} -1.00000 q^{44} +3.72393 q^{45} -6.46590 q^{46} -4.81591 q^{47} -1.02498 q^{48} -6.52118 q^{49} +1.35080 q^{50} +7.17165 q^{51} -0.362562 q^{52} -6.54593 q^{53} -5.07306 q^{54} +1.91029 q^{55} -0.691966 q^{56} -2.33718 q^{57} -1.00614 q^{58} +3.27790 q^{59} +1.95801 q^{60} +4.54021 q^{61} +8.75557 q^{62} -1.34893 q^{63} +1.00000 q^{64} +0.692598 q^{65} -1.02498 q^{66} +1.96939 q^{67} -6.99685 q^{68} -6.62743 q^{69} +1.32185 q^{70} +3.77947 q^{71} +1.94941 q^{72} -6.91125 q^{73} +2.02334 q^{74} +1.38455 q^{75} +2.28021 q^{76} -0.691966 q^{77} -0.371620 q^{78} +15.8015 q^{79} -1.91029 q^{80} +0.648434 q^{81} +3.00686 q^{82} +6.39196 q^{83} -0.709254 q^{84} +13.3660 q^{85} +5.96518 q^{86} -1.03128 q^{87} +1.00000 q^{88} -6.83793 q^{89} -3.72393 q^{90} -0.250881 q^{91} +6.46590 q^{92} +8.97431 q^{93} +4.81591 q^{94} -4.35586 q^{95} +1.02498 q^{96} +3.87509 q^{97} +6.52118 q^{98} +1.94941 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{2} - 4 q^{3} + 24 q^{4} - 4 q^{5} + 4 q^{6} + 7 q^{7} - 24 q^{8} + 28 q^{9}+O(q^{10}) $ 24 * q - 24 * q^2 - 4 * q^3 + 24 * q^4 - 4 * q^5 + 4 * q^6 + 7 * q^7 - 24 * q^8 + 28 * q^9	\( 24 q - 24 q^{2} - 4 q^{3} + 24 q^{4} - 4 q^{5} + 4 q^{6} + 7 q^{7} - 24 q^{8} + 28 q^{9} + 4 q^{10} - 24 q^{11} - 4 q^{12} + 21 q^{13} - 7 q^{14} - 2 q^{15} + 24 q^{16} + 15 q^{17} - 28 q^{18} + 21 q^{19} - 4 q^{20} + 15 q^{21} + 24 q^{22} - 17 q^{23} + 4 q^{24} + 46 q^{25} - 21 q^{26} - 19 q^{27} + 7 q^{28} + 9 q^{29} + 2 q^{30} + 27 q^{31} - 24 q^{32} + 4 q^{33} - 15 q^{34} - 2 q^{35} + 28 q^{36} + 5 q^{37} - 21 q^{38} + 17 q^{39} + 4 q^{40} + 16 q^{41} - 15 q^{42} + 3 q^{43} - 24 q^{44} - 21 q^{45} + 17 q^{46} - 24 q^{47} - 4 q^{48} + 55 q^{49} - 46 q^{50} - 12 q^{51} + 21 q^{52} - 26 q^{53} + 19 q^{54} + 4 q^{55} - 7 q^{56} + 30 q^{57} - 9 q^{58} - 17 q^{59} - 2 q^{60} + 44 q^{61} - 27 q^{62} + 4 q^{63} + 24 q^{64} + 35 q^{65} - 4 q^{66} + 10 q^{67} + 15 q^{68} + 3 q^{69} + 2 q^{70} - 6 q^{71} - 28 q^{72} + 77 q^{73} - 5 q^{74} - 32 q^{75} + 21 q^{76} - 7 q^{77} - 17 q^{78} + 43 q^{79} - 4 q^{80} + 48 q^{81} - 16 q^{82} - 20 q^{83} + 15 q^{84} + 35 q^{85} - 3 q^{86} + 36 q^{87} + 24 q^{88} + 3 q^{89} + 21 q^{90} + 63 q^{91} - 17 q^{92} + 36 q^{93} + 24 q^{94} - 3 q^{95} + 4 q^{96} + 16 q^{97} - 55 q^{98} - 28 q^{99}+O(q^{100}) \) 24 * q - 24 * q^2 - 4 * q^3 + 24 * q^4 - 4 * q^5 + 4 * q^6 + 7 * q^7 - 24 * q^8 + 28 * q^9 + 4 * q^10 - 24 * q^11 - 4 * q^12 + 21 * q^13 - 7 * q^14 - 2 * q^15 + 24 * q^16 + 15 * q^17 - 28 * q^18 + 21 * q^19 - 4 * q^20 + 15 * q^21 + 24 * q^22 - 17 * q^23 + 4 * q^24 + 46 * q^25 - 21 * q^26 - 19 * q^27 + 7 * q^28 + 9 * q^29 + 2 * q^30 + 27 * q^31 - 24 * q^32 + 4 * q^33 - 15 * q^34 - 2 * q^35 + 28 * q^36 + 5 * q^37 - 21 * q^38 + 17 * q^39 + 4 * q^40 + 16 * q^41 - 15 * q^42 + 3 * q^43 - 24 * q^44 - 21 * q^45 + 17 * q^46 - 24 * q^47 - 4 * q^48 + 55 * q^49 - 46 * q^50 - 12 * q^51 + 21 * q^52 - 26 * q^53 + 19 * q^54 + 4 * q^55 - 7 * q^56 + 30 * q^57 - 9 * q^58 - 17 * q^59 - 2 * q^60 + 44 * q^61 - 27 * q^62 + 4 * q^63 + 24 * q^64 + 35 * q^65 - 4 * q^66 + 10 * q^67 + 15 * q^68 + 3 * q^69 + 2 * q^70 - 6 * q^71 - 28 * q^72 + 77 * q^73 - 5 * q^74 - 32 * q^75 + 21 * q^76 - 7 * q^77 - 17 * q^78 + 43 * q^79 - 4 * q^80 + 48 * q^81 - 16 * q^82 - 20 * q^83 + 15 * q^84 + 35 * q^85 - 3 * q^86 + 36 * q^87 + 24 * q^88 + 3 * q^89 + 21 * q^90 + 63 * q^91 - 17 * q^92 + 36 * q^93 + 24 * q^94 - 3 * q^95 + 4 * q^96 + 16 * q^97 - 55 * q^98 - 28 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.02498	−0.591774	−0.295887	−	0.955223i	$-0.595615\pi$
$3$	−1.02498	−0.591774	−0.295887	+	0.955223i	$0.595615\pi$
$4$	1.00000	0.500000
$5$	−1.91029	−0.854306	−0.427153	−	0.904179i	$-0.640484\pi$
$5$	−1.91029	−0.854306	−0.427153	+	0.904179i	$0.640484\pi$
$6$	1.02498	0.418447
$7$	0.691966	0.261539	0.130769	−	0.991413i	$-0.458255\pi$
$7$	0.691966	0.261539	0.130769	+	0.991413i	$0.458255\pi$
$8$	−1.00000	−0.353553
$9$	−1.94941	−0.649804
$10$	1.91029	0.604086
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−1.02498	−0.295887
$13$	−0.362562	−0.100557	−0.0502783	−	0.998735i	$-0.516011\pi$
$13$	−0.362562	−0.100557	−0.0502783	+	0.998735i	$0.516011\pi$
$14$	−0.691966	−0.184936
$15$	1.95801	0.505556
$16$	1.00000	0.250000
$17$	−6.99685	−1.69699	−0.848493	−	0.529207i	$-0.822489\pi$
$17$	−6.99685	−1.69699	−0.848493	+	0.529207i	$0.822489\pi$
$18$	1.94941	0.459481
$19$	2.28021	0.523117	0.261558	−	0.965188i	$-0.415764\pi$
$19$	2.28021	0.523117	0.261558	+	0.965188i	$0.415764\pi$
$20$	−1.91029	−0.427153
$21$	−0.709254	−0.154772
$22$	1.00000	0.213201
$23$	6.46590	1.34823	0.674117	−	0.738625i	$-0.264525\pi$
$23$	6.46590	1.34823	0.674117	+	0.738625i	$0.264525\pi$
$24$	1.02498	0.209224
$25$	−1.35080	−0.270160
$26$	0.362562	0.0711042
$27$	5.07306	0.976311
$28$	0.691966	0.130769
$29$	1.00614	0.186836	0.0934179	−	0.995627i	$-0.470221\pi$
$29$	1.00614	0.186836	0.0934179	+	0.995627i	$0.470221\pi$
$30$	−1.95801	−0.357482
$31$	−8.75557	−1.57255	−0.786274	−	0.617878i	$-0.787992\pi$
$31$	−8.75557	−1.57255	−0.786274	+	0.617878i	$0.787992\pi$
$32$	−1.00000	−0.176777
$33$	1.02498	0.178427
$34$	6.99685	1.19995
$35$	−1.32185	−0.223434
$36$	−1.94941	−0.324902
$37$	−2.02334	−0.332635	−0.166317	−	0.986072i	$-0.553188\pi$
$37$	−2.02334	−0.332635	−0.166317	+	0.986072i	$0.553188\pi$
$38$	−2.28021	−0.369899
$39$	0.371620	0.0595068
$40$	1.91029	0.302043
$41$	−3.00686	−0.469593	−0.234796	−	0.972045i	$-0.575442\pi$
$41$	−3.00686	−0.469593	−0.234796	+	0.972045i	$0.575442\pi$
$42$	0.709254	0.109440
$43$	−5.96518	−0.909681	−0.454841	−	0.890573i	$-0.650304\pi$
$43$	−5.96518	−0.909681	−0.454841	+	0.890573i	$0.650304\pi$
$44$	−1.00000	−0.150756
$45$	3.72393	0.555131
$46$	−6.46590	−0.953345
$47$	−4.81591	−0.702473	−0.351237	−	0.936287i	$-0.614239\pi$
$47$	−4.81591	−0.702473	−0.351237	+	0.936287i	$0.614239\pi$
$48$	−1.02498	−0.147943
$49$	−6.52118	−0.931597
$50$	1.35080	0.191032
$51$	7.17165	1.00423
$52$	−0.362562	−0.0502783
$53$	−6.54593	−0.899152	−0.449576	−	0.893242i	$-0.648425\pi$
$53$	−6.54593	−0.899152	−0.449576	+	0.893242i	$0.648425\pi$
$54$	−5.07306	−0.690356
$55$	1.91029	0.257583
$56$	−0.691966	−0.0924679
$57$	−2.33718	−0.309567
$58$	−1.00614	−0.132113
$59$	3.27790	0.426746	0.213373	−	0.976971i	$-0.431555\pi$
$59$	3.27790	0.426746	0.213373	+	0.976971i	$0.431555\pi$
$60$	1.95801	0.252778
$61$	4.54021	0.581315	0.290657	−	0.956827i	$-0.406126\pi$
$61$	4.54021	0.581315	0.290657	+	0.956827i	$0.406126\pi$
$62$	8.75557	1.11196
$63$	−1.34893	−0.169949
$64$	1.00000	0.125000
$65$	0.692598	0.0859061
$66$	−1.02498	−0.126167
$67$	1.96939	0.240599	0.120300	−	0.992738i	$-0.461615\pi$
$67$	1.96939	0.240599	0.120300	+	0.992738i	$0.461615\pi$
$68$	−6.99685	−0.848493
$69$	−6.62743	−0.797849
$70$	1.32185	0.157992
$71$	3.77947	0.448541	0.224270	−	0.974527i	$-0.428000\pi$
$71$	3.77947	0.448541	0.224270	+	0.974527i	$0.428000\pi$
$72$	1.94941	0.229740
$73$	−6.91125	−0.808901	−0.404450	−	0.914560i	$-0.632537\pi$
$73$	−6.91125	−0.808901	−0.404450	+	0.914560i	$0.632537\pi$
$74$	2.02334	0.235208
$75$	1.38455	0.159874
$76$	2.28021	0.261558
$77$	−0.691966	−0.0788569
$78$	−0.371620	−0.0420776
$79$	15.8015	1.77781	0.888904	−	0.458093i	$-0.151467\pi$
$79$	15.8015	1.77781	0.888904	+	0.458093i	$0.151467\pi$
$80$	−1.91029	−0.213577
$81$	0.648434	0.0720482
$82$	3.00686	0.332052
$83$	6.39196	0.701609	0.350804	−	0.936449i	$-0.385908\pi$
$83$	6.39196	0.701609	0.350804	+	0.936449i	$0.385908\pi$
$84$	−0.709254	−0.0773859
$85$	13.3660	1.44975
$86$	5.96518	0.643242
$87$	−1.03128	−0.110565
$88$	1.00000	0.106600
$89$	−6.83793	−0.724819	−0.362409	−	0.932019i	$-0.618046\pi$
$89$	−6.83793	−0.724819	−0.362409	+	0.932019i	$0.618046\pi$
$90$	−3.72393	−0.392537
$91$	−0.250881	−0.0262994
$92$	6.46590	0.674117
$93$	8.97431	0.930593
$94$	4.81591	0.496724
$95$	−4.35586	−0.446902
$96$	1.02498	0.104612
$97$	3.87509	0.393456	0.196728	−	0.980458i	$-0.436968\pi$
$97$	3.87509	0.393456	0.196728	+	0.980458i	$0.436968\pi$
$98$	6.52118	0.658739
$99$	1.94941	0.195923
$100$	−1.35080	−0.135080
$101$	−13.6335	−1.35659	−0.678293	−	0.734791i	$-0.737280\pi$
$101$	−13.6335	−1.35659	−0.678293	+	0.734791i	$0.737280\pi$
$102$	−7.17165	−0.710099
$103$	14.7954	1.45783	0.728917	−	0.684602i	$-0.240024\pi$
$103$	14.7954	1.45783	0.728917	+	0.684602i	$0.240024\pi$
$104$	0.362562	0.0355521
$105$	1.35488	0.132223
$106$	6.54593	0.635797
$107$	−15.0190	−1.45194	−0.725970	−	0.687727i	$-0.758609\pi$
$107$	−15.0190	−1.45194	−0.725970	+	0.687727i	$0.758609\pi$
$108$	5.07306	0.488155
$109$	−3.07263	−0.294305	−0.147152	−	0.989114i	$-0.547011\pi$
$109$	−3.07263	−0.294305	−0.147152	+	0.989114i	$0.547011\pi$
$110$	−1.91029	−0.182139
$111$	2.07389	0.196845
$112$	0.691966	0.0653847
$113$	−5.93046	−0.557890	−0.278945	−	0.960307i	$-0.589985\pi$
$113$	−5.93046	−0.557890	−0.278945	+	0.960307i	$0.589985\pi$
$114$	2.33718	0.218897
$115$	−12.3517	−1.15180
$116$	1.00614	0.0934179
$117$	0.706782	0.0653420
$118$	−3.27790	−0.301755
$119$	−4.84158	−0.443827
$120$	−1.95801	−0.178741
$121$	1.00000	0.0909091
$122$	−4.54021	−0.411052
$123$	3.08198	0.277893
$124$	−8.75557	−0.786274
$125$	12.1319	1.08511
$126$	1.34893	0.120172
$127$	0.249734	0.0221603	0.0110802	−	0.999939i	$-0.496473\pi$
$127$	0.249734	0.0221603	0.0110802	+	0.999939i	$0.496473\pi$
$128$	−1.00000	−0.0883883
$129$	6.11420	0.538326
$130$	−0.692598	−0.0607448
$131$	−14.1909	−1.23987	−0.619933	−	0.784655i	$-0.712840\pi$
$131$	−14.1909	−1.23987	−0.619933	+	0.784655i	$0.712840\pi$
$132$	1.02498	0.0892133
$133$	1.57783	0.136815
$134$	−1.96939	−0.170129
$135$	−9.69100	−0.834069
$136$	6.99685	0.599975
$137$	−5.71820	−0.488539	−0.244269	−	0.969707i	$-0.578548\pi$
$137$	−5.71820	−0.488539	−0.244269	+	0.969707i	$0.578548\pi$
$138$	6.62743	0.564165
$139$	14.8186	1.25690	0.628448	−	0.777852i	$-0.283690\pi$
$139$	14.8186	1.25690	0.628448	+	0.777852i	$0.283690\pi$
$140$	−1.32185	−0.111717
$141$	4.93623	0.415705
$142$	−3.77947	−0.317166
$143$	0.362562	0.0303190
$144$	−1.94941	−0.162451
$145$	−1.92202	−0.159615
$146$	6.91125	0.571979
$147$	6.68410	0.551295
$148$	−2.02334	−0.166317
$149$	−1.86274	−0.152602	−0.0763008	−	0.997085i	$-0.524311\pi$
$149$	−1.86274	−0.152602	−0.0763008	+	0.997085i	$0.524311\pi$
$150$	−1.38455	−0.113048
$151$	5.50265	0.447799	0.223900	−	0.974612i	$-0.428121\pi$
$151$	5.50265	0.447799	0.223900	+	0.974612i	$0.428121\pi$
$152$	−2.28021	−0.184950
$153$	13.6397	1.10271
$154$	0.691966	0.0557602
$155$	16.7257	1.34344
$156$	0.371620	0.0297534
$157$	2.78723	0.222445	0.111222	−	0.993796i	$-0.464523\pi$
$157$	2.78723	0.222445	0.111222	+	0.993796i	$0.464523\pi$
$158$	−15.8015	−1.25710
$159$	6.70946	0.532095
$160$	1.91029	0.151021
$161$	4.47419	0.352615
$162$	−0.648434	−0.0509458
$163$	−3.30645	−0.258981	−0.129491	−	0.991581i	$-0.541334\pi$
$163$	−3.30645	−0.258981	−0.129491	+	0.991581i	$0.541334\pi$
$164$	−3.00686	−0.234796
$165$	−1.95801	−0.152431
$166$	−6.39196	−0.496112
$167$	−0.130271	−0.0100807	−0.00504035	−	0.999987i	$-0.501604\pi$
$167$	−0.130271	−0.0100807	−0.00504035	+	0.999987i	$0.501604\pi$
$168$	0.709254	0.0547201
$169$	−12.8685	−0.989888
$170$	−13.3660	−1.02512
$171$	−4.44507	−0.339923
$172$	−5.96518	−0.454841
$173$	−2.52845	−0.192234	−0.0961171	−	0.995370i	$-0.530642\pi$
$173$	−2.52845	−0.192234	−0.0961171	+	0.995370i	$0.530642\pi$
$174$	1.03128	0.0781810
$175$	−0.934710	−0.0706574
$176$	−1.00000	−0.0753778
$177$	−3.35979	−0.252537
$178$	6.83793	0.512524
$179$	−0.794054	−0.0593504	−0.0296752	−	0.999560i	$-0.509447\pi$
$179$	−0.794054	−0.0593504	−0.0296752	+	0.999560i	$0.509447\pi$
$180$	3.72393	0.277566
$181$	10.4185	0.774404	0.387202	−	0.921995i	$-0.373442\pi$
$181$	10.4185	0.774404	0.387202	+	0.921995i	$0.373442\pi$
$182$	0.250881	0.0185965
$183$	−4.65364	−0.344007
$184$	−6.46590	−0.476672
$185$	3.86516	0.284172
$186$	−8.97431	−0.658028
$187$	6.99685	0.511660
$188$	−4.81591	−0.351237
$189$	3.51039	0.255343
$190$	4.35586	0.316007
$191$	−18.5990	−1.34578	−0.672888	−	0.739744i	$-0.734947\pi$
$191$	−18.5990	−1.34578	−0.672888	+	0.739744i	$0.734947\pi$
$192$	−1.02498	−0.0739717
$193$	−0.168161	−0.0121045	−0.00605226	−	0.999982i	$-0.501927\pi$
$193$	−0.168161	−0.0121045	−0.00605226	+	0.999982i	$0.501927\pi$
$194$	−3.87509	−0.278216
$195$	−0.709900	−0.0508370
$196$	−6.52118	−0.465799
$197$	1.00000	0.0712470
$197$	1.00000	0.0712470
$198$	−1.94941	−0.138539
$199$	26.9662	1.91158	0.955791	−	0.294047i	$-0.0950024\pi$
$199$	26.9662	1.91158	0.955791	+	0.294047i	$0.0950024\pi$
$200$	1.35080	0.0955161
$201$	−2.01859	−0.142380
$202$	13.6335	0.959251
$203$	0.696216	0.0488648
$204$	7.17165	0.502116
$205$	5.74397	0.401176
$206$	−14.7954	−1.03084
$207$	−12.6047	−0.876087
$208$	−0.362562	−0.0251391
$209$	−2.28021	−0.157726
$210$	−1.35488	−0.0934955
$211$	27.9221	1.92224	0.961118	−	0.276138i	$-0.0890547\pi$
$211$	27.9221	1.92224	0.961118	+	0.276138i	$0.0890547\pi$
$212$	−6.54593	−0.449576
$213$	−3.87389	−0.265435
$214$	15.0190	1.02668
$215$	11.3952	0.777146
$216$	−5.07306	−0.345178
$217$	−6.05856	−0.411282
$218$	3.07263	0.208105
$219$	7.08391	0.478686
$220$	1.91029	0.128792
$221$	2.53679	0.170643
$222$	−2.07389	−0.139190
$223$	21.4897	1.43906	0.719528	−	0.694463i	$-0.244358\pi$
$223$	21.4897	1.43906	0.719528	+	0.694463i	$0.244358\pi$
$224$	−0.691966	−0.0462340
$225$	2.63327	0.175551
$226$	5.93046	0.394488
$227$	16.7313	1.11049	0.555247	−	0.831686i	$-0.312624\pi$
$227$	16.7313	1.11049	0.555247	+	0.831686i	$0.312624\pi$
$228$	−2.33718	−0.154783
$229$	5.79858	0.383181	0.191591	−	0.981475i	$-0.438635\pi$
$229$	5.79858	0.383181	0.191591	+	0.981475i	$0.438635\pi$
$230$	12.3517	0.814449
$231$	0.709254	0.0466655
$232$	−1.00614	−0.0660565
$233$	2.18494	0.143140	0.0715699	−	0.997436i	$-0.477199\pi$
$233$	2.18494	0.143140	0.0715699	+	0.997436i	$0.477199\pi$
$234$	−0.706782	−0.0462038
$235$	9.19978	0.600128
$236$	3.27790	0.213373
$237$	−16.1963	−1.05206
$238$	4.84158	0.313833
$239$	−20.8204	−1.34676	−0.673379	−	0.739297i	$-0.735158\pi$
$239$	−20.8204	−1.34676	−0.673379	+	0.739297i	$0.735158\pi$
$240$	1.95801	0.126389
$241$	21.4563	1.38212	0.691059	−	0.722798i	$-0.257144\pi$
$241$	21.4563	1.38212	0.691059	+	0.722798i	$0.257144\pi$
$242$	−1.00000	−0.0642824
$243$	−15.8838	−1.01895
$244$	4.54021	0.290657
$245$	12.4573	0.795870
$246$	−3.08198	−0.196500
$247$	−0.826718	−0.0526028
$248$	8.75557	0.555979
$249$	−6.55165	−0.415194
$250$	−12.1319	−0.767286
$251$	−13.6654	−0.862552	−0.431276	−	0.902220i	$-0.641936\pi$
$251$	−13.6654	−0.862552	−0.431276	+	0.902220i	$0.641936\pi$
$252$	−1.34893	−0.0849744
$253$	−6.46590	−0.406508
$254$	−0.249734	−0.0156697
$255$	−13.6999	−0.857922
$256$	1.00000	0.0625000
$257$	15.1219	0.943277	0.471639	−	0.881792i	$-0.343663\pi$
$257$	15.1219	0.943277	0.471639	+	0.881792i	$0.343663\pi$
$258$	−6.11420	−0.380654
$259$	−1.40008	−0.0869968
$260$	0.692598	0.0429531
$261$	−1.96138	−0.121407
$262$	14.1909	0.876718
$263$	8.99530	0.554674	0.277337	−	0.960773i	$-0.410548\pi$
$263$	8.99530	0.554674	0.277337	+	0.960773i	$0.410548\pi$
$264$	−1.02498	−0.0630833
$265$	12.5046	0.768151
$266$	−1.57783	−0.0967430
$267$	7.00876	0.428929
$268$	1.96939	0.120300
$269$	0.237819	0.0145001	0.00725004	−	0.999974i	$-0.497692\pi$
$269$	0.237819	0.0145001	0.00725004	+	0.999974i	$0.497692\pi$
$270$	9.69100	0.589776
$271$	18.7921	1.14154	0.570768	−	0.821111i	$-0.306645\pi$
$271$	18.7921	1.14154	0.570768	+	0.821111i	$0.306645\pi$
$272$	−6.99685	−0.424246
$273$	0.257148	0.0155633
$274$	5.71820	0.345449
$275$	1.35080	0.0814564
$276$	−6.62743	−0.398925
$277$	11.1676	0.670994	0.335497	−	0.942041i	$-0.391096\pi$
$277$	11.1676	0.670994	0.335497	+	0.942041i	$0.391096\pi$
$278$	−14.8186	−0.888760
$279$	17.0682	1.02185
$280$	1.32185	0.0789959
$281$	−22.3586	−1.33380	−0.666901	−	0.745146i	$-0.732380\pi$
$281$	−22.3586	−1.33380	−0.666901	+	0.745146i	$0.732380\pi$
$282$	−4.93623	−0.293948
$283$	−5.95205	−0.353813	−0.176906	−	0.984228i	$-0.556609\pi$
$283$	−5.95205	−0.353813	−0.176906	+	0.984228i	$0.556609\pi$
$284$	3.77947	0.224270
$285$	4.46468	0.264465
$286$	−0.362562	−0.0214387
$287$	−2.08065	−0.122817
$288$	1.94941	0.114870
$289$	31.9559	1.87976
$290$	1.92202	0.112865
$291$	−3.97190	−0.232837
$292$	−6.91125	−0.404450
$293$	11.6230	0.679023	0.339511	−	0.940602i	$-0.389738\pi$
$293$	11.6230	0.679023	0.339511	+	0.940602i	$0.389738\pi$
$294$	−6.68410	−0.389825
$295$	−6.26172	−0.364572
$296$	2.02334	0.117604
$297$	−5.07306	−0.294369
$298$	1.86274	0.107906
$299$	−2.34429	−0.135574
$300$	1.38455	0.0799369
$301$	−4.12770	−0.237917
$302$	−5.50265	−0.316642
$303$	13.9741	0.802793
$304$	2.28021	0.130779
$305$	−8.67311	−0.496621
$306$	−13.6397	−0.779732
$307$	26.0116	1.48456	0.742281	−	0.670088i	$-0.233744\pi$
$307$	26.0116	1.48456	0.742281	+	0.670088i	$0.233744\pi$
$308$	−0.691966	−0.0394284
$309$	−15.1650	−0.862708
$310$	−16.7257	−0.949954
$311$	24.6144	1.39575	0.697876	−	0.716218i	$-0.254128\pi$
$311$	24.6144	1.39575	0.697876	+	0.716218i	$0.254128\pi$
$312$	−0.371620	−0.0210388
$313$	−2.58669	−0.146208	−0.0731042	−	0.997324i	$-0.523291\pi$
$313$	−2.58669	−0.146208	−0.0731042	+	0.997324i	$0.523291\pi$
$314$	−2.78723	−0.157292
$315$	2.57684	0.145188
$316$	15.8015	0.888904
$317$	4.09194	0.229826	0.114913	−	0.993376i	$-0.463341\pi$
$317$	4.09194	0.229826	0.114913	+	0.993376i	$0.463341\pi$
$318$	−6.70946	−0.376248
$319$	−1.00614	−0.0563331
$320$	−1.91029	−0.106788
$321$	15.3942	0.859220
$322$	−4.47419	−0.249337
$323$	−15.9543	−0.887721
$324$	0.648434	0.0360241
$325$	0.489749	0.0271664
$326$	3.30645	0.183127
$327$	3.14939	0.174162
$328$	3.00686	0.166026
$329$	−3.33245	−0.183724
$330$	1.95801	0.107785
$331$	−5.48792	−0.301643	−0.150822	−	0.988561i	$-0.548192\pi$
$331$	−5.48792	−0.301643	−0.150822	+	0.988561i	$0.548192\pi$
$332$	6.39196	0.350804
$333$	3.94432	0.216147
$334$	0.130271	0.00712813
$335$	−3.76210	−0.205545
$336$	−0.709254	−0.0386930
$337$	−22.5668	−1.22929	−0.614647	−	0.788802i	$-0.710702\pi$
$337$	−22.5668	−1.22929	−0.614647	+	0.788802i	$0.710702\pi$
$338$	12.8685	0.699957
$339$	6.07861	0.330145
$340$	13.3660	0.724873
$341$	8.75557	0.474141
$342$	4.44507	0.240362
$343$	−9.35620	−0.505188
$344$	5.96518	0.321621
$345$	12.6603	0.681608
$346$	2.52845	0.135930
$347$	−9.26970	−0.497623	−0.248812	−	0.968552i	$-0.580040\pi$
$347$	−9.26970	−0.497623	−0.248812	+	0.968552i	$0.580040\pi$
$348$	−1.03128	−0.0552823
$349$	12.1446	0.650086	0.325043	−	0.945699i	$-0.394621\pi$
$349$	12.1446	0.650086	0.325043	+	0.945699i	$0.394621\pi$
$350$	0.934710	0.0499623
$351$	−1.83930	−0.0981745
$352$	1.00000	0.0533002
$353$	35.6822	1.89917	0.949586	−	0.313506i	$-0.101504\pi$
$353$	35.6822	1.89917	0.949586	+	0.313506i	$0.101504\pi$
$354$	3.35979	0.178571
$355$	−7.21988	−0.383191
$356$	−6.83793	−0.362409
$357$	4.96254	0.262645
$358$	0.794054	0.0419671
$359$	−2.48268	−0.131031	−0.0655154	−	0.997852i	$-0.520869\pi$
$359$	−2.48268	−0.131031	−0.0655154	+	0.997852i	$0.520869\pi$
$360$	−3.72393	−0.196269
$361$	−13.8006	−0.726349
$362$	−10.4185	−0.547586
$363$	−1.02498	−0.0537976
$364$	−0.250881	−0.0131497
$365$	13.2025	0.691049
$366$	4.65364	0.243250
$367$	−3.36478	−0.175640	−0.0878200	−	0.996136i	$-0.527990\pi$
$367$	−3.36478	−0.175640	−0.0878200	+	0.996136i	$0.527990\pi$
$368$	6.46590	0.337058
$369$	5.86161	0.305143
$370$	−3.86516	−0.200940
$371$	−4.52956	−0.235163
$372$	8.97431	0.465296
$373$	−8.91296	−0.461495	−0.230748	−	0.973014i	$-0.574117\pi$
$373$	−8.91296	−0.461495	−0.230748	+	0.973014i	$0.574117\pi$
$374$	−6.99685	−0.361798
$375$	−12.4349	−0.642138
$376$	4.81591	0.248362
$377$	−0.364789	−0.0187876
$378$	−3.51039	−0.180555
$379$	7.52346	0.386454	0.193227	−	0.981154i	$-0.438105\pi$
$379$	7.52346	0.386454	0.193227	+	0.981154i	$0.438105\pi$
$380$	−4.35586	−0.223451
$381$	−0.255973	−0.0131139
$382$	18.5990	0.951608
$383$	1.94618	0.0994454	0.0497227	−	0.998763i	$-0.484166\pi$
$383$	1.94618	0.0994454	0.0497227	+	0.998763i	$0.484166\pi$
$384$	1.02498	0.0523059
$385$	1.32185	0.0673680
$386$	0.168161	0.00855919
$387$	11.6286	0.591114
$388$	3.87509	0.196728
$389$	35.7615	1.81318	0.906589	−	0.422014i	$-0.138677\pi$
$389$	35.7615	1.81318	0.906589	+	0.422014i	$0.138677\pi$
$390$	0.709900	0.0359472
$391$	−45.2409	−2.28793
$392$	6.52118	0.329369
$393$	14.5454	0.733721
$394$	−1.00000	−0.0503793
$395$	−30.1854	−1.51879
$396$	1.94941	0.0979616
$397$	−0.205376	−0.0103075	−0.00515376	−	0.999987i	$-0.501640\pi$
$397$	−0.205376	−0.0103075	−0.00515376	+	0.999987i	$0.501640\pi$
$398$	−26.9662	−1.35169
$399$	−1.61725	−0.0809637
$400$	−1.35080	−0.0675401
$401$	16.8203	0.839964	0.419982	−	0.907532i	$-0.362036\pi$
$401$	16.8203	0.839964	0.419982	+	0.907532i	$0.362036\pi$
$402$	2.01859	0.100678
$403$	3.17444	0.158130
$404$	−13.6335	−0.678293
$405$	−1.23870	−0.0615513
$406$	−0.696216	−0.0345526
$407$	2.02334	0.100293
$408$	−7.17165	−0.355049
$409$	22.7072	1.12280	0.561399	−	0.827545i	$-0.310263\pi$
$409$	22.7072	1.12280	0.561399	+	0.827545i	$0.310263\pi$
$410$	−5.74397	−0.283674
$411$	5.86106	0.289105
$412$	14.7954	0.728917
$413$	2.26819	0.111611
$414$	12.6047	0.619487
$415$	−12.2105	−0.599389
$416$	0.362562	0.0177761
$417$	−15.1888	−0.743798
$418$	2.28021	0.111529
$419$	9.05718	0.442472	0.221236	−	0.975220i	$-0.428991\pi$
$419$	9.05718	0.442472	0.221236	+	0.975220i	$0.428991\pi$
$420$	1.35488	0.0661113
$421$	0.134894	0.00657435	0.00328718	−	0.999995i	$-0.498954\pi$
$421$	0.134894	0.00657435	0.00328718	+	0.999995i	$0.498954\pi$
$422$	−27.9221	−1.35923
$423$	9.38819	0.456470
$424$	6.54593	0.317898
$425$	9.45136	0.458458
$426$	3.87389	0.187691
$427$	3.14168	0.152036
$428$	−15.0190	−0.725970
$429$	−0.371620	−0.0179420
$430$	−11.3952	−0.549525
$431$	−11.6429	−0.560818	−0.280409	−	0.959881i	$-0.590470\pi$
$431$	−11.6429	−0.560818	−0.280409	+	0.959881i	$0.590470\pi$
$432$	5.07306	0.244078
$433$	−14.2522	−0.684917	−0.342458	−	0.939533i	$-0.611260\pi$
$433$	−14.2522	−0.684917	−0.342458	+	0.939533i	$0.611260\pi$
$434$	6.05856	0.290820
$435$	1.97004	0.0944561
$436$	−3.07263	−0.147152
$437$	14.7436	0.705283
$438$	−7.08391	−0.338482
$439$	18.1677	0.867095	0.433548	−	0.901131i	$-0.357262\pi$
$439$	18.1677	0.867095	0.433548	+	0.901131i	$0.357262\pi$
$440$	−1.91029	−0.0910694
$441$	12.7125	0.605355
$442$	−2.53679	−0.120663
$443$	−15.2289	−0.723549	−0.361774	−	0.932266i	$-0.617829\pi$
$443$	−15.2289	−0.723549	−0.361774	+	0.932266i	$0.617829\pi$
$444$	2.07389	0.0984223
$445$	13.0624	0.619217
$446$	−21.4897	−1.01757
$447$	1.90928	0.0903056
$448$	0.691966	0.0326923
$449$	17.2722	0.815126	0.407563	−	0.913177i	$-0.366379\pi$
$449$	17.2722	0.815126	0.407563	+	0.913177i	$0.366379\pi$
$450$	−2.63327	−0.124133
$451$	3.00686	0.141588
$452$	−5.93046	−0.278945
$453$	−5.64012	−0.264996
$454$	−16.7313	−0.785238
$455$	0.479254	0.0224678
$456$	2.33718	0.109448
$457$	14.4024	0.673715	0.336858	−	0.941556i	$-0.390636\pi$
$457$	14.4024	0.673715	0.336858	+	0.941556i	$0.390636\pi$
$458$	−5.79858	−0.270950
$459$	−35.4954	−1.65678
$460$	−12.3517	−0.575902
$461$	19.8128	0.922774	0.461387	−	0.887199i	$-0.347352\pi$
$461$	19.8128	0.922774	0.461387	+	0.887199i	$0.347352\pi$
$462$	−0.709254	−0.0329975
$463$	−26.3608	−1.22509	−0.612545	−	0.790436i	$-0.709854\pi$
$463$	−26.3608	−1.22509	−0.612545	+	0.790436i	$0.709854\pi$
$464$	1.00614	0.0467090
$465$	−17.1435	−0.795011
$466$	−2.18494	−0.101215
$467$	31.3089	1.44880	0.724402	−	0.689378i	$-0.242116\pi$
$467$	31.3089	1.44880	0.724402	+	0.689378i	$0.242116\pi$
$468$	0.706782	0.0326710
$469$	1.36275	0.0629260
$470$	−9.19978	−0.424354
$471$	−2.85686	−0.131637
$472$	−3.27790	−0.150877
$473$	5.96518	0.274279
$474$	16.1963	0.743919
$475$	−3.08012	−0.141325
$476$	−4.84158	−0.221914
$477$	12.7607	0.584272
$478$	20.8204	0.952302
$479$	−16.8451	−0.769671	−0.384836	−	0.922985i	$-0.625742\pi$
$479$	−16.8451	−0.769671	−0.384836	+	0.922985i	$0.625742\pi$
$480$	−1.95801	−0.0893706
$481$	0.733585	0.0334486
$482$	−21.4563	−0.977305
$483$	−4.58596	−0.208669
$484$	1.00000	0.0454545
$485$	−7.40254	−0.336132
$486$	15.8838	0.720504
$487$	−15.5316	−0.703806	−0.351903	−	0.936037i	$-0.614465\pi$
$487$	−15.5316	−0.703806	−0.351903	+	0.936037i	$0.614465\pi$
$488$	−4.54021	−0.205526
$489$	3.38905	0.153258
$490$	−12.4573	−0.562765
$491$	−19.3385	−0.872736	−0.436368	−	0.899768i	$-0.643735\pi$
$491$	−19.3385	−0.872736	−0.436368	+	0.899768i	$0.643735\pi$
$492$	3.08198	0.138946
$493$	−7.03982	−0.317058
$494$	0.826718	0.0371958
$495$	−3.72393	−0.167378
$496$	−8.75557	−0.393137
$497$	2.61527	0.117311
$498$	6.55165	0.293586
$499$	41.4759	1.85672	0.928358	−	0.371687i	$-0.121221\pi$
$499$	41.4759	1.85672	0.928358	+	0.371687i	$0.121221\pi$
$500$	12.1319	0.542553
$501$	0.133526	0.00596550
$502$	13.6654	0.609916
$503$	10.0770	0.449312	0.224656	−	0.974438i	$-0.427874\pi$
$503$	10.0770	0.449312	0.224656	+	0.974438i	$0.427874\pi$
$504$	1.34893	0.0600860
$505$	26.0440	1.15894
$506$	6.46590	0.287444
$507$	13.1900	0.585790
$508$	0.249734	0.0110802
$509$	23.3691	1.03582	0.517909	−	0.855435i	$-0.326710\pi$
$509$	23.3691	1.03582	0.517909	+	0.855435i	$0.326710\pi$
$510$	13.6999	0.606642
$511$	−4.78235	−0.211559
$512$	−1.00000	−0.0441942
$513$	11.5677	0.510724
$514$	−15.1219	−0.666998
$515$	−28.2635	−1.24544
$516$	6.11420	0.269163
$517$	4.81591	0.211804
$518$	1.40008	0.0615161
$519$	2.59161	0.113759
$520$	−0.692598	−0.0303724
$521$	23.4057	1.02542	0.512710	−	0.858562i	$-0.328641\pi$
$521$	23.4057	1.02542	0.512710	+	0.858562i	$0.328641\pi$
$522$	1.96138	0.0858474
$523$	−33.4846	−1.46418	−0.732090	−	0.681208i	$-0.761455\pi$
$523$	−33.4846	−1.46418	−0.732090	+	0.681208i	$0.761455\pi$
$524$	−14.1909	−0.619933
$525$	0.958061	0.0418132
$526$	−8.99530	−0.392214
$527$	61.2614	2.66859
$528$	1.02498	0.0446066
$529$	18.8079	0.817733
$530$	−12.5046	−0.543165
$531$	−6.38997	−0.277301
$532$	1.57783	0.0684076
$533$	1.09017	0.0472207
$534$	−7.00876	−0.303299
$535$	28.6906	1.24040
$536$	−1.96939	−0.0850646
$537$	0.813892	0.0351220
$538$	−0.237819	−0.0102531
$539$	6.52118	0.280887
$540$	−9.69100	−0.417034
$541$	3.98772	0.171446	0.0857228	−	0.996319i	$-0.472680\pi$
$541$	3.98772	0.171446	0.0857228	+	0.996319i	$0.472680\pi$
$542$	−18.7921	−0.807188
$543$	−10.6788	−0.458272
$544$	6.99685	0.299987
$545$	5.86961	0.251426
$546$	−0.257148	−0.0110049
$547$	44.7076	1.91156	0.955779	−	0.294086i	$-0.0950152\pi$
$547$	44.7076	1.91156	0.955779	+	0.294086i	$0.0950152\pi$
$548$	−5.71820	−0.244269
$549$	−8.85074	−0.377740
$550$	−1.35080	−0.0575984
$551$	2.29422	0.0977369
$552$	6.62743	0.282082
$553$	10.9341	0.464966
$554$	−11.1676	−0.474464
$555$	−3.96172	−0.168166
$556$	14.8186	0.628448
$557$	16.2217	0.687336	0.343668	−	0.939091i	$-0.388331\pi$
$557$	16.2217	0.687336	0.343668	+	0.939091i	$0.388331\pi$
$558$	−17.0682	−0.722555
$559$	2.16275	0.0914744
$560$	−1.32185	−0.0558586
$561$	−7.17165	−0.302787
$562$	22.3586	0.943140
$563$	−18.8588	−0.794805	−0.397403	−	0.917644i	$-0.630088\pi$
$563$	−18.8588	−0.794805	−0.397403	+	0.917644i	$0.630088\pi$
$564$	4.93623	0.207853
$565$	11.3289	0.476609
$566$	5.95205	0.250183
$567$	0.448695	0.0188434
$568$	−3.77947	−0.158583
$569$	−12.6474	−0.530207	−0.265103	−	0.964220i	$-0.585406\pi$
$569$	−12.6474	−0.530207	−0.265103	+	0.964220i	$0.585406\pi$
$570$	−4.46468	−0.187005
$571$	−11.6076	−0.485761	−0.242881	−	0.970056i	$-0.578092\pi$
$571$	−11.6076	−0.485761	−0.242881	+	0.970056i	$0.578092\pi$
$572$	0.362562	0.0151595
$573$	19.0637	0.796396
$574$	2.08065	0.0868445
$575$	−8.73415	−0.364239
$576$	−1.94941	−0.0812254
$577$	−35.9438	−1.49636	−0.748179	−	0.663496i	$-0.769072\pi$
$577$	−35.9438	−1.49636	−0.748179	+	0.663496i	$0.769072\pi$
$578$	−31.9559	−1.32919
$579$	0.172363	0.00716314
$580$	−1.92202	−0.0798076
$581$	4.42302	0.183498
$582$	3.97190	0.164641
$583$	6.54593	0.271105
$584$	6.91125	0.285990
$585$	−1.35016	−0.0558221
$586$	−11.6230	−0.480141
$587$	−34.1928	−1.41129	−0.705643	−	0.708567i	$-0.749342\pi$
$587$	−34.1928	−1.41129	−0.705643	+	0.708567i	$0.749342\pi$
$588$	6.68410	0.275648
$589$	−19.9646	−0.822626
$590$	6.26172	0.257791
$591$	−1.02498	−0.0421622
$592$	−2.02334	−0.0831587
$593$	1.16524	0.0478506	0.0239253	−	0.999714i	$-0.492384\pi$
$593$	1.16524	0.0478506	0.0239253	+	0.999714i	$0.492384\pi$
$594$	5.07306	0.208150
$595$	9.24882	0.379165
$596$	−1.86274	−0.0763008
$597$	−27.6399	−1.13122
$598$	2.34429	0.0958651
$599$	35.0565	1.43237	0.716184	−	0.697912i	$-0.245887\pi$
$599$	35.0565	1.43237	0.716184	+	0.697912i	$0.245887\pi$
$600$	−1.38455	−0.0565240
$601$	40.7014	1.66025	0.830123	−	0.557581i	$-0.188270\pi$
$601$	40.7014	1.66025	0.830123	+	0.557581i	$0.188270\pi$
$602$	4.12770	0.168233
$603$	−3.83915	−0.156342
$604$	5.50265	0.223900
$605$	−1.91029	−0.0776642
$606$	−13.9741	−0.567660
$607$	−0.799270	−0.0324414	−0.0162207	−	0.999868i	$-0.505163\pi$
$607$	−0.799270	−0.0324414	−0.0162207	+	0.999868i	$0.505163\pi$
$608$	−2.28021	−0.0924748
$609$	−0.713610	−0.0289169
$610$	8.67311	0.351164
$611$	1.74607	0.0706383
$612$	13.6397	0.551353
$613$	−15.8737	−0.641132	−0.320566	−	0.947226i	$-0.603873\pi$
$613$	−15.8737	−0.641132	−0.320566	+	0.947226i	$0.603873\pi$
$614$	−26.0116	−1.04974
$615$	−5.88747	−0.237406
$616$	0.691966	0.0278801
$617$	9.73678	0.391988	0.195994	−	0.980605i	$-0.437207\pi$
$617$	9.73678	0.391988	0.195994	+	0.980605i	$0.437207\pi$
$618$	15.1650	0.610027
$619$	−46.4288	−1.86613	−0.933066	−	0.359704i	$-0.882878\pi$
$619$	−46.4288	−1.86613	−0.933066	+	0.359704i	$0.882878\pi$
$620$	16.7257	0.671719
$621$	32.8019	1.31629
$622$	−24.6144	−0.986946
$623$	−4.73162	−0.189568
$624$	0.371620	0.0148767
$625$	−16.4213	−0.656853
$626$	2.58669	0.103385
$627$	2.33718	0.0933379
$628$	2.78723	0.111222
$629$	14.1570	0.564476
$630$	−2.57684	−0.102664
$631$	34.7260	1.38242	0.691209	−	0.722655i	$-0.257078\pi$
$631$	34.7260	1.38242	0.691209	+	0.722655i	$0.257078\pi$
$632$	−15.8015	−0.628550
$633$	−28.6197	−1.13753
$634$	−4.09194	−0.162512
$635$	−0.477064	−0.0189317
$636$	6.70946	0.266047
$637$	2.36433	0.0936783
$638$	1.00614	0.0398335
$639$	−7.36774	−0.291463
$640$	1.91029	0.0755107
$641$	−22.8655	−0.903133	−0.451567	−	0.892237i	$-0.649135\pi$
$641$	−22.8655	−0.903133	−0.451567	+	0.892237i	$0.649135\pi$
$642$	−15.3942	−0.607560
$643$	−24.5896	−0.969720	−0.484860	−	0.874592i	$-0.661129\pi$
$643$	−24.5896	−0.969720	−0.484860	+	0.874592i	$0.661129\pi$
$644$	4.47419	0.176308
$645$	−11.6799	−0.459895
$646$	15.9543	0.627714
$647$	−30.3736	−1.19411	−0.597054	−	0.802201i	$-0.703662\pi$
$647$	−30.3736	−1.19411	−0.597054	+	0.802201i	$0.703662\pi$
$648$	−0.648434	−0.0254729
$649$	−3.27790	−0.128669
$650$	−0.489749	−0.0192096
$651$	6.20992	0.243386
$652$	−3.30645	−0.129491
$653$	4.98952	0.195255	0.0976275	−	0.995223i	$-0.468875\pi$
$653$	4.98952	0.195255	0.0976275	+	0.995223i	$0.468875\pi$
$654$	−3.14939	−0.123151
$655$	27.1087	1.05923
$656$	−3.00686	−0.117398
$657$	13.4729	0.525627
$658$	3.33245	0.129912
$659$	−22.5351	−0.877845	−0.438922	−	0.898525i	$-0.644640\pi$
$659$	−22.5351	−0.877845	−0.438922	+	0.898525i	$0.644640\pi$
$660$	−1.95801	−0.0762155
$661$	−22.6937	−0.882683	−0.441341	−	0.897339i	$-0.645497\pi$
$661$	−22.6937	−0.882683	−0.441341	+	0.897339i	$0.645497\pi$
$662$	5.48792	0.213294
$663$	−2.60017	−0.100982
$664$	−6.39196	−0.248056
$665$	−3.01411	−0.116882
$666$	−3.94432	−0.152839
$667$	6.50561	0.251898
$668$	−0.130271	−0.00504035
$669$	−22.0266	−0.851596
$670$	3.76210	0.145342
$671$	−4.54021	−0.175273
$672$	0.709254	0.0273601
$673$	37.9396	1.46246	0.731231	−	0.682130i	$-0.238946\pi$
$673$	37.9396	1.46246	0.731231	+	0.682130i	$0.238946\pi$
$674$	22.5668	0.869242
$675$	−6.85270	−0.263761
$676$	−12.8685	−0.494944
$677$	−32.5413	−1.25066	−0.625332	−	0.780359i	$-0.715037\pi$
$677$	−32.5413	−1.25066	−0.625332	+	0.780359i	$0.715037\pi$
$678$	−6.07861	−0.233448
$679$	2.68144	0.102904
$680$	−13.3660	−0.512562
$681$	−17.1493	−0.657161
$682$	−8.75557	−0.335268
$683$	−44.5617	−1.70511	−0.852553	−	0.522640i	$-0.824947\pi$
$683$	−44.5617	−1.70511	−0.852553	+	0.522640i	$0.824947\pi$
$684$	−4.44507	−0.169962
$685$	10.9234	0.417362
$686$	9.35620	0.357222
$687$	−5.94345	−0.226757
$688$	−5.96518	−0.227420
$689$	2.37330	0.0904157
$690$	−12.6603	−0.481970
$691$	−16.3396	−0.621589	−0.310794	−	0.950477i	$-0.600595\pi$
$691$	−16.3396	−0.621589	−0.310794	+	0.950477i	$0.600595\pi$
$692$	−2.52845	−0.0961171
$693$	1.34893	0.0512415
$694$	9.26970	0.351873
$695$	−28.3078	−1.07377
$696$	1.03128	0.0390905
$697$	21.0386	0.796892
$698$	−12.1446	−0.459680
$699$	−2.23952	−0.0847065
$700$	−0.934710	−0.0353287
$701$	41.6919	1.57468	0.787342	−	0.616517i	$-0.211457\pi$
$701$	41.6919	1.57468	0.787342	+	0.616517i	$0.211457\pi$
$702$	1.83930	0.0694198
$703$	−4.61364	−0.174007
$704$	−1.00000	−0.0376889
$705$	−9.42961	−0.355140
$706$	−35.6822	−1.34292
$707$	−9.43394	−0.354800
$708$	−3.35979	−0.126269
$709$	−33.2556	−1.24894	−0.624471	−	0.781048i	$-0.714685\pi$
$709$	−33.2556	−1.24894	−0.624471	+	0.781048i	$0.714685\pi$
$710$	7.21988	0.270957
$711$	−30.8036	−1.15523
$712$	6.83793	0.256262
$713$	−56.6127	−2.12016
$714$	−4.96254	−0.185718
$715$	−0.692598	−0.0259017
$716$	−0.794054	−0.0296752
$717$	21.3405	0.796976
$718$	2.48268	0.0926528
$719$	34.5617	1.28893	0.644466	−	0.764633i	$-0.277080\pi$
$719$	34.5617	1.28893	0.644466	+	0.764633i	$0.277080\pi$
$720$	3.72393	0.138783
$721$	10.2379	0.381280
$722$	13.8006	0.513606
$723$	−21.9923	−0.817902
$724$	10.4185	0.387202
$725$	−1.35910	−0.0504757
$726$	1.02498	0.0380407
$727$	−18.8906	−0.700613	−0.350307	−	0.936635i	$-0.613923\pi$
$727$	−18.8906	−0.700613	−0.350307	+	0.936635i	$0.613923\pi$
$728$	0.250881	0.00929826
$729$	14.3353	0.530938
$730$	−13.2025	−0.488645
$731$	41.7374	1.54372
$732$	−4.65364	−0.172004
$733$	20.2386	0.747531	0.373766	−	0.927523i	$-0.378066\pi$
$733$	20.2386	0.747531	0.373766	+	0.927523i	$0.378066\pi$
$734$	3.36478	0.124196
$735$	−12.7685	−0.470975
$736$	−6.46590	−0.238336
$737$	−1.96939	−0.0725433
$738$	−5.86161	−0.215769
$739$	−14.6182	−0.537741	−0.268870	−	0.963176i	$-0.586650\pi$
$739$	−14.6182	−0.537741	−0.268870	+	0.963176i	$0.586650\pi$
$740$	3.86516	0.142086
$741$	0.847372	0.0311290
$742$	4.52956	0.166285
$743$	47.3240	1.73615	0.868074	−	0.496435i	$-0.165358\pi$
$743$	47.3240	1.73615	0.868074	+	0.496435i	$0.165358\pi$
$744$	−8.97431	−0.329014
$745$	3.55837	0.130369
$746$	8.91296	0.326327
$747$	−12.4606	−0.455908
$748$	6.99685	0.255830
$749$	−10.3926	−0.379738
$750$	12.4349	0.454060
$751$	52.1862	1.90430	0.952150	−	0.305631i	$-0.0988674\pi$
$751$	52.1862	1.90430	0.952150	+	0.305631i	$0.0988674\pi$
$752$	−4.81591	−0.175618
$753$	14.0068	0.510436
$754$	0.364789	0.0132848
$755$	−10.5116	−0.382558
$756$	3.51039	0.127672
$757$	10.0667	0.365882	0.182941	−	0.983124i	$-0.441438\pi$
$757$	10.0667	0.365882	0.182941	+	0.983124i	$0.441438\pi$
$758$	−7.52346	−0.273264
$759$	6.62743	0.240561
$760$	4.35586	0.158004
$761$	26.5757	0.963367	0.481684	−	0.876345i	$-0.340025\pi$
$761$	26.5757	0.963367	0.481684	+	0.876345i	$0.340025\pi$
$762$	0.255973	0.00927292
$763$	−2.12616	−0.0769721
$764$	−18.5990	−0.672888
$765$	−26.0558	−0.942050
$766$	−1.94618	−0.0703185
$767$	−1.18844	−0.0429121
$768$	−1.02498	−0.0369859
$769$	−46.4123	−1.67367	−0.836836	−	0.547454i	$-0.815597\pi$
$769$	−46.4123	−1.67367	−0.836836	+	0.547454i	$0.815597\pi$
$770$	−1.32185	−0.0476363
$771$	−15.4997	−0.558207
$772$	−0.168161	−0.00605226
$773$	−22.5241	−0.810137	−0.405068	−	0.914286i	$-0.632752\pi$
$773$	−22.5241	−0.810137	−0.405068	+	0.914286i	$0.632752\pi$
$774$	−11.6286	−0.417981
$775$	11.8270	0.424840
$776$	−3.87509	−0.139108
$777$	1.43506	0.0514825
$778$	−35.7615	−1.28211
$779$	−6.85628	−0.245652
$780$	−0.709900	−0.0254185
$781$	−3.77947	−0.135240
$782$	45.2409	1.61781
$783$	5.10422	0.182410
$784$	−6.52118	−0.232899
$785$	−5.32440	−0.190036
$786$	−14.5454	−0.518819
$787$	−14.1067	−0.502848	−0.251424	−	0.967877i	$-0.580899\pi$
$787$	−14.1067	−0.502848	−0.251424	+	0.967877i	$0.580899\pi$
$788$	1.00000	0.0356235
$789$	−9.22002	−0.328241
$790$	30.1854	1.07395
$791$	−4.10368	−0.145910
$792$	−1.94941	−0.0692693
$793$	−1.64611	−0.0584550
$794$	0.205376	0.00728851
$795$	−12.8170	−0.454572
$796$	26.9662	0.955791
$797$	−40.9772	−1.45149	−0.725743	−	0.687966i	$-0.758504\pi$
$797$	−40.9772	−1.45149	−0.725743	+	0.687966i	$0.758504\pi$
$798$	1.61725	0.0572500
$799$	33.6962	1.19209
$800$	1.35080	0.0477581
$801$	13.3299	0.470990
$802$	−16.8203	−0.593944
$803$	6.91125	0.243893
$804$	−2.01859	−0.0711901
$805$	−8.54698	−0.301241
$806$	−3.17444	−0.111815
$807$	−0.243760	−0.00858077
$808$	13.6335	0.479626
$809$	−3.26870	−0.114921	−0.0574607	−	0.998348i	$-0.518300\pi$
$809$	−3.26870	−0.114921	−0.0574607	+	0.998348i	$0.518300\pi$
$810$	1.23870	0.0435233
$811$	29.8085	1.04672	0.523359	−	0.852112i	$-0.324679\pi$
$811$	29.8085	1.04672	0.523359	+	0.852112i	$0.324679\pi$
$812$	0.696216	0.0244324
$813$	−19.2615	−0.675532
$814$	−2.02334	−0.0709179
$815$	6.31627	0.221249
$816$	7.17165	0.251058
$817$	−13.6019	−0.475869
$818$	−22.7072	−0.793939
$819$	0.489070	0.0170895
$820$	5.74397	0.200588
$821$	36.4044	1.27052	0.635261	−	0.772298i	$-0.280893\pi$
$821$	36.4044	1.27052	0.635261	+	0.772298i	$0.280893\pi$
$822$	−5.86106	−0.204428
$823$	−37.5358	−1.30842	−0.654208	−	0.756315i	$-0.726998\pi$
$823$	−37.5358	−1.30842	−0.654208	+	0.756315i	$0.726998\pi$
$824$	−14.7954	−0.515422
$825$	−1.38455	−0.0482038
$826$	−2.26819	−0.0789206
$827$	−27.8943	−0.969980	−0.484990	−	0.874520i	$-0.661177\pi$
$827$	−27.8943	−0.969980	−0.484990	+	0.874520i	$0.661177\pi$
$828$	−12.6047	−0.438043
$829$	−49.2427	−1.71027	−0.855136	−	0.518404i	$-0.826526\pi$
$829$	−49.2427	−1.71027	−0.855136	+	0.518404i	$0.826526\pi$
$830$	12.2105	0.423832
$831$	−11.4466	−0.397076
$832$	−0.362562	−0.0125696
$833$	45.6277	1.58091
$834$	15.1888	0.525945
$835$	0.248856	0.00861201
$836$	−2.28021	−0.0788628
$837$	−44.4175	−1.53530
$838$	−9.05718	−0.312875
$839$	21.6872	0.748726	0.374363	−	0.927282i	$-0.377861\pi$
$839$	21.6872	0.748726	0.374363	+	0.927282i	$0.377861\pi$
$840$	−1.35488	−0.0467477
$841$	−27.9877	−0.965092
$842$	−0.134894	−0.00464877
$843$	22.9172	0.789309
$844$	27.9221	0.961118
$845$	24.5826	0.845668
$846$	−9.38819	−0.322773
$847$	0.691966	0.0237762
$848$	−6.54593	−0.224788
$849$	6.10075	0.209377
$850$	−9.45136	−0.324179
$851$	−13.0827	−0.448469
$852$	−3.87389	−0.132717
$853$	−50.0271	−1.71289	−0.856447	−	0.516235i	$-0.827333\pi$
$853$	−50.0271	−1.71289	−0.856447	+	0.516235i	$0.827333\pi$
$854$	−3.14168	−0.107506
$855$	8.49136	0.290398
$856$	15.0190	0.513338
$857$	5.59674	0.191181	0.0955905	−	0.995421i	$-0.469526\pi$
$857$	5.59674	0.191181	0.0955905	+	0.995421i	$0.469526\pi$
$858$	0.371620	0.0126869
$859$	−25.0179	−0.853601	−0.426801	−	0.904346i	$-0.640359\pi$
$859$	−25.0179	−0.853601	−0.426801	+	0.904346i	$0.640359\pi$
$860$	11.3952	0.388573
$861$	2.13263	0.0726797
$862$	11.6429	0.396558
$863$	21.3223	0.725818	0.362909	−	0.931825i	$-0.381784\pi$
$863$	21.3223	0.725818	0.362909	+	0.931825i	$0.381784\pi$
$864$	−5.07306	−0.172589
$865$	4.83006	0.164227
$866$	14.2522	0.484309
$867$	−32.7542	−1.11239
$868$	−6.05856	−0.205641
$869$	−15.8015	−0.536029
$870$	−1.97004	−0.0667905
$871$	−0.714025	−0.0241938
$872$	3.07263	0.104052
$873$	−7.55415	−0.255669
$874$	−14.7436	−0.498711
$875$	8.39484	0.283797
$876$	7.08391	0.239343
$877$	19.6472	0.663437	0.331719	−	0.943378i	$-0.392372\pi$
$877$	19.6472	0.663437	0.331719	+	0.943378i	$0.392372\pi$
$878$	−18.1677	−0.613129
$879$	−11.9134	−0.401828
$880$	1.91029	0.0643958
$881$	0.0397180	0.00133813	0.000669067	−	1.00000i	$-0.499787\pi$
$881$	0.0397180	0.00133813	0.000669067		1.00000i	$0.499787\pi$
$882$	−12.7125	−0.428051
$883$	−16.5142	−0.555747	−0.277873	−	0.960618i	$-0.589630\pi$
$883$	−16.5142	−0.555747	−0.277873	+	0.960618i	$0.589630\pi$
$884$	2.53679	0.0853215
$885$	6.41816	0.215744
$886$	15.2289	0.511626
$887$	−14.1661	−0.475650	−0.237825	−	0.971308i	$-0.576435\pi$
$887$	−14.1661	−0.475650	−0.237825	+	0.971308i	$0.576435\pi$
$888$	−2.07389	−0.0695951
$889$	0.172808	0.00579578
$890$	−13.0624	−0.437853
$891$	−0.648434	−0.0217234
$892$	21.4897	0.719528
$893$	−10.9813	−0.367475
$894$	−1.90928	−0.0638557
$895$	1.51687	0.0507034
$896$	−0.691966	−0.0231170
$897$	2.40286	0.0802290
$898$	−17.2722	−0.576381
$899$	−8.80935	−0.293808
$900$	2.63327	0.0877756
$901$	45.8009	1.52585
$902$	−3.00686	−0.100118
$903$	4.23082	0.140793
$904$	5.93046	0.197244
$905$	−19.9024	−0.661578
$906$	5.64012	0.187381
$907$	−9.10246	−0.302242	−0.151121	−	0.988515i	$-0.548288\pi$
$907$	−9.10246	−0.302242	−0.151121	+	0.988515i	$0.548288\pi$
$908$	16.7313	0.555247
$909$	26.5773	0.881515
$910$	−0.479254	−0.0158871
$911$	−18.1230	−0.600441	−0.300220	−	0.953870i	$-0.597060\pi$
$911$	−18.1230	−0.600441	−0.300220	+	0.953870i	$0.597060\pi$
$912$	−2.33718	−0.0773917
$913$	−6.39196	−0.211543
$914$	−14.4024	−0.476389
$915$	8.88979	0.293887
$916$	5.79858	0.191591
$917$	−9.81964	−0.324273
$918$	35.4954	1.17152
$919$	41.7718	1.37792	0.688962	−	0.724797i	$-0.258067\pi$
$919$	41.7718	1.37792	0.688962	+	0.724797i	$0.258067\pi$
$920$	12.3517	0.407224
$921$	−26.6615	−0.878525
$922$	−19.8128	−0.652500
$923$	−1.37029	−0.0451037
$924$	0.709254	0.0233327
$925$	2.73313	0.0898647
$926$	26.3608	0.866270
$927$	−28.8423	−0.947305
$928$	−1.00614	−0.0330282
$929$	−5.08891	−0.166962	−0.0834808	−	0.996509i	$-0.526604\pi$
$929$	−5.08891	−0.166962	−0.0834808	+	0.996509i	$0.526604\pi$
$930$	17.1435	0.562158
$931$	−14.8697	−0.487334
$932$	2.18494	0.0715699
$933$	−25.2293	−0.825970
$934$	−31.3089	−1.02446
$935$	−13.3660	−0.437115
$936$	−0.706782	−0.0231019
$937$	41.1809	1.34532	0.672660	−	0.739952i	$-0.265152\pi$
$937$	41.1809	1.34532	0.672660	+	0.739952i	$0.265152\pi$
$938$	−1.36275	−0.0444954
$939$	2.65131	0.0865223
$940$	9.19978	0.300064
$941$	9.64924	0.314556	0.157278	−	0.987554i	$-0.449728\pi$
$941$	9.64924	0.314556	0.157278	+	0.987554i	$0.449728\pi$
$942$	2.85686	0.0930815
$943$	−19.4421	−0.633121
$944$	3.27790	0.106686
$945$	−6.70585	−0.218141
$946$	−5.96518	−0.193945
$947$	−17.5460	−0.570170	−0.285085	−	0.958502i	$-0.592022\pi$
$947$	−17.5460	−0.570170	−0.285085	+	0.958502i	$0.592022\pi$
$948$	−16.1963	−0.526030
$949$	2.50576	0.0813403
$950$	3.08012	0.0999321
$951$	−4.19417	−0.136005
$952$	4.84158	0.156917
$953$	−25.7850	−0.835258	−0.417629	−	0.908618i	$-0.637139\pi$
$953$	−25.7850	−0.835258	−0.417629	+	0.908618i	$0.637139\pi$
$954$	−12.7607	−0.413143
$955$	35.5295	1.14971
$956$	−20.8204	−0.673379
$957$	1.03128	0.0333365
$958$	16.8451	0.544240
$959$	−3.95680	−0.127772
$960$	1.95801	0.0631945
$961$	45.6601	1.47291
$962$	−0.733585	−0.0236517
$963$	29.2782	0.943475
$964$	21.4563	0.691059
$965$	0.321237	0.0103410
$966$	4.58596	0.147551
$967$	−9.73680	−0.313114	−0.156557	−	0.987669i	$-0.550040\pi$
$967$	−9.73680	−0.313114	−0.156557	+	0.987669i	$0.550040\pi$
$968$	−1.00000	−0.0321412
$969$	16.3529	0.525330
$970$	7.40254	0.237681
$971$	51.7020	1.65919	0.829597	−	0.558362i	$-0.188570\pi$
$971$	51.7020	1.65919	0.829597	+	0.558362i	$0.188570\pi$
$972$	−15.8838	−0.509474
$973$	10.2540	0.328727
$974$	15.5316	0.497666
$975$	−0.501985	−0.0160764
$976$	4.54021	0.145329
$977$	4.91080	0.157110	0.0785552	−	0.996910i	$-0.474969\pi$
$977$	4.91080	0.157110	0.0785552	+	0.996910i	$0.474969\pi$
$978$	−3.38905	−0.108370
$979$	6.83793	0.218541
$980$	12.4573	0.397935
$981$	5.98982	0.191240
$982$	19.3385	0.617117
$983$	−26.0148	−0.829743	−0.414872	−	0.909880i	$-0.636174\pi$
$983$	−26.0148	−0.829743	−0.414872	+	0.909880i	$0.636174\pi$
$984$	−3.08198	−0.0982499
$985$	−1.91029	−0.0608668
$986$	7.03982	0.224194
$987$	3.41570	0.108723
$988$	−0.826718	−0.0263014
$989$	−38.5702	−1.22646
$990$	3.72393	0.118354
$991$	−21.4810	−0.682365	−0.341183	−	0.939997i	$-0.610827\pi$
$991$	−21.4810	−0.682365	−0.341183	+	0.939997i	$0.610827\pi$
$992$	8.75557	0.277990
$993$	5.62502	0.178505
$994$	−2.61527	−0.0829512
$995$	−51.5132	−1.63308
$996$	−6.55165	−0.207597
$997$	26.9749	0.854302	0.427151	−	0.904180i	$-0.359517\pi$
$997$	26.9749	0.854302	0.427151	+	0.904180i	$0.359517\pi$
$998$	−41.4759	−1.31290
$999$	−10.2645	−0.324755

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4334.2.a.e.1.10	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4334.2.a.e.1.10	✓	24	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:	\( N \)	\(=\)	\( 4334 = 2 \cdot 11 \cdot 197 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4334.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.02498 q^{3} +1.00000 q^{4} -1.91029 q^{5} +1.02498 q^{6} +0.691966 q^{7} -1.00000 q^{8} -1.94941 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.02498 q^{3} +1.00000 q^{4} -1.91029 q^{5} +1.02498 q^{6} +0.691966 q^{7} -1.00000 q^{8} -1.94941 q^{9} +1.91029 q^{10} -1.00000 q^{11} -1.02498 q^{12} -0.362562 q^{13} -0.691966 q^{14} +1.95801 q^{15} +1.00000 q^{16} -6.99685 q^{17} +1.94941 q^{18} +2.28021 q^{19} -1.91029 q^{20} -0.709254 q^{21} +1.00000 q^{22} +6.46590 q^{23} +1.02498 q^{24} -1.35080 q^{25} +0.362562 q^{26} +5.07306 q^{27} +0.691966 q^{28} +1.00614 q^{29} -1.95801 q^{30} -8.75557 q^{31} -1.00000 q^{32} +1.02498 q^{33} +6.99685 q^{34} -1.32185 q^{35} -1.94941 q^{36} -2.02334 q^{37} -2.28021 q^{38} +0.371620 q^{39} +1.91029 q^{40} -3.00686 q^{41} +0.709254 q^{42} -5.96518 q^{43} -1.00000 q^{44} +3.72393 q^{45} -6.46590 q^{46} -4.81591 q^{47} -1.02498 q^{48} -6.52118 q^{49} +1.35080 q^{50} +7.17165 q^{51} -0.362562 q^{52} -6.54593 q^{53} -5.07306 q^{54} +1.91029 q^{55} -0.691966 q^{56} -2.33718 q^{57} -1.00614 q^{58} +3.27790 q^{59} +1.95801 q^{60} +4.54021 q^{61} +8.75557 q^{62} -1.34893 q^{63} +1.00000 q^{64} +0.692598 q^{65} -1.02498 q^{66} +1.96939 q^{67} -6.99685 q^{68} -6.62743 q^{69} +1.32185 q^{70} +3.77947 q^{71} +1.94941 q^{72} -6.91125 q^{73} +2.02334 q^{74} +1.38455 q^{75} +2.28021 q^{76} -0.691966 q^{77} -0.371620 q^{78} +15.8015 q^{79} -1.91029 q^{80} +0.648434 q^{81} +3.00686 q^{82} +6.39196 q^{83} -0.709254 q^{84} +13.3660 q^{85} +5.96518 q^{86} -1.03128 q^{87} +1.00000 q^{88} -6.83793 q^{89} -3.72393 q^{90} -0.250881 q^{91} +6.46590 q^{92} +8.97431 q^{93} +4.81591 q^{94} -4.35586 q^{95} +1.02498 q^{96} +3.87509 q^{97} +6.52118 q^{98} +1.94941 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{2} - 4 q^{3} + 24 q^{4} - 4 q^{5} + 4 q^{6} + 7 q^{7} - 24 q^{8} + 28 q^{9}+O(q^{10}) \) 24 * q - 24 * q^2 - 4 * q^3 + 24 * q^4 - 4 * q^5 + 4 * q^6 + 7 * q^7 - 24 * q^8 + 28 * q^9	\( 24 q - 24 q^{2} - 4 q^{3} + 24 q^{4} - 4 q^{5} + 4 q^{6} + 7 q^{7} - 24 q^{8} + 28 q^{9} + 4 q^{10} - 24 q^{11} - 4 q^{12} + 21 q^{13} - 7 q^{14} - 2 q^{15} + 24 q^{16} + 15 q^{17} - 28 q^{18} + 21 q^{19} - 4 q^{20} + 15 q^{21} + 24 q^{22} - 17 q^{23} + 4 q^{24} + 46 q^{25} - 21 q^{26} - 19 q^{27} + 7 q^{28} + 9 q^{29} + 2 q^{30} + 27 q^{31} - 24 q^{32} + 4 q^{33} - 15 q^{34} - 2 q^{35} + 28 q^{36} + 5 q^{37} - 21 q^{38} + 17 q^{39} + 4 q^{40} + 16 q^{41} - 15 q^{42} + 3 q^{43} - 24 q^{44} - 21 q^{45} + 17 q^{46} - 24 q^{47} - 4 q^{48} + 55 q^{49} - 46 q^{50} - 12 q^{51} + 21 q^{52} - 26 q^{53} + 19 q^{54} + 4 q^{55} - 7 q^{56} + 30 q^{57} - 9 q^{58} - 17 q^{59} - 2 q^{60} + 44 q^{61} - 27 q^{62} + 4 q^{63} + 24 q^{64} + 35 q^{65} - 4 q^{66} + 10 q^{67} + 15 q^{68} + 3 q^{69} + 2 q^{70} - 6 q^{71} - 28 q^{72} + 77 q^{73} - 5 q^{74} - 32 q^{75} + 21 q^{76} - 7 q^{77} - 17 q^{78} + 43 q^{79} - 4 q^{80} + 48 q^{81} - 16 q^{82} - 20 q^{83} + 15 q^{84} + 35 q^{85} - 3 q^{86} + 36 q^{87} + 24 q^{88} + 3 q^{89} + 21 q^{90} + 63 q^{91} - 17 q^{92} + 36 q^{93} + 24 q^{94} - 3 q^{95} + 4 q^{96} + 16 q^{97} - 55 q^{98} - 28 q^{99}+O(q^{100}) \) 24 * q - 24 * q^2 - 4 * q^3 + 24 * q^4 - 4 * q^5 + 4 * q^6 + 7 * q^7 - 24 * q^8 + 28 * q^9 + 4 * q^10 - 24 * q^11 - 4 * q^12 + 21 * q^13 - 7 * q^14 - 2 * q^15 + 24 * q^16 + 15 * q^17 - 28 * q^18 + 21 * q^19 - 4 * q^20 + 15 * q^21 + 24 * q^22 - 17 * q^23 + 4 * q^24 + 46 * q^25 - 21 * q^26 - 19 * q^27 + 7 * q^28 + 9 * q^29 + 2 * q^30 + 27 * q^31 - 24 * q^32 + 4 * q^33 - 15 * q^34 - 2 * q^35 + 28 * q^36 + 5 * q^37 - 21 * q^38 + 17 * q^39 + 4 * q^40 + 16 * q^41 - 15 * q^42 + 3 * q^43 - 24 * q^44 - 21 * q^45 + 17 * q^46 - 24 * q^47 - 4 * q^48 + 55 * q^49 - 46 * q^50 - 12 * q^51 + 21 * q^52 - 26 * q^53 + 19 * q^54 + 4 * q^55 - 7 * q^56 + 30 * q^57 - 9 * q^58 - 17 * q^59 - 2 * q^60 + 44 * q^61 - 27 * q^62 + 4 * q^63 + 24 * q^64 + 35 * q^65 - 4 * q^66 + 10 * q^67 + 15 * q^68 + 3 * q^69 + 2 * q^70 - 6 * q^71 - 28 * q^72 + 77 * q^73 - 5 * q^74 - 32 * q^75 + 21 * q^76 - 7 * q^77 - 17 * q^78 + 43 * q^79 - 4 * q^80 + 48 * q^81 - 16 * q^82 - 20 * q^83 + 15 * q^84 + 35 * q^85 - 3 * q^86 + 36 * q^87 + 24 * q^88 + 3 * q^89 + 21 * q^90 + 63 * q^91 - 17 * q^92 + 36 * q^93 + 24 * q^94 - 3 * q^95 + 4 * q^96 + 16 * q^97 - 55 * q^98 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more