LMFDB - Embedded newform 4031.2.a.d.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4031,2,Mod(1,4031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4031, 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("4031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4031 = 29 \cdot 139 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4031.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$32.1876970548$
Analytic rank:	$0$
Dimension:	$98$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	4031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.97068 q^{2} +0.422540 q^{3} +1.88358 q^{4} +2.63171 q^{5} -0.832693 q^{6} -1.79791 q^{7} +0.229417 q^{8} -2.82146 q^{9} +O(q^{10})$	\(q-1.97068 q^{2} +0.422540 q^{3} +1.88358 q^{4} +2.63171 q^{5} -0.832693 q^{6} -1.79791 q^{7} +0.229417 q^{8} -2.82146 q^{9} -5.18627 q^{10} +1.31235 q^{11} +0.795891 q^{12} -6.34236 q^{13} +3.54310 q^{14} +1.11200 q^{15} -4.21928 q^{16} -2.80565 q^{17} +5.56020 q^{18} +5.51508 q^{19} +4.95705 q^{20} -0.759688 q^{21} -2.58622 q^{22} +7.51430 q^{23} +0.0969381 q^{24} +1.92591 q^{25} +12.4988 q^{26} -2.45980 q^{27} -3.38651 q^{28} +1.00000 q^{29} -2.19141 q^{30} -9.31843 q^{31} +7.85602 q^{32} +0.554519 q^{33} +5.52905 q^{34} -4.73157 q^{35} -5.31446 q^{36} -7.08458 q^{37} -10.8685 q^{38} -2.67990 q^{39} +0.603760 q^{40} -4.71396 q^{41} +1.49710 q^{42} +1.91932 q^{43} +2.47192 q^{44} -7.42527 q^{45} -14.8083 q^{46} +10.6458 q^{47} -1.78282 q^{48} -3.76753 q^{49} -3.79535 q^{50} -1.18550 q^{51} -11.9464 q^{52} -1.19051 q^{53} +4.84749 q^{54} +3.45372 q^{55} -0.412471 q^{56} +2.33035 q^{57} -1.97068 q^{58} +3.68636 q^{59} +2.09456 q^{60} +5.35080 q^{61} +18.3636 q^{62} +5.07272 q^{63} -7.04315 q^{64} -16.6913 q^{65} -1.09278 q^{66} +14.4069 q^{67} -5.28469 q^{68} +3.17509 q^{69} +9.32442 q^{70} +13.9490 q^{71} -0.647291 q^{72} -7.01255 q^{73} +13.9615 q^{74} +0.813774 q^{75} +10.3881 q^{76} -2.35948 q^{77} +5.28124 q^{78} +2.61837 q^{79} -11.1039 q^{80} +7.42501 q^{81} +9.28972 q^{82} +14.9430 q^{83} -1.43094 q^{84} -7.38368 q^{85} -3.78238 q^{86} +0.422540 q^{87} +0.301075 q^{88} +11.3483 q^{89} +14.6328 q^{90} +11.4030 q^{91} +14.1538 q^{92} -3.93741 q^{93} -20.9794 q^{94} +14.5141 q^{95} +3.31949 q^{96} +2.60665 q^{97} +7.42460 q^{98} -3.70273 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 98 q + 6 q^{2} + 6 q^{3} + 116 q^{4} + q^{5} + 7 q^{6} + 12 q^{7} + 15 q^{8} + 136 q^{9}+O(q^{10}) $ 98 * q + 6 * q^2 + 6 * q^3 + 116 * q^4 + q^5 + 7 * q^6 + 12 * q^7 + 15 * q^8 + 136 * q^9	\( 98 q + 6 q^{2} + 6 q^{3} + 116 q^{4} + q^{5} + 7 q^{6} + 12 q^{7} + 15 q^{8} + 136 q^{9} + 16 q^{10} + 25 q^{11} + 8 q^{12} + 18 q^{13} + 34 q^{14} + 14 q^{15} + 136 q^{16} + 35 q^{17} + 20 q^{18} + 48 q^{19} - 20 q^{20} + 62 q^{21} + 32 q^{22} + q^{23} + 8 q^{24} + 173 q^{25} + 15 q^{26} + 18 q^{27} + 37 q^{28} + 98 q^{29} - 6 q^{30} + 20 q^{31} + 16 q^{32} + 24 q^{33} - 6 q^{34} + 11 q^{35} + 155 q^{36} + 62 q^{37} - 5 q^{38} + 45 q^{39} + 38 q^{40} + 50 q^{41} + 7 q^{42} + 72 q^{43} + 92 q^{44} - 13 q^{45} + 32 q^{46} - 16 q^{47} - 13 q^{48} + 194 q^{49} + 50 q^{50} + 2 q^{51} + 83 q^{52} - 11 q^{53} + 58 q^{54} + 32 q^{55} + 106 q^{56} + 84 q^{57} + 6 q^{58} + 20 q^{59} + 62 q^{60} + 142 q^{61} - 27 q^{62} + 26 q^{63} + 213 q^{64} + 13 q^{65} - 35 q^{66} + 5 q^{67} + 69 q^{68} + 68 q^{69} - 2 q^{70} - 10 q^{71} - 9 q^{72} + 74 q^{73} + 21 q^{74} + 19 q^{75} + 116 q^{76} + 41 q^{77} - 162 q^{78} + 104 q^{79} - 86 q^{80} + 230 q^{81} + 53 q^{82} - 19 q^{83} + 76 q^{84} + 125 q^{85} + 9 q^{86} + 6 q^{87} + 68 q^{88} + 120 q^{89} + 122 q^{90} + 30 q^{91} + 3 q^{92} - 56 q^{93} + 22 q^{94} + 32 q^{95} - 55 q^{96} + 98 q^{97} - 15 q^{98} + 69 q^{99}+O(q^{100}) \) 98 * q + 6 * q^2 + 6 * q^3 + 116 * q^4 + q^5 + 7 * q^6 + 12 * q^7 + 15 * q^8 + 136 * q^9 + 16 * q^10 + 25 * q^11 + 8 * q^12 + 18 * q^13 + 34 * q^14 + 14 * q^15 + 136 * q^16 + 35 * q^17 + 20 * q^18 + 48 * q^19 - 20 * q^20 + 62 * q^21 + 32 * q^22 + q^23 + 8 * q^24 + 173 * q^25 + 15 * q^26 + 18 * q^27 + 37 * q^28 + 98 * q^29 - 6 * q^30 + 20 * q^31 + 16 * q^32 + 24 * q^33 - 6 * q^34 + 11 * q^35 + 155 * q^36 + 62 * q^37 - 5 * q^38 + 45 * q^39 + 38 * q^40 + 50 * q^41 + 7 * q^42 + 72 * q^43 + 92 * q^44 - 13 * q^45 + 32 * q^46 - 16 * q^47 - 13 * q^48 + 194 * q^49 + 50 * q^50 + 2 * q^51 + 83 * q^52 - 11 * q^53 + 58 * q^54 + 32 * q^55 + 106 * q^56 + 84 * q^57 + 6 * q^58 + 20 * q^59 + 62 * q^60 + 142 * q^61 - 27 * q^62 + 26 * q^63 + 213 * q^64 + 13 * q^65 - 35 * q^66 + 5 * q^67 + 69 * q^68 + 68 * q^69 - 2 * q^70 - 10 * q^71 - 9 * q^72 + 74 * q^73 + 21 * q^74 + 19 * q^75 + 116 * q^76 + 41 * q^77 - 162 * q^78 + 104 * q^79 - 86 * q^80 + 230 * q^81 + 53 * q^82 - 19 * q^83 + 76 * q^84 + 125 * q^85 + 9 * q^86 + 6 * q^87 + 68 * q^88 + 120 * q^89 + 122 * q^90 + 30 * q^91 + 3 * q^92 - 56 * q^93 + 22 * q^94 + 32 * q^95 - 55 * q^96 + 98 * q^97 - 15 * q^98 + 69 * 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 $

$2$	−1.97068	−1.39348	−0.696741	−	0.717323i	$-0.745367\pi$
$2$	−1.97068	−1.39348	−0.696741	+	0.717323i	$0.745367\pi$
$3$	0.422540	0.243954	0.121977	−	0.992533i	$-0.461077\pi$
$3$	0.422540	0.243954	0.121977	+	0.992533i	$0.461077\pi$
$4$	1.88358	0.941792
$5$	2.63171	1.17694	0.588469	−	0.808520i	$-0.299731\pi$
$5$	2.63171	1.17694	0.588469	+	0.808520i	$0.299731\pi$
$6$	−0.832693	−0.339945
$7$	−1.79791	−0.679545	−0.339772	−	0.940508i	$-0.610350\pi$
$7$	−1.79791	−0.679545	−0.339772	+	0.940508i	$0.610350\pi$
$8$	0.229417	0.0811112
$9$	−2.82146	−0.940487
$10$	−5.18627	−1.64004
$11$	1.31235	0.395687	0.197844	−	0.980234i	$-0.436606\pi$
$11$	1.31235	0.395687	0.197844	+	0.980234i	$0.436606\pi$
$12$	0.795891	0.229754
$13$	−6.34236	−1.75905	−0.879527	−	0.475849i	$-0.842141\pi$
$13$	−6.34236	−1.75905	−0.879527	+	0.475849i	$0.842141\pi$
$14$	3.54310	0.946934
$15$	1.11200	0.287118
$16$	−4.21928	−1.05482
$17$	−2.80565	−0.680471	−0.340236	−	0.940340i	$-0.610507\pi$
$17$	−2.80565	−0.680471	−0.340236	+	0.940340i	$0.610507\pi$
$18$	5.56020	1.31055
$19$	5.51508	1.26525	0.632623	−	0.774460i	$-0.281978\pi$
$19$	5.51508	1.26525	0.632623	+	0.774460i	$0.281978\pi$
$20$	4.95705	1.10843
$21$	−0.759688	−0.165778
$22$	−2.58622	−0.551383
$23$	7.51430	1.56684	0.783420	−	0.621493i	$-0.213474\pi$
$23$	7.51430	1.56684	0.783420	+	0.621493i	$0.213474\pi$
$24$	0.0969381	0.0197874
$25$	1.92591	0.385182
$26$	12.4988	2.45121
$27$	−2.45980	−0.473389
$28$	−3.38651	−0.639990
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	−2.19141	−0.400094
$31$	−9.31843	−1.67364	−0.836819	−	0.547479i	$-0.815588\pi$
$31$	−9.31843	−1.67364	−0.836819	+	0.547479i	$0.815588\pi$
$32$	7.85602	1.38876
$33$	0.554519	0.0965295
$34$	5.52905	0.948224
$35$	−4.73157	−0.799782
$36$	−5.31446	−0.885743
$37$	−7.08458	−1.16470	−0.582349	−	0.812939i	$-0.697866\pi$
$37$	−7.08458	−1.16470	−0.582349	+	0.812939i	$0.697866\pi$
$38$	−10.8685	−1.76310
$39$	−2.67990	−0.429128
$40$	0.603760	0.0954628
$41$	−4.71396	−0.736197	−0.368099	−	0.929787i	$-0.619991\pi$
$41$	−4.71396	−0.736197	−0.368099	+	0.929787i	$0.619991\pi$
$42$	1.49710	0.231008
$43$	1.91932	0.292694	0.146347	−	0.989233i	$-0.453248\pi$
$43$	1.91932	0.292694	0.146347	+	0.989233i	$0.453248\pi$
$44$	2.47192	0.372655
$45$	−7.42527	−1.10689
$46$	−14.8083	−2.18336
$47$	10.6458	1.55285	0.776423	−	0.630212i	$-0.217032\pi$
$47$	10.6458	1.55285	0.776423	+	0.630212i	$0.217032\pi$
$48$	−1.78282	−0.257327
$49$	−3.76753	−0.538219
$50$	−3.79535	−0.536744
$51$	−1.18550	−0.166004
$52$	−11.9464	−1.65666
$53$	−1.19051	−0.163530	−0.0817648	−	0.996652i	$-0.526056\pi$
$53$	−1.19051	−0.163530	−0.0817648	+	0.996652i	$0.526056\pi$
$54$	4.84749	0.659659
$55$	3.45372	0.465699
$56$	−0.412471	−0.0551187
$57$	2.33035	0.308662
$58$	−1.97068	−0.258763
$59$	3.68636	0.479923	0.239961	−	0.970782i	$-0.422865\pi$
$59$	3.68636	0.479923	0.239961	+	0.970782i	$0.422865\pi$
$60$	2.09456	0.270406
$61$	5.35080	0.685100	0.342550	−	0.939500i	$-0.388709\pi$
$61$	5.35080	0.685100	0.342550	+	0.939500i	$0.388709\pi$
$62$	18.3636	2.33219
$63$	5.07272	0.639103
$64$	−7.04315	−0.880394
$65$	−16.6913	−2.07030
$66$	−1.09278	−0.134512
$67$	14.4069	1.76008	0.880040	−	0.474899i	$-0.157515\pi$
$67$	14.4069	1.76008	0.880040	+	0.474899i	$0.157515\pi$
$68$	−5.28469	−0.640863
$69$	3.17509	0.382236
$70$	9.32442	1.11448
$71$	13.9490	1.65544	0.827718	−	0.561144i	$-0.189639\pi$
$71$	13.9490	1.65544	0.827718	+	0.561144i	$0.189639\pi$
$72$	−0.647291	−0.0762840
$73$	−7.01255	−0.820757	−0.410379	−	0.911915i	$-0.634603\pi$
$73$	−7.01255	−0.820757	−0.410379	+	0.911915i	$0.634603\pi$
$74$	13.9615	1.62299
$75$	0.813774	0.0939666
$76$	10.3881	1.19160
$77$	−2.35948	−0.268887
$78$	5.28124	0.597982
$79$	2.61837	0.294590	0.147295	−	0.989093i	$-0.452943\pi$
$79$	2.61837	0.294590	0.147295	+	0.989093i	$0.452943\pi$
$80$	−11.1039	−1.24146
$81$	7.42501	0.825001
$82$	9.28972	1.02588
$83$	14.9430	1.64021	0.820103	−	0.572216i	$-0.193916\pi$
$83$	14.9430	1.64021	0.820103	+	0.572216i	$0.193916\pi$
$84$	−1.43094	−0.156128
$85$	−7.38368	−0.800872
$86$	−3.78238	−0.407864
$87$	0.422540	0.0453011
$88$	0.301075	0.0320947
$89$	11.3483	1.20292	0.601460	−	0.798903i	$-0.294586\pi$
$89$	11.3483	1.20292	0.601460	+	0.798903i	$0.294586\pi$
$90$	14.6328	1.54244
$91$	11.4030	1.19536
$92$	14.1538	1.47564
$93$	−3.93741	−0.408291
$94$	−20.9794	−2.16386
$95$	14.5141	1.48912
$96$	3.31949	0.338794
$97$	2.60665	0.264665	0.132332	−	0.991205i	$-0.457753\pi$
$97$	2.60665	0.264665	0.132332	+	0.991205i	$0.457753\pi$
$98$	7.42460	0.749998
$99$	−3.70273	−0.372139
$100$	3.62761	0.362761
$101$	4.85432	0.483023	0.241511	−	0.970398i	$-0.422357\pi$
$101$	4.85432	0.483023	0.241511	+	0.970398i	$0.422357\pi$
$102$	2.33625	0.231323
$103$	−19.8699	−1.95784	−0.978922	−	0.204235i	$-0.934529\pi$
$103$	−19.8699	−1.95784	−0.978922	+	0.204235i	$0.934529\pi$
$104$	−1.45505	−0.142679
$105$	−1.99928	−0.195110
$106$	2.34612	0.227876
$107$	11.1460	1.07752	0.538761	−	0.842459i	$-0.318893\pi$
$107$	11.1460	1.07752	0.538761	+	0.842459i	$0.318893\pi$
$108$	−4.63325	−0.445834
$109$	2.20680	0.211373	0.105687	−	0.994399i	$-0.466296\pi$
$109$	2.20680	0.211373	0.105687	+	0.994399i	$0.466296\pi$
$110$	−6.80618	−0.648944
$111$	−2.99352	−0.284133
$112$	7.58587	0.716797
$113$	11.2498	1.05829	0.529145	−	0.848532i	$-0.322513\pi$
$113$	11.2498	1.05829	0.529145	+	0.848532i	$0.322513\pi$
$114$	−4.59237	−0.430115
$115$	19.7755	1.84407
$116$	1.88358	0.174886
$117$	17.8947	1.65437
$118$	−7.26464	−0.668764
$119$	5.04431	0.462411
$120$	0.255113	0.0232885
$121$	−9.27775	−0.843432
$122$	−10.5447	−0.954675
$123$	−1.99184	−0.179598
$124$	−17.5520	−1.57622
$125$	−8.09012	−0.723603
$126$	−9.99672	−0.890578
$127$	−6.85476	−0.608261	−0.304131	−	0.952630i	$-0.598366\pi$
$127$	−6.85476	−0.608261	−0.304131	+	0.952630i	$0.598366\pi$
$128$	−1.83223	−0.161948
$129$	0.810992	0.0714039
$130$	32.8932	2.88492
$131$	15.8624	1.38590	0.692951	−	0.720984i	$-0.256310\pi$
$131$	15.8624	1.38590	0.692951	+	0.720984i	$0.256310\pi$
$132$	1.04448	0.0909107
$133$	−9.91560	−0.859792
$134$	−28.3914	−2.45264
$135$	−6.47349	−0.557149
$136$	−0.643665	−0.0551939
$137$	−21.0437	−1.79788	−0.898940	−	0.438071i	$-0.855662\pi$
$137$	−21.0437	−1.79788	−0.898940	+	0.438071i	$0.855662\pi$
$138$	−6.25710	−0.532640
$139$	−1.00000	−0.0848189
$139$	−1.00000	−0.0848189
$140$	−8.91232	−0.753228
$141$	4.49827	0.378823
$142$	−27.4889	−2.30682
$143$	−8.32337	−0.696035
$144$	11.9045	0.992043
$145$	2.63171	0.218552
$146$	13.8195	1.14371
$147$	−1.59193	−0.131301
$148$	−13.3444	−1.09690
$149$	15.9766	1.30885	0.654426	−	0.756126i	$-0.272910\pi$
$149$	15.9766	1.30885	0.654426	+	0.756126i	$0.272910\pi$
$150$	−1.60369	−0.130941
$151$	6.11827	0.497897	0.248949	−	0.968517i	$-0.419915\pi$
$151$	6.11827	0.497897	0.248949	+	0.968517i	$0.419915\pi$
$152$	1.26525	0.102626
$153$	7.91604	0.639974
$154$	4.64978	0.374690
$155$	−24.5234	−1.96977
$156$	−5.04783	−0.404150
$157$	9.05147	0.722386	0.361193	−	0.932491i	$-0.382370\pi$
$157$	9.05147	0.722386	0.361193	+	0.932491i	$0.382370\pi$
$158$	−5.15997	−0.410505
$159$	−0.503040	−0.0398937
$160$	20.6748	1.63448
$161$	−13.5100	−1.06474
$162$	−14.6323	−1.14962
$163$	−19.4039	−1.51983	−0.759914	−	0.650024i	$-0.774759\pi$
$163$	−19.4039	−1.51983	−0.759914	+	0.650024i	$0.774759\pi$
$164$	−8.87915	−0.693345
$165$	1.45934	0.113609
$166$	−29.4479	−2.28560
$167$	18.0728	1.39852	0.699259	−	0.714868i	$-0.253513\pi$
$167$	18.0728	1.39852	0.699259	+	0.714868i	$0.253513\pi$
$168$	−0.174286	−0.0134464
$169$	27.2255	2.09427
$170$	14.5509	1.11600
$171$	−15.5606	−1.18995
$172$	3.61521	0.275657
$173$	−12.9286	−0.982947	−0.491473	−	0.870893i	$-0.663541\pi$
$173$	−12.9286	−0.982947	−0.491473	+	0.870893i	$0.663541\pi$
$174$	−0.832693	−0.0631263
$175$	−3.46260	−0.261748
$176$	−5.53715	−0.417379
$177$	1.55764	0.117079
$178$	−22.3639	−1.67625
$179$	−17.4242	−1.30235	−0.651175	−	0.758928i	$-0.725723\pi$
$179$	−17.4242	−1.30235	−0.651175	+	0.758928i	$0.725723\pi$
$180$	−13.9861	−1.04246
$181$	20.2553	1.50556	0.752782	−	0.658270i	$-0.228711\pi$
$181$	20.2553	1.50556	0.752782	+	0.658270i	$0.228711\pi$
$182$	−22.4716	−1.66571
$183$	2.26093	0.167133
$184$	1.72391	0.127088
$185$	−18.6446	−1.37078
$186$	7.75938	0.568946
$187$	−3.68199	−0.269254
$188$	20.0522	1.46246
$189$	4.42250	0.321689
$190$	−28.6027	−2.07506
$191$	7.40594	0.535875	0.267937	−	0.963436i	$-0.413658\pi$
$191$	7.40594	0.535875	0.267937	+	0.963436i	$0.413658\pi$
$192$	−2.97602	−0.214775
$193$	20.4258	1.47028	0.735141	−	0.677915i	$-0.237116\pi$
$193$	20.4258	1.47028	0.735141	+	0.677915i	$0.237116\pi$
$194$	−5.13687	−0.368806
$195$	−7.05273	−0.505057
$196$	−7.09646	−0.506890
$197$	27.8758	1.98607	0.993034	−	0.117827i	$-0.0375929\pi$
$197$	27.8758	1.98607	0.993034	+	0.117827i	$0.0375929\pi$
$198$	7.29691	0.518568
$199$	−12.4339	−0.881413	−0.440707	−	0.897651i	$-0.645272\pi$
$199$	−12.4339	−0.881413	−0.440707	+	0.897651i	$0.645272\pi$
$200$	0.441837	0.0312426
$201$	6.08749	0.429378
$202$	−9.56631	−0.673083
$203$	−1.79791	−0.126188
$204$	−2.23299	−0.156341
$205$	−12.4058	−0.866458
$206$	39.1573	2.72822
$207$	−21.2013	−1.47359
$208$	26.7602	1.85548
$209$	7.23770	0.500642
$210$	3.93995	0.271882
$211$	2.78561	0.191769	0.0958846	−	0.995392i	$-0.469432\pi$
$211$	2.78561	0.191769	0.0958846	+	0.995392i	$0.469432\pi$
$212$	−2.24243	−0.154011
$213$	5.89400	0.403850
$214$	−21.9652	−1.50151
$215$	5.05111	0.344483
$216$	−0.564321	−0.0383972
$217$	16.7537	1.13731
$218$	−4.34890	−0.294545
$219$	−2.96309	−0.200227
$220$	6.50537	0.438592
$221$	17.7945	1.19699
$222$	5.89928	0.395934
$223$	−4.37378	−0.292890	−0.146445	−	0.989219i	$-0.546783\pi$
$223$	−4.37378	−0.292890	−0.146445	+	0.989219i	$0.546783\pi$
$224$	−14.1244	−0.943725
$225$	−5.43387	−0.362258
$226$	−22.1697	−1.47471
$227$	−2.15202	−0.142835	−0.0714174	−	0.997447i	$-0.522752\pi$
$227$	−2.15202	−0.142835	−0.0714174	+	0.997447i	$0.522752\pi$
$228$	4.38940	0.290695
$229$	−10.3877	−0.686437	−0.343219	−	0.939256i	$-0.611517\pi$
$229$	−10.3877	−0.686437	−0.343219	+	0.939256i	$0.611517\pi$
$230$	−38.9711	−2.56968
$231$	−0.996974	−0.0655961
$232$	0.229417	0.0150620
$233$	−15.9933	−1.04776	−0.523879	−	0.851793i	$-0.675516\pi$
$233$	−15.9933	−1.04776	−0.523879	+	0.851793i	$0.675516\pi$
$234$	−35.2648	−2.30533
$235$	28.0166	1.82760
$236$	6.94357	0.451988
$237$	1.10637	0.0718662
$238$	−9.94072	−0.644361
$239$	−8.87818	−0.574282	−0.287141	−	0.957888i	$-0.592705\pi$
$239$	−8.87818	−0.574282	−0.287141	+	0.957888i	$0.592705\pi$
$240$	−4.69186	−0.302858
$241$	22.3652	1.44067	0.720335	−	0.693626i	$-0.243988\pi$
$241$	22.3652	1.44067	0.720335	+	0.693626i	$0.243988\pi$
$242$	18.2835	1.17531
$243$	10.5168	0.674651
$244$	10.0787	0.645222
$245$	−9.91506	−0.633450
$246$	3.92528	0.250267
$247$	−34.9786	−2.22564
$248$	−2.13781	−0.135751
$249$	6.31402	0.400134
$250$	15.9431	1.00833
$251$	21.5739	1.36173	0.680866	−	0.732408i	$-0.261603\pi$
$251$	21.5739	1.36173	0.680866	+	0.732408i	$0.261603\pi$
$252$	9.55490	0.601902
$253$	9.86136	0.619978
$254$	13.5085	0.847601
$255$	−3.11990	−0.195376
$256$	17.6970	1.10607
$257$	−14.5193	−0.905688	−0.452844	−	0.891590i	$-0.649590\pi$
$257$	−14.5193	−0.905688	−0.452844	+	0.891590i	$0.649590\pi$
$258$	−1.59821	−0.0995000
$259$	12.7374	0.791465
$260$	−31.4394	−1.94979
$261$	−2.82146	−0.174644
$262$	−31.2597	−1.93123
$263$	2.45099	0.151134	0.0755672	−	0.997141i	$-0.475923\pi$
$263$	2.45099	0.151134	0.0755672	+	0.997141i	$0.475923\pi$
$264$	0.127216	0.00782962
$265$	−3.13309	−0.192464
$266$	19.5405	1.19810
$267$	4.79512	0.293457
$268$	27.1366	1.65763
$269$	−10.4717	−0.638471	−0.319235	−	0.947675i	$-0.603426\pi$
$269$	−10.4717	−0.638471	−0.319235	+	0.947675i	$0.603426\pi$
$270$	12.7572	0.776378
$271$	−11.2913	−0.685895	−0.342948	−	0.939354i	$-0.611425\pi$
$271$	−11.2913	−0.685895	−0.342948	+	0.939354i	$0.611425\pi$
$272$	11.8378	0.717774
$273$	4.81822	0.291612
$274$	41.4703	2.50531
$275$	2.52746	0.152412
$276$	5.98056	0.359987
$277$	−1.80905	−0.108695	−0.0543476	−	0.998522i	$-0.517308\pi$
$277$	−1.80905	−0.108695	−0.0543476	+	0.998522i	$0.517308\pi$
$278$	1.97068	0.118194
$279$	26.2916	1.57403
$280$	−1.08550	−0.0648713
$281$	6.95566	0.414940	0.207470	−	0.978241i	$-0.433477\pi$
$281$	6.95566	0.414940	0.207470	+	0.978241i	$0.433477\pi$
$282$	−8.86466	−0.527883
$283$	6.14017	0.364995	0.182497	−	0.983206i	$-0.441582\pi$
$283$	6.14017	0.364995	0.182497	+	0.983206i	$0.441582\pi$
$284$	26.2740	1.55908
$285$	6.13280	0.363276
$286$	16.4027	0.969913
$287$	8.47526	0.500279
$288$	−22.1654	−1.30611
$289$	−9.12830	−0.536959
$290$	−5.18627	−0.304548
$291$	1.10141	0.0645660
$292$	−13.2087	−0.772983
$293$	−11.1631	−0.652153	−0.326076	−	0.945343i	$-0.605727\pi$
$293$	−11.1631	−0.652153	−0.326076	+	0.945343i	$0.605727\pi$
$294$	3.13720	0.182965
$295$	9.70143	0.564839
$296$	−1.62532	−0.0944701
$297$	−3.22811	−0.187314
$298$	−31.4847	−1.82386
$299$	−47.6584	−2.75615
$300$	1.53281	0.0884970
$301$	−3.45077	−0.198899
$302$	−12.0572	−0.693811
$303$	2.05115	0.117835
$304$	−23.2697	−1.33461
$305$	14.0818	0.806320
$306$	−15.6000	−0.891792
$307$	12.3300	0.703710	0.351855	−	0.936055i	$-0.385551\pi$
$307$	12.3300	0.703710	0.351855	+	0.936055i	$0.385551\pi$
$308$	−4.44427	−0.253236
$309$	−8.39586	−0.477624
$310$	48.3278	2.74484
$311$	28.2150	1.59992	0.799962	−	0.600050i	$-0.204853\pi$
$311$	28.2150	1.59992	0.799962	+	0.600050i	$0.204853\pi$
$312$	−0.614816	−0.0348071
$313$	4.81695	0.272270	0.136135	−	0.990690i	$-0.456532\pi$
$313$	4.81695	0.272270	0.136135	+	0.990690i	$0.456532\pi$
$314$	−17.8376	−1.00663
$315$	13.3499	0.752184
$316$	4.93192	0.277442
$317$	−22.9537	−1.28921	−0.644603	−	0.764517i	$-0.722977\pi$
$317$	−22.9537	−1.28921	−0.644603	+	0.764517i	$0.722977\pi$
$318$	0.991332	0.0555911
$319$	1.31235	0.0734773
$320$	−18.5355	−1.03617
$321$	4.70962	0.262866
$322$	26.6239	1.48369
$323$	−15.4734	−0.860964
$324$	13.9856	0.776980
$325$	−12.2148	−0.677555
$326$	38.2388	2.11785
$327$	0.932463	0.0515653
$328$	−1.08146	−0.0597139
$329$	−19.1401	−1.05523
$330$	−2.87589	−0.158312
$331$	4.62448	0.254184	0.127092	−	0.991891i	$-0.459436\pi$
$331$	4.62448	0.254184	0.127092	+	0.991891i	$0.459436\pi$
$332$	28.1464	1.54473
$333$	19.9889	1.09538
$334$	−35.6158	−1.94881
$335$	37.9148	2.07150
$336$	3.20534	0.174865
$337$	17.2220	0.938141	0.469071	−	0.883161i	$-0.344589\pi$
$337$	17.2220	0.938141	0.469071	+	0.883161i	$0.344589\pi$
$338$	−53.6528	−2.91833
$339$	4.75348	0.258174
$340$	−13.9078	−0.754255
$341$	−12.2290	−0.662238
$342$	30.6649	1.65817
$343$	19.3590	1.04529
$344$	0.440326	0.0237408
$345$	8.35593	0.449868
$346$	25.4782	1.36972
$347$	−16.5353	−0.887664	−0.443832	−	0.896110i	$-0.646381\pi$
$347$	−16.5353	−0.887664	−0.443832	+	0.896110i	$0.646381\pi$
$348$	0.795891	0.0426642
$349$	−3.86038	−0.206641	−0.103321	−	0.994648i	$-0.532947\pi$
$349$	−3.86038	−0.206641	−0.103321	+	0.994648i	$0.532947\pi$
$350$	6.82369	0.364742
$351$	15.6010	0.832717
$352$	10.3098	0.549515
$353$	−27.2193	−1.44874	−0.724369	−	0.689412i	$-0.757869\pi$
$353$	−27.2193	−1.44874	−0.724369	+	0.689412i	$0.757869\pi$
$354$	−3.06960	−0.163148
$355$	36.7096	1.94835
$356$	21.3755	1.13290
$357$	2.13142	0.112807
$358$	34.3376	1.81480
$359$	15.1073	0.797335	0.398668	−	0.917096i	$-0.369473\pi$
$359$	15.1073	0.797335	0.398668	+	0.917096i	$0.369473\pi$
$360$	−1.70348	−0.0897815
$361$	11.4161	0.600848
$362$	−39.9167	−2.09798
$363$	−3.92022	−0.205758
$364$	21.4785	1.12578
$365$	−18.4550	−0.965980
$366$	−4.45557	−0.232897
$367$	9.98222	0.521068	0.260534	−	0.965465i	$-0.416101\pi$
$367$	9.98222	0.521068	0.260534	+	0.965465i	$0.416101\pi$
$368$	−31.7049	−1.65273
$369$	13.3003	0.692384
$370$	36.7425	1.91015
$371$	2.14043	0.111126
$372$	−7.41645	−0.384525
$373$	−2.12201	−0.109874	−0.0549369	−	0.998490i	$-0.517496\pi$
$373$	−2.12201	−0.109874	−0.0549369	+	0.998490i	$0.517496\pi$
$374$	7.25603	0.375200
$375$	−3.41840	−0.176526
$376$	2.44232	0.125953
$377$	−6.34236	−0.326648
$378$	−8.71533	−0.448268
$379$	0.347475	0.0178486	0.00892430	−	0.999960i	$-0.497159\pi$
$379$	0.347475	0.0178486	0.00892430	+	0.999960i	$0.497159\pi$
$380$	27.3386	1.40244
$381$	−2.89641	−0.148388
$382$	−14.5947	−0.746732
$383$	−14.9243	−0.762599	−0.381299	−	0.924452i	$-0.624523\pi$
$383$	−14.9243	−0.762599	−0.381299	+	0.924452i	$0.624523\pi$
$384$	−0.774191	−0.0395078
$385$	−6.20946	−0.316464
$386$	−40.2527	−2.04881
$387$	−5.41530	−0.275275
$388$	4.90984	0.249259
$389$	11.4881	0.582470	0.291235	−	0.956652i	$-0.405934\pi$
$389$	11.4881	0.582470	0.291235	+	0.956652i	$0.405934\pi$
$390$	13.8987	0.703788
$391$	−21.0825	−1.06619
$392$	−0.864336	−0.0436556
$393$	6.70250	0.338096
$394$	−54.9343	−2.76755
$395$	6.89079	0.346713
$396$	−6.97441	−0.350477
$397$	34.6105	1.73705	0.868526	−	0.495644i	$-0.165068\pi$
$397$	34.6105	1.73705	0.868526	+	0.495644i	$0.165068\pi$
$398$	24.5032	1.22823
$399$	−4.18974	−0.209750
$400$	−8.12594	−0.406297
$401$	31.0420	1.55016	0.775081	−	0.631862i	$-0.217709\pi$
$401$	31.0420	1.55016	0.775081	+	0.631862i	$0.217709\pi$
$402$	−11.9965	−0.598331
$403$	59.1008	2.94402
$404$	9.14352	0.454907
$405$	19.5405	0.970975
$406$	3.54310	0.175841
$407$	−9.29742	−0.460856
$408$	−0.271975	−0.0134648
$409$	19.3834	0.958445	0.479223	−	0.877693i	$-0.340919\pi$
$409$	19.3834	0.958445	0.479223	+	0.877693i	$0.340919\pi$
$410$	24.4479	1.20739
$411$	−8.89179	−0.438600
$412$	−37.4267	−1.84388
$413$	−6.62773	−0.326129
$414$	41.7810	2.05342
$415$	39.3256	1.93042
$416$	−49.8257	−2.44291
$417$	−0.422540	−0.0206919
$418$	−14.2632	−0.697636
$419$	−3.41342	−0.166757	−0.0833783	−	0.996518i	$-0.526571\pi$
$419$	−3.41342	−0.166757	−0.0833783	+	0.996518i	$0.526571\pi$
$420$	−3.76582	−0.183753
$421$	−10.2775	−0.500895	−0.250448	−	0.968130i	$-0.580578\pi$
$421$	−10.2775	−0.500895	−0.250448	+	0.968130i	$0.580578\pi$
$422$	−5.48955	−0.267227
$423$	−30.0366	−1.46043
$424$	−0.273124	−0.0132641
$425$	−5.40343	−0.262105
$426$	−11.6152	−0.562758
$427$	−9.62024	−0.465556
$428$	20.9944	1.01480
$429$	−3.51696	−0.169801
$430$	−9.95413	−0.480031
$431$	−32.2924	−1.55547	−0.777736	−	0.628591i	$-0.783632\pi$
$431$	−32.2924	−1.55547	−0.777736	+	0.628591i	$0.783632\pi$
$432$	10.3786	0.499340
$433$	−24.9959	−1.20123	−0.600613	−	0.799540i	$-0.705077\pi$
$433$	−24.9959	−1.20123	−0.600613	+	0.799540i	$0.705077\pi$
$434$	−33.0161	−1.58482
$435$	1.11200	0.0533166
$436$	4.15670	0.199070
$437$	41.4420	1.98244
$438$	5.83930	0.279013
$439$	30.5883	1.45990	0.729949	−	0.683502i	$-0.239544\pi$
$439$	30.5883	1.45990	0.729949	+	0.683502i	$0.239544\pi$
$440$	0.792342	0.0377734
$441$	10.6299	0.506187
$442$	−35.0672	−1.66798
$443$	21.5316	1.02300	0.511498	−	0.859285i	$-0.329091\pi$
$443$	21.5316	1.02300	0.511498	+	0.859285i	$0.329091\pi$
$444$	−5.63855	−0.267594
$445$	29.8655	1.41576
$446$	8.61932	0.408137
$447$	6.75075	0.319300
$448$	12.6629	0.598267
$449$	35.6728	1.68350	0.841751	−	0.539866i	$-0.181525\pi$
$449$	35.6728	1.68350	0.841751	+	0.539866i	$0.181525\pi$
$450$	10.7084	0.504800
$451$	−6.18635	−0.291304
$452$	21.1899	0.996689
$453$	2.58522	0.121464
$454$	4.24095	0.199038
$455$	30.0093	1.40686
$456$	0.534621	0.0250359
$457$	21.6409	1.01232	0.506160	−	0.862440i	$-0.331065\pi$
$457$	21.6409	1.01232	0.506160	+	0.862440i	$0.331065\pi$
$458$	20.4708	0.956538
$459$	6.90136	0.322128
$460$	37.2488	1.73673
$461$	−9.25730	−0.431155	−0.215578	−	0.976487i	$-0.569163\pi$
$461$	−9.25730	−0.431155	−0.215578	+	0.976487i	$0.569163\pi$
$462$	1.96472	0.0914070
$463$	−24.3740	−1.13276	−0.566379	−	0.824145i	$-0.691656\pi$
$463$	−24.3740	−1.13276	−0.566379	+	0.824145i	$0.691656\pi$
$464$	−4.21928	−0.195875
$465$	−10.3621	−0.480532
$466$	31.5177	1.46003
$467$	−11.0270	−0.510270	−0.255135	−	0.966905i	$-0.582120\pi$
$467$	−11.0270	−0.510270	−0.255135	+	0.966905i	$0.582120\pi$
$468$	33.7062	1.55807
$469$	−25.9022	−1.19605
$470$	−55.2118	−2.54673
$471$	3.82461	0.176229
$472$	0.845714	0.0389271
$473$	2.51882	0.115815
$474$	−2.18030	−0.100144
$475$	10.6215	0.487350
$476$	9.50138	0.435495
$477$	3.35899	0.153797
$478$	17.4961	0.800252
$479$	7.45038	0.340417	0.170208	−	0.985408i	$-0.445556\pi$
$479$	7.45038	0.340417	0.170208	+	0.985408i	$0.445556\pi$
$480$	8.73593	0.398739
$481$	44.9330	2.04877
$482$	−44.0747	−2.00755
$483$	−5.70852	−0.259747
$484$	−17.4754	−0.794337
$485$	6.85994	0.311494
$486$	−20.7252	−0.940115
$487$	−34.6451	−1.56992	−0.784960	−	0.619546i	$-0.787317\pi$
$487$	−34.6451	−1.56992	−0.784960	+	0.619546i	$0.787317\pi$
$488$	1.22757	0.0555693
$489$	−8.19892	−0.370768
$490$	19.5394	0.882701
$491$	22.8251	1.03008	0.515041	−	0.857166i	$-0.327777\pi$
$491$	22.8251	1.03008	0.515041	+	0.857166i	$0.327777\pi$
$492$	−3.75180	−0.169144
$493$	−2.80565	−0.126360
$494$	68.9317	3.10139
$495$	−9.74453	−0.437984
$496$	39.3170	1.76539
$497$	−25.0789	−1.12494
$498$	−12.4429	−0.557580
$499$	36.5200	1.63486	0.817430	−	0.576027i	$-0.195398\pi$
$499$	36.5200	1.63486	0.817430	+	0.576027i	$0.195398\pi$
$500$	−15.2384	−0.681483
$501$	7.63651	0.341174
$502$	−42.5153	−1.89755
$503$	13.4921	0.601584	0.300792	−	0.953690i	$-0.402749\pi$
$503$	13.4921	0.601584	0.300792	+	0.953690i	$0.402749\pi$
$504$	1.16377	0.0518384
$505$	12.7752	0.568487
$506$	−19.4336	−0.863929
$507$	11.5039	0.510906
$508$	−12.9115	−0.572856
$509$	7.08951	0.314237	0.157119	−	0.987580i	$-0.449780\pi$
$509$	7.08951	0.314237	0.157119	+	0.987580i	$0.449780\pi$
$510$	6.14833	0.272253
$511$	12.6079	0.557741
$512$	−31.2108	−1.37933
$513$	−13.5660	−0.598954
$514$	28.6129	1.26206
$515$	−52.2920	−2.30426
$516$	1.52757	0.0672476
$517$	13.9709	0.614442
$518$	−25.1014	−1.10289
$519$	−5.46288	−0.239794
$520$	−3.82926	−0.167924
$521$	1.55816	0.0682641	0.0341320	−	0.999417i	$-0.489133\pi$
$521$	1.55816	0.0682641	0.0341320	+	0.999417i	$0.489133\pi$
$522$	5.56020	0.243363
$523$	9.58832	0.419268	0.209634	−	0.977780i	$-0.432773\pi$
$523$	9.58832	0.419268	0.209634	+	0.977780i	$0.432773\pi$
$524$	29.8781	1.30523
$525$	−1.46309	−0.0638545
$526$	−4.83011	−0.210603
$527$	26.1443	1.13886
$528$	−2.33967	−0.101821
$529$	33.4647	1.45499
$530$	6.17432	0.268195
$531$	−10.4009	−0.451361
$532$	−18.6769	−0.809745
$533$	29.8976	1.29501
$534$	−9.44966	−0.408927
$535$	29.3330	1.26818
$536$	3.30519	0.142762
$537$	−7.36245	−0.317713
$538$	20.6364	0.889698
$539$	−4.94431	−0.212966
$540$	−12.1934	−0.524719
$541$	10.7337	0.461476	0.230738	−	0.973016i	$-0.425886\pi$
$541$	10.7337	0.461476	0.230738	+	0.973016i	$0.425886\pi$
$542$	22.2515	0.955783
$543$	8.55868	0.367288
$544$	−22.0413	−0.945012
$545$	5.80766	0.248773
$546$	−9.49517	−0.406356
$547$	−16.8092	−0.718709	−0.359355	−	0.933201i	$-0.617003\pi$
$547$	−16.8092	−0.718709	−0.359355	+	0.933201i	$0.617003\pi$
$548$	−39.6375	−1.69323
$549$	−15.0971	−0.644327
$550$	−4.98082	−0.212383
$551$	5.51508	0.234950
$552$	0.728421	0.0310037
$553$	−4.70758	−0.200187
$554$	3.56506	0.151465
$555$	−7.87809	−0.334406
$556$	−1.88358	−0.0798818
$557$	−9.40910	−0.398676	−0.199338	−	0.979931i	$-0.563879\pi$
$557$	−9.40910	−0.398676	−0.199338	+	0.979931i	$0.563879\pi$
$558$	−51.8123	−2.19339
$559$	−12.1730	−0.514865
$560$	19.9638	0.843625
$561$	−1.55579	−0.0656855
$562$	−13.7074	−0.578211
$563$	25.0665	1.05643	0.528214	−	0.849111i	$-0.322862\pi$
$563$	25.0665	1.05643	0.528214	+	0.849111i	$0.322862\pi$
$564$	8.47288	0.356772
$565$	29.6062	1.24554
$566$	−12.1003	−0.508614
$567$	−13.3495	−0.560625
$568$	3.20013	0.134274
$569$	14.6522	0.614252	0.307126	−	0.951669i	$-0.400633\pi$
$569$	14.6522	0.614252	0.307126	+	0.951669i	$0.400633\pi$
$570$	−12.0858	−0.506218
$571$	−27.0577	−1.13233	−0.566165	−	0.824292i	$-0.691573\pi$
$571$	−27.0577	−1.13233	−0.566165	+	0.824292i	$0.691573\pi$
$572$	−15.6778	−0.655521
$573$	3.12931	0.130729
$574$	−16.7020	−0.697130
$575$	14.4718	0.603518
$576$	19.8720	0.827999
$577$	−46.5126	−1.93634	−0.968172	−	0.250287i	$-0.919475\pi$
$577$	−46.5126	−1.93634	−0.968172	+	0.250287i	$0.919475\pi$
$578$	17.9890	0.748243
$579$	8.63073	0.358681
$580$	4.95705	0.205830
$581$	−26.8661	−1.11459
$582$	−2.17054	−0.0899716
$583$	−1.56237	−0.0647066
$584$	−1.60880	−0.0665726
$585$	47.0937	1.94709
$586$	21.9988	0.908763
$587$	30.1657	1.24507	0.622536	−	0.782591i	$-0.286102\pi$
$587$	30.1657	1.24507	0.622536	+	0.782591i	$0.286102\pi$
$588$	−2.99854	−0.123658
$589$	−51.3919	−2.11757
$590$	−19.1184	−0.787093
$591$	11.7787	0.484509
$592$	29.8918	1.22855
$593$	23.9060	0.981703	0.490852	−	0.871243i	$-0.336686\pi$
$593$	23.9060	0.981703	0.490852	+	0.871243i	$0.336686\pi$
$594$	6.36158	0.261019
$595$	13.2752	0.544228
$596$	30.0932	1.23267
$597$	−5.25381	−0.215024
$598$	93.9195	3.84065
$599$	−21.7287	−0.887812	−0.443906	−	0.896073i	$-0.646408\pi$
$599$	−21.7287	−0.887812	−0.443906	+	0.896073i	$0.646408\pi$
$600$	0.186694	0.00762174
$601$	−31.4554	−1.28309	−0.641546	−	0.767085i	$-0.721706\pi$
$601$	−31.4554	−1.28309	−0.641546	+	0.767085i	$0.721706\pi$
$602$	6.80036	0.277162
$603$	−40.6484	−1.65533
$604$	11.5243	0.468916
$605$	−24.4164	−0.992666
$606$	−4.04215	−0.164201
$607$	−2.49507	−0.101272	−0.0506359	−	0.998717i	$-0.516125\pi$
$607$	−2.49507	−0.101272	−0.0506359	+	0.998717i	$0.516125\pi$
$608$	43.3266	1.75712
$609$	−0.759688	−0.0307841
$610$	−27.7507	−1.12359
$611$	−67.5193	−2.73154
$612$	14.9105	0.602723
$613$	−0.586942	−0.0237064	−0.0118532	−	0.999930i	$-0.503773\pi$
$613$	−0.586942	−0.0237064	−0.0118532	+	0.999930i	$0.503773\pi$
$614$	−24.2985	−0.980607
$615$	−5.24195	−0.211376
$616$	−0.541304	−0.0218098
$617$	37.5560	1.51195	0.755974	−	0.654602i	$-0.227164\pi$
$617$	37.5560	1.51195	0.755974	+	0.654602i	$0.227164\pi$
$618$	16.5456	0.665560
$619$	27.7154	1.11398	0.556988	−	0.830521i	$-0.311957\pi$
$619$	27.7154	1.11398	0.556988	+	0.830521i	$0.311957\pi$
$620$	−46.1919	−1.85511
$621$	−18.4837	−0.741725
$622$	−55.6027	−2.22947
$623$	−20.4032	−0.817438
$624$	11.3073	0.452653
$625$	−30.9204	−1.23682
$626$	−9.49267	−0.379403
$627$	3.05822	0.122134
$628$	17.0492	0.680337
$629$	19.8769	0.792543
$630$	−26.3085	−1.04815
$631$	29.0367	1.15593	0.577966	−	0.816061i	$-0.303847\pi$
$631$	29.0367	1.15593	0.577966	+	0.816061i	$0.303847\pi$
$632$	0.600699	0.0238945
$633$	1.17703	0.0467828
$634$	45.2343	1.79649
$635$	−18.0397	−0.715886
$636$	−0.947519	−0.0375716
$637$	23.8950	0.946756
$638$	−2.58622	−0.102389
$639$	−39.3564	−1.55692
$640$	−4.82190	−0.190602
$641$	−9.98696	−0.394461	−0.197231	−	0.980357i	$-0.563195\pi$
$641$	−9.98696	−0.394461	−0.197231	+	0.980357i	$0.563195\pi$
$642$	−9.28117	−0.366298
$643$	−7.24343	−0.285653	−0.142826	−	0.989748i	$-0.545619\pi$
$643$	−7.24343	−0.285653	−0.142826	+	0.989748i	$0.545619\pi$
$644$	−25.4472	−1.00276
$645$	2.13430	0.0840379
$646$	30.4932	1.19974
$647$	7.74040	0.304307	0.152153	−	0.988357i	$-0.451379\pi$
$647$	7.74040	0.304307	0.152153	+	0.988357i	$0.451379\pi$
$648$	1.70343	0.0669169
$649$	4.83778	0.189899
$650$	24.0715	0.944161
$651$	7.07910	0.277452
$652$	−36.5488	−1.43136
$653$	35.7431	1.39874	0.699368	−	0.714762i	$-0.253465\pi$
$653$	35.7431	1.39874	0.699368	+	0.714762i	$0.253465\pi$
$654$	−1.83759	−0.0718553
$655$	41.7452	1.63112
$656$	19.8895	0.776555
$657$	19.7856	0.771911
$658$	37.7191	1.47044
$659$	−3.07831	−0.119914	−0.0599570	−	0.998201i	$-0.519096\pi$
$659$	−3.07831	−0.119914	−0.0599570	+	0.998201i	$0.519096\pi$
$660$	2.74878	0.106996
$661$	10.7294	0.417325	0.208662	−	0.977988i	$-0.433089\pi$
$661$	10.7294	0.417325	0.208662	+	0.977988i	$0.433089\pi$
$662$	−9.11338	−0.354201
$663$	7.51888	0.292009
$664$	3.42818	0.133039
$665$	−26.0950	−1.01192
$666$	−39.3917	−1.52640
$667$	7.51430	0.290955
$668$	34.0417	1.31711
$669$	−1.84810	−0.0714516
$670$	−74.7179	−2.88661
$671$	7.02211	0.271085
$672$	−5.96813	−0.230225
$673$	−43.5608	−1.67914	−0.839572	−	0.543249i	$-0.817194\pi$
$673$	−43.5608	−1.67914	−0.839572	+	0.543249i	$0.817194\pi$
$674$	−33.9390	−1.30728
$675$	−4.73735	−0.182341
$676$	51.2816	1.97237
$677$	−3.00802	−0.115608	−0.0578038	−	0.998328i	$-0.518410\pi$
$677$	−3.00802	−0.115608	−0.0578038	+	0.998328i	$0.518410\pi$
$678$	−9.36760	−0.359761
$679$	−4.68651	−0.179852
$680$	−1.69394	−0.0649597
$681$	−0.909316	−0.0348451
$682$	24.0995	0.922816
$683$	−4.62614	−0.177014	−0.0885072	−	0.996076i	$-0.528210\pi$
$683$	−4.62614	−0.177014	−0.0885072	+	0.996076i	$0.528210\pi$
$684$	−29.3097	−1.12068
$685$	−55.3808	−2.11599
$686$	−38.1505	−1.45659
$687$	−4.38921	−0.167459
$688$	−8.09816	−0.308740
$689$	7.55067	0.287658
$690$	−16.4669	−0.626884
$691$	−10.5560	−0.401569	−0.200785	−	0.979635i	$-0.564349\pi$
$691$	−10.5560	−0.401569	−0.200785	+	0.979635i	$0.564349\pi$
$692$	−24.3522	−0.925732
$693$	6.65717	0.252885
$694$	32.5859	1.23694
$695$	−2.63171	−0.0998265
$696$	0.0969381	0.00367443
$697$	13.2257	0.500961
$698$	7.60758	0.287951
$699$	−6.75783	−0.255604
$700$	−6.52211	−0.246513
$701$	50.9360	1.92382	0.961912	−	0.273358i	$-0.0881344\pi$
$701$	50.9360	1.92382	0.961912	+	0.273358i	$0.0881344\pi$
$702$	−30.7445	−1.16038
$703$	−39.0720	−1.47363
$704$	−9.24305	−0.348361
$705$	11.8382	0.445851
$706$	53.6406	2.01879
$707$	−8.72761	−0.328236
$708$	2.93394	0.110264
$709$	18.8420	0.707625	0.353812	−	0.935316i	$-0.384885\pi$
$709$	18.8420	0.707625	0.353812	+	0.935316i	$0.384885\pi$
$710$	−72.3430	−2.71498
$711$	−7.38762	−0.277057
$712$	2.60350	0.0975703
$713$	−70.0214	−2.62232
$714$	−4.20036	−0.157194
$715$	−21.9047	−0.819190
$716$	−32.8200	−1.22654
$717$	−3.75139	−0.140098
$718$	−29.7718	−1.11107
$719$	−4.66233	−0.173876	−0.0869378	−	0.996214i	$-0.527708\pi$
$719$	−4.66233	−0.173876	−0.0869378	+	0.996214i	$0.527708\pi$
$720$	31.3293	1.16757
$721$	35.7243	1.33044
$722$	−22.4975	−0.837272
$723$	9.45021	0.351457
$724$	38.1526	1.41793
$725$	1.92591	0.0715265
$726$	7.72551	0.286721
$727$	−45.1857	−1.67585	−0.837923	−	0.545789i	$-0.816230\pi$
$727$	−45.1857	−1.67585	−0.837923	+	0.545789i	$0.816230\pi$
$728$	2.61604	0.0969568
$729$	−17.8313	−0.660418
$730$	36.3690	1.34608
$731$	−5.38496	−0.199170
$732$	4.25865	0.157404
$733$	0.0694421	0.00256490	0.00128245	−	0.999999i	$-0.499592\pi$
$733$	0.0694421	0.00256490	0.00128245	+	0.999999i	$0.499592\pi$
$734$	−19.6718	−0.726099
$735$	−4.18951	−0.154533
$736$	59.0324	2.17596
$737$	18.9068	0.696442
$738$	−26.2106	−0.964824
$739$	33.3158	1.22554	0.612770	−	0.790261i	$-0.290055\pi$
$739$	33.3158	1.22554	0.612770	+	0.790261i	$0.290055\pi$
$740$	−35.1186	−1.29099
$741$	−14.7799	−0.542953
$742$	−4.21811	−0.154852
$743$	31.0233	1.13814	0.569068	−	0.822291i	$-0.307304\pi$
$743$	31.0233	1.13814	0.569068	+	0.822291i	$0.307304\pi$
$744$	−0.903310	−0.0331170
$745$	42.0457	1.54044
$746$	4.18182	0.153107
$747$	−42.1610	−1.54259
$748$	−6.93534	−0.253581
$749$	−20.0394	−0.732224
$750$	6.73659	0.245985
$751$	−31.6985	−1.15669	−0.578346	−	0.815791i	$-0.696302\pi$
$751$	−31.6985	−1.15669	−0.578346	+	0.815791i	$0.696302\pi$
$752$	−44.9175	−1.63797
$753$	9.11585	0.332200
$754$	12.4988	0.455178
$755$	16.1015	0.585994
$756$	8.33014	0.302964
$757$	−2.46893	−0.0897347	−0.0448674	−	0.998993i	$-0.514287\pi$
$757$	−2.46893	−0.0897347	−0.0448674	+	0.998993i	$0.514287\pi$
$758$	−0.684763	−0.0248717
$759$	4.16682	0.151246
$760$	3.32979	0.120784
$761$	−33.7452	−1.22326	−0.611632	−	0.791143i	$-0.709487\pi$
$761$	−33.7452	−1.22326	−0.611632	+	0.791143i	$0.709487\pi$
$762$	5.70791	0.206776
$763$	−3.96762	−0.143638
$764$	13.9497	0.504683
$765$	20.8327	0.753209
$766$	29.4111	1.06267
$767$	−23.3802	−0.844210
$768$	7.47772	0.269829
$769$	42.3230	1.52621	0.763103	−	0.646277i	$-0.223675\pi$
$769$	42.3230	1.52621	0.763103	+	0.646277i	$0.223675\pi$
$770$	12.2369	0.440986
$771$	−6.13498	−0.220946
$772$	38.4737	1.38470
$773$	2.34499	0.0843435	0.0421717	−	0.999110i	$-0.486572\pi$
$773$	2.34499	0.0843435	0.0421717	+	0.999110i	$0.486572\pi$
$774$	10.6718	0.383591
$775$	−17.9464	−0.644655
$776$	0.598010	0.0214673
$777$	5.38207	0.193081
$778$	−22.6394	−0.811661
$779$	−25.9979	−0.931471
$780$	−13.2844	−0.475659
$781$	18.3059	0.655035
$782$	41.5469	1.48572
$783$	−2.45980	−0.0879062
$784$	15.8963	0.567724
$785$	23.8209	0.850203
$786$	−13.2085	−0.471131
$787$	28.1401	1.00309	0.501543	−	0.865133i	$-0.332766\pi$
$787$	28.1401	1.00309	0.501543	+	0.865133i	$0.332766\pi$
$788$	52.5064	1.87046
$789$	1.03564	0.0368698
$790$	−13.5796	−0.483139
$791$	−20.2260	−0.719155
$792$	−0.849470	−0.0301846
$793$	−33.9367	−1.20513
$794$	−68.2063	−2.42055
$795$	−1.32386	−0.0469524
$796$	−23.4202	−0.830108
$797$	−40.0990	−1.42038	−0.710190	−	0.704010i	$-0.751391\pi$
$797$	−40.0990	−1.42038	−0.710190	+	0.704010i	$0.751391\pi$
$798$	8.25665	0.292282
$799$	−29.8684	−1.05667
$800$	15.1300	0.534925
$801$	−32.0188	−1.13133
$802$	−61.1738	−2.16012
$803$	−9.20290	−0.324763
$804$	11.4663	0.404385
$805$	−35.5544	−1.25313
$806$	−116.469	−4.10244
$807$	−4.42472	−0.155757
$808$	1.11366	0.0391786
$809$	−21.7592	−0.765011	−0.382506	−	0.923953i	$-0.624939\pi$
$809$	−21.7592	−0.765011	−0.382506	+	0.923953i	$0.624939\pi$
$810$	−38.5081	−1.35304
$811$	−55.3151	−1.94238	−0.971189	−	0.238312i	$-0.923406\pi$
$811$	−55.3151	−1.94238	−0.971189	+	0.238312i	$0.923406\pi$
$812$	−3.38651	−0.118843
$813$	−4.77102	−0.167327
$814$	18.3223	0.642195
$815$	−51.0654	−1.78874
$816$	5.00196	0.175104
$817$	10.5852	0.370330
$818$	−38.1984	−1.33558
$819$	−32.1730	−1.12422
$820$	−23.3674	−0.816024
$821$	−2.79121	−0.0974138	−0.0487069	−	0.998813i	$-0.515510\pi$
$821$	−2.79121	−0.0974138	−0.0487069	+	0.998813i	$0.515510\pi$
$822$	17.5229	0.611181
$823$	53.4822	1.86427	0.932136	−	0.362108i	$-0.117943\pi$
$823$	53.4822	1.86427	0.932136	+	0.362108i	$0.117943\pi$
$824$	−4.55851	−0.158803
$825$	1.06795	0.0371814
$826$	13.0611	0.454455
$827$	11.4821	0.399271	0.199636	−	0.979870i	$-0.436024\pi$
$827$	11.4821	0.399271	0.199636	+	0.979870i	$0.436024\pi$
$828$	−39.9344	−1.38782
$829$	−19.3225	−0.671098	−0.335549	−	0.942023i	$-0.608922\pi$
$829$	−19.3225	−0.671098	−0.335549	+	0.942023i	$0.608922\pi$
$830$	−77.4983	−2.69000
$831$	−0.764396	−0.0265166
$832$	44.6702	1.54866
$833$	10.5704	0.366242
$834$	0.832693	0.0288338
$835$	47.5625	1.64597
$836$	13.6328	0.471501
$837$	22.9215	0.792282
$838$	6.72676	0.232372
$839$	−28.3712	−0.979482	−0.489741	−	0.871868i	$-0.662909\pi$
$839$	−28.3712	−0.979482	−0.489741	+	0.871868i	$0.662909\pi$
$840$	−0.458669	−0.0158256
$841$	1.00000	0.0344828
$842$	20.2537	0.697988
$843$	2.93905	0.101226
$844$	5.24693	0.180607
$845$	71.6497	2.46483
$846$	59.1926	2.03508
$847$	16.6805	0.573150
$848$	5.02311	0.172494
$849$	2.59447	0.0890419
$850$	10.6484	0.365239
$851$	−53.2356	−1.82489
$852$	11.1018	0.380343
$853$	0.292998	0.0100321	0.00501603	−	0.999987i	$-0.498403\pi$
$853$	0.292998	0.0100321	0.00501603	+	0.999987i	$0.498403\pi$
$854$	18.9584	0.648744
$855$	−40.9510	−1.40049
$856$	2.55708	0.0873991
$857$	−28.3080	−0.966983	−0.483491	−	0.875349i	$-0.660632\pi$
$857$	−28.3080	−0.966983	−0.483491	+	0.875349i	$0.660632\pi$
$858$	6.93081	0.236614
$859$	−11.4618	−0.391071	−0.195535	−	0.980697i	$-0.562644\pi$
$859$	−11.4618	−0.391071	−0.195535	+	0.980697i	$0.562644\pi$
$860$	9.51419	0.324431
$861$	3.58114	0.122045
$862$	63.6381	2.16752
$863$	29.1847	0.993459	0.496730	−	0.867905i	$-0.334534\pi$
$863$	29.1847	0.993459	0.496730	+	0.867905i	$0.334534\pi$
$864$	−19.3243	−0.657424
$865$	−34.0245	−1.15687
$866$	49.2589	1.67389
$867$	−3.85708	−0.130993
$868$	31.5569	1.07111
$869$	3.43621	0.116565
$870$	−2.19141	−0.0742957
$871$	−91.3736	−3.09608
$872$	0.506278	0.0171447
$873$	−7.35455	−0.248914
$874$	−81.6689	−2.76249
$875$	14.5453	0.491720
$876$	−5.58123	−0.188572
$877$	11.9069	0.402066	0.201033	−	0.979584i	$-0.435570\pi$
$877$	11.9069	0.402066	0.201033	+	0.979584i	$0.435570\pi$
$878$	−60.2797	−2.03434
$879$	−4.71684	−0.159095
$880$	−14.5722	−0.491229
$881$	8.91821	0.300462	0.150231	−	0.988651i	$-0.451998\pi$
$881$	8.91821	0.300462	0.150231	+	0.988651i	$0.451998\pi$
$882$	−20.9482	−0.705363
$883$	5.39313	0.181493	0.0907466	−	0.995874i	$-0.471075\pi$
$883$	5.39313	0.181493	0.0907466	+	0.995874i	$0.471075\pi$
$884$	33.5174	1.12731
$885$	4.09925	0.137795
$886$	−42.4318	−1.42553
$887$	15.9140	0.534341	0.267171	−	0.963649i	$-0.413911\pi$
$887$	15.9140	0.534341	0.267171	+	0.963649i	$0.413911\pi$
$888$	−0.686765	−0.0230463
$889$	12.3242	0.413341
$890$	−58.8554	−1.97284
$891$	9.74419	0.326443
$892$	−8.23838	−0.275841
$893$	58.7123	1.96473
$894$	−13.3036	−0.444938
$895$	−45.8556	−1.53278
$896$	3.29418	0.110051
$897$	−20.1376	−0.672375
$898$	−70.2996	−2.34593
$899$	−9.31843	−0.310787
$900$	−10.2352	−0.341172
$901$	3.34017	0.111277
$902$	12.1913	0.405927
$903$	−1.45809	−0.0485221
$904$	2.58089	0.0858392
$905$	53.3061	1.77195
$906$	−5.09464	−0.169258
$907$	−16.1896	−0.537565	−0.268783	−	0.963201i	$-0.586621\pi$
$907$	−16.1896	−0.537565	−0.268783	+	0.963201i	$0.586621\pi$
$908$	−4.05352	−0.134521
$909$	−13.6963	−0.454276
$910$	−59.1388	−1.96043
$911$	−15.7953	−0.523322	−0.261661	−	0.965160i	$-0.584270\pi$
$911$	−15.7953	−0.523322	−0.261661	+	0.965160i	$0.584270\pi$
$912$	−9.83237	−0.325582
$913$	19.6104	0.649009
$914$	−42.6473	−1.41065
$915$	5.95012	0.196705
$916$	−19.5661	−0.646481
$917$	−28.5191	−0.941783
$918$	−13.6004	−0.448879
$919$	34.6474	1.14291	0.571456	−	0.820633i	$-0.306379\pi$
$919$	34.6474	1.14291	0.571456	+	0.820633i	$0.306379\pi$
$920$	4.53683	0.149575
$921$	5.20992	0.171673
$922$	18.2432	0.600807
$923$	−88.4693	−2.91200
$924$	−1.87789	−0.0617779
$925$	−13.6443	−0.448620
$926$	48.0335	1.57848
$927$	56.0622	1.84133
$928$	7.85602	0.257886
$929$	20.4733	0.671706	0.335853	−	0.941914i	$-0.390975\pi$
$929$	20.4733	0.671706	0.335853	+	0.941914i	$0.390975\pi$
$930$	20.4205	0.669613
$931$	−20.7782	−0.680979
$932$	−30.1248	−0.986770
$933$	11.9220	0.390308
$934$	21.7308	0.711052
$935$	−9.68994	−0.316895
$936$	4.10535	0.134188
$937$	−21.4168	−0.699656	−0.349828	−	0.936814i	$-0.613760\pi$
$937$	−21.4168	−0.699656	−0.349828	+	0.936814i	$0.613760\pi$
$938$	51.0450	1.66668
$939$	2.03536	0.0664213
$940$	52.7717	1.72122
$941$	53.2739	1.73668	0.868340	−	0.495970i	$-0.165187\pi$
$941$	53.2739	1.73668	0.868340	+	0.495970i	$0.165187\pi$
$942$	−7.53709	−0.245572
$943$	−35.4221	−1.15350
$944$	−15.5538	−0.506232
$945$	11.6387	0.378608
$946$	−4.96379	−0.161387
$947$	53.1372	1.72673	0.863364	−	0.504582i	$-0.168353\pi$
$947$	53.1372	1.72673	0.863364	+	0.504582i	$0.168353\pi$
$948$	2.08394	0.0676831
$949$	44.4761	1.44376
$950$	−20.9317	−0.679113
$951$	−9.69885	−0.314507
$952$	1.15725	0.0375067
$953$	−46.5363	−1.50746	−0.753729	−	0.657185i	$-0.771747\pi$
$953$	−46.5363	−1.50746	−0.753729	+	0.657185i	$0.771747\pi$
$954$	−6.61949	−0.214314
$955$	19.4903	0.630691
$956$	−16.7228	−0.540854
$957$	0.554519	0.0179251
$958$	−14.6823	−0.474365
$959$	37.8345	1.22174
$960$	−7.83202	−0.252777
$961$	55.8330	1.80107
$962$	−88.5485	−2.85492
$963$	−31.4479	−1.01339
$964$	42.1268	1.35681
$965$	53.7548	1.73043
$966$	11.2497	0.361953
$967$	27.7293	0.891713	0.445856	−	0.895104i	$-0.352899\pi$
$967$	27.7293	0.891713	0.445856	+	0.895104i	$0.352899\pi$
$968$	−2.12847	−0.0684118
$969$	−6.53814	−0.210035
$970$	−13.5188	−0.434061
$971$	58.8444	1.88841	0.944203	−	0.329363i	$-0.106834\pi$
$971$	58.8444	1.88841	0.944203	+	0.329363i	$0.106834\pi$
$972$	19.8092	0.635382
$973$	1.79791	0.0576382
$974$	68.2745	2.18766
$975$	−5.16125	−0.165292
$976$	−22.5765	−0.722657
$977$	−23.2768	−0.744691	−0.372346	−	0.928094i	$-0.621446\pi$
$977$	−23.2768	−0.744691	−0.372346	+	0.928094i	$0.621446\pi$
$978$	16.1575	0.516658
$979$	14.8929	0.475980
$980$	−18.6759	−0.596578
$981$	−6.22640	−0.198794
$982$	−44.9809	−1.43540
$983$	23.2380	0.741176	0.370588	−	0.928797i	$-0.379156\pi$
$983$	23.2380	0.741176	0.370588	+	0.928797i	$0.379156\pi$
$984$	−0.456962	−0.0145674
$985$	73.3611	2.33748
$986$	5.52905	0.176081
$987$	−8.08747	−0.257427
$988$	−65.8852	−2.09609
$989$	14.4224	0.458605
$990$	19.2034	0.610323
$991$	−22.1552	−0.703784	−0.351892	−	0.936041i	$-0.614462\pi$
$991$	−22.1552	−0.703784	−0.351892	+	0.936041i	$0.614462\pi$
$992$	−73.2057	−2.32428
$993$	1.95403	0.0620093
$994$	49.4226	1.56759
$995$	−32.7224	−1.03737
$996$	11.8930	0.376844
$997$	29.0482	0.919966	0.459983	−	0.887928i	$-0.347856\pi$
$997$	29.0482	0.919966	0.459983	+	0.887928i	$0.347856\pi$
$998$	−71.9693	−2.27815
$999$	17.4267	0.551355

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4031.2.a.d.1.17	✓	98

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4031.2.a.d.1.17	✓	98	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:	\( N \)	\(=\)	\( 4031 = 29 \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4031.a (trivial)

\(f(q)\)	\(=\)	\(q-1.97068 q^{2} +0.422540 q^{3} +1.88358 q^{4} +2.63171 q^{5} -0.832693 q^{6} -1.79791 q^{7} +0.229417 q^{8} -2.82146 q^{9} +O(q^{10})\)	\(q-1.97068 q^{2} +0.422540 q^{3} +1.88358 q^{4} +2.63171 q^{5} -0.832693 q^{6} -1.79791 q^{7} +0.229417 q^{8} -2.82146 q^{9} -5.18627 q^{10} +1.31235 q^{11} +0.795891 q^{12} -6.34236 q^{13} +3.54310 q^{14} +1.11200 q^{15} -4.21928 q^{16} -2.80565 q^{17} +5.56020 q^{18} +5.51508 q^{19} +4.95705 q^{20} -0.759688 q^{21} -2.58622 q^{22} +7.51430 q^{23} +0.0969381 q^{24} +1.92591 q^{25} +12.4988 q^{26} -2.45980 q^{27} -3.38651 q^{28} +1.00000 q^{29} -2.19141 q^{30} -9.31843 q^{31} +7.85602 q^{32} +0.554519 q^{33} +5.52905 q^{34} -4.73157 q^{35} -5.31446 q^{36} -7.08458 q^{37} -10.8685 q^{38} -2.67990 q^{39} +0.603760 q^{40} -4.71396 q^{41} +1.49710 q^{42} +1.91932 q^{43} +2.47192 q^{44} -7.42527 q^{45} -14.8083 q^{46} +10.6458 q^{47} -1.78282 q^{48} -3.76753 q^{49} -3.79535 q^{50} -1.18550 q^{51} -11.9464 q^{52} -1.19051 q^{53} +4.84749 q^{54} +3.45372 q^{55} -0.412471 q^{56} +2.33035 q^{57} -1.97068 q^{58} +3.68636 q^{59} +2.09456 q^{60} +5.35080 q^{61} +18.3636 q^{62} +5.07272 q^{63} -7.04315 q^{64} -16.6913 q^{65} -1.09278 q^{66} +14.4069 q^{67} -5.28469 q^{68} +3.17509 q^{69} +9.32442 q^{70} +13.9490 q^{71} -0.647291 q^{72} -7.01255 q^{73} +13.9615 q^{74} +0.813774 q^{75} +10.3881 q^{76} -2.35948 q^{77} +5.28124 q^{78} +2.61837 q^{79} -11.1039 q^{80} +7.42501 q^{81} +9.28972 q^{82} +14.9430 q^{83} -1.43094 q^{84} -7.38368 q^{85} -3.78238 q^{86} +0.422540 q^{87} +0.301075 q^{88} +11.3483 q^{89} +14.6328 q^{90} +11.4030 q^{91} +14.1538 q^{92} -3.93741 q^{93} -20.9794 q^{94} +14.5141 q^{95} +3.31949 q^{96} +2.60665 q^{97} +7.42460 q^{98} -3.70273 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 98 q + 6 q^{2} + 6 q^{3} + 116 q^{4} + q^{5} + 7 q^{6} + 12 q^{7} + 15 q^{8} + 136 q^{9}+O(q^{10}) \) 98 * q + 6 * q^2 + 6 * q^3 + 116 * q^4 + q^5 + 7 * q^6 + 12 * q^7 + 15 * q^8 + 136 * q^9	\( 98 q + 6 q^{2} + 6 q^{3} + 116 q^{4} + q^{5} + 7 q^{6} + 12 q^{7} + 15 q^{8} + 136 q^{9} + 16 q^{10} + 25 q^{11} + 8 q^{12} + 18 q^{13} + 34 q^{14} + 14 q^{15} + 136 q^{16} + 35 q^{17} + 20 q^{18} + 48 q^{19} - 20 q^{20} + 62 q^{21} + 32 q^{22} + q^{23} + 8 q^{24} + 173 q^{25} + 15 q^{26} + 18 q^{27} + 37 q^{28} + 98 q^{29} - 6 q^{30} + 20 q^{31} + 16 q^{32} + 24 q^{33} - 6 q^{34} + 11 q^{35} + 155 q^{36} + 62 q^{37} - 5 q^{38} + 45 q^{39} + 38 q^{40} + 50 q^{41} + 7 q^{42} + 72 q^{43} + 92 q^{44} - 13 q^{45} + 32 q^{46} - 16 q^{47} - 13 q^{48} + 194 q^{49} + 50 q^{50} + 2 q^{51} + 83 q^{52} - 11 q^{53} + 58 q^{54} + 32 q^{55} + 106 q^{56} + 84 q^{57} + 6 q^{58} + 20 q^{59} + 62 q^{60} + 142 q^{61} - 27 q^{62} + 26 q^{63} + 213 q^{64} + 13 q^{65} - 35 q^{66} + 5 q^{67} + 69 q^{68} + 68 q^{69} - 2 q^{70} - 10 q^{71} - 9 q^{72} + 74 q^{73} + 21 q^{74} + 19 q^{75} + 116 q^{76} + 41 q^{77} - 162 q^{78} + 104 q^{79} - 86 q^{80} + 230 q^{81} + 53 q^{82} - 19 q^{83} + 76 q^{84} + 125 q^{85} + 9 q^{86} + 6 q^{87} + 68 q^{88} + 120 q^{89} + 122 q^{90} + 30 q^{91} + 3 q^{92} - 56 q^{93} + 22 q^{94} + 32 q^{95} - 55 q^{96} + 98 q^{97} - 15 q^{98} + 69 q^{99}+O(q^{100}) \) 98 * q + 6 * q^2 + 6 * q^3 + 116 * q^4 + q^5 + 7 * q^6 + 12 * q^7 + 15 * q^8 + 136 * q^9 + 16 * q^10 + 25 * q^11 + 8 * q^12 + 18 * q^13 + 34 * q^14 + 14 * q^15 + 136 * q^16 + 35 * q^17 + 20 * q^18 + 48 * q^19 - 20 * q^20 + 62 * q^21 + 32 * q^22 + q^23 + 8 * q^24 + 173 * q^25 + 15 * q^26 + 18 * q^27 + 37 * q^28 + 98 * q^29 - 6 * q^30 + 20 * q^31 + 16 * q^32 + 24 * q^33 - 6 * q^34 + 11 * q^35 + 155 * q^36 + 62 * q^37 - 5 * q^38 + 45 * q^39 + 38 * q^40 + 50 * q^41 + 7 * q^42 + 72 * q^43 + 92 * q^44 - 13 * q^45 + 32 * q^46 - 16 * q^47 - 13 * q^48 + 194 * q^49 + 50 * q^50 + 2 * q^51 + 83 * q^52 - 11 * q^53 + 58 * q^54 + 32 * q^55 + 106 * q^56 + 84 * q^57 + 6 * q^58 + 20 * q^59 + 62 * q^60 + 142 * q^61 - 27 * q^62 + 26 * q^63 + 213 * q^64 + 13 * q^65 - 35 * q^66 + 5 * q^67 + 69 * q^68 + 68 * q^69 - 2 * q^70 - 10 * q^71 - 9 * q^72 + 74 * q^73 + 21 * q^74 + 19 * q^75 + 116 * q^76 + 41 * q^77 - 162 * q^78 + 104 * q^79 - 86 * q^80 + 230 * q^81 + 53 * q^82 - 19 * q^83 + 76 * q^84 + 125 * q^85 + 9 * q^86 + 6 * q^87 + 68 * q^88 + 120 * q^89 + 122 * q^90 + 30 * q^91 + 3 * q^92 - 56 * q^93 + 22 * q^94 + 32 * q^95 - 55 * q^96 + 98 * q^97 - 15 * q^98 + 69 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more