LMFDB - Embedded newform 8029.2.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8029,2,Mod(1,8029)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8029, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8029.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8029 = 7 \cdot 31 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8029.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.1118877829$
Analytic rank:	$0$
Dimension:	$71$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	8029.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.69431 q^{2} +0.251057 q^{3} +5.25931 q^{4} +0.178464 q^{5} -0.676426 q^{6} -1.00000 q^{7} -8.78160 q^{8} -2.93697 q^{9} +O(q^{10})$	\(q-2.69431 q^{2} +0.251057 q^{3} +5.25931 q^{4} +0.178464 q^{5} -0.676426 q^{6} -1.00000 q^{7} -8.78160 q^{8} -2.93697 q^{9} -0.480839 q^{10} +3.66197 q^{11} +1.32039 q^{12} +2.67742 q^{13} +2.69431 q^{14} +0.0448048 q^{15} +13.1417 q^{16} +4.46232 q^{17} +7.91311 q^{18} +4.96953 q^{19} +0.938600 q^{20} -0.251057 q^{21} -9.86649 q^{22} -1.77881 q^{23} -2.20468 q^{24} -4.96815 q^{25} -7.21379 q^{26} -1.49052 q^{27} -5.25931 q^{28} +9.19377 q^{29} -0.120718 q^{30} -1.00000 q^{31} -17.8447 q^{32} +0.919364 q^{33} -12.0229 q^{34} -0.178464 q^{35} -15.4464 q^{36} +1.00000 q^{37} -13.3895 q^{38} +0.672184 q^{39} -1.56720 q^{40} +2.12262 q^{41} +0.676426 q^{42} +10.2905 q^{43} +19.2594 q^{44} -0.524145 q^{45} +4.79266 q^{46} -7.05947 q^{47} +3.29932 q^{48} +1.00000 q^{49} +13.3857 q^{50} +1.12030 q^{51} +14.0814 q^{52} +2.62856 q^{53} +4.01592 q^{54} +0.653532 q^{55} +8.78160 q^{56} +1.24764 q^{57} -24.7709 q^{58} +14.2418 q^{59} +0.235642 q^{60} +4.95002 q^{61} +2.69431 q^{62} +2.93697 q^{63} +21.7957 q^{64} +0.477824 q^{65} -2.47705 q^{66} +5.53909 q^{67} +23.4687 q^{68} -0.446582 q^{69} +0.480839 q^{70} +6.97179 q^{71} +25.7913 q^{72} -9.60295 q^{73} -2.69431 q^{74} -1.24729 q^{75} +26.1363 q^{76} -3.66197 q^{77} -1.81107 q^{78} -10.3299 q^{79} +2.34533 q^{80} +8.43671 q^{81} -5.71899 q^{82} +0.0564772 q^{83} -1.32039 q^{84} +0.796366 q^{85} -27.7257 q^{86} +2.30816 q^{87} -32.1580 q^{88} -10.2245 q^{89} +1.41221 q^{90} -2.67742 q^{91} -9.35530 q^{92} -0.251057 q^{93} +19.0204 q^{94} +0.886885 q^{95} -4.48004 q^{96} -16.4804 q^{97} -2.69431 q^{98} -10.7551 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 71 q + 6 q^{2} + 8 q^{3} + 78 q^{4} + 5 q^{6} - 71 q^{7} + 18 q^{8} + 87 q^{9}+O(q^{10}) $ 71 * q + 6 * q^2 + 8 * q^3 + 78 * q^4 + 5 * q^6 - 71 * q^7 + 18 * q^8 + 87 * q^9	\( 71 q + 6 q^{2} + 8 q^{3} + 78 q^{4} + 5 q^{6} - 71 q^{7} + 18 q^{8} + 87 q^{9} + 4 q^{10} + 57 q^{11} + 21 q^{12} - 20 q^{13} - 6 q^{14} + 22 q^{15} + 88 q^{16} - 19 q^{17} + q^{18} + 23 q^{19} + 25 q^{20} - 8 q^{21} + 18 q^{22} + 34 q^{23} + 15 q^{24} + 81 q^{25} - 13 q^{26} + 20 q^{27} - 78 q^{28} + 16 q^{29} + 6 q^{30} - 71 q^{31} + 47 q^{32} - 16 q^{33} + 32 q^{34} + 125 q^{36} + 71 q^{37} + 13 q^{38} + 30 q^{39} + 31 q^{40} + 17 q^{41} - 5 q^{42} + 38 q^{43} + 80 q^{44} - q^{45} + 26 q^{46} + 32 q^{47} + 61 q^{48} + 71 q^{49} + 47 q^{50} + 73 q^{51} - 23 q^{52} + 31 q^{53} + 47 q^{54} + 11 q^{55} - 18 q^{56} + 17 q^{57} - 2 q^{58} + 97 q^{59} + 103 q^{60} - q^{61} - 6 q^{62} - 87 q^{63} + 100 q^{64} + 46 q^{65} + 43 q^{66} + 75 q^{67} - 43 q^{68} + 10 q^{69} - 4 q^{70} + 131 q^{71} - 11 q^{72} - 15 q^{73} + 6 q^{74} + 76 q^{75} + 41 q^{76} - 57 q^{77} + 89 q^{78} + 8 q^{79} + 10 q^{80} + 171 q^{81} + 14 q^{82} + 18 q^{83} - 21 q^{84} + 47 q^{85} + 90 q^{86} - 59 q^{87} + 13 q^{88} + 18 q^{89} + 69 q^{90} + 20 q^{91} + 110 q^{92} - 8 q^{93} + 39 q^{94} + 72 q^{95} + 100 q^{96} + 23 q^{97} + 6 q^{98} + 168 q^{99}+O(q^{100}) \) 71 * q + 6 * q^2 + 8 * q^3 + 78 * q^4 + 5 * q^6 - 71 * q^7 + 18 * q^8 + 87 * q^9 + 4 * q^10 + 57 * q^11 + 21 * q^12 - 20 * q^13 - 6 * q^14 + 22 * q^15 + 88 * q^16 - 19 * q^17 + q^18 + 23 * q^19 + 25 * q^20 - 8 * q^21 + 18 * q^22 + 34 * q^23 + 15 * q^24 + 81 * q^25 - 13 * q^26 + 20 * q^27 - 78 * q^28 + 16 * q^29 + 6 * q^30 - 71 * q^31 + 47 * q^32 - 16 * q^33 + 32 * q^34 + 125 * q^36 + 71 * q^37 + 13 * q^38 + 30 * q^39 + 31 * q^40 + 17 * q^41 - 5 * q^42 + 38 * q^43 + 80 * q^44 - q^45 + 26 * q^46 + 32 * q^47 + 61 * q^48 + 71 * q^49 + 47 * q^50 + 73 * q^51 - 23 * q^52 + 31 * q^53 + 47 * q^54 + 11 * q^55 - 18 * q^56 + 17 * q^57 - 2 * q^58 + 97 * q^59 + 103 * q^60 - q^61 - 6 * q^62 - 87 * q^63 + 100 * q^64 + 46 * q^65 + 43 * q^66 + 75 * q^67 - 43 * q^68 + 10 * q^69 - 4 * q^70 + 131 * q^71 - 11 * q^72 - 15 * q^73 + 6 * q^74 + 76 * q^75 + 41 * q^76 - 57 * q^77 + 89 * q^78 + 8 * q^79 + 10 * q^80 + 171 * q^81 + 14 * q^82 + 18 * q^83 - 21 * q^84 + 47 * q^85 + 90 * q^86 - 59 * q^87 + 13 * q^88 + 18 * q^89 + 69 * q^90 + 20 * q^91 + 110 * q^92 - 8 * q^93 + 39 * q^94 + 72 * q^95 + 100 * q^96 + 23 * q^97 + 6 * q^98 + 168 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.69431	−1.90517	−0.952583	−	0.304280i	$-0.901584\pi$
$2$	−2.69431	−1.90517	−0.952583	+	0.304280i	$0.901584\pi$
$3$	0.251057	0.144948	0.0724739	−	0.997370i	$-0.476911\pi$
$3$	0.251057	0.144948	0.0724739	+	0.997370i	$0.476911\pi$
$4$	5.25931	2.62966
$5$	0.178464	0.0798117	0.0399059	−	0.999203i	$-0.487294\pi$
$5$	0.178464	0.0798117	0.0399059	+	0.999203i	$0.487294\pi$
$6$	−0.676426	−0.276150
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−8.78160	−3.10476
$9$	−2.93697	−0.978990
$10$	−0.480839	−0.152055
$11$	3.66197	1.10413	0.552063	−	0.833802i	$-0.313841\pi$
$11$	3.66197	1.10413	0.552063	+	0.833802i	$0.313841\pi$
$12$	1.32039	0.381163
$13$	2.67742	0.742582	0.371291	−	0.928517i	$-0.378915\pi$
$13$	2.67742	0.742582	0.371291	+	0.928517i	$0.378915\pi$
$14$	2.69431	0.720085
$15$	0.0448048	0.0115685
$16$	13.1417	3.28543
$17$	4.46232	1.08227	0.541136	−	0.840935i	$-0.317994\pi$
$17$	4.46232	1.08227	0.541136	+	0.840935i	$0.317994\pi$
$18$	7.91311	1.86514
$19$	4.96953	1.14009	0.570045	−	0.821614i	$-0.306926\pi$
$19$	4.96953	1.14009	0.570045	+	0.821614i	$0.306926\pi$
$20$	0.938600	0.209877
$21$	−0.251057	−0.0547851
$22$	−9.86649	−2.10354
$23$	−1.77881	−0.370907	−0.185454	−	0.982653i	$-0.559375\pi$
$23$	−1.77881	−0.370907	−0.185454	+	0.982653i	$0.559375\pi$
$24$	−2.20468	−0.450029
$25$	−4.96815	−0.993630
$26$	−7.21379	−1.41474
$27$	−1.49052	−0.286850
$28$	−5.25931	−0.993916
$29$	9.19377	1.70724	0.853621	−	0.520895i	$-0.174402\pi$
$29$	9.19377	1.70724	0.853621	+	0.520895i	$0.174402\pi$
$30$	−0.120718	−0.0220400
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−17.8447	−3.15453
$33$	0.919364	0.160041
$34$	−12.0229	−2.06191
$35$	−0.178464	−0.0301660
$36$	−15.4464	−2.57441
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	−13.3895	−2.17206
$39$	0.672184	0.107636
$40$	−1.56720	−0.247797
$41$	2.12262	0.331497	0.165748	−	0.986168i	$-0.446996\pi$
$41$	2.12262	0.331497	0.165748	+	0.986168i	$0.446996\pi$
$42$	0.676426	0.104375
$43$	10.2905	1.56928	0.784641	−	0.619951i	$-0.212847\pi$
$43$	10.2905	1.56928	0.784641	+	0.619951i	$0.212847\pi$
$44$	19.2594	2.90347
$45$	−0.524145	−0.0781349
$46$	4.79266	0.706639
$47$	−7.05947	−1.02973	−0.514864	−	0.857272i	$-0.672158\pi$
$47$	−7.05947	−1.02973	−0.514864	+	0.857272i	$0.672158\pi$
$48$	3.29932	0.476216
$49$	1.00000	0.142857
$50$	13.3857	1.89303
$51$	1.12030	0.156873
$52$	14.0814	1.95273
$53$	2.62856	0.361060	0.180530	−	0.983569i	$-0.442219\pi$
$53$	2.62856	0.361060	0.180530	+	0.983569i	$0.442219\pi$
$54$	4.01592	0.546497
$55$	0.653532	0.0881222
$56$	8.78160	1.17349
$57$	1.24764	0.165253
$58$	−24.7709	−3.25258
$59$	14.2418	1.85413	0.927063	−	0.374905i	$-0.122325\pi$
$59$	14.2418	1.85413	0.927063	+	0.374905i	$0.122325\pi$
$60$	0.235642	0.0304213
$61$	4.95002	0.633785	0.316893	−	0.948461i	$-0.397361\pi$
$61$	4.95002	0.633785	0.316893	+	0.948461i	$0.397361\pi$
$62$	2.69431	0.342178
$63$	2.93697	0.370023
$64$	21.7957	2.72447
$65$	0.477824	0.0592667
$66$	−2.47705	−0.304904
$67$	5.53909	0.676707	0.338354	−	0.941019i	$-0.390130\pi$
$67$	5.53909	0.676707	0.338354	+	0.941019i	$0.390130\pi$
$68$	23.4687	2.84600
$69$	−0.446582	−0.0537622
$70$	0.480839	0.0574712
$71$	6.97179	0.827400	0.413700	−	0.910413i	$-0.364236\pi$
$71$	6.97179	0.827400	0.413700	+	0.910413i	$0.364236\pi$
$72$	25.7913	3.03953
$73$	−9.60295	−1.12394	−0.561970	−	0.827158i	$-0.689956\pi$
$73$	−9.60295	−1.12394	−0.561970	+	0.827158i	$0.689956\pi$
$74$	−2.69431	−0.313207
$75$	−1.24729	−0.144025
$76$	26.1363	2.99804
$77$	−3.66197	−0.417320
$78$	−1.81107	−0.205064
$79$	−10.3299	−1.16221	−0.581105	−	0.813829i	$-0.697379\pi$
$79$	−10.3299	−1.16221	−0.581105	+	0.813829i	$0.697379\pi$
$80$	2.34533	0.262216
$81$	8.43671	0.937412
$82$	−5.71899	−0.631556
$83$	0.0564772	0.00619918	0.00309959	−	0.999995i	$-0.499013\pi$
$83$	0.0564772	0.00619918	0.00309959	+	0.999995i	$0.499013\pi$
$84$	−1.32039	−0.144066
$85$	0.796366	0.0863780
$86$	−27.7257	−2.98974
$87$	2.30816	0.247461
$88$	−32.1580	−3.42805
$89$	−10.2245	−1.08379	−0.541897	−	0.840445i	$-0.682294\pi$
$89$	−10.2245	−1.08379	−0.541897	+	0.840445i	$0.682294\pi$
$90$	1.41221	0.148860
$91$	−2.67742	−0.280670
$92$	−9.35530	−0.975358
$93$	−0.251057	−0.0260334
$94$	19.0204	1.96180
$95$	0.886885	0.0909925
$96$	−4.48004	−0.457242
$97$	−16.4804	−1.67333	−0.836665	−	0.547714i	$-0.815498\pi$
$97$	−16.4804	−1.67333	−0.836665	+	0.547714i	$0.815498\pi$
$98$	−2.69431	−0.272166
$99$	−10.7551	−1.08093
$100$	−26.1290	−2.61290
$101$	11.7195	1.16614	0.583069	−	0.812423i	$-0.301852\pi$
$101$	11.7195	1.16614	0.583069	+	0.812423i	$0.301852\pi$
$102$	−3.01843	−0.298869
$103$	1.40605	0.138542	0.0692709	−	0.997598i	$-0.477933\pi$
$103$	1.40605	0.138542	0.0692709	+	0.997598i	$0.477933\pi$
$104$	−23.5120	−2.30554
$105$	−0.0448048	−0.00437250
$106$	−7.08216	−0.687880
$107$	−18.2428	−1.76360	−0.881800	−	0.471623i	$-0.843668\pi$
$107$	−18.2428	−1.76360	−0.881800	+	0.471623i	$0.843668\pi$
$108$	−7.83910	−0.754318
$109$	10.0749	0.965003	0.482501	−	0.875895i	$-0.339728\pi$
$109$	10.0749	0.965003	0.482501	+	0.875895i	$0.339728\pi$
$110$	−1.76082	−0.167887
$111$	0.251057	0.0238293
$112$	−13.1417	−1.24178
$113$	−4.56902	−0.429817	−0.214908	−	0.976634i	$-0.568945\pi$
$113$	−4.56902	−0.429817	−0.214908	+	0.976634i	$0.568945\pi$
$114$	−3.36152	−0.314835
$115$	−0.317454	−0.0296027
$116$	48.3529	4.48946
$117$	−7.86349	−0.726980
$118$	−38.3719	−3.53242
$119$	−4.46232	−0.409061
$120$	−0.393457	−0.0359176
$121$	2.41004	0.219095
$122$	−13.3369	−1.20747
$123$	0.532898	0.0480498
$124$	−5.25931	−0.472300
$125$	−1.77896	−0.159115
$126$	−7.91311	−0.704956
$127$	8.46035	0.750734	0.375367	−	0.926876i	$-0.377517\pi$
$127$	8.46035	0.750734	0.375367	+	0.926876i	$0.377517\pi$
$128$	−23.0351	−2.03603
$129$	2.58349	0.227464
$130$	−1.28741	−0.112913
$131$	12.5494	1.09645	0.548223	−	0.836332i	$-0.315304\pi$
$131$	12.5494	1.09645	0.548223	+	0.836332i	$0.315304\pi$
$132$	4.83522	0.420852
$133$	−4.96953	−0.430913
$134$	−14.9240	−1.28924
$135$	−0.266005	−0.0228940
$136$	−39.1863	−3.36020
$137$	7.41292	0.633328	0.316664	−	0.948538i	$-0.397437\pi$
$137$	7.41292	0.633328	0.316664	+	0.948538i	$0.397437\pi$
$138$	1.20323	0.102426
$139$	−14.8282	−1.25771	−0.628857	−	0.777521i	$-0.716477\pi$
$139$	−14.8282	−1.25771	−0.628857	+	0.777521i	$0.716477\pi$
$140$	−0.938600	−0.0793262
$141$	−1.77233	−0.149257
$142$	−18.7842	−1.57633
$143$	9.80463	0.819904
$144$	−38.5969	−3.21641
$145$	1.64076	0.136258
$146$	25.8733	2.14129
$147$	0.251057	0.0207068
$148$	5.25931	0.432313
$149$	17.0936	1.40036	0.700179	−	0.713967i	$-0.253103\pi$
$149$	17.0936	1.40036	0.700179	+	0.713967i	$0.253103\pi$
$150$	3.36058	0.274391
$151$	3.87739	0.315538	0.157769	−	0.987476i	$-0.449570\pi$
$151$	3.87739	0.315538	0.157769	+	0.987476i	$0.449570\pi$
$152$	−43.6404	−3.53971
$153$	−13.1057	−1.05953
$154$	9.86649	0.795064
$155$	−0.178464	−0.0143346
$156$	3.53523	0.283045
$157$	−1.40374	−0.112031	−0.0560154	−	0.998430i	$-0.517840\pi$
$157$	−1.40374	−0.112031	−0.0560154	+	0.998430i	$0.517840\pi$
$158$	27.8321	2.21420
$159$	0.659918	0.0523349
$160$	−3.18465	−0.251768
$161$	1.77881	0.140190
$162$	−22.7311	−1.78592
$163$	−12.9645	−1.01546	−0.507730	−	0.861516i	$-0.669515\pi$
$163$	−12.9645	−1.01546	−0.507730	+	0.861516i	$0.669515\pi$
$164$	11.1635	0.871723
$165$	0.164074	0.0127731
$166$	−0.152167	−0.0118105
$167$	1.63568	0.126573	0.0632863	−	0.997995i	$-0.479842\pi$
$167$	1.63568	0.126573	0.0632863	+	0.997995i	$0.479842\pi$
$168$	2.20468	0.170095
$169$	−5.83144	−0.448572
$170$	−2.14566	−0.164564
$171$	−14.5954	−1.11614
$172$	54.1208	4.12667
$173$	8.28553	0.629937	0.314968	−	0.949102i	$-0.398006\pi$
$173$	8.28553	0.629937	0.314968	+	0.949102i	$0.398006\pi$
$174$	−6.21890	−0.471454
$175$	4.96815	0.375557
$176$	48.1246	3.62753
$177$	3.57551	0.268752
$178$	27.5480	2.06481
$179$	−3.26741	−0.244218	−0.122109	−	0.992517i	$-0.538966\pi$
$179$	−3.26741	−0.244218	−0.122109	+	0.992517i	$0.538966\pi$
$180$	−2.75664	−0.205468
$181$	−14.7212	−1.09421	−0.547107	−	0.837062i	$-0.684271\pi$
$181$	−14.7212	−1.09421	−0.547107	+	0.837062i	$0.684271\pi$
$182$	7.21379	0.534722
$183$	1.24274	0.0918658
$184$	15.6208	1.15158
$185$	0.178464	0.0131210
$186$	0.676426	0.0495979
$187$	16.3409	1.19497
$188$	−37.1279	−2.70783
$189$	1.49052	0.108419
$190$	−2.38954	−0.173356
$191$	−21.3203	−1.54268	−0.771341	−	0.636422i	$-0.780414\pi$
$191$	−21.3203	−1.54268	−0.771341	+	0.636422i	$0.780414\pi$
$192$	5.47197	0.394905
$193$	−0.222014	−0.0159809	−0.00799044	−	0.999968i	$-0.502543\pi$
$193$	−0.222014	−0.0159809	−0.00799044	+	0.999968i	$0.502543\pi$
$194$	44.4033	3.18797
$195$	0.119961	0.00859059
$196$	5.25931	0.375665
$197$	23.4663	1.67191	0.835954	−	0.548800i	$-0.184915\pi$
$197$	23.4663	1.67191	0.835954	+	0.548800i	$0.184915\pi$
$198$	28.9776	2.05935
$199$	14.9999	1.06331	0.531657	−	0.846960i	$-0.321570\pi$
$199$	14.9999	1.06331	0.531657	+	0.846960i	$0.321570\pi$
$200$	43.6283	3.08499
$201$	1.39063	0.0980872
$202$	−31.5761	−2.22169
$203$	−9.19377	−0.645276
$204$	5.89199	0.412522
$205$	0.378812	0.0264573
$206$	−3.78833	−0.263945
$207$	5.22431	0.363114
$208$	35.1859	2.43970
$209$	18.1983	1.25880
$210$	0.120718	0.00833033
$211$	−10.2685	−0.706915	−0.353457	−	0.935451i	$-0.614994\pi$
$211$	−10.2685	−0.706915	−0.353457	+	0.935451i	$0.614994\pi$
$212$	13.8244	0.949465
$213$	1.75032	0.119930
$214$	49.1518	3.35995
$215$	1.83648	0.125247
$216$	13.0891	0.890602
$217$	1.00000	0.0678844
$218$	−27.1450	−1.83849
$219$	−2.41089	−0.162913
$220$	3.43713	0.231731
$221$	11.9475	0.803676
$222$	−0.676426	−0.0453987
$223$	9.76360	0.653819	0.326909	−	0.945056i	$-0.393993\pi$
$223$	9.76360	0.653819	0.326909	+	0.945056i	$0.393993\pi$
$224$	17.8447	1.19230
$225$	14.5913	0.972754
$226$	12.3104	0.818872
$227$	−12.3276	−0.818210	−0.409105	−	0.912487i	$-0.634159\pi$
$227$	−12.3276	−0.818210	−0.409105	+	0.912487i	$0.634159\pi$
$228$	6.56171	0.434560
$229$	−13.3595	−0.882818	−0.441409	−	0.897306i	$-0.645521\pi$
$229$	−13.3595	−0.882818	−0.441409	+	0.897306i	$0.645521\pi$
$230$	0.855320	0.0563981
$231$	−0.919364	−0.0604897
$232$	−80.7360	−5.30058
$233$	27.3040	1.78874	0.894371	−	0.447325i	$-0.147623\pi$
$233$	27.3040	1.78874	0.894371	+	0.447325i	$0.147623\pi$
$234$	21.1867	1.38502
$235$	−1.25986	−0.0821844
$236$	74.9021	4.87571
$237$	−2.59341	−0.168460
$238$	12.0229	0.779328
$239$	−14.1941	−0.918139	−0.459069	−	0.888400i	$-0.651817\pi$
$239$	−14.1941	−0.918139	−0.459069	+	0.888400i	$0.651817\pi$
$240$	0.588812	0.0380076
$241$	−1.35010	−0.0869678	−0.0434839	−	0.999054i	$-0.513846\pi$
$241$	−1.35010	−0.0869678	−0.0434839	+	0.999054i	$0.513846\pi$
$242$	−6.49340	−0.417411
$243$	6.58965	0.422726
$244$	26.0337	1.66664
$245$	0.178464	0.0114017
$246$	−1.43579	−0.0915427
$247$	13.3055	0.846609
$248$	8.78160	0.557632
$249$	0.0141790	0.000898557 0
$250$	4.79307	0.303141
$251$	−13.8546	−0.874492	−0.437246	−	0.899342i	$-0.644046\pi$
$251$	−13.8546	−0.874492	−0.437246	+	0.899342i	$0.644046\pi$
$252$	15.4464	0.973034
$253$	−6.51394	−0.409528
$254$	−22.7948	−1.43027
$255$	0.199933	0.0125203
$256$	18.4721	1.15451
$257$	23.5301	1.46777	0.733883	−	0.679276i	$-0.237706\pi$
$257$	23.5301	1.46777	0.733883	+	0.679276i	$0.237706\pi$
$258$	−6.96074	−0.433356
$259$	−1.00000	−0.0621370
$260$	2.51302	0.155851
$261$	−27.0018	−1.67137
$262$	−33.8120	−2.08891
$263$	−3.46948	−0.213937	−0.106969	−	0.994262i	$-0.534114\pi$
$263$	−3.46948	−0.213937	−0.106969	+	0.994262i	$0.534114\pi$
$264$	−8.07348	−0.496888
$265$	0.469105	0.0288169
$266$	13.3895	0.820961
$267$	−2.56693	−0.157094
$268$	29.1318	1.77951
$269$	−20.8782	−1.27297	−0.636483	−	0.771291i	$-0.719611\pi$
$269$	−20.8782	−1.27297	−0.636483	+	0.771291i	$0.719611\pi$
$270$	0.716699	0.0436169
$271$	31.9346	1.93989	0.969946	−	0.243321i	$-0.0782369\pi$
$271$	31.9346	1.93989	0.969946	+	0.243321i	$0.0782369\pi$
$272$	58.6426	3.55573
$273$	−0.672184	−0.0406824
$274$	−19.9727	−1.20660
$275$	−18.1932	−1.09709
$276$	−2.34871	−0.141376
$277$	−31.4435	−1.88926	−0.944630	−	0.328139i	$-0.893579\pi$
$277$	−31.4435	−1.88926	−0.944630	+	0.328139i	$0.893579\pi$
$278$	39.9519	2.39615
$279$	2.93697	0.175832
$280$	1.56720	0.0936583
$281$	−6.49411	−0.387406	−0.193703	−	0.981060i	$-0.562050\pi$
$281$	−6.49411	−0.387406	−0.193703	+	0.981060i	$0.562050\pi$
$282$	4.77520	0.284359
$283$	14.8968	0.885523	0.442762	−	0.896639i	$-0.353999\pi$
$283$	14.8968	0.885523	0.442762	+	0.896639i	$0.353999\pi$
$284$	36.6668	2.17578
$285$	0.222659	0.0131892
$286$	−26.4167	−1.56205
$287$	−2.12262	−0.125294
$288$	52.4094	3.08825
$289$	2.91233	0.171314
$290$	−4.42072	−0.259594
$291$	−4.13752	−0.242546
$292$	−50.5049	−2.95558
$293$	5.35259	0.312702	0.156351	−	0.987702i	$-0.450027\pi$
$293$	5.35259	0.312702	0.156351	+	0.987702i	$0.450027\pi$
$294$	−0.676426	−0.0394499
$295$	2.54166	0.147981
$296$	−8.78160	−0.510420
$297$	−5.45824	−0.316719
$298$	−46.0553	−2.66791
$299$	−4.76261	−0.275429
$300$	−6.55988	−0.378735
$301$	−10.2905	−0.593133
$302$	−10.4469	−0.601151
$303$	2.94227	0.169029
$304$	65.3083	3.74569
$305$	0.883403	0.0505835
$306$	35.3109	2.01859
$307$	9.86897	0.563252	0.281626	−	0.959524i	$-0.409126\pi$
$307$	9.86897	0.563252	0.281626	+	0.959524i	$0.409126\pi$
$308$	−19.2594	−1.09741
$309$	0.352998	0.0200813
$310$	0.480839	0.0273098
$311$	8.89263	0.504255	0.252127	−	0.967694i	$-0.418870\pi$
$311$	8.89263	0.504255	0.252127	+	0.967694i	$0.418870\pi$
$312$	−5.90285	−0.334183
$313$	3.89005	0.219879	0.109939	−	0.993938i	$-0.464934\pi$
$313$	3.89005	0.219879	0.109939	+	0.993938i	$0.464934\pi$
$314$	3.78211	0.213437
$315$	0.524145	0.0295322
$316$	−54.3284	−3.05621
$317$	−2.84496	−0.159789	−0.0798945	−	0.996803i	$-0.525458\pi$
$317$	−2.84496	−0.159789	−0.0798945	+	0.996803i	$0.525458\pi$
$318$	−1.77803	−0.0997067
$319$	33.6673	1.88501
$320$	3.88976	0.217444
$321$	−4.57999	−0.255630
$322$	−4.79266	−0.267085
$323$	22.1757	1.23389
$324$	44.3713	2.46507
$325$	−13.3018	−0.737852
$326$	34.9304	1.93462
$327$	2.52938	0.139875
$328$	−18.6400	−1.02922
$329$	7.05947	0.389201
$330$	−0.442066	−0.0243349
$331$	−3.23889	−0.178025	−0.0890127	−	0.996030i	$-0.528371\pi$
$331$	−3.23889	−0.178025	−0.0890127	+	0.996030i	$0.528371\pi$
$332$	0.297031	0.0163017
$333$	−2.93697	−0.160945
$334$	−4.40703	−0.241142
$335$	0.988530	0.0540092
$336$	−3.29932	−0.179993
$337$	12.8106	0.697839	0.348920	−	0.937153i	$-0.386549\pi$
$337$	12.8106	0.697839	0.348920	+	0.937153i	$0.386549\pi$
$338$	15.7117	0.854604
$339$	−1.14708	−0.0623010
$340$	4.18834	0.227144
$341$	−3.66197	−0.198307
$342$	39.3245	2.12642
$343$	−1.00000	−0.0539949
$344$	−90.3667	−4.87225
$345$	−0.0796991	−0.00429085
$346$	−22.3238	−1.20013
$347$	2.50700	0.134583	0.0672915	−	0.997733i	$-0.478564\pi$
$347$	2.50700	0.134583	0.0672915	+	0.997733i	$0.478564\pi$
$348$	12.1393	0.650737
$349$	−4.11264	−0.220144	−0.110072	−	0.993924i	$-0.535108\pi$
$349$	−4.11264	−0.220144	−0.110072	+	0.993924i	$0.535108\pi$
$350$	−13.3857	−0.715498
$351$	−3.99074	−0.213010
$352$	−65.3468	−3.48300
$353$	16.8364	0.896112	0.448056	−	0.894005i	$-0.352116\pi$
$353$	16.8364	0.896112	0.448056	+	0.894005i	$0.352116\pi$
$354$	−9.63353	−0.512016
$355$	1.24422	0.0660362
$356$	−53.7738	−2.85001
$357$	−1.12030	−0.0592924
$358$	8.80342	0.465275
$359$	8.63293	0.455628	0.227814	−	0.973705i	$-0.426842\pi$
$359$	8.63293	0.455628	0.227814	+	0.973705i	$0.426842\pi$
$360$	4.60283	0.242590
$361$	5.69626	0.299803
$362$	39.6634	2.08466
$363$	0.605057	0.0317573
$364$	−14.0814	−0.738064
$365$	−1.71379	−0.0897036
$366$	−3.34832	−0.175019
$367$	23.5229	1.22789	0.613944	−	0.789350i	$-0.289582\pi$
$367$	23.5229	1.22789	0.613944	+	0.789350i	$0.289582\pi$
$368$	−23.3766	−1.21859
$369$	−6.23406	−0.324532
$370$	−0.480839	−0.0249976
$371$	−2.62856	−0.136468
$372$	−1.32039	−0.0684589
$373$	−22.0977	−1.14418	−0.572088	−	0.820192i	$-0.693867\pi$
$373$	−22.0977	−1.14418	−0.572088	+	0.820192i	$0.693867\pi$
$374$	−44.0275	−2.27661
$375$	−0.446621	−0.0230634
$376$	61.9934	3.19706
$377$	24.6156	1.26777
$378$	−4.01592	−0.206557
$379$	7.65737	0.393332	0.196666	−	0.980470i	$-0.436988\pi$
$379$	7.65737	0.393332	0.196666	+	0.980470i	$0.436988\pi$
$380$	4.66440	0.239279
$381$	2.12403	0.108817
$382$	57.4435	2.93907
$383$	−3.82347	−0.195370	−0.0976852	−	0.995217i	$-0.531144\pi$
$383$	−3.82347	−0.195370	−0.0976852	+	0.995217i	$0.531144\pi$
$384$	−5.78311	−0.295118
$385$	−0.653532	−0.0333071
$386$	0.598174	0.0304462
$387$	−30.2228	−1.53631
$388$	−86.6755	−4.40028
$389$	23.4442	1.18867	0.594334	−	0.804218i	$-0.297416\pi$
$389$	23.4442	1.18867	0.594334	+	0.804218i	$0.297416\pi$
$390$	−0.323212	−0.0163665
$391$	−7.93762	−0.401423
$392$	−8.78160	−0.443538
$393$	3.15062	0.158928
$394$	−63.2256	−3.18526
$395$	−1.84353	−0.0927580
$396$	−56.5644	−2.84247
$397$	−20.2982	−1.01874	−0.509368	−	0.860549i	$-0.670121\pi$
$397$	−20.2982	−1.01874	−0.509368	+	0.860549i	$0.670121\pi$
$398$	−40.4143	−2.02579
$399$	−1.24764	−0.0624599
$400$	−65.2901	−3.26450
$401$	−25.3918	−1.26801	−0.634003	−	0.773330i	$-0.718589\pi$
$401$	−25.3918	−1.26801	−0.634003	+	0.773330i	$0.718589\pi$
$402$	−3.74678	−0.186872
$403$	−2.67742	−0.133372
$404$	61.6367	3.06654
$405$	1.50565	0.0748165
$406$	24.7709	1.22936
$407$	3.66197	0.181517
$408$	−9.83800	−0.487054
$409$	7.86515	0.388907	0.194453	−	0.980912i	$-0.437707\pi$
$409$	7.86515	0.388907	0.194453	+	0.980912i	$0.437707\pi$
$410$	−1.02064	−0.0504056
$411$	1.86107	0.0917996
$412$	7.39484	0.364317
$413$	−14.2418	−0.700794
$414$	−14.0759	−0.691793
$415$	0.0100792	0.000494767 0
$416$	−47.7777	−2.34249
$417$	−3.72273	−0.182303
$418$	−49.0319	−2.39823
$419$	14.7585	0.721001	0.360500	−	0.932759i	$-0.382606\pi$
$419$	14.7585	0.721001	0.360500	+	0.932759i	$0.382606\pi$
$420$	−0.235642	−0.0114982
$421$	−28.8854	−1.40779	−0.703894	−	0.710305i	$-0.748557\pi$
$421$	−28.8854	−1.40779	−0.703894	+	0.710305i	$0.748557\pi$
$422$	27.6666	1.34679
$423$	20.7334	1.00809
$424$	−23.0830	−1.12101
$425$	−22.1695	−1.07538
$426$	−4.71590	−0.228486
$427$	−4.95002	−0.239548
$428$	−95.9447	−4.63766
$429$	2.46152	0.118843
$430$	−4.94806	−0.238616
$431$	32.6594	1.57315	0.786575	−	0.617495i	$-0.211852\pi$
$431$	32.6594	1.57315	0.786575	+	0.617495i	$0.211852\pi$
$432$	−19.5880	−0.942427
$433$	14.3364	0.688966	0.344483	−	0.938793i	$-0.388054\pi$
$433$	14.3364	0.688966	0.344483	+	0.938793i	$0.388054\pi$
$434$	−2.69431	−0.129331
$435$	0.411925	0.0197503
$436$	52.9872	2.53763
$437$	−8.83985	−0.422867
$438$	6.49568	0.310376
$439$	−17.3154	−0.826421	−0.413211	−	0.910635i	$-0.635593\pi$
$439$	−17.3154	−0.826421	−0.413211	+	0.910635i	$0.635593\pi$
$440$	−5.73905	−0.273599
$441$	−2.93697	−0.139856
$442$	−32.1903	−1.53114
$443$	−17.4383	−0.828520	−0.414260	−	0.910159i	$-0.635960\pi$
$443$	−17.4383	−0.828520	−0.414260	+	0.910159i	$0.635960\pi$
$444$	1.32039	0.0626628
$445$	−1.82471	−0.0864995
$446$	−26.3062	−1.24563
$447$	4.29146	0.202979
$448$	−21.7957	−1.02975
$449$	−3.82243	−0.180392	−0.0901959	−	0.995924i	$-0.528749\pi$
$449$	−3.82243	−0.180392	−0.0901959	+	0.995924i	$0.528749\pi$
$450$	−39.3135	−1.85326
$451$	7.77296	0.366014
$452$	−24.0299	−1.13027
$453$	0.973446	0.0457365
$454$	33.2143	1.55883
$455$	−0.477824	−0.0224007
$456$	−10.9562	−0.513073
$457$	−2.76990	−0.129571	−0.0647853	−	0.997899i	$-0.520636\pi$
$457$	−2.76990	−0.129571	−0.0647853	+	0.997899i	$0.520636\pi$
$458$	35.9945	1.68191
$459$	−6.65117	−0.310450
$460$	−1.66959	−0.0778450
$461$	−3.98753	−0.185718	−0.0928589	−	0.995679i	$-0.529601\pi$
$461$	−3.98753	−0.185718	−0.0928589	+	0.995679i	$0.529601\pi$
$462$	2.47705	0.115243
$463$	−12.7326	−0.591734	−0.295867	−	0.955229i	$-0.595608\pi$
$463$	−12.7326	−0.591734	−0.295867	+	0.955229i	$0.595608\pi$
$464$	120.822	5.60902
$465$	−0.0448048	−0.00207777
$466$	−73.5654	−3.40785
$467$	−3.02487	−0.139974	−0.0699872	−	0.997548i	$-0.522296\pi$
$467$	−3.02487	−0.139974	−0.0699872	+	0.997548i	$0.522296\pi$
$468$	−41.3566	−1.91171
$469$	−5.53909	−0.255771
$470$	3.39446	0.156575
$471$	−0.352419	−0.0162386
$472$	−125.066	−5.75662
$473$	37.6834	1.73268
$474$	6.98744	0.320944
$475$	−24.6894	−1.13283
$476$	−23.4687	−1.07569
$477$	−7.72000	−0.353475
$478$	38.2433	1.74921
$479$	9.28028	0.424027	0.212013	−	0.977267i	$-0.431998\pi$
$479$	9.28028	0.424027	0.212013	+	0.977267i	$0.431998\pi$
$480$	−0.799528	−0.0364933
$481$	2.67742	0.122080
$482$	3.63760	0.165688
$483$	0.446582	0.0203202
$484$	12.6751	0.576143
$485$	−2.94117	−0.133551
$486$	−17.7546	−0.805363
$487$	−19.2271	−0.871264	−0.435632	−	0.900125i	$-0.643475\pi$
$487$	−19.2271	−0.871264	−0.435632	+	0.900125i	$0.643475\pi$
$488$	−43.4691	−1.96775
$489$	−3.25483	−0.147189
$490$	−0.480839	−0.0217221
$491$	16.8385	0.759910	0.379955	−	0.925005i	$-0.375940\pi$
$491$	16.8385	0.759910	0.379955	+	0.925005i	$0.375940\pi$
$492$	2.80267	0.126354
$493$	41.0256	1.84770
$494$	−35.8492	−1.61293
$495$	−1.91940	−0.0862708
$496$	−13.1417	−0.590081
$497$	−6.97179	−0.312728
$498$	−0.0382026	−0.00171190
$499$	24.6499	1.10348	0.551741	−	0.834015i	$-0.313964\pi$
$499$	24.6499	1.10348	0.551741	+	0.834015i	$0.313964\pi$
$500$	−9.35611	−0.418418
$501$	0.410649	0.0183464
$502$	37.3285	1.66605
$503$	0.925447	0.0412637	0.0206318	−	0.999787i	$-0.493432\pi$
$503$	0.925447	0.0412637	0.0206318	+	0.999787i	$0.493432\pi$
$504$	−25.7913	−1.14884
$505$	2.09152	0.0930715
$506$	17.5506	0.780219
$507$	−1.46402	−0.0650196
$508$	44.4956	1.97417
$509$	41.2831	1.82984	0.914920	−	0.403635i	$-0.132253\pi$
$509$	41.2831	1.82984	0.914920	+	0.403635i	$0.132253\pi$
$510$	−0.538683	−0.0238533
$511$	9.60295	0.424809
$512$	−3.69957	−0.163500
$513$	−7.40718	−0.327035
$514$	−63.3974	−2.79634
$515$	0.250929	0.0110573
$516$	13.5874	0.598152
$517$	−25.8516	−1.13695
$518$	2.69431	0.118381
$519$	2.08014	0.0913080
$520$	−4.19605	−0.184009
$521$	34.5280	1.51270	0.756349	−	0.654168i	$-0.226981\pi$
$521$	34.5280	1.51270	0.756349	+	0.654168i	$0.226981\pi$
$522$	72.7514	3.18424
$523$	16.1455	0.705991	0.352996	−	0.935625i	$-0.385163\pi$
$523$	16.1455	0.705991	0.352996	+	0.935625i	$0.385163\pi$
$524$	66.0012	2.88328
$525$	1.24729	0.0544362
$526$	9.34786	0.407586
$527$	−4.46232	−0.194382
$528$	12.0820	0.525803
$529$	−19.8358	−0.862428
$530$	−1.26391	−0.0549009
$531$	−41.8278	−1.81517
$532$	−26.1363	−1.13315
$533$	5.68313	0.246164
$534$	6.91611	0.299289
$535$	−3.25570	−0.140756
$536$	−48.6420	−2.10102
$537$	−0.820306	−0.0353988
$538$	56.2524	2.42521
$539$	3.66197	0.157732
$540$	−1.39900	−0.0602034
$541$	15.0906	0.648794	0.324397	−	0.945921i	$-0.394839\pi$
$541$	15.0906	0.648794	0.324397	+	0.945921i	$0.394839\pi$
$542$	−86.0419	−3.69581
$543$	−3.69585	−0.158604
$544$	−79.6288	−3.41406
$545$	1.79802	0.0770186
$546$	1.81107	0.0775068
$547$	−32.8676	−1.40532	−0.702659	−	0.711527i	$-0.748004\pi$
$547$	−32.8676	−1.40532	−0.702659	+	0.711527i	$0.748004\pi$
$548$	38.9868	1.66544
$549$	−14.5381	−0.620469
$550$	49.0182	2.09014
$551$	45.6888	1.94641
$552$	3.92170	0.166919
$553$	10.3299	0.439274
$554$	84.7187	3.59935
$555$	0.0448048	0.00190186
$556$	−77.9863	−3.30736
$557$	13.7036	0.580640	0.290320	−	0.956930i	$-0.406238\pi$
$557$	13.7036	0.580640	0.290320	+	0.956930i	$0.406238\pi$
$558$	−7.91311	−0.334989
$559$	27.5519	1.16532
$560$	−2.34533	−0.0991083
$561$	4.10250	0.173208
$562$	17.4971	0.738073
$563$	5.11925	0.215751	0.107875	−	0.994164i	$-0.465595\pi$
$563$	5.11925	0.215751	0.107875	+	0.994164i	$0.465595\pi$
$564$	−9.32122	−0.392494
$565$	−0.815407	−0.0343044
$566$	−40.1366	−1.68707
$567$	−8.43671	−0.354308
$568$	−61.2235	−2.56888
$569$	20.9523	0.878367	0.439184	−	0.898397i	$-0.355268\pi$
$569$	20.9523	0.878367	0.439184	+	0.898397i	$0.355268\pi$
$570$	−0.599912	−0.0251275
$571$	−32.9140	−1.37741	−0.688704	−	0.725042i	$-0.741820\pi$
$571$	−32.9140	−1.37741	−0.688704	+	0.725042i	$0.741820\pi$
$572$	51.5656	2.15606
$573$	−5.35261	−0.223608
$574$	5.71899	0.238706
$575$	8.83739	0.368544
$576$	−64.0134	−2.66723
$577$	−21.5461	−0.896978	−0.448489	−	0.893788i	$-0.648038\pi$
$577$	−21.5461	−0.896978	−0.448489	+	0.893788i	$0.648038\pi$
$578$	−7.84673	−0.326381
$579$	−0.0557381	−0.00231640
$580$	8.62928	0.358311
$581$	−0.0564772	−0.00234307
$582$	11.1478	0.462090
$583$	9.62571	0.398656
$584$	84.3292	3.48957
$585$	−1.40335	−0.0580216
$586$	−14.4216	−0.595749
$587$	−1.32271	−0.0545942	−0.0272971	−	0.999627i	$-0.508690\pi$
$587$	−1.32271	−0.0545942	−0.0272971	+	0.999627i	$0.508690\pi$
$588$	1.32039	0.0544518
$589$	−4.96953	−0.204766
$590$	−6.84802	−0.281928
$591$	5.89139	0.242339
$592$	13.1417	0.540122
$593$	−17.2230	−0.707264	−0.353632	−	0.935385i	$-0.615053\pi$
$593$	−17.2230	−0.707264	−0.353632	+	0.935385i	$0.615053\pi$
$594$	14.7062	0.603402
$595$	−0.796366	−0.0326478
$596$	89.9003	3.68246
$597$	3.76582	0.154125
$598$	12.8320	0.524737
$599$	−9.67484	−0.395303	−0.197652	−	0.980272i	$-0.563331\pi$
$599$	−9.67484	−0.395303	−0.197652	+	0.980272i	$0.563331\pi$
$600$	10.9532	0.447162
$601$	27.5006	1.12177	0.560886	−	0.827893i	$-0.310461\pi$
$601$	27.5006	1.12177	0.560886	+	0.827893i	$0.310461\pi$
$602$	27.7257	1.13002
$603$	−16.2681	−0.662490
$604$	20.3924	0.829755
$605$	0.430106	0.0174863
$606$	−7.92740	−0.322029
$607$	−39.9500	−1.62152	−0.810761	−	0.585377i	$-0.800947\pi$
$607$	−39.9500	−1.62152	−0.810761	+	0.585377i	$0.800947\pi$
$608$	−88.6799	−3.59644
$609$	−2.30816	−0.0935314
$610$	−2.38016	−0.0963699
$611$	−18.9011	−0.764658
$612$	−68.9270	−2.78621
$613$	−12.1820	−0.492026	−0.246013	−	0.969267i	$-0.579121\pi$
$613$	−12.1820	−0.492026	−0.246013	+	0.969267i	$0.579121\pi$
$614$	−26.5901	−1.07309
$615$	0.0951033	0.00383493
$616$	32.1580	1.29568
$617$	9.28903	0.373962	0.186981	−	0.982364i	$-0.440130\pi$
$617$	9.28903	0.373962	0.186981	+	0.982364i	$0.440130\pi$
$618$	−0.951086	−0.0382583
$619$	26.9927	1.08493	0.542465	−	0.840078i	$-0.317491\pi$
$619$	26.9927	1.08493	0.542465	+	0.840078i	$0.317491\pi$
$620$	−0.938600	−0.0376951
$621$	2.65135	0.106395
$622$	−23.9595	−0.960689
$623$	10.2245	0.409636
$624$	8.83366	0.353629
$625$	24.5233	0.980931
$626$	−10.4810	−0.418905
$627$	4.56881	0.182461
$628$	−7.38271	−0.294602
$629$	4.46232	0.177924
$630$	−1.41221	−0.0562638
$631$	23.1780	0.922703	0.461351	−	0.887217i	$-0.347365\pi$
$631$	23.1780	0.922703	0.461351	+	0.887217i	$0.347365\pi$
$632$	90.7134	3.60839
$633$	−2.57799	−0.102466
$634$	7.66522	0.304425
$635$	1.50987	0.0599174
$636$	3.47072	0.137623
$637$	2.67742	0.106083
$638$	−90.7103	−3.59125
$639$	−20.4759	−0.810016
$640$	−4.11094	−0.162499
$641$	17.8433	0.704767	0.352383	−	0.935856i	$-0.385371\pi$
$641$	17.8433	0.704767	0.352383	+	0.935856i	$0.385371\pi$
$642$	12.3399	0.487018
$643$	−16.7184	−0.659308	−0.329654	−	0.944102i	$-0.606932\pi$
$643$	−16.7184	−0.659308	−0.329654	+	0.944102i	$0.606932\pi$
$644$	9.35530	0.368651
$645$	0.461062	0.0181543
$646$	−59.7481	−2.35076
$647$	−24.1380	−0.948962	−0.474481	−	0.880266i	$-0.657364\pi$
$647$	−24.1380	−0.948962	−0.474481	+	0.880266i	$0.657364\pi$
$648$	−74.0877	−2.91044
$649$	52.1531	2.04719
$650$	35.8392	1.40573
$651$	0.251057	0.00983970
$652$	−68.1844	−2.67031
$653$	−47.3406	−1.85258	−0.926291	−	0.376810i	$-0.877021\pi$
$653$	−47.3406	−1.85258	−0.926291	+	0.376810i	$0.877021\pi$
$654$	−6.81494	−0.266485
$655$	2.23962	0.0875093
$656$	27.8948	1.08911
$657$	28.2036	1.10033
$658$	−19.0204	−0.741492
$659$	0.00202863	7.90240e−5 0	3.95120e−5	−	1.00000i	$-0.499987\pi$
$659$	0.00202863	7.90240e−5 0	3.95120e−5		1.00000i	$0.499987\pi$
$660$	0.862915	0.0335889
$661$	19.3540	0.752783	0.376392	−	0.926461i	$-0.377165\pi$
$661$	19.3540	0.752783	0.376392	+	0.926461i	$0.377165\pi$
$662$	8.72657	0.339168
$663$	2.99950	0.116491
$664$	−0.495960	−0.0192470
$665$	−0.886885	−0.0343919
$666$	7.91311	0.306627
$667$	−16.3540	−0.633228
$668$	8.60254	0.332842
$669$	2.45122	0.0947696
$670$	−2.66341	−0.102896
$671$	18.1268	0.699779
$672$	4.48004	0.172821
$673$	7.06381	0.272290	0.136145	−	0.990689i	$-0.456529\pi$
$673$	7.06381	0.272290	0.136145	+	0.990689i	$0.456529\pi$
$674$	−34.5158	−1.32950
$675$	7.40512	0.285023
$676$	−30.6694	−1.17959
$677$	7.80024	0.299788	0.149894	−	0.988702i	$-0.452107\pi$
$677$	7.80024	0.299788	0.149894	+	0.988702i	$0.452107\pi$
$678$	3.09060	0.118694
$679$	16.4804	0.632460
$680$	−6.99337	−0.268183
$681$	−3.09493	−0.118598
$682$	9.86649	0.377807
$683$	29.9541	1.14616	0.573081	−	0.819498i	$-0.305748\pi$
$683$	29.9541	1.14616	0.573081	+	0.819498i	$0.305748\pi$
$684$	−76.7616	−2.93505
$685$	1.32294	0.0505470
$686$	2.69431	0.102869
$687$	−3.35399	−0.127963
$688$	135.235	5.15577
$689$	7.03775	0.268117
$690$	0.214734	0.00817478
$691$	14.2316	0.541394	0.270697	−	0.962665i	$-0.412746\pi$
$691$	14.2316	0.541394	0.270697	+	0.962665i	$0.412746\pi$
$692$	43.5762	1.65652
$693$	10.7551	0.408553
$694$	−6.75464	−0.256403
$695$	−2.64631	−0.100380
$696$	−20.2693	−0.768307
$697$	9.47180	0.358770
$698$	11.0807	0.419411
$699$	6.85485	0.259274
$700$	26.1290	0.987585
$701$	5.38225	0.203285	0.101642	−	0.994821i	$-0.467590\pi$
$701$	5.38225	0.203285	0.101642	+	0.994821i	$0.467590\pi$
$702$	10.7523	0.405819
$703$	4.96953	0.187430
$704$	79.8153	3.00815
$705$	−0.316298	−0.0119125
$706$	−45.3626	−1.70724
$707$	−11.7195	−0.440759
$708$	18.8047	0.706724
$709$	10.4232	0.391451	0.195725	−	0.980659i	$-0.437294\pi$
$709$	10.4232	0.391451	0.195725	+	0.980659i	$0.437294\pi$
$710$	−3.35231	−0.125810
$711$	30.3388	1.13779
$712$	89.7874	3.36492
$713$	1.77881	0.0666169
$714$	3.01843	0.112962
$715$	1.74978	0.0654380
$716$	−17.1843	−0.642209
$717$	−3.56352	−0.133082
$718$	−23.2598	−0.868048
$719$	28.3091	1.05575	0.527875	−	0.849322i	$-0.322989\pi$
$719$	28.3091	1.05575	0.527875	+	0.849322i	$0.322989\pi$
$720$	−6.88817	−0.256707
$721$	−1.40605	−0.0523639
$722$	−15.3475	−0.571175
$723$	−0.338953	−0.0126058
$724$	−77.4231	−2.87741
$725$	−45.6761	−1.69637
$726$	−1.63021	−0.0605029
$727$	4.42240	0.164018	0.0820088	−	0.996632i	$-0.473866\pi$
$727$	4.42240	0.164018	0.0820088	+	0.996632i	$0.473866\pi$
$728$	23.5120	0.871412
$729$	−23.6557	−0.876139
$730$	4.61747	0.170900
$731$	45.9194	1.69839
$732$	6.53594	0.241575
$733$	−26.4098	−0.975469	−0.487735	−	0.872992i	$-0.662177\pi$
$733$	−26.4098	−0.975469	−0.487735	+	0.872992i	$0.662177\pi$
$734$	−63.3781	−2.33933
$735$	0.0448048	0.00165265
$736$	31.7423	1.17004
$737$	20.2840	0.747170
$738$	16.7965	0.618288
$739$	25.9483	0.954522	0.477261	−	0.878762i	$-0.341630\pi$
$739$	25.9483	0.954522	0.477261	+	0.878762i	$0.341630\pi$
$740$	0.938600	0.0345036
$741$	3.34044	0.122714
$742$	7.08216	0.259994
$743$	−16.6151	−0.609547	−0.304774	−	0.952425i	$-0.598581\pi$
$743$	−16.6151	−0.609547	−0.304774	+	0.952425i	$0.598581\pi$
$744$	2.20468	0.0808275
$745$	3.05059	0.111765
$746$	59.5381	2.17985
$747$	−0.165872	−0.00606893
$748$	85.9419	3.14235
$749$	18.2428	0.666578
$750$	1.20333	0.0439396
$751$	20.2353	0.738398	0.369199	−	0.929350i	$-0.379632\pi$
$751$	20.2353	0.738398	0.369199	+	0.929350i	$0.379632\pi$
$752$	−92.7736	−3.38310
$753$	−3.47828	−0.126756
$754$	−66.3220	−2.41530
$755$	0.691977	0.0251836
$756$	7.83910	0.285105
$757$	2.06517	0.0750597	0.0375299	−	0.999296i	$-0.488051\pi$
$757$	2.06517	0.0750597	0.0375299	+	0.999296i	$0.488051\pi$
$758$	−20.6313	−0.749363
$759$	−1.63537	−0.0593602
$760$	−7.78827	−0.282510
$761$	32.0730	1.16265	0.581324	−	0.813672i	$-0.302535\pi$
$761$	32.0730	1.16265	0.581324	+	0.813672i	$0.302535\pi$
$762$	−5.72280	−0.207315
$763$	−10.0749	−0.364737
$764$	−112.130	−4.05672
$765$	−2.33890	−0.0845633
$766$	10.3016	0.372213
$767$	38.1313	1.37684
$768$	4.63756	0.167344
$769$	−16.8162	−0.606408	−0.303204	−	0.952926i	$-0.598056\pi$
$769$	−16.8162	−0.606408	−0.303204	+	0.952926i	$0.598056\pi$
$770$	1.76082	0.0634555
$771$	5.90739	0.212750
$772$	−1.16764	−0.0420242
$773$	−40.4245	−1.45397	−0.726985	−	0.686654i	$-0.759079\pi$
$773$	−40.4245	−1.45397	−0.726985	+	0.686654i	$0.759079\pi$
$774$	81.4296	2.92693
$775$	4.96815	0.178461
$776$	144.724	5.19530
$777$	−0.251057	−0.00900662
$778$	−63.1659	−2.26461
$779$	10.5484	0.377936
$780$	0.630912	0.0225903
$781$	25.5305	0.913553
$782$	21.3864	0.764776
$783$	−13.7035	−0.489723
$784$	13.1417	0.469347
$785$	−0.250518	−0.00894137
$786$	−8.48874	−0.302783
$787$	37.4797	1.33601	0.668004	−	0.744157i	$-0.267149\pi$
$787$	37.4797	1.33601	0.668004	+	0.744157i	$0.267149\pi$
$788$	123.417	4.39654
$789$	−0.871037	−0.0310097
$790$	4.96704	0.176719
$791$	4.56902	0.162456
$792$	94.4470	3.35603
$793$	13.2533	0.470637
$794$	54.6896	1.94086
$795$	0.117772	0.00417694
$796$	78.8890	2.79615
$797$	−10.2577	−0.363346	−0.181673	−	0.983359i	$-0.558151\pi$
$797$	−10.2577	−0.363346	−0.181673	+	0.983359i	$0.558151\pi$
$798$	3.36152	0.118997
$799$	−31.5016	−1.11445
$800$	88.6552	3.13443
$801$	30.0290	1.06102
$802$	68.4134	2.41576
$803$	−35.1657	−1.24097
$804$	7.31374	0.257936
$805$	0.317454	0.0111888
$806$	7.21379	0.254095
$807$	−5.24162	−0.184514
$808$	−102.916	−3.62058
$809$	13.4253	0.472007	0.236004	−	0.971752i	$-0.424162\pi$
$809$	13.4253	0.472007	0.236004	+	0.971752i	$0.424162\pi$
$810$	−4.05670	−0.142538
$811$	2.93197	0.102955	0.0514777	−	0.998674i	$-0.483607\pi$
$811$	2.93197	0.102955	0.0514777	+	0.998674i	$0.483607\pi$
$812$	−48.3529	−1.69685
$813$	8.01742	0.281183
$814$	−9.86649	−0.345820
$815$	−2.31371	−0.0810456
$816$	14.7226	0.515396
$817$	51.1388	1.78912
$818$	−21.1912	−0.740932
$819$	7.86349	0.274773
$820$	1.99229	0.0695737
$821$	−0.967206	−0.0337557	−0.0168779	−	0.999858i	$-0.505373\pi$
$821$	−0.967206	−0.0337557	−0.0168779	+	0.999858i	$0.505373\pi$
$822$	−5.01429	−0.174893
$823$	−33.4575	−1.16626	−0.583128	−	0.812380i	$-0.698171\pi$
$823$	−33.4575	−1.16626	−0.583128	+	0.812380i	$0.698171\pi$
$824$	−12.3473	−0.430140
$825$	−4.56754	−0.159021
$826$	38.3719	1.33513
$827$	24.8260	0.863284	0.431642	−	0.902045i	$-0.357934\pi$
$827$	24.8260	0.863284	0.431642	+	0.902045i	$0.357934\pi$
$828$	27.4762	0.954866
$829$	51.9693	1.80497	0.902485	−	0.430722i	$-0.141741\pi$
$829$	51.9693	1.80497	0.902485	+	0.430722i	$0.141741\pi$
$830$	−0.0271564	−0.000942613 0
$831$	−7.89412	−0.273844
$832$	58.3562	2.02314
$833$	4.46232	0.154610
$834$	10.0302	0.347317
$835$	0.291911	0.0101020
$836$	95.7105	3.31022
$837$	1.49052	0.0515198
$838$	−39.7640	−1.37363
$839$	32.0290	1.10576	0.552882	−	0.833260i	$-0.313528\pi$
$839$	32.0290	1.10576	0.552882	+	0.833260i	$0.313528\pi$
$840$	0.393457	0.0135756
$841$	55.5255	1.91467
$842$	77.8263	2.68207
$843$	−1.63039	−0.0561537
$844$	−54.0054	−1.85894
$845$	−1.04070	−0.0358013
$846$	−55.8623	−1.92059
$847$	−2.41004	−0.0828099
$848$	34.5438	1.18624
$849$	3.73995	0.128355
$850$	59.7315	2.04877
$851$	−1.77881	−0.0609767
$852$	9.20546	0.315374
$853$	20.3896	0.698127	0.349063	−	0.937099i	$-0.386500\pi$
$853$	20.3896	0.698127	0.349063	+	0.937099i	$0.386500\pi$
$854$	13.3369	0.456379
$855$	−2.60476	−0.0890808
$856$	160.201	5.47556
$857$	28.7801	0.983109	0.491554	−	0.870847i	$-0.336429\pi$
$857$	28.7801	0.983109	0.491554	+	0.870847i	$0.336429\pi$
$858$	−6.63210	−0.226416
$859$	−7.78592	−0.265652	−0.132826	−	0.991139i	$-0.542405\pi$
$859$	−7.78592	−0.265652	−0.132826	+	0.991139i	$0.542405\pi$
$860$	9.65863	0.329357
$861$	−0.532898	−0.0181611
$862$	−87.9947	−2.99711
$863$	−4.21370	−0.143436	−0.0717180	−	0.997425i	$-0.522848\pi$
$863$	−4.21370	−0.143436	−0.0717180	+	0.997425i	$0.522848\pi$
$864$	26.5979	0.904877
$865$	1.47867	0.0502764
$866$	−38.6268	−1.31259
$867$	0.731161	0.0248315
$868$	5.25931	0.178513
$869$	−37.8280	−1.28323
$870$	−1.10985	−0.0376276
$871$	14.8304	0.502510
$872$	−88.4739	−2.99611
$873$	48.4024	1.63817
$874$	23.8173	0.805632
$875$	1.77896	0.0601398
$876$	−12.6796	−0.428404
$877$	2.17778	0.0735384	0.0367692	−	0.999324i	$-0.488293\pi$
$877$	2.17778	0.0735384	0.0367692	+	0.999324i	$0.488293\pi$
$878$	46.6532	1.57447
$879$	1.34381	0.0453255
$880$	8.58854	0.289520
$881$	1.18433	0.0399010	0.0199505	−	0.999801i	$-0.493649\pi$
$881$	1.18433	0.0399010	0.0199505	+	0.999801i	$0.493649\pi$
$882$	7.91311	0.266448
$883$	56.4191	1.89865	0.949327	−	0.314289i	$-0.101766\pi$
$883$	56.4191	1.89865	0.949327	+	0.314289i	$0.101766\pi$
$884$	62.8356	2.11339
$885$	0.638101	0.0214495
$886$	46.9843	1.57847
$887$	−45.5257	−1.52860	−0.764301	−	0.644860i	$-0.776916\pi$
$887$	−45.5257	−1.52860	−0.764301	+	0.644860i	$0.776916\pi$
$888$	−2.20468	−0.0739843
$889$	−8.46035	−0.283751
$890$	4.91633	0.164796
$891$	30.8950	1.03502
$892$	51.3498	1.71932
$893$	−35.0822	−1.17398
$894$	−11.5625	−0.386708
$895$	−0.583117	−0.0194914
$896$	23.0351	0.769547
$897$	−1.19569	−0.0399228
$898$	10.2988	0.343676
$899$	−9.19377	−0.306630
$900$	76.7402	2.55801
$901$	11.7295	0.390766
$902$	−20.9428	−0.697318
$903$	−2.58349	−0.0859733
$904$	40.1233	1.33448
$905$	−2.62720	−0.0873312
$906$	−2.62277	−0.0871356
$907$	−11.8334	−0.392920	−0.196460	−	0.980512i	$-0.562945\pi$
$907$	−11.8334	−0.392920	−0.196460	+	0.980512i	$0.562945\pi$
$908$	−64.8346	−2.15161
$909$	−34.4199	−1.14164
$910$	1.28741	0.0426771
$911$	−2.64070	−0.0874903	−0.0437452	−	0.999043i	$-0.513929\pi$
$911$	−2.64070	−0.0874903	−0.0437452	+	0.999043i	$0.513929\pi$
$912$	16.3961	0.542929
$913$	0.206818	0.00684467
$914$	7.46298	0.246853
$915$	0.221784	0.00733197
$916$	−70.2616	−2.32151
$917$	−12.5494	−0.414418
$918$	17.9203	0.591459
$919$	−24.6791	−0.814088	−0.407044	−	0.913409i	$-0.633440\pi$
$919$	−24.6791	−0.814088	−0.407044	+	0.913409i	$0.633440\pi$
$920$	2.78775	0.0919095
$921$	2.47767	0.0816422
$922$	10.7437	0.353823
$923$	18.6664	0.614412
$924$	−4.83522	−0.159067
$925$	−4.96815	−0.163352
$926$	34.3056	1.12735
$927$	−4.12952	−0.135631
$928$	−164.060	−5.38554
$929$	−49.9616	−1.63919	−0.819593	−	0.572947i	$-0.805800\pi$
$929$	−49.9616	−1.63919	−0.819593	+	0.572947i	$0.805800\pi$
$930$	0.120718	0.00395850
$931$	4.96953	0.162870
$932$	143.600	4.70378
$933$	2.23256	0.0730907
$934$	8.14994	0.266674
$935$	2.91627	0.0953723
$936$	69.0540	2.25710
$937$	−17.0283	−0.556291	−0.278146	−	0.960539i	$-0.589720\pi$
$937$	−17.0283	−0.556291	−0.278146	+	0.960539i	$0.589720\pi$
$938$	14.9240	0.487287
$939$	0.976624	0.0318709
$940$	−6.62601	−0.216117
$941$	−8.38286	−0.273273	−0.136637	−	0.990621i	$-0.543629\pi$
$941$	−8.38286	−0.273273	−0.136637	+	0.990621i	$0.543629\pi$
$942$	0.949526	0.0309372
$943$	−3.77573	−0.122955
$944$	187.162	6.09161
$945$	0.266005	0.00865313
$946$	−101.531	−3.30105
$947$	53.6014	1.74181	0.870905	−	0.491451i	$-0.163533\pi$
$947$	53.6014	1.74181	0.870905	+	0.491451i	$0.163533\pi$
$948$	−13.6395	−0.442991
$949$	−25.7111	−0.834617
$950$	66.5209	2.15822
$951$	−0.714248	−0.0231611
$952$	39.1863	1.27004
$953$	41.3747	1.34026	0.670128	−	0.742245i	$-0.266239\pi$
$953$	41.3747	1.34026	0.670128	+	0.742245i	$0.266239\pi$
$954$	20.8001	0.673428
$955$	−3.80492	−0.123124
$956$	−74.6511	−2.41439
$957$	8.45242	0.273228
$958$	−25.0039	−0.807841
$959$	−7.41292	−0.239376
$960$	0.976552	0.0315181
$961$	1.00000	0.0322581
$962$	−7.21379	−0.232582
$963$	53.5786	1.72655
$964$	−7.10062	−0.228695
$965$	−0.0396215	−0.00127546
$966$	−1.20323	−0.0387133
$967$	−46.7824	−1.50442	−0.752210	−	0.658923i	$-0.771012\pi$
$967$	−46.7824	−1.50442	−0.752210	+	0.658923i	$0.771012\pi$
$968$	−21.1640	−0.680237
$969$	5.56736	0.178849
$970$	7.92441	0.254438
$971$	45.4403	1.45825	0.729124	−	0.684382i	$-0.239928\pi$
$971$	45.4403	1.45825	0.729124	+	0.684382i	$0.239928\pi$
$972$	34.6570	1.11162
$973$	14.8282	0.475372
$974$	51.8038	1.65990
$975$	−3.33951	−0.106950
$976$	65.0518	2.08226
$977$	−24.8823	−0.796056	−0.398028	−	0.917373i	$-0.630305\pi$
$977$	−24.8823	−0.796056	−0.398028	+	0.917373i	$0.630305\pi$
$978$	8.76953	0.280419
$979$	−37.4418	−1.19665
$980$	0.938600	0.0299825
$981$	−29.5898	−0.944728
$982$	−45.3681	−1.44775
$983$	−43.7462	−1.39529	−0.697644	−	0.716445i	$-0.745768\pi$
$983$	−43.7462	−1.39529	−0.697644	+	0.716445i	$0.745768\pi$
$984$	−4.67969	−0.149183
$985$	4.18791	0.133438
$986$	−110.536	−3.52017
$987$	1.77233	0.0564138
$988$	69.9778	2.22629
$989$	−18.3048	−0.582058
$990$	5.17147	0.164360
$991$	17.9494	0.570183	0.285091	−	0.958500i	$-0.407976\pi$
$991$	17.9494	0.570183	0.285091	+	0.958500i	$0.407976\pi$
$992$	17.8447	0.566570
$993$	−0.813145	−0.0258044
$994$	18.7842	0.595798
$995$	2.67695	0.0848649
$996$	0.0745717	0.00236290
$997$	4.12101	0.130514	0.0652568	−	0.997869i	$-0.479213\pi$
$997$	4.12101	0.130514	0.0652568	+	0.997869i	$0.479213\pi$
$998$	−66.4146	−2.10232
$999$	−1.49052	−0.0471579

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8029.2.a.h.1.2	✓	71

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8029.2.a.h.1.2	✓	71	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:	\( N \)	\(=\)	\( 8029 = 7 \cdot 31 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8029.a (trivial)

\(f(q)\)	\(=\)	\(q-2.69431 q^{2} +0.251057 q^{3} +5.25931 q^{4} +0.178464 q^{5} -0.676426 q^{6} -1.00000 q^{7} -8.78160 q^{8} -2.93697 q^{9} +O(q^{10})\)	\(q-2.69431 q^{2} +0.251057 q^{3} +5.25931 q^{4} +0.178464 q^{5} -0.676426 q^{6} -1.00000 q^{7} -8.78160 q^{8} -2.93697 q^{9} -0.480839 q^{10} +3.66197 q^{11} +1.32039 q^{12} +2.67742 q^{13} +2.69431 q^{14} +0.0448048 q^{15} +13.1417 q^{16} +4.46232 q^{17} +7.91311 q^{18} +4.96953 q^{19} +0.938600 q^{20} -0.251057 q^{21} -9.86649 q^{22} -1.77881 q^{23} -2.20468 q^{24} -4.96815 q^{25} -7.21379 q^{26} -1.49052 q^{27} -5.25931 q^{28} +9.19377 q^{29} -0.120718 q^{30} -1.00000 q^{31} -17.8447 q^{32} +0.919364 q^{33} -12.0229 q^{34} -0.178464 q^{35} -15.4464 q^{36} +1.00000 q^{37} -13.3895 q^{38} +0.672184 q^{39} -1.56720 q^{40} +2.12262 q^{41} +0.676426 q^{42} +10.2905 q^{43} +19.2594 q^{44} -0.524145 q^{45} +4.79266 q^{46} -7.05947 q^{47} +3.29932 q^{48} +1.00000 q^{49} +13.3857 q^{50} +1.12030 q^{51} +14.0814 q^{52} +2.62856 q^{53} +4.01592 q^{54} +0.653532 q^{55} +8.78160 q^{56} +1.24764 q^{57} -24.7709 q^{58} +14.2418 q^{59} +0.235642 q^{60} +4.95002 q^{61} +2.69431 q^{62} +2.93697 q^{63} +21.7957 q^{64} +0.477824 q^{65} -2.47705 q^{66} +5.53909 q^{67} +23.4687 q^{68} -0.446582 q^{69} +0.480839 q^{70} +6.97179 q^{71} +25.7913 q^{72} -9.60295 q^{73} -2.69431 q^{74} -1.24729 q^{75} +26.1363 q^{76} -3.66197 q^{77} -1.81107 q^{78} -10.3299 q^{79} +2.34533 q^{80} +8.43671 q^{81} -5.71899 q^{82} +0.0564772 q^{83} -1.32039 q^{84} +0.796366 q^{85} -27.7257 q^{86} +2.30816 q^{87} -32.1580 q^{88} -10.2245 q^{89} +1.41221 q^{90} -2.67742 q^{91} -9.35530 q^{92} -0.251057 q^{93} +19.0204 q^{94} +0.886885 q^{95} -4.48004 q^{96} -16.4804 q^{97} -2.69431 q^{98} -10.7551 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 71 q + 6 q^{2} + 8 q^{3} + 78 q^{4} + 5 q^{6} - 71 q^{7} + 18 q^{8} + 87 q^{9}+O(q^{10}) \) 71 * q + 6 * q^2 + 8 * q^3 + 78 * q^4 + 5 * q^6 - 71 * q^7 + 18 * q^8 + 87 * q^9	\( 71 q + 6 q^{2} + 8 q^{3} + 78 q^{4} + 5 q^{6} - 71 q^{7} + 18 q^{8} + 87 q^{9} + 4 q^{10} + 57 q^{11} + 21 q^{12} - 20 q^{13} - 6 q^{14} + 22 q^{15} + 88 q^{16} - 19 q^{17} + q^{18} + 23 q^{19} + 25 q^{20} - 8 q^{21} + 18 q^{22} + 34 q^{23} + 15 q^{24} + 81 q^{25} - 13 q^{26} + 20 q^{27} - 78 q^{28} + 16 q^{29} + 6 q^{30} - 71 q^{31} + 47 q^{32} - 16 q^{33} + 32 q^{34} + 125 q^{36} + 71 q^{37} + 13 q^{38} + 30 q^{39} + 31 q^{40} + 17 q^{41} - 5 q^{42} + 38 q^{43} + 80 q^{44} - q^{45} + 26 q^{46} + 32 q^{47} + 61 q^{48} + 71 q^{49} + 47 q^{50} + 73 q^{51} - 23 q^{52} + 31 q^{53} + 47 q^{54} + 11 q^{55} - 18 q^{56} + 17 q^{57} - 2 q^{58} + 97 q^{59} + 103 q^{60} - q^{61} - 6 q^{62} - 87 q^{63} + 100 q^{64} + 46 q^{65} + 43 q^{66} + 75 q^{67} - 43 q^{68} + 10 q^{69} - 4 q^{70} + 131 q^{71} - 11 q^{72} - 15 q^{73} + 6 q^{74} + 76 q^{75} + 41 q^{76} - 57 q^{77} + 89 q^{78} + 8 q^{79} + 10 q^{80} + 171 q^{81} + 14 q^{82} + 18 q^{83} - 21 q^{84} + 47 q^{85} + 90 q^{86} - 59 q^{87} + 13 q^{88} + 18 q^{89} + 69 q^{90} + 20 q^{91} + 110 q^{92} - 8 q^{93} + 39 q^{94} + 72 q^{95} + 100 q^{96} + 23 q^{97} + 6 q^{98} + 168 q^{99}+O(q^{100}) \) 71 * q + 6 * q^2 + 8 * q^3 + 78 * q^4 + 5 * q^6 - 71 * q^7 + 18 * q^8 + 87 * q^9 + 4 * q^10 + 57 * q^11 + 21 * q^12 - 20 * q^13 - 6 * q^14 + 22 * q^15 + 88 * q^16 - 19 * q^17 + q^18 + 23 * q^19 + 25 * q^20 - 8 * q^21 + 18 * q^22 + 34 * q^23 + 15 * q^24 + 81 * q^25 - 13 * q^26 + 20 * q^27 - 78 * q^28 + 16 * q^29 + 6 * q^30 - 71 * q^31 + 47 * q^32 - 16 * q^33 + 32 * q^34 + 125 * q^36 + 71 * q^37 + 13 * q^38 + 30 * q^39 + 31 * q^40 + 17 * q^41 - 5 * q^42 + 38 * q^43 + 80 * q^44 - q^45 + 26 * q^46 + 32 * q^47 + 61 * q^48 + 71 * q^49 + 47 * q^50 + 73 * q^51 - 23 * q^52 + 31 * q^53 + 47 * q^54 + 11 * q^55 - 18 * q^56 + 17 * q^57 - 2 * q^58 + 97 * q^59 + 103 * q^60 - q^61 - 6 * q^62 - 87 * q^63 + 100 * q^64 + 46 * q^65 + 43 * q^66 + 75 * q^67 - 43 * q^68 + 10 * q^69 - 4 * q^70 + 131 * q^71 - 11 * q^72 - 15 * q^73 + 6 * q^74 + 76 * q^75 + 41 * q^76 - 57 * q^77 + 89 * q^78 + 8 * q^79 + 10 * q^80 + 171 * q^81 + 14 * q^82 + 18 * q^83 - 21 * q^84 + 47 * q^85 + 90 * q^86 - 59 * q^87 + 13 * q^88 + 18 * q^89 + 69 * q^90 + 20 * q^91 + 110 * q^92 - 8 * q^93 + 39 * q^94 + 72 * q^95 + 100 * q^96 + 23 * q^97 + 6 * q^98 + 168 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more