LMFDB - Embedded newform 8041.2.a.e.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8041,2,Mod(1,8041)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8041, 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("8041.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8041 = 11 \cdot 17 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8041.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:	$64.2077082653$
Analytic rank:	$0$
Dimension:	$66$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	8041.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.64902 q^{2} -1.36167 q^{3} +0.719266 q^{4} -2.44108 q^{5} +2.24541 q^{6} -1.20008 q^{7} +2.11196 q^{8} -1.14587 q^{9} +O(q^{10})$	\(q-1.64902 q^{2} -1.36167 q^{3} +0.719266 q^{4} -2.44108 q^{5} +2.24541 q^{6} -1.20008 q^{7} +2.11196 q^{8} -1.14587 q^{9} +4.02539 q^{10} -1.00000 q^{11} -0.979400 q^{12} -4.81268 q^{13} +1.97895 q^{14} +3.32394 q^{15} -4.92119 q^{16} -1.00000 q^{17} +1.88956 q^{18} +7.74061 q^{19} -1.75579 q^{20} +1.63411 q^{21} +1.64902 q^{22} +7.20110 q^{23} -2.87578 q^{24} +0.958881 q^{25} +7.93620 q^{26} +5.64528 q^{27} -0.863177 q^{28} +4.21941 q^{29} -5.48124 q^{30} -3.58819 q^{31} +3.89123 q^{32} +1.36167 q^{33} +1.64902 q^{34} +2.92949 q^{35} -0.824184 q^{36} +6.40440 q^{37} -12.7644 q^{38} +6.55326 q^{39} -5.15546 q^{40} -4.98878 q^{41} -2.69467 q^{42} +1.00000 q^{43} -0.719266 q^{44} +2.79716 q^{45} -11.8748 q^{46} -2.55209 q^{47} +6.70101 q^{48} -5.55981 q^{49} -1.58121 q^{50} +1.36167 q^{51} -3.46160 q^{52} +2.65531 q^{53} -9.30918 q^{54} +2.44108 q^{55} -2.53451 q^{56} -10.5401 q^{57} -6.95790 q^{58} -14.2073 q^{59} +2.39079 q^{60} +0.0305325 q^{61} +5.91699 q^{62} +1.37513 q^{63} +3.42567 q^{64} +11.7481 q^{65} -2.24541 q^{66} +11.4336 q^{67} -0.719266 q^{68} -9.80548 q^{69} -4.83079 q^{70} -12.9261 q^{71} -2.42002 q^{72} +4.42425 q^{73} -10.5610 q^{74} -1.30567 q^{75} +5.56756 q^{76} +1.20008 q^{77} -10.8064 q^{78} -12.1158 q^{79} +12.0130 q^{80} -4.24938 q^{81} +8.22659 q^{82} -14.2675 q^{83} +1.17536 q^{84} +2.44108 q^{85} -1.64902 q^{86} -5.74543 q^{87} -2.11196 q^{88} -1.88533 q^{89} -4.61257 q^{90} +5.77560 q^{91} +5.17951 q^{92} +4.88591 q^{93} +4.20845 q^{94} -18.8955 q^{95} -5.29855 q^{96} -9.71630 q^{97} +9.16824 q^{98} +1.14587 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 66 q + 7 q^{2} + 3 q^{3} + 61 q^{4} + 4 q^{5} + 10 q^{6} + 14 q^{7} + 21 q^{8} + 65 q^{9}+O(q^{10}) $ 66 * q + 7 * q^2 + 3 * q^3 + 61 * q^4 + 4 * q^5 + 10 * q^6 + 14 * q^7 + 21 * q^8 + 65 * q^9	\( 66 q + 7 q^{2} + 3 q^{3} + 61 q^{4} + 4 q^{5} + 10 q^{6} + 14 q^{7} + 21 q^{8} + 65 q^{9} + 13 q^{10} - 66 q^{11} + 9 q^{12} - 12 q^{13} + 25 q^{14} + 13 q^{15} + 47 q^{16} - 66 q^{17} + 37 q^{18} + 19 q^{20} + 26 q^{21} - 7 q^{22} + 47 q^{23} + 15 q^{24} + 52 q^{25} + 16 q^{26} + 9 q^{27} + 3 q^{28} + 57 q^{29} + 2 q^{30} + 31 q^{31} + 39 q^{32} - 3 q^{33} - 7 q^{34} + 36 q^{35} + 39 q^{36} - 14 q^{37} + 18 q^{38} + 71 q^{39} + 29 q^{40} + 62 q^{41} - 3 q^{42} + 66 q^{43} - 61 q^{44} - 2 q^{45} + 19 q^{46} + 32 q^{47} + 26 q^{48} + 42 q^{49} + 10 q^{50} - 3 q^{51} - 7 q^{52} + 33 q^{53} + 100 q^{54} - 4 q^{55} + 61 q^{56} + 35 q^{57} - 16 q^{58} + 59 q^{59} + 50 q^{60} + 26 q^{61} + 29 q^{62} + 62 q^{63} + 29 q^{64} + 55 q^{65} - 10 q^{66} + 5 q^{67} - 61 q^{68} - 36 q^{69} - 35 q^{70} + 128 q^{71} + 87 q^{72} + 23 q^{73} + 64 q^{74} - 11 q^{75} + 74 q^{76} - 14 q^{77} + 45 q^{78} + 39 q^{79} + 95 q^{80} + 54 q^{81} - 6 q^{82} + 48 q^{83} + 38 q^{84} - 4 q^{85} + 7 q^{86} + 14 q^{87} - 21 q^{88} + 28 q^{89} + 135 q^{90} - 18 q^{91} + 108 q^{92} - 9 q^{93} + 37 q^{94} + 149 q^{95} + 104 q^{96} + 19 q^{97} + 30 q^{98} - 65 q^{99}+O(q^{100}) \) 66 * q + 7 * q^2 + 3 * q^3 + 61 * q^4 + 4 * q^5 + 10 * q^6 + 14 * q^7 + 21 * q^8 + 65 * q^9 + 13 * q^10 - 66 * q^11 + 9 * q^12 - 12 * q^13 + 25 * q^14 + 13 * q^15 + 47 * q^16 - 66 * q^17 + 37 * q^18 + 19 * q^20 + 26 * q^21 - 7 * q^22 + 47 * q^23 + 15 * q^24 + 52 * q^25 + 16 * q^26 + 9 * q^27 + 3 * q^28 + 57 * q^29 + 2 * q^30 + 31 * q^31 + 39 * q^32 - 3 * q^33 - 7 * q^34 + 36 * q^35 + 39 * q^36 - 14 * q^37 + 18 * q^38 + 71 * q^39 + 29 * q^40 + 62 * q^41 - 3 * q^42 + 66 * q^43 - 61 * q^44 - 2 * q^45 + 19 * q^46 + 32 * q^47 + 26 * q^48 + 42 * q^49 + 10 * q^50 - 3 * q^51 - 7 * q^52 + 33 * q^53 + 100 * q^54 - 4 * q^55 + 61 * q^56 + 35 * q^57 - 16 * q^58 + 59 * q^59 + 50 * q^60 + 26 * q^61 + 29 * q^62 + 62 * q^63 + 29 * q^64 + 55 * q^65 - 10 * q^66 + 5 * q^67 - 61 * q^68 - 36 * q^69 - 35 * q^70 + 128 * q^71 + 87 * q^72 + 23 * q^73 + 64 * q^74 - 11 * q^75 + 74 * q^76 - 14 * q^77 + 45 * q^78 + 39 * q^79 + 95 * q^80 + 54 * q^81 - 6 * q^82 + 48 * q^83 + 38 * q^84 - 4 * q^85 + 7 * q^86 + 14 * q^87 - 21 * q^88 + 28 * q^89 + 135 * q^90 - 18 * q^91 + 108 * q^92 - 9 * q^93 + 37 * q^94 + 149 * q^95 + 104 * q^96 + 19 * q^97 + 30 * q^98 - 65 * 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 $

$2$	−1.64902	−1.16603	−0.583017	−	0.812460i	$-0.698128\pi$
$2$	−1.64902	−1.16603	−0.583017	+	0.812460i	$0.698128\pi$
$3$	−1.36167	−0.786158	−0.393079	−	0.919505i	$-0.628590\pi$
$3$	−1.36167	−0.786158	−0.393079	+	0.919505i	$0.628590\pi$
$4$	0.719266	0.359633
$5$	−2.44108	−1.09169	−0.545843	−	0.837888i	$-0.683790\pi$
$5$	−2.44108	−1.09169	−0.545843	+	0.837888i	$0.683790\pi$
$6$	2.24541	0.916686
$7$	−1.20008	−0.453587	−0.226794	−	0.973943i	$-0.572824\pi$
$7$	−1.20008	−0.453587	−0.226794	+	0.973943i	$0.572824\pi$
$8$	2.11196	0.746689
$9$	−1.14587	−0.381956
$10$	4.02539	1.27294
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−0.979400	−0.282728
$13$	−4.81268	−1.33480	−0.667398	−	0.744701i	$-0.732592\pi$
$13$	−4.81268	−1.33480	−0.667398	+	0.744701i	$0.732592\pi$
$14$	1.97895	0.528898
$15$	3.32394	0.858237
$16$	−4.92119	−1.23030
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	1.88956	0.445373
$19$	7.74061	1.77582	0.887908	−	0.460020i	$-0.152158\pi$
$19$	7.74061	1.77582	0.887908	+	0.460020i	$0.152158\pi$
$20$	−1.75579	−0.392606
$21$	1.63411	0.356591
$22$	1.64902	0.351572
$23$	7.20110	1.50153	0.750766	−	0.660568i	$-0.229684\pi$
$23$	7.20110	1.50153	0.750766	+	0.660568i	$0.229684\pi$
$24$	−2.87578	−0.587015
$25$	0.958881	0.191776
$26$	7.93620	1.55642
$27$	5.64528	1.08644
$28$	−0.863177	−0.163125
$29$	4.21941	0.783526	0.391763	−	0.920066i	$-0.371865\pi$
$29$	4.21941	0.783526	0.391763	+	0.920066i	$0.371865\pi$
$30$	−5.48124	−1.00073
$31$	−3.58819	−0.644457	−0.322229	−	0.946662i	$-0.604432\pi$
$31$	−3.58819	−0.644457	−0.322229	+	0.946662i	$0.604432\pi$
$32$	3.89123	0.687878
$33$	1.36167	0.237035
$34$	1.64902	0.282805
$35$	2.92949	0.495175
$36$	−0.824184	−0.137364
$37$	6.40440	1.05288	0.526439	−	0.850213i	$-0.323527\pi$
$37$	6.40440	1.05288	0.526439	+	0.850213i	$0.323527\pi$
$38$	−12.7644	−2.07066
$39$	6.55326	1.04936
$40$	−5.15546	−0.815149
$41$	−4.98878	−0.779116	−0.389558	−	0.921002i	$-0.627372\pi$
$41$	−4.98878	−0.779116	−0.389558	+	0.921002i	$0.627372\pi$
$42$	−2.69467	−0.415797
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	−0.719266	−0.108433
$45$	2.79716	0.416976
$46$	−11.8748	−1.75084
$47$	−2.55209	−0.372261	−0.186130	−	0.982525i	$-0.559595\pi$
$47$	−2.55209	−0.372261	−0.186130	+	0.982525i	$0.559595\pi$
$48$	6.70101	0.967208
$49$	−5.55981	−0.794259
$50$	−1.58121	−0.223617
$51$	1.36167	0.190671
$52$	−3.46160	−0.480037
$53$	2.65531	0.364735	0.182368	−	0.983230i	$-0.441624\pi$
$53$	2.65531	0.364735	0.182368	+	0.983230i	$0.441624\pi$
$54$	−9.30918	−1.26682
$55$	2.44108	0.329155
$56$	−2.53451	−0.338689
$57$	−10.5401	−1.39607
$58$	−6.95790	−0.913617
$59$	−14.2073	−1.84964	−0.924819	−	0.380408i	$-0.875784\pi$
$59$	−14.2073	−1.84964	−0.924819	+	0.380408i	$0.875784\pi$
$60$	2.39079	0.308650
$61$	0.0305325	0.00390929	0.00195464	−	0.999998i	$-0.499378\pi$
$61$	0.0305325	0.00390929	0.00195464	+	0.999998i	$0.499378\pi$
$62$	5.91699	0.751458
$63$	1.37513	0.173250
$64$	3.42567	0.428208
$65$	11.7481	1.45718
$66$	−2.24541	−0.276391
$67$	11.4336	1.39684	0.698421	−	0.715687i	$-0.253886\pi$
$67$	11.4336	1.39684	0.698421	+	0.715687i	$0.253886\pi$
$68$	−0.719266	−0.0872238
$69$	−9.80548	−1.18044
$70$	−4.83079	−0.577390
$71$	−12.9261	−1.53405	−0.767026	−	0.641617i	$-0.778264\pi$
$71$	−12.9261	−1.53405	−0.767026	+	0.641617i	$0.778264\pi$
$72$	−2.42002	−0.285202
$73$	4.42425	0.517819	0.258910	−	0.965902i	$-0.416637\pi$
$73$	4.42425	0.517819	0.258910	+	0.965902i	$0.416637\pi$
$74$	−10.5610	−1.22769
$75$	−1.30567	−0.150766
$76$	5.56756	0.638643
$77$	1.20008	0.136762
$78$	−10.8064	−1.22359
$79$	−12.1158	−1.36313	−0.681566	−	0.731757i	$-0.738701\pi$
$79$	−12.1158	−1.36313	−0.681566	+	0.731757i	$0.738701\pi$
$80$	12.0130	1.34310
$81$	−4.24938	−0.472154
$82$	8.22659	0.908475
$83$	−14.2675	−1.56607	−0.783034	−	0.621980i	$-0.786329\pi$
$83$	−14.2675	−1.56607	−0.783034	+	0.621980i	$0.786329\pi$
$84$	1.17536	0.128242
$85$	2.44108	0.264773
$86$	−1.64902	−0.177818
$87$	−5.74543	−0.615975
$88$	−2.11196	−0.225135
$89$	−1.88533	−0.199845	−0.0999225	−	0.994995i	$-0.531860\pi$
$89$	−1.88533	−0.199845	−0.0999225	+	0.994995i	$0.531860\pi$
$90$	−4.61257	−0.486207
$91$	5.77560	0.605447
$92$	5.17951	0.540001
$93$	4.88591	0.506645
$94$	4.20845	0.434068
$95$	−18.8955	−1.93863
$96$	−5.29855	−0.540781
$97$	−9.71630	−0.986541	−0.493271	−	0.869876i	$-0.664199\pi$
$97$	−9.71630	−0.986541	−0.493271	+	0.869876i	$0.664199\pi$
$98$	9.16824	0.926132
$99$	1.14587	0.115164
$100$	0.689691	0.0689691
$101$	12.5021	1.24400	0.622001	−	0.783016i	$-0.286320\pi$
$101$	12.5021	1.24400	0.622001	+	0.783016i	$0.286320\pi$
$102$	−2.24541	−0.222329
$103$	0.891733	0.0878651	0.0439325	−	0.999035i	$-0.486011\pi$
$103$	0.891733	0.0878651	0.0439325	+	0.999035i	$0.486011\pi$
$104$	−10.1642	−0.996678
$105$	−3.98899	−0.389285
$106$	−4.37866	−0.425293
$107$	8.75685	0.846557	0.423278	−	0.906000i	$-0.360879\pi$
$107$	8.75685	0.846557	0.423278	+	0.906000i	$0.360879\pi$
$108$	4.06046	0.390718
$109$	−6.26524	−0.600101	−0.300050	−	0.953923i	$-0.597003\pi$
$109$	−6.26524	−0.600101	−0.300050	+	0.953923i	$0.597003\pi$
$110$	−4.02539	−0.383806
$111$	−8.72065	−0.827728
$112$	5.90582	0.558047
$113$	3.56183	0.335069	0.167534	−	0.985866i	$-0.446419\pi$
$113$	3.56183	0.335069	0.167534	+	0.985866i	$0.446419\pi$
$114$	17.3809	1.62787
$115$	−17.5785	−1.63920
$116$	3.03488	0.281782
$117$	5.51470	0.509834
$118$	23.4282	2.15674
$119$	1.20008	0.110011
$120$	7.02000	0.640836
$121$	1.00000	0.0909091
$122$	−0.0503487	−0.00455836
$123$	6.79304	0.612508
$124$	−2.58086	−0.231768
$125$	9.86470	0.882326
$126$	−2.26762	−0.202016
$127$	−12.2418	−1.08628	−0.543142	−	0.839641i	$-0.682765\pi$
$127$	−12.2418	−1.08628	−0.543142	+	0.839641i	$0.682765\pi$
$128$	−13.4314	−1.18718
$129$	−1.36167	−0.119888
$130$	−19.3729	−1.69912
$131$	−3.08258	−0.269326	−0.134663	−	0.990891i	$-0.542995\pi$
$131$	−3.08258	−0.269326	−0.134663	+	0.990891i	$0.542995\pi$
$132$	0.979400	0.0852458
$133$	−9.28934	−0.805488
$134$	−18.8543	−1.62876
$135$	−13.7806	−1.18605
$136$	−2.11196	−0.181099
$137$	1.93081	0.164960	0.0824801	−	0.996593i	$-0.473716\pi$
$137$	1.93081	0.164960	0.0824801	+	0.996593i	$0.473716\pi$
$138$	16.1694	1.37643
$139$	4.32672	0.366988	0.183494	−	0.983021i	$-0.441259\pi$
$139$	4.32672	0.366988	0.183494	+	0.983021i	$0.441259\pi$
$140$	2.10708	0.178081
$141$	3.47509	0.292656
$142$	21.3155	1.78875
$143$	4.81268	0.402456
$144$	5.63903	0.469919
$145$	−10.2999	−0.855363
$146$	−7.29567	−0.603794
$147$	7.57060	0.624412
$148$	4.60647	0.378650
$149$	22.0215	1.80407	0.902035	−	0.431662i	$-0.142073\pi$
$149$	22.0215	1.80407	0.902035	+	0.431662i	$0.142073\pi$
$150$	2.15308	0.175799
$151$	15.7852	1.28458	0.642291	−	0.766461i	$-0.277984\pi$
$151$	15.7852	1.28458	0.642291	+	0.766461i	$0.277984\pi$
$152$	16.3478	1.32598
$153$	1.14587	0.0926379
$154$	−1.97895	−0.159469
$155$	8.75906	0.703544
$156$	4.71354	0.377385
$157$	−12.6128	−1.00661	−0.503307	−	0.864108i	$-0.667884\pi$
$157$	−12.6128	−1.00661	−0.503307	+	0.864108i	$0.667884\pi$
$158$	19.9792	1.58946
$159$	−3.61565	−0.286739
$160$	−9.49880	−0.750946
$161$	−8.64189	−0.681076
$162$	7.00731	0.550547
$163$	−2.25582	−0.176690	−0.0883449	−	0.996090i	$-0.528158\pi$
$163$	−2.25582	−0.176690	−0.0883449	+	0.996090i	$0.528158\pi$
$164$	−3.58826	−0.280196
$165$	−3.32394	−0.258768
$166$	23.5275	1.82609
$167$	−21.3485	−1.65200	−0.826000	−	0.563671i	$-0.809389\pi$
$167$	−21.3485	−1.65200	−0.826000	+	0.563671i	$0.809389\pi$
$168$	3.45116	0.266263
$169$	10.1619	0.781683
$170$	−4.02539	−0.308733
$171$	−8.86971	−0.678284
$172$	0.719266	0.0548435
$173$	−20.5129	−1.55957	−0.779784	−	0.626049i	$-0.784671\pi$
$173$	−20.5129	−1.55957	−0.779784	+	0.626049i	$0.784671\pi$
$174$	9.47433	0.718247
$175$	−1.15073	−0.0869872
$176$	4.92119	0.370949
$177$	19.3456	1.45411
$178$	3.10895	0.233026
$179$	2.43228	0.181797	0.0908986	−	0.995860i	$-0.471026\pi$
$179$	2.43228	0.181797	0.0908986	+	0.995860i	$0.471026\pi$
$180$	2.01190	0.149958
$181$	−8.23984	−0.612463	−0.306231	−	0.951957i	$-0.599068\pi$
$181$	−8.23984	−0.612463	−0.306231	+	0.951957i	$0.599068\pi$
$182$	−9.52407	−0.705971
$183$	−0.0415750	−0.00307332
$184$	15.2084	1.12118
$185$	−15.6337	−1.14941
$186$	−8.05696	−0.590765
$187$	1.00000	0.0731272
$188$	−1.83563	−0.133877
$189$	−6.77479	−0.492793
$190$	31.1590	2.26051
$191$	0.166372	0.0120382	0.00601911	−	0.999982i	$-0.498084\pi$
$191$	0.166372	0.0120382	0.00601911	+	0.999982i	$0.498084\pi$
$192$	−4.66461	−0.336639
$193$	−5.67257	−0.408320	−0.204160	−	0.978937i	$-0.565446\pi$
$193$	−5.67257	−0.408320	−0.204160	+	0.978937i	$0.565446\pi$
$194$	16.0224	1.15034
$195$	−15.9970	−1.14557
$196$	−3.99898	−0.285642
$197$	−8.94688	−0.637439	−0.318719	−	0.947849i	$-0.603253\pi$
$197$	−8.94688	−0.637439	−0.318719	+	0.947849i	$0.603253\pi$
$198$	−1.88956	−0.134285
$199$	4.82734	0.342201	0.171101	−	0.985254i	$-0.445268\pi$
$199$	4.82734	0.342201	0.171101	+	0.985254i	$0.445268\pi$
$200$	2.02511	0.143197
$201$	−15.5688	−1.09814
$202$	−20.6162	−1.45055
$203$	−5.06363	−0.355397
$204$	0.979400	0.0685717
$205$	12.1780	0.850549
$206$	−1.47049	−0.102454
$207$	−8.25151	−0.573519
$208$	23.6841	1.64220
$209$	−7.74061	−0.535429
$210$	6.57792	0.453919
$211$	14.9261	1.02755	0.513777	−	0.857924i	$-0.328246\pi$
$211$	14.9261	1.02755	0.513777	+	0.857924i	$0.328246\pi$
$212$	1.90988	0.131171
$213$	17.6011	1.20601
$214$	−14.4402	−0.987113
$215$	−2.44108	−0.166480
$216$	11.9226	0.811229
$217$	4.30611	0.292318
$218$	10.3315	0.699737
$219$	−6.02435	−0.407088
$220$	1.75579	0.118375
$221$	4.81268	0.323736
$222$	14.3805	0.965158
$223$	3.35268	0.224512	0.112256	−	0.993679i	$-0.464192\pi$
$223$	3.35268	0.224512	0.112256	+	0.993679i	$0.464192\pi$
$224$	−4.66978	−0.312013
$225$	−1.09875	−0.0732501
$226$	−5.87353	−0.390701
$227$	−8.29014	−0.550236	−0.275118	−	0.961411i	$-0.588717\pi$
$227$	−8.29014	−0.550236	−0.275118	+	0.961411i	$0.588717\pi$
$228$	−7.58115	−0.502074
$229$	−14.8889	−0.983886	−0.491943	−	0.870627i	$-0.663713\pi$
$229$	−14.8889	−0.983886	−0.491943	+	0.870627i	$0.663713\pi$
$230$	28.9872	1.91136
$231$	−1.63411	−0.107516
$232$	8.91121	0.585050
$233$	−13.7489	−0.900718	−0.450359	−	0.892848i	$-0.648704\pi$
$233$	−13.7489	−0.900718	−0.450359	+	0.892848i	$0.648704\pi$
$234$	−9.09384	−0.594483
$235$	6.22987	0.406392
$236$	−10.2189	−0.665191
$237$	16.4976	1.07164
$238$	−1.97895	−0.128277
$239$	1.53481	0.0992783	0.0496392	−	0.998767i	$-0.484193\pi$
$239$	1.53481	0.0992783	0.0496392	+	0.998767i	$0.484193\pi$
$240$	−16.3577	−1.05589
$241$	−24.1821	−1.55771	−0.778854	−	0.627205i	$-0.784199\pi$
$241$	−24.1821	−1.55771	−0.778854	+	0.627205i	$0.784199\pi$
$242$	−1.64902	−0.106003
$243$	−11.1496	−0.715248
$244$	0.0219610	0.00140591
$245$	13.5720	0.867080
$246$	−11.2019	−0.714205
$247$	−37.2530	−2.37035
$248$	−7.57809	−0.481209
$249$	19.4276	1.23118
$250$	−16.2671	−1.02882
$251$	−15.1938	−0.959025	−0.479512	−	0.877535i	$-0.659186\pi$
$251$	−15.1938	−0.959025	−0.479512	+	0.877535i	$0.659186\pi$
$252$	0.989086	0.0623066
$253$	−7.20110	−0.452729
$254$	20.1870	1.26664
$255$	−3.32394	−0.208153
$256$	15.2974	0.956087
$257$	−1.81762	−0.113380	−0.0566899	−	0.998392i	$-0.518055\pi$
$257$	−1.81762	−0.113380	−0.0566899	+	0.998392i	$0.518055\pi$
$258$	2.24541	0.139793
$259$	−7.68579	−0.477572
$260$	8.45004	0.524049
$261$	−4.83489	−0.299272
$262$	5.08323	0.314043
$263$	−27.4961	−1.69548	−0.847742	−	0.530409i	$-0.822038\pi$
$263$	−27.4961	−1.69548	−0.847742	+	0.530409i	$0.822038\pi$
$264$	2.87578	0.176992
$265$	−6.48183	−0.398176
$266$	15.3183	0.939226
$267$	2.56719	0.157110
$268$	8.22383	0.502351
$269$	25.4497	1.55169	0.775847	−	0.630921i	$-0.217323\pi$
$269$	25.4497	1.55169	0.775847	+	0.630921i	$0.217323\pi$
$270$	22.7245	1.38297
$271$	−19.0323	−1.15613	−0.578066	−	0.815990i	$-0.696192\pi$
$271$	−19.0323	−1.15613	−0.578066	+	0.815990i	$0.696192\pi$
$272$	4.92119	0.298391
$273$	−7.86443	−0.475977
$274$	−3.18394	−0.192349
$275$	−0.958881	−0.0578227
$276$	−7.05275	−0.424526
$277$	−24.5754	−1.47659	−0.738297	−	0.674476i	$-0.764369\pi$
$277$	−24.5754	−1.47659	−0.738297	+	0.674476i	$0.764369\pi$
$278$	−7.13485	−0.427920
$279$	4.11159	0.246154
$280$	6.18696	0.369741
$281$	32.6451	1.94744	0.973721	−	0.227745i	$-0.0731353\pi$
$281$	32.6451	1.94744	0.973721	+	0.227745i	$0.0731353\pi$
$282$	−5.73050	−0.341246
$283$	−23.8814	−1.41960	−0.709801	−	0.704402i	$-0.751215\pi$
$283$	−23.8814	−1.41960	−0.709801	+	0.704402i	$0.751215\pi$
$284$	−9.29734	−0.551696
$285$	25.7293	1.52407
$286$	−7.93620	−0.469277
$287$	5.98693	0.353397
$288$	−4.45883	−0.262739
$289$	1.00000	0.0588235
$290$	16.9848	0.997382
$291$	13.2304	0.775577
$292$	3.18221	0.186225
$293$	15.3138	0.894641	0.447320	−	0.894374i	$-0.352378\pi$
$293$	15.3138	0.894641	0.447320	+	0.894374i	$0.352378\pi$
$294$	−12.4841	−0.728086
$295$	34.6813	2.01922
$296$	13.5258	0.786172
$297$	−5.64528	−0.327573
$298$	−36.3139	−2.10361
$299$	−34.6566	−2.00424
$300$	−0.939128	−0.0542206
$301$	−1.20008	−0.0691714
$302$	−26.0301	−1.49787
$303$	−17.0236	−0.977982
$304$	−38.0930	−2.18478
$305$	−0.0745323	−0.00426771
$306$	−1.88956	−0.108019
$307$	−0.914062	−0.0521683	−0.0260841	−	0.999660i	$-0.508304\pi$
$307$	−0.914062	−0.0521683	−0.0260841	+	0.999660i	$0.508304\pi$
$308$	0.863177	0.0491840
$309$	−1.21424	−0.0690758
$310$	−14.4439	−0.820356
$311$	−1.51353	−0.0858246	−0.0429123	−	0.999079i	$-0.513664\pi$
$311$	−1.51353	−0.0858246	−0.0429123	+	0.999079i	$0.513664\pi$
$312$	13.8402	0.783546
$313$	−14.9906	−0.847321	−0.423660	−	0.905821i	$-0.639255\pi$
$313$	−14.9906	−0.847321	−0.423660	+	0.905821i	$0.639255\pi$
$314$	20.7988	1.17375
$315$	−3.35681	−0.189135
$316$	−8.71447	−0.490227
$317$	33.1137	1.85985	0.929926	−	0.367748i	$-0.119871\pi$
$317$	33.1137	1.85985	0.929926	+	0.367748i	$0.119871\pi$
$318$	5.96227	0.334348
$319$	−4.21941	−0.236242
$320$	−8.36234	−0.467469
$321$	−11.9239	−0.665527
$322$	14.2506	0.794157
$323$	−7.74061	−0.430699
$324$	−3.05644	−0.169802
$325$	−4.61479	−0.255982
$326$	3.71990	0.206026
$327$	8.53115	0.471774
$328$	−10.5361	−0.581757
$329$	3.06271	0.168853
$330$	5.48124	0.301732
$331$	−25.5955	−1.40685	−0.703427	−	0.710767i	$-0.748348\pi$
$331$	−25.5955	−1.40685	−0.703427	+	0.710767i	$0.748348\pi$
$332$	−10.2622	−0.563210
$333$	−7.33860	−0.402153
$334$	35.2042	1.92629
$335$	−27.9105	−1.52491
$336$	−8.04174	−0.438713
$337$	12.4906	0.680409	0.340205	−	0.940351i	$-0.389504\pi$
$337$	12.4906	0.680409	0.340205	+	0.940351i	$0.389504\pi$
$338$	−16.7571	−0.911468
$339$	−4.85002	−0.263417
$340$	1.75579	0.0952210
$341$	3.58819	0.194311
$342$	14.6263	0.790902
$343$	15.0728	0.813853
$344$	2.11196	0.113869
$345$	23.9360	1.28867
$346$	33.8262	1.81851
$347$	6.71579	0.360523	0.180261	−	0.983619i	$-0.442306\pi$
$347$	6.71579	0.360523	0.180261	+	0.983619i	$0.442306\pi$
$348$	−4.13249	−0.221525
$349$	−3.77397	−0.202016	−0.101008	−	0.994886i	$-0.532207\pi$
$349$	−3.77397	−0.202016	−0.101008	+	0.994886i	$0.532207\pi$
$350$	1.89758	0.101430
$351$	−27.1689	−1.45017
$352$	−3.89123	−0.207403
$353$	6.49045	0.345452	0.172726	−	0.984970i	$-0.444743\pi$
$353$	6.49045	0.345452	0.172726	+	0.984970i	$0.444743\pi$
$354$	−31.9013	−1.69554
$355$	31.5538	1.67470
$356$	−1.35606	−0.0718709
$357$	−1.63411	−0.0864861
$358$	−4.01088	−0.211982
$359$	26.9402	1.42185	0.710925	−	0.703268i	$-0.248277\pi$
$359$	26.9402	1.42185	0.710925	+	0.703268i	$0.248277\pi$
$360$	5.90747	0.311351
$361$	40.9170	2.15353
$362$	13.5877	0.714152
$363$	−1.36167	−0.0714689
$364$	4.15419	0.217739
$365$	−10.8000	−0.565295
$366$	0.0685581	0.00358359
$367$	12.6303	0.659297	0.329648	−	0.944104i	$-0.393070\pi$
$367$	12.6303	0.659297	0.329648	+	0.944104i	$0.393070\pi$
$368$	−35.4380	−1.84733
$369$	5.71648	0.297588
$370$	25.7802	1.34025
$371$	−3.18658	−0.165439
$372$	3.51427	0.182206
$373$	−18.6726	−0.966833	−0.483416	−	0.875391i	$-0.660604\pi$
$373$	−18.6726	−0.966833	−0.483416	+	0.875391i	$0.660604\pi$
$374$	−1.64902	−0.0852688
$375$	−13.4324	−0.693647
$376$	−5.38990	−0.277963
$377$	−20.3067	−1.04585
$378$	11.1718	0.574613
$379$	−5.15481	−0.264785	−0.132392	−	0.991197i	$-0.542266\pi$
$379$	−5.15481	−0.264785	−0.132392	+	0.991197i	$0.542266\pi$
$380$	−13.5909	−0.697196
$381$	16.6692	0.853990
$382$	−0.274350	−0.0140370
$383$	24.9863	1.27674	0.638371	−	0.769729i	$-0.279609\pi$
$383$	24.9863	1.27674	0.638371	+	0.769729i	$0.279609\pi$
$384$	18.2891	0.933313
$385$	−2.92949	−0.149301
$386$	9.35418	0.476115
$387$	−1.14587	−0.0582478
$388$	−6.98861	−0.354793
$389$	−19.5422	−0.990828	−0.495414	−	0.868657i	$-0.664984\pi$
$389$	−19.5422	−0.990828	−0.495414	+	0.868657i	$0.664984\pi$
$390$	26.3794	1.33577
$391$	−7.20110	−0.364175
$392$	−11.7421	−0.593064
$393$	4.19744	0.211733
$394$	14.7536	0.743275
$395$	29.5756	1.48811
$396$	0.824184	0.0414168
$397$	23.6165	1.18528	0.592640	−	0.805468i	$-0.298086\pi$
$397$	23.6165	1.18528	0.592640	+	0.805468i	$0.298086\pi$
$398$	−7.96039	−0.399018
$399$	12.6490	0.633241
$400$	−4.71883	−0.235942
$401$	−10.6549	−0.532083	−0.266041	−	0.963962i	$-0.585716\pi$
$401$	−10.6549	−0.532083	−0.266041	+	0.963962i	$0.585716\pi$
$402$	25.6732	1.28047
$403$	17.2688	0.860219
$404$	8.99232	0.447385
$405$	10.3731	0.515443
$406$	8.35003	0.414405
$407$	−6.40440	−0.317455
$408$	2.87578	0.142372
$409$	29.1362	1.44069	0.720347	−	0.693614i	$-0.243983\pi$
$409$	29.1362	1.44069	0.720347	+	0.693614i	$0.243983\pi$
$410$	−20.0818	−0.991768
$411$	−2.62912	−0.129685
$412$	0.641393	0.0315992
$413$	17.0499	0.838972
$414$	13.6069	0.668743
$415$	34.8283	1.70965
$416$	−18.7272	−0.918178
$417$	−5.89155	−0.288510
$418$	12.7644	0.624328
$419$	−10.1585	−0.496277	−0.248139	−	0.968725i	$-0.579819\pi$
$419$	−10.1585	−0.496277	−0.248139	+	0.968725i	$0.579819\pi$
$420$	−2.86914	−0.140000
$421$	21.4202	1.04396	0.521978	−	0.852959i	$-0.325194\pi$
$421$	21.4202	1.04396	0.521978	+	0.852959i	$0.325194\pi$
$422$	−24.6134	−1.19816
$423$	2.92436	0.142187
$424$	5.60790	0.272344
$425$	−0.958881	−0.0465126
$426$	−29.0245	−1.40624
$427$	−0.0366414	−0.00177320
$428$	6.29851	0.304450
$429$	−6.55326	−0.316394
$430$	4.02539	0.194122
$431$	12.3018	0.592555	0.296277	−	0.955102i	$-0.404255\pi$
$431$	12.3018	0.592555	0.296277	+	0.955102i	$0.404255\pi$
$432$	−27.7815	−1.33664
$433$	19.5599	0.939987	0.469994	−	0.882670i	$-0.344256\pi$
$433$	19.5599	0.939987	0.469994	+	0.882670i	$0.344256\pi$
$434$	−7.10086	−0.340852
$435$	14.0251	0.672450
$436$	−4.50637	−0.215816
$437$	55.7409	2.66645
$438$	9.93426	0.474678
$439$	−37.5430	−1.79183	−0.895914	−	0.444227i	$-0.853478\pi$
$439$	−37.5430	−1.79183	−0.895914	+	0.444227i	$0.853478\pi$
$440$	5.15546	0.245777
$441$	6.37081	0.303372
$442$	−7.93620	−0.377487
$443$	−1.50708	−0.0716033	−0.0358017	−	0.999359i	$-0.511398\pi$
$443$	−1.50708	−0.0716033	−0.0358017	+	0.999359i	$0.511398\pi$
$444$	−6.27247	−0.297678
$445$	4.60226	0.218168
$446$	−5.52864	−0.261789
$447$	−29.9859	−1.41828
$448$	−4.11107	−0.194230
$449$	30.1439	1.42258	0.711290	−	0.702899i	$-0.248111\pi$
$449$	30.1439	1.42258	0.711290	+	0.702899i	$0.248111\pi$
$450$	1.81186	0.0854120
$451$	4.98878	0.234912
$452$	2.56190	0.120502
$453$	−21.4942	−1.00988
$454$	13.6706	0.641593
$455$	−14.0987	−0.660957
$456$	−22.2602	−1.04243
$457$	−41.0760	−1.92145	−0.960727	−	0.277495i	$-0.910496\pi$
$457$	−41.0760	−1.92145	−0.960727	+	0.277495i	$0.910496\pi$
$458$	24.5521	1.14724
$459$	−5.64528	−0.263499
$460$	−12.6436	−0.589511
$461$	−7.45270	−0.347107	−0.173553	−	0.984824i	$-0.555525\pi$
$461$	−7.45270	−0.347107	−0.173553	+	0.984824i	$0.555525\pi$
$462$	2.69467	0.125368
$463$	−26.7653	−1.24389	−0.621945	−	0.783061i	$-0.713657\pi$
$463$	−26.7653	−1.24389	−0.621945	+	0.783061i	$0.713657\pi$
$464$	−20.7645	−0.963969
$465$	−11.9269	−0.553097
$466$	22.6721	1.05027
$467$	15.0121	0.694677	0.347339	−	0.937740i	$-0.387085\pi$
$467$	15.0121	0.694677	0.347339	+	0.937740i	$0.387085\pi$
$468$	3.96653	0.183353
$469$	−13.7213	−0.633590
$470$	−10.2732	−0.473866
$471$	17.1745	0.791358
$472$	−30.0053	−1.38110
$473$	−1.00000	−0.0459800
$474$	−27.2049	−1.24956
$475$	7.42232	0.340559
$476$	0.863177	0.0395636
$477$	−3.04264	−0.139313
$478$	−2.53092	−0.115762
$479$	35.8064	1.63604	0.818018	−	0.575193i	$-0.195073\pi$
$479$	35.8064	1.63604	0.818018	+	0.575193i	$0.195073\pi$
$480$	12.9342	0.590362
$481$	−30.8223	−1.40538
$482$	39.8768	1.81634
$483$	11.7674	0.535433
$484$	0.719266	0.0326939
$485$	23.7183	1.07699
$486$	18.3859	0.834003
$487$	−34.6977	−1.57230	−0.786151	−	0.618034i	$-0.787929\pi$
$487$	−34.6977	−1.57230	−0.786151	+	0.618034i	$0.787929\pi$
$488$	0.0644833	0.00291902
$489$	3.07168	0.138906
$490$	−22.3804	−1.01104
$491$	3.04674	0.137498	0.0687488	−	0.997634i	$-0.478099\pi$
$491$	3.04674	0.137498	0.0687488	+	0.997634i	$0.478099\pi$
$492$	4.88601	0.220278
$493$	−4.21941	−0.190033
$494$	61.4310	2.76391
$495$	−2.79716	−0.125723
$496$	17.6581	0.792874
$497$	15.5124	0.695826
$498$	−32.0365	−1.43559
$499$	8.21108	0.367578	0.183789	−	0.982966i	$-0.441164\pi$
$499$	8.21108	0.367578	0.183789	+	0.982966i	$0.441164\pi$
$500$	7.09535	0.317314
$501$	29.0696	1.29873
$502$	25.0549	1.11825
$503$	−12.4621	−0.555658	−0.277829	−	0.960630i	$-0.589615\pi$
$503$	−12.4621	−0.555658	−0.277829	+	0.960630i	$0.589615\pi$
$504$	2.90422	0.129364
$505$	−30.5186	−1.35806
$506$	11.8748	0.527897
$507$	−13.8371	−0.614526
$508$	−8.80511	−0.390663
$509$	−1.21002	−0.0536330	−0.0268165	−	0.999640i	$-0.508537\pi$
$509$	−1.21002	−0.0536330	−0.0268165	+	0.999640i	$0.508537\pi$
$510$	5.48124	0.242713
$511$	−5.30945	−0.234876
$512$	1.63720	0.0723548
$513$	43.6979	1.92931
$514$	2.99729	0.132205
$515$	−2.17679	−0.0959210
$516$	−0.979400	−0.0431157
$517$	2.55209	0.112241
$518$	12.6740	0.556865
$519$	27.9317	1.22607
$520$	24.8116	1.08806
$521$	37.7008	1.65170	0.825852	−	0.563887i	$-0.190695\pi$
$521$	37.7008	1.65170	0.825852	+	0.563887i	$0.190695\pi$
$522$	7.97283	0.348961
$523$	−32.1816	−1.40720	−0.703600	−	0.710596i	$-0.748425\pi$
$523$	−32.1816	−1.40720	−0.703600	+	0.710596i	$0.748425\pi$
$524$	−2.21719	−0.0968585
$525$	1.56691	0.0683857
$526$	45.3417	1.97699
$527$	3.58819	0.156304
$528$	−6.70101	−0.291624
$529$	28.8558	1.25460
$530$	10.6887	0.464286
$531$	16.2797	0.706480
$532$	−6.68151	−0.289680
$533$	24.0094	1.03996
$534$	−4.23335	−0.183195
$535$	−21.3762	−0.924173
$536$	24.1473	1.04301
$537$	−3.31195	−0.142921
$538$	−41.9670	−1.80933
$539$	5.55981	0.239478
$540$	−9.91192	−0.426541
$541$	34.2708	1.47342	0.736708	−	0.676211i	$-0.236379\pi$
$541$	34.2708	1.47342	0.736708	+	0.676211i	$0.236379\pi$
$542$	31.3847	1.34809
$543$	11.2199	0.481492
$544$	−3.89123	−0.166835
$545$	15.2940	0.655121
$546$	12.9686	0.555005
$547$	6.87630	0.294010	0.147005	−	0.989136i	$-0.453037\pi$
$547$	6.87630	0.294010	0.147005	+	0.989136i	$0.453037\pi$
$548$	1.38877	0.0593251
$549$	−0.0349862	−0.00149318
$550$	1.58121	0.0674232
$551$	32.6608	1.39140
$552$	−20.7087	−0.881423
$553$	14.5399	0.618299
$554$	40.5253	1.72176
$555$	21.2878	0.903618
$556$	3.11207	0.131981
$557$	6.22171	0.263622	0.131811	−	0.991275i	$-0.457921\pi$
$557$	6.22171	0.263622	0.131811	+	0.991275i	$0.457921\pi$
$558$	−6.78009	−0.287024
$559$	−4.81268	−0.203555
$560$	−14.4166	−0.609212
$561$	−1.36167	−0.0574895
$562$	−53.8324	−2.27078
$563$	−14.5465	−0.613062	−0.306531	−	0.951861i	$-0.599168\pi$
$563$	−14.5465	−0.613062	−0.306531	+	0.951861i	$0.599168\pi$
$564$	2.49952	0.105249
$565$	−8.69472	−0.365790
$566$	39.3809	1.65530
$567$	5.09960	0.214163
$568$	−27.2994	−1.14546
$569$	−7.29276	−0.305728	−0.152864	−	0.988247i	$-0.548850\pi$
$569$	−7.29276	−0.305728	−0.152864	+	0.988247i	$0.548850\pi$
$570$	−42.4281	−1.77712
$571$	22.5224	0.942535	0.471267	−	0.881990i	$-0.343797\pi$
$571$	22.5224	0.942535	0.471267	+	0.881990i	$0.343797\pi$
$572$	3.46160	0.144737
$573$	−0.226542	−0.00946394
$574$	−9.87256	−0.412073
$575$	6.90499	0.287958
$576$	−3.92536	−0.163557
$577$	19.0231	0.791943	0.395972	−	0.918263i	$-0.370408\pi$
$577$	19.0231	0.791943	0.395972	+	0.918263i	$0.370408\pi$
$578$	−1.64902	−0.0685902
$579$	7.72414	0.321004
$580$	−7.40840	−0.307617
$581$	17.1222	0.710348
$582$	−21.8171	−0.904348
$583$	−2.65531	−0.109972
$584$	9.34382	0.386650
$585$	−13.4618	−0.556578
$586$	−25.2527	−1.04318
$587$	−2.17553	−0.0897936	−0.0448968	−	0.998992i	$-0.514296\pi$
$587$	−2.17553	−0.0897936	−0.0448968	+	0.998992i	$0.514296\pi$
$588$	5.44528	0.224559
$589$	−27.7747	−1.14444
$590$	−57.1901	−2.35448
$591$	12.1827	0.501128
$592$	−31.5173	−1.29535
$593$	32.5581	1.33700	0.668500	−	0.743712i	$-0.266937\pi$
$593$	32.5581	1.33700	0.668500	+	0.743712i	$0.266937\pi$
$594$	9.30918	0.381960
$595$	−2.92949	−0.120097
$596$	15.8393	0.648804
$597$	−6.57323	−0.269024
$598$	57.1494	2.33701
$599$	22.4164	0.915908	0.457954	−	0.888976i	$-0.348582\pi$
$599$	22.4164	0.915908	0.457954	+	0.888976i	$0.348582\pi$
$600$	−2.75753	−0.112576
$601$	−14.8338	−0.605084	−0.302542	−	0.953136i	$-0.597835\pi$
$601$	−14.8338	−0.605084	−0.302542	+	0.953136i	$0.597835\pi$
$602$	1.97895	0.0806562
$603$	−13.1014	−0.533532
$604$	11.3538	0.461978
$605$	−2.44108	−0.0992441
$606$	28.0723	1.14036
$607$	39.1155	1.58765	0.793826	−	0.608145i	$-0.208086\pi$
$607$	39.1155	1.58765	0.793826	+	0.608145i	$0.208086\pi$
$608$	30.1205	1.22155
$609$	6.89497	0.279398
$610$	0.122905	0.00497629
$611$	12.2824	0.496893
$612$	0.824184	0.0333157
$613$	−41.1418	−1.66170	−0.830852	−	0.556494i	$-0.812146\pi$
$613$	−41.1418	−1.66170	−0.830852	+	0.556494i	$0.812146\pi$
$614$	1.50731	0.0608299
$615$	−16.5824	−0.668666
$616$	2.53451	0.102118
$617$	8.37911	0.337330	0.168665	−	0.985673i	$-0.446054\pi$
$617$	8.37911	0.337330	0.168665	+	0.985673i	$0.446054\pi$
$618$	2.00231	0.0805447
$619$	−15.8526	−0.637170	−0.318585	−	0.947894i	$-0.603208\pi$
$619$	−15.8526	−0.637170	−0.318585	+	0.947894i	$0.603208\pi$
$620$	6.30009	0.253018
$621$	40.6522	1.63132
$622$	2.49585	0.100074
$623$	2.26255	0.0906472
$624$	−32.2498	−1.29103
$625$	−28.8750	−1.15500
$626$	24.7198	0.988004
$627$	10.5401	0.420932
$628$	−9.07199	−0.362012
$629$	−6.40440	−0.255360
$630$	5.53545	0.220538
$631$	−22.1822	−0.883060	−0.441530	−	0.897247i	$-0.645564\pi$
$631$	−22.1822	−0.883060	−0.441530	+	0.897247i	$0.645564\pi$
$632$	−25.5880	−1.01784
$633$	−20.3243	−0.807819
$634$	−54.6052	−2.16865
$635$	29.8832	1.18588
$636$	−2.60061	−0.103121
$637$	26.7576	1.06017
$638$	6.95790	0.275466
$639$	14.8117	0.585940
$640$	32.7873	1.29603
$641$	−6.85948	−0.270933	−0.135467	−	0.990782i	$-0.543253\pi$
$641$	−6.85948	−0.270933	−0.135467	+	0.990782i	$0.543253\pi$
$642$	19.6627	0.776026
$643$	−7.08081	−0.279240	−0.139620	−	0.990205i	$-0.544588\pi$
$643$	−7.08081	−0.279240	−0.139620	+	0.990205i	$0.544588\pi$
$644$	−6.21582	−0.244938
$645$	3.32394	0.130880
$646$	12.7644	0.502209
$647$	−4.38075	−0.172225	−0.0861125	−	0.996285i	$-0.527444\pi$
$647$	−4.38075	−0.172225	−0.0861125	+	0.996285i	$0.527444\pi$
$648$	−8.97450	−0.352552
$649$	14.2073	0.557687
$650$	7.60987	0.298484
$651$	−5.86348	−0.229808
$652$	−1.62254	−0.0635435
$653$	−25.0373	−0.979784	−0.489892	−	0.871783i	$-0.662964\pi$
$653$	−25.0373	−0.979784	−0.489892	+	0.871783i	$0.662964\pi$
$654$	−14.0680	−0.550104
$655$	7.52482	0.294019
$656$	24.5507	0.958544
$657$	−5.06961	−0.197784
$658$	−5.05047	−0.196888
$659$	34.6926	1.35143	0.675715	−	0.737163i	$-0.263835\pi$
$659$	34.6926	1.35143	0.675715	+	0.737163i	$0.263835\pi$
$660$	−2.39079	−0.0930616
$661$	−4.18354	−0.162721	−0.0813604	−	0.996685i	$-0.525926\pi$
$661$	−4.18354	−0.162721	−0.0813604	+	0.996685i	$0.525926\pi$
$662$	42.2074	1.64044
$663$	−6.55326	−0.254507
$664$	−30.1324	−1.16936
$665$	22.6760	0.879339
$666$	12.1015	0.468924
$667$	30.3844	1.17649
$668$	−15.3553	−0.594114
$669$	−4.56523	−0.176502
$670$	46.0249	1.77810
$671$	−0.0305325	−0.00117869
$672$	6.35868	0.245291
$673$	−36.2822	−1.39858	−0.699289	−	0.714839i	$-0.746500\pi$
$673$	−36.2822	−1.39858	−0.699289	+	0.714839i	$0.746500\pi$
$674$	−20.5973	−0.793379
$675$	5.41315	0.208352
$676$	7.30909	0.281119
$677$	−15.4233	−0.592765	−0.296382	−	0.955069i	$-0.595780\pi$
$677$	−15.4233	−0.592765	−0.296382	+	0.955069i	$0.595780\pi$
$678$	7.99778	0.307153
$679$	11.6603	0.447483
$680$	5.15546	0.197703
$681$	11.2884	0.432572
$682$	−5.91699	−0.226573
$683$	14.6856	0.561930	0.280965	−	0.959718i	$-0.409346\pi$
$683$	14.6856	0.561930	0.280965	+	0.959718i	$0.409346\pi$
$684$	−6.37969	−0.243933
$685$	−4.71326	−0.180085
$686$	−24.8553	−0.948979
$687$	20.2737	0.773489
$688$	−4.92119	−0.187619
$689$	−12.7792	−0.486847
$690$	−39.4709	−1.50263
$691$	27.0815	1.03023	0.515114	−	0.857122i	$-0.327750\pi$
$691$	27.0815	1.03023	0.515114	+	0.857122i	$0.327750\pi$
$692$	−14.7542	−0.560872
$693$	−1.37513	−0.0522370
$694$	−11.0745	−0.420381
$695$	−10.5619	−0.400635
$696$	−12.1341	−0.459941
$697$	4.98878	0.188963
$698$	6.22334	0.235557
$699$	18.7213	0.708106
$700$	−0.827683	−0.0312835
$701$	−13.8923	−0.524706	−0.262353	−	0.964972i	$-0.584498\pi$
$701$	−13.8923	−0.524706	−0.262353	+	0.964972i	$0.584498\pi$
$702$	44.8021	1.69095
$703$	49.5740	1.86972
$704$	−3.42567	−0.129110
$705$	−8.48299	−0.319488
$706$	−10.7029	−0.402808
$707$	−15.0035	−0.564264
$708$	13.9147	0.522945
$709$	−27.2235	−1.02240	−0.511201	−	0.859461i	$-0.670799\pi$
$709$	−27.2235	−1.02240	−0.511201	+	0.859461i	$0.670799\pi$
$710$	−52.0328	−1.95276
$711$	13.8831	0.520656
$712$	−3.98174	−0.149222
$713$	−25.8389	−0.967673
$714$	2.69467	0.100846
$715$	−11.7481	−0.439356
$716$	1.74946	0.0653803
$717$	−2.08989	−0.0780484
$718$	−44.4249	−1.65792
$719$	39.2696	1.46451	0.732255	−	0.681031i	$-0.238468\pi$
$719$	39.2696	1.46451	0.732255	+	0.681031i	$0.238468\pi$
$720$	−13.7653	−0.513004
$721$	−1.07015	−0.0398545
$722$	−67.4729	−2.51108
$723$	32.9280	1.22460
$724$	−5.92664	−0.220262
$725$	4.04592	0.150262
$726$	2.24541	0.0833351
$727$	4.15608	0.154141	0.0770703	−	0.997026i	$-0.475443\pi$
$727$	4.15608	0.154141	0.0770703	+	0.997026i	$0.475443\pi$
$728$	12.1978	0.452081
$729$	27.9302	1.03445
$730$	17.8093	0.659153
$731$	−1.00000	−0.0369863
$732$	−0.0299035	−0.00110527
$733$	17.1123	0.632057	0.316028	−	0.948750i	$-0.397651\pi$
$733$	17.1123	0.632057	0.316028	+	0.948750i	$0.397651\pi$
$734$	−20.8276	−0.768762
$735$	−18.4805	−0.681662
$736$	28.0211	1.03287
$737$	−11.4336	−0.421164
$738$	−9.42659	−0.346997
$739$	10.1070	0.371793	0.185896	−	0.982569i	$-0.440481\pi$
$739$	10.1070	0.371793	0.185896	+	0.982569i	$0.440481\pi$
$740$	−11.2448	−0.413366
$741$	50.7262	1.86347
$742$	5.25474	0.192908
$743$	28.2265	1.03553	0.517764	−	0.855523i	$-0.326765\pi$
$743$	28.2265	1.03553	0.517764	+	0.855523i	$0.326765\pi$
$744$	10.3188	0.378306
$745$	−53.7563	−1.96948
$746$	30.7916	1.12736
$747$	16.3487	0.598169
$748$	0.719266	0.0262990
$749$	−10.5089	−0.383987
$750$	22.1503	0.808816
$751$	8.33856	0.304278	0.152139	−	0.988359i	$-0.451384\pi$
$751$	8.33856	0.304278	0.152139	+	0.988359i	$0.451384\pi$
$752$	12.5593	0.457991
$753$	20.6889	0.753945
$754$	33.4861	1.21949
$755$	−38.5330	−1.40236
$756$	−4.87288	−0.177225
$757$	−28.3720	−1.03120	−0.515598	−	0.856830i	$-0.672430\pi$
$757$	−28.3720	−1.03120	−0.515598	+	0.856830i	$0.672430\pi$
$758$	8.50039	0.308748
$759$	9.80548	0.355916
$760$	−39.9064	−1.44756
$761$	−25.0030	−0.906357	−0.453179	−	0.891420i	$-0.649710\pi$
$761$	−25.0030	−0.906357	−0.453179	+	0.891420i	$0.649710\pi$
$762$	−27.4879	−0.995781
$763$	7.51878	0.272198
$764$	0.119665	0.00432934
$765$	−2.79716	−0.101131
$766$	−41.2030	−1.48872
$767$	68.3753	2.46889
$768$	−20.8299	−0.751635
$769$	−31.8340	−1.14796	−0.573982	−	0.818868i	$-0.694602\pi$
$769$	−31.8340	−1.14796	−0.573982	+	0.818868i	$0.694602\pi$
$770$	4.83079	0.174090
$771$	2.47499	0.0891345
$772$	−4.08009	−0.146846
$773$	26.8949	0.967341	0.483671	−	0.875250i	$-0.339303\pi$
$773$	26.8949	0.967341	0.483671	+	0.875250i	$0.339303\pi$
$774$	1.88956	0.0679188
$775$	−3.44064	−0.123592
$776$	−20.5204	−0.736639
$777$	10.4655	0.375447
$778$	32.2255	1.15534
$779$	−38.6161	−1.38357
$780$	−11.5061	−0.411985
$781$	12.9261	0.462534
$782$	11.8748	0.424640
$783$	23.8198	0.851250
$784$	27.3609	0.977174
$785$	30.7890	1.09891
$786$	−6.92166	−0.246887
$787$	32.8918	1.17247	0.586234	−	0.810142i	$-0.300610\pi$
$787$	32.8918	1.17247	0.586234	+	0.810142i	$0.300610\pi$
$788$	−6.43519	−0.229244
$789$	37.4405	1.33292
$790$	−48.7708	−1.73519
$791$	−4.27448	−0.151983
$792$	2.42002	0.0859917
$793$	−0.146943	−0.00521810
$794$	−38.9441	−1.38207
$795$	8.82609	0.313029
$796$	3.47215	0.123067
$797$	−47.6730	−1.68866	−0.844332	−	0.535820i	$-0.820003\pi$
$797$	−47.6730	−1.68866	−0.844332	+	0.535820i	$0.820003\pi$
$798$	−20.8584	−0.738380
$799$	2.55209	0.0902865
$800$	3.73122	0.131919
$801$	2.16034	0.0763320
$802$	17.5702	0.620426
$803$	−4.42425	−0.156128
$804$	−11.1981	−0.394927
$805$	21.0956	0.743521
$806$	−28.4766	−1.00304
$807$	−34.6539	−1.21988
$808$	26.4038	0.928883
$809$	13.4211	0.471861	0.235930	−	0.971770i	$-0.424186\pi$
$809$	13.4211	0.471861	0.235930	+	0.971770i	$0.424186\pi$
$810$	−17.1054	−0.601024
$811$	9.24238	0.324544	0.162272	−	0.986746i	$-0.448118\pi$
$811$	9.24238	0.324544	0.162272	+	0.986746i	$0.448118\pi$
$812$	−3.64210	−0.127813
$813$	25.9157	0.908902
$814$	10.5610	0.370162
$815$	5.50665	0.192890
$816$	−6.70101	−0.234582
$817$	7.74061	0.270810
$818$	−48.0462	−1.67990
$819$	−6.61807	−0.231254
$820$	8.75923	0.305886
$821$	11.9019	0.415380	0.207690	−	0.978195i	$-0.433405\pi$
$821$	11.9019	0.415380	0.207690	+	0.978195i	$0.433405\pi$
$822$	4.33546	0.151217
$823$	44.3055	1.54439	0.772196	−	0.635385i	$-0.219158\pi$
$823$	44.3055	1.54439	0.772196	+	0.635385i	$0.219158\pi$
$824$	1.88330	0.0656079
$825$	1.30567	0.0454578
$826$	−28.1157	−0.978269
$827$	−17.7742	−0.618068	−0.309034	−	0.951051i	$-0.600006\pi$
$827$	−17.7742	−0.618068	−0.309034	+	0.951051i	$0.600006\pi$
$828$	−5.93503	−0.206257
$829$	−50.3916	−1.75017	−0.875087	−	0.483966i	$-0.839196\pi$
$829$	−50.3916	−1.75017	−0.875087	+	0.483966i	$0.839196\pi$
$830$	−57.4325	−1.99351
$831$	33.4635	1.16084
$832$	−16.4866	−0.571571
$833$	5.55981	0.192636
$834$	9.71528	0.336413
$835$	52.1135	1.80346
$836$	−5.56756	−0.192558
$837$	−20.2563	−0.700161
$838$	16.7516	0.578676
$839$	−11.7357	−0.405162	−0.202581	−	0.979265i	$-0.564933\pi$
$839$	−11.7357	−0.405162	−0.202581	+	0.979265i	$0.564933\pi$
$840$	−8.42456	−0.290675
$841$	−11.1965	−0.386088
$842$	−35.3223	−1.21729
$843$	−44.4517	−1.53100
$844$	10.7358	0.369542
$845$	−24.8060	−0.853351
$846$	−4.82233	−0.165795
$847$	−1.20008	−0.0412352
$848$	−13.0673	−0.448733
$849$	32.5185	1.11603
$850$	1.58121	0.0542352
$851$	46.1187	1.58093
$852$	12.6599	0.433720
$853$	−44.1736	−1.51248	−0.756238	−	0.654297i	$-0.772965\pi$
$853$	−44.1736	−1.51248	−0.756238	+	0.654297i	$0.772965\pi$
$854$	0.0604224	0.00206761
$855$	21.6517	0.740472
$856$	18.4941	0.632114
$857$	16.6292	0.568041	0.284021	−	0.958818i	$-0.408332\pi$
$857$	16.6292	0.568041	0.284021	+	0.958818i	$0.408332\pi$
$858$	10.8064	0.368926
$859$	17.6318	0.601590	0.300795	−	0.953689i	$-0.402748\pi$
$859$	17.6318	0.601590	0.300795	+	0.953689i	$0.402748\pi$
$860$	−1.75579	−0.0598719
$861$	−8.15219	−0.277826
$862$	−20.2858	−0.690938
$863$	−40.4523	−1.37701	−0.688506	−	0.725230i	$-0.741733\pi$
$863$	−40.4523	−1.37701	−0.688506	+	0.725230i	$0.741733\pi$
$864$	21.9671	0.747335
$865$	50.0737	1.70256
$866$	−32.2546	−1.09606
$867$	−1.36167	−0.0462446
$868$	3.09724	0.105127
$869$	12.1158	0.411000
$870$	−23.1276	−0.784099
$871$	−55.0265	−1.86450
$872$	−13.2319	−0.448089
$873$	11.1336	0.376815
$874$	−91.9178	−3.10916
$875$	−11.8384	−0.400212
$876$	−4.33311	−0.146402
$877$	9.15070	0.308997	0.154499	−	0.987993i	$-0.450624\pi$
$877$	9.15070	0.308997	0.154499	+	0.987993i	$0.450624\pi$
$878$	61.9091	2.08933
$879$	−20.8522	−0.703329
$880$	−12.0130	−0.404959
$881$	8.67574	0.292293	0.146146	−	0.989263i	$-0.453313\pi$
$881$	8.67574	0.292293	0.146146	+	0.989263i	$0.453313\pi$
$882$	−10.5056	−0.353742
$883$	−15.4808	−0.520969	−0.260485	−	0.965478i	$-0.583882\pi$
$883$	−15.4808	−0.520969	−0.260485	+	0.965478i	$0.583882\pi$
$884$	3.46160	0.116426
$885$	−47.2243	−1.58743
$886$	2.48520	0.0834918
$887$	−27.8878	−0.936381	−0.468191	−	0.883628i	$-0.655094\pi$
$887$	−27.8878	−0.936381	−0.468191	+	0.883628i	$0.655094\pi$
$888$	−18.4176	−0.618055
$889$	14.6911	0.492724
$890$	−7.58921	−0.254391
$891$	4.24938	0.142360
$892$	2.41147	0.0807421
$893$	−19.7547	−0.661067
$894$	49.4473	1.65377
$895$	−5.93740	−0.198465
$896$	16.1188	0.538491
$897$	47.1906	1.57565
$898$	−49.7079	−1.65877
$899$	−15.1400	−0.504949
$900$	−0.790294	−0.0263431
$901$	−2.65531	−0.0884613
$902$	−8.22659	−0.273915
$903$	1.63411	0.0543796
$904$	7.52243	0.250192
$905$	20.1141	0.668616
$906$	35.4443	1.17756
$907$	−8.75489	−0.290701	−0.145351	−	0.989380i	$-0.546431\pi$
$907$	−8.75489	−0.290701	−0.145351	+	0.989380i	$0.546431\pi$
$908$	−5.96281	−0.197883
$909$	−14.3257	−0.475154
$910$	23.2490	0.770698
$911$	41.5373	1.37619	0.688097	−	0.725619i	$-0.258446\pi$
$911$	41.5373	1.37619	0.688097	+	0.725619i	$0.258446\pi$
$912$	51.8699	1.71758
$913$	14.2675	0.472187
$914$	67.7351	2.24048
$915$	0.101488	0.00335509
$916$	−10.7091	−0.353838
$917$	3.69934	0.122163
$918$	9.30918	0.307249
$919$	−11.0927	−0.365915	−0.182958	−	0.983121i	$-0.558567\pi$
$919$	−11.0927	−0.365915	−0.182958	+	0.983121i	$0.558567\pi$
$920$	−37.1249	−1.22397
$921$	1.24465	0.0410125
$922$	12.2896	0.404738
$923$	62.2094	2.04765
$924$	−1.17536	−0.0386664
$925$	6.14106	0.201917
$926$	44.1365	1.45042
$927$	−1.02181	−0.0335606
$928$	16.4187	0.538970
$929$	−15.1845	−0.498188	−0.249094	−	0.968479i	$-0.580133\pi$
$929$	−15.1845	−0.498188	−0.249094	+	0.968479i	$0.580133\pi$
$930$	19.6677	0.644929
$931$	−43.0363	−1.41046
$932$	−9.88909	−0.323928
$933$	2.06093	0.0674717
$934$	−24.7552	−0.810017
$935$	−2.44108	−0.0798319
$936$	11.6468	0.380687
$937$	−0.534726	−0.0174687	−0.00873436	−	0.999962i	$-0.502780\pi$
$937$	−0.534726	−0.0174687	−0.00873436	+	0.999962i	$0.502780\pi$
$938$	22.6267	0.738787
$939$	20.4122	0.666128
$940$	4.48093	0.146152
$941$	−24.7041	−0.805332	−0.402666	−	0.915347i	$-0.631916\pi$
$941$	−24.7041	−0.805332	−0.402666	+	0.915347i	$0.631916\pi$
$942$	−28.3210	−0.922749
$943$	−35.9247	−1.16987
$944$	69.9170	2.27560
$945$	16.5378	0.537975
$946$	1.64902	0.0536143
$947$	−12.3240	−0.400478	−0.200239	−	0.979747i	$-0.564172\pi$
$947$	−12.3240	−0.400478	−0.200239	+	0.979747i	$0.564172\pi$
$948$	11.8662	0.385396
$949$	−21.2925	−0.691183
$950$	−12.2396	−0.397103
$951$	−45.0898	−1.46214
$952$	2.53451	0.0821441
$953$	12.2957	0.398296	0.199148	−	0.979969i	$-0.436183\pi$
$953$	12.2957	0.398296	0.199148	+	0.979969i	$0.436183\pi$
$954$	5.01737	0.162443
$955$	−0.406127	−0.0131419
$956$	1.10393	0.0357038
$957$	5.74543	0.185723
$958$	−59.0454	−1.90767
$959$	−2.31712	−0.0748238
$960$	11.3867	0.367504
$961$	−18.1249	−0.584675
$962$	50.8266	1.63872
$963$	−10.0342	−0.323347
$964$	−17.3934	−0.560203
$965$	13.8472	0.445757
$966$	−19.4046	−0.624333
$967$	−6.75339	−0.217174	−0.108587	−	0.994087i	$-0.534633\pi$
$967$	−6.75339	−0.217174	−0.108587	+	0.994087i	$0.534633\pi$
$968$	2.11196	0.0678808
$969$	10.5401	0.338597
$970$	−39.1119	−1.25581
$971$	22.6836	0.727952	0.363976	−	0.931408i	$-0.381419\pi$
$971$	22.6836	0.727952	0.363976	+	0.931408i	$0.381419\pi$
$972$	−8.01954	−0.257227
$973$	−5.19241	−0.166461
$974$	57.2172	1.83336
$975$	6.28379	0.201242
$976$	−0.150256	−0.00480958
$977$	18.7546	0.600013	0.300007	−	0.953937i	$-0.403011\pi$
$977$	18.7546	0.600013	0.300007	+	0.953937i	$0.403011\pi$
$978$	−5.06526	−0.161969
$979$	1.88533	0.0602556
$980$	9.76185	0.311831
$981$	7.17913	0.229212
$982$	−5.02414	−0.160327
$983$	22.5212	0.718316	0.359158	−	0.933277i	$-0.383064\pi$
$983$	22.5212	0.718316	0.359158	+	0.933277i	$0.383064\pi$
$984$	14.3466	0.457353
$985$	21.8401	0.695883
$986$	6.95790	0.221585
$987$	−4.17039	−0.132745
$988$	−26.7949	−0.852458
$989$	7.20110	0.228982
$990$	4.61257	0.146597
$991$	61.7579	1.96180	0.980902	−	0.194500i	$-0.0623085\pi$
$991$	61.7579	1.96180	0.980902	+	0.194500i	$0.0623085\pi$
$992$	−13.9624	−0.443308
$993$	34.8525	1.10601
$994$	−25.5803	−0.811356
$995$	−11.7839	−0.373576
$996$	13.9736	0.442772
$997$	−1.04172	−0.0329917	−0.0164959	−	0.999864i	$-0.505251\pi$
$997$	−1.04172	−0.0329917	−0.0164959	+	0.999864i	$0.505251\pi$
$998$	−13.5402	−0.428609
$999$	36.1547	1.14388

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8041.2.a.e.1.15	✓	66

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8041.2.a.e.1.15	✓	66	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:	\( N \)	\(=\)	\( 8041 = 11 \cdot 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8041.a (trivial)

\(f(q)\)	\(=\)	\(q-1.64902 q^{2} -1.36167 q^{3} +0.719266 q^{4} -2.44108 q^{5} +2.24541 q^{6} -1.20008 q^{7} +2.11196 q^{8} -1.14587 q^{9} +O(q^{10})\)	\(q-1.64902 q^{2} -1.36167 q^{3} +0.719266 q^{4} -2.44108 q^{5} +2.24541 q^{6} -1.20008 q^{7} +2.11196 q^{8} -1.14587 q^{9} +4.02539 q^{10} -1.00000 q^{11} -0.979400 q^{12} -4.81268 q^{13} +1.97895 q^{14} +3.32394 q^{15} -4.92119 q^{16} -1.00000 q^{17} +1.88956 q^{18} +7.74061 q^{19} -1.75579 q^{20} +1.63411 q^{21} +1.64902 q^{22} +7.20110 q^{23} -2.87578 q^{24} +0.958881 q^{25} +7.93620 q^{26} +5.64528 q^{27} -0.863177 q^{28} +4.21941 q^{29} -5.48124 q^{30} -3.58819 q^{31} +3.89123 q^{32} +1.36167 q^{33} +1.64902 q^{34} +2.92949 q^{35} -0.824184 q^{36} +6.40440 q^{37} -12.7644 q^{38} +6.55326 q^{39} -5.15546 q^{40} -4.98878 q^{41} -2.69467 q^{42} +1.00000 q^{43} -0.719266 q^{44} +2.79716 q^{45} -11.8748 q^{46} -2.55209 q^{47} +6.70101 q^{48} -5.55981 q^{49} -1.58121 q^{50} +1.36167 q^{51} -3.46160 q^{52} +2.65531 q^{53} -9.30918 q^{54} +2.44108 q^{55} -2.53451 q^{56} -10.5401 q^{57} -6.95790 q^{58} -14.2073 q^{59} +2.39079 q^{60} +0.0305325 q^{61} +5.91699 q^{62} +1.37513 q^{63} +3.42567 q^{64} +11.7481 q^{65} -2.24541 q^{66} +11.4336 q^{67} -0.719266 q^{68} -9.80548 q^{69} -4.83079 q^{70} -12.9261 q^{71} -2.42002 q^{72} +4.42425 q^{73} -10.5610 q^{74} -1.30567 q^{75} +5.56756 q^{76} +1.20008 q^{77} -10.8064 q^{78} -12.1158 q^{79} +12.0130 q^{80} -4.24938 q^{81} +8.22659 q^{82} -14.2675 q^{83} +1.17536 q^{84} +2.44108 q^{85} -1.64902 q^{86} -5.74543 q^{87} -2.11196 q^{88} -1.88533 q^{89} -4.61257 q^{90} +5.77560 q^{91} +5.17951 q^{92} +4.88591 q^{93} +4.20845 q^{94} -18.8955 q^{95} -5.29855 q^{96} -9.71630 q^{97} +9.16824 q^{98} +1.14587 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q + 7 q^{2} + 3 q^{3} + 61 q^{4} + 4 q^{5} + 10 q^{6} + 14 q^{7} + 21 q^{8} + 65 q^{9}+O(q^{10}) \) 66 * q + 7 * q^2 + 3 * q^3 + 61 * q^4 + 4 * q^5 + 10 * q^6 + 14 * q^7 + 21 * q^8 + 65 * q^9	\( 66 q + 7 q^{2} + 3 q^{3} + 61 q^{4} + 4 q^{5} + 10 q^{6} + 14 q^{7} + 21 q^{8} + 65 q^{9} + 13 q^{10} - 66 q^{11} + 9 q^{12} - 12 q^{13} + 25 q^{14} + 13 q^{15} + 47 q^{16} - 66 q^{17} + 37 q^{18} + 19 q^{20} + 26 q^{21} - 7 q^{22} + 47 q^{23} + 15 q^{24} + 52 q^{25} + 16 q^{26} + 9 q^{27} + 3 q^{28} + 57 q^{29} + 2 q^{30} + 31 q^{31} + 39 q^{32} - 3 q^{33} - 7 q^{34} + 36 q^{35} + 39 q^{36} - 14 q^{37} + 18 q^{38} + 71 q^{39} + 29 q^{40} + 62 q^{41} - 3 q^{42} + 66 q^{43} - 61 q^{44} - 2 q^{45} + 19 q^{46} + 32 q^{47} + 26 q^{48} + 42 q^{49} + 10 q^{50} - 3 q^{51} - 7 q^{52} + 33 q^{53} + 100 q^{54} - 4 q^{55} + 61 q^{56} + 35 q^{57} - 16 q^{58} + 59 q^{59} + 50 q^{60} + 26 q^{61} + 29 q^{62} + 62 q^{63} + 29 q^{64} + 55 q^{65} - 10 q^{66} + 5 q^{67} - 61 q^{68} - 36 q^{69} - 35 q^{70} + 128 q^{71} + 87 q^{72} + 23 q^{73} + 64 q^{74} - 11 q^{75} + 74 q^{76} - 14 q^{77} + 45 q^{78} + 39 q^{79} + 95 q^{80} + 54 q^{81} - 6 q^{82} + 48 q^{83} + 38 q^{84} - 4 q^{85} + 7 q^{86} + 14 q^{87} - 21 q^{88} + 28 q^{89} + 135 q^{90} - 18 q^{91} + 108 q^{92} - 9 q^{93} + 37 q^{94} + 149 q^{95} + 104 q^{96} + 19 q^{97} + 30 q^{98} - 65 q^{99}+O(q^{100}) \) 66 * q + 7 * q^2 + 3 * q^3 + 61 * q^4 + 4 * q^5 + 10 * q^6 + 14 * q^7 + 21 * q^8 + 65 * q^9 + 13 * q^10 - 66 * q^11 + 9 * q^12 - 12 * q^13 + 25 * q^14 + 13 * q^15 + 47 * q^16 - 66 * q^17 + 37 * q^18 + 19 * q^20 + 26 * q^21 - 7 * q^22 + 47 * q^23 + 15 * q^24 + 52 * q^25 + 16 * q^26 + 9 * q^27 + 3 * q^28 + 57 * q^29 + 2 * q^30 + 31 * q^31 + 39 * q^32 - 3 * q^33 - 7 * q^34 + 36 * q^35 + 39 * q^36 - 14 * q^37 + 18 * q^38 + 71 * q^39 + 29 * q^40 + 62 * q^41 - 3 * q^42 + 66 * q^43 - 61 * q^44 - 2 * q^45 + 19 * q^46 + 32 * q^47 + 26 * q^48 + 42 * q^49 + 10 * q^50 - 3 * q^51 - 7 * q^52 + 33 * q^53 + 100 * q^54 - 4 * q^55 + 61 * q^56 + 35 * q^57 - 16 * q^58 + 59 * q^59 + 50 * q^60 + 26 * q^61 + 29 * q^62 + 62 * q^63 + 29 * q^64 + 55 * q^65 - 10 * q^66 + 5 * q^67 - 61 * q^68 - 36 * q^69 - 35 * q^70 + 128 * q^71 + 87 * q^72 + 23 * q^73 + 64 * q^74 - 11 * q^75 + 74 * q^76 - 14 * q^77 + 45 * q^78 + 39 * q^79 + 95 * q^80 + 54 * q^81 - 6 * q^82 + 48 * q^83 + 38 * q^84 - 4 * q^85 + 7 * q^86 + 14 * q^87 - 21 * q^88 + 28 * q^89 + 135 * q^90 - 18 * q^91 + 108 * q^92 - 9 * q^93 + 37 * q^94 + 149 * q^95 + 104 * q^96 + 19 * q^97 + 30 * q^98 - 65 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more