LMFDB - Embedded newform 8041.2.a.i.1.17

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:	$78$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
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)

$f(q)$	$=$	$q-1.73352 q^{2} +2.13034 q^{3} +1.00511 q^{4} -4.15912 q^{5} -3.69300 q^{6} -1.81917 q^{7} +1.72467 q^{8} +1.53836 q^{9} +O(q^{10})$	\(q-1.73352 q^{2} +2.13034 q^{3} +1.00511 q^{4} -4.15912 q^{5} -3.69300 q^{6} -1.81917 q^{7} +1.72467 q^{8} +1.53836 q^{9} +7.20994 q^{10} +1.00000 q^{11} +2.14123 q^{12} +3.34135 q^{13} +3.15357 q^{14} -8.86036 q^{15} -4.99997 q^{16} -1.00000 q^{17} -2.66679 q^{18} -0.395669 q^{19} -4.18037 q^{20} -3.87545 q^{21} -1.73352 q^{22} +3.74793 q^{23} +3.67414 q^{24} +12.2983 q^{25} -5.79231 q^{26} -3.11379 q^{27} -1.82846 q^{28} +6.92498 q^{29} +15.3597 q^{30} -4.77160 q^{31} +5.21824 q^{32} +2.13034 q^{33} +1.73352 q^{34} +7.56615 q^{35} +1.54622 q^{36} +12.0076 q^{37} +0.685902 q^{38} +7.11821 q^{39} -7.17311 q^{40} +3.93128 q^{41} +6.71820 q^{42} -1.00000 q^{43} +1.00511 q^{44} -6.39824 q^{45} -6.49714 q^{46} +0.482571 q^{47} -10.6517 q^{48} -3.69062 q^{49} -21.3194 q^{50} -2.13034 q^{51} +3.35841 q^{52} -5.31367 q^{53} +5.39783 q^{54} -4.15912 q^{55} -3.13746 q^{56} -0.842911 q^{57} -12.0046 q^{58} -5.33453 q^{59} -8.90562 q^{60} -14.4626 q^{61} +8.27168 q^{62} -2.79854 q^{63} +0.953998 q^{64} -13.8971 q^{65} -3.69300 q^{66} -9.55002 q^{67} -1.00511 q^{68} +7.98439 q^{69} -13.1161 q^{70} -10.8443 q^{71} +2.65317 q^{72} +0.787245 q^{73} -20.8154 q^{74} +26.1996 q^{75} -0.397690 q^{76} -1.81917 q^{77} -12.3396 q^{78} -13.0934 q^{79} +20.7955 q^{80} -11.2485 q^{81} -6.81497 q^{82} +6.37930 q^{83} -3.89525 q^{84} +4.15912 q^{85} +1.73352 q^{86} +14.7526 q^{87} +1.72467 q^{88} -6.94533 q^{89} +11.0915 q^{90} -6.07847 q^{91} +3.76708 q^{92} -10.1651 q^{93} -0.836549 q^{94} +1.64564 q^{95} +11.1166 q^{96} +9.60390 q^{97} +6.39779 q^{98} +1.53836 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 78 q + 7 q^{2} + 10 q^{3} + 91 q^{4} + 17 q^{5} + 12 q^{6} + 11 q^{7} + 33 q^{8} + 102 q^{9}+O(q^{10}) $ 78 * q + 7 * q^2 + 10 * q^3 + 91 * q^4 + 17 * q^5 + 12 * q^6 + 11 * q^7 + 33 * q^8 + 102 * q^9	\( 78 q + 7 q^{2} + 10 q^{3} + 91 q^{4} + 17 q^{5} + 12 q^{6} + 11 q^{7} + 33 q^{8} + 102 q^{9} + 3 q^{10} + 78 q^{11} + 31 q^{12} - 16 q^{13} + 31 q^{14} + 38 q^{15} + 121 q^{16} - 78 q^{17} + 11 q^{18} + 51 q^{20} + 6 q^{21} + 7 q^{22} + 48 q^{23} + 11 q^{24} + 101 q^{25} + 18 q^{26} + 46 q^{27} + 27 q^{28} + 22 q^{29} + 14 q^{30} + 56 q^{31} + 83 q^{32} + 10 q^{33} - 7 q^{34} + 24 q^{35} + 139 q^{36} + 53 q^{37} + 10 q^{38} + 79 q^{39} - q^{40} + 23 q^{41} + 17 q^{42} - 78 q^{43} + 91 q^{44} + 76 q^{45} + 21 q^{46} + 57 q^{47} + 78 q^{48} + 115 q^{49} + 58 q^{50} - 10 q^{51} - 63 q^{52} + 22 q^{53} - 18 q^{54} + 17 q^{55} + 111 q^{56} - 11 q^{57} + 36 q^{58} + 71 q^{59} + 36 q^{60} + 4 q^{61} - 5 q^{62} + 71 q^{63} + 183 q^{64} + 47 q^{65} + 12 q^{66} + 11 q^{67} - 91 q^{68} + 31 q^{69} + 33 q^{70} + 159 q^{71} + 59 q^{72} + 2 q^{73} - 4 q^{74} + 83 q^{75} - 44 q^{76} + 11 q^{77} + 101 q^{78} + 35 q^{79} + 85 q^{80} + 170 q^{81} + 98 q^{82} - 32 q^{83} + 44 q^{84} - 17 q^{85} - 7 q^{86} - 6 q^{87} + 33 q^{88} + 50 q^{89} - 5 q^{90} + 86 q^{91} + 106 q^{92} + 68 q^{93} - q^{94} + 109 q^{95} - 50 q^{96} + 40 q^{97} + 106 q^{98} + 102 q^{99}+O(q^{100}) \) 78 * q + 7 * q^2 + 10 * q^3 + 91 * q^4 + 17 * q^5 + 12 * q^6 + 11 * q^7 + 33 * q^8 + 102 * q^9 + 3 * q^10 + 78 * q^11 + 31 * q^12 - 16 * q^13 + 31 * q^14 + 38 * q^15 + 121 * q^16 - 78 * q^17 + 11 * q^18 + 51 * q^20 + 6 * q^21 + 7 * q^22 + 48 * q^23 + 11 * q^24 + 101 * q^25 + 18 * q^26 + 46 * q^27 + 27 * q^28 + 22 * q^29 + 14 * q^30 + 56 * q^31 + 83 * q^32 + 10 * q^33 - 7 * q^34 + 24 * q^35 + 139 * q^36 + 53 * q^37 + 10 * q^38 + 79 * q^39 - q^40 + 23 * q^41 + 17 * q^42 - 78 * q^43 + 91 * q^44 + 76 * q^45 + 21 * q^46 + 57 * q^47 + 78 * q^48 + 115 * q^49 + 58 * q^50 - 10 * q^51 - 63 * q^52 + 22 * q^53 - 18 * q^54 + 17 * q^55 + 111 * q^56 - 11 * q^57 + 36 * q^58 + 71 * q^59 + 36 * q^60 + 4 * q^61 - 5 * q^62 + 71 * q^63 + 183 * q^64 + 47 * q^65 + 12 * q^66 + 11 * q^67 - 91 * q^68 + 31 * q^69 + 33 * q^70 + 159 * q^71 + 59 * q^72 + 2 * q^73 - 4 * q^74 + 83 * q^75 - 44 * q^76 + 11 * q^77 + 101 * q^78 + 35 * q^79 + 85 * q^80 + 170 * q^81 + 98 * q^82 - 32 * q^83 + 44 * q^84 - 17 * q^85 - 7 * q^86 - 6 * q^87 + 33 * q^88 + 50 * q^89 - 5 * q^90 + 86 * q^91 + 106 * q^92 + 68 * q^93 - q^94 + 109 * q^95 - 50 * q^96 + 40 * q^97 + 106 * q^98 + 102 * 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 $

$2$	−1.73352	−1.22579	−0.612894	−	0.790165i	$-0.709995\pi$
$2$	−1.73352	−1.22579	−0.612894	+	0.790165i	$0.709995\pi$
$3$	2.13034	1.22995	0.614977	−	0.788545i	$-0.289165\pi$
$3$	2.13034	1.22995	0.614977	+	0.788545i	$0.289165\pi$
$4$	1.00511	0.502554
$5$	−4.15912	−1.86002	−0.930008	−	0.367539i	$-0.880200\pi$
$5$	−4.15912	−1.86002	−0.930008	+	0.367539i	$0.880200\pi$
$6$	−3.69300	−1.50766
$7$	−1.81917	−0.687581	−0.343791	−	0.939046i	$-0.611711\pi$
$7$	−1.81917	−0.687581	−0.343791	+	0.939046i	$0.611711\pi$
$8$	1.72467	0.609763
$9$	1.53836	0.512788
$10$	7.20994	2.27998
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	2.14123	0.618119
$13$	3.34135	0.926722	0.463361	−	0.886170i	$-0.346643\pi$
$13$	3.34135	0.926722	0.463361	+	0.886170i	$0.346643\pi$
$14$	3.15357	0.842828
$15$	−8.86036	−2.28774
$16$	−4.99997	−1.24999
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−2.66679	−0.628569
$19$	−0.395669	−0.0907727	−0.0453864	−	0.998970i	$-0.514452\pi$
$19$	−0.395669	−0.0907727	−0.0453864	+	0.998970i	$0.514452\pi$
$20$	−4.18037	−0.934759
$21$	−3.87545	−0.845694
$22$	−1.73352	−0.369589
$23$	3.74793	0.781498	0.390749	−	0.920497i	$-0.372216\pi$
$23$	3.74793	0.781498	0.390749	+	0.920497i	$0.372216\pi$
$24$	3.67414	0.749980
$25$	12.2983	2.45966
$26$	−5.79231	−1.13596
$27$	−3.11379	−0.599249
$28$	−1.82846	−0.345547
$29$	6.92498	1.28594	0.642969	−	0.765893i	$-0.277703\pi$
$29$	6.92498	1.28594	0.642969	+	0.765893i	$0.277703\pi$
$30$	15.3597	2.80428
$31$	−4.77160	−0.857004	−0.428502	−	0.903541i	$-0.640959\pi$
$31$	−4.77160	−0.857004	−0.428502	+	0.903541i	$0.640959\pi$
$32$	5.21824	0.922463
$33$	2.13034	0.370845
$34$	1.73352	0.297297
$35$	7.56615	1.27891
$36$	1.54622	0.257704
$37$	12.0076	1.97403	0.987015	−	0.160628i	$-0.0513518\pi$
$37$	12.0076	1.97403	0.987015	+	0.160628i	$0.0513518\pi$
$38$	0.685902	0.111268
$39$	7.11821	1.13983
$40$	−7.17311	−1.13417
$41$	3.93128	0.613963	0.306981	−	0.951716i	$-0.400681\pi$
$41$	3.93128	0.613963	0.306981	+	0.951716i	$0.400681\pi$
$42$	6.71820	1.03664
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	1.00511	0.151526
$45$	−6.39824	−0.953794
$46$	−6.49714	−0.957951
$47$	0.482571	0.0703902	0.0351951	−	0.999380i	$-0.488795\pi$
$47$	0.482571	0.0703902	0.0351951	+	0.999380i	$0.488795\pi$
$48$	−10.6517	−1.53743
$49$	−3.69062	−0.527232
$50$	−21.3194	−3.01502
$51$	−2.13034	−0.298308
$52$	3.35841	0.465728
$53$	−5.31367	−0.729888	−0.364944	−	0.931029i	$-0.618912\pi$
$53$	−5.31367	−0.729888	−0.364944	+	0.931029i	$0.618912\pi$
$54$	5.39783	0.734552
$55$	−4.15912	−0.560816
$56$	−3.13746	−0.419261
$57$	−0.842911	−0.111646
$58$	−12.0046	−1.57629
$59$	−5.33453	−0.694496	−0.347248	−	0.937773i	$-0.612884\pi$
$59$	−5.33453	−0.694496	−0.347248	+	0.937773i	$0.612884\pi$
$60$	−8.90562	−1.14971
$61$	−14.4626	−1.85175	−0.925873	−	0.377834i	$-0.876669\pi$
$61$	−14.4626	−1.85175	−0.925873	+	0.377834i	$0.876669\pi$
$62$	8.27168	1.05050
$63$	−2.79854	−0.352583
$64$	0.953998	0.119250
$65$	−13.8971	−1.72372
$66$	−3.69300	−0.454577
$67$	−9.55002	−1.16672	−0.583360	−	0.812213i	$-0.698262\pi$
$67$	−9.55002	−1.16672	−0.583360	+	0.812213i	$0.698262\pi$
$68$	−1.00511	−0.121887
$69$	7.98439	0.961207
$70$	−13.1161	−1.56767
$71$	−10.8443	−1.28698	−0.643489	−	0.765456i	$-0.722514\pi$
$71$	−10.8443	−1.28698	−0.643489	+	0.765456i	$0.722514\pi$
$72$	2.65317	0.312679
$73$	0.787245	0.0921400	0.0460700	−	0.998938i	$-0.485330\pi$
$73$	0.787245	0.0921400	0.0460700	+	0.998938i	$0.485330\pi$
$74$	−20.8154	−2.41974
$75$	26.1996	3.02527
$76$	−0.397690	−0.0456182
$77$	−1.81917	−0.207314
$78$	−12.3396	−1.39718
$79$	−13.0934	−1.47312	−0.736559	−	0.676373i	$-0.763551\pi$
$79$	−13.0934	−1.47312	−0.736559	+	0.676373i	$0.763551\pi$
$80$	20.7955	2.32501
$81$	−11.2485	−1.24984
$82$	−6.81497	−0.752588
$83$	6.37930	0.700219	0.350110	−	0.936709i	$-0.386144\pi$
$83$	6.37930	0.700219	0.350110	+	0.936709i	$0.386144\pi$
$84$	−3.89525	−0.425007
$85$	4.15912	0.451120
$86$	1.73352	0.186931
$87$	14.7526	1.58164
$88$	1.72467	0.183850
$89$	−6.94533	−0.736204	−0.368102	−	0.929785i	$-0.619992\pi$
$89$	−6.94533	−0.736204	−0.368102	+	0.929785i	$0.619992\pi$
$90$	11.0915	1.16915
$91$	−6.07847	−0.637197
$92$	3.76708	0.392745
$93$	−10.1651	−1.05408
$94$	−0.836549	−0.0862834
$95$	1.64564	0.168839
$96$	11.1166	1.13459
$97$	9.60390	0.975129	0.487564	−	0.873087i	$-0.337885\pi$
$97$	9.60390	0.975129	0.487564	+	0.873087i	$0.337885\pi$
$98$	6.39779	0.646274
$99$	1.53836	0.154611
$100$	12.3611	1.23611
$101$	0.280169	0.0278778	0.0139389	−	0.999903i	$-0.495563\pi$
$101$	0.280169	0.0278778	0.0139389	+	0.999903i	$0.495563\pi$
$102$	3.69300	0.365662
$103$	−6.91930	−0.681778	−0.340889	−	0.940103i	$-0.610728\pi$
$103$	−6.91930	−0.681778	−0.340889	+	0.940103i	$0.610728\pi$
$104$	5.76272	0.565081
$105$	16.1185	1.57300
$106$	9.21137	0.894688
$107$	−5.02302	−0.485594	−0.242797	−	0.970077i	$-0.578065\pi$
$107$	−5.02302	−0.485594	−0.242797	+	0.970077i	$0.578065\pi$
$108$	−3.12970	−0.301155
$109$	13.0521	1.25016	0.625081	−	0.780560i	$-0.285066\pi$
$109$	13.0521	1.25016	0.625081	+	0.780560i	$0.285066\pi$
$110$	7.20994	0.687441
$111$	25.5802	2.42797
$112$	9.09580	0.859472
$113$	4.90543	0.461464	0.230732	−	0.973017i	$-0.425888\pi$
$113$	4.90543	0.461464	0.230732	+	0.973017i	$0.425888\pi$
$114$	1.46121	0.136855
$115$	−15.5881	−1.45360
$116$	6.96036	0.646253
$117$	5.14020	0.475212
$118$	9.24753	0.851304
$119$	1.81917	0.166763
$120$	−15.2812	−1.39498
$121$	1.00000	0.0909091
$122$	25.0713	2.26985
$123$	8.37498	0.755146
$124$	−4.79597	−0.430691
$125$	−30.3545	−2.71499
$126$	4.85134	0.432192
$127$	18.8511	1.67277	0.836384	−	0.548144i	$-0.184665\pi$
$127$	18.8511	1.67277	0.836384	+	0.548144i	$0.184665\pi$
$128$	−12.0903	−1.06864
$129$	−2.13034	−0.187566
$130$	24.0909	2.11291
$131$	−0.920618	−0.0804348	−0.0402174	−	0.999191i	$-0.512805\pi$
$131$	−0.920618	−0.0804348	−0.0402174	+	0.999191i	$0.512805\pi$
$132$	2.14123	0.186370
$133$	0.719789	0.0624136
$134$	16.5552	1.43015
$135$	12.9506	1.11461
$136$	−1.72467	−0.147889
$137$	−6.23598	−0.532776	−0.266388	−	0.963866i	$-0.585830\pi$
$137$	−6.23598	−0.532776	−0.266388	+	0.963866i	$0.585830\pi$
$138$	−13.8411	−1.17824
$139$	−5.62858	−0.477410	−0.238705	−	0.971092i	$-0.576723\pi$
$139$	−5.62858	−0.477410	−0.238705	+	0.971092i	$0.576723\pi$
$140$	7.60480	0.642723
$141$	1.02804	0.0865768
$142$	18.7988	1.57756
$143$	3.34135	0.279417
$144$	−7.69178	−0.640981
$145$	−28.8019	−2.39186
$146$	−1.36471	−0.112944
$147$	−7.86230	−0.648471
$148$	12.0689	0.992057
$149$	22.5681	1.84885	0.924425	−	0.381363i	$-0.124545\pi$
$149$	22.5681	1.84885	0.924425	+	0.381363i	$0.124545\pi$
$150$	−45.4177	−3.70834
$151$	−7.25551	−0.590445	−0.295223	−	0.955429i	$-0.595394\pi$
$151$	−7.25551	−0.590445	−0.295223	+	0.955429i	$0.595394\pi$
$152$	−0.682398	−0.0553498
$153$	−1.53836	−0.124369
$154$	3.15357	0.254122
$155$	19.8457	1.59404
$156$	7.15458	0.572825
$157$	12.6197	1.00716	0.503582	−	0.863947i	$-0.332015\pi$
$157$	12.6197	1.00716	0.503582	+	0.863947i	$0.332015\pi$
$158$	22.6977	1.80573
$159$	−11.3199	−0.897729
$160$	−21.7033	−1.71580
$161$	−6.81813	−0.537344
$162$	19.4996	1.53203
$163$	22.8361	1.78866	0.894330	−	0.447409i	$-0.147653\pi$
$163$	22.8361	1.78866	0.894330	+	0.447409i	$0.147653\pi$
$164$	3.95136	0.308550
$165$	−8.86036	−0.689778
$166$	−11.0587	−0.858320
$167$	18.9126	1.46350	0.731749	−	0.681574i	$-0.238704\pi$
$167$	18.9126	1.46350	0.731749	+	0.681574i	$0.238704\pi$
$168$	−6.68388	−0.515672
$169$	−1.83541	−0.141186
$170$	−7.20994	−0.552977
$171$	−0.608683	−0.0465471
$172$	−1.00511	−0.0766388
$173$	7.37677	0.560846	0.280423	−	0.959877i	$-0.409525\pi$
$173$	7.37677	0.560846	0.280423	+	0.959877i	$0.409525\pi$
$174$	−25.5740	−1.93876
$175$	−22.3727	−1.69122
$176$	−4.99997	−0.376887
$177$	−11.3644	−0.854199
$178$	12.0399	0.902429
$179$	2.42944	0.181585	0.0907925	−	0.995870i	$-0.471060\pi$
$179$	2.42944	0.181585	0.0907925	+	0.995870i	$0.471060\pi$
$180$	−6.43093	−0.479333
$181$	−7.53724	−0.560239	−0.280119	−	0.959965i	$-0.590374\pi$
$181$	−7.53724	−0.560239	−0.280119	+	0.959965i	$0.590374\pi$
$182$	10.5372	0.781068
$183$	−30.8103	−2.27756
$184$	6.46395	0.476529
$185$	−49.9409	−3.67173
$186$	17.6215	1.29207
$187$	−1.00000	−0.0731272
$188$	0.485036	0.0353749
$189$	5.66451	0.412032
$190$	−2.85275	−0.206960
$191$	24.2995	1.75825	0.879125	−	0.476591i	$-0.158128\pi$
$191$	24.2995	1.75825	0.879125	+	0.476591i	$0.158128\pi$
$192$	2.03234	0.146672
$193$	−15.6469	−1.12629	−0.563144	−	0.826359i	$-0.690408\pi$
$193$	−15.6469	−1.12629	−0.563144	+	0.826359i	$0.690408\pi$
$194$	−16.6486	−1.19530
$195$	−29.6055	−2.12010
$196$	−3.70948	−0.264963
$197$	11.5970	0.826252	0.413126	−	0.910674i	$-0.364437\pi$
$197$	11.5970	0.826252	0.413126	+	0.910674i	$0.364437\pi$
$198$	−2.66679	−0.189521
$199$	−16.7890	−1.19014	−0.595069	−	0.803674i	$-0.702875\pi$
$199$	−16.7890	−1.19014	−0.595069	+	0.803674i	$0.702875\pi$
$200$	21.2105	1.49981
$201$	−20.3448	−1.43501
$202$	−0.485679	−0.0341723
$203$	−12.5977	−0.884186
$204$	−2.14123	−0.149916
$205$	−16.3507	−1.14198
$206$	11.9948	0.835715
$207$	5.76568	0.400743
$208$	−16.7066	−1.15840
$209$	−0.395669	−0.0273690
$210$	−27.9418	−1.92817
$211$	25.0401	1.72383	0.861917	−	0.507049i	$-0.169264\pi$
$211$	25.0401	1.72383	0.861917	+	0.507049i	$0.169264\pi$
$212$	−5.34081	−0.366808
$213$	−23.1020	−1.58292
$214$	8.70753	0.595234
$215$	4.15912	0.283650
$216$	−5.37026	−0.365400
$217$	8.68034	0.589260
$218$	−22.6261	−1.53243
$219$	1.67710	0.113328
$220$	−4.18037	−0.281840
$221$	−3.34135	−0.224763
$222$	−44.3439	−2.97617
$223$	−1.55714	−0.104274	−0.0521368	−	0.998640i	$-0.516603\pi$
$223$	−1.55714	−0.104274	−0.0521368	+	0.998640i	$0.516603\pi$
$224$	−9.49286	−0.634268
$225$	18.9193	1.26128
$226$	−8.50369	−0.565657
$227$	19.2961	1.28072	0.640362	−	0.768073i	$-0.278784\pi$
$227$	19.2961	1.28072	0.640362	+	0.768073i	$0.278784\pi$
$228$	−0.847217	−0.0561083
$229$	16.6755	1.10195	0.550974	−	0.834523i	$-0.314256\pi$
$229$	16.6755	1.10195	0.550974	+	0.834523i	$0.314256\pi$
$230$	27.0224	1.78180
$231$	−3.87545	−0.254986
$232$	11.9433	0.784116
$233$	16.9746	1.11204	0.556022	−	0.831168i	$-0.312327\pi$
$233$	16.9746	1.11204	0.556022	+	0.831168i	$0.312327\pi$
$234$	−8.91067	−0.582509
$235$	−2.00707	−0.130927
$236$	−5.36178	−0.349022
$237$	−27.8934	−1.81187
$238$	−3.15357	−0.204416
$239$	−2.40411	−0.155509	−0.0777544	−	0.996973i	$-0.524775\pi$
$239$	−2.40411	−0.155509	−0.0777544	+	0.996973i	$0.524775\pi$
$240$	44.3016	2.85965
$241$	−16.4944	−1.06250	−0.531249	−	0.847216i	$-0.678277\pi$
$241$	−16.4944	−1.06250	−0.531249	+	0.847216i	$0.678277\pi$
$242$	−1.73352	−0.111435
$243$	−14.6219	−0.937993
$244$	−14.5365	−0.930603
$245$	15.3498	0.980660
$246$	−14.5182	−0.925648
$247$	−1.32207	−0.0841211
$248$	−8.22943	−0.522569
$249$	13.5901	0.861238
$250$	52.6203	3.32800
$251$	19.0017	1.19938	0.599689	−	0.800233i	$-0.295291\pi$
$251$	19.0017	1.19938	0.599689	+	0.800233i	$0.295291\pi$
$252$	−2.81284	−0.177192
$253$	3.74793	0.235631
$254$	−32.6789	−2.05046
$255$	8.86036	0.554857
$256$	19.0508	1.19067
$257$	17.0767	1.06522	0.532608	−	0.846362i	$-0.321212\pi$
$257$	17.0767	1.06522	0.532608	+	0.846362i	$0.321212\pi$
$258$	3.69300	0.229916
$259$	−21.8438	−1.35731
$260$	−13.9681	−0.866262
$261$	10.6531	0.659413
$262$	1.59591	0.0985959
$263$	−6.92502	−0.427015	−0.213508	−	0.976941i	$-0.568489\pi$
$263$	−6.92502	−0.427015	−0.213508	+	0.976941i	$0.568489\pi$
$264$	3.67414	0.226128
$265$	22.1002	1.35760
$266$	−1.24777	−0.0765058
$267$	−14.7959	−0.905497
$268$	−9.59881	−0.586340
$269$	−4.89486	−0.298445	−0.149223	−	0.988804i	$-0.547677\pi$
$269$	−4.89486	−0.298445	−0.149223	+	0.988804i	$0.547677\pi$
$270$	−22.4502	−1.36628
$271$	−8.59353	−0.522020	−0.261010	−	0.965336i	$-0.584055\pi$
$271$	−8.59353	−0.522020	−0.261010	+	0.965336i	$0.584055\pi$
$272$	4.99997	0.303168
$273$	−12.9492	−0.783723
$274$	10.8102	0.653070
$275$	12.2983	0.741616
$276$	8.02518	0.483059
$277$	−30.3758	−1.82511	−0.912553	−	0.408959i	$-0.865892\pi$
$277$	−30.3758	−1.82511	−0.912553	+	0.408959i	$0.865892\pi$
$278$	9.75728	0.585203
$279$	−7.34045	−0.439461
$280$	13.0491	0.779833
$281$	−10.8956	−0.649977	−0.324989	−	0.945718i	$-0.605360\pi$
$281$	−10.8956	−0.649977	−0.324989	+	0.945718i	$0.605360\pi$
$282$	−1.78214	−0.106125
$283$	−8.28555	−0.492525	−0.246262	−	0.969203i	$-0.579202\pi$
$283$	−8.28555	−0.492525	−0.246262	+	0.969203i	$0.579202\pi$
$284$	−10.8997	−0.646776
$285$	3.50577	0.207664
$286$	−5.79231	−0.342506
$287$	−7.15166	−0.422149
$288$	8.02755	0.473028
$289$	1.00000	0.0588235
$290$	49.9287	2.93192
$291$	20.4596	1.19936
$292$	0.791267	0.0463054
$293$	−13.1064	−0.765683	−0.382841	−	0.923814i	$-0.625054\pi$
$293$	−13.1064	−0.765683	−0.382841	+	0.923814i	$0.625054\pi$
$294$	13.6295	0.794888
$295$	22.1869	1.29177
$296$	20.7091	1.20369
$297$	−3.11379	−0.180680
$298$	−39.1224	−2.26630
$299$	12.5231	0.724232
$300$	26.3334	1.52036
$301$	1.81917	0.104855
$302$	12.5776	0.723760
$303$	0.596856	0.0342885
$304$	1.97834	0.113465
$305$	60.1517	3.44428
$306$	2.66679	0.152450
$307$	−3.52390	−0.201120	−0.100560	−	0.994931i	$-0.532063\pi$
$307$	−3.52390	−0.201120	−0.100560	+	0.994931i	$0.532063\pi$
$308$	−1.82846	−0.104186
$309$	−14.7405	−0.838556
$310$	−34.4029	−1.95396
$311$	6.21116	0.352203	0.176101	−	0.984372i	$-0.443651\pi$
$311$	6.21116	0.352203	0.176101	+	0.984372i	$0.443651\pi$
$312$	12.2766	0.695024
$313$	−22.4968	−1.27160	−0.635798	−	0.771855i	$-0.719329\pi$
$313$	−22.4968	−1.27160	−0.635798	+	0.771855i	$0.719329\pi$
$314$	−21.8766	−1.23457
$315$	11.6395	0.655811
$316$	−13.1603	−0.740322
$317$	−4.91928	−0.276294	−0.138147	−	0.990412i	$-0.544115\pi$
$317$	−4.91928	−0.276294	−0.138147	+	0.990412i	$0.544115\pi$
$318$	19.6234	1.10042
$319$	6.92498	0.387725
$320$	−3.96780	−0.221807
$321$	−10.7008	−0.597258
$322$	11.8194	0.658669
$323$	0.395669	0.0220156
$324$	−11.3060	−0.628111
$325$	41.0929	2.27942
$326$	−39.5869	−2.19252
$327$	27.8054	1.53764
$328$	6.78016	0.374372
$329$	−0.877878	−0.0483990
$330$	15.3597	0.845521
$331$	29.0547	1.59699	0.798496	−	0.602000i	$-0.205629\pi$
$331$	29.0547	1.59699	0.798496	+	0.602000i	$0.205629\pi$
$332$	6.41189	0.351898
$333$	18.4720	1.01226
$334$	−32.7854	−1.79394
$335$	39.7197	2.17012
$336$	19.3772	1.05711
$337$	9.54892	0.520163	0.260081	−	0.965587i	$-0.416251\pi$
$337$	9.54892	0.520163	0.260081	+	0.965587i	$0.416251\pi$
$338$	3.18173	0.173063
$339$	10.4503	0.567580
$340$	4.18037	0.226712
$341$	−4.77160	−0.258396
$342$	1.05517	0.0570569
$343$	19.4481	1.05010
$344$	−1.72467	−0.0929879
$345$	−33.2080	−1.78786
$346$	−12.7878	−0.687478
$347$	18.7629	1.00725	0.503624	−	0.863923i	$-0.332000\pi$
$347$	18.7629	1.00725	0.503624	+	0.863923i	$0.332000\pi$
$348$	14.8280	0.794862
$349$	5.88659	0.315102	0.157551	−	0.987511i	$-0.449640\pi$
$349$	5.88659	0.315102	0.157551	+	0.987511i	$0.449640\pi$
$350$	38.7836	2.07307
$351$	−10.4042	−0.555337
$352$	5.21824	0.278133
$353$	18.7976	1.00049	0.500247	−	0.865883i	$-0.333243\pi$
$353$	18.7976	1.00049	0.500247	+	0.865883i	$0.333243\pi$
$354$	19.7004	1.04707
$355$	45.1026	2.39380
$356$	−6.98081	−0.369982
$357$	3.87545	0.205111
$358$	−4.21150	−0.222585
$359$	23.5778	1.24439	0.622193	−	0.782864i	$-0.286242\pi$
$359$	23.5778	1.24439	0.622193	+	0.782864i	$0.286242\pi$
$360$	−11.0349	−0.581588
$361$	−18.8434	−0.991760
$362$	13.0660	0.686733
$363$	2.13034	0.111814
$364$	−6.10952	−0.320226
$365$	−3.27425	−0.171382
$366$	53.4104	2.79181
$367$	30.9896	1.61764	0.808822	−	0.588054i	$-0.200106\pi$
$367$	30.9896	1.61764	0.808822	+	0.588054i	$0.200106\pi$
$368$	−18.7396	−0.976868
$369$	6.04774	0.314833
$370$	86.5738	4.50076
$371$	9.66646	0.501857
$372$	−10.2171	−0.529730
$373$	−12.5384	−0.649213	−0.324607	−	0.945849i	$-0.605232\pi$
$373$	−12.5384	−0.649213	−0.324607	+	0.945849i	$0.605232\pi$
$374$	1.73352	0.0896384
$375$	−64.6656	−3.33932
$376$	0.832275	0.0429213
$377$	23.1388	1.19171
$378$	−9.81956	−0.505064
$379$	7.02346	0.360771	0.180385	−	0.983596i	$-0.442265\pi$
$379$	7.02346	0.360771	0.180385	+	0.983596i	$0.442265\pi$
$380$	1.65404	0.0848506
$381$	40.1594	2.05743
$382$	−42.1238	−2.15524
$383$	3.15574	0.161251	0.0806254	−	0.996744i	$-0.474308\pi$
$383$	3.15574	0.161251	0.0806254	+	0.996744i	$0.474308\pi$
$384$	−25.7564	−1.31438
$385$	7.56615	0.385607
$386$	27.1243	1.38059
$387$	−1.53836	−0.0781994
$388$	9.65297	0.490055
$389$	19.4912	0.988241	0.494121	−	0.869393i	$-0.335490\pi$
$389$	19.4912	0.988241	0.494121	+	0.869393i	$0.335490\pi$
$390$	51.3219	2.59879
$391$	−3.74793	−0.189541
$392$	−6.36511	−0.321486
$393$	−1.96123	−0.0989311
$394$	−20.1037	−1.01281
$395$	54.4569	2.74002
$396$	1.54622	0.0777006
$397$	4.50829	0.226264	0.113132	−	0.993580i	$-0.463912\pi$
$397$	4.50829	0.226264	0.113132	+	0.993580i	$0.463912\pi$
$398$	29.1041	1.45886
$399$	1.53340	0.0767659
$400$	−61.4912	−3.07456
$401$	−19.4741	−0.972492	−0.486246	−	0.873822i	$-0.661634\pi$
$401$	−19.4741	−0.972492	−0.486246	+	0.873822i	$0.661634\pi$
$402$	35.2683	1.75902
$403$	−15.9436	−0.794205
$404$	0.281600	0.0140101
$405$	46.7840	2.32472
$406$	21.8385	1.08382
$407$	12.0076	0.595192
$408$	−3.67414	−0.181897
$409$	−37.4564	−1.85210	−0.926051	−	0.377398i	$-0.876819\pi$
$409$	−37.4564	−1.85210	−0.926051	+	0.377398i	$0.876819\pi$
$410$	28.3443	1.39983
$411$	−13.2848	−0.655290
$412$	−6.95464	−0.342631
$413$	9.70440	0.477523
$414$	−9.99496	−0.491225
$415$	−26.5323	−1.30242
$416$	17.4359	0.854867
$417$	−11.9908	−0.587192
$418$	0.685902	0.0335486
$419$	−5.37835	−0.262750	−0.131375	−	0.991333i	$-0.541939\pi$
$419$	−5.37835	−0.262750	−0.131375	+	0.991333i	$0.541939\pi$
$420$	16.2008	0.790520
$421$	−10.7539	−0.524112	−0.262056	−	0.965053i	$-0.584400\pi$
$421$	−10.7539	−0.524112	−0.262056	+	0.965053i	$0.584400\pi$
$422$	−43.4077	−2.11305
$423$	0.742370	0.0360952
$424$	−9.16432	−0.445059
$425$	−12.2983	−0.596555
$426$	40.0479	1.94033
$427$	26.3099	1.27323
$428$	−5.04868	−0.244037
$429$	7.11821	0.343671
$430$	−7.20994	−0.347694
$431$	31.0868	1.49740	0.748700	−	0.662909i	$-0.230678\pi$
$431$	31.0868	1.49740	0.748700	+	0.662909i	$0.230678\pi$
$432$	15.5689	0.749057
$433$	24.9861	1.20075	0.600377	−	0.799717i	$-0.295017\pi$
$433$	24.9861	1.20075	0.600377	+	0.799717i	$0.295017\pi$
$434$	−15.0476	−0.722307
$435$	−61.3578	−2.94188
$436$	13.1187	0.628274
$437$	−1.48294	−0.0709387
$438$	−2.90730	−0.138916
$439$	−14.7450	−0.703742	−0.351871	−	0.936049i	$-0.614454\pi$
$439$	−14.7450	−0.703742	−0.351871	+	0.936049i	$0.614454\pi$
$440$	−7.17311	−0.341965
$441$	−5.67752	−0.270358
$442$	5.79231	0.275512
$443$	7.33150	0.348330	0.174165	−	0.984716i	$-0.444277\pi$
$443$	7.33150	0.348330	0.174165	+	0.984716i	$0.444277\pi$
$444$	25.7109	1.22019
$445$	28.8865	1.36935
$446$	2.69933	0.127817
$447$	48.0778	2.27400
$448$	−1.73548	−0.0819939
$449$	8.64688	0.408071	0.204036	−	0.978963i	$-0.434594\pi$
$449$	8.64688	0.408071	0.204036	+	0.978963i	$0.434594\pi$
$450$	−32.7970	−1.54607
$451$	3.93128	0.185117
$452$	4.93049	0.231911
$453$	−15.4567	−0.726220
$454$	−33.4502	−1.56989
$455$	25.2811	1.18520
$456$	−1.45374	−0.0680777
$457$	1.01564	0.0475096	0.0237548	−	0.999718i	$-0.492438\pi$
$457$	1.01564	0.0475096	0.0237548	+	0.999718i	$0.492438\pi$
$458$	−28.9074	−1.35075
$459$	3.11379	0.145339
$460$	−15.6678	−0.730513
$461$	−18.7447	−0.873027	−0.436514	−	0.899698i	$-0.643787\pi$
$461$	−18.7447	−0.873027	−0.436514	+	0.899698i	$0.643787\pi$
$462$	6.71820	0.312559
$463$	−3.26052	−0.151529	−0.0757646	−	0.997126i	$-0.524140\pi$
$463$	−3.26052	−0.151529	−0.0757646	+	0.997126i	$0.524140\pi$
$464$	−34.6247	−1.60741
$465$	42.2781	1.96060
$466$	−29.4259	−1.36313
$467$	−23.2079	−1.07393	−0.536967	−	0.843603i	$-0.680430\pi$
$467$	−23.2079	−1.07393	−0.536967	+	0.843603i	$0.680430\pi$
$468$	5.16646	0.238820
$469$	17.3731	0.802215
$470$	3.47931	0.160489
$471$	26.8844	1.23877
$472$	−9.20029	−0.423478
$473$	−1.00000	−0.0459800
$474$	48.3538	2.22097
$475$	−4.86606	−0.223270
$476$	1.82846	0.0838074
$477$	−8.17435	−0.374278
$478$	4.16758	0.190621
$479$	−20.1309	−0.919803	−0.459902	−	0.887970i	$-0.652115\pi$
$479$	−20.1309	−0.919803	−0.459902	+	0.887970i	$0.652115\pi$
$480$	−46.2355	−2.11035
$481$	40.1214	1.82938
$482$	28.5935	1.30240
$483$	−14.5249	−0.660908
$484$	1.00511	0.0456867
$485$	−39.9438	−1.81376
$486$	25.3474	1.14978
$487$	16.0837	0.728822	0.364411	−	0.931238i	$-0.381270\pi$
$487$	16.0837	0.728822	0.364411	+	0.931238i	$0.381270\pi$
$488$	−24.9432	−1.12913
$489$	48.6487	2.19997
$490$	−26.6092	−1.20208
$491$	24.4966	1.10552	0.552759	−	0.833341i	$-0.313575\pi$
$491$	24.4966	1.10552	0.552759	+	0.833341i	$0.313575\pi$
$492$	8.41776	0.379502
$493$	−6.92498	−0.311886
$494$	2.29184	0.103115
$495$	−6.39824	−0.287580
$496$	23.8579	1.07125
$497$	19.7276	0.884902
$498$	−23.5588	−1.05569
$499$	−10.6609	−0.477249	−0.238625	−	0.971112i	$-0.576697\pi$
$499$	−10.6609	−0.477249	−0.238625	+	0.971112i	$0.576697\pi$
$500$	−30.5096	−1.36443
$501$	40.2903	1.80004
$502$	−32.9400	−1.47018
$503$	3.94210	0.175770	0.0878849	−	0.996131i	$-0.471989\pi$
$503$	3.94210	0.175770	0.0878849	+	0.996131i	$0.471989\pi$
$504$	−4.82656	−0.214992
$505$	−1.16526	−0.0518532
$506$	−6.49714	−0.288833
$507$	−3.91006	−0.173652
$508$	18.9474	0.840657
$509$	24.0289	1.06506	0.532531	−	0.846410i	$-0.321241\pi$
$509$	24.0289	1.06506	0.532531	+	0.846410i	$0.321241\pi$
$510$	−15.3597	−0.680137
$511$	−1.43213	−0.0633538
$512$	−8.84446	−0.390874
$513$	1.23203	0.0543955
$514$	−29.6029	−1.30573
$515$	28.7782	1.26812
$516$	−2.14123	−0.0942622
$517$	0.482571	0.0212235
$518$	37.8667	1.66377
$519$	15.7151	0.689815
$520$	−23.9678	−1.05106
$521$	−2.06632	−0.0905270	−0.0452635	−	0.998975i	$-0.514413\pi$
$521$	−2.06632	−0.0905270	−0.0452635	+	0.998975i	$0.514413\pi$
$522$	−18.4675	−0.808300
$523$	−40.6133	−1.77589	−0.887947	−	0.459945i	$-0.847869\pi$
$523$	−40.6133	−1.77589	−0.887947	+	0.459945i	$0.847869\pi$
$524$	−0.925321	−0.0404228
$525$	−47.6615	−2.08012
$526$	12.0047	0.523430
$527$	4.77160	0.207854
$528$	−10.6517	−0.463554
$529$	−8.95299	−0.389260
$530$	−38.3112	−1.66413
$531$	−8.20644	−0.356129
$532$	0.723466	0.0313662
$533$	13.1358	0.568973
$534$	25.6491	1.10995
$535$	20.8914	0.903212
$536$	−16.4706	−0.711423
$537$	5.17555	0.223341
$538$	8.48537	0.365830
$539$	−3.69062	−0.158966
$540$	13.0168	0.560153
$541$	26.4953	1.13912	0.569561	−	0.821949i	$-0.307113\pi$
$541$	26.4953	1.13912	0.569561	+	0.821949i	$0.307113\pi$
$542$	14.8971	0.639885
$543$	−16.0569	−0.689068
$544$	−5.21824	−0.223730
$545$	−54.2851	−2.32532
$546$	22.4478	0.960678
$547$	−3.85418	−0.164793	−0.0823965	−	0.996600i	$-0.526257\pi$
$547$	−3.85418	−0.164793	−0.0823965	+	0.996600i	$0.526257\pi$
$548$	−6.26784	−0.267749
$549$	−22.2487	−0.949553
$550$	−21.3194	−0.909063
$551$	−2.74000	−0.116728
$552$	13.7704	0.586108
$553$	23.8190	1.01289
$554$	52.6572	2.23719
$555$	−106.391	−4.51606
$556$	−5.65733	−0.239924
$557$	17.8754	0.757403	0.378702	−	0.925519i	$-0.376371\pi$
$557$	17.8754	0.757403	0.378702	+	0.925519i	$0.376371\pi$
$558$	12.7249	0.538686
$559$	−3.34135	−0.141324
$560$	−37.8305	−1.59863
$561$	−2.13034	−0.0899432
$562$	18.8878	0.796734
$563$	7.87441	0.331867	0.165933	−	0.986137i	$-0.446936\pi$
$563$	7.87441	0.331867	0.165933	+	0.986137i	$0.446936\pi$
$564$	1.03329	0.0435095
$565$	−20.4023	−0.858331
$566$	14.3632	0.603731
$567$	20.4630	0.859364
$568$	−18.7028	−0.784751
$569$	−27.4812	−1.15207	−0.576036	−	0.817424i	$-0.695401\pi$
$569$	−27.4812	−1.15207	−0.576036	+	0.817424i	$0.695401\pi$
$570$	−6.07734	−0.254552
$571$	−44.0173	−1.84207	−0.921034	−	0.389482i	$-0.872654\pi$
$571$	−44.0173	−1.84207	−0.921034	+	0.389482i	$0.872654\pi$
$572$	3.35841	0.140422
$573$	51.7663	2.16257
$574$	12.3976	0.517465
$575$	46.0932	1.92222
$576$	1.46760	0.0611498
$577$	12.4992	0.520347	0.260174	−	0.965562i	$-0.416220\pi$
$577$	12.4992	0.520347	0.260174	+	0.965562i	$0.416220\pi$
$578$	−1.73352	−0.0721051
$579$	−33.3333	−1.38528
$580$	−28.9490	−1.20204
$581$	−11.6050	−0.481458
$582$	−35.4673	−1.47016
$583$	−5.31367	−0.220070
$584$	1.35774	0.0561836
$585$	−21.3787	−0.883902
$586$	22.7202	0.938564
$587$	39.4789	1.62947	0.814735	−	0.579834i	$-0.196882\pi$
$587$	39.4789	1.62947	0.814735	+	0.579834i	$0.196882\pi$
$588$	−7.90246	−0.325892
$589$	1.88797	0.0777926
$590$	−38.4616	−1.58344
$591$	24.7056	1.01625
$592$	−60.0375	−2.46752
$593$	37.1415	1.52522	0.762610	−	0.646858i	$-0.223917\pi$
$593$	37.1415	1.52522	0.762610	+	0.646858i	$0.223917\pi$
$594$	5.39783	0.221476
$595$	−7.56615	−0.310182
$596$	22.6834	0.929148
$597$	−35.7663	−1.46382
$598$	−21.7092	−0.887754
$599$	15.8187	0.646334	0.323167	−	0.946342i	$-0.395253\pi$
$599$	15.8187	0.646334	0.323167	+	0.946342i	$0.395253\pi$
$600$	45.1857	1.84470
$601$	39.6107	1.61575	0.807877	−	0.589351i	$-0.200616\pi$
$601$	39.6107	1.61575	0.807877	+	0.589351i	$0.200616\pi$
$602$	−3.15357	−0.128530
$603$	−14.6914	−0.598280
$604$	−7.29257	−0.296731
$605$	−4.15912	−0.169092
$606$	−1.03466	−0.0420304
$607$	14.5160	0.589188	0.294594	−	0.955622i	$-0.404816\pi$
$607$	14.5160	0.589188	0.294594	+	0.955622i	$0.404816\pi$
$608$	−2.06470	−0.0837345
$609$	−26.8375	−1.08751
$610$	−104.275	−4.22195
$611$	1.61244	0.0652322
$612$	−1.54622	−0.0625023
$613$	−32.4613	−1.31110	−0.655550	−	0.755152i	$-0.727563\pi$
$613$	−32.4613	−1.31110	−0.655550	+	0.755152i	$0.727563\pi$
$614$	6.10878	0.246530
$615$	−34.8326	−1.40458
$616$	−3.13746	−0.126412
$617$	−10.0335	−0.403935	−0.201967	−	0.979392i	$-0.564734\pi$
$617$	−10.0335	−0.403935	−0.201967	+	0.979392i	$0.564734\pi$
$618$	25.5530	1.02789
$619$	14.0314	0.563970	0.281985	−	0.959419i	$-0.409007\pi$
$619$	14.0314	0.563970	0.281985	+	0.959419i	$0.409007\pi$
$620$	19.9470	0.801092
$621$	−11.6703	−0.468312
$622$	−10.7672	−0.431725
$623$	12.6347	0.506200
$624$	−35.5909	−1.42478
$625$	64.7567	2.59027
$626$	38.9988	1.55871
$627$	−0.842911	−0.0336626
$628$	12.6842	0.506155
$629$	−12.0076	−0.478773
$630$	−20.1773	−0.803884
$631$	30.5604	1.21659	0.608294	−	0.793712i	$-0.291854\pi$
$631$	30.5604	1.21659	0.608294	+	0.793712i	$0.291854\pi$
$632$	−22.5817	−0.898253
$633$	53.3441	2.12024
$634$	8.52769	0.338678
$635$	−78.4042	−3.11138
$636$	−11.3778	−0.451158
$637$	−12.3317	−0.488598
$638$	−12.0046	−0.475268
$639$	−16.6824	−0.659946
$640$	50.2849	1.98768
$641$	−37.1632	−1.46786	−0.733929	−	0.679226i	$-0.762315\pi$
$641$	−37.1632	−1.46786	−0.733929	+	0.679226i	$0.762315\pi$
$642$	18.5500	0.732111
$643$	48.4321	1.90997	0.954987	−	0.296648i	$-0.0958689\pi$
$643$	48.4321	1.90997	0.954987	+	0.296648i	$0.0958689\pi$
$644$	−6.85296	−0.270044
$645$	8.86036	0.348876
$646$	−0.685902	−0.0269865
$647$	−6.47105	−0.254403	−0.127202	−	0.991877i	$-0.540600\pi$
$647$	−6.47105	−0.254403	−0.127202	+	0.991877i	$0.540600\pi$
$648$	−19.4000	−0.762104
$649$	−5.33453	−0.209398
$650$	−71.2355	−2.79409
$651$	18.4921	0.724763
$652$	22.9527	0.898898
$653$	−6.13116	−0.239931	−0.119965	−	0.992778i	$-0.538278\pi$
$653$	−6.13116	−0.239931	−0.119965	+	0.992778i	$0.538278\pi$
$654$	−48.2013	−1.88482
$655$	3.82896	0.149610
$656$	−19.6563	−0.767449
$657$	1.21107	0.0472483
$658$	1.52182	0.0593269
$659$	16.9276	0.659407	0.329703	−	0.944085i	$-0.393051\pi$
$659$	16.9276	0.659407	0.329703	+	0.944085i	$0.393051\pi$
$660$	−8.90562	−0.346651
$661$	−8.40237	−0.326814	−0.163407	−	0.986559i	$-0.552248\pi$
$661$	−8.40237	−0.326814	−0.163407	+	0.986559i	$0.552248\pi$
$662$	−50.3671	−1.95757
$663$	−7.11821	−0.276448
$664$	11.0022	0.426968
$665$	−2.99369	−0.116090
$666$	−32.0216	−1.24081
$667$	25.9544	1.00496
$668$	19.0092	0.735487
$669$	−3.31723	−0.128252
$670$	−68.8551	−2.66010
$671$	−14.4626	−0.558323
$672$	−20.2231	−0.780121
$673$	−7.19800	−0.277462	−0.138731	−	0.990330i	$-0.544302\pi$
$673$	−7.19800	−0.277462	−0.138731	+	0.990330i	$0.544302\pi$
$674$	−16.5533	−0.637609
$675$	−38.2943	−1.47395
$676$	−1.84479	−0.0709534
$677$	4.59262	0.176509	0.0882543	−	0.996098i	$-0.471871\pi$
$677$	4.59262	0.176509	0.0882543	+	0.996098i	$0.471871\pi$
$678$	−18.1158	−0.695732
$679$	−17.4711	−0.670480
$680$	7.17311	0.275076
$681$	41.1072	1.57523
$682$	8.27168	0.316739
$683$	20.3470	0.778557	0.389278	−	0.921120i	$-0.372724\pi$
$683$	20.3470	0.778557	0.389278	+	0.921120i	$0.372724\pi$
$684$	−0.611792	−0.0233925
$685$	25.9362	0.990972
$686$	−33.7137	−1.28719
$687$	35.5245	1.35534
$688$	4.99997	0.190622
$689$	−17.7548	−0.676404
$690$	57.5670	2.19154
$691$	−40.6621	−1.54686	−0.773429	−	0.633883i	$-0.781460\pi$
$691$	−40.6621	−1.54686	−0.773429	+	0.633883i	$0.781460\pi$
$692$	7.41446	0.281855
$693$	−2.79854	−0.106308
$694$	−32.5260	−1.23467
$695$	23.4099	0.887990
$696$	25.4433	0.964427
$697$	−3.93128	−0.148908
$698$	−10.2045	−0.386248
$699$	36.1618	1.36776
$700$	−22.4870	−0.849928
$701$	−3.56567	−0.134673	−0.0673367	−	0.997730i	$-0.521450\pi$
$701$	−3.56567	−0.134673	−0.0673367	+	0.997730i	$0.521450\pi$
$702$	18.0360	0.680725
$703$	−4.75102	−0.179188
$704$	0.953998	0.0359552
$705$	−4.27575	−0.161034
$706$	−32.5861	−1.22639
$707$	−0.509674	−0.0191683
$708$	−11.4224	−0.429281
$709$	3.29202	0.123634	0.0618172	−	0.998087i	$-0.480310\pi$
$709$	3.29202	0.123634	0.0618172	+	0.998087i	$0.480310\pi$
$710$	−78.1865	−2.93429
$711$	−20.1423	−0.755397
$712$	−11.9784	−0.448910
$713$	−17.8836	−0.669747
$714$	−6.71820	−0.251422
$715$	−13.8971	−0.519721
$716$	2.44185	0.0912563
$717$	−5.12158	−0.191269
$718$	−40.8726	−1.52535
$719$	22.9828	0.857115	0.428558	−	0.903514i	$-0.359022\pi$
$719$	22.9828	0.857115	0.428558	+	0.903514i	$0.359022\pi$
$720$	31.9910	1.19224
$721$	12.5874	0.468778
$722$	32.6656	1.21569
$723$	−35.1387	−1.30682
$724$	−7.57574	−0.281550
$725$	85.1655	3.16297
$726$	−3.69300	−0.137060
$727$	25.2996	0.938311	0.469155	−	0.883116i	$-0.344558\pi$
$727$	25.2996	0.938311	0.469155	+	0.883116i	$0.344558\pi$
$728$	−10.4834	−0.388539
$729$	2.59599	0.0961479
$730$	5.67599	0.210078
$731$	1.00000	0.0369863
$732$	−30.9677	−1.14460
$733$	9.34073	0.345008	0.172504	−	0.985009i	$-0.444814\pi$
$733$	9.34073	0.345008	0.172504	+	0.985009i	$0.444814\pi$
$734$	−53.7212	−1.98289
$735$	32.7003	1.20617
$736$	19.5576	0.720904
$737$	−9.55002	−0.351780
$738$	−10.4839	−0.385918
$739$	−15.6028	−0.573957	−0.286978	−	0.957937i	$-0.592651\pi$
$739$	−15.6028	−0.573957	−0.286978	+	0.957937i	$0.592651\pi$
$740$	−50.1960	−1.84524
$741$	−2.81646	−0.103465
$742$	−16.7570	−0.615170
$743$	−2.48960	−0.0913346	−0.0456673	−	0.998957i	$-0.514541\pi$
$743$	−2.48960	−0.0913346	−0.0456673	+	0.998957i	$0.514541\pi$
$744$	−17.5315	−0.642736
$745$	−93.8635	−3.43889
$746$	21.7356	0.795797
$747$	9.81368	0.359064
$748$	−1.00511	−0.0367504
$749$	9.13772	0.333885
$750$	112.099	4.09329
$751$	16.8503	0.614877	0.307438	−	0.951568i	$-0.400528\pi$
$751$	16.8503	0.614877	0.307438	+	0.951568i	$0.400528\pi$
$752$	−2.41284	−0.0879873
$753$	40.4802	1.47518
$754$	−40.1116	−1.46078
$755$	30.1766	1.09824
$756$	5.69344	0.207069
$757$	25.7264	0.935042	0.467521	−	0.883982i	$-0.345147\pi$
$757$	25.7264	0.935042	0.467521	+	0.883982i	$0.345147\pi$
$758$	−12.1753	−0.442228
$759$	7.98439	0.289815
$760$	2.83818	0.102952
$761$	8.56334	0.310421	0.155210	−	0.987881i	$-0.450394\pi$
$761$	8.56334	0.310421	0.155210	+	0.987881i	$0.450394\pi$
$762$	−69.6173	−2.52197
$763$	−23.7439	−0.859587
$764$	24.4236	0.883616
$765$	6.39824	0.231329
$766$	−5.47055	−0.197659
$767$	−17.8245	−0.643605
$768$	40.5847	1.46447
$769$	12.4303	0.448248	0.224124	−	0.974561i	$-0.428048\pi$
$769$	12.4303	0.448248	0.224124	+	0.974561i	$0.428048\pi$
$770$	−13.1161	−0.472672
$771$	36.3793	1.31017
$772$	−15.7268	−0.566021
$773$	33.4064	1.20154	0.600772	−	0.799421i	$-0.294860\pi$
$773$	33.4064	1.20154	0.600772	+	0.799421i	$0.294860\pi$
$774$	2.66679	0.0958558
$775$	−58.6826	−2.10794
$776$	16.5636	0.594597
$777$	−46.5347	−1.66942
$778$	−33.7884	−1.21137
$779$	−1.55549	−0.0557311
$780$	−29.7568	−1.06546
$781$	−10.8443	−0.388038
$782$	6.49714	0.232337
$783$	−21.5629	−0.770596
$784$	18.4530	0.659037
$785$	−52.4870	−1.87334
$786$	3.39985	0.121268
$787$	−6.95976	−0.248089	−0.124044	−	0.992277i	$-0.539587\pi$
$787$	−6.95976	−0.248089	−0.124044	+	0.992277i	$0.539587\pi$
$788$	11.6562	0.415236
$789$	−14.7527	−0.525209
$790$	−94.4024	−3.35869
$791$	−8.92381	−0.317294
$792$	2.65317	0.0942762
$793$	−48.3245	−1.71606
$794$	−7.81523	−0.277352
$795$	47.0810	1.66979
$796$	−16.8747	−0.598109
$797$	21.1866	0.750469	0.375234	−	0.926930i	$-0.377562\pi$
$797$	21.1866	0.750469	0.375234	+	0.926930i	$0.377562\pi$
$798$	−2.65818	−0.0940987
$799$	−0.482571	−0.0170721
$800$	64.1755	2.26895
$801$	−10.6844	−0.377516
$802$	33.7589	1.19207
$803$	0.787245	0.0277813
$804$	−20.4488	−0.721172
$805$	28.3574	0.999468
$806$	27.6385	0.973526
$807$	−10.4277	−0.367074
$808$	0.483198	0.0169989
$809$	−52.0591	−1.83030	−0.915151	−	0.403111i	$-0.867929\pi$
$809$	−52.0591	−1.83030	−0.915151	+	0.403111i	$0.867929\pi$
$810$	−81.1012	−2.84961
$811$	18.3140	0.643092	0.321546	−	0.946894i	$-0.395798\pi$
$811$	18.3140	0.643092	0.321546	+	0.946894i	$0.395798\pi$
$812$	−12.6621	−0.444352
$813$	−18.3072	−0.642060
$814$	−20.8154	−0.729579
$815$	−94.9780	−3.32694
$816$	10.6517	0.372883
$817$	0.395669	0.0138427
$818$	64.9317	2.27028
$819$	−9.35090	−0.326747
$820$	−16.4342	−0.573907
$821$	−21.6929	−0.757087	−0.378543	−	0.925584i	$-0.623575\pi$
$821$	−21.6929	−0.757087	−0.378543	+	0.925584i	$0.623575\pi$
$822$	23.0295	0.803246
$823$	−34.8886	−1.21614	−0.608070	−	0.793884i	$-0.708056\pi$
$823$	−34.8886	−1.21614	−0.608070	+	0.793884i	$0.708056\pi$
$824$	−11.9335	−0.415723
$825$	26.1996	0.912153
$826$	−16.8228	−0.585341
$827$	18.4110	0.640213	0.320107	−	0.947382i	$-0.396281\pi$
$827$	18.4110	0.640213	0.320107	+	0.947382i	$0.396281\pi$
$828$	5.79514	0.201395
$829$	3.47139	0.120566	0.0602831	−	0.998181i	$-0.480800\pi$
$829$	3.47139	0.120566	0.0602831	+	0.998181i	$0.480800\pi$
$830$	45.9944	1.59649
$831$	−64.7109	−2.24480
$832$	3.18764	0.110511
$833$	3.69062	0.127873
$834$	20.7864	0.719773
$835$	−78.6597	−2.72213
$836$	−0.397690	−0.0137544
$837$	14.8577	0.513559
$838$	9.32350	0.322075
$839$	32.6769	1.12813	0.564066	−	0.825729i	$-0.309236\pi$
$839$	32.6769	1.12813	0.564066	+	0.825729i	$0.309236\pi$
$840$	27.7991	0.959159
$841$	18.9554	0.653634
$842$	18.6421	0.642450
$843$	−23.2114	−0.799442
$844$	25.1681	0.866320
$845$	7.63370	0.262607
$846$	−1.28692	−0.0442451
$847$	−1.81917	−0.0625074
$848$	26.5682	0.912355
$849$	−17.6511	−0.605783
$850$	21.3194	0.731250
$851$	45.0035	1.54270
$852$	−23.2200	−0.795505
$853$	14.5448	0.498004	0.249002	−	0.968503i	$-0.419897\pi$
$853$	14.5448	0.498004	0.249002	+	0.968503i	$0.419897\pi$
$854$	−45.6089	−1.56070
$855$	2.53159	0.0865784
$856$	−8.66305	−0.296097
$857$	−57.2896	−1.95698	−0.978488	−	0.206303i	$-0.933857\pi$
$857$	−57.2896	−1.95698	−0.978488	+	0.206303i	$0.933857\pi$
$858$	−12.3396	−0.421267
$859$	23.2202	0.792263	0.396131	−	0.918194i	$-0.370352\pi$
$859$	23.2202	0.792263	0.396131	+	0.918194i	$0.370352\pi$
$860$	4.18037	0.142549
$861$	−15.2355	−0.519224
$862$	−53.8898	−1.83549
$863$	41.9492	1.42797	0.713983	−	0.700163i	$-0.246889\pi$
$863$	41.9492	1.42797	0.713983	+	0.700163i	$0.246889\pi$
$864$	−16.2485	−0.552785
$865$	−30.6809	−1.04318
$866$	−43.3139	−1.47187
$867$	2.13034	0.0723503
$868$	8.72468	0.296135
$869$	−13.0934	−0.444162
$870$	106.365	3.60612
$871$	−31.9099	−1.08123
$872$	22.5105	0.762302
$873$	14.7743	0.500034
$874$	2.57072	0.0869558
$875$	55.2200	1.86678
$876$	1.68567	0.0569535
$877$	0.187881	0.00634430	0.00317215	−	0.999995i	$-0.498990\pi$
$877$	0.187881	0.00634430	0.00317215	+	0.999995i	$0.498990\pi$
$878$	25.5609	0.862638
$879$	−27.9211	−0.941755
$880$	20.7955	0.701016
$881$	−17.9792	−0.605736	−0.302868	−	0.953033i	$-0.597944\pi$
$881$	−17.9792	−0.605736	−0.302868	+	0.953033i	$0.597944\pi$
$882$	9.84212	0.331402
$883$	12.2957	0.413783	0.206891	−	0.978364i	$-0.433665\pi$
$883$	12.2957	0.413783	0.206891	+	0.978364i	$0.433665\pi$
$884$	−3.35841	−0.112956
$885$	47.2658	1.58882
$886$	−12.7093	−0.426978
$887$	15.8416	0.531909	0.265954	−	0.963986i	$-0.414313\pi$
$887$	15.8416	0.531909	0.265954	+	0.963986i	$0.414313\pi$
$888$	44.1174	1.48048
$889$	−34.2934	−1.15016
$890$	−50.0755	−1.67853
$891$	−11.2485	−0.376840
$892$	−1.56509	−0.0524031
$893$	−0.190938	−0.00638951
$894$	−83.3441	−2.78744
$895$	−10.1043	−0.337751
$896$	21.9942	0.734776
$897$	26.6786	0.890772
$898$	−14.9896	−0.500209
$899$	−33.0432	−1.10205
$900$	19.0159	0.633863
$901$	5.31367	0.177024
$902$	−6.81497	−0.226914
$903$	3.87545	0.128967
$904$	8.46025	0.281384
$905$	31.3483	1.04205
$906$	26.7946	0.890192
$907$	−0.113375	−0.00376457	−0.00188228	−	0.999998i	$-0.500599\pi$
$907$	−0.113375	−0.00376457	−0.00188228	+	0.999998i	$0.500599\pi$
$908$	19.3946	0.643633
$909$	0.431001	0.0142954
$910$	−43.8254	−1.45280
$911$	50.7226	1.68051	0.840257	−	0.542188i	$-0.182404\pi$
$911$	50.7226	1.68051	0.840257	+	0.542188i	$0.182404\pi$
$912$	4.21453	0.139557
$913$	6.37930	0.211124
$914$	−1.76064	−0.0582367
$915$	128.144	4.23631
$916$	16.7607	0.553788
$917$	1.67476	0.0553054
$918$	−5.39783	−0.178155
$919$	1.42520	0.0470129	0.0235064	−	0.999724i	$-0.492517\pi$
$919$	1.42520	0.0470129	0.0235064	+	0.999724i	$0.492517\pi$
$920$	−26.8844	−0.886351
$921$	−7.50713	−0.247368
$922$	32.4944	1.07015
$923$	−36.2344	−1.19267
$924$	−3.89525	−0.128144
$925$	147.673	4.85544
$926$	5.65219	0.185743
$927$	−10.6444	−0.349608
$928$	36.1362	1.18623
$929$	−6.45400	−0.211749	−0.105874	−	0.994380i	$-0.533764\pi$
$929$	−6.45400	−0.211749	−0.105874	+	0.994380i	$0.533764\pi$
$930$	−73.2901	−2.40328
$931$	1.46027	0.0478583
$932$	17.0613	0.558862
$933$	13.2319	0.433193
$934$	40.2315	1.31641
$935$	4.15912	0.136018
$936$	8.86515	0.289766
$937$	−36.7454	−1.20042	−0.600210	−	0.799843i	$-0.704916\pi$
$937$	−36.7454	−1.20042	−0.600210	+	0.799843i	$0.704916\pi$
$938$	−30.1167	−0.983345
$939$	−47.9260	−1.56401
$940$	−2.01733	−0.0657979
$941$	−51.8476	−1.69018	−0.845092	−	0.534622i	$-0.820454\pi$
$941$	−51.8476	−1.69018	−0.845092	+	0.534622i	$0.820454\pi$
$942$	−46.6047	−1.51846
$943$	14.7342	0.479811
$944$	26.6725	0.868116
$945$	−23.5594	−0.766387
$946$	1.73352	0.0563618
$947$	19.7023	0.640239	0.320120	−	0.947377i	$-0.396277\pi$
$947$	19.7023	0.640239	0.320120	+	0.947377i	$0.396277\pi$
$948$	−28.0359	−0.910562
$949$	2.63046	0.0853882
$950$	8.43543	0.273682
$951$	−10.4798	−0.339829
$952$	3.13746	0.101686
$953$	11.1653	0.361681	0.180841	−	0.983512i	$-0.442118\pi$
$953$	11.1653	0.361681	0.180841	+	0.983512i	$0.442118\pi$
$954$	14.1704	0.458785
$955$	−101.065	−3.27037
$956$	−2.41639	−0.0781516
$957$	14.7526	0.476884
$958$	34.8974	1.12748
$959$	11.3443	0.366327
$960$	−8.45277	−0.272812
$961$	−8.23186	−0.265544
$962$	−69.5514	−2.24243
$963$	−7.72723	−0.249006
$964$	−16.5787	−0.533963
$965$	65.0773	2.09491
$966$	25.1794	0.810133
$967$	47.2611	1.51982	0.759908	−	0.650031i	$-0.225244\pi$
$967$	47.2611	1.51982	0.759908	+	0.650031i	$0.225244\pi$
$968$	1.72467	0.0554330
$969$	0.842911	0.0270782
$970$	69.2436	2.22328
$971$	−53.2219	−1.70797	−0.853986	−	0.520295i	$-0.825822\pi$
$971$	−53.2219	−1.70797	−0.853986	+	0.520295i	$0.825822\pi$
$972$	−14.6966	−0.471392
$973$	10.2393	0.328258
$974$	−27.8815	−0.893381
$975$	87.5419	2.80359
$976$	72.3126	2.31467
$977$	39.1537	1.25264	0.626319	−	0.779567i	$-0.284561\pi$
$977$	39.1537	1.25264	0.626319	+	0.779567i	$0.284561\pi$
$978$	−84.3337	−2.69669
$979$	−6.94533	−0.221974
$980$	15.4282	0.492835
$981$	20.0788	0.641067
$982$	−42.4655	−1.35513
$983$	20.0267	0.638752	0.319376	−	0.947628i	$-0.396527\pi$
$983$	20.0267	0.638752	0.319376	+	0.947628i	$0.396527\pi$
$984$	14.4441	0.460460
$985$	−48.2334	−1.53684
$986$	12.0046	0.382305
$987$	−1.87018	−0.0595286
$988$	−1.32882	−0.0422754
$989$	−3.74793	−0.119177
$990$	11.0915	0.352511
$991$	17.0975	0.543120	0.271560	−	0.962421i	$-0.412460\pi$
$991$	17.0975	0.543120	0.271560	+	0.962421i	$0.412460\pi$
$992$	−24.8993	−0.790555
$993$	61.8965	1.96423
$994$	−34.1982	−1.08470
$995$	69.8274	2.21368
$996$	13.6595	0.432819
$997$	−14.4961	−0.459097	−0.229549	−	0.973297i	$-0.573725\pi$
$997$	−14.4961	−0.459097	−0.229549	+	0.973297i	$0.573725\pi$
$998$	18.4810	0.585006
$999$	−37.3890	−1.18294

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8041.2.a.i.1.17	✓	78

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8041.2.a.i.1.17	✓	78	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)

\(f(q)\)	\(=\)	\(q-1.73352 q^{2} +2.13034 q^{3} +1.00511 q^{4} -4.15912 q^{5} -3.69300 q^{6} -1.81917 q^{7} +1.72467 q^{8} +1.53836 q^{9} +O(q^{10})\)	\(q-1.73352 q^{2} +2.13034 q^{3} +1.00511 q^{4} -4.15912 q^{5} -3.69300 q^{6} -1.81917 q^{7} +1.72467 q^{8} +1.53836 q^{9} +7.20994 q^{10} +1.00000 q^{11} +2.14123 q^{12} +3.34135 q^{13} +3.15357 q^{14} -8.86036 q^{15} -4.99997 q^{16} -1.00000 q^{17} -2.66679 q^{18} -0.395669 q^{19} -4.18037 q^{20} -3.87545 q^{21} -1.73352 q^{22} +3.74793 q^{23} +3.67414 q^{24} +12.2983 q^{25} -5.79231 q^{26} -3.11379 q^{27} -1.82846 q^{28} +6.92498 q^{29} +15.3597 q^{30} -4.77160 q^{31} +5.21824 q^{32} +2.13034 q^{33} +1.73352 q^{34} +7.56615 q^{35} +1.54622 q^{36} +12.0076 q^{37} +0.685902 q^{38} +7.11821 q^{39} -7.17311 q^{40} +3.93128 q^{41} +6.71820 q^{42} -1.00000 q^{43} +1.00511 q^{44} -6.39824 q^{45} -6.49714 q^{46} +0.482571 q^{47} -10.6517 q^{48} -3.69062 q^{49} -21.3194 q^{50} -2.13034 q^{51} +3.35841 q^{52} -5.31367 q^{53} +5.39783 q^{54} -4.15912 q^{55} -3.13746 q^{56} -0.842911 q^{57} -12.0046 q^{58} -5.33453 q^{59} -8.90562 q^{60} -14.4626 q^{61} +8.27168 q^{62} -2.79854 q^{63} +0.953998 q^{64} -13.8971 q^{65} -3.69300 q^{66} -9.55002 q^{67} -1.00511 q^{68} +7.98439 q^{69} -13.1161 q^{70} -10.8443 q^{71} +2.65317 q^{72} +0.787245 q^{73} -20.8154 q^{74} +26.1996 q^{75} -0.397690 q^{76} -1.81917 q^{77} -12.3396 q^{78} -13.0934 q^{79} +20.7955 q^{80} -11.2485 q^{81} -6.81497 q^{82} +6.37930 q^{83} -3.89525 q^{84} +4.15912 q^{85} +1.73352 q^{86} +14.7526 q^{87} +1.72467 q^{88} -6.94533 q^{89} +11.0915 q^{90} -6.07847 q^{91} +3.76708 q^{92} -10.1651 q^{93} -0.836549 q^{94} +1.64564 q^{95} +11.1166 q^{96} +9.60390 q^{97} +6.39779 q^{98} +1.53836 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 78 q + 7 q^{2} + 10 q^{3} + 91 q^{4} + 17 q^{5} + 12 q^{6} + 11 q^{7} + 33 q^{8} + 102 q^{9}+O(q^{10}) \) 78 * q + 7 * q^2 + 10 * q^3 + 91 * q^4 + 17 * q^5 + 12 * q^6 + 11 * q^7 + 33 * q^8 + 102 * q^9	\( 78 q + 7 q^{2} + 10 q^{3} + 91 q^{4} + 17 q^{5} + 12 q^{6} + 11 q^{7} + 33 q^{8} + 102 q^{9} + 3 q^{10} + 78 q^{11} + 31 q^{12} - 16 q^{13} + 31 q^{14} + 38 q^{15} + 121 q^{16} - 78 q^{17} + 11 q^{18} + 51 q^{20} + 6 q^{21} + 7 q^{22} + 48 q^{23} + 11 q^{24} + 101 q^{25} + 18 q^{26} + 46 q^{27} + 27 q^{28} + 22 q^{29} + 14 q^{30} + 56 q^{31} + 83 q^{32} + 10 q^{33} - 7 q^{34} + 24 q^{35} + 139 q^{36} + 53 q^{37} + 10 q^{38} + 79 q^{39} - q^{40} + 23 q^{41} + 17 q^{42} - 78 q^{43} + 91 q^{44} + 76 q^{45} + 21 q^{46} + 57 q^{47} + 78 q^{48} + 115 q^{49} + 58 q^{50} - 10 q^{51} - 63 q^{52} + 22 q^{53} - 18 q^{54} + 17 q^{55} + 111 q^{56} - 11 q^{57} + 36 q^{58} + 71 q^{59} + 36 q^{60} + 4 q^{61} - 5 q^{62} + 71 q^{63} + 183 q^{64} + 47 q^{65} + 12 q^{66} + 11 q^{67} - 91 q^{68} + 31 q^{69} + 33 q^{70} + 159 q^{71} + 59 q^{72} + 2 q^{73} - 4 q^{74} + 83 q^{75} - 44 q^{76} + 11 q^{77} + 101 q^{78} + 35 q^{79} + 85 q^{80} + 170 q^{81} + 98 q^{82} - 32 q^{83} + 44 q^{84} - 17 q^{85} - 7 q^{86} - 6 q^{87} + 33 q^{88} + 50 q^{89} - 5 q^{90} + 86 q^{91} + 106 q^{92} + 68 q^{93} - q^{94} + 109 q^{95} - 50 q^{96} + 40 q^{97} + 106 q^{98} + 102 q^{99}+O(q^{100}) \) 78 * q + 7 * q^2 + 10 * q^3 + 91 * q^4 + 17 * q^5 + 12 * q^6 + 11 * q^7 + 33 * q^8 + 102 * q^9 + 3 * q^10 + 78 * q^11 + 31 * q^12 - 16 * q^13 + 31 * q^14 + 38 * q^15 + 121 * q^16 - 78 * q^17 + 11 * q^18 + 51 * q^20 + 6 * q^21 + 7 * q^22 + 48 * q^23 + 11 * q^24 + 101 * q^25 + 18 * q^26 + 46 * q^27 + 27 * q^28 + 22 * q^29 + 14 * q^30 + 56 * q^31 + 83 * q^32 + 10 * q^33 - 7 * q^34 + 24 * q^35 + 139 * q^36 + 53 * q^37 + 10 * q^38 + 79 * q^39 - q^40 + 23 * q^41 + 17 * q^42 - 78 * q^43 + 91 * q^44 + 76 * q^45 + 21 * q^46 + 57 * q^47 + 78 * q^48 + 115 * q^49 + 58 * q^50 - 10 * q^51 - 63 * q^52 + 22 * q^53 - 18 * q^54 + 17 * q^55 + 111 * q^56 - 11 * q^57 + 36 * q^58 + 71 * q^59 + 36 * q^60 + 4 * q^61 - 5 * q^62 + 71 * q^63 + 183 * q^64 + 47 * q^65 + 12 * q^66 + 11 * q^67 - 91 * q^68 + 31 * q^69 + 33 * q^70 + 159 * q^71 + 59 * q^72 + 2 * q^73 - 4 * q^74 + 83 * q^75 - 44 * q^76 + 11 * q^77 + 101 * q^78 + 35 * q^79 + 85 * q^80 + 170 * q^81 + 98 * q^82 - 32 * q^83 + 44 * q^84 - 17 * q^85 - 7 * q^86 - 6 * q^87 + 33 * q^88 + 50 * q^89 - 5 * q^90 + 86 * q^91 + 106 * q^92 + 68 * q^93 - q^94 + 109 * q^95 - 50 * q^96 + 40 * q^97 + 106 * q^98 + 102 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more