LMFDB - Embedded newform 241.2.a.b.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [241,2,Mod(1,241)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(241, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("241.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	241.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:	$1.92439468871$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 14 x^{10} + 44 x^{9} + 65 x^{8} - 219 x^{7} - 123 x^{6} + 444 x^{5} + 105 x^{4} + \cdots - 1 $ x^12 - 3x^11 - 14x^10 + 44x^9 + 65x^8 - 219x^7 - 123x^6 + 444x^5 + 105x^4 - 328x^3 - 45x^2 + 18*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$2.70063$ of defining polynomial
Character	$\chi$	$=$	241.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.70063 q^{2} -2.50808 q^{3} +5.29342 q^{4} +0.533570 q^{5} -6.77340 q^{6} +0.354992 q^{7} +8.89432 q^{8} +3.29045 q^{9} +O(q^{10})$	\(q+2.70063 q^{2} -2.50808 q^{3} +5.29342 q^{4} +0.533570 q^{5} -6.77340 q^{6} +0.354992 q^{7} +8.89432 q^{8} +3.29045 q^{9} +1.44098 q^{10} +4.18781 q^{11} -13.2763 q^{12} -3.72447 q^{13} +0.958703 q^{14} -1.33823 q^{15} +13.4334 q^{16} -6.46259 q^{17} +8.88631 q^{18} -1.31002 q^{19} +2.82441 q^{20} -0.890347 q^{21} +11.3097 q^{22} -4.10799 q^{23} -22.3076 q^{24} -4.71530 q^{25} -10.0584 q^{26} -0.728479 q^{27} +1.87912 q^{28} -8.85227 q^{29} -3.61408 q^{30} +5.11371 q^{31} +18.4902 q^{32} -10.5034 q^{33} -17.4531 q^{34} +0.189413 q^{35} +17.4177 q^{36} +5.41403 q^{37} -3.53789 q^{38} +9.34127 q^{39} +4.74574 q^{40} +11.8244 q^{41} -2.40450 q^{42} +0.673253 q^{43} +22.1678 q^{44} +1.75569 q^{45} -11.0942 q^{46} +5.22965 q^{47} -33.6921 q^{48} -6.87398 q^{49} -12.7343 q^{50} +16.2087 q^{51} -19.7152 q^{52} -9.92404 q^{53} -1.96735 q^{54} +2.23449 q^{55} +3.15741 q^{56} +3.28564 q^{57} -23.9067 q^{58} -1.23315 q^{59} -7.08384 q^{60} +4.04837 q^{61} +13.8102 q^{62} +1.16808 q^{63} +23.0683 q^{64} -1.98727 q^{65} -28.3657 q^{66} -14.0522 q^{67} -34.2092 q^{68} +10.3032 q^{69} +0.511535 q^{70} +13.0525 q^{71} +29.2663 q^{72} +7.76916 q^{73} +14.6213 q^{74} +11.8263 q^{75} -6.93450 q^{76} +1.48664 q^{77} +25.2273 q^{78} -1.17673 q^{79} +7.16768 q^{80} -8.04428 q^{81} +31.9333 q^{82} +6.25297 q^{83} -4.71298 q^{84} -3.44824 q^{85} +1.81821 q^{86} +22.2022 q^{87} +37.2477 q^{88} +3.80839 q^{89} +4.74147 q^{90} -1.32216 q^{91} -21.7453 q^{92} -12.8256 q^{93} +14.1234 q^{94} -0.698989 q^{95} -46.3748 q^{96} -9.91591 q^{97} -18.5641 q^{98} +13.7798 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 3 q^{2} + q^{3} + 13 q^{4} + 6 q^{5} - q^{6} + 3 q^{7} + 9 q^{8} + 15 q^{9}+O(q^{10}) $ 12 * q + 3 * q^2 + q^3 + 13 * q^4 + 6 * q^5 - q^6 + 3 * q^7 + 9 * q^8 + 15 * q^9	\( 12 q + 3 q^{2} + q^{3} + 13 q^{4} + 6 q^{5} - q^{6} + 3 q^{7} + 9 q^{8} + 15 q^{9} - 7 q^{10} + 22 q^{11} - 7 q^{12} - 5 q^{13} + 6 q^{14} + 13 q^{15} + 15 q^{16} - 4 q^{17} - q^{18} - 6 q^{19} + 10 q^{20} - 14 q^{21} - 12 q^{22} + 32 q^{23} - 15 q^{24} + 4 q^{25} + 8 q^{26} - 5 q^{27} - 11 q^{28} + 6 q^{29} - 19 q^{30} + 8 q^{31} + q^{32} - 24 q^{33} - 19 q^{34} + 15 q^{35} - 8 q^{36} - 8 q^{37} - 10 q^{38} + 31 q^{39} - 52 q^{40} - q^{41} - 49 q^{42} - 2 q^{43} + 42 q^{44} - 15 q^{45} - 25 q^{46} + 34 q^{47} - 49 q^{48} - 9 q^{49} - 27 q^{50} - 3 q^{51} - 41 q^{52} + 5 q^{53} - 40 q^{54} - 3 q^{55} + q^{56} - 22 q^{57} - 33 q^{58} + 26 q^{59} - 57 q^{60} - 26 q^{61} - 17 q^{62} - 4 q^{63} + 13 q^{64} - 25 q^{65} - 2 q^{66} + 6 q^{67} - 35 q^{68} - 2 q^{69} - 4 q^{70} + 94 q^{71} + 17 q^{72} - 22 q^{73} + 26 q^{74} - 20 q^{76} - 7 q^{77} + 54 q^{78} + 9 q^{79} + 19 q^{80} + 4 q^{81} + 15 q^{82} - 8 q^{83} + 2 q^{84} + 4 q^{85} + 9 q^{86} + 4 q^{87} + 6 q^{88} - 3 q^{89} + 11 q^{90} - 20 q^{91} + 36 q^{92} + 12 q^{93} + 48 q^{94} + 33 q^{95} - 23 q^{96} - 29 q^{97} + 28 q^{98} + 36 q^{99}+O(q^{100}) \) 12 * q + 3 * q^2 + q^3 + 13 * q^4 + 6 * q^5 - q^6 + 3 * q^7 + 9 * q^8 + 15 * q^9 - 7 * q^10 + 22 * q^11 - 7 * q^12 - 5 * q^13 + 6 * q^14 + 13 * q^15 + 15 * q^16 - 4 * q^17 - q^18 - 6 * q^19 + 10 * q^20 - 14 * q^21 - 12 * q^22 + 32 * q^23 - 15 * q^24 + 4 * q^25 + 8 * q^26 - 5 * q^27 - 11 * q^28 + 6 * q^29 - 19 * q^30 + 8 * q^31 + q^32 - 24 * q^33 - 19 * q^34 + 15 * q^35 - 8 * q^36 - 8 * q^37 - 10 * q^38 + 31 * q^39 - 52 * q^40 - q^41 - 49 * q^42 - 2 * q^43 + 42 * q^44 - 15 * q^45 - 25 * q^46 + 34 * q^47 - 49 * q^48 - 9 * q^49 - 27 * q^50 - 3 * q^51 - 41 * q^52 + 5 * q^53 - 40 * q^54 - 3 * q^55 + q^56 - 22 * q^57 - 33 * q^58 + 26 * q^59 - 57 * q^60 - 26 * q^61 - 17 * q^62 - 4 * q^63 + 13 * q^64 - 25 * q^65 - 2 * q^66 + 6 * q^67 - 35 * q^68 - 2 * q^69 - 4 * q^70 + 94 * q^71 + 17 * q^72 - 22 * q^73 + 26 * q^74 - 20 * q^76 - 7 * q^77 + 54 * q^78 + 9 * q^79 + 19 * q^80 + 4 * q^81 + 15 * q^82 - 8 * q^83 + 2 * q^84 + 4 * q^85 + 9 * q^86 + 4 * q^87 + 6 * q^88 - 3 * q^89 + 11 * q^90 - 20 * q^91 + 36 * q^92 + 12 * q^93 + 48 * q^94 + 33 * q^95 - 23 * q^96 - 29 * q^97 + 28 * q^98 + 36 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.70063	1.90964	0.954818	−	0.297191i	$-0.0960499\pi$
$2$	2.70063	1.90964	0.954818	+	0.297191i	$0.0960499\pi$
$3$	−2.50808	−1.44804	−0.724020	−	0.689779i	$-0.757708\pi$
$3$	−2.50808	−1.44804	−0.724020	+	0.689779i	$0.757708\pi$
$4$	5.29342	2.64671
$5$	0.533570	0.238620	0.119310	−	0.992857i	$-0.461932\pi$
$5$	0.533570	0.238620	0.119310	+	0.992857i	$0.461932\pi$
$6$	−6.77340	−2.76523
$7$	0.354992	0.134174	0.0670872	−	0.997747i	$-0.478629\pi$
$7$	0.354992	0.134174	0.0670872	+	0.997747i	$0.478629\pi$
$8$	8.89432	3.14462
$9$	3.29045	1.09682
$10$	1.44098	0.455677
$11$	4.18781	1.26267	0.631336	−	0.775509i	$-0.282507\pi$
$11$	4.18781	1.26267	0.631336	+	0.775509i	$0.282507\pi$
$12$	−13.2763	−3.83254
$13$	−3.72447	−1.03298	−0.516492	−	0.856292i	$-0.672762\pi$
$13$	−3.72447	−1.03298	−0.516492	+	0.856292i	$0.672762\pi$
$14$	0.958703	0.256224
$15$	−1.33823	−0.345531
$16$	13.4334	3.35836
$17$	−6.46259	−1.56741	−0.783704	−	0.621134i	$-0.786672\pi$
$17$	−6.46259	−1.56741	−0.783704	+	0.621134i	$0.786672\pi$
$18$	8.88631	2.09452
$19$	−1.31002	−0.300540	−0.150270	−	0.988645i	$-0.548014\pi$
$19$	−1.31002	−0.300540	−0.150270	+	0.988645i	$0.548014\pi$
$20$	2.82441	0.631557
$21$	−0.890347	−0.194290
$22$	11.3097	2.41124
$23$	−4.10799	−0.856576	−0.428288	−	0.903642i	$-0.640883\pi$
$23$	−4.10799	−0.856576	−0.428288	+	0.903642i	$0.640883\pi$
$24$	−22.3076	−4.55353
$25$	−4.71530	−0.943061
$26$	−10.0584	−1.97262
$27$	−0.728479	−0.140196
$28$	1.87912	0.355121
$29$	−8.85227	−1.64382	−0.821912	−	0.569614i	$-0.807093\pi$
$29$	−8.85227	−1.64382	−0.821912	+	0.569614i	$0.807093\pi$
$30$	−3.61408	−0.659838
$31$	5.11371	0.918449	0.459225	−	0.888320i	$-0.348127\pi$
$31$	5.11371	0.918449	0.459225	+	0.888320i	$0.348127\pi$
$32$	18.4902	3.26863
$33$	−10.5034	−1.82840
$34$	−17.4531	−2.99318
$35$	0.189413	0.0320166
$36$	17.4177	2.90296
$37$	5.41403	0.890061	0.445030	−	0.895515i	$-0.353193\pi$
$37$	5.41403	0.890061	0.445030	+	0.895515i	$0.353193\pi$
$38$	−3.53789	−0.573922
$39$	9.34127	1.49580
$40$	4.74574	0.750367
$41$	11.8244	1.84665	0.923327	−	0.384014i	$-0.125459\pi$
$41$	11.8244	1.84665	0.923327	+	0.384014i	$0.125459\pi$
$42$	−2.40450	−0.371023
$43$	0.673253	0.102670	0.0513350	−	0.998681i	$-0.483652\pi$
$43$	0.673253	0.102670	0.0513350	+	0.998681i	$0.483652\pi$
$44$	22.1678	3.34193
$45$	1.75569	0.261722
$46$	−11.0942	−1.63575
$47$	5.22965	0.762823	0.381412	−	0.924405i	$-0.375438\pi$
$47$	5.22965	0.762823	0.381412	+	0.924405i	$0.375438\pi$
$48$	−33.6921	−4.86304
$49$	−6.87398	−0.981997
$50$	−12.7343	−1.80090
$51$	16.2087	2.26967
$52$	−19.7152	−2.73401
$53$	−9.92404	−1.36317	−0.681586	−	0.731738i	$-0.738709\pi$
$53$	−9.92404	−1.36317	−0.681586	+	0.731738i	$0.738709\pi$
$54$	−1.96735	−0.267723
$55$	2.23449	0.301298
$56$	3.15741	0.421927
$57$	3.28564	0.435194
$58$	−23.9067	−3.13911
$59$	−1.23315	−0.160542	−0.0802710	−	0.996773i	$-0.525579\pi$
$59$	−1.23315	−0.160542	−0.0802710	+	0.996773i	$0.525579\pi$
$60$	−7.08384	−0.914519
$61$	4.04837	0.518341	0.259170	−	0.965832i	$-0.416551\pi$
$61$	4.04837	0.518341	0.259170	+	0.965832i	$0.416551\pi$
$62$	13.8102	1.75390
$63$	1.16808	0.147165
$64$	23.0683	2.88354
$65$	−1.98727	−0.246490
$66$	−28.3657	−3.49158
$67$	−14.0522	−1.71675	−0.858375	−	0.513022i	$-0.828526\pi$
$67$	−14.0522	−1.71675	−0.858375	+	0.513022i	$0.828526\pi$
$68$	−34.2092	−4.14847
$69$	10.3032	1.24036
$70$	0.511535	0.0611401
$71$	13.0525	1.54905	0.774526	−	0.632542i	$-0.217989\pi$
$71$	13.0525	1.54905	0.774526	+	0.632542i	$0.217989\pi$
$72$	29.2663	3.44907
$73$	7.76916	0.909312	0.454656	−	0.890667i	$-0.349762\pi$
$73$	7.76916	0.909312	0.454656	+	0.890667i	$0.349762\pi$
$74$	14.6213	1.69969
$75$	11.8263	1.36559
$76$	−6.93450	−0.795442
$77$	1.48664	0.169418
$78$	25.2273	2.85643
$79$	−1.17673	−0.132393	−0.0661963	−	0.997807i	$-0.521086\pi$
$79$	−1.17673	−0.132393	−0.0661963	+	0.997807i	$0.521086\pi$
$80$	7.16768	0.801371
$81$	−8.04428	−0.893809
$82$	31.9333	3.52644
$83$	6.25297	0.686353	0.343176	−	0.939271i	$-0.388497\pi$
$83$	6.25297	0.686353	0.343176	+	0.939271i	$0.388497\pi$
$84$	−4.71298	−0.514228
$85$	−3.44824	−0.374014
$86$	1.81821	0.196062
$87$	22.2022	2.38032
$88$	37.2477	3.97062
$89$	3.80839	0.403688	0.201844	−	0.979418i	$-0.435307\pi$
$89$	3.80839	0.403688	0.201844	+	0.979418i	$0.435307\pi$
$90$	4.74147	0.499794
$91$	−1.32216	−0.138600
$92$	−21.7453	−2.26711
$93$	−12.8256	−1.32995
$94$	14.1234	1.45671
$95$	−0.698989	−0.0717147
$96$	−46.3748	−4.73311
$97$	−9.91591	−1.00681	−0.503404	−	0.864051i	$-0.667919\pi$
$97$	−9.91591	−1.00681	−0.503404	+	0.864051i	$0.667919\pi$
$98$	−18.5641	−1.87526
$99$	13.7798	1.38492
$100$	−24.9601	−2.49601
$101$	7.46224	0.742521	0.371260	−	0.928529i	$-0.378926\pi$
$101$	7.46224	0.742521	0.371260	+	0.928529i	$0.378926\pi$
$102$	43.7737	4.33424
$103$	7.83988	0.772486	0.386243	−	0.922397i	$-0.373773\pi$
$103$	7.83988	0.772486	0.386243	+	0.922397i	$0.373773\pi$
$104$	−33.1267	−3.24834
$105$	−0.475062	−0.0463613
$106$	−26.8012	−2.60316
$107$	−0.890206	−0.0860595	−0.0430297	−	0.999074i	$-0.513701\pi$
$107$	−0.890206	−0.0860595	−0.0430297	+	0.999074i	$0.513701\pi$
$108$	−3.85614	−0.371058
$109$	3.92555	0.375999	0.188000	−	0.982169i	$-0.439800\pi$
$109$	3.92555	0.375999	0.188000	+	0.982169i	$0.439800\pi$
$110$	6.03454	0.575370
$111$	−13.5788	−1.28884
$112$	4.76877	0.450606
$113$	17.3995	1.63681	0.818404	−	0.574643i	$-0.194859\pi$
$113$	17.3995	1.63681	0.818404	+	0.574643i	$0.194859\pi$
$114$	8.87331	0.831061
$115$	−2.19190	−0.204396
$116$	−46.8588	−4.35073
$117$	−12.2552	−1.13299
$118$	−3.33028	−0.306577
$119$	−2.29417	−0.210306
$120$	−11.9027	−1.08656
$121$	6.53776	0.594342
$122$	10.9332	0.989842
$123$	−29.6564	−2.67403
$124$	27.0690	2.43087
$125$	−5.18379	−0.463653
$126$	3.15457	0.281031
$127$	18.7331	1.66230	0.831148	−	0.556052i	$-0.187684\pi$
$127$	18.7331	1.66230	0.831148	+	0.556052i	$0.187684\pi$
$128$	25.3187	2.23787
$129$	−1.68857	−0.148670
$130$	−5.36688	−0.470707
$131$	−1.40627	−0.122867	−0.0614333	−	0.998111i	$-0.519567\pi$
$131$	−1.40627	−0.122867	−0.0614333	+	0.998111i	$0.519567\pi$
$132$	−55.5987	−4.83924
$133$	−0.465048	−0.0403247
$134$	−37.9499	−3.27837
$135$	−0.388694	−0.0334535
$136$	−57.4803	−4.92890
$137$	5.54974	0.474146	0.237073	−	0.971492i	$-0.423812\pi$
$137$	5.54974	0.474146	0.237073	+	0.971492i	$0.423812\pi$
$138$	27.8251	2.36863
$139$	−11.1540	−0.946072	−0.473036	−	0.881043i	$-0.656842\pi$
$139$	−11.1540	−0.946072	−0.473036	+	0.881043i	$0.656842\pi$
$140$	1.00264	0.0847387
$141$	−13.1164	−1.10460
$142$	35.2501	2.95812
$143$	−15.5974	−1.30432
$144$	44.2021	3.68351
$145$	−4.72330	−0.392249
$146$	20.9817	1.73645
$147$	17.2405	1.42197
$148$	28.6587	2.35573
$149$	8.27009	0.677512	0.338756	−	0.940874i	$-0.389994\pi$
$149$	8.27009	0.677512	0.338756	+	0.940874i	$0.389994\pi$
$150$	31.9386	2.60778
$151$	−11.8990	−0.968323	−0.484162	−	0.874979i	$-0.660875\pi$
$151$	−11.8990	−0.968323	−0.484162	+	0.874979i	$0.660875\pi$
$152$	−11.6518	−0.945083
$153$	−21.2648	−1.71916
$154$	4.01487	0.323527
$155$	2.72852	0.219160
$156$	49.4473	3.95895
$157$	−5.88156	−0.469399	−0.234700	−	0.972068i	$-0.575411\pi$
$157$	−5.88156	−0.469399	−0.234700	+	0.972068i	$0.575411\pi$
$158$	−3.17792	−0.252822
$159$	24.8903	1.97393
$160$	9.86580	0.779960
$161$	−1.45830	−0.114930
$162$	−21.7246	−1.70685
$163$	−3.21033	−0.251452	−0.125726	−	0.992065i	$-0.540126\pi$
$163$	−3.21033	−0.251452	−0.125726	+	0.992065i	$0.540126\pi$
$164$	62.5913	4.88756
$165$	−5.60427	−0.436292
$166$	16.8870	1.31068
$167$	−9.62511	−0.744813	−0.372407	−	0.928070i	$-0.621467\pi$
$167$	−9.62511	−0.744813	−0.372407	+	0.928070i	$0.621467\pi$
$168$	−7.91903	−0.610967
$169$	0.871711	0.0670547
$170$	−9.31244	−0.714231
$171$	−4.31057	−0.329637
$172$	3.56381	0.271738
$173$	−11.7677	−0.894682	−0.447341	−	0.894364i	$-0.647629\pi$
$173$	−11.7677	−0.894682	−0.447341	+	0.894364i	$0.647629\pi$
$174$	59.9599	4.54555
$175$	−1.67389	−0.126535
$176$	56.2567	4.24051
$177$	3.09283	0.232471
$178$	10.2851	0.770898
$179$	−4.74606	−0.354737	−0.177369	−	0.984144i	$-0.556759\pi$
$179$	−4.74606	−0.354737	−0.177369	+	0.984144i	$0.556759\pi$
$180$	9.29358	0.692703
$181$	−9.56167	−0.710713	−0.355357	−	0.934731i	$-0.615641\pi$
$181$	−9.56167	−0.710713	−0.355357	+	0.934731i	$0.615641\pi$
$182$	−3.57066	−0.264675
$183$	−10.1536	−0.750578
$184$	−36.5378	−2.69360
$185$	2.88876	0.212386
$186$	−34.6372	−2.53972
$187$	−27.0641	−1.97912
$188$	27.6827	2.01897
$189$	−0.258604	−0.0188107
$190$	−1.88771	−0.136949
$191$	23.1412	1.67444	0.837218	−	0.546870i	$-0.184181\pi$
$191$	23.1412	1.67444	0.837218	+	0.546870i	$0.184181\pi$
$192$	−57.8571	−4.17548
$193$	9.15643	0.659095	0.329547	−	0.944139i	$-0.393104\pi$
$193$	9.15643	0.659095	0.329547	+	0.944139i	$0.393104\pi$
$194$	−26.7792	−1.92264
$195$	4.98422	0.356927
$196$	−36.3869	−2.59906
$197$	−6.75045	−0.480950	−0.240475	−	0.970655i	$-0.577303\pi$
$197$	−6.75045	−0.480950	−0.240475	+	0.970655i	$0.577303\pi$
$198$	37.2142	2.64470
$199$	3.26300	0.231308	0.115654	−	0.993290i	$-0.463104\pi$
$199$	3.26300	0.231308	0.115654	+	0.993290i	$0.463104\pi$
$200$	−41.9394	−2.96556
$201$	35.2440	2.48592
$202$	20.1528	1.41794
$203$	−3.14248	−0.220559
$204$	85.7993	6.00715
$205$	6.30912	0.440648
$206$	21.1726	1.47517
$207$	−13.5172	−0.939507
$208$	−50.0325	−3.46913
$209$	−5.48613	−0.379483
$210$	−1.28297	−0.0885333
$211$	−23.1474	−1.59354	−0.796768	−	0.604285i	$-0.793459\pi$
$211$	−23.1474	−1.59354	−0.796768	+	0.604285i	$0.793459\pi$
$212$	−52.5321	−3.60792
$213$	−32.7368	−2.24309
$214$	−2.40412	−0.164342
$215$	0.359227	0.0244991
$216$	−6.47932	−0.440862
$217$	1.81532	0.123232
$218$	10.6015	0.718022
$219$	−19.4857	−1.31672
$220$	11.8281	0.797450
$221$	24.0698	1.61911
$222$	−36.6714	−2.46122
$223$	−18.2948	−1.22511	−0.612557	−	0.790427i	$-0.709859\pi$
$223$	−18.2948	−1.22511	−0.612557	+	0.790427i	$0.709859\pi$
$224$	6.56387	0.438567
$225$	−15.5155	−1.03437
$226$	46.9897	3.12571
$227$	10.4809	0.695642	0.347821	−	0.937561i	$-0.386922\pi$
$227$	10.4809	0.695642	0.347821	+	0.937561i	$0.386922\pi$
$228$	17.3923	1.15183
$229$	−5.51626	−0.364524	−0.182262	−	0.983250i	$-0.558342\pi$
$229$	−5.51626	−0.364524	−0.182262	+	0.983250i	$0.558342\pi$
$230$	−5.91952	−0.390322
$231$	−3.72861	−0.245324
$232$	−78.7349	−5.16920
$233$	27.7343	1.81693	0.908466	−	0.417959i	$-0.137254\pi$
$233$	27.7343	1.81693	0.908466	+	0.417959i	$0.137254\pi$
$234$	−33.0968	−2.16361
$235$	2.79039	0.182025
$236$	−6.52756	−0.424908
$237$	2.95133	0.191710
$238$	−6.19570	−0.401608
$239$	6.60568	0.427286	0.213643	−	0.976912i	$-0.431467\pi$
$239$	6.60568	0.427286	0.213643	+	0.976912i	$0.431467\pi$
$240$	−17.9771	−1.16042
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	17.6561	1.13498
$243$	22.3611	1.43447
$244$	21.4297	1.37190
$245$	−3.66775	−0.234324
$246$	−80.0911	−5.10642
$247$	4.87915	0.310453
$248$	45.4829	2.88817
$249$	−15.6829	−0.993865
$250$	−13.9995	−0.885408
$251$	21.2628	1.34210	0.671049	−	0.741413i	$-0.265844\pi$
$251$	21.2628	1.34210	0.671049	+	0.741413i	$0.265844\pi$
$252$	6.18316	0.389502
$253$	−17.2035	−1.08157
$254$	50.5913	3.17438
$255$	8.64846	0.541588
$256$	22.2398	1.38999
$257$	7.33227	0.457374	0.228687	−	0.973500i	$-0.426557\pi$
$257$	7.33227	0.457374	0.228687	+	0.973500i	$0.426557\pi$
$258$	−4.56021	−0.283906
$259$	1.92194	0.119423
$260$	−10.5194	−0.652388
$261$	−29.1280	−1.80298
$262$	−3.79782	−0.234630
$263$	20.6204	1.27151	0.635753	−	0.771892i	$-0.280690\pi$
$263$	20.6204	1.27151	0.635753	+	0.771892i	$0.280690\pi$
$264$	−93.4202	−5.74961
$265$	−5.29517	−0.325280
$266$	−1.25592	−0.0770056
$267$	−9.55173	−0.584557
$268$	−74.3842	−4.54374
$269$	−0.178772	−0.0108999	−0.00544997	−	0.999985i	$-0.501735\pi$
$269$	−0.178772	−0.0108999	−0.00544997	+	0.999985i	$0.501735\pi$
$270$	−1.04972	−0.0638840
$271$	−17.9311	−1.08923	−0.544617	−	0.838685i	$-0.683325\pi$
$271$	−17.9311	−1.08923	−0.544617	+	0.838685i	$0.683325\pi$
$272$	−86.8149	−5.26392
$273$	3.31608	0.200698
$274$	14.9878	0.905446
$275$	−19.7468	−1.19078
$276$	54.5390	3.28286
$277$	2.56354	0.154028	0.0770140	−	0.997030i	$-0.475461\pi$
$277$	2.56354	0.154028	0.0770140	+	0.997030i	$0.475461\pi$
$278$	−30.1229	−1.80665
$279$	16.8264	1.00737
$280$	1.68470	0.100680
$281$	−22.4531	−1.33944	−0.669720	−	0.742614i	$-0.733586\pi$
$281$	−22.4531	−1.33944	−0.669720	+	0.742614i	$0.733586\pi$
$282$	−35.4225	−2.10938
$283$	13.1240	0.780141	0.390070	−	0.920785i	$-0.372451\pi$
$283$	13.1240	0.780141	0.390070	+	0.920785i	$0.372451\pi$
$284$	69.0926	4.09989
$285$	1.75312	0.103846
$286$	−42.1228	−2.49078
$287$	4.19755	0.247774
$288$	60.8411	3.58509
$289$	24.7651	1.45677
$290$	−12.7559	−0.749053
$291$	24.8699	1.45790
$292$	41.1254	2.40668
$293$	−32.3935	−1.89245	−0.946225	−	0.323509i	$-0.895137\pi$
$293$	−32.3935	−1.89245	−0.946225	+	0.323509i	$0.895137\pi$
$294$	46.5602	2.71545
$295$	−0.657970	−0.0383085
$296$	48.1541	2.79890
$297$	−3.05073	−0.177021
$298$	22.3345	1.29380
$299$	15.3001	0.884829
$300$	62.6018	3.61432
$301$	0.238999	0.0137757
$302$	−32.1347	−1.84914
$303$	−18.7159	−1.07520
$304$	−17.5981	−1.00932
$305$	2.16009	0.123686
$306$	−57.4286	−3.28297
$307$	12.6847	0.723952	0.361976	−	0.932187i	$-0.382102\pi$
$307$	12.6847	0.723952	0.361976	+	0.932187i	$0.382102\pi$
$308$	7.86940	0.448401
$309$	−19.6630	−1.11859
$310$	7.36873	0.418516
$311$	22.1050	1.25346	0.626730	−	0.779236i	$-0.284393\pi$
$311$	22.1050	1.25346	0.626730	+	0.779236i	$0.284393\pi$
$312$	83.0842	4.70372
$313$	10.8859	0.615305	0.307653	−	0.951499i	$-0.400457\pi$
$313$	10.8859	0.615305	0.307653	+	0.951499i	$0.400457\pi$
$314$	−15.8839	−0.896382
$315$	0.623255	0.0351164
$316$	−6.22893	−0.350405
$317$	−7.76634	−0.436201	−0.218101	−	0.975926i	$-0.569986\pi$
$317$	−7.76634	−0.436201	−0.218101	+	0.975926i	$0.569986\pi$
$318$	67.2195	3.76948
$319$	−37.0716	−2.07561
$320$	12.3085	0.688069
$321$	2.23271	0.124618
$322$	−3.93835	−0.219475
$323$	8.46614	0.471069
$324$	−42.5817	−2.36565
$325$	17.5620	0.974166
$326$	−8.66992	−0.480182
$327$	−9.84558	−0.544462
$328$	105.170	5.80702
$329$	1.85648	0.102351
$330$	−15.1351	−0.833159
$331$	0.317941	0.0174756	0.00873781	−	0.999962i	$-0.497219\pi$
$331$	0.317941	0.0174756	0.00873781	+	0.999962i	$0.497219\pi$
$332$	33.0996	1.81658
$333$	17.8146	0.976235
$334$	−25.9939	−1.42232
$335$	−7.49784	−0.409651
$336$	−11.9604	−0.652495
$337$	−10.0901	−0.549641	−0.274821	−	0.961496i	$-0.588618\pi$
$337$	−10.0901	−0.549641	−0.274821	+	0.961496i	$0.588618\pi$
$338$	2.35417	0.128050
$339$	−43.6393	−2.37016
$340$	−18.2530	−0.989908
$341$	21.4152	1.15970
$342$	−11.6413	−0.629488
$343$	−4.92515	−0.265933
$344$	5.98812	0.322858
$345$	5.49746	0.295973
$346$	−31.7802	−1.70852
$347$	−12.2330	−0.656700	−0.328350	−	0.944556i	$-0.606492\pi$
$347$	−12.2330	−0.656700	−0.328350	+	0.944556i	$0.606492\pi$
$348$	117.525	6.30002
$349$	0.0471917	0.00252612	0.00126306	−	0.999999i	$-0.499598\pi$
$349$	0.0471917	0.00252612	0.00126306	+	0.999999i	$0.499598\pi$
$350$	−4.52058	−0.241635
$351$	2.71320	0.144820
$352$	77.4334	4.12721
$353$	9.38275	0.499393	0.249697	−	0.968324i	$-0.419669\pi$
$353$	9.38275	0.499393	0.249697	+	0.968324i	$0.419669\pi$
$354$	8.35259	0.443935
$355$	6.96444	0.369634
$356$	20.1594	1.06845
$357$	5.75395	0.304531
$358$	−12.8174	−0.677419
$359$	4.83057	0.254948	0.127474	−	0.991842i	$-0.459313\pi$
$359$	4.83057	0.254948	0.127474	+	0.991842i	$0.459313\pi$
$360$	15.6156	0.823016
$361$	−17.2838	−0.909676
$362$	−25.8226	−1.35720
$363$	−16.3972	−0.860630
$364$	−6.99874	−0.366834
$365$	4.14539	0.216980
$366$	−27.4212	−1.43333
$367$	−26.6792	−1.39264	−0.696322	−	0.717730i	$-0.745181\pi$
$367$	−26.6792	−1.39264	−0.696322	+	0.717730i	$0.745181\pi$
$368$	−55.1845	−2.87669
$369$	38.9075	2.02544
$370$	7.80149	0.405580
$371$	−3.52295	−0.182903
$372$	−67.8911	−3.51999
$373$	−1.30441	−0.0675400	−0.0337700	−	0.999430i	$-0.510751\pi$
$373$	−1.30441	−0.0675400	−0.0337700	+	0.999430i	$0.510751\pi$
$374$	−73.0902	−3.77940
$375$	13.0014	0.671387
$376$	46.5142	2.39879
$377$	32.9700	1.69804
$378$	−0.698395	−0.0359216
$379$	−3.62474	−0.186191	−0.0930953	−	0.995657i	$-0.529676\pi$
$379$	−3.62474	−0.186191	−0.0930953	+	0.995657i	$0.529676\pi$
$380$	−3.70004	−0.189808
$381$	−46.9841	−2.40707
$382$	62.4958	3.19756
$383$	−28.5501	−1.45884	−0.729420	−	0.684066i	$-0.760210\pi$
$383$	−28.5501	−1.45884	−0.729420	+	0.684066i	$0.760210\pi$
$384$	−63.5011	−3.24053
$385$	0.793226	0.0404265
$386$	24.7282	1.25863
$387$	2.21531	0.112610
$388$	−52.4891	−2.66473
$389$	29.3393	1.48756	0.743780	−	0.668424i	$-0.233031\pi$
$389$	29.3393	1.48756	0.743780	+	0.668424i	$0.233031\pi$
$390$	13.4606	0.681602
$391$	26.5483	1.34260
$392$	−61.1394	−3.08800
$393$	3.52704	0.177916
$394$	−18.2305	−0.918439
$395$	−0.627868	−0.0315915
$396$	72.9422	3.66548
$397$	3.94029	0.197758	0.0988788	−	0.995099i	$-0.468474\pi$
$397$	3.94029	0.197758	0.0988788	+	0.995099i	$0.468474\pi$
$398$	8.81218	0.441715
$399$	1.16638	0.0583918
$400$	−63.3428	−3.16714
$401$	0.0336542	0.00168061	0.000840304	−	1.00000i	$-0.499733\pi$
$401$	0.0336542	0.00168061	0.000840304		1.00000i	$0.499733\pi$
$402$	95.1812	4.74721
$403$	−19.0459	−0.948743
$404$	39.5008	1.96524
$405$	−4.29218	−0.213280
$406$	−8.48669	−0.421188
$407$	22.6729	1.12386
$408$	144.165	7.13724
$409$	−8.89629	−0.439893	−0.219947	−	0.975512i	$-0.570588\pi$
$409$	−8.89629	−0.439893	−0.219947	+	0.975512i	$0.570588\pi$
$410$	17.0386	0.841477
$411$	−13.9192	−0.686582
$412$	41.4998	2.04455
$413$	−0.437757	−0.0215406
$414$	−36.5049	−1.79412
$415$	3.33640	0.163777
$416$	−68.8662	−3.37644
$417$	27.9752	1.36995
$418$	−14.8160	−0.724675
$419$	−29.8619	−1.45885	−0.729425	−	0.684061i	$-0.760212\pi$
$419$	−29.8619	−1.45885	−0.729425	+	0.684061i	$0.760212\pi$
$420$	−2.51470	−0.122705
$421$	15.9683	0.778247	0.389124	−	0.921186i	$-0.372778\pi$
$421$	15.9683	0.778247	0.389124	+	0.921186i	$0.372778\pi$
$422$	−62.5128	−3.04307
$423$	17.2079	0.836678
$424$	−88.2676	−4.28665
$425$	30.4731	1.47816
$426$	−88.4100	−4.28348
$427$	1.43714	0.0695480
$428$	−4.71223	−0.227774
$429$	39.1195	1.88871
$430$	0.970141	0.0467844
$431$	24.8345	1.19624	0.598118	−	0.801408i	$-0.295915\pi$
$431$	24.8345	1.19624	0.598118	+	0.801408i	$0.295915\pi$
$432$	−9.78598	−0.470828
$433$	−31.1730	−1.49808	−0.749040	−	0.662524i	$-0.769485\pi$
$433$	−31.1730	−1.49808	−0.749040	+	0.662524i	$0.769485\pi$
$434$	4.90253	0.235329
$435$	11.8464	0.567992
$436$	20.7796	0.995161
$437$	5.38157	0.257435
$438$	−52.6236	−2.51445
$439$	−0.959384	−0.0457889	−0.0228944	−	0.999738i	$-0.507288\pi$
$439$	−0.959384	−0.0457889	−0.0228944	+	0.999738i	$0.507288\pi$
$440$	19.8743	0.947468
$441$	−22.6185	−1.07707
$442$	65.0036	3.09190
$443$	6.52446	0.309986	0.154993	−	0.987916i	$-0.450464\pi$
$443$	6.52446	0.309986	0.154993	+	0.987916i	$0.450464\pi$
$444$	−71.8783	−3.41119
$445$	2.03204	0.0963280
$446$	−49.4077	−2.33952
$447$	−20.7420	−0.981065
$448$	8.18906	0.386897
$449$	−9.39811	−0.443524	−0.221762	−	0.975101i	$-0.571181\pi$
$449$	−9.39811	−0.443524	−0.221762	+	0.975101i	$0.571181\pi$
$450$	−41.9016	−1.97526
$451$	49.5182	2.33172
$452$	92.1029	4.33216
$453$	29.8435	1.40217
$454$	28.3051	1.32842
$455$	−0.705464	−0.0330727
$456$	29.2235	1.36852
$457$	−27.0904	−1.26724	−0.633618	−	0.773646i	$-0.718431\pi$
$457$	−27.0904	−1.26724	−0.633618	+	0.773646i	$0.718431\pi$
$458$	−14.8974	−0.696109
$459$	4.70786	0.219744
$460$	−11.6027	−0.540976
$461$	−32.1068	−1.49536	−0.747682	−	0.664057i	$-0.768833\pi$
$461$	−32.1068	−1.49536	−0.747682	+	0.664057i	$0.768833\pi$
$462$	−10.0696	−0.468480
$463$	17.5754	0.816797	0.408399	−	0.912804i	$-0.366087\pi$
$463$	17.5754	0.816797	0.408399	+	0.912804i	$0.366087\pi$
$464$	−118.916	−5.52056
$465$	−6.84334	−0.317352
$466$	74.9001	3.46968
$467$	12.9732	0.600329	0.300164	−	0.953887i	$-0.402958\pi$
$467$	12.9732	0.600329	0.300164	+	0.953887i	$0.402958\pi$
$468$	−64.8720	−2.99871
$469$	−4.98842	−0.230344
$470$	7.53581	0.347601
$471$	14.7514	0.679708
$472$	−10.9680	−0.504843
$473$	2.81945	0.129639
$474$	7.97046	0.366096
$475$	6.17716	0.283427
$476$	−12.1440	−0.556619
$477$	−32.6546	−1.49515
$478$	17.8395	0.815960
$479$	−6.68222	−0.305319	−0.152659	−	0.988279i	$-0.548784\pi$
$479$	−6.68222	−0.305319	−0.152659	+	0.988279i	$0.548784\pi$
$480$	−24.7442	−1.12941
$481$	−20.1644	−0.919418
$482$	2.70063	0.123010
$483$	3.65754	0.166424
$484$	34.6071	1.57305
$485$	−5.29083	−0.240244
$486$	60.3892	2.73931
$487$	19.6021	0.888256	0.444128	−	0.895963i	$-0.353514\pi$
$487$	19.6021	0.888256	0.444128	+	0.895963i	$0.353514\pi$
$488$	36.0075	1.62998
$489$	8.05175	0.364113
$490$	−9.90524	−0.447473
$491$	−6.61946	−0.298732	−0.149366	−	0.988782i	$-0.547723\pi$
$491$	−6.61946	−0.298732	−0.149366	+	0.988782i	$0.547723\pi$
$492$	−156.984	−7.07738
$493$	57.2086	2.57654
$494$	13.1768	0.592852
$495$	7.35248	0.330470
$496$	68.6947	3.08448
$497$	4.63355	0.207843
$498$	−42.3538	−1.89792
$499$	39.6828	1.77645	0.888223	−	0.459413i	$-0.151940\pi$
$499$	39.6828	1.77645	0.888223	+	0.459413i	$0.151940\pi$
$500$	−27.4400	−1.22715
$501$	24.1405	1.07852
$502$	57.4231	2.56292
$503$	−13.9042	−0.619960	−0.309980	−	0.950743i	$-0.600322\pi$
$503$	−13.9042	−0.619960	−0.309980	+	0.950743i	$0.600322\pi$
$504$	10.3893	0.462777
$505$	3.98163	0.177180
$506$	−46.4603	−2.06541
$507$	−2.18632	−0.0970979
$508$	99.1622	4.39961
$509$	33.7260	1.49488	0.747439	−	0.664331i	$-0.231283\pi$
$509$	33.7260	1.49488	0.747439	+	0.664331i	$0.231283\pi$
$510$	23.3563	1.03424
$511$	2.75799	0.122006
$512$	9.42420	0.416495
$513$	0.954324	0.0421344
$514$	19.8018	0.873418
$515$	4.18312	0.184330
$516$	−8.93831	−0.393487
$517$	21.9008	0.963196
$518$	5.19045	0.228055
$519$	29.5143	1.29553
$520$	−17.6754	−0.775117
$521$	−37.2520	−1.63204	−0.816020	−	0.578023i	$-0.803824\pi$
$521$	−37.2520	−1.63204	−0.816020	+	0.578023i	$0.803824\pi$
$522$	−78.6639	−3.44303
$523$	−14.7673	−0.645730	−0.322865	−	0.946445i	$-0.604646\pi$
$523$	−14.7673	−0.645730	−0.322865	+	0.946445i	$0.604646\pi$
$524$	−7.44399	−0.325192
$525$	4.19826	0.183227
$526$	55.6880	2.42811
$527$	−33.0478	−1.43958
$528$	−141.096	−6.14043
$529$	−6.12439	−0.266278
$530$	−14.3003	−0.621166
$531$	−4.05761	−0.176085
$532$	−2.46169	−0.106728
$533$	−44.0395	−1.90756
$534$	−25.7957	−1.11629
$535$	−0.474987	−0.0205355
$536$	−124.985	−5.39852
$537$	11.9035	0.513674
$538$	−0.482799	−0.0208149
$539$	−28.7869	−1.23994
$540$	−2.05752	−0.0885417
$541$	0.345044	0.0148346	0.00741730	−	0.999972i	$-0.497639\pi$
$541$	0.345044	0.0148346	0.00741730	+	0.999972i	$0.497639\pi$
$542$	−48.4252	−2.08004
$543$	23.9814	1.02914
$544$	−119.494	−5.12328
$545$	2.09455	0.0897209
$546$	8.95550	0.383260
$547$	−6.16271	−0.263498	−0.131749	−	0.991283i	$-0.542059\pi$
$547$	−6.16271	−0.263498	−0.131749	+	0.991283i	$0.542059\pi$
$548$	29.3771	1.25493
$549$	13.3210	0.568525
$550$	−53.3289	−2.27395
$551$	11.5967	0.494035
$552$	91.6396	3.90044
$553$	−0.417730	−0.0177637
$554$	6.92318	0.294138
$555$	−7.24524	−0.307543
$556$	−59.0429	−2.50398
$557$	7.45145	0.315728	0.157864	−	0.987461i	$-0.449539\pi$
$557$	7.45145	0.315728	0.157864	+	0.987461i	$0.449539\pi$
$558$	45.4420	1.92371
$559$	−2.50751	−0.106056
$560$	2.54447	0.107523
$561$	67.8789	2.86585
$562$	−60.6376	−2.55784
$563$	−14.5393	−0.612760	−0.306380	−	0.951909i	$-0.599118\pi$
$563$	−14.5393	−0.612760	−0.306380	+	0.951909i	$0.599118\pi$
$564$	−69.4305	−2.92355
$565$	9.28385	0.390575
$566$	35.4431	1.48978
$567$	−2.85565	−0.119926
$568$	116.093	4.87117
$569$	12.1696	0.510175	0.255088	−	0.966918i	$-0.417896\pi$
$569$	12.1696	0.510175	0.255088	+	0.966918i	$0.417896\pi$
$570$	4.73453	0.198308
$571$	−16.0026	−0.669687	−0.334843	−	0.942274i	$-0.608683\pi$
$571$	−16.0026	−0.669687	−0.334843	+	0.942274i	$0.608683\pi$
$572$	−82.5635	−3.45216
$573$	−58.0398	−2.42465
$574$	11.3360	0.473157
$575$	19.3704	0.807803
$576$	75.9052	3.16271
$577$	0.0727829	0.00302999	0.00151500	−	0.999999i	$-0.499518\pi$
$577$	0.0727829	0.00302999	0.00151500	+	0.999999i	$0.499518\pi$
$578$	66.8814	2.78190
$579$	−22.9650	−0.954395
$580$	−25.0024	−1.03817
$581$	2.21975	0.0920909
$582$	67.1644	2.78405
$583$	−41.5600	−1.72124
$584$	69.1014	2.85944
$585$	−6.53901	−0.270355
$586$	−87.4830	−3.61389
$587$	0.902377	0.0372451	0.0186225	−	0.999827i	$-0.494072\pi$
$587$	0.902377	0.0372451	0.0186225	+	0.999827i	$0.494072\pi$
$588$	91.2611	3.76354
$589$	−6.69907	−0.276031
$590$	−1.77694	−0.0731553
$591$	16.9307	0.696434
$592$	72.7291	2.98915
$593$	−23.7787	−0.976476	−0.488238	−	0.872711i	$-0.662360\pi$
$593$	−23.7787	−0.976476	−0.488238	+	0.872711i	$0.662360\pi$
$594$	−8.23891	−0.338046
$595$	−1.22410	−0.0501831
$596$	43.7771	1.79318
$597$	−8.18387	−0.334943
$598$	41.3200	1.68970
$599$	48.7217	1.99072	0.995358	−	0.0962439i	$-0.0306829\pi$
$599$	48.7217	1.99072	0.995358	+	0.0962439i	$0.0306829\pi$
$600$	105.187	4.29425
$601$	−3.90565	−0.159315	−0.0796573	−	0.996822i	$-0.525383\pi$
$601$	−3.90565	−0.159315	−0.0796573	+	0.996822i	$0.525383\pi$
$602$	0.645449	0.0263066
$603$	−46.2381	−1.88296
$604$	−62.9862	−2.56287
$605$	3.48835	0.141822
$606$	−50.5447	−2.05324
$607$	−15.1915	−0.616605	−0.308302	−	0.951288i	$-0.599761\pi$
$607$	−15.1915	−0.616605	−0.308302	+	0.951288i	$0.599761\pi$
$608$	−24.2226	−0.982355
$609$	7.88159	0.319378
$610$	5.83360	0.236196
$611$	−19.4777	−0.787984
$612$	−112.564	−4.55012
$613$	−29.1406	−1.17698	−0.588490	−	0.808505i	$-0.700277\pi$
$613$	−29.1406	−1.17698	−0.588490	+	0.808505i	$0.700277\pi$
$614$	34.2566	1.38248
$615$	−15.8238	−0.638076
$616$	13.2226	0.532755
$617$	−31.1143	−1.25261	−0.626307	−	0.779576i	$-0.715435\pi$
$617$	−31.1143	−1.25261	−0.626307	+	0.779576i	$0.715435\pi$
$618$	−53.1026	−2.13610
$619$	40.3950	1.62361	0.811806	−	0.583927i	$-0.198484\pi$
$619$	40.3950	1.62361	0.811806	+	0.583927i	$0.198484\pi$
$620$	14.4432	0.580053
$621$	2.99259	0.120088
$622$	59.6976	2.39365
$623$	1.35195	0.0541646
$624$	125.485	5.02344
$625$	20.8106	0.832424
$626$	29.3987	1.17501
$627$	13.7596	0.549507
$628$	−31.1335	−1.24236
$629$	−34.9886	−1.39509
$630$	1.68318	0.0670596
$631$	7.99234	0.318170	0.159085	−	0.987265i	$-0.449146\pi$
$631$	7.99234	0.318170	0.159085	+	0.987265i	$0.449146\pi$
$632$	−10.4662	−0.416324
$633$	58.0556	2.30750
$634$	−20.9740	−0.832986
$635$	9.99542	0.396656
$636$	131.755	5.22441
$637$	25.6020	1.01439
$638$	−100.117	−3.96366
$639$	42.9488	1.69903
$640$	13.5093	0.534001
$641$	−25.0212	−0.988277	−0.494139	−	0.869383i	$-0.664516\pi$
$641$	−25.0212	−0.988277	−0.494139	+	0.869383i	$0.664516\pi$
$642$	6.02972	0.237974
$643$	−9.88861	−0.389969	−0.194984	−	0.980806i	$-0.562466\pi$
$643$	−9.88861	−0.389969	−0.194984	+	0.980806i	$0.562466\pi$
$644$	−7.71942	−0.304188
$645$	−0.900970	−0.0354757
$646$	22.8639	0.899570
$647$	43.3237	1.70323	0.851615	−	0.524168i	$-0.175624\pi$
$647$	43.3237	1.70323	0.851615	+	0.524168i	$0.175624\pi$
$648$	−71.5484	−2.81068
$649$	−5.16419	−0.202712
$650$	47.4286	1.86030
$651$	−4.55298	−0.178445
$652$	−16.9936	−0.665521
$653$	−13.6441	−0.533936	−0.266968	−	0.963705i	$-0.586022\pi$
$653$	−13.6441	−0.533936	−0.266968	+	0.963705i	$0.586022\pi$
$654$	−26.5893	−1.03972
$655$	−0.750344	−0.0293184
$656$	158.842	6.20173
$657$	25.5641	0.997349
$658$	5.01368	0.195454
$659$	−18.6052	−0.724756	−0.362378	−	0.932031i	$-0.618035\pi$
$659$	−18.6052	−0.724756	−0.362378	+	0.932031i	$0.618035\pi$
$660$	−29.6658	−1.15474
$661$	−39.9082	−1.55225	−0.776124	−	0.630580i	$-0.782817\pi$
$661$	−39.9082	−1.55225	−0.776124	+	0.630580i	$0.782817\pi$
$662$	0.858643	0.0333721
$663$	−60.3688	−2.34453
$664$	55.6159	2.15832
$665$	−0.248135	−0.00962228
$666$	48.1107	1.86425
$667$	36.3650	1.40806
$668$	−50.9497	−1.97130
$669$	45.8849	1.77401
$670$	−20.2489	−0.782283
$671$	16.9538	0.654494
$672$	−16.4627	−0.635062
$673$	−0.931169	−0.0358939	−0.0179470	−	0.999839i	$-0.505713\pi$
$673$	−0.931169	−0.0358939	−0.0179470	+	0.999839i	$0.505713\pi$
$674$	−27.2496	−1.04961
$675$	3.43500	0.132213
$676$	4.61433	0.177474
$677$	35.1023	1.34909	0.674546	−	0.738233i	$-0.264339\pi$
$677$	35.1023	1.34909	0.674546	+	0.738233i	$0.264339\pi$
$678$	−117.854	−4.52615
$679$	−3.52007	−0.135088
$680$	−30.6698	−1.17613
$681$	−26.2869	−1.00732
$682$	57.8347	2.21461
$683$	−17.5204	−0.670399	−0.335199	−	0.942147i	$-0.608804\pi$
$683$	−17.5204	−0.670399	−0.335199	+	0.942147i	$0.608804\pi$
$684$	−22.8177	−0.872455
$685$	2.96117	0.113141
$686$	−13.3010	−0.507836
$687$	13.8352	0.527846
$688$	9.04410	0.344803
$689$	36.9618	1.40813
$690$	14.8466	0.565201
$691$	33.6421	1.27981	0.639903	−	0.768455i	$-0.278974\pi$
$691$	33.6421	1.27981	0.639903	+	0.768455i	$0.278974\pi$
$692$	−62.2914	−2.36796
$693$	4.89172	0.185821
$694$	−33.0367	−1.25406
$695$	−5.95145	−0.225751
$696$	197.473	7.48520
$697$	−76.4160	−2.89446
$698$	0.127448	0.00482396
$699$	−69.5597	−2.63099
$700$	−8.86063	−0.334900
$701$	−20.5783	−0.777232	−0.388616	−	0.921400i	$-0.627047\pi$
$701$	−20.5783	−0.777232	−0.388616	+	0.921400i	$0.627047\pi$
$702$	7.32736	0.276553
$703$	−7.09250	−0.267499
$704$	96.6057	3.64096
$705$	−6.99850	−0.263579
$706$	25.3394	0.953660
$707$	2.64904	0.0996272
$708$	16.3716	0.615284
$709$	−16.3548	−0.614216	−0.307108	−	0.951675i	$-0.599361\pi$
$709$	−16.3548	−0.614216	−0.307108	+	0.951675i	$0.599361\pi$
$710$	18.8084	0.705867
$711$	−3.87198	−0.145210
$712$	33.8730	1.26944
$713$	−21.0071	−0.786721
$714$	15.5393	0.581544
$715$	−8.32230	−0.311236
$716$	−25.1229	−0.938887
$717$	−16.5675	−0.618726
$718$	13.0456	0.486858
$719$	21.1731	0.789624	0.394812	−	0.918762i	$-0.370810\pi$
$719$	21.1731	0.789624	0.394812	+	0.918762i	$0.370810\pi$
$720$	23.5849	0.878958
$721$	2.78309	0.103648
$722$	−46.6773	−1.73715
$723$	−2.50808	−0.0932764
$724$	−50.6139	−1.88105
$725$	41.7411	1.55023
$726$	−44.2828	−1.64349
$727$	−15.8288	−0.587059	−0.293530	−	0.955950i	$-0.594830\pi$
$727$	−15.8288	−0.587059	−0.293530	+	0.955950i	$0.594830\pi$
$728$	−11.7597	−0.435843
$729$	−31.9506	−1.18335
$730$	11.1952	0.414352
$731$	−4.35096	−0.160926
$732$	−53.7474	−1.98656
$733$	−8.66500	−0.320049	−0.160025	−	0.987113i	$-0.551157\pi$
$733$	−8.66500	−0.320049	−0.160025	+	0.987113i	$0.551157\pi$
$734$	−72.0507	−2.65944
$735$	9.19900	0.339310
$736$	−75.9575	−2.79983
$737$	−58.8480	−2.16769
$738$	105.075	3.86786
$739$	−51.3157	−1.88768	−0.943840	−	0.330403i	$-0.892815\pi$
$739$	−51.3157	−1.88768	−0.943840	+	0.330403i	$0.892815\pi$
$740$	15.2914	0.562124
$741$	−12.2373	−0.449548
$742$	−9.51421	−0.349278
$743$	19.9955	0.733563	0.366781	−	0.930307i	$-0.380460\pi$
$743$	19.9955	0.733563	0.366781	+	0.930307i	$0.380460\pi$
$744$	−114.075	−4.18218
$745$	4.41267	0.161668
$746$	−3.52274	−0.128977
$747$	20.5751	0.752804
$748$	−143.262	−5.23816
$749$	−0.316016	−0.0115470
$750$	35.1119	1.28210
$751$	−30.4912	−1.11264	−0.556320	−	0.830968i	$-0.687787\pi$
$751$	−30.4912	−1.11264	−0.556320	+	0.830968i	$0.687787\pi$
$752$	70.2523	2.56184
$753$	−53.3288	−1.94341
$754$	89.0400	3.24264
$755$	−6.34892	−0.231061
$756$	−1.36890	−0.0497864
$757$	10.7919	0.392238	0.196119	−	0.980580i	$-0.437166\pi$
$757$	10.7919	0.392238	0.196119	+	0.980580i	$0.437166\pi$
$758$	−9.78910	−0.355556
$759$	43.1477	1.56616
$760$	−6.21703	−0.225515
$761$	14.3354	0.519657	0.259828	−	0.965655i	$-0.416334\pi$
$761$	14.3354	0.519657	0.259828	+	0.965655i	$0.416334\pi$
$762$	−126.887	−4.59662
$763$	1.39354	0.0504495
$764$	122.496	4.43174
$765$	−11.3463	−0.410226
$766$	−77.1032	−2.78585
$767$	4.59282	0.165837
$768$	−55.7791	−2.01276
$769$	20.7301	0.747547	0.373773	−	0.927520i	$-0.378064\pi$
$769$	20.7301	0.747547	0.373773	+	0.927520i	$0.378064\pi$
$770$	2.14221	0.0772000
$771$	−18.3899	−0.662296
$772$	48.4688	1.74443
$773$	40.3752	1.45220	0.726098	−	0.687591i	$-0.241332\pi$
$773$	40.3752	1.45220	0.726098	+	0.687591i	$0.241332\pi$
$774$	5.98273	0.215045
$775$	−24.1127	−0.866153
$776$	−88.1952	−3.16602
$777$	−4.82037	−0.172930
$778$	79.2346	2.84070
$779$	−15.4902	−0.554993
$780$	26.3836	0.944683
$781$	54.6616	1.95594
$782$	71.6971	2.56388
$783$	6.44869	0.230457
$784$	−92.3413	−3.29790
$785$	−3.13822	−0.112008
$786$	9.52524	0.339754
$787$	−18.3264	−0.653266	−0.326633	−	0.945151i	$-0.605914\pi$
$787$	−18.3264	−0.653266	−0.326633	+	0.945151i	$0.605914\pi$
$788$	−35.7330	−1.27293
$789$	−51.7175	−1.84119
$790$	−1.69564	−0.0603282
$791$	6.17668	0.219618
$792$	122.562	4.35505
$793$	−15.0780	−0.535437
$794$	10.6413	0.377645
$795$	13.2807	0.471018
$796$	17.2725	0.612206
$797$	−36.1893	−1.28189	−0.640945	−	0.767587i	$-0.721457\pi$
$797$	−36.1893	−1.28189	−0.640945	+	0.767587i	$0.721457\pi$
$798$	3.14995	0.111507
$799$	−33.7971	−1.19566
$800$	−87.1868	−3.08252
$801$	12.5313	0.442773
$802$	0.0908875	0.00320935
$803$	32.5358	1.14816
$804$	186.561	6.57952
$805$	−0.778107	−0.0274247
$806$	−51.4359	−1.81175
$807$	0.448375	0.0157835
$808$	66.3715	2.33494
$809$	−0.129006	−0.00453561	−0.00226781	−	0.999997i	$-0.500722\pi$
$809$	−0.129006	−0.00453561	−0.00226781	+	0.999997i	$0.500722\pi$
$810$	−11.5916	−0.407288
$811$	−22.3392	−0.784434	−0.392217	−	0.919873i	$-0.628292\pi$
$811$	−22.3392	−0.784434	−0.392217	+	0.919873i	$0.628292\pi$
$812$	−16.6345	−0.583756
$813$	44.9725	1.57725
$814$	61.2313	2.14615
$815$	−1.71293	−0.0600015
$816$	217.738	7.62237
$817$	−0.881977	−0.0308565
$818$	−24.0256	−0.840036
$819$	−4.35050	−0.152019
$820$	33.3968	1.16627
$821$	27.9841	0.976653	0.488327	−	0.872661i	$-0.337607\pi$
$821$	27.9841	0.976653	0.488327	+	0.872661i	$0.337607\pi$
$822$	−37.5906	−1.31112
$823$	4.52041	0.157572	0.0787859	−	0.996892i	$-0.474896\pi$
$823$	4.52041	0.157572	0.0787859	+	0.996892i	$0.474896\pi$
$824$	69.7303	2.42917
$825$	49.5265	1.72429
$826$	−1.18222	−0.0411347
$827$	−10.2432	−0.356189	−0.178095	−	0.984013i	$-0.556993\pi$
$827$	−10.2432	−0.356189	−0.178095	+	0.984013i	$0.556993\pi$
$828$	−71.5520	−2.48660
$829$	15.8251	0.549630	0.274815	−	0.961497i	$-0.411383\pi$
$829$	15.8251	0.549630	0.274815	+	0.961497i	$0.411383\pi$
$830$	9.01038	0.312755
$831$	−6.42955	−0.223039
$832$	−85.9173	−2.97865
$833$	44.4237	1.53919
$834$	75.5506	2.61610
$835$	−5.13567	−0.177727
$836$	−29.0404	−1.00438
$837$	−3.72523	−0.128763
$838$	−80.6461	−2.78587
$839$	33.9655	1.17262	0.586309	−	0.810087i	$-0.300580\pi$
$839$	33.9655	1.17262	0.586309	+	0.810087i	$0.300580\pi$
$840$	−4.22536	−0.145789
$841$	49.3626	1.70216
$842$	43.1245	1.48617
$843$	56.3141	1.93956
$844$	−122.529	−4.21763
$845$	0.465119	0.0160006
$846$	46.4723	1.59775
$847$	2.32085	0.0797454
$848$	−133.314	−4.57802
$849$	−32.9160	−1.12967
$850$	82.2966	2.82275
$851$	−22.2408	−0.762405
$852$	−173.290	−5.93680
$853$	51.3501	1.75819	0.879097	−	0.476643i	$-0.158147\pi$
$853$	51.3501	1.75819	0.879097	+	0.476643i	$0.158147\pi$
$854$	3.88118	0.132811
$855$	−2.29999	−0.0786580
$856$	−7.91778	−0.270624
$857$	−27.2848	−0.932030	−0.466015	−	0.884777i	$-0.654311\pi$
$857$	−27.2848	−0.932030	−0.466015	+	0.884777i	$0.654311\pi$
$858$	105.647	3.60674
$859$	−6.65604	−0.227101	−0.113551	−	0.993532i	$-0.536222\pi$
$859$	−6.65604	−0.227101	−0.113551	+	0.993532i	$0.536222\pi$
$860$	1.90154	0.0648420
$861$	−10.5278	−0.358786
$862$	67.0689	2.28438
$863$	43.1149	1.46765	0.733824	−	0.679340i	$-0.237734\pi$
$863$	43.1149	1.46765	0.733824	+	0.679340i	$0.237734\pi$
$864$	−13.4697	−0.458249
$865$	−6.27889	−0.213489
$866$	−84.1870	−2.86079
$867$	−62.1127	−2.10946
$868$	9.60928	0.326160
$869$	−4.92793	−0.167168
$870$	31.9928	1.08466
$871$	52.3371	1.77338
$872$	34.9151	1.18237
$873$	−32.6278	−1.10429
$874$	14.5336	0.491607
$875$	−1.84020	−0.0622103
$876$	−103.146	−3.48497
$877$	30.7736	1.03915	0.519575	−	0.854425i	$-0.326090\pi$
$877$	30.7736	1.03915	0.519575	+	0.854425i	$0.326090\pi$
$878$	−2.59094	−0.0874401
$879$	81.2455	2.74034
$880$	30.0169	1.01187
$881$	−35.5730	−1.19849	−0.599243	−	0.800567i	$-0.704532\pi$
$881$	−35.5730	−1.19849	−0.599243	+	0.800567i	$0.704532\pi$
$882$	−61.0843	−2.05682
$883$	−39.0447	−1.31396	−0.656979	−	0.753909i	$-0.728166\pi$
$883$	−39.0447	−1.31396	−0.656979	+	0.753909i	$0.728166\pi$
$884$	127.411	4.28531
$885$	1.65024	0.0554722
$886$	17.6202	0.591961
$887$	−35.4372	−1.18987	−0.594933	−	0.803775i	$-0.702821\pi$
$887$	−35.4372	−1.18987	−0.594933	+	0.803775i	$0.702821\pi$
$888$	−120.774	−4.05292
$889$	6.65010	0.223037
$890$	5.48780	0.183951
$891$	−33.6879	−1.12859
$892$	−96.8423	−3.24252
$893$	−6.85097	−0.229259
$894$	−56.0166	−1.87348
$895$	−2.53236	−0.0846473
$896$	8.98792	0.300265
$897$	−38.3739	−1.28127
$898$	−25.3808	−0.846970
$899$	−45.2679	−1.50977
$900$	−82.1300	−2.73767
$901$	64.1350	2.13665
$902$	133.730	4.45274
$903$	−0.599429	−0.0199477
$904$	154.757	5.14713
$905$	−5.10182	−0.169590
$906$	80.5964	2.67763
$907$	−6.19074	−0.205560	−0.102780	−	0.994704i	$-0.532774\pi$
$907$	−6.19074	−0.205560	−0.102780	+	0.994704i	$0.532774\pi$
$908$	55.4798	1.84116
$909$	24.5542	0.814410
$910$	−1.90520	−0.0631567
$911$	0.700721	0.0232159	0.0116080	−	0.999933i	$-0.496305\pi$
$911$	0.700721	0.0232159	0.0116080	+	0.999933i	$0.496305\pi$
$912$	44.1375	1.46154
$913$	26.1863	0.866638
$914$	−73.1612	−2.41996
$915$	−5.41767	−0.179103
$916$	−29.1999	−0.964790
$917$	−0.499215	−0.0164855
$918$	12.7142	0.419631
$919$	−29.3520	−0.968233	−0.484117	−	0.875004i	$-0.660859\pi$
$919$	−29.3520	−0.968233	−0.484117	+	0.875004i	$0.660859\pi$
$920$	−19.4955	−0.642746
$921$	−31.8141	−1.04831
$922$	−86.7088	−2.85560
$923$	−48.6139	−1.60014
$924$	−19.7371	−0.649302
$925$	−25.5288	−0.839381
$926$	47.4646	1.55978
$927$	25.7967	0.847276
$928$	−163.680	−5.37306
$929$	52.4498	1.72082	0.860411	−	0.509601i	$-0.170207\pi$
$929$	52.4498	1.72082	0.860411	+	0.509601i	$0.170207\pi$
$930$	−18.4814	−0.606027
$931$	9.00507	0.295129
$932$	146.809	4.80889
$933$	−55.4411	−1.81506
$934$	35.0359	1.14641
$935$	−14.4406	−0.472258
$936$	−109.002	−3.56283
$937$	−25.6384	−0.837569	−0.418784	−	0.908086i	$-0.637544\pi$
$937$	−25.6384	−0.837569	−0.418784	+	0.908086i	$0.637544\pi$
$938$	−13.4719	−0.439873
$939$	−27.3026	−0.890986
$940$	14.7707	0.481766
$941$	−12.3906	−0.403921	−0.201960	−	0.979394i	$-0.564731\pi$
$941$	−12.3906	−0.403921	−0.201960	+	0.979394i	$0.564731\pi$
$942$	39.8381	1.29800
$943$	−48.5744	−1.58180
$944$	−16.5654	−0.539158
$945$	−0.137983	−0.00448860
$946$	7.61431	0.247563
$947$	−7.84757	−0.255012	−0.127506	−	0.991838i	$-0.540697\pi$
$947$	−7.84757	−0.255012	−0.127506	+	0.991838i	$0.540697\pi$
$948$	15.6226	0.507400
$949$	−28.9361	−0.939304
$950$	16.6822	0.541243
$951$	19.4786	0.631637
$952$	−20.4051	−0.661331
$953$	−31.9352	−1.03448	−0.517242	−	0.855839i	$-0.673041\pi$
$953$	−31.9352	−1.03448	−0.517242	+	0.855839i	$0.673041\pi$
$954$	−88.1881	−2.85519
$955$	12.3474	0.399553
$956$	34.9666	1.13090
$957$	92.9785	3.00557
$958$	−18.0462	−0.583047
$959$	1.97011	0.0636182
$960$	−30.8708	−0.996351
$961$	−4.84999	−0.156451
$962$	−54.4567	−1.75575
$963$	−2.92918	−0.0943916
$964$	5.29342	0.170490
$965$	4.88560	0.157273
$966$	9.87767	0.317809
$967$	8.42368	0.270887	0.135444	−	0.990785i	$-0.456754\pi$
$967$	8.42368	0.270887	0.135444	+	0.990785i	$0.456754\pi$
$968$	58.1489	1.86898
$969$	−21.2337	−0.682126
$970$	−14.2886	−0.458779
$971$	−44.1565	−1.41705	−0.708525	−	0.705686i	$-0.750639\pi$
$971$	−44.1565	−1.41705	−0.708525	+	0.705686i	$0.750639\pi$
$972$	118.367	3.79661
$973$	−3.95959	−0.126939
$974$	52.9381	1.69625
$975$	−44.0469	−1.41063
$976$	54.3836	1.74078
$977$	−5.28247	−0.169001	−0.0845006	−	0.996423i	$-0.526929\pi$
$977$	−5.28247	−0.169001	−0.0845006	+	0.996423i	$0.526929\pi$
$978$	21.7448	0.695323
$979$	15.9488	0.509726
$980$	−19.4149	−0.620187
$981$	12.9168	0.412403
$982$	−17.8767	−0.570469
$983$	−3.33673	−0.106425	−0.0532125	−	0.998583i	$-0.516946\pi$
$983$	−3.33673	−0.106425	−0.0532125	+	0.998583i	$0.516946\pi$
$984$	−263.773	−8.40879
$985$	−3.60184	−0.114764
$986$	154.499	4.92026
$987$	−4.65621	−0.148209
$988$	25.8274	0.821678
$989$	−2.76572	−0.0879447
$990$	19.8564	0.631076
$991$	−21.6985	−0.689274	−0.344637	−	0.938736i	$-0.611998\pi$
$991$	−21.6985	−0.689274	−0.344637	+	0.938736i	$0.611998\pi$
$992$	94.5534	3.00207
$993$	−0.797421	−0.0253054
$994$	12.5135	0.396904
$995$	1.74104	0.0551947
$996$	−83.0163	−2.63047
$997$	−10.6834	−0.338347	−0.169174	−	0.985586i	$-0.554110\pi$
$997$	−10.6834	−0.338347	−0.169174	+	0.985586i	$0.554110\pi$
$998$	107.169	3.39236
$999$	−3.94401	−0.124783

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	241.2.a.b.1.12	✓	12
3.2	odd	2		2169.2.a.h.1.1		12
4.3	odd	2		3856.2.a.n.1.11		12
5.4	even	2		6025.2.a.h.1.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
241.2.a.b.1.12	✓	12	1.1	even	1	trivial
2169.2.a.h.1.1		12	3.2	odd	2
3856.2.a.n.1.11		12	4.3	odd	2
6025.2.a.h.1.1		12	5.4	even	2

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:	\( N \)	\(=\)	\( 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	241.a (trivial)

\(f(q)\)	\(=\)	\(q+2.70063 q^{2} -2.50808 q^{3} +5.29342 q^{4} +0.533570 q^{5} -6.77340 q^{6} +0.354992 q^{7} +8.89432 q^{8} +3.29045 q^{9} +O(q^{10})\)	\(q+2.70063 q^{2} -2.50808 q^{3} +5.29342 q^{4} +0.533570 q^{5} -6.77340 q^{6} +0.354992 q^{7} +8.89432 q^{8} +3.29045 q^{9} +1.44098 q^{10} +4.18781 q^{11} -13.2763 q^{12} -3.72447 q^{13} +0.958703 q^{14} -1.33823 q^{15} +13.4334 q^{16} -6.46259 q^{17} +8.88631 q^{18} -1.31002 q^{19} +2.82441 q^{20} -0.890347 q^{21} +11.3097 q^{22} -4.10799 q^{23} -22.3076 q^{24} -4.71530 q^{25} -10.0584 q^{26} -0.728479 q^{27} +1.87912 q^{28} -8.85227 q^{29} -3.61408 q^{30} +5.11371 q^{31} +18.4902 q^{32} -10.5034 q^{33} -17.4531 q^{34} +0.189413 q^{35} +17.4177 q^{36} +5.41403 q^{37} -3.53789 q^{38} +9.34127 q^{39} +4.74574 q^{40} +11.8244 q^{41} -2.40450 q^{42} +0.673253 q^{43} +22.1678 q^{44} +1.75569 q^{45} -11.0942 q^{46} +5.22965 q^{47} -33.6921 q^{48} -6.87398 q^{49} -12.7343 q^{50} +16.2087 q^{51} -19.7152 q^{52} -9.92404 q^{53} -1.96735 q^{54} +2.23449 q^{55} +3.15741 q^{56} +3.28564 q^{57} -23.9067 q^{58} -1.23315 q^{59} -7.08384 q^{60} +4.04837 q^{61} +13.8102 q^{62} +1.16808 q^{63} +23.0683 q^{64} -1.98727 q^{65} -28.3657 q^{66} -14.0522 q^{67} -34.2092 q^{68} +10.3032 q^{69} +0.511535 q^{70} +13.0525 q^{71} +29.2663 q^{72} +7.76916 q^{73} +14.6213 q^{74} +11.8263 q^{75} -6.93450 q^{76} +1.48664 q^{77} +25.2273 q^{78} -1.17673 q^{79} +7.16768 q^{80} -8.04428 q^{81} +31.9333 q^{82} +6.25297 q^{83} -4.71298 q^{84} -3.44824 q^{85} +1.81821 q^{86} +22.2022 q^{87} +37.2477 q^{88} +3.80839 q^{89} +4.74147 q^{90} -1.32216 q^{91} -21.7453 q^{92} -12.8256 q^{93} +14.1234 q^{94} -0.698989 q^{95} -46.3748 q^{96} -9.91591 q^{97} -18.5641 q^{98} +13.7798 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 3 q^{2} + q^{3} + 13 q^{4} + 6 q^{5} - q^{6} + 3 q^{7} + 9 q^{8} + 15 q^{9}+O(q^{10}) \) 12 * q + 3 * q^2 + q^3 + 13 * q^4 + 6 * q^5 - q^6 + 3 * q^7 + 9 * q^8 + 15 * q^9	\( 12 q + 3 q^{2} + q^{3} + 13 q^{4} + 6 q^{5} - q^{6} + 3 q^{7} + 9 q^{8} + 15 q^{9} - 7 q^{10} + 22 q^{11} - 7 q^{12} - 5 q^{13} + 6 q^{14} + 13 q^{15} + 15 q^{16} - 4 q^{17} - q^{18} - 6 q^{19} + 10 q^{20} - 14 q^{21} - 12 q^{22} + 32 q^{23} - 15 q^{24} + 4 q^{25} + 8 q^{26} - 5 q^{27} - 11 q^{28} + 6 q^{29} - 19 q^{30} + 8 q^{31} + q^{32} - 24 q^{33} - 19 q^{34} + 15 q^{35} - 8 q^{36} - 8 q^{37} - 10 q^{38} + 31 q^{39} - 52 q^{40} - q^{41} - 49 q^{42} - 2 q^{43} + 42 q^{44} - 15 q^{45} - 25 q^{46} + 34 q^{47} - 49 q^{48} - 9 q^{49} - 27 q^{50} - 3 q^{51} - 41 q^{52} + 5 q^{53} - 40 q^{54} - 3 q^{55} + q^{56} - 22 q^{57} - 33 q^{58} + 26 q^{59} - 57 q^{60} - 26 q^{61} - 17 q^{62} - 4 q^{63} + 13 q^{64} - 25 q^{65} - 2 q^{66} + 6 q^{67} - 35 q^{68} - 2 q^{69} - 4 q^{70} + 94 q^{71} + 17 q^{72} - 22 q^{73} + 26 q^{74} - 20 q^{76} - 7 q^{77} + 54 q^{78} + 9 q^{79} + 19 q^{80} + 4 q^{81} + 15 q^{82} - 8 q^{83} + 2 q^{84} + 4 q^{85} + 9 q^{86} + 4 q^{87} + 6 q^{88} - 3 q^{89} + 11 q^{90} - 20 q^{91} + 36 q^{92} + 12 q^{93} + 48 q^{94} + 33 q^{95} - 23 q^{96} - 29 q^{97} + 28 q^{98} + 36 q^{99}+O(q^{100}) \) 12 * q + 3 * q^2 + q^3 + 13 * q^4 + 6 * q^5 - q^6 + 3 * q^7 + 9 * q^8 + 15 * q^9 - 7 * q^10 + 22 * q^11 - 7 * q^12 - 5 * q^13 + 6 * q^14 + 13 * q^15 + 15 * q^16 - 4 * q^17 - q^18 - 6 * q^19 + 10 * q^20 - 14 * q^21 - 12 * q^22 + 32 * q^23 - 15 * q^24 + 4 * q^25 + 8 * q^26 - 5 * q^27 - 11 * q^28 + 6 * q^29 - 19 * q^30 + 8 * q^31 + q^32 - 24 * q^33 - 19 * q^34 + 15 * q^35 - 8 * q^36 - 8 * q^37 - 10 * q^38 + 31 * q^39 - 52 * q^40 - q^41 - 49 * q^42 - 2 * q^43 + 42 * q^44 - 15 * q^45 - 25 * q^46 + 34 * q^47 - 49 * q^48 - 9 * q^49 - 27 * q^50 - 3 * q^51 - 41 * q^52 + 5 * q^53 - 40 * q^54 - 3 * q^55 + q^56 - 22 * q^57 - 33 * q^58 + 26 * q^59 - 57 * q^60 - 26 * q^61 - 17 * q^62 - 4 * q^63 + 13 * q^64 - 25 * q^65 - 2 * q^66 + 6 * q^67 - 35 * q^68 - 2 * q^69 - 4 * q^70 + 94 * q^71 + 17 * q^72 - 22 * q^73 + 26 * q^74 - 20 * q^76 - 7 * q^77 + 54 * q^78 + 9 * q^79 + 19 * q^80 + 4 * q^81 + 15 * q^82 - 8 * q^83 + 2 * q^84 + 4 * q^85 + 9 * q^86 + 4 * q^87 + 6 * q^88 - 3 * q^89 + 11 * q^90 - 20 * q^91 + 36 * q^92 + 12 * q^93 + 48 * q^94 + 33 * q^95 - 23 * q^96 - 29 * q^97 + 28 * q^98 + 36 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more