LMFDB - Embedded newform 4001.2.a.a.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4001,2,Mod(1,4001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4001, 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("4001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4001.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:	$31.9481458487$
Analytic rank:	$1$
Dimension:	$149$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	4001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.46316 q^{2} +2.96188 q^{3} +4.06716 q^{4} -3.16778 q^{5} -7.29560 q^{6} +1.16261 q^{7} -5.09175 q^{8} +5.77276 q^{9} +O(q^{10})$	\(q-2.46316 q^{2} +2.96188 q^{3} +4.06716 q^{4} -3.16778 q^{5} -7.29560 q^{6} +1.16261 q^{7} -5.09175 q^{8} +5.77276 q^{9} +7.80275 q^{10} +5.60133 q^{11} +12.0465 q^{12} -3.28150 q^{13} -2.86370 q^{14} -9.38259 q^{15} +4.40748 q^{16} -4.14371 q^{17} -14.2192 q^{18} -3.55436 q^{19} -12.8839 q^{20} +3.44352 q^{21} -13.7970 q^{22} +5.83809 q^{23} -15.0812 q^{24} +5.03483 q^{25} +8.08285 q^{26} +8.21258 q^{27} +4.72853 q^{28} -4.97539 q^{29} +23.1108 q^{30} -9.32871 q^{31} -0.672838 q^{32} +16.5905 q^{33} +10.2066 q^{34} -3.68290 q^{35} +23.4787 q^{36} -2.76942 q^{37} +8.75497 q^{38} -9.71941 q^{39} +16.1296 q^{40} -0.775448 q^{41} -8.48195 q^{42} -10.8023 q^{43} +22.7815 q^{44} -18.2868 q^{45} -14.3801 q^{46} +2.49802 q^{47} +13.0545 q^{48} -5.64833 q^{49} -12.4016 q^{50} -12.2732 q^{51} -13.3464 q^{52} -8.56338 q^{53} -20.2289 q^{54} -17.7438 q^{55} -5.91974 q^{56} -10.5276 q^{57} +12.2552 q^{58} +1.99615 q^{59} -38.1605 q^{60} +6.01539 q^{61} +22.9781 q^{62} +6.71148 q^{63} -7.15766 q^{64} +10.3951 q^{65} -40.8650 q^{66} -5.97218 q^{67} -16.8532 q^{68} +17.2917 q^{69} +9.07158 q^{70} -9.41463 q^{71} -29.3935 q^{72} +7.38071 q^{73} +6.82153 q^{74} +14.9126 q^{75} -14.4562 q^{76} +6.51217 q^{77} +23.9405 q^{78} -2.56921 q^{79} -13.9619 q^{80} +7.00645 q^{81} +1.91005 q^{82} +17.8721 q^{83} +14.0054 q^{84} +13.1264 q^{85} +26.6078 q^{86} -14.7365 q^{87} -28.5206 q^{88} +7.49153 q^{89} +45.0434 q^{90} -3.81511 q^{91} +23.7444 q^{92} -27.6306 q^{93} -6.15304 q^{94} +11.2594 q^{95} -1.99287 q^{96} -15.7245 q^{97} +13.9128 q^{98} +32.3351 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 149 q - 6 q^{2} - 28 q^{3} + 116 q^{4} - 19 q^{5} - 31 q^{6} - 47 q^{7} - 15 q^{8} + 115 q^{9}+O(q^{10}) $ 149 * q - 6 * q^2 - 28 * q^3 + 116 * q^4 - 19 * q^5 - 31 * q^6 - 47 * q^7 - 15 * q^8 + 115 * q^9	\( 149 q - 6 q^{2} - 28 q^{3} + 116 q^{4} - 19 q^{5} - 31 q^{6} - 47 q^{7} - 15 q^{8} + 115 q^{9} - 48 q^{10} - 31 q^{11} - 61 q^{12} - 54 q^{13} - 44 q^{14} - 65 q^{15} + 58 q^{16} - 26 q^{17} - 23 q^{18} - 86 q^{19} - 52 q^{20} - 30 q^{21} - 56 q^{22} - 63 q^{23} - 90 q^{24} + 92 q^{25} - 38 q^{26} - 103 q^{27} - 77 q^{28} - 51 q^{29} - 22 q^{30} - 256 q^{31} - 21 q^{32} - 36 q^{33} - 124 q^{34} - 50 q^{35} + 45 q^{36} - 42 q^{37} - 14 q^{38} - 119 q^{39} - 131 q^{40} - 55 q^{41} - 5 q^{42} - 55 q^{43} - 54 q^{44} - 68 q^{45} - 59 q^{46} - 82 q^{47} - 89 q^{48} + 30 q^{49} + 13 q^{50} - 83 q^{51} - 126 q^{52} - 23 q^{53} - 83 q^{54} - 244 q^{55} - 94 q^{56} - 14 q^{57} - 60 q^{58} - 93 q^{59} - 70 q^{60} - 139 q^{61} - 10 q^{62} - 120 q^{63} - 45 q^{64} - 37 q^{65} - 28 q^{66} - 110 q^{67} - 27 q^{68} - 79 q^{69} - 78 q^{70} - 123 q^{71} - 74 q^{73} - 25 q^{74} - 146 q^{75} - 192 q^{76} - q^{77} + 26 q^{78} - 273 q^{79} - 51 q^{80} + 41 q^{81} - 120 q^{82} - 27 q^{83} - 14 q^{84} - 71 q^{85} + 3 q^{86} - 98 q^{87} - 143 q^{88} - 70 q^{89} - 24 q^{90} - 256 q^{91} - 47 q^{92} + 16 q^{93} - 151 q^{94} - 83 q^{95} - 137 q^{96} - 108 q^{97} + 17 q^{98} - 131 q^{99}+O(q^{100}) \) 149 * q - 6 * q^2 - 28 * q^3 + 116 * q^4 - 19 * q^5 - 31 * q^6 - 47 * q^7 - 15 * q^8 + 115 * q^9 - 48 * q^10 - 31 * q^11 - 61 * q^12 - 54 * q^13 - 44 * q^14 - 65 * q^15 + 58 * q^16 - 26 * q^17 - 23 * q^18 - 86 * q^19 - 52 * q^20 - 30 * q^21 - 56 * q^22 - 63 * q^23 - 90 * q^24 + 92 * q^25 - 38 * q^26 - 103 * q^27 - 77 * q^28 - 51 * q^29 - 22 * q^30 - 256 * q^31 - 21 * q^32 - 36 * q^33 - 124 * q^34 - 50 * q^35 + 45 * q^36 - 42 * q^37 - 14 * q^38 - 119 * q^39 - 131 * q^40 - 55 * q^41 - 5 * q^42 - 55 * q^43 - 54 * q^44 - 68 * q^45 - 59 * q^46 - 82 * q^47 - 89 * q^48 + 30 * q^49 + 13 * q^50 - 83 * q^51 - 126 * q^52 - 23 * q^53 - 83 * q^54 - 244 * q^55 - 94 * q^56 - 14 * q^57 - 60 * q^58 - 93 * q^59 - 70 * q^60 - 139 * q^61 - 10 * q^62 - 120 * q^63 - 45 * q^64 - 37 * q^65 - 28 * q^66 - 110 * q^67 - 27 * q^68 - 79 * q^69 - 78 * q^70 - 123 * q^71 - 74 * q^73 - 25 * q^74 - 146 * q^75 - 192 * q^76 - q^77 + 26 * q^78 - 273 * q^79 - 51 * q^80 + 41 * q^81 - 120 * q^82 - 27 * q^83 - 14 * q^84 - 71 * q^85 + 3 * q^86 - 98 * q^87 - 143 * q^88 - 70 * q^89 - 24 * q^90 - 256 * q^91 - 47 * q^92 + 16 * q^93 - 151 * q^94 - 83 * q^95 - 137 * q^96 - 108 * q^97 + 17 * q^98 - 131 * 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 $

$2$	−2.46316	−1.74172	−0.870859	−	0.491533i	$-0.836437\pi$
$2$	−2.46316	−1.74172	−0.870859	+	0.491533i	$0.836437\pi$
$3$	2.96188	1.71004	0.855022	−	0.518591i	$-0.173543\pi$
$3$	2.96188	1.71004	0.855022	+	0.518591i	$0.173543\pi$
$4$	4.06716	2.03358
$5$	−3.16778	−1.41667	−0.708337	−	0.705874i	$-0.750554\pi$
$5$	−3.16778	−1.41667	−0.708337	+	0.705874i	$0.750554\pi$
$6$	−7.29560	−2.97842
$7$	1.16261	0.439426	0.219713	−	0.975565i	$-0.429488\pi$
$7$	1.16261	0.439426	0.219713	+	0.975565i	$0.429488\pi$
$8$	−5.09175	−1.80021
$9$	5.77276	1.92425
$10$	7.80275	2.46745
$11$	5.60133	1.68886	0.844432	−	0.535663i	$-0.179938\pi$
$11$	5.60133	1.68886	0.844432	+	0.535663i	$0.179938\pi$
$12$	12.0465	3.47751
$13$	−3.28150	−0.910123	−0.455062	−	0.890460i	$-0.650383\pi$
$13$	−3.28150	−0.910123	−0.455062	+	0.890460i	$0.650383\pi$
$14$	−2.86370	−0.765357
$15$	−9.38259	−2.42258
$16$	4.40748	1.10187
$17$	−4.14371	−1.00500	−0.502499	−	0.864578i	$-0.667586\pi$
$17$	−4.14371	−1.00500	−0.502499	+	0.864578i	$0.667586\pi$
$18$	−14.2192	−3.35150
$19$	−3.55436	−0.815427	−0.407713	−	0.913110i	$-0.633674\pi$
$19$	−3.55436	−0.815427	−0.407713	+	0.913110i	$0.633674\pi$
$20$	−12.8839	−2.88092
$21$	3.44352	0.751439
$22$	−13.7970	−2.94152
$23$	5.83809	1.21733	0.608663	−	0.793429i	$-0.291706\pi$
$23$	5.83809	1.21733	0.608663	+	0.793429i	$0.291706\pi$
$24$	−15.0812	−3.07843
$25$	5.03483	1.00697
$26$	8.08285	1.58518
$27$	8.21258	1.58051
$28$	4.72853	0.893609
$29$	−4.97539	−0.923906	−0.461953	−	0.886904i	$-0.652851\pi$
$29$	−4.97539	−0.923906	−0.461953	+	0.886904i	$0.652851\pi$
$30$	23.1108	4.21944
$31$	−9.32871	−1.67549	−0.837743	−	0.546065i	$-0.816125\pi$
$31$	−9.32871	−1.67549	−0.837743	+	0.546065i	$0.816125\pi$
$32$	−0.672838	−0.118942
$33$	16.5905	2.88803
$34$	10.2066	1.75042
$35$	−3.68290	−0.622524
$36$	23.4787	3.91312
$37$	−2.76942	−0.455290	−0.227645	−	0.973744i	$-0.573103\pi$
$37$	−2.76942	−0.455290	−0.227645	+	0.973744i	$0.573103\pi$
$38$	8.75497	1.42024
$39$	−9.71941	−1.55635
$40$	16.1296	2.55031
$41$	−0.775448	−0.121105	−0.0605523	−	0.998165i	$-0.519286\pi$
$41$	−0.775448	−0.121105	−0.0605523	+	0.998165i	$0.519286\pi$
$42$	−8.48195	−1.30879
$43$	−10.8023	−1.64734	−0.823669	−	0.567071i	$-0.808077\pi$
$43$	−10.8023	−1.64734	−0.823669	+	0.567071i	$0.808077\pi$
$44$	22.7815	3.43444
$45$	−18.2868	−2.72604
$46$	−14.3801	−2.12024
$47$	2.49802	0.364374	0.182187	−	0.983264i	$-0.441682\pi$
$47$	2.49802	0.364374	0.182187	+	0.983264i	$0.441682\pi$
$48$	13.0545	1.88425
$49$	−5.64833	−0.806905
$50$	−12.4016	−1.75385
$51$	−12.2732	−1.71859
$52$	−13.3464	−1.85081
$53$	−8.56338	−1.17627	−0.588135	−	0.808763i	$-0.700138\pi$
$53$	−8.56338	−1.17627	−0.588135	+	0.808763i	$0.700138\pi$
$54$	−20.2289	−2.75281
$55$	−17.7438	−2.39257
$56$	−5.91974	−0.791058
$57$	−10.5276	−1.39442
$58$	12.2552	1.60918
$59$	1.99615	0.259877	0.129938	−	0.991522i	$-0.458522\pi$
$59$	1.99615	0.259877	0.129938	+	0.991522i	$0.458522\pi$
$60$	−38.1605	−4.92650
$61$	6.01539	0.770192	0.385096	−	0.922877i	$-0.374168\pi$
$61$	6.01539	0.770192	0.385096	+	0.922877i	$0.374168\pi$
$62$	22.9781	2.91822
$63$	6.71148	0.845567
$64$	−7.15766	−0.894708
$65$	10.3951	1.28935
$66$	−40.8650	−5.03014
$67$	−5.97218	−0.729618	−0.364809	−	0.931082i	$-0.618866\pi$
$67$	−5.97218	−0.729618	−0.364809	+	0.931082i	$0.618866\pi$
$68$	−16.8532	−2.04374
$69$	17.2917	2.08168
$70$	9.07158	1.08426
$71$	−9.41463	−1.11731	−0.558656	−	0.829400i	$-0.688683\pi$
$71$	−9.41463	−1.11731	−0.558656	+	0.829400i	$0.688683\pi$
$72$	−29.3935	−3.46405
$73$	7.38071	0.863846	0.431923	−	0.901910i	$-0.357835\pi$
$73$	7.38071	0.863846	0.431923	+	0.901910i	$0.357835\pi$
$74$	6.82153	0.792986
$75$	14.9126	1.72196
$76$	−14.4562	−1.65824
$77$	6.51217	0.742131
$78$	23.9405	2.71072
$79$	−2.56921	−0.289059	−0.144530	−	0.989500i	$-0.546167\pi$
$79$	−2.56921	−0.289059	−0.144530	+	0.989500i	$0.546167\pi$
$80$	−13.9619	−1.56099
$81$	7.00645	0.778494
$82$	1.91005	0.210930
$83$	17.8721	1.96171	0.980857	−	0.194730i	$-0.0623831\pi$
$83$	17.8721	1.96171	0.980857	+	0.194730i	$0.0623831\pi$
$84$	14.0054	1.52811
$85$	13.1264	1.42375
$86$	26.6078	2.86920
$87$	−14.7365	−1.57992
$88$	−28.5206	−3.04030
$89$	7.49153	0.794101	0.397050	−	0.917797i	$-0.370034\pi$
$89$	7.49153	0.794101	0.397050	+	0.917797i	$0.370034\pi$
$90$	45.0434	4.74799
$91$	−3.81511	−0.399932
$92$	23.7444	2.47553
$93$	−27.6306	−2.86516
$94$	−6.15304	−0.634637
$95$	11.2594	1.15519
$96$	−1.99287	−0.203396
$97$	−15.7245	−1.59658	−0.798289	−	0.602275i	$-0.794261\pi$
$97$	−15.7245	−1.59658	−0.798289	+	0.602275i	$0.794261\pi$
$98$	13.9128	1.40540
$99$	32.3351	3.24980
$100$	20.4775	2.04775
$101$	−6.57524	−0.654260	−0.327130	−	0.944979i	$-0.606082\pi$
$101$	−6.57524	−0.654260	−0.327130	+	0.944979i	$0.606082\pi$
$102$	30.2309	2.99330
$103$	−11.1101	−1.09471	−0.547356	−	0.836900i	$-0.684366\pi$
$103$	−11.1101	−1.09471	−0.547356	+	0.836900i	$0.684366\pi$
$104$	16.7086	1.63841
$105$	−10.9083	−1.06454
$106$	21.0930	2.04873
$107$	−0.912121	−0.0881780	−0.0440890	−	0.999028i	$-0.514039\pi$
$107$	−0.912121	−0.0881780	−0.0440890	+	0.999028i	$0.514039\pi$
$108$	33.4019	3.21410
$109$	17.9323	1.71760	0.858801	−	0.512310i	$-0.171210\pi$
$109$	17.9323	1.71760	0.858801	+	0.512310i	$0.171210\pi$
$110$	43.7058	4.16718
$111$	−8.20270	−0.778566
$112$	5.12420	0.484191
$113$	−14.2417	−1.33974	−0.669872	−	0.742477i	$-0.733651\pi$
$113$	−14.2417	−1.33974	−0.669872	+	0.742477i	$0.733651\pi$
$114$	25.9312	2.42868
$115$	−18.4938	−1.72455
$116$	−20.2357	−1.87884
$117$	−18.9433	−1.75131
$118$	−4.91684	−0.452632
$119$	−4.81753	−0.441623
$120$	47.7739	4.36114
$121$	20.3749	1.85226
$122$	−14.8169	−1.34146
$123$	−2.29679	−0.207094
$124$	−37.9414	−3.40724
$125$	−0.110322	−0.00986752
$126$	−16.5315	−1.47274
$127$	9.05309	0.803332	0.401666	−	0.915786i	$-0.368431\pi$
$127$	9.05309	0.803332	0.401666	+	0.915786i	$0.368431\pi$
$128$	18.9762	1.67727
$129$	−31.9952	−2.81702
$130$	−25.6047	−2.24568
$131$	−4.07072	−0.355661	−0.177830	−	0.984061i	$-0.556908\pi$
$131$	−4.07072	−0.355661	−0.177830	+	0.984061i	$0.556908\pi$
$132$	67.4762	5.87305
$133$	−4.13235	−0.358320
$134$	14.7105	1.27079
$135$	−26.0157	−2.23907
$136$	21.0988	1.80920
$137$	16.4338	1.40403	0.702017	−	0.712160i	$-0.252283\pi$
$137$	16.4338	1.40403	0.702017	+	0.712160i	$0.252283\pi$
$138$	−42.5923	−3.62570
$139$	−2.48151	−0.210479	−0.105239	−	0.994447i	$-0.533561\pi$
$139$	−2.48151	−0.210479	−0.105239	+	0.994447i	$0.533561\pi$
$140$	−14.9790	−1.26595
$141$	7.39886	0.623096
$142$	23.1898	1.94604
$143$	−18.3807	−1.53707
$144$	25.4433	2.12028
$145$	15.7609	1.30887
$146$	−18.1799	−1.50458
$147$	−16.7297	−1.37984
$148$	−11.2637	−0.925869
$149$	−14.2573	−1.16801	−0.584004	−	0.811751i	$-0.698515\pi$
$149$	−14.2573	−1.16801	−0.584004	+	0.811751i	$0.698515\pi$
$150$	−36.7321	−2.99916
$151$	3.31505	0.269775	0.134887	−	0.990861i	$-0.456933\pi$
$151$	3.31505	0.269775	0.134887	+	0.990861i	$0.456933\pi$
$152$	18.0979	1.46794
$153$	−23.9206	−1.93387
$154$	−16.0405	−1.29258
$155$	29.5513	2.37362
$156$	−39.5304	−3.16497
$157$	−12.2788	−0.979954	−0.489977	−	0.871735i	$-0.662995\pi$
$157$	−12.2788	−0.979954	−0.489977	+	0.871735i	$0.662995\pi$
$158$	6.32839	0.503459
$159$	−25.3637	−2.01147
$160$	2.13140	0.168502
$161$	6.78743	0.534925
$162$	−17.2580	−1.35592
$163$	0.0392046	0.00307074	0.00153537	−	0.999999i	$-0.499511\pi$
$163$	0.0392046	0.00307074	0.00153537	+	0.999999i	$0.499511\pi$
$164$	−3.15387	−0.246276
$165$	−52.5550	−4.09140
$166$	−44.0218	−3.41675
$167$	3.93356	0.304388	0.152194	−	0.988351i	$-0.451366\pi$
$167$	3.93356	0.304388	0.152194	+	0.988351i	$0.451366\pi$
$168$	−17.5336	−1.35274
$169$	−2.23178	−0.171676
$170$	−32.3324	−2.47978
$171$	−20.5185	−1.56909
$172$	−43.9348	−3.35000
$173$	21.7294	1.65205	0.826026	−	0.563632i	$-0.190596\pi$
$173$	21.7294	1.65205	0.826026	+	0.563632i	$0.190596\pi$
$174$	36.2984	2.75178
$175$	5.85355	0.442487
$176$	24.6878	1.86091
$177$	5.91237	0.444401
$178$	−18.4529	−1.38310
$179$	−26.3499	−1.96949	−0.984743	−	0.174014i	$-0.944326\pi$
$179$	−26.3499	−1.96949	−0.984743	+	0.174014i	$0.944326\pi$
$180$	−74.3755	−5.54362
$181$	−14.8889	−1.10668	−0.553342	−	0.832954i	$-0.686648\pi$
$181$	−14.8889	−1.10668	−0.553342	+	0.832954i	$0.686648\pi$
$182$	9.39723	0.696569
$183$	17.8169	1.31706
$184$	−29.7261	−2.19144
$185$	8.77291	0.644997
$186$	68.0585	4.99029
$187$	−23.2103	−1.69730
$188$	10.1599	0.740985
$189$	9.54805	0.694519
$190$	−27.7338	−2.01202
$191$	4.68410	0.338929	0.169465	−	0.985536i	$-0.445796\pi$
$191$	4.68410	0.338929	0.169465	+	0.985536i	$0.445796\pi$
$192$	−21.2002	−1.52999
$193$	11.3626	0.817899	0.408950	−	0.912557i	$-0.365895\pi$
$193$	11.3626	0.817899	0.408950	+	0.912557i	$0.365895\pi$
$194$	38.7319	2.78079
$195$	30.7889	2.20484
$196$	−22.9727	−1.64091
$197$	13.7294	0.978176	0.489088	−	0.872234i	$-0.337330\pi$
$197$	13.7294	0.978176	0.489088	+	0.872234i	$0.337330\pi$
$198$	−79.6465	−5.66023
$199$	−11.1648	−0.791455	−0.395727	−	0.918368i	$-0.629507\pi$
$199$	−11.1648	−0.791455	−0.395727	+	0.918368i	$0.629507\pi$
$200$	−25.6361	−1.81275
$201$	−17.6889	−1.24768
$202$	16.1959	1.13954
$203$	−5.78445	−0.405989
$204$	−49.9171	−3.49489
$205$	2.45645	0.171566
$206$	27.3660	1.90668
$207$	33.7018	2.34244
$208$	−14.4631	−1.00284
$209$	−19.9092	−1.37714
$210$	26.8690	1.85413
$211$	4.51290	0.310681	0.155340	−	0.987861i	$-0.450353\pi$
$211$	4.51290	0.310681	0.155340	+	0.987861i	$0.450353\pi$
$212$	−34.8286	−2.39204
$213$	−27.8850	−1.91065
$214$	2.24670	0.153581
$215$	34.2194	2.33374
$216$	−41.8165	−2.84525
$217$	−10.8457	−0.736253
$218$	−44.1701	−2.99158
$219$	21.8608	1.47722
$220$	−72.1668	−4.86548
$221$	13.5976	0.914672
$222$	20.2046	1.35604
$223$	8.68502	0.581592	0.290796	−	0.956785i	$-0.406080\pi$
$223$	8.68502	0.581592	0.290796	+	0.956785i	$0.406080\pi$
$224$	−0.782250	−0.0522662
$225$	29.0648	1.93766
$226$	35.0795	2.33346
$227$	−4.69398	−0.311551	−0.155775	−	0.987793i	$-0.549788\pi$
$227$	−4.69398	−0.311551	−0.155775	+	0.987793i	$0.549788\pi$
$228$	−42.8175	−2.83566
$229$	−22.0672	−1.45824	−0.729122	−	0.684383i	$-0.760072\pi$
$229$	−22.0672	−1.45824	−0.729122	+	0.684383i	$0.760072\pi$
$230$	45.5531	3.00368
$231$	19.2883	1.26908
$232$	25.3334	1.66322
$233$	−9.31592	−0.610306	−0.305153	−	0.952303i	$-0.598708\pi$
$233$	−9.31592	−0.610306	−0.305153	+	0.952303i	$0.598708\pi$
$234$	46.6603	3.05028
$235$	−7.91319	−0.516200
$236$	8.11867	0.528480
$237$	−7.60971	−0.494304
$238$	11.8664	0.769182
$239$	−7.44456	−0.481548	−0.240774	−	0.970581i	$-0.577401\pi$
$239$	−7.44456	−0.481548	−0.240774	+	0.970581i	$0.577401\pi$
$240$	−41.3536	−2.66937
$241$	0.0522324	0.00336458	0.00168229	−	0.999999i	$-0.499465\pi$
$241$	0.0522324	0.00336458	0.00168229	+	0.999999i	$0.499465\pi$
$242$	−50.1866	−3.22611
$243$	−3.88546	−0.249253
$244$	24.4656	1.56625
$245$	17.8927	1.14312
$246$	5.65736	0.360700
$247$	11.6636	0.742139
$248$	47.4995	3.01622
$249$	52.9350	3.35462
$250$	0.271741	0.0171864
$251$	26.1831	1.65266	0.826331	−	0.563184i	$-0.190424\pi$
$251$	26.1831	1.65266	0.826331	+	0.563184i	$0.190424\pi$
$252$	27.2967	1.71953
$253$	32.7010	2.05590
$254$	−22.2992	−1.39918
$255$	38.8788	2.43468
$256$	−32.4260	−2.02662
$257$	−13.2452	−0.826212	−0.413106	−	0.910683i	$-0.635556\pi$
$257$	−13.2452	−0.826212	−0.413106	+	0.910683i	$0.635556\pi$
$258$	78.8094	4.90646
$259$	−3.21976	−0.200066
$260$	42.2784	2.62199
$261$	−28.7217	−1.77783
$262$	10.0268	0.619460
$263$	5.58617	0.344458	0.172229	−	0.985057i	$-0.444903\pi$
$263$	5.58617	0.344458	0.172229	+	0.985057i	$0.444903\pi$
$264$	−84.4746	−5.19905
$265$	27.1269	1.66639
$266$	10.1786	0.624092
$267$	22.1890	1.35795
$268$	−24.2898	−1.48374
$269$	−23.1915	−1.41401	−0.707004	−	0.707209i	$-0.749954\pi$
$269$	−23.1915	−1.41401	−0.707004	+	0.707209i	$0.749954\pi$
$270$	64.0807	3.89983
$271$	−13.6336	−0.828183	−0.414092	−	0.910235i	$-0.635901\pi$
$271$	−13.6336	−0.828183	−0.414092	+	0.910235i	$0.635901\pi$
$272$	−18.2634	−1.10738
$273$	−11.2999	−0.683902
$274$	−40.4791	−2.44543
$275$	28.2017	1.70063
$276$	70.3283	4.23327
$277$	−5.82265	−0.349849	−0.174924	−	0.984582i	$-0.555968\pi$
$277$	−5.82265	−0.349849	−0.174924	+	0.984582i	$0.555968\pi$
$278$	6.11235	0.366595
$279$	−53.8524	−3.22406
$280$	18.7524	1.12067
$281$	−0.642865	−0.0383501	−0.0191750	−	0.999816i	$-0.506104\pi$
$281$	−0.642865	−0.0383501	−0.0191750	+	0.999816i	$0.506104\pi$
$282$	−18.2246	−1.08526
$283$	33.0088	1.96217	0.981085	−	0.193579i	$-0.0620096\pi$
$283$	33.0088	1.96217	0.981085	+	0.193579i	$0.0620096\pi$
$284$	−38.2908	−2.27214
$285$	33.3492	1.97543
$286$	45.2747	2.67715
$287$	−0.901546	−0.0532166
$288$	−3.88413	−0.228874
$289$	0.170355	0.0100209
$290$	−38.8217	−2.27969
$291$	−46.5740	−2.73022
$292$	30.0185	1.75670
$293$	−14.3468	−0.838147	−0.419074	−	0.907952i	$-0.637645\pi$
$293$	−14.3468	−0.838147	−0.419074	+	0.907952i	$0.637645\pi$
$294$	41.2080	2.40330
$295$	−6.32337	−0.368161
$296$	14.1012	0.819616
$297$	46.0014	2.66927
$298$	35.1181	2.03434
$299$	−19.1577	−1.10792
$300$	60.6518	3.50174
$301$	−12.5589	−0.723884
$302$	−8.16550	−0.469872
$303$	−19.4751	−1.11881
$304$	−15.6658	−0.898495
$305$	−19.0554	−1.09111
$306$	58.9204	3.36826
$307$	−5.53470	−0.315882	−0.157941	−	0.987449i	$-0.550486\pi$
$307$	−5.53470	−0.315882	−0.157941	+	0.987449i	$0.550486\pi$
$308$	26.4861	1.50918
$309$	−32.9069	−1.87201
$310$	−72.7896	−4.13417
$311$	−24.1070	−1.36698	−0.683492	−	0.729958i	$-0.739539\pi$
$311$	−24.1070	−1.36698	−0.683492	+	0.729958i	$0.739539\pi$
$312$	49.4888	2.80175
$313$	20.1259	1.13758	0.568792	−	0.822481i	$-0.307411\pi$
$313$	20.1259	1.13758	0.568792	+	0.822481i	$0.307411\pi$
$314$	30.2446	1.70680
$315$	−21.2605	−1.19789
$316$	−10.4494	−0.587825
$317$	−27.1745	−1.52627	−0.763136	−	0.646238i	$-0.776341\pi$
$317$	−27.1745	−1.52627	−0.763136	+	0.646238i	$0.776341\pi$
$318$	62.4749	3.50342
$319$	−27.8688	−1.56035
$320$	22.6739	1.26751
$321$	−2.70160	−0.150788
$322$	−16.7185	−0.931688
$323$	14.7283	0.819502
$324$	28.4964	1.58313
$325$	−16.5218	−0.916462
$326$	−0.0965672	−0.00534836
$327$	53.1134	2.93717
$328$	3.94839	0.218013
$329$	2.90424	0.160116
$330$	129.451	7.12606
$331$	26.9585	1.48177	0.740886	−	0.671631i	$-0.234406\pi$
$331$	26.9585	1.48177	0.740886	+	0.671631i	$0.234406\pi$
$332$	72.6886	3.98930
$333$	−15.9872	−0.876092
$334$	−9.68899	−0.530158
$335$	18.9186	1.03363
$336$	15.1773	0.827989
$337$	8.26426	0.450183	0.225092	−	0.974338i	$-0.427732\pi$
$337$	8.26426	0.450183	0.225092	+	0.974338i	$0.427732\pi$
$338$	5.49724	0.299011
$339$	−42.1822	−2.29102
$340$	53.3871	2.89532
$341$	−52.2531	−2.82967
$342$	50.5403	2.73291
$343$	−14.7051	−0.794001
$344$	55.0027	2.96555
$345$	−54.7764	−2.94906
$346$	−53.5229	−2.87741
$347$	28.2924	1.51882	0.759409	−	0.650614i	$-0.225488\pi$
$347$	28.2924	1.51882	0.759409	+	0.650614i	$0.225488\pi$
$348$	−59.9358	−3.21290
$349$	2.51157	0.134441	0.0672207	−	0.997738i	$-0.478587\pi$
$349$	2.51157	0.134441	0.0672207	+	0.997738i	$0.478587\pi$
$350$	−14.4182	−0.770688
$351$	−26.9496	−1.43846
$352$	−3.76878	−0.200877
$353$	24.7631	1.31800	0.659002	−	0.752141i	$-0.270979\pi$
$353$	24.7631	1.31800	0.659002	+	0.752141i	$0.270979\pi$
$354$	−14.5631	−0.774021
$355$	29.8235	1.58287
$356$	30.4693	1.61487
$357$	−14.2690	−0.755194
$358$	64.9041	3.43029
$359$	−11.2984	−0.596304	−0.298152	−	0.954518i	$-0.596370\pi$
$359$	−11.2984	−0.596304	−0.298152	+	0.954518i	$0.596370\pi$
$360$	93.1120	4.90743
$361$	−6.36650	−0.335079
$362$	36.6738	1.92753
$363$	60.3480	3.16745
$364$	−15.5167	−0.813294
$365$	−23.3805	−1.22379
$366$	−43.8859	−2.29395
$367$	−5.66855	−0.295896	−0.147948	−	0.988995i	$-0.547267\pi$
$367$	−5.66855	−0.295896	−0.147948	+	0.988995i	$0.547267\pi$
$368$	25.7313	1.34134
$369$	−4.47647	−0.233036
$370$	−21.6091	−1.12340
$371$	−9.95589	−0.516884
$372$	−112.378	−5.82653
$373$	−15.5423	−0.804752	−0.402376	−	0.915474i	$-0.631816\pi$
$373$	−15.5423	−0.804752	−0.402376	+	0.915474i	$0.631816\pi$
$374$	57.1707	2.95623
$375$	−0.326761	−0.0168739
$376$	−12.7193	−0.655949
$377$	16.3267	0.840868
$378$	−23.5184	−1.20966
$379$	−4.81684	−0.247424	−0.123712	−	0.992318i	$-0.539480\pi$
$379$	−4.81684	−0.247424	−0.123712	+	0.992318i	$0.539480\pi$
$380$	45.7940	2.34918
$381$	26.8142	1.37373
$382$	−11.5377	−0.590319
$383$	−35.7947	−1.82902	−0.914512	−	0.404558i	$-0.867425\pi$
$383$	−35.7947	−1.82902	−0.914512	+	0.404558i	$0.867425\pi$
$384$	56.2052	2.86821
$385$	−20.6291	−1.05136
$386$	−27.9880	−1.42455
$387$	−62.3591	−3.16989
$388$	−63.9539	−3.24677
$389$	−24.1140	−1.22263	−0.611315	−	0.791388i	$-0.709359\pi$
$389$	−24.1140	−1.22263	−0.611315	+	0.791388i	$0.709359\pi$
$390$	−75.8381	−3.84021
$391$	−24.1914	−1.22341
$392$	28.7599	1.45259
$393$	−12.0570	−0.608195
$394$	−33.8176	−1.70371
$395$	8.13870	0.409502
$396$	131.512	6.60873
$397$	−2.02344	−0.101554	−0.0507768	−	0.998710i	$-0.516170\pi$
$397$	−2.02344	−0.101554	−0.0507768	+	0.998710i	$0.516170\pi$
$398$	27.5008	1.37849
$399$	−12.2395	−0.612743
$400$	22.1909	1.10955
$401$	9.81027	0.489902	0.244951	−	0.969536i	$-0.421228\pi$
$401$	9.81027	0.489902	0.244951	+	0.969536i	$0.421228\pi$
$402$	43.5707	2.17311
$403$	30.6121	1.52490
$404$	−26.7425	−1.33049
$405$	−22.1949	−1.10287
$406$	14.2480	0.707118
$407$	−15.5124	−0.768922
$408$	62.4921	3.09382
$409$	−8.61996	−0.426230	−0.213115	−	0.977027i	$-0.568361\pi$
$409$	−8.61996	−0.426230	−0.213115	+	0.977027i	$0.568361\pi$
$410$	−6.05063	−0.298819
$411$	48.6750	2.40096
$412$	−45.1866	−2.22619
$413$	2.32075	0.114197
$414$	−83.0131	−4.07987
$415$	−56.6148	−2.77911
$416$	2.20791	0.108252
$417$	−7.34994	−0.359928
$418$	49.0394	2.39860
$419$	9.07204	0.443198	0.221599	−	0.975138i	$-0.428872\pi$
$419$	9.07204	0.443198	0.221599	+	0.975138i	$0.428872\pi$
$420$	−44.3659	−2.16484
$421$	27.4047	1.33562	0.667812	−	0.744330i	$-0.267231\pi$
$421$	27.4047	1.33562	0.667812	+	0.744330i	$0.267231\pi$
$422$	−11.1160	−0.541118
$423$	14.4205	0.701148
$424$	43.6026	2.11753
$425$	−20.8629	−1.01200
$426$	68.6854	3.32782
$427$	6.99357	0.338443
$428$	−3.70974	−0.179317
$429$	−54.4416	−2.62846
$430$	−84.2878	−4.06472
$431$	32.6484	1.57262	0.786309	−	0.617833i	$-0.211989\pi$
$431$	32.6484	1.57262	0.786309	+	0.617833i	$0.211989\pi$
$432$	36.1968	1.74152
$433$	−12.6075	−0.605876	−0.302938	−	0.953010i	$-0.597967\pi$
$433$	−12.6075	−0.605876	−0.302938	+	0.953010i	$0.597967\pi$
$434$	26.7146	1.28234
$435$	46.6820	2.23823
$436$	72.9335	3.49288
$437$	−20.7507	−0.992640
$438$	−53.8467	−2.57289
$439$	−12.1464	−0.579715	−0.289857	−	0.957070i	$-0.593608\pi$
$439$	−12.1464	−0.579715	−0.289857	+	0.957070i	$0.593608\pi$
$440$	90.3469	4.30712
$441$	−32.6064	−1.55269
$442$	−33.4930	−1.59310
$443$	30.2366	1.43658	0.718292	−	0.695742i	$-0.244924\pi$
$443$	30.2366	1.43658	0.718292	+	0.695742i	$0.244924\pi$
$444$	−33.3617	−1.58328
$445$	−23.7315	−1.12498
$446$	−21.3926	−1.01297
$447$	−42.2286	−1.99734
$448$	−8.32159	−0.393158
$449$	4.49165	0.211974	0.105987	−	0.994368i	$-0.466200\pi$
$449$	4.49165	0.211974	0.105987	+	0.994368i	$0.466200\pi$
$450$	−71.5913	−3.37485
$451$	−4.34354	−0.204529
$452$	−57.9232	−2.72448
$453$	9.81879	0.461327
$454$	11.5620	0.542633
$455$	12.0854	0.566573
$456$	53.6040	2.51024
$457$	−4.84914	−0.226833	−0.113417	−	0.993548i	$-0.536180\pi$
$457$	−4.84914	−0.226833	−0.113417	+	0.993548i	$0.536180\pi$
$458$	54.3552	2.53985
$459$	−34.0306	−1.58841
$460$	−75.2172	−3.50702
$461$	13.1340	0.611714	0.305857	−	0.952078i	$-0.401057\pi$
$461$	13.1340	0.611714	0.305857	+	0.952078i	$0.401057\pi$
$462$	−47.5102	−2.21037
$463$	24.5980	1.14317	0.571583	−	0.820544i	$-0.306329\pi$
$463$	24.5980	1.14317	0.571583	+	0.820544i	$0.306329\pi$
$464$	−21.9289	−1.01803
$465$	87.5275	4.05899
$466$	22.9466	1.06298
$467$	24.5163	1.13448	0.567241	−	0.823552i	$-0.308011\pi$
$467$	24.5163	1.13448	0.567241	+	0.823552i	$0.308011\pi$
$468$	−77.0454	−3.56142
$469$	−6.94334	−0.320614
$470$	19.4915	0.899074
$471$	−36.3683	−1.67576
$472$	−10.1639	−0.467832
$473$	−60.5073	−2.78213
$474$	18.7439	0.860938
$475$	−17.8956	−0.821107
$476$	−19.5937	−0.898075
$477$	−49.4343	−2.26344
$478$	18.3371	0.838721
$479$	−27.4212	−1.25291	−0.626453	−	0.779459i	$-0.715494\pi$
$479$	−27.4212	−1.25291	−0.626453	+	0.779459i	$0.715494\pi$
$480$	6.31296	0.288146
$481$	9.08784	0.414370
$482$	−0.128657	−0.00586015
$483$	20.1036	0.914745
$484$	82.8679	3.76672
$485$	49.8116	2.26183
$486$	9.57052	0.434128
$487$	−16.7478	−0.758914	−0.379457	−	0.925209i	$-0.623889\pi$
$487$	−16.7478	−0.758914	−0.379457	+	0.925209i	$0.623889\pi$
$488$	−30.6289	−1.38650
$489$	0.116119	0.00525110
$490$	−44.0725	−1.99099
$491$	20.4493	0.922866	0.461433	−	0.887175i	$-0.347335\pi$
$491$	20.4493	0.922866	0.461433	+	0.887175i	$0.347335\pi$
$492$	−9.34141	−0.421143
$493$	20.6166	0.928524
$494$	−28.7294	−1.29260
$495$	−102.430	−4.60391
$496$	−41.1161	−1.84617
$497$	−10.9456	−0.490976
$498$	−130.387	−5.84280
$499$	22.4778	1.00625	0.503123	−	0.864215i	$-0.332184\pi$
$499$	22.4778	1.00625	0.503123	+	0.864215i	$0.332184\pi$
$500$	−0.448698	−0.0200664
$501$	11.6507	0.520517
$502$	−64.4932	−2.87847
$503$	−24.0251	−1.07122	−0.535612	−	0.844464i	$-0.679919\pi$
$503$	−24.0251	−1.07122	−0.535612	+	0.844464i	$0.679919\pi$
$504$	−34.1732	−1.52220
$505$	20.8289	0.926874
$506$	−80.5479	−3.58079
$507$	−6.61029	−0.293573
$508$	36.8204	1.63364
$509$	−28.6472	−1.26976	−0.634882	−	0.772609i	$-0.718951\pi$
$509$	−28.6472	−1.26976	−0.634882	+	0.772609i	$0.718951\pi$
$510$	−95.7647	−4.24053
$511$	8.58090	0.379597
$512$	41.9181	1.85254
$513$	−29.1905	−1.28879
$514$	32.6250	1.43903
$515$	35.1944	1.55085
$516$	−130.130	−5.72864
$517$	13.9923	0.615379
$518$	7.93079	0.348459
$519$	64.3598	2.82508
$520$	−52.9291	−2.32109
$521$	15.1033	0.661688	0.330844	−	0.943685i	$-0.392667\pi$
$521$	15.1033	0.661688	0.330844	+	0.943685i	$0.392667\pi$
$522$	70.7462	3.09648
$523$	10.1616	0.444334	0.222167	−	0.975009i	$-0.428687\pi$
$523$	10.1616	0.444334	0.222167	+	0.975009i	$0.428687\pi$
$524$	−16.5563	−0.723265
$525$	17.3375	0.756673
$526$	−13.7596	−0.599948
$527$	38.6555	1.68386
$528$	73.1223	3.18224
$529$	11.0832	0.481880
$530$	−66.8179	−2.90238
$531$	11.5233	0.500068
$532$	−16.8069	−0.728673
$533$	2.54463	0.110220
$534$	−54.6552	−2.36516
$535$	2.88940	0.124920
$536$	30.4089	1.31346
$537$	−78.0454	−3.36791
$538$	57.1243	2.46280
$539$	−31.6381	−1.36275
$540$	−105.810	−4.55333
$541$	−22.7402	−0.977678	−0.488839	−	0.872374i	$-0.662580\pi$
$541$	−22.7402	−0.977678	−0.488839	+	0.872374i	$0.662580\pi$
$542$	33.5818	1.44246
$543$	−44.0992	−1.89248
$544$	2.78805	0.119536
$545$	−56.8055	−2.43328
$546$	27.8335	1.19116
$547$	−35.8645	−1.53346	−0.766728	−	0.641973i	$-0.778116\pi$
$547$	−35.8645	−1.53346	−0.766728	+	0.641973i	$0.778116\pi$
$548$	66.8389	2.85522
$549$	34.7254	1.48204
$550$	−69.4653	−2.96201
$551$	17.6843	0.753378
$552$	−88.0452	−3.74745
$553$	−2.98700	−0.127020
$554$	14.3421	0.609338
$555$	25.9843	1.10297
$556$	−10.0927	−0.428026
$557$	−13.6077	−0.576578	−0.288289	−	0.957543i	$-0.593086\pi$
$557$	−13.6077	−0.576578	−0.288289	+	0.957543i	$0.593086\pi$
$558$	132.647	5.61540
$559$	35.4478	1.49928
$560$	−16.2323	−0.685941
$561$	−68.7462	−2.90247
$562$	1.58348	0.0667950
$563$	−8.71683	−0.367370	−0.183685	−	0.982985i	$-0.558803\pi$
$563$	−8.71683	−0.367370	−0.183685	+	0.982985i	$0.558803\pi$
$564$	30.0924	1.26712
$565$	45.1145	1.89798
$566$	−81.3060	−3.41755
$567$	8.14579	0.342091
$568$	47.9370	2.01139
$569$	46.1303	1.93388	0.966941	−	0.254999i	$-0.0820753\pi$
$569$	46.1303	1.93388	0.966941	+	0.254999i	$0.0820753\pi$
$570$	−82.1443	−3.44065
$571$	−38.0648	−1.59296	−0.796481	−	0.604664i	$-0.793307\pi$
$571$	−38.0648	−1.59296	−0.796481	+	0.604664i	$0.793307\pi$
$572$	−74.7574	−3.12576
$573$	13.8737	0.579584
$574$	2.22065	0.0926883
$575$	29.3937	1.22580
$576$	−41.3194	−1.72164
$577$	−2.14277	−0.0892047	−0.0446023	−	0.999005i	$-0.514202\pi$
$577$	−2.14277	−0.0892047	−0.0446023	+	0.999005i	$0.514202\pi$
$578$	−0.419612	−0.0174536
$579$	33.6548	1.39864
$580$	64.1023	2.66170
$581$	20.7783	0.862029
$582$	114.719	4.75527
$583$	−47.9663	−1.98656
$584$	−37.5807	−1.55510
$585$	60.0081	2.48103
$586$	35.3384	1.45982
$587$	35.3650	1.45967	0.729835	−	0.683623i	$-0.239597\pi$
$587$	35.3650	1.45967	0.729835	+	0.683623i	$0.239597\pi$
$588$	−68.0424	−2.80602
$589$	33.1576	1.36624
$590$	15.5755	0.641232
$591$	40.6648	1.67272
$592$	−12.2062	−0.501671
$593$	−0.428967	−0.0176156	−0.00880778	−	0.999961i	$-0.502804\pi$
$593$	−0.428967	−0.0176156	−0.00880778	+	0.999961i	$0.502804\pi$
$594$	−113.309	−4.64911
$595$	15.2609	0.625635
$596$	−57.9870	−2.37524
$597$	−33.0690	−1.35342
$598$	47.1884	1.92968
$599$	15.0062	0.613137	0.306568	−	0.951849i	$-0.400819\pi$
$599$	15.0062	0.613137	0.306568	+	0.951849i	$0.400819\pi$
$600$	−75.9311	−3.09988
$601$	−43.5620	−1.77693	−0.888465	−	0.458945i	$-0.848227\pi$
$601$	−43.5620	−1.77693	−0.888465	+	0.458945i	$0.848227\pi$
$602$	30.9346	1.26080
$603$	−34.4760	−1.40397
$604$	13.4828	0.548609
$605$	−64.5431	−2.62405
$606$	47.9703	1.94866
$607$	−9.92949	−0.403025	−0.201513	−	0.979486i	$-0.564586\pi$
$607$	−9.92949	−0.403025	−0.201513	+	0.979486i	$0.564586\pi$
$608$	2.39151	0.0969885
$609$	−17.1329	−0.694259
$610$	46.9366	1.90041
$611$	−8.19726	−0.331626
$612$	−97.2891	−3.93268
$613$	4.06607	0.164227	0.0821134	−	0.996623i	$-0.473833\pi$
$613$	4.06607	0.164227	0.0821134	+	0.996623i	$0.473833\pi$
$614$	13.6328	0.550177
$615$	7.27572	0.293385
$616$	−33.1584	−1.33599
$617$	−35.2558	−1.41934	−0.709672	−	0.704532i	$-0.751157\pi$
$617$	−35.2558	−1.41934	−0.709672	+	0.704532i	$0.751157\pi$
$618$	81.0549	3.26051
$619$	3.36140	0.135106	0.0675531	−	0.997716i	$-0.478481\pi$
$619$	3.36140	0.135106	0.0675531	+	0.997716i	$0.478481\pi$
$620$	120.190	4.82694
$621$	47.9458	1.92400
$622$	59.3795	2.38090
$623$	8.70975	0.348949
$624$	−42.8382	−1.71490
$625$	−24.8247	−0.992986
$626$	−49.5734	−1.98135
$627$	−58.9686	−2.35498
$628$	−49.9398	−1.99282
$629$	11.4757	0.457565
$630$	52.3680	2.08639
$631$	−35.7237	−1.42214	−0.711069	−	0.703123i	$-0.751789\pi$
$631$	−35.7237	−1.42214	−0.711069	+	0.703123i	$0.751789\pi$
$632$	13.0818	0.520366
$633$	13.3667	0.531278
$634$	66.9352	2.65833
$635$	−28.6782	−1.13806
$636$	−103.158	−4.09050
$637$	18.5350	0.734383
$638$	68.6453	2.71769
$639$	−54.3484	−2.14999
$640$	−60.1123	−2.37615
$641$	−0.258993	−0.0102296	−0.00511480	−	0.999987i	$-0.501628\pi$
$641$	−0.258993	−0.0102296	−0.00511480	+	0.999987i	$0.501628\pi$
$642$	6.65446	0.262631
$643$	2.32862	0.0918316	0.0459158	−	0.998945i	$-0.485379\pi$
$643$	2.32862	0.0918316	0.0459158	+	0.998945i	$0.485379\pi$
$644$	27.6056	1.08781
$645$	101.354	3.99080
$646$	−36.2781	−1.42734
$647$	−34.2023	−1.34463	−0.672315	−	0.740265i	$-0.734700\pi$
$647$	−34.2023	−1.34463	−0.672315	+	0.740265i	$0.734700\pi$
$648$	−35.6751	−1.40145
$649$	11.1811	0.438896
$650$	40.6958	1.59622
$651$	−32.1236	−1.25902
$652$	0.159451	0.00624460
$653$	−27.6661	−1.08266	−0.541328	−	0.840811i	$-0.682078\pi$
$653$	−27.6661	−1.08266	−0.541328	+	0.840811i	$0.682078\pi$
$654$	−130.827	−5.11573
$655$	12.8951	0.503855
$656$	−3.41778	−0.133442
$657$	42.6070	1.66226
$658$	−7.15360	−0.278876
$659$	32.6543	1.27203	0.636016	−	0.771676i	$-0.280581\pi$
$659$	32.6543	1.27203	0.636016	+	0.771676i	$0.280581\pi$
$660$	−213.750	−8.32019
$661$	12.4814	0.485470	0.242735	−	0.970093i	$-0.421956\pi$
$661$	12.4814	0.485470	0.242735	+	0.970093i	$0.421956\pi$
$662$	−66.4030	−2.58083
$663$	40.2744	1.56413
$664$	−91.0001	−3.53149
$665$	13.0904	0.507623
$666$	39.3790	1.52591
$667$	−29.0467	−1.12469
$668$	15.9984	0.618998
$669$	25.7240	0.994549
$670$	−46.5995	−1.80029
$671$	33.6942	1.30075
$672$	−2.31693	−0.0893776
$673$	12.4537	0.480054	0.240027	−	0.970766i	$-0.422844\pi$
$673$	12.4537	0.480054	0.240027	+	0.970766i	$0.422844\pi$
$674$	−20.3562	−0.784092
$675$	41.3489	1.59152
$676$	−9.07703	−0.349117
$677$	−3.74736	−0.144023	−0.0720114	−	0.997404i	$-0.522942\pi$
$677$	−3.74736	−0.144023	−0.0720114	+	0.997404i	$0.522942\pi$
$678$	103.902	3.99031
$679$	−18.2815	−0.701578
$680$	−66.8362	−2.56305
$681$	−13.9030	−0.532765
$682$	128.708	4.92848
$683$	−19.5659	−0.748668	−0.374334	−	0.927294i	$-0.622129\pi$
$683$	−19.5659	−0.748668	−0.374334	+	0.927294i	$0.622129\pi$
$684$	−83.4520	−3.19087
$685$	−52.0587	−1.98906
$686$	36.2211	1.38293
$687$	−65.3606	−2.49366
$688$	−47.6111	−1.81515
$689$	28.1007	1.07055
$690$	134.923	5.13643
$691$	−26.5530	−1.01012	−0.505061	−	0.863083i	$-0.668530\pi$
$691$	−26.5530	−1.01012	−0.505061	+	0.863083i	$0.668530\pi$
$692$	88.3768	3.35958
$693$	37.5932	1.42805
$694$	−69.6888	−2.64535
$695$	7.86087	0.298180
$696$	75.0347	2.84418
$697$	3.21324	0.121710
$698$	−6.18641	−0.234159
$699$	−27.5927	−1.04365
$700$	23.8074	0.899833
$701$	36.1463	1.36523	0.682614	−	0.730779i	$-0.260843\pi$
$701$	36.1463	1.36523	0.682614	+	0.730779i	$0.260843\pi$
$702$	66.3811	2.50539
$703$	9.84352	0.371255
$704$	−40.0924	−1.51104
$705$	−23.4380	−0.882724
$706$	−60.9954	−2.29559
$707$	−7.64445	−0.287499
$708$	24.0466	0.903725
$709$	39.2523	1.47415	0.737076	−	0.675810i	$-0.236206\pi$
$709$	39.2523	1.47415	0.737076	+	0.675810i	$0.236206\pi$
$710$	−73.4600	−2.75691
$711$	−14.8314	−0.556223
$712$	−38.1450	−1.42955
$713$	−54.4618	−2.03961
$714$	35.1468	1.31534
$715$	58.2261	2.17753
$716$	−107.169	−4.00511
$717$	−22.0499	−0.823469
$718$	27.8297	1.03859
$719$	32.1891	1.20045	0.600225	−	0.799831i	$-0.295078\pi$
$719$	32.1891	1.20045	0.600225	+	0.799831i	$0.295078\pi$
$720$	−80.5989	−3.00374
$721$	−12.9168	−0.481045
$722$	15.6817	0.583613
$723$	0.154706	0.00575359
$724$	−60.5556	−2.25053
$725$	−25.0502	−0.930341
$726$	−148.647	−5.51680
$727$	16.4174	0.608887	0.304443	−	0.952530i	$-0.401530\pi$
$727$	16.4174	0.608887	0.304443	+	0.952530i	$0.401530\pi$
$728$	19.4256	0.719960
$729$	−32.5276	−1.20473
$730$	57.5898	2.13149
$731$	44.7617	1.65557
$732$	72.4642	2.67835
$733$	5.52578	0.204100	0.102050	−	0.994779i	$-0.467460\pi$
$733$	5.52578	0.204100	0.102050	+	0.994779i	$0.467460\pi$
$734$	13.9626	0.515367
$735$	52.9960	1.95479
$736$	−3.92808	−0.144791
$737$	−33.4522	−1.23223
$738$	11.0263	0.405883
$739$	−39.2232	−1.44285	−0.721425	−	0.692492i	$-0.756513\pi$
$739$	−39.2232	−1.44285	−0.721425	+	0.692492i	$0.756513\pi$
$740$	35.6808	1.31165
$741$	34.5463	1.26909
$742$	24.5230	0.900266
$743$	−8.22646	−0.301799	−0.150900	−	0.988549i	$-0.548217\pi$
$743$	−8.22646	−0.301799	−0.150900	+	0.988549i	$0.548217\pi$
$744$	140.688	5.15787
$745$	45.1641	1.65469
$746$	38.2833	1.40165
$747$	103.171	3.77483
$748$	−94.4000	−3.45161
$749$	−1.06044	−0.0387477
$750$	0.804866	0.0293896
$751$	45.5294	1.66139	0.830696	−	0.556727i	$-0.187943\pi$
$751$	45.5294	1.66139	0.830696	+	0.556727i	$0.187943\pi$
$752$	11.0100	0.401494
$753$	77.5513	2.82613
$754$	−40.2153	−1.46456
$755$	−10.5013	−0.382183
$756$	38.8335	1.41236
$757$	−31.8693	−1.15831	−0.579154	−	0.815218i	$-0.696617\pi$
$757$	−31.8693	−1.15831	−0.579154	+	0.815218i	$0.696617\pi$
$758$	11.8647	0.430944
$759$	96.8566	3.51567
$760$	−57.3303	−2.07959
$761$	−0.0354660	−0.00128564	−0.000642821	−	1.00000i	$-0.500205\pi$
$761$	−0.0354660	−0.00128564	−0.000642821		1.00000i	$0.500205\pi$
$762$	−66.0477	−2.39266
$763$	20.8483	0.754759
$764$	19.0510	0.689240
$765$	75.7753	2.73966
$766$	88.1681	3.18564
$767$	−6.55036	−0.236520
$768$	−96.0420	−3.46562
$769$	−3.95631	−0.142668	−0.0713341	−	0.997452i	$-0.522726\pi$
$769$	−3.95631	−0.142668	−0.0713341	+	0.997452i	$0.522726\pi$
$770$	50.8129	1.83117
$771$	−39.2307	−1.41286
$772$	46.2136	1.66326
$773$	−11.7835	−0.423822	−0.211911	−	0.977289i	$-0.567969\pi$
$773$	−11.7835	−0.423822	−0.211911	+	0.977289i	$0.567969\pi$
$774$	153.601	5.52106
$775$	−46.9684	−1.68716
$776$	80.0651	2.87417
$777$	−9.53656	−0.342122
$778$	59.3967	2.12947
$779$	2.75623	0.0987520
$780$	125.224	4.48373
$781$	−52.7344	−1.88699
$782$	59.5872	2.13083
$783$	−40.8608	−1.46025
$784$	−24.8949	−0.889105
$785$	38.8965	1.38827
$786$	29.6983	1.05930
$787$	40.7813	1.45369	0.726847	−	0.686799i	$-0.240985\pi$
$787$	40.7813	1.45369	0.726847	+	0.686799i	$0.240985\pi$
$788$	55.8395	1.98920
$789$	16.5456	0.589038
$790$	−20.0469	−0.713238
$791$	−16.5576	−0.588719
$792$	−164.642	−5.85031
$793$	−19.7395	−0.700969
$794$	4.98406	0.176878
$795$	80.3467	2.84960
$796$	−45.4092	−1.60949
$797$	−48.1379	−1.70513	−0.852565	−	0.522621i	$-0.824954\pi$
$797$	−48.1379	−1.70513	−0.852565	+	0.522621i	$0.824954\pi$
$798$	30.1479	1.06723
$799$	−10.3511	−0.366195
$800$	−3.38762	−0.119770
$801$	43.2468	1.52805
$802$	−24.1643	−0.853270
$803$	41.3418	1.45892
$804$	−71.9437	−2.53726
$805$	−21.5011	−0.757814
$806$	−75.4026	−2.65594
$807$	−68.6904	−2.41802
$808$	33.4795	1.17780
$809$	26.7217	0.939483	0.469742	−	0.882804i	$-0.344347\pi$
$809$	26.7217	0.939483	0.469742	+	0.882804i	$0.344347\pi$
$810$	54.6696	1.92089
$811$	−51.7456	−1.81703	−0.908516	−	0.417849i	$-0.862784\pi$
$811$	−51.7456	−1.81703	−0.908516	+	0.417849i	$0.862784\pi$
$812$	−23.5263	−0.825611
$813$	−40.3812	−1.41623
$814$	38.2096	1.33925
$815$	−0.124191	−0.00435024
$816$	−54.0939	−1.89367
$817$	38.3954	1.34328
$818$	21.2324	0.742372
$819$	−22.0237	−0.769570
$820$	9.99078	0.348893
$821$	−36.8719	−1.28684	−0.643418	−	0.765515i	$-0.722484\pi$
$821$	−36.8719	−1.28684	−0.643418	+	0.765515i	$0.722484\pi$
$822$	−119.894	−4.18180
$823$	8.75831	0.305296	0.152648	−	0.988281i	$-0.451220\pi$
$823$	8.75831	0.305296	0.152648	+	0.988281i	$0.451220\pi$
$824$	56.5700	1.97071
$825$	83.5302	2.90815
$826$	−5.71638	−0.198898
$827$	30.1612	1.04881	0.524404	−	0.851469i	$-0.324288\pi$
$827$	30.1612	1.04881	0.524404	+	0.851469i	$0.324288\pi$
$828$	137.071	4.76354
$829$	−31.7605	−1.10309	−0.551544	−	0.834146i	$-0.685961\pi$
$829$	−31.7605	−1.10309	−0.551544	+	0.834146i	$0.685961\pi$
$830$	139.451	4.84042
$831$	−17.2460	−0.598257
$832$	23.4878	0.814294
$833$	23.4051	0.810937
$834$	18.1041	0.626893
$835$	−12.4607	−0.431219
$836$	−80.9738	−2.80054
$837$	−76.6128	−2.64813
$838$	−22.3459	−0.771926
$839$	24.9896	0.862736	0.431368	−	0.902176i	$-0.358031\pi$
$839$	24.9896	0.862736	0.431368	+	0.902176i	$0.358031\pi$
$840$	55.5425	1.91640
$841$	−4.24552	−0.146397
$842$	−67.5022	−2.32628
$843$	−1.90409	−0.0655803
$844$	18.3547	0.631795
$845$	7.06980	0.243209
$846$	−35.5200	−1.22120
$847$	23.6881	0.813932
$848$	−37.7430	−1.29610
$849$	97.7682	3.35540
$850$	51.3886	1.76262
$851$	−16.1681	−0.554236
$852$	−113.413	−3.88547
$853$	29.8485	1.02199	0.510996	−	0.859583i	$-0.329277\pi$
$853$	29.8485	1.02199	0.510996	+	0.859583i	$0.329277\pi$
$854$	−17.2263	−0.589471
$855$	64.9980	2.22288
$856$	4.64429	0.158739
$857$	−49.4141	−1.68795	−0.843977	−	0.536380i	$-0.819791\pi$
$857$	−49.4141	−1.68795	−0.843977	+	0.536380i	$0.819791\pi$
$858$	134.098	4.57804
$859$	18.5564	0.633136	0.316568	−	0.948570i	$-0.397469\pi$
$859$	18.5564	0.633136	0.316568	+	0.948570i	$0.397469\pi$
$860$	139.176	4.74585
$861$	−2.67028	−0.0910027
$862$	−80.4183	−2.73906
$863$	34.1512	1.16252	0.581261	−	0.813717i	$-0.302560\pi$
$863$	34.1512	1.16252	0.581261	+	0.813717i	$0.302560\pi$
$864$	−5.52573	−0.187989
$865$	−68.8338	−2.34042
$866$	31.0542	1.05526
$867$	0.504572	0.0171362
$868$	−44.1111	−1.49723
$869$	−14.3910	−0.488181
$870$	−114.985	−3.89837
$871$	19.5977	0.664043
$872$	−91.3068	−3.09204
$873$	−90.7735	−3.07222
$874$	51.1123	1.72890
$875$	−0.128262	−0.00433605
$876$	88.9114	3.00404
$877$	54.3101	1.83392	0.916960	−	0.398979i	$-0.130635\pi$
$877$	54.3101	1.83392	0.916960	+	0.398979i	$0.130635\pi$
$878$	29.9185	1.00970
$879$	−42.4935	−1.43327
$880$	−78.2054	−2.63630
$881$	−33.2565	−1.12044	−0.560220	−	0.828344i	$-0.689284\pi$
$881$	−33.2565	−1.12044	−0.560220	+	0.828344i	$0.689284\pi$
$882$	80.3149	2.70434
$883$	39.2760	1.32174	0.660872	−	0.750499i	$-0.270187\pi$
$883$	39.2760	1.32174	0.660872	+	0.750499i	$0.270187\pi$
$884$	55.3036	1.86006
$885$	−18.7291	−0.629571
$886$	−74.4776	−2.50212
$887$	26.9900	0.906237	0.453118	−	0.891450i	$-0.350311\pi$
$887$	26.9900	0.906237	0.453118	+	0.891450i	$0.350311\pi$
$888$	41.7661	1.40158
$889$	10.5252	0.353005
$890$	58.4546	1.95940
$891$	39.2454	1.31477
$892$	35.3234	1.18272
$893$	−8.87889	−0.297121
$894$	104.016	3.47881
$895$	83.4708	2.79012
$896$	22.0619	0.737037
$897$	−56.7428	−1.89459
$898$	−11.0637	−0.369199
$899$	46.4139	1.54799
$900$	118.211	3.94038
$901$	35.4842	1.18215
$902$	10.6988	0.356232
$903$	−37.1980	−1.23787
$904$	72.5151	2.41182
$905$	47.1648	1.56781
$906$	−24.1853	−0.803502
$907$	50.1897	1.66652	0.833261	−	0.552880i	$-0.186471\pi$
$907$	50.1897	1.66652	0.833261	+	0.552880i	$0.186471\pi$
$908$	−19.0912	−0.633564
$909$	−37.9572	−1.25896
$910$	−29.7683	−0.986811
$911$	−29.7271	−0.984903	−0.492451	−	0.870340i	$-0.663899\pi$
$911$	−29.7271	−0.984903	−0.492451	+	0.870340i	$0.663899\pi$
$912$	−46.4003	−1.53647
$913$	100.107	3.31307
$914$	11.9442	0.395080
$915$	−56.4400	−1.86585
$916$	−89.7511	−2.96546
$917$	−4.73267	−0.156287
$918$	83.8228	2.76657
$919$	−16.5694	−0.546575	−0.273287	−	0.961932i	$-0.588111\pi$
$919$	−16.5694	−0.546575	−0.273287	+	0.961932i	$0.588111\pi$
$920$	94.1657	3.10455
$921$	−16.3931	−0.540172
$922$	−32.3513	−1.06543
$923$	30.8941	1.01689
$924$	78.4487	2.58077
$925$	−13.9435	−0.458461
$926$	−60.5888	−1.99107
$927$	−64.1360	−2.10650
$928$	3.34763	0.109891
$929$	22.4711	0.737254	0.368627	−	0.929577i	$-0.379828\pi$
$929$	22.4711	0.737254	0.368627	+	0.929577i	$0.379828\pi$
$930$	−215.594	−7.06962
$931$	20.0762	0.657972
$932$	−37.8894	−1.24111
$933$	−71.4022	−2.33760
$934$	−60.3877	−1.97595
$935$	73.5251	2.40453
$936$	96.4545	3.15271
$937$	−10.5879	−0.345891	−0.172945	−	0.984931i	$-0.555328\pi$
$937$	−10.5879	−0.345891	−0.172945	+	0.984931i	$0.555328\pi$
$938$	17.1026	0.558418
$939$	59.6106	1.94532
$940$	−32.1842	−1.04973
$941$	−30.4446	−0.992467	−0.496234	−	0.868189i	$-0.665284\pi$
$941$	−30.4446	−0.992467	−0.496234	+	0.868189i	$0.665284\pi$
$942$	89.5811	2.91871
$943$	−4.52713	−0.147424
$944$	8.79800	0.286351
$945$	−30.2461	−0.983907
$946$	149.039	4.84568
$947$	13.8778	0.450967	0.225484	−	0.974247i	$-0.427604\pi$
$947$	13.8778	0.450967	0.225484	+	0.974247i	$0.427604\pi$
$948$	−30.9499	−1.00521
$949$	−24.2198	−0.786207
$950$	44.0798	1.43014
$951$	−80.4877	−2.60999
$952$	24.5297	0.795012
$953$	25.2870	0.819127	0.409564	−	0.912282i	$-0.365681\pi$
$953$	25.2870	0.819127	0.409564	+	0.912282i	$0.365681\pi$
$954$	121.765	3.94227
$955$	−14.8382	−0.480152
$956$	−30.2782	−0.979268
$957$	−82.5441	−2.66827
$958$	67.5428	2.18221
$959$	19.1061	0.616970
$960$	67.1574	2.16750
$961$	56.0248	1.80725
$962$	−22.3848	−0.721715
$963$	−5.26545	−0.169677
$964$	0.212438	0.00684215
$965$	−35.9943	−1.15870
$966$	−49.5184	−1.59323
$967$	42.7879	1.37597	0.687983	−	0.725727i	$-0.258496\pi$
$967$	42.7879	1.37597	0.687983	+	0.725727i	$0.258496\pi$
$968$	−103.744	−3.33445
$969$	43.6234	1.40139
$970$	−122.694	−3.93947
$971$	6.67962	0.214359	0.107180	−	0.994240i	$-0.465818\pi$
$971$	6.67962	0.214359	0.107180	+	0.994240i	$0.465818\pi$
$972$	−15.8028	−0.506875
$973$	−2.88503	−0.0924899
$974$	41.2524	1.32181
$975$	−48.9355	−1.56719
$976$	26.5127	0.848652
$977$	−27.5831	−0.882463	−0.441231	−	0.897393i	$-0.645458\pi$
$977$	−27.5831	−0.882463	−0.441231	+	0.897393i	$0.645458\pi$
$978$	−0.286021	−0.00914594
$979$	41.9625	1.34113
$980$	72.7724	2.32463
$981$	103.519	3.30510
$982$	−50.3700	−1.60737
$983$	1.05501	0.0336495	0.0168248	−	0.999858i	$-0.494644\pi$
$983$	1.05501	0.0336495	0.0168248	+	0.999858i	$0.494644\pi$
$984$	11.6947	0.372813
$985$	−43.4916	−1.38576
$986$	−50.7819	−1.61723
$987$	8.60201	0.273805
$988$	47.4379	1.50920
$989$	−63.0649	−2.00535
$990$	252.303	8.01871
$991$	40.3621	1.28214	0.641072	−	0.767481i	$-0.278490\pi$
$991$	40.3621	1.28214	0.641072	+	0.767481i	$0.278490\pi$
$992$	6.27671	0.199286
$993$	79.8478	2.53389
$994$	26.9607	0.855142
$995$	35.3678	1.12123
$996$	215.295	6.82189
$997$	−19.8142	−0.627521	−0.313761	−	0.949502i	$-0.601589\pi$
$997$	−19.8142	−0.627521	−0.313761	+	0.949502i	$0.601589\pi$
$998$	−55.3665	−1.75260
$999$	−22.7441	−0.719591

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4001.2.a.a.1.10	✓	149

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4001.2.a.a.1.10	✓	149	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:	\( N \)	\(=\)	\( 4001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4001.a (trivial)

\(f(q)\)	\(=\)	\(q-2.46316 q^{2} +2.96188 q^{3} +4.06716 q^{4} -3.16778 q^{5} -7.29560 q^{6} +1.16261 q^{7} -5.09175 q^{8} +5.77276 q^{9} +O(q^{10})\)	\(q-2.46316 q^{2} +2.96188 q^{3} +4.06716 q^{4} -3.16778 q^{5} -7.29560 q^{6} +1.16261 q^{7} -5.09175 q^{8} +5.77276 q^{9} +7.80275 q^{10} +5.60133 q^{11} +12.0465 q^{12} -3.28150 q^{13} -2.86370 q^{14} -9.38259 q^{15} +4.40748 q^{16} -4.14371 q^{17} -14.2192 q^{18} -3.55436 q^{19} -12.8839 q^{20} +3.44352 q^{21} -13.7970 q^{22} +5.83809 q^{23} -15.0812 q^{24} +5.03483 q^{25} +8.08285 q^{26} +8.21258 q^{27} +4.72853 q^{28} -4.97539 q^{29} +23.1108 q^{30} -9.32871 q^{31} -0.672838 q^{32} +16.5905 q^{33} +10.2066 q^{34} -3.68290 q^{35} +23.4787 q^{36} -2.76942 q^{37} +8.75497 q^{38} -9.71941 q^{39} +16.1296 q^{40} -0.775448 q^{41} -8.48195 q^{42} -10.8023 q^{43} +22.7815 q^{44} -18.2868 q^{45} -14.3801 q^{46} +2.49802 q^{47} +13.0545 q^{48} -5.64833 q^{49} -12.4016 q^{50} -12.2732 q^{51} -13.3464 q^{52} -8.56338 q^{53} -20.2289 q^{54} -17.7438 q^{55} -5.91974 q^{56} -10.5276 q^{57} +12.2552 q^{58} +1.99615 q^{59} -38.1605 q^{60} +6.01539 q^{61} +22.9781 q^{62} +6.71148 q^{63} -7.15766 q^{64} +10.3951 q^{65} -40.8650 q^{66} -5.97218 q^{67} -16.8532 q^{68} +17.2917 q^{69} +9.07158 q^{70} -9.41463 q^{71} -29.3935 q^{72} +7.38071 q^{73} +6.82153 q^{74} +14.9126 q^{75} -14.4562 q^{76} +6.51217 q^{77} +23.9405 q^{78} -2.56921 q^{79} -13.9619 q^{80} +7.00645 q^{81} +1.91005 q^{82} +17.8721 q^{83} +14.0054 q^{84} +13.1264 q^{85} +26.6078 q^{86} -14.7365 q^{87} -28.5206 q^{88} +7.49153 q^{89} +45.0434 q^{90} -3.81511 q^{91} +23.7444 q^{92} -27.6306 q^{93} -6.15304 q^{94} +11.2594 q^{95} -1.99287 q^{96} -15.7245 q^{97} +13.9128 q^{98} +32.3351 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 149 q - 6 q^{2} - 28 q^{3} + 116 q^{4} - 19 q^{5} - 31 q^{6} - 47 q^{7} - 15 q^{8} + 115 q^{9}+O(q^{10}) \) 149 * q - 6 * q^2 - 28 * q^3 + 116 * q^4 - 19 * q^5 - 31 * q^6 - 47 * q^7 - 15 * q^8 + 115 * q^9	\( 149 q - 6 q^{2} - 28 q^{3} + 116 q^{4} - 19 q^{5} - 31 q^{6} - 47 q^{7} - 15 q^{8} + 115 q^{9} - 48 q^{10} - 31 q^{11} - 61 q^{12} - 54 q^{13} - 44 q^{14} - 65 q^{15} + 58 q^{16} - 26 q^{17} - 23 q^{18} - 86 q^{19} - 52 q^{20} - 30 q^{21} - 56 q^{22} - 63 q^{23} - 90 q^{24} + 92 q^{25} - 38 q^{26} - 103 q^{27} - 77 q^{28} - 51 q^{29} - 22 q^{30} - 256 q^{31} - 21 q^{32} - 36 q^{33} - 124 q^{34} - 50 q^{35} + 45 q^{36} - 42 q^{37} - 14 q^{38} - 119 q^{39} - 131 q^{40} - 55 q^{41} - 5 q^{42} - 55 q^{43} - 54 q^{44} - 68 q^{45} - 59 q^{46} - 82 q^{47} - 89 q^{48} + 30 q^{49} + 13 q^{50} - 83 q^{51} - 126 q^{52} - 23 q^{53} - 83 q^{54} - 244 q^{55} - 94 q^{56} - 14 q^{57} - 60 q^{58} - 93 q^{59} - 70 q^{60} - 139 q^{61} - 10 q^{62} - 120 q^{63} - 45 q^{64} - 37 q^{65} - 28 q^{66} - 110 q^{67} - 27 q^{68} - 79 q^{69} - 78 q^{70} - 123 q^{71} - 74 q^{73} - 25 q^{74} - 146 q^{75} - 192 q^{76} - q^{77} + 26 q^{78} - 273 q^{79} - 51 q^{80} + 41 q^{81} - 120 q^{82} - 27 q^{83} - 14 q^{84} - 71 q^{85} + 3 q^{86} - 98 q^{87} - 143 q^{88} - 70 q^{89} - 24 q^{90} - 256 q^{91} - 47 q^{92} + 16 q^{93} - 151 q^{94} - 83 q^{95} - 137 q^{96} - 108 q^{97} + 17 q^{98} - 131 q^{99}+O(q^{100}) \) 149 * q - 6 * q^2 - 28 * q^3 + 116 * q^4 - 19 * q^5 - 31 * q^6 - 47 * q^7 - 15 * q^8 + 115 * q^9 - 48 * q^10 - 31 * q^11 - 61 * q^12 - 54 * q^13 - 44 * q^14 - 65 * q^15 + 58 * q^16 - 26 * q^17 - 23 * q^18 - 86 * q^19 - 52 * q^20 - 30 * q^21 - 56 * q^22 - 63 * q^23 - 90 * q^24 + 92 * q^25 - 38 * q^26 - 103 * q^27 - 77 * q^28 - 51 * q^29 - 22 * q^30 - 256 * q^31 - 21 * q^32 - 36 * q^33 - 124 * q^34 - 50 * q^35 + 45 * q^36 - 42 * q^37 - 14 * q^38 - 119 * q^39 - 131 * q^40 - 55 * q^41 - 5 * q^42 - 55 * q^43 - 54 * q^44 - 68 * q^45 - 59 * q^46 - 82 * q^47 - 89 * q^48 + 30 * q^49 + 13 * q^50 - 83 * q^51 - 126 * q^52 - 23 * q^53 - 83 * q^54 - 244 * q^55 - 94 * q^56 - 14 * q^57 - 60 * q^58 - 93 * q^59 - 70 * q^60 - 139 * q^61 - 10 * q^62 - 120 * q^63 - 45 * q^64 - 37 * q^65 - 28 * q^66 - 110 * q^67 - 27 * q^68 - 79 * q^69 - 78 * q^70 - 123 * q^71 - 74 * q^73 - 25 * q^74 - 146 * q^75 - 192 * q^76 - q^77 + 26 * q^78 - 273 * q^79 - 51 * q^80 + 41 * q^81 - 120 * q^82 - 27 * q^83 - 14 * q^84 - 71 * q^85 + 3 * q^86 - 98 * q^87 - 143 * q^88 - 70 * q^89 - 24 * q^90 - 256 * q^91 - 47 * q^92 + 16 * q^93 - 151 * q^94 - 83 * q^95 - 137 * q^96 - 108 * q^97 + 17 * q^98 - 131 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more