LMFDB - Embedded newform 8047.2.a.b.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8047,2,Mod(1,8047)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8047, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8047.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8047 = 13 \cdot 619 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8047.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.2556185065$
Analytic rank:	$1$
Dimension:	$142$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	8047.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56842 q^{2} +0.368799 q^{3} +4.59680 q^{4} -0.479572 q^{5} -0.947232 q^{6} +4.01594 q^{7} -6.66967 q^{8} -2.86399 q^{9} +O(q^{10})$	\(q-2.56842 q^{2} +0.368799 q^{3} +4.59680 q^{4} -0.479572 q^{5} -0.947232 q^{6} +4.01594 q^{7} -6.66967 q^{8} -2.86399 q^{9} +1.23174 q^{10} +2.41061 q^{11} +1.69529 q^{12} +1.00000 q^{13} -10.3146 q^{14} -0.176866 q^{15} +7.93693 q^{16} -0.131894 q^{17} +7.35593 q^{18} -3.90661 q^{19} -2.20449 q^{20} +1.48107 q^{21} -6.19146 q^{22} +6.17118 q^{23} -2.45977 q^{24} -4.77001 q^{25} -2.56842 q^{26} -2.16263 q^{27} +18.4604 q^{28} -1.74972 q^{29} +0.454266 q^{30} -5.54174 q^{31} -7.04607 q^{32} +0.889030 q^{33} +0.338759 q^{34} -1.92593 q^{35} -13.1652 q^{36} +5.17598 q^{37} +10.0338 q^{38} +0.368799 q^{39} +3.19858 q^{40} -8.47513 q^{41} -3.80402 q^{42} +6.79359 q^{43} +11.0811 q^{44} +1.37349 q^{45} -15.8502 q^{46} +1.05395 q^{47} +2.92713 q^{48} +9.12774 q^{49} +12.2514 q^{50} -0.0486423 q^{51} +4.59680 q^{52} -7.74698 q^{53} +5.55456 q^{54} -1.15606 q^{55} -26.7850 q^{56} -1.44075 q^{57} +4.49403 q^{58} -7.30694 q^{59} -0.813015 q^{60} +0.0733858 q^{61} +14.2335 q^{62} -11.5016 q^{63} +2.22341 q^{64} -0.479572 q^{65} -2.28340 q^{66} +10.6328 q^{67} -0.606288 q^{68} +2.27593 q^{69} +4.94660 q^{70} -6.14530 q^{71} +19.1018 q^{72} -3.87192 q^{73} -13.2941 q^{74} -1.75918 q^{75} -17.9579 q^{76} +9.68084 q^{77} -0.947232 q^{78} -3.76051 q^{79} -3.80633 q^{80} +7.79438 q^{81} +21.7677 q^{82} -6.77790 q^{83} +6.80819 q^{84} +0.0632525 q^{85} -17.4488 q^{86} -0.645296 q^{87} -16.0779 q^{88} -7.48139 q^{89} -3.52770 q^{90} +4.01594 q^{91} +28.3677 q^{92} -2.04379 q^{93} -2.70698 q^{94} +1.87350 q^{95} -2.59858 q^{96} -4.17090 q^{97} -23.4439 q^{98} -6.90395 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9}+O(q^{10}) $ 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9	\( 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9} - 25 q^{10} - 25 q^{11} - 62 q^{12} + 142 q^{13} - 57 q^{14} - 14 q^{15} + 111 q^{16} - 141 q^{17} - 29 q^{18} - 3 q^{19} - 87 q^{20} - 19 q^{21} - 24 q^{22} - 69 q^{23} - 40 q^{24} + 87 q^{25} - 13 q^{26} - 95 q^{27} - 34 q^{28} - 147 q^{29} - 2 q^{30} - 21 q^{31} - 66 q^{32} - 62 q^{33} - 6 q^{34} - 59 q^{35} + 74 q^{36} - 37 q^{37} - 76 q^{38} - 26 q^{39} - 61 q^{40} - 97 q^{41} - 29 q^{42} - 33 q^{43} - 57 q^{44} - 86 q^{45} - q^{46} - 102 q^{47} - 141 q^{48} + 70 q^{49} - 28 q^{50} - 13 q^{51} + 129 q^{52} - 137 q^{53} - 29 q^{54} - 24 q^{55} - 130 q^{56} - 65 q^{57} - 15 q^{58} - 56 q^{59} + 11 q^{60} - 77 q^{61} - 150 q^{62} - 32 q^{63} + 73 q^{64} - 37 q^{65} - 32 q^{66} - 9 q^{67} - 226 q^{68} - 113 q^{69} + 6 q^{70} - 18 q^{71} - 82 q^{72} - 117 q^{73} - 70 q^{74} - 83 q^{75} + 40 q^{76} - 214 q^{77} - 15 q^{78} - 52 q^{79} - 161 q^{80} - 10 q^{81} - 36 q^{82} - 74 q^{83} + 53 q^{84} + 2 q^{85} + 17 q^{86} - 49 q^{87} - 29 q^{88} - 171 q^{89} - 57 q^{90} - 14 q^{91} - 187 q^{92} - 39 q^{93} + 13 q^{94} - 150 q^{95} - 47 q^{96} - 126 q^{97} - 85 q^{98}+O(q^{100}) \) 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9 - 25 * q^10 - 25 * q^11 - 62 * q^12 + 142 * q^13 - 57 * q^14 - 14 * q^15 + 111 * q^16 - 141 * q^17 - 29 * q^18 - 3 * q^19 - 87 * q^20 - 19 * q^21 - 24 * q^22 - 69 * q^23 - 40 * q^24 + 87 * q^25 - 13 * q^26 - 95 * q^27 - 34 * q^28 - 147 * q^29 - 2 * q^30 - 21 * q^31 - 66 * q^32 - 62 * q^33 - 6 * q^34 - 59 * q^35 + 74 * q^36 - 37 * q^37 - 76 * q^38 - 26 * q^39 - 61 * q^40 - 97 * q^41 - 29 * q^42 - 33 * q^43 - 57 * q^44 - 86 * q^45 - q^46 - 102 * q^47 - 141 * q^48 + 70 * q^49 - 28 * q^50 - 13 * q^51 + 129 * q^52 - 137 * q^53 - 29 * q^54 - 24 * q^55 - 130 * q^56 - 65 * q^57 - 15 * q^58 - 56 * q^59 + 11 * q^60 - 77 * q^61 - 150 * q^62 - 32 * q^63 + 73 * q^64 - 37 * q^65 - 32 * q^66 - 9 * q^67 - 226 * q^68 - 113 * q^69 + 6 * q^70 - 18 * q^71 - 82 * q^72 - 117 * q^73 - 70 * q^74 - 83 * q^75 + 40 * q^76 - 214 * q^77 - 15 * q^78 - 52 * q^79 - 161 * q^80 - 10 * q^81 - 36 * q^82 - 74 * q^83 + 53 * q^84 + 2 * q^85 + 17 * q^86 - 49 * q^87 - 29 * q^88 - 171 * q^89 - 57 * q^90 - 14 * q^91 - 187 * q^92 - 39 * q^93 + 13 * q^94 - 150 * q^95 - 47 * q^96 - 126 * q^97 - 85 * q^98

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 $

$2$	−2.56842	−1.81615	−0.908075	−	0.418809i	$-0.862448\pi$
$2$	−2.56842	−1.81615	−0.908075	+	0.418809i	$0.862448\pi$
$3$	0.368799	0.212926	0.106463	−	0.994317i	$-0.466047\pi$
$3$	0.368799	0.212926	0.106463	+	0.994317i	$0.466047\pi$
$4$	4.59680	2.29840
$5$	−0.479572	−0.214471	−0.107235	−	0.994234i	$-0.534200\pi$
$5$	−0.479572	−0.214471	−0.107235	+	0.994234i	$0.534200\pi$
$6$	−0.947232	−0.386706
$7$	4.01594	1.51788	0.758940	−	0.651160i	$-0.225717\pi$
$7$	4.01594	1.51788	0.758940	+	0.651160i	$0.225717\pi$
$8$	−6.66967	−2.35808
$9$	−2.86399	−0.954662
$10$	1.23174	0.389511
$11$	2.41061	0.726825	0.363413	−	0.931628i	$-0.381612\pi$
$11$	2.41061	0.726825	0.363413	+	0.931628i	$0.381612\pi$
$12$	1.69529	0.489389
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−10.3146	−2.75670
$15$	−0.176866	−0.0456665
$16$	7.93693	1.98423
$17$	−0.131894	−0.0319889	−0.0159945	−	0.999872i	$-0.505091\pi$
$17$	−0.131894	−0.0319889	−0.0159945	+	0.999872i	$0.505091\pi$
$18$	7.35593	1.73381
$19$	−3.90661	−0.896237	−0.448118	−	0.893974i	$-0.647906\pi$
$19$	−3.90661	−0.896237	−0.448118	+	0.893974i	$0.647906\pi$
$20$	−2.20449	−0.492940
$21$	1.48107	0.323197
$22$	−6.19146	−1.32002
$23$	6.17118	1.28678	0.643390	−	0.765538i	$-0.277527\pi$
$23$	6.17118	1.28678	0.643390	+	0.765538i	$0.277527\pi$
$24$	−2.45977	−0.502098
$25$	−4.77001	−0.954002
$26$	−2.56842	−0.503709
$27$	−2.16263	−0.416199
$28$	18.4604	3.48869
$29$	−1.74972	−0.324915	−0.162458	−	0.986716i	$-0.551942\pi$
$29$	−1.74972	−0.324915	−0.162458	+	0.986716i	$0.551942\pi$
$30$	0.454266	0.0829372
$31$	−5.54174	−0.995325	−0.497663	−	0.867371i	$-0.665808\pi$
$31$	−5.54174	−0.995325	−0.497663	+	0.867371i	$0.665808\pi$
$32$	−7.04607	−1.24558
$33$	0.889030	0.154760
$34$	0.338759	0.0580966
$35$	−1.92593	−0.325541
$36$	−13.1652	−2.19419
$37$	5.17598	0.850926	0.425463	−	0.904976i	$-0.360111\pi$
$37$	5.17598	0.850926	0.425463	+	0.904976i	$0.360111\pi$
$38$	10.0338	1.62770
$39$	0.368799	0.0590551
$40$	3.19858	0.505740
$41$	−8.47513	−1.32359	−0.661796	−	0.749684i	$-0.730206\pi$
$41$	−8.47513	−1.32359	−0.661796	+	0.749684i	$0.730206\pi$
$42$	−3.80402	−0.586973
$43$	6.79359	1.03601	0.518006	−	0.855377i	$-0.326674\pi$
$43$	6.79359	1.03601	0.518006	+	0.855377i	$0.326674\pi$
$44$	11.0811	1.67053
$45$	1.37349	0.204747
$46$	−15.8502	−2.33698
$47$	1.05395	0.153734	0.0768670	−	0.997041i	$-0.475508\pi$
$47$	1.05395	0.153734	0.0768670	+	0.997041i	$0.475508\pi$
$48$	2.92713	0.422495
$49$	9.12774	1.30396
$50$	12.2514	1.73261
$51$	−0.0486423	−0.00681128
$52$	4.59680	0.637461
$53$	−7.74698	−1.06413	−0.532065	−	0.846704i	$-0.678584\pi$
$53$	−7.74698	−1.06413	−0.532065	+	0.846704i	$0.678584\pi$
$54$	5.55456	0.755879
$55$	−1.15606	−0.155883
$56$	−26.7850	−3.57929
$57$	−1.44075	−0.190832
$58$	4.49403	0.590095
$59$	−7.30694	−0.951282	−0.475641	−	0.879639i	$-0.657784\pi$
$59$	−7.30694	−0.951282	−0.475641	+	0.879639i	$0.657784\pi$
$60$	−0.813015	−0.104960
$61$	0.0733858	0.00939608	0.00469804	−	0.999989i	$-0.498505\pi$
$61$	0.0733858	0.00939608	0.00469804	+	0.999989i	$0.498505\pi$
$62$	14.2335	1.80766
$63$	−11.5016	−1.44906
$64$	2.22341	0.277927
$65$	−0.479572	−0.0594835
$66$	−2.28340	−0.281068
$67$	10.6328	1.29901	0.649505	−	0.760358i	$-0.274976\pi$
$67$	10.6328	1.29901	0.649505	+	0.760358i	$0.274976\pi$
$68$	−0.606288	−0.0735232
$69$	2.27593	0.273989
$70$	4.94660	0.591232
$71$	−6.14530	−0.729313	−0.364657	−	0.931142i	$-0.618814\pi$
$71$	−6.14530	−0.729313	−0.364657	+	0.931142i	$0.618814\pi$
$72$	19.1018	2.25117
$73$	−3.87192	−0.453174	−0.226587	−	0.973991i	$-0.572757\pi$
$73$	−3.87192	−0.453174	−0.226587	+	0.973991i	$0.572757\pi$
$74$	−13.2941	−1.54541
$75$	−1.75918	−0.203132
$76$	−17.9579	−2.05991
$77$	9.68084	1.10323
$78$	−0.947232	−0.107253
$79$	−3.76051	−0.423090	−0.211545	−	0.977368i	$-0.567850\pi$
$79$	−3.76051	−0.423090	−0.211545	+	0.977368i	$0.567850\pi$
$80$	−3.80633	−0.425560
$81$	7.79438	0.866043
$82$	21.7677	2.40384
$83$	−6.77790	−0.743972	−0.371986	−	0.928238i	$-0.621323\pi$
$83$	−6.77790	−0.743972	−0.371986	+	0.928238i	$0.621323\pi$
$84$	6.80819	0.742834
$85$	0.0632525	0.00686069
$86$	−17.4488	−1.88155
$87$	−0.645296	−0.0691830
$88$	−16.0779	−1.71391
$89$	−7.48139	−0.793025	−0.396513	−	0.918029i	$-0.629780\pi$
$89$	−7.48139	−0.793025	−0.396513	+	0.918029i	$0.629780\pi$
$90$	−3.52770	−0.371852
$91$	4.01594	0.420984
$92$	28.3677	2.95753
$93$	−2.04379	−0.211931
$94$	−2.70698	−0.279204
$95$	1.87350	0.192217
$96$	−2.59858	−0.265217
$97$	−4.17090	−0.423491	−0.211745	−	0.977325i	$-0.567915\pi$
$97$	−4.17090	−0.423491	−0.211745	+	0.977325i	$0.567915\pi$
$98$	−23.4439	−2.36819
$99$	−6.90395	−0.693873
$100$	−21.9268	−2.19268
$101$	−12.2721	−1.22112	−0.610561	−	0.791969i	$-0.709056\pi$
$101$	−12.2721	−1.22112	−0.610561	+	0.791969i	$0.709056\pi$
$102$	0.124934	0.0123703
$103$	5.15437	0.507876	0.253938	−	0.967221i	$-0.418274\pi$
$103$	5.15437	0.507876	0.253938	+	0.967221i	$0.418274\pi$
$104$	−6.66967	−0.654015
$105$	−0.710281	−0.0693163
$106$	19.8975	1.93262
$107$	4.40399	0.425749	0.212875	−	0.977080i	$-0.431717\pi$
$107$	4.40399	0.425749	0.212875	+	0.977080i	$0.431717\pi$
$108$	−9.94118	−0.956591
$109$	17.6203	1.68772	0.843858	−	0.536566i	$-0.180279\pi$
$109$	17.6203	1.68772	0.843858	+	0.536566i	$0.180279\pi$
$110$	2.96925	0.283107
$111$	1.90890	0.181185
$112$	31.8742	3.01183
$113$	8.27697	0.778632	0.389316	−	0.921104i	$-0.372711\pi$
$113$	8.27697	0.778632	0.389316	+	0.921104i	$0.372711\pi$
$114$	3.70046	0.346580
$115$	−2.95952	−0.275977
$116$	−8.04312	−0.746785
$117$	−2.86399	−0.264776
$118$	18.7673	1.72767
$119$	−0.529677	−0.0485554
$120$	1.17963	0.107685
$121$	−5.18898	−0.471725
$122$	−0.188486	−0.0170647
$123$	−3.12562	−0.281828
$124$	−25.4742	−2.28765
$125$	4.68542	0.419077
$126$	29.5409	2.63172
$127$	−3.99235	−0.354264	−0.177132	−	0.984187i	$-0.556682\pi$
$127$	−3.99235	−0.354264	−0.177132	+	0.984187i	$0.556682\pi$
$128$	8.38147	0.740825
$129$	2.50547	0.220594
$130$	1.23174	0.108031
$131$	−13.3045	−1.16242	−0.581211	−	0.813753i	$-0.697421\pi$
$131$	−13.3045	−1.16242	−0.581211	+	0.813753i	$0.697421\pi$
$132$	4.08669	0.355700
$133$	−15.6887	−1.36038
$134$	−27.3096	−2.35919
$135$	1.03714	0.0892626
$136$	0.879687	0.0754325
$137$	−14.6116	−1.24836	−0.624179	−	0.781282i	$-0.714566\pi$
$137$	−14.6116	−1.24836	−0.624179	+	0.781282i	$0.714566\pi$
$138$	−5.84554	−0.497605
$139$	−7.35278	−0.623654	−0.311827	−	0.950139i	$-0.600941\pi$
$139$	−7.35278	−0.623654	−0.311827	+	0.950139i	$0.600941\pi$
$140$	−8.85310	−0.748223
$141$	0.388695	0.0327340
$142$	15.7837	1.32454
$143$	2.41061	0.201585
$144$	−22.7313	−1.89427
$145$	0.839118	0.0696849
$146$	9.94473	0.823032
$147$	3.36630	0.277648
$148$	23.7929	1.95577
$149$	−7.00050	−0.573503	−0.286751	−	0.958005i	$-0.592575\pi$
$149$	−7.00050	−0.573503	−0.286751	+	0.958005i	$0.592575\pi$
$150$	4.51831	0.368918
$151$	9.06306	0.737541	0.368771	−	0.929520i	$-0.379779\pi$
$151$	9.06306	0.737541	0.368771	+	0.929520i	$0.379779\pi$
$152$	26.0558	2.11340
$153$	0.377742	0.0305386
$154$	−24.8645	−2.00364
$155$	2.65766	0.213468
$156$	1.69529	0.135732
$157$	−10.6148	−0.847157	−0.423578	−	0.905859i	$-0.639226\pi$
$157$	−10.6148	−0.847157	−0.423578	+	0.905859i	$0.639226\pi$
$158$	9.65858	0.768395
$159$	−2.85708	−0.226581
$160$	3.37909	0.267141
$161$	24.7831	1.95318
$162$	−20.0193	−1.57286
$163$	5.56277	0.435710	0.217855	−	0.975981i	$-0.430094\pi$
$163$	5.56277	0.435710	0.217855	+	0.975981i	$0.430094\pi$
$164$	−38.9584	−3.04214
$165$	−0.426353	−0.0331916
$166$	17.4085	1.35116
$167$	−2.96439	−0.229392	−0.114696	−	0.993401i	$-0.536589\pi$
$167$	−2.96439	−0.229392	−0.114696	+	0.993401i	$0.536589\pi$
$168$	−9.87827	−0.762125
$169$	1.00000	0.0769231
$170$	−0.162459	−0.0124600
$171$	11.1885	0.855604
$172$	31.2287	2.38117
$173$	−10.0208	−0.761868	−0.380934	−	0.924602i	$-0.624397\pi$
$173$	−10.0208	−0.761868	−0.380934	+	0.924602i	$0.624397\pi$
$174$	1.65739	0.125647
$175$	−19.1561	−1.44806
$176$	19.1328	1.44219
$177$	−2.69479	−0.202553
$178$	19.2154	1.44025
$179$	−12.1357	−0.907066	−0.453533	−	0.891239i	$-0.649837\pi$
$179$	−12.1357	−0.907066	−0.453533	+	0.891239i	$0.649837\pi$
$180$	6.31364	0.470591
$181$	2.15195	0.159953	0.0799767	−	0.996797i	$-0.474515\pi$
$181$	2.15195	0.159953	0.0799767	+	0.996797i	$0.474515\pi$
$182$	−10.3146	−0.764570
$183$	0.0270646	0.00200067
$184$	−41.1597	−3.03434
$185$	−2.48225	−0.182499
$186$	5.24931	0.384898
$187$	−0.317944	−0.0232504
$188$	4.84478	0.353342
$189$	−8.68499	−0.631740
$190$	−4.81193	−0.349094
$191$	−18.0945	−1.30927	−0.654635	−	0.755945i	$-0.727178\pi$
$191$	−18.0945	−1.30927	−0.654635	+	0.755945i	$0.727178\pi$
$192$	0.819993	0.0591779
$193$	−11.6503	−0.838607	−0.419304	−	0.907846i	$-0.637726\pi$
$193$	−11.6503	−0.838607	−0.419304	+	0.907846i	$0.637726\pi$
$194$	10.7126	0.769122
$195$	−0.176866	−0.0126656
$196$	41.9583	2.99702
$197$	−7.06627	−0.503451	−0.251725	−	0.967799i	$-0.580998\pi$
$197$	−7.06627	−0.503451	−0.251725	+	0.967799i	$0.580998\pi$
$198$	17.7323	1.26018
$199$	−2.51466	−0.178259	−0.0891297	−	0.996020i	$-0.528409\pi$
$199$	−2.51466	−0.178259	−0.0891297	+	0.996020i	$0.528409\pi$
$200$	31.8144	2.24962
$201$	3.92138	0.276593
$202$	31.5200	2.21774
$203$	−7.02678	−0.493183
$204$	−0.223599	−0.0156550
$205$	4.06443	0.283872
$206$	−13.2386	−0.922378
$207$	−17.6742	−1.22844
$208$	7.93693	0.550327
$209$	−9.41729	−0.651408
$210$	1.82430	0.125889
$211$	11.0007	0.757316	0.378658	−	0.925537i	$-0.376386\pi$
$211$	11.0007	0.757316	0.378658	+	0.925537i	$0.376386\pi$
$212$	−35.6113	−2.44579
$213$	−2.26638	−0.155290
$214$	−11.3113	−0.773224
$215$	−3.25801	−0.222195
$216$	14.4240	0.981432
$217$	−22.2553	−1.51079
$218$	−45.2563	−3.06514
$219$	−1.42796	−0.0964927
$220$	−5.31416	−0.358281
$221$	−0.131894	−0.00887213
$222$	−4.90286	−0.329058
$223$	21.8612	1.46393	0.731967	−	0.681340i	$-0.238603\pi$
$223$	21.8612	1.46393	0.731967	+	0.681340i	$0.238603\pi$
$224$	−28.2966	−1.89064
$225$	13.6613	0.910750
$226$	−21.2588	−1.41411
$227$	−20.6778	−1.37243	−0.686217	−	0.727397i	$-0.740730\pi$
$227$	−20.6778	−1.37243	−0.686217	+	0.727397i	$0.740730\pi$
$228$	−6.62284	−0.438609
$229$	−11.1303	−0.735512	−0.367756	−	0.929922i	$-0.619874\pi$
$229$	−11.1303	−0.735512	−0.367756	+	0.929922i	$0.619874\pi$
$230$	7.60131	0.501215
$231$	3.57029	0.234908
$232$	11.6701	0.766178
$233$	9.68517	0.634497	0.317248	−	0.948342i	$-0.397241\pi$
$233$	9.68517	0.634497	0.317248	+	0.948342i	$0.397241\pi$
$234$	7.35593	0.480872
$235$	−0.505443	−0.0329715
$236$	−33.5885	−2.18642
$237$	−1.38687	−0.0900871
$238$	1.36043	0.0881838
$239$	−2.97535	−0.192460	−0.0962298	−	0.995359i	$-0.530678\pi$
$239$	−2.97535	−0.192460	−0.0962298	+	0.995359i	$0.530678\pi$
$240$	−1.40377	−0.0906130
$241$	15.2082	0.979648	0.489824	−	0.871821i	$-0.337061\pi$
$241$	15.2082	0.979648	0.489824	+	0.871821i	$0.337061\pi$
$242$	13.3275	0.856723
$243$	9.36246	0.600602
$244$	0.337339	0.0215959
$245$	−4.37740	−0.279662
$246$	8.02791	0.511841
$247$	−3.90661	−0.248571
$248$	36.9615	2.34706
$249$	−2.49968	−0.158411
$250$	−12.0341	−0.761106
$251$	−15.2988	−0.965649	−0.482825	−	0.875717i	$-0.660389\pi$
$251$	−15.2988	−0.965649	−0.482825	+	0.875717i	$0.660389\pi$
$252$	−52.8704	−3.33052
$253$	14.8763	0.935264
$254$	10.2540	0.643396
$255$	0.0233275	0.00146082
$256$	−25.9740	−1.62337
$257$	−1.13315	−0.0706843	−0.0353421	−	0.999375i	$-0.511252\pi$
$257$	−1.13315	−0.0706843	−0.0353421	+	0.999375i	$0.511252\pi$
$258$	−6.43510	−0.400632
$259$	20.7864	1.29160
$260$	−2.20449	−0.136717
$261$	5.01119	0.310185
$262$	34.1717	2.11113
$263$	18.6583	1.15052	0.575261	−	0.817970i	$-0.304901\pi$
$263$	18.6583	1.15052	0.575261	+	0.817970i	$0.304901\pi$
$264$	−5.92953	−0.364937
$265$	3.71523	0.228225
$266$	40.2952	2.47065
$267$	−2.75913	−0.168856
$268$	48.8770	2.98564
$269$	−13.0785	−0.797408	−0.398704	−	0.917080i	$-0.630540\pi$
$269$	−13.0785	−0.797408	−0.398704	+	0.917080i	$0.630540\pi$
$270$	−2.66381	−0.162114
$271$	−4.51980	−0.274558	−0.137279	−	0.990532i	$-0.543836\pi$
$271$	−4.51980	−0.274558	−0.137279	+	0.990532i	$0.543836\pi$
$272$	−1.04683	−0.0634735
$273$	1.48107	0.0896386
$274$	37.5289	2.26720
$275$	−11.4986	−0.693393
$276$	10.4620	0.629736
$277$	16.8578	1.01289	0.506445	−	0.862272i	$-0.330959\pi$
$277$	16.8578	1.01289	0.506445	+	0.862272i	$0.330959\pi$
$278$	18.8850	1.13265
$279$	15.8715	0.950200
$280$	12.8453	0.767654
$281$	13.6399	0.813686	0.406843	−	0.913498i	$-0.366630\pi$
$281$	13.6399	0.813686	0.406843	+	0.913498i	$0.366630\pi$
$282$	−0.998333	−0.0594498
$283$	−11.0997	−0.659811	−0.329906	−	0.944014i	$-0.607017\pi$
$283$	−11.0997	−0.659811	−0.329906	+	0.944014i	$0.607017\pi$
$284$	−28.2487	−1.67625
$285$	0.690944	0.0409280
$286$	−6.19146	−0.366109
$287$	−34.0356	−2.00906
$288$	20.1799	1.18911
$289$	−16.9826	−0.998977
$290$	−2.15521	−0.126558
$291$	−1.53822	−0.0901723
$292$	−17.7984	−1.04157
$293$	−9.09262	−0.531197	−0.265598	−	0.964084i	$-0.585570\pi$
$293$	−9.09262	−0.531197	−0.265598	+	0.964084i	$0.585570\pi$
$294$	−8.64608	−0.504250
$295$	3.50420	0.204022
$296$	−34.5221	−2.00656
$297$	−5.21326	−0.302504
$298$	17.9802	1.04157
$299$	6.17118	0.356889
$300$	−8.08657	−0.466878
$301$	27.2826	1.57254
$302$	−23.2778	−1.33948
$303$	−4.52595	−0.260009
$304$	−31.0065	−1.77834
$305$	−0.0351937	−0.00201519
$306$	−0.970201	−0.0554627
$307$	−21.7706	−1.24251	−0.621257	−	0.783607i	$-0.713378\pi$
$307$	−21.7706	−1.24251	−0.621257	+	0.783607i	$0.713378\pi$
$308$	44.5008	2.53567
$309$	1.90093	0.108140
$310$	−6.82599	−0.387690
$311$	−16.9656	−0.962030	−0.481015	−	0.876712i	$-0.659732\pi$
$311$	−16.9656	−0.962030	−0.481015	+	0.876712i	$0.659732\pi$
$312$	−2.45977	−0.139257
$313$	−6.47578	−0.366033	−0.183016	−	0.983110i	$-0.558586\pi$
$313$	−6.47578	−0.366033	−0.183016	+	0.983110i	$0.558586\pi$
$314$	27.2634	1.53856
$315$	5.51584	0.310782
$316$	−17.2863	−0.972430
$317$	1.61941	0.0909551	0.0454776	−	0.998965i	$-0.485519\pi$
$317$	1.61941	0.0909551	0.0454776	+	0.998965i	$0.485519\pi$
$318$	7.33818	0.411505
$319$	−4.21789	−0.236157
$320$	−1.06629	−0.0596072
$321$	1.62419	0.0906532
$322$	−63.6534	−3.54726
$323$	0.515257	0.0286696
$324$	35.8292	1.99051
$325$	−4.77001	−0.264593
$326$	−14.2876	−0.791315
$327$	6.49834	0.359359
$328$	56.5263	3.12114
$329$	4.23258	0.233350
$330$	1.09506	0.0602808
$331$	−20.9317	−1.15051	−0.575254	−	0.817975i	$-0.695097\pi$
$331$	−20.9317	−1.15051	−0.575254	+	0.817975i	$0.695097\pi$
$332$	−31.1566	−1.70994
$333$	−14.8239	−0.812347
$334$	7.61382	0.416610
$335$	−5.09921	−0.278600
$336$	11.7552	0.641298
$337$	19.9951	1.08920	0.544600	−	0.838696i	$-0.316681\pi$
$337$	19.9951	1.08920	0.544600	+	0.838696i	$0.316681\pi$
$338$	−2.56842	−0.139704
$339$	3.05254	0.165791
$340$	0.290759	0.0157686
$341$	−13.3589	−0.723428
$342$	−28.7367	−1.55390
$343$	8.54485	0.461379
$344$	−45.3110	−2.44300
$345$	−1.09147	−0.0587627
$346$	25.7377	1.38367
$347$	1.66682	0.0894797	0.0447399	−	0.998999i	$-0.485754\pi$
$347$	1.66682	0.0894797	0.0447399	+	0.998999i	$0.485754\pi$
$348$	−2.96630	−0.159010
$349$	13.5560	0.725638	0.362819	−	0.931860i	$-0.381814\pi$
$349$	13.5560	0.725638	0.362819	+	0.931860i	$0.381814\pi$
$350$	49.2008	2.62990
$351$	−2.16263	−0.115433
$352$	−16.9853	−0.905319
$353$	24.5343	1.30583	0.652915	−	0.757431i	$-0.273546\pi$
$353$	24.5343	1.30583	0.652915	+	0.757431i	$0.273546\pi$
$354$	6.92137	0.367866
$355$	2.94711	0.156417
$356$	−34.3904	−1.82269
$357$	−0.195344	−0.0103387
$358$	31.1697	1.64737
$359$	11.8041	0.622994	0.311497	−	0.950247i	$-0.399170\pi$
$359$	11.8041	0.622994	0.311497	+	0.950247i	$0.399170\pi$
$360$	−9.16070	−0.482811
$361$	−3.73843	−0.196760
$362$	−5.52713	−0.290499
$363$	−1.91369	−0.100443
$364$	18.4604	0.967590
$365$	1.85686	0.0971927
$366$	−0.0695133	−0.00363352
$367$	−22.7889	−1.18957	−0.594786	−	0.803884i	$-0.702763\pi$
$367$	−22.7889	−1.18957	−0.594786	+	0.803884i	$0.702763\pi$
$368$	48.9803	2.55327
$369$	24.2727	1.26358
$370$	6.37548	0.331445
$371$	−31.1114	−1.61522
$372$	−9.39487	−0.487101
$373$	12.4489	0.644580	0.322290	−	0.946641i	$-0.395547\pi$
$373$	12.4489	0.644580	0.322290	+	0.946641i	$0.395547\pi$
$374$	0.816614	0.0422261
$375$	1.72798	0.0892324
$376$	−7.02948	−0.362518
$377$	−1.74972	−0.0901153
$378$	22.3067	1.14733
$379$	35.2004	1.80812	0.904062	−	0.427401i	$-0.140571\pi$
$379$	35.2004	1.80812	0.904062	+	0.427401i	$0.140571\pi$
$380$	8.61208	0.441791
$381$	−1.47238	−0.0754321
$382$	46.4743	2.37783
$383$	25.5242	1.30423	0.652114	−	0.758121i	$-0.273882\pi$
$383$	25.5242	1.30423	0.652114	+	0.758121i	$0.273882\pi$
$384$	3.09108	0.157741
$385$	−4.64266	−0.236612
$386$	29.9229	1.52304
$387$	−19.4567	−0.989042
$388$	−19.1728	−0.973350
$389$	15.6986	0.795952	0.397976	−	0.917396i	$-0.369713\pi$
$389$	15.6986	0.795952	0.397976	+	0.917396i	$0.369713\pi$
$390$	0.454266	0.0230026
$391$	−0.813940	−0.0411627
$392$	−60.8790	−3.07485
$393$	−4.90670	−0.247510
$394$	18.1492	0.914341
$395$	1.80343	0.0907406
$396$	−31.7360	−1.59480
$397$	18.1259	0.909714	0.454857	−	0.890565i	$-0.349690\pi$
$397$	18.1259	0.909714	0.454857	+	0.890565i	$0.349690\pi$
$398$	6.45871	0.323746
$399$	−5.78597	−0.289661
$400$	−37.8593	−1.89296
$401$	16.0335	0.800676	0.400338	−	0.916367i	$-0.368893\pi$
$401$	16.0335	0.800676	0.400338	+	0.916367i	$0.368893\pi$
$402$	−10.0718	−0.502334
$403$	−5.54174	−0.276054
$404$	−56.4125	−2.80663
$405$	−3.73797	−0.185741
$406$	18.0477	0.895694
$407$	12.4773	0.618475
$408$	0.324428	0.0160616
$409$	7.30829	0.361372	0.180686	−	0.983541i	$-0.442168\pi$
$409$	7.30829	0.361372	0.180686	+	0.983541i	$0.442168\pi$
$410$	−10.4392	−0.515554
$411$	−5.38876	−0.265808
$412$	23.6936	1.16730
$413$	−29.3442	−1.44393
$414$	45.3948	2.23103
$415$	3.25049	0.159560
$416$	−7.04607	−0.345462
$417$	−2.71170	−0.132792
$418$	24.1876	1.18305
$419$	16.3226	0.797413	0.398707	−	0.917078i	$-0.369459\pi$
$419$	16.3226	0.797413	0.398707	+	0.917078i	$0.369459\pi$
$420$	−3.26501	−0.159316
$421$	10.0501	0.489812	0.244906	−	0.969547i	$-0.421243\pi$
$421$	10.0501	0.489812	0.244906	+	0.969547i	$0.421243\pi$
$422$	−28.2543	−1.37540
$423$	−3.01849	−0.146764
$424$	51.6698	2.50931
$425$	0.629134	0.0305175
$426$	5.82103	0.282030
$427$	0.294712	0.0142621
$428$	20.2442	0.978541
$429$	0.889030	0.0429227
$430$	8.36795	0.403538
$431$	19.3204	0.930630	0.465315	−	0.885145i	$-0.345941\pi$
$431$	19.3204	0.930630	0.465315	+	0.885145i	$0.345941\pi$
$432$	−17.1647	−0.825836
$433$	−7.51856	−0.361319	−0.180660	−	0.983546i	$-0.557823\pi$
$433$	−7.51856	−0.361319	−0.180660	+	0.983546i	$0.557823\pi$
$434$	57.1609	2.74381
$435$	0.309466	0.0148378
$436$	80.9968	3.87904
$437$	−24.1084	−1.15326
$438$	3.66761	0.175245
$439$	−14.9288	−0.712511	−0.356256	−	0.934389i	$-0.615947\pi$
$439$	−14.9288	−0.712511	−0.356256	+	0.934389i	$0.615947\pi$
$440$	7.71053	0.367585
$441$	−26.1417	−1.24484
$442$	0.338759	0.0161131
$443$	−40.9005	−1.94324	−0.971621	−	0.236542i	$-0.923986\pi$
$443$	−40.9005	−1.94324	−0.971621	+	0.236542i	$0.923986\pi$
$444$	8.77481	0.416434
$445$	3.58786	0.170081
$446$	−56.1488	−2.65872
$447$	−2.58178	−0.122114
$448$	8.92908	0.421859
$449$	8.13521	0.383924	0.191962	−	0.981402i	$-0.438515\pi$
$449$	8.13521	0.383924	0.191962	+	0.981402i	$0.438515\pi$
$450$	−35.0879	−1.65406
$451$	−20.4302	−0.962021
$452$	38.0475	1.78961
$453$	3.34245	0.157042
$454$	53.1093	2.49254
$455$	−1.92593	−0.0902889
$456$	9.60934	0.449999
$457$	8.32422	0.389391	0.194695	−	0.980864i	$-0.437628\pi$
$457$	8.32422	0.389391	0.194695	+	0.980864i	$0.437628\pi$
$458$	28.5874	1.33580
$459$	0.285238	0.0133138
$460$	−13.6043	−0.634305
$461$	−4.81487	−0.224251	−0.112125	−	0.993694i	$-0.535766\pi$
$461$	−4.81487	−0.224251	−0.112125	+	0.993694i	$0.535766\pi$
$462$	−9.17000	−0.426627
$463$	9.14969	0.425222	0.212611	−	0.977137i	$-0.431803\pi$
$463$	9.14969	0.425222	0.212611	+	0.977137i	$0.431803\pi$
$464$	−13.8874	−0.644708
$465$	0.980143	0.0454530
$466$	−24.8756	−1.15234
$467$	7.61442	0.352353	0.176177	−	0.984359i	$-0.443627\pi$
$467$	7.61442	0.352353	0.176177	+	0.984359i	$0.443627\pi$
$468$	−13.1652	−0.608560
$469$	42.7008	1.97174
$470$	1.29819	0.0598811
$471$	−3.91475	−0.180382
$472$	48.7348	2.24320
$473$	16.3767	0.753000
$474$	3.56208	0.163612
$475$	18.6346	0.855012
$476$	−2.43481	−0.111600
$477$	22.1872	1.01588
$478$	7.64196	0.349535
$479$	−32.2113	−1.47177	−0.735886	−	0.677105i	$-0.763234\pi$
$479$	−32.2113	−1.47177	−0.735886	+	0.677105i	$0.763234\pi$
$480$	1.24621	0.0568813
$481$	5.17598	0.236004
$482$	−39.0611	−1.77919
$483$	9.13997	0.415883
$484$	−23.8527	−1.08421
$485$	2.00025	0.0908265
$486$	−24.0468	−1.09078
$487$	24.9603	1.13106	0.565529	−	0.824728i	$-0.308672\pi$
$487$	24.9603	1.13106	0.565529	+	0.824728i	$0.308672\pi$
$488$	−0.489459	−0.0221567
$489$	2.05155	0.0927741
$490$	11.2430	0.507908
$491$	−14.0313	−0.633225	−0.316613	−	0.948555i	$-0.602546\pi$
$491$	−14.0313	−0.633225	−0.316613	+	0.948555i	$0.602546\pi$
$492$	−14.3678	−0.647752
$493$	0.230777	0.0103937
$494$	10.0338	0.451443
$495$	3.31094	0.148816
$496$	−43.9844	−1.97496
$497$	−24.6791	−1.10701
$498$	6.42025	0.287698
$499$	−26.5386	−1.18803	−0.594015	−	0.804454i	$-0.702458\pi$
$499$	−26.5386	−1.18803	−0.594015	+	0.804454i	$0.702458\pi$
$500$	21.5379	0.963205
$501$	−1.09327	−0.0488435
$502$	39.2937	1.75376
$503$	−38.4138	−1.71279	−0.856394	−	0.516323i	$-0.827300\pi$
$503$	−38.4138	−1.71279	−0.856394	+	0.516323i	$0.827300\pi$
$504$	76.7118	3.41701
$505$	5.88537	0.261895
$506$	−38.2086	−1.69858
$507$	0.368799	0.0163789
$508$	−18.3520	−0.814239
$509$	5.83637	0.258692	0.129346	−	0.991599i	$-0.458712\pi$
$509$	5.83637	0.258692	0.129346	+	0.991599i	$0.458712\pi$
$510$	−0.0599148	−0.00265307
$511$	−15.5494	−0.687865
$512$	49.9492	2.20747
$513$	8.44856	0.373013
$514$	2.91042	0.128373
$515$	−2.47189	−0.108925
$516$	11.5171	0.507013
$517$	2.54065	0.111738
$518$	−53.3883	−2.34575
$519$	−3.69567	−0.162222
$520$	3.19858	0.140267
$521$	33.2484	1.45664	0.728320	−	0.685237i	$-0.240302\pi$
$521$	33.2484	1.45664	0.728320	+	0.685237i	$0.240302\pi$
$522$	−12.8708	−0.563341
$523$	−7.99914	−0.349778	−0.174889	−	0.984588i	$-0.555957\pi$
$523$	−7.99914	−0.349778	−0.174889	+	0.984588i	$0.555957\pi$
$524$	−61.1582	−2.67171
$525$	−7.06474	−0.308330
$526$	−47.9225	−2.08952
$527$	0.730920	0.0318394
$528$	7.05617	0.307080
$529$	15.0835	0.655804
$530$	−9.54228	−0.414490
$531$	20.9270	0.908153
$532$	−72.1176	−3.12670
$533$	−8.47513	−0.367099
$534$	7.08661	0.306667
$535$	−2.11203	−0.0913109
$536$	−70.9176	−3.06317
$537$	−4.47564	−0.193138
$538$	33.5910	1.44821
$539$	22.0034	0.947753
$540$	4.76751	0.205161
$541$	−33.3220	−1.43263	−0.716313	−	0.697779i	$-0.754172\pi$
$541$	−33.3220	−1.43263	−0.716313	+	0.697779i	$0.754172\pi$
$542$	11.6088	0.498639
$543$	0.793638	0.0340583
$544$	0.929332	0.0398448
$545$	−8.45018	−0.361966
$546$	−3.80402	−0.162797
$547$	2.05136	0.0877097	0.0438548	−	0.999038i	$-0.486036\pi$
$547$	2.05136	0.0877097	0.0438548	+	0.999038i	$0.486036\pi$
$548$	−67.1667	−2.86922
$549$	−0.210176	−0.00897009
$550$	29.5333	1.25930
$551$	6.83548	0.291201
$552$	−15.1797	−0.646090
$553$	−15.1020	−0.642201
$554$	−43.2981	−1.83956
$555$	−0.915453	−0.0388588
$556$	−33.7992	−1.43341
$557$	−32.5707	−1.38007	−0.690033	−	0.723778i	$-0.742404\pi$
$557$	−32.5707	−1.38007	−0.690033	+	0.723778i	$0.742404\pi$
$558$	−40.7646	−1.72570
$559$	6.79359	0.287338
$560$	−15.2860	−0.645950
$561$	−0.117257	−0.00495061
$562$	−35.0329	−1.47778
$563$	5.63834	0.237628	0.118814	−	0.992917i	$-0.462091\pi$
$563$	5.63834	0.237628	0.118814	+	0.992917i	$0.462091\pi$
$564$	1.78675	0.0752358
$565$	−3.96940	−0.166994
$566$	28.5088	1.19832
$567$	31.3017	1.31455
$568$	40.9871	1.71978
$569$	17.5228	0.734594	0.367297	−	0.930104i	$-0.380283\pi$
$569$	17.5228	0.734594	0.367297	+	0.930104i	$0.380283\pi$
$570$	−1.77464	−0.0743313
$571$	19.0269	0.796253	0.398126	−	0.917331i	$-0.369661\pi$
$571$	19.0269	0.796253	0.398126	+	0.917331i	$0.369661\pi$
$572$	11.0811	0.463323
$573$	−6.67323	−0.278778
$574$	87.4177	3.64875
$575$	−29.4366	−1.22759
$576$	−6.36783	−0.265326
$577$	−0.0625611	−0.00260445	−0.00130223	−	0.999999i	$-0.500415\pi$
$577$	−0.0625611	−0.00260445	−0.00130223	+	0.999999i	$0.500415\pi$
$578$	43.6185	1.81429
$579$	−4.29662	−0.178561
$580$	3.85725	0.160164
$581$	−27.2196	−1.12926
$582$	3.95081	0.163766
$583$	−18.6749	−0.773436
$584$	25.8244	1.06862
$585$	1.37349	0.0567867
$586$	23.3537	0.964733
$587$	20.4159	0.842652	0.421326	−	0.906909i	$-0.361565\pi$
$587$	20.4159	0.842652	0.421326	+	0.906909i	$0.361565\pi$
$588$	15.4742	0.638145
$589$	21.6494	0.892047
$590$	−9.00027	−0.370535
$591$	−2.60603	−0.107198
$592$	41.0814	1.68844
$593$	−31.3283	−1.28650	−0.643250	−	0.765656i	$-0.722414\pi$
$593$	−31.3283	−1.28650	−0.643250	+	0.765656i	$0.722414\pi$
$594$	13.3898	0.549392
$595$	0.254018	0.0104137
$596$	−32.1798	−1.31814
$597$	−0.927404	−0.0379561
$598$	−15.8502	−0.648163
$599$	6.56814	0.268367	0.134183	−	0.990957i	$-0.457159\pi$
$599$	6.56814	0.268367	0.134183	+	0.990957i	$0.457159\pi$
$600$	11.7331	0.479002
$601$	6.96170	0.283973	0.141987	−	0.989869i	$-0.454651\pi$
$601$	6.96170	0.283973	0.141987	+	0.989869i	$0.454651\pi$
$602$	−70.0733	−2.85597
$603$	−30.4523	−1.24012
$604$	41.6610	1.69516
$605$	2.48849	0.101171
$606$	11.6246	0.472215
$607$	−22.9850	−0.932933	−0.466466	−	0.884539i	$-0.654473\pi$
$607$	−22.9850	−0.932933	−0.466466	+	0.884539i	$0.654473\pi$
$608$	27.5262	1.11634
$609$	−2.59147	−0.105012
$610$	0.0903924	0.00365988
$611$	1.05395	0.0426381
$612$	1.73640	0.0701899
$613$	−16.7191	−0.675278	−0.337639	−	0.941276i	$-0.609628\pi$
$613$	−16.7191	−0.675278	−0.337639	+	0.941276i	$0.609628\pi$
$614$	55.9161	2.25659
$615$	1.49896	0.0604438
$616$	−64.5680	−2.60152
$617$	38.7760	1.56106	0.780532	−	0.625116i	$-0.214948\pi$
$617$	38.7760	1.56106	0.780532	+	0.625116i	$0.214948\pi$
$618$	−4.88239	−0.196398
$619$	1.00000	0.0401934
$619$	1.00000	0.0401934
$620$	12.2167	0.490635
$621$	−13.3460	−0.535557
$622$	43.5748	1.74719
$623$	−30.0448	−1.20372
$624$	2.92713	0.117179
$625$	21.6031	0.864122
$626$	16.6325	0.664770
$627$	−3.47309	−0.138702
$628$	−48.7943	−1.94710
$629$	−0.682679	−0.0272202
$630$	−14.1670	−0.564427
$631$	25.4322	1.01244	0.506221	−	0.862404i	$-0.331042\pi$
$631$	25.4322	1.01244	0.506221	+	0.862404i	$0.331042\pi$
$632$	25.0814	0.997683
$633$	4.05703	0.161253
$634$	−4.15933	−0.165188
$635$	1.91462	0.0759793
$636$	−13.1334	−0.520773
$637$	9.12774	0.361654
$638$	10.8333	0.428896
$639$	17.6001	0.696248
$640$	−4.01952	−0.158885
$641$	35.6339	1.40745	0.703727	−	0.710471i	$-0.251518\pi$
$641$	35.6339	1.40745	0.703727	+	0.710471i	$0.251518\pi$
$642$	−4.17160	−0.164640
$643$	41.3555	1.63090	0.815452	−	0.578825i	$-0.196489\pi$
$643$	41.3555	1.63090	0.815452	+	0.578825i	$0.196489\pi$
$644$	113.923	4.48918
$645$	−1.20155	−0.0473111
$646$	−1.32340	−0.0520684
$647$	−14.6576	−0.576251	−0.288126	−	0.957593i	$-0.593032\pi$
$647$	−14.6576	−0.576251	−0.288126	+	0.957593i	$0.593032\pi$
$648$	−51.9860	−2.04220
$649$	−17.6142	−0.691416
$650$	12.2514	0.480540
$651$	−8.20772	−0.321686
$652$	25.5709	1.00144
$653$	17.9419	0.702120	0.351060	−	0.936353i	$-0.385821\pi$
$653$	17.9419	0.702120	0.351060	+	0.936353i	$0.385821\pi$
$654$	−16.6905	−0.652650
$655$	6.38047	0.249306
$656$	−67.2665	−2.62632
$657$	11.0891	0.432628
$658$	−10.8711	−0.423798
$659$	−5.32562	−0.207457	−0.103728	−	0.994606i	$-0.533077\pi$
$659$	−5.32562	−0.207457	−0.103728	+	0.994606i	$0.533077\pi$
$660$	−1.95986	−0.0762874
$661$	−23.0577	−0.896839	−0.448420	−	0.893823i	$-0.648013\pi$
$661$	−23.0577	−0.896839	−0.448420	+	0.893823i	$0.648013\pi$
$662$	53.7613	2.08949
$663$	−0.0486423	−0.00188911
$664$	45.2064	1.75435
$665$	7.52384	0.291762
$666$	38.0742	1.47534
$667$	−10.7979	−0.418095
$668$	−13.6267	−0.527233
$669$	8.06239	0.311710
$670$	13.0969	0.505979
$671$	0.176904	0.00682931
$672$	−10.4357	−0.402568
$673$	−9.88896	−0.381191	−0.190596	−	0.981669i	$-0.561042\pi$
$673$	−9.88896	−0.381191	−0.190596	+	0.981669i	$0.561042\pi$
$674$	−51.3558	−1.97815
$675$	10.3158	0.397055
$676$	4.59680	0.176800
$677$	−24.4486	−0.939637	−0.469818	−	0.882763i	$-0.655681\pi$
$677$	−24.4486	−0.939637	−0.469818	+	0.882763i	$0.655681\pi$
$678$	−7.84021	−0.301101
$679$	−16.7501	−0.642808
$680$	−0.421873	−0.0161781
$681$	−7.62595	−0.292227
$682$	34.3114	1.31385
$683$	6.00016	0.229590	0.114795	−	0.993389i	$-0.463379\pi$
$683$	6.00016	0.229590	0.114795	+	0.993389i	$0.463379\pi$
$684$	51.4311	1.96652
$685$	7.00733	0.267736
$686$	−21.9468	−0.837932
$687$	−4.10485	−0.156610
$688$	53.9203	2.05569
$689$	−7.74698	−0.295136
$690$	2.80336	0.106722
$691$	−40.0836	−1.52485	−0.762426	−	0.647076i	$-0.775992\pi$
$691$	−40.0836	−1.52485	−0.762426	+	0.647076i	$0.775992\pi$
$692$	−46.0636	−1.75108
$693$	−27.7258	−1.05322
$694$	−4.28110	−0.162508
$695$	3.52618	0.133756
$696$	4.30391	0.163139
$697$	1.11782	0.0423403
$698$	−34.8176	−1.31787
$699$	3.57188	0.135101
$700$	−88.0565	−3.32822
$701$	−15.1824	−0.573430	−0.286715	−	0.958016i	$-0.592563\pi$
$701$	−15.1824	−0.573430	−0.286715	+	0.958016i	$0.592563\pi$
$702$	5.55456	0.209643
$703$	−20.2205	−0.762631
$704$	5.35977	0.202004
$705$	−0.186407	−0.00702049
$706$	−63.0145	−2.37158
$707$	−49.2841	−1.85352
$708$	−12.3874	−0.465547
$709$	47.3939	1.77992	0.889958	−	0.456043i	$-0.150734\pi$
$709$	47.3939	1.77992	0.889958	+	0.456043i	$0.150734\pi$
$710$	−7.56943	−0.284076
$711$	10.7701	0.403909
$712$	49.8984	1.87002
$713$	−34.1991	−1.28077
$714$	0.501727	0.0187766
$715$	−1.15606	−0.0432341
$716$	−55.7854	−2.08480
$717$	−1.09731	−0.0409797
$718$	−30.3178	−1.13145
$719$	−38.0865	−1.42039	−0.710194	−	0.704006i	$-0.751393\pi$
$719$	−38.0865	−1.42039	−0.710194	+	0.704006i	$0.751393\pi$
$720$	10.9013	0.406267
$721$	20.6996	0.770895
$722$	9.60187	0.357345
$723$	5.60878	0.208593
$724$	9.89209	0.367637
$725$	8.34620	0.309970
$726$	4.91516	0.182419
$727$	−50.4663	−1.87169	−0.935846	−	0.352409i	$-0.885362\pi$
$727$	−50.4663	−1.87169	−0.935846	+	0.352409i	$0.885362\pi$
$728$	−26.7850	−0.992716
$729$	−19.9303	−0.738159
$730$	−4.76921	−0.176516
$731$	−0.896031	−0.0331409
$732$	0.124410	0.00459834
$733$	0.741731	0.0273965	0.0136982	−	0.999906i	$-0.495640\pi$
$733$	0.741731	0.0273965	0.0136982	+	0.999906i	$0.495640\pi$
$734$	58.5316	2.16044
$735$	−1.61438	−0.0595474
$736$	−43.4826	−1.60279
$737$	25.6316	0.944153
$738$	−62.3424	−2.29486
$739$	5.69387	0.209452	0.104726	−	0.994501i	$-0.466603\pi$
$739$	5.69387	0.209452	0.104726	+	0.994501i	$0.466603\pi$
$740$	−11.4104	−0.419455
$741$	−1.44075	−0.0529274
$742$	79.9071	2.93348
$743$	31.9298	1.17139	0.585695	−	0.810532i	$-0.300822\pi$
$743$	31.9298	1.17139	0.585695	+	0.810532i	$0.300822\pi$
$744$	13.6314	0.499751
$745$	3.35724	0.123000
$746$	−31.9741	−1.17065
$747$	19.4118	0.710242
$748$	−1.46152	−0.0534386
$749$	17.6861	0.646237
$750$	−4.43818	−0.162059
$751$	−48.9484	−1.78615	−0.893076	−	0.449906i	$-0.851457\pi$
$751$	−48.9484	−1.78615	−0.893076	+	0.449906i	$0.851457\pi$
$752$	8.36511	0.305044
$753$	−5.64217	−0.205612
$754$	4.49403	0.163663
$755$	−4.34638	−0.158181
$756$	−39.9231	−1.45199
$757$	−27.6448	−1.00477	−0.502383	−	0.864645i	$-0.667543\pi$
$757$	−27.6448	−1.00477	−0.502383	+	0.864645i	$0.667543\pi$
$758$	−90.4096	−3.28382
$759$	5.48636	0.199142
$760$	−12.4956	−0.453263
$761$	0.685304	0.0248423	0.0124211	−	0.999923i	$-0.496046\pi$
$761$	0.685304	0.0248423	0.0124211	+	0.999923i	$0.496046\pi$
$762$	3.78168	0.136996
$763$	70.7619	2.56175
$764$	−83.1766	−3.00923
$765$	−0.181154	−0.00654965
$766$	−65.5570	−2.36867
$767$	−7.30694	−0.263838
$768$	−9.57918	−0.345659
$769$	−11.5405	−0.416161	−0.208080	−	0.978112i	$-0.566722\pi$
$769$	−11.5405	−0.416161	−0.208080	+	0.978112i	$0.566722\pi$
$770$	11.9243	0.429722
$771$	−0.417907	−0.0150505
$772$	−53.5541	−1.92745
$773$	30.9551	1.11338	0.556689	−	0.830721i	$-0.312071\pi$
$773$	30.9551	1.11338	0.556689	+	0.830721i	$0.312071\pi$
$774$	49.9732	1.79625
$775$	26.4341	0.949543
$776$	27.8185	0.998626
$777$	7.66601	0.275017
$778$	−40.3207	−1.44557
$779$	33.1090	1.18625
$780$	−0.813015	−0.0291106
$781$	−14.8139	−0.530083
$782$	2.09054	0.0747576
$783$	3.78401	0.135229
$784$	72.4462	2.58737
$785$	5.09058	0.181691
$786$	12.6025	0.449515
$787$	−36.9734	−1.31796	−0.658979	−	0.752161i	$-0.729011\pi$
$787$	−36.9734	−1.31796	−0.658979	+	0.752161i	$0.729011\pi$
$788$	−32.4822	−1.15713
$789$	6.88117	0.244976
$790$	−4.63198	−0.164798
$791$	33.2398	1.18187
$792$	46.0470	1.63621
$793$	0.0733858	0.00260600
$794$	−46.5550	−1.65218
$795$	1.37017	0.0485951
$796$	−11.5594	−0.409711
$797$	1.25261	0.0443699	0.0221849	−	0.999754i	$-0.492938\pi$
$797$	1.25261	0.0443699	0.0221849	+	0.999754i	$0.492938\pi$
$798$	14.8608	0.526067
$799$	−0.139009	−0.00491779
$800$	33.6098	1.18829
$801$	21.4266	0.757071
$802$	−41.1809	−1.45415
$803$	−9.33368	−0.329378
$804$	18.0258	0.635721
$805$	−11.8853	−0.418900
$806$	14.2335	0.501354
$807$	−4.82332	−0.169789
$808$	81.8510	2.87951
$809$	35.3858	1.24410	0.622050	−	0.782978i	$-0.286300\pi$
$809$	35.3858	1.24410	0.622050	+	0.782978i	$0.286300\pi$
$810$	9.60068	0.337333
$811$	14.1638	0.497357	0.248678	−	0.968586i	$-0.420004\pi$
$811$	14.1638	0.497357	0.248678	+	0.968586i	$0.420004\pi$
$812$	−32.3006	−1.13353
$813$	−1.66690	−0.0584607
$814$	−32.0469	−1.12324
$815$	−2.66775	−0.0934472
$816$	−0.386071	−0.0135152
$817$	−26.5399	−0.928513
$818$	−18.7708	−0.656305
$819$	−11.5016	−0.401898
$820$	18.6834	0.652451
$821$	−27.4236	−0.957090	−0.478545	−	0.878063i	$-0.658836\pi$
$821$	−27.4236	−0.957090	−0.478545	+	0.878063i	$0.658836\pi$
$822$	13.8406	0.482747
$823$	3.22768	0.112510	0.0562549	−	0.998416i	$-0.482084\pi$
$823$	3.22768	0.112510	0.0562549	+	0.998416i	$0.482084\pi$
$824$	−34.3780	−1.19761
$825$	−4.24068	−0.147642
$826$	75.3683	2.62240
$827$	−48.2241	−1.67692	−0.838458	−	0.544967i	$-0.816542\pi$
$827$	−48.2241	−1.67692	−0.838458	+	0.544967i	$0.816542\pi$
$828$	−81.2446	−2.82345
$829$	−41.9537	−1.45711	−0.728557	−	0.684985i	$-0.759809\pi$
$829$	−41.9537	−1.45711	−0.728557	+	0.684985i	$0.759809\pi$
$830$	−8.34863	−0.289785
$831$	6.21716	0.215671
$832$	2.22341	0.0770830
$833$	−1.20389	−0.0417123
$834$	6.96478	0.241171
$835$	1.42164	0.0491979
$836$	−43.2893	−1.49719
$837$	11.9847	0.414253
$838$	−41.9235	−1.44822
$839$	−4.87087	−0.168161	−0.0840805	−	0.996459i	$-0.526795\pi$
$839$	−4.87087	−0.168161	−0.0840805	+	0.996459i	$0.526795\pi$
$840$	4.73734	0.163454
$841$	−25.9385	−0.894430
$842$	−25.8129	−0.889571
$843$	5.03037	0.173255
$844$	50.5678	1.74061
$845$	−0.479572	−0.0164978
$846$	7.75276	0.266545
$847$	−20.8386	−0.716022
$848$	−61.4873	−2.11148
$849$	−4.09358	−0.140491
$850$	−1.61588	−0.0554243
$851$	31.9419	1.09496
$852$	−10.4181	−0.356918
$853$	43.2328	1.48026	0.740131	−	0.672462i	$-0.234763\pi$
$853$	43.2328	1.48026	0.740131	+	0.672462i	$0.234763\pi$
$854$	−0.756946	−0.0259022
$855$	−5.36567	−0.183502
$856$	−29.3731	−1.00395
$857$	−1.47221	−0.0502899	−0.0251449	−	0.999684i	$-0.508005\pi$
$857$	−1.47221	−0.0502899	−0.0251449	+	0.999684i	$0.508005\pi$
$858$	−2.28340	−0.0779541
$859$	0.204429	0.00697501	0.00348751	−	0.999994i	$-0.498890\pi$
$859$	0.204429	0.00697501	0.00348751	+	0.999994i	$0.498890\pi$
$860$	−14.9764	−0.510691
$861$	−12.5523	−0.427781
$862$	−49.6229	−1.69016
$863$	−12.9219	−0.439866	−0.219933	−	0.975515i	$-0.570584\pi$
$863$	−12.9219	−0.439866	−0.219933	+	0.975515i	$0.570584\pi$
$864$	15.2381	0.518409
$865$	4.80570	0.163399
$866$	19.3109	0.656209
$867$	−6.26317	−0.212708
$868$	−102.303	−3.47239
$869$	−9.06511	−0.307513
$870$	−0.794839	−0.0269476
$871$	10.6328	0.360280
$872$	−117.521	−3.97978
$873$	11.9454	0.404291
$874$	61.9205	2.09449
$875$	18.8163	0.636109
$876$	−6.56405	−0.221779
$877$	15.6956	0.530004	0.265002	−	0.964248i	$-0.414627\pi$
$877$	15.6956	0.530004	0.265002	+	0.964248i	$0.414627\pi$
$878$	38.3434	1.29403
$879$	−3.35335	−0.113106
$880$	−9.17556	−0.309308
$881$	−38.5668	−1.29935	−0.649675	−	0.760212i	$-0.725095\pi$
$881$	−38.5668	−1.29935	−0.649675	+	0.760212i	$0.725095\pi$
$882$	67.1430	2.26082
$883$	−5.15729	−0.173557	−0.0867783	−	0.996228i	$-0.527657\pi$
$883$	−5.15729	−0.173557	−0.0867783	+	0.996228i	$0.527657\pi$
$884$	−0.606288	−0.0203917
$885$	1.29235	0.0434417
$886$	105.050	3.52922
$887$	3.02348	0.101518	0.0507592	−	0.998711i	$-0.483836\pi$
$887$	3.02348	0.101518	0.0507592	+	0.998711i	$0.483836\pi$
$888$	−12.7317	−0.427248
$889$	−16.0330	−0.537730
$890$	−9.21514	−0.308892
$891$	18.7892	0.629462
$892$	100.491	3.36470
$893$	−4.11736	−0.137782
$894$	6.63109	0.221777
$895$	5.81995	0.194539
$896$	33.6595	1.12448
$897$	2.27593	0.0759910
$898$	−20.8947	−0.697264
$899$	9.69651	0.323397
$900$	62.7980	2.09327
$901$	1.02178	0.0340403
$902$	52.4734	1.74717
$903$	10.0618	0.334836
$904$	−55.2046	−1.83608
$905$	−1.03202	−0.0343054
$906$	−8.58482	−0.285211
$907$	−35.6825	−1.18482	−0.592409	−	0.805637i	$-0.701823\pi$
$907$	−35.6825	−1.18482	−0.592409	+	0.805637i	$0.701823\pi$
$908$	−95.0516	−3.15440
$909$	35.1472	1.16576
$910$	4.94660	0.163978
$911$	−20.0467	−0.664178	−0.332089	−	0.943248i	$-0.607753\pi$
$911$	−20.0467	−0.664178	−0.332089	+	0.943248i	$0.607753\pi$
$912$	−11.4352	−0.378656
$913$	−16.3389	−0.540737
$914$	−21.3801	−0.707192
$915$	−0.0129794	−0.000429086 0
$916$	−51.1638	−1.69050
$917$	−53.4301	−1.76442
$918$	−0.732611	−0.0241798
$919$	−44.8216	−1.47853	−0.739264	−	0.673416i	$-0.764826\pi$
$919$	−44.8216	−1.47853	−0.739264	+	0.673416i	$0.764826\pi$
$920$	19.7390	0.650777
$921$	−8.02898	−0.264564
$922$	12.3666	0.407273
$923$	−6.14530	−0.202275
$924$	16.4119	0.539911
$925$	−24.6895	−0.811786
$926$	−23.5003	−0.772267
$927$	−14.7621	−0.484850
$928$	12.3287	0.404708
$929$	31.8255	1.04416	0.522081	−	0.852896i	$-0.325156\pi$
$929$	31.8255	1.04416	0.522081	+	0.852896i	$0.325156\pi$
$930$	−2.51742	−0.0825495
$931$	−35.6585	−1.16866
$932$	44.5208	1.45833
$933$	−6.25689	−0.204841
$934$	−19.5570	−0.639926
$935$	0.152477	0.00498653
$936$	19.1018	0.624363
$937$	−50.4418	−1.64786	−0.823930	−	0.566691i	$-0.808223\pi$
$937$	−50.4418	−1.64786	−0.823930	+	0.566691i	$0.808223\pi$
$938$	−109.674	−3.58098
$939$	−2.38826	−0.0779380
$940$	−2.32342	−0.0757816
$941$	−27.4312	−0.894232	−0.447116	−	0.894476i	$-0.647549\pi$
$941$	−27.4312	−0.894232	−0.447116	+	0.894476i	$0.647549\pi$
$942$	10.0547	0.327601
$943$	−52.3016	−1.70317
$944$	−57.9947	−1.88757
$945$	4.16508	0.135490
$946$	−42.0622	−1.36756
$947$	7.05281	0.229186	0.114593	−	0.993413i	$-0.463444\pi$
$947$	7.05281	0.229186	0.114593	+	0.993413i	$0.463444\pi$
$948$	−6.37517	−0.207056
$949$	−3.87192	−0.125688
$950$	−47.8614	−1.55283
$951$	0.597237	0.0193667
$952$	3.53277	0.114498
$953$	11.4772	0.371783	0.185892	−	0.982570i	$-0.440483\pi$
$953$	11.4772	0.371783	0.185892	+	0.982570i	$0.440483\pi$
$954$	−56.9862	−1.84500
$955$	8.67760	0.280801
$956$	−13.6771	−0.442349
$957$	−1.55556	−0.0502840
$958$	82.7323	2.67296
$959$	−58.6794	−1.89486
$960$	−0.393245	−0.0126919
$961$	−0.289150	−0.00932741
$962$	−13.2941	−0.428619
$963$	−12.6130	−0.406447
$964$	69.9091	2.25162
$965$	5.58715	0.179857
$966$	−23.4753	−0.755306
$967$	7.94626	0.255534	0.127767	−	0.991804i	$-0.459219\pi$
$967$	7.94626	0.255534	0.127767	+	0.991804i	$0.459219\pi$
$968$	34.6087	1.11237
$969$	0.190026	0.00610452
$970$	−5.13748	−0.164954
$971$	2.67528	0.0858538	0.0429269	−	0.999078i	$-0.486332\pi$
$971$	2.67528	0.0858538	0.0429269	+	0.999078i	$0.486332\pi$
$972$	43.0373	1.38042
$973$	−29.5283	−0.946633
$974$	−64.1085	−2.05417
$975$	−1.75918	−0.0563387
$976$	0.582458	0.0186440
$977$	−39.8335	−1.27439	−0.637193	−	0.770704i	$-0.719905\pi$
$977$	−39.8335	−1.27439	−0.637193	+	0.770704i	$0.719905\pi$
$978$	−5.26924	−0.168492
$979$	−18.0347	−0.576391
$980$	−20.1220	−0.642775
$981$	−50.4642	−1.61120
$982$	36.0384	1.15003
$983$	59.7713	1.90641	0.953205	−	0.302326i	$-0.0977630\pi$
$983$	59.7713	1.90641	0.953205	+	0.302326i	$0.0977630\pi$
$984$	20.8468	0.664573
$985$	3.38878	0.107976
$986$	−0.592734	−0.0188765
$987$	1.56097	0.0496863
$988$	−17.9579	−0.571316
$989$	41.9245	1.33312
$990$	−8.50389	−0.270271
$991$	−60.1957	−1.91218	−0.956089	−	0.293076i	$-0.905321\pi$
$991$	−60.1957	−1.91218	−0.956089	+	0.293076i	$0.905321\pi$
$992$	39.0475	1.23976
$993$	−7.71958	−0.244973
$994$	63.3865	2.01050
$995$	1.20596	0.0382315
$996$	−11.4905	−0.364092
$997$	−46.5265	−1.47351	−0.736755	−	0.676160i	$-0.763643\pi$
$997$	−46.5265	−1.47351	−0.736755	+	0.676160i	$0.763643\pi$
$998$	68.1623	2.15764
$999$	−11.1938	−0.354155

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.b.1.8	✓	142

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.b.1.8	✓	142	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:	\( N \)	\(=\)	\( 8047 = 13 \cdot 619 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8047.a (trivial)

\(f(q)\)	\(=\)	\(q-2.56842 q^{2} +0.368799 q^{3} +4.59680 q^{4} -0.479572 q^{5} -0.947232 q^{6} +4.01594 q^{7} -6.66967 q^{8} -2.86399 q^{9} +O(q^{10})\)	\(q-2.56842 q^{2} +0.368799 q^{3} +4.59680 q^{4} -0.479572 q^{5} -0.947232 q^{6} +4.01594 q^{7} -6.66967 q^{8} -2.86399 q^{9} +1.23174 q^{10} +2.41061 q^{11} +1.69529 q^{12} +1.00000 q^{13} -10.3146 q^{14} -0.176866 q^{15} +7.93693 q^{16} -0.131894 q^{17} +7.35593 q^{18} -3.90661 q^{19} -2.20449 q^{20} +1.48107 q^{21} -6.19146 q^{22} +6.17118 q^{23} -2.45977 q^{24} -4.77001 q^{25} -2.56842 q^{26} -2.16263 q^{27} +18.4604 q^{28} -1.74972 q^{29} +0.454266 q^{30} -5.54174 q^{31} -7.04607 q^{32} +0.889030 q^{33} +0.338759 q^{34} -1.92593 q^{35} -13.1652 q^{36} +5.17598 q^{37} +10.0338 q^{38} +0.368799 q^{39} +3.19858 q^{40} -8.47513 q^{41} -3.80402 q^{42} +6.79359 q^{43} +11.0811 q^{44} +1.37349 q^{45} -15.8502 q^{46} +1.05395 q^{47} +2.92713 q^{48} +9.12774 q^{49} +12.2514 q^{50} -0.0486423 q^{51} +4.59680 q^{52} -7.74698 q^{53} +5.55456 q^{54} -1.15606 q^{55} -26.7850 q^{56} -1.44075 q^{57} +4.49403 q^{58} -7.30694 q^{59} -0.813015 q^{60} +0.0733858 q^{61} +14.2335 q^{62} -11.5016 q^{63} +2.22341 q^{64} -0.479572 q^{65} -2.28340 q^{66} +10.6328 q^{67} -0.606288 q^{68} +2.27593 q^{69} +4.94660 q^{70} -6.14530 q^{71} +19.1018 q^{72} -3.87192 q^{73} -13.2941 q^{74} -1.75918 q^{75} -17.9579 q^{76} +9.68084 q^{77} -0.947232 q^{78} -3.76051 q^{79} -3.80633 q^{80} +7.79438 q^{81} +21.7677 q^{82} -6.77790 q^{83} +6.80819 q^{84} +0.0632525 q^{85} -17.4488 q^{86} -0.645296 q^{87} -16.0779 q^{88} -7.48139 q^{89} -3.52770 q^{90} +4.01594 q^{91} +28.3677 q^{92} -2.04379 q^{93} -2.70698 q^{94} +1.87350 q^{95} -2.59858 q^{96} -4.17090 q^{97} -23.4439 q^{98} -6.90395 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9}+O(q^{10}) \) 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9	\( 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9} - 25 q^{10} - 25 q^{11} - 62 q^{12} + 142 q^{13} - 57 q^{14} - 14 q^{15} + 111 q^{16} - 141 q^{17} - 29 q^{18} - 3 q^{19} - 87 q^{20} - 19 q^{21} - 24 q^{22} - 69 q^{23} - 40 q^{24} + 87 q^{25} - 13 q^{26} - 95 q^{27} - 34 q^{28} - 147 q^{29} - 2 q^{30} - 21 q^{31} - 66 q^{32} - 62 q^{33} - 6 q^{34} - 59 q^{35} + 74 q^{36} - 37 q^{37} - 76 q^{38} - 26 q^{39} - 61 q^{40} - 97 q^{41} - 29 q^{42} - 33 q^{43} - 57 q^{44} - 86 q^{45} - q^{46} - 102 q^{47} - 141 q^{48} + 70 q^{49} - 28 q^{50} - 13 q^{51} + 129 q^{52} - 137 q^{53} - 29 q^{54} - 24 q^{55} - 130 q^{56} - 65 q^{57} - 15 q^{58} - 56 q^{59} + 11 q^{60} - 77 q^{61} - 150 q^{62} - 32 q^{63} + 73 q^{64} - 37 q^{65} - 32 q^{66} - 9 q^{67} - 226 q^{68} - 113 q^{69} + 6 q^{70} - 18 q^{71} - 82 q^{72} - 117 q^{73} - 70 q^{74} - 83 q^{75} + 40 q^{76} - 214 q^{77} - 15 q^{78} - 52 q^{79} - 161 q^{80} - 10 q^{81} - 36 q^{82} - 74 q^{83} + 53 q^{84} + 2 q^{85} + 17 q^{86} - 49 q^{87} - 29 q^{88} - 171 q^{89} - 57 q^{90} - 14 q^{91} - 187 q^{92} - 39 q^{93} + 13 q^{94} - 150 q^{95} - 47 q^{96} - 126 q^{97} - 85 q^{98}+O(q^{100}) \) 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9 - 25 * q^10 - 25 * q^11 - 62 * q^12 + 142 * q^13 - 57 * q^14 - 14 * q^15 + 111 * q^16 - 141 * q^17 - 29 * q^18 - 3 * q^19 - 87 * q^20 - 19 * q^21 - 24 * q^22 - 69 * q^23 - 40 * q^24 + 87 * q^25 - 13 * q^26 - 95 * q^27 - 34 * q^28 - 147 * q^29 - 2 * q^30 - 21 * q^31 - 66 * q^32 - 62 * q^33 - 6 * q^34 - 59 * q^35 + 74 * q^36 - 37 * q^37 - 76 * q^38 - 26 * q^39 - 61 * q^40 - 97 * q^41 - 29 * q^42 - 33 * q^43 - 57 * q^44 - 86 * q^45 - q^46 - 102 * q^47 - 141 * q^48 + 70 * q^49 - 28 * q^50 - 13 * q^51 + 129 * q^52 - 137 * q^53 - 29 * q^54 - 24 * q^55 - 130 * q^56 - 65 * q^57 - 15 * q^58 - 56 * q^59 + 11 * q^60 - 77 * q^61 - 150 * q^62 - 32 * q^63 + 73 * q^64 - 37 * q^65 - 32 * q^66 - 9 * q^67 - 226 * q^68 - 113 * q^69 + 6 * q^70 - 18 * q^71 - 82 * q^72 - 117 * q^73 - 70 * q^74 - 83 * q^75 + 40 * q^76 - 214 * q^77 - 15 * q^78 - 52 * q^79 - 161 * q^80 - 10 * q^81 - 36 * q^82 - 74 * q^83 + 53 * q^84 + 2 * q^85 + 17 * q^86 - 49 * q^87 - 29 * q^88 - 171 * q^89 - 57 * q^90 - 14 * q^91 - 187 * q^92 - 39 * q^93 + 13 * q^94 - 150 * q^95 - 47 * q^96 - 126 * q^97 - 85 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more