LMFDB - Embedded newform 8002.2.a.e.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8002,2,Mod(1,8002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8002, 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("8002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8002 = 2 \cdot 4001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8002.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:	$63.8962916974$
Analytic rank:	$0$
Dimension:	$77$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	8002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.73843 q^{3} +1.00000 q^{4} +4.07186 q^{5} +1.73843 q^{6} -2.38793 q^{7} -1.00000 q^{8} +0.0221381 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.73843 q^{3} +1.00000 q^{4} +4.07186 q^{5} +1.73843 q^{6} -2.38793 q^{7} -1.00000 q^{8} +0.0221381 q^{9} -4.07186 q^{10} +4.57210 q^{11} -1.73843 q^{12} -3.31788 q^{13} +2.38793 q^{14} -7.07864 q^{15} +1.00000 q^{16} +7.27985 q^{17} -0.0221381 q^{18} +6.96966 q^{19} +4.07186 q^{20} +4.15124 q^{21} -4.57210 q^{22} +7.61639 q^{23} +1.73843 q^{24} +11.5800 q^{25} +3.31788 q^{26} +5.17680 q^{27} -2.38793 q^{28} +2.53097 q^{29} +7.07864 q^{30} +5.36879 q^{31} -1.00000 q^{32} -7.94827 q^{33} -7.27985 q^{34} -9.72330 q^{35} +0.0221381 q^{36} +8.55921 q^{37} -6.96966 q^{38} +5.76790 q^{39} -4.07186 q^{40} +9.12763 q^{41} -4.15124 q^{42} +2.29493 q^{43} +4.57210 q^{44} +0.0901431 q^{45} -7.61639 q^{46} +13.1787 q^{47} -1.73843 q^{48} -1.29780 q^{49} -11.5800 q^{50} -12.6555 q^{51} -3.31788 q^{52} -1.72202 q^{53} -5.17680 q^{54} +18.6169 q^{55} +2.38793 q^{56} -12.1163 q^{57} -2.53097 q^{58} -10.7237 q^{59} -7.07864 q^{60} -7.61107 q^{61} -5.36879 q^{62} -0.0528641 q^{63} +1.00000 q^{64} -13.5099 q^{65} +7.94827 q^{66} -5.01292 q^{67} +7.27985 q^{68} -13.2406 q^{69} +9.72330 q^{70} -14.2437 q^{71} -0.0221381 q^{72} -0.562149 q^{73} -8.55921 q^{74} -20.1311 q^{75} +6.96966 q^{76} -10.9178 q^{77} -5.76790 q^{78} -5.14917 q^{79} +4.07186 q^{80} -9.06592 q^{81} -9.12763 q^{82} +0.591408 q^{83} +4.15124 q^{84} +29.6425 q^{85} -2.29493 q^{86} -4.39992 q^{87} -4.57210 q^{88} -7.78532 q^{89} -0.0901431 q^{90} +7.92286 q^{91} +7.61639 q^{92} -9.33326 q^{93} -13.1787 q^{94} +28.3794 q^{95} +1.73843 q^{96} -14.4588 q^{97} +1.29780 q^{98} +0.101217 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9}+O(q^{10}) $ 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9} - 18 q^{10} + 30 q^{11} + 10 q^{12} - 2 q^{13} - 21 q^{14} + 21 q^{15} + 77 q^{16} + 60 q^{17} - 71 q^{18} - 3 q^{19} + 18 q^{20} + 10 q^{21} - 30 q^{22} + 53 q^{23} - 10 q^{24} + 59 q^{25} + 2 q^{26} + 43 q^{27} + 21 q^{28} + 30 q^{29} - 21 q^{30} + 22 q^{31} - 77 q^{32} + 31 q^{33} - 60 q^{34} + 41 q^{35} + 71 q^{36} - 3 q^{37} + 3 q^{38} + 44 q^{39} - 18 q^{40} + 48 q^{41} - 10 q^{42} + 21 q^{43} + 30 q^{44} + 33 q^{45} - 53 q^{46} + 107 q^{47} + 10 q^{48} + 24 q^{49} - 59 q^{50} + 18 q^{51} - 2 q^{52} + 42 q^{53} - 43 q^{54} + 49 q^{55} - 21 q^{56} + 32 q^{57} - 30 q^{58} + 42 q^{59} + 21 q^{60} - 31 q^{61} - 22 q^{62} + 109 q^{63} + 77 q^{64} + 39 q^{65} - 31 q^{66} - q^{67} + 60 q^{68} - 33 q^{69} - 41 q^{70} + 58 q^{71} - 71 q^{72} + 35 q^{73} + 3 q^{74} + 34 q^{75} - 3 q^{76} + 86 q^{77} - 44 q^{78} + 25 q^{79} + 18 q^{80} + 53 q^{81} - 48 q^{82} + 107 q^{83} + 10 q^{84} + 21 q^{85} - 21 q^{86} + 100 q^{87} - 30 q^{88} + 34 q^{89} - 33 q^{90} - 51 q^{91} + 53 q^{92} + 48 q^{93} - 107 q^{94} + 118 q^{95} - 10 q^{96} - 13 q^{97} - 24 q^{98} + 63 q^{99}+O(q^{100}) \) 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9 - 18 * q^10 + 30 * q^11 + 10 * q^12 - 2 * q^13 - 21 * q^14 + 21 * q^15 + 77 * q^16 + 60 * q^17 - 71 * q^18 - 3 * q^19 + 18 * q^20 + 10 * q^21 - 30 * q^22 + 53 * q^23 - 10 * q^24 + 59 * q^25 + 2 * q^26 + 43 * q^27 + 21 * q^28 + 30 * q^29 - 21 * q^30 + 22 * q^31 - 77 * q^32 + 31 * q^33 - 60 * q^34 + 41 * q^35 + 71 * q^36 - 3 * q^37 + 3 * q^38 + 44 * q^39 - 18 * q^40 + 48 * q^41 - 10 * q^42 + 21 * q^43 + 30 * q^44 + 33 * q^45 - 53 * q^46 + 107 * q^47 + 10 * q^48 + 24 * q^49 - 59 * q^50 + 18 * q^51 - 2 * q^52 + 42 * q^53 - 43 * q^54 + 49 * q^55 - 21 * q^56 + 32 * q^57 - 30 * q^58 + 42 * q^59 + 21 * q^60 - 31 * q^61 - 22 * q^62 + 109 * q^63 + 77 * q^64 + 39 * q^65 - 31 * q^66 - q^67 + 60 * q^68 - 33 * q^69 - 41 * q^70 + 58 * q^71 - 71 * q^72 + 35 * q^73 + 3 * q^74 + 34 * q^75 - 3 * q^76 + 86 * q^77 - 44 * q^78 + 25 * q^79 + 18 * q^80 + 53 * q^81 - 48 * q^82 + 107 * q^83 + 10 * q^84 + 21 * q^85 - 21 * q^86 + 100 * q^87 - 30 * q^88 + 34 * q^89 - 33 * q^90 - 51 * q^91 + 53 * q^92 + 48 * q^93 - 107 * q^94 + 118 * q^95 - 10 * q^96 - 13 * q^97 - 24 * q^98 + 63 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.73843	−1.00368	−0.501841	−	0.864960i	$-0.667344\pi$
$3$	−1.73843	−1.00368	−0.501841	+	0.864960i	$0.667344\pi$
$4$	1.00000	0.500000
$5$	4.07186	1.82099	0.910495	−	0.413520i	$-0.135701\pi$
$5$	4.07186	1.82099	0.910495	+	0.413520i	$0.135701\pi$
$6$	1.73843	0.709711
$7$	−2.38793	−0.902552	−0.451276	−	0.892385i	$-0.649031\pi$
$7$	−2.38793	−0.902552	−0.451276	+	0.892385i	$0.649031\pi$
$8$	−1.00000	−0.353553
$9$	0.0221381	0.00737936
$10$	−4.07186	−1.28763
$11$	4.57210	1.37854	0.689270	−	0.724505i	$-0.257931\pi$
$11$	4.57210	1.37854	0.689270	+	0.724505i	$0.257931\pi$
$12$	−1.73843	−0.501841
$13$	−3.31788	−0.920214	−0.460107	−	0.887863i	$-0.652189\pi$
$13$	−3.31788	−0.920214	−0.460107	+	0.887863i	$0.652189\pi$
$14$	2.38793	0.638200
$15$	−7.07864	−1.82770
$16$	1.00000	0.250000
$17$	7.27985	1.76562	0.882811	−	0.469727i	$-0.155648\pi$
$17$	7.27985	1.76562	0.882811	+	0.469727i	$0.155648\pi$
$18$	−0.0221381	−0.00521800
$19$	6.96966	1.59895	0.799474	−	0.600700i	$-0.205111\pi$
$19$	6.96966	1.59895	0.799474	+	0.600700i	$0.205111\pi$
$20$	4.07186	0.910495
$21$	4.15124	0.905876
$22$	−4.57210	−0.974775
$23$	7.61639	1.58813	0.794064	−	0.607834i	$-0.207962\pi$
$23$	7.61639	1.58813	0.794064	+	0.607834i	$0.207962\pi$
$24$	1.73843	0.354855
$25$	11.5800	2.31601
$26$	3.31788	0.650690
$27$	5.17680	0.996276
$28$	−2.38793	−0.451276
$29$	2.53097	0.469990	0.234995	−	0.971997i	$-0.424493\pi$
$29$	2.53097	0.469990	0.234995	+	0.971997i	$0.424493\pi$
$30$	7.07864	1.29238
$31$	5.36879	0.964263	0.482131	−	0.876099i	$-0.339863\pi$
$31$	5.36879	0.964263	0.482131	+	0.876099i	$0.339863\pi$
$32$	−1.00000	−0.176777
$33$	−7.94827	−1.38362
$34$	−7.27985	−1.24848
$35$	−9.72330	−1.64354
$36$	0.0221381	0.00368968
$37$	8.55921	1.40713	0.703563	−	0.710633i	$-0.251592\pi$
$37$	8.55921	1.40713	0.703563	+	0.710633i	$0.251592\pi$
$38$	−6.96966	−1.13063
$39$	5.76790	0.923603
$40$	−4.07186	−0.643817
$41$	9.12763	1.42550	0.712748	−	0.701420i	$-0.247450\pi$
$41$	9.12763	1.42550	0.712748	+	0.701420i	$0.247450\pi$
$42$	−4.15124	−0.640551
$43$	2.29493	0.349973	0.174986	−	0.984571i	$-0.444012\pi$
$43$	2.29493	0.349973	0.174986	+	0.984571i	$0.444012\pi$
$44$	4.57210	0.689270
$45$	0.0901431	0.0134377
$46$	−7.61639	−1.12298
$47$	13.1787	1.92231	0.961156	−	0.276005i	$-0.0890107\pi$
$47$	13.1787	1.92231	0.961156	+	0.276005i	$0.0890107\pi$
$48$	−1.73843	−0.250921
$49$	−1.29780	−0.185401
$50$	−11.5800	−1.63766
$51$	−12.6555	−1.77213
$52$	−3.31788	−0.460107
$53$	−1.72202	−0.236538	−0.118269	−	0.992982i	$-0.537735\pi$
$53$	−1.72202	−0.236538	−0.118269	+	0.992982i	$0.537735\pi$
$54$	−5.17680	−0.704474
$55$	18.6169	2.51031
$56$	2.38793	0.319100
$57$	−12.1163	−1.60484
$58$	−2.53097	−0.332333
$59$	−10.7237	−1.39610	−0.698052	−	0.716047i	$-0.745950\pi$
$59$	−10.7237	−1.39610	−0.698052	+	0.716047i	$0.745950\pi$
$60$	−7.07864	−0.913848
$61$	−7.61107	−0.974498	−0.487249	−	0.873263i	$-0.662000\pi$
$61$	−7.61107	−0.974498	−0.487249	+	0.873263i	$0.662000\pi$
$62$	−5.36879	−0.681837
$63$	−0.0528641	−0.00666025
$64$	1.00000	0.125000
$65$	−13.5099	−1.67570
$66$	7.94827	0.978365
$67$	−5.01292	−0.612426	−0.306213	−	0.951963i	$-0.599062\pi$
$67$	−5.01292	−0.612426	−0.306213	+	0.951963i	$0.599062\pi$
$68$	7.27985	0.882811
$69$	−13.2406	−1.59398
$70$	9.72330	1.16216
$71$	−14.2437	−1.69042	−0.845211	−	0.534433i	$-0.820525\pi$
$71$	−14.2437	−1.69042	−0.845211	+	0.534433i	$0.820525\pi$
$72$	−0.0221381	−0.00260900
$73$	−0.562149	−0.0657945	−0.0328973	−	0.999459i	$-0.510473\pi$
$73$	−0.562149	−0.0657945	−0.0328973	+	0.999459i	$0.510473\pi$
$74$	−8.55921	−0.994988
$75$	−20.1311	−2.32454
$76$	6.96966	0.799474
$77$	−10.9178	−1.24420
$78$	−5.76790	−0.653086
$79$	−5.14917	−0.579327	−0.289663	−	0.957129i	$-0.593543\pi$
$79$	−5.14917	−0.579327	−0.289663	+	0.957129i	$0.593543\pi$
$80$	4.07186	0.455248
$81$	−9.06592	−1.00732
$82$	−9.12763	−1.00798
$83$	0.591408	0.0649155	0.0324578	−	0.999473i	$-0.489667\pi$
$83$	0.591408	0.0649155	0.0324578	+	0.999473i	$0.489667\pi$
$84$	4.15124	0.452938
$85$	29.6425	3.21518
$86$	−2.29493	−0.247468
$87$	−4.39992	−0.471721
$88$	−4.57210	−0.487387
$89$	−7.78532	−0.825242	−0.412621	−	0.910903i	$-0.635387\pi$
$89$	−7.78532	−0.825242	−0.412621	+	0.910903i	$0.635387\pi$
$90$	−0.0901431	−0.00950192
$91$	7.92286	0.830541
$92$	7.61639	0.794064
$93$	−9.33326	−0.967814
$94$	−13.1787	−1.35928
$95$	28.3794	2.91167
$96$	1.73843	0.177428
$97$	−14.4588	−1.46807	−0.734033	−	0.679114i	$-0.762364\pi$
$97$	−14.4588	−1.46807	−0.734033	+	0.679114i	$0.762364\pi$
$98$	1.29780	0.131098
$99$	0.101217	0.0101727
$100$	11.5800	1.15800
$101$	−5.51473	−0.548736	−0.274368	−	0.961625i	$-0.588469\pi$
$101$	−5.51473	−0.548736	−0.274368	+	0.961625i	$0.588469\pi$
$102$	12.6555	1.25308
$103$	11.7681	1.15954	0.579770	−	0.814780i	$-0.303142\pi$
$103$	11.7681	1.15954	0.579770	+	0.814780i	$0.303142\pi$
$104$	3.31788	0.325345
$105$	16.9033	1.64959
$106$	1.72202	0.167258
$107$	1.15542	0.111699	0.0558493	−	0.998439i	$-0.482213\pi$
$107$	1.15542	0.111699	0.0558493	+	0.998439i	$0.482213\pi$
$108$	5.17680	0.498138
$109$	−1.06109	−0.101634	−0.0508168	−	0.998708i	$-0.516182\pi$
$109$	−1.06109	−0.101634	−0.0508168	+	0.998708i	$0.516182\pi$
$110$	−18.6169	−1.77506
$111$	−14.8796	−1.41231
$112$	−2.38793	−0.225638
$113$	14.5492	1.36868	0.684339	−	0.729164i	$-0.260091\pi$
$113$	14.5492	1.36868	0.684339	+	0.729164i	$0.260091\pi$
$114$	12.1163	1.13479
$115$	31.0129	2.89197
$116$	2.53097	0.234995
$117$	−0.0734515	−0.00679059
$118$	10.7237	0.987195
$119$	−17.3838	−1.59357
$120$	7.07864	0.646188
$121$	9.90409	0.900371
$122$	7.61107	0.689074
$123$	−15.8677	−1.43075
$124$	5.36879	0.482131
$125$	26.7929	2.39643
$126$	0.0528641	0.00470951
$127$	−6.25091	−0.554679	−0.277339	−	0.960772i	$-0.589453\pi$
$127$	−6.25091	−0.554679	−0.277339	+	0.960772i	$0.589453\pi$
$128$	−1.00000	−0.0883883
$129$	−3.98957	−0.351262
$130$	13.5099	1.18490
$131$	−2.42136	−0.211556	−0.105778	−	0.994390i	$-0.533733\pi$
$131$	−2.42136	−0.211556	−0.105778	+	0.994390i	$0.533733\pi$
$132$	−7.94827	−0.691808
$133$	−16.6430	−1.44313
$134$	5.01292	0.433051
$135$	21.0792	1.81421
$136$	−7.27985	−0.624242
$137$	−8.96139	−0.765623	−0.382812	−	0.923826i	$-0.625044\pi$
$137$	−8.96139	−0.765623	−0.382812	+	0.923826i	$0.625044\pi$
$138$	13.2406	1.12711
$139$	−0.542216	−0.0459902	−0.0229951	−	0.999736i	$-0.507320\pi$
$139$	−0.542216	−0.0459902	−0.0229951	+	0.999736i	$0.507320\pi$
$140$	−9.72330	−0.821769
$141$	−22.9103	−1.92939
$142$	14.2437	1.19531
$143$	−15.1697	−1.26855
$144$	0.0221381	0.00184484
$145$	10.3058	0.855847
$146$	0.562149	0.0465238
$147$	2.25614	0.186083
$148$	8.55921	0.703563
$149$	−22.2401	−1.82198	−0.910988	−	0.412432i	$-0.864680\pi$
$149$	−22.2401	−1.82198	−0.910988	+	0.412432i	$0.864680\pi$
$150$	20.1311	1.64369
$151$	21.9221	1.78399	0.891997	−	0.452042i	$-0.149304\pi$
$151$	21.9221	1.78399	0.891997	+	0.452042i	$0.149304\pi$
$152$	−6.96966	−0.565314
$153$	0.161162	0.0130292
$154$	10.9178	0.879785
$155$	21.8609	1.75591
$156$	5.76790	0.461802
$157$	−8.14095	−0.649719	−0.324859	−	0.945762i	$-0.605317\pi$
$157$	−8.14095	−0.649719	−0.324859	+	0.945762i	$0.605317\pi$
$158$	5.14917	0.409646
$159$	2.99362	0.237409
$160$	−4.07186	−0.321909
$161$	−18.1874	−1.43337
$162$	9.06592	0.712286
$163$	1.82196	0.142707	0.0713533	−	0.997451i	$-0.477268\pi$
$163$	1.82196	0.142707	0.0713533	+	0.997451i	$0.477268\pi$
$164$	9.12763	0.712748
$165$	−32.3642	−2.51955
$166$	−0.591408	−0.0459022
$167$	15.8130	1.22365	0.611824	−	0.790994i	$-0.290436\pi$
$167$	15.8130	1.22365	0.611824	+	0.790994i	$0.290436\pi$
$168$	−4.15124	−0.320275
$169$	−1.99167	−0.153206
$170$	−29.6425	−2.27348
$171$	0.154295	0.0117992
$172$	2.29493	0.174986
$173$	−7.99538	−0.607877	−0.303939	−	0.952692i	$-0.598302\pi$
$173$	−7.99538	−0.607877	−0.303939	+	0.952692i	$0.598302\pi$
$174$	4.39992	0.333557
$175$	−27.6523	−2.09031
$176$	4.57210	0.344635
$177$	18.6424	1.40125
$178$	7.78532	0.583534
$179$	−4.44776	−0.332441	−0.166221	−	0.986089i	$-0.553156\pi$
$179$	−4.44776	−0.332441	−0.166221	+	0.986089i	$0.553156\pi$
$180$	0.0901431	0.00671887
$181$	−7.70175	−0.572467	−0.286233	−	0.958160i	$-0.592403\pi$
$181$	−7.70175	−0.572467	−0.286233	+	0.958160i	$0.592403\pi$
$182$	−7.92286	−0.587281
$183$	13.2313	0.978087
$184$	−7.61639	−0.561488
$185$	34.8519	2.56236
$186$	9.33326	0.684348
$187$	33.2842	2.43398
$188$	13.1787	0.961156
$189$	−12.3618	−0.899191
$190$	−28.3794	−2.05886
$191$	−3.76241	−0.272238	−0.136119	−	0.990692i	$-0.543463\pi$
$191$	−3.76241	−0.272238	−0.136119	+	0.990692i	$0.543463\pi$
$192$	−1.73843	−0.125460
$193$	19.2791	1.38774	0.693871	−	0.720099i	$-0.255904\pi$
$193$	19.2791	1.38774	0.693871	+	0.720099i	$0.255904\pi$
$194$	14.4588	1.03808
$195$	23.4861	1.68187
$196$	−1.29780	−0.0927003
$197$	−18.9201	−1.34800	−0.674001	−	0.738730i	$-0.735426\pi$
$197$	−18.9201	−1.34800	−0.674001	+	0.738730i	$0.735426\pi$
$198$	−0.101217	−0.00719321
$199$	8.76145	0.621083	0.310541	−	0.950560i	$-0.399490\pi$
$199$	8.76145	0.621083	0.310541	+	0.950560i	$0.399490\pi$
$200$	−11.5800	−0.818832
$201$	8.71462	0.614682
$202$	5.51473	0.388015
$203$	−6.04378	−0.424190
$204$	−12.6555	−0.886063
$205$	37.1664	2.59581
$206$	−11.7681	−0.819919
$207$	0.168612	0.0117194
$208$	−3.31788	−0.230054
$209$	31.8660	2.20421
$210$	−16.9033	−1.16644
$211$	−19.7342	−1.35856	−0.679280	−	0.733879i	$-0.737708\pi$
$211$	−19.7342	−1.35856	−0.679280	+	0.733879i	$0.737708\pi$
$212$	−1.72202	−0.118269
$213$	24.7618	1.69665
$214$	−1.15542	−0.0789829
$215$	9.34461	0.637297
$216$	−5.17680	−0.352237
$217$	−12.8203	−0.870297
$218$	1.06109	0.0718659
$219$	0.977256	0.0660369
$220$	18.6169	1.25515
$221$	−24.1537	−1.62475
$222$	14.8796	0.998652
$223$	9.22087	0.617475	0.308738	−	0.951147i	$-0.400093\pi$
$223$	9.22087	0.617475	0.308738	+	0.951147i	$0.400093\pi$
$224$	2.38793	0.159550
$225$	0.256360	0.0170906
$226$	−14.5492	−0.967801
$227$	−3.74748	−0.248729	−0.124365	−	0.992237i	$-0.539689\pi$
$227$	−3.74748	−0.248729	−0.124365	+	0.992237i	$0.539689\pi$
$228$	−12.1163	−0.802419
$229$	−2.10228	−0.138923	−0.0694614	−	0.997585i	$-0.522128\pi$
$229$	−2.10228	−0.138923	−0.0694614	+	0.997585i	$0.522128\pi$
$230$	−31.0129	−2.04493
$231$	18.9799	1.24879
$232$	−2.53097	−0.166166
$233$	24.8250	1.62634	0.813170	−	0.582026i	$-0.197740\pi$
$233$	24.8250	1.62634	0.813170	+	0.582026i	$0.197740\pi$
$234$	0.0734515	0.00480167
$235$	53.6618	3.50051
$236$	−10.7237	−0.698052
$237$	8.95146	0.581460
$238$	17.3838	1.12682
$239$	3.60732	0.233338	0.116669	−	0.993171i	$-0.462778\pi$
$239$	3.60732	0.233338	0.116669	+	0.993171i	$0.462778\pi$
$240$	−7.07864	−0.456924
$241$	20.2807	1.30640	0.653198	−	0.757187i	$-0.273427\pi$
$241$	20.2807	1.30640	0.653198	+	0.757187i	$0.273427\pi$
$242$	−9.90409	−0.636659
$243$	0.230061	0.0147584
$244$	−7.61107	−0.487249
$245$	−5.28447	−0.337613
$246$	15.8677	1.01169
$247$	−23.1245	−1.47138
$248$	−5.36879	−0.340918
$249$	−1.02812	−0.0651546
$250$	−26.7929	−1.69453
$251$	−22.7440	−1.43559	−0.717793	−	0.696256i	$-0.754848\pi$
$251$	−22.7440	−1.43559	−0.717793	+	0.696256i	$0.754848\pi$
$252$	−0.0528641	−0.00333013
$253$	34.8229	2.18930
$254$	6.25091	0.392217
$255$	−51.5314	−3.22702
$256$	1.00000	0.0625000
$257$	−17.8225	−1.11174	−0.555869	−	0.831270i	$-0.687614\pi$
$257$	−17.8225	−1.11174	−0.555869	+	0.831270i	$0.687614\pi$
$258$	3.98957	0.248380
$259$	−20.4388	−1.27000
$260$	−13.5099	−0.837851
$261$	0.0560309	0.00346822
$262$	2.42136	0.149592
$263$	26.6279	1.64195	0.820974	−	0.570965i	$-0.193431\pi$
$263$	26.6279	1.64195	0.820974	+	0.570965i	$0.193431\pi$
$264$	7.94827	0.489182
$265$	−7.01183	−0.430733
$266$	16.6430	1.02045
$267$	13.5342	0.828281
$268$	−5.01292	−0.306213
$269$	−30.1938	−1.84095	−0.920475	−	0.390801i	$-0.872198\pi$
$269$	−30.1938	−1.84095	−0.920475	+	0.390801i	$0.872198\pi$
$270$	−21.0792	−1.28284
$271$	20.1446	1.22370	0.611849	−	0.790974i	$-0.290426\pi$
$271$	20.1446	1.22370	0.611849	+	0.790974i	$0.290426\pi$
$272$	7.27985	0.441406
$273$	−13.7733	−0.833600
$274$	8.96139	0.541377
$275$	52.9450	3.19271
$276$	−13.2406	−0.796988
$277$	−0.493290	−0.0296389	−0.0148195	−	0.999890i	$-0.504717\pi$
$277$	−0.493290	−0.0296389	−0.0148195	+	0.999890i	$0.504717\pi$
$278$	0.542216	0.0325200
$279$	0.118855	0.00711564
$280$	9.72330	0.581078
$281$	−27.6877	−1.65171	−0.825856	−	0.563881i	$-0.809308\pi$
$281$	−27.6877	−1.65171	−0.825856	+	0.563881i	$0.809308\pi$
$282$	22.9103	1.36429
$283$	17.4944	1.03994	0.519968	−	0.854185i	$-0.325944\pi$
$283$	17.4944	1.03994	0.519968	+	0.854185i	$0.325944\pi$
$284$	−14.2437	−0.845211
$285$	−49.3357	−2.92239
$286$	15.1697	0.897002
$287$	−21.7961	−1.28658
$288$	−0.0221381	−0.00130450
$289$	35.9962	2.11742
$290$	−10.3058	−0.605175
$291$	25.1356	1.47347
$292$	−0.562149	−0.0328973
$293$	−3.38270	−0.197620	−0.0988098	−	0.995106i	$-0.531504\pi$
$293$	−3.38270	−0.197620	−0.0988098	+	0.995106i	$0.531504\pi$
$294$	−2.25614	−0.131581
$295$	−43.6653	−2.54229
$296$	−8.55921	−0.497494
$297$	23.6689	1.37341
$298$	22.2401	1.28833
$299$	−25.2703	−1.46142
$300$	−20.1311	−1.16227
$301$	−5.48011	−0.315869
$302$	−21.9221	−1.26147
$303$	9.58698	0.550757
$304$	6.96966	0.399737
$305$	−30.9912	−1.77455
$306$	−0.161162	−0.00921301
$307$	−20.6600	−1.17913	−0.589563	−	0.807722i	$-0.700700\pi$
$307$	−20.6600	−1.17913	−0.589563	+	0.807722i	$0.700700\pi$
$308$	−10.9178	−0.622102
$309$	−20.4579	−1.16381
$310$	−21.8609	−1.24162
$311$	−6.65115	−0.377152	−0.188576	−	0.982059i	$-0.560387\pi$
$311$	−6.65115	−0.377152	−0.188576	+	0.982059i	$0.560387\pi$
$312$	−5.76790	−0.326543
$313$	−8.33752	−0.471265	−0.235632	−	0.971842i	$-0.575716\pi$
$313$	−8.33752	−0.471265	−0.235632	+	0.971842i	$0.575716\pi$
$314$	8.14095	0.459420
$315$	−0.215255	−0.0121283
$316$	−5.14917	−0.289663
$317$	−20.7472	−1.16528	−0.582638	−	0.812732i	$-0.697980\pi$
$317$	−20.7472	−1.16528	−0.582638	+	0.812732i	$0.697980\pi$
$318$	−2.99362	−0.167874
$319$	11.5719	0.647899
$320$	4.07186	0.227624
$321$	−2.00862	−0.112110
$322$	18.1874	1.01354
$323$	50.7380	2.82314
$324$	−9.06592	−0.503662
$325$	−38.4211	−2.13122
$326$	−1.82196	−0.100909
$327$	1.84462	0.102008
$328$	−9.12763	−0.503989
$329$	−31.4698	−1.73499
$330$	32.3642	1.78159
$331$	14.7031	0.808157	0.404078	−	0.914724i	$-0.367592\pi$
$331$	14.7031	0.808157	0.404078	+	0.914724i	$0.367592\pi$
$332$	0.591408	0.0324578
$333$	0.189484	0.0103837
$334$	−15.8130	−0.865250
$335$	−20.4119	−1.11522
$336$	4.15124	0.226469
$337$	−20.5633	−1.12015	−0.560077	−	0.828441i	$-0.689228\pi$
$337$	−20.5633	−1.12015	−0.560077	+	0.828441i	$0.689228\pi$
$338$	1.99167	0.108333
$339$	−25.2928	−1.37372
$340$	29.6425	1.60759
$341$	24.5466	1.32927
$342$	−0.154295	−0.00834331
$343$	19.8146	1.06989
$344$	−2.29493	−0.123734
$345$	−53.9137	−2.90262
$346$	7.99538	0.429834
$347$	29.9834	1.60959	0.804797	−	0.593550i	$-0.202274\pi$
$347$	29.9834	1.60959	0.804797	+	0.593550i	$0.202274\pi$
$348$	−4.39992	−0.235860
$349$	−16.4215	−0.879021	−0.439510	−	0.898237i	$-0.644848\pi$
$349$	−16.4215	−0.879021	−0.439510	+	0.898237i	$0.644848\pi$
$350$	27.6523	1.47808
$351$	−17.1760	−0.916788
$352$	−4.57210	−0.243694
$353$	3.84501	0.204649	0.102325	−	0.994751i	$-0.467372\pi$
$353$	3.84501	0.204649	0.102325	+	0.994751i	$0.467372\pi$
$354$	−18.6424	−0.990831
$355$	−57.9985	−3.07824
$356$	−7.78532	−0.412621
$357$	30.2204	1.59943
$358$	4.44776	0.235071
$359$	−15.2603	−0.805409	−0.402704	−	0.915330i	$-0.631930\pi$
$359$	−15.2603	−0.805409	−0.402704	+	0.915330i	$0.631930\pi$
$360$	−0.0901431	−0.00475096
$361$	29.5761	1.55664
$362$	7.70175	0.404795
$363$	−17.2176	−0.903687
$364$	7.92286	0.415270
$365$	−2.28899	−0.119811
$366$	−13.2313	−0.691612
$367$	−11.7180	−0.611674	−0.305837	−	0.952084i	$-0.598936\pi$
$367$	−11.7180	−0.611674	−0.305837	+	0.952084i	$0.598936\pi$
$368$	7.61639	0.397032
$369$	0.202068	0.0105192
$370$	−34.8519	−1.81186
$371$	4.11206	0.213488
$372$	−9.33326	−0.483907
$373$	31.8546	1.64937	0.824684	−	0.565593i	$-0.191353\pi$
$373$	31.8546	1.64937	0.824684	+	0.565593i	$0.191353\pi$
$374$	−33.2842	−1.72108
$375$	−46.5777	−2.40526
$376$	−13.1787	−0.679640
$377$	−8.39746	−0.432491
$378$	12.3618	0.635824
$379$	19.8770	1.02101	0.510506	−	0.859874i	$-0.329458\pi$
$379$	19.8770	1.02101	0.510506	+	0.859874i	$0.329458\pi$
$380$	28.3794	1.45583
$381$	10.8668	0.556722
$382$	3.76241	0.192501
$383$	−3.26449	−0.166807	−0.0834037	−	0.996516i	$-0.526579\pi$
$383$	−3.26449	−0.166807	−0.0834037	+	0.996516i	$0.526579\pi$
$384$	1.73843	0.0887139
$385$	−44.4559	−2.26568
$386$	−19.2791	−0.981282
$387$	0.0508052	0.00258258
$388$	−14.4588	−0.734033
$389$	1.07557	0.0545334	0.0272667	−	0.999628i	$-0.491320\pi$
$389$	1.07557	0.0545334	0.0272667	+	0.999628i	$0.491320\pi$
$390$	−23.4861	−1.18926
$391$	55.4462	2.80403
$392$	1.29780	0.0655490
$393$	4.20937	0.212335
$394$	18.9201	0.953182
$395$	−20.9667	−1.05495
$396$	0.101217	0.00508637
$397$	7.64431	0.383657	0.191828	−	0.981428i	$-0.438558\pi$
$397$	7.64431	0.383657	0.191828	+	0.981428i	$0.438558\pi$
$398$	−8.76145	−0.439172
$399$	28.9327	1.44845
$400$	11.5800	0.579001
$401$	−11.0183	−0.550229	−0.275114	−	0.961411i	$-0.588716\pi$
$401$	−11.0183	−0.550229	−0.275114	+	0.961411i	$0.588716\pi$
$402$	−8.71462	−0.434646
$403$	−17.8130	−0.887328
$404$	−5.51473	−0.274368
$405$	−36.9152	−1.83433
$406$	6.04378	0.299948
$407$	39.1335	1.93978
$408$	12.6555	0.626541
$409$	−4.12639	−0.204037	−0.102018	−	0.994783i	$-0.532530\pi$
$409$	−4.12639	−0.204037	−0.102018	+	0.994783i	$0.532530\pi$
$410$	−37.1664	−1.83552
$411$	15.5787	0.768443
$412$	11.7681	0.579770
$413$	25.6074	1.26006
$414$	−0.168612	−0.00828684
$415$	2.40813	0.118211
$416$	3.31788	0.162672
$417$	0.942605	0.0461596
$418$	−31.8660	−1.55861
$419$	3.62912	0.177294	0.0886472	−	0.996063i	$-0.471746\pi$
$419$	3.62912	0.177294	0.0886472	+	0.996063i	$0.471746\pi$
$420$	16.9033	0.824795
$421$	18.3450	0.894080	0.447040	−	0.894514i	$-0.352478\pi$
$421$	18.3450	0.894080	0.447040	+	0.894514i	$0.352478\pi$
$422$	19.7342	0.960647
$423$	0.291751	0.0141854
$424$	1.72202	0.0836288
$425$	84.3009	4.08919
$426$	−24.7618	−1.19971
$427$	18.1747	0.879535
$428$	1.15542	0.0558493
$429$	26.3714	1.27322
$430$	−9.34461	−0.450637
$431$	−28.1164	−1.35432	−0.677159	−	0.735837i	$-0.736789\pi$
$431$	−28.1164	−1.35432	−0.677159	+	0.735837i	$0.736789\pi$
$432$	5.17680	0.249069
$433$	−32.9621	−1.58406	−0.792029	−	0.610484i	$-0.790975\pi$
$433$	−32.9621	−1.58406	−0.792029	+	0.610484i	$0.790975\pi$
$434$	12.8203	0.615393
$435$	−17.9158	−0.858999
$436$	−1.06109	−0.0508168
$437$	53.0836	2.53933
$438$	−0.977256	−0.0466951
$439$	−38.8435	−1.85390	−0.926951	−	0.375184i	$-0.877580\pi$
$439$	−38.8435	−1.85390	−0.926951	+	0.375184i	$0.877580\pi$
$440$	−18.6169	−0.887528
$441$	−0.0287309	−0.00136814
$442$	24.1537	1.14887
$443$	−3.92747	−0.186600	−0.0932999	−	0.995638i	$-0.529742\pi$
$443$	−3.92747	−0.186600	−0.0932999	+	0.995638i	$0.529742\pi$
$444$	−14.8796	−0.706154
$445$	−31.7007	−1.50276
$446$	−9.22087	−0.436621
$447$	38.6628	1.82869
$448$	−2.38793	−0.112819
$449$	−8.74236	−0.412578	−0.206289	−	0.978491i	$-0.566139\pi$
$449$	−8.74236	−0.412578	−0.206289	+	0.978491i	$0.566139\pi$
$450$	−0.256360	−0.0120849
$451$	41.7324	1.96510
$452$	14.5492	0.684339
$453$	−38.1100	−1.79056
$454$	3.74748	0.175878
$455$	32.2607	1.51241
$456$	12.1163	0.567396
$457$	−21.4303	−1.00247	−0.501233	−	0.865313i	$-0.667120\pi$
$457$	−21.4303	−1.00247	−0.501233	+	0.865313i	$0.667120\pi$
$458$	2.10228	0.0982332
$459$	37.6864	1.75905
$460$	31.0129	1.44598
$461$	−2.07110	−0.0964605	−0.0482303	−	0.998836i	$-0.515358\pi$
$461$	−2.07110	−0.0964605	−0.0482303	+	0.998836i	$0.515358\pi$
$462$	−18.9799	−0.883025
$463$	−21.6822	−1.00766	−0.503829	−	0.863804i	$-0.668076\pi$
$463$	−21.6822	−1.00766	−0.503829	+	0.863804i	$0.668076\pi$
$464$	2.53097	0.117497
$465$	−38.0037	−1.76238
$466$	−24.8250	−1.15000
$467$	−36.1603	−1.67330	−0.836650	−	0.547738i	$-0.815489\pi$
$467$	−36.1603	−1.67330	−0.836650	+	0.547738i	$0.815489\pi$
$468$	−0.0734515	−0.00339530
$469$	11.9705	0.552746
$470$	−53.6618	−2.47524
$471$	14.1525	0.652111
$472$	10.7237	0.493598
$473$	10.4926	0.482451
$474$	−8.95146	−0.411154
$475$	80.7088	3.70317
$476$	−17.3838	−0.796783
$477$	−0.0381223	−0.00174550
$478$	−3.60732	−0.164995
$479$	20.7692	0.948967	0.474483	−	0.880264i	$-0.342635\pi$
$479$	20.7692	0.948967	0.474483	+	0.880264i	$0.342635\pi$
$480$	7.07864	0.323094
$481$	−28.3984	−1.29486
$482$	−20.2807	−0.923762
$483$	31.6175	1.43865
$484$	9.90409	0.450186
$485$	−58.8741	−2.67333
$486$	−0.230061	−0.0104358
$487$	−27.1702	−1.23120	−0.615599	−	0.788059i	$-0.711086\pi$
$487$	−27.1702	−1.23120	−0.615599	+	0.788059i	$0.711086\pi$
$488$	7.61107	0.344537
$489$	−3.16734	−0.143232
$490$	5.28447	0.238728
$491$	−10.4613	−0.472113	−0.236056	−	0.971739i	$-0.575855\pi$
$491$	−10.4613	−0.472113	−0.236056	+	0.971739i	$0.575855\pi$
$492$	−15.8677	−0.715373
$493$	18.4251	0.829825
$494$	23.1245	1.04042
$495$	0.412143	0.0185245
$496$	5.36879	0.241066
$497$	34.0130	1.52569
$498$	1.02812	0.0460712
$499$	19.8841	0.890133	0.445066	−	0.895498i	$-0.353180\pi$
$499$	19.8841	0.890133	0.445066	+	0.895498i	$0.353180\pi$
$500$	26.7929	1.19822
$501$	−27.4898	−1.22815
$502$	22.7440	1.01511
$503$	26.5016	1.18165	0.590824	−	0.806801i	$-0.298803\pi$
$503$	26.5016	1.18165	0.590824	+	0.806801i	$0.298803\pi$
$504$	0.0528641	0.00235476
$505$	−22.4552	−0.999244
$506$	−34.8229	−1.54807
$507$	3.46238	0.153770
$508$	−6.25091	−0.277339
$509$	−26.8185	−1.18871	−0.594355	−	0.804203i	$-0.702593\pi$
$509$	−26.8185	−1.18871	−0.594355	+	0.804203i	$0.702593\pi$
$510$	51.5314	2.28185
$511$	1.34237	0.0593830
$512$	−1.00000	−0.0441942
$513$	36.0805	1.59299
$514$	17.8225	0.786117
$515$	47.9178	2.11151
$516$	−3.98957	−0.175631
$517$	60.2544	2.64998
$518$	20.4388	0.898028
$519$	13.8994	0.610116
$520$	13.5099	0.592450
$521$	23.2460	1.01843	0.509213	−	0.860640i	$-0.329936\pi$
$521$	23.2460	1.01843	0.509213	+	0.860640i	$0.329936\pi$
$522$	−0.0560309	−0.00245240
$523$	18.1821	0.795049	0.397525	−	0.917591i	$-0.369869\pi$
$523$	18.1821	0.795049	0.397525	+	0.917591i	$0.369869\pi$
$524$	−2.42136	−0.105778
$525$	48.0715	2.09801
$526$	−26.6279	−1.16103
$527$	39.0840	1.70252
$528$	−7.94827	−0.345904
$529$	35.0094	1.52215
$530$	7.01183	0.304574
$531$	−0.237402	−0.0103024
$532$	−16.6430	−0.721567
$533$	−30.2844	−1.31176
$534$	−13.5342	−0.585683
$535$	4.70471	0.203402
$536$	5.01292	0.216525
$537$	7.73212	0.333665
$538$	30.1938	1.30175
$539$	−5.93369	−0.255582
$540$	21.0792	0.907105
$541$	−19.7616	−0.849617	−0.424809	−	0.905283i	$-0.639659\pi$
$541$	−19.7616	−0.849617	−0.424809	+	0.905283i	$0.639659\pi$
$542$	−20.1446	−0.865286
$543$	13.3890	0.574575
$544$	−7.27985	−0.312121
$545$	−4.32059	−0.185074
$546$	13.7733	0.589444
$547$	−35.6462	−1.52412	−0.762060	−	0.647506i	$-0.775812\pi$
$547$	−35.6462	−1.52412	−0.762060	+	0.647506i	$0.775812\pi$
$548$	−8.96139	−0.382812
$549$	−0.168495	−0.00719117
$550$	−52.9450	−2.25758
$551$	17.6400	0.751489
$552$	13.2406	0.563556
$553$	12.2958	0.522872
$554$	0.493290	0.0209579
$555$	−60.5875	−2.57180
$556$	−0.542216	−0.0229951
$557$	1.24815	0.0528859	0.0264430	−	0.999650i	$-0.491582\pi$
$557$	1.24815	0.0528859	0.0264430	+	0.999650i	$0.491582\pi$
$558$	−0.118855	−0.00503152
$559$	−7.61429	−0.322050
$560$	−9.72330	−0.410884
$561$	−57.8622	−2.44295
$562$	27.6877	1.16794
$563$	27.9056	1.17608	0.588040	−	0.808832i	$-0.299900\pi$
$563$	27.9056	1.17608	0.588040	+	0.808832i	$0.299900\pi$
$564$	−22.9103	−0.964696
$565$	59.2424	2.49235
$566$	−17.4944	−0.735346
$567$	21.6488	0.909163
$568$	14.2437	0.597654
$569$	−14.5305	−0.609150	−0.304575	−	0.952488i	$-0.598514\pi$
$569$	−14.5305	−0.609150	−0.304575	+	0.952488i	$0.598514\pi$
$570$	49.3357	2.06644
$571$	−1.31868	−0.0551852	−0.0275926	−	0.999619i	$-0.508784\pi$
$571$	−1.31868	−0.0551852	−0.0275926	+	0.999619i	$0.508784\pi$
$572$	−15.1697	−0.634276
$573$	6.54068	0.273241
$574$	21.7961	0.909752
$575$	88.1980	3.67811
$576$	0.0221381	0.000922420 0
$577$	8.24906	0.343413	0.171706	−	0.985148i	$-0.445072\pi$
$577$	8.24906	0.343413	0.171706	+	0.985148i	$0.445072\pi$
$578$	−35.9962	−1.49725
$579$	−33.5154	−1.39285
$580$	10.3058	0.427923
$581$	−1.41224	−0.0585896
$582$	−25.1356	−1.04190
$583$	−7.87326	−0.326077
$584$	0.562149	0.0232619
$585$	−0.299084	−0.0123656
$586$	3.38270	0.139738
$587$	4.26028	0.175841	0.0879203	−	0.996128i	$-0.471978\pi$
$587$	4.26028	0.175841	0.0879203	+	0.996128i	$0.471978\pi$
$588$	2.25614	0.0930417
$589$	37.4186	1.54181
$590$	43.6653	1.79767
$591$	32.8913	1.35297
$592$	8.55921	0.351781
$593$	−2.96489	−0.121753	−0.0608766	−	0.998145i	$-0.519390\pi$
$593$	−2.96489	−0.121753	−0.0608766	+	0.998145i	$0.519390\pi$
$594$	−23.6689	−0.971145
$595$	−70.7842	−2.90187
$596$	−22.2401	−0.910988
$597$	−15.2312	−0.623370
$598$	25.2703	1.03338
$599$	33.0778	1.35152	0.675761	−	0.737121i	$-0.263815\pi$
$599$	33.0778	1.35152	0.675761	+	0.737121i	$0.263815\pi$
$600$	20.1311	0.821847
$601$	2.09323	0.0853846	0.0426923	−	0.999088i	$-0.486406\pi$
$601$	2.09323	0.0853846	0.0426923	+	0.999088i	$0.486406\pi$
$602$	5.48011	0.223353
$603$	−0.110977	−0.00451931
$604$	21.9221	0.891997
$605$	40.3280	1.63957
$606$	−9.58698	−0.389444
$607$	−39.5427	−1.60499	−0.802495	−	0.596659i	$-0.796495\pi$
$607$	−39.5427	−1.60499	−0.802495	+	0.596659i	$0.796495\pi$
$608$	−6.96966	−0.282657
$609$	10.5067	0.425752
$610$	30.9912	1.25480
$611$	−43.7254	−1.76894
$612$	0.161162	0.00651458
$613$	−6.65960	−0.268979	−0.134489	−	0.990915i	$-0.542939\pi$
$613$	−6.65960	−0.268979	−0.134489	+	0.990915i	$0.542939\pi$
$614$	20.6600	0.833769
$615$	−64.6112	−2.60537
$616$	10.9178	0.439892
$617$	32.9038	1.32466	0.662329	−	0.749213i	$-0.269568\pi$
$617$	32.9038	1.32466	0.662329	+	0.749213i	$0.269568\pi$
$618$	20.4579	0.822939
$619$	−2.84125	−0.114200	−0.0570998	−	0.998368i	$-0.518185\pi$
$619$	−2.84125	−0.114200	−0.0570998	+	0.998368i	$0.518185\pi$
$620$	21.8609	0.877956
$621$	39.4286	1.58221
$622$	6.65115	0.266687
$623$	18.5908	0.744823
$624$	5.76790	0.230901
$625$	51.1969	2.04788
$626$	8.33752	0.333234
$627$	−55.3967	−2.21233
$628$	−8.14095	−0.324859
$629$	62.3097	2.48445
$630$	0.215255	0.00857597
$631$	−34.2621	−1.36395	−0.681977	−	0.731374i	$-0.738880\pi$
$631$	−34.2621	−1.36395	−0.681977	+	0.731374i	$0.738880\pi$
$632$	5.14917	0.204823
$633$	34.3066	1.36356
$634$	20.7472	0.823975
$635$	−25.4528	−1.01006
$636$	2.99362	0.118705
$637$	4.30596	0.170608
$638$	−11.5719	−0.458134
$639$	−0.315329	−0.0124742
$640$	−4.07186	−0.160954
$641$	−43.8522	−1.73206	−0.866030	−	0.499992i	$-0.833336\pi$
$641$	−43.8522	−1.73206	−0.866030	+	0.499992i	$0.833336\pi$
$642$	2.00862	0.0792738
$643$	−12.8328	−0.506078	−0.253039	−	0.967456i	$-0.581430\pi$
$643$	−12.8328	−0.506078	−0.253039	+	0.967456i	$0.581430\pi$
$644$	−18.1874	−0.716684
$645$	−16.2449	−0.639644
$646$	−50.7380	−1.99626
$647$	−4.37468	−0.171986	−0.0859932	−	0.996296i	$-0.527406\pi$
$647$	−4.37468	−0.171986	−0.0859932	+	0.996296i	$0.527406\pi$
$648$	9.06592	0.356143
$649$	−49.0297	−1.92459
$650$	38.4211	1.50700
$651$	22.2871	0.873502
$652$	1.82196	0.0713533
$653$	15.5406	0.608151	0.304075	−	0.952648i	$-0.401653\pi$
$653$	15.5406	0.608151	0.304075	+	0.952648i	$0.401653\pi$
$654$	−1.84462	−0.0721305
$655$	−9.85945	−0.385241
$656$	9.12763	0.356374
$657$	−0.0124449	−0.000485522 0
$658$	31.4698	1.22682
$659$	−42.1126	−1.64048	−0.820238	−	0.572023i	$-0.806159\pi$
$659$	−42.1126	−1.64048	−0.820238	+	0.572023i	$0.806159\pi$
$660$	−32.3642	−1.25978
$661$	29.4406	1.14511	0.572553	−	0.819868i	$-0.305953\pi$
$661$	29.4406	1.14511	0.572553	+	0.819868i	$0.305953\pi$
$662$	−14.7031	−0.571453
$663$	41.9895	1.63074
$664$	−0.591408	−0.0229511
$665$	−67.7681	−2.62793
$666$	−0.189484	−0.00734237
$667$	19.2769	0.746404
$668$	15.8130	0.611824
$669$	−16.0298	−0.619749
$670$	20.4119	0.788581
$671$	−34.7986	−1.34338
$672$	−4.15124	−0.160138
$673$	−32.8780	−1.26735	−0.633677	−	0.773597i	$-0.718455\pi$
$673$	−32.8780	−1.26735	−0.633677	+	0.773597i	$0.718455\pi$
$674$	20.5633	0.792068
$675$	59.9475	2.30738
$676$	−1.99167	−0.0766028
$677$	−16.1654	−0.621286	−0.310643	−	0.950527i	$-0.600544\pi$
$677$	−16.1654	−0.621286	−0.310643	+	0.950527i	$0.600544\pi$
$678$	25.2928	0.971365
$679$	34.5265	1.32501
$680$	−29.6425	−1.13674
$681$	6.51474	0.249645
$682$	−24.5466	−0.939939
$683$	−17.7512	−0.679231	−0.339615	−	0.940564i	$-0.610297\pi$
$683$	−17.7512	−0.679231	−0.339615	+	0.940564i	$0.610297\pi$
$684$	0.154295	0.00589961
$685$	−36.4895	−1.39419
$686$	−19.8146	−0.756523
$687$	3.65467	0.139434
$688$	2.29493	0.0874932
$689$	5.71346	0.217666
$690$	53.9137	2.05246
$691$	−7.42356	−0.282406	−0.141203	−	0.989981i	$-0.545097\pi$
$691$	−7.42356	−0.282406	−0.141203	+	0.989981i	$0.545097\pi$
$692$	−7.99538	−0.303939
$693$	−0.241700	−0.00918142
$694$	−29.9834	−1.13815
$695$	−2.20783	−0.0837477
$696$	4.39992	0.166778
$697$	66.4478	2.51689
$698$	16.4215	0.621562
$699$	−43.1565	−1.63233
$700$	−27.6523	−1.04516
$701$	50.8533	1.92070	0.960351	−	0.278794i	$-0.0899345\pi$
$701$	50.8533	1.92070	0.960351	+	0.278794i	$0.0899345\pi$
$702$	17.1760	0.648267
$703$	59.6547	2.24992
$704$	4.57210	0.172317
$705$	−93.2873	−3.51340
$706$	−3.84501	−0.144709
$707$	13.1688	0.495263
$708$	18.6424	0.700623
$709$	19.5614	0.734643	0.367321	−	0.930094i	$-0.380275\pi$
$709$	19.5614	0.734643	0.367321	+	0.930094i	$0.380275\pi$
$710$	57.9985	2.17665
$711$	−0.113993	−0.00427506
$712$	7.78532	0.291767
$713$	40.8908	1.53137
$714$	−30.2204	−1.13097
$715$	−61.7688	−2.31002
$716$	−4.44776	−0.166221
$717$	−6.27107	−0.234197
$718$	15.2603	0.569510
$719$	5.22916	0.195015	0.0975074	−	0.995235i	$-0.468913\pi$
$719$	5.22916	0.195015	0.0975074	+	0.995235i	$0.468913\pi$
$720$	0.0901431	0.00335944
$721$	−28.1013	−1.04655
$722$	−29.5761	−1.10071
$723$	−35.2566	−1.31121
$724$	−7.70175	−0.286233
$725$	29.3087	1.08850
$726$	17.2176	0.639004
$727$	−29.8793	−1.10816	−0.554080	−	0.832463i	$-0.686930\pi$
$727$	−29.8793	−1.10816	−0.554080	+	0.832463i	$0.686930\pi$
$728$	−7.92286	−0.293641
$729$	26.7978	0.992512
$730$	2.28899	0.0847193
$731$	16.7067	0.617920
$732$	13.2313	0.489043
$733$	−49.8669	−1.84188	−0.920938	−	0.389710i	$-0.872575\pi$
$733$	−49.8669	−1.84188	−0.920938	+	0.389710i	$0.872575\pi$
$734$	11.7180	0.432519
$735$	9.18668	0.338856
$736$	−7.61639	−0.280744
$737$	−22.9196	−0.844254
$738$	−0.202068	−0.00743823
$739$	−27.8292	−1.02371	−0.511857	−	0.859070i	$-0.671042\pi$
$739$	−27.8292	−1.02371	−0.511857	+	0.859070i	$0.671042\pi$
$740$	34.8519	1.28118
$741$	40.2003	1.47679
$742$	−4.11206	−0.150959
$743$	6.47096	0.237396	0.118698	−	0.992930i	$-0.462128\pi$
$743$	6.47096	0.237396	0.118698	+	0.992930i	$0.462128\pi$
$744$	9.33326	0.342174
$745$	−90.5584	−3.31780
$746$	−31.8546	−1.16628
$747$	0.0130926	0.000479035 0
$748$	33.2842	1.21699
$749$	−2.75906	−0.100814
$750$	46.5777	1.70078
$751$	33.4418	1.22031	0.610155	−	0.792282i	$-0.291107\pi$
$751$	33.4418	1.22031	0.610155	+	0.792282i	$0.291107\pi$
$752$	13.1787	0.480578
$753$	39.5388	1.44087
$754$	8.39746	0.305817
$755$	89.2636	3.24864
$756$	−12.3618	−0.449595
$757$	−27.7711	−1.00936	−0.504680	−	0.863307i	$-0.668389\pi$
$757$	−27.7711	−1.00936	−0.504680	+	0.863307i	$0.668389\pi$
$758$	−19.8770	−0.721965
$759$	−60.5372	−2.19736
$760$	−28.3794	−1.02943
$761$	−41.0460	−1.48792	−0.743958	−	0.668226i	$-0.767054\pi$
$761$	−41.0460	−1.48792	−0.743958	+	0.668226i	$0.767054\pi$
$762$	−10.8668	−0.393662
$763$	2.53380	0.0917296
$764$	−3.76241	−0.136119
$765$	0.656228	0.0237260
$766$	3.26449	0.117951
$767$	35.5799	1.28472
$768$	−1.73843	−0.0627302
$769$	−30.1552	−1.08742	−0.543712	−	0.839272i	$-0.682982\pi$
$769$	−30.1552	−1.08742	−0.543712	+	0.839272i	$0.682982\pi$
$770$	44.4559	1.60208
$771$	30.9832	1.11583
$772$	19.2791	0.693871
$773$	28.6642	1.03098	0.515490	−	0.856896i	$-0.327610\pi$
$773$	28.6642	1.03098	0.515490	+	0.856896i	$0.327610\pi$
$774$	−0.0508052	−0.00182616
$775$	62.1707	2.23324
$776$	14.4588	0.519040
$777$	35.5314	1.27468
$778$	−1.07557	−0.0385609
$779$	63.6164	2.27929
$780$	23.4861	0.840936
$781$	−65.1238	−2.33031
$782$	−55.4462	−1.98275
$783$	13.1023	0.468240
$784$	−1.29780	−0.0463501
$785$	−33.1488	−1.18313
$786$	−4.20937	−0.150143
$787$	15.8650	0.565527	0.282764	−	0.959190i	$-0.408749\pi$
$787$	15.8650	0.565527	0.282764	+	0.959190i	$0.408749\pi$
$788$	−18.9201	−0.674001
$789$	−46.2908	−1.64800
$790$	20.9667	0.745961
$791$	−34.7425	−1.23530
$792$	−0.101217	−0.00359661
$793$	25.2526	0.896747
$794$	−7.64431	−0.271286
$795$	12.1896	0.432320
$796$	8.76145	0.310541
$797$	−18.2564	−0.646673	−0.323337	−	0.946284i	$-0.604805\pi$
$797$	−18.2564	−0.646673	−0.323337	+	0.946284i	$0.604805\pi$
$798$	−28.9327	−1.02421
$799$	95.9390	3.39408
$800$	−11.5800	−0.409416
$801$	−0.172352	−0.00608976
$802$	11.0183	0.389071
$803$	−2.57020	−0.0907004
$804$	8.71462	0.307341
$805$	−74.0565	−2.61015
$806$	17.8130	0.627436
$807$	52.4898	1.84773
$808$	5.51473	0.194008
$809$	40.5243	1.42476	0.712379	−	0.701795i	$-0.247618\pi$
$809$	40.5243	1.42476	0.712379	+	0.701795i	$0.247618\pi$
$810$	36.9152	1.29707
$811$	6.46438	0.226995	0.113498	−	0.993538i	$-0.463795\pi$
$811$	6.46438	0.226995	0.113498	+	0.993538i	$0.463795\pi$
$812$	−6.04378	−0.212095
$813$	−35.0200	−1.22821
$814$	−39.1335	−1.37163
$815$	7.41874	0.259867
$816$	−12.6555	−0.443031
$817$	15.9948	0.559589
$818$	4.12639	0.144276
$819$	0.175397	0.00612886
$820$	37.1664	1.29791
$821$	−8.72261	−0.304421	−0.152211	−	0.988348i	$-0.548639\pi$
$821$	−8.72261	−0.304421	−0.152211	+	0.988348i	$0.548639\pi$
$822$	−15.5787	−0.543371
$823$	3.79429	0.132261	0.0661303	−	0.997811i	$-0.478935\pi$
$823$	3.79429	0.132261	0.0661303	+	0.997811i	$0.478935\pi$
$824$	−11.7681	−0.409960
$825$	−92.0412	−3.20446
$826$	−25.6074	−0.890995
$827$	−11.3093	−0.393264	−0.196632	−	0.980477i	$-0.563000\pi$
$827$	−11.3093	−0.393264	−0.196632	+	0.980477i	$0.563000\pi$
$828$	0.168612	0.00585968
$829$	−17.1890	−0.596999	−0.298500	−	0.954410i	$-0.596486\pi$
$829$	−17.1890	−0.596999	−0.298500	+	0.954410i	$0.596486\pi$
$830$	−2.40813	−0.0835875
$831$	0.857550	0.0297481
$832$	−3.31788	−0.115027
$833$	−9.44782	−0.327347
$834$	−0.942605	−0.0326397
$835$	64.3884	2.22825
$836$	31.8660	1.10211
$837$	27.7932	0.960672
$838$	−3.62912	−0.125366
$839$	−6.04137	−0.208571	−0.104286	−	0.994547i	$-0.533256\pi$
$839$	−6.04137	−0.208571	−0.104286	+	0.994547i	$0.533256\pi$
$840$	−16.9033	−0.583218
$841$	−22.5942	−0.779110
$842$	−18.3450	−0.632210
$843$	48.1332	1.65780
$844$	−19.7342	−0.679280
$845$	−8.10981	−0.278986
$846$	−0.291751	−0.0100306
$847$	−23.6502	−0.812632
$848$	−1.72202	−0.0591345
$849$	−30.4129	−1.04377
$850$	−84.3009	−2.89150
$851$	65.1903	2.23469
$852$	24.7618	0.848324
$853$	38.1037	1.30465	0.652323	−	0.757941i	$-0.273795\pi$
$853$	38.1037	1.30465	0.652323	+	0.757941i	$0.273795\pi$
$854$	−18.1747	−0.621925
$855$	0.628267	0.0214863
$856$	−1.15542	−0.0394914
$857$	37.8607	1.29330	0.646648	−	0.762788i	$-0.276170\pi$
$857$	37.8607	1.29330	0.646648	+	0.762788i	$0.276170\pi$
$858$	−26.3714	−0.900305
$859$	−45.0060	−1.53558	−0.767792	−	0.640699i	$-0.778645\pi$
$859$	−45.0060	−1.53558	−0.767792	+	0.640699i	$0.778645\pi$
$860$	9.34461	0.318649
$861$	37.8910	1.29132
$862$	28.1164	0.957647
$863$	−37.8986	−1.29008	−0.645042	−	0.764148i	$-0.723160\pi$
$863$	−37.8986	−1.29008	−0.645042	+	0.764148i	$0.723160\pi$
$864$	−5.17680	−0.176118
$865$	−32.5561	−1.10694
$866$	32.9621	1.12010
$867$	−62.5769	−2.12522
$868$	−12.8203	−0.435148
$869$	−23.5425	−0.798625
$870$	17.9158	0.607404
$871$	16.6323	0.563563
$872$	1.06109	0.0359329
$873$	−0.320089	−0.0108334
$874$	−53.0836	−1.79558
$875$	−63.9796	−2.16291
$876$	0.977256	0.0330184
$877$	27.7815	0.938114	0.469057	−	0.883168i	$-0.344594\pi$
$877$	27.7815	0.938114	0.469057	+	0.883168i	$0.344594\pi$
$878$	38.8435	1.31091
$879$	5.88059	0.198347
$880$	18.6169	0.627577
$881$	12.6032	0.424613	0.212307	−	0.977203i	$-0.431902\pi$
$881$	12.6032	0.424613	0.212307	+	0.977203i	$0.431902\pi$
$882$	0.0287309	0.000967419 0
$883$	51.2580	1.72497	0.862485	−	0.506083i	$-0.168907\pi$
$883$	51.2580	1.72497	0.862485	+	0.506083i	$0.168907\pi$
$884$	−24.1537	−0.812376
$885$	75.9091	2.55166
$886$	3.92747	0.131946
$887$	35.0683	1.17748	0.588739	−	0.808323i	$-0.299625\pi$
$887$	35.0683	1.17748	0.588739	+	0.808323i	$0.299625\pi$
$888$	14.8796	0.499326
$889$	14.9267	0.500626
$890$	31.7007	1.06261
$891$	−41.4503	−1.38864
$892$	9.22087	0.308738
$893$	91.8511	3.07368
$894$	−38.6628	−1.29308
$895$	−18.1106	−0.605372
$896$	2.38793	0.0797750
$897$	43.9306	1.46680
$898$	8.74236	0.291736
$899$	13.5882	0.453193
$900$	0.256360	0.00854532
$901$	−12.5361	−0.417637
$902$	−41.7324	−1.38954
$903$	9.52679	0.317032
$904$	−14.5492	−0.483900
$905$	−31.3604	−1.04246
$906$	38.1100	1.26612
$907$	7.80443	0.259142	0.129571	−	0.991570i	$-0.458640\pi$
$907$	7.80443	0.259142	0.129571	+	0.991570i	$0.458640\pi$
$908$	−3.74748	−0.124365
$909$	−0.122086	−0.00404932
$910$	−32.2607	−1.06943
$911$	18.8363	0.624074	0.312037	−	0.950070i	$-0.398989\pi$
$911$	18.8363	0.624074	0.312037	+	0.950070i	$0.398989\pi$
$912$	−12.1163	−0.401209
$913$	2.70398	0.0894886
$914$	21.4303	0.708850
$915$	53.8760	1.78109
$916$	−2.10228	−0.0694614
$917$	5.78204	0.190940
$918$	−37.6864	−1.24384
$919$	48.3188	1.59389	0.796945	−	0.604052i	$-0.206448\pi$
$919$	48.3188	1.59389	0.796945	+	0.604052i	$0.206448\pi$
$920$	−31.0129	−1.02246
$921$	35.9159	1.18347
$922$	2.07110	0.0682079
$923$	47.2590	1.55555
$924$	18.9799	0.624393
$925$	99.1159	3.25891
$926$	21.6822	0.712521
$927$	0.260522	0.00855667
$928$	−2.53097	−0.0830832
$929$	−34.3210	−1.12604	−0.563018	−	0.826445i	$-0.690360\pi$
$929$	−34.3210	−1.12604	−0.563018	+	0.826445i	$0.690360\pi$
$930$	38.0037	1.24619
$931$	−9.04524	−0.296446
$932$	24.8250	0.813170
$933$	11.5626	0.378541
$934$	36.1603	1.18320
$935$	135.529	4.43226
$936$	0.0734515	0.00240084
$937$	12.1778	0.397830	0.198915	−	0.980017i	$-0.436258\pi$
$937$	12.1778	0.397830	0.198915	+	0.980017i	$0.436258\pi$
$938$	−11.9705	−0.390851
$939$	14.4942	0.473000
$940$	53.6618	1.75026
$941$	42.8898	1.39817	0.699083	−	0.715040i	$-0.253592\pi$
$941$	42.8898	1.39817	0.699083	+	0.715040i	$0.253592\pi$
$942$	−14.1525	−0.461112
$943$	69.5196	2.26387
$944$	−10.7237	−0.349026
$945$	−50.3356	−1.63742
$946$	−10.4926	−0.341145
$947$	−11.8894	−0.386354	−0.193177	−	0.981164i	$-0.561879\pi$
$947$	−11.8894	−0.386354	−0.193177	+	0.981164i	$0.561879\pi$
$948$	8.95146	0.290730
$949$	1.86514	0.0605451
$950$	−80.7088	−2.61854
$951$	36.0675	1.16957
$952$	17.3838	0.563411
$953$	−20.3147	−0.658059	−0.329029	−	0.944320i	$-0.606722\pi$
$953$	−20.3147	−0.658059	−0.329029	+	0.944320i	$0.606722\pi$
$954$	0.0381223	0.00123425
$955$	−15.3200	−0.495743
$956$	3.60732	0.116669
$957$	−20.1169	−0.650286
$958$	−20.7692	−0.671021
$959$	21.3992	0.691015
$960$	−7.07864	−0.228462
$961$	−2.17613	−0.0701976
$962$	28.3984	0.915602
$963$	0.0255788	0.000824265 0
$964$	20.2807	0.653198
$965$	78.5019	2.52707
$966$	−31.6175	−1.01728
$967$	−9.76555	−0.314039	−0.157019	−	0.987596i	$-0.550189\pi$
$967$	−9.76555	−0.314039	−0.157019	+	0.987596i	$0.550189\pi$
$968$	−9.90409	−0.318329
$969$	−88.2045	−2.83354
$970$	58.8741	1.89033
$971$	−27.3994	−0.879290	−0.439645	−	0.898172i	$-0.644896\pi$
$971$	−27.3994	−0.879290	−0.439645	+	0.898172i	$0.644896\pi$
$972$	0.230061	0.00737921
$973$	1.29477	0.0415085
$974$	27.1702	0.870589
$975$	66.7925	2.13907
$976$	−7.61107	−0.243624
$977$	−41.6786	−1.33342	−0.666709	−	0.745318i	$-0.732297\pi$
$977$	−41.6786	−1.33342	−0.666709	+	0.745318i	$0.732297\pi$
$978$	3.16734	0.101280
$979$	−35.5952	−1.13763
$980$	−5.28447	−0.168806
$981$	−0.0234904	−0.000749992 0
$982$	10.4613	0.333834
$983$	−34.6888	−1.10640	−0.553200	−	0.833048i	$-0.686594\pi$
$983$	−34.6888	−1.10640	−0.553200	+	0.833048i	$0.686594\pi$
$984$	15.8677	0.505845
$985$	−77.0400	−2.45470
$986$	−18.4251	−0.586775
$987$	54.7080	1.74138
$988$	−23.1245	−0.735688
$989$	17.4791	0.555801
$990$	−0.412143	−0.0130988
$991$	18.8669	0.599326	0.299663	−	0.954045i	$-0.403126\pi$
$991$	18.8669	0.599326	0.299663	+	0.954045i	$0.403126\pi$
$992$	−5.36879	−0.170459
$993$	−25.5603	−0.811133
$994$	−34.0130	−1.07883
$995$	35.6754	1.13099
$996$	−1.02812	−0.0325773
$997$	12.5787	0.398371	0.199186	−	0.979962i	$-0.436170\pi$
$997$	12.5787	0.398371	0.199186	+	0.979962i	$0.436170\pi$
$998$	−19.8841	−0.629419
$999$	44.3093	1.40189

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.e.1.17	✓	77

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.e.1.17	✓	77	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:	\( N \)	\(=\)	\( 8002 = 2 \cdot 4001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8002.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.73843 q^{3} +1.00000 q^{4} +4.07186 q^{5} +1.73843 q^{6} -2.38793 q^{7} -1.00000 q^{8} +0.0221381 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.73843 q^{3} +1.00000 q^{4} +4.07186 q^{5} +1.73843 q^{6} -2.38793 q^{7} -1.00000 q^{8} +0.0221381 q^{9} -4.07186 q^{10} +4.57210 q^{11} -1.73843 q^{12} -3.31788 q^{13} +2.38793 q^{14} -7.07864 q^{15} +1.00000 q^{16} +7.27985 q^{17} -0.0221381 q^{18} +6.96966 q^{19} +4.07186 q^{20} +4.15124 q^{21} -4.57210 q^{22} +7.61639 q^{23} +1.73843 q^{24} +11.5800 q^{25} +3.31788 q^{26} +5.17680 q^{27} -2.38793 q^{28} +2.53097 q^{29} +7.07864 q^{30} +5.36879 q^{31} -1.00000 q^{32} -7.94827 q^{33} -7.27985 q^{34} -9.72330 q^{35} +0.0221381 q^{36} +8.55921 q^{37} -6.96966 q^{38} +5.76790 q^{39} -4.07186 q^{40} +9.12763 q^{41} -4.15124 q^{42} +2.29493 q^{43} +4.57210 q^{44} +0.0901431 q^{45} -7.61639 q^{46} +13.1787 q^{47} -1.73843 q^{48} -1.29780 q^{49} -11.5800 q^{50} -12.6555 q^{51} -3.31788 q^{52} -1.72202 q^{53} -5.17680 q^{54} +18.6169 q^{55} +2.38793 q^{56} -12.1163 q^{57} -2.53097 q^{58} -10.7237 q^{59} -7.07864 q^{60} -7.61107 q^{61} -5.36879 q^{62} -0.0528641 q^{63} +1.00000 q^{64} -13.5099 q^{65} +7.94827 q^{66} -5.01292 q^{67} +7.27985 q^{68} -13.2406 q^{69} +9.72330 q^{70} -14.2437 q^{71} -0.0221381 q^{72} -0.562149 q^{73} -8.55921 q^{74} -20.1311 q^{75} +6.96966 q^{76} -10.9178 q^{77} -5.76790 q^{78} -5.14917 q^{79} +4.07186 q^{80} -9.06592 q^{81} -9.12763 q^{82} +0.591408 q^{83} +4.15124 q^{84} +29.6425 q^{85} -2.29493 q^{86} -4.39992 q^{87} -4.57210 q^{88} -7.78532 q^{89} -0.0901431 q^{90} +7.92286 q^{91} +7.61639 q^{92} -9.33326 q^{93} -13.1787 q^{94} +28.3794 q^{95} +1.73843 q^{96} -14.4588 q^{97} +1.29780 q^{98} +0.101217 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9}+O(q^{10}) \) 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9} - 18 q^{10} + 30 q^{11} + 10 q^{12} - 2 q^{13} - 21 q^{14} + 21 q^{15} + 77 q^{16} + 60 q^{17} - 71 q^{18} - 3 q^{19} + 18 q^{20} + 10 q^{21} - 30 q^{22} + 53 q^{23} - 10 q^{24} + 59 q^{25} + 2 q^{26} + 43 q^{27} + 21 q^{28} + 30 q^{29} - 21 q^{30} + 22 q^{31} - 77 q^{32} + 31 q^{33} - 60 q^{34} + 41 q^{35} + 71 q^{36} - 3 q^{37} + 3 q^{38} + 44 q^{39} - 18 q^{40} + 48 q^{41} - 10 q^{42} + 21 q^{43} + 30 q^{44} + 33 q^{45} - 53 q^{46} + 107 q^{47} + 10 q^{48} + 24 q^{49} - 59 q^{50} + 18 q^{51} - 2 q^{52} + 42 q^{53} - 43 q^{54} + 49 q^{55} - 21 q^{56} + 32 q^{57} - 30 q^{58} + 42 q^{59} + 21 q^{60} - 31 q^{61} - 22 q^{62} + 109 q^{63} + 77 q^{64} + 39 q^{65} - 31 q^{66} - q^{67} + 60 q^{68} - 33 q^{69} - 41 q^{70} + 58 q^{71} - 71 q^{72} + 35 q^{73} + 3 q^{74} + 34 q^{75} - 3 q^{76} + 86 q^{77} - 44 q^{78} + 25 q^{79} + 18 q^{80} + 53 q^{81} - 48 q^{82} + 107 q^{83} + 10 q^{84} + 21 q^{85} - 21 q^{86} + 100 q^{87} - 30 q^{88} + 34 q^{89} - 33 q^{90} - 51 q^{91} + 53 q^{92} + 48 q^{93} - 107 q^{94} + 118 q^{95} - 10 q^{96} - 13 q^{97} - 24 q^{98} + 63 q^{99}+O(q^{100}) \) 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9 - 18 * q^10 + 30 * q^11 + 10 * q^12 - 2 * q^13 - 21 * q^14 + 21 * q^15 + 77 * q^16 + 60 * q^17 - 71 * q^18 - 3 * q^19 + 18 * q^20 + 10 * q^21 - 30 * q^22 + 53 * q^23 - 10 * q^24 + 59 * q^25 + 2 * q^26 + 43 * q^27 + 21 * q^28 + 30 * q^29 - 21 * q^30 + 22 * q^31 - 77 * q^32 + 31 * q^33 - 60 * q^34 + 41 * q^35 + 71 * q^36 - 3 * q^37 + 3 * q^38 + 44 * q^39 - 18 * q^40 + 48 * q^41 - 10 * q^42 + 21 * q^43 + 30 * q^44 + 33 * q^45 - 53 * q^46 + 107 * q^47 + 10 * q^48 + 24 * q^49 - 59 * q^50 + 18 * q^51 - 2 * q^52 + 42 * q^53 - 43 * q^54 + 49 * q^55 - 21 * q^56 + 32 * q^57 - 30 * q^58 + 42 * q^59 + 21 * q^60 - 31 * q^61 - 22 * q^62 + 109 * q^63 + 77 * q^64 + 39 * q^65 - 31 * q^66 - q^67 + 60 * q^68 - 33 * q^69 - 41 * q^70 + 58 * q^71 - 71 * q^72 + 35 * q^73 + 3 * q^74 + 34 * q^75 - 3 * q^76 + 86 * q^77 - 44 * q^78 + 25 * q^79 + 18 * q^80 + 53 * q^81 - 48 * q^82 + 107 * q^83 + 10 * q^84 + 21 * q^85 - 21 * q^86 + 100 * q^87 - 30 * q^88 + 34 * q^89 - 33 * q^90 - 51 * q^91 + 53 * q^92 + 48 * q^93 - 107 * q^94 + 118 * q^95 - 10 * q^96 - 13 * q^97 - 24 * q^98 + 63 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more