Properties

Label 8005.2.a.e
Level 8005
Weight 2
Character orbit 8005.a
Self dual Yes
Analytic conductor 63.920
Analytic rank 1
Dimension 126
CM No

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Newspace parameters

Level: \( N \) = \( 8005 = 5 \cdot 1601 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8005.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(63.920246818\)
Analytic rank: \(1\)
Dimension: \(126\)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \(126q \) \(\mathstrut -\mathstrut 15q^{2} \) \(\mathstrut -\mathstrut 46q^{3} \) \(\mathstrut +\mathstrut 119q^{4} \) \(\mathstrut +\mathstrut 126q^{5} \) \(\mathstrut -\mathstrut 10q^{6} \) \(\mathstrut -\mathstrut 60q^{7} \) \(\mathstrut -\mathstrut 57q^{8} \) \(\mathstrut +\mathstrut 112q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(126q \) \(\mathstrut -\mathstrut 15q^{2} \) \(\mathstrut -\mathstrut 46q^{3} \) \(\mathstrut +\mathstrut 119q^{4} \) \(\mathstrut +\mathstrut 126q^{5} \) \(\mathstrut -\mathstrut 10q^{6} \) \(\mathstrut -\mathstrut 60q^{7} \) \(\mathstrut -\mathstrut 57q^{8} \) \(\mathstrut +\mathstrut 112q^{9} \) \(\mathstrut -\mathstrut 15q^{10} \) \(\mathstrut -\mathstrut 35q^{11} \) \(\mathstrut -\mathstrut 86q^{12} \) \(\mathstrut -\mathstrut 36q^{13} \) \(\mathstrut -\mathstrut 31q^{14} \) \(\mathstrut -\mathstrut 46q^{15} \) \(\mathstrut +\mathstrut 101q^{16} \) \(\mathstrut -\mathstrut 62q^{17} \) \(\mathstrut -\mathstrut 45q^{18} \) \(\mathstrut -\mathstrut 29q^{19} \) \(\mathstrut +\mathstrut 119q^{20} \) \(\mathstrut -\mathstrut 16q^{21} \) \(\mathstrut -\mathstrut 67q^{22} \) \(\mathstrut -\mathstrut 107q^{23} \) \(\mathstrut -\mathstrut 9q^{24} \) \(\mathstrut +\mathstrut 126q^{25} \) \(\mathstrut -\mathstrut 53q^{26} \) \(\mathstrut -\mathstrut 181q^{27} \) \(\mathstrut -\mathstrut 100q^{28} \) \(\mathstrut -\mathstrut 39q^{29} \) \(\mathstrut -\mathstrut 10q^{30} \) \(\mathstrut -\mathstrut 29q^{31} \) \(\mathstrut -\mathstrut 99q^{32} \) \(\mathstrut -\mathstrut 72q^{33} \) \(\mathstrut -\mathstrut 18q^{34} \) \(\mathstrut -\mathstrut 60q^{35} \) \(\mathstrut +\mathstrut 93q^{36} \) \(\mathstrut -\mathstrut 72q^{37} \) \(\mathstrut -\mathstrut 93q^{38} \) \(\mathstrut -\mathstrut 8q^{39} \) \(\mathstrut -\mathstrut 57q^{40} \) \(\mathstrut -\mathstrut 28q^{41} \) \(\mathstrut -\mathstrut 22q^{42} \) \(\mathstrut -\mathstrut 103q^{43} \) \(\mathstrut -\mathstrut 47q^{44} \) \(\mathstrut +\mathstrut 112q^{45} \) \(\mathstrut +\mathstrut q^{46} \) \(\mathstrut -\mathstrut 130q^{47} \) \(\mathstrut -\mathstrut 134q^{48} \) \(\mathstrut +\mathstrut 116q^{49} \) \(\mathstrut -\mathstrut 15q^{50} \) \(\mathstrut -\mathstrut 46q^{51} \) \(\mathstrut -\mathstrut 117q^{52} \) \(\mathstrut -\mathstrut 103q^{53} \) \(\mathstrut +\mathstrut 11q^{54} \) \(\mathstrut -\mathstrut 35q^{55} \) \(\mathstrut -\mathstrut 84q^{56} \) \(\mathstrut -\mathstrut 70q^{57} \) \(\mathstrut -\mathstrut 77q^{58} \) \(\mathstrut -\mathstrut 219q^{59} \) \(\mathstrut -\mathstrut 86q^{60} \) \(\mathstrut -\mathstrut 4q^{61} \) \(\mathstrut -\mathstrut 77q^{62} \) \(\mathstrut -\mathstrut 145q^{63} \) \(\mathstrut +\mathstrut 51q^{64} \) \(\mathstrut -\mathstrut 36q^{65} \) \(\mathstrut +\mathstrut 16q^{66} \) \(\mathstrut -\mathstrut 150q^{67} \) \(\mathstrut -\mathstrut 130q^{68} \) \(\mathstrut -\mathstrut 21q^{69} \) \(\mathstrut -\mathstrut 31q^{70} \) \(\mathstrut -\mathstrut 92q^{71} \) \(\mathstrut -\mathstrut 115q^{72} \) \(\mathstrut -\mathstrut 79q^{73} \) \(\mathstrut -\mathstrut 57q^{74} \) \(\mathstrut -\mathstrut 46q^{75} \) \(\mathstrut -\mathstrut 38q^{76} \) \(\mathstrut -\mathstrut 75q^{77} \) \(\mathstrut -\mathstrut 23q^{78} \) \(\mathstrut -\mathstrut 20q^{79} \) \(\mathstrut +\mathstrut 101q^{80} \) \(\mathstrut +\mathstrut 142q^{81} \) \(\mathstrut -\mathstrut 63q^{82} \) \(\mathstrut -\mathstrut 243q^{83} \) \(\mathstrut -\mathstrut 2q^{84} \) \(\mathstrut -\mathstrut 62q^{85} \) \(\mathstrut -\mathstrut 14q^{86} \) \(\mathstrut -\mathstrut 107q^{87} \) \(\mathstrut -\mathstrut 121q^{88} \) \(\mathstrut -\mathstrut 84q^{89} \) \(\mathstrut -\mathstrut 45q^{90} \) \(\mathstrut -\mathstrut 82q^{91} \) \(\mathstrut -\mathstrut 228q^{92} \) \(\mathstrut -\mathstrut 149q^{93} \) \(\mathstrut -\mathstrut 29q^{95} \) \(\mathstrut +\mathstrut 38q^{96} \) \(\mathstrut -\mathstrut 85q^{97} \) \(\mathstrut -\mathstrut 48q^{98} \) \(\mathstrut -\mathstrut 33q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.77468 −2.03994 5.69885 1.00000 5.66018 −0.694570 −10.2631 1.16135 −2.77468
1.2 −2.76854 −3.01214 5.66482 1.00000 8.33923 −2.50261 −10.1462 6.07298 −2.76854
1.3 −2.76679 0.0783895 5.65511 1.00000 −0.216887 3.44587 −10.1129 −2.99386 −2.76679
1.4 −2.75799 0.297267 5.60652 1.00000 −0.819859 −0.723666 −9.94674 −2.91163 −2.75799
1.5 −2.69183 −0.785166 5.24594 1.00000 2.11353 −4.09093 −8.73752 −2.38351 −2.69183
1.6 −2.68284 2.48419 5.19763 1.00000 −6.66470 0.389624 −8.57874 3.17122 −2.68284
1.7 −2.65121 2.20817 5.02889 1.00000 −5.85432 1.13274 −8.03022 1.87603 −2.65121
1.8 −2.60227 −2.65544 4.77178 1.00000 6.91016 2.75161 −7.21291 4.05137 −2.60227
1.9 −2.59696 −3.31155 4.74420 1.00000 8.59996 3.53678 −7.12657 7.96636 −2.59696
1.10 −2.55329 −1.12143 4.51928 1.00000 2.86335 −4.52576 −6.43246 −1.74239 −2.55329
1.11 −2.45517 1.07973 4.02786 1.00000 −2.65092 3.17620 −4.97873 −1.83418 −2.45517
1.12 −2.44890 0.480075 3.99711 1.00000 −1.17566 −2.53410 −4.89071 −2.76953 −2.44890
1.13 −2.42783 −3.02578 3.89435 1.00000 7.34609 −3.63249 −4.59917 6.15537 −2.42783
1.14 −2.42700 2.14529 3.89034 1.00000 −5.20661 0.410687 −4.58786 1.60225 −2.42700
1.15 −2.36778 2.79221 3.60639 1.00000 −6.61135 −4.83880 −3.80359 4.79644 −2.36778
1.16 −2.32335 0.155880 3.39795 1.00000 −0.362164 0.705093 −3.24794 −2.97570 −2.32335
1.17 −2.30625 1.53084 3.31877 1.00000 −3.53050 −2.53384 −3.04142 −0.656523 −2.30625
1.18 −2.30618 −3.20957 3.31845 1.00000 7.40183 −1.98846 −3.04058 7.30133 −2.30618
1.19 −2.23772 −0.862309 3.00741 1.00000 1.92961 4.70956 −2.25429 −2.25642 −2.23772
1.20 −2.18184 −0.884770 2.76044 1.00000 1.93043 1.24169 −1.65916 −2.21718 −2.18184
See next 80 embeddings (of 126 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.126
Significant digits:
Format:

Inner twists

This newform does not have CM; other inner twists have not been computed.

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(1601\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8005))\):

\(T_{2}^{126} + \cdots\)
\(T_{3}^{126} + \cdots\)