Properties

Label 4005.2.a.r
Level 4005
Weight 2
Character orbit 4005.a
Self dual Yes
Analytic conductor 31.980
Analytic rank 0
Dimension 10
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4005.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(31.9800860095\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{2} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} - q^{5} + ( 1 + \beta_{2} + \beta_{3} + \beta_{7} ) q^{7} + ( -2 + \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{2} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} - q^{5} + ( 1 + \beta_{2} + \beta_{3} + \beta_{7} ) q^{7} + ( -2 + \beta_{1} + \beta_{3} ) q^{8} + ( 1 - \beta_{1} ) q^{10} + ( 1 + \beta_{2} + \beta_{5} - \beta_{9} ) q^{11} + ( 3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{13} + ( 1 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{6} - \beta_{9} ) q^{14} + ( 3 - 3 \beta_{1} + \beta_{2} + \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{16} + ( -1 + \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{17} + ( 1 + \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} ) q^{19} + ( -2 + \beta_{1} - \beta_{2} ) q^{20} + ( 1 + \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} ) q^{22} + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} ) q^{23} + q^{25} + ( 2 - 2 \beta_{1} + 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{26} + ( 1 + \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} ) q^{28} + ( 1 - \beta_{1} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{29} + ( 1 + \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} ) q^{31} + ( -3 + 4 \beta_{1} - \beta_{2} - 2 \beta_{4} - 3 \beta_{5} - \beta_{6} + \beta_{7} ) q^{32} + ( -3 \beta_{1} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{34} + ( -1 - \beta_{2} - \beta_{3} - \beta_{7} ) q^{35} + ( -2 + \beta_{1} - \beta_{2} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{37} + ( -4 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + \beta_{4} + 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} ) q^{38} + ( 2 - \beta_{1} - \beta_{3} ) q^{40} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{41} + ( -\beta_{1} - \beta_{2} + 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} ) q^{43} + ( 3 - 2 \beta_{1} + 5 \beta_{2} - \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} - \beta_{9} ) q^{44} + ( 1 - 4 \beta_{1} + \beta_{2} + \beta_{4} + 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + \beta_{9} ) q^{46} + ( -2 + \beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{47} + ( 4 + 2 \beta_{1} + \beta_{3} - \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{49} + ( -1 + \beta_{1} ) q^{50} + ( -5 + 4 \beta_{1} - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} - 4 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{52} + ( -\beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{53} + ( -1 - \beta_{2} - \beta_{5} + \beta_{9} ) q^{55} + ( 5 - \beta_{1} + \beta_{3} + \beta_{4} - 3 \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{56} + ( -5 + 3 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{58} + ( 3 - \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} - 2 \beta_{8} ) q^{59} + ( \beta_{1} + 2 \beta_{2} - \beta_{4} + \beta_{6} - \beta_{8} ) q^{61} + ( 3 - 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} + 3 \beta_{9} ) q^{62} + ( 1 - 4 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} - \beta_{7} + \beta_{9} ) q^{64} + ( -3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} ) q^{65} + ( -1 + 2 \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{5} + 2 \beta_{7} + \beta_{9} ) q^{67} + ( -1 + 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - 3 \beta_{5} + \beta_{7} - \beta_{9} ) q^{68} + ( -1 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} + \beta_{9} ) q^{70} + ( \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{6} + \beta_{8} - 2 \beta_{9} ) q^{71} + ( -\beta_{1} + 2 \beta_{5} + \beta_{6} + \beta_{8} + 2 \beta_{9} ) q^{73} + ( 2 - 4 \beta_{1} + \beta_{4} + \beta_{7} - \beta_{8} ) q^{74} + ( -1 + 8 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 7 \beta_{5} - 4 \beta_{6} + 2 \beta_{8} - 3 \beta_{9} ) q^{76} + ( 4 + 2 \beta_{2} + \beta_{4} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{77} + ( 2 + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{79} + ( -3 + 3 \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{80} + ( 1 - 2 \beta_{1} + \beta_{2} + 4 \beta_{3} + 3 \beta_{4} - 3 \beta_{8} + 3 \beta_{9} ) q^{82} + ( 2 - \beta_{1} + 2 \beta_{5} - \beta_{6} - \beta_{8} + 2 \beta_{9} ) q^{83} + ( 1 - \beta_{1} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{85} + ( 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - \beta_{4} - 4 \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} ) q^{86} + ( 7 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - 4 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{88} - q^{89} + ( 3 \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{5} - 3 \beta_{6} + 4 \beta_{7} ) q^{91} + ( -2 + 5 \beta_{1} - 2 \beta_{2} - \beta_{4} - 4 \beta_{5} - \beta_{6} - \beta_{8} - 2 \beta_{9} ) q^{92} + ( 3 + \beta_{1} - \beta_{2} - 2 \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{94} + ( -1 - \beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{95} + ( 3 - \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} + 3 \beta_{8} - \beta_{9} ) q^{97} + ( 4 + 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 6q^{2} + 14q^{4} - 10q^{5} + 7q^{7} - 15q^{8} + O(q^{10}) \) \( 10q - 6q^{2} + 14q^{4} - 10q^{5} + 7q^{7} - 15q^{8} + 6q^{10} + 10q^{11} + 7q^{13} + 7q^{14} + 22q^{16} - 11q^{17} + 10q^{19} - 14q^{20} + 8q^{22} - 6q^{23} + 10q^{25} + 14q^{26} + 16q^{28} + 3q^{29} + 12q^{31} - 21q^{32} - 6q^{34} - 7q^{35} - 19q^{37} - 6q^{38} + 15q^{40} - 13q^{41} + 9q^{43} + 26q^{44} + 8q^{46} - 21q^{47} + 53q^{49} - 6q^{50} - 43q^{52} - 7q^{53} - 10q^{55} + 53q^{56} - 42q^{58} + 19q^{59} + 4q^{61} + 28q^{62} + 5q^{64} - 7q^{65} - 6q^{67} - 2q^{68} - 7q^{70} + 6q^{71} + 6q^{73} - 2q^{76} + 40q^{77} + 25q^{79} - 22q^{80} + q^{82} + 22q^{83} + 11q^{85} + 2q^{86} + 20q^{88} - 10q^{89} - 10q^{91} - 10q^{92} + 25q^{94} - 10q^{95} + 40q^{97} + 56q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 4 x^{9} - 8 x^{8} + 35 x^{7} + 29 x^{6} - 103 x^{5} - 57 x^{4} + 106 x^{3} + 29 x^{2} - 39 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 3 \nu^{2} - 2 \nu + 5 \)
\(\beta_{4}\)\(=\)\( \nu^{9} - 5 \nu^{8} - 2 \nu^{7} + 32 \nu^{6} - 6 \nu^{5} - 62 \nu^{4} + 6 \nu^{3} + 23 \nu^{2} - 5 \nu + 2 \)
\(\beta_{5}\)\(=\)\( \nu^{8} - 5 \nu^{7} - 2 \nu^{6} + 32 \nu^{5} - 5 \nu^{4} - 66 \nu^{3} + 4 \nu^{2} + 36 \nu - 3 \)
\(\beta_{6}\)\(=\)\( -\nu^{9} + 4 \nu^{8} + 7 \nu^{7} - 30 \nu^{6} - 27 \nu^{5} + 71 \nu^{4} + 62 \nu^{3} - 41 \nu^{2} - 31 \nu + 7 \)
\(\beta_{7}\)\(=\)\( \nu^{9} - 3 \nu^{8} - 12 \nu^{7} + 28 \nu^{6} + 58 \nu^{5} - 73 \nu^{4} - 122 \nu^{3} + 32 \nu^{2} + 55 \nu - 3 \)
\(\beta_{8}\)\(=\)\( \nu^{9} - 2 \nu^{8} - 16 \nu^{7} + 21 \nu^{6} + 90 \nu^{5} - 54 \nu^{4} - 199 \nu^{3} + 6 \nu^{2} + 99 \nu - 3 \)
\(\beta_{9}\)\(=\)\( 2 \nu^{9} - 8 \nu^{8} - 14 \nu^{7} + 61 \nu^{6} + 49 \nu^{5} - 142 \nu^{4} - 101 \nu^{3} + 72 \nu^{2} + 39 \nu - 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 3 \beta_{2} + 5 \beta_{1} + 4\)
\(\nu^{4}\)\(=\)\(-\beta_{7} + 2 \beta_{5} + \beta_{4} + 4 \beta_{3} + 13 \beta_{2} + 9 \beta_{1} + 20\)
\(\nu^{5}\)\(=\)\(-4 \beta_{7} - \beta_{6} + 7 \beta_{5} + 3 \beta_{4} + 18 \beta_{3} + 44 \beta_{2} + 32 \beta_{1} + 52\)
\(\nu^{6}\)\(=\)\(\beta_{9} - 20 \beta_{7} - 3 \beta_{6} + 35 \beta_{5} + 15 \beta_{4} + 67 \beta_{3} + 161 \beta_{2} + 78 \beta_{1} + 189\)
\(\nu^{7}\)\(=\)\(5 \beta_{9} + \beta_{8} - 77 \beta_{7} - 15 \beta_{6} + 126 \beta_{5} + 51 \beta_{4} + 250 \beta_{3} + 556 \beta_{2} + 251 \beta_{1} + 596\)
\(\nu^{8}\)\(=\)\(27 \beta_{9} + 5 \beta_{8} - 302 \beta_{7} - 49 \beta_{6} + 487 \beta_{5} + 194 \beta_{4} + 894 \beta_{3} + 1953 \beta_{2} + 722 \beta_{1} + 2049\)
\(\nu^{9}\)\(=\)\(113 \beta_{9} + 27 \beta_{8} - 1110 \beta_{7} - 185 \beta_{6} + 1733 \beta_{5} + 673 \beta_{4} + 3176 \beta_{3} + 6754 \beta_{2} + 2318 \beta_{1} + 6846\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.71694
−1.53398
−1.32759
−1.05100
0.0538492
0.704585
0.770724
2.21375
2.43743
3.44917
−2.71694 0 5.38176 −1.00000 0 0.704196 −9.18805 0 2.71694
1.2 −2.53398 0 4.42108 −1.00000 0 −3.60443 −6.13497 0 2.53398
1.3 −2.32759 0 3.41766 −1.00000 0 −0.231668 −3.29972 0 2.32759
1.4 −2.05100 0 2.20661 −1.00000 0 4.21605 −0.423758 0 2.05100
1.5 −0.946151 0 −1.10480 −1.00000 0 2.86767 2.93761 0 0.946151
1.6 −0.295415 0 −1.91273 −1.00000 0 3.54012 1.15588 0 0.295415
1.7 −0.229276 0 −1.94743 −1.00000 0 −3.83075 0.905050 0 0.229276
1.8 1.21375 0 −0.526821 −1.00000 0 −4.72892 −3.06692 0 −1.21375
1.9 1.43743 0 0.0662179 −1.00000 0 3.21347 −2.77969 0 −1.43743
1.10 2.44917 0 3.99846 −1.00000 0 4.85425 4.89457 0 −2.44917
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(89\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4005))\):

\(T_{2}^{10} + \cdots\)
\(T_{7}^{10} - \cdots\)
\(T_{11}^{10} - \cdots\)