Properties

Label 4005.2.a.q
Level 4005
Weight 2
Character orbit 4005.a
Self dual yes
Analytic conductor 31.980
Analytic rank 1
Dimension 9
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: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1335)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{2} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} + q^{5} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{8} ) q^{7} + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} ) q^{8} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{2} + ( 2 - \beta_{1} + \beta_{2} ) q^{4} + q^{5} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{8} ) q^{7} + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} ) q^{8} + ( -1 + \beta_{1} ) q^{10} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{11} + ( 1 - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{13} + ( 1 - \beta_{2} - \beta_{4} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{14} + ( 3 - 4 \beta_{1} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{16} + ( -3 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{17} + ( -1 - \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{8} ) q^{19} + ( 2 - \beta_{1} + \beta_{2} ) q^{20} + ( 1 - \beta_{1} + \beta_{3} + \beta_{4} + 2 \beta_{7} ) q^{22} + ( -3 + \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{6} ) q^{23} + q^{25} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - 2 \beta_{6} + \beta_{8} ) q^{26} + ( -2 + \beta_{3} + \beta_{4} - \beta_{5} - 4 \beta_{6} + 2 \beta_{7} + 4 \beta_{8} ) q^{28} + ( -1 + 2 \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} ) q^{29} + ( 1 - 3 \beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{31} + ( -7 + 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{32} + ( 4 - 5 \beta_{1} - \beta_{3} - 2 \beta_{5} - \beta_{7} ) q^{34} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{8} ) q^{35} + ( \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{37} + ( -2 - 5 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{8} ) q^{38} + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} ) q^{40} + ( 2 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} ) q^{41} + ( -2 + \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + 3 \beta_{7} ) q^{43} + ( -5 + \beta_{2} - \beta_{3} + \beta_{6} + 3 \beta_{8} ) q^{44} + ( 3 + 2 \beta_{1} - \beta_{2} - \beta_{3} + 4 \beta_{5} + \beta_{6} - 2 \beta_{7} - 5 \beta_{8} ) q^{46} + ( -1 - \beta_{1} - \beta_{2} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} ) q^{47} + ( 3 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{49} + ( -1 + \beta_{1} ) q^{50} + ( 4 - \beta_{2} - 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} - 5 \beta_{8} ) q^{52} + ( -5 + \beta_{1} + \beta_{2} + \beta_{5} ) q^{53} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{55} + ( 2 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + 3 \beta_{6} - \beta_{7} - 8 \beta_{8} ) q^{56} + ( 3 - 2 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + 4 \beta_{8} ) q^{58} + ( -2 - \beta_{2} + 3 \beta_{3} + \beta_{4} - 3 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} ) q^{59} + ( -4 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{8} ) q^{61} + ( -5 - 3 \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} ) q^{62} + ( 10 - 8 \beta_{1} + 2 \beta_{2} - 4 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} ) q^{64} + ( 1 - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{65} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{67} + ( -9 + 4 \beta_{1} - 3 \beta_{2} + 2 \beta_{4} - \beta_{6} - \beta_{8} ) q^{68} + ( 1 - \beta_{2} - \beta_{4} + 2 \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{70} + ( 2 - \beta_{1} - \beta_{2} - \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} - 3 \beta_{8} ) q^{71} + ( 2 + \beta_{1} - 3 \beta_{2} - 3 \beta_{4} + 3 \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{73} + ( 3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - 3 \beta_{7} - 2 \beta_{8} ) q^{74} + ( -3 + 5 \beta_{1} - 5 \beta_{2} - \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + \beta_{6} + 2 \beta_{7} - 4 \beta_{8} ) q^{76} + ( -3 \beta_{1} + 4 \beta_{2} - \beta_{4} + 3 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} - \beta_{8} ) q^{77} + ( -1 - 3 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + 4 \beta_{8} ) q^{79} + ( 3 - 4 \beta_{1} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{80} + ( -9 + 7 \beta_{1} - 2 \beta_{2} + 6 \beta_{4} + 2 \beta_{5} - \beta_{6} + 3 \beta_{7} + \beta_{8} ) q^{82} + ( -2 + \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - \beta_{6} + 2 \beta_{7} - \beta_{8} ) q^{83} + ( -3 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{85} + ( -5 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} - 5 \beta_{5} - \beta_{6} - 3 \beta_{7} + 2 \beta_{8} ) q^{86} + ( 12 - 4 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + 3 \beta_{6} - 4 \beta_{7} - 4 \beta_{8} ) q^{88} - q^{89} + ( -9 + 5 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - 3 \beta_{6} + \beta_{7} + \beta_{8} ) q^{91} + ( -2 - 4 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} + 6 \beta_{8} ) q^{92} + ( -3 - 5 \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{5} - 4 \beta_{6} + 5 \beta_{7} + 7 \beta_{8} ) q^{94} + ( -1 - \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{8} ) q^{95} + ( 3 + \beta_{1} + 4 \beta_{2} - 3 \beta_{3} + \beta_{4} + 2 \beta_{6} - 4 \beta_{7} - 2 \beta_{8} ) q^{97} + ( -12 + 4 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{8} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q - 5q^{2} + 11q^{4} + 9q^{5} - 3q^{7} - 18q^{8} + O(q^{10}) \) \( 9q - 5q^{2} + 11q^{4} + 9q^{5} - 3q^{7} - 18q^{8} - 5q^{10} - 4q^{11} + 5q^{13} + q^{14} + 15q^{16} - 21q^{17} - 18q^{19} + 11q^{20} + 6q^{22} - 16q^{23} + 9q^{25} - 8q^{26} + 6q^{28} - 3q^{29} - 6q^{31} - 46q^{32} + 12q^{34} - 3q^{35} + 11q^{37} - 20q^{38} - 18q^{40} + q^{41} - 3q^{43} - 38q^{44} + 16q^{46} - 27q^{47} + 24q^{49} - 5q^{50} + 17q^{52} - 43q^{53} - 4q^{55} - 5q^{56} + 34q^{58} - 3q^{59} - 30q^{61} - 36q^{62} + 50q^{64} + 5q^{65} - 12q^{67} - 64q^{68} + q^{70} + 4q^{71} + 26q^{73} - 2q^{74} - 12q^{76} - 34q^{77} + q^{79} + 15q^{80} - 51q^{82} - 24q^{83} - 21q^{85} - 18q^{86} + 64q^{88} - 9q^{89} - 50q^{91} - 10q^{92} - 11q^{94} - 18q^{95} - 4q^{97} - 75q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 4 x^{8} - 6 x^{7} + 31 x^{6} + 13 x^{5} - 75 x^{4} - 17 x^{3} + 52 x^{2} + 11 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - 2 \nu^{3} - 4 \nu^{2} + 4 \nu + 2 \)
\(\beta_{4}\)\(=\)\( \nu^{6} - 3 \nu^{5} - 4 \nu^{4} + 13 \nu^{3} + 3 \nu^{2} - 12 \nu \)
\(\beta_{5}\)\(=\)\( -\nu^{6} + 3 \nu^{5} + 4 \nu^{4} - 12 \nu^{3} - 5 \nu^{2} + 8 \nu + 3 \)
\(\beta_{6}\)\(=\)\( \nu^{8} - 5 \nu^{7} + \nu^{6} + 23 \nu^{5} - 16 \nu^{4} - 28 \nu^{3} + 12 \nu^{2} + 10 \nu + 1 \)
\(\beta_{7}\)\(=\)\( \nu^{8} - 5 \nu^{7} + 27 \nu^{5} - 15 \nu^{4} - 44 \nu^{3} + 19 \nu^{2} + 22 \nu - 2 \)
\(\beta_{8}\)\(=\)\( -\nu^{8} + 6 \nu^{7} - 5 \nu^{6} - 25 \nu^{5} + 35 \nu^{4} + 24 \nu^{3} - 35 \nu^{2} - 5 \nu + 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{5} + \beta_{4} + 2 \beta_{2} + 6 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(2 \beta_{5} + 2 \beta_{4} + \beta_{3} + 8 \beta_{2} + 12 \beta_{1} + 16\)
\(\nu^{5}\)\(=\)\(\beta_{7} - \beta_{6} + 9 \beta_{5} + 10 \beta_{4} + 3 \beta_{3} + 20 \beta_{2} + 44 \beta_{1} + 30\)
\(\nu^{6}\)\(=\)\(3 \beta_{7} - 3 \beta_{6} + 22 \beta_{5} + 26 \beta_{4} + 13 \beta_{3} + 63 \beta_{2} + 111 \beta_{1} + 106\)
\(\nu^{7}\)\(=\)\(\beta_{8} + 14 \beta_{7} - 13 \beta_{6} + 72 \beta_{5} + 90 \beta_{4} + 39 \beta_{3} + 171 \beta_{2} + 346 \beta_{1} + 256\)
\(\nu^{8}\)\(=\)\(5 \beta_{8} + 44 \beta_{7} - 38 \beta_{6} + 191 \beta_{5} + 254 \beta_{4} + 129 \beta_{3} + 504 \beta_{2} + 945 \beta_{1} + 787\)

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
−1.72546
−1.70432
−1.02867
−0.286244
0.0690386
1.09195
1.89144
2.81142
2.88084
−2.72546 0 5.42813 1.00000 0 −4.89499 −9.34325 0 −2.72546
1.2 −2.70432 0 5.31333 1.00000 0 3.91511 −8.96030 0 −2.70432
1.3 −2.02867 0 2.11549 1.00000 0 2.64395 −0.234298 0 −2.02867
1.4 −1.28624 0 −0.345575 1.00000 0 −0.646623 3.01698 0 −1.28624
1.5 −0.930961 0 −1.13331 1.00000 0 −1.89313 2.91699 0 −0.930961
1.6 0.0919478 0 −1.99155 1.00000 0 −4.79397 −0.367014 0 0.0919478
1.7 0.891444 0 −1.20533 1.00000 0 3.64096 −2.85737 0 0.891444
1.8 1.81142 0 1.28123 1.00000 0 −0.549078 −1.30199 0 1.81142
1.9 1.88084 0 1.53757 1.00000 0 −0.422231 −0.869761 0 1.88084
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4005.2.a.q 9
3.b odd 2 1 1335.2.a.h 9
15.d odd 2 1 6675.2.a.x 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1335.2.a.h 9 3.b odd 2 1
4005.2.a.q 9 1.a even 1 1 trivial
6675.2.a.x 9 15.d odd 2 1

Atkin-Lehner signs

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

Hecke kernels

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

\(T_{2}^{9} + \cdots\)
\(T_{7}^{9} + \cdots\)
\(T_{11}^{9} + \cdots\)