Properties

Label 4010.2.a.h
Level 4010
Weight 2
Character orbit 4010.a
Self dual Yes
Analytic conductor 32.020
Analytic rank 1
Dimension 9
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4010 = 2 \cdot 5 \cdot 401 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4010.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0200112105\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} -\beta_{7} q^{3} + q^{4} + q^{5} -\beta_{7} q^{6} + ( -1 + \beta_{5} + \beta_{7} ) q^{7} + q^{8} + ( -1 - \beta_{1} + \beta_{7} ) q^{9} +O(q^{10})\) \( q + q^{2} -\beta_{7} q^{3} + q^{4} + q^{5} -\beta_{7} q^{6} + ( -1 + \beta_{5} + \beta_{7} ) q^{7} + q^{8} + ( -1 - \beta_{1} + \beta_{7} ) q^{9} + q^{10} + ( -\beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} ) q^{11} -\beta_{7} q^{12} + ( -1 - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{13} + ( -1 + \beta_{5} + \beta_{7} ) q^{14} -\beta_{7} q^{15} + q^{16} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{8} ) q^{17} + ( -1 - \beta_{1} + \beta_{7} ) q^{18} + ( -5 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{6} + 3 \beta_{7} ) q^{19} + q^{20} + ( -2 + \beta_{1} + \beta_{4} + \beta_{7} ) q^{21} + ( -\beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} ) q^{22} + ( -2 + \beta_{1} + \beta_{4} - \beta_{6} - \beta_{8} ) q^{23} -\beta_{7} q^{24} + q^{25} + ( -1 - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{26} + ( -3 + 2 \beta_{1} + \beta_{2} + \beta_{4} + 3 \beta_{7} ) q^{27} + ( -1 + \beta_{5} + \beta_{7} ) q^{28} + ( -3 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{29} -\beta_{7} q^{30} + ( -1 + 3 \beta_{2} + 3 \beta_{3} - \beta_{5} + 2 \beta_{6} - \beta_{7} - 3 \beta_{8} ) q^{31} + q^{32} + ( \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{33} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{8} ) q^{34} + ( -1 + \beta_{5} + \beta_{7} ) q^{35} + ( -1 - \beta_{1} + \beta_{7} ) q^{36} + ( -6 + \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{37} + ( -5 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{6} + 3 \beta_{7} ) q^{38} + ( \beta_{1} - 2 \beta_{4} + 2 \beta_{6} + \beta_{7} - \beta_{8} ) q^{39} + q^{40} + ( -4 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{6} + \beta_{8} ) q^{41} + ( -2 + \beta_{1} + \beta_{4} + \beta_{7} ) q^{42} + ( -1 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{5} - 2 \beta_{6} + 2 \beta_{8} ) q^{43} + ( -\beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} ) q^{44} + ( -1 - \beta_{1} + \beta_{7} ) q^{45} + ( -2 + \beta_{1} + \beta_{4} - \beta_{6} - \beta_{8} ) q^{46} + ( -\beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{47} -\beta_{7} q^{48} + ( -2 + \beta_{1} - \beta_{3} - \beta_{4} - 3 \beta_{6} - 2 \beta_{7} + \beta_{8} ) q^{49} + q^{50} + ( -2 - 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 4 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{51} + ( -1 - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{52} + ( 2 - 3 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + \beta_{6} - 2 \beta_{7} ) q^{53} + ( -3 + 2 \beta_{1} + \beta_{2} + \beta_{4} + 3 \beta_{7} ) q^{54} + ( -\beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} ) q^{55} + ( -1 + \beta_{5} + \beta_{7} ) q^{56} + ( -3 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{57} + ( -3 + 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{58} + ( 1 - 2 \beta_{1} + 3 \beta_{3} - 2 \beta_{4} - \beta_{7} + \beta_{8} ) q^{59} -\beta_{7} q^{60} + ( 6 - \beta_{1} - 4 \beta_{2} - \beta_{3} - 6 \beta_{4} - \beta_{5} + 3 \beta_{6} - 5 \beta_{7} - \beta_{8} ) q^{61} + ( -1 + 3 \beta_{2} + 3 \beta_{3} - \beta_{5} + 2 \beta_{6} - \beta_{7} - 3 \beta_{8} ) q^{62} + ( 4 - \beta_{1} + \beta_{3} - 2 \beta_{4} - 3 \beta_{5} + \beta_{6} - 3 \beta_{7} - \beta_{8} ) q^{63} + q^{64} + ( -1 - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{65} + ( \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{66} + ( 3 - 2 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + \beta_{5} + 3 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} ) q^{67} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{8} ) q^{68} + ( 1 - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} ) q^{69} + ( -1 + \beta_{5} + \beta_{7} ) q^{70} + ( 1 + \beta_{1} - 2 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} + 3 \beta_{8} ) q^{71} + ( -1 - \beta_{1} + \beta_{7} ) q^{72} + ( -2 - 3 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + 4 \beta_{8} ) q^{73} + ( -6 + \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{74} -\beta_{7} q^{75} + ( -5 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{6} + 3 \beta_{7} ) q^{76} + ( -3 + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} ) q^{77} + ( \beta_{1} - 2 \beta_{4} + 2 \beta_{6} + \beta_{7} - \beta_{8} ) q^{78} + ( -5 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{6} + 3 \beta_{7} ) q^{79} + q^{80} + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - 4 \beta_{7} ) q^{81} + ( -4 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{6} + \beta_{8} ) q^{82} + ( 1 - 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{8} ) q^{83} + ( -2 + \beta_{1} + \beta_{4} + \beta_{7} ) q^{84} + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{8} ) q^{85} + ( -1 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{5} - 2 \beta_{6} + 2 \beta_{8} ) q^{86} + ( 1 - 3 \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - 3 \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{87} + ( -\beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} ) q^{88} + ( -7 + \beta_{1} - 6 \beta_{2} - 7 \beta_{3} + 5 \beta_{4} + 8 \beta_{5} - 8 \beta_{6} + 4 \beta_{7} + 3 \beta_{8} ) q^{89} + ( -1 - \beta_{1} + \beta_{7} ) q^{90} + ( 1 - 3 \beta_{1} - 2 \beta_{3} - \beta_{4} - 3 \beta_{5} - \beta_{6} - 4 \beta_{7} ) q^{91} + ( -2 + \beta_{1} + \beta_{4} - \beta_{6} - \beta_{8} ) q^{92} + ( -2 - 3 \beta_{2} - 4 \beta_{3} + 5 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + 4 \beta_{7} + 3 \beta_{8} ) q^{93} + ( -\beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{94} + ( -5 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{6} + 3 \beta_{7} ) q^{95} -\beta_{7} q^{96} + ( -3 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} + 3 \beta_{6} + \beta_{7} + 3 \beta_{8} ) q^{97} + ( -2 + \beta_{1} - \beta_{3} - \beta_{4} - 3 \beta_{6} - 2 \beta_{7} + \beta_{8} ) q^{98} + ( -2 + 2 \beta_{1} + \beta_{2} + 3 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q + 9q^{2} - 4q^{3} + 9q^{4} + 9q^{5} - 4q^{6} - 7q^{7} + 9q^{8} - 7q^{9} + O(q^{10}) \) \( 9q + 9q^{2} - 4q^{3} + 9q^{4} + 9q^{5} - 4q^{6} - 7q^{7} + 9q^{8} - 7q^{9} + 9q^{10} - 11q^{11} - 4q^{12} - 14q^{13} - 7q^{14} - 4q^{15} + 9q^{16} - 13q^{17} - 7q^{18} - 11q^{19} + 9q^{20} - 8q^{21} - 11q^{22} - 9q^{23} - 4q^{24} + 9q^{25} - 14q^{26} - 4q^{27} - 7q^{28} - 20q^{29} - 4q^{30} - 11q^{31} + 9q^{32} + 4q^{33} - 13q^{34} - 7q^{35} - 7q^{36} - 25q^{37} - 11q^{38} - 8q^{39} + 9q^{40} - 29q^{41} - 8q^{42} - 11q^{43} - 11q^{44} - 7q^{45} - 9q^{46} - 3q^{47} - 4q^{48} - 18q^{49} + 9q^{50} + q^{51} - 14q^{52} - 9q^{53} - 4q^{54} - 11q^{55} - 7q^{56} - 17q^{57} - 20q^{58} - 10q^{59} - 4q^{60} - 10q^{61} - 11q^{62} + 16q^{63} + 9q^{64} - 14q^{65} + 4q^{66} - 16q^{67} - 13q^{68} + 5q^{69} - 7q^{70} - 8q^{71} - 7q^{72} - 22q^{73} - 25q^{74} - 4q^{75} - 11q^{76} - 15q^{77} - 8q^{78} - 9q^{79} + 9q^{80} - 15q^{81} - 29q^{82} + 11q^{83} - 8q^{84} - 13q^{85} - 11q^{86} + 12q^{87} - 11q^{88} - 28q^{89} - 7q^{90} - 6q^{91} - 9q^{92} + 16q^{93} - 3q^{94} - 11q^{95} - 4q^{96} - 28q^{97} - 18q^{98} + 9q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 2 x^{8} - 8 x^{7} + 16 x^{6} + 17 x^{5} - 36 x^{4} - 4 x^{3} + 17 x^{2} - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{4} - \nu^{3} - 4 \nu^{2} + 3 \nu + 1 \)
\(\beta_{3}\)\(=\)\( -\nu^{4} + \nu^{3} + 5 \nu^{2} - 3 \nu - 3 \)
\(\beta_{4}\)\(=\)\( -\nu^{5} + \nu^{4} + 5 \nu^{3} - 4 \nu^{2} - 4 \nu + 2 \)
\(\beta_{5}\)\(=\)\( \nu^{7} - \nu^{6} - 8 \nu^{5} + 8 \nu^{4} + 17 \nu^{3} - 17 \nu^{2} - 5 \nu + 3 \)
\(\beta_{6}\)\(=\)\( \nu^{8} - \nu^{7} - 9 \nu^{6} + 7 \nu^{5} + 25 \nu^{4} - 13 \nu^{3} - 21 \nu^{2} + 4 \nu + 4 \)
\(\beta_{7}\)\(=\)\( \nu^{8} - 2 \nu^{7} - 8 \nu^{6} + 16 \nu^{5} + 16 \nu^{4} - 34 \nu^{3} + 9 \nu \)
\(\beta_{8}\)\(=\)\( \nu^{8} - 2 \nu^{7} - 7 \nu^{6} + 15 \nu^{5} + 10 \nu^{4} - 30 \nu^{3} + 9 \nu^{2} + 8 \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 4 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 4 \beta_{3} + 5 \beta_{2} + \beta_{1} + 6\)
\(\nu^{5}\)\(=\)\(6 \beta_{7} - 6 \beta_{6} + 6 \beta_{5} + 5 \beta_{4} + \beta_{2} + 17 \beta_{1} - 5\)
\(\nu^{6}\)\(=\)\(\beta_{8} + 7 \beta_{7} - 8 \beta_{6} + 8 \beta_{5} + 7 \beta_{4} + 15 \beta_{3} + 22 \beta_{2} + 8 \beta_{1} + 21\)
\(\nu^{7}\)\(=\)\(\beta_{8} + 30 \beta_{7} - 31 \beta_{6} + 32 \beta_{5} + 22 \beta_{4} + 7 \beta_{2} + 73 \beta_{1} - 19\)
\(\nu^{8}\)\(=\)\(10 \beta_{8} + 39 \beta_{7} - 48 \beta_{6} + 50 \beta_{5} + 38 \beta_{4} + 56 \beta_{3} + 94 \beta_{2} + 49 \beta_{1} + 80\)

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
−1.87174
−0.589646
0.446506
2.07126
2.23442
1.34139
0.793701
−0.408238
−2.01764
1.00000 −2.53021 1.00000 1.00000 −2.53021 1.34700 1.00000 3.40195 1.00000
1.2 1.00000 −2.18512 1.00000 1.00000 −2.18512 −0.791922 1.00000 1.77477 1.00000
1.3 1.00000 −1.84294 1.00000 1.00000 −1.84294 −0.0938987 1.00000 0.396437 1.00000
1.4 1.00000 −0.922780 1.00000 1.00000 −0.922780 −2.45091 1.00000 −2.14848 1.00000
1.5 1.00000 −0.375168 1.00000 1.00000 −0.375168 3.43840 1.00000 −2.85925 1.00000
1.6 1.00000 0.453213 1.00000 1.00000 0.453213 −1.57101 1.00000 −2.79460 1.00000
1.7 1.00000 0.706772 1.00000 1.00000 0.706772 −4.28122 1.00000 −2.50047 1.00000
1.8 1.00000 1.13041 1.00000 1.00000 1.13041 −0.772637 1.00000 −1.72217 1.00000
1.9 1.00000 1.56583 1.00000 1.00000 1.56583 −1.82380 1.00000 −0.548185 1.00000
\(n\): 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.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(401\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4010))\):

\(T_{3}^{9} + \cdots\)
\(T_{7}^{9} + \cdots\)
\(T_{11}^{9} + \cdots\)