Properties

Label 4025.2.a.m
Level 4025
Weight 2
Character orbit 4025.a
Self dual Yes
Analytic conductor 32.140
Analytic rank 0
Dimension 4
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4025.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.1397868136\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.7537.1
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(x^{3}\mathstrut -\mathstrut \) \(5\) \(x^{2}\mathstrut +\mathstrut \) \(4\) \(x\mathstrut +\mathstrut \) \(3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 4 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\)

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
2.15976
1.37933
−0.491918
−2.04717
−2.15976 −2.43525 2.66454 0 5.25956 −1.00000 −1.43525 2.93047 0
1.2 −1.37933 1.89307 −0.0974383 0 −2.61117 −1.00000 2.89307 0.583705 0
1.3 0.491918 −2.84864 −1.75802 0 −1.40130 −1.00000 −1.84864 5.11474 0
1.4 2.04717 −0.609175 2.19091 0 −1.24709 −1.00000 0.390825 −2.62891 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(7\) \(1\)
\(23\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4025))\):

\(T_{2}^{4} \) \(\mathstrut +\mathstrut T_{2}^{3} \) \(\mathstrut -\mathstrut 5 T_{2}^{2} \) \(\mathstrut -\mathstrut 4 T_{2} \) \(\mathstrut +\mathstrut 3 \)
\(T_{3}^{4} \) \(\mathstrut +\mathstrut 4 T_{3}^{3} \) \(\mathstrut -\mathstrut T_{3}^{2} \) \(\mathstrut -\mathstrut 15 T_{3} \) \(\mathstrut -\mathstrut 8 \)
\(T_{11}^{4} \) \(\mathstrut -\mathstrut 11 T_{11}^{3} \) \(\mathstrut +\mathstrut 40 T_{11}^{2} \) \(\mathstrut -\mathstrut 55 T_{11} \) \(\mathstrut +\mathstrut 24 \)