Properties

Label 4025.2.a.o
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: 4.4.2777.1
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.

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

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

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

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
−0.679643
2.36234
−1.50848
0.825785
−1.26308 0.320357 −0.404635 0 −0.404635 −1.00000 3.03724 −2.89737 0
1.2 −0.515722 3.36234 −1.73403 0 −1.73403 −1.00000 1.92572 8.30533 0
1.3 1.18264 −0.508481 −0.601352 0 −0.601352 −1.00000 −3.07647 −2.74145 0
1.4 2.59615 1.82578 4.74002 0 4.74002 −1.00000 7.11351 0.333489 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4025))\):

\(T_{2}^{4} \) \(\mathstrut -\mathstrut 2 T_{2}^{3} \) \(\mathstrut -\mathstrut 3 T_{2}^{2} \) \(\mathstrut +\mathstrut 3 T_{2} \) \(\mathstrut +\mathstrut 2 \)
\(T_{3}^{4} \) \(\mathstrut -\mathstrut 5 T_{3}^{3} \) \(\mathstrut +\mathstrut 5 T_{3}^{2} \) \(\mathstrut +\mathstrut 2 T_{3} \) \(\mathstrut -\mathstrut 1 \)
\(T_{11}^{4} \) \(\mathstrut +\mathstrut 8 T_{11}^{3} \) \(\mathstrut +\mathstrut 3 T_{11}^{2} \) \(\mathstrut -\mathstrut 51 T_{11} \) \(\mathstrut +\mathstrut 41 \)