Properties

Label 4025.2.a.u
Level 4025
Weight 2
Character orbit 4025.a
Self dual Yes
Analytic conductor 32.140
Analytic rank 1
Dimension 8
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: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(x^{7}\mathstrut -\mathstrut \) \(10\) \(x^{6}\mathstrut +\mathstrut \) \(9\) \(x^{5}\mathstrut +\mathstrut \) \(28\) \(x^{4}\mathstrut -\mathstrut \) \(22\) \(x^{3}\mathstrut -\mathstrut \) \(16\) \(x^{2}\mathstrut +\mathstrut \) \(7\) \(x\mathstrut +\mathstrut \) \(3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 4 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} - \nu^{3} - 5 \nu^{2} + 4 \nu + 1 \)
\(\beta_{5}\)\(=\)\( \nu^{6} - \nu^{5} - 8 \nu^{4} + 6 \nu^{3} + 17 \nu^{2} - 7 \nu - 7 \)
\(\beta_{6}\)\(=\)\( \nu^{7} - \nu^{6} - 9 \nu^{5} + 8 \nu^{4} + 21 \nu^{3} - 17 \nu^{2} - 5 \nu + 3 \)
\(\beta_{7}\)\(=\)\( \nu^{7} - \nu^{6} - 10 \nu^{5} + 9 \nu^{4} + 27 \nu^{3} - 21 \nu^{2} - 10 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(5\) \(\beta_{2}\mathstrut +\mathstrut \) \(14\)
\(\nu^{5}\)\(=\)\(-\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(7\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(19\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{6}\)\(=\)\(-\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(9\) \(\beta_{4}\mathstrut +\mathstrut \) \(9\) \(\beta_{3}\mathstrut +\mathstrut \) \(24\) \(\beta_{2}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(70\)
\(\nu^{7}\)\(=\)\(-\)\(10\) \(\beta_{7}\mathstrut +\mathstrut \) \(11\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(10\) \(\beta_{4}\mathstrut +\mathstrut \) \(43\) \(\beta_{3}\mathstrut +\mathstrut \) \(10\) \(\beta_{2}\mathstrut +\mathstrut \) \(94\) \(\beta_{1}\mathstrut +\mathstrut \) \(24\)

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
2.39863
2.02014
1.09801
0.651939
−0.333224
−0.571357
−2.07739
−2.18675
−2.39863 −2.12900 3.75344 0 5.10670 1.00000 −4.20585 1.53265 0
1.2 −2.02014 1.91409 2.08098 0 −3.86674 1.00000 −0.163592 0.663753 0
1.3 −1.09801 −0.493654 −0.794374 0 0.542037 1.00000 3.06825 −2.75631 0
1.4 −0.651939 −2.38619 −1.57498 0 1.55565 1.00000 2.33067 2.69390 0
1.5 0.333224 −0.161242 −1.88896 0 −0.0537298 1.00000 −1.29590 −2.97400 0
1.6 0.571357 1.62458 −1.67355 0 0.928216 1.00000 −2.09891 −0.360734 0
1.7 2.07739 0.298210 2.31556 0 0.619500 1.00000 0.655538 −2.91107 0
1.8 2.18675 −2.66680 2.78189 0 −5.83163 1.00000 1.70979 4.11181 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4025))\):

\(T_{2}^{8} + \cdots\)
\(T_{3}^{8} + \cdots\)
\(T_{11}^{8} - \cdots\)