Properties

Label 4025.2.a.h
Level 4025
Weight 2
Character orbit 4025.a
Self dual Yes
Analytic conductor 32.140
Analytic rank 0
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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
1.61803
−0.618034
−2.61803 2.23607 4.85410 0 −5.85410 1.00000 −7.47214 2.00000 0
1.2 −0.381966 −2.23607 −1.85410 0 0.854102 1.00000 1.47214 2.00000 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4025))\):

\(T_{2}^{2} \) \(\mathstrut +\mathstrut 3 T_{2} \) \(\mathstrut +\mathstrut 1 \)
\(T_{3}^{2} \) \(\mathstrut -\mathstrut 5 \)
\(T_{11} \) \(\mathstrut -\mathstrut 6 \)