Properties

Label 6025.2.a.b
Level 6025
Weight 2
Character orbit 6025.a
Self dual Yes
Analytic conductor 48.110
Analytic rank 1
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6025 = 5^{2} \cdot 241 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6025.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.1098672178\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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
−2.23607 −0.381966 3.00000 0 0.854102 −1.23607 −2.23607 −2.85410 0
1.2 2.23607 −2.61803 3.00000 0 −5.85410 3.23607 2.23607 3.85410 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
\(5\) \(-1\)
\(241\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6025))\):

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