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: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1205)
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.

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

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:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(241\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6025.2.a.b 2
5.b even 2 1 6025.2.a.c 2
5.c odd 4 2 1205.2.b.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1205.2.b.a 4 5.c odd 4 2
6025.2.a.b 2 1.a even 1 1 trivial
6025.2.a.c 2 5.b even 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -5 + T^{2} \)
$3$ \( 1 + 3 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( -4 - 2 T + T^{2} \)
$11$ \( -1 + T + T^{2} \)
$13$ \( -11 - T + T^{2} \)
$17$ \( -31 + T + T^{2} \)
$19$ \( 41 + 13 T + T^{2} \)
$23$ \( -20 + T^{2} \)
$29$ \( -29 + 3 T + T^{2} \)
$31$ \( -16 + 4 T + T^{2} \)
$37$ \( -20 + T^{2} \)
$41$ \( 11 - 7 T + T^{2} \)
$43$ \( 16 + 12 T + T^{2} \)
$47$ \( 19 - 9 T + T^{2} \)
$53$ \( 16 - 12 T + T^{2} \)
$59$ \( -44 - 2 T + T^{2} \)
$61$ \( 79 - 19 T + T^{2} \)
$67$ \( -29 + 3 T + T^{2} \)
$71$ \( -95 + 5 T + T^{2} \)
$73$ \( 41 - 13 T + T^{2} \)
$79$ \( -100 - 10 T + T^{2} \)
$83$ \( 11 + 13 T + T^{2} \)
$89$ \( -44 + 12 T + T^{2} \)
$97$ \( 20 + 10 T + T^{2} \)
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