Properties

Label 6025.2.a.d
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.

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

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
−0.618034
1.61803
1.00000 −1.61803 −1.00000 0 −1.61803 −1.23607 −3.00000 −0.381966 0
1.2 1.00000 0.618034 −1.00000 0 0.618034 3.23607 −3.00000 −2.61803 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.d 2
5.b even 2 1 6025.2.a.a 2
5.c odd 4 2 1205.2.b.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1205.2.b.b 4 5.c odd 4 2
6025.2.a.a 2 5.b even 2 1
6025.2.a.d 2 1.a even 1 1 trivial

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( -1 + T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( -4 - 2 T + T^{2} \)
$11$ \( 5 - 5 T + T^{2} \)
$13$ \( 1 + 3 T + T^{2} \)
$17$ \( -11 + T + T^{2} \)
$19$ \( -1 + T + T^{2} \)
$23$ \( -4 - 8 T + T^{2} \)
$29$ \( -29 + 3 T + T^{2} \)
$31$ \( 16 + 12 T + T^{2} \)
$37$ \( -20 + T^{2} \)
$41$ \( -29 - 3 T + T^{2} \)
$43$ \( -16 - 4 T + T^{2} \)
$47$ \( 9 - 9 T + T^{2} \)
$53$ \( ( 4 + T )^{2} \)
$59$ \( 20 + 10 T + T^{2} \)
$61$ \( -9 - 3 T + T^{2} \)
$67$ \( -151 - T + T^{2} \)
$71$ \( -145 + 5 T + T^{2} \)
$73$ \( -11 - 9 T + T^{2} \)
$79$ \( -244 + 2 T + T^{2} \)
$83$ \( 209 + 29 T + T^{2} \)
$89$ \( 124 - 24 T + T^{2} \)
$97$ \( 220 + 30 T + T^{2} \)
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