Properties

Label 3025.2.a.i
Level $3025$
Weight $2$
Character orbit 3025.a
Self dual yes
Analytic conductor $24.155$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3025 = 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(24.1547466114\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 275)
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.

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

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
−1.61803 2.61803 0.618034 0 −4.23607 2.85410 2.23607 3.85410 0
1.2 0.618034 0.381966 −1.61803 0 0.236068 −3.85410 −2.23607 −2.85410 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(11\) \(-1\)

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 3025.2.a.i 2
5.b even 2 1 3025.2.a.m 2
11.b odd 2 1 275.2.a.g yes 2
33.d even 2 1 2475.2.a.n 2
44.c even 2 1 4400.2.a.bg 2
55.d odd 2 1 275.2.a.d 2
55.e even 4 2 275.2.b.e 4
165.d even 2 1 2475.2.a.s 2
165.l odd 4 2 2475.2.c.p 4
220.g even 2 1 4400.2.a.bv 2
220.i odd 4 2 4400.2.b.x 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.a.d 2 55.d odd 2 1
275.2.a.g yes 2 11.b odd 2 1
275.2.b.e 4 55.e even 4 2
2475.2.a.n 2 33.d even 2 1
2475.2.a.s 2 165.d even 2 1
2475.2.c.p 4 165.l odd 4 2
3025.2.a.i 2 1.a even 1 1 trivial
3025.2.a.m 2 5.b even 2 1
4400.2.a.bg 2 44.c even 2 1
4400.2.a.bv 2 220.g even 2 1
4400.2.b.x 4 220.i odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3025))\):

\( T_{2}^{2} + T_{2} - 1 \)
\( T_{3}^{2} - 3 T_{3} + 1 \)
\( T_{19}^{2} - 45 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T + T^{2} \)
$3$ \( 1 - 3 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( -11 + T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 11 + 8 T + T^{2} \)
$17$ \( -1 + T + T^{2} \)
$19$ \( -45 + T^{2} \)
$23$ \( -29 - 3 T + T^{2} \)
$29$ \( 5 - 5 T + T^{2} \)
$31$ \( ( 3 + T )^{2} \)
$37$ \( 59 - 16 T + T^{2} \)
$41$ \( ( -3 + T )^{2} \)
$43$ \( ( -6 + T )^{2} \)
$47$ \( -71 - 6 T + T^{2} \)
$53$ \( -59 - 3 T + T^{2} \)
$59$ \( 5 - 10 T + T^{2} \)
$61$ \( -1 - 11 T + T^{2} \)
$67$ \( ( -8 + T )^{2} \)
$71$ \( -116 + 6 T + T^{2} \)
$73$ \( 131 + 23 T + T^{2} \)
$79$ \( -5 + 5 T + T^{2} \)
$83$ \( 171 - 27 T + T^{2} \)
$89$ \( 125 + 25 T + T^{2} \)
$97$ \( -1 - T + T^{2} \)
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