Properties

Label 1225.2.a.n
Level $1225$
Weight $2$
Character orbit 1225.a
Self dual yes
Analytic conductor $9.782$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.78167424761\)
Analytic rank: \(1\)
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 175)
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.

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

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
−1.61803 1.23607 0.618034 0 −2.00000 0 2.23607 −1.47214 0
1.2 0.618034 −3.23607 −1.61803 0 −2.00000 0 −2.23607 7.47214 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(7\) \(-1\)

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 1225.2.a.n 2
5.b even 2 1 1225.2.a.u 2
5.c odd 4 2 1225.2.b.k 4
7.b odd 2 1 175.2.a.d 2
21.c even 2 1 1575.2.a.s 2
28.d even 2 1 2800.2.a.bh 2
35.c odd 2 1 175.2.a.e yes 2
35.f even 4 2 175.2.b.c 4
105.g even 2 1 1575.2.a.n 2
105.k odd 4 2 1575.2.d.k 4
140.c even 2 1 2800.2.a.bp 2
140.j odd 4 2 2800.2.g.s 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.2.a.d 2 7.b odd 2 1
175.2.a.e yes 2 35.c odd 2 1
175.2.b.c 4 35.f even 4 2
1225.2.a.n 2 1.a even 1 1 trivial
1225.2.a.u 2 5.b even 2 1
1225.2.b.k 4 5.c odd 4 2
1575.2.a.n 2 105.g even 2 1
1575.2.a.s 2 21.c even 2 1
1575.2.d.k 4 105.k odd 4 2
2800.2.a.bh 2 28.d even 2 1
2800.2.a.bp 2 140.c even 2 1
2800.2.g.s 4 140.j odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1225))\):

\( T_{2}^{2} + T_{2} - 1 \)
\( T_{3}^{2} + 2 T_{3} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T + T^{2} \)
$3$ \( -4 + 2 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( -1 - 4 T + T^{2} \)
$13$ \( -4 + 2 T + T^{2} \)
$17$ \( -16 + 4 T + T^{2} \)
$19$ \( -20 + T^{2} \)
$23$ \( 11 + 8 T + T^{2} \)
$29$ \( ( -5 + T )^{2} \)
$31$ \( -36 - 6 T + T^{2} \)
$37$ \( ( 3 + T )^{2} \)
$41$ \( 44 + 14 T + T^{2} \)
$43$ \( 11 + 8 T + T^{2} \)
$47$ \( ( 2 + T )^{2} \)
$53$ \( -4 + 8 T + T^{2} \)
$59$ \( -20 + 10 T + T^{2} \)
$61$ \( -36 - 6 T + T^{2} \)
$67$ \( -1 - 4 T + T^{2} \)
$71$ \( -41 - 4 T + T^{2} \)
$73$ \( 116 + 22 T + T^{2} \)
$79$ \( -125 + T^{2} \)
$83$ \( -44 + 2 T + T^{2} \)
$89$ \( 220 + 30 T + T^{2} \)
$97$ \( 4 - 6 T + T^{2} \)
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