Properties

Label 1225.2.b.k
Level $1225$
Weight $2$
Character orbit 1225.b
Analytic conductor $9.782$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1225.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.78167424761\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 175)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{3} + ( 1 + \beta_{2} ) q^{4} -2 q^{6} + ( 2 \beta_{1} + \beta_{3} ) q^{8} + ( -5 - 4 \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{3} + ( 1 + \beta_{2} ) q^{4} -2 q^{6} + ( 2 \beta_{1} + \beta_{3} ) q^{8} + ( -5 - 4 \beta_{2} ) q^{9} + ( 1 - 2 \beta_{2} ) q^{11} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{12} -2 \beta_{1} q^{13} + 3 \beta_{2} q^{16} + 4 \beta_{1} q^{17} + ( -\beta_{1} - 4 \beta_{3} ) q^{18} + ( -2 - 4 \beta_{2} ) q^{19} + ( 3 \beta_{1} - 2 \beta_{3} ) q^{22} + ( 2 \beta_{1} + 5 \beta_{3} ) q^{23} + ( -6 - 2 \beta_{2} ) q^{24} + ( 2 - 2 \beta_{2} ) q^{26} + ( -4 \beta_{1} - 12 \beta_{3} ) q^{27} -5 q^{29} -6 \beta_{2} q^{31} + ( \beta_{1} + 5 \beta_{3} ) q^{32} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{33} + ( -4 + 4 \beta_{2} ) q^{34} + ( -9 - 5 \beta_{2} ) q^{36} -3 \beta_{3} q^{37} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{38} + 4 q^{39} + ( -6 + 2 \beta_{2} ) q^{41} + ( -2 \beta_{1} + 3 \beta_{3} ) q^{43} + ( -1 + \beta_{2} ) q^{44} + ( -2 - 3 \beta_{2} ) q^{46} -2 \beta_{3} q^{47} + 6 \beta_{3} q^{48} -8 q^{51} -2 \beta_{3} q^{52} + ( 4 \beta_{1} + 6 \beta_{3} ) q^{53} + ( 4 + 8 \beta_{2} ) q^{54} + ( -4 \beta_{1} - 12 \beta_{3} ) q^{57} -5 \beta_{1} q^{58} + ( 8 + 6 \beta_{2} ) q^{59} + ( 6 + 6 \beta_{2} ) q^{61} + ( 6 \beta_{1} - 6 \beta_{3} ) q^{62} + ( -1 + 2 \beta_{2} ) q^{64} + ( -2 + 4 \beta_{2} ) q^{66} + ( 2 \beta_{1} + 3 \beta_{3} ) q^{67} + 4 \beta_{3} q^{68} + ( -14 - 10 \beta_{2} ) q^{69} + ( 5 + 6 \beta_{2} ) q^{71} + ( -6 \beta_{1} - 13 \beta_{3} ) q^{72} + ( -2 \beta_{1} + 10 \beta_{3} ) q^{73} + 3 \beta_{2} q^{74} + ( -6 - 2 \beta_{2} ) q^{76} + 4 \beta_{1} q^{78} + ( 5 + 10 \beta_{2} ) q^{79} + ( 17 + 12 \beta_{2} ) q^{81} + ( -8 \beta_{1} + 2 \beta_{3} ) q^{82} + ( 6 \beta_{1} + 4 \beta_{3} ) q^{83} + ( 2 - 5 \beta_{2} ) q^{86} + ( -10 \beta_{1} - 10 \beta_{3} ) q^{87} + ( 4 \beta_{1} - 3 \beta_{3} ) q^{88} + ( 16 + 2 \beta_{2} ) q^{89} + ( 5 \beta_{1} + 7 \beta_{3} ) q^{92} -12 \beta_{3} q^{93} + 2 \beta_{2} q^{94} + ( -12 - 10 \beta_{2} ) q^{96} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{97} + ( 3 - 2 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} - 8q^{6} - 12q^{9} + O(q^{10}) \) \( 4q + 2q^{4} - 8q^{6} - 12q^{9} + 8q^{11} - 6q^{16} - 20q^{24} + 12q^{26} - 20q^{29} + 12q^{31} - 24q^{34} - 26q^{36} + 16q^{39} - 28q^{41} - 6q^{44} - 2q^{46} - 32q^{51} + 20q^{59} + 12q^{61} - 8q^{64} - 16q^{66} - 36q^{69} + 8q^{71} - 6q^{74} - 20q^{76} + 44q^{81} + 18q^{86} + 60q^{89} - 4q^{94} - 28q^{96} + 16q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 1 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 2 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 1\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 2 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(1177\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
1.61803i
0.618034i
0.618034i
1.61803i
1.61803i 1.23607i −0.618034 0 −2.00000 0 2.23607i 1.47214 0
99.2 0.618034i 3.23607i 1.61803 0 −2.00000 0 2.23607i −7.47214 0
99.3 0.618034i 3.23607i 1.61803 0 −2.00000 0 2.23607i −7.47214 0
99.4 1.61803i 1.23607i −0.618034 0 −2.00000 0 2.23607i 1.47214 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

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

Hecke kernels

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

\( T_{2}^{4} + 3 T_{2}^{2} + 1 \)
\( T_{19}^{2} - 20 \)
\( T_{31}^{2} - 6 T_{31} - 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T^{2} + T^{4} \)
$3$ \( 16 + 12 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( -1 - 4 T + T^{2} )^{2} \)
$13$ \( 16 + 12 T^{2} + T^{4} \)
$17$ \( 256 + 48 T^{2} + T^{4} \)
$19$ \( ( -20 + T^{2} )^{2} \)
$23$ \( 121 + 42 T^{2} + T^{4} \)
$29$ \( ( 5 + T )^{4} \)
$31$ \( ( -36 - 6 T + T^{2} )^{2} \)
$37$ \( ( 9 + T^{2} )^{2} \)
$41$ \( ( 44 + 14 T + T^{2} )^{2} \)
$43$ \( 121 + 42 T^{2} + T^{4} \)
$47$ \( ( 4 + T^{2} )^{2} \)
$53$ \( 16 + 72 T^{2} + T^{4} \)
$59$ \( ( -20 - 10 T + T^{2} )^{2} \)
$61$ \( ( -36 - 6 T + T^{2} )^{2} \)
$67$ \( 1 + 18 T^{2} + T^{4} \)
$71$ \( ( -41 - 4 T + T^{2} )^{2} \)
$73$ \( 13456 + 252 T^{2} + T^{4} \)
$79$ \( ( -125 + T^{2} )^{2} \)
$83$ \( 1936 + 92 T^{2} + T^{4} \)
$89$ \( ( 220 - 30 T + T^{2} )^{2} \)
$97$ \( 16 + 28 T^{2} + T^{4} \)
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