Properties

Label 1225.2.a.m
Level $1225$
Weight $2$
Character orbit 1225.a
Self dual yes
Analytic conductor $9.782$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.41421
1.41421
−2.41421 −0.414214 3.82843 0 1.00000 0 −4.41421 −2.82843 0
1.2 0.414214 2.41421 −1.82843 0 1.00000 0 −1.58579 2.82843 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.m 2
5.b even 2 1 245.2.a.g 2
5.c odd 4 2 1225.2.b.h 4
7.b odd 2 1 1225.2.a.k 2
7.d odd 6 2 175.2.e.c 4
15.d odd 2 1 2205.2.a.q 2
20.d odd 2 1 3920.2.a.bv 2
35.c odd 2 1 245.2.a.h 2
35.f even 4 2 1225.2.b.g 4
35.i odd 6 2 35.2.e.a 4
35.j even 6 2 245.2.e.e 4
35.k even 12 4 175.2.k.a 8
105.g even 2 1 2205.2.a.n 2
105.p even 6 2 315.2.j.e 4
140.c even 2 1 3920.2.a.bq 2
140.s even 6 2 560.2.q.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.e.a 4 35.i odd 6 2
175.2.e.c 4 7.d odd 6 2
175.2.k.a 8 35.k even 12 4
245.2.a.g 2 5.b even 2 1
245.2.a.h 2 35.c odd 2 1
245.2.e.e 4 35.j even 6 2
315.2.j.e 4 105.p even 6 2
560.2.q.k 4 140.s even 6 2
1225.2.a.k 2 7.b odd 2 1
1225.2.a.m 2 1.a even 1 1 trivial
1225.2.b.g 4 35.f even 4 2
1225.2.b.h 4 5.c odd 4 2
2205.2.a.n 2 105.g even 2 1
2205.2.a.q 2 15.d odd 2 1
3920.2.a.bq 2 140.c even 2 1
3920.2.a.bv 2 20.d odd 2 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + 2 T + T^{2} \)
$3$ \( -1 - 2 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( -4 - 4 T + T^{2} \)
$13$ \( -4 + 4 T + T^{2} \)
$17$ \( -4 - 4 T + T^{2} \)
$19$ \( -8 + T^{2} \)
$23$ \( -1 - 2 T + T^{2} \)
$29$ \( ( 1 + T )^{2} \)
$31$ \( ( -6 + T )^{2} \)
$37$ \( T^{2} \)
$41$ \( 17 - 10 T + T^{2} \)
$43$ \( 23 + 10 T + T^{2} \)
$47$ \( ( -2 + T )^{2} \)
$53$ \( 8 - 8 T + T^{2} \)
$59$ \( -56 - 8 T + T^{2} \)
$61$ \( -63 - 6 T + T^{2} \)
$67$ \( 119 + 22 T + T^{2} \)
$71$ \( -56 + 8 T + T^{2} \)
$73$ \( -4 - 4 T + T^{2} \)
$79$ \( 136 - 24 T + T^{2} \)
$83$ \( -161 - 2 T + T^{2} \)
$89$ \( -23 - 6 T + T^{2} \)
$97$ \( 4 - 12 T + T^{2} \)
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