Properties

Label 245.2.a.f
Level $245$
Weight $2$
Character orbit 245.a
Self dual yes
Analytic conductor $1.956$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 245.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.95633484952\)
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: yes
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.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( 1 + \beta ) q^{3} + q^{5} + ( 2 + \beta ) q^{6} -2 \beta q^{8} + 2 \beta q^{9} +O(q^{10})\) \( q + \beta q^{2} + ( 1 + \beta ) q^{3} + q^{5} + ( 2 + \beta ) q^{6} -2 \beta q^{8} + 2 \beta q^{9} + \beta q^{10} + ( -3 - 2 \beta ) q^{11} + ( 3 - \beta ) q^{13} + ( 1 + \beta ) q^{15} -4 q^{16} + ( -1 - 3 \beta ) q^{17} + 4 q^{18} + 6 q^{19} + ( -4 - 3 \beta ) q^{22} + ( 6 - \beta ) q^{23} + ( -4 - 2 \beta ) q^{24} + q^{25} + ( -2 + 3 \beta ) q^{26} + ( 1 - \beta ) q^{27} + ( -3 + 4 \beta ) q^{29} + ( 2 + \beta ) q^{30} + ( 6 - 3 \beta ) q^{31} + ( -7 - 5 \beta ) q^{33} + ( -6 - \beta ) q^{34} + ( -2 - 3 \beta ) q^{37} + 6 \beta q^{38} + ( 1 + 2 \beta ) q^{39} -2 \beta q^{40} + ( -2 + 3 \beta ) q^{41} + 2 q^{43} + 2 \beta q^{45} + ( -2 + 6 \beta ) q^{46} + ( -3 + 3 \beta ) q^{47} + ( -4 - 4 \beta ) q^{48} + \beta q^{50} + ( -7 - 4 \beta ) q^{51} -3 \beta q^{53} + ( -2 + \beta ) q^{54} + ( -3 - 2 \beta ) q^{55} + ( 6 + 6 \beta ) q^{57} + ( 8 - 3 \beta ) q^{58} + ( 2 + 3 \beta ) q^{59} + 2 \beta q^{61} + ( -6 + 6 \beta ) q^{62} + 8 q^{64} + ( 3 - \beta ) q^{65} + ( -10 - 7 \beta ) q^{66} + ( -4 + 3 \beta ) q^{67} + ( 4 + 5 \beta ) q^{69} + ( -6 - 2 \beta ) q^{71} -8 q^{72} + 6 \beta q^{73} + ( -6 - 2 \beta ) q^{74} + ( 1 + \beta ) q^{75} + ( 4 + \beta ) q^{78} + ( -7 - 6 \beta ) q^{79} -4 q^{80} + ( -1 - 6 \beta ) q^{81} + ( 6 - 2 \beta ) q^{82} + ( -1 - 3 \beta ) q^{85} + 2 \beta q^{86} + ( 5 + \beta ) q^{87} + ( 8 + 6 \beta ) q^{88} -8 q^{89} + 4 q^{90} + 3 \beta q^{93} + ( 6 - 3 \beta ) q^{94} + 6 q^{95} + ( 9 - 3 \beta ) q^{97} + ( -8 - 6 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} + 2q^{5} + 4q^{6} + O(q^{10}) \) \( 2q + 2q^{3} + 2q^{5} + 4q^{6} - 6q^{11} + 6q^{13} + 2q^{15} - 8q^{16} - 2q^{17} + 8q^{18} + 12q^{19} - 8q^{22} + 12q^{23} - 8q^{24} + 2q^{25} - 4q^{26} + 2q^{27} - 6q^{29} + 4q^{30} + 12q^{31} - 14q^{33} - 12q^{34} - 4q^{37} + 2q^{39} - 4q^{41} + 4q^{43} - 4q^{46} - 6q^{47} - 8q^{48} - 14q^{51} - 4q^{54} - 6q^{55} + 12q^{57} + 16q^{58} + 4q^{59} - 12q^{62} + 16q^{64} + 6q^{65} - 20q^{66} - 8q^{67} + 8q^{69} - 12q^{71} - 16q^{72} - 12q^{74} + 2q^{75} + 8q^{78} - 14q^{79} - 8q^{80} - 2q^{81} + 12q^{82} - 2q^{85} + 10q^{87} + 16q^{88} - 16q^{89} + 8q^{90} + 12q^{94} + 12q^{95} + 18q^{97} - 16q^{99} + O(q^{100}) \)

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
−1.41421 −0.414214 0 1.00000 0.585786 0 2.82843 −2.82843 −1.41421
1.2 1.41421 2.41421 0 1.00000 3.41421 0 −2.82843 2.82843 1.41421
\(n\): 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 245.2.a.f yes 2
3.b odd 2 1 2205.2.a.t 2
4.b odd 2 1 3920.2.a.br 2
5.b even 2 1 1225.2.a.p 2
5.c odd 4 2 1225.2.b.j 4
7.b odd 2 1 245.2.a.e 2
7.c even 3 2 245.2.e.f 4
7.d odd 6 2 245.2.e.g 4
21.c even 2 1 2205.2.a.v 2
28.d even 2 1 3920.2.a.bw 2
35.c odd 2 1 1225.2.a.r 2
35.f even 4 2 1225.2.b.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.2.a.e 2 7.b odd 2 1
245.2.a.f yes 2 1.a even 1 1 trivial
245.2.e.f 4 7.c even 3 2
245.2.e.g 4 7.d odd 6 2
1225.2.a.p 2 5.b even 2 1
1225.2.a.r 2 35.c odd 2 1
1225.2.b.i 4 35.f even 4 2
1225.2.b.j 4 5.c odd 4 2
2205.2.a.t 2 3.b odd 2 1
2205.2.a.v 2 21.c even 2 1
3920.2.a.br 2 4.b odd 2 1
3920.2.a.bw 2 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(245))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2 + T^{2} \)
$3$ \( -1 - 2 T + T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( 1 + 6 T + T^{2} \)
$13$ \( 7 - 6 T + T^{2} \)
$17$ \( -17 + 2 T + T^{2} \)
$19$ \( ( -6 + T )^{2} \)
$23$ \( 34 - 12 T + T^{2} \)
$29$ \( -23 + 6 T + T^{2} \)
$31$ \( 18 - 12 T + T^{2} \)
$37$ \( -14 + 4 T + T^{2} \)
$41$ \( -14 + 4 T + T^{2} \)
$43$ \( ( -2 + T )^{2} \)
$47$ \( -9 + 6 T + T^{2} \)
$53$ \( -18 + T^{2} \)
$59$ \( -14 - 4 T + T^{2} \)
$61$ \( -8 + T^{2} \)
$67$ \( -2 + 8 T + T^{2} \)
$71$ \( 28 + 12 T + T^{2} \)
$73$ \( -72 + T^{2} \)
$79$ \( -23 + 14 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( ( 8 + T )^{2} \)
$97$ \( 63 - 18 T + T^{2} \)
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