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: \( x^{2} - 2 \) Copy content Toggle raw display
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.

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

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(245))\):

\( T_{2}^{2} - 2 \) Copy content Toggle raw display
\( T_{3}^{2} - 2T_{3} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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