Properties

Label 245.2.a.h
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: 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.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{2} + ( - \beta + 1) q^{3} + (2 \beta + 1) q^{4} + q^{5} - q^{6} + (\beta + 3) q^{8} - 2 \beta q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{2} + ( - \beta + 1) q^{3} + (2 \beta + 1) q^{4} + q^{5} - q^{6} + (\beta + 3) q^{8} - 2 \beta q^{9} + (\beta + 1) q^{10} + ( - 2 \beta + 2) q^{11} + (\beta - 3) q^{12} + ( - 2 \beta - 2) q^{13} + ( - \beta + 1) q^{15} + 3 q^{16} + (2 \beta + 2) q^{17} + ( - 2 \beta - 4) q^{18} + 2 \beta q^{19} + (2 \beta + 1) q^{20} - 2 q^{22} + (\beta - 1) q^{23} + ( - 2 \beta + 1) q^{24} + q^{25} + ( - 4 \beta - 6) q^{26} + (\beta + 1) q^{27} - q^{29} - q^{30} - 6 q^{31} + (\beta - 3) q^{32} + ( - 4 \beta + 6) q^{33} + (4 \beta + 6) q^{34} + ( - 2 \beta - 8) q^{36} + (2 \beta + 4) q^{38} + 2 q^{39} + (\beta + 3) q^{40} + ( - 2 \beta - 5) q^{41} + ( - \beta + 5) q^{43} + (2 \beta - 6) q^{44} - 2 \beta q^{45} + q^{46} + 2 q^{47} + ( - 3 \beta + 3) q^{48} + (\beta + 1) q^{50} - 2 q^{51} + ( - 6 \beta - 10) q^{52} + (2 \beta - 4) q^{53} + (2 \beta + 3) q^{54} + ( - 2 \beta + 2) q^{55} + (2 \beta - 4) q^{57} + ( - \beta - 1) q^{58} + (6 \beta - 4) q^{59} + (\beta - 3) q^{60} + (6 \beta - 3) q^{61} + ( - 6 \beta - 6) q^{62} + ( - 2 \beta - 7) q^{64} + ( - 2 \beta - 2) q^{65} + (2 \beta - 2) q^{66} + ( - \beta + 11) q^{67} + (6 \beta + 10) q^{68} + (2 \beta - 3) q^{69} + (6 \beta - 4) q^{71} + ( - 6 \beta - 4) q^{72} + ( - 2 \beta + 2) q^{73} + ( - \beta + 1) q^{75} + (2 \beta + 8) q^{76} + (2 \beta + 2) q^{78} + (2 \beta + 12) q^{79} + 3 q^{80} + (6 \beta - 1) q^{81} + ( - 7 \beta - 9) q^{82} + (9 \beta + 1) q^{83} + (2 \beta + 2) q^{85} + (4 \beta + 3) q^{86} + (\beta - 1) q^{87} + ( - 4 \beta + 2) q^{88} + ( - 4 \beta - 3) q^{89} + ( - 2 \beta - 4) q^{90} + ( - \beta + 3) q^{92} + (6 \beta - 6) q^{93} + (2 \beta + 2) q^{94} + 2 \beta q^{95} + (4 \beta - 5) q^{96} + (4 \beta + 6) q^{97} + ( - 4 \beta + 8) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} + 6 q^{8} + 2 q^{10} + 4 q^{11} - 6 q^{12} - 4 q^{13} + 2 q^{15} + 6 q^{16} + 4 q^{17} - 8 q^{18} + 2 q^{20} - 4 q^{22} - 2 q^{23} + 2 q^{24} + 2 q^{25} - 12 q^{26} + 2 q^{27} - 2 q^{29} - 2 q^{30} - 12 q^{31} - 6 q^{32} + 12 q^{33} + 12 q^{34} - 16 q^{36} + 8 q^{38} + 4 q^{39} + 6 q^{40} - 10 q^{41} + 10 q^{43} - 12 q^{44} + 2 q^{46} + 4 q^{47} + 6 q^{48} + 2 q^{50} - 4 q^{51} - 20 q^{52} - 8 q^{53} + 6 q^{54} + 4 q^{55} - 8 q^{57} - 2 q^{58} - 8 q^{59} - 6 q^{60} - 6 q^{61} - 12 q^{62} - 14 q^{64} - 4 q^{65} - 4 q^{66} + 22 q^{67} + 20 q^{68} - 6 q^{69} - 8 q^{71} - 8 q^{72} + 4 q^{73} + 2 q^{75} + 16 q^{76} + 4 q^{78} + 24 q^{79} + 6 q^{80} - 2 q^{81} - 18 q^{82} + 2 q^{83} + 4 q^{85} + 6 q^{86} - 2 q^{87} + 4 q^{88} - 6 q^{89} - 8 q^{90} + 6 q^{92} - 12 q^{93} + 4 q^{94} - 10 q^{96} + 12 q^{97} + 16 q^{99}+O(q^{100}) \) 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
−0.414214 2.41421 −1.82843 1.00000 −1.00000 0 1.58579 2.82843 −0.414214
1.2 2.41421 −0.414214 3.82843 1.00000 −1.00000 0 4.41421 −2.82843 2.41421
\(n\): 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.h 2
3.b odd 2 1 2205.2.a.n 2
4.b odd 2 1 3920.2.a.bq 2
5.b even 2 1 1225.2.a.k 2
5.c odd 4 2 1225.2.b.g 4
7.b odd 2 1 245.2.a.g 2
7.c even 3 2 35.2.e.a 4
7.d odd 6 2 245.2.e.e 4
21.c even 2 1 2205.2.a.q 2
21.h odd 6 2 315.2.j.e 4
28.d even 2 1 3920.2.a.bv 2
28.g odd 6 2 560.2.q.k 4
35.c odd 2 1 1225.2.a.m 2
35.f even 4 2 1225.2.b.h 4
35.j even 6 2 175.2.e.c 4
35.l odd 12 4 175.2.k.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.e.a 4 7.c even 3 2
175.2.e.c 4 35.j even 6 2
175.2.k.a 8 35.l odd 12 4
245.2.a.g 2 7.b odd 2 1
245.2.a.h 2 1.a even 1 1 trivial
245.2.e.e 4 7.d odd 6 2
315.2.j.e 4 21.h odd 6 2
560.2.q.k 4 28.g odd 6 2
1225.2.a.k 2 5.b even 2 1
1225.2.a.m 2 35.c odd 2 1
1225.2.b.g 4 5.c odd 4 2
1225.2.b.h 4 35.f even 4 2
2205.2.a.n 2 3.b odd 2 1
2205.2.a.q 2 21.c even 2 1
3920.2.a.bq 2 4.b odd 2 1
3920.2.a.bv 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(245))\):

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

Hecke characteristic polynomials

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