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: \(x^{2} - 2\)
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.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{2} + ( 1 - \beta ) q^{3} + ( 1 + 2 \beta ) q^{4} + q^{5} - q^{6} + ( 3 + \beta ) q^{8} -2 \beta q^{9} +O(q^{10})\) \( q + ( 1 + \beta ) q^{2} + ( 1 - \beta ) q^{3} + ( 1 + 2 \beta ) q^{4} + q^{5} - q^{6} + ( 3 + \beta ) q^{8} -2 \beta q^{9} + ( 1 + \beta ) q^{10} + ( 2 - 2 \beta ) q^{11} + ( -3 + \beta ) q^{12} + ( -2 - 2 \beta ) q^{13} + ( 1 - \beta ) q^{15} + 3 q^{16} + ( 2 + 2 \beta ) q^{17} + ( -4 - 2 \beta ) q^{18} + 2 \beta q^{19} + ( 1 + 2 \beta ) q^{20} -2 q^{22} + ( -1 + \beta ) q^{23} + ( 1 - 2 \beta ) q^{24} + q^{25} + ( -6 - 4 \beta ) q^{26} + ( 1 + \beta ) q^{27} - q^{29} - q^{30} -6 q^{31} + ( -3 + \beta ) q^{32} + ( 6 - 4 \beta ) q^{33} + ( 6 + 4 \beta ) q^{34} + ( -8 - 2 \beta ) q^{36} + ( 4 + 2 \beta ) q^{38} + 2 q^{39} + ( 3 + \beta ) q^{40} + ( -5 - 2 \beta ) q^{41} + ( 5 - \beta ) q^{43} + ( -6 + 2 \beta ) q^{44} -2 \beta q^{45} + q^{46} + 2 q^{47} + ( 3 - 3 \beta ) q^{48} + ( 1 + \beta ) q^{50} -2 q^{51} + ( -10 - 6 \beta ) q^{52} + ( -4 + 2 \beta ) q^{53} + ( 3 + 2 \beta ) q^{54} + ( 2 - 2 \beta ) q^{55} + ( -4 + 2 \beta ) q^{57} + ( -1 - \beta ) q^{58} + ( -4 + 6 \beta ) q^{59} + ( -3 + \beta ) q^{60} + ( -3 + 6 \beta ) q^{61} + ( -6 - 6 \beta ) q^{62} + ( -7 - 2 \beta ) q^{64} + ( -2 - 2 \beta ) q^{65} + ( -2 + 2 \beta ) q^{66} + ( 11 - \beta ) q^{67} + ( 10 + 6 \beta ) q^{68} + ( -3 + 2 \beta ) q^{69} + ( -4 + 6 \beta ) q^{71} + ( -4 - 6 \beta ) q^{72} + ( 2 - 2 \beta ) q^{73} + ( 1 - \beta ) q^{75} + ( 8 + 2 \beta ) q^{76} + ( 2 + 2 \beta ) q^{78} + ( 12 + 2 \beta ) q^{79} + 3 q^{80} + ( -1 + 6 \beta ) q^{81} + ( -9 - 7 \beta ) q^{82} + ( 1 + 9 \beta ) q^{83} + ( 2 + 2 \beta ) q^{85} + ( 3 + 4 \beta ) q^{86} + ( -1 + \beta ) q^{87} + ( 2 - 4 \beta ) q^{88} + ( -3 - 4 \beta ) q^{89} + ( -4 - 2 \beta ) q^{90} + ( 3 - \beta ) q^{92} + ( -6 + 6 \beta ) q^{93} + ( 2 + 2 \beta ) q^{94} + 2 \beta q^{95} + ( -5 + 4 \beta ) q^{96} + ( 6 + 4 \beta ) q^{97} + ( 8 - 4 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 6q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 6q^{8} + 2q^{10} + 4q^{11} - 6q^{12} - 4q^{13} + 2q^{15} + 6q^{16} + 4q^{17} - 8q^{18} + 2q^{20} - 4q^{22} - 2q^{23} + 2q^{24} + 2q^{25} - 12q^{26} + 2q^{27} - 2q^{29} - 2q^{30} - 12q^{31} - 6q^{32} + 12q^{33} + 12q^{34} - 16q^{36} + 8q^{38} + 4q^{39} + 6q^{40} - 10q^{41} + 10q^{43} - 12q^{44} + 2q^{46} + 4q^{47} + 6q^{48} + 2q^{50} - 4q^{51} - 20q^{52} - 8q^{53} + 6q^{54} + 4q^{55} - 8q^{57} - 2q^{58} - 8q^{59} - 6q^{60} - 6q^{61} - 12q^{62} - 14q^{64} - 4q^{65} - 4q^{66} + 22q^{67} + 20q^{68} - 6q^{69} - 8q^{71} - 8q^{72} + 4q^{73} + 2q^{75} + 16q^{76} + 4q^{78} + 24q^{79} + 6q^{80} - 2q^{81} - 18q^{82} + 2q^{83} + 4q^{85} + 6q^{86} - 2q^{87} + 4q^{88} - 6q^{89} - 8q^{90} + 6q^{92} - 12q^{93} + 4q^{94} - 10q^{96} + 12q^{97} + 16q^{99} + O(q^{100}) \)

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

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

Hecke characteristic polynomials

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