Properties

Label 2205.2.a.t
Level $2205$
Weight $2$
Character orbit 2205.a
Self dual yes
Analytic conductor $17.607$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2205.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.6070136457\)
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 245)
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.

\(f(q)\) \(=\) \( q + \beta q^{2} - q^{5} -2 \beta q^{8} +O(q^{10})\) \( q + \beta q^{2} - q^{5} -2 \beta q^{8} -\beta q^{10} + ( 3 - 2 \beta ) q^{11} + ( 3 + \beta ) q^{13} -4 q^{16} + ( 1 - 3 \beta ) q^{17} + 6 q^{19} + ( -4 + 3 \beta ) q^{22} + ( -6 - \beta ) q^{23} + q^{25} + ( 2 + 3 \beta ) q^{26} + ( 3 + 4 \beta ) q^{29} + ( 6 + 3 \beta ) q^{31} + ( -6 + \beta ) q^{34} + ( -2 + 3 \beta ) q^{37} + 6 \beta q^{38} + 2 \beta q^{40} + ( 2 + 3 \beta ) q^{41} + 2 q^{43} + ( -2 - 6 \beta ) q^{46} + ( 3 + 3 \beta ) q^{47} + \beta q^{50} -3 \beta q^{53} + ( -3 + 2 \beta ) q^{55} + ( 8 + 3 \beta ) q^{58} + ( -2 + 3 \beta ) q^{59} -2 \beta q^{61} + ( 6 + 6 \beta ) q^{62} + 8 q^{64} + ( -3 - \beta ) q^{65} + ( -4 - 3 \beta ) q^{67} + ( 6 - 2 \beta ) q^{71} -6 \beta q^{73} + ( 6 - 2 \beta ) q^{74} + ( -7 + 6 \beta ) q^{79} + 4 q^{80} + ( 6 + 2 \beta ) q^{82} + ( -1 + 3 \beta ) q^{85} + 2 \beta q^{86} + ( 8 - 6 \beta ) q^{88} + 8 q^{89} + ( 6 + 3 \beta ) q^{94} -6 q^{95} + ( 9 + 3 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} + O(q^{10}) \) \( 2q - 2q^{5} + 6q^{11} + 6q^{13} - 8q^{16} + 2q^{17} + 12q^{19} - 8q^{22} - 12q^{23} + 2q^{25} + 4q^{26} + 6q^{29} + 12q^{31} - 12q^{34} - 4q^{37} + 4q^{41} + 4q^{43} - 4q^{46} + 6q^{47} - 6q^{55} + 16q^{58} - 4q^{59} + 12q^{62} + 16q^{64} - 6q^{65} - 8q^{67} + 12q^{71} + 12q^{74} - 14q^{79} + 8q^{80} + 12q^{82} - 2q^{85} + 16q^{88} + 16q^{89} + 12q^{94} - 12q^{95} + 18q^{97} + O(q^{100}) \)

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 0 −1.00000 0 0 2.82843 0 1.41421
1.2 1.41421 0 0 −1.00000 0 0 −2.82843 0 −1.41421
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(7\) \(1\)

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 2205.2.a.t 2
3.b odd 2 1 245.2.a.f yes 2
7.b odd 2 1 2205.2.a.v 2
12.b even 2 1 3920.2.a.br 2
15.d odd 2 1 1225.2.a.p 2
15.e even 4 2 1225.2.b.j 4
21.c even 2 1 245.2.a.e 2
21.g even 6 2 245.2.e.g 4
21.h odd 6 2 245.2.e.f 4
84.h odd 2 1 3920.2.a.bw 2
105.g even 2 1 1225.2.a.r 2
105.k odd 4 2 1225.2.b.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.2.a.e 2 21.c even 2 1
245.2.a.f yes 2 3.b odd 2 1
245.2.e.f 4 21.h odd 6 2
245.2.e.g 4 21.g even 6 2
1225.2.a.p 2 15.d odd 2 1
1225.2.a.r 2 105.g even 2 1
1225.2.b.i 4 105.k odd 4 2
1225.2.b.j 4 15.e even 4 2
2205.2.a.t 2 1.a even 1 1 trivial
2205.2.a.v 2 7.b odd 2 1
3920.2.a.br 2 12.b even 2 1
3920.2.a.bw 2 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2205))\):

\( T_{2}^{2} - 2 \)
\( T_{11}^{2} - 6 T_{11} + 1 \)
\( T_{13}^{2} - 6 T_{13} + 7 \)
\( T_{17}^{2} - 2 T_{17} - 17 \)

Hecke characteristic polynomials

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