Properties

Label 2205.2.a.n
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 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.

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{2} + ( 1 - 2 \beta ) q^{4} - q^{5} + ( -3 + \beta ) q^{8} +O(q^{10})\) \( q + ( -1 + \beta ) q^{2} + ( 1 - 2 \beta ) q^{4} - q^{5} + ( -3 + \beta ) q^{8} + ( 1 - \beta ) q^{10} + ( -2 - 2 \beta ) q^{11} + ( -2 + 2 \beta ) q^{13} + 3 q^{16} + ( -2 + 2 \beta ) q^{17} -2 \beta q^{19} + ( -1 + 2 \beta ) q^{20} -2 q^{22} + ( 1 + \beta ) q^{23} + q^{25} + ( 6 - 4 \beta ) q^{26} + q^{29} -6 q^{31} + ( 3 + \beta ) q^{32} + ( 6 - 4 \beta ) q^{34} + ( -4 + 2 \beta ) q^{38} + ( 3 - \beta ) q^{40} + ( 5 - 2 \beta ) q^{41} + ( 5 + \beta ) q^{43} + ( 6 + 2 \beta ) q^{44} + q^{46} -2 q^{47} + ( -1 + \beta ) q^{50} + ( -10 + 6 \beta ) q^{52} + ( 4 + 2 \beta ) q^{53} + ( 2 + 2 \beta ) q^{55} + ( -1 + \beta ) q^{58} + ( 4 + 6 \beta ) q^{59} + ( -3 - 6 \beta ) q^{61} + ( 6 - 6 \beta ) q^{62} + ( -7 + 2 \beta ) q^{64} + ( 2 - 2 \beta ) q^{65} + ( 11 + \beta ) q^{67} + ( -10 + 6 \beta ) q^{68} + ( 4 + 6 \beta ) q^{71} + ( 2 + 2 \beta ) q^{73} + ( 8 - 2 \beta ) q^{76} + ( 12 - 2 \beta ) q^{79} -3 q^{80} + ( -9 + 7 \beta ) q^{82} + ( -1 + 9 \beta ) q^{83} + ( 2 - 2 \beta ) q^{85} + ( -3 + 4 \beta ) q^{86} + ( 2 + 4 \beta ) q^{88} + ( 3 - 4 \beta ) q^{89} + ( -3 - \beta ) q^{92} + ( 2 - 2 \beta ) q^{94} + 2 \beta q^{95} + ( 6 - 4 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} - 2q^{5} - 6q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} - 2q^{5} - 6q^{8} + 2q^{10} - 4q^{11} - 4q^{13} + 6q^{16} - 4q^{17} - 2q^{20} - 4q^{22} + 2q^{23} + 2q^{25} + 12q^{26} + 2q^{29} - 12q^{31} + 6q^{32} + 12q^{34} - 8q^{38} + 6q^{40} + 10q^{41} + 10q^{43} + 12q^{44} + 2q^{46} - 4q^{47} - 2q^{50} - 20q^{52} + 8q^{53} + 4q^{55} - 2q^{58} + 8q^{59} - 6q^{61} + 12q^{62} - 14q^{64} + 4q^{65} + 22q^{67} - 20q^{68} + 8q^{71} + 4q^{73} + 16q^{76} + 24q^{79} - 6q^{80} - 18q^{82} - 2q^{83} + 4q^{85} - 6q^{86} + 4q^{88} + 6q^{89} - 6q^{92} + 4q^{94} + 12q^{97} + O(q^{100}) \)

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
−2.41421 0 3.82843 −1.00000 0 0 −4.41421 0 2.41421
1.2 0.414214 0 −1.82843 −1.00000 0 0 −1.58579 0 −0.414214
\(n\): 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.n 2
3.b odd 2 1 245.2.a.h 2
7.b odd 2 1 2205.2.a.q 2
7.c even 3 2 315.2.j.e 4
12.b even 2 1 3920.2.a.bq 2
15.d odd 2 1 1225.2.a.k 2
15.e even 4 2 1225.2.b.g 4
21.c even 2 1 245.2.a.g 2
21.g even 6 2 245.2.e.e 4
21.h odd 6 2 35.2.e.a 4
84.h odd 2 1 3920.2.a.bv 2
84.n even 6 2 560.2.q.k 4
105.g even 2 1 1225.2.a.m 2
105.k odd 4 2 1225.2.b.h 4
105.o odd 6 2 175.2.e.c 4
105.x even 12 4 175.2.k.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.e.a 4 21.h odd 6 2
175.2.e.c 4 105.o odd 6 2
175.2.k.a 8 105.x even 12 4
245.2.a.g 2 21.c even 2 1
245.2.a.h 2 3.b odd 2 1
245.2.e.e 4 21.g even 6 2
315.2.j.e 4 7.c even 3 2
560.2.q.k 4 84.n even 6 2
1225.2.a.k 2 15.d odd 2 1
1225.2.a.m 2 105.g even 2 1
1225.2.b.g 4 15.e even 4 2
1225.2.b.h 4 105.k odd 4 2
2205.2.a.n 2 1.a even 1 1 trivial
2205.2.a.q 2 7.b odd 2 1
3920.2.a.bq 2 12.b even 2 1
3920.2.a.bv 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2205))\):

\( T_{2}^{2} + 2 T_{2} - 1 \)
\( T_{11}^{2} + 4 T_{11} - 4 \)
\( T_{13}^{2} + 4 T_{13} - 4 \)
\( T_{17}^{2} + 4 T_{17} - 4 \)

Hecke characteristic polynomials

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