Properties

Label 2205.2.a.x
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: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( 2 + \beta ) q^{4} + q^{5} + ( 4 + \beta ) q^{8} +O(q^{10})\) \( q + \beta q^{2} + ( 2 + \beta ) q^{4} + q^{5} + ( 4 + \beta ) q^{8} + \beta q^{10} + ( -1 + \beta ) q^{11} + ( -3 + \beta ) q^{13} + 3 \beta q^{16} + ( -3 + \beta ) q^{17} + ( 2 + 2 \beta ) q^{19} + ( 2 + \beta ) q^{20} + 4 q^{22} + ( 2 - 2 \beta ) q^{23} + q^{25} + ( 4 - 2 \beta ) q^{26} + ( 1 - 3 \beta ) q^{29} + ( 4 + \beta ) q^{32} + ( 4 - 2 \beta ) q^{34} + 6 q^{37} + ( 8 + 4 \beta ) q^{38} + ( 4 + \beta ) q^{40} + 2 \beta q^{41} + ( 6 - 2 \beta ) q^{43} + ( 2 + 2 \beta ) q^{44} -8 q^{46} + ( -1 - 3 \beta ) q^{47} + \beta q^{50} -2 q^{52} + 2 \beta q^{53} + ( -1 + \beta ) q^{55} + ( -12 - 2 \beta ) q^{58} -4 q^{59} -6 \beta q^{61} + ( 4 - \beta ) q^{64} + ( -3 + \beta ) q^{65} + 4 \beta q^{67} -2 q^{68} -8 q^{71} + ( 2 + 4 \beta ) q^{73} + 6 \beta q^{74} + ( 12 + 8 \beta ) q^{76} + ( -5 + \beta ) q^{79} + 3 \beta q^{80} + ( 8 + 2 \beta ) q^{82} + 4 q^{83} + ( -3 + \beta ) q^{85} + ( -8 + 4 \beta ) q^{86} + 4 \beta q^{88} + ( 4 - 2 \beta ) q^{89} + ( -4 - 4 \beta ) q^{92} + ( -12 - 4 \beta ) q^{94} + ( 2 + 2 \beta ) q^{95} + ( 7 - 5 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + 5q^{4} + 2q^{5} + 9q^{8} + O(q^{10}) \) \( 2q + q^{2} + 5q^{4} + 2q^{5} + 9q^{8} + q^{10} - q^{11} - 5q^{13} + 3q^{16} - 5q^{17} + 6q^{19} + 5q^{20} + 8q^{22} + 2q^{23} + 2q^{25} + 6q^{26} - q^{29} + 9q^{32} + 6q^{34} + 12q^{37} + 20q^{38} + 9q^{40} + 2q^{41} + 10q^{43} + 6q^{44} - 16q^{46} - 5q^{47} + q^{50} - 4q^{52} + 2q^{53} - q^{55} - 26q^{58} - 8q^{59} - 6q^{61} + 7q^{64} - 5q^{65} + 4q^{67} - 4q^{68} - 16q^{71} + 8q^{73} + 6q^{74} + 32q^{76} - 9q^{79} + 3q^{80} + 18q^{82} + 8q^{83} - 5q^{85} - 12q^{86} + 4q^{88} + 6q^{89} - 12q^{92} - 28q^{94} + 6q^{95} + 9q^{97} + O(q^{100}) \)

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
−1.56155
2.56155
−1.56155 0 0.438447 1.00000 0 0 2.43845 0 −1.56155
1.2 2.56155 0 4.56155 1.00000 0 0 6.56155 0 2.56155
\(n\): 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.x 2
3.b odd 2 1 245.2.a.d 2
7.b odd 2 1 315.2.a.e 2
12.b even 2 1 3920.2.a.bs 2
15.d odd 2 1 1225.2.a.s 2
15.e even 4 2 1225.2.b.f 4
21.c even 2 1 35.2.a.b 2
21.g even 6 2 245.2.e.i 4
21.h odd 6 2 245.2.e.h 4
28.d even 2 1 5040.2.a.bt 2
35.c odd 2 1 1575.2.a.p 2
35.f even 4 2 1575.2.d.e 4
84.h odd 2 1 560.2.a.i 2
105.g even 2 1 175.2.a.f 2
105.k odd 4 2 175.2.b.b 4
168.e odd 2 1 2240.2.a.bd 2
168.i even 2 1 2240.2.a.bh 2
231.h odd 2 1 4235.2.a.m 2
273.g even 2 1 5915.2.a.l 2
420.o odd 2 1 2800.2.a.bi 2
420.w even 4 2 2800.2.g.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.b 2 21.c even 2 1
175.2.a.f 2 105.g even 2 1
175.2.b.b 4 105.k odd 4 2
245.2.a.d 2 3.b odd 2 1
245.2.e.h 4 21.h odd 6 2
245.2.e.i 4 21.g even 6 2
315.2.a.e 2 7.b odd 2 1
560.2.a.i 2 84.h odd 2 1
1225.2.a.s 2 15.d odd 2 1
1225.2.b.f 4 15.e even 4 2
1575.2.a.p 2 35.c odd 2 1
1575.2.d.e 4 35.f even 4 2
2205.2.a.x 2 1.a even 1 1 trivial
2240.2.a.bd 2 168.e odd 2 1
2240.2.a.bh 2 168.i even 2 1
2800.2.a.bi 2 420.o odd 2 1
2800.2.g.t 4 420.w even 4 2
3920.2.a.bs 2 12.b even 2 1
4235.2.a.m 2 231.h odd 2 1
5040.2.a.bt 2 28.d even 2 1
5915.2.a.l 2 273.g even 2 1

Hecke kernels

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

\( T_{2}^{2} - T_{2} - 4 \)
\( T_{11}^{2} + T_{11} - 4 \)
\( T_{13}^{2} + 5 T_{13} + 2 \)
\( T_{17}^{2} + 5 T_{17} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 - T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( -4 + T + T^{2} \)
$13$ \( 2 + 5 T + T^{2} \)
$17$ \( 2 + 5 T + T^{2} \)
$19$ \( -8 - 6 T + T^{2} \)
$23$ \( -16 - 2 T + T^{2} \)
$29$ \( -38 + T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( ( -6 + T )^{2} \)
$41$ \( -16 - 2 T + T^{2} \)
$43$ \( 8 - 10 T + T^{2} \)
$47$ \( -32 + 5 T + T^{2} \)
$53$ \( -16 - 2 T + T^{2} \)
$59$ \( ( 4 + T )^{2} \)
$61$ \( -144 + 6 T + T^{2} \)
$67$ \( -64 - 4 T + T^{2} \)
$71$ \( ( 8 + T )^{2} \)
$73$ \( -52 - 8 T + T^{2} \)
$79$ \( 16 + 9 T + T^{2} \)
$83$ \( ( -4 + T )^{2} \)
$89$ \( -8 - 6 T + T^{2} \)
$97$ \( -86 - 9 T + T^{2} \)
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