Properties

Label 4410.2.a.ca
Level $4410$
Weight $2$
Character orbit 4410.a
Self dual yes
Analytic conductor $35.214$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4410.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(35.2140272914\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \(x^{2} - 7\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 630)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{5} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + q^{5} + q^{8} + q^{10} + ( 2 + \beta ) q^{11} + ( 2 - \beta ) q^{13} + q^{16} + ( 2 - 2 \beta ) q^{17} + ( -1 + 2 \beta ) q^{19} + q^{20} + ( 2 + \beta ) q^{22} + 3 q^{23} + q^{25} + ( 2 - \beta ) q^{26} -2 q^{31} + q^{32} + ( 2 - 2 \beta ) q^{34} + ( 2 - 3 \beta ) q^{37} + ( -1 + 2 \beta ) q^{38} + q^{40} + ( -4 + \beta ) q^{41} + ( 6 + 2 \beta ) q^{43} + ( 2 + \beta ) q^{44} + 3 q^{46} + ( -1 + 4 \beta ) q^{47} + q^{50} + ( 2 - \beta ) q^{52} + ( 5 - 2 \beta ) q^{53} + ( 2 + \beta ) q^{55} + ( 4 + 2 \beta ) q^{59} + ( -6 - 2 \beta ) q^{61} -2 q^{62} + q^{64} + ( 2 - \beta ) q^{65} + ( 10 - 2 \beta ) q^{67} + ( 2 - 2 \beta ) q^{68} + ( -8 + 2 \beta ) q^{71} -2 q^{73} + ( 2 - 3 \beta ) q^{74} + ( -1 + 2 \beta ) q^{76} + ( 6 + 2 \beta ) q^{79} + q^{80} + ( -4 + \beta ) q^{82} + ( -10 - 2 \beta ) q^{83} + ( 2 - 2 \beta ) q^{85} + ( 6 + 2 \beta ) q^{86} + ( 2 + \beta ) q^{88} + ( 4 - 4 \beta ) q^{89} + 3 q^{92} + ( -1 + 4 \beta ) q^{94} + ( -1 + 2 \beta ) q^{95} + 4 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + 2q^{5} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + 2q^{5} + 2q^{8} + 2q^{10} + 4q^{11} + 4q^{13} + 2q^{16} + 4q^{17} - 2q^{19} + 2q^{20} + 4q^{22} + 6q^{23} + 2q^{25} + 4q^{26} - 4q^{31} + 2q^{32} + 4q^{34} + 4q^{37} - 2q^{38} + 2q^{40} - 8q^{41} + 12q^{43} + 4q^{44} + 6q^{46} - 2q^{47} + 2q^{50} + 4q^{52} + 10q^{53} + 4q^{55} + 8q^{59} - 12q^{61} - 4q^{62} + 2q^{64} + 4q^{65} + 20q^{67} + 4q^{68} - 16q^{71} - 4q^{73} + 4q^{74} - 2q^{76} + 12q^{79} + 2q^{80} - 8q^{82} - 20q^{83} + 4q^{85} + 12q^{86} + 4q^{88} + 8q^{89} + 6q^{92} - 2q^{94} - 2q^{95} + 8q^{97} + O(q^{100}) \)

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
−2.64575
2.64575
1.00000 0 1.00000 1.00000 0 0 1.00000 0 1.00000
1.2 1.00000 0 1.00000 1.00000 0 0 1.00000 0 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(-1\)
\(7\) \(-1\)

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 4410.2.a.ca 2
3.b odd 2 1 4410.2.a.bo 2
7.b odd 2 1 4410.2.a.bv 2
7.d odd 6 2 630.2.k.i 4
21.c even 2 1 4410.2.a.bq 2
21.g even 6 2 630.2.k.j yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.k.i 4 7.d odd 6 2
630.2.k.j yes 4 21.g even 6 2
4410.2.a.bo 2 3.b odd 2 1
4410.2.a.bq 2 21.c even 2 1
4410.2.a.bv 2 7.b odd 2 1
4410.2.a.ca 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4410))\):

\( T_{11}^{2} - 4 T_{11} - 3 \)
\( T_{13}^{2} - 4 T_{13} - 3 \)
\( T_{17}^{2} - 4 T_{17} - 24 \)
\( T_{19}^{2} + 2 T_{19} - 27 \)
\( T_{29} \)
\( T_{31} + 2 \)