Properties

Label 4410.2.a.by
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: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 490)
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{5} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + q^{5} + q^{8} + q^{10} + ( -2 + 2 \beta ) q^{11} + ( 2 + 2 \beta ) q^{13} + q^{16} + ( 4 - \beta ) q^{17} + ( -2 + \beta ) q^{19} + q^{20} + ( -2 + 2 \beta ) q^{22} + ( 4 - 2 \beta ) q^{23} + q^{25} + ( 2 + 2 \beta ) q^{26} + ( 2 + 2 \beta ) q^{29} + 2 \beta q^{31} + q^{32} + ( 4 - \beta ) q^{34} + ( -2 - 4 \beta ) q^{37} + ( -2 + \beta ) q^{38} + q^{40} + ( 4 - 5 \beta ) q^{41} + ( -6 - 2 \beta ) q^{43} + ( -2 + 2 \beta ) q^{44} + ( 4 - 2 \beta ) q^{46} + ( 8 - 2 \beta ) q^{47} + q^{50} + ( 2 + 2 \beta ) q^{52} + ( 2 - 6 \beta ) q^{53} + ( -2 + 2 \beta ) q^{55} + ( 2 + 2 \beta ) q^{58} + ( 10 - \beta ) q^{59} + ( 2 - 8 \beta ) q^{61} + 2 \beta q^{62} + q^{64} + ( 2 + 2 \beta ) q^{65} + ( -4 + 4 \beta ) q^{67} + ( 4 - \beta ) q^{68} + ( -4 + 6 \beta ) q^{71} + ( 8 + \beta ) q^{73} + ( -2 - 4 \beta ) q^{74} + ( -2 + \beta ) q^{76} + ( -4 - 2 \beta ) q^{79} + q^{80} + ( 4 - 5 \beta ) q^{82} + ( 2 - 3 \beta ) q^{83} + ( 4 - \beta ) q^{85} + ( -6 - 2 \beta ) q^{86} + ( -2 + 2 \beta ) q^{88} + 9 \beta q^{89} + ( 4 - 2 \beta ) q^{92} + ( 8 - 2 \beta ) q^{94} + ( -2 + \beta ) q^{95} + ( 12 - 3 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + 2q^{5} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + 2q^{5} + 2q^{8} + 2q^{10} - 4q^{11} + 4q^{13} + 2q^{16} + 8q^{17} - 4q^{19} + 2q^{20} - 4q^{22} + 8q^{23} + 2q^{25} + 4q^{26} + 4q^{29} + 2q^{32} + 8q^{34} - 4q^{37} - 4q^{38} + 2q^{40} + 8q^{41} - 12q^{43} - 4q^{44} + 8q^{46} + 16q^{47} + 2q^{50} + 4q^{52} + 4q^{53} - 4q^{55} + 4q^{58} + 20q^{59} + 4q^{61} + 2q^{64} + 4q^{65} - 8q^{67} + 8q^{68} - 8q^{71} + 16q^{73} - 4q^{74} - 4q^{76} - 8q^{79} + 2q^{80} + 8q^{82} + 4q^{83} + 8q^{85} - 12q^{86} - 4q^{88} + 8q^{92} + 16q^{94} - 4q^{95} + 24q^{97} + O(q^{100}) \)

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
1.00000 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.by 2
3.b odd 2 1 490.2.a.l 2
7.b odd 2 1 4410.2.a.bt 2
12.b even 2 1 3920.2.a.ca 2
15.d odd 2 1 2450.2.a.bs 2
15.e even 4 2 2450.2.c.w 4
21.c even 2 1 490.2.a.m yes 2
21.g even 6 2 490.2.e.i 4
21.h odd 6 2 490.2.e.j 4
84.h odd 2 1 3920.2.a.bm 2
105.g even 2 1 2450.2.a.bn 2
105.k odd 4 2 2450.2.c.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.2.a.l 2 3.b odd 2 1
490.2.a.m yes 2 21.c even 2 1
490.2.e.i 4 21.g even 6 2
490.2.e.j 4 21.h odd 6 2
2450.2.a.bn 2 105.g even 2 1
2450.2.a.bs 2 15.d odd 2 1
2450.2.c.t 4 105.k odd 4 2
2450.2.c.w 4 15.e even 4 2
3920.2.a.bm 2 84.h odd 2 1
3920.2.a.ca 2 12.b even 2 1
4410.2.a.bt 2 7.b odd 2 1
4410.2.a.by 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4410))\):

\( T_{11}^{2} + 4 T_{11} - 4 \)
\( T_{13}^{2} - 4 T_{13} - 4 \)
\( T_{17}^{2} - 8 T_{17} + 14 \)
\( T_{19}^{2} + 4 T_{19} + 2 \)
\( T_{29}^{2} - 4 T_{29} - 4 \)
\( T_{31}^{2} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{2} \)
$3$ 1
$5$ \( ( 1 - T )^{2} \)
$7$ 1
$11$ \( 1 + 4 T + 18 T^{2} + 44 T^{3} + 121 T^{4} \)
$13$ \( 1 - 4 T + 22 T^{2} - 52 T^{3} + 169 T^{4} \)
$17$ \( 1 - 8 T + 48 T^{2} - 136 T^{3} + 289 T^{4} \)
$19$ \( 1 + 4 T + 40 T^{2} + 76 T^{3} + 361 T^{4} \)
$23$ \( 1 - 8 T + 54 T^{2} - 184 T^{3} + 529 T^{4} \)
$29$ \( 1 - 4 T + 54 T^{2} - 116 T^{3} + 841 T^{4} \)
$31$ \( 1 + 54 T^{2} + 961 T^{4} \)
$37$ \( 1 + 4 T + 46 T^{2} + 148 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 8 T + 48 T^{2} - 328 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 12 T + 114 T^{2} + 516 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 16 T + 150 T^{2} - 752 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 4 T + 38 T^{2} - 212 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 20 T + 216 T^{2} - 1180 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 4 T - 2 T^{2} - 244 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 8 T + 118 T^{2} + 536 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 8 T + 86 T^{2} + 568 T^{3} + 5041 T^{4} \)
$73$ \( 1 - 16 T + 208 T^{2} - 1168 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 8 T + 166 T^{2} + 632 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 4 T + 152 T^{2} - 332 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 16 T^{2} + 7921 T^{4} \)
$97$ \( 1 - 24 T + 320 T^{2} - 2328 T^{3} + 9409 T^{4} \)
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