Properties

Label 495.2.k.a
Level 495
Weight 2
Character orbit 495.k
Analytic conductor 3.953
Analytic rank 0
Dimension 4
CM discriminant -11
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 495.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.95259490005\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{11})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 \beta_{2} q^{4} + ( -\beta_{1} - \beta_{2} ) q^{5} +O(q^{10})\) \( q -2 \beta_{2} q^{4} + ( -\beta_{1} - \beta_{2} ) q^{5} + ( -2 \beta_{1} + \beta_{2} ) q^{11} -4 q^{16} + ( -2 + 2 \beta_{3} ) q^{20} + ( -4 + \beta_{1} - 5 \beta_{2} + \beta_{3} ) q^{23} + ( 2 + 3 \beta_{3} ) q^{25} + ( -6 \beta_{1} + 3 \beta_{2} ) q^{31} + ( -5 + 3 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{37} + ( 2 + 4 \beta_{3} ) q^{44} + ( 5 + 2 \beta_{1} - 7 \beta_{2} - 2 \beta_{3} ) q^{47} -7 \beta_{2} q^{49} + ( 1 - 4 \beta_{1} + 5 \beta_{2} - 4 \beta_{3} ) q^{53} + ( 7 + 3 \beta_{3} ) q^{55} + ( -1 - 2 \beta_{3} ) q^{59} + 8 \beta_{2} q^{64} + ( 5 + 3 \beta_{1} - 8 \beta_{2} - 3 \beta_{3} ) q^{67} + 3 q^{71} + ( 4 \beta_{1} + 4 \beta_{2} ) q^{80} + 9 \beta_{2} q^{89} + ( -10 + 2 \beta_{1} + 8 \beta_{2} - 2 \beta_{3} ) q^{92} + ( -10 + 3 \beta_{1} + 7 \beta_{2} - 3 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 16q^{16} - 12q^{20} - 18q^{23} + 2q^{25} - 14q^{37} + 24q^{47} + 12q^{53} + 22q^{55} + 26q^{67} + 12q^{71} - 36q^{92} - 34q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 5 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 2 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\( \nu^{2} - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 3\)
\(\nu^{3}\)\(=\)\(3 \beta_{2} + 2 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/495\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\) \(397\)
\(\chi(n)\) \(-1\) \(1\) \(-\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
208.1
1.65831 + 0.500000i
−1.65831 + 0.500000i
1.65831 0.500000i
−1.65831 0.500000i
0 0 2.00000i −1.65831 1.50000i 0 0 0 0 0
208.2 0 0 2.00000i 1.65831 1.50000i 0 0 0 0 0
307.1 0 0 2.00000i −1.65831 + 1.50000i 0 0 0 0 0
307.2 0 0 2.00000i 1.65831 + 1.50000i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
5.c odd 4 1 inner
55.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.2.k.a 4
3.b odd 2 1 55.2.e.b 4
5.c odd 4 1 inner 495.2.k.a 4
11.b odd 2 1 CM 495.2.k.a 4
12.b even 2 1 880.2.bd.d 4
15.d odd 2 1 275.2.e.a 4
15.e even 4 1 55.2.e.b 4
15.e even 4 1 275.2.e.a 4
33.d even 2 1 55.2.e.b 4
33.f even 10 4 605.2.m.a 16
33.h odd 10 4 605.2.m.a 16
55.e even 4 1 inner 495.2.k.a 4
60.l odd 4 1 880.2.bd.d 4
132.d odd 2 1 880.2.bd.d 4
165.d even 2 1 275.2.e.a 4
165.l odd 4 1 55.2.e.b 4
165.l odd 4 1 275.2.e.a 4
165.u odd 20 4 605.2.m.a 16
165.v even 20 4 605.2.m.a 16
660.q even 4 1 880.2.bd.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.e.b 4 3.b odd 2 1
55.2.e.b 4 15.e even 4 1
55.2.e.b 4 33.d even 2 1
55.2.e.b 4 165.l odd 4 1
275.2.e.a 4 15.d odd 2 1
275.2.e.a 4 15.e even 4 1
275.2.e.a 4 165.d even 2 1
275.2.e.a 4 165.l odd 4 1
495.2.k.a 4 1.a even 1 1 trivial
495.2.k.a 4 5.c odd 4 1 inner
495.2.k.a 4 11.b odd 2 1 CM
495.2.k.a 4 55.e even 4 1 inner
605.2.m.a 16 33.f even 10 4
605.2.m.a 16 33.h odd 10 4
605.2.m.a 16 165.u odd 20 4
605.2.m.a 16 165.v even 20 4
880.2.bd.d 4 12.b even 2 1
880.2.bd.d 4 60.l odd 4 1
880.2.bd.d 4 132.d odd 2 1
880.2.bd.d 4 660.q even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(495, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T + 2 T^{2} )^{2}( 1 + 2 T + 2 T^{2} )^{2} \)
$3$ \( \)
$5$ \( 1 - T^{2} + 25 T^{4} \)
$7$ \( ( 1 + 49 T^{4} )^{2} \)
$11$ \( ( 1 - 11 T^{2} )^{2} \)
$13$ \( ( 1 + 169 T^{4} )^{2} \)
$17$ \( ( 1 + 289 T^{4} )^{2} \)
$19$ \( ( 1 + 19 T^{2} )^{4} \)
$23$ \( ( 1 + 9 T + 23 T^{2} )^{2}( 1 + 35 T^{2} + 529 T^{4} ) \)
$29$ \( ( 1 + 29 T^{2} )^{4} \)
$31$ \( ( 1 - 37 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 + 7 T + 37 T^{2} )^{2}( 1 - 25 T^{2} + 1369 T^{4} ) \)
$41$ \( ( 1 - 41 T^{2} )^{4} \)
$43$ \( ( 1 + 1849 T^{4} )^{2} \)
$47$ \( ( 1 - 12 T + 47 T^{2} )^{2}( 1 + 50 T^{2} + 2209 T^{4} ) \)
$53$ \( ( 1 - 6 T + 53 T^{2} )^{2}( 1 - 70 T^{2} + 2809 T^{4} ) \)
$59$ \( ( 1 - 15 T + 59 T^{2} )^{2}( 1 + 15 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 61 T^{2} )^{4} \)
$67$ \( ( 1 - 13 T + 67 T^{2} )^{2}( 1 + 35 T^{2} + 4489 T^{4} ) \)
$71$ \( ( 1 - 3 T + 71 T^{2} )^{4} \)
$73$ \( ( 1 + 5329 T^{4} )^{2} \)
$79$ \( ( 1 + 79 T^{2} )^{4} \)
$83$ \( ( 1 + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 97 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 + 17 T + 97 T^{2} )^{2}( 1 + 95 T^{2} + 9409 T^{4} ) \)
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