Properties

Label 495.2.d.a
Level $495$
Weight $2$
Character orbit 495.d
Analytic conductor $3.953$
Analytic rank $0$
Dimension $4$
CM no
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.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.95259490005\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\zeta_{8}^{2} q^{2} + q^{4} + ( 2 \zeta_{8} - \zeta_{8}^{3} ) q^{5} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{7} -3 \zeta_{8}^{2} q^{8} +O(q^{10})\) \( q -\zeta_{8}^{2} q^{2} + q^{4} + ( 2 \zeta_{8} - \zeta_{8}^{3} ) q^{5} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{7} -3 \zeta_{8}^{2} q^{8} + ( -\zeta_{8} - 2 \zeta_{8}^{3} ) q^{10} + ( -3 + \zeta_{8} + \zeta_{8}^{3} ) q^{11} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{13} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{14} - q^{16} -2 \zeta_{8}^{2} q^{17} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{19} + ( 2 \zeta_{8} - \zeta_{8}^{3} ) q^{20} + ( \zeta_{8} + 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{22} + ( 4 + 3 \zeta_{8}^{2} ) q^{25} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{26} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{28} -6 q^{29} -5 \zeta_{8}^{2} q^{32} -2 q^{34} + ( 3 + \zeta_{8}^{2} ) q^{35} -6 \zeta_{8}^{2} q^{37} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{38} + ( -3 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{40} + 6 q^{41} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{43} + ( -3 + \zeta_{8} + \zeta_{8}^{3} ) q^{44} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{47} -5 q^{49} + ( 3 - 4 \zeta_{8}^{2} ) q^{50} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{52} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{53} + ( -1 - 6 \zeta_{8} + 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{55} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{56} + 6 \zeta_{8}^{2} q^{58} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{59} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{61} -7 q^{64} + ( 6 + 2 \zeta_{8}^{2} ) q^{65} -2 \zeta_{8}^{2} q^{68} + ( 1 - 3 \zeta_{8}^{2} ) q^{70} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{71} + ( -8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{73} -6 q^{74} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{76} + ( -3 \zeta_{8} + 2 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{77} + ( 9 \zeta_{8} + 9 \zeta_{8}^{3} ) q^{79} + ( -2 \zeta_{8} + \zeta_{8}^{3} ) q^{80} -6 \zeta_{8}^{2} q^{82} + 14 \zeta_{8}^{2} q^{83} + ( -2 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{85} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{86} + ( 3 \zeta_{8} + 9 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{88} + ( 5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{89} + 4 q^{91} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{94} + ( -3 + 9 \zeta_{8}^{2} ) q^{95} -12 \zeta_{8}^{2} q^{97} + 5 \zeta_{8}^{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} + O(q^{10}) \) \( 4 q + 4 q^{4} - 12 q^{11} - 4 q^{16} + 16 q^{25} - 24 q^{29} - 8 q^{34} + 12 q^{35} + 24 q^{41} - 12 q^{44} - 20 q^{49} + 12 q^{50} - 4 q^{55} - 28 q^{64} + 24 q^{65} + 4 q^{70} - 24 q^{74} + 16 q^{91} - 12 q^{95} + O(q^{100}) \)

Character values

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

\(n\) \(46\) \(56\) \(397\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
494.1
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
1.00000i 0 1.00000 −2.12132 0.707107i 0 −1.41421 3.00000i 0 −0.707107 + 2.12132i
494.2 1.00000i 0 1.00000 2.12132 + 0.707107i 0 1.41421 3.00000i 0 0.707107 2.12132i
494.3 1.00000i 0 1.00000 −2.12132 + 0.707107i 0 −1.41421 3.00000i 0 −0.707107 2.12132i
494.4 1.00000i 0 1.00000 2.12132 0.707107i 0 1.41421 3.00000i 0 0.707107 + 2.12132i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
33.d even 2 1 inner
165.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.2.d.a 4
3.b odd 2 1 495.2.d.b yes 4
5.b even 2 1 inner 495.2.d.a 4
5.c odd 4 1 2475.2.f.a 2
5.c odd 4 1 2475.2.f.c 2
11.b odd 2 1 495.2.d.b yes 4
15.d odd 2 1 495.2.d.b yes 4
15.e even 4 1 2475.2.f.b 2
15.e even 4 1 2475.2.f.d 2
33.d even 2 1 inner 495.2.d.a 4
55.d odd 2 1 495.2.d.b yes 4
55.e even 4 1 2475.2.f.b 2
55.e even 4 1 2475.2.f.d 2
165.d even 2 1 inner 495.2.d.a 4
165.l odd 4 1 2475.2.f.a 2
165.l odd 4 1 2475.2.f.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.2.d.a 4 1.a even 1 1 trivial
495.2.d.a 4 5.b even 2 1 inner
495.2.d.a 4 33.d even 2 1 inner
495.2.d.a 4 165.d even 2 1 inner
495.2.d.b yes 4 3.b odd 2 1
495.2.d.b yes 4 11.b odd 2 1
495.2.d.b yes 4 15.d odd 2 1
495.2.d.b yes 4 55.d odd 2 1
2475.2.f.a 2 5.c odd 4 1
2475.2.f.a 2 165.l odd 4 1
2475.2.f.b 2 15.e even 4 1
2475.2.f.b 2 55.e even 4 1
2475.2.f.c 2 5.c odd 4 1
2475.2.f.c 2 165.l odd 4 1
2475.2.f.d 2 15.e even 4 1
2475.2.f.d 2 55.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(495, [\chi])\):

\( T_{2}^{2} + 1 \)
\( T_{29} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( T^{4} \)
$5$ \( 25 - 8 T^{2} + T^{4} \)
$7$ \( ( -2 + T^{2} )^{2} \)
$11$ \( ( 11 + 6 T + T^{2} )^{2} \)
$13$ \( ( -8 + T^{2} )^{2} \)
$17$ \( ( 4 + T^{2} )^{2} \)
$19$ \( ( 18 + T^{2} )^{2} \)
$23$ \( T^{4} \)
$29$ \( ( 6 + T )^{4} \)
$31$ \( T^{4} \)
$37$ \( ( 36 + T^{2} )^{2} \)
$41$ \( ( -6 + T )^{4} \)
$43$ \( ( -2 + T^{2} )^{2} \)
$47$ \( ( -72 + T^{2} )^{2} \)
$53$ \( ( -18 + T^{2} )^{2} \)
$59$ \( ( 72 + T^{2} )^{2} \)
$61$ \( ( 72 + T^{2} )^{2} \)
$67$ \( T^{4} \)
$71$ \( ( 72 + T^{2} )^{2} \)
$73$ \( ( -128 + T^{2} )^{2} \)
$79$ \( ( 162 + T^{2} )^{2} \)
$83$ \( ( 196 + T^{2} )^{2} \)
$89$ \( ( 50 + T^{2} )^{2} \)
$97$ \( ( 144 + T^{2} )^{2} \)
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