Properties

Label 476.2.i.a
Level $476$
Weight $2$
Character orbit 476.i
Analytic conductor $3.801$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 476 = 2^{2} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 476.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.80087913621\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{3} + 4 \zeta_{6} q^{5} + (2 \zeta_{6} - 3) q^{7} + 2 \zeta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 1) q^{3} + 4 \zeta_{6} q^{5} + (2 \zeta_{6} - 3) q^{7} + 2 \zeta_{6} q^{9} + ( - 3 \zeta_{6} + 3) q^{11} - 5 q^{13} - 4 q^{15} + ( - \zeta_{6} + 1) q^{17} - 6 \zeta_{6} q^{19} + ( - 3 \zeta_{6} + 1) q^{21} + 4 \zeta_{6} q^{23} + (11 \zeta_{6} - 11) q^{25} - 5 q^{27} + 8 q^{29} + 3 \zeta_{6} q^{33} + ( - 4 \zeta_{6} - 8) q^{35} + 8 \zeta_{6} q^{37} + ( - 5 \zeta_{6} + 5) q^{39} - 8 q^{41} + 10 q^{43} + (8 \zeta_{6} - 8) q^{45} + 2 \zeta_{6} q^{47} + ( - 8 \zeta_{6} + 5) q^{49} + \zeta_{6} q^{51} + (3 \zeta_{6} - 3) q^{53} + 12 q^{55} + 6 q^{57} + (2 \zeta_{6} - 2) q^{59} + 8 \zeta_{6} q^{61} + ( - 2 \zeta_{6} - 4) q^{63} - 20 \zeta_{6} q^{65} + (14 \zeta_{6} - 14) q^{67} - 4 q^{69} + 7 q^{71} + ( - 8 \zeta_{6} + 8) q^{73} - 11 \zeta_{6} q^{75} + (9 \zeta_{6} - 3) q^{77} - 15 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} + 12 q^{83} + 4 q^{85} + (8 \zeta_{6} - 8) q^{87} - \zeta_{6} q^{89} + ( - 10 \zeta_{6} + 15) q^{91} + ( - 24 \zeta_{6} + 24) q^{95} + 18 q^{97} + 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 4 q^{5} - 4 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} + 4 q^{5} - 4 q^{7} + 2 q^{9} + 3 q^{11} - 10 q^{13} - 8 q^{15} + q^{17} - 6 q^{19} - q^{21} + 4 q^{23} - 11 q^{25} - 10 q^{27} + 16 q^{29} + 3 q^{33} - 20 q^{35} + 8 q^{37} + 5 q^{39} - 16 q^{41} + 20 q^{43} - 8 q^{45} + 2 q^{47} + 2 q^{49} + q^{51} - 3 q^{53} + 24 q^{55} + 12 q^{57} - 2 q^{59} + 8 q^{61} - 10 q^{63} - 20 q^{65} - 14 q^{67} - 8 q^{69} + 14 q^{71} + 8 q^{73} - 11 q^{75} + 3 q^{77} - 15 q^{79} - q^{81} + 24 q^{83} + 8 q^{85} - 8 q^{87} - q^{89} + 20 q^{91} + 24 q^{95} + 36 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/476\mathbb{Z}\right)^\times\).

\(n\) \(239\) \(309\) \(409\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
137.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −0.500000 + 0.866025i 0 2.00000 + 3.46410i 0 −2.00000 + 1.73205i 0 1.00000 + 1.73205i 0
205.1 0 −0.500000 0.866025i 0 2.00000 3.46410i 0 −2.00000 1.73205i 0 1.00000 1.73205i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 476.2.i.a 2
7.c even 3 1 inner 476.2.i.a 2
7.c even 3 1 3332.2.a.d 1
7.d odd 6 1 3332.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
476.2.i.a 2 1.a even 1 1 trivial
476.2.i.a 2 7.c even 3 1 inner
3332.2.a.c 1 7.d odd 6 1
3332.2.a.d 1 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(476, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$7$ \( T^{2} + 4T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$13$ \( (T + 5)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$19$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$23$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$29$ \( (T - 8)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$41$ \( (T + 8)^{2} \) Copy content Toggle raw display
$43$ \( (T - 10)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$53$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$59$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$61$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$67$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$71$ \( (T - 7)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$79$ \( T^{2} + 15T + 225 \) Copy content Toggle raw display
$83$ \( (T - 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$97$ \( (T - 18)^{2} \) Copy content Toggle raw display
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