Properties

Label 567.2.f.i
Level $567$
Weight $2$
Character orbit 567.f
Analytic conductor $4.528$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.f (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.52751779461\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial: \( x^{4} + 7x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 189)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 5 \beta_{2} q^{4} + (\beta_{3} + \beta_1) q^{5} + (\beta_{2} + 1) q^{7} + 3 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 5 \beta_{2} q^{4} + (\beta_{3} + \beta_1) q^{5} + (\beta_{2} + 1) q^{7} + 3 \beta_{3} q^{8} - 7 q^{10} - \beta_1 q^{11} - 2 \beta_{2} q^{13} + (\beta_{3} + \beta_1) q^{14} + ( - 11 \beta_{2} - 11) q^{16} + 7 q^{19} - 5 \beta_1 q^{20} - 7 \beta_{2} q^{22} + (3 \beta_{3} + 3 \beta_1) q^{23} + ( - 2 \beta_{2} - 2) q^{25} - 2 \beta_{3} q^{26} - 5 q^{28} + 2 \beta_1 q^{29} + 3 \beta_{2} q^{31} + ( - 5 \beta_{3} - 5 \beta_1) q^{32} + \beta_{3} q^{35} - 3 q^{37} + 7 \beta_1 q^{38} - 21 \beta_{2} q^{40} + ( - \beta_{3} - \beta_1) q^{41} + ( - 8 \beta_{2} - 8) q^{43} - 5 \beta_{3} q^{44} - 21 q^{46} + \beta_{2} q^{49} + ( - 2 \beta_{3} - 2 \beta_1) q^{50} + (10 \beta_{2} + 10) q^{52} + 7 q^{55} - 3 \beta_1 q^{56} + 14 \beta_{2} q^{58} + (8 \beta_{2} + 8) q^{61} + 3 \beta_{3} q^{62} + 13 q^{64} + 2 \beta_1 q^{65} - 2 \beta_{2} q^{67} + ( - 7 \beta_{2} - 7) q^{70} + 3 \beta_{3} q^{71} - 3 \beta_1 q^{74} + 35 \beta_{2} q^{76} + ( - \beta_{3} - \beta_1) q^{77} + (4 \beta_{2} + 4) q^{79} - 11 \beta_{3} q^{80} + 7 q^{82} + 6 \beta_1 q^{83} + ( - 8 \beta_{3} - 8 \beta_1) q^{86} + (21 \beta_{2} + 21) q^{88} - 7 \beta_{3} q^{89} + 2 q^{91} - 15 \beta_1 q^{92} + (7 \beta_{3} + 7 \beta_1) q^{95} + (12 \beta_{2} + 12) q^{97} + \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{4} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 10 q^{4} + 2 q^{7} - 28 q^{10} + 4 q^{13} - 22 q^{16} + 28 q^{19} + 14 q^{22} - 4 q^{25} - 20 q^{28} - 6 q^{31} - 12 q^{37} + 42 q^{40} - 16 q^{43} - 84 q^{46} - 2 q^{49} + 20 q^{52} + 28 q^{55} - 28 q^{58} + 16 q^{61} + 52 q^{64} + 4 q^{67} - 14 q^{70} - 70 q^{76} + 8 q^{79} + 28 q^{82} + 42 q^{88} + 8 q^{91} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 7\beta_{3} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
190.1
−1.32288 2.29129i
1.32288 + 2.29129i
−1.32288 + 2.29129i
1.32288 2.29129i
−1.32288 2.29129i 0 −2.50000 + 4.33013i 1.32288 2.29129i 0 0.500000 + 0.866025i 7.93725 0 −7.00000
190.2 1.32288 + 2.29129i 0 −2.50000 + 4.33013i −1.32288 + 2.29129i 0 0.500000 + 0.866025i −7.93725 0 −7.00000
379.1 −1.32288 + 2.29129i 0 −2.50000 4.33013i 1.32288 + 2.29129i 0 0.500000 0.866025i 7.93725 0 −7.00000
379.2 1.32288 2.29129i 0 −2.50000 4.33013i −1.32288 2.29129i 0 0.500000 0.866025i −7.93725 0 −7.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 567.2.f.i 4
3.b odd 2 1 inner 567.2.f.i 4
9.c even 3 1 189.2.a.f 2
9.c even 3 1 inner 567.2.f.i 4
9.d odd 6 1 189.2.a.f 2
9.d odd 6 1 inner 567.2.f.i 4
36.f odd 6 1 3024.2.a.bi 2
36.h even 6 1 3024.2.a.bi 2
45.h odd 6 1 4725.2.a.bb 2
45.j even 6 1 4725.2.a.bb 2
63.l odd 6 1 1323.2.a.w 2
63.o even 6 1 1323.2.a.w 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.a.f 2 9.c even 3 1
189.2.a.f 2 9.d odd 6 1
567.2.f.i 4 1.a even 1 1 trivial
567.2.f.i 4 3.b odd 2 1 inner
567.2.f.i 4 9.c even 3 1 inner
567.2.f.i 4 9.d odd 6 1 inner
1323.2.a.w 2 63.l odd 6 1
1323.2.a.w 2 63.o even 6 1
3024.2.a.bi 2 36.f odd 6 1
3024.2.a.bi 2 36.h even 6 1
4725.2.a.bb 2 45.h odd 6 1
4725.2.a.bb 2 45.j even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(567, [\chi])\):

\( T_{2}^{4} + 7T_{2}^{2} + 49 \) Copy content Toggle raw display
\( T_{5}^{4} + 7T_{5}^{2} + 49 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 7T^{2} + 49 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 7T^{2} + 49 \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 7T^{2} + 49 \) Copy content Toggle raw display
$13$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T - 7)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 63T^{2} + 3969 \) Copy content Toggle raw display
$29$ \( T^{4} + 28T^{2} + 784 \) Copy content Toggle raw display
$31$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$37$ \( (T + 3)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + 7T^{2} + 49 \) Copy content Toggle raw display
$43$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 63)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 252 T^{2} + 63504 \) Copy content Toggle raw display
$89$ \( (T^{2} - 343)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 12 T + 144)^{2} \) Copy content Toggle raw display
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