Properties

Label 1008.3.dc.a
Level $1008$
Weight $3$
Character orbit 1008.dc
Analytic conductor $27.466$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1008.dc (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(27.4660106475\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + ( - 3 \beta_{2} - 5) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} + ( - 3 \beta_{2} - 5) q^{7} + ( - 5 \beta_{3} + 5 \beta_1) q^{11} + 15 q^{13} + (8 \beta_{3} - 8 \beta_1) q^{17} + (13 \beta_{2} - 13) q^{19} + 16 \beta_1 q^{23} - 23 \beta_{2} q^{25} - 16 \beta_{3} q^{29} + 3 \beta_{2} q^{31} + ( - 3 \beta_{3} - 5 \beta_1) q^{35} + (17 \beta_{2} - 17) q^{37} - 57 \beta_{3} q^{41} + 85 q^{43} - 51 \beta_1 q^{47} + (39 \beta_{2} + 16) q^{49} + ( - 24 \beta_{3} + 24 \beta_1) q^{53} + 10 q^{55} + ( - 64 \beta_{3} + 64 \beta_1) q^{59} + ( - 72 \beta_{2} + 72) q^{61} + 15 \beta_1 q^{65} + 43 \beta_{2} q^{67} - 37 \beta_{3} q^{71} + 95 \beta_{2} q^{73} + (25 \beta_{3} - 40 \beta_1) q^{77} + ( - 69 \beta_{2} + 69) q^{79} - 43 \beta_{3} q^{83} - 16 q^{85} + 96 \beta_1 q^{89} + ( - 45 \beta_{2} - 75) q^{91} + (13 \beta_{3} - 13 \beta_1) q^{95} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 26 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 26 q^{7} + 60 q^{13} - 26 q^{19} - 46 q^{25} + 6 q^{31} - 34 q^{37} + 340 q^{43} + 142 q^{49} + 40 q^{55} + 144 q^{61} + 86 q^{67} + 190 q^{73} + 138 q^{79} - 64 q^{85} - 390 q^{91} + 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(-1 + \beta_{2}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
305.1
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i
1.22474 + 0.707107i
0 0 0 −1.22474 + 0.707107i 0 −6.50000 + 2.59808i 0 0 0
305.2 0 0 0 1.22474 0.707107i 0 −6.50000 + 2.59808i 0 0 0
737.1 0 0 0 −1.22474 0.707107i 0 −6.50000 2.59808i 0 0 0
737.2 0 0 0 1.22474 + 0.707107i 0 −6.50000 2.59808i 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
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.3.dc.a 4
3.b odd 2 1 inner 1008.3.dc.a 4
4.b odd 2 1 126.3.s.b 4
7.c even 3 1 inner 1008.3.dc.a 4
12.b even 2 1 126.3.s.b 4
21.h odd 6 1 inner 1008.3.dc.a 4
28.d even 2 1 882.3.s.c 4
28.f even 6 1 882.3.b.d 2
28.f even 6 1 882.3.s.c 4
28.g odd 6 1 126.3.s.b 4
28.g odd 6 1 882.3.b.c 2
84.h odd 2 1 882.3.s.c 4
84.j odd 6 1 882.3.b.d 2
84.j odd 6 1 882.3.s.c 4
84.n even 6 1 126.3.s.b 4
84.n even 6 1 882.3.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.s.b 4 4.b odd 2 1
126.3.s.b 4 12.b even 2 1
126.3.s.b 4 28.g odd 6 1
126.3.s.b 4 84.n even 6 1
882.3.b.c 2 28.g odd 6 1
882.3.b.c 2 84.n even 6 1
882.3.b.d 2 28.f even 6 1
882.3.b.d 2 84.j odd 6 1
882.3.s.c 4 28.d even 2 1
882.3.s.c 4 28.f even 6 1
882.3.s.c 4 84.h odd 2 1
882.3.s.c 4 84.j odd 6 1
1008.3.dc.a 4 1.a even 1 1 trivial
1008.3.dc.a 4 3.b odd 2 1 inner
1008.3.dc.a 4 7.c even 3 1 inner
1008.3.dc.a 4 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1008, [\chi])\):

\( T_{5}^{4} - 2T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{4} - 50T_{11}^{2} + 2500 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 2T^{2} + 4 \) Copy content Toggle raw display
$7$ \( (T^{2} + 13 T + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 50T^{2} + 2500 \) Copy content Toggle raw display
$13$ \( (T - 15)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 128 T^{2} + 16384 \) Copy content Toggle raw display
$19$ \( (T^{2} + 13 T + 169)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 512 T^{2} + 262144 \) Copy content Toggle raw display
$29$ \( (T^{2} + 512)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 17 T + 289)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 6498)^{2} \) Copy content Toggle raw display
$43$ \( (T - 85)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 5202 T^{2} + \cdots + 27060804 \) Copy content Toggle raw display
$53$ \( T^{4} - 1152 T^{2} + \cdots + 1327104 \) Copy content Toggle raw display
$59$ \( T^{4} - 8192 T^{2} + \cdots + 67108864 \) Copy content Toggle raw display
$61$ \( (T^{2} - 72 T + 5184)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 43 T + 1849)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 2738)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 95 T + 9025)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 69 T + 4761)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 3698)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 18432 T^{2} + \cdots + 339738624 \) Copy content Toggle raw display
$97$ \( (T - 16)^{4} \) Copy content Toggle raw display
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