Properties

Label 3564.2.i.a
Level $3564$
Weight $2$
Character orbit 3564.i
Analytic conductor $28.459$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3564 = 2^{2} \cdot 3^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3564.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(28.4586832804\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 44)
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.

\(f(q)\) \(=\) \( q - 3 \zeta_{6} q^{5} + (2 \zeta_{6} - 2) q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - 3 \zeta_{6} q^{5} + (2 \zeta_{6} - 2) q^{7} + (\zeta_{6} - 1) q^{11} + 4 \zeta_{6} q^{13} - 6 q^{17} + 8 q^{19} - 3 \zeta_{6} q^{23} + (4 \zeta_{6} - 4) q^{25} - 5 \zeta_{6} q^{31} + 6 q^{35} - q^{37} + ( - 10 \zeta_{6} + 10) q^{43} + 3 \zeta_{6} q^{49} + 6 q^{53} + 3 q^{55} + 3 \zeta_{6} q^{59} + ( - 4 \zeta_{6} + 4) q^{61} + ( - 12 \zeta_{6} + 12) q^{65} + \zeta_{6} q^{67} - 15 q^{71} - 4 q^{73} - 2 \zeta_{6} q^{77} + (2 \zeta_{6} - 2) q^{79} + ( - 6 \zeta_{6} + 6) q^{83} + 18 \zeta_{6} q^{85} + 9 q^{89} - 8 q^{91} - 24 \zeta_{6} q^{95} + ( - 7 \zeta_{6} + 7) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{5} - 2 q^{7} - q^{11} + 4 q^{13} - 12 q^{17} + 16 q^{19} - 3 q^{23} - 4 q^{25} - 5 q^{31} + 12 q^{35} - 2 q^{37} + 10 q^{43} + 3 q^{49} + 12 q^{53} + 6 q^{55} + 3 q^{59} + 4 q^{61} + 12 q^{65} + q^{67} - 30 q^{71} - 8 q^{73} - 2 q^{77} - 2 q^{79} + 6 q^{83} + 18 q^{85} + 18 q^{89} - 16 q^{91} - 24 q^{95} + 7 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3564\mathbb{Z}\right)^\times\).

\(n\) \(1541\) \(1783\) \(2917\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1189.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 −1.50000 2.59808i 0 −1.00000 + 1.73205i 0 0 0
2377.1 0 0 0 −1.50000 + 2.59808i 0 −1.00000 1.73205i 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
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3564.2.i.a 2
3.b odd 2 1 3564.2.i.j 2
9.c even 3 1 396.2.a.c 1
9.c even 3 1 inner 3564.2.i.a 2
9.d odd 6 1 44.2.a.a 1
9.d odd 6 1 3564.2.i.j 2
36.f odd 6 1 1584.2.a.p 1
36.h even 6 1 176.2.a.a 1
45.h odd 6 1 1100.2.a.b 1
45.j even 6 1 9900.2.a.h 1
45.k odd 12 2 9900.2.c.g 2
45.l even 12 2 1100.2.b.c 2
63.i even 6 1 2156.2.i.c 2
63.j odd 6 1 2156.2.i.b 2
63.n odd 6 1 2156.2.i.b 2
63.o even 6 1 2156.2.a.a 1
63.s even 6 1 2156.2.i.c 2
72.j odd 6 1 704.2.a.f 1
72.l even 6 1 704.2.a.i 1
72.n even 6 1 6336.2.a.j 1
72.p odd 6 1 6336.2.a.i 1
99.g even 6 1 484.2.a.a 1
99.h odd 6 1 4356.2.a.j 1
99.n odd 30 4 484.2.e.a 4
99.p even 30 4 484.2.e.b 4
117.n odd 6 1 7436.2.a.d 1
144.u even 12 2 2816.2.c.k 2
144.w odd 12 2 2816.2.c.e 2
180.n even 6 1 4400.2.a.v 1
180.v odd 12 2 4400.2.b.k 2
252.s odd 6 1 8624.2.a.w 1
396.o odd 6 1 1936.2.a.c 1
792.s odd 6 1 7744.2.a.bc 1
792.w even 6 1 7744.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.2.a.a 1 9.d odd 6 1
176.2.a.a 1 36.h even 6 1
396.2.a.c 1 9.c even 3 1
484.2.a.a 1 99.g even 6 1
484.2.e.a 4 99.n odd 30 4
484.2.e.b 4 99.p even 30 4
704.2.a.f 1 72.j odd 6 1
704.2.a.i 1 72.l even 6 1
1100.2.a.b 1 45.h odd 6 1
1100.2.b.c 2 45.l even 12 2
1584.2.a.p 1 36.f odd 6 1
1936.2.a.c 1 396.o odd 6 1
2156.2.a.a 1 63.o even 6 1
2156.2.i.b 2 63.j odd 6 1
2156.2.i.b 2 63.n odd 6 1
2156.2.i.c 2 63.i even 6 1
2156.2.i.c 2 63.s even 6 1
2816.2.c.e 2 144.w odd 12 2
2816.2.c.k 2 144.u even 12 2
3564.2.i.a 2 1.a even 1 1 trivial
3564.2.i.a 2 9.c even 3 1 inner
3564.2.i.j 2 3.b odd 2 1
3564.2.i.j 2 9.d odd 6 1
4356.2.a.j 1 99.h odd 6 1
4400.2.a.v 1 180.n even 6 1
4400.2.b.k 2 180.v odd 12 2
6336.2.a.i 1 72.p odd 6 1
6336.2.a.j 1 72.n even 6 1
7436.2.a.d 1 117.n odd 6 1
7744.2.a.m 1 792.w even 6 1
7744.2.a.bc 1 792.s odd 6 1
8624.2.a.w 1 252.s odd 6 1
9900.2.a.h 1 45.j even 6 1
9900.2.c.g 2 45.k odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3564, [\chi])\):

\( T_{5}^{2} + 3T_{5} + 9 \) Copy content Toggle raw display
\( T_{7}^{2} + 2T_{7} + 4 \) Copy content Toggle raw display
\( T_{17} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$7$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$11$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$13$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$17$ \( (T + 6)^{2} \) Copy content Toggle raw display
$19$ \( (T - 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$37$ \( (T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$61$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$67$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$71$ \( (T + 15)^{2} \) Copy content Toggle raw display
$73$ \( (T + 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$83$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$89$ \( (T - 9)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
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