Properties

Label 1053.2.e.d
Level $1053$
Weight $2$
Character orbit 1053.e
Analytic conductor $8.408$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1053 = 3^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1053.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.40824733284\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{6} + 1) q^{2} + \zeta_{6} q^{4} + 2 \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{6} + 1) q^{2} + \zeta_{6} q^{4} + 2 \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{7} + 3 q^{8} + 2 q^{10} + ( - 4 \zeta_{6} + 4) q^{11} - \zeta_{6} q^{13} - 4 \zeta_{6} q^{14} + ( - \zeta_{6} + 1) q^{16} - 2 q^{17} + (2 \zeta_{6} - 2) q^{20} - 4 \zeta_{6} q^{22} + ( - \zeta_{6} + 1) q^{25} - q^{26} + 4 q^{28} + (10 \zeta_{6} - 10) q^{29} - 4 \zeta_{6} q^{31} + 5 \zeta_{6} q^{32} + (2 \zeta_{6} - 2) q^{34} + 8 q^{35} - 2 q^{37} + 6 \zeta_{6} q^{40} + 6 \zeta_{6} q^{41} + ( - 12 \zeta_{6} + 12) q^{43} + 4 q^{44} - 9 \zeta_{6} q^{49} - \zeta_{6} q^{50} + ( - \zeta_{6} + 1) q^{52} - 6 q^{53} + 8 q^{55} + ( - 12 \zeta_{6} + 12) q^{56} + 10 \zeta_{6} q^{58} + 12 \zeta_{6} q^{59} + ( - 2 \zeta_{6} + 2) q^{61} - 4 q^{62} + 7 q^{64} + ( - 2 \zeta_{6} + 2) q^{65} + 8 \zeta_{6} q^{67} - 2 \zeta_{6} q^{68} + ( - 8 \zeta_{6} + 8) q^{70} + 2 q^{73} + (2 \zeta_{6} - 2) q^{74} - 16 \zeta_{6} q^{77} + (8 \zeta_{6} - 8) q^{79} + 2 q^{80} + 6 q^{82} + ( - 4 \zeta_{6} + 4) q^{83} - 4 \zeta_{6} q^{85} - 12 \zeta_{6} q^{86} + ( - 12 \zeta_{6} + 12) q^{88} + 2 q^{89} - 4 q^{91} + (10 \zeta_{6} - 10) q^{97} - 9 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + q^{4} + 2 q^{5} + 4 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + q^{4} + 2 q^{5} + 4 q^{7} + 6 q^{8} + 4 q^{10} + 4 q^{11} - q^{13} - 4 q^{14} + q^{16} - 4 q^{17} - 2 q^{20} - 4 q^{22} + q^{25} - 2 q^{26} + 8 q^{28} - 10 q^{29} - 4 q^{31} + 5 q^{32} - 2 q^{34} + 16 q^{35} - 4 q^{37} + 6 q^{40} + 6 q^{41} + 12 q^{43} + 8 q^{44} - 9 q^{49} - q^{50} + q^{52} - 12 q^{53} + 16 q^{55} + 12 q^{56} + 10 q^{58} + 12 q^{59} + 2 q^{61} - 8 q^{62} + 14 q^{64} + 2 q^{65} + 8 q^{67} - 2 q^{68} + 8 q^{70} + 4 q^{73} - 2 q^{74} - 16 q^{77} - 8 q^{79} + 4 q^{80} + 12 q^{82} + 4 q^{83} - 4 q^{85} - 12 q^{86} + 12 q^{88} + 4 q^{89} - 8 q^{91} - 10 q^{97} - 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1053\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(730\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
352.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 + 0.866025i 0 0.500000 0.866025i 1.00000 1.73205i 0 2.00000 + 3.46410i 3.00000 0 2.00000
703.1 0.500000 0.866025i 0 0.500000 + 0.866025i 1.00000 + 1.73205i 0 2.00000 3.46410i 3.00000 0 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 1053.2.e.d 2
3.b odd 2 1 1053.2.e.b 2
9.c even 3 1 117.2.a.a 1
9.c even 3 1 inner 1053.2.e.d 2
9.d odd 6 1 39.2.a.a 1
9.d odd 6 1 1053.2.e.b 2
36.f odd 6 1 1872.2.a.h 1
36.h even 6 1 624.2.a.i 1
45.h odd 6 1 975.2.a.f 1
45.j even 6 1 2925.2.a.p 1
45.k odd 12 2 2925.2.c.e 2
45.l even 12 2 975.2.c.f 2
63.l odd 6 1 5733.2.a.e 1
63.o even 6 1 1911.2.a.f 1
72.j odd 6 1 2496.2.a.q 1
72.l even 6 1 2496.2.a.e 1
72.n even 6 1 7488.2.a.bl 1
72.p odd 6 1 7488.2.a.by 1
99.g even 6 1 4719.2.a.c 1
117.k odd 6 1 507.2.e.a 2
117.m odd 6 1 507.2.e.b 2
117.n odd 6 1 507.2.a.a 1
117.t even 6 1 1521.2.a.e 1
117.u odd 6 1 507.2.e.a 2
117.v odd 6 1 507.2.e.b 2
117.x even 12 2 507.2.j.e 4
117.y odd 12 2 1521.2.b.b 2
117.z even 12 2 507.2.b.a 2
117.bc even 12 2 507.2.j.e 4
468.x even 6 1 8112.2.a.s 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.a 1 9.d odd 6 1
117.2.a.a 1 9.c even 3 1
507.2.a.a 1 117.n odd 6 1
507.2.b.a 2 117.z even 12 2
507.2.e.a 2 117.k odd 6 1
507.2.e.a 2 117.u odd 6 1
507.2.e.b 2 117.m odd 6 1
507.2.e.b 2 117.v odd 6 1
507.2.j.e 4 117.x even 12 2
507.2.j.e 4 117.bc even 12 2
624.2.a.i 1 36.h even 6 1
975.2.a.f 1 45.h odd 6 1
975.2.c.f 2 45.l even 12 2
1053.2.e.b 2 3.b odd 2 1
1053.2.e.b 2 9.d odd 6 1
1053.2.e.d 2 1.a even 1 1 trivial
1053.2.e.d 2 9.c even 3 1 inner
1521.2.a.e 1 117.t even 6 1
1521.2.b.b 2 117.y odd 12 2
1872.2.a.h 1 36.f odd 6 1
1911.2.a.f 1 63.o even 6 1
2496.2.a.e 1 72.l even 6 1
2496.2.a.q 1 72.j odd 6 1
2925.2.a.p 1 45.j even 6 1
2925.2.c.e 2 45.k odd 12 2
4719.2.a.c 1 99.g even 6 1
5733.2.a.e 1 63.l odd 6 1
7488.2.a.bl 1 72.n even 6 1
7488.2.a.by 1 72.p odd 6 1
8112.2.a.s 1 468.x even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1053, [\chi])\):

\( T_{2}^{2} - T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} - 2T_{5} + 4 \) Copy content Toggle raw display
\( T_{7}^{2} - 4T_{7} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$11$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$13$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$17$ \( (T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$31$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$37$ \( (T + 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$43$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$61$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T - 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$83$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$89$ \( (T - 2)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
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