Properties

Label 1521.2.b.f
Level $1521$
Weight $2$
Character orbit 1521.b
Analytic conductor $12.145$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.1452461474\)
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: \( 2 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{4} + 2 \beta q^{5} + \beta q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{4} + 2 \beta q^{5} + \beta q^{7} + 2 \beta q^{11} + 4 q^{16} - 2 \beta q^{19} + 4 \beta q^{20} + 6 q^{23} - 7 q^{25} + 2 \beta q^{28} - 6 q^{29} - \beta q^{31} - 6 q^{35} + 4 \beta q^{41} - q^{43} + 4 \beta q^{44} + 2 \beta q^{47} + 4 q^{49} - 12 q^{53} - 12 q^{55} - 2 \beta q^{59} + q^{61} + 8 q^{64} + 5 \beta q^{67} - 6 \beta q^{71} - \beta q^{73} - 4 \beta q^{76} - 6 q^{77} - 11 q^{79} + 8 \beta q^{80} - 8 \beta q^{83} - 4 \beta q^{89} + 12 q^{92} + 12 q^{95} + 3 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{4} + 8 q^{16} + 12 q^{23} - 14 q^{25} - 12 q^{29} - 12 q^{35} - 2 q^{43} + 8 q^{49} - 24 q^{53} - 24 q^{55} + 2 q^{61} + 16 q^{64} - 12 q^{77} - 22 q^{79} + 24 q^{92} + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1351.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 2.00000 3.46410i 0 1.73205i 0 0 0
1351.2 0 0 2.00000 3.46410i 0 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
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1521.2.b.f 2
3.b odd 2 1 507.2.b.c 2
13.b even 2 1 inner 1521.2.b.f 2
13.c even 3 1 117.2.q.a 2
13.d odd 4 2 1521.2.a.h 2
13.e even 6 1 117.2.q.a 2
39.d odd 2 1 507.2.b.c 2
39.f even 4 2 507.2.a.e 2
39.h odd 6 1 39.2.j.a 2
39.h odd 6 1 507.2.j.b 2
39.i odd 6 1 39.2.j.a 2
39.i odd 6 1 507.2.j.b 2
39.k even 12 4 507.2.e.f 4
52.i odd 6 1 1872.2.by.f 2
52.j odd 6 1 1872.2.by.f 2
156.l odd 4 2 8112.2.a.bu 2
156.p even 6 1 624.2.bv.b 2
156.r even 6 1 624.2.bv.b 2
195.x odd 6 1 975.2.bc.c 2
195.y odd 6 1 975.2.bc.c 2
195.bf even 12 2 975.2.w.d 4
195.bl even 12 2 975.2.w.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.j.a 2 39.h odd 6 1
39.2.j.a 2 39.i odd 6 1
117.2.q.a 2 13.c even 3 1
117.2.q.a 2 13.e even 6 1
507.2.a.e 2 39.f even 4 2
507.2.b.c 2 3.b odd 2 1
507.2.b.c 2 39.d odd 2 1
507.2.e.f 4 39.k even 12 4
507.2.j.b 2 39.h odd 6 1
507.2.j.b 2 39.i odd 6 1
624.2.bv.b 2 156.p even 6 1
624.2.bv.b 2 156.r even 6 1
975.2.w.d 4 195.bf even 12 2
975.2.w.d 4 195.bl even 12 2
975.2.bc.c 2 195.x odd 6 1
975.2.bc.c 2 195.y odd 6 1
1521.2.a.h 2 13.d odd 4 2
1521.2.b.f 2 1.a even 1 1 trivial
1521.2.b.f 2 13.b even 2 1 inner
1872.2.by.f 2 52.i odd 6 1
1872.2.by.f 2 52.j odd 6 1
8112.2.a.bu 2 156.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1521, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{5}^{2} + 12 \) Copy content Toggle raw display
\( T_{7}^{2} + 3 \) 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
$5$ \( T^{2} + 12 \) Copy content Toggle raw display
$7$ \( T^{2} + 3 \) Copy content Toggle raw display
$11$ \( T^{2} + 12 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 12 \) Copy content Toggle raw display
$23$ \( (T - 6)^{2} \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 3 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 48 \) Copy content Toggle raw display
$43$ \( (T + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 12 \) Copy content Toggle raw display
$53$ \( (T + 12)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 12 \) Copy content Toggle raw display
$61$ \( (T - 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 75 \) Copy content Toggle raw display
$71$ \( T^{2} + 108 \) Copy content Toggle raw display
$73$ \( T^{2} + 3 \) Copy content Toggle raw display
$79$ \( (T + 11)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 192 \) Copy content Toggle raw display
$89$ \( T^{2} + 48 \) Copy content Toggle raw display
$97$ \( T^{2} + 27 \) Copy content Toggle raw display
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