Properties

Label 1521.2.b.a
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 13)
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.

\(f(q)\) \(=\) \( q - \beta q^{2} - q^{4} + \beta q^{5} - \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} - q^{4} + \beta q^{5} - \beta q^{8} + 3 q^{10} - 5 q^{16} + 3 q^{17} - 2 \beta q^{19} - \beta q^{20} + 6 q^{23} + 2 q^{25} - 3 q^{29} + 2 \beta q^{31} + 3 \beta q^{32} - 3 \beta q^{34} - 5 \beta q^{37} - 6 q^{38} + 3 q^{40} - 3 \beta q^{41} + 8 q^{43} - 6 \beta q^{46} + 2 \beta q^{47} + 7 q^{49} - 2 \beta q^{50} + 3 q^{53} + 3 \beta q^{58} - 4 \beta q^{59} + q^{61} + 6 q^{62} - q^{64} - 2 \beta q^{67} - 3 q^{68} - 2 \beta q^{71} + \beta q^{73} - 15 q^{74} + 2 \beta q^{76} + 4 q^{79} - 5 \beta q^{80} - 9 q^{82} - 8 \beta q^{83} + 3 \beta q^{85} - 8 \beta q^{86} - 4 \beta q^{89} - 6 q^{92} + 6 q^{94} + 6 q^{95} + 4 \beta q^{97} - 7 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 6 q^{10} - 10 q^{16} + 6 q^{17} + 12 q^{23} + 4 q^{25} - 6 q^{29} - 12 q^{38} + 6 q^{40} + 16 q^{43} + 14 q^{49} + 6 q^{53} + 2 q^{61} + 12 q^{62} - 2 q^{64} - 6 q^{68} - 30 q^{74} + 8 q^{79} - 18 q^{82} - 12 q^{92} + 12 q^{94} + 12 q^{95}+O(q^{100}) \) 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
1.73205i 0 −1.00000 1.73205i 0 0 1.73205i 0 3.00000
1351.2 1.73205i 0 −1.00000 1.73205i 0 0 1.73205i 0 3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.a 2
3.b odd 2 1 169.2.b.a 2
12.b even 2 1 2704.2.f.b 2
13.b even 2 1 inner 1521.2.b.a 2
13.c even 3 1 117.2.q.c 2
13.d odd 4 2 1521.2.a.k 2
13.e even 6 1 117.2.q.c 2
39.d odd 2 1 169.2.b.a 2
39.f even 4 2 169.2.a.a 2
39.h odd 6 1 13.2.e.a 2
39.h odd 6 1 169.2.e.a 2
39.i odd 6 1 13.2.e.a 2
39.i odd 6 1 169.2.e.a 2
39.k even 12 4 169.2.c.a 4
52.i odd 6 1 1872.2.by.d 2
52.j odd 6 1 1872.2.by.d 2
156.h even 2 1 2704.2.f.b 2
156.l odd 4 2 2704.2.a.o 2
156.p even 6 1 208.2.w.b 2
156.r even 6 1 208.2.w.b 2
195.n even 4 2 4225.2.a.v 2
195.x odd 6 1 325.2.n.a 2
195.y odd 6 1 325.2.n.a 2
195.bf even 12 2 325.2.m.a 4
195.bl even 12 2 325.2.m.a 4
273.o odd 4 2 8281.2.a.q 2
273.r even 6 1 637.2.k.c 2
273.s odd 6 1 637.2.k.a 2
273.u even 6 1 637.2.q.a 2
273.x odd 6 1 637.2.u.c 2
273.y even 6 1 637.2.u.b 2
273.bf even 6 1 637.2.u.b 2
273.bm odd 6 1 637.2.u.c 2
273.bn even 6 1 637.2.q.a 2
273.bp odd 6 1 637.2.k.a 2
273.br even 6 1 637.2.k.c 2
312.ba even 6 1 832.2.w.a 2
312.bg odd 6 1 832.2.w.d 2
312.bh odd 6 1 832.2.w.d 2
312.bn even 6 1 832.2.w.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.2.e.a 2 39.h odd 6 1
13.2.e.a 2 39.i odd 6 1
117.2.q.c 2 13.c even 3 1
117.2.q.c 2 13.e even 6 1
169.2.a.a 2 39.f even 4 2
169.2.b.a 2 3.b odd 2 1
169.2.b.a 2 39.d odd 2 1
169.2.c.a 4 39.k even 12 4
169.2.e.a 2 39.h odd 6 1
169.2.e.a 2 39.i odd 6 1
208.2.w.b 2 156.p even 6 1
208.2.w.b 2 156.r even 6 1
325.2.m.a 4 195.bf even 12 2
325.2.m.a 4 195.bl even 12 2
325.2.n.a 2 195.x odd 6 1
325.2.n.a 2 195.y odd 6 1
637.2.k.a 2 273.s odd 6 1
637.2.k.a 2 273.bp odd 6 1
637.2.k.c 2 273.r even 6 1
637.2.k.c 2 273.br even 6 1
637.2.q.a 2 273.u even 6 1
637.2.q.a 2 273.bn even 6 1
637.2.u.b 2 273.y even 6 1
637.2.u.b 2 273.bf even 6 1
637.2.u.c 2 273.x odd 6 1
637.2.u.c 2 273.bm odd 6 1
832.2.w.a 2 312.ba even 6 1
832.2.w.a 2 312.bn even 6 1
832.2.w.d 2 312.bg odd 6 1
832.2.w.d 2 312.bh odd 6 1
1521.2.a.k 2 13.d odd 4 2
1521.2.b.a 2 1.a even 1 1 trivial
1521.2.b.a 2 13.b even 2 1 inner
1872.2.by.d 2 52.i odd 6 1
1872.2.by.d 2 52.j odd 6 1
2704.2.a.o 2 156.l odd 4 2
2704.2.f.b 2 12.b even 2 1
2704.2.f.b 2 156.h even 2 1
4225.2.a.v 2 195.n even 4 2
8281.2.a.q 2 273.o odd 4 2

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 3 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 12 \) Copy content Toggle raw display
$23$ \( (T - 6)^{2} \) Copy content Toggle raw display
$29$ \( (T + 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 12 \) Copy content Toggle raw display
$37$ \( T^{2} + 75 \) Copy content Toggle raw display
$41$ \( T^{2} + 27 \) Copy content Toggle raw display
$43$ \( (T - 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 12 \) Copy content Toggle raw display
$53$ \( (T - 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 48 \) Copy content Toggle raw display
$61$ \( (T - 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 12 \) Copy content Toggle raw display
$71$ \( T^{2} + 12 \) Copy content Toggle raw display
$73$ \( T^{2} + 3 \) Copy content Toggle raw display
$79$ \( (T - 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 192 \) Copy content Toggle raw display
$89$ \( T^{2} + 48 \) Copy content Toggle raw display
$97$ \( T^{2} + 48 \) Copy content Toggle raw display
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