Properties

Label 1521.2.b.c
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: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) 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: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} + q^{4} - i q^{5} + 2 i q^{7} + 3 i q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} + q^{4} - i q^{5} + 2 i q^{7} + 3 i q^{8} + q^{10} + 2 i q^{11} - 2 q^{14} - q^{16} - 7 q^{17} + 6 i q^{19} - i q^{20} - 2 q^{22} - 6 q^{23} + 4 q^{25} + 2 i q^{28} + q^{29} - 4 i q^{31} + 5 i q^{32} - 7 i q^{34} + 2 q^{35} + i q^{37} - 6 q^{38} + 3 q^{40} + 9 i q^{41} - 6 q^{43} + 2 i q^{44} - 6 i q^{46} - 6 i q^{47} + 3 q^{49} + 4 i q^{50} + 9 q^{53} + 2 q^{55} - 6 q^{56} + i q^{58} + q^{61} + 4 q^{62} - 7 q^{64} + 2 i q^{67} - 7 q^{68} + 2 i q^{70} + 6 i q^{71} + 11 i q^{73} - q^{74} + 6 i q^{76} - 4 q^{77} - 4 q^{79} + i q^{80} - 9 q^{82} - 14 i q^{83} + 7 i q^{85} - 6 i q^{86} - 6 q^{88} + 14 i q^{89} - 6 q^{92} + 6 q^{94} + 6 q^{95} + 2 i q^{97} + 3 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} + 2 q^{10} - 4 q^{14} - 2 q^{16} - 14 q^{17} - 4 q^{22} - 12 q^{23} + 8 q^{25} + 2 q^{29} + 4 q^{35} - 12 q^{38} + 6 q^{40} - 12 q^{43} + 6 q^{49} + 18 q^{53} + 4 q^{55} - 12 q^{56} + 2 q^{61} + 8 q^{62} - 14 q^{64} - 14 q^{68} - 2 q^{74} - 8 q^{77} - 8 q^{79} - 18 q^{82} - 12 q^{88} - 12 q^{92} + 12 q^{94} + 12 q^{95}+O(q^{100}) \) 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
1.00000i
1.00000i
1.00000i 0 1.00000 1.00000i 0 2.00000i 3.00000i 0 1.00000
1351.2 1.00000i 0 1.00000 1.00000i 0 2.00000i 3.00000i 0 1.00000
\(n\): 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.c 2
3.b odd 2 1 507.2.b.b 2
13.b even 2 1 inner 1521.2.b.c 2
13.d odd 4 1 1521.2.a.a 1
13.d odd 4 1 1521.2.a.d 1
13.f odd 12 2 117.2.g.b 2
39.d odd 2 1 507.2.b.b 2
39.f even 4 1 507.2.a.b 1
39.f even 4 1 507.2.a.c 1
39.h odd 6 2 507.2.j.d 4
39.i odd 6 2 507.2.j.d 4
39.k even 12 2 39.2.e.a 2
39.k even 12 2 507.2.e.c 2
52.l even 12 2 1872.2.t.j 2
156.l odd 4 1 8112.2.a.w 1
156.l odd 4 1 8112.2.a.bc 1
156.v odd 12 2 624.2.q.c 2
195.bc odd 12 2 975.2.bb.d 4
195.bh even 12 2 975.2.i.f 2
195.bn odd 12 2 975.2.bb.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.a 2 39.k even 12 2
117.2.g.b 2 13.f odd 12 2
507.2.a.b 1 39.f even 4 1
507.2.a.c 1 39.f even 4 1
507.2.b.b 2 3.b odd 2 1
507.2.b.b 2 39.d odd 2 1
507.2.e.c 2 39.k even 12 2
507.2.j.d 4 39.h odd 6 2
507.2.j.d 4 39.i odd 6 2
624.2.q.c 2 156.v odd 12 2
975.2.i.f 2 195.bh even 12 2
975.2.bb.d 4 195.bc odd 12 2
975.2.bb.d 4 195.bn odd 12 2
1521.2.a.a 1 13.d odd 4 1
1521.2.a.d 1 13.d odd 4 1
1521.2.b.c 2 1.a even 1 1 trivial
1521.2.b.c 2 13.b even 2 1 inner
1872.2.t.j 2 52.l even 12 2
8112.2.a.w 1 156.l odd 4 1
8112.2.a.bc 1 156.l odd 4 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( T^{2} + 4 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 7)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 36 \) Copy content Toggle raw display
$23$ \( (T + 6)^{2} \) Copy content Toggle raw display
$29$ \( (T - 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 1 \) Copy content Toggle raw display
$41$ \( T^{2} + 81 \) Copy content Toggle raw display
$43$ \( (T + 6)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 36 \) Copy content Toggle raw display
$53$ \( (T - 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T - 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 4 \) Copy content Toggle raw display
$71$ \( T^{2} + 36 \) Copy content Toggle raw display
$73$ \( T^{2} + 121 \) Copy content Toggle raw display
$79$ \( (T + 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 196 \) Copy content Toggle raw display
$89$ \( T^{2} + 196 \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
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