Properties

Label 1521.2.b.b
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.

\(f(q)\) \(=\) \( q + i q^{2} + q^{4} + 2 i q^{5} - 4 i q^{7} + 3 i q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} + q^{4} + 2 i q^{5} - 4 i q^{7} + 3 i q^{8} - 2 q^{10} - 4 i q^{11} + 4 q^{14} - q^{16} + 2 q^{17} + 2 i q^{20} + 4 q^{22} + q^{25} - 4 i q^{28} + 10 q^{29} - 4 i q^{31} + 5 i q^{32} + 2 i q^{34} + 8 q^{35} - 2 i q^{37} - 6 q^{40} + 6 i q^{41} + 12 q^{43} - 4 i q^{44} - 9 q^{49} + i q^{50} - 6 q^{53} + 8 q^{55} + 12 q^{56} + 10 i q^{58} - 12 i q^{59} - 2 q^{61} + 4 q^{62} - 7 q^{64} + 8 i q^{67} + 2 q^{68} + 8 i q^{70} + 2 i q^{73} + 2 q^{74} - 16 q^{77} + 8 q^{79} - 2 i q^{80} - 6 q^{82} + 4 i q^{83} + 4 i q^{85} + 12 i q^{86} + 12 q^{88} + 2 i q^{89} - 10 i q^{97} - 9 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} - 4 q^{10} + 8 q^{14} - 2 q^{16} + 4 q^{17} + 8 q^{22} + 2 q^{25} + 20 q^{29} + 16 q^{35} - 12 q^{40} + 24 q^{43} - 18 q^{49} - 12 q^{53} + 16 q^{55} + 24 q^{56} - 4 q^{61} + 8 q^{62} - 14 q^{64} + 4 q^{68} + 4 q^{74} - 32 q^{77} + 16 q^{79} - 12 q^{82} + 24 q^{88}+O(q^{100}) \) 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 2.00000i 0 4.00000i 3.00000i 0 −2.00000
1351.2 1.00000i 0 1.00000 2.00000i 0 4.00000i 3.00000i 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
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.b 2
3.b odd 2 1 507.2.b.a 2
13.b even 2 1 inner 1521.2.b.b 2
13.d odd 4 1 117.2.a.a 1
13.d odd 4 1 1521.2.a.e 1
39.d odd 2 1 507.2.b.a 2
39.f even 4 1 39.2.a.a 1
39.f even 4 1 507.2.a.a 1
39.h odd 6 2 507.2.j.e 4
39.i odd 6 2 507.2.j.e 4
39.k even 12 2 507.2.e.a 2
39.k even 12 2 507.2.e.b 2
52.f even 4 1 1872.2.a.h 1
65.f even 4 1 2925.2.c.e 2
65.g odd 4 1 2925.2.a.p 1
65.k even 4 1 2925.2.c.e 2
91.i even 4 1 5733.2.a.e 1
104.j odd 4 1 7488.2.a.bl 1
104.m even 4 1 7488.2.a.by 1
117.y odd 12 2 1053.2.e.d 2
117.z even 12 2 1053.2.e.b 2
156.l odd 4 1 624.2.a.i 1
156.l odd 4 1 8112.2.a.s 1
195.j odd 4 1 975.2.c.f 2
195.n even 4 1 975.2.a.f 1
195.u odd 4 1 975.2.c.f 2
273.o odd 4 1 1911.2.a.f 1
312.w odd 4 1 2496.2.a.e 1
312.y even 4 1 2496.2.a.q 1
429.l odd 4 1 4719.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.a 1 39.f even 4 1
117.2.a.a 1 13.d odd 4 1
507.2.a.a 1 39.f even 4 1
507.2.b.a 2 3.b odd 2 1
507.2.b.a 2 39.d odd 2 1
507.2.e.a 2 39.k even 12 2
507.2.e.b 2 39.k even 12 2
507.2.j.e 4 39.h odd 6 2
507.2.j.e 4 39.i odd 6 2
624.2.a.i 1 156.l odd 4 1
975.2.a.f 1 195.n even 4 1
975.2.c.f 2 195.j odd 4 1
975.2.c.f 2 195.u odd 4 1
1053.2.e.b 2 117.z even 12 2
1053.2.e.d 2 117.y odd 12 2
1521.2.a.e 1 13.d odd 4 1
1521.2.b.b 2 1.a even 1 1 trivial
1521.2.b.b 2 13.b even 2 1 inner
1872.2.a.h 1 52.f even 4 1
1911.2.a.f 1 273.o odd 4 1
2496.2.a.e 1 312.w odd 4 1
2496.2.a.q 1 312.y even 4 1
2925.2.a.p 1 65.g odd 4 1
2925.2.c.e 2 65.f even 4 1
2925.2.c.e 2 65.k even 4 1
4719.2.a.c 1 429.l odd 4 1
5733.2.a.e 1 91.i even 4 1
7488.2.a.bl 1 104.j odd 4 1
7488.2.a.by 1 104.m even 4 1
8112.2.a.s 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
\( T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{2} + 16 \) 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
$5$ \( T^{2} + 4 \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{2} + 16 \) Copy content Toggle raw display
$13$ \( T^{2} \) 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
$29$ \( (T - 10)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( T^{2} + 36 \) Copy content Toggle raw display
$43$ \( (T - 12)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 144 \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 64 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 4 \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16 \) Copy content Toggle raw display
$89$ \( T^{2} + 4 \) Copy content Toggle raw display
$97$ \( T^{2} + 100 \) Copy content Toggle raw display
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