Properties

Label 1521.2.b.j
Level $1521$
Weight $2$
Character orbit 1521.b
Analytic conductor $12.145$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3} \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} - 1) q^{4} + (\beta_{2} - \beta_1) q^{5} + ( - \beta_{2} + \beta_1) q^{7} + ( - \beta_{2} - 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{3} - 1) q^{4} + (\beta_{2} - \beta_1) q^{5} + ( - \beta_{2} + \beta_1) q^{7} + ( - \beta_{2} - 2 \beta_1) q^{8} + (\beta_{3} + 4) q^{10} + ( - \beta_{2} - \beta_1) q^{11} + ( - \beta_{3} - 4) q^{14} + 3 q^{16} + ( - 2 \beta_{3} + 2) q^{17} + ( - \beta_{2} + \beta_1) q^{19} + (3 \beta_{2} + 5 \beta_1) q^{20} + (\beta_{3} + 2) q^{22} - 4 q^{23} - 3 q^{25} + ( - 3 \beta_{2} - 5 \beta_1) q^{28} - 2 q^{29} + ( - \beta_{2} - 3 \beta_1) q^{31} + ( - 2 \beta_{2} - \beta_1) q^{32} + ( - 2 \beta_{2} - 4 \beta_1) q^{34} + 8 q^{35} + (3 \beta_{2} - \beta_1) q^{37} + ( - \beta_{3} - 4) q^{38} + ( - 3 \beta_{3} - 4) q^{40} + ( - 3 \beta_{2} - 5 \beta_1) q^{41} + ( - 2 \beta_{3} - 4) q^{43} + ( - \beta_{2} + 3 \beta_1) q^{44} - 4 \beta_1 q^{46} + ( - 5 \beta_{2} - \beta_1) q^{47} - q^{49} - 3 \beta_1 q^{50} + 2 q^{53} - 2 \beta_{3} q^{55} + (3 \beta_{3} + 4) q^{56} - 2 \beta_1 q^{58} + (3 \beta_{2} - \beta_1) q^{59} + ( - 4 \beta_{3} + 2) q^{61} + (3 \beta_{3} + 8) q^{62} + (\beta_{3} + 7) q^{64} + (3 \beta_{2} + \beta_1) q^{67} + 14 q^{68} + 8 \beta_1 q^{70} + ( - \beta_{2} - \beta_1) q^{71} + ( - \beta_{2} - 5 \beta_1) q^{73} + (\beta_{3} + 6) q^{74} + ( - 3 \beta_{2} - 5 \beta_1) q^{76} + 2 \beta_{3} q^{77} + 4 \beta_{3} q^{79} + (3 \beta_{2} - 3 \beta_1) q^{80} + (5 \beta_{3} + 12) q^{82} + ( - \beta_{2} + 3 \beta_1) q^{83} + (10 \beta_{2} + 6 \beta_1) q^{85} + ( - 2 \beta_{2} - 10 \beta_1) q^{86} + ( - \beta_{3} - 6) q^{88} + (7 \beta_{2} + 5 \beta_1) q^{89} + (4 \beta_{3} + 4) q^{92} + (\beta_{3} - 2) q^{94} + 8 q^{95} + (\beta_{2} - 3 \beta_1) q^{97} - \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 16 q^{10} - 16 q^{14} + 12 q^{16} + 8 q^{17} + 8 q^{22} - 16 q^{23} - 12 q^{25} - 8 q^{29} + 32 q^{35} - 16 q^{38} - 16 q^{40} - 16 q^{43} - 4 q^{49} + 8 q^{53} + 16 q^{56} + 8 q^{61} + 32 q^{62} + 28 q^{64} + 56 q^{68} + 24 q^{74} + 48 q^{82} - 24 q^{88} + 16 q^{92} - 8 q^{94} + 32 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8}^{2} - \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\zeta_{8}^{3} + 2\zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} - \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} - \beta_{2} + \beta_1 ) / 4 \) 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
0.707107 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
2.41421i 0 −3.82843 2.82843i 0 2.82843i 4.41421i 0 6.82843
1351.2 0.414214i 0 1.82843 2.82843i 0 2.82843i 1.58579i 0 1.17157
1351.3 0.414214i 0 1.82843 2.82843i 0 2.82843i 1.58579i 0 1.17157
1351.4 2.41421i 0 −3.82843 2.82843i 0 2.82843i 4.41421i 0 6.82843
\(n\): 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.j 4
3.b odd 2 1 507.2.b.e 4
13.b even 2 1 inner 1521.2.b.j 4
13.d odd 4 1 117.2.a.c 2
13.d odd 4 1 1521.2.a.f 2
39.d odd 2 1 507.2.b.e 4
39.f even 4 1 39.2.a.b 2
39.f even 4 1 507.2.a.h 2
39.h odd 6 2 507.2.j.f 8
39.i odd 6 2 507.2.j.f 8
39.k even 12 2 507.2.e.d 4
39.k even 12 2 507.2.e.h 4
52.f even 4 1 1872.2.a.w 2
65.f even 4 1 2925.2.c.u 4
65.g odd 4 1 2925.2.a.v 2
65.k even 4 1 2925.2.c.u 4
91.i even 4 1 5733.2.a.u 2
104.j odd 4 1 7488.2.a.cl 2
104.m even 4 1 7488.2.a.co 2
117.y odd 12 2 1053.2.e.e 4
117.z even 12 2 1053.2.e.m 4
156.l odd 4 1 624.2.a.k 2
156.l odd 4 1 8112.2.a.bm 2
195.j odd 4 1 975.2.c.h 4
195.n even 4 1 975.2.a.l 2
195.u odd 4 1 975.2.c.h 4
273.o odd 4 1 1911.2.a.h 2
312.w odd 4 1 2496.2.a.bi 2
312.y even 4 1 2496.2.a.bf 2
429.l odd 4 1 4719.2.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.b 2 39.f even 4 1
117.2.a.c 2 13.d odd 4 1
507.2.a.h 2 39.f even 4 1
507.2.b.e 4 3.b odd 2 1
507.2.b.e 4 39.d odd 2 1
507.2.e.d 4 39.k even 12 2
507.2.e.h 4 39.k even 12 2
507.2.j.f 8 39.h odd 6 2
507.2.j.f 8 39.i odd 6 2
624.2.a.k 2 156.l odd 4 1
975.2.a.l 2 195.n even 4 1
975.2.c.h 4 195.j odd 4 1
975.2.c.h 4 195.u odd 4 1
1053.2.e.e 4 117.y odd 12 2
1053.2.e.m 4 117.z even 12 2
1521.2.a.f 2 13.d odd 4 1
1521.2.b.j 4 1.a even 1 1 trivial
1521.2.b.j 4 13.b even 2 1 inner
1872.2.a.w 2 52.f even 4 1
1911.2.a.h 2 273.o odd 4 1
2496.2.a.bf 2 312.y even 4 1
2496.2.a.bi 2 312.w odd 4 1
2925.2.a.v 2 65.g odd 4 1
2925.2.c.u 4 65.f even 4 1
2925.2.c.u 4 65.k even 4 1
4719.2.a.p 2 429.l odd 4 1
5733.2.a.u 2 91.i even 4 1
7488.2.a.cl 2 104.j odd 4 1
7488.2.a.co 2 104.m even 4 1
8112.2.a.bm 2 156.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 \(S_{2}^{\mathrm{new}}(1521, [\chi])\):

\( T_{2}^{4} + 6T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} + 8 \) Copy content Toggle raw display
\( T_{7}^{2} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 6T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 4 T - 28)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$23$ \( (T + 4)^{4} \) Copy content Toggle raw display
$29$ \( (T + 2)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 48T^{2} + 64 \) Copy content Toggle raw display
$37$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
$41$ \( T^{4} + 144T^{2} + 3136 \) Copy content Toggle raw display
$43$ \( (T^{2} + 8 T - 16)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 136T^{2} + 16 \) Copy content Toggle raw display
$53$ \( (T - 2)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T - 124)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 48T^{2} + 64 \) Copy content Toggle raw display
$71$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 136T^{2} + 16 \) Copy content Toggle raw display
$79$ \( (T^{2} - 128)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
$89$ \( T^{4} + 304 T^{2} + 18496 \) Copy content Toggle raw display
$97$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
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