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: \(x^{4} + 1\)
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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

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

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 \(S_{2}^{\mathrm{new}}(1521, [\chi])\):

\( T_{2}^{4} + 6 T_{2}^{2} + 1 \)
\( T_{5}^{2} + 8 \)
\( T_{7}^{2} + 8 \)

Hecke characteristic polynomials

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