Properties

Label 1521.2.b.m
Level $1521$
Weight $2$
Character orbit 1521.b
Analytic conductor $12.145$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(6\)
Coefficient field: 6.0.153664.1
Defining polynomial: \(x^{6} + 5 x^{4} + 6 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 507)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( -\beta_{3} + \beta_{5} ) q^{5} + ( -\beta_{3} + 3 \beta_{5} ) q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( -\beta_{3} + \beta_{5} ) q^{5} + ( -\beta_{3} + 3 \beta_{5} ) q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} + ( 1 - 2 \beta_{4} ) q^{10} + ( 3 \beta_{1} - 4 \beta_{3} - 2 \beta_{5} ) q^{11} + ( 1 - 4 \beta_{4} ) q^{14} + ( -3 + 3 \beta_{2} + \beta_{4} ) q^{16} + ( -2 - \beta_{2} ) q^{17} + ( -3 \beta_{1} - \beta_{3} - 3 \beta_{5} ) q^{19} -\beta_{1} q^{20} + ( -2 + 3 \beta_{2} - 2 \beta_{4} ) q^{22} + ( -2 + 5 \beta_{2} + 3 \beta_{4} ) q^{23} + ( 2 \beta_{2} + 3 \beta_{4} ) q^{25} + ( -3 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} ) q^{28} + ( 1 + 2 \beta_{2} + 3 \beta_{4} ) q^{29} + ( 5 \beta_{1} - 2 \beta_{3} - 5 \beta_{5} ) q^{31} + ( -3 \beta_{1} + 4 \beta_{3} + \beta_{5} ) q^{32} + ( -\beta_{1} - \beta_{3} ) q^{34} + ( -9 + 4 \beta_{2} + 5 \beta_{4} ) q^{35} + ( \beta_{1} - \beta_{3} + 4 \beta_{5} ) q^{37} + ( 7 - 3 \beta_{2} + 2 \beta_{4} ) q^{38} + ( 4 - \beta_{2} - 4 \beta_{4} ) q^{40} + \beta_{1} q^{41} + ( 1 - 2 \beta_{2} + 2 \beta_{4} ) q^{43} + ( -\beta_{1} - 3 \beta_{3} - 6 \beta_{5} ) q^{44} + ( -4 \beta_{1} + 2 \beta_{3} + 3 \beta_{5} ) q^{46} + ( -9 \beta_{1} + 3 \beta_{3} + 7 \beta_{5} ) q^{47} + ( -10 + 6 \beta_{2} + 7 \beta_{4} ) q^{49} + ( \beta_{1} - \beta_{3} + 3 \beta_{5} ) q^{50} + ( 6 - 2 \beta_{2} - 3 \beta_{4} ) q^{53} + ( -5 + 2 \beta_{2} ) q^{55} + ( 6 - 3 \beta_{2} - 8 \beta_{4} ) q^{56} + ( 2 \beta_{1} - \beta_{3} + 3 \beta_{5} ) q^{58} + ( -4 \beta_{1} + 6 \beta_{3} + 8 \beta_{5} ) q^{59} + ( -4 - 3 \beta_{2} + 2 \beta_{4} ) q^{61} + ( -8 + 5 \beta_{2} + 3 \beta_{4} ) q^{62} + ( -4 + 3 \beta_{2} + 5 \beta_{4} ) q^{64} + ( 4 \beta_{1} - 3 \beta_{3} - 4 \beta_{5} ) q^{67} + ( -1 - 3 \beta_{2} - \beta_{4} ) q^{68} + ( -8 \beta_{1} - \beta_{3} + 5 \beta_{5} ) q^{70} + ( -5 \beta_{1} - 2 \beta_{3} - \beta_{5} ) q^{71} + ( -2 \beta_{1} + \beta_{3} + 7 \beta_{5} ) q^{73} + ( -1 + \beta_{2} - 5 \beta_{4} ) q^{74} + ( 6 \beta_{1} - 7 \beta_{3} - 4 \beta_{5} ) q^{76} + ( -9 + 10 \beta_{2} + 2 \beta_{4} ) q^{77} + ( -5 + 5 \beta_{2} + \beta_{4} ) q^{79} + ( -\beta_{1} + 3 \beta_{3} - 4 \beta_{5} ) q^{80} + ( -2 + \beta_{2} ) q^{82} + ( -\beta_{1} - 3 \beta_{3} - 6 \beta_{5} ) q^{83} + ( \beta_{1} + 2 \beta_{3} - 2 \beta_{5} ) q^{85} + ( 5 \beta_{1} - 4 \beta_{3} + 2 \beta_{5} ) q^{86} + ( 1 + 5 \beta_{2} - \beta_{4} ) q^{88} + ( -2 \beta_{1} - \beta_{3} + 2 \beta_{5} ) q^{89} + ( 2 + 6 \beta_{2} + 5 \beta_{4} ) q^{92} + ( 15 - 9 \beta_{2} - 4 \beta_{4} ) q^{94} + ( -2 \beta_{2} + 5 \beta_{4} ) q^{95} + ( -4 \beta_{1} + 10 \beta_{3} + 3 \beta_{5} ) q^{97} + ( -9 \beta_{1} - \beta_{3} + 7 \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 2q^{4} + O(q^{10}) \) \( 6q + 2q^{4} + 2q^{10} - 2q^{14} - 10q^{16} - 14q^{17} - 10q^{22} + 4q^{23} + 10q^{25} + 16q^{29} - 36q^{35} + 40q^{38} + 14q^{40} + 6q^{43} - 34q^{49} + 26q^{53} - 26q^{55} + 14q^{56} - 26q^{61} - 32q^{62} - 8q^{64} - 14q^{68} - 14q^{74} - 30q^{77} - 18q^{79} - 10q^{82} + 14q^{88} + 34q^{92} + 64q^{94} + 6q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 5 x^{4} + 6 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 2 \)
\(\beta_{3}\)\(=\)\( \nu^{3} + 3 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} + 3 \nu^{2} + 1 \)
\(\beta_{5}\)\(=\)\( \nu^{5} + 4 \nu^{3} + 3 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 2\)
\(\nu^{3}\)\(=\)\(\beta_{3} - 3 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4} - 3 \beta_{2} + 5\)
\(\nu^{5}\)\(=\)\(\beta_{5} - 4 \beta_{3} + 9 \beta_{1}\)

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
1.80194i
1.24698i
0.445042i
0.445042i
1.24698i
1.80194i
1.80194i 0 −1.24698 1.44504i 0 3.44504i 1.35690i 0 −2.60388
1351.2 1.24698i 0 0.445042 2.80194i 0 4.80194i 3.04892i 0 3.49396
1351.3 0.445042i 0 1.80194 0.246980i 0 1.75302i 1.69202i 0 0.109916
1351.4 0.445042i 0 1.80194 0.246980i 0 1.75302i 1.69202i 0 0.109916
1351.5 1.24698i 0 0.445042 2.80194i 0 4.80194i 3.04892i 0 3.49396
1351.6 1.80194i 0 −1.24698 1.44504i 0 3.44504i 1.35690i 0 −2.60388
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1351.6
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.m 6
3.b odd 2 1 507.2.b.g 6
13.b even 2 1 inner 1521.2.b.m 6
13.d odd 4 1 1521.2.a.p 3
13.d odd 4 1 1521.2.a.q 3
39.d odd 2 1 507.2.b.g 6
39.f even 4 1 507.2.a.j 3
39.f even 4 1 507.2.a.k yes 3
39.h odd 6 2 507.2.j.h 12
39.i odd 6 2 507.2.j.h 12
39.k even 12 2 507.2.e.j 6
39.k even 12 2 507.2.e.k 6
156.l odd 4 1 8112.2.a.by 3
156.l odd 4 1 8112.2.a.cf 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.a.j 3 39.f even 4 1
507.2.a.k yes 3 39.f even 4 1
507.2.b.g 6 3.b odd 2 1
507.2.b.g 6 39.d odd 2 1
507.2.e.j 6 39.k even 12 2
507.2.e.k 6 39.k even 12 2
507.2.j.h 12 39.h odd 6 2
507.2.j.h 12 39.i odd 6 2
1521.2.a.p 3 13.d odd 4 1
1521.2.a.q 3 13.d odd 4 1
1521.2.b.m 6 1.a even 1 1 trivial
1521.2.b.m 6 13.b even 2 1 inner
8112.2.a.by 3 156.l odd 4 1
8112.2.a.cf 3 156.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 \(S_{2}^{\mathrm{new}}(1521, [\chi])\):

\( T_{2}^{6} + 5 T_{2}^{4} + 6 T_{2}^{2} + 1 \)
\( T_{5}^{6} + 10 T_{5}^{4} + 17 T_{5}^{2} + 1 \)
\( T_{7}^{6} + 38 T_{7}^{4} + 381 T_{7}^{2} + 841 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 6 T^{2} + 5 T^{4} + T^{6} \)
$3$ \( T^{6} \)
$5$ \( 1 + 17 T^{2} + 10 T^{4} + T^{6} \)
$7$ \( 841 + 381 T^{2} + 38 T^{4} + T^{6} \)
$11$ \( 1849 + 986 T^{2} + 61 T^{4} + T^{6} \)
$13$ \( T^{6} \)
$17$ \( ( 7 + 14 T + 7 T^{2} + T^{3} )^{2} \)
$19$ \( 12769 + 2586 T^{2} + 101 T^{4} + T^{6} \)
$23$ \( ( -83 - 43 T - 2 T^{2} + T^{3} )^{2} \)
$29$ \( ( 43 + 5 T - 8 T^{2} + T^{3} )^{2} \)
$31$ \( 38809 + 3681 T^{2} + 110 T^{4} + T^{6} \)
$37$ \( 8281 + 1421 T^{2} + 70 T^{4} + T^{6} \)
$41$ \( 1 + 6 T^{2} + 5 T^{4} + T^{6} \)
$43$ \( ( -29 - 25 T - 3 T^{2} + T^{3} )^{2} \)
$47$ \( 829921 + 30798 T^{2} + 321 T^{4} + T^{6} \)
$53$ \( ( -29 + 40 T - 13 T^{2} + T^{3} )^{2} \)
$59$ \( 3136 + 1568 T^{2} + 196 T^{4} + T^{6} \)
$61$ \( ( -223 + 12 T + 13 T^{2} + T^{3} )^{2} \)
$67$ \( 9409 + 1454 T^{2} + 69 T^{4} + T^{6} \)
$71$ \( 212521 + 11773 T^{2} + 194 T^{4} + T^{6} \)
$73$ \( 27889 + 4189 T^{2} + 122 T^{4} + T^{6} \)
$79$ \( ( -169 - 22 T + 9 T^{2} + T^{3} )^{2} \)
$83$ \( 1849 + 4401 T^{2} + 146 T^{4} + T^{6} \)
$89$ \( 1 + 54 T^{2} + 41 T^{4} + T^{6} \)
$97$ \( 1413721 + 40451 T^{2} + 363 T^{4} + T^{6} \)
show more
show less