Properties

Label 1521.2.b.k
Level $1521$
Weight $2$
Character orbit 1521.b
Analytic conductor $12.145$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.0.153664.1
Defining polynomial: \( x^{6} + 5x^{4} + 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_{3} + \beta_1) q^{2} + ( - \beta_{4} - \beta_{2} - 3) q^{4} + (\beta_{5} - 2 \beta_{3} + \beta_1) q^{5} + (\beta_{5} - \beta_1) q^{7} + ( - 5 \beta_{5} - 3 \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + \beta_{3} + \beta_1) q^{2} + ( - \beta_{4} - \beta_{2} - 3) q^{4} + (\beta_{5} - 2 \beta_{3} + \beta_1) q^{5} + (\beta_{5} - \beta_1) q^{7} + ( - 5 \beta_{5} - 3 \beta_{3}) q^{8} + ( - 4 \beta_{4} + 2 \beta_{2} + 1) q^{10} + ( - 3 \beta_{5} - \beta_{3} + 3 \beta_1) q^{11} + ( - 2 \beta_{4} - 2 \beta_{2} + 3) q^{14} + (5 \beta_{4} + 6 \beta_{2}) q^{16} + ( - \beta_{4} - 3 \beta_{2} + 1) q^{17} + (2 \beta_{5} - \beta_{3}) q^{19} + ( - \beta_{5} + 7 \beta_{3} - 5 \beta_1) q^{20} + (5 \beta_{4} + 7 \beta_{2} - 7) q^{22} + ( - \beta_{4} + 4 \beta_{2} - 1) q^{23} + (2 \beta_{4} + 5 \beta_{2} - 6) q^{25} + ( - 3 \beta_{5} - \beta_{3} + 3 \beta_1) q^{28} + ( - 2 \beta_{4} + \beta_{2} + 1) q^{29} + (8 \beta_{5} + 3 \beta_{3} - 5 \beta_1) q^{31} + (12 \beta_{5} + 7 \beta_{3} - 7 \beta_1) q^{32} + ( - 7 \beta_{5} - 7 \beta_{3} + 6 \beta_1) q^{34} + (4 \beta_{4} + \beta_{2} - 3) q^{35} + (7 \beta_{5} - \beta_{3}) q^{37} + ( - 5 \beta_{4} - \beta_{2} + 2) q^{38} + (\beta_{4} - 7 \beta_{2} + 3) q^{40} + (5 \beta_{5} + \beta_{3} - 3 \beta_1) q^{41} + (2 \beta_{4} + 4 \beta_{2} + 3) q^{43} + (11 \beta_{5} + 7 \beta_{3} - 10 \beta_1) q^{44} + (5 \beta_{5} + 12 \beta_{3} - 10 \beta_1) q^{46} + ( - 2 \beta_{5} + \beta_{3}) q^{47} + (2 \beta_{4} + \beta_{2} + 4) q^{49} + (8 \beta_{5} + 7 \beta_{3} - 14 \beta_1) q^{50} + (4 \beta_{4} + \beta_{2} + 4) q^{53} + ( - 10 \beta_{4} - 2 \beta_{2} + 5) q^{55} + (\beta_{4} + 3 \beta_{2} - 1) q^{56} + ( - \beta_{5} + 6 \beta_{3} - 3 \beta_1) q^{58} + ( - 4 \beta_{3} + 2 \beta_1) q^{59} + ( - 5 \beta_{4} + \beta_{2} - 3) q^{61} + ( - 13 \beta_{4} - 16 \beta_{2} + 9) q^{62} + ( - 7 \beta_{4} - 14 \beta_{2} + 7) q^{64} + (4 \beta_{3} - 7 \beta_1) q^{67} + (5 \beta_{4} + 14 \beta_{2} - 2) q^{68} + (7 \beta_{5} - 4 \beta_{3} - \beta_1) q^{70} + ( - 4 \beta_{5} - 7 \beta_{3} + 5 \beta_1) q^{71} + (7 \beta_{5} + 6 \beta_{3} - 9 \beta_1) q^{73} + ( - 15 \beta_{4} - 6 \beta_{2} + 2) q^{74} + ( - 6 \beta_{5} + 2 \beta_{3} - \beta_1) q^{76} + ( - 4 \beta_{4} - 2 \beta_{2} + 7) q^{77} + 3 \beta_{4} q^{79} + ( - 11 \beta_{5} - 5 \beta_{3} + 8 \beta_1) q^{80} + ( - 9 \beta_{4} - 9 \beta_{2} + 7) q^{82} + (3 \beta_{5} - \beta_{3} + 2 \beta_1) q^{83} + (5 \beta_{5} - \beta_{3} + 3 \beta_1) q^{85} + (15 \beta_{5} + 13 \beta_{3} - 3 \beta_1) q^{86} + ( - 5 \beta_{4} - 14 \beta_{2} + 2) q^{88} + ( - 2 \beta_{5} - 6 \beta_{3} - \beta_1) q^{89} + ( - 19 \beta_{2} + 4) q^{92} + (5 \beta_{4} + \beta_{2} - 2) q^{94} + (4 \beta_{4} + 5 \beta_{2} - 10) q^{95} + ( - 3 \beta_{5} - 12 \beta_{3} + 2 \beta_1) q^{97} + (10 \beta_{5} + 5 \beta_{3} + 4 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 22 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 22 q^{4} + 2 q^{10} + 10 q^{14} + 22 q^{16} - 2 q^{17} - 18 q^{22} - 22 q^{25} + 4 q^{29} - 8 q^{35} + 6 q^{40} + 30 q^{43} + 30 q^{49} + 34 q^{53} + 6 q^{55} + 2 q^{56} - 26 q^{61} - 4 q^{62} + 26 q^{68} - 30 q^{74} + 30 q^{77} + 6 q^{79} + 6 q^{82} - 26 q^{88} - 14 q^{92} - 42 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 3\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} + 3\nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} + 4\nu^{3} + 3\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - 3\beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 4\beta_{3} + 9\beta_1 \) 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
0.445042i
1.80194i
1.24698i
1.24698i
1.80194i
0.445042i
2.69202i 0 −5.24698 1.04892i 0 0.554958i 8.74094i 0 2.82371
1351.2 2.35690i 0 −3.55496 3.69202i 0 0.801938i 3.66487i 0 −8.70171
1351.3 2.04892i 0 −2.19806 3.35690i 0 2.24698i 0.405813i 0 6.87800
1351.4 2.04892i 0 −2.19806 3.35690i 0 2.24698i 0.405813i 0 6.87800
1351.5 2.35690i 0 −3.55496 3.69202i 0 0.801938i 3.66487i 0 −8.70171
1351.6 2.69202i 0 −5.24698 1.04892i 0 0.554958i 8.74094i 0 2.82371
\(n\): 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.k 6
3.b odd 2 1 507.2.b.f 6
13.b even 2 1 inner 1521.2.b.k 6
13.d odd 4 1 1521.2.a.n 3
13.d odd 4 1 1521.2.a.s 3
39.d odd 2 1 507.2.b.f 6
39.f even 4 1 507.2.a.i 3
39.f even 4 1 507.2.a.l yes 3
39.h odd 6 2 507.2.j.i 12
39.i odd 6 2 507.2.j.i 12
39.k even 12 2 507.2.e.i 6
39.k even 12 2 507.2.e.l 6
156.l odd 4 1 8112.2.a.cg 3
156.l odd 4 1 8112.2.a.cp 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.a.i 3 39.f even 4 1
507.2.a.l yes 3 39.f even 4 1
507.2.b.f 6 3.b odd 2 1
507.2.b.f 6 39.d odd 2 1
507.2.e.i 6 39.k even 12 2
507.2.e.l 6 39.k even 12 2
507.2.j.i 12 39.h odd 6 2
507.2.j.i 12 39.i odd 6 2
1521.2.a.n 3 13.d odd 4 1
1521.2.a.s 3 13.d odd 4 1
1521.2.b.k 6 1.a even 1 1 trivial
1521.2.b.k 6 13.b even 2 1 inner
8112.2.a.cg 3 156.l odd 4 1
8112.2.a.cp 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 \(S_{2}^{\mathrm{new}}(1521, [\chi])\):

\( T_{2}^{6} + 17T_{2}^{4} + 94T_{2}^{2} + 169 \) Copy content Toggle raw display
\( T_{5}^{6} + 26T_{5}^{4} + 181T_{5}^{2} + 169 \) Copy content Toggle raw display
\( T_{7}^{6} + 6T_{7}^{4} + 5T_{7}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 17 T^{4} + 94 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 26 T^{4} + 181 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$7$ \( T^{6} + 6 T^{4} + 5 T^{2} + 1 \) Copy content Toggle raw display
$11$ \( T^{6} + 41 T^{4} + 474 T^{2} + \cdots + 1681 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( (T^{3} + T^{2} - 16 T + 13)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 21 T^{4} + 98 T^{2} + 49 \) Copy content Toggle raw display
$23$ \( (T^{3} - 49 T + 91)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 2 T^{2} - 15 T + 29)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 174 T^{4} + 7985 T^{2} + \cdots + 38809 \) Copy content Toggle raw display
$37$ \( T^{6} + 166 T^{4} + 8693 T^{2} + \cdots + 142129 \) Copy content Toggle raw display
$41$ \( T^{6} + 73 T^{4} + 1214 T^{2} + \cdots + 841 \) Copy content Toggle raw display
$43$ \( (T^{3} - 15 T^{2} + 47 T - 41)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 21 T^{4} + 98 T^{2} + 49 \) Copy content Toggle raw display
$53$ \( (T^{3} - 17 T^{2} + 66 T + 41)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 68 T^{4} + 1504 T^{2} + \cdots + 10816 \) Copy content Toggle raw display
$61$ \( (T^{3} + 13 T^{2} - 16 T - 167)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 213 T^{4} + 1214 T^{2} + \cdots + 1681 \) Copy content Toggle raw display
$71$ \( T^{6} + 182 T^{4} + 8281 T^{2} + \cdots + 41209 \) Copy content Toggle raw display
$73$ \( T^{6} + 306 T^{4} + 29301 T^{2} + \cdots + 851929 \) Copy content Toggle raw display
$79$ \( (T^{3} - 3 T^{2} - 18 T + 27)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 62 T^{4} + 649 T^{2} + \cdots + 1849 \) Copy content Toggle raw display
$89$ \( T^{6} + 201 T^{4} + 10226 T^{2} + \cdots + 12769 \) Copy content Toggle raw display
$97$ \( T^{6} + 587 T^{4} + 95331 T^{2} + \cdots + 2679769 \) Copy content Toggle raw display
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