Properties

Label 1512.2.s.m
Level $1512$
Weight $2$
Character orbit 1512.s
Analytic conductor $12.073$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1512.s (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.0733807856\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.9391935744.3
Defining polynomial: \(x^{8} - 4 x^{7} + 5 x^{6} + 12 x^{5} - 76 x^{4} + 84 x^{3} + 245 x^{2} - 1372 x + 2401\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{3} + \beta_{5} - \beta_{6} ) q^{5} + ( -\beta_{1} - \beta_{3} - \beta_{4} ) q^{7} +O(q^{10})\) \( q + ( \beta_{3} + \beta_{5} - \beta_{6} ) q^{5} + ( -\beta_{1} - \beta_{3} - \beta_{4} ) q^{7} + ( -1 + \beta_{1} - \beta_{3} - \beta_{7} ) q^{11} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{5} + \beta_{7} ) q^{13} + ( 1 + \beta_{3} + 3 \beta_{6} ) q^{17} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{19} + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{23} + ( 1 + \beta_{3} + 2 \beta_{6} ) q^{25} + ( 3 - 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{29} + ( -2 + \beta_{1} - 2 \beta_{3} + 2 \beta_{6} - \beta_{7} ) q^{31} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{7} ) q^{35} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{37} + ( -1 - 2 \beta_{2} - 2 \beta_{4} - 3 \beta_{5} ) q^{41} + ( -1 + 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{43} + ( 5 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{47} + ( -\beta_{2} + 2 \beta_{3} + \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{49} + ( -2 + \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 3 \beta_{6} + \beta_{7} ) q^{53} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{7} ) q^{55} + ( 5 + \beta_{1} + 5 \beta_{3} - 2 \beta_{6} - \beta_{7} ) q^{59} + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} + 4 \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{61} + ( -1 + \beta_{1} + \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} ) q^{65} + ( \beta_{2} - \beta_{3} - \beta_{4} - 4 \beta_{6} - \beta_{7} ) q^{67} + ( 6 - \beta_{2} - \beta_{4} + 2 \beta_{5} ) q^{71} + ( -3 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{7} ) q^{73} + ( 7 - \beta_{1} - \beta_{3} - 2 \beta_{5} + 4 \beta_{6} - \beta_{7} ) q^{77} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{7} ) q^{79} + ( -4 - 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{83} + ( 8 - 2 \beta_{5} ) q^{85} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} + \beta_{5} - \beta_{6} ) q^{89} + ( -7 + \beta_{1} + 2 \beta_{2} - 5 \beta_{3} + \beta_{4} - 2 \beta_{5} + 6 \beta_{6} + \beta_{7} ) q^{91} + ( -8 + \beta_{1} - 8 \beta_{3} + 3 \beta_{6} - \beta_{7} ) q^{95} + ( 1 - 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{5} - 2q^{7} + O(q^{10}) \) \( 8q - 4q^{5} - 2q^{7} - 2q^{11} - 8q^{13} + 4q^{17} - 6q^{19} + 2q^{23} + 4q^{25} + 16q^{29} - 6q^{31} - 2q^{35} - 16q^{41} - 20q^{47} - 6q^{49} - 10q^{53} + 16q^{55} + 22q^{59} + 2q^{61} - 14q^{65} + 2q^{67} + 44q^{71} - 10q^{73} + 54q^{77} + 8q^{79} - 40q^{83} + 64q^{85} - 16q^{89} - 24q^{91} - 30q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} + 5 x^{6} + 12 x^{5} - 76 x^{4} + 84 x^{3} + 245 x^{2} - 1372 x + 2401\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} - 3 \nu^{6} - 26 \nu^{5} - 194 \nu^{4} - 1282 \nu^{3} + 3290 \nu^{2} - 1911 \nu + 343 \)\()/12348\)
\(\beta_{3}\)\(=\)\((\)\( 13 \nu^{7} + 18 \nu^{6} + 128 \nu^{5} + 506 \nu^{4} - 1520 \nu^{3} + 574 \nu^{2} + 2205 \nu - 42532 \)\()/37044\)
\(\beta_{4}\)\(=\)\((\)\( 19 \nu^{7} + 15 \nu^{6} + 74 \nu^{5} - 346 \nu^{4} + 334 \nu^{3} - 518 \nu^{2} - 8967 \nu + 3773 \)\()/12348\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{7} - 4 \nu^{6} - 10 \nu^{5} + 48 \nu^{4} - 146 \nu^{3} - 468 \nu^{2} + 1169 \nu - 1960 \)\()/1764\)
\(\beta_{6}\)\(=\)\((\)\( -43 \nu^{7} + 53 \nu^{6} + 114 \nu^{5} - 866 \nu^{4} + 762 \nu^{3} - 2506 \nu^{2} - 5929 \nu + 27783 \)\()/24696\)
\(\beta_{7}\)\(=\)\((\)\( -124 \nu^{7} + 405 \nu^{6} - 746 \nu^{5} - 2384 \nu^{4} + 5882 \nu^{3} + 224 \nu^{2} - 34398 \nu + 154693 \)\()/37044\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-2 \beta_{6} - 2 \beta_{5} + \beta_{2} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(-4 \beta_{6} - 4 \beta_{5} + \beta_{4} - 6 \beta_{3} - 5 \beta_{2} + 2 \beta_{1} - 7\)
\(\nu^{4}\)\(=\)\(\beta_{7} - 10 \beta_{6} - 4 \beta_{5} - 11 \beta_{4} + 25 \beta_{3} - 11 \beta_{2} - 10 \beta_{1} + 35\)
\(\nu^{5}\)\(=\)\(-17 \beta_{7} + 32 \beta_{6} - 28 \beta_{5} + 14 \beta_{4} + 67 \beta_{3} - 17 \beta_{2} + 14 \beta_{1} + 77\)
\(\nu^{6}\)\(=\)\(74 \beta_{7} - 24 \beta_{6} + 50 \beta_{4} + 362 \beta_{3} - 24 \beta_{2} + 74 \beta_{1} + 119\)
\(\nu^{7}\)\(=\)\(26 \beta_{7} - 272 \beta_{6} + 52 \beta_{5} + 338 \beta_{4} + 14 \beta_{3} + 169 \beta_{1} + 168\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1081\) \(1135\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{3}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
865.1
2.61033 0.431486i
−2.47635 + 0.931486i
1.57052 + 2.12920i
0.295509 2.62920i
2.61033 + 0.431486i
−2.47635 0.931486i
1.57052 2.12920i
0.295509 + 2.62920i
0 0 0 −1.36603 2.36603i 0 −0.931486 + 2.47635i 0 0 0
865.2 0 0 0 −1.36603 2.36603i 0 0.431486 2.61033i 0 0 0
865.3 0 0 0 0.366025 + 0.633975i 0 −2.62920 + 0.295509i 0 0 0
865.4 0 0 0 0.366025 + 0.633975i 0 2.12920 + 1.57052i 0 0 0
1297.1 0 0 0 −1.36603 + 2.36603i 0 −0.931486 2.47635i 0 0 0
1297.2 0 0 0 −1.36603 + 2.36603i 0 0.431486 + 2.61033i 0 0 0
1297.3 0 0 0 0.366025 0.633975i 0 −2.62920 0.295509i 0 0 0
1297.4 0 0 0 0.366025 0.633975i 0 2.12920 1.57052i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1297.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1512.2.s.m 8
3.b odd 2 1 1512.2.s.p yes 8
7.c even 3 1 inner 1512.2.s.m 8
21.h odd 6 1 1512.2.s.p yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.s.m 8 1.a even 1 1 trivial
1512.2.s.m 8 7.c even 3 1 inner
1512.2.s.p yes 8 3.b odd 2 1
1512.2.s.p yes 8 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1512, [\chi])\):

\( T_{5}^{4} + 2 T_{5}^{3} + 6 T_{5}^{2} - 4 T_{5} + 4 \)
\(T_{11}^{8} + \cdots\)
\( T_{13}^{4} + 4 T_{13}^{3} - 35 T_{13}^{2} - 204 T_{13} - 264 \)