Properties

Label 912.2.bn.h
Level $912$
Weight $2$
Character orbit 912.bn
Analytic conductor $7.282$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.bn (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{3} + \beta_{1} q^{5} + ( -2 - 2 \beta_{1} + \beta_{3} ) q^{7} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{9} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{3} + \beta_{1} q^{5} + ( -2 - 2 \beta_{1} + \beta_{3} ) q^{7} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{9} + ( 1 - 2 \beta_{2} + \beta_{3} ) q^{11} + ( -4 + 2 \beta_{2} ) q^{13} + ( 2 - \beta_{3} ) q^{15} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{17} + ( 3 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{19} + ( -4 - \beta_{1} + 4 \beta_{2} + 3 \beta_{3} ) q^{21} + ( -5 \beta_{1} + 5 \beta_{3} ) q^{23} -3 \beta_{2} q^{25} + ( -5 + \beta_{3} ) q^{27} + ( -\beta_{1} - \beta_{3} ) q^{29} -3 \beta_{3} q^{31} + ( -2 + 3 \beta_{2} - 2 \beta_{3} ) q^{33} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{35} + ( -2 + 4 \beta_{2} - 3 \beta_{3} ) q^{37} + ( 2 - 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{39} + ( -3 + 3 \beta_{2} ) q^{41} + ( -4 + 2 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{43} + ( \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{45} + ( -4 + 7 \beta_{1} + 2 \beta_{2} - 7 \beta_{3} ) q^{47} + ( 3 + 8 \beta_{1} - 4 \beta_{3} ) q^{49} + ( -6 - 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{51} + ( 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} ) q^{53} + ( -2 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{55} + ( 7 - 2 \beta_{1} - 7 \beta_{2} ) q^{57} + ( -9 + \beta_{1} + 9 \beta_{2} - 2 \beta_{3} ) q^{59} + ( \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{61} + ( 2 + 3 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{63} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{65} + ( 2 + 3 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{67} + ( -10 + 5 \beta_{1} + 10 \beta_{2} ) q^{69} + ( -6 + 6 \beta_{2} ) q^{71} + ( 1 + 4 \beta_{1} - \beta_{2} - 8 \beta_{3} ) q^{73} + ( -3 - 3 \beta_{1} + 3 \beta_{2} ) q^{75} + \beta_{3} q^{77} -6 \beta_{1} q^{79} + ( -4 \beta_{1} + 7 \beta_{2} + 4 \beta_{3} ) q^{81} + ( 9 - 18 \beta_{2} + \beta_{3} ) q^{83} + ( -2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{85} + ( -2 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{87} + ( 2 \beta_{1} + 12 \beta_{2} + 2 \beta_{3} ) q^{89} + ( 8 + 6 \beta_{1} - 4 \beta_{2} - 6 \beta_{3} ) q^{91} + ( -3 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{93} + ( 6 + \beta_{3} ) q^{95} + ( -3 - 6 \beta_{1} - 3 \beta_{2} ) q^{97} + ( 3 - \beta_{1} - 5 \beta_{2} + 4 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{3} - 8q^{7} + 2q^{9} + O(q^{10}) \) \( 4q - 2q^{3} - 8q^{7} + 2q^{9} - 12q^{13} + 8q^{15} - 12q^{17} + 2q^{19} - 8q^{21} - 6q^{25} - 20q^{27} - 2q^{33} - 12q^{35} + 12q^{39} - 6q^{41} - 8q^{43} - 8q^{45} - 12q^{47} + 12q^{49} - 16q^{51} - 12q^{53} - 4q^{55} + 14q^{57} - 18q^{59} + 8q^{61} + 20q^{63} + 6q^{67} - 20q^{69} - 12q^{71} + 2q^{73} - 6q^{75} + 14q^{81} - 8q^{85} - 12q^{87} + 24q^{89} + 24q^{91} - 12q^{93} + 24q^{95} - 18q^{97} + 2q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i
1.22474 0.707107i
0 −1.72474 0.158919i 0 −1.22474 0.707107i 0 0.449490 0 2.94949 + 0.548188i 0
65.2 0 0.724745 1.57313i 0 1.22474 + 0.707107i 0 −4.44949 0 −1.94949 2.28024i 0
449.1 0 −1.72474 + 0.158919i 0 −1.22474 + 0.707107i 0 0.449490 0 2.94949 0.548188i 0
449.2 0 0.724745 + 1.57313i 0 1.22474 0.707107i 0 −4.44949 0 −1.94949 + 2.28024i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.bn.h 4
3.b odd 2 1 912.2.bn.g 4
4.b odd 2 1 114.2.h.f yes 4
12.b even 2 1 114.2.h.e 4
19.d odd 6 1 912.2.bn.g 4
57.f even 6 1 inner 912.2.bn.h 4
76.f even 6 1 114.2.h.e 4
228.n odd 6 1 114.2.h.f yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.h.e 4 12.b even 2 1
114.2.h.e 4 76.f even 6 1
114.2.h.f yes 4 4.b odd 2 1
114.2.h.f yes 4 228.n odd 6 1
912.2.bn.g 4 3.b odd 2 1
912.2.bn.g 4 19.d odd 6 1
912.2.bn.h 4 1.a even 1 1 trivial
912.2.bn.h 4 57.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(912, [\chi])\):

\( T_{5}^{4} - 2 T_{5}^{2} + 4 \)
\( T_{7}^{2} + 4 T_{7} - 2 \)
\( T_{17}^{4} + 12 T_{17}^{3} + 52 T_{17}^{2} + 48 T_{17} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 9 + 6 T + T^{2} + 2 T^{3} + T^{4} \)
$5$ \( 4 - 2 T^{2} + T^{4} \)
$7$ \( ( -2 + 4 T + T^{2} )^{2} \)
$11$ \( 1 + 10 T^{2} + T^{4} \)
$13$ \( ( 12 + 6 T + T^{2} )^{2} \)
$17$ \( 16 + 48 T + 52 T^{2} + 12 T^{3} + T^{4} \)
$19$ \( 361 - 38 T - 15 T^{2} - 2 T^{3} + T^{4} \)
$23$ \( 2500 - 50 T^{2} + T^{4} \)
$29$ \( 36 + 6 T^{2} + T^{4} \)
$31$ \( ( 18 + T^{2} )^{2} \)
$37$ \( 36 + 60 T^{2} + T^{4} \)
$41$ \( ( 9 + 3 T + T^{2} )^{2} \)
$43$ \( 64 - 64 T + 72 T^{2} + 8 T^{3} + T^{4} \)
$47$ \( 7396 - 1032 T - 38 T^{2} + 12 T^{3} + T^{4} \)
$53$ \( 144 + 144 T + 132 T^{2} + 12 T^{3} + T^{4} \)
$59$ \( 5625 + 1350 T + 249 T^{2} + 18 T^{3} + T^{4} \)
$61$ \( 100 - 80 T + 54 T^{2} - 8 T^{3} + T^{4} \)
$67$ \( 225 + 90 T - 3 T^{2} - 6 T^{3} + T^{4} \)
$71$ \( ( 36 + 6 T + T^{2} )^{2} \)
$73$ \( 9025 + 190 T + 99 T^{2} - 2 T^{3} + T^{4} \)
$79$ \( 5184 - 72 T^{2} + T^{4} \)
$83$ \( 58081 + 490 T^{2} + T^{4} \)
$89$ \( 14400 - 2880 T + 456 T^{2} - 24 T^{3} + T^{4} \)
$97$ \( 2025 - 810 T + 63 T^{2} + 18 T^{3} + T^{4} \)
show more
show less