Properties

Label 672.2.i.c
Level 672
Weight 2
Character orbit 672.i
Analytic conductor 5.366
Analytic rank 0
Dimension 4
CM no
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.i (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{3} + \beta_{1} q^{5} + ( 2 - \beta_{2} ) q^{7} + ( -1 - 2 \beta_{1} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{3} + \beta_{1} q^{5} + ( 2 - \beta_{2} ) q^{7} + ( -1 - 2 \beta_{1} ) q^{9} + \beta_{3} q^{11} + 2 q^{13} + ( 2 + \beta_{1} ) q^{15} -3 \beta_{3} q^{17} + 4 q^{19} + ( 2 - 2 \beta_{1} - \beta_{2} - \beta_{3} ) q^{21} -\beta_{1} q^{23} + 3 q^{25} + ( -5 - \beta_{1} ) q^{27} + 2 \beta_{3} q^{29} -4 \beta_{2} q^{31} + ( -2 \beta_{2} + \beta_{3} ) q^{33} + ( 2 \beta_{1} + \beta_{3} ) q^{35} + 6 \beta_{2} q^{37} + ( 2 - 2 \beta_{1} ) q^{39} + \beta_{3} q^{41} -2 \beta_{2} q^{43} + ( 4 - \beta_{1} ) q^{45} -2 \beta_{3} q^{47} + ( 1 - 4 \beta_{2} ) q^{49} + ( 6 \beta_{2} - 3 \beta_{3} ) q^{51} + 2 \beta_{2} q^{55} + ( 4 - 4 \beta_{1} ) q^{57} -4 \beta_{1} q^{59} -10 q^{61} + ( -2 - 4 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{63} + 2 \beta_{1} q^{65} + 2 \beta_{2} q^{67} + ( -2 - \beta_{1} ) q^{69} -\beta_{1} q^{71} + ( 3 - 3 \beta_{1} ) q^{75} + ( -3 \beta_{1} + 2 \beta_{3} ) q^{77} -8 q^{79} + ( -7 + 4 \beta_{1} ) q^{81} + 8 \beta_{1} q^{83} -6 \beta_{2} q^{85} + ( -4 \beta_{2} + 2 \beta_{3} ) q^{87} + 3 \beta_{3} q^{89} + ( 4 - 2 \beta_{2} ) q^{91} + ( -4 \beta_{2} - 4 \beta_{3} ) q^{93} + 4 \beta_{1} q^{95} + 8 \beta_{2} q^{97} + ( -4 \beta_{2} - \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 8q^{7} - 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 8q^{7} - 4q^{9} + 8q^{13} + 8q^{15} + 16q^{19} + 8q^{21} + 12q^{25} - 20q^{27} + 8q^{39} + 16q^{45} + 4q^{49} + 16q^{57} - 40q^{61} - 8q^{63} - 8q^{69} + 12q^{75} - 32q^{79} - 28q^{81} + 16q^{91} + O(q^{100}) \)

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
1.22474 + 0.707107i
−1.22474 + 0.707107i
−1.22474 0.707107i
1.22474 0.707107i
0 1.00000 1.41421i 0 1.41421i 0 2.00000 1.73205i 0 −1.00000 2.82843i 0
209.2 0 1.00000 1.41421i 0 1.41421i 0 2.00000 + 1.73205i 0 −1.00000 2.82843i 0
209.3 0 1.00000 + 1.41421i 0 1.41421i 0 2.00000 1.73205i 0 −1.00000 + 2.82843i 0
209.4 0 1.00000 + 1.41421i 0 1.41421i 0 2.00000 + 1.73205i 0 −1.00000 + 2.82843i 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
3.b odd 2 1 inner
56.h odd 2 1 inner
168.i even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.2.i.c 4
3.b odd 2 1 inner 672.2.i.c 4
4.b odd 2 1 168.2.i.a 4
7.b odd 2 1 672.2.i.a 4
8.b even 2 1 672.2.i.a 4
8.d odd 2 1 168.2.i.c yes 4
12.b even 2 1 168.2.i.a 4
21.c even 2 1 672.2.i.a 4
24.f even 2 1 168.2.i.c yes 4
24.h odd 2 1 672.2.i.a 4
28.d even 2 1 168.2.i.c yes 4
56.e even 2 1 168.2.i.a 4
56.h odd 2 1 inner 672.2.i.c 4
84.h odd 2 1 168.2.i.c yes 4
168.e odd 2 1 168.2.i.a 4
168.i even 2 1 inner 672.2.i.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.i.a 4 4.b odd 2 1
168.2.i.a 4 12.b even 2 1
168.2.i.a 4 56.e even 2 1
168.2.i.a 4 168.e odd 2 1
168.2.i.c yes 4 8.d odd 2 1
168.2.i.c yes 4 24.f even 2 1
168.2.i.c yes 4 28.d even 2 1
168.2.i.c yes 4 84.h odd 2 1
672.2.i.a 4 7.b odd 2 1
672.2.i.a 4 8.b even 2 1
672.2.i.a 4 21.c even 2 1
672.2.i.a 4 24.h odd 2 1
672.2.i.c 4 1.a even 1 1 trivial
672.2.i.c 4 3.b odd 2 1 inner
672.2.i.c 4 56.h odd 2 1 inner
672.2.i.c 4 168.i even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(672, [\chi])\):

\( T_{5}^{2} + 2 \)
\( T_{11}^{2} - 6 \)
\( T_{13} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 2 T + 3 T^{2} )^{2} \)
$5$ \( ( 1 - 8 T^{2} + 25 T^{4} )^{2} \)
$7$ \( ( 1 - 4 T + 7 T^{2} )^{2} \)
$11$ \( ( 1 + 16 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 2 T + 13 T^{2} )^{4} \)
$17$ \( ( 1 - 20 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 4 T + 19 T^{2} )^{4} \)
$23$ \( ( 1 - 44 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 + 34 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 14 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 + 34 T^{2} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 + 76 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 74 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 + 70 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 + 53 T^{2} )^{4} \)
$59$ \( ( 1 - 86 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 10 T + 61 T^{2} )^{4} \)
$67$ \( ( 1 - 16 T + 67 T^{2} )^{2}( 1 + 16 T + 67 T^{2} )^{2} \)
$71$ \( ( 1 - 140 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 73 T^{2} )^{4} \)
$79$ \( ( 1 + 8 T + 79 T^{2} )^{4} \)
$83$ \( ( 1 - 38 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 + 124 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 14 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2} \)
show more
show less