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Properties

Label 672.2.bl.a
Level $672$
Weight $2$
Character orbit 672.bl
Analytic conductor $5.366$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [672,2,Mod(31,672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(672, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("672.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.bl (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 24x^{14} + 218x^{12} + 968x^{10} + 2241x^{8} + 2672x^{6} + 1512x^{4} + 320x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{3} - \beta_{6} q^{5} + (\beta_{13} - \beta_{9}) q^{7} + \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{3} - \beta_{6} q^{5} + (\beta_{13} - \beta_{9}) q^{7} + \beta_1 q^{9} + (\beta_{12} - \beta_{9} + \beta_{8} + \cdots + 1) q^{11}+ \cdots + ( - \beta_{13} - \beta_{12} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{3} - 4 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{3} - 4 q^{7} - 8 q^{9} + 12 q^{11} + 4 q^{19} + 8 q^{21} + 4 q^{25} + 16 q^{27} + 4 q^{31} - 12 q^{33} + 8 q^{35} + 4 q^{37} + 12 q^{39} + 8 q^{47} + 24 q^{49} + 8 q^{53} - 16 q^{55} - 8 q^{57} + 4 q^{59} + 24 q^{61} - 4 q^{63} + 8 q^{65} - 12 q^{67} + 12 q^{73} + 4 q^{75} - 32 q^{77} + 12 q^{79} - 8 q^{81} + 8 q^{83} + 32 q^{85} - 4 q^{91} + 4 q^{93} - 48 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 24x^{14} + 218x^{12} + 968x^{10} + 2241x^{8} + 2672x^{6} + 1512x^{4} + 320x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{15} + 22\nu^{13} + 174\nu^{11} + 612\nu^{9} + 865\nu^{7} - 26\nu^{5} - 1020\nu^{3} - 488\nu - 64 ) / 128 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5 \nu^{15} + 9 \nu^{14} + 108 \nu^{13} + 198 \nu^{12} + 814 \nu^{11} + 1558 \nu^{10} + 2512 \nu^{9} + \cdots - 168 ) / 128 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 5 \nu^{15} + 31 \nu^{14} - 108 \nu^{13} + 714 \nu^{12} - 814 \nu^{11} + 6074 \nu^{10} - 2512 \nu^{9} + \cdots + 872 ) / 128 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5 \nu^{15} - 35 \nu^{14} + 132 \nu^{13} - 794 \nu^{12} + 1358 \nu^{11} - 6594 \nu^{10} + 7008 \nu^{9} + \cdots - 104 ) / 128 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 17 \nu^{15} + 34 \nu^{14} - 382 \nu^{13} + 764 \nu^{12} - 3110 \nu^{11} + 6236 \nu^{10} + \cdots - 400 ) / 128 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5 \nu^{15} + 31 \nu^{14} + 108 \nu^{13} + 714 \nu^{12} + 814 \nu^{11} + 6074 \nu^{10} + 2512 \nu^{9} + \cdots + 872 ) / 128 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7 \nu^{15} + 10 \nu^{14} + 164 \nu^{13} + 228 \nu^{12} + 1434 \nu^{11} + 1908 \nu^{10} + 6000 \nu^{9} + \cdots + 176 ) / 64 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11 \nu^{15} + 12 \nu^{14} - 254 \nu^{13} + 272 \nu^{12} - 2170 \nu^{11} + 2248 \nu^{10} - 8740 \nu^{9} + \cdots - 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5 \nu^{15} + 35 \nu^{14} + 132 \nu^{13} + 794 \nu^{12} + 1358 \nu^{11} + 6594 \nu^{10} + 7008 \nu^{9} + \cdots + 232 ) / 128 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 11 \nu^{15} - 12 \nu^{14} - 254 \nu^{13} - 272 \nu^{12} - 2170 \nu^{11} - 2248 \nu^{10} - 8740 \nu^{9} + \cdots + 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 14 \nu^{15} - 67 \nu^{14} + 330 \nu^{13} - 1522 \nu^{12} + 2916 \nu^{11} - 12658 \nu^{10} + \cdots - 1224 ) / 128 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 33 \nu^{15} - 34 \nu^{14} + 766 \nu^{13} - 780 \nu^{12} + 6598 \nu^{11} - 6588 \nu^{10} + 26924 \nu^{9} + \cdots - 816 ) / 128 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 17 \nu^{15} + 17 \nu^{14} + 394 \nu^{13} + 390 \nu^{12} + 3386 \nu^{11} + 3294 \nu^{10} + 13768 \nu^{9} + \cdots + 440 ) / 64 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 41 \nu^{15} + 35 \nu^{14} + 928 \nu^{13} + 802 \nu^{12} + 7654 \nu^{11} + 6754 \nu^{10} + 28872 \nu^{9} + \cdots + 8 ) / 128 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 48 \nu^{15} - 29 \nu^{14} - 1094 \nu^{13} - 662 \nu^{12} - 9144 \nu^{11} - 5550 \nu^{10} + \cdots - 1112 ) / 128 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{15} + \beta_{13} + 2\beta_{12} - \beta_{11} - \beta_{9} - 2\beta_{7} + 2\beta_{3} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{15} - \beta_{14} - \beta_{13} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{15} - \beta_{14} - \beta_{13} - 6 \beta_{12} + 4 \beta_{11} + 2 \beta_{10} + 3 \beta_{9} + \cdots + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4 \beta_{15} + 5 \beta_{14} + 3 \beta_{13} + \beta_{12} - \beta_{11} + 5 \beta_{10} - 6 \beta_{9} + \cdots + 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 47 \beta_{15} + 18 \beta_{14} - 5 \beta_{13} + 78 \beta_{12} - 65 \beta_{11} - 64 \beta_{10} - 43 \beta_{9} + \cdots - 48 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 69 \beta_{15} - 93 \beta_{14} - 37 \beta_{13} - 32 \beta_{12} + 24 \beta_{11} - 91 \beta_{10} + \cdots - 274 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 207 \beta_{15} - 77 \beta_{14} + 75 \beta_{13} - 286 \beta_{12} + 284 \beta_{11} + 370 \beta_{10} + \cdots + 234 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 309 \beta_{15} + 427 \beta_{14} + 130 \beta_{13} + 179 \beta_{12} - 118 \beta_{11} + 413 \beta_{10} + \cdots + 1154 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3793 \beta_{15} + 1350 \beta_{14} - 1867 \beta_{13} + 4626 \beta_{12} - 5143 \beta_{11} - 7608 \beta_{10} + \cdots - 4400 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 5641 \beta_{15} - 7863 \beta_{14} - 2049 \beta_{13} - 3592 \beta_{12} + 2222 \beta_{11} + \cdots - 20422 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 17579 \beta_{15} - 6067 \beta_{14} + 9787 \beta_{13} - 19926 \beta_{12} + 23646 \beta_{11} + \cdots + 20542 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 25998 \beta_{15} + 36353 \beta_{14} + 8713 \beta_{13} + 17285 \beta_{12} - 10355 \beta_{11} + \cdots + 92630 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 326943 \beta_{15} + 110842 \beta_{14} - 192613 \beta_{13} + 356014 \beta_{12} - 437785 \beta_{11} + \cdots - 383088 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 481657 \beta_{15} - 674301 \beta_{14} - 154825 \beta_{13} - 326832 \beta_{12} + 192644 \beta_{11} + \cdots - 1702050 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 1521435 \beta_{15} - 510885 \beta_{14} + 921391 \beta_{13} - 1621814 \beta_{12} + 2032320 \beta_{11} + \cdots + 1785194 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-\beta_{1}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
− 1.06676i
3.05093i
0.266470i
− 1.11907i
− 1.60698i
− 2.19832i
2.12462i
0.549125i
1.06676i
− 3.05093i
− 0.266470i
1.11907i
1.60698i
2.19832i
− 2.12462i
− 0.549125i
0 −0.500000 + 0.866025i 0 −3.44269 + 1.98764i 0 −1.72692 − 2.00443i 0 −0.500000 − 0.866025i 0
31.2 0 −0.500000 + 0.866025i 0 −1.43254 + 0.827079i 0 1.65159 − 2.06694i 0 −0.500000 − 0.866025i 0
31.3 0 −0.500000 + 0.866025i 0 −1.15442 + 0.666502i 0 2.64320 − 0.116219i 0 −0.500000 − 0.866025i 0
31.4 0 −0.500000 + 0.866025i 0 −0.658228 + 0.380028i 0 −1.09610 + 2.40802i 0 −0.500000 − 0.866025i 0
31.5 0 −0.500000 + 0.866025i 0 −0.247599 + 0.142951i 0 −1.36578 − 2.26597i 0 −0.500000 − 0.866025i 0
31.6 0 −0.500000 + 0.866025i 0 1.51314 − 0.873609i 0 −2.46664 − 0.956913i 0 −0.500000 − 0.866025i 0
31.7 0 −0.500000 + 0.866025i 0 2.33837 − 1.35006i 0 2.63871 + 0.192843i 0 −0.500000 − 0.866025i 0
31.8 0 −0.500000 + 0.866025i 0 3.08397 − 1.78053i 0 −2.27807 + 1.34552i 0 −0.500000 − 0.866025i 0
607.1 0 −0.500000 − 0.866025i 0 −3.44269 − 1.98764i 0 −1.72692 + 2.00443i 0 −0.500000 + 0.866025i 0
607.2 0 −0.500000 − 0.866025i 0 −1.43254 − 0.827079i 0 1.65159 + 2.06694i 0 −0.500000 + 0.866025i 0
607.3 0 −0.500000 − 0.866025i 0 −1.15442 − 0.666502i 0 2.64320 + 0.116219i 0 −0.500000 + 0.866025i 0
607.4 0 −0.500000 − 0.866025i 0 −0.658228 − 0.380028i 0 −1.09610 − 2.40802i 0 −0.500000 + 0.866025i 0
607.5 0 −0.500000 − 0.866025i 0 −0.247599 − 0.142951i 0 −1.36578 + 2.26597i 0 −0.500000 + 0.866025i 0
607.6 0 −0.500000 − 0.866025i 0 1.51314 + 0.873609i 0 −2.46664 + 0.956913i 0 −0.500000 + 0.866025i 0
607.7 0 −0.500000 − 0.866025i 0 2.33837 + 1.35006i 0 2.63871 − 0.192843i 0 −0.500000 + 0.866025i 0
607.8 0 −0.500000 − 0.866025i 0 3.08397 + 1.78053i 0 −2.27807 − 1.34552i 0 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.2.bl.a ✓ 16
3.b odd 2 1 2016.2.cs.a 16
4.b odd 2 1 672.2.bl.b yes 16
7.c even 3 1 4704.2.b.e 16
7.d odd 6 1 672.2.bl.b yes 16
7.d odd 6 1 4704.2.b.d 16
8.b even 2 1 1344.2.bl.l 16
8.d odd 2 1 1344.2.bl.k 16
12.b even 2 1 2016.2.cs.c 16
21.g even 6 1 2016.2.cs.c 16
28.f even 6 1 inner 672.2.bl.a ✓ 16
28.f even 6 1 4704.2.b.e 16
28.g odd 6 1 4704.2.b.d 16
56.j odd 6 1 1344.2.bl.k 16
56.m even 6 1 1344.2.bl.l 16
84.j odd 6 1 2016.2.cs.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.bl.a ✓ 16 1.a even 1 1 trivial
672.2.bl.a ✓ 16 28.f even 6 1 inner
672.2.bl.b yes 16 4.b odd 2 1
672.2.bl.b yes 16 7.d odd 6 1
1344.2.bl.k 16 8.d odd 2 1
1344.2.bl.k 16 56.j odd 6 1
1344.2.bl.l 16 8.b even 2 1
1344.2.bl.l 16 56.m even 6 1
2016.2.cs.a 16 3.b odd 2 1
2016.2.cs.a 16 84.j odd 6 1
2016.2.cs.c 16 12.b even 2 1
2016.2.cs.c 16 21.g even 6 1
4704.2.b.d 16 7.d odd 6 1
4704.2.b.d 16 28.g odd 6 1
4704.2.b.e 16 7.c even 3 1
4704.2.b.e 16 28.f even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{16} - 12 T_{11}^{15} + 34 T_{11}^{14} + 168 T_{11}^{13} - 813 T_{11}^{12} - 2448 T_{11}^{11} + \cdots + 135424 \) acting on \(S_{2}^{\mathrm{new}}(672, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} - 22 T^{14} + \cdots + 1024 \) Copy content Toggle raw display
$7$ \( T^{16} + 4 T^{15} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{16} - 12 T^{15} + \cdots + 135424 \) Copy content Toggle raw display
$13$ \( T^{16} + 124 T^{14} + \cdots + 16384 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 3743481856 \) Copy content Toggle raw display
$19$ \( T^{16} - 4 T^{15} + \cdots + 29073664 \) Copy content Toggle raw display
$23$ \( T^{16} - 96 T^{14} + \cdots + 4194304 \) Copy content Toggle raw display
$29$ \( (T^{8} - 106 T^{6} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 20535749809 \) Copy content Toggle raw display
$37$ \( T^{16} - 4 T^{15} + \cdots + 135424 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 57538576384 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 17573803020544 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 39464206336 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 1630259097856 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 47901450496 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 644513529856 \) Copy content Toggle raw display
$67$ \( T^{16} + 12 T^{15} + \cdots + 21827584 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 2218786816 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 362073308594176 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 157558981969 \) Copy content Toggle raw display
$83$ \( (T^{8} - 4 T^{7} + \cdots - 314144)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 16211639271424 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 135706771456 \) Copy content Toggle raw display
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