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Properties

Label 672.3.bh.a
Level $672$
Weight $3$
Character orbit 672.bh
Analytic conductor $18.311$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [672,3,Mod(481,672)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(672, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 5])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("672.481"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 672.bh (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,-24,0,0,0,-12,0,24,0,-12,0,0,0,0,0,-48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.3106737650\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 76 x^{14} - 392 x^{13} + 1982 x^{12} - 7160 x^{11} + 23796 x^{10} - 61736 x^{9} + \cdots + 16807 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{18}\cdot 7 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + (\beta_{5} - 1) q^{3} + ( - \beta_{6} - \beta_{3}) q^{5} + (\beta_{14} - \beta_{5} - 1) q^{7} - 3 \beta_{5} q^{9} + ( - \beta_{15} + \beta_{12} - \beta_{11} + \cdots - 1) q^{11} + ( - \beta_{13} - \beta_{7} - \beta_{5} + \cdots - 2) q^{13}+ \cdots + ( - 3 \beta_{11} + 3 \beta_{10} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 24 q^{3} - 12 q^{7} + 24 q^{9} - 12 q^{11} - 48 q^{17} - 60 q^{19} + 24 q^{21} - 48 q^{23} + 52 q^{25} - 64 q^{29} - 60 q^{31} + 36 q^{33} - 4 q^{37} + 12 q^{39} + 72 q^{43} + 120 q^{47} - 8 q^{49}+ \cdots - 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 76 x^{14} - 392 x^{13} + 1982 x^{12} - 7160 x^{11} + 23796 x^{10} - 61736 x^{9} + \cdots + 16807 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 121 \nu^{14} - 847 \nu^{13} + 7810 \nu^{12} - 35849 \nu^{11} + 169477 \nu^{10} - 538956 \nu^{9} + \cdots - 3311665 ) / 123480 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 121 \nu^{14} + 847 \nu^{13} - 8230 \nu^{12} + 38369 \nu^{11} - 192997 \nu^{10} + 633456 \nu^{9} + \cdots - 1895075 ) / 30870 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 88971 \nu^{15} + 2295330 \nu^{14} - 18051137 \nu^{13} + 143013023 \nu^{12} + \cdots + 29711629661 ) / 525572040 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 467002 \nu^{15} + 335803 \nu^{14} - 10201243 \nu^{13} - 53901724 \nu^{12} + \cdots - 17724196749 ) / 1576716120 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 43440 \nu^{15} + 325800 \nu^{14} - 3087464 \nu^{13} + 15127216 \nu^{12} - 75010228 \nu^{11} + \cdots - 85865591 ) / 131393010 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 177942 \nu^{15} - 1334565 \nu^{14} + 13309609 \nu^{13} - 66271556 \nu^{12} + 350688857 \nu^{11} + \cdots - 10391767757 ) / 525572040 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1064411 \nu^{15} + 7989467 \nu^{14} - 75131996 \nu^{13} + 365062609 \nu^{12} + \cdots - 22813411572 ) / 1576716120 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 334815 \nu^{15} - 1738588 \nu^{14} + 19022131 \nu^{13} - 68164075 \nu^{12} + 374466902 \nu^{11} + \cdots + 2782681825 ) / 394179030 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 490224 \nu^{15} + 5221729 \nu^{14} - 46949091 \nu^{13} + 283658954 \nu^{12} + \cdots + 37593246243 ) / 525572040 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1070353 \nu^{15} + 9796154 \nu^{14} - 87200249 \nu^{13} + 484508367 \nu^{12} + \cdots + 14785153229 ) / 788358060 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3738943 \nu^{15} + 18318479 \nu^{14} - 196592384 \nu^{13} + 636688977 \nu^{12} + \cdots - 97661008396 ) / 1576716120 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 4205945 \nu^{15} + 24987706 \nu^{14} - 251127595 \nu^{13} + 1005134697 \nu^{12} + \cdots - 65788723637 ) / 1576716120 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1678603 \nu^{15} - 11536080 \nu^{14} + 111679971 \nu^{13} - 511876407 \nu^{12} + \cdots + 19122039741 ) / 525572040 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 7168175 \nu^{15} + 51890654 \nu^{14} - 498660953 \nu^{13} + 2388972379 \nu^{12} + \cdots + 75260069073 ) / 1576716120 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 10925061 \nu^{15} + 78075335 \nu^{14} - 750833930 \nu^{13} + 3553403495 \nu^{12} + \cdots + 73089848734 ) / 1576716120 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 12 \beta_{15} - 8 \beta_{14} + 6 \beta_{13} - 16 \beta_{12} + 10 \beta_{11} - 2 \beta_{10} - 4 \beta_{9} + \cdots + 36 ) / 56 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 6 \beta_{15} + 10 \beta_{14} + 10 \beta_{13} - 8 \beta_{12} - 2 \beta_{11} - 8 \beta_{10} - 16 \beta_{9} + \cdots - 150 ) / 28 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 128 \beta_{15} + 146 \beta_{14} - 64 \beta_{13} + 152 \beta_{12} - 102 \beta_{11} - 2 \beta_{10} + \cdots - 762 ) / 56 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 60 \beta_{15} - 30 \beta_{14} - 72 \beta_{13} + 80 \beta_{12} + 20 \beta_{11} + 38 \beta_{10} + \cdots + 548 ) / 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1412 \beta_{15} - 2122 \beta_{14} + 678 \beta_{13} - 1598 \beta_{12} + 1494 \beta_{11} + 152 \beta_{10} + \cdots + 12076 ) / 56 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2050 \beta_{15} - 256 \beta_{14} + 2026 \beta_{13} - 2738 \beta_{12} - 254 \beta_{11} - 778 \beta_{10} + \cdots - 5974 ) / 28 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 15878 \beta_{15} + 28888 \beta_{14} - 7218 \beta_{13} + 16518 \beta_{12} - 22502 \beta_{11} + \cdots - 172472 ) / 56 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 8493 \beta_{15} + 4388 \beta_{14} - 7400 \beta_{13} + 11135 \beta_{12} - 991 \beta_{11} + 2161 \beta_{10} + \cdots - 5620 ) / 7 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 172156 \beta_{15} - 378302 \beta_{14} + 69264 \beta_{13} - 150530 \beta_{12} + 329976 \beta_{11} + \cdots + 2309756 ) / 56 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 555932 \beta_{15} - 434942 \beta_{14} + 443628 \beta_{13} - 698062 \beta_{12} + 190328 \beta_{11} + \cdots + 1594940 ) / 28 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1603478 \beta_{15} + 4685498 \beta_{14} - 434358 \beta_{13} + 879982 \beta_{12} - 4637056 \beta_{11} + \cdots - 29202858 ) / 56 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 4445536 \beta_{15} + 4334916 \beta_{14} - 3333234 \beta_{13} + 5266264 \beta_{12} - 2425718 \beta_{11} + \cdots - 19126196 ) / 14 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 8385424 \beta_{15} - 52674868 \beta_{14} - 3380770 \beta_{13} + 6815464 \beta_{12} + 61430274 \beta_{11} + \cdots + 343055084 ) / 56 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 137200762 \beta_{15} - 155091822 \beta_{14} + 98305294 \beta_{13} - 152222064 \beta_{12} + \cdots + 734019514 ) / 28 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 124416388 \beta_{15} + 485908030 \beta_{14} + 210509676 \beta_{13} - 390430336 \beta_{12} + \cdots - 3560039090 ) / 56 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\beta_{5}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
481.1
0.500000 + 3.21278i
0.500000 − 1.12361i
0.500000 + 0.311996i
0.500000 − 1.58706i
0.500000 + 0.920550i
0.500000 − 3.06833i
0.500000 − 2.54626i
0.500000 + 3.87994i
0.500000 − 3.21278i
0.500000 + 1.12361i
0.500000 − 0.311996i
0.500000 + 1.58706i
0.500000 − 0.920550i
0.500000 + 3.06833i
0.500000 + 2.54626i
0.500000 − 3.87994i
0 −1.50000 + 0.866025i 0 −7.21243 − 4.16410i 0 −5.15876 − 4.73151i 0 1.50000 − 2.59808i 0
481.2 0 −1.50000 + 0.866025i 0 −5.81789 − 3.35896i 0 −5.35254 + 4.51113i 0 1.50000 − 2.59808i 0
481.3 0 −1.50000 + 0.866025i 0 −4.27656 − 2.46907i 0 4.80010 − 5.09500i 0 1.50000 − 2.59808i 0
481.4 0 −1.50000 + 0.866025i 0 −0.452956 − 0.261514i 0 6.50765 − 2.57886i 0 1.50000 − 2.59808i 0
481.5 0 −1.50000 + 0.866025i 0 3.54218 + 2.04508i 0 −3.04344 + 6.30377i 0 1.50000 − 2.59808i 0
481.6 0 −1.50000 + 0.866025i 0 3.60807 + 2.08312i 0 3.89168 + 5.81849i 0 1.50000 − 2.59808i 0
481.7 0 −1.50000 + 0.866025i 0 4.12320 + 2.38053i 0 −6.96231 − 0.725445i 0 1.50000 − 2.59808i 0
481.8 0 −1.50000 + 0.866025i 0 6.48637 + 3.74491i 0 −0.682383 − 6.96666i 0 1.50000 − 2.59808i 0
577.1 0 −1.50000 − 0.866025i 0 −7.21243 + 4.16410i 0 −5.15876 + 4.73151i 0 1.50000 + 2.59808i 0
577.2 0 −1.50000 − 0.866025i 0 −5.81789 + 3.35896i 0 −5.35254 − 4.51113i 0 1.50000 + 2.59808i 0
577.3 0 −1.50000 − 0.866025i 0 −4.27656 + 2.46907i 0 4.80010 + 5.09500i 0 1.50000 + 2.59808i 0
577.4 0 −1.50000 − 0.866025i 0 −0.452956 + 0.261514i 0 6.50765 + 2.57886i 0 1.50000 + 2.59808i 0
577.5 0 −1.50000 − 0.866025i 0 3.54218 − 2.04508i 0 −3.04344 − 6.30377i 0 1.50000 + 2.59808i 0
577.6 0 −1.50000 − 0.866025i 0 3.60807 − 2.08312i 0 3.89168 − 5.81849i 0 1.50000 + 2.59808i 0
577.7 0 −1.50000 − 0.866025i 0 4.12320 − 2.38053i 0 −6.96231 + 0.725445i 0 1.50000 + 2.59808i 0
577.8 0 −1.50000 − 0.866025i 0 6.48637 − 3.74491i 0 −0.682383 + 6.96666i 0 1.50000 + 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 481.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.3.bh.a ✓ 16
4.b odd 2 1 672.3.bh.c yes 16
7.d odd 6 1 inner 672.3.bh.a ✓ 16
28.f even 6 1 672.3.bh.c yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.3.bh.a ✓ 16 1.a even 1 1 trivial
672.3.bh.a ✓ 16 7.d odd 6 1 inner
672.3.bh.c yes 16 4.b odd 2 1
672.3.bh.c yes 16 28.f even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(672, [\chi])\):

\( T_{5}^{16} - 126 T_{5}^{14} + 10859 T_{5}^{12} - 9792 T_{5}^{11} - 488718 T_{5}^{10} + 726720 T_{5}^{9} + \cdots + 7710244864 \) Copy content Toggle raw display
\( T_{11}^{16} + 12 T_{11}^{15} + 730 T_{11}^{14} + 4416 T_{11}^{13} + 323499 T_{11}^{12} + \cdots + 270161556574464 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 3)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 7710244864 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 33232930569601 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 270161556574464 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 25401600000000 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 61\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 62\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 74\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( (T^{8} + 32 T^{7} + \cdots + 1851015168)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 59\!\cdots\!69 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 73\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( (T^{8} - 36 T^{7} + \cdots + 7365366288)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 32\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 87\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 79\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 57891189981184)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 46\!\cdots\!81 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 23\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 10\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 60\!\cdots\!96 \) Copy content Toggle raw display
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