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Properties

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

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Show commands: Magma / Pari/GP / SageMath

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 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);
 
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")
 
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: \( x^{16} - 8 x^{15} + 120 x^{14} - 700 x^{13} + 5060 x^{12} - 21624 x^{11} + 95002 x^{10} - 292520 x^{9} + \cdots + 76783 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{3} - 1) q^{3} - \beta_{8} q^{5} + (\beta_{10} + \beta_{6} - 1) q^{7} - 3 \beta_{3} q^{9} + ( - \beta_{15} - 2 \beta_{3} - 2) q^{11} + ( - \beta_{14} + \beta_{13} + \cdots + \beta_1) q^{13}+ \cdots + ( - 3 \beta_{2} - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 24 q^{3} - 12 q^{7} + 24 q^{9} - 12 q^{11} + 48 q^{17} + 36 q^{19} + 24 q^{21} + 48 q^{23} + 20 q^{25} + 64 q^{29} - 60 q^{31} + 36 q^{33} - 36 q^{37} + 12 q^{39} + 72 q^{43} - 72 q^{47} - 40 q^{49}+ \cdots - 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 120 x^{14} - 700 x^{13} + 5060 x^{12} - 21624 x^{11} + 95002 x^{10} - 292520 x^{9} + \cdots + 76783 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 21592812 \nu^{14} + 151149684 \nu^{13} - 2422903266 \nu^{12} + 12572473704 \nu^{11} + \cdots - 3015082827988 ) / 101086184101 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2047017258 \nu^{14} + 14329120806 \nu^{13} - 230248297055 \nu^{12} + \cdots - 336208105164239 ) / 5372008640796 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 263361216 \nu^{15} + 1975209120 \nu^{14} - 30613310770 \nu^{13} + 169029181685 \nu^{12} + \cdots + 40893255565715 ) / 1240000512492 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 751304187678 \nu^{15} - 4845657559931 \nu^{14} + 82165333497982 \nu^{13} + \cdots - 11\!\cdots\!53 ) / 33\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 751304187678 \nu^{15} + 6423905255239 \nu^{14} - 93213067365138 \nu^{13} + \cdots + 26\!\cdots\!20 ) / 33\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1448351779610 \nu^{15} - 10636348214969 \nu^{14} + 166968087036478 \nu^{13} + \cdots - 18\!\cdots\!09 ) / 33\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1448351779610 \nu^{15} + 11088928479181 \nu^{14} - 170136148885962 \nu^{13} + \cdots + 33\!\cdots\!82 ) / 33\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3209575805662 \nu^{15} - 23792625159943 \nu^{14} + 371351526749854 \nu^{13} + \cdots - 53\!\cdots\!07 ) / 33\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3209575805662 \nu^{15} + 24351011924987 \nu^{14} - 375260234105162 \nu^{13} + \cdots + 52\!\cdots\!38 ) / 33\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4686917492632 \nu^{15} - 35070084588185 \nu^{14} + 544169587870622 \nu^{13} + \cdots - 71\!\cdots\!83 ) / 33\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1229135808992 \nu^{15} - 9307880419902 \nu^{14} + 143433872965208 \nu^{13} + \cdots - 18\!\cdots\!10 ) / 836690345803977 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3067634636121 \nu^{15} - 23161303140238 \nu^{14} + 357725444624177 \nu^{13} + \cdots - 49\!\cdots\!62 ) / 16\!\cdots\!54 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 8855003427238 \nu^{15} + 65897495204958 \nu^{14} + \cdots + 12\!\cdots\!35 ) / 33\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 8855003427238 \nu^{15} - 66927556203612 \nu^{14} + \cdots - 15\!\cdots\!87 ) / 33\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1727235234248 \nu^{15} - 13045356524841 \nu^{14} + 201321302743302 \nu^{13} + \cdots - 26\!\cdots\!40 ) / 478108769030844 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{11} + 2\beta_{9} - 2\beta_{8} - 2\beta_{5} + 2\beta_{4} - \beta _1 + 4 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 4 \beta_{12} + \beta_{11} + 4 \beta_{10} - 2 \beta_{9} - 4 \beta_{8} - 4 \beta_{7} - 2 \beta_{5} + \cdots - 48 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 12 \beta_{15} + 8 \beta_{14} - 8 \beta_{13} - 30 \beta_{12} - 61 \beta_{11} - 6 \beta_{10} - 72 \beta_{9} + \cdots - 86 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6 \beta_{15} + 8 \beta_{14} + 64 \beta_{12} - 31 \beta_{11} - 82 \beta_{10} + 14 \beta_{9} + 78 \beta_{8} + \cdots + 636 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 736 \beta_{15} - 240 \beta_{14} + 320 \beta_{13} + 1426 \beta_{12} + 2215 \beta_{11} - 74 \beta_{10} + \cdots + 2370 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 1134 \beta_{15} - 854 \beta_{14} + 26 \beta_{13} - 3922 \beta_{12} + 3478 \beta_{11} + 6040 \beta_{10} + \cdots - 41118 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 31880 \beta_{15} + 6028 \beta_{14} - 12104 \beta_{13} - 61928 \beta_{12} - 77598 \beta_{11} + \cdots - 97724 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 17270 \beta_{15} + 8912 \beta_{14} - 1190 \beta_{13} + 28333 \beta_{12} - 42893 \beta_{11} + \cdots + 359801 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1183456 \beta_{15} - 138864 \beta_{14} + 453648 \beta_{13} + 2575926 \beta_{12} + 2612003 \beta_{11} + \cdots + 4993050 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 3484738 \beta_{15} - 1409046 \beta_{14} + 381534 \beta_{13} - 2933954 \beta_{12} + 7841455 \beta_{11} + \cdots - 51507506 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 39777532 \beta_{15} + 2346460 \beta_{14} - 16602328 \beta_{13} - 103771318 \beta_{12} - 83735439 \beta_{11} + \cdots - 259367034 ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 79414840 \beta_{15} + 27221954 \beta_{14} - 11392596 \beta_{13} + 29019872 \beta_{12} - 170185402 \beta_{11} + \cdots + 914672050 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 1218150152 \beta_{15} + 17010624 \beta_{14} + 587633000 \beta_{13} + 4063127668 \beta_{12} + \cdots + 12835379344 ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 6793770080 \beta_{15} - 2060996984 \beta_{14} + 1170213508 \beta_{13} - 140073732 \beta_{12} + \cdots - 63437853844 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 32910756660 \beta_{15} - 4639763376 \beta_{14} - 19990727440 \beta_{13} - 154811673466 \beta_{12} + \cdots - 601136805810 ) / 8 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\beta_{3}\)

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 − 2.86275i
0.500000 + 4.23762i
0.500000 − 0.957705i
0.500000 − 6.14236i
0.500000 + 1.70210i
0.500000 + 4.68078i
0.500000 − 0.627490i
0.500000 − 0.0302085i
0.500000 + 2.86275i
0.500000 − 4.23762i
0.500000 + 0.957705i
0.500000 + 6.14236i
0.500000 − 1.70210i
0.500000 − 4.68078i
0.500000 + 0.627490i
0.500000 + 0.0302085i
0 −1.50000 + 0.866025i 0 −7.17494 − 4.14245i 0 5.68961 + 4.07778i 0 1.50000 − 2.59808i 0
481.2 0 −1.50000 + 0.866025i 0 −2.62483 − 1.51545i 0 −6.96833 + 0.665103i 0 1.50000 − 2.59808i 0
481.3 0 −1.50000 + 0.866025i 0 −2.55189 − 1.47333i 0 −2.27125 + 6.62128i 0 1.50000 − 2.59808i 0
481.4 0 −1.50000 + 0.866025i 0 −2.13902 − 1.23496i 0 −3.79376 − 5.88280i 0 1.50000 − 2.59808i 0
481.5 0 −1.50000 + 0.866025i 0 −0.803356 − 0.463818i 0 0.377407 − 6.98982i 0 1.50000 − 2.59808i 0
481.6 0 −1.50000 + 0.866025i 0 1.77068 + 1.02231i 0 6.69221 + 2.05287i 0 1.50000 − 2.59808i 0
481.7 0 −1.50000 + 0.866025i 0 6.36048 + 3.67223i 0 −6.34133 + 2.96437i 0 1.50000 − 2.59808i 0
481.8 0 −1.50000 + 0.866025i 0 7.16286 + 4.13548i 0 0.615444 − 6.97289i 0 1.50000 − 2.59808i 0
577.1 0 −1.50000 − 0.866025i 0 −7.17494 + 4.14245i 0 5.68961 − 4.07778i 0 1.50000 + 2.59808i 0
577.2 0 −1.50000 − 0.866025i 0 −2.62483 + 1.51545i 0 −6.96833 − 0.665103i 0 1.50000 + 2.59808i 0
577.3 0 −1.50000 − 0.866025i 0 −2.55189 + 1.47333i 0 −2.27125 − 6.62128i 0 1.50000 + 2.59808i 0
577.4 0 −1.50000 − 0.866025i 0 −2.13902 + 1.23496i 0 −3.79376 + 5.88280i 0 1.50000 + 2.59808i 0
577.5 0 −1.50000 − 0.866025i 0 −0.803356 + 0.463818i 0 0.377407 + 6.98982i 0 1.50000 + 2.59808i 0
577.6 0 −1.50000 − 0.866025i 0 1.77068 − 1.02231i 0 6.69221 − 2.05287i 0 1.50000 + 2.59808i 0
577.7 0 −1.50000 − 0.866025i 0 6.36048 − 3.67223i 0 −6.34133 − 2.96437i 0 1.50000 + 2.59808i 0
577.8 0 −1.50000 − 0.866025i 0 7.16286 − 4.13548i 0 0.615444 + 6.97289i 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.b ✓ 16
4.b odd 2 1 672.3.bh.d yes 16
7.d odd 6 1 inner 672.3.bh.b ✓ 16
28.f even 6 1 672.3.bh.d yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.3.bh.b ✓ 16 1.a even 1 1 trivial
672.3.bh.b ✓ 16 7.d odd 6 1 inner
672.3.bh.d yes 16 4.b odd 2 1
672.3.bh.d 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 \(S_{3}^{\mathrm{new}}(672, [\chi])\):

\( T_{5}^{16} - 110 T_{5}^{14} + 9179 T_{5}^{12} + 7104 T_{5}^{11} - 298622 T_{5}^{10} - 505152 T_{5}^{9} + \cdots + 443355136 \) Copy content Toggle raw display
\( T_{11}^{16} + 12 T_{11}^{15} + 602 T_{11}^{14} + 5184 T_{11}^{13} + 205867 T_{11}^{12} + \cdots + 32\!\cdots\!84 \) 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
$5$ \( T^{16} + \cdots + 443355136 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 33232930569601 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 32\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 23\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 29\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 41\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( (T^{8} - 32 T^{7} + \cdots + 91213545472)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 58\!\cdots\!89 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 63\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( (T^{8} - 36 T^{7} + \cdots + 221126535952)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 82\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 97\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 65\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 96\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 15\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots - 14681443340288)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 28\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 83\!\cdots\!09 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 40\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 54\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 13\!\cdots\!04 \) Copy content Toggle raw display
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