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Properties

Label 672.4.k.d
Level $672$
Weight $4$
Character orbit 672.k
Analytic conductor $39.649$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [672,4,Mod(545,672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(672, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("672.545");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 672.k (of order \(2\), degree \(1\), 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: \(39.6492835239\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 473x^{12} + 41664x^{8} + 295625x^{4} + 390625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{9} q^{3} + \beta_{13} q^{5} + ( - \beta_{15} - \beta_1) q^{7} + (\beta_{6} - \beta_{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{9} q^{3} + \beta_{13} q^{5} + ( - \beta_{15} - \beta_1) q^{7} + (\beta_{6} - \beta_{2} + 3) q^{9} + ( - \beta_{7} + 11 \beta_{4}) q^{11} + ( - \beta_{14} + \beta_{13} + \cdots - 2 \beta_{5}) q^{13}+ \cdots + ( - 36 \beta_{7} + 201 \beta_{4} + \cdots - 10 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 48 q^{9} + 96 q^{21} - 1360 q^{25} - 2976 q^{37} + 1136 q^{49} - 2208 q^{57} + 5040 q^{81} + 7552 q^{85} + 10560 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 473x^{12} + 41664x^{8} + 295625x^{4} + 390625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1331\nu^{12} + 617688\nu^{8} + 50669784\nu^{4} + 146096875 ) / 7095000 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{12} - 1892\nu^{8} - 164156\nu^{4} - 591250 ) / 20625 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1132 \nu^{14} + 94875 \nu^{12} - 537936 \nu^{10} + 44829000 \nu^{8} - 48408648 \nu^{6} + \cdots + 27070921875 ) / 532125000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 283\nu^{14} + 134484\nu^{10} + 12102162\nu^{6} + 121514375\nu^{2} ) / 66515625 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 10009 \nu^{15} - 239675 \nu^{13} - 3868632 \nu^{11} - 113069400 \nu^{9} + \cdots - 8086796875 \nu ) / 5321250000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 387 \nu^{14} + 400 \nu^{12} + 181176 \nu^{10} + 189200 \nu^{8} + 15283968 \nu^{6} + \cdots + 59125000 ) / 4125000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -233\nu^{14} - 109584\nu^{10} - 9396462\nu^{6} - 39090625\nu^{2} ) / 2015625 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5029 \nu^{15} - 13535 \nu^{13} + 2377592 \nu^{11} - 6342680 \nu^{9} + 208699256 \nu^{7} + \cdots - 1853859375 \nu ) / 354750000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12759 \nu^{15} - 8325 \nu^{13} + 5990632 \nu^{11} - 3984600 \nu^{9} + 510835976 \nu^{7} + \cdots - 2787203125 \nu ) / 591250000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 927 \nu^{14} - 400 \nu^{12} - 439096 \nu^{10} - 189200 \nu^{8} - 38902528 \nu^{6} + \cdots - 59125000 ) / 4125000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 134209 \nu^{15} + 532525 \nu^{13} + 64430232 \nu^{11} + 249556200 \nu^{9} + \cdots - 6188171875 \nu ) / 5321250000 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 5029 \nu^{15} - 13535 \nu^{13} - 2377592 \nu^{11} - 6342680 \nu^{9} - 208699256 \nu^{7} + \cdots - 1853859375 \nu ) / 177375000 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 7649 \nu^{15} - 12065 \nu^{13} + 3596352 \nu^{11} - 5653620 \nu^{9} + 308724936 \nu^{7} + \cdots - 2034315625 \nu ) / 266062500 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 12759 \nu^{15} + 8325 \nu^{13} + 5990632 \nu^{11} + 3984600 \nu^{9} + 510835976 \nu^{7} + \cdots + 2787203125 \nu ) / 295625000 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 66641 \nu^{15} + 88625 \nu^{13} + 31400568 \nu^{11} + 41716500 \nu^{9} + 2720490624 \nu^{7} + \cdots + 19662390625 \nu ) / 1330312500 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{15} + \beta_{14} + 2\beta_{13} - 2\beta_{12} - \beta_{11} - \beta_{9} - 4\beta_{8} + 2\beta_{5} ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{10} - 3\beta_{7} - \beta_{6} + 99\beta_{4} - 2\beta_{2} ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 22 \beta_{15} + 35 \beta_{14} - 44 \beta_{13} + 8 \beta_{12} + 5 \beta_{11} + 38 \beta_{9} + \cdots + 10 \beta_{5} ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 19\beta_{4} + 38\beta_{3} + 33\beta_{2} - 2\beta _1 - 946 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 704 \beta_{15} - 134 \beta_{14} - 1408 \beta_{13} + 949 \beta_{12} + 137 \beta_{11} + \cdots - 274 \beta_{5} ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -258\beta_{10} + 567\beta_{7} + 350\beta_{6} - 5940\beta_{4} + 304\beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 15976 \beta_{15} - 21974 \beta_{14} + 31952 \beta_{13} - 767 \beta_{12} - 2141 \beta_{11} + \cdots - 4282 \beta_{5} ) / 24 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -6325\beta_{4} - 12650\beta_{3} - 15609\beta_{2} - 4114\beta _1 + 280802 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 133574 \beta_{15} + 38699 \beta_{14} + 267148 \beta_{13} - 165772 \beta_{12} + \cdots + 36082 \beta_{5} ) / 12 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 353469\beta_{10} - 926547\beta_{7} - 643711\beta_{6} + 7690419\beta_{4} - 498590\beta_{2} ) / 24 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 5752307 \beta_{15} + 7519597 \beta_{14} - 11504614 \beta_{13} - 303458 \beta_{12} + \cdots + 1276106 \beta_{5} ) / 24 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 276498\beta_{4} + 552996\beta_{3} + 748440\beta_{2} + 253500\beta _1 - 11897369 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 97462673 \beta_{15} - 31103153 \beta_{14} - 194925346 \beta_{13} + 117922426 \beta_{12} + \cdots - 23246386 \beta_{5} ) / 24 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( -125127339\beta_{10} + 344602779\beta_{7} + 246456953\beta_{6} - 2675351547\beta_{4} + 185792146\beta_{2} ) / 24 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 1034151386 \beta_{15} - 1330930135 \beta_{14} + 2068302772 \beta_{13} + 85302836 \beta_{12} + \cdots - 215474570 \beta_{5} ) / 12 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
545.1
−3.07864 + 3.07864i
−0.812046 − 0.812046i
−3.07864 − 3.07864i
−0.812046 + 0.812046i
2.26906 + 2.26906i
1.10178 − 1.10178i
2.26906 − 2.26906i
1.10178 + 1.10178i
−1.10178 − 1.10178i
−2.26906 + 2.26906i
−1.10178 + 1.10178i
−2.26906 − 2.26906i
0.812046 − 0.812046i
3.07864 + 3.07864i
0.812046 + 0.812046i
3.07864 − 3.07864i
0 −5.13077 − 0.821735i 0 −7.42292 0 8.43226 + 16.4893i 0 25.6495 + 8.43226i 0
545.2 0 −5.13077 − 0.821735i 0 7.42292 0 −8.43226 + 16.4893i 0 25.6495 + 8.43226i 0
545.3 0 −5.13077 + 0.821735i 0 −7.42292 0 8.43226 − 16.4893i 0 25.6495 − 8.43226i 0
545.4 0 −5.13077 + 0.821735i 0 7.42292 0 −8.43226 − 16.4893i 0 25.6495 − 8.43226i 0
545.5 0 −1.91709 − 4.82957i 0 −4.99002 0 −18.5175 − 0.320905i 0 −19.6495 + 18.5175i 0
545.6 0 −1.91709 − 4.82957i 0 4.99002 0 18.5175 − 0.320905i 0 −19.6495 + 18.5175i 0
545.7 0 −1.91709 + 4.82957i 0 −4.99002 0 −18.5175 + 0.320905i 0 −19.6495 − 18.5175i 0
545.8 0 −1.91709 + 4.82957i 0 4.99002 0 18.5175 + 0.320905i 0 −19.6495 − 18.5175i 0
545.9 0 1.91709 − 4.82957i 0 −4.99002 0 18.5175 − 0.320905i 0 −19.6495 − 18.5175i 0
545.10 0 1.91709 − 4.82957i 0 4.99002 0 −18.5175 − 0.320905i 0 −19.6495 − 18.5175i 0
545.11 0 1.91709 + 4.82957i 0 −4.99002 0 18.5175 + 0.320905i 0 −19.6495 + 18.5175i 0
545.12 0 1.91709 + 4.82957i 0 4.99002 0 −18.5175 + 0.320905i 0 −19.6495 + 18.5175i 0
545.13 0 5.13077 − 0.821735i 0 −7.42292 0 −8.43226 + 16.4893i 0 25.6495 − 8.43226i 0
545.14 0 5.13077 − 0.821735i 0 7.42292 0 8.43226 + 16.4893i 0 25.6495 − 8.43226i 0
545.15 0 5.13077 + 0.821735i 0 −7.42292 0 −8.43226 − 16.4893i 0 25.6495 + 8.43226i 0
545.16 0 5.13077 + 0.821735i 0 7.42292 0 8.43226 − 16.4893i 0 25.6495 + 8.43226i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 545.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
7.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner
84.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.4.k.d ✓ 16
3.b odd 2 1 inner 672.4.k.d ✓ 16
4.b odd 2 1 inner 672.4.k.d ✓ 16
7.b odd 2 1 inner 672.4.k.d ✓ 16
12.b even 2 1 inner 672.4.k.d ✓ 16
21.c even 2 1 inner 672.4.k.d ✓ 16
28.d even 2 1 inner 672.4.k.d ✓ 16
84.h odd 2 1 inner 672.4.k.d ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.4.k.d ✓ 16 1.a even 1 1 trivial
672.4.k.d ✓ 16 3.b odd 2 1 inner
672.4.k.d ✓ 16 4.b odd 2 1 inner
672.4.k.d ✓ 16 7.b odd 2 1 inner
672.4.k.d ✓ 16 12.b even 2 1 inner
672.4.k.d ✓ 16 21.c even 2 1 inner
672.4.k.d ✓ 16 28.d even 2 1 inner
672.4.k.d ✓ 16 84.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 80T_{5}^{2} + 1372 \) acting on \(S_{4}^{\mathrm{new}}(672, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} - 12 T^{6} + \cdots + 531441)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - 80 T^{2} + 1372)^{4} \) Copy content Toggle raw display
$7$ \( (T^{8} - 284 T^{6} + \cdots + 13841287201)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 1424 T^{2} + 65536)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 2952 T^{2} + 682668)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 12416 T^{2} + 35879872)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 3440 T^{2} + 149212)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 20744 T^{2} + 9461776)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 94392 T^{2} + 1267421904)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 82176 T^{2} + 1246436352)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 372 T + 20004)^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} - 92288 T^{2} + 1884557248)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 53712 T^{2} + 399458304)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 403488 T^{2} + 38761944768)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + 55224 T^{2} + 655751376)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 199944 T^{2} + 8716987692)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 247320 T^{2} + 1601079372)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 1347984 T^{2} + 313175310336)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 359552 T^{2} + 31587241984)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 1573920 T^{2} + 587902758912)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 1025032243536)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 1694520 T^{2} + 330288167052)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} - 542720 T^{2} + 37392990208)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 376416 T^{2} + 20196050112)^{4} \) Copy content Toggle raw display
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