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Properties

Label 336.4.k.c
Level $336$
Weight $4$
Character orbit 336.k
Analytic conductor $19.825$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [336,4,Mod(209,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, 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("336.209");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 336.k (of order \(2\), degree \(1\), not 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: \(19.8246417619\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.116876510171136.13
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 15x^{6} + 455x^{4} + 5097x^{2} + 21904 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 42)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} + ( - \beta_{7} - \beta_{6} + \beta_{2}) q^{5} + ( - \beta_{7} + \beta_{6} - \beta_{5} - 1) q^{7} + ( - \beta_{7} - \beta_{5} - \beta_{4} + \cdots - 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{3} + ( - \beta_{7} - \beta_{6} + \beta_{2}) q^{5} + ( - \beta_{7} + \beta_{6} - \beta_{5} - 1) q^{7} + ( - \beta_{7} - \beta_{5} - \beta_{4} + \cdots - 8) q^{9}+ \cdots + (15 \beta_{7} - 129 \beta_{4} + \cdots - 504) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{7} - 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{7} - 60 q^{9} - 252 q^{15} - 168 q^{21} + 320 q^{25} - 592 q^{37} + 804 q^{39} + 1696 q^{43} + 2192 q^{49} - 504 q^{51} + 2532 q^{57} + 2496 q^{63} - 496 q^{67} - 2824 q^{79} - 3936 q^{85} + 264 q^{91} - 1512 q^{93} - 4032 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 15x^{6} + 455x^{4} + 5097x^{2} + 21904 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 317 \nu^{7} - 4329 \nu^{6} + 20776 \nu^{5} - 33152 \nu^{4} + 328643 \nu^{3} - 1590519 \nu^{2} + \cdots - 9683936 ) / 1444184 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 595 \nu^{7} - 6623 \nu^{6} + 4284 \nu^{5} - 44548 \nu^{4} - 76993 \nu^{3} - 2007509 \nu^{2} + \cdots - 12161752 ) / 1444184 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 595 \nu^{7} + 6623 \nu^{6} + 4284 \nu^{5} + 44548 \nu^{4} - 76993 \nu^{3} + 2007509 \nu^{2} + \cdots + 12161752 ) / 1444184 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -36\nu^{7} - 651\nu^{5} - 11940\nu^{3} - 168729\nu ) / 51578 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1411 \nu^{7} + 555 \nu^{6} + 10472 \nu^{5} - 125356 \nu^{4} + 553945 \nu^{3} - 73815 \nu^{2} + \cdots - 22717112 ) / 1444184 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1411 \nu^{7} + 2035 \nu^{6} - 10472 \nu^{5} + 21756 \nu^{4} - 553945 \nu^{3} + 1173529 \nu^{2} + \cdots + 7206120 ) / 1444184 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1411 \nu^{7} - 4329 \nu^{6} - 10472 \nu^{5} - 33152 \nu^{4} - 553945 \nu^{3} - 1590519 \nu^{2} + \cdots - 9683936 ) / 722092 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + \beta_{4} - \beta_{3} - 2\beta_{2} + 2\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -5\beta_{7} + 11\beta_{6} + \beta_{5} - 5\beta_{3} + 5\beta_{2} - 22 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -28\beta_{7} - 21\beta_{6} + 75\beta_{4} + 21\beta_{3} + 42\beta_{2} + 14\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -13\beta_{7} - 43\beta_{6} - 69\beta_{5} + \beta_{3} - \beta_{2} - 1062 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 224\beta_{7} + 13\beta_{6} - 1435\beta_{4} + 563\beta_{3} + 550\beta_{2} - 422\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1603\beta_{7} - 3045\beta_{6} + 161\beta_{5} + 2163\beta_{3} - 2163\beta_{2} + 2794 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 5236\beta_{7} + 2043\beta_{6} - 20805\beta_{4} - 12459\beta_{3} - 14502\beta_{2} - 6386\beta_1 ) / 12 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
−0.696949 + 2.67617i
−0.696949 − 2.67617i
−3.04373 + 3.17617i
−3.04373 − 3.17617i
3.04373 − 3.17617i
3.04373 + 3.17617i
0.696949 − 2.67617i
0.696949 + 2.67617i
0 −4.30448 − 2.91058i 0 11.3967 0 17.0570 − 7.21506i 0 10.0570 + 25.0570i 0
209.2 0 −4.30448 + 2.91058i 0 11.3967 0 17.0570 + 7.21506i 0 10.0570 − 25.0570i 0
209.3 0 −0.985634 − 5.10182i 0 14.1462 0 −18.0570 − 4.11618i 0 −25.0570 + 10.0570i 0
209.4 0 −0.985634 + 5.10182i 0 14.1462 0 −18.0570 + 4.11618i 0 −25.0570 − 10.0570i 0
209.5 0 0.985634 − 5.10182i 0 −14.1462 0 −18.0570 − 4.11618i 0 −25.0570 − 10.0570i 0
209.6 0 0.985634 + 5.10182i 0 −14.1462 0 −18.0570 + 4.11618i 0 −25.0570 + 10.0570i 0
209.7 0 4.30448 − 2.91058i 0 −11.3967 0 17.0570 − 7.21506i 0 10.0570 − 25.0570i 0
209.8 0 4.30448 + 2.91058i 0 −11.3967 0 17.0570 + 7.21506i 0 10.0570 + 25.0570i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.4.k.c 8
3.b odd 2 1 inner 336.4.k.c 8
4.b odd 2 1 42.4.d.a ✓ 8
7.b odd 2 1 inner 336.4.k.c 8
12.b even 2 1 42.4.d.a ✓ 8
21.c even 2 1 inner 336.4.k.c 8
28.d even 2 1 42.4.d.a ✓ 8
28.f even 6 2 294.4.f.b 16
28.g odd 6 2 294.4.f.b 16
84.h odd 2 1 42.4.d.a ✓ 8
84.j odd 6 2 294.4.f.b 16
84.n even 6 2 294.4.f.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.d.a ✓ 8 4.b odd 2 1
42.4.d.a ✓ 8 12.b even 2 1
42.4.d.a ✓ 8 28.d even 2 1
42.4.d.a ✓ 8 84.h odd 2 1
294.4.f.b 16 28.f even 6 2
294.4.f.b 16 28.g odd 6 2
294.4.f.b 16 84.j odd 6 2
294.4.f.b 16 84.n even 6 2
336.4.k.c 8 1.a even 1 1 trivial
336.4.k.c 8 3.b odd 2 1 inner
336.4.k.c 8 7.b odd 2 1 inner
336.4.k.c 8 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 330T_{5}^{2} + 25992 \) acting on \(S_{4}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 30 T^{6} + \cdots + 531441 \) Copy content Toggle raw display
$5$ \( (T^{4} - 330 T^{2} + 25992)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 2 T^{3} + \cdots + 117649)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 2916 T^{2} + 1016064)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 2958 T^{2} + 1002528)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 16116 T^{2} + 40069152)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 18762 T^{2} + 43022088)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 23976 T^{2} + 4511376)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 19764 T^{2} + 54997056)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 112860 T^{2} + 1170505728)^{2} \) Copy content Toggle raw display
$37$ \( (T + 74)^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} - 94740 T^{2} + 10071072)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 424 T - 33968)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 255480 T^{2} + 9930350592)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 209268 T^{2} + 9046292544)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 580830 T^{2} + 4021968672)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 183678 T^{2} + 8432329248)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 124 T - 1088)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 607716 T^{2} + 12745506816)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 1099680 T^{2} + 148979386368)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 706 T + 24736)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 1502862 T^{2} + 422927723808)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 392796 T^{2} + 35407798272)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 1338360 T^{2} + 94643822592)^{2} \) Copy content Toggle raw display
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