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Properties

Label 336.6.k.e
Level $336$
Weight $6$
Character orbit 336.k
Analytic conductor $53.889$
Analytic rank $0$
Dimension $12$
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] = [336,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.209");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(53.8889634572\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 480x^{10} + 90258x^{8} + 8362760x^{6} + 391768761x^{4} + 8420565480x^{2} + 61757220100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{8}\cdot 5^{4} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + ( - \beta_{3} - \beta_{2}) q^{5} + ( - \beta_{5} - \beta_{4} - \beta_{2} + 21) q^{7} + ( - \beta_{11} + \beta_{8} + \beta_{4} + 50) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + ( - \beta_{3} - \beta_{2}) q^{5} + ( - \beta_{5} - \beta_{4} - \beta_{2} + 21) q^{7} + ( - \beta_{11} + \beta_{8} + \beta_{4} + 50) q^{9} + (3 \beta_{11} - \beta_{10} + \cdots + \beta_{7}) q^{11}+ \cdots + ( - 153 \beta_{11} + 87 \beta_{10} + \cdots + 68880) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 252 q^{7} + 600 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 252 q^{7} + 600 q^{9} + 3516 q^{15} - 3444 q^{21} + 2412 q^{25} - 43944 q^{37} - 20724 q^{39} - 77568 q^{43} - 1260 q^{49} + 36888 q^{51} - 97764 q^{57} - 101640 q^{63} + 102384 q^{67} - 364200 q^{79} + 107172 q^{81} + 440736 q^{85} + 369768 q^{91} - 357840 q^{93} + 826560 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 480x^{10} + 90258x^{8} + 8362760x^{6} + 391768761x^{4} + 8420565480x^{2} + 61757220100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 15587208 \nu^{11} + 2988363480 \nu^{9} + 2178294086736 \nu^{7} + \cdots + 18\!\cdots\!80 \nu ) / 14\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 160125716784 \nu^{11} + 135749208775 \nu^{10} + 61408308112530 \nu^{9} + \cdots + 56\!\cdots\!00 ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 160125716784 \nu^{11} + 1047064951465 \nu^{10} - 61408308112530 \nu^{9} + \cdots + 50\!\cdots\!00 ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 163463865471 \nu^{11} - 4026412325395 \nu^{10} + 62781069676995 \nu^{9} + \cdots - 31\!\cdots\!00 ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 109629520939 \nu^{11} - 45083316888305 \nu^{9} + \cdots + 47\!\cdots\!40 \nu ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 49447262512 \nu^{11} + 5939145335945 \nu^{10} - 21222503634470 \nu^{9} + \cdots + 48\!\cdots\!00 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 52262522699 \nu^{11} + 20011848849355 \nu^{9} + \cdots + 58\!\cdots\!60 \nu ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 515272922469 \nu^{11} + 22294642366855 \nu^{10} - 209868833384565 \nu^{9} + \cdots + 14\!\cdots\!00 ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 515272922469 \nu^{11} - 22023143949305 \nu^{10} - 209868833384565 \nu^{9} + \cdots - 13\!\cdots\!00 ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 320251433568 \nu^{11} - 135749208775 \nu^{10} + 122816616225060 \nu^{9} + \cdots - 56\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 8952252624 \nu^{11} - 3661624166585 \nu^{9} - 546037501896092 \nu^{7} + \cdots - 88\!\cdots\!60 \nu ) / 26\!\cdots\!50 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -24\beta_{10} + 56\beta_{7} - 24\beta_{5} - 48\beta_{2} - 5\beta_1 ) / 480 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 92 \beta_{10} - 12 \beta_{9} + 12 \beta_{8} + 48 \beta_{7} - 6 \beta_{6} + 96 \beta_{4} + \cdots - 19200 ) / 240 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 72 \beta_{11} + 1632 \beta_{10} + 144 \beta_{9} + 144 \beta_{8} - 2736 \beta_{7} + 1728 \beta_{5} + \cdots + 601 \beta_1 ) / 240 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3208 \beta_{10} + 531 \beta_{9} - 531 \beta_{8} - 1584 \beta_{7} + 396 \beta_{6} - 3168 \beta_{4} + \cdots + 498840 ) / 60 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 10380 \beta_{11} - 114396 \beta_{10} - 15360 \beta_{9} - 15360 \beta_{8} + 154564 \beta_{7} + \cdots - 50735 \beta_1 ) / 120 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 142736 \beta_{10} - 26781 \beta_{9} + 26781 \beta_{8} + 61764 \beta_{7} - 23998 \beta_{6} + \cdots - 19029200 ) / 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1052478 \beta_{11} + 7905618 \beta_{10} + 1190196 \beta_{9} + 1190196 \beta_{8} - 9158614 \beta_{7} + \cdots + 3884179 \beta_1 ) / 60 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 57409328 \beta_{10} + 11694351 \beta_{9} - 11694351 \beta_{8} - 21640944 \beta_{7} + 11542176 \beta_{6} + \cdots + 6848244840 ) / 60 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 92792655 \beta_{11} - 543249549 \beta_{10} - 83357640 \beta_{9} - 83357640 \beta_{8} + \cdots - 281658305 \beta_1 ) / 30 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 7695231148 \beta_{10} - 1680161103 \beta_{9} + 1680161103 \beta_{8} + 2572880652 \beta_{7} + \cdots - 846762342000 ) / 60 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 15170178639 \beta_{11} + 74646665289 \beta_{10} + 11196456018 \beta_{9} + 11196456018 \beta_{8} + \cdots + 39582821642 \beta_1 ) / 30 \) 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
− 3.83030i
3.83030i
8.91475i
− 8.91475i
− 9.74085i
9.74085i
− 11.7408i
11.7408i
10.9148i
− 10.9148i
− 5.83030i
5.83030i
0 −15.4743 − 1.88284i 0 24.8913 0 −102.591 + 79.2601i 0 235.910 + 58.2714i 0
209.2 0 −15.4743 + 1.88284i 0 24.8913 0 −102.591 − 79.2601i 0 235.910 − 58.2714i 0
209.3 0 −13.1536 − 8.36554i 0 −96.7263 0 103.019 + 78.7019i 0 103.035 + 220.074i 0
209.4 0 −13.1536 + 8.36554i 0 −96.7263 0 103.019 − 78.7019i 0 103.035 − 220.074i 0
209.5 0 −5.19879 − 14.6960i 0 1.56269 0 62.5711 − 113.542i 0 −188.945 + 152.803i 0
209.6 0 −5.19879 + 14.6960i 0 1.56269 0 62.5711 + 113.542i 0 −188.945 − 152.803i 0
209.7 0 5.19879 − 14.6960i 0 −1.56269 0 62.5711 − 113.542i 0 −188.945 − 152.803i 0
209.8 0 5.19879 + 14.6960i 0 −1.56269 0 62.5711 + 113.542i 0 −188.945 + 152.803i 0
209.9 0 13.1536 − 8.36554i 0 96.7263 0 103.019 + 78.7019i 0 103.035 − 220.074i 0
209.10 0 13.1536 + 8.36554i 0 96.7263 0 103.019 − 78.7019i 0 103.035 + 220.074i 0
209.11 0 15.4743 − 1.88284i 0 −24.8913 0 −102.591 + 79.2601i 0 235.910 − 58.2714i 0
209.12 0 15.4743 + 1.88284i 0 −24.8913 0 −102.591 − 79.2601i 0 235.910 + 58.2714i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.12
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 336.6.k.e 12
3.b odd 2 1 inner 336.6.k.e 12
4.b odd 2 1 42.6.d.a ✓ 12
7.b odd 2 1 inner 336.6.k.e 12
12.b even 2 1 42.6.d.a ✓ 12
21.c even 2 1 inner 336.6.k.e 12
28.d even 2 1 42.6.d.a ✓ 12
84.h odd 2 1 42.6.d.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.d.a ✓ 12 4.b odd 2 1
42.6.d.a ✓ 12 12.b even 2 1
42.6.d.a ✓ 12 28.d even 2 1
42.6.d.a ✓ 12 84.h odd 2 1
336.6.k.e 12 1.a even 1 1 trivial
336.6.k.e 12 3.b odd 2 1 inner
336.6.k.e 12 7.b odd 2 1 inner
336.6.k.e 12 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 9978T_{5}^{4} + 5821128T_{5}^{2} - 14155776 \) acting on \(S_{6}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 205891132094649 \) Copy content Toggle raw display
$5$ \( (T^{6} - 9978 T^{4} + \cdots - 14155776)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots + 4747561509943)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 210691031040000)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 2411249112576)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots - 11\!\cdots\!84)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 59\!\cdots\!36)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 16\!\cdots\!56)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 25\!\cdots\!16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 10986 T^{2} + \cdots - 271247986792)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 93\!\cdots\!84)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 19392 T^{2} + \cdots + 242532114304)^{4} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 14\!\cdots\!24)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 50\!\cdots\!36)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 43\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + \cdots + 46777678533632)^{4} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 34\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 33\!\cdots\!56)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + \cdots - 274338290742400)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 45\!\cdots\!76)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 34\!\cdots\!04)^{2} \) Copy content Toggle raw display
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