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Properties

Label 336.6.h.a
Level $336$
Weight $6$
Character orbit 336.h
Analytic conductor $53.889$
Analytic rank $0$
Dimension $20$
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(239,336)]
 
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([1, 0, 1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.239");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 336.h (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: \(53.8889634572\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 444940 x^{16} + 56262171366 x^{12} + \cdots + 77\!\cdots\!16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{35}\cdot 3^{14}\cdot 7^{12} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + \beta_{5} q^{5} - \beta_{3} q^{7} + ( - \beta_{10} + 2 \beta_{5} - 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + \beta_{5} q^{5} - \beta_{3} q^{7} + ( - \beta_{10} + 2 \beta_{5} - 7) q^{9} + (\beta_{14} + \beta_{4} + 6 \beta_{2}) q^{11} + (\beta_1 + 52) q^{13} + ( - \beta_{14} - \beta_{7} + \cdots + 2 \beta_{3}) q^{15}+ \cdots + (55 \beta_{17} + 14 \beta_{16} + \cdots + 1187 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 140 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 140 q^{9} + 1048 q^{13} - 980 q^{21} + 5916 q^{25} - 26056 q^{33} + 61360 q^{37} - 92512 q^{45} - 48020 q^{49} - 20720 q^{57} + 46680 q^{61} - 28360 q^{69} - 54280 q^{73} + 152660 q^{81} - 150536 q^{85} + 41688 q^{93} - 421352 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 444940 x^{16} + 56262171366 x^{12} + \cdots + 77\!\cdots\!16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 75\!\cdots\!03 \nu^{16} + \cdots - 12\!\cdots\!76 ) / 56\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 91\!\cdots\!13 \nu^{19} + \cdots - 10\!\cdots\!64 \nu ) / 20\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 35218191863629 \nu^{18} + \cdots - 51\!\cdots\!92 \nu^{2} ) / 79\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 91\!\cdots\!13 \nu^{19} + \cdots - 10\!\cdots\!64 \nu ) / 22\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13\!\cdots\!49 \nu^{19} + \cdots + 12\!\cdots\!76 \nu ) / 14\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 43\!\cdots\!02 \nu^{19} + \cdots - 18\!\cdots\!80 \nu ) / 41\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 43\!\cdots\!02 \nu^{19} + \cdots + 18\!\cdots\!80 \nu ) / 41\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 50\!\cdots\!66 \nu^{19} + \cdots + 26\!\cdots\!72 \nu ) / 41\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 92\!\cdots\!11 \nu^{19} + \cdots - 23\!\cdots\!28 ) / 29\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 10\!\cdots\!19 \nu^{19} + \cdots + 67\!\cdots\!08 ) / 29\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 14\!\cdots\!55 \nu^{19} + \cdots - 66\!\cdots\!44 ) / 29\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 38\!\cdots\!81 \nu^{19} + \cdots + 34\!\cdots\!64 \nu ) / 41\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 10\!\cdots\!95 \nu^{19} + \cdots - 35\!\cdots\!04 ) / 97\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 61\!\cdots\!49 \nu^{19} + \cdots - 62\!\cdots\!04 \nu ) / 41\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 23\!\cdots\!15 \nu^{19} + \cdots - 67\!\cdots\!08 ) / 14\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 11\!\cdots\!12 \nu^{19} + \cdots + 11\!\cdots\!16 \nu ) / 69\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 26\!\cdots\!71 \nu^{19} + \cdots + 26\!\cdots\!60 \nu ) / 13\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 27\!\cdots\!31 \nu^{19} + \cdots + 31\!\cdots\!48 ) / 13\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 15\!\cdots\!10 \nu^{19} + \cdots - 16\!\cdots\!28 ) / 14\!\cdots\!36 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} + \beta_{13} - \beta_{9} - 3\beta_{5} - 7\beta_{4} - 63\beta_{2} ) / 126 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 6\beta_{16} + 6\beta_{14} - 21\beta_{8} + 12\beta_{7} + 3\beta_{6} - 4\beta_{4} - 600\beta_{3} + 6\beta_{2} ) / 126 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 90 \beta_{19} - 198 \beta_{18} + 105 \beta_{16} + 896 \beta_{15} - 399 \beta_{14} + 722 \beta_{13} + \cdots + 45 ) / 252 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 791 \beta_{15} + 742 \beta_{13} + 63 \beta_{11} + 6195 \beta_{10} - 184 \beta_{9} - 1668 \beta_{5} + \cdots - 1601883 ) / 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 26316 \beta_{19} + 81936 \beta_{18} + 12474 \beta_{17} + 31836 \beta_{16} - 191989 \beta_{15} + \cdots - 13158 ) / 126 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 1027032 \beta_{16} - 1027032 \beta_{14} + 5086011 \beta_{8} - 2518752 \beta_{7} + \cdots - 51984078 \beta_{2} ) / 126 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 27531342 \beta_{19} + 100876302 \beta_{18} - 28479276 \beta_{17} - 31260327 \beta_{16} + \cdots - 13765671 ) / 252 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 157625129 \beta_{15} - 182638630 \beta_{13} - 31545351 \beta_{11} - 1392600387 \beta_{10} + \cdots + 307668465639 ) / 18 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 7007616360 \beta_{19} - 28281832902 \beta_{18} - 10704429288 \beta_{17} - 7257218808 \beta_{16} + \cdots + 3503808180 ) / 126 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 187107857754 \beta_{16} + 187107857754 \beta_{14} - 1116029343009 \beta_{8} + \cdots + 10533873932070 \beta_{2} ) / 126 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 7055976919914 \beta_{19} - 30386250590670 \beta_{18} + 13636981098216 \beta_{17} + \cdots + 3527988459957 ) / 252 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 34693729364471 \beta_{15} + 40263277672078 \beta_{13} + 9393437122371 \beta_{11} + \cdots - 65\!\cdots\!59 ) / 18 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 17\!\cdots\!84 \beta_{19} + \cdots - 882081772883442 ) / 126 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 36\!\cdots\!80 \beta_{16} + \cdots - 19\!\cdots\!18 \beta_{2} ) / 126 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 17\!\cdots\!82 \beta_{19} + \cdots - 87\!\cdots\!91 ) / 252 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 78\!\cdots\!77 \beta_{15} + \cdots + 14\!\cdots\!23 ) / 18 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 43\!\cdots\!08 \beta_{19} + \cdots + 21\!\cdots\!04 ) / 126 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 75\!\cdots\!62 \beta_{16} + \cdots + 37\!\cdots\!58 \beta_{2} ) / 126 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 42\!\cdots\!54 \beta_{19} + \cdots + 21\!\cdots\!77 ) / 252 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1
−15.5877 + 15.5877i
−15.5877 − 15.5877i
−14.3532 − 14.3532i
−14.3532 + 14.3532i
−9.02465 + 9.02465i
−9.02465 − 9.02465i
−7.46608 + 7.46608i
−7.46608 − 7.46608i
−1.95479 − 1.95479i
−1.95479 + 1.95479i
1.95479 + 1.95479i
1.95479 − 1.95479i
7.46608 − 7.46608i
7.46608 + 7.46608i
9.02465 − 9.02465i
9.02465 + 9.02465i
14.3532 + 14.3532i
14.3532 − 14.3532i
15.5877 − 15.5877i
15.5877 + 15.5877i
0 −15.5877 − 0.152257i 0 − 4.09696i 0 − 49.0000i 0 242.954 + 4.74667i 0
239.2 0 −15.5877 + 0.152257i 0 4.09696i 0 49.0000i 0 242.954 − 4.74667i 0
239.3 0 −14.3532 − 6.08151i 0 63.6624i 0 49.0000i 0 169.030 + 174.579i 0
239.4 0 −14.3532 + 6.08151i 0 − 63.6624i 0 − 49.0000i 0 169.030 − 174.579i 0
239.5 0 −9.02465 − 12.7105i 0 89.1061i 0 − 49.0000i 0 −80.1112 + 229.415i 0
239.6 0 −9.02465 + 12.7105i 0 − 89.1061i 0 49.0000i 0 −80.1112 − 229.415i 0
239.7 0 −7.46608 − 13.6842i 0 − 31.0107i 0 − 49.0000i 0 −131.515 + 204.335i 0
239.8 0 −7.46608 + 13.6842i 0 31.0107i 0 49.0000i 0 −131.515 − 204.335i 0
239.9 0 −1.95479 − 15.4654i 0 − 34.2747i 0 49.0000i 0 −235.358 + 60.4632i 0
239.10 0 −1.95479 + 15.4654i 0 34.2747i 0 − 49.0000i 0 −235.358 − 60.4632i 0
239.11 0 1.95479 − 15.4654i 0 34.2747i 0 49.0000i 0 −235.358 − 60.4632i 0
239.12 0 1.95479 + 15.4654i 0 − 34.2747i 0 − 49.0000i 0 −235.358 + 60.4632i 0
239.13 0 7.46608 − 13.6842i 0 31.0107i 0 − 49.0000i 0 −131.515 − 204.335i 0
239.14 0 7.46608 + 13.6842i 0 − 31.0107i 0 49.0000i 0 −131.515 + 204.335i 0
239.15 0 9.02465 − 12.7105i 0 − 89.1061i 0 − 49.0000i 0 −80.1112 − 229.415i 0
239.16 0 9.02465 + 12.7105i 0 89.1061i 0 49.0000i 0 −80.1112 + 229.415i 0
239.17 0 14.3532 − 6.08151i 0 − 63.6624i 0 49.0000i 0 169.030 − 174.579i 0
239.18 0 14.3532 + 6.08151i 0 63.6624i 0 − 49.0000i 0 169.030 + 174.579i 0
239.19 0 15.5877 − 0.152257i 0 4.09696i 0 − 49.0000i 0 242.954 − 4.74667i 0
239.20 0 15.5877 + 0.152257i 0 − 4.09696i 0 49.0000i 0 242.954 + 4.74667i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 239.20
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
12.b 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.h.a ✓ 20
3.b odd 2 1 inner 336.6.h.a ✓ 20
4.b odd 2 1 inner 336.6.h.a ✓ 20
12.b even 2 1 inner 336.6.h.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.6.h.a ✓ 20 1.a even 1 1 trivial
336.6.h.a ✓ 20 3.b odd 2 1 inner
336.6.h.a ✓ 20 4.b odd 2 1 inner
336.6.h.a ✓ 20 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} + 14146T_{5}^{8} + 59168104T_{5}^{6} + 83286671696T_{5}^{4} + 37735204528784T_{5}^{2} + 610203077456672 \) acting on \(S_{6}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 71\!\cdots\!49 \) Copy content Toggle raw display
$5$ \( (T^{10} + \cdots + 610203077456672)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 2401)^{10} \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots - 22\!\cdots\!68)^{2} \) Copy content Toggle raw display
$13$ \( (T^{5} + \cdots + 26963334853336)^{4} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots + 53\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 51\!\cdots\!04)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 34\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 30\!\cdots\!72)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 62\!\cdots\!76)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots - 45\!\cdots\!48)^{4} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 52\!\cdots\!12)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 44\!\cdots\!36)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots - 27\!\cdots\!52)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 84\!\cdots\!48)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots - 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 52\!\cdots\!68)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 14\!\cdots\!76)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots + 98\!\cdots\!36)^{4} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 41\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots + 12\!\cdots\!92)^{4} \) Copy content Toggle raw display
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