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Properties

Label 336.6.bl.d
Level $336$
Weight $6$
Character orbit 336.bl
Analytic conductor $53.889$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [336,6,Mod(31,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.31");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 336.bl (of order \(6\), degree \(2\), 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: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2 x^{9} + 1095 x^{8} - 13422 x^{7} + 1015213 x^{6} - 8859054 x^{5} + 266674836 x^{4} + \cdots + 1231567257600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 9 \beta_1 q^{3} + (\beta_{5} + \beta_{2} + 6 \beta_1 + 6) q^{5} + (2 \beta_{4} - \beta_{2} + 18 \beta_1 - 26) q^{7} + (81 \beta_1 - 81) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 \beta_1 q^{3} + (\beta_{5} + \beta_{2} + 6 \beta_1 + 6) q^{5} + (2 \beta_{4} - \beta_{2} + 18 \beta_1 - 26) q^{7} + (81 \beta_1 - 81) q^{9} + (\beta_{7} - \beta_{3} + 104 \beta_1 - 208) q^{11} + ( - \beta_{9} - \beta_{8} + \beta_{7} + \cdots + 13) q^{13}+ \cdots + ( - 81 \beta_{7} - 16848 \beta_1 + 8424) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 45 q^{3} + 93 q^{5} - 169 q^{7} - 405 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 45 q^{3} + 93 q^{5} - 169 q^{7} - 405 q^{9} - 1557 q^{11} - 198 q^{17} - 2864 q^{19} - 1935 q^{21} + 2136 q^{23} - 1408 q^{25} - 7290 q^{27} - 6666 q^{29} + 7421 q^{31} - 14013 q^{33} - 1182 q^{35} - 1364 q^{37} + 1782 q^{39} - 7533 q^{45} + 42192 q^{47} + 62959 q^{49} - 1782 q^{51} + 35883 q^{53} - 7122 q^{55} - 51552 q^{57} - 29619 q^{59} + 120096 q^{61} - 3726 q^{63} - 23916 q^{65} - 139134 q^{67} + 196050 q^{73} + 12672 q^{75} - 1377 q^{77} + 144207 q^{79} - 32805 q^{81} - 159546 q^{83} - 99084 q^{85} - 29997 q^{87} - 274602 q^{89} + 44850 q^{91} - 66789 q^{93} - 282414 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2 x^{9} + 1095 x^{8} - 13422 x^{7} + 1015213 x^{6} - 8859054 x^{5} + 266674836 x^{4} + \cdots + 1231567257600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 85\!\cdots\!35 \nu^{9} + \cdots + 37\!\cdots\!00 ) / 17\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 16\!\cdots\!57 \nu^{9} + \cdots + 34\!\cdots\!60 ) / 62\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 86\!\cdots\!15 \nu^{9} + \cdots - 25\!\cdots\!80 ) / 10\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 59\!\cdots\!73 \nu^{9} + \cdots - 87\!\cdots\!60 ) / 52\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12\!\cdots\!85 \nu^{9} + \cdots - 54\!\cdots\!20 ) / 10\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 97\!\cdots\!77 \nu^{9} + \cdots + 32\!\cdots\!20 ) / 66\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 11\!\cdots\!05 \nu^{9} + \cdots + 60\!\cdots\!40 ) / 62\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 23\!\cdots\!71 \nu^{9} + \cdots + 72\!\cdots\!00 ) / 10\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 37\!\cdots\!41 \nu^{9} + \cdots - 10\!\cdots\!80 ) / 10\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{6} + \beta_{5} - \beta_{4} - 2\beta_{2} + 3\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 6 \beta_{9} - 3 \beta_{8} - 2 \beta_{7} - 12 \beta_{6} - 30 \beta_{5} - 12 \beta_{4} + \beta_{3} + \cdots - 5238 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 23 \beta_{9} - 23 \beta_{8} + 10 \beta_{7} - 415 \beta_{6} + 1180 \beta_{5} + 1595 \beta_{4} + \cdots + 19643 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2909 \beta_{9} + 5818 \beta_{8} + 1595 \beta_{7} + 53478 \beta_{6} + 17262 \beta_{5} - 17262 \beta_{4} + \cdots + 2909 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 67774 \beta_{9} - 33887 \beta_{8} - 26432 \beta_{7} - 1107304 \beta_{6} - 1422353 \beta_{5} + \cdots - 32045782 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2825047 \beta_{9} - 2825047 \beta_{8} + 1652649 \beta_{7} - 18396690 \beta_{6} + 48987576 \beta_{5} + \cdots + 3282358051 ) / 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 39443583 \beta_{9} + 78887166 \beta_{8} + 16526502 \beta_{7} + 1372252295 \beta_{6} + 303544111 \beta_{5} + \cdots + 39443583 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 5694821770 \beta_{9} - 2847410885 \beta_{8} - 3233678950 \beta_{7} - 56924573544 \beta_{6} + \cdots - 3165525107470 ) / 12 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 43130598531 \beta_{9} - 43130598531 \beta_{8} + 19455603204 \beta_{7} - 313722595225 \beta_{6} + \cdots + 44698058474043 ) / 6 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1 - \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
11.4770 − 19.8788i
−6.85605 + 11.8750i
9.84453 − 17.0512i
−16.2248 + 28.1022i
2.75932 − 4.77928i
11.4770 + 19.8788i
−6.85605 − 11.8750i
9.84453 + 17.0512i
−16.2248 − 28.1022i
2.75932 + 4.77928i
0 4.50000 − 7.79423i 0 −43.9185 + 25.3564i 0 −58.7120 − 115.585i 0 −40.5000 − 70.1481i 0
31.2 0 4.50000 − 7.79423i 0 −40.6759 + 23.4843i 0 120.295 − 48.3332i 0 −40.5000 − 70.1481i 0
31.3 0 4.50000 − 7.79423i 0 16.8315 − 9.71764i 0 −123.387 − 39.7819i 0 −40.5000 − 70.1481i 0
31.4 0 4.50000 − 7.79423i 0 45.5543 − 26.3008i 0 99.0090 + 83.6912i 0 −40.5000 − 70.1481i 0
31.5 0 4.50000 − 7.79423i 0 68.7087 − 39.6690i 0 −121.705 + 44.6647i 0 −40.5000 − 70.1481i 0
271.1 0 4.50000 + 7.79423i 0 −43.9185 − 25.3564i 0 −58.7120 + 115.585i 0 −40.5000 + 70.1481i 0
271.2 0 4.50000 + 7.79423i 0 −40.6759 − 23.4843i 0 120.295 + 48.3332i 0 −40.5000 + 70.1481i 0
271.3 0 4.50000 + 7.79423i 0 16.8315 + 9.71764i 0 −123.387 + 39.7819i 0 −40.5000 + 70.1481i 0
271.4 0 4.50000 + 7.79423i 0 45.5543 + 26.3008i 0 99.0090 − 83.6912i 0 −40.5000 + 70.1481i 0
271.5 0 4.50000 + 7.79423i 0 68.7087 + 39.6690i 0 −121.705 − 44.6647i 0 −40.5000 + 70.1481i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.6.bl.d yes 10
4.b odd 2 1 336.6.bl.c ✓ 10
7.d odd 6 1 336.6.bl.c ✓ 10
28.f even 6 1 inner 336.6.bl.d yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.6.bl.c ✓ 10 4.b odd 2 1
336.6.bl.c ✓ 10 7.d odd 6 1
336.6.bl.d yes 10 1.a even 1 1 trivial
336.6.bl.d yes 10 28.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(336, [\chi])\):

\( T_{5}^{10} - 93 T_{5}^{9} - 2784 T_{5}^{8} + 527031 T_{5}^{7} + 12366612 T_{5}^{6} + \cdots + 37\!\cdots\!48 \) Copy content Toggle raw display
\( T_{11}^{10} + 1557 T_{11}^{9} + 807828 T_{11}^{8} - 397035 T_{11}^{7} - 82961610192 T_{11}^{6} + \cdots + 31\!\cdots\!32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( (T^{2} - 9 T + 81)^{5} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 37\!\cdots\!48 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 13\!\cdots\!07 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 31\!\cdots\!32 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 24\!\cdots\!68 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots + 311855416492032)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 50\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 76\!\cdots\!08 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 11\!\cdots\!92 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 38\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 48\!\cdots\!72 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 37\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 21\!\cdots\!48 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 42\!\cdots\!27 \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots + 87\!\cdots\!92)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 90\!\cdots\!92 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 10\!\cdots\!88 \) Copy content Toggle raw display
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