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Properties

Label 336.5.d.b
Level $336$
Weight $5$
Character orbit 336.d
Analytic conductor $34.732$
Analytic rank $0$
Dimension $8$
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] = [336,5,Mod(113,336)]
 
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, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.113");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 336.d (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: \(34.7323075962\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 82x^{6} + 2017x^{4} + 13020x^{2} + 756 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 21)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} - \beta_{7} q^{5} - \beta_{4} q^{7} + (2 \beta_{7} - \beta_{6} + \beta_{5} + \cdots + 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} - \beta_{7} q^{5} - \beta_{4} q^{7} + (2 \beta_{7} - \beta_{6} + \beta_{5} + \cdots + 9) q^{9}+ \cdots + (202 \beta_{7} + 187 \beta_{6} + \cdots + 3747) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} + 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{3} + 64 q^{9} + 420 q^{13} - 76 q^{15} + 372 q^{19} + 98 q^{21} + 1056 q^{25} + 1862 q^{27} + 2776 q^{31} + 1396 q^{33} - 2560 q^{37} + 2540 q^{39} - 4720 q^{43} + 9700 q^{45} + 2744 q^{49} - 4764 q^{51} - 184 q^{55} - 14144 q^{57} + 972 q^{61} + 6076 q^{63} - 10200 q^{67} - 5760 q^{69} - 32008 q^{73} - 2114 q^{75} + 23168 q^{79} - 17216 q^{81} + 32016 q^{85} - 50764 q^{87} - 11956 q^{91} + 31848 q^{93} + 28112 q^{97} + 32432 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 82x^{6} + 2017x^{4} + 13020x^{2} + 756 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + \nu + 21 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 4\nu^{5} - 44\nu^{4} + 188\nu^{3} - 285\nu^{2} + 1560\nu + 198 ) / 48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 48\nu^{4} + 441\nu^{2} + 322 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} - 4\nu^{5} - 44\nu^{4} - 188\nu^{3} - 285\nu^{2} - 1560\nu + 182 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} - 20\nu^{5} - 44\nu^{4} - 892\nu^{3} - 285\nu^{2} - 6312\nu + 198 ) / 48 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 64\nu^{5} + 1177\nu^{3} + 6114\nu ) / 48 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{2} - \beta _1 - 42 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{6} - 2\beta_{5} + 4\beta_{3} - 31\beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{5} + 8\beta_{4} + 12\beta_{3} - 78\beta_{2} + 39\beta _1 + 1382 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -94\beta_{6} + 90\beta_{5} - 176\beta_{3} + 1067\beta _1 + 90 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -192\beta_{5} - 352\beta_{4} - 576\beta_{3} + 2862\beta_{2} - 1431\beta _1 - 48458 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 96\beta_{7} + 3662\beta_{6} - 3406\beta_{5} + 6556\beta_{3} - 37915\beta _1 - 3406 ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1
− 3.15624i
3.15624i
5.97075i
− 5.97075i
− 0.242064i
0.242064i
− 6.02741i
6.02741i
0 −7.09942 − 5.53157i 0 − 16.3361i 0 18.5203 0 19.8035 + 78.5418i 0
113.2 0 −7.09942 + 5.53157i 0 16.3361i 0 18.5203 0 19.8035 − 78.5418i 0
113.3 0 −5.29264 − 7.27928i 0 − 22.9681i 0 −18.5203 0 −24.9759 + 77.0533i 0
113.4 0 −5.29264 + 7.27928i 0 22.9681i 0 −18.5203 0 −24.9759 − 77.0533i 0
113.5 0 4.46977 − 7.81161i 0 30.4863i 0 −18.5203 0 −41.0424 − 69.8321i 0
113.6 0 4.46977 + 7.81161i 0 − 30.4863i 0 −18.5203 0 −41.0424 + 69.8321i 0
113.7 0 8.92230 − 1.18010i 0 − 15.7540i 0 18.5203 0 78.2147 − 21.0583i 0
113.8 0 8.92230 + 1.18010i 0 15.7540i 0 18.5203 0 78.2147 + 21.0583i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.5.d.b 8
3.b odd 2 1 inner 336.5.d.b 8
4.b odd 2 1 21.5.b.a ✓ 8
12.b even 2 1 21.5.b.a ✓ 8
28.d even 2 1 147.5.b.e 8
28.f even 6 2 147.5.h.c 16
28.g odd 6 2 147.5.h.e 16
84.h odd 2 1 147.5.b.e 8
84.j odd 6 2 147.5.h.c 16
84.n even 6 2 147.5.h.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.5.b.a ✓ 8 4.b odd 2 1
21.5.b.a ✓ 8 12.b even 2 1
147.5.b.e 8 28.d even 2 1
147.5.b.e 8 84.h odd 2 1
147.5.h.c 16 28.f even 6 2
147.5.h.c 16 84.j odd 6 2
147.5.h.e 16 28.g odd 6 2
147.5.h.e 16 84.n even 6 2
336.5.d.b 8 1.a even 1 1 trivial
336.5.d.b 8 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 1972T_{5}^{6} + 1306936T_{5}^{4} + 349027728T_{5}^{2} + 32473916496 \) acting on \(S_{5}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{7} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 32473916496 \) Copy content Toggle raw display
$7$ \( (T^{2} - 343)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( (T^{4} - 210 T^{3} + \cdots + 137696008)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 33\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( (T^{4} - 186 T^{3} + \cdots + 5829791872)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 26\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( (T^{4} - 1388 T^{3} + \cdots + 74477967264)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 1280 T^{3} + \cdots - 319722971328)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 25\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( (T^{4} + 2360 T^{3} + \cdots + 117096670336)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 34\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 86\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 13501614097288)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 16869420316288)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 62\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 737570944880496)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 301006877602048)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 66\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 39\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 23\!\cdots\!48)^{2} \) Copy content Toggle raw display
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