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Properties

Label 336.7.be.b
Level $336$
Weight $7$
Character orbit 336.be
Analytic conductor $77.298$
Analytic rank $0$
Dimension $16$
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] = [336,7,Mod(79,336)]
 
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, 2]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.79");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 336.be (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: \(77.2981720963\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 73142 x^{14} + 1026048 x^{13} + 3832175263 x^{12} + 62397711008 x^{11} + \cdots + 67\!\cdots\!36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{2}\cdot 7^{3}\cdot 43^{2} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (9 \beta_{2} - 9) q^{3} + ( - 7 \beta_{2} - \beta_1) q^{5} + ( - \beta_{4} + 7 \beta_{2} - 26) q^{7} - 243 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (9 \beta_{2} - 9) q^{3} + ( - 7 \beta_{2} - \beta_1) q^{5} + ( - \beta_{4} + 7 \beta_{2} - 26) q^{7} - 243 \beta_{2} q^{9} + (\beta_{9} + \beta_{6} - \beta_{5} + \cdots + 80) q^{11}+ \cdots + (243 \beta_{6} - 243 \beta_{5} + \cdots - 19683) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 216 q^{3} + 56 q^{5} - 468 q^{7} + 1944 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 216 q^{3} + 56 q^{5} - 468 q^{7} + 1944 q^{9} + 1932 q^{11} - 728 q^{13} + 3136 q^{17} + 5628 q^{19} + 5616 q^{21} - 12000 q^{23} - 21604 q^{25} - 43984 q^{29} + 79236 q^{31} - 17388 q^{33} - 29520 q^{35} - 29764 q^{37} + 9828 q^{39} + 183856 q^{41} - 13608 q^{45} + 82440 q^{47} - 19760 q^{49} - 84672 q^{51} - 136480 q^{53} - 101304 q^{57} - 77148 q^{59} + 56096 q^{61} - 37908 q^{63} - 492824 q^{65} - 258348 q^{67} + 216000 q^{69} + 66868 q^{73} + 583308 q^{75} - 1712504 q^{77} + 972372 q^{79} - 472392 q^{81} + 2204256 q^{85} + 593784 q^{87} + 1464160 q^{89} - 65940 q^{91} - 713124 q^{93} + 31488 q^{95} + 1585096 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 73142 x^{14} + 1026048 x^{13} + 3832175263 x^{12} + 62397711008 x^{11} + \cdots + 67\!\cdots\!36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 90\!\cdots\!54 \nu^{15} + \cdots - 23\!\cdots\!60 ) / 35\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 90\!\cdots\!54 \nu^{15} + \cdots - 23\!\cdots\!60 ) / 35\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 28\!\cdots\!87 \nu^{15} + \cdots - 59\!\cdots\!68 ) / 35\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11\!\cdots\!94 \nu^{15} + \cdots + 43\!\cdots\!12 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10\!\cdots\!78 \nu^{15} + \cdots - 96\!\cdots\!92 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 30\!\cdots\!84 \nu^{15} + \cdots - 24\!\cdots\!52 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 47\!\cdots\!20 \nu^{15} + \cdots + 58\!\cdots\!24 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 17\!\cdots\!58 \nu^{15} + \cdots + 47\!\cdots\!36 ) / 68\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 12\!\cdots\!18 \nu^{15} + \cdots + 24\!\cdots\!48 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 63\!\cdots\!76 \nu^{15} + \cdots + 27\!\cdots\!68 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 63\!\cdots\!79 \nu^{15} + \cdots + 30\!\cdots\!56 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 64\!\cdots\!44 \nu^{15} + \cdots + 86\!\cdots\!88 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 17\!\cdots\!44 \nu^{15} + \cdots + 24\!\cdots\!76 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 42\!\cdots\!71 \nu^{15} + \cdots + 82\!\cdots\!84 ) / 55\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 16\!\cdots\!87 \nu^{15} + \cdots - 48\!\cdots\!20 ) / 11\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2 \beta_{15} - 2 \beta_{14} + 2 \beta_{13} - 3 \beta_{12} + 2 \beta_{11} + \beta_{10} - 6 \beta_{9} + \cdots - 18290 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 236 \beta_{14} - 294 \beta_{13} + 313 \beta_{11} + 236 \beta_{10} - 448 \beta_{9} + 964 \beta_{8} + \cdots - 304040 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 62534 \beta_{15} + 46063 \beta_{14} + 153888 \beta_{12} - 46063 \beta_{11} - 92126 \beta_{10} + \cdots + 112675 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 17007306 \beta_{15} - 28089286 \beta_{14} + 17007306 \beta_{13} - 12065724 \beta_{12} + \cdots + 15496527081 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1784459695 \beta_{14} - 2150010506 \beta_{13} - 3262272793 \beta_{11} + 1784459695 \beta_{10} + \cdots + 22904048714767 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 797776007652 \beta_{15} + 671943743822 \beta_{14} + 591652236525 \beta_{12} - 671943743822 \beta_{11} + \cdots + 1785351914039 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 79261912717550 \beta_{15} - 135504575423822 \beta_{14} + 79261912717550 \beta_{13} + \cdots - 96\!\cdots\!92 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 30\!\cdots\!09 \beta_{14} + \cdots - 41\!\cdots\!04 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 30\!\cdots\!94 \beta_{15} + \cdots + 12\!\cdots\!43 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 15\!\cdots\!60 \beta_{15} + \cdots + 21\!\cdots\!79 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 98\!\cdots\!71 \beta_{14} + \cdots + 18\!\cdots\!09 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 67\!\cdots\!30 \beta_{15} + \cdots + 12\!\cdots\!83 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 43\!\cdots\!42 \beta_{15} + \cdots - 82\!\cdots\!66 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 25\!\cdots\!46 \beta_{14} + \cdots - 56\!\cdots\!14 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
107.132 − 185.558i
68.7531 − 119.084i
55.4887 − 96.1092i
10.0711 − 17.4436i
−25.2918 + 43.8067i
−51.8806 + 89.8599i
−57.8261 + 100.158i
−102.446 + 177.442i
107.132 + 185.558i
68.7531 + 119.084i
55.4887 + 96.1092i
10.0711 + 17.4436i
−25.2918 − 43.8067i
−51.8806 − 89.8599i
−57.8261 − 100.158i
−102.446 − 177.442i
0 −13.5000 + 7.79423i 0 −103.132 + 178.630i 0 −186.554 + 287.831i 0 121.500 − 210.444i 0
79.2 0 −13.5000 + 7.79423i 0 −64.7531 + 112.156i 0 341.298 + 34.1242i 0 121.500 − 210.444i 0
79.3 0 −13.5000 + 7.79423i 0 −51.4887 + 89.1810i 0 −142.507 − 311.995i 0 121.500 − 210.444i 0
79.4 0 −13.5000 + 7.79423i 0 −6.07109 + 10.5154i 0 −56.8171 − 338.261i 0 121.500 − 210.444i 0
79.5 0 −13.5000 + 7.79423i 0 29.2918 − 50.7349i 0 −258.745 + 225.167i 0 121.500 − 210.444i 0
79.6 0 −13.5000 + 7.79423i 0 55.8806 − 96.7881i 0 317.334 − 130.185i 0 121.500 − 210.444i 0
79.7 0 −13.5000 + 7.79423i 0 61.8261 − 107.086i 0 90.9703 + 330.716i 0 121.500 − 210.444i 0
79.8 0 −13.5000 + 7.79423i 0 106.446 − 184.370i 0 −338.979 − 52.3633i 0 121.500 − 210.444i 0
319.1 0 −13.5000 − 7.79423i 0 −103.132 − 178.630i 0 −186.554 − 287.831i 0 121.500 + 210.444i 0
319.2 0 −13.5000 − 7.79423i 0 −64.7531 − 112.156i 0 341.298 − 34.1242i 0 121.500 + 210.444i 0
319.3 0 −13.5000 − 7.79423i 0 −51.4887 − 89.1810i 0 −142.507 + 311.995i 0 121.500 + 210.444i 0
319.4 0 −13.5000 − 7.79423i 0 −6.07109 − 10.5154i 0 −56.8171 + 338.261i 0 121.500 + 210.444i 0
319.5 0 −13.5000 − 7.79423i 0 29.2918 + 50.7349i 0 −258.745 − 225.167i 0 121.500 + 210.444i 0
319.6 0 −13.5000 − 7.79423i 0 55.8806 + 96.7881i 0 317.334 + 130.185i 0 121.500 + 210.444i 0
319.7 0 −13.5000 − 7.79423i 0 61.8261 + 107.086i 0 90.9703 − 330.716i 0 121.500 + 210.444i 0
319.8 0 −13.5000 − 7.79423i 0 106.446 + 184.370i 0 −338.979 + 52.3633i 0 121.500 + 210.444i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.7.be.b ✓ 16
4.b odd 2 1 336.7.be.f yes 16
7.c even 3 1 336.7.be.f yes 16
28.g odd 6 1 inner 336.7.be.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.7.be.b ✓ 16 1.a even 1 1 trivial
336.7.be.b ✓ 16 28.g odd 6 1 inner
336.7.be.f yes 16 4.b odd 2 1
336.7.be.f yes 16 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(336, [\chi])\):

\( T_{5}^{16} - 56 T_{5}^{15} + 74870 T_{5}^{14} - 3403968 T_{5}^{13} + 3925943647 T_{5}^{12} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
\( T_{11}^{16} - 1932 T_{11}^{15} - 3550830 T_{11}^{14} + 9264013416 T_{11}^{13} + 12240806783235 T_{11}^{12} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{2} + 27 T + 243)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 36\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 34\!\cdots\!12)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 84\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 24\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots - 30\!\cdots\!08)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 40\!\cdots\!01 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 41\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 22\!\cdots\!60)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 33\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 18\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 94\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 17\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 19\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 84\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 11\!\cdots\!80)^{2} \) Copy content Toggle raw display
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