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Properties

Label 336.7.be.c
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: \( x^{16} - 8 x^{15} - 146176 x^{14} - 825756 x^{13} + 7859971066 x^{12} + 145878273860 x^{11} + \cdots + 24\!\cdots\!21 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{2}\cdot 7^{3} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (9 \beta_{2} - 9) q^{3} + ( - \beta_{3} - 7 \beta_{2}) q^{5} + (\beta_{8} - \beta_{6} + 30 \beta_{2} + \cdots - 9) q^{7}+ \cdots - 243 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (9 \beta_{2} - 9) q^{3} + ( - \beta_{3} - 7 \beta_{2}) q^{5} + (\beta_{8} - \beta_{6} + 30 \beta_{2} + \cdots - 9) q^{7}+ \cdots + (243 \beta_{11} - 243 \beta_{5} + \cdots + 14580) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 216 q^{3} + 56 q^{5} - 396 q^{7} + 1944 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 216 q^{3} + 56 q^{5} - 396 q^{7} + 1944 q^{9} - 1428 q^{11} - 3752 q^{13} - 4928 q^{17} + 1932 q^{19} + 2160 q^{21} + 5952 q^{23} - 21604 q^{25} + 34640 q^{29} - 92628 q^{31} + 12852 q^{33} - 168336 q^{35} - 52444 q^{37} + 50652 q^{39} - 70736 q^{41} - 13608 q^{45} - 214920 q^{47} + 223456 q^{49} + 133056 q^{51} + 219344 q^{53} - 34776 q^{57} + 596196 q^{59} - 325792 q^{61} + 37908 q^{63} + 1073608 q^{65} - 266316 q^{67} - 107136 q^{69} + 259108 q^{73} + 583308 q^{75} + 55960 q^{77} + 206988 q^{79} - 472392 q^{81} - 807648 q^{85} - 467640 q^{87} + 749632 q^{89} - 2530164 q^{91} + 833652 q^{93} - 1373136 q^{95} - 4770488 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} - 146176 x^{14} - 825756 x^{13} + 7859971066 x^{12} + 145878273860 x^{11} + \cdots + 24\!\cdots\!21 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 30\!\cdots\!27 \nu^{15} + \cdots + 26\!\cdots\!63 ) / 12\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 30\!\cdots\!27 \nu^{15} + \cdots + 26\!\cdots\!63 ) / 12\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 50\!\cdots\!51 \nu^{15} + \cdots + 71\!\cdots\!64 ) / 12\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 50\!\cdots\!01 \nu^{15} + \cdots + 16\!\cdots\!69 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 41\!\cdots\!13 \nu^{15} + \cdots + 15\!\cdots\!61 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 35\!\cdots\!38 \nu^{15} + \cdots + 13\!\cdots\!95 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 42\!\cdots\!20 \nu^{15} + \cdots + 11\!\cdots\!39 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 62\!\cdots\!93 \nu^{15} + \cdots - 30\!\cdots\!02 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 39\!\cdots\!77 \nu^{15} + \cdots - 80\!\cdots\!07 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 34\!\cdots\!65 \nu^{15} + \cdots - 77\!\cdots\!91 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 70\!\cdots\!49 \nu^{15} + \cdots + 38\!\cdots\!35 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 13\!\cdots\!18 \nu^{15} + \cdots - 19\!\cdots\!69 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 71\!\cdots\!91 \nu^{15} + \cdots - 16\!\cdots\!05 ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 18\!\cdots\!27 \nu^{15} + \cdots + 74\!\cdots\!51 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 77\!\cdots\!57 \nu^{15} + \cdots + 98\!\cdots\!02 ) / 22\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( - 2 \beta_{14} - 4 \beta_{12} + 2 \beta_{11} - 3 \beta_{10} + 2 \beta_{9} + 6 \beta_{8} - 11 \beta_{7} + \cdots + 18285 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 39 \beta_{15} - 224 \beta_{14} + 239 \beta_{12} + 372 \beta_{11} + 667 \beta_{10} + 447 \beta_{9} + \cdots + 344863 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 1338 \beta_{15} - 116528 \beta_{14} - 2820 \beta_{13} - 247237 \beta_{12} + 26254 \beta_{11} + \cdots + 709294635 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2543165 \beta_{15} - 14945603 \beta_{14} + 49875 \beta_{13} + 5777639 \beta_{12} + 21570658 \beta_{11} + \cdots + 24018091009 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 118870812 \beta_{15} - 6263548679 \beta_{14} - 224498475 \beta_{13} - 13052641507 \beta_{12} + \cdots + 30858806128413 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 150832698835 \beta_{15} - 853357605875 \beta_{14} + 898849560 \beta_{13} + 8302933040 \beta_{12} + \cdots + 14\!\cdots\!15 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 9089455102686 \beta_{15} - 322174547497682 \beta_{14} - 14796054433950 \beta_{13} + \cdots + 14\!\cdots\!86 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 87\!\cdots\!61 \beta_{15} + \cdots + 85\!\cdots\!61 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 64\!\cdots\!56 \beta_{15} + \cdots + 67\!\cdots\!98 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 50\!\cdots\!56 \beta_{15} + \cdots + 48\!\cdots\!52 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 43\!\cdots\!20 \beta_{15} + \cdots + 32\!\cdots\!73 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 28\!\cdots\!54 \beta_{15} + \cdots + 27\!\cdots\!06 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 28\!\cdots\!28 \beta_{15} + \cdots + 15\!\cdots\!71 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 16\!\cdots\!43 \beta_{15} + \cdots + 15\!\cdots\!47 \) 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
−211.449 − 0.866025i
−143.336 − 0.866025i
−57.3420 − 0.866025i
−15.2218 − 0.866025i
18.6598 − 0.866025i
27.8514 − 0.866025i
158.769 − 0.866025i
226.068 − 0.866025i
−211.449 + 0.866025i
−143.336 + 0.866025i
−57.3420 + 0.866025i
−15.2218 + 0.866025i
18.6598 + 0.866025i
27.8514 + 0.866025i
158.769 + 0.866025i
226.068 + 0.866025i
0 −13.5000 + 7.79423i 0 −102.474 + 177.491i 0 178.996 + 292.591i 0 121.500 − 210.444i 0
79.2 0 −13.5000 + 7.79423i 0 −68.4181 + 118.504i 0 −251.999 − 232.691i 0 121.500 − 210.444i 0
79.3 0 −13.5000 + 7.79423i 0 −25.4210 + 44.0305i 0 223.929 + 259.817i 0 121.500 − 210.444i 0
79.4 0 −13.5000 + 7.79423i 0 −4.36091 + 7.55332i 0 −319.125 + 125.731i 0 121.500 − 210.444i 0
79.5 0 −13.5000 + 7.79423i 0 12.5799 − 21.7890i 0 154.877 − 306.043i 0 121.500 − 210.444i 0
79.6 0 −13.5000 + 7.79423i 0 17.1757 − 29.7492i 0 −250.610 + 234.187i 0 121.500 − 210.444i 0
79.7 0 −13.5000 + 7.79423i 0 82.6346 − 143.127i 0 341.160 + 35.4827i 0 121.500 − 210.444i 0
79.8 0 −13.5000 + 7.79423i 0 116.284 − 201.410i 0 −275.227 − 204.693i 0 121.500 − 210.444i 0
319.1 0 −13.5000 − 7.79423i 0 −102.474 − 177.491i 0 178.996 − 292.591i 0 121.500 + 210.444i 0
319.2 0 −13.5000 − 7.79423i 0 −68.4181 − 118.504i 0 −251.999 + 232.691i 0 121.500 + 210.444i 0
319.3 0 −13.5000 − 7.79423i 0 −25.4210 − 44.0305i 0 223.929 − 259.817i 0 121.500 + 210.444i 0
319.4 0 −13.5000 − 7.79423i 0 −4.36091 − 7.55332i 0 −319.125 − 125.731i 0 121.500 + 210.444i 0
319.5 0 −13.5000 − 7.79423i 0 12.5799 + 21.7890i 0 154.877 + 306.043i 0 121.500 + 210.444i 0
319.6 0 −13.5000 − 7.79423i 0 17.1757 + 29.7492i 0 −250.610 − 234.187i 0 121.500 + 210.444i 0
319.7 0 −13.5000 − 7.79423i 0 82.6346 + 143.127i 0 341.160 − 35.4827i 0 121.500 + 210.444i 0
319.8 0 −13.5000 − 7.79423i 0 116.284 + 201.410i 0 −275.227 + 204.693i 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.c ✓ 16
4.b odd 2 1 336.7.be.e yes 16
7.c even 3 1 336.7.be.e yes 16
28.g odd 6 1 inner 336.7.be.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.7.be.c ✓ 16 1.a even 1 1 trivial
336.7.be.c ✓ 16 28.g odd 6 1 inner
336.7.be.e yes 16 4.b odd 2 1
336.7.be.e 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 \(S_{7}^{\mathrm{new}}(336, [\chi])\):

\( T_{5}^{16} - 56 T_{5}^{15} + 74870 T_{5}^{14} - 236256 T_{5}^{13} + 4032396511 T_{5}^{12} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
\( T_{11}^{16} + 1428 T_{11}^{15} - 7518318 T_{11}^{14} - 11706809688 T_{11}^{13} + 42071072310339 T_{11}^{12} + \cdots + 11\!\cdots\!24 \) 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
$5$ \( T^{16} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 36\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 31\!\cdots\!52)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 24\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 77\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 87\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots - 18\!\cdots\!64)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 16\!\cdots\!61 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 42\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 30\!\cdots\!64)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 48\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 64\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 84\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 28\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 86\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 66\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 17\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 90\!\cdots\!41 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 19\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 21\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 20\!\cdots\!08)^{2} \) Copy content Toggle raw display
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