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Properties

Label 336.7.be.f
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: \( x^{16} + 62820 x^{14} + 1608823374 x^{12} + 21497406035987 x^{10} + \cdots + 32\!\cdots\!24 \) 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.

\(f(q)\) \(=\) \( q + ( - 9 \beta_{2} + 18) q^{3} + (\beta_{6} - 7 \beta_{2} + \beta_1 + 7) q^{5} + ( - \beta_{9} - 7 \beta_{2} + 33) q^{7} + ( - 243 \beta_{2} + 243) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 9 \beta_{2} + 18) q^{3} + (\beta_{6} - 7 \beta_{2} + \beta_1 + 7) q^{5} + ( - \beta_{9} - 7 \beta_{2} + 33) q^{7} + ( - 243 \beta_{2} + 243) q^{9} + (\beta_{9} + \beta_{8} - \beta_{6} + \cdots - 162) q^{11}+ \cdots + ( - 243 \beta_{11} + 243 \beta_{8} + \cdots - 19683) q^{99}+O(q^{100}) \) 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 \( x^{16} + 62820 x^{14} + 1608823374 x^{12} + 21497406035987 x^{10} + \cdots + 32\!\cdots\!24 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 37\!\cdots\!11 \nu^{14} + \cdots + 41\!\cdots\!12 ) / 45\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 34\!\cdots\!11 \nu^{15} + \cdots + 14\!\cdots\!16 ) / 29\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 91\!\cdots\!02 \nu^{15} + \cdots + 59\!\cdots\!64 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 22\!\cdots\!60 \nu^{15} + \cdots + 17\!\cdots\!12 ) / 47\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 78\!\cdots\!81 \nu^{15} + \cdots - 31\!\cdots\!36 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 19\!\cdots\!03 \nu^{15} + \cdots - 12\!\cdots\!16 ) / 26\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 33\!\cdots\!85 \nu^{15} + \cdots + 78\!\cdots\!32 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10\!\cdots\!07 \nu^{15} + \cdots + 16\!\cdots\!20 ) / 95\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 14\!\cdots\!81 \nu^{15} + \cdots - 19\!\cdots\!96 ) / 68\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 20\!\cdots\!10 \nu^{15} + \cdots + 11\!\cdots\!64 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 14\!\cdots\!73 \nu^{15} + \cdots + 59\!\cdots\!88 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 72\!\cdots\!79 \nu^{15} + \cdots + 75\!\cdots\!48 ) / 95\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 19\!\cdots\!07 \nu^{15} + \cdots + 11\!\cdots\!44 ) / 94\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 16\!\cdots\!45 \nu^{15} + \cdots + 15\!\cdots\!68 ) / 78\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 16\!\cdots\!45 \nu^{15} + \cdots - 20\!\cdots\!28 ) / 78\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} + 2\beta_{8} - \beta_{4} - 8\beta_{2} + 3 ) / 14 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 24 \beta_{12} + 14 \beta_{10} + 31 \beta_{9} + 99 \beta_{8} + 63 \beta_{7} + 55 \beta_{4} + \cdots - 219923 ) / 28 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 1365 \beta_{15} - 1365 \beta_{14} - 280 \beta_{13} - 1616 \beta_{12} + 441 \beta_{11} + \cdots + 2228453 ) / 56 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 86541 \beta_{15} + 86541 \beta_{14} - 357236 \beta_{12} + 142443 \beta_{11} - 178514 \beta_{10} + \cdots + 2550873577 ) / 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 7169778 \beta_{15} + 7169778 \beta_{14} - 2566704 \beta_{13} + 25572728 \beta_{12} + 15163344 \beta_{11} + \cdots - 7893818883 ) / 28 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 4539306471 \beta_{15} - 4539306471 \beta_{14} + 7451810054 \beta_{12} - 5251745079 \beta_{11} + \cdots - 64424438418819 ) / 56 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 38407774335 \beta_{15} - 38407774335 \beta_{14} + 103318435934 \beta_{13} - 495574076088 \beta_{12} + \cdots - 17587502027404 ) / 28 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 43173421449144 \beta_{15} + 43173421449144 \beta_{14} - 30917333454732 \beta_{12} + \cdots + 42\!\cdots\!71 ) / 28 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 788678515136253 \beta_{15} - 788678515136253 \beta_{14} + \cdots + 37\!\cdots\!95 ) / 56 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 72\!\cdots\!33 \beta_{15} + \cdots - 57\!\cdots\!23 ) / 28 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 18\!\cdots\!76 \beta_{15} + \cdots - 48\!\cdots\!72 ) / 28 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 23\!\cdots\!21 \beta_{15} + \cdots + 15\!\cdots\!89 ) / 56 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 44\!\cdots\!45 \beta_{15} + \cdots + 96\!\cdots\!69 ) / 28 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 17\!\cdots\!62 \beta_{15} + \cdots - 11\!\cdots\!41 ) / 28 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 17\!\cdots\!99 \beta_{15} + \cdots - 35\!\cdots\!95 ) / 56 \) Copy content Toggle raw display

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

Character values

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

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

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
− 42.1392i
− 100.261i
118.340i
104.643i
− 6.37271i
− 47.3863i
− 123.062i
92.7744i
42.1392i
100.261i
− 118.340i
− 104.643i
6.37271i
47.3863i
123.062i
− 92.7744i
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.f yes 16
4.b odd 2 1 336.7.be.b ✓ 16
7.c even 3 1 336.7.be.b ✓ 16
28.g odd 6 1 inner 336.7.be.f yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.7.be.b ✓ 16 4.b odd 2 1
336.7.be.b ✓ 16 7.c even 3 1
336.7.be.f yes 16 1.a even 1 1 trivial
336.7.be.f yes 16 28.g odd 6 1 inner

Hecke kernels

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

\( T_{5}^{16} - 56 T_{5}^{15} + 74870 T_{5}^{14} - 3403968 T_{5}^{13} + 3925943647 T_{5}^{12} + \cdots + 33\!\cdots\!00 \) 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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