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Properties

Label 336.2.bc.f
Level $336$
Weight $2$
Character orbit 336.bc
Analytic conductor $2.683$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [336,2,Mod(17,336)]
 
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([0, 0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 336.bc (of order \(6\), degree \(2\), 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: \(2.68297350792\)
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: \( x^{16} - 6 x^{15} + 19 x^{14} - 42 x^{13} + 65 x^{12} - 48 x^{11} - 94 x^{10} + 444 x^{9} - 962 x^{8} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 168)
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.

\(f(q)\) \(=\) \( q + (\beta_{8} - \beta_{5}) q^{3} - \beta_{14} q^{5} + \beta_{10} q^{7} - \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{8} - \beta_{5}) q^{3} - \beta_{14} q^{5} + \beta_{10} q^{7} - \beta_1 q^{9} + ( - \beta_{15} + \beta_{14} + \cdots + \beta_{9}) q^{11}+ \cdots + ( - 3 \beta_{15} + 4 \beta_{14} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{7} + 2 q^{9} - 8 q^{15} + 6 q^{19} + 14 q^{21} - 18 q^{25} + 48 q^{31} - 12 q^{33} - 2 q^{37} + 22 q^{39} - 20 q^{43} - 42 q^{45} - 28 q^{49} - 6 q^{51} - 8 q^{57} + 36 q^{61} + 32 q^{63} - 14 q^{67} + 30 q^{73} - 54 q^{75} - 28 q^{79} + 30 q^{81} + 16 q^{85} - 78 q^{87} - 66 q^{91} + 16 q^{93} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{15} + 19 x^{14} - 42 x^{13} + 65 x^{12} - 48 x^{11} - 94 x^{10} + 444 x^{9} - 962 x^{8} + \cdots + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{15} - 15 \nu^{14} + 71 \nu^{13} + 48 \nu^{12} - 110 \nu^{11} + 384 \nu^{10} - 266 \nu^{9} + \cdots - 52488 ) / 34992 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11 \nu^{15} - 30 \nu^{14} + 142 \nu^{13} - 363 \nu^{12} + 662 \nu^{11} - 258 \nu^{10} + \cdots + 19683 ) / 69984 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 22 \nu^{15} + 75 \nu^{14} - 238 \nu^{13} + 489 \nu^{12} - 737 \nu^{11} + 429 \nu^{10} + \cdots + 131220 ) / 34992 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 25 \nu^{15} + 111 \nu^{14} - 295 \nu^{13} + 606 \nu^{12} - 770 \nu^{11} - 66 \nu^{10} + \cdots + 69984 ) / 23328 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 19 \nu^{15} + 95 \nu^{14} - 274 \nu^{13} + 500 \nu^{12} - 653 \nu^{11} + 145 \nu^{10} + \cdots + 85293 ) / 11664 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 125 \nu^{15} + 669 \nu^{14} - 2069 \nu^{13} + 4116 \nu^{12} - 5794 \nu^{11} + 1302 \nu^{10} + \cdots + 599238 ) / 69984 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 19 \nu^{15} - 82 \nu^{14} + 226 \nu^{13} - 433 \nu^{12} + 542 \nu^{11} - 74 \nu^{10} - 1912 \nu^{9} + \cdots - 75087 ) / 7776 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 193 \nu^{15} - 753 \nu^{14} + 1975 \nu^{13} - 3624 \nu^{12} + 4292 \nu^{11} - 84 \nu^{10} + \cdots - 708588 ) / 69984 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 100 \nu^{15} - 429 \nu^{14} + 1207 \nu^{13} - 2301 \nu^{12} + 2648 \nu^{11} + 510 \nu^{10} + \cdots - 347733 ) / 34992 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 61 \nu^{15} + 302 \nu^{14} - 784 \nu^{13} + 1463 \nu^{12} - 1772 \nu^{11} + 100 \nu^{10} + \cdots + 255879 ) / 23328 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 74 \nu^{15} + 351 \nu^{14} - 1037 \nu^{13} + 2043 \nu^{12} - 2470 \nu^{11} + 234 \nu^{10} + \cdots + 321489 ) / 23328 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 41 \nu^{15} + 179 \nu^{14} - 482 \nu^{13} + 962 \nu^{12} - 1225 \nu^{11} + 205 \nu^{10} + \cdots + 165483 ) / 11664 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 275 \nu^{15} + 1101 \nu^{14} - 2921 \nu^{13} + 5466 \nu^{12} - 7066 \nu^{11} + 6 \nu^{10} + \cdots + 883548 ) / 69984 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2 \nu^{15} + 8 \nu^{14} - 21 \nu^{13} + 38 \nu^{12} - 41 \nu^{11} - 17 \nu^{10} + 195 \nu^{9} + \cdots + 4617 ) / 432 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} - 2 \beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} + \beta_{10} - 4 \beta_{9} + \beta_{8} + \cdots + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{15} - 6 \beta_{13} + \beta_{12} + 6 \beta_{11} + \beta_{10} - \beta_{8} + \beta_{7} - 6 \beta_{6} + \cdots - 5 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} - 2\beta_{14} + \beta_{13} - 2\beta_{12} - \beta_{10} + \beta_{6} + 3\beta_{4} - 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2 \beta_{15} - 6 \beta_{14} + 17 \beta_{13} - 14 \beta_{12} - 7 \beta_{11} - 16 \beta_{10} + \cdots - 15 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{15} + 8 \beta_{14} - 2 \beta_{13} + 2 \beta_{12} + 4 \beta_{11} - 6 \beta_{10} + 8 \beta_{9} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 13 \beta_{15} + 2 \beta_{14} - 37 \beta_{13} - 11 \beta_{12} + 13 \beta_{11} - 37 \beta_{10} + \cdots + 22 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2 \beta_{14} + 12 \beta_{13} - 18 \beta_{11} + 4 \beta_{10} - 14 \beta_{9} - 4 \beta_{8} - 14 \beta_{7} + \cdots + 12 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 37 \beta_{15} - 10 \beta_{13} - 37 \beta_{12} - 94 \beta_{11} - 21 \beta_{10} - 299 \beta_{8} + \cdots + 97 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 23 \beta_{15} - 28 \beta_{14} - 19 \beta_{13} - 12 \beta_{12} + 16 \beta_{11} + 19 \beta_{10} + \cdots + 44 \beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 6 \beta_{15} - 682 \beta_{14} + 179 \beta_{13} - 346 \beta_{12} - 173 \beta_{11} - 432 \beta_{10} + \cdots - 349 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 44 \beta_{15} + 48 \beta_{14} + 64 \beta_{13} + 172 \beta_{12} + 24 \beta_{11} - 32 \beta_{10} + \cdots - 135 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 855 \beta_{15} + 1182 \beta_{14} - 615 \beta_{13} + 327 \beta_{12} + 855 \beta_{11} - 615 \beta_{10} + \cdots + 2418 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 196 \beta_{14} + 324 \beta_{13} + 372 \beta_{11} + 460 \beta_{10} - 224 \beta_{9} + 416 \beta_{8} + \cdots + 984 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 2919 \beta_{15} + 2858 \beta_{13} + 153 \beta_{12} - 490 \beta_{11} - 1463 \beta_{10} - 9 \beta_{8} + \cdots + 6899 ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1 - \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
0.247636 + 1.71426i
−0.601642 − 1.62420i
1.60841 − 0.642670i
−1.70742 + 0.291063i
1.73018 + 0.0805675i
1.22961 − 1.21986i
0.934861 + 1.45809i
−0.441628 + 1.67480i
0.247636 − 1.71426i
−0.601642 + 1.62420i
1.60841 + 0.642670i
−1.70742 − 0.291063i
1.73018 − 0.0805675i
1.22961 + 1.21986i
0.934861 − 1.45809i
−0.441628 − 1.67480i
0 −1.71426 + 0.247636i 0 −1.28955 − 2.23357i 0 0.203402 + 2.63792i 0 2.87735 − 0.849022i 0
17.2 0 −1.62420 + 0.601642i 0 0.0726693 + 0.125867i 0 −1.05451 − 2.42652i 0 2.27605 − 1.95437i 0
17.3 0 −0.642670 − 1.60841i 0 1.28955 + 2.23357i 0 0.203402 + 2.63792i 0 −2.17395 + 2.06735i 0
17.4 0 −0.291063 − 1.70742i 0 −0.0726693 − 0.125867i 0 −1.05451 − 2.42652i 0 −2.83056 + 0.993934i 0
17.5 0 −0.0805675 + 1.73018i 0 1.90017 + 3.29119i 0 −2.23495 + 1.41598i 0 −2.98702 − 0.278792i 0
17.6 0 1.21986 + 1.22961i 0 −1.40397 − 2.43175i 0 2.08606 − 1.62738i 0 −0.0238727 + 2.99991i 0
17.7 0 1.45809 − 0.934861i 0 −1.90017 − 3.29119i 0 −2.23495 + 1.41598i 0 1.25207 − 2.72623i 0
17.8 0 1.67480 + 0.441628i 0 1.40397 + 2.43175i 0 2.08606 − 1.62738i 0 2.60993 + 1.47928i 0
257.1 0 −1.71426 − 0.247636i 0 −1.28955 + 2.23357i 0 0.203402 − 2.63792i 0 2.87735 + 0.849022i 0
257.2 0 −1.62420 − 0.601642i 0 0.0726693 − 0.125867i 0 −1.05451 + 2.42652i 0 2.27605 + 1.95437i 0
257.3 0 −0.642670 + 1.60841i 0 1.28955 − 2.23357i 0 0.203402 − 2.63792i 0 −2.17395 − 2.06735i 0
257.4 0 −0.291063 + 1.70742i 0 −0.0726693 + 0.125867i 0 −1.05451 + 2.42652i 0 −2.83056 − 0.993934i 0
257.5 0 −0.0805675 − 1.73018i 0 1.90017 − 3.29119i 0 −2.23495 − 1.41598i 0 −2.98702 + 0.278792i 0
257.6 0 1.21986 − 1.22961i 0 −1.40397 + 2.43175i 0 2.08606 + 1.62738i 0 −0.0238727 − 2.99991i 0
257.7 0 1.45809 + 0.934861i 0 −1.90017 + 3.29119i 0 −2.23495 − 1.41598i 0 1.25207 + 2.72623i 0
257.8 0 1.67480 − 0.441628i 0 1.40397 − 2.43175i 0 2.08606 + 1.62738i 0 2.60993 − 1.47928i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.2.bc.f 16
3.b odd 2 1 inner 336.2.bc.f 16
4.b odd 2 1 168.2.u.a ✓ 16
7.c even 3 1 2352.2.k.i 16
7.d odd 6 1 inner 336.2.bc.f 16
7.d odd 6 1 2352.2.k.i 16
12.b even 2 1 168.2.u.a ✓ 16
21.g even 6 1 inner 336.2.bc.f 16
21.g even 6 1 2352.2.k.i 16
21.h odd 6 1 2352.2.k.i 16
28.d even 2 1 1176.2.u.b 16
28.f even 6 1 168.2.u.a ✓ 16
28.f even 6 1 1176.2.k.a 16
28.g odd 6 1 1176.2.k.a 16
28.g odd 6 1 1176.2.u.b 16
84.h odd 2 1 1176.2.u.b 16
84.j odd 6 1 168.2.u.a ✓ 16
84.j odd 6 1 1176.2.k.a 16
84.n even 6 1 1176.2.k.a 16
84.n even 6 1 1176.2.u.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.u.a ✓ 16 4.b odd 2 1
168.2.u.a ✓ 16 12.b even 2 1
168.2.u.a ✓ 16 28.f even 6 1
168.2.u.a ✓ 16 84.j odd 6 1
336.2.bc.f 16 1.a even 1 1 trivial
336.2.bc.f 16 3.b odd 2 1 inner
336.2.bc.f 16 7.d odd 6 1 inner
336.2.bc.f 16 21.g even 6 1 inner
1176.2.k.a 16 28.f even 6 1
1176.2.k.a 16 28.g odd 6 1
1176.2.k.a 16 84.j odd 6 1
1176.2.k.a 16 84.n even 6 1
1176.2.u.b 16 28.d even 2 1
1176.2.u.b 16 28.g odd 6 1
1176.2.u.b 16 84.h odd 2 1
1176.2.u.b 16 84.n even 6 1
2352.2.k.i 16 7.c even 3 1
2352.2.k.i 16 7.d odd 6 1
2352.2.k.i 16 21.g even 6 1
2352.2.k.i 16 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(336, [\chi])\):

\( T_{5}^{16} + 29 T_{5}^{14} + 578 T_{5}^{12} + 6101 T_{5}^{10} + 47026 T_{5}^{8} + 199741 T_{5}^{6} + \cdots + 256 \) Copy content Toggle raw display
\( T_{13}^{8} + 55T_{13}^{6} + 836T_{13}^{4} + 3584T_{13}^{2} + 4096 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - T^{14} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{16} + 29 T^{14} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( (T^{8} + 2 T^{7} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} - 39 T^{14} + \cdots + 4096 \) Copy content Toggle raw display
$13$ \( (T^{8} + 55 T^{6} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 94 T^{14} + \cdots + 4096 \) Copy content Toggle raw display
$19$ \( (T^{8} - 3 T^{7} + \cdots + 3844)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} - 82 T^{14} + \cdots + 4096 \) Copy content Toggle raw display
$29$ \( (T^{8} + 129 T^{6} + \cdots + 262144)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 24 T^{7} + \cdots + 368449)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + T^{7} + \cdots + 232324)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 88 T^{6} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 5 T^{3} + \cdots - 128)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + 218 T^{14} + \cdots + 65536 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 4347792138496 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 20352513413376 \) Copy content Toggle raw display
$61$ \( (T^{8} - 18 T^{7} + \cdots + 82944)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 7 T^{7} + \cdots + 51076)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 344 T^{6} + \cdots + 2166784)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 15 T^{7} + \cdots + 16)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 14 T^{7} + \cdots + 160801)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 523 T^{6} + \cdots + 131239936)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 968381956096 \) Copy content Toggle raw display
$97$ \( (T^{8} + 347 T^{6} + \cdots + 5914624)^{2} \) Copy content Toggle raw display
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