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Properties

Label 336.4.bc.d
Level $336$
Weight $4$
Character orbit 336.bc
Analytic conductor $19.825$
Analytic rank $0$
Dimension $12$
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] = [336,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.17");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(19.8246417619\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} - 29x^{9} + 6x^{8} - 49x^{7} + 1564x^{6} - 441x^{5} + 486x^{4} - 21141x^{3} - 59049x + 531441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 21)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{4} q^{5} + (3 \beta_{11} - \beta_{10} + \beta_{7} + \cdots + 4) q^{7}+ \cdots + (\beta_{11} + 2 \beta_{9} + \cdots + 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{4} q^{5} + (3 \beta_{11} - \beta_{10} + \beta_{7} + \cdots + 4) q^{7}+ \cdots + (27 \beta_{11} - 87 \beta_{10} + \cdots + 303) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{3} + 56 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 3 q^{3} + 56 q^{7} - 3 q^{9} - 6 q^{15} - 300 q^{19} + 357 q^{21} - 42 q^{25} + 930 q^{31} - 855 q^{33} + 764 q^{37} + 426 q^{39} + 1012 q^{43} + 2367 q^{45} - 336 q^{49} + 1341 q^{51} + 270 q^{57} + 2358 q^{61} - 1071 q^{63} - 792 q^{67} - 2904 q^{73} + 2418 q^{75} - 1674 q^{79} + 837 q^{81} + 348 q^{85} - 1638 q^{87} + 1218 q^{91} - 1479 q^{93} + 3354 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} - 29x^{9} + 6x^{8} - 49x^{7} + 1564x^{6} - 441x^{5} + 486x^{4} - 21141x^{3} - 59049x + 531441 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 538 \nu^{11} + 22601 \nu^{10} - 146502 \nu^{9} + 1327 \nu^{8} - 161148 \nu^{7} + \cdots - 142839531 ) / 489398112 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 109 \nu^{11} + 3322 \nu^{10} + 26109 \nu^{9} - 69172 \nu^{8} - 83625 \nu^{7} + \cdots + 271271106 ) / 69914016 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 857 \nu^{11} - 5426 \nu^{10} + 627 \nu^{9} - 6088 \nu^{8} + 24405 \nu^{7} - 289700 \nu^{6} + \cdots - 336854862 ) / 163132704 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4301 \nu^{11} - 145853 \nu^{10} + 231543 \nu^{9} + 1385435 \nu^{8} + 5417499 \nu^{7} + \cdots - 8693961417 ) / 489398112 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4280 \nu^{11} + 50035 \nu^{10} + 126396 \nu^{9} + 330857 \nu^{8} - 3202098 \nu^{7} + \cdots - 3237833817 ) / 244699056 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1523 \nu^{11} + 209 \nu^{10} + 6255 \nu^{9} + 6421 \nu^{8} - 82569 \nu^{7} - 227449 \nu^{6} + \cdots - 151814979 ) / 54377568 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1523 \nu^{11} + 209 \nu^{10} + 6255 \nu^{9} + 6421 \nu^{8} - 82569 \nu^{7} - 227449 \nu^{6} + \cdots - 151814979 ) / 54377568 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1228 \nu^{11} + 12349 \nu^{10} + 50088 \nu^{9} + 39527 \nu^{8} - 725238 \nu^{7} + \cdots - 585313371 ) / 40783176 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 22376 \nu^{11} + 116845 \nu^{10} - 61380 \nu^{9} - 2138089 \nu^{8} - 3679350 \nu^{7} + \cdots + 5512873689 ) / 489398112 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 8011 \nu^{11} - 4808 \nu^{10} - 1005 \nu^{9} - 51046 \nu^{8} - 321723 \nu^{7} + \cdots + 864004968 ) / 163132704 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 25402 \nu^{11} - 2263 \nu^{10} - 146502 \nu^{9} - 719729 \nu^{8} - 11964 \nu^{7} + \cdots - 1611033867 ) / 489398112 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{6} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{11} + 2\beta_{9} - \beta_{7} + \beta_{6} + \beta_{4} - 3\beta_{2} - \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4 \beta_{11} + 3 \beta_{10} - \beta_{9} + 6 \beta_{8} - 6 \beta_{7} - 3 \beta_{6} - 6 \beta_{5} + \cdots + 21 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 5 \beta_{11} - 27 \beta_{10} - 7 \beta_{9} + 2 \beta_{7} + 25 \beta_{6} - 6 \beta_{5} - 14 \beta_{4} + \cdots + 48 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 33\beta_{11} + 12\beta_{9} - 10\beta_{7} + 52\beta_{6} + 6\beta_{4} - 216\beta_{3} - 72\beta_{2} - 105\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 44 \beta_{11} + 114 \beta_{10} + 8 \beta_{9} + 90 \beta_{8} - 280 \beta_{7} - 140 \beta_{6} + \cdots - 1629 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 32 \beta_{11} - 552 \beta_{10} - 10 \beta_{9} + 681 \beta_{7} - 129 \beta_{6} - 228 \beta_{5} + \cdots - 984 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1282 \beta_{11} - 1582 \beta_{9} + 294 \beta_{8} - 493 \beta_{7} - 1013 \beta_{6} - 791 \beta_{4} + \cdots - 1255 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 3930 \beta_{11} - 2709 \beta_{10} + 681 \beta_{9} - 2106 \beta_{8} + 974 \beta_{7} + 487 \beta_{6} + \cdots - 29619 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 7579 \beta_{11} + 4467 \beta_{10} + 3905 \beta_{9} + 8606 \beta_{7} - 13073 \beta_{6} + 2772 \beta_{5} + \cdots - 114192 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 3799 \beta_{11} - 20608 \beta_{9} + 10020 \beta_{8} - 25200 \beta_{7} - 96240 \beta_{6} + \cdots + 42041 \beta_1 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-\beta_{3}\)

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
−2.59957 + 1.49740i
0.00299931 + 3.00000i
−2.23014 − 2.00661i
−0.232749 − 2.99096i
2.85284 + 0.928053i
2.70662 − 1.29391i
−2.59957 − 1.49740i
0.00299931 − 3.00000i
−2.23014 + 2.00661i
−0.232749 + 2.99096i
2.85284 − 0.928053i
2.70662 + 1.29391i
0 −5.19615 + 0.00519496i 0 8.05907 + 13.9587i 0 5.67909 + 17.6280i 0 26.9999 − 0.0539876i 0
17.2 0 −2.59358 − 4.50260i 0 −8.05907 − 13.9587i 0 5.67909 + 17.6280i 0 −13.5467 + 23.3556i 0
17.3 0 −1.60743 + 4.94127i 0 −0.623706 − 1.08029i 0 −10.0808 − 15.5363i 0 −21.8323 − 15.8855i 0
17.4 0 2.24112 + 4.68800i 0 −5.80193 − 10.0492i 0 18.4018 − 2.09174i 0 −16.9548 + 21.0128i 0
17.5 0 3.47555 − 3.86271i 0 0.623706 + 1.08029i 0 −10.0808 − 15.5363i 0 −2.84113 − 26.8501i 0
17.6 0 5.18049 − 0.403134i 0 5.80193 + 10.0492i 0 18.4018 − 2.09174i 0 26.6750 − 4.17686i 0
257.1 0 −5.19615 − 0.00519496i 0 8.05907 − 13.9587i 0 5.67909 − 17.6280i 0 26.9999 + 0.0539876i 0
257.2 0 −2.59358 + 4.50260i 0 −8.05907 + 13.9587i 0 5.67909 − 17.6280i 0 −13.5467 − 23.3556i 0
257.3 0 −1.60743 − 4.94127i 0 −0.623706 + 1.08029i 0 −10.0808 + 15.5363i 0 −21.8323 + 15.8855i 0
257.4 0 2.24112 − 4.68800i 0 −5.80193 + 10.0492i 0 18.4018 + 2.09174i 0 −16.9548 − 21.0128i 0
257.5 0 3.47555 + 3.86271i 0 0.623706 − 1.08029i 0 −10.0808 + 15.5363i 0 −2.84113 + 26.8501i 0
257.6 0 5.18049 + 0.403134i 0 5.80193 − 10.0492i 0 18.4018 + 2.09174i 0 26.6750 + 4.17686i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.6
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 336.4.bc.d 12
3.b odd 2 1 inner 336.4.bc.d 12
4.b odd 2 1 21.4.g.a ✓ 12
7.d odd 6 1 inner 336.4.bc.d 12
12.b even 2 1 21.4.g.a ✓ 12
21.g even 6 1 inner 336.4.bc.d 12
28.d even 2 1 147.4.g.d 12
28.f even 6 1 21.4.g.a ✓ 12
28.f even 6 1 147.4.c.a 12
28.g odd 6 1 147.4.c.a 12
28.g odd 6 1 147.4.g.d 12
84.h odd 2 1 147.4.g.d 12
84.j odd 6 1 21.4.g.a ✓ 12
84.j odd 6 1 147.4.c.a 12
84.n even 6 1 147.4.c.a 12
84.n even 6 1 147.4.g.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.g.a ✓ 12 4.b odd 2 1
21.4.g.a ✓ 12 12.b even 2 1
21.4.g.a ✓ 12 28.f even 6 1
21.4.g.a ✓ 12 84.j odd 6 1
147.4.c.a 12 28.f even 6 1
147.4.c.a 12 28.g odd 6 1
147.4.c.a 12 84.j odd 6 1
147.4.c.a 12 84.n even 6 1
147.4.g.d 12 28.d even 2 1
147.4.g.d 12 28.g odd 6 1
147.4.g.d 12 84.h odd 2 1
147.4.g.d 12 84.n even 6 1
336.4.bc.d 12 1.a even 1 1 trivial
336.4.bc.d 12 3.b odd 2 1 inner
336.4.bc.d 12 7.d odd 6 1 inner
336.4.bc.d 12 21.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(336, [\chi])\):

\( T_{5}^{12} + 396T_{5}^{10} + 121221T_{5}^{8} + 13986756T_{5}^{6} + 1245448953T_{5}^{4} + 1937507040T_{5}^{2} + 2962842624 \) Copy content Toggle raw display
\( T_{13}^{6} + 4335T_{13}^{4} + 1731204T_{13}^{2} + 82121472 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 387420489 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 2962842624 \) Copy content Toggle raw display
$7$ \( (T^{6} - 28 T^{5} + \cdots + 40353607)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 45\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( (T^{6} + 4335 T^{4} + \cdots + 82121472)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 17\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( (T^{6} + 150 T^{5} + \cdots + 243972972)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 89\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 14683734245376)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 465 T^{5} + \cdots + 33414175107)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 3418867564324)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 4591113633792)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 253 T^{2} + \cdots + 6662944)^{4} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 38\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 64\!\cdots\!12)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 96\!\cdots\!44)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 6720226523136)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 40\!\cdots\!92)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 363201760969609)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 388952511994368)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 88\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 98\!\cdots\!48)^{2} \) Copy content Toggle raw display
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