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Properties

Label 336.3.d.d
Level $336$
Weight $3$
Character orbit 336.d
Analytic conductor $9.155$
Analytic rank $0$
Dimension $12$
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] = [336,3,Mod(113,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.113");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 336.d (of order \(2\), degree \(1\), 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: \(9.15533688251\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 12 x^{10} + 56 x^{9} + 58 x^{8} + 540 x^{7} + 1594 x^{6} + 3248 x^{5} + 10437 x^{4} + \cdots + 13122 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + (\beta_{9} + \beta_{3}) q^{5} + \beta_1 q^{7} + ( - \beta_{10} - \beta_{6} + \cdots + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + (\beta_{9} + \beta_{3}) q^{5} + \beta_1 q^{7} + ( - \beta_{10} - \beta_{6} + \cdots + \beta_{2}) q^{9}+ \cdots + (3 \beta_{11} + 3 \beta_{10} + \cdots + 24) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{3} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{3} + 12 q^{9} - 32 q^{13} + 56 q^{15} - 24 q^{19} - 12 q^{25} - 116 q^{27} - 16 q^{31} - 16 q^{37} + 8 q^{39} + 112 q^{43} - 72 q^{45} + 84 q^{49} - 40 q^{51} - 16 q^{55} + 96 q^{57} + 208 q^{61} + 56 q^{63} + 64 q^{67} - 152 q^{69} - 136 q^{73} + 92 q^{75} - 128 q^{79} - 132 q^{81} - 552 q^{85} + 40 q^{87} - 56 q^{91} + 296 q^{93} + 584 q^{97} + 192 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 12 x^{10} + 56 x^{9} + 58 x^{8} + 540 x^{7} + 1594 x^{6} + 3248 x^{5} + 10437 x^{4} + \cdots + 13122 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 6102030 \nu^{11} - 14932 \nu^{10} - 107476918 \nu^{9} + 1163728204 \nu^{8} + \cdots - 132004253259 ) / 124545548241 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 215429421643 \nu^{11} - 554468550787 \nu^{10} + 1382968381260 \nu^{9} + \cdots - 43\!\cdots\!48 ) / 17\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 468077717150 \nu^{11} - 3141540878504 \nu^{10} + 12535410489657 \nu^{9} + \cdots + 28\!\cdots\!99 ) / 17\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 200825517743 \nu^{11} + 278386362695 \nu^{10} - 371981474790 \nu^{9} + \cdots + 62\!\cdots\!19 ) / 578763162675927 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 228876215852 \nu^{11} - 1130770044830 \nu^{10} + 4187248194684 \nu^{9} + \cdots - 28\!\cdots\!61 ) / 578763162675927 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 141295592 \nu^{11} + 985343624 \nu^{10} - 3809453316 \nu^{9} - 650600392 \nu^{8} + \cdots - 2269812994488 ) / 324722178423 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 785140940230 \nu^{11} - 2646886107547 \nu^{10} + 5053574543097 \nu^{9} + \cdots + 24\!\cdots\!62 ) / 17\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1053739188850 \nu^{11} + 5814052053712 \nu^{10} - 23061651275955 \nu^{9} + \cdots - 62\!\cdots\!00 ) / 17\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 440022853873 \nu^{11} - 1958120229376 \nu^{10} + 6640472425809 \nu^{9} + \cdots + 11\!\cdots\!03 ) / 578763162675927 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1351030326460 \nu^{11} - 6339321787834 \nu^{10} + 20896591881072 \nu^{9} + \cdots + 42\!\cdots\!92 ) / 17\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1424537600876 \nu^{11} + 7501968208850 \nu^{10} - 27389171626971 \nu^{9} + \cdots + 86\!\cdots\!66 ) / 17\!\cdots\!81 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - \beta_{11} + \beta_{10} - 2 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{4} - 3 \beta_{3} + \cdots + 6 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4 \beta_{11} + 14 \beta_{10} - 12 \beta_{9} - 10 \beta_{8} - 4 \beta_{7} + 7 \beta_{6} + 8 \beta_{5} + \cdots + 6 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8 \beta_{11} + 76 \beta_{10} - 18 \beta_{9} - 20 \beta_{8} - 20 \beta_{7} + 26 \beta_{6} + 25 \beta_{5} + \cdots - 219 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 4 \beta_{11} + 61 \beta_{10} - 6 \beta_{9} + \beta_{8} + 40 \beta_{7} - 37 \beta_{6} - 11 \beta_{5} + \cdots - 726 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 304 \beta_{11} - 725 \beta_{10} + 603 \beta_{9} + 400 \beta_{8} + 859 \beta_{7} - 397 \beta_{6} + \cdots - 5961 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 1244 \beta_{11} - 9076 \beta_{10} + 4470 \beta_{9} + 3506 \beta_{8} + 5330 \beta_{7} - 2027 \beta_{6} + \cdots - 6078 ) / 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1777 \beta_{11} - 46742 \beta_{10} + 17037 \beta_{9} + 16426 \beta_{8} + 15697 \beta_{7} + \cdots + 107880 ) / 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 10748 \beta_{11} - 54533 \beta_{10} + 5376 \beta_{9} + 14683 \beta_{8} - 13934 \beta_{7} - 5788 \beta_{6} + \cdots + 463764 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 190529 \beta_{11} + 429313 \beta_{10} - 357516 \beta_{9} - 218054 \beta_{8} - 631694 \beta_{7} + \cdots + 4051086 ) / 12 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 797968 \beta_{11} + 5959166 \beta_{10} - 2887428 \beta_{9} - 2375638 \beta_{8} - 3848032 \beta_{7} + \cdots + 5605278 ) / 12 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 848234 \beta_{11} + 32249848 \beta_{10} - 11881764 \beta_{9} - 11732972 \beta_{8} - 11437322 \beta_{7} + \cdots - 64494939 ) / 12 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1
3.63757 − 3.56639i
3.63757 + 3.56639i
−2.08107 − 0.688714i
−2.08107 + 0.688714i
−1.55835 + 0.517533i
−1.55835 − 0.517533i
−0.261929 − 2.50063i
−0.261929 + 2.50063i
0.697249 − 0.397698i
0.697249 + 0.397698i
1.56653 + 2.66971i
1.56653 − 2.66971i
0 −2.84763 − 0.943923i 0 6.54693i 0 2.64575 0 7.21802 + 5.37589i 0
113.2 0 −2.84763 + 0.943923i 0 − 6.54693i 0 2.64575 0 7.21802 − 5.37589i 0
113.3 0 −2.48987 − 1.67348i 0 2.14450i 0 −2.64575 0 3.39892 + 8.33351i 0
113.4 0 −2.48987 + 1.67348i 0 − 2.14450i 0 −2.64575 0 3.39892 − 8.33351i 0
113.5 0 0.890713 − 2.86472i 0 8.14934i 0 2.64575 0 −7.41326 − 5.10329i 0
113.6 0 0.890713 + 2.86472i 0 − 8.14934i 0 2.64575 0 −7.41326 + 5.10329i 0
113.7 0 1.56439 − 2.55982i 0 3.00081i 0 −2.64575 0 −4.10535 − 8.00913i 0
113.8 0 1.56439 + 2.55982i 0 − 3.00081i 0 −2.64575 0 −4.10535 + 8.00913i 0
113.9 0 1.92548 − 2.30055i 0 − 5.71377i 0 −2.64575 0 −1.58507 − 8.85932i 0
113.10 0 1.92548 + 2.30055i 0 5.71377i 0 −2.64575 0 −1.58507 + 8.85932i 0
113.11 0 2.95692 − 0.506586i 0 0.689173i 0 2.64575 0 8.48674 − 2.99586i 0
113.12 0 2.95692 + 0.506586i 0 − 0.689173i 0 2.64575 0 8.48674 + 2.99586i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.3.d.d 12
3.b odd 2 1 inner 336.3.d.d 12
4.b odd 2 1 168.3.d.a ✓ 12
8.b even 2 1 1344.3.d.i 12
8.d odd 2 1 1344.3.d.k 12
12.b even 2 1 168.3.d.a ✓ 12
24.f even 2 1 1344.3.d.k 12
24.h odd 2 1 1344.3.d.i 12
28.d even 2 1 1176.3.d.e 12
84.h odd 2 1 1176.3.d.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.3.d.a ✓ 12 4.b odd 2 1
168.3.d.a ✓ 12 12.b even 2 1
336.3.d.d 12 1.a even 1 1 trivial
336.3.d.d 12 3.b odd 2 1 inner
1176.3.d.e 12 28.d even 2 1
1176.3.d.e 12 84.h odd 2 1
1344.3.d.i 12 8.b even 2 1
1344.3.d.i 12 24.h odd 2 1
1344.3.d.k 12 8.d odd 2 1
1344.3.d.k 12 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 156T_{5}^{10} + 8460T_{5}^{8} + 190048T_{5}^{6} + 1618224T_{5}^{4} + 4575168T_{5}^{2} + 1827904 \) acting on \(S_{3}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 4 T^{11} + \cdots + 531441 \) Copy content Toggle raw display
$5$ \( T^{12} + 156 T^{10} + \cdots + 1827904 \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 416551486464 \) Copy content Toggle raw display
$13$ \( (T^{6} + 16 T^{5} + \cdots - 1415088)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 23184918365184 \) Copy content Toggle raw display
$19$ \( (T^{6} + 12 T^{5} + \cdots - 65542144)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 510288582426624 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 8487946174464 \) Copy content Toggle raw display
$31$ \( (T^{6} + 8 T^{5} + \cdots - 33797888)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 8 T^{5} + \cdots - 2100489984)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 13\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( (T^{6} - 56 T^{5} + \cdots + 137654784)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 58\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 32\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( (T^{6} - 104 T^{5} + \cdots - 672025392)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 32 T^{5} + \cdots - 870096384)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 23\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( (T^{6} + 68 T^{5} + \cdots + 218937905216)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 64 T^{5} + \cdots - 35101536256)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 31\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( (T^{6} - 292 T^{5} + \cdots - 290198476224)^{2} \) Copy content Toggle raw display
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