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Properties

Label 336.7.d.c
Level $336$
Weight $7$
Character orbit 336.d
Analytic conductor $77.298$
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] = [336,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.113");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
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: \(77.2981720963\)
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: \( x^{12} + 532x^{10} + 74137x^{8} + 4103612x^{6} + 100648268x^{4} + 983303384x^{2} + 1835779716 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{10}\cdot 7^{6} \)
Twist minimal: no (minimal twist has level 42)
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.

\(f(q)\) \(=\) \( q + (\beta_1 + 7) q^{3} + ( - \beta_{7} - \beta_{3} - \beta_1) q^{5} + ( - \beta_{2} + \beta_1) q^{7} + ( - \beta_{11} + 2 \beta_{9} + \cdots - 86) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 7) q^{3} + ( - \beta_{7} - \beta_{3} - \beta_1) q^{5} + ( - \beta_{2} + \beta_1) q^{7} + ( - \beta_{11} + 2 \beta_{9} + \cdots - 86) q^{9}+ \cdots + (2751 \beta_{11} - 197 \beta_{10} + \cdots + 325867) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 84 q^{3} - 1012 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 84 q^{3} - 1012 q^{9} + 1440 q^{13} + 3736 q^{15} + 11400 q^{19} + 10976 q^{21} + 6132 q^{25} + 72828 q^{27} + 65328 q^{31} + 62656 q^{33} - 182160 q^{37} - 5336 q^{39} + 209328 q^{43} - 247688 q^{45} + 201684 q^{49} - 484168 q^{51} - 907728 q^{55} - 448032 q^{57} - 146160 q^{61} + 57624 q^{63} + 657600 q^{67} - 920984 q^{69} - 199176 q^{73} - 1584884 q^{75} - 99456 q^{79} + 271484 q^{81} + 1772952 q^{85} - 3538040 q^{87} - 2033304 q^{91} + 23912 q^{93} - 1694136 q^{97} + 3919552 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 532x^{10} + 74137x^{8} + 4103612x^{6} + 100648268x^{4} + 983303384x^{2} + 1835779716 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 529666689486 \nu^{11} - 39695648290178 \nu^{10} - 304634313057903 \nu^{9} + \cdots + 40\!\cdots\!68 ) / 93\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 529666689486 \nu^{11} + 938303053717394 \nu^{10} - 304634313057903 \nu^{9} + \cdots + 49\!\cdots\!96 ) / 93\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 204905672 \nu^{11} + 103568818246 \nu^{9} + 12424440435780 \nu^{7} + \cdots - 20\!\cdots\!92 \nu ) / 659527432664715 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 16206676772566 \nu^{11} - 189648519118598 \nu^{10} + \cdots - 15\!\cdots\!12 ) / 31\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 678195546829 \nu^{11} + 9075932281366 \nu^{10} + 347381055217967 \nu^{9} + \cdots + 77\!\cdots\!64 ) / 12\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 24692345115836 \nu^{11} - 44209520649576 \nu^{10} + \cdots - 56\!\cdots\!44 ) / 23\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 100425816975820 \nu^{11} + 39695648290178 \nu^{10} + \cdots - 40\!\cdots\!68 ) / 93\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 28344593962009 \nu^{11} - 187752571350360 \nu^{10} + \cdots - 14\!\cdots\!50 ) / 23\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 87464080899364 \nu^{11} + 9027744718796 \nu^{10} + \cdots + 12\!\cdots\!24 ) / 46\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 180067256772169 \nu^{11} - 259574186731100 \nu^{10} + \cdots - 47\!\cdots\!60 ) / 93\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 241132495874995 \nu^{11} + 190637575868394 \nu^{10} + \cdots + 18\!\cdots\!16 ) / 93\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 69 \beta_{11} - 23 \beta_{10} - 46 \beta_{9} - 173 \beta_{8} - 69 \beta_{6} - 196 \beta_{5} + \cdots - 23 ) / 3528 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 63 \beta_{11} - 707 \beta_{10} - 504 \beta_{9} - 721 \beta_{8} + 259 \beta_{6} - 168 \beta_{5} + \cdots - 155687 ) / 1764 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 24546 \beta_{11} + 3422 \beta_{10} + 15356 \beta_{9} + 39558 \beta_{8} - 16632 \beta_{7} + \cdots + 10562 ) / 3528 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 3675 \beta_{11} + 48069 \beta_{10} + 33908 \beta_{9} + 46060 \beta_{8} - 13377 \beta_{6} + \cdots + 6561835 ) / 294 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 4215261 \beta_{11} - 574467 \beta_{10} - 2887706 \beta_{9} - 6277175 \beta_{8} + 3297420 \beta_{7} + \cdots - 1820397 ) / 1764 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 4387572 \beta_{11} - 52348142 \beta_{10} - 36059604 \beta_{9} - 48881455 \beta_{8} + 13000204 \beta_{6} + \cdots - 6507582998 ) / 882 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 210896556 \beta_{11} + 29543816 \beta_{10} + 148874560 \beta_{9} + 308174474 \beta_{8} + \cdots + 90676370 ) / 252 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1609270425 \beta_{11} + 18576782483 \beta_{10} + 12700840698 \beta_{9} + 17214926932 \beta_{8} + \cdots + 2262940394351 ) / 882 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 259775855601 \beta_{11} - 36747900757 \beta_{10} - 184875244276 \beta_{9} - 377810912394 \beta_{8} + \cdots - 111513977422 ) / 882 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 191126251995 \beta_{11} - 2186106582401 \beta_{10} - 1491655201414 \beta_{9} - 2021740134162 \beta_{8} + \cdots - 265033587359527 ) / 294 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 91526824956468 \beta_{11} + 12982438381446 \beta_{10} + 65273523442460 \beta_{9} + \cdots + 39272193287511 ) / 882 \) 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
9.39357i
− 9.39357i
− 5.38801i
5.38801i
− 4.78309i
4.78309i
− 6.08251i
6.08251i
− 18.7737i
18.7737i
− 1.54992i
1.54992i
0 −17.7163 − 20.3748i 0 25.2202i 0 −129.642 0 −101.264 + 721.933i 0
113.2 0 −17.7163 + 20.3748i 0 − 25.2202i 0 −129.642 0 −101.264 − 721.933i 0
113.3 0 −6.11719 − 26.2979i 0 156.019i 0 −129.642 0 −654.160 + 321.739i 0
113.4 0 −6.11719 + 26.2979i 0 − 156.019i 0 −129.642 0 −654.160 − 321.739i 0
113.5 0 −1.87668 − 26.9347i 0 − 33.0748i 0 129.642 0 −721.956 + 101.095i 0
113.6 0 −1.87668 + 26.9347i 0 33.0748i 0 129.642 0 −721.956 − 101.095i 0
113.7 0 17.0635 − 20.9245i 0 − 111.171i 0 129.642 0 −146.671 − 714.093i 0
113.8 0 17.0635 + 20.9245i 0 111.171i 0 129.642 0 −146.671 + 714.093i 0
113.9 0 23.6675 − 12.9942i 0 54.1650i 0 −129.642 0 391.302 − 615.080i 0
113.10 0 23.6675 + 12.9942i 0 − 54.1650i 0 −129.642 0 391.302 + 615.080i 0
113.11 0 26.9791 − 1.06089i 0 − 222.080i 0 129.642 0 726.749 − 57.2438i 0
113.12 0 26.9791 + 1.06089i 0 222.080i 0 129.642 0 726.749 + 57.2438i 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 336.7.d.c 12
3.b odd 2 1 inner 336.7.d.c 12
4.b odd 2 1 42.7.b.a ✓ 12
12.b even 2 1 42.7.b.a ✓ 12
28.d even 2 1 294.7.b.c 12
84.h odd 2 1 294.7.b.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.7.b.a ✓ 12 4.b odd 2 1
42.7.b.a ✓ 12 12.b even 2 1
294.7.b.c 12 28.d even 2 1
294.7.b.c 12 84.h odd 2 1
336.7.d.c 12 1.a even 1 1 trivial
336.7.d.c 12 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 90684 T_{5}^{10} + 2517855756 T_{5}^{8} + 25180615215200 T_{5}^{6} + \cdots + 30\!\cdots\!00 \) acting on \(S_{7}^{\mathrm{new}}(336, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 15\!\cdots\!21 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{2} - 16807)^{6} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 22\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 77\!\cdots\!04)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots - 14\!\cdots\!96)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 48\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 46\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 29\!\cdots\!64)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots - 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 27\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 88\!\cdots\!64)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 98\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 29\!\cdots\!88)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 10\!\cdots\!48)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 30\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 28\!\cdots\!24)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 47\!\cdots\!92)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 25\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 30\!\cdots\!04)^{2} \) Copy content Toggle raw display
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