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Properties

Label 336.3.z.e
Level $336$
Weight $3$
Character orbit 336.z
Analytic conductor $9.155$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [336,3,Mod(47,336)]
 
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, 3, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.47");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 336.z (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: \(9.15533688251\)
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: \( x^{16} - 17x^{14} + 116x^{12} - 1797x^{10} + 19529x^{8} - 75972x^{6} + 210236x^{4} - 386672x^{2} + 456976 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{4} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{9} + \beta_{7}) q^{3} + \beta_{8} q^{5} + ( - 2 \beta_{13} + \beta_{9}) q^{7} + ( - \beta_{6} - \beta_{4} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{9} + \beta_{7}) q^{3} + \beta_{8} q^{5} + ( - 2 \beta_{13} + \beta_{9}) q^{7} + ( - \beta_{6} - \beta_{4} + \cdots + \beta_1) q^{9}+ \cdots + ( - 2 \beta_{14} - 8 \beta_{13} + \cdots + 15 \beta_{7}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 14 q^{9} - 38 q^{21} - 88 q^{25} - 30 q^{33} + 220 q^{37} + 318 q^{45} + 52 q^{49} - 220 q^{57} + 204 q^{61} - 900 q^{73} + 14 q^{81} + 120 q^{85} - 170 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 17x^{14} + 116x^{12} - 1797x^{10} + 19529x^{8} - 75972x^{6} + 210236x^{4} - 386672x^{2} + 456976 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1159 \nu^{14} - 115026 \nu^{12} + 1280396 \nu^{10} - 8154743 \nu^{8} + 163701682 \nu^{6} + \cdots - 6989231600 ) / 792728976 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 571271 \nu^{14} - 8472408 \nu^{12} + 51370060 \nu^{10} - 947637727 \nu^{8} + \cdots - 177082383088 ) / 103054766880 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 4235785 \nu^{14} + 33647932 \nu^{12} + 25679220 \nu^{10} + 5107072865 \nu^{8} + \cdots + 185514715152 ) / 309164300640 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10016107 \nu^{14} - 102224656 \nu^{12} + 162758820 \nu^{10} - 12434681939 \nu^{8} + \cdots + 1549131661584 ) / 309164300640 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5300261 \nu^{14} + 82965916 \nu^{12} - 485371700 \nu^{10} + 8424516037 \nu^{8} + \cdots + 859260514736 ) / 154582150320 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5601727 \nu^{14} + 86599296 \nu^{12} - 513680660 \nu^{10} + 9127082279 \nu^{8} + \cdots + 475333182896 ) / 103054766880 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9648029 \nu^{15} - 180908238 \nu^{13} + 1490928880 \nu^{11} - 20257857493 \nu^{9} + \cdots - 9315862960768 \nu ) / 6028703862480 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 26373751 \nu^{14} - 393154220 \nu^{12} + 2155254100 \nu^{10} - 41418311327 \nu^{8} + \cdots - 2214023234800 ) / 309164300640 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 20201191 \nu^{15} + 80134809 \nu^{13} + 1348208950 \nu^{11} + 17686055027 \nu^{9} + \cdots - 6485301380896 \nu ) / 6028703862480 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 23936417 \nu^{15} + 380167728 \nu^{13} - 2305039960 \nu^{11} + 39465340129 \nu^{9} + \cdots - 2706990523952 \nu ) / 6028703862480 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3421 \nu^{15} + 76458 \nu^{13} - 642680 \nu^{11} + 7083917 \nu^{9} - 92703946 \nu^{7} + \cdots + 1349477168 \nu ) / 749792160 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 70559297 \nu^{15} - 915085989 \nu^{13} + 3714282400 \nu^{11} - 100377938449 \nu^{9} + \cdots - 810079755784 \nu ) / 6028703862480 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 148919231 \nu^{15} + 2561118012 \nu^{13} - 16316703280 \nu^{11} + \cdots + 25790702727712 \nu ) / 12057407724960 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 3192713 \nu^{15} + 58253378 \nu^{13} - 425883030 \nu^{11} + 6172916101 \nu^{9} + \cdots + 2830131647928 \nu ) / 154582150320 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 244178549 \nu^{15} + 3675293794 \nu^{13} - 21339267920 \nu^{11} + \cdots + 1937998200944 \nu ) / 4019135908320 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{13} + \beta_{12} + \beta_{11} - 3\beta_{10} + 2\beta_{9} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{8} + \beta_{6} - \beta_{5} + 2\beta_{4} + 2\beta_{3} + 12\beta_{2} + \beta _1 + 12 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 11\beta_{13} + 17\beta_{12} + 5\beta_{11} - 6\beta_{10} + 22\beta_{9} + 6\beta_{7} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{8} - 8\beta_{6} - 5\beta_{5} + 3\beta_{4} - 3\beta_{3} - 12\beta_{2} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -11\beta_{15} - 22\beta_{14} + 168\beta_{13} + 168\beta_{12} - 9\beta_{11} + 216\beta_{9} - 207\beta_{7} ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -56\beta_{8} + 65\beta_{6} - 209\beta_{5} - 56\beta_{4} - 209\beta_{3} + 65\beta _1 + 1440 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 161 \beta_{15} - 161 \beta_{14} + 1353 \beta_{13} + 1500 \beta_{12} + 300 \beta_{11} + \cdots + 1653 \beta_{9} ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -147\beta_{8} + 843\beta_{6} - 843\beta_{5} + 504\beta_{4} + 852\beta_{3} + 4472\beta_{2} + 9\beta _1 + 4472 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1668 \beta_{15} + 834 \beta_{14} + 14857 \beta_{13} + 23785 \beta_{12} - 1529 \beta_{11} + \cdots + 16386 \beta_{7} ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -28552\beta_{8} - 23564\beta_{6} - 24979\beta_{5} + 8561\beta_{4} + 1415\beta_{3} + 7656\beta_{2} - 1415\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 7383 \beta_{15} - 14766 \beta_{14} + 245228 \beta_{13} + 245228 \beta_{12} - 99289 \beta_{11} + \cdots - 144447 \beta_{7} ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -44224\beta_{8} - 9925\beta_{6} - 97531\beta_{5} - 44224\beta_{4} - 97531\beta_{3} - 9925\beta _1 + 463824 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 185137 \beta_{15} - 185137 \beta_{14} + 2235585 \beta_{13} + 1994964 \beta_{12} + \cdots + 1631829 \beta_{9} ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 237761 \beta_{8} + 3451337 \beta_{6} - 3451337 \beta_{5} + 790312 \beta_{4} + 2397100 \beta_{3} + \cdots + 18279000 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 3003716 \beta_{15} + 1501858 \beta_{14} + 20618433 \beta_{13} + 28439313 \beta_{12} + \cdots + 29424786 \beta_{7} ) / 3 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
3.23994 + 0.184247i
1.15687 + 1.06326i
1.77953 + 2.71374i
1.49924 + 0.470253i
−1.49924 − 0.470253i
−1.77953 − 2.71374i
−1.15687 − 1.06326i
−3.23994 − 0.184247i
3.23994 − 0.184247i
1.15687 − 1.06326i
1.77953 − 2.71374i
1.49924 − 0.470253i
−1.49924 + 0.470253i
−1.77953 + 2.71374i
−1.15687 + 1.06326i
−3.23994 + 0.184247i
0 −2.89756 + 0.777251i 0 −1.71852 + 2.97656i 0 4.12977 − 5.65199i 0 7.79176 − 4.50427i 0
47.2 0 −2.61728 − 1.46624i 0 3.87901 − 6.71864i 0 5.93254 + 3.71550i 0 4.70029 + 7.67511i 0
47.3 0 −2.12190 + 2.12074i 0 1.71852 − 2.97656i 0 −4.12977 + 5.65199i 0 0.00493169 − 9.00000i 0
47.4 0 −0.0388393 + 2.99975i 0 −3.87901 + 6.71864i 0 −5.93254 − 3.71550i 0 −8.99698 − 0.233016i 0
47.5 0 0.0388393 − 2.99975i 0 −3.87901 + 6.71864i 0 5.93254 + 3.71550i 0 −8.99698 − 0.233016i 0
47.6 0 2.12190 − 2.12074i 0 1.71852 − 2.97656i 0 4.12977 − 5.65199i 0 0.00493169 − 9.00000i 0
47.7 0 2.61728 + 1.46624i 0 3.87901 − 6.71864i 0 −5.93254 − 3.71550i 0 4.70029 + 7.67511i 0
47.8 0 2.89756 − 0.777251i 0 −1.71852 + 2.97656i 0 −4.12977 + 5.65199i 0 7.79176 − 4.50427i 0
143.1 0 −2.89756 − 0.777251i 0 −1.71852 − 2.97656i 0 4.12977 + 5.65199i 0 7.79176 + 4.50427i 0
143.2 0 −2.61728 + 1.46624i 0 3.87901 + 6.71864i 0 5.93254 − 3.71550i 0 4.70029 − 7.67511i 0
143.3 0 −2.12190 − 2.12074i 0 1.71852 + 2.97656i 0 −4.12977 − 5.65199i 0 0.00493169 + 9.00000i 0
143.4 0 −0.0388393 − 2.99975i 0 −3.87901 − 6.71864i 0 −5.93254 + 3.71550i 0 −8.99698 + 0.233016i 0
143.5 0 0.0388393 + 2.99975i 0 −3.87901 − 6.71864i 0 5.93254 − 3.71550i 0 −8.99698 + 0.233016i 0
143.6 0 2.12190 + 2.12074i 0 1.71852 + 2.97656i 0 4.12977 + 5.65199i 0 0.00493169 + 9.00000i 0
143.7 0 2.61728 − 1.46624i 0 3.87901 + 6.71864i 0 −5.93254 + 3.71550i 0 4.70029 − 7.67511i 0
143.8 0 2.89756 + 0.777251i 0 −1.71852 − 2.97656i 0 −4.12977 − 5.65199i 0 7.79176 + 4.50427i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
7.d odd 6 1 inner
12.b even 2 1 inner
21.g even 6 1 inner
28.f even 6 1 inner
84.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.3.z.e ✓ 16
3.b odd 2 1 inner 336.3.z.e ✓ 16
4.b odd 2 1 inner 336.3.z.e ✓ 16
7.d odd 6 1 inner 336.3.z.e ✓ 16
12.b even 2 1 inner 336.3.z.e ✓ 16
21.g even 6 1 inner 336.3.z.e ✓ 16
28.f even 6 1 inner 336.3.z.e ✓ 16
84.j odd 6 1 inner 336.3.z.e ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.3.z.e ✓ 16 1.a even 1 1 trivial
336.3.z.e ✓ 16 3.b odd 2 1 inner
336.3.z.e ✓ 16 4.b odd 2 1 inner
336.3.z.e ✓ 16 7.d odd 6 1 inner
336.3.z.e ✓ 16 12.b even 2 1 inner
336.3.z.e ✓ 16 21.g even 6 1 inner
336.3.z.e ✓ 16 28.f even 6 1 inner
336.3.z.e ✓ 16 84.j 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 \(S_{3}^{\mathrm{new}}(336, [\chi])\):

\( T_{5}^{8} + 72T_{5}^{6} + 4473T_{5}^{4} + 51192T_{5}^{2} + 505521 \) Copy content Toggle raw display
\( T_{13}^{4} + 540T_{13}^{2} + 14400 \) Copy content Toggle raw display
\( T_{19}^{8} + 185T_{19}^{6} + 29325T_{19}^{4} + 906500T_{19}^{2} + 24010000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 7 T^{14} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( (T^{8} + 72 T^{6} + \cdots + 505521)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} - 13 T^{6} + \cdots + 5764801)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 360 T^{6} + \cdots + 315950625)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 540 T^{2} + 14400)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 645 T^{6} + \cdots + 5055210000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 185 T^{6} + \cdots + 24010000)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 129 T^{6} + \cdots + 8088336)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 2061 T^{2} + 102384)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 2480703750625)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 55 T^{3} + \cdots + 372100)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 3972 T^{2} + 2912256)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 1728 T^{2} + 147456)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} - 4887 T^{6} + \cdots + 167719735296)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 15995000390625)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 167909116100625)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 51 T^{3} + \cdots + 13927824)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} - 2643 T^{6} + \cdots + 17944209936)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 17460 T^{2} + 4550400)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 225 T^{3} + \cdots + 14288400)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 100344796700625)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 20673 T^{2} + 104841216)^{4} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 13\!\cdots\!16)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 1035 T^{2} + 176400)^{4} \) Copy content Toggle raw display
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