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Properties

Label 1344.4.c.f
Level $1344$
Weight $4$
Character orbit 1344.c
Analytic conductor $79.299$
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] = [1344,4,Mod(673,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.673");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.c (of order \(2\), degree \(1\), 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: \(79.2985670477\)
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: \( x^{12} - 2 x^{11} - x^{10} - 861 x^{8} - 2158 x^{7} + 8654 x^{6} + 118244 x^{5} + 707300 x^{4} + \cdots + 43264 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - 3 \beta_{2} q^{3} + (\beta_{11} + \beta_{2}) q^{5} - 7 q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 \beta_{2} q^{3} + (\beta_{11} + \beta_{2}) q^{5} - 7 q^{7} - 9 q^{9} + (2 \beta_{11} + \beta_{7} - 4 \beta_{2}) q^{11} + ( - \beta_{11} + \beta_{10} + \cdots - 14 \beta_{2}) q^{13}+ \cdots + ( - 18 \beta_{11} + \cdots + 36 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 84 q^{7} - 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 84 q^{7} - 108 q^{9} + 24 q^{15} + 376 q^{17} - 336 q^{23} - 180 q^{25} + 192 q^{31} - 168 q^{33} - 504 q^{39} + 488 q^{41} - 448 q^{47} + 588 q^{49} - 3600 q^{55} - 432 q^{57} + 756 q^{63} + 1408 q^{65} - 5104 q^{71} - 1752 q^{73} - 1632 q^{79} + 972 q^{81} - 336 q^{87} - 3688 q^{89} - 2496 q^{95} - 1944 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} - x^{10} - 861 x^{8} - 2158 x^{7} + 8654 x^{6} + 118244 x^{5} + 707300 x^{4} + \cdots + 43264 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 24\!\cdots\!81 \nu^{11} + \cdots + 82\!\cdots\!20 ) / 51\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 15\!\cdots\!91 \nu^{11} + \cdots - 12\!\cdots\!36 ) / 21\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 21\!\cdots\!07 \nu^{11} + \cdots - 25\!\cdots\!68 ) / 15\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 28327139191123 \nu^{11} - 10132937675420 \nu^{10} + 291510454010223 \nu^{9} + \cdots - 64\!\cdots\!64 ) / 97\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 263835326032171 \nu^{11} - 754400875389294 \nu^{10} + 205725021722285 \nu^{9} + \cdots - 10\!\cdots\!56 ) / 83\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 48\!\cdots\!21 \nu^{11} + \cdots + 51\!\cdots\!76 ) / 15\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 26\!\cdots\!43 \nu^{11} + \cdots + 76\!\cdots\!80 ) / 65\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 40\!\cdots\!89 \nu^{11} + \cdots + 15\!\cdots\!88 ) / 65\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 16\!\cdots\!37 \nu^{11} + \cdots - 18\!\cdots\!08 ) / 19\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 14\!\cdots\!95 \nu^{11} + \cdots + 44\!\cdots\!60 ) / 14\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 10\!\cdots\!45 \nu^{11} + \cdots - 93\!\cdots\!36 ) / 10\!\cdots\!72 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 8 \beta_{11} - 4 \beta_{10} + 2 \beta_{9} + 4 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} + \beta_{5} + \cdots + 6 ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 14 \beta_{11} - \beta_{10} - \beta_{9} + 9 \beta_{8} - 2 \beta_{7} - 6 \beta_{6} + 27 \beta_{5} + \cdots + 18 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 74 \beta_{11} + 31 \beta_{10} - 56 \beta_{9} + 37 \beta_{8} - 37 \beta_{7} - 31 \beta_{6} + 82 \beta_{5} + \cdots + 63 ) / 24 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 158 \beta_{11} + 79 \beta_{10} - 75 \beta_{9} + 43 \beta_{8} - 112 \beta_{7} + 32 \beta_{6} + \cdots + 2248 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2698 \beta_{11} - 749 \beta_{10} - 1073 \beta_{9} + 599 \beta_{8} - 2122 \beta_{7} + 1924 \beta_{6} + \cdots + 36540 ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -20518\beta_{11} - 4715\beta_{10} - 1235\beta_{9} - 5746\beta_{7} - 2770\beta_{4} + 235917\beta_{2} ) / 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 105796 \beta_{11} - 23048 \beta_{10} - 39929 \beta_{9} - 19370 \beta_{8} - 80425 \beta_{7} + \cdots - 1510923 ) / 24 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 234362 \beta_{11} + 112369 \beta_{10} - 144041 \beta_{9} - 78557 \beta_{8} - 161212 \beta_{7} + \cdots - 2906740 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3585422 \beta_{11} + 1697149 \beta_{10} - 2359082 \beta_{9} - 2409673 \beta_{8} - 2409673 \beta_{7} + \cdots - 7015461 ) / 24 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 29818874 \beta_{11} - 7736629 \beta_{10} - 1980361 \beta_{9} - 18708813 \beta_{8} - 9394298 \beta_{7} + \cdots - 54979650 ) / 24 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 292246246 \beta_{11} - 93014579 \beta_{10} + 29706397 \beta_{9} - 120706703 \beta_{8} + \cdots - 2613730860 ) / 24 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
673.1
1.59395 + 5.94871i
6.03643 + 1.61746i
−3.79295 − 1.01632i
−1.04508 − 3.90029i
−1.74348 − 0.467165i
−0.0488747 − 0.182403i
−0.0488747 + 0.182403i
−1.74348 + 0.467165i
−1.04508 + 3.90029i
−3.79295 + 1.01632i
6.03643 − 1.61746i
1.59395 − 5.94871i
0 − 3.00000i 0 − 17.4720i 0 −7.00000 0 −9.00000 0
673.2 0 − 3.00000i 0 − 6.24978i 0 −7.00000 0 −9.00000 0
673.3 0 − 3.00000i 0 − 2.27324i 0 −7.00000 0 −9.00000 0
673.4 0 − 3.00000i 0 − 1.12662i 0 −7.00000 0 −9.00000 0
673.5 0 − 3.00000i 0 13.9871i 0 −7.00000 0 −9.00000 0
673.6 0 − 3.00000i 0 17.1345i 0 −7.00000 0 −9.00000 0
673.7 0 3.00000i 0 − 17.1345i 0 −7.00000 0 −9.00000 0
673.8 0 3.00000i 0 − 13.9871i 0 −7.00000 0 −9.00000 0
673.9 0 3.00000i 0 1.12662i 0 −7.00000 0 −9.00000 0
673.10 0 3.00000i 0 2.27324i 0 −7.00000 0 −9.00000 0
673.11 0 3.00000i 0 6.24978i 0 −7.00000 0 −9.00000 0
673.12 0 3.00000i 0 17.4720i 0 −7.00000 0 −9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 673.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.4.c.f ✓ 12
4.b odd 2 1 1344.4.c.g yes 12
8.b even 2 1 inner 1344.4.c.f ✓ 12
8.d odd 2 1 1344.4.c.g yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1344.4.c.f ✓ 12 1.a even 1 1 trivial
1344.4.c.f ✓ 12 8.b even 2 1 inner
1344.4.c.g yes 12 4.b odd 2 1
1344.4.c.g yes 12 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1344, [\chi])\):

\( T_{5}^{12} + 840T_{5}^{10} + 243192T_{5}^{8} + 27147584T_{5}^{6} + 851300112T_{5}^{4} + 4576494720T_{5}^{2} + 4492216576 \) Copy content Toggle raw display
\( T_{23}^{6} + 168T_{23}^{5} - 23688T_{23}^{4} - 4379760T_{23}^{3} - 39519900T_{23}^{2} + 11108216448T_{23} + 157604189184 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 4492216576 \) Copy content Toggle raw display
$7$ \( (T + 7)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 63360835842304 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( (T^{6} - 188 T^{5} + \cdots + 3823312)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( (T^{6} + 168 T^{5} + \cdots + 157604189184)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 16\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 3105857137664)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 52\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 1874674085200)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 68\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 2057937719296)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 53\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 46\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 20\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 22\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 270319500291328)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 903469003040448)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 15\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 32\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 432185216528816)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 11\!\cdots\!68)^{2} \) Copy content Toggle raw display
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