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Properties

Label 338.4.e.h
Level $338$
Weight $4$
Character orbit 338.e
Analytic conductor $19.943$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [338,4,Mod(23,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.23");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 338.e (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: \(19.9426455819\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.17213603549184.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_{6} q^{2} + ( - 3 \beta_{9} - 2 \beta_{7} + \cdots + 2) q^{3}+ \cdots + ( - 21 \beta_{9} + 14 \beta_{7} + \cdots - 21) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_{6} q^{2} + ( - 3 \beta_{9} - 2 \beta_{7} + \cdots + 2) q^{3}+ \cdots + ( - 20 \beta_{11} + \cdots - 20 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 24 q^{3} + 24 q^{4} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 24 q^{3} + 24 q^{4} - 18 q^{9} - 48 q^{10} + 192 q^{12} + 216 q^{14} - 96 q^{16} - 180 q^{17} + 328 q^{22} + 38 q^{23} + 244 q^{25} - 276 q^{27} + 202 q^{29} + 360 q^{30} + 916 q^{35} + 72 q^{36} + 1040 q^{38} - 384 q^{40} + 936 q^{42} - 2410 q^{43} + 384 q^{48} - 368 q^{49} + 828 q^{51} - 4380 q^{53} + 1052 q^{55} + 432 q^{56} + 2216 q^{61} - 2076 q^{62} - 768 q^{64} + 1168 q^{66} + 720 q^{68} - 628 q^{69} + 336 q^{74} - 1010 q^{75} - 1920 q^{77} - 7844 q^{79} - 1830 q^{81} - 748 q^{82} - 3272 q^{87} - 1312 q^{88} + 5160 q^{90} + 304 q^{92} - 2144 q^{94} + 3658 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -25\nu^{11} + 95\nu^{9} - 361\nu^{7} + 155\nu^{5} - 30\nu^{3} - 1563\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 25\nu^{10} - 95\nu^{8} + 361\nu^{6} - 155\nu^{4} + 30\nu^{2} + 1004 ) / 559 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{10} - 20\nu^{8} + 76\nu^{6} - 139\nu^{4} + 124\nu^{2} - 24 ) / 43 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 45\nu^{10} - 171\nu^{8} + 538\nu^{6} - 279\nu^{4} + 54\nu^{2} + 242 ) / 559 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 70\nu^{11} - 266\nu^{9} + 899\nu^{7} - 434\nu^{5} + 84\nu^{3} + 1246\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 114\nu^{10} - 545\nu^{8} + 2071\nu^{6} - 2831\nu^{4} + 3379\nu^{2} - 95 ) / 559 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 114\nu^{11} - 545\nu^{9} + 2071\nu^{7} - 2831\nu^{5} + 3379\nu^{3} - 95\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -128\nu^{10} + 710\nu^{8} - 2698\nu^{6} + 4483\nu^{4} - 4402\nu^{2} + 852 ) / 559 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -242\nu^{11} + 1255\nu^{9} - 4769\nu^{7} + 7314\nu^{5} - 7781\nu^{3} + 1506\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -317\nu^{11} + 1540\nu^{9} - 5852\nu^{7} + 8338\nu^{5} - 9548\nu^{3} + 1848\nu ) / 559 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + \beta_{7} + \beta_{4} - \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + 3\beta_{8} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{9} + 2\beta_{7} + 4\beta_{4} - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{11} - \beta_{10} + 9\beta_{8} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -5\beta_{5} + 9\beta_{3} - 14 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -5\beta_{6} - 14\beta_{2} - 28\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -28\beta_{9} - 14\beta_{7} - 19\beta_{5} - 47\beta_{4} + 28\beta_{3} - 28 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -47\beta_{11} + 19\beta_{10} - 89\beta_{8} - 19\beta_{6} - 47\beta_{2} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -89\beta_{9} - 42\beta_{7} - 155\beta_{4} + 42 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -155\beta_{11} + 66\beta_{10} - 286\beta_{8} + 286\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/338\mathbb{Z}\right)^\times\).

\(n\) \(171\)
\(\chi(n)\) \(\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
−0.385418 + 0.222521i
1.07992 − 0.623490i
−1.56052 + 0.900969i
0.385418 − 0.222521i
−1.07992 + 0.623490i
1.56052 − 0.900969i
−0.385418 − 0.222521i
1.07992 + 0.623490i
−1.56052 − 0.900969i
0.385418 + 0.222521i
−1.07992 − 0.623490i
1.56052 + 0.900969i
−1.73205 + 1.00000i −0.202907 − 0.351445i 2.00000 − 3.46410i 6.36227i 0.702889 + 0.405813i −2.20892 − 1.27532i 8.00000i 13.4177 − 23.2401i −6.36227 − 11.0198i
23.2 −1.73205 + 1.00000i 1.83244 + 3.17387i 2.00000 − 3.46410i − 8.53079i −6.34775 − 3.66487i 3.63821 + 2.10052i 8.00000i 6.78435 − 11.7508i 8.53079 + 14.7758i
23.3 −1.73205 + 1.00000i 4.37047 + 7.56988i 2.00000 − 3.46410i 14.1685i −15.1398 − 8.74094i −24.8120 − 14.3252i 8.00000i −24.7020 + 42.7851i −14.1685 − 24.5406i
23.4 1.73205 − 1.00000i −0.202907 − 0.351445i 2.00000 − 3.46410i − 6.36227i −0.702889 − 0.405813i 2.20892 + 1.27532i − 8.00000i 13.4177 − 23.2401i −6.36227 − 11.0198i
23.5 1.73205 − 1.00000i 1.83244 + 3.17387i 2.00000 − 3.46410i 8.53079i 6.34775 + 3.66487i −3.63821 − 2.10052i − 8.00000i 6.78435 − 11.7508i 8.53079 + 14.7758i
23.6 1.73205 − 1.00000i 4.37047 + 7.56988i 2.00000 − 3.46410i − 14.1685i 15.1398 + 8.74094i 24.8120 + 14.3252i − 8.00000i −24.7020 + 42.7851i −14.1685 − 24.5406i
147.1 −1.73205 − 1.00000i −0.202907 + 0.351445i 2.00000 + 3.46410i − 6.36227i 0.702889 − 0.405813i −2.20892 + 1.27532i − 8.00000i 13.4177 + 23.2401i −6.36227 + 11.0198i
147.2 −1.73205 − 1.00000i 1.83244 − 3.17387i 2.00000 + 3.46410i 8.53079i −6.34775 + 3.66487i 3.63821 − 2.10052i − 8.00000i 6.78435 + 11.7508i 8.53079 − 14.7758i
147.3 −1.73205 − 1.00000i 4.37047 − 7.56988i 2.00000 + 3.46410i − 14.1685i −15.1398 + 8.74094i −24.8120 + 14.3252i − 8.00000i −24.7020 − 42.7851i −14.1685 + 24.5406i
147.4 1.73205 + 1.00000i −0.202907 + 0.351445i 2.00000 + 3.46410i 6.36227i −0.702889 + 0.405813i 2.20892 − 1.27532i 8.00000i 13.4177 + 23.2401i −6.36227 + 11.0198i
147.5 1.73205 + 1.00000i 1.83244 − 3.17387i 2.00000 + 3.46410i − 8.53079i 6.34775 − 3.66487i −3.63821 + 2.10052i 8.00000i 6.78435 + 11.7508i 8.53079 − 14.7758i
147.6 1.73205 + 1.00000i 4.37047 − 7.56988i 2.00000 + 3.46410i 14.1685i 15.1398 − 8.74094i 24.8120 − 14.3252i 8.00000i −24.7020 − 42.7851i −14.1685 + 24.5406i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 338.4.e.h 12
13.b even 2 1 inner 338.4.e.h 12
13.c even 3 1 338.4.b.f 6
13.c even 3 1 inner 338.4.e.h 12
13.d odd 4 1 338.4.c.k 6
13.d odd 4 1 338.4.c.l 6
13.e even 6 1 338.4.b.f 6
13.e even 6 1 inner 338.4.e.h 12
13.f odd 12 1 338.4.a.j ✓ 3
13.f odd 12 1 338.4.a.k yes 3
13.f odd 12 1 338.4.c.k 6
13.f odd 12 1 338.4.c.l 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
338.4.a.j ✓ 3 13.f odd 12 1
338.4.a.k yes 3 13.f odd 12 1
338.4.b.f 6 13.c even 3 1
338.4.b.f 6 13.e even 6 1
338.4.c.k 6 13.d odd 4 1
338.4.c.k 6 13.f odd 12 1
338.4.c.l 6 13.d odd 4 1
338.4.c.l 6 13.f odd 12 1
338.4.e.h 12 1.a even 1 1 trivial
338.4.e.h 12 13.b even 2 1 inner
338.4.e.h 12 13.c even 3 1 inner
338.4.e.h 12 13.e 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 \(S_{4}^{\mathrm{new}}(338, [\chi])\):

\( T_{3}^{6} - 12T_{3}^{5} + 117T_{3}^{4} - 350T_{3}^{3} + 885T_{3}^{2} + 351T_{3} + 169 \) Copy content Toggle raw display
\( T_{5}^{6} + 314T_{5}^{4} + 25681T_{5}^{2} + 591361 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 4 T^{2} + 16)^{3} \) Copy content Toggle raw display
$3$ \( (T^{6} - 12 T^{5} + \cdots + 169)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} + 314 T^{4} + \cdots + 591361)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 8882874001 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 69\!\cdots\!41 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( (T^{6} + 90 T^{5} + \cdots + 5954745889)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 34\!\cdots\!61 \) Copy content Toggle raw display
$23$ \( (T^{6} - 19 T^{5} + \cdots + 365005680649)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 13714793815801)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 11573311429849)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 91\!\cdots\!21 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 16\!\cdots\!81 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 41\!\cdots\!09)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 682686545449441)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} + 1095 T^{2} + \cdots + 45870749)^{4} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 28\!\cdots\!81 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 805285400757601)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 14\!\cdots\!01 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 32\!\cdots\!81 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 31\!\cdots\!69)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 1961 T^{2} + \cdots + 214064899)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 899211352843609)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 13\!\cdots\!01 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 27\!\cdots\!61 \) Copy content Toggle raw display
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