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Properties

Label 399.2.k.c
Level $399$
Weight $2$
Character orbit 399.k
Analytic conductor $3.186$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [399,2,Mod(64,399)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(399, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("399.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 399 = 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 399.k (of order \(3\), 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: \(3.18603104065\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + 11 x^{10} - 10 x^{9} + 90 x^{8} - 79 x^{7} + 275 x^{6} - 177 x^{5} + 560 x^{4} + \cdots + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{7} q^{3} + ( - \beta_{10} + \beta_{7} - 1) q^{4} + ( - \beta_{11} + \beta_{4}) q^{5} + (\beta_{2} + \beta_1) q^{6} - q^{7} + (\beta_{9} - \beta_{8} - 2 \beta_{4} + \cdots - 1) q^{8}+ \cdots + ( - \beta_{9} + \beta_{7} - \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{2} - 6 q^{3} - 9 q^{4} - q^{6} - 12 q^{7} - 6 q^{8} - 6 q^{9} - 4 q^{10} + 8 q^{11} + 18 q^{12} - 2 q^{13} + q^{14} - 19 q^{16} + 3 q^{17} + 2 q^{18} + 3 q^{19} - 26 q^{20} + 6 q^{21} + 2 q^{22}+ \cdots - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 11 x^{10} - 10 x^{9} + 90 x^{8} - 79 x^{7} + 275 x^{6} - 177 x^{5} + 560 x^{4} + \cdots + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5576235 \nu^{11} + 4006708 \nu^{10} + 52642890 \nu^{9} + 38804014 \nu^{8} + 439627424 \nu^{7} + \cdots + 566549244 ) / 3919460469 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9582943 \nu^{11} - 8695695 \nu^{10} + 94566364 \nu^{9} - 62233726 \nu^{8} + 755727611 \nu^{7} + \cdots + 15627655761 ) / 3919460469 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 45555061 \nu^{11} + 74797351 \nu^{10} - 419624797 \nu^{9} + 572692130 \nu^{8} + \cdots - 42131100258 ) / 11758381407 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 88687263 \nu^{11} + 62160086 \nu^{10} + 1066737554 \nu^{9} + 801080883 \nu^{8} + \cdots + 41824379079 ) / 11758381407 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 149438023 \nu^{11} + 3720477 \nu^{10} - 1301282363 \nu^{9} + 76165883 \nu^{8} + \cdots - 35591284287 ) / 11758381407 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 62949916 \nu^{11} + 68526151 \nu^{10} - 688442368 \nu^{9} + 682142050 \nu^{8} + \cdots + 550554462 ) / 3919460469 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 296159965 \nu^{11} - 421316621 \nu^{10} + 3495458979 \nu^{9} - 4062702137 \nu^{8} + \cdots + 20612935080 ) / 11758381407 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 317442197 \nu^{11} - 237708384 \nu^{10} + 3729551827 \nu^{9} - 2521958845 \nu^{8} + \cdots + 3490321914 ) / 11758381407 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 62949916 \nu^{11} + 68526151 \nu^{10} - 688442368 \nu^{9} + 682142050 \nu^{8} + \cdots - 3368906007 ) / 1306486823 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 693732367 \nu^{11} + 743763428 \nu^{10} - 7473534885 \nu^{9} + 7455115610 \nu^{8} + \cdots - 36576464796 ) / 11758381407 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{10} + 3\beta_{7} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} + \beta_{8} + 2\beta_{4} + \beta_{3} + 5\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{10} + 2\beta_{9} + \beta_{8} - 18\beta_{7} - \beta_{6} - 2\beta_{5} + 8\beta_{3} - \beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 19 \beta_{11} + 9 \beta_{10} - \beta_{9} - 10 \beta_{8} + 19 \beta_{7} + 10 \beta_{6} + 9 \beta_{5} + \cdots - 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -11\beta_{9} + 11\beta_{8} - 9\beta_{6} + 9\beta_{5} + \beta_{4} - 59\beta_{3} - 10\beta_{2} + 165 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 149 \beta_{11} - 68 \beta_{10} + 80 \beta_{9} + 11 \beta_{8} - 149 \beta_{7} - 69 \beta_{6} - 80 \beta_{5} + \cdots + 79 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 13 \beta_{11} - 430 \beta_{10} - 69 \beta_{9} - 161 \beta_{8} + 892 \beta_{7} + 161 \beta_{6} + \cdots - 823 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -512\beta_{9} + 512\beta_{8} - 92\beta_{6} + 92\beta_{5} + 1113\beta_{4} + 495\beta_{3} + 1522\beta_{2} + 443 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 126 \beta_{11} + 3130 \beta_{10} + 1222 \beta_{9} + 512 \beta_{8} - 6501 \beta_{7} - 710 \beta_{6} + \cdots - 2618 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 8192 \beta_{11} + 3568 \beta_{10} - 710 \beta_{9} - 4478 \beta_{8} + 8360 \beta_{7} + 4478 \beta_{6} + \cdots - 7650 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/399\mathbb{Z}\right)^\times\).

\(n\) \(115\) \(134\) \(211\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
1.34639 + 2.33202i
0.777141 + 1.34605i
0.597305 + 1.03456i
−0.0695651 − 0.120490i
−0.794855 − 1.37673i
−1.35642 − 2.34939i
1.34639 − 2.33202i
0.777141 − 1.34605i
0.597305 − 1.03456i
−0.0695651 + 0.120490i
−0.794855 + 1.37673i
−1.35642 + 2.34939i
−1.34639 − 2.33202i −0.500000 − 0.866025i −2.62554 + 4.54758i −1.03963 − 1.80069i −1.34639 + 2.33202i −1.00000 8.75448 −0.500000 + 0.866025i −2.79950 + 4.84888i
64.2 −0.777141 − 1.34605i −0.500000 − 0.866025i −0.207897 + 0.360089i 2.07326 + 3.59099i −0.777141 + 1.34605i −1.00000 −2.46230 −0.500000 + 0.866025i 3.22243 − 5.58142i
64.3 −0.597305 − 1.03456i −0.500000 − 0.866025i 0.286453 − 0.496151i 0.0760004 + 0.131637i −0.597305 + 1.03456i −1.00000 −3.07362 −0.500000 + 0.866025i 0.0907909 − 0.157254i
64.4 0.0695651 + 0.120490i −0.500000 − 0.866025i 0.990321 − 1.71529i −1.78910 − 3.09882i 0.0695651 − 0.120490i −1.00000 0.553828 −0.500000 + 0.866025i 0.248918 − 0.431139i
64.5 0.794855 + 1.37673i −0.500000 − 0.866025i −0.263589 + 0.456549i −0.818550 − 1.41777i 0.794855 − 1.37673i −1.00000 2.34136 −0.500000 + 0.866025i 1.30126 − 2.25384i
64.6 1.35642 + 2.34939i −0.500000 − 0.866025i −2.67974 + 4.64145i 1.49802 + 2.59465i 1.35642 − 2.34939i −1.00000 −9.11374 −0.500000 + 0.866025i −4.06389 + 7.03887i
106.1 −1.34639 + 2.33202i −0.500000 + 0.866025i −2.62554 − 4.54758i −1.03963 + 1.80069i −1.34639 − 2.33202i −1.00000 8.75448 −0.500000 − 0.866025i −2.79950 − 4.84888i
106.2 −0.777141 + 1.34605i −0.500000 + 0.866025i −0.207897 − 0.360089i 2.07326 − 3.59099i −0.777141 − 1.34605i −1.00000 −2.46230 −0.500000 − 0.866025i 3.22243 + 5.58142i
106.3 −0.597305 + 1.03456i −0.500000 + 0.866025i 0.286453 + 0.496151i 0.0760004 − 0.131637i −0.597305 − 1.03456i −1.00000 −3.07362 −0.500000 − 0.866025i 0.0907909 + 0.157254i
106.4 0.0695651 − 0.120490i −0.500000 + 0.866025i 0.990321 + 1.71529i −1.78910 + 3.09882i 0.0695651 + 0.120490i −1.00000 0.553828 −0.500000 − 0.866025i 0.248918 + 0.431139i
106.5 0.794855 − 1.37673i −0.500000 + 0.866025i −0.263589 − 0.456549i −0.818550 + 1.41777i 0.794855 + 1.37673i −1.00000 2.34136 −0.500000 − 0.866025i 1.30126 + 2.25384i
106.6 1.35642 − 2.34939i −0.500000 + 0.866025i −2.67974 − 4.64145i 1.49802 − 2.59465i 1.35642 + 2.34939i −1.00000 −9.11374 −0.500000 − 0.866025i −4.06389 − 7.03887i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 399.2.k.c ✓ 12
3.b odd 2 1 1197.2.k.i 12
19.c even 3 1 inner 399.2.k.c ✓ 12
19.c even 3 1 7581.2.a.bc 6
19.d odd 6 1 7581.2.a.ba 6
57.h odd 6 1 1197.2.k.i 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
399.2.k.c ✓ 12 1.a even 1 1 trivial
399.2.k.c ✓ 12 19.c even 3 1 inner
1197.2.k.i 12 3.b odd 2 1
1197.2.k.i 12 57.h odd 6 1
7581.2.a.ba 6 19.d odd 6 1
7581.2.a.bc 6 19.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + T_{2}^{11} + 11 T_{2}^{10} + 10 T_{2}^{9} + 90 T_{2}^{8} + 79 T_{2}^{7} + 275 T_{2}^{6} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(399, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + T^{11} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} + 23 T^{10} + \cdots + 529 \) Copy content Toggle raw display
$7$ \( (T + 1)^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} - 4 T^{5} - 28 T^{4} + \cdots - 99)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 2 T^{11} + \cdots + 3297856 \) Copy content Toggle raw display
$17$ \( T^{12} - 3 T^{11} + \cdots + 1252161 \) Copy content Toggle raw display
$19$ \( T^{12} - 3 T^{11} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( T^{12} - 5 T^{11} + \cdots + 144 \) Copy content Toggle raw display
$29$ \( T^{12} - 11 T^{11} + \cdots + 7529536 \) Copy content Toggle raw display
$31$ \( (T^{6} + 5 T^{5} + \cdots - 2592)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 5 T^{5} + \cdots + 1179)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} - 5 T^{11} + \cdots + 4888521 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 1055990016 \) Copy content Toggle raw display
$47$ \( T^{12} + 37 T^{11} + \cdots + 36869184 \) Copy content Toggle raw display
$53$ \( T^{12} - 9 T^{11} + \cdots + 52186176 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 983951424 \) Copy content Toggle raw display
$61$ \( T^{12} - 19 T^{11} + \cdots + 37503376 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 47248847424 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 11490982416 \) Copy content Toggle raw display
$73$ \( T^{12} - 24 T^{11} + \cdots + 104976 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 30685229584 \) Copy content Toggle raw display
$83$ \( (T^{6} - 35 T^{5} + \cdots - 54252)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + 4 T^{11} + \cdots + 93083904 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 1660725504 \) Copy content Toggle raw display
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