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Properties

Label 2394.2.o.v
Level $2394$
Weight $2$
Character orbit 2394.o
Analytic conductor $19.116$
Analytic rank $0$
Dimension $8$
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] = [2394,2,Mod(505,2394)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2394, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2394.505");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2394.o (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: \(19.1161862439\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 11x^{6} - 6x^{5} + 104x^{4} - 72x^{3} + 104x^{2} + 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 266)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} - 1) q^{2} - \beta_{4} q^{4} + (\beta_{7} - \beta_{6} - \beta_{3} - \beta_1) q^{5} + q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} - 1) q^{2} - \beta_{4} q^{4} + (\beta_{7} - \beta_{6} - \beta_{3} - \beta_1) q^{5} + q^{7} + q^{8} + ( - \beta_{7} + \beta_1) q^{10} + ( - \beta_{3} - 2) q^{11} + ( - \beta_{5} + \beta_{4} + \beta_{2}) q^{13} + (\beta_{4} - 1) q^{14} + (\beta_{4} - 1) q^{16} + (\beta_{7} - \beta_{6} - \beta_{4} + \cdots + 1) q^{17}+ \cdots + (\beta_{4} - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} - 4 q^{4} + q^{5} + 8 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} - 4 q^{4} + q^{5} + 8 q^{7} + 8 q^{8} + q^{10} - 14 q^{11} + 5 q^{13} - 4 q^{14} - 4 q^{16} + 2 q^{17} + 6 q^{19} - 2 q^{20} + 7 q^{22} + 5 q^{23} - 15 q^{25} - 10 q^{26} - 4 q^{28} + 4 q^{29} - 12 q^{31} - 4 q^{32} + 2 q^{34} + q^{35} + 20 q^{37} - 15 q^{38} + q^{40} - 17 q^{41} - 18 q^{43} + 7 q^{44} - 10 q^{46} - 4 q^{47} + 8 q^{49} + 30 q^{50} + 5 q^{52} - 10 q^{53} + 10 q^{55} + 8 q^{56} - 8 q^{58} - 20 q^{59} - 9 q^{61} + 6 q^{62} + 8 q^{64} + 6 q^{65} + 7 q^{67} - 4 q^{68} + q^{70} + 21 q^{71} - 21 q^{73} - 10 q^{74} + 9 q^{76} - 14 q^{77} - 8 q^{79} + q^{80} - 17 q^{82} + 24 q^{83} - 10 q^{85} - 18 q^{86} - 14 q^{88} + 34 q^{89} + 5 q^{91} + 5 q^{92} + 8 q^{94} - 31 q^{95} - 5 q^{97} - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 11x^{6} - 6x^{5} + 104x^{4} - 72x^{3} + 104x^{2} + 32x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -25\nu^{7} - 465\nu^{6} - 245\nu^{5} - 4652\nu^{4} - 4168\nu^{3} - 41488\nu^{2} - 9264\nu - 3152 ) / 9144 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 245\nu^{7} - 15\nu^{6} + 2401\nu^{5} + 784\nu^{4} + 26216\nu^{3} + 3332\nu^{2} + 1176\nu + 8944 ) / 27432 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -559\nu^{7} + 804\nu^{6} - 6164\nu^{5} + 5755\nu^{4} - 57352\nu^{3} + 66464\nu^{2} - 54804\nu + 10720 ) / 27432 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -185\nu^{7} - 12\nu^{6} - 1813\nu^{5} - 592\nu^{4} - 17813\nu^{3} - 2516\nu^{2} - 888\nu + 1364 ) / 4572 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -670\nu^{7} + 111\nu^{6} - 6566\nu^{5} - 2144\nu^{4} - 63925\nu^{3} - 9112\nu^{2} - 3216\nu - 62528 ) / 13716 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 170\nu^{7} - 267\nu^{6} + 2047\nu^{5} - 2123\nu^{4} + 19046\nu^{3} - 22072\nu^{2} + 32820\nu - 3560 ) / 3048 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{6} + 5\beta_{4} + \beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} + 10\beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -9\beta_{7} + 11\beta_{5} - 45\beta_{4} - 11\beta_{2} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{7} - 9\beta_{6} + 5\beta_{4} - 98\beta_{3} - 13\beta_{2} - 98\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 85\beta_{6} - 111\beta_{5} - 38\beta_{3} + 433 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -73\beta_{7} - 149\beta_{5} + 27\beta_{4} + 149\beta_{2} + 962\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2394\mathbb{Z}\right)^\times\).

\(n\) \(533\) \(1009\) \(1711\)
\(\chi(n)\) \(1\) \(-\beta_{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
505.1
−1.51459 − 2.62334i
0.582831 + 1.00949i
1.60789 + 2.78495i
−0.176135 − 0.305076i
−1.51459 + 2.62334i
0.582831 − 1.00949i
1.60789 − 2.78495i
−0.176135 + 0.305076i
−0.500000 + 0.866025i 0 −0.500000 − 0.866025i −1.68446 + 2.91758i 0 1.00000 1.00000 0 −1.68446 − 2.91758i
505.2 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −0.775050 + 1.34243i 0 1.00000 1.00000 0 −0.775050 − 1.34243i
505.3 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0.796924 − 1.38031i 0 1.00000 1.00000 0 0.796924 + 1.38031i
505.4 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 2.16259 − 3.74571i 0 1.00000 1.00000 0 2.16259 + 3.74571i
1261.1 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −1.68446 − 2.91758i 0 1.00000 1.00000 0 −1.68446 + 2.91758i
1261.2 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −0.775050 − 1.34243i 0 1.00000 1.00000 0 −0.775050 + 1.34243i
1261.3 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 0.796924 + 1.38031i 0 1.00000 1.00000 0 0.796924 − 1.38031i
1261.4 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 2.16259 + 3.74571i 0 1.00000 1.00000 0 2.16259 − 3.74571i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 505.4
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 2394.2.o.v 8
3.b odd 2 1 266.2.f.d ✓ 8
19.c even 3 1 inner 2394.2.o.v 8
57.f even 6 1 5054.2.a.x 4
57.h odd 6 1 266.2.f.d ✓ 8
57.h odd 6 1 5054.2.a.w 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
266.2.f.d ✓ 8 3.b odd 2 1
266.2.f.d ✓ 8 57.h odd 6 1
2394.2.o.v 8 1.a even 1 1 trivial
2394.2.o.v 8 19.c even 3 1 inner
5054.2.a.w 4 57.h odd 6 1
5054.2.a.x 4 57.f even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2394, [\chi])\):

\( T_{5}^{8} - T_{5}^{7} + 18T_{5}^{6} + 11T_{5}^{5} + 256T_{5}^{4} + 21T_{5}^{3} + 621T_{5}^{2} + 108T_{5} + 1296 \) Copy content Toggle raw display
\( T_{11}^{4} + 7T_{11}^{3} + 8T_{11}^{2} - 12T_{11} - 12 \) Copy content Toggle raw display
\( T_{13}^{8} - 5T_{13}^{7} + 50T_{13}^{6} - 165T_{13}^{5} + 1448T_{13}^{4} - 4605T_{13}^{3} + 18575T_{13}^{2} - 14210T_{13} + 9604 \) Copy content Toggle raw display
\( T_{17}^{8} - 2T_{17}^{7} + 56T_{17}^{6} - 208T_{17}^{5} + 2992T_{17}^{4} - 8016T_{17}^{3} + 25584T_{17}^{2} + 3744T_{17} + 576 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - T^{7} + \cdots + 1296 \) Copy content Toggle raw display
$7$ \( (T - 1)^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 7 T^{3} + 8 T^{2} + \cdots - 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 5 T^{7} + \cdots + 9604 \) Copy content Toggle raw display
$17$ \( T^{8} - 2 T^{7} + \cdots + 576 \) Copy content Toggle raw display
$19$ \( T^{8} - 6 T^{7} + \cdots + 130321 \) Copy content Toggle raw display
$23$ \( T^{8} - 5 T^{7} + \cdots + 213444 \) Copy content Toggle raw display
$29$ \( T^{8} - 4 T^{7} + \cdots + 2304 \) Copy content Toggle raw display
$31$ \( (T^{4} + 6 T^{3} + \cdots - 328)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 10 T^{3} + \cdots + 288)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 17 T^{7} + \cdots + 17424 \) Copy content Toggle raw display
$43$ \( T^{8} + 18 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$47$ \( T^{8} + 4 T^{7} + \cdots + 2304 \) Copy content Toggle raw display
$53$ \( T^{8} + 10 T^{7} + \cdots + 5184 \) Copy content Toggle raw display
$59$ \( T^{8} + 20 T^{7} + \cdots + 103041 \) Copy content Toggle raw display
$61$ \( T^{8} + 9 T^{7} + \cdots + 6512704 \) Copy content Toggle raw display
$67$ \( T^{8} - 7 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$71$ \( T^{8} - 21 T^{7} + \cdots + 4356 \) Copy content Toggle raw display
$73$ \( T^{8} + 21 T^{7} + \cdots + 70425664 \) Copy content Toggle raw display
$79$ \( T^{8} + 8 T^{7} + \cdots + 30294016 \) Copy content Toggle raw display
$83$ \( (T^{4} - 12 T^{3} + \cdots - 11061)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 34 T^{7} + \cdots + 4460544 \) Copy content Toggle raw display
$97$ \( T^{8} + 5 T^{7} + \cdots + 150544 \) Copy content Toggle raw display
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