Properties

Label 714.2.t.d
Level $714$
Weight $2$
Character orbit 714.t
Analytic conductor $5.701$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [714,2,Mod(67,714)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(714, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("714.67");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 714 = 2 \cdot 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 714.t (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: \(5.70131870432\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{2} + (\beta_{3} - \beta_1) q^{3} - \beta_{2} q^{4} + (\beta_{6} - \beta_1) q^{5} + \beta_{3} q^{6} + (3 \beta_{3} - 2 \beta_1) q^{7} - q^{8} + ( - \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 1) q^{2} + (\beta_{3} - \beta_1) q^{3} - \beta_{2} q^{4} + (\beta_{6} - \beta_1) q^{5} + \beta_{3} q^{6} + (3 \beta_{3} - 2 \beta_1) q^{7} - q^{8} + ( - \beta_{2} + 1) q^{9} + (\beta_{5} + \beta_{3} - \beta_1) q^{10} + ( - \beta_{5} + 2 \beta_{3} - 2 \beta_1) q^{11} + \beta_1 q^{12} + ( - \beta_{7} - 1) q^{13} + (2 \beta_{3} + \beta_1) q^{14} + ( - \beta_{7} + 1) q^{15} + (\beta_{2} - 1) q^{16} + ( - \beta_{4} - 3 \beta_{3} + 3 \beta_1) q^{17} - \beta_{2} q^{18} + (4 \beta_{2} - 4) q^{19} + ( - \beta_{6} + \beta_{5} + \beta_{3}) q^{20} + ( - 3 \beta_{2} + 2) q^{21} + (\beta_{6} - \beta_{5} + 2 \beta_{3}) q^{22} - 3 \beta_{6} q^{23} + ( - \beta_{3} + \beta_1) q^{24} + ( - 2 \beta_{4} + 4 \beta_{2}) q^{25} + ( - \beta_{7} + \beta_{4} + \beta_{2} - 1) q^{26} + \beta_{3} q^{27} + ( - \beta_{3} + 3 \beta_1) q^{28} + (\beta_{6} - \beta_{5} + 3 \beta_{3}) q^{29} + ( - \beta_{7} + \beta_{4} - \beta_{2} + 1) q^{30} + (2 \beta_{5} + 5 \beta_{3} - 5 \beta_1) q^{31} + \beta_{2} q^{32} + (\beta_{7} - \beta_{4} - 2 \beta_{2} + 2) q^{33} + ( - \beta_{7} - 3 \beta_{3}) q^{34} + ( - 3 \beta_{7} + \beta_{4} - \beta_{2} + 3) q^{35} - q^{36} + 6 \beta_1 q^{37} + 4 \beta_{2} q^{38} + (\beta_{5} - \beta_{3} + \beta_1) q^{39} + ( - \beta_{6} + \beta_1) q^{40} + (2 \beta_{6} - 2 \beta_{5} + \beta_{3}) q^{41} + ( - 2 \beta_{2} - 1) q^{42} - 6 q^{43} + (\beta_{6} + 2 \beta_1) q^{44} + (\beta_{5} + \beta_{3} - \beta_1) q^{45} - 3 \beta_{5} q^{46} + (2 \beta_{7} - 2 \beta_{4} - \beta_{2} + 1) q^{47} - \beta_{3} q^{48} + ( - 8 \beta_{2} + 3) q^{49} + ( - 2 \beta_{7} + 4) q^{50} + (\beta_{6} + 3 \beta_{2} - 3) q^{51} + (\beta_{4} + \beta_{2}) q^{52} + (2 \beta_{4} + 6 \beta_{2}) q^{53} + \beta_1 q^{54} + ( - \beta_{7} - 6) q^{55} + ( - 3 \beta_{3} + 2 \beta_1) q^{56} - 4 \beta_{3} q^{57} + (\beta_{6} + 3 \beta_1) q^{58} + (\beta_{4} + \beta_{2}) q^{59} + (\beta_{4} - \beta_{2}) q^{60} + ( - 3 \beta_{6} - 4 \beta_1) q^{61} + ( - 2 \beta_{6} + 2 \beta_{5} + 5 \beta_{3}) q^{62} + (2 \beta_{3} + \beta_1) q^{63} + q^{64} - 7 \beta_1 q^{65} + ( - \beta_{4} - 2 \beta_{2}) q^{66} + (3 \beta_{4} + 6 \beta_{2}) q^{67} + ( - \beta_{7} + \beta_{4} - 3 \beta_1) q^{68} + 3 \beta_{7} q^{69} + ( - 2 \beta_{7} + 3 \beta_{4} - 3 \beta_{2} + 2) q^{70} + (\beta_{6} - \beta_{5} + 12 \beta_{3}) q^{71} + (\beta_{2} - 1) q^{72} + (\beta_{5} + 2 \beta_{3} - 2 \beta_1) q^{73} + ( - 6 \beta_{3} + 6 \beta_1) q^{74} + (2 \beta_{6} - 4 \beta_1) q^{75} + 4 q^{76} + (2 \beta_{7} - 3 \beta_{4} - 6 \beta_{2} + 4) q^{77} + ( - \beta_{6} + \beta_{5} - \beta_{3}) q^{78} + ( - 4 \beta_{6} + 4 \beta_1) q^{79} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{80} - \beta_{2} q^{81} + (2 \beta_{6} + \beta_1) q^{82} + (3 \beta_{7} + 9) q^{83} + (\beta_{2} - 3) q^{84} + (3 \beta_{7} + \beta_{6} - \beta_{5} - 8 \beta_{3} - 3) q^{85} + (6 \beta_{2} - 6) q^{86} + ( - \beta_{4} - 3 \beta_{2}) q^{87} + (\beta_{5} - 2 \beta_{3} + 2 \beta_1) q^{88} + ( - \beta_{7} + \beta_{4} + 8 \beta_{2} - 8) q^{89} + ( - \beta_{6} + \beta_{5} + \beta_{3}) q^{90} + ( - \beta_{6} + 3 \beta_{5} - 3 \beta_{3} + 2 \beta_1) q^{91} + (3 \beta_{6} - 3 \beta_{5}) q^{92} + ( - 2 \beta_{7} + 2 \beta_{4} - 5 \beta_{2} + 5) q^{93} + ( - 2 \beta_{4} - \beta_{2}) q^{94} + ( - 4 \beta_{5} - 4 \beta_{3} + 4 \beta_1) q^{95} - \beta_1 q^{96} + ( - 4 \beta_{6} + 4 \beta_{5} - 4 \beta_{3}) q^{97} + ( - 3 \beta_{2} - 5) q^{98} + (\beta_{6} - \beta_{5} + 2 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 4 q^{4} - 8 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} - 4 q^{4} - 8 q^{8} + 4 q^{9} - 8 q^{13} + 8 q^{15} - 4 q^{16} - 4 q^{18} - 16 q^{19} + 4 q^{21} + 16 q^{25} - 4 q^{26} + 4 q^{30} + 4 q^{32} + 8 q^{33} + 20 q^{35} - 8 q^{36} + 16 q^{38} - 16 q^{42} - 48 q^{43} + 4 q^{47} - 8 q^{49} + 32 q^{50} - 12 q^{51} + 4 q^{52} + 24 q^{53} - 48 q^{55} + 4 q^{59} - 4 q^{60} + 8 q^{64} - 8 q^{66} + 24 q^{67} + 4 q^{70} - 4 q^{72} + 32 q^{76} + 8 q^{77} - 4 q^{81} + 72 q^{83} - 20 q^{84} - 24 q^{85} - 24 q^{86} - 12 q^{87} - 32 q^{89} + 20 q^{93} - 4 q^{94} - 52 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 2\zeta_{24}^{7} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -2\zeta_{24}^{7} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -2\zeta_{24}^{7} + 2\zeta_{24}^{5} + 2\zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -2\zeta_{24}^{5} + 2\zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{5} + \beta_{4} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{4} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{5} + \beta_{4} ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/714\mathbb{Z}\right)^\times\).

\(n\) \(239\) \(409\) \(547\)
\(\chi(n)\) \(1\) \(-1 + \beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
−0.965926 + 0.258819i
0.965926 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
−0.965926 0.258819i
0.965926 + 0.258819i
−0.258819 + 0.965926i
0.258819 0.965926i
0.500000 + 0.866025i −0.866025 0.500000i −0.500000 + 0.866025i −3.31552 + 1.91421i 1.00000i −1.73205 2.00000i −1.00000 0.500000 + 0.866025i −3.31552 1.91421i
67.2 0.500000 + 0.866025i −0.866025 0.500000i −0.500000 + 0.866025i 1.58346 0.914214i 1.00000i −1.73205 2.00000i −1.00000 0.500000 + 0.866025i 1.58346 + 0.914214i
67.3 0.500000 + 0.866025i 0.866025 + 0.500000i −0.500000 + 0.866025i −1.58346 + 0.914214i 1.00000i 1.73205 + 2.00000i −1.00000 0.500000 + 0.866025i −1.58346 0.914214i
67.4 0.500000 + 0.866025i 0.866025 + 0.500000i −0.500000 + 0.866025i 3.31552 1.91421i 1.00000i 1.73205 + 2.00000i −1.00000 0.500000 + 0.866025i 3.31552 + 1.91421i
373.1 0.500000 0.866025i −0.866025 + 0.500000i −0.500000 0.866025i −3.31552 1.91421i 1.00000i −1.73205 + 2.00000i −1.00000 0.500000 0.866025i −3.31552 + 1.91421i
373.2 0.500000 0.866025i −0.866025 + 0.500000i −0.500000 0.866025i 1.58346 + 0.914214i 1.00000i −1.73205 + 2.00000i −1.00000 0.500000 0.866025i 1.58346 0.914214i
373.3 0.500000 0.866025i 0.866025 0.500000i −0.500000 0.866025i −1.58346 0.914214i 1.00000i 1.73205 2.00000i −1.00000 0.500000 0.866025i −1.58346 + 0.914214i
373.4 0.500000 0.866025i 0.866025 0.500000i −0.500000 0.866025i 3.31552 + 1.91421i 1.00000i 1.73205 2.00000i −1.00000 0.500000 0.866025i 3.31552 1.91421i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
17.b even 2 1 inner
119.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 714.2.t.d 8
7.c even 3 1 inner 714.2.t.d 8
17.b even 2 1 inner 714.2.t.d 8
119.j even 6 1 inner 714.2.t.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
714.2.t.d 8 1.a even 1 1 trivial
714.2.t.d 8 7.c even 3 1 inner
714.2.t.d 8 17.b even 2 1 inner
714.2.t.d 8 119.j 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 \(S_{2}^{\mathrm{new}}(714, [\chi])\):

\( T_{5}^{8} - 18T_{5}^{6} + 275T_{5}^{4} - 882T_{5}^{2} + 2401 \) Copy content Toggle raw display
\( T_{11}^{8} - 24T_{11}^{6} + 560T_{11}^{4} - 384T_{11}^{2} + 256 \) Copy content Toggle raw display
\( T_{13}^{2} + 2T_{13} - 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 18 T^{6} + 275 T^{4} + \cdots + 2401 \) Copy content Toggle raw display
$7$ \( (T^{4} + 2 T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 24 T^{6} + 560 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( (T^{2} + 2 T - 7)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} - 2 T^{6} - 285 T^{4} + \cdots + 83521 \) Copy content Toggle raw display
$19$ \( (T^{2} + 4 T + 16)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 72 T^{2} + 5184)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 34 T^{2} + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 114 T^{6} + 12947 T^{4} + \cdots + 2401 \) Copy content Toggle raw display
$37$ \( (T^{4} - 36 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 66 T^{2} + 961)^{2} \) Copy content Toggle raw display
$43$ \( (T + 6)^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} - 2 T^{3} + 35 T^{2} + 62 T + 961)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 12 T^{3} + 140 T^{2} - 48 T + 16)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 2 T^{3} + 11 T^{2} + 14 T + 49)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} - 176 T^{6} + 27840 T^{4} + \cdots + 9834496 \) Copy content Toggle raw display
$67$ \( (T^{4} - 12 T^{3} + 180 T^{2} + 432 T + 1296)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 304 T^{2} + 18496)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 24 T^{6} + 560 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$79$ \( T^{8} - 288 T^{6} + \cdots + 157351936 \) Copy content Toggle raw display
$83$ \( (T^{2} - 18 T + 9)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 16 T^{3} + 200 T^{2} + 896 T + 3136)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 288 T^{2} + 12544)^{2} \) Copy content Toggle raw display
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