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Properties

Label 714.2.m.d
Level $714$
Weight $2$
Character orbit 714.m
Analytic conductor $5.701$
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] = [714,2,Mod(421,714)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(714, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("714.421");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 714 = 2 \cdot 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 714.m (of order \(4\), 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: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 38x^{10} + 509x^{8} + 2748x^{6} + 4804x^{4} + 2496x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} + \beta_{2} q^{3} - q^{4} + \beta_{6} q^{5} - \beta_{3} q^{6} - \beta_{3} q^{7} - \beta_{7} q^{8} + \beta_{7} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{2} + \beta_{2} q^{3} - q^{4} + \beta_{6} q^{5} - \beta_{3} q^{6} - \beta_{3} q^{7} - \beta_{7} q^{8} + \beta_{7} q^{9} - \beta_{5} q^{10} + (\beta_{7} - \beta_{5} - 1) q^{11} - \beta_{2} q^{12} + ( - \beta_{9} + \beta_{8} + \beta_{4} + 2) q^{13} - \beta_{2} q^{14} - \beta_1 q^{15} + q^{16} + (\beta_{10} - \beta_{7} + \beta_{4} + \cdots - 1) q^{17}+ \cdots + ( - \beta_{7} - \beta_{6} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{4} - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{4} - 4 q^{5} + 4 q^{10} - 8 q^{11} + 12 q^{13} + 12 q^{16} - 12 q^{17} - 12 q^{18} + 4 q^{20} - 12 q^{21} - 8 q^{22} + 8 q^{23} + 12 q^{29} - 4 q^{30} + 12 q^{31} - 4 q^{33} + 4 q^{34} - 4 q^{35} - 24 q^{38} + 4 q^{39} - 4 q^{40} + 4 q^{41} + 8 q^{44} + 4 q^{45} + 8 q^{46} + 8 q^{47} - 16 q^{50} - 8 q^{51} - 12 q^{52} - 68 q^{55} + 8 q^{57} - 12 q^{58} + 40 q^{61} - 12 q^{62} - 12 q^{64} + 20 q^{65} - 12 q^{67} + 12 q^{68} + 24 q^{71} + 12 q^{72} - 16 q^{75} - 4 q^{78} - 8 q^{79} - 4 q^{80} - 12 q^{81} + 4 q^{82} + 12 q^{84} - 12 q^{85} + 20 q^{86} + 8 q^{88} - 44 q^{89} + 4 q^{90} - 4 q^{91} - 8 q^{92} + 92 q^{95} - 32 q^{97} + 12 q^{98} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 38x^{10} + 509x^{8} + 2748x^{6} + 4804x^{4} + 2496x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} - 34 \nu^{10} + 46 \nu^{9} - 1156 \nu^{8} + 877 \nu^{7} - 13498 \nu^{6} + 8132 \nu^{5} + \cdots - 18496 ) / 13056 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{11} - 34 \nu^{10} - 46 \nu^{9} - 1156 \nu^{8} - 877 \nu^{7} - 13498 \nu^{6} - 8132 \nu^{5} + \cdots - 18496 ) / 13056 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{10} - 29\nu^{8} - 265\nu^{6} - 839\nu^{4} - 1146\nu^{2} - 240 ) / 272 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 17 \nu^{11} + 4 \nu^{10} - 578 \nu^{9} + 184 \nu^{8} - 6749 \nu^{7} + 2692 \nu^{6} - 30328 \nu^{5} + \cdots - 128 ) / 6528 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 17 \nu^{11} + 4 \nu^{10} + 578 \nu^{9} + 184 \nu^{8} + 6749 \nu^{7} + 2692 \nu^{6} + 30328 \nu^{5} + \cdots - 128 ) / 6528 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -15\nu^{11} - 554\nu^{9} - 7171\nu^{7} - 36980\nu^{5} - 58636\nu^{3} - 19104\nu ) / 4352 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 101 \nu^{11} + 52 \nu^{10} - 3830 \nu^{9} + 1984 \nu^{8} - 51041 \nu^{7} + 26020 \nu^{6} + \cdots - 8192 ) / 13056 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 101 \nu^{11} - 52 \nu^{10} - 3830 \nu^{9} - 1984 \nu^{8} - 51041 \nu^{7} - 26020 \nu^{6} + \cdots + 8192 ) / 13056 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 143 \nu^{11} + 56 \nu^{10} - 5354 \nu^{9} + 2168 \nu^{8} - 69923 \nu^{7} + 29528 \nu^{6} + \cdots + 96128 ) / 13056 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 143 \nu^{11} + 56 \nu^{10} + 5354 \nu^{9} + 2168 \nu^{8} + 69923 \nu^{7} + 29528 \nu^{6} + \cdots + 96128 ) / 13056 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - \beta_{8} + \beta_{6} + \beta_{5} + \beta_{4} - 2\beta_{3} - 2\beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} - \beta_{8} + 6\beta_{7} + \beta_{6} - \beta_{5} - 11\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + \beta_{10} - 12 \beta_{9} + 12 \beta_{8} - 11 \beta_{6} - 11 \beta_{5} - 15 \beta_{4} + \cdots + 66 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 4 \beta_{11} + 4 \beta_{10} + 11 \beta_{9} + 11 \beta_{8} - 98 \beta_{7} - 15 \beta_{6} + \cdots + 127 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 15 \beta_{11} - 15 \beta_{10} + 142 \beta_{9} - 142 \beta_{8} + 107 \beta_{6} + 107 \beta_{5} + \cdots - 794 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 96 \beta_{11} - 96 \beta_{10} - 107 \beta_{9} - 107 \beta_{8} + 1418 \beta_{7} + 207 \beta_{6} + \cdots - 1495 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 207 \beta_{11} + 207 \beta_{10} - 1702 \beta_{9} + 1702 \beta_{8} - 967 \beta_{6} - 967 \beta_{5} + \cdots + 9754 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1724 \beta_{11} + 1724 \beta_{10} + 967 \beta_{9} + 967 \beta_{8} - 19914 \beta_{7} + \cdots + 17783 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2867 \beta_{11} - 2867 \beta_{10} + 20650 \beta_{9} - 20650 \beta_{8} + 7771 \beta_{6} + \cdots - 121194 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 27640 \beta_{11} - 27640 \beta_{10} - 7771 \beta_{9} - 7771 \beta_{8} + 275450 \beta_{7} + \cdots - 213447 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(239\) \(409\) \(547\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
421.1
− 3.38067i
− 0.367423i
1.33388i
3.26061i
0.809999i
− 3.65639i
3.38067i
0.367423i
− 1.33388i
− 3.26061i
− 0.809999i
3.65639i
1.00000i −0.707107 − 0.707107i −1.00000 −2.39049 − 2.39049i 0.707107 − 0.707107i 0.707107 − 0.707107i − 1.00000i 1.00000i 2.39049 − 2.39049i
421.2 1.00000i −0.707107 − 0.707107i −1.00000 −0.259807 − 0.259807i 0.707107 − 0.707107i 0.707107 − 0.707107i − 1.00000i 1.00000i 0.259807 − 0.259807i
421.3 1.00000i −0.707107 − 0.707107i −1.00000 0.943194 + 0.943194i 0.707107 − 0.707107i 0.707107 − 0.707107i − 1.00000i 1.00000i −0.943194 + 0.943194i
421.4 1.00000i 0.707107 + 0.707107i −1.00000 −2.30560 − 2.30560i −0.707107 + 0.707107i −0.707107 + 0.707107i − 1.00000i 1.00000i 2.30560 − 2.30560i
421.5 1.00000i 0.707107 + 0.707107i −1.00000 −0.572756 − 0.572756i −0.707107 + 0.707107i −0.707107 + 0.707107i − 1.00000i 1.00000i 0.572756 − 0.572756i
421.6 1.00000i 0.707107 + 0.707107i −1.00000 2.58546 + 2.58546i −0.707107 + 0.707107i −0.707107 + 0.707107i − 1.00000i 1.00000i −2.58546 + 2.58546i
463.1 − 1.00000i −0.707107 + 0.707107i −1.00000 −2.39049 + 2.39049i 0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i − 1.00000i 2.39049 + 2.39049i
463.2 − 1.00000i −0.707107 + 0.707107i −1.00000 −0.259807 + 0.259807i 0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i − 1.00000i 0.259807 + 0.259807i
463.3 − 1.00000i −0.707107 + 0.707107i −1.00000 0.943194 − 0.943194i 0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i − 1.00000i −0.943194 − 0.943194i
463.4 − 1.00000i 0.707107 − 0.707107i −1.00000 −2.30560 + 2.30560i −0.707107 − 0.707107i −0.707107 − 0.707107i 1.00000i − 1.00000i 2.30560 + 2.30560i
463.5 − 1.00000i 0.707107 − 0.707107i −1.00000 −0.572756 + 0.572756i −0.707107 − 0.707107i −0.707107 − 0.707107i 1.00000i − 1.00000i 0.572756 + 0.572756i
463.6 − 1.00000i 0.707107 − 0.707107i −1.00000 2.58546 − 2.58546i −0.707107 − 0.707107i −0.707107 − 0.707107i 1.00000i − 1.00000i −2.58546 − 2.58546i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 421.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 714.2.m.d ✓ 12
3.b odd 2 1 2142.2.p.g 12
17.c even 4 1 inner 714.2.m.d ✓ 12
51.f odd 4 1 2142.2.p.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
714.2.m.d ✓ 12 1.a even 1 1 trivial
714.2.m.d ✓ 12 17.c even 4 1 inner
2142.2.p.g 12 3.b odd 2 1
2142.2.p.g 12 51.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 4 T_{5}^{11} + 8 T_{5}^{10} + 181 T_{5}^{8} + 732 T_{5}^{7} + 1480 T_{5}^{6} - 184 T_{5}^{5} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(714, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{12} + 4 T^{11} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( (T^{4} + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{12} + 8 T^{11} + \cdots + 1024 \) Copy content Toggle raw display
$13$ \( (T^{6} - 6 T^{5} + \cdots - 2176)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 12 T^{11} + \cdots + 24137569 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 2680754176 \) Copy content Toggle raw display
$23$ \( T^{12} - 8 T^{11} + \cdots + 4734976 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 154157056 \) Copy content Toggle raw display
$31$ \( T^{12} - 12 T^{11} + \cdots + 65536 \) Copy content Toggle raw display
$37$ \( T^{12} + 80 T^{9} + \cdots + 14776336 \) Copy content Toggle raw display
$41$ \( T^{12} - 4 T^{11} + \cdots + 75759616 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 6879707136 \) Copy content Toggle raw display
$47$ \( (T^{6} - 4 T^{5} + \cdots - 8704)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 52877922304 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 98024095744 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 115949824 \) Copy content Toggle raw display
$67$ \( (T^{6} + 6 T^{5} + \cdots - 2944)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 3421782016 \) Copy content Toggle raw display
$73$ \( T^{12} + 308 T^{9} + \cdots + 7974976 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 81622204416 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 349630959616 \) Copy content Toggle raw display
$89$ \( (T^{6} + 22 T^{5} + \cdots - 2176)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 792535104 \) Copy content Toggle raw display
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