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Properties

Label 1071.2.n.a
Level $1071$
Weight $2$
Character orbit 1071.n
Analytic conductor $8.552$
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] = [1071,2,Mod(64,1071)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1071, 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("1071.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1071 = 3^{2} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1071.n (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: \(8.55197805648\)
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: \( x^{12} + 27x^{8} + 107x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 357)
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.

\(f(q)\) \(=\) \( q + \beta_{10} q^{2} + ( - \beta_{8} + \beta_{7} - \beta_{2} + 1) q^{4} + (\beta_{11} + \beta_{10} + \beta_{8} + \cdots - 1) q^{5}+ \cdots + ( - \beta_{6} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{10} q^{2} + ( - \beta_{8} + \beta_{7} - \beta_{2} + 1) q^{4} + (\beta_{11} + \beta_{10} + \beta_{8} + \cdots - 1) q^{5}+ \cdots + \beta_{11} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8 q^{5} - 16 q^{10} - 4 q^{11} + 4 q^{13} + 4 q^{14} - 16 q^{16} - 20 q^{17} - 12 q^{20} + 32 q^{22} - 4 q^{23} + 32 q^{29} - 8 q^{31} + 20 q^{34} + 12 q^{35} - 16 q^{37} + 16 q^{38} + 12 q^{40} + 20 q^{41} + 8 q^{44} - 12 q^{46} - 32 q^{47} - 48 q^{50} - 16 q^{52} + 76 q^{55} - 4 q^{56} - 28 q^{58} - 20 q^{61} + 28 q^{62} + 8 q^{64} + 44 q^{65} - 56 q^{67} - 12 q^{68} + 16 q^{71} + 8 q^{73} - 8 q^{74} - 24 q^{79} + 8 q^{80} + 12 q^{82} - 24 q^{85} + 32 q^{86} + 12 q^{88} - 64 q^{89} + 4 q^{91} - 24 q^{92} + 36 q^{95} - 52 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 27x^{8} + 107x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{8} + 19\nu^{4} + 12 ) / 38 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{9} + 76\nu^{5} + 245\nu ) / 76 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{10} - 76\nu^{6} - 245\nu^{2} ) / 76 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7\nu^{10} + 190\nu^{6} + 787\nu^{2} ) / 76 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7\nu^{11} + 190\nu^{7} + 787\nu^{3} ) / 76 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -21\nu^{10} - \nu^{8} - 570\nu^{6} - 38\nu^{4} - 2285\nu^{2} - 221 ) / 152 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -21\nu^{10} + \nu^{8} - 570\nu^{6} + 38\nu^{4} - 2285\nu^{2} + 221 ) / 152 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -12\nu^{11} - 323\nu^{7} - 1265\nu^{3} ) / 38 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 69\nu^{11} + 7\nu^{9} + 1862\nu^{7} + 190\nu^{5} + 7345\nu^{3} + 711\nu ) / 152 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 69\nu^{11} - 7\nu^{9} + 1862\nu^{7} - 190\nu^{5} + 7345\nu^{3} - 711\nu ) / 152 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + \beta_{7} + 3\beta_{5} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} - \beta_{10} - 2\beta_{9} + 3\beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{8} - 4\beta_{7} - 2\beta_{2} - 11 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -6\beta_{11} + 6\beta_{10} - 14\beta_{3} - 11\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -17\beta_{8} - 17\beta_{7} - 45\beta_{5} + 14\beta_{4} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 31\beta_{11} + 31\beta_{10} + 76\beta_{9} - 45\beta_{6} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -76\beta_{8} + 76\beta_{7} + 76\beta_{2} + 197 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 152\beta_{11} - 152\beta_{10} + 380\beta_{3} + 197\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 349\beta_{8} + 349\beta_{7} + 895\beta_{5} - 380\beta_{4} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -729\beta_{11} - 729\beta_{10} - 1838\beta_{9} + 895\beta_{6} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1071\mathbb{Z}\right)^\times\).

\(n\) \(190\) \(596\) \(766\)
\(\chi(n)\) \(-\beta_{5}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
0.219986 − 0.219986i
−1.53448 + 1.53448i
−1.04736 + 1.04736i
1.04736 − 1.04736i
1.53448 − 1.53448i
−0.219986 + 0.219986i
−0.219986 − 0.219986i
1.53448 + 1.53448i
1.04736 + 1.04736i
−1.04736 − 1.04736i
−1.53448 − 1.53448i
0.219986 + 0.219986i
− 2.05288i 0 −2.21432 −1.82126 − 1.82126i 0 −0.707107 + 0.707107i 0.439973i 0 −3.73883 + 3.73883i
64.2 − 1.20864i 0 0.539189 −1.52880 − 1.52880i 0 0.707107 − 0.707107i − 3.06897i 0 −1.84776 + 1.84776i
64.3 − 0.569973i 0 1.67513 −0.119579 − 0.119579i 0 −0.707107 + 0.707107i − 2.09473i 0 −0.0681566 + 0.0681566i
64.4 0.569973i 0 1.67513 −1.07436 − 1.07436i 0 0.707107 − 0.707107i 2.09473i 0 0.612355 − 0.612355i
64.5 1.20864i 0 0.539189 −2.18048 − 2.18048i 0 −0.707107 + 0.707107i 3.06897i 0 2.63542 − 2.63542i
64.6 2.05288i 0 −2.21432 2.72447 + 2.72447i 0 0.707107 − 0.707107i − 0.439973i 0 −5.59302 + 5.59302i
820.1 − 2.05288i 0 −2.21432 2.72447 − 2.72447i 0 0.707107 + 0.707107i 0.439973i 0 −5.59302 − 5.59302i
820.2 − 1.20864i 0 0.539189 −2.18048 + 2.18048i 0 −0.707107 − 0.707107i − 3.06897i 0 2.63542 + 2.63542i
820.3 − 0.569973i 0 1.67513 −1.07436 + 1.07436i 0 0.707107 + 0.707107i − 2.09473i 0 0.612355 + 0.612355i
820.4 0.569973i 0 1.67513 −0.119579 + 0.119579i 0 −0.707107 − 0.707107i 2.09473i 0 −0.0681566 − 0.0681566i
820.5 1.20864i 0 0.539189 −1.52880 + 1.52880i 0 0.707107 + 0.707107i 3.06897i 0 −1.84776 − 1.84776i
820.6 2.05288i 0 −2.21432 −1.82126 + 1.82126i 0 −0.707107 − 0.707107i − 0.439973i 0 −3.73883 − 3.73883i
\(n\): 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
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 1071.2.n.a 12
3.b odd 2 1 357.2.k.a ✓ 12
17.c even 4 1 inner 1071.2.n.a 12
51.f odd 4 1 357.2.k.a ✓ 12
51.g odd 8 1 6069.2.a.x 6
51.g odd 8 1 6069.2.a.y 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
357.2.k.a ✓ 12 3.b odd 2 1
357.2.k.a ✓ 12 51.f odd 4 1
1071.2.n.a 12 1.a even 1 1 trivial
1071.2.n.a 12 17.c even 4 1 inner
6069.2.a.x 6 51.g odd 8 1
6069.2.a.y 6 51.g odd 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 6T_{2}^{4} + 8T_{2}^{2} + 2 \) acting on \(S_{2}^{\mathrm{new}}(1071, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 6 T^{4} + 8 T^{2} + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 8 T^{11} + \cdots + 289 \) Copy content Toggle raw display
$7$ \( (T^{4} + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{12} + 4 T^{11} + \cdots + 289 \) Copy content Toggle raw display
$13$ \( (T^{6} - 2 T^{5} + \cdots + 167)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 20 T^{11} + \cdots + 24137569 \) Copy content Toggle raw display
$19$ \( T^{12} + 86 T^{10} + \cdots + 22801 \) Copy content Toggle raw display
$23$ \( T^{12} + 4 T^{11} + \cdots + 58081 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 106090000 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 178115716 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 711502276 \) Copy content Toggle raw display
$41$ \( T^{12} - 20 T^{11} + \cdots + 361201 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 12697007761 \) Copy content Toggle raw display
$47$ \( (T^{6} + 16 T^{5} + \cdots - 2558)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 1863303556 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 585405553924 \) Copy content Toggle raw display
$61$ \( T^{12} + 20 T^{11} + \cdots + 26357956 \) Copy content Toggle raw display
$67$ \( (T^{6} + 28 T^{5} + \cdots - 412)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 491597584 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 4187752960000 \) Copy content Toggle raw display
$79$ \( T^{12} + 24 T^{11} + \cdots + 49308484 \) Copy content Toggle raw display
$83$ \( T^{12} + 692 T^{10} + \cdots + 38440000 \) Copy content Toggle raw display
$89$ \( (T^{6} + 32 T^{5} + \cdots + 29824)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 3735746764864 \) Copy content Toggle raw display
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