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Properties

Label 1911.2.c.n
Level $1911$
Weight $2$
Character orbit 1911.c
Analytic conductor $15.259$
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] = [1911,2,Mod(883,1911)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1911, base_ring=CyclotomicField(2))
 
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("1911.883");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1911.c (of order \(2\), degree \(1\), 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: \(15.2594118263\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 11x^{6} + 33x^{4} + 30x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 273)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} - 1) q^{4} - \beta_{7} q^{5} + \beta_1 q^{6} + (\beta_{5} + \beta_{4} - \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} - 1) q^{4} - \beta_{7} q^{5} + \beta_1 q^{6} + (\beta_{5} + \beta_{4} - \beta_1) q^{8} + q^{9} + \beta_{6} q^{10} - \beta_{4} q^{11} + (\beta_{2} - 1) q^{12} + ( - \beta_{7} - \beta_{6} - \beta_{2} + 1) q^{13} - \beta_{7} q^{15} + (\beta_{3} - \beta_{2} + 2) q^{16} + ( - \beta_{6} + \beta_{3} + 2) q^{17} + \beta_1 q^{18} + ( - \beta_{7} - \beta_1) q^{19} + ( - \beta_{7} - \beta_{5} + \beta_1) q^{20} + (\beta_{6} - \beta_{3} + 2 \beta_{2} - 2) q^{22} + ( - \beta_{6} - \beta_{3} + 1) q^{23} + (\beta_{5} + \beta_{4} - \beta_1) q^{24} + ( - \beta_{2} + 1) q^{25} + ( - \beta_{7} + \beta_{6} + \cdots + 2 \beta_1) q^{26}+ \cdots - \beta_{4} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3} - 6 q^{4} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{3} - 6 q^{4} + 8 q^{9} + 4 q^{10} - 6 q^{12} + 2 q^{13} + 10 q^{16} + 8 q^{17} - 4 q^{22} + 8 q^{23} + 6 q^{25} + 4 q^{26} + 8 q^{27} - 18 q^{29} + 4 q^{30} - 6 q^{36} + 26 q^{38} + 2 q^{39} - 6 q^{40} + 16 q^{43} + 10 q^{48} + 8 q^{51} - 40 q^{52} - 36 q^{53} - 4 q^{55} - 12 q^{61} - 40 q^{62} - 28 q^{64} - 34 q^{65} - 4 q^{66} - 10 q^{68} + 8 q^{69} - 18 q^{74} + 6 q^{75} + 4 q^{78} - 8 q^{79} + 8 q^{81} - 12 q^{82} - 18 q^{87} + 98 q^{88} + 4 q^{90} + 20 q^{92} - 46 q^{94} - 38 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 11x^{6} + 33x^{4} + 30x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} + 7\nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{5} - 8\nu^{3} - 10\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} + 9\nu^{3} + 15\nu \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} + 9\nu^{4} + 15\nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{7} + 10\nu^{5} + 24\nu^{3} + 15\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{3} - 7\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{5} - 9\beta_{4} + 30\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{6} - 9\beta_{3} + 48\beta_{2} - 100 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{7} + 56\beta_{5} + 66\beta_{4} - 195\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1911\mathbb{Z}\right)^\times\).

\(n\) \(638\) \(1471\) \(1522\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
883.1
− 2.60472i
− 1.55846i
− 1.32363i
− 0.186113i
0.186113i
1.32363i
1.55846i
2.60472i
− 2.60472i 1.00000 −4.78456 0.464160i − 2.60472i 0 7.25298i 1.00000 1.20901
883.2 − 1.55846i 1.00000 −0.428808 2.13803i − 1.55846i 0 − 2.44865i 1.00000 3.33205
883.3 − 1.32363i 1.00000 0.248001 − 2.29085i − 1.32363i 0 − 2.97552i 1.00000 −3.03224
883.4 − 0.186113i 1.00000 1.96536 2.63920i − 0.186113i 0 − 0.738004i 1.00000 0.491188
883.5 0.186113i 1.00000 1.96536 − 2.63920i 0.186113i 0 0.738004i 1.00000 0.491188
883.6 1.32363i 1.00000 0.248001 2.29085i 1.32363i 0 2.97552i 1.00000 −3.03224
883.7 1.55846i 1.00000 −0.428808 − 2.13803i 1.55846i 0 2.44865i 1.00000 3.33205
883.8 2.60472i 1.00000 −4.78456 − 0.464160i 2.60472i 0 − 7.25298i 1.00000 1.20901
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 883.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1911.2.c.n 8
7.b odd 2 1 1911.2.c.k 8
7.c even 3 2 273.2.bj.c ✓ 16
13.b even 2 1 inner 1911.2.c.n 8
21.h odd 6 2 819.2.dl.f 16
91.b odd 2 1 1911.2.c.k 8
91.r even 6 2 273.2.bj.c ✓ 16
273.w odd 6 2 819.2.dl.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bj.c ✓ 16 7.c even 3 2
273.2.bj.c ✓ 16 91.r even 6 2
819.2.dl.f 16 21.h odd 6 2
819.2.dl.f 16 273.w odd 6 2
1911.2.c.k 8 7.b odd 2 1
1911.2.c.k 8 91.b odd 2 1
1911.2.c.n 8 1.a even 1 1 trivial
1911.2.c.n 8 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1911, [\chi])\):

\( T_{2}^{8} + 11T_{2}^{6} + 33T_{2}^{4} + 30T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{8} + 17T_{5}^{6} + 96T_{5}^{4} + 187T_{5}^{2} + 36 \) Copy content Toggle raw display
\( T_{17}^{4} - 4T_{17}^{3} - 49T_{17}^{2} + 245T_{17} - 169 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 11 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 17 T^{6} + \cdots + 36 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 56 T^{6} + \cdots + 3249 \) Copy content Toggle raw display
$13$ \( T^{8} - 2 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( (T^{4} - 4 T^{3} + \cdots - 169)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 32 T^{6} + \cdots + 961 \) Copy content Toggle raw display
$23$ \( (T^{4} - 4 T^{3} + \cdots + 494)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 9 T^{3} + 13 T^{2} + \cdots + 11)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 108 T^{6} + \cdots + 3364 \) Copy content Toggle raw display
$37$ \( T^{8} + 103 T^{6} + \cdots + 21316 \) Copy content Toggle raw display
$41$ \( T^{8} + 100 T^{6} + \cdots + 7396 \) Copy content Toggle raw display
$43$ \( (T^{4} - 8 T^{3} + \cdots - 3552)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 255 T^{6} + \cdots + 1201216 \) Copy content Toggle raw display
$53$ \( (T^{4} + 18 T^{3} + \cdots - 2349)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 308 T^{6} + \cdots + 30151081 \) Copy content Toggle raw display
$61$ \( (T^{4} + 6 T^{3} + \cdots - 1081)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 180 T^{6} + \cdots + 37249 \) Copy content Toggle raw display
$71$ \( T^{8} + 332 T^{6} + \cdots + 4004001 \) Copy content Toggle raw display
$73$ \( T^{8} + 284 T^{6} + \cdots + 1633284 \) Copy content Toggle raw display
$79$ \( (T^{4} + 4 T^{3} + \cdots + 14124)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 252 T^{6} + \cdots + 26244 \) Copy content Toggle raw display
$89$ \( T^{8} + 53 T^{6} + \cdots + 36 \) Copy content Toggle raw display
$97$ \( T^{8} + 63 T^{6} + \cdots + 3364 \) Copy content Toggle raw display
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