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Properties

Label 1911.2.c.p
Level $1911$
Weight $2$
Character orbit 1911.c
Analytic conductor $15.259$
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] = [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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 16x^{10} + 92x^{8} + 228x^{6} + 233x^{4} + 84x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} - 1) q^{4} + \beta_{4} q^{5} + \beta_1 q^{6} + ( - \beta_{5} + \beta_{4}) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} - 1) q^{4} + \beta_{4} q^{5} + \beta_1 q^{6} + ( - \beta_{5} + \beta_{4}) q^{8} + q^{9} + (\beta_{7} - \beta_{2} + 1) q^{10} + ( - \beta_{11} - \beta_{5} + \cdots + \beta_1) q^{11}+ \cdots + ( - \beta_{11} - \beta_{5} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{3} - 8 q^{4} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{3} - 8 q^{4} + 12 q^{9} + 8 q^{10} - 8 q^{12} + 8 q^{17} - 4 q^{23} + 8 q^{25} - 16 q^{26} + 12 q^{27} - 4 q^{29} + 8 q^{30} - 8 q^{36} + 8 q^{38} - 48 q^{40} + 28 q^{43} + 8 q^{51} + 12 q^{53} - 32 q^{55} - 16 q^{61} + 56 q^{62} + 40 q^{64} - 4 q^{65} - 16 q^{68} - 4 q^{69} + 8 q^{75} - 16 q^{78} - 28 q^{79} + 12 q^{81} - 24 q^{82} - 4 q^{87} - 56 q^{88} + 8 q^{90} + 16 q^{92} + 8 q^{94} + 36 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 16x^{10} + 92x^{8} + 228x^{6} + 233x^{4} + 84x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{6} + 8\nu^{4} + 15\nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} + 14\nu^{9} + 64\nu^{7} + 92\nu^{5} - 23\nu^{3} - 46\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{11} + 14\nu^{9} + 64\nu^{7} + 92\nu^{5} - 31\nu^{3} - 78\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{7} + 9\nu^{5} + 21\nu^{3} + 10\nu \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{10} - 14\nu^{8} - 68\nu^{6} - 128\nu^{4} - 61\nu^{2} + 6 ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{11} + 14\nu^{9} + 68\nu^{7} + 132\nu^{5} + 85\nu^{3} + 10\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{10} + 14\nu^{8} + 68\nu^{6} + 132\nu^{4} + 85\nu^{2} + 10 ) / 4 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{10} + 18\nu^{8} + 108\nu^{6} + 248\nu^{4} + 181\nu^{2} + 22 ) / 4 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{11} - 16\nu^{9} - 90\nu^{7} - 208\nu^{5} - 175\nu^{3} - 36\nu ) / 2 \) 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
\(\nu^{3}\)\(=\) \( -\beta_{5} + \beta_{4} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{7} - 6\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{8} - \beta_{6} + 6\beta_{5} - 8\beta_{4} + 20\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -8\beta_{9} - 8\beta_{7} + \beta_{3} + 33\beta_{2} - 72 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -9\beta_{8} + 10\beta_{6} - 33\beta_{5} + 51\beta_{4} - 106\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( \beta_{10} + 50\beta_{9} + 51\beta_{7} - 10\beta_{3} - 180\beta_{2} + 383 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -\beta_{11} + 59\beta_{8} - 72\beta_{6} + 180\beta_{5} - 302\beta_{4} + 573\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -14\beta_{10} - 284\beta_{9} - 302\beta_{7} + 72\beta_{3} + 983\beta_{2} - 2069 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 14\beta_{11} - 342\beta_{8} + 460\beta_{6} - 983\beta_{5} + 1731\beta_{4} - 3124\beta_1 \) 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
− 2.36207i
− 2.32134i
− 1.78249i
− 1.06835i
− 0.810907i
− 0.236204i
0.236204i
0.810907i
1.06835i
1.78249i
2.32134i
2.36207i
− 2.36207i 1.00000 −3.57937 2.81395i − 2.36207i 0 3.73059i 1.00000 6.64676
883.2 − 2.32134i 1.00000 −3.38863 1.15449i − 2.32134i 0 3.22348i 1.00000 2.67996
883.3 − 1.78249i 1.00000 −1.17727 − 1.33358i − 1.78249i 0 − 1.46651i 1.00000 −2.37709
883.4 − 1.06835i 1.00000 0.858635 − 3.57405i − 1.06835i 0 − 3.05401i 1.00000 −3.81833
883.5 − 0.810907i 1.00000 1.34243 0.689060i − 0.810907i 0 − 2.71040i 1.00000 0.558763
883.6 − 0.236204i 1.00000 1.94421 1.31216i − 0.236204i 0 − 0.931639i 1.00000 0.309937
883.7 0.236204i 1.00000 1.94421 − 1.31216i 0.236204i 0 0.931639i 1.00000 0.309937
883.8 0.810907i 1.00000 1.34243 − 0.689060i 0.810907i 0 2.71040i 1.00000 0.558763
883.9 1.06835i 1.00000 0.858635 3.57405i 1.06835i 0 3.05401i 1.00000 −3.81833
883.10 1.78249i 1.00000 −1.17727 1.33358i 1.78249i 0 1.46651i 1.00000 −2.37709
883.11 2.32134i 1.00000 −3.38863 − 1.15449i 2.32134i 0 − 3.22348i 1.00000 2.67996
883.12 2.36207i 1.00000 −3.57937 − 2.81395i 2.36207i 0 − 3.73059i 1.00000 6.64676
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 883.12
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.p yes 12
7.b odd 2 1 1911.2.c.o ✓ 12
13.b even 2 1 inner 1911.2.c.p yes 12
91.b odd 2 1 1911.2.c.o ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1911.2.c.o ✓ 12 7.b odd 2 1
1911.2.c.o ✓ 12 91.b odd 2 1
1911.2.c.p yes 12 1.a even 1 1 trivial
1911.2.c.p yes 12 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 \(S_{2}^{\mathrm{new}}(1911, [\chi])\):

\( T_{2}^{12} + 16T_{2}^{10} + 92T_{2}^{8} + 228T_{2}^{6} + 233T_{2}^{4} + 84T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{5}^{12} + 26T_{5}^{10} + 221T_{5}^{8} + 752T_{5}^{6} + 1176T_{5}^{4} + 824T_{5}^{2} + 196 \) Copy content Toggle raw display
\( T_{17}^{6} - 4T_{17}^{5} - 38T_{17}^{4} + 192T_{17}^{3} - 80T_{17}^{2} - 576T_{17} + 576 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 16 T^{10} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( (T - 1)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 26 T^{10} + \cdots + 196 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 56 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( T^{12} - 2 T^{10} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( (T^{6} - 4 T^{5} + \cdots + 576)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 102 T^{10} + \cdots + 82944 \) Copy content Toggle raw display
$23$ \( (T^{6} + 2 T^{5} + \cdots + 1008)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 2 T^{5} + \cdots - 5904)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 214 T^{10} + \cdots + 331776 \) Copy content Toggle raw display
$37$ \( T^{12} + 288 T^{10} + \cdots + 84934656 \) Copy content Toggle raw display
$41$ \( T^{12} + 72 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$43$ \( (T^{6} - 14 T^{5} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 314 T^{10} + \cdots + 11303044 \) Copy content Toggle raw display
$53$ \( (T^{6} - 6 T^{5} + \cdots + 14832)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 1799795776 \) Copy content Toggle raw display
$61$ \( (T^{6} + 8 T^{5} + \cdots + 36864)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 76 T^{10} + \cdots + 331776 \) Copy content Toggle raw display
$71$ \( T^{12} + 440 T^{10} + \cdots + 27709696 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 339450725376 \) Copy content Toggle raw display
$79$ \( (T^{6} + 14 T^{5} + \cdots - 3196)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 226303809796 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 80611702084 \) Copy content Toggle raw display
$97$ \( T^{12} + 422 T^{10} + \cdots + 21233664 \) Copy content Toggle raw display
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