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Properties

Label 1911.2.a.y
Level $1911$
Weight $2$
Character orbit 1911.a
Self dual yes
Analytic conductor $15.259$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1911,2,Mod(1,1911)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1911.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1911.a (trivial)

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: yes
Analytic conductor: \(15.2594118263\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4x^{9} - 10x^{8} + 52x^{7} + 16x^{6} - 212x^{5} + 64x^{4} + 300x^{3} - 159x^{2} - 80x + 46 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4x^{9} - 10x^{8} + 52x^{7} + 16x^{6} - 212x^{5} + 64x^{4} + 300x^{3} - 159x^{2} - 80x + 46 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 9 \nu^{9} + 16 \nu^{8} + 149 \nu^{7} - 301 \nu^{6} - 766 \nu^{5} + 1800 \nu^{4} + 1314 \nu^{3} + \cdots + 1178 ) / 211 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 22 \nu^{9} - 86 \nu^{8} - 247 \nu^{7} + 1064 \nu^{6} + 794 \nu^{5} - 3978 \nu^{4} - 891 \nu^{3} + \cdots - 1426 ) / 211 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 10 \nu^{9} + 135 \nu^{8} - 22 \nu^{7} - 1788 \nu^{6} + 1001 \nu^{5} + 7486 \nu^{4} - 2971 \nu^{3} + \cdots + 2950 ) / 211 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 31 \nu^{9} + 102 \nu^{8} + 396 \nu^{7} - 1365 \nu^{6} - 1560 \nu^{5} + 5778 \nu^{4} + 1994 \nu^{3} + \cdots + 2182 ) / 211 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 72 \nu^{9} + 128 \nu^{8} + 981 \nu^{7} - 1564 \nu^{6} - 4229 \nu^{5} + 5960 \nu^{4} + 6292 \nu^{3} + \cdots + 2672 ) / 211 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 78 \nu^{9} - 209 \nu^{8} - 1010 \nu^{7} + 2679 \nu^{6} + 4177 \nu^{5} - 10747 \nu^{4} - 6535 \nu^{3} + \cdots - 4442 ) / 211 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 113 \nu^{9} - 365 \nu^{8} - 1355 \nu^{7} + 4717 \nu^{6} + 4788 \nu^{5} - 19013 \nu^{4} - 5315 \nu^{3} + \cdots - 6538 ) / 211 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + 5\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{7} + \beta_{6} + \beta_{5} + 7\beta_{2} + 2\beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{8} + \beta_{7} - 7\beta_{6} - 10\beta_{4} + 9\beta_{3} + 9\beta_{2} + 30\beta _1 + 20 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11\beta_{9} + \beta_{8} + 11\beta_{7} + 11\beta_{6} + 10\beta_{5} - 4\beta_{4} + 49\beta_{2} + 20\beta _1 + 138 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4\beta_{9} + 22\beta_{8} + 12\beta_{7} - 39\beta_{6} - 86\beta_{4} + 69\beta_{3} + 73\beta_{2} + 193\beta _1 + 164 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 96 \beta_{9} + 16 \beta_{8} + 93 \beta_{7} + 100 \beta_{6} + 79 \beta_{5} - 67 \beta_{4} + 4 \beta_{3} + \cdots + 928 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 69 \beta_{9} + 189 \beta_{8} + 111 \beta_{7} - 186 \beta_{6} + 6 \beta_{5} - 704 \beta_{4} + 506 \beta_{3} + \cdots + 1280 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.58460
−1.83272
−1.56844
−0.583659
0.556045
0.691507
1.67947
2.36375
2.51105
2.76760
−2.58460 1.00000 4.68016 −3.63737 −2.58460 0 −6.92714 1.00000 9.40115
1.2 −1.83272 1.00000 1.35888 3.20067 −1.83272 0 1.17500 1.00000 −5.86594
1.3 −1.56844 1.00000 0.460010 −4.07203 −1.56844 0 2.41539 1.00000 6.38674
1.4 −0.583659 1.00000 −1.65934 −0.00312810 −0.583659 0 2.13581 1.00000 0.00182574
1.5 0.556045 1.00000 −1.69081 −3.27818 0.556045 0 −2.05226 1.00000 −1.82282
1.6 0.691507 1.00000 −1.52182 2.34977 0.691507 0 −2.43536 1.00000 1.62488
1.7 1.67947 1.00000 0.820610 1.01562 1.67947 0 −1.98075 1.00000 1.70571
1.8 2.36375 1.00000 3.58734 −4.08271 2.36375 0 3.75207 1.00000 −9.65053
1.9 2.51105 1.00000 4.30539 2.80810 2.51105 0 5.78895 1.00000 7.05129
1.10 2.76760 1.00000 5.65960 −0.300737 2.76760 0 10.1283 1.00000 −0.832320
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)
\(13\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1911.2.a.y yes 10
3.b odd 2 1 5733.2.a.bx 10
7.b odd 2 1 1911.2.a.x ✓ 10
21.c even 2 1 5733.2.a.bw 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1911.2.a.x ✓ 10 7.b odd 2 1
1911.2.a.y yes 10 1.a even 1 1 trivial
5733.2.a.bw 10 21.c even 2 1
5733.2.a.bx 10 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1911))\):

\( T_{2}^{10} - 4T_{2}^{9} - 10T_{2}^{8} + 52T_{2}^{7} + 16T_{2}^{6} - 212T_{2}^{5} + 64T_{2}^{4} + 300T_{2}^{3} - 159T_{2}^{2} - 80T_{2} + 46 \) Copy content Toggle raw display
\( T_{5}^{10} + 6 T_{5}^{9} - 23 T_{5}^{8} - 160 T_{5}^{7} + 196 T_{5}^{6} + 1456 T_{5}^{5} - 956 T_{5}^{4} + \cdots + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 4 T^{9} + \cdots + 46 \) Copy content Toggle raw display
$3$ \( (T - 1)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + 6 T^{9} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( T^{10} - 12 T^{9} + \cdots + 101776 \) Copy content Toggle raw display
$13$ \( (T + 1)^{10} \) Copy content Toggle raw display
$17$ \( T^{10} - 98 T^{8} + \cdots + 60928 \) Copy content Toggle raw display
$19$ \( T^{10} - 10 T^{9} + \cdots + 2243584 \) Copy content Toggle raw display
$23$ \( T^{10} - 14 T^{9} + \cdots - 21787136 \) Copy content Toggle raw display
$29$ \( T^{10} - 18 T^{9} + \cdots + 15341312 \) Copy content Toggle raw display
$31$ \( T^{10} - 14 T^{9} + \cdots - 38642688 \) Copy content Toggle raw display
$37$ \( T^{10} - 24 T^{9} + \cdots + 33918976 \) Copy content Toggle raw display
$41$ \( T^{10} + 24 T^{9} + \cdots + 150752 \) Copy content Toggle raw display
$43$ \( T^{10} - 2 T^{9} + \cdots - 868096 \) Copy content Toggle raw display
$47$ \( T^{10} + 18 T^{9} + \cdots - 2080316 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots - 702861568 \) Copy content Toggle raw display
$59$ \( T^{10} + 12 T^{9} + \cdots + 30888592 \) Copy content Toggle raw display
$61$ \( T^{10} + 4 T^{9} + \cdots + 131072 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 224530432 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots - 152918752 \) Copy content Toggle raw display
$73$ \( T^{10} + 18 T^{9} + \cdots - 1751296 \) Copy content Toggle raw display
$79$ \( T^{10} - 34 T^{9} + \cdots + 8925328 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 225377852 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 221391652 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 867344128 \) Copy content Toggle raw display
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