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Properties

Label 2169.2.a.j
Level $2169$
Weight $2$
Character orbit 2169.a
Self dual yes
Analytic conductor $17.320$
Analytic rank $1$
Dimension $14$
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] = [2169,2,Mod(1,2169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2169, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2169.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2169 = 3^{2} \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2169.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: \(17.3195521984\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 16x^{12} + 98x^{10} - 289x^{8} + 420x^{6} - 271x^{4} + 60x^{2} - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{4} - \beta_{12} q^{5} + ( - \beta_{8} - 1) q^{7} + (\beta_{13} + \beta_{12}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{2} q^{4} - \beta_{12} q^{5} + ( - \beta_{8} - 1) q^{7} + (\beta_{13} + \beta_{12}) q^{8} + (\beta_{5} - \beta_{4} - \beta_{2}) q^{10} + ( - \beta_{13} + \beta_{10} + \cdots - \beta_1) q^{11}+ \cdots + (2 \beta_{10} + \beta_{9} + \cdots - \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 4 q^{4} - 14 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 4 q^{4} - 14 q^{7} - 6 q^{10} - 8 q^{13} - 16 q^{16} - 18 q^{19} - 16 q^{22} - 14 q^{25} - 18 q^{28} - 42 q^{31} - 14 q^{34} - 14 q^{37} - 32 q^{40} - 26 q^{43} - 14 q^{46} - 24 q^{49} - 38 q^{55} - 56 q^{58} - 22 q^{61} - 26 q^{64} - 54 q^{67} + 4 q^{70} - 12 q^{73} - 58 q^{76} - 54 q^{79} - 6 q^{82} - 40 q^{85} + 22 q^{88} - 36 q^{91} - 22 q^{94} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 16x^{12} + 98x^{10} - 289x^{8} + 420x^{6} - 271x^{4} + 60x^{2} - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{11} - 13\nu^{9} + 58\nu^{7} - 102\nu^{5} + 56\nu^{3} - \nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{12} - 13\nu^{10} + 58\nu^{8} - 102\nu^{6} + 56\nu^{4} - \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{10} - 13\nu^{8} + 58\nu^{6} - 103\nu^{4} + 62\nu^{2} - 7 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{10} - 13\nu^{8} + 59\nu^{6} - 111\nu^{4} + 77\nu^{2} - 10 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{12} - 14\nu^{10} + 72\nu^{8} - 171\nu^{6} + 196\nu^{4} - 101\nu^{2} + 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( -\nu^{12} + 14\nu^{10} - 71\nu^{8} + 160\nu^{6} - 158\nu^{4} + 58\nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{13} - 18\nu^{11} + 124\nu^{9} - 405\nu^{7} + 624\nu^{5} - 381\nu^{3} + 52\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -3\nu^{13} + 50\nu^{11} - 318\nu^{9} + 959\nu^{7} - 1366\nu^{5} + 765\nu^{3} - 90\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -5\nu^{13} + 78\nu^{11} - 462\nu^{9} + 1301\nu^{7} - 1758\nu^{5} + 963\nu^{3} - 98\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3\nu^{13} - 46\nu^{11} + 266\nu^{9} - 725\nu^{7} + 940\nu^{5} - 495\nu^{3} + 56\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -3\nu^{13} + 46\nu^{11} - 266\nu^{9} + 725\nu^{7} - 940\nu^{5} + 497\nu^{3} - 64\nu ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} + \beta_{12} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} - \beta_{5} + \beta_{4} + 5\beta_{2} + 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6\beta_{13} + 7\beta_{12} + \beta_{11} + \beta_{10} + \beta_{9} + 17\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{8} + \beta_{6} - 9\beta_{5} + 8\beta_{4} + 25\beta_{2} + 29 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 31\beta_{13} + 41\beta_{12} + 9\beta_{11} + 11\beta_{10} + 9\beta_{9} - 2\beta_{3} + 76\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 51\beta_{8} + 2\beta_{7} + 11\beta_{6} - 61\beta_{5} + 50\beta_{4} + 128\beta_{2} + 132 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 155\beta_{13} + 226\beta_{12} + 59\beta_{11} + 85\beta_{10} + 62\beta_{9} - 22\beta_{3} + 356\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 302\beta_{8} + 26\beta_{7} + 85\beta_{6} - 373\beta_{5} + 289\beta_{4} + 667\beta_{2} + 638 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 773\beta_{13} + 1218\beta_{12} + 347\beta_{11} + 569\beta_{10} + 386\beta_{9} - 169\beta_{3} + 1731\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1728\beta_{8} + 222\beta_{7} + 569\beta_{6} - 2173\beta_{5} + 1618\beta_{4} + 3518\beta_{2} + 3205 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 3886\beta_{13} + 6518\beta_{12} + 1951\beta_{11} + 3533\beta_{10} + 2283\beta_{9} - 1124\beta_{3} + 8658\beta_1 \) 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
−2.32868
−1.99148
−1.64933
−1.58543
−1.02096
−0.460272
−0.350967
0.350967
0.460272
1.02096
1.58543
1.64933
1.99148
2.32868
−2.32868 0 3.42277 2.80618 0 −3.18068 −3.31319 0 −6.53470
1.2 −1.99148 0 1.96597 0.153668 0 −1.55903 0.0677634 0 −0.306025
1.3 −1.64933 0 0.720286 0.491742 0 2.29223 2.11067 0 −0.811044
1.4 −1.58543 0 0.513574 −2.87384 0 −2.82002 2.35662 0 4.55627
1.5 −1.02096 0 −0.957634 0.778672 0 1.87793 3.01964 0 −0.794995
1.6 −0.460272 0 −1.78815 −3.00443 0 −2.58034 1.74358 0 1.38286
1.7 −0.350967 0 −1.87682 1.40287 0 −1.03010 1.36064 0 −0.492360
1.8 0.350967 0 −1.87682 −1.40287 0 −1.03010 −1.36064 0 −0.492360
1.9 0.460272 0 −1.78815 3.00443 0 −2.58034 −1.74358 0 1.38286
1.10 1.02096 0 −0.957634 −0.778672 0 1.87793 −3.01964 0 −0.794995
1.11 1.58543 0 0.513574 2.87384 0 −2.82002 −2.35662 0 4.55627
1.12 1.64933 0 0.720286 −0.491742 0 2.29223 −2.11067 0 −0.811044
1.13 1.99148 0 1.96597 −0.153668 0 −1.55903 −0.0677634 0 −0.306025
1.14 2.32868 0 3.42277 −2.80618 0 −3.18068 3.31319 0 −6.53470
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(241\) \(1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2169.2.a.j ✓ 14
3.b odd 2 1 inner 2169.2.a.j ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2169.2.a.j ✓ 14 1.a even 1 1 trivial
2169.2.a.j ✓ 14 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} - 16T_{2}^{12} + 98T_{2}^{10} - 289T_{2}^{8} + 420T_{2}^{6} - 271T_{2}^{4} + 60T_{2}^{2} - 4 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2169))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} - 16 T^{12} + \cdots - 4 \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( T^{14} - 28 T^{12} + \cdots - 4 \) Copy content Toggle raw display
$7$ \( (T^{7} + 7 T^{6} + \cdots + 160)^{2} \) Copy content Toggle raw display
$11$ \( T^{14} - 60 T^{12} + \cdots - 625 \) Copy content Toggle raw display
$13$ \( (T^{7} + 4 T^{6} + \cdots - 334)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} - 116 T^{12} + \cdots - 39816100 \) Copy content Toggle raw display
$19$ \( (T^{7} + 9 T^{6} + \cdots + 29900)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} - 154 T^{12} + \cdots - 2128681 \) Copy content Toggle raw display
$29$ \( T^{14} - 167 T^{12} + \cdots - 763876 \) Copy content Toggle raw display
$31$ \( (T^{7} + 21 T^{6} + \cdots - 278)^{2} \) Copy content Toggle raw display
$37$ \( (T^{7} + 7 T^{6} + \cdots + 12088)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} - 184 T^{12} + \cdots - 52128400 \) Copy content Toggle raw display
$43$ \( (T^{7} + 13 T^{6} + \cdots - 950)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 3514402606276 \) Copy content Toggle raw display
$53$ \( T^{14} - 177 T^{12} + \cdots - 58491904 \) Copy content Toggle raw display
$59$ \( T^{14} - 239 T^{12} + \cdots - 33154564 \) Copy content Toggle raw display
$61$ \( (T^{7} + 11 T^{6} + \cdots - 29123)^{2} \) Copy content Toggle raw display
$67$ \( (T^{7} + 27 T^{6} + \cdots + 21905)^{2} \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 443111223556 \) Copy content Toggle raw display
$73$ \( (T^{7} + 6 T^{6} + \cdots - 5014)^{2} \) Copy content Toggle raw display
$79$ \( (T^{7} + 27 T^{6} + \cdots - 10429)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} - 175 T^{12} + \cdots - 53173264 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 59388227809 \) Copy content Toggle raw display
$97$ \( (T^{7} + 2 T^{6} + \cdots - 357829)^{2} \) Copy content Toggle raw display
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