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Properties

Label 867.2.d.g
Level $867$
Weight $2$
Character orbit 867.d
Analytic conductor $6.923$
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] = [867,2,Mod(577,867)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(867, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("867.577");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 867 = 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 867.d (of order \(2\), degree \(1\), not 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: \(6.92302985525\)
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: \( x^{12} + 18x^{10} + 117x^{8} + 342x^{6} + 438x^{4} + 180x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + (\beta_{6} - \beta_{2} - 1) q^{2} + \beta_{3} q^{3} + (2 \beta_{8} - \beta_{5} + \beta_{2} + 2) q^{4} + (\beta_{11} + \beta_{10} + \cdots + \beta_1) q^{5}+ \cdots - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} - \beta_{2} - 1) q^{2} + \beta_{3} q^{3} + (2 \beta_{8} - \beta_{5} + \beta_{2} + 2) q^{4} + (\beta_{11} + \beta_{10} + \cdots + \beta_1) q^{5}+ \cdots + ( - \beta_{11} - \beta_{7} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{2} + 18 q^{4} - 24 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{2} + 18 q^{4} - 24 q^{8} - 12 q^{9} + 18 q^{13} + 6 q^{15} + 30 q^{16} + 6 q^{18} - 18 q^{19} + 6 q^{21} - 30 q^{25} + 24 q^{26} + 24 q^{30} - 84 q^{32} - 18 q^{33} - 18 q^{36} - 12 q^{38} + 12 q^{42} + 48 q^{47} - 42 q^{49} + 24 q^{50} - 36 q^{52} - 48 q^{53} - 48 q^{55} + 18 q^{59} - 12 q^{60} + 48 q^{64} + 36 q^{66} - 12 q^{67} - 18 q^{69} + 6 q^{70} + 24 q^{72} + 6 q^{76} - 66 q^{77} + 12 q^{81} - 12 q^{83} - 30 q^{84} + 12 q^{86} + 12 q^{87} - 48 q^{93} + 30 q^{94} - 114 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 18x^{10} + 117x^{8} + 342x^{6} + 438x^{4} + 180x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 13\nu^{4} + 42\nu^{2} + 17 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{9} - 15\nu^{7} - 72\nu^{5} - 125\nu^{3} - 54\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{9} + 31\nu^{7} + 157\nu^{5} + 300\nu^{3} + 165\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{10} - 14\nu^{8} - 58\nu^{6} - 66\nu^{4} + 25\nu^{2} + 13 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{10} + 15\nu^{8} + 71\nu^{6} + 116\nu^{4} + 40\nu^{2} + 3 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{11} + 16\nu^{9} + 87\nu^{7} + 197\nu^{5} + 175\nu^{3} + 38\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{10} + 16\nu^{8} + 86\nu^{6} + 184\nu^{4} + 133\nu^{2} + 13 ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{10} - 16\nu^{8} - 86\nu^{6} - 184\nu^{4} - 125\nu^{2} + 11 ) / 8 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{11} + 18\nu^{9} + 116\nu^{7} + 328\nu^{5} + 383\nu^{3} + 121\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -2\nu^{11} - 35\nu^{9} - 218\nu^{7} - 601\nu^{5} - 723\nu^{3} - 297\nu ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + \beta_{8} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - \beta_{7} + \beta_{4} + 2\beta_{3} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{9} - 8\beta_{8} + 2\beta_{6} + \beta_{5} + 2\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{11} - 12\beta_{10} + 8\beta_{7} - 10\beta_{4} - 19\beta_{3} + 19\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 49\beta_{9} + 62\beta_{8} - 26\beta_{6} - 13\beta_{5} - 18\beta_{2} - 99 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 26\beta_{11} + 106\beta_{10} - 54\beta_{7} + 88\beta_{4} + 155\beta_{3} - 104\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -352\beta_{9} - 471\beta_{8} + 246\beta_{6} + 127\beta_{5} + 134\beta_{2} + 666 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -246\beta_{11} - 851\beta_{10} + 359\beta_{7} - 725\beta_{4} - 1211\beta_{3} + 638\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2573\beta_{9} + 3551\beta_{8} - 2068\beta_{6} - 1098\beta_{5} - 964\beta_{2} - 4700 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2068\beta_{11} + 6583\beta_{10} - 2439\beta_{7} + 5739\beta_{4} + 9284\beta_{3} - 4241\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/867\mathbb{Z}\right)^\times\).

\(n\) \(290\) \(292\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
0.0750494i
− 0.0750494i
− 2.06104i
2.06104i
2.73700i
− 2.73700i
1.71374i
− 1.71374i
− 0.857616i
0.857616i
− 1.60714i
1.60714i
−2.74819 − 1.00000i 5.55252 0.973936i 2.74819i − 1.60333i −9.76298 −1.00000 − 2.67656i
577.2 −2.74819 1.00000i 5.55252 − 0.973936i − 2.74819i 1.60333i −9.76298 −1.00000 2.67656i
577.3 −2.44395 − 1.00000i 3.97290 − 2.87349i 2.44395i − 1.56252i −4.82168 −1.00000 7.02266i
577.4 −2.44395 1.00000i 3.97290 2.87349i − 2.44395i 1.56252i −4.82168 −1.00000 − 7.02266i
577.5 −0.907065 − 1.00000i −1.17723 − 3.19333i 0.907065i 3.56234i 2.88196 −1.00000 2.89656i
577.6 −0.907065 1.00000i −1.17723 3.19333i − 0.907065i − 3.56234i 2.88196 −1.00000 − 2.89656i
577.7 −0.435433 − 1.00000i −1.81040 4.22078i 0.435433i 2.90981i 1.65917 −1.00000 − 1.83787i
577.8 −0.435433 1.00000i −1.81040 − 4.22078i − 0.435433i − 2.90981i 1.65917 −1.00000 1.83787i
577.9 1.43915 − 1.00000i 0.0711653 2.31394i − 1.43915i − 4.44173i −2.77589 −1.00000 3.33012i
577.10 1.43915 1.00000i 0.0711653 − 2.31394i 1.43915i 4.44173i −2.77589 −1.00000 − 3.33012i
577.11 2.09548 − 1.00000i 2.39104 1.55815i − 2.09548i 4.13541i 0.819422 −1.00000 3.26508i
577.12 2.09548 1.00000i 2.39104 − 1.55815i 2.09548i − 4.13541i 0.819422 −1.00000 − 3.26508i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 577.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 867.2.d.g 12
17.b even 2 1 inner 867.2.d.g 12
17.c even 4 1 867.2.a.o ✓ 6
17.c even 4 1 867.2.a.p yes 6
17.d even 8 4 867.2.e.k 24
17.e odd 16 8 867.2.h.m 48
51.f odd 4 1 2601.2.a.bh 6
51.f odd 4 1 2601.2.a.bi 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
867.2.a.o ✓ 6 17.c even 4 1
867.2.a.p yes 6 17.c even 4 1
867.2.d.g 12 1.a even 1 1 trivial
867.2.d.g 12 17.b even 2 1 inner
867.2.e.k 24 17.d even 8 4
867.2.h.m 48 17.e odd 16 8
2601.2.a.bh 6 51.f odd 4 1
2601.2.a.bi 6 51.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 3T_{2}^{5} - 6T_{2}^{4} - 19T_{2}^{3} + 6T_{2}^{2} + 24T_{2} + 8 \) acting on \(S_{2}^{\mathrm{new}}(867, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 3 T^{5} - 6 T^{4} + \cdots + 8)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} + 45 T^{10} + \cdots + 18496 \) Copy content Toggle raw display
$7$ \( T^{12} + 63 T^{10} + \cdots + 227529 \) Copy content Toggle raw display
$11$ \( T^{12} + 93 T^{10} + \cdots + 87616 \) Copy content Toggle raw display
$13$ \( (T^{6} - 9 T^{5} + \cdots - 109)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( (T^{6} + 9 T^{5} + \cdots + 333)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 153 T^{10} + \cdots + 2483776 \) Copy content Toggle raw display
$29$ \( T^{12} + 42 T^{10} + \cdots + 18496 \) Copy content Toggle raw display
$31$ \( T^{12} + 204 T^{10} + \cdots + 16129 \) Copy content Toggle raw display
$37$ \( T^{12} + 159 T^{10} + \cdots + 11881 \) Copy content Toggle raw display
$41$ \( T^{12} + 198 T^{10} + \cdots + 2534464 \) Copy content Toggle raw display
$43$ \( (T^{6} - 117 T^{4} + \cdots - 11736)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 24 T^{5} + \cdots + 20312)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 24 T^{5} + \cdots - 5608)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 9 T^{5} + \cdots + 1432)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 1013849281 \) Copy content Toggle raw display
$67$ \( (T^{6} + 6 T^{5} + \cdots - 14552)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + 315 T^{10} + \cdots + 75759616 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 8543120041 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 987153561 \) Copy content Toggle raw display
$83$ \( (T^{6} + 6 T^{5} + \cdots - 205336)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 327 T^{4} + \cdots - 39896)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 302377321 \) Copy content Toggle raw display
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