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Properties

Label 867.2.e.f
Level $867$
Weight $2$
Character orbit 867.e
Analytic conductor $6.923$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [867,2,Mod(616,867)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(867, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("867.616");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 867 = 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 867.e (of order \(4\), degree \(2\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.5473632256.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 49x^{4} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 51)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + \beta_{7} q^{3} + ( - \beta_{4} - 2) q^{4} + (2 \beta_{7} + \beta_{6}) q^{5} + (\beta_{2} + \beta_1) q^{6} + ( - \beta_{5} + 4 \beta_{3}) q^{8} - \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + \beta_{7} q^{3} + ( - \beta_{4} - 2) q^{4} + (2 \beta_{7} + \beta_{6}) q^{5} + (\beta_{2} + \beta_1) q^{6} + ( - \beta_{5} + 4 \beta_{3}) q^{8} - \beta_{3} q^{9} + (6 \beta_{2} + 2 \beta_1) q^{10} + \beta_1 q^{11} + ( - 3 \beta_{7} - \beta_{6}) q^{12} + (\beta_{4} - 3) q^{13} + (\beta_{5} - \beta_{3}) q^{15} + 3 \beta_{4} q^{16} - \beta_{4} q^{18} + (3 \beta_{5} + 3 \beta_{3}) q^{19} + ( - 10 \beta_{7} - 4 \beta_{6}) q^{20} - 4 \beta_{7} q^{22} + ( - 4 \beta_{2} + \beta_1) q^{23} + ( - 5 \beta_{2} - \beta_1) q^{24} + 3 \beta_{5} q^{25} + ( - 2 \beta_{5} - 4 \beta_{3}) q^{26} + \beta_{2} q^{27} + ( - 2 \beta_{7} - 4 \beta_{6}) q^{29} + ( - 2 \beta_{4} - 4) q^{30} - 2 \beta_{6} q^{31} + (\beta_{5} - 4 \beta_{3}) q^{32} + ( - \beta_{4} + 1) q^{33} + ( - \beta_{5} + 2 \beta_{3}) q^{36} + (2 \beta_{7} + 2 \beta_{6}) q^{37} - 12 q^{38} + ( - 2 \beta_{7} + \beta_{6}) q^{39} + ( - 14 \beta_{2} - 6 \beta_1) q^{40} + (2 \beta_{2} + \beta_1) q^{41} + (3 \beta_{5} + 3 \beta_{3}) q^{43} + ( - 4 \beta_{2} - 2 \beta_1) q^{44} + (2 \beta_{2} + \beta_1) q^{45} + 4 \beta_{6} q^{46} + (2 \beta_{4} + 6) q^{47} + (3 \beta_{7} + 3 \beta_{6}) q^{48} - 7 \beta_{3} q^{49} + ( - 3 \beta_{4} - 12) q^{50} + 2 q^{52} + ( - 4 \beta_{5} + 2 \beta_{3}) q^{53} + ( - \beta_{7} - \beta_{6}) q^{54} + ( - \beta_{4} - 3) q^{55} + 3 \beta_1 q^{57} + ( - 18 \beta_{2} - 2 \beta_1) q^{58} + (2 \beta_{5} - 2 \beta_{3}) q^{59} + ( - 4 \beta_{5} + 6 \beta_{3}) q^{60} + ( - 6 \beta_{2} - 2 \beta_1) q^{61} - 8 \beta_{2} q^{62} + (\beta_{4} - 4) q^{64} - \beta_{6} q^{65} + 4 \beta_{3} q^{66} + 4 q^{67} + ( - \beta_{4} + 5) q^{69} + 4 \beta_{6} q^{71} + (\beta_{4} + 4) q^{72} + ( - 6 \beta_{7} - 4 \beta_{6}) q^{73} + (10 \beta_{2} + 2 \beta_1) q^{74} + (3 \beta_{2} + 3 \beta_1) q^{75} + ( - 6 \beta_{5} + 6 \beta_{3}) q^{76} + (2 \beta_{2} - 2 \beta_1) q^{78} - 6 \beta_1 q^{79} + (18 \beta_{7} + 6 \beta_{6}) q^{80} - q^{81} + ( - 6 \beta_{7} - 2 \beta_{6}) q^{82} + ( - 2 \beta_{5} - 6 \beta_{3}) q^{83} - 12 q^{86} + ( - 4 \beta_{5} - 2 \beta_{3}) q^{87} + (4 \beta_{7} + 4 \beta_{6}) q^{88} + (2 \beta_{4} - 4) q^{89} + ( - 6 \beta_{7} - 2 \beta_{6}) q^{90} + (8 \beta_{2} + 2 \beta_1) q^{92} + ( - 2 \beta_{5} - 2 \beta_{3}) q^{93} + (8 \beta_{5} - 8 \beta_{3}) q^{94} + (12 \beta_{2} + 3 \beta_1) q^{95} + (5 \beta_{2} + \beta_1) q^{96} + ( - 6 \beta_{7} + 2 \beta_{6}) q^{97} - 7 \beta_{4} q^{98} - \beta_{6} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 20 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 20 q^{4} - 20 q^{13} + 12 q^{16} - 4 q^{18} - 40 q^{30} + 4 q^{33} - 96 q^{38} + 56 q^{47} - 108 q^{50} + 16 q^{52} - 28 q^{55} - 28 q^{64} + 32 q^{67} + 36 q^{69} + 36 q^{72} - 8 q^{81} - 96 q^{86} - 24 q^{89} - 28 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 49x^{4} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 29\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 65\nu^{2} ) / 144 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} + 29 ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{6} - 181\nu^{2} ) / 144 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + 65\nu^{3} ) / 144 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{7} + 181\nu^{3} ) / 576 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 5\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{7} + 5\beta_{6} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{4} - 29 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 36\beta_{2} - 29\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -65\beta_{5} - 181\beta_{3} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 260\beta_{7} - 181\beta_{6} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(290\) \(292\)
\(\chi(n)\) \(1\) \(-\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
616.1
1.10418 + 1.10418i
−1.10418 − 1.10418i
−1.81129 − 1.81129i
1.81129 + 1.81129i
−1.81129 + 1.81129i
1.81129 − 1.81129i
1.10418 − 1.10418i
−1.10418 + 1.10418i
− 2.56155i −0.707107 + 0.707107i −4.56155 −2.51840 + 2.51840i 1.81129 + 1.81129i 0 6.56155i − 1.00000i 6.45101 + 6.45101i
616.2 − 2.56155i 0.707107 − 0.707107i −4.56155 2.51840 − 2.51840i −1.81129 − 1.81129i 0 6.56155i − 1.00000i −6.45101 − 6.45101i
616.3 1.56155i −0.707107 + 0.707107i −0.438447 0.397078 − 0.397078i −1.10418 − 1.10418i 0 2.43845i − 1.00000i 0.620058 + 0.620058i
616.4 1.56155i 0.707107 − 0.707107i −0.438447 −0.397078 + 0.397078i 1.10418 + 1.10418i 0 2.43845i − 1.00000i −0.620058 − 0.620058i
829.1 − 1.56155i −0.707107 − 0.707107i −0.438447 0.397078 + 0.397078i −1.10418 + 1.10418i 0 − 2.43845i 1.00000i 0.620058 − 0.620058i
829.2 − 1.56155i 0.707107 + 0.707107i −0.438447 −0.397078 − 0.397078i 1.10418 − 1.10418i 0 − 2.43845i 1.00000i −0.620058 + 0.620058i
829.3 2.56155i −0.707107 − 0.707107i −4.56155 −2.51840 − 2.51840i 1.81129 − 1.81129i 0 − 6.56155i 1.00000i 6.45101 − 6.45101i
829.4 2.56155i 0.707107 + 0.707107i −4.56155 2.51840 + 2.51840i −1.81129 + 1.81129i 0 − 6.56155i 1.00000i −6.45101 + 6.45101i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 616.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner
17.c even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 867.2.e.f 8
17.b even 2 1 inner 867.2.e.f 8
17.c even 4 2 inner 867.2.e.f 8
17.d even 8 1 51.2.a.b ✓ 2
17.d even 8 1 867.2.a.f 2
17.d even 8 2 867.2.d.c 4
17.e odd 16 8 867.2.h.j 16
51.g odd 8 1 153.2.a.e 2
51.g odd 8 1 2601.2.a.t 2
68.g odd 8 1 816.2.a.m 2
85.k odd 8 1 1275.2.b.d 4
85.m even 8 1 1275.2.a.n 2
85.n odd 8 1 1275.2.b.d 4
119.l odd 8 1 2499.2.a.o 2
136.o even 8 1 3264.2.a.bl 2
136.p odd 8 1 3264.2.a.bg 2
187.i odd 8 1 6171.2.a.p 2
204.p even 8 1 2448.2.a.v 2
221.p even 8 1 8619.2.a.q 2
255.y odd 8 1 3825.2.a.s 2
357.w even 8 1 7497.2.a.v 2
408.bd even 8 1 9792.2.a.cz 2
408.be odd 8 1 9792.2.a.cy 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.a.b ✓ 2 17.d even 8 1
153.2.a.e 2 51.g odd 8 1
816.2.a.m 2 68.g odd 8 1
867.2.a.f 2 17.d even 8 1
867.2.d.c 4 17.d even 8 2
867.2.e.f 8 1.a even 1 1 trivial
867.2.e.f 8 17.b even 2 1 inner
867.2.e.f 8 17.c even 4 2 inner
867.2.h.j 16 17.e odd 16 8
1275.2.a.n 2 85.m even 8 1
1275.2.b.d 4 85.k odd 8 1
1275.2.b.d 4 85.n odd 8 1
2448.2.a.v 2 204.p even 8 1
2499.2.a.o 2 119.l odd 8 1
2601.2.a.t 2 51.g odd 8 1
3264.2.a.bg 2 136.p odd 8 1
3264.2.a.bl 2 136.o even 8 1
3825.2.a.s 2 255.y odd 8 1
6171.2.a.p 2 187.i odd 8 1
7497.2.a.v 2 357.w even 8 1
8619.2.a.q 2 221.p even 8 1
9792.2.a.cy 2 408.be odd 8 1
9792.2.a.cz 2 408.bd even 8 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(867, [\chi])\):

\( T_{2}^{4} + 9T_{2}^{2} + 16 \) Copy content Toggle raw display
\( T_{5}^{8} + 161T_{5}^{4} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 9 T^{2} + 16)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 161T^{4} + 16 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 49T^{4} + 256 \) Copy content Toggle raw display
$13$ \( (T^{2} + 5 T + 2)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} + 81 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 1889 T^{4} + 65536 \) Copy content Toggle raw display
$29$ \( (T^{4} + 4624)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 784 T^{4} + 65536 \) Copy content Toggle raw display
$37$ \( T^{8} + 784 T^{4} + 65536 \) Copy content Toggle raw display
$41$ \( T^{8} + 161T^{4} + 16 \) Copy content Toggle raw display
$43$ \( (T^{4} + 81 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 14 T + 32)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + 168 T^{2} + 2704)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 52 T^{2} + 64)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 6928 T^{4} + 4096 \) Copy content Toggle raw display
$67$ \( (T - 4)^{8} \) Copy content Toggle raw display
$71$ \( T^{8} + 12544 T^{4} + 16777216 \) Copy content Toggle raw display
$73$ \( T^{8} + 22816 T^{4} + 7311616 \) Copy content Toggle raw display
$79$ \( T^{8} + 63504 T^{4} + 429981696 \) Copy content Toggle raw display
$83$ \( (T^{4} + 84 T^{2} + 64)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 6 T - 8)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} + 15376 T^{4} + 1048576 \) Copy content Toggle raw display
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