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Properties

Label 867.2.e.a
Level $867$
Weight $2$
Character orbit 867.e
Analytic conductor $6.923$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) 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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \zeta_{8}^{2} q^{2} - \zeta_{8} q^{3} - 2 q^{4} + 3 \zeta_{8} q^{5} + 2 \zeta_{8}^{3} q^{6} + 2 \zeta_{8}^{3} q^{7} + \zeta_{8}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 \zeta_{8}^{2} q^{2} - \zeta_{8} q^{3} - 2 q^{4} + 3 \zeta_{8} q^{5} + 2 \zeta_{8}^{3} q^{6} + 2 \zeta_{8}^{3} q^{7} + \zeta_{8}^{2} q^{9} - 6 \zeta_{8}^{3} q^{10} + 5 \zeta_{8}^{3} q^{11} + 2 \zeta_{8} q^{12} + q^{13} + 4 \zeta_{8} q^{14} - 3 \zeta_{8}^{2} q^{15} - 4 q^{16} + 2 q^{18} + 5 \zeta_{8}^{2} q^{19} - 6 \zeta_{8} q^{20} + 2 q^{21} + 10 \zeta_{8} q^{22} + \zeta_{8}^{3} q^{23} + 4 \zeta_{8}^{2} q^{25} - 2 \zeta_{8}^{2} q^{26} - \zeta_{8}^{3} q^{27} - 4 \zeta_{8}^{3} q^{28} + 6 \zeta_{8} q^{29} - 6 q^{30} + 10 \zeta_{8} q^{31} + 8 \zeta_{8}^{2} q^{32} + 5 q^{33} - 6 q^{35} - 2 \zeta_{8}^{2} q^{36} - 2 \zeta_{8} q^{37} + 10 q^{38} - \zeta_{8} q^{39} - 5 \zeta_{8}^{3} q^{41} - 4 \zeta_{8}^{2} q^{42} + \zeta_{8}^{2} q^{43} - 10 \zeta_{8}^{3} q^{44} + 3 \zeta_{8}^{3} q^{45} + 2 \zeta_{8} q^{46} + 2 q^{47} + 4 \zeta_{8} q^{48} + 3 \zeta_{8}^{2} q^{49} + 8 q^{50} - 2 q^{52} - 6 \zeta_{8}^{2} q^{53} - 2 \zeta_{8} q^{54} - 15 q^{55} - 5 \zeta_{8}^{3} q^{57} - 12 \zeta_{8}^{3} q^{58} + 6 \zeta_{8}^{2} q^{60} + 10 \zeta_{8}^{3} q^{61} - 20 \zeta_{8}^{3} q^{62} - 2 \zeta_{8} q^{63} + 8 q^{64} + 3 \zeta_{8} q^{65} - 10 \zeta_{8}^{2} q^{66} - 12 q^{67} + q^{69} + 12 \zeta_{8}^{2} q^{70} - 6 \zeta_{8} q^{73} + 4 \zeta_{8}^{3} q^{74} - 4 \zeta_{8}^{3} q^{75} - 10 \zeta_{8}^{2} q^{76} - 10 \zeta_{8}^{2} q^{77} + 2 \zeta_{8}^{3} q^{78} + 4 \zeta_{8}^{3} q^{79} - 12 \zeta_{8} q^{80} - q^{81} - 10 \zeta_{8} q^{82} - 6 \zeta_{8}^{2} q^{83} - 4 q^{84} + 2 q^{86} - 6 \zeta_{8}^{2} q^{87} + 10 q^{89} + 6 \zeta_{8} q^{90} + 2 \zeta_{8}^{3} q^{91} - 2 \zeta_{8}^{3} q^{92} - 10 \zeta_{8}^{2} q^{93} - 4 \zeta_{8}^{2} q^{94} + 15 \zeta_{8}^{3} q^{95} - 8 \zeta_{8}^{3} q^{96} - 8 \zeta_{8} q^{97} + 6 q^{98} - 5 \zeta_{8} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} + 4 q^{13} - 16 q^{16} + 8 q^{18} + 8 q^{21} - 24 q^{30} + 20 q^{33} - 24 q^{35} + 40 q^{38} + 8 q^{47} + 32 q^{50} - 8 q^{52} - 60 q^{55} + 32 q^{64} - 48 q^{67} + 4 q^{69} - 4 q^{81} - 16 q^{84} + 8 q^{86} + 40 q^{89} + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(290\) \(292\)
\(\chi(n)\) \(1\) \(\zeta_{8}^{2}\)

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
0.707107 − 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
−0.707107 − 0.707107i
2.00000i −0.707107 + 0.707107i −2.00000 2.12132 − 2.12132i −1.41421 − 1.41421i −1.41421 − 1.41421i 0 − 1.00000i 4.24264 + 4.24264i
616.2 2.00000i 0.707107 − 0.707107i −2.00000 −2.12132 + 2.12132i 1.41421 + 1.41421i 1.41421 + 1.41421i 0 − 1.00000i −4.24264 − 4.24264i
829.1 − 2.00000i −0.707107 − 0.707107i −2.00000 2.12132 + 2.12132i −1.41421 + 1.41421i −1.41421 + 1.41421i 0 1.00000i 4.24264 − 4.24264i
829.2 − 2.00000i 0.707107 + 0.707107i −2.00000 −2.12132 − 2.12132i 1.41421 − 1.41421i 1.41421 − 1.41421i 0 1.00000i −4.24264 + 4.24264i
\(n\): e.g. 2-40 or 990-1000
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.a 4
17.b even 2 1 inner 867.2.e.a 4
17.c even 4 2 inner 867.2.e.a 4
17.d even 8 2 51.2.d.a ✓ 2
17.d even 8 1 867.2.a.d 1
17.d even 8 1 867.2.a.e 1
17.e odd 16 8 867.2.h.e 8
51.g odd 8 2 153.2.d.c 2
51.g odd 8 1 2601.2.a.a 1
51.g odd 8 1 2601.2.a.c 1
68.g odd 8 2 816.2.c.b 2
85.k odd 8 1 1275.2.d.a 2
85.k odd 8 1 1275.2.d.c 2
85.m even 8 2 1275.2.g.b 2
85.n odd 8 1 1275.2.d.a 2
85.n odd 8 1 1275.2.d.c 2
136.o even 8 2 3264.2.c.g 2
136.p odd 8 2 3264.2.c.h 2
204.p even 8 2 2448.2.c.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.d.a ✓ 2 17.d even 8 2
153.2.d.c 2 51.g odd 8 2
816.2.c.b 2 68.g odd 8 2
867.2.a.d 1 17.d even 8 1
867.2.a.e 1 17.d even 8 1
867.2.e.a 4 1.a even 1 1 trivial
867.2.e.a 4 17.b even 2 1 inner
867.2.e.a 4 17.c even 4 2 inner
867.2.h.e 8 17.e odd 16 8
1275.2.d.a 2 85.k odd 8 1
1275.2.d.a 2 85.n odd 8 1
1275.2.d.c 2 85.k odd 8 1
1275.2.d.c 2 85.n odd 8 1
1275.2.g.b 2 85.m even 8 2
2448.2.c.f 2 204.p even 8 2
2601.2.a.a 1 51.g odd 8 1
2601.2.a.c 1 51.g odd 8 1
3264.2.c.g 2 136.o even 8 2
3264.2.c.h 2 136.p odd 8 2

Hecke kernels

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

\( T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{5}^{4} + 81 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} + 81 \) Copy content Toggle raw display
$7$ \( T^{4} + 16 \) Copy content Toggle raw display
$11$ \( T^{4} + 625 \) Copy content Toggle raw display
$13$ \( (T - 1)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 1 \) Copy content Toggle raw display
$29$ \( T^{4} + 1296 \) Copy content Toggle raw display
$31$ \( T^{4} + 10000 \) Copy content Toggle raw display
$37$ \( T^{4} + 16 \) Copy content Toggle raw display
$41$ \( T^{4} + 625 \) Copy content Toggle raw display
$43$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$47$ \( (T - 2)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} + 10000 \) Copy content Toggle raw display
$67$ \( (T + 12)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 1296 \) Copy content Toggle raw display
$79$ \( T^{4} + 256 \) Copy content Toggle raw display
$83$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$89$ \( (T - 10)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 4096 \) Copy content Toggle raw display
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