Properties

Label 294.2.d.b
Level $294$
Weight $2$
Character orbit 294.d
Analytic conductor $2.348$
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] = [294,2,Mod(293,294)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(294, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("294.293");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 294.d (of order \(2\), degree \(1\), 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: \(2.34760181943\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 primitive root of unity \(\zeta_{16}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{16}^{4} q^{2} + (\zeta_{16}^{7} + \zeta_{16}^{5} + \zeta_{16}) q^{3} - q^{4} + ( - 2 \zeta_{16}^{7} + 2 \zeta_{16}) q^{5} + ( - \zeta_{16}^{5} + \zeta_{16}^{3} + \zeta_{16}) q^{6} + \zeta_{16}^{4} q^{8} + (\zeta_{16}^{6} - 2 \zeta_{16}^{4} - 2) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{16}^{4} q^{2} + (\zeta_{16}^{7} + \zeta_{16}^{5} + \zeta_{16}) q^{3} - q^{4} + ( - 2 \zeta_{16}^{7} + 2 \zeta_{16}) q^{5} + ( - \zeta_{16}^{5} + \zeta_{16}^{3} + \zeta_{16}) q^{6} + \zeta_{16}^{4} q^{8} + (\zeta_{16}^{6} - 2 \zeta_{16}^{4} - 2) q^{9} + ( - 2 \zeta_{16}^{5} - 2 \zeta_{16}^{3}) q^{10} + (3 \zeta_{16}^{6} - 2 \zeta_{16}^{4} + 3 \zeta_{16}^{2}) q^{11} + ( - \zeta_{16}^{7} - \zeta_{16}^{5} - \zeta_{16}) q^{12} + ( - 2 \zeta_{16}^{7} + 2 \zeta_{16}^{5} + 2 \zeta_{16}^{3} - 2 \zeta_{16}) q^{13} + (4 \zeta_{16}^{6} + 2 \zeta_{16}^{4} + 2 \zeta_{16}^{2}) q^{15} + q^{16} + ( - 3 \zeta_{16}^{7} + 3 \zeta_{16}) q^{17} + (2 \zeta_{16}^{4} + \zeta_{16}^{2} - 2) q^{18} + (\zeta_{16}^{7} + \zeta_{16}) q^{19} + (2 \zeta_{16}^{7} - 2 \zeta_{16}) q^{20} + ( - 3 \zeta_{16}^{6} + 3 \zeta_{16}^{2} - 2) q^{22} + ( - 2 \zeta_{16}^{6} - 2 \zeta_{16}^{2}) q^{23} + (\zeta_{16}^{5} - \zeta_{16}^{3} - \zeta_{16}) q^{24} + ( - 4 \zeta_{16}^{6} + 4 \zeta_{16}^{2} + 3) q^{25} + ( - 2 \zeta_{16}^{7} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{3} + 2 \zeta_{16}) q^{26} + ( - \zeta_{16}^{7} - 5 \zeta_{16}^{5} + \zeta_{16}^{3}) q^{27} + ( - 2 \zeta_{16}^{6} - 2 \zeta_{16}^{4} - 2 \zeta_{16}^{2}) q^{29} + ( - 2 \zeta_{16}^{6} + 4 \zeta_{16}^{2} + 2) q^{30} + ( - 2 \zeta_{16}^{7} + 4 \zeta_{16}^{5} + 4 \zeta_{16}^{3} - 2 \zeta_{16}) q^{31} - \zeta_{16}^{4} q^{32} + (6 \zeta_{16}^{7} - 5 \zeta_{16}^{5} + 2 \zeta_{16}^{3} - \zeta_{16}) q^{33} + ( - 3 \zeta_{16}^{5} - 3 \zeta_{16}^{3}) q^{34} + ( - \zeta_{16}^{6} + 2 \zeta_{16}^{4} + 2) q^{36} - 2 q^{37} + ( - \zeta_{16}^{5} + \zeta_{16}^{3}) q^{38} + (2 \zeta_{16}^{6} + 2 \zeta_{16}^{4} - 6 \zeta_{16}^{2} + 2) q^{39} + (2 \zeta_{16}^{5} + 2 \zeta_{16}^{3}) q^{40} + (5 \zeta_{16}^{5} - 5 \zeta_{16}^{3}) q^{41} + (5 \zeta_{16}^{6} - 5 \zeta_{16}^{2} - 2) q^{43} + ( - 3 \zeta_{16}^{6} + 2 \zeta_{16}^{4} - 3 \zeta_{16}^{2}) q^{44} + (6 \zeta_{16}^{7} - 2 \zeta_{16}^{5} - 4 \zeta_{16}^{3} - 4 \zeta_{16}) q^{45} + (2 \zeta_{16}^{6} - 2 \zeta_{16}^{2}) q^{46} + (2 \zeta_{16}^{5} - 2 \zeta_{16}^{3}) q^{47} + (\zeta_{16}^{7} + \zeta_{16}^{5} + \zeta_{16}) q^{48} + ( - 4 \zeta_{16}^{6} - 3 \zeta_{16}^{4} - 4 \zeta_{16}^{2}) q^{50} + (6 \zeta_{16}^{6} + 3 \zeta_{16}^{4} + 3 \zeta_{16}^{2}) q^{51} + (2 \zeta_{16}^{7} - 2 \zeta_{16}^{5} - 2 \zeta_{16}^{3} + 2 \zeta_{16}) q^{52} + ( - 6 \zeta_{16}^{6} - 2 \zeta_{16}^{4} - 6 \zeta_{16}^{2}) q^{53} + ( - \zeta_{16}^{7} - \zeta_{16}^{3} - 5 \zeta_{16}) q^{54} + (6 \zeta_{16}^{7} + 2 \zeta_{16}^{5} + 2 \zeta_{16}^{3} + 6 \zeta_{16}) q^{55} + ( - \zeta_{16}^{4} + \zeta_{16}^{2} - 2) q^{57} + (2 \zeta_{16}^{6} - 2 \zeta_{16}^{2} - 2) q^{58} + (\zeta_{16}^{7} - 2 \zeta_{16}^{5} + 2 \zeta_{16}^{3} - \zeta_{16}) q^{59} + ( - 4 \zeta_{16}^{6} - 2 \zeta_{16}^{4} - 2 \zeta_{16}^{2}) q^{60} + ( - 4 \zeta_{16}^{7} - 2 \zeta_{16}^{5} - 2 \zeta_{16}^{3} - 4 \zeta_{16}) q^{61} + ( - 4 \zeta_{16}^{7} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{3} + 4 \zeta_{16}) q^{62} - q^{64} + 8 \zeta_{16}^{4} q^{65} + ( - 2 \zeta_{16}^{7} + \zeta_{16}^{5} + 6 \zeta_{16}^{3} - 5 \zeta_{16}) q^{66} + (6 \zeta_{16}^{6} - 6 \zeta_{16}^{2} - 2) q^{67} + (3 \zeta_{16}^{7} - 3 \zeta_{16}) q^{68} + ( - 4 \zeta_{16}^{7} + 2 \zeta_{16}^{5} + 2 \zeta_{16}) q^{69} + ( - 2 \zeta_{16}^{6} - 8 \zeta_{16}^{4} - 2 \zeta_{16}^{2}) q^{71} + ( - 2 \zeta_{16}^{4} - \zeta_{16}^{2} + 2) q^{72} + (\zeta_{16}^{5} + \zeta_{16}^{3}) q^{73} + 2 \zeta_{16}^{4} q^{74} + (3 \zeta_{16}^{7} + 7 \zeta_{16}^{5} + 8 \zeta_{16}^{3} - \zeta_{16}) q^{75} + ( - \zeta_{16}^{7} - \zeta_{16}) q^{76} + (6 \zeta_{16}^{6} - 2 \zeta_{16}^{4} + 2 \zeta_{16}^{2} + 2) q^{78} + ( - 2 \zeta_{16}^{6} + 2 \zeta_{16}^{2} + 12) q^{79} + ( - 2 \zeta_{16}^{7} + 2 \zeta_{16}) q^{80} + ( - 4 \zeta_{16}^{6} + 7 \zeta_{16}^{4} + 4 \zeta_{16}^{2}) q^{81} + (5 \zeta_{16}^{7} + 5 \zeta_{16}) q^{82} + (5 \zeta_{16}^{7} + 2 \zeta_{16}^{5} - 2 \zeta_{16}^{3} - 5 \zeta_{16}) q^{83} + ( - 6 \zeta_{16}^{6} + 6 \zeta_{16}^{2} + 12) q^{85} + (5 \zeta_{16}^{6} + 2 \zeta_{16}^{4} + 5 \zeta_{16}^{2}) q^{86} + ( - 4 \zeta_{16}^{7} + 2 \zeta_{16}^{3} + 4 \zeta_{16}) q^{87} + (3 \zeta_{16}^{6} - 3 \zeta_{16}^{2} + 2) q^{88} + (4 \zeta_{16}^{7} + 5 \zeta_{16}^{5} - 5 \zeta_{16}^{3} - 4 \zeta_{16}) q^{89} + (4 \zeta_{16}^{7} + 4 \zeta_{16}^{5} + 6 \zeta_{16}^{3} - 2 \zeta_{16}) q^{90} + (2 \zeta_{16}^{6} + 2 \zeta_{16}^{2}) q^{92} + (4 \zeta_{16}^{6} + 2 \zeta_{16}^{4} - 10 \zeta_{16}^{2}) q^{93} + (2 \zeta_{16}^{7} + 2 \zeta_{16}) q^{94} + (2 \zeta_{16}^{6} + 2 \zeta_{16}^{2}) q^{95} + ( - \zeta_{16}^{5} + \zeta_{16}^{3} + \zeta_{16}) q^{96} + ( - 3 \zeta_{16}^{7} - 4 \zeta_{16}^{5} - 4 \zeta_{16}^{3} - 3 \zeta_{16}) q^{97} + ( - 12 \zeta_{16}^{6} + \zeta_{16}^{4} + 2 \zeta_{16}^{2} - 7) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} - 16 q^{9} + 8 q^{16} - 16 q^{18} - 16 q^{22} + 24 q^{25} + 16 q^{30} + 16 q^{36} - 16 q^{37} + 16 q^{39} - 16 q^{43} - 16 q^{57} - 16 q^{58} - 8 q^{64} - 16 q^{67} + 16 q^{72} + 16 q^{78} + 96 q^{79} + 96 q^{85} + 16 q^{88} - 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/294\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(199\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
293.1
−0.382683 + 0.923880i
0.923880 + 0.382683i
−0.923880 0.382683i
0.382683 0.923880i
−0.382683 0.923880i
0.923880 0.382683i
−0.923880 + 0.382683i
0.382683 + 0.923880i
1.00000i −0.923880 + 1.46508i −1.00000 −1.53073 1.46508 + 0.923880i 0 1.00000i −1.29289 2.70711i 1.53073i
293.2 1.00000i −0.382683 + 1.68925i −1.00000 3.69552 1.68925 + 0.382683i 0 1.00000i −2.70711 1.29289i 3.69552i
293.3 1.00000i 0.382683 1.68925i −1.00000 −3.69552 −1.68925 0.382683i 0 1.00000i −2.70711 1.29289i 3.69552i
293.4 1.00000i 0.923880 1.46508i −1.00000 1.53073 −1.46508 0.923880i 0 1.00000i −1.29289 2.70711i 1.53073i
293.5 1.00000i −0.923880 1.46508i −1.00000 −1.53073 1.46508 0.923880i 0 1.00000i −1.29289 + 2.70711i 1.53073i
293.6 1.00000i −0.382683 1.68925i −1.00000 3.69552 1.68925 0.382683i 0 1.00000i −2.70711 + 1.29289i 3.69552i
293.7 1.00000i 0.382683 + 1.68925i −1.00000 −3.69552 −1.68925 + 0.382683i 0 1.00000i −2.70711 + 1.29289i 3.69552i
293.8 1.00000i 0.923880 + 1.46508i −1.00000 1.53073 −1.46508 + 0.923880i 0 1.00000i −1.29289 + 2.70711i 1.53073i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 293.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 294.2.d.b 8
3.b odd 2 1 inner 294.2.d.b 8
4.b odd 2 1 2352.2.k.f 8
7.b odd 2 1 inner 294.2.d.b 8
7.c even 3 2 294.2.f.c 16
7.d odd 6 2 294.2.f.c 16
12.b even 2 1 2352.2.k.f 8
21.c even 2 1 inner 294.2.d.b 8
21.g even 6 2 294.2.f.c 16
21.h odd 6 2 294.2.f.c 16
28.d even 2 1 2352.2.k.f 8
84.h odd 2 1 2352.2.k.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.2.d.b 8 1.a even 1 1 trivial
294.2.d.b 8 3.b odd 2 1 inner
294.2.d.b 8 7.b odd 2 1 inner
294.2.d.b 8 21.c even 2 1 inner
294.2.f.c 16 7.c even 3 2
294.2.f.c 16 7.d odd 6 2
294.2.f.c 16 21.g even 6 2
294.2.f.c 16 21.h odd 6 2
2352.2.k.f 8 4.b odd 2 1
2352.2.k.f 8 12.b even 2 1
2352.2.k.f 8 28.d even 2 1
2352.2.k.f 8 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 16T_{5}^{2} + 32 \) acting on \(S_{2}^{\mathrm{new}}(294, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + 8 T^{6} + 32 T^{4} + 72 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{4} - 16 T^{2} + 32)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 44 T^{2} + 196)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 32 T^{2} + 128)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 36 T^{2} + 162)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 24 T^{2} + 16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 80 T^{2} + 1568)^{2} \) Copy content Toggle raw display
$37$ \( (T + 2)^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} - 100 T^{2} + 1250)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4 T - 46)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 16 T^{2} + 32)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 152 T^{2} + 4624)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 20 T^{2} + 2)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 80 T^{2} + 1568)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 4 T - 68)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 144 T^{2} + 3136)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 24 T + 136)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 116 T^{2} + 2)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 164 T^{2} + 4802)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 100 T^{2} + 578)^{2} \) Copy content Toggle raw display
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