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Properties

Label 294.3.h.e
Level $294$
Weight $3$
Character orbit 294.h
Analytic conductor $8.011$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [294,3,Mod(263,294)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(294, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("294.263");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 294.h (of order \(6\), 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: \(8.01091977219\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + (2 \beta_{7} - 2 \beta_{4} + \beta_1) q^{3} + ( - 2 \beta_{2} + 2) q^{4} + ( - 4 \beta_{3} + 4 \beta_1) q^{5} + (\beta_{7} - 4 \beta_{3}) q^{6} + ( - 2 \beta_{6} + 2 \beta_{5}) q^{8} + (4 \beta_{5} - 7 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + (2 \beta_{7} - 2 \beta_{4} + \beta_1) q^{3} + ( - 2 \beta_{2} + 2) q^{4} + ( - 4 \beta_{3} + 4 \beta_1) q^{5} + (\beta_{7} - 4 \beta_{3}) q^{6} + ( - 2 \beta_{6} + 2 \beta_{5}) q^{8} + (4 \beta_{5} - 7 \beta_{2}) q^{9} + (4 \beta_{7} - 4 \beta_{4}) q^{10} - 7 \beta_{6} q^{11} + ( - 4 \beta_{4} - 2 \beta_{3} + 2 \beta_1) q^{12} - 9 \beta_{7} q^{13} + ( - 8 \beta_{6} + 8 \beta_{5} + 4) q^{15} - 4 \beta_{2} q^{16} + 32 \beta_1 q^{17} + ( - 7 \beta_{6} - 8 \beta_{2} + 8) q^{18} + 20 \beta_{4} q^{19} - 8 \beta_{3} q^{20} - 14 q^{22} - \beta_{5} q^{23} + (2 \beta_{7} - 2 \beta_{4} - 8 \beta_1) q^{24} + (9 \beta_{2} - 9) q^{25} + (18 \beta_{3} - 18 \beta_1) q^{26} + ( - 10 \beta_{7} - 23 \beta_{3}) q^{27} + ( - 40 \beta_{6} + 40 \beta_{5}) q^{29} + (4 \beta_{5} - 16 \beta_{2}) q^{30} - 4 \beta_{6} q^{32} + ( - 7 \beta_{4} + 28 \beta_{3} - 28 \beta_1) q^{33} + 32 \beta_{7} q^{34} + ( - 8 \beta_{6} + 8 \beta_{5} - 14) q^{36} + 50 \beta_{2} q^{37} + 40 \beta_1 q^{38} + ( - 9 \beta_{6} + 36 \beta_{2} - 36) q^{39} - 8 \beta_{4} q^{40} + 16 \beta_{3} q^{41} - 52 q^{43} - 14 \beta_{5} q^{44} + (16 \beta_{7} - 16 \beta_{4} - 28 \beta_1) q^{45} + (2 \beta_{2} - 2) q^{46} + ( - 34 \beta_{3} + 34 \beta_1) q^{47} + ( - 8 \beta_{7} - 4 \beta_{3}) q^{48} + (9 \beta_{6} - 9 \beta_{5}) q^{50} + (64 \beta_{5} + 32 \beta_{2}) q^{51} + ( - 18 \beta_{7} + 18 \beta_{4}) q^{52} + 32 \beta_{6} q^{53} + ( - 23 \beta_{4} + 20 \beta_{3} - 20 \beta_1) q^{54} - 28 \beta_{7} q^{55} + (20 \beta_{6} - 20 \beta_{5} + 80) q^{57} - 80 \beta_{2} q^{58} + 32 \beta_1 q^{59} + ( - 16 \beta_{6} - 8 \beta_{2} + 8) q^{60} - 23 \beta_{4} q^{61} - 8 q^{64} - 36 \beta_{5} q^{65} + ( - 28 \beta_{7} + 28 \beta_{4} - 14 \beta_1) q^{66} + ( - 12 \beta_{2} + 12) q^{67} + ( - 64 \beta_{3} + 64 \beta_1) q^{68} + ( - \beta_{7} + 4 \beta_{3}) q^{69} + (63 \beta_{6} - 63 \beta_{5}) q^{71} + ( - 14 \beta_{5} - 16 \beta_{2}) q^{72} + ( - 23 \beta_{7} + 23 \beta_{4}) q^{73} + 50 \beta_{6} q^{74} + (18 \beta_{4} + 9 \beta_{3} - 9 \beta_1) q^{75} + 40 \beta_{7} q^{76} + (36 \beta_{6} - 36 \beta_{5} - 18) q^{78} + 40 \beta_{2} q^{79} - 16 \beta_1 q^{80} + ( - 56 \beta_{6} + 17 \beta_{2} - 17) q^{81} + 16 \beta_{4} q^{82} + 62 \beta_{3} q^{83} + 128 q^{85} - 52 \beta_{5} q^{86} + (40 \beta_{7} - 40 \beta_{4} - 160 \beta_1) q^{87} + (28 \beta_{2} - 28) q^{88} + ( - 56 \beta_{3} + 56 \beta_1) q^{89} + ( - 28 \beta_{7} - 32 \beta_{3}) q^{90} + (2 \beta_{6} - 2 \beta_{5}) q^{92} + (34 \beta_{7} - 34 \beta_{4}) q^{94} + 80 \beta_{6} q^{95} + ( - 4 \beta_{4} + 16 \beta_{3} - 16 \beta_1) q^{96} - 17 \beta_{7} q^{97} + (49 \beta_{6} - 49 \beta_{5} - 56) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{4} - 28 q^{9} + 32 q^{15} - 16 q^{16} + 32 q^{18} - 112 q^{22} - 36 q^{25} - 64 q^{30} - 112 q^{36} + 200 q^{37} - 144 q^{39} - 416 q^{43} - 8 q^{46} + 128 q^{51} + 640 q^{57} - 320 q^{58} + 32 q^{60} - 64 q^{64} + 48 q^{67} - 64 q^{72} - 144 q^{78} + 160 q^{79} - 68 q^{81} + 1024 q^{85} - 112 q^{88} - 448 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

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

\(n\) \(197\) \(199\)
\(\chi(n)\) \(-1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
263.1
−0.965926 − 0.258819i
−0.258819 + 0.965926i
0.258819 − 0.965926i
0.965926 + 0.258819i
−0.965926 + 0.258819i
−0.258819 − 0.965926i
0.258819 + 0.965926i
0.965926 − 0.258819i
−1.22474 + 0.707107i −0.548188 + 2.94949i 1.00000 − 1.73205i 3.46410 − 2.00000i −1.41421 − 4.00000i 0 2.82843i −8.39898 − 3.23375i −2.82843 + 4.89898i
263.2 −1.22474 + 0.707107i 0.548188 − 2.94949i 1.00000 − 1.73205i −3.46410 + 2.00000i 1.41421 + 4.00000i 0 2.82843i −8.39898 − 3.23375i 2.82843 − 4.89898i
263.3 1.22474 − 0.707107i −2.28024 + 1.94949i 1.00000 − 1.73205i −3.46410 + 2.00000i −1.41421 + 4.00000i 0 − 2.82843i 1.39898 − 8.89060i −2.82843 + 4.89898i
263.4 1.22474 − 0.707107i 2.28024 − 1.94949i 1.00000 − 1.73205i 3.46410 − 2.00000i 1.41421 − 4.00000i 0 − 2.82843i 1.39898 − 8.89060i 2.82843 − 4.89898i
275.1 −1.22474 − 0.707107i −0.548188 − 2.94949i 1.00000 + 1.73205i 3.46410 + 2.00000i −1.41421 + 4.00000i 0 − 2.82843i −8.39898 + 3.23375i −2.82843 − 4.89898i
275.2 −1.22474 − 0.707107i 0.548188 + 2.94949i 1.00000 + 1.73205i −3.46410 − 2.00000i 1.41421 − 4.00000i 0 − 2.82843i −8.39898 + 3.23375i 2.82843 + 4.89898i
275.3 1.22474 + 0.707107i −2.28024 − 1.94949i 1.00000 + 1.73205i −3.46410 − 2.00000i −1.41421 − 4.00000i 0 2.82843i 1.39898 + 8.89060i −2.82843 − 4.89898i
275.4 1.22474 + 0.707107i 2.28024 + 1.94949i 1.00000 + 1.73205i 3.46410 + 2.00000i 1.41421 + 4.00000i 0 2.82843i 1.39898 + 8.89060i 2.82843 + 4.89898i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 263.4
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
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 294.3.h.e 8
3.b odd 2 1 inner 294.3.h.e 8
7.b odd 2 1 inner 294.3.h.e 8
7.c even 3 1 294.3.b.g ✓ 4
7.c even 3 1 inner 294.3.h.e 8
7.d odd 6 1 294.3.b.g ✓ 4
7.d odd 6 1 inner 294.3.h.e 8
21.c even 2 1 inner 294.3.h.e 8
21.g even 6 1 294.3.b.g ✓ 4
21.g even 6 1 inner 294.3.h.e 8
21.h odd 6 1 294.3.b.g ✓ 4
21.h odd 6 1 inner 294.3.h.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.3.b.g ✓ 4 7.c even 3 1
294.3.b.g ✓ 4 7.d odd 6 1
294.3.b.g ✓ 4 21.g even 6 1
294.3.b.g ✓ 4 21.h odd 6 1
294.3.h.e 8 1.a even 1 1 trivial
294.3.h.e 8 3.b odd 2 1 inner
294.3.h.e 8 7.b odd 2 1 inner
294.3.h.e 8 7.c even 3 1 inner
294.3.h.e 8 7.d odd 6 1 inner
294.3.h.e 8 21.c even 2 1 inner
294.3.h.e 8 21.g even 6 1 inner
294.3.h.e 8 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(294, [\chi])\):

\( T_{5}^{4} - 16T_{5}^{2} + 256 \) Copy content Toggle raw display
\( T_{13}^{2} - 162 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 14 T^{6} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( (T^{4} - 16 T^{2} + 256)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 98 T^{2} + 9604)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 162)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 1024 T^{2} + 1048576)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 800 T^{2} + 640000)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 3200)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{2} - 50 T + 2500)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 256)^{4} \) Copy content Toggle raw display
$43$ \( (T + 52)^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} - 1156 T^{2} + 1336336)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 2048 T^{2} + 4194304)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 1024 T^{2} + 1048576)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 1058 T^{2} + 1119364)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 12 T + 144)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 7938)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 1058 T^{2} + 1119364)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 40 T + 1600)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 3844)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} - 3136 T^{2} + 9834496)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 578)^{4} \) Copy content Toggle raw display
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