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Properties

Label 294.3.h.h
Level $294$
Weight $3$
Character orbit 294.h
Analytic conductor $8.011$
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] = [294,3,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: 8.0.4857532416.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 7x^{6} - 2x^{5} + 98x^{4} - 98x^{3} + 67x^{2} - 30x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{7} - \beta_{6} - \beta_{4} + \cdots + 1) q^{3}+ \cdots + ( - 2 \beta_{6} - 3 \beta_{5} + \cdots - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{7} - \beta_{6} - \beta_{4} + \cdots + 1) q^{3}+ \cdots + ( - 29 \beta_{7} - 33 \beta_{6} + \cdots - 32) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} + 8 q^{4} + 8 q^{6} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{3} + 8 q^{4} + 8 q^{6} - 10 q^{9} - 16 q^{10} - 4 q^{12} + 64 q^{13} + 28 q^{15} - 16 q^{16} - 40 q^{18} - 4 q^{19} - 16 q^{22} + 8 q^{24} - 24 q^{25} - 160 q^{27} + 52 q^{30} - 20 q^{31} + 106 q^{33} - 32 q^{34} - 40 q^{36} + 4 q^{37} - 72 q^{39} + 32 q^{40} + 208 q^{43} + 58 q^{45} + 56 q^{46} - 16 q^{48} + 14 q^{51} + 64 q^{52} + 32 q^{54} + 472 q^{55} - 268 q^{57} - 80 q^{58} + 28 q^{60} - 212 q^{61} - 64 q^{64} - 224 q^{66} + 156 q^{67} - 164 q^{69} + 80 q^{72} - 132 q^{73} - 164 q^{75} - 16 q^{76} + 240 q^{78} - 52 q^{79} + 98 q^{81} - 48 q^{82} + 152 q^{85} + 260 q^{87} - 16 q^{88} + 256 q^{90} - 210 q^{93} + 360 q^{94} - 16 q^{96} - 576 q^{97} - 140 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 7x^{6} - 2x^{5} + 98x^{4} - 98x^{3} + 67x^{2} - 30x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 2272 \nu^{7} + 10231 \nu^{6} + 9128 \nu^{5} - 44054 \nu^{4} - 273322 \nu^{3} + 767308 \nu^{2} + \cdots + 94527 ) / 84870 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3823 \nu^{7} - 12079 \nu^{6} - 22382 \nu^{5} + 30236 \nu^{4} + 414148 \nu^{3} - 781972 \nu^{2} + \cdots - 135243 ) / 84870 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 4433 \nu^{7} + 4379 \nu^{6} + 37882 \nu^{5} + 39494 \nu^{4} - 407318 \nu^{3} + 41912 \nu^{2} + \cdots - 204147 ) / 84870 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4487 \nu^{7} + 6851 \nu^{6} + 30628 \nu^{5} + 27116 \nu^{4} - 392522 \nu^{3} + 276458 \nu^{2} + \cdots + 39897 ) / 84870 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1508 \nu^{7} + 2499 \nu^{6} + 11172 \nu^{5} + 7434 \nu^{4} - 143178 \nu^{3} + 100842 \nu^{2} + \cdots + 14553 ) / 28290 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -209\nu^{7} + 227\nu^{6} + 1786\nu^{5} + 1862\nu^{4} - 19784\nu^{3} + 1976\nu^{2} - 969\nu - 261 ) / 2070 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -88\nu^{7} + 97\nu^{6} + 752\nu^{5} + 784\nu^{4} - 8278\nu^{3} + 832\nu^{2} - 408\nu - 477 ) / 738 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{7} - \beta_{6} + \beta_{2} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{6} + 5\beta_{5} - 2\beta_{4} + 2\beta_{3} + 2\beta_{2} + \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 8\beta_{7} - 11\beta_{6} + 3\beta_{3} + 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 17\beta_{7} - 28\beta_{6} + 46\beta_{5} - 16\beta_{4} + 28\beta_{2} + 17\beta _1 + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -45\beta_{6} + 75\beta_{5} - 45\beta_{4} + 45\beta_{3} - 49\beta_{2} - 61\beta _1 + 75 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 226\beta_{7} - 210\beta_{6} - 110\beta_{3} - 145 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -365\beta_{7} + 155\beta_{6} + 1098\beta_{5} - 546\beta_{4} - 155\beta_{2} - 365\beta _1 - 365 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(197\) \(199\)
\(\chi(n)\) \(-1\) \(-1 - \beta_{5}\)

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
−2.41089 − 1.96928i
0.461396 − 0.310963i
0.0386042 − 0.555062i
2.91089 + 1.10325i
−2.41089 + 1.96928i
0.461396 + 0.310963i
0.0386042 + 0.555062i
2.91089 − 1.10325i
−1.22474 + 0.707107i −2.97112 + 0.415287i 1.00000 − 1.73205i −0.422792 + 0.244099i 3.34521 − 2.60952i 0 2.82843i 8.65507 − 2.46773i 0.345208 − 0.597918i
263.2 −1.22474 + 0.707107i 2.24637 − 1.98842i 1.00000 − 1.73205i 5.32177 − 3.07253i −1.34521 + 4.02373i 0 2.82843i 1.09238 − 8.93346i −4.34521 + 7.52612i
263.3 1.22474 − 0.707107i 0.598836 − 2.93963i 1.00000 − 1.73205i −5.32177 + 3.07253i −1.34521 − 4.02373i 0 − 2.82843i −8.28279 − 3.52071i −4.34521 + 7.52612i
263.4 1.22474 − 0.707107i 1.12591 + 2.78071i 1.00000 − 1.73205i 0.422792 − 0.244099i 3.34521 + 2.60952i 0 − 2.82843i −6.46466 + 6.26165i 0.345208 − 0.597918i
275.1 −1.22474 − 0.707107i −2.97112 − 0.415287i 1.00000 + 1.73205i −0.422792 − 0.244099i 3.34521 + 2.60952i 0 − 2.82843i 8.65507 + 2.46773i 0.345208 + 0.597918i
275.2 −1.22474 − 0.707107i 2.24637 + 1.98842i 1.00000 + 1.73205i 5.32177 + 3.07253i −1.34521 − 4.02373i 0 − 2.82843i 1.09238 + 8.93346i −4.34521 − 7.52612i
275.3 1.22474 + 0.707107i 0.598836 + 2.93963i 1.00000 + 1.73205i −5.32177 − 3.07253i −1.34521 + 4.02373i 0 2.82843i −8.28279 + 3.52071i −4.34521 − 7.52612i
275.4 1.22474 + 0.707107i 1.12591 − 2.78071i 1.00000 + 1.73205i 0.422792 + 0.244099i 3.34521 − 2.60952i 0 2.82843i −6.46466 − 6.26165i 0.345208 + 0.597918i
\(n\): 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.c even 3 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.h 8
3.b odd 2 1 inner 294.3.h.h 8
7.b odd 2 1 42.3.h.b ✓ 8
7.c even 3 1 294.3.b.e 4
7.c even 3 1 inner 294.3.h.h 8
7.d odd 6 1 42.3.h.b ✓ 8
7.d odd 6 1 294.3.b.i 4
21.c even 2 1 42.3.h.b ✓ 8
21.g even 6 1 42.3.h.b ✓ 8
21.g even 6 1 294.3.b.i 4
21.h odd 6 1 294.3.b.e 4
21.h odd 6 1 inner 294.3.h.h 8
28.d even 2 1 336.3.bn.g 8
28.f even 6 1 336.3.bn.g 8
84.h odd 2 1 336.3.bn.g 8
84.j odd 6 1 336.3.bn.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.h.b ✓ 8 7.b odd 2 1
42.3.h.b ✓ 8 7.d odd 6 1
42.3.h.b ✓ 8 21.c even 2 1
42.3.h.b ✓ 8 21.g even 6 1
294.3.b.e 4 7.c even 3 1
294.3.b.e 4 21.h odd 6 1
294.3.b.i 4 7.d odd 6 1
294.3.b.i 4 21.g even 6 1
294.3.h.h 8 1.a even 1 1 trivial
294.3.h.h 8 3.b odd 2 1 inner
294.3.h.h 8 7.c even 3 1 inner
294.3.h.h 8 21.h odd 6 1 inner
336.3.bn.g 8 28.d even 2 1
336.3.bn.g 8 28.f even 6 1
336.3.bn.g 8 84.h odd 2 1
336.3.bn.g 8 84.j odd 6 1

Hecke kernels

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

\( T_{5}^{8} - 38T_{5}^{6} + 1435T_{5}^{4} - 342T_{5}^{2} + 81 \) Copy content Toggle raw display
\( T_{13}^{2} - 16T_{13} - 24 \) 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
$3$ \( T^{8} - 2 T^{7} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{8} - 38 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 5554571841 \) Copy content Toggle raw display
$13$ \( (T^{2} - 16 T - 24)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} - 38 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( (T^{4} + 2 T^{3} + \cdots + 38809)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 746 T^{6} + \cdots + 981506241 \) Copy content Toggle raw display
$29$ \( (T^{4} + 2600 T^{2} + 810000)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 10 T^{3} + \cdots + 275625)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 936 T^{2} + 104976)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 52 T - 116)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 243684619380801 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 38626834072401 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 922404990796641 \) Copy content Toggle raw display
$61$ \( (T^{4} + 106 T^{3} + \cdots + 6036849)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 78 T^{3} + \cdots + 4826809)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 18848 T^{2} + 82301184)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 66 T^{3} + \cdots + 543169)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 26 T^{3} + \cdots + 38303721)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 9152 T^{2} + 10036224)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 77\!\cdots\!61 \) Copy content Toggle raw display
$97$ \( (T^{2} + 144 T + 2984)^{4} \) Copy content Toggle raw display
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