Properties

Label 294.6.e.b
Level $294$
Weight $6$
Character orbit 294.e
Analytic conductor $47.153$
Analytic rank $0$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [294,6,Mod(67,294)]
 
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([0, 4]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("294.67");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 294.e (of order \(3\), 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: \(47.1528430250\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) 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 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4 \zeta_{6} q^{2} + (9 \zeta_{6} - 9) q^{3} + (16 \zeta_{6} - 16) q^{4} - 24 \zeta_{6} q^{5} + 36 q^{6} + 64 q^{8} - 81 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 4 \zeta_{6} q^{2} + (9 \zeta_{6} - 9) q^{3} + (16 \zeta_{6} - 16) q^{4} - 24 \zeta_{6} q^{5} + 36 q^{6} + 64 q^{8} - 81 \zeta_{6} q^{9} + (96 \zeta_{6} - 96) q^{10} + (66 \zeta_{6} - 66) q^{11} - 144 \zeta_{6} q^{12} + 98 q^{13} + 216 q^{15} - 256 \zeta_{6} q^{16} + ( - 216 \zeta_{6} + 216) q^{17} + (324 \zeta_{6} - 324) q^{18} + 340 \zeta_{6} q^{19} + 384 q^{20} + 264 q^{22} + 1038 \zeta_{6} q^{23} + (576 \zeta_{6} - 576) q^{24} + ( - 2549 \zeta_{6} + 2549) q^{25} - 392 \zeta_{6} q^{26} + 729 q^{27} - 2490 q^{29} - 864 \zeta_{6} q^{30} + ( - 7048 \zeta_{6} + 7048) q^{31} + (1024 \zeta_{6} - 1024) q^{32} - 594 \zeta_{6} q^{33} - 864 q^{34} + 1296 q^{36} + 12238 \zeta_{6} q^{37} + ( - 1360 \zeta_{6} + 1360) q^{38} + (882 \zeta_{6} - 882) q^{39} - 1536 \zeta_{6} q^{40} + 6468 q^{41} - 15412 q^{43} - 1056 \zeta_{6} q^{44} + (1944 \zeta_{6} - 1944) q^{45} + ( - 4152 \zeta_{6} + 4152) q^{46} - 20604 \zeta_{6} q^{47} + 2304 q^{48} - 10196 q^{50} + 1944 \zeta_{6} q^{51} + (1568 \zeta_{6} - 1568) q^{52} + (32490 \zeta_{6} - 32490) q^{53} - 2916 \zeta_{6} q^{54} + 1584 q^{55} - 3060 q^{57} + 9960 \zeta_{6} q^{58} + (34224 \zeta_{6} - 34224) q^{59} + (3456 \zeta_{6} - 3456) q^{60} - 35654 \zeta_{6} q^{61} - 28192 q^{62} + 4096 q^{64} - 2352 \zeta_{6} q^{65} + (2376 \zeta_{6} - 2376) q^{66} + (12680 \zeta_{6} - 12680) q^{67} + 3456 \zeta_{6} q^{68} - 9342 q^{69} - 42642 q^{71} - 5184 \zeta_{6} q^{72} + (33734 \zeta_{6} - 33734) q^{73} + ( - 48952 \zeta_{6} + 48952) q^{74} + 22941 \zeta_{6} q^{75} - 5440 q^{76} + 3528 q^{78} + 85108 \zeta_{6} q^{79} + (6144 \zeta_{6} - 6144) q^{80} + (6561 \zeta_{6} - 6561) q^{81} - 25872 \zeta_{6} q^{82} - 106764 q^{83} - 5184 q^{85} + 61648 \zeta_{6} q^{86} + ( - 22410 \zeta_{6} + 22410) q^{87} + (4224 \zeta_{6} - 4224) q^{88} - 34884 \zeta_{6} q^{89} + 7776 q^{90} - 16608 q^{92} + 63432 \zeta_{6} q^{93} + (82416 \zeta_{6} - 82416) q^{94} + ( - 8160 \zeta_{6} + 8160) q^{95} - 9216 \zeta_{6} q^{96} + 18662 q^{97} + 5346 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} - 9 q^{3} - 16 q^{4} - 24 q^{5} + 72 q^{6} + 128 q^{8} - 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} - 9 q^{3} - 16 q^{4} - 24 q^{5} + 72 q^{6} + 128 q^{8} - 81 q^{9} - 96 q^{10} - 66 q^{11} - 144 q^{12} + 196 q^{13} + 432 q^{15} - 256 q^{16} + 216 q^{17} - 324 q^{18} + 340 q^{19} + 768 q^{20} + 528 q^{22} + 1038 q^{23} - 576 q^{24} + 2549 q^{25} - 392 q^{26} + 1458 q^{27} - 4980 q^{29} - 864 q^{30} + 7048 q^{31} - 1024 q^{32} - 594 q^{33} - 1728 q^{34} + 2592 q^{36} + 12238 q^{37} + 1360 q^{38} - 882 q^{39} - 1536 q^{40} + 12936 q^{41} - 30824 q^{43} - 1056 q^{44} - 1944 q^{45} + 4152 q^{46} - 20604 q^{47} + 4608 q^{48} - 20392 q^{50} + 1944 q^{51} - 1568 q^{52} - 32490 q^{53} - 2916 q^{54} + 3168 q^{55} - 6120 q^{57} + 9960 q^{58} - 34224 q^{59} - 3456 q^{60} - 35654 q^{61} - 56384 q^{62} + 8192 q^{64} - 2352 q^{65} - 2376 q^{66} - 12680 q^{67} + 3456 q^{68} - 18684 q^{69} - 85284 q^{71} - 5184 q^{72} - 33734 q^{73} + 48952 q^{74} + 22941 q^{75} - 10880 q^{76} + 7056 q^{78} + 85108 q^{79} - 6144 q^{80} - 6561 q^{81} - 25872 q^{82} - 213528 q^{83} - 10368 q^{85} + 61648 q^{86} + 22410 q^{87} - 4224 q^{88} - 34884 q^{89} + 15552 q^{90} - 33216 q^{92} + 63432 q^{93} - 82416 q^{94} + 8160 q^{95} - 9216 q^{96} + 37324 q^{97} + 10692 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(197\) \(199\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0.500000 + 0.866025i
0.500000 0.866025i
−2.00000 3.46410i −4.50000 + 7.79423i −8.00000 + 13.8564i −12.0000 20.7846i 36.0000 0 64.0000 −40.5000 70.1481i −48.0000 + 83.1384i
79.1 −2.00000 + 3.46410i −4.50000 7.79423i −8.00000 13.8564i −12.0000 + 20.7846i 36.0000 0 64.0000 −40.5000 + 70.1481i −48.0000 83.1384i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 294.6.e.b 2
7.b odd 2 1 294.6.e.f 2
7.c even 3 1 42.6.a.f 1
7.c even 3 1 inner 294.6.e.b 2
7.d odd 6 1 294.6.a.i 1
7.d odd 6 1 294.6.e.f 2
21.g even 6 1 882.6.a.i 1
21.h odd 6 1 126.6.a.b 1
28.g odd 6 1 336.6.a.g 1
35.j even 6 1 1050.6.a.a 1
35.l odd 12 2 1050.6.g.m 2
84.n even 6 1 1008.6.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.a.f 1 7.c even 3 1
126.6.a.b 1 21.h odd 6 1
294.6.a.i 1 7.d odd 6 1
294.6.e.b 2 1.a even 1 1 trivial
294.6.e.b 2 7.c even 3 1 inner
294.6.e.f 2 7.b odd 2 1
294.6.e.f 2 7.d odd 6 1
336.6.a.g 1 28.g odd 6 1
882.6.a.i 1 21.g even 6 1
1008.6.a.k 1 84.n even 6 1
1050.6.a.a 1 35.j even 6 1
1050.6.g.m 2 35.l odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(294, [\chi])\):

\( T_{5}^{2} + 24T_{5} + 576 \) Copy content Toggle raw display
\( T_{11}^{2} + 66T_{11} + 4356 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$3$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$5$ \( T^{2} + 24T + 576 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 66T + 4356 \) Copy content Toggle raw display
$13$ \( (T - 98)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 216T + 46656 \) Copy content Toggle raw display
$19$ \( T^{2} - 340T + 115600 \) Copy content Toggle raw display
$23$ \( T^{2} - 1038 T + 1077444 \) Copy content Toggle raw display
$29$ \( (T + 2490)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 7048 T + 49674304 \) Copy content Toggle raw display
$37$ \( T^{2} - 12238 T + 149768644 \) Copy content Toggle raw display
$41$ \( (T - 6468)^{2} \) Copy content Toggle raw display
$43$ \( (T + 15412)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 20604 T + 424524816 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 1055600100 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 1171282176 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 1271207716 \) Copy content Toggle raw display
$67$ \( T^{2} + 12680 T + 160782400 \) Copy content Toggle raw display
$71$ \( (T + 42642)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 1137982756 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 7243371664 \) Copy content Toggle raw display
$83$ \( (T + 106764)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 1216893456 \) Copy content Toggle raw display
$97$ \( (T - 18662)^{2} \) Copy content Toggle raw display
show more
show less