Properties

Label 294.6.e.e
Level $294$
Weight $6$
Character orbit 294.e
Analytic conductor $47.153$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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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.

\(f(q)\) \(=\) \( q - 4 \zeta_{6} q^{2} + ( - 9 \zeta_{6} + 9) q^{3} + (16 \zeta_{6} - 16) q^{4} - 6 \zeta_{6} q^{5} - 36 q^{6} + 64 q^{8} - 81 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 4 \zeta_{6} q^{2} + ( - 9 \zeta_{6} + 9) q^{3} + (16 \zeta_{6} - 16) q^{4} - 6 \zeta_{6} q^{5} - 36 q^{6} + 64 q^{8} - 81 \zeta_{6} q^{9} + (24 \zeta_{6} - 24) q^{10} + ( - 666 \zeta_{6} + 666) q^{11} + 144 \zeta_{6} q^{12} + 559 q^{13} - 54 q^{15} - 256 \zeta_{6} q^{16} + (1740 \zeta_{6} - 1740) q^{17} + (324 \zeta_{6} - 324) q^{18} + 1157 \zeta_{6} q^{19} + 96 q^{20} - 2664 q^{22} + 3468 \zeta_{6} q^{23} + ( - 576 \zeta_{6} + 576) q^{24} + ( - 3089 \zeta_{6} + 3089) q^{25} - 2236 \zeta_{6} q^{26} - 729 q^{27} + 3372 q^{29} + 216 \zeta_{6} q^{30} + ( - 6293 \zeta_{6} + 6293) q^{31} + (1024 \zeta_{6} - 1024) q^{32} - 5994 \zeta_{6} q^{33} + 6960 q^{34} + 1296 q^{36} - 3131 \zeta_{6} q^{37} + ( - 4628 \zeta_{6} + 4628) q^{38} + ( - 5031 \zeta_{6} + 5031) q^{39} - 384 \zeta_{6} q^{40} + 4866 q^{41} - 11407 q^{43} + 10656 \zeta_{6} q^{44} + (486 \zeta_{6} - 486) q^{45} + ( - 13872 \zeta_{6} + 13872) q^{46} + 2310 \zeta_{6} q^{47} - 2304 q^{48} - 12356 q^{50} + 15660 \zeta_{6} q^{51} + (8944 \zeta_{6} - 8944) q^{52} + ( - 28296 \zeta_{6} + 28296) q^{53} + 2916 \zeta_{6} q^{54} - 3996 q^{55} + 10413 q^{57} - 13488 \zeta_{6} q^{58} + ( - 20544 \zeta_{6} + 20544) q^{59} + ( - 864 \zeta_{6} + 864) q^{60} - 4630 \zeta_{6} q^{61} - 25172 q^{62} + 4096 q^{64} - 3354 \zeta_{6} q^{65} + (23976 \zeta_{6} - 23976) q^{66} + ( - 18745 \zeta_{6} + 18745) q^{67} - 27840 \zeta_{6} q^{68} + 31212 q^{69} - 38226 q^{71} - 5184 \zeta_{6} q^{72} + ( - 70589 \zeta_{6} + 70589) q^{73} + (12524 \zeta_{6} - 12524) q^{74} - 27801 \zeta_{6} q^{75} - 18512 q^{76} - 20124 q^{78} + 62293 \zeta_{6} q^{79} + (1536 \zeta_{6} - 1536) q^{80} + (6561 \zeta_{6} - 6561) q^{81} - 19464 \zeta_{6} q^{82} - 79818 q^{83} + 10440 q^{85} + 45628 \zeta_{6} q^{86} + ( - 30348 \zeta_{6} + 30348) q^{87} + ( - 42624 \zeta_{6} + 42624) q^{88} - 18120 \zeta_{6} q^{89} + 1944 q^{90} - 55488 q^{92} - 56637 \zeta_{6} q^{93} + ( - 9240 \zeta_{6} + 9240) q^{94} + ( - 6942 \zeta_{6} + 6942) q^{95} + 9216 \zeta_{6} q^{96} - 124754 q^{97} - 53946 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 9 q^{3} - 16 q^{4} - 6 q^{5} - 72 q^{6} + 128 q^{8} - 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} + 9 q^{3} - 16 q^{4} - 6 q^{5} - 72 q^{6} + 128 q^{8} - 81 q^{9} - 24 q^{10} + 666 q^{11} + 144 q^{12} + 1118 q^{13} - 108 q^{15} - 256 q^{16} - 1740 q^{17} - 324 q^{18} + 1157 q^{19} + 192 q^{20} - 5328 q^{22} + 3468 q^{23} + 576 q^{24} + 3089 q^{25} - 2236 q^{26} - 1458 q^{27} + 6744 q^{29} + 216 q^{30} + 6293 q^{31} - 1024 q^{32} - 5994 q^{33} + 13920 q^{34} + 2592 q^{36} - 3131 q^{37} + 4628 q^{38} + 5031 q^{39} - 384 q^{40} + 9732 q^{41} - 22814 q^{43} + 10656 q^{44} - 486 q^{45} + 13872 q^{46} + 2310 q^{47} - 4608 q^{48} - 24712 q^{50} + 15660 q^{51} - 8944 q^{52} + 28296 q^{53} + 2916 q^{54} - 7992 q^{55} + 20826 q^{57} - 13488 q^{58} + 20544 q^{59} + 864 q^{60} - 4630 q^{61} - 50344 q^{62} + 8192 q^{64} - 3354 q^{65} - 23976 q^{66} + 18745 q^{67} - 27840 q^{68} + 62424 q^{69} - 76452 q^{71} - 5184 q^{72} + 70589 q^{73} - 12524 q^{74} - 27801 q^{75} - 37024 q^{76} - 40248 q^{78} + 62293 q^{79} - 1536 q^{80} - 6561 q^{81} - 19464 q^{82} - 159636 q^{83} + 20880 q^{85} + 45628 q^{86} + 30348 q^{87} + 42624 q^{88} - 18120 q^{89} + 3888 q^{90} - 110976 q^{92} - 56637 q^{93} + 9240 q^{94} + 6942 q^{95} + 9216 q^{96} - 249508 q^{97} - 107892 q^{99}+O(q^{100}) \) 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 −3.00000 5.19615i −36.0000 0 64.0000 −40.5000 70.1481i −12.0000 + 20.7846i
79.1 −2.00000 + 3.46410i 4.50000 + 7.79423i −8.00000 13.8564i −3.00000 + 5.19615i −36.0000 0 64.0000 −40.5000 + 70.1481i −12.0000 20.7846i
\(n\): 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.e 2
7.b odd 2 1 42.6.e.a 2
7.c even 3 1 294.6.a.j 1
7.c even 3 1 inner 294.6.e.e 2
7.d odd 6 1 42.6.e.a 2
7.d odd 6 1 294.6.a.l 1
21.c even 2 1 126.6.g.c 2
21.g even 6 1 126.6.g.c 2
21.g even 6 1 882.6.a.f 1
21.h odd 6 1 882.6.a.e 1
28.d even 2 1 336.6.q.c 2
28.f even 6 1 336.6.q.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.e.a 2 7.b odd 2 1
42.6.e.a 2 7.d odd 6 1
126.6.g.c 2 21.c even 2 1
126.6.g.c 2 21.g even 6 1
294.6.a.j 1 7.c even 3 1
294.6.a.l 1 7.d odd 6 1
294.6.e.e 2 1.a even 1 1 trivial
294.6.e.e 2 7.c even 3 1 inner
336.6.q.c 2 28.d even 2 1
336.6.q.c 2 28.f even 6 1
882.6.a.e 1 21.h odd 6 1
882.6.a.f 1 21.g even 6 1

Hecke kernels

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

\( T_{5}^{2} + 6T_{5} + 36 \) Copy content Toggle raw display
\( T_{11}^{2} - 666T_{11} + 443556 \) 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
$5$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 666T + 443556 \) Copy content Toggle raw display
$13$ \( (T - 559)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 1740 T + 3027600 \) Copy content Toggle raw display
$19$ \( T^{2} - 1157 T + 1338649 \) Copy content Toggle raw display
$23$ \( T^{2} - 3468 T + 12027024 \) Copy content Toggle raw display
$29$ \( (T - 3372)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 6293 T + 39601849 \) Copy content Toggle raw display
$37$ \( T^{2} + 3131 T + 9803161 \) Copy content Toggle raw display
$41$ \( (T - 4866)^{2} \) Copy content Toggle raw display
$43$ \( (T + 11407)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 2310 T + 5336100 \) Copy content Toggle raw display
$53$ \( T^{2} - 28296 T + 800663616 \) Copy content Toggle raw display
$59$ \( T^{2} - 20544 T + 422055936 \) Copy content Toggle raw display
$61$ \( T^{2} + 4630 T + 21436900 \) Copy content Toggle raw display
$67$ \( T^{2} - 18745 T + 351375025 \) Copy content Toggle raw display
$71$ \( (T + 38226)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 70589 T + 4982806921 \) Copy content Toggle raw display
$79$ \( T^{2} - 62293 T + 3880417849 \) Copy content Toggle raw display
$83$ \( (T + 79818)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 18120 T + 328334400 \) Copy content Toggle raw display
$97$ \( (T + 124754)^{2} \) Copy content Toggle raw display
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