Properties

Label 294.6.e.s
Level $294$
Weight $6$
Character orbit 294.e
Analytic conductor $47.153$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{9601})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 2401x^{2} + 2400x + 5760000 \) 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: $\mathrm{SU}(2)[C_{3}]$

$q$-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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4 \beta_{2} q^{2} + (9 \beta_{2} - 9) q^{3} + (16 \beta_{2} - 16) q^{4} + ( - 26 \beta_{2} - \beta_1) q^{5} + 36 q^{6} + 64 q^{8} - 81 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 \beta_{2} q^{2} + (9 \beta_{2} - 9) q^{3} + (16 \beta_{2} - 16) q^{4} + ( - 26 \beta_{2} - \beta_1) q^{5} + 36 q^{6} + 64 q^{8} - 81 \beta_{2} q^{9} + (4 \beta_{3} + 104 \beta_{2} + 4 \beta_1 - 108) q^{10} + ( - 5 \beta_{3} + 98 \beta_{2} - 5 \beta_1 - 93) q^{11} - 144 \beta_{2} q^{12} + (11 \beta_{3} + 184) q^{13} + ( - 9 \beta_{3} + 243) q^{15} - 256 \beta_{2} q^{16} + (20 \beta_{3} + 160 \beta_{2} + 20 \beta_1 - 180) q^{17} + (324 \beta_{2} - 324) q^{18} + ( - 865 \beta_{2} - 39 \beta_1) q^{19} + ( - 16 \beta_{3} + 432) q^{20} + (20 \beta_{3} + 372) q^{22} + ( - 1616 \beta_{2} - 4 \beta_1) q^{23} + (576 \beta_{2} - 576) q^{24} + (53 \beta_{3} - 49 \beta_{2} + 53 \beta_1 - 4) q^{25} + ( - 780 \beta_{2} + 44 \beta_1) q^{26} + 729 q^{27} + (61 \beta_{3} + 2199) q^{29} + ( - 936 \beta_{2} - 36 \beta_1) q^{30} + ( - 164 \beta_{3} - 915 \beta_{2} - 164 \beta_1 + 1079) q^{31} + (1024 \beta_{2} - 1024) q^{32} + ( - 882 \beta_{2} + 45 \beta_1) q^{33} + ( - 80 \beta_{3} + 720) q^{34} + 1296 q^{36} + ( - 10319 \beta_{2} + 51 \beta_1) q^{37} + (156 \beta_{3} + 3460 \beta_{2} + 156 \beta_1 - 3616) q^{38} + ( - 99 \beta_{3} + 1755 \beta_{2} - 99 \beta_1 - 1656) q^{39} + ( - 1664 \beta_{2} - 64 \beta_1) q^{40} + (66 \beta_{3} - 4440) q^{41} + ( - 87 \beta_{3} + 7970) q^{43} + ( - 1568 \beta_{2} + 80 \beta_1) q^{44} + (81 \beta_{3} + 2106 \beta_{2} + 81 \beta_1 - 2187) q^{45} + (16 \beta_{3} + 6464 \beta_{2} + 16 \beta_1 - 6480) q^{46} + (16878 \beta_{2} + 156 \beta_1) q^{47} + 2304 q^{48} + ( - 212 \beta_{3} + 16) q^{50} + ( - 1440 \beta_{2} - 180 \beta_1) q^{51} + ( - 176 \beta_{3} + 3120 \beta_{2} - 176 \beta_1 - 2944) q^{52} + ( - 225 \beta_{3} + 24732 \beta_{2} - 225 \beta_1 - 24507) q^{53} - 2916 \beta_{2} q^{54} + (37 \beta_{3} - 9489) q^{55} + ( - 351 \beta_{3} + 8136) q^{57} + ( - 9040 \beta_{2} + 244 \beta_1) q^{58} + ( - 41 \beta_{3} + 28388 \beta_{2} - 41 \beta_1 - 28347) q^{59} + (144 \beta_{3} + 3744 \beta_{2} + 144 \beta_1 - 3888) q^{60} + (33738 \beta_{2} + 32 \beta_1) q^{61} + (656 \beta_{3} - 4316) q^{62} + 4096 q^{64} + (21330 \beta_{2} + 102 \beta_1) q^{65} + ( - 180 \beta_{3} + 3528 \beta_{2} - 180 \beta_1 - 3348) q^{66} + (337 \beta_{3} + 37693 \beta_{2} + 337 \beta_1 - 38030) q^{67} + ( - 2560 \beta_{2} - 320 \beta_1) q^{68} + ( - 36 \beta_{3} + 14580) q^{69} + (1340 \beta_{3} - 5166) q^{71} - 5184 \beta_{2} q^{72} + (875 \beta_{3} + 1163 \beta_{2} + 875 \beta_1 - 2038) q^{73} + ( - 204 \beta_{3} + 41276 \beta_{2} - 204 \beta_1 - 41072) q^{74} + (441 \beta_{2} - 477 \beta_1) q^{75} + ( - 624 \beta_{3} + 14464) q^{76} + (396 \beta_{3} + 6624) q^{78} + ( - 12813 \beta_{2} - 986 \beta_1) q^{79} + (256 \beta_{3} + 6656 \beta_{2} + 256 \beta_1 - 6912) q^{80} + (6561 \beta_{2} - 6561) q^{81} + (17496 \beta_{2} + 264 \beta_1) q^{82} + ( - 1769 \beta_{3} + 1359) q^{83} + ( - 700 \beta_{3} + 52860) q^{85} + ( - 31532 \beta_{2} - 348 \beta_1) q^{86} + ( - 549 \beta_{3} + 20340 \beta_{2} - 549 \beta_1 - 19791) q^{87} + ( - 320 \beta_{3} + 6272 \beta_{2} - 320 \beta_1 - 5952) q^{88} + (88176 \beta_{2} + 210 \beta_1) q^{89} + ( - 324 \beta_{3} + 8748) q^{90} + ( - 64 \beta_{3} + 25920) q^{92} + (8235 \beta_{2} + 1476 \beta_1) q^{93} + ( - 624 \beta_{3} - 67512 \beta_{2} - 624 \beta_1 + 68136) q^{94} + (1918 \beta_{3} + 116090 \beta_{2} + 1918 \beta_1 - 118008) q^{95} - 9216 \beta_{2} q^{96} + (1981 \beta_{3} + 63721) q^{97} + (405 \beta_{3} + 7533) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{2} - 18 q^{3} - 32 q^{4} - 53 q^{5} + 144 q^{6} + 256 q^{8} - 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{2} - 18 q^{3} - 32 q^{4} - 53 q^{5} + 144 q^{6} + 256 q^{8} - 162 q^{9} - 212 q^{10} - 191 q^{11} - 288 q^{12} + 758 q^{13} + 954 q^{15} - 512 q^{16} - 340 q^{17} - 648 q^{18} - 1769 q^{19} + 1696 q^{20} + 1528 q^{22} - 3236 q^{23} - 1152 q^{24} + 45 q^{25} - 1516 q^{26} + 2916 q^{27} + 8918 q^{29} - 1908 q^{30} + 1994 q^{31} - 2048 q^{32} - 1719 q^{33} + 2720 q^{34} + 5184 q^{36} - 20587 q^{37} - 7076 q^{38} - 3411 q^{39} - 3392 q^{40} - 17628 q^{41} + 31706 q^{43} - 3056 q^{44} - 4293 q^{45} - 12944 q^{46} + 33912 q^{47} + 9216 q^{48} - 360 q^{50} - 3060 q^{51} - 6064 q^{52} - 49239 q^{53} - 5832 q^{54} - 37882 q^{55} + 31842 q^{57} - 17836 q^{58} - 56735 q^{59} - 7632 q^{60} + 67508 q^{61} - 15952 q^{62} + 16384 q^{64} + 42762 q^{65} - 6876 q^{66} - 75723 q^{67} - 5440 q^{68} + 58248 q^{69} - 17984 q^{71} - 10368 q^{72} - 3201 q^{73} - 82348 q^{74} + 405 q^{75} + 56608 q^{76} + 27288 q^{78} - 26612 q^{79} - 13568 q^{80} - 13122 q^{81} + 35256 q^{82} + 1898 q^{83} + 210040 q^{85} - 63412 q^{86} - 40131 q^{87} - 12224 q^{88} + 176562 q^{89} + 34344 q^{90} + 103552 q^{92} + 17946 q^{93} + 135648 q^{94} - 234098 q^{95} - 18432 q^{96} + 258846 q^{97} + 30942 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 2401x^{2} + 2400x + 5760000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 2401\nu^{2} - 2401\nu + 5760000 ) / 5762400 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 4801 ) / 2401 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 2400\beta_{2} + \beta _1 - 2401 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2401\beta_{3} - 4801 \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
24.7462 + 42.8616i
−24.2462 41.9956i
24.7462 42.8616i
−24.2462 + 41.9956i
−2.00000 3.46410i −4.50000 + 7.79423i −8.00000 + 13.8564i −37.7462 65.3783i 36.0000 0 64.0000 −40.5000 70.1481i −150.985 + 261.513i
67.2 −2.00000 3.46410i −4.50000 + 7.79423i −8.00000 + 13.8564i 11.2462 + 19.4789i 36.0000 0 64.0000 −40.5000 70.1481i 44.9847 77.9158i
79.1 −2.00000 + 3.46410i −4.50000 7.79423i −8.00000 13.8564i −37.7462 + 65.3783i 36.0000 0 64.0000 −40.5000 + 70.1481i −150.985 261.513i
79.2 −2.00000 + 3.46410i −4.50000 7.79423i −8.00000 13.8564i 11.2462 19.4789i 36.0000 0 64.0000 −40.5000 + 70.1481i 44.9847 + 77.9158i
\(n\): 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.s 4
7.b odd 2 1 42.6.e.c 4
7.c even 3 1 294.6.a.w 2
7.c even 3 1 inner 294.6.e.s 4
7.d odd 6 1 42.6.e.c 4
7.d odd 6 1 294.6.a.r 2
21.c even 2 1 126.6.g.h 4
21.g even 6 1 126.6.g.h 4
21.g even 6 1 882.6.a.bh 2
21.h odd 6 1 882.6.a.bb 2
28.d even 2 1 336.6.q.f 4
28.f even 6 1 336.6.q.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.e.c 4 7.b odd 2 1
42.6.e.c 4 7.d odd 6 1
126.6.g.h 4 21.c even 2 1
126.6.g.h 4 21.g even 6 1
294.6.a.r 2 7.d odd 6 1
294.6.a.w 2 7.c even 3 1
294.6.e.s 4 1.a even 1 1 trivial
294.6.e.s 4 7.c even 3 1 inner
336.6.q.f 4 28.d even 2 1
336.6.q.f 4 28.f even 6 1
882.6.a.bb 2 21.h odd 6 1
882.6.a.bh 2 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 \(S_{6}^{\mathrm{new}}(294, [\chi])\):

\( T_{5}^{4} + 53T_{5}^{3} + 4507T_{5}^{2} - 89994T_{5} + 2883204 \) Copy content Toggle raw display
\( T_{11}^{4} + 191T_{11}^{3} + 87367T_{11}^{2} - 9719226T_{11} + 2589384996 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9 T + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 53 T^{3} + 4507 T^{2} + \cdots + 2883204 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 191 T^{3} + \cdots + 2589384996 \) Copy content Toggle raw display
$13$ \( (T^{2} - 379 T - 254520)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 340 T^{3} + \cdots + 867133440000 \) Copy content Toggle raw display
$19$ \( T^{4} + 1769 T^{3} + \cdots + 8227948033600 \) Copy content Toggle raw display
$23$ \( T^{4} + 3236 T^{3} + \cdots + 6653923430400 \) Copy content Toggle raw display
$29$ \( (T^{2} - 4459 T - 3960660)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 1994 T^{3} + \cdots + 40\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{4} + 20587 T^{3} + \cdots + 99\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( (T^{2} + 8814 T + 8966160)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 15853 T + 44661910)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 33912 T^{3} + \cdots + 52\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{4} + 49239 T^{3} + \cdots + 23\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{4} + 56735 T^{3} + \cdots + 64\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{4} - 67508 T^{3} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{4} + 75723 T^{3} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{2} + 8992 T - 4289674884)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 3201 T^{3} + \cdots + 33\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{4} + 26612 T^{3} + \cdots + 46\!\cdots\!69 \) Copy content Toggle raw display
$83$ \( (T^{2} - 949 T - 7511023590)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 176562 T^{3} + \cdots + 59\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( (T^{2} - 129423 T - 5231869258)^{2} \) Copy content Toggle raw display
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