Properties

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

\(f(q)\) \(=\) \( q + 4 \zeta_{6} q^{2} + (9 \zeta_{6} - 9) q^{3} + (16 \zeta_{6} - 16) q^{4} - 54 \zeta_{6} q^{5} - 36 q^{6} - 64 q^{8} - 81 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 4 \zeta_{6} q^{2} + (9 \zeta_{6} - 9) q^{3} + (16 \zeta_{6} - 16) q^{4} - 54 \zeta_{6} q^{5} - 36 q^{6} - 64 q^{8} - 81 \zeta_{6} q^{9} + ( - 216 \zeta_{6} + 216) q^{10} + (216 \zeta_{6} - 216) q^{11} - 144 \zeta_{6} q^{12} - 998 q^{13} + 486 q^{15} - 256 \zeta_{6} q^{16} + ( - 1302 \zeta_{6} + 1302) q^{17} + ( - 324 \zeta_{6} + 324) q^{18} + 884 \zeta_{6} q^{19} + 864 q^{20} - 864 q^{22} + 2268 \zeta_{6} q^{23} + ( - 576 \zeta_{6} + 576) q^{24} + ( - 209 \zeta_{6} + 209) q^{25} - 3992 \zeta_{6} q^{26} + 729 q^{27} - 1482 q^{29} + 1944 \zeta_{6} q^{30} + ( - 8360 \zeta_{6} + 8360) q^{31} + ( - 1024 \zeta_{6} + 1024) q^{32} - 1944 \zeta_{6} q^{33} + 5208 q^{34} + 1296 q^{36} + 4714 \zeta_{6} q^{37} + (3536 \zeta_{6} - 3536) q^{38} + ( - 8982 \zeta_{6} + 8982) q^{39} + 3456 \zeta_{6} q^{40} + 9786 q^{41} + 19436 q^{43} - 3456 \zeta_{6} q^{44} + (4374 \zeta_{6} - 4374) q^{45} + (9072 \zeta_{6} - 9072) q^{46} + 22200 \zeta_{6} q^{47} + 2304 q^{48} + 836 q^{50} + 11718 \zeta_{6} q^{51} + ( - 15968 \zeta_{6} + 15968) q^{52} + (26790 \zeta_{6} - 26790) q^{53} + 2916 \zeta_{6} q^{54} + 11664 q^{55} - 7956 q^{57} - 5928 \zeta_{6} q^{58} + ( - 28092 \zeta_{6} + 28092) q^{59} + (7776 \zeta_{6} - 7776) q^{60} - 38866 \zeta_{6} q^{61} + 33440 q^{62} + 4096 q^{64} + 53892 \zeta_{6} q^{65} + ( - 7776 \zeta_{6} + 7776) q^{66} + (23948 \zeta_{6} - 23948) q^{67} + 20832 \zeta_{6} q^{68} - 20412 q^{69} - 20628 q^{71} + 5184 \zeta_{6} q^{72} + ( - 290 \zeta_{6} + 290) q^{73} + (18856 \zeta_{6} - 18856) q^{74} + 1881 \zeta_{6} q^{75} - 14144 q^{76} + 35928 q^{78} + 99544 \zeta_{6} q^{79} + (13824 \zeta_{6} - 13824) q^{80} + (6561 \zeta_{6} - 6561) q^{81} + 39144 \zeta_{6} q^{82} - 19308 q^{83} - 70308 q^{85} + 77744 \zeta_{6} q^{86} + ( - 13338 \zeta_{6} + 13338) q^{87} + ( - 13824 \zeta_{6} + 13824) q^{88} + 36390 \zeta_{6} q^{89} - 17496 q^{90} - 36288 q^{92} + 75240 \zeta_{6} q^{93} + (88800 \zeta_{6} - 88800) q^{94} + ( - 47736 \zeta_{6} + 47736) q^{95} + 9216 \zeta_{6} q^{96} + 79078 q^{97} + 17496 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} - 9 q^{3} - 16 q^{4} - 54 q^{5} - 72 q^{6} - 128 q^{8} - 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} - 9 q^{3} - 16 q^{4} - 54 q^{5} - 72 q^{6} - 128 q^{8} - 81 q^{9} + 216 q^{10} - 216 q^{11} - 144 q^{12} - 1996 q^{13} + 972 q^{15} - 256 q^{16} + 1302 q^{17} + 324 q^{18} + 884 q^{19} + 1728 q^{20} - 1728 q^{22} + 2268 q^{23} + 576 q^{24} + 209 q^{25} - 3992 q^{26} + 1458 q^{27} - 2964 q^{29} + 1944 q^{30} + 8360 q^{31} + 1024 q^{32} - 1944 q^{33} + 10416 q^{34} + 2592 q^{36} + 4714 q^{37} - 3536 q^{38} + 8982 q^{39} + 3456 q^{40} + 19572 q^{41} + 38872 q^{43} - 3456 q^{44} - 4374 q^{45} - 9072 q^{46} + 22200 q^{47} + 4608 q^{48} + 1672 q^{50} + 11718 q^{51} + 15968 q^{52} - 26790 q^{53} + 2916 q^{54} + 23328 q^{55} - 15912 q^{57} - 5928 q^{58} + 28092 q^{59} - 7776 q^{60} - 38866 q^{61} + 66880 q^{62} + 8192 q^{64} + 53892 q^{65} + 7776 q^{66} - 23948 q^{67} + 20832 q^{68} - 40824 q^{69} - 41256 q^{71} + 5184 q^{72} + 290 q^{73} - 18856 q^{74} + 1881 q^{75} - 28288 q^{76} + 71856 q^{78} + 99544 q^{79} - 13824 q^{80} - 6561 q^{81} + 39144 q^{82} - 38616 q^{83} - 140616 q^{85} + 77744 q^{86} + 13338 q^{87} + 13824 q^{88} + 36390 q^{89} - 34992 q^{90} - 72576 q^{92} + 75240 q^{93} - 88800 q^{94} + 47736 q^{95} + 9216 q^{96} + 158156 q^{97} + 34992 q^{99}+O(q^{100}) \) 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 −27.0000 46.7654i −36.0000 0 −64.0000 −40.5000 70.1481i 108.000 187.061i
79.1 2.00000 3.46410i −4.50000 7.79423i −8.00000 13.8564i −27.0000 + 46.7654i −36.0000 0 −64.0000 −40.5000 + 70.1481i 108.000 + 187.061i
\(n\): 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.h 2
7.b odd 2 1 294.6.e.r 2
7.c even 3 1 294.6.a.h 1
7.c even 3 1 inner 294.6.e.h 2
7.d odd 6 1 42.6.a.a 1
7.d odd 6 1 294.6.e.r 2
21.g even 6 1 126.6.a.k 1
21.h odd 6 1 882.6.a.o 1
28.f even 6 1 336.6.a.j 1
35.i odd 6 1 1050.6.a.n 1
35.k even 12 2 1050.6.g.o 2
84.j odd 6 1 1008.6.a.x 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.a.a 1 7.d odd 6 1
126.6.a.k 1 21.g even 6 1
294.6.a.h 1 7.c even 3 1
294.6.e.h 2 1.a even 1 1 trivial
294.6.e.h 2 7.c even 3 1 inner
294.6.e.r 2 7.b odd 2 1
294.6.e.r 2 7.d odd 6 1
336.6.a.j 1 28.f even 6 1
882.6.a.o 1 21.h odd 6 1
1008.6.a.x 1 84.j odd 6 1
1050.6.a.n 1 35.i odd 6 1
1050.6.g.o 2 35.k even 12 2

Hecke kernels

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

\( T_{5}^{2} + 54T_{5} + 2916 \) Copy content Toggle raw display
\( T_{11}^{2} + 216T_{11} + 46656 \) 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
$5$ \( T^{2} + 54T + 2916 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 216T + 46656 \) Copy content Toggle raw display
$13$ \( (T + 998)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 1302 T + 1695204 \) Copy content Toggle raw display
$19$ \( T^{2} - 884T + 781456 \) Copy content Toggle raw display
$23$ \( T^{2} - 2268 T + 5143824 \) Copy content Toggle raw display
$29$ \( (T + 1482)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 8360 T + 69889600 \) Copy content Toggle raw display
$37$ \( T^{2} - 4714 T + 22221796 \) Copy content Toggle raw display
$41$ \( (T - 9786)^{2} \) Copy content Toggle raw display
$43$ \( (T - 19436)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 22200 T + 492840000 \) Copy content Toggle raw display
$53$ \( T^{2} + 26790 T + 717704100 \) Copy content Toggle raw display
$59$ \( T^{2} - 28092 T + 789160464 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 1510565956 \) Copy content Toggle raw display
$67$ \( T^{2} + 23948 T + 573506704 \) Copy content Toggle raw display
$71$ \( (T + 20628)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 290T + 84100 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 9909007936 \) Copy content Toggle raw display
$83$ \( (T + 19308)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 1324232100 \) Copy content Toggle raw display
$97$ \( (T - 79078)^{2} \) Copy content Toggle raw display
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