Properties

Label 294.6.e.i
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.

\(f(q)\) \(=\) \( q + 4 \zeta_{6} q^{2} + (9 \zeta_{6} - 9) q^{3} + (16 \zeta_{6} - 16) q^{4} - 26 \zeta_{6} q^{5} - 36 q^{6} - 64 q^{8} - 81 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 4 \zeta_{6} q^{2} + (9 \zeta_{6} - 9) q^{3} + (16 \zeta_{6} - 16) q^{4} - 26 \zeta_{6} q^{5} - 36 q^{6} - 64 q^{8} - 81 \zeta_{6} q^{9} + ( - 104 \zeta_{6} + 104) q^{10} + (664 \zeta_{6} - 664) q^{11} - 144 \zeta_{6} q^{12} + 318 q^{13} + 234 q^{15} - 256 \zeta_{6} q^{16} + (1582 \zeta_{6} - 1582) q^{17} + ( - 324 \zeta_{6} + 324) q^{18} - 236 \zeta_{6} q^{19} + 416 q^{20} - 2656 q^{22} - 2212 \zeta_{6} q^{23} + ( - 576 \zeta_{6} + 576) q^{24} + ( - 2449 \zeta_{6} + 2449) q^{25} + 1272 \zeta_{6} q^{26} + 729 q^{27} - 4954 q^{29} + 936 \zeta_{6} q^{30} + ( - 7128 \zeta_{6} + 7128) q^{31} + ( - 1024 \zeta_{6} + 1024) q^{32} - 5976 \zeta_{6} q^{33} - 6328 q^{34} + 1296 q^{36} - 4358 \zeta_{6} q^{37} + ( - 944 \zeta_{6} + 944) q^{38} + (2862 \zeta_{6} - 2862) q^{39} + 1664 \zeta_{6} q^{40} + 10542 q^{41} - 8452 q^{43} - 10624 \zeta_{6} q^{44} + (2106 \zeta_{6} - 2106) q^{45} + ( - 8848 \zeta_{6} + 8848) q^{46} - 5352 \zeta_{6} q^{47} + 2304 q^{48} + 9796 q^{50} - 14238 \zeta_{6} q^{51} + (5088 \zeta_{6} - 5088) q^{52} + ( - 33354 \zeta_{6} + 33354) q^{53} + 2916 \zeta_{6} q^{54} + 17264 q^{55} + 2124 q^{57} - 19816 \zeta_{6} q^{58} + ( - 15436 \zeta_{6} + 15436) q^{59} + (3744 \zeta_{6} - 3744) q^{60} + 36762 \zeta_{6} q^{61} + 28512 q^{62} + 4096 q^{64} - 8268 \zeta_{6} q^{65} + ( - 23904 \zeta_{6} + 23904) q^{66} + (40972 \zeta_{6} - 40972) q^{67} - 25312 \zeta_{6} q^{68} + 19908 q^{69} - 9092 q^{71} + 5184 \zeta_{6} q^{72} + ( - 73454 \zeta_{6} + 73454) q^{73} + ( - 17432 \zeta_{6} + 17432) q^{74} + 22041 \zeta_{6} q^{75} + 3776 q^{76} - 11448 q^{78} - 89400 \zeta_{6} q^{79} + (6656 \zeta_{6} - 6656) q^{80} + (6561 \zeta_{6} - 6561) q^{81} + 42168 \zeta_{6} q^{82} - 6428 q^{83} + 41132 q^{85} - 33808 \zeta_{6} q^{86} + ( - 44586 \zeta_{6} + 44586) q^{87} + ( - 42496 \zeta_{6} + 42496) q^{88} + 122658 \zeta_{6} q^{89} - 8424 q^{90} + 35392 q^{92} + 64152 \zeta_{6} q^{93} + ( - 21408 \zeta_{6} + 21408) q^{94} + (6136 \zeta_{6} - 6136) q^{95} + 9216 \zeta_{6} q^{96} + 21370 q^{97} + 53784 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} - 9 q^{3} - 16 q^{4} - 26 q^{5} - 72 q^{6} - 128 q^{8} - 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} - 9 q^{3} - 16 q^{4} - 26 q^{5} - 72 q^{6} - 128 q^{8} - 81 q^{9} + 104 q^{10} - 664 q^{11} - 144 q^{12} + 636 q^{13} + 468 q^{15} - 256 q^{16} - 1582 q^{17} + 324 q^{18} - 236 q^{19} + 832 q^{20} - 5312 q^{22} - 2212 q^{23} + 576 q^{24} + 2449 q^{25} + 1272 q^{26} + 1458 q^{27} - 9908 q^{29} + 936 q^{30} + 7128 q^{31} + 1024 q^{32} - 5976 q^{33} - 12656 q^{34} + 2592 q^{36} - 4358 q^{37} + 944 q^{38} - 2862 q^{39} + 1664 q^{40} + 21084 q^{41} - 16904 q^{43} - 10624 q^{44} - 2106 q^{45} + 8848 q^{46} - 5352 q^{47} + 4608 q^{48} + 19592 q^{50} - 14238 q^{51} - 5088 q^{52} + 33354 q^{53} + 2916 q^{54} + 34528 q^{55} + 4248 q^{57} - 19816 q^{58} + 15436 q^{59} - 3744 q^{60} + 36762 q^{61} + 57024 q^{62} + 8192 q^{64} - 8268 q^{65} + 23904 q^{66} - 40972 q^{67} - 25312 q^{68} + 39816 q^{69} - 18184 q^{71} + 5184 q^{72} + 73454 q^{73} + 17432 q^{74} + 22041 q^{75} + 7552 q^{76} - 22896 q^{78} - 89400 q^{79} - 6656 q^{80} - 6561 q^{81} + 42168 q^{82} - 12856 q^{83} + 82264 q^{85} - 33808 q^{86} + 44586 q^{87} + 42496 q^{88} + 122658 q^{89} - 16848 q^{90} + 70784 q^{92} + 64152 q^{93} + 21408 q^{94} - 6136 q^{95} + 9216 q^{96} + 42740 q^{97} + 107568 q^{99}+O(q^{100}) \) 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 −13.0000 22.5167i −36.0000 0 −64.0000 −40.5000 70.1481i 52.0000 90.0666i
79.1 2.00000 3.46410i −4.50000 7.79423i −8.00000 13.8564i −13.0000 + 22.5167i −36.0000 0 −64.0000 −40.5000 + 70.1481i 52.0000 + 90.0666i
\(n\): 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.i 2
7.b odd 2 1 294.6.e.p 2
7.c even 3 1 42.6.a.d 1
7.c even 3 1 inner 294.6.e.i 2
7.d odd 6 1 294.6.a.b 1
7.d odd 6 1 294.6.e.p 2
21.g even 6 1 882.6.a.s 1
21.h odd 6 1 126.6.a.i 1
28.g odd 6 1 336.6.a.h 1
35.j even 6 1 1050.6.a.k 1
35.l odd 12 2 1050.6.g.i 2
84.n even 6 1 1008.6.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.a.d 1 7.c even 3 1
126.6.a.i 1 21.h odd 6 1
294.6.a.b 1 7.d odd 6 1
294.6.e.i 2 1.a even 1 1 trivial
294.6.e.i 2 7.c even 3 1 inner
294.6.e.p 2 7.b odd 2 1
294.6.e.p 2 7.d odd 6 1
336.6.a.h 1 28.g odd 6 1
882.6.a.s 1 21.g even 6 1
1008.6.a.j 1 84.n even 6 1
1050.6.a.k 1 35.j even 6 1
1050.6.g.i 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 \(S_{6}^{\mathrm{new}}(294, [\chi])\):

\( T_{5}^{2} + 26T_{5} + 676 \) Copy content Toggle raw display
\( T_{11}^{2} + 664T_{11} + 440896 \) 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
$5$ \( T^{2} + 26T + 676 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 664T + 440896 \) Copy content Toggle raw display
$13$ \( (T - 318)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 1582 T + 2502724 \) Copy content Toggle raw display
$19$ \( T^{2} + 236T + 55696 \) Copy content Toggle raw display
$23$ \( T^{2} + 2212 T + 4892944 \) Copy content Toggle raw display
$29$ \( (T + 4954)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 7128 T + 50808384 \) Copy content Toggle raw display
$37$ \( T^{2} + 4358 T + 18992164 \) Copy content Toggle raw display
$41$ \( (T - 10542)^{2} \) Copy content Toggle raw display
$43$ \( (T + 8452)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 5352 T + 28643904 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 1112489316 \) Copy content Toggle raw display
$59$ \( T^{2} - 15436 T + 238270096 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 1351444644 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 1678704784 \) Copy content Toggle raw display
$71$ \( (T + 9092)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 5395490116 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 7992360000 \) Copy content Toggle raw display
$83$ \( (T + 6428)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 15044984964 \) Copy content Toggle raw display
$97$ \( (T - 21370)^{2} \) Copy content Toggle raw display
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