Properties

Label 294.6.e.k
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.

\(f(q)\) \(=\) \( q + 4 \zeta_{6} q^{2} + (9 \zeta_{6} - 9) q^{3} + (16 \zeta_{6} - 16) q^{4} + 44 \zeta_{6} q^{5} - 36 q^{6} - 64 q^{8} - 81 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 4 \zeta_{6} q^{2} + (9 \zeta_{6} - 9) q^{3} + (16 \zeta_{6} - 16) q^{4} + 44 \zeta_{6} q^{5} - 36 q^{6} - 64 q^{8} - 81 \zeta_{6} q^{9} + (176 \zeta_{6} - 176) q^{10} + ( - 470 \zeta_{6} + 470) q^{11} - 144 \zeta_{6} q^{12} + 1158 q^{13} - 396 q^{15} - 256 \zeta_{6} q^{16} + ( - 1204 \zeta_{6} + 1204) q^{17} + ( - 324 \zeta_{6} + 324) q^{18} - 2644 \zeta_{6} q^{19} - 704 q^{20} + 1880 q^{22} + 1190 \zeta_{6} q^{23} + ( - 576 \zeta_{6} + 576) q^{24} + ( - 1189 \zeta_{6} + 1189) q^{25} + 4632 \zeta_{6} q^{26} + 729 q^{27} + 3614 q^{29} - 1584 \zeta_{6} q^{30} + ( - 5616 \zeta_{6} + 5616) q^{31} + ( - 1024 \zeta_{6} + 1024) q^{32} + 4230 \zeta_{6} q^{33} + 4816 q^{34} + 1296 q^{36} + 6478 \zeta_{6} q^{37} + ( - 10576 \zeta_{6} + 10576) q^{38} + (10422 \zeta_{6} - 10422) q^{39} - 2816 \zeta_{6} q^{40} - 2856 q^{41} - 13492 q^{43} + 7520 \zeta_{6} q^{44} + ( - 3564 \zeta_{6} + 3564) q^{45} + (4760 \zeta_{6} - 4760) q^{46} - 18372 \zeta_{6} q^{47} + 2304 q^{48} + 4756 q^{50} + 10836 \zeta_{6} q^{51} + (18528 \zeta_{6} - 18528) q^{52} + ( - 4374 \zeta_{6} + 4374) q^{53} + 2916 \zeta_{6} q^{54} + 20680 q^{55} + 23796 q^{57} + 14456 \zeta_{6} q^{58} + ( - 30248 \zeta_{6} + 30248) q^{59} + ( - 6336 \zeta_{6} + 6336) q^{60} + 19542 \zeta_{6} q^{61} + 22464 q^{62} + 4096 q^{64} + 50952 \zeta_{6} q^{65} + (16920 \zeta_{6} - 16920) q^{66} + (54328 \zeta_{6} - 54328) q^{67} + 19264 \zeta_{6} q^{68} - 10710 q^{69} - 10730 q^{71} + 5184 \zeta_{6} q^{72} + ( - 35374 \zeta_{6} + 35374) q^{73} + (25912 \zeta_{6} - 25912) q^{74} + 10701 \zeta_{6} q^{75} + 42304 q^{76} - 41688 q^{78} + 49956 \zeta_{6} q^{79} + ( - 11264 \zeta_{6} + 11264) q^{80} + (6561 \zeta_{6} - 6561) q^{81} - 11424 \zeta_{6} q^{82} + 26948 q^{83} + 52976 q^{85} - 53968 \zeta_{6} q^{86} + (32526 \zeta_{6} - 32526) q^{87} + (30080 \zeta_{6} - 30080) q^{88} + 100776 \zeta_{6} q^{89} + 14256 q^{90} - 19040 q^{92} + 50544 \zeta_{6} q^{93} + ( - 73488 \zeta_{6} + 73488) q^{94} + ( - 116336 \zeta_{6} + 116336) q^{95} + 9216 \zeta_{6} q^{96} - 77134 q^{97} - 38070 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} - 9 q^{3} - 16 q^{4} + 44 q^{5} - 72 q^{6} - 128 q^{8} - 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} - 9 q^{3} - 16 q^{4} + 44 q^{5} - 72 q^{6} - 128 q^{8} - 81 q^{9} - 176 q^{10} + 470 q^{11} - 144 q^{12} + 2316 q^{13} - 792 q^{15} - 256 q^{16} + 1204 q^{17} + 324 q^{18} - 2644 q^{19} - 1408 q^{20} + 3760 q^{22} + 1190 q^{23} + 576 q^{24} + 1189 q^{25} + 4632 q^{26} + 1458 q^{27} + 7228 q^{29} - 1584 q^{30} + 5616 q^{31} + 1024 q^{32} + 4230 q^{33} + 9632 q^{34} + 2592 q^{36} + 6478 q^{37} + 10576 q^{38} - 10422 q^{39} - 2816 q^{40} - 5712 q^{41} - 26984 q^{43} + 7520 q^{44} + 3564 q^{45} - 4760 q^{46} - 18372 q^{47} + 4608 q^{48} + 9512 q^{50} + 10836 q^{51} - 18528 q^{52} + 4374 q^{53} + 2916 q^{54} + 41360 q^{55} + 47592 q^{57} + 14456 q^{58} + 30248 q^{59} + 6336 q^{60} + 19542 q^{61} + 44928 q^{62} + 8192 q^{64} + 50952 q^{65} - 16920 q^{66} - 54328 q^{67} + 19264 q^{68} - 21420 q^{69} - 21460 q^{71} + 5184 q^{72} + 35374 q^{73} - 25912 q^{74} + 10701 q^{75} + 84608 q^{76} - 83376 q^{78} + 49956 q^{79} + 11264 q^{80} - 6561 q^{81} - 11424 q^{82} + 53896 q^{83} + 105952 q^{85} - 53968 q^{86} - 32526 q^{87} - 30080 q^{88} + 100776 q^{89} + 28512 q^{90} - 38080 q^{92} + 50544 q^{93} + 73488 q^{94} + 116336 q^{95} + 9216 q^{96} - 154268 q^{97} - 76140 q^{99}+O(q^{100}) \) 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 22.0000 + 38.1051i −36.0000 0 −64.0000 −40.5000 70.1481i −88.0000 + 152.420i
79.1 2.00000 3.46410i −4.50000 7.79423i −8.00000 13.8564i 22.0000 38.1051i −36.0000 0 −64.0000 −40.5000 + 70.1481i −88.0000 152.420i
\(n\): 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.k 2
7.b odd 2 1 294.6.e.o 2
7.c even 3 1 294.6.a.f 1
7.c even 3 1 inner 294.6.e.k 2
7.d odd 6 1 42.6.a.b 1
7.d odd 6 1 294.6.e.o 2
21.g even 6 1 126.6.a.h 1
21.h odd 6 1 882.6.a.v 1
28.f even 6 1 336.6.a.o 1
35.i odd 6 1 1050.6.a.o 1
35.k even 12 2 1050.6.g.l 2
84.j odd 6 1 1008.6.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.a.b 1 7.d odd 6 1
126.6.a.h 1 21.g even 6 1
294.6.a.f 1 7.c even 3 1
294.6.e.k 2 1.a even 1 1 trivial
294.6.e.k 2 7.c even 3 1 inner
294.6.e.o 2 7.b odd 2 1
294.6.e.o 2 7.d odd 6 1
336.6.a.o 1 28.f even 6 1
882.6.a.v 1 21.h odd 6 1
1008.6.a.g 1 84.j odd 6 1
1050.6.a.o 1 35.i odd 6 1
1050.6.g.l 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 \(S_{6}^{\mathrm{new}}(294, [\chi])\):

\( T_{5}^{2} - 44T_{5} + 1936 \) Copy content Toggle raw display
\( T_{11}^{2} - 470T_{11} + 220900 \) 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
$5$ \( T^{2} - 44T + 1936 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 470T + 220900 \) Copy content Toggle raw display
$13$ \( (T - 1158)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 1204 T + 1449616 \) Copy content Toggle raw display
$19$ \( T^{2} + 2644 T + 6990736 \) Copy content Toggle raw display
$23$ \( T^{2} - 1190 T + 1416100 \) Copy content Toggle raw display
$29$ \( (T - 3614)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 5616 T + 31539456 \) Copy content Toggle raw display
$37$ \( T^{2} - 6478 T + 41964484 \) Copy content Toggle raw display
$41$ \( (T + 2856)^{2} \) Copy content Toggle raw display
$43$ \( (T + 13492)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 18372 T + 337530384 \) Copy content Toggle raw display
$53$ \( T^{2} - 4374 T + 19131876 \) Copy content Toggle raw display
$59$ \( T^{2} - 30248 T + 914941504 \) Copy content Toggle raw display
$61$ \( T^{2} - 19542 T + 381889764 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 2951531584 \) Copy content Toggle raw display
$71$ \( (T + 10730)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 1251319876 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 2495601936 \) Copy content Toggle raw display
$83$ \( (T - 26948)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 10155802176 \) Copy content Toggle raw display
$97$ \( (T + 77134)^{2} \) Copy content Toggle raw display
show more
show less