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Properties

Label 294.4.e.g
Level $294$
Weight $4$
Character orbit 294.e
Analytic conductor $17.347$
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] = [294,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("294.67");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(17.3465615417\)
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 6)
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.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{2} + (3 \zeta_{6} - 3) q^{3} + (4 \zeta_{6} - 4) q^{4} + 6 \zeta_{6} q^{5} - 6 q^{6} - 8 q^{8} - 9 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 \zeta_{6} q^{2} + (3 \zeta_{6} - 3) q^{3} + (4 \zeta_{6} - 4) q^{4} + 6 \zeta_{6} q^{5} - 6 q^{6} - 8 q^{8} - 9 \zeta_{6} q^{9} + (12 \zeta_{6} - 12) q^{10} + (12 \zeta_{6} - 12) q^{11} - 12 \zeta_{6} q^{12} - 38 q^{13} - 18 q^{15} - 16 \zeta_{6} q^{16} + (126 \zeta_{6} - 126) q^{17} + ( - 18 \zeta_{6} + 18) q^{18} + 20 \zeta_{6} q^{19} - 24 q^{20} - 24 q^{22} - 168 \zeta_{6} q^{23} + ( - 24 \zeta_{6} + 24) q^{24} + ( - 89 \zeta_{6} + 89) q^{25} - 76 \zeta_{6} q^{26} + 27 q^{27} + 30 q^{29} - 36 \zeta_{6} q^{30} + (88 \zeta_{6} - 88) q^{31} + ( - 32 \zeta_{6} + 32) q^{32} - 36 \zeta_{6} q^{33} - 252 q^{34} + 36 q^{36} - 254 \zeta_{6} q^{37} + (40 \zeta_{6} - 40) q^{38} + ( - 114 \zeta_{6} + 114) q^{39} - 48 \zeta_{6} q^{40} - 42 q^{41} - 52 q^{43} - 48 \zeta_{6} q^{44} + ( - 54 \zeta_{6} + 54) q^{45} + ( - 336 \zeta_{6} + 336) q^{46} - 96 \zeta_{6} q^{47} + 48 q^{48} + 178 q^{50} - 378 \zeta_{6} q^{51} + ( - 152 \zeta_{6} + 152) q^{52} + (198 \zeta_{6} - 198) q^{53} + 54 \zeta_{6} q^{54} - 72 q^{55} - 60 q^{57} + 60 \zeta_{6} q^{58} + (660 \zeta_{6} - 660) q^{59} + ( - 72 \zeta_{6} + 72) q^{60} - 538 \zeta_{6} q^{61} - 176 q^{62} + 64 q^{64} - 228 \zeta_{6} q^{65} + ( - 72 \zeta_{6} + 72) q^{66} + (884 \zeta_{6} - 884) q^{67} - 504 \zeta_{6} q^{68} + 504 q^{69} + 792 q^{71} + 72 \zeta_{6} q^{72} + ( - 218 \zeta_{6} + 218) q^{73} + ( - 508 \zeta_{6} + 508) q^{74} + 267 \zeta_{6} q^{75} - 80 q^{76} + 228 q^{78} + 520 \zeta_{6} q^{79} + ( - 96 \zeta_{6} + 96) q^{80} + (81 \zeta_{6} - 81) q^{81} - 84 \zeta_{6} q^{82} + 492 q^{83} - 756 q^{85} - 104 \zeta_{6} q^{86} + (90 \zeta_{6} - 90) q^{87} + ( - 96 \zeta_{6} + 96) q^{88} + 810 \zeta_{6} q^{89} + 108 q^{90} + 672 q^{92} - 264 \zeta_{6} q^{93} + ( - 192 \zeta_{6} + 192) q^{94} + (120 \zeta_{6} - 120) q^{95} + 96 \zeta_{6} q^{96} - 1154 q^{97} + 108 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 3 q^{3} - 4 q^{4} + 6 q^{5} - 12 q^{6} - 16 q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 3 q^{3} - 4 q^{4} + 6 q^{5} - 12 q^{6} - 16 q^{8} - 9 q^{9} - 12 q^{10} - 12 q^{11} - 12 q^{12} - 76 q^{13} - 36 q^{15} - 16 q^{16} - 126 q^{17} + 18 q^{18} + 20 q^{19} - 48 q^{20} - 48 q^{22} - 168 q^{23} + 24 q^{24} + 89 q^{25} - 76 q^{26} + 54 q^{27} + 60 q^{29} - 36 q^{30} - 88 q^{31} + 32 q^{32} - 36 q^{33} - 504 q^{34} + 72 q^{36} - 254 q^{37} - 40 q^{38} + 114 q^{39} - 48 q^{40} - 84 q^{41} - 104 q^{43} - 48 q^{44} + 54 q^{45} + 336 q^{46} - 96 q^{47} + 96 q^{48} + 356 q^{50} - 378 q^{51} + 152 q^{52} - 198 q^{53} + 54 q^{54} - 144 q^{55} - 120 q^{57} + 60 q^{58} - 660 q^{59} + 72 q^{60} - 538 q^{61} - 352 q^{62} + 128 q^{64} - 228 q^{65} + 72 q^{66} - 884 q^{67} - 504 q^{68} + 1008 q^{69} + 1584 q^{71} + 72 q^{72} + 218 q^{73} + 508 q^{74} + 267 q^{75} - 160 q^{76} + 456 q^{78} + 520 q^{79} + 96 q^{80} - 81 q^{81} - 84 q^{82} + 984 q^{83} - 1512 q^{85} - 104 q^{86} - 90 q^{87} + 96 q^{88} + 810 q^{89} + 216 q^{90} + 1344 q^{92} - 264 q^{93} + 192 q^{94} - 120 q^{95} + 96 q^{96} - 2308 q^{97} + 216 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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
1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i 3.00000 + 5.19615i −6.00000 0 −8.00000 −4.50000 − 7.79423i −6.00000 + 10.3923i
79.1 1.00000 − 1.73205i −1.50000 − 2.59808i −2.00000 − 3.46410i 3.00000 − 5.19615i −6.00000 0 −8.00000 −4.50000 + 7.79423i −6.00000 − 10.3923i
\(n\): 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 294.4.e.g 2
3.b odd 2 1 882.4.g.f 2
7.b odd 2 1 294.4.e.h 2
7.c even 3 1 294.4.a.e 1
7.c even 3 1 inner 294.4.e.g 2
7.d odd 6 1 6.4.a.a ✓ 1
7.d odd 6 1 294.4.e.h 2
21.c even 2 1 882.4.g.i 2
21.g even 6 1 18.4.a.a 1
21.g even 6 1 882.4.g.i 2
21.h odd 6 1 882.4.a.n 1
21.h odd 6 1 882.4.g.f 2
28.f even 6 1 48.4.a.c 1
28.g odd 6 1 2352.4.a.e 1
35.i odd 6 1 150.4.a.i 1
35.k even 12 2 150.4.c.d 2
56.j odd 6 1 192.4.a.i 1
56.m even 6 1 192.4.a.c 1
63.i even 6 1 162.4.c.c 2
63.k odd 6 1 162.4.c.f 2
63.s even 6 1 162.4.c.c 2
63.t odd 6 1 162.4.c.f 2
77.i even 6 1 726.4.a.f 1
84.j odd 6 1 144.4.a.c 1
91.s odd 6 1 1014.4.a.g 1
91.bb even 12 2 1014.4.b.d 2
105.p even 6 1 450.4.a.h 1
105.w odd 12 2 450.4.c.e 2
112.v even 12 2 768.4.d.c 2
112.x odd 12 2 768.4.d.n 2
119.h odd 6 1 1734.4.a.d 1
133.o even 6 1 2166.4.a.i 1
140.s even 6 1 1200.4.a.b 1
140.x odd 12 2 1200.4.f.j 2
168.ba even 6 1 576.4.a.q 1
168.be odd 6 1 576.4.a.r 1
231.k odd 6 1 2178.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a ✓ 1 7.d odd 6 1
18.4.a.a 1 21.g even 6 1
48.4.a.c 1 28.f even 6 1
144.4.a.c 1 84.j odd 6 1
150.4.a.i 1 35.i odd 6 1
150.4.c.d 2 35.k even 12 2
162.4.c.c 2 63.i even 6 1
162.4.c.c 2 63.s even 6 1
162.4.c.f 2 63.k odd 6 1
162.4.c.f 2 63.t odd 6 1
192.4.a.c 1 56.m even 6 1
192.4.a.i 1 56.j odd 6 1
294.4.a.e 1 7.c even 3 1
294.4.e.g 2 1.a even 1 1 trivial
294.4.e.g 2 7.c even 3 1 inner
294.4.e.h 2 7.b odd 2 1
294.4.e.h 2 7.d odd 6 1
450.4.a.h 1 105.p even 6 1
450.4.c.e 2 105.w odd 12 2
576.4.a.q 1 168.ba even 6 1
576.4.a.r 1 168.be odd 6 1
726.4.a.f 1 77.i even 6 1
768.4.d.c 2 112.v even 12 2
768.4.d.n 2 112.x odd 12 2
882.4.a.n 1 21.h odd 6 1
882.4.g.f 2 3.b odd 2 1
882.4.g.f 2 21.h odd 6 1
882.4.g.i 2 21.c even 2 1
882.4.g.i 2 21.g even 6 1
1014.4.a.g 1 91.s odd 6 1
1014.4.b.d 2 91.bb even 12 2
1200.4.a.b 1 140.s even 6 1
1200.4.f.j 2 140.x odd 12 2
1734.4.a.d 1 119.h odd 6 1
2166.4.a.i 1 133.o even 6 1
2178.4.a.e 1 231.k odd 6 1
2352.4.a.e 1 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(294, [\chi])\):

\( T_{5}^{2} - 6T_{5} + 36 \) Copy content Toggle raw display
\( T_{11}^{2} + 12T_{11} + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$13$ \( (T + 38)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 126T + 15876 \) Copy content Toggle raw display
$19$ \( T^{2} - 20T + 400 \) Copy content Toggle raw display
$23$ \( T^{2} + 168T + 28224 \) Copy content Toggle raw display
$29$ \( (T - 30)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 88T + 7744 \) Copy content Toggle raw display
$37$ \( T^{2} + 254T + 64516 \) Copy content Toggle raw display
$41$ \( (T + 42)^{2} \) Copy content Toggle raw display
$43$ \( (T + 52)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 96T + 9216 \) Copy content Toggle raw display
$53$ \( T^{2} + 198T + 39204 \) Copy content Toggle raw display
$59$ \( T^{2} + 660T + 435600 \) Copy content Toggle raw display
$61$ \( T^{2} + 538T + 289444 \) Copy content Toggle raw display
$67$ \( T^{2} + 884T + 781456 \) Copy content Toggle raw display
$71$ \( (T - 792)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 218T + 47524 \) Copy content Toggle raw display
$79$ \( T^{2} - 520T + 270400 \) Copy content Toggle raw display
$83$ \( (T - 492)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 810T + 656100 \) Copy content Toggle raw display
$97$ \( (T + 1154)^{2} \) Copy content Toggle raw display
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