Properties

Label 294.6.a.w
Level $294$
Weight $6$
Character orbit 294.a
Self dual yes
Analytic conductor $47.153$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [294,6,Mod(1,294)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(294, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("294.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 294.a (trivial)

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: yes
Analytic conductor: \(47.1528430250\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{9601}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 2400 \) 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)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{9601})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 q^{2} + 9 q^{3} + 16 q^{4} + ( - \beta + 27) q^{5} + 36 q^{6} + 64 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} + 9 q^{3} + 16 q^{4} + ( - \beta + 27) q^{5} + 36 q^{6} + 64 q^{8} + 81 q^{9} + ( - 4 \beta + 108) q^{10} + (5 \beta + 93) q^{11} + 144 q^{12} + (11 \beta + 184) q^{13} + ( - 9 \beta + 243) q^{15} + 256 q^{16} + ( - 20 \beta + 180) q^{17} + 324 q^{18} + ( - 39 \beta + 904) q^{19} + ( - 16 \beta + 432) q^{20} + (20 \beta + 372) q^{22} + ( - 4 \beta + 1620) q^{23} + 576 q^{24} + ( - 53 \beta + 4) q^{25} + (44 \beta + 736) q^{26} + 729 q^{27} + (61 \beta + 2199) q^{29} + ( - 36 \beta + 972) q^{30} + (164 \beta - 1079) q^{31} + 1024 q^{32} + (45 \beta + 837) q^{33} + ( - 80 \beta + 720) q^{34} + 1296 q^{36} + (51 \beta + 10268) q^{37} + ( - 156 \beta + 3616) q^{38} + (99 \beta + 1656) q^{39} + ( - 64 \beta + 1728) q^{40} + (66 \beta - 4440) q^{41} + ( - 87 \beta + 7970) q^{43} + (80 \beta + 1488) q^{44} + ( - 81 \beta + 2187) q^{45} + ( - 16 \beta + 6480) q^{46} + (156 \beta - 17034) q^{47} + 2304 q^{48} + ( - 212 \beta + 16) q^{50} + ( - 180 \beta + 1620) q^{51} + (176 \beta + 2944) q^{52} + (225 \beta + 24507) q^{53} + 2916 q^{54} + (37 \beta - 9489) q^{55} + ( - 351 \beta + 8136) q^{57} + (244 \beta + 8796) q^{58} + (41 \beta + 28347) q^{59} + ( - 144 \beta + 3888) q^{60} + (32 \beta - 33770) q^{61} + (656 \beta - 4316) q^{62} + 4096 q^{64} + (102 \beta - 21432) q^{65} + (180 \beta + 3348) q^{66} + ( - 337 \beta + 38030) q^{67} + ( - 320 \beta + 2880) q^{68} + ( - 36 \beta + 14580) q^{69} + (1340 \beta - 5166) q^{71} + 5184 q^{72} + ( - 875 \beta + 2038) q^{73} + (204 \beta + 41072) q^{74} + ( - 477 \beta + 36) q^{75} + ( - 624 \beta + 14464) q^{76} + (396 \beta + 6624) q^{78} + ( - 986 \beta + 13799) q^{79} + ( - 256 \beta + 6912) q^{80} + 6561 q^{81} + (264 \beta - 17760) q^{82} + ( - 1769 \beta + 1359) q^{83} + ( - 700 \beta + 52860) q^{85} + ( - 348 \beta + 31880) q^{86} + (549 \beta + 19791) q^{87} + (320 \beta + 5952) q^{88} + (210 \beta - 88386) q^{89} + ( - 324 \beta + 8748) q^{90} + ( - 64 \beta + 25920) q^{92} + (1476 \beta - 9711) q^{93} + (624 \beta - 68136) q^{94} + ( - 1918 \beta + 118008) q^{95} + 9216 q^{96} + (1981 \beta + 63721) q^{97} + (405 \beta + 7533) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{2} + 18 q^{3} + 32 q^{4} + 53 q^{5} + 72 q^{6} + 128 q^{8} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{2} + 18 q^{3} + 32 q^{4} + 53 q^{5} + 72 q^{6} + 128 q^{8} + 162 q^{9} + 212 q^{10} + 191 q^{11} + 288 q^{12} + 379 q^{13} + 477 q^{15} + 512 q^{16} + 340 q^{17} + 648 q^{18} + 1769 q^{19} + 848 q^{20} + 764 q^{22} + 3236 q^{23} + 1152 q^{24} - 45 q^{25} + 1516 q^{26} + 1458 q^{27} + 4459 q^{29} + 1908 q^{30} - 1994 q^{31} + 2048 q^{32} + 1719 q^{33} + 1360 q^{34} + 2592 q^{36} + 20587 q^{37} + 7076 q^{38} + 3411 q^{39} + 3392 q^{40} - 8814 q^{41} + 15853 q^{43} + 3056 q^{44} + 4293 q^{45} + 12944 q^{46} - 33912 q^{47} + 4608 q^{48} - 180 q^{50} + 3060 q^{51} + 6064 q^{52} + 49239 q^{53} + 5832 q^{54} - 18941 q^{55} + 15921 q^{57} + 17836 q^{58} + 56735 q^{59} + 7632 q^{60} - 67508 q^{61} - 7976 q^{62} + 8192 q^{64} - 42762 q^{65} + 6876 q^{66} + 75723 q^{67} + 5440 q^{68} + 29124 q^{69} - 8992 q^{71} + 10368 q^{72} + 3201 q^{73} + 82348 q^{74} - 405 q^{75} + 28304 q^{76} + 13644 q^{78} + 26612 q^{79} + 13568 q^{80} + 13122 q^{81} - 35256 q^{82} + 949 q^{83} + 105020 q^{85} + 63412 q^{86} + 40131 q^{87} + 12224 q^{88} - 176562 q^{89} + 17172 q^{90} + 51776 q^{92} - 17946 q^{93} - 135648 q^{94} + 234098 q^{95} + 18432 q^{96} + 129423 q^{97} + 15471 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
49.4923
−48.4923
4.00000 9.00000 16.0000 −22.4923 36.0000 0 64.0000 81.0000 −89.9694
1.2 4.00000 9.00000 16.0000 75.4923 36.0000 0 64.0000 81.0000 301.969
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(7\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 294.6.a.w 2
3.b odd 2 1 882.6.a.bb 2
7.b odd 2 1 294.6.a.r 2
7.c even 3 2 294.6.e.s 4
7.d odd 6 2 42.6.e.c 4
21.c even 2 1 882.6.a.bh 2
21.g even 6 2 126.6.g.h 4
28.f even 6 2 336.6.q.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.e.c 4 7.d odd 6 2
126.6.g.h 4 21.g even 6 2
294.6.a.r 2 7.b odd 2 1
294.6.a.w 2 1.a even 1 1 trivial
294.6.e.s 4 7.c even 3 2
336.6.q.f 4 28.f even 6 2
882.6.a.bb 2 3.b odd 2 1
882.6.a.bh 2 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(294))\):

\( T_{5}^{2} - 53T_{5} - 1698 \) Copy content Toggle raw display
\( T_{11}^{2} - 191T_{11} - 50886 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 4)^{2} \) Copy content Toggle raw display
$3$ \( (T - 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 53T - 1698 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 191T - 50886 \) Copy content Toggle raw display
$13$ \( T^{2} - 379T - 254520 \) Copy content Toggle raw display
$17$ \( T^{2} - 340T - 931200 \) Copy content Toggle raw display
$19$ \( T^{2} - 1769 T - 2868440 \) Copy content Toggle raw display
$23$ \( T^{2} - 3236 T + 2579520 \) Copy content Toggle raw display
$29$ \( T^{2} - 4459 T - 3960660 \) Copy content Toggle raw display
$31$ \( T^{2} + 1994 T - 63563115 \) Copy content Toggle raw display
$37$ \( T^{2} - 20587 T + 99713092 \) Copy content Toggle raw display
$41$ \( T^{2} + 8814 T + 8966160 \) Copy content Toggle raw display
$43$ \( T^{2} - 15853 T + 44661910 \) Copy content Toggle raw display
$47$ \( T^{2} + 33912 T + 229093452 \) Copy content Toggle raw display
$53$ \( T^{2} - 49239 T + 484607124 \) Copy content Toggle raw display
$59$ \( T^{2} - 56735 T + 800680236 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 1136874660 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 1160899190 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 4289674884 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 1835129806 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 2156463813 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 7511023590 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 7687683936 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 5231869258 \) Copy content Toggle raw display
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