Properties

Label 294.6.a.v
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: \(\Q(\sqrt{4705}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1176 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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 \(\beta = \sqrt{4705}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 q^{2} + 9 q^{3} + 16 q^{4} + ( - \beta + 9) q^{5} + 36 q^{6} + 64 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} + 9 q^{3} + 16 q^{4} + ( - \beta + 9) q^{5} + 36 q^{6} + 64 q^{8} + 81 q^{9} + ( - 4 \beta + 36) q^{10} + ( - 9 \beta + 1) q^{11} + 144 q^{12} + (4 \beta + 144) q^{13} + ( - 9 \beta + 81) q^{15} + 256 q^{16} + (15 \beta + 765) q^{17} + 324 q^{18} + (10 \beta + 594) q^{19} + ( - 16 \beta + 144) q^{20} + ( - 36 \beta + 4) q^{22} + (45 \beta + 1695) q^{23} + 576 q^{24} + ( - 18 \beta + 1661) q^{25} + (16 \beta + 576) q^{26} + 729 q^{27} + (54 \beta - 1988) q^{29} + ( - 36 \beta + 324) q^{30} + ( - 46 \beta + 3798) q^{31} + 1024 q^{32} + ( - 81 \beta + 9) q^{33} + (60 \beta + 3060) q^{34} + 1296 q^{36} + ( - 54 \beta + 1344) q^{37} + (40 \beta + 2376) q^{38} + (36 \beta + 1296) q^{39} + ( - 64 \beta + 576) q^{40} + ( - 39 \beta + 18315) q^{41} + (144 \beta - 11516) q^{43} + ( - 144 \beta + 16) q^{44} + ( - 81 \beta + 729) q^{45} + (180 \beta + 6780) q^{46} + (296 \beta + 432) q^{47} + 2304 q^{48} + ( - 72 \beta + 6644) q^{50} + (135 \beta + 6885) q^{51} + (64 \beta + 2304) q^{52} + (162 \beta - 16460) q^{53} + 2916 q^{54} + ( - 82 \beta + 42354) q^{55} + (90 \beta + 5346) q^{57} + (216 \beta - 7952) q^{58} + (440 \beta - 13356) q^{59} + ( - 144 \beta + 1296) q^{60} + ( - 346 \beta + 10206) q^{61} + ( - 184 \beta + 15192) q^{62} + 4096 q^{64} + ( - 108 \beta - 17524) q^{65} + ( - 324 \beta + 36) q^{66} + ( - 414 \beta - 18086) q^{67} + (240 \beta + 12240) q^{68} + (405 \beta + 15255) q^{69} + ( - 585 \beta + 36853) q^{71} + 5184 q^{72} + ( - 574 \beta - 37386) q^{73} + ( - 216 \beta + 5376) q^{74} + ( - 162 \beta + 14949) q^{75} + (160 \beta + 9504) q^{76} + (144 \beta + 5184) q^{78} + ( - 90 \beta - 11558) q^{79} + ( - 256 \beta + 2304) q^{80} + 6561 q^{81} + ( - 156 \beta + 73260) q^{82} + (72 \beta + 73908) q^{83} + ( - 630 \beta - 63690) q^{85} + (576 \beta - 46064) q^{86} + (486 \beta - 17892) q^{87} + ( - 576 \beta + 64) q^{88} + ( - 119 \beta + 82323) q^{89} + ( - 324 \beta + 2916) q^{90} + (720 \beta + 27120) q^{92} + ( - 414 \beta + 34182) q^{93} + (1184 \beta + 1728) q^{94} + ( - 504 \beta - 41704) q^{95} + 9216 q^{96} + ( - 826 \beta + 81018) q^{97} + ( - 729 \beta + 81) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{2} + 18 q^{3} + 32 q^{4} + 18 q^{5} + 72 q^{6} + 128 q^{8} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{2} + 18 q^{3} + 32 q^{4} + 18 q^{5} + 72 q^{6} + 128 q^{8} + 162 q^{9} + 72 q^{10} + 2 q^{11} + 288 q^{12} + 288 q^{13} + 162 q^{15} + 512 q^{16} + 1530 q^{17} + 648 q^{18} + 1188 q^{19} + 288 q^{20} + 8 q^{22} + 3390 q^{23} + 1152 q^{24} + 3322 q^{25} + 1152 q^{26} + 1458 q^{27} - 3976 q^{29} + 648 q^{30} + 7596 q^{31} + 2048 q^{32} + 18 q^{33} + 6120 q^{34} + 2592 q^{36} + 2688 q^{37} + 4752 q^{38} + 2592 q^{39} + 1152 q^{40} + 36630 q^{41} - 23032 q^{43} + 32 q^{44} + 1458 q^{45} + 13560 q^{46} + 864 q^{47} + 4608 q^{48} + 13288 q^{50} + 13770 q^{51} + 4608 q^{52} - 32920 q^{53} + 5832 q^{54} + 84708 q^{55} + 10692 q^{57} - 15904 q^{58} - 26712 q^{59} + 2592 q^{60} + 20412 q^{61} + 30384 q^{62} + 8192 q^{64} - 35048 q^{65} + 72 q^{66} - 36172 q^{67} + 24480 q^{68} + 30510 q^{69} + 73706 q^{71} + 10368 q^{72} - 74772 q^{73} + 10752 q^{74} + 29898 q^{75} + 19008 q^{76} + 10368 q^{78} - 23116 q^{79} + 4608 q^{80} + 13122 q^{81} + 146520 q^{82} + 147816 q^{83} - 127380 q^{85} - 92128 q^{86} - 35784 q^{87} + 128 q^{88} + 164646 q^{89} + 5832 q^{90} + 54240 q^{92} + 68364 q^{93} + 3456 q^{94} - 83408 q^{95} + 18432 q^{96} + 162036 q^{97} + 162 q^{99}+O(q^{100}) \) 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
34.7965
−33.7965
4.00000 9.00000 16.0000 −59.5930 36.0000 0 64.0000 81.0000 −238.372
1.2 4.00000 9.00000 16.0000 77.5930 36.0000 0 64.0000 81.0000 310.372
\(n\): 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.v yes 2
3.b odd 2 1 882.6.a.bc 2
7.b odd 2 1 294.6.a.s 2
7.c even 3 2 294.6.e.t 4
7.d odd 6 2 294.6.e.w 4
21.c even 2 1 882.6.a.bg 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.6.a.s 2 7.b odd 2 1
294.6.a.v yes 2 1.a even 1 1 trivial
294.6.e.t 4 7.c even 3 2
294.6.e.w 4 7.d odd 6 2
882.6.a.bc 2 3.b odd 2 1
882.6.a.bg 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(294))\):

\( T_{5}^{2} - 18T_{5} - 4624 \) Copy content Toggle raw display
\( T_{11}^{2} - 2T_{11} - 381104 \) 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
$5$ \( T^{2} - 18T - 4624 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 2T - 381104 \) Copy content Toggle raw display
$13$ \( T^{2} - 288T - 54544 \) Copy content Toggle raw display
$17$ \( T^{2} - 1530 T - 473400 \) Copy content Toggle raw display
$19$ \( T^{2} - 1188 T - 117664 \) Copy content Toggle raw display
$23$ \( T^{2} - 3390 T - 6654600 \) Copy content Toggle raw display
$29$ \( T^{2} + 3976 T - 9767636 \) Copy content Toggle raw display
$31$ \( T^{2} - 7596 T + 4469024 \) Copy content Toggle raw display
$37$ \( T^{2} - 2688 T - 11913444 \) Copy content Toggle raw display
$41$ \( T^{2} - 36630 T + 328282920 \) Copy content Toggle raw display
$43$ \( T^{2} + 23032 T + 35055376 \) Copy content Toggle raw display
$47$ \( T^{2} - 864 T - 412046656 \) Copy content Toggle raw display
$53$ \( T^{2} + 32920 T + 147453580 \) Copy content Toggle raw display
$59$ \( T^{2} + 26712 T - 732505264 \) Copy content Toggle raw display
$61$ \( T^{2} - 20412 T - 459101344 \) Copy content Toggle raw display
$67$ \( T^{2} + 36172 T - 479314784 \) Copy content Toggle raw display
$71$ \( T^{2} - 73706 T - 252025016 \) Copy content Toggle raw display
$73$ \( T^{2} + 74772 T - 152471584 \) Copy content Toggle raw display
$79$ \( T^{2} + 23116 T + 95476864 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 5438001744 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 6710448824 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 3353807744 \) Copy content Toggle raw display
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