Properties

Label 294.6.a.o
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: \(\Q(\sqrt{505}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 126 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 7 \)
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 \(\beta = \frac{1}{2}(-1 + 7\sqrt{505})\). 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} + ( - 7 \beta + 69) q^{11} - 144 q^{12} + (\beta + 358) q^{13} + (9 \beta + 81) q^{15} + 256 q^{16} + ( - 20 \beta - 696) q^{17} - 324 q^{18} + ( - 27 \beta - 554) q^{19} + ( - 16 \beta - 144) q^{20} + (28 \beta - 276) q^{22} + ( - 4 \beta - 2256) q^{23} + 576 q^{24} + (17 \beta + 3142) q^{25} + ( - 4 \beta - 1432) q^{26} - 729 q^{27} + ( - 59 \beta + 3903) q^{29} + ( - 36 \beta - 324) q^{30} + ( - 14 \beta - 4415) q^{31} - 1024 q^{32} + (63 \beta - 621) q^{33} + (80 \beta + 2784) q^{34} + 1296 q^{36} + (81 \beta - 7246) q^{37} + (108 \beta + 2216) q^{38} + ( - 9 \beta - 3222) q^{39} + (64 \beta + 576) q^{40} + ( - 66 \beta - 3708) q^{41} + ( - 63 \beta - 2992) q^{43} + ( - 112 \beta + 1104) q^{44} + ( - 81 \beta - 729) q^{45} + (16 \beta + 9024) q^{46} + (36 \beta - 22386) q^{47} - 2304 q^{48} + ( - 68 \beta - 12568) q^{50} + (180 \beta + 6264) q^{51} + (16 \beta + 5728) q^{52} + (45 \beta + 4731) q^{53} + 2916 q^{54} + ( - 13 \beta + 42681) q^{55} + (243 \beta + 4986) q^{57} + (236 \beta - 15612) q^{58} + (377 \beta + 2727) q^{59} + (144 \beta + 1296) q^{60} + ( - 92 \beta - 21230) q^{61} + (56 \beta + 17660) q^{62} + 4096 q^{64} + ( - 366 \beta - 9408) q^{65} + ( - 252 \beta + 2484) q^{66} + ( - 317 \beta + 15092) q^{67} + ( - 320 \beta - 11136) q^{68} + (36 \beta + 20304) q^{69} + ( - 220 \beta + 45762) q^{71} - 5184 q^{72} + ( - 505 \beta + 42580) q^{73} + ( - 324 \beta + 28984) q^{74} + ( - 153 \beta - 28278) q^{75} + ( - 432 \beta - 8864) q^{76} + (36 \beta + 12888) q^{78} + ( - 688 \beta + 46979) q^{79} + ( - 256 \beta - 2304) q^{80} + 6561 q^{81} + (264 \beta + 14832) q^{82} + (31 \beta - 16905) q^{83} + (856 \beta + 129984) q^{85} + (252 \beta + 11968) q^{86} + (531 \beta - 35127) q^{87} + (448 \beta - 4416) q^{88} + ( - 174 \beta - 13866) q^{89} + (324 \beta + 2916) q^{90} + ( - 64 \beta - 36096) q^{92} + (126 \beta + 39735) q^{93} + ( - 144 \beta + 89544) q^{94} + (770 \beta + 172008) q^{95} + 9216 q^{96} + (863 \beta + 23767) q^{97} + ( - 567 \beta + 5589) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{2} - 18 q^{3} + 32 q^{4} - 17 q^{5} + 72 q^{6} - 128 q^{8} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{2} - 18 q^{3} + 32 q^{4} - 17 q^{5} + 72 q^{6} - 128 q^{8} + 162 q^{9} + 68 q^{10} + 145 q^{11} - 288 q^{12} + 715 q^{13} + 153 q^{15} + 512 q^{16} - 1372 q^{17} - 648 q^{18} - 1081 q^{19} - 272 q^{20} - 580 q^{22} - 4508 q^{23} + 1152 q^{24} + 6267 q^{25} - 2860 q^{26} - 1458 q^{27} + 7865 q^{29} - 612 q^{30} - 8816 q^{31} - 2048 q^{32} - 1305 q^{33} + 5488 q^{34} + 2592 q^{36} - 14573 q^{37} + 4324 q^{38} - 6435 q^{39} + 1088 q^{40} - 7350 q^{41} - 5921 q^{43} + 2320 q^{44} - 1377 q^{45} + 18032 q^{46} - 44808 q^{47} - 4608 q^{48} - 25068 q^{50} + 12348 q^{51} + 11440 q^{52} + 9417 q^{53} + 5832 q^{54} + 85375 q^{55} + 9729 q^{57} - 31460 q^{58} + 5077 q^{59} + 2448 q^{60} - 42368 q^{61} + 35264 q^{62} + 8192 q^{64} - 18450 q^{65} + 5220 q^{66} + 30501 q^{67} - 21952 q^{68} + 40572 q^{69} + 91744 q^{71} - 10368 q^{72} + 85665 q^{73} + 58292 q^{74} - 56403 q^{75} - 17296 q^{76} + 25740 q^{78} + 94646 q^{79} - 4352 q^{80} + 13122 q^{81} + 29400 q^{82} - 33841 q^{83} + 259112 q^{85} + 23684 q^{86} - 70785 q^{87} - 9280 q^{88} - 27558 q^{89} + 5508 q^{90} - 72128 q^{92} + 79344 q^{93} + 179232 q^{94} + 343246 q^{95} + 18432 q^{96} + 46671 q^{97} + 11745 q^{99}+O(q^{100}) \) 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
11.7361
−10.7361
−4.00000 −9.00000 16.0000 −87.1527 36.0000 0 −64.0000 81.0000 348.611
1.2 −4.00000 −9.00000 16.0000 70.1527 36.0000 0 −64.0000 81.0000 −280.611
\(n\): 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.o 2
3.b odd 2 1 882.6.a.bs 2
7.b odd 2 1 294.6.a.p 2
7.c even 3 2 294.6.e.y 4
7.d odd 6 2 42.6.e.d 4
21.c even 2 1 882.6.a.bm 2
21.g even 6 2 126.6.g.g 4
28.f even 6 2 336.6.q.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.e.d 4 7.d odd 6 2
126.6.g.g 4 21.g even 6 2
294.6.a.o 2 1.a even 1 1 trivial
294.6.a.p 2 7.b odd 2 1
294.6.e.y 4 7.c even 3 2
336.6.q.h 4 28.f even 6 2
882.6.a.bm 2 21.c even 2 1
882.6.a.bs 2 3.b odd 2 1

Hecke kernels

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

\( T_{5}^{2} + 17T_{5} - 6114 \) Copy content Toggle raw display
\( T_{11}^{2} - 145T_{11} - 297870 \) 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
$5$ \( T^{2} + 17T - 6114 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 145T - 297870 \) Copy content Toggle raw display
$13$ \( T^{2} - 715T + 121620 \) Copy content Toggle raw display
$17$ \( T^{2} + 1372 T - 2003904 \) Copy content Toggle raw display
$19$ \( T^{2} + 1081 T - 4217636 \) Copy content Toggle raw display
$23$ \( T^{2} + 4508 T + 4981536 \) Copy content Toggle raw display
$29$ \( T^{2} - 7865 T - 6069780 \) Copy content Toggle raw display
$31$ \( T^{2} + 8816 T + 18217959 \) Copy content Toggle raw display
$37$ \( T^{2} + 14573 T + 12505096 \) Copy content Toggle raw display
$41$ \( T^{2} + 7350 T - 13441680 \) Copy content Toggle raw display
$43$ \( T^{2} + 5921 T - 15788666 \) Copy content Toggle raw display
$47$ \( T^{2} + 44808 T + 493921836 \) Copy content Toggle raw display
$53$ \( T^{2} - 9417 T + 9642816 \) Copy content Toggle raw display
$59$ \( T^{2} - 5077 T - 872801544 \) Copy content Toggle raw display
$61$ \( T^{2} + 42368 T + 396401436 \) Copy content Toggle raw display
$67$ \( T^{2} - 30501 T - 389072326 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 1804825884 \) Copy content Toggle raw display
$73$ \( T^{2} - 85665 T + 256974650 \) Copy content Toggle raw display
$79$ \( T^{2} - 94646 T - 688757991 \) Copy content Toggle raw display
$83$ \( T^{2} + 33841 T + 280358334 \) Copy content Toggle raw display
$89$ \( T^{2} + 27558 T + 2565936 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 4062781666 \) Copy content Toggle raw display
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