Properties

Label 294.6.e.z
Level $294$
Weight $6$
Character orbit 294.e
Analytic conductor $47.153$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 7^{2} \)
Twist minimal: yes
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4 \beta_{2} q^{2} + (9 \beta_{2} + 9) q^{3} + ( - 16 \beta_{2} - 16) q^{4} + (5 \beta_{3} - 54 \beta_{2} + 5 \beta_1) q^{5} + 36 q^{6} - 64 q^{8} + 81 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 \beta_{2} q^{2} + (9 \beta_{2} + 9) q^{3} + ( - 16 \beta_{2} - 16) q^{4} + (5 \beta_{3} - 54 \beta_{2} + 5 \beta_1) q^{5} + 36 q^{6} - 64 q^{8} + 81 \beta_{2} q^{9} + ( - 216 \beta_{2} + 20 \beta_1 - 216) q^{10} + ( - 62 \beta_{2} + 18 \beta_1 - 62) q^{11} - 144 \beta_{2} q^{12} + ( - 45 \beta_{3} - 360) q^{13} + (45 \beta_{3} + 486) q^{15} + 256 \beta_{2} q^{16} + ( - 630 \beta_{2} + 67 \beta_1 - 630) q^{17} + (324 \beta_{2} + 324) q^{18} + ( - 46 \beta_{3} - 180 \beta_{2} - 46 \beta_1) q^{19} + ( - 80 \beta_{3} - 864) q^{20} + ( - 72 \beta_{3} - 248) q^{22} + ( - 54 \beta_{3} + 3262 \beta_{2} - 54 \beta_1) q^{23} + ( - 576 \beta_{2} - 576) q^{24} + ( - 2241 \beta_{2} + 540 \beta_1 - 2241) q^{25} + ( - 180 \beta_{3} + 1440 \beta_{2} - 180 \beta_1) q^{26} - 729 q^{27} + ( - 234 \beta_{3} + 3544) q^{29} + (180 \beta_{3} - 1944 \beta_{2} + 180 \beta_1) q^{30} + (2952 \beta_{2} - 270 \beta_1 + 2952) q^{31} + (1024 \beta_{2} + 1024) q^{32} + (162 \beta_{3} - 558 \beta_{2} + 162 \beta_1) q^{33} + ( - 268 \beta_{3} - 2520) q^{34} + 1296 q^{36} + (612 \beta_{3} - 3020 \beta_{2} + 612 \beta_1) q^{37} + ( - 720 \beta_{2} - 184 \beta_1 - 720) q^{38} + ( - 3240 \beta_{2} + 405 \beta_1 - 3240) q^{39} + ( - 320 \beta_{3} + 3456 \beta_{2} - 320 \beta_1) q^{40} + (961 \beta_{3} + 8694) q^{41} + ( - 1152 \beta_{3} - 304) q^{43} + ( - 288 \beta_{3} + 992 \beta_{2} - 288 \beta_1) q^{44} + (4374 \beta_{2} - 405 \beta_1 + 4374) q^{45} + (13048 \beta_{2} - 216 \beta_1 + 13048) q^{46} + ( - 790 \beta_{3} + 15228 \beta_{2} - 790 \beta_1) q^{47} - 2304 q^{48} + ( - 2160 \beta_{3} - 8964) q^{50} + (603 \beta_{3} - 5670 \beta_{2} + 603 \beta_1) q^{51} + (5760 \beta_{2} - 720 \beta_1 + 5760) q^{52} + ( - 1982 \beta_{2} + 1584 \beta_1 - 1982) q^{53} + 2916 \beta_{2} q^{54} + ( - 1282 \beta_{3} - 12168) q^{55} + ( - 414 \beta_{3} + 1620) q^{57} + ( - 936 \beta_{3} - 14176 \beta_{2} - 936 \beta_1) q^{58} + (20376 \beta_{2} + 202 \beta_1 + 20376) q^{59} + ( - 7776 \beta_{2} + 720 \beta_1 - 7776) q^{60} + (1213 \beta_{3} + 684 \beta_{2} + 1213 \beta_1) q^{61} + (1080 \beta_{3} + 11808) q^{62} + 4096 q^{64} + ( - 4230 \beta_{3} + 41490 \beta_{2} - 4230 \beta_1) q^{65} + ( - 2232 \beta_{2} + 648 \beta_1 - 2232) q^{66} + (8112 \beta_{2} + 4464 \beta_1 + 8112) q^{67} + ( - 1072 \beta_{3} + 10080 \beta_{2} - 1072 \beta_1) q^{68} + ( - 486 \beta_{3} - 29358) q^{69} + ( - 6246 \beta_{3} - 1602) q^{71} - 5184 \beta_{2} q^{72} + (11988 \beta_{2} + 6563 \beta_1 + 11988) q^{73} + ( - 12080 \beta_{2} + 2448 \beta_1 - 12080) q^{74} + (4860 \beta_{3} - 20169 \beta_{2} + 4860 \beta_1) q^{75} + (736 \beta_{3} - 2880) q^{76} + ( - 1620 \beta_{3} - 12960) q^{78} + (756 \beta_{3} - 41080 \beta_{2} + 756 \beta_1) q^{79} + (13824 \beta_{2} - 1280 \beta_1 + 13824) q^{80} + ( - 6561 \beta_{2} - 6561) q^{81} + (3844 \beta_{3} - 34776 \beta_{2} + 3844 \beta_1) q^{82} + ( - 2656 \beta_{3} - 86868) q^{83} + ( - 6768 \beta_{3} - 66850) q^{85} + ( - 4608 \beta_{3} + 1216 \beta_{2} - 4608 \beta_1) q^{86} + (31896 \beta_{2} + 2106 \beta_1 + 31896) q^{87} + (3968 \beta_{2} - 1152 \beta_1 + 3968) q^{88} + (817 \beta_{3} - 100278 \beta_{2} + 817 \beta_1) q^{89} + (1620 \beta_{3} + 17496) q^{90} + (864 \beta_{3} + 52192) q^{92} + ( - 2430 \beta_{3} + 26568 \beta_{2} - 2430 \beta_1) q^{93} + (60912 \beta_{2} - 3160 \beta_1 + 60912) q^{94} + (12820 \beta_{2} - 1584 \beta_1 + 12820) q^{95} + 9216 \beta_{2} q^{96} + (2659 \beta_{3} - 125964) q^{97} + (1458 \beta_{3} + 5022) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{2} + 18 q^{3} - 32 q^{4} + 108 q^{5} + 144 q^{6} - 256 q^{8} - 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{2} + 18 q^{3} - 32 q^{4} + 108 q^{5} + 144 q^{6} - 256 q^{8} - 162 q^{9} - 432 q^{10} - 124 q^{11} + 288 q^{12} - 1440 q^{13} + 1944 q^{15} - 512 q^{16} - 1260 q^{17} + 648 q^{18} + 360 q^{19} - 3456 q^{20} - 992 q^{22} - 6524 q^{23} - 1152 q^{24} - 4482 q^{25} - 2880 q^{26} - 2916 q^{27} + 14176 q^{29} + 3888 q^{30} + 5904 q^{31} + 2048 q^{32} + 1116 q^{33} - 10080 q^{34} + 5184 q^{36} + 6040 q^{37} - 1440 q^{38} - 6480 q^{39} - 6912 q^{40} + 34776 q^{41} - 1216 q^{43} - 1984 q^{44} + 8748 q^{45} + 26096 q^{46} - 30456 q^{47} - 9216 q^{48} - 35856 q^{50} + 11340 q^{51} + 11520 q^{52} - 3964 q^{53} - 5832 q^{54} - 48672 q^{55} + 6480 q^{57} + 28352 q^{58} + 40752 q^{59} - 15552 q^{60} - 1368 q^{61} + 47232 q^{62} + 16384 q^{64} - 82980 q^{65} - 4464 q^{66} + 16224 q^{67} - 20160 q^{68} - 117432 q^{69} - 6408 q^{71} + 10368 q^{72} + 23976 q^{73} - 24160 q^{74} + 40338 q^{75} - 11520 q^{76} - 51840 q^{78} + 82160 q^{79} + 27648 q^{80} - 13122 q^{81} + 69552 q^{82} - 347472 q^{83} - 267400 q^{85} - 2432 q^{86} + 63792 q^{87} + 7936 q^{88} + 200556 q^{89} + 69984 q^{90} + 208768 q^{92} - 53136 q^{93} + 121824 q^{94} + 25640 q^{95} - 18432 q^{96} - 503856 q^{97} + 20088 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 7\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{3} ) / 7 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/294\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(199\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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
0.707107 1.22474i
−0.707107 + 1.22474i
0.707107 + 1.22474i
−0.707107 1.22474i
2.00000 + 3.46410i 4.50000 7.79423i −8.00000 + 13.8564i 2.25126 + 3.89930i 36.0000 0 −64.0000 −40.5000 70.1481i −9.00505 + 15.5972i
67.2 2.00000 + 3.46410i 4.50000 7.79423i −8.00000 + 13.8564i 51.7487 + 89.6314i 36.0000 0 −64.0000 −40.5000 70.1481i −206.995 + 358.526i
79.1 2.00000 3.46410i 4.50000 + 7.79423i −8.00000 13.8564i 2.25126 3.89930i 36.0000 0 −64.0000 −40.5000 + 70.1481i −9.00505 15.5972i
79.2 2.00000 3.46410i 4.50000 + 7.79423i −8.00000 13.8564i 51.7487 89.6314i 36.0000 0 −64.0000 −40.5000 + 70.1481i −206.995 358.526i
\(n\): 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.z 4
7.b odd 2 1 294.6.e.x 4
7.c even 3 1 294.6.a.n 2
7.c even 3 1 inner 294.6.e.z 4
7.d odd 6 1 294.6.a.q yes 2
7.d odd 6 1 294.6.e.x 4
21.g even 6 1 882.6.a.bk 2
21.h odd 6 1 882.6.a.bu 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.6.a.n 2 7.c even 3 1
294.6.a.q yes 2 7.d odd 6 1
294.6.e.x 4 7.b odd 2 1
294.6.e.x 4 7.d odd 6 1
294.6.e.z 4 1.a even 1 1 trivial
294.6.e.z 4 7.c even 3 1 inner
882.6.a.bk 2 21.g even 6 1
882.6.a.bu 2 21.h odd 6 1

Hecke kernels

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

\( T_{5}^{4} - 108T_{5}^{3} + 11198T_{5}^{2} - 50328T_{5} + 217156 \) Copy content Toggle raw display
\( T_{11}^{4} + 124T_{11}^{3} + 43284T_{11}^{2} - 3460592T_{11} + 778856464 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - 9 T + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - 108 T^{3} + \cdots + 217156 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 124 T^{3} + \cdots + 778856464 \) Copy content Toggle raw display
$13$ \( (T^{2} + 720 T - 68850)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 1850892484 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 30613801024 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 107223456975376 \) Copy content Toggle raw display
$29$ \( (T^{2} - 7088 T + 7193848)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 2465226570816 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 760927370047744 \) Copy content Toggle raw display
$41$ \( (T^{2} - 17388 T - 14919422)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 608 T - 129963776)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 29\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 58\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 16\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 20\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 35\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{2} + 3204 T - 3820660164)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 16\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 26\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( (T^{2} + 173736 T + 6854724496)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 99\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( (T^{2} + 251928 T + 15174041758)^{2} \) Copy content Toggle raw display
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