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Properties

Label 66.6.e.a
Level $66$
Weight $6$
Character orbit 66.e
Analytic conductor $10.585$
Analytic rank $0$
Dimension $8$
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] = [66,6,Mod(25,66)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(66, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 8]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("66.25");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 66 = 2 \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 66.e (of order \(5\), degree \(4\), 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: \(10.5853321077\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 89x^{6} + 22551x^{4} + 4006069x^{2} + 405257161 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4 \beta_{2} q^{2} + (9 \beta_{5} - 9 \beta_{3} + 9 \beta_{2} - 9) q^{3} + 16 \beta_{3} q^{4} + (\beta_{7} - \beta_{6} + \cdots + 2 \beta_1) q^{5}+ \cdots - 81 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 \beta_{2} q^{2} + (9 \beta_{5} - 9 \beta_{3} + 9 \beta_{2} - 9) q^{3} + 16 \beta_{3} q^{4} + (\beta_{7} - \beta_{6} + \cdots + 2 \beta_1) q^{5}+ \cdots + (324 \beta_{7} + 1134 \beta_{6} + \cdots - 2511) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} - 18 q^{3} - 32 q^{4} + 150 q^{5} - 72 q^{6} + 474 q^{7} - 128 q^{8} - 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} - 18 q^{3} - 32 q^{4} + 150 q^{5} - 72 q^{6} + 474 q^{7} - 128 q^{8} - 162 q^{9} - 1200 q^{10} - 582 q^{11} + 1152 q^{12} + 510 q^{13} + 1896 q^{14} - 512 q^{16} + 2004 q^{17} - 648 q^{18} - 540 q^{19} + 2400 q^{20} - 1224 q^{21} + 672 q^{22} - 9944 q^{23} - 1152 q^{24} + 12228 q^{25} - 600 q^{26} - 1458 q^{27} - 6496 q^{28} - 11964 q^{29} - 9160 q^{31} + 8192 q^{32} + 4302 q^{33} + 2896 q^{34} + 42634 q^{35} - 2592 q^{36} - 718 q^{37} + 7560 q^{38} + 4590 q^{39} - 1666 q^{41} - 14616 q^{42} - 70528 q^{43} - 32 q^{44} - 24300 q^{45} + 56664 q^{46} + 51914 q^{47} - 4608 q^{48} - 23052 q^{49} - 52888 q^{50} - 21294 q^{51} - 2400 q^{52} - 53636 q^{53} + 23328 q^{54} + 104980 q^{55} - 8704 q^{56} + 17010 q^{57} - 47856 q^{58} - 17600 q^{59} + 21600 q^{60} - 10618 q^{61} + 77640 q^{62} + 38394 q^{63} - 8192 q^{64} + 116324 q^{65} - 2232 q^{66} - 182364 q^{67} + 32064 q^{68} - 82746 q^{69} - 79104 q^{70} + 29954 q^{71} - 10368 q^{72} + 127228 q^{73} - 2872 q^{74} - 118998 q^{75} - 43200 q^{76} - 33046 q^{77} - 25920 q^{78} + 39938 q^{79} - 13122 q^{81} - 51624 q^{82} + 208842 q^{83} + 68256 q^{84} + 125910 q^{85} + 51528 q^{86} + 36864 q^{87} + 30592 q^{88} - 344008 q^{89} + 48600 q^{90} + 282622 q^{91} - 147104 q^{92} - 82440 q^{93} - 106544 q^{94} + 31306 q^{95} - 18432 q^{96} - 26202 q^{97} + 323072 q^{98} + 13608 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 89x^{6} + 22551x^{4} + 4006069x^{2} + 405257161 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -139392\nu^{6} + 56240822\nu^{4} - 62654832109\nu^{2} + 1185782453086 ) / 9162533275391 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -982080\nu^{6} + 396242155\nu^{4} - 24952986432\nu^{2} - 808156901376 ) / 9162533275391 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -1964160\nu^{7} + 792484310\nu^{5} - 49905972864\nu^{3} - 1616313802752\nu ) / 9162533275391 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2816815\nu^{6} + 309599841\nu^{4} + 71570489651\nu^{2} + 11480499716159 ) / 9162533275391 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5633630\nu^{7} + 619199682\nu^{5} + 143140979302\nu^{3} + 22960999432318\nu ) / 9162533275391 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7319006\nu^{7} - 60802984\nu^{5} + 67737287948\nu^{3} + 8623811590460\nu ) / 9162533275391 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 22\beta_{3} - 155\beta_{2} + 22 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -155\beta_{7} + 155\beta_{6} - 133\beta_{4} - 133\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6336\beta_{5} + 18173\beta_{3} - 6336 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6336\beta_{6} + 18173\beta_{4} - 6336\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2556401\beta_{5} - 2556401\beta_{3} + 3938287\beta_{2} - 3938287 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3938287\beta_{7} - 1381886\beta_{6} + 1381886\beta_{4} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/66\mathbb{Z}\right)^\times\).

\(n\) \(13\) \(23\)
\(\chi(n)\) \(\beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
−3.37668 − 10.3924i
3.37668 + 10.3924i
−10.5047 + 7.63209i
10.5047 − 7.63209i
−3.37668 + 10.3924i
3.37668 − 10.3924i
−10.5047 − 7.63209i
10.5047 + 7.63209i
−3.23607 + 2.35114i 2.78115 + 8.55951i 4.94427 − 15.2169i −2.38251 − 1.73100i −29.1246 − 21.1603i −19.5441 + 60.1504i 19.7771 + 60.8676i −65.5304 + 47.6106i 11.7798
25.2 −3.23607 + 2.35114i 2.78115 + 8.55951i 4.94427 − 15.2169i 90.1940 + 65.5298i −29.1246 − 21.1603i 69.8440 − 214.958i 19.7771 + 60.8676i −65.5304 + 47.6106i −445.944
31.1 1.23607 − 3.80423i −7.28115 + 5.29007i −12.9443 − 9.40456i −9.47098 − 29.1487i 11.1246 + 34.2380i 1.28786 + 0.935685i −51.7771 + 37.6183i 25.0304 − 77.0356i −122.595
31.2 1.23607 − 3.80423i −7.28115 + 5.29007i −12.9443 − 9.40456i −3.34054 − 10.2811i 11.1246 + 34.2380i 185.412 + 134.710i −51.7771 + 37.6183i 25.0304 − 77.0356i −43.2409
37.1 −3.23607 − 2.35114i 2.78115 − 8.55951i 4.94427 + 15.2169i −2.38251 + 1.73100i −29.1246 + 21.1603i −19.5441 − 60.1504i 19.7771 − 60.8676i −65.5304 − 47.6106i 11.7798
37.2 −3.23607 − 2.35114i 2.78115 − 8.55951i 4.94427 + 15.2169i 90.1940 − 65.5298i −29.1246 + 21.1603i 69.8440 + 214.958i 19.7771 − 60.8676i −65.5304 − 47.6106i −445.944
49.1 1.23607 + 3.80423i −7.28115 − 5.29007i −12.9443 + 9.40456i −9.47098 + 29.1487i 11.1246 − 34.2380i 1.28786 − 0.935685i −51.7771 − 37.6183i 25.0304 + 77.0356i −122.595
49.2 1.23607 + 3.80423i −7.28115 − 5.29007i −12.9443 + 9.40456i −3.34054 + 10.2811i 11.1246 − 34.2380i 185.412 − 134.710i −51.7771 − 37.6183i 25.0304 + 77.0356i −43.2409
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 66.6.e.a ✓ 8
3.b odd 2 1 198.6.f.c 8
11.c even 5 1 inner 66.6.e.a ✓ 8
11.c even 5 1 726.6.a.be 4
11.d odd 10 1 726.6.a.bb 4
33.h odd 10 1 198.6.f.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.6.e.a ✓ 8 1.a even 1 1 trivial
66.6.e.a ✓ 8 11.c even 5 1 inner
198.6.f.c 8 3.b odd 2 1
198.6.f.c 8 33.h odd 10 1
726.6.a.bb 4 11.d odd 10 1
726.6.a.be 4 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 150 T_{5}^{7} + 8261 T_{5}^{6} + 155100 T_{5}^{5} + 13898346 T_{5}^{4} + 149974650 T_{5}^{3} + \cdots + 11832870841 \) acting on \(S_{6}^{\mathrm{new}}(66, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 4 T^{3} + \cdots + 256)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} + 9 T^{3} + \cdots + 6561)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 11832870841 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 27198258888025 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 67\!\cdots\!01 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 50\!\cdots\!01 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 47\!\cdots\!61 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 14\!\cdots\!81 \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots - 3587654924095)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 27\!\cdots\!81 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 28\!\cdots\!25 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 12\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 878847087788595)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 40\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 81\!\cdots\!21 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 45\!\cdots\!21 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 19\!\cdots\!61 \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots - 20\!\cdots\!51)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 78\!\cdots\!61 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 15\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 84\!\cdots\!01 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 84\!\cdots\!99)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 52\!\cdots\!21 \) Copy content Toggle raw display
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