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Properties

Label 66.2.h.b
Level $66$
Weight $2$
Character orbit 66.h
Analytic conductor $0.527$
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] = [66,2,Mod(17,66)]
 
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([5, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("66.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 66 = 2 \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 66.h (of order \(10\), 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: \(0.527012653340\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.185640625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + x^{6} + x^{5} + 4x^{4} + 3x^{3} + 9x^{2} - 81x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} - \beta_{7} q^{3} + ( - \beta_{6} + \beta_{4} + \beta_{2} - 1) q^{4} + (\beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{3} + \beta_1 + 1) q^{5} + \beta_{3} q^{6} + ( - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{7} + \beta_{4} q^{8} + (\beta_{6} + \beta_{5} + \beta_1 - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} - \beta_{7} q^{3} + ( - \beta_{6} + \beta_{4} + \beta_{2} - 1) q^{4} + (\beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{3} + \beta_1 + 1) q^{5} + \beta_{3} q^{6} + ( - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{7} + \beta_{4} q^{8} + (\beta_{6} + \beta_{5} + \beta_1 - 2) q^{9} + (\beta_{7} + \beta_{6} - \beta_{4} - \beta_{2} + 1) q^{10} + ( - 2 \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{11} - \beta_1 q^{12} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{13} + ( - \beta_{7} - \beta_{6} - \beta_{3} + \beta_{2} - 1) q^{14} + (3 \beta_{2} + \beta_1) q^{15} - \beta_{2} q^{16} + ( - \beta_{7} + \beta_{5} + 3 \beta_{4} + \beta_{3} + 3 \beta_{2} - \beta_1 - 2) q^{17} + (\beta_{7} + 3 \beta_{6} + \beta_{3} + \beta_1) q^{18} + (2 \beta_{7} + 2 \beta_{6} + \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 1) q^{19} + ( - \beta_{4} - \beta_{3}) q^{20} + ( - \beta_{6} - \beta_{5} - 3 \beta_{2} - \beta_1 + 2) q^{21} + ( - 2 \beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - \beta_1) q^{22} + ( - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{2} - 2 \beta_1) q^{23} + (\beta_{6} + \beta_{5} + 1) q^{24} + ( - \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{25} + (2 \beta_1 - 2) q^{26} + (2 \beta_{7} - 2 \beta_{6} + \beta_{5} + 3 \beta_{2} + \beta_1 - 2) q^{27} + (\beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{28} + (2 \beta_{7} + 2 \beta_{5} + 2 \beta_{4} - 3 \beta_{2} + \beta_1 + 4) q^{29} + ( - \beta_{6} - \beta_{5} + 2) q^{30} + ( - \beta_{7} - 3 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + 4 \beta_{2} + 3 \beta_1 - 1) q^{31} - q^{32} + ( - 2 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} - \beta_{3} - 3 \beta_{2} - \beta_1 + 1) q^{33} + (\beta_{7} + 4 \beta_{6} + 2 \beta_{5} + \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 4) q^{34} + (\beta_{7} + \beta_{6} - 3 \beta_{4} - 4 \beta_{2} - \beta_1 + 1) q^{35} + (2 \beta_{6} - \beta_{5} - 3 \beta_{4} - \beta_{3} - 3 \beta_{2} - \beta_1 + 2) q^{36} + (2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 2) q^{37} + (\beta_{7} + 4 \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - 3 \beta_{2} - \beta_1 + 4) q^{38} + ( - 4 \beta_{6} + 2 \beta_{5} + 2) q^{39} + (\beta_{2} + \beta_1) q^{40} + (\beta_{7} + 2 \beta_{6} + 2 \beta_{5} + \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + 4 \beta_1) q^{41} + ( - \beta_{7} - 3 \beta_{6} - \beta_{3} - \beta_1 - 3) q^{42} + ( - 2 \beta_{7} - \beta_{5} + 2 \beta_{4} + 4 \beta_{2} - \beta_1 - 2) q^{43} + ( - \beta_{7} - \beta_{6} + \beta_{3} + 2 \beta_{2} + \beta_1) q^{44} + ( - 3 \beta_{7} - 5 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + \cdots - 5) q^{45}+ \cdots + (\beta_{7} + 5 \beta_{6} + 5 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} - 3 \beta_{2} + \beta_1 + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{6} + 2 q^{8} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{6} + 2 q^{8} - 18 q^{9} - q^{11} - 3 q^{12} + 9 q^{15} - 2 q^{16} - 10 q^{17} - 7 q^{18} - 15 q^{19} + 12 q^{21} + 6 q^{22} + 3 q^{24} - 6 q^{25} - 10 q^{26} - 10 q^{27} + 5 q^{28} + 23 q^{29} + 21 q^{30} + 13 q^{31} - 8 q^{32} + 17 q^{33} + 10 q^{34} - 13 q^{35} + 2 q^{36} + 6 q^{37} + 10 q^{38} + 18 q^{39} + 5 q^{40} + 2 q^{41} - 17 q^{42} + 9 q^{44} - 8 q^{45} - 10 q^{46} + 10 q^{47} + 2 q^{48} - 18 q^{49} + q^{50} - 30 q^{51} - 10 q^{52} + 15 q^{53} + 15 q^{54} - 14 q^{55} + 20 q^{57} - 23 q^{58} - 25 q^{59} + 4 q^{60} - 10 q^{61} + 2 q^{62} + 16 q^{63} - 2 q^{64} - 32 q^{65} - 22 q^{66} - 2 q^{67} - 10 q^{68} - 26 q^{69} - 17 q^{70} + 40 q^{71} - 2 q^{72} + 5 q^{73} - 6 q^{74} - 34 q^{75} - 12 q^{77} - 8 q^{78} + 10 q^{79} + 5 q^{80} + 14 q^{81} + 3 q^{82} - 21 q^{83} - 8 q^{84} + 30 q^{85} + 25 q^{86} + 42 q^{87} + 6 q^{88} + 8 q^{90} + 24 q^{91} + 10 q^{92} - 53 q^{93} + 40 q^{94} + 20 q^{95} - 2 q^{96} + 9 q^{97} - 22 q^{98} + 34 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + x^{6} + x^{5} + 4x^{4} + 3x^{3} + 9x^{2} - 81x + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + \nu^{5} + 4\nu^{4} + 16\nu^{3} + 51\nu^{2} - 54\nu - 27 ) / 216 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} - \nu^{5} - 4\nu^{4} - 16\nu^{3} + 21\nu^{2} - 18\nu + 27 ) / 72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{7} + 6\nu^{6} + 7\nu^{5} + 16\nu^{4} + 28\nu^{3} - 69\nu^{2} - 216\nu + 351 ) / 216 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 3\nu^{6} + \nu^{5} + \nu^{4} + 4\nu^{3} + 3\nu^{2} + 9\nu - 81 ) / 27 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 19\nu^{7} - 30\nu^{6} - 35\nu^{5} - 8\nu^{4} + 76\nu^{3} + 165\nu^{2} + 360\nu - 1107 ) / 216 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{7} - 2\nu^{6} - 3\nu^{5} - 8\nu^{4} + 4\nu^{3} + 13\nu^{2} + 60\nu - 99 ) / 24 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 3\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 3\beta_{4} - \beta_{3} + 3\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -4\beta_{7} + 2\beta_{6} - \beta_{5} - 6\beta_{4} - 4\beta_{3} + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{7} - 6\beta_{6} + 6\beta_{5} + 12\beta_{2} + 6\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{6} - 8\beta_{5} - 6\beta_{4} - 8\beta_{3} + 12\beta_{2} + \beta _1 - 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -2\beta_{7} - 2\beta_{6} - 2\beta_{5} - 24\beta_{4} - 19\beta_{3} + 3\beta_{2} - 19\beta _1 + 22 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(13\) \(23\)
\(\chi(n)\) \(1 - \beta_{2} - \beta_{4} + \beta_{6}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−0.245684 − 1.71454i
1.55470 + 0.763481i
−1.52536 + 0.820539i
1.71634 − 0.232753i
1.55470 − 0.763481i
−0.245684 + 1.71454i
1.71634 + 0.232753i
−1.52536 − 0.820539i
−0.309017 − 0.951057i 0.809017 − 1.53150i −0.809017 + 0.587785i −0.897526 − 0.291624i −1.70654 − 0.296161i 0.897526 + 1.23534i 0.809017 + 0.587785i −1.69098 − 2.47802i 0.943715i
17.2 −0.309017 − 0.951057i 0.809017 + 1.53150i −0.809017 + 0.587785i 2.01556 + 0.654895i 1.20654 − 1.24268i −2.01556 − 2.77418i 0.809017 + 0.587785i −1.69098 + 2.47802i − 2.11929i
29.1 0.809017 + 0.587785i −0.309017 − 1.70426i 0.309017 + 0.951057i 0.442723 + 0.609356i 0.751740 − 1.56041i −0.442723 + 0.143849i −0.309017 + 0.951057i −2.80902 + 1.05329i 0.753205i
29.2 0.809017 + 0.587785i −0.309017 + 1.70426i 0.309017 + 0.951057i −1.56076 − 2.14820i −1.25174 + 1.19714i 1.56076 − 0.507121i −0.309017 + 0.951057i −2.80902 − 1.05329i − 2.65532i
35.1 −0.309017 + 0.951057i 0.809017 − 1.53150i −0.809017 − 0.587785i 2.01556 − 0.654895i 1.20654 + 1.24268i −2.01556 + 2.77418i 0.809017 − 0.587785i −1.69098 − 2.47802i 2.11929i
35.2 −0.309017 + 0.951057i 0.809017 + 1.53150i −0.809017 − 0.587785i −0.897526 + 0.291624i −1.70654 + 0.296161i 0.897526 − 1.23534i 0.809017 − 0.587785i −1.69098 + 2.47802i − 0.943715i
41.1 0.809017 − 0.587785i −0.309017 − 1.70426i 0.309017 − 0.951057i −1.56076 + 2.14820i −1.25174 − 1.19714i 1.56076 + 0.507121i −0.309017 − 0.951057i −2.80902 + 1.05329i 2.65532i
41.2 0.809017 − 0.587785i −0.309017 + 1.70426i 0.309017 − 0.951057i 0.442723 − 0.609356i 0.751740 + 1.56041i −0.442723 − 0.143849i −0.309017 − 0.951057i −2.80902 − 1.05329i − 0.753205i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.f even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 66.2.h.b yes 8
3.b odd 2 1 66.2.h.a ✓ 8
4.b odd 2 1 528.2.bn.a 8
11.b odd 2 1 726.2.h.d 8
11.c even 5 1 726.2.b.c 8
11.c even 5 1 726.2.h.f 8
11.c even 5 1 726.2.h.h 8
11.c even 5 1 726.2.h.j 8
11.d odd 10 1 66.2.h.a ✓ 8
11.d odd 10 1 726.2.b.e 8
11.d odd 10 1 726.2.h.a 8
11.d odd 10 1 726.2.h.c 8
12.b even 2 1 528.2.bn.b 8
33.d even 2 1 726.2.h.j 8
33.f even 10 1 inner 66.2.h.b yes 8
33.f even 10 1 726.2.b.c 8
33.f even 10 1 726.2.h.f 8
33.f even 10 1 726.2.h.h 8
33.h odd 10 1 726.2.b.e 8
33.h odd 10 1 726.2.h.a 8
33.h odd 10 1 726.2.h.c 8
33.h odd 10 1 726.2.h.d 8
44.g even 10 1 528.2.bn.b 8
132.n odd 10 1 528.2.bn.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.h.a ✓ 8 3.b odd 2 1
66.2.h.a ✓ 8 11.d odd 10 1
66.2.h.b yes 8 1.a even 1 1 trivial
66.2.h.b yes 8 33.f even 10 1 inner
528.2.bn.a 8 4.b odd 2 1
528.2.bn.a 8 132.n odd 10 1
528.2.bn.b 8 12.b even 2 1
528.2.bn.b 8 44.g even 10 1
726.2.b.c 8 11.c even 5 1
726.2.b.c 8 33.f even 10 1
726.2.b.e 8 11.d odd 10 1
726.2.b.e 8 33.h odd 10 1
726.2.h.a 8 11.d odd 10 1
726.2.h.a 8 33.h odd 10 1
726.2.h.c 8 11.d odd 10 1
726.2.h.c 8 33.h odd 10 1
726.2.h.d 8 11.b odd 2 1
726.2.h.d 8 33.h odd 10 1
726.2.h.f 8 11.c even 5 1
726.2.h.f 8 33.f even 10 1
726.2.h.h 8 11.c even 5 1
726.2.h.h 8 33.f even 10 1
726.2.h.j 8 11.c even 5 1
726.2.h.j 8 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 2T_{5}^{6} - 15T_{5}^{5} + 19T_{5}^{4} + 30T_{5}^{3} - 8T_{5}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(66, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{3} + 5 T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 2 T^{6} - 15 T^{5} + 19 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{8} + 2 T^{6} - 25 T^{5} + 59 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{8} + T^{7} + 15 T^{6} - 11 T^{5} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} - 8 T^{6} - 120 T^{5} + \cdots + 4096 \) Copy content Toggle raw display
$17$ \( T^{8} + 10 T^{7} + 60 T^{6} + \cdots + 144400 \) Copy content Toggle raw display
$19$ \( T^{8} + 15 T^{7} + 100 T^{6} + \cdots + 400 \) Copy content Toggle raw display
$23$ \( T^{8} + 88 T^{6} + 2544 T^{4} + \cdots + 30976 \) Copy content Toggle raw display
$29$ \( T^{8} - 23 T^{7} + 328 T^{6} + \cdots + 15376 \) Copy content Toggle raw display
$31$ \( T^{8} - 13 T^{7} + 168 T^{6} + \cdots + 1175056 \) Copy content Toggle raw display
$37$ \( T^{8} - 6 T^{7} + 32 T^{6} + \cdots + 30976 \) Copy content Toggle raw display
$41$ \( T^{8} - 2 T^{7} - 32 T^{6} + 91 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$43$ \( T^{8} + 113 T^{6} + 4549 T^{4} + \cdots + 430336 \) Copy content Toggle raw display
$47$ \( T^{8} - 10 T^{7} - 8 T^{6} + \cdots + 1048576 \) Copy content Toggle raw display
$53$ \( T^{8} - 15 T^{7} + 100 T^{6} + \cdots + 2310400 \) Copy content Toggle raw display
$59$ \( T^{8} + 25 T^{7} + 182 T^{6} + \cdots + 844561 \) Copy content Toggle raw display
$61$ \( T^{8} + 10 T^{7} + 112 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$67$ \( (T^{4} + T^{3} - 149 T^{2} - 284 T + 3076)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 20 T^{3} + 120 T^{2} - 120 T + 80)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 5 T^{7} - 50 T^{6} + \cdots + 24025 \) Copy content Toggle raw display
$79$ \( T^{8} - 10 T^{7} - 48 T^{6} + \cdots + 55696 \) Copy content Toggle raw display
$83$ \( T^{8} + 21 T^{7} + 252 T^{6} + \cdots + 737881 \) Copy content Toggle raw display
$89$ \( T^{8} + 513 T^{6} + \cdots + 12702096 \) Copy content Toggle raw display
$97$ \( T^{8} - 9 T^{7} + 282 T^{6} + \cdots + 2070721 \) Copy content Toggle raw display
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