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Properties

Label 66.4.e.c
Level $66$
Weight $4$
Character orbit 66.e
Analytic conductor $3.894$
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] = [66,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("66.25");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 66 = 2 \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(3.89412606038\)
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: \( x^{8} - x^{7} + 239x^{6} - 477x^{5} + 34439x^{4} + 39246x^{3} + 3687596x^{2} + 7369432x + 958769296 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q - 2 \beta_{3} q^{2} - 3 \beta_{2} q^{3} + (4 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} - 4) q^{4} + ( - \beta_{5} - 6 \beta_{4} - \beta_{2} + 1) q^{5} - 6 \beta_{4} q^{6} + (\beta_{6} - \beta_{5} + 5 \beta_{4} + \cdots - 5) q^{7}+ \cdots - 9 \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_{3} q^{2} - 3 \beta_{2} q^{3} + (4 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} - 4) q^{4} + ( - \beta_{5} - 6 \beta_{4} - \beta_{2} + 1) q^{5} - 6 \beta_{4} q^{6} + (\beta_{6} - \beta_{5} + 5 \beta_{4} + \cdots - 5) q^{7}+ \cdots + ( - 9 \beta_{7} + 27 \beta_{5} + \cdots - 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} - 6 q^{3} - 8 q^{4} - 5 q^{5} - 12 q^{6} - 13 q^{7} - 16 q^{8} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} - 6 q^{3} - 8 q^{4} - 5 q^{5} - 12 q^{6} - 13 q^{7} - 16 q^{8} - 18 q^{9} + 80 q^{10} + 17 q^{11} + 96 q^{12} - 89 q^{13} - 26 q^{14} - 45 q^{15} - 32 q^{16} - 80 q^{17} - 36 q^{18} - 110 q^{19} - 20 q^{20} + 126 q^{21} - 66 q^{22} + 632 q^{23} - 48 q^{24} - 89 q^{25} + 102 q^{26} - 54 q^{27} - 32 q^{28} - 549 q^{29} - 90 q^{30} + 467 q^{31} + 256 q^{32} + 66 q^{33} + 340 q^{34} - 728 q^{35} - 72 q^{36} + 757 q^{37} + 210 q^{38} - 267 q^{39} - 120 q^{40} - 43 q^{41} - 48 q^{42} - 1360 q^{43} - 32 q^{44} + 360 q^{45} - 266 q^{46} + 85 q^{47} - 96 q^{48} + 893 q^{49} - 458 q^{50} - 15 q^{51} + 204 q^{52} + 1115 q^{53} + 432 q^{54} - 190 q^{55} + 336 q^{56} + 315 q^{57} - 1098 q^{58} + 2348 q^{59} - 60 q^{60} - 575 q^{61} - 1806 q^{62} - 117 q^{63} - 128 q^{64} - 4038 q^{65} + 12 q^{66} - 1738 q^{67} - 320 q^{68} - 549 q^{69} + 1714 q^{70} + 2063 q^{71} - 144 q^{72} - 185 q^{73} + 1514 q^{74} - 687 q^{75} + 40 q^{76} + 1133 q^{77} + 456 q^{78} + 2711 q^{79} - 240 q^{80} - 162 q^{81} + 874 q^{82} - 1213 q^{83} - 156 q^{84} + 2263 q^{85} + 670 q^{86} + 738 q^{87} + 176 q^{88} - 516 q^{89} - 90 q^{90} - 2587 q^{91} - 732 q^{92} + 1401 q^{93} - 2340 q^{94} - 3695 q^{95} - 192 q^{96} - 91 q^{97} - 1244 q^{98} - 297 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 239x^{6} - 477x^{5} + 34439x^{4} + 39246x^{3} + 3687596x^{2} + 7369432x + 958769296 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 76808189723274 \nu^{7} + \cdots + 58\!\cdots\!44 ) / 16\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 169665281069249 \nu^{7} + \cdots - 43\!\cdots\!68 ) / 33\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 749572068360521 \nu^{7} + \cdots + 45\!\cdots\!68 ) / 66\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 54\!\cdots\!75 \nu^{7} + \cdots - 78\!\cdots\!76 ) / 66\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 448858228263 \nu^{7} - 625262648901 \nu^{6} - 383647123479290 \nu^{5} + \cdots - 32\!\cdots\!61 ) / 10\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7901350497103 \nu^{7} - 4417914539743 \nu^{6} + \cdots + 46\!\cdots\!88 ) / 42\!\cdots\!24 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - \beta_{6} + \beta_{5} - 23\beta_{4} - 190\beta_{3} - 24\beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 24\beta_{6} + 167\beta_{5} + 333\beta_{4} - 24\beta_{2} - 24\beta _1 + 48 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 381\beta_{7} + 48\beta_{6} - 48\beta_{5} + 28250\beta_{4} + 36866\beta_{3} + 36866\beta_{2} - 28250 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8616\beta_{7} - 37247\beta_{6} - 17112\beta_{4} - 17112\beta_{3} + 8616\beta _1 - 93677 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -25728\beta_{7} + 25728\beta_{5} + 25728\beta_{4} - 2324184\beta_{3} - 7283714\beta_{2} - 85061\beta _1 + 2324184 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 7394503 \beta_{7} + 7394503 \beta_{6} - 5044591 \beta_{5} + 1885193 \beta_{4} + 21667382 \beta_{3} + \cdots + 7394503 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(13\) \(23\)
\(\chi(n)\) \(-1 + \beta_{2} + \beta_{3} + \beta_{4}\) \(1\)

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
−4.57985 − 14.0953i
4.27084 + 13.1443i
−9.55058 + 6.93890i
10.3596 − 7.52669i
−4.57985 + 14.0953i
4.27084 − 13.1443i
−9.55058 − 6.93890i
10.3596 + 7.52669i
−1.61803 + 1.17557i 0.927051 + 2.85317i 1.23607 − 3.80423i −14.7263 − 10.6993i −4.85410 − 3.52671i 4.18461 − 12.8789i 2.47214 + 7.60845i −7.28115 + 5.29007i 36.4054
25.2 −1.61803 + 1.17557i 0.927051 + 2.85317i 1.23607 − 3.80423i 8.44512 + 6.13574i −4.85410 − 3.52671i −1.28542 + 3.95611i 2.47214 + 7.60845i −7.28115 + 5.29007i −20.8775
31.1 0.618034 − 1.90211i −2.42705 + 1.76336i −3.23607 − 2.35114i −1.91193 − 5.88432i 1.85410 + 5.70634i −20.8073 − 15.1174i −6.47214 + 4.70228i 2.78115 − 8.55951i −12.3743
31.2 0.618034 − 1.90211i −2.42705 + 1.76336i −3.23607 − 2.35114i 5.69308 + 17.5215i 1.85410 + 5.70634i 11.4081 + 8.28846i −6.47214 + 4.70228i 2.78115 − 8.55951i 36.8464
37.1 −1.61803 − 1.17557i 0.927051 − 2.85317i 1.23607 + 3.80423i −14.7263 + 10.6993i −4.85410 + 3.52671i 4.18461 + 12.8789i 2.47214 − 7.60845i −7.28115 − 5.29007i 36.4054
37.2 −1.61803 − 1.17557i 0.927051 − 2.85317i 1.23607 + 3.80423i 8.44512 − 6.13574i −4.85410 + 3.52671i −1.28542 − 3.95611i 2.47214 − 7.60845i −7.28115 − 5.29007i −20.8775
49.1 0.618034 + 1.90211i −2.42705 − 1.76336i −3.23607 + 2.35114i −1.91193 + 5.88432i 1.85410 − 5.70634i −20.8073 + 15.1174i −6.47214 − 4.70228i 2.78115 + 8.55951i −12.3743
49.2 0.618034 + 1.90211i −2.42705 − 1.76336i −3.23607 + 2.35114i 5.69308 − 17.5215i 1.85410 − 5.70634i 11.4081 − 8.28846i −6.47214 − 4.70228i 2.78115 + 8.55951i 36.8464
\(n\): 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 66.4.e.c ✓ 8
3.b odd 2 1 198.4.f.f 8
11.c even 5 1 inner 66.4.e.c ✓ 8
11.c even 5 1 726.4.a.z 4
11.d odd 10 1 726.4.a.w 4
33.f even 10 1 2178.4.a.bz 4
33.h odd 10 1 198.4.f.f 8
33.h odd 10 1 2178.4.a.bu 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.4.e.c ✓ 8 1.a even 1 1 trivial
66.4.e.c ✓ 8 11.c even 5 1 inner
198.4.f.f 8 3.b odd 2 1
198.4.f.f 8 33.h odd 10 1
726.4.a.w 4 11.d odd 10 1
726.4.a.z 4 11.c even 5 1
2178.4.a.bu 4 33.h odd 10 1
2178.4.a.bz 4 33.f even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 5 T_{5}^{7} + 182 T_{5}^{6} + 3105 T_{5}^{5} + 58879 T_{5}^{4} - 956715 T_{5}^{3} + \cdots + 469112281 \) acting on \(S_{4}^{\mathrm{new}}(66, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} + 3 T^{3} + 9 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 5 T^{7} + \cdots + 469112281 \) Copy content Toggle raw display
$7$ \( T^{8} + 13 T^{7} + \cdots + 417344041 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 3138428376721 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 2583207559696 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 5360817838336 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 2828787610000 \) Copy content Toggle raw display
$23$ \( (T^{4} - 316 T^{3} + \cdots - 1312124)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 78\!\cdots\!21 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 22\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 24\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( (T^{4} + 680 T^{3} + \cdots - 117394596)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 93\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 12\!\cdots\!25 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 14\!\cdots\!41 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( (T^{4} + 869 T^{3} + \cdots - 39260618944)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 30\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 52\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 68\!\cdots\!81 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 72\!\cdots\!81 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 1183601607284)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 25\!\cdots\!41 \) Copy content Toggle raw display
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