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Properties

Label 66.4.h.b
Level $66$
Weight $4$
Character orbit 66.h
Analytic conductor $3.894$
Analytic rank $0$
Dimension $24$
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(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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("66.17");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 66 = 2 \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(3.89412606038\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{2} - q^{3} - 24 q^{4} - 18 q^{6} + 48 q^{8} + 107 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{2} - q^{3} - 24 q^{4} - 18 q^{6} + 48 q^{8} + 107 q^{9} + 111 q^{11} + 36 q^{12} - 288 q^{15} - 96 q^{16} - 66 q^{17} - 4 q^{18} + 225 q^{19} - 36 q^{21} - 12 q^{22} + 128 q^{24} + 534 q^{25} + 240 q^{26} - 97 q^{27} - 60 q^{28} - 591 q^{29} - 204 q^{30} - 531 q^{31} - 768 q^{32} - 457 q^{33} + 12 q^{34} + 147 q^{35} - 452 q^{36} - 66 q^{37} + 120 q^{38} - 347 q^{39} + 120 q^{40} + 318 q^{41} + 1252 q^{42} - 156 q^{44} + 274 q^{45} - 180 q^{46} + 810 q^{47} - 16 q^{48} + 354 q^{49} - 198 q^{50} + 1644 q^{51} + 480 q^{52} - 1755 q^{53} - 186 q^{54} + 588 q^{55} - 246 q^{57} + 1182 q^{58} + 4395 q^{59} + 28 q^{60} - 210 q^{61} - 768 q^{62} - 3237 q^{63} - 384 q^{64} + 408 q^{65} + 624 q^{66} - 4386 q^{67} - 264 q^{68} - 3042 q^{69} + 246 q^{70} - 840 q^{71} + 264 q^{72} - 705 q^{73} + 132 q^{74} + 9076 q^{75} + 480 q^{77} + 1384 q^{78} + 1170 q^{79} + 240 q^{80} - 2249 q^{81} + 1794 q^{82} + 1983 q^{83} + 2696 q^{84} + 4170 q^{85} - 5070 q^{86} - 1854 q^{87} - 48 q^{88} - 3028 q^{90} + 1260 q^{91} + 1080 q^{92} - 1518 q^{93} - 3120 q^{94} - 7128 q^{95} + 32 q^{96} - 5139 q^{97} + 3852 q^{98} - 3723 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −0.618034 − 1.90211i −5.03313 + 1.29136i −3.23607 + 2.35114i 16.5254 + 5.36942i 5.56696 + 8.77547i −4.86030 − 6.68962i 6.47214 + 4.70228i 23.6648 − 12.9992i − 34.7516i
17.2 −0.618034 − 1.90211i −4.64059 − 2.33771i −3.23607 + 2.35114i −9.32256 − 3.02908i −1.57855 + 10.2717i 18.1216 + 24.9422i 6.47214 + 4.70228i 16.0702 + 21.6968i 19.6046i
17.3 −0.618034 − 1.90211i −1.45259 − 4.98899i −3.23607 + 2.35114i 1.14202 + 0.371066i −8.59186 + 5.84636i −15.8100 − 21.7606i 6.47214 + 4.70228i −22.7800 + 14.4939i − 2.40159i
17.4 −0.618034 − 1.90211i 2.73492 + 4.41817i −3.23607 + 2.35114i −0.263485 − 0.0856114i 6.71357 − 7.93271i 12.5039 + 17.2102i 6.47214 + 4.70228i −12.0404 + 24.1667i 0.554088i
17.5 −0.618034 − 1.90211i 4.73800 − 2.13339i −3.23607 + 2.35114i 16.0767 + 5.22364i −6.98619 − 7.69370i 4.34702 + 5.98315i 6.47214 + 4.70228i 17.8973 − 20.2160i − 33.8081i
17.6 −0.618034 − 1.90211i 5.08044 − 1.09048i −3.23607 + 2.35114i −20.8040 − 6.75962i −5.21410 − 8.98962i −10.9481 − 15.0687i 6.47214 + 4.70228i 24.6217 − 11.0802i 43.7492i
29.1 1.61803 + 1.17557i −4.82606 − 1.92590i 1.23607 + 3.80423i −9.10721 − 12.5350i −5.54470 − 8.78956i −30.4774 + 9.90272i −2.47214 + 7.60845i 19.5818 + 18.5891i − 30.9882i
29.2 1.61803 + 1.17557i −4.02093 − 3.29122i 1.23607 + 3.80423i 9.82162 + 13.5183i −2.63695 − 10.0522i 13.9517 − 4.53319i −2.47214 + 7.60845i 5.33578 + 26.4675i 33.4191i
29.3 1.61803 + 1.17557i −2.83539 + 4.35437i 1.23607 + 3.80423i 2.83713 + 3.90497i −9.70663 + 3.71232i −16.7863 + 5.45421i −2.47214 + 7.60845i −10.9211 − 24.6927i 9.65361i
29.4 1.61803 + 1.17557i 1.72956 − 4.89986i 1.23607 + 3.80423i −11.3394 − 15.6073i 8.55862 − 5.89492i 26.3937 − 8.57584i −2.47214 + 7.60845i −21.0172 − 16.9492i − 38.5834i
29.5 1.61803 + 1.17557i 3.19526 + 4.09760i 1.23607 + 3.80423i 0.0960686 + 0.132227i 0.353014 + 10.3863i 10.0627 − 3.26957i −2.47214 + 7.60845i −6.58068 + 26.1858i 0.326883i
29.6 1.61803 + 1.17557i 4.83052 − 1.91470i 1.23607 + 3.80423i 4.33767 + 5.97029i 10.0668 + 2.58056i −6.49850 + 2.11149i −2.47214 + 7.60845i 19.6678 − 18.4980i 14.7594i
35.1 −0.618034 + 1.90211i −5.03313 − 1.29136i −3.23607 − 2.35114i 16.5254 − 5.36942i 5.56696 − 8.77547i −4.86030 + 6.68962i 6.47214 − 4.70228i 23.6648 + 12.9992i 34.7516i
35.2 −0.618034 + 1.90211i −4.64059 + 2.33771i −3.23607 − 2.35114i −9.32256 + 3.02908i −1.57855 − 10.2717i 18.1216 − 24.9422i 6.47214 − 4.70228i 16.0702 − 21.6968i − 19.6046i
35.3 −0.618034 + 1.90211i −1.45259 + 4.98899i −3.23607 − 2.35114i 1.14202 − 0.371066i −8.59186 − 5.84636i −15.8100 + 21.7606i 6.47214 − 4.70228i −22.7800 − 14.4939i 2.40159i
35.4 −0.618034 + 1.90211i 2.73492 − 4.41817i −3.23607 − 2.35114i −0.263485 + 0.0856114i 6.71357 + 7.93271i 12.5039 − 17.2102i 6.47214 − 4.70228i −12.0404 − 24.1667i − 0.554088i
35.5 −0.618034 + 1.90211i 4.73800 + 2.13339i −3.23607 − 2.35114i 16.0767 − 5.22364i −6.98619 + 7.69370i 4.34702 − 5.98315i 6.47214 − 4.70228i 17.8973 + 20.2160i 33.8081i
35.6 −0.618034 + 1.90211i 5.08044 + 1.09048i −3.23607 − 2.35114i −20.8040 + 6.75962i −5.21410 + 8.98962i −10.9481 + 15.0687i 6.47214 − 4.70228i 24.6217 + 11.0802i − 43.7492i
41.1 1.61803 − 1.17557i −4.82606 + 1.92590i 1.23607 − 3.80423i −9.10721 + 12.5350i −5.54470 + 8.78956i −30.4774 − 9.90272i −2.47214 − 7.60845i 19.5818 − 18.5891i 30.9882i
41.2 1.61803 − 1.17557i −4.02093 + 3.29122i 1.23607 − 3.80423i 9.82162 − 13.5183i −2.63695 + 10.0522i 13.9517 + 4.53319i −2.47214 − 7.60845i 5.33578 − 26.4675i − 33.4191i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.6
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 66.4.h.b yes 24
3.b odd 2 1 66.4.h.a ✓ 24
11.d odd 10 1 66.4.h.a ✓ 24
33.f even 10 1 inner 66.4.h.b yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.4.h.a ✓ 24 3.b odd 2 1
66.4.h.a ✓ 24 11.d odd 10 1
66.4.h.b yes 24 1.a even 1 1 trivial
66.4.h.b yes 24 33.f even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - 642 T_{5}^{22} - 3105 T_{5}^{21} + 254874 T_{5}^{20} + 1993410 T_{5}^{19} + \cdots + 37\!\cdots\!76 \) acting on \(S_{4}^{\mathrm{new}}(66, [\chi])\). Copy content Toggle raw display