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Properties

Label 66.5.f.a
Level $66$
Weight $5$
Character orbit 66.f
Analytic conductor $6.822$
Analytic rank $0$
Dimension $32$
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] = [66,5,Mod(7,66)]
 
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, 7]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("66.7");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 66 = 2 \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 66.f (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: \(6.82241756353\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 64 q^{4} - 72 q^{5} - 300 q^{7} - 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 64 q^{4} - 72 q^{5} - 300 q^{7} - 216 q^{9} + 540 q^{11} + 1020 q^{13} + 192 q^{14} + 720 q^{15} - 512 q^{16} - 3540 q^{17} + 576 q^{20} + 192 q^{22} + 2472 q^{23} - 2052 q^{25} + 1344 q^{26} - 1600 q^{28} - 960 q^{29} - 1340 q^{31} + 216 q^{33} + 3712 q^{34} - 960 q^{35} + 1728 q^{36} + 7908 q^{37} - 2304 q^{38} + 1280 q^{40} - 3120 q^{41} + 3168 q^{42} + 1440 q^{44} - 1944 q^{45} - 4480 q^{46} - 1692 q^{47} + 6660 q^{49} - 13440 q^{50} - 11160 q^{51} - 4000 q^{52} - 5460 q^{53} - 11288 q^{55} + 6144 q^{56} + 12600 q^{57} - 1376 q^{58} + 684 q^{59} + 7200 q^{60} + 42860 q^{61} + 39360 q^{62} + 8100 q^{63} + 4096 q^{64} - 34904 q^{67} - 28320 q^{68} - 7740 q^{69} - 38336 q^{70} - 27768 q^{71} - 4540 q^{73} - 11520 q^{74} - 15408 q^{75} + 3156 q^{77} + 16180 q^{79} + 6912 q^{80} - 5832 q^{81} + 11520 q^{82} + 49140 q^{83} + 57180 q^{85} + 25920 q^{86} + 6144 q^{88} + 14616 q^{89} + 20052 q^{91} - 6336 q^{92} + 15264 q^{93} - 62720 q^{94} - 61620 q^{95} - 35844 q^{97} + 3240 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −1.66251 + 2.28825i −1.60570 + 4.94183i −2.47214 − 7.60845i 8.24269 − 5.98866i −8.63864 − 11.8901i −68.6895 + 22.3186i 21.5200 + 6.99226i −21.8435 − 15.8702i 28.8175i
7.2 −1.66251 + 2.28825i −1.60570 + 4.94183i −2.47214 − 7.60845i 18.5313 − 13.4638i −8.63864 − 11.8901i 49.8176 − 16.1867i 21.5200 + 6.99226i −21.8435 − 15.8702i 64.7877i
7.3 −1.66251 + 2.28825i 1.60570 − 4.94183i −2.47214 − 7.60845i −24.4019 + 17.7291i 8.63864 + 11.8901i 61.9962 − 20.1438i 21.5200 + 6.99226i −21.8435 − 15.8702i − 85.3123i
7.4 −1.66251 + 2.28825i 1.60570 − 4.94183i −2.47214 − 7.60845i −2.66700 + 1.93769i 8.63864 + 11.8901i −42.7542 + 13.8917i 21.5200 + 6.99226i −21.8435 − 15.8702i − 9.32419i
7.5 1.66251 − 2.28825i −1.60570 + 4.94183i −2.47214 − 7.60845i −14.9471 + 10.8597i 8.63864 + 11.8901i −45.0509 + 14.6379i −21.5200 − 6.99226i −21.8435 − 15.8702i 52.2569i
7.6 1.66251 − 2.28825i −1.60570 + 4.94183i −2.47214 − 7.60845i 14.3792 − 10.4471i 8.63864 + 11.8901i 32.0130 − 10.4016i −21.5200 − 6.99226i −21.8435 − 15.8702i − 50.2716i
7.7 1.66251 − 2.28825i 1.60570 − 4.94183i −2.47214 − 7.60845i −32.7359 + 23.7840i −8.63864 − 11.8901i −72.4195 + 23.5305i −21.5200 − 6.99226i −21.8435 − 15.8702i 114.449i
7.8 1.66251 − 2.28825i 1.60570 − 4.94183i −2.47214 − 7.60845i 15.5988 − 11.3332i −8.63864 − 11.8901i 21.2677 − 6.91030i −21.5200 − 6.99226i −21.8435 − 15.8702i − 54.5353i
13.1 −2.68999 − 0.874032i −4.20378 + 3.05422i 6.47214 + 4.70228i −5.62699 − 17.3181i 13.9776 − 4.54160i −20.5736 + 28.3171i −13.3001 − 18.3060i 8.34346 − 25.6785i 51.5037i
13.2 −2.68999 − 0.874032i −4.20378 + 3.05422i 6.47214 + 4.70228i 3.86597 + 11.8982i 13.9776 − 4.54160i 9.46992 − 13.0342i −13.3001 − 18.3060i 8.34346 − 25.6785i − 35.3852i
13.3 −2.68999 − 0.874032i 4.20378 − 3.05422i 6.47214 + 4.70228i −10.9172 − 33.5997i −13.9776 + 4.54160i −26.2177 + 36.0856i −13.3001 − 18.3060i 8.34346 − 25.6785i 99.9250i
13.4 −2.68999 − 0.874032i 4.20378 − 3.05422i 6.47214 + 4.70228i 5.73319 + 17.6449i −13.9776 + 4.54160i −25.7189 + 35.3990i −13.3001 − 18.3060i 8.34346 − 25.6785i − 52.4758i
13.5 2.68999 + 0.874032i −4.20378 + 3.05422i 6.47214 + 4.70228i −14.6389 − 45.0540i −13.9776 + 4.54160i 21.8594 − 30.0869i 13.3001 + 18.3060i 8.34346 − 25.6785i − 133.990i
13.6 2.68999 + 0.874032i −4.20378 + 3.05422i 6.47214 + 4.70228i −0.695240 − 2.13973i −13.9776 + 4.54160i −53.8459 + 74.1125i 13.3001 + 18.3060i 8.34346 − 25.6785i − 6.36352i
13.7 2.68999 + 0.874032i 4.20378 − 3.05422i 6.47214 + 4.70228i −6.44362 − 19.8314i 13.9776 − 4.54160i 35.1156 − 48.3325i 13.3001 + 18.3060i 8.34346 − 25.6785i − 58.9784i
13.8 2.68999 + 0.874032i 4.20378 − 3.05422i 6.47214 + 4.70228i 10.7228 + 33.0015i 13.9776 − 4.54160i −26.2692 + 36.1565i 13.3001 + 18.3060i 8.34346 − 25.6785i 98.1459i
19.1 −1.66251 − 2.28825i −1.60570 − 4.94183i −2.47214 + 7.60845i 8.24269 + 5.98866i −8.63864 + 11.8901i −68.6895 − 22.3186i 21.5200 − 6.99226i −21.8435 + 15.8702i − 28.8175i
19.2 −1.66251 − 2.28825i −1.60570 − 4.94183i −2.47214 + 7.60845i 18.5313 + 13.4638i −8.63864 + 11.8901i 49.8176 + 16.1867i 21.5200 − 6.99226i −21.8435 + 15.8702i − 64.7877i
19.3 −1.66251 − 2.28825i 1.60570 + 4.94183i −2.47214 + 7.60845i −24.4019 − 17.7291i 8.63864 − 11.8901i 61.9962 + 20.1438i 21.5200 − 6.99226i −21.8435 + 15.8702i 85.3123i
19.4 −1.66251 − 2.28825i 1.60570 + 4.94183i −2.47214 + 7.60845i −2.66700 − 1.93769i 8.63864 − 11.8901i −42.7542 − 13.8917i 21.5200 − 6.99226i −21.8435 + 15.8702i 9.32419i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 66.5.f.a ✓ 32
3.b odd 2 1 198.5.j.c 32
11.c even 5 1 726.5.d.g 32
11.d odd 10 1 inner 66.5.f.a ✓ 32
11.d odd 10 1 726.5.d.g 32
33.f even 10 1 198.5.j.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.5.f.a ✓ 32 1.a even 1 1 trivial
66.5.f.a ✓ 32 11.d odd 10 1 inner
198.5.j.c 32 3.b odd 2 1
198.5.j.c 32 33.f even 10 1
726.5.d.g 32 11.c even 5 1
726.5.d.g 32 11.d odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(66, [\chi])\).