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Properties

Label 165.6.c.b
Level $165$
Weight $6$
Character orbit 165.c
Analytic conductor $26.463$
Analytic rank $0$
Dimension $26$
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] = [165,6,Mod(34,165)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(165, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("165.34");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 165.c (of order \(2\), degree \(1\), 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: \(26.4633302691\)
Analytic rank: \(0\)
Dimension: \(26\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 26 q - 418 q^{4} - 98 q^{5} + 234 q^{6} - 2106 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 26 q - 418 q^{4} - 98 q^{5} + 234 q^{6} - 2106 q^{9} + 236 q^{10} + 3146 q^{11} + 1220 q^{14} + 7002 q^{16} - 540 q^{19} + 4930 q^{20} + 5472 q^{21} - 7182 q^{24} + 218 q^{25} + 5304 q^{26} - 23904 q^{29} + 12114 q^{30} + 38192 q^{31} + 2604 q^{34} - 11988 q^{35} + 33858 q^{36} - 17748 q^{39} - 41096 q^{40} + 70368 q^{41} - 50578 q^{44} + 7938 q^{45} - 8240 q^{46} - 29114 q^{49} - 133876 q^{50} + 26568 q^{51} - 18954 q^{54} - 11858 q^{55} + 119604 q^{56} + 18384 q^{59} - 14148 q^{60} + 10876 q^{61} - 213114 q^{64} - 117068 q^{65} + 28314 q^{66} - 163512 q^{69} - 58660 q^{70} + 203400 q^{71} - 27352 q^{74} - 35352 q^{75} + 279932 q^{76} - 187908 q^{79} - 256654 q^{80} + 170586 q^{81} - 196560 q^{84} + 37396 q^{85} + 741860 q^{86} + 36836 q^{89} - 19116 q^{90} + 349072 q^{91} - 129040 q^{94} - 208284 q^{95} + 209898 q^{96} - 254826 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1 − 10.7531i 9.00000i −83.6283 8.68170 + 55.2234i 96.7775 − 41.3813i 555.162i −81.0000 593.821 − 93.3549i
34.2 − 10.4306i − 9.00000i −76.7973 −52.2792 + 19.7960i −93.8753 21.9028i 467.262i −81.0000 206.484 + 545.303i
34.3 − 10.0460i 9.00000i −68.9227 24.7608 − 50.1189i 90.4143 29.0524i 370.927i −81.0000 −503.496 − 248.748i
34.4 − 8.57523i 9.00000i −41.5346 −51.2624 − 22.2971i 77.1771 − 178.462i 81.7612i −81.0000 −191.203 + 439.587i
34.5 − 7.81495i − 9.00000i −29.0734 48.1851 − 28.3407i −70.3345 11.0326i − 22.8710i −81.0000 −221.481 − 376.564i
34.6 − 7.39855i − 9.00000i −22.7385 −50.7560 + 23.4272i −66.5869 150.852i − 68.5217i −81.0000 173.327 + 375.520i
34.7 − 5.72893i 9.00000i −0.820586 −32.0393 + 45.8092i 51.5603 158.171i − 178.625i −81.0000 262.437 + 183.551i
34.8 − 5.09558i 9.00000i 6.03505 54.3767 + 12.9683i 45.8602 − 170.277i − 193.811i −81.0000 66.0813 − 277.081i
34.9 − 4.97394i 9.00000i 7.25990 47.6857 − 29.1732i 44.7655 155.412i − 195.276i −81.0000 −145.106 − 237.186i
34.10 − 3.62206i − 9.00000i 18.8807 5.26063 − 55.6536i −32.5985 − 168.040i − 184.293i −81.0000 −201.581 − 19.0543i
34.11 − 3.33747i − 9.00000i 20.8613 21.4823 + 51.6092i −30.0373 103.473i − 176.423i −81.0000 172.244 − 71.6967i
34.12 − 1.31741i 9.00000i 30.2644 −35.6376 − 43.0693i 11.8567 87.4786i − 82.0279i −81.0000 −56.7399 + 46.9494i
34.13 − 0.886559i − 9.00000i 31.2140 −37.4584 − 41.4954i −7.97903 224.774i − 56.0430i −81.0000 −36.7881 + 33.2091i
34.14 0.886559i 9.00000i 31.2140 −37.4584 + 41.4954i −7.97903 − 224.774i 56.0430i −81.0000 −36.7881 − 33.2091i
34.15 1.31741i − 9.00000i 30.2644 −35.6376 + 43.0693i 11.8567 − 87.4786i 82.0279i −81.0000 −56.7399 − 46.9494i
34.16 3.33747i 9.00000i 20.8613 21.4823 − 51.6092i −30.0373 − 103.473i 176.423i −81.0000 172.244 + 71.6967i
34.17 3.62206i 9.00000i 18.8807 5.26063 + 55.6536i −32.5985 168.040i 184.293i −81.0000 −201.581 + 19.0543i
34.18 4.97394i − 9.00000i 7.25990 47.6857 + 29.1732i 44.7655 − 155.412i 195.276i −81.0000 −145.106 + 237.186i
34.19 5.09558i − 9.00000i 6.03505 54.3767 − 12.9683i 45.8602 170.277i 193.811i −81.0000 66.0813 + 277.081i
34.20 5.72893i − 9.00000i −0.820586 −32.0393 − 45.8092i 51.5603 − 158.171i 178.625i −81.0000 262.437 − 183.551i
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.6.c.b ✓ 26
5.b even 2 1 inner 165.6.c.b ✓ 26
5.c odd 4 1 825.6.a.v 13
5.c odd 4 1 825.6.a.y 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.6.c.b ✓ 26 1.a even 1 1 trivial
165.6.c.b ✓ 26 5.b even 2 1 inner
825.6.a.v 13 5.c odd 4 1
825.6.a.y 13 5.c odd 4 1