↑

Properties

Label 165.4.f.a
Level $165$
Weight $4$
Character orbit 165.f
Analytic conductor $9.735$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [165,4,Mod(131,165)]
 
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([1, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("165.131");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 165.f (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: \(9.73531515095\)
Analytic rank: \(0\)
Dimension: \(48\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{3} + 192 q^{4} - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{3} + 192 q^{4} - 28 q^{9} - 48 q^{12} - 40 q^{15} + 1248 q^{16} + 372 q^{22} - 1200 q^{25} - 628 q^{27} - 24 q^{31} - 608 q^{33} - 1464 q^{34} + 444 q^{36} - 264 q^{37} + 2392 q^{42} + 584 q^{48} - 1848 q^{49} + 60 q^{55} - 5040 q^{58} - 1620 q^{60} + 7920 q^{64} - 4236 q^{66} - 168 q^{67} + 3872 q^{69} - 1080 q^{70} + 100 q^{75} + 1048 q^{78} + 8164 q^{81} + 9720 q^{82} - 948 q^{88} - 2256 q^{91} - 3584 q^{93} - 864 q^{97} - 3184 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
131.1 −5.53529 1.32319 − 5.02486i 22.6394 − 5.00000i −7.32423 + 27.8140i − 26.3911i −81.0335 −23.4983 − 13.2977i 27.6765i
131.2 −5.53529 1.32319 + 5.02486i 22.6394 5.00000i −7.32423 − 27.8140i 26.3911i −81.0335 −23.4983 + 13.2977i − 27.6765i
131.3 −5.23426 −5.14577 − 0.721827i 19.3975 5.00000i 26.9343 + 3.77823i − 28.0885i −59.6575 25.9579 + 7.42872i − 26.1713i
131.4 −5.23426 −5.14577 + 0.721827i 19.3975 − 5.00000i 26.9343 − 3.77823i 28.0885i −59.6575 25.9579 − 7.42872i 26.1713i
131.5 −4.68846 2.77931 − 4.39038i 13.9816 5.00000i −13.0307 + 20.5841i 13.2938i −28.0446 −11.5508 − 24.4045i − 23.4423i
131.6 −4.68846 2.77931 + 4.39038i 13.9816 − 5.00000i −13.0307 − 20.5841i − 13.2938i −28.0446 −11.5508 + 24.4045i 23.4423i
131.7 −4.45233 5.18230 − 0.379109i 11.8233 − 5.00000i −23.0733 + 1.68792i 13.4456i −17.0224 26.7126 − 3.92932i 22.2617i
131.8 −4.45233 5.18230 + 0.379109i 11.8233 5.00000i −23.0733 − 1.68792i − 13.4456i −17.0224 26.7126 + 3.92932i − 22.2617i
131.9 −3.86514 −2.30310 − 4.65787i 6.93930 − 5.00000i 8.90178 + 18.0033i 7.27480i 4.09974 −16.3915 + 21.4550i 19.3257i
131.10 −3.86514 −2.30310 + 4.65787i 6.93930 5.00000i 8.90178 − 18.0033i − 7.27480i 4.09974 −16.3915 − 21.4550i − 19.3257i
131.11 −3.21233 −4.99978 − 1.41499i 2.31906 − 5.00000i 16.0609 + 4.54543i − 13.7977i 18.2491 22.9956 + 14.1493i 16.0616i
131.12 −3.21233 −4.99978 + 1.41499i 2.31906 5.00000i 16.0609 − 4.54543i 13.7977i 18.2491 22.9956 − 14.1493i − 16.0616i
131.13 −2.69954 −2.04842 − 4.77535i −0.712494 5.00000i 5.52979 + 12.8912i − 33.6257i 23.5197 −18.6079 + 19.5639i − 13.4977i
131.14 −2.69954 −2.04842 + 4.77535i −0.712494 − 5.00000i 5.52979 − 12.8912i 33.6257i 23.5197 −18.6079 − 19.5639i 13.4977i
131.15 −2.45914 3.48830 − 3.85120i −1.95264 − 5.00000i −8.57821 + 9.47064i 10.3212i 24.4749 −2.66352 − 26.8683i 12.2957i
131.16 −2.45914 3.48830 + 3.85120i −1.95264 5.00000i −8.57821 − 9.47064i − 10.3212i 24.4749 −2.66352 + 26.8683i − 12.2957i
131.17 −1.86247 1.05045 − 5.08887i −4.53119 5.00000i −1.95644 + 9.47788i 11.2183i 23.3390 −24.7931 − 10.6912i − 9.31237i
131.18 −1.86247 1.05045 + 5.08887i −4.53119 − 5.00000i −1.95644 − 9.47788i − 11.2183i 23.3390 −24.7931 + 10.6912i 9.31237i
131.19 −1.31619 −4.76484 − 2.07276i −6.26764 5.00000i 6.27143 + 2.72815i 19.1367i 18.7789 18.4073 + 19.7527i − 6.58095i
131.20 −1.31619 −4.76484 + 2.07276i −6.26764 − 5.00000i 6.27143 − 2.72815i − 19.1367i 18.7789 18.4073 − 19.7527i 6.58095i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 131.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.4.f.a ✓ 48
3.b odd 2 1 inner 165.4.f.a ✓ 48
11.b odd 2 1 inner 165.4.f.a ✓ 48
33.d even 2 1 inner 165.4.f.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.f.a ✓ 48 1.a even 1 1 trivial
165.4.f.a ✓ 48 3.b odd 2 1 inner
165.4.f.a ✓ 48 11.b odd 2 1 inner
165.4.f.a ✓ 48 33.d even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(165, [\chi])\).