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Properties

Label 50.4.e.a
Level $50$
Weight $4$
Character orbit 50.e
Analytic conductor $2.950$
Analytic rank $0$
Dimension $32$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [50,4,Mod(9,50)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(50, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([7]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("50.9");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 50.e (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: \(2.95009550029\)
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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 32 q^{4} - 30 q^{5} - 12 q^{6} + 26 q^{9} - 40 q^{10} - 106 q^{11} + 80 q^{12} + 56 q^{14} + 260 q^{15} - 128 q^{16} + 320 q^{17} + 110 q^{19} - 160 q^{20} - 36 q^{21} - 360 q^{22} - 370 q^{23} - 192 q^{24}+ \cdots - 2108 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −1.17557 + 1.61803i −9.37926 + 3.04751i −1.23607 − 3.80423i 9.19742 − 6.35669i 6.09501 − 18.7585i − 2.82989i 7.60845 + 2.47214i 56.8397 − 41.2965i −0.526876 + 22.3545i
9.2 −1.17557 + 1.61803i −1.91394 + 0.621877i −1.23607 − 3.80423i −11.1347 + 1.00973i 1.24375 − 3.82788i − 29.3318i 7.60845 + 2.47214i −18.5670 + 13.4897i 11.4558 − 19.2033i
9.3 −1.17557 + 1.61803i 0.234870 − 0.0763140i −1.23607 − 3.80423i 5.37427 + 9.80394i −0.152628 + 0.469741i 19.9138i 7.60845 + 2.47214i −21.7941 + 15.8344i −22.1809 − 2.82947i
9.4 −1.17557 + 1.61803i 5.96909 − 1.93948i −1.23607 − 3.80423i 5.14683 − 9.92523i −3.87895 + 11.9382i − 9.80956i 7.60845 + 2.47214i 10.0250 − 7.28362i 10.0089 + 19.9956i
9.5 1.17557 − 1.61803i −6.75819 + 2.19587i −1.23607 − 3.80423i −8.51031 + 7.25084i −4.39174 + 13.5164i 23.3487i −7.60845 − 2.47214i 19.0078 − 13.8100i 1.72764 + 22.2938i
9.6 1.17557 − 1.61803i −4.16531 + 1.35339i −1.23607 − 3.80423i −1.15597 − 11.1204i −2.70679 + 8.33063i − 31.2051i −7.60845 − 2.47214i −6.32528 + 4.59559i −19.3521 − 11.2024i
9.7 1.17557 − 1.61803i 4.78125 − 1.55352i −1.23607 − 3.80423i 7.44908 + 8.33734i 3.10704 − 9.56250i − 11.4048i −7.60845 − 2.47214i −1.39654 + 1.01465i 22.2470 − 2.25173i
9.8 1.17557 − 1.61803i 6.75935 − 2.19625i −1.23607 − 3.80423i −6.04044 − 9.40814i 4.39249 − 13.5187i 25.2037i −7.60845 − 2.47214i 19.0219 − 13.8202i −22.3237 − 1.28629i
19.1 −1.90211 + 0.618034i −4.41051 − 6.07054i 3.23607 − 2.35114i −3.40245 + 10.6500i 12.1411 + 8.82101i 17.5556i −4.70228 + 6.47214i −9.05545 + 27.8698i −0.110239 − 22.3604i
19.2 −1.90211 + 0.618034i −0.184196 − 0.253524i 3.23607 − 2.35114i −9.32973 − 6.16085i 0.507047 + 0.368391i − 8.77601i −4.70228 + 6.47214i 8.31311 − 25.5851i 21.5538 + 5.95254i
19.3 −1.90211 + 0.618034i 2.86980 + 3.94994i 3.23607 − 2.35114i 7.54709 − 8.24872i −7.89989 − 5.73960i 32.3828i −4.70228 + 6.47214i 0.977169 − 3.00742i −9.25744 + 20.3544i
19.4 −1.90211 + 0.618034i 5.72432 + 7.87886i 3.23607 − 2.35114i −3.73936 + 10.5365i −15.7577 − 11.4486i − 26.8849i −4.70228 + 6.47214i −20.9650 + 64.5237i 0.600782 − 22.3526i
19.5 1.90211 − 0.618034i −5.11745 − 7.04356i 3.23607 − 2.35114i −8.52044 + 7.23893i −14.0871 − 10.2349i − 24.3678i 4.70228 − 6.47214i −15.0801 + 46.4116i −11.7329 + 19.0352i
19.6 1.90211 − 0.618034i −1.78443 − 2.45606i 3.23607 − 2.35114i 3.46177 − 10.6309i −4.91212 − 3.56886i 1.93842i 4.70228 − 6.47214i 5.49543 − 16.9132i 0.0144147 − 22.3607i
19.7 1.90211 − 0.618034i 1.87704 + 2.58353i 3.23607 − 2.35114i 4.46514 + 10.2500i 5.16706 + 3.75409i − 2.29951i 4.70228 − 6.47214i 5.19213 − 15.9797i 14.8281 + 16.7370i
19.8 1.90211 − 0.618034i 5.49755 + 7.56672i 3.23607 − 2.35114i −5.80826 − 9.55322i 15.1334 + 10.9951i 11.0064i 4.70228 − 6.47214i −18.6888 + 57.5183i −16.9522 − 14.5816i
29.1 −1.90211 − 0.618034i −4.41051 + 6.07054i 3.23607 + 2.35114i −3.40245 − 10.6500i 12.1411 − 8.82101i − 17.5556i −4.70228 − 6.47214i −9.05545 − 27.8698i −0.110239 + 22.3604i
29.2 −1.90211 − 0.618034i −0.184196 + 0.253524i 3.23607 + 2.35114i −9.32973 + 6.16085i 0.507047 − 0.368391i 8.77601i −4.70228 − 6.47214i 8.31311 + 25.5851i 21.5538 − 5.95254i
29.3 −1.90211 − 0.618034i 2.86980 − 3.94994i 3.23607 + 2.35114i 7.54709 + 8.24872i −7.89989 + 5.73960i − 32.3828i −4.70228 − 6.47214i 0.977169 + 3.00742i −9.25744 − 20.3544i
29.4 −1.90211 − 0.618034i 5.72432 − 7.87886i 3.23607 + 2.35114i −3.73936 − 10.5365i −15.7577 + 11.4486i 26.8849i −4.70228 − 6.47214i −20.9650 − 64.5237i 0.600782 + 22.3526i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 50.4.e.a ✓ 32
5.b even 2 1 250.4.e.b 32
5.c odd 4 1 250.4.d.c 32
5.c odd 4 1 250.4.d.d 32
25.d even 5 1 250.4.e.b 32
25.e even 10 1 inner 50.4.e.a ✓ 32
25.f odd 20 1 250.4.d.c 32
25.f odd 20 1 250.4.d.d 32
25.f odd 20 1 1250.4.a.m 16
25.f odd 20 1 1250.4.a.n 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.4.e.a ✓ 32 1.a even 1 1 trivial
50.4.e.a ✓ 32 25.e even 10 1 inner
250.4.d.c 32 5.c odd 4 1
250.4.d.c 32 25.f odd 20 1
250.4.d.d 32 5.c odd 4 1
250.4.d.d 32 25.f odd 20 1
250.4.e.b 32 5.b even 2 1
250.4.e.b 32 25.d even 5 1
1250.4.a.m 16 25.f odd 20 1
1250.4.a.n 16 25.f odd 20 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(50, [\chi])\).