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Properties

Label 160.5.v.a
Level $160$
Weight $5$
Character orbit 160.v
Analytic conductor $16.539$
Analytic rank $0$
Dimension $376$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [160,5,Mod(13,160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(160, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 7, 6]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("160.13");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 160.v (of order \(8\), 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: \(16.5391940934\)
Analytic rank: \(0\)
Dimension: \(376\)
Relative dimension: \(94\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 376 q - 4 q^{2} - 4 q^{3} - 4 q^{5} - 8 q^{6} - 184 q^{8} - 400 q^{10} - 8 q^{11} - 724 q^{12} - 4 q^{13} + 128 q^{14} - 8 q^{16} - 68 q^{18} + 1408 q^{19} - 2608 q^{20} - 8 q^{21} - 564 q^{22} - 1504 q^{24}+ \cdots + 10856 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −3.98499 − 0.346213i 0.128089 − 0.0530562i 15.7603 + 2.75931i 16.9985 + 18.3317i −0.528802 + 0.167082i − 62.4933i −61.8492 − 16.4522i −57.2621 + 57.2621i −61.3920 − 78.9368i
13.2 −3.97805 + 0.418478i −7.76843 + 3.21779i 15.6498 − 3.32945i −14.6131 + 20.2844i 29.5566 − 16.0514i 40.1362i −60.8622 + 19.7938i −7.28137 + 7.28137i 49.6429 − 86.8078i
13.3 −3.97203 − 0.472207i 5.15973 − 2.13723i 15.5540 + 3.75124i −24.8599 + 2.64309i −21.5038 + 6.05269i − 95.6435i −60.0098 − 22.2448i −35.2206 + 35.2206i 99.9923 + 1.24058i
13.4 −3.95509 + 0.597740i −3.51515 + 1.45602i 15.2854 − 4.72822i −5.09280 − 24.4758i 13.0324 − 7.85985i 11.5838i −57.6289 + 27.8372i −47.0394 + 47.0394i 34.7726 + 93.7596i
13.5 −3.94656 + 0.651675i 9.21091 − 3.81529i 15.1506 − 5.14375i −20.6597 + 14.0776i −33.8651 + 21.0598i 59.0380i −56.4408 + 30.1734i 13.0089 − 13.0089i 72.3606 − 69.0214i
13.6 −3.94440 + 0.664616i −14.9165 + 6.17861i 15.1166 − 5.24302i 24.6823 − 3.97312i 54.7302 − 34.2846i 79.7323i −56.1412 + 30.7273i 127.051 − 127.051i −94.7161 + 32.0758i
13.7 −3.94424 − 0.665580i 10.6791 − 4.42342i 15.1140 + 5.25041i −9.86317 − 22.9721i −45.0650 + 10.3392i 58.1318i −56.1187 − 30.7684i 37.2005 − 37.2005i 23.6129 + 97.1722i
13.8 −3.89698 − 0.901970i 12.8021 − 5.30278i 14.3729 + 7.02992i 23.9599 − 7.13615i −54.6723 + 9.11777i 0.919552i −49.6701 − 40.3594i 78.4974 − 78.4974i −99.8077 + 6.19836i
13.9 −3.82954 + 1.15524i −14.6759 + 6.07894i 13.3308 − 8.84811i −18.9760 − 16.2761i 49.1792 − 40.2337i − 60.5714i −40.8293 + 49.2846i 121.152 − 121.152i 91.4724 + 40.4080i
13.10 −3.79738 + 1.25694i 3.53513 − 1.46430i 12.8402 − 9.54613i 23.8622 − 7.45626i −11.5837 + 10.0039i 6.88899i −36.7603 + 52.3897i −46.9227 + 46.9227i −81.2418 + 58.3075i
13.11 −3.74573 − 1.40339i −9.09653 + 3.76791i 12.0610 + 10.5134i 14.4866 − 20.3750i 39.3610 − 1.34758i − 23.5684i −30.4228 − 56.3068i 11.2741 − 11.2741i −82.8568 + 55.9888i
13.12 −3.72533 − 1.45668i −4.91200 + 2.03462i 11.7562 + 10.8532i −22.0636 − 11.7558i 21.2626 − 0.424433i 7.00115i −27.9861 − 57.5567i −37.2875 + 37.2875i 65.0699 + 75.9336i
13.13 −3.70265 − 1.51340i −1.99002 + 0.824294i 11.4193 + 11.2072i 15.0993 + 19.9251i 8.61584 − 0.0403829i 76.5279i −25.3207 − 58.7781i −53.9949 + 53.9949i −25.7531 − 96.6270i
13.14 −3.69548 + 1.53083i 16.1614 − 6.69428i 11.3131 − 11.3143i 7.00934 + 23.9973i −49.4763 + 49.4790i − 45.3574i −24.4869 + 59.1303i 159.102 − 159.102i −62.6387 − 77.9512i
13.15 −3.59482 + 1.75421i −8.90051 + 3.68671i 9.84548 − 12.6122i 7.03388 + 23.9901i 25.5285 − 28.8664i − 33.4308i −13.2684 + 62.6095i 8.35154 − 8.35154i −67.3692 − 73.9012i
13.16 −3.56430 − 1.81543i −13.9928 + 5.79600i 9.40840 + 12.9415i −13.0943 + 21.2965i 60.3966 + 4.74431i − 29.5921i −10.0399 − 63.2076i 104.929 − 104.929i 85.3343 − 52.1350i
13.17 −3.43108 + 2.05614i 11.0076 − 4.55950i 7.54456 − 14.1096i −10.1237 − 22.8585i −28.3930 + 38.2772i − 41.7696i 3.12533 + 63.9236i 43.1029 − 43.1029i 81.7354 + 57.6136i
13.18 −3.32287 − 2.22678i 10.4116 − 4.31263i 6.08287 + 14.7986i −4.07857 + 24.6651i −44.1997 − 8.85412i 5.71715i 12.7407 − 62.7190i 32.5272 − 32.5272i 68.4763 − 72.8766i
13.19 −3.23036 + 2.35897i −5.32912 + 2.20739i 4.87049 − 15.2407i −21.7190 − 12.3808i 12.0078 − 19.7019i 58.3034i 20.2189 + 60.7223i −33.7488 + 33.7488i 99.3663 − 11.2400i
13.20 −3.07306 + 2.56052i 1.12258 − 0.464989i 2.88745 − 15.7373i −20.4558 + 14.3722i −2.25915 + 4.30334i − 34.3270i 31.4224 + 55.7551i −56.2317 + 56.2317i 26.0616 − 96.5443i
See next 80 embeddings (of 376 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.94
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
160.v odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.5.v.a ✓ 376
5.c odd 4 1 160.5.bb.a yes 376
32.g even 8 1 160.5.bb.a yes 376
160.v odd 8 1 inner 160.5.v.a ✓ 376
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.5.v.a ✓ 376 1.a even 1 1 trivial
160.5.v.a ✓ 376 160.v odd 8 1 inner
160.5.bb.a yes 376 5.c odd 4 1
160.5.bb.a yes 376 32.g even 8 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(160, [\chi])\).