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Properties

Label 177.4.d.c
Level $177$
Weight $4$
Character orbit 177.d
Analytic conductor $10.443$
Analytic rank $0$
Dimension $52$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [177,4,Mod(176,177)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(177, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("177.176");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 177.d (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: \(10.4433380710\)
Analytic rank: \(0\)
Dimension: \(52\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 52 q - 8 q^{3} + 268 q^{4} - 16 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 52 q - 8 q^{3} + 268 q^{4} - 16 q^{7} - 4 q^{9} + 28 q^{12} + 114 q^{15} + 484 q^{16} - 184 q^{19} - 758 q^{21} - 60 q^{22} + 36 q^{25} + 742 q^{27} - 4 q^{28} - 888 q^{36} + 1402 q^{45} - 660 q^{46} - 488 q^{48} - 924 q^{49} - 1772 q^{51} - 630 q^{57} - 1880 q^{60} - 212 q^{63} + 7648 q^{64} + 1316 q^{66} - 1556 q^{75} - 5680 q^{76} + 3224 q^{78} - 1504 q^{79} - 276 q^{81} + 1228 q^{84} - 848 q^{85} + 3598 q^{87} + 5760 q^{88} + 888 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
176.1 −5.56568 −1.93387 − 4.82288i 22.9768 − 10.3188i 10.7633 + 26.8426i −5.13468 −83.3558 −19.5203 + 18.6536i 57.4309i
176.2 −5.56568 −1.93387 + 4.82288i 22.9768 10.3188i 10.7633 − 26.8426i −5.13468 −83.3558 −19.5203 − 18.6536i − 57.4309i
176.3 −5.02351 4.35259 − 2.83813i 17.2357 15.7890i −21.8653 + 14.2574i −18.9623 −46.3956 10.8900 − 24.7064i − 79.3161i
176.4 −5.02351 4.35259 + 2.83813i 17.2357 − 15.7890i −21.8653 − 14.2574i −18.9623 −46.3956 10.8900 + 24.7064i 79.3161i
176.5 −4.72011 −4.94573 − 1.59367i 14.2794 15.2708i 23.3444 + 7.52230i 9.03916 −29.6395 21.9204 + 15.7637i − 72.0799i
176.6 −4.72011 −4.94573 + 1.59367i 14.2794 − 15.2708i 23.3444 − 7.52230i 9.03916 −29.6395 21.9204 − 15.7637i 72.0799i
176.7 −4.62765 4.25578 − 2.98133i 13.4152 − 5.89622i −19.6943 + 13.7966i 10.0941 −25.0596 9.22335 − 25.3758i 27.2857i
176.8 −4.62765 4.25578 + 2.98133i 13.4152 5.89622i −19.6943 − 13.7966i 10.0941 −25.0596 9.22335 + 25.3758i − 27.2857i
176.9 −4.28707 0.191639 − 5.19262i 10.3790 4.91692i −0.821572 + 22.2611i 31.8159 −10.1988 −26.9265 − 1.99022i − 21.0792i
176.10 −4.28707 0.191639 + 5.19262i 10.3790 − 4.91692i −0.821572 − 22.2611i 31.8159 −10.1988 −26.9265 + 1.99022i 21.0792i
176.11 −3.85434 −4.18856 − 3.07505i 6.85594 − 0.146910i 16.1441 + 11.8523i −24.4419 4.40959 8.08809 + 25.7601i 0.566243i
176.12 −3.85434 −4.18856 + 3.07505i 6.85594 0.146910i 16.1441 − 11.8523i −24.4419 4.40959 8.08809 − 25.7601i − 0.566243i
176.13 −3.37014 1.51919 − 4.96911i 3.35783 − 10.3951i −5.11987 + 16.7466i −22.3857 15.6447 −22.3841 − 15.0980i 35.0329i
176.14 −3.37014 1.51919 + 4.96911i 3.35783 10.3951i −5.11987 − 16.7466i −22.3857 15.6447 −22.3841 + 15.0980i − 35.0329i
176.15 −2.68371 −3.19645 − 4.09667i −0.797724 − 15.8157i 8.57832 + 10.9943i 25.4256 23.6105 −6.56545 + 26.1896i 42.4446i
176.16 −2.68371 −3.19645 + 4.09667i −0.797724 15.8157i 8.57832 − 10.9943i 25.4256 23.6105 −6.56545 − 26.1896i − 42.4446i
176.17 −2.62059 5.14935 − 0.695815i −1.13250 − 14.1294i −13.4944 + 1.82345i 10.3672 23.9326 26.0317 − 7.16599i 37.0275i
176.18 −2.62059 5.14935 + 0.695815i −1.13250 14.1294i −13.4944 − 1.82345i 10.3672 23.9326 26.0317 + 7.16599i − 37.0275i
176.19 −2.52893 0.100502 − 5.19518i −1.60453 17.9211i −0.254162 + 13.1382i −4.91298 24.2892 −26.9798 − 1.04425i − 45.3212i
176.20 −2.52893 0.100502 + 5.19518i −1.60453 − 17.9211i −0.254162 − 13.1382i −4.91298 24.2892 −26.9798 + 1.04425i 45.3212i
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 176.52
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 177.4.d.c ✓ 52
3.b odd 2 1 inner 177.4.d.c ✓ 52
59.b odd 2 1 inner 177.4.d.c ✓ 52
177.d even 2 1 inner 177.4.d.c ✓ 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.4.d.c ✓ 52 1.a even 1 1 trivial
177.4.d.c ✓ 52 3.b odd 2 1 inner
177.4.d.c ✓ 52 59.b odd 2 1 inner
177.4.d.c ✓ 52 177.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(177, [\chi])\):

\( T_{2}^{26} - 171 T_{2}^{24} + 12924 T_{2}^{22} - 569475 T_{2}^{20} + 16251156 T_{2}^{18} + \cdots - 2324803738752 \) Copy content Toggle raw display
\( T_{5}^{26} + 1616 T_{5}^{24} + 1130418 T_{5}^{22} + 449254663 T_{5}^{20} + 111997256480 T_{5}^{18} + \cdots + 21\!\cdots\!92 \) Copy content Toggle raw display