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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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 177.d (of order \(2\), degree \(1\), minimal)

Newform invariants

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.

\(\operatorname{Tr}(f)(q) = \) \( 52q - 8q^{3} + 268q^{4} - 16q^{7} - 4q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 52q - 8q^{3} + 268q^{4} - 16q^{7} - 4q^{9} + 28q^{12} + 114q^{15} + 484q^{16} - 184q^{19} - 758q^{21} - 60q^{22} + 36q^{25} + 742q^{27} - 4q^{28} - 888q^{36} + 1402q^{45} - 660q^{46} - 488q^{48} - 924q^{49} - 1772q^{51} - 630q^{57} - 1880q^{60} - 212q^{63} + 7648q^{64} + 1316q^{66} - 1556q^{75} - 5680q^{76} + 3224q^{78} - 1504q^{79} - 276q^{81} + 1228q^{84} - 848q^{85} + 3598q^{87} + 5760q^{88} + 888q^{94} + O(q^{100}) \)

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.

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:
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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 \(S_{4}^{\mathrm{new}}(177, [\chi])\):

\(11\!\cdots\!22\)\( T_{2}^{8} + \)\(34\!\cdots\!96\)\( T_{2}^{6} - \)\(61\!\cdots\!48\)\( T_{2}^{4} + \)\(59\!\cdots\!64\)\( T_{2}^{2} - \)\(23\!\cdots\!52\)\( \)">\(T_{2}^{26} - \cdots\)
\(18\!\cdots\!91\)\( T_{5}^{16} + \)\(19\!\cdots\!20\)\( T_{5}^{14} + \)\(14\!\cdots\!35\)\( T_{5}^{12} + \)\(66\!\cdots\!19\)\( T_{5}^{10} + \)\(19\!\cdots\!97\)\( T_{5}^{8} + \)\(33\!\cdots\!10\)\( T_{5}^{6} + \)\(30\!\cdots\!54\)\( T_{5}^{4} + \)\(10\!\cdots\!96\)\( T_{5}^{2} + \)\(21\!\cdots\!92\)\( \)">\(T_{5}^{26} + \cdots\)