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Properties

Label 177.6.d.b
Level $177$
Weight $6$
Character orbit 177.d
Analytic conductor $28.388$
Analytic rank $0$
Dimension $92$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 177.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(28.3879361069\)
Analytic rank: \(0\)
Dimension: \(92\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 92q + 22q^{3} + 1724q^{4} - 80q^{7} + 2q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 92q + 22q^{3} + 1724q^{4} - 80q^{7} + 2q^{9} - 1244q^{12} + 1116q^{15} + 14724q^{16} + 1784q^{19} + 6388q^{21} - 8140q^{22} - 48208q^{25} - 6458q^{27} - 19092q^{28} - 20832q^{36} - 134984q^{45} + 51180q^{46} + 61720q^{48} + 174556q^{49} + 8332q^{51} + 236784q^{57} + 375208q^{60} - 429890q^{63} + 561472q^{64} - 11596q^{66} + 169948q^{75} + 111488q^{76} + 356264q^{78} + 180260q^{79} + 79554q^{81} + 269308q^{84} + 111028q^{85} - 318764q^{87} - 1242976q^{88} - 513608q^{94} + O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
176.1 −10.9379 4.58836 + 14.8979i 87.6366 − 82.7497i −50.1868 − 162.951i 211.698 −608.545 −200.894 + 136.714i 905.104i
176.2 −10.9379 4.58836 − 14.8979i 87.6366 82.7497i −50.1868 + 162.951i 211.698 −608.545 −200.894 − 136.714i − 905.104i
176.3 −10.7490 9.69295 + 12.2085i 83.5412 33.1454i −104.190 − 131.229i −174.539 −554.017 −55.0934 + 236.672i − 356.281i
176.4 −10.7490 9.69295 − 12.2085i 83.5412 − 33.1454i −104.190 + 131.229i −174.539 −554.017 −55.0934 − 236.672i 356.281i
176.5 −10.3821 −12.6016 − 9.17604i 75.7873 − 9.35226i 130.831 + 95.2663i 67.1723 −454.603 74.6006 + 231.266i 97.0958i
176.6 −10.3821 −12.6016 + 9.17604i 75.7873 9.35226i 130.831 − 95.2663i 67.1723 −454.603 74.6006 − 231.266i − 97.0958i
176.7 −9.95283 −15.5523 − 1.06177i 67.0588 − 97.9414i 154.789 + 10.5677i −155.750 −348.934 240.745 + 33.0259i 974.794i
176.8 −9.95283 −15.5523 + 1.06177i 67.0588 97.9414i 154.789 − 10.5677i −155.750 −348.934 240.745 − 33.0259i − 974.794i
176.9 −9.63720 15.3454 − 2.74211i 60.8757 − 48.0995i −147.887 + 26.4263i 55.2643 −278.281 227.962 − 84.1575i 463.545i
176.10 −9.63720 15.3454 + 2.74211i 60.8757 48.0995i −147.887 − 26.4263i 55.2643 −278.281 227.962 + 84.1575i − 463.545i
176.11 −9.16184 −4.65820 + 14.8762i 51.9393 − 80.4460i 42.6777 − 136.293i −176.825 −182.680 −199.602 − 138.593i 737.033i
176.12 −9.16184 −4.65820 − 14.8762i 51.9393 80.4460i 42.6777 + 136.293i −176.825 −182.680 −199.602 + 138.593i − 737.033i
176.13 −8.73168 1.48415 − 15.5176i 44.2422 − 95.2512i −12.9591 + 135.495i 144.410 −106.895 −238.595 − 46.0609i 831.703i
176.14 −8.73168 1.48415 + 15.5176i 44.2422 95.2512i −12.9591 − 135.495i 144.410 −106.895 −238.595 + 46.0609i − 831.703i
176.15 −8.62331 −4.57233 + 14.9028i 42.3615 15.6511i 39.4286 − 128.512i −96.1896 −89.3504 −201.188 − 136.281i − 134.964i
176.16 −8.62331 −4.57233 − 14.9028i 42.3615 − 15.6511i 39.4286 + 128.512i −96.1896 −89.3504 −201.188 + 136.281i 134.964i
176.17 −7.56324 11.2364 + 10.8048i 25.2027 − 51.3528i −84.9838 − 81.7190i 90.7178 51.4099 9.51431 + 242.814i 388.394i
176.18 −7.56324 11.2364 − 10.8048i 25.2027 51.3528i −84.9838 + 81.7190i 90.7178 51.4099 9.51431 − 242.814i − 388.394i
176.19 −7.43533 −12.1962 − 9.70834i 23.2842 26.1059i 90.6831 + 72.1847i 171.796 64.8050 54.4964 + 236.810i − 194.106i
176.20 −7.43533 −12.1962 + 9.70834i 23.2842 − 26.1059i 90.6831 − 72.1847i 171.796 64.8050 54.4964 − 236.810i 194.106i
See all 92 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 176.92
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 177.6.d.b ✓ 92
3.b odd 2 1 inner 177.6.d.b ✓ 92
59.b odd 2 1 inner 177.6.d.b ✓ 92
177.d even 2 1 inner 177.6.d.b ✓ 92
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.6.d.b ✓ 92 1.a even 1 1 trivial
177.6.d.b ✓ 92 3.b odd 2 1 inner
177.6.d.b ✓ 92 59.b odd 2 1 inner
177.6.d.b ✓ 92 177.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(86\!\cdots\!83\)\( T_{2}^{36} + \)\(11\!\cdots\!84\)\( T_{2}^{34} - \)\(11\!\cdots\!61\)\( T_{2}^{32} + \)\(96\!\cdots\!75\)\( T_{2}^{30} - \)\(63\!\cdots\!14\)\( T_{2}^{28} + \)\(33\!\cdots\!84\)\( T_{2}^{26} - \)\(14\!\cdots\!60\)\( T_{2}^{24} + \)\(49\!\cdots\!72\)\( T_{2}^{22} - \)\(13\!\cdots\!68\)\( T_{2}^{20} + \)\(30\!\cdots\!28\)\( T_{2}^{18} - \)\(53\!\cdots\!84\)\( T_{2}^{16} + \)\(72\!\cdots\!96\)\( T_{2}^{14} - \)\(74\!\cdots\!80\)\( T_{2}^{12} + \)\(57\!\cdots\!44\)\( T_{2}^{10} - \)\(31\!\cdots\!00\)\( T_{2}^{8} + \)\(11\!\cdots\!40\)\( T_{2}^{6} - \)\(28\!\cdots\!72\)\( T_{2}^{4} + \)\(38\!\cdots\!44\)\( T_{2}^{2} - \)\(21\!\cdots\!32\)\( \)">\(T_{2}^{46} - \cdots\) acting on \(S_{6}^{\mathrm{new}}(177, [\chi])\).