Properties

Label 177.4.f.a
Level $177$
Weight $4$
Character orbit 177.f
Analytic conductor $10.443$
Analytic rank $0$
Dimension $1624$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 177.f (of order \(58\), degree \(28\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.4433380710\)
Analytic rank: \(0\)
Dimension: \(1624\)
Relative dimension: \(58\) over \(\Q(\zeta_{58})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{58}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 1624q - 21q^{3} - 278q^{4} - 29q^{6} - 42q^{7} - 25q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 1624q - 21q^{3} - 278q^{4} - 29q^{6} - 42q^{7} - 25q^{9} - 58q^{10} - 57q^{12} - 58q^{13} - 11q^{15} - 926q^{16} - 29q^{18} + 126q^{19} + 159q^{21} + 2q^{22} - 29q^{24} + 656q^{25} - 99q^{27} - 54q^{28} - 29q^{30} - 58q^{31} - 29q^{33} - 58q^{34} + 859q^{36} - 58q^{37} - 29q^{39} - 58q^{40} - 29q^{42} - 58q^{43} + 1703q^{45} + 602q^{46} + 9507q^{48} - 1192q^{49} + 1511q^{51} - 58q^{52} - 7743q^{54} - 58q^{55} - 7441q^{57} - 18722q^{60} - 58q^{61} - 3251q^{63} - 4634q^{64} - 1751q^{66} - 58q^{67} + 6003q^{69} - 58q^{70} + 21547q^{72} - 58q^{73} + 3869q^{75} + 5622q^{76} - 3253q^{78} + 1446q^{79} + 247q^{81} - 58q^{82} + 3303q^{84} + 790q^{85} - 2199q^{87} - 5818q^{88} - 29q^{90} - 58q^{91} - 29q^{93} - 946q^{94} - 29q^{96} - 58q^{97} - 29q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −0.301320 + 5.55751i 4.73924 2.13064i −22.8421 2.48423i −5.05728 + 15.0095i 10.4130 + 26.9804i −9.02219 13.3067i 13.4855 82.2577i 17.9208 20.1952i −81.8915 32.6285i
2.2 −0.285117 + 5.25867i 5.18186 + 0.385093i −19.6192 2.13372i 6.02049 17.8682i −3.50252 + 27.1399i 18.4360 + 27.1910i 9.99825 60.9866i 26.7034 + 3.99100i 92.2463 + 36.7543i
2.3 −0.282358 + 5.20779i −0.412985 + 5.17971i −19.0882 2.07597i −2.66012 + 7.89495i −26.8583 3.61327i 2.27648 + 3.35756i 9.45083 57.6475i −26.6589 4.27829i −40.3641 16.0825i
2.4 −0.281900 + 5.19934i −5.12170 + 0.876440i −19.0005 2.06643i 2.67889 7.95067i −3.11310 26.8765i 1.88189 + 2.77557i 9.36119 57.1007i 25.4637 8.97773i 40.5830 + 16.1698i
2.5 −0.273845 + 5.05077i −1.98491 4.80210i −17.4822 1.90130i −0.350795 + 1.04112i 24.7978 8.71029i −5.00566 7.38279i 7.84382 47.8452i −19.1203 + 19.0635i −5.16240 2.05689i
2.6 −0.254048 + 4.68564i 1.63609 + 4.93186i −13.9375 1.51580i 5.44329 16.1551i −23.5245 + 6.41321i −18.4501 27.2119i 4.56997 27.8756i −21.6464 + 16.1380i 74.3141 + 29.6094i
2.7 −0.247929 + 4.57278i 0.343239 5.18480i −12.8957 1.40250i 2.38761 7.08619i 23.6239 + 2.85502i 6.25613 + 9.22711i 3.68350 22.4684i −26.7644 3.55926i 31.8116 + 12.6749i
2.8 −0.238865 + 4.40561i −4.39778 2.76759i −11.3992 1.23974i −6.88505 + 20.4341i 13.2434 18.7138i 12.3388 + 18.1984i 2.47433 15.0928i 11.6809 + 24.3425i −88.3800 35.2138i
2.9 −0.216913 + 4.00073i 4.38754 2.78379i −8.00566 0.870667i 2.33628 6.93383i 10.1855 + 18.1572i −13.7785 20.3217i 0.0342646 0.209005i 11.5010 24.4280i 27.2336 + 10.8508i
2.10 −0.214445 + 3.95521i 4.30093 + 2.91582i −7.64463 0.831403i −4.87721 + 14.4750i −12.4550 + 16.3858i 8.97536 + 13.2377i −0.198845 + 1.21290i 9.99602 + 25.0815i −56.2060 22.3945i
2.11 −0.212337 + 3.91633i 4.58888 + 2.43766i −7.33945 0.798213i −0.421405 + 1.25069i −10.5211 + 17.4540i −5.56806 8.21227i −0.391673 + 2.38910i 15.1156 + 22.3723i −4.80862 1.91593i
2.12 −0.206723 + 3.81279i −4.04421 + 3.26257i −6.54149 0.711430i −3.85784 + 11.4497i −11.6035 16.0941i −12.2513 18.0693i −0.877159 + 5.35043i 5.71121 26.3891i −42.8576 17.0760i
2.13 −0.194915 + 3.59500i 3.72957 3.61806i −4.93296 0.536492i −1.29349 + 3.83895i 12.2800 + 14.1130i 12.2031 + 17.9982i −1.76949 + 10.7934i 0.819319 26.9876i −13.5489 5.39838i
2.14 −0.191601 + 3.53386i −0.887175 + 5.11986i −4.49838 0.489228i 2.01755 5.98787i −17.9229 4.11612i 16.3407 + 24.1007i −1.98968 + 12.1365i −25.4258 9.08442i 20.7738 + 8.27702i
2.15 −0.181075 + 3.33974i −4.07453 3.22462i −3.16798 0.344538i 5.90079 17.5129i 11.5072 13.0240i −10.9419 16.1380i −2.60452 + 15.8869i 6.20361 + 26.2777i 57.4202 + 22.8783i
2.16 −0.176293 + 3.25153i −5.15278 0.670007i −2.58825 0.281489i −0.0532670 + 0.158091i 3.08694 16.6363i 11.3529 + 16.7443i −2.84293 + 17.3411i 26.1022 + 6.90479i −0.504646 0.201069i
2.17 −0.150879 + 2.78279i −4.15765 + 3.11672i 0.231945 + 0.0252256i 4.42533 13.1339i −8.04587 12.0401i −0.713636 1.05253i −3.71213 + 22.6430i 7.57215 25.9165i 35.8813 + 14.2964i
2.18 −0.142776 + 2.63336i 0.601836 5.16118i 1.03892 + 0.112990i −5.43994 + 16.1452i 13.5053 + 2.32174i −11.5153 16.9838i −3.85912 + 23.5396i −26.2756 6.21237i −41.7393 16.6305i
2.19 −0.123545 + 2.27865i −4.04503 3.26155i 2.77611 + 0.301920i −0.776377 + 2.30421i 7.93169 8.81427i −10.4076 15.3501i −3.98444 + 24.3040i 5.72454 + 26.3862i −5.15457 2.05377i
2.20 −0.100013 + 1.84463i 2.00907 4.79204i 4.56043 + 0.495977i 6.12126 18.1673i 8.63862 + 4.18527i 1.32430 + 1.95320i −3.76193 + 22.9468i −18.9273 19.2551i 32.8997 + 13.1084i
See next 80 embeddings (of 1624 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 173.58
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
59.d odd 58 1 inner
177.f even 58 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 177.4.f.a 1624
3.b odd 2 1 inner 177.4.f.a 1624
59.d odd 58 1 inner 177.4.f.a 1624
177.f even 58 1 inner 177.4.f.a 1624
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.4.f.a 1624 1.a even 1 1 trivial
177.4.f.a 1624 3.b odd 2 1 inner
177.4.f.a 1624 59.d odd 58 1 inner
177.4.f.a 1624 177.f even 58 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(177, [\chi])\).