Properties

Label 177.10.a.d
Level $177$
Weight $10$
Character orbit 177.a
Self dual yes
Analytic conductor $91.161$
Analytic rank $0$
Dimension $22$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(91.1613430010\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 22q + 46q^{2} + 1782q^{3} + 5974q^{4} + 5786q^{5} + 3726q^{6} + 7641q^{7} + 61395q^{8} + 144342q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 22q + 46q^{2} + 1782q^{3} + 5974q^{4} + 5786q^{5} + 3726q^{6} + 7641q^{7} + 61395q^{8} + 144342q^{9} + 45337q^{10} + 111769q^{11} + 483894q^{12} + 189121q^{13} + 251053q^{14} + 468666q^{15} + 2311074q^{16} + 1113841q^{17} + 301806q^{18} + 476068q^{19} - 42495q^{20} + 618921q^{21} - 2252022q^{22} + 7103062q^{23} + 4972995q^{24} + 10628442q^{25} + 6871048q^{26} + 11691702q^{27} + 8112650q^{28} + 15279316q^{29} + 3672297q^{30} + 17610338q^{31} + 32378276q^{32} + 9053289q^{33} + 29339436q^{34} + 7134904q^{35} + 39195414q^{36} + 21961411q^{37} + 65195131q^{38} + 15318801q^{39} + 75185084q^{40} + 52781575q^{41} + 20335293q^{42} + 76191313q^{43} + 61127768q^{44} + 37961946q^{45} + 290208769q^{46} + 160572396q^{47} + 187196994q^{48} + 156292703q^{49} + 169504821q^{50} + 90221121q^{51} + 65465920q^{52} - 8762038q^{53} + 24446286q^{54} + 147125140q^{55} + 9671794q^{56} + 38561508q^{57} - 37665424q^{58} - 266581942q^{59} - 3442095q^{60} + 120750754q^{61} - 152465186q^{62} + 50132601q^{63} - 40658803q^{64} + 331055798q^{65} - 182413782q^{66} + 41371828q^{67} + 145606631q^{68} + 575348022q^{69} - 920887614q^{70} + 261018751q^{71} + 402812595q^{72} + 178388q^{73} - 303908734q^{74} + 860903802q^{75} - 94541144q^{76} + 299640561q^{77} + 556554888q^{78} - 905381353q^{79} + 939128289q^{80} + 947027862q^{81} - 551739753q^{82} + 1173257869q^{83} + 657124650q^{84} - 1546633210q^{85} + 1384869460q^{86} + 1237624596q^{87} + 189740713q^{88} + 898004974q^{89} + 297456057q^{90} + 591272339q^{91} + 4328210270q^{92} + 1426437378q^{93} + 122568068q^{94} + 2487967134q^{95} + 2622640356q^{96} + 3175709684q^{97} + 5095778404q^{98} + 733316409q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −39.7571 81.0000 1068.63 −1871.54 −3220.33 −43.1074 −22129.9 6561.00 74407.2
1.2 −39.1029 81.0000 1017.03 1845.13 −3167.33 8719.40 −19748.3 6561.00 −72149.9
1.3 −38.0945 81.0000 939.188 −310.159 −3085.65 −4492.16 −16273.5 6561.00 11815.4
1.4 −32.9800 81.0000 575.677 −958.538 −2671.38 −5513.62 −2100.07 6561.00 31612.5
1.5 −28.6453 81.0000 308.554 1354.22 −2320.27 4084.12 5827.78 6561.00 −38792.0
1.6 −20.0625 81.0000 −109.497 2619.85 −1625.06 −4642.93 12468.8 6561.00 −52560.7
1.7 −16.3468 81.0000 −244.784 −1401.93 −1324.09 3361.75 12371.0 6561.00 22917.0
1.8 −12.8477 81.0000 −346.936 −288.160 −1040.66 −11158.8 11035.4 6561.00 3702.19
1.9 −9.57853 81.0000 −420.252 279.985 −775.861 7953.06 8929.60 6561.00 −2681.84
1.10 −6.00717 81.0000 −475.914 2084.99 −486.580 −6970.40 5934.56 6561.00 −12524.9
1.11 −1.53527 81.0000 −509.643 521.717 −124.357 11357.2 1568.49 6561.00 −800.975
1.12 1.23437 81.0000 −510.476 −1722.16 99.9839 4577.43 −1262.11 6561.00 −2125.78
1.13 10.2821 81.0000 −406.278 1021.11 832.853 −5408.24 −9441.86 6561.00 10499.2
1.14 14.9387 81.0000 −288.835 −659.121 1210.04 −8231.50 −11963.4 6561.00 −9846.43
1.15 15.4806 81.0000 −272.350 2705.59 1253.93 6167.99 −12142.2 6561.00 41884.2
1.16 22.7532 81.0000 5.71029 −1397.96 1843.01 −3353.38 −11519.7 6561.00 −31808.2
1.17 26.8820 81.0000 210.640 1004.40 2177.44 2540.49 −8101.16 6561.00 27000.3
1.18 35.4666 81.0000 745.883 1622.90 2872.80 5254.11 8295.05 6561.00 57558.7
1.19 37.7126 81.0000 910.242 −2684.70 3054.72 8718.73 15018.8 6561.00 −101247.
1.20 40.5682 81.0000 1133.78 −281.912 3286.03 −11944.6 25224.6 6561.00 −11436.7
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.22
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(59\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 177.10.a.d 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.10.a.d 22 1.a even 1 1 trivial