↑

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 $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [177,10,Mod(1,177)]
 
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([0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("177.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

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: 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.

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