Properties

Label 177.12.a.b
Level $177$
Weight $12$
Character orbit 177.a
Self dual yes
Analytic conductor $135.997$
Analytic rank $1$
Dimension $27$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [177,12,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, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("177.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 12 \)
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: \(135.996742959\)
Analytic rank: \(1\)
Dimension: \(27\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 27 q - 128 q^{2} + 6561 q^{3} + 26142 q^{4} - 17188 q^{5} - 31104 q^{6} - 126579 q^{7} - 355797 q^{8} + 1594323 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 27 q - 128 q^{2} + 6561 q^{3} + 26142 q^{4} - 17188 q^{5} - 31104 q^{6} - 126579 q^{7} - 355797 q^{8} + 1594323 q^{9} - 383719 q^{10} - 1816556 q^{11} + 6352506 q^{12} - 3951804 q^{13} - 6207867 q^{14} - 4176684 q^{15} + 28295194 q^{16} - 17723275 q^{17} - 7558272 q^{18} - 19573013 q^{19} - 48468099 q^{20} - 30758697 q^{21} - 1729910 q^{22} - 88593797 q^{23} - 86458671 q^{24} + 345714963 q^{25} - 6676346 q^{26} + 387420489 q^{27} + 126954286 q^{28} - 276632427 q^{29} - 93243717 q^{30} - 357680917 q^{31} - 859842334 q^{32} - 441423108 q^{33} + 232730000 q^{34} - 510315139 q^{35} + 1543658958 q^{36} - 660238257 q^{37} - 2067286961 q^{38} - 960288372 q^{39} - 3388951110 q^{40} - 1671147569 q^{41} - 1508511681 q^{42} - 1883107790 q^{43} - 3895687630 q^{44} - 1014934212 q^{45} - 1720344243 q^{46} - 5818572501 q^{47} + 6875732142 q^{48} - 18858180 q^{49} - 21474519647 q^{50} - 4306755825 q^{51} - 42214560062 q^{52} - 11444513368 q^{53} - 1836660096 q^{54} - 24401486484 q^{55} - 50583585764 q^{56} - 4756242159 q^{57} - 45017395090 q^{58} - 19302956073 q^{59} - 11777748057 q^{60} + 408637955 q^{61} - 28543084070 q^{62} - 7474363371 q^{63} + 33067284293 q^{64} - 21656714730 q^{65} - 420368130 q^{66} - 49803132690 q^{67} - 16500749319 q^{68} - 21528292671 q^{69} - 45808890782 q^{70} - 34127492216 q^{71} - 21009457053 q^{72} - 55734362153 q^{73} - 40367816298 q^{74} + 84008736009 q^{75} - 14840406404 q^{76} - 99723443615 q^{77} - 1622352078 q^{78} - 76484916442 q^{79} + 93882788915 q^{80} + 94143178827 q^{81} + 52951239205 q^{82} - 140433865655 q^{83} + 30849891498 q^{84} + 34329063335 q^{85} + 175223869508 q^{86} - 67221679761 q^{87} + 268823645069 q^{88} - 1191878597 q^{89} - 22658223231 q^{90} + 201632581559 q^{91} - 206501888812 q^{92} - 86916462831 q^{93} + 319770144384 q^{94} - 81387074885 q^{95} - 208941687162 q^{96} - 144896178730 q^{97} + 135739195260 q^{98} - 107265815244 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −88.8229 243.000 5841.51 11395.4 −21584.0 23008.9 −336951. 59049.0 −1.01218e6
1.2 −86.7335 243.000 5474.69 −11727.0 −21076.2 24236.6 −297209. 59049.0 1.01712e6
1.3 −78.8162 243.000 4163.99 6894.11 −19152.3 41129.0 −166774. 59049.0 −543367.
1.4 −71.6191 243.000 3081.30 −9710.01 −17403.4 37645.0 −74003.9 59049.0 695422.
1.5 −68.7593 243.000 2679.84 7324.59 −16708.5 −63928.8 −43445.2 59049.0 −503634.
1.6 −63.4822 243.000 1982.00 −1374.17 −15426.2 −16112.8 4190.11 59049.0 87235.4
1.7 −54.1540 243.000 884.652 −9497.19 −13159.4 −21091.6 62999.9 59049.0 514310.
1.8 −47.2991 243.000 189.204 4204.59 −11493.7 8235.98 87919.4 59049.0 −198873.
1.9 −45.9608 243.000 64.3958 −11937.5 −11168.5 −79874.8 91168.1 59049.0 548659.
1.10 −45.6492 243.000 35.8459 −2292.40 −11092.7 55379.8 91853.1 59049.0 104646.
1.11 −19.9033 243.000 −1651.86 11169.4 −4836.50 −27660.5 73639.4 59049.0 −222307.
1.12 −18.7578 243.000 −1696.14 −6326.10 −4558.15 −59668.2 70231.9 59049.0 118664.
1.13 −17.2441 243.000 −1750.64 7994.91 −4190.32 1325.62 65504.2 59049.0 −137865.
1.14 −4.91496 243.000 −2023.84 −1117.68 −1194.33 −21416.7 20012.9 59049.0 5493.36
1.15 −2.16187 243.000 −2043.33 654.400 −525.335 79533.7 8844.92 59049.0 −1414.73
1.16 10.6976 243.000 −1933.56 12608.0 2599.53 −75685.5 −42593.3 59049.0 134876.
1.17 13.5218 243.000 −1865.16 −12002.6 3285.79 −19689.8 −52912.9 59049.0 −162297.
1.18 32.2053 243.000 −1010.82 −5509.44 7825.90 −36737.1 −98510.2 59049.0 −177433.
1.19 33.0193 243.000 −957.729 −9842.24 8023.68 66020.7 −99246.9 59049.0 −324984.
1.20 35.0313 243.000 −820.805 1509.53 8512.61 260.533 −100498. 59049.0 52880.9
See all 27 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.27
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 177.12.a.b 27
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.12.a.b 27 1.a even 1 1 trivial