Properties

Label 177.12.a.a
Level $177$
Weight $12$
Character orbit 177.a
Self dual yes
Analytic conductor $135.997$
Analytic rank $1$
Dimension $26$
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: \(26\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 26 q - 78 q^{2} - 6318 q^{3} + 23070 q^{4} + 3808 q^{5} + 18954 q^{6} - 98819 q^{7} - 117645 q^{8} + 1535274 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 26 q - 78 q^{2} - 6318 q^{3} + 23070 q^{4} + 3808 q^{5} + 18954 q^{6} - 98819 q^{7} - 117645 q^{8} + 1535274 q^{9} - 859751 q^{10} + 579094 q^{11} - 5606010 q^{12} - 2018538 q^{13} + 4157413 q^{14} - 925344 q^{15} + 20190274 q^{16} - 13084493 q^{17} - 4605822 q^{18} + 9917231 q^{19} + 10165633 q^{20} + 24013017 q^{21} - 89820518 q^{22} - 63513223 q^{23} + 28587735 q^{24} + 218986852 q^{25} - 77999532 q^{26} - 373071582 q^{27} - 444601862 q^{28} + 81530981 q^{29} + 208919493 q^{30} - 408861231 q^{31} - 26253128 q^{32} - 140719842 q^{33} - 508910076 q^{34} - 75731421 q^{35} + 1362260430 q^{36} - 802381301 q^{37} + 732704675 q^{38} + 490504734 q^{39} - 646130800 q^{40} - 1354472849 q^{41} - 1010251359 q^{42} + 282952194 q^{43} + 1846047996 q^{44} + 224858592 q^{45} + 9629305849 q^{46} - 1196794197 q^{47} - 4906236582 q^{48} + 10889725683 q^{49} - 6236232091 q^{50} + 3179531799 q^{51} - 1968200812 q^{52} - 8276044236 q^{53} + 1119214746 q^{54} - 6672895076 q^{55} + 2579741342 q^{56} - 2409887133 q^{57} - 9401656060 q^{58} + 18588031774 q^{59} - 2470248819 q^{60} - 21181559029 q^{61} - 6117706514 q^{62} - 5835163131 q^{63} + 42975855037 q^{64} + 25680681860 q^{65} + 21826385874 q^{66} + 26234163394 q^{67} + 19707344091 q^{68} + 15433713189 q^{69} + 129203099090 q^{70} + 52088830406 q^{71} - 6946819605 q^{72} + 20943384867 q^{73} + 41969200146 q^{74} - 53213805036 q^{75} + 223987219368 q^{76} + 94604773153 q^{77} + 18953886276 q^{78} + 68965662774 q^{79} + 218947784293 q^{80} + 90656394426 q^{81} + 11938614923 q^{82} + 17947446393 q^{83} + 108038252466 q^{84} - 52849386709 q^{85} + 384986147852 q^{86} - 19812028383 q^{87} - 49061112607 q^{88} + 38570593981 q^{89} - 50767436799 q^{90} - 226268806999 q^{91} - 79559686310 q^{92} + 99353279133 q^{93} - 16709400108 q^{94} - 252795831501 q^{95} + 6379510104 q^{96} - 186894587836 q^{97} - 252443311612 q^{98} + 34194921606 q^{99}+O(q^{100}) \) 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 −87.0020 −243.000 5521.36 −497.667 21141.5 10880.7 −302189. 59049.0 43298.0
1.2 −86.0142 −243.000 5350.44 12869.1 20901.4 −72810.3 −284057. 59049.0 −1.10693e6
1.3 −70.0902 −243.000 2864.64 3425.74 17031.9 −19416.7 −57238.6 59049.0 −240111.
1.4 −68.1728 −243.000 2599.53 −12352.0 16566.0 47034.4 −37599.4 59049.0 842068.
1.5 −66.9000 −243.000 2427.61 −5686.53 16256.7 26687.5 −25395.7 59049.0 380429.
1.6 −60.8905 −243.000 1659.66 −4960.99 14796.4 −87058.0 23646.5 59049.0 302077.
1.7 −49.4794 −243.000 400.209 6608.87 12023.5 −17933.1 81531.7 59049.0 −327003.
1.8 −43.6565 −243.000 −142.110 7124.43 10608.5 50312.1 95612.5 59049.0 −311028.
1.9 −42.1259 −243.000 −273.404 −6338.85 10236.6 −22948.3 97791.4 59049.0 267030.
1.10 −35.6107 −243.000 −779.875 −1849.70 8653.41 64152.0 100703. 59049.0 65869.2
1.11 −26.3367 −243.000 −1354.38 12487.3 6399.81 −48376.8 89607.3 59049.0 −328874.
1.12 −16.5456 −243.000 −1774.24 2844.36 4020.58 41460.7 63241.3 59049.0 −47061.6
1.13 −10.8994 −243.000 −1929.20 3570.42 2648.56 −66491.5 43349.2 59049.0 −38915.5
1.14 9.61747 −243.000 −1955.50 −8978.47 −2337.04 1935.43 −38503.6 59049.0 −86350.1
1.15 10.8639 −243.000 −1929.97 8237.50 −2639.94 85962.3 −43216.5 59049.0 89491.8
1.16 16.4601 −243.000 −1777.06 −11170.6 −3999.81 −64730.0 −62961.0 59049.0 −183870.
1.17 20.9819 −243.000 −1607.76 5799.05 −5098.61 22173.1 −76704.8 59049.0 121675.
1.18 27.2209 −243.000 −1307.02 −8750.00 −6614.68 −68225.0 −91326.7 59049.0 −238183.
1.19 38.1800 −243.000 −590.285 118.373 −9277.75 −24998.6 −100730. 59049.0 4519.50
1.20 40.1266 −243.000 −437.857 10858.4 −9750.76 −2158.10 −99749.0 59049.0 435712.
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.26
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.a 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.12.a.a 26 1.a even 1 1 trivial