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Properties

Label 177.14.a.a
Level $177$
Weight $14$
Character orbit 177.a
Self dual yes
Analytic conductor $189.799$
Analytic rank $1$
Dimension $30$
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] = [177,14,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, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("177.1");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 14 \)
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: \(189.798744245\)
Analytic rank: \(1\)
Dimension: \(30\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 138 q^{2} + 21870 q^{3} + 114598 q^{4} - 137742 q^{5} - 100602 q^{6} - 879443 q^{7} - 872301 q^{8} + 15943230 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 30 q - 138 q^{2} + 21870 q^{3} + 114598 q^{4} - 137742 q^{5} - 100602 q^{6} - 879443 q^{7} - 872301 q^{8} + 15943230 q^{9} - 5352519 q^{10} - 13950782 q^{11} + 83541942 q^{12} - 17256988 q^{13} + 33780109 q^{14} - 100413918 q^{15} + 499996762 q^{16} - 317583695 q^{17} - 73338858 q^{18} - 863401469 q^{19} - 1841280623 q^{20} - 641113947 q^{21} - 2723764842 q^{22} - 3142075981 q^{23} - 635907429 q^{24} + 5435751692 q^{25} - 6441414040 q^{26} + 11622614670 q^{27} - 7538400046 q^{28} - 4604589283 q^{29} - 3901986351 q^{30} + 4308675373 q^{31} + 6094556360 q^{32} - 10170120078 q^{33} + 38097713432 q^{34} - 15447827315 q^{35} + 60902075718 q^{36} - 19633376949 q^{37} - 18152222923 q^{38} - 12580344252 q^{39} + 14680384170 q^{40} - 103644439493 q^{41} + 24625699461 q^{42} - 64494894924 q^{43} - 199714496208 q^{44} - 73201746222 q^{45} - 265425792847 q^{46} - 293365585139 q^{47} + 364497639498 q^{48} + 414396765797 q^{49} - 126058522207 q^{50} - 231518513655 q^{51} + 156029960316 q^{52} - 76747013118 q^{53} - 53464027482 q^{54} - 433465885754 q^{55} - 502955241518 q^{56} - 629419670901 q^{57} - 1755031845830 q^{58} + 1265416009230 q^{59} - 1342293574167 q^{60} - 2022612531219 q^{61} - 3816005187046 q^{62} - 467372067363 q^{63} - 3570205594131 q^{64} - 3889749040576 q^{65} - 1985624569818 q^{66} - 502618987776 q^{67} - 8953998390517 q^{68} - 2290573390149 q^{69} - 6805178272420 q^{70} - 1599540605456 q^{71} - 463576515741 q^{72} - 3826795087235 q^{73} - 7573387813210 q^{74} + 3962662983468 q^{75} - 19498723328388 q^{76} - 9088623115219 q^{77} - 4695790835160 q^{78} - 8595482172338 q^{79} - 17452527463963 q^{80} + 8472886094430 q^{81} - 11181116792901 q^{82} - 13548556984389 q^{83} - 5495493633534 q^{84} - 12851795888367 q^{85} + 8539949468848 q^{86} - 3356745587307 q^{87} - 25134826741387 q^{88} - 21826401667403 q^{89} - 2844548049879 q^{90} - 26577050621355 q^{91} - 34908210763168 q^{92} + 3141024346917 q^{93} - 26426808959500 q^{94} - 29105233533993 q^{95} + 4442931586440 q^{96} + 417815797414 q^{97} + 29159956938360 q^{98} - 7414017536862 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 −169.531 729.000 20548.9 −21413.6 −123588. 361154. −2.09488e6 531441. 3.63027e6
1.2 −158.048 729.000 16787.0 −61907.4 −115217. 140047. −1.35842e6 531441. 9.78431e6
1.3 −153.212 729.000 15282.0 58431.3 −111692. 57012.1 −1.08627e6 531441. −8.95239e6
1.4 −152.210 729.000 14976.0 −42581.4 −110961. −571520. −1.03259e6 531441. 6.48132e6
1.5 −150.711 729.000 14521.7 40403.2 −109868. −111720. −953949. 531441. −6.08919e6
1.6 −141.008 729.000 11691.3 −32545.4 −102795. −339369. −493432. 531441. 4.58917e6
1.7 −112.878 729.000 4549.48 37606.3 −82288.2 −60488.5 411161. 531441. −4.24493e6
1.8 −108.701 729.000 3623.83 −15679.8 −79242.8 −222995. 496563. 531441. 1.70441e6
1.9 −102.341 729.000 2281.63 1529.46 −74606.4 −106408. 604872. 531441. −156526.
1.10 −73.9227 729.000 −2727.43 −22721.6 −53889.7 405591. 807194. 531441. 1.67964e6
1.11 −49.4802 729.000 −5743.71 45167.9 −36071.0 116012. 689542. 531441. −2.23492e6
1.12 −44.2174 729.000 −6236.82 14897.9 −32234.5 473501. 638005. 531441. −658745.
1.13 −41.9317 729.000 −6433.74 −10428.5 −30568.2 −484911. 613281. 531441. 437283.
1.14 −28.5091 729.000 −7379.23 −13887.1 −20783.1 351614. 443921. 531441. 395908.
1.15 −17.8149 729.000 −7874.63 5823.96 −12987.1 −495710. 286225. 531441. −103753.
1.16 9.64157 729.000 −8099.04 −58650.0 7028.71 −68208.9 −157071. 531441. −565478.
1.17 20.2681 729.000 −7781.20 37235.2 14775.5 −260203. −323747. 531441. 754688.
1.18 23.4632 729.000 −7641.48 −42940.2 17104.7 465285. −371505. 531441. −1.00752e6
1.19 31.8985 729.000 −7174.48 61104.8 23254.0 −236717. −490168. 531441. 1.94915e6
1.20 43.5631 729.000 −6294.26 −53819.8 31757.5 −600256. −631066. 531441. −2.34456e6
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.30
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 177.14.a.a ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.14.a.a ✓ 30 1.a even 1 1 trivial