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Properties

Label 177.14.a.c
Level $177$
Weight $14$
Character orbit 177.a
Self dual yes
Analytic conductor $189.799$
Analytic rank $0$
Dimension $31$
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: \(0\)
Dimension: \(31\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 31 q + 310 q^{2} + 22599 q^{3} + 126886 q^{4} + 81008 q^{5} + 225990 q^{6} + 1002941 q^{7} + 4632723 q^{8} + 16474671 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 31 q + 310 q^{2} + 22599 q^{3} + 126886 q^{4} + 81008 q^{5} + 225990 q^{6} + 1002941 q^{7} + 4632723 q^{8} + 16474671 q^{9} + 4647481 q^{10} + 17937316 q^{11} + 92499894 q^{12} + 40664720 q^{13} + 139193613 q^{14} + 59054832 q^{15} + 370110498 q^{16} + 213442823 q^{17} + 164746710 q^{18} - 62592329 q^{19} + 1637085153 q^{20} + 731143989 q^{21} + 4142028314 q^{22} + 1873486387 q^{23} + 3377255067 q^{24} + 8307272395 q^{25} - 534777728 q^{26} + 12010035159 q^{27} + 766416778 q^{28} + 13765513563 q^{29} + 3388013649 q^{30} + 14274077235 q^{31} + 30574460156 q^{32} + 13076303364 q^{33} - 677551028 q^{34} + 36023610185 q^{35} + 67432422726 q^{36} - 18278838391 q^{37} - 23650502933 q^{38} + 29644580880 q^{39} + 10045447572 q^{40} + 34748006725 q^{41} + 101472143877 q^{42} + 40350158146 q^{43} + 163101196592 q^{44} + 43050972528 q^{45} + 296118466353 q^{46} + 233954631099 q^{47} + 269810553042 q^{48} + 324065402790 q^{49} - 102960745787 q^{50} + 155599817967 q^{51} + 668297695096 q^{52} + 500927963876 q^{53} + 120100351590 q^{54} + 884972340924 q^{55} + 1392234478810 q^{56} - 45629807841 q^{57} + 689262776200 q^{58} - 1307596542871 q^{59} + 1193435076537 q^{60} + 1716832157925 q^{61} + 1816094290366 q^{62} + 533003967981 q^{63} + 4381780009133 q^{64} + 1457007885906 q^{65} + 3019538640906 q^{66} + 1212131702006 q^{67} + 6552992665503 q^{68} + 1365771576123 q^{69} + 8806714081634 q^{70} + 6074000239936 q^{71} + 2462018943843 q^{72} + 3756145185973 q^{73} + 8066450143602 q^{74} + 6056001575955 q^{75} + 7913230001992 q^{76} + 6031241575915 q^{77} - 389852963712 q^{78} + 11377744190862 q^{79} + 16473302366969 q^{80} + 8755315630911 q^{81} + 10413363680159 q^{82} + 19915461517429 q^{83} + 558717831162 q^{84} + 15280981141573 q^{85} + 7573325358452 q^{86} + 10035059387427 q^{87} + 19271409121081 q^{88} + 14115863121241 q^{89} + 2469861950121 q^{90} + 18296287784699 q^{91} + 15158951168774 q^{92} + 10405802304315 q^{93} - 18637923572412 q^{94} - 2294034679397 q^{95} + 22288781453724 q^{96} + 38558536599054 q^{97} - 1998410212380 q^{98} + 9532625152356 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 −170.157 729.000 20761.4 7033.25 −124044. −166219. −2.13877e6 531441. −1.19676e6
1.2 −166.716 729.000 19602.3 43558.7 −121536. −395203. −1.90228e6 531441. −7.26193e6
1.3 −138.949 729.000 11114.9 −41433.2 −101294. −70948.7 −406128. 531441. 5.75710e6
1.4 −126.798 729.000 7885.84 57088.2 −92436.0 543807. 38821.0 531441. −7.23869e6
1.5 −121.100 729.000 6473.28 −61050.1 −88282.1 274195. 208137. 531441. 7.39319e6
1.6 −118.157 729.000 5769.07 218.169 −86136.4 217680. 286287. 531441. −25778.2
1.7 −107.990 729.000 3469.94 5614.85 −78725.0 −441395. 509938. 531441. −606350.
1.8 −102.312 729.000 2275.81 −9860.18 −74585.7 439885. 605299. 531441. 1.00882e6
1.9 −83.5640 729.000 −1209.07 55860.4 −60918.1 −515369. 785590. 531441. −4.66791e6
1.10 −60.0941 729.000 −4580.70 −50851.6 −43808.6 −259334. 767564. 531441. 3.05588e6
1.11 −59.5570 729.000 −4644.96 −45189.8 −43417.1 −134821. 764531. 531441. 2.69137e6
1.12 −55.7627 729.000 −5082.53 35389.9 −40651.0 112882. 740223. 531441. −1.97344e6
1.13 −50.5493 729.000 −5636.77 37474.1 −36850.4 131476. 699034. 531441. −1.89429e6
1.14 −7.03559 729.000 −8142.50 12466.0 −5128.95 −179621. 114923. 531441. −87705.8
1.15 −1.12446 729.000 −8190.74 −53521.6 −819.732 445275. 18421.7 531441. 60183.0
1.16 15.7636 729.000 −7943.51 −18232.0 11491.7 −139545. −254354. 531441. −287402.
1.17 28.6118 729.000 −7373.36 60045.3 20858.0 428045. −445354. 531441. 1.71801e6
1.18 28.6646 729.000 −7370.34 −9309.96 20896.5 216532. −446089. 531441. −266867.
1.19 66.7973 729.000 −3730.11 36075.7 48695.3 −269744. −796366. 531441. 2.40976e6
1.20 68.2207 729.000 −3537.94 −14307.6 49732.9 −16077.5 −800224. 531441. −976072.
See all 31 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.31
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.c ✓ 31
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.14.a.c ✓ 31 1.a even 1 1 trivial