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Properties

Label 177.14.a.d
Level $177$
Weight $14$
Character orbit 177.a
Self dual yes
Analytic conductor $189.799$
Analytic rank $0$
Dimension $32$
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: \(32\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 12 q^{2} - 23328 q^{3} + 139174 q^{4} + 2236 q^{5} - 8748 q^{6} + 746845 q^{7} - 733317 q^{8} + 17006112 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 12 q^{2} - 23328 q^{3} + 139174 q^{4} + 2236 q^{5} - 8748 q^{6} + 746845 q^{7} - 733317 q^{8} + 17006112 q^{9} + 6145337 q^{10} + 400846 q^{11} - 101457846 q^{12} + 9411686 q^{13} - 36368387 q^{14} - 1630044 q^{15} + 734877786 q^{16} + 228113833 q^{17} + 6377292 q^{18} + 524233755 q^{19} - 420745331 q^{20} - 544450005 q^{21} - 1844479318 q^{22} - 399937087 q^{23} + 534588093 q^{24} + 8617402914 q^{25} - 499433574 q^{26} - 12397455648 q^{27} + 12648993070 q^{28} - 225284149 q^{29} - 4479950673 q^{30} + 9454638761 q^{31} + 11648295118 q^{32} - 292216734 q^{33} + 39279537096 q^{34} + 17608963479 q^{35} + 73962769734 q^{36} + 37463929597 q^{37} + 65554547351 q^{38} - 6861119094 q^{39} + 144414252742 q^{40} + 22650227173 q^{41} + 26512554123 q^{42} + 96253617602 q^{43} - 132186868002 q^{44} + 1188302076 q^{45} + 327853892309 q^{46} + 239981844027 q^{47} - 535725905994 q^{48} + 286262776863 q^{49} - 671840368399 q^{50} - 166294984257 q^{51} - 952971648498 q^{52} - 47446514136 q^{53} - 4649045868 q^{54} - 474454082548 q^{55} - 1167728875984 q^{56} - 382166407395 q^{57} + 547596592762 q^{58} + 1349777076512 q^{59} + 306723346299 q^{60} + 661498471821 q^{61} + 555821093242 q^{62} + 396904053645 q^{63} + 3522679273173 q^{64} + 1269187682756 q^{65} + 1344625422822 q^{66} + 2838711491386 q^{67} + 1395029358261 q^{68} + 291554136423 q^{69} + 5677102514386 q^{70} + 1912914480734 q^{71} - 389714719797 q^{72} + 2403595726697 q^{73} - 742136417562 q^{74} - 6282086724306 q^{75} - 4020161987188 q^{76} - 4878303804101 q^{77} + 364087075446 q^{78} - 1705546365970 q^{79} - 4347383766449 q^{80} + 9037745167392 q^{81} - 6943720239935 q^{82} - 2549647313691 q^{83} - 9221115948030 q^{84} - 8455706309615 q^{85} - 33993832711012 q^{86} + 164232144621 q^{87} - 42970239360587 q^{88} - 17356719361241 q^{89} + 3265884040617 q^{90} - 30776775043291 q^{91} - 13184590997480 q^{92} - 6892431656769 q^{93} - 35604563339520 q^{94} + 219501126195 q^{95} - 8491607141022 q^{96} - 4427131429152 q^{97} - 32707332037060 q^{98} + 213025999086 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 −173.834 −729.000 22026.2 30477.2 126725. 24653.9 −2.40484e6 531441. −5.29796e6
1.2 −172.182 −729.000 21454.7 −57212.9 125521. 531130. −2.28361e6 531441. 9.85105e6
1.3 −163.348 −729.000 18490.4 −43774.8 119080. −199445. −1.68222e6 531441. 7.15050e6
1.4 −147.251 −729.000 13490.8 17719.1 107346. −561918. −780255. 531441. −2.60916e6
1.5 −140.074 −729.000 11428.7 13186.6 102114. 367255. −453372. 531441. −1.84710e6
1.6 −133.010 −729.000 9499.76 8413.32 96964.6 460911. −173946. 531441. −1.11906e6
1.7 −120.018 −729.000 6212.27 26582.2 87493.0 −250845. 237603. 531441. −3.19033e6
1.8 −110.535 −729.000 4025.95 49561.8 80579.9 306205. 460493. 531441. −5.47830e6
1.9 −106.076 −729.000 3060.11 −8049.61 77329.4 81507.3 544370. 531441. 853871.
1.10 −96.0044 −729.000 1024.84 −68768.7 69987.2 4010.99 688079. 531441. 6.60210e6
1.11 −61.5481 −729.000 −4403.84 −24160.8 44868.5 −453377. 775249. 531441. 1.48705e6
1.12 −51.5372 −729.000 −5535.92 63378.5 37570.6 469956. 707499. 531441. −3.26635e6
1.13 −36.3263 −729.000 −6872.40 50926.0 26481.8 −312829. 547233. 531441. −1.84995e6
1.14 −27.7758 −729.000 −7420.51 −62792.1 20248.5 −98510.4 433649. 531441. 1.74410e6
1.15 −22.2942 −729.000 −7694.97 −29584.7 16252.4 −211098. 354187. 531441. 659567.
1.16 −6.98718 −729.000 −8143.18 −4313.69 5093.66 −10944.0 114137. 531441. 30140.5
1.17 13.1411 −729.000 −8019.31 66164.2 −9579.89 −314365. −213035. 531441. 869472.
1.18 30.7253 −729.000 −7247.96 −61215.1 −22398.7 283335. −474397. 531441. −1.88085e6
1.19 30.9448 −729.000 −7234.42 −9684.64 −22558.7 −52292.6 −477367. 531441. −299689.
1.20 39.6367 −729.000 −6620.93 −3479.31 −28895.2 585971. −587136. 531441. −137908.
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.32
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.d ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.14.a.d ✓ 32 1.a even 1 1 trivial