↑

Properties

Label 334.8.a.b
Level $334$
Weight $8$
Character orbit 334.a
Self dual yes
Analytic conductor $104.337$
Analytic rank $1$
Dimension $23$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [334,8,Mod(1,334)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(334, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("334.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 334 = 2 \cdot 167 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 334.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: \(104.336598512\)
Analytic rank: \(1\)
Dimension: \(23\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 23 q + 184 q^{2} - 79 q^{3} + 1472 q^{4} - 750 q^{5} - 632 q^{6} - 3252 q^{7} + 11776 q^{8} + 11584 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 23 q + 184 q^{2} - 79 q^{3} + 1472 q^{4} - 750 q^{5} - 632 q^{6} - 3252 q^{7} + 11776 q^{8} + 11584 q^{9} - 6000 q^{10} - 15833 q^{11} - 5056 q^{12} - 19564 q^{13} - 26016 q^{14} - 30653 q^{15} + 94208 q^{16} - 29529 q^{17} + 92672 q^{18} - 65342 q^{19} - 48000 q^{20} - 170781 q^{21} - 126664 q^{22} - 196630 q^{23} - 40448 q^{24} + 355269 q^{25} - 156512 q^{26} - 217912 q^{27} - 208128 q^{28} - 603408 q^{29} - 245224 q^{30} - 506696 q^{31} + 753664 q^{32} - 664676 q^{33} - 236232 q^{34} - 1218632 q^{35} + 741376 q^{36} - 939879 q^{37} - 522736 q^{38} - 760171 q^{39} - 384000 q^{40} - 1376362 q^{41} - 1366248 q^{42} - 182469 q^{43} - 1013312 q^{44} + 243677 q^{45} - 1573040 q^{46} - 933172 q^{47} - 323584 q^{48} + 2414647 q^{49} + 2842152 q^{50} - 3857806 q^{51} - 1252096 q^{52} - 4990399 q^{53} - 1743296 q^{54} - 4670505 q^{55} - 1665024 q^{56} - 5592433 q^{57} - 4827264 q^{58} - 7371158 q^{59} - 1961792 q^{60} - 5460761 q^{61} - 4053568 q^{62} - 15065741 q^{63} + 6029312 q^{64} - 19808210 q^{65} - 5317408 q^{66} - 10645093 q^{67} - 1889856 q^{68} - 16063557 q^{69} - 9749056 q^{70} - 22326906 q^{71} + 5931008 q^{72} - 17047847 q^{73} - 7519032 q^{74} - 32862896 q^{75} - 4181888 q^{76} - 16200203 q^{77} - 6081368 q^{78} - 29584376 q^{79} - 3072000 q^{80} - 26810129 q^{81} - 11010896 q^{82} - 28183333 q^{83} - 10929984 q^{84} + 4585711 q^{85} - 1459752 q^{86} + 19585005 q^{87} - 8106496 q^{88} - 4006456 q^{89} + 1949416 q^{90} - 1377562 q^{91} - 12584320 q^{92} - 12636283 q^{93} - 7465376 q^{94} - 41916902 q^{95} - 2588672 q^{96} + 5741246 q^{97} + 19317176 q^{98} - 14983720 q^{99}+O(q^{100}) \) 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 8.00000 −81.5520 64.0000 252.399 −652.416 −1487.32 512.000 4463.73 2019.19
1.2 8.00000 −81.0086 64.0000 517.338 −648.069 454.770 512.000 4375.40 4138.70
1.3 8.00000 −65.7400 64.0000 111.201 −525.920 908.591 512.000 2134.75 889.611
1.4 8.00000 −65.3361 64.0000 −41.7637 −522.689 −577.635 512.000 2081.81 −334.110
1.5 8.00000 −64.6822 64.0000 −472.397 −517.458 −1512.80 512.000 1996.79 −3779.18
1.6 8.00000 −52.5954 64.0000 −528.651 −420.763 1692.09 512.000 579.277 −4229.21
1.7 8.00000 −47.7286 64.0000 −405.581 −381.828 −214.943 512.000 91.0156 −3244.65
1.8 8.00000 −35.8467 64.0000 −356.426 −286.774 263.967 512.000 −902.014 −2851.41
1.9 8.00000 −30.1715 64.0000 96.4413 −241.372 1414.02 512.000 −1276.68 771.530
1.10 8.00000 −22.7247 64.0000 316.685 −181.798 386.924 512.000 −1670.59 2533.48
1.11 8.00000 −19.5844 64.0000 217.801 −156.675 −1113.83 512.000 −1803.45 1742.40
1.12 8.00000 −13.6088 64.0000 260.670 −108.870 −513.572 512.000 −2001.80 2085.36
1.13 8.00000 10.8053 64.0000 −248.415 86.4425 −358.451 512.000 −2070.25 −1987.32
1.14 8.00000 22.7002 64.0000 212.591 181.601 308.382 512.000 −1671.70 1700.73
1.15 8.00000 27.7944 64.0000 −334.367 222.355 1446.65 512.000 −1414.47 −2674.94
1.16 8.00000 32.3826 64.0000 −9.69015 259.061 140.569 512.000 −1138.37 −77.5212
1.17 8.00000 36.2644 64.0000 243.181 290.115 −795.491 512.000 −871.896 1945.45
1.18 8.00000 50.5268 64.0000 −399.615 404.215 −760.344 512.000 365.960 −3196.92
1.19 8.00000 51.3912 64.0000 457.453 411.129 −1590.02 512.000 454.051 3659.63
1.20 8.00000 55.6030 64.0000 −256.389 444.824 867.115 512.000 904.695 −2051.11
See all 23 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.23
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(167\) \( +1 \)

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 334.8.a.b ✓ 23
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
334.8.a.b ✓ 23 1.a even 1 1 trivial