↑

Properties

Label 1334.4.a.f
Level $1334$
Weight $4$
Character orbit 1334.a
Self dual yes
Analytic conductor $78.709$
Analytic rank $0$
Dimension $21$
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] = [1334,4,Mod(1,1334)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1334, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1334.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1334 = 2 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1334.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: \(78.7085479477\)
Analytic rank: \(0\)
Dimension: \(21\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 21 q - 42 q^{2} - 2 q^{3} + 84 q^{4} + 25 q^{5} + 4 q^{6} + 29 q^{7} - 168 q^{8} + 211 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 21 q - 42 q^{2} - 2 q^{3} + 84 q^{4} + 25 q^{5} + 4 q^{6} + 29 q^{7} - 168 q^{8} + 211 q^{9} - 50 q^{10} + 41 q^{11} - 8 q^{12} + 154 q^{13} - 58 q^{14} + 129 q^{15} + 336 q^{16} - 28 q^{17} - 422 q^{18} - 6 q^{19} + 100 q^{20} + 55 q^{21} - 82 q^{22} + 483 q^{23} + 16 q^{24} + 652 q^{25} - 308 q^{26} - 425 q^{27} + 116 q^{28} + 609 q^{29} - 258 q^{30} + 253 q^{31} - 672 q^{32} + 288 q^{33} + 56 q^{34} + 347 q^{35} + 844 q^{36} + 577 q^{37} + 12 q^{38} - 279 q^{39} - 200 q^{40} + 425 q^{41} - 110 q^{42} + 268 q^{43} + 164 q^{44} + 64 q^{45} - 966 q^{46} - 272 q^{47} - 32 q^{48} + 2008 q^{49} - 1304 q^{50} + 302 q^{51} + 616 q^{52} + 290 q^{53} + 850 q^{54} - 136 q^{55} - 232 q^{56} + 470 q^{57} - 1218 q^{58} + 191 q^{59} + 516 q^{60} + 918 q^{61} - 506 q^{62} + 75 q^{63} + 1344 q^{64} + 262 q^{65} - 576 q^{66} + 887 q^{67} - 112 q^{68} - 46 q^{69} - 694 q^{70} - 67 q^{71} - 1688 q^{72} + 2942 q^{73} - 1154 q^{74} + 707 q^{75} - 24 q^{76} + 649 q^{77} + 558 q^{78} + 1496 q^{79} + 400 q^{80} + 221 q^{81} - 850 q^{82} + 204 q^{83} + 220 q^{84} - 376 q^{85} - 536 q^{86} - 58 q^{87} - 328 q^{88} + 1146 q^{89} - 128 q^{90} + 2137 q^{91} + 1932 q^{92} + 1999 q^{93} + 544 q^{94} + 246 q^{95} + 64 q^{96} + 1656 q^{97} - 4016 q^{98} - 1944 q^{99}+O(q^{100}) \) 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 −2.00000 −10.2601 4.00000 −14.9907 20.5203 −16.7096 −8.00000 78.2705 29.9814
1.2 −2.00000 −8.81815 4.00000 13.0821 17.6363 −17.5004 −8.00000 50.7597 −26.1642
1.3 −2.00000 −7.78430 4.00000 19.3504 15.5686 32.6658 −8.00000 33.5953 −38.7009
1.4 −2.00000 −7.46538 4.00000 −14.5114 14.9308 −0.279705 −8.00000 28.7320 29.0229
1.5 −2.00000 −6.12676 4.00000 −2.93997 12.2535 27.4874 −8.00000 10.5372 5.87994
1.6 −2.00000 −5.66640 4.00000 4.17537 11.3328 −12.8882 −8.00000 5.10813 −8.35074
1.7 −2.00000 −3.76494 4.00000 −15.6897 7.52989 17.2448 −8.00000 −12.8252 31.3794
1.8 −2.00000 −3.50405 4.00000 0.811634 7.00809 −26.7251 −8.00000 −14.7217 −1.62327
1.9 −2.00000 −2.97383 4.00000 15.7702 5.94765 12.7618 −8.00000 −18.1564 −31.5405
1.10 −2.00000 −2.73250 4.00000 1.15704 5.46499 14.5255 −8.00000 −19.5335 −2.31407
1.11 −2.00000 1.50961 4.00000 3.30145 −3.01921 −5.19395 −8.00000 −24.7211 −6.60291
1.12 −2.00000 2.12484 4.00000 6.15733 −4.24968 1.91553 −8.00000 −22.4851 −12.3147
1.13 −2.00000 3.47041 4.00000 −11.1755 −6.94082 −29.9676 −8.00000 −14.9563 22.3510
1.14 −2.00000 3.57922 4.00000 −3.08329 −7.15845 −23.9788 −8.00000 −14.1892 6.16658
1.15 −2.00000 4.24462 4.00000 −17.6491 −8.48924 25.5685 −8.00000 −8.98321 35.2982
1.16 −2.00000 4.28641 4.00000 19.5282 −8.57283 −18.6560 −8.00000 −8.62666 −39.0563
1.17 −2.00000 5.05622 4.00000 −3.66722 −10.1124 27.2686 −8.00000 −1.43466 7.33443
1.18 −2.00000 7.04145 4.00000 14.9162 −14.0829 35.8936 −8.00000 22.5821 −29.8324
1.19 −2.00000 7.61965 4.00000 19.5107 −15.2393 −19.4762 −8.00000 31.0591 −39.0213
1.20 −2.00000 8.97413 4.00000 −15.4706 −17.9483 −10.1628 −8.00000 53.5351 30.9413
See all 21 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.21
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(23\) \(-1\)
\(29\) \(-1\)

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 1334.4.a.f ✓ 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1334.4.a.f ✓ 21 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{21} + 2 T_{3}^{20} - 387 T_{3}^{19} - 599 T_{3}^{18} + 63256 T_{3}^{17} + 70264 T_{3}^{16} + \cdots - 317366433260992 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1334))\). Copy content Toggle raw display