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Properties

Label 1334.4.a.g
Level $1334$
Weight $4$
Character orbit 1334.a
Self dual yes
Analytic conductor $78.709$
Analytic rank $0$
Dimension $21$
CM no
Inner twists $1$

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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.

\(\operatorname{Tr}(f)(q) = \) \( 21 q + 42 q^{2} + 18 q^{3} + 84 q^{4} + 25 q^{5} + 36 q^{6} + 57 q^{7} + 168 q^{8} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 21 q + 42 q^{2} + 18 q^{3} + 84 q^{4} + 25 q^{5} + 36 q^{6} + 57 q^{7} + 168 q^{8} + 243 q^{9} + 50 q^{10} + 83 q^{11} + 72 q^{12} + 30 q^{13} + 114 q^{14} + 247 q^{15} + 336 q^{16} + 238 q^{17} + 486 q^{18} + 604 q^{19} + 100 q^{20} + 279 q^{21} + 166 q^{22} - 483 q^{23} + 144 q^{24} + 444 q^{25} + 60 q^{26} + 699 q^{27} + 228 q^{28} + 609 q^{29} + 494 q^{30} + 699 q^{31} + 672 q^{32} + 252 q^{33} + 476 q^{34} + 497 q^{35} + 972 q^{36} + 281 q^{37} + 1208 q^{38} + 649 q^{39} + 200 q^{40} - 59 q^{41} + 558 q^{42} + 966 q^{43} + 332 q^{44} + 1558 q^{45} - 966 q^{46} + 730 q^{47} + 288 q^{48} + 1948 q^{49} + 888 q^{50} + 6 q^{51} + 120 q^{52} + 478 q^{53} + 1398 q^{54} + 900 q^{55} + 456 q^{56} + 1232 q^{57} + 1218 q^{58} + 1459 q^{59} + 988 q^{60} + 1884 q^{61} + 1398 q^{62} + 2525 q^{63} + 1344 q^{64} + 782 q^{65} + 504 q^{66} + 573 q^{67} + 952 q^{68} - 414 q^{69} + 994 q^{70} + 639 q^{71} + 1944 q^{72} + 1520 q^{73} + 562 q^{74} + 4067 q^{75} + 2416 q^{76} - 837 q^{77} + 1298 q^{78} + 4434 q^{79} + 400 q^{80} + 4189 q^{81} - 118 q^{82} + 5058 q^{83} + 1116 q^{84} + 1936 q^{85} + 1932 q^{86} + 522 q^{87} + 664 q^{88} + 4796 q^{89} + 3116 q^{90} + 1457 q^{91} - 1932 q^{92} + 2103 q^{93} + 1460 q^{94} - 40 q^{95} + 576 q^{96} + 4762 q^{97} + 3896 q^{98} + 4480 q^{99}+O(q^{100}) \) 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 −9.77364 4.00000 −2.47889 −19.5473 −14.0135 8.00000 68.5241 −4.95778
1.2 2.00000 −8.28995 4.00000 −9.52361 −16.5799 25.6451 8.00000 41.7233 −19.0472
1.3 2.00000 −6.91295 4.00000 4.70577 −13.8259 0.772761 8.00000 20.7889 9.41155
1.4 2.00000 −6.51579 4.00000 17.5689 −13.0316 20.5091 8.00000 15.4555 35.1379
1.5 2.00000 −5.97772 4.00000 1.05484 −11.9554 −5.45369 8.00000 8.73311 2.10968
1.6 2.00000 −4.02648 4.00000 −5.21670 −8.05297 −32.6082 8.00000 −10.7874 −10.4334
1.7 2.00000 −2.29092 4.00000 −13.7068 −4.58184 5.13272 8.00000 −21.7517 −27.4137
1.8 2.00000 −1.51318 4.00000 9.37320 −3.02635 32.3233 8.00000 −24.7103 18.7464
1.9 2.00000 −0.826545 4.00000 −15.3987 −1.65309 26.9565 8.00000 −26.3168 −30.7975
1.10 2.00000 −0.798537 4.00000 18.5826 −1.59707 −14.9180 8.00000 −26.3623 37.1653
1.11 2.00000 0.0693464 4.00000 −11.6493 0.138693 −19.8096 8.00000 −26.9952 −23.2985
1.12 2.00000 2.07134 4.00000 0.912709 4.14268 −34.3538 8.00000 −22.7095 1.82542
1.13 2.00000 2.25310 4.00000 8.64162 4.50620 8.26190 8.00000 −21.9235 17.2832
1.14 2.00000 4.00880 4.00000 −21.4952 8.01761 −10.6964 8.00000 −10.9295 −42.9903
1.15 2.00000 5.72412 4.00000 14.3181 11.4482 18.6712 8.00000 5.76552 28.6362
1.16 2.00000 7.25468 4.00000 −8.32407 14.5094 18.1196 8.00000 25.6304 −16.6481
1.17 2.00000 7.41683 4.00000 13.2636 14.8337 2.21591 8.00000 28.0094 26.5272
1.18 2.00000 7.71133 4.00000 18.2680 15.4227 17.3472 8.00000 32.4645 36.5360
1.19 2.00000 9.16623 4.00000 −10.0344 18.3325 3.71669 8.00000 57.0197 −20.0688
1.20 2.00000 9.41102 4.00000 2.29252 18.8220 35.9622 8.00000 61.5674 4.58503
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.g ✓ 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1334.4.a.g ✓ 21 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{21} - 18 T_{3}^{20} - 243 T_{3}^{19} + 5761 T_{3}^{18} + 17180 T_{3}^{17} - 746392 T_{3}^{16} + \cdots - 525632236704 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1334))\). Copy content Toggle raw display