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Properties

Label 529.6.a.k
Level $529$
Weight $6$
Character orbit 529.a
Self dual yes
Analytic conductor $84.843$
Analytic rank $0$
Dimension $45$
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] = [529,6,Mod(1,529)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(529, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("529.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 529 = 23^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 529.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: \(84.8430406811\)
Analytic rank: \(0\)
Dimension: \(45\)
Twist minimal: no (minimal twist has level 23)
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.

\(\operatorname{Tr}(f)(q) = \) \( 45 q + 2 q^{3} + 640 q^{4} + 267 q^{5} - 382 q^{6} + 548 q^{7} + 1155 q^{8} + 2677 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 45 q + 2 q^{3} + 640 q^{4} + 267 q^{5} - 382 q^{6} + 548 q^{7} + 1155 q^{8} + 2677 q^{9} + 1017 q^{10} + 2701 q^{11} + 2305 q^{12} + 162 q^{13} + 3828 q^{14} + 4745 q^{15} + 9336 q^{16} + 5710 q^{17} - 1954 q^{18} + 10138 q^{19} + 10390 q^{20} + 6443 q^{21} + 5025 q^{22} - 10021 q^{24} + 15878 q^{25} - 18418 q^{26} + 8315 q^{27} + 16901 q^{28} - 13505 q^{29} + 16663 q^{30} - 7619 q^{31} + 30860 q^{32} + 29493 q^{33} + 11713 q^{34} + 10137 q^{35} - 12276 q^{36} + 32241 q^{37} + 49002 q^{38} + 56679 q^{39} + 59635 q^{40} + 34925 q^{41} + 37304 q^{42} + 23272 q^{43} + 83463 q^{44} + 48953 q^{45} - 36261 q^{47} - 32516 q^{48} + 51857 q^{49} + 17956 q^{50} + 103797 q^{51} - 120873 q^{52} + 110012 q^{53} - 93972 q^{54} + 12048 q^{55} + 250769 q^{56} + 26318 q^{57} - 24792 q^{58} + 69296 q^{59} + 200115 q^{60} + 100662 q^{61} - 109136 q^{62} + 197501 q^{63} + 169417 q^{64} + 189877 q^{65} + 177787 q^{66} + 85847 q^{67} + 309649 q^{68} - 308157 q^{70} + 194121 q^{71} + 329967 q^{72} - 106792 q^{73} + 238219 q^{74} - 99472 q^{75} + 423596 q^{76} - 127932 q^{77} - 455516 q^{78} + 390789 q^{79} + 404751 q^{80} - 135339 q^{81} + 485075 q^{82} + 421445 q^{83} + 264219 q^{84} + 258029 q^{85} + 452061 q^{86} - 100797 q^{87} + 117190 q^{88} + 544607 q^{89} + 350405 q^{90} + 315066 q^{91} - 700704 q^{93} + 986987 q^{94} + 620544 q^{95} - 774244 q^{96} + 199878 q^{97} - 353401 q^{98} + 897462 q^{99}+O(q^{100}) \) 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 −10.7675 7.33530 83.9388 −18.8852 −78.9828 −114.433 −559.251 −189.193 203.346
1.2 −10.1010 −8.27161 70.0295 −4.29871 83.5512 101.555 −384.135 −174.581 43.4211
1.3 −9.96336 7.19977 67.2686 −75.3518 −71.7339 −44.4829 −351.393 −191.163 750.757
1.4 −9.72969 25.9606 62.6669 56.1365 −252.588 −143.530 −298.379 430.950 −546.191
1.5 −9.07197 −7.87072 50.3006 82.7284 71.4029 180.228 −166.022 −181.052 −750.509
1.6 −8.59543 −11.0813 41.8814 −18.2773 95.2481 64.4318 −84.9350 −120.206 157.101
1.7 −8.32411 14.8468 37.2908 −37.1593 −123.587 −115.037 −44.0416 −22.5712 309.318
1.8 −7.95023 −23.4780 31.2061 47.2112 186.656 −41.0931 6.31178 308.217 −375.339
1.9 −7.11115 29.8757 18.5685 10.7320 −212.451 93.0172 95.5134 649.559 −76.3166
1.10 −6.96765 23.2714 16.5481 102.097 −162.147 194.136 107.663 298.560 −711.375
1.11 −6.28205 0.981810 7.46412 79.0908 −6.16777 142.297 154.136 −242.036 −496.852
1.12 −5.95909 2.06317 3.51077 −10.5948 −12.2946 −243.382 169.770 −238.743 63.1355
1.13 −5.77967 −18.9002 1.40464 −44.5562 109.237 −85.5341 176.831 114.218 257.520
1.14 −5.37229 −26.5393 −3.13849 −31.5076 142.577 −184.051 188.774 461.335 169.268
1.15 −4.39354 26.1201 −12.6968 −80.0710 −114.759 191.236 196.377 439.258 351.795
1.16 −3.25790 15.1293 −21.3861 −86.3504 −49.2898 −20.6296 173.927 −14.1044 281.321
1.17 −2.90121 −17.5696 −23.5830 61.6514 50.9730 −157.399 161.258 65.6896 −178.864
1.18 −2.63571 −29.5450 −25.0530 −86.6223 77.8721 33.8421 150.375 629.908 228.311
1.19 −2.55964 7.65268 −25.4482 −4.28580 −19.5881 −154.839 147.047 −184.436 10.9701
1.20 −2.31986 −13.1515 −26.6183 −49.4101 30.5097 199.042 135.986 −70.0368 114.624
See all 45 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.45
Significant digits:
Format:

Atkin-Lehner signs

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

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 529.6.a.k 45
23.b odd 2 1 529.6.a.j 45
23.d odd 22 2 23.6.c.a ✓ 90
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.6.c.a ✓ 90 23.d odd 22 2
529.6.a.j 45 23.b odd 2 1
529.6.a.k 45 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(529))\):

\( T_{2}^{45} - 1040 T_{2}^{43} - 385 T_{2}^{42} + 500066 T_{2}^{41} + 364660 T_{2}^{40} + \cdots - 36\!\cdots\!92 \) Copy content Toggle raw display
\( T_{5}^{45} - 267 T_{5}^{44} - 42607 T_{5}^{43} + 16617306 T_{5}^{42} + 547553285 T_{5}^{41} + \cdots - 51\!\cdots\!13 \) Copy content Toggle raw display