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Properties

Label 619.6.a.b
Level $619$
Weight $6$
Character orbit 619.a
Self dual yes
Analytic conductor $99.278$
Analytic rank $0$
Dimension $134$
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] = [619,6,Mod(1,619)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(619, 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("619.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 619 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 619.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: \(99.2775844642\)
Analytic rank: \(0\)
Dimension: \(134\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 134 q + 39 q^{2} + 45 q^{3} + 2261 q^{4} + 617 q^{5} + 485 q^{6} + 354 q^{7} + 1887 q^{8} + 12475 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 134 q + 39 q^{2} + 45 q^{3} + 2261 q^{4} + 617 q^{5} + 485 q^{6} + 354 q^{7} + 1887 q^{8} + 12475 q^{9} + 1017 q^{10} + 3259 q^{11} + 1371 q^{12} + 2729 q^{13} + 4712 q^{14} + 2654 q^{15} + 39865 q^{16} + 6116 q^{17} + 11125 q^{18} + 5259 q^{19} + 21195 q^{20} + 24682 q^{21} + 6971 q^{22} + 13652 q^{23} + 18229 q^{24} + 94699 q^{25} + 21548 q^{26} + 17187 q^{27} + 9030 q^{28} + 84436 q^{29} + 14817 q^{30} + 27034 q^{31} + 72923 q^{32} + 23174 q^{33} + 18741 q^{34} + 36441 q^{35} + 266682 q^{36} + 60319 q^{37} + 42772 q^{38} + 65200 q^{39} + 51840 q^{40} + 143437 q^{41} + 30701 q^{42} + 45408 q^{43} + 172339 q^{44} + 126908 q^{45} + 94512 q^{46} + 36660 q^{47} + 40735 q^{48} + 380406 q^{49} + 178761 q^{50} + 72896 q^{51} + 117173 q^{52} + 167860 q^{53} + 139660 q^{54} + 59731 q^{55} + 187401 q^{56} + 145264 q^{57} + 130547 q^{58} + 117257 q^{59} + 203658 q^{60} + 123038 q^{61} + 146443 q^{62} + 94912 q^{63} + 795543 q^{64} + 424989 q^{65} + 83271 q^{66} + 28249 q^{67} + 189268 q^{68} + 274250 q^{69} + 69456 q^{70} + 465098 q^{71} + 495597 q^{72} + 218896 q^{73} + 369192 q^{74} + 175672 q^{75} + 195547 q^{76} + 396391 q^{77} + 248560 q^{78} + 180623 q^{79} + 684479 q^{80} + 1530534 q^{81} + 101928 q^{82} + 262497 q^{83} + 624408 q^{84} + 626681 q^{85} + 535924 q^{86} + 268820 q^{87} + 453406 q^{88} + 615798 q^{89} + 554061 q^{90} + 265281 q^{91} + 548490 q^{92} + 692675 q^{93} + 112342 q^{94} + 732355 q^{95} + 1001575 q^{96} + 372693 q^{97} + 578755 q^{98} + 854868 q^{99}+O(q^{100}) \) 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 −11.1595 22.9280 92.5336 27.2400 −255.865 17.5133 −675.523 282.695 −303.984
1.2 −11.0350 3.61338 89.7709 −50.2698 −39.8735 −128.729 −637.501 −229.944 554.727
1.3 −10.9823 −19.5067 88.6110 −0.425123 214.229 32.2620 −621.719 137.513 4.66883
1.4 −10.6321 17.5445 81.0415 −10.1840 −186.535 −198.901 −521.414 64.8093 108.278
1.5 −10.4263 −20.5209 76.7073 −27.4991 213.956 −208.404 −466.131 178.106 286.713
1.6 −10.4008 15.5628 76.1771 13.9434 −161.866 44.3501 −459.478 −0.800315 −145.022
1.7 −10.2876 −20.2022 73.8340 55.5662 207.831 229.608 −430.371 165.127 −571.641
1.8 −9.99882 8.97797 67.9764 −55.8283 −89.7691 175.605 −359.721 −162.396 558.217
1.9 −9.94264 −14.3022 66.8562 −102.120 142.202 58.8978 −346.563 −38.4461 1015.35
1.10 −9.94126 −28.2815 66.8286 59.3422 281.153 103.295 −346.240 556.841 −589.936
1.11 −9.74222 25.7566 62.9109 90.5569 −250.926 −204.944 −301.141 420.402 −882.226
1.12 −9.69886 7.70654 62.0679 42.8967 −74.7447 −192.982 −291.624 −183.609 −416.049
1.13 −9.49392 20.5489 58.1346 102.340 −195.090 253.234 −248.119 179.259 −971.604
1.14 −9.37153 −12.8286 55.8256 −1.47780 120.224 −33.6218 −223.282 −78.4267 13.8492
1.15 −9.04900 −27.2788 49.8845 82.8586 246.846 −161.618 −161.836 501.136 −749.787
1.16 −9.00629 −1.16850 49.1132 37.1575 10.5238 85.4338 −154.126 −241.635 −334.651
1.17 −8.88837 23.3449 47.0032 −54.4649 −207.498 −41.5856 −133.354 301.983 484.104
1.18 −8.87777 −27.2645 46.8148 −48.5781 242.048 −233.212 −131.522 500.354 431.265
1.19 −8.77645 −19.5505 45.0261 −50.2247 171.584 95.9853 −114.323 139.221 440.795
1.20 −8.73472 1.88498 44.2953 −64.5134 −16.4647 −118.955 −107.396 −239.447 563.506
See next 80 embeddings (of 134 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.134
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(619\) \(-1\)

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 619.6.a.b ✓ 134
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
619.6.a.b ✓ 134 1.a even 1 1 trivial