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Properties

Label 619.8.a.a
Level $619$
Weight $8$
Character orbit 619.a
Self dual yes
Analytic conductor $193.366$
Analytic rank $1$
Dimension $175$
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] = [619,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("619.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 619 \)
Weight: \( k \) \(=\) \( 8 \)
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: \(193.366330775\)
Analytic rank: \(1\)
Dimension: \(175\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 175 q - 81 q^{2} - 135 q^{3} + 10597 q^{4} - 2931 q^{5} - 2431 q^{6} - 2070 q^{7} - 15489 q^{8} + 110938 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 175 q - 81 q^{2} - 135 q^{3} + 10597 q^{4} - 2931 q^{5} - 2431 q^{6} - 2070 q^{7} - 15489 q^{8} + 110938 q^{9} - 6879 q^{10} - 37705 q^{11} - 24261 q^{12} - 41051 q^{13} - 77492 q^{14} - 59530 q^{15} + 593369 q^{16} - 75866 q^{17} - 227663 q^{18} - 109521 q^{19} - 403093 q^{20} - 527162 q^{21} - 156233 q^{22} - 330356 q^{23} - 319043 q^{24} + 2312588 q^{25} - 396008 q^{26} - 346665 q^{27} - 366666 q^{28} - 2320794 q^{29} - 620967 q^{30} - 690262 q^{31} - 2328413 q^{32} - 885454 q^{33} - 665067 q^{34} - 925587 q^{35} + 4396050 q^{36} - 1733161 q^{37} - 1769560 q^{38} - 2466296 q^{39} - 1936440 q^{40} - 5786251 q^{41} - 1151083 q^{42} - 1068240 q^{43} - 6208541 q^{44} - 5727082 q^{45} - 4716468 q^{46} - 2829836 q^{47} - 2137145 q^{48} + 16156983 q^{49} - 8084711 q^{50} - 4718920 q^{51} - 7968683 q^{52} - 8853746 q^{53} - 4483676 q^{54} - 1432741 q^{55} - 12992095 q^{56} - 9167456 q^{57} - 4782617 q^{58} - 11161835 q^{59} - 16715022 q^{60} - 10604588 q^{61} - 9544457 q^{62} - 8372648 q^{63} + 29332791 q^{64} - 22260761 q^{65} - 7325049 q^{66} - 4800931 q^{67} - 9504388 q^{68} - 20864842 q^{69} + 119328 q^{70} - 36689358 q^{71} - 41793123 q^{72} - 14163790 q^{73} - 34955672 q^{74} - 24913232 q^{75} - 11514613 q^{76} - 38389075 q^{77} - 22527104 q^{78} - 19342745 q^{79} - 52753929 q^{80} + 37999239 q^{81} + 2608836 q^{82} - 27045851 q^{83} - 59014152 q^{84} - 50380409 q^{85} - 49157576 q^{86} - 26354980 q^{87} - 44740546 q^{88} - 60808268 q^{89} - 42098403 q^{90} - 25795083 q^{91} - 56656318 q^{92} - 58204591 q^{93} - 48679042 q^{94} - 74484589 q^{95} - 49787897 q^{96} - 27393867 q^{97} - 60608337 q^{98} - 95326620 q^{99}+O(q^{100}) \) 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 −22.4276 74.2749 374.997 −216.567 −1665.81 −1063.11 −5539.54 3329.76 4857.08
1.2 −22.2315 20.2159 366.238 −411.565 −449.430 −470.991 −5296.39 −1778.32 9149.70
1.3 −22.1191 −65.3168 361.254 −178.696 1444.75 1247.01 −5159.36 2079.29 3952.58
1.4 −21.9189 −21.2623 352.436 −369.904 466.046 965.108 −4919.38 −1734.91 8107.88
1.5 −21.7996 −67.4715 347.222 226.004 1470.85 538.934 −4778.95 2365.40 −4926.79
1.6 −20.8728 43.9033 307.675 161.006 −916.387 689.323 −3750.34 −259.497 −3360.65
1.7 −20.7393 −52.0857 302.118 506.704 1080.22 539.971 −3611.09 525.918 −10508.7
1.8 −20.6954 81.5875 300.299 382.502 −1688.48 −989.428 −3565.79 4469.52 −7916.03
1.9 −20.5973 −58.3221 296.248 230.191 1201.28 −757.483 −3465.45 1214.47 −4741.31
1.10 −20.4533 56.0549 290.337 −206.160 −1146.51 1257.43 −3320.34 955.157 4216.64
1.11 −20.2159 52.2408 280.681 −334.875 −1056.09 1499.80 −3086.58 542.099 6769.78
1.12 −20.1936 73.0405 279.781 43.9142 −1474.95 −445.100 −3065.01 3147.91 −886.786
1.13 −20.1632 42.3858 278.553 −513.639 −854.633 −1254.95 −3035.63 −390.440 10356.6
1.14 −20.1418 6.25775 277.692 331.554 −126.042 1355.02 −3015.06 −2147.84 −6678.09
1.15 −20.1409 42.7510 277.658 47.2118 −861.045 −664.952 −3014.25 −359.355 −950.890
1.16 −20.0616 −67.4825 274.467 −518.870 1353.81 413.041 −2938.35 2366.89 10409.4
1.17 −19.8196 −30.3408 264.815 49.9653 601.341 −306.396 −2711.61 −1266.44 −990.290
1.18 −19.2445 −44.7695 242.352 −270.856 861.567 −940.858 −2200.64 −182.696 5212.50
1.19 −19.2194 −44.1801 241.385 489.573 849.114 966.752 −2179.18 −235.120 −9409.28
1.20 −19.1136 −77.4908 237.330 −128.348 1481.13 −450.667 −2089.68 3817.83 2453.19
See next 80 embeddings (of 175 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.175
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 619.8.a.a ✓ 175
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
619.8.a.a ✓ 175 1.a even 1 1 trivial