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Properties

Label 61.6.c.a
Level $61$
Weight $6$
Character orbit 61.c
Analytic conductor $9.783$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [61,6,Mod(13,61)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(61, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("61.13");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 61 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 61.c (of order \(3\), degree \(2\), minimal)

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: no
Analytic conductor: \(9.78341300859\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 5 q^{2} - 22 q^{3} - 317 q^{4} + 46 q^{5} - 173 q^{6} - 113 q^{7} + 444 q^{8} + 2694 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 5 q^{2} - 22 q^{3} - 317 q^{4} + 46 q^{5} - 173 q^{6} - 113 q^{7} + 444 q^{8} + 2694 q^{9} - 469 q^{10} - 162 q^{11} - 41 q^{12} + 109 q^{13} + 1184 q^{14} + 1144 q^{15} - 737 q^{16} + 2178 q^{17} + 1245 q^{18} + 3009 q^{19} - 1444 q^{20} - 4278 q^{21} - 53 q^{22} - 9848 q^{23} + 946 q^{24} - 8188 q^{25} + 11021 q^{26} + 5138 q^{27} + 10296 q^{28} + 4340 q^{29} - 15068 q^{30} + 1139 q^{31} - 13339 q^{32} - 2682 q^{33} + 17772 q^{34} + 18493 q^{35} - 12406 q^{36} + 23400 q^{37} - 33528 q^{38} + 8888 q^{39} - 19134 q^{40} - 30096 q^{41} + 25871 q^{42} + 18162 q^{43} - 8774 q^{44} + 45303 q^{45} - 7633 q^{46} - 1732 q^{47} - 7606 q^{48} - 9825 q^{49} - 56554 q^{50} - 8528 q^{51} + 10688 q^{52} + 45878 q^{53} - 18644 q^{54} - 5276 q^{55} + 47642 q^{56} - 53016 q^{57} - 174620 q^{58} - 40359 q^{59} - 109996 q^{60} + 36316 q^{61} + 121834 q^{62} - 60138 q^{63} - 2688 q^{64} + 90910 q^{65} - 115368 q^{66} + 101367 q^{67} - 33257 q^{68} - 107882 q^{69} + 266988 q^{70} - 47411 q^{71} + 112468 q^{72} + 43384 q^{73} + 20327 q^{74} - 126563 q^{75} + 77509 q^{76} + 71493 q^{77} + 230993 q^{78} + 31548 q^{79} + 42895 q^{80} - 166688 q^{81} + 28357 q^{82} + 113600 q^{83} + 55246 q^{84} + 144188 q^{85} + 93236 q^{86} + 29347 q^{87} - 190380 q^{88} - 108604 q^{89} - 94928 q^{90} + 168112 q^{91} + 286627 q^{92} - 154496 q^{93} + 453614 q^{94} - 197954 q^{95} - 206026 q^{96} - 192309 q^{97} - 920168 q^{98} + 211334 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −5.33226 + 9.23574i −8.81130 −40.8660 − 70.7819i 20.9793 − 36.3372i 46.9841 − 81.3789i −30.3495 + 52.5668i 530.367 −165.361 223.734 + 387.519i
13.2 −5.09491 + 8.82465i 12.3904 −35.9163 − 62.2088i −41.8777 + 72.5343i −63.1279 + 109.341i 32.5184 − 56.3234i 405.887 −89.4787 −426.727 − 739.112i
13.3 −4.44481 + 7.69864i 27.5113 −23.5127 − 40.7252i 23.4977 − 40.6993i −122.283 + 211.800i 9.29538 − 16.1001i 133.570 513.872 208.886 + 361.801i
13.4 −4.24051 + 7.34478i −25.8010 −19.9639 − 34.5784i 1.83935 − 3.18585i 109.409 − 189.503i 63.4052 − 109.821i 67.2354 422.692 15.5996 + 27.0193i
13.5 −3.81838 + 6.61363i 9.92500 −13.1601 − 22.7940i 9.19598 − 15.9279i −37.8975 + 65.6403i −9.09220 + 15.7482i −43.3751 −144.494 70.2276 + 121.638i
13.6 −3.57469 + 6.19155i −15.6957 −9.55685 − 16.5530i −40.3229 + 69.8413i 56.1072 − 97.1806i −89.8442 + 155.615i −92.1291 3.35454 −288.284 − 499.323i
13.7 −2.75776 + 4.77659i −3.67197 0.789479 + 1.36742i 5.34428 − 9.25656i 10.1264 − 17.5395i 116.888 − 202.457i −185.206 −229.517 29.4765 + 51.0548i
13.8 −2.68007 + 4.64202i −2.57412 1.63445 + 2.83095i 42.4565 − 73.5368i 6.89883 − 11.9491i −75.5297 + 130.821i −189.046 −236.374 227.573 + 394.168i
13.9 −2.00978 + 3.48104i 13.7991 7.92157 + 13.7206i −38.5968 + 66.8515i −27.7331 + 48.0351i 22.5401 − 39.0406i −192.308 −52.5856 −155.142 − 268.714i
13.10 −1.43577 + 2.48682i 23.6988 11.8772 + 20.5718i −11.7606 + 20.3699i −34.0259 + 58.9345i −98.1579 + 170.015i −160.100 318.631 −33.7709 − 58.4929i
13.11 −0.915549 + 1.58578i −22.4767 14.3235 + 24.8091i 41.0158 − 71.0415i 20.5785 − 35.6431i 4.36744 − 7.56462i −111.051 262.203 75.1041 + 130.084i
13.12 −0.456419 + 0.790540i −21.1511 15.5834 + 26.9912i −19.6304 + 34.0009i 9.65376 − 16.7208i −10.0006 + 17.3216i −57.6610 204.369 −17.9194 − 31.0373i
13.13 −0.111858 + 0.193743i 20.3127 15.9750 + 27.6695i 36.2457 − 62.7793i −2.27214 + 3.93546i 71.0282 − 123.025i −14.3066 169.608 8.10872 + 14.0447i
13.14 0.438897 − 0.760192i −2.70161 15.6147 + 27.0455i −29.2820 + 50.7179i −1.18573 + 2.05374i 58.3441 − 101.055i 55.5025 −235.701 25.7036 + 44.5199i
13.15 1.00054 − 1.73298i 1.32608 13.9979 + 24.2450i 4.26814 − 7.39264i 1.32680 − 2.29808i −86.4714 + 149.773i 120.056 −241.242 −8.54086 − 14.7932i
13.16 2.46221 − 4.26468i 1.37515 3.87500 + 6.71170i 23.1832 − 40.1545i 3.38591 − 5.86456i 20.4171 − 35.3635i 195.746 −241.109 −114.164 − 197.738i
13.17 2.52581 − 4.37484i 24.4261 3.24053 + 5.61276i −26.0235 + 45.0739i 61.6958 − 106.860i 21.3398 − 36.9617i 194.392 353.634 131.461 + 227.697i
13.18 2.93441 − 5.08255i −28.1052 −1.22152 − 2.11573i 7.76825 − 13.4550i −82.4723 + 142.846i −70.1442 + 121.493i 173.464 546.904 −45.5904 − 78.9650i
13.19 3.22017 − 5.57751i −20.7324 −4.73905 − 8.20827i −41.0854 + 71.1620i −66.7620 + 115.635i 103.137 − 178.638i 145.049 186.833 264.604 + 458.308i
13.20 3.35472 − 5.81055i −12.5226 −6.50835 − 11.2728i 37.6745 − 65.2541i −42.0097 + 72.7630i 54.5996 − 94.5693i 127.367 −86.1855 −252.775 − 437.819i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.24
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 61.6.c.a ✓ 48
61.c even 3 1 inner 61.6.c.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
61.6.c.a ✓ 48 1.a even 1 1 trivial
61.6.c.a ✓ 48 61.c even 3 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(61, [\chi])\).