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Properties

Label 61.6.e.a
Level $61$
Weight $6$
Character orbit 61.e
Analytic conductor $9.783$
Analytic rank $0$
Dimension $100$
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(9,61)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(61, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("61.9");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 61 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 61.e (of order \(5\), degree \(4\), 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: \(100\)
Relative dimension: \(25\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 100 q - 7 q^{2} + 15 q^{3} - 383 q^{4} + 93 q^{5} + 251 q^{6} - 262 q^{7} - 131 q^{8} - 1296 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 100 q - 7 q^{2} + 15 q^{3} - 383 q^{4} + 93 q^{5} + 251 q^{6} - 262 q^{7} - 131 q^{8} - 1296 q^{9} + 335 q^{10} - 166 q^{11} - 1016 q^{12} - 2022 q^{13} - 3994 q^{14} + 651 q^{15} - 8631 q^{16} + 168 q^{17} + 8350 q^{18} + 8479 q^{19} + 2237 q^{20} + 4748 q^{21} + 8633 q^{22} - 3127 q^{23} + 11110 q^{24} - 3384 q^{25} - 4692 q^{26} - 3552 q^{27} - 16513 q^{28} - 12860 q^{29} + 20853 q^{30} - 2709 q^{31} + 35466 q^{32} + 17977 q^{33} - 47470 q^{34} + 23730 q^{35} - 57323 q^{36} + 31655 q^{37} - 15439 q^{38} - 4891 q^{39} - 51974 q^{40} - 10986 q^{41} - 40481 q^{42} + 15841 q^{43} + 89760 q^{44} - 17096 q^{45} + 102919 q^{46} - 136974 q^{47} - 150956 q^{48} - 88989 q^{49} + 31844 q^{50} + 40232 q^{51} + 188700 q^{52} + 14663 q^{53} + 201822 q^{54} - 75368 q^{55} + 15346 q^{56} + 149353 q^{57} + 33289 q^{58} + 30787 q^{59} + 422730 q^{60} - 170328 q^{61} - 345672 q^{62} - 4317 q^{63} - 213483 q^{64} + 134132 q^{65} - 25937 q^{66} + 142767 q^{67} - 5868 q^{68} - 11753 q^{69} + 211669 q^{70} - 75788 q^{71} + 333834 q^{72} - 152901 q^{73} - 27012 q^{74} - 455784 q^{75} - 136863 q^{76} - 443425 q^{77} - 195730 q^{78} - 142414 q^{79} + 572417 q^{80} - 257264 q^{81} + 158584 q^{82} + 85963 q^{83} - 167383 q^{84} - 264858 q^{85} + 91171 q^{86} - 41459 q^{87} - 458551 q^{88} + 138953 q^{89} - 1069308 q^{90} + 216893 q^{91} + 135185 q^{92} + 521194 q^{93} + 939883 q^{94} + 317480 q^{95} + 1127157 q^{96} - 140211 q^{97} - 201760 q^{98} - 127773 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −8.92287 − 6.48285i 21.8828 + 15.8988i 27.7018 + 85.2574i −23.0166 − 70.8378i −92.1879 − 283.725i −51.3857 + 37.3339i 196.468 − 604.665i 150.994 + 464.711i −253.856 + 781.289i
9.2 −8.39128 − 6.09662i −11.4049 − 8.28617i 23.3562 + 71.8831i −1.12864 − 3.47359i 45.1843 + 139.063i 177.668 − 129.083i 139.689 − 429.919i −13.6793 − 42.1004i −11.7064 + 36.0287i
9.3 −8.00378 − 5.81509i 4.48709 + 3.26006i 20.3567 + 62.6515i 28.2451 + 86.9294i −16.9561 − 52.1856i −82.2366 + 59.7484i 103.564 − 318.737i −65.5852 − 201.850i 279.435 − 860.012i
9.4 −7.36266 − 5.34928i −16.6835 − 12.1213i 15.7053 + 48.3361i −22.8845 − 70.4312i 57.9949 + 178.490i −182.642 + 132.697i 52.9373 − 162.924i 56.3232 + 173.345i −208.266 + 640.977i
9.5 −6.36758 − 4.62632i 2.73752 + 1.98892i 9.25471 + 28.4831i −11.8229 − 36.3871i −8.22997 − 25.3292i 1.64430 − 1.19465i −4.98872 + 15.3537i −71.5529 − 220.217i −93.0550 + 286.394i
9.6 −5.38220 − 3.91040i 10.4563 + 7.59698i 3.78833 + 11.6593i −1.82135 − 5.60554i −26.5709 − 81.7769i 16.4649 − 11.9625i −40.5833 + 124.903i −23.4701 − 72.2334i −12.1170 + 37.2924i
9.7 −5.29370 − 3.84610i −20.9327 − 15.2085i 3.34224 + 10.2864i 17.2337 + 53.0399i 52.3182 + 161.019i 35.8230 − 26.0269i −42.8349 + 131.832i 131.789 + 405.605i 112.767 − 347.060i
9.8 −5.21633 − 3.78989i 23.2165 + 16.8678i 2.95834 + 9.10483i 19.7132 + 60.6711i −57.1780 − 175.976i 166.986 − 121.323i −44.6842 + 137.524i 179.393 + 552.115i 127.106 − 391.191i
9.9 −3.14471 − 2.28477i −7.49561 − 5.44588i −5.21950 − 16.0640i 13.0467 + 40.1536i 11.1290 + 34.2514i −97.9495 + 71.1644i −58.7261 + 180.740i −48.5645 − 149.466i 50.7135 − 156.080i
9.10 −2.79679 − 2.03199i −9.15681 − 6.65281i −6.19547 − 19.0677i −28.8540 − 88.8034i 12.0913 + 37.2131i 168.944 − 122.745i −55.6029 + 171.128i −35.5039 − 109.270i −99.7489 + 306.996i
9.11 −2.04573 − 1.48631i 16.2601 + 11.8137i −7.91265 − 24.3526i −22.5484 − 69.3969i −15.7050 − 48.3351i −102.666 + 74.5914i −45.0132 + 138.536i 49.7375 + 153.076i −57.0173 + 175.481i
9.12 −0.507937 − 0.369038i 17.6949 + 12.8561i −9.76673 − 30.0589i 24.2348 + 74.5870i −4.24351 − 13.0602i −168.348 + 122.312i −12.3405 + 37.9800i 72.7391 + 223.868i 15.2157 − 46.8290i
9.13 −0.400635 − 0.291078i −3.70802 − 2.69404i −9.81276 − 30.2006i 18.8653 + 58.0613i 0.701388 + 2.15865i 93.8372 − 68.1767i −9.75632 + 30.0269i −68.5995 − 211.128i 9.34231 − 28.7527i
9.14 −0.150049 − 0.109017i −14.4194 − 10.4763i −9.87791 − 30.4011i −16.9404 − 52.1371i 1.02152 + 3.14391i −104.934 + 76.2390i −3.66610 + 11.2831i 23.0747 + 71.0168i −3.14194 + 9.66990i
9.15 1.42858 + 1.03793i 12.5330 + 9.10579i −8.92498 − 27.4683i −0.0921352 − 0.283563i 8.45336 + 26.0168i 115.145 − 83.6579i 33.2214 − 102.245i −0.929401 − 2.86040i 0.162695 − 0.500723i
9.16 1.99713 + 1.45100i −23.9803 − 17.4227i −8.00542 − 24.6382i −5.58453 − 17.1874i −22.6114 − 69.5907i 35.3404 − 25.6763i 44.1728 − 135.950i 196.413 + 604.497i 13.7859 − 42.4286i
9.17 3.97225 + 2.88601i −3.59945 − 2.61515i −2.43882 − 7.50590i −2.66509 − 8.20231i −6.75056 − 20.7761i −142.516 + 103.544i 60.5270 − 186.283i −68.9741 − 212.281i 13.0855 − 40.2731i
9.18 4.13265 + 3.00255i 7.26826 + 5.28070i −1.82501 − 5.61681i −26.2755 − 80.8677i 14.1816 + 43.6466i 17.5414 − 12.7446i 59.8357 − 184.155i −50.1493 − 154.344i 134.222 − 413.092i
9.19 4.63563 + 3.36798i −14.3242 − 10.4072i 0.257216 + 0.791631i 22.8173 + 70.2244i −31.3507 − 96.4876i 64.6677 − 46.9838i 55.1871 − 169.848i 21.7836 + 67.0431i −130.742 + 402.383i
9.20 5.73378 + 4.16584i 24.6013 + 17.8739i 5.63352 + 17.3382i −5.92740 − 18.2427i 66.5989 + 204.970i −10.7328 + 7.79785i 30.1568 − 92.8131i 210.658 + 648.337i 42.0096 − 129.292i
See all 100 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.25
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.e even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 61.6.e.a ✓ 100
61.e even 5 1 inner 61.6.e.a ✓ 100
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
61.6.e.a ✓ 100 1.a even 1 1 trivial
61.6.e.a ✓ 100 61.e even 5 1 inner

Hecke kernels

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