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Properties

Label 61.6.f.a
Level $61$
Weight $6$
Character orbit 61.f
Analytic conductor $9.783$
Analytic rank $0$
Dimension $50$
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(14,61)]
 
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([5]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("61.14");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 61 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 61.f (of order \(6\), 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: \(50\)
Relative dimension: \(25\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 50 q - 3 q^{2} - 22 q^{3} + 403 q^{4} + 11 q^{5} + 21 q^{6} + 207 q^{7} + 4264 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 50 q - 3 q^{2} - 22 q^{3} + 403 q^{4} + 11 q^{5} + 21 q^{6} + 207 q^{7} + 4264 q^{9} + 1353 q^{10} - 447 q^{12} + 255 q^{13} + 1232 q^{14} - 656 q^{15} - 3745 q^{16} + 2196 q^{17} - 10251 q^{18} - 123 q^{19} + 4092 q^{20} + 3774 q^{21} + 51 q^{22} - 11708 q^{25} - 3219 q^{26} - 10966 q^{27} + 3525 q^{29} - 6546 q^{30} + 23451 q^{31} - 14901 q^{32} + 40380 q^{34} - 6939 q^{35} + 9470 q^{36} + 10246 q^{39} + 60732 q^{40} + 11506 q^{41} - 3179 q^{42} + 8334 q^{43} - 61800 q^{44} - 22456 q^{45} - 25133 q^{46} - 35556 q^{47} + 20110 q^{48} + 60692 q^{49} + 96396 q^{51} - 108516 q^{52} - 44076 q^{54} - 121236 q^{55} - 26486 q^{56} - 135704 q^{57} + 81740 q^{58} - 53313 q^{59} + 108368 q^{60} - 25195 q^{61} - 111022 q^{62} - 124398 q^{63} + 148432 q^{64} + 121464 q^{65} - 19298 q^{66} - 198237 q^{67} - 300081 q^{68} + 294224 q^{70} + 44181 q^{71} + 51555 q^{73} - 63547 q^{74} + 150149 q^{75} - 11447 q^{76} - 199859 q^{77} - 504507 q^{78} + 388956 q^{79} - 171475 q^{80} + 900394 q^{81} + 562101 q^{82} + 74128 q^{83} - 34924 q^{86} - 84471 q^{87} + 454960 q^{88} + 33756 q^{90} + 20244 q^{91} + 430305 q^{92} - 295140 q^{93} + 657034 q^{95} - 219156 q^{96} + 478832 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1 −8.82463 + 5.09490i 7.45682 35.9160 − 62.2084i −44.2216 − 76.5941i −65.8036 + 37.9917i 52.4002 − 30.2533i 405.881i −187.396 780.479 + 450.610i
14.2 −8.66618 + 5.00342i −4.30325 34.0684 − 59.0082i 29.5085 + 51.1102i 37.2927 − 21.5310i −79.2178 + 45.7364i 361.615i −224.482 −511.452 − 295.287i
14.3 −8.53217 + 4.92605i −29.1038 32.5319 − 56.3470i −22.4307 − 38.8511i 248.319 − 143.367i −49.6296 + 28.6537i 325.749i 604.033 382.765 + 220.990i
14.4 −7.84297 + 4.52814i 17.1199 25.0081 − 43.3154i 19.5582 + 33.8758i −134.271 + 77.5214i 156.523 − 90.3684i 163.161i 50.0917 −306.789 − 177.125i
14.5 −7.27118 + 4.19802i 29.5204 19.2467 − 33.3363i 3.62206 + 6.27359i −214.648 + 123.927i −183.284 + 105.819i 54.5194i 628.451 −52.6733 − 30.4110i
14.6 −6.13315 + 3.54098i −20.4110 9.07704 − 15.7219i 33.6596 + 58.3001i 125.184 − 72.2750i 182.780 − 105.528i − 98.0562i 173.610 −412.879 − 238.376i
14.7 −5.30528 + 3.06301i −9.50617 2.76401 − 4.78741i −9.04682 − 15.6696i 50.4329 − 29.1175i −139.867 + 80.7525i − 162.168i −152.633 95.9919 + 55.4210i
14.8 −4.85971 + 2.80576i −12.6609 −0.255465 + 0.442478i −31.8526 − 55.1703i 61.5281 − 35.5233i 102.473 − 59.1627i − 182.435i −82.7026 309.589 + 178.741i
14.9 −4.17180 + 2.40859i 11.7686 −4.39740 + 7.61653i −15.4722 − 26.7986i −49.0963 + 28.3458i −6.23439 + 3.59943i − 196.516i −104.499 129.094 + 74.5322i
14.10 −2.56603 + 1.48150i 13.1716 −11.6103 + 20.1097i 44.2550 + 76.6519i −33.7986 + 19.5136i −9.52914 + 5.50165i − 163.618i −69.5101 −227.119 − 131.127i
14.11 −1.65849 + 0.957530i −25.2946 −14.1663 + 24.5367i 37.7930 + 65.4593i 41.9509 − 24.2203i −194.301 + 112.180i − 115.540i 396.817 −125.358 − 72.3758i
14.12 −1.33428 + 0.770348i 28.3419 −14.8131 + 25.6571i −34.6172 − 59.9588i −37.8161 + 21.8331i 169.921 − 98.1041i − 94.9473i 560.265 92.3783 + 53.3346i
14.13 0.532088 − 0.307201i −12.3461 −15.8113 + 27.3859i 11.6874 + 20.2431i −6.56920 + 3.79273i 70.4812 − 40.6924i 39.0898i −90.5741 12.4374 + 7.18075i
14.14 0.541363 − 0.312556i −24.3819 −15.8046 + 27.3744i −46.8737 − 81.1877i −13.1994 + 7.62070i −21.4453 + 12.3814i 39.7629i 351.477 −50.7514 − 29.3013i
14.15 1.52349 − 0.879585i 11.5259 −14.4527 + 25.0327i −32.6357 − 56.5268i 17.5596 − 10.1380i −167.355 + 96.6222i 107.143i −110.153 −99.4402 − 57.4118i
14.16 2.37669 − 1.37218i 5.01039 −12.2342 + 21.1903i 7.66611 + 13.2781i 11.9081 − 6.87517i 137.587 − 79.4359i 154.970i −217.896 36.4400 + 21.0386i
14.17 3.69873 − 2.13546i 26.4579 −6.87958 + 11.9158i 23.9372 + 41.4604i 97.8607 − 56.4999i −33.9660 + 19.6103i 195.434i 457.020 177.074 + 102.234i
14.18 5.36111 − 3.09524i −2.54764 3.16098 − 5.47497i 39.7841 + 68.9082i −13.6582 + 7.88554i −162.430 + 93.7789i 158.959i −236.510 426.574 + 246.283i
14.19 5.97052 − 3.44708i −13.9645 7.76473 − 13.4489i −7.99258 − 13.8436i −83.3754 + 48.1368i −72.5498 + 41.8866i 113.551i −47.9923 −95.4397 − 55.1022i
14.20 6.07673 − 3.50840i −30.1249 8.61775 − 14.9264i 16.7440 + 29.0014i −183.061 + 105.690i 118.902 − 68.6479i 103.600i 664.511 203.497 + 117.489i
See all 50 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 14.25
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 61.6.f.a ✓ 50
61.f even 6 1 inner 61.6.f.a ✓ 50
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
61.6.f.a ✓ 50 1.a even 1 1 trivial
61.6.f.a ✓ 50 61.f even 6 1 inner

Hecke kernels

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