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Properties

Label 61.4.f.a
Level $61$
Weight $4$
Character orbit 61.f
Analytic conductor $3.599$
Analytic rank $0$
Dimension $28$
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] = [61,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("61.14");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 61 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(3.59911651035\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 3 q^{2} + 8 q^{3} + 35 q^{4} - 22 q^{5} - 15 q^{6} + 33 q^{7} + 148 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 3 q^{2} + 8 q^{3} + 35 q^{4} - 22 q^{5} - 15 q^{6} + 33 q^{7} + 148 q^{9} - 27 q^{10} - 39 q^{12} + 41 q^{13} + 20 q^{14} - 56 q^{15} + 71 q^{16} + 99 q^{17} - 27 q^{18} + 95 q^{19} - 420 q^{20} - 276 q^{21} + 103 q^{22} - 198 q^{25} + 345 q^{26} + 140 q^{27} - 516 q^{29} + 186 q^{30} - 435 q^{31} - 141 q^{32} - 1028 q^{34} - 27 q^{35} - 538 q^{36} + 112 q^{39} + 492 q^{40} + 1042 q^{41} - 203 q^{42} - 273 q^{43} + 1560 q^{44} - 400 q^{45} - 17 q^{46} + 435 q^{47} + 1006 q^{48} + 265 q^{49} - 1038 q^{51} + 572 q^{52} + 1176 q^{54} + 300 q^{55} - 470 q^{56} + 634 q^{57} - 2476 q^{58} + 615 q^{59} - 688 q^{60} + 2073 q^{61} - 742 q^{62} - 927 q^{63} + 1632 q^{64} - 927 q^{65} + 262 q^{66} + 2943 q^{67} + 4479 q^{68} - 1192 q^{70} - 1917 q^{71} - 754 q^{73} - 619 q^{74} - 2350 q^{75} - 63 q^{76} + 2884 q^{77} + 1545 q^{78} - 1437 q^{79} + 245 q^{80} - 4892 q^{81} - 2715 q^{82} + 637 q^{83} + 320 q^{86} - 6 q^{87} - 3232 q^{88} - 4872 q^{90} + 5463 q^{91} + 969 q^{92} - 2064 q^{93} - 3386 q^{95} - 5748 q^{96} - 2894 q^{97}+O(q^{100}) \) 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 −4.43820 + 2.56239i −4.68764 9.13172 − 15.8166i −2.08398 − 3.60956i 20.8047 − 12.0116i 7.86940 − 4.54340i 52.5979i −5.02605 18.4982 + 10.6799i
14.2 −3.53939 + 2.04347i 5.72153 4.35151 − 7.53704i −9.61254 − 16.6494i −20.2507 + 11.6917i −17.1431 + 9.89759i 2.87320i 5.73585 68.0450 + 39.2858i
14.3 −3.23879 + 1.86991i −1.12848 2.99316 − 5.18431i 6.72919 + 11.6553i 3.65491 − 2.11017i −13.5116 + 7.80093i − 7.53079i −25.7265 −43.5888 − 25.1660i
14.4 −3.22991 + 1.86479i 6.71607 2.95490 − 5.11803i 3.09865 + 5.36702i −21.6923 + 12.5241i 22.5744 − 13.0333i − 7.79559i 18.1057 −20.0167 − 11.5567i
14.5 −1.83581 + 1.05991i −9.15042 −1.75320 + 3.03663i 0.600663 + 1.04038i 16.7985 − 9.69859i 5.31718 − 3.06987i − 24.3914i 56.7303 −2.20541 − 1.27329i
14.6 −1.04129 + 0.601190i −0.645273 −3.27714 + 5.67618i −8.04437 − 13.9333i 0.671917 − 0.387931i 25.4233 − 14.6782i − 17.4998i −26.5836 16.7531 + 9.67238i
14.7 −0.577098 + 0.333187i 8.27690 −3.77797 + 6.54364i 4.71952 + 8.17444i −4.77658 + 2.75776i −15.3639 + 8.87036i − 10.3661i 41.5071 −5.44724 − 3.14497i
14.8 −0.293912 + 0.169690i −0.384601 −3.94241 + 6.82845i −1.12328 − 1.94557i 0.113039 − 0.0652632i −14.5356 + 8.39213i − 5.39100i −26.8521 0.660290 + 0.381219i
14.9 1.68555 − 0.973153i 4.84463 −2.10595 + 3.64761i 3.50198 + 6.06561i 8.16587 − 4.71457i 21.1976 − 12.2385i 23.7681i −3.52955 11.8055 + 6.81592i
14.10 1.86681 − 1.07780i −5.62312 −1.67669 + 2.90411i 8.44841 + 14.6331i −10.4973 + 6.06061i 4.98836 − 2.88003i 24.4734i 4.61953 31.5431 + 18.2114i
14.11 2.33469 − 1.34794i −6.87295 −0.366141 + 0.634175i −6.84861 − 11.8621i −16.0462 + 9.26429i −17.6222 + 10.1742i 23.5411i 20.2374 −31.9788 − 18.4630i
14.12 2.60095 − 1.50166i 8.14876 0.509972 − 0.883297i −9.60204 − 16.6312i 21.1945 − 12.2367i −2.79342 + 1.61278i 20.9634i 39.4023 −49.9489 − 28.8380i
14.13 4.01689 − 2.31915i 3.17654 6.75695 − 11.7034i 3.05169 + 5.28568i 12.7598 − 7.36690i −14.7839 + 8.53550i − 25.5751i −16.9096 24.5166 + 14.1547i
14.14 4.18950 − 2.41881i −4.39195 7.70129 − 13.3390i −3.83528 − 6.64291i −18.4001 + 10.6233i 24.8834 − 14.3664i − 35.8109i −7.71075 −32.1359 − 18.5536i
48.1 −4.43820 − 2.56239i −4.68764 9.13172 + 15.8166i −2.08398 + 3.60956i 20.8047 + 12.0116i 7.86940 + 4.54340i − 52.5979i −5.02605 18.4982 − 10.6799i
48.2 −3.53939 − 2.04347i 5.72153 4.35151 + 7.53704i −9.61254 + 16.6494i −20.2507 − 11.6917i −17.1431 − 9.89759i − 2.87320i 5.73585 68.0450 − 39.2858i
48.3 −3.23879 − 1.86991i −1.12848 2.99316 + 5.18431i 6.72919 − 11.6553i 3.65491 + 2.11017i −13.5116 − 7.80093i 7.53079i −25.7265 −43.5888 + 25.1660i
48.4 −3.22991 − 1.86479i 6.71607 2.95490 + 5.11803i 3.09865 − 5.36702i −21.6923 − 12.5241i 22.5744 + 13.0333i 7.79559i 18.1057 −20.0167 + 11.5567i
48.5 −1.83581 − 1.05991i −9.15042 −1.75320 − 3.03663i 0.600663 − 1.04038i 16.7985 + 9.69859i 5.31718 + 3.06987i 24.3914i 56.7303 −2.20541 + 1.27329i
48.6 −1.04129 − 0.601190i −0.645273 −3.27714 − 5.67618i −8.04437 + 13.9333i 0.671917 + 0.387931i 25.4233 + 14.6782i 17.4998i −26.5836 16.7531 − 9.67238i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 14.14
Significant digits:
Format:

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 61.4.f.a ✓ 28
61.f even 6 1 inner 61.4.f.a ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
61.4.f.a ✓ 28 1.a even 1 1 trivial
61.4.f.a ✓ 28 61.f even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(61, [\chi])\).