Properties

Label 61.9.d.a
Level $61$
Weight $9$
Character orbit 61.d
Analytic conductor $24.850$
Analytic rank $0$
Dimension $82$
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] = [61,9,Mod(11,61)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(61, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("61.11");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 61 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 61.d (of order \(4\), 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: \(24.8500952137\)
Analytic rank: \(0\)
Dimension: \(82\)
Relative dimension: \(41\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 82 q - 2 q^{2} - 2270 q^{6} - 7502 q^{7} + 510 q^{8} - 179338 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 82 q - 2 q^{2} - 2270 q^{6} - 7502 q^{7} + 510 q^{8} - 179338 q^{9} + 21818 q^{10} + 6988 q^{11} + 26240 q^{12} + 85676 q^{13} - 161792 q^{15} - 1394812 q^{16} - 211544 q^{17} - 435962 q^{18} + 1134588 q^{20} - 934050 q^{21} - 4 q^{22} - 752504 q^{23} - 1434126 q^{24} - 7323234 q^{25} + 1630028 q^{26} + 2682748 q^{28} + 1119640 q^{29} - 1771234 q^{30} + 217798 q^{31} - 2640152 q^{32} + 152570 q^{33} + 2194524 q^{34} + 1225608 q^{35} + 6121344 q^{37} - 2905674 q^{38} + 14452216 q^{40} + 17600500 q^{42} - 7154326 q^{43} + 5789590 q^{44} + 9958520 q^{47} - 3239078 q^{50} + 599092 q^{51} + 29185330 q^{53} + 54420904 q^{54} - 55341774 q^{55} - 39127944 q^{56} - 143253120 q^{57} - 45618356 q^{58} - 2951618 q^{59} - 23685838 q^{61} - 181502272 q^{62} + 72576018 q^{63} + 40494886 q^{67} - 12094154 q^{68} - 123694458 q^{69} - 31827960 q^{70} + 20412526 q^{71} + 12172606 q^{72} + 147831820 q^{73} + 128794088 q^{74} - 80600916 q^{76} - 66161428 q^{77} - 95314566 q^{78} + 77000096 q^{79} + 448656258 q^{81} + 215741808 q^{82} + 199990616 q^{83} - 422006946 q^{84} + 356820486 q^{85} + 483920696 q^{86} + 273809288 q^{87} - 131953574 q^{89} + 346422210 q^{90} - 170218934 q^{91} + 453516746 q^{92} - 24122328 q^{93} + 111270468 q^{94} + 254683332 q^{95} - 332833526 q^{96} - 134644432 q^{98} - 356022224 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −21.2881 21.2881i 12.3702i 650.367i 793.137i 263.338 263.338i −2412.03 2412.03i 8395.32 8395.32i 6407.98 16884.4 16884.4i
11.2 −20.8664 20.8664i 38.5690i 614.817i 69.4108i −804.797 + 804.797i 1997.03 + 1997.03i 7487.23 7487.23i 5073.43 1448.36 1448.36i
11.3 −20.5638 20.5638i 121.719i 589.743i 1083.18i −2503.01 + 2503.01i −1781.46 1781.46i 6863.04 6863.04i −8254.53 −22274.3 + 22274.3i
11.4 −20.1232 20.1232i 127.522i 553.885i 671.220i 2566.14 2566.14i 536.994 + 536.994i 5994.39 5994.39i −9700.77 −13507.1 + 13507.1i
11.5 −18.6699 18.6699i 93.9195i 441.127i 616.035i 1753.46 1753.46i 1452.34 + 1452.34i 3456.30 3456.30i −2259.87 11501.3 11501.3i
11.6 −18.5138 18.5138i 9.00826i 429.525i 775.655i 166.778 166.778i −581.583 581.583i 3212.61 3212.61i 6479.85 −14360.4 + 14360.4i
11.7 −17.4953 17.4953i 150.121i 356.170i 862.462i −2626.41 + 2626.41i −1749.89 1749.89i 1752.51 1752.51i −15975.3 15089.0 15089.0i
11.8 −16.0682 16.0682i 98.3603i 260.374i 156.143i −1580.47 + 1580.47i 1884.54 + 1884.54i 70.2833 70.2833i −3113.75 2508.94 2508.94i
11.9 −13.9825 13.9825i 129.045i 135.022i 482.619i 1804.38 1804.38i −1709.24 1709.24i −1691.58 + 1691.58i −10091.7 6748.23 6748.23i
11.10 −13.2958 13.2958i 37.3980i 97.5592i 131.019i −497.239 + 497.239i −1655.82 1655.82i −2106.60 + 2106.60i 5162.39 1742.01 1742.01i
11.11 −13.0804 13.0804i 48.4247i 86.1913i 475.622i 633.412 633.412i −1359.87 1359.87i −2221.16 + 2221.16i 4216.05 −6221.30 + 6221.30i
11.12 −10.9363 10.9363i 25.8158i 16.7929i 1164.26i 282.331 282.331i 1657.56 + 1657.56i −2983.36 + 2983.36i 5894.54 12732.8 12732.8i
11.13 −10.9355 10.9355i 40.9506i 16.8293i 742.032i 447.815 447.815i 2481.46 + 2481.46i −2983.53 + 2983.53i 4884.05 −8114.50 + 8114.50i
11.14 −8.83447 8.83447i 119.831i 99.9043i 948.313i −1058.64 + 1058.64i 1107.39 + 1107.39i −3144.23 + 3144.23i −7798.38 −8377.84 + 8377.84i
11.15 −6.85491 6.85491i 130.247i 162.020i 18.3560i 892.832 892.832i 2331.62 + 2331.62i −2865.49 + 2865.49i −10403.3 −125.829 + 125.829i
11.16 −6.15613 6.15613i 88.4223i 180.204i 287.135i −544.339 + 544.339i −2555.37 2555.37i −2685.33 + 2685.33i −1257.50 −1767.64 + 1767.64i
11.17 −6.14056 6.14056i 66.5883i 180.587i 857.795i −408.890 + 408.890i −216.769 216.769i −2680.89 + 2680.89i 2127.00 5267.34 5267.34i
11.18 −3.70176 3.70176i 117.318i 228.594i 1215.69i 434.282 434.282i −2288.13 2288.13i −1793.85 + 1793.85i −7202.43 −4500.21 + 4500.21i
11.19 −2.43956 2.43956i 26.2435i 244.097i 56.8987i 64.0225 64.0225i 1007.09 + 1007.09i −1220.02 + 1220.02i 5872.28 −138.808 + 138.808i
11.20 −1.16699 1.16699i 110.695i 253.276i 625.680i 129.181 129.181i −1738.72 1738.72i −594.320 + 594.320i −5692.48 730.163 730.163i
See all 82 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.41
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 61.9.d.a 82
61.d odd 4 1 inner 61.9.d.a 82
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
61.9.d.a 82 1.a even 1 1 trivial
61.9.d.a 82 61.d odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(61, [\chi])\).