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Properties

Label 103.6.c.a
Level $103$
Weight $6$
Character orbit 103.c
Analytic conductor $16.520$
Analytic rank $0$
Dimension $84$
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] = [103,6,Mod(46,103)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(103, 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("103.46");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 103 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 103.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: \(16.5195334407\)
Analytic rank: \(0\)
Dimension: \(84\)
Relative dimension: \(42\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 84 q - q^{2} - 20 q^{3} - 641 q^{4} - 9 q^{5} + 79 q^{6} + 139 q^{7} + 306 q^{8} + 5556 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 84 q - q^{2} - 20 q^{3} - 641 q^{4} - 9 q^{5} + 79 q^{6} + 139 q^{7} + 306 q^{8} + 5556 q^{9} + 2288 q^{10} + 574 q^{11} + 342 q^{12} + 344 q^{13} - 254 q^{14} - 1007 q^{15} - 11801 q^{16} + 838 q^{17} - 168 q^{18} - 5352 q^{19} + 1533 q^{20} + 2917 q^{21} + 3672 q^{22} - 7498 q^{23} - 14406 q^{24} - 12529 q^{25} + 3018 q^{26} - 578 q^{27} + 4874 q^{28} + 1707 q^{29} - 21172 q^{30} - 28614 q^{31} + 1593 q^{32} + 3259 q^{33} - 14042 q^{34} + 17343 q^{35} - 20590 q^{36} - 39938 q^{37} - 21000 q^{38} + 11194 q^{39} - 55561 q^{40} - 25984 q^{41} - 43020 q^{42} + 16728 q^{43} + 60747 q^{44} - 19244 q^{45} - 34618 q^{46} + 8975 q^{47} + 26382 q^{48} - 81577 q^{49} - 43634 q^{50} + 13350 q^{51} + 20517 q^{52} + 28215 q^{53} + 73355 q^{54} - 101828 q^{55} + 25242 q^{56} - 24546 q^{57} + 38622 q^{58} - 50831 q^{59} - 40734 q^{60} + 85524 q^{61} + 95493 q^{62} + 19324 q^{63} + 466446 q^{64} + 37191 q^{65} - 62984 q^{66} + 37597 q^{67} - 44063 q^{68} - 160706 q^{69} + 255972 q^{70} + 108938 q^{71} - 11032 q^{72} - 120018 q^{73} + 216497 q^{74} + 79102 q^{75} + 369234 q^{76} + 124901 q^{77} + 221446 q^{78} + 192536 q^{79} + 111940 q^{80} + 145684 q^{81} + 91881 q^{82} - 37934 q^{83} + 12529 q^{84} + 66904 q^{85} - 204721 q^{86} - 184103 q^{87} - 272177 q^{88} - 4668 q^{89} - 225494 q^{90} + 182886 q^{91} + 25419 q^{92} - 23228 q^{93} - 245028 q^{94} - 231598 q^{95} - 254675 q^{96} - 376094 q^{97} - 258782 q^{98} + 64530 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1 −5.48333 − 9.49741i −5.19739 −44.1338 + 76.4420i −19.0260 + 32.9540i 28.4990 + 49.3617i −111.395 + 192.942i 617.068 −215.987 417.304
46.2 −5.39353 − 9.34186i 10.6714 −42.1803 + 73.0584i −32.1514 + 55.6878i −57.5564 − 99.6907i 125.314 − 217.051i 564.816 −129.121 693.637
46.3 −5.02232 − 8.69891i −23.6250 −34.4473 + 59.6645i 29.9863 − 51.9378i 118.652 + 205.512i −12.1955 + 21.1233i 370.593 315.143 −602.402
46.4 −4.84433 − 8.39062i 25.3014 −30.9350 + 53.5810i 27.9026 − 48.3287i −122.568 − 212.294i 8.46889 − 14.6685i 289.400 397.159 −540.678
46.5 −4.79551 − 8.30607i −1.86665 −29.9938 + 51.9509i 31.6496 − 54.8186i 8.95156 + 15.5046i 15.9853 − 27.6874i 268.431 −239.516 −607.103
46.6 −4.68374 − 8.11247i −26.2294 −27.8747 + 48.2805i −34.1195 + 59.0967i 122.852 + 212.785i 73.9606 − 128.103i 222.473 444.983 639.227
46.7 −4.30827 − 7.46215i 18.0600 −21.1224 + 36.5851i −7.56827 + 13.1086i −77.8072 − 134.766i −59.6606 + 103.335i 88.2751 83.1624 130.425
46.8 −3.67474 − 6.36483i −9.95262 −11.0074 + 19.0654i 11.2513 − 19.4879i 36.5733 + 63.3468i 62.5213 − 108.290i −73.3858 −143.945 −165.383
46.9 −3.60923 − 6.25136i 0.871914 −10.0530 + 17.4124i −35.6308 + 61.7144i −3.14694 − 5.45065i 0.267625 − 0.463541i −85.8558 −242.240 514.399
46.10 −3.43733 − 5.95363i −18.7524 −7.63046 + 13.2163i −32.1856 + 55.7470i 64.4580 + 111.645i −88.5779 + 153.421i −115.075 108.651 442.529
46.11 −3.20169 − 5.54549i 22.7098 −4.50164 + 7.79706i −44.7996 + 77.5952i −72.7098 − 125.937i 26.2389 − 45.4472i −147.257 272.736 573.738
46.12 −2.71716 − 4.70625i 8.10585 1.23413 − 2.13758i 49.8234 − 86.2966i −22.0249 − 38.1482i −107.254 + 185.769i −187.311 −177.295 −541.512
46.13 −2.65323 − 4.59554i −24.3000 1.92070 − 3.32675i 30.3026 − 52.4856i 64.4735 + 111.671i −49.7664 + 86.1979i −190.191 347.488 −321.599
46.14 −2.23845 − 3.87711i 21.5933 5.97865 − 10.3553i 23.0996 − 40.0096i −48.3356 − 83.7197i 101.551 − 175.891i −196.793 223.270 −206.829
46.15 −2.22458 − 3.85308i 3.45424 6.10251 − 10.5699i 24.4359 − 42.3242i −7.68422 − 13.3095i 56.4404 − 97.7577i −196.675 −231.068 −217.438
46.16 −1.31424 − 2.27634i −11.1528 12.5455 − 21.7295i −30.6331 + 53.0581i 14.6575 + 25.3876i 70.4337 − 121.995i −150.063 −118.615 161.038
46.17 −1.22868 − 2.12813i 9.45582 12.9807 − 22.4832i −2.01016 + 3.48170i −11.6182 − 20.1232i −74.2337 + 128.577i −142.432 −153.587 9.87936
46.18 −1.17105 − 2.02831i 29.6370 13.2573 − 22.9623i −14.3554 + 24.8642i −34.7063 − 60.1131i −52.3046 + 90.5941i −137.047 635.352 67.2433
46.19 −0.899242 − 1.55753i −27.8133 14.3827 − 24.9116i −0.586076 + 1.01511i 25.0109 + 43.3202i 80.2919 − 139.070i −109.286 530.581 2.10810
46.20 −0.782795 − 1.35584i −15.1018 14.7745 − 25.5901i 1.57500 − 2.72799i 11.8216 + 20.4757i −57.3127 + 99.2686i −96.3604 −14.9347 −4.93163
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.42
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 103.6.c.a ✓ 84
103.c even 3 1 inner 103.6.c.a ✓ 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
103.6.c.a ✓ 84 1.a even 1 1 trivial
103.6.c.a ✓ 84 103.c even 3 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(103, [\chi])\).