Properties

Label 87.6.g.b
Level $87$
Weight $6$
Character orbit 87.g
Analytic conductor $13.953$
Analytic rank $0$
Dimension $78$
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] = [87,6,Mod(7,87)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(87, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 6]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("87.7");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 87 = 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 87.g (of order \(7\), degree \(6\), 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: \(13.9533923237\)
Analytic rank: \(0\)
Dimension: \(78\)
Relative dimension: \(13\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 78 q + 10 q^{2} + 117 q^{3} - 202 q^{4} + 49 q^{5} - 90 q^{6} - 60 q^{7} + 273 q^{8} - 1053 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 78 q + 10 q^{2} + 117 q^{3} - 202 q^{4} + 49 q^{5} - 90 q^{6} - 60 q^{7} + 273 q^{8} - 1053 q^{9} + 288 q^{10} + 460 q^{11} - 12294 q^{12} + 1883 q^{13} - 4565 q^{14} - 441 q^{15} - 7794 q^{16} - 9592 q^{17} + 810 q^{18} + 2538 q^{19} - 116 q^{20} + 540 q^{21} + 2812 q^{22} - 8224 q^{23} - 8064 q^{24} + 11226 q^{25} - 12824 q^{26} + 9477 q^{27} + 58598 q^{28} + 9286 q^{29} + 10008 q^{30} - 7224 q^{31} - 31647 q^{32} - 360 q^{33} - 3153 q^{34} + 27153 q^{35} - 16362 q^{36} + 17228 q^{37} - 25396 q^{38} + 11340 q^{39} - 32434 q^{40} + 57300 q^{41} + 5049 q^{42} + 33473 q^{43} + 33742 q^{44} - 6804 q^{45} - 28276 q^{46} + 87241 q^{47} - 30402 q^{48} - 72481 q^{49} - 155243 q^{50} + 2475 q^{51} - 198520 q^{52} + 161593 q^{53} - 2187 q^{54} - 123661 q^{55} - 253890 q^{56} + 31590 q^{57} + 301566 q^{58} + 64628 q^{59} + 49302 q^{60} + 87396 q^{61} - 128679 q^{62} - 15633 q^{63} + 42865 q^{64} - 136204 q^{65} + 207 q^{66} + 23066 q^{67} - 28244 q^{68} - 135144 q^{69} + 121526 q^{70} - 276862 q^{71} + 72576 q^{72} + 30459 q^{73} + 481517 q^{74} - 538758 q^{75} + 441053 q^{76} + 70464 q^{77} + 123795 q^{78} - 86879 q^{79} + 281480 q^{80} - 85293 q^{81} + 597939 q^{82} + 262084 q^{83} - 66285 q^{84} + 98649 q^{85} + 53476 q^{86} - 276039 q^{87} + 1616024 q^{88} + 37596 q^{89} + 23328 q^{90} + 87713 q^{91} - 190625 q^{92} + 65016 q^{93} - 377368 q^{94} - 169368 q^{95} - 535626 q^{96} + 683983 q^{97} - 29629 q^{98} - 133974 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −2.26086 9.90549i 8.10872 + 3.90495i −64.1762 + 30.9056i 3.70316 + 16.2246i 20.3478 89.1494i 99.3933 + 47.8653i 248.515 + 311.628i 50.5027 + 63.3284i 152.340 73.3632i
7.2 −1.93840 8.49270i 8.10872 + 3.90495i −39.5375 + 19.0403i −7.82896 34.3009i 17.4456 76.4343i −90.1162 43.3977i 64.5416 + 80.9326i 50.5027 + 63.3284i −276.131 + 132.978i
7.3 −1.60675 7.03963i 8.10872 + 3.90495i −18.1438 + 8.73759i 13.2265 + 57.9493i 14.4608 63.3567i −17.0594 8.21538i −53.4025 66.9646i 50.5027 + 63.3284i 386.690 186.220i
7.4 −0.991659 4.34474i 8.10872 + 3.90495i 10.9376 5.26727i 11.7683 + 51.5605i 8.92493 39.1027i −152.983 73.6729i −122.645 153.793i 50.5027 + 63.3284i 212.347 102.261i
7.5 −0.852092 3.73326i 8.10872 + 3.90495i 15.6198 7.52212i −11.0422 48.3789i 7.66883 33.5993i 218.824 + 105.380i −117.792 147.706i 50.5027 + 63.3284i −171.202 + 82.4466i
7.6 −0.682925 2.99209i 8.10872 + 3.90495i 20.3448 9.79754i −17.6030 77.1239i 6.14632 26.9288i −99.2164 47.7801i −104.441 130.965i 50.5027 + 63.3284i −218.740 + 105.340i
7.7 0.0330315 + 0.144721i 8.10872 + 3.90495i 28.8112 13.8747i 15.6670 + 68.6416i −0.297284 + 1.30249i 67.3517 + 32.4349i 5.92131 + 7.42508i 50.5027 + 63.3284i −9.41635 + 4.53467i
7.8 0.621155 + 2.72146i 8.10872 + 3.90495i 21.8105 10.5034i −4.60003 20.1541i −5.59040 + 24.4931i 28.2944 + 13.6259i 97.8263 + 122.670i 50.5027 + 63.3284i 51.9911 25.0376i
7.9 1.38807 + 6.08151i 8.10872 + 3.90495i −6.22706 + 2.99879i 14.1607 + 62.0419i −12.4926 + 54.7336i −214.780 103.433i 97.5760 + 122.356i 50.5027 + 63.3284i −357.653 + 172.236i
7.10 1.72031 + 7.53718i 8.10872 + 3.90495i −25.0187 + 12.0484i −9.36503 41.0309i −15.4828 + 67.8347i 139.550 + 67.2038i 20.3961 + 25.5759i 50.5027 + 63.3284i 293.146 141.172i
7.11 1.73808 + 7.61502i 8.10872 + 3.90495i −26.1367 + 12.5868i −21.6674 94.9309i −15.6427 + 68.5352i −141.588 68.1854i 14.5636 + 18.2622i 50.5027 + 63.3284i 685.241 329.995i
7.12 1.77168 + 7.76224i 8.10872 + 3.90495i −28.2825 + 13.6201i 19.5439 + 85.6275i −15.9451 + 69.8601i 133.220 + 64.1553i 3.02217 + 3.78968i 50.5027 + 63.3284i −630.035 + 303.409i
7.13 2.43957 + 10.6884i 8.10872 + 3.90495i −79.4602 + 38.2660i 1.75680 + 7.69705i −21.9561 + 96.1959i −66.8029 32.1706i −384.116 481.666i 50.5027 + 63.3284i −77.9836 + 37.5549i
16.1 −8.99529 4.33190i −5.61141 + 7.03648i 42.1982 + 52.9148i 44.5360 + 21.4474i 80.9576 38.9871i 87.7619 110.050i −79.2699 347.304i −18.0242 78.9692i −307.706 385.851i
16.2 −7.92947 3.81863i −5.61141 + 7.03648i 28.3429 + 35.5409i −91.4078 44.0197i 71.3652 34.3677i 111.730 140.105i −26.3573 115.479i −18.0242 78.9692i 556.721 + 698.106i
16.3 −6.29108 3.02962i −5.61141 + 7.03648i 10.4473 + 13.1006i 74.0869 + 35.6784i 56.6197 27.2666i −44.0006 + 55.1750i 23.6853 + 103.772i −18.0242 78.9692i −357.994 448.911i
16.4 −5.63420 2.71329i −5.61141 + 7.03648i 4.43057 + 5.55576i −10.4355 5.02547i 50.7078 24.4196i −39.0334 + 48.9464i 34.6407 + 151.771i −18.0242 78.9692i 45.1601 + 56.6289i
16.5 −1.55088 0.746863i −5.61141 + 7.03648i −18.1043 22.7020i −73.4400 35.3668i 13.9579 6.72176i −101.424 + 127.182i 23.3793 + 102.431i −18.0242 78.9692i 87.4822 + 109.699i
16.6 −0.807834 0.389033i −5.61141 + 7.03648i −19.4504 24.3901i 43.7893 + 21.0878i 7.27051 3.50129i 16.5251 20.7218i 12.6088 + 55.2427i −18.0242 78.9692i −27.1707 34.0709i
16.7 −0.507638 0.244466i −5.61141 + 7.03648i −19.7537 24.7704i −38.1099 18.3528i 4.56874 2.20019i 66.6920 83.6292i 7.98428 + 34.9814i −18.0242 78.9692i 14.8594 + 18.6331i
See all 78 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.13
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.d even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 87.6.g.b 78
29.d even 7 1 inner 87.6.g.b 78
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.6.g.b 78 1.a even 1 1 trivial
87.6.g.b 78 29.d even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{78} - 10 T_{2}^{77} + 359 T_{2}^{76} - 3561 T_{2}^{75} + 81951 T_{2}^{74} - 789996 T_{2}^{73} + 14705941 T_{2}^{72} - 126989374 T_{2}^{71} + 2167074599 T_{2}^{70} - 16945371146 T_{2}^{69} + \cdots + 30\!\cdots\!44 \) acting on \(S_{6}^{\mathrm{new}}(87, [\chi])\). Copy content Toggle raw display