↑

Properties

Label 91.10.k.a
Level $91$
Weight $10$
Character orbit 91.k
Analytic conductor $46.868$
Analytic rank $0$
Dimension $164$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [91,10,Mod(4,91)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(91, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 1]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("91.4");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 91.k (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: \(46.8682610909\)
Analytic rank: \(0\)
Dimension: \(164\)
Relative dimension: \(82\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 164 q - 161 q^{3} - 40962 q^{4} - 6 q^{6} - 2055 q^{7} - 510909 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 164 q - 161 q^{3} - 40962 q^{4} - 6 q^{6} - 2055 q^{7} - 510909 q^{9} + 41025 q^{10} - 85389 q^{11} + 43064 q^{12} - 232462 q^{13} + 82708 q^{14} + 338445 q^{15} + 9425286 q^{16} - 1277590 q^{17} + 1176873 q^{18} - 1344159 q^{19} + 571062 q^{20} - 1367523 q^{21} - 1417332 q^{22} - 2148066 q^{23} + 62118 q^{24} + 29660776 q^{25} + 3184908 q^{26} + 7350574 q^{27} + 4828755 q^{28} + 996812 q^{29} - 9177016 q^{30} - 21474090 q^{31} - 15796239 q^{33} + 10592613 q^{35} + 131881331 q^{36} - 8950274 q^{38} - 35258971 q^{39} - 7901342 q^{40} + 24649770 q^{41} - 51088299 q^{42} + 15892214 q^{43} - 1176612 q^{44} - 69222114 q^{47} - 15455476 q^{48} - 57160671 q^{49} + 309166224 q^{50} + 52823362 q^{51} + 177629625 q^{52} - 155327528 q^{53} - 77925569 q^{55} - 92078463 q^{56} - 597924735 q^{58} - 290928939 q^{60} - 468761537 q^{61} + 238770500 q^{62} - 358980429 q^{63} - 2035250696 q^{64} - 542245158 q^{65} - 154929237 q^{66} - 353402973 q^{67} + 1196053594 q^{68} + 127053056 q^{69} - 1180478502 q^{70} - 269829840 q^{71} - 485282307 q^{72} - 35353098 q^{73} + 920062914 q^{74} - 1606576144 q^{75} + 1140965370 q^{76} - 1032378781 q^{77} - 835494667 q^{78} - 683323974 q^{79} - 432598794 q^{80} - 1724339502 q^{81} - 324994534 q^{82} + 3803919051 q^{84} - 196261605 q^{85} + 572477022 q^{86} + 3699935866 q^{87} + 37403221 q^{88} - 9086462948 q^{90} + 2603275581 q^{91} - 2683122798 q^{92} - 2561821264 q^{94} - 397964902 q^{95} + 392112 q^{96} + 6155159124 q^{97} + 2492693148 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 − 44.3894i 105.590 + 182.888i −1458.42 −1212.41 + 699.984i 8118.27 − 4687.09i −4644.37 + 4333.98i 42010.8i −12457.1 + 21576.4i 31071.9 + 53818.0i
4.2 − 43.4880i −42.3730 − 73.3922i −1379.21 −1069.03 + 617.207i −3191.68 + 1842.72i 5803.48 + 2583.26i 37713.2i 6250.56 − 10826.3i 26841.1 + 46490.2i
4.3 − 42.9063i −3.86494 − 6.69428i −1328.95 1575.27 − 909.485i −287.227 + 165.831i −4344.78 − 4634.27i 35052.4i 9811.62 − 16994.2i −39022.6 − 67589.2i
4.4 − 40.9925i −66.0336 − 114.374i −1168.38 −1239.03 + 715.356i −4688.46 + 2706.88i −6342.27 + 359.552i 26906.8i 1120.63 − 1940.99i 29324.2 + 50791.0i
4.5 − 40.5568i 73.1837 + 126.758i −1132.85 436.798 − 252.185i 5140.89 − 2968.10i 958.384 − 6279.74i 25179.8i −870.206 + 1507.24i −10227.8 − 17715.1i
4.6 − 40.4599i −134.562 − 233.068i −1125.00 63.4069 − 36.6080i −9429.90 + 5444.36i 479.713 − 6334.31i 24802.0i −26372.3 + 45678.1i −1481.16 − 2565.44i
4.7 − 40.2060i −97.8710 − 169.518i −1104.52 1750.16 − 1010.46i −6815.62 + 3935.00i −1251.01 + 6228.05i 23822.8i −9315.97 + 16135.7i −40626.4 − 70367.0i
4.8 − 37.7981i 106.418 + 184.321i −916.698 1677.20 − 968.331i 6967.00 − 4022.40i 5714.50 + 2774.55i 15296.8i −12808.1 + 22184.2i −36601.1 − 63394.9i
4.9 − 37.3816i 73.2483 + 126.870i −885.387 −1496.61 + 864.068i 4742.60 − 2738.14i 6349.05 + 207.735i 13957.8i −889.129 + 1540.02i 32300.3 + 55945.7i
4.10 − 37.2316i 47.4988 + 82.2703i −874.190 1072.58 − 619.257i 3063.05 − 1768.45i −3140.13 + 5522.06i 13484.9i 5329.24 − 9230.51i −23055.9 − 39934.0i
4.11 − 37.1692i −35.2205 − 61.0038i −869.552 714.561 − 412.552i −2267.46 + 1309.12i 5892.38 − 2373.50i 13289.9i 7360.53 − 12748.8i −15334.2 − 26559.7i
4.12 − 33.6268i 40.5181 + 70.1794i −618.759 −1658.21 + 957.369i 2359.91 − 1362.49i −1160.12 − 6245.62i 3589.95i 6558.07 − 11358.9i 32193.2 + 55760.3i
4.13 − 32.8878i −47.2530 − 81.8447i −569.605 −1116.08 + 644.368i −2691.69 + 1554.05i −6165.78 − 1528.64i 1894.52i 5375.80 − 9311.16i 21191.8 + 36705.3i
4.14 − 32.6641i −10.6449 − 18.4375i −554.947 −611.718 + 353.175i −602.244 + 347.706i −1517.64 + 6168.50i 1402.82i 9614.87 − 16653.4i 11536.2 + 19981.2i
4.15 − 31.6219i −129.144 − 223.684i −487.947 −2120.68 + 1224.37i −7073.31 + 4083.78i 1134.35 + 6250.35i − 760.607i −23514.7 + 40728.7i 38717.1 + 67060.0i
4.16 − 30.8117i 127.442 + 220.735i −437.364 −455.574 + 263.026i 6801.24 − 3926.70i −6167.30 − 1522.51i − 2299.67i −22641.2 + 39215.7i 8104.29 + 14037.0i
4.17 − 30.3899i −91.7985 − 159.000i −411.547 1465.14 − 845.901i −4831.98 + 2789.75i 4172.98 + 4789.55i − 3052.77i −7012.42 + 12145.9i −25706.9 − 44525.6i
4.18 − 27.5646i −90.0611 − 155.990i −247.805 −97.1768 + 56.1050i −4299.81 + 2482.50i 3557.14 − 5263.12i − 7282.43i −6380.52 + 11051.4i 1546.51 + 2678.63i
4.19 − 27.3770i 126.131 + 218.466i −237.502 1023.39 − 590.853i 5980.94 − 3453.10i −891.449 − 6289.59i − 7514.94i −21976.7 + 38064.7i −16175.8 − 28017.3i
4.20 − 25.6952i −88.2071 − 152.779i −148.244 1048.20 − 605.176i −3925.69 + 2266.50i −6088.90 − 1810.76i − 9346.80i −5719.49 + 9906.44i −15550.1 − 26933.6i
See next 80 embeddings (of 164 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.82
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.k even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.10.k.a ✓ 164
7.c even 3 1 91.10.u.a yes 164
13.e even 6 1 91.10.u.a yes 164
91.k even 6 1 inner 91.10.k.a ✓ 164
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.10.k.a ✓ 164 1.a even 1 1 trivial
91.10.k.a ✓ 164 91.k even 6 1 inner
91.10.u.a yes 164 7.c even 3 1
91.10.u.a yes 164 13.e even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(91, [\chi])\).