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Properties

Label 96.8.n.a
Level $96$
Weight $8$
Character orbit 96.n
Analytic conductor $29.989$
Analytic rank $0$
Dimension $224$
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] = [96,8,Mod(13,96)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(96, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 7, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("96.13");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 96 = 2^{5} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 96.n (of order \(8\), degree \(4\), 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: \(29.9889624465\)
Analytic rank: \(0\)
Dimension: \(224\)
Relative dimension: \(56\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 224 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 224 q + 13000 q^{10} - 30672 q^{12} + 52384 q^{14} + 21112 q^{16} - 40824 q^{18} - 164000 q^{20} + 511464 q^{22} + 286832 q^{23} - 81864 q^{24} + 727960 q^{26} + 390920 q^{28} + 1429968 q^{31} - 1071320 q^{32} + 403160 q^{34} - 1633008 q^{35} + 1244920 q^{38} - 4089528 q^{40} - 732368 q^{43} + 3370472 q^{44} + 2715808 q^{46} + 2317416 q^{50} - 2993328 q^{51} - 8882216 q^{52} + 3631264 q^{53} + 157464 q^{54} + 4191008 q^{55} + 10737272 q^{56} + 7269664 q^{58} + 3671872 q^{59} - 2168208 q^{60} + 4559776 q^{61} - 10347504 q^{62} - 4000752 q^{63} - 22412520 q^{64} - 3757104 q^{66} + 776272 q^{67} + 17922136 q^{68} - 9580896 q^{69} - 1696200 q^{70} + 24697408 q^{71} - 21862880 q^{74} - 11260512 q^{75} + 5188008 q^{76} + 23828896 q^{77} - 14742648 q^{78} + 69474376 q^{80} - 6997080 q^{82} - 39909568 q^{86} - 37691824 q^{88} + 3406992 q^{91} - 684560 q^{92} + 50770592 q^{94} + 54100440 q^{96} - 52426160 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −11.3130 + 0.130905i 10.3325 − 24.9447i 127.966 − 2.96185i 9.42355 − 3.90336i −113.625 + 283.551i 1262.09 − 1262.09i −1447.28 + 50.2587i −515.481 − 515.481i −106.097 + 45.3921i
13.2 −11.3112 + 0.239110i −10.3325 + 24.9447i 127.886 − 5.40923i −358.388 + 148.449i 110.908 − 284.625i 265.643 − 265.643i −1445.24 + 91.7636i −515.481 − 515.481i 4018.29 − 1764.83i
13.3 −11.2323 + 1.35442i −10.3325 + 24.9447i 124.331 − 30.4267i 329.611 − 136.529i 82.2719 − 294.182i −308.975 + 308.975i −1355.32 + 510.161i −515.481 − 515.481i −3517.38 + 1979.97i
13.4 −11.1907 − 1.66402i 10.3325 − 24.9447i 122.462 + 37.2430i 470.765 − 194.997i −157.136 + 261.955i −244.434 + 244.434i −1308.46 − 620.553i −515.481 − 515.481i −5592.65 + 1398.79i
13.5 −11.1751 + 1.76561i 10.3325 − 24.9447i 121.765 − 39.4616i −155.733 + 64.5067i −71.4235 + 297.003i −859.738 + 859.738i −1291.06 + 655.976i −515.481 − 515.481i 1626.44 − 995.832i
13.6 −10.5653 − 4.04648i −10.3325 + 24.9447i 95.2520 + 85.5047i −127.982 + 53.0119i 210.104 − 221.739i −1234.44 + 1234.44i −660.376 − 1288.82i −515.481 − 515.481i 1566.68 − 42.2112i
13.7 −10.5140 + 4.17792i −10.3325 + 24.9447i 93.0900 − 87.8536i 171.995 − 71.2427i 4.41861 − 305.438i −141.601 + 141.601i −611.706 + 1312.62i −515.481 − 515.481i −1510.72 + 1467.63i
13.8 −9.98351 − 5.32254i −10.3325 + 24.9447i 71.3411 + 106.275i −30.4035 + 12.5936i 235.924 − 194.041i 727.985 − 727.985i −146.581 − 1440.72i −515.481 − 515.481i 370.564 + 36.0961i
13.9 −9.93025 − 5.42127i 10.3325 − 24.9447i 69.2196 + 107.669i −17.9528 + 7.43630i −237.836 + 191.692i 43.6415 − 43.6415i −103.664 − 1444.44i −515.481 − 515.481i 218.590 + 23.4828i
13.10 −9.62483 − 5.94666i −10.3325 + 24.9447i 57.2746 + 114.471i 385.998 − 159.886i 247.786 − 178.645i 392.334 − 392.334i 129.462 − 1442.36i −515.481 − 515.481i −4665.95 − 756.526i
13.11 −9.60858 + 5.97288i 10.3325 − 24.9447i 56.6495 − 114.782i 43.1508 − 17.8736i 49.7118 + 301.398i 429.488 − 429.488i 141.257 + 1441.25i −515.481 − 515.481i −307.860 + 429.474i
13.12 −8.86062 + 7.03487i −10.3325 + 24.9447i 29.0213 − 124.667i −115.244 + 47.7355i −83.9310 − 293.713i 1052.64 − 1052.64i 619.867 + 1308.78i −515.481 − 515.481i 685.318 − 1233.69i
13.13 −8.70026 − 7.23225i 10.3325 − 24.9447i 23.3891 + 125.845i −187.919 + 77.8384i −270.302 + 142.299i −37.5079 + 37.5079i 706.650 − 1264.04i −515.481 − 515.481i 2197.89 + 681.859i
13.14 −8.68559 + 7.24987i 10.3325 − 24.9447i 22.8788 − 125.939i −438.852 + 181.778i 91.1027 + 291.569i 343.163 − 343.163i 714.323 + 1259.72i −515.481 − 515.481i 2493.81 − 4760.47i
13.15 −7.32149 − 8.62530i −10.3325 + 24.9447i −20.7914 + 126.300i −212.878 + 88.1768i 290.805 − 93.5124i −343.062 + 343.062i 1241.60 − 745.373i −515.481 − 515.481i 2319.13 + 1190.55i
13.16 −7.13292 + 8.78188i 10.3325 − 24.9447i −26.2430 − 125.281i 191.101 − 79.1566i 145.361 + 268.667i −823.986 + 823.986i 1287.39 + 663.156i −515.481 − 515.481i −667.963 + 2242.84i
13.17 −6.68730 + 9.12579i −10.3325 + 24.9447i −38.5600 − 122.054i 267.512 − 110.807i −158.544 − 261.105i −652.006 + 652.006i 1371.70 + 464.321i −515.481 − 515.481i −777.732 + 3182.26i
13.18 −5.94886 − 9.62346i 10.3325 − 24.9447i −57.2221 + 114.497i 299.793 − 124.178i −301.521 + 48.9588i −154.444 + 154.444i 1442.27 − 130.453i −515.481 − 515.481i −2978.46 − 2146.33i
13.19 −4.41371 − 10.4173i 10.3325 − 24.9447i −89.0383 + 91.9575i −325.274 + 134.733i −305.460 + 2.46312i −991.087 + 991.087i 1350.93 + 521.661i −515.481 − 515.481i 2839.21 + 2793.79i
13.20 −4.23380 − 10.4917i −10.3325 + 24.9447i −92.1498 + 88.8392i 86.8933 − 35.9924i 305.457 + 2.79344i −22.5564 + 22.5564i 1322.22 + 590.677i −515.481 − 515.481i −745.509 − 759.270i
See next 80 embeddings (of 224 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 96.8.n.a ✓ 224
32.g even 8 1 inner 96.8.n.a ✓ 224
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.8.n.a ✓ 224 1.a even 1 1 trivial
96.8.n.a ✓ 224 32.g even 8 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(96, [\chi])\).