# Properties

 Label 96.9.p.a Level $96$ Weight $9$ Character orbit 96.p Analytic conductor $39.108$ Analytic rank $0$ Dimension $504$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [96,9,Mod(5,96)]

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, 1, 4]))

N = Newforms(chi, 9, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("96.5");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$96 = 2^{5} \cdot 3$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 96.p (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: $$39.1083465659$$ Analytic rank: $$0$$ Dimension: $$504$$ Relative dimension: $$126$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$504 q - 4 q^{3} - 8 q^{4} - 4 q^{6} - 8 q^{7} - 4 q^{9}+O(q^{10})$$ 504 * q - 4 * q^3 - 8 * q^4 - 4 * q^6 - 8 * q^7 - 4 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$504 q - 4 q^{3} - 8 q^{4} - 4 q^{6} - 8 q^{7} - 4 q^{9} - 35008 q^{10} - 4 q^{12} - 8 q^{13} + 445400 q^{16} - 4 q^{18} - 8 q^{19} - 4 q^{21} + 528072 q^{22} + 161136 q^{24} - 8 q^{25} + 51068 q^{27} - 8 q^{28} + 3966132 q^{30} - 16 q^{31} - 8 q^{33} - 2056 q^{34} + 4835584 q^{36} - 8 q^{37} - 7650052 q^{39} - 15547624 q^{40} - 18128864 q^{42} - 8 q^{43} - 4 q^{45} + 11790680 q^{46} - 21819984 q^{48} - 26248 q^{51} + 12444944 q^{52} + 42722368 q^{54} - 46326792 q^{55} - 4 q^{57} + 39078992 q^{58} - 85332824 q^{60} - 48952072 q^{61} - 8 q^{63} - 48383960 q^{64} - 113022436 q^{66} - 149408776 q^{67} - 4 q^{69} + 46444936 q^{70} + 53585696 q^{72} - 8 q^{73} - 1562504 q^{75} - 139390896 q^{76} + 10120308 q^{78} - 34899808 q^{82} + 688968232 q^{84} - 3125008 q^{85} - 149712644 q^{87} - 22793968 q^{88} - 931912864 q^{90} + 350400760 q^{91} - 26248 q^{93} - 463124896 q^{94} + 710546072 q^{96} - 16 q^{97} - 630017028 q^{99}+O(q^{100})$$ 504 * q - 4 * q^3 - 8 * q^4 - 4 * q^6 - 8 * q^7 - 4 * q^9 - 35008 * q^10 - 4 * q^12 - 8 * q^13 + 445400 * q^16 - 4 * q^18 - 8 * q^19 - 4 * q^21 + 528072 * q^22 + 161136 * q^24 - 8 * q^25 + 51068 * q^27 - 8 * q^28 + 3966132 * q^30 - 16 * q^31 - 8 * q^33 - 2056 * q^34 + 4835584 * q^36 - 8 * q^37 - 7650052 * q^39 - 15547624 * q^40 - 18128864 * q^42 - 8 * q^43 - 4 * q^45 + 11790680 * q^46 - 21819984 * q^48 - 26248 * q^51 + 12444944 * q^52 + 42722368 * q^54 - 46326792 * q^55 - 4 * q^57 + 39078992 * q^58 - 85332824 * q^60 - 48952072 * q^61 - 8 * q^63 - 48383960 * q^64 - 113022436 * q^66 - 149408776 * q^67 - 4 * q^69 + 46444936 * q^70 + 53585696 * q^72 - 8 * q^73 - 1562504 * q^75 - 139390896 * q^76 + 10120308 * q^78 - 34899808 * q^82 + 688968232 * q^84 - 3125008 * q^85 - 149712644 * q^87 - 22793968 * q^88 - 931912864 * q^90 + 350400760 * q^91 - 26248 * q^93 - 463124896 * q^94 + 710546072 * q^96 - 16 * q^97 - 630017028 * q^99

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −15.9624 1.09635i −12.1267 + 80.0871i 253.596 + 35.0006i −427.511 + 1032.10i 281.375 1265.09i −1564.27 1564.27i −4009.63 836.723i −6266.88 1942.39i 7955.64 16006.1i
5.2 −15.9374 + 1.41393i −38.9920 + 70.9973i 252.002 45.0687i 151.743 366.339i 521.047 1186.64i −1906.71 1906.71i −3952.53 + 1074.59i −3520.24 5536.66i −1900.41 + 6053.05i
5.3 −15.9262 1.53449i 80.3356 + 10.3531i 251.291 + 48.8774i −275.745 + 665.708i −1263.56 288.161i −126.185 126.185i −3927.12 1164.04i 6346.63 + 1663.45i 5413.11 10179.1i
5.4 −15.9048 + 1.74295i 28.8437 75.6904i 249.924 55.4423i 171.678 414.468i −326.828 + 1254.11i 178.029 + 178.029i −3878.36 + 1317.40i −4897.09 4366.38i −2008.11 + 6891.24i
5.5 −15.8932 1.84560i −78.6850 + 19.2269i 249.188 + 58.6649i 132.823 320.664i 1286.04 160.356i 1465.28 + 1465.28i −3852.12 1392.27i 5821.65 3025.74i −2702.81 + 4851.24i
5.6 −15.8488 + 2.19416i −34.8569 73.1163i 246.371 69.5499i −234.083 + 565.127i 712.870 + 1082.33i −2941.80 2941.80i −3752.09 + 1642.86i −4131.00 + 5097.21i 2469.97 9470.23i
5.7 −15.7422 2.86081i 70.5846 39.7343i 239.632 + 90.0707i 417.839 1008.75i −1224.83 + 423.574i 2133.41 + 2133.41i −3514.64 2103.45i 3403.38 5609.26i −9463.54 + 14684.6i
5.8 −15.6274 + 3.43262i −27.5426 + 76.1735i 232.434 107.286i 77.5335 187.182i 168.946 1284.94i 3212.54 + 3212.54i −3264.08 + 2474.46i −5043.81 4196.03i −569.126 + 3191.33i
5.9 −15.5929 3.58626i −26.0548 76.6952i 230.277 + 111.840i −4.98443 + 12.0335i 131.222 + 1289.34i 1516.58 + 1516.58i −3189.61 2569.75i −5203.29 + 3996.56i 120.877 169.761i
5.10 −15.5569 + 3.73952i −75.6669 28.9054i 228.032 116.350i 439.766 1061.69i 1285.23 + 166.719i −900.352 900.352i −3112.37 + 2662.78i 4889.96 + 4374.36i −2871.17 + 18161.1i
5.11 −15.5155 3.90752i −77.2928 + 24.2245i 225.463 + 121.255i −353.603 + 853.673i 1293.90 73.8322i 736.773 + 736.773i −3024.36 2762.33i 5387.35 3744.76i 8822.08 11863.5i
5.12 −15.5106 3.92712i 50.6220 + 63.2330i 225.155 + 121.824i 184.231 444.772i −536.851 1179.58i −2659.89 2659.89i −3013.87 2773.77i −1435.84 + 6401.96i −4604.20 + 6175.17i
5.13 −15.3778 + 4.41869i 54.0804 60.3018i 216.950 135.899i −346.021 + 835.368i −565.180 + 1166.27i 742.497 + 742.497i −2735.71 + 3048.46i −711.622 6522.29i 1629.79 14375.0i
5.14 −15.2868 + 4.72377i 50.9538 + 62.9659i 211.372 144.423i −131.848 + 318.310i −1076.36 721.852i 1010.64 + 1010.64i −2548.98 + 3206.23i −1368.41 + 6416.71i 511.913 5488.75i
5.15 −15.1511 + 5.14246i 80.9573 + 2.63085i 203.110 155.828i 228.408 551.426i −1240.12 + 376.460i −1274.74 1274.74i −2276.00 + 3405.44i 6547.16 + 425.973i −624.940 + 9529.28i
5.16 −15.1188 5.23657i 42.6326 + 68.8728i 201.157 + 158.341i 291.478 703.690i −283.898 1264.52i 1230.25 + 1230.25i −2212.08 3447.30i −2925.92 + 5872.45i −8091.72 + 9112.61i
5.17 −14.9250 5.76571i −67.4612 44.8329i 189.513 + 172.107i 110.800 267.496i 748.367 + 1058.09i −831.309 831.309i −1836.17 3661.38i 2541.02 + 6048.96i −3196.00 + 3353.54i
5.18 −14.6951 + 6.32884i −73.8028 33.3788i 175.891 186.006i −269.566 + 650.791i 1295.79 + 23.4180i 1492.34 + 1492.34i −1407.54 + 3846.56i 4332.71 + 4926.90i −157.451 11269.5i
5.19 −14.5987 6.54815i 59.7468 54.6930i 170.243 + 191.189i −39.6613 + 95.7507i −1230.36 + 407.216i −2369.22 2369.22i −1233.40 3905.89i 578.349 6535.46i 1205.99 1138.13i
5.20 −14.2660 7.24436i 78.6822 + 19.2383i 151.038 + 206.696i −161.861 + 390.767i −983.113 844.456i 2604.83 + 2604.83i −657.334 4042.91i 5820.78 + 3027.42i 5139.97 4402.11i
See next 80 embeddings (of 504 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.126 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
32.g even 8 1 inner
96.p odd 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 96.9.p.a 504
3.b odd 2 1 inner 96.9.p.a 504
32.g even 8 1 inner 96.9.p.a 504
96.p odd 8 1 inner 96.9.p.a 504

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.9.p.a 504 1.a even 1 1 trivial
96.9.p.a 504 3.b odd 2 1 inner
96.9.p.a 504 32.g even 8 1 inner
96.9.p.a 504 96.p odd 8 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{9}^{\mathrm{new}}(96, [\chi])$$.