# Properties

 Label 96.4.n.a Level $96$ Weight $4$ Character orbit 96.n Analytic conductor $5.664$ Analytic rank $0$ Dimension $96$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [96,4,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, 4, names="a")

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

chi := DirichletCharacter("96.13");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$96 = 2^{5} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$5.66418336055$$ Analytic rank: $$0$$ Dimension: $$96$$ Relative dimension: $$24$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$96 q+O(q^{10})$$ 96 * q $$\operatorname{Tr}(f)(q) =$$ $$96 q - 120 q^{10} + 48 q^{12} + 416 q^{14} + 120 q^{16} + 72 q^{18} - 160 q^{20} + 40 q^{22} - 656 q^{23} - 456 q^{24} - 40 q^{26} - 760 q^{28} + 1488 q^{31} - 1240 q^{32} - 1000 q^{34} + 912 q^{35} + 440 q^{38} + 1352 q^{40} - 1616 q^{43} + 1000 q^{44} + 1440 q^{46} + 2856 q^{50} + 1488 q^{51} + 2520 q^{52} - 1504 q^{53} + 216 q^{54} + 288 q^{55} - 392 q^{56} - 2016 q^{58} - 2752 q^{59} - 2448 q^{60} + 1824 q^{61} - 2928 q^{62} - 1008 q^{63} - 6888 q^{64} - 2736 q^{66} - 816 q^{67} - 5032 q^{68} + 1056 q^{69} - 1800 q^{70} - 448 q^{71} + 6944 q^{74} + 2208 q^{75} + 5032 q^{76} - 3808 q^{77} + 3528 q^{78} + 11336 q^{80} + 9320 q^{82} + 2368 q^{86} - 944 q^{88} + 3600 q^{91} - 7696 q^{92} - 16480 q^{94} - 5160 q^{96} - 10288 q^{98}+O(q^{100})$$ 96 * q - 120 * q^10 + 48 * q^12 + 416 * q^14 + 120 * q^16 + 72 * q^18 - 160 * q^20 + 40 * q^22 - 656 * q^23 - 456 * q^24 - 40 * q^26 - 760 * q^28 + 1488 * q^31 - 1240 * q^32 - 1000 * q^34 + 912 * q^35 + 440 * q^38 + 1352 * q^40 - 1616 * q^43 + 1000 * q^44 + 1440 * q^46 + 2856 * q^50 + 1488 * q^51 + 2520 * q^52 - 1504 * q^53 + 216 * q^54 + 288 * q^55 - 392 * q^56 - 2016 * q^58 - 2752 * q^59 - 2448 * q^60 + 1824 * q^61 - 2928 * q^62 - 1008 * q^63 - 6888 * q^64 - 2736 * q^66 - 816 * q^67 - 5032 * q^68 + 1056 * q^69 - 1800 * q^70 - 448 * q^71 + 6944 * q^74 + 2208 * q^75 + 5032 * q^76 - 3808 * q^77 + 3528 * q^78 + 11336 * q^80 + 9320 * q^82 + 2368 * q^86 - 944 * q^88 + 3600 * q^91 - 7696 * q^92 - 16480 * q^94 - 5160 * q^96 - 10288 * q^98

## 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 −2.82755 0.0702733i −1.14805 + 2.77164i 7.99012 + 0.397403i 1.11467 0.461711i 3.44095 7.75628i 10.3263 10.3263i −22.5646 1.68517i −6.36396 6.36396i −3.18423 + 1.22718i
13.2 −2.82068 0.209163i 1.14805 2.77164i 7.91250 + 1.17997i −11.2604 + 4.66423i −3.81801 + 7.57778i 19.4197 19.4197i −22.0718 4.98332i −6.36396 6.36396i 32.7377 10.8010i
13.3 −2.75464 + 0.641854i 1.14805 2.77164i 7.17605 3.53615i 14.0875 5.83525i −1.38347 + 8.37174i −4.38635 + 4.38635i −17.4977 + 14.3468i −6.36396 6.36396i −35.0607 + 25.1161i
13.4 −2.43224 1.44367i 1.14805 2.77164i 3.83163 + 7.02272i 1.29231 0.535293i −6.79367 + 5.08390i −24.0002 + 24.0002i 0.819034 22.6126i −6.36396 6.36396i −3.91600 0.563708i
13.5 −2.41430 + 1.47348i −1.14805 + 2.77164i 3.65771 7.11486i 4.44682 1.84193i −1.31222 8.38320i −17.8659 + 17.8659i 1.65281 + 22.5670i −6.36396 6.36396i −8.02191 + 10.9993i
13.6 −2.10989 1.88371i −1.14805 + 2.77164i 0.903279 + 7.94884i −2.45258 + 1.01589i 7.64322 3.68526i −1.25402 + 1.25402i 13.0675 18.4727i −6.36396 6.36396i 7.08832 + 2.47652i
13.7 −1.99600 + 2.00399i 1.14805 2.77164i −0.0319341 7.99994i −6.77784 + 2.80747i 3.26282 + 7.83288i −5.88177 + 5.88177i 16.0955 + 15.9039i −6.36396 6.36396i 7.90246 19.1864i
13.8 −1.75107 + 2.22121i −1.14805 + 2.77164i −1.86751 7.77897i −20.2950 + 8.40645i −4.14606 7.40339i 16.9467 16.9467i 20.5488 + 9.47341i −6.36396 6.36396i 16.8655 59.7996i
13.9 −1.02540 + 2.63601i −1.14805 + 2.77164i −5.89712 5.40592i 18.3205 7.58861i −6.12886 5.86831i 9.93733 9.93733i 20.2970 10.0017i −6.36396 6.36396i 1.21784 + 56.0744i
13.10 −0.789922 2.71588i 1.14805 2.77164i −6.75205 + 4.29067i 15.1675 6.28259i −8.43432 0.928592i 12.6068 12.6068i 16.9866 + 14.9485i −6.36396 6.36396i −29.0439 36.2304i
13.11 −0.609387 + 2.76200i 1.14805 2.77164i −7.25730 3.36625i −5.00405 + 2.07274i 6.95566 + 4.85991i −5.72490 + 5.72490i 13.7201 17.9933i −6.36396 6.36396i −2.67552 15.0843i
13.12 −0.598743 2.76433i −1.14805 + 2.77164i −7.28301 + 3.31024i 12.1847 5.04706i 8.34910 + 1.51409i −21.3715 + 21.3715i 13.5113 + 18.1506i −6.36396 6.36396i −21.2472 30.6606i
13.13 0.285711 2.81396i 1.14805 2.77164i −7.83674 1.60796i −9.00970 + 3.73194i −7.47127 4.02245i −14.1767 + 14.1767i −6.76377 + 21.5929i −6.36396 6.36396i 7.92736 + 26.4192i
13.14 0.540520 + 2.77630i 1.14805 2.77164i −7.41568 + 3.00129i 8.98718 3.72261i 8.31544 + 1.68921i 21.9910 21.9910i −12.3408 18.9659i −6.36396 6.36396i 15.1928 + 22.9390i
13.15 0.733576 + 2.73164i −1.14805 + 2.77164i −6.92373 + 4.00773i −5.53124 + 2.29112i −8.41331 1.10285i −1.36712 + 1.36712i −16.0268 15.9732i −6.36396 6.36396i −10.3161 13.4287i
13.16 0.803238 2.71197i −1.14805 + 2.77164i −6.70962 4.35672i 7.39764 3.06420i 6.59446 + 5.33977i 17.8317 17.8317i −17.2047 + 14.6968i −6.36396 6.36396i −2.36797 22.5235i
13.17 1.76523 + 2.20997i 1.14805 2.77164i −1.76791 + 7.80221i −18.0829 + 7.49018i 8.15181 2.35543i −6.45609 + 6.45609i −20.3634 + 9.86570i −6.36396 6.36396i −48.4736 26.7407i
13.18 1.77134 2.20508i −1.14805 + 2.77164i −1.72473 7.81187i −14.5482 + 6.02605i 4.07809 + 7.44105i −14.8835 + 14.8835i −20.2809 10.0343i −6.36396 6.36396i −12.4818 + 42.7540i
13.19 2.14640 1.84200i 1.14805 2.77164i 1.21409 7.90734i 11.0630 4.58245i −2.64117 8.06376i −5.90766 + 5.90766i −11.9594 19.2087i −6.36396 6.36396i 15.3048 30.2138i
13.20 2.27724 1.67756i 1.14805 2.77164i 2.37161 7.64038i −14.5962 + 6.04595i −2.03520 8.23759i 19.3791 19.3791i −7.41646 21.3775i −6.36396 6.36396i −23.0966 + 38.2541i
See all 96 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 13.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 96.4.n.a 96
4.b odd 2 1 384.4.n.a 96
32.g even 8 1 inner 96.4.n.a 96
32.h odd 8 1 384.4.n.a 96

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.4.n.a 96 1.a even 1 1 trivial
96.4.n.a 96 32.g even 8 1 inner
384.4.n.a 96 4.b odd 2 1
384.4.n.a 96 32.h odd 8 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(96, [\chi])$$.