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 $2$

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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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q - 120 q^{10} + 48 q^{12} + 416 q^{14} + 120 q^{16} + 72 q^{18} - 160 q^{20} + 40 q^{22} - 656 q^{23} - 456 q^{24} - 40 q^{26} - 760 q^{28} + 1488 q^{31} - 1240 q^{32} - 1000 q^{34} + 912 q^{35} + 440 q^{38} + 1352 q^{40} - 1616 q^{43} + 1000 q^{44} + 1440 q^{46} + 2856 q^{50} + 1488 q^{51} + 2520 q^{52} - 1504 q^{53} + 216 q^{54} + 288 q^{55} - 392 q^{56} - 2016 q^{58} - 2752 q^{59} - 2448 q^{60} + 1824 q^{61} - 2928 q^{62} - 1008 q^{63} - 6888 q^{64} - 2736 q^{66} - 816 q^{67} - 5032 q^{68} + 1056 q^{69} - 1800 q^{70} - 448 q^{71} + 6944 q^{74} + 2208 q^{75} + 5032 q^{76} - 3808 q^{77} + 3528 q^{78} + 11336 q^{80} + 9320 q^{82} + 2368 q^{86} - 944 q^{88} + 3600 q^{91} - 7696 q^{92} - 16480 q^{94} - 5160 q^{96} - 10288 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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:

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 \(S_{4}^{\mathrm{new}}(96, [\chi])\).