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Properties

Label 192.2.r.a
Level $192$
Weight $2$
Character orbit 192.r
Analytic conductor $1.533$
Analytic rank $0$
Dimension $128$
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] = [192,2,Mod(13,192)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(192, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([0, 15, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 192.r (of order \(16\), degree \(8\), 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: \(1.53312771881\)
Analytic rank: \(0\)
Dimension: \(128\)
Relative dimension: \(16\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 128 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 128 q - 16 q^{22} - 80 q^{26} - 80 q^{28} - 80 q^{32} - 80 q^{34} - 80 q^{38} - 80 q^{40} - 16 q^{44} + 48 q^{50} - 32 q^{51} + 48 q^{52} + 16 q^{54} - 64 q^{55} + 112 q^{56} - 128 q^{59} + 96 q^{60} + 96 q^{62} - 32 q^{63} + 96 q^{64} + 96 q^{66} - 32 q^{67} + 96 q^{68} + 96 q^{70} - 128 q^{71} + 112 q^{74} - 64 q^{75} + 16 q^{76} + 48 q^{78} - 32 q^{79} + 48 q^{80} - 80 q^{82} - 80 q^{88} - 96 q^{94}+O(q^{100}) \) 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 −1.40448 + 0.165612i 0.831470 − 0.555570i 1.94515 − 0.465199i 3.01271 − 0.599265i −1.07578 + 0.917991i −1.64361 − 3.96803i −2.65488 + 0.975505i 0.382683 − 0.923880i −4.13205 + 1.34060i
13.2 −1.37379 + 0.335725i −0.831470 + 0.555570i 1.77458 − 0.922428i −1.61488 + 0.321219i 0.955743 − 1.04238i 0.265757 + 0.641595i −2.12821 + 1.86299i 0.382683 − 0.923880i 2.11065 − 0.983439i
13.3 −1.34188 + 0.446493i 0.831470 − 0.555570i 1.60129 − 1.19828i −1.15816 + 0.230372i −0.867675 + 1.11675i 1.66636 + 4.02294i −1.61371 + 2.32291i 0.382683 − 0.923880i 1.45125 − 0.826242i
13.4 −1.21525 − 0.723297i −0.831470 + 0.555570i 0.953684 + 1.75798i −1.45351 + 0.289121i 1.41229 − 0.0737597i −0.968670 − 2.33858i 0.112572 − 2.82619i 0.382683 − 0.923880i 1.97550 + 0.699962i
13.5 −0.968999 + 1.03007i −0.831470 + 0.555570i −0.122081 − 1.99627i 2.48770 − 0.494834i 0.233418 − 1.39482i −1.03562 − 2.50021i 2.17459 + 1.80863i 0.382683 − 0.923880i −1.90087 + 3.04199i
13.6 −0.744400 + 1.20244i 0.831470 − 0.555570i −0.891737 − 1.79020i −4.01045 + 0.797728i 0.0490954 + 1.41336i −1.73589 − 4.19080i 2.81642 + 0.260360i 0.382683 − 0.923880i 2.02616 − 5.41616i
13.7 −0.643008 − 1.25958i 0.831470 − 0.555570i −1.17308 + 1.61984i 1.10082 − 0.218966i −1.23443 − 0.690066i −0.899420 − 2.17139i 2.79462 + 0.436020i 0.382683 − 0.923880i −0.983641 − 1.24577i
13.8 −0.392875 − 1.35855i −0.831470 + 0.555570i −1.69130 + 1.06748i −0.00368381 0.000732755i 1.08143 + 0.911321i 1.78909 + 4.31925i 2.11469 + 1.87832i 0.382683 − 0.923880i 0.00244276 + 0.00471675i
13.9 −0.187202 + 1.40177i 0.831470 − 0.555570i −1.92991 − 0.524828i 2.51755 − 0.500772i 0.623128 + 1.26953i 0.484569 + 1.16985i 1.09697 − 2.60704i 0.382683 − 0.923880i 0.230676 + 3.62277i
13.10 0.431051 − 1.34692i −0.831470 + 0.555570i −1.62839 − 1.16118i −2.72905 + 0.542841i 0.389904 + 1.35940i −1.23163 − 2.97341i −2.26594 + 1.69279i 0.382683 − 0.923880i −0.445193 + 3.90980i
13.11 0.688951 − 1.23505i 0.831470 − 0.555570i −1.05069 − 1.70178i 2.41656 − 0.480684i −0.113315 − 1.40967i 1.33165 + 3.21488i −2.82565 + 0.125218i 0.382683 − 0.923880i 1.07122 − 3.31574i
13.12 0.711584 + 1.22215i −0.831470 + 0.555570i −0.987297 + 1.73932i 3.50591 − 0.697369i −1.27065 − 0.620845i 0.519328 + 1.25377i −2.82826 + 0.0310494i 0.382683 − 0.923880i 3.34704 + 3.78851i
13.13 0.929504 + 1.06584i −0.831470 + 0.555570i −0.272044 + 1.98141i −4.03340 + 0.802293i −1.36501 − 0.369811i −0.0612023 − 0.147755i −2.36474 + 1.55177i 0.382683 − 0.923880i −4.60418 − 3.55324i
13.14 1.11696 + 0.867416i 0.831470 − 0.555570i 0.495179 + 1.93773i −0.668642 + 0.133001i 1.41063 + 0.100683i 0.440731 + 1.06402i −1.12772 + 2.59388i 0.382683 − 0.923880i −0.862210 − 0.431434i
13.15 1.14999 − 0.823123i −0.831470 + 0.555570i 0.644937 − 1.89316i 1.87933 − 0.373823i −0.498876 + 1.32330i −0.0424280 − 0.102430i −0.816635 − 2.70797i 0.382683 − 0.923880i 1.85351 − 1.97681i
13.16 1.39610 − 0.225635i 0.831470 − 0.555570i 1.89818 − 0.630017i −1.24882 + 0.248406i 1.03546 − 0.963239i −0.409753 − 0.989232i 2.50789 − 1.30786i 0.382683 − 0.923880i −1.68743 + 0.628577i
37.1 −1.40493 − 0.161763i −0.555570 + 0.831470i 1.94767 + 0.454533i 0.573343 − 2.88239i 0.915039 − 1.07829i −0.457164 + 1.10369i −2.66281 − 0.953649i −0.382683 − 0.923880i −1.27177 + 3.95682i
37.2 −1.35486 − 0.405391i 0.555570 − 0.831470i 1.67132 + 1.09850i −0.550464 + 2.76737i −1.08979 + 0.901306i −1.44187 + 3.48098i −1.81909 − 2.16585i −0.382683 − 0.923880i 1.86767 − 3.52626i
37.3 −1.27221 + 0.617652i 0.555570 − 0.831470i 1.23701 − 1.57156i −0.108368 + 0.544803i −0.193240 + 1.40095i 0.796770 − 1.92357i −0.603056 + 2.76339i −0.382683 − 0.923880i −0.198632 − 0.760034i
37.4 −1.27201 + 0.618052i −0.555570 + 0.831470i 1.23602 − 1.57234i −0.391022 + 1.96580i 0.192800 − 1.40101i −0.156914 + 0.378825i −0.600450 + 2.76396i −0.382683 − 0.923880i −0.717582 − 2.74219i
See next 80 embeddings (of 128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
64.i even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.2.r.a ✓ 128
3.b odd 2 1 576.2.bd.b 128
4.b odd 2 1 768.2.r.a 128
64.i even 16 1 inner 192.2.r.a ✓ 128
64.j odd 16 1 768.2.r.a 128
192.q odd 16 1 576.2.bd.b 128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.2.r.a ✓ 128 1.a even 1 1 trivial
192.2.r.a ✓ 128 64.i even 16 1 inner
576.2.bd.b 128 3.b odd 2 1
576.2.bd.b 128 192.q odd 16 1
768.2.r.a 128 4.b odd 2 1
768.2.r.a 128 64.j odd 16 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(192, [\chi])\).