# 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 $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 192.r (of order $$16$$, degree $$8$$, minimal)

## Newform invariants

 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.

 $$\operatorname{Tr}(f)(q) =$$ $$128 q+O(q^{10})$$ 128 * q $$\operatorname{Tr}(f)(q) =$$ $$128 q - 16 q^{22} - 80 q^{26} - 80 q^{28} - 80 q^{32} - 80 q^{34} - 80 q^{38} - 80 q^{40} - 16 q^{44} + 48 q^{50} - 32 q^{51} + 48 q^{52} + 16 q^{54} - 64 q^{55} + 112 q^{56} - 128 q^{59} + 96 q^{60} + 96 q^{62} - 32 q^{63} + 96 q^{64} + 96 q^{66} - 32 q^{67} + 96 q^{68} + 96 q^{70} - 128 q^{71} + 112 q^{74} - 64 q^{75} + 16 q^{76} + 48 q^{78} - 32 q^{79} + 48 q^{80} - 80 q^{82} - 80 q^{88} - 96 q^{94}+O(q^{100})$$ 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

## 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.

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 181.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{2}^{\mathrm{new}}(192, [\chi])$$.