# Properties

 Label 336.2.w.b Level 336 Weight 2 Character orbit 336.w Analytic conductor 2.683 Analytic rank 0 Dimension 28 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 336.w (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.68297350792$$ Analytic rank: $$0$$ Dimension: $$28$$ Relative dimension: $$14$$ over $$\Q(i)$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$28q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$28q - 4q^{10} - 4q^{11} - 8q^{12} + 8q^{15} + 4q^{16} + 8q^{19} - 28q^{20} - 8q^{22} - 16q^{24} - 20q^{26} + 4q^{28} + 4q^{29} + 8q^{30} + 24q^{33} + 12q^{34} + 4q^{36} + 4q^{37} + 60q^{38} - 56q^{40} - 4q^{42} + 20q^{43} + 56q^{44} - 44q^{46} + 16q^{48} - 28q^{49} + 20q^{50} - 8q^{51} - 32q^{52} + 20q^{53} - 4q^{54} - 12q^{56} - 12q^{58} - 12q^{60} - 40q^{61} - 60q^{62} - 28q^{63} + 60q^{64} + 16q^{65} + 24q^{66} + 4q^{67} - 108q^{68} - 16q^{69} - 4q^{70} - 12q^{72} + 28q^{74} - 16q^{75} - 8q^{76} - 4q^{77} + 12q^{78} + 24q^{79} - 72q^{80} - 28q^{81} + 36q^{82} + 40q^{83} + 48q^{85} + 24q^{86} + 4q^{88} - 4q^{90} + 52q^{92} - 8q^{94} + 40q^{96} + 72q^{97} - 4q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
85.1 −1.40848 + 0.127230i −0.707107 + 0.707107i 1.96762 0.358402i −0.805240 0.805240i 0.905980 1.08591i 1.00000i −2.72576 + 0.755143i 1.00000i 1.23661 + 1.03171i
85.2 −1.40007 0.199481i 0.707107 0.707107i 1.92041 + 0.558576i 1.25389 + 1.25389i −1.13106 + 0.848947i 1.00000i −2.57730 1.16513i 1.00000i −1.50541 2.00566i
85.3 −1.14710 0.827134i −0.707107 + 0.707107i 0.631698 + 1.89762i 2.39875 + 2.39875i 1.39600 0.226253i 1.00000i 0.844961 2.69927i 1.00000i −0.767531 4.73571i
85.4 −0.884433 1.10353i 0.707107 0.707107i −0.435558 + 1.95200i −1.84737 1.84737i −1.40570 0.154926i 1.00000i 2.53931 1.24576i 1.00000i −0.404756 + 3.67251i
85.5 −0.819262 + 1.15274i −0.707107 + 0.707107i −0.657621 1.88879i −0.979857 0.979857i −0.235805 1.39442i 1.00000i 2.71605 + 0.789348i 1.00000i 1.93228 0.326762i
85.6 −0.394485 + 1.35808i 0.707107 0.707107i −1.68876 1.07148i −0.646579 0.646579i 0.681365 + 1.23925i 1.00000i 2.12135 1.87079i 1.00000i 1.13317 0.623041i
85.7 −0.371814 1.36446i −0.707107 + 0.707107i −1.72351 + 1.01465i 0.116928 + 0.116928i 1.22773 + 0.701908i 1.00000i 2.02528 + 1.97440i 1.00000i 0.116068 0.203020i
85.8 0.421626 + 1.34990i 0.707107 0.707107i −1.64446 + 1.13831i 2.44528 + 2.44528i 1.25266 + 0.656389i 1.00000i −2.22995 1.73992i 1.00000i −2.26989 + 4.33188i
85.9 0.428680 + 1.34768i −0.707107 + 0.707107i −1.63247 + 1.15545i −1.77230 1.77230i −1.25607 0.649829i 1.00000i −2.25697 1.70472i 1.00000i 1.62874 3.14824i
85.10 0.647888 1.25708i 0.707107 0.707107i −1.16048 1.62889i 2.52107 + 2.52107i −0.430761 1.34701i 1.00000i −2.79950 + 0.403476i 1.00000i 4.80254 1.53580i
85.11 0.919506 1.07448i 0.707107 0.707107i −0.309018 1.97598i −1.77267 1.77267i −0.109584 1.40996i 1.00000i −2.40730 1.48489i 1.00000i −3.53468 + 0.274720i
85.12 1.25104 + 0.659465i −0.707107 + 0.707107i 1.13021 + 1.65004i 2.66347 + 2.66347i −1.35093 + 0.418308i 1.00000i 0.325801 + 2.80960i 1.00000i 1.57565 + 5.08857i
85.13 1.35983 0.388411i −0.707107 + 0.707107i 1.69827 1.05634i −3.03597 3.03597i −0.686897 + 1.23619i 1.00000i 1.89907 2.09608i 1.00000i −5.30760 2.94920i
85.14 1.39708 + 0.219481i 0.707107 0.707107i 1.90366 + 0.613265i −0.539395 0.539395i 1.14308 0.832687i 1.00000i 2.52496 + 1.27460i 1.00000i −0.635190 0.871964i
253.1 −1.40848 0.127230i −0.707107 0.707107i 1.96762 + 0.358402i −0.805240 + 0.805240i 0.905980 + 1.08591i 1.00000i −2.72576 0.755143i 1.00000i 1.23661 1.03171i
253.2 −1.40007 + 0.199481i 0.707107 + 0.707107i 1.92041 0.558576i 1.25389 1.25389i −1.13106 0.848947i 1.00000i −2.57730 + 1.16513i 1.00000i −1.50541 + 2.00566i
253.3 −1.14710 + 0.827134i −0.707107 0.707107i 0.631698 1.89762i 2.39875 2.39875i 1.39600 + 0.226253i 1.00000i 0.844961 + 2.69927i 1.00000i −0.767531 + 4.73571i
253.4 −0.884433 + 1.10353i 0.707107 + 0.707107i −0.435558 1.95200i −1.84737 + 1.84737i −1.40570 + 0.154926i 1.00000i 2.53931 + 1.24576i 1.00000i −0.404756 3.67251i
253.5 −0.819262 1.15274i −0.707107 0.707107i −0.657621 + 1.88879i −0.979857 + 0.979857i −0.235805 + 1.39442i 1.00000i 2.71605 0.789348i 1.00000i 1.93228 + 0.326762i
253.6 −0.394485 1.35808i 0.707107 + 0.707107i −1.68876 + 1.07148i −0.646579 + 0.646579i 0.681365 1.23925i 1.00000i 2.12135 + 1.87079i 1.00000i 1.13317 + 0.623041i
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 253.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.2.w.b 28
4.b odd 2 1 1344.2.w.b 28
8.b even 2 1 2688.2.w.d 28
8.d odd 2 1 2688.2.w.c 28
16.e even 4 1 inner 336.2.w.b 28
16.e even 4 1 2688.2.w.d 28
16.f odd 4 1 1344.2.w.b 28
16.f odd 4 1 2688.2.w.c 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.w.b 28 1.a even 1 1 trivial
336.2.w.b 28 16.e even 4 1 inner
1344.2.w.b 28 4.b odd 2 1
1344.2.w.b 28 16.f odd 4 1
2688.2.w.c 28 8.d odd 2 1
2688.2.w.c 28 16.f odd 4 1
2688.2.w.d 28 8.b even 2 1
2688.2.w.d 28 16.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{28} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(336, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database