# Properties

 Label 336.2.bo.a Level 336 Weight 2 Character orbit 336.bo Analytic conductor 2.683 Analytic rank 0 Dimension 240 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ = $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 336.bo (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.68297350792$$ Analytic rank: $$0$$ Dimension: $$240$$ Relative dimension: $$60$$ over $$\Q(\zeta_{12})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$240q - 6q^{3} - 4q^{4} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$240q - 6q^{3} - 4q^{4} - 12q^{10} - 6q^{12} - 16q^{15} - 20q^{16} - 4q^{18} - 12q^{19} + 2q^{21} - 40q^{22} - 6q^{24} - 12q^{28} + 22q^{30} - 24q^{31} - 12q^{33} - 64q^{36} - 4q^{37} + 48q^{40} - 18q^{42} - 16q^{43} - 6q^{45} + 12q^{46} - 16q^{49} - 10q^{51} - 48q^{52} - 90q^{54} - 4q^{58} - 18q^{60} - 12q^{61} - 36q^{63} + 32q^{64} - 66q^{66} - 36q^{67} - 76q^{70} - 46q^{72} + 24q^{75} - 76q^{78} - 8q^{79} - 4q^{81} + 72q^{82} - 24q^{84} + 24q^{85} + 12q^{88} - 88q^{91} - 14q^{93} + 24q^{94} + 96q^{96} + 28q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −1.41011 0.107640i −1.45943 + 0.932779i 1.97683 + 0.303568i 4.12446 1.10515i 2.15836 1.15823i 2.48663 + 0.903685i −2.75487 0.640850i 1.25985 2.72264i −5.93491 + 1.11442i
5.2 −1.40544 0.157286i 0.268319 + 1.71114i 1.95052 + 0.442112i −1.46545 + 0.392666i −0.107968 2.44711i 2.54285 0.730693i −2.67180 0.928152i −2.85601 + 0.918264i 2.12136 0.321373i
5.3 −1.40387 0.170759i −1.18100 1.26698i 1.94168 + 0.479446i 0.813225 0.217903i 1.44161 + 1.98034i −2.57502 + 0.607683i −2.64399 1.00464i −0.210487 + 2.99261i −1.17887 + 0.167041i
5.4 −1.40230 0.183144i 1.05826 1.37116i 1.93292 + 0.513647i 0.478799 0.128294i −1.73512 + 1.72898i 1.18449 2.36579i −2.61647 1.07429i −0.760185 2.90209i −0.694919 + 0.0922180i
5.5 −1.38159 + 0.302030i −1.35899 + 1.07385i 1.81756 0.834561i −3.09219 + 0.828549i 1.55322 1.89407i −2.16452 + 1.52147i −2.25904 + 1.70197i 0.693695 2.91870i 4.02187 2.07865i
5.6 −1.35186 + 0.415301i 1.14217 + 1.30209i 1.65505 1.12286i 0.248314 0.0665356i −2.08481 1.28590i 0.154609 + 2.64123i −1.77107 + 2.20529i −0.390898 + 2.97442i −0.308054 + 0.193072i
5.7 −1.30302 + 0.549667i 1.61139 + 0.635165i 1.39573 1.43246i 2.20923 0.591960i −2.44880 + 0.0580921i −1.46235 2.20489i −1.03130 + 2.63371i 2.19313 + 2.04699i −2.55329 + 1.98568i
5.8 −1.28250 + 0.595974i −0.372920 1.69143i 1.28963 1.52868i 1.30488 0.349641i 1.48632 + 1.94701i 1.99747 + 1.73496i −0.742906 + 2.72912i −2.72186 + 1.26154i −1.46514 + 1.22609i
5.9 −1.27506 0.611746i 1.71517 0.241204i 1.25153 + 1.56002i 2.72073 0.729016i −2.33450 0.741703i −0.617966 + 2.57257i −0.641437 2.75473i 2.88364 0.827414i −3.91505 0.734858i
5.10 −1.26134 0.639550i 1.52248 + 0.825872i 1.18195 + 1.61338i −3.47557 + 0.931276i −1.39217 2.01540i −2.05754 1.66329i −0.459004 2.79093i 1.63587 + 2.51474i 4.97947 + 1.04815i
5.11 −1.15775 0.812169i −1.44132 + 0.960512i 0.680764 + 1.88057i −0.0464448 + 0.0124448i 2.44879 + 0.0585683i −1.71843 2.01172i 0.739191 2.73013i 1.15484 2.76882i 0.0638787 + 0.0233130i
5.12 −1.15113 + 0.821523i 0.961162 1.44089i 0.650200 1.89136i −1.84766 + 0.495078i 0.0773035 + 2.44827i −2.64062 + 0.164729i 0.805330 + 2.71135i −1.15234 2.76986i 1.72017 2.08779i
5.13 −1.14504 0.829988i 0.380888 1.68965i 0.622238 + 1.90074i −3.47185 + 0.930279i −1.83852 + 1.61859i −0.200208 + 2.63817i 0.865105 2.69288i −2.70985 1.28714i 4.74753 + 1.81639i
5.14 −1.05796 0.938469i −1.55615 0.760516i 0.238553 + 1.98572i −0.772302 + 0.206938i 0.932627 + 2.26500i 1.91339 1.82728i 1.61116 2.32469i 1.84323 + 2.36696i 1.01127 + 0.505849i
5.15 −1.02837 + 0.970804i −0.556326 + 1.64027i 0.115079 1.99669i −1.19193 + 0.319377i −1.02028 2.22689i 1.17942 2.36832i 1.82005 + 2.16505i −2.38100 1.82506i 0.915691 1.48557i
5.16 −0.902393 + 1.08889i −0.669822 + 1.59729i −0.371375 1.96522i 3.29199 0.882085i −1.13484 2.17075i −2.31271 + 1.28506i 2.47504 + 1.36901i −2.10268 2.13980i −2.01017 + 4.38061i
5.17 −0.857643 + 1.12448i −1.72911 0.100921i −0.528898 1.92880i −0.536765 + 0.143826i 1.59644 1.85779i 1.79545 + 1.94328i 2.62250 + 1.05949i 2.97963 + 0.349007i 0.298624 0.726931i
5.18 −0.840481 + 1.13736i 1.71872 + 0.214511i −0.587184 1.91186i −2.81798 + 0.755075i −1.68852 + 1.77451i 2.00119 1.73067i 2.66800 + 0.939042i 2.90797 + 0.737366i 1.50966 3.83969i
5.19 −0.832268 1.14338i 0.838260 + 1.51569i −0.614658 + 1.90321i 3.73210 1.00001i 1.03536 2.21992i 0.762031 2.53364i 2.68766 0.881188i −1.59464 + 2.54109i −4.24951 3.43495i
5.20 −0.813125 1.15708i −0.426736 + 1.67866i −0.677657 + 1.88170i −1.11305 + 0.298242i 2.28933 0.871192i 1.13651 + 2.38922i 2.72829 0.745953i −2.63579 1.43269i 1.25014 + 1.04538i
See next 80 embeddings (of 240 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 269.60 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
16.e even 4 1 inner
21.g even 6 1 inner
48.i odd 4 1 inner
112.x odd 12 1 inner
336.bo even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.2.bo.a 240
3.b odd 2 1 inner 336.2.bo.a 240
7.d odd 6 1 inner 336.2.bo.a 240
16.e even 4 1 inner 336.2.bo.a 240
21.g even 6 1 inner 336.2.bo.a 240
48.i odd 4 1 inner 336.2.bo.a 240
112.x odd 12 1 inner 336.2.bo.a 240
336.bo even 12 1 inner 336.2.bo.a 240

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.bo.a 240 1.a even 1 1 trivial
336.2.bo.a 240 3.b odd 2 1 inner
336.2.bo.a 240 7.d odd 6 1 inner
336.2.bo.a 240 16.e even 4 1 inner
336.2.bo.a 240 21.g even 6 1 inner
336.2.bo.a 240 48.i odd 4 1 inner
336.2.bo.a 240 112.x odd 12 1 inner
336.2.bo.a 240 336.bo even 12 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(336, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database