# Properties

 Label 276.2.m.a Level $276$ Weight $2$ Character orbit 276.m Analytic conductor $2.204$ Analytic rank $0$ Dimension $240$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$276 = 2^{2} \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 276.m (of order $$22$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.20387109579$$ Analytic rank: $$0$$ Dimension: $$240$$ Relative dimension: $$24$$ over $$\Q(\zeta_{22})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$240q - 4q^{2} - 4q^{6} - 4q^{8} + 24q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$240q - 4q^{2} - 4q^{6} - 4q^{8} + 24q^{9} - 8q^{16} + 4q^{18} + 4q^{24} + 24q^{25} - 40q^{26} + 32q^{29} + 36q^{32} - 22q^{34} - 22q^{36} - 110q^{38} - 22q^{40} - 16q^{41} - 110q^{42} - 154q^{44} - 88q^{46} - 56q^{48} - 40q^{49} - 142q^{50} - 70q^{52} - 18q^{54} - 110q^{56} - 46q^{58} - 22q^{60} + 40q^{62} - 48q^{64} - 16q^{69} - 72q^{70} + 4q^{72} + 22q^{74} + 110q^{76} - 192q^{77} + 198q^{80} - 24q^{81} + 172q^{82} - 200q^{85} + 220q^{86} + 176q^{88} - 88q^{89} + 154q^{92} - 16q^{93} + 126q^{94} - 44q^{96} - 88q^{97} + 228q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 −1.40160 + 0.188444i 0.540641 + 0.841254i 1.92898 0.528247i −1.29084 + 0.589505i −0.916292 1.07722i 1.16908 0.343272i −2.60411 + 1.10390i −0.415415 + 0.909632i 1.69815 1.06950i
7.2 −1.39487 0.233083i −0.540641 0.841254i 1.89135 + 0.650241i −1.34962 + 0.616349i 0.558044 + 1.29946i 3.09157 0.907767i −2.48663 1.34784i −0.415415 + 0.909632i 2.02620 0.545157i
7.3 −1.35202 + 0.414766i −0.540641 0.841254i 1.65594 1.12155i 3.59483 1.64171i 1.07988 + 0.913155i 0.696898 0.204628i −1.77369 + 2.20319i −0.415415 + 0.909632i −4.17938 + 3.71064i
7.4 −1.24424 0.672203i −0.540641 0.841254i 1.09629 + 1.67277i 1.03264 0.471590i 0.107195 + 1.41015i −4.96264 + 1.45716i −0.239604 2.81826i −0.415415 + 0.909632i −1.60186 0.107370i
7.5 −1.14571 0.829070i 0.540641 + 0.841254i 0.625286 + 1.89974i −3.51516 + 1.60532i 0.0780426 1.41206i 0.708179 0.207940i 0.858625 2.69495i −0.415415 + 0.909632i 5.35826 + 1.07509i
7.6 −1.03590 + 0.962761i 0.540641 + 0.841254i 0.146182 1.99465i 0.0885609 0.0404444i −1.36998 0.350947i 3.48963 1.02465i 1.76894 + 2.20700i −0.415415 + 0.909632i −0.0528020 + 0.127159i
7.7 −0.916755 1.07683i 0.540641 + 0.841254i −0.319120 + 1.97438i 1.92635 0.879733i 0.410251 1.35340i −0.906673 + 0.266223i 2.41862 1.46638i −0.415415 + 0.909632i −2.71331 1.26785i
7.8 −0.805537 + 1.16237i −0.540641 0.841254i −0.702219 1.87267i 0.0885609 0.0404444i 1.41336 + 0.0492354i −3.48963 + 1.02465i 2.74240 + 0.692265i −0.415415 + 0.909632i −0.0243277 + 0.135520i
7.9 −0.723978 1.21485i −0.540641 0.841254i −0.951711 + 1.75905i −2.68981 + 1.22840i −0.630583 + 1.26585i 0.0149354 0.00438542i 2.82599 0.117329i −0.415415 + 0.909632i 3.43968 + 2.37838i
7.10 −0.416968 1.35135i −0.540641 0.841254i −1.65228 + 1.12694i 3.16176 1.44393i −0.911395 + 1.08137i 3.54109 1.03976i 2.21183 + 1.76290i −0.415415 + 0.909632i −3.26960 3.67057i
7.11 −0.218131 + 1.39729i 0.540641 + 0.841254i −1.90484 0.609585i 3.59483 1.64171i −1.29341 + 0.571928i −0.696898 + 0.204628i 1.26727 2.52864i −0.415415 + 0.909632i 1.50979 + 5.38113i
7.12 0.0129431 + 1.41415i −0.540641 0.841254i −1.99966 + 0.0366070i −1.29084 + 0.589505i 1.18266 0.775438i −1.16908 + 0.343272i −0.0776497 2.82736i −0.415415 + 0.909632i −0.850358 1.81781i
7.13 0.101854 1.41054i 0.540641 + 0.841254i −1.97925 0.287338i −1.98780 + 0.907796i 1.24169 0.676911i −4.66467 + 1.36967i −0.606897 + 2.76255i −0.415415 + 0.909632i 1.07802 + 2.89633i
7.14 0.190297 1.40135i −0.540641 0.841254i −1.92757 0.533346i −0.953526 + 0.435461i −1.28177 + 0.597540i −1.56668 + 0.460018i −1.11422 + 2.59972i −0.415415 + 0.909632i 0.428780 + 1.41909i
7.15 0.429221 + 1.34750i 0.540641 + 0.841254i −1.63154 + 1.15676i −1.34962 + 0.616349i −0.901539 + 1.08960i −3.09157 + 0.907767i −2.25902 1.70200i −0.415415 + 0.909632i −1.40982 1.55406i
7.16 0.740078 1.20511i 0.540641 + 0.841254i −0.904569 1.78375i 1.98260 0.905423i 1.41392 0.0289372i 1.01964 0.299394i −2.81906 0.230009i −0.415415 + 0.909632i 0.376146 3.05933i
7.17 0.842436 + 1.13591i 0.540641 + 0.841254i −0.580604 + 1.91387i 1.03264 0.471590i −0.500137 + 1.32282i 4.96264 1.45716i −2.66311 + 0.952796i −0.415415 + 0.909632i 1.40562 + 0.775704i
7.18 0.983682 + 1.01606i −0.540641 0.841254i −0.0647382 + 1.99895i −3.51516 + 1.60532i 0.322942 1.37685i −0.708179 + 0.207940i −2.09473 + 1.90056i −0.415415 + 0.909632i −5.08889 1.99247i
7.19 1.08752 0.904050i −0.540641 0.841254i 0.365388 1.96634i 1.98260 0.905423i −1.34849 0.426112i −1.01964 + 0.299394i −1.38030 2.46876i −0.415415 + 0.909632i 1.33756 2.77703i
7.20 1.19634 + 0.754175i −0.540641 0.841254i 0.862440 + 1.80449i 1.92635 0.879733i −0.0123356 1.41416i 0.906673 0.266223i −0.329137 + 2.80921i −0.415415 + 0.909632i 2.96803 + 0.400347i
See next 80 embeddings (of 240 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 247.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
23.d odd 22 1 inner
92.h even 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 276.2.m.a 240
3.b odd 2 1 828.2.u.c 240
4.b odd 2 1 inner 276.2.m.a 240
12.b even 2 1 828.2.u.c 240
23.d odd 22 1 inner 276.2.m.a 240
69.g even 22 1 828.2.u.c 240
92.h even 22 1 inner 276.2.m.a 240
276.j odd 22 1 828.2.u.c 240

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
276.2.m.a 240 1.a even 1 1 trivial
276.2.m.a 240 4.b odd 2 1 inner
276.2.m.a 240 23.d odd 22 1 inner
276.2.m.a 240 92.h even 22 1 inner
828.2.u.c 240 3.b odd 2 1
828.2.u.c 240 12.b even 2 1
828.2.u.c 240 69.g even 22 1
828.2.u.c 240 276.j odd 22 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(276, [\chi])$$.