# Properties

 Label 287.3.p.a Level 287 Weight 3 Character orbit 287.p Analytic conductor 7.820 Analytic rank 0 Dimension 216 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$287 = 7 \cdot 41$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 287.p (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.82018358714$$ Analytic rank: $$0$$ Dimension: $$216$$ Relative dimension: $$54$$ over $$\Q(\zeta_{10})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$216q - 6q^{2} - 110q^{4} - 5q^{7} + 22q^{8} + 552q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$216q - 6q^{2} - 110q^{4} - 5q^{7} + 22q^{8} + 552q^{9} - 10q^{11} - 100q^{15} - 182q^{16} - 84q^{18} - 61q^{21} + 30q^{22} + 54q^{23} + 204q^{25} + 90q^{28} - 390q^{29} - 300q^{30} + 152q^{32} + 180q^{35} - 556q^{36} - 28q^{37} - 160q^{39} - 180q^{42} - 44q^{43} - 480q^{46} - 11q^{49} - 20q^{50} + 246q^{51} - 460q^{53} + 525q^{56} + 152q^{57} - 10q^{58} + 690q^{60} - 215q^{63} - 850q^{64} - 580q^{65} - 370q^{67} - 425q^{70} + 700q^{71} + 234q^{72} + 632q^{74} + 378q^{77} + 350q^{78} + 1096q^{81} - 160q^{84} + 184q^{86} - 600q^{88} + 838q^{91} + 534q^{92} - 780q^{93} + 1430q^{95} - 5q^{98} - 1110q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
146.1 −1.21309 3.73351i −3.29511 −9.23144 + 6.70704i 4.29860 + 5.91651i 3.99727 + 12.3023i −6.05523 3.51200i 23.5357 + 17.0997i 1.85777 16.8748 23.2261i
146.2 −1.21309 3.73351i 3.29511 −9.23144 + 6.70704i −4.29860 5.91651i −3.99727 12.3023i −6.96309 0.717909i 23.5357 + 17.0997i 1.85777 −16.8748 + 23.2261i
146.3 −1.10630 3.40483i −1.16868 −7.13292 + 5.18237i −1.31989 1.81667i 1.29291 + 3.97916i 4.70124 + 5.18636i 13.9509 + 10.1359i −7.63418 −4.72528 + 6.50378i
146.4 −1.10630 3.40483i 1.16868 −7.13292 + 5.18237i 1.31989 + 1.81667i −1.29291 3.97916i 6.85185 1.43253i 13.9509 + 10.1359i −7.63418 4.72528 6.50378i
146.5 −1.07676 3.31392i −5.20007 −6.58656 + 4.78542i −3.83595 5.27973i 5.59921 + 17.2326i −2.02913 + 6.69945i 11.6747 + 8.48214i 18.0408 −13.3662 + 18.3970i
146.6 −1.07676 3.31392i 5.20007 −6.58656 + 4.78542i 3.83595 + 5.27973i −5.59921 17.2326i 2.29624 6.61266i 11.6747 + 8.48214i 18.0408 13.3662 18.3970i
146.7 −0.974724 2.99989i −2.12660 −4.81319 + 3.49699i −2.94240 4.04986i 2.07284 + 6.37956i 1.62289 6.80927i 4.97468 + 3.61432i −4.47759 −9.28112 + 12.7744i
146.8 −0.974724 2.99989i 2.12660 −4.81319 + 3.49699i 2.94240 + 4.04986i −2.07284 6.37956i −2.68944 + 6.46273i 4.97468 + 3.61432i −4.47759 9.28112 12.7744i
146.9 −0.843522 2.59609i −5.83087 −2.79211 + 2.02859i 1.53896 + 2.11819i 4.91847 + 15.1375i 5.20818 4.67706i −1.21187 0.880473i 24.9991 4.20089 5.78202i
146.10 −0.843522 2.59609i 5.83087 −2.79211 + 2.02859i −1.53896 2.11819i −4.91847 15.1375i 1.46440 + 6.84511i −1.21187 0.880473i 24.9991 −4.20089 + 5.78202i
146.11 −0.831194 2.55815i −2.81846 −2.61719 + 1.90150i −0.0250128 0.0344271i 2.34268 + 7.21004i −6.96961 0.651603i −1.66465 1.20944i −1.05631 −0.0672794 + 0.0926022i
146.12 −0.831194 2.55815i 2.81846 −2.61719 + 1.90150i 0.0250128 + 0.0344271i −2.34268 7.21004i −6.02153 3.56947i −1.66465 1.20944i −1.05631 0.0672794 0.0926022i
146.13 −0.707080 2.17617i −3.12138 −0.999682 + 0.726311i 4.21845 + 5.80620i 2.20707 + 6.79266i −1.42526 + 6.85337i −5.11721 3.71787i 0.743042 9.65249 13.2855i
146.14 −0.707080 2.17617i 3.12138 −0.999682 + 0.726311i −4.21845 5.80620i −2.20707 6.79266i 2.87524 6.38224i −5.11721 3.71787i 0.743042 −9.65249 + 13.2855i
146.15 −0.698858 2.15086i −1.93902 −0.901748 + 0.655158i 4.75547 + 6.54535i 1.35510 + 4.17056i 6.03058 3.55416i −5.27919 3.83556i −5.24021 10.7548 14.8027i
146.16 −0.698858 2.15086i 1.93902 −0.901748 + 0.655158i −4.75547 6.54535i −1.35510 4.17056i 2.78976 + 6.42006i −5.27919 3.83556i −5.24021 −10.7548 + 14.8027i
146.17 −0.517693 1.59330i −2.17790 0.965480 0.701463i −3.00150 4.13121i 1.12748 + 3.47003i 6.85258 + 1.42904i −7.03882 5.11400i −4.25677 −5.02839 + 6.92098i
146.18 −0.517693 1.59330i 2.17790 0.965480 0.701463i 3.00150 + 4.13121i −1.12748 3.47003i 6.38382 + 2.87173i −7.03882 5.11400i −4.25677 5.02839 6.92098i
146.19 −0.358368 1.10294i −0.0821187 2.14801 1.56062i −1.73706 2.39085i 0.0294287 + 0.0905723i −6.11447 + 3.40783i −6.24394 4.53649i −8.99326 −2.01447 + 2.77268i
146.20 −0.358368 1.10294i 0.0821187 2.14801 1.56062i 1.73706 + 2.39085i −0.0294287 0.0905723i −2.94364 6.35098i −6.24394 4.53649i −8.99326 2.01447 2.77268i
See next 80 embeddings (of 216 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 230.54 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
41.f even 10 1 inner
287.p odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 287.3.p.a 216
7.b odd 2 1 inner 287.3.p.a 216
41.f even 10 1 inner 287.3.p.a 216
287.p odd 10 1 inner 287.3.p.a 216

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.3.p.a 216 1.a even 1 1 trivial
287.3.p.a 216 7.b odd 2 1 inner
287.3.p.a 216 41.f even 10 1 inner
287.3.p.a 216 287.p odd 10 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(287, [\chi])$$.