# Properties

 Label 96.3.p.a Level $96$ Weight $3$ Character orbit 96.p Analytic conductor $2.616$ Analytic rank $0$ Dimension $120$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$96 = 2^{5} \cdot 3$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 96.p (of order $$8$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.61581053786$$ Analytic rank: $$0$$ Dimension: $$120$$ Relative dimension: $$30$$ over $$\Q(\zeta_{8})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$120q - 4q^{3} - 8q^{4} - 4q^{6} - 8q^{7} - 4q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$120q - 4q^{3} - 8q^{4} - 4q^{6} - 8q^{7} - 4q^{9} + 32q^{10} - 4q^{12} - 8q^{13} - 40q^{16} - 4q^{18} - 8q^{19} - 4q^{21} - 120q^{22} - 144q^{24} - 8q^{25} + 92q^{27} - 8q^{28} - 60q^{30} - 16q^{31} - 8q^{33} - 40q^{34} + 64q^{36} - 8q^{37} - 196q^{39} - 232q^{40} + 16q^{42} - 8q^{43} - 4q^{45} - 40q^{46} + 48q^{48} - 40q^{51} - 112q^{52} + 304q^{54} - 264q^{55} - 4q^{57} - 16q^{58} + 424q^{60} + 56q^{61} - 8q^{63} + 40q^{64} + 428q^{66} + 248q^{67} - 4q^{69} + 328q^{70} + 416q^{72} - 8q^{73} - 104q^{75} + 720q^{76} + 276q^{78} + 512q^{82} + 232q^{84} - 208q^{85} - 452q^{87} + 272q^{88} - 304q^{90} - 200q^{91} - 40q^{93} + 416q^{94} - 616q^{96} - 16q^{97} + 252q^{99} + O(q^{100})$$

## 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 −1.99935 0.0509440i 2.77972 + 1.12834i 3.99481 + 0.203710i 2.35810 5.69297i −5.50016 2.39756i −1.55454 1.55454i −7.97665 0.610799i 6.45370 + 6.27294i −5.00470 + 11.2621i
5.2 −1.99160 0.183158i 2.25564 1.97790i 3.93291 + 0.729554i −3.05886 + 7.38474i −4.85460 + 3.52603i 6.52353 + 6.52353i −7.69913 2.17332i 1.17585 8.92286i 7.44458 14.1472i
5.3 −1.99034 + 0.196320i −2.18088 2.06004i 3.92292 0.781488i 0.400040 0.965783i 4.74513 + 3.67202i −1.74071 1.74071i −7.65452 + 2.32558i 0.512501 + 8.98540i −0.606614 + 2.00077i
5.4 −1.84984 0.760319i −0.317768 + 2.98312i 2.84383 + 2.81294i −1.77368 + 4.28204i 2.85595 5.27670i −7.55140 7.55140i −3.12190 7.36571i −8.79805 1.89588i 6.53674 6.57253i
5.5 −1.76181 + 0.946587i −2.17853 + 2.06252i 2.20795 3.33541i 2.46183 5.94339i 1.88580 5.69594i 0.814968 + 0.814968i −0.732725 + 7.96637i 0.491998 8.98654i 1.28865 + 12.8015i
5.6 −1.50790 1.31386i −2.99990 + 0.0247672i 0.547547 + 3.96235i −0.0391005 + 0.0943970i 4.55610 + 3.90410i 2.42911 + 2.42911i 4.38032 6.69424i 8.99877 0.148598i 0.182984 0.0909690i
5.7 −1.50510 + 1.31707i 1.22667 + 2.73775i 0.530655 3.96464i −1.77994 + 4.29715i −5.45207 2.50497i 2.62887 + 2.62887i 4.42302 + 6.66610i −5.99054 + 6.71665i −2.98066 8.81195i
5.8 −1.44213 + 1.38573i 1.71367 2.46239i 0.159500 3.99682i −0.0258289 + 0.0623564i 0.940865 + 5.92577i −8.71239 8.71239i 5.30849 + 5.98497i −3.12668 8.43942i −0.0491605 0.125718i
5.9 −1.43387 1.39428i 0.960376 2.84213i 0.111984 + 3.99843i 2.53071 6.10968i −5.33977 + 2.73622i 1.09521 + 1.09521i 5.41435 5.88938i −7.15536 5.45902i −12.1473 + 5.23199i
5.10 −1.05135 + 1.70137i −1.64571 2.50831i −1.78934 3.57746i −1.03584 + 2.50074i 5.99779 0.162867i 7.44249 + 7.44249i 7.96782 + 0.716807i −3.58326 + 8.25592i −3.16566 4.39149i
5.11 −0.896196 1.78797i 2.31396 + 1.90935i −2.39366 + 3.20474i −0.794982 + 1.91926i 1.34009 5.84843i 4.69711 + 4.69711i 7.87517 + 1.40772i 1.70880 + 8.83629i 4.14403 0.298628i
5.12 −0.576838 1.91501i −0.719266 2.91250i −3.33452 + 2.20930i −3.63719 + 8.78095i −5.16256 + 3.05744i −5.54083 5.54083i 6.15430 + 5.11122i −7.96531 + 4.18972i 18.9137 + 1.90006i
5.13 −0.503078 + 1.93569i 2.97132 0.413808i −3.49383 1.94761i 2.54567 6.14580i −0.693801 + 5.95975i 7.32134 + 7.32134i 5.52764 5.78318i 8.65753 2.45911i 10.6157 + 8.01946i
5.14 −0.369569 + 1.96556i −2.74887 + 1.20154i −3.72684 1.45282i −0.710186 + 1.71454i −1.34580 5.84712i −3.31956 3.31956i 4.23292 6.78840i 6.11260 6.60576i −3.10757 2.02955i
5.15 −0.310806 1.97570i −1.34666 + 2.68076i −3.80680 + 1.22812i 2.72057 6.56803i 5.71494 + 1.82741i −5.53321 5.53321i 3.60958 + 7.13939i −5.37299 7.22018i −13.8221 3.33365i
5.16 0.310806 + 1.97570i 2.84782 + 0.943351i −3.80680 + 1.22812i −2.72057 + 6.56803i −0.978660 + 5.91965i −5.53321 5.53321i −3.60958 7.13939i 7.22018 + 5.37299i −13.8221 3.33365i
5.17 0.369569 1.96556i 2.79336 1.09413i −3.72684 1.45282i 0.710186 1.71454i −1.11823 5.89488i −3.31956 3.31956i −4.23292 + 6.78840i 6.60576 6.11260i −3.10757 2.02955i
5.18 0.503078 1.93569i −2.39365 + 1.80844i −3.49383 1.94761i −2.54567 + 6.14580i 2.29639 + 5.54316i 7.32134 + 7.32134i −5.52764 + 5.78318i 2.45911 8.65753i 10.6157 + 8.01946i
5.19 0.576838 + 1.91501i −1.55085 2.56805i −3.33452 + 2.20930i 3.63719 8.78095i 4.02324 4.45124i −5.54083 5.54083i −6.15430 5.11122i −4.18972 + 7.96531i 18.9137 + 1.90006i
5.20 0.896196 + 1.78797i −0.286103 + 2.98633i −2.39366 + 3.20474i 0.794982 1.91926i −5.59586 + 2.16479i 4.69711 + 4.69711i −7.87517 1.40772i −8.83629 1.70880i 4.14403 0.298628i
See next 80 embeddings (of 120 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 77.30 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
32.g even 8 1 inner
96.p odd 8 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 96.3.p.a 120
3.b odd 2 1 inner 96.3.p.a 120
4.b odd 2 1 384.3.p.a 120
12.b even 2 1 384.3.p.a 120
32.g even 8 1 inner 96.3.p.a 120
32.h odd 8 1 384.3.p.a 120
96.o even 8 1 384.3.p.a 120
96.p odd 8 1 inner 96.3.p.a 120

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.3.p.a 120 1.a even 1 1 trivial
96.3.p.a 120 3.b odd 2 1 inner
96.3.p.a 120 32.g even 8 1 inner
96.3.p.a 120 96.p odd 8 1 inner
384.3.p.a 120 4.b odd 2 1
384.3.p.a 120 12.b even 2 1
384.3.p.a 120 32.h odd 8 1
384.3.p.a 120 96.o even 8 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(96, [\chi])$$.