# Properties

 Label 144.3.w.a Level $144$ Weight $3$ Character orbit 144.w Analytic conductor $3.924$ Analytic rank $0$ Dimension $184$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 144.w (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.92371580679$$ Analytic rank: $$0$$ Dimension: $$184$$ Relative dimension: $$46$$ over $$\Q(\zeta_{12})$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$184q - 6q^{2} - 4q^{3} - 2q^{4} - 6q^{5} - 10q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$184q - 6q^{2} - 4q^{3} - 2q^{4} - 6q^{5} - 10q^{6} - 8q^{10} - 6q^{11} - 64q^{12} - 2q^{13} - 6q^{14} - 8q^{15} - 2q^{16} + 54q^{18} - 8q^{19} + 120q^{20} - 22q^{21} - 2q^{22} - 160q^{24} + 44q^{27} - 72q^{28} - 6q^{29} - 90q^{30} - 4q^{31} - 6q^{32} - 8q^{33} + 6q^{34} - 202q^{36} - 8q^{37} - 6q^{38} - 2q^{40} + 44q^{42} - 2q^{43} + 46q^{45} - 160q^{46} - 12q^{47} - 118q^{48} + 472q^{49} + 228q^{50} - 48q^{51} - 2q^{52} + 206q^{54} - 300q^{56} - 92q^{58} - 438q^{59} - 90q^{60} - 2q^{61} - 204q^{63} + 244q^{64} - 12q^{65} - 508q^{66} - 2q^{67} - 144q^{68} + 14q^{69} + 96q^{70} + 6q^{72} + 246q^{74} + 152q^{75} - 158q^{76} - 6q^{77} + 304q^{78} - 4q^{79} - 8q^{81} - 388q^{82} - 726q^{83} + 542q^{84} + 48q^{85} + 894q^{86} + 22q^{88} - 528q^{90} - 204q^{91} - 348q^{92} + 62q^{93} - 18q^{94} - 12q^{95} + 262q^{96} - 4q^{97} + 286q^{99} + O(q^{100})$$

## 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 −1.99959 + 0.0402970i 0.538615 + 2.95125i 3.99675 0.161155i −2.08673 7.78777i −1.19594 5.87960i −2.60739 + 1.50538i −7.98539 + 0.483303i −8.41979 + 3.17918i 4.48643 + 15.4883i
5.2 −1.99458 + 0.147176i −2.86084 + 0.903108i 3.95668 0.587107i −0.0992523 0.370415i 5.57325 2.22237i 3.11010 1.79562i −7.80549 + 1.75336i 7.36879 5.16729i 0.252482 + 0.724213i
5.3 −1.98351 + 0.256277i −1.40595 2.65015i 3.86864 1.01666i 1.00259 + 3.74171i 3.46789 + 4.89630i 4.65369 2.68681i −7.41296 + 3.00800i −5.04662 + 7.45195i −2.94756 7.16478i
5.4 −1.88582 0.666085i 2.99994 + 0.0192045i 3.11266 + 2.51224i −0.345472 1.28932i −5.64456 2.03443i 4.11923 2.37824i −4.19657 6.81093i 8.99926 + 0.115225i −0.207297 + 2.66154i
5.5 −1.88357 + 0.672432i 2.93495 + 0.621365i 3.09567 2.53315i 1.03307 + 3.85549i −5.94600 + 0.803167i −11.2715 + 6.50760i −4.12754 + 6.85298i 8.22781 + 3.64735i −4.53842 6.56740i
5.6 −1.84760 0.765753i −0.120474 + 2.99758i 2.82724 + 2.82961i 2.37709 + 8.87143i 2.51799 5.44607i 0.152137 0.0878365i −3.05683 7.39295i −8.97097 0.722263i 2.40141 18.2111i
5.7 −1.79350 0.885072i 0.352522 2.97922i 2.43329 + 3.17476i 0.899691 + 3.35769i −3.26907 + 5.03122i −9.56176 + 5.52049i −1.55423 7.84757i −8.75146 2.10048i 1.35820 6.81832i
5.8 −1.73513 + 0.994646i 1.85623 2.35678i 2.02136 3.45168i −0.777454 2.90150i −0.876653 + 5.93561i 2.72697 1.57442i −0.0741259 + 7.99966i −2.10879 8.74946i 4.23494 + 4.26119i
5.9 −1.61016 1.18633i −2.90827 + 0.736170i 1.18523 + 3.82037i −0.295811 1.10398i 5.55613 + 2.26483i −7.30360 + 4.21674i 2.62382 7.55748i 7.91611 4.28197i −0.833388 + 2.12852i
5.10 −1.59704 1.20393i −1.75135 2.43573i 1.10109 + 3.84547i −2.49590 9.31484i −0.135487 + 5.99847i 7.55275 4.36058i 2.87120 7.46701i −2.86558 + 8.53161i −7.22838 + 17.8811i
5.11 −1.58134 + 1.22449i 1.74774 + 2.43832i 1.00126 3.87266i 0.973349 + 3.63259i −5.74946 1.71573i 11.7048 6.75777i 3.15868 + 7.35002i −2.89081 + 8.52310i −5.98725 4.55250i
5.12 −1.51447 + 1.30628i −2.22089 2.01684i 0.587245 3.95666i −1.88738 7.04379i 5.99804 + 0.153334i −8.89672 + 5.13652i 4.27915 + 6.75935i 0.864706 + 8.95836i 12.0596 + 8.20216i
5.13 −1.26432 + 1.54968i −2.98386 0.310733i −0.803013 3.91857i 2.23236 + 8.33129i 4.25408 4.23117i −3.69610 + 2.13394i 7.08778 + 3.70989i 8.80689 + 1.85437i −15.7332 7.07393i
5.14 −1.16958 + 1.62237i −1.89172 + 2.32839i −1.26417 3.79498i −1.06962 3.99189i −1.56500 5.79230i 2.82785 1.63266i 7.63541 + 2.38759i −1.84282 8.80931i 7.72733 + 2.93351i
5.15 −1.09867 1.67120i 0.894041 + 2.86368i −1.58584 + 3.67221i −0.835695 3.11886i 3.80354 4.64037i 0.108974 0.0629164i 7.87933 1.38428i −7.40138 + 5.12050i −4.29409 + 4.82321i
5.16 −1.08605 1.67944i 1.99667 2.23905i −1.64101 + 3.64789i 1.20857 + 4.51043i −5.92880 0.921569i 7.69955 4.44534i 7.90861 1.20579i −1.02665 8.94125i 6.26242 6.92824i
5.17 −0.786053 + 1.83905i 1.51963 2.58664i −2.76424 2.89119i 1.99100 + 7.43050i 3.56247 + 4.82792i −1.50577 + 0.869355i 7.48989 2.81097i −4.38144 7.86149i −15.2301 2.17922i
5.18 −0.762335 1.84901i −2.07303 + 2.16854i −2.83769 + 2.81913i −0.115759 0.432017i 5.58999 + 2.17990i 7.23679 4.17816i 7.37588 + 3.09780i −0.405106 8.99088i −0.710557 + 0.543380i
5.19 −0.702398 + 1.87260i 1.41045 + 2.64776i −3.01327 2.63062i −0.226081 0.843746i −5.94890 + 0.781440i −7.54519 + 4.35622i 7.04263 3.79492i −5.02124 + 7.46908i 1.73880 + 0.169286i
5.20 −0.608739 1.90511i 2.61769 1.46550i −3.25887 + 2.31943i −2.13498 7.96785i −4.38543 4.09487i −10.2471 + 5.91616i 6.40256 + 4.79658i 4.70460 7.67247i −13.8800 + 8.91771i
See next 80 embeddings (of 184 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 101.46 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
16.e even 4 1 inner
144.w odd 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.3.w.a 184
3.b odd 2 1 432.3.x.a 184
9.c even 3 1 432.3.x.a 184
9.d odd 6 1 inner 144.3.w.a 184
16.e even 4 1 inner 144.3.w.a 184
48.i odd 4 1 432.3.x.a 184
144.w odd 12 1 inner 144.3.w.a 184
144.x even 12 1 432.3.x.a 184

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.3.w.a 184 1.a even 1 1 trivial
144.3.w.a 184 9.d odd 6 1 inner
144.3.w.a 184 16.e even 4 1 inner
144.3.w.a 184 144.w odd 12 1 inner
432.3.x.a 184 3.b odd 2 1
432.3.x.a 184 9.c even 3 1
432.3.x.a 184 48.i odd 4 1
432.3.x.a 184 144.x even 12 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(144, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database