# Properties

 Label 143.4.g.a Level $143$ Weight $4$ Character orbit 143.g Analytic conductor $8.437$ Analytic rank $0$ Dimension $80$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$143 = 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 143.g (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.43727313082$$ Analytic rank: $$0$$ Dimension: $$80$$ Relative dimension: $$40$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$80 q - 8 q^{3} - 4 q^{5} + 640 q^{9}+O(q^{10})$$ 80 * q - 8 * q^3 - 4 * q^5 + 640 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$80 q - 8 q^{3} - 4 q^{5} + 640 q^{9} + 14 q^{11} + 280 q^{14} - 80 q^{15} - 952 q^{16} - 200 q^{20} - 424 q^{22} - 508 q^{26} - 848 q^{27} + 208 q^{31} + 860 q^{33} - 232 q^{34} - 340 q^{37} + 1308 q^{42} - 644 q^{44} - 1148 q^{45} - 280 q^{47} + 2420 q^{48} + 2976 q^{53} + 1652 q^{55} - 1972 q^{58} - 84 q^{59} + 1484 q^{60} - 4924 q^{66} + 2468 q^{67} - 3540 q^{70} + 1704 q^{71} + 2368 q^{78} - 3544 q^{80} + 6160 q^{81} + 32 q^{86} + 424 q^{89} - 5868 q^{91} - 164 q^{92} + 3944 q^{93} - 4936 q^{97} - 3750 q^{99}+O(q^{100})$$ 80 * q - 8 * q^3 - 4 * q^5 + 640 * q^9 + 14 * q^11 + 280 * q^14 - 80 * q^15 - 952 * q^16 - 200 * q^20 - 424 * q^22 - 508 * q^26 - 848 * q^27 + 208 * q^31 + 860 * q^33 - 232 * q^34 - 340 * q^37 + 1308 * q^42 - 644 * q^44 - 1148 * q^45 - 280 * q^47 + 2420 * q^48 + 2976 * q^53 + 1652 * q^55 - 1972 * q^58 - 84 * q^59 + 1484 * q^60 - 4924 * q^66 + 2468 * q^67 - 3540 * q^70 + 1704 * q^71 + 2368 * q^78 - 3544 * q^80 + 6160 * q^81 + 32 * q^86 + 424 * q^89 - 5868 * q^91 - 164 * q^92 + 3944 * q^93 - 4936 * q^97 - 3750 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
21.1 −3.84534 + 3.84534i 4.01609 21.5733i 8.40231 + 8.40231i −15.4432 + 15.4432i −18.8709 18.8709i 52.1938 + 52.1938i −10.8710 −64.6194
21.2 −3.74271 + 3.74271i −8.58845 20.0158i −0.720115 0.720115i 32.1441 32.1441i 5.95160 + 5.95160i 44.9718 + 44.9718i 46.7614 5.39037
21.3 −3.52796 + 3.52796i 5.29006 16.8930i −5.19660 5.19660i −18.6631 + 18.6631i 9.17347 + 9.17347i 31.3742 + 31.3742i 0.984731 36.6668
21.4 −3.45073 + 3.45073i −2.54445 15.8151i −13.2855 13.2855i 8.78022 8.78022i −17.9229 17.9229i 26.9677 + 26.9677i −20.5258 91.6893
21.5 −3.38682 + 3.38682i −2.36084 14.9411i 3.11894 + 3.11894i 7.99572 7.99572i 15.8418 + 15.8418i 23.5081 + 23.5081i −21.4265 −21.1266
21.6 −2.94074 + 2.94074i 7.57281 9.29587i 4.79920 + 4.79920i −22.2696 + 22.2696i 1.82819 + 1.82819i 3.81081 + 3.81081i 30.3474 −28.2264
21.7 −2.78445 + 2.78445i −4.21766 7.50635i 11.6430 + 11.6430i 11.7439 11.7439i −7.67226 7.67226i −1.37456 1.37456i −9.21134 −64.8389
21.8 −2.57365 + 2.57365i −0.422811 5.24731i −8.59350 8.59350i 1.08817 1.08817i 13.9093 + 13.9093i −7.08444 7.08444i −26.8212 44.2333
21.9 −2.51606 + 2.51606i −7.60421 4.66116i 2.32460 + 2.32460i 19.1327 19.1327i −13.4109 13.4109i −8.40074 8.40074i 30.8240 −11.6977
21.10 −2.48360 + 2.48360i 9.63275 4.33653i −13.2589 13.2589i −23.9239 + 23.9239i −7.66310 7.66310i −9.09859 9.09859i 65.7898 65.8596
21.11 −2.06691 + 2.06691i 5.96848 0.544246i 12.2628 + 12.2628i −12.3363 + 12.3363i 19.3939 + 19.3939i −15.4104 15.4104i 8.62277 −50.6922
21.12 −1.95429 + 1.95429i 2.44343 0.361505i 5.52317 + 5.52317i −4.77517 + 4.77517i −18.1442 18.1442i −16.3408 16.3408i −21.0296 −21.5877
21.13 −1.74793 + 1.74793i −3.12343 1.88950i −2.44120 2.44120i 5.45953 5.45953i 8.07208 + 8.07208i −17.2861 17.2861i −17.2442 8.53410
21.14 −1.57816 + 1.57816i −8.34617 3.01884i −13.3430 13.3430i 13.1716 13.1716i −5.61955 5.61955i −17.3895 17.3895i 42.6586 42.1147
21.15 −1.47926 + 1.47926i 4.19980 3.62357i −3.33873 3.33873i −6.21261 + 6.21261i −14.0682 14.0682i −17.1943 17.1943i −9.36165 9.87770
21.16 −1.20788 + 1.20788i −9.63914 5.08205i 9.61526 + 9.61526i 11.6429 11.6429i 19.2678 + 19.2678i −15.8016 15.8016i 65.9130 −23.2282
21.17 −0.629214 + 0.629214i 3.19940 7.20818i −11.3086 11.3086i −2.01311 + 2.01311i 15.7918 + 15.7918i −9.56920 9.56920i −16.7639 14.2311
21.18 −0.289795 + 0.289795i −0.816938 7.83204i 11.7413 + 11.7413i 0.236745 0.236745i 9.28117 + 9.28117i −4.58805 4.58805i −26.3326 −6.80516
21.19 −0.248253 + 0.248253i 8.60259 7.87674i 3.46248 + 3.46248i −2.13562 + 2.13562i 6.79614 + 6.79614i −3.94145 3.94145i 47.0046 −1.71914
21.20 −0.177069 + 0.177069i −5.26131 7.93729i −2.40696 2.40696i 0.931617 0.931617i −14.4545 14.4545i −2.82201 2.82201i 0.681391 0.852396
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 109.40 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
13.d odd 4 1 inner
143.g even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 143.4.g.a 80
11.b odd 2 1 inner 143.4.g.a 80
13.d odd 4 1 inner 143.4.g.a 80
143.g even 4 1 inner 143.4.g.a 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
143.4.g.a 80 1.a even 1 1 trivial
143.4.g.a 80 11.b odd 2 1 inner
143.4.g.a 80 13.d odd 4 1 inner
143.4.g.a 80 143.g even 4 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(143, [\chi])$$.