# Properties

 Label 231.2.ba.a Level 231 Weight 2 Character orbit 231.ba Analytic conductor 1.845 Analytic rank 0 Dimension 128 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$231 = 3 \cdot 7 \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 231.ba (of order $$30$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.84454428669$$ Analytic rank: $$0$$ Dimension: $$128$$ Relative dimension: $$16$$ over $$\Q(\zeta_{30})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$128q - 12q^{4} + 12q^{5} - 10q^{7} - 40q^{8} - 16q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$128q - 12q^{4} + 12q^{5} - 10q^{7} - 40q^{8} - 16q^{9} - 2q^{11} + 12q^{14} + 12q^{15} + 40q^{16} - 60q^{17} - 10q^{18} + 52q^{22} - 24q^{23} - 90q^{24} - 20q^{25} + 24q^{26} + 30q^{28} + 40q^{29} - 18q^{31} + 18q^{33} - 80q^{35} - 24q^{36} - 8q^{37} - 24q^{38} - 90q^{40} + 14q^{42} - 82q^{44} + 12q^{45} + 70q^{46} - 24q^{47} - 94q^{49} - 20q^{51} + 4q^{53} - 104q^{56} - 32q^{58} + 48q^{59} + 30q^{61} - 10q^{63} - 48q^{64} + 36q^{66} - 40q^{67} + 180q^{68} + 146q^{70} - 32q^{71} + 10q^{72} + 90q^{73} + 40q^{74} - 24q^{75} - 72q^{78} + 50q^{79} + 228q^{80} + 16q^{81} + 168q^{82} - 60q^{84} - 20q^{85} + 146q^{86} + 16q^{88} + 48q^{91} - 204q^{92} + 44q^{93} + 10q^{95} - 44q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1 −2.64148 + 0.277631i 0.743145 + 0.669131i 4.94404 1.05089i 1.44910 + 3.25473i −2.14877 1.56118i 1.42250 + 2.23081i −7.71576 + 2.50700i 0.104528 + 0.994522i −4.73138 8.19500i
19.2 −2.63866 + 0.277334i −0.743145 0.669131i 4.92930 1.04776i 0.00599929 + 0.0134746i 2.14648 + 1.55951i 0.918955 2.48103i −7.66950 + 2.49197i 0.104528 + 0.994522i −0.0195671 0.0338911i
19.3 −2.47627 + 0.260267i 0.743145 + 0.669131i 4.10789 0.873159i −1.61981 3.63814i −2.01438 1.46353i −2.49816 + 0.871324i −5.20890 + 1.69247i 0.104528 + 0.994522i 4.95797 + 8.58745i
19.4 −2.08559 + 0.219204i −0.743145 0.669131i 2.34532 0.498514i −0.216464 0.486186i 1.69657 + 1.23263i −0.578918 + 2.58164i −0.793223 + 0.257734i 0.104528 + 0.994522i 0.558028 + 0.966533i
19.5 −1.62945 + 0.171263i 0.743145 + 0.669131i 0.669495 0.142305i 0.520050 + 1.16805i −1.32552 0.963045i 1.22463 2.34527i 2.04994 0.666066i 0.104528 + 0.994522i −1.04744 1.81422i
19.6 −1.58179 + 0.166253i −0.743145 0.669131i 0.518133 0.110133i 1.31309 + 2.94925i 1.28675 + 0.934876i −2.10118 1.60780i 2.22405 0.722639i 0.104528 + 0.994522i −2.56736 4.44679i
19.7 −0.554274 + 0.0582566i −0.743145 0.669131i −1.65247 + 0.351243i −0.990165 2.22395i 0.450887 + 0.327589i 0.582413 + 2.58085i 1.95556 0.635400i 0.104528 + 0.994522i 0.678382 + 1.17499i
19.8 −0.373095 + 0.0392138i −0.743145 0.669131i −1.81863 + 0.386562i 0.294587 + 0.661654i 0.303503 + 0.220507i 2.60364 0.470147i 1.37694 0.447395i 0.104528 + 0.994522i −0.135855 0.235308i
19.9 −0.0764050 + 0.00803049i 0.743145 + 0.669131i −1.95052 + 0.414596i 0.750633 + 1.68595i −0.0621535 0.0451571i −1.66664 + 2.05482i 0.291832 0.0948219i 0.104528 + 0.994522i −0.0708911 0.122787i
19.10 −0.0732542 + 0.00769933i 0.743145 + 0.669131i −1.95099 + 0.414695i −1.46180 3.28325i −0.0595904 0.0432949i 0.114944 2.64325i 0.279831 0.0909225i 0.104528 + 0.994522i 0.132362 + 0.229257i
19.11 0.788039 0.0828262i 0.743145 + 0.669131i −1.34215 + 0.285283i 0.490588 + 1.10188i 0.641048 + 0.465749i 2.62289 + 0.347026i −2.54123 + 0.825697i 0.104528 + 0.994522i 0.477866 + 0.827689i
19.12 0.884754 0.0929913i −0.743145 0.669131i −1.18215 + 0.251275i 1.63822 + 3.67950i −0.719723 0.522910i −1.92178 + 1.81845i −2.71472 + 0.882066i 0.104528 + 0.994522i 1.79158 + 3.10311i
19.13 1.80279 0.189480i −0.743145 0.669131i 1.25784 0.267361i −1.26266 2.83599i −1.46652 1.06549i 1.48247 2.19141i −1.23104 + 0.399989i 0.104528 + 0.994522i −2.81368 4.87343i
19.14 1.95149 0.205109i 0.743145 + 0.669131i 1.80993 0.384713i 0.888231 + 1.99500i 1.58748 + 1.15337i −1.14019 2.38746i −0.279246 + 0.0907326i 0.104528 + 0.994522i 2.14256 + 3.71103i
19.15 2.26565 0.238129i 0.743145 + 0.669131i 3.12016 0.663211i −0.832978 1.87090i 1.84304 + 1.33905i −1.73794 + 1.99489i 2.57800 0.837642i 0.104528 + 0.994522i −2.33275 4.04044i
19.16 2.65417 0.278965i −0.743145 0.669131i 5.01052 1.06502i −0.0958279 0.215233i −2.15910 1.56868i −2.64356 0.107664i 7.92533 2.57510i 0.104528 + 0.994522i −0.314386 0.544533i
40.1 −1.08550 2.43806i 0.207912 + 0.978148i −3.42759 + 3.80673i −0.234311 + 0.0246271i 2.15910 1.56868i −2.20197 + 1.46674i 7.92533 + 2.57510i −0.913545 + 0.406737i 0.314386 + 0.544533i
40.2 −0.926598 2.08117i −0.207912 0.978148i −2.13444 + 2.37053i −2.03674 + 0.214070i −1.84304 + 1.33905i −0.233458 + 2.63543i 2.57800 + 0.837642i −0.913545 + 0.406737i 2.33275 + 4.04044i
40.3 −0.798113 1.79259i −0.207912 0.978148i −1.23814 + 1.37509i 2.17184 0.228269i −1.58748 + 1.15337i −2.32575 1.26131i −0.279246 0.0907326i −0.913545 + 0.406737i −2.14256 3.71103i
40.4 −0.737298 1.65600i 0.207912 + 0.978148i −0.860460 + 0.955638i −3.08737 + 0.324496i 1.46652 1.06549i −0.0887369 2.64426i −1.23104 0.399989i −0.913545 + 0.406737i 2.81368 + 4.87343i
See next 80 embeddings (of 128 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 178.16 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.d odd 6 1 inner
11.d odd 10 1 inner
77.n even 30 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 231.2.ba.a 128
3.b odd 2 1 693.2.cg.c 128
7.d odd 6 1 inner 231.2.ba.a 128
11.d odd 10 1 inner 231.2.ba.a 128
21.g even 6 1 693.2.cg.c 128
33.f even 10 1 693.2.cg.c 128
77.n even 30 1 inner 231.2.ba.a 128
231.bf odd 30 1 693.2.cg.c 128

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.ba.a 128 1.a even 1 1 trivial
231.2.ba.a 128 7.d odd 6 1 inner
231.2.ba.a 128 11.d odd 10 1 inner
231.2.ba.a 128 77.n even 30 1 inner
693.2.cg.c 128 3.b odd 2 1
693.2.cg.c 128 21.g even 6 1
693.2.cg.c 128 33.f even 10 1
693.2.cg.c 128 231.bf odd 30 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(231, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database