Properties

 Label 231.2.y.a Level 231 Weight 2 Character orbit 231.y Analytic conductor 1.845 Analytic rank 0 Dimension 64 CM no Inner twists 4

Related objects

Newspace parameters

 Level: $$N$$ = $$231 = 3 \cdot 7 \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 231.y (of order $$15$$, degree $$8$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$1.84454428669$$ Analytic rank: $$0$$ Dimension: $$64$$ Relative dimension: $$8$$ over $$\Q(\zeta_{15})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$64q - 8q^{3} + 10q^{4} - q^{7} + 8q^{8} + 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$64q - 8q^{3} + 10q^{4} - q^{7} + 8q^{8} + 8q^{9} - 22q^{10} - 13q^{11} + 30q^{12} - 8q^{13} - 26q^{14} + 4q^{17} - 5q^{18} + 10q^{19} + 24q^{20} + 2q^{21} - 38q^{22} - 8q^{23} + 14q^{24} - 2q^{25} + 4q^{26} + 16q^{27} - 67q^{28} + 2q^{29} - 3q^{30} + 25q^{31} + 72q^{32} - 12q^{33} - 56q^{34} + 19q^{35} - 20q^{36} - 12q^{37} - 37q^{38} - 4q^{39} - 9q^{40} + 20q^{41} + 9q^{42} - 100q^{43} - 5q^{44} - 33q^{46} - 18q^{47} + 10q^{48} + 29q^{49} - 46q^{50} + 6q^{51} + 26q^{52} - 49q^{53} - 10q^{54} - 24q^{55} - 48q^{56} + 20q^{57} - 40q^{58} + q^{59} + 12q^{60} + 3q^{61} + 4q^{62} - 7q^{63} - 24q^{64} + 82q^{65} + 11q^{66} + 76q^{67} - 39q^{68} - 16q^{69} + 59q^{70} + 70q^{71} - 14q^{72} - 3q^{73} + 32q^{74} - 28q^{75} + 104q^{76} + 38q^{77} - 12q^{78} - 15q^{79} - 83q^{80} + 8q^{81} + 42q^{82} + 68q^{83} + 27q^{84} + 62q^{85} - 47q^{86} - 54q^{87} - 64q^{88} + 82q^{89} - 16q^{90} - 10q^{91} + 190q^{92} - 25q^{93} - 6q^{94} - 53q^{95} + 8q^{96} - 32q^{97} - 152q^{98} + 16q^{99} + O(q^{100})$$

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1 −0.210352 + 2.00137i −0.669131 0.743145i −2.00492 0.426159i 0.455221 + 0.202678i 1.62806 1.18285i 1.13813 + 2.38845i 0.0309144 0.0951446i −0.104528 + 0.994522i −0.501389 + 0.868431i
4.2 −0.176917 + 1.68325i −0.669131 0.743145i −0.845742 0.179768i 2.77196 + 1.23415i 1.36928 0.994840i −0.180307 2.63960i −0.593816 + 1.82758i −0.104528 + 0.994522i −2.56780 + 4.44756i
4.3 −0.133110 + 1.26646i −0.669131 0.743145i 0.370096 + 0.0786664i −3.47779 1.54841i 1.03023 0.748506i −1.94739 1.79100i −0.935917 + 2.88046i −0.104528 + 0.994522i 2.42393 4.19837i
4.4 −0.00677966 + 0.0645042i −0.669131 0.743145i 1.95218 + 0.414949i −1.02829 0.457823i 0.0524724 0.0381234i 2.43175 + 1.04239i −0.0800864 + 0.246481i −0.104528 + 0.994522i 0.0365029 0.0632249i
4.5 0.0140677 0.133845i −0.669131 0.743145i 1.93858 + 0.412058i 3.31825 + 1.47738i −0.108880 + 0.0791057i −1.93235 + 1.80722i 0.165600 0.509665i −0.104528 + 0.994522i 0.244421 0.423350i
4.6 0.176448 1.67879i −0.669131 0.743145i −0.830899 0.176613i 2.44652 + 1.08926i −1.36565 + 0.992202i 2.46827 0.952696i 0.600157 1.84709i −0.104528 + 0.994522i 2.26032 3.91499i
4.7 0.194840 1.85378i −0.669131 0.743145i −1.44224 0.306558i −3.17278 1.41261i −1.50800 + 1.09563i −2.04677 + 1.67652i 0.302713 0.931655i −0.104528 + 0.994522i −3.23685 + 5.60640i
4.8 0.286258 2.72356i −0.669131 0.743145i −5.37956 1.14346i −1.31309 0.584627i −2.21555 + 1.60969i 2.06489 1.65416i −2.96170 + 9.11518i −0.104528 + 0.994522i −1.96815 + 3.40894i
16.1 −2.26770 0.482014i 0.104528 0.994522i 3.08302 + 1.37265i −0.445594 0.494882i −0.716412 + 2.20489i 2.34578 + 1.22364i −2.57853 1.87341i −0.978148 0.207912i 0.771931 + 1.33702i
16.2 −1.27562 0.271142i 0.104528 0.994522i −0.273395 0.121723i −0.103450 0.114892i −0.402996 + 1.24029i 1.44851 2.21401i 2.42586 + 1.76249i −0.978148 0.207912i 0.100810 + 0.174609i
16.3 −0.957375 0.203496i 0.104528 0.994522i −0.951935 0.423829i 2.18684 + 2.42873i −0.302455 + 0.930859i −0.413889 + 2.61318i 2.40878 + 1.75008i −0.978148 0.207912i −1.59938 2.77022i
16.4 0.417062 + 0.0886493i 0.104528 0.994522i −1.66101 0.739529i −0.589560 0.654772i 0.131759 0.405511i −2.62927 0.294825i −1.31708 0.956916i −0.978148 0.207912i −0.187838 0.325345i
16.5 1.17765 + 0.250317i 0.104528 0.994522i −0.502893 0.223903i −2.18435 2.42597i 0.372043 1.14503i 1.90032 1.84087i −2.48423 1.80490i −0.978148 0.207912i −1.96514 3.40372i
16.6 1.72647 + 0.366972i 0.104528 0.994522i 1.01892 + 0.453654i 0.597146 + 0.663197i 0.545426 1.67865i 2.29418 + 1.31786i −1.26323 0.917790i −0.978148 0.207912i 0.787577 + 1.36412i
16.7 2.04772 + 0.435257i 0.104528 0.994522i 2.17664 + 0.969101i 2.51828 + 2.79684i 0.646918 1.99101i −2.47353 0.938955i 0.648035 + 0.470825i −0.978148 0.207912i 3.93941 + 6.82326i
16.8 2.67076 + 0.567688i 0.104528 0.994522i 4.98362 + 2.21885i −1.97931 2.19824i 0.843750 2.59679i −1.73835 + 1.99452i 7.63253 + 5.54536i −0.978148 0.207912i −4.03835 6.99462i
25.1 −2.50180 1.11387i 0.978148 + 0.207912i 3.68004 + 4.08710i 0.150245 1.42949i −2.21555 1.60969i −2.64282 + 0.124535i −2.96170 9.11518i 0.913545 + 0.406737i −1.96815 + 3.40894i
25.2 −1.70284 0.758153i 0.978148 + 0.207912i 0.986607 + 1.09574i 0.363031 3.45401i −1.50800 1.09563i 2.64131 0.153270i 0.302713 + 0.931655i 0.913545 + 0.406737i −3.23685 + 5.60640i
25.3 −1.54210 0.686586i 0.978148 + 0.207912i 0.568401 + 0.631273i −0.279932 + 2.66338i −1.36565 0.992202i −2.55686 0.680067i 0.600157 + 1.84709i 0.913545 + 0.406737i 2.26032 3.91499i
25.4 −0.122947 0.0547397i 0.978148 + 0.207912i −1.32614 1.47283i −0.379677 + 3.61238i −0.108880 0.0791057i 2.62556 0.326267i 0.165600 + 0.509665i 0.913545 + 0.406737i 0.244421 0.423350i
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 214.8 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.c even 3 1 inner
11.c even 5 1 inner
77.m even 15 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 231.2.y.a 64
3.b odd 2 1 693.2.by.d 64
7.c even 3 1 inner 231.2.y.a 64
11.c even 5 1 inner 231.2.y.a 64
21.h odd 6 1 693.2.by.d 64
33.h odd 10 1 693.2.by.d 64
77.m even 15 1 inner 231.2.y.a 64
231.z odd 30 1 693.2.by.d 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.y.a 64 1.a even 1 1 trivial
231.2.y.a 64 7.c even 3 1 inner
231.2.y.a 64 11.c even 5 1 inner
231.2.y.a 64 77.m even 15 1 inner
693.2.by.d 64 3.b odd 2 1
693.2.by.d 64 21.h odd 6 1
693.2.by.d 64 33.h odd 10 1
693.2.by.d 64 231.z odd 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{64} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(231, [\chi])$$.

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database