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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 $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [231,2,Mod(4,231)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(231, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 20, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("231.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 231.y (of order \(15\), degree \(8\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.84454428669\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(8\) over \(\Q(\zeta_{15})\)
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.

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

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 4.8
Significant digits:
Format:

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 \( T_{2}^{64} - 13 T_{2}^{62} - 16 T_{2}^{61} + 61 T_{2}^{60} + 96 T_{2}^{59} + 238 T_{2}^{58} + 614 T_{2}^{57} + \cdots + 625 \) acting on \(S_{2}^{\mathrm{new}}(231, [\chi])\). Copy content Toggle raw display