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Properties

Label 315.2.l.b
Level $315$
Weight $2$
Character orbit 315.l
Analytic conductor $2.515$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [315,2,Mod(121,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.l (of order \(3\), degree \(2\), 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: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{2} - 5 q^{3} + 14 q^{4} + 12 q^{5} + 3 q^{6} - 11 q^{7} + 12 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{2} - 5 q^{3} + 14 q^{4} + 12 q^{5} + 3 q^{6} - 11 q^{7} + 12 q^{8} - 3 q^{9} + q^{10} + q^{11} - 7 q^{12} - 4 q^{13} + 8 q^{14} - q^{15} + 10 q^{16} - 7 q^{17} + 18 q^{18} - 2 q^{19} + 7 q^{20} - 17 q^{21} + 19 q^{22} + q^{23} + 18 q^{24} - 12 q^{25} + 11 q^{26} - 11 q^{27} - 28 q^{28} - 16 q^{31} - 34 q^{32} + 7 q^{33} + q^{34} - 7 q^{35} - 25 q^{36} + 17 q^{37} - 35 q^{38} - 17 q^{39} + 6 q^{40} + 20 q^{41} - 36 q^{42} + 31 q^{43} - 7 q^{44} + 6 q^{45} - 10 q^{46} + 62 q^{47} - 36 q^{48} - 11 q^{49} - q^{50} + 14 q^{51} - 4 q^{52} + 8 q^{53} - 51 q^{54} + 2 q^{55} + 5 q^{57} + 45 q^{58} + 42 q^{59} - 23 q^{60} - 10 q^{61} + 14 q^{62} + 18 q^{63} - 56 q^{64} - 8 q^{65} + 4 q^{66} - 86 q^{67} - 48 q^{68} + 26 q^{69} - 5 q^{70} + 24 q^{71} - 6 q^{72} - 18 q^{73} + 9 q^{74} + 4 q^{75} - 13 q^{76} + 35 q^{77} + 19 q^{78} - 80 q^{79} + 5 q^{80} + 21 q^{81} + 5 q^{82} - 60 q^{83} + 35 q^{84} + 7 q^{85} + 12 q^{86} + 68 q^{87} + 50 q^{88} - 4 q^{89} + 12 q^{90} - 33 q^{91} - 18 q^{92} + 7 q^{93} + 22 q^{94} - 4 q^{95} + 6 q^{97} - 15 q^{98} - q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1 −2.56008 −1.18219 + 1.26587i 4.55403 0.500000 + 0.866025i 3.02651 − 3.24073i −0.992363 − 2.45259i −6.53853 −0.204852 − 2.99300i −1.28004 − 2.21710i
121.2 −1.71927 −0.324473 − 1.70139i 0.955889 0.500000 + 0.866025i 0.557856 + 2.92514i −2.52983 − 0.774581i 1.79511 −2.78943 + 1.10411i −0.859635 − 1.48893i
121.3 −1.61038 0.678306 + 1.59371i 0.593327 0.500000 + 0.866025i −1.09233 − 2.56648i −2.03107 + 1.69552i 2.26528 −2.07980 + 2.16204i −0.805191 − 1.39463i
121.4 −1.03554 −0.388944 − 1.68782i −0.927661 0.500000 + 0.866025i 0.402766 + 1.74780i 2.63139 + 0.275284i 3.03170 −2.69744 + 1.31293i −0.517769 − 0.896802i
121.5 −0.609814 1.67160 − 0.453588i −1.62813 0.500000 + 0.866025i −1.01937 + 0.276604i 0.731085 + 2.54274i 2.21248 2.58852 − 1.51644i −0.304907 − 0.528114i
121.6 −0.308078 −1.71752 + 0.223878i −1.90509 0.500000 + 0.866025i 0.529131 − 0.0689719i −2.36933 + 1.17741i 1.20307 2.89976 − 0.769029i −0.154039 − 0.266804i
121.7 0.297462 −1.63105 − 0.582822i −1.91152 0.500000 + 0.866025i −0.485175 − 0.173367i 1.24794 − 2.33295i −1.16353 2.32064 + 1.90122i 0.148731 + 0.257610i
121.8 0.518491 1.09609 − 1.34112i −1.73117 0.500000 + 0.866025i 0.568312 − 0.695356i −0.619045 − 2.57231i −1.93458 −0.597182 − 2.93996i 0.259245 + 0.449026i
121.9 1.19807 −0.773328 + 1.54983i −0.564635 0.500000 + 0.866025i −0.926499 + 1.85680i 0.433740 + 2.60996i −3.07261 −1.80393 − 2.39705i 0.599034 + 1.03756i
121.10 2.16352 1.43178 − 0.974681i 2.68083 0.500000 + 0.866025i 3.09769 − 2.10874i −2.41085 + 1.08987i 1.47299 1.09999 − 2.79106i 1.08176 + 1.87367i
121.11 2.32555 −1.66925 − 0.462165i 3.40817 0.500000 + 0.866025i −3.88192 − 1.07479i 2.02670 + 1.70073i 3.27475 2.57281 + 1.54294i 1.16277 + 2.01398i
121.12 2.34008 0.308978 + 1.70427i 3.47596 0.500000 + 0.866025i 0.723033 + 3.98812i −1.61838 − 2.09305i 3.45385 −2.80906 + 1.05316i 1.17004 + 2.02657i
151.1 −2.56008 −1.18219 − 1.26587i 4.55403 0.500000 − 0.866025i 3.02651 + 3.24073i −0.992363 + 2.45259i −6.53853 −0.204852 + 2.99300i −1.28004 + 2.21710i
151.2 −1.71927 −0.324473 + 1.70139i 0.955889 0.500000 − 0.866025i 0.557856 − 2.92514i −2.52983 + 0.774581i 1.79511 −2.78943 − 1.10411i −0.859635 + 1.48893i
151.3 −1.61038 0.678306 − 1.59371i 0.593327 0.500000 − 0.866025i −1.09233 + 2.56648i −2.03107 − 1.69552i 2.26528 −2.07980 − 2.16204i −0.805191 + 1.39463i
151.4 −1.03554 −0.388944 + 1.68782i −0.927661 0.500000 − 0.866025i 0.402766 − 1.74780i 2.63139 − 0.275284i 3.03170 −2.69744 − 1.31293i −0.517769 + 0.896802i
151.5 −0.609814 1.67160 + 0.453588i −1.62813 0.500000 − 0.866025i −1.01937 − 0.276604i 0.731085 − 2.54274i 2.21248 2.58852 + 1.51644i −0.304907 + 0.528114i
151.6 −0.308078 −1.71752 − 0.223878i −1.90509 0.500000 − 0.866025i 0.529131 + 0.0689719i −2.36933 − 1.17741i 1.20307 2.89976 + 0.769029i −0.154039 + 0.266804i
151.7 0.297462 −1.63105 + 0.582822i −1.91152 0.500000 − 0.866025i −0.485175 + 0.173367i 1.24794 + 2.33295i −1.16353 2.32064 − 1.90122i 0.148731 − 0.257610i
151.8 0.518491 1.09609 + 1.34112i −1.73117 0.500000 − 0.866025i 0.568312 + 0.695356i −0.619045 + 2.57231i −1.93458 −0.597182 + 2.93996i 0.259245 − 0.449026i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.l.b yes 24
3.b odd 2 1 945.2.l.b 24
7.c even 3 1 315.2.k.b ✓ 24
9.c even 3 1 315.2.k.b ✓ 24
9.d odd 6 1 945.2.k.b 24
21.h odd 6 1 945.2.k.b 24
63.h even 3 1 inner 315.2.l.b yes 24
63.j odd 6 1 945.2.l.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.k.b ✓ 24 7.c even 3 1
315.2.k.b ✓ 24 9.c even 3 1
315.2.l.b yes 24 1.a even 1 1 trivial
315.2.l.b yes 24 63.h even 3 1 inner
945.2.k.b 24 9.d odd 6 1
945.2.k.b 24 21.h odd 6 1
945.2.l.b 24 3.b odd 2 1
945.2.l.b 24 63.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - T_{2}^{11} - 15 T_{2}^{10} + 12 T_{2}^{9} + 80 T_{2}^{8} - 42 T_{2}^{7} - 186 T_{2}^{6} + \cdots + 3 \) acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\). Copy content Toggle raw display