Properties

Label 315.2.l.c
Level $315$
Weight $2$
Character orbit 315.l
Analytic conductor $2.515$
Analytic rank $0$
Dimension $36$
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: \(36\)
Relative dimension: \(18\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - q^{3} + 44 q^{4} - 18 q^{5} - 4 q^{6} - q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - q^{3} + 44 q^{4} - 18 q^{5} - 4 q^{6} - q^{7} - 9 q^{9} + q^{11} + 8 q^{12} + 2 q^{13} + 9 q^{14} - q^{15} + 60 q^{16} - 5 q^{17} - 21 q^{18} - 2 q^{19} - 22 q^{20} - 23 q^{21} - 19 q^{22} - 3 q^{23} - 32 q^{24} - 18 q^{25} - 4 q^{26} + 17 q^{27} + 5 q^{28} - 8 q^{29} + 2 q^{30} - 20 q^{32} - 35 q^{33} + 10 q^{34} - q^{35} - 44 q^{36} - 15 q^{37} - 22 q^{38} + 7 q^{39} - 4 q^{41} + 57 q^{42} - 29 q^{43} - 7 q^{44} + 6 q^{45} - 24 q^{46} + 46 q^{47} - 19 q^{48} - 7 q^{49} + 42 q^{51} - 7 q^{52} + 21 q^{54} - 2 q^{55} - 12 q^{56} + 21 q^{57} - 20 q^{58} + 10 q^{59} - 13 q^{60} + 6 q^{61} - 12 q^{62} + 2 q^{63} + 128 q^{64} - 4 q^{65} - 12 q^{66} + 70 q^{67} - 17 q^{68} - 50 q^{69} - 3 q^{70} + 24 q^{71} - 10 q^{72} - 10 q^{73} + 22 q^{74} + 2 q^{75} + 10 q^{76} + 35 q^{77} + 66 q^{78} + 56 q^{79} - 30 q^{80} - 49 q^{81} - 8 q^{82} - 22 q^{83} - 86 q^{84} - 5 q^{85} + 19 q^{86} - 42 q^{87} - 50 q^{88} - 4 q^{89} + 3 q^{90} + 7 q^{91} - 50 q^{92} - q^{93} + 4 q^{94} + 4 q^{95} - 179 q^{96} + 16 q^{97} + 16 q^{98} - 89 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −2.69765 −0.273945 1.71025i 5.27730 −0.500000 0.866025i 0.739006 + 4.61365i −0.230272 2.63571i −8.84100 −2.84991 + 0.937027i 1.34882 + 2.33623i
121.2 −2.65219 1.69529 + 0.354953i 5.03414 −0.500000 0.866025i −4.49624 0.941405i −1.76710 + 1.96910i −8.04712 2.74802 + 1.20350i 1.32610 + 2.29687i
121.3 −2.35938 −0.431498 + 1.67744i 3.56668 −0.500000 0.866025i 1.01807 3.95772i 2.33019 + 1.25307i −3.69640 −2.62762 1.44762i 1.17969 + 2.04328i
121.4 −1.69039 −1.59654 0.671613i 0.857411 −0.500000 0.866025i 2.69877 + 1.13529i 2.40616 1.10017i 1.93142 2.09787 + 2.14451i 0.845194 + 1.46392i
121.5 −1.58441 1.09319 1.34348i 0.510363 −0.500000 0.866025i −1.73206 + 2.12863i −1.17061 + 2.37269i 2.36020 −0.609890 2.93735i 0.792206 + 1.37214i
121.6 −1.42579 −1.59516 + 0.674891i 0.0328702 −0.500000 0.866025i 2.27435 0.962251i −1.98240 1.75217i 2.80471 2.08904 2.15311i 0.712894 + 1.23477i
121.7 −1.17766 1.55400 + 0.764908i −0.613115 −0.500000 0.866025i −1.83009 0.900802i −1.48383 2.19049i 3.07736 1.82983 + 2.37733i 0.588830 + 1.01988i
121.8 −0.390993 0.919343 + 1.46793i −1.84712 −0.500000 0.866025i −0.359457 0.573950i 2.26118 + 1.37370i 1.50420 −1.30962 + 2.69906i 0.195497 + 0.338610i
121.9 −0.259663 −1.21719 1.23225i −1.93258 −0.500000 0.866025i 0.316059 + 0.319971i −0.593390 + 2.57835i 1.02114 −0.0368964 + 2.99977i 0.129832 + 0.224875i
121.10 0.0255806 0.676102 1.59464i −1.99935 −0.500000 0.866025i 0.0172951 0.0407920i −2.37158 1.17286i −0.102306 −2.08577 2.15628i −0.0127903 0.0221535i
121.11 0.699049 −1.56820 + 0.735367i −1.51133 −0.500000 0.866025i −1.09625 + 0.514057i 2.62064 0.363625i −2.45459 1.91847 2.30640i −0.349525 0.605394i
121.12 0.831231 −0.611026 + 1.62069i −1.30905 −0.500000 0.866025i −0.507903 + 1.34717i −2.57526 0.606656i −2.75059 −2.25330 1.98057i −0.415616 0.719867i
121.13 1.50060 −1.11516 1.32530i 0.251799 −0.500000 0.866025i −1.67341 1.98874i 0.0793460 2.64456i −2.62335 −0.512840 + 2.95584i −0.750300 1.29956i
121.14 1.85256 1.72871 0.107594i 1.43197 −0.500000 0.866025i 3.20253 0.199324i 0.370146 2.61973i −1.05231 2.97685 0.371996i −0.926279 1.60436i
121.15 1.89985 0.960253 1.44150i 1.60945 −0.500000 0.866025i 1.82434 2.73864i 1.72052 + 2.00992i −0.741992 −1.15583 2.76840i −0.949927 1.64532i
121.16 2.08660 0.878572 + 1.49269i 2.35391 −0.500000 0.866025i 1.83323 + 3.11464i −0.122839 + 2.64290i 0.738471 −1.45622 + 2.62286i −1.04330 1.80705i
121.17 2.58565 −1.01529 + 1.40328i 4.68556 −0.500000 0.866025i −2.62518 + 3.62838i 2.41754 1.07494i 6.94392 −0.938372 2.84947i −1.29282 2.23923i
121.18 2.75701 −0.581455 1.63154i 5.60109 −0.500000 0.866025i −1.60307 4.49816i −2.40845 + 1.09515i 9.92824 −2.32382 + 1.89733i −1.37850 2.38764i
151.1 −2.69765 −0.273945 + 1.71025i 5.27730 −0.500000 + 0.866025i 0.739006 4.61365i −0.230272 + 2.63571i −8.84100 −2.84991 0.937027i 1.34882 2.33623i
151.2 −2.65219 1.69529 0.354953i 5.03414 −0.500000 + 0.866025i −4.49624 + 0.941405i −1.76710 1.96910i −8.04712 2.74802 1.20350i 1.32610 2.29687i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.18
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.c yes 36
3.b odd 2 1 945.2.l.c 36
7.c even 3 1 315.2.k.c 36
9.c even 3 1 315.2.k.c 36
9.d odd 6 1 945.2.k.c 36
21.h odd 6 1 945.2.k.c 36
63.h even 3 1 inner 315.2.l.c yes 36
63.j odd 6 1 945.2.l.c 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.k.c 36 7.c even 3 1
315.2.k.c 36 9.c even 3 1
315.2.l.c yes 36 1.a even 1 1 trivial
315.2.l.c yes 36 63.h even 3 1 inner
945.2.k.c 36 9.d odd 6 1
945.2.k.c 36 21.h odd 6 1
945.2.l.c 36 3.b odd 2 1
945.2.l.c 36 63.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} - 29 T_{2}^{16} + 344 T_{2}^{14} + 2 T_{2}^{13} - 2159 T_{2}^{12} - 42 T_{2}^{11} + 7749 T_{2}^{10} + 312 T_{2}^{9} - 16013 T_{2}^{8} - 1003 T_{2}^{7} + 18068 T_{2}^{6} + 1417 T_{2}^{5} - 9642 T_{2}^{4} - 839 T_{2}^{3} + \cdots - 9 \) acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\). Copy content Toggle raw display