Properties

Label 315.2.cg.c
Level $315$
Weight $2$
Character orbit 315.cg
Analytic conductor $2.515$
Analytic rank $0$
Dimension $4$
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(157,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([4, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.157");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.cg (of order \(12\), degree \(4\), 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: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{3} + \zeta_{12} + 1) q^{2} + ( - \zeta_{12}^{3} + 2 \zeta_{12}) q^{3} + ( - \zeta_{12}^{2} + 2) q^{4} + ( - \zeta_{12}^{3} + 2) q^{5} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + 2 \zeta_{12} + 2) q^{6} + (\zeta_{12}^{2} - 3) q^{7} + (\zeta_{12}^{2} - \zeta_{12}) q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{3} + \zeta_{12} + 1) q^{2} + ( - \zeta_{12}^{3} + 2 \zeta_{12}) q^{3} + ( - \zeta_{12}^{2} + 2) q^{4} + ( - \zeta_{12}^{3} + 2) q^{5} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + 2 \zeta_{12} + 2) q^{6} + (\zeta_{12}^{2} - 3) q^{7} + (\zeta_{12}^{2} - \zeta_{12}) q^{8} + 3 q^{9} + ( - 3 \zeta_{12}^{3} - \zeta_{12}^{2} + 2 \zeta_{12} + 2) q^{10} + (2 \zeta_{12}^{3} - 4 \zeta_{12} - 2) q^{11} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{12} + (2 \zeta_{12}^{3} - 2 \zeta_{12} - 2) q^{13} + (3 \zeta_{12}^{3} + \zeta_{12}^{2} - 2 \zeta_{12} - 3) q^{14} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 4 \zeta_{12} + 1) q^{15} + (4 \zeta_{12}^{3} + \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{16} + ( - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 2) q^{17} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12} + 3) q^{18} + ( - \zeta_{12}^{3} + 5 \zeta_{12}^{2} - \zeta_{12}) q^{19} + ( - \zeta_{12}^{3} - 2 \zeta_{12}^{2} - \zeta_{12} + 4) q^{20} + (4 \zeta_{12}^{3} - 5 \zeta_{12}) q^{21} + (4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 6 \zeta_{12} - 6) q^{22} + ( - 3 \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} + 3) q^{23} + (\zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} - 1) q^{24} + ( - 4 \zeta_{12}^{3} + 3) q^{25} + (4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 4 \zeta_{12} - 4) q^{26} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}) q^{27} + (4 \zeta_{12}^{2} - 5) q^{28} + ( - 4 \zeta_{12}^{2} + 8) q^{29} + ( - 3 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 3 \zeta_{12} + 5) q^{30} + ( - 5 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 5 \zeta_{12} - 6) q^{31} + (5 \zeta_{12}^{3} + 5 \zeta_{12}^{2} - 4 \zeta_{12} - 1) q^{32} + (2 \zeta_{12}^{3} - 4 \zeta_{12} - 6) q^{33} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12}) q^{34} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \zeta_{12} - 6) q^{35} + ( - 3 \zeta_{12}^{2} + 6) q^{36} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3 \zeta_{12}) q^{37} + ( - \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 4 \zeta_{12} - 1) q^{38} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 4 \zeta_{12} - 4) q^{39} + ( - \zeta_{12}^{3} + 3 \zeta_{12}^{2} - \zeta_{12} - 1) q^{40} + ( - 2 \zeta_{12}^{2} + 6 \zeta_{12} - 2) q^{41} + (4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 5 \zeta_{12} - 5) q^{42} + (3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3) q^{43} + (6 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 6 \zeta_{12} - 4) q^{44} + ( - 3 \zeta_{12}^{3} + 6) q^{45} + ( - 6 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 3 \zeta_{12} + 4) q^{46} + (9 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 3 \zeta_{12} + 3) q^{47} + (2 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - \zeta_{12} - 6) q^{48} + ( - 5 \zeta_{12}^{2} + 8) q^{49} + ( - 7 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 3 \zeta_{12} + 3) q^{50} + ( - 4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{51} + (4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 4) q^{52} + ( - 4 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + \zeta_{12} - 1) q^{53} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 6 \zeta_{12} + 6) q^{54} + (6 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 8 \zeta_{12} - 6) q^{55} + ( - \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 3 \zeta_{12} - 1) q^{56} + (5 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 5 \zeta_{12}) q^{57} + ( - 8 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 4 \zeta_{12} + 8) q^{58} + (2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{59} + ( - 6 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 6 \zeta_{12}) q^{60} + ( - 6 \zeta_{12}^{2} - \zeta_{12} - 6) q^{61} + (\zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} - 1) q^{62} + (3 \zeta_{12}^{2} - 9) q^{63} + (4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{64} + (6 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 4 \zeta_{12} - 4) q^{65} + (8 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 10 \zeta_{12} - 10) q^{66} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 5 \zeta_{12} - 3) q^{67} + (2 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 4 \zeta_{12} + 2) q^{68} + ( - 4 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 5 \zeta_{12} + 4) q^{69} + (8 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 3 \zeta_{12} - 5) q^{70} - 6 q^{71} + (3 \zeta_{12}^{2} - 3 \zeta_{12}) q^{72} + (4 \zeta_{12}^{3} + \zeta_{12}^{2} - 5 \zeta_{12} - 5) q^{73} + ( - 3 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 3) q^{74} + ( - 3 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 6 \zeta_{12} + 4) q^{75} + (5 \zeta_{12}^{2} - 3 \zeta_{12} + 5) q^{76} + ( - 8 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 10 \zeta_{12} + 6) q^{77} + (6 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 6 \zeta_{12} - 8) q^{78} + ( - 2 \zeta_{12}^{2} + 8 \zeta_{12} - 2) q^{79} + (8 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 3 \zeta_{12}) q^{80} + 9 q^{81} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 4) q^{82} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{83} + (9 \zeta_{12}^{3} - 6 \zeta_{12}) q^{84} + ( - 2 \zeta_{12}^{2} - 6 \zeta_{12} + 2) q^{85} + 3 q^{86} + ( - 12 \zeta_{12}^{3} + 12 \zeta_{12}) q^{87} + ( - 2 \zeta_{12}^{3} + 2) q^{88} + (3 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 3 \zeta_{12}) q^{89} + ( - 9 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 6 \zeta_{12} + 6) q^{90} + ( - 6 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 4 \zeta_{12} + 6) q^{91} + ( - 4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - \zeta_{12} + 5) q^{92} + (9 \zeta_{12}^{3} - 5 \zeta_{12}^{2} - 9 \zeta_{12} + 10) q^{93} + (6 \zeta_{12}^{3} + 15 \zeta_{12}^{2} + 6 \zeta_{12}) q^{94} + ( - 7 \zeta_{12}^{3} + 11 \zeta_{12}^{2} + 3 \zeta_{12} - 2) q^{95} + (6 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 3 \zeta_{12} - 9) q^{96} + (3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 5 \zeta_{12} - 2) q^{97} + ( - 8 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 3 \zeta_{12} + 8) q^{98} + (6 \zeta_{12}^{3} - 12 \zeta_{12} - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 6 q^{4} + 8 q^{5} + 6 q^{6} - 10 q^{7} + 2 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 6 q^{4} + 8 q^{5} + 6 q^{6} - 10 q^{7} + 2 q^{8} + 12 q^{9} + 6 q^{10} - 8 q^{11} - 8 q^{13} - 10 q^{14} - 2 q^{16} + 4 q^{17} + 12 q^{18} + 10 q^{19} + 12 q^{20} - 20 q^{22} + 10 q^{23} - 6 q^{24} + 12 q^{25} - 12 q^{26} - 12 q^{28} + 24 q^{29} + 12 q^{30} - 18 q^{31} + 6 q^{32} - 24 q^{33} - 4 q^{34} - 20 q^{35} + 18 q^{36} - 6 q^{37} + 4 q^{38} - 12 q^{39} + 2 q^{40} - 12 q^{41} - 12 q^{42} + 6 q^{43} - 12 q^{44} + 24 q^{45} + 8 q^{46} + 24 q^{47} - 12 q^{48} + 22 q^{49} + 4 q^{50} - 12 q^{51} - 12 q^{52} - 10 q^{53} + 18 q^{54} - 16 q^{55} - 8 q^{56} - 6 q^{57} + 24 q^{58} - 6 q^{60} - 36 q^{61} - 8 q^{62} - 30 q^{63} - 12 q^{65} - 36 q^{66} - 16 q^{67} + 6 q^{69} - 12 q^{70} - 24 q^{71} + 6 q^{72} - 18 q^{73} + 30 q^{76} + 20 q^{77} - 24 q^{78} - 12 q^{79} + 8 q^{80} + 36 q^{81} + 12 q^{82} - 2 q^{83} + 4 q^{85} + 12 q^{86} + 8 q^{88} + 8 q^{89} + 18 q^{90} + 20 q^{91} + 12 q^{92} + 30 q^{93} + 30 q^{94} + 14 q^{95} - 24 q^{96} - 14 q^{97} + 22 q^{98} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(\zeta_{12}^{3}\) \(\zeta_{12}^{2}\) \(-1 + \zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1
0.866025 + 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
1.86603 0.500000i 1.73205 1.50000 0.866025i 2.00000 1.00000i 3.23205 0.866025i −2.50000 + 0.866025i −0.366025 + 0.366025i 3.00000 3.23205 2.86603i
187.1 0.133975 0.500000i −1.73205 1.50000 + 0.866025i 2.00000 1.00000i −0.232051 + 0.866025i −2.50000 0.866025i 1.36603 1.36603i 3.00000 −0.232051 1.13397i
283.1 0.133975 + 0.500000i −1.73205 1.50000 0.866025i 2.00000 + 1.00000i −0.232051 0.866025i −2.50000 + 0.866025i 1.36603 + 1.36603i 3.00000 −0.232051 + 1.13397i
313.1 1.86603 + 0.500000i 1.73205 1.50000 + 0.866025i 2.00000 + 1.00000i 3.23205 + 0.866025i −2.50000 0.866025i −0.366025 0.366025i 3.00000 3.23205 + 2.86603i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
315.cg even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.cg.c yes 4
3.b odd 2 1 945.2.cj.a 4
5.c odd 4 1 315.2.cg.a yes 4
7.d odd 6 1 315.2.bs.b 4
9.c even 3 1 315.2.bs.c yes 4
9.d odd 6 1 945.2.bv.d 4
15.e even 4 1 945.2.cj.d 4
21.g even 6 1 945.2.bv.a 4
35.k even 12 1 315.2.bs.c yes 4
45.k odd 12 1 315.2.bs.b 4
45.l even 12 1 945.2.bv.a 4
63.k odd 6 1 315.2.cg.a yes 4
63.s even 6 1 945.2.cj.d 4
105.w odd 12 1 945.2.bv.d 4
315.bw odd 12 1 945.2.cj.a 4
315.cg even 12 1 inner 315.2.cg.c yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.bs.b 4 7.d odd 6 1
315.2.bs.b 4 45.k odd 12 1
315.2.bs.c yes 4 9.c even 3 1
315.2.bs.c yes 4 35.k even 12 1
315.2.cg.a yes 4 5.c odd 4 1
315.2.cg.a yes 4 63.k odd 6 1
315.2.cg.c yes 4 1.a even 1 1 trivial
315.2.cg.c yes 4 315.cg even 12 1 inner
945.2.bv.a 4 21.g even 6 1
945.2.bv.a 4 45.l even 12 1
945.2.bv.d 4 9.d odd 6 1
945.2.bv.d 4 105.w odd 12 1
945.2.cj.a 4 3.b odd 2 1
945.2.cj.a 4 315.bw odd 12 1
945.2.cj.d 4 15.e even 4 1
945.2.cj.d 4 63.s even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\):

\( T_{2}^{4} - 4T_{2}^{3} + 5T_{2}^{2} - 2T_{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 4 T^{3} + 5 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 4 T + 5)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 5 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 4 T - 8)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 8 T^{3} + 20 T^{2} + 16 T + 16 \) Copy content Toggle raw display
$17$ \( T^{4} - 4 T^{3} + 8 T^{2} - 32 T + 64 \) Copy content Toggle raw display
$19$ \( T^{4} - 10 T^{3} + 78 T^{2} + \cdots + 484 \) Copy content Toggle raw display
$23$ \( T^{4} - 10 T^{3} + 50 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$29$ \( (T^{2} - 12 T + 48)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 18 T^{3} + 110 T^{2} + 36 T + 4 \) Copy content Toggle raw display
$37$ \( T^{4} + 6 T^{3} + 18 T^{2} + 108 T + 324 \) Copy content Toggle raw display
$41$ \( T^{4} + 12 T^{3} + 24 T^{2} + \cdots + 576 \) Copy content Toggle raw display
$43$ \( T^{4} - 6 T^{3} + 45 T^{2} - 108 T + 81 \) Copy content Toggle raw display
$47$ \( T^{4} - 24 T^{3} + 369 T^{2} + \cdots + 13689 \) Copy content Toggle raw display
$53$ \( T^{4} + 10 T^{3} + 74 T^{2} + \cdots + 484 \) Copy content Toggle raw display
$59$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$61$ \( T^{4} + 36 T^{3} + 539 T^{2} + \cdots + 11449 \) Copy content Toggle raw display
$67$ \( T^{4} + 16 T^{3} + 65 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$71$ \( (T + 6)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 18 T^{3} + 90 T^{2} + 36 T + 36 \) Copy content Toggle raw display
$79$ \( T^{4} + 12 T^{3} - 4 T^{2} + \cdots + 2704 \) Copy content Toggle raw display
$83$ \( T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$89$ \( T^{4} - 8 T^{3} + 75 T^{2} + 88 T + 121 \) Copy content Toggle raw display
$97$ \( T^{4} + 14 T^{3} + 50 T^{2} + \cdots + 676 \) Copy content Toggle raw display
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