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Properties

Label 945.2.cj.b
Level $945$
Weight $2$
Character orbit 945.cj
Analytic conductor $7.546$
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] = [945,2,Mod(208,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([8, 9, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.208");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.cj (of order \(12\), degree \(4\), not 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: \(7.54586299101\)
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: no (minimal twist has level 315)
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.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} - \zeta_{12} - 1) q^{2} + ( - \zeta_{12}^{2} + 2) q^{4} + ( - 2 \zeta_{12}^{3} + 1) q^{5} + ( - 3 \zeta_{12}^{2} + 2) q^{7} + ( - \zeta_{12}^{2} + \zeta_{12}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{3} - \zeta_{12} - 1) q^{2} + ( - \zeta_{12}^{2} + 2) q^{4} + ( - 2 \zeta_{12}^{3} + 1) q^{5} + ( - 3 \zeta_{12}^{2} + 2) q^{7} + ( - \zeta_{12}^{2} + \zeta_{12}) q^{8} + (3 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \cdots - 1) q^{10}+ \cdots + ( - 5 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + \cdots + 5) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 6 q^{4} + 4 q^{5} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 6 q^{4} + 4 q^{5} + 2 q^{7} - 2 q^{8} + 8 q^{11} + 10 q^{13} - 2 q^{14} - 2 q^{16} + 8 q^{17} - 2 q^{19} + 6 q^{20} - 8 q^{22} - 4 q^{23} - 12 q^{25} - 18 q^{26} - 6 q^{28} + 24 q^{29} - 12 q^{31} - 6 q^{32} - 16 q^{34} + 2 q^{35} + 18 q^{37} + 8 q^{38} + 2 q^{40} + 6 q^{43} + 12 q^{44} - 10 q^{46} - 6 q^{47} - 26 q^{49} + 20 q^{50} + 12 q^{52} + 4 q^{53} + 8 q^{55} - 10 q^{56} - 18 q^{58} - 6 q^{59} + 24 q^{61} + 8 q^{62} + 6 q^{65} - 10 q^{67} + 24 q^{68} + 18 q^{70} - 12 q^{71} - 6 q^{76} + 4 q^{77} - 12 q^{79} + 22 q^{80} + 12 q^{82} + 26 q^{83} + 40 q^{85} - 12 q^{86} - 4 q^{88} + 16 q^{89} - 4 q^{91} - 18 q^{92} - 12 q^{94} - 14 q^{95} + 34 q^{97} + 26 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/945\mathbb{Z}\right)^\times\).

\(n\) \(136\) \(596\) \(757\)
\(\chi(n)\) \(\zeta_{12}^{2}\) \(-1 + \zeta_{12}^{2}\) \(\zeta_{12}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
208.1
0.866025 − 0.500000i
−0.866025 − 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−1.86603 − 0.500000i 0 1.50000 + 0.866025i 1.00000 + 2.00000i 0 0.500000 + 2.59808i 0.366025 + 0.366025i 0 −0.866025 − 4.23205i
388.1 −0.133975 − 0.500000i 0 1.50000 − 0.866025i 1.00000 + 2.00000i 0 0.500000 − 2.59808i −1.36603 − 1.36603i 0 0.866025 − 0.767949i
397.1 −0.133975 + 0.500000i 0 1.50000 + 0.866025i 1.00000 − 2.00000i 0 0.500000 + 2.59808i −1.36603 + 1.36603i 0 0.866025 + 0.767949i
577.1 −1.86603 + 0.500000i 0 1.50000 − 0.866025i 1.00000 − 2.00000i 0 0.500000 − 2.59808i 0.366025 − 0.366025i 0 −0.866025 + 4.23205i
\(n\): 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 945.2.cj.b 4
3.b odd 2 1 315.2.cg.d yes 4
5.c odd 4 1 945.2.cj.c 4
7.d odd 6 1 945.2.bv.c 4
9.c even 3 1 945.2.bv.b 4
9.d odd 6 1 315.2.bs.a ✓ 4
15.e even 4 1 315.2.cg.b yes 4
21.g even 6 1 315.2.bs.d yes 4
35.k even 12 1 945.2.bv.b 4
45.k odd 12 1 945.2.bv.c 4
45.l even 12 1 315.2.bs.d yes 4
63.k odd 6 1 945.2.cj.c 4
63.s even 6 1 315.2.cg.b yes 4
105.w odd 12 1 315.2.bs.a ✓ 4
315.bw odd 12 1 315.2.cg.d yes 4
315.cg even 12 1 inner 945.2.cj.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.bs.a ✓ 4 9.d odd 6 1
315.2.bs.a ✓ 4 105.w odd 12 1
315.2.bs.d yes 4 21.g even 6 1
315.2.bs.d yes 4 45.l even 12 1
315.2.cg.b yes 4 15.e even 4 1
315.2.cg.b yes 4 63.s even 6 1
315.2.cg.d yes 4 3.b odd 2 1
315.2.cg.d yes 4 315.bw odd 12 1
945.2.bv.b 4 9.c even 3 1
945.2.bv.b 4 35.k even 12 1
945.2.bv.c 4 7.d odd 6 1
945.2.bv.c 4 45.k odd 12 1
945.2.cj.b 4 1.a even 1 1 trivial
945.2.cj.b 4 315.cg even 12 1 inner
945.2.cj.c 4 5.c odd 4 1
945.2.cj.c 4 63.k odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 4 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 2 T + 5)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 7)^{2} \) Copy content Toggle raw display
$11$ \( (T - 2)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 10 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$17$ \( T^{4} - 8 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( T^{4} + 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$23$ \( T^{4} + 4 T^{3} + \cdots + 484 \) Copy content Toggle raw display
$29$ \( T^{4} - 24 T^{3} + \cdots + 1521 \) Copy content Toggle raw display
$31$ \( T^{4} + 12 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$37$ \( T^{4} - 18 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$41$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$43$ \( T^{4} - 6 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$47$ \( T^{4} + 6 T^{3} + \cdots + 1521 \) Copy content Toggle raw display
$53$ \( T^{4} - 4 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$59$ \( T^{4} + 6 T^{3} + \cdots + 19044 \) Copy content Toggle raw display
$61$ \( T^{4} - 24 T^{3} + \cdots + 1024 \) Copy content Toggle raw display
$67$ \( T^{4} + 10 T^{3} + \cdots + 484 \) Copy content Toggle raw display
$71$ \( (T^{2} + 6 T + 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 144 T^{2} + \cdots + 2304 \) Copy content Toggle raw display
$79$ \( T^{4} + 12 T^{3} + \cdots + 2704 \) Copy content Toggle raw display
$83$ \( T^{4} - 26 T^{3} + \cdots + 14641 \) Copy content Toggle raw display
$89$ \( T^{4} - 16 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$97$ \( T^{4} - 34 T^{3} + \cdots + 20164 \) Copy content Toggle raw display
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