Properties

Label 3150.2.bf.c
Level $3150$
Weight $2$
Character orbit 3150.bf
Analytic conductor $25.153$
Analytic rank $0$
Dimension $8$
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] = [3150,2,Mod(1151,3150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3150, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3150.1151");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3150.bf (of order \(6\), 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: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 630)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{24}^{2} q^{2} + \zeta_{24}^{4} q^{4} + (3 \zeta_{24}^{5} - \zeta_{24}) q^{7} - \zeta_{24}^{6} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{24}^{2} q^{2} + \zeta_{24}^{4} q^{4} + (3 \zeta_{24}^{5} - \zeta_{24}) q^{7} - \zeta_{24}^{6} q^{8} + (\zeta_{24}^{5} - 2 \zeta_{24}^{4} - \zeta_{24}^{3} + 4) q^{11} + (\zeta_{24}^{7} + 2 \zeta_{24}^{6} + \zeta_{24}^{5}) q^{13} + ( - 3 \zeta_{24}^{7} + \zeta_{24}^{3}) q^{14} + (\zeta_{24}^{4} - 1) q^{16} + ( - 2 \zeta_{24}^{6} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{2}) q^{17} + (2 \zeta_{24}^{7} - 2 \zeta_{24}^{5} - \zeta_{24}^{2} + 2 \zeta_{24}) q^{19} + ( - \zeta_{24}^{7} + 2 \zeta_{24}^{6} + \zeta_{24}^{5} - 4 \zeta_{24}^{2}) q^{22} + ( - \zeta_{24}^{4} - 2 \zeta_{24}^{2} - 1) q^{23} + ( - \zeta_{24}^{7} - \zeta_{24}^{5} - 2 \zeta_{24}^{4} + \zeta_{24} + 2) q^{26} + (2 \zeta_{24}^{5} - 3 \zeta_{24}) q^{28} + (4 \zeta_{24}^{7} + 4 \zeta_{24}^{6} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{3} + 2 \zeta_{24}) q^{29} + ( - 2 \zeta_{24}^{7} + 2 \zeta_{24}^{6} - 2 \zeta_{24}^{2} + 2 \zeta_{24}) q^{31} + ( - \zeta_{24}^{6} + \zeta_{24}^{2}) q^{32} + ( - 2 \zeta_{24}^{7} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{4} - 2) q^{34} + (\zeta_{24}^{7} + \zeta_{24}^{5} + 6 \zeta_{24}^{4} - \zeta_{24} - 6) q^{37} + (2 \zeta_{24}^{7} - 2 \zeta_{24}^{5} + \zeta_{24}^{4} - 2 \zeta_{24}^{3} + 2 \zeta_{24}) q^{38} + (\zeta_{24}^{7} - 3 \zeta_{24}^{5} + 2 \zeta_{24}^{3} + 2 \zeta_{24} + 4) q^{41} + ( - 2 \zeta_{24}^{7} + 2 \zeta_{24}^{5} + 2) q^{43} + ( - \zeta_{24}^{7} + \zeta_{24}^{5} + 2 \zeta_{24}^{4} - \zeta_{24} + 2) q^{44} + (\zeta_{24}^{6} + 2 \zeta_{24}^{4} + \zeta_{24}^{2}) q^{46} + ( - 2 \zeta_{24}^{6} - 2 \zeta_{24}^{4} - 4 \zeta_{24}^{3} + \zeta_{24}^{2} + 4 \zeta_{24} + 2) q^{47} + (3 \zeta_{24}^{6} - 8 \zeta_{24}^{2}) q^{49} + (\zeta_{24}^{7} + 2 \zeta_{24}^{6} + \zeta_{24}^{5} - \zeta_{24}^{3} - 2 \zeta_{24}^{2} - \zeta_{24}) q^{52} + ( - 4 \zeta_{24}^{7} + \zeta_{24}^{6} - 6 \zeta_{24}^{5} + 2 \zeta_{24}^{4} + 6 \zeta_{24}^{3} - \zeta_{24}^{2} + 4 \zeta_{24} - 4) q^{53} + ( - 2 \zeta_{24}^{7} + 3 \zeta_{24}^{3}) q^{56} + ( - 2 \zeta_{24}^{7} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{4} - 2 \zeta_{24}^{3} + 4 \zeta_{24} + 4) q^{58} + (3 \zeta_{24}^{7} + 2 \zeta_{24}^{6} + 2 \zeta_{24}^{5} - 6 \zeta_{24}^{4} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{2} + 3 \zeta_{24}) q^{59} + (4 \zeta_{24}^{7} - 4 \zeta_{24}^{5} - 6 \zeta_{24}^{3} - 2 \zeta_{24}^{2} - 2 \zeta_{24}) q^{61} + (2 \zeta_{24}^{5} - 2 \zeta_{24}^{3} - 2 \zeta_{24} + 2) q^{62} - q^{64} + ( - 4 \zeta_{24}^{6} - 6 \zeta_{24}^{5} + 6 \zeta_{24}^{4} - 6 \zeta_{24}^{3} - 4 \zeta_{24}^{2}) q^{67} + (2 \zeta_{24}^{7} - 4 \zeta_{24}^{6} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{2} - 2 \zeta_{24}) q^{68} + (6 \zeta_{24}^{7} - 4 \zeta_{24}^{6} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{3} + 4 \zeta_{24}) q^{71} + (4 \zeta_{24}^{7} - 6 \zeta_{24}^{6} + 6 \zeta_{24}^{2} - 4 \zeta_{24}) q^{73} + ( - \zeta_{24}^{7} - 6 \zeta_{24}^{6} - \zeta_{24}^{5} + \zeta_{24}^{3} + 6 \zeta_{24}^{2} + \zeta_{24}) q^{74} + (2 \zeta_{24}^{7} - \zeta_{24}^{6} - 2 \zeta_{24}^{3} + 2 \zeta_{24}) q^{76} + (2 \zeta_{24}^{6} + 8 \zeta_{24}^{5} - 2 \zeta_{24}^{4} - 3 \zeta_{24}^{2} + 2 \zeta_{24} + 3) q^{77} + ( - 4 \zeta_{24}^{7} + 8 \zeta_{24}^{6} - 4 \zeta_{24}^{5} + 6 \zeta_{24}^{4} + 6 \zeta_{24}^{3} - 4 \zeta_{24}^{2} + \cdots - 6) q^{79} + \cdots + (5 \zeta_{24}^{4} + 3) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} + 24 q^{11} - 4 q^{16} - 12 q^{23} + 8 q^{26} - 24 q^{37} + 4 q^{38} + 32 q^{41} + 16 q^{43} + 24 q^{44} + 8 q^{46} + 8 q^{47} - 24 q^{53} + 16 q^{58} - 24 q^{59} + 16 q^{62} - 8 q^{64} + 24 q^{67} + 16 q^{77} - 24 q^{79} + 16 q^{83} - 16 q^{89} - 20 q^{91} - 12 q^{94} + 44 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(1\) \(1 - \zeta_{24}^{4}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1151.1
0.965926 0.258819i
−0.965926 + 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
−0.258819 + 0.965926i
0.258819 0.965926i
−0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 −0.189469 2.63896i 1.00000i 0 0
1151.2 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 0.189469 + 2.63896i 1.00000i 0 0
1151.3 0.866025 0.500000i 0 0.500000 0.866025i 0 0 −2.63896 + 0.189469i 1.00000i 0 0
1151.4 0.866025 0.500000i 0 0.500000 0.866025i 0 0 2.63896 0.189469i 1.00000i 0 0
1601.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 −0.189469 + 2.63896i 1.00000i 0 0
1601.2 −0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 0.189469 2.63896i 1.00000i 0 0
1601.3 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 −2.63896 0.189469i 1.00000i 0 0
1601.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 2.63896 + 0.189469i 1.00000i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1151.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3150.2.bf.c 8
3.b odd 2 1 3150.2.bf.b 8
5.b even 2 1 630.2.be.b yes 8
5.c odd 4 1 3150.2.bp.c 8
5.c odd 4 1 3150.2.bp.f 8
7.d odd 6 1 3150.2.bf.b 8
15.d odd 2 1 630.2.be.a 8
15.e even 4 1 3150.2.bp.a 8
15.e even 4 1 3150.2.bp.d 8
21.g even 6 1 inner 3150.2.bf.c 8
35.i odd 6 1 630.2.be.a 8
35.i odd 6 1 4410.2.b.e 8
35.j even 6 1 4410.2.b.b 8
35.k even 12 1 3150.2.bp.a 8
35.k even 12 1 3150.2.bp.d 8
105.o odd 6 1 4410.2.b.e 8
105.p even 6 1 630.2.be.b yes 8
105.p even 6 1 4410.2.b.b 8
105.w odd 12 1 3150.2.bp.c 8
105.w odd 12 1 3150.2.bp.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.be.a 8 15.d odd 2 1
630.2.be.a 8 35.i odd 6 1
630.2.be.b yes 8 5.b even 2 1
630.2.be.b yes 8 105.p even 6 1
3150.2.bf.b 8 3.b odd 2 1
3150.2.bf.b 8 7.d odd 6 1
3150.2.bf.c 8 1.a even 1 1 trivial
3150.2.bf.c 8 21.g even 6 1 inner
3150.2.bp.a 8 15.e even 4 1
3150.2.bp.a 8 35.k even 12 1
3150.2.bp.c 8 5.c odd 4 1
3150.2.bp.c 8 105.w odd 12 1
3150.2.bp.d 8 15.e even 4 1
3150.2.bp.d 8 35.k even 12 1
3150.2.bp.f 8 5.c odd 4 1
3150.2.bp.f 8 105.w odd 12 1
4410.2.b.b 8 35.j even 6 1
4410.2.b.b 8 105.p even 6 1
4410.2.b.e 8 35.i odd 6 1
4410.2.b.e 8 105.o odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3150, [\chi])\):

\( T_{11}^{8} - 24 T_{11}^{7} + 260 T_{11}^{6} - 1632 T_{11}^{5} + 6447 T_{11}^{4} - 16320 T_{11}^{3} + 25796 T_{11}^{2} - 23280 T_{11} + 9409 \) Copy content Toggle raw display
\( T_{37}^{8} + 24 T_{37}^{7} + 364 T_{37}^{6} + 3456 T_{37}^{5} + 24207 T_{37}^{4} + 117648 T_{37}^{3} + 421420 T_{37}^{2} + 940848 T_{37} + 1329409 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 94T^{4} + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} - 24 T^{7} + 260 T^{6} + \cdots + 9409 \) Copy content Toggle raw display
$13$ \( T^{8} + 24 T^{6} + 146 T^{4} + 216 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{8} + 40 T^{6} - 192 T^{5} + \cdots + 1024 \) Copy content Toggle raw display
$19$ \( T^{8} - 18 T^{6} + 323 T^{4} + 864 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( (T^{4} + 6 T^{3} + 11 T^{2} - 6 T + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 80 T^{2} + 64)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 24 T^{6} + 560 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$37$ \( T^{8} + 24 T^{7} + 364 T^{6} + \cdots + 1329409 \) Copy content Toggle raw display
$41$ \( (T^{4} - 16 T^{3} + 68 T^{2} - 32 T - 71)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 8 T^{3} + 8 T^{2} + 32 T - 32)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 8 T^{7} + 110 T^{6} + \cdots + 330625 \) Copy content Toggle raw display
$53$ \( T^{8} + 24 T^{7} + 150 T^{6} + \cdots + 2745649 \) Copy content Toggle raw display
$59$ \( T^{8} + 24 T^{7} + 460 T^{6} + \cdots + 22090000 \) Copy content Toggle raw display
$61$ \( T^{8} - 120 T^{6} + 12896 T^{4} + \cdots + 2262016 \) Copy content Toggle raw display
$67$ \( T^{8} - 24 T^{7} + \cdots + 442008576 \) Copy content Toggle raw display
$71$ \( T^{8} + 288 T^{6} + 23072 T^{4} + \cdots + 1364224 \) Copy content Toggle raw display
$73$ \( T^{8} - 136 T^{6} + 18480 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$79$ \( T^{8} + 24 T^{7} + \cdots + 171295744 \) Copy content Toggle raw display
$83$ \( (T^{4} - 8 T^{3} - 184 T^{2} + 800 T + 7648)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 16 T^{7} + 308 T^{6} + \cdots + 2062096 \) Copy content Toggle raw display
$97$ \( T^{8} + 800 T^{6} + \cdots + 610287616 \) Copy content Toggle raw display
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