Properties

Label 3150.2.m.k
Level $3150$
Weight $2$
Character orbit 3150.m
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(1457,3150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3150, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3150.1457");
 
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.m (of order \(4\), 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 \(\Q(i)\)
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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} - \beta_{3} q^{4} - \beta_1 q^{7} - \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} - \beta_{3} q^{4} - \beta_1 q^{7} - \beta_1 q^{8} + ( - \beta_{5} + 2 \beta_{3} + \beta_1) q^{11} + ( - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 1) q^{13} - q^{14} - q^{16} + ( - \beta_{7} + \beta_{6} - 2 \beta_{5} - \beta_{4}) q^{17} + (\beta_{5} + 2 \beta_{4} - \beta_1) q^{19} + (\beta_{3} + 2 \beta_1 + 1) q^{22} + ( - 2 \beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{23} + (\beta_{7} + \beta_{5} + \beta_{3} + \beta_{2} - \beta_1) q^{26} - \beta_{5} q^{28} + (2 \beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{2} - \beta_1) q^{29} + (\beta_{7} + \beta_{6} - 4 \beta_{5} - \beta_{2} - 4 \beta_1) q^{31} - \beta_{5} q^{32} + ( - \beta_{7} - \beta_{4} + 2 \beta_{3} - \beta_{2}) q^{34} + (2 \beta_{6} + 2 \beta_{4} - 2 \beta_{3} - 2 \beta_1 - 2) q^{37} + ( - \beta_{3} + 2 \beta_{2} - 1) q^{38} + (2 \beta_{7} + \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1) q^{41} + ( - \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + \beta_{3} - 1) q^{43} + (\beta_{5} + \beta_1 + 2) q^{44} + (2 \beta_{6} - 2 \beta_{5} - 2 \beta_1 + 1) q^{46} + ( - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - \beta_{3} + 1) q^{47} + \beta_{3} q^{49} + (\beta_{6} + \beta_{4} - \beta_{3} + \beta_1 - 1) q^{52} + (\beta_{6} + \beta_{4} - 4 \beta_{3} - \beta_1 - 4) q^{53} + \beta_{3} q^{56} + (\beta_{7} - 2 \beta_{6} + 2 \beta_{4} + \beta_{3} - 1) q^{58} + (\beta_{7} + 2 \beta_{6} + 3 \beta_{5} - \beta_{2} + 3 \beta_1 - 1) q^{59} + ( - \beta_{7} + \beta_{6} - 5 \beta_{5} + \beta_{2} - 5 \beta_1 - 4) q^{61} + ( - \beta_{7} - \beta_{6} + \beta_{4} + 4 \beta_{3} - 4) q^{62} + \beta_{3} q^{64} + (2 \beta_{6} + 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 4 \beta_1 + 2) q^{67} + ( - \beta_{6} - \beta_{4} - \beta_{2} + 2 \beta_1) q^{68} + ( - 4 \beta_{4} + 4 \beta_{3}) q^{71} + ( - \beta_{6} - 6 \beta_{5} + \beta_{4}) q^{73} + ( - 2 \beta_{7} - 2 \beta_{5} + 2 \beta_{2} - 2 \beta_1 - 2) q^{74} + (2 \beta_{6} - \beta_{5} - \beta_1) q^{76} + (2 \beta_{5} - \beta_{3} + 1) q^{77} + (3 \beta_{7} - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1) q^{79} + (2 \beta_{6} + 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{82} + ( - \beta_{6} - \beta_{4} + \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 1) q^{83} + (2 \beta_{7} - \beta_{5} - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \beta_1) q^{86} + (2 \beta_{5} - \beta_{3} + 1) q^{88} + (2 \beta_{7} - 2 \beta_{5} - 2 \beta_{2} - 2 \beta_1 + 2) q^{89} + ( - \beta_{7} - \beta_{5} + \beta_{2} - \beta_1 + 1) q^{91} + ( - 2 \beta_{7} + \beta_{5} + 2 \beta_{3} - 2) q^{92} + (2 \beta_{7} + \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - \beta_1) q^{94} + (3 \beta_{6} + 3 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 2) q^{97} + \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{13} - 8 q^{14} - 8 q^{16} + 8 q^{22} - 16 q^{23} - 16 q^{37} - 8 q^{38} - 8 q^{43} + 16 q^{44} + 8 q^{46} + 8 q^{47} - 8 q^{52} - 32 q^{53} - 8 q^{58} - 8 q^{59} - 32 q^{61} - 32 q^{62} + 16 q^{67} - 16 q^{74} + 8 q^{77} - 8 q^{82} + 8 q^{83} + 8 q^{88} + 16 q^{89} + 8 q^{91} - 16 q^{92} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{5} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 2\zeta_{24}^{4} - 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{5} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{6} + 2\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 2\zeta_{24}^{7} - \zeta_{24}^{3} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{5} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{6} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta_{4} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{5} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} + \beta_1 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(\beta_{3}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1457.1
−0.965926 0.258819i
0.258819 + 0.965926i
0.965926 + 0.258819i
−0.258819 0.965926i
−0.965926 + 0.258819i
0.258819 0.965926i
0.965926 0.258819i
−0.258819 + 0.965926i
−0.707107 + 0.707107i 0 1.00000i 0 0 0.707107 + 0.707107i 0.707107 + 0.707107i 0 0
1457.2 −0.707107 + 0.707107i 0 1.00000i 0 0 0.707107 + 0.707107i 0.707107 + 0.707107i 0 0
1457.3 0.707107 0.707107i 0 1.00000i 0 0 −0.707107 0.707107i −0.707107 0.707107i 0 0
1457.4 0.707107 0.707107i 0 1.00000i 0 0 −0.707107 0.707107i −0.707107 0.707107i 0 0
2843.1 −0.707107 0.707107i 0 1.00000i 0 0 0.707107 0.707107i 0.707107 0.707107i 0 0
2843.2 −0.707107 0.707107i 0 1.00000i 0 0 0.707107 0.707107i 0.707107 0.707107i 0 0
2843.3 0.707107 + 0.707107i 0 1.00000i 0 0 −0.707107 + 0.707107i −0.707107 + 0.707107i 0 0
2843.4 0.707107 + 0.707107i 0 1.00000i 0 0 −0.707107 + 0.707107i −0.707107 + 0.707107i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1457.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3150.2.m.k yes 8
3.b odd 2 1 3150.2.m.l yes 8
5.b even 2 1 3150.2.m.g 8
5.c odd 4 1 3150.2.m.h yes 8
5.c odd 4 1 3150.2.m.l yes 8
15.d odd 2 1 3150.2.m.h yes 8
15.e even 4 1 3150.2.m.g 8
15.e even 4 1 inner 3150.2.m.k yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3150.2.m.g 8 5.b even 2 1
3150.2.m.g 8 15.e even 4 1
3150.2.m.h yes 8 5.c odd 4 1
3150.2.m.h yes 8 15.d odd 2 1
3150.2.m.k yes 8 1.a even 1 1 trivial
3150.2.m.k yes 8 15.e even 4 1 inner
3150.2.m.l yes 8 3.b odd 2 1
3150.2.m.l yes 8 5.c odd 4 1

Hecke kernels

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

\( T_{11}^{4} + 12T_{11}^{2} + 4 \) Copy content Toggle raw display
\( T_{13}^{8} - 8T_{13}^{7} + 32T_{13}^{6} - 24T_{13}^{5} + 2T_{13}^{4} - 72T_{13}^{3} + 800T_{13}^{2} + 40T_{13} + 1 \) Copy content Toggle raw display
\( T_{17}^{8} - 96T_{17}^{5} + 770T_{17}^{4} - 2496T_{17}^{3} + 4608T_{17}^{2} - 4512T_{17} + 2209 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 12 T^{2} + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 8 T^{7} + 32 T^{6} - 24 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{8} - 96 T^{5} + 770 T^{4} + \cdots + 2209 \) Copy content Toggle raw display
$19$ \( (T^{4} + 28 T^{2} + 100)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 529 \) Copy content Toggle raw display
$29$ \( (T^{4} - 58 T^{2} + 96 T + 337)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 82 T^{2} - 192 T + 457)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$41$ \( T^{8} + 156 T^{6} + 7094 T^{4} + \cdots + 69169 \) Copy content Toggle raw display
$43$ \( T^{8} + 8 T^{7} + 32 T^{6} + \cdots + 5041 \) Copy content Toggle raw display
$47$ \( T^{8} - 8 T^{7} + 32 T^{6} + \cdots + 446224 \) Copy content Toggle raw display
$53$ \( T^{8} + 32 T^{7} + 512 T^{6} + \cdots + 361201 \) Copy content Toggle raw display
$59$ \( (T^{4} + 4 T^{3} - 66 T^{2} + 148 T - 71)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 16 T^{3} - 22 T^{2} - 448 T + 937)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 16 T^{7} + 128 T^{6} + \cdots + 9339136 \) Copy content Toggle raw display
$71$ \( (T^{4} + 128 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 5256 T^{4} + 810000 \) Copy content Toggle raw display
$79$ \( T^{8} + 312 T^{6} + 31640 T^{4} + \cdots + 913936 \) Copy content Toggle raw display
$83$ \( T^{8} - 8 T^{7} + 32 T^{6} + \cdots + 113569 \) Copy content Toggle raw display
$89$ \( (T^{4} - 8 T^{3} - 40 T^{2} + 224 T + 16)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 16 T^{7} + 128 T^{6} + \cdots + 595984 \) Copy content Toggle raw display
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