Properties

Label 3150.2.m.b
Level $3150$
Weight $2$
Character orbit 3150.m
Analytic conductor $25.153$
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] = [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: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 630)
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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} - \zeta_{8}^{3} q^{7} + \zeta_{8}^{3} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{8} q^{2} + \zeta_{8}^{2} q^{4} - \zeta_{8}^{3} q^{7} + \zeta_{8}^{3} q^{8} + ( - \zeta_{8}^{3} + 2 \zeta_{8}^{2} - \zeta_{8}) q^{11} + ( - 4 \zeta_{8}^{2} - 4) q^{13} + q^{14} - q^{16} + ( - 2 \zeta_{8}^{2} + 2 \zeta_{8} - 2) q^{17} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{19} + (2 \zeta_{8}^{3} - \zeta_{8}^{2} + 1) q^{22} + (4 \zeta_{8}^{3} + 2 \zeta_{8}^{2} - 2) q^{23} + ( - 4 \zeta_{8}^{3} - 4 \zeta_{8}) q^{26} + \zeta_{8} q^{28} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8} - 2) q^{29} + (3 \zeta_{8}^{3} - 3 \zeta_{8} + 6) q^{31} - \zeta_{8} q^{32} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}^{2} - 2 \zeta_{8}) q^{34} + (6 \zeta_{8}^{3} + 2 \zeta_{8}^{2} - 2) q^{37} + ( - 2 \zeta_{8}^{2} + 2) q^{38} + (2 \zeta_{8}^{3} - 6 \zeta_{8}^{2} + 2 \zeta_{8}) q^{41} + (\zeta_{8}^{2} - 10 \zeta_{8} + 1) q^{43} + ( - \zeta_{8}^{3} + \zeta_{8} - 2) q^{44} + (2 \zeta_{8}^{3} - 2 \zeta_{8} - 4) q^{46} + ( - 5 \zeta_{8}^{2} - 6 \zeta_{8} - 5) q^{47} - \zeta_{8}^{2} q^{49} + ( - 4 \zeta_{8}^{2} + 4) q^{52} + (2 \zeta_{8}^{3} - 4 \zeta_{8}^{2} + 4) q^{53} + \zeta_{8}^{2} q^{56} + (2 \zeta_{8}^{2} - 2 \zeta_{8} + 2) q^{58} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{59} + (7 \zeta_{8}^{3} - 7 \zeta_{8}) q^{61} + ( - 3 \zeta_{8}^{2} + 6 \zeta_{8} - 3) q^{62} - \zeta_{8}^{2} q^{64} + (2 \zeta_{8}^{3} + 7 \zeta_{8}^{2} - 7) q^{67} + (2 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 2) q^{68} + ( - 2 \zeta_{8}^{3} + 4 \zeta_{8}^{2} - 2 \zeta_{8}) q^{71} - 10 \zeta_{8} q^{73} + (2 \zeta_{8}^{3} - 2 \zeta_{8} - 6) q^{74} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{76} + ( - \zeta_{8}^{2} + 2 \zeta_{8} - 1) q^{77} + ( - 4 \zeta_{8}^{3} - 4 \zeta_{8}) q^{79} + ( - 6 \zeta_{8}^{3} + 2 \zeta_{8}^{2} - 2) q^{82} + ( - 4 \zeta_{8}^{3} + 2 \zeta_{8}^{2} - 2) q^{83} + (\zeta_{8}^{3} - 10 \zeta_{8}^{2} + \zeta_{8}) q^{86} + (\zeta_{8}^{2} - 2 \zeta_{8} + 1) q^{88} + (2 \zeta_{8}^{3} - 2 \zeta_{8} + 6) q^{89} + (4 \zeta_{8}^{3} - 4 \zeta_{8}) q^{91} + ( - 2 \zeta_{8}^{2} - 4 \zeta_{8} - 2) q^{92} + ( - 5 \zeta_{8}^{3} - 6 \zeta_{8}^{2} - 5 \zeta_{8}) q^{94} + (2 \zeta_{8}^{3} - 6 \zeta_{8}^{2} + 6) q^{97} - \zeta_{8}^{3} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{13} + 4 q^{14} - 4 q^{16} - 8 q^{17} + 4 q^{22} - 8 q^{23} - 8 q^{29} + 24 q^{31} - 8 q^{37} + 8 q^{38} + 4 q^{43} - 8 q^{44} - 16 q^{46} - 20 q^{47} + 16 q^{52} + 16 q^{53} + 8 q^{58} - 12 q^{62} - 28 q^{67} + 8 q^{68} - 24 q^{74} - 4 q^{77} - 8 q^{82} - 8 q^{83} + 4 q^{88} + 24 q^{89} - 8 q^{92} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(-\zeta_{8}^{2}\) \(1\) \(-1\)

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
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
−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
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
\(n\): e.g. 2-40 or 990-1000
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.b 4
3.b odd 2 1 3150.2.m.a 4
5.b even 2 1 630.2.m.a 4
5.c odd 4 1 630.2.m.b yes 4
5.c odd 4 1 3150.2.m.a 4
15.d odd 2 1 630.2.m.b yes 4
15.e even 4 1 630.2.m.a 4
15.e even 4 1 inner 3150.2.m.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.m.a 4 5.b even 2 1
630.2.m.a 4 15.e even 4 1
630.2.m.b yes 4 5.c odd 4 1
630.2.m.b yes 4 15.d odd 2 1
3150.2.m.a 4 3.b odd 2 1
3150.2.m.a 4 5.c odd 4 1
3150.2.m.b 4 1.a even 1 1 trivial
3150.2.m.b 4 15.e even 4 1 inner

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
\( T_{13}^{2} + 8T_{13} + 32 \) Copy content Toggle raw display
\( T_{17}^{4} + 8T_{17}^{3} + 32T_{17}^{2} + 32T_{17} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 1 \) Copy content Toggle raw display
$11$ \( T^{4} + 12T^{2} + 4 \) Copy content Toggle raw display
$13$ \( (T^{2} + 8 T + 32)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 8 T^{3} + 32 T^{2} + 32 T + 16 \) Copy content Toggle raw display
$19$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 8 T^{3} + 32 T^{2} - 64 T + 64 \) Copy content Toggle raw display
$29$ \( (T^{2} + 4 T - 4)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 12 T + 18)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 8 T^{3} + 32 T^{2} - 224 T + 784 \) Copy content Toggle raw display
$41$ \( T^{4} + 88T^{2} + 784 \) Copy content Toggle raw display
$43$ \( T^{4} - 4 T^{3} + 8 T^{2} + 392 T + 9604 \) Copy content Toggle raw display
$47$ \( T^{4} + 20 T^{3} + 200 T^{2} + \cdots + 196 \) Copy content Toggle raw display
$53$ \( T^{4} - 16 T^{3} + 128 T^{2} + \cdots + 784 \) Copy content Toggle raw display
$59$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 98)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 28 T^{3} + 392 T^{2} + \cdots + 8836 \) Copy content Toggle raw display
$71$ \( T^{4} + 48T^{2} + 64 \) Copy content Toggle raw display
$73$ \( T^{4} + 10000 \) Copy content Toggle raw display
$79$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 8 T^{3} + 32 T^{2} - 64 T + 64 \) Copy content Toggle raw display
$89$ \( (T^{2} - 12 T + 28)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 24 T^{3} + 288 T^{2} + \cdots + 4624 \) Copy content Toggle raw display
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