Properties

Label 2400.2.w.c
Level $2400$
Weight $2$
Character orbit 2400.w
Analytic conductor $19.164$
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] = [2400,2,Mod(607,2400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2400, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 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("2400.607");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2400.w (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: \(19.1640964851\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 480)
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.

\(f(q)\) \(=\) \( q - \zeta_{8} q^{3} + (\zeta_{8}^{2} - 1) q^{7} + \zeta_{8}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{8} q^{3} + (\zeta_{8}^{2} - 1) q^{7} + \zeta_{8}^{2} q^{9} + ( - \zeta_{8}^{3} - 4 \zeta_{8}^{2} - \zeta_{8}) q^{11} + (2 \zeta_{8}^{3} + 2 \zeta_{8}^{2} - 2) q^{13} + ( - 4 \zeta_{8}^{2} - 2 \zeta_{8} - 4) q^{17} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8} + 2) q^{19} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{21} + 6 \zeta_{8} q^{23} - \zeta_{8}^{3} q^{27} + (3 \zeta_{8}^{3} - 4 \zeta_{8}^{2} + 3 \zeta_{8}) q^{29} + (4 \zeta_{8}^{3} - 4 \zeta_{8}^{2} + 4 \zeta_{8}) q^{31} + (4 \zeta_{8}^{3} + \zeta_{8}^{2} - 1) q^{33} + (2 \zeta_{8}^{2} - 2 \zeta_{8} + 2) q^{37} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8} + 2) q^{39} + (6 \zeta_{8}^{3} - 6 \zeta_{8} + 2) q^{41} + (2 \zeta_{8}^{2} + 4 \zeta_{8} + 2) q^{43} + (6 \zeta_{8}^{3} + 2 \zeta_{8}^{2} - 2) q^{47} + 5 \zeta_{8}^{2} q^{49} + (4 \zeta_{8}^{3} + 2 \zeta_{8}^{2} + 4 \zeta_{8}) q^{51} - 4 \zeta_{8}^{3} q^{53} + ( - 2 \zeta_{8}^{2} - 2 \zeta_{8} - 2) q^{57} + (\zeta_{8}^{3} - \zeta_{8}) q^{59} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8} - 6) q^{61} + ( - \zeta_{8}^{2} - 1) q^{63} + (8 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 2) q^{67} - 6 \zeta_{8}^{2} q^{69} + ( - 4 \zeta_{8}^{3} + 8 \zeta_{8}^{2} - 4 \zeta_{8}) q^{71} + (12 \zeta_{8}^{3} - \zeta_{8}^{2} + 1) q^{73} + (4 \zeta_{8}^{2} + 2 \zeta_{8} + 4) q^{77} + ( - 4 \zeta_{8}^{3} + 4 \zeta_{8} - 8) q^{79} - q^{81} + ( - 4 \zeta_{8}^{2} + 10 \zeta_{8} - 4) q^{83} + (4 \zeta_{8}^{3} - 3 \zeta_{8}^{2} + 3) q^{87} + (4 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 4 \zeta_{8}) q^{89} + ( - 2 \zeta_{8}^{3} - 4 \zeta_{8}^{2} - 2 \zeta_{8}) q^{91} + (4 \zeta_{8}^{3} - 4 \zeta_{8}^{2} + 4) q^{93} + (5 \zeta_{8}^{2} - 8 \zeta_{8} + 5) q^{97} + ( - \zeta_{8}^{3} + \zeta_{8} + 4) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{7} - 8 q^{13} - 16 q^{17} + 8 q^{19} - 4 q^{33} + 8 q^{37} + 8 q^{39} + 8 q^{41} + 8 q^{43} - 8 q^{47} - 8 q^{57} - 24 q^{61} - 4 q^{63} + 8 q^{67} + 4 q^{73} + 16 q^{77} - 32 q^{79} - 4 q^{81} - 16 q^{83} + 12 q^{87} + 16 q^{93} + 20 q^{97} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1601\) \(1951\)
\(\chi(n)\) \(\zeta_{8}^{2}\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
607.1
0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
0 −0.707107 0.707107i 0 0 0 −1.00000 + 1.00000i 0 1.00000i 0
607.2 0 0.707107 + 0.707107i 0 0 0 −1.00000 + 1.00000i 0 1.00000i 0
2143.1 0 −0.707107 + 0.707107i 0 0 0 −1.00000 1.00000i 0 1.00000i 0
2143.2 0 0.707107 0.707107i 0 0 0 −1.00000 1.00000i 0 1.00000i 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
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2400.2.w.c 4
4.b odd 2 1 2400.2.w.d 4
5.b even 2 1 480.2.w.b yes 4
5.c odd 4 1 480.2.w.a 4
5.c odd 4 1 2400.2.w.d 4
15.d odd 2 1 1440.2.x.n 4
15.e even 4 1 1440.2.x.m 4
20.d odd 2 1 480.2.w.a 4
20.e even 4 1 480.2.w.b yes 4
20.e even 4 1 inner 2400.2.w.c 4
40.e odd 2 1 960.2.w.a 4
40.f even 2 1 960.2.w.b 4
40.i odd 4 1 960.2.w.a 4
40.k even 4 1 960.2.w.b 4
60.h even 2 1 1440.2.x.m 4
60.l odd 4 1 1440.2.x.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.w.a 4 5.c odd 4 1
480.2.w.a 4 20.d odd 2 1
480.2.w.b yes 4 5.b even 2 1
480.2.w.b yes 4 20.e even 4 1
960.2.w.a 4 40.e odd 2 1
960.2.w.a 4 40.i odd 4 1
960.2.w.b 4 40.f even 2 1
960.2.w.b 4 40.k even 4 1
1440.2.x.m 4 15.e even 4 1
1440.2.x.m 4 60.h even 2 1
1440.2.x.n 4 15.d odd 2 1
1440.2.x.n 4 60.l odd 4 1
2400.2.w.c 4 1.a even 1 1 trivial
2400.2.w.c 4 20.e even 4 1 inner
2400.2.w.d 4 4.b odd 2 1
2400.2.w.d 4 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 \(S_{2}^{\mathrm{new}}(2400, [\chi])\):

\( T_{7}^{2} + 2T_{7} + 2 \) Copy content Toggle raw display
\( T_{17}^{4} + 16T_{17}^{3} + 128T_{17}^{2} + 448T_{17} + 784 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 36T^{2} + 196 \) Copy content Toggle raw display
$13$ \( T^{4} + 8 T^{3} + 32 T^{2} + 32 T + 16 \) Copy content Toggle raw display
$17$ \( T^{4} + 16 T^{3} + 128 T^{2} + \cdots + 784 \) Copy content Toggle raw display
$19$ \( (T^{2} - 4 T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 1296 \) Copy content Toggle raw display
$29$ \( T^{4} + 68T^{2} + 4 \) Copy content Toggle raw display
$31$ \( T^{4} + 96T^{2} + 256 \) Copy content Toggle raw display
$37$ \( T^{4} - 8 T^{3} + 32 T^{2} - 32 T + 16 \) Copy content Toggle raw display
$41$ \( (T^{2} - 4 T - 68)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 8 T^{3} + 32 T^{2} + 64 T + 64 \) Copy content Toggle raw display
$47$ \( T^{4} + 8 T^{3} + 32 T^{2} - 224 T + 784 \) Copy content Toggle raw display
$53$ \( T^{4} + 256 \) Copy content Toggle raw display
$59$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 12 T + 28)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 8 T^{3} + 32 T^{2} + \cdots + 3136 \) Copy content Toggle raw display
$71$ \( T^{4} + 192T^{2} + 1024 \) Copy content Toggle raw display
$73$ \( T^{4} - 4 T^{3} + 8 T^{2} + \cdots + 20164 \) Copy content Toggle raw display
$79$ \( (T^{2} + 16 T + 32)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 16 T^{3} + 128 T^{2} + \cdots + 4624 \) Copy content Toggle raw display
$89$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
$97$ \( T^{4} - 20 T^{3} + 200 T^{2} + \cdots + 196 \) Copy content Toggle raw display
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