Properties

Label 1200.4.f.b
Level $1200$
Weight $4$
Character orbit 1200.f
Analytic conductor $70.802$
Analytic rank $0$
Dimension $2$
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] = [1200,4,Mod(49,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1200.f (of order \(2\), degree \(1\), 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: \(70.8022920069\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 i q^{3} + 24 i q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 i q^{3} + 24 i q^{7} - 9 q^{9} - 52 q^{11} - 22 i q^{13} - 14 i q^{17} - 20 q^{19} - 72 q^{21} - 168 i q^{23} - 27 i q^{27} - 230 q^{29} + 288 q^{31} - 156 i q^{33} - 34 i q^{37} + 66 q^{39} + 122 q^{41} - 188 i q^{43} - 256 i q^{47} - 233 q^{49} + 42 q^{51} + 338 i q^{53} - 60 i q^{57} + 100 q^{59} + 742 q^{61} - 216 i q^{63} + 84 i q^{67} + 504 q^{69} + 328 q^{71} + 38 i q^{73} - 1248 i q^{77} - 240 q^{79} + 81 q^{81} + 1212 i q^{83} - 690 i q^{87} - 330 q^{89} + 528 q^{91} + 864 i q^{93} + 866 i q^{97} + 468 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 18 q^{9} - 104 q^{11} - 40 q^{19} - 144 q^{21} - 460 q^{29} + 576 q^{31} + 132 q^{39} + 244 q^{41} - 466 q^{49} + 84 q^{51} + 200 q^{59} + 1484 q^{61} + 1008 q^{69} + 656 q^{71} - 480 q^{79} + 162 q^{81} - 660 q^{89} + 1056 q^{91} + 936 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.00000i
1.00000i
0 3.00000i 0 0 0 24.0000i 0 −9.00000 0
49.2 0 3.00000i 0 0 0 24.0000i 0 −9.00000 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.4.f.b 2
4.b odd 2 1 75.4.b.b 2
5.b even 2 1 inner 1200.4.f.b 2
5.c odd 4 1 240.4.a.e 1
5.c odd 4 1 1200.4.a.t 1
12.b even 2 1 225.4.b.e 2
15.e even 4 1 720.4.a.n 1
20.d odd 2 1 75.4.b.b 2
20.e even 4 1 15.4.a.a 1
20.e even 4 1 75.4.a.b 1
40.i odd 4 1 960.4.a.ba 1
40.k even 4 1 960.4.a.b 1
60.h even 2 1 225.4.b.e 2
60.l odd 4 1 45.4.a.c 1
60.l odd 4 1 225.4.a.f 1
140.j odd 4 1 735.4.a.e 1
180.v odd 12 2 405.4.e.i 2
180.x even 12 2 405.4.e.g 2
220.i odd 4 1 1815.4.a.e 1
420.w even 4 1 2205.4.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.a 1 20.e even 4 1
45.4.a.c 1 60.l odd 4 1
75.4.a.b 1 20.e even 4 1
75.4.b.b 2 4.b odd 2 1
75.4.b.b 2 20.d odd 2 1
225.4.a.f 1 60.l odd 4 1
225.4.b.e 2 12.b even 2 1
225.4.b.e 2 60.h even 2 1
240.4.a.e 1 5.c odd 4 1
405.4.e.g 2 180.x even 12 2
405.4.e.i 2 180.v odd 12 2
720.4.a.n 1 15.e even 4 1
735.4.a.e 1 140.j odd 4 1
960.4.a.b 1 40.k even 4 1
960.4.a.ba 1 40.i odd 4 1
1200.4.a.t 1 5.c odd 4 1
1200.4.f.b 2 1.a even 1 1 trivial
1200.4.f.b 2 5.b even 2 1 inner
1815.4.a.e 1 220.i odd 4 1
2205.4.a.l 1 420.w even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7}^{2} + 576 \) Copy content Toggle raw display
\( T_{11} + 52 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 576 \) Copy content Toggle raw display
$11$ \( (T + 52)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 484 \) Copy content Toggle raw display
$17$ \( T^{2} + 196 \) Copy content Toggle raw display
$19$ \( (T + 20)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 28224 \) Copy content Toggle raw display
$29$ \( (T + 230)^{2} \) Copy content Toggle raw display
$31$ \( (T - 288)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 1156 \) Copy content Toggle raw display
$41$ \( (T - 122)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 35344 \) Copy content Toggle raw display
$47$ \( T^{2} + 65536 \) Copy content Toggle raw display
$53$ \( T^{2} + 114244 \) Copy content Toggle raw display
$59$ \( (T - 100)^{2} \) Copy content Toggle raw display
$61$ \( (T - 742)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 7056 \) Copy content Toggle raw display
$71$ \( (T - 328)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 1444 \) Copy content Toggle raw display
$79$ \( (T + 240)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1468944 \) Copy content Toggle raw display
$89$ \( (T + 330)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 749956 \) Copy content Toggle raw display
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