Properties

Label 1200.4.f.v
Level $1200$
Weight $4$
Character orbit 1200.f
Analytic conductor $70.802$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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: \(4\)
Coefficient field: \(\Q(i, \sqrt{19})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 9x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 75)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 \beta_1 q^{3} + (\beta_{3} - 13 \beta_1) q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 \beta_1 q^{3} + (\beta_{3} - 13 \beta_1) q^{7} - 9 q^{9} + (\beta_{2} - 14) q^{11} + ( - 4 \beta_{3} - 9 \beta_1) q^{13} + (5 \beta_{3} - 34 \beta_1) q^{17} + (\beta_{2} + 3) q^{19} + ( - 3 \beta_{2} + 39) q^{21} + (3 \beta_{3} + 66 \beta_1) q^{23} - 27 \beta_1 q^{27} + ( - 7 \beta_{2} - 46) q^{29} + (7 \beta_{2} - 61) q^{31} + (3 \beta_{3} - 42 \beta_1) q^{33} + ( - 6 \beta_{3} - 142 \beta_1) q^{37} + (12 \beta_{2} + 27) q^{39} + (13 \beta_{2} + 196) q^{41} + (\beta_{3} + 345 \beta_1) q^{43} + ( - 8 \beta_{3} + 310 \beta_1) q^{47} + (26 \beta_{2} - 130) q^{49} + ( - 15 \beta_{2} + 102) q^{51} + (7 \beta_{3} + 424 \beta_1) q^{53} + (3 \beta_{3} + 9 \beta_1) q^{57} + ( - 16 \beta_{2} + 62) q^{59} + ( - 14 \beta_{2} + 375) q^{61} + ( - 9 \beta_{3} + 117 \beta_1) q^{63} + (25 \beta_{3} + 179 \beta_1) q^{67} + ( - 9 \beta_{2} - 198) q^{69} + (5 \beta_{2} - 412) q^{71} + (2 \beta_{3} + 54 \beta_1) q^{73} + ( - 27 \beta_{3} + 486 \beta_1) q^{77} + ( - 40 \beta_{2} - 440) q^{79} + 81 q^{81} + ( - 48 \beta_{3} + 78 \beta_1) q^{83} + ( - 21 \beta_{3} - 138 \beta_1) q^{87} + ( - 36 \beta_{2} + 432) q^{89} + ( - 43 \beta_{2} + 1099) q^{91} + (21 \beta_{3} - 183 \beta_1) q^{93} + 521 \beta_1 q^{97} + ( - 9 \beta_{2} + 126) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 36 q^{9} - 56 q^{11} + 12 q^{19} + 156 q^{21} - 184 q^{29} - 244 q^{31} + 108 q^{39} + 784 q^{41} - 520 q^{49} + 408 q^{51} + 248 q^{59} + 1500 q^{61} - 792 q^{69} - 1648 q^{71} - 1760 q^{79} + 324 q^{81} + 1728 q^{89} + 4396 q^{91} + 504 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 9x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - 4\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{3} + 56\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 8\nu^{2} - 36 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 4\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 36 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{2} + 14\beta_1 ) / 2 \) 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
2.17945 0.500000i
−2.17945 0.500000i
−2.17945 + 0.500000i
2.17945 + 0.500000i
0 3.00000i 0 0 0 4.43560i 0 −9.00000 0
49.2 0 3.00000i 0 0 0 30.4356i 0 −9.00000 0
49.3 0 3.00000i 0 0 0 30.4356i 0 −9.00000 0
49.4 0 3.00000i 0 0 0 4.43560i 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.v 4
4.b odd 2 1 75.4.b.c 4
5.b even 2 1 inner 1200.4.f.v 4
5.c odd 4 1 1200.4.a.bl 2
5.c odd 4 1 1200.4.a.bu 2
12.b even 2 1 225.4.b.h 4
20.d odd 2 1 75.4.b.c 4
20.e even 4 1 75.4.a.d 2
20.e even 4 1 75.4.a.e yes 2
60.h even 2 1 225.4.b.h 4
60.l odd 4 1 225.4.a.j 2
60.l odd 4 1 225.4.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.4.a.d 2 20.e even 4 1
75.4.a.e yes 2 20.e even 4 1
75.4.b.c 4 4.b odd 2 1
75.4.b.c 4 20.d odd 2 1
225.4.a.j 2 60.l odd 4 1
225.4.a.n 2 60.l odd 4 1
225.4.b.h 4 12.b even 2 1
225.4.b.h 4 60.h even 2 1
1200.4.a.bl 2 5.c odd 4 1
1200.4.a.bu 2 5.c odd 4 1
1200.4.f.v 4 1.a even 1 1 trivial
1200.4.f.v 4 5.b even 2 1 inner

Hecke kernels

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

\( T_{7}^{4} + 946T_{7}^{2} + 18225 \) Copy content Toggle raw display
\( T_{11}^{2} + 28T_{11} - 108 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 946 T^{2} + 18225 \) Copy content Toggle raw display
$11$ \( (T^{2} + 28 T - 108)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 9890 T^{2} + 22877089 \) Copy content Toggle raw display
$17$ \( T^{4} + 17512 T^{2} + 41525136 \) Copy content Toggle raw display
$19$ \( (T^{2} - 6 T - 295)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 14184 T^{2} + 2624400 \) Copy content Toggle raw display
$29$ \( (T^{2} + 92 T - 12780)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 122 T - 11175)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 62216 T^{2} + 85008400 \) Copy content Toggle raw display
$41$ \( (T^{2} - 392 T - 12960)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 14094675841 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 5874302736 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 27185414400 \) Copy content Toggle raw display
$59$ \( (T^{2} - 124 T - 73980)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 750 T + 81041)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 24951045681 \) Copy content Toggle raw display
$71$ \( (T^{2} + 824 T + 162144)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 8264 T^{2} + 2890000 \) Copy content Toggle raw display
$79$ \( (T^{2} + 880 T - 292800)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 482096926224 \) Copy content Toggle raw display
$89$ \( (T^{2} - 864 T - 207360)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 271441)^{2} \) Copy content Toggle raw display
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