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Properties

Label 1600.3.b.m
Level $1600$
Weight $3$
Character orbit 1600.b
Analytic conductor $43.597$
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] = [1600,3,Mod(1151,1600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.1151");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1600.b (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: \(43.5968422976\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 800)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} - 2 \beta_{2} q^{7} + ( - \beta_1 - 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} - 2 \beta_{2} q^{7} + ( - \beta_1 - 6) q^{9} + ( - \beta_{3} - 2 \beta_{2}) q^{11} + 4 q^{13} + ( - \beta_1 - 1) q^{17} + ( - 2 \beta_{3} + 7 \beta_{2}) q^{19} + ( - 2 \beta_1 - 30) q^{21} + ( - 4 \beta_{3} - 2 \beta_{2}) q^{23} + ( - 7 \beta_{3} + 12 \beta_{2}) q^{27} + ( - 2 \beta_1 - 24) q^{29} + (10 \beta_{3} - 8 \beta_{2}) q^{31} + ( - 2 \beta_1 - 37) q^{33} + (4 \beta_1 + 6) q^{37} - 4 \beta_{2} q^{39} + ( - 2 \beta_1 + 33) q^{41} + (5 \beta_{3} + 3 \beta_{2}) q^{43} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{47} + ( - 4 \beta_1 - 11) q^{49} + ( - 7 \beta_{3} + 16 \beta_{2}) q^{51} + ( - 2 \beta_1 + 26) q^{53} + (7 \beta_1 + 91) q^{57} + ( - 17 \beta_{3} - 3 \beta_{2}) q^{59} + (2 \beta_1 - 54) q^{61} + ( - 14 \beta_{3} + 42 \beta_{2}) q^{63} + ( - 29 \beta_{3} + 22 \beta_{2}) q^{67} + ( - 2 \beta_1 - 58) q^{69} + (30 \beta_{3} - 26 \beta_{2}) q^{71} + ( - \beta_1 - 43) q^{73} + ( - 4 \beta_1 - 74) q^{77} + ( - 10 \beta_{3} + 16 \beta_{2}) q^{79} + (3 \beta_1 + 77) q^{81} + (\beta_{3} - 12 \beta_{2}) q^{83} + ( - 14 \beta_{3} + 54 \beta_{2}) q^{87} + (5 \beta_1 + 55) q^{89} - 8 \beta_{2} q^{91} + ( - 8 \beta_1 - 50) q^{93} + 46 q^{97} + ( - 23 \beta_{3} + 49 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 24 q^{9} + 16 q^{13} - 4 q^{17} - 120 q^{21} - 96 q^{29} - 148 q^{33} + 24 q^{37} + 132 q^{41} - 44 q^{49} + 104 q^{53} + 364 q^{57} - 216 q^{61} - 232 q^{69} - 172 q^{73} - 296 q^{77} + 308 q^{81} + 220 q^{89} - 200 q^{93} + 184 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -4\nu^{3} + 32\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{3} + 6\nu^{2} - 4\nu - 15 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{3} + 6\nu^{2} + 4\nu - 15 ) / 3 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} + \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 10 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{3} + 4\beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1151.1
1.65831 + 0.500000i
−1.65831 − 0.500000i
−1.65831 + 0.500000i
1.65831 − 0.500000i
0 − 5.31662i 0 0 0 − 10.6332i 0 −19.2665 0
1151.2 0 − 1.31662i 0 0 0 − 2.63325i 0 7.26650 0
1151.3 0 1.31662i 0 0 0 2.63325i 0 7.26650 0
1151.4 0 5.31662i 0 0 0 10.6332i 0 −19.2665 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
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.3.b.m 4
4.b odd 2 1 inner 1600.3.b.m 4
5.b even 2 1 1600.3.b.l 4
5.c odd 4 1 1600.3.h.e 4
5.c odd 4 1 1600.3.h.l 4
8.b even 2 1 800.3.b.b ✓ 4
8.d odd 2 1 800.3.b.b ✓ 4
20.d odd 2 1 1600.3.b.l 4
20.e even 4 1 1600.3.h.e 4
20.e even 4 1 1600.3.h.l 4
40.e odd 2 1 800.3.b.c yes 4
40.f even 2 1 800.3.b.c yes 4
40.i odd 4 1 800.3.h.d 4
40.i odd 4 1 800.3.h.i 4
40.k even 4 1 800.3.h.d 4
40.k even 4 1 800.3.h.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.3.b.b ✓ 4 8.b even 2 1
800.3.b.b ✓ 4 8.d odd 2 1
800.3.b.c yes 4 40.e odd 2 1
800.3.b.c yes 4 40.f even 2 1
800.3.h.d 4 40.i odd 4 1
800.3.h.d 4 40.k even 4 1
800.3.h.i 4 40.i odd 4 1
800.3.h.i 4 40.k even 4 1
1600.3.b.l 4 5.b even 2 1
1600.3.b.l 4 20.d odd 2 1
1600.3.b.m 4 1.a even 1 1 trivial
1600.3.b.m 4 4.b odd 2 1 inner
1600.3.h.e 4 5.c odd 4 1
1600.3.h.e 4 20.e even 4 1
1600.3.h.l 4 5.c odd 4 1
1600.3.h.l 4 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1600, [\chi])\):

\( T_{3}^{4} + 30T_{3}^{2} + 49 \) Copy content Toggle raw display
\( T_{7}^{4} + 120T_{7}^{2} + 784 \) Copy content Toggle raw display
\( T_{13} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 30T^{2} + 49 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 120T^{2} + 784 \) Copy content Toggle raw display
$11$ \( T^{4} + 206T^{2} + 9025 \) Copy content Toggle raw display
$13$ \( (T - 4)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 2 T - 175)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 1198 T^{2} + 2401 \) Copy content Toggle raw display
$23$ \( T^{4} + 824 T^{2} + 144400 \) Copy content Toggle raw display
$29$ \( (T^{2} + 48 T - 128)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 2680 T^{2} + 1567504 \) Copy content Toggle raw display
$37$ \( (T^{2} - 12 T - 2780)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 66 T + 385)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 1440 T^{2} + 473344 \) Copy content Toggle raw display
$47$ \( (T^{2} + 176)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 52 T - 28)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 10368 T^{2} + 13075456 \) Copy content Toggle raw display
$61$ \( (T^{2} + 108 T + 2212)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 21886 T^{2} + 97318225 \) Copy content Toggle raw display
$71$ \( T^{4} + 25440 T^{2} + 152967424 \) Copy content Toggle raw display
$73$ \( (T^{2} + 86 T + 1673)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 6200 T^{2} + 5326864 \) Copy content Toggle raw display
$83$ \( T^{4} + 4014 T^{2} + 429025 \) Copy content Toggle raw display
$89$ \( (T^{2} - 110 T - 1375)^{2} \) Copy content Toggle raw display
$97$ \( (T - 46)^{4} \) Copy content Toggle raw display
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