Properties

Label 400.3.p.b
Level $400$
Weight $3$
Character orbit 400.p
Analytic conductor $10.899$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [400,3,Mod(193,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.193");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 400.p (of order \(4\), degree \(2\), 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: \(10.8992105744\)
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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 10)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 i - 2) q^{3} + (2 i + 2) q^{7} + i q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 i - 2) q^{3} + (2 i + 2) q^{7} + i q^{9} + 8 q^{11} + (3 i - 3) q^{13} + ( - 7 i - 7) q^{17} + 20 i q^{19} - 8 q^{21} + (2 i - 2) q^{23} + ( - 20 i - 20) q^{27} + 40 i q^{29} - 52 q^{31} + (16 i - 16) q^{33} + (3 i + 3) q^{37} - 12 i q^{39} - 8 q^{41} + (42 i - 42) q^{43} + ( - 18 i - 18) q^{47} - 41 i q^{49} + 28 q^{51} + (53 i - 53) q^{53} + ( - 40 i - 40) q^{57} + 20 i q^{59} - 48 q^{61} + (2 i - 2) q^{63} + (62 i + 62) q^{67} - 8 i q^{69} + 28 q^{71} + ( - 47 i + 47) q^{73} + (16 i + 16) q^{77} + 71 q^{81} + ( - 18 i + 18) q^{83} + ( - 80 i - 80) q^{87} + 80 i q^{89} - 12 q^{91} + ( - 104 i + 104) q^{93} + (63 i + 63) q^{97} + 8 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{3} + 4 q^{7} + 16 q^{11} - 6 q^{13} - 14 q^{17} - 16 q^{21} - 4 q^{23} - 40 q^{27} - 104 q^{31} - 32 q^{33} + 6 q^{37} - 16 q^{41} - 84 q^{43} - 36 q^{47} + 56 q^{51} - 106 q^{53} - 80 q^{57} - 96 q^{61} - 4 q^{63} + 124 q^{67} + 56 q^{71} + 94 q^{73} + 32 q^{77} + 142 q^{81} + 36 q^{83} - 160 q^{87} - 24 q^{91} + 208 q^{93} + 126 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(i\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
1.00000i
1.00000i
0 −2.00000 2.00000i 0 0 0 2.00000 2.00000i 0 1.00000i 0
257.1 0 −2.00000 + 2.00000i 0 0 0 2.00000 + 2.00000i 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
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.3.p.b 2
4.b odd 2 1 50.3.c.c 2
5.b even 2 1 80.3.p.c 2
5.c odd 4 1 80.3.p.c 2
5.c odd 4 1 inner 400.3.p.b 2
12.b even 2 1 450.3.g.b 2
15.d odd 2 1 720.3.bh.c 2
15.e even 4 1 720.3.bh.c 2
20.d odd 2 1 10.3.c.a 2
20.e even 4 1 10.3.c.a 2
20.e even 4 1 50.3.c.c 2
40.e odd 2 1 320.3.p.h 2
40.f even 2 1 320.3.p.a 2
40.i odd 4 1 320.3.p.a 2
40.k even 4 1 320.3.p.h 2
60.h even 2 1 90.3.g.b 2
60.l odd 4 1 90.3.g.b 2
60.l odd 4 1 450.3.g.b 2
140.c even 2 1 490.3.f.b 2
140.j odd 4 1 490.3.f.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.3.c.a 2 20.d odd 2 1
10.3.c.a 2 20.e even 4 1
50.3.c.c 2 4.b odd 2 1
50.3.c.c 2 20.e even 4 1
80.3.p.c 2 5.b even 2 1
80.3.p.c 2 5.c odd 4 1
90.3.g.b 2 60.h even 2 1
90.3.g.b 2 60.l odd 4 1
320.3.p.a 2 40.f even 2 1
320.3.p.a 2 40.i odd 4 1
320.3.p.h 2 40.e odd 2 1
320.3.p.h 2 40.k even 4 1
400.3.p.b 2 1.a even 1 1 trivial
400.3.p.b 2 5.c odd 4 1 inner
450.3.g.b 2 12.b even 2 1
450.3.g.b 2 60.l odd 4 1
490.3.f.b 2 140.c even 2 1
490.3.f.b 2 140.j odd 4 1
720.3.bh.c 2 15.d odd 2 1
720.3.bh.c 2 15.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 \(S_{3}^{\mathrm{new}}(400, [\chi])\):

\( T_{3}^{2} + 4T_{3} + 8 \) Copy content Toggle raw display
\( T_{7}^{2} - 4T_{7} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 8 \) Copy content Toggle raw display
$11$ \( (T - 8)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$17$ \( T^{2} + 14T + 98 \) Copy content Toggle raw display
$19$ \( T^{2} + 400 \) Copy content Toggle raw display
$23$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$29$ \( T^{2} + 1600 \) Copy content Toggle raw display
$31$ \( (T + 52)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$41$ \( (T + 8)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 84T + 3528 \) Copy content Toggle raw display
$47$ \( T^{2} + 36T + 648 \) Copy content Toggle raw display
$53$ \( T^{2} + 106T + 5618 \) Copy content Toggle raw display
$59$ \( T^{2} + 400 \) Copy content Toggle raw display
$61$ \( (T + 48)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 124T + 7688 \) Copy content Toggle raw display
$71$ \( (T - 28)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 94T + 4418 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 36T + 648 \) Copy content Toggle raw display
$89$ \( T^{2} + 6400 \) Copy content Toggle raw display
$97$ \( T^{2} - 126T + 7938 \) Copy content Toggle raw display
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