Properties

Label 400.4.c.m
Level $400$
Weight $4$
Character orbit 400.c
Analytic conductor $23.601$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [400,4,Mod(49,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("400.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 400.c (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: \(23.6007640023\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 200)
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.

\(f(q)\) \(=\) \( q + i q^{3} + 6 i q^{7} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + i q^{3} + 6 i q^{7} + 26 q^{9} + 19 q^{11} - 12 i q^{13} - 75 i q^{17} - 91 q^{19} - 6 q^{21} + 174 i q^{23} + 53 i q^{27} + 272 q^{29} + 230 q^{31} + 19 i q^{33} - 182 i q^{37} + 12 q^{39} + 117 q^{41} + 372 i q^{43} + 52 i q^{47} + 307 q^{49} + 75 q^{51} + 402 i q^{53} - 91 i q^{57} + 312 q^{59} + 170 q^{61} + 156 i q^{63} - 763 i q^{67} - 174 q^{69} + 52 q^{71} + 981 i q^{73} + 114 i q^{77} + 1054 q^{79} + 649 q^{81} + 351 i q^{83} + 272 i q^{87} - 799 q^{89} + 72 q^{91} + 230 i q^{93} + 962 i q^{97} + 494 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 52 q^{9} + 38 q^{11} - 182 q^{19} - 12 q^{21} + 544 q^{29} + 460 q^{31} + 24 q^{39} + 234 q^{41} + 614 q^{49} + 150 q^{51} + 624 q^{59} + 340 q^{61} - 348 q^{69} + 104 q^{71} + 2108 q^{79} + 1298 q^{81} - 1598 q^{89} + 144 q^{91} + 988 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 1.00000i 0 0 0 6.00000i 0 26.0000 0
49.2 0 1.00000i 0 0 0 6.00000i 0 26.0000 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 400.4.c.m 2
4.b odd 2 1 200.4.c.g 2
5.b even 2 1 inner 400.4.c.m 2
5.c odd 4 1 400.4.a.j 1
5.c odd 4 1 400.4.a.k 1
12.b even 2 1 1800.4.f.p 2
20.d odd 2 1 200.4.c.g 2
20.e even 4 1 200.4.a.e 1
20.e even 4 1 200.4.a.f yes 1
40.i odd 4 1 1600.4.a.v 1
40.i odd 4 1 1600.4.a.be 1
40.k even 4 1 1600.4.a.w 1
40.k even 4 1 1600.4.a.bf 1
60.h even 2 1 1800.4.f.p 2
60.l odd 4 1 1800.4.a.l 1
60.l odd 4 1 1800.4.a.w 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.4.a.e 1 20.e even 4 1
200.4.a.f yes 1 20.e even 4 1
200.4.c.g 2 4.b odd 2 1
200.4.c.g 2 20.d odd 2 1
400.4.a.j 1 5.c odd 4 1
400.4.a.k 1 5.c odd 4 1
400.4.c.m 2 1.a even 1 1 trivial
400.4.c.m 2 5.b even 2 1 inner
1600.4.a.v 1 40.i odd 4 1
1600.4.a.w 1 40.k even 4 1
1600.4.a.be 1 40.i odd 4 1
1600.4.a.bf 1 40.k even 4 1
1800.4.a.l 1 60.l odd 4 1
1800.4.a.w 1 60.l odd 4 1
1800.4.f.p 2 12.b even 2 1
1800.4.f.p 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(400, [\chi])\):

\( T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 36 \) Copy content Toggle raw display
\( T_{11} - 19 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 36 \) Copy content Toggle raw display
$11$ \( (T - 19)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 144 \) Copy content Toggle raw display
$17$ \( T^{2} + 5625 \) Copy content Toggle raw display
$19$ \( (T + 91)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 30276 \) Copy content Toggle raw display
$29$ \( (T - 272)^{2} \) Copy content Toggle raw display
$31$ \( (T - 230)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 33124 \) Copy content Toggle raw display
$41$ \( (T - 117)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 138384 \) Copy content Toggle raw display
$47$ \( T^{2} + 2704 \) Copy content Toggle raw display
$53$ \( T^{2} + 161604 \) Copy content Toggle raw display
$59$ \( (T - 312)^{2} \) Copy content Toggle raw display
$61$ \( (T - 170)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 582169 \) Copy content Toggle raw display
$71$ \( (T - 52)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 962361 \) Copy content Toggle raw display
$79$ \( (T - 1054)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 123201 \) Copy content Toggle raw display
$89$ \( (T + 799)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 925444 \) Copy content Toggle raw display
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