Properties

Label 400.8.c.l
Level $400$
Weight $8$
Character orbit 400.c
Analytic conductor $124.954$
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] = [400,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.49");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
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: \(124.954010194\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 20)
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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 \beta q^{3} - 353 \beta q^{7} + 2151 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 3 \beta q^{3} - 353 \beta q^{7} + 2151 q^{9} + 3840 q^{11} - 2027 \beta q^{13} - 429 \beta q^{17} + 21044 q^{19} + 4236 q^{21} - 42669 \beta q^{23} + 13014 \beta q^{27} + 83106 q^{29} + 145564 q^{31} + 11520 \beta q^{33} + 249443 \beta q^{37} + 24324 q^{39} - 689514 q^{41} - 433945 \beta q^{43} + 117819 \beta q^{47} + 325107 q^{49} + 5148 q^{51} + 917721 \beta q^{53} + 63132 \beta q^{57} + 629508 q^{59} - 2667958 q^{61} - 759303 \beta q^{63} - 1686653 \beta q^{67} + 512028 q^{69} + 2600052 q^{71} - 814247 \beta q^{73} - 1355520 \beta q^{77} - 4243528 q^{79} + 4548069 q^{81} - 625689 \beta q^{83} + 249318 \beta q^{87} - 6299466 q^{89} - 2862124 q^{91} + 436692 \beta q^{93} - 1988257 \beta q^{97} + 8259840 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4302 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4302 q^{9} + 7680 q^{11} + 42088 q^{19} + 8472 q^{21} + 166212 q^{29} + 291128 q^{31} + 48648 q^{39} - 1379028 q^{41} + 650214 q^{49} + 10296 q^{51} + 1259016 q^{59} - 5335916 q^{61} + 1024056 q^{69} + 5200104 q^{71} - 8487056 q^{79} + 9096138 q^{81} - 12598932 q^{89} - 5724248 q^{91} + 16519680 q^{99}+O(q^{100}) \) 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 6.00000i 0 0 0 706.000i 0 2151.00 0
49.2 0 6.00000i 0 0 0 706.000i 0 2151.00 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 400.8.c.l 2
4.b odd 2 1 100.8.c.a 2
5.b even 2 1 inner 400.8.c.l 2
5.c odd 4 1 80.8.a.b 1
5.c odd 4 1 400.8.a.j 1
20.d odd 2 1 100.8.c.a 2
20.e even 4 1 20.8.a.a 1
20.e even 4 1 100.8.a.a 1
40.i odd 4 1 320.8.a.d 1
40.k even 4 1 320.8.a.e 1
60.l odd 4 1 180.8.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.8.a.a 1 20.e even 4 1
80.8.a.b 1 5.c odd 4 1
100.8.a.a 1 20.e even 4 1
100.8.c.a 2 4.b odd 2 1
100.8.c.a 2 20.d odd 2 1
180.8.a.c 1 60.l odd 4 1
320.8.a.d 1 40.i odd 4 1
320.8.a.e 1 40.k even 4 1
400.8.a.j 1 5.c odd 4 1
400.8.c.l 2 1.a even 1 1 trivial
400.8.c.l 2 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 36 \) acting on \(S_{8}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 36 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 498436 \) Copy content Toggle raw display
$11$ \( (T - 3840)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 16434916 \) Copy content Toggle raw display
$17$ \( T^{2} + 736164 \) Copy content Toggle raw display
$19$ \( (T - 21044)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 7282574244 \) Copy content Toggle raw display
$29$ \( (T - 83106)^{2} \) Copy content Toggle raw display
$31$ \( (T - 145564)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 248887240996 \) Copy content Toggle raw display
$41$ \( (T + 689514)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 753233052100 \) Copy content Toggle raw display
$47$ \( T^{2} + 55525267044 \) Copy content Toggle raw display
$53$ \( T^{2} + 3368847335364 \) Copy content Toggle raw display
$59$ \( (T - 629508)^{2} \) Copy content Toggle raw display
$61$ \( (T + 2667958)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 11379193369636 \) Copy content Toggle raw display
$71$ \( (T - 2600052)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 2651992708036 \) Copy content Toggle raw display
$79$ \( (T + 4243528)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 1565946898884 \) Copy content Toggle raw display
$89$ \( (T + 6299466)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 15812663592196 \) Copy content Toggle raw display
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