Properties

Label 400.8.c.b
Level $400$
Weight $8$
Character orbit 400.c
Analytic conductor $124.954$
Analytic rank $0$
Dimension $2$
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 8)
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.

\(f(q)\) \(=\) \( q + 42 \beta q^{3} - 228 \beta q^{7} - 4869 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 42 \beta q^{3} - 228 \beta q^{7} - 4869 q^{9} + 2524 q^{11} - 5389 \beta q^{13} + 5575 \beta q^{17} + 4124 q^{19} + 38304 q^{21} - 40852 \beta q^{23} - 112644 \beta q^{27} - 99798 q^{29} + 40480 q^{31} + 106008 \beta q^{33} + 209721 \beta q^{37} + 905352 q^{39} + 141402 q^{41} + 345214 \beta q^{43} - 341016 \beta q^{47} + 615607 q^{49} - 936600 q^{51} + 906559 \beta q^{53} + 173208 \beta q^{57} - 966028 q^{59} + 1887670 q^{61} + 1110132 \beta q^{63} + 1482934 \beta q^{67} + 6863136 q^{69} + 2548232 q^{71} - 840163 \beta q^{73} - 575472 \beta q^{77} + 4038064 q^{79} + 8275689 q^{81} + 2692882 \beta q^{83} - 4191516 \beta q^{87} + 6473046 q^{89} - 4914768 q^{91} + 1700160 \beta q^{93} + 3032879 \beta q^{97} - 12289356 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 9738 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 9738 q^{9} + 5048 q^{11} + 8248 q^{19} + 76608 q^{21} - 199596 q^{29} + 80960 q^{31} + 1810704 q^{39} + 282804 q^{41} + 1231214 q^{49} - 1873200 q^{51} - 1932056 q^{59} + 3775340 q^{61} + 13726272 q^{69} + 5096464 q^{71} + 8076128 q^{79} + 16551378 q^{81} + 12946092 q^{89} - 9829536 q^{91} - 24578712 q^{99}+O(q^{100}) \) 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 84.0000i 0 0 0 456.000i 0 −4869.00 0
49.2 0 84.0000i 0 0 0 456.000i 0 −4869.00 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.b 2
4.b odd 2 1 200.8.c.a 2
5.b even 2 1 inner 400.8.c.b 2
5.c odd 4 1 16.8.a.c 1
5.c odd 4 1 400.8.a.b 1
15.e even 4 1 144.8.a.g 1
20.d odd 2 1 200.8.c.a 2
20.e even 4 1 8.8.a.a 1
20.e even 4 1 200.8.a.i 1
40.i odd 4 1 64.8.a.a 1
40.k even 4 1 64.8.a.g 1
60.l odd 4 1 72.8.a.d 1
80.i odd 4 1 256.8.b.c 2
80.j even 4 1 256.8.b.e 2
80.s even 4 1 256.8.b.e 2
80.t odd 4 1 256.8.b.c 2
120.q odd 4 1 576.8.a.j 1
120.w even 4 1 576.8.a.k 1
140.j odd 4 1 392.8.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.8.a.a 1 20.e even 4 1
16.8.a.c 1 5.c odd 4 1
64.8.a.a 1 40.i odd 4 1
64.8.a.g 1 40.k even 4 1
72.8.a.d 1 60.l odd 4 1
144.8.a.g 1 15.e even 4 1
200.8.a.i 1 20.e even 4 1
200.8.c.a 2 4.b odd 2 1
200.8.c.a 2 20.d odd 2 1
256.8.b.c 2 80.i odd 4 1
256.8.b.c 2 80.t odd 4 1
256.8.b.e 2 80.j even 4 1
256.8.b.e 2 80.s even 4 1
392.8.a.d 1 140.j odd 4 1
400.8.a.b 1 5.c odd 4 1
400.8.c.b 2 1.a even 1 1 trivial
400.8.c.b 2 5.b even 2 1 inner
576.8.a.j 1 120.q odd 4 1
576.8.a.k 1 120.w even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 7056 \) 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
$3$ \( T^{2} + 7056 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 207936 \) Copy content Toggle raw display
$11$ \( (T - 2524)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 116165284 \) Copy content Toggle raw display
$17$ \( T^{2} + 124322500 \) Copy content Toggle raw display
$19$ \( (T - 4124)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 6675543616 \) Copy content Toggle raw display
$29$ \( (T + 99798)^{2} \) Copy content Toggle raw display
$31$ \( (T - 40480)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 175931591364 \) Copy content Toggle raw display
$41$ \( (T - 141402)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 476690823184 \) Copy content Toggle raw display
$47$ \( T^{2} + 465167649024 \) Copy content Toggle raw display
$53$ \( T^{2} + 3287396881924 \) Copy content Toggle raw display
$59$ \( (T + 966028)^{2} \) Copy content Toggle raw display
$61$ \( (T - 1887670)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 8796372993424 \) Copy content Toggle raw display
$71$ \( (T - 2548232)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 2823495466276 \) Copy content Toggle raw display
$79$ \( (T - 4038064)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 29006453863696 \) Copy content Toggle raw display
$89$ \( (T - 6473046)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 36793420114564 \) Copy content Toggle raw display
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