Properties

Label 400.4.n.d
Level $400$
Weight $4$
Character orbit 400.n
Analytic conductor $23.601$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [400,4,Mod(143,400)]
 
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([2, 0, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.143");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 400.n (of order \(4\), degree \(2\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.1731891456.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{22} \)
Twist minimal: yes
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 a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} - \beta_{5} q^{7} + 7 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} - \beta_{5} q^{7} + 7 \beta_{2} q^{9} + \beta_1 q^{11} + \beta_{6} q^{13} - \beta_{7} q^{17} - 7 \beta_{3} q^{19} + 34 q^{21} + 21 \beta_{4} q^{23} - 20 \beta_{5} q^{27} + 174 \beta_{2} q^{29} + 14 \beta_1 q^{31} + \beta_{6} q^{33} - 34 \beta_{3} q^{39} + 252 q^{41} - 49 \beta_{4} q^{43} + 69 \beta_{5} q^{47} + 309 \beta_{2} q^{49} + 34 \beta_1 q^{51} - 7 \beta_{6} q^{53} + 7 \beta_{7} q^{57} - 63 \beta_{3} q^{59} + 56 q^{61} + 7 \beta_{4} q^{63} - 77 \beta_{5} q^{67} + 714 \beta_{2} q^{69} + 28 \beta_1 q^{71} - 7 \beta_{6} q^{73} - \beta_{7} q^{77} - 50 \beta_{3} q^{79} + 869 q^{81} - 117 \beta_{4} q^{83} + 174 \beta_{5} q^{87} - 42 \beta_{2} q^{89} + 34 \beta_1 q^{91} + 14 \beta_{6} q^{93} - 13 \beta_{7} q^{97} - 7 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 272 q^{21} + 2016 q^{41} + 448 q^{61} + 6952 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 9x^{6} + 65x^{4} - 144x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 9\nu^{6} - 65\nu^{4} + 585\nu^{2} - 776 ) / 65 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -9\nu^{7} + 65\nu^{5} - 377\nu^{3} + 256\nu ) / 832 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 25\nu^{5} + 145\nu^{3} - 544\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -117\nu^{7} + 128\nu^{6} + 845\nu^{5} - 6565\nu^{3} + 3328\nu + 19008 ) / 4160 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -117\nu^{7} - 128\nu^{6} + 845\nu^{5} - 6565\nu^{3} + 3328\nu - 19008 ) / 4160 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -17\nu^{7} + 104\nu^{6} + 169\nu^{5} - 936\nu^{4} - 1313\nu^{3} + 5096\nu^{2} + 5152\nu - 7488 ) / 104 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -17\nu^{7} - 104\nu^{6} + 169\nu^{5} + 936\nu^{4} - 1313\nu^{3} - 5096\nu^{2} + 5152\nu + 7488 ) / 104 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} - 8\beta_{5} - 8\beta_{4} + 2\beta_{3} + 16\beta_{2} ) / 64 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{6} + 8\beta_{5} - 8\beta_{4} + 18\beta _1 + 144 ) / 64 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{5} - 5\beta_{4} + 26\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9\beta_{7} - 9\beta_{6} - 72\beta_{5} + 72\beta_{4} + 98\beta _1 - 784 ) / 64 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -29\beta_{7} - 29\beta_{6} - 232\beta_{5} - 232\beta_{4} - 202\beta_{3} + 1616\beta_{2} ) / 64 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -65\beta_{5} + 65\beta_{4} - 594 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -181\beta_{7} - 181\beta_{6} + 1448\beta_{5} + 1448\beta_{4} - 1402\beta_{3} - 11216\beta_{2} ) / 64 \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1
2.21837 1.28078i
−2.21837 1.28078i
−1.35234 + 0.780776i
1.35234 + 0.780776i
−2.21837 + 1.28078i
2.21837 + 1.28078i
1.35234 0.780776i
−1.35234 0.780776i
0 −4.12311 + 4.12311i 0 0 0 −4.12311 4.12311i 0 7.00000i 0
143.2 0 −4.12311 + 4.12311i 0 0 0 −4.12311 4.12311i 0 7.00000i 0
143.3 0 4.12311 4.12311i 0 0 0 4.12311 + 4.12311i 0 7.00000i 0
143.4 0 4.12311 4.12311i 0 0 0 4.12311 + 4.12311i 0 7.00000i 0
207.1 0 −4.12311 4.12311i 0 0 0 −4.12311 + 4.12311i 0 7.00000i 0
207.2 0 −4.12311 4.12311i 0 0 0 −4.12311 + 4.12311i 0 7.00000i 0
207.3 0 4.12311 + 4.12311i 0 0 0 4.12311 4.12311i 0 7.00000i 0
207.4 0 4.12311 + 4.12311i 0 0 0 4.12311 4.12311i 0 7.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 143.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
20.d odd 2 1 inner
20.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.4.n.d 8
4.b odd 2 1 inner 400.4.n.d 8
5.b even 2 1 inner 400.4.n.d 8
5.c odd 4 2 inner 400.4.n.d 8
20.d odd 2 1 inner 400.4.n.d 8
20.e even 4 2 inner 400.4.n.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
400.4.n.d 8 1.a even 1 1 trivial
400.4.n.d 8 4.b odd 2 1 inner
400.4.n.d 8 5.b even 2 1 inner
400.4.n.d 8 5.c odd 4 2 inner
400.4.n.d 8 20.d odd 2 1 inner
400.4.n.d 8 20.e even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 1156 \) acting on \(S_{4}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1156)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 1156)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 192)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 42614784)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 42614784)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 9408)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 224820036)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 30276)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 37632)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T - 252)^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} + 6664109956)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 26203191876)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 102318096384)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 762048)^{4} \) Copy content Toggle raw display
$61$ \( (T - 56)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} + 40636915396)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 150528)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 102318096384)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 480000)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 216621361476)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 1764)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 1217120845824)^{2} \) Copy content Toggle raw display
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