Properties

Label 400.9.h.b
Level $400$
Weight $9$
Character orbit 400.h
Analytic conductor $162.951$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [400,9,Mod(399,400)]
 
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([1, 0, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.399");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 400.h (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: \(162.951444024\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{35})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 17x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 16)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} - 18 \beta_{2} q^{7} + 13599 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} - 18 \beta_{2} q^{7} + 13599 q^{9} - 27 \beta_{3} q^{11} + 2771 \beta_1 q^{13} + 5037 \beta_1 q^{17} + 153 \beta_{3} q^{19} + 362880 q^{21} - 1242 \beta_{2} q^{23} - 7038 \beta_{2} q^{27} - 54978 q^{29} - 1656 \beta_{3} q^{31} + 272160 \beta_1 q^{33} + 79373 \beta_1 q^{37} - 5542 \beta_{3} q^{39} - 75582 q^{41} - 3519 \beta_{2} q^{43} + 20196 \beta_{2} q^{47} + 767039 q^{49} - 10074 \beta_{3} q^{51} - 1116621 \beta_1 q^{53} - 1542240 \beta_1 q^{57} - 30753 \beta_{3} q^{59} - 23826622 q^{61} - 244782 \beta_{2} q^{63} + 52785 \beta_{2} q^{67} + 25038720 q^{69} + 14202 \beta_{3} q^{71} - 651661 \beta_1 q^{73} + 4898880 \beta_1 q^{77} - 68724 \beta_{3} q^{79} + 52663041 q^{81} - 517293 \beta_{2} q^{83} + 54978 \beta_{2} q^{87} - 86795778 q^{89} - 99756 \beta_{3} q^{91} + 16692480 \beta_1 q^{93} - 4667027 \beta_1 q^{97} - 367173 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 54396 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 54396 q^{9} + 1451520 q^{21} - 219912 q^{29} - 302328 q^{41} + 3068156 q^{49} - 95306488 q^{61} + 100154880 q^{69} + 210652164 q^{81} - 347183112 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 17x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 10\nu^{3} - 80\nu ) / 9 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -8\nu^{3} + 208\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 240\nu^{2} - 2040 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 5\beta_{2} + 12\beta_1 ) / 240 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 2040 ) / 240 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{2} + 39\beta_1 ) / 30 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
399.1
2.95804 + 0.500000i
2.95804 0.500000i
−2.95804 0.500000i
−2.95804 + 0.500000i
0 −141.986 0 0 0 −2555.75 0 13599.0 0
399.2 0 −141.986 0 0 0 −2555.75 0 13599.0 0
399.3 0 141.986 0 0 0 2555.75 0 13599.0 0
399.4 0 141.986 0 0 0 2555.75 0 13599.0 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
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.9.h.b 4
4.b odd 2 1 inner 400.9.h.b 4
5.b even 2 1 inner 400.9.h.b 4
5.c odd 4 1 16.9.c.a 2
5.c odd 4 1 400.9.b.c 2
15.e even 4 1 144.9.g.g 2
20.d odd 2 1 inner 400.9.h.b 4
20.e even 4 1 16.9.c.a 2
20.e even 4 1 400.9.b.c 2
40.i odd 4 1 64.9.c.d 2
40.k even 4 1 64.9.c.d 2
60.l odd 4 1 144.9.g.g 2
80.i odd 4 1 256.9.d.f 4
80.j even 4 1 256.9.d.f 4
80.s even 4 1 256.9.d.f 4
80.t odd 4 1 256.9.d.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.9.c.a 2 5.c odd 4 1
16.9.c.a 2 20.e even 4 1
64.9.c.d 2 40.i odd 4 1
64.9.c.d 2 40.k even 4 1
144.9.g.g 2 15.e even 4 1
144.9.g.g 2 60.l odd 4 1
256.9.d.f 4 80.i odd 4 1
256.9.d.f 4 80.j even 4 1
256.9.d.f 4 80.s even 4 1
256.9.d.f 4 80.t odd 4 1
400.9.b.c 2 5.c odd 4 1
400.9.b.c 2 20.e even 4 1
400.9.h.b 4 1.a even 1 1 trivial
400.9.h.b 4 4.b odd 2 1 inner
400.9.h.b 4 5.b even 2 1 inner
400.9.h.b 4 20.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 20160 \) acting on \(S_{9}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 20160)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 6531840)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 367416000)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 767844100)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 2537136900)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 11798136000)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 31098090240)^{2} \) Copy content Toggle raw display
$29$ \( (T + 54978)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 1382137344000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 630007312900)^{2} \) Copy content Toggle raw display
$41$ \( (T + 75582)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 249648557760)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 8222828866560)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 124684245764100)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 476656492536000)^{2} \) Copy content Toggle raw display
$61$ \( (T + 23826622)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 56170925496000)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 101655189216000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 42466205892100)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 53\!\cdots\!40)^{2} \) Copy content Toggle raw display
$89$ \( (T + 86795778)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
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