Properties

Label 400.9.b.c
Level $400$
Weight $9$
Character orbit 400.b
Analytic conductor $162.951$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [400,9,Mod(351,400)]
 
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, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.351");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 400.b (of order \(2\), degree \(1\), 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: \(2\)
Coefficient field: \(\Q(\sqrt{-35}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
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 \(\beta = 24\sqrt{-35}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{3} + 18 \beta q^{7} - 13599 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{3} + 18 \beta q^{7} - 13599 q^{9} + 135 \beta q^{11} + 27710 q^{13} - 50370 q^{17} + 765 \beta q^{19} + 362880 q^{21} - 1242 \beta q^{23} + 7038 \beta q^{27} + 54978 q^{29} + 8280 \beta q^{31} + 2721600 q^{33} - 793730 q^{37} - 27710 \beta q^{39} - 75582 q^{41} - 3519 \beta q^{43} - 20196 \beta q^{47} - 767039 q^{49} + 50370 \beta q^{51} - 11166210 q^{53} + 15422400 q^{57} - 153765 \beta q^{59} - 23826622 q^{61} - 244782 \beta q^{63} - 52785 \beta q^{67} - 25038720 q^{69} - 71010 \beta q^{71} - 6516610 q^{73} - 48988800 q^{77} - 343620 \beta q^{79} + 52663041 q^{81} - 517293 \beta q^{83} - 54978 \beta q^{87} + 86795778 q^{89} + 498780 \beta q^{91} + 166924800 q^{93} + 46670270 q^{97} - 1835865 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 27198 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 27198 q^{9} + 55420 q^{13} - 100740 q^{17} + 725760 q^{21} + 109956 q^{29} + 5443200 q^{33} - 1587460 q^{37} - 151164 q^{41} - 1534078 q^{49} - 22332420 q^{53} + 30844800 q^{57} - 47653244 q^{61} - 50077440 q^{69} - 13033220 q^{73} - 97977600 q^{77} + 105326082 q^{81} + 173591556 q^{89} + 333849600 q^{93} + 93340540 q^{97}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
351.1
0.500000 + 2.95804i
0.500000 2.95804i
0 141.986i 0 0 0 2555.75i 0 −13599.0 0
351.2 0 141.986i 0 0 0 2555.75i 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

Twists

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{9}^{\mathrm{new}}(400, [\chi])\):

\( T_{3}^{2} + 20160 \) Copy content Toggle raw display
\( T_{13} - 27710 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 20160 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 6531840 \) Copy content Toggle raw display
$11$ \( T^{2} + 367416000 \) Copy content Toggle raw display
$13$ \( (T - 27710)^{2} \) Copy content Toggle raw display
$17$ \( (T + 50370)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 11798136000 \) Copy content Toggle raw display
$23$ \( T^{2} + 31098090240 \) Copy content Toggle raw display
$29$ \( (T - 54978)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 1382137344000 \) Copy content Toggle raw display
$37$ \( (T + 793730)^{2} \) Copy content Toggle raw display
$41$ \( (T + 75582)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 249648557760 \) Copy content Toggle raw display
$47$ \( T^{2} + 8222828866560 \) Copy content Toggle raw display
$53$ \( (T + 11166210)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 476656492536000 \) Copy content Toggle raw display
$61$ \( (T + 23826622)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 56170925496000 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 101655189216000 \) Copy content Toggle raw display
$73$ \( (T + 6516610)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 23\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + 53\!\cdots\!40 \) Copy content Toggle raw display
$89$ \( (T - 86795778)^{2} \) Copy content Toggle raw display
$97$ \( (T - 46670270)^{2} \) Copy content Toggle raw display
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