Properties

Label 450.9.d.b
Level $450$
Weight $9$
Character orbit 450.d
Analytic conductor $183.320$
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] = [450,9,Mod(251,450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("450.251");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 450.d (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: \(183.320374528\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 18)
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 \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 8 \beta q^{2} - 128 q^{4} + 3532 q^{7} + 1024 \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - 8 \beta q^{2} - 128 q^{4} + 3532 q^{7} + 1024 \beta q^{8} - 14268 \beta q^{11} + 41824 q^{13} - 28256 \beta q^{14} + 16384 q^{16} - 67023 \beta q^{17} - 36304 q^{19} - 228288 q^{22} - 292476 \beta q^{23} - 334592 \beta q^{26} - 452096 q^{28} + 190347 \beta q^{29} - 471196 q^{31} - 131072 \beta q^{32} - 1072368 q^{34} + 3007402 q^{37} + 290432 \beta q^{38} - 1212927 \beta q^{41} - 3623720 q^{43} + 1826304 \beta q^{44} - 4679616 q^{46} + 4252980 \beta q^{47} + 6710223 q^{49} - 5353472 q^{52} - 7266699 \beta q^{53} + 3616768 \beta q^{56} + 3045552 q^{58} + 1900776 \beta q^{59} - 5440630 q^{61} + 3769568 \beta q^{62} - 2097152 q^{64} + 6121576 q^{67} + 8578944 \beta q^{68} - 14986476 \beta q^{71} + 49031152 q^{73} - 24059216 \beta q^{74} + 4646912 q^{76} - 50394576 \beta q^{77} + 8357756 q^{79} - 19406832 q^{82} + 36339492 \beta q^{83} + 28989760 \beta q^{86} + 29220864 q^{88} + 75898881 \beta q^{89} + 147722368 q^{91} + 37436928 \beta q^{92} + 68047680 q^{94} - 20431328 q^{97} - 53681784 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 256 q^{4} + 7064 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 256 q^{4} + 7064 q^{7} + 83648 q^{13} + 32768 q^{16} - 72608 q^{19} - 456576 q^{22} - 904192 q^{28} - 942392 q^{31} - 2144736 q^{34} + 6014804 q^{37} - 7247440 q^{43} - 9359232 q^{46} + 13420446 q^{49} - 10706944 q^{52} + 6091104 q^{58} - 10881260 q^{61} - 4194304 q^{64} + 12243152 q^{67} + 98062304 q^{73} + 9293824 q^{76} + 16715512 q^{79} - 38813664 q^{82} + 58441728 q^{88} + 295444736 q^{91} + 136095360 q^{94} - 40862656 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
1.41421i
1.41421i
11.3137i 0 −128.000 0 0 3532.00 1448.15i 0 0
251.2 11.3137i 0 −128.000 0 0 3532.00 1448.15i 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
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.9.d.b 2
3.b odd 2 1 inner 450.9.d.b 2
5.b even 2 1 18.9.b.a 2
5.c odd 4 2 450.9.b.a 4
15.d odd 2 1 18.9.b.a 2
15.e even 4 2 450.9.b.a 4
20.d odd 2 1 144.9.e.d 2
45.h odd 6 2 162.9.d.d 4
45.j even 6 2 162.9.d.d 4
60.h even 2 1 144.9.e.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.9.b.a 2 5.b even 2 1
18.9.b.a 2 15.d odd 2 1
144.9.e.d 2 20.d odd 2 1
144.9.e.d 2 60.h even 2 1
162.9.d.d 4 45.h odd 6 2
162.9.d.d 4 45.j even 6 2
450.9.b.a 4 5.c odd 4 2
450.9.b.a 4 15.e even 4 2
450.9.d.b 2 1.a even 1 1 trivial
450.9.d.b 2 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 3532 \) acting on \(S_{9}^{\mathrm{new}}(450, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 128 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 3532)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 407151648 \) Copy content Toggle raw display
$13$ \( (T - 41824)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 8984165058 \) Copy content Toggle raw display
$19$ \( (T + 36304)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 171084421152 \) Copy content Toggle raw display
$29$ \( T^{2} + 72463960818 \) Copy content Toggle raw display
$31$ \( (T + 471196)^{2} \) Copy content Toggle raw display
$37$ \( (T - 3007402)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 2942383814658 \) Copy content Toggle raw display
$43$ \( (T + 3623720)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 36175677760800 \) Copy content Toggle raw display
$53$ \( T^{2} + 105609828713202 \) Copy content Toggle raw display
$59$ \( T^{2} + 7225898804352 \) Copy content Toggle raw display
$61$ \( (T + 5440630)^{2} \) Copy content Toggle raw display
$67$ \( (T - 6121576)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 449188925797152 \) Copy content Toggle raw display
$73$ \( (T - 49031152)^{2} \) Copy content Toggle raw display
$79$ \( (T - 8357756)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 26\!\cdots\!28 \) Copy content Toggle raw display
$89$ \( T^{2} + 11\!\cdots\!22 \) Copy content Toggle raw display
$97$ \( (T + 20431328)^{2} \) Copy content Toggle raw display
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