Properties

Label 450.9.b.a
Level $450$
Weight $9$
Character orbit 450.b
Analytic conductor $183.320$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [450,9,Mod(449,450)]
 
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, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("450.449");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 450.b (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: \(183.320374528\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{2} \)
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 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.

\(f(q)\) \(=\) \( q + 8 \beta_{3} q^{2} + 128 q^{4} - 1766 \beta_1 q^{7} + 1024 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 8 \beta_{3} q^{2} + 128 q^{4} - 1766 \beta_1 q^{7} + 1024 \beta_{3} q^{8} + 14268 \beta_{2} q^{11} + 20912 \beta_1 q^{13} - 28256 \beta_{2} q^{14} + 16384 q^{16} + 67023 \beta_{3} q^{17} + 36304 q^{19} + 114144 \beta_1 q^{22} - 292476 \beta_{3} q^{23} + 334592 \beta_{2} q^{26} - 226048 \beta_1 q^{28} + 190347 \beta_{2} q^{29} - 471196 q^{31} + 131072 \beta_{3} q^{32} + 1072368 q^{34} - 1503701 \beta_1 q^{37} + 290432 \beta_{3} q^{38} + 1212927 \beta_{2} q^{41} - 1811860 \beta_1 q^{43} + 1826304 \beta_{2} q^{44} - 4679616 q^{46} - 4252980 \beta_{3} q^{47} - 6710223 q^{49} + 2676736 \beta_1 q^{52} - 7266699 \beta_{3} q^{53} - 3616768 \beta_{2} q^{56} + 1522776 \beta_1 q^{58} + 1900776 \beta_{2} q^{59} - 5440630 q^{61} - 3769568 \beta_{3} q^{62} + 2097152 q^{64} - 3060788 \beta_1 q^{67} + 8578944 \beta_{3} q^{68} + 14986476 \beta_{2} q^{71} + 24515576 \beta_1 q^{73} - 24059216 \beta_{2} q^{74} + 4646912 q^{76} + 50394576 \beta_{3} q^{77} - 8357756 q^{79} + 9703416 \beta_1 q^{82} + 36339492 \beta_{3} q^{83} - 28989760 \beta_{2} q^{86} + 14610432 \beta_1 q^{88} + 75898881 \beta_{2} q^{89} + 147722368 q^{91} - 37436928 \beta_{3} q^{92} - 68047680 q^{94} + 10215664 \beta_1 q^{97} - 53681784 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 512 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 512 q^{4} + 65536 q^{16} + 145216 q^{19} - 1884784 q^{31} + 4289472 q^{34} - 18718464 q^{46} - 26840892 q^{49} - 21762520 q^{61} + 8388608 q^{64} + 18587648 q^{76} - 33431024 q^{79} + 590889472 q^{91} - 272190720 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
0.707107 0.707107i
−11.3137 0 128.000 0 0 3532.00i −1448.15 0 0
449.2 −11.3137 0 128.000 0 0 3532.00i −1448.15 0 0
449.3 11.3137 0 128.000 0 0 3532.00i 1448.15 0 0
449.4 11.3137 0 128.000 0 0 3532.00i 1448.15 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
5.b even 2 1 inner
15.d 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.b.a 4
3.b odd 2 1 inner 450.9.b.a 4
5.b even 2 1 inner 450.9.b.a 4
5.c odd 4 1 18.9.b.a 2
5.c odd 4 1 450.9.d.b 2
15.d odd 2 1 inner 450.9.b.a 4
15.e even 4 1 18.9.b.a 2
15.e even 4 1 450.9.d.b 2
20.e even 4 1 144.9.e.d 2
45.k odd 12 2 162.9.d.d 4
45.l even 12 2 162.9.d.d 4
60.l odd 4 1 144.9.e.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.9.b.a 2 5.c odd 4 1
18.9.b.a 2 15.e even 4 1
144.9.e.d 2 20.e even 4 1
144.9.e.d 2 60.l odd 4 1
162.9.d.d 4 45.k odd 12 2
162.9.d.d 4 45.l even 12 2
450.9.b.a 4 1.a even 1 1 trivial
450.9.b.a 4 3.b odd 2 1 inner
450.9.b.a 4 5.b even 2 1 inner
450.9.b.a 4 15.d odd 2 1 inner
450.9.d.b 2 5.c odd 4 1
450.9.d.b 2 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 12475024 \) acting on \(S_{9}^{\mathrm{new}}(450, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 128)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 12475024)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 407151648)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1749246976)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 8984165058)^{2} \) Copy content Toggle raw display
$19$ \( (T - 36304)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 171084421152)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 72463960818)^{2} \) Copy content Toggle raw display
$31$ \( (T + 471196)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 9044466789604)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 2942383814658)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 13131346638400)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 36175677760800)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 105609828713202)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 7225898804352)^{2} \) Copy content Toggle raw display
$61$ \( (T + 5440630)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 37473692723776)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 449188925797152)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 24\!\cdots\!04)^{2} \) Copy content Toggle raw display
$79$ \( (T + 8357756)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 26\!\cdots\!28)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 11\!\cdots\!22)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 417439163843584)^{2} \) Copy content Toggle raw display
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