Properties

Label 450.4.c.a
Level $450$
Weight $4$
Character orbit 450.c
Analytic conductor $26.551$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [450,4,Mod(199,450)]
 
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([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("450.199");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 450.c (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: \(26.5508595026\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 150)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 i q^{2} - 4 q^{4} + i q^{7} + 8 i q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 i q^{2} - 4 q^{4} + i q^{7} + 8 i q^{8} - 42 q^{11} - 67 i q^{13} + 2 q^{14} + 16 q^{16} + 54 i q^{17} + 115 q^{19} + 84 i q^{22} + 162 i q^{23} - 134 q^{26} - 4 i q^{28} - 210 q^{29} - 193 q^{31} - 32 i q^{32} + 108 q^{34} + 286 i q^{37} - 230 i q^{38} - 12 q^{41} + 263 i q^{43} + 168 q^{44} + 324 q^{46} + 414 i q^{47} + 342 q^{49} + 268 i q^{52} + 192 i q^{53} - 8 q^{56} + 420 i q^{58} + 690 q^{59} - 733 q^{61} + 386 i q^{62} - 64 q^{64} - 299 i q^{67} - 216 i q^{68} + 228 q^{71} + 938 i q^{73} + 572 q^{74} - 460 q^{76} - 42 i q^{77} + 160 q^{79} + 24 i q^{82} + 462 i q^{83} + 526 q^{86} - 336 i q^{88} - 240 q^{89} + 67 q^{91} - 648 i q^{92} + 828 q^{94} + 511 i q^{97} - 684 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} - 84 q^{11} + 4 q^{14} + 32 q^{16} + 230 q^{19} - 268 q^{26} - 420 q^{29} - 386 q^{31} + 216 q^{34} - 24 q^{41} + 336 q^{44} + 648 q^{46} + 684 q^{49} - 16 q^{56} + 1380 q^{59} - 1466 q^{61} - 128 q^{64} + 456 q^{71} + 1144 q^{74} - 920 q^{76} + 320 q^{79} + 1052 q^{86} - 480 q^{89} + 134 q^{91} + 1656 q^{94}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
1.00000i
1.00000i
2.00000i 0 −4.00000 0 0 1.00000i 8.00000i 0 0
199.2 2.00000i 0 −4.00000 0 0 1.00000i 8.00000i 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.4.c.a 2
3.b odd 2 1 150.4.c.e 2
5.b even 2 1 inner 450.4.c.a 2
5.c odd 4 1 450.4.a.f 1
5.c odd 4 1 450.4.a.o 1
12.b even 2 1 1200.4.f.c 2
15.d odd 2 1 150.4.c.e 2
15.e even 4 1 150.4.a.a 1
15.e even 4 1 150.4.a.h yes 1
60.h even 2 1 1200.4.f.c 2
60.l odd 4 1 1200.4.a.i 1
60.l odd 4 1 1200.4.a.bb 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.4.a.a 1 15.e even 4 1
150.4.a.h yes 1 15.e even 4 1
150.4.c.e 2 3.b odd 2 1
150.4.c.e 2 15.d odd 2 1
450.4.a.f 1 5.c odd 4 1
450.4.a.o 1 5.c odd 4 1
450.4.c.a 2 1.a even 1 1 trivial
450.4.c.a 2 5.b even 2 1 inner
1200.4.a.i 1 60.l odd 4 1
1200.4.a.bb 1 60.l odd 4 1
1200.4.f.c 2 12.b even 2 1
1200.4.f.c 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(450, [\chi])\):

\( T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{11} + 42 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T + 42)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4489 \) Copy content Toggle raw display
$17$ \( T^{2} + 2916 \) Copy content Toggle raw display
$19$ \( (T - 115)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 26244 \) Copy content Toggle raw display
$29$ \( (T + 210)^{2} \) Copy content Toggle raw display
$31$ \( (T + 193)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 81796 \) Copy content Toggle raw display
$41$ \( (T + 12)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 69169 \) Copy content Toggle raw display
$47$ \( T^{2} + 171396 \) Copy content Toggle raw display
$53$ \( T^{2} + 36864 \) Copy content Toggle raw display
$59$ \( (T - 690)^{2} \) Copy content Toggle raw display
$61$ \( (T + 733)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 89401 \) Copy content Toggle raw display
$71$ \( (T - 228)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 879844 \) Copy content Toggle raw display
$79$ \( (T - 160)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 213444 \) Copy content Toggle raw display
$89$ \( (T + 240)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 261121 \) Copy content Toggle raw display
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