Properties

Label 450.4.c.k
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 30)
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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} - 4 q^{4} + 16 \beta q^{7} - 4 \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} - 4 q^{4} + 16 \beta q^{7} - 4 \beta q^{8} + 60 q^{11} + 17 \beta q^{13} - 64 q^{14} + 16 q^{16} - 21 \beta q^{17} + 76 q^{19} + 60 \beta q^{22} - 68 q^{26} - 64 \beta q^{28} + 6 q^{29} - 232 q^{31} + 16 \beta q^{32} + 84 q^{34} + 67 \beta q^{37} + 76 \beta q^{38} - 234 q^{41} + 206 \beta q^{43} - 240 q^{44} + 180 \beta q^{47} - 681 q^{49} - 68 \beta q^{52} + 111 \beta q^{53} + 256 q^{56} + 6 \beta q^{58} + 660 q^{59} - 490 q^{61} - 232 \beta q^{62} - 64 q^{64} + 406 \beta q^{67} + 84 \beta q^{68} - 120 q^{71} - 373 \beta q^{73} - 268 q^{74} - 304 q^{76} + 960 \beta q^{77} - 152 q^{79} - 234 \beta q^{82} - 402 \beta q^{83} - 824 q^{86} - 240 \beta q^{88} - 678 q^{89} - 1088 q^{91} - 720 q^{94} + 97 \beta q^{97} - 681 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} + 120 q^{11} - 128 q^{14} + 32 q^{16} + 152 q^{19} - 136 q^{26} + 12 q^{29} - 464 q^{31} + 168 q^{34} - 468 q^{41} - 480 q^{44} - 1362 q^{49} + 512 q^{56} + 1320 q^{59} - 980 q^{61} - 128 q^{64} - 240 q^{71} - 536 q^{74} - 608 q^{76} - 304 q^{79} - 1648 q^{86} - 1356 q^{89} - 2176 q^{91} - 1440 q^{94}+O(q^{100}) \) 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 32.0000i 8.00000i 0 0
199.2 2.00000i 0 −4.00000 0 0 32.0000i 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.k 2
3.b odd 2 1 150.4.c.a 2
5.b even 2 1 inner 450.4.c.k 2
5.c odd 4 1 90.4.a.d 1
5.c odd 4 1 450.4.a.b 1
12.b even 2 1 1200.4.f.u 2
15.d odd 2 1 150.4.c.a 2
15.e even 4 1 30.4.a.a 1
15.e even 4 1 150.4.a.e 1
20.e even 4 1 720.4.a.b 1
45.k odd 12 2 810.4.e.e 2
45.l even 12 2 810.4.e.m 2
60.h even 2 1 1200.4.f.u 2
60.l odd 4 1 240.4.a.c 1
60.l odd 4 1 1200.4.a.bk 1
105.k odd 4 1 1470.4.a.a 1
120.q odd 4 1 960.4.a.s 1
120.w even 4 1 960.4.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.4.a.a 1 15.e even 4 1
90.4.a.d 1 5.c odd 4 1
150.4.a.e 1 15.e even 4 1
150.4.c.a 2 3.b odd 2 1
150.4.c.a 2 15.d odd 2 1
240.4.a.c 1 60.l odd 4 1
450.4.a.b 1 5.c odd 4 1
450.4.c.k 2 1.a even 1 1 trivial
450.4.c.k 2 5.b even 2 1 inner
720.4.a.b 1 20.e even 4 1
810.4.e.e 2 45.k odd 12 2
810.4.e.m 2 45.l even 12 2
960.4.a.j 1 120.w even 4 1
960.4.a.s 1 120.q odd 4 1
1200.4.a.bk 1 60.l odd 4 1
1200.4.f.u 2 12.b even 2 1
1200.4.f.u 2 60.h even 2 1
1470.4.a.a 1 105.k odd 4 1

Hecke kernels

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

\( T_{7}^{2} + 1024 \) Copy content Toggle raw display
\( T_{11} - 60 \) 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
$7$ \( T^{2} + 1024 \) Copy content Toggle raw display
$11$ \( (T - 60)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1156 \) Copy content Toggle raw display
$17$ \( T^{2} + 1764 \) Copy content Toggle raw display
$19$ \( (T - 76)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( (T + 232)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 17956 \) Copy content Toggle raw display
$41$ \( (T + 234)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 169744 \) Copy content Toggle raw display
$47$ \( T^{2} + 129600 \) Copy content Toggle raw display
$53$ \( T^{2} + 49284 \) Copy content Toggle raw display
$59$ \( (T - 660)^{2} \) Copy content Toggle raw display
$61$ \( (T + 490)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 659344 \) Copy content Toggle raw display
$71$ \( (T + 120)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 556516 \) Copy content Toggle raw display
$79$ \( (T + 152)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 646416 \) Copy content Toggle raw display
$89$ \( (T + 678)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 37636 \) Copy content Toggle raw display
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