Properties

Label 450.4.c.e
Level $450$
Weight $4$
Character orbit 450.c
Analytic conductor $26.551$
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] = [450,4,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 6)
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.

\(f(q)\) \(=\) \( q - \beta q^{2} - 4 q^{4} + 8 \beta q^{7} + 4 \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} - 4 q^{4} + 8 \beta q^{7} + 4 \beta q^{8} - 12 q^{11} + 19 \beta q^{13} + 32 q^{14} + 16 q^{16} - 63 \beta q^{17} - 20 q^{19} + 12 \beta q^{22} - 84 \beta q^{23} + 76 q^{26} - 32 \beta q^{28} + 30 q^{29} - 88 q^{31} - 16 \beta q^{32} - 252 q^{34} - 127 \beta q^{37} + 20 \beta q^{38} - 42 q^{41} - 26 \beta q^{43} + 48 q^{44} - 336 q^{46} - 48 \beta q^{47} + 87 q^{49} - 76 \beta q^{52} - 99 \beta q^{53} - 128 q^{56} - 30 \beta q^{58} - 660 q^{59} - 538 q^{61} + 88 \beta q^{62} - 64 q^{64} - 442 \beta q^{67} + 252 \beta q^{68} - 792 q^{71} + 109 \beta q^{73} - 508 q^{74} + 80 q^{76} - 96 \beta q^{77} + 520 q^{79} + 42 \beta q^{82} + 246 \beta q^{83} - 104 q^{86} - 48 \beta q^{88} + 810 q^{89} - 608 q^{91} + 336 \beta q^{92} - 192 q^{94} - 577 \beta q^{97} - 87 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} - 24 q^{11} + 64 q^{14} + 32 q^{16} - 40 q^{19} + 152 q^{26} + 60 q^{29} - 176 q^{31} - 504 q^{34} - 84 q^{41} + 96 q^{44} - 672 q^{46} + 174 q^{49} - 256 q^{56} - 1320 q^{59} - 1076 q^{61} - 128 q^{64} - 1584 q^{71} - 1016 q^{74} + 160 q^{76} + 1040 q^{79} - 208 q^{86} + 1620 q^{89} - 1216 q^{91} - 384 q^{94}+O(q^{100}) \) 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 16.0000i 8.00000i 0 0
199.2 2.00000i 0 −4.00000 0 0 16.0000i 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.e 2
3.b odd 2 1 150.4.c.d 2
5.b even 2 1 inner 450.4.c.e 2
5.c odd 4 1 18.4.a.a 1
5.c odd 4 1 450.4.a.h 1
12.b even 2 1 1200.4.f.j 2
15.d odd 2 1 150.4.c.d 2
15.e even 4 1 6.4.a.a 1
15.e even 4 1 150.4.a.i 1
20.e even 4 1 144.4.a.c 1
35.f even 4 1 882.4.a.n 1
35.k even 12 2 882.4.g.f 2
35.l odd 12 2 882.4.g.i 2
40.i odd 4 1 576.4.a.q 1
40.k even 4 1 576.4.a.r 1
45.k odd 12 2 162.4.c.c 2
45.l even 12 2 162.4.c.f 2
55.e even 4 1 2178.4.a.e 1
60.h even 2 1 1200.4.f.j 2
60.l odd 4 1 48.4.a.c 1
60.l odd 4 1 1200.4.a.b 1
105.k odd 4 1 294.4.a.e 1
105.w odd 12 2 294.4.e.g 2
105.x even 12 2 294.4.e.h 2
120.q odd 4 1 192.4.a.c 1
120.w even 4 1 192.4.a.i 1
165.l odd 4 1 726.4.a.f 1
195.j odd 4 1 1014.4.b.d 2
195.s even 4 1 1014.4.a.g 1
195.u odd 4 1 1014.4.b.d 2
240.z odd 4 1 768.4.d.c 2
240.bb even 4 1 768.4.d.n 2
240.bd odd 4 1 768.4.d.c 2
240.bf even 4 1 768.4.d.n 2
255.o even 4 1 1734.4.a.d 1
285.j odd 4 1 2166.4.a.i 1
420.w even 4 1 2352.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.4.a.a 1 15.e even 4 1
18.4.a.a 1 5.c odd 4 1
48.4.a.c 1 60.l odd 4 1
144.4.a.c 1 20.e even 4 1
150.4.a.i 1 15.e even 4 1
150.4.c.d 2 3.b odd 2 1
150.4.c.d 2 15.d odd 2 1
162.4.c.c 2 45.k odd 12 2
162.4.c.f 2 45.l even 12 2
192.4.a.c 1 120.q odd 4 1
192.4.a.i 1 120.w even 4 1
294.4.a.e 1 105.k odd 4 1
294.4.e.g 2 105.w odd 12 2
294.4.e.h 2 105.x even 12 2
450.4.a.h 1 5.c odd 4 1
450.4.c.e 2 1.a even 1 1 trivial
450.4.c.e 2 5.b even 2 1 inner
576.4.a.q 1 40.i odd 4 1
576.4.a.r 1 40.k even 4 1
726.4.a.f 1 165.l odd 4 1
768.4.d.c 2 240.z odd 4 1
768.4.d.c 2 240.bd odd 4 1
768.4.d.n 2 240.bb even 4 1
768.4.d.n 2 240.bf even 4 1
882.4.a.n 1 35.f even 4 1
882.4.g.f 2 35.k even 12 2
882.4.g.i 2 35.l odd 12 2
1014.4.a.g 1 195.s even 4 1
1014.4.b.d 2 195.j odd 4 1
1014.4.b.d 2 195.u odd 4 1
1200.4.a.b 1 60.l odd 4 1
1200.4.f.j 2 12.b even 2 1
1200.4.f.j 2 60.h even 2 1
1734.4.a.d 1 255.o even 4 1
2166.4.a.i 1 285.j odd 4 1
2178.4.a.e 1 55.e even 4 1
2352.4.a.e 1 420.w even 4 1

Hecke kernels

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

\( T_{7}^{2} + 256 \) Copy content Toggle raw display
\( T_{11} + 12 \) 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
$7$ \( T^{2} + 256 \) Copy content Toggle raw display
$11$ \( (T + 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1444 \) Copy content Toggle raw display
$17$ \( T^{2} + 15876 \) Copy content Toggle raw display
$19$ \( (T + 20)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 28224 \) Copy content Toggle raw display
$29$ \( (T - 30)^{2} \) Copy content Toggle raw display
$31$ \( (T + 88)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 64516 \) Copy content Toggle raw display
$41$ \( (T + 42)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 2704 \) Copy content Toggle raw display
$47$ \( T^{2} + 9216 \) Copy content Toggle raw display
$53$ \( T^{2} + 39204 \) Copy content Toggle raw display
$59$ \( (T + 660)^{2} \) Copy content Toggle raw display
$61$ \( (T + 538)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 781456 \) Copy content Toggle raw display
$71$ \( (T + 792)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 47524 \) Copy content Toggle raw display
$79$ \( (T - 520)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 242064 \) Copy content Toggle raw display
$89$ \( (T - 810)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 1331716 \) Copy content Toggle raw display
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