Properties

Label 1450.2.d.h
Level $1450$
Weight $2$
Character orbit 1450.d
Analytic conductor $11.578$
Analytic rank $0$
Dimension $4$
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] = [1450,2,Mod(1449,1450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1450.1449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1450 = 2 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1450.d (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: \(11.5783082931\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 290)
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.

\(f(q)\) \(=\) \( q + q^{2} + ( - \beta_{2} + 1) q^{3} + q^{4} + ( - \beta_{2} + 1) q^{6} - \beta_1 q^{7} + q^{8} + ( - 3 \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + ( - \beta_{2} + 1) q^{3} + q^{4} + ( - \beta_{2} + 1) q^{6} - \beta_1 q^{7} + q^{8} + ( - 3 \beta_{2} - 1) q^{9} + ( - 2 \beta_{3} - 4 \beta_1) q^{11} + ( - \beta_{2} + 1) q^{12} + ( - 3 \beta_{3} - 3 \beta_1) q^{13} - \beta_1 q^{14} + q^{16} - \beta_{2} q^{17} + ( - 3 \beta_{2} - 1) q^{18} + 2 \beta_1 q^{19} + (\beta_{3} - 2 \beta_1) q^{21} + ( - 2 \beta_{3} - 4 \beta_1) q^{22} + ( - \beta_{3} + \beta_1) q^{23} + ( - \beta_{2} + 1) q^{24} + ( - 3 \beta_{3} - 3 \beta_1) q^{26} + ( - 2 \beta_{2} - 1) q^{27} - \beta_1 q^{28} + ( - 3 \beta_{3} + 4 \beta_{2} + 2) q^{29} + ( - 4 \beta_{3} - 3 \beta_1) q^{31} + q^{32} + (2 \beta_{3} - 6 \beta_1) q^{33} - \beta_{2} q^{34} + ( - 3 \beta_{2} - 1) q^{36} + ( - 4 \beta_{2} - 4) q^{37} + 2 \beta_1 q^{38} - 3 \beta_1 q^{39} + ( - 4 \beta_{3} + 2 \beta_1) q^{41} + (\beta_{3} - 2 \beta_1) q^{42} + 7 \beta_{2} q^{43} + ( - 2 \beta_{3} - 4 \beta_1) q^{44} + ( - \beta_{3} + \beta_1) q^{46} - 6 \beta_{2} q^{47} + ( - \beta_{2} + 1) q^{48} + (\beta_{2} + 6) q^{49} + ( - 2 \beta_{2} + 1) q^{51} + ( - 3 \beta_{3} - 3 \beta_1) q^{52} + (4 \beta_{3} + \beta_1) q^{53} + ( - 2 \beta_{2} - 1) q^{54} - \beta_1 q^{56} + ( - 2 \beta_{3} + 4 \beta_1) q^{57} + ( - 3 \beta_{3} + 4 \beta_{2} + 2) q^{58} + (7 \beta_{2} + 1) q^{59} + (5 \beta_{3} + 5 \beta_1) q^{61} + ( - 4 \beta_{3} - 3 \beta_1) q^{62} + (3 \beta_{3} - 2 \beta_1) q^{63} + q^{64} + (2 \beta_{3} - 6 \beta_1) q^{66} + ( - 4 \beta_{3} + 6 \beta_1) q^{67} - \beta_{2} q^{68} + ( - 2 \beta_{3} + 3 \beta_1) q^{69} + ( - 4 \beta_{2} + 10) q^{71} + ( - 3 \beta_{2} - 1) q^{72} + (3 \beta_{2} + 8) q^{73} + ( - 4 \beta_{2} - 4) q^{74} + 2 \beta_1 q^{76} + (2 \beta_{2} - 4) q^{77} - 3 \beta_1 q^{78} + ( - 5 \beta_{3} - 3 \beta_1) q^{79} + (6 \beta_{2} + 4) q^{81} + ( - 4 \beta_{3} + 2 \beta_1) q^{82} + 8 \beta_1 q^{83} + (\beta_{3} - 2 \beta_1) q^{84} + 7 \beta_{2} q^{86} + ( - 3 \beta_{3} + 6 \beta_{2} + 3 \beta_1 - 2) q^{87} + ( - 2 \beta_{3} - 4 \beta_1) q^{88} + (8 \beta_{3} - 2 \beta_1) q^{89} - 3 q^{91} + ( - \beta_{3} + \beta_1) q^{92} + ( - \beta_{3} - 2 \beta_1) q^{93} - 6 \beta_{2} q^{94} + ( - \beta_{2} + 1) q^{96} + ( - 9 \beta_{2} - 9) q^{97} + (\beta_{2} + 6) q^{98} + (14 \beta_{3} - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 6 q^{3} + 4 q^{4} + 6 q^{6} + 4 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 6 q^{3} + 4 q^{4} + 6 q^{6} + 4 q^{8} + 2 q^{9} + 6 q^{12} + 4 q^{16} + 2 q^{17} + 2 q^{18} + 6 q^{24} + 4 q^{32} + 2 q^{34} + 2 q^{36} - 8 q^{37} - 14 q^{43} + 12 q^{47} + 6 q^{48} + 22 q^{49} + 8 q^{51} - 10 q^{59} + 4 q^{64} + 2 q^{68} + 48 q^{71} + 2 q^{72} + 26 q^{73} - 8 q^{74} - 20 q^{77} + 4 q^{81} - 14 q^{86} - 20 q^{87} - 12 q^{91} + 12 q^{94} + 6 q^{96} - 18 q^{97} + 22 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 2\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1450\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1277\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1449.1
0.618034i
0.618034i
1.61803i
1.61803i
1.00000 0.381966 1.00000 0 0.381966 0.618034i 1.00000 −2.85410 0
1449.2 1.00000 0.381966 1.00000 0 0.381966 0.618034i 1.00000 −2.85410 0
1449.3 1.00000 2.61803 1.00000 0 2.61803 1.61803i 1.00000 3.85410 0
1449.4 1.00000 2.61803 1.00000 0 2.61803 1.61803i 1.00000 3.85410 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
145.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1450.2.d.h 4
5.b even 2 1 1450.2.d.e 4
5.c odd 4 1 290.2.c.c 4
5.c odd 4 1 1450.2.c.d 4
15.e even 4 1 2610.2.f.d 4
20.e even 4 1 2320.2.g.g 4
29.b even 2 1 1450.2.d.e 4
145.d even 2 1 inner 1450.2.d.h 4
145.e even 4 1 8410.2.a.p 2
145.h odd 4 1 290.2.c.c 4
145.h odd 4 1 1450.2.c.d 4
145.j even 4 1 8410.2.a.q 2
435.p even 4 1 2610.2.f.d 4
580.o even 4 1 2320.2.g.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
290.2.c.c 4 5.c odd 4 1
290.2.c.c 4 145.h odd 4 1
1450.2.c.d 4 5.c odd 4 1
1450.2.c.d 4 145.h odd 4 1
1450.2.d.e 4 5.b even 2 1
1450.2.d.e 4 29.b even 2 1
1450.2.d.h 4 1.a even 1 1 trivial
1450.2.d.h 4 145.d even 2 1 inner
2320.2.g.g 4 20.e even 4 1
2320.2.g.g 4 580.o even 4 1
2610.2.f.d 4 15.e even 4 1
2610.2.f.d 4 435.p even 4 1
8410.2.a.p 2 145.e even 4 1
8410.2.a.q 2 145.j even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1450, [\chi])\):

\( T_{3}^{2} - 3T_{3} + 1 \) Copy content Toggle raw display
\( T_{17}^{2} - T_{17} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 3T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 27T^{2} + 81 \) Copy content Toggle raw display
$17$ \( (T^{2} - T - 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$23$ \( T^{4} + 7T^{2} + 1 \) Copy content Toggle raw display
$29$ \( T^{4} - 22T^{2} + 841 \) Copy content Toggle raw display
$31$ \( T^{4} + 35T^{2} + 25 \) Copy content Toggle raw display
$37$ \( (T^{2} + 4 T - 16)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 60T^{2} + 400 \) Copy content Toggle raw display
$43$ \( (T^{2} + 7 T - 49)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 6 T - 36)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 27T^{2} + 121 \) Copy content Toggle raw display
$59$ \( (T^{2} + 5 T - 55)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 75T^{2} + 625 \) Copy content Toggle raw display
$67$ \( T^{4} + 188T^{2} + 16 \) Copy content Toggle raw display
$71$ \( (T^{2} - 24 T + 124)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 13 T + 31)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 47T^{2} + 1 \) Copy content Toggle raw display
$83$ \( T^{4} + 192T^{2} + 4096 \) Copy content Toggle raw display
$89$ \( T^{4} + 172T^{2} + 5776 \) Copy content Toggle raw display
$97$ \( (T^{2} + 9 T - 81)^{2} \) Copy content Toggle raw display
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