Properties

Label 8450.2.a.cg
Level $8450$
Weight $2$
Character orbit 8450.a
Self dual yes
Analytic conductor $67.474$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8450,2,Mod(1,8450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8450.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8450.a (trivial)

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: yes
Analytic conductor: \(67.4735897080\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1690)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta_1 q^{3} + q^{4} + \beta_1 q^{6} + 2 \beta_1 q^{7} + q^{8} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta_1 q^{3} + q^{4} + \beta_1 q^{6} + 2 \beta_1 q^{7} + q^{8} + (\beta_{2} - 1) q^{9} + (\beta_{2} - \beta_1 + 5) q^{11} + \beta_1 q^{12} + 2 \beta_1 q^{14} + q^{16} + ( - \beta_{2} + 3 \beta_1 - 1) q^{17} + (\beta_{2} - 1) q^{18} + (\beta_{2} - 2 \beta_1 + 2) q^{19} + (2 \beta_{2} + 4) q^{21} + (\beta_{2} - \beta_1 + 5) q^{22} + (2 \beta_{2} + 4) q^{23} + \beta_1 q^{24} + (\beta_{2} - 4 \beta_1 + 1) q^{27} + 2 \beta_1 q^{28} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{29} + ( - 4 \beta_{2} + 6 \beta_1 - 2) q^{31} + q^{32} + (5 \beta_1 - 1) q^{33} + ( - \beta_{2} + 3 \beta_1 - 1) q^{34} + (\beta_{2} - 1) q^{36} + 6 \beta_{2} q^{37} + (\beta_{2} - 2 \beta_1 + 2) q^{38} + (\beta_{2} - 4 \beta_1 + 8) q^{41} + (2 \beta_{2} + 4) q^{42} + ( - 9 \beta_{2} + 4 \beta_1 - 6) q^{43} + (\beta_{2} - \beta_1 + 5) q^{44} + (2 \beta_{2} + 4) q^{46} + ( - 2 \beta_{2} - 4 \beta_1 - 2) q^{47} + \beta_1 q^{48} + (4 \beta_{2} + 1) q^{49} + (2 \beta_{2} - \beta_1 + 5) q^{51} + (8 \beta_{2} - 2 \beta_1 + 6) q^{53} + (\beta_{2} - 4 \beta_1 + 1) q^{54} + 2 \beta_1 q^{56} + ( - \beta_{2} + 2 \beta_1 - 3) q^{57} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{58} + ( - 4 \beta_{2} - 5 \beta_1 + 2) q^{59} + ( - 8 \beta_{2} + 2 \beta_1 - 6) q^{61} + ( - 4 \beta_{2} + 6 \beta_1 - 2) q^{62} + (2 \beta_{2} - 2 \beta_1 + 2) q^{63} + q^{64} + (5 \beta_1 - 1) q^{66} + ( - 4 \beta_{2} - 5 \beta_1) q^{67} + ( - \beta_{2} + 3 \beta_1 - 1) q^{68} + (2 \beta_{2} + 4 \beta_1 + 2) q^{69} + (6 \beta_1 + 2) q^{71} + (\beta_{2} - 1) q^{72} + ( - 8 \beta_{2} + 5 \beta_1 - 6) q^{73} + 6 \beta_{2} q^{74} + (\beta_{2} - 2 \beta_1 + 2) q^{76} + (10 \beta_1 - 2) q^{77} + ( - 6 \beta_{2} + 2 \beta_1 - 2) q^{79} + ( - 6 \beta_{2} + \beta_1 - 4) q^{81} + (\beta_{2} - 4 \beta_1 + 8) q^{82} + (3 \beta_{2} - 8 \beta_1 + 6) q^{83} + (2 \beta_{2} + 4) q^{84} + ( - 9 \beta_{2} + 4 \beta_1 - 6) q^{86} + ( - 4 \beta_1 + 2) q^{87} + (\beta_{2} - \beta_1 + 5) q^{88} + ( - 4 \beta_{2} + 5 \beta_1 + 4) q^{89} + (2 \beta_{2} + 4) q^{92} + (2 \beta_{2} - 2 \beta_1 + 8) q^{93} + ( - 2 \beta_{2} - 4 \beta_1 - 2) q^{94} + \beta_1 q^{96} + (\beta_{2} + \beta_1 - 7) q^{97} + (4 \beta_{2} + 1) q^{98} + (2 \beta_{2} + 2 \beta_1 - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + q^{3} + 3 q^{4} + q^{6} + 2 q^{7} + 3 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + q^{3} + 3 q^{4} + q^{6} + 2 q^{7} + 3 q^{8} - 4 q^{9} + 13 q^{11} + q^{12} + 2 q^{14} + 3 q^{16} + q^{17} - 4 q^{18} + 3 q^{19} + 10 q^{21} + 13 q^{22} + 10 q^{23} + q^{24} - 2 q^{27} + 2 q^{28} - 8 q^{29} + 4 q^{31} + 3 q^{32} + 2 q^{33} + q^{34} - 4 q^{36} - 6 q^{37} + 3 q^{38} + 19 q^{41} + 10 q^{42} - 5 q^{43} + 13 q^{44} + 10 q^{46} - 8 q^{47} + q^{48} - q^{49} + 12 q^{51} + 8 q^{53} - 2 q^{54} + 2 q^{56} - 6 q^{57} - 8 q^{58} + 5 q^{59} - 8 q^{61} + 4 q^{62} + 2 q^{63} + 3 q^{64} + 2 q^{66} - q^{67} + q^{68} + 8 q^{69} + 12 q^{71} - 4 q^{72} - 5 q^{73} - 6 q^{74} + 3 q^{76} + 4 q^{77} + 2 q^{79} - 5 q^{81} + 19 q^{82} + 7 q^{83} + 10 q^{84} - 5 q^{86} + 2 q^{87} + 13 q^{88} + 21 q^{89} + 10 q^{92} + 20 q^{93} - 8 q^{94} + q^{96} - 21 q^{97} - q^{98} - 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.24698
0.445042
1.80194
1.00000 −1.24698 1.00000 0 −1.24698 −2.49396 1.00000 −1.44504 0
1.2 1.00000 0.445042 1.00000 0 0.445042 0.890084 1.00000 −2.80194 0
1.3 1.00000 1.80194 1.00000 0 1.80194 3.60388 1.00000 0.246980 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(13\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8450.2.a.cg 3
5.b even 2 1 1690.2.a.p 3
13.b even 2 1 8450.2.a.bv 3
65.d even 2 1 1690.2.a.r yes 3
65.g odd 4 2 1690.2.d.i 6
65.l even 6 2 1690.2.e.p 6
65.n even 6 2 1690.2.e.r 6
65.s odd 12 4 1690.2.l.m 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1690.2.a.p 3 5.b even 2 1
1690.2.a.r yes 3 65.d even 2 1
1690.2.d.i 6 65.g odd 4 2
1690.2.e.p 6 65.l even 6 2
1690.2.e.r 6 65.n even 6 2
1690.2.l.m 12 65.s odd 12 4
8450.2.a.bv 3 13.b even 2 1
8450.2.a.cg 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8450))\):

\( T_{3}^{3} - T_{3}^{2} - 2T_{3} + 1 \) Copy content Toggle raw display
\( T_{7}^{3} - 2T_{7}^{2} - 8T_{7} + 8 \) Copy content Toggle raw display
\( T_{11}^{3} - 13T_{11}^{2} + 54T_{11} - 71 \) Copy content Toggle raw display
\( T_{17}^{3} - T_{17}^{2} - 16T_{17} + 29 \) Copy content Toggle raw display
\( T_{31}^{3} - 4T_{31}^{2} - 60T_{31} + 232 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - T^{2} - 2T + 1 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 2 T^{2} - 8 T + 8 \) Copy content Toggle raw display
$11$ \( T^{3} - 13 T^{2} + 54 T - 71 \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - T^{2} - 16 T + 29 \) Copy content Toggle raw display
$19$ \( T^{3} - 3 T^{2} - 4 T - 1 \) Copy content Toggle raw display
$23$ \( T^{3} - 10 T^{2} + 24 T - 8 \) Copy content Toggle raw display
$29$ \( T^{3} + 8 T^{2} + 12 T - 8 \) Copy content Toggle raw display
$31$ \( T^{3} - 4 T^{2} - 60 T + 232 \) Copy content Toggle raw display
$37$ \( T^{3} + 6 T^{2} - 72 T - 216 \) Copy content Toggle raw display
$41$ \( T^{3} - 19 T^{2} + 90 T - 113 \) Copy content Toggle raw display
$43$ \( T^{3} + 5 T^{2} - 134 T - 839 \) Copy content Toggle raw display
$47$ \( T^{3} + 8 T^{2} - 44 T - 8 \) Copy content Toggle raw display
$53$ \( T^{3} - 8 T^{2} - 100 T + 568 \) Copy content Toggle raw display
$59$ \( T^{3} - 5 T^{2} - 134 T + 839 \) Copy content Toggle raw display
$61$ \( T^{3} + 8 T^{2} - 100 T - 568 \) Copy content Toggle raw display
$67$ \( T^{3} + T^{2} - 142 T + 559 \) Copy content Toggle raw display
$71$ \( T^{3} - 12 T^{2} - 36 T + 328 \) Copy content Toggle raw display
$73$ \( T^{3} + 5 T^{2} - 106 T - 643 \) Copy content Toggle raw display
$79$ \( T^{3} - 2 T^{2} - 64 T - 104 \) Copy content Toggle raw display
$83$ \( T^{3} - 7 T^{2} - 98 T - 203 \) Copy content Toggle raw display
$89$ \( T^{3} - 21 T^{2} + 98 T + 49 \) Copy content Toggle raw display
$97$ \( T^{3} + 21 T^{2} + 140 T + 287 \) Copy content Toggle raw display
show more
show less