Properties

Label 8450.2.a.cs
Level $8450$
Weight $2$
Character orbit 8450.a
Self dual yes
Analytic conductor $67.474$
Analytic rank $0$
Dimension $8$
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] = [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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 20x^{6} + 132x^{4} - 332x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 130)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta_1 q^{3} + q^{4} + \beta_1 q^{6} + (\beta_{7} + 1) q^{7} + q^{8} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta_1 q^{3} + q^{4} + \beta_1 q^{6} + (\beta_{7} + 1) q^{7} + q^{8} + (\beta_{2} + 2) q^{9} + ( - \beta_{6} + \beta_1) q^{11} + \beta_1 q^{12} + (\beta_{7} + 1) q^{14} + q^{16} + ( - \beta_{6} + \beta_{3} + \beta_1) q^{17} + (\beta_{2} + 2) q^{18} + ( - \beta_{5} + \beta_{3} + \beta_1) q^{19} + (2 \beta_{6} + \beta_1) q^{21} + ( - \beta_{6} + \beta_1) q^{22} + (\beta_{6} - \beta_{5} + \beta_{3}) q^{23} + \beta_1 q^{24} + (2 \beta_{5} - 2 \beta_{3}) q^{27} + (\beta_{7} + 1) q^{28} + (\beta_{7} - \beta_{2} - 1) q^{29} + ( - \beta_{6} + \beta_{5} - \beta_{3} + \beta_1) q^{31} + q^{32} + ( - 2 \beta_{7} - \beta_{4} + 3) q^{33} + ( - \beta_{6} + \beta_{3} + \beta_1) q^{34} + (\beta_{2} + 2) q^{36} + (\beta_{4} + \beta_{2} + 5) q^{37} + ( - \beta_{5} + \beta_{3} + \beta_1) q^{38} + ( - \beta_{6} + 2 \beta_{5} + \beta_{3} - 2 \beta_1) q^{41} + (2 \beta_{6} + \beta_1) q^{42} + ( - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_1) q^{43} + ( - \beta_{6} + \beta_1) q^{44} + (\beta_{6} - \beta_{5} + \beta_{3}) q^{46} + (\beta_{7} + 2 \beta_{2} - 1) q^{47} + \beta_1 q^{48} + (3 \beta_{7} - \beta_{2} + 3) q^{49} + ( - 2 \beta_{7} - 2 \beta_{4} - \beta_{2} + 3) q^{51} + ( - \beta_{6} - \beta_{5} - 4 \beta_{3} - \beta_1) q^{53} + (2 \beta_{5} - 2 \beta_{3}) q^{54} + (\beta_{7} + 1) q^{56} + ( - \beta_{4} - 2 \beta_{2} + 3) q^{57} + (\beta_{7} - \beta_{2} - 1) q^{58} + ( - 2 \beta_{5} - 2 \beta_{3}) q^{59} + (\beta_{7} + 3 \beta_{4} + 2 \beta_{2} + 1) q^{61} + ( - \beta_{6} + \beta_{5} - \beta_{3} + \beta_1) q^{62} + (\beta_{7} + 2 \beta_{4} + 3 \beta_{2} + 6) q^{63} + q^{64} + ( - 2 \beta_{7} - \beta_{4} + 3) q^{66} + 4 q^{67} + ( - \beta_{6} + \beta_{3} + \beta_1) q^{68} + (2 \beta_{7} - 2 \beta_{2}) q^{69} + 4 \beta_1 q^{71} + (\beta_{2} + 2) q^{72} + ( - 3 \beta_{7} + \beta_{4} + \beta_{2} + 4) q^{73} + (\beta_{4} + \beta_{2} + 5) q^{74} + ( - \beta_{5} + \beta_{3} + \beta_1) q^{76} + (\beta_{5} - \beta_{3} - 3 \beta_1) q^{77} + (2 \beta_{7} - \beta_{4} - 4 \beta_{2} - 1) q^{79} + (2 \beta_{4} + 3 \beta_{2} - 2) q^{81} + ( - \beta_{6} + 2 \beta_{5} + \beta_{3} - 2 \beta_1) q^{82} + ( - \beta_{4} - 3 \beta_{2} + 6) q^{83} + (2 \beta_{6} + \beta_1) q^{84} + ( - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_1) q^{86} + (2 \beta_{6} - 2 \beta_{5} + 2 \beta_{3} - 2 \beta_1) q^{87} + ( - \beta_{6} + \beta_1) q^{88} + (3 \beta_{6} - 2 \beta_{5} - 2 \beta_{3}) q^{89} + (\beta_{6} - \beta_{5} + \beta_{3}) q^{92} + ( - 2 \beta_{7} + 3 \beta_{2} + 5) q^{93} + (\beta_{7} + 2 \beta_{2} - 1) q^{94} + \beta_1 q^{96} + ( - \beta_{4} - 2 \beta_{2} + 1) q^{97} + (3 \beta_{7} - \beta_{2} + 3) q^{98} + ( - 2 \beta_{6} + \beta_{5} + 3 \beta_{3} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 8 q^{4} + 10 q^{7} + 8 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} + 8 q^{4} + 10 q^{7} + 8 q^{8} + 16 q^{9} + 10 q^{14} + 8 q^{16} + 16 q^{18} + 10 q^{28} - 6 q^{29} + 8 q^{32} + 20 q^{33} + 16 q^{36} + 40 q^{37} - 6 q^{47} + 30 q^{49} + 20 q^{51} + 10 q^{56} + 24 q^{57} - 6 q^{58} + 10 q^{61} + 50 q^{63} + 8 q^{64} + 20 q^{66} + 32 q^{67} + 4 q^{69} + 16 q^{72} + 26 q^{73} + 40 q^{74} - 4 q^{79} - 16 q^{81} + 48 q^{83} + 36 q^{93} - 6 q^{94} + 8 q^{97} + 30 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 20x^{6} + 132x^{4} - 332x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 20\nu^{5} + 116\nu^{3} - 172\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} - 12\nu^{2} + 26 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 20\nu^{5} + 132\nu^{3} - 268\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 16\nu^{5} + 72\nu^{3} - 84\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{6} - 16\nu^{4} + 72\nu^{2} - 84 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} - 2\beta_{3} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{4} + 12\beta_{2} + 34 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{6} + 22\beta_{5} - 30\beta_{3} + 44\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{7} + 32\beta_{4} + 120\beta_{2} + 268 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 40\beta_{6} + 208\beta_{5} - 336\beta_{3} + 356\beta_1 \) 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
−3.07108
−2.40987
−1.83766
−1.17644
1.17644
1.83766
2.40987
3.07108
1.00000 −3.07108 1.00000 0 −3.07108 3.69510 1.00000 6.43154 0
1.2 1.00000 −2.40987 1.00000 0 −2.40987 −1.40561 1.00000 2.80745 0
1.3 1.00000 −1.83766 1.00000 0 −1.83766 4.79742 1.00000 0.376989 0
1.4 1.00000 −1.17644 1.00000 0 −1.17644 −2.08692 1.00000 −1.61598 0
1.5 1.00000 1.17644 1.00000 0 1.17644 −2.08692 1.00000 −1.61598 0
1.6 1.00000 1.83766 1.00000 0 1.83766 4.79742 1.00000 0.376989 0
1.7 1.00000 2.40987 1.00000 0 2.40987 −1.40561 1.00000 2.80745 0
1.8 1.00000 3.07108 1.00000 0 3.07108 3.69510 1.00000 6.43154 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8450.2.a.cs 8
5.b even 2 1 8450.2.a.cr 8
5.c odd 4 2 1690.2.b.e 16
13.b even 2 1 8450.2.a.cr 8
13.f odd 12 2 650.2.m.e 16
65.d even 2 1 inner 8450.2.a.cs 8
65.f even 4 2 1690.2.c.e 8
65.h odd 4 2 1690.2.b.e 16
65.k even 4 2 1690.2.c.f 8
65.o even 12 2 130.2.m.a 8
65.s odd 12 2 650.2.m.e 16
65.t even 12 2 130.2.m.b yes 8
195.bc odd 12 2 1170.2.bj.a 8
195.bn odd 12 2 1170.2.bj.b 8
260.be odd 12 2 1040.2.df.c 8
260.bl odd 12 2 1040.2.df.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.m.a 8 65.o even 12 2
130.2.m.b yes 8 65.t even 12 2
650.2.m.e 16 13.f odd 12 2
650.2.m.e 16 65.s odd 12 2
1040.2.df.a 8 260.bl odd 12 2
1040.2.df.c 8 260.be odd 12 2
1170.2.bj.a 8 195.bc odd 12 2
1170.2.bj.b 8 195.bn odd 12 2
1690.2.b.e 16 5.c odd 4 2
1690.2.b.e 16 65.h odd 4 2
1690.2.c.e 8 65.f even 4 2
1690.2.c.f 8 65.k even 4 2
8450.2.a.cr 8 5.b even 2 1
8450.2.a.cr 8 13.b even 2 1
8450.2.a.cs 8 1.a even 1 1 trivial
8450.2.a.cs 8 65.d even 2 1 inner

Hecke kernels

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

\( T_{3}^{8} - 20T_{3}^{6} + 132T_{3}^{4} - 332T_{3}^{2} + 256 \) Copy content Toggle raw display
\( T_{7}^{4} - 5T_{7}^{3} - 9T_{7}^{2} + 37T_{7} + 52 \) Copy content Toggle raw display
\( T_{11}^{8} - 41T_{11}^{6} + 399T_{11}^{4} - 1091T_{11}^{2} + 784 \) Copy content Toggle raw display
\( T_{17}^{8} - 59T_{17}^{6} + 480T_{17}^{4} - 620T_{17}^{2} + 4 \) Copy content Toggle raw display
\( T_{31}^{8} - 60T_{31}^{6} + 1176T_{31}^{4} - 7596T_{31}^{2} + 576 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 20 T^{6} + 132 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 5 T^{3} - 9 T^{2} + 37 T + 52)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 41 T^{6} + 399 T^{4} + \cdots + 784 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} - 59 T^{6} + 480 T^{4} - 620 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$19$ \( T^{8} - 45 T^{6} + 699 T^{4} + \cdots + 9216 \) Copy content Toggle raw display
$23$ \( T^{8} - 44 T^{6} + 336 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( (T^{4} + 3 T^{3} - 18 T^{2} - 60 T - 24)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 60 T^{6} + 1176 T^{4} + \cdots + 576 \) Copy content Toggle raw display
$37$ \( (T^{4} - 20 T^{3} + 120 T^{2} - 218 T + 109)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 203 T^{6} + 11448 T^{4} + \cdots + 1106704 \) Copy content Toggle raw display
$43$ \( T^{8} - 180 T^{6} + 11184 T^{4} + \cdots + 2359296 \) Copy content Toggle raw display
$47$ \( (T^{4} + 3 T^{3} - 117 T^{2} - 267 T - 96)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} - 356 T^{6} + \cdots + 17783089 \) Copy content Toggle raw display
$59$ \( T^{8} - 212 T^{6} + 9840 T^{4} + \cdots + 369664 \) Copy content Toggle raw display
$61$ \( (T^{4} - 5 T^{3} - 246 T^{2} + 1024 T + 5362)^{2} \) Copy content Toggle raw display
$67$ \( (T - 4)^{8} \) Copy content Toggle raw display
$71$ \( T^{8} - 320 T^{6} + \cdots + 16777216 \) Copy content Toggle raw display
$73$ \( (T^{4} - 13 T^{3} - 114 T^{2} + 1298 T - 1406)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 2 T^{3} - 216 T^{2} - 502 T + 1384)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 24 T^{3} + 78 T^{2} + 1602 T - 9744)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 425 T^{6} + \cdots + 59474944 \) Copy content Toggle raw display
$97$ \( (T^{4} - 4 T^{3} - 60 T^{2} + 386 T - 572)^{2} \) Copy content Toggle raw display
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