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Properties

Label 8450.2.a.da
Level $8450$
Weight $2$
Character orbit 8450.a
Self dual yes
Analytic conductor $67.474$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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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: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - x^{8} - 14x^{7} + 17x^{6} + 53x^{5} - 69x^{4} - 33x^{3} + 26x^{2} + 8x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + (\beta_{4} + 1) q^{3} + q^{4} + (\beta_{4} + 1) q^{6} + ( - \beta_{8} + \beta_{6} - \beta_{3}) q^{7} + q^{8} + ( - \beta_{8} + \beta_{4} - \beta_{3} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + (\beta_{4} + 1) q^{3} + q^{4} + (\beta_{4} + 1) q^{6} + ( - \beta_{8} + \beta_{6} - \beta_{3}) q^{7} + q^{8} + ( - \beta_{8} + \beta_{4} - \beta_{3} + \cdots + 1) q^{9}+ \cdots + (2 \beta_{8} - 9 \beta_{7} + 3 \beta_{6} + \cdots - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{2} + 7 q^{3} + 9 q^{4} + 7 q^{6} + q^{7} + 9 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 9 q^{2} + 7 q^{3} + 9 q^{4} + 7 q^{6} + q^{7} + 9 q^{8} + 8 q^{9} + 4 q^{11} + 7 q^{12} + q^{14} + 9 q^{16} + 12 q^{17} + 8 q^{18} + 6 q^{19} + 8 q^{21} + 4 q^{22} + 11 q^{23} + 7 q^{24} + 34 q^{27} + q^{28} + 15 q^{29} - 6 q^{31} + 9 q^{32} - 14 q^{33} + 12 q^{34} + 8 q^{36} - 10 q^{37} + 6 q^{38} + 27 q^{41} + 8 q^{42} + 41 q^{43} + 4 q^{44} + 11 q^{46} + 5 q^{47} + 7 q^{48} - 10 q^{49} + 12 q^{51} + 18 q^{53} + 34 q^{54} + q^{56} - 12 q^{57} + 15 q^{58} + 20 q^{59} - 19 q^{61} - 6 q^{62} + 39 q^{63} + 9 q^{64} - 14 q^{66} - q^{67} + 12 q^{68} - 14 q^{69} - 10 q^{71} + 8 q^{72} - 12 q^{73} - 10 q^{74} + 6 q^{76} + 30 q^{77} - 14 q^{79} + 53 q^{81} + 27 q^{82} + q^{83} + 8 q^{84} + 41 q^{86} + 33 q^{87} + 4 q^{88} + 43 q^{89} + 11 q^{92} - 28 q^{93} + 5 q^{94} + 7 q^{96} + 6 q^{97} - 10 q^{98} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - x^{8} - 14x^{7} + 17x^{6} + 53x^{5} - 69x^{4} - 33x^{3} + 26x^{2} + 8x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{7} + \nu^{6} - 11\nu^{5} - 4\nu^{4} + 33\nu^{3} - 7\nu^{2} - 8\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{8} - 12\nu^{6} + 6\nu^{5} + 35\nu^{4} - 34\nu^{3} + 9\nu^{2} + 3\nu - 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{8} + 12\nu^{6} - 6\nu^{5} - 35\nu^{4} + 35\nu^{3} - 8\nu^{2} - 9\nu + 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{8} + 13\nu^{6} - 5\nu^{5} - 46\nu^{4} + 30\nu^{3} + 24\nu^{2} - 10\nu - 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{8} + 2\nu^{7} - 11\nu^{6} - 18\nu^{5} + 35\nu^{4} + 42\nu^{3} - 23\nu^{2} - 14\nu + 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\nu^{8} + \nu^{7} + 13\nu^{6} - 18\nu^{5} - 42\nu^{4} + 73\nu^{3} - 19\nu \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( -\nu^{8} - 3\nu^{7} + 10\nu^{6} + 29\nu^{5} - 30\nu^{4} - 74\nu^{3} + 24\nu^{2} + 19\nu \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} + 6\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} + 7\beta_{7} - 6\beta_{6} - 7\beta_{5} - \beta_{4} + 6\beta_{3} + 8\beta_{2} - 3\beta _1 + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{8} - 12\beta_{7} + 8\beta_{6} + 11\beta_{5} + 9\beta_{4} - 3\beta_{3} - 13\beta_{2} + 37\beta _1 - 25 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14\beta_{8} + 52\beta_{7} - 37\beta_{6} - 50\beta_{5} - 16\beta_{4} + 37\beta_{3} + 64\beta_{2} - 39\beta _1 + 157 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 43 \beta_{8} - 116 \beta_{7} + 61 \beta_{6} + 103 \beta_{5} + 78 \beta_{4} - 39 \beta_{3} + \cdots - 242 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 151 \beta_{8} + 408 \beta_{7} - 239 \beta_{6} - 378 \beta_{5} - 177 \beta_{4} + 244 \beta_{3} + \cdots + 1095 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
0.671265
2.08395
−2.90009
−0.506152
2.20982
2.17115
−0.430845
0.0982708
−2.39737
1.00000 −1.92714 1.00000 0 −1.92714 1.36403 1.00000 0.713864 0
1.2 1.00000 −1.49535 1.00000 0 −1.49535 1.01867 1.00000 −0.763924 0
1.3 1.00000 −0.303663 1.00000 0 −0.303663 −1.08593 1.00000 −2.90779 0
1.4 1.00000 0.0670024 1.00000 0 0.0670024 −5.03568 1.00000 −2.99551 0
1.5 1.00000 0.666723 1.00000 0 0.666723 2.41461 1.00000 −2.55548 0
1.6 1.00000 0.956446 1.00000 0 0.956446 −2.15124 1.00000 −2.08521 0
1.7 1.00000 2.55203 1.00000 0 2.55203 1.86919 1.00000 3.51288 0
1.8 1.00000 3.06821 1.00000 0 3.06821 3.06611 1.00000 6.41393 0
1.9 1.00000 3.41573 1.00000 0 3.41573 −0.459770 1.00000 8.66724 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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.da 9
5.b even 2 1 8450.2.a.ct 9
5.c odd 4 2 1690.2.b.g yes 18
13.b even 2 1 8450.2.a.cw 9
65.d even 2 1 8450.2.a.cx 9
65.f even 4 2 1690.2.c.g 18
65.h odd 4 2 1690.2.b.f ✓ 18
65.k even 4 2 1690.2.c.h 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1690.2.b.f ✓ 18 65.h odd 4 2
1690.2.b.g yes 18 5.c odd 4 2
1690.2.c.g 18 65.f even 4 2
1690.2.c.h 18 65.k even 4 2
8450.2.a.ct 9 5.b even 2 1
8450.2.a.cw 9 13.b even 2 1
8450.2.a.cx 9 65.d even 2 1
8450.2.a.da 9 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8450))\):

\( T_{3}^{9} - 7T_{3}^{8} + 7T_{3}^{7} + 40T_{3}^{6} - 70T_{3}^{5} - 49T_{3}^{4} + 111T_{3}^{3} - 21T_{3}^{2} - 14T_{3} + 1 \) Copy content Toggle raw display
\( T_{7}^{9} - T_{7}^{8} - 26T_{7}^{7} + 55T_{7}^{6} + 108T_{7}^{5} - 310T_{7}^{4} - 24T_{7}^{3} + 380T_{7}^{2} - 80T_{7} - 104 \) Copy content Toggle raw display
\( T_{11}^{9} - 4 T_{11}^{8} - 45 T_{11}^{7} + 126 T_{11}^{6} + 672 T_{11}^{5} - 938 T_{11}^{4} + \cdots - 1688 \) Copy content Toggle raw display
\( T_{17}^{9} - 12 T_{17}^{8} - 11 T_{17}^{7} + 552 T_{17}^{6} - 994 T_{17}^{5} - 7138 T_{17}^{4} + \cdots + 28664 \) Copy content Toggle raw display
\( T_{31}^{9} + 6 T_{31}^{8} - 152 T_{31}^{7} - 1052 T_{31}^{6} + 6852 T_{31}^{5} + 58096 T_{31}^{4} + \cdots + 3017216 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{9} \) Copy content Toggle raw display
$3$ \( T^{9} - 7 T^{8} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{9} \) Copy content Toggle raw display
$7$ \( T^{9} - T^{8} + \cdots - 104 \) Copy content Toggle raw display
$11$ \( T^{9} - 4 T^{8} + \cdots - 1688 \) Copy content Toggle raw display
$13$ \( T^{9} \) Copy content Toggle raw display
$17$ \( T^{9} - 12 T^{8} + \cdots + 28664 \) Copy content Toggle raw display
$19$ \( T^{9} - 6 T^{8} + \cdots + 1912 \) Copy content Toggle raw display
$23$ \( T^{9} - 11 T^{8} + \cdots + 568 \) Copy content Toggle raw display
$29$ \( T^{9} - 15 T^{8} + \cdots - 369496 \) Copy content Toggle raw display
$31$ \( T^{9} + 6 T^{8} + \cdots + 3017216 \) Copy content Toggle raw display
$37$ \( T^{9} + 10 T^{8} + \cdots + 229312 \) Copy content Toggle raw display
$41$ \( T^{9} - 27 T^{8} + \cdots + 138557 \) Copy content Toggle raw display
$43$ \( T^{9} - 41 T^{8} + \cdots + 2743369 \) Copy content Toggle raw display
$47$ \( T^{9} - 5 T^{8} + \cdots + 1112 \) Copy content Toggle raw display
$53$ \( T^{9} - 18 T^{8} + \cdots + 884416 \) Copy content Toggle raw display
$59$ \( T^{9} - 20 T^{8} + \cdots + 1232008 \) Copy content Toggle raw display
$61$ \( T^{9} + 19 T^{8} + \cdots - 17534728 \) Copy content Toggle raw display
$67$ \( T^{9} + T^{8} + \cdots - 116700191 \) Copy content Toggle raw display
$71$ \( T^{9} + 10 T^{8} + \cdots + 5913536 \) Copy content Toggle raw display
$73$ \( T^{9} + 12 T^{8} + \cdots + 6936152 \) Copy content Toggle raw display
$79$ \( T^{9} + 14 T^{8} + \cdots + 487936 \) Copy content Toggle raw display
$83$ \( T^{9} - T^{8} + \cdots - 6817999 \) Copy content Toggle raw display
$89$ \( T^{9} - 43 T^{8} + \cdots + 263242657 \) Copy content Toggle raw display
$97$ \( T^{9} - 6 T^{8} + \cdots + 10724792 \) Copy content Toggle raw display
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