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Properties

Label 8450.2.a.cv
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: \( x^{9} - 21x^{7} - 3x^{6} + 133x^{5} + 28x^{4} - 249x^{3} + 21x^{2} + 126x - 43 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - q^{2} - \beta_1 q^{3} + q^{4} + \beta_1 q^{6} - \beta_{8} q^{7} - q^{8} + ( - \beta_{6} - \beta_{4} + \beta_{3} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - \beta_1 q^{3} + q^{4} + \beta_1 q^{6} - \beta_{8} q^{7} - q^{8} + ( - \beta_{6} - \beta_{4} + \beta_{3} + \cdots + 1) q^{9}+ \cdots + ( - 2 \beta_{7} - \beta_{6} - 7 \beta_{5} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 9 q^{2} + 9 q^{4} + 4 q^{7} - 9 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 9 q^{2} + 9 q^{4} + 4 q^{7} - 9 q^{8} + 15 q^{9} + q^{11} - 4 q^{14} + 9 q^{16} + 2 q^{17} - 15 q^{18} - 10 q^{19} - 3 q^{21} - q^{22} - q^{23} - 9 q^{27} + 4 q^{28} - q^{29} + 4 q^{31} - 9 q^{32} - 30 q^{33} - 2 q^{34} + 15 q^{36} - 18 q^{37} + 10 q^{38} + 2 q^{41} + 3 q^{42} - 5 q^{43} + q^{44} + q^{46} + 10 q^{47} + 39 q^{49} + 38 q^{51} + 3 q^{53} + 9 q^{54} - 4 q^{56} - 14 q^{57} + q^{58} + 20 q^{59} + 40 q^{61} - 4 q^{62} + 45 q^{63} + 9 q^{64} + 30 q^{66} - 15 q^{67} + 2 q^{68} + 68 q^{69} - 46 q^{71} - 15 q^{72} - 13 q^{73} + 18 q^{74} - 10 q^{76} + 13 q^{77} + 37 q^{79} + 53 q^{81} - 2 q^{82} - 2 q^{83} - 3 q^{84} + 5 q^{86} + q^{87} - q^{88} - 14 q^{89} - q^{92} - 2 q^{93} - 10 q^{94} - 23 q^{97} - 39 q^{98} + 62 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 21x^{7} - 3x^{6} + 133x^{5} + 28x^{4} - 249x^{3} + 21x^{2} + 126x - 43 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 92 \nu^{8} + 30 \nu^{7} - 3055 \nu^{6} - 989 \nu^{5} + 31454 \nu^{4} + 8868 \nu^{3} - 102143 \nu^{2} + \cdots + 44265 ) / 13027 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1104 \nu^{8} - 360 \nu^{7} + 23633 \nu^{6} + 11868 \nu^{5} - 155989 \nu^{4} - 93389 \nu^{3} + \cdots - 192478 ) / 13027 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 298 \nu^{8} + 259 \nu^{7} - 6214 \nu^{6} - 5995 \nu^{5} + 37315 \nu^{4} + 36735 \nu^{3} + \cdots + 22051 ) / 1861 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2235 \nu^{8} - 1012 \nu^{7} + 46605 \nu^{6} + 27283 \nu^{5} - 286376 \nu^{4} - 187115 \nu^{3} + \cdots - 190506 ) / 13027 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3098 \nu^{8} - 2143 \nu^{7} + 64076 \nu^{6} + 52844 \nu^{5} - 385740 \nu^{4} - 341666 \nu^{3} + \cdots - 250462 ) / 13027 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10604 \nu^{8} + 5157 \nu^{7} - 219020 \nu^{6} - 140047 \nu^{5} + 1319070 \nu^{4} + 956432 \nu^{3} + \cdots + 789519 ) / 13027 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11559 \nu^{8} + 6318 \nu^{7} - 239546 \nu^{6} - 166597 \nu^{5} + 1449888 \nu^{4} + 1132878 \nu^{3} + \cdots + 932821 ) / 13027 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} + 9\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{8} - 14\beta_{6} - \beta_{5} - 12\beta_{4} + 9\beta_{3} + 10\beta_{2} + 13\beta _1 + 29 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10\beta_{8} - 13\beta_{7} - 2\beta_{6} - 24\beta_{5} - 11\beta_{4} + 13\beta_{3} - 3\beta_{2} + 84\beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -16\beta_{8} - \beta_{7} - 169\beta_{6} - 18\beta_{5} - 136\beta_{4} + 84\beta_{3} + 89\beta_{2} + 147\beta _1 + 244 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 88 \beta_{8} - 152 \beta_{7} - 68 \beta_{6} - 354 \beta_{5} - 115 \beta_{4} + 147 \beta_{3} - 55 \beta_{2} + \cdots + 148 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 207 \beta_{8} - 27 \beta_{7} - 1935 \beta_{6} - 302 \beta_{5} - 1508 \beta_{4} + 818 \beta_{3} + \cdots + 2164 \) 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
3.28584
2.97980
1.09230
0.567463
0.517382
−0.943657
−1.85524
−2.44632
−3.19756
−1.00000 −3.28584 1.00000 0 3.28584 3.31873 −1.00000 7.79673 0
1.2 −1.00000 −2.97980 1.00000 0 2.97980 −2.14281 −1.00000 5.87918 0
1.3 −1.00000 −1.09230 1.00000 0 1.09230 2.53775 −1.00000 −1.80689 0
1.4 −1.00000 −0.567463 1.00000 0 0.567463 4.77673 −1.00000 −2.67799 0
1.5 −1.00000 −0.517382 1.00000 0 0.517382 −1.91546 −1.00000 −2.73232 0
1.6 −1.00000 0.943657 1.00000 0 −0.943657 −4.61826 −1.00000 −2.10951 0
1.7 −1.00000 1.85524 1.00000 0 −1.85524 −2.88091 −1.00000 0.441905 0
1.8 −1.00000 2.44632 1.00000 0 −2.44632 0.0417661 −1.00000 2.98450 0
1.9 −1.00000 3.19756 1.00000 0 −3.19756 4.88245 −1.00000 7.22438 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.cv yes 9
5.b even 2 1 8450.2.a.cz yes 9
13.b even 2 1 8450.2.a.cy yes 9
65.d even 2 1 8450.2.a.cu ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8450.2.a.cu ✓ 9 65.d even 2 1
8450.2.a.cv yes 9 1.a even 1 1 trivial
8450.2.a.cy yes 9 13.b even 2 1
8450.2.a.cz yes 9 5.b even 2 1

Hecke kernels

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

\( T_{3}^{9} - 21T_{3}^{7} + 3T_{3}^{6} + 133T_{3}^{5} - 28T_{3}^{4} - 249T_{3}^{3} - 21T_{3}^{2} + 126T_{3} + 43 \) Copy content Toggle raw display
\( T_{7}^{9} - 4T_{7}^{8} - 43T_{7}^{7} + 147T_{7}^{6} + 636T_{7}^{5} - 1544T_{7}^{4} - 4192T_{7}^{3} + 4928T_{7}^{2} + 10528T_{7} - 448 \) Copy content Toggle raw display
\( T_{11}^{9} - T_{11}^{8} - 63 T_{11}^{7} + 22 T_{11}^{6} + 1319 T_{11}^{5} + 142 T_{11}^{4} - 9983 T_{11}^{3} + \cdots - 839 \) Copy content Toggle raw display
\( T_{17}^{9} - 2 T_{17}^{8} - 99 T_{17}^{7} + 267 T_{17}^{6} + 2499 T_{17}^{5} - 8386 T_{17}^{4} + \cdots - 2107 \) Copy content Toggle raw display
\( T_{31}^{9} - 4 T_{31}^{8} - 125 T_{31}^{7} + 567 T_{31}^{6} + 3398 T_{31}^{5} - 18860 T_{31}^{4} + \cdots - 832 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{9} \) Copy content Toggle raw display
$3$ \( T^{9} - 21 T^{7} + \cdots + 43 \) Copy content Toggle raw display
$5$ \( T^{9} \) Copy content Toggle raw display
$7$ \( T^{9} - 4 T^{8} + \cdots - 448 \) Copy content Toggle raw display
$11$ \( T^{9} - T^{8} + \cdots - 839 \) Copy content Toggle raw display
$13$ \( T^{9} \) Copy content Toggle raw display
$17$ \( T^{9} - 2 T^{8} + \cdots - 2107 \) Copy content Toggle raw display
$19$ \( T^{9} + 10 T^{8} + \cdots + 4969 \) Copy content Toggle raw display
$23$ \( T^{9} + T^{8} + \cdots + 250432 \) Copy content Toggle raw display
$29$ \( T^{9} + T^{8} + \cdots - 1386944 \) Copy content Toggle raw display
$31$ \( T^{9} - 4 T^{8} + \cdots - 832 \) Copy content Toggle raw display
$37$ \( T^{9} + 18 T^{8} + \cdots - 74304 \) Copy content Toggle raw display
$41$ \( T^{9} - 2 T^{8} + \cdots + 288701 \) Copy content Toggle raw display
$43$ \( T^{9} + 5 T^{8} + \cdots - 321469 \) Copy content Toggle raw display
$47$ \( T^{9} - 10 T^{8} + \cdots + 758848 \) Copy content Toggle raw display
$53$ \( T^{9} - 3 T^{8} + \cdots - 1571648 \) Copy content Toggle raw display
$59$ \( T^{9} - 20 T^{8} + \cdots - 2282111 \) Copy content Toggle raw display
$61$ \( T^{9} - 40 T^{8} + \cdots + 352192 \) Copy content Toggle raw display
$67$ \( T^{9} + 15 T^{8} + \cdots - 6249256 \) Copy content Toggle raw display
$71$ \( T^{9} + 46 T^{8} + \cdots + 22789312 \) Copy content Toggle raw display
$73$ \( T^{9} + 13 T^{8} + \cdots - 2759477 \) Copy content Toggle raw display
$79$ \( T^{9} - 37 T^{8} + \cdots + 121546048 \) Copy content Toggle raw display
$83$ \( T^{9} + 2 T^{8} + \cdots - 40330723 \) Copy content Toggle raw display
$89$ \( T^{9} + 14 T^{8} + \cdots - 11073173 \) Copy content Toggle raw display
$97$ \( T^{9} + 23 T^{8} + \cdots - 15715253 \) Copy content Toggle raw display
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