↑

Properties

Label 8410.2.a.bx
Level $8410$
Weight $2$
Character orbit 8410.a
Self dual yes
Analytic conductor $67.154$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8410,2,Mod(1,8410)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8410.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8410, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8410 = 2 \cdot 5 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8410.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,12,-4,12,-12,-4,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(67.1541880999\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} - 13 x^{10} + 56 x^{9} + 41 x^{8} - 234 x^{7} - 8 x^{6} + 298 x^{5} + 41 x^{4} + \cdots + 1 \) 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)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + ( - \beta_{4} + \beta_1 - 1) q^{3} + q^{4} - q^{5} + ( - \beta_{4} + \beta_1 - 1) q^{6} + ( - \beta_{11} - \beta_{10} + \cdots + \beta_{5}) q^{7} + q^{8} + (\beta_{10} - \beta_{5} + 2 \beta_{4} + \cdots + 1) q^{9}+ \cdots + ( - 2 \beta_{11} - \beta_{10} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{2} - 4 q^{3} + 12 q^{4} - 12 q^{5} - 4 q^{6} - 8 q^{7} + 12 q^{8} + 6 q^{9} - 12 q^{10} - 14 q^{11} - 4 q^{12} - 4 q^{13} - 8 q^{14} + 4 q^{15} + 12 q^{16} - 8 q^{17} + 6 q^{18} + 16 q^{19}+ \cdots + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} - 13 x^{10} + 56 x^{9} + 41 x^{8} - 234 x^{7} - 8 x^{6} + 298 x^{5} + 41 x^{4} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 57 \nu^{11} + 218 \nu^{10} + 737 \nu^{9} - 2952 \nu^{8} - 2158 \nu^{7} + 11555 \nu^{6} - 1063 \nu^{5} + \cdots - 181 ) / 123 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 254 \nu^{11} + 1140 \nu^{10} + 2858 \nu^{9} - 15956 \nu^{8} - 4347 \nu^{7} + 65880 \nu^{6} + \cdots - 1213 ) / 369 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 844 \nu^{11} - 3630 \nu^{10} - 9832 \nu^{9} + 50122 \nu^{8} + 18648 \nu^{7} - 201843 \nu^{6} + \cdots + 2603 ) / 369 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1219 \nu^{11} + 5433 \nu^{10} + 13459 \nu^{9} - 74716 \nu^{8} - 17214 \nu^{7} + 297030 \nu^{6} + \cdots - 3143 ) / 369 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1216 \nu^{11} + 5283 \nu^{10} + 13972 \nu^{9} - 72859 \nu^{8} - 24546 \nu^{7} + 292371 \nu^{6} + \cdots - 2954 ) / 369 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1186 \nu^{11} - 4965 \nu^{10} - 14479 \nu^{9} + 69157 \nu^{8} + 35232 \nu^{7} - 284718 \nu^{6} + \cdots + 3851 ) / 369 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 545 \nu^{11} + 2428 \nu^{10} + 5979 \nu^{9} - 33245 \nu^{8} - 7196 \nu^{7} + 130921 \nu^{6} + \cdots - 1761 ) / 123 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1756 \nu^{11} - 7818 \nu^{10} - 19189 \nu^{9} + 106726 \nu^{8} + 22092 \nu^{7} - 417297 \nu^{6} + \cdots + 3713 ) / 369 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2600 \nu^{11} + 11412 \nu^{10} + 29336 \nu^{9} - 156929 \nu^{8} - 45240 \nu^{7} + 624819 \nu^{6} + \cdots - 7207 ) / 369 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2851 \nu^{11} - 12540 \nu^{10} - 32053 \nu^{9} + 172378 \nu^{8} + 48033 \nu^{7} - 685611 \nu^{6} + \cdots + 6599 ) / 369 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - \beta_{6} - \beta_{5} + \beta_{4} + 2\beta_{3} + \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{11} + \beta_{10} - 3\beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} + 3\beta_{3} + 7\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6 \beta_{11} + 11 \beta_{10} - 7 \beta_{9} - 3 \beta_{8} - 3 \beta_{7} - 7 \beta_{6} - 8 \beta_{5} + \cdots + 38 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 46 \beta_{11} + 27 \beta_{10} - 43 \beta_{9} - 20 \beta_{8} - 9 \beta_{7} + 16 \beta_{6} - 7 \beta_{5} + \cdots + 62 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 130 \beta_{11} + 152 \beta_{10} - 128 \beta_{9} - 64 \beta_{8} - 40 \beta_{7} - 34 \beta_{6} + \cdots + 441 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 672 \beta_{11} + 498 \beta_{10} - 596 \beta_{9} - 322 \beta_{8} - 92 \beta_{7} + 212 \beta_{6} + \cdots + 1093 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2229 \beta_{11} + 2249 \beta_{10} - 2029 \beta_{9} - 1112 \beta_{8} - 477 \beta_{7} + 25 \beta_{6} + \cdots + 5763 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 9902 \beta_{11} + 8151 \beta_{10} - 8563 \beta_{9} - 4927 \beta_{8} - 1188 \beta_{7} + 2764 \beta_{6} + \cdots + 17838 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 35543 \beta_{11} + 33792 \beta_{10} - 31222 \beta_{9} - 17907 \beta_{8} - 5990 \beta_{7} + 4329 \beta_{6} + \cdots + 80786 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 147400 \beta_{11} + 127374 \beta_{10} - 126234 \beta_{9} - 74405 \beta_{8} - 17417 \beta_{7} + \cdots + 280304 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−2.74180
−0.773746
0.0921653
−2.25814
1.28705
−0.345759
−0.493658
−0.400832
1.73309
3.88849
2.46438
1.54875
1.00000 −3.29676 1.00000 −1.00000 −3.29676 3.18978 1.00000 7.86860 −1.00000
1.2 1.00000 −3.02073 1.00000 −1.00000 −3.02073 −4.64841 1.00000 6.12478 −1.00000
1.3 1.00000 −2.15481 1.00000 −1.00000 −2.15481 −2.14386 1.00000 1.64322 −1.00000
1.4 1.00000 −1.45620 1.00000 −1.00000 −1.45620 −5.07194 1.00000 −0.879489 −1.00000
1.5 1.00000 −0.959933 1.00000 −1.00000 −0.959933 4.10940 1.00000 −2.07853 −1.00000
1.6 1.00000 −0.900717 1.00000 −1.00000 −0.900717 −1.72171 1.00000 −2.18871 −1.00000
1.7 1.00000 0.308280 1.00000 −1.00000 0.308280 2.44943 1.00000 −2.90496 −1.00000
1.8 1.00000 0.401106 1.00000 −1.00000 0.401106 1.20419 1.00000 −2.83911 −1.00000
1.9 1.00000 1.17814 1.00000 −1.00000 1.17814 0.356400 1.00000 −1.61200 −1.00000
1.10 1.00000 1.64151 1.00000 −1.00000 1.64151 −2.92101 1.00000 −0.305434 −1.00000
1.11 1.00000 1.90942 1.00000 −1.00000 1.90942 −1.33050 1.00000 0.645893 −1.00000
1.12 1.00000 2.35069 1.00000 −1.00000 2.35069 −1.47177 1.00000 2.52573 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( +1 \)
\(29\) \( -1 \)

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 8410.2.a.bx 12
29.b even 2 1 8410.2.a.bw 12
29.f odd 28 2 290.2.m.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
290.2.m.a ✓ 24 29.f odd 28 2
8410.2.a.bw 12 29.b even 2 1
8410.2.a.bx 12 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8410))\):

\( T_{3}^{12} + 4 T_{3}^{11} - 13 T_{3}^{10} - 56 T_{3}^{9} + 62 T_{3}^{8} + 276 T_{3}^{7} - 134 T_{3}^{6} + \cdots + 29 \) Copy content Toggle raw display
\( T_{7}^{12} + 8 T_{7}^{11} - 19 T_{7}^{10} - 270 T_{7}^{9} - 156 T_{7}^{8} + 2836 T_{7}^{7} + 4563 T_{7}^{6} + \cdots - 6859 \) Copy content Toggle raw display
\( T_{11}^{12} + 14 T_{11}^{11} + 55 T_{11}^{10} - 56 T_{11}^{9} - 783 T_{11}^{8} - 938 T_{11}^{7} + \cdots + 2017 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 4 T^{11} + \cdots + 29 \) Copy content Toggle raw display
$5$ \( (T + 1)^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 8 T^{11} + \cdots - 6859 \) Copy content Toggle raw display
$11$ \( T^{12} + 14 T^{11} + \cdots + 2017 \) Copy content Toggle raw display
$13$ \( T^{12} + 4 T^{11} + \cdots - 8231 \) Copy content Toggle raw display
$17$ \( T^{12} + 8 T^{11} + \cdots - 27383 \) Copy content Toggle raw display
$19$ \( T^{12} - 16 T^{11} + \cdots - 25160659 \) Copy content Toggle raw display
$23$ \( T^{12} - 4 T^{11} + \cdots - 1399 \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} - 2 T^{11} + \cdots + 18019933 \) Copy content Toggle raw display
$37$ \( T^{12} + 18 T^{11} + \cdots + 464857 \) Copy content Toggle raw display
$41$ \( T^{12} + 4 T^{11} + \cdots + 2747977 \) Copy content Toggle raw display
$43$ \( T^{12} + 28 T^{11} + \cdots + 1448881 \) Copy content Toggle raw display
$47$ \( T^{12} + 36 T^{11} + \cdots - 39840751 \) Copy content Toggle raw display
$53$ \( T^{12} - 305 T^{10} + \cdots + 17104361 \) Copy content Toggle raw display
$59$ \( T^{12} + 12 T^{11} + \cdots - 5707211 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots - 1459133003 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 3475662793 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots - 26318719699 \) Copy content Toggle raw display
$73$ \( T^{12} + 48 T^{11} + \cdots - 61899599 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots - 545276647 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots - 12572314159 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 2079144677 \) Copy content Toggle raw display
$97$ \( T^{12} + 16 T^{11} + \cdots - 44948903 \) Copy content Toggle raw display
show more
show less