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Properties

Label 650.2.w.h
Level $650$
Weight $2$
Character orbit 650.w
Analytic conductor $5.190$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [650,2,Mod(193,650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(650, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([9, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("650.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 650.w (of order \(12\), degree \(4\), minimal)

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: no
Analytic conductor: \(5.19027613138\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 28x^{14} + 294x^{12} + 1516x^{10} + 4147x^{8} + 6012x^{6} + 4338x^{4} + 1296x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{15} q^{2} + \beta_{2} q^{3} + (\beta_{15} - 1) q^{4} + ( - \beta_{7} - \beta_{5} + \cdots - \beta_1) q^{6} + ( - \beta_{14} + \beta_{12} - \beta_{10} + \cdots + 2) q^{7} - q^{8} + ( - \beta_{15} - \beta_{13} + \beta_{12} + \cdots - 1) q^{9}+ \cdots + (\beta_{14} + 3 \beta_{13} - \beta_{12} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{2} - 8 q^{4} + 12 q^{7} - 16 q^{8} - 24 q^{9} - 4 q^{11} + 8 q^{13} - 8 q^{16} - 8 q^{17} + 16 q^{19} - 4 q^{21} + 4 q^{22} - 4 q^{23} + 4 q^{26} + 36 q^{27} - 12 q^{28} + 36 q^{29} - 8 q^{31}+ \cdots - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 28x^{14} + 294x^{12} + 1516x^{10} + 4147x^{8} + 6012x^{6} + 4338x^{4} + 1296x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4 \nu^{15} - 8 \nu^{14} + 145 \nu^{13} - 221 \nu^{12} + 2010 \nu^{11} - 2295 \nu^{10} + 13435 \nu^{9} + \cdots + 756 ) / 4968 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 53 \nu^{15} - 138 \nu^{14} + 1490 \nu^{13} - 3657 \nu^{12} + 15489 \nu^{11} - 35397 \nu^{10} + \cdots - 9315 ) / 14904 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 37\nu^{14} + 979\nu^{12} + 9381\nu^{10} + 41836\nu^{8} + 89761\nu^{6} + 83688\nu^{4} + 26658\nu^{2} + 4887 ) / 2484 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 89 \nu^{15} - 132 \nu^{14} + 2381 \nu^{13} - 3750 \nu^{12} + 23229 \nu^{11} - 39834 \nu^{10} + \cdots - 35964 ) / 14904 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 44 \nu^{15} - 37 \nu^{14} + 1250 \nu^{13} - 979 \nu^{12} + 13278 \nu^{11} - 9381 \nu^{10} + \cdots - 2403 ) / 4968 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 89 \nu^{15} + 132 \nu^{14} + 2381 \nu^{13} + 3750 \nu^{12} + 23229 \nu^{11} + 39834 \nu^{10} + \cdots + 35964 ) / 14904 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 29 \nu^{15} - 81 \nu^{14} + 758 \nu^{13} - 2229 \nu^{12} + 7086 \nu^{11} - 22659 \nu^{10} + \cdots - 6939 ) / 4968 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 61 \nu^{15} - 77 \nu^{14} + 1711 \nu^{13} - 2015 \nu^{12} + 17991 \nu^{11} - 18924 \nu^{10} + \cdots + 7587 ) / 4968 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 29 \nu^{15} - 81 \nu^{14} - 758 \nu^{13} - 2229 \nu^{12} - 7086 \nu^{11} - 22659 \nu^{10} + \cdots - 6939 ) / 4968 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 73 \nu^{15} - 38 \nu^{14} - 1939 \nu^{13} - 998 \nu^{12} - 18639 \nu^{11} - 9504 \nu^{10} + \cdots - 3861 ) / 4968 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -2\nu^{14} - 53\nu^{12} - 510\nu^{10} - 2303\nu^{8} - 5132\nu^{6} - 5331\nu^{4} - 2142\nu^{2} + 36\nu - 162 ) / 72 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 19 \nu^{15} - 23 \nu^{14} - 522 \nu^{13} - 598 \nu^{12} - 5327 \nu^{11} - 5543 \nu^{10} + \cdots + 2277 ) / 1656 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 77 \nu^{15} - 46 \nu^{14} - 2084 \nu^{13} - 1219 \nu^{12} - 20649 \nu^{11} - 11799 \nu^{10} + \cdots - 3105 ) / 4968 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 2\nu^{15} + 54\nu^{13} + 535\nu^{11} + 2522\nu^{9} + 5991\nu^{7} + 6892\nu^{5} + 3345\nu^{3} + 450\nu + 36 ) / 72 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 2 \beta_{15} + 2 \beta_{14} - \beta_{13} - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \cdots - 5 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{14} + 2 \beta_{13} - 4 \beta_{12} + 2 \beta_{11} - 9 \beta_{10} + 2 \beta_{9} - 9 \beta_{8} + \cdots + 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 38 \beta_{15} - 24 \beta_{14} + 21 \beta_{13} + 15 \beta_{11} + 17 \beta_{10} - 21 \beta_{9} - 17 \beta_{8} + \cdots - 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 22 \beta_{14} - 27 \beta_{13} + 62 \beta_{12} - 40 \beta_{11} + 88 \beta_{10} - 27 \beta_{9} + \cdots - 259 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 524 \beta_{15} + 265 \beta_{14} - 293 \beta_{13} - 182 \beta_{11} - 222 \beta_{10} + 293 \beta_{9} + \cdots + 128 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 234 \beta_{14} + 316 \beta_{13} - 792 \beta_{12} + 546 \beta_{11} - 923 \beta_{10} + 316 \beta_{9} + \cdots + 2713 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 6506 \beta_{15} - 2942 \beta_{14} + 3637 \beta_{13} + 2101 \beta_{11} + 2695 \beta_{10} - 3637 \beta_{9} + \cdots - 1749 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2594 \beta_{14} - 3605 \beta_{13} + 9514 \beta_{12} - 6680 \beta_{11} + 10101 \beta_{10} + \cdots - 29935 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 77366 \beta_{15} + 33081 \beta_{14} - 43200 \beta_{13} - 23991 \beta_{11} - 31763 \beta_{10} + \cdots + 21658 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 29416 \beta_{14} + 41016 \beta_{13} - 111416 \beta_{12} + 78628 \beta_{11} - 113122 \beta_{10} + \cdots + 337525 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 902396 \beta_{15} - 375304 \beta_{14} + 503534 \beta_{13} + 273782 \beta_{11} + 369198 \beta_{10} + \cdots - 257408 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 336384 \beta_{14} - 467572 \beta_{13} + 1289976 \beta_{12} - 911652 \beta_{11} + 1281923 \beta_{10} + \cdots - 3840844 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 10431278 \beta_{15} + 4279454 \beta_{14} - 5818687 \beta_{13} - 3128719 \beta_{11} - 4262047 \beta_{10} + \cdots + 3002670 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/650\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(\beta_{5} + \beta_{7}\) \(\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
3.38585i
− 0.779057i
1.77906i
− 2.38585i
1.28827i
− 1.05387i
− 0.288273i
2.05387i
− 3.38585i
0.779057i
− 1.77906i
2.38585i
− 1.28827i
1.05387i
0.288273i
− 2.05387i
0.500000 + 0.866025i −0.876322 + 3.27048i −0.500000 + 0.866025i 0 −3.27048 + 0.876322i 3.61605 + 2.08773i −1.00000 −7.33001 − 4.23198i 0
193.2 0.500000 + 0.866025i −0.201635 + 0.752511i −0.500000 + 0.866025i 0 −0.752511 + 0.201635i −2.09844 − 1.21153i −1.00000 2.07246 + 1.19654i 0
193.3 0.500000 + 0.866025i 0.460454 − 1.71844i −0.500000 + 0.866025i 0 1.71844 − 0.460454i 0.670111 + 0.386889i −1.00000 −0.142931 − 0.0825211i 0
193.4 0.500000 + 0.866025i 0.617503 − 2.30455i −0.500000 + 0.866025i 0 2.30455 − 0.617503i 2.54433 + 1.46897i −1.00000 −2.33157 − 1.34613i 0
293.1 0.500000 − 0.866025i −1.24438 + 0.333429i −0.500000 − 0.866025i 0 −0.333429 + 1.24438i 2.21606 − 1.27944i −1.00000 −1.16078 + 0.670177i 0
293.2 0.500000 − 0.866025i −1.01796 + 0.272763i −0.500000 − 0.866025i 0 −0.272763 + 1.01796i 0.524968 − 0.303090i −1.00000 −1.63622 + 0.944675i 0
293.3 0.500000 − 0.866025i 0.278450 − 0.0746104i −0.500000 − 0.866025i 0 0.0746104 − 0.278450i −4.15170 + 2.39698i −1.00000 −2.52611 + 1.45845i 0
293.4 0.500000 − 0.866025i 1.98389 − 0.531582i −0.500000 − 0.866025i 0 0.531582 − 1.98389i 2.67861 − 1.54650i −1.00000 1.05516 − 0.609199i 0
357.1 0.500000 − 0.866025i −0.876322 − 3.27048i −0.500000 − 0.866025i 0 −3.27048 − 0.876322i 3.61605 − 2.08773i −1.00000 −7.33001 + 4.23198i 0
357.2 0.500000 − 0.866025i −0.201635 − 0.752511i −0.500000 − 0.866025i 0 −0.752511 − 0.201635i −2.09844 + 1.21153i −1.00000 2.07246 − 1.19654i 0
357.3 0.500000 − 0.866025i 0.460454 + 1.71844i −0.500000 − 0.866025i 0 1.71844 + 0.460454i 0.670111 − 0.386889i −1.00000 −0.142931 + 0.0825211i 0
357.4 0.500000 − 0.866025i 0.617503 + 2.30455i −0.500000 − 0.866025i 0 2.30455 + 0.617503i 2.54433 − 1.46897i −1.00000 −2.33157 + 1.34613i 0
457.1 0.500000 + 0.866025i −1.24438 − 0.333429i −0.500000 + 0.866025i 0 −0.333429 − 1.24438i 2.21606 + 1.27944i −1.00000 −1.16078 − 0.670177i 0
457.2 0.500000 + 0.866025i −1.01796 − 0.272763i −0.500000 + 0.866025i 0 −0.272763 − 1.01796i 0.524968 + 0.303090i −1.00000 −1.63622 − 0.944675i 0
457.3 0.500000 + 0.866025i 0.278450 + 0.0746104i −0.500000 + 0.866025i 0 0.0746104 + 0.278450i −4.15170 − 2.39698i −1.00000 −2.52611 − 1.45845i 0
457.4 0.500000 + 0.866025i 1.98389 + 0.531582i −0.500000 + 0.866025i 0 0.531582 + 1.98389i 2.67861 + 1.54650i −1.00000 1.05516 + 0.609199i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.o even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 650.2.w.h yes 16
5.b even 2 1 650.2.w.f yes 16
5.c odd 4 1 650.2.t.f ✓ 16
5.c odd 4 1 650.2.t.h yes 16
13.f odd 12 1 650.2.t.f ✓ 16
65.o even 12 1 inner 650.2.w.h yes 16
65.s odd 12 1 650.2.t.h yes 16
65.t even 12 1 650.2.w.f yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.2.t.f ✓ 16 5.c odd 4 1
650.2.t.f ✓ 16 13.f odd 12 1
650.2.t.h yes 16 5.c odd 4 1
650.2.t.h yes 16 65.s odd 12 1
650.2.w.f yes 16 5.b even 2 1
650.2.w.f yes 16 65.t even 12 1
650.2.w.h yes 16 1.a even 1 1 trivial
650.2.w.h yes 16 65.o even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 12 T_{3}^{14} - 12 T_{3}^{13} + 23 T_{3}^{12} - 12 T_{3}^{11} - 228 T_{3}^{10} + 48 T_{3}^{9} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(650, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} + 12 T^{14} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - 12 T^{15} + \cdots + 279841 \) Copy content Toggle raw display
$11$ \( T^{16} + 4 T^{15} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{16} + 8 T^{15} + \cdots + 9199089 \) Copy content Toggle raw display
$19$ \( T^{16} - 16 T^{15} + \cdots + 81 \) Copy content Toggle raw display
$23$ \( T^{16} + 4 T^{15} + \cdots + 11881809 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 7001170929 \) Copy content Toggle raw display
$31$ \( T^{16} + 8 T^{15} + \cdots + 20736 \) Copy content Toggle raw display
$37$ \( T^{16} + 36 T^{15} + \cdots + 187489 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 5749430625 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 45270647361 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 4512867171201 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 178024081041 \) Copy content Toggle raw display
$59$ \( T^{16} + 24 T^{15} + \cdots + 3470769 \) Copy content Toggle raw display
$61$ \( T^{16} + 20 T^{15} + \cdots + 42849 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 36051756129 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 10\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( (T^{8} - 304 T^{6} + \cdots + 90000)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 211086032481 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 175707680625 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 709755295760289 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 45\!\cdots\!89 \) Copy content Toggle raw display
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