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Properties

Label 1690.2.e.u
Level $1690$
Weight $2$
Character orbit 1690.e
Analytic conductor $13.495$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1690,2,Mod(191,1690)] 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("1690.191"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1690, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1690 = 2 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1690.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,-6,2,-6,12,2,-3] 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: no
Analytic conductor: \(13.4947179416\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
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: \( x^{12} - 2 x^{11} + 19 x^{10} - 14 x^{9} + 199 x^{8} - 106 x^{7} + 1240 x^{6} + 42 x^{5} + 4767 x^{4} + \cdots + 8281 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{8} - 1) q^{2} + ( - \beta_{5} - \beta_1) q^{3} + \beta_{8} q^{4} + q^{5} + \beta_1 q^{6} + (\beta_{8} + \beta_{6} - 2 \beta_{2}) q^{7} + q^{8} + ( - \beta_{10} + 2 \beta_{8} + \cdots - \beta_1) q^{9}+ \cdots + ( - \beta_{11} - \beta_{9} - 5 \beta_{7} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{2} + 2 q^{3} - 6 q^{4} + 12 q^{5} + 2 q^{6} - 3 q^{7} + 12 q^{8} - 16 q^{9} - 6 q^{10} - 15 q^{11} - 4 q^{12} + 6 q^{14} + 2 q^{15} - 6 q^{16} + 3 q^{17} + 32 q^{18} - q^{19} - 6 q^{20} - 4 q^{21}+ \cdots + 110 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 19 x^{10} - 14 x^{9} + 199 x^{8} - 106 x^{7} + 1240 x^{6} + 42 x^{5} + 4767 x^{4} + \cdots + 8281 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 181315362 \nu^{11} - 72015751 \nu^{10} - 6549692070 \nu^{9} + 23053269170 \nu^{8} + \cdots + 4532615817184 ) / 8593381770428 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 226710165 \nu^{11} + 1013859543 \nu^{10} - 529474083 \nu^{9} + 22879467934 \nu^{8} + \cdots + 3538034344754 ) / 8593381770428 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 989620 \nu^{11} - 16733693 \nu^{10} + 31104657 \nu^{9} - 189723443 \nu^{8} + \cdots - 14774681194 ) / 30581429788 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 189386578 \nu^{11} + 32854886 \nu^{10} + 2709665393 \nu^{9} + 3957050916 \nu^{8} + \cdots + 2073076984342 ) / 2148345442607 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 785685821 \nu^{11} + 6789841047 \nu^{10} - 18619653005 \nu^{9} + 178826124385 \nu^{8} + \cdots + 29982621762548 ) / 8593381770428 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 222241464 \nu^{11} - 921534475 \nu^{10} + 3898797615 \nu^{9} - 7305537785 \nu^{8} + \cdots + 7100339976275 ) / 2148345442607 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 250341382 \nu^{11} - 690069342 \nu^{10} + 4723631372 \nu^{9} - 6214444741 \nu^{8} + \cdots - 1697652781370 ) / 2148345442607 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5508710 \nu^{11} - 2461033 \nu^{10} + 81375286 \nu^{9} + 72575867 \nu^{8} + 953512894 \nu^{7} + \cdots + 46329171161 ) / 30581429788 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3405710275 \nu^{11} - 9304824692 \nu^{10} + 60959609255 \nu^{9} - 64962210240 \nu^{8} + \cdots - 19609164455790 ) / 8593381770428 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 4112782583 \nu^{11} + 12275935050 \nu^{10} - 99697380855 \nu^{9} + 127985856157 \nu^{8} + \cdots - 19199626970160 ) / 8593381770428 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - 5\beta_{8} + \beta_{7} + \beta_{5} - \beta_{3} - \beta_{2} + \beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{9} + \beta_{7} + 7\beta_{5} - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -11\beta_{10} + 38\beta_{8} + 3\beta_{3} + 7\beta_{2} - 12\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4 \beta_{11} - 16 \beta_{10} + 52 \beta_{9} + 57 \beta_{8} - 16 \beta_{7} - 57 \beta_{5} + 4 \beta_{4} + \cdots + 57 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{11} + 136\beta_{9} - 105\beta_{7} - 8\beta_{6} - 126\beta_{5} + 68\beta_{4} + 8\beta_{2} + 337 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 194\beta_{10} - 693\beta_{8} - 76\beta_{6} + 402\beta_{3} - 10\beta_{2} + 500\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 184 \beta_{11} + 988 \beta_{10} - 1752 \beta_{9} - 3206 \beta_{8} + 988 \beta_{7} + 1279 \beta_{5} + \cdots - 3206 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1052 \beta_{11} - 6624 \beta_{9} + 2163 \beta_{7} + 1052 \beta_{6} + 4605 \beta_{5} - 1804 \beta_{4} + \cdots - 7895 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -9425\beta_{10} + 31694\beta_{8} + 2856\beta_{6} - 11111\beta_{3} - 3511\beta_{2} - 12859\beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 12936 \beta_{11} - 23315 \beta_{10} + 72516 \beta_{9} + 86830 \beta_{8} - 23315 \beta_{7} - 43898 \beta_{5} + \cdots + 86830 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1690\mathbb{Z}\right)^\times\).

\(n\) \(171\) \(677\)
\(\chi(n)\) \(\beta_{8}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
−1.28627 − 2.22789i
−1.11864 − 1.93753i
−0.494225 − 0.856023i
0.885303 + 1.53339i
1.39612 + 2.41814i
1.61772 + 2.80196i
−1.28627 + 2.22789i
−1.11864 + 1.93753i
−0.494225 + 0.856023i
0.885303 − 1.53339i
1.39612 − 2.41814i
1.61772 − 2.80196i
−0.500000 + 0.866025i −1.28627 + 2.22789i −0.500000 − 0.866025i 1.00000 −1.28627 − 2.22789i −0.952014 − 1.64894i 1.00000 −1.80899 − 3.13326i −0.500000 + 0.866025i
191.2 −0.500000 + 0.866025i −1.11864 + 1.93753i −0.500000 − 0.866025i 1.00000 −1.11864 − 1.93753i −0.716469 − 1.24096i 1.00000 −1.00269 − 1.73672i −0.500000 + 0.866025i
191.3 −0.500000 + 0.866025i −0.494225 + 0.856023i −0.500000 − 0.866025i 1.00000 −0.494225 − 0.856023i 1.73657 + 3.00783i 1.00000 1.01148 + 1.75194i −0.500000 + 0.866025i
191.4 −0.500000 + 0.866025i 0.885303 − 1.53339i −0.500000 − 0.866025i 1.00000 0.885303 + 1.53339i −1.91846 − 3.32286i 1.00000 −0.0675215 − 0.116951i −0.500000 + 0.866025i
191.5 −0.500000 + 0.866025i 1.39612 − 2.41814i −0.500000 − 0.866025i 1.00000 1.39612 + 2.41814i 2.41938 + 4.19048i 1.00000 −2.39828 − 4.15393i −0.500000 + 0.866025i
191.6 −0.500000 + 0.866025i 1.61772 − 2.80196i −0.500000 − 0.866025i 1.00000 1.61772 + 2.80196i −2.06901 − 3.58363i 1.00000 −3.73400 − 6.46748i −0.500000 + 0.866025i
991.1 −0.500000 − 0.866025i −1.28627 − 2.22789i −0.500000 + 0.866025i 1.00000 −1.28627 + 2.22789i −0.952014 + 1.64894i 1.00000 −1.80899 + 3.13326i −0.500000 − 0.866025i
991.2 −0.500000 − 0.866025i −1.11864 − 1.93753i −0.500000 + 0.866025i 1.00000 −1.11864 + 1.93753i −0.716469 + 1.24096i 1.00000 −1.00269 + 1.73672i −0.500000 − 0.866025i
991.3 −0.500000 − 0.866025i −0.494225 − 0.856023i −0.500000 + 0.866025i 1.00000 −0.494225 + 0.856023i 1.73657 − 3.00783i 1.00000 1.01148 − 1.75194i −0.500000 − 0.866025i
991.4 −0.500000 − 0.866025i 0.885303 + 1.53339i −0.500000 + 0.866025i 1.00000 0.885303 − 1.53339i −1.91846 + 3.32286i 1.00000 −0.0675215 + 0.116951i −0.500000 − 0.866025i
991.5 −0.500000 − 0.866025i 1.39612 + 2.41814i −0.500000 + 0.866025i 1.00000 1.39612 − 2.41814i 2.41938 − 4.19048i 1.00000 −2.39828 + 4.15393i −0.500000 − 0.866025i
991.6 −0.500000 − 0.866025i 1.61772 + 2.80196i −0.500000 + 0.866025i 1.00000 1.61772 − 2.80196i −2.06901 + 3.58363i 1.00000 −3.73400 + 6.46748i −0.500000 − 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1690.2.e.u 12
13.b even 2 1 1690.2.e.v 12
13.c even 3 1 1690.2.a.w yes 6
13.c even 3 1 inner 1690.2.e.u 12
13.d odd 4 2 1690.2.l.n 24
13.e even 6 1 1690.2.a.v ✓ 6
13.e even 6 1 1690.2.e.v 12
13.f odd 12 2 1690.2.d.l 12
13.f odd 12 2 1690.2.l.n 24
65.l even 6 1 8450.2.a.cq 6
65.n even 6 1 8450.2.a.cp 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1690.2.a.v ✓ 6 13.e even 6 1
1690.2.a.w yes 6 13.c even 3 1
1690.2.d.l 12 13.f odd 12 2
1690.2.e.u 12 1.a even 1 1 trivial
1690.2.e.u 12 13.c even 3 1 inner
1690.2.e.v 12 13.b even 2 1
1690.2.e.v 12 13.e even 6 1
1690.2.l.n 24 13.d odd 4 2
1690.2.l.n 24 13.f odd 12 2
8450.2.a.cp 6 65.n even 6 1
8450.2.a.cq 6 65.l even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1690, [\chi])\):

\( T_{3}^{12} - 2 T_{3}^{11} + 19 T_{3}^{10} - 14 T_{3}^{9} + 199 T_{3}^{8} - 106 T_{3}^{7} + 1240 T_{3}^{6} + \cdots + 8281 \) Copy content Toggle raw display
\( T_{7}^{12} + 3 T_{7}^{11} + 41 T_{7}^{10} + 126 T_{7}^{9} + 1175 T_{7}^{8} + 3356 T_{7}^{7} + \cdots + 529984 \) Copy content Toggle raw display
\( T_{11}^{12} + 15 T_{11}^{11} + 171 T_{11}^{10} + 1176 T_{11}^{9} + 7201 T_{11}^{8} + 33546 T_{11}^{7} + \cdots + 529984 \) Copy content Toggle raw display
\( T_{19}^{12} + T_{19}^{11} + 77 T_{19}^{10} + 230 T_{19}^{9} + 4745 T_{19}^{8} + 11368 T_{19}^{7} + \cdots + 9096256 \) Copy content Toggle raw display
\( T_{31}^{6} - 104T_{31}^{4} + 2000T_{31}^{2} - 896T_{31} - 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} - 2 T^{11} + \cdots + 8281 \) Copy content Toggle raw display
$5$ \( (T - 1)^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 3 T^{11} + \cdots + 529984 \) Copy content Toggle raw display
$11$ \( T^{12} + 15 T^{11} + \cdots + 529984 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( T^{12} - 3 T^{11} + \cdots + 4032064 \) Copy content Toggle raw display
$19$ \( T^{12} + T^{11} + \cdots + 9096256 \) Copy content Toggle raw display
$23$ \( T^{12} - 3 T^{11} + \cdots + 1827904 \) Copy content Toggle raw display
$29$ \( T^{12} + 7 T^{11} + \cdots + 54523456 \) Copy content Toggle raw display
$31$ \( (T^{6} - 104 T^{4} + \cdots - 64)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 220463104 \) Copy content Toggle raw display
$41$ \( T^{12} + 2 T^{11} + \cdots + 8281 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 222099409 \) Copy content Toggle raw display
$47$ \( (T^{6} - 7 T^{5} + \cdots + 36392)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 16 T^{5} + \cdots - 832)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 354342976 \) Copy content Toggle raw display
$61$ \( T^{12} + 33 T^{11} + \cdots + 45265984 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 130210921 \) Copy content Toggle raw display
$71$ \( (T^{6} + 20 T^{5} + \cdots + 64)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 21 T^{5} + 148 T^{4} + \cdots - 8)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 20 T^{5} + \cdots - 77888)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 22 T^{5} + \cdots - 186641)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + 20 T^{11} + \cdots + 12752041 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 97608755776 \) Copy content Toggle raw display
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