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Properties

Label 990.2.f.c
Level $990$
Weight $2$
Character orbit 990.f
Analytic conductor $7.905$
Analytic rank $0$
Dimension $16$
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [990,2,Mod(989,990)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(990, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("990.989"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 990 = 2 \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 990.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,-16,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.90518980011\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 38x^{14} + 517x^{12} + 3164x^{10} + 9372x^{8} + 14032x^{6} + 10320x^{4} + 3200x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

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

\(f(q)\) \(=\) \( q + \beta_{8} q^{2} - q^{4} + \beta_{10} q^{5} - \beta_{4} q^{7} - \beta_{8} q^{8} + \beta_{2} q^{10} + (\beta_{14} + \beta_{3}) q^{11} + (\beta_{4} - \beta_{2} - \beta_1) q^{13} + \beta_{3} q^{14} + q^{16}+ \cdots - 5 \beta_{8} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} + 16 q^{16} + 8 q^{25} - 32 q^{31} - 32 q^{34} - 80 q^{49} + 24 q^{55} - 16 q^{64} - 8 q^{70} - 16 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 38x^{14} + 517x^{12} + 3164x^{10} + 9372x^{8} + 14032x^{6} + 10320x^{4} + 3200x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 257 \nu^{15} + 142 \nu^{14} + 8898 \nu^{13} + 5228 \nu^{12} + 101197 \nu^{11} + 68022 \nu^{10} + \cdots + 371328 ) / 140800 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 257 \nu^{15} + 142 \nu^{14} - 8898 \nu^{13} + 5228 \nu^{12} - 101197 \nu^{11} + 68022 \nu^{10} + \cdots + 371328 ) / 140800 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 61 \nu^{15} + 2314 \nu^{13} + 31241 \nu^{11} + 185880 \nu^{9} + 499372 \nu^{7} + 542144 \nu^{5} + \cdots - 72256 \nu ) / 19200 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 683 \nu^{14} - 25022 \nu^{12} - 318903 \nu^{10} - 1723680 \nu^{8} - 4024796 \nu^{6} + \cdots - 174272 ) / 35200 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 313 \nu^{15} + 754 \nu^{14} - 11062 \nu^{13} + 27636 \nu^{12} - 131053 \nu^{11} + 352914 \nu^{10} + \cdots + 359936 ) / 70400 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 467 \nu^{15} + 247 \nu^{14} + 17893 \nu^{13} + 8598 \nu^{12} + 246487 \nu^{11} + 99527 \nu^{10} + \cdots + 147648 ) / 52800 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 467 \nu^{15} - 247 \nu^{14} + 17893 \nu^{13} - 8598 \nu^{12} + 246487 \nu^{11} - 99527 \nu^{10} + \cdots - 147648 ) / 52800 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 729 \nu^{15} + 26710 \nu^{13} + 340533 \nu^{11} + 1842572 \nu^{9} + 4315252 \nu^{7} + \cdots - 6592 \nu ) / 42240 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 934 \nu^{15} + 5635 \nu^{14} - 35786 \nu^{13} + 207990 \nu^{12} - 492974 \nu^{11} + \cdots + 1688640 ) / 105600 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 12703 \nu^{15} + 13282 \nu^{14} + 470102 \nu^{13} + 486548 \nu^{12} + 6102323 \nu^{11} + \cdots + 1629568 ) / 422400 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2173 \nu^{15} - 2128 \nu^{14} - 79842 \nu^{13} - 77392 \nu^{12} - 1023353 \nu^{11} - 973488 \nu^{10} + \cdots - 99072 ) / 70400 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2025 \nu^{15} + 2128 \nu^{14} - 73070 \nu^{13} + 77392 \nu^{12} - 905445 \nu^{11} + 973488 \nu^{10} + \cdots + 28672 ) / 70400 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 12703 \nu^{15} - 13282 \nu^{14} + 470102 \nu^{13} - 486548 \nu^{12} + 6102323 \nu^{11} + \cdots - 1629568 ) / 422400 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 3008 \nu^{15} + 4441 \nu^{14} - 111162 \nu^{13} + 164294 \nu^{12} - 1439668 \nu^{11} + \cdots + 1318144 ) / 105600 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2173 \nu^{15} + 2128 \nu^{14} - 79842 \nu^{13} + 77392 \nu^{12} - 1023353 \nu^{11} + 973488 \nu^{10} + \cdots + 28672 ) / 70400 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{15} - 2\beta_{13} - \beta_{11} - 2\beta_{10} + 2\beta_{8} + \beta_{7} + \beta_{6} + 2\beta_{3} - 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} - 4 \beta_{14} + 2 \beta_{13} - \beta_{11} - 6 \beta_{10} + 4 \beta_{9} + 3 \beta_{7} + \cdots - 19 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 13 \beta_{15} + 20 \beta_{13} + 13 \beta_{11} + 20 \beta_{10} - 6 \beta_{8} - 11 \beta_{7} - 11 \beta_{6} + \cdots + 13 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 19 \beta_{15} + 68 \beta_{14} - 22 \beta_{13} + 19 \beta_{11} + 90 \beta_{10} - 56 \beta_{9} + \cdots + 205 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 161 \beta_{15} - 252 \beta_{13} - 40 \beta_{12} - 201 \beta_{11} - 252 \beta_{10} - 90 \beta_{8} + \cdots - 201 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 335 \beta_{15} - 996 \beta_{14} + 342 \beta_{13} - 335 \beta_{11} - 1338 \beta_{10} + 752 \beta_{9} + \cdots - 2713 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2129 \beta_{15} + 3484 \beta_{13} + 1064 \beta_{12} + 3193 \beta_{11} + 3484 \beta_{10} + 2954 \beta_{8} + \cdots + 3193 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 5815 \beta_{15} + 14404 \beta_{14} - 5918 \beta_{13} + 5815 \beta_{11} + 20322 \beta_{10} + \cdots + 38481 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 29825 \beta_{15} - 50420 \beta_{13} - 21144 \beta_{12} - 50969 \beta_{11} - 50420 \beta_{10} + \cdots - 50969 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 98559 \beta_{15} - 210660 \beta_{14} + 102246 \beta_{13} - 98559 \beta_{11} - 312906 \beta_{10} + \cdots - 565289 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 433985 \beta_{15} + 749564 \beta_{13} + 378312 \beta_{12} + 812297 \beta_{11} + 749564 \beta_{10} + \cdots + 812297 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 1635095 \beta_{15} + 3128580 \beta_{14} - 1727310 \beta_{13} + 1635095 \beta_{11} + 4855890 \beta_{10} + \cdots + 8484625 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 6471553 \beta_{15} - 11338884 \beta_{13} - 6436664 \beta_{12} - 12908217 \beta_{11} - 11338884 \beta_{10} + \cdots - 12908217 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 26693743 \beta_{15} - 47127204 \beta_{14} + 28584630 \beta_{13} - 26693743 \beta_{11} - 75711834 \beta_{10} + \cdots - 129180665 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 98090145 \beta_{15} + 173565900 \beta_{13} + 106495016 \beta_{12} + 204585161 \beta_{11} + \cdots + 204585161 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/990\mathbb{Z}\right)^\times\).

\(n\) \(397\) \(541\) \(551\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
989.1
1.21715i
0.344399i
0.750561i
− 3.31211i
− 1.10374i
− 1.45781i
3.96698i
− 2.40543i
− 0.344399i
− 1.21715i
3.31211i
− 0.750561i
1.45781i
1.10374i
2.40543i
− 3.96698i
− 1.00000i 0 −1.00000 −1.94442 − 1.10418i 0 −1.41421 1.00000i 0 −1.10418 + 1.94442i
989.2 − 1.00000i 0 −1.00000 −1.94442 + 1.10418i 0 1.41421 1.00000i 0 1.10418 + 1.94442i
989.3 − 1.00000i 0 −1.00000 −1.31119 − 1.81129i 0 1.41421 1.00000i 0 −1.81129 + 1.31119i
989.4 − 1.00000i 0 −1.00000 −1.31119 + 1.81129i 0 −1.41421 1.00000i 0 1.81129 + 1.31119i
989.5 − 1.00000i 0 −1.00000 1.31119 − 1.81129i 0 1.41421 1.00000i 0 −1.81129 − 1.31119i
989.6 − 1.00000i 0 −1.00000 1.31119 + 1.81129i 0 −1.41421 1.00000i 0 1.81129 − 1.31119i
989.7 − 1.00000i 0 −1.00000 1.94442 − 1.10418i 0 −1.41421 1.00000i 0 −1.10418 − 1.94442i
989.8 − 1.00000i 0 −1.00000 1.94442 + 1.10418i 0 1.41421 1.00000i 0 1.10418 − 1.94442i
989.9 1.00000i 0 −1.00000 −1.94442 − 1.10418i 0 1.41421 − 1.00000i 0 1.10418 − 1.94442i
989.10 1.00000i 0 −1.00000 −1.94442 + 1.10418i 0 −1.41421 − 1.00000i 0 −1.10418 − 1.94442i
989.11 1.00000i 0 −1.00000 −1.31119 − 1.81129i 0 −1.41421 − 1.00000i 0 1.81129 − 1.31119i
989.12 1.00000i 0 −1.00000 −1.31119 + 1.81129i 0 1.41421 − 1.00000i 0 −1.81129 − 1.31119i
989.13 1.00000i 0 −1.00000 1.31119 − 1.81129i 0 −1.41421 − 1.00000i 0 1.81129 + 1.31119i
989.14 1.00000i 0 −1.00000 1.31119 + 1.81129i 0 1.41421 − 1.00000i 0 −1.81129 + 1.31119i
989.15 1.00000i 0 −1.00000 1.94442 − 1.10418i 0 1.41421 − 1.00000i 0 1.10418 + 1.94442i
989.16 1.00000i 0 −1.00000 1.94442 + 1.10418i 0 −1.41421 − 1.00000i 0 −1.10418 + 1.94442i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 989.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
11.b odd 2 1 inner
15.d odd 2 1 inner
33.d even 2 1 inner
55.d odd 2 1 inner
165.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 990.2.f.c ✓ 16
3.b odd 2 1 inner 990.2.f.c ✓ 16
5.b even 2 1 inner 990.2.f.c ✓ 16
5.c odd 4 1 4950.2.d.g 8
5.c odd 4 1 4950.2.d.n 8
11.b odd 2 1 inner 990.2.f.c ✓ 16
15.d odd 2 1 inner 990.2.f.c ✓ 16
15.e even 4 1 4950.2.d.g 8
15.e even 4 1 4950.2.d.n 8
33.d even 2 1 inner 990.2.f.c ✓ 16
55.d odd 2 1 inner 990.2.f.c ✓ 16
55.e even 4 1 4950.2.d.g 8
55.e even 4 1 4950.2.d.n 8
165.d even 2 1 inner 990.2.f.c ✓ 16
165.l odd 4 1 4950.2.d.g 8
165.l odd 4 1 4950.2.d.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
990.2.f.c ✓ 16 1.a even 1 1 trivial
990.2.f.c ✓ 16 3.b odd 2 1 inner
990.2.f.c ✓ 16 5.b even 2 1 inner
990.2.f.c ✓ 16 11.b odd 2 1 inner
990.2.f.c ✓ 16 15.d odd 2 1 inner
990.2.f.c ✓ 16 33.d even 2 1 inner
990.2.f.c ✓ 16 55.d odd 2 1 inner
990.2.f.c ✓ 16 165.d even 2 1 inner
4950.2.d.g 8 5.c odd 4 1
4950.2.d.g 8 15.e even 4 1
4950.2.d.g 8 55.e even 4 1
4950.2.d.g 8 165.l odd 4 1
4950.2.d.n 8 5.c odd 4 1
4950.2.d.n 8 15.e even 4 1
4950.2.d.n 8 55.e even 4 1
4950.2.d.n 8 165.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(990, [\chi])\):

\( T_{7}^{2} - 2 \) Copy content Toggle raw display
\( T_{29}^{4} - 44T_{29}^{2} + 416 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 2 T^{6} + \cdots + 625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 2)^{8} \) Copy content Toggle raw display
$11$ \( (T^{8} - 2 T^{6} + \cdots + 14641)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 18 T^{2} + 64)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 4)^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} + 82 T^{2} + 1664)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 88 T^{2} + 1664)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} - 44 T^{2} + 416)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 4 T - 64)^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} + 46 T^{2} + 104)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 158 T^{2} + 104)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 154 T^{2} + 5776)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 116 T^{2} + 1664)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} - 82 T^{2} + 1664)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + 26 T^{2} + 16)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 116 T^{2} + 1664)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 184 T^{2} + 1664)^{4} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( (T^{4} - 264 T^{2} + 4096)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 82 T^{2} + 1664)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 356 T^{2} + 16384)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 132 T^{2} + 4)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 176 T^{2} + 6656)^{4} \) Copy content Toggle raw display
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