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Properties

Label 990.4.c.i
Level $990$
Weight $4$
Character orbit 990.c
Analytic conductor $58.412$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [990,4,Mod(199,990)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(990, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("990.199");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 990 = 2 \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 990.c (of order \(2\), degree \(1\), 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: \(58.4118909057\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 151x^{6} + 7935x^{4} + 171721x^{2} + 1308736 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 110)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - 4 q^{4} + (\beta_{6} + \beta_{2} - 2) q^{5} + ( - \beta_{6} - \beta_{5} + \cdots + \beta_1) q^{7} - 4 \beta_{2} q^{8} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \cdots - 4) q^{10}+ \cdots + (28 \beta_{7} + 50 \beta_{6} + \cdots + 14) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 32 q^{4} - 16 q^{5} - 28 q^{10} + 88 q^{11} - 44 q^{14} + 128 q^{16} - 302 q^{19} + 64 q^{20} - 162 q^{25} - 256 q^{26} + 58 q^{29} + 1022 q^{31} - 660 q^{34} + 1058 q^{35} + 112 q^{40} - 452 q^{41}+ \cdots - 1268 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 151x^{6} + 7935x^{4} + 171721x^{2} + 1308736 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8\nu^{7} + 1065\nu^{5} + 43460\nu^{3} + 522203\nu ) / 31460 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 140\nu^{4} + 5955\nu^{2} + 72996 ) / 220 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 140\nu^{4} + 6175\nu^{2} + 81356 ) / 220 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 80 \nu^{7} + 1001 \nu^{6} - 10650 \nu^{5} + 124410 \nu^{4} - 450330 \nu^{3} + 4686825 \nu^{2} + \cdots + 52336856 ) / 188760 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 80 \nu^{7} - 1001 \nu^{6} - 10650 \nu^{5} - 124410 \nu^{4} - 450330 \nu^{3} - 4686825 \nu^{2} + \cdots - 52336856 ) / 188760 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 323 \nu^{7} - 429 \nu^{6} - 40050 \nu^{5} - 60060 \nu^{4} - 1499085 \nu^{3} - 2554695 \nu^{2} + \cdots - 31409664 ) / 188760 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} - 38 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -6\beta_{6} - 6\beta_{5} - 20\beta_{2} - 45\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{6} - 6\beta_{5} - 81\beta_{4} + 95\beta_{3} + 1760 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 64\beta_{7} + 520\beta_{6} + 520\beta_{5} + 32\beta_{3} + 2164\beta_{2} + 2329\beta _1 + 32 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -840\beta_{6} + 840\beta_{5} + 5385\beta_{4} - 7125\beta_{3} - 93106 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -8520\beta_{7} - 36630\beta_{6} - 36630\beta_{5} - 4260\beta_{3} - 175500\beta_{2} - 130861\beta _1 - 4260 \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
6.63700i
− 4.48251i
− 7.97576i
4.82127i
− 6.63700i
4.48251i
7.97576i
− 4.82127i
− 2.00000i 0 −4.00000 −11.1511 + 0.807757i 0 − 0.978517i 8.00000i 0 1.61551 + 22.3022i
199.2 − 2.00000i 0 −4.00000 −3.44238 + 10.6372i 0 − 31.7569i 8.00000i 0 21.2744 + 6.88475i
199.3 − 2.00000i 0 −4.00000 −1.78092 − 11.0376i 0 8.09941i 8.00000i 0 −22.0752 + 3.56184i
199.4 − 2.00000i 0 −4.00000 8.37442 − 7.40737i 0 13.6360i 8.00000i 0 −14.8147 − 16.7488i
199.5 2.00000i 0 −4.00000 −11.1511 − 0.807757i 0 0.978517i − 8.00000i 0 1.61551 − 22.3022i
199.6 2.00000i 0 −4.00000 −3.44238 − 10.6372i 0 31.7569i − 8.00000i 0 21.2744 − 6.88475i
199.7 2.00000i 0 −4.00000 −1.78092 + 11.0376i 0 − 8.09941i − 8.00000i 0 −22.0752 − 3.56184i
199.8 2.00000i 0 −4.00000 8.37442 + 7.40737i 0 − 13.6360i − 8.00000i 0 −14.8147 + 16.7488i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 990.4.c.i 8
3.b odd 2 1 110.4.b.c ✓ 8
5.b even 2 1 inner 990.4.c.i 8
12.b even 2 1 880.4.b.h 8
15.d odd 2 1 110.4.b.c ✓ 8
15.e even 4 1 550.4.a.ba 4
15.e even 4 1 550.4.a.bb 4
60.h even 2 1 880.4.b.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.4.b.c ✓ 8 3.b odd 2 1
110.4.b.c ✓ 8 15.d odd 2 1
550.4.a.ba 4 15.e even 4 1
550.4.a.bb 4 15.e even 4 1
880.4.b.h 8 12.b even 2 1
880.4.b.h 8 60.h even 2 1
990.4.c.i 8 1.a even 1 1 trivial
990.4.c.i 8 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(990, [\chi])\):

\( T_{7}^{8} + 1261T_{7}^{6} + 267084T_{7}^{4} + 12556080T_{7}^{2} + 11778624 \) Copy content Toggle raw display
\( T_{29}^{4} - 29T_{29}^{3} - 17130T_{29}^{2} - 58688T_{29} + 15788512 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 16 T^{7} + \cdots + 244140625 \) Copy content Toggle raw display
$7$ \( T^{8} + 1261 T^{6} + \cdots + 11778624 \) Copy content Toggle raw display
$11$ \( (T - 11)^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 5498837401600 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 136200103430400 \) Copy content Toggle raw display
$19$ \( (T^{4} + 151 T^{3} + \cdots - 390272)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 73\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( (T^{4} - 29 T^{3} + \cdots + 15788512)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 511 T^{3} + \cdots - 256907000)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 49077590691600 \) Copy content Toggle raw display
$41$ \( (T^{4} + 226 T^{3} + \cdots + 3452698624)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( (T^{4} + 134 T^{3} + \cdots + 1066184592)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 245 T^{3} + \cdots + 22378892000)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 80\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( (T^{4} + 1017 T^{3} + \cdots - 50573201976)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{4} - 136 T^{3} + \cdots + 145164269184)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 1272726658650)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 77\!\cdots\!96 \) Copy content Toggle raw display
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