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Properties

Label 1100.3.f.d
Level $1100$
Weight $3$
Character orbit 1100.f
Analytic conductor $29.973$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1100,3,Mod(901,1100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1100.901");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1100.f (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: \(29.9728290796\)
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: \( x^{8} - 4x^{7} + 80x^{6} - 226x^{5} + 1925x^{4} - 3478x^{3} + 11768x^{2} - 10066x + 67277 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 220)
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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + \beta_{4} q^{7} + ( - \beta_1 - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{3} + \beta_{4} q^{7} + ( - \beta_1 - 3) q^{9} + (\beta_{2} + \beta_1 - 5) q^{11} - \beta_{7} q^{13} + \beta_{4} q^{17} + (\beta_{3} - \beta_{2}) q^{19} + ( - \beta_{3} - \beta_{2}) q^{21} + ( - 2 \beta_{6} - 4 \beta_{5}) q^{23} + ( - \beta_{6} - 10 \beta_{5}) q^{27} + ( - \beta_{3} - 3 \beta_{2}) q^{29} + (5 \beta_1 - 4) q^{31} + (\beta_{7} + \beta_{6} + \cdots - \beta_{4}) q^{33}+ \cdots + (\beta_{3} - 4 \beta_{2} + 3 \beta_1 + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 20 q^{9} - 44 q^{11} - 52 q^{31} - 284 q^{49} - 384 q^{59} - 200 q^{69} - 60 q^{71} - 336 q^{81} - 380 q^{89} - 32 q^{91} + 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 80x^{6} - 226x^{5} + 1925x^{4} - 3478x^{3} + 11768x^{2} - 10066x + 67277 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 5\nu^{6} - 15\nu^{5} + 287\nu^{4} - 549\nu^{3} + 1761\nu^{2} - 1489\nu - 49826 ) / 16898 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 25\nu^{6} - 75\nu^{5} + 1435\nu^{4} - 2745\nu^{3} + 25703\nu^{2} - 24343\nu + 105728 ) / 16898 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 43\nu^{6} - 129\nu^{5} + 4158\nu^{4} - 8101\nu^{3} + 104704\nu^{2} - 100675\nu + 394429 ) / 16898 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 7296 \nu^{7} + 25536 \nu^{6} - 872462 \nu^{5} + 2117315 \nu^{4} - 33738080 \nu^{3} + \cdots + 184346659 ) / 68859350 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11026 \nu^{7} + 38591 \nu^{6} - 771097 \nu^{5} + 1831265 \nu^{4} - 15046555 \nu^{3} + \cdots + 5712154 ) / 68859350 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11238 \nu^{7} - 39333 \nu^{6} + 1060711 \nu^{5} - 2553445 \nu^{4} + 23879265 \nu^{3} + \cdots - 5085052 ) / 68859350 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11026 \nu^{7} - 38591 \nu^{6} + 771097 \nu^{5} - 1831265 \nu^{4} + 15046555 \nu^{3} + \cdots - 74571504 ) / 34429675 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + 2\beta_{5} + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 2\beta_{5} + 4\beta_{2} - 20\beta _1 - 82 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -28\beta_{7} - 29\beta_{6} - 71\beta_{5} - 22\beta_{4} + 6\beta_{2} - 30\beta _1 - 124 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -57\beta_{7} - 58\beta_{6} - 144\beta_{5} - 44\beta_{4} + 40\beta_{3} - 200\beta_{2} + 656\beta _1 + 2254 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 763\beta_{7} + 1759\beta_{6} + 2939\beta_{5} + 574\beta_{4} + 100\beta_{3} - 510\beta_{2} + 1690\beta _1 + 5842 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1216 \beta_{7} + 2711 \beta_{6} + 4589 \beta_{5} + 916 \beta_{4} - 998 \beta_{3} + 4600 \beta_{2} + \cdots - 28066 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 15875 \beta_{7} - 74096 \beta_{6} - 124962 \beta_{5} - 11016 \beta_{4} - 7336 \beta_{3} + \cdots - 217054 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(551\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
901.1
−1.05737 + 2.16046i
−1.05737 − 2.16046i
−0.408080 − 5.92304i
−0.408080 + 5.92304i
1.40808 − 5.92304i
1.40808 + 5.92304i
2.05737 + 2.16046i
2.05737 − 2.16046i
0 −3.11473 0 0 0 − 12.3180i 0 0.701562 0
901.2 0 −3.11473 0 0 0 12.3180i 0 0.701562 0
901.3 0 −1.81616 0 0 0 − 4.15538i 0 −5.70156 0
901.4 0 −1.81616 0 0 0 4.15538i 0 −5.70156 0
901.5 0 1.81616 0 0 0 − 4.15538i 0 −5.70156 0
901.6 0 1.81616 0 0 0 4.15538i 0 −5.70156 0
901.7 0 3.11473 0 0 0 − 12.3180i 0 0.701562 0
901.8 0 3.11473 0 0 0 12.3180i 0 0.701562 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 901.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1100.3.f.d 8
5.b even 2 1 inner 1100.3.f.d 8
5.c odd 4 2 220.3.e.b ✓ 8
11.b odd 2 1 inner 1100.3.f.d 8
15.e even 4 2 1980.3.p.b 8
20.e even 4 2 880.3.i.h 8
55.d odd 2 1 inner 1100.3.f.d 8
55.e even 4 2 220.3.e.b ✓ 8
165.l odd 4 2 1980.3.p.b 8
220.i odd 4 2 880.3.i.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.3.e.b ✓ 8 5.c odd 4 2
220.3.e.b ✓ 8 55.e even 4 2
880.3.i.h 8 20.e even 4 2
880.3.i.h 8 220.i odd 4 2
1100.3.f.d 8 1.a even 1 1 trivial
1100.3.f.d 8 5.b even 2 1 inner
1100.3.f.d 8 11.b odd 2 1 inner
1100.3.f.d 8 55.d odd 2 1 inner
1980.3.p.b 8 15.e even 4 2
1980.3.p.b 8 165.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 13T_{3}^{2} + 32 \) acting on \(S_{3}^{\mathrm{new}}(1100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 13 T^{2} + 32)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 169 T^{2} + 2620)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 22 T^{3} + \cdots + 14641)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 636 T^{2} + 41920)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 169 T^{2} + 2620)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 1465 T^{2} + 524000)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 596 T^{2} + 2048)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 2881 T^{2} + 524000)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 13 T - 214)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 2197 T^{2} + 800)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 5860 T^{2} + 8384000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 6496 T^{2} + 10071280)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 1300 T^{2} + 320000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 6629 T^{2} + 10580000)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 96 T + 2140)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 5201 T^{2} + 20960)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 3988 T^{2} + 236672)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 15 T - 11106)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 16416 T^{2} + 5543920)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 22084 T^{2} + 85852160)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 8236 T^{2} + 4192000)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 95 T - 3166)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 20852 T^{2} + 108339200)^{2} \) Copy content Toggle raw display
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