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Properties

Label 44.3.f.a
Level $44$
Weight $3$
Character orbit 44.f
Analytic conductor $1.199$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [44,3,Mod(13,44)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(44, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([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("44.13");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 44 = 2^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 44.f (of order \(10\), 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: \(1.19891316319\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 19x^{6} - 37x^{5} + 229x^{4} + 196x^{3} + 1496x^{2} + 2952x + 26896 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{7} + \beta_{6} - \beta_{5} + \cdots - 2) q^{3}+ \cdots + (\beta_{7} - 2 \beta_{6} + 5 \beta_{5} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} + \beta_{6} - \beta_{5} + \cdots - 2) q^{3}+ \cdots + (8 \beta_{7} - \beta_{6} + 46 \beta_{5} + \cdots - 40) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 5 q^{3} - q^{5} + 15 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 5 q^{3} - q^{5} + 15 q^{7} - q^{9} + 17 q^{11} - 15 q^{13} - 63 q^{15} - 75 q^{17} - 30 q^{19} + 100 q^{23} + 51 q^{25} + 100 q^{27} + 125 q^{29} + 73 q^{31} - 20 q^{33} - 155 q^{35} - 75 q^{37} - 185 q^{39} - 155 q^{41} + 332 q^{45} + 45 q^{47} + 87 q^{49} + 150 q^{51} - 85 q^{53} - 89 q^{55} - 310 q^{57} - 178 q^{59} - 145 q^{61} - 110 q^{63} + 10 q^{67} + 124 q^{69} + 207 q^{71} + 65 q^{73} + 192 q^{75} + 365 q^{77} + 235 q^{79} - 12 q^{81} + 210 q^{83} + 105 q^{85} + 18 q^{89} - 195 q^{91} - 335 q^{93} - 255 q^{95} - 90 q^{97} - 419 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 19x^{6} - 37x^{5} + 229x^{4} + 196x^{3} + 1496x^{2} + 2952x + 26896 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 6255681 \nu^{7} - 139767705 \nu^{6} - 156710697 \nu^{5} + 206843551 \nu^{4} + \cdots + 150931987344 ) / 492550409912 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6940435 \nu^{7} + 14747663 \nu^{6} - 86739061 \nu^{5} + 563477539 \nu^{4} + \cdots + 50387442328 ) / 492550409912 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 264489 \nu^{7} + 2665943 \nu^{6} - 10003337 \nu^{5} + 46080023 \nu^{4} - 567915711 \nu^{3} + \cdots + 2276462680 ) / 6006712316 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 31369083 \nu^{7} + 69559599 \nu^{6} - 532742689 \nu^{5} + 3872519071 \nu^{4} + \cdots - 106099357856 ) / 492550409912 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 232869 \nu^{7} + 385792 \nu^{6} + 16535750 \nu^{5} + 3703796 \nu^{4} + 42541302 \nu^{3} + \cdots + 8147885770 ) / 3003356158 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 627750 \nu^{7} - 358185 \nu^{6} + 10843760 \nu^{5} - 12665997 \nu^{4} + 123254968 \nu^{3} + \cdots + 1490183294 ) / 1501678079 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + \beta_{6} - 2\beta_{5} - \beta_{4} + 12\beta_{3} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{6} + 12\beta_{5} - 13\beta_{4} + 12\beta_{3} - 14\beta_{2} - 2\beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 29\beta_{7} - 4\beta_{6} + 160\beta_{5} + 4\beta_{4} - 54\beta_{3} + 54\beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -58\beta_{7} + 243\beta_{6} - 106\beta_{5} + 106\beta_{2} + 58\beta _1 - 657 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -164\beta_{7} - 164\beta_{4} + 1182\beta_{3} - 3518\beta_{2} - 493\beta _1 + 1182 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4175\beta_{7} - 4175\beta_{6} + 3282\beta_{5} + 2829\beta_{4} - 6244\beta_{3} - 2829\beta _1 + 7457 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/44\mathbb{Z}\right)^\times\).

\(n\) \(13\) \(23\)
\(\chi(n)\) \(1 - \beta_{2} + \beta_{3} + \beta_{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
−1.37056 − 4.21816i
1.06155 + 3.26710i
−1.37056 + 4.21816i
1.06155 − 3.26710i
−2.28040 + 1.65680i
3.08941 − 2.24459i
−2.28040 − 1.65680i
3.08941 + 2.24459i
0 −4.08818 + 2.97024i 0 1.46509 + 4.50908i 0 −3.91877 + 5.39373i 0 5.10976 − 15.7262i 0
13.2 0 2.27916 − 1.65591i 0 −0.0380374 − 0.117067i 0 1.51959 − 2.09153i 0 −0.328603 + 1.01134i 0
17.1 0 −4.08818 − 2.97024i 0 1.46509 − 4.50908i 0 −3.91877 − 5.39373i 0 5.10976 + 15.7262i 0
17.2 0 2.27916 + 1.65591i 0 −0.0380374 + 0.117067i 0 1.51959 + 2.09153i 0 −0.328603 − 1.01134i 0
29.1 0 −1.37103 + 4.21961i 0 −5.30779 + 3.85634i 0 10.9532 − 3.55892i 0 −8.64421 − 6.28038i 0
29.2 0 0.680051 − 2.09298i 0 3.38074 − 2.45625i 0 −1.05404 + 0.342477i 0 3.36305 + 2.44340i 0
41.1 0 −1.37103 − 4.21961i 0 −5.30779 − 3.85634i 0 10.9532 + 3.55892i 0 −8.64421 + 6.28038i 0
41.2 0 0.680051 + 2.09298i 0 3.38074 + 2.45625i 0 −1.05404 − 0.342477i 0 3.36305 − 2.44340i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 44.3.f.a ✓ 8
3.b odd 2 1 396.3.t.a 8
4.b odd 2 1 176.3.n.c 8
11.b odd 2 1 484.3.f.a 8
11.c even 5 1 484.3.d.c 8
11.c even 5 1 484.3.f.a 8
11.c even 5 1 484.3.f.d 8
11.c even 5 1 484.3.f.e 8
11.d odd 10 1 inner 44.3.f.a ✓ 8
11.d odd 10 1 484.3.d.c 8
11.d odd 10 1 484.3.f.d 8
11.d odd 10 1 484.3.f.e 8
33.f even 10 1 396.3.t.a 8
33.f even 10 1 4356.3.f.g 8
33.h odd 10 1 4356.3.f.g 8
44.g even 10 1 176.3.n.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.3.f.a ✓ 8 1.a even 1 1 trivial
44.3.f.a ✓ 8 11.d odd 10 1 inner
176.3.n.c 8 4.b odd 2 1
176.3.n.c 8 44.g even 10 1
396.3.t.a 8 3.b odd 2 1
396.3.t.a 8 33.f even 10 1
484.3.d.c 8 11.c even 5 1
484.3.d.c 8 11.d odd 10 1
484.3.f.a 8 11.b odd 2 1
484.3.f.a 8 11.c even 5 1
484.3.f.d 8 11.c even 5 1
484.3.f.d 8 11.d odd 10 1
484.3.f.e 8 11.c even 5 1
484.3.f.e 8 11.d odd 10 1
4356.3.f.g 8 33.f even 10 1
4356.3.f.g 8 33.h odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(44, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 5 T^{7} + \cdots + 19321 \) Copy content Toggle raw display
$5$ \( T^{8} + T^{7} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( T^{8} - 15 T^{7} + \cdots + 48400 \) Copy content Toggle raw display
$11$ \( T^{8} - 17 T^{7} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( T^{8} + 15 T^{7} + \cdots + 302064400 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 1835694025 \) Copy content Toggle raw display
$19$ \( T^{8} + 30 T^{7} + \cdots + 506025025 \) Copy content Toggle raw display
$23$ \( (T^{4} - 50 T^{3} + \cdots - 18416)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 9844608400 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 405723137296 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 5733898437136 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 21147167946025 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 21950349414400 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 101060410000 \) Copy content Toggle raw display
$53$ \( T^{8} + 85 T^{7} + \cdots + 62410000 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 174163494241 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 263471980512400 \) Copy content Toggle raw display
$67$ \( (T^{4} - 5 T^{3} + \cdots + 1267204)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 81838149045136 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 32317576371025 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 8370721968400 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 26701194025 \) Copy content Toggle raw display
$89$ \( (T^{4} - 9 T^{3} + \cdots + 466236)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 27189041881 \) Copy content Toggle raw display
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