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Properties

Label 44.4.e.a
Level $44$
Weight $4$
Character orbit 44.e
Analytic conductor $2.596$
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] = [44,4,Mod(5,44)] 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("44.5"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(44, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 44 = 2^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 44.e (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.59608404025\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
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: \( x^{12} - x^{11} + 70 x^{10} - 84 x^{9} + 2459 x^{8} - 8514 x^{7} + 54995 x^{6} - 432951 x^{5} + \cdots + 40896025 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{7} q^{3} + (\beta_{9} - \beta_{5} + 3 \beta_{4} + \cdots + 1) q^{5} + (\beta_{11} - 2 \beta_{7} + \cdots + 2 \beta_{3}) q^{7} + (\beta_{11} - \beta_{10} - \beta_{7} + \cdots + 7) q^{9} + ( - 2 \beta_{11} + \beta_{10} + \cdots + 4) q^{11}+ \cdots + ( - 10 \beta_{11} + 23 \beta_{9} + \cdots + 202) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{3} - 4 q^{5} + 6 q^{7} + 47 q^{9} + 39 q^{11} - 10 q^{13} + 74 q^{15} - 56 q^{17} - 141 q^{19} - 304 q^{21} - 388 q^{23} - 203 q^{25} - 331 q^{27} + 772 q^{29} + 882 q^{31} + 981 q^{33} + 412 q^{35}+ \cdots + 3563 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 70 x^{10} - 84 x^{9} + 2459 x^{8} - 8514 x^{7} + 54995 x^{6} - 432951 x^{5} + \cdots + 40896025 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 33\!\cdots\!66 \nu^{11} + \cdots - 23\!\cdots\!65 ) / 89\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 43\!\cdots\!67 \nu^{11} + \cdots - 31\!\cdots\!75 ) / 89\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 69\!\cdots\!24 \nu^{11} + \cdots + 80\!\cdots\!60 ) / 89\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 69\!\cdots\!24 \nu^{11} + \cdots - 80\!\cdots\!60 ) / 89\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 86\!\cdots\!80 \nu^{11} + \cdots - 24\!\cdots\!35 ) / 89\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 15\!\cdots\!47 \nu^{11} + \cdots - 49\!\cdots\!05 ) / 89\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13\!\cdots\!71 \nu^{11} + \cdots + 13\!\cdots\!25 ) / 69\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 18\!\cdots\!23 \nu^{11} + \cdots + 49\!\cdots\!00 ) / 69\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 81\!\cdots\!38 \nu^{11} + \cdots + 42\!\cdots\!40 ) / 89\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 72\!\cdots\!41 \nu^{11} + \cdots - 38\!\cdots\!60 ) / 69\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 25\!\cdots\!10 \nu^{11} + \cdots - 26\!\cdots\!00 ) / 89\!\cdots\!55 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - 7\beta_{5} + 6\beta_{4} - 2\beta_{3} + 33\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{11} + 3 \beta_{10} + \beta_{9} + 9 \beta_{8} - 47 \beta_{7} - 3 \beta_{6} - 34 \beta_{5} + \cdots + 40 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 51 \beta_{11} - 12 \beta_{9} + 51 \beta_{8} - 51 \beta_{6} + 982 \beta_{5} - 1546 \beta_{4} + \cdots - 982 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 177 \beta_{10} - 138 \beta_{9} - 2320 \beta_{8} + 2320 \beta_{7} + 138 \beta_{6} - 2064 \beta_{5} + \cdots + 1878 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2536 \beta_{11} - 2536 \beta_{10} - 2008 \beta_{7} + 3460 \beta_{6} + 77349 \beta_{4} + 5444 \beta_{3} + \cdots + 30028 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2140 \beta_{11} + 7820 \beta_{10} + 7820 \beta_{9} + 116945 \beta_{8} - 42648 \beta_{7} + \cdots + 116945 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 56148 \beta_{11} + 185193 \beta_{10} + 56148 \beta_{9} - 65685 \beta_{8} - 183609 \beta_{7} + \cdots - 1693515 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 424950 \beta_{11} - 136257 \beta_{9} - 2472945 \beta_{8} - 424950 \beta_{6} + 3881226 \beta_{5} + \cdots - 3881226 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 9842947 \beta_{10} - 6656359 \beta_{9} - 9871627 \beta_{8} + 9871627 \beta_{7} + 6656359 \beta_{6} + \cdots + 98863474 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 22901162 \beta_{11} - 22901162 \beta_{10} + 138864783 \beta_{7} + 31527909 \beta_{6} + 334605086 \beta_{4} + \cdots - 143502376 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(13\) \(23\)
\(\chi(n)\) \(\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
2.26056 + 6.95728i
0.747054 + 2.29920i
−2.19860 − 6.76658i
2.26056 − 6.95728i
0.747054 − 2.29920i
−2.19860 + 6.76658i
3.41047 − 2.47785i
1.50918 − 1.09648i
−5.22867 + 3.79885i
3.41047 + 2.47785i
1.50918 + 1.09648i
−5.22867 − 3.79885i
0 −5.91822 − 4.29984i 0 −5.81848 + 17.9074i 0 −13.5120 + 9.81708i 0 8.19325 + 25.2162i 0
5.2 0 −1.95581 − 1.42098i 0 4.44180 − 13.6705i 0 14.2851 − 10.3787i 0 −6.53744 − 20.1202i 0
5.3 0 5.75600 + 4.18198i 0 −1.85939 + 5.72260i 0 −0.391096 + 0.284148i 0 7.29911 + 22.4644i 0
9.1 0 −5.91822 + 4.29984i 0 −5.81848 − 17.9074i 0 −13.5120 − 9.81708i 0 8.19325 − 25.2162i 0
9.2 0 −1.95581 + 1.42098i 0 4.44180 + 13.6705i 0 14.2851 + 10.3787i 0 −6.53744 + 20.1202i 0
9.3 0 5.75600 − 4.18198i 0 −1.85939 − 5.72260i 0 −0.391096 − 0.284148i 0 7.29911 − 22.4644i 0
25.1 0 −1.30268 − 4.00925i 0 −13.7488 − 9.98909i 0 1.14865 − 3.53519i 0 7.46635 − 5.42462i 0
25.2 0 −0.576456 − 1.77415i 0 14.9139 + 10.8356i 0 9.02047 − 27.7622i 0 19.0282 − 13.8248i 0
25.3 0 1.99717 + 6.14667i 0 0.0709832 + 0.0515723i 0 −7.55109 + 23.2399i 0 −11.9494 + 8.68176i 0
37.1 0 −1.30268 + 4.00925i 0 −13.7488 + 9.98909i 0 1.14865 + 3.53519i 0 7.46635 + 5.42462i 0
37.2 0 −0.576456 + 1.77415i 0 14.9139 − 10.8356i 0 9.02047 + 27.7622i 0 19.0282 + 13.8248i 0
37.3 0 1.99717 − 6.14667i 0 0.0709832 − 0.0515723i 0 −7.55109 − 23.2399i 0 −11.9494 − 8.68176i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 44.4.e.a ✓ 12
3.b odd 2 1 396.4.j.d 12
4.b odd 2 1 176.4.m.d 12
11.c even 5 1 inner 44.4.e.a ✓ 12
11.c even 5 1 484.4.a.i 6
11.d odd 10 1 484.4.a.h 6
33.h odd 10 1 396.4.j.d 12
44.g even 10 1 1936.4.a.bs 6
44.h odd 10 1 176.4.m.d 12
44.h odd 10 1 1936.4.a.br 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.4.e.a ✓ 12 1.a even 1 1 trivial
44.4.e.a ✓ 12 11.c even 5 1 inner
176.4.m.d 12 4.b odd 2 1
176.4.m.d 12 44.h odd 10 1
396.4.j.d 12 3.b odd 2 1
396.4.j.d 12 33.h odd 10 1
484.4.a.h 6 11.d odd 10 1
484.4.a.i 6 11.c even 5 1
1936.4.a.br 6 44.h odd 10 1
1936.4.a.bs 6 44.g even 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(44, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 4 T^{11} + \cdots + 40896025 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 2003815696 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 142887024016 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 55\!\cdots\!81 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 37\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 50\!\cdots\!41 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 24\!\cdots\!61 \) Copy content Toggle raw display
$23$ \( (T^{6} + 194 T^{5} + \cdots + 294777143296)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 67\!\cdots\!96 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 22\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 31\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 43\!\cdots\!01 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 13245040385280)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 31\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 27\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 20\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 53\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 92\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 10\!\cdots\!81 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 11\!\cdots\!76)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 13\!\cdots\!61 \) Copy content Toggle raw display
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