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Properties

Label 55.3.i.d
Level $55$
Weight $3$
Character orbit 55.i
Analytic conductor $1.499$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [55,3,Mod(6,55)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(55, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 9]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("55.6");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 55 = 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 55.i (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: \(1.49864145398\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 25x^{10} + 235x^{8} + 1025x^{6} + 2090x^{4} + 1880x^{2} + 605 \) 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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{10} - \beta_{4} - \beta_{3} + \cdots + 1) q^{2} + (\beta_{11} - \beta_{10} - \beta_{9} + \cdots - 1) q^{3} + ( - \beta_{11} - 3 \beta_{10} + \cdots + 2) q^{4} + (2 \beta_{10} - \beta_{3} - 2) q^{5}+ \cdots + (32 \beta_{11} - 2 \beta_{10} + \cdots + 76) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 5 q^{2} - 13 q^{3} + 13 q^{4} - 15 q^{5} + 20 q^{6} - 5 q^{7} - 35 q^{8} + 4 q^{9} + 12 q^{11} - 2 q^{12} + 70 q^{13} - 60 q^{14} + 35 q^{15} - 43 q^{16} - 15 q^{17} - 30 q^{18} - 80 q^{19} + 20 q^{20}+ \cdots + 669 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 25x^{10} + 235x^{8} + 1025x^{6} + 2090x^{4} + 1880x^{2} + 605 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} + 33 \nu^{10} - 74 \nu^{9} + 616 \nu^{8} - 1613 \nu^{7} + 3344 \nu^{6} - 10536 \nu^{5} + \cdots - 14069 ) / 3058 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{11} + 77 \nu^{10} + 74 \nu^{9} + 1947 \nu^{8} + 1613 \nu^{7} + 17996 \nu^{6} + 10536 \nu^{5} + \cdots + 57893 ) / 3058 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{11} - 77 \nu^{10} + 74 \nu^{9} - 1947 \nu^{8} + 1613 \nu^{7} - 17996 \nu^{6} + 10536 \nu^{5} + \cdots - 57893 ) / 3058 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 48 \nu^{11} + 77 \nu^{10} + 1035 \nu^{9} + 1947 \nu^{8} + 8200 \nu^{7} + 17996 \nu^{6} + 29422 \nu^{5} + \cdots + 76241 ) / 3058 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 48 \nu^{11} - 77 \nu^{10} + 1035 \nu^{9} - 1947 \nu^{8} + 8200 \nu^{7} - 17996 \nu^{6} + 29422 \nu^{5} + \cdots - 76241 ) / 3058 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7 \nu^{11} + 9 \nu^{10} + 177 \nu^{9} + 168 \nu^{8} + 1636 \nu^{7} + 1051 \nu^{6} + 6590 \nu^{5} + \cdots + 55 ) / 278 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 7 \nu^{11} - 9 \nu^{10} + 177 \nu^{9} - 168 \nu^{8} + 1636 \nu^{7} - 1051 \nu^{6} + 6590 \nu^{5} + \cdots - 55 ) / 278 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 50\nu^{10} + 1165\nu^{8} + 9978\nu^{6} + 37957\nu^{4} + 60559\nu^{2} + 139\nu + 30484 ) / 278 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 125 \nu^{11} + 77 \nu^{10} - 2982 \nu^{9} + 1947 \nu^{8} - 26196 \nu^{7} + 17996 \nu^{6} + \cdots + 59422 ) / 3058 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 483 \nu^{11} + 330 \nu^{10} + 11657 \nu^{9} + 7689 \nu^{8} + 103154 \nu^{7} + 65549 \nu^{6} + \cdots + 189574 ) / 3058 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - 2\beta_{4} + \beta_{3} - \beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{10} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - 6\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{9} + 9\beta_{8} - 9\beta_{7} + 8\beta_{6} - 8\beta_{5} + 19\beta_{4} - 10\beta_{3} + 9\beta_{2} - 9\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{11} - 12 \beta_{10} - \beta_{9} - 11 \beta_{8} - 11 \beta_{7} - 8 \beta_{6} - 8 \beta_{5} + \cdots + 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 72 \beta_{9} - 73 \beta_{8} + 73 \beta_{7} - 59 \beta_{6} + 59 \beta_{5} - 160 \beta_{4} + 82 \beta_{3} + \cdots - 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 32 \beta_{11} + 40 \beta_{10} + 16 \beta_{9} + 97 \beta_{8} + 97 \beta_{7} + 57 \beta_{6} + 57 \beta_{5} + \cdots - 20 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 566 \beta_{9} + 586 \beta_{8} - 586 \beta_{7} + 437 \beta_{6} - 437 \beta_{5} + 1311 \beta_{4} + \cdots - 34 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 364 \beta_{11} + 106 \beta_{10} - 182 \beta_{9} - 801 \beta_{8} - 801 \beta_{7} - 404 \beta_{6} + \cdots - 53 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 4435 \beta_{9} - 4707 \beta_{8} + 4707 \beta_{7} - 3270 \beta_{6} + 3270 \beta_{5} - 10618 \beta_{4} + \cdots + 1243 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 3608 \beta_{11} - 3776 \beta_{10} + 1804 \beta_{9} + 6437 \beta_{8} + 6437 \beta_{7} + 2903 \beta_{6} + \cdots + 1888 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/55\mathbb{Z}\right)^\times\).

\(n\) \(12\) \(46\)
\(\chi(n)\) \(1\) \(1 + \beta_{3} - \beta_{4} - \beta_{10}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1
1.48398i
− 0.914937i
− 2.47115i
− 2.81976i
− 0.987790i
2.63198i
− 1.48398i
0.914937i
2.47115i
2.81976i
0.987790i
− 2.63198i
−3.40164 + 1.10526i −3.07249 − 2.23230i 7.11349 − 5.16825i −0.690983 + 2.12663i 12.9188 + 4.19757i 4.78108 + 6.58059i −10.0760 + 13.8684i 1.67591 + 5.15793i − 7.99774i
6.2 0.289909 − 0.0941972i −4.17687 − 3.03467i −3.16089 + 2.29652i −0.690983 + 2.12663i −1.49677 − 0.486330i −5.94293 − 8.17974i −1.41674 + 1.94998i 5.45583 + 16.7913i 0.681617i
6.3 2.68468 − 0.872305i 0.0862408 + 0.0626576i 3.21052 − 2.33258i −0.690983 + 2.12663i 0.286186 + 0.0929874i −1.76520 − 2.42959i −0.0523905 + 0.0721094i −2.77764 − 8.54870i 6.31206i
41.1 0.0936958 + 0.128961i −0.602908 − 1.85556i 1.22822 − 3.78006i −1.80902 − 1.31433i 0.182805 − 0.251610i 0.0125693 + 0.00408401i 1.20897 − 0.392819i 4.20154 − 3.05260i − 0.356440i
41.2 0.759198 + 1.04495i 1.55259 + 4.77837i 0.720536 − 2.21758i −1.80902 − 1.31433i −3.81442 + 5.25010i −3.31915 − 1.07846i 7.77792 − 2.52720i −13.1411 + 9.54757i − 2.88816i
41.3 2.07416 + 2.85483i −0.286558 − 0.881935i −2.61187 + 8.03851i −1.80902 − 1.31433i 1.92341 − 2.64735i 3.73363 + 1.21313i −14.9418 + 4.85489i 6.58546 − 4.78462i − 7.89056i
46.1 −3.40164 − 1.10526i −3.07249 + 2.23230i 7.11349 + 5.16825i −0.690983 − 2.12663i 12.9188 − 4.19757i 4.78108 − 6.58059i −10.0760 − 13.8684i 1.67591 − 5.15793i 7.99774i
46.2 0.289909 + 0.0941972i −4.17687 + 3.03467i −3.16089 − 2.29652i −0.690983 − 2.12663i −1.49677 + 0.486330i −5.94293 + 8.17974i −1.41674 − 1.94998i 5.45583 − 16.7913i − 0.681617i
46.3 2.68468 + 0.872305i 0.0862408 − 0.0626576i 3.21052 + 2.33258i −0.690983 − 2.12663i 0.286186 − 0.0929874i −1.76520 + 2.42959i −0.0523905 − 0.0721094i −2.77764 + 8.54870i − 6.31206i
51.1 0.0936958 − 0.128961i −0.602908 + 1.85556i 1.22822 + 3.78006i −1.80902 + 1.31433i 0.182805 + 0.251610i 0.0125693 − 0.00408401i 1.20897 + 0.392819i 4.20154 + 3.05260i 0.356440i
51.2 0.759198 − 1.04495i 1.55259 − 4.77837i 0.720536 + 2.21758i −1.80902 + 1.31433i −3.81442 − 5.25010i −3.31915 + 1.07846i 7.77792 + 2.52720i −13.1411 − 9.54757i 2.88816i
51.3 2.07416 − 2.85483i −0.286558 + 0.881935i −2.61187 − 8.03851i −1.80902 + 1.31433i 1.92341 + 2.64735i 3.73363 − 1.21313i −14.9418 − 4.85489i 6.58546 + 4.78462i 7.89056i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.3
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 55.3.i.d ✓ 12
5.b even 2 1 275.3.x.f 12
5.c odd 4 2 275.3.q.f 24
11.c even 5 1 605.3.c.d 12
11.d odd 10 1 inner 55.3.i.d ✓ 12
11.d odd 10 1 605.3.c.d 12
55.h odd 10 1 275.3.x.f 12
55.l even 20 2 275.3.q.f 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.3.i.d ✓ 12 1.a even 1 1 trivial
55.3.i.d ✓ 12 11.d odd 10 1 inner
275.3.q.f 24 5.c odd 4 2
275.3.q.f 24 55.l even 20 2
275.3.x.f 12 5.b even 2 1
275.3.x.f 12 55.h odd 10 1
605.3.c.d 12 11.c even 5 1
605.3.c.d 12 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 5 T_{2}^{11} + 80 T_{2}^{9} - 215 T_{2}^{8} - 300 T_{2}^{7} + 2425 T_{2}^{6} - 4660 T_{2}^{5} + \cdots + 5 \) acting on \(S_{3}^{\mathrm{new}}(55, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 5 T^{11} + \cdots + 5 \) Copy content Toggle raw display
$3$ \( T^{12} + 13 T^{11} + \cdots + 361 \) Copy content Toggle raw display
$5$ \( (T^{4} + 5 T^{3} + 15 T^{2} + \cdots + 25)^{3} \) Copy content Toggle raw display
$7$ \( T^{12} + 5 T^{11} + \cdots + 2000 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 3138428376721 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 2507904080 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 605990405 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 373311161071805 \) Copy content Toggle raw display
$23$ \( (T^{6} - 21 T^{5} + \cdots + 8843956)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 13\!\cdots\!80 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 4020067216 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 25\!\cdots\!05 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 32774983803136 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 17\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 36\!\cdots\!01 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 33\!\cdots\!80 \) Copy content Toggle raw display
$67$ \( (T^{6} + 106 T^{5} + \cdots - 31173869579)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 29\!\cdots\!05 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 12\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 20\!\cdots\!05 \) Copy content Toggle raw display
$89$ \( (T^{6} + 57 T^{5} + \cdots - 494725464851)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 24\!\cdots\!41 \) Copy content Toggle raw display
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