Properties

Label 55.5.i.a
Level $55$
Weight $5$
Character orbit 55.i
Analytic conductor $5.685$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [55,5,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, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("55.6");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 55 = 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 5 \)
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: \(5.68534796961\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 2) q^{2} + (4 \zeta_{10}^{3} + 9 \zeta_{10}^{2} + \cdots - 4) q^{3}+ \cdots + ( - 104 \zeta_{10}^{3} + 65 \zeta_{10} - 65) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 2) q^{2} + (4 \zeta_{10}^{3} + 9 \zeta_{10}^{2} + \cdots - 4) q^{3}+ \cdots + (429 \zeta_{10}^{3} - 4147 \zeta_{10}^{2} + \cdots - 16588) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 5 q^{2} - 30 q^{3} - q^{4} - 25 q^{5} - 85 q^{6} + 150 q^{7} - 125 q^{8} - 299 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 5 q^{2} - 30 q^{3} - q^{4} - 25 q^{5} - 85 q^{6} + 150 q^{7} - 125 q^{8} - 299 q^{9} + 11 q^{11} - 370 q^{12} - 90 q^{13} + 290 q^{14} + 100 q^{15} - 441 q^{16} + 70 q^{17} + 520 q^{18} - 825 q^{19} - 325 q^{20} - 715 q^{22} - 280 q^{23} + 2045 q^{24} - 125 q^{25} - 360 q^{26} + 3405 q^{27} - 30 q^{28} + 590 q^{29} + 750 q^{30} - 838 q^{31} + 935 q^{33} - 2290 q^{34} - 1850 q^{35} + 1326 q^{36} - 20 q^{37} + 1010 q^{38} + 2700 q^{39} - 425 q^{40} + 6520 q^{41} - 2230 q^{42} + 1496 q^{44} + 3250 q^{45} - 3470 q^{46} + 1740 q^{47} - 310 q^{48} - 2101 q^{49} - 625 q^{50} - 11845 q^{51} + 1980 q^{52} - 2590 q^{53} - 2475 q^{55} - 10440 q^{56} + 6685 q^{57} + 8390 q^{58} - 7883 q^{59} + 4225 q^{60} + 10060 q^{61} - 130 q^{62} - 20800 q^{63} - 7901 q^{64} + 7590 q^{66} - 21750 q^{67} - 5490 q^{68} + 6260 q^{69} - 3650 q^{70} + 5018 q^{71} - 2795 q^{72} + 15970 q^{73} - 2080 q^{74} + 4375 q^{75} - 2640 q^{77} + 4500 q^{78} + 10280 q^{79} - 375 q^{80} + 24904 q^{81} + 765 q^{82} - 9595 q^{83} - 14640 q^{84} - 6775 q^{85} - 3345 q^{86} - 165 q^{88} + 40818 q^{89} - 6175 q^{90} - 6480 q^{91} + 9430 q^{92} + 10610 q^{93} + 7880 q^{94} + 14225 q^{95} + 6340 q^{96} + 20045 q^{97} - 45331 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(12\) \(46\)
\(\chi(n)\) \(1\) \(\zeta_{10}\)

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
0.809017 0.587785i
−0.309017 + 0.951057i
0.809017 + 0.587785i
−0.309017 0.951057i
2.92705 0.951057i −9.73607 7.07367i −5.28115 + 3.83698i −3.45492 + 10.6331i −35.2254 11.4454i 36.3820 + 50.0755i −40.7533 + 56.0921i 19.7239 + 60.7038i 34.4095i
41.1 −0.427051 0.587785i −5.26393 16.2007i 4.78115 14.7149i −9.04508 6.57164i −7.27458 + 10.0126i 38.6180 + 12.5478i −21.7467 + 7.06593i −169.224 + 122.948i 8.12299i
46.1 2.92705 + 0.951057i −9.73607 + 7.07367i −5.28115 3.83698i −3.45492 10.6331i −35.2254 + 11.4454i 36.3820 50.0755i −40.7533 56.0921i 19.7239 60.7038i 34.4095i
51.1 −0.427051 + 0.587785i −5.26393 + 16.2007i 4.78115 + 14.7149i −9.04508 + 6.57164i −7.27458 10.0126i 38.6180 12.5478i −21.7467 7.06593i −169.224 122.948i 8.12299i
\(n\): e.g. 2-40 or 990-1000
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 55.5.i.a 4
11.d odd 10 1 inner 55.5.i.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.5.i.a 4 1.a even 1 1 trivial
55.5.i.a 4 11.d odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 5T_{2}^{3} + 5T_{2}^{2} + 5T_{2} + 5 \) acting on \(S_{5}^{\mathrm{new}}(55, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 5 T^{3} + \cdots + 5 \) Copy content Toggle raw display
$3$ \( T^{4} + 30 T^{3} + \cdots + 42025 \) Copy content Toggle raw display
$5$ \( T^{4} + 25 T^{3} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( T^{4} - 150 T^{3} + \cdots + 6316880 \) Copy content Toggle raw display
$11$ \( T^{4} - 11 T^{3} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( T^{4} + 90 T^{3} + 13122000 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 15374067005 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 11002271405 \) Copy content Toggle raw display
$23$ \( (T^{2} + 140 T - 211420)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 1262209670480 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 8056139536 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 46751088400 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 17606617786805 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 1360442338205 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 3011196678400 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 32536756810000 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 132885308152921 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 1324868697680 \) Copy content Toggle raw display
$67$ \( (T^{2} + 10875 T + 24059155)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 259640499396496 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 317879444778005 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 138787779399680 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 12\!\cdots\!05 \) Copy content Toggle raw display
$89$ \( (T^{2} - 20409 T + 83252359)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 95\!\cdots\!25 \) Copy content Toggle raw display
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