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Properties

Label 1100.2.n.a
Level $1100$
Weight $2$
Character orbit 1100.n
Analytic conductor $8.784$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1100,2,Mod(201,1100)] 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("1100.201"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1100, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 8])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1100.n (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.78354422234\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Copy content comment:defining polynomial
 
Copy content 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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 44)
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 primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{10}^{3} + \zeta_{10}^{2} + \zeta_{10}) q^{3} + ( - 2 \zeta_{10}^{3} - 3 \zeta_{10} + 3) q^{7} + (2 \zeta_{10}^{3} + \zeta_{10}^{2} + \cdots - 2) q^{9} + ( - 4 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + \cdots + 1) q^{11}+ \cdots + (8 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{3} + 7 q^{7} - 8 q^{9} - 4 q^{11} + q^{13} + q^{17} + 7 q^{19} + 18 q^{21} - 8 q^{23} - 5 q^{27} - 5 q^{29} - 5 q^{31} + 19 q^{33} + 9 q^{37} + 9 q^{39} - 5 q^{41} + 5 q^{47} - 12 q^{49} - q^{51}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(177\) \(551\)
\(\chi(n)\) \(-\zeta_{10}^{3}\) \(1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
201.1
0.809017 + 0.587785i
0.809017 − 0.587785i
−0.309017 + 0.951057i
−0.309017 − 0.951057i
0 0.809017 + 2.48990i 0 0 0 1.19098 − 3.66547i 0 −3.11803 + 2.26538i 0
301.1 0 0.809017 − 2.48990i 0 0 0 1.19098 + 3.66547i 0 −3.11803 − 2.26538i 0
401.1 0 −0.309017 − 0.224514i 0 0 0 2.30902 − 1.67760i 0 −0.881966 − 2.71441i 0
801.1 0 −0.309017 + 0.224514i 0 0 0 2.30902 + 1.67760i 0 −0.881966 + 2.71441i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1100.2.n.a 4
5.b even 2 1 44.2.e.a ✓ 4
5.c odd 4 2 1100.2.cb.a 8
11.c even 5 1 inner 1100.2.n.a 4
15.d odd 2 1 396.2.j.a 4
20.d odd 2 1 176.2.m.b 4
40.e odd 2 1 704.2.m.d 4
40.f even 2 1 704.2.m.e 4
55.d odd 2 1 484.2.e.c 4
55.h odd 10 1 484.2.a.c 2
55.h odd 10 1 484.2.e.c 4
55.h odd 10 2 484.2.e.d 4
55.j even 10 1 44.2.e.a ✓ 4
55.j even 10 1 484.2.a.b 2
55.j even 10 2 484.2.e.e 4
55.k odd 20 2 1100.2.cb.a 8
165.o odd 10 1 396.2.j.a 4
165.o odd 10 1 4356.2.a.t 2
165.r even 10 1 4356.2.a.u 2
220.n odd 10 1 176.2.m.b 4
220.n odd 10 1 1936.2.a.ba 2
220.o even 10 1 1936.2.a.z 2
440.ba odd 10 1 7744.2.a.db 2
440.bd even 10 1 704.2.m.e 4
440.bd even 10 1 7744.2.a.da 2
440.bh odd 10 1 704.2.m.d 4
440.bh odd 10 1 7744.2.a.bp 2
440.bm even 10 1 7744.2.a.bo 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.2.e.a ✓ 4 5.b even 2 1
44.2.e.a ✓ 4 55.j even 10 1
176.2.m.b 4 20.d odd 2 1
176.2.m.b 4 220.n odd 10 1
396.2.j.a 4 15.d odd 2 1
396.2.j.a 4 165.o odd 10 1
484.2.a.b 2 55.j even 10 1
484.2.a.c 2 55.h odd 10 1
484.2.e.c 4 55.d odd 2 1
484.2.e.c 4 55.h odd 10 1
484.2.e.d 4 55.h odd 10 2
484.2.e.e 4 55.j even 10 2
704.2.m.d 4 40.e odd 2 1
704.2.m.d 4 440.bh odd 10 1
704.2.m.e 4 40.f even 2 1
704.2.m.e 4 440.bd even 10 1
1100.2.n.a 4 1.a even 1 1 trivial
1100.2.n.a 4 11.c even 5 1 inner
1100.2.cb.a 8 5.c odd 4 2
1100.2.cb.a 8 55.k odd 20 2
1936.2.a.z 2 220.o even 10 1
1936.2.a.ba 2 220.n odd 10 1
4356.2.a.t 2 165.o odd 10 1
4356.2.a.u 2 165.r even 10 1
7744.2.a.bo 2 440.bm even 10 1
7744.2.a.bp 2 440.bh odd 10 1
7744.2.a.da 2 440.bd even 10 1
7744.2.a.db 2 440.ba odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - T_{3}^{3} + 6T_{3}^{2} + 4T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - T^{3} + 6 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 7 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$11$ \( T^{4} + 4 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$13$ \( T^{4} - T^{3} + \cdots + 121 \) Copy content Toggle raw display
$17$ \( T^{4} - T^{3} + \cdots + 121 \) Copy content Toggle raw display
$19$ \( T^{4} - 7 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$23$ \( (T^{2} + 4 T - 16)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 5 T^{3} + \cdots + 3025 \) Copy content Toggle raw display
$31$ \( T^{4} + 5 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$37$ \( T^{4} - 9 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$41$ \( T^{4} + 5 T^{3} + \cdots + 3025 \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 5 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$53$ \( T^{4} + 11 T^{3} + \cdots + 841 \) Copy content Toggle raw display
$59$ \( T^{4} - 17 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$61$ \( T^{4} + T^{3} + \cdots + 121 \) Copy content Toggle raw display
$67$ \( (T^{2} + 8 T - 64)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 5 T^{3} + \cdots + 3025 \) Copy content Toggle raw display
$73$ \( T^{4} + 19 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$79$ \( T^{4} - 3 T^{3} + \cdots + 9801 \) Copy content Toggle raw display
$83$ \( T^{4} - 31 T^{3} + \cdots + 43681 \) Copy content Toggle raw display
$89$ \( (T^{2} + 8 T - 4)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 27 T^{3} + \cdots + 6561 \) Copy content Toggle raw display
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