Properties

Label 368.2.m.b
Level 368368
Weight 22
Character orbit 368.m
Analytic conductor 2.9382.938
Analytic rank 00
Dimension 1010
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [368,2,Mod(49,368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(368, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 0, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("368.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 368=2423 368 = 2^{4} \cdot 23
Weight: k k == 2 2
Character orbit: [χ][\chi] == 368.m (of order 1111, degree 1010, not 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: 2.938494794382.93849479438
Analytic rank: 00
Dimension: 1010
Coefficient field: Q(ζ22)\Q(\zeta_{22})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x10x9+x8x7+x6x5+x4x3+x2x+1 x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 46)
Sato-Tate group: SU(2)[C11]\mathrm{SU}(2)[C_{11}]

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the qq-expansion are expressed in terms of a primitive root of unity ζ22\zeta_{22}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(ζ226++ζ223)q3+(ζ229+ζ226+ζ22)q5+(ζ228++ζ222)q7+(2ζ229+ζ227+1)q9++(4ζ229++4ζ22)q99+O(q100) q + ( - \zeta_{22}^{6} + \cdots + \zeta_{22}^{3}) q^{3} + ( - \zeta_{22}^{9} + \zeta_{22}^{6} + \cdots - \zeta_{22}) q^{5} + (\zeta_{22}^{8} + \cdots + \zeta_{22}^{2}) q^{7} + (2 \zeta_{22}^{9} + \zeta_{22}^{7} + \cdots - 1) q^{9}+ \cdots + (4 \zeta_{22}^{9} + \cdots + 4 \zeta_{22}) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 10q+4q36q53q7+9q9+12q1114q13+13q15+15q172q19+q21+q23+13q2526q278q29+21q3115q337q35+28q37+20q99+O(q100) 10 q + 4 q^{3} - 6 q^{5} - 3 q^{7} + 9 q^{9} + 12 q^{11} - 14 q^{13} + 13 q^{15} + 15 q^{17} - 2 q^{19} + q^{21} + q^{23} + 13 q^{25} - 26 q^{27} - 8 q^{29} + 21 q^{31} - 15 q^{33} - 7 q^{35} + 28 q^{37}+ \cdots - 20 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/368Z)×\left(\mathbb{Z}/368\mathbb{Z}\right)^\times.

nn 4747 9797 277277
χ(n)\chi(n) 11 ζ224\zeta_{22}^{4} 11

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\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   ιm(ν)\iota_m(\nu) a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
49.1
−0.415415 0.909632i
0.142315 + 0.989821i
−0.841254 0.540641i
0.654861 0.755750i
0.142315 0.989821i
0.654861 + 0.755750i
0.959493 + 0.281733i
−0.841254 + 0.540641i
0.959493 0.281733i
−0.415415 + 0.909632i
0 −0.580699 + 1.27155i 0 −1.66741 + 1.92429i 0 −1.75667 + 1.12894i 0 0.684944 + 0.790468i 0
81.1 0 0.0530529 0.368991i 0 −3.26024 + 0.957293i 0 0.297176 + 0.342959i 0 2.74514 + 0.806046i 0
177.1 0 2.71616 1.74557i 0 0.985691 + 2.15836i 0 −0.381761 + 0.112095i 0 3.08427 6.75361i 0
193.1 0 −0.712591 0.822373i 0 0.174863 + 1.21620i 0 −0.260554 + 0.570534i 0 0.258432 1.79743i 0
209.1 0 0.0530529 + 0.368991i 0 −3.26024 0.957293i 0 0.297176 0.342959i 0 2.74514 0.806046i 0
225.1 0 −0.712591 + 0.822373i 0 0.174863 1.21620i 0 −0.260554 0.570534i 0 0.258432 + 1.79743i 0
257.1 0 0.524075 0.153882i 0 0.767092 + 0.492980i 0 0.601808 + 4.18567i 0 −2.27279 + 1.46063i 0
289.1 0 2.71616 + 1.74557i 0 0.985691 2.15836i 0 −0.381761 0.112095i 0 3.08427 + 6.75361i 0
305.1 0 0.524075 + 0.153882i 0 0.767092 0.492980i 0 0.601808 4.18567i 0 −2.27279 1.46063i 0
353.1 0 −0.580699 1.27155i 0 −1.66741 1.92429i 0 −1.75667 1.12894i 0 0.684944 0.790468i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.1
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 368.2.m.b 10
4.b odd 2 1 46.2.c.a 10
12.b even 2 1 414.2.i.f 10
23.c even 11 1 inner 368.2.m.b 10
23.c even 11 1 8464.2.a.bx 5
23.d odd 22 1 8464.2.a.bw 5
92.g odd 22 1 46.2.c.a 10
92.g odd 22 1 1058.2.a.m 5
92.h even 22 1 1058.2.a.l 5
276.j odd 22 1 9522.2.a.bu 5
276.o even 22 1 414.2.i.f 10
276.o even 22 1 9522.2.a.bp 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
46.2.c.a 10 4.b odd 2 1
46.2.c.a 10 92.g odd 22 1
368.2.m.b 10 1.a even 1 1 trivial
368.2.m.b 10 23.c even 11 1 inner
414.2.i.f 10 12.b even 2 1
414.2.i.f 10 276.o even 22 1
1058.2.a.l 5 92.h even 22 1
1058.2.a.m 5 92.g odd 22 1
8464.2.a.bw 5 23.d odd 22 1
8464.2.a.bx 5 23.c even 11 1
9522.2.a.bp 5 276.o even 22 1
9522.2.a.bu 5 276.j odd 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3104T39+5T38+2T37+25T36T35+4T3416T33+9T323T3+1 T_{3}^{10} - 4T_{3}^{9} + 5T_{3}^{8} + 2T_{3}^{7} + 25T_{3}^{6} - T_{3}^{5} + 4T_{3}^{4} - 16T_{3}^{3} + 9T_{3}^{2} - 3T_{3} + 1 acting on S2new(368,[χ])S_{2}^{\mathrm{new}}(368, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T10 T^{10} Copy content Toggle raw display
33 T104T9++1 T^{10} - 4 T^{9} + \cdots + 1 Copy content Toggle raw display
55 T10+6T9++529 T^{10} + 6 T^{9} + \cdots + 529 Copy content Toggle raw display
77 T10+3T9++1 T^{10} + 3 T^{9} + \cdots + 1 Copy content Toggle raw display
1111 T1012T9++109561 T^{10} - 12 T^{9} + \cdots + 109561 Copy content Toggle raw display
1313 T10+14T9++1 T^{10} + 14 T^{9} + \cdots + 1 Copy content Toggle raw display
1717 T1015T9++214369 T^{10} - 15 T^{9} + \cdots + 214369 Copy content Toggle raw display
1919 T10+2T9++4489 T^{10} + 2 T^{9} + \cdots + 4489 Copy content Toggle raw display
2323 T10T9++6436343 T^{10} - T^{9} + \cdots + 6436343 Copy content Toggle raw display
2929 T10+8T9++4489 T^{10} + 8 T^{9} + \cdots + 4489 Copy content Toggle raw display
3131 T1021T9++20529961 T^{10} - 21 T^{9} + \cdots + 20529961 Copy content Toggle raw display
3737 T1028T9++49857721 T^{10} - 28 T^{9} + \cdots + 49857721 Copy content Toggle raw display
4141 T10++172475689 T^{10} + \cdots + 172475689 Copy content Toggle raw display
4343 T10+11T9++7027801 T^{10} + 11 T^{9} + \cdots + 7027801 Copy content Toggle raw display
4747 (T5+9T4++529)2 (T^{5} + 9 T^{4} + \cdots + 529)^{2} Copy content Toggle raw display
5353 T10+21T9++31236921 T^{10} + 21 T^{9} + \cdots + 31236921 Copy content Toggle raw display
5959 T105T9++4489 T^{10} - 5 T^{9} + \cdots + 4489 Copy content Toggle raw display
6161 T10++349727401 T^{10} + \cdots + 349727401 Copy content Toggle raw display
6767 T1013T9++94109401 T^{10} - 13 T^{9} + \cdots + 94109401 Copy content Toggle raw display
7171 T10+49T9++8300161 T^{10} + 49 T^{9} + \cdots + 8300161 Copy content Toggle raw display
7373 T10+8T9++7921 T^{10} + 8 T^{9} + \cdots + 7921 Copy content Toggle raw display
7979 T10+8T9++17161 T^{10} + 8 T^{9} + \cdots + 17161 Copy content Toggle raw display
8383 T10++667137241 T^{10} + \cdots + 667137241 Copy content Toggle raw display
8989 T10+13T9++26739241 T^{10} + 13 T^{9} + \cdots + 26739241 Copy content Toggle raw display
9797 T10+32T9++5031049 T^{10} + 32 T^{9} + \cdots + 5031049 Copy content Toggle raw display
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