Properties

Label 1120.2.g.b
Level 11201120
Weight 22
Character orbit 1120.g
Analytic conductor 8.9438.943
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] = [1120,2,Mod(449,1120)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1120, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1120.449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 1120=2557 1120 = 2^{5} \cdot 5 \cdot 7
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1120.g (of order 22, degree 11, 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: 8.943245026388.94324502638
Analytic rank: 00
Dimension: 1010
Coefficient field: 10.0.65174749855744.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x10+13x8+56x6+97x4+61x2+4 x^{10} + 13x^{8} + 56x^{6} + 97x^{4} + 61x^{2} + 4 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 23 2^{3}
Twist minimal: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{2}]

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 basis 1,β1,,β91,\beta_1,\ldots,\beta_{9} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β1q3+β4q5+β5q7+(β9β8β31)q9+(β9β8+β21)q11+(β7+β52β1)q13+(β9+β7+β6+1)q15++(4β3+2β2+6)q99+O(q100) q + \beta_1 q^{3} + \beta_{4} q^{5} + \beta_{5} q^{7} + (\beta_{9} - \beta_{8} - \beta_{3} - 1) q^{9} + (\beta_{9} - \beta_{8} + \beta_{2} - 1) q^{11} + (\beta_{7} + \beta_{5} - 2 \beta_1) q^{13} + ( - \beta_{9} + \beta_{7} + \beta_{6} + \cdots - 1) q^{15}+ \cdots + (4 \beta_{3} + 2 \beta_{2} + 6) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 10q2q514q98q114q15+24q194q21+6q25+24q2924q31+64q394q41+10q4510q4924q5116q55+32q5920q618q65++80q99+O(q100) 10 q - 2 q^{5} - 14 q^{9} - 8 q^{11} - 4 q^{15} + 24 q^{19} - 4 q^{21} + 6 q^{25} + 24 q^{29} - 24 q^{31} + 64 q^{39} - 4 q^{41} + 10 q^{45} - 10 q^{49} - 24 q^{51} - 16 q^{55} + 32 q^{59} - 20 q^{61} - 8 q^{65}+ \cdots + 80 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x10+13x8+56x6+97x4+61x2+4 x^{10} + 13x^{8} + 56x^{6} + 97x^{4} + 61x^{2} + 4 : Copy content Toggle raw display

β1\beta_{1}== (ν9+12ν7+44ν5+55ν3+20ν)/2 ( \nu^{9} + 12\nu^{7} + 44\nu^{5} + 55\nu^{3} + 20\nu ) / 2 Copy content Toggle raw display
β2\beta_{2}== (3ν834ν6112ν4107ν22)/2 ( -3\nu^{8} - 34\nu^{6} - 112\nu^{4} - 107\nu^{2} - 2 ) / 2 Copy content Toggle raw display
β3\beta_{3}== (3ν834ν6112ν4111ν212)/2 ( -3\nu^{8} - 34\nu^{6} - 112\nu^{4} - 111\nu^{2} - 12 ) / 2 Copy content Toggle raw display
β4\beta_{4}== (3ν93ν834ν734ν6110ν5110ν493ν397ν2+16ν)/4 ( -3\nu^{9} - 3\nu^{8} - 34\nu^{7} - 34\nu^{6} - 110\nu^{5} - 110\nu^{4} - 93\nu^{3} - 97\nu^{2} + 16\nu ) / 4 Copy content Toggle raw display
β5\beta_{5}== (2ν923ν778ν582ν313ν)/2 ( -2\nu^{9} - 23\nu^{7} - 78\nu^{5} - 82\nu^{3} - 13\nu ) / 2 Copy content Toggle raw display
β6\beta_{6}== (3ν9+3ν834ν7+34ν6110ν5+110ν493ν3+97ν2+16ν)/4 ( -3\nu^{9} + 3\nu^{8} - 34\nu^{7} + 34\nu^{6} - 110\nu^{5} + 110\nu^{4} - 93\nu^{3} + 97\nu^{2} + 16\nu ) / 4 Copy content Toggle raw display
β7\beta_{7}== (3ν9+35ν7+122ν5+137ν3+29ν)/2 ( 3\nu^{9} + 35\nu^{7} + 122\nu^{5} + 137\nu^{3} + 29\nu ) / 2 Copy content Toggle raw display
β8\beta_{8}== (5ν9+5ν8+56ν7+58ν6+178ν5+198ν4+151ν3+203ν214ν+20)/4 ( 5\nu^{9} + 5\nu^{8} + 56\nu^{7} + 58\nu^{6} + 178\nu^{5} + 198\nu^{4} + 151\nu^{3} + 203\nu^{2} - 14\nu + 20 ) / 4 Copy content Toggle raw display
β9\beta_{9}== (5ν95ν8+56ν758ν6+178ν5198ν4+151ν3203ν214ν20)/4 ( 5\nu^{9} - 5\nu^{8} + 56\nu^{7} - 58\nu^{6} + 178\nu^{5} - 198\nu^{4} + 151\nu^{3} - 203\nu^{2} - 14\nu - 20 ) / 4 Copy content Toggle raw display
ν\nu== (β7β5+β1)/2 ( -\beta_{7} - \beta_{5} + \beta_1 ) / 2 Copy content Toggle raw display
ν2\nu^{2}== (β3+β25)/2 ( -\beta_{3} + \beta_{2} - 5 ) / 2 Copy content Toggle raw display
ν3\nu^{3}== (β9+β8+4β7+β6+6β5+β42β1)/2 ( \beta_{9} + \beta_{8} + 4\beta_{7} + \beta_{6} + 6\beta_{5} + \beta_{4} - 2\beta_1 ) / 2 Copy content Toggle raw display
ν4\nu^{4}== (2β6+2β4+5β37β2+23)/2 ( -2\beta_{6} + 2\beta_{4} + 5\beta_{3} - 7\beta_{2} + 23 ) / 2 Copy content Toggle raw display
ν5\nu^{5}== (8β98β821β76β641β56β4+3β1)/2 ( -8\beta_{9} - 8\beta_{8} - 21\beta_{7} - 6\beta_{6} - 41\beta_{5} - 6\beta_{4} + 3\beta_1 ) / 2 Copy content Toggle raw display
ν6\nu^{6}== (3β9+3β8+17β617β424β3+46β2128)/2 ( -3\beta_{9} + 3\beta_{8} + 17\beta_{6} - 17\beta_{4} - 24\beta_{3} + 46\beta_{2} - 128 ) / 2 Copy content Toggle raw display
ν7\nu^{7}== (52β9+52β8+125β7+32β6+273β5+32β4+7β1)/2 ( 52\beta_{9} + 52\beta_{8} + 125\beta_{7} + 32\beta_{6} + 273\beta_{5} + 32\beta_{4} + 7\beta_1 ) / 2 Copy content Toggle raw display
ν8\nu^{8}== (34β934β8118β6+118β4+121β3297β2+769)/2 ( 34\beta_{9} - 34\beta_{8} - 118\beta_{6} + 118\beta_{4} + 121\beta_{3} - 297\beta_{2} + 769 ) / 2 Copy content Toggle raw display
ν9\nu^{9}== (327β9327β8776β7175β61782β5175β4122β1)/2 ( -327\beta_{9} - 327\beta_{8} - 776\beta_{7} - 175\beta_{6} - 1782\beta_{5} - 175\beta_{4} - 122\beta_1 ) / 2 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/1120Z)×\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times.

nn 351351 421421 801801 897897
χ(n)\chi(n) 11 11 11 1-1

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}
449.1
1.84576i
0.271831i
2.52064i
1.28447i
1.23118i
1.23118i
1.28447i
2.52064i
0.271831i
1.84576i
0 3.25260i 0 −1.49436 + 1.66340i 0 1.00000i 0 −7.57939 0
449.2 0 2.19794i 0 1.64514 1.51444i 0 1.00000i 0 −1.83094 0
449.3 0 1.83297i 0 1.86302 1.23660i 0 1.00000i 0 −0.359777 0
449.4 0 1.63460i 0 −2.23081 0.153266i 0 1.00000i 0 0.328072 0
449.5 0 0.746976i 0 −0.782984 2.09450i 0 1.00000i 0 2.44203 0
449.6 0 0.746976i 0 −0.782984 + 2.09450i 0 1.00000i 0 2.44203 0
449.7 0 1.63460i 0 −2.23081 + 0.153266i 0 1.00000i 0 0.328072 0
449.8 0 1.83297i 0 1.86302 + 1.23660i 0 1.00000i 0 −0.359777 0
449.9 0 2.19794i 0 1.64514 + 1.51444i 0 1.00000i 0 −1.83094 0
449.10 0 3.25260i 0 −1.49436 1.66340i 0 1.00000i 0 −7.57939 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1120.2.g.b 10
4.b odd 2 1 1120.2.g.c yes 10
5.b even 2 1 inner 1120.2.g.b 10
5.c odd 4 1 5600.2.a.bu 5
5.c odd 4 1 5600.2.a.bw 5
8.b even 2 1 2240.2.g.o 10
8.d odd 2 1 2240.2.g.n 10
20.d odd 2 1 1120.2.g.c yes 10
20.e even 4 1 5600.2.a.bv 5
20.e even 4 1 5600.2.a.bx 5
40.e odd 2 1 2240.2.g.n 10
40.f even 2 1 2240.2.g.o 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1120.2.g.b 10 1.a even 1 1 trivial
1120.2.g.b 10 5.b even 2 1 inner
1120.2.g.c yes 10 4.b odd 2 1
1120.2.g.c yes 10 20.d odd 2 1
2240.2.g.n 10 8.d odd 2 1
2240.2.g.n 10 40.e odd 2 1
2240.2.g.o 10 8.b even 2 1
2240.2.g.o 10 40.f even 2 1
5600.2.a.bu 5 5.c odd 4 1
5600.2.a.bv 5 20.e even 4 1
5600.2.a.bw 5 5.c odd 4 1
5600.2.a.bx 5 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(1120,[χ])S_{2}^{\mathrm{new}}(1120, [\chi]):

T310+22T38+165T36+532T34+708T32+256 T_{3}^{10} + 22T_{3}^{8} + 165T_{3}^{6} + 532T_{3}^{4} + 708T_{3}^{2} + 256 Copy content Toggle raw display
T115+4T11425T113152T112248T11128 T_{11}^{5} + 4T_{11}^{4} - 25T_{11}^{3} - 152T_{11}^{2} - 248T_{11} - 128 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T10 T^{10} Copy content Toggle raw display
33 T10+22T8++256 T^{10} + 22 T^{8} + \cdots + 256 Copy content Toggle raw display
55 T10+2T9++3125 T^{10} + 2 T^{9} + \cdots + 3125 Copy content Toggle raw display
77 (T2+1)5 (T^{2} + 1)^{5} Copy content Toggle raw display
1111 (T5+4T4+128)2 (T^{5} + 4 T^{4} + \cdots - 128)^{2} Copy content Toggle raw display
1313 T10+94T8++678976 T^{10} + 94 T^{8} + \cdots + 678976 Copy content Toggle raw display
1717 T10+66T8++1024 T^{10} + 66 T^{8} + \cdots + 1024 Copy content Toggle raw display
1919 (T512T4+256)2 (T^{5} - 12 T^{4} + \cdots - 256)^{2} Copy content Toggle raw display
2323 T10+104T8++262144 T^{10} + 104 T^{8} + \cdots + 262144 Copy content Toggle raw display
2929 (T512T4+4744)2 (T^{5} - 12 T^{4} + \cdots - 4744)^{2} Copy content Toggle raw display
3131 (T5+12T4+4096)2 (T^{5} + 12 T^{4} + \cdots - 4096)^{2} Copy content Toggle raw display
3737 T10+136T8++65536 T^{10} + 136 T^{8} + \cdots + 65536 Copy content Toggle raw display
4141 (T5+2T4+128)2 (T^{5} + 2 T^{4} + \cdots - 128)^{2} Copy content Toggle raw display
4343 T10++228130816 T^{10} + \cdots + 228130816 Copy content Toggle raw display
4747 T10+122T8++4096 T^{10} + 122 T^{8} + \cdots + 4096 Copy content Toggle raw display
5353 T10+208T8++3444736 T^{10} + 208 T^{8} + \cdots + 3444736 Copy content Toggle raw display
5959 (T516T4+58112)2 (T^{5} - 16 T^{4} + \cdots - 58112)^{2} Copy content Toggle raw display
6161 (T5+10T4++1648)2 (T^{5} + 10 T^{4} + \cdots + 1648)^{2} Copy content Toggle raw display
6767 T10+232T8++262144 T^{10} + 232 T^{8} + \cdots + 262144 Copy content Toggle raw display
7171 (T5+4T4+1024)2 (T^{5} + 4 T^{4} + \cdots - 1024)^{2} Copy content Toggle raw display
7373 T10+224T8++4194304 T^{10} + 224 T^{8} + \cdots + 4194304 Copy content Toggle raw display
7979 (T532T4+3904)2 (T^{5} - 32 T^{4} + \cdots - 3904)^{2} Copy content Toggle raw display
8383 T10++750321664 T^{10} + \cdots + 750321664 Copy content Toggle raw display
8989 (T5+2T4++3104)2 (T^{5} + 2 T^{4} + \cdots + 3104)^{2} Copy content Toggle raw display
9797 T10+274T8++11723776 T^{10} + 274 T^{8} + \cdots + 11723776 Copy content Toggle raw display
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