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Properties

Label 1100.6.a.i
Level 11001100
Weight 66
Character orbit 1100.a
Self dual yes
Analytic conductor 176.422176.422
Analytic rank 11
Dimension 88
CM no
Inner twists 11

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1100,6,Mod(1,1100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1100.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: N N == 1100=22⋅52⋅11 1100 = 2^{2} \cdot 5^{2} \cdot 11
Weight: k k == 6 6
Character orbit: [χ][\chi] == 1100.a (trivial)

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: yes
Analytic conductor: 176.422201794176.422201794
Analytic rank: 11
Dimension: 88
Coefficient field: Q[x]/(x8−⋯ )\mathbb{Q}[x]/(x^{8} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x8−x7−1155x6+1122x5+365994x4−476334x3−28296355x2+54177943x+132029745 x^{8} - x^{7} - 1155x^{6} + 1122x^{5} + 365994x^{4} - 476334x^{3} - 28296355x^{2} + 54177943x + 132029745 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a13]\Z[a_1, \ldots, a_{13}]
Coefficient ring index: 27⋅3⋅52 2^{7}\cdot 3\cdot 5^{2}
Twist minimal: yes
Fricke sign: +1+1
Sato-Tate group: SU(2)\mathrm{SU}(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,…,β71,\beta_1,\ldots,\beta_{7} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(−β1−1)q3+(β5−2β1−22)q7+(−β5+β4+2β1+47)q9−121q11+(−β5−β2+6β1+134)q13+(β7+β6+2β5+⋯−2)q17+⋯+(121β5−121β4+⋯−5687)q99+O(q100) q + ( - \beta_1 - 1) q^{3} + (\beta_{5} - 2 \beta_1 - 22) q^{7} + ( - \beta_{5} + \beta_{4} + 2 \beta_1 + 47) q^{9} - 121 q^{11} + ( - \beta_{5} - \beta_{2} + 6 \beta_1 + 134) q^{13} + (\beta_{7} + \beta_{6} + 2 \beta_{5} + \cdots - 2) q^{17}+ \cdots + (121 \beta_{5} - 121 \beta_{4} + \cdots - 5687) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 8q−9q3−175q7+377q9−968q11+1072q13−37q17−1356q19+5462q21−1327q23−2670q27−11785q29+3532q31+1089q33+8176q37−16831q39+⋯−45617q99+O(q100) 8 q - 9 q^{3} - 175 q^{7} + 377 q^{9} - 968 q^{11} + 1072 q^{13} - 37 q^{17} - 1356 q^{19} + 5462 q^{21} - 1327 q^{23} - 2670 q^{27} - 11785 q^{29} + 3532 q^{31} + 1089 q^{33} + 8176 q^{37} - 16831 q^{39}+ \cdots - 45617 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x8−x7−1155x6+1122x5+365994x4−476334x3−28296355x2+54177943x+132029745 x^{8} - x^{7} - 1155x^{6} + 1122x^{5} + 365994x^{4} - 476334x^{3} - 28296355x^{2} + 54177943x + 132029745 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (−29993519ν7+972671008ν6+55968546197ν5−1193739185525ν4+⋯−21 ⁣⋯ ⁣05)/25734183149700 ( - 29993519 \nu^{7} + 972671008 \nu^{6} + 55968546197 \nu^{5} - 1193739185525 \nu^{4} + \cdots - 21\!\cdots\!05 ) / 25734183149700 Copy content Toggle raw display
β3\beta_{3}== (−37037373ν7−921476764ν6+45229096399ν5+773761442225ν4+⋯−40 ⁣⋯ ⁣35)/25734183149700 ( - 37037373 \nu^{7} - 921476764 \nu^{6} + 45229096399 \nu^{5} + 773761442225 \nu^{4} + \cdots - 40\!\cdots\!35 ) / 25734183149700 Copy content Toggle raw display
β4\beta_{4}== (−21022391ν7−265098288ν6+23951024733ν5+274101152025ν4+⋯−17 ⁣⋯ ⁣95)/12867091574850 ( - 21022391 \nu^{7} - 265098288 \nu^{6} + 23951024733 \nu^{5} + 274101152025 \nu^{4} + \cdots - 17\!\cdots\!95 ) / 12867091574850 Copy content Toggle raw display
β5\beta_{5}== (−21022391ν7−265098288ν6+23951024733ν5+274101152025ν4+⋯+20 ⁣⋯ ⁣55)/12867091574850 ( - 21022391 \nu^{7} - 265098288 \nu^{6} + 23951024733 \nu^{5} + 274101152025 \nu^{4} + \cdots + 20\!\cdots\!55 ) / 12867091574850 Copy content Toggle raw display
β6\beta_{6}== (27803583ν7+99527944ν6−33424966629ν5−121794895375ν4+⋯−482867449744815)/4289030524950 ( 27803583 \nu^{7} + 99527944 \nu^{6} - 33424966629 \nu^{5} - 121794895375 \nu^{4} + \cdots - 482867449744815 ) / 4289030524950 Copy content Toggle raw display
β7\beta_{7}== (−75377227ν7−27532536ν6+83784812576ν5+28968909500ν4+⋯−12 ⁣⋯ ⁣15)/2144515262475 ( - 75377227 \nu^{7} - 27532536 \nu^{6} + 83784812576 \nu^{5} + 28968909500 \nu^{4} + \cdots - 12\!\cdots\!15 ) / 2144515262475 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== −β5+β4+289 -\beta_{5} + \beta_{4} + 289 Copy content Toggle raw display
ν3\nu^{3}== −3β6−11β5−3β4+4β3−2β2+516β1−47 -3\beta_{6} - 11\beta_{5} - 3\beta_{4} + 4\beta_{3} - 2\beta_{2} + 516\beta _1 - 47 Copy content Toggle raw display
ν4\nu^{4}== −8β7−73β6−719β5+680β4−60β3−36β2+329β1+150562 -8\beta_{7} - 73\beta_{6} - 719\beta_{5} + 680\beta_{4} - 60\beta_{3} - 36\beta_{2} + 329\beta _1 + 150562 Copy content Toggle raw display
ν5\nu^{5}== −187β7−3587β6−9365β5−3106β4+3588β3−1260β2+⋯+58422 - 187 \beta_{7} - 3587 \beta_{6} - 9365 \beta_{5} - 3106 \beta_{4} + 3588 \beta_{3} - 1260 \beta_{2} + \cdots + 58422 Copy content Toggle raw display
ν6\nu^{6}== −6368β7−74971β6−521509β5+443087β4−65339β3+⋯+89500577 - 6368 \beta_{7} - 74971 \beta_{6} - 521509 \beta_{5} + 443087 \beta_{4} - 65339 \beta_{3} + \cdots + 89500577 Copy content Toggle raw display
ν7\nu^{7}== −237057β7−3019971β6−6969672β5−2362314β4+2698620β3+⋯+95589645 - 237057 \beta_{7} - 3019971 \beta_{6} - 6969672 \beta_{5} - 2362314 \beta_{4} + 2698620 \beta_{3} + \cdots + 95589645 Copy content Toggle raw display

Display βi\beta_i in terms of νj\nu^j

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}
1.1
26.2245
18.9133
9.67677
3.57844
−1.42686
−10.7235
−19.5686
−25.6740
0 −27.2245 0 0 0 −247.835 0 498.172 0
1.2 0 −19.9133 0 0 0 63.9161 0 153.541 0
1.3 0 −10.6768 0 0 0 −123.211 0 −129.007 0
1.4 0 −4.57844 0 0 0 209.239 0 −222.038 0
1.5 0 0.426864 0 0 0 59.7938 0 −242.818 0
1.6 0 9.72346 0 0 0 −220.259 0 −148.454 0
1.7 0 18.5686 0 0 0 50.8505 0 101.795 0
1.8 0 24.6740 0 0 0 32.5053 0 365.809 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 −1 -1
55 −1 -1
1111 +1 +1

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1100.6.a.i ✓ 8
5.b even 2 1 1100.6.a.j yes 8
5.c odd 4 2 1100.6.b.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1100.6.a.i ✓ 8 1.a even 1 1 trivial
1100.6.a.j yes 8 5.b even 2 1
1100.6.b.h 16 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T38+9T37−1120T36−7975T35+343164T34+1906081T33+⋯+50395500 T_{3}^{8} + 9 T_{3}^{7} - 1120 T_{3}^{6} - 7975 T_{3}^{5} + 343164 T_{3}^{4} + 1906081 T_{3}^{3} + \cdots + 50395500 acting on S6new(Γ0(1100))S_{6}^{\mathrm{new}}(\Gamma_0(1100)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T8 T^{8} Copy content Toggle raw display
33 T8+9T7+⋯+50395500 T^{8} + 9 T^{7} + \cdots + 50395500 Copy content Toggle raw display
55 T8 T^{8} Copy content Toggle raw display
77 T8+⋯−88 ⁣⋯ ⁣76 T^{8} + \cdots - 88\!\cdots\!76 Copy content Toggle raw display
1111 (T+121)8 (T + 121)^{8} Copy content Toggle raw display
1313 T8+⋯+48 ⁣⋯ ⁣89 T^{8} + \cdots + 48\!\cdots\!89 Copy content Toggle raw display
1717 T8+⋯+28 ⁣⋯ ⁣00 T^{8} + \cdots + 28\!\cdots\!00 Copy content Toggle raw display
1919 T8+⋯−71 ⁣⋯ ⁣03 T^{8} + \cdots - 71\!\cdots\!03 Copy content Toggle raw display
2323 T8+⋯−45 ⁣⋯ ⁣53 T^{8} + \cdots - 45\!\cdots\!53 Copy content Toggle raw display
2929 T8+⋯+37 ⁣⋯ ⁣75 T^{8} + \cdots + 37\!\cdots\!75 Copy content Toggle raw display
3131 T8+⋯+19 ⁣⋯ ⁣25 T^{8} + \cdots + 19\!\cdots\!25 Copy content Toggle raw display
3737 T8+⋯+36 ⁣⋯ ⁣72 T^{8} + \cdots + 36\!\cdots\!72 Copy content Toggle raw display
4141 T8+⋯+16 ⁣⋯ ⁣00 T^{8} + \cdots + 16\!\cdots\!00 Copy content Toggle raw display
4343 T8+⋯+11 ⁣⋯ ⁣00 T^{8} + \cdots + 11\!\cdots\!00 Copy content Toggle raw display
4747 T8+⋯+10 ⁣⋯ ⁣00 T^{8} + \cdots + 10\!\cdots\!00 Copy content Toggle raw display
5353 T8+⋯+61 ⁣⋯ ⁣48 T^{8} + \cdots + 61\!\cdots\!48 Copy content Toggle raw display
5959 T8+⋯+19 ⁣⋯ ⁣00 T^{8} + \cdots + 19\!\cdots\!00 Copy content Toggle raw display
6161 T8+⋯+45 ⁣⋯ ⁣72 T^{8} + \cdots + 45\!\cdots\!72 Copy content Toggle raw display
6767 T8+⋯−74 ⁣⋯ ⁣00 T^{8} + \cdots - 74\!\cdots\!00 Copy content Toggle raw display
7171 T8+⋯+18 ⁣⋯ ⁣00 T^{8} + \cdots + 18\!\cdots\!00 Copy content Toggle raw display
7373 T8+⋯+54 ⁣⋯ ⁣00 T^{8} + \cdots + 54\!\cdots\!00 Copy content Toggle raw display
7979 T8+⋯+16 ⁣⋯ ⁣68 T^{8} + \cdots + 16\!\cdots\!68 Copy content Toggle raw display
8383 T8+⋯+72 ⁣⋯ ⁣25 T^{8} + \cdots + 72\!\cdots\!25 Copy content Toggle raw display
8989 T8+⋯+11 ⁣⋯ ⁣73 T^{8} + \cdots + 11\!\cdots\!73 Copy content Toggle raw display
9797 T8+⋯−22 ⁣⋯ ⁣93 T^{8} + \cdots - 22\!\cdots\!93 Copy content Toggle raw display
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