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Properties

Label 9075.2.a.dt
Level 90759075
Weight 22
Character orbit 9075.a
Self dual yes
Analytic conductor 72.46472.464
Analytic rank 00
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] = [9075,2,Mod(1,9075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9075, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9075.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 9075=3⋅52⋅112 9075 = 3 \cdot 5^{2} \cdot 11^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 9075.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: 72.464239834372.4642398343
Analytic rank: 00
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−x7−11x6+9x5+38x4−25x3−41x2+20x+5 x^{8} - x^{7} - 11x^{6} + 9x^{5} + 38x^{4} - 25x^{3} - 41x^{2} + 20x + 5 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 825)
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−β1q2−q3+(β2+1)q4+β1q6+(−β6−1)q7+(−β5−β4−β1)q8+q9+(−β2−1)q12+(−2β7+β6+β5+⋯−1)q13+⋯+(−2β7+β4−2β3+⋯+3)q98+O(q100) q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + \beta_1 q^{6} + ( - \beta_{6} - 1) q^{7} + ( - \beta_{5} - \beta_{4} - \beta_1) q^{8} + q^{9} + ( - \beta_{2} - 1) q^{12} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} + \cdots - 1) q^{13}+ \cdots + ( - 2 \beta_{7} + \beta_{4} - 2 \beta_{3} + \cdots + 3) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 8q−q2−8q3+7q4+q6−8q7−3q8+8q9−7q12−7q13−8q14−7q16+q17−q18+6q19+8q21−7q23+3q24−q26−8q27−11q28+⋯+27q98+O(q100) 8 q - q^{2} - 8 q^{3} + 7 q^{4} + q^{6} - 8 q^{7} - 3 q^{8} + 8 q^{9} - 7 q^{12} - 7 q^{13} - 8 q^{14} - 7 q^{16} + q^{17} - q^{18} + 6 q^{19} + 8 q^{21} - 7 q^{23} + 3 q^{24} - q^{26} - 8 q^{27} - 11 q^{28}+ \cdots + 27 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x8−x7−11x6+9x5+38x4−25x3−41x2+20x+5 x^{8} - x^{7} - 11x^{6} + 9x^{5} + 38x^{4} - 25x^{3} - 41x^{2} + 20x + 5 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== ν2−3 \nu^{2} - 3 Copy content Toggle raw display
β3\beta_{3}== (ν6−2ν5−6ν4+11ν3+7ν2−11ν−1)/2 ( \nu^{6} - 2\nu^{5} - 6\nu^{4} + 11\nu^{3} + 7\nu^{2} - 11\nu - 1 ) / 2 Copy content Toggle raw display
β4\beta_{4}== (ν7−2ν6−6ν5+11ν4+7ν3−11ν2−ν)/2 ( \nu^{7} - 2\nu^{6} - 6\nu^{5} + 11\nu^{4} + 7\nu^{3} - 11\nu^{2} - \nu ) / 2 Copy content Toggle raw display
β5\beta_{5}== (−ν7+2ν6+6ν5−11ν4−5ν3+11ν2−9ν)/2 ( -\nu^{7} + 2\nu^{6} + 6\nu^{5} - 11\nu^{4} - 5\nu^{3} + 11\nu^{2} - 9\nu ) / 2 Copy content Toggle raw display
β6\beta_{6}== (−ν6+2ν5+8ν4−13ν3−17ν2+17ν+5)/2 ( -\nu^{6} + 2\nu^{5} + 8\nu^{4} - 13\nu^{3} - 17\nu^{2} + 17\nu + 5 ) / 2 Copy content Toggle raw display
β7\beta_{7}== (−ν7+2ν6+8ν5−13ν4−19ν3+19ν2+13ν−2)/2 ( -\nu^{7} + 2\nu^{6} + 8\nu^{5} - 13\nu^{4} - 19\nu^{3} + 19\nu^{2} + 13\nu - 2 ) / 2 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}== β2+3 \beta_{2} + 3 Copy content Toggle raw display
ν3\nu^{3}== β5+β4+5β1 \beta_{5} + \beta_{4} + 5\beta_1 Copy content Toggle raw display
ν4\nu^{4}== β6+β5+β4+β3+5β2+2β1+13 \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 5\beta_{2} + 2\beta _1 + 13 Copy content Toggle raw display
ν5\nu^{5}== β7+β6+7β5+8β4+β3+β2+26β1+2 \beta_{7} + \beta_{6} + 7\beta_{5} + 8\beta_{4} + \beta_{3} + \beta_{2} + 26\beta _1 + 2 Copy content Toggle raw display
ν6\nu^{6}== 2β7+8β6+9β5+11β4+10β3+25β2+20β1+62 2\beta_{7} + 8\beta_{6} + 9\beta_{5} + 11\beta_{4} + 10\beta_{3} + 25\beta_{2} + 20\beta _1 + 62 Copy content Toggle raw display
ν7\nu^{7}== 10β7+11β6+42β5+54β4+15β3+12β2+140β1+26 10\beta_{7} + 11\beta_{6} + 42\beta_{5} + 54\beta_{4} + 15\beta_{3} + 12\beta_{2} + 140\beta _1 + 26 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
2.49311
2.02189
1.40736
0.684625
−0.188059
−1.27571
−2.04884
−2.09438
−2.49311 −1.00000 4.21558 0 2.49311 −1.92071 −5.52367 1.00000 0
1.2 −2.02189 −1.00000 2.08806 0 2.02189 1.30998 −0.178044 1.00000 0
1.3 −1.40736 −1.00000 −0.0193259 0 1.40736 2.16377 2.84193 1.00000 0
1.4 −0.684625 −1.00000 −1.53129 0 0.684625 −4.22715 2.41761 1.00000 0
1.5 0.188059 −1.00000 −1.96463 0 −0.188059 −1.64886 −0.745587 1.00000 0
1.6 1.27571 −1.00000 −0.372560 0 −1.27571 2.62160 −3.02670 1.00000 0
1.7 2.04884 −1.00000 2.19776 0 −2.04884 −3.70442 0.405171 1.00000 0
1.8 2.09438 −1.00000 2.38641 0 −2.09438 −2.59420 0.809299 1.00000 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
33 +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 9075.2.a.dt 8
5.b even 2 1 9075.2.a.dw 8
11.b odd 2 1 9075.2.a.dv 8
11.d odd 10 2 825.2.n.n yes 16
55.d odd 2 1 9075.2.a.du 8
55.h odd 10 2 825.2.n.m ✓ 16
55.l even 20 4 825.2.bx.j 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.2.n.m ✓ 16 55.h odd 10 2
825.2.n.n yes 16 11.d odd 10 2
825.2.bx.j 32 55.l even 20 4
9075.2.a.dt 8 1.a even 1 1 trivial
9075.2.a.du 8 55.d odd 2 1
9075.2.a.dv 8 11.b odd 2 1
9075.2.a.dw 8 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(Γ0(9075))S_{2}^{\mathrm{new}}(\Gamma_0(9075)):

T28+T27−11T26−9T25+38T24+25T23−41T22−20T2+5 T_{2}^{8} + T_{2}^{7} - 11T_{2}^{6} - 9T_{2}^{5} + 38T_{2}^{4} + 25T_{2}^{3} - 41T_{2}^{2} - 20T_{2} + 5 Copy content Toggle raw display
T78+8T77+3T76−105T75−165T74+401T73+807T72−394T7−956 T_{7}^{8} + 8T_{7}^{7} + 3T_{7}^{6} - 105T_{7}^{5} - 165T_{7}^{4} + 401T_{7}^{3} + 807T_{7}^{2} - 394T_{7} - 956 Copy content Toggle raw display
T138+7T137−69T136−541T135+908T134+11144T133+9019T132−44622T13−62716 T_{13}^{8} + 7T_{13}^{7} - 69T_{13}^{6} - 541T_{13}^{5} + 908T_{13}^{4} + 11144T_{13}^{3} + 9019T_{13}^{2} - 44622T_{13} - 62716 Copy content Toggle raw display
T178−T177−79T176+25T175+2019T174+598T173−17295T172−10532T17+31364 T_{17}^{8} - T_{17}^{7} - 79T_{17}^{6} + 25T_{17}^{5} + 2019T_{17}^{4} + 598T_{17}^{3} - 17295T_{17}^{2} - 10532T_{17} + 31364 Copy content Toggle raw display
T198−6T197−55T196+503T195−809T194−2617T193+9969T192−10970T19+3980 T_{19}^{8} - 6T_{19}^{7} - 55T_{19}^{6} + 503T_{19}^{5} - 809T_{19}^{4} - 2617T_{19}^{3} + 9969T_{19}^{2} - 10970T_{19} + 3980 Copy content Toggle raw display
T238+7T237−80T236−548T235+431T234+4440T233−1036T232−7409T23+3169 T_{23}^{8} + 7T_{23}^{7} - 80T_{23}^{6} - 548T_{23}^{5} + 431T_{23}^{4} + 4440T_{23}^{3} - 1036T_{23}^{2} - 7409T_{23} + 3169 Copy content Toggle raw display
T378+14T377−71T376−1657T375−3023T374+34702T373+⋯−542461 T_{37}^{8} + 14 T_{37}^{7} - 71 T_{37}^{6} - 1657 T_{37}^{5} - 3023 T_{37}^{4} + 34702 T_{37}^{3} + \cdots - 542461 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T8+T7−11T6+⋯+5 T^{8} + T^{7} - 11 T^{6} + \cdots + 5 Copy content Toggle raw display
33 (T+1)8 (T + 1)^{8} Copy content Toggle raw display
55 T8 T^{8} Copy content Toggle raw display
77 T8+8T7+⋯−956 T^{8} + 8 T^{7} + \cdots - 956 Copy content Toggle raw display
1111 T8 T^{8} Copy content Toggle raw display
1313 T8+7T7+⋯−62716 T^{8} + 7 T^{7} + \cdots - 62716 Copy content Toggle raw display
1717 T8−T7+⋯+31364 T^{8} - T^{7} + \cdots + 31364 Copy content Toggle raw display
1919 T8−6T7+⋯+3980 T^{8} - 6 T^{7} + \cdots + 3980 Copy content Toggle raw display
2323 T8+7T7+⋯+3169 T^{8} + 7 T^{7} + \cdots + 3169 Copy content Toggle raw display
2929 T8−28T7+⋯−49849 T^{8} - 28 T^{7} + \cdots - 49849 Copy content Toggle raw display
3131 T8−10T7+⋯+30689 T^{8} - 10 T^{7} + \cdots + 30689 Copy content Toggle raw display
3737 T8+14T7+⋯−542461 T^{8} + 14 T^{7} + \cdots - 542461 Copy content Toggle raw display
4141 T8−14T7+⋯−2253521 T^{8} - 14 T^{7} + \cdots - 2253521 Copy content Toggle raw display
4343 T8+11T7+⋯+5045 T^{8} + 11 T^{7} + \cdots + 5045 Copy content Toggle raw display
4747 T8+5T7+⋯+4632625 T^{8} + 5 T^{7} + \cdots + 4632625 Copy content Toggle raw display
5353 T8−5T7+⋯−16004 T^{8} - 5 T^{7} + \cdots - 16004 Copy content Toggle raw display
5959 T8+2T7+⋯−59695 T^{8} + 2 T^{7} + \cdots - 59695 Copy content Toggle raw display
6161 T8−36T7+⋯+180080 T^{8} - 36 T^{7} + \cdots + 180080 Copy content Toggle raw display
6767 T8+6T7+⋯+45905 T^{8} + 6 T^{7} + \cdots + 45905 Copy content Toggle raw display
7171 T8+17T7+⋯+123920 T^{8} + 17 T^{7} + \cdots + 123920 Copy content Toggle raw display
7373 T8+32T7+⋯+994769 T^{8} + 32 T^{7} + \cdots + 994769 Copy content Toggle raw display
7979 T8−41T7+⋯+1546669 T^{8} - 41 T^{7} + \cdots + 1546669 Copy content Toggle raw display
8383 T8−6T7+⋯+498884 T^{8} - 6 T^{7} + \cdots + 498884 Copy content Toggle raw display
8989 T8−21T7+⋯−60824801 T^{8} - 21 T^{7} + \cdots - 60824801 Copy content Toggle raw display
9797 T8−4T7+⋯+45758221 T^{8} - 4 T^{7} + \cdots + 45758221 Copy content Toggle raw display
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