Properties

Label 9075.2.a.dr
Level 90759075
Weight 22
Character orbit 9075.a
Self dual yes
Analytic conductor 72.46472.464
Analytic rank 11
Dimension 66
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=352112 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: 11
Dimension: 66
Coefficient field: 6.6.860280160.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x6x510x4+9x3+23x222x2 x^{6} - x^{5} - 10x^{4} + 9x^{3} + 23x^{2} - 22x - 2 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 1815)
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,,β51,\beta_1,\ldots,\beta_{5} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β1q2q3+(β2+2)q4β1q6+(β41)q7+(β3+β2+β1)q8+q9+(β22)q12+(β5β4)q13++(β5+3β2+2β1+10)q98+O(q100) q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 2) q^{4} - \beta_1 q^{6} + ( - \beta_{4} - 1) q^{7} + (\beta_{3} + \beta_{2} + \beta_1) q^{8} + q^{9} + ( - \beta_{2} - 2) q^{12} + (\beta_{5} - \beta_{4}) q^{13}+ \cdots + (\beta_{5} + 3 \beta_{2} + 2 \beta_1 + 10) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 6q+q26q3+9q4q64q7+6q99q12+2q1310q14+11q16+2q17+q18+2q19+4q2112q236q2724q2812q29+18q31++53q98+O(q100) 6 q + q^{2} - 6 q^{3} + 9 q^{4} - q^{6} - 4 q^{7} + 6 q^{9} - 9 q^{12} + 2 q^{13} - 10 q^{14} + 11 q^{16} + 2 q^{17} + q^{18} + 2 q^{19} + 4 q^{21} - 12 q^{23} - 6 q^{27} - 24 q^{28} - 12 q^{29} + 18 q^{31}+ \cdots + 53 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x6x510x4+9x3+23x222x2 x^{6} - x^{5} - 10x^{4} + 9x^{3} + 23x^{2} - 22x - 2 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== ν24 \nu^{2} - 4 Copy content Toggle raw display
β3\beta_{3}== ν3ν25ν+4 \nu^{3} - \nu^{2} - 5\nu + 4 Copy content Toggle raw display
β4\beta_{4}== ν59ν3ν2+16ν2 \nu^{5} - 9\nu^{3} - \nu^{2} + 16\nu - 2 Copy content Toggle raw display
β5\beta_{5}== ν5+ν410ν38ν2+20ν+4 \nu^{5} + \nu^{4} - 10\nu^{3} - 8\nu^{2} + 20\nu + 4 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β2+4 \beta_{2} + 4 Copy content Toggle raw display
ν3\nu^{3}== β3+β2+5β1 \beta_{3} + \beta_{2} + 5\beta_1 Copy content Toggle raw display
ν4\nu^{4}== β5β4+β3+8β2+β1+22 \beta_{5} - \beta_{4} + \beta_{3} + 8\beta_{2} + \beta _1 + 22 Copy content Toggle raw display
ν5\nu^{5}== β4+9β3+10β2+29β1+6 \beta_{4} + 9\beta_{3} + 10\beta_{2} + 29\beta _1 + 6 Copy content Toggle raw display

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
−2.42911
−1.95455
−0.0838261
1.19557
1.53672
2.73519
−2.42911 −1.00000 3.90056 0 2.42911 1.34168 −4.61665 1.00000 0
1.2 −1.95455 −1.00000 1.82026 0 1.95455 −2.58348 0.351308 1.00000 0
1.3 −0.0838261 −1.00000 −1.99297 0 0.0838261 2.34295 0.334715 1.00000 0
1.4 1.19557 −1.00000 −0.570614 0 −1.19557 −3.76208 −3.07335 1.00000 0
1.5 1.53672 −1.00000 0.361523 0 −1.53672 2.86503 −2.51789 1.00000 0
1.6 2.73519 −1.00000 5.48125 0 −2.73519 −4.20410 9.52186 1.00000 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.dr 6
5.b even 2 1 9075.2.a.dp 6
5.c odd 4 2 1815.2.c.h 12
11.b odd 2 1 9075.2.a.do 6
55.d odd 2 1 9075.2.a.ds 6
55.e even 4 2 1815.2.c.i yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.c.h 12 5.c odd 4 2
1815.2.c.i yes 12 55.e even 4 2
9075.2.a.do 6 11.b odd 2 1
9075.2.a.dp 6 5.b even 2 1
9075.2.a.dr 6 1.a even 1 1 trivial
9075.2.a.ds 6 55.d odd 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)):

T26T2510T24+9T23+23T2222T22 T_{2}^{6} - T_{2}^{5} - 10T_{2}^{4} + 9T_{2}^{3} + 23T_{2}^{2} - 22T_{2} - 2 Copy content Toggle raw display
T76+4T7519T7462T73+136T72+232T7368 T_{7}^{6} + 4T_{7}^{5} - 19T_{7}^{4} - 62T_{7}^{3} + 136T_{7}^{2} + 232T_{7} - 368 Copy content Toggle raw display
T1362T13547T134+110T133+300T132432T13+32 T_{13}^{6} - 2T_{13}^{5} - 47T_{13}^{4} + 110T_{13}^{3} + 300T_{13}^{2} - 432T_{13} + 32 Copy content Toggle raw display
T1762T17568T174+106T173+1025T1722036T17+46 T_{17}^{6} - 2T_{17}^{5} - 68T_{17}^{4} + 106T_{17}^{3} + 1025T_{17}^{2} - 2036T_{17} + 46 Copy content Toggle raw display
T1962T19547T194+64T193+368T192128T19512 T_{19}^{6} - 2T_{19}^{5} - 47T_{19}^{4} + 64T_{19}^{3} + 368T_{19}^{2} - 128T_{19} - 512 Copy content Toggle raw display
T236+12T23521T234564T233528T232+4656T231072 T_{23}^{6} + 12T_{23}^{5} - 21T_{23}^{4} - 564T_{23}^{3} - 528T_{23}^{2} + 4656T_{23} - 1072 Copy content Toggle raw display
T376+24T375+126T374858T3737687T372+4710T37+98540 T_{37}^{6} + 24T_{37}^{5} + 126T_{37}^{4} - 858T_{37}^{3} - 7687T_{37}^{2} + 4710T_{37} + 98540 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T6T510T4+2 T^{6} - T^{5} - 10 T^{4} + \cdots - 2 Copy content Toggle raw display
33 (T+1)6 (T + 1)^{6} Copy content Toggle raw display
55 T6 T^{6} Copy content Toggle raw display
77 T6+4T5+368 T^{6} + 4 T^{5} + \cdots - 368 Copy content Toggle raw display
1111 T6 T^{6} Copy content Toggle raw display
1313 T62T5++32 T^{6} - 2 T^{5} + \cdots + 32 Copy content Toggle raw display
1717 T62T5++46 T^{6} - 2 T^{5} + \cdots + 46 Copy content Toggle raw display
1919 T62T5+512 T^{6} - 2 T^{5} + \cdots - 512 Copy content Toggle raw display
2323 T6+12T5+1072 T^{6} + 12 T^{5} + \cdots - 1072 Copy content Toggle raw display
2929 T6+12T5+7688 T^{6} + 12 T^{5} + \cdots - 7688 Copy content Toggle raw display
3131 T618T5+2656 T^{6} - 18 T^{5} + \cdots - 2656 Copy content Toggle raw display
3737 T6+24T5++98540 T^{6} + 24 T^{5} + \cdots + 98540 Copy content Toggle raw display
4141 T66T5+52544 T^{6} - 6 T^{5} + \cdots - 52544 Copy content Toggle raw display
4343 T610T5++16960 T^{6} - 10 T^{5} + \cdots + 16960 Copy content Toggle raw display
4747 T6+8T5+160 T^{6} + 8 T^{5} + \cdots - 160 Copy content Toggle raw display
5353 T6+18T5+9188 T^{6} + 18 T^{5} + \cdots - 9188 Copy content Toggle raw display
5959 T6+18T5+8224 T^{6} + 18 T^{5} + \cdots - 8224 Copy content Toggle raw display
6161 T6+4T5+104284 T^{6} + 4 T^{5} + \cdots - 104284 Copy content Toggle raw display
6767 T6+12T5+464 T^{6} + 12 T^{5} + \cdots - 464 Copy content Toggle raw display
7171 T6130T4++11744 T^{6} - 130 T^{4} + \cdots + 11744 Copy content Toggle raw display
7373 T614T5++44288 T^{6} - 14 T^{5} + \cdots + 44288 Copy content Toggle raw display
7979 T6+2T5+196624 T^{6} + 2 T^{5} + \cdots - 196624 Copy content Toggle raw display
8383 T6+20T5++266240 T^{6} + 20 T^{5} + \cdots + 266240 Copy content Toggle raw display
8989 T6+16T5+80776 T^{6} + 16 T^{5} + \cdots - 80776 Copy content Toggle raw display
9797 T6+12T5++23572 T^{6} + 12 T^{5} + \cdots + 23572 Copy content Toggle raw display
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