Properties

Label 2744.2.a.h
Level 27442744
Weight 22
Character orbit 2744.a
Self dual yes
Analytic conductor 21.91121.911
Analytic rank 00
Dimension 1212
Inner twists 22

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2744,2,Mod(1,2744)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2744, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2744.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 2744=2373 2744 = 2^{3} \cdot 7^{3}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 2744.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,0,0,0,22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 21.910950314621.9109503146
Analytic rank: 00
Dimension: 1212
Coefficient field: Q[x]/(x12)\mathbb{Q}[x]/(x^{12} - \cdots)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x1229x10+304x81393x6+2574x41164x2+8 x^{12} - 29x^{10} + 304x^{8} - 1393x^{6} + 2574x^{4} - 1164x^{2} + 8 Copy content Toggle raw display
Coefficient ring: Z[a1,,a11]\Z[a_1, \ldots, a_{11}]
Coefficient ring index: 1 1
Twist minimal: yes
Fricke sign: 1-1
Sato-Tate group: SU(2)\mathrm{SU}(2)

qq-expansion

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

Coefficients of the qq-expansion are expressed in terms of a basis 1,β1,,β111,\beta_1,\ldots,\beta_{11} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β1q3β10q5+(β3+β2+2)q9+(β7β6+β3+2)q11+(β102β8+β5)q13+(β62β4+β2+3)q15++(β77β62β4++8)q99+O(q100) q + \beta_1 q^{3} - \beta_{10} q^{5} + (\beta_{3} + \beta_{2} + 2) q^{9} + ( - \beta_{7} - \beta_{6} + \beta_{3} + 2) q^{11} + ( - \beta_{10} - 2 \beta_{8} + \beta_{5}) q^{13} + (\beta_{6} - 2 \beta_{4} + \beta_{2} + 3) q^{15}+ \cdots + ( - \beta_{7} - 7 \beta_{6} - 2 \beta_{4} + \cdots + 8) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 12q+22q9+18q11+28q15+20q23+14q2518q2918q37+36q39+10q43+48q5138q53+12q57+8q65+42q67+56q71+56q79+52q81++90q99+O(q100) 12 q + 22 q^{9} + 18 q^{11} + 28 q^{15} + 20 q^{23} + 14 q^{25} - 18 q^{29} - 18 q^{37} + 36 q^{39} + 10 q^{43} + 48 q^{51} - 38 q^{53} + 12 q^{57} + 8 q^{65} + 42 q^{67} + 56 q^{71} + 56 q^{79} + 52 q^{81}+ \cdots + 90 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x1229x10+304x81393x6+2574x41164x2+8 x^{12} - 29x^{10} + 304x^{8} - 1393x^{6} + 2574x^{4} - 1164x^{2} + 8 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (9ν10+680ν813379ν6+97733ν4226709ν2+1782)/18452 ( -9\nu^{10} + 680\nu^{8} - 13379\nu^{6} + 97733\nu^{4} - 226709\nu^{2} + 1782 ) / 18452 Copy content Toggle raw display
β3\beta_{3}== (9ν10680ν8+13379ν697733ν4+245161ν294042)/18452 ( 9\nu^{10} - 680\nu^{8} + 13379\nu^{6} - 97733\nu^{4} + 245161\nu^{2} - 94042 ) / 18452 Copy content Toggle raw display
β4\beta_{4}== (37ν10+2283ν836038ν6+196769ν4310296ν2+35004)/73808 ( -37\nu^{10} + 2283\nu^{8} - 36038\nu^{6} + 196769\nu^{4} - 310296\nu^{2} + 35004 ) / 73808 Copy content Toggle raw display
β5\beta_{5}== (37ν11+2283ν936038ν7+196769ν5310296ν3+35004ν)/73808 ( -37\nu^{11} + 2283\nu^{9} - 36038\nu^{7} + 196769\nu^{5} - 310296\nu^{3} + 35004\nu ) / 73808 Copy content Toggle raw display
β6\beta_{6}== (30ν10729ν8+6155ν624394ν4+52983ν233618)/18452 ( 30\nu^{10} - 729\nu^{8} + 6155\nu^{6} - 24394\nu^{4} + 52983\nu^{2} - 33618 ) / 18452 Copy content Toggle raw display
β7\beta_{7}== (1395ν1036205ν8+325418ν61198903ν4+1543416ν2174724)/73808 ( 1395\nu^{10} - 36205\nu^{8} + 325418\nu^{6} - 1198903\nu^{4} + 1543416\nu^{2} - 174724 ) / 73808 Copy content Toggle raw display
β8\beta_{8}== (2299ν11+65553ν9670806ν7+2962367ν55110644ν3+1866780ν)/73808 ( -2299\nu^{11} + 65553\nu^{9} - 670806\nu^{7} + 2962367\nu^{5} - 5110644\nu^{3} + 1866780\nu ) / 73808 Copy content Toggle raw display
β9\beta_{9}== (2419ν1168469ν9+695426ν73059943ν5+5322576ν31927444ν)/73808 ( 2419\nu^{11} - 68469\nu^{9} + 695426\nu^{7} - 3059943\nu^{5} + 5322576\nu^{3} - 1927444\nu ) / 73808 Copy content Toggle raw display
β10\beta_{10}== (3009ν1187419ν9+919498ν74234717ν5+7845348ν33428164ν)/73808 ( 3009\nu^{11} - 87419\nu^{9} + 919498\nu^{7} - 4234717\nu^{5} + 7845348\nu^{3} - 3428164\nu ) / 73808 Copy content Toggle raw display
β11\beta_{11}== (1643ν1147767ν9+502388ν72313319ν5+4326658ν32087480ν)/36904 ( 1643\nu^{11} - 47767\nu^{9} + 502388\nu^{7} - 2313319\nu^{5} + 4326658\nu^{3} - 2087480\nu ) / 36904 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β3+β2+5 \beta_{3} + \beta_{2} + 5 Copy content Toggle raw display
ν3\nu^{3}== β11β102β92β8+β5+8β1 \beta_{11} - \beta_{10} - 2\beta_{9} - 2\beta_{8} + \beta_{5} + 8\beta_1 Copy content Toggle raw display
ν4\nu^{4}== β711β6+3β4+11β3+10β2+36 \beta_{7} - 11\beta_{6} + 3\beta_{4} + 11\beta_{3} + 10\beta_{2} + 36 Copy content Toggle raw display
ν5\nu^{5}== 9β119β1035β936β8+16β5+80β1 9\beta_{11} - 9\beta_{10} - 35\beta_{9} - 36\beta_{8} + 16\beta_{5} + 80\beta_1 Copy content Toggle raw display
ν6\nu^{6}== 17β7186β6+63β4+124β3+98β2+294 17\beta_{7} - 186\beta_{6} + 63\beta_{4} + 124\beta_{3} + 98\beta_{2} + 294 Copy content Toggle raw display
ν7\nu^{7}== 72β1163β10477β9485β8+221β5+852β1 72\beta_{11} - 63\beta_{10} - 477\beta_{9} - 485\beta_{8} + 221\beta_{5} + 852\beta_1 Copy content Toggle raw display
ν8\nu^{8}== 229β72482β6+987β4+1401β3+987β2+2597 229\beta_{7} - 2482\beta_{6} + 987\beta_{4} + 1401\beta_{3} + 987\beta_{2} + 2597 Copy content Toggle raw display
ν9\nu^{9}== 573β11388β105927β95971β8+2846β5+9282β1 573\beta_{11} - 388\beta_{10} - 5927\beta_{9} - 5971\beta_{8} + 2846\beta_{5} + 9282\beta_1 Copy content Toggle raw display
ν10\nu^{10}== 2890β730481β6+13498β4+15782β3+10243β2+24351 2890\beta_{7} - 30481\beta_{6} + 13498\beta_{4} + 15782\beta_{3} + 10243\beta_{2} + 24351 Copy content Toggle raw display
ν11\nu^{11}== 4704β112055β1070474β970715β8+35060β5+102178β1 4704\beta_{11} - 2055\beta_{10} - 70474\beta_{9} - 70715\beta_{8} + 35060\beta_{5} + 102178\beta_1 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−3.35261
−2.74923
−2.46901
−1.85982
−0.799876
−0.0835475
0.0835475
0.799876
1.85982
2.46901
2.74923
3.35261
0 −3.35261 0 −0.758778 0 0 0 8.24000 0
1.2 0 −2.74923 0 −3.64717 0 0 0 4.55826 0
1.3 0 −2.46901 0 −2.10686 0 0 0 3.09602 0
1.4 0 −1.85982 0 1.80700 0 0 0 0.458937 0
1.5 0 −0.799876 0 0.913902 0 0 0 −2.36020 0
1.6 0 −0.0835475 0 −3.81878 0 0 0 −2.99302 0
1.7 0 0.0835475 0 3.81878 0 0 0 −2.99302 0
1.8 0 0.799876 0 −0.913902 0 0 0 −2.36020 0
1.9 0 1.85982 0 −1.80700 0 0 0 0.458937 0
1.10 0 2.46901 0 2.10686 0 0 0 3.09602 0
1.11 0 2.74923 0 3.64717 0 0 0 4.55826 0
1.12 0 3.35261 0 0.758778 0 0 0 8.24000 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
22 +1 +1
77 1 -1

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2744.2.a.h 12
4.b odd 2 1 5488.2.a.w 12
7.b odd 2 1 inner 2744.2.a.h 12
28.d even 2 1 5488.2.a.w 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2744.2.a.h 12 1.a even 1 1 trivial
2744.2.a.h 12 7.b odd 2 1 inner
5488.2.a.w 12 4.b odd 2 1
5488.2.a.w 12 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T31229T310+304T381393T36+2574T341164T32+8 T_{3}^{12} - 29T_{3}^{10} + 304T_{3}^{8} - 1393T_{3}^{6} + 2574T_{3}^{4} - 1164T_{3}^{2} + 8 acting on S2new(Γ0(2744))S_{2}^{\mathrm{new}}(\Gamma_0(2744)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T12 T^{12} Copy content Toggle raw display
33 T1229T10++8 T^{12} - 29 T^{10} + \cdots + 8 Copy content Toggle raw display
55 T1237T10++1352 T^{12} - 37 T^{10} + \cdots + 1352 Copy content Toggle raw display
77 T12 T^{12} Copy content Toggle raw display
1111 (T69T5++728)2 (T^{6} - 9 T^{5} + \cdots + 728)^{2} Copy content Toggle raw display
1313 T12129T10++724808 T^{12} - 129 T^{10} + \cdots + 724808 Copy content Toggle raw display
1717 T12108T10++86528 T^{12} - 108 T^{10} + \cdots + 86528 Copy content Toggle raw display
1919 T12162T10++73544192 T^{12} - 162 T^{10} + \cdots + 73544192 Copy content Toggle raw display
2323 (T610T5++2639)2 (T^{6} - 10 T^{5} + \cdots + 2639)^{2} Copy content Toggle raw display
2929 (T6+9T5++9304)2 (T^{6} + 9 T^{5} + \cdots + 9304)^{2} Copy content Toggle raw display
3131 T12++5776835072 T^{12} + \cdots + 5776835072 Copy content Toggle raw display
3737 (T6+9T5+28904)2 (T^{6} + 9 T^{5} + \cdots - 28904)^{2} Copy content Toggle raw display
4141 T12276T10++512 T^{12} - 276 T^{10} + \cdots + 512 Copy content Toggle raw display
4343 (T65T5+5384)2 (T^{6} - 5 T^{5} + \cdots - 5384)^{2} Copy content Toggle raw display
4747 T1274T10++86528 T^{12} - 74 T^{10} + \cdots + 86528 Copy content Toggle raw display
5353 (T6+19T5++24408)2 (T^{6} + 19 T^{5} + \cdots + 24408)^{2} Copy content Toggle raw display
5959 T12++42838986632 T^{12} + \cdots + 42838986632 Copy content Toggle raw display
6161 T12193T10++3623432 T^{12} - 193 T^{10} + \cdots + 3623432 Copy content Toggle raw display
6767 (T621T5++5144)2 (T^{6} - 21 T^{5} + \cdots + 5144)^{2} Copy content Toggle raw display
7171 (T628T5++323743)2 (T^{6} - 28 T^{5} + \cdots + 323743)^{2} Copy content Toggle raw display
7373 T12++97067704832 T^{12} + \cdots + 97067704832 Copy content Toggle raw display
7979 (T628T5++41957)2 (T^{6} - 28 T^{5} + \cdots + 41957)^{2} Copy content Toggle raw display
8383 T12++5051733128 T^{12} + \cdots + 5051733128 Copy content Toggle raw display
8989 T12++45858455552 T^{12} + \cdots + 45858455552 Copy content Toggle raw display
9797 T12++796164608 T^{12} + \cdots + 796164608 Copy content Toggle raw display
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