Properties

Label 605.4.a.o
Level 605605
Weight 44
Character orbit 605.a
Self dual yes
Analytic conductor 35.69635.696
Analytic rank 11
Dimension 88
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] = [605,4,Mod(1,605)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(605, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("605.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: N N == 605=5112 605 = 5 \cdot 11^{2}
Weight: k k == 4 4
Character orbit: [χ][\chi] == 605.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,-24,18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 35.696155553535.6961555535
Analytic rank: 11
Dimension: 88
Coefficient field: Q[x]/(x8)\mathbb{Q}[x]/(x^{8} - \cdots)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x835x6+376x41452x2+1764 x^{8} - 35x^{6} + 376x^{4} - 1452x^{2} + 1764 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 23 2\cdot 3
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,,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+(β3+β1)q2+(β43)q3+(β5+β4+2)q4+5q5+(β66β3+7β1)q6+(β64β3β2)q7++(54β6+143β3++β1)q98+O(q100) q + (\beta_{3} + \beta_1) q^{2} + ( - \beta_{4} - 3) q^{3} + ( - \beta_{5} + \beta_{4} + 2) q^{4} + 5 q^{5} + (\beta_{6} - 6 \beta_{3} + \cdots - 7 \beta_1) q^{6} + (\beta_{6} - 4 \beta_{3} - \beta_{2}) q^{7}+ \cdots + ( - 54 \beta_{6} + 143 \beta_{3} + \cdots + \beta_1) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q24q3+18q4+40q5+104q9362q1270q14120q1542q16+90q20572q23+200q25+692q261044q27492q31600q34+2172q36+1864q97+O(q100) 8 q - 24 q^{3} + 18 q^{4} + 40 q^{5} + 104 q^{9} - 362 q^{12} - 70 q^{14} - 120 q^{15} - 42 q^{16} + 90 q^{20} - 572 q^{23} + 200 q^{25} + 692 q^{26} - 1044 q^{27} - 492 q^{31} - 600 q^{34} + 2172 q^{36}+ \cdots - 1864 q^{97}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x835x6+376x41452x2+1764 x^{8} - 35x^{6} + 376x^{4} - 1452x^{2} + 1764 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (2ν7+49ν5143ν31086ν)/126 ( -2\nu^{7} + 49\nu^{5} - 143\nu^{3} - 1086\nu ) / 126 Copy content Toggle raw display
β3\beta_{3}== (5ν7+154ν51271ν3+2640ν)/378 ( -5\nu^{7} + 154\nu^{5} - 1271\nu^{3} + 2640\nu ) / 378 Copy content Toggle raw display
β4\beta_{4}== (ν6+29ν4202ν2+276)/18 ( -\nu^{6} + 29\nu^{4} - 202\nu^{2} + 276 ) / 18 Copy content Toggle raw display
β5\beta_{5}== (ν629ν4+220ν2438)/18 ( \nu^{6} - 29\nu^{4} + 220\nu^{2} - 438 ) / 18 Copy content Toggle raw display
β6\beta_{6}== (19ν7+623ν55548ν3+11544ν)/378 ( -19\nu^{7} + 623\nu^{5} - 5548\nu^{3} + 11544\nu ) / 378 Copy content Toggle raw display
β7\beta_{7}== (2ν6+67ν4620ν2+1380)/9 ( -2\nu^{6} + 67\nu^{4} - 620\nu^{2} + 1380 ) / 9 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β5+β4+9 \beta_{5} + \beta_{4} + 9 Copy content Toggle raw display
ν3\nu^{3}== β65β3+β2+13β1 \beta_{6} - 5\beta_{3} + \beta_{2} + 13\beta_1 Copy content Toggle raw display
ν4\nu^{4}== β7+24β5+20β4+124 \beta_{7} + 24\beta_{5} + 20\beta_{4} + 124 Copy content Toggle raw display
ν5\nu^{5}== 29β6133β3+19β2+207β1 29\beta_{6} - 133\beta_{3} + 19\beta_{2} + 207\beta_1 Copy content Toggle raw display
ν6\nu^{6}== 29β7+494β5+360β4+2054 29\beta_{7} + 494\beta_{5} + 360\beta_{4} + 2054 Copy content Toggle raw display
ν7\nu^{7}== 639β62901β3+331β2+3599β1 639\beta_{6} - 2901\beta_{3} + 331\beta_{2} + 3599\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
−3.10041
−1.54002
−4.35061
−2.02187
2.02187
4.35061
1.54002
3.10041
−4.83246 −9.98262 15.3527 5.00000 48.2407 23.0737 −35.5316 72.6528 −24.1623
1.2 −3.27207 −0.0390529 2.70645 5.00000 0.127784 −14.1165 17.3209 −26.9985 −16.3604
1.3 −2.61856 −6.39235 −1.14314 5.00000 16.7387 −10.2514 23.9419 13.8621 −13.0928
1.4 −0.289819 4.41402 −7.91600 5.00000 −1.27927 −11.9683 4.61277 −7.51640 −1.44910
1.5 0.289819 4.41402 −7.91600 5.00000 1.27927 11.9683 −4.61277 −7.51640 1.44910
1.6 2.61856 −6.39235 −1.14314 5.00000 −16.7387 10.2514 −23.9419 13.8621 13.0928
1.7 3.27207 −0.0390529 2.70645 5.00000 −0.127784 14.1165 −17.3209 −26.9985 16.3604
1.8 4.83246 −9.98262 15.3527 5.00000 −48.2407 −23.0737 35.5316 72.6528 24.1623
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
55 1 -1
1111 +1 +1

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.4.a.o 8
11.b odd 2 1 inner 605.4.a.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.4.a.o 8 1.a even 1 1 trivial
605.4.a.o 8 11.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S4new(Γ0(605))S_{4}^{\mathrm{new}}(\Gamma_0(605)):

T2841T26+487T241755T22+144 T_{2}^{8} - 41T_{2}^{6} + 487T_{2}^{4} - 1755T_{2}^{2} + 144 Copy content Toggle raw display
T34+12T338T32282T311 T_{3}^{4} + 12T_{3}^{3} - 8T_{3}^{2} - 282T_{3} - 11 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T841T6++144 T^{8} - 41 T^{6} + \cdots + 144 Copy content Toggle raw display
33 (T4+12T3+11)2 (T^{4} + 12 T^{3} + \cdots - 11)^{2} Copy content Toggle raw display
55 (T5)8 (T - 5)^{8} Copy content Toggle raw display
77 T8++1597041369 T^{8} + \cdots + 1597041369 Copy content Toggle raw display
1111 T8 T^{8} Copy content Toggle raw display
1313 T8++6219956192256 T^{8} + \cdots + 6219956192256 Copy content Toggle raw display
1717 T8++20351214157824 T^{8} + \cdots + 20351214157824 Copy content Toggle raw display
1919 T8++11606940354816 T^{8} + \cdots + 11606940354816 Copy content Toggle raw display
2323 (T4+286T3++5778720)2 (T^{4} + 286 T^{3} + \cdots + 5778720)^{2} Copy content Toggle raw display
2929 T8++13 ⁣ ⁣00 T^{8} + \cdots + 13\!\cdots\!00 Copy content Toggle raw display
3131 (T4+246T3+84685544)2 (T^{4} + 246 T^{3} + \cdots - 84685544)^{2} Copy content Toggle raw display
3737 (T4+16T3+446406528)2 (T^{4} + 16 T^{3} + \cdots - 446406528)^{2} Copy content Toggle raw display
4141 T8++21 ⁣ ⁣49 T^{8} + \cdots + 21\!\cdots\!49 Copy content Toggle raw display
4343 T8++10 ⁣ ⁣25 T^{8} + \cdots + 10\!\cdots\!25 Copy content Toggle raw display
4747 (T4+236T3++6087784941)2 (T^{4} + 236 T^{3} + \cdots + 6087784941)^{2} Copy content Toggle raw display
5353 (T4+756T3+8094368016)2 (T^{4} + 756 T^{3} + \cdots - 8094368016)^{2} Copy content Toggle raw display
5959 (T454T3++2928268152)2 (T^{4} - 54 T^{3} + \cdots + 2928268152)^{2} Copy content Toggle raw display
6161 T8++61 ⁣ ⁣25 T^{8} + \cdots + 61\!\cdots\!25 Copy content Toggle raw display
6767 (T4+2392T3++91567885245)2 (T^{4} + 2392 T^{3} + \cdots + 91567885245)^{2} Copy content Toggle raw display
7171 (T4+1454T3++903758616)2 (T^{4} + 1454 T^{3} + \cdots + 903758616)^{2} Copy content Toggle raw display
7373 T8++68 ⁣ ⁣00 T^{8} + \cdots + 68\!\cdots\!00 Copy content Toggle raw display
7979 T8++56 ⁣ ⁣96 T^{8} + \cdots + 56\!\cdots\!96 Copy content Toggle raw display
8383 T8++19 ⁣ ⁣96 T^{8} + \cdots + 19\!\cdots\!96 Copy content Toggle raw display
8989 (T4+160T3++339659726877)2 (T^{4} + 160 T^{3} + \cdots + 339659726877)^{2} Copy content Toggle raw display
9797 (T4+932T3+237654925680)2 (T^{4} + 932 T^{3} + \cdots - 237654925680)^{2} Copy content Toggle raw display
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