Properties

Label 504.2.cc.a
Level 504504
Weight 22
Character orbit 504.cc
Analytic conductor 4.0244.024
Analytic rank 00
Dimension 1616
CM discriminant -56
Inner twists 88

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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] = [504,2,Mod(293,504)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(504, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 5, 3])) 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("504.293"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 504=23327 504 = 2^{3} \cdot 3^{2} \cdot 7
Weight: k k == 2 2
Character orbit: [χ][\chi] == 504.cc (of order 66, degree 22, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 4.024460261874.02446026187
Analytic rank: 00
Dimension: 1616
Relative dimension: 88 over Q(ζ6)\Q(\zeta_{6})
Coefficient field: Q[x]/(x16+)\mathbb{Q}[x]/(x^{16} + \cdots)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x16+10x12+19x8+810x4+6561 x^{16} + 10x^{12} + 19x^{8} + 810x^{4} + 6561 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 263 2^{6}\cdot 3
Twist minimal: yes
Sato-Tate group: U(1)[D6]\mathrm{U}(1)[D_{6}]

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,,β151,\beta_1,\ldots,\beta_{15} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β11q2β14q3+(2β4+2)q4+(β15+β10++β8)q5+β1q6β12q7+2β2q8+(β13β12+β2)q9+7β2q98+O(q100) q + \beta_{11} q^{2} - \beta_{14} q^{3} + ( - 2 \beta_{4} + 2) q^{4} + (\beta_{15} + \beta_{10} + \cdots + \beta_{8}) q^{5} + \beta_1 q^{6} - \beta_{12} q^{7} + 2 \beta_{2} q^{8} + (\beta_{13} - \beta_{12} + \beta_{2}) q^{9}+ \cdots - 7 \beta_{2} q^{98}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 16q+16q416q1532q1616q18+72q23+40q25+32q3040q3956q49144q50+8q5716q60+56q63128q6472q6564q72+128q78++144q92+O(q100) 16 q + 16 q^{4} - 16 q^{15} - 32 q^{16} - 16 q^{18} + 72 q^{23} + 40 q^{25} + 32 q^{30} - 40 q^{39} - 56 q^{49} - 144 q^{50} + 8 q^{57} - 16 q^{60} + 56 q^{63} - 128 q^{64} - 72 q^{65} - 64 q^{72} + 128 q^{78}+ \cdots + 144 q^{92}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x16+10x12+19x8+810x4+6561 x^{16} + 10x^{12} + 19x^{8} + 810x^{4} + 6561 : Copy content Toggle raw display

β1\beta_{1}== (10ν13+19ν91349ν5+6561ν)/9234 ( 10\nu^{13} + 19\nu^{9} - 1349\nu^{5} + 6561\nu ) / 9234 Copy content Toggle raw display
β2\beta_{2}== (10ν14+19ν101349ν6+6561ν2)/27702 ( 10\nu^{14} + 19\nu^{10} - 1349\nu^{6} + 6561\nu^{2} ) / 27702 Copy content Toggle raw display
β3\beta_{3}== (11ν13+133ν9209ν5+324ν)/9234 ( -11\nu^{13} + 133\nu^{9} - 209\nu^{5} + 324\nu ) / 9234 Copy content Toggle raw display
β4\beta_{4}== (10ν12+19ν8+190ν4+8100)/1539 ( 10\nu^{12} + 19\nu^{8} + 190\nu^{4} + 8100 ) / 1539 Copy content Toggle raw display
β5\beta_{5}== (31ν12+95ν8589ν425110)/3078 ( -31\nu^{12} + 95\nu^{8} - 589\nu^{4} - 25110 ) / 3078 Copy content Toggle raw display
β6\beta_{6}== (ν1510ν1119ν7810ν3)/2187 ( -\nu^{15} - 10\nu^{11} - 19\nu^{7} - 810\nu^{3} ) / 2187 Copy content Toggle raw display
β7\beta_{7}== (50ν12+95ν8589ν4+32805)/3078 ( 50\nu^{12} + 95\nu^{8} - 589\nu^{4} + 32805 ) / 3078 Copy content Toggle raw display
β8\beta_{8}== (70ν13133ν9+209ν536693ν)/9234 ( -70\nu^{13} - 133\nu^{9} + 209\nu^{5} - 36693\nu ) / 9234 Copy content Toggle raw display
β9\beta_{9}== (ν151133ν3+1026ν)/1026 ( -\nu^{15} - 1133\nu^{3} + 1026\nu ) / 1026 Copy content Toggle raw display
β10\beta_{10}== (ν13+791ν)/114 ( \nu^{13} + 791\nu ) / 114 Copy content Toggle raw display
β11\beta_{11}== (ν14+791ν2)/342 ( \nu^{14} + 791\nu^{2} ) / 342 Copy content Toggle raw display
β12\beta_{12}== (ν14449ν2)/342 ( -\nu^{14} - 449\nu^{2} ) / 342 Copy content Toggle raw display
β13\beta_{13}== (109ν14+361ν10+2071ν6+88290ν2)/27702 ( 109\nu^{14} + 361\nu^{10} + 2071\nu^{6} + 88290\nu^{2} ) / 27702 Copy content Toggle raw display
β14\beta_{14}== (ν15620ν3)/513 ( -\nu^{15} - 620\nu^{3} ) / 513 Copy content Toggle raw display
β15\beta_{15}== (170ν15+323ν11+4769ν7+111537ν383106ν)/83106 ( 170\nu^{15} + 323\nu^{11} + 4769\nu^{7} + 111537\nu^{3} - 83106\nu ) / 83106 Copy content Toggle raw display
ν\nu== (β10+β8+β3)/3 ( \beta_{10} + \beta_{8} + \beta_{3} ) / 3 Copy content Toggle raw display
ν2\nu^{2}== β12+β11 \beta_{12} + \beta_{11} Copy content Toggle raw display
ν3\nu^{3}== (3β14+2β106β9+2β8+2β3)/3 ( 3\beta_{14} + 2\beta_{10} - 6\beta_{9} + 2\beta_{8} + 2\beta_{3} ) / 3 Copy content Toggle raw display
ν4\nu^{4}== 2β7+5β45 -2\beta_{7} + 5\beta_{4} - 5 Copy content Toggle raw display
ν5\nu^{5}== (β102β8+β321β1)/3 ( \beta_{10} - 2\beta_{8} + \beta_{3} - 21\beta_1 ) / 3 Copy content Toggle raw display
ν6\nu^{6}== β13β1219β2 \beta_{13} - \beta_{12} - 19\beta_{2} Copy content Toggle raw display
ν7\nu^{7}== (60β15+51β14+20β10+20β8+51β6+20β3)/3 ( 60\beta_{15} + 51\beta_{14} + 20\beta_{10} + 20\beta_{8} + 51\beta_{6} + 20\beta_{3} ) / 3 Copy content Toggle raw display
ν8\nu^{8}== 20β5+31β4 20\beta_{5} + 31\beta_{4} Copy content Toggle raw display
ν9\nu^{9}== (38β1071β8+142β333β1)/3 ( -38\beta_{10} - 71\beta_{8} + 142\beta_{3} - 33\beta_1 ) / 3 Copy content Toggle raw display
ν10\nu^{10}== 71β13109β11+109β2 71\beta_{13} - 109\beta_{11} + 109\beta_{2} Copy content Toggle raw display
ν11\nu^{11}== (114β1576β10+114β976β8753β676β3)/3 ( -114\beta_{15} - 76\beta_{10} + 114\beta_{9} - 76\beta_{8} - 753\beta_{6} - 76\beta_{3} ) / 3 Copy content Toggle raw display
ν12\nu^{12}== 38β738β5715 38\beta_{7} - 38\beta_{5} - 715 Copy content Toggle raw display
ν13\nu^{13}== (449β10791β8791β3)/3 ( -449\beta_{10} - 791\beta_{8} - 791\beta_{3} ) / 3 Copy content Toggle raw display
ν14\nu^{14}== 791β12449β11 -791\beta_{12} - 449\beta_{11} Copy content Toggle raw display
ν15\nu^{15}== (3399β141240β10+3720β91240β81240β3)/3 ( -3399\beta_{14} - 1240\beta_{10} + 3720\beta_{9} - 1240\beta_{8} - 1240\beta_{3} ) / 3 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/504Z)×\left(\mathbb{Z}/504\mathbb{Z}\right)^\times.

nn 7373 127127 253253 281281
χ(n)\chi(n) 1-1 11 1-1 1β41 - \beta_{4}

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}
293.1
−0.475594 + 1.66548i
1.24461 + 1.20455i
−1.24461 1.20455i
0.475594 1.66548i
−1.20455 + 1.24461i
−1.66548 0.475594i
1.66548 + 0.475594i
1.20455 1.24461i
−0.475594 1.66548i
1.24461 1.20455i
−1.24461 + 1.20455i
0.475594 + 1.66548i
−1.20455 1.24461i
−1.66548 + 0.475594i
1.66548 0.475594i
1.20455 + 1.24461i
−1.22474 + 0.707107i −1.68014 0.420861i 1.00000 1.73205i 3.32070 + 1.91721i 2.35534 0.672592i 1.32288 + 2.29129i 2.82843i 2.64575 + 1.41421i −5.42268
293.2 −1.22474 + 0.707107i −0.420861 1.68014i 1.00000 1.73205i −3.87242 2.23574i 1.70349 + 1.76015i −1.32288 2.29129i 2.82843i −2.64575 + 1.41421i 6.32364
293.3 −1.22474 + 0.707107i 0.420861 + 1.68014i 1.00000 1.73205i 3.87242 + 2.23574i −1.70349 1.76015i −1.32288 2.29129i 2.82843i −2.64575 + 1.41421i −6.32364
293.4 −1.22474 + 0.707107i 1.68014 + 0.420861i 1.00000 1.73205i −3.32070 1.91721i −2.35534 + 0.672592i 1.32288 + 2.29129i 2.82843i 2.64575 + 1.41421i 5.42268
293.5 1.22474 0.707107i −1.68014 + 0.420861i 1.00000 1.73205i −0.0658376 0.0380114i −1.76015 + 1.70349i 1.32288 + 2.29129i 2.82843i 2.64575 1.41421i −0.107512
293.6 1.22474 0.707107i −0.420861 + 1.68014i 1.00000 1.73205i −1.99323 1.15079i 0.672592 + 2.35534i −1.32288 2.29129i 2.82843i −2.64575 1.41421i −3.25493
293.7 1.22474 0.707107i 0.420861 1.68014i 1.00000 1.73205i 1.99323 + 1.15079i −0.672592 2.35534i −1.32288 2.29129i 2.82843i −2.64575 1.41421i 3.25493
293.8 1.22474 0.707107i 1.68014 0.420861i 1.00000 1.73205i 0.0658376 + 0.0380114i 1.76015 1.70349i 1.32288 + 2.29129i 2.82843i 2.64575 1.41421i 0.107512
461.1 −1.22474 0.707107i −1.68014 + 0.420861i 1.00000 + 1.73205i 3.32070 1.91721i 2.35534 + 0.672592i 1.32288 2.29129i 2.82843i 2.64575 1.41421i −5.42268
461.2 −1.22474 0.707107i −0.420861 + 1.68014i 1.00000 + 1.73205i −3.87242 + 2.23574i 1.70349 1.76015i −1.32288 + 2.29129i 2.82843i −2.64575 1.41421i 6.32364
461.3 −1.22474 0.707107i 0.420861 1.68014i 1.00000 + 1.73205i 3.87242 2.23574i −1.70349 + 1.76015i −1.32288 + 2.29129i 2.82843i −2.64575 1.41421i −6.32364
461.4 −1.22474 0.707107i 1.68014 0.420861i 1.00000 + 1.73205i −3.32070 + 1.91721i −2.35534 0.672592i 1.32288 2.29129i 2.82843i 2.64575 1.41421i 5.42268
461.5 1.22474 + 0.707107i −1.68014 0.420861i 1.00000 + 1.73205i −0.0658376 + 0.0380114i −1.76015 1.70349i 1.32288 2.29129i 2.82843i 2.64575 + 1.41421i −0.107512
461.6 1.22474 + 0.707107i −0.420861 1.68014i 1.00000 + 1.73205i −1.99323 + 1.15079i 0.672592 2.35534i −1.32288 + 2.29129i 2.82843i −2.64575 + 1.41421i −3.25493
461.7 1.22474 + 0.707107i 0.420861 + 1.68014i 1.00000 + 1.73205i 1.99323 1.15079i −0.672592 + 2.35534i −1.32288 + 2.29129i 2.82843i −2.64575 + 1.41421i 3.25493
461.8 1.22474 + 0.707107i 1.68014 + 0.420861i 1.00000 + 1.73205i 0.0658376 0.0380114i 1.76015 + 1.70349i 1.32288 2.29129i 2.82843i 2.64575 + 1.41421i 0.107512
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 293.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by Q(14)\Q(\sqrt{-14})
7.b odd 2 1 inner
8.b even 2 1 inner
9.d odd 6 1 inner
63.o even 6 1 inner
72.j odd 6 1 inner
504.cc even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.cc.a 16
7.b odd 2 1 inner 504.2.cc.a 16
8.b even 2 1 inner 504.2.cc.a 16
9.d odd 6 1 inner 504.2.cc.a 16
56.h odd 2 1 CM 504.2.cc.a 16
63.o even 6 1 inner 504.2.cc.a 16
72.j odd 6 1 inner 504.2.cc.a 16
504.cc even 6 1 inner 504.2.cc.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.cc.a 16 1.a even 1 1 trivial
504.2.cc.a 16 7.b odd 2 1 inner
504.2.cc.a 16 8.b even 2 1 inner
504.2.cc.a 16 9.d odd 6 1 inner
504.2.cc.a 16 56.h odd 2 1 CM
504.2.cc.a 16 63.o even 6 1 inner
504.2.cc.a 16 72.j odd 6 1 inner
504.2.cc.a 16 504.cc even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T51640T514+1122T51216000T510+166075T58744960T56++81 T_{5}^{16} - 40 T_{5}^{14} + 1122 T_{5}^{12} - 16000 T_{5}^{10} + 166075 T_{5}^{8} - 744960 T_{5}^{6} + \cdots + 81 acting on S2new(504,[χ])S_{2}^{\mathrm{new}}(504, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T42T2+4)4 (T^{4} - 2 T^{2} + 4)^{4} Copy content Toggle raw display
33 (T810T4+81)2 (T^{8} - 10 T^{4} + 81)^{2} Copy content Toggle raw display
55 T1640T14++81 T^{16} - 40 T^{14} + \cdots + 81 Copy content Toggle raw display
77 (T4+7T2+49)4 (T^{4} + 7 T^{2} + 49)^{4} Copy content Toggle raw display
1111 T16 T^{16} Copy content Toggle raw display
1313 (T8+52T6++419904)2 (T^{8} + 52 T^{6} + \cdots + 419904)^{2} Copy content Toggle raw display
1717 T16 T^{16} Copy content Toggle raw display
1919 (T8152T6++1022121)2 (T^{8} - 152 T^{6} + \cdots + 1022121)^{2} Copy content Toggle raw display
2323 (T418T3++169)4 (T^{4} - 18 T^{3} + \cdots + 169)^{4} Copy content Toggle raw display
2929 T16 T^{16} Copy content Toggle raw display
3131 T16 T^{16} Copy content Toggle raw display
3737 T16 T^{16} Copy content Toggle raw display
4141 T16 T^{16} Copy content Toggle raw display
4343 T16 T^{16} Copy content Toggle raw display
4747 T16 T^{16} Copy content Toggle raw display
5353 T16 T^{16} Copy content Toggle raw display
5959 (T8236T6++34012224)2 (T^{8} - 236 T^{6} + \cdots + 34012224)^{2} Copy content Toggle raw display
6161 T16++15 ⁣ ⁣61 T^{16} + \cdots + 15\!\cdots\!61 Copy content Toggle raw display
6767 T16 T^{16} Copy content Toggle raw display
7171 (T4+394T2+32761)4 (T^{4} + 394 T^{2} + 32761)^{4} Copy content Toggle raw display
7373 T16 T^{16} Copy content Toggle raw display
7979 (T8+446T6++1908029761)2 (T^{8} + 446 T^{6} + \cdots + 1908029761)^{2} Copy content Toggle raw display
8383 (T8332T6++419904)2 (T^{8} - 332 T^{6} + \cdots + 419904)^{2} Copy content Toggle raw display
8989 T16 T^{16} Copy content Toggle raw display
9797 T16 T^{16} Copy content Toggle raw display
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