Properties

Label 520.6.a.e
Level 520520
Weight 66
Character orbit 520.a
Self dual yes
Analytic conductor 83.40083.400
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] = [520,6,Mod(1,520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(520, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("520.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: N N == 520=23513 520 = 2^{3} \cdot 5 \cdot 13
Weight: k k == 6 6
Character orbit: [χ][\chi] == 520.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: 83.399586302783.3995863027
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: x81392x6960x5+541704x4+955392x349992640x2201007872x+5544720 x^{8} - 1392x^{6} - 960x^{5} + 541704x^{4} + 955392x^{3} - 49992640x^{2} - 201007872x + 5544720 Copy content Toggle raw display
Coefficient ring: Z[a1,,a11]\Z[a_1, \ldots, a_{11}]
Coefficient ring index: 2153 2^{15}\cdot 3
Twist minimal: yes
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+(β12)q325q5+(β2+23)q7+(β3β23β1+109)q9+(β7β44β1+16)q11+169q13+(25β1+50)q15++(169β737β6++39118)q99+O(q100) q + (\beta_1 - 2) q^{3} - 25 q^{5} + ( - \beta_{2} + 23) q^{7} + (\beta_{3} - \beta_{2} - 3 \beta_1 + 109) q^{9} + (\beta_{7} - \beta_{4} - 4 \beta_1 + 16) q^{11} + 169 q^{13} + ( - 25 \beta_1 + 50) q^{15}+ \cdots + ( - 169 \beta_{7} - 37 \beta_{6} + \cdots + 39118) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q16q3200q5+184q7+872q9+128q11+1352q13+400q15+1656q172112q191352q211144q23+5000q256112q276672q292984q31++312944q99+O(q100) 8 q - 16 q^{3} - 200 q^{5} + 184 q^{7} + 872 q^{9} + 128 q^{11} + 1352 q^{13} + 400 q^{15} + 1656 q^{17} - 2112 q^{19} - 1352 q^{21} - 1144 q^{23} + 5000 q^{25} - 6112 q^{27} - 6672 q^{29} - 2984 q^{31}+ \cdots + 312944 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x81392x6960x5+541704x4+955392x349992640x2201007872x+5544720 x^{8} - 1392x^{6} - 960x^{5} + 541704x^{4} + 955392x^{3} - 49992640x^{2} - 201007872x + 5544720 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (163ν7+574ν6+250262ν5902522ν4114612364ν3++17405786520)/283757760 ( - 163 \nu^{7} + 574 \nu^{6} + 250262 \nu^{5} - 902522 \nu^{4} - 114612364 \nu^{3} + \cdots + 17405786520 ) / 283757760 Copy content Toggle raw display
β3\beta_{3}== (163ν7+574ν6+250262ν5902522ν4114612364ν3+81341913960)/283757760 ( - 163 \nu^{7} + 574 \nu^{6} + 250262 \nu^{5} - 902522 \nu^{4} - 114612364 \nu^{3} + \cdots - 81341913960 ) / 283757760 Copy content Toggle raw display
β4\beta_{4}== (20ν7+167ν6+21286ν5160462ν46054464ν3++2326668840)/21827520 ( - 20 \nu^{7} + 167 \nu^{6} + 21286 \nu^{5} - 160462 \nu^{4} - 6054464 \nu^{3} + \cdots + 2326668840 ) / 21827520 Copy content Toggle raw display
β5\beta_{5}== (406ν7+12310ν6+547190ν514254391ν4198999712ν3+70426804860)/141878880 ( - 406 \nu^{7} + 12310 \nu^{6} + 547190 \nu^{5} - 14254391 \nu^{4} - 198999712 \nu^{3} + \cdots - 70426804860 ) / 141878880 Copy content Toggle raw display
β6\beta_{6}== (250ν7+1119ν6359938ν52053654ν4+152417592ν3+89164763640)/94585920 ( 250 \nu^{7} + 1119 \nu^{6} - 359938 \nu^{5} - 2053654 \nu^{4} + 152417592 \nu^{3} + \cdots - 89164763640 ) / 94585920 Copy content Toggle raw display
β7\beta_{7}== (1376ν75543ν61880389ν5+6171619ν4+696868448ν3+73845429300)/141878880 ( 1376 \nu^{7} - 5543 \nu^{6} - 1880389 \nu^{5} + 6171619 \nu^{4} + 696868448 \nu^{3} + \cdots - 73845429300 ) / 141878880 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β3β2+β1+348 \beta_{3} - \beta_{2} + \beta _1 + 348 Copy content Toggle raw display
ν3\nu^{3}== 2β7+β67β44β314β2+620β1+360 -2\beta_{7} + \beta_{6} - 7\beta_{4} - 4\beta_{3} - 14\beta_{2} + 620\beta _1 + 360 Copy content Toggle raw display
ν4\nu^{4}== 16β7104β6+32β540β4+760β31064β2+776β1+213564 16\beta_{7} - 104\beta_{6} + 32\beta_{5} - 40\beta_{4} + 760\beta_{3} - 1064\beta_{2} + 776\beta _1 + 213564 Copy content Toggle raw display
ν5\nu^{5}== 1552β7+152β6+352β58552β42928β310688β2++238080 - 1552 \beta_{7} + 152 \beta_{6} + 352 \beta_{5} - 8552 \beta_{4} - 2928 \beta_{3} - 10688 \beta_{2} + \cdots + 238080 Copy content Toggle raw display
ν6\nu^{6}== 22688β7121648β6+50816β548368β4+567420β3++146608176 22688 \beta_{7} - 121648 \beta_{6} + 50816 \beta_{5} - 48368 \beta_{4} + 567420 \beta_{3} + \cdots + 146608176 Copy content Toggle raw display
ν7\nu^{7}== 985272β7322308β6+542208β58157156β42077200β3++184820832 - 985272 \beta_{7} - 322308 \beta_{6} + 542208 \beta_{5} - 8157156 \beta_{4} - 2077200 \beta_{3} + \cdots + 184820832 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
−27.7866
−21.5235
−9.07452
−4.68805
0.0273980
12.7436
22.2538
28.0480
0 −29.7866 0 −25.0000 0 138.753 0 644.243 0
1.2 0 −23.5235 0 −25.0000 0 −122.557 0 310.356 0
1.3 0 −11.0745 0 −25.0000 0 109.586 0 −120.355 0
1.4 0 −6.68805 0 −25.0000 0 140.231 0 −198.270 0
1.5 0 −1.97260 0 −25.0000 0 −39.7415 0 −239.109 0
1.6 0 10.7436 0 −25.0000 0 −213.035 0 −127.575 0
1.7 0 20.2538 0 −25.0000 0 32.4777 0 167.214 0
1.8 0 26.0480 0 −25.0000 0 138.286 0 435.496 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
22 +1 +1
55 +1 +1
1313 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 520.6.a.e 8
4.b odd 2 1 1040.6.a.z 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.6.a.e 8 1.a even 1 1 trivial
1040.6.a.z 8 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T38+16T371280T3617216T35+449704T34+5029696T33+580250736 T_{3}^{8} + 16 T_{3}^{7} - 1280 T_{3}^{6} - 17216 T_{3}^{5} + 449704 T_{3}^{4} + 5029696 T_{3}^{3} + \cdots - 580250736 acting on S6new(Γ0(520))S_{6}^{\mathrm{new}}(\Gamma_0(520)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T8 T^{8} Copy content Toggle raw display
33 T8+16T7+580250736 T^{8} + 16 T^{7} + \cdots - 580250736 Copy content Toggle raw display
55 (T+25)8 (T + 25)^{8} Copy content Toggle raw display
77 T8+99 ⁣ ⁣80 T^{8} + \cdots - 99\!\cdots\!80 Copy content Toggle raw display
1111 T8+22 ⁣ ⁣08 T^{8} + \cdots - 22\!\cdots\!08 Copy content Toggle raw display
1313 (T169)8 (T - 169)^{8} Copy content Toggle raw display
1717 T8++22 ⁣ ⁣40 T^{8} + \cdots + 22\!\cdots\!40 Copy content Toggle raw display
1919 T8++11 ⁣ ⁣36 T^{8} + \cdots + 11\!\cdots\!36 Copy content Toggle raw display
2323 T8++41 ⁣ ⁣12 T^{8} + \cdots + 41\!\cdots\!12 Copy content Toggle raw display
2929 T8++23 ⁣ ⁣36 T^{8} + \cdots + 23\!\cdots\!36 Copy content Toggle raw display
3131 T8++35 ⁣ ⁣40 T^{8} + \cdots + 35\!\cdots\!40 Copy content Toggle raw display
3737 T8++58 ⁣ ⁣00 T^{8} + \cdots + 58\!\cdots\!00 Copy content Toggle raw display
4141 T8+49 ⁣ ⁣64 T^{8} + \cdots - 49\!\cdots\!64 Copy content Toggle raw display
4343 T8++73 ⁣ ⁣88 T^{8} + \cdots + 73\!\cdots\!88 Copy content Toggle raw display
4747 T8++51 ⁣ ⁣28 T^{8} + \cdots + 51\!\cdots\!28 Copy content Toggle raw display
5353 T8++12 ⁣ ⁣32 T^{8} + \cdots + 12\!\cdots\!32 Copy content Toggle raw display
5959 T8++64 ⁣ ⁣12 T^{8} + \cdots + 64\!\cdots\!12 Copy content Toggle raw display
6161 T8+22 ⁣ ⁣00 T^{8} + \cdots - 22\!\cdots\!00 Copy content Toggle raw display
6767 T8+89 ⁣ ⁣04 T^{8} + \cdots - 89\!\cdots\!04 Copy content Toggle raw display
7171 T8+21 ⁣ ⁣52 T^{8} + \cdots - 21\!\cdots\!52 Copy content Toggle raw display
7373 T8+39 ⁣ ⁣76 T^{8} + \cdots - 39\!\cdots\!76 Copy content Toggle raw display
7979 T8+95 ⁣ ⁣32 T^{8} + \cdots - 95\!\cdots\!32 Copy content Toggle raw display
8383 T8++47 ⁣ ⁣92 T^{8} + \cdots + 47\!\cdots\!92 Copy content Toggle raw display
8989 T8++88 ⁣ ⁣00 T^{8} + \cdots + 88\!\cdots\!00 Copy content Toggle raw display
9797 T8+18 ⁣ ⁣00 T^{8} + \cdots - 18\!\cdots\!00 Copy content Toggle raw display
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