Properties

Label 637.6.a.e
Level 637637
Weight 66
Character orbit 637.a
Self dual yes
Analytic conductor 102.164102.164
Analytic rank 11
Dimension 88
CM no
Inner twists 11

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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] = [637,6,Mod(1,637)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(637, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("637.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Level: N N == 637=7213 637 = 7^{2} \cdot 13
Weight: k k == 6 6
Character orbit: [χ][\chi] == 637.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-1,-28,245,-219] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 102.164493221102.164493221
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: x8x7250x6+210x5+20076x412252x3544784x2+65648x+2393792 x^{8} - x^{7} - 250x^{6} + 210x^{5} + 20076x^{4} - 12252x^{3} - 544784x^{2} + 65648x + 2393792 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 22 2^{2}
Twist minimal: no (minimal twist has level 91)
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) == qβ1q2+(β4+β14)q3+(β2+31)q4+(β32β127)q5+(2β7β62β5+53)q6+(3β74β65β4+1)q8++(25β7+570β6++10210)q99+O(q100) q - \beta_1 q^{2} + (\beta_{4} + \beta_1 - 4) q^{3} + (\beta_{2} + 31) q^{4} + ( - \beta_{3} - 2 \beta_1 - 27) q^{5} + (2 \beta_{7} - \beta_{6} - 2 \beta_{5} + \cdots - 53) q^{6} + (3 \beta_{7} - 4 \beta_{6} - 5 \beta_{4} + \cdots - 1) q^{8}+ \cdots + ( - 25 \beta_{7} + 570 \beta_{6} + \cdots + 10210) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 8qq228q3+245q4219q5435q657q8+930q9+948q10+184q11463q12+1352q13256q15+4953q162278q175308q184959q19++90754q99+O(q100) 8 q - q^{2} - 28 q^{3} + 245 q^{4} - 219 q^{5} - 435 q^{6} - 57 q^{8} + 930 q^{9} + 948 q^{10} + 184 q^{11} - 463 q^{12} + 1352 q^{13} - 256 q^{15} + 4953 q^{16} - 2278 q^{17} - 5308 q^{18} - 4959 q^{19}+ \cdots + 90754 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x8x7250x6+210x5+20076x412252x3544784x2+65648x+2393792 x^{8} - x^{7} - 250x^{6} + 210x^{5} + 20076x^{4} - 12252x^{3} - 544784x^{2} + 65648x + 2393792 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== ν263 \nu^{2} - 63 Copy content Toggle raw display
β3\beta_{3}== (2501ν7+35467ν6+1254008ν58124970ν4138426240ν3+2702698464)/99699232 ( - 2501 \nu^{7} + 35467 \nu^{6} + 1254008 \nu^{5} - 8124970 \nu^{4} - 138426240 \nu^{3} + \cdots - 2702698464 ) / 99699232 Copy content Toggle raw display
β4\beta_{4}== (29683ν7461047ν6+5659660ν5+89483730ν4226396840ν3++27309929696)/598195392 ( - 29683 \nu^{7} - 461047 \nu^{6} + 5659660 \nu^{5} + 89483730 \nu^{4} - 226396840 \nu^{3} + \cdots + 27309929696 ) / 598195392 Copy content Toggle raw display
β5\beta_{5}== (109811ν752255ν6+21329812ν5+32657314ν41151406808ν3++15868421856)/598195392 ( - 109811 \nu^{7} - 52255 \nu^{6} + 21329812 \nu^{5} + 32657314 \nu^{4} - 1151406808 \nu^{3} + \cdots + 15868421856 ) / 598195392 Copy content Toggle raw display
β6\beta_{6}== (91705ν7+702673ν617428720ν5131929806ν4+849745264ν3+30023872448)/299097696 ( 91705 \nu^{7} + 702673 \nu^{6} - 17428720 \nu^{5} - 131929806 \nu^{4} + 849745264 \nu^{3} + \cdots - 30023872448 ) / 299097696 Copy content Toggle raw display
β7\beta_{7}== (190073ν7+1176317ν634535804ν5218923206ν4+1412408360ν3+14628902368)/598195392 ( 190073 \nu^{7} + 1176317 \nu^{6} - 34535804 \nu^{5} - 218923206 \nu^{4} + 1412408360 \nu^{3} + \cdots - 14628902368 ) / 598195392 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β2+63 \beta_{2} + 63 Copy content Toggle raw display
ν3\nu^{3}== 3β7+4β6+5β4+β32β2+88β1+1 -3\beta_{7} + 4\beta_{6} + 5\beta_{4} + \beta_{3} - 2\beta_{2} + 88\beta _1 + 1 Copy content Toggle raw display
ν4\nu^{4}== 9β7+8β6+18β5+43β45β3+114β22β1+5629 9\beta_{7} + 8\beta_{6} + 18\beta_{5} + 43\beta_{4} - 5\beta_{3} + 114\beta_{2} - 2\beta _1 + 5629 Copy content Toggle raw display
ν5\nu^{5}== 389β7+578β6+18β5+885β4+255β3234β2+8650β1203 -389\beta_{7} + 578\beta_{6} + 18\beta_{5} + 885\beta_{4} + 255\beta_{3} - 234\beta_{2} + 8650\beta _1 - 203 Copy content Toggle raw display
ν6\nu^{6}== 1147β7+2066β6+3606β5+7161β4773β3+12090β2++553569 1147 \beta_{7} + 2066 \beta_{6} + 3606 \beta_{5} + 7161 \beta_{4} - 773 \beta_{3} + 12090 \beta_{2} + \cdots + 553569 Copy content Toggle raw display
ν7\nu^{7}== 41973β7+71726β6+1686β5+128857β4+37927β3++17009 - 41973 \beta_{7} + 71726 \beta_{6} + 1686 \beta_{5} + 128857 \beta_{4} + 37927 \beta_{3} + \cdots + 17009 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
10.9127
8.37806
7.87682
2.33047
−2.31470
−6.17463
−9.46332
−10.5454
−10.9127 26.7530 87.0880 −91.1424 −291.949 0 −601.161 472.725 994.614
1.2 −8.37806 5.43213 38.1920 5.58680 −45.5107 0 −51.8766 −213.492 −46.8066
1.3 −7.87682 −28.4534 30.0443 −12.9458 224.122 0 15.4049 566.593 101.971
1.4 −2.33047 −3.45560 −26.5689 −94.0769 8.05316 0 136.493 −231.059 219.243
1.5 2.31470 16.6968 −26.6421 48.4304 38.6483 0 −135.739 35.7848 112.102
1.6 6.17463 −24.2913 6.12609 −68.1144 −149.990 0 −159.762 347.066 −420.581
1.7 9.46332 0.254216 57.5544 −54.0690 2.40573 0 241.829 −242.935 −511.672
1.8 10.5454 −20.9360 79.2064 47.3313 −220.779 0 497.812 195.317 499.129
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
77 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 637.6.a.e 8
7.b odd 2 1 91.6.a.c 8
21.c even 2 1 819.6.a.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.6.a.c 8 7.b odd 2 1
637.6.a.e 8 1.a even 1 1 trivial
819.6.a.j 8 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S6new(Γ0(637))S_{6}^{\mathrm{new}}(\Gamma_0(637)):

T28+T27250T26210T25+20076T24+12252T23544784T2265648T2+2393792 T_{2}^{8} + T_{2}^{7} - 250T_{2}^{6} - 210T_{2}^{5} + 20076T_{2}^{4} + 12252T_{2}^{3} - 544784T_{2}^{2} - 65648T_{2} + 2393792 Copy content Toggle raw display
T38+28T371045T3629036T35+259264T34+6632208T33++30844800 T_{3}^{8} + 28 T_{3}^{7} - 1045 T_{3}^{6} - 29036 T_{3}^{5} + 259264 T_{3}^{4} + 6632208 T_{3}^{3} + \cdots + 30844800 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T8+T7++2393792 T^{8} + T^{7} + \cdots + 2393792 Copy content Toggle raw display
33 T8+28T7++30844800 T^{8} + 28 T^{7} + \cdots + 30844800 Copy content Toggle raw display
55 T8+5235380906112 T^{8} + \cdots - 5235380906112 Copy content Toggle raw display
77 T8 T^{8} Copy content Toggle raw display
1111 T8++96 ⁣ ⁣00 T^{8} + \cdots + 96\!\cdots\!00 Copy content Toggle raw display
1313 (T169)8 (T - 169)^{8} Copy content Toggle raw display
1717 T8+98 ⁣ ⁣72 T^{8} + \cdots - 98\!\cdots\!72 Copy content Toggle raw display
1919 T8+75 ⁣ ⁣80 T^{8} + \cdots - 75\!\cdots\!80 Copy content Toggle raw display
2323 T8++65 ⁣ ⁣40 T^{8} + \cdots + 65\!\cdots\!40 Copy content Toggle raw display
2929 T8++77 ⁣ ⁣52 T^{8} + \cdots + 77\!\cdots\!52 Copy content Toggle raw display
3131 T8+75 ⁣ ⁣60 T^{8} + \cdots - 75\!\cdots\!60 Copy content Toggle raw display
3737 T8+59 ⁣ ⁣00 T^{8} + \cdots - 59\!\cdots\!00 Copy content Toggle raw display
4141 T8+33 ⁣ ⁣80 T^{8} + \cdots - 33\!\cdots\!80 Copy content Toggle raw display
4343 T8+13 ⁣ ⁣96 T^{8} + \cdots - 13\!\cdots\!96 Copy content Toggle raw display
4747 T8++17 ⁣ ⁣80 T^{8} + \cdots + 17\!\cdots\!80 Copy content Toggle raw display
5353 T8+31 ⁣ ⁣76 T^{8} + \cdots - 31\!\cdots\!76 Copy content Toggle raw display
5959 T8++10 ⁣ ⁣40 T^{8} + \cdots + 10\!\cdots\!40 Copy content Toggle raw display
6161 T8++34 ⁣ ⁣20 T^{8} + \cdots + 34\!\cdots\!20 Copy content Toggle raw display
6767 T8+20 ⁣ ⁣92 T^{8} + \cdots - 20\!\cdots\!92 Copy content Toggle raw display
7171 T8++42 ⁣ ⁣04 T^{8} + \cdots + 42\!\cdots\!04 Copy content Toggle raw display
7373 T8++29 ⁣ ⁣22 T^{8} + \cdots + 29\!\cdots\!22 Copy content Toggle raw display
7979 T8++25 ⁣ ⁣36 T^{8} + \cdots + 25\!\cdots\!36 Copy content Toggle raw display
8383 T8++19 ⁣ ⁣96 T^{8} + \cdots + 19\!\cdots\!96 Copy content Toggle raw display
8989 T8++77 ⁣ ⁣00 T^{8} + \cdots + 77\!\cdots\!00 Copy content Toggle raw display
9797 T8+28 ⁣ ⁣50 T^{8} + \cdots - 28\!\cdots\!50 Copy content Toggle raw display
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