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Properties

Label 1682.2.a.u
Level 16821682
Weight 22
Character orbit 1682.a
Self dual yes
Analytic conductor 13.43113.431
Analytic rank 00
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] = [1682,2,Mod(1,1682)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1682, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("1682.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 1682=2⋅292 1682 = 2 \cdot 29^{2}
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1682.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-8,-1,8,5] 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: 13.430837620013.4308376200
Analytic rank: 00
Dimension: 88
Coefficient field: 8.8.32836640625.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x8−x7−18x6+17x5+95x4−77x3−128x2+51x+31 x^{8} - x^{7} - 18x^{6} + 17x^{5} + 95x^{4} - 77x^{3} - 128x^{2} + 51x + 31 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a7]\Z[a_1, \ldots, a_{7}]
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,…,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−q2−β1q3+q4+(−β5−β4−β1+1)q5+β1q6+(−β7−β4−β3+⋯+1)q7−q8+(β6−β4+β3+⋯+2)q9+⋯+(−3β7−3β6+⋯−4)q99+O(q100) q - q^{2} - \beta_1 q^{3} + q^{4} + ( - \beta_{5} - \beta_{4} - \beta_1 + 1) q^{5} + \beta_1 q^{6} + ( - \beta_{7} - \beta_{4} - \beta_{3} + \cdots + 1) q^{7} - q^{8} + (\beta_{6} - \beta_{4} + \beta_{3} + \cdots + 2) q^{9}+ \cdots + ( - 3 \beta_{7} - 3 \beta_{6} + \cdots - 4) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 8q−8q2−q3+8q4+5q5+q6+7q7−8q8+13q9−5q10+7q11−q12+13q13−7q14+20q15+8q16−9q17−13q18+13q19+5q20+⋯+2q99+O(q100) 8 q - 8 q^{2} - q^{3} + 8 q^{4} + 5 q^{5} + q^{6} + 7 q^{7} - 8 q^{8} + 13 q^{9} - 5 q^{10} + 7 q^{11} - q^{12} + 13 q^{13} - 7 q^{14} + 20 q^{15} + 8 q^{16} - 9 q^{17} - 13 q^{18} + 13 q^{19} + 5 q^{20}+ \cdots + 2 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x8−x7−18x6+17x5+95x4−77x3−128x2+51x+31 x^{8} - x^{7} - 18x^{6} + 17x^{5} + 95x^{4} - 77x^{3} - 128x^{2} + 51x + 31 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (18ν7−151ν6−168ν5+2008ν4+386ν3−6004ν2−1858ν+1364)/2073 ( 18\nu^{7} - 151\nu^{6} - 168\nu^{5} + 2008\nu^{4} + 386\nu^{3} - 6004\nu^{2} - 1858\nu + 1364 ) / 2073 Copy content Toggle raw display
β3\beta_{3}== (−28ν7−149ν6+31ν5+1867ν4+2010ν3−5325ν2−3252ν+28)/2073 ( -28\nu^{7} - 149\nu^{6} + 31\nu^{5} + 1867\nu^{4} + 2010\nu^{3} - 5325\nu^{2} - 3252\nu + 28 ) / 2073 Copy content Toggle raw display
β4\beta_{4}== (44ν7−62ν6−641ν5+916ν4+2172ν3−3774ν2+372ν+4102)/2073 ( 44\nu^{7} - 62\nu^{6} - 641\nu^{5} + 916\nu^{4} + 2172\nu^{3} - 3774\nu^{2} + 372\nu + 4102 ) / 2073 Copy content Toggle raw display
β5\beta_{5}== (45ν7−32ν6−420ν5+1565ν4+274ν3−11555ν2+883ν+8938)/2073 ( 45\nu^{7} - 32\nu^{6} - 420\nu^{5} + 1565\nu^{4} + 274\nu^{3} - 11555\nu^{2} + 883\nu + 8938 ) / 2073 Copy content Toggle raw display
β6\beta_{6}== (54ν7+238ν6−504ν5−2959ν4−224ν3+9628ν2+5482ν−7655)/2073 ( 54\nu^{7} + 238\nu^{6} - 504\nu^{5} - 2959\nu^{4} - 224\nu^{3} + 9628\nu^{2} + 5482\nu - 7655 ) / 2073 Copy content Toggle raw display
β7\beta_{7}== (140ν7+54ν6−2228ν5−352ν4+9989ν3−1015ν2−9307ν+1242)/2073 ( 140\nu^{7} + 54\nu^{6} - 2228\nu^{5} - 352\nu^{4} + 9989\nu^{3} - 1015\nu^{2} - 9307\nu + 1242 ) / 2073 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β6−β4+β3+β2+5 \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + 5 Copy content Toggle raw display
ν3\nu^{3}== β7−β6−3β4−β3+β2+7β1+1 \beta_{7} - \beta_{6} - 3\beta_{4} - \beta_{3} + \beta_{2} + 7\beta _1 + 1 Copy content Toggle raw display
ν4\nu^{4}== 9β6+2β5−10β4+10β3+8β2+39 9\beta_{6} + 2\beta_{5} - 10\beta_{4} + 10\beta_{3} + 8\beta_{2} + 39 Copy content Toggle raw display
ν5\nu^{5}== 8β7−10β6−27β4−14β3+12β2+56β1+4 8\beta_{7} - 10\beta_{6} - 27\beta_{4} - 14\beta_{3} + 12\beta_{2} + 56\beta _1 + 4 Copy content Toggle raw display
ν6\nu^{6}== 2β7+78β6+26β5−96β4+88β3+57β2−2β1+326 2\beta_{7} + 78\beta_{6} + 26\beta_{5} - 96\beta_{4} + 88\beta_{3} + 57\beta_{2} - 2\beta _1 + 326 Copy content Toggle raw display
ν7\nu^{7}== 70β7−88β6−5β5−211β4−153β3+125β2+459β1−8 70\beta_{7} - 88\beta_{6} - 5\beta_{5} - 211\beta_{4} - 153\beta_{3} + 125\beta_{2} + 459\beta _1 - 8 Copy content Toggle raw display

Display βi\beta_i in terms of νj\nu^j

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
2.88972
2.79588
1.70955
0.710020
−0.371284
−1.06263
−2.66631
−3.00493
−1.00000 −2.88972 1.00000 −2.81297 2.88972 3.85101 −1.00000 5.35050 2.81297
1.2 −1.00000 −2.79588 1.00000 0.869361 2.79588 2.55676 −1.00000 4.81692 −0.869361
1.3 −1.00000 −1.70955 1.00000 4.23180 1.70955 1.69226 −1.00000 −0.0774536 −4.23180
1.4 −1.00000 −0.710020 1.00000 −3.33469 0.710020 4.40617 −1.00000 −2.49587 3.33469
1.5 −1.00000 0.371284 1.00000 −3.64078 −0.371284 −3.75542 −1.00000 −2.86215 3.64078
1.6 −1.00000 1.06263 1.00000 4.09274 −1.06263 2.34135 −1.00000 −1.87081 −4.09274
1.7 −1.00000 2.66631 1.00000 1.88957 −2.66631 −4.43319 −1.00000 4.10924 −1.88957
1.8 −1.00000 3.00493 1.00000 3.70497 −3.00493 0.341047 −1.00000 6.02962 −3.70497
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
2929 −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 1682.2.a.u ✓ 8
29.b even 2 1 1682.2.a.v yes 8
29.c odd 4 2 1682.2.b.k 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1682.2.a.u ✓ 8 1.a even 1 1 trivial
1682.2.a.v yes 8 29.b even 2 1
1682.2.b.k 16 29.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S2new(Γ0(1682))S_{2}^{\mathrm{new}}(\Gamma_0(1682)):

T38+T37−18T36−17T35+95T34+77T33−128T32−51T3+31 T_{3}^{8} + T_{3}^{7} - 18T_{3}^{6} - 17T_{3}^{5} + 95T_{3}^{4} + 77T_{3}^{3} - 128T_{3}^{2} - 51T_{3} + 31 Copy content Toggle raw display
T58−5T57−30T56+160T55+265T54−1650T53−300T52+5400T5−3600 T_{5}^{8} - 5T_{5}^{7} - 30T_{5}^{6} + 160T_{5}^{5} + 265T_{5}^{4} - 1650T_{5}^{3} - 300T_{5}^{2} + 5400T_{5} - 3600 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T+1)8 (T + 1)^{8} Copy content Toggle raw display
33 T8+T7+⋯+31 T^{8} + T^{7} + \cdots + 31 Copy content Toggle raw display
55 T8−5T7+⋯−3600 T^{8} - 5 T^{7} + \cdots - 3600 Copy content Toggle raw display
77 T8−7T7+⋯+976 T^{8} - 7 T^{7} + \cdots + 976 Copy content Toggle raw display
1111 T8−7T7+⋯−1629 T^{8} - 7 T^{7} + \cdots - 1629 Copy content Toggle raw display
1313 T8−13T7+⋯−464 T^{8} - 13 T^{7} + \cdots - 464 Copy content Toggle raw display
1717 T8+9T7+⋯+2511 T^{8} + 9 T^{7} + \cdots + 2511 Copy content Toggle raw display
1919 T8−13T7+⋯+1611 T^{8} - 13 T^{7} + \cdots + 1611 Copy content Toggle raw display
2323 T8−12T7+⋯−144 T^{8} - 12 T^{7} + \cdots - 144 Copy content Toggle raw display
2929 T8 T^{8} Copy content Toggle raw display
3131 T8+15T7+⋯−26480 T^{8} + 15 T^{7} + \cdots - 26480 Copy content Toggle raw display
3737 T8+4T7+⋯+50896 T^{8} + 4 T^{7} + \cdots + 50896 Copy content Toggle raw display
4141 T8+8T7+⋯−279 T^{8} + 8 T^{7} + \cdots - 279 Copy content Toggle raw display
4343 T8+12T7+⋯−3689264 T^{8} + 12 T^{7} + \cdots - 3689264 Copy content Toggle raw display
4747 T8−13T7+⋯−2471184 T^{8} - 13 T^{7} + \cdots - 2471184 Copy content Toggle raw display
5353 T8−4T7+⋯−924624 T^{8} - 4 T^{7} + \cdots - 924624 Copy content Toggle raw display
5959 T8−8T7+⋯+5024961 T^{8} - 8 T^{7} + \cdots + 5024961 Copy content Toggle raw display
6161 T8+9T7+⋯+416656 T^{8} + 9 T^{7} + \cdots + 416656 Copy content Toggle raw display
6767 T8−34T7+⋯+967471 T^{8} - 34 T^{7} + \cdots + 967471 Copy content Toggle raw display
7171 T8+11T7+⋯+6717456 T^{8} + 11 T^{7} + \cdots + 6717456 Copy content Toggle raw display
7373 T8−13T7+⋯+1321 T^{8} - 13 T^{7} + \cdots + 1321 Copy content Toggle raw display
7979 T8−8T7+⋯−3550544 T^{8} - 8 T^{7} + \cdots - 3550544 Copy content Toggle raw display
8383 T8−10T7+⋯+110386845 T^{8} - 10 T^{7} + \cdots + 110386845 Copy content Toggle raw display
8989 T8+6T7+⋯−2505339 T^{8} + 6 T^{7} + \cdots - 2505339 Copy content Toggle raw display
9797 T8−11T7+⋯+49865776 T^{8} - 11 T^{7} + \cdots + 49865776 Copy content Toggle raw display
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