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Properties

Label 110.4.g.a
Level 110110
Weight 44
Character orbit 110.g
Analytic conductor 6.4906.490
Analytic rank 00
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] = [110,4,Mod(31,110)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(110, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 6])) 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("110.31"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: N N == 110=2⋅5⋅11 110 = 2 \cdot 5 \cdot 11
Weight: k k == 4 4
Character orbit: [χ][\chi] == 110.g (of order 55, degree 44, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 6.490210100636.49021010063
Analytic rank: 00
Dimension: 88
Relative dimension: 22 over Q(ζ5)\Q(\zeta_{5})
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: x8−x7+26x6−51x5+301x4−125x3+250x2+3125x+15625 x^{8} - x^{7} + 26x^{6} - 51x^{5} + 301x^{4} - 125x^{3} + 250x^{2} + 3125x + 15625 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 1 1
Twist minimal: yes
Sato-Tate group: SU(2)[C5]\mathrm{SU}(2)[C_{5}]

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+2β4q2+(β7−2β3−2β2)q3−4β3q4−5β2q5+(2β7−2β6−2β4+⋯−2)q6+(β7−β5+β4+⋯−1)q7+⋯+(−204β7−32β6+⋯−369)q99+O(q100) q + 2 \beta_{4} q^{2} + (\beta_{7} - 2 \beta_{3} - 2 \beta_{2}) q^{3} - 4 \beta_{3} q^{4} - 5 \beta_{2} q^{5} + (2 \beta_{7} - 2 \beta_{6} - 2 \beta_{4} + \cdots - 2) q^{6} + (\beta_{7} - \beta_{5} + \beta_{4} + \cdots - 1) q^{7}+ \cdots + ( - 204 \beta_{7} - 32 \beta_{6} + \cdots - 369) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 8q+4q2−7q3−8q4−10q5−26q6−15q7+16q8−41q9−80q10+23q11−48q12−32q13+30q14−35q15−32q16+81q17−138q18+157q19+⋯−2906q99+O(q100) 8 q + 4 q^{2} - 7 q^{3} - 8 q^{4} - 10 q^{5} - 26 q^{6} - 15 q^{7} + 16 q^{8} - 41 q^{9} - 80 q^{10} + 23 q^{11} - 48 q^{12} - 32 q^{13} + 30 q^{14} - 35 q^{15} - 32 q^{16} + 81 q^{17} - 138 q^{18} + 157 q^{19}+ \cdots - 2906 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x8−x7+26x6−51x5+301x4−125x3+250x2+3125x+15625 x^{8} - x^{7} + 26x^{6} - 51x^{5} + 301x^{4} - 125x^{3} + 250x^{2} + 3125x + 15625 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (818163ν7−6311083ν6+40474033ν5−243871108ν4+737106733ν3+⋯−7908965625)/42767069375 ( 818163 \nu^{7} - 6311083 \nu^{6} + 40474033 \nu^{5} - 243871108 \nu^{4} + 737106733 \nu^{3} + \cdots - 7908965625 ) / 42767069375 Copy content Toggle raw display
β3\beta_{3}== (−1765046ν7−5788314ν6−46020836ν5+150326111ν4+⋯+26734970000)/42767069375 ( - 1765046 \nu^{7} - 5788314 \nu^{6} - 46020836 \nu^{5} + 150326111 \nu^{4} + \cdots + 26734970000 ) / 42767069375 Copy content Toggle raw display
β4\beta_{4}== (−440023ν7+4939127ν6−12667902ν5+123620452ν4−266223677ν3+⋯+501539125)/8553413875 ( - 440023 \nu^{7} + 4939127 \nu^{6} - 12667902 \nu^{5} + 123620452 \nu^{4} - 266223677 \nu^{3} + \cdots + 501539125 ) / 8553413875 Copy content Toggle raw display
β5\beta_{5}== (−1538607ν7+8779486ν6−53096861ν5+221788386ν4−1418690236ν3+⋯−2055220250)/8553413875 ( - 1538607 \nu^{7} + 8779486 \nu^{6} - 53096861 \nu^{5} + 221788386 \nu^{4} - 1418690236 \nu^{3} + \cdots - 2055220250 ) / 8553413875 Copy content Toggle raw display
β6\beta_{6}== (−22705ν7+41009ν6−833834ν5+1460809ν4−11790909ν3+⋯−135384125)/56645125 ( - 22705 \nu^{7} + 41009 \nu^{6} - 833834 \nu^{5} + 1460809 \nu^{4} - 11790909 \nu^{3} + \cdots - 135384125 ) / 56645125 Copy content Toggle raw display
β7\beta_{7}== (−4499104ν7+1227304ν6−101179279ν5+133776754ν4+⋯−6875359375)/8553413875 ( - 4499104 \nu^{7} + 1227304 \nu^{6} - 101179279 \nu^{5} + 133776754 \nu^{4} + \cdots - 6875359375 ) / 8553413875 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}== β7−β6−4β4−5β3−15β2+β1−1 \beta_{7} - \beta_{6} - 4\beta_{4} - 5\beta_{3} - 15\beta_{2} + \beta _1 - 1 Copy content Toggle raw display
ν3\nu^{3}== 5β6−16β5+21β4−5β3−5β1+10 5\beta_{6} - 16\beta_{5} + 21\beta_{4} - 5\beta_{3} - 5\beta _1 + 10 Copy content Toggle raw display
ν4\nu^{4}== −31β7+10β6+125β4+265β3+265β2−125 -31\beta_{7} + 10\beta_{6} + 125\beta_{4} + 265\beta_{3} + 265\beta_{2} - 125 Copy content Toggle raw display
ν5\nu^{5}== 140β7−296β6+296β5−255β4−255β2+140β1−556 140\beta_{7} - 296\beta_{6} + 296\beta_{5} - 255\beta_{4} - 255\beta_{2} + 140\beta _1 - 556 Copy content Toggle raw display
ν6\nu^{6}== 395β7−395β5+395β4−5140β3−2880β2−416β1+2880 395\beta_{7} - 395\beta_{5} + 395\beta_{4} - 5140\beta_{3} - 2880\beta_{2} - 416\beta _1 + 2880 Copy content Toggle raw display
ν7\nu^{7}== −5951β7+5951β6−3275β5+5329β4+8005β3+14140β2+⋯+5951 - 5951 \beta_{7} + 5951 \beta_{6} - 3275 \beta_{5} + 5329 \beta_{4} + 8005 \beta_{3} + 14140 \beta_{2} + \cdots + 5951 Copy content Toggle raw display

Display βi\beta_i in terms of νj\nu^j

Character values

We give the values of χ\chi on generators for (Z/110Z)×\left(\mathbb{Z}/110\mathbb{Z}\right)^\times.

nn 6767 101101
χ(n)\chi(n) 11 −β3-\beta_{3}

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}
31.1
−1.76025 − 1.27889i
2.56926 + 1.86668i
−1.76025 + 1.27889i
2.56926 − 1.86668i
−1.47789 − 4.54847i
1.16887 + 3.59741i
−1.47789 + 4.54847i
1.16887 − 3.59741i
−0.618034 + 1.90211i −2.76025 + 2.00544i −3.23607 − 2.35114i 1.54508 + 4.75528i −2.10864 − 6.48973i −13.3408 − 9.69264i 6.47214 − 4.70228i −4.74627 + 14.6075i −10.0000
31.2 −0.618034 + 1.90211i 1.56926 − 1.14014i −3.23607 − 2.35114i 1.54508 + 4.75528i 1.19881 + 3.68956i 2.32355 + 1.68816i 6.47214 − 4.70228i −7.18078 + 22.1002i −10.0000
71.1 −0.618034 − 1.90211i −2.76025 − 2.00544i −3.23607 + 2.35114i 1.54508 − 4.75528i −2.10864 + 6.48973i −13.3408 + 9.69264i 6.47214 + 4.70228i −4.74627 − 14.6075i −10.0000
71.2 −0.618034 − 1.90211i 1.56926 + 1.14014i −3.23607 + 2.35114i 1.54508 − 4.75528i 1.19881 − 3.68956i 2.32355 − 1.68816i 6.47214 + 4.70228i −7.18078 − 22.1002i −10.0000
81.1 1.61803 + 1.17557i −2.47789 + 7.62615i 1.23607 + 3.80423i −4.04508 + 2.93893i −12.9744 + 9.42644i −0.0702535 − 0.216218i −2.47214 + 7.60845i −30.1748 − 21.9233i −10.0000
81.2 1.61803 + 1.17557i 0.168870 − 0.519728i 1.23607 + 3.80423i −4.04508 + 2.93893i 0.884214 − 0.642419i 3.58747 + 11.0411i −2.47214 + 7.60845i 21.6019 + 15.6947i −10.0000
91.1 1.61803 − 1.17557i −2.47789 − 7.62615i 1.23607 − 3.80423i −4.04508 − 2.93893i −12.9744 − 9.42644i −0.0702535 + 0.216218i −2.47214 − 7.60845i −30.1748 + 21.9233i −10.0000
91.2 1.61803 − 1.17557i 0.168870 + 0.519728i 1.23607 − 3.80423i −4.04508 − 2.93893i 0.884214 + 0.642419i 3.58747 − 11.0411i −2.47214 − 7.60845i 21.6019 − 15.6947i −10.0000
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 110.4.g.a ✓ 8
11.c even 5 1 inner 110.4.g.a ✓ 8
11.c even 5 1 1210.4.a.y 4
11.d odd 10 1 1210.4.a.ba 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.4.g.a ✓ 8 1.a even 1 1 trivial
110.4.g.a ✓ 8 11.c even 5 1 inner
1210.4.a.y 4 11.c even 5 1
1210.4.a.ba 4 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T38+7T37+72T36+105T35−179T34−705T33+3038T32−1189T3+841 T_{3}^{8} + 7T_{3}^{7} + 72T_{3}^{6} + 105T_{3}^{5} - 179T_{3}^{4} - 705T_{3}^{3} + 3038T_{3}^{2} - 1189T_{3} + 841 acting on S4new(110,[χ])S_{4}^{\mathrm{new}}(110, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T4−2T3+4T2+⋯+16)2 (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 16)^{2} Copy content Toggle raw display
33 T8+7T7+⋯+841 T^{8} + 7 T^{7} + \cdots + 841 Copy content Toggle raw display
55 (T4+5T3+⋯+625)2 (T^{4} + 5 T^{3} + \cdots + 625)^{2} Copy content Toggle raw display
77 T8+15T7+⋯+15625 T^{8} + 15 T^{7} + \cdots + 15625 Copy content Toggle raw display
1111 T8+⋯+3138428376721 T^{8} + \cdots + 3138428376721 Copy content Toggle raw display
1313 T8+⋯+20645945775961 T^{8} + \cdots + 20645945775961 Copy content Toggle raw display
1717 T8+⋯+7677825850321 T^{8} + \cdots + 7677825850321 Copy content Toggle raw display
1919 T8+⋯+556487581020121 T^{8} + \cdots + 556487581020121 Copy content Toggle raw display
2323 (T4+259T3+⋯−3351821)2 (T^{4} + 259 T^{3} + \cdots - 3351821)^{2} Copy content Toggle raw display
2929 T8+⋯+60 ⁣⋯ ⁣25 T^{8} + \cdots + 60\!\cdots\!25 Copy content Toggle raw display
3131 T8+⋯+15211411832761 T^{8} + \cdots + 15211411832761 Copy content Toggle raw display
3737 T8+⋯+31 ⁣⋯ ⁣21 T^{8} + \cdots + 31\!\cdots\!21 Copy content Toggle raw display
4141 T8+⋯+54 ⁣⋯ ⁣81 T^{8} + \cdots + 54\!\cdots\!81 Copy content Toggle raw display
4343 (T4+147T3+⋯+959787745)2 (T^{4} + 147 T^{3} + \cdots + 959787745)^{2} Copy content Toggle raw display
4747 T8+⋯+31 ⁣⋯ ⁣81 T^{8} + \cdots + 31\!\cdots\!81 Copy content Toggle raw display
5353 T8+⋯+61 ⁣⋯ ⁣25 T^{8} + \cdots + 61\!\cdots\!25 Copy content Toggle raw display
5959 T8+⋯+18 ⁣⋯ ⁣01 T^{8} + \cdots + 18\!\cdots\!01 Copy content Toggle raw display
6161 T8+⋯+15 ⁣⋯ ⁣25 T^{8} + \cdots + 15\!\cdots\!25 Copy content Toggle raw display
6767 (T4+689T3+⋯−58505616781)2 (T^{4} + 689 T^{3} + \cdots - 58505616781)^{2} Copy content Toggle raw display
7171 T8+⋯+10 ⁣⋯ ⁣25 T^{8} + \cdots + 10\!\cdots\!25 Copy content Toggle raw display
7373 T8+⋯+22 ⁣⋯ ⁣61 T^{8} + \cdots + 22\!\cdots\!61 Copy content Toggle raw display
7979 T8+⋯+29 ⁣⋯ ⁣25 T^{8} + \cdots + 29\!\cdots\!25 Copy content Toggle raw display
8383 T8+⋯+43 ⁣⋯ ⁣81 T^{8} + \cdots + 43\!\cdots\!81 Copy content Toggle raw display
8989 (T4+1684T3+⋯−24048117271)2 (T^{4} + 1684 T^{3} + \cdots - 24048117271)^{2} Copy content Toggle raw display
9797 T8+⋯+17 ⁣⋯ ⁣41 T^{8} + \cdots + 17\!\cdots\!41 Copy content Toggle raw display
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