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Properties

Label 8.10.b.a
Level 88
Weight 1010
Character orbit 8.b
Analytic conductor 4.1204.120
Analytic rank 00
Dimension 88
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8,10,Mod(5,8)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.5");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: N N == 8=23 8 = 2^{3}
Weight: k k == 10 10
Character orbit: [χ][\chi] == 8.b (of order 22, degree 11, minimal)

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: no
Analytic conductor: 4.120286689314.12028668931
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: x8−x7+59x6−313x5−315x4−92091x3+1261649x2−16074123x+251007534 x^{8} - x^{7} + 59x^{6} - 313x^{5} - 315x^{4} - 92091x^{3} + 1261649x^{2} - 16074123x + 251007534 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 228⋅32 2^{28}\cdot 3^{2}
Twist minimal: yes
Sato-Tate group: SU(2)[C2]\mathrm{SU}(2)[C_{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+(−β1−2)q2+(β3+β1)q3+(β2+3β1−54)q4+(β4−β2+6β1−2)q5+(−β6+β4−2β3+⋯+585)q6+⋯+(−66624β7−148128β5+⋯−3435648)q99+O(q100) q + ( - \beta_1 - 2) q^{2} + (\beta_{3} + \beta_1) q^{3} + (\beta_{2} + 3 \beta_1 - 54) q^{4} + (\beta_{4} - \beta_{2} + 6 \beta_1 - 2) q^{5} + ( - \beta_{6} + \beta_{4} - 2 \beta_{3} + \cdots + 585) q^{6}+ \cdots + ( - 66624 \beta_{7} - 148128 \beta_{5} + \cdots - 3435648) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 8q−18q2−428q4+4684q6+4800q7−3384q8−39368q9+26392q10+54760q12−72336q14−163136q15+185616q16−102000q17−23614q18+1245264q20+⋯−3062604162q98+O(q100) 8 q - 18 q^{2} - 428 q^{4} + 4684 q^{6} + 4800 q^{7} - 3384 q^{8} - 39368 q^{9} + 26392 q^{10} + 54760 q^{12} - 72336 q^{14} - 163136 q^{15} + 185616 q^{16} - 102000 q^{17} - 23614 q^{18} + 1245264 q^{20}+ \cdots - 3062604162 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x8−x7+59x6−313x5−315x4−92091x3+1261649x2−16074123x+251007534 x^{8} - x^{7} + 59x^{6} - 313x^{5} - 315x^{4} - 92091x^{3} + 1261649x^{2} - 16074123x + 251007534 : Copy content Toggle raw display

β1\beta_{1}== 2ν 2\nu Copy content Toggle raw display
β2\beta_{2}== 4ν2+2ν+58 4\nu^{2} + 2\nu + 58 Copy content Toggle raw display
β3\beta_{3}== (−ν7−318ν6+2947ν5+70006ν4+49989ν3−742730ν2+⋯−529423278)/8912896 ( - \nu^{7} - 318 \nu^{6} + 2947 \nu^{5} + 70006 \nu^{4} + 49989 \nu^{3} - 742730 \nu^{2} + \cdots - 529423278 ) / 8912896 Copy content Toggle raw display
β4\beta_{4}== (−55ν7−2258ν6−22875ν5−188326ν4−3365165ν3+4878490ν2+⋯+120340126)/4456448 ( - 55 \nu^{7} - 2258 \nu^{6} - 22875 \nu^{5} - 188326 \nu^{4} - 3365165 \nu^{3} + 4878490 \nu^{2} + \cdots + 120340126 ) / 4456448 Copy content Toggle raw display
β5\beta_{5}== (−545ν7+770ν6−30237ν5+273462ν4+17893ν3+49386870ν2+⋯+7650083474)/8912896 ( - 545 \nu^{7} + 770 \nu^{6} - 30237 \nu^{5} + 273462 \nu^{4} + 17893 \nu^{3} + 49386870 \nu^{2} + \cdots + 7650083474 ) / 8912896 Copy content Toggle raw display
β6\beta_{6}== (−11ν7+22ν6+1377ν5−4078ν4−123529ν3+501650ν2+⋯+79996934)/131072 ( - 11 \nu^{7} + 22 \nu^{6} + 1377 \nu^{5} - 4078 \nu^{4} - 123529 \nu^{3} + 501650 \nu^{2} + \cdots + 79996934 ) / 131072 Copy content Toggle raw display
β7\beta_{7}== (703ν7+10306ν6−39357ν5+642294ν4−14156923ν3+⋯+4806907090)/8912896 ( 703 \nu^{7} + 10306 \nu^{6} - 39357 \nu^{5} + 642294 \nu^{4} - 14156923 \nu^{3} + \cdots + 4806907090 ) / 8912896 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== (β1)/2 ( \beta_1 ) / 2 Copy content Toggle raw display
ν2\nu^{2}== (β2−β1−58)/4 ( \beta_{2} - \beta _1 - 58 ) / 4 Copy content Toggle raw display
ν3\nu^{3}== (−2β7−2β6+β5−4β4−15β3−2β2−57β1+774)/8 ( -2\beta_{7} - 2\beta_{6} + \beta_{5} - 4\beta_{4} - 15\beta_{3} - 2\beta_{2} - 57\beta _1 + 774 ) / 8 Copy content Toggle raw display
ν4\nu^{4}== (14β7−10β6+33β5−12β4+657β3+4β2+301β1+9346)/8 ( 14\beta_{7} - 10\beta_{6} + 33\beta_{5} - 12\beta_{4} + 657\beta_{3} + 4\beta_{2} + 301\beta _1 + 9346 ) / 8 Copy content Toggle raw display
ν5\nu^{5}== (−16β7+88β6−128β5−72β4+608β3+99β2+574β1+97102)/2 ( -16\beta_{7} + 88\beta_{6} - 128\beta_{5} - 72\beta_{4} + 608\beta_{3} + 99\beta_{2} + 574\beta _1 + 97102 ) / 2 Copy content Toggle raw display
ν6\nu^{6}== (1066β7+410β6+1467β5−2972β4−29877β3−469β2+⋯−3751528)/4 ( 1066 \beta_{7} + 410 \beta_{6} + 1467 \beta_{5} - 2972 \beta_{4} - 29877 \beta_{3} - 469 \beta_{2} + \cdots - 3751528 ) / 4 Copy content Toggle raw display
ν7\nu^{7}== (13522β7−23454β6−81689β5+1428β4+109815β3+⋯+74348202)/8 ( 13522 \beta_{7} - 23454 \beta_{6} - 81689 \beta_{5} + 1428 \beta_{4} + 109815 \beta_{3} + \cdots + 74348202 ) / 8 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/8Z)×\left(\mathbb{Z}/8\mathbb{Z}\right)^\times.

nn 55 77
χ(n)\chi(n) −1-1 11

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}
5.1
9.73909 + 3.55976i
9.73909 − 3.55976i
3.68032 + 10.3002i
3.68032 − 10.3002i
−2.43481 + 11.2224i
−2.43481 − 11.2224i
−10.4846 + 6.16784i
−10.4846 − 6.16784i
−21.4782 − 7.11952i 100.481i 410.625 + 305.829i − 2583.09i 715.380 − 2158.16i 6967.65 −6642.12 − 9492.10i 9586.48 −18390.4 + 55480.1i
5.2 −21.4782 + 7.11952i − 100.481i 410.625 − 305.829i 2583.09i 715.380 + 2158.16i 6967.65 −6642.12 + 9492.10i 9586.48 −18390.4 − 55480.1i
5.3 −9.36065 − 20.6004i − 150.106i −336.757 + 385.667i 292.339i −3092.24 + 1405.08i −9955.46 11097.2 + 3327.24i −2848.66 6022.31 − 2736.48i
5.4 −9.36065 + 20.6004i 150.106i −336.757 − 385.667i − 292.339i −3092.24 − 1405.08i −9955.46 11097.2 − 3327.24i −2848.66 6022.31 + 2736.48i
5.5 2.86961 − 22.4447i 247.414i −495.531 − 128.815i 1417.55i 5553.14 + 709.983i 5087.57 −4313.20 + 10752.4i −41530.8 31816.6 + 4067.83i
5.6 2.86961 + 22.4447i − 247.414i −495.531 + 128.815i − 1417.55i 5553.14 − 709.983i 5087.57 −4313.20 − 10752.4i −41530.8 31816.6 − 4067.83i
5.7 18.9692 − 12.3357i − 67.6316i 207.662 − 467.996i − 506.862i −834.282 − 1282.92i 300.249 −1833.85 − 11439.2i 15109.0 −6252.48 − 9614.77i
5.8 18.9692 + 12.3357i 67.6316i 207.662 + 467.996i 506.862i −834.282 + 1282.92i 300.249 −1833.85 + 11439.2i 15109.0 −6252.48 + 9614.77i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.10.b.a ✓ 8
3.b odd 2 1 72.10.d.b 8
4.b odd 2 1 32.10.b.a 8
8.b even 2 1 inner 8.10.b.a ✓ 8
8.d odd 2 1 32.10.b.a 8
12.b even 2 1 288.10.d.b 8
16.e even 4 2 256.10.a.p 8
16.f odd 4 2 256.10.a.s 8
24.f even 2 1 288.10.d.b 8
24.h odd 2 1 72.10.d.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.10.b.a ✓ 8 1.a even 1 1 trivial
8.10.b.a ✓ 8 8.b even 2 1 inner
32.10.b.a 8 4.b odd 2 1
32.10.b.a 8 8.d odd 2 1
72.10.d.b 8 3.b odd 2 1
72.10.d.b 8 24.h odd 2 1
256.10.a.p 8 16.e even 4 2
256.10.a.s 8 16.f odd 4 2
288.10.d.b 8 12.b even 2 1
288.10.d.b 8 24.f even 2 1

Hecke kernels

This newform subspace is the entire newspace S10new(8,[χ])S_{10}^{\mathrm{new}}(8, [\chi]).

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T8+⋯+68719476736 T^{8} + \cdots + 68719476736 Copy content Toggle raw display
33 T8+⋯+63 ⁣⋯ ⁣88 T^{8} + \cdots + 63\!\cdots\!88 Copy content Toggle raw display
55 T8+⋯+29 ⁣⋯ ⁣00 T^{8} + \cdots + 29\!\cdots\!00 Copy content Toggle raw display
77 (T4+⋯−105959154151424)2 (T^{4} + \cdots - 105959154151424)^{2} Copy content Toggle raw display
1111 T8+⋯+18 ⁣⋯ ⁣00 T^{8} + \cdots + 18\!\cdots\!00 Copy content Toggle raw display
1313 T8+⋯+18 ⁣⋯ ⁣00 T^{8} + \cdots + 18\!\cdots\!00 Copy content Toggle raw display
1717 (T4+⋯+49 ⁣⋯ ⁣48)2 (T^{4} + \cdots + 49\!\cdots\!48)^{2} Copy content Toggle raw display
1919 T8+⋯+48 ⁣⋯ ⁣72 T^{8} + \cdots + 48\!\cdots\!72 Copy content Toggle raw display
2323 (T4+⋯−42 ⁣⋯ ⁣76)2 (T^{4} + \cdots - 42\!\cdots\!76)^{2} Copy content Toggle raw display
2929 T8+⋯+46 ⁣⋯ ⁣00 T^{8} + \cdots + 46\!\cdots\!00 Copy content Toggle raw display
3131 (T4+⋯+67 ⁣⋯ ⁣56)2 (T^{4} + \cdots + 67\!\cdots\!56)^{2} Copy content Toggle raw display
3737 T8+⋯+17 ⁣⋯ ⁣48 T^{8} + \cdots + 17\!\cdots\!48 Copy content Toggle raw display
4141 (T4+⋯+74 ⁣⋯ ⁣80)2 (T^{4} + \cdots + 74\!\cdots\!80)^{2} Copy content Toggle raw display
4343 T8+⋯+94 ⁣⋯ ⁣08 T^{8} + \cdots + 94\!\cdots\!08 Copy content Toggle raw display
4747 (T4+⋯+54 ⁣⋯ ⁣12)2 (T^{4} + \cdots + 54\!\cdots\!12)^{2} Copy content Toggle raw display
5353 T8+⋯+11 ⁣⋯ ⁣88 T^{8} + \cdots + 11\!\cdots\!88 Copy content Toggle raw display
5959 T8+⋯+77 ⁣⋯ ⁣08 T^{8} + \cdots + 77\!\cdots\!08 Copy content Toggle raw display
6161 T8+⋯+10 ⁣⋯ ⁣00 T^{8} + \cdots + 10\!\cdots\!00 Copy content Toggle raw display
6767 T8+⋯+13 ⁣⋯ ⁣68 T^{8} + \cdots + 13\!\cdots\!68 Copy content Toggle raw display
7171 (T4+⋯+60 ⁣⋯ ⁣52)2 (T^{4} + \cdots + 60\!\cdots\!52)^{2} Copy content Toggle raw display
7373 (T4+⋯−27 ⁣⋯ ⁣12)2 (T^{4} + \cdots - 27\!\cdots\!12)^{2} Copy content Toggle raw display
7979 (T4+⋯+14 ⁣⋯ ⁣40)2 (T^{4} + \cdots + 14\!\cdots\!40)^{2} Copy content Toggle raw display
8383 T8+⋯+37 ⁣⋯ ⁣68 T^{8} + \cdots + 37\!\cdots\!68 Copy content Toggle raw display
8989 (T4+⋯+17 ⁣⋯ ⁣20)2 (T^{4} + \cdots + 17\!\cdots\!20)^{2} Copy content Toggle raw display
9797 (T4+⋯+41 ⁣⋯ ⁣72)2 (T^{4} + \cdots + 41\!\cdots\!72)^{2} Copy content Toggle raw display
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