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Properties

Label 8.10.b.a
Level $8$
Weight $10$
Character orbit 8.b
Analytic conductor $4.120$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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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: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 8.b (of order \(2\), degree \(1\), 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: \(4.12028668931\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 59x^{6} - 313x^{5} - 315x^{4} - 92091x^{3} + 1261649x^{2} - 16074123x + 251007534 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{28}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 2) q^{2} + (\beta_{3} + \beta_1) q^{3} + (\beta_{2} + 3 \beta_1 - 54) q^{4} + (\beta_{4} - \beta_{2} + 6 \beta_1 - 2) q^{5} + ( - \beta_{6} + \beta_{4} - 2 \beta_{3} + \cdots + 585) q^{6}+ \cdots + ( - 2 \beta_{7} + 4 \beta_{6} + \cdots - 4923) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 2) q^{2} + (\beta_{3} + \beta_1) q^{3} + (\beta_{2} + 3 \beta_1 - 54) q^{4} + (\beta_{4} - \beta_{2} + 6 \beta_1 - 2) q^{5} + ( - \beta_{6} + \beta_{4} - 2 \beta_{3} + \cdots + 585) q^{6}+ \cdots + ( - 66624 \beta_{7} - 148128 \beta_{5} + \cdots - 3435648) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 18 q^{2} - 428 q^{4} + 4684 q^{6} + 4800 q^{7} - 3384 q^{8} - 39368 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 18 q^{2} - 428 q^{4} + 4684 q^{6} + 4800 q^{7} - 3384 q^{8} - 39368 q^{9} + 26392 q^{10} + 54760 q^{12} - 72336 q^{14} - 163136 q^{15} + 185616 q^{16} - 102000 q^{17} - 23614 q^{18} + 1245264 q^{20} - 2373124 q^{22} + 3412032 q^{23} - 3961456 q^{24} - 2423384 q^{25} + 4551240 q^{26} + 7509920 q^{28} - 15284368 q^{30} + 803584 q^{31} - 14113248 q^{32} + 58272 q^{33} + 27757244 q^{34} + 57226188 q^{36} - 63661140 q^{38} - 17590208 q^{39} - 93063648 q^{40} - 2180784 q^{41} + 127541344 q^{42} + 114013320 q^{44} - 131840944 q^{46} + 7432320 q^{47} - 217917408 q^{48} + 24436680 q^{49} + 231784902 q^{50} + 219270896 q^{52} - 362934280 q^{54} + 7056832 q^{55} - 358503360 q^{56} + 134003744 q^{57} + 375425192 q^{58} + 516952992 q^{60} - 344291904 q^{62} - 223198400 q^{63} - 316815296 q^{64} - 146501760 q^{65} + 239713176 q^{66} + 79875048 q^{68} - 56202048 q^{70} + 560234688 q^{71} + 112273016 q^{72} - 523987120 q^{73} - 65773608 q^{74} - 87532760 q^{76} + 318117968 q^{78} - 248943744 q^{79} + 890441280 q^{80} + 231960296 q^{81} - 1051981172 q^{82} - 1275608768 q^{84} + 1492810428 q^{86} + 540527424 q^{87} + 1544767952 q^{88} + 744827856 q^{89} - 3218579800 q^{90} - 2959012128 q^{92} + 3068552352 q^{94} - 1465245504 q^{95} + 4296343616 q^{96} - 9932784 q^{97} - 3062604162 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 59x^{6} - 313x^{5} - 315x^{4} - 92091x^{3} + 1261649x^{2} - 16074123x + 251007534 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\nu^{2} + 2\nu + 58 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{7} - 318 \nu^{6} + 2947 \nu^{5} + 70006 \nu^{4} + 49989 \nu^{3} - 742730 \nu^{2} + \cdots - 529423278 ) / 8912896 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 55 \nu^{7} - 2258 \nu^{6} - 22875 \nu^{5} - 188326 \nu^{4} - 3365165 \nu^{3} + 4878490 \nu^{2} + \cdots + 120340126 ) / 4456448 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 545 \nu^{7} + 770 \nu^{6} - 30237 \nu^{5} + 273462 \nu^{4} + 17893 \nu^{3} + 49386870 \nu^{2} + \cdots + 7650083474 ) / 8912896 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11 \nu^{7} + 22 \nu^{6} + 1377 \nu^{5} - 4078 \nu^{4} - 123529 \nu^{3} + 501650 \nu^{2} + \cdots + 79996934 ) / 131072 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 703 \nu^{7} + 10306 \nu^{6} - 39357 \nu^{5} + 642294 \nu^{4} - 14156923 \nu^{3} + \cdots + 4806907090 ) / 8912896 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} - \beta _1 - 58 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{7} - 2\beta_{6} + \beta_{5} - 4\beta_{4} - 15\beta_{3} - 2\beta_{2} - 57\beta _1 + 774 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 14\beta_{7} - 10\beta_{6} + 33\beta_{5} - 12\beta_{4} + 657\beta_{3} + 4\beta_{2} + 301\beta _1 + 9346 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -16\beta_{7} + 88\beta_{6} - 128\beta_{5} - 72\beta_{4} + 608\beta_{3} + 99\beta_{2} + 574\beta _1 + 97102 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1066 \beta_{7} + 410 \beta_{6} + 1467 \beta_{5} - 2972 \beta_{4} - 29877 \beta_{3} - 469 \beta_{2} + \cdots - 3751528 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 13522 \beta_{7} - 23454 \beta_{6} - 81689 \beta_{5} + 1428 \beta_{4} + 109815 \beta_{3} + \cdots + 74348202 ) / 8 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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
\(n\): 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 \(S_{10}^{\mathrm{new}}(8, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + \cdots + 68719476736 \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 63\!\cdots\!88 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{4} + \cdots - 105959154151424)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 49\!\cdots\!48)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 48\!\cdots\!72 \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots - 42\!\cdots\!76)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 67\!\cdots\!56)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 17\!\cdots\!48 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 74\!\cdots\!80)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 94\!\cdots\!08 \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 54\!\cdots\!12)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 11\!\cdots\!88 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 77\!\cdots\!08 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 13\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 60\!\cdots\!52)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 27\!\cdots\!12)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 14\!\cdots\!40)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 37\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 17\!\cdots\!20)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 41\!\cdots\!72)^{2} \) Copy content Toggle raw display
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