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Properties

Label 8.11.d.b
Level $8$
Weight $11$
Character orbit 8.d
Analytic conductor $5.083$
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] = [8,11,Mod(3,8)]
 
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([1, 1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.3");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 8.d (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: \(5.08285802139\)
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: \( x^{8} - 3x^{7} + 5x^{6} - 51x^{5} + 30855x^{4} - 121569x^{3} + 12144527x^{2} + 279415575x + 3348211684 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{28}\cdot 3^{3} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 5) q^{2} + (\beta_{2} + \beta_1 + 60) q^{3} + (\beta_{3} + \beta_{2} - 5 \beta_1 + 25) q^{4} + ( - \beta_{6} + \beta_1) q^{5} + ( - \beta_{7} + \beta_{6} + \beta_{4} + \cdots - 697) q^{6}+ \cdots + (18 \beta_{4} + 150 \beta_{2} + \cdots - 4581) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 5) q^{2} + (\beta_{2} + \beta_1 + 60) q^{3} + (\beta_{3} + \beta_{2} - 5 \beta_1 + 25) q^{4} + ( - \beta_{6} + \beta_1) q^{5} + ( - \beta_{7} + \beta_{6} + \beta_{4} + \cdots - 697) q^{6}+ \cdots + ( - 355968 \beta_{7} + \cdots - 1687350924) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 42 q^{2} + 480 q^{3} + 212 q^{4} - 5412 q^{6} - 72 q^{8} - 35496 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 42 q^{2} + 480 q^{3} + 212 q^{4} - 5412 q^{6} - 72 q^{8} - 35496 q^{9} + 9120 q^{10} + 143328 q^{11} + 180120 q^{12} - 400320 q^{14} - 1987312 q^{16} - 370800 q^{17} + 3067758 q^{18} + 1753312 q^{19} + 2557440 q^{20} + 1549180 q^{22} - 19794288 q^{24} - 18792760 q^{25} + 1891680 q^{26} + 28949184 q^{27} + 18286080 q^{28} + 67026240 q^{30} - 64016928 q^{32} - 74308704 q^{33} - 115705388 q^{34} + 53736960 q^{35} + 50631516 q^{36} + 375128220 q^{38} - 357584640 q^{40} + 92669328 q^{41} - 576155520 q^{42} - 10190624 q^{43} + 727831512 q^{44} + 851947200 q^{46} - 1142760480 q^{48} - 80432248 q^{49} - 1669361190 q^{50} - 276714816 q^{51} + 2420759040 q^{52} + 1749768696 q^{54} - 2928529920 q^{56} + 1400712864 q^{57} - 3245264160 q^{58} - 965642016 q^{59} + 5250055680 q^{60} + 4060980480 q^{62} - 5804705728 q^{64} - 839028480 q^{65} - 5970262296 q^{66} - 1225582880 q^{67} + 5258111400 q^{68} + 7723829760 q^{70} - 4343608728 q^{72} + 2800072720 q^{73} - 7669373280 q^{74} + 4485554400 q^{75} + 3753451288 q^{76} + 7176312000 q^{78} - 6408629760 q^{80} - 11909065176 q^{81} - 3638778380 q^{82} + 1853422560 q^{83} - 2916218880 q^{84} + 3022180476 q^{86} + 5815578640 q^{88} + 6162596112 q^{89} + 12545940960 q^{90} + 7645985280 q^{91} - 8903892480 q^{92} - 14182177920 q^{94} + 7877525568 q^{96} - 9697863536 q^{97} + 34759868298 q^{98} - 13646747232 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 5x^{6} - 51x^{5} + 30855x^{4} - 121569x^{3} + 12144527x^{2} + 279415575x + 3348211684 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 39 \nu^{7} + 640 \nu^{6} - 6467 \nu^{5} - 38788 \nu^{4} + 639715 \nu^{3} + 3220592 \nu^{2} + \cdots - 5385260908 ) / 29360128 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 39 \nu^{7} - 640 \nu^{6} + 6467 \nu^{5} + 38788 \nu^{4} - 639715 \nu^{3} + 114219920 \nu^{2} + \cdots + 5561421676 ) / 29360128 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 73 \nu^{7} + 640 \nu^{6} - 5907 \nu^{5} - 42820 \nu^{4} + 4083379 \nu^{3} - 64144 \nu^{2} + \cdots + 25027780436 ) / 29360128 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 127 \nu^{7} + 2496 \nu^{6} - 17723 \nu^{5} + 547420 \nu^{4} - 1923813 \nu^{3} + \cdots - 3601739852 ) / 3670016 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 51 \nu^{7} - 2688 \nu^{6} + 81793 \nu^{5} - 1067988 \nu^{4} + 10212255 \nu^{3} + \cdots - 56323662812 ) / 14680064 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 147 \nu^{7} - 2560 \nu^{6} + 1247 \nu^{5} + 270164 \nu^{4} + 7125889 \nu^{3} - 75156144 \nu^{2} + \cdots + 21257762140 ) / 7340032 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 7\beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{7} + 3\beta_{6} + \beta_{5} + 11\beta_{4} + 11\beta_{3} + 58\beta_{2} + 46\beta _1 + 131 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 36\beta_{7} + 9\beta_{6} + 43\beta_{5} - 31\beta_{4} + 67\beta_{3} - 592\beta_{2} - 412\beta _1 - 122793 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 300 \beta_{7} + 629 \beta_{6} + 303 \beta_{5} + 1773 \beta_{4} - 159 \beta_{3} - 10902 \beta_{2} + \cdots + 67607 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 7556 \beta_{7} + 2137 \beta_{6} + 2883 \beta_{5} + 64497 \beta_{4} - 18502 \beta_{3} + \cdots - 35764058 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 118692 \beta_{7} - 98207 \beta_{6} - 32229 \beta_{5} + 1740089 \beta_{4} - 275559 \beta_{3} + \cdots - 2229652087 ) / 8 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
16.5295 + 8.54130i
16.5295 − 8.54130i
4.83019 + 15.8950i
4.83019 − 15.8950i
−8.43092 + 11.1953i
−8.43092 − 11.1953i
−11.4287 + 6.91461i
−11.4287 − 6.91461i
−27.0589 − 17.0826i 203.867 440.370 + 924.473i − 2088.87i −5516.42 − 3482.58i − 25367.2i 3876.45 − 32537.9i −17487.2 −35683.3 + 56522.5i
3.2 −27.0589 + 17.0826i 203.867 440.370 − 924.473i 2088.87i −5516.42 + 3482.58i 25367.2i 3876.45 + 32537.9i −17487.2 −35683.3 − 56522.5i
3.3 −3.66038 − 31.7900i −119.281 −997.203 + 232.727i 948.910i 436.614 + 3791.94i 15458.1i 11048.5 + 30849.2i −44821.0 30165.8 − 3473.37i
3.4 −3.66038 + 31.7900i −119.281 −997.203 − 232.727i − 948.910i 436.614 − 3791.94i − 15458.1i 11048.5 − 30849.2i −44821.0 30165.8 + 3473.37i
3.5 22.8618 − 22.3905i 352.099 21.3279 − 1023.78i 3773.58i 8049.62 − 7883.68i − 15618.2i −22435.3 − 23883.0i 64924.4 84492.5 + 86271.1i
3.6 22.8618 + 22.3905i 352.099 21.3279 + 1023.78i − 3773.58i 8049.62 + 7883.68i 15618.2i −22435.3 + 23883.0i 64924.4 84492.5 − 86271.1i
3.7 28.8575 − 13.8292i −196.684 641.505 − 798.152i − 5381.00i −5675.81 + 2719.99i 6613.83i 7474.38 − 31904.2i −20364.2 −74415.0 − 155282.i
3.8 28.8575 + 13.8292i −196.684 641.505 + 798.152i 5381.00i −5675.81 − 2719.99i − 6613.83i 7474.38 + 31904.2i −20364.2 −74415.0 + 155282.i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.11.d.b ✓ 8
3.b odd 2 1 72.11.b.b 8
4.b odd 2 1 32.11.d.b 8
8.b even 2 1 32.11.d.b 8
8.d odd 2 1 inner 8.11.d.b ✓ 8
12.b even 2 1 288.11.b.b 8
16.e even 4 2 256.11.c.m 16
16.f odd 4 2 256.11.c.m 16
24.f even 2 1 72.11.b.b 8
24.h odd 2 1 288.11.b.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.11.d.b ✓ 8 1.a even 1 1 trivial
8.11.d.b ✓ 8 8.d odd 2 1 inner
32.11.d.b 8 4.b odd 2 1
32.11.d.b 8 8.b even 2 1
72.11.b.b 8 3.b odd 2 1
72.11.b.b 8 24.f even 2 1
256.11.c.m 16 16.e even 4 2
256.11.c.m 16 16.f odd 4 2
288.11.b.b 8 12.b even 2 1
288.11.b.b 8 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 240T_{3}^{3} - 80424T_{3}^{2} + 9637056T_{3} + 1684044432 \) acting on \(S_{11}^{\mathrm{new}}(8, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + \cdots + 1099511627776 \) Copy content Toggle raw display
$3$ \( (T^{4} - 240 T^{3} + \cdots + 1684044432)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 56\!\cdots\!48)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 29\!\cdots\!32)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 20\!\cdots\!48)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 68\!\cdots\!88)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 51\!\cdots\!16)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 47\!\cdots\!56)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots - 13\!\cdots\!88)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 23\!\cdots\!12)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 77\!\cdots\!72)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 20\!\cdots\!28)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 57\!\cdots\!84)^{2} \) Copy content Toggle raw display
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