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Properties

Label 8.18.b.a
Level $8$
Weight $18$
Character orbit 8.b
Analytic conductor $14.658$
Analytic rank $0$
Dimension $16$
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] = [8,18,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, 18, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.5");
 
S:= CuspForms(chi, 18);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 18 \)
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: \(14.6577669876\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 7 x^{15} + 4022 x^{14} - 1102776 x^{13} - 373411968 x^{12} + 2100004864 x^{11} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{120}\cdot 3^{14}\cdot 7 \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 17) q^{2} + ( - \beta_{2} + 3 \beta_1) q^{3} + ( - \beta_{3} - \beta_{2} + \cdots - 1712) q^{4}+ \cdots + (\beta_{11} - 2 \beta_{8} + \cdots - 37672113) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 17) q^{2} + ( - \beta_{2} + 3 \beta_1) q^{3} + ( - \beta_{3} - \beta_{2} + \cdots - 1712) q^{4}+ \cdots + (43468656 \beta_{15} + \cdots + 3728874040020) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 270 q^{2} - 27436 q^{4} + 5839948 q^{6} + 11529600 q^{7} + 24334920 q^{8} - 602654096 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 270 q^{2} - 27436 q^{4} + 5839948 q^{6} + 11529600 q^{7} + 24334920 q^{8} - 602654096 q^{9} + 131002712 q^{10} - 2795125400 q^{12} + 16363788528 q^{14} - 9993282176 q^{15} + 26500434192 q^{16} - 7489125600 q^{17} - 113450563870 q^{18} - 209445719856 q^{20} + 223126527100 q^{22} + 746845345920 q^{23} - 1099415493232 q^{24} - 1809682431664 q^{25} + 2467726531080 q^{26} + 3220542267040 q^{28} - 1188624268048 q^{30} - 318979758592 q^{31} + 1455647316000 q^{32} + 5633526177600 q^{33} - 4461251980292 q^{34} - 33088278002484 q^{36} + 24076283913900 q^{38} - 18457706051456 q^{39} + 60626292962592 q^{40} + 7482251536032 q^{41} - 51630378688160 q^{42} + 193654716236040 q^{44} - 195097141003568 q^{46} - 376698804821760 q^{47} - 329350060416480 q^{48} + 127691292101520 q^{49} + 474997408872102 q^{50} - 272251877663120 q^{52} + 735354219382520 q^{54} + 22\!\cdots\!52 q^{55}+ \cdots + 33\!\cdots\!90 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 7 x^{15} + 4022 x^{14} - 1102776 x^{13} - 373411968 x^{12} + 2100004864 x^{11} + \cdots + 12\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{15} - 15 \nu^{14} + 4142 \nu^{13} - 1135912 \nu^{12} - 364324672 \nu^{11} + \cdots - 50\!\cdots\!00 ) / 20\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 202665796662607 \nu^{15} + \cdots - 13\!\cdots\!96 ) / 56\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 113144783158307 \nu^{15} + \cdots + 18\!\cdots\!96 ) / 70\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 69\!\cdots\!03 \nu^{15} + \cdots - 42\!\cdots\!68 ) / 56\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 23\!\cdots\!21 \nu^{15} + \cdots - 16\!\cdots\!04 ) / 16\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 79\!\cdots\!57 \nu^{15} + \cdots - 21\!\cdots\!28 ) / 56\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13\!\cdots\!51 \nu^{15} + \cdots - 18\!\cdots\!64 ) / 84\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 90\!\cdots\!77 \nu^{15} + \cdots + 33\!\cdots\!28 ) / 42\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 63\!\cdots\!13 \nu^{15} + \cdots - 17\!\cdots\!80 ) / 21\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 63\!\cdots\!43 \nu^{15} + \cdots - 44\!\cdots\!16 ) / 16\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 14\!\cdots\!19 \nu^{15} + \cdots - 12\!\cdots\!32 ) / 16\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 44\!\cdots\!49 \nu^{15} + \cdots - 13\!\cdots\!88 ) / 16\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 67\!\cdots\!19 \nu^{15} + \cdots - 14\!\cdots\!28 ) / 16\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 33\!\cdots\!91 \nu^{15} + \cdots + 37\!\cdots\!20 ) / 52\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 45\!\cdots\!27 \nu^{15} + \cdots + 18\!\cdots\!96 ) / 42\!\cdots\!96 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} - 35\beta_{2} + 258\beta _1 + 14317 ) / 32768 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4 \beta_{9} + 4 \beta_{8} - 20 \beta_{6} - 128 \beta_{5} - 11 \beta_{4} - 821 \beta_{3} + \cdots - 16364119 ) / 32768 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 128 \beta_{14} - 256 \beta_{11} + 32 \beta_{10} - 68 \beta_{9} - 420 \beta_{8} + 320 \beta_{7} + \cdots + 6608796329 ) / 32768 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2048 \beta_{15} - 6784 \beta_{14} + 45056 \beta_{13} + 10240 \beta_{12} - 133888 \beta_{11} + \cdots + 3186731777169 ) / 32768 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 604160 \beta_{15} - 132224 \beta_{14} + 3198976 \beta_{13} + 1906688 \beta_{12} + \cdots - 38440802803255 ) / 32768 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 451729408 \beta_{15} - 98987136 \beta_{14} - 1729851392 \beta_{13} + 254359552 \beta_{12} + \cdots + 34\!\cdots\!89 ) / 32768 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 37029533696 \beta_{15} - 7822938240 \beta_{14} - 44582662144 \beta_{13} + 3116251136 \beta_{12} + \cdots - 98\!\cdots\!87 ) / 32768 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 10629839611904 \beta_{15} + 3261967807360 \beta_{14} + 2968419635200 \beta_{13} + \cdots + 18\!\cdots\!17 ) / 32768 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 306966240221184 \beta_{15} - 117737012369536 \beta_{14} + \cdots - 23\!\cdots\!55 ) / 32768 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 48\!\cdots\!12 \beta_{15} + \cdots - 35\!\cdots\!15 ) / 32768 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 23\!\cdots\!88 \beta_{15} + \cdots - 48\!\cdots\!95 ) / 32768 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 94\!\cdots\!16 \beta_{15} + \cdots + 13\!\cdots\!41 ) / 32768 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 21\!\cdots\!32 \beta_{15} + \cdots - 99\!\cdots\!19 ) / 32768 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 21\!\cdots\!08 \beta_{15} + \cdots + 17\!\cdots\!69 ) / 32768 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 50\!\cdots\!96 \beta_{15} + \cdots - 10\!\cdots\!87 ) / 32768 \) 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
−189.013 + 1.56793i
−189.013 − 1.56793i
−136.895 + 127.099i
−136.895 − 127.099i
−108.197 + 150.760i
−108.197 − 150.760i
1.16697 + 180.787i
1.16697 − 180.787i
13.1473 + 179.780i
13.1473 − 179.780i
99.0623 + 145.965i
99.0623 − 145.965i
156.321 + 75.9390i
156.321 − 75.9390i
167.908 + 42.7144i
167.908 − 42.7144i
−362.025 − 3.13586i 13867.2i 131052. + 2270.52i − 665059.i 43485.7 − 5.02028e6i −7.57536e6 −4.74371e7 − 1.23295e6i −6.31593e7 −2.08553e6 + 2.40768e8i
5.2 −362.025 + 3.13586i − 13867.2i 131052. − 2270.52i 665059.i 43485.7 + 5.02028e6i −7.57536e6 −4.74371e7 + 1.23295e6i −6.31593e7 −2.08553e6 − 2.40768e8i
5.3 −257.790 − 254.197i − 4834.14i 1839.67 + 131059.i − 524871.i −1.22882e6 + 1.24619e6i 1.57495e7 3.28406e7 − 3.42534e7i 1.05771e8 −1.33421e8 + 1.35307e8i
5.4 −257.790 + 254.197i 4834.14i 1839.67 − 131059.i 524871.i −1.22882e6 − 1.24619e6i 1.57495e7 3.28406e7 + 3.42534e7i 1.05771e8 −1.33421e8 − 1.35307e8i
5.5 −200.394 − 301.520i 13481.8i −50756.5 + 120846.i 1.59197e6i 4.06504e6 − 2.70168e6i −1.66055e7 4.66086e7 − 8.91263e6i −5.26193e7 4.80011e8 − 3.19021e8i
5.6 −200.394 + 301.520i − 13481.8i −50756.5 − 120846.i − 1.59197e6i 4.06504e6 + 2.70168e6i −1.66055e7 4.66086e7 + 8.91263e6i −5.26193e7 4.80011e8 + 3.19021e8i
5.7 18.3339 − 361.574i − 13786.7i −130400. − 13258.2i − 96356.3i −4.98490e6 − 252764.i −1.47728e7 −7.18455e6 + 4.69061e7i −6.09318e7 −3.48399e7 − 1.76659e6i
5.8 18.3339 + 361.574i 13786.7i −130400. + 13258.2i 96356.3i −4.98490e6 + 252764.i −1.47728e7 −7.18455e6 − 4.69061e7i −6.09318e7 −3.48399e7 + 1.76659e6i
5.9 42.2945 − 359.560i 16002.3i −127494. − 30414.8i − 1.27253e6i 5.75379e6 + 676810.i 5.50569e6 −1.63282e7 + 4.45555e7i −1.26934e8 −4.57551e8 − 5.38211e7i
5.10 42.2945 + 359.560i − 16002.3i −127494. + 30414.8i 1.27253e6i 5.75379e6 − 676810.i 5.50569e6 −1.63282e7 − 4.45555e7i −1.26934e8 −4.57551e8 + 5.38211e7i
5.11 214.125 − 291.929i 1638.94i −39373.4 − 125018.i 1.21254e6i 478455. + 350938.i 1.76580e7 −4.49273e7 − 1.52753e7i 1.26454e8 3.53976e8 + 2.59635e8i
5.12 214.125 + 291.929i − 1638.94i −39373.4 + 125018.i − 1.21254e6i 478455. − 350938.i 1.76580e7 −4.49273e7 + 1.52753e7i 1.26454e8 3.53976e8 − 2.59635e8i
5.13 328.641 − 151.878i 4248.51i 84938.1 − 99826.8i − 663971.i 645256. + 1.39624e6i −1.66742e7 1.27527e7 − 4.57074e7i 1.11090e8 −1.00843e8 − 2.18208e8i
5.14 328.641 + 151.878i − 4248.51i 84938.1 + 99826.8i 663971.i 645256. − 1.39624e6i −1.66742e7 1.27527e7 + 4.57074e7i 1.11090e8 −1.00843e8 + 2.18208e8i
5.15 351.815 − 85.4288i − 21682.7i 116476. − 60110.3i − 465248.i −1.85232e6 − 7.62829e6i 2.24795e7 3.58428e7 − 3.10981e7i −3.40998e8 −3.97456e7 − 1.63681e8i
5.16 351.815 + 85.4288i 21682.7i 116476. + 60110.3i 465248.i −1.85232e6 + 7.62829e6i 2.24795e7 3.58428e7 + 3.10981e7i −3.40998e8 −3.97456e7 + 1.63681e8i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.16
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 8.18.b.a ✓ 16
3.b odd 2 1 72.18.d.b 16
4.b odd 2 1 32.18.b.a 16
8.b even 2 1 inner 8.18.b.a ✓ 16
8.d odd 2 1 32.18.b.a 16
24.h odd 2 1 72.18.d.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.18.b.a ✓ 16 1.a even 1 1 trivial
8.18.b.a ✓ 16 8.b even 2 1 inner
32.18.b.a 16 4.b odd 2 1
32.18.b.a 16 8.d odd 2 1
72.18.d.b 16 3.b odd 2 1
72.18.d.b 16 24.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{18}^{\mathrm{new}}(8, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 87\!\cdots\!36 \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 90\!\cdots\!84 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 10\!\cdots\!76)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots - 41\!\cdots\!36)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 11\!\cdots\!16)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots - 53\!\cdots\!24)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 14\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 94\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 17\!\cdots\!04)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 15\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 22\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 95\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots - 21\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 18\!\cdots\!24)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 13\!\cdots\!36)^{2} \) Copy content Toggle raw display
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