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Properties

Label 8.17.d.b
Level $8$
Weight $17$
Character orbit 8.d
Analytic conductor $12.986$
Analytic rank $0$
Dimension $14$
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] = [8,17,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, 17, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.3");
 
S:= CuspForms(chi, 17);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 17 \)
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: \(12.9859635085\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 7 x^{13} + 7894 x^{12} + 219192 x^{11} - 135772320 x^{10} + 11331786624 x^{9} + \cdots + 27\!\cdots\!20 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{91}\cdot 3^{10} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 25) q^{2} + (\beta_{3} - 6 \beta_1 + 855) q^{3} + (\beta_{2} + 27 \beta_1 - 3873) q^{4} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \cdots + 1) q^{5}+ \cdots + ( - \beta_{10} + \beta_{7} + \cdots + 6172027) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 25) q^{2} + (\beta_{3} - 6 \beta_1 + 855) q^{3} + (\beta_{2} + 27 \beta_1 - 3873) q^{4} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \cdots + 1) q^{5}+ \cdots + ( - 60570360 \beta_{13} + \cdots - 13\!\cdots\!63) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 350 q^{2} + 11964 q^{3} - 54220 q^{4} + 4868124 q^{6} + 10899160 q^{8} + 86397354 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 350 q^{2} + 11964 q^{3} - 54220 q^{4} + 4868124 q^{6} + 10899160 q^{8} + 86397354 q^{9} + 87155280 q^{10} - 49140292 q^{11} + 31616184 q^{12} - 2675193504 q^{14} + 9716374544 q^{16} - 9985136356 q^{17} - 7692086106 q^{18} + 62922133180 q^{19} + 31953450720 q^{20} - 51493048676 q^{22} + 54962419344 q^{24} - 429906632050 q^{25} - 518273679696 q^{26} + 655936606584 q^{27} + 102542376000 q^{28} - 364331826720 q^{30} + 95751080800 q^{32} + 4034287537464 q^{33} + 1015818512644 q^{34} - 3057653406720 q^{35} - 9569289445092 q^{36} + 9010383442844 q^{38} - 4027933291200 q^{40} + 8719629072668 q^{41} - 6653698130880 q^{42} + 16716309178300 q^{43} - 4351929975880 q^{44} + 21043605267744 q^{46} + 46562301973344 q^{48} - 104740771400434 q^{49} - 54483696687710 q^{50} + 40911296041848 q^{51} - 73249356722400 q^{52} - 18031398151752 q^{54} + 110873799752064 q^{56} - 310730528383176 q^{57} + 256627273576560 q^{58} + 347007330293180 q^{59} - 83822619107520 q^{60} + 346701622780800 q^{62} - 332173589517760 q^{64} + 220877370432000 q^{65} + 875905832132808 q^{66} - 614258765968196 q^{67} - 12\!\cdots\!76 q^{68}+ \cdots - 18\!\cdots\!92 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 7 x^{13} + 7894 x^{12} + 219192 x^{11} - 135772320 x^{10} + 11331786624 x^{9} + \cdots + 27\!\cdots\!20 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\nu^{2} + 42\nu + 4476 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 58320936057 \nu^{13} + 5986305819397 \nu^{12} + \cdots - 17\!\cdots\!20 ) / 35\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 59559682613073 \nu^{13} + \cdots + 27\!\cdots\!20 ) / 46\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11214439470053 \nu^{13} + \cdots + 13\!\cdots\!20 ) / 35\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 375239388039 \nu^{13} + 14223951977221 \nu^{12} - 953736685017366 \nu^{11} + \cdots + 10\!\cdots\!08 ) / 10\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 22798506203329 \nu^{13} + \cdots - 32\!\cdots\!20 ) / 35\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 218134071226359 \nu^{13} + \cdots - 55\!\cdots\!60 ) / 15\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 238078010965047 \nu^{13} + \cdots + 18\!\cdots\!80 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 41\!\cdots\!01 \nu^{13} + \cdots + 37\!\cdots\!40 ) / 46\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 49\!\cdots\!19 \nu^{13} + \cdots - 46\!\cdots\!80 ) / 23\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 35\!\cdots\!51 \nu^{13} + \cdots - 12\!\cdots\!20 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 16\!\cdots\!21 \nu^{13} + \cdots - 43\!\cdots\!20 ) / 46\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} - 21\beta _1 - 4497 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{8} + 3\beta_{6} + \beta_{4} - 109\beta_{3} - 44\beta_{2} - 3785\beta _1 - 470331 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 16 \beta_{13} + 6 \beta_{12} - 20 \beta_{11} - 15 \beta_{10} - 182 \beta_{9} - 47 \beta_{8} + \cdots + 377177286 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 32 \beta_{13} - 1470 \beta_{12} - 516 \beta_{11} - 5077 \beta_{10} - 9810 \beta_{9} + \cdots - 23517462772 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 14144 \beta_{13} + 180298 \beta_{12} + 35228 \beta_{11} + 350839 \beta_{10} + 621158 \beta_{9} + \cdots - 2068502352968 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 12430464 \beta_{13} - 34697334 \beta_{12} - 41746180 \beta_{11} - 74505833 \beta_{10} + \cdots - 17\!\cdots\!92 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 3852040192 \beta_{13} + 2045631546 \beta_{12} + 5476328156 \beta_{11} - 2541798177 \beta_{10} + \cdots + 61\!\cdots\!60 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 891841605376 \beta_{13} - 327529170870 \beta_{12} - 171527044420 \beta_{11} + 2564970646839 \beta_{10} + \cdots - 16\!\cdots\!88 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 83704357255936 \beta_{13} + 40286808739482 \beta_{12} + 68537342963612 \beta_{11} + \cdots - 16\!\cdots\!08 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 90574744598272 \beta_{13} + \cdots + 52\!\cdots\!48 ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 17\!\cdots\!00 \beta_{13} + \cdots - 98\!\cdots\!84 ) / 8 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 11\!\cdots\!08 \beta_{13} + \cdots + 97\!\cdots\!00 ) / 8 \) 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
109.380 + 40.6323i
109.380 − 40.6323i
101.867 + 58.4667i
101.867 − 58.4667i
41.8643 + 116.115i
41.8643 − 116.115i
12.1331 + 125.704i
12.1331 − 125.704i
−30.5698 + 126.646i
−30.5698 − 126.646i
−93.5934 + 98.6231i
−93.5934 − 98.6231i
−137.581 + 24.7690i
−137.581 − 24.7690i
−242.759 − 81.2647i −7667.81 52328.1 + 39455.5i − 557358.i 1.86143e6 + 623122.i − 8.55454e6i −9.49679e6 − 1.38306e7i 1.57485e7 −4.52935e7 + 1.35304e8i
3.2 −242.759 + 81.2647i −7667.81 52328.1 − 39455.5i 557358.i 1.86143e6 − 623122.i 8.55454e6i −9.49679e6 + 1.38306e7i 1.57485e7 −4.52935e7 − 1.35304e8i
3.3 −227.734 − 116.933i 6146.29 38189.2 + 53259.3i 351107.i −1.39972e6 − 718707.i − 1.24931e6i −2.46917e6 − 1.65945e7i −5.26981e6 4.10562e7 − 7.99589e7i
3.4 −227.734 + 116.933i 6146.29 38189.2 − 53259.3i − 351107.i −1.39972e6 + 718707.i 1.24931e6i −2.46917e6 + 1.65945e7i −5.26981e6 4.10562e7 + 7.99589e7i
3.5 −107.729 − 232.229i 693.074 −42325.1 + 50035.5i − 333015.i −74663.9 − 160952.i 1.01327e7i 1.61794e7 + 4.43887e6i −4.25664e7 −7.73360e7 + 3.58753e7i
3.6 −107.729 + 232.229i 693.074 −42325.1 − 50035.5i 333015.i −74663.9 + 160952.i − 1.01327e7i 1.61794e7 − 4.43887e6i −4.25664e7 −7.73360e7 − 3.58753e7i
3.7 −48.2661 − 251.409i −8648.56 −60876.8 + 24269.1i 474083.i 417432. + 2.17432e6i − 5.25940e6i 9.03974e6 + 1.41336e7i 3.17508e7 1.19189e8 − 2.28822e7i
3.8 −48.2661 + 251.409i −8648.56 −60876.8 − 24269.1i − 474083.i 417432. − 2.17432e6i 5.25940e6i 9.03974e6 − 1.41336e7i 3.17508e7 1.19189e8 + 2.28822e7i
3.9 37.1396 − 253.292i 11400.0 −62777.3 − 18814.3i 8041.89i 423391. − 2.88752e6i − 7.58607e6i −7.09702e6 + 1.52022e7i 8.69132e7 2.03694e6 + 298672.i
3.10 37.1396 + 253.292i 11400.0 −62777.3 + 18814.3i − 8041.89i 423391. + 2.88752e6i 7.58607e6i −7.09702e6 − 1.52022e7i 8.69132e7 2.03694e6 − 298672.i
3.11 163.187 − 197.246i −2122.49 −12276.1 − 64376.0i − 155753.i −346362. + 418652.i 829393.i −1.47012e7 − 8.08389e6i −3.85418e7 −3.07217e7 − 2.54169e7i
3.12 163.187 + 197.246i −2122.49 −12276.1 + 64376.0i 155753.i −346362. − 418652.i − 829393.i −1.47012e7 + 8.08389e6i −3.85418e7 −3.07217e7 + 2.54169e7i
3.13 251.161 − 49.5380i 6181.49 60628.0 − 24884.1i 699404.i 1.55255e6 − 306219.i 4.65744e6i 1.39947e7 − 9.25331e6i −4.83590e6 3.46471e7 + 1.75663e8i
3.14 251.161 + 49.5380i 6181.49 60628.0 + 24884.1i − 699404.i 1.55255e6 + 306219.i − 4.65744e6i 1.39947e7 + 9.25331e6i −4.83590e6 3.46471e7 − 1.75663e8i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.14
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 8.17.d.b ✓ 14
3.b odd 2 1 72.17.b.b 14
4.b odd 2 1 32.17.d.b 14
8.b even 2 1 32.17.d.b 14
8.d odd 2 1 inner 8.17.d.b ✓ 14
24.f even 2 1 72.17.b.b 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.17.d.b ✓ 14 1.a even 1 1 trivial
8.17.d.b ✓ 14 8.d odd 2 1 inner
32.17.d.b 14 4.b odd 2 1
32.17.d.b 14 8.b even 2 1
72.17.b.b 14 3.b odd 2 1
72.17.b.b 14 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{7} - 5982 T_{3}^{6} - 154370700 T_{3}^{5} + 713806378344 T_{3}^{4} + \cdots + 42\!\cdots\!80 \) acting on \(S_{17}^{\mathrm{new}}(8, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots + 51\!\cdots\!96 \) Copy content Toggle raw display
$3$ \( (T^{7} + \cdots + 42\!\cdots\!80)^{2} \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{7} + \cdots - 78\!\cdots\!76)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{7} + \cdots + 14\!\cdots\!80)^{2} \) Copy content Toggle raw display
$19$ \( (T^{7} + \cdots + 15\!\cdots\!96)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{7} + \cdots - 84\!\cdots\!96)^{2} \) Copy content Toggle raw display
$43$ \( (T^{7} + \cdots + 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{7} + \cdots + 13\!\cdots\!08)^{2} \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{7} + \cdots + 57\!\cdots\!80)^{2} \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{7} + \cdots + 66\!\cdots\!40)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{7} + \cdots + 91\!\cdots\!20)^{2} \) Copy content Toggle raw display
$89$ \( (T^{7} + \cdots - 21\!\cdots\!24)^{2} \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots - 35\!\cdots\!40)^{2} \) Copy content Toggle raw display
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