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Properties

Label 8.19.d.b
Level $8$
Weight $19$
Character orbit 8.d
Analytic conductor $16.431$
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] = [8,19,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, 19, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.3");
 
S:= CuspForms(chi, 19);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 19 \)
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: \(16.4308910168\)
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: \( x^{16} - 5 x^{15} + 56971 x^{14} - 14457953 x^{13} + 5222713963 x^{12} - 1887111404561 x^{11} + \cdots + 32\!\cdots\!30 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{120}\cdot 3^{13}\cdot 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 27) q^{2} + ( - \beta_{2} - 4 \beta_1 + 203) q^{3} + (\beta_{3} - \beta_{2} + 29 \beta_1 - 27759) q^{4} + ( - \beta_{4} - \beta_{3} + 376 \beta_1 + 141) q^{5} + (\beta_{6} - 4 \beta_{3} + \cdots - 959222) q^{6}+ \cdots + (\beta_{10} - \beta_{8} + \cdots + 144517361) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 27) q^{2} + ( - \beta_{2} - 4 \beta_1 + 203) q^{3} + (\beta_{3} - \beta_{2} + 29 \beta_1 - 27759) q^{4} + ( - \beta_{4} - \beta_{3} + 376 \beta_1 + 141) q^{5} + (\beta_{6} - 4 \beta_{3} + \cdots - 959222) q^{6}+ \cdots + ( - 60568344 \beta_{14} + \cdots - 50\!\cdots\!19) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 426 q^{2} + 3264 q^{3} - 444332 q^{4} - 15348708 q^{6} + 304914744 q^{8} + 2313856176 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 426 q^{2} + 3264 q^{3} - 444332 q^{4} - 15348708 q^{6} + 304914744 q^{8} + 2313856176 q^{9} - 1569837600 q^{10} - 2471571264 q^{11} + 4764364056 q^{12} + 26163923904 q^{14} - 192320075504 q^{16} - 176439301344 q^{17} - 1044291511122 q^{18} - 833365634368 q^{19} + 1256486405760 q^{20} + 59927319356 q^{22} - 1968891910512 q^{24} - 15320509140080 q^{25} + 4184514840864 q^{26} + 11565649473408 q^{27} - 12301604294400 q^{28} - 36336510039360 q^{30} + 141481742931936 q^{32} - 900457491648 q^{33} + 80653465357268 q^{34} + 20487495736320 q^{35} + 277183098847068 q^{36} + 493456694265564 q^{38} - 519930573603840 q^{40} - 594931562445024 q^{41} - 222362598288000 q^{42} - 25\!\cdots\!92 q^{43}+ \cdots - 81\!\cdots\!68 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 5 x^{15} + 56971 x^{14} - 14457953 x^{13} + 5222713963 x^{12} - 1887111404561 x^{11} + \cdots + 32\!\cdots\!30 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 24\!\cdots\!33 \nu^{15} + \cdots + 25\!\cdots\!30 ) / 17\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 24\!\cdots\!33 \nu^{15} + \cdots + 52\!\cdots\!10 ) / 17\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 98\!\cdots\!21 \nu^{15} + \cdots + 12\!\cdots\!10 ) / 93\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10\!\cdots\!79 \nu^{15} + \cdots - 86\!\cdots\!10 ) / 29\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 98\!\cdots\!79 \nu^{15} + \cdots + 17\!\cdots\!90 ) / 17\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 56\!\cdots\!15 \nu^{15} + \cdots - 92\!\cdots\!50 ) / 93\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 12\!\cdots\!85 \nu^{15} + \cdots - 44\!\cdots\!50 ) / 46\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 12\!\cdots\!73 \nu^{15} + \cdots - 71\!\cdots\!10 ) / 93\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 14\!\cdots\!79 \nu^{15} + \cdots - 15\!\cdots\!50 ) / 93\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 58\!\cdots\!57 \nu^{15} + \cdots + 51\!\cdots\!70 ) / 31\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 26\!\cdots\!55 \nu^{15} + \cdots - 12\!\cdots\!90 ) / 93\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 28\!\cdots\!85 \nu^{15} + \cdots - 50\!\cdots\!90 ) / 93\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 37\!\cdots\!81 \nu^{15} + \cdots + 43\!\cdots\!50 ) / 93\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 48\!\cdots\!73 \nu^{15} + \cdots - 75\!\cdots\!50 ) / 31\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - \beta_{2} - 23\beta _1 - 28487 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{8} + \beta_{6} + 7\beta_{5} - 7\beta_{4} - 40\beta_{3} + 364\beta_{2} - 29939\beta _1 + 21248286 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2 \beta_{15} - 14 \beta_{14} - 26 \beta_{13} - 20 \beta_{12} + 21 \beta_{11} - 37 \beta_{10} + \cdots - 14107139986 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 49 \beta_{15} + 685 \beta_{14} - 574 \beta_{13} + 443 \beta_{12} + 294 \beta_{11} + 154 \beta_{10} + \cdots + 658313469092 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1596 \beta_{15} + 585492 \beta_{14} + 55590 \beta_{13} - 469138 \beta_{12} - 350217 \beta_{11} + \cdots - 314443661819751 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 53764410 \beta_{15} - 282908826 \beta_{14} - 46722882 \beta_{13} - 108646456 \beta_{12} + \cdots - 54\!\cdots\!96 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 18960265230 \beta_{15} + 1916947458 \beta_{14} + 63902236750 \beta_{13} + 85424542612 \beta_{12} + \cdots + 11\!\cdots\!22 ) / 16 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 521684003538 \beta_{15} + 11486427320766 \beta_{14} - 573087780438 \beta_{13} - 24966914404044 \beta_{12} + \cdots - 73\!\cdots\!26 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 290548649560470 \beta_{15} + \cdots - 73\!\cdots\!49 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 20\!\cdots\!16 \beta_{15} + \cdots - 16\!\cdots\!22 ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 30\!\cdots\!42 \beta_{15} + \cdots + 27\!\cdots\!30 ) / 16 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 38\!\cdots\!16 \beta_{15} + \cdots + 67\!\cdots\!26 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 18\!\cdots\!92 \beta_{15} + \cdots - 26\!\cdots\!15 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 14\!\cdots\!30 \beta_{15} + \cdots - 21\!\cdots\!72 ) / 8 \) 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
−226.199 − 141.712i
−226.199 + 141.712i
−195.992 − 179.025i
−195.992 + 179.025i
−162.812 − 207.587i
−162.812 + 207.587i
−13.0246 − 256.000i
−13.0246 + 256.000i
24.2392 − 253.277i
24.2392 + 253.277i
135.604 − 208.454i
135.604 + 208.454i
202.692 − 137.887i
202.692 + 137.887i
237.992 − 50.3904i
237.992 + 50.3904i
−426.398 − 283.424i 35846.9 101486. + 241703.i − 3.47680e6i −1.52851e7 − 1.01599e7i 1.08296e7i 2.52310e7 − 1.31825e8i 8.97583e8 −9.85407e8 + 1.48250e9i
3.2 −426.398 + 283.424i 35846.9 101486. − 241703.i 3.47680e6i −1.52851e7 + 1.01599e7i − 1.08296e7i 2.52310e7 + 1.31825e8i 8.97583e8 −9.85407e8 − 1.48250e9i
3.3 −365.984 − 358.050i −33476.4 5744.83 + 262081.i − 436282.i 1.22518e7 + 1.19862e7i 5.56713e7i 9.17355e7 − 9.79745e7i 7.33247e8 −1.56211e8 + 1.59672e8i
3.4 −365.984 + 358.050i −33476.4 5744.83 − 262081.i 436282.i 1.22518e7 − 1.19862e7i − 5.56713e7i 9.17355e7 + 9.79745e7i 7.33247e8 −1.56211e8 − 1.59672e8i
3.5 −299.623 − 415.174i 5937.76 −82595.7 + 248792.i 2.28157e6i −1.77909e6 − 2.46521e6i − 3.15838e7i 1.28040e8 − 4.02522e7i −3.52164e8 9.47248e8 − 6.83611e8i
3.6 −299.623 + 415.174i 5937.76 −82595.7 − 248792.i − 2.28157e6i −1.77909e6 + 2.46521e6i 3.15838e7i 1.28040e8 + 4.02522e7i −3.52164e8 9.47248e8 + 6.83611e8i
3.7 −0.0491315 − 512.000i −9978.36 −262144. + 50.3107i − 3.00573e6i 490.252 + 5.10892e6i − 5.44315e7i 38638.6 + 1.34218e8i −2.87853e8 −1.53893e9 + 147676.i
3.8 −0.0491315 + 512.000i −9978.36 −262144. − 50.3107i 3.00573e6i 490.252 − 5.10892e6i 5.44315e7i 38638.6 − 1.34218e8i −2.87853e8 −1.53893e9 − 147676.i
3.9 74.4785 − 506.554i 19252.8 −251050. − 75454.8i 901770.i 1.43392e6 − 9.75260e6i 6.41362e7i −5.69197e7 + 1.21551e8i −1.67489e7 4.56795e8 + 6.71625e7i
3.10 74.4785 + 506.554i 19252.8 −251050. + 75454.8i − 901770.i 1.43392e6 + 9.75260e6i − 6.41362e7i −5.69197e7 − 1.21551e8i −1.67489e7 4.56795e8 − 6.71625e7i
3.11 297.207 − 416.907i −25925.7 −85479.4 − 247816.i 1.90314e6i −7.70530e6 + 1.08086e7i 9.33592e6i −1.28721e8 − 3.80157e7i 2.84720e8 7.93431e8 + 5.65626e8i
3.12 297.207 + 416.907i −25925.7 −85479.4 + 247816.i − 1.90314e6i −7.70530e6 − 1.08086e7i − 9.33592e6i −1.28721e8 + 3.80157e7i 2.84720e8 7.93431e8 − 5.65626e8i
3.13 431.385 − 275.773i 22640.1 110042. − 237929.i − 113148.i 9.76661e6 − 6.24355e6i − 3.55480e7i −1.81440e7 − 1.32986e8i 1.25155e8 −3.12032e7 − 4.88104e7i
3.14 431.385 + 275.773i 22640.1 110042. + 237929.i 113148.i 9.76661e6 + 6.24355e6i 3.55480e7i −1.81440e7 + 1.32986e8i 1.25155e8 −3.12032e7 + 4.88104e7i
3.15 501.983 − 100.781i −12665.3 241830. − 101181.i − 2.68543e6i −6.35775e6 + 1.27642e6i 4.44900e7i 1.11198e8 − 7.51629e7i −2.27011e8 −2.70640e8 − 1.34804e9i
3.16 501.983 + 100.781i −12665.3 241830. + 101181.i 2.68543e6i −6.35775e6 − 1.27642e6i − 4.44900e7i 1.11198e8 + 7.51629e7i −2.27011e8 −2.70640e8 + 1.34804e9i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.16
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 8.19.d.b ✓ 16
3.b odd 2 1 72.19.b.b 16
4.b odd 2 1 32.19.d.b 16
8.b even 2 1 32.19.d.b 16
8.d odd 2 1 inner 8.19.d.b ✓ 16
24.f even 2 1 72.19.b.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.19.d.b ✓ 16 1.a even 1 1 trivial
8.19.d.b ✓ 16 8.d odd 2 1 inner
32.19.d.b 16 4.b odd 2 1
32.19.d.b 16 8.b even 2 1
72.19.b.b 16 3.b odd 2 1
72.19.b.b 16 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 1632 T_{3}^{7} - 2126814288 T_{3}^{6} + 1123288083072 T_{3}^{5} + \cdots + 10\!\cdots\!20 \) acting on \(S_{19}^{\mathrm{new}}(8, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
$3$ \( (T^{8} + \cdots + 10\!\cdots\!20)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 96\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 43\!\cdots\!84)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots - 19\!\cdots\!80)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 60\!\cdots\!16)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 80\!\cdots\!96)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots - 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots - 30\!\cdots\!76)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 98\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 83\!\cdots\!20)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 45\!\cdots\!40)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots - 87\!\cdots\!20)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 55\!\cdots\!36)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 14\!\cdots\!20)^{2} \) Copy content Toggle raw display
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