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Properties

Label 17.12.b.a
Level $17$
Weight $12$
Character orbit 17.b
Analytic conductor $13.062$
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] = [17,12,Mod(16,17)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(17, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("17.16");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 17 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 17.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: \(13.0618340695\)
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: \( x^{16} + 2012924 x^{14} + 1580196076372 x^{12} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{29}\cdot 3^{4}\cdot 17^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_1 q^{3} + (\beta_{3} + 1271) q^{4} + \beta_{7} q^{5} + ( - \beta_{4} + 9 \beta_1) q^{6} + (\beta_{10} - 2 \beta_{7} - 16 \beta_1) q^{7} + ( - \beta_{6} + 9 \beta_{3} + \cdots - 77) q^{8}+ \cdots + (\beta_{6} + \beta_{5} - 9 \beta_{3} + \cdots - 74551) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + \beta_1 q^{3} + (\beta_{3} + 1271) q^{4} + \beta_{7} q^{5} + ( - \beta_{4} + 9 \beta_1) q^{6} + (\beta_{10} - 2 \beta_{7} - 16 \beta_1) q^{7} + ( - \beta_{6} + 9 \beta_{3} + \cdots - 77) q^{8}+ \cdots + ( - 11556 \beta_{15} + \cdots + 127441925 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{2} + 20338 q^{4} - 4098 q^{8} - 1191496 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{2} + 20338 q^{4} - 4098 q^{8} - 1191496 q^{9} - 1045192 q^{13} - 928176 q^{15} + 34826050 q^{16} + 3554632 q^{17} - 35173654 q^{18} + 4588736 q^{19} + 66662344 q^{21} - 58748400 q^{25} - 317977540 q^{26} - 131021808 q^{30} + 1067460734 q^{32} + 724766552 q^{33} - 775893498 q^{34} + 1999765296 q^{35} - 2520870986 q^{36} - 1607971816 q^{38} + 1845301744 q^{42} - 2666979472 q^{43} + 1869667792 q^{47} - 5944064168 q^{49} + 15444320726 q^{50} + 8437689968 q^{51} - 7784119948 q^{52} - 5942183760 q^{53} - 11128140752 q^{55} + 7494118800 q^{59} - 2434494672 q^{60} + 80595388930 q^{64} - 86599472704 q^{66} + 17007290816 q^{67} + 73491523226 q^{68} + 13676754040 q^{69} - 91280536608 q^{70} - 229207542918 q^{72} + 149151579272 q^{76} + 32130668824 q^{77} + 145538020840 q^{81} - 112706231184 q^{83} + 424712287520 q^{84} + 77452876928 q^{85} + 64143446456 q^{86} - 368269123632 q^{87} - 89466414808 q^{89} - 57312497768 q^{93} - 672691463040 q^{94} + 274175066082 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 2012924 x^{14} + 1580196076372 x^{12} + \cdots + 11\!\cdots\!16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 95\!\cdots\!95 \nu^{14} + \cdots + 10\!\cdots\!80 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 69\!\cdots\!89 \nu^{14} + \cdots - 46\!\cdots\!76 ) / 26\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 95\!\cdots\!95 \nu^{15} + \cdots - 89\!\cdots\!80 \nu ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15\!\cdots\!93 \nu^{14} + \cdots + 15\!\cdots\!96 ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 34\!\cdots\!61 \nu^{14} + \cdots + 98\!\cdots\!68 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 83\!\cdots\!51 \nu^{15} + \cdots + 33\!\cdots\!20 \nu ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 49\!\cdots\!21 \nu^{14} + \cdots - 21\!\cdots\!76 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 62\!\cdots\!97 \nu^{14} + \cdots - 55\!\cdots\!76 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 14\!\cdots\!31 \nu^{15} + \cdots + 58\!\cdots\!28 \nu ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 38\!\cdots\!03 \nu^{14} + \cdots - 12\!\cdots\!96 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 16\!\cdots\!05 \nu^{15} + \cdots + 10\!\cdots\!80 \nu ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 77\!\cdots\!85 \nu^{15} + \cdots + 66\!\cdots\!00 \nu ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 46\!\cdots\!03 \nu^{15} + \cdots + 14\!\cdots\!64 \nu ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 23\!\cdots\!55 \nu^{15} + \cdots - 23\!\cdots\!24 \nu ) / 21\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} - 9\beta_{3} - 666\beta_{2} - 251698 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -108\beta_{14} - 44\beta_{13} + 99\beta_{12} + 422\beta_{10} + 4495\beta_{7} - 354\beta_{4} - 443716\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 6999 \beta_{11} - 2988 \beta_{9} - 13035 \beta_{8} - 627706 \beta_{6} - 573061 \beta_{5} + \cdots + 111454118563 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 163944 \beta_{15} + 75875076 \beta_{14} + 37115924 \beta_{13} - 56270826 \beta_{12} + \cdots + 224133808960 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7049632074 \beta_{11} + 2452501752 \beta_{9} + 12050041458 \beta_{8} + 387115418404 \beta_{6} + \cdots - 56\!\cdots\!42 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 164368022688 \beta_{15} - 48167930005848 \beta_{14} - 25002192314648 \beta_{13} + \cdots - 12\!\cdots\!88 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 54\!\cdots\!24 \beta_{11} + \cdots + 30\!\cdots\!12 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 16\!\cdots\!92 \beta_{15} + \cdots + 67\!\cdots\!56 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 37\!\cdots\!56 \beta_{11} + \cdots - 16\!\cdots\!56 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 15\!\cdots\!60 \beta_{15} + \cdots - 38\!\cdots\!64 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 25\!\cdots\!84 \beta_{11} + \cdots + 96\!\cdots\!20 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 12\!\cdots\!76 \beta_{15} + \cdots + 22\!\cdots\!52 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 16\!\cdots\!08 \beta_{11} + \cdots - 56\!\cdots\!16 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 91\!\cdots\!64 \beta_{15} + \cdots - 13\!\cdots\!84 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/17\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
− 177.753i
177.753i
− 787.938i
787.938i
− 260.983i
260.983i
− 316.038i
316.038i
− 584.483i
584.483i
− 586.850i
586.850i
− 37.5132i
37.5132i
− 710.682i
710.682i
−84.8485 − 177.753i 5151.27 − 11112.5i 15082.1i 74471.4i −263307. 145551. 942877.i
16.2 −84.8485 177.753i 5151.27 11112.5i − 15082.1i − 74471.4i −263307. 145551. − 942877.i
16.3 −61.5364 − 787.938i 1738.73 5583.64i 48486.9i 6866.36i 19031.1 −443699. − 343597.i
16.4 −61.5364 787.938i 1738.73 − 5583.64i − 48486.9i − 6866.36i 19031.1 −443699. 343597.i
16.5 −49.4086 − 260.983i 393.209 − 5284.72i 12894.8i − 52066.0i 81760.9 109035. 261110.i
16.6 −49.4086 260.983i 393.209 5284.72i − 12894.8i 52066.0i 81760.9 109035. − 261110.i
16.7 −6.79604 − 316.038i −2001.81 5903.88i 2147.81i 24317.5i 27522.7 77267.0 − 40123.0i
16.8 −6.79604 316.038i −2001.81 − 5903.88i − 2147.81i − 24317.5i 27522.7 77267.0 40123.0i
16.9 14.8104 − 584.483i −1828.65 − 10659.5i − 8656.42i 45403.9i −57414.7 −164474. − 157871.i
16.10 14.8104 584.483i −1828.65 10659.5i 8656.42i − 45403.9i −57414.7 −164474. 157871.i
16.11 37.8318 − 586.850i −616.752 6253.69i − 22201.6i − 63445.7i −100812. −167246. 236588.i
16.12 37.8318 586.850i −616.752 − 6253.69i 22201.6i 63445.7i −100812. −167246. − 236588.i
16.13 61.5054 − 37.5132i 1734.91 − 7015.33i − 2307.27i − 40939.8i −19256.6 175740. − 431481.i
16.14 61.5054 37.5132i 1734.91 7015.33i 2307.27i 40939.8i −19256.6 175740. 431481.i
16.15 87.4420 − 710.682i 5598.10 − 776.017i − 62143.4i 46177.6i 310427. −327922. − 67856.4i
16.16 87.4420 710.682i 5598.10 776.017i 62143.4i − 46177.6i 310427. −327922. 67856.4i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 17.12.b.a ✓ 16
3.b odd 2 1 153.12.d.b 16
4.b odd 2 1 272.12.b.c 16
17.b even 2 1 inner 17.12.b.a ✓ 16
51.c odd 2 1 153.12.d.b 16
68.d odd 2 1 272.12.b.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.12.b.a ✓ 16 1.a even 1 1 trivial
17.12.b.a ✓ 16 17.b even 2 1 inner
153.12.d.b 16 3.b odd 2 1
153.12.d.b 16 51.c odd 2 1
272.12.b.c 16 4.b odd 2 1
272.12.b.c 16 68.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(17, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + \cdots + 5283138650112)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 90\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 19\!\cdots\!41 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots - 12\!\cdots\!80)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 78\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 94\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 31\!\cdots\!28)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 15\!\cdots\!64)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 29\!\cdots\!56)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 46\!\cdots\!60)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots - 21\!\cdots\!80)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 21\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 10\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots - 51\!\cdots\!24)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 10\!\cdots\!40)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 69\!\cdots\!96 \) Copy content Toggle raw display
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