↑

Properties

Label 169.8.b.d
Level $169$
Weight $8$
Character orbit 169.b
Analytic conductor $52.793$
Analytic rank $0$
Dimension $14$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [169,8,Mod(168,169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(169, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.168");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 169.b (of order \(2\), degree \(1\), not 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: \(52.7930693068\)
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: \( x^{14} + 1279 x^{12} + 629380 x^{10} + 148562016 x^{8} + 16872573312 x^{6} + 790180980480 x^{4} + \cdots + 4669637050368 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{17}\cdot 3^{3}\cdot 13^{6} \)
Twist minimal: no (minimal twist has level 13)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{5} - 4) q^{3} + ( - \beta_{5} + \beta_{4} - 55) q^{4} + ( - \beta_{11} + 2 \beta_{6}) q^{5} + ( - \beta_{12} + \beta_{10} + \cdots - 18 \beta_1) q^{6} + (\beta_{13} - \beta_{11} + \cdots + 20 \beta_1) q^{7}+ \cdots + (3405 \beta_{13} - 8249 \beta_{12} + \cdots - 518286 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 52 q^{3} - 766 q^{4} + 6982 q^{9} + 1018 q^{10} + 38380 q^{12} - 47916 q^{14} + 1266 q^{16} + 76806 q^{17} + 251764 q^{22} + 137100 q^{23} + 39380 q^{25} - 432400 q^{27} - 443166 q^{29} + 315780 q^{30}+ \cdots + 86840772 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 1279 x^{12} + 629380 x^{10} + 148562016 x^{8} + 16872573312 x^{6} + 790180980480 x^{4} + \cdots + 4669637050368 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1054513 \nu^{12} + 1163482171 \nu^{10} + 463409859688 \nu^{8} + 80046208978752 \nu^{6} + \cdots - 82\!\cdots\!68 ) / 62\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5396519 \nu^{12} + 5174922173 \nu^{10} + 1685539808744 \nu^{8} + 206024928720576 \nu^{6} + \cdots - 10\!\cdots\!84 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 38514673 \nu^{12} - 38644774891 \nu^{10} - 13645656502648 \nu^{8} + \cdots + 57\!\cdots\!28 ) / 56\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 38514673 \nu^{12} - 38644774891 \nu^{10} - 13645656502648 \nu^{8} + \cdots - 46\!\cdots\!72 ) / 56\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 124535171 \nu^{13} + 172984934657 \nu^{11} + 93500560218296 \nu^{9} + \cdots + 27\!\cdots\!44 \nu ) / 81\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3022147 \nu^{12} + 2957499199 \nu^{10} + 997245242122 \nu^{8} + 133494994574088 \nu^{6} + \cdots - 40\!\cdots\!92 ) / 260996426496000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 9427621 \nu^{12} + 9415249057 \nu^{10} + 3290899516246 \nu^{8} + 474416172380784 \nu^{6} + \cdots + 78\!\cdots\!44 ) / 587241959616000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1375915123 \nu^{13} + 1217501705841 \nu^{11} + 324347726567448 \nu^{9} + \cdots - 51\!\cdots\!28 \nu ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1155834321 \nu^{13} + 1208454152107 \nu^{11} + 458751774263896 \nu^{9} + \cdots + 39\!\cdots\!44 \nu ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 9081478603 \nu^{13} - 10552047478201 \nu^{11} + \cdots - 49\!\cdots\!92 \nu ) / 81\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 590711549 \nu^{13} - 594051400575 \nu^{11} - 210556973921352 \nu^{9} + \cdots + 85\!\cdots\!48 \nu ) / 10\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 5756404813 \nu^{13} + 6241529176271 \nu^{11} + \cdots + 14\!\cdots\!32 \nu ) / 90\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} - 183 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{13} - \beta_{12} - \beta_{11} + 3\beta_{10} - 5\beta_{9} - 14\beta_{6} - 297\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{8} - 3\beta_{7} + 450\beta_{5} - 355\beta_{4} - 17\beta_{3} - 37\beta_{2} + 53993 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 449\beta_{13} + 581\beta_{12} + 125\beta_{11} - 1677\beta_{10} + 3379\beta_{9} + 10408\beta_{6} + 97719\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -4925\beta_{8} + 2265\beta_{7} - 191526\beta_{5} + 124985\beta_{4} + 8299\beta_{3} + 24959\beta_{2} - 17621467 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 179827 \beta_{13} - 267391 \beta_{12} + 51833 \beta_{11} + 814887 \beta_{10} - 1699481 \beta_{9} + \cdots - 34044789 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2522455 \beta_{8} - 1221675 \beta_{7} + 78908034 \beta_{5} - 45462427 \beta_{4} - 3425633 \beta_{3} + \cdots + 6092145761 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 70953161 \beta_{13} + 114440429 \beta_{12} - 49175419 \beta_{11} - 370751685 \beta_{10} + \cdots + 12375223503 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1153547669 \beta_{8} + 577854801 \beta_{7} - 32091175014 \beta_{5} + 17051372705 \beta_{4} + \cdots - 2199771683731 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 28059663691 \beta_{13} - 47572113271 \beta_{12} + 27336567617 \beta_{11} + 162346576767 \beta_{10} + \cdots - 4647783047421 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 501083296495 \beta_{8} - 256767707235 \beta_{7} + 12997279964370 \beta_{5} - 6552433029715 \beta_{4} + \cdots + 821720441319689 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 11162205575345 \beta_{13} + 19529218262645 \beta_{12} - 13039218072115 \beta_{11} + \cdots + 17\!\cdots\!19 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
168.1
− 20.2132i
− 16.7657i
− 16.7213i
− 14.7644i
− 8.41902i
− 4.51724i
− 0.679146i
0.679146i
4.51724i
8.41902i
14.7644i
16.7213i
16.7657i
20.2132i
− 20.2132i −50.3967 −280.575 − 228.046i 1018.68i − 1526.32i 3084.03i 352.832 −4609.54
168.2 − 16.7657i −81.0229 −153.090 223.318i 1358.41i 666.907i 420.649i 4377.70 3744.09
168.3 − 16.7213i 42.5564 −151.602 94.5127i − 711.598i − 1415.38i 394.655i −375.955 1580.37
168.4 − 14.7644i 45.3963 −89.9867 248.787i − 670.248i 573.906i − 561.243i −126.176 3673.19
168.5 − 8.41902i −1.14749 57.1202 − 399.024i 9.66073i 944.231i − 1558.53i −2185.68 −3359.39
168.6 − 4.51724i −43.8390 107.595 − 49.0275i 198.031i 850.411i − 1064.24i −265.146 −221.469
168.7 − 0.679146i 62.4534 127.539 − 439.155i − 42.4150i − 1302.66i − 173.548i 1713.42 −298.251
168.8 0.679146i 62.4534 127.539 439.155i 42.4150i 1302.66i 173.548i 1713.42 −298.251
168.9 4.51724i −43.8390 107.595 49.0275i − 198.031i − 850.411i 1064.24i −265.146 −221.469
168.10 8.41902i −1.14749 57.1202 399.024i − 9.66073i − 944.231i 1558.53i −2185.68 −3359.39
168.11 14.7644i 45.3963 −89.9867 − 248.787i 670.248i − 573.906i 561.243i −126.176 3673.19
168.12 16.7213i 42.5564 −151.602 − 94.5127i 711.598i 1415.38i − 394.655i −375.955 1580.37
168.13 16.7657i −81.0229 −153.090 − 223.318i − 1358.41i − 666.907i − 420.649i 4377.70 3744.09
168.14 20.2132i −50.3967 −280.575 228.046i − 1018.68i 1526.32i − 3084.03i 352.832 −4609.54
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 168.14
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.8.b.d 14
13.b even 2 1 inner 169.8.b.d 14
13.c even 3 1 13.8.e.a ✓ 14
13.d odd 4 2 169.8.a.g 14
13.e even 6 1 13.8.e.a ✓ 14
39.h odd 6 1 117.8.q.b 14
39.i odd 6 1 117.8.q.b 14
52.i odd 6 1 208.8.w.a 14
52.j odd 6 1 208.8.w.a 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.8.e.a ✓ 14 13.c even 3 1
13.8.e.a ✓ 14 13.e even 6 1
117.8.q.b 14 39.h odd 6 1
117.8.q.b 14 39.i odd 6 1
169.8.a.g 14 13.d odd 4 2
169.8.b.d 14 1.a even 1 1 trivial
169.8.b.d 14 13.b even 2 1 inner
208.8.w.a 14 52.i odd 6 1
208.8.w.a 14 52.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 1279 T_{2}^{12} + 629380 T_{2}^{10} + 148562016 T_{2}^{8} + 16872573312 T_{2}^{6} + \cdots + 4669637050368 \) acting on \(S_{8}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots + 4669637050368 \) Copy content Toggle raw display
$3$ \( (T^{7} + 26 T^{6} + \cdots - 24783330888)^{2} \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{14} \) Copy content Toggle raw display
$17$ \( (T^{7} + \cdots - 17\!\cdots\!73)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 27\!\cdots\!72 \) Copy content Toggle raw display
$23$ \( (T^{7} + \cdots + 13\!\cdots\!84)^{2} \) Copy content Toggle raw display
$29$ \( (T^{7} + \cdots + 89\!\cdots\!09)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 25\!\cdots\!75 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 15\!\cdots\!83 \) Copy content Toggle raw display
$43$ \( (T^{7} + \cdots + 81\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 12\!\cdots\!48 \) Copy content Toggle raw display
$53$ \( (T^{7} + \cdots - 32\!\cdots\!76)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 15\!\cdots\!28 \) Copy content Toggle raw display
$61$ \( (T^{7} + \cdots + 47\!\cdots\!75)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{7} + \cdots - 60\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 20\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 35\!\cdots\!68 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 28\!\cdots\!28 \) Copy content Toggle raw display
show more
show less