Properties

Label 169.4.b.g
Level $169$
Weight $4$
Character orbit 169.b
Analytic conductor $9.971$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [169,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.168");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(9.97132279097\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 108 x^{16} + 4636 x^{14} + 101999 x^{12} + 1237806 x^{10} + 8358937 x^{8} + 30682857 x^{6} + \cdots + 16777216 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 13^{10} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{9} q^{2} - \beta_{2} q^{3} + (\beta_1 - 4) q^{4} + (\beta_{14} - \beta_{9}) q^{5} + \beta_{15} q^{6} + (\beta_{16} - \beta_{11} + 2 \beta_{9}) q^{7} + ( - \beta_{17} + \beta_{16} + \cdots + \beta_{3}) q^{8}+ \cdots + ( - \beta_{10} + \beta_{5} + 2 \beta_{2} + \cdots + 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{2} - \beta_{2} q^{3} + (\beta_1 - 4) q^{4} + (\beta_{14} - \beta_{9}) q^{5} + \beta_{15} q^{6} + (\beta_{16} - \beta_{11} + 2 \beta_{9}) q^{7} + ( - \beta_{17} + \beta_{16} + \cdots + \beta_{3}) q^{8}+ \cdots + (23 \beta_{17} - 48 \beta_{16} + \cdots + 4 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 2 q^{3} - 74 q^{4} + 132 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 2 q^{3} - 74 q^{4} + 132 q^{9} + 294 q^{10} - 78 q^{12} - 294 q^{14} + 538 q^{16} + 110 q^{17} + 680 q^{22} + 408 q^{23} - 614 q^{25} - 1336 q^{27} + 560 q^{29} - 1042 q^{30} + 40 q^{35} + 1818 q^{36} + 1478 q^{38} + 26 q^{40} - 8 q^{42} + 1066 q^{43} - 264 q^{48} - 806 q^{49} - 940 q^{51} - 556 q^{53} - 500 q^{55} - 500 q^{56} - 272 q^{61} - 4070 q^{62} - 568 q^{64} + 6558 q^{66} - 3072 q^{68} + 4100 q^{69} - 3980 q^{74} - 4786 q^{75} + 1436 q^{77} + 824 q^{79} - 1670 q^{81} - 5514 q^{82} - 1572 q^{87} + 1272 q^{88} + 2560 q^{90} + 8020 q^{92} - 5062 q^{94} + 3228 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 108 x^{16} + 4636 x^{14} + 101999 x^{12} + 1237806 x^{10} + 8358937 x^{8} + 30682857 x^{6} + \cdots + 16777216 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 50549041 \nu^{16} + 5349047468 \nu^{14} + 222746127708 \nu^{12} + 4677870448703 \nu^{10} + \cdots + 588138790938624 ) / 8717951776768 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 90555493054153 \nu^{16} + \cdots - 84\!\cdots\!56 ) / 87\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 78286664463129 \nu^{17} + \cdots + 64\!\cdots\!16 \nu ) / 70\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 175545556853629 \nu^{16} + \cdots + 23\!\cdots\!28 ) / 87\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 25144259567411 \nu^{16} + \cdots - 31\!\cdots\!76 ) / 10\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4939795233473 \nu^{16} - 519748721664620 \nu^{14} + \cdots - 18\!\cdots\!36 ) / 21\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 208173859301893 \nu^{16} + \cdots - 21\!\cdots\!20 ) / 87\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 112780372656063 \nu^{16} + \cdots - 83\!\cdots\!16 ) / 43\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 7511006491 \nu^{17} - 798248146532 \nu^{15} - 33451669940468 \nu^{13} + \cdots - 82\!\cdots\!08 \nu ) / 44\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 27775644003 \nu^{16} + 2945501792196 \nu^{14} + 123050610758868 \nu^{12} + \cdots + 26\!\cdots\!92 ) / 742699117641728 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 274130340825971 \nu^{17} + \cdots + 71\!\cdots\!76 \nu ) / 70\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 43687270743959 \nu^{17} + \cdots - 17\!\cdots\!64 \nu ) / 10\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 10\!\cdots\!43 \nu^{17} + \cdots + 11\!\cdots\!20 \nu ) / 14\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 14\!\cdots\!13 \nu^{17} + \cdots + 20\!\cdots\!84 \nu ) / 14\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 11\!\cdots\!61 \nu^{17} + \cdots + 13\!\cdots\!92 \nu ) / 70\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 12\!\cdots\!95 \nu^{17} + \cdots + 10\!\cdots\!28 \nu ) / 70\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 22533019473 \nu^{17} + 2394744439596 \nu^{15} + 100355009821404 \nu^{13} + \cdots + 26\!\cdots\!76 \nu ) / 11\!\cdots\!04 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{17} + 12\beta_{9} ) / 13 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} + 2\beta_{6} + 2\beta_{5} - 2\beta_{4} - 2\beta_{2} + 11\beta _1 - 155 ) / 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 29 \beta_{17} - 5 \beta_{16} + 15 \beta_{15} + \beta_{14} + 10 \beta_{13} - 10 \beta_{12} + \cdots + \beta_{3} ) / 13 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 13 \beta_{10} - 11 \beta_{8} + 64 \beta_{7} - 67 \beta_{6} - 56 \beta_{5} + 95 \beta_{4} + 213 \beta_{2} + \cdots + 3433 ) / 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 800 \beta_{17} + 395 \beta_{16} - 622 \beta_{15} - 240 \beta_{14} - 235 \beta_{13} + \cdots - 49 \beta_{3} ) / 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 592 \beta_{10} + 411 \beta_{8} - 2022 \beta_{7} + 2237 \beta_{6} + 1228 \beta_{5} - 3748 \beta_{4} + \cdots - 91240 ) / 13 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 22472 \beta_{17} - 18739 \beta_{16} + 20926 \beta_{15} + 14941 \beta_{14} + 4792 \beta_{13} + \cdots + 2730 \beta_{3} ) / 13 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 24894 \beta_{10} - 9876 \beta_{8} + 64001 \beta_{7} - 74915 \beta_{6} - 22893 \beta_{5} + \cdots + 2648506 ) / 13 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 655882 \beta_{17} + 746536 \beta_{16} - 678543 \beta_{15} - 670357 \beta_{14} - 89149 \beta_{13} + \cdots - 132364 \beta_{3} ) / 13 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1003243 \beta_{10} + 175888 \beta_{8} - 2038962 \beta_{7} + 2509454 \beta_{6} + 305956 \beta_{5} + \cdots - 81227644 ) / 13 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 19864911 \beta_{17} - 27615119 \beta_{16} + 22008411 \beta_{15} + 26366746 \beta_{14} + \cdots + 5700712 \beta_{3} ) / 13 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2978855 \beta_{10} - 113438 \beta_{8} + 5055767 \beta_{7} - 6474363 \beta_{6} + 78819 \beta_{5} + \cdots + 198704833 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 620757008 \beta_{17} + 985301460 \beta_{16} - 720342389 \beta_{15} - 973171784 \beta_{14} + \cdots - 227187857 \beta_{3} ) / 13 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 1443120887 \beta_{10} - 64075219 \beta_{8} - 2146467637 \beta_{7} + 2829568380 \beta_{6} + \cdots - 84141504784 ) / 13 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 19882655549 \beta_{17} - 34491056384 \beta_{16} + 23812343388 \beta_{15} + \cdots + 8610177876 \beta_{3} ) / 13 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 52395301244 \beta_{10} + 4944996916 \beta_{8} + 70922367710 \beta_{7} - 95386070574 \beta_{6} + \cdots + 2784290372163 ) / 13 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 648776665817 \beta_{17} + 1194630259737 \beta_{16} - 793864643895 \beta_{15} + \cdots - 315655806853 \beta_{3} ) / 13 \) Copy content Toggle raw display

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
4.42835i
5.84018i
3.82555i
4.83438i
2.16135i
1.22799i
2.72763i
1.39012i
0.850942i
0.850942i
1.39012i
2.72763i
1.22799i
2.16135i
4.83438i
3.82555i
5.84018i
4.42835i
5.42835i 1.67510 −21.4670 7.70909i 9.09301i 15.0250i 73.1038i −24.1941 −41.8477
168.2 4.84018i −6.19662 −15.4273 15.2399i 29.9927i 4.31620i 35.9495i 11.3981 73.7636
168.3 4.82555i 4.44352 −15.2860 12.7712i 21.4425i 26.1871i 35.1589i −7.25513 61.6281
168.4 3.83438i −0.279163 −6.70249 11.3710i 1.07042i 31.0623i 4.97517i −26.9221 43.6008
168.5 3.16135i 7.08883 −1.99415 13.6039i 22.4103i 14.3315i 18.9866i 23.2516 −43.0068
168.6 2.22799i −9.74867 3.03607 8.20685i 21.7199i 8.35495i 24.5882i 68.0366 18.2848
168.7 1.72763i 6.89591 5.01528 20.8281i 11.9136i 7.56566i 22.4856i 20.5536 35.9833
168.8 0.390115i 3.60967 7.84781 7.52136i 1.40819i 19.5446i 6.18247i −13.9703 −2.93420
168.9 0.149058i −6.48858 7.97778 10.2526i 0.967177i 29.6743i 2.38162i 15.1017 1.52823
168.10 0.149058i −6.48858 7.97778 10.2526i 0.967177i 29.6743i 2.38162i 15.1017 1.52823
168.11 0.390115i 3.60967 7.84781 7.52136i 1.40819i 19.5446i 6.18247i −13.9703 −2.93420
168.12 1.72763i 6.89591 5.01528 20.8281i 11.9136i 7.56566i 22.4856i 20.5536 35.9833
168.13 2.22799i −9.74867 3.03607 8.20685i 21.7199i 8.35495i 24.5882i 68.0366 18.2848
168.14 3.16135i 7.08883 −1.99415 13.6039i 22.4103i 14.3315i 18.9866i 23.2516 −43.0068
168.15 3.83438i −0.279163 −6.70249 11.3710i 1.07042i 31.0623i 4.97517i −26.9221 43.6008
168.16 4.82555i 4.44352 −15.2860 12.7712i 21.4425i 26.1871i 35.1589i −7.25513 61.6281
168.17 4.84018i −6.19662 −15.4273 15.2399i 29.9927i 4.31620i 35.9495i 11.3981 73.7636
168.18 5.42835i 1.67510 −21.4670 7.70909i 9.09301i 15.0250i 73.1038i −24.1941 −41.8477
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 168.18
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 169.4.b.g 18
13.b even 2 1 inner 169.4.b.g 18
13.c even 3 2 169.4.e.h 36
13.d odd 4 1 169.4.a.k 9
13.d odd 4 1 169.4.a.l yes 9
13.e even 6 2 169.4.e.h 36
13.f odd 12 2 169.4.c.k 18
13.f odd 12 2 169.4.c.l 18
39.f even 4 1 1521.4.a.bg 9
39.f even 4 1 1521.4.a.bh 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.4.a.k 9 13.d odd 4 1
169.4.a.l yes 9 13.d odd 4 1
169.4.b.g 18 1.a even 1 1 trivial
169.4.b.g 18 13.b even 2 1 inner
169.4.c.k 18 13.f odd 12 2
169.4.c.l 18 13.f odd 12 2
169.4.e.h 36 13.c even 3 2
169.4.e.h 36 13.e even 6 2
1521.4.a.bg 9 39.f even 4 1
1521.4.a.bh 9 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} + 109 T_{2}^{16} + 4786 T_{2}^{14} + 108463 T_{2}^{12} + 1350792 T_{2}^{10} + 9102500 T_{2}^{8} + \cdots + 118336 \) acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + 109 T^{16} + \cdots + 118336 \) Copy content Toggle raw display
$3$ \( (T^{9} - T^{8} + \cdots - 143717)^{2} \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 93\!\cdots\!04 \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 76\!\cdots\!76 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 76\!\cdots\!61 \) Copy content Toggle raw display
$13$ \( T^{18} \) Copy content Toggle raw display
$17$ \( (T^{9} + \cdots - 5572934105557)^{2} \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 74\!\cdots\!61 \) Copy content Toggle raw display
$23$ \( (T^{9} + \cdots - 68\!\cdots\!92)^{2} \) Copy content Toggle raw display
$29$ \( (T^{9} + \cdots + 56\!\cdots\!96)^{2} \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 95\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 96\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 38\!\cdots\!09 \) Copy content Toggle raw display
$43$ \( (T^{9} + \cdots + 35\!\cdots\!77)^{2} \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( (T^{9} + \cdots - 12\!\cdots\!76)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 75\!\cdots\!29 \) Copy content Toggle raw display
$61$ \( (T^{9} + \cdots - 27\!\cdots\!32)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 10\!\cdots\!01 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 19\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 10\!\cdots\!41 \) Copy content Toggle raw display
$79$ \( (T^{9} + \cdots + 63\!\cdots\!88)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 20\!\cdots\!21 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 19\!\cdots\!29 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 43\!\cdots\!21 \) Copy content Toggle raw display
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