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Properties

Label 169.4.c.k
Level $169$
Weight $4$
Character orbit 169.c
Analytic conductor $9.971$
Analytic rank $0$
Dimension $18$
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] = [169,4,Mod(22,169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(169, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.22");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 169.c (of order \(3\), degree \(2\), 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: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 4 x^{17} + 62 x^{16} - 106 x^{15} + 2016 x^{14} - 2731 x^{13} + 39895 x^{12} - 21896 x^{11} + \cdots + 16777216 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 13^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_{2} - \beta_1) q^{2} + ( - \beta_{14} + \beta_{11}) q^{3} + (\beta_{12} - \beta_{10} - \beta_{6} + \cdots - 5) q^{4}+ \cdots + ( - 2 \beta_{14} + \beta_{10} + \cdots - 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_{2} - \beta_1) q^{2} + ( - \beta_{14} + \beta_{11}) q^{3} + (\beta_{12} - \beta_{10} - \beta_{6} + \cdots - 5) q^{4}+ \cdots + ( - 18 \beta_{17} - 100 \beta_{16} + \cdots + 121) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 5 q^{2} - q^{3} - 37 q^{4} + 60 q^{5} - 48 q^{6} - 38 q^{7} + 120 q^{8} - 66 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 5 q^{2} - q^{3} - 37 q^{4} + 60 q^{5} - 48 q^{6} - 38 q^{7} + 120 q^{8} - 66 q^{9} + 147 q^{10} - 181 q^{11} + 78 q^{12} - 294 q^{14} - 218 q^{15} - 269 q^{16} + 55 q^{17} + 158 q^{18} - 161 q^{19} - 370 q^{20} + 376 q^{21} - 340 q^{22} + 204 q^{23} - 798 q^{24} + 614 q^{25} - 1336 q^{27} - 344 q^{28} - 280 q^{29} - 521 q^{30} + 1412 q^{31} - 680 q^{32} - 500 q^{33} + 432 q^{34} - 20 q^{35} + 909 q^{36} - 298 q^{37} - 1478 q^{38} + 26 q^{40} - 1201 q^{41} + 4 q^{42} + 533 q^{43} + 710 q^{44} + 90 q^{45} + 840 q^{46} + 1912 q^{47} + 132 q^{48} - 403 q^{49} + 1156 q^{50} + 940 q^{51} - 556 q^{53} + 2555 q^{54} + 250 q^{55} - 250 q^{56} - 1620 q^{57} + 2877 q^{58} - 1377 q^{59} - 6314 q^{60} + 136 q^{61} - 2035 q^{62} + 944 q^{63} + 568 q^{64} + 6558 q^{66} + 931 q^{67} + 1536 q^{68} + 2050 q^{69} - 9708 q^{70} - 2046 q^{71} + 4342 q^{72} - 90 q^{73} + 1990 q^{74} - 2393 q^{75} + 3608 q^{76} - 1436 q^{77} + 824 q^{79} + 787 q^{80} + 835 q^{81} - 2757 q^{82} + 7418 q^{83} + 1539 q^{84} + 2106 q^{85} + 250 q^{86} + 786 q^{87} + 636 q^{88} - 1663 q^{89} - 2560 q^{90} + 8020 q^{92} + 1186 q^{93} + 2531 q^{94} + 1614 q^{95} - 6168 q^{96} + 1087 q^{97} + 282 q^{98} + 2714 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 4 x^{17} + 62 x^{16} - 106 x^{15} + 2016 x^{14} - 2731 x^{13} + 39895 x^{12} - 21896 x^{11} + \cdots + 16777216 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 71\!\cdots\!85 \nu^{17} + \cdots + 26\!\cdots\!56 ) / 32\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 12\!\cdots\!53 \nu^{17} + \cdots + 61\!\cdots\!76 ) / 25\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\!\cdots\!41 \nu^{17} + \cdots + 25\!\cdots\!24 ) / 20\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10\!\cdots\!51 \nu^{17} + \cdots - 56\!\cdots\!96 ) / 16\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 17\!\cdots\!07 \nu^{17} + \cdots + 13\!\cdots\!44 ) / 20\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 28\!\cdots\!09 \nu^{17} + \cdots + 74\!\cdots\!32 ) / 31\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 12\!\cdots\!01 \nu^{17} + \cdots + 90\!\cdots\!72 ) / 10\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11\!\cdots\!13 \nu^{17} + \cdots - 33\!\cdots\!56 ) / 81\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 38\!\cdots\!47 \nu^{17} + \cdots - 14\!\cdots\!88 ) / 20\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 28\!\cdots\!69 \nu^{17} + \cdots + 11\!\cdots\!84 ) / 12\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 54\!\cdots\!69 \nu^{17} + \cdots + 17\!\cdots\!56 ) / 20\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 25\!\cdots\!97 \nu^{17} + \cdots - 63\!\cdots\!36 ) / 81\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 82\!\cdots\!31 \nu^{17} + \cdots + 18\!\cdots\!88 ) / 20\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 65\!\cdots\!29 \nu^{17} + \cdots + 10\!\cdots\!20 ) / 12\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 11\!\cdots\!21 \nu^{17} + \cdots - 19\!\cdots\!44 ) / 16\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 12\!\cdots\!63 \nu^{17} + \cdots - 14\!\cdots\!88 ) / 16\!\cdots\!28 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{17} + \beta_{16} - \beta_{13} + \beta_{10} + \beta_{6} - 12\beta_{3} - 2\beta_{2} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{17} + 3\beta_{16} - 3\beta_{13} + 2\beta_{12} - \beta_{11} - 4\beta_{9} + 4\beta_{7} - 23\beta_{2} - 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{14} + 27\beta_{12} - 37\beta_{10} + 8\beta_{7} - 32\beta_{6} - 5\beta_{4} + 260\beta_{3} - 77\beta _1 - 260 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 75 \beta_{17} - 128 \beta_{16} + 10 \beta_{15} - 45 \beta_{14} + 166 \beta_{13} + 45 \beta_{11} + \cdots - 636 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 748 \beta_{17} - 1004 \beta_{16} + 46 \beta_{15} + 1315 \beta_{13} - 748 \beta_{12} + 339 \beta_{11} + \cdots + 6752 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1762 \beta_{14} - 2684 \beta_{12} + 6896 \beta_{10} + 644 \beta_{8} - 4588 \beta_{7} + 4677 \beta_{6} + \cdots + 23360 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 21858 \beta_{17} + 32278 \beta_{16} - 3242 \beta_{15} + 12178 \beta_{14} - 46550 \beta_{13} + \cdots + 95033 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 93585 \beta_{17} + 164491 \beta_{16} - 29122 \beta_{15} - 260223 \beta_{13} + 93585 \beta_{12} + \cdots - 831272 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 423443 \beta_{14} + 667892 \beta_{12} - 1637813 \beta_{10} - 155970 \beta_{8} + 812858 \beta_{7} + \cdots - 5853008 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3212681 \beta_{17} - 5697380 \beta_{16} + 1156558 \beta_{15} - 2341040 \beta_{14} + \cdots - 19896385 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 21117643 \beta_{17} - 35204052 \beta_{16} + 6439744 \beta_{15} + 57218756 \beta_{13} - 21117643 \beta_{12} + \cdots + 185118648 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 82492865 \beta_{14} - 109440828 \beta_{12} + 332264719 \beta_{10} + 43215104 \beta_{8} + \cdots + 974986692 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 684437956 \beta_{17} + 1181688405 \beta_{16} - 246219578 \beta_{15} + 501405476 \beta_{14} + \cdots + 3738644514 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 3716482924 \beta_{17} + 6705745868 \beta_{16} - 1562310104 \beta_{15} - 11588149060 \beta_{13} + \cdots - 33051801880 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 17193582024 \beta_{14} + 22567596581 \beta_{12} - 68620566389 \beta_{10} - 9017389320 \beta_{8} + \cdots - 198745293060 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 126112059954 \beta_{17} - 229119790391 \beta_{16} + 55380969056 \beta_{15} - 99346555769 \beta_{14} + \cdots - 745830358799 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-1 + \beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1
−2.21418 − 3.83507i
−1.91278 − 3.31303i
−1.08068 − 1.87179i
−0.613994 − 1.06347i
0.425471 + 0.736937i
0.695058 + 1.20388i
1.36382 + 2.36220i
2.41719 + 4.18670i
2.92009 + 5.05774i
−2.21418 + 3.83507i
−1.91278 + 3.31303i
−1.08068 + 1.87179i
−0.613994 + 1.06347i
0.425471 − 0.736937i
0.695058 − 1.20388i
1.36382 − 2.36220i
2.41719 − 4.18670i
2.92009 − 5.05774i
−2.71418 + 4.70109i −0.837548 + 1.45068i −10.7335 − 18.5910i 7.70909 −4.54651 − 7.87478i 7.51249 + 13.0120i 73.1038 12.0970 + 20.9527i −20.9238 + 36.2412i
22.2 −2.41278 + 4.17905i −2.22176 + 3.84820i −7.64299 − 13.2380i −12.7712 −10.7212 − 18.5697i −13.0936 − 22.6787i 35.1589 3.62756 + 6.28312i 30.8140 − 53.3715i
22.3 −1.58068 + 2.73781i −3.54442 + 6.13911i −0.997073 − 1.72698i 13.6039 −11.2051 − 19.4079i 7.16574 + 12.4114i −18.9866 −11.6258 − 20.1364i −21.5034 + 37.2450i
22.4 −1.11399 + 1.92949i 4.87434 − 8.44260i 1.51803 + 2.62931i −8.20685 10.8600 + 18.8100i −4.17747 − 7.23560i −24.5882 −34.0183 − 58.9214i 9.14239 − 15.8351i
22.5 −0.0745292 + 0.129088i 3.24429 − 5.61927i 3.98889 + 6.90896i −10.2526 0.483588 + 0.837600i 14.8372 + 25.6987i −2.38162 −7.55083 − 13.0784i 0.764114 − 1.32348i
22.6 0.195058 − 0.337850i −1.80483 + 3.12606i 3.92391 + 6.79640i −7.52136 0.704093 + 1.21953i −9.77228 − 16.9261i 6.18247 6.98515 + 12.0986i −1.46710 + 2.54109i
22.7 0.863817 − 1.49617i −3.44796 + 5.97204i 2.50764 + 4.34336i 20.8281 5.95681 + 10.3175i −3.78283 − 6.55206i 22.4856 −10.2768 − 17.8000i 17.9916 − 31.1624i
22.8 1.91719 − 3.32067i 0.139581 − 0.241762i −3.35124 − 5.80452i 11.3710 −0.535209 − 0.927008i −15.5311 − 26.9007i 4.97517 13.4610 + 23.3152i 21.8004 − 37.7594i
22.9 2.42009 − 4.19172i 3.09831 − 5.36643i −7.71365 − 13.3604i 15.2399 −14.9964 − 25.9745i −2.15810 − 3.73794i −35.9495 −5.69903 − 9.87102i 36.8818 − 63.8811i
146.1 −2.71418 − 4.70109i −0.837548 − 1.45068i −10.7335 + 18.5910i 7.70909 −4.54651 + 7.87478i 7.51249 − 13.0120i 73.1038 12.0970 − 20.9527i −20.9238 − 36.2412i
146.2 −2.41278 − 4.17905i −2.22176 − 3.84820i −7.64299 + 13.2380i −12.7712 −10.7212 + 18.5697i −13.0936 + 22.6787i 35.1589 3.62756 − 6.28312i 30.8140 + 53.3715i
146.3 −1.58068 − 2.73781i −3.54442 − 6.13911i −0.997073 + 1.72698i 13.6039 −11.2051 + 19.4079i 7.16574 − 12.4114i −18.9866 −11.6258 + 20.1364i −21.5034 − 37.2450i
146.4 −1.11399 − 1.92949i 4.87434 + 8.44260i 1.51803 − 2.62931i −8.20685 10.8600 − 18.8100i −4.17747 + 7.23560i −24.5882 −34.0183 + 58.9214i 9.14239 + 15.8351i
146.5 −0.0745292 − 0.129088i 3.24429 + 5.61927i 3.98889 − 6.90896i −10.2526 0.483588 − 0.837600i 14.8372 − 25.6987i −2.38162 −7.55083 + 13.0784i 0.764114 + 1.32348i
146.6 0.195058 + 0.337850i −1.80483 − 3.12606i 3.92391 − 6.79640i −7.52136 0.704093 − 1.21953i −9.77228 + 16.9261i 6.18247 6.98515 − 12.0986i −1.46710 − 2.54109i
146.7 0.863817 + 1.49617i −3.44796 − 5.97204i 2.50764 − 4.34336i 20.8281 5.95681 − 10.3175i −3.78283 + 6.55206i 22.4856 −10.2768 + 17.8000i 17.9916 + 31.1624i
146.8 1.91719 + 3.32067i 0.139581 + 0.241762i −3.35124 + 5.80452i 11.3710 −0.535209 + 0.927008i −15.5311 + 26.9007i 4.97517 13.4610 − 23.3152i 21.8004 + 37.7594i
146.9 2.42009 + 4.19172i 3.09831 + 5.36643i −7.71365 + 13.3604i 15.2399 −14.9964 + 25.9745i −2.15810 + 3.73794i −35.9495 −5.69903 + 9.87102i 36.8818 + 63.8811i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 22.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.4.c.k 18
13.b even 2 1 169.4.c.l 18
13.c even 3 1 169.4.a.l yes 9
13.c even 3 1 inner 169.4.c.k 18
13.d odd 4 2 169.4.e.h 36
13.e even 6 1 169.4.a.k ✓ 9
13.e even 6 1 169.4.c.l 18
13.f odd 12 2 169.4.b.g 18
13.f odd 12 2 169.4.e.h 36
39.h odd 6 1 1521.4.a.bh 9
39.i odd 6 1 1521.4.a.bg 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
169.4.a.k ✓ 9 13.e even 6 1
169.4.a.l yes 9 13.c even 3 1
169.4.b.g 18 13.f odd 12 2
169.4.c.k 18 1.a even 1 1 trivial
169.4.c.k 18 13.c even 3 1 inner
169.4.c.l 18 13.b even 2 1
169.4.c.l 18 13.e even 6 1
169.4.e.h 36 13.d odd 4 2
169.4.e.h 36 13.f odd 12 2
1521.4.a.bg 9 39.i odd 6 1
1521.4.a.bh 9 39.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{18} + 5 T_{2}^{17} + 67 T_{2}^{16} + 200 T_{2}^{15} + 2303 T_{2}^{14} + 6060 T_{2}^{13} + \cdots + 118336 \) acting on \(S_{4}^{\mathrm{new}}(169, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + 5 T^{17} + \cdots + 118336 \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots + 20654576089 \) Copy content Toggle raw display
$5$ \( (T^{9} - 30 T^{8} + \cdots - 3059376152)^{2} \) 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
$17$ \( T^{18} + \cdots + 31\!\cdots\!49 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 74\!\cdots\!61 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 46\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 31\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( (T^{9} + \cdots + 30\!\cdots\!56)^{2} \) 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
$43$ \( T^{18} + \cdots + 12\!\cdots\!29 \) Copy content Toggle raw display
$47$ \( (T^{9} + \cdots - 11\!\cdots\!84)^{2} \) 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
$61$ \( T^{18} + \cdots + 75\!\cdots\!24 \) 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
$73$ \( (T^{9} + \cdots + 31\!\cdots\!21)^{2} \) Copy content Toggle raw display
$79$ \( (T^{9} + \cdots + 63\!\cdots\!88)^{2} \) Copy content Toggle raw display
$83$ \( (T^{9} + \cdots - 14\!\cdots\!61)^{2} \) 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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