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Properties

Label 169.4.a.k
Level $169$
Weight $4$
Character orbit 169.a
Self dual yes
Analytic conductor $9.971$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [169,4,Mod(1,169)]
 
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([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("169.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 169.a (trivial)

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: yes
Analytic conductor: \(9.97132279097\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 4x^{8} - 46x^{7} + 145x^{6} + 680x^{5} - 1501x^{4} - 3203x^{3} + 4784x^{2} + 3584x - 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 13^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + \beta_{2} q^{3} + (\beta_{8} + \beta_{6} + \beta_{5} + 5) q^{4} + ( - \beta_{8} - \beta_{5} + \beta_{4} + \cdots - 2) q^{5}+ \cdots + ( - \beta_{8} - \beta_{7} - \beta_{6} + \cdots + 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + \beta_{2} q^{3} + (\beta_{8} + \beta_{6} + \beta_{5} + 5) q^{4} + ( - \beta_{8} - \beta_{5} + \beta_{4} + \cdots - 2) q^{5}+ \cdots + ( - 100 \beta_{8} - 14 \beta_{7} + \cdots - 121) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 5 q^{2} + q^{3} + 37 q^{4} - 30 q^{5} - 48 q^{6} - 38 q^{7} - 60 q^{8} + 66 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 5 q^{2} + q^{3} + 37 q^{4} - 30 q^{5} - 48 q^{6} - 38 q^{7} - 60 q^{8} + 66 q^{9} - 147 q^{10} - 181 q^{11} + 39 q^{12} - 147 q^{14} - 218 q^{15} + 269 q^{16} - 55 q^{17} - 79 q^{18} - 161 q^{19} - 370 q^{20} - 188 q^{21} + 340 q^{22} - 204 q^{23} - 798 q^{24} + 307 q^{25} - 668 q^{27} - 344 q^{28} + 280 q^{29} + 521 q^{30} - 706 q^{31} - 680 q^{32} - 500 q^{33} - 216 q^{34} + 20 q^{35} - 909 q^{36} - 298 q^{37} - 739 q^{38} + 13 q^{40} - 1201 q^{41} - 4 q^{42} - 533 q^{43} - 355 q^{44} + 90 q^{45} + 840 q^{46} - 956 q^{47} - 132 q^{48} + 403 q^{49} + 1156 q^{50} + 470 q^{51} - 278 q^{53} + 2555 q^{54} - 250 q^{55} + 250 q^{56} + 810 q^{57} + 2877 q^{58} - 1377 q^{59} + 3157 q^{60} - 136 q^{61} + 2035 q^{62} + 944 q^{63} + 284 q^{64} + 3279 q^{66} + 931 q^{67} - 1536 q^{68} - 2050 q^{69} + 4854 q^{70} - 2046 q^{71} + 4342 q^{72} + 45 q^{73} - 1990 q^{74} + 2393 q^{75} + 3608 q^{76} - 718 q^{77} + 412 q^{79} + 787 q^{80} - 835 q^{81} + 2757 q^{82} - 3709 q^{83} + 1539 q^{84} + 2106 q^{85} - 125 q^{86} - 786 q^{87} - 636 q^{88} - 1663 q^{89} - 1280 q^{90} + 4010 q^{92} + 1186 q^{93} - 2531 q^{94} - 1614 q^{95} + 3084 q^{96} + 1087 q^{97} + 282 q^{98} - 1357 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 4x^{8} - 46x^{7} + 145x^{6} + 680x^{5} - 1501x^{4} - 3203x^{3} + 4784x^{2} + 3584x - 4096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 6901 \nu^{8} - 12396 \nu^{7} + 501446 \nu^{6} + 728955 \nu^{5} - 11530440 \nu^{4} + \cdots - 149293824 ) / 8071424 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3843 \nu^{8} + 11372 \nu^{7} + 195178 \nu^{6} - 384275 \nu^{5} - 3297016 \nu^{4} + \cdots - 6050816 ) / 4035712 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 595 \nu^{8} + 2737 \nu^{7} + 19422 \nu^{6} - 72705 \nu^{5} - 171803 \nu^{4} + 485407 \nu^{3} + \cdots - 646464 ) / 504464 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 23843 \nu^{8} + 103372 \nu^{7} + 1059978 \nu^{6} - 3803155 \nu^{5} - 14845688 \nu^{4} + \cdots - 92826880 ) / 8071424 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 32107 \nu^{8} + 90940 \nu^{7} + 1548474 \nu^{6} - 2570427 \nu^{5} - 23878504 \nu^{4} + \cdots - 122506496 ) / 8071424 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 53515 \nu^{8} + 88524 \nu^{7} + 2837370 \nu^{6} - 2210427 \nu^{5} - 46218904 \nu^{4} + \cdots - 202361088 ) / 8071424 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 27975 \nu^{8} - 97156 \nu^{7} - 1304226 \nu^{6} + 3186791 \nu^{5} + 19362096 \nu^{4} + \cdots + 59238144 ) / 4035712 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + \beta_{6} + \beta_{5} + 2\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{8} + 3\beta_{6} + 2\beta_{5} + 4\beta_{3} - \beta_{2} + 23\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 32\beta_{8} + 37\beta_{6} + 27\beta_{5} - 5\beta_{4} + 8\beta_{3} - 8\beta_{2} + 77\beta _1 + 260 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 128 \beta_{8} - 10 \beta_{7} + 166 \beta_{6} + 75 \beta_{5} - 18 \beta_{4} + 138 \beta_{3} - 45 \beta_{2} + \cdots + 604 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1004 \beta_{8} - 46 \beta_{7} + 1315 \beta_{6} + 748 \beta_{5} - 299 \beta_{4} + 490 \beta_{3} + \cdots + 6752 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4677 \beta_{8} - 644 \beta_{7} + 6896 \beta_{6} + 2684 \beta_{5} - 1299 \beta_{4} + 4588 \beta_{3} + \cdots + 23360 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 32278 \beta_{8} - 3242 \beta_{7} + 46550 \beta_{6} + 21858 \beta_{5} - 12940 \beta_{4} + 21190 \beta_{3} + \cdots + 192772 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−4.42835
−3.82555
−2.16135
−1.22799
0.850942
1.39012
2.72763
4.83438
5.84018
−5.42835 1.67510 21.4670 −7.70909 −9.09301 15.0250 −73.1038 −24.1941 41.8477
1.2 −4.82555 4.44352 15.2860 12.7712 −21.4425 −26.1871 −35.1589 −7.25513 −61.6281
1.3 −3.16135 7.08883 1.99415 −13.6039 −22.4103 14.3315 18.9866 23.2516 43.0068
1.4 −2.22799 −9.74867 −3.03607 8.20685 21.7199 −8.35495 24.5882 68.0366 −18.2848
1.5 −0.149058 −6.48858 −7.97778 10.2526 0.967177 29.6743 2.38162 15.1017 −1.52823
1.6 0.390115 3.60967 −7.84781 7.52136 1.40819 −19.5446 −6.18247 −13.9703 2.93420
1.7 1.72763 6.89591 −5.01528 −20.8281 11.9136 −7.56566 −22.4856 20.5536 −35.9833
1.8 3.83438 −0.279163 6.70249 −11.3710 −1.07042 −31.0623 −4.97517 −26.9221 −43.6008
1.9 4.84018 −6.19662 15.4273 −15.2399 −29.9927 −4.31620 35.9495 11.3981 −73.7636
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(13\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{9} + 5T_{2}^{8} - 42T_{2}^{7} - 205T_{2}^{6} + 486T_{2}^{5} + 2310T_{2}^{4} - 1257T_{2}^{3} - 5898T_{2}^{2} + 1464T_{2} + 344 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(169))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} + 5 T^{8} + \cdots + 344 \) Copy content Toggle raw display
$3$ \( T^{9} - T^{8} + \cdots - 143717 \) Copy content Toggle raw display
$5$ \( T^{9} + \cdots + 3059376152 \) Copy content Toggle raw display
$7$ \( T^{9} + \cdots - 27715644424 \) Copy content Toggle raw display
$11$ \( T^{9} + \cdots + 276199564381 \) Copy content Toggle raw display
$13$ \( T^{9} \) Copy content Toggle raw display
$17$ \( T^{9} + \cdots + 5572934105557 \) Copy content Toggle raw display
$19$ \( T^{9} + \cdots + 865058822963419 \) Copy content Toggle raw display
$23$ \( T^{9} + \cdots + 68\!\cdots\!92 \) Copy content Toggle raw display
$29$ \( T^{9} + \cdots + 56\!\cdots\!96 \) Copy content Toggle raw display
$31$ \( T^{9} + \cdots - 30\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( T^{9} + \cdots - 31\!\cdots\!48 \) Copy content Toggle raw display
$41$ \( T^{9} + \cdots + 61\!\cdots\!53 \) Copy content Toggle raw display
$43$ \( T^{9} + \cdots - 35\!\cdots\!77 \) Copy content Toggle raw display
$47$ \( T^{9} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{9} + \cdots - 12\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{9} + \cdots + 27\!\cdots\!23 \) Copy content Toggle raw display
$61$ \( T^{9} + \cdots - 27\!\cdots\!32 \) Copy content Toggle raw display
$67$ \( T^{9} + \cdots + 32\!\cdots\!99 \) Copy content Toggle raw display
$71$ \( T^{9} + \cdots + 44\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( T^{9} + \cdots - 31\!\cdots\!21 \) Copy content Toggle raw display
$79$ \( T^{9} + \cdots + 63\!\cdots\!88 \) Copy content Toggle raw display
$83$ \( T^{9} + \cdots + 14\!\cdots\!61 \) Copy content Toggle raw display
$89$ \( T^{9} + \cdots + 43\!\cdots\!23 \) Copy content Toggle raw display
$97$ \( T^{9} + \cdots + 20\!\cdots\!89 \) Copy content Toggle raw display
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