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Properties

Label 59.8.a.a
Level $59$
Weight $8$
Character orbit 59.a
Self dual yes
Analytic conductor $18.431$
Analytic rank $1$
Dimension $14$
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] = [59,8,Mod(1,59)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(59, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("59.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 59.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: \(18.4307165036\)
Analytic rank: \(1\)
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: \( x^{14} - 5 x^{13} - 1169 x^{12} + 5113 x^{11} + 509966 x^{10} - 1844082 x^{9} - 104172650 x^{8} + \cdots - 143083653176 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9}\cdot 5 \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{3} - 4) q^{3} + (\beta_{2} - \beta_1 + 41) q^{4} + (\beta_{4} - \beta_{2} - 3 \beta_1 - 29) q^{5} + (\beta_{7} - \beta_{4} - 2 \beta_{3} + \cdots - 21) q^{6}+ \cdots + (\beta_{11} - 2 \beta_{10} + \cdots + 335) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_{3} - 4) q^{3} + (\beta_{2} - \beta_1 + 41) q^{4} + (\beta_{4} - \beta_{2} - 3 \beta_1 - 29) q^{5} + (\beta_{7} - \beta_{4} - 2 \beta_{3} + \cdots - 21) q^{6}+ \cdots + (6132 \beta_{13} - 4941 \beta_{12} + \cdots - 3989748) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 9 q^{2} - 54 q^{3} + 575 q^{4} - 430 q^{5} - 346 q^{6} - 2390 q^{7} - 2463 q^{8} + 4504 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 9 q^{2} - 54 q^{3} + 575 q^{4} - 430 q^{5} - 346 q^{6} - 2390 q^{7} - 2463 q^{8} + 4504 q^{9} - 5362 q^{10} - 5030 q^{11} - 10368 q^{12} - 24364 q^{13} - 12717 q^{14} + 5828 q^{15} + 33859 q^{16} + 14504 q^{17} - 47889 q^{18} - 80234 q^{19} - 190220 q^{20} - 240088 q^{21} - 266687 q^{22} - 113272 q^{23} - 458658 q^{24} - 62580 q^{25} - 386729 q^{26} - 522072 q^{27} - 617413 q^{28} - 490250 q^{29} - 904656 q^{30} - 379844 q^{31} - 700263 q^{32} - 380708 q^{33} - 709263 q^{34} - 611196 q^{35} - 856631 q^{36} - 1203748 q^{37} - 340 q^{38} - 187728 q^{39} - 927130 q^{40} - 681860 q^{41} + 403450 q^{42} - 967090 q^{43} + 218679 q^{44} - 863386 q^{45} - 136632 q^{46} + 287456 q^{47} + 1478400 q^{48} + 341754 q^{49} + 2153697 q^{50} - 1897268 q^{51} + 1661397 q^{52} - 1227618 q^{53} + 2748476 q^{54} + 975320 q^{55} + 4449921 q^{56} + 2583684 q^{57} - 265424 q^{58} + 2875306 q^{59} + 9108150 q^{60} - 4711840 q^{61} + 17057148 q^{62} + 11582976 q^{63} + 2117595 q^{64} + 4072956 q^{65} + 14904718 q^{66} + 7298936 q^{67} + 13001951 q^{68} + 3830620 q^{69} + 12397514 q^{70} + 5657790 q^{71} + 13868181 q^{72} + 4750028 q^{73} + 1875491 q^{74} - 1265960 q^{75} + 3128138 q^{76} - 8640944 q^{77} + 11930154 q^{78} - 17385506 q^{79} + 20096996 q^{80} - 13129322 q^{81} - 941109 q^{82} - 15067470 q^{83} + 13172408 q^{84} - 28577148 q^{85} + 3286963 q^{86} - 8604448 q^{87} - 20655117 q^{88} - 15451868 q^{89} + 23198450 q^{90} - 24287002 q^{91} + 13921944 q^{92} - 32910288 q^{93} - 12765942 q^{94} - 43655474 q^{95} - 18493194 q^{96} + 2400932 q^{97} + 19642950 q^{98} - 56609186 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 5 x^{13} - 1169 x^{12} + 5113 x^{11} + 509966 x^{10} - 1844082 x^{9} - 104172650 x^{8} + \cdots - 143083653176 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 168 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 44\!\cdots\!91 \nu^{13} + \cdots + 13\!\cdots\!36 ) / 17\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 94\!\cdots\!13 \nu^{13} + \cdots - 24\!\cdots\!40 ) / 17\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 39\!\cdots\!31 \nu^{13} + \cdots - 54\!\cdots\!36 ) / 17\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 66\!\cdots\!89 \nu^{13} + \cdots - 12\!\cdots\!68 ) / 17\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 67\!\cdots\!33 \nu^{13} + \cdots - 36\!\cdots\!56 ) / 17\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 75\!\cdots\!71 \nu^{13} + \cdots - 13\!\cdots\!36 ) / 17\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 22\!\cdots\!23 \nu^{13} + \cdots + 20\!\cdots\!08 ) / 44\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 10\!\cdots\!55 \nu^{13} + \cdots - 74\!\cdots\!68 ) / 17\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 12\!\cdots\!25 \nu^{13} + \cdots - 38\!\cdots\!60 ) / 17\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 16\!\cdots\!87 \nu^{13} + \cdots + 55\!\cdots\!48 ) / 17\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 25\!\cdots\!09 \nu^{13} + \cdots - 11\!\cdots\!56 ) / 17\!\cdots\!76 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 168 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{10} + \beta_{8} - 3\beta_{7} - 2\beta_{4} - 16\beta_{3} - \beta_{2} + 297\beta _1 + 60 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 14 \beta_{13} - 9 \beta_{12} - 21 \beta_{11} + 8 \beta_{10} + 9 \beta_{9} + 6 \beta_{8} + \cdots + 50279 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 70 \beta_{13} + 147 \beta_{12} - 133 \beta_{11} - 1034 \beta_{10} + \beta_{9} + 457 \beta_{8} + \cdots + 23391 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 8900 \beta_{13} - 8804 \beta_{12} - 9908 \beta_{11} + 4496 \beta_{10} + 4240 \beta_{9} + \cdots + 17707160 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 53292 \beta_{13} + 88366 \beta_{12} - 86082 \beta_{11} - 443138 \beta_{10} + 23114 \beta_{9} + \cdots + 11623570 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 4391658 \beta_{13} - 5431319 \beta_{12} - 3672843 \beta_{11} + 1803432 \beta_{10} + 1589487 \beta_{9} + \cdots + 6684735793 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 29485426 \beta_{13} + 39207577 \beta_{12} - 42925871 \beta_{11} - 181448570 \beta_{10} + \cdots + 5998187573 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1984576400 \beta_{13} - 2845578842 \beta_{12} - 1243198034 \beta_{11} + 641846176 \beta_{10} + \cdots + 2610417674202 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 14454971544 \beta_{13} + 15492444388 \beta_{12} - 19655125884 \beta_{11} - 73529474802 \beta_{10} + \cdots + 2897583067704 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 863853025078 \beta_{13} - 1380963457325 \beta_{12} - 398154438793 \beta_{11} + 214629149880 \beta_{10} + \cdots + 10\!\cdots\!67 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 6679729613342 \beta_{13} + 5741064352015 \beta_{12} - 8649426088873 \beta_{11} - 29817592060970 \beta_{10} + \cdots + 13\!\cdots\!39 \) 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
−20.4619
−17.9966
−14.1713
−9.40618
−7.18182
−6.20543
−0.0470033
0.479526
4.29939
10.7353
12.6389
12.6908
19.0516
20.5748
−21.4619 42.8448 332.613 139.384 −919.531 −548.392 −4391.37 −351.320 −2991.44
1.2 −18.9966 −42.6129 232.872 −324.125 809.502 −536.491 −1992.22 −371.141 6157.28
1.3 −15.1713 36.5399 102.167 105.628 −554.356 −580.307 391.915 −851.837 −1602.51
1.4 −10.4062 −81.1806 −19.7113 −351.654 844.780 1011.39 1537.11 4403.29 3659.38
1.5 −8.18182 −63.4426 −61.0578 131.051 519.076 −311.388 1546.84 1837.96 −1072.23
1.6 −7.20543 20.8246 −76.0817 −133.759 −150.050 1302.79 1470.50 −1753.34 963.791
1.7 −1.04700 74.2971 −126.904 −45.6259 −77.7893 −399.043 266.885 3333.05 47.7705
1.8 −0.520474 26.5428 −127.729 477.166 −13.8148 −1635.27 133.100 −1482.48 −248.352
1.9 3.29939 −70.0200 −117.114 344.524 −231.023 1322.34 −808.726 2715.80 1136.72
1.10 9.73528 53.8112 −33.2243 −440.340 523.867 384.373 −1569.56 708.642 −4286.84
1.11 11.6389 −13.7545 7.46330 102.369 −160.087 222.706 −1402.91 −1997.81 1191.46
1.12 11.6908 24.1619 8.67453 56.7681 282.472 −1471.33 −1395.01 −1603.20 663.664
1.13 18.0516 3.40096 197.860 −418.904 61.3927 −1086.58 1261.08 −2175.43 −7561.89
1.14 19.5748 −65.4126 255.172 −72.4807 −1280.44 −64.7916 2489.37 2091.81 −1418.79
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(59\) \(-1\)

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 59.8.a.a ✓ 14
3.b odd 2 1 531.8.a.a 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
59.8.a.a ✓ 14 1.a even 1 1 trivial
531.8.a.a 14 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 9 T_{2}^{13} - 1143 T_{2}^{12} - 8941 T_{2}^{11} + 488626 T_{2}^{10} + 3278040 T_{2}^{9} + \cdots + 3193476509696 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(59))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots + 3193476509696 \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 39\!\cdots\!24 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots - 53\!\cdots\!53 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 11\!\cdots\!92 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 60\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots - 11\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots - 46\!\cdots\!40 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 18\!\cdots\!92 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots - 50\!\cdots\!20 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots - 35\!\cdots\!48 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 22\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 83\!\cdots\!21 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 11\!\cdots\!72 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 18\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 15\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( (T - 205379)^{14} \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots - 25\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 84\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 17\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 60\!\cdots\!12 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 55\!\cdots\!05 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 69\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 15\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
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