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Properties

Label 59.6.a.a
Level $59$
Weight $6$
Character orbit 59.a
Self dual yes
Analytic conductor $9.463$
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] = [59,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("59.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 59 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(9.46264536897\)
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: \( x^{9} - 171x^{7} - 44x^{6} + 9767x^{5} + 2200x^{4} - 215105x^{3} - 33724x^{2} + 1380292x + 1109072 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{5} - \beta_1) q^{3} + (\beta_{2} - 2 \beta_1 + 7) q^{4} + ( - 2 \beta_{5} - \beta_{4} - \beta_{2} + \cdots - 8) q^{5}+ \cdots + (4 \beta_{8} - 4 \beta_{7} - 4 \beta_{6} + \cdots + 17) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_{5} - \beta_1) q^{3} + (\beta_{2} - 2 \beta_1 + 7) q^{4} + ( - 2 \beta_{5} - \beta_{4} - \beta_{2} + \cdots - 8) q^{5}+ \cdots + ( - 891 \beta_{8} + 28 \beta_{7} + \cdots - 5912) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 9 q^{2} + 63 q^{4} - 72 q^{5} - 274 q^{6} - 208 q^{7} - 327 q^{8} + 163 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 9 q^{2} + 63 q^{4} - 72 q^{5} - 274 q^{6} - 208 q^{7} - 327 q^{8} + 163 q^{9} - 402 q^{10} - 1176 q^{11} - 1928 q^{13} - 89 q^{14} - 2953 q^{15} - 3533 q^{16} - 3031 q^{17} - 8061 q^{18} - 4516 q^{19} - 5540 q^{20} - 9211 q^{21} - 2075 q^{22} - 2192 q^{23} - 5190 q^{24} - 5611 q^{25} + 3155 q^{26} + 9189 q^{27} - 10477 q^{28} - 3096 q^{29} + 17460 q^{30} - 10148 q^{31} + 15377 q^{32} - 5774 q^{33} + 13101 q^{34} + 9947 q^{35} + 42673 q^{36} - 11554 q^{37} + 21836 q^{38} - 1890 q^{39} + 24298 q^{40} + 11294 q^{41} + 90454 q^{42} - 12526 q^{43} + 24051 q^{44} - 7759 q^{45} + 22376 q^{46} + 11716 q^{47} + 41664 q^{48} + 19137 q^{49} + 66081 q^{50} - 680 q^{51} + 23609 q^{52} + 16552 q^{53} - 19576 q^{54} - 9668 q^{55} + 51401 q^{56} - 18393 q^{57} + 62124 q^{58} - 31329 q^{59} - 7626 q^{60} - 143662 q^{61} + 96300 q^{62} - 169464 q^{63} - 79981 q^{64} - 69180 q^{65} + 88366 q^{66} - 119488 q^{67} - 19761 q^{68} - 73496 q^{69} - 119574 q^{70} - 38295 q^{71} - 59631 q^{72} - 197332 q^{73} + 50783 q^{74} - 99200 q^{75} - 104806 q^{76} - 67070 q^{77} + 85986 q^{78} - 75556 q^{79} + 45772 q^{80} + 31417 q^{81} - 100205 q^{82} - 49800 q^{83} - 251176 q^{84} - 189532 q^{85} + 23879 q^{86} - 112633 q^{87} + 43035 q^{88} - 85950 q^{89} + 269762 q^{90} - 117668 q^{91} + 167480 q^{92} - 214422 q^{93} - 5734 q^{94} + 169216 q^{95} + 283530 q^{96} - 23462 q^{97} - 70298 q^{98} - 49934 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 171x^{7} - 44x^{6} + 9767x^{5} + 2200x^{4} - 215105x^{3} - 33724x^{2} + 1380292x + 1109072 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 38 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1331 \nu^{8} + 16757 \nu^{7} + 131826 \nu^{6} - 2052130 \nu^{5} - 2424511 \nu^{4} + \cdots - 277353808 ) / 4935872 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3819 \nu^{8} + 35101 \nu^{7} + 450790 \nu^{6} - 4086070 \nu^{5} - 14813507 \nu^{4} + \cdots - 857818768 ) / 9871744 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5909 \nu^{8} - 43567 \nu^{7} - 718978 \nu^{6} + 5100538 \nu^{5} + 24372309 \nu^{4} + \cdots + 1238548912 ) / 9871744 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3611 \nu^{8} - 40015 \nu^{7} - 396350 \nu^{6} + 4841506 \nu^{5} + 10692851 \nu^{4} + \cdots + 748560880 ) / 4935872 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5319 \nu^{8} - 38225 \nu^{7} - 665410 \nu^{6} + 4541090 \nu^{5} + 23670971 \nu^{4} + \cdots + 1351814544 ) / 4935872 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 12293 \nu^{8} + 46991 \nu^{7} + 1583042 \nu^{6} - 4766602 \nu^{5} - 56778421 \nu^{4} + \cdots - 337773488 ) / 9871744 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 38 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - 2\beta_{6} - 2\beta_{5} - 2\beta_{4} - 3\beta_{3} + 2\beta_{2} + 56\beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{8} - 20\beta_{6} + 16\beta_{5} - 12\beta_{4} - 20\beta_{3} + 83\beta_{2} + 46\beta _1 + 2150 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12 \beta_{8} + 107 \beta_{7} - 218 \beta_{6} - 142 \beta_{5} - 146 \beta_{4} - 325 \beta_{3} + \cdots + 2890 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 440 \beta_{8} - 86 \beta_{7} - 2420 \beta_{6} + 2020 \beta_{5} - 1268 \beta_{4} - 2638 \beta_{3} + \cdots + 140802 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1704 \beta_{8} + 8977 \beta_{7} - 20890 \beta_{6} - 7314 \beta_{5} - 10058 \beta_{4} - 30475 \beta_{3} + \cdots + 393406 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 39244 \beta_{8} - 7716 \beta_{7} - 235652 \beta_{6} + 192264 \beta_{5} - 110764 \beta_{4} + \cdots + 10120654 \) 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
−7.92155
−7.01083
−6.93831
−2.46796
−0.891598
4.31542
4.80409
6.71539
9.39535
−8.92155 29.3483 47.5941 −38.9836 −261.832 −235.760 −139.124 618.323 347.794
1.2 −8.01083 −23.7213 32.1734 9.58620 190.027 72.0914 −1.38934 319.699 −76.7934
1.3 −7.93831 8.77773 31.0168 −31.0319 −69.6804 51.4370 7.80461 −165.951 246.341
1.4 −3.46796 −11.7945 −19.9733 77.1268 40.9029 107.152 180.241 −103.889 −267.473
1.5 −1.89160 11.9185 −28.4219 20.9706 −22.5451 −70.9632 114.294 −100.948 −39.6679
1.6 3.31542 12.9877 −21.0080 −53.3327 43.0597 −148.172 −175.744 −74.3196 −176.821
1.7 3.80409 0.664530 −17.5289 −42.9333 2.52793 214.202 −188.412 −242.558 −163.322
1.8 5.71539 −14.9739 0.665728 59.5320 −85.5816 −137.853 −179.088 −18.7832 340.249
1.9 8.39535 −13.2071 38.4819 −72.9341 −110.878 −60.1345 54.4174 −68.5720 −612.307
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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 59.6.a.a ✓ 9
3.b odd 2 1 531.6.a.a 9
4.b odd 2 1 944.6.a.e 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
59.6.a.a ✓ 9 1.a even 1 1 trivial
531.6.a.a 9 3.b odd 2 1
944.6.a.e 9 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{9} + 9 T_{2}^{8} - 135 T_{2}^{7} - 1157 T_{2}^{6} + 6038 T_{2}^{5} + 44516 T_{2}^{4} + \cdots + 2252288 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(59))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} + 9 T^{8} + \cdots + 2252288 \) Copy content Toggle raw display
$3$ \( T^{9} + \cdots - 1466210484 \) Copy content Toggle raw display
$5$ \( T^{9} + \cdots + 186475584425516 \) Copy content Toggle raw display
$7$ \( T^{9} + \cdots + 17\!\cdots\!41 \) Copy content Toggle raw display
$11$ \( T^{9} + \cdots - 10\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( T^{9} + \cdots + 26\!\cdots\!12 \) Copy content Toggle raw display
$17$ \( T^{9} + \cdots - 43\!\cdots\!48 \) Copy content Toggle raw display
$19$ \( T^{9} + \cdots + 32\!\cdots\!60 \) Copy content Toggle raw display
$23$ \( T^{9} + \cdots + 15\!\cdots\!12 \) Copy content Toggle raw display
$29$ \( T^{9} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{9} + \cdots + 87\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( T^{9} + \cdots - 30\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( T^{9} + \cdots + 10\!\cdots\!13 \) Copy content Toggle raw display
$43$ \( T^{9} + \cdots - 74\!\cdots\!72 \) Copy content Toggle raw display
$47$ \( T^{9} + \cdots - 51\!\cdots\!52 \) Copy content Toggle raw display
$53$ \( T^{9} + \cdots + 11\!\cdots\!88 \) Copy content Toggle raw display
$59$ \( (T + 3481)^{9} \) Copy content Toggle raw display
$61$ \( T^{9} + \cdots - 93\!\cdots\!72 \) Copy content Toggle raw display
$67$ \( T^{9} + \cdots - 14\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{9} + \cdots - 11\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{9} + \cdots - 22\!\cdots\!48 \) Copy content Toggle raw display
$79$ \( T^{9} + \cdots + 13\!\cdots\!95 \) Copy content Toggle raw display
$83$ \( T^{9} + \cdots - 70\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{9} + \cdots - 96\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{9} + \cdots - 15\!\cdots\!44 \) Copy content Toggle raw display
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