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Properties

Label 59.6.a.b
Level $59$
Weight $6$
Character orbit 59.a
Self dual yes
Analytic conductor $9.463$
Analytic rank $0$
Dimension $15$
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: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 3 x^{14} - 387 x^{13} + 1023 x^{12} + 57328 x^{11} - 124838 x^{10} - 4067604 x^{9} + \cdots - 6425465344 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{7} + 1) q^{3} + (\beta_{2} + 20) q^{4} + ( - \beta_{7} - \beta_{3} + \beta_1 + 8) q^{5} + (\beta_{14} - \beta_{13} + \cdots + \beta_1) q^{6}+ \cdots + ( - 3 \beta_{14} + 2 \beta_{13} + \cdots + 106) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{7} + 1) q^{3} + (\beta_{2} + 20) q^{4} + ( - \beta_{7} - \beta_{3} + \beta_1 + 8) q^{5} + (\beta_{14} - \beta_{13} + \cdots + \beta_1) q^{6}+ \cdots + ( - 220 \beta_{14} + 445 \beta_{13} + \cdots - 1176) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 3 q^{2} + 18 q^{3} + 303 q^{4} + 128 q^{5} + 14 q^{6} + 282 q^{7} + 249 q^{8} + 1621 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 3 q^{2} + 18 q^{3} + 303 q^{4} + 128 q^{5} + 14 q^{6} + 282 q^{7} + 249 q^{8} + 1621 q^{9} + 598 q^{10} + 34 q^{11} + 864 q^{12} + 1790 q^{13} + 1087 q^{14} + 2900 q^{15} + 12155 q^{16} + 6130 q^{17} + 10417 q^{18} + 5342 q^{19} + 11772 q^{20} + 7976 q^{21} + 7657 q^{22} + 1552 q^{23} + 612 q^{24} + 10587 q^{25} - 7885 q^{26} + 3072 q^{27} - 3541 q^{28} + 7476 q^{29} - 25854 q^{30} - 3468 q^{31} - 9479 q^{32} - 11228 q^{33} + 2177 q^{34} - 15556 q^{35} - 39367 q^{36} + 22158 q^{37} - 60264 q^{38} + 8000 q^{39} - 11660 q^{40} + 4670 q^{41} - 130868 q^{42} + 20134 q^{43} - 74355 q^{44} - 22660 q^{45} - 15144 q^{46} - 23192 q^{47} - 77896 q^{48} + 75323 q^{49} - 110939 q^{50} - 22092 q^{51} - 104973 q^{52} + 22148 q^{53} - 122246 q^{54} - 27480 q^{55} - 103031 q^{56} - 65580 q^{57} + 60642 q^{58} + 52215 q^{59} - 1822 q^{60} + 156158 q^{61} - 262068 q^{62} + 283004 q^{63} + 209263 q^{64} + 148264 q^{65} - 32770 q^{66} + 166884 q^{67} + 290919 q^{68} + 221972 q^{69} - 50302 q^{70} - 3954 q^{71} + 62839 q^{72} + 32606 q^{73} + 105727 q^{74} + 305348 q^{75} + 96994 q^{76} + 143452 q^{77} - 71054 q^{78} + 352558 q^{79} + 696 q^{80} + 405359 q^{81} - 76879 q^{82} + 153906 q^{83} - 157160 q^{84} + 327528 q^{85} - 81369 q^{86} - 82912 q^{87} + 252111 q^{88} + 20806 q^{89} - 162642 q^{90} + 88714 q^{91} - 163940 q^{92} + 16120 q^{93} + 109102 q^{94} - 87254 q^{95} - 118508 q^{96} + 45926 q^{97} - 601532 q^{98} - 13706 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 3 x^{14} - 387 x^{13} + 1023 x^{12} + 57328 x^{11} - 124838 x^{10} - 4067604 x^{9} + \cdots - 6425465344 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 52 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 35\!\cdots\!89 \nu^{14} + \cdots + 77\!\cdots\!80 ) / 33\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 32\!\cdots\!55 \nu^{14} + \cdots + 92\!\cdots\!76 ) / 67\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 21\!\cdots\!79 \nu^{14} + \cdots - 24\!\cdots\!56 ) / 33\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 54\!\cdots\!29 \nu^{14} + \cdots + 97\!\cdots\!56 ) / 67\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 85\!\cdots\!13 \nu^{14} + \cdots + 70\!\cdots\!64 ) / 84\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 26\!\cdots\!85 \nu^{14} + \cdots + 27\!\cdots\!20 ) / 16\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11\!\cdots\!89 \nu^{14} + \cdots - 31\!\cdots\!00 ) / 67\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 15\!\cdots\!27 \nu^{14} + \cdots - 11\!\cdots\!60 ) / 67\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 16\!\cdots\!11 \nu^{14} + \cdots - 16\!\cdots\!52 ) / 67\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 10\!\cdots\!05 \nu^{14} + \cdots - 10\!\cdots\!16 ) / 33\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 28\!\cdots\!61 \nu^{14} + \cdots + 25\!\cdots\!60 ) / 67\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 14\!\cdots\!79 \nu^{14} + \cdots + 16\!\cdots\!44 ) / 33\!\cdots\!16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 52 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} + \beta_{9} - \beta_{7} + \beta_{5} - 2\beta_{4} - \beta_{2} + 93\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{14} - 10 \beta_{13} + 8 \beta_{12} - 10 \beta_{11} + 2 \beta_{9} - 6 \beta_{8} + 32 \beta_{7} + \cdots + 4774 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 24 \beta_{14} + 157 \beta_{13} + 40 \beta_{12} + 4 \beta_{11} + 24 \beta_{10} + 121 \beta_{9} + \cdots + 549 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 558 \beta_{14} - 2454 \beta_{13} + 1518 \beta_{12} - 2450 \beta_{11} + 112 \beta_{10} - 122 \beta_{9} + \cdots + 490852 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 5444 \beta_{14} + 21671 \beta_{13} + 9406 \beta_{12} + 748 \beta_{11} + 5076 \beta_{10} + 12779 \beta_{9} + \cdots - 52335 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 107494 \beta_{14} - 431278 \beta_{13} + 228100 \beta_{12} - 424350 \beta_{11} + 33540 \beta_{10} + \cdots + 52944538 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 929296 \beta_{14} + 2875053 \beta_{13} + 1614380 \beta_{12} + 111508 \beta_{11} + 760736 \beta_{10} + \cdots - 19500603 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 17592014 \beta_{14} - 66346210 \beta_{13} + 31565910 \beta_{12} - 63944850 \beta_{11} + \cdots + 5887135744 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 141333500 \beta_{14} + 375364315 \beta_{13} + 244312318 \beta_{12} + 16658180 \beta_{11} + \cdots - 3804582195 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2633189470 \beta_{14} - 9517095674 \beta_{13} + 4188776148 \beta_{12} - 8973525894 \beta_{11} + \cdots + 669787426670 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 20165333592 \beta_{14} + 48668951681 \beta_{13} + 34574949460 \beta_{12} + 2622550156 \beta_{11} + \cdots - 624493274127 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 373138244774 \beta_{14} - 1309105571798 \beta_{13} + 542077166726 \beta_{12} - 1208501571930 \beta_{11} + \cdots + 77601886140028 \) 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
−11.2255
−9.85927
−9.10832
−5.95319
−4.17665
−3.22135
−0.828056
−0.534888
1.53130
2.82525
4.86011
8.15629
9.40628
10.0953
11.0327
−11.2255 −8.63404 94.0128 −50.1430 96.9218 −21.1197 −696.127 −168.453 562.883
1.2 −9.85927 18.6293 65.2052 81.5820 −183.671 251.046 −327.379 104.050 −804.338
1.3 −9.10832 −3.63483 50.9615 90.5422 33.1072 −227.055 −172.707 −229.788 −824.687
1.4 −5.95319 0.492952 3.44050 −84.8993 −2.93464 32.9427 170.020 −242.757 505.422
1.5 −4.17665 18.0893 −14.5556 45.9772 −75.5525 1.10481 194.446 84.2217 −192.030
1.6 −3.22135 −18.1848 −21.6229 −24.3897 58.5795 −257.943 172.738 87.6865 78.5677
1.7 −0.828056 29.9124 −31.3143 −58.3082 −24.7692 219.523 52.4278 651.754 48.2825
1.8 −0.534888 −17.0946 −31.7139 −82.2494 9.14369 79.8225 34.0798 49.2248 43.9942
1.9 1.53130 −6.64194 −29.6551 56.9386 −10.1708 76.1602 −94.4128 −198.885 87.1904
1.10 2.82525 −31.0313 −24.0180 −1.02175 −87.6710 57.2437 −158.265 719.940 −2.88669
1.11 4.86011 25.3526 −8.37935 94.2198 123.216 −55.6504 −196.248 399.754 457.919
1.12 8.15629 8.76727 34.5251 28.8952 71.5084 142.726 20.5954 −166.135 235.677
1.13 9.40628 24.3201 56.4781 −54.9900 228.762 −13.5015 230.248 348.468 −517.251
1.14 10.0953 −25.6337 69.9160 29.7197 −258.781 167.892 382.775 414.087 300.031
1.15 11.0327 3.29128 89.7201 56.1267 36.3117 −171.191 636.807 −232.167 619.228
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
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.b ✓ 15
3.b odd 2 1 531.6.a.f 15
4.b odd 2 1 944.6.a.h 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
59.6.a.b ✓ 15 1.a even 1 1 trivial
531.6.a.f 15 3.b odd 2 1
944.6.a.h 15 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{15} - 3 T_{2}^{14} - 387 T_{2}^{13} + 1023 T_{2}^{12} + 57328 T_{2}^{11} - 124838 T_{2}^{10} + \cdots - 6425465344 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(59))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} + \cdots - 6425465344 \) Copy content Toggle raw display
$3$ \( T^{15} + \cdots + 45\!\cdots\!96 \) Copy content Toggle raw display
$5$ \( T^{15} + \cdots + 24\!\cdots\!04 \) Copy content Toggle raw display
$7$ \( T^{15} + \cdots - 26\!\cdots\!68 \) Copy content Toggle raw display
$11$ \( T^{15} + \cdots + 61\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( T^{15} + \cdots + 26\!\cdots\!48 \) Copy content Toggle raw display
$17$ \( T^{15} + \cdots + 15\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots - 17\!\cdots\!88 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots + 45\!\cdots\!88 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots + 50\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 85\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots - 10\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots + 12\!\cdots\!34 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 10\!\cdots\!72 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 23\!\cdots\!88 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T - 3481)^{15} \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 50\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 77\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 91\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 49\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots - 18\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 38\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 10\!\cdots\!64 \) Copy content Toggle raw display
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