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Properties

Label 57.5.g.b
Level $57$
Weight $5$
Character orbit 57.g
Analytic conductor $5.892$
Analytic rank $0$
Dimension $14$
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] = [57,5,Mod(31,57)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(57, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("57.31");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 57 = 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 57.g (of order \(6\), degree \(2\), 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: \(5.89208789578\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 169x^{12} + 10623x^{10} + 308395x^{8} + 4150816x^{6} + 22978776x^{4} + 52992144x^{2} + 42865200 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (3 \beta_{7} + 3) q^{3} + (\beta_{12} + 8 \beta_{7} - \beta_{3}) q^{4} + (\beta_{11} - 2 \beta_{7} - \beta_{5} + 2) q^{5} + (6 \beta_{2} - 3 \beta_1) q^{6} + (\beta_{4} + \beta_{2} + \beta_1 - 13) q^{7} + ( - 2 \beta_{12} + 2 \beta_{10} + \cdots + 15) q^{8}+ \cdots + 27 \beta_{7} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (3 \beta_{7} + 3) q^{3} + (\beta_{12} + 8 \beta_{7} - \beta_{3}) q^{4} + (\beta_{11} - 2 \beta_{7} - \beta_{5} + 2) q^{5} + (6 \beta_{2} - 3 \beta_1) q^{6} + (\beta_{4} + \beta_{2} + \beta_1 - 13) q^{7} + ( - 2 \beta_{12} + 2 \beta_{10} + \cdots + 15) q^{8}+ \cdots + (27 \beta_{13} - 54 \beta_{12} + \cdots - 27 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 3 q^{2} + 63 q^{3} + 57 q^{4} + 12 q^{5} + 9 q^{6} - 174 q^{7} + 189 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 3 q^{2} + 63 q^{3} + 57 q^{4} + 12 q^{5} + 9 q^{6} - 174 q^{7} + 189 q^{9} - 126 q^{10} + 204 q^{11} + 549 q^{13} + 615 q^{14} + 108 q^{15} - 995 q^{16} - 342 q^{17} - 610 q^{19} + 120 q^{20} - 783 q^{21} + 1512 q^{22} - 822 q^{23} + 855 q^{24} - 749 q^{25} - 186 q^{26} - 1697 q^{28} - 1686 q^{29} - 756 q^{30} + 8673 q^{32} + 918 q^{33} + 1986 q^{34} - 2220 q^{35} - 1539 q^{36} + 2319 q^{38} + 3294 q^{39} - 8274 q^{40} - 5004 q^{41} + 1845 q^{42} - 1921 q^{43} - 6930 q^{44} + 648 q^{45} - 4566 q^{47} - 8955 q^{48} + 8520 q^{49} - 3078 q^{51} + 10179 q^{52} - 2430 q^{53} - 243 q^{54} - 8548 q^{55} + 549 q^{57} + 37868 q^{58} + 5112 q^{59} + 540 q^{60} + 6803 q^{61} - 3105 q^{62} - 2349 q^{63} - 23690 q^{64} + 4536 q^{66} - 5043 q^{67} - 16308 q^{68} + 11874 q^{70} + 40296 q^{71} + 7695 q^{72} - 4353 q^{73} + 867 q^{74} - 6613 q^{76} + 18660 q^{77} - 837 q^{78} - 2451 q^{79} + 34404 q^{80} - 5103 q^{81} - 21524 q^{82} - 21924 q^{83} - 3596 q^{85} - 39879 q^{86} - 10116 q^{87} - 12912 q^{89} - 3402 q^{90} - 53493 q^{91} + 13998 q^{92} - 2439 q^{93} + 31788 q^{95} + 52038 q^{96} - 23712 q^{97} - 14916 q^{98} + 2754 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 169x^{12} + 10623x^{10} + 308395x^{8} + 4150816x^{6} + 22978776x^{4} + 52992144x^{2} + 42865200 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 77 \nu^{12} + 12815 \nu^{10} + 785409 \nu^{8} + 21786593 \nu^{6} + 266597810 \nu^{4} + \cdots + 1436982120 ) / 7748352 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 77 \nu^{12} + 12815 \nu^{10} + 785409 \nu^{8} + 21786593 \nu^{6} + 266597810 \nu^{4} + \cdots + 1436982120 ) / 7748352 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 77 \nu^{12} - 12815 \nu^{10} - 785409 \nu^{8} - 21786593 \nu^{6} - 266597810 \nu^{4} + \cdots - 1251021672 ) / 7748352 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 647 \nu^{12} - 107993 \nu^{10} - 6649791 \nu^{8} - 185932151 \nu^{6} - 2309785850 \nu^{4} + \cdots - 13432107528 ) / 3670272 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 17209 \nu^{12} - 2871007 \nu^{10} - 176551257 \nu^{8} - 4919730193 \nu^{6} + \cdots - 331842999960 ) / 69735168 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3113 \nu^{12} + 518483 \nu^{10} + 31812765 \nu^{8} + 883808477 \nu^{6} + 10833225914 \nu^{4} + \cdots + 58594213512 ) / 11622528 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 63359 \nu^{13} - 10562141 \nu^{11} - 648842307 \nu^{9} - 18055175795 \nu^{7} + \cdots + 7322192640 ) / 14644385280 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 18182 \nu^{13} - 340725 \nu^{12} + 3024563 \nu^{11} - 56839755 \nu^{10} + 185028891 \nu^{9} + \cdots - 6633297914040 ) / 1220365440 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 18182 \nu^{13} + 340725 \nu^{12} + 3024563 \nu^{11} + 56839755 \nu^{10} + 185028891 \nu^{9} + \cdots + 6633297914040 ) / 1220365440 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 47189 \nu^{13} - 217910 \nu^{12} + 7870991 \nu^{11} - 36293810 \nu^{10} + 483906417 \nu^{9} + \cdots - 4101594945840 ) / 1627153920 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 33951 \nu^{13} - 120463 \nu^{12} + 5670057 \nu^{11} - 20097049 \nu^{10} + 349265151 \nu^{9} + \cdots - 2322900999720 ) / 976292352 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 229181 \nu^{13} - 24255 \nu^{12} + 38211839 \nu^{11} - 4036725 \nu^{10} + 2347965393 \nu^{9} + \cdots - 423360597240 ) / 2440730880 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 438898 \nu^{13} + 205185 \nu^{12} + 73241182 \nu^{11} + 34238055 \nu^{10} + \cdots + 4216135817880 ) / 2092055040 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta _1 - 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{12} + 2\beta_{10} - 30\beta_{7} + \beta_{6} + \beta_{3} + 43\beta_{2} - 44\beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{9} + 3\beta_{8} + 3\beta_{6} - 3\beta_{5} - 3\beta_{4} - 58\beta_{3} - 20\beta_{2} - 78\beta _1 + 1044 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 18 \beta_{13} + 204 \beta_{12} + 30 \beta_{11} - 140 \beta_{10} - 15 \beta_{9} - 15 \beta_{8} + \cdots - 1377 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 252 \beta_{9} - 252 \beta_{8} - 318 \beta_{6} + 240 \beta_{5} + 240 \beta_{4} + 3271 \beta_{3} + \cdots - 52716 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1644 \beta_{13} - 15658 \beta_{12} - 2340 \beta_{11} + 8510 \beta_{10} + 1434 \beta_{9} + 1434 \beta_{8} + \cdots + 105999 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 16401 \beta_{9} + 16401 \beta_{8} + 25311 \beta_{6} - 14049 \beta_{5} - 15633 \beta_{4} + \cdots + 2860386 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 116226 \beta_{13} + 1087796 \beta_{12} + 136782 \beta_{11} - 500048 \beta_{10} - 108783 \beta_{9} + \cdots - 7485531 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 992802 \beta_{9} - 992802 \beta_{8} - 1818426 \beta_{6} + 725286 \beta_{5} + 967638 \beta_{4} + \cdots - 162104382 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 7608060 \beta_{13} - 72122610 \beta_{12} - 7206708 \beta_{11} + 29225138 \beta_{10} + 7632906 \beta_{9} + \cdots + 502912593 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 58853151 \beta_{9} + 58853151 \beta_{8} + 124051755 \beta_{6} - 34926207 \beta_{5} + \cdots + 9439804524 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 483516114 \beta_{13} + 4661104180 \beta_{12} + 362426526 \beta_{11} - 1716051020 \beta_{10} + \cdots - 32790437097 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/57\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(40\)
\(\chi(n)\) \(1\) \(\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
− 7.84915i
− 4.75956i
− 1.50786i
1.48002i
2.10894i
5.54842i
6.71124i
7.84915i
4.75956i
1.50786i
− 1.48002i
− 2.10894i
− 5.54842i
− 6.71124i
−6.79756 + 3.92458i 4.50000 − 2.59808i 22.8046 − 39.4987i 6.46744 + 11.2019i −20.3927 + 35.3212i −58.1044 232.407i 13.5000 − 23.3827i −87.9257 − 50.7639i
31.2 −4.12190 + 2.37978i 4.50000 − 2.59808i 3.32670 − 5.76201i −2.83035 − 4.90231i −12.3657 + 21.4180i 30.0245 − 44.4857i 13.5000 − 23.3827i 23.3328 + 13.4712i
31.3 −1.30584 + 0.753928i 4.50000 − 2.59808i −6.86318 + 11.8874i −11.5987 − 20.0896i −3.91753 + 6.78535i −39.6528 − 44.8231i 13.5000 − 23.3827i 30.2922 + 17.4892i
31.4 1.28173 − 0.740009i 4.50000 − 2.59808i −6.90477 + 11.9594i 22.1939 + 38.4409i 3.84520 − 6.66008i −89.3245 44.1187i 13.5000 − 23.3827i 56.8932 + 32.8473i
31.5 1.82639 − 1.05447i 4.50000 − 2.59808i −5.77619 + 10.0047i 4.34081 + 7.51850i 5.47918 − 9.49022i 82.3572 58.1062i 13.5000 − 23.3827i 15.8560 + 9.15449i
31.6 4.80507 − 2.77421i 4.50000 − 2.59808i 7.39248 − 12.8041i −22.1953 − 38.4434i 14.4152 − 24.9679i −19.9310 6.74158i 13.5000 − 23.3827i −213.300 − 123.149i
31.7 5.81211 − 3.35562i 4.50000 − 2.59808i 14.5204 − 25.1500i 9.62230 + 16.6663i 17.4363 − 30.2006i 7.63109 − 87.5198i 13.5000 − 23.3827i 111.852 + 64.5776i
46.1 −6.79756 − 3.92458i 4.50000 + 2.59808i 22.8046 + 39.4987i 6.46744 − 11.2019i −20.3927 − 35.3212i −58.1044 − 232.407i 13.5000 + 23.3827i −87.9257 + 50.7639i
46.2 −4.12190 − 2.37978i 4.50000 + 2.59808i 3.32670 + 5.76201i −2.83035 + 4.90231i −12.3657 − 21.4180i 30.0245 44.4857i 13.5000 + 23.3827i 23.3328 − 13.4712i
46.3 −1.30584 − 0.753928i 4.50000 + 2.59808i −6.86318 − 11.8874i −11.5987 + 20.0896i −3.91753 − 6.78535i −39.6528 44.8231i 13.5000 + 23.3827i 30.2922 − 17.4892i
46.4 1.28173 + 0.740009i 4.50000 + 2.59808i −6.90477 − 11.9594i 22.1939 − 38.4409i 3.84520 + 6.66008i −89.3245 − 44.1187i 13.5000 + 23.3827i 56.8932 − 32.8473i
46.5 1.82639 + 1.05447i 4.50000 + 2.59808i −5.77619 − 10.0047i 4.34081 − 7.51850i 5.47918 + 9.49022i 82.3572 − 58.1062i 13.5000 + 23.3827i 15.8560 − 9.15449i
46.6 4.80507 + 2.77421i 4.50000 + 2.59808i 7.39248 + 12.8041i −22.1953 + 38.4434i 14.4152 + 24.9679i −19.9310 − 6.74158i 13.5000 + 23.3827i −213.300 + 123.149i
46.7 5.81211 + 3.35562i 4.50000 + 2.59808i 14.5204 + 25.1500i 9.62230 − 16.6663i 17.4363 + 30.2006i 7.63109 87.5198i 13.5000 + 23.3827i 111.852 − 64.5776i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 57.5.g.b ✓ 14
3.b odd 2 1 171.5.p.b 14
19.d odd 6 1 inner 57.5.g.b ✓ 14
57.f even 6 1 171.5.p.b 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.5.g.b ✓ 14 1.a even 1 1 trivial
57.5.g.b ✓ 14 19.d odd 6 1 inner
171.5.p.b 14 3.b odd 2 1
171.5.p.b 14 57.f even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} - 3 T_{2}^{13} - 80 T_{2}^{12} + 249 T_{2}^{11} + 5022 T_{2}^{10} - 20433 T_{2}^{9} + \cdots + 42865200 \) acting on \(S_{5}^{\mathrm{new}}(57, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} - 3 T^{13} + \cdots + 42865200 \) Copy content Toggle raw display
$3$ \( (T^{2} - 9 T + 27)^{7} \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 31\!\cdots\!76 \) Copy content Toggle raw display
$7$ \( (T^{7} + 87 T^{6} + \cdots - 77401088899)^{2} \) Copy content Toggle raw display
$11$ \( (T^{7} + \cdots + 24067875877080)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 45\!\cdots\!75 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 20\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 63\!\cdots\!41 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 69\!\cdots\!75 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 43\!\cdots\!03 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 10\!\cdots\!08 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 51\!\cdots\!69 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 12\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 30\!\cdots\!28 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 41\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 87\!\cdots\!75 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 41\!\cdots\!09 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 68\!\cdots\!03 \) Copy content Toggle raw display
$83$ \( (T^{7} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 12\!\cdots\!12 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 11\!\cdots\!08 \) Copy content Toggle raw display
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