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Properties

Label 4719.2.a.br
Level $4719$
Weight $2$
Character orbit 4719.a
Self dual yes
Analytic conductor $37.681$
Analytic rank $0$
Dimension $18$
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] = [4719,2,Mod(1,4719)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4719, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4719.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4719 = 3 \cdot 11^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4719.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: \(37.6814047138\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - x^{17} - 31 x^{16} + 29 x^{15} + 396 x^{14} - 348 x^{13} - 2689 x^{12} + 2242 x^{11} + \cdots + 55 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 429)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 2) q^{4} + ( - \beta_{7} + 1) q^{5} + \beta_1 q^{6} + \beta_{16} q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 2) q^{4} + ( - \beta_{7} + 1) q^{5} + \beta_1 q^{6} + \beta_{16} q^{7} + (\beta_{3} + 2 \beta_1) q^{8} + q^{9} + ( - \beta_{17} - \beta_{16} + \cdots - \beta_{2}) q^{10}+ \cdots + (\beta_{17} + 2 \beta_{16} - \beta_{15} + \cdots + 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + q^{2} + 18 q^{3} + 27 q^{4} + 10 q^{5} + q^{6} + 7 q^{7} + 3 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + q^{2} + 18 q^{3} + 27 q^{4} + 10 q^{5} + q^{6} + 7 q^{7} + 3 q^{8} + 18 q^{9} - 3 q^{10} + 27 q^{12} + 18 q^{13} + 10 q^{14} + 10 q^{15} + 41 q^{16} - 5 q^{17} + q^{18} + 10 q^{19} + 16 q^{20} + 7 q^{21} + 9 q^{23} + 3 q^{24} + 52 q^{25} + q^{26} + 18 q^{27} + 4 q^{28} - 3 q^{30} + 36 q^{31} + 37 q^{32} + 20 q^{34} - 23 q^{35} + 27 q^{36} + 37 q^{37} + 6 q^{38} + 18 q^{39} - 54 q^{40} + 12 q^{41} + 10 q^{42} - 2 q^{43} + 10 q^{45} + 6 q^{46} + 41 q^{48} + 61 q^{49} - 48 q^{50} - 5 q^{51} + 27 q^{52} + 18 q^{53} + q^{54} + 36 q^{56} + 10 q^{57} + 40 q^{58} + 10 q^{59} + 16 q^{60} - 12 q^{61} - 25 q^{62} + 7 q^{63} + 67 q^{64} + 10 q^{65} + 52 q^{67} - 18 q^{68} + 9 q^{69} - 13 q^{70} + q^{71} + 3 q^{72} + 29 q^{73} - 2 q^{74} + 52 q^{75} + 26 q^{76} + q^{78} - 18 q^{79} + 48 q^{80} + 18 q^{81} - 6 q^{82} - 26 q^{83} + 4 q^{84} + 15 q^{85} + 9 q^{86} + 59 q^{89} - 3 q^{90} + 7 q^{91} + 4 q^{92} + 36 q^{93} - 53 q^{94} - 5 q^{95} + 37 q^{96} + 70 q^{97} + 93 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - x^{17} - 31 x^{16} + 29 x^{15} + 396 x^{14} - 348 x^{13} - 2689 x^{12} + 2242 x^{11} + \cdots + 55 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 6\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1794316 \nu^{17} + 40117477 \nu^{16} - 121295666 \nu^{15} - 1140544478 \nu^{14} + \cdots - 2013758985 ) / 1855265315 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7494672 \nu^{17} + 9572099 \nu^{16} - 220222849 \nu^{15} - 225723188 \nu^{14} + \cdots - 1885514235 ) / 3710530630 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 34282077 \nu^{17} + 26787405 \nu^{16} + 1053172288 \nu^{15} - 773957384 \nu^{14} + \cdots + 12988761180 ) / 3710530630 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 22768834 \nu^{17} - 83197933 \nu^{16} - 578141728 \nu^{15} + 2298516130 \nu^{14} + \cdots - 7596659310 ) / 1855265315 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 58233332 \nu^{17} - 169924489 \nu^{16} - 1608424113 \nu^{15} + 4611815144 \nu^{14} + \cdots - 6979046035 ) / 3710530630 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 62700640 \nu^{17} + 30228543 \nu^{16} - 2251325551 \nu^{15} - 529501840 \nu^{14} + \cdots - 21931031955 ) / 3710530630 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 63851057 \nu^{17} + 8776947 \nu^{16} - 1945929542 \nu^{15} - 574209510 \nu^{14} + \cdots + 8111535620 ) / 3710530630 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 32589642 \nu^{17} + 67028098 \nu^{16} + 937238049 \nu^{15} - 1806452155 \nu^{14} + \cdots + 11821918765 ) / 1855265315 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 72628004 \nu^{17} - 33453225 \nu^{16} + 2425890163 \nu^{15} + 893540784 \nu^{14} + \cdots + 18353930655 ) / 3710530630 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 94494963 \nu^{17} - 185484818 \nu^{16} - 2805678553 \nu^{15} + 5218759922 \nu^{14} + \cdots - 56440849885 ) / 3710530630 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 2631396 \nu^{17} + 2929809 \nu^{16} - 84836171 \nu^{15} - 83694562 \nu^{14} + 1120419802 \nu^{13} + \cdots - 479252305 ) / 86291410 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 120061537 \nu^{17} + 119261603 \nu^{16} + 3769620476 \nu^{15} - 3438745238 \nu^{14} + \cdots + 36700715240 ) / 3710530630 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 121444634 \nu^{17} - 87841057 \nu^{16} + 3911217537 \nu^{15} + 2506477964 \nu^{14} + \cdots + 29096747885 ) / 3710530630 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 209426356 \nu^{17} + 219910667 \nu^{16} + 6192440073 \nu^{15} - 5835704426 \nu^{14} + \cdots + 33473638275 ) / 3710530630 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} + \beta_{11} + \beta_{10} + \beta_{9} + \beta_{7} + \beta_{6} + 7\beta_{2} + \beta _1 + 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{17} + \beta_{14} + \beta_{8} + \beta_{7} - \beta_{5} + 9\beta_{3} + 38\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{17} - \beta_{15} + \beta_{14} - \beta_{13} + 11 \beta_{12} + 10 \beta_{11} + 10 \beta_{10} + \cdots + 122 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 14 \beta_{17} + 13 \beta_{14} + 2 \beta_{11} + \beta_{10} + 15 \beta_{8} + 14 \beta_{7} - 3 \beta_{6} + \cdots + 26 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 16 \beta_{17} + 4 \beta_{16} - 17 \beta_{15} + 17 \beta_{14} - 16 \beta_{13} + 95 \beta_{12} + \cdots + 757 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 142 \beta_{17} - \beta_{16} + 3 \beta_{15} + 123 \beta_{14} + 2 \beta_{13} + \beta_{12} + 35 \beta_{11} + \cdots + 247 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 179 \beta_{17} + 81 \beta_{16} - 196 \beta_{15} + 204 \beta_{14} - 174 \beta_{13} + 755 \beta_{12} + \cdots + 4920 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1270 \beta_{17} - 11 \beta_{16} + 67 \beta_{15} + 1041 \beta_{14} + 42 \beta_{13} + 19 \beta_{12} + \cdots + 2117 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1730 \beta_{17} + 1069 \beta_{16} - 1917 \beta_{15} + 2098 \beta_{14} - 1612 \beta_{13} + 5772 \beta_{12} + \cdots + 33081 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 10668 \beta_{17} - 21 \beta_{16} + 951 \beta_{15} + 8392 \beta_{14} + 570 \beta_{13} + 227 \beta_{12} + \cdots + 17476 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 15488 \beta_{17} + 11709 \beta_{16} - 17161 \beta_{15} + 19779 \beta_{14} - 13724 \beta_{13} + \cdots + 228164 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 86523 \beta_{17} + 1130 \beta_{16} + 10994 \beta_{15} + 66128 \beta_{14} + 6378 \beta_{13} + \cdots + 142382 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 132548 \beta_{17} + 115815 \beta_{16} - 145567 \beta_{15} + 176614 \beta_{14} - 111113 \beta_{13} + \cdots + 1604495 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 687197 \beta_{17} + 24430 \beta_{16} + 113127 \beta_{15} + 515665 \beta_{14} + 64120 \beta_{13} + \cdots + 1154806 \) 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
−2.71545
−2.58601
−2.24423
−1.99876
−1.95234
−1.37009
−1.28332
−0.265598
0.110220
0.207338
0.855437
1.00961
1.47269
1.61608
2.30337
2.33995
2.71642
2.78469
−2.71545 1.00000 5.37368 4.27193 −2.71545 2.45705 −9.16108 1.00000 −11.6002
1.2 −2.58601 1.00000 4.68746 −0.367017 −2.58601 −1.96460 −6.94980 1.00000 0.949111
1.3 −2.24423 1.00000 3.03656 0.599319 −2.24423 −4.24044 −2.32627 1.00000 −1.34501
1.4 −1.99876 1.00000 1.99506 −3.43571 −1.99876 4.07684 0.00987225 1.00000 6.86718
1.5 −1.95234 1.00000 1.81162 3.98212 −1.95234 −1.61365 0.367789 1.00000 −7.77443
1.6 −1.37009 1.00000 −0.122856 −4.35041 −1.37009 1.02695 2.90850 1.00000 5.96045
1.7 −1.28332 1.00000 −0.353096 0.613240 −1.28332 −0.715509 3.01977 1.00000 −0.786982
1.8 −0.265598 1.00000 −1.92946 3.84309 −0.265598 2.81061 1.04366 1.00000 −1.02072
1.9 0.110220 1.00000 −1.98785 2.75251 0.110220 2.87193 −0.439542 1.00000 0.303382
1.10 0.207338 1.00000 −1.95701 −0.942442 0.207338 −0.527618 −0.820440 1.00000 −0.195404
1.11 0.855437 1.00000 −1.26823 −2.89210 0.855437 3.88648 −2.79576 1.00000 −2.47401
1.12 1.00961 1.00000 −0.980693 3.79468 1.00961 −5.10168 −3.00933 1.00000 3.83114
1.13 1.47269 1.00000 0.168814 1.13349 1.47269 4.08081 −2.69677 1.00000 1.66928
1.14 1.61608 1.00000 0.611726 −2.09951 1.61608 −4.41014 −2.24357 1.00000 −3.39298
1.15 2.30337 1.00000 3.30550 2.22133 2.30337 4.39340 3.00706 1.00000 5.11655
1.16 2.33995 1.00000 3.47535 3.43996 2.33995 −1.93374 3.45224 1.00000 8.04932
1.17 2.71642 1.00000 5.37894 0.225569 2.71642 −2.19622 9.17863 1.00000 0.612740
1.18 2.78469 1.00000 5.75448 −2.79004 2.78469 4.09953 10.4550 1.00000 −7.76939
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(11\) \(-1\)
\(13\) \(-1\)

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 4719.2.a.br 18
11.b odd 2 1 4719.2.a.bq 18
11.d odd 10 2 429.2.n.d ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.n.d ✓ 36 11.d odd 10 2
4719.2.a.bq 18 11.b odd 2 1
4719.2.a.br 18 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4719))\):

\( T_{2}^{18} - T_{2}^{17} - 31 T_{2}^{16} + 29 T_{2}^{15} + 396 T_{2}^{14} - 348 T_{2}^{13} - 2689 T_{2}^{12} + \cdots + 55 \) Copy content Toggle raw display
\( T_{5}^{18} - 10 T_{5}^{17} - 21 T_{5}^{16} + 494 T_{5}^{15} - 590 T_{5}^{14} - 8923 T_{5}^{13} + \cdots - 42944 \) Copy content Toggle raw display
\( T_{7}^{18} - 7 T_{7}^{17} - 69 T_{7}^{16} + 569 T_{7}^{15} + 1618 T_{7}^{14} - 18070 T_{7}^{13} + \cdots - 11501824 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} - T^{17} + \cdots + 55 \) Copy content Toggle raw display
$3$ \( (T - 1)^{18} \) Copy content Toggle raw display
$5$ \( T^{18} - 10 T^{17} + \cdots - 42944 \) Copy content Toggle raw display
$7$ \( T^{18} - 7 T^{17} + \cdots - 11501824 \) Copy content Toggle raw display
$11$ \( T^{18} \) Copy content Toggle raw display
$13$ \( (T - 1)^{18} \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots - 567398656 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots - 117633024 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 12466253824 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots - 2567147264 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 29764267264 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots - 627994624 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 385454080 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 5808956218736 \) Copy content Toggle raw display
$47$ \( T^{18} - 388 T^{16} + \cdots - 3117031 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots - 13263190902016 \) Copy content Toggle raw display
$59$ \( T^{18} - 10 T^{17} + \cdots + 647429 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 180385112801839 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 44563124224 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 217467806374144 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 57371318726656 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 1752198906176 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 9652886263955 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots - 14\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots - 31840696422400 \) Copy content Toggle raw display
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