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Properties

Label 285.10.a.h
Level $285$
Weight $10$
Character orbit 285.a
Self dual yes
Analytic conductor $146.785$
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] = [285,10,Mod(1,285)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(285, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("285.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 285 = 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 285.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: \(146.785213307\)
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: \( x^{15} - 2 x^{14} - 6356 x^{13} + 18436 x^{12} + 15858707 x^{11} - 49616078 x^{10} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{6}\cdot 5^{6} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{2} - 81 q^{3} + (\beta_{2} + \beta_1 + 337) q^{4} + 625 q^{5} + (81 \beta_1 + 81) q^{6} + (\beta_{4} + 2 \beta_{2} + 45 \beta_1 + 84) q^{7} + ( - \beta_{3} + 5 \beta_{2} + \cdots - 195) q^{8}+ \cdots + 6561 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{2} - 81 q^{3} + (\beta_{2} + \beta_1 + 337) q^{4} + 625 q^{5} + (81 \beta_1 + 81) q^{6} + (\beta_{4} + 2 \beta_{2} + 45 \beta_1 + 84) q^{7} + ( - \beta_{3} + 5 \beta_{2} + \cdots - 195) q^{8}+ \cdots + (6561 \beta_{8} - 6561 \beta_{4} + \cdots + 60420249) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 17 q^{2} - 1215 q^{3} + 5055 q^{4} + 9375 q^{5} + 1377 q^{6} + 1352 q^{7} - 3597 q^{8} + 98415 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - 17 q^{2} - 1215 q^{3} + 5055 q^{4} + 9375 q^{5} + 1377 q^{6} + 1352 q^{7} - 3597 q^{8} + 98415 q^{9} - 10625 q^{10} + 138230 q^{11} - 409455 q^{12} - 176712 q^{13} - 555994 q^{14} - 759375 q^{15} + 1695731 q^{16} - 274992 q^{17} - 111537 q^{18} - 1954815 q^{19} + 3159375 q^{20} - 109512 q^{21} - 1031106 q^{22} + 1714212 q^{23} + 291357 q^{24} + 5859375 q^{25} + 9500004 q^{26} - 7971615 q^{27} + 14545598 q^{28} + 1754340 q^{29} + 860625 q^{30} + 8442914 q^{31} + 35638859 q^{32} - 11196630 q^{33} + 47218266 q^{34} + 845000 q^{35} + 33165855 q^{36} + 2956096 q^{37} + 2215457 q^{38} + 14313672 q^{39} - 2248125 q^{40} - 38550502 q^{41} + 45035514 q^{42} + 50753570 q^{43} + 212125630 q^{44} + 61509375 q^{45} - 117130008 q^{46} - 40252876 q^{47} - 137354211 q^{48} + 110123035 q^{49} - 6640625 q^{50} + 22274352 q^{51} - 87136648 q^{52} + 65532542 q^{53} + 9034497 q^{54} + 86393750 q^{55} - 377288898 q^{56} + 158340015 q^{57} + 211630876 q^{58} + 175407418 q^{59} - 255909375 q^{60} + 151231854 q^{61} - 30983940 q^{62} + 8870472 q^{63} + 836879575 q^{64} - 110445000 q^{65} + 83519586 q^{66} + 40009476 q^{67} - 124850430 q^{68} - 138851172 q^{69} - 347496250 q^{70} + 87578500 q^{71} - 23599917 q^{72} - 360657638 q^{73} + 1373397084 q^{74} - 474609375 q^{75} - 658772655 q^{76} - 304618172 q^{77} - 769500324 q^{78} + 205798286 q^{79} + 1059831875 q^{80} + 645700815 q^{81} - 2327138772 q^{82} - 63321462 q^{83} - 1178193438 q^{84} - 171870000 q^{85} - 848405762 q^{86} - 142101540 q^{87} - 3211126502 q^{88} - 381069174 q^{89} - 69710625 q^{90} + 1476892872 q^{91} - 2382818588 q^{92} - 683876034 q^{93} - 5137318040 q^{94} - 1221759375 q^{95} - 2886747579 q^{96} - 3915268828 q^{97} - 8273557437 q^{98} + 906927030 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 2 x^{14} - 6356 x^{13} + 18436 x^{12} + 15858707 x^{11} - 49616078 x^{10} + \cdots + 15\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + \nu - 848 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 8\nu^{2} - 1350\nu - 5458 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10\!\cdots\!63 \nu^{14} + \cdots + 83\!\cdots\!12 ) / 11\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 64\!\cdots\!31 \nu^{14} + \cdots - 81\!\cdots\!16 ) / 37\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 19\!\cdots\!19 \nu^{14} + \cdots + 28\!\cdots\!04 ) / 55\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13\!\cdots\!09 \nu^{14} + \cdots - 32\!\cdots\!32 ) / 37\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 34\!\cdots\!69 \nu^{14} + \cdots - 38\!\cdots\!48 ) / 55\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 36\!\cdots\!63 \nu^{14} + \cdots - 68\!\cdots\!56 ) / 55\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 50\!\cdots\!43 \nu^{14} + \cdots - 12\!\cdots\!12 ) / 37\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 20\!\cdots\!27 \nu^{14} + \cdots + 41\!\cdots\!20 ) / 11\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 27\!\cdots\!63 \nu^{14} + \cdots + 20\!\cdots\!68 ) / 11\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 12\!\cdots\!73 \nu^{14} + \cdots - 43\!\cdots\!96 ) / 42\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 33\!\cdots\!31 \nu^{14} + \cdots + 29\!\cdots\!84 ) / 11\!\cdots\!36 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - \beta _1 + 848 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 8\beta_{2} + 1358\beta _1 - 1326 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{14} - 2 \beta_{12} + \beta_{11} + \beta_{7} + 3 \beta_{6} - \beta_{5} - 3 \beta_{4} + \cdots + 1155587 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{14} + 4 \beta_{13} + 33 \beta_{12} + 37 \beta_{11} - 5 \beta_{10} - 13 \beta_{9} + 53 \beta_{8} + \cdots - 6457399 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 6393 \beta_{14} - 352 \beta_{13} - 6087 \beta_{12} + 3560 \beta_{11} + 629 \beta_{10} + \cdots + 1834170224 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 21354 \beta_{14} + 26512 \beta_{13} + 98962 \beta_{12} + 108460 \beta_{11} - 9798 \beta_{10} + \cdots - 18335135982 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 14653854 \beta_{14} - 1567216 \beta_{13} - 14160998 \beta_{12} + 8791335 \beta_{11} + \cdots + 3166138344941 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 86995799 \beta_{14} + 102501340 \beta_{13} + 225956063 \beta_{12} + 229071591 \beta_{11} + \cdots - 43688326956721 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 29856116751 \beta_{14} - 4991595808 \beta_{13} - 30142612289 \beta_{12} + 19112113766 \beta_{11} + \cdots + 57\!\cdots\!26 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 250602871540 \beta_{14} + 310321656312 \beta_{13} + 480069438484 \beta_{12} + 426239605650 \beta_{11} + \cdots - 96\!\cdots\!44 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 57582048662440 \beta_{14} - 13812899364560 \beta_{13} - 61747963465724 \beta_{12} + \cdots + 10\!\cdots\!95 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 621880725470449 \beta_{14} + 826412541410916 \beta_{13} + \cdots - 20\!\cdots\!91 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 10\!\cdots\!85 \beta_{14} + \cdots + 20\!\cdots\!88 \) 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
43.0696
35.4527
32.1152
31.1284
30.2020
16.4829
5.41213
0.751797
−10.1020
−14.0251
−21.1073
−28.5474
−30.5956
−43.1627
−45.0746
−44.0696 −81.0000 1430.13 625.000 3569.64 9758.51 −40461.6 6561.00 −27543.5
1.2 −36.4527 −81.0000 816.797 625.000 2952.67 −11814.3 −11110.7 6561.00 −22782.9
1.3 −33.1152 −81.0000 584.616 625.000 2682.33 17.0326 −2404.69 6561.00 −20697.0
1.4 −32.1284 −81.0000 520.237 625.000 2602.40 7291.16 −264.644 6561.00 −20080.3
1.5 −31.2020 −81.0000 461.564 625.000 2527.36 9737.89 1573.71 6561.00 −19501.2
1.6 −17.4829 −81.0000 −206.350 625.000 1416.11 −3269.06 12558.8 6561.00 −10926.8
1.7 −6.41213 −81.0000 −470.885 625.000 519.382 −7663.31 6302.38 6561.00 −4007.58
1.8 −1.75180 −81.0000 −508.931 625.000 141.896 2356.35 1788.46 6561.00 −1094.87
1.9 9.10202 −81.0000 −429.153 625.000 −737.264 6714.13 −8566.40 6561.00 5688.76
1.10 13.0251 −81.0000 −342.346 625.000 −1055.03 −10702.7 −11128.0 6561.00 8140.70
1.11 20.1073 −81.0000 −107.695 625.000 −1628.69 −2841.09 −12460.4 6561.00 12567.1
1.12 27.5474 −81.0000 246.858 625.000 −2231.34 −1319.66 −7303.96 6561.00 17217.1
1.13 29.5956 −81.0000 363.897 625.000 −2397.24 5260.19 −4383.19 6561.00 18497.2
1.14 42.1627 −81.0000 1265.69 625.000 −3415.18 4307.69 31777.6 6561.00 26351.7
1.15 44.0746 −81.0000 1430.57 625.000 −3570.04 −6480.81 40485.6 6561.00 27546.6
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( -1 \)
\(19\) \( +1 \)

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 285.10.a.h ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.10.a.h ✓ 15 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{15} + 17 T_{2}^{14} - 6223 T_{2}^{13} - 100427 T_{2}^{12} + 15143800 T_{2}^{11} + \cdots - 37\!\cdots\!48 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(285))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} + \cdots - 37\!\cdots\!48 \) Copy content Toggle raw display
$3$ \( (T + 81)^{15} \) Copy content Toggle raw display
$5$ \( (T - 625)^{15} \) Copy content Toggle raw display
$7$ \( T^{15} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{15} + \cdots + 86\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{15} + \cdots - 35\!\cdots\!32 \) Copy content Toggle raw display
$17$ \( T^{15} + \cdots - 21\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( (T + 130321)^{15} \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots - 82\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots - 55\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots - 21\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots - 44\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 44\!\cdots\!52 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 89\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots - 87\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 24\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 60\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots - 87\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
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