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Properties

Label 279.2.y.b
Level $279$
Weight $2$
Character orbit 279.y
Analytic conductor $2.228$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [279,2,Mod(10,279)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(279, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("279.10");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 279 = 3^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 279.y (of order \(15\), degree \(8\), 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: \(2.22782621639\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{15})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 7 x^{15} + 24 x^{14} - 36 x^{13} + 17 x^{12} + 18 x^{11} - 52 x^{10} + 59 x^{9} + 51 x^{8} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 93)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{13} - \beta_{12} + \beta_{7}) q^{2} + (\beta_{13} + \beta_{12} + \cdots - \beta_1) q^{4} + (\beta_{15} + \beta_{14} + \cdots - \beta_{2}) q^{5} + (\beta_{15} + \beta_{11} + \beta_{7} + \cdots - 1) q^{7}+ \cdots + (2 \beta_{15} - \beta_{14} - 3 \beta_{12} + \cdots + 6) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{5} - 9 q^{7} - 10 q^{8} - 16 q^{10} - 10 q^{11} + 3 q^{13} + 12 q^{14} + 6 q^{16} - 3 q^{17} - 32 q^{19} + 10 q^{20} - 9 q^{22} + 22 q^{23} + 12 q^{25} - 18 q^{26} + 30 q^{28} - 38 q^{29} - 8 q^{31}+ \cdots + 32 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 7 x^{15} + 24 x^{14} - 36 x^{13} + 17 x^{12} + 18 x^{11} - 52 x^{10} + 59 x^{9} + 51 x^{8} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4069 \nu^{15} - 19456 \nu^{14} + 27208 \nu^{13} + 124275 \nu^{12} - 454832 \nu^{11} + 578461 \nu^{10} + \cdots + 47017 ) / 42775 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 281230 \nu^{15} - 2206991 \nu^{14} + 8820835 \nu^{13} - 18950566 \nu^{12} + 25208438 \nu^{11} + \cdots + 3961979 ) / 2917255 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 414244 \nu^{15} - 4245520 \nu^{14} + 20135556 \nu^{13} - 52884723 \nu^{12} + 75994945 \nu^{11} + \cdots - 6297446 ) / 2917255 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 438059 \nu^{15} - 4006616 \nu^{14} + 17503636 \nu^{13} - 41291889 \nu^{12} + 51732233 \nu^{11} + \cdots - 4047292 ) / 2917255 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2218133 \nu^{15} - 16532307 \nu^{14} + 61500181 \nu^{13} - 113668840 \nu^{12} + 111579296 \nu^{11} + \cdots + 7548029 ) / 14586275 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 658973 \nu^{15} + 4783229 \nu^{14} - 16700930 \nu^{13} + 25334664 \nu^{12} - 7519577 \nu^{11} + \cdots + 3523920 ) / 2917255 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3520873 \nu^{15} + 20695527 \nu^{14} - 55347436 \nu^{13} + 21687685 \nu^{12} + 115751194 \nu^{11} + \cdots - 63757429 ) / 14586275 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3931773 \nu^{15} - 28151292 \nu^{14} + 97933481 \nu^{13} - 150119825 \nu^{12} + 64974376 \nu^{11} + \cdots - 6770211 ) / 14586275 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 4000716 \nu^{15} - 28055144 \nu^{14} + 94918812 \nu^{13} - 133880845 \nu^{12} + 26286982 \nu^{11} + \cdots + 479843 ) / 14586275 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 4505391 \nu^{15} + 38673179 \nu^{14} - 162462327 \nu^{13} + 365941565 \nu^{12} + \cdots + 12436352 ) / 14586275 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 4542542 \nu^{15} + 33615153 \nu^{14} - 122783349 \nu^{13} + 214743570 \nu^{12} + \cdots - 14630136 ) / 14586275 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 9695294 \nu^{15} - 73319806 \nu^{14} + 272694298 \nu^{13} - 492497640 \nu^{12} + 402855243 \nu^{11} + \cdots - 1270948 ) / 14586275 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 2250154 \nu^{15} + 15774893 \nu^{14} - 53764792 \nu^{13} + 78373624 \nu^{12} - 26659784 \nu^{11} + \cdots - 7504933 ) / 2917255 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 13223986 \nu^{15} + 95918539 \nu^{14} - 340634242 \nu^{13} + 554715415 \nu^{12} + \cdots - 22728483 ) / 14586275 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 3452553 \nu^{15} - 24597524 \nu^{14} + 85480923 \nu^{13} - 131493548 \nu^{12} + 61829267 \nu^{11} + \cdots + 10474376 ) / 2917255 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{15} + \beta_{13} + \beta_{10} - \beta_{4} + \beta_{3} - \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} + 3 \beta_{13} + 2 \beta_{12} - \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{15} + 5 \beta_{13} + 6 \beta_{12} - 3 \beta_{11} + \beta_{8} - 5 \beta_{7} + \cdots - 8 \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 12 \beta_{15} - 3 \beta_{14} - 2 \beta_{13} + 10 \beta_{12} - 3 \beta_{11} - 10 \beta_{10} + 10 \beta_{9} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 39 \beta_{15} - 12 \beta_{14} - 48 \beta_{13} - 6 \beta_{12} + 23 \beta_{11} - 46 \beta_{10} + \cdots - 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 42 \beta_{15} - 13 \beta_{14} - 183 \beta_{13} - 125 \beta_{12} + 144 \beta_{11} - 102 \beta_{10} + \cdots + 125 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 244 \beta_{15} + 77 \beta_{14} - 286 \beta_{13} - 499 \beta_{12} + 390 \beta_{11} + 45 \beta_{10} + \cdots + 133 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1544 \beta_{15} + 477 \beta_{14} + 721 \beta_{13} - 876 \beta_{12} + 160 \beta_{11} + 1275 \beta_{10} + \cdots + 581 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4167 \beta_{15} + 1275 \beta_{14} + 6233 \beta_{13} + 1476 \beta_{12} - 3471 \beta_{11} + 5052 \beta_{10} + \cdots + 1114 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1741 \beta_{15} + 522 \beta_{14} + 19510 \beta_{13} + 15705 \beta_{12} - 16872 \beta_{11} + 8436 \beta_{10} + \cdots - 1368 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 36800 \beta_{15} - 11291 \beta_{14} + 18665 \beta_{13} + 52610 \beta_{12} - 36800 \beta_{11} + \cdots - 17986 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 179278 \beta_{15} - 54958 \beta_{14} - 126439 \beta_{13} + 62160 \beta_{12} + 18665 \beta_{11} + \cdots - 64279 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 391308 \beta_{15} - 119973 \beta_{14} - 751071 \beta_{13} - 288789 \beta_{12} + 472833 \beta_{11} + \cdots - 89639 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 199150 \beta_{15} + 60945 \beta_{14} - 1934123 \beta_{13} - 1922450 \beta_{12} + 1865497 \beta_{11} + \cdots + 292348 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 5034075 \beta_{15} + 1542815 \beta_{14} - 450830 \beta_{13} - 5319310 \beta_{12} + 3111625 \beta_{11} + \cdots + 2230242 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/279\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(218\)
\(\chi(n)\) \(\beta_{9}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
−0.205080 + 0.227764i
2.18323 − 2.42472i
1.15747 + 0.246029i
−0.826604 − 0.175700i
−0.205080 − 0.227764i
2.18323 + 2.42472i
1.61977 + 0.721167i
−0.515238 − 0.229399i
−0.0698868 − 0.664929i
0.156341 + 1.48749i
−0.0698868 + 0.664929i
0.156341 − 1.48749i
1.15747 − 0.246029i
−0.826604 + 0.175700i
1.61977 − 0.721167i
−0.515238 + 0.229399i
−1.05697 − 0.767934i 0 −0.0905704 − 0.278747i −0.841054 − 1.45675i 0 −2.39861 − 0.509841i −0.925782 + 2.84927i 0 −0.229717 + 2.18561i
10.2 1.83063 + 1.33003i 0 0.964190 + 2.96747i −0.467963 − 0.810536i 0 3.25038 + 0.690890i −0.783275 + 2.41067i 0 0.221370 − 2.10620i
19.1 −1.76635 − 1.28333i 0 0.855032 + 2.63152i 0.272011 − 0.471137i 0 0.385694 − 0.428357i 0.517444 − 1.59253i 0 −1.08509 + 0.483114i
19.2 −0.125340 − 0.0910649i 0 −0.610617 − 1.87928i −1.58103 + 2.73842i 0 −1.81041 + 2.01066i −0.190353 + 0.585848i 0 0.447540 − 0.199258i
28.1 −1.05697 + 0.767934i 0 −0.0905704 + 0.278747i −0.841054 + 1.45675i 0 −2.39861 + 0.509841i −0.925782 − 2.84927i 0 −0.229717 − 2.18561i
28.2 1.83063 − 1.33003i 0 0.964190 − 2.96747i −0.467963 + 0.810536i 0 3.25038 − 0.690890i −0.783275 − 2.41067i 0 0.221370 + 2.10620i
82.1 −0.238887 − 0.735219i 0 1.13455 − 0.824302i −0.877388 − 1.51968i 0 −0.136396 − 1.29772i −2.12790 − 1.54601i 0 −0.907702 + 1.00811i
82.2 0.483302 + 1.48745i 0 −0.360895 + 0.262205i 0.686405 + 1.18889i 0 0.0145831 + 0.138749i 1.96616 + 1.42850i 0 −1.43667 + 1.59559i
100.1 0.102411 + 0.315188i 0 1.52918 − 1.11101i 0.962998 − 1.66796i 0 −1.29164 − 0.575076i 1.04301 + 0.757794i 0 0.624344 + 0.132708i
100.2 0.771208 + 2.37353i 0 −3.42087 + 2.48541i −1.15398 + 1.99875i 0 −2.51360 − 1.11913i −4.49931 − 3.26894i 0 −5.63407 − 1.19756i
226.1 0.102411 − 0.315188i 0 1.52918 + 1.11101i 0.962998 + 1.66796i 0 −1.29164 + 0.575076i 1.04301 − 0.757794i 0 0.624344 − 0.132708i
226.2 0.771208 − 2.37353i 0 −3.42087 − 2.48541i −1.15398 − 1.99875i 0 −2.51360 + 1.11913i −4.49931 + 3.26894i 0 −5.63407 + 1.19756i
235.1 −1.76635 + 1.28333i 0 0.855032 − 2.63152i 0.272011 + 0.471137i 0 0.385694 + 0.428357i 0.517444 + 1.59253i 0 −1.08509 − 0.483114i
235.2 −0.125340 + 0.0910649i 0 −0.610617 + 1.87928i −1.58103 − 2.73842i 0 −1.81041 − 2.01066i −0.190353 − 0.585848i 0 0.447540 + 0.199258i
262.1 −0.238887 + 0.735219i 0 1.13455 + 0.824302i −0.877388 + 1.51968i 0 −0.136396 + 1.29772i −2.12790 + 1.54601i 0 −0.907702 − 1.00811i
262.2 0.483302 − 1.48745i 0 −0.360895 − 0.262205i 0.686405 − 1.18889i 0 0.0145831 − 0.138749i 1.96616 − 1.42850i 0 −1.43667 − 1.59559i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.g even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 279.2.y.b 16
3.b odd 2 1 93.2.m.a ✓ 16
31.g even 15 1 inner 279.2.y.b 16
31.g even 15 1 8649.2.a.bj 8
31.h odd 30 1 8649.2.a.bi 8
93.o odd 30 1 93.2.m.a ✓ 16
93.o odd 30 1 2883.2.a.m 8
93.p even 30 1 2883.2.a.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
93.2.m.a ✓ 16 3.b odd 2 1
93.2.m.a ✓ 16 93.o odd 30 1
279.2.y.b 16 1.a even 1 1 trivial
279.2.y.b 16 31.g even 15 1 inner
2883.2.a.m 8 93.o odd 30 1
2883.2.a.n 8 93.p even 30 1
8649.2.a.bi 8 31.h odd 30 1
8649.2.a.bj 8 31.g even 15 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 4 T_{2}^{14} + 10 T_{2}^{13} + 22 T_{2}^{12} + 157 T_{2}^{10} + 210 T_{2}^{9} + 400 T_{2}^{8} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(279, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 6 T^{15} + \cdots + 841 \) Copy content Toggle raw display
$7$ \( T^{16} + 9 T^{15} + \cdots + 81 \) Copy content Toggle raw display
$11$ \( T^{16} + 10 T^{15} + \cdots + 44521 \) Copy content Toggle raw display
$13$ \( T^{16} - 3 T^{15} + \cdots + 81 \) Copy content Toggle raw display
$17$ \( T^{16} + 3 T^{15} + \cdots + 434281 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 611127841 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 276191161 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 2703064081 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 852891037441 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 150583578601 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 2740352470801 \) Copy content Toggle raw display
$43$ \( T^{16} - T^{15} + \cdots + 96255721 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 274367487601 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 443304287721 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 158703279867001 \) Copy content Toggle raw display
$61$ \( (T^{8} - 13 T^{7} + \cdots - 452849)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 1656598093921 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 397962601 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 23\!\cdots\!81 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 72100834070841 \) Copy content Toggle raw display
$83$ \( T^{16} - 17 T^{15} + \cdots + 39325441 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 13843640281 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 8032683971601 \) Copy content Toggle raw display
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