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Properties

Label 1287.2.b.c
Level $1287$
Weight $2$
Character orbit 1287.b
Analytic conductor $10.277$
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] = [1287,2,Mod(298,1287)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1287, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1287.298");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1287 = 3^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1287.b (of order \(2\), degree \(1\), 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: \(10.2767467401\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 23x^{12} + 201x^{10} + 835x^{8} + 1695x^{6} + 1565x^{4} + 511x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 429)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{6} - \beta_{5} - 1) q^{4} + ( - \beta_{3} + \beta_{2}) q^{5} + (\beta_{13} - \beta_{3}) q^{7} + ( - \beta_{9} + \beta_{7} + \cdots - 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{6} - \beta_{5} - 1) q^{4} + ( - \beta_{3} + \beta_{2}) q^{5} + (\beta_{13} - \beta_{3}) q^{7} + ( - \beta_{9} + \beta_{7} + \cdots - 2 \beta_1) q^{8}+ \cdots + ( - 3 \beta_{13} + \beta_{10} + \cdots - 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 18 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 18 q^{4} - 16 q^{14} + 34 q^{16} - 4 q^{17} + 6 q^{22} + 8 q^{23} - 26 q^{25} + 6 q^{26} + 24 q^{29} + 8 q^{35} + 32 q^{38} - 20 q^{40} + 32 q^{43} - 46 q^{49} + 4 q^{52} - 20 q^{53} + 12 q^{55} + 32 q^{56} - 20 q^{61} - 72 q^{62} - 58 q^{64} - 12 q^{65} + 20 q^{68} + 12 q^{77} + 12 q^{79} + 20 q^{82} - 30 q^{88} + 16 q^{91} + 24 q^{92} + 64 q^{94} + 36 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 23x^{12} + 201x^{10} + 835x^{8} + 1695x^{6} + 1565x^{4} + 511x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{13} - 2\nu^{11} + 233\nu^{9} + 2504\nu^{7} + 10009\nu^{5} + 16390\nu^{3} + 8491\nu ) / 784 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{13} + 6\nu^{11} - 699\nu^{9} - 7120\nu^{7} - 24931\nu^{5} - 29962\nu^{3} - 6265\nu ) / 784 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{12} - 62\nu^{10} - 449\nu^{8} - 1280\nu^{6} - 885\nu^{4} + 954\nu^{2} + 413 ) / 56 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -9\nu^{12} - 186\nu^{10} - 1375\nu^{8} - 4288\nu^{6} - 4951\nu^{4} - 1114\nu^{2} - 21 ) / 112 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -9\nu^{12} - 186\nu^{10} - 1375\nu^{8} - 4288\nu^{6} - 4951\nu^{4} - 1002\nu^{2} + 315 ) / 112 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -57\nu^{13} - 1290\nu^{11} - 11023\nu^{9} - 44256\nu^{7} - 84519\nu^{5} - 66938\nu^{3} - 8757\nu ) / 784 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 51 \nu^{13} - 7 \nu^{12} - 1082 \nu^{11} - 210 \nu^{10} - 8305 \nu^{9} - 2485 \nu^{8} - 27528 \nu^{7} + \cdots - 7399 ) / 784 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -27\nu^{13} - 642\nu^{11} - 5861\nu^{9} - 25688\nu^{7} - 54725\nu^{5} - 48842\nu^{3} - 9863\nu ) / 392 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 51 \nu^{13} - 7 \nu^{12} + 1082 \nu^{11} - 210 \nu^{10} + 8305 \nu^{9} - 2485 \nu^{8} + 27528 \nu^{7} + \cdots - 7399 ) / 784 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{12} - 22\nu^{10} - 179\nu^{8} - 660\nu^{6} - 1091\nu^{4} - 714\nu^{2} - 117 ) / 8 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3\nu^{12} + 62\nu^{10} + 456\nu^{8} + 1392\nu^{6} + 1473\nu^{4} + 138\nu^{2} - 14 ) / 14 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 219\nu^{13} + 4750\nu^{11} + 37957\nu^{9} + 136448\nu^{7} + 216869\nu^{5} + 135374\nu^{3} + 25207\nu ) / 784 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{5} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} + \beta_{7} + \beta_{3} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} - 7\beta_{6} + 9\beta_{5} + \beta_{4} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} - \beta_{10} + 11\beta_{9} + \beta_{8} - 9\beta_{7} - 12\beta_{3} + 40\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -13\beta_{12} - 2\beta_{11} + 2\beta_{10} + 2\beta_{8} + 48\beta_{6} - 74\beta_{5} - 9\beta_{4} - 87 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -13\beta_{13} + 13\beta_{10} - 94\beta_{9} - 13\beta_{8} + 68\beta_{7} + 109\beta_{3} + 6\beta_{2} - 275\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 126\beta_{12} + 32\beta_{11} - 32\beta_{10} - 32\beta_{8} - 336\beta_{6} + 584\beta_{5} + 68\beta_{4} + 543 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 126 \beta_{13} - 132 \beta_{10} + 746 \beta_{9} + 132 \beta_{8} - 504 \beta_{7} - 890 \beta_{3} + \cdots + 1935 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1094 \beta_{12} - 360 \beta_{11} + 354 \beta_{10} + 354 \beta_{8} + 2397 \beta_{6} - 4497 \beta_{5} + \cdots - 3545 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1094 \beta_{13} + 1218 \beta_{10} - 5747 \beta_{9} - 1218 \beta_{8} + 3765 \beta_{7} + \cdots - 13856 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 9003 \beta_{12} + 3504 \beta_{11} - 3380 \beta_{10} - 3380 \beta_{8} - 17347 \beta_{6} + 34133 \beta_{5} + \cdots + 23873 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 9003 \beta_{13} - 10649 \beta_{10} + 43645 \beta_{9} + 10649 \beta_{8} - 28381 \beta_{7} + \cdots + 100478 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1287\mathbb{Z}\right)^\times\).

\(n\) \(496\) \(937\) \(1145\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
298.1
− 2.73878i
− 2.53441i
− 2.15754i
− 1.42819i
− 1.36814i
− 0.584778i
− 0.409068i
0.409068i
0.584778i
1.36814i
1.42819i
2.15754i
2.53441i
2.73878i
− 2.73878i 0 −5.50093 2.84154i 0 − 3.93129i 9.58828i 0 7.78236
298.2 − 2.53441i 0 −4.42325 − 3.70100i 0 0.957295i 6.14151i 0 −9.37985
298.3 − 2.15754i 0 −2.65498 0.710210i 0 2.30964i 1.41315i 0 1.53231
298.4 − 1.42819i 0 −0.0397381 − 0.0606573i 0 1.70646i − 2.79963i 0 −0.0866304
298.5 − 1.36814i 0 0.128197 2.18365i 0 − 4.27070i − 2.91167i 0 2.98753
298.6 − 0.584778i 0 1.65803 − 1.95350i 0 1.51078i − 2.13914i 0 −1.14237
298.7 − 0.409068i 0 1.83266 − 4.13953i 0 − 5.18273i − 1.56782i 0 −1.69335
298.8 0.409068i 0 1.83266 4.13953i 0 5.18273i 1.56782i 0 −1.69335
298.9 0.584778i 0 1.65803 1.95350i 0 − 1.51078i 2.13914i 0 −1.14237
298.10 1.36814i 0 0.128197 − 2.18365i 0 4.27070i 2.91167i 0 2.98753
298.11 1.42819i 0 −0.0397381 0.0606573i 0 − 1.70646i 2.79963i 0 −0.0866304
298.12 2.15754i 0 −2.65498 − 0.710210i 0 − 2.30964i − 1.41315i 0 1.53231
298.13 2.53441i 0 −4.42325 3.70100i 0 − 0.957295i − 6.14151i 0 −9.37985
298.14 2.73878i 0 −5.50093 − 2.84154i 0 3.93129i − 9.58828i 0 7.78236
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 298.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1287.2.b.c 14
3.b odd 2 1 429.2.b.b ✓ 14
13.b even 2 1 inner 1287.2.b.c 14
39.d odd 2 1 429.2.b.b ✓ 14
39.f even 4 1 5577.2.a.x 7
39.f even 4 1 5577.2.a.y 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.b.b ✓ 14 3.b odd 2 1
429.2.b.b ✓ 14 39.d odd 2 1
1287.2.b.c 14 1.a even 1 1 trivial
1287.2.b.c 14 13.b even 2 1 inner
5577.2.a.x 7 39.f even 4 1
5577.2.a.y 7 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 23T_{2}^{12} + 201T_{2}^{10} + 835T_{2}^{8} + 1695T_{2}^{6} + 1565T_{2}^{4} + 511T_{2}^{2} + 49 \) acting on \(S_{2}^{\mathrm{new}}(1287, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 23 T^{12} + \cdots + 49 \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + 48 T^{12} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( T^{14} + 72 T^{12} + \cdots + 246016 \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{7} \) Copy content Toggle raw display
$13$ \( T^{14} + 7 T^{12} + \cdots + 62748517 \) Copy content Toggle raw display
$17$ \( (T^{7} + 2 T^{6} + \cdots - 1256)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + 108 T^{12} + \cdots + 6400 \) Copy content Toggle raw display
$23$ \( (T^{7} - 4 T^{6} + \cdots - 1024)^{2} \) Copy content Toggle raw display
$29$ \( (T^{7} - 12 T^{6} + \cdots - 104)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + 224 T^{12} + \cdots + 153664 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 15831678976 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 1580857600 \) Copy content Toggle raw display
$43$ \( (T^{7} - 16 T^{6} + \cdots + 692704)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 14973927424 \) Copy content Toggle raw display
$53$ \( (T^{7} + 10 T^{6} + \cdots - 11456)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 21143486464 \) Copy content Toggle raw display
$61$ \( (T^{7} + 10 T^{6} + \cdots + 32)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 250296087616 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 352284983296 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 308886962176 \) Copy content Toggle raw display
$79$ \( (T^{7} - 6 T^{6} + \cdots + 55424)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 1284505600 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 7722894400 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 46960623616 \) Copy content Toggle raw display
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