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Properties

Label 2763.2.a.q
Level $2763$
Weight $2$
Character orbit 2763.a
Self dual yes
Analytic conductor $22.063$
Analytic rank $0$
Dimension $13$
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] = [2763,2,Mod(1,2763)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2763, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("2763.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2763 = 3^{2} \cdot 307 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2763.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: \(22.0626660785\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 2 x^{12} - 17 x^{11} + 33 x^{10} + 104 x^{9} - 197 x^{8} - 275 x^{7} + 526 x^{6} + 273 x^{5} + \cdots - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 921)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (\beta_{11} - \beta_{10} + \beta_{9} + \cdots + 1) q^{5}+ \cdots + ( - \beta_{11} + \beta_{10} + \cdots + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (\beta_{11} - \beta_{10} + \beta_{9} + \cdots + 1) q^{5}+ \cdots + (\beta_{12} - 2 \beta_{11} - 2 \beta_{10} + \cdots - 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 8 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 13 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 8 q^{7} + 3 q^{8} - 9 q^{10} + 16 q^{11} + 8 q^{13} + 12 q^{14} + 10 q^{16} + 8 q^{17} - 17 q^{19} + 32 q^{20} + 4 q^{22} + 18 q^{23} + 22 q^{25} + 23 q^{26} - 15 q^{28} + 20 q^{29} - 42 q^{31} + 18 q^{32} - 20 q^{34} + 6 q^{35} + 27 q^{37} + 17 q^{38} - 18 q^{40} + 24 q^{41} + 7 q^{43} + 43 q^{44} - 13 q^{46} + 29 q^{47} + 3 q^{49} + 5 q^{50} - q^{52} + 16 q^{53} - 9 q^{55} + 47 q^{56} + 19 q^{58} + 34 q^{59} + 5 q^{61} + 9 q^{62} + 3 q^{64} + 42 q^{65} - 2 q^{67} + 7 q^{68} + 9 q^{70} + 67 q^{71} + 11 q^{73} + 41 q^{74} - 48 q^{76} + 12 q^{77} - 25 q^{79} + 58 q^{80} + 41 q^{82} + 13 q^{83} + 33 q^{85} - 5 q^{86} + 12 q^{88} + 44 q^{89} - 6 q^{91} + 22 q^{92} - 23 q^{94} + 28 q^{95} + 7 q^{97} - 30 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 2 x^{12} - 17 x^{11} + 33 x^{10} + 104 x^{9} - 197 x^{8} - 275 x^{7} + 526 x^{6} + 273 x^{5} + \cdots - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10 \nu^{12} + 31 \nu^{11} + 434 \nu^{10} - 699 \nu^{9} - 4851 \nu^{8} + 5382 \nu^{7} + 20461 \nu^{6} + \cdots - 420 ) / 1084 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 33 \nu^{12} - 21 \nu^{11} - 565 \nu^{10} + 220 \nu^{9} + 3461 \nu^{8} - 525 \nu^{7} - 8904 \nu^{6} + \cdots - 1866 ) / 542 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 30 \nu^{12} - 93 \nu^{11} - 489 \nu^{10} + 1555 \nu^{9} + 2900 \nu^{8} - 9371 \nu^{7} - 7725 \nu^{6} + \cdots - 1450 ) / 542 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 120 \nu^{12} + 101 \nu^{11} + 1956 \nu^{10} - 1071 \nu^{9} - 11329 \nu^{8} + 1712 \nu^{7} + \cdots - 704 ) / 1084 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 64 \nu^{12} - 90 \nu^{11} - 989 \nu^{10} + 1330 \nu^{9} + 5193 \nu^{8} - 6857 \nu^{7} - 10518 \nu^{6} + \cdots - 1648 ) / 542 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 185 \nu^{12} + 438 \nu^{11} + 3151 \nu^{10} - 7105 \nu^{9} - 19690 \nu^{8} + 41031 \nu^{7} + \cdots - 1808 ) / 1084 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 194 \nu^{12} + 49 \nu^{11} - 3650 \nu^{10} - 965 \nu^{9} + 25167 \nu^{8} + 6374 \nu^{7} - 77597 \nu^{6} + \cdots + 5980 ) / 1084 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 269 \nu^{12} + 319 \nu^{11} + 4737 \nu^{10} - 4684 \nu^{9} - 30249 \nu^{8} + 23693 \nu^{7} + \cdots - 1000 ) / 1084 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 399 \nu^{12} + 451 \nu^{11} + 7127 \nu^{10} - 6454 \nu^{9} - 46429 \nu^{8} + 30787 \nu^{7} + \cdots - 2124 ) / 1084 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 595 \nu^{12} - 625 \nu^{11} - 10647 \nu^{10} + 9748 \nu^{9} + 69531 \nu^{8} - 54107 \nu^{7} + \cdots - 484 ) / 1084 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + \beta_{10} + \beta_{6} + \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} - \beta_{8} + \beta_{7} - 2\beta_{5} + \beta_{4} - \beta_{3} + 5\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{12} - 8 \beta_{11} + 9 \beta_{10} - \beta_{9} - \beta_{7} + 8 \beta_{6} - \beta_{5} + \beta_{4} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{12} + 13 \beta_{10} - \beta_{9} - 10 \beta_{8} + 11 \beta_{7} - \beta_{6} - 21 \beta_{5} + \cdots + 87 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 13 \beta_{12} - 57 \beta_{11} + 72 \beta_{10} - 13 \beta_{9} - 10 \beta_{7} + 55 \beta_{6} - 14 \beta_{5} + \cdots + 22 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 13 \beta_{12} + 122 \beta_{10} - 12 \beta_{9} - 83 \beta_{8} + 91 \beta_{7} - 14 \beta_{6} - 174 \beta_{5} + \cdots + 540 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 123 \beta_{12} - 395 \beta_{11} + 552 \beta_{10} - 122 \beta_{9} - 4 \beta_{8} - 74 \beta_{7} + \cdots + 183 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 127 \beta_{12} - 5 \beta_{11} + 1016 \beta_{10} - 112 \beta_{9} - 640 \beta_{8} + 687 \beta_{7} + \cdots + 3484 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1026 \beta_{12} - 2714 \beta_{11} + 4129 \beta_{10} - 1006 \beta_{9} - 75 \beta_{8} - 489 \beta_{7} + \cdots + 1408 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1114 \beta_{12} - 103 \beta_{11} + 7982 \beta_{10} - 956 \beta_{9} - 4743 \beta_{8} + 4987 \beta_{7} + \cdots + 23033 \) 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
−2.59298
−2.03480
−1.73257
−1.58609
−0.696890
−0.141533
0.337424
0.824704
1.16601
1.26083
2.24705
2.28095
2.66789
−2.59298 0 4.72355 4.45782 0 −2.90349 −7.06211 0 −11.5590
1.2 −2.03480 0 2.14040 1.13405 0 −3.30812 −0.285694 0 −2.30755
1.3 −1.73257 0 1.00180 2.10628 0 2.50052 1.72945 0 −3.64927
1.4 −1.58609 0 0.515676 −1.75873 0 −2.86354 2.35427 0 2.78950
1.5 −0.696890 0 −1.51434 −1.25153 0 −1.24512 2.44911 0 0.872182
1.6 −0.141533 0 −1.97997 1.47166 0 3.34172 0.563296 0 −0.208288
1.7 0.337424 0 −1.88615 3.32193 0 2.38440 −1.31128 0 1.12090
1.8 0.824704 0 −1.31986 −3.66965 0 −2.82944 −2.73790 0 −3.02637
1.9 1.16601 0 −0.640413 0.531770 0 −5.26130 −3.07876 0 0.620051
1.10 1.26083 0 −0.410314 0.0665454 0 0.570129 −3.03899 0 0.0839022
1.11 2.24705 0 3.04924 4.33653 0 −1.18316 2.35771 0 9.74440
1.12 2.28095 0 3.20273 −3.04825 0 1.87102 2.74336 0 −6.95290
1.13 2.66789 0 5.11764 1.30159 0 0.926372 8.31754 0 3.47249
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(307\) \( +1 \)

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 2763.2.a.q 13
3.b odd 2 1 921.2.a.g ✓ 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
921.2.a.g ✓ 13 3.b odd 2 1
2763.2.a.q 13 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2763))\):

\( T_{2}^{13} - 2 T_{2}^{12} - 17 T_{2}^{11} + 33 T_{2}^{10} + 104 T_{2}^{9} - 197 T_{2}^{8} - 275 T_{2}^{7} + \cdots - 8 \) Copy content Toggle raw display
\( T_{5}^{13} - 9 T_{5}^{12} - 3 T_{5}^{11} + 224 T_{5}^{10} - 423 T_{5}^{9} - 1492 T_{5}^{8} + 4811 T_{5}^{7} + \cdots - 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} - 2 T^{12} + \cdots - 8 \) Copy content Toggle raw display
$3$ \( T^{13} \) Copy content Toggle raw display
$5$ \( T^{13} - 9 T^{12} + \cdots - 256 \) Copy content Toggle raw display
$7$ \( T^{13} + 8 T^{12} + \cdots + 11876 \) Copy content Toggle raw display
$11$ \( T^{13} - 16 T^{12} + \cdots - 177640 \) Copy content Toggle raw display
$13$ \( T^{13} - 8 T^{12} + \cdots - 25856 \) Copy content Toggle raw display
$17$ \( T^{13} - 8 T^{12} + \cdots + 144862 \) Copy content Toggle raw display
$19$ \( T^{13} + 17 T^{12} + \cdots - 64876748 \) Copy content Toggle raw display
$23$ \( T^{13} - 18 T^{12} + \cdots - 7924608 \) Copy content Toggle raw display
$29$ \( T^{13} + \cdots - 299441664 \) Copy content Toggle raw display
$31$ \( T^{13} + \cdots + 230285056 \) Copy content Toggle raw display
$37$ \( T^{13} - 27 T^{12} + \cdots + 53507824 \) Copy content Toggle raw display
$41$ \( T^{13} - 24 T^{12} + \cdots + 14247328 \) Copy content Toggle raw display
$43$ \( T^{13} - 7 T^{12} + \cdots + 361024 \) Copy content Toggle raw display
$47$ \( T^{13} - 29 T^{12} + \cdots - 2078464 \) Copy content Toggle raw display
$53$ \( T^{13} + \cdots + 286364492 \) Copy content Toggle raw display
$59$ \( T^{13} + \cdots + 4926577856 \) Copy content Toggle raw display
$61$ \( T^{13} - 5 T^{12} + \cdots - 78105088 \) Copy content Toggle raw display
$67$ \( T^{13} + 2 T^{12} + \cdots - 87139328 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots - 2841632800 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots + 170606928896 \) Copy content Toggle raw display
$79$ \( T^{13} + 25 T^{12} + \cdots + 4365040 \) Copy content Toggle raw display
$83$ \( T^{13} - 13 T^{12} + \cdots + 25004272 \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots + 38778364604 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots + 1390786816 \) Copy content Toggle raw display
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