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Properties

Label 5577.2.a.bg
Level $5577$
Weight $2$
Character orbit 5577.a
Self dual yes
Analytic conductor $44.533$
Analytic rank $0$
Dimension $14$
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] = [5577,2,Mod(1,5577)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5577, 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("5577.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5577 = 3 \cdot 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5577.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: \(44.5325692073\)
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: \( x^{14} - 6 x^{13} - 5 x^{12} + 90 x^{11} - 84 x^{10} - 450 x^{9} + 761 x^{8} + 782 x^{7} - 2061 x^{6} + \cdots + 64 \) 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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{9} + 1) q^{5} + \beta_1 q^{6} + (\beta_{13} + 1) q^{7} + (\beta_{3} + \beta_{2} + \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{9} + 1) q^{5} + \beta_1 q^{6} + (\beta_{13} + 1) q^{7} + (\beta_{3} + \beta_{2} + \beta_1) q^{8} + q^{9} + (\beta_{7} + \beta_{6} + \beta_{3} + \cdots - 1) q^{10}+ \cdots + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 6 q^{2} + 14 q^{3} + 18 q^{4} + 12 q^{5} + 6 q^{6} + 12 q^{7} + 12 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 6 q^{2} + 14 q^{3} + 18 q^{4} + 12 q^{5} + 6 q^{6} + 12 q^{7} + 12 q^{8} + 14 q^{9} + 14 q^{11} + 18 q^{12} - 2 q^{14} + 12 q^{15} + 22 q^{16} + 2 q^{17} + 6 q^{18} - 2 q^{19} + 44 q^{20} + 12 q^{21} + 6 q^{22} + 2 q^{23} + 12 q^{24} + 20 q^{25} + 14 q^{27} + 24 q^{28} + 28 q^{31} + 30 q^{32} + 14 q^{33} - 16 q^{34} - 2 q^{35} + 18 q^{36} - 10 q^{38} + 10 q^{40} + 40 q^{41} - 2 q^{42} - 2 q^{43} + 18 q^{44} + 12 q^{45} + 32 q^{46} + 48 q^{47} + 22 q^{48} + 10 q^{49} + 2 q^{51} + 8 q^{53} + 6 q^{54} + 12 q^{55} - 10 q^{56} - 2 q^{57} - 16 q^{58} + 40 q^{59} + 44 q^{60} + 4 q^{61} - 6 q^{62} + 12 q^{63} + 16 q^{64} + 6 q^{66} + 40 q^{67} + 22 q^{68} + 2 q^{69} - 40 q^{70} + 36 q^{71} + 12 q^{72} + 10 q^{73} - 48 q^{74} + 20 q^{75} - 4 q^{76} + 12 q^{77} - 24 q^{79} + 68 q^{80} + 14 q^{81} + 46 q^{82} + 12 q^{83} + 24 q^{84} + 34 q^{85} + 48 q^{86} + 12 q^{88} + 20 q^{89} + 36 q^{92} + 28 q^{93} - 50 q^{94} - 60 q^{95} + 30 q^{96} - 16 q^{97} + 44 q^{98} + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 6 x^{13} - 5 x^{12} + 90 x^{11} - 84 x^{10} - 450 x^{9} + 761 x^{8} + 782 x^{7} - 2061 x^{6} + \cdots + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 5\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 83 \nu^{13} - 170 \nu^{12} - 1495 \nu^{11} + 2670 \nu^{10} + 10332 \nu^{9} - 14702 \nu^{8} + \cdots - 872 ) / 968 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 175 \nu^{13} - 1266 \nu^{12} - 651 \nu^{11} + 19806 \nu^{10} - 17452 \nu^{9} - 107678 \nu^{8} + \cdots - 11720 ) / 1936 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 393 \nu^{13} + 2302 \nu^{12} + 3229 \nu^{11} - 37506 \nu^{10} + 12756 \nu^{9} + 219634 \nu^{8} + \cdots + 46248 ) / 1936 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 887 \nu^{13} + 4690 \nu^{12} + 8115 \nu^{11} - 74670 \nu^{10} + 14940 \nu^{9} + 419822 \nu^{8} + \cdots + 37816 ) / 1936 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 981 \nu^{13} - 4662 \nu^{12} - 10633 \nu^{11} + 74810 \nu^{10} + 9516 \nu^{9} - 426362 \nu^{8} + \cdots - 42120 ) / 1936 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 687 \nu^{13} - 3482 \nu^{12} - 6931 \nu^{11} + 56158 \nu^{10} - 1620 \nu^{9} - 322998 \nu^{8} + \cdots - 47088 ) / 968 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1629 \nu^{13} + 7974 \nu^{12} + 16817 \nu^{11} - 127418 \nu^{10} - 2492 \nu^{9} + 721194 \nu^{8} + \cdots + 84904 ) / 1936 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 225 \nu^{13} + 1084 \nu^{12} + 2399 \nu^{11} - 17416 \nu^{10} - 1482 \nu^{9} + 99456 \nu^{8} + \cdots + 13032 ) / 242 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 202 \nu^{13} + 986 \nu^{12} + 2082 \nu^{11} - 15750 \nu^{10} - 222 \nu^{9} + 89093 \nu^{8} + \cdots + 10903 ) / 121 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 3353 \nu^{13} - 16590 \nu^{12} - 34621 \nu^{11} + 266498 \nu^{10} + 4652 \nu^{9} - 1521650 \nu^{8} + \cdots - 202872 ) / 1936 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} + \beta_{8} - \beta_{6} + \beta_{3} + 7\beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{12} + \beta_{11} + \beta_{10} + 2 \beta_{8} + 2 \beta_{7} + \beta_{5} - \beta_{4} + 10 \beta_{3} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{13} - 2 \beta_{12} + 2 \beta_{11} + 3 \beta_{10} - 11 \beta_{9} + 12 \beta_{8} + 3 \beta_{7} + \cdots + 87 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3 \beta_{13} - 14 \beta_{12} + 16 \beta_{11} + 16 \beta_{10} - 4 \beta_{9} + 27 \beta_{8} + 26 \beta_{7} + \cdots + 32 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 16 \beta_{13} - 33 \beta_{12} + 32 \beta_{11} + 54 \beta_{10} - 93 \beta_{9} + 115 \beta_{8} + \cdots + 544 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 48 \beta_{13} - 146 \beta_{12} + 177 \beta_{11} + 185 \beta_{10} - 63 \beta_{9} + 275 \beta_{8} + \cdots + 356 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 178 \beta_{13} - 378 \beta_{12} + 365 \beta_{11} + 646 \beta_{10} - 720 \beta_{9} + 1015 \beta_{8} + \cdots + 3581 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 532 \beta_{13} - 1370 \beta_{12} + 1691 \beta_{11} + 1877 \beta_{10} - 695 \beta_{9} + 2522 \beta_{8} + \cdots + 3440 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1712 \beta_{13} - 3736 \beta_{12} + 3635 \beta_{11} + 6515 \beta_{10} - 5395 \beta_{9} + 8595 \beta_{8} + \cdots + 24521 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 5087 \beta_{13} - 12203 \beta_{12} + 15004 \beta_{11} + 17714 \beta_{10} - 6657 \beta_{9} + 21955 \beta_{8} + \cdots + 30960 \) 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
−2.45072
−2.27294
−1.81638
−0.939913
−0.565458
−0.291537
0.528994
1.09120
1.11176
1.81001
2.08864
2.40348
2.48071
2.82215
−2.45072 1.00000 4.00602 4.42067 −2.45072 1.04233 −4.91619 1.00000 −10.8338
1.2 −2.27294 1.00000 3.16626 1.61338 −2.27294 3.20457 −2.65083 1.00000 −3.66711
1.3 −1.81638 1.00000 1.29925 −0.705953 −1.81638 1.56901 1.27283 1.00000 1.28228
1.4 −0.939913 1.00000 −1.11656 0.163878 −0.939913 −2.77415 2.92930 1.00000 −0.154031
1.5 −0.565458 1.00000 −1.68026 −1.14515 −0.565458 3.18153 2.08103 1.00000 0.647533
1.6 −0.291537 1.00000 −1.91501 3.62473 −0.291537 1.35287 1.14137 1.00000 −1.05674
1.7 0.528994 1.00000 −1.72017 −3.29610 0.528994 −0.703911 −1.96794 1.00000 −1.74362
1.8 1.09120 1.00000 −0.809275 −1.08345 1.09120 3.23091 −3.06549 1.00000 −1.18226
1.9 1.11176 1.00000 −0.763990 2.79143 1.11176 −4.28650 −3.07289 1.00000 3.10340
1.10 1.81001 1.00000 1.27615 2.99723 1.81001 4.54162 −1.31019 1.00000 5.42503
1.11 2.08864 1.00000 2.36242 −2.92121 2.08864 0.790904 0.756961 1.00000 −6.10137
1.12 2.40348 1.00000 3.77673 3.20365 2.40348 −2.45990 4.27033 1.00000 7.69992
1.13 2.48071 1.00000 4.15393 0.0418579 2.48071 4.14960 5.34327 1.00000 0.103837
1.14 2.82215 1.00000 5.96451 2.29504 2.82215 −0.838881 11.1884 1.00000 6.47693
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
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 5577.2.a.bg 14
13.b even 2 1 5577.2.a.bf 14
13.f odd 12 2 429.2.s.b ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.s.b ✓ 28 13.f odd 12 2
5577.2.a.bf 14 13.b even 2 1
5577.2.a.bg 14 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5577))\):

\( T_{2}^{14} - 6 T_{2}^{13} - 5 T_{2}^{12} + 90 T_{2}^{11} - 84 T_{2}^{10} - 450 T_{2}^{9} + 761 T_{2}^{8} + \cdots + 64 \) Copy content Toggle raw display
\( T_{5}^{14} - 12 T_{5}^{13} + 27 T_{5}^{12} + 196 T_{5}^{11} - 949 T_{5}^{10} - 192 T_{5}^{9} + 7213 T_{5}^{8} + \cdots - 92 \) Copy content Toggle raw display
\( T_{7}^{14} - 12 T_{7}^{13} + 18 T_{7}^{12} + 308 T_{7}^{11} - 1313 T_{7}^{10} - 1000 T_{7}^{9} + \cdots - 18764 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} - 6 T^{13} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( (T - 1)^{14} \) Copy content Toggle raw display
$5$ \( T^{14} - 12 T^{13} + \cdots - 92 \) Copy content Toggle raw display
$7$ \( T^{14} - 12 T^{13} + \cdots - 18764 \) Copy content Toggle raw display
$11$ \( (T - 1)^{14} \) Copy content Toggle raw display
$13$ \( T^{14} \) Copy content Toggle raw display
$17$ \( T^{14} - 2 T^{13} + \cdots - 71928956 \) Copy content Toggle raw display
$19$ \( T^{14} + 2 T^{13} + \cdots + 874816 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots - 168702656 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots - 121025696 \) Copy content Toggle raw display
$31$ \( T^{14} - 28 T^{13} + \cdots - 45957356 \) Copy content Toggle raw display
$37$ \( T^{14} - 308 T^{12} + \cdots + 2064448 \) Copy content Toggle raw display
$41$ \( T^{14} - 40 T^{13} + \cdots + 77810704 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 3106508176 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 366875467264 \) Copy content Toggle raw display
$53$ \( T^{14} - 8 T^{13} + \cdots + 3031636 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 34756825408 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots - 151477926791 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots - 86900968304 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 101122736 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots - 13705170707 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 406233491372 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 158307664 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 12800542421248 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 430554032 \) Copy content Toggle raw display
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