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Properties

Label 5577.2.a.z
Level $5577$
Weight $2$
Character orbit 5577.a
Self dual yes
Analytic conductor $44.533$
Analytic rank $1$
Dimension $12$
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: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} - 17 x^{10} + 30 x^{9} + 108 x^{8} - 158 x^{7} - 319 x^{6} + 354 x^{5} + 435 x^{4} + \cdots + 33 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} - \beta_{3} q^{5} + \beta_1 q^{6} + ( - \beta_{11} - 1) q^{7} + ( - \beta_{10} + \beta_{7} + \beta_{5} - 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} - \beta_{3} q^{5} + \beta_1 q^{6} + ( - \beta_{11} - 1) q^{7} + ( - \beta_{10} + \beta_{7} + \beta_{5} - 1) q^{8} + q^{9} + ( - \beta_{11} + \beta_{10}) q^{10} + q^{11} + ( - \beta_{2} - 1) q^{12} + (\beta_{11} - \beta_{9} + \cdots - \beta_{3}) q^{14}+ \cdots + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} - 12 q^{3} + 14 q^{4} + 4 q^{5} + 2 q^{6} - 12 q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{2} - 12 q^{3} + 14 q^{4} + 4 q^{5} + 2 q^{6} - 12 q^{7} - 12 q^{8} + 12 q^{9} + 12 q^{11} - 14 q^{12} + 10 q^{14} - 4 q^{15} + 14 q^{16} + 10 q^{17} - 2 q^{18} - 18 q^{19} - 12 q^{20} + 12 q^{21} - 2 q^{22} - 14 q^{23} + 12 q^{24} - 2 q^{25} - 12 q^{27} - 24 q^{28} - 4 q^{29} + 4 q^{31} - 26 q^{32} - 12 q^{33} - 32 q^{34} - 6 q^{35} + 14 q^{36} - 16 q^{37} - 6 q^{38} + 10 q^{40} - 10 q^{42} - 4 q^{43} + 14 q^{44} + 4 q^{45} - 28 q^{46} - 16 q^{47} - 14 q^{48} + 18 q^{49} + 16 q^{50} - 10 q^{51} - 28 q^{53} + 2 q^{54} + 4 q^{55} + 26 q^{56} + 18 q^{57} - 64 q^{58} + 12 q^{60} + 2 q^{61} + 18 q^{62} - 12 q^{63} + 24 q^{64} + 2 q^{66} - 40 q^{67} + 18 q^{68} + 14 q^{69} + 48 q^{70} - 28 q^{71} - 12 q^{72} - 30 q^{73} + 2 q^{75} + 4 q^{76} - 12 q^{77} - 10 q^{79} - 52 q^{80} + 12 q^{81} - 10 q^{82} + 4 q^{83} + 24 q^{84} + 6 q^{85} + 40 q^{86} + 4 q^{87} - 12 q^{88} + 28 q^{89} + 4 q^{92} - 4 q^{93} + 22 q^{94} + 4 q^{95} + 26 q^{96} - 40 q^{97} - 60 q^{98} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} - 17 x^{10} + 30 x^{9} + 108 x^{8} - 158 x^{7} - 319 x^{6} + 354 x^{5} + 435 x^{4} + \cdots + 33 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 21 \nu^{11} + 2365 \nu^{10} - 3173 \nu^{9} - 41301 \nu^{8} + 49290 \nu^{7} + 258583 \nu^{6} + \cdots - 123531 ) / 27235 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 33 \nu^{11} - 125 \nu^{10} - 401 \nu^{9} + 1453 \nu^{8} + 1855 \nu^{7} - 4404 \nu^{6} - 5510 \nu^{5} + \cdots - 6102 ) / 2095 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 59 \nu^{11} + 160 \nu^{10} + 463 \nu^{9} - 1709 \nu^{8} + 810 \nu^{7} + 5017 \nu^{6} - 12305 \nu^{5} + \cdots - 1089 ) / 2095 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 912 \nu^{11} + 1550 \nu^{10} + 13939 \nu^{9} - 15587 \nu^{8} - 88785 \nu^{7} + 40196 \nu^{6} + \cdots - 65622 ) / 27235 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 413 \nu^{11} + 1120 \nu^{10} + 6593 \nu^{9} - 16991 \nu^{8} - 38325 \nu^{7} + 88751 \nu^{6} + \cdots - 16422 ) / 5447 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 475 \nu^{11} - 2396 \nu^{10} - 3969 \nu^{9} + 33878 \nu^{8} - 4483 \nu^{7} - 164827 \nu^{6} + \cdots + 40306 ) / 5447 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2773 \nu^{11} + 7520 \nu^{10} + 42711 \nu^{9} - 111748 \nu^{8} - 230090 \nu^{7} + 570999 \nu^{6} + \cdots - 153838 ) / 27235 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2832 \nu^{11} + 7680 \nu^{10} + 38984 \nu^{9} - 107172 \nu^{8} - 181095 \nu^{7} + 508976 \nu^{6} + \cdots - 123502 ) / 27235 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1031 \nu^{11} + 2242 \nu^{10} + 15931 \nu^{9} - 31746 \nu^{8} - 87272 \nu^{7} + 152828 \nu^{6} + \cdots - 24839 ) / 5447 \) 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
\(\nu^{3}\)\(=\) \( \beta_{10} - \beta_{7} - \beta_{5} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - \beta_{9} - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} + 6\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} + 8 \beta_{10} - 2 \beta_{9} - \beta_{8} - 8 \beta_{7} - 10 \beta_{5} + \beta_{4} - \beta_{3} + \cdots + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12 \beta_{11} - \beta_{10} - 12 \beta_{9} - 2 \beta_{8} - 12 \beta_{7} + 2 \beta_{6} - 14 \beta_{5} + \cdots + 87 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 16 \beta_{11} + 53 \beta_{10} - 27 \beta_{9} - 15 \beta_{8} - 59 \beta_{7} - 85 \beta_{5} + 14 \beta_{4} + \cdots + 69 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 110 \beta_{11} - 14 \beta_{10} - 111 \beta_{9} - 32 \beta_{8} - 110 \beta_{7} + 26 \beta_{6} + \cdots + 538 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 175 \beta_{11} + 331 \beta_{10} - 265 \beta_{9} - 157 \beta_{8} - 434 \beta_{7} + 7 \beta_{6} + \cdots + 510 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 915 \beta_{11} - 141 \beta_{10} - 935 \beta_{9} - 356 \beta_{8} - 912 \beta_{7} + 246 \beta_{6} + \cdots + 3455 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1637 \beta_{11} + 2005 \beta_{10} - 2311 \beta_{9} - 1438 \beta_{8} - 3202 \beta_{7} + 139 \beta_{6} + \cdots + 3748 \) 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
2.74492
2.49211
2.21629
1.50314
1.02160
0.611022
−0.253034
−0.879614
−1.16411
−1.80429
−1.97314
−2.51491
−2.74492 −1.00000 5.53457 0.261284 2.74492 −4.23356 −9.70209 1.00000 −0.717202
1.2 −2.49211 −1.00000 4.21063 −1.90483 2.49211 −1.59006 −5.50915 1.00000 4.74706
1.3 −2.21629 −1.00000 2.91194 −0.692237 2.21629 3.24553 −2.02113 1.00000 1.53420
1.4 −1.50314 −1.00000 0.259442 4.13566 1.50314 −4.18047 2.61631 1.00000 −6.21650
1.5 −1.02160 −1.00000 −0.956325 −0.892170 1.02160 0.482605 3.02019 1.00000 0.911445
1.6 −0.611022 −1.00000 −1.62665 2.30641 0.611022 −3.35178 2.21596 1.00000 −1.40927
1.7 0.253034 −1.00000 −1.93597 0.770885 −0.253034 2.75699 −0.995935 1.00000 0.195060
1.8 0.879614 −1.00000 −1.22628 −3.40975 −0.879614 −5.00114 −2.83788 1.00000 −2.99926
1.9 1.16411 −1.00000 −0.644850 1.30774 −1.16411 2.39248 −3.07889 1.00000 1.52235
1.10 1.80429 −1.00000 1.25545 3.28340 −1.80429 −1.56962 −1.34338 1.00000 5.92420
1.11 1.97314 −1.00000 1.89328 1.03130 −1.97314 0.164532 −0.210579 1.00000 2.03490
1.12 2.51491 −1.00000 4.32477 −2.19769 −2.51491 −1.11551 5.84658 1.00000 −5.52699
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.z 12
13.b even 2 1 5577.2.a.be 12
13.f odd 12 2 429.2.s.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.s.a ✓ 24 13.f odd 12 2
5577.2.a.z 12 1.a even 1 1 trivial
5577.2.a.be 12 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5577))\):

\( T_{2}^{12} + 2 T_{2}^{11} - 17 T_{2}^{10} - 30 T_{2}^{9} + 108 T_{2}^{8} + 158 T_{2}^{7} - 319 T_{2}^{6} + \cdots + 33 \) Copy content Toggle raw display
\( T_{5}^{12} - 4 T_{5}^{11} - 21 T_{5}^{10} + 84 T_{5}^{9} + 131 T_{5}^{8} - 544 T_{5}^{7} - 259 T_{5}^{6} + \cdots - 75 \) Copy content Toggle raw display
\( T_{7}^{12} + 12 T_{7}^{11} + 21 T_{7}^{10} - 240 T_{7}^{9} - 898 T_{7}^{8} + 1096 T_{7}^{7} + \cdots - 1404 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 2 T^{11} + \cdots + 33 \) Copy content Toggle raw display
$3$ \( (T + 1)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 4 T^{11} + \cdots - 75 \) Copy content Toggle raw display
$7$ \( T^{12} + 12 T^{11} + \cdots - 1404 \) Copy content Toggle raw display
$11$ \( (T - 1)^{12} \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( T^{12} - 10 T^{11} + \cdots - 22275 \) Copy content Toggle raw display
$19$ \( T^{12} + 18 T^{11} + \cdots + 1141008 \) Copy content Toggle raw display
$23$ \( T^{12} + 14 T^{11} + \cdots + 22932 \) Copy content Toggle raw display
$29$ \( T^{12} + 4 T^{11} + \cdots - 83386179 \) Copy content Toggle raw display
$31$ \( T^{12} - 4 T^{11} + \cdots - 54400188 \) Copy content Toggle raw display
$37$ \( T^{12} + 16 T^{11} + \cdots + 38699925 \) Copy content Toggle raw display
$41$ \( T^{12} - 193 T^{10} + \cdots + 1102257 \) Copy content Toggle raw display
$43$ \( T^{12} + 4 T^{11} + \cdots - 42006896 \) Copy content Toggle raw display
$47$ \( T^{12} + 16 T^{11} + \cdots - 14282400 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots - 143260479 \) Copy content Toggle raw display
$59$ \( T^{12} - 154 T^{10} + \cdots + 148164 \) Copy content Toggle raw display
$61$ \( T^{12} - 2 T^{11} + \cdots + 23583201 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 6743317776 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 535645476 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 115198593 \) Copy content Toggle raw display
$79$ \( T^{12} + 10 T^{11} + \cdots + 15592852 \) Copy content Toggle raw display
$83$ \( T^{12} - 4 T^{11} + \cdots + 49789428 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 107762389776 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots - 59793279600 \) Copy content Toggle raw display
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