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Properties

Label 5577.2.a.bi
Level $5577$
Weight $2$
Character orbit 5577.a
Self dual yes
Analytic conductor $44.533$
Analytic rank $0$
Dimension $18$
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: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - x^{17} - 29 x^{16} + 26 x^{15} + 347 x^{14} - 277 x^{13} - 2215 x^{12} + 1560 x^{11} + \cdots - 71 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} - \beta_{10} q^{5} - \beta_1 q^{6} + \beta_{6} q^{7} + (\beta_{16} - \beta_{14} + \cdots + \beta_{10}) q^{8}+ \cdots + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} - \beta_{10} q^{5} - \beta_1 q^{6} + \beta_{6} q^{7} + (\beta_{16} - \beta_{14} + \cdots + \beta_{10}) q^{8}+ \cdots + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - q^{2} + 18 q^{3} + 23 q^{4} + 6 q^{5} - q^{6} - q^{7} - 6 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - q^{2} + 18 q^{3} + 23 q^{4} + 6 q^{5} - q^{6} - q^{7} - 6 q^{8} + 18 q^{9} - 7 q^{10} + 18 q^{11} + 23 q^{12} + 7 q^{14} + 6 q^{15} + 25 q^{16} + 19 q^{17} - q^{18} + 2 q^{20} - q^{21} - q^{22} + 38 q^{23} - 6 q^{24} + 36 q^{25} + 18 q^{27} - 33 q^{28} + 47 q^{29} - 7 q^{30} + 22 q^{31} - 33 q^{32} + 18 q^{33} - 4 q^{34} + 20 q^{35} + 23 q^{36} + 41 q^{37} + 14 q^{38} - 28 q^{40} - 8 q^{41} + 7 q^{42} + 28 q^{43} + 23 q^{44} + 6 q^{45} - 6 q^{46} - 11 q^{47} + 25 q^{48} + 9 q^{49} + q^{50} + 19 q^{51} + 49 q^{53} - q^{54} + 6 q^{55} + 35 q^{56} + 27 q^{58} + 2 q^{59} + 2 q^{60} - 13 q^{61} + 46 q^{62} - q^{63} + 40 q^{64} - q^{66} - 4 q^{67} + 30 q^{68} + 38 q^{69} + 83 q^{70} - 2 q^{71} - 6 q^{72} - 39 q^{73} + 42 q^{74} + 36 q^{75} - 20 q^{76} - q^{77} + 18 q^{79} - 18 q^{80} + 18 q^{81} - 12 q^{82} - 5 q^{83} - 33 q^{84} - 2 q^{85} - 57 q^{86} + 47 q^{87} - 6 q^{88} + 9 q^{89} - 7 q^{90} + 86 q^{92} + 22 q^{93} - 27 q^{94} + 74 q^{95} - 33 q^{96} + 17 q^{97} - 4 q^{98} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - x^{17} - 29 x^{16} + 26 x^{15} + 347 x^{14} - 277 x^{13} - 2215 x^{12} + 1560 x^{11} + \cdots - 71 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9169652 \nu^{17} - 298655945 \nu^{16} + 91161897 \nu^{15} + 8016562390 \nu^{14} + \cdots - 14658701393 ) / 7295400308 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 27572835 \nu^{17} + 105364698 \nu^{16} - 903297117 \nu^{15} - 3018440343 \nu^{14} + \cdots + 15131792621 ) / 7295400308 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 26732937 \nu^{17} - 98119845 \nu^{16} + 880992952 \nu^{15} + 2774252573 \nu^{14} + \cdots - 17978406510 ) / 3647700154 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 97595545 \nu^{17} + 149055033 \nu^{16} + 2355255394 \nu^{15} - 3507340629 \nu^{14} + \cdots - 20804262314 ) / 7295400308 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 123767881 \nu^{17} - 194971043 \nu^{16} + 3826495950 \nu^{15} + 5508091979 \nu^{14} + \cdots - 16616372678 ) / 7295400308 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 132204970 \nu^{17} - 143301579 \nu^{16} - 3925050699 \nu^{15} + 4102477818 \nu^{14} + \cdots - 63245672679 ) / 7295400308 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 141923903 \nu^{17} + 223718878 \nu^{16} - 4109252799 \nu^{15} - 6800254571 \nu^{14} + \cdots + 28146090083 ) / 7295400308 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 72896031 \nu^{17} - 180033962 \nu^{16} + 2314805315 \nu^{15} + 5022331689 \nu^{14} + \cdots - 11127190983 ) / 3647700154 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 38657025 \nu^{17} - 85411175 \nu^{16} - 998780061 \nu^{15} + 2107953476 \nu^{14} + \cdots - 3449956046 ) / 1823850077 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 234033418 \nu^{17} + 110265537 \nu^{16} + 6591998079 \nu^{15} - 2258372918 \nu^{14} + \cdots - 10291473061 ) / 7295400308 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 242057647 \nu^{17} - 54319682 \nu^{16} - 6533188643 \nu^{15} + 403831705 \nu^{14} + \cdots + 13117752491 ) / 7295400308 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 250717424 \nu^{17} - 81292203 \nu^{16} + 7361452993 \nu^{15} + 3272948420 \nu^{14} + \cdots - 51281911291 ) / 7295400308 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 258105913 \nu^{17} + 147375237 \nu^{16} + 6993886564 \nu^{15} - 2997370697 \nu^{14} + \cdots - 1416724680 ) / 7295400308 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 70934039 \nu^{17} - 43283629 \nu^{16} - 1963818990 \nu^{15} + 979617966 \nu^{14} + \cdots - 8133970112 ) / 1823850077 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 82473864 \nu^{17} - 72452225 \nu^{16} - 2321987862 \nu^{15} + 1797579040 \nu^{14} + \cdots - 8939028578 ) / 1823850077 \) 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
\(\nu^{3}\)\(=\) \( -\beta_{16} + \beta_{14} - \beta_{12} + \beta_{11} - \beta_{10} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} - \beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} + 7\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{17} - 8 \beta_{16} + 9 \beta_{14} - 9 \beta_{12} + 9 \beta_{11} - 9 \beta_{10} - \beta_{9} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{17} + 10 \beta_{15} + 2 \beta_{13} + \beta_{10} - \beta_{9} - 13 \beta_{7} - 8 \beta_{6} + \cdots + 99 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 14 \beta_{17} - 54 \beta_{16} + 2 \beta_{15} + 66 \beta_{14} + \beta_{13} - 71 \beta_{12} + 65 \beta_{11} + \cdots + 29 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 14 \beta_{17} + 80 \beta_{15} + 3 \beta_{14} + 27 \beta_{13} - 5 \beta_{12} + 3 \beta_{11} + \cdots + 651 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 141 \beta_{17} - 351 \beta_{16} + 31 \beta_{15} + 459 \beta_{14} + 16 \beta_{13} - 544 \beta_{12} + \cdots + 302 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 145 \beta_{17} + 3 \beta_{16} + 600 \beta_{15} + 57 \beta_{14} + 260 \beta_{13} - 101 \beta_{12} + \cdots + 4413 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1250 \beta_{17} - 2265 \beta_{16} + 333 \beta_{15} + 3146 \beta_{14} + 177 \beta_{13} - 4112 \beta_{12} + \cdots + 2791 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1347 \beta_{17} + 63 \beta_{16} + 4395 \beta_{15} + 713 \beta_{14} + 2201 \beta_{13} - 1322 \beta_{12} + \cdots + 30481 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 10379 \beta_{17} - 14631 \beta_{16} + 3098 \beta_{15} + 21533 \beta_{14} + 1697 \beta_{13} + \cdots + 24361 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 11851 \beta_{17} + 825 \beta_{16} + 31885 \beta_{15} + 7458 \beta_{14} + 17502 \beta_{13} + \cdots + 213444 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 82911 \beta_{17} - 94773 \beta_{16} + 26830 \beta_{15} + 147952 \beta_{14} + 15141 \beta_{13} + \cdots + 205805 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 100861 \beta_{17} + 8721 \beta_{16} + 230376 \beta_{15} + 70840 \beta_{14} + 134482 \beta_{13} + \cdots + 1511511 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 646222 \beta_{17} - 615447 \beta_{16} + 223091 \beta_{15} + 1022797 \beta_{14} + 129472 \beta_{13} + \cdots + 1702420 \) 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
2.77128
2.62086
2.46325
1.88763
1.65040
1.37427
1.13032
1.02759
0.107855
−0.262680
−0.470705
−0.845375
−1.54432
−1.59135
−1.93876
−2.34193
−2.47230
−2.56603
−2.77128 1.00000 5.67998 2.69848 −2.77128 −0.464313 −10.1983 1.00000 −7.47824
1.2 −2.62086 1.00000 4.86892 −0.892565 −2.62086 −3.59391 −7.51906 1.00000 2.33929
1.3 −2.46325 1.00000 4.06758 −0.813446 −2.46325 −2.39287 −5.09296 1.00000 2.00372
1.4 −1.88763 1.00000 1.56314 −2.17541 −1.88763 3.72459 0.824626 1.00000 4.10637
1.5 −1.65040 1.00000 0.723807 4.43233 −1.65040 −3.18712 2.10622 1.00000 −7.31510
1.6 −1.37427 1.00000 −0.111377 2.03304 −1.37427 3.75026 2.90161 1.00000 −2.79395
1.7 −1.13032 1.00000 −0.722371 3.03040 −1.13032 1.92857 3.07716 1.00000 −3.42533
1.8 −1.02759 1.00000 −0.944062 −4.21828 −1.02759 0.280543 3.02528 1.00000 4.33466
1.9 −0.107855 1.00000 −1.98837 −2.20520 −0.107855 −1.50837 0.430164 1.00000 0.237841
1.10 0.262680 1.00000 −1.93100 2.67012 0.262680 −3.43929 −1.03259 1.00000 0.701385
1.11 0.470705 1.00000 −1.77844 1.57480 0.470705 0.714161 −1.77853 1.00000 0.741265
1.12 0.845375 1.00000 −1.28534 −1.18975 0.845375 4.74381 −2.77735 1.00000 −1.00578
1.13 1.54432 1.00000 0.384914 −2.83501 1.54432 −2.51573 −2.49420 1.00000 −4.37816
1.14 1.59135 1.00000 0.532402 2.92304 1.59135 1.64740 −2.33547 1.00000 4.65159
1.15 1.93876 1.00000 1.75880 3.38961 1.93876 0.793974 −0.467622 1.00000 6.57166
1.16 2.34193 1.00000 3.48463 1.89586 2.34193 3.05607 3.47690 1.00000 4.43996
1.17 2.47230 1.00000 4.11225 −3.72355 2.47230 −4.24669 5.22211 1.00000 −9.20573
1.18 2.56603 1.00000 4.58452 −0.594474 2.56603 −0.291094 6.63196 1.00000 −1.52544
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
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.bi ✓ 18
13.b even 2 1 5577.2.a.bk yes 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5577.2.a.bi ✓ 18 1.a even 1 1 trivial
5577.2.a.bk yes 18 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5577))\):

\( T_{2}^{18} + T_{2}^{17} - 29 T_{2}^{16} - 26 T_{2}^{15} + 347 T_{2}^{14} + 277 T_{2}^{13} - 2215 T_{2}^{12} + \cdots - 71 \) Copy content Toggle raw display
\( T_{5}^{18} - 6 T_{5}^{17} - 45 T_{5}^{16} + 321 T_{5}^{15} + 671 T_{5}^{14} - 6729 T_{5}^{13} + \cdots - 638464 \) Copy content Toggle raw display
\( T_{7}^{18} + T_{7}^{17} - 67 T_{7}^{16} - 75 T_{7}^{15} + 1782 T_{7}^{14} + 2088 T_{7}^{13} + \cdots - 21013 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + T^{17} + \cdots - 71 \) Copy content Toggle raw display
$3$ \( (T - 1)^{18} \) Copy content Toggle raw display
$5$ \( T^{18} - 6 T^{17} + \cdots - 638464 \) Copy content Toggle raw display
$7$ \( T^{18} + T^{17} + \cdots - 21013 \) Copy content Toggle raw display
$11$ \( (T - 1)^{18} \) Copy content Toggle raw display
$13$ \( T^{18} \) Copy content Toggle raw display
$17$ \( T^{18} - 19 T^{17} + \cdots - 5850887 \) Copy content Toggle raw display
$19$ \( T^{18} - 145 T^{16} + \cdots - 1655177 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots - 288824059 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots - 39440260019 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 126846784 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 79431152269 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots - 1825618237016 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 3099738349 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots - 1669912733 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 313999730752 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 908640743563 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 58893074393152 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 23235725248 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots - 5642239763 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots - 10668483133952 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 39866740577971 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots - 2111052611584 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 6318061382464 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots - 911937000869629 \) Copy content Toggle raw display
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