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Properties

Label 9027.2.a.n
Level $9027$
Weight $2$
Character orbit 9027.a
Self dual yes
Analytic conductor $72.081$
Analytic rank $0$
Dimension $16$
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] = [9027,2,Mod(1,9027)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9027, 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("9027.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9027 = 3^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9027.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: \(72.0809579046\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6 x^{15} - 5 x^{14} + 90 x^{13} - 82 x^{12} - 456 x^{11} + 723 x^{10} + 951 x^{9} - 2105 x^{8} + \cdots - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1003)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (\beta_{4} + 1) q^{5} + ( - \beta_{11} + \beta_{10} + \beta_{5} + \cdots - 1) q^{7}+ \cdots + (\beta_{7} - \beta_{6} + 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (\beta_{4} + 1) q^{5} + ( - \beta_{11} + \beta_{10} + \beta_{5} + \cdots - 1) q^{7}+ \cdots + ( - \beta_{15} - 4 \beta_{14} + \cdots + 6) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 6 q^{2} + 14 q^{4} + 21 q^{5} - 11 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 6 q^{2} + 14 q^{4} + 21 q^{5} - 11 q^{7} + 12 q^{8} + 12 q^{10} + 7 q^{11} - 16 q^{13} - 11 q^{14} + 22 q^{16} - 16 q^{17} + q^{19} + 43 q^{20} - 18 q^{22} + 8 q^{23} + 23 q^{25} + 49 q^{26} - 15 q^{28} + 39 q^{29} - 3 q^{31} - 6 q^{34} - 9 q^{35} - 28 q^{37} + 27 q^{38} + 26 q^{40} + 31 q^{41} + 5 q^{43} + 19 q^{44} - 39 q^{46} + 47 q^{47} + 35 q^{49} + 13 q^{50} + 9 q^{52} + 36 q^{53} + 32 q^{55} - 3 q^{56} + 22 q^{58} - 16 q^{59} - 22 q^{61} - q^{62} + 32 q^{64} + 19 q^{65} + 4 q^{67} - 14 q^{68} - 11 q^{70} + 5 q^{71} - 35 q^{73} - 31 q^{74} + q^{76} + 79 q^{77} - 48 q^{79} + 127 q^{80} - 7 q^{82} + 42 q^{83} - 21 q^{85} - 58 q^{86} - 2 q^{88} - 20 q^{89} + 49 q^{91} + 21 q^{92} + 22 q^{95} - 2 q^{97} + 78 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{15} - 5 x^{14} + 90 x^{13} - 82 x^{12} - 456 x^{11} + 723 x^{10} + 951 x^{9} - 2105 x^{8} + \cdots - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 21607 \nu^{15} + 206853 \nu^{14} - 210704 \nu^{13} - 2846302 \nu^{12} + 6450296 \nu^{11} + \cdots - 1629531 ) / 796522 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 46479 \nu^{15} + 448281 \nu^{14} - 494240 \nu^{13} - 6086430 \nu^{12} + 14620862 \nu^{11} + \cdots - 252191 ) / 796522 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 73855 \nu^{15} + 331382 \nu^{14} + 929938 \nu^{13} - 5685137 \nu^{12} - 2256821 \nu^{11} + \cdots + 621523 ) / 398261 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 169225 \nu^{15} - 821071 \nu^{14} - 1814188 \nu^{13} + 13610466 \nu^{12} + 173262 \nu^{11} + \cdots - 1684399 ) / 796522 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 169225 \nu^{15} - 821071 \nu^{14} - 1814188 \nu^{13} + 13610466 \nu^{12} + 173262 \nu^{11} + \cdots - 1684399 ) / 796522 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 171663 \nu^{15} - 912757 \nu^{14} - 1423874 \nu^{13} + 14140202 \nu^{12} - 4762104 \nu^{11} + \cdots + 623575 ) / 796522 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 252191 \nu^{15} + 1466667 \nu^{14} + 1709236 \nu^{13} - 23191430 \nu^{12} + 14593232 \nu^{11} + \cdots - 5345561 ) / 796522 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 174999 \nu^{15} - 976139 \nu^{14} - 1206377 \nu^{13} + 14819972 \nu^{12} - 8664781 \nu^{11} + \cdots - 317590 ) / 398261 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 454985 \nu^{15} - 2624923 \nu^{14} - 2947570 \nu^{13} + 40413834 \nu^{12} - 26534880 \nu^{11} + \cdots - 180195 ) / 796522 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 493455 \nu^{15} + 2751753 \nu^{14} + 3520236 \nu^{13} - 42280306 \nu^{12} + 22964484 \nu^{11} + \cdots - 1226399 ) / 796522 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 710147 \nu^{15} + 3849819 \nu^{14} + 5665410 \nu^{13} - 60313288 \nu^{12} + 25543250 \nu^{11} + \cdots + 303563 ) / 796522 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 360387 \nu^{15} - 1988496 \nu^{14} - 2756504 \nu^{13} + 31115804 \nu^{12} - 14693575 \nu^{11} + \cdots + 1585084 ) / 398261 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 488289 \nu^{15} + 2675091 \nu^{14} + 3747450 \nu^{13} - 41578626 \nu^{12} + 19291176 \nu^{11} + \cdots - 1545655 ) / 398261 \) 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
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{6} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{14} - \beta_{12} - \beta_{9} + \beta_{4} - \beta_{3} + 7\beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} + \beta_{10} + 10\beta_{7} - 9\beta_{6} - \beta_{5} - 2\beta_{3} - \beta_{2} + 41\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 12 \beta_{14} - \beta_{13} - 11 \beta_{12} + \beta_{11} - 10 \beta_{9} - 2 \beta_{8} - \beta_{7} + \cdots + 88 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2 \beta_{15} - \beta_{14} + 9 \beta_{13} + 14 \beta_{10} - \beta_{9} - \beta_{8} + 81 \beta_{7} + \cdots - 38 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( \beta_{15} - 107 \beta_{14} - 17 \beta_{13} - 91 \beta_{12} + 11 \beta_{11} + 4 \beta_{10} - 81 \beta_{9} + \cdots + 560 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 30 \beta_{15} - 19 \beta_{14} + 54 \beta_{13} + \beta_{12} - \beta_{11} + 137 \beta_{10} - 13 \beta_{9} + \cdots - 348 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 22 \beta_{15} - 865 \beta_{14} - 201 \beta_{13} - 688 \beta_{12} + 92 \beta_{11} + 67 \beta_{10} + \cdots + 3732 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 314 \beta_{15} - 237 \beta_{14} + 232 \beta_{13} + 19 \beta_{12} - 12 \beta_{11} + 1173 \beta_{10} + \cdots - 2828 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 299 \beta_{15} - 6703 \beta_{14} - 2027 \beta_{13} - 5027 \beta_{12} + 711 \beta_{11} + 754 \beta_{10} + \cdots + 25562 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 2868 \beta_{15} - 2464 \beta_{14} + 304 \beta_{13} + 211 \beta_{12} - 87 \beta_{11} + 9428 \beta_{10} + \cdots - 21700 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 3294 \beta_{15} - 50861 \beta_{14} - 18715 \beta_{13} - 36261 \beta_{12} + 5341 \beta_{11} + \cdots + 178066 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 24544 \beta_{15} - 23195 \beta_{14} - 8241 \beta_{13} + 1770 \beta_{12} - 452 \beta_{11} + 73300 \beta_{10} + \cdots - 161242 \) 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
−2.67178
−1.94258
−1.38820
−1.35393
−0.896165
−0.446420
−0.0187805
0.377947
0.431919
0.886486
1.62876
1.79724
1.98811
2.33579
2.52760
2.74400
−2.67178 0 5.13841 3.81472 0 0.767070 −8.38515 0 −10.1921
1.2 −1.94258 0 1.77364 −1.30751 0 −3.17249 0.439731 0 2.53995
1.3 −1.38820 0 −0.0729129 3.88583 0 −3.47551 2.87761 0 −5.39429
1.4 −1.35393 0 −0.166881 −1.77529 0 2.41510 2.93380 0 2.40362
1.5 −0.896165 0 −1.19689 0.516029 0 −1.20609 2.86494 0 −0.462447
1.6 −0.446420 0 −1.80071 2.85809 0 4.83764 1.69671 0 −1.27591
1.7 −0.0187805 0 −1.99965 1.12710 0 −0.878091 0.0751153 0 −0.0211674
1.8 0.377947 0 −1.85716 −0.658559 0 −1.52100 −1.45780 0 −0.248900
1.9 0.431919 0 −1.81345 0.726091 0 1.71462 −1.64710 0 0.313613
1.10 0.886486 0 −1.21414 3.79855 0 −4.56031 −2.84929 0 3.36736
1.11 1.62876 0 0.652867 −1.81397 0 −1.88099 −2.19416 0 −2.95453
1.12 1.79724 0 1.23006 2.26932 0 −0.408229 −1.38377 0 4.07850
1.13 1.98811 0 1.95260 −1.43913 0 −4.79587 −0.0942457 0 −2.86115
1.14 2.33579 0 3.45592 4.30332 0 1.79793 3.40073 0 10.0517
1.15 2.52760 0 4.38874 1.05573 0 3.96203 6.03776 0 2.66845
1.16 2.74400 0 5.52956 3.63969 0 −4.59580 9.68512 0 9.98732
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(17\) \(1\)
\(59\) \(1\)

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 9027.2.a.n 16
3.b odd 2 1 1003.2.a.h ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1003.2.a.h ✓ 16 3.b odd 2 1
9027.2.a.n 16 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9027))\):

\( T_{2}^{16} - 6 T_{2}^{15} - 5 T_{2}^{14} + 90 T_{2}^{13} - 82 T_{2}^{12} - 456 T_{2}^{11} + 723 T_{2}^{10} + \cdots - 1 \) Copy content Toggle raw display
\( T_{5}^{16} - 21 T_{5}^{15} + 169 T_{5}^{14} - 568 T_{5}^{13} - 23 T_{5}^{12} + 5510 T_{5}^{11} + \cdots - 10177 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 6 T^{15} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 21 T^{15} + \cdots - 10177 \) Copy content Toggle raw display
$7$ \( T^{16} + 11 T^{15} + \cdots + 150053 \) Copy content Toggle raw display
$11$ \( T^{16} - 7 T^{15} + \cdots - 1246272 \) Copy content Toggle raw display
$13$ \( T^{16} + 16 T^{15} + \cdots + 39156176 \) Copy content Toggle raw display
$17$ \( (T + 1)^{16} \) Copy content Toggle raw display
$19$ \( T^{16} - T^{15} + \cdots - 45241463 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots - 2243683088 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots - 18054172293 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots - 2089303136 \) Copy content Toggle raw display
$37$ \( T^{16} + 28 T^{15} + \cdots + 93871312 \) Copy content Toggle raw display
$41$ \( T^{16} - 31 T^{15} + \cdots - 4051605 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots - 9106747489312 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots - 484870704 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 2659210263 \) Copy content Toggle raw display
$59$ \( (T + 1)^{16} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 74808359200 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots - 26547138880 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots - 1232446234112 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots - 521350404720 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 364375937949 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 1273873429168 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 58933363171168 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 1403955611216 \) Copy content Toggle raw display
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