Properties

Label 9036.2.a.l
Level $9036$
Weight $2$
Character orbit 9036.a
Self dual yes
Analytic conductor $72.153$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9036,2,Mod(1,9036)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9036, 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("9036.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9036 = 2^{2} \cdot 3^{2} \cdot 251 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9036.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: \(72.1528232664\)
Analytic rank: \(0\)
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: \( x^{12} - 5 x^{11} - 26 x^{10} + 136 x^{9} + 267 x^{8} - 1337 x^{7} - 1553 x^{6} + 5791 x^{5} + 5621 x^{4} - 10299 x^{3} - 9634 x^{2} + 4990 x + 4088 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 3012)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{5} + \beta_{6} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{5} + \beta_{6} q^{7} + ( - \beta_{9} - \beta_{8} - \beta_{6} + \beta_{4} + 1) q^{11} + ( - \beta_{3} + 1) q^{13} + ( - \beta_{8} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{2} + \beta_1 - 1) q^{17} + (\beta_{11} + \beta_{6}) q^{19} + ( - \beta_{9} + 1) q^{23} + ( - \beta_{2} - \beta_1 + 2) q^{25} + (\beta_{11} - \beta_{10} + \beta_1 - 2) q^{29} + ( - \beta_{11} + \beta_{7} - \beta_{5}) q^{31} + (\beta_{10} + \beta_{9} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - 1) q^{35} + ( - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{6} + \beta_{4} + 2) q^{37} + ( - \beta_{11} + \beta_{10} + \beta_{8} + \beta_{7} + \beta_{6} - \beta_{4} + \beta_{2} - 1) q^{41} + (\beta_{10} - \beta_{9} - 2 \beta_{8} - \beta_{7} + 2) q^{43} + ( - \beta_{10} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_1) q^{47} + (\beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + \beta_{3} + \beta_{2} - \beta_1 + 3) q^{49} + (\beta_{10} - \beta_{9} - \beta_{8} - 2 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} - 1) q^{53} + ( - \beta_{10} + \beta_{9} + 2 \beta_{8} - \beta_{7} + \beta_{4} + \beta_1 - 1) q^{55} + (\beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} - \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - \beta_{2} + 2 \beta_1) q^{59} + (\beta_{9} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_1 + 1) q^{61} + (\beta_{9} + \beta_{7} + \beta_{6} + \beta_{4} + 2 \beta_{3} + 2 \beta_1 - 1) q^{65} + (\beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - 2 \beta_{2} + 2 \beta_1) q^{67} + ( - \beta_{11} - \beta_{7} + \beta_{6} + \beta_{3} + \beta_1 + 2) q^{71} + (\beta_{10} + 2 \beta_{8} - \beta_{7} + 2 \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 4) q^{73} + (\beta_{11} - 3 \beta_{10} - \beta_{9} + \beta_{8} + \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} - \beta_1 - 4) q^{77} + ( - \beta_{11} + \beta_{9} + \beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{3} + \beta_1 - 1) q^{79} + (\beta_{10} - \beta_{9} - 2 \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2} + 1) q^{83} + (2 \beta_{8} + \beta_{7} + 2 \beta_{6} + \beta_{5} - \beta_{4} + 3 \beta_{2} - 2 \beta_1 + 2) q^{85} + ( - \beta_{11} + \beta_{10} + \beta_{7} + \beta_{6} + \beta_{3} - \beta_1 - 1) q^{89} + (\beta_{11} - \beta_{9} - 2 \beta_{8} + \beta_{7} + 2 \beta_{5} + 2 \beta_{2} + \beta_1 + 2) q^{91} + ( - 2 \beta_{11} + \beta_{10} + \beta_{9} + 3 \beta_{7} - \beta_{6} - \beta_{4}) q^{95} + (\beta_{11} + \beta_{10} - 2 \beta_{8} - \beta_{7} - 2 \beta_{5} + \beta_{3} + 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 7 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 7 q^{5} + q^{11} + 17 q^{13} - 17 q^{17} + 3 q^{19} + 8 q^{23} + 19 q^{25} - 15 q^{29} - 2 q^{31} + 7 q^{35} + 14 q^{37} - 5 q^{41} + 9 q^{43} + 5 q^{47} + 22 q^{49} - 20 q^{53} - 4 q^{55} + q^{59} + 21 q^{61} - 8 q^{65} + 21 q^{67} + 17 q^{71} + 45 q^{73} - 40 q^{77} + 6 q^{79} + q^{83} + 31 q^{85} - 22 q^{89} + 32 q^{91} + 13 q^{95} + 29 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5 x^{11} - 26 x^{10} + 136 x^{9} + 267 x^{8} - 1337 x^{7} - 1553 x^{6} + 5791 x^{5} + 5621 x^{4} - 10299 x^{3} - 9634 x^{2} + 4990 x + 4088 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{2} + \nu + 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 7126353 \nu^{11} + 45468870 \nu^{10} + 116166372 \nu^{9} - 1110023356 \nu^{8} - 193383023 \nu^{7} + 9351167288 \nu^{6} + \cdots - 16616876824 ) / 186594976 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9111803 \nu^{11} - 57189042 \nu^{10} - 152198396 \nu^{9} + 1390057348 \nu^{8} + 353069493 \nu^{7} - 11641595816 \nu^{6} + 3681506813 \nu^{5} + \cdots + 20662670920 ) / 186594976 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12779401 \nu^{11} + 84773334 \nu^{10} + 196928436 \nu^{9} - 2071626668 \nu^{8} - 109081671 \nu^{7} + 17531461976 \nu^{6} + \cdots - 32877029272 ) / 186594976 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 9425689 \nu^{11} + 61624806 \nu^{10} + 152387372 \nu^{9} - 1522474988 \nu^{8} - 232061751 \nu^{7} + 13093832192 \nu^{6} + \cdots - 26137754024 ) / 93297488 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 6589065 \nu^{11} + 43887146 \nu^{10} + 100728564 \nu^{9} - 1071183884 \nu^{8} - 40359615 \nu^{7} + 9053974080 \nu^{6} - 4229274599 \nu^{5} + \cdots - 17543307008 ) / 46648744 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3975487 \nu^{11} - 25889352 \nu^{10} - 64147390 \nu^{9} + 637057852 \nu^{8} + 96608525 \nu^{7} - 5444625654 \nu^{6} + 2075212809 \nu^{5} + \cdots + 10435206628 ) / 23324372 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 33385083 \nu^{11} - 217959218 \nu^{10} - 530817628 \nu^{9} + 5340298292 \nu^{8} + 642642165 \nu^{7} - 45376779608 \nu^{6} + \cdots + 89270410920 ) / 186594976 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 18364375 \nu^{11} - 119721978 \nu^{10} - 293109828 \nu^{9} + 2932646164 \nu^{8} + 385468217 \nu^{7} - 24897549136 \nu^{6} + \cdots + 47538334360 ) / 93297488 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 81946099 \nu^{11} - 538736066 \nu^{10} - 1280069804 \nu^{9} + 13149805716 \nu^{8} + 1121934957 \nu^{7} - 111091047592 \nu^{6} + \cdots + 211095944072 ) / 186594976 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{2} + \beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + 2\beta_{8} + 3\beta_{6} + 2\beta_{5} - 2\beta_{4} + \beta_{2} + 8\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{11} + 5 \beta_{10} - \beta_{8} - \beta_{7} + 2 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} - \beta_{3} - 12 \beta_{2} + 14 \beta _1 + 59 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 5 \beta_{11} + 29 \beta_{10} + 2 \beta_{9} + 30 \beta_{8} - 3 \beta_{7} + 55 \beta_{6} + 30 \beta_{5} - 36 \beta_{4} + \beta_{3} + 11 \beta_{2} + 82 \beta _1 + 77 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 27 \beta_{11} + 131 \beta_{10} - 2 \beta_{9} - 11 \beta_{8} - 20 \beta_{7} + 70 \beta_{6} + 62 \beta_{5} - 82 \beta_{4} - 21 \beta_{3} - 132 \beta_{2} + 173 \beta _1 + 690 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 124 \beta_{11} + 595 \beta_{10} + 44 \beta_{9} + 383 \beta_{8} - 56 \beta_{7} + 858 \beta_{6} + 403 \beta_{5} - 581 \beta_{4} + 28 \beta_{3} + 100 \beta_{2} + 918 \beta _1 + 1256 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 527 \beta_{11} + 2543 \beta_{10} - 6 \beta_{9} - 40 \beta_{8} - 272 \beta_{7} + 1591 \beta_{6} + 1007 \beta_{5} - 1663 \beta_{4} - 295 \beta_{3} - 1457 \beta_{2} + 2137 \beta _1 + 8704 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2288 \beta_{11} + 10677 \beta_{10} + 814 \beta_{9} + 4790 \beta_{8} - 685 \beta_{7} + 13013 \beta_{6} + 5383 \beta_{5} - 9279 \beta_{4} + 502 \beta_{3} + 811 \beta_{2} + 10751 \beta _1 + 19523 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 9074 \beta_{11} + 44233 \beta_{10} + 848 \beta_{9} + 1248 \beta_{8} - 2794 \beta_{7} + 30697 \beta_{6} + 15166 \beta_{5} - 30150 \beta_{4} - 3442 \beta_{3} - 16349 \beta_{2} + 26958 \beta _1 + 114742 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 37953 \beta_{11} + 179745 \beta_{10} + 14766 \beta_{9} + 60587 \beta_{8} - 5163 \beta_{7} + 197048 \beta_{6} + 72651 \beta_{5} - 147907 \beta_{4} + 7957 \beta_{3} + 5340 \beta_{2} + \cdots + 297561 \) Copy content Toggle raw display

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
−3.29592
−2.63471
−1.63780
−1.57283
−1.52087
−0.621922
0.818252
1.75299
2.93793
3.20830
3.63244
3.93413
0 0 0 −4.29592 0 −1.75706 0 0 0
1.2 0 0 0 −3.63471 0 3.32358 0 0 0
1.3 0 0 0 −2.63780 0 −2.18828 0 0 0
1.4 0 0 0 −2.57283 0 −1.16447 0 0 0
1.5 0 0 0 −2.52087 0 4.52057 0 0 0
1.6 0 0 0 −1.62192 0 −4.17725 0 0 0
1.7 0 0 0 −0.181748 0 1.78679 0 0 0
1.8 0 0 0 0.752995 0 −3.70216 0 0 0
1.9 0 0 0 1.93793 0 −2.29269 0 0 0
1.10 0 0 0 2.20830 0 2.75433 0 0 0
1.11 0 0 0 2.63244 0 −1.18801 0 0 0
1.12 0 0 0 2.93413 0 4.08465 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(251\) \(-1\)

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 9036.2.a.l 12
3.b odd 2 1 3012.2.a.i 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3012.2.a.i 12 3.b odd 2 1
9036.2.a.l 12 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9036))\):

\( T_{5}^{12} + 7 T_{5}^{11} - 15 T_{5}^{10} - 179 T_{5}^{9} - 9 T_{5}^{8} + 1717 T_{5}^{7} + 1142 T_{5}^{6} - 7586 T_{5}^{5} - 6303 T_{5}^{4} + 14891 T_{5}^{3} + 10726 T_{5}^{2} - 9356 T_{5} - 1960 \) Copy content Toggle raw display
\( T_{17}^{12} + 17 T_{17}^{11} + 30 T_{17}^{10} - 914 T_{17}^{9} - 4937 T_{17}^{8} + 7019 T_{17}^{7} + 99275 T_{17}^{6} + 160149 T_{17}^{5} - 245161 T_{17}^{4} - 783129 T_{17}^{3} - 381478 T_{17}^{2} + 212976 T_{17} + 68320 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 7 T^{11} - 15 T^{10} + \cdots - 1960 \) Copy content Toggle raw display
$7$ \( T^{12} - 53 T^{10} - 23 T^{9} + \cdots - 56960 \) Copy content Toggle raw display
$11$ \( T^{12} - T^{11} - 90 T^{10} + \cdots + 3761408 \) Copy content Toggle raw display
$13$ \( T^{12} - 17 T^{11} + 45 T^{10} + \cdots - 121280 \) Copy content Toggle raw display
$17$ \( T^{12} + 17 T^{11} + 30 T^{10} + \cdots + 68320 \) Copy content Toggle raw display
$19$ \( T^{12} - 3 T^{11} - 110 T^{10} + 273 T^{9} + \cdots - 40 \) Copy content Toggle raw display
$23$ \( T^{12} - 8 T^{11} - 72 T^{10} + \cdots - 22952 \) Copy content Toggle raw display
$29$ \( T^{12} + 15 T^{11} - 26 T^{10} + \cdots - 290528 \) Copy content Toggle raw display
$31$ \( T^{12} + 2 T^{11} - 181 T^{10} + \cdots + 785776 \) Copy content Toggle raw display
$37$ \( T^{12} - 14 T^{11} - 67 T^{10} + \cdots - 21728 \) Copy content Toggle raw display
$41$ \( T^{12} + 5 T^{11} - 254 T^{10} + \cdots + 31840 \) Copy content Toggle raw display
$43$ \( T^{12} - 9 T^{11} - 266 T^{10} + \cdots + 314888 \) Copy content Toggle raw display
$47$ \( T^{12} - 5 T^{11} - 149 T^{10} + \cdots - 1146880 \) Copy content Toggle raw display
$53$ \( T^{12} + 20 T^{11} + \cdots - 581717024 \) Copy content Toggle raw display
$59$ \( T^{12} - T^{11} - 563 T^{10} + \cdots - 8144520896 \) Copy content Toggle raw display
$61$ \( T^{12} - 21 T^{11} - 214 T^{10} + \cdots - 203840 \) Copy content Toggle raw display
$67$ \( T^{12} - 21 T^{11} + \cdots - 201489907712 \) Copy content Toggle raw display
$71$ \( T^{12} - 17 T^{11} + \cdots - 1076397632 \) Copy content Toggle raw display
$73$ \( T^{12} - 45 T^{11} + \cdots - 19825280350 \) Copy content Toggle raw display
$79$ \( T^{12} - 6 T^{11} + \cdots + 1782086848 \) Copy content Toggle raw display
$83$ \( T^{12} - T^{11} - 607 T^{10} + \cdots - 9606022132 \) Copy content Toggle raw display
$89$ \( T^{12} + 22 T^{11} - 75 T^{10} + \cdots + 476288 \) Copy content Toggle raw display
$97$ \( T^{12} - 29 T^{11} + \cdots - 227151311744 \) Copy content Toggle raw display
show more
show less