Properties

Label 4536.2.a.x
Level 4536
Weight 2
Character orbit 4536.a
Self dual yes
Analytic conductor 36.220
Analytic rank 0
Dimension 4
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 4536 = 2^{3} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4536.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.2201423569\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.22545.1
Defining polynomial: \(x^{4} - x^{3} - 6 x^{2} + 5 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 504)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} + \beta_{3} ) q^{5} + q^{7} +O(q^{10})\) \( q + ( -1 + \beta_{1} + \beta_{3} ) q^{5} + q^{7} + ( 2 - \beta_{1} + \beta_{3} ) q^{11} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{13} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{17} + ( -\beta_{2} + \beta_{3} ) q^{19} + ( 1 + 2 \beta_{2} + \beta_{3} ) q^{23} + ( 3 - \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{25} + ( 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{29} + ( -3 + \beta_{2} ) q^{31} + ( -1 + \beta_{1} + \beta_{3} ) q^{35} + ( 7 - \beta_{2} ) q^{37} + ( -2 + 3 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{41} + ( 3 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{43} + ( -2 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{47} + q^{49} + ( -1 - 3 \beta_{1} + 3 \beta_{2} ) q^{53} + ( 1 + 4 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{55} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{59} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{61} + ( 1 + 6 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{65} + ( 3 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{67} + ( -4 + 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{71} + ( 4 - 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{73} + ( 2 - \beta_{1} + \beta_{3} ) q^{77} + ( 4 + 5 \beta_{2} - \beta_{3} ) q^{79} + ( 1 + 2 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{83} + ( 3 - \beta_{1} - 5 \beta_{2} - 5 \beta_{3} ) q^{85} + ( -4 - \beta_{1} - 3 \beta_{3} ) q^{89} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{91} + ( 6 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{95} + ( 2 - 6 \beta_{1} + \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{5} + 4q^{7} + O(q^{10}) \) \( 4q - 4q^{5} + 4q^{7} + 6q^{11} + 3q^{13} - 8q^{17} - 2q^{19} + 5q^{23} + 14q^{25} - q^{29} - 11q^{31} - 4q^{35} + 27q^{37} - 2q^{41} + 11q^{43} - 7q^{47} + 4q^{49} - 4q^{53} + 6q^{55} - 9q^{59} + 7q^{61} + 9q^{65} + 12q^{67} - 12q^{71} + 13q^{73} + 6q^{77} + 22q^{79} + 6q^{83} + 11q^{85} - 14q^{89} + 3q^{91} + 23q^{95} + q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 6 x^{2} + 5 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 5 \beta_{1}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
1.45106
−2.27060
−0.519120
2.33866
0 0 0 −3.74893 0 1.00000 0 0 0
1.2 0 0 0 −3.62393 0 1.00000 0 0 0
1.3 0 0 0 0.936586 0 1.00000 0 0 0
1.4 0 0 0 2.43628 0 1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(7\) \(-1\)

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 4536.2.a.x 4
3.b odd 2 1 4536.2.a.ba 4
4.b odd 2 1 9072.2.a.ce 4
9.c even 3 2 504.2.r.d 8
9.d odd 6 2 1512.2.r.d 8
12.b even 2 1 9072.2.a.cl 4
36.f odd 6 2 1008.2.r.m 8
36.h even 6 2 3024.2.r.l 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.r.d 8 9.c even 3 2
1008.2.r.m 8 36.f odd 6 2
1512.2.r.d 8 9.d odd 6 2
3024.2.r.l 8 36.h even 6 2
4536.2.a.x 4 1.a even 1 1 trivial
4536.2.a.ba 4 3.b odd 2 1
9072.2.a.ce 4 4.b odd 2 1
9072.2.a.cl 4 12.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4536))\):

\( T_{5}^{4} + 4 T_{5}^{3} - 9 T_{5}^{2} - 29 T_{5} + 31 \)
\( T_{11}^{4} - 6 T_{11}^{3} - 9 T_{11}^{2} + 81 T_{11} - 54 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 + 4 T + 11 T^{2} + 31 T^{3} + 91 T^{4} + 155 T^{5} + 275 T^{6} + 500 T^{7} + 625 T^{8} \)
$7$ \( ( 1 - T )^{4} \)
$11$ \( 1 - 6 T + 35 T^{2} - 117 T^{3} + 474 T^{4} - 1287 T^{5} + 4235 T^{6} - 7986 T^{7} + 14641 T^{8} \)
$13$ \( 1 - 3 T + 25 T^{2} - 18 T^{3} + 234 T^{4} - 234 T^{5} + 4225 T^{6} - 6591 T^{7} + 28561 T^{8} \)
$17$ \( 1 + 8 T + 35 T^{2} + 167 T^{3} + 886 T^{4} + 2839 T^{5} + 10115 T^{6} + 39304 T^{7} + 83521 T^{8} \)
$19$ \( 1 + 2 T + 55 T^{2} + 41 T^{3} + 1309 T^{4} + 779 T^{5} + 19855 T^{6} + 13718 T^{7} + 130321 T^{8} \)
$23$ \( 1 - 5 T + 53 T^{2} - 221 T^{3} + 1729 T^{4} - 5083 T^{5} + 28037 T^{6} - 60835 T^{7} + 279841 T^{8} \)
$29$ \( 1 + T + 50 T^{2} + 172 T^{3} + 1192 T^{4} + 4988 T^{5} + 42050 T^{6} + 24389 T^{7} + 707281 T^{8} \)
$31$ \( 1 + 11 T + 160 T^{2} + 1058 T^{3} + 8008 T^{4} + 32798 T^{5} + 153760 T^{6} + 327701 T^{7} + 923521 T^{8} \)
$37$ \( 1 - 27 T + 412 T^{2} - 4104 T^{3} + 29436 T^{4} - 151848 T^{5} + 564028 T^{6} - 1367631 T^{7} + 1874161 T^{8} \)
$41$ \( 1 + 2 T + 41 T^{2} + 257 T^{3} + 2008 T^{4} + 10537 T^{5} + 68921 T^{6} + 137842 T^{7} + 2825761 T^{8} \)
$43$ \( 1 - 11 T + 148 T^{2} - 1112 T^{3} + 9532 T^{4} - 47816 T^{5} + 273652 T^{6} - 874577 T^{7} + 3418801 T^{8} \)
$47$ \( 1 + 7 T + 110 T^{2} + 298 T^{3} + 4906 T^{4} + 14006 T^{5} + 242990 T^{6} + 726761 T^{7} + 4879681 T^{8} \)
$53$ \( 1 + 4 T + 83 T^{2} - 197 T^{3} + 2722 T^{4} - 10441 T^{5} + 233147 T^{6} + 595508 T^{7} + 7890481 T^{8} \)
$59$ \( 1 + 9 T + 200 T^{2} + 1458 T^{3} + 16542 T^{4} + 86022 T^{5} + 696200 T^{6} + 1848411 T^{7} + 12117361 T^{8} \)
$61$ \( 1 - 7 T + 193 T^{2} - 1333 T^{3} + 16135 T^{4} - 81313 T^{5} + 718153 T^{6} - 1588867 T^{7} + 13845841 T^{8} \)
$67$ \( 1 - 12 T + 283 T^{2} - 2367 T^{3} + 28956 T^{4} - 158589 T^{5} + 1270387 T^{6} - 3609156 T^{7} + 20151121 T^{8} \)
$71$ \( 1 + 12 T + 167 T^{2} + 591 T^{3} + 7587 T^{4} + 41961 T^{5} + 841847 T^{6} + 4294932 T^{7} + 25411681 T^{8} \)
$73$ \( 1 - 13 T + 232 T^{2} - 1504 T^{3} + 18820 T^{4} - 109792 T^{5} + 1236328 T^{6} - 5057221 T^{7} + 28398241 T^{8} \)
$79$ \( 1 - 22 T + 247 T^{2} - 1909 T^{3} + 16129 T^{4} - 150811 T^{5} + 1541527 T^{6} - 10846858 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 6 T + 149 T^{2} - 1299 T^{3} + 11256 T^{4} - 107817 T^{5} + 1026461 T^{6} - 3430722 T^{7} + 47458321 T^{8} \)
$89$ \( 1 + 14 T + 323 T^{2} + 2963 T^{3} + 41152 T^{4} + 263707 T^{5} + 2558483 T^{6} + 9869566 T^{7} + 62742241 T^{8} \)
$97$ \( 1 - T + 124 T^{2} - 490 T^{3} + 19024 T^{4} - 47530 T^{5} + 1166716 T^{6} - 912673 T^{7} + 88529281 T^{8} \)
show more
show less