Properties

Label 9072.2.a.cj
Level 9072
Weight 2
Character orbit 9072.a
Self dual yes
Analytic conductor 72.440
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 9072 = 2^{4} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9072.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.4402847137\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.45729.1
Defining polynomial: \(x^{4} - x^{3} - 11 x^{2} + 12 x - 3\)
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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{5} + q^{7} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{5} + q^{7} + ( 2 - \beta_{1} ) q^{11} + ( -1 - \beta_{2} ) q^{13} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{17} + ( 1 - \beta_{1} + \beta_{3} ) q^{19} + ( -\beta_{2} + \beta_{3} ) q^{23} + ( 2 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{25} + ( 3 + 2 \beta_{2} - \beta_{3} ) q^{29} + ( 1 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{31} + ( 1 - \beta_{1} ) q^{35} + ( 1 - \beta_{1} ) q^{37} + ( -2 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{41} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{43} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{47} + q^{49} + ( 4 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{53} + ( 8 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{55} + ( 3 - \beta_{2} - 3 \beta_{3} ) q^{59} + ( -6 - 3 \beta_{2} + \beta_{3} ) q^{61} + ( -1 + 3 \beta_{1} ) q^{65} + ( 4 - \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{67} + ( 1 - \beta_{3} ) q^{71} + ( -5 - \beta_{1} - 3 \beta_{3} ) q^{73} + ( 2 - \beta_{1} ) q^{77} + ( 5 + \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{79} + ( 2 + \beta_{1} - \beta_{3} ) q^{83} + ( -3 + 2 \beta_{1} - \beta_{2} ) q^{85} + ( 3 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{89} + ( -1 - \beta_{2} ) q^{91} + ( 10 - 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{95} + ( -3 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 3q^{5} + 4q^{7} + O(q^{10}) \) \( 4q + 3q^{5} + 4q^{7} + 7q^{11} - 3q^{13} + 3q^{17} + 4q^{19} + 2q^{23} + 5q^{25} + 9q^{29} + 3q^{31} + 3q^{35} + 3q^{37} - 9q^{41} + 8q^{43} + 3q^{47} + 4q^{49} + 6q^{53} + 28q^{55} + 10q^{59} - 20q^{61} - q^{65} + 11q^{67} + 3q^{71} - 24q^{73} + 7q^{77} + 21q^{79} + 8q^{83} - 9q^{85} + 6q^{89} - 3q^{91} + 36q^{95} - 16q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 11 x^{2} + 12 x - 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - 11 \nu + 3 \)
\(\beta_{3}\)\(=\)\( -2 \nu^{3} + \nu^{2} + 22 \nu - 12 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 2 \beta_{2} + 6\)
\(\nu^{3}\)\(=\)\(\beta_{2} + 11 \beta_{1} - 3\)

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
3.31050
0.670984
0.399463
−3.38095
0 0 0 −2.31050 0 1.00000 0 0 0
1.2 0 0 0 0.329016 0 1.00000 0 0 0
1.3 0 0 0 0.600537 0 1.00000 0 0 0
1.4 0 0 0 4.38095 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 9072.2.a.cj 4
3.b odd 2 1 9072.2.a.cg 4
4.b odd 2 1 4536.2.a.z 4
9.c even 3 2 1008.2.r.l 8
9.d odd 6 2 3024.2.r.m 8
12.b even 2 1 4536.2.a.y 4
36.f odd 6 2 504.2.r.e 8
36.h even 6 2 1512.2.r.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.r.e 8 36.f odd 6 2
1008.2.r.l 8 9.c even 3 2
1512.2.r.e 8 36.h even 6 2
3024.2.r.m 8 9.d odd 6 2
4536.2.a.y 4 12.b even 2 1
4536.2.a.z 4 4.b odd 2 1
9072.2.a.cg 4 3.b odd 2 1
9072.2.a.cj 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9072))\):

\( T_{5}^{4} - 3 T_{5}^{3} - 8 T_{5}^{2} + 9 T_{5} - 2 \)
\( T_{11}^{4} - 7 T_{11}^{3} + 7 T_{11}^{2} + 12 T_{11} - 15 \)
\( T_{13}^{4} + 3 T_{13}^{3} - 11 T_{13}^{2} - 27 T_{13} + 10 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 3 T + 12 T^{2} - 36 T^{3} + 68 T^{4} - 180 T^{5} + 300 T^{6} - 375 T^{7} + 625 T^{8} \)
$7$ \( ( 1 - T )^{4} \)
$11$ \( 1 - 7 T + 51 T^{2} - 219 T^{3} + 865 T^{4} - 2409 T^{5} + 6171 T^{6} - 9317 T^{7} + 14641 T^{8} \)
$13$ \( 1 + 3 T + 41 T^{2} + 90 T^{3} + 738 T^{4} + 1170 T^{5} + 6929 T^{6} + 6591 T^{7} + 28561 T^{8} \)
$17$ \( 1 - 3 T + 41 T^{2} - 189 T^{3} + 807 T^{4} - 3213 T^{5} + 11849 T^{6} - 14739 T^{7} + 83521 T^{8} \)
$19$ \( 1 - 4 T + 49 T^{2} - 193 T^{3} + 1297 T^{4} - 3667 T^{5} + 17689 T^{6} - 27436 T^{7} + 130321 T^{8} \)
$23$ \( 1 - 2 T + 63 T^{2} - 159 T^{3} + 1876 T^{4} - 3657 T^{5} + 33327 T^{6} - 24334 T^{7} + 279841 T^{8} \)
$29$ \( 1 - 9 T + 68 T^{2} - 270 T^{3} + 1290 T^{4} - 7830 T^{5} + 57188 T^{6} - 219501 T^{7} + 707281 T^{8} \)
$31$ \( 1 - 3 T + 50 T^{2} - 252 T^{3} + 1872 T^{4} - 7812 T^{5} + 48050 T^{6} - 89373 T^{7} + 923521 T^{8} \)
$37$ \( 1 - 3 T + 140 T^{2} - 324 T^{3} + 7620 T^{4} - 11988 T^{5} + 191660 T^{6} - 151959 T^{7} + 1874161 T^{8} \)
$41$ \( 1 + 9 T + 117 T^{2} + 717 T^{3} + 6341 T^{4} + 29397 T^{5} + 196677 T^{6} + 620289 T^{7} + 2825761 T^{8} \)
$43$ \( 1 - 8 T + 137 T^{2} - 1005 T^{3} + 8087 T^{4} - 43215 T^{5} + 253313 T^{6} - 636056 T^{7} + 3418801 T^{8} \)
$47$ \( 1 - 3 T + 150 T^{2} - 468 T^{3} + 9674 T^{4} - 21996 T^{5} + 331350 T^{6} - 311469 T^{7} + 4879681 T^{8} \)
$53$ \( 1 - 6 T + 59 T^{2} - 621 T^{3} + 4704 T^{4} - 32913 T^{5} + 165731 T^{6} - 893262 T^{7} + 7890481 T^{8} \)
$59$ \( 1 - 10 T + 171 T^{2} - 1623 T^{3} + 13357 T^{4} - 95757 T^{5} + 595251 T^{6} - 2053790 T^{7} + 12117361 T^{8} \)
$61$ \( 1 + 20 T + 239 T^{2} + 2109 T^{3} + 15872 T^{4} + 128649 T^{5} + 889319 T^{6} + 4539620 T^{7} + 13845841 T^{8} \)
$67$ \( 1 - 11 T + 203 T^{2} - 1461 T^{3} + 17099 T^{4} - 97887 T^{5} + 911267 T^{6} - 3308393 T^{7} + 20151121 T^{8} \)
$71$ \( 1 - 3 T + 276 T^{2} - 630 T^{3} + 29126 T^{4} - 44730 T^{5} + 1391316 T^{6} - 1073733 T^{7} + 25411681 T^{8} \)
$73$ \( 1 + 24 T + 425 T^{2} + 4923 T^{3} + 48495 T^{4} + 359379 T^{5} + 2264825 T^{6} + 9336408 T^{7} + 28398241 T^{8} \)
$79$ \( 1 - 21 T + 308 T^{2} - 3360 T^{3} + 34200 T^{4} - 265440 T^{5} + 1922228 T^{6} - 10353819 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 8 T + 323 T^{2} - 1865 T^{3} + 39832 T^{4} - 154795 T^{5} + 2225147 T^{6} - 4574296 T^{7} + 47458321 T^{8} \)
$89$ \( 1 - 6 T + 263 T^{2} - 1377 T^{3} + 32646 T^{4} - 122553 T^{5} + 2083223 T^{6} - 4229814 T^{7} + 62742241 T^{8} \)
$97$ \( 1 + 16 T + 431 T^{2} + 4491 T^{3} + 64415 T^{4} + 435627 T^{5} + 4055279 T^{6} + 14602768 T^{7} + 88529281 T^{8} \)
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