Properties

Label 1323.4.a.ba
Level $1323$
Weight $4$
Character orbit 1323.a
Self dual yes
Analytic conductor $78.060$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1323.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(78.0595269376\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{5}, \sqrt{13})\)
Defining polynomial: \(x^{4} - 9 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 189)
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.

\(f(q)\) \(=\) \( q + ( -\beta_{1} - \beta_{2} ) q^{2} + ( 6 - \beta_{3} ) q^{4} + ( 4 \beta_{1} + 3 \beta_{2} ) q^{5} + ( -5 \beta_{1} - 13 \beta_{2} ) q^{8} +O(q^{10})\) \( q + ( -\beta_{1} - \beta_{2} ) q^{2} + ( 6 - \beta_{3} ) q^{4} + ( 4 \beta_{1} + 3 \beta_{2} ) q^{5} + ( -5 \beta_{1} - 13 \beta_{2} ) q^{8} + ( -50 + 3 \beta_{3} ) q^{10} + ( -6 \beta_{1} + 11 \beta_{2} ) q^{11} + ( 14 - 6 \beta_{3} ) q^{13} + ( 70 - 5 \beta_{3} ) q^{16} + ( 4 \beta_{1} + 2 \beta_{2} ) q^{17} + ( -45 - 4 \beta_{3} ) q^{19} + ( 39 \beta_{1} + 71 \beta_{2} ) q^{20} + ( -18 + 11 \beta_{3} ) q^{22} + ( -8 \beta_{1} - 43 \beta_{2} ) q^{23} + ( 64 - 8 \beta_{3} ) q^{25} + ( -56 \beta_{1} - 104 \beta_{2} ) q^{26} + ( 22 \beta_{1} + 28 \beta_{2} ) q^{29} + ( -107 - 10 \beta_{3} ) q^{31} + ( -65 \beta_{1} - 41 \beta_{2} ) q^{32} + ( -44 + 2 \beta_{3} ) q^{34} + ( 327 - 6 \beta_{3} ) q^{37} + ( 17 \beta_{1} - 15 \beta_{2} ) q^{38} + ( -338 + 47 \beta_{3} ) q^{40} + ( -22 \beta_{1} - 111 \beta_{2} ) q^{41} + ( 28 - 2 \beta_{3} ) q^{43} + ( 143 \beta_{1} + 95 \beta_{2} ) q^{44} + ( 322 - 43 \beta_{3} ) q^{46} + ( -114 \beta_{1} + 6 \beta_{2} ) q^{47} + ( -120 \beta_{1} - 184 \beta_{2} ) q^{50} + ( 960 - 56 \beta_{3} ) q^{52} + ( 148 \beta_{1} + 20 \beta_{2} ) q^{53} + ( -113 - 50 \beta_{3} ) q^{55} + ( -344 + 28 \beta_{3} ) q^{58} + ( 94 \beta_{1} + 74 \beta_{2} ) q^{59} + ( 672 + 8 \beta_{3} ) q^{61} + ( 37 \beta_{1} - 43 \beta_{2} ) q^{62} + ( 206 - \beta_{3} ) q^{64} + ( 146 \beta_{1} + 360 \beta_{2} ) q^{65} + ( 342 - 38 \beta_{3} ) q^{67} + ( 26 \beta_{1} + 58 \beta_{2} ) q^{68} + ( -136 \beta_{1} + 61 \beta_{2} ) q^{71} + ( 104 + 10 \beta_{3} ) q^{73} + ( -369 \beta_{1} - 417 \beta_{2} ) q^{74} + ( 314 + 17 \beta_{3} ) q^{76} + ( 292 - 26 \beta_{3} ) q^{79} + ( 355 \beta_{1} + 475 \beta_{2} ) q^{80} + ( 842 - 111 \beta_{3} ) q^{82} + ( -158 \beta_{1} - 4 \beta_{2} ) q^{83} + ( 178 - 4 \beta_{3} ) q^{85} + ( -42 \beta_{1} - 58 \beta_{2} ) q^{86} + ( -1570 + 7 \beta_{3} ) q^{88} + ( -110 \beta_{1} - 87 \beta_{2} ) q^{89} + ( -559 \beta_{1} - 623 \beta_{2} ) q^{92} + ( 876 + 6 \beta_{3} ) q^{94} + ( -120 \beta_{1} + 77 \beta_{2} ) q^{95} + ( 152 - 8 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 26q^{4} + O(q^{10}) \) \( 4q + 26q^{4} - 206q^{10} + 68q^{13} + 290q^{16} - 172q^{19} - 94q^{22} + 272q^{25} - 408q^{31} - 180q^{34} + 1320q^{37} - 1446q^{40} + 116q^{43} + 1374q^{46} + 3952q^{52} - 352q^{55} - 1432q^{58} + 2672q^{61} + 826q^{64} + 1444q^{67} + 396q^{73} + 1222q^{76} + 1220q^{79} + 3590q^{82} + 720q^{85} - 6294q^{88} + 3492q^{94} + 624q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 9 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - 5 \nu \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 11 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\( 3 \nu^{2} - 14 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} + \beta_{1}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 14\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-5 \beta_{2} + 11 \beta_{1}\)\()/3\)

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
−0.684742
2.92081
−2.92081
0.684742
−5.15688 0 18.5934 17.0220 0 0 −54.6288 0 −87.7802
1.2 −1.55133 0 −5.59339 9.81086 0 0 21.0878 0 −15.2198
1.3 1.55133 0 −5.59339 −9.81086 0 0 −21.0878 0 −15.2198
1.4 5.15688 0 18.5934 −17.0220 0 0 54.6288 0 −87.7802
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.4.a.ba 4
3.b odd 2 1 inner 1323.4.a.ba 4
7.b odd 2 1 189.4.a.l 4
21.c even 2 1 189.4.a.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.4.a.l 4 7.b odd 2 1
189.4.a.l 4 21.c even 2 1
1323.4.a.ba 4 1.a even 1 1 trivial
1323.4.a.ba 4 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1323))\):

\( T_{2}^{4} - 29 T_{2}^{2} + 64 \)
\( T_{5}^{4} - 386 T_{5}^{2} + 27889 \)
\( T_{13}^{2} - 34 T_{13} - 4976 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 64 - 29 T^{2} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( 27889 - 386 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 4592449 - 5906 T^{2} + T^{4} \)
$13$ \( ( -4976 - 34 T + T^{2} )^{2} \)
$17$ \( ( -180 + T^{2} )^{2} \)
$19$ \( ( -491 + 86 T + T^{2} )^{2} \)
$23$ \( 363016809 - 40986 T^{2} + T^{4} \)
$29$ \( 2849344 - 18404 T^{2} + T^{4} \)
$31$ \( ( -4221 + 204 T + T^{2} )^{2} \)
$37$ \( ( 103635 - 660 T + T^{2} )^{2} \)
$41$ \( 15513948025 - 270890 T^{2} + T^{4} \)
$43$ \( ( 256 - 58 T + T^{2} )^{2} \)
$47$ \( 8950673664 - 395604 T^{2} + T^{4} \)
$53$ \( 43477254144 - 568656 T^{2} + T^{4} \)
$59$ \( 8087045184 - 217764 T^{2} + T^{4} \)
$61$ \( ( 436864 - 1336 T + T^{2} )^{2} \)
$67$ \( ( -80864 - 722 T + T^{2} )^{2} \)
$71$ \( 68112009 - 848826 T^{2} + T^{4} \)
$73$ \( ( -4824 - 198 T + T^{2} )^{2} \)
$79$ \( ( -5840 - 610 T + T^{2} )^{2} \)
$83$ \( 43147598400 - 707940 T^{2} + T^{4} \)
$89$ \( 15083032969 - 298874 T^{2} + T^{4} \)
$97$ \( ( 14976 - 312 T + T^{2} )^{2} \)
show more
show less