Properties

Label 1764.4.a.ba
Level $1764$
Weight $4$
Character orbit 1764.a
Self dual yes
Analytic conductor $104.079$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1764.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(104.079369250\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.136768.1
Defining polynomial: \(x^{4} - 2 x^{3} - 23 x^{2} + 18 x + 119\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 7 \)
Twist minimal: no (minimal twist has level 588)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{5} +O(q^{10})\) \( q -\beta_{1} q^{5} + ( -3 \beta_{1} + \beta_{3} ) q^{11} -3 \beta_{3} q^{13} + ( -12 - \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{17} + ( 48 - 6 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} ) q^{19} + ( -48 - 3 \beta_{1} - 6 \beta_{2} - \beta_{3} ) q^{23} + ( 81 - 6 \beta_{2} + 6 \beta_{3} ) q^{25} + ( -24 + 9 \beta_{2} - \beta_{3} ) q^{29} + ( 12 - 6 \beta_{1} + 9 \beta_{2} - 3 \beta_{3} ) q^{31} + ( 64 - 6 \beta_{1} + 24 \beta_{2} - 6 \beta_{3} ) q^{37} + ( -252 - \beta_{1} + 15 \beta_{2} - 3 \beta_{3} ) q^{41} + ( -28 - 6 \beta_{1} - 12 \beta_{2} - 6 \beta_{3} ) q^{43} + ( -216 - 10 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{47} + ( 162 + 24 \beta_{1} + 18 \beta_{2} + 10 \beta_{3} ) q^{53} + ( 588 - 6 \beta_{1} + 5 \beta_{2} + 21 \beta_{3} ) q^{55} + ( -84 - 10 \beta_{1} - 15 \beta_{2} + 21 \beta_{3} ) q^{59} + ( 240 - 30 \beta_{1} + 14 \beta_{2} - 15 \beta_{3} ) q^{61} + ( 90 + 18 \beta_{1} - 69 \beta_{2} - 9 \beta_{3} ) q^{65} + ( 180 - 6 \beta_{1} - 48 \beta_{2} + 30 \beta_{3} ) q^{67} + ( 336 - 3 \beta_{1} + 30 \beta_{2} - 9 \beta_{3} ) q^{71} + ( 168 - 30 \beta_{1} - 49 \beta_{2} + 18 \beta_{3} ) q^{73} + ( -496 + 36 \beta_{1} + 96 \beta_{2} - 24 \beta_{3} ) q^{79} + ( -780 - 8 \beta_{1} + 18 \beta_{2} + 18 \beta_{3} ) q^{83} + ( 170 + 30 \beta_{1} - 66 \beta_{2} - 24 \beta_{3} ) q^{85} + ( -540 - 11 \beta_{1} - 69 \beta_{2} - 15 \beta_{3} ) q^{89} + ( 936 - 66 \beta_{1} + 48 \beta_{2} + 10 \beta_{3} ) q^{95} + ( 504 - 18 \beta_{1} + 49 \beta_{2} - 48 \beta_{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 48q^{17} + 192q^{19} - 192q^{23} + 324q^{25} - 96q^{29} + 48q^{31} + 256q^{37} - 1008q^{41} - 112q^{43} - 864q^{47} + 648q^{53} + 2352q^{55} - 336q^{59} + 960q^{61} + 360q^{65} + 720q^{67} + 1344q^{71} + 672q^{73} - 1984q^{79} - 3120q^{83} + 680q^{85} - 2160q^{89} + 3744q^{95} + 2016q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 23 x^{2} + 18 x + 119\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - 3 \nu^{2} + 25 \nu + 2 \)\()/9\)
\(\beta_{2}\)\(=\)\((\)\( -7 \nu^{3} + 21 \nu^{2} + 77 \nu - 140 \)\()/9\)
\(\beta_{3}\)\(=\)\( -\nu^{3} + 7 \nu^{2} + 7 \nu - 68 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 7 \beta_{1} + 14\)\()/28\)
\(\nu^{2}\)\(=\)\((\)\(7 \beta_{3} - 8 \beta_{2} + 7 \beta_{1} + 350\)\()/28\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{3} - 7 \beta_{2} + 14 \beta_{1} + 92\)\()/4\)

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
4.93153
2.89590
−2.51732
−3.31012
0 0 0 −19.1403 0 0 0 0 0
1.2 0 0 0 −8.16940 0 0 0 0 0
1.3 0 0 0 10.6550 0 0 0 0 0
1.4 0 0 0 16.6547 0 0 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 1764.4.a.ba 4
3.b odd 2 1 588.4.a.k yes 4
7.b odd 2 1 1764.4.a.bc 4
7.c even 3 2 1764.4.k.bd 8
7.d odd 6 2 1764.4.k.bb 8
12.b even 2 1 2352.4.a.cl 4
21.c even 2 1 588.4.a.j 4
21.g even 6 2 588.4.i.l 8
21.h odd 6 2 588.4.i.k 8
84.h odd 2 1 2352.4.a.cq 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.4.a.j 4 21.c even 2 1
588.4.a.k yes 4 3.b odd 2 1
588.4.i.k 8 21.h odd 6 2
588.4.i.l 8 21.g even 6 2
1764.4.a.ba 4 1.a even 1 1 trivial
1764.4.a.bc 4 7.b odd 2 1
1764.4.k.bb 8 7.d odd 6 2
1764.4.k.bd 8 7.c even 3 2
2352.4.a.cl 4 12.b even 2 1
2352.4.a.cq 4 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1764))\):

\( T_{5}^{4} - 412 T_{5}^{2} + 576 T_{5} + 27748 \)
\( T_{11}^{4} - 4136 T_{11}^{2} - 82944 T_{11} + 733072 \)
\( T_{13}^{4} - 7092 T_{13}^{2} - 31104 T_{13} + 10008036 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 27748 + 576 T - 412 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( 733072 - 82944 T - 4136 T^{2} + T^{4} \)
$13$ \( 10008036 - 31104 T - 7092 T^{2} + T^{4} \)
$17$ \( -4865084 - 590496 T - 9772 T^{2} + 48 T^{3} + T^{4} \)
$19$ \( -41971136 + 1549824 T - 3280 T^{2} - 192 T^{3} + T^{4} \)
$23$ \( -40554608 - 908544 T + 3928 T^{2} + 192 T^{3} + T^{4} \)
$29$ \( 38719552 - 648960 T - 11696 T^{2} + 96 T^{3} + T^{4} \)
$31$ \( -189895104 + 5705856 T - 43632 T^{2} - 48 T^{3} + T^{4} \)
$37$ \( -1479272192 + 41160704 T - 135840 T^{2} - 256 T^{3} + T^{4} \)
$41$ \( 1611829828 + 41616864 T + 334100 T^{2} + 1008 T^{3} + T^{4} \)
$43$ \( -789373952 - 14991872 T - 71040 T^{2} + 112 T^{3} + T^{4} \)
$47$ \( -1168478144 + 13356288 T + 226928 T^{2} + 864 T^{3} + T^{4} \)
$53$ \( -20504773616 + 174760416 T - 208616 T^{2} - 648 T^{3} + T^{4} \)
$59$ \( 18986185792 - 47665536 T - 287152 T^{2} + 336 T^{3} + T^{4} \)
$61$ \( -106656271196 + 499716480 T - 330196 T^{2} - 960 T^{3} + T^{4} \)
$67$ \( 14336621568 + 330683904 T - 669312 T^{2} - 720 T^{3} + T^{4} \)
$71$ \( -17989567344 + 15945984 T + 460440 T^{2} - 1344 T^{3} + T^{4} \)
$73$ \( -90986816444 + 459082560 T - 467428 T^{2} - 672 T^{3} + T^{4} \)
$79$ \( -1013049875456 - 2268354560 T - 246720 T^{2} + 1984 T^{3} + T^{4} \)
$83$ \( 74256064768 + 1203548928 T + 3243872 T^{2} + 3120 T^{3} + T^{4} \)
$89$ \( -311467391228 - 994684320 T + 523124 T^{2} + 2160 T^{3} + T^{4} \)
$97$ \( -580580611196 + 2679572160 T - 752260 T^{2} - 2016 T^{3} + T^{4} \)
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