Properties

Label 441.4.a.s
Level $441$
Weight $4$
Character orbit 441.a
Self dual yes
Analytic conductor $26.020$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.0198423125\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.57516.1
Defining polynomial: \(x^{3} - x^{2} - 24 x + 6\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 8 + \beta_{1} + \beta_{2} ) q^{4} + ( -4 + \beta_{1} - \beta_{2} ) q^{5} + ( -10 - 9 \beta_{1} - \beta_{2} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 8 + \beta_{1} + \beta_{2} ) q^{4} + ( -4 + \beta_{1} - \beta_{2} ) q^{5} + ( -10 - 9 \beta_{1} - \beta_{2} ) q^{8} + ( -22 + 11 \beta_{1} - \beta_{2} ) q^{10} + ( -12 + \beta_{1} + 3 \beta_{2} ) q^{11} + ( 19 + 5 \beta_{1} - \beta_{2} ) q^{13} + ( 74 + 19 \beta_{1} + \beta_{2} ) q^{16} + ( -16 - 4 \beta_{2} ) q^{17} + ( -65 - 7 \beta_{1} - \beta_{2} ) q^{19} + ( -150 + 11 \beta_{1} - 3 \beta_{2} ) q^{20} + ( 2 - 13 \beta_{1} - \beta_{2} ) q^{22} + ( -80 + 24 \beta_{1} + 4 \beta_{2} ) q^{23} + ( 53 - 29 \beta_{1} + \beta_{2} ) q^{25} + ( -86 - 16 \beta_{1} - 5 \beta_{2} ) q^{26} + ( -26 + 25 \beta_{1} - 5 \beta_{2} ) q^{29} + ( -39 + 22 \beta_{1} - 2 \beta_{2} ) q^{31} + ( -218 - 29 \beta_{1} - 11 \beta_{2} ) q^{32} + ( -24 + 48 \beta_{1} ) q^{34} + ( 81 + 19 \beta_{1} + \beta_{2} ) q^{37} + ( 106 + 80 \beta_{1} + 7 \beta_{2} ) q^{38} + ( -18 + 75 \beta_{1} - 3 \beta_{2} ) q^{40} + ( -82 + 2 \beta_{1} + 14 \beta_{2} ) q^{41} + ( 143 - 69 \beta_{1} - 3 \beta_{2} ) q^{43} + ( 298 + 11 \beta_{1} - 11 \beta_{2} ) q^{44} + ( -360 + 24 \beta_{1} - 24 \beta_{2} ) q^{46} + ( 46 + 72 \beta_{1} + 28 \beta_{2} ) q^{47} + ( 470 - 32 \beta_{1} + 29 \beta_{2} ) q^{50} + ( 74 + 102 \beta_{1} + 24 \beta_{2} ) q^{52} + ( -154 + 69 \beta_{1} + 11 \beta_{2} ) q^{53} + ( -350 + 19 \beta_{1} + 25 \beta_{2} ) q^{55} + ( -430 + 41 \beta_{1} - 25 \beta_{2} ) q^{58} + ( -358 - 69 \beta_{1} + 29 \beta_{2} ) q^{59} + ( 10 - 100 \beta_{1} + 20 \beta_{2} ) q^{61} + ( -364 + 33 \beta_{1} - 22 \beta_{2} ) q^{62} + ( -194 + 183 \beta_{1} + 21 \beta_{2} ) q^{64} + ( 174 - 50 \beta_{1} - 18 \beta_{2} ) q^{65} + ( -215 + 17 \beta_{1} + 47 \beta_{2} ) q^{67} + ( -640 - 24 \beta_{1} - 16 \beta_{2} ) q^{68} + ( -66 - 120 \beta_{1} - 12 \beta_{2} ) q^{71} + ( 363 - 101 \beta_{1} - 23 \beta_{2} ) q^{73} + ( -298 - 108 \beta_{1} - 19 \beta_{2} ) q^{74} + ( -718 - 186 \beta_{1} - 72 \beta_{2} ) q^{76} + ( 299 - 36 \beta_{1} + 48 \beta_{2} ) q^{79} + ( -18 - 121 \beta_{1} - 51 \beta_{2} ) q^{80} + ( 52 - 32 \beta_{1} - 2 \beta_{2} ) q^{82} + ( -156 - 51 \beta_{1} + 27 \beta_{2} ) q^{83} + ( 624 - 72 \beta_{1} ) q^{85} + ( 1086 - 50 \beta_{1} + 69 \beta_{2} ) q^{86} + ( -258 - 117 \beta_{1} - 3 \beta_{2} ) q^{88} + ( -532 - 170 \beta_{1} - 22 \beta_{2} ) q^{89} + ( 112 + 336 \beta_{1} - 56 \beta_{2} ) q^{92} + ( -984 - 342 \beta_{1} - 72 \beta_{2} ) q^{94} + ( 246 - 2 \beta_{1} + 54 \beta_{2} ) q^{95} + ( 24 - 53 \beta_{1} + 49 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{2} + 25q^{4} - 11q^{5} - 39q^{8} + O(q^{10}) \) \( 3q - q^{2} + 25q^{4} - 11q^{5} - 39q^{8} - 55q^{10} - 35q^{11} + 62q^{13} + 241q^{16} - 48q^{17} - 202q^{19} - 439q^{20} - 7q^{22} - 216q^{23} + 130q^{25} - 274q^{26} - 53q^{29} - 95q^{31} - 683q^{32} - 24q^{34} + 262q^{37} + 398q^{38} + 21q^{40} - 244q^{41} + 360q^{43} + 905q^{44} - 1056q^{46} + 210q^{47} + 1378q^{50} + 324q^{52} - 393q^{53} - 1031q^{55} - 1249q^{58} - 1143q^{59} - 70q^{61} - 1059q^{62} - 399q^{64} + 472q^{65} - 628q^{67} - 1944q^{68} - 318q^{71} + 988q^{73} - 1002q^{74} - 2340q^{76} + 861q^{79} - 175q^{80} + 124q^{82} - 519q^{83} + 1800q^{85} + 3208q^{86} - 891q^{88} - 1766q^{89} + 672q^{92} - 3294q^{94} + 736q^{95} + 19q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 24 x + 6\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 16 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 16\)

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
5.30829
0.248072
−4.55637
−5.30829 0 20.1780 −5.56140 0 0 −64.6443 0 29.5215
1.2 −0.248072 0 −7.93846 12.4346 0 0 3.95388 0 −3.08468
1.3 4.55637 0 12.7605 −17.8732 0 0 21.6905 0 −81.4369
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 441.4.a.s 3
3.b odd 2 1 147.4.a.l 3
7.b odd 2 1 441.4.a.t 3
7.c even 3 2 63.4.e.c 6
7.d odd 6 2 441.4.e.w 6
12.b even 2 1 2352.4.a.ci 3
21.c even 2 1 147.4.a.m 3
21.g even 6 2 147.4.e.n 6
21.h odd 6 2 21.4.e.b 6
84.h odd 2 1 2352.4.a.cg 3
84.n even 6 2 336.4.q.k 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.e.b 6 21.h odd 6 2
63.4.e.c 6 7.c even 3 2
147.4.a.l 3 3.b odd 2 1
147.4.a.m 3 21.c even 2 1
147.4.e.n 6 21.g even 6 2
336.4.q.k 6 84.n even 6 2
441.4.a.s 3 1.a even 1 1 trivial
441.4.a.t 3 7.b odd 2 1
441.4.e.w 6 7.d odd 6 2
2352.4.a.cg 3 84.h odd 2 1
2352.4.a.ci 3 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2}^{3} + T_{2}^{2} - 24 T_{2} - 6 \)
\( T_{5}^{3} + 11 T_{5}^{2} - 192 T_{5} - 1236 \)
\( T_{13}^{3} - 62 T_{13}^{2} + 425 T_{13} + 18452 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -6 - 24 T + T^{2} + T^{3} \)
$3$ \( T^{3} \)
$5$ \( -1236 - 192 T + 11 T^{2} + T^{3} \)
$7$ \( T^{3} \)
$11$ \( 9564 - 1368 T + 35 T^{2} + T^{3} \)
$13$ \( 18452 + 425 T - 62 T^{2} + T^{3} \)
$17$ \( -112896 - 2400 T + 48 T^{2} + T^{3} \)
$19$ \( 233804 + 12281 T + 202 T^{2} + T^{3} \)
$23$ \( -1580544 - 672 T + 216 T^{2} + T^{3} \)
$29$ \( 824976 - 20472 T + 53 T^{2} + T^{3} \)
$31$ \( -11823 - 10001 T + 95 T^{2} + T^{3} \)
$37$ \( -49152 + 14089 T - 262 T^{2} + T^{3} \)
$41$ \( 300384 - 18780 T + 244 T^{2} + T^{3} \)
$43$ \( 18269746 - 72363 T - 360 T^{2} + T^{3} \)
$47$ \( -5119128 - 246516 T - 210 T^{2} + T^{3} \)
$53$ \( -33169392 - 80736 T + 393 T^{2} + T^{3} \)
$59$ \( -100468944 + 133104 T + 1143 T^{2} + T^{3} \)
$61$ \( -84631000 - 340900 T + 70 T^{2} + T^{3} \)
$67$ \( 27993002 - 304963 T + 628 T^{2} + T^{3} \)
$71$ \( 28535976 - 330804 T + 318 T^{2} + T^{3} \)
$73$ \( 143207118 - 4355 T - 988 T^{2} + T^{3} \)
$79$ \( 193956337 - 257901 T - 861 T^{2} + T^{3} \)
$83$ \( -47916036 - 131616 T + 519 T^{2} + T^{3} \)
$89$ \( -13004544 + 277920 T + 1766 T^{2} + T^{3} \)
$97$ \( 44776452 - 569600 T - 19 T^{2} + T^{3} \)
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