Properties

Label 441.4.a.r
Level $441$
Weight $4$
Character orbit 441.a
Self dual yes
Analytic conductor $26.020$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{2} + ( 7 + 3 \beta ) q^{4} + ( 2 + 2 \beta ) q^{5} + ( 41 + 5 \beta ) q^{8} +O(q^{10})\) \( q + ( 1 + \beta ) q^{2} + ( 7 + 3 \beta ) q^{4} + ( 2 + 2 \beta ) q^{5} + ( 41 + 5 \beta ) q^{8} + ( 30 + 6 \beta ) q^{10} + ( 8 - 10 \beta ) q^{11} + ( -14 + 12 \beta ) q^{13} + ( 55 + 27 \beta ) q^{16} + ( -2 - 2 \beta ) q^{17} + ( -44 + 24 \beta ) q^{19} + ( 98 + 26 \beta ) q^{20} + ( -132 - 12 \beta ) q^{22} + ( -20 + 34 \beta ) q^{23} + ( -65 + 12 \beta ) q^{25} + ( 154 + 10 \beta ) q^{26} + ( 138 - 24 \beta ) q^{29} + ( 16 - 72 \beta ) q^{31} + ( 105 + 69 \beta ) q^{32} + ( -30 - 6 \beta ) q^{34} + ( -106 - 36 \beta ) q^{37} + ( 292 + 4 \beta ) q^{38} + ( 222 + 102 \beta ) q^{40} + ( -210 - 30 \beta ) q^{41} + ( 212 - 48 \beta ) q^{43} + ( -364 - 76 \beta ) q^{44} + ( 456 + 48 \beta ) q^{46} + ( -40 + 68 \beta ) q^{47} + ( 103 - 41 \beta ) q^{50} + ( 406 + 78 \beta ) q^{52} + ( 554 - 4 \beta ) q^{53} + ( -264 - 24 \beta ) q^{55} + ( -198 + 90 \beta ) q^{58} + ( 460 - 116 \beta ) q^{59} + ( 250 - 72 \beta ) q^{61} + ( -992 - 128 \beta ) q^{62} + ( 631 + 27 \beta ) q^{64} + ( 308 + 20 \beta ) q^{65} + ( 20 + 108 \beta ) q^{67} + ( -98 - 26 \beta ) q^{68} + ( -492 + 30 \beta ) q^{71} + ( -530 - 12 \beta ) q^{73} + ( -610 - 178 \beta ) q^{74} + ( 700 + 108 \beta ) q^{76} + ( -232 - 108 \beta ) q^{79} + ( 866 + 218 \beta ) q^{80} + ( -630 - 270 \beta ) q^{82} + ( 924 + 96 \beta ) q^{83} + ( -60 - 12 \beta ) q^{85} + ( -460 + 116 \beta ) q^{86} + ( -372 - 420 \beta ) q^{88} + ( 254 - 142 \beta ) q^{89} + ( 1288 + 280 \beta ) q^{92} + ( 912 + 96 \beta ) q^{94} + ( 584 + 8 \beta ) q^{95} + ( -266 - 276 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{2} + 17q^{4} + 6q^{5} + 87q^{8} + O(q^{10}) \) \( 2q + 3q^{2} + 17q^{4} + 6q^{5} + 87q^{8} + 66q^{10} + 6q^{11} - 16q^{13} + 137q^{16} - 6q^{17} - 64q^{19} + 222q^{20} - 276q^{22} - 6q^{23} - 118q^{25} + 318q^{26} + 252q^{29} - 40q^{31} + 279q^{32} - 66q^{34} - 248q^{37} + 588q^{38} + 546q^{40} - 450q^{41} + 376q^{43} - 804q^{44} + 960q^{46} - 12q^{47} + 165q^{50} + 890q^{52} + 1104q^{53} - 552q^{55} - 306q^{58} + 804q^{59} + 428q^{61} - 2112q^{62} + 1289q^{64} + 636q^{65} + 148q^{67} - 222q^{68} - 954q^{71} - 1072q^{73} - 1398q^{74} + 1508q^{76} - 572q^{79} + 1950q^{80} - 1530q^{82} + 1944q^{83} - 132q^{85} - 804q^{86} - 1164q^{88} + 366q^{89} + 2856q^{92} + 1920q^{94} + 1176q^{95} - 808q^{97} + O(q^{100}) \)

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
−3.27492
4.27492
−2.27492 0 −2.82475 −4.54983 0 0 24.6254 0 10.3505
1.2 5.27492 0 19.8248 10.5498 0 0 62.3746 0 55.6495
\(n\): 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.r 2
3.b odd 2 1 147.4.a.i 2
7.b odd 2 1 63.4.a.e 2
7.c even 3 2 441.4.e.p 4
7.d odd 6 2 441.4.e.q 4
12.b even 2 1 2352.4.a.bz 2
21.c even 2 1 21.4.a.c 2
21.g even 6 2 147.4.e.l 4
21.h odd 6 2 147.4.e.m 4
28.d even 2 1 1008.4.a.ba 2
35.c odd 2 1 1575.4.a.p 2
84.h odd 2 1 336.4.a.m 2
105.g even 2 1 525.4.a.n 2
105.k odd 4 2 525.4.d.g 4
168.e odd 2 1 1344.4.a.bo 2
168.i even 2 1 1344.4.a.bg 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.c 2 21.c even 2 1
63.4.a.e 2 7.b odd 2 1
147.4.a.i 2 3.b odd 2 1
147.4.e.l 4 21.g even 6 2
147.4.e.m 4 21.h odd 6 2
336.4.a.m 2 84.h odd 2 1
441.4.a.r 2 1.a even 1 1 trivial
441.4.e.p 4 7.c even 3 2
441.4.e.q 4 7.d odd 6 2
525.4.a.n 2 105.g even 2 1
525.4.d.g 4 105.k odd 4 2
1008.4.a.ba 2 28.d even 2 1
1344.4.a.bg 2 168.i even 2 1
1344.4.a.bo 2 168.e odd 2 1
1575.4.a.p 2 35.c odd 2 1
2352.4.a.bz 2 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2}^{2} - 3 T_{2} - 12 \)
\( T_{5}^{2} - 6 T_{5} - 48 \)
\( T_{13}^{2} + 16 T_{13} - 1988 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -12 - 3 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( -48 - 6 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( -1416 - 6 T + T^{2} \)
$13$ \( -1988 + 16 T + T^{2} \)
$17$ \( -48 + 6 T + T^{2} \)
$19$ \( -7184 + 64 T + T^{2} \)
$23$ \( -16464 + 6 T + T^{2} \)
$29$ \( 7668 - 252 T + T^{2} \)
$31$ \( -73472 + 40 T + T^{2} \)
$37$ \( -3092 + 248 T + T^{2} \)
$41$ \( 37800 + 450 T + T^{2} \)
$43$ \( 2512 - 376 T + T^{2} \)
$47$ \( -65856 + 12 T + T^{2} \)
$53$ \( 304476 - 1104 T + T^{2} \)
$59$ \( -30144 - 804 T + T^{2} \)
$61$ \( -28076 - 428 T + T^{2} \)
$67$ \( -160736 - 148 T + T^{2} \)
$71$ \( 214704 + 954 T + T^{2} \)
$73$ \( 285244 + 1072 T + T^{2} \)
$79$ \( -84416 + 572 T + T^{2} \)
$83$ \( 813456 - 1944 T + T^{2} \)
$89$ \( -253848 - 366 T + T^{2} \)
$97$ \( -922292 + 808 T + T^{2} \)
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