Properties

Label 441.6.a.l
Level 441
Weight 6
Character orbit 441.a
Self dual yes
Analytic conductor 70.729
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(70.7292645375\)
Analytic rank: \(1\)
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 7)
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.

\(f(q)\) \(=\) \( q + ( -4 - \beta ) q^{2} + ( -2 + 9 \beta ) q^{4} + ( -14 + 10 \beta ) q^{5} + ( 10 - 11 \beta ) q^{8} +O(q^{10})\) \( q + ( -4 - \beta ) q^{2} + ( -2 + 9 \beta ) q^{4} + ( -14 + 10 \beta ) q^{5} + ( 10 - 11 \beta ) q^{8} + ( -84 - 36 \beta ) q^{10} + ( -260 + 124 \beta ) q^{11} + ( 238 - 126 \beta ) q^{13} + ( 178 - 243 \beta ) q^{16} + ( 938 - 76 \beta ) q^{17} + ( 1624 + 18 \beta ) q^{19} + ( 1288 - 56 \beta ) q^{20} + ( -696 - 360 \beta ) q^{22} + ( -760 - 568 \beta ) q^{23} + ( -1529 - 180 \beta ) q^{25} + ( 812 + 392 \beta ) q^{26} + ( -3222 - 252 \beta ) q^{29} + ( 280 - 540 \beta ) q^{31} + ( 2370 + 1389 \beta ) q^{32} + ( -2688 - 558 \beta ) q^{34} + ( 2846 + 540 \beta ) q^{37} + ( -6748 - 1714 \beta ) q^{38} + ( -1680 + 144 \beta ) q^{40} + ( -2478 - 1092 \beta ) q^{41} + ( 884 - 4788 \beta ) q^{43} + ( 16144 - 1472 \beta ) q^{44} + ( 10992 + 3600 \beta ) q^{46} + ( 3976 + 3748 \beta ) q^{47} + ( 8636 + 2429 \beta ) q^{50} + ( -16352 + 1260 \beta ) q^{52} + ( -4838 + 208 \beta ) q^{53} + ( 21000 - 3096 \beta ) q^{55} + ( 16416 + 4482 \beta ) q^{58} + ( -20944 - 2050 \beta ) q^{59} + ( 29974 + 4806 \beta ) q^{61} + ( 6440 + 2420 \beta ) q^{62} + ( -34622 - 1539 \beta ) q^{64} + ( -20972 + 2884 \beta ) q^{65} + ( 13364 - 1944 \beta ) q^{67} + ( -11452 + 7910 \beta ) q^{68} + ( -50808 + 4200 \beta ) q^{71} + ( -11354 + 5256 \beta ) q^{73} + ( -18944 - 5546 \beta ) q^{74} + ( -980 + 14742 \beta ) q^{76} + ( 18176 + 14904 \beta ) q^{79} + ( -36512 + 2752 \beta ) q^{80} + ( 25200 + 7938 \beta ) q^{82} + ( 50904 + 15750 \beta ) q^{83} + ( -23772 + 9684 \beta ) q^{85} + ( 63496 + 23056 \beta ) q^{86} + ( -21696 + 2736 \beta ) q^{88} + ( 53242 - 22208 \beta ) q^{89} + ( -70048 - 10816 \beta ) q^{92} + ( -68376 - 22716 \beta ) q^{94} + ( -20216 + 16168 \beta ) q^{95} + ( -5978 - 8820 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 9q^{2} + 5q^{4} - 18q^{5} + 9q^{8} + O(q^{10}) \) \( 2q - 9q^{2} + 5q^{4} - 18q^{5} + 9q^{8} - 204q^{10} - 396q^{11} + 350q^{13} + 113q^{16} + 1800q^{17} + 3266q^{19} + 2520q^{20} - 1752q^{22} - 2088q^{23} - 3238q^{25} + 2016q^{26} - 6696q^{29} + 20q^{31} + 6129q^{32} - 5934q^{34} + 6232q^{37} - 15210q^{38} - 3216q^{40} - 6048q^{41} - 3020q^{43} + 30816q^{44} + 25584q^{46} + 11700q^{47} + 19701q^{50} - 31444q^{52} - 9468q^{53} + 38904q^{55} + 37314q^{58} - 43938q^{59} + 64754q^{61} + 15300q^{62} - 70783q^{64} - 39060q^{65} + 24784q^{67} - 14994q^{68} - 97416q^{71} - 17452q^{73} - 43434q^{74} + 12782q^{76} + 51256q^{79} - 70272q^{80} + 58338q^{82} + 117558q^{83} - 37860q^{85} + 150048q^{86} - 40656q^{88} + 84276q^{89} - 150912q^{92} - 159468q^{94} - 24264q^{95} - 20776q^{97} + O(q^{100}) \)

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
4.27492
−3.27492
−8.27492 0 36.4743 28.7492 0 0 −37.0241 0 −237.897
1.2 −0.725083 0 −31.4743 −46.7492 0 0 46.0241 0 33.8970
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 441.6.a.l 2
3.b odd 2 1 49.6.a.f 2
7.b odd 2 1 63.6.a.f 2
12.b even 2 1 784.6.a.v 2
21.c even 2 1 7.6.a.b 2
21.g even 6 2 49.6.c.e 4
21.h odd 6 2 49.6.c.d 4
28.d even 2 1 1008.6.a.bq 2
84.h odd 2 1 112.6.a.h 2
105.g even 2 1 175.6.a.c 2
105.k odd 4 2 175.6.b.c 4
168.e odd 2 1 448.6.a.u 2
168.i even 2 1 448.6.a.w 2
231.h odd 2 1 847.6.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.a.b 2 21.c even 2 1
49.6.a.f 2 3.b odd 2 1
49.6.c.d 4 21.h odd 6 2
49.6.c.e 4 21.g even 6 2
63.6.a.f 2 7.b odd 2 1
112.6.a.h 2 84.h odd 2 1
175.6.a.c 2 105.g even 2 1
175.6.b.c 4 105.k odd 4 2
441.6.a.l 2 1.a even 1 1 trivial
448.6.a.u 2 168.e odd 2 1
448.6.a.w 2 168.i even 2 1
784.6.a.v 2 12.b even 2 1
847.6.a.c 2 231.h odd 2 1
1008.6.a.bq 2 28.d even 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2}^{2} + 9 T_{2} + 6 \)
\( T_{5}^{2} + 18 T_{5} - 1344 \)
\( T_{13}^{2} - 350 T_{13} - 195608 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 9 T + 70 T^{2} + 288 T^{3} + 1024 T^{4} \)
$3$ 1
$5$ \( 1 + 18 T + 4906 T^{2} + 56250 T^{3} + 9765625 T^{4} \)
$7$ 1
$11$ \( 1 + 396 T + 142198 T^{2} + 63776196 T^{3} + 25937424601 T^{4} \)
$13$ \( 1 - 350 T + 546978 T^{2} - 129952550 T^{3} + 137858491849 T^{4} \)
$17$ \( 1 - 1800 T + 3567406 T^{2} - 2555742600 T^{3} + 2015993900449 T^{4} \)
$19$ \( 1 - 3266 T + 7614270 T^{2} - 8086939334 T^{3} + 6131066257801 T^{4} \)
$23$ \( 1 + 2088 T + 9365230 T^{2} + 13439084184 T^{3} + 41426511213649 T^{4} \)
$29$ \( 1 + 6696 T + 51326470 T^{2} + 137342653704 T^{3} + 420707233300201 T^{4} \)
$31$ \( 1 - 20 T + 53103102 T^{2} - 572583020 T^{3} + 819628286980801 T^{4} \)
$37$ \( 1 - 6232 T + 144242070 T^{2} - 432151540024 T^{3} + 4808584372417849 T^{4} \)
$41$ \( 1 + 6048 T + 223864366 T^{2} + 700698303648 T^{3} + 13422659310152401 T^{4} \)
$43$ \( 1 + 3020 T - 30383466 T^{2} + 443965497860 T^{3} + 21611482313284249 T^{4} \)
$47$ \( 1 - 11700 T + 292735582 T^{2} - 2683336581900 T^{3} + 52599132235830049 T^{4} \)
$53$ \( 1 + 9468 T + 858185230 T^{2} + 3959474927724 T^{3} + 174887470365513049 T^{4} \)
$59$ \( 1 + 43938 T + 1852599934 T^{2} + 31412343849462 T^{3} + 511116753300641401 T^{4} \)
$61$ \( 1 - 64754 T + 2408321418 T^{2} - 54690988874954 T^{3} + 713342911662882601 T^{4} \)
$67$ \( 1 - 24784 T + 2799959190 T^{2} - 33461500651888 T^{3} + 1822837804551761449 T^{4} \)
$71$ \( 1 + 97416 T + 5729557966 T^{2} + 175760806457016 T^{3} + 3255243551009881201 T^{4} \)
$73$ \( 1 + 17452 T + 3828622374 T^{2} + 36179245441036 T^{3} + 4297625829703557649 T^{4} \)
$79$ \( 1 - 51256 T + 3645565854 T^{2} - 157717602787144 T^{3} + 9468276082626847201 T^{4} \)
$83$ \( 1 - 117558 T + 7798161502 T^{2} - 463065739909794 T^{3} + 15516041187205853449 T^{4} \)
$89$ \( 1 - 84276 T + 5915697430 T^{2} - 470602194123924 T^{3} + 31181719929966183601 T^{4} \)
$97$ \( 1 + 20776 T + 16174049358 T^{2} + 178410581179432 T^{3} + 73742412689492826049 T^{4} \)
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