Properties

Label 441.6.a.s
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: \(\Q(\sqrt{249}) \)
Defining polynomial: \( x^{2} - x - 62 \) Copy content Toggle raw display
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 \(\beta = \frac{1}{2}(1 + \sqrt{249})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{2} + (3 \beta + 31) q^{4} + ( - 7 \beta - 13) q^{5} + (5 \beta + 185) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{2} + (3 \beta + 31) q^{4} + ( - 7 \beta - 13) q^{5} + (5 \beta + 185) q^{8} + ( - 27 \beta - 447) q^{10} + ( - \beta + 569) q^{11} + ( - 9 \beta - 458) q^{13} + (99 \beta - 497) q^{16} + (148 \beta - 236) q^{17} + ( - 27 \beta - 1142) q^{19} + ( - 277 \beta - 1705) q^{20} + (567 \beta + 507) q^{22} + ( - 308 \beta - 644) q^{23} + (231 \beta + 82) q^{25} + ( - 476 \beta - 1016) q^{26} + ( - 45 \beta + 1131) q^{29} + ( - 768 \beta - 1763) q^{31} + ( - 459 \beta - 279) q^{32} + (60 \beta + 8940) q^{34} + (855 \beta - 9982) q^{37} + ( - 1196 \beta - 2816) q^{38} + ( - 1395 \beta - 4575) q^{40} + (846 \beta - 6852) q^{41} + ( - 2043 \beta - 364) q^{43} + (1673 \beta + 17453) q^{44} + ( - 1260 \beta - 19740) q^{46} + ( - 604 \beta - 11278) q^{47} + (544 \beta + 14404) q^{50} + ( - 1680 \beta - 15872) q^{52} + (1751 \beta + 14951) q^{53} + ( - 3963 \beta - 6963) q^{55} + (1041 \beta - 1659) q^{58} + ( - 3917 \beta + 22507) q^{59} + (2544 \beta - 22298) q^{61} + ( - 3299 \beta - 49379) q^{62} + ( - 4365 \beta - 12833) q^{64} + (3386 \beta + 9860) q^{65} + ( - 4461 \beta + 17612) q^{67} + (4324 \beta + 20212) q^{68} + ( - 1404 \beta - 50346) q^{71} + ( - 5247 \beta + 16912) q^{73} + ( - 8272 \beta + 43028) q^{74} + ( - 4344 \beta - 40424) q^{76} + (6834 \beta - 12649) q^{79} + (1499 \beta - 36505) q^{80} + ( - 5160 \beta + 45600) q^{82} + ( - 1899 \beta + 31539) q^{83} + ( - 1308 \beta - 61164) q^{85} + ( - 4450 \beta - 127030) q^{86} + (2655 \beta + 104955) q^{88} + ( - 130 \beta + 14726) q^{89} + ( - 12404 \beta - 77252) q^{92} + ( - 12486 \beta - 48726) q^{94} + (8534 \beta + 26564) q^{95} + (1017 \beta + 4387) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} + 65 q^{4} - 33 q^{5} + 375 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{2} + 65 q^{4} - 33 q^{5} + 375 q^{8} - 921 q^{10} + 1137 q^{11} - 925 q^{13} - 895 q^{16} - 324 q^{17} - 2311 q^{19} - 3687 q^{20} + 1581 q^{22} - 1596 q^{23} + 395 q^{25} - 2508 q^{26} + 2217 q^{29} - 4294 q^{31} - 1017 q^{32} + 17940 q^{34} - 19109 q^{37} - 6828 q^{38} - 10545 q^{40} - 12858 q^{41} - 2771 q^{43} + 36579 q^{44} - 40740 q^{46} - 23160 q^{47} + 29352 q^{50} - 33424 q^{52} + 31653 q^{53} - 17889 q^{55} - 2277 q^{58} + 41097 q^{59} - 42052 q^{61} - 102057 q^{62} - 30031 q^{64} + 23106 q^{65} + 30763 q^{67} + 44748 q^{68} - 102096 q^{71} + 28577 q^{73} + 77784 q^{74} - 85192 q^{76} - 18464 q^{79} - 71511 q^{80} + 86040 q^{82} + 61179 q^{83} - 123636 q^{85} - 258510 q^{86} + 212565 q^{88} + 29322 q^{89} - 166908 q^{92} - 109938 q^{94} + 61662 q^{95} + 9791 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
−7.38987
8.38987
−6.38987 0 8.83040 38.7291 0 0 148.051 0 −247.474
1.2 9.38987 0 56.1696 −71.7291 0 0 226.949 0 −673.526
\(n\): 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 441.6.a.s 2
3.b odd 2 1 147.6.a.k 2
7.b odd 2 1 441.6.a.t 2
7.d odd 6 2 63.6.e.c 4
21.c even 2 1 147.6.a.i 2
21.g even 6 2 21.6.e.b 4
21.h odd 6 2 147.6.e.l 4
84.j odd 6 2 336.6.q.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.b 4 21.g even 6 2
63.6.e.c 4 7.d odd 6 2
147.6.a.i 2 21.c even 2 1
147.6.a.k 2 3.b odd 2 1
147.6.e.l 4 21.h odd 6 2
336.6.q.e 4 84.j odd 6 2
441.6.a.s 2 1.a even 1 1 trivial
441.6.a.t 2 7.b odd 2 1

Hecke kernels

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

\( T_{2}^{2} - 3T_{2} - 60 \) Copy content Toggle raw display
\( T_{5}^{2} + 33T_{5} - 2778 \) Copy content Toggle raw display
\( T_{13}^{2} + 925T_{13} + 208864 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3T - 60 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 33T - 2778 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 1137 T + 323130 \) Copy content Toggle raw display
$13$ \( T^{2} + 925T + 208864 \) Copy content Toggle raw display
$17$ \( T^{2} + 324 T - 1337280 \) Copy content Toggle raw display
$19$ \( T^{2} + 2311 T + 1289800 \) Copy content Toggle raw display
$23$ \( T^{2} + 1596 T - 5268480 \) Copy content Toggle raw display
$29$ \( T^{2} - 2217 T + 1102716 \) Copy content Toggle raw display
$31$ \( T^{2} + 4294 T - 32106935 \) Copy content Toggle raw display
$37$ \( T^{2} + 19109 T + 45782164 \) Copy content Toggle raw display
$41$ \( T^{2} + 12858 T - 3221280 \) Copy content Toggle raw display
$43$ \( T^{2} + 2771 T - 257902490 \) Copy content Toggle raw display
$47$ \( T^{2} + 23160 T + 111386604 \) Copy content Toggle raw display
$53$ \( T^{2} - 31653 T + 59619540 \) Copy content Toggle raw display
$59$ \( T^{2} - 41097 T - 532853988 \) Copy content Toggle raw display
$61$ \( T^{2} + 42052 T + 39214660 \) Copy content Toggle raw display
$67$ \( T^{2} - 30763 T - 1002216890 \) Copy content Toggle raw display
$71$ \( T^{2} + 102096 T + 2483190108 \) Copy content Toggle raw display
$73$ \( T^{2} - 28577 T - 1509644078 \) Copy content Toggle raw display
$79$ \( T^{2} + 18464 T - 2822066537 \) Copy content Toggle raw display
$83$ \( T^{2} - 61179 T + 711231498 \) Copy content Toggle raw display
$89$ \( T^{2} - 29322 T + 213892896 \) Copy content Toggle raw display
$97$ \( T^{2} - 9791 T - 40418570 \) Copy content Toggle raw display
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