Properties

Label 441.4.a.n
Level $441$
Weight $4$
Character orbit 441.a
Self dual yes
Analytic conductor $26.020$
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.0198423125\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 147)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta ) q^{2} + ( -5 - 2 \beta ) q^{4} + ( -10 + 7 \beta ) q^{5} + ( 9 - 11 \beta ) q^{8} +O(q^{10})\) \( q + ( -1 + \beta ) q^{2} + ( -5 - 2 \beta ) q^{4} + ( -10 + 7 \beta ) q^{5} + ( 9 - 11 \beta ) q^{8} + ( 24 - 17 \beta ) q^{10} + ( 10 + 24 \beta ) q^{11} + ( 52 - 25 \beta ) q^{13} + ( 9 + 36 \beta ) q^{16} + ( -58 - 45 \beta ) q^{17} + ( 96 + 22 \beta ) q^{19} + ( 22 - 15 \beta ) q^{20} + ( 38 - 14 \beta ) q^{22} + ( -14 - 28 \beta ) q^{23} + ( 73 - 140 \beta ) q^{25} + ( -102 + 77 \beta ) q^{26} + ( -148 - 62 \beta ) q^{29} + ( -52 + 50 \beta ) q^{31} + ( -9 + 61 \beta ) q^{32} + ( -32 - 13 \beta ) q^{34} + ( -124 - 48 \beta ) q^{37} + ( -52 + 74 \beta ) q^{38} + ( -244 + 173 \beta ) q^{40} + ( -10 - 219 \beta ) q^{41} + ( -360 + 100 \beta ) q^{43} + ( -146 - 140 \beta ) q^{44} + ( -42 + 14 \beta ) q^{46} + ( 48 + 250 \beta ) q^{47} + ( -353 + 213 \beta ) q^{50} + ( -160 + 21 \beta ) q^{52} + ( -134 - 360 \beta ) q^{53} + ( 236 - 170 \beta ) q^{55} + ( 24 - 86 \beta ) q^{58} + ( 308 - 226 \beta ) q^{59} + ( 8 + 3 \beta ) q^{61} + ( 152 - 102 \beta ) q^{62} + ( 59 - 358 \beta ) q^{64} + ( -870 + 614 \beta ) q^{65} + ( -72 + 524 \beta ) q^{67} + ( 470 + 341 \beta ) q^{68} + ( -494 - 232 \beta ) q^{71} + ( 52 - 401 \beta ) q^{73} + ( 28 - 76 \beta ) q^{74} + ( -568 - 302 \beta ) q^{76} + ( -472 - 236 \beta ) q^{79} + ( 414 - 297 \beta ) q^{80} + ( -428 + 209 \beta ) q^{82} + ( -508 + 80 \beta ) q^{83} + ( -50 + 44 \beta ) q^{85} + ( 560 - 460 \beta ) q^{86} + ( -438 + 106 \beta ) q^{88} + ( 194 + 339 \beta ) q^{89} + ( 182 + 168 \beta ) q^{92} + ( 452 - 202 \beta ) q^{94} + ( -652 + 452 \beta ) q^{95} + ( 244 + 599 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 10q^{4} - 20q^{5} + 18q^{8} + O(q^{10}) \) \( 2q - 2q^{2} - 10q^{4} - 20q^{5} + 18q^{8} + 48q^{10} + 20q^{11} + 104q^{13} + 18q^{16} - 116q^{17} + 192q^{19} + 44q^{20} + 76q^{22} - 28q^{23} + 146q^{25} - 204q^{26} - 296q^{29} - 104q^{31} - 18q^{32} - 64q^{34} - 248q^{37} - 104q^{38} - 488q^{40} - 20q^{41} - 720q^{43} - 292q^{44} - 84q^{46} + 96q^{47} - 706q^{50} - 320q^{52} - 268q^{53} + 472q^{55} + 48q^{58} + 616q^{59} + 16q^{61} + 304q^{62} + 118q^{64} - 1740q^{65} - 144q^{67} + 940q^{68} - 988q^{71} + 104q^{73} + 56q^{74} - 1136q^{76} - 944q^{79} + 828q^{80} - 856q^{82} - 1016q^{83} - 100q^{85} + 1120q^{86} - 876q^{88} + 388q^{89} + 364q^{92} + 904q^{94} - 1304q^{95} + 488q^{97} + O(q^{100}) \)

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
−1.41421
1.41421
−2.41421 0 −2.17157 −19.8995 0 0 24.5563 0 48.0416
1.2 0.414214 0 −7.82843 −0.100505 0 0 −6.55635 0 −0.0416306
\(n\): 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.n 2
3.b odd 2 1 147.4.a.j 2
7.b odd 2 1 441.4.a.o 2
7.c even 3 2 441.4.e.v 4
7.d odd 6 2 441.4.e.u 4
12.b even 2 1 2352.4.a.cf 2
21.c even 2 1 147.4.a.k yes 2
21.g even 6 2 147.4.e.j 4
21.h odd 6 2 147.4.e.k 4
84.h odd 2 1 2352.4.a.bl 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.4.a.j 2 3.b odd 2 1
147.4.a.k yes 2 21.c even 2 1
147.4.e.j 4 21.g even 6 2
147.4.e.k 4 21.h odd 6 2
441.4.a.n 2 1.a even 1 1 trivial
441.4.a.o 2 7.b odd 2 1
441.4.e.u 4 7.d odd 6 2
441.4.e.v 4 7.c even 3 2
2352.4.a.bl 2 84.h odd 2 1
2352.4.a.cf 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(441))\):

\( T_{2}^{2} + 2 T_{2} - 1 \)
\( T_{5}^{2} + 20 T_{5} + 2 \)
\( T_{13}^{2} - 104 T_{13} + 1454 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + 2 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 2 + 20 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( -1052 - 20 T + T^{2} \)
$13$ \( 1454 - 104 T + T^{2} \)
$17$ \( -686 + 116 T + T^{2} \)
$19$ \( 8248 - 192 T + T^{2} \)
$23$ \( -1372 + 28 T + T^{2} \)
$29$ \( 14216 + 296 T + T^{2} \)
$31$ \( -2296 + 104 T + T^{2} \)
$37$ \( 10768 + 248 T + T^{2} \)
$41$ \( -95822 + 20 T + T^{2} \)
$43$ \( 109600 + 720 T + T^{2} \)
$47$ \( -122696 - 96 T + T^{2} \)
$53$ \( -241244 + 268 T + T^{2} \)
$59$ \( -7288 - 616 T + T^{2} \)
$61$ \( 46 - 16 T + T^{2} \)
$67$ \( -543968 + 144 T + T^{2} \)
$71$ \( 136388 + 988 T + T^{2} \)
$73$ \( -318898 - 104 T + T^{2} \)
$79$ \( 111392 + 944 T + T^{2} \)
$83$ \( 245264 + 1016 T + T^{2} \)
$89$ \( -192206 - 388 T + T^{2} \)
$97$ \( -658066 - 488 T + T^{2} \)
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