Properties

Label 441.6.a.be
Level $441$
Weight $6$
Character orbit 441.a
Self dual yes
Analytic conductor $70.729$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $4$

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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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 146x^{6} + 5453x^{4} - 40868x^{2} + 3844 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5}\cdot 7^{4} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + ( - \beta_{2} + 3) q^{4} + \beta_1 q^{5} + (\beta_{6} - \beta_{4}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + ( - \beta_{2} + 3) q^{4} + \beta_1 q^{5} + (\beta_{6} - \beta_{4}) q^{8} + ( - 3 \beta_{7} - \beta_{5}) q^{10} + ( - 2 \beta_{6} - 8 \beta_{4}) q^{11} + (4 \beta_{7} + 3 \beta_{5}) q^{13} + (27 \beta_{2} - 117) q^{16} + (4 \beta_{3} - 25 \beta_1) q^{17} + (24 \beta_{7} - 4 \beta_{5}) q^{19} + ( - 13 \beta_{3} - 26 \beta_1) q^{20} + (20 \beta_{2} - 308) q^{22} + ( - 24 \beta_{6} - 40 \beta_{4}) q^{23} + (80 \beta_{2} - 339) q^{25} + (19 \beta_{3} - 8 \beta_1) q^{26} + (69 \beta_{6} - 344 \beta_{4}) q^{29} + ( - 184 \beta_{7} - 16 \beta_{5}) q^{31} + ( - 59 \beta_{6} - 841 \beta_{4}) q^{32} + (103 \beta_{7} + 165 \beta_{5}) q^{34} + (264 \beta_{2} - 2524) q^{37} + (92 \beta_{3} - 48 \beta_1) q^{38} + (83 \beta_{7} - 397 \beta_{5}) q^{40} + ( - 236 \beta_{3} - 13 \beta_1) q^{41} + ( - 16 \beta_{2} - 5140) q^{43} + (44 \beta_{6} - 612 \beta_{4}) q^{44} + (184 \beta_{2} - 1736) q^{46} + (152 \beta_{3} - 100 \beta_1) q^{47} + ( - 80 \beta_{6} - 2579 \beta_{4}) q^{50} + (29 \beta_{7} + 577 \beta_{5}) q^{52} + (238 \beta_{6} - 4328 \beta_{4}) q^{53} + ( - 136 \beta_{7} + 804 \beta_{5}) q^{55} + ( - 70 \beta_{2} - 11074) q^{58} + (440 \beta_{3} + 288 \beta_1) q^{59} + (116 \beta_{7} - 2399 \beta_{5}) q^{61} + ( - 752 \beta_{3} + 368 \beta_1) q^{62} + (331 \beta_{2} - 26517) q^{64} + (95 \beta_{6} - 728 \beta_{4}) q^{65} + ( - 800 \beta_{2} - 4892) q^{67} + (449 \beta_{3} + 594 \beta_1) q^{68} + ( - 636 \beta_{6} - 5128 \beta_{4}) q^{71} + (784 \beta_{7} + 2605 \beta_{5}) q^{73} + ( - 264 \beta_{6} - 9916 \beta_{4}) q^{74} + (20 \beta_{7} + 3396 \beta_{5}) q^{76} + ( - 2128 \beta_{2} - 30320) q^{79} + (351 \beta_{3} + 666 \beta_1) q^{80} + ( - 1613 \beta_{7} - 8247 \beta_{5}) q^{82} + ( - 1168 \beta_{3} - 420 \beta_1) q^{83} + ( - 1832 \beta_{2} - 70042) q^{85} + (16 \beta_{6} - 4692 \beta_{4}) q^{86} + ( - 292 \beta_{2} - 10948) q^{88} + (340 \beta_{3} - 895 \beta_1) q^{89} + (584 \beta_{6} - 5608 \beta_{4}) q^{92} + (1364 \beta_{7} + 5420 \beta_{5}) q^{94} + (724 \beta_{6} - 4368 \beta_{4}) q^{95} + ( - 1736 \beta_{7} + 5613 \beta_{5}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 20 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 20 q^{4} - 828 q^{16} - 2384 q^{22} - 2392 q^{25} - 19136 q^{37} - 41184 q^{43} - 13152 q^{46} - 88872 q^{58} - 210812 q^{64} - 42336 q^{67} - 251072 q^{79} - 567664 q^{85} - 88752 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 146x^{6} + 5453x^{4} - 40868x^{2} + 3844 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 13\nu^{6} + 2930\nu^{4} - 324365\nu^{2} + 1463438 ) / 82236 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{6} + 227\nu^{4} - 2060\nu^{2} - 90946 ) / 2937 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 61\nu^{6} - 8392\nu^{4} + 271357\nu^{2} - 970822 ) / 11748 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2911\nu^{7} - 428788\nu^{5} + 16393987\nu^{3} - 135790882\nu ) / 5098632 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -343\nu^{7} + 49210\nu^{5} - 1771861\nu^{3} + 11849026\nu ) / 364188 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3931\nu^{7} - 597424\nu^{5} + 24375907\nu^{3} - 205795858\nu ) / 2549316 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -722\nu^{7} + 105443\nu^{5} - 3895061\nu^{3} + 26260934\nu ) / 182094 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} - 2\beta_{5} - 6\beta_{4} ) / 14 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 6\beta_{3} + 49\beta_{2} + 14\beta _1 + 1764 ) / 49 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 42\beta_{7} + 75\beta_{6} - 208\beta_{5} - 254\beta_{4} ) / 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 460\beta_{3} + 3969\beta_{2} + 1988\beta _1 + 125538 ) / 49 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5110\beta_{7} + 5965\beta_{6} - 22868\beta_{5} - 18346\beta_{4} ) / 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 46030\beta_{3} + 328055\beta_{2} + 211218\beta _1 + 10203466 ) / 49 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 516166\beta_{7} + 502907\beta_{6} - 2290332\beta_{5} - 1527254\beta_{4} ) / 14 \) 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
3.13708
0.308653
6.71178
9.54020
−6.71178
−9.54020
−3.13708
−0.308653
−8.12599 0 34.0317 −17.4201 0 0 −16.5098 0 141.556
1.2 −8.12599 0 34.0317 17.4201 0 0 −16.5098 0 −141.556
1.3 −1.72287 0 −29.0317 −73.1337 0 0 105.150 0 126.000
1.4 −1.72287 0 −29.0317 73.1337 0 0 105.150 0 −126.000
1.5 1.72287 0 −29.0317 −73.1337 0 0 −105.150 0 −126.000
1.6 1.72287 0 −29.0317 73.1337 0 0 −105.150 0 126.000
1.7 8.12599 0 34.0317 −17.4201 0 0 16.5098 0 −141.556
1.8 8.12599 0 34.0317 17.4201 0 0 16.5098 0 141.556
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.6.a.be 8
3.b odd 2 1 inner 441.6.a.be 8
7.b odd 2 1 inner 441.6.a.be 8
21.c even 2 1 inner 441.6.a.be 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.6.a.be 8 1.a even 1 1 trivial
441.6.a.be 8 3.b odd 2 1 inner
441.6.a.be 8 7.b odd 2 1 inner
441.6.a.be 8 21.c even 2 1 inner

Hecke kernels

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

\( T_{2}^{4} - 69T_{2}^{2} + 196 \) Copy content Toggle raw display
\( T_{5}^{4} - 5652T_{5}^{2} + 1623076 \) Copy content Toggle raw display
\( T_{13}^{4} - 65076T_{13}^{2} + 966836836 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 69 T^{2} + 196)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 5652 T^{2} + 1623076)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 50336 T^{2} + \cdots + 486820096)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 65076 T^{2} + \cdots + 966836836)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 3671492 T^{2} + \cdots + 1776617078404)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 2297152 T^{2} + \cdots + 1304566014976)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 6663744 T^{2} + \cdots + 4889971523584)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 60939144 T^{2} + \cdots + 71690156736016)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 134646912 T^{2} + \cdots + 45\!\cdots\!36)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4784 T - 63573584)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 376626596 T^{2} + \cdots + 87\!\cdots\!16)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 10296 T + 26247376)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 217430848 T^{2} + \cdots + 43\!\cdots\!44)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 1887460256 T^{2} + \cdots + 88\!\cdots\!96)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 1738892928 T^{2} + \cdots + 75\!\cdots\!96)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 1189339316 T^{2} + \cdots + 29\!\cdots\!16)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 10584 T - 608314736)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 6530375232 T^{2} + \cdots + 10\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 3717977508 T^{2} + \cdots + 34\!\cdots\!64)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 62768 T - 3517390336)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 10070803008 T^{2} + \cdots + 19\!\cdots\!44)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 5402804900 T^{2} + \cdots + 65\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 18436450116 T^{2} + \cdots + 76\!\cdots\!56)^{2} \) Copy content Toggle raw display
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