Properties

Label 448.6.a.s
Level $448$
Weight $6$
Character orbit 448.a
Self dual yes
Analytic conductor $71.852$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 448.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(71.8519512762\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{61}) \)
Defining polynomial: \( x^{2} - x - 15 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 224)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{61}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 7) q^{3} + (7 \beta - 17) q^{5} + 49 q^{7} + (14 \beta - 133) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 7) q^{3} + (7 \beta - 17) q^{5} + 49 q^{7} + (14 \beta - 133) q^{9} + ( - 46 \beta + 210) q^{11} + (49 \beta + 245) q^{13} + ( - 32 \beta - 308) q^{15} + ( - 14 \beta - 528) q^{17} + ( - 247 \beta + 623) q^{19} + ( - 49 \beta - 343) q^{21} + (16 \beta + 252) q^{23} + ( - 238 \beta + 153) q^{25} + (278 \beta + 1778) q^{27} + (546 \beta + 1952) q^{29} + ( - 642 \beta - 1022) q^{31} + (112 \beta + 1336) q^{33} + (343 \beta - 833) q^{35} + ( - 14 \beta + 3744) q^{37} + ( - 588 \beta - 4704) q^{39} + (1134 \beta + 3916) q^{41} + ( - 1090 \beta + 5166) q^{43} + ( - 1169 \beta + 8239) q^{45} + (850 \beta - 20986) q^{47} + 2401 q^{49} + (626 \beta + 4550) q^{51} + (2688 \beta - 16406) q^{53} + (2252 \beta - 23212) q^{55} + (1106 \beta + 10706) q^{57} + ( - 1639 \beta + 24199) q^{59} + ( - 441 \beta + 359) q^{61} + (686 \beta - 6517) q^{63} + (882 \beta + 16758) q^{65} + (3696 \beta + 6412) q^{67} + ( - 364 \beta - 2740) q^{69} + (396 \beta - 51996) q^{71} + (5404 \beta - 27050) q^{73} + (1513 \beta + 13447) q^{75} + ( - 2254 \beta + 10290) q^{77} + (1108 \beta - 32284) q^{79} + ( - 7126 \beta + 2915) q^{81} + ( - 527 \beta - 23905) q^{83} + ( - 3458 \beta + 2998) q^{85} + ( - 5774 \beta - 46970) q^{87} + ( - 14448 \beta - 8694) q^{89} + (2401 \beta + 12005) q^{91} + (5516 \beta + 46316) q^{93} + (8560 \beta - 116060) q^{95} + (6146 \beta - 48648) q^{97} + (9058 \beta - 67214) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 14 q^{3} - 34 q^{5} + 98 q^{7} - 266 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 14 q^{3} - 34 q^{5} + 98 q^{7} - 266 q^{9} + 420 q^{11} + 490 q^{13} - 616 q^{15} - 1056 q^{17} + 1246 q^{19} - 686 q^{21} + 504 q^{23} + 306 q^{25} + 3556 q^{27} + 3904 q^{29} - 2044 q^{31} + 2672 q^{33} - 1666 q^{35} + 7488 q^{37} - 9408 q^{39} + 7832 q^{41} + 10332 q^{43} + 16478 q^{45} - 41972 q^{47} + 4802 q^{49} + 9100 q^{51} - 32812 q^{53} - 46424 q^{55} + 21412 q^{57} + 48398 q^{59} + 718 q^{61} - 13034 q^{63} + 33516 q^{65} + 12824 q^{67} - 5480 q^{69} - 103992 q^{71} - 54100 q^{73} + 26894 q^{75} + 20580 q^{77} - 64568 q^{79} + 5830 q^{81} - 47810 q^{83} + 5996 q^{85} - 93940 q^{87} - 17388 q^{89} + 24010 q^{91} + 92632 q^{93} - 232120 q^{95} - 97296 q^{97} - 134428 q^{99}+O(q^{100}) \) 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
4.40512
−3.40512
0 −14.8102 0 37.6717 0 49.0000 0 −23.6565 0
1.2 0 0.810250 0 −71.6717 0 49.0000 0 −242.343 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)

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 448.6.a.s 2
4.b odd 2 1 448.6.a.y 2
8.b even 2 1 224.6.a.d yes 2
8.d odd 2 1 224.6.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.6.a.c 2 8.d odd 2 1
224.6.a.d yes 2 8.b even 2 1
448.6.a.s 2 1.a even 1 1 trivial
448.6.a.y 2 4.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 \(S_{6}^{\mathrm{new}}(\Gamma_0(448))\):

\( T_{3}^{2} + 14T_{3} - 12 \) Copy content Toggle raw display
\( T_{5}^{2} + 34T_{5} - 2700 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 14T - 12 \) Copy content Toggle raw display
$5$ \( T^{2} + 34T - 2700 \) Copy content Toggle raw display
$7$ \( (T - 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 420T - 84976 \) Copy content Toggle raw display
$13$ \( T^{2} - 490T - 86436 \) Copy content Toggle raw display
$17$ \( T^{2} + 1056 T + 266828 \) Copy content Toggle raw display
$19$ \( T^{2} - 1246 T - 3333420 \) Copy content Toggle raw display
$23$ \( T^{2} - 504T + 47888 \) Copy content Toggle raw display
$29$ \( T^{2} - 3904 T - 14374772 \) Copy content Toggle raw display
$31$ \( T^{2} + 2044 T - 24097520 \) Copy content Toggle raw display
$37$ \( T^{2} - 7488 T + 14005580 \) Copy content Toggle raw display
$41$ \( T^{2} - 7832 T - 63108260 \) Copy content Toggle raw display
$43$ \( T^{2} - 10332 T - 45786544 \) Copy content Toggle raw display
$47$ \( T^{2} + 41972 T + 396339696 \) Copy content Toggle raw display
$53$ \( T^{2} + 32812 T - 171589148 \) Copy content Toggle raw display
$59$ \( T^{2} - 48398 T + 421726020 \) Copy content Toggle raw display
$61$ \( T^{2} - 718 T - 11734460 \) Copy content Toggle raw display
$67$ \( T^{2} - 12824 T - 792171632 \) Copy content Toggle raw display
$71$ \( T^{2} + 103992 T + 2694018240 \) Copy content Toggle raw display
$73$ \( T^{2} + 54100 T - 1049693676 \) Copy content Toggle raw display
$79$ \( T^{2} + 64568 T + 967369152 \) Copy content Toggle raw display
$83$ \( T^{2} + 47810 T + 554507556 \) Copy content Toggle raw display
$89$ \( T^{2} + 17388 T - 12657841308 \) Copy content Toggle raw display
$97$ \( T^{2} + 97296 T + 62455628 \) Copy content Toggle raw display
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