Properties

Label 448.6.a.r
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: \(\Q(\sqrt{193}) \)
Defining polynomial: \( x^{2} - x - 48 \) 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 56)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{193}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 7) q^{3} + ( - 5 \beta - 21) q^{5} + 49 q^{7} + (14 \beta - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 7) q^{3} + ( - 5 \beta - 21) q^{5} + 49 q^{7} + (14 \beta - 1) q^{9} + ( - 14 \beta - 358) q^{11} + (17 \beta + 357) q^{13} + (56 \beta + 1112) q^{15} + ( - 54 \beta - 672) q^{17} + (93 \beta - 973) q^{19} + ( - 49 \beta - 343) q^{21} + ( - 112 \beta + 896) q^{23} + (210 \beta + 2141) q^{25} + (146 \beta - 994) q^{27} + (546 \beta + 600) q^{29} + (446 \beta + 3402) q^{31} + (456 \beta + 5208) q^{33} + ( - 245 \beta - 1029) q^{35} + (154 \beta - 7320) q^{37} + ( - 476 \beta - 5780) q^{39} + ( - 442 \beta + 3948) q^{41} + (518 \beta + 262) q^{43} + ( - 289 \beta - 13489) q^{45} + (746 \beta + 9198) q^{47} + 2401 q^{49} + (1050 \beta + 15126) q^{51} + ( - 1008 \beta - 22566) q^{53} + (2084 \beta + 21028) q^{55} + (322 \beta - 11138) q^{57} + (365 \beta + 11291) q^{59} + ( - 1117 \beta + 26411) q^{61} + (686 \beta - 49) q^{63} + ( - 2142 \beta - 23902) q^{65} + (224 \beta + 4924) q^{67} + ( - 112 \beta + 15344) q^{69} + ( - 4060 \beta + 420) q^{71} + ( - 1204 \beta - 61026) q^{73} + ( - 3611 \beta - 55517) q^{75} + ( - 686 \beta - 17542) q^{77} + (1372 \beta - 15852) q^{79} + ( - 3430 \beta - 20977) q^{81} + (4185 \beta + 18487) q^{83} + (4494 \beta + 66222) q^{85} + ( - 4422 \beta - 109578) q^{87} + (1512 \beta - 105294) q^{89} + (833 \beta + 17493) q^{91} + ( - 6524 \beta - 109892) q^{93} + (2912 \beta - 69312) q^{95} + (8882 \beta - 22120) q^{97} + ( - 4998 \beta - 37470) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 14 q^{3} - 42 q^{5} + 98 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 14 q^{3} - 42 q^{5} + 98 q^{7} - 2 q^{9} - 716 q^{11} + 714 q^{13} + 2224 q^{15} - 1344 q^{17} - 1946 q^{19} - 686 q^{21} + 1792 q^{23} + 4282 q^{25} - 1988 q^{27} + 1200 q^{29} + 6804 q^{31} + 10416 q^{33} - 2058 q^{35} - 14640 q^{37} - 11560 q^{39} + 7896 q^{41} + 524 q^{43} - 26978 q^{45} + 18396 q^{47} + 4802 q^{49} + 30252 q^{51} - 45132 q^{53} + 42056 q^{55} - 22276 q^{57} + 22582 q^{59} + 52822 q^{61} - 98 q^{63} - 47804 q^{65} + 9848 q^{67} + 30688 q^{69} + 840 q^{71} - 122052 q^{73} - 111034 q^{75} - 35084 q^{77} - 31704 q^{79} - 41954 q^{81} + 36974 q^{83} + 132444 q^{85} - 219156 q^{87} - 210588 q^{89} + 34986 q^{91} - 219784 q^{93} - 138624 q^{95} - 44240 q^{97} - 74940 q^{99}+O(q^{100}) \) 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
7.44622
−6.44622
0 −20.8924 0 −90.4622 0 49.0000 0 193.494 0
1.2 0 6.89244 0 48.4622 0 49.0000 0 −195.494 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.r 2
4.b odd 2 1 448.6.a.x 2
8.b even 2 1 112.6.a.j 2
8.d odd 2 1 56.6.a.d 2
24.f even 2 1 504.6.a.m 2
24.h odd 2 1 1008.6.a.bi 2
56.e even 2 1 392.6.a.e 2
56.h odd 2 1 784.6.a.q 2
56.k odd 6 2 392.6.i.k 4
56.m even 6 2 392.6.i.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.6.a.d 2 8.d odd 2 1
112.6.a.j 2 8.b even 2 1
392.6.a.e 2 56.e even 2 1
392.6.i.h 4 56.m even 6 2
392.6.i.k 4 56.k odd 6 2
448.6.a.r 2 1.a even 1 1 trivial
448.6.a.x 2 4.b odd 2 1
504.6.a.m 2 24.f even 2 1
784.6.a.q 2 56.h odd 2 1
1008.6.a.bi 2 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(448))\):

\( T_{3}^{2} + 14T_{3} - 144 \) Copy content Toggle raw display
\( T_{5}^{2} + 42T_{5} - 4384 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 14T - 144 \) Copy content Toggle raw display
$5$ \( T^{2} + 42T - 4384 \) Copy content Toggle raw display
$7$ \( (T - 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 716T + 90336 \) Copy content Toggle raw display
$13$ \( T^{2} - 714T + 71672 \) Copy content Toggle raw display
$17$ \( T^{2} + 1344 T - 111204 \) Copy content Toggle raw display
$19$ \( T^{2} + 1946 T - 722528 \) Copy content Toggle raw display
$23$ \( T^{2} - 1792 T - 1618176 \) Copy content Toggle raw display
$29$ \( T^{2} - 1200 T - 57176388 \) Copy content Toggle raw display
$31$ \( T^{2} - 6804 T - 26817184 \) Copy content Toggle raw display
$37$ \( T^{2} + 14640 T + 49005212 \) Copy content Toggle raw display
$41$ \( T^{2} - 7896 T - 22118548 \) Copy content Toggle raw display
$43$ \( T^{2} - 524 T - 51717888 \) Copy content Toggle raw display
$47$ \( T^{2} - 18396 T - 22804384 \) Copy content Toggle raw display
$53$ \( T^{2} + 45132 T + 313124004 \) Copy content Toggle raw display
$59$ \( T^{2} - 22582 T + 101774256 \) Copy content Toggle raw display
$61$ \( T^{2} - 52822 T + 456736944 \) Copy content Toggle raw display
$67$ \( T^{2} - 9848 T + 14561808 \) Copy content Toggle raw display
$71$ \( T^{2} - 840 T - 3181158400 \) Copy content Toggle raw display
$73$ \( T^{2} + 122052 T + 3444396788 \) Copy content Toggle raw display
$79$ \( T^{2} + 31704 T - 112014208 \) Copy content Toggle raw display
$83$ \( T^{2} - 36974 T - 3038476256 \) Copy content Toggle raw display
$89$ \( T^{2} + 210588 T + 10645600644 \) Copy content Toggle raw display
$97$ \( T^{2} + 44240 T - 14736460932 \) Copy content Toggle raw display
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