Properties

Label 448.6.a.t
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: \(\Q(\sqrt{345}) \)
Defining polynomial: \( x^{2} - x - 86 \) 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 \(\beta = \sqrt{345}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 3) q^{3} + (3 \beta - 41) q^{5} + 49 q^{7} + (6 \beta + 111) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 3) q^{3} + (3 \beta - 41) q^{5} + 49 q^{7} + (6 \beta + 111) q^{9} + ( - 6 \beta + 170) q^{11} + ( - 39 \beta - 455) q^{13} + (32 \beta - 912) q^{15} + ( - 6 \beta + 1608) q^{17} + (45 \beta - 337) q^{19} + ( - 49 \beta - 147) q^{21} + (72 \beta + 552) q^{23} + ( - 246 \beta + 1661) q^{25} + (114 \beta - 1674) q^{27} + (114 \beta - 4032) q^{29} + ( - 210 \beta + 3106) q^{31} + ( - 152 \beta + 1560) q^{33} + (147 \beta - 2009) q^{35} + ( - 390 \beta + 4256) q^{37} + (572 \beta + 14820) q^{39} + (678 \beta - 652) q^{41} + (798 \beta - 5002) q^{43} + (87 \beta + 1659) q^{45} + (714 \beta + 6374) q^{47} + 2401 q^{49} + ( - 1590 \beta - 2754) q^{51} + (768 \beta + 5610) q^{53} + (756 \beta - 13180) q^{55} + (202 \beta - 14514) q^{57} + (2253 \beta - 6009) q^{59} + (75 \beta - 51369) q^{61} + (294 \beta + 5439) q^{63} + (234 \beta - 21710) q^{65} + (1944 \beta + 12068) q^{67} + ( - 768 \beta - 26496) q^{69} + ( - 1740 \beta - 44860) q^{71} + ( - 1716 \beta - 27794) q^{73} + ( - 923 \beta + 79887) q^{75} + ( - 294 \beta + 8330) q^{77} + ( - 1956 \beta - 24412) q^{79} + ( - 126 \beta - 61281) q^{81} + ( - 3447 \beta + 17891) q^{83} + (5070 \beta - 72138) q^{85} + (3690 \beta - 27234) q^{87} + (4728 \beta - 9150) q^{89} + ( - 1911 \beta - 22295) q^{91} + ( - 2476 \beta + 63132) q^{93} + ( - 2856 \beta + 60392) q^{95} + ( - 462 \beta - 34992) q^{97} + (354 \beta + 6450) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 82 q^{5} + 98 q^{7} + 222 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 82 q^{5} + 98 q^{7} + 222 q^{9} + 340 q^{11} - 910 q^{13} - 1824 q^{15} + 3216 q^{17} - 674 q^{19} - 294 q^{21} + 1104 q^{23} + 3322 q^{25} - 3348 q^{27} - 8064 q^{29} + 6212 q^{31} + 3120 q^{33} - 4018 q^{35} + 8512 q^{37} + 29640 q^{39} - 1304 q^{41} - 10004 q^{43} + 3318 q^{45} + 12748 q^{47} + 4802 q^{49} - 5508 q^{51} + 11220 q^{53} - 26360 q^{55} - 29028 q^{57} - 12018 q^{59} - 102738 q^{61} + 10878 q^{63} - 43420 q^{65} + 24136 q^{67} - 52992 q^{69} - 89720 q^{71} - 55588 q^{73} + 159774 q^{75} + 16660 q^{77} - 48824 q^{79} - 122562 q^{81} + 35782 q^{83} - 144276 q^{85} - 54468 q^{87} - 18300 q^{89} - 44590 q^{91} + 126264 q^{93} + 120784 q^{95} - 69984 q^{97} + 12900 q^{99}+O(q^{100}) \) 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
9.78709
−8.78709
0 −21.5742 0 14.7225 0 49.0000 0 222.445 0
1.2 0 15.5742 0 −96.7225 0 49.0000 0 −0.445054 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.t 2
4.b odd 2 1 448.6.a.v 2
8.b even 2 1 112.6.a.i 2
8.d odd 2 1 56.6.a.e 2
24.f even 2 1 504.6.a.i 2
24.h odd 2 1 1008.6.a.bd 2
56.e even 2 1 392.6.a.d 2
56.h odd 2 1 784.6.a.u 2
56.k odd 6 2 392.6.i.j 4
56.m even 6 2 392.6.i.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.6.a.e 2 8.d odd 2 1
112.6.a.i 2 8.b even 2 1
392.6.a.d 2 56.e even 2 1
392.6.i.i 4 56.m even 6 2
392.6.i.j 4 56.k odd 6 2
448.6.a.t 2 1.a even 1 1 trivial
448.6.a.v 2 4.b odd 2 1
504.6.a.i 2 24.f even 2 1
784.6.a.u 2 56.h odd 2 1
1008.6.a.bd 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(448))\):

\( T_{3}^{2} + 6T_{3} - 336 \) Copy content Toggle raw display
\( T_{5}^{2} + 82T_{5} - 1424 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 6T - 336 \) Copy content Toggle raw display
$5$ \( T^{2} + 82T - 1424 \) Copy content Toggle raw display
$7$ \( (T - 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 340T + 16480 \) Copy content Toggle raw display
$13$ \( T^{2} + 910T - 317720 \) Copy content Toggle raw display
$17$ \( T^{2} - 3216 T + 2573244 \) Copy content Toggle raw display
$19$ \( T^{2} + 674T - 585056 \) Copy content Toggle raw display
$23$ \( T^{2} - 1104 T - 1483776 \) Copy content Toggle raw display
$29$ \( T^{2} + 8064 T + 11773404 \) Copy content Toggle raw display
$31$ \( T^{2} - 6212 T - 5567264 \) Copy content Toggle raw display
$37$ \( T^{2} - 8512 T - 34360964 \) Copy content Toggle raw display
$41$ \( T^{2} + 1304 T - 158165876 \) Copy content Toggle raw display
$43$ \( T^{2} + 10004 T - 194677376 \) Copy content Toggle raw display
$47$ \( T^{2} - 12748 T - 135251744 \) Copy content Toggle raw display
$53$ \( T^{2} - 11220 T - 172017180 \) Copy content Toggle raw display
$59$ \( T^{2} + 12018 T - 1715115024 \) Copy content Toggle raw display
$61$ \( T^{2} + 102738 T + 2636833536 \) Copy content Toggle raw display
$67$ \( T^{2} - 24136 T - 1158165296 \) Copy content Toggle raw display
$71$ \( T^{2} + 89720 T + 967897600 \) Copy content Toggle raw display
$73$ \( T^{2} + 55588 T - 243399884 \) Copy content Toggle raw display
$79$ \( T^{2} + 48824 T - 724002176 \) Copy content Toggle raw display
$83$ \( T^{2} - 35782 T - 3779136224 \) Copy content Toggle raw display
$89$ \( T^{2} + 18300 T - 7628401980 \) Copy content Toggle raw display
$97$ \( T^{2} + 69984 T + 1150801884 \) Copy content Toggle raw display
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