Properties

Label 448.3.r.c
Level 448
Weight 3
Character orbit 448.r
Analytic conductor 12.207
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 448.r (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.2071158433\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 224)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 5 \zeta_{12} q^{3} + 9 \zeta_{12}^{2} q^{5} + ( -8 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{7} + 16 \zeta_{12}^{2} q^{9} +O(q^{10})\) \( q + 5 \zeta_{12} q^{3} + 9 \zeta_{12}^{2} q^{5} + ( -8 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{7} + 16 \zeta_{12}^{2} q^{9} + 3 \zeta_{12} q^{11} -16 q^{13} + 45 \zeta_{12}^{3} q^{15} + ( 7 - 7 \zeta_{12}^{2} ) q^{17} + ( 11 \zeta_{12} - 11 \zeta_{12}^{3} ) q^{19} + ( -15 - 25 \zeta_{12}^{2} ) q^{21} + ( 19 \zeta_{12} - 19 \zeta_{12}^{3} ) q^{23} + ( -56 + 56 \zeta_{12}^{2} ) q^{25} + 35 \zeta_{12}^{3} q^{27} + 32 q^{29} -11 \zeta_{12} q^{31} + 15 \zeta_{12}^{2} q^{33} + ( -27 \zeta_{12} - 45 \zeta_{12}^{3} ) q^{35} -\zeta_{12}^{2} q^{37} -80 \zeta_{12} q^{39} -40 q^{41} -40 \zeta_{12}^{3} q^{43} + ( -144 + 144 \zeta_{12}^{2} ) q^{45} + ( 85 \zeta_{12} - 85 \zeta_{12}^{3} ) q^{47} + ( 39 + 16 \zeta_{12}^{2} ) q^{49} + ( 35 \zeta_{12} - 35 \zeta_{12}^{3} ) q^{51} + ( 7 - 7 \zeta_{12}^{2} ) q^{53} + 27 \zeta_{12}^{3} q^{55} + 55 q^{57} + 53 \zeta_{12} q^{59} + 79 \zeta_{12}^{2} q^{61} + ( -48 \zeta_{12} - 80 \zeta_{12}^{3} ) q^{63} -144 \zeta_{12}^{2} q^{65} -11 \zeta_{12} q^{67} + 95 q^{69} + 48 \zeta_{12}^{3} q^{71} + ( -143 + 143 \zeta_{12}^{2} ) q^{73} + ( -280 \zeta_{12} + 280 \zeta_{12}^{3} ) q^{75} + ( -9 - 15 \zeta_{12}^{2} ) q^{77} + ( 35 \zeta_{12} - 35 \zeta_{12}^{3} ) q^{79} + ( -31 + 31 \zeta_{12}^{2} ) q^{81} -8 \zeta_{12}^{3} q^{83} + 63 q^{85} + 160 \zeta_{12} q^{87} + 97 \zeta_{12}^{2} q^{89} + ( 128 \zeta_{12} - 48 \zeta_{12}^{3} ) q^{91} -55 \zeta_{12}^{2} q^{93} + 99 \zeta_{12} q^{95} -88 q^{97} + 48 \zeta_{12}^{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 18q^{5} + 32q^{9} + O(q^{10}) \) \( 4q + 18q^{5} + 32q^{9} - 64q^{13} + 14q^{17} - 110q^{21} - 112q^{25} + 128q^{29} + 30q^{33} - 2q^{37} - 160q^{41} - 288q^{45} + 188q^{49} + 14q^{53} + 220q^{57} + 158q^{61} - 288q^{65} + 380q^{69} - 286q^{73} - 66q^{77} - 62q^{81} + 252q^{85} + 194q^{89} - 110q^{93} - 352q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(-\zeta_{12}^{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
0 −4.33013 + 2.50000i 0 4.50000 7.79423i 0 6.92820 1.00000i 0 8.00000 13.8564i 0
191.2 0 4.33013 2.50000i 0 4.50000 7.79423i 0 −6.92820 + 1.00000i 0 8.00000 13.8564i 0
319.1 0 −4.33013 2.50000i 0 4.50000 + 7.79423i 0 6.92820 + 1.00000i 0 8.00000 + 13.8564i 0
319.2 0 4.33013 + 2.50000i 0 4.50000 + 7.79423i 0 −6.92820 1.00000i 0 8.00000 + 13.8564i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.c even 3 1 inner
28.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.3.r.c 4
4.b odd 2 1 inner 448.3.r.c 4
7.c even 3 1 inner 448.3.r.c 4
8.b even 2 1 224.3.r.a 4
8.d odd 2 1 224.3.r.a 4
28.g odd 6 1 inner 448.3.r.c 4
56.j odd 6 1 1568.3.d.a 2
56.k odd 6 1 224.3.r.a 4
56.k odd 6 1 1568.3.d.e 2
56.m even 6 1 1568.3.d.a 2
56.p even 6 1 224.3.r.a 4
56.p even 6 1 1568.3.d.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.3.r.a 4 8.b even 2 1
224.3.r.a 4 8.d odd 2 1
224.3.r.a 4 56.k odd 6 1
224.3.r.a 4 56.p even 6 1
448.3.r.c 4 1.a even 1 1 trivial
448.3.r.c 4 4.b odd 2 1 inner
448.3.r.c 4 7.c even 3 1 inner
448.3.r.c 4 28.g odd 6 1 inner
1568.3.d.a 2 56.j odd 6 1
1568.3.d.a 2 56.m even 6 1
1568.3.d.e 2 56.k odd 6 1
1568.3.d.e 2 56.p even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 25 T_{3}^{2} + 625 \) acting on \(S_{3}^{\mathrm{new}}(448, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 7 T^{2} - 32 T^{4} - 567 T^{6} + 6561 T^{8} \)
$5$ \( ( 1 - 9 T + 56 T^{2} - 225 T^{3} + 625 T^{4} )^{2} \)
$7$ \( 1 - 94 T^{2} + 2401 T^{4} \)
$11$ \( 1 + 233 T^{2} + 39648 T^{4} + 3411353 T^{6} + 214358881 T^{8} \)
$13$ \( ( 1 + 16 T + 169 T^{2} )^{4} \)
$17$ \( ( 1 - 7 T - 240 T^{2} - 2023 T^{3} + 83521 T^{4} )^{2} \)
$19$ \( ( 1 - 46 T^{2} + 130321 T^{4} )( 1 + 647 T^{2} + 130321 T^{4} ) \)
$23$ \( 1 + 697 T^{2} + 205968 T^{4} + 195049177 T^{6} + 78310985281 T^{8} \)
$29$ \( ( 1 - 32 T + 841 T^{2} )^{4} \)
$31$ \( 1 + 1801 T^{2} + 2320080 T^{4} + 1663261321 T^{6} + 852891037441 T^{8} \)
$37$ \( ( 1 + T - 1368 T^{2} + 1369 T^{3} + 1874161 T^{4} )^{2} \)
$41$ \( ( 1 + 40 T + 1681 T^{2} )^{4} \)
$43$ \( ( 1 - 2098 T^{2} + 3418801 T^{4} )^{2} \)
$47$ \( 1 - 2807 T^{2} + 2999568 T^{4} - 13697264567 T^{6} + 23811286661761 T^{8} \)
$53$ \( ( 1 - 7 T - 2760 T^{2} - 19663 T^{3} + 7890481 T^{4} )^{2} \)
$59$ \( 1 + 4153 T^{2} + 5130048 T^{4} + 50323400233 T^{6} + 146830437604321 T^{8} \)
$61$ \( ( 1 - 79 T + 2520 T^{2} - 293959 T^{3} + 13845841 T^{4} )^{2} \)
$67$ \( 1 + 8857 T^{2} + 58295328 T^{4} + 178478478697 T^{6} + 406067677556641 T^{8} \)
$71$ \( ( 1 - 7778 T^{2} + 25411681 T^{4} )^{2} \)
$73$ \( ( 1 + 46 T + 5329 T^{2} )^{2}( 1 + 97 T + 5329 T^{2} )^{2} \)
$79$ \( 1 + 11257 T^{2} + 87769968 T^{4} + 438461061817 T^{6} + 1517108809906561 T^{8} \)
$83$ \( ( 1 - 13714 T^{2} + 47458321 T^{4} )^{2} \)
$89$ \( ( 1 - 97 T + 1488 T^{2} - 768337 T^{3} + 62742241 T^{4} )^{2} \)
$97$ \( ( 1 + 88 T + 9409 T^{2} )^{4} \)
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