Properties

Label 448.4.i.e
Level $448$
Weight $4$
Character orbit 448.i
Analytic conductor $26.433$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 448.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.4328556826\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 5 \zeta_{6} + 5) q^{3} - 9 \zeta_{6} q^{5} + ( - 14 \zeta_{6} + 21) q^{7} + 2 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 5 \zeta_{6} + 5) q^{3} - 9 \zeta_{6} q^{5} + ( - 14 \zeta_{6} + 21) q^{7} + 2 \zeta_{6} q^{9} + ( - 57 \zeta_{6} + 57) q^{11} + 70 q^{13} - 45 q^{15} + (51 \zeta_{6} - 51) q^{17} - 5 \zeta_{6} q^{19} + ( - 105 \zeta_{6} + 35) q^{21} + 69 \zeta_{6} q^{23} + ( - 44 \zeta_{6} + 44) q^{25} + 145 q^{27} - 114 q^{29} + ( - 23 \zeta_{6} + 23) q^{31} - 285 \zeta_{6} q^{33} + ( - 63 \zeta_{6} - 126) q^{35} - 253 \zeta_{6} q^{37} + ( - 350 \zeta_{6} + 350) q^{39} - 42 q^{41} - 124 q^{43} + ( - 18 \zeta_{6} + 18) q^{45} + 201 \zeta_{6} q^{47} + ( - 392 \zeta_{6} + 245) q^{49} + 255 \zeta_{6} q^{51} + (393 \zeta_{6} - 393) q^{53} - 513 q^{55} - 25 q^{57} + (219 \zeta_{6} - 219) q^{59} - 709 \zeta_{6} q^{61} + (14 \zeta_{6} + 28) q^{63} - 630 \zeta_{6} q^{65} + (419 \zeta_{6} - 419) q^{67} + 345 q^{69} + 96 q^{71} + ( - 313 \zeta_{6} + 313) q^{73} - 220 \zeta_{6} q^{75} + ( - 1197 \zeta_{6} + 399) q^{77} + 461 \zeta_{6} q^{79} + ( - 671 \zeta_{6} + 671) q^{81} - 588 q^{83} + 459 q^{85} + (570 \zeta_{6} - 570) q^{87} + 1017 \zeta_{6} q^{89} + ( - 980 \zeta_{6} + 1470) q^{91} - 115 \zeta_{6} q^{93} + (45 \zeta_{6} - 45) q^{95} - 1834 q^{97} + 114 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{3} - 9 q^{5} + 28 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 5 q^{3} - 9 q^{5} + 28 q^{7} + 2 q^{9} + 57 q^{11} + 140 q^{13} - 90 q^{15} - 51 q^{17} - 5 q^{19} - 35 q^{21} + 69 q^{23} + 44 q^{25} + 290 q^{27} - 228 q^{29} + 23 q^{31} - 285 q^{33} - 315 q^{35} - 253 q^{37} + 350 q^{39} - 84 q^{41} - 248 q^{43} + 18 q^{45} + 201 q^{47} + 98 q^{49} + 255 q^{51} - 393 q^{53} - 1026 q^{55} - 50 q^{57} - 219 q^{59} - 709 q^{61} + 70 q^{63} - 630 q^{65} - 419 q^{67} + 690 q^{69} + 192 q^{71} + 313 q^{73} - 220 q^{75} - 399 q^{77} + 461 q^{79} + 671 q^{81} - 1176 q^{83} + 918 q^{85} - 570 q^{87} + 1017 q^{89} + 1960 q^{91} - 115 q^{93} - 45 q^{95} - 3668 q^{97} + 228 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
0.500000 0.866025i
0.500000 + 0.866025i
0 2.50000 + 4.33013i 0 −4.50000 + 7.79423i 0 14.0000 + 12.1244i 0 1.00000 1.73205i 0
193.1 0 2.50000 4.33013i 0 −4.50000 7.79423i 0 14.0000 12.1244i 0 1.00000 + 1.73205i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.4.i.e 2
4.b odd 2 1 448.4.i.b 2
7.c even 3 1 inner 448.4.i.e 2
8.b even 2 1 112.4.i.a 2
8.d odd 2 1 14.4.c.a 2
24.f even 2 1 126.4.g.d 2
28.g odd 6 1 448.4.i.b 2
40.e odd 2 1 350.4.e.e 2
40.k even 4 2 350.4.j.b 4
56.e even 2 1 98.4.c.a 2
56.j odd 6 1 784.4.a.c 1
56.k odd 6 1 14.4.c.a 2
56.k odd 6 1 98.4.a.d 1
56.m even 6 1 98.4.a.f 1
56.m even 6 1 98.4.c.a 2
56.p even 6 1 112.4.i.a 2
56.p even 6 1 784.4.a.p 1
168.e odd 2 1 882.4.g.u 2
168.v even 6 1 126.4.g.d 2
168.v even 6 1 882.4.a.f 1
168.be odd 6 1 882.4.a.c 1
168.be odd 6 1 882.4.g.u 2
280.ba even 6 1 2450.4.a.d 1
280.bi odd 6 1 350.4.e.e 2
280.bi odd 6 1 2450.4.a.q 1
280.br even 12 2 350.4.j.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.a 2 8.d odd 2 1
14.4.c.a 2 56.k odd 6 1
98.4.a.d 1 56.k odd 6 1
98.4.a.f 1 56.m even 6 1
98.4.c.a 2 56.e even 2 1
98.4.c.a 2 56.m even 6 1
112.4.i.a 2 8.b even 2 1
112.4.i.a 2 56.p even 6 1
126.4.g.d 2 24.f even 2 1
126.4.g.d 2 168.v even 6 1
350.4.e.e 2 40.e odd 2 1
350.4.e.e 2 280.bi odd 6 1
350.4.j.b 4 40.k even 4 2
350.4.j.b 4 280.br even 12 2
448.4.i.b 2 4.b odd 2 1
448.4.i.b 2 28.g odd 6 1
448.4.i.e 2 1.a even 1 1 trivial
448.4.i.e 2 7.c even 3 1 inner
784.4.a.c 1 56.j odd 6 1
784.4.a.p 1 56.p even 6 1
882.4.a.c 1 168.be odd 6 1
882.4.a.f 1 168.v even 6 1
882.4.g.u 2 168.e odd 2 1
882.4.g.u 2 168.be odd 6 1
2450.4.a.d 1 280.ba even 6 1
2450.4.a.q 1 280.bi odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(448, [\chi])\):

\( T_{3}^{2} - 5T_{3} + 25 \) Copy content Toggle raw display
\( T_{11}^{2} - 57T_{11} + 3249 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$5$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$7$ \( T^{2} - 28T + 343 \) Copy content Toggle raw display
$11$ \( T^{2} - 57T + 3249 \) Copy content Toggle raw display
$13$ \( (T - 70)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 51T + 2601 \) Copy content Toggle raw display
$19$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$23$ \( T^{2} - 69T + 4761 \) Copy content Toggle raw display
$29$ \( (T + 114)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 23T + 529 \) Copy content Toggle raw display
$37$ \( T^{2} + 253T + 64009 \) Copy content Toggle raw display
$41$ \( (T + 42)^{2} \) Copy content Toggle raw display
$43$ \( (T + 124)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 201T + 40401 \) Copy content Toggle raw display
$53$ \( T^{2} + 393T + 154449 \) Copy content Toggle raw display
$59$ \( T^{2} + 219T + 47961 \) Copy content Toggle raw display
$61$ \( T^{2} + 709T + 502681 \) Copy content Toggle raw display
$67$ \( T^{2} + 419T + 175561 \) Copy content Toggle raw display
$71$ \( (T - 96)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 313T + 97969 \) Copy content Toggle raw display
$79$ \( T^{2} - 461T + 212521 \) Copy content Toggle raw display
$83$ \( (T + 588)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 1017 T + 1034289 \) Copy content Toggle raw display
$97$ \( (T + 1834)^{2} \) Copy content Toggle raw display
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