Properties

Label 224.3.o.a
Level $224$
Weight $3$
Character orbit 224.o
Analytic conductor $6.104$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{3} + ( - 3 \zeta_{6} - 3) q^{5} + (8 \zeta_{6} - 3) q^{7} + ( - 8 \zeta_{6} + 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{3} + ( - 3 \zeta_{6} - 3) q^{5} + (8 \zeta_{6} - 3) q^{7} + ( - 8 \zeta_{6} + 8) q^{9} + 17 \zeta_{6} q^{11} + (16 \zeta_{6} - 8) q^{13} + ( - 6 \zeta_{6} + 3) q^{15} + 25 \zeta_{6} q^{17} + (7 \zeta_{6} - 7) q^{19} + (5 \zeta_{6} - 8) q^{21} + ( - 3 \zeta_{6} - 3) q^{23} + 2 \zeta_{6} q^{25} + 17 q^{27} + ( - 16 \zeta_{6} + 8) q^{29} + (19 \zeta_{6} - 38) q^{31} + (17 \zeta_{6} - 17) q^{33} + ( - 39 \zeta_{6} + 33) q^{35} + (5 \zeta_{6} + 5) q^{37} + (8 \zeta_{6} - 16) q^{39} + 26 q^{41} - 14 q^{43} + (24 \zeta_{6} - 48) q^{45} + (29 \zeta_{6} + 29) q^{47} + (16 \zeta_{6} - 55) q^{49} + (25 \zeta_{6} - 25) q^{51} + ( - 53 \zeta_{6} + 106) q^{53} + ( - 102 \zeta_{6} + 51) q^{55} - 7 q^{57} - 55 \zeta_{6} q^{59} + (13 \zeta_{6} + 13) q^{61} + (24 \zeta_{6} + 40) q^{63} + ( - 72 \zeta_{6} + 72) q^{65} + 17 \zeta_{6} q^{67} + ( - 6 \zeta_{6} + 3) q^{69} - 119 \zeta_{6} q^{73} + (2 \zeta_{6} - 2) q^{75} + (85 \zeta_{6} - 136) q^{77} + ( - 43 \zeta_{6} - 43) q^{79} - 55 \zeta_{6} q^{81} - 110 q^{83} + ( - 150 \zeta_{6} + 75) q^{85} + ( - 8 \zeta_{6} + 16) q^{87} + (71 \zeta_{6} - 71) q^{89} + (16 \zeta_{6} - 104) q^{91} + ( - 19 \zeta_{6} - 19) q^{93} + ( - 21 \zeta_{6} + 42) q^{95} - 22 q^{97} + 136 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - 9 q^{5} + 2 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} - 9 q^{5} + 2 q^{7} + 8 q^{9} + 17 q^{11} + 25 q^{17} - 7 q^{19} - 11 q^{21} - 9 q^{23} + 2 q^{25} + 34 q^{27} - 57 q^{31} - 17 q^{33} + 27 q^{35} + 15 q^{37} - 24 q^{39} + 52 q^{41} - 28 q^{43} - 72 q^{45} + 87 q^{47} - 94 q^{49} - 25 q^{51} + 159 q^{53} - 14 q^{57} - 55 q^{59} + 39 q^{61} + 104 q^{63} + 72 q^{65} + 17 q^{67} - 119 q^{73} - 2 q^{75} - 187 q^{77} - 129 q^{79} - 55 q^{81} - 220 q^{83} + 24 q^{87} - 71 q^{89} - 192 q^{91} - 57 q^{93} + 63 q^{95} - 44 q^{97} + 272 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0.500000 + 0.866025i 0 −4.50000 2.59808i 0 1.00000 + 6.92820i 0 4.00000 6.92820i 0
207.1 0 0.500000 0.866025i 0 −4.50000 + 2.59808i 0 1.00000 6.92820i 0 4.00000 + 6.92820i 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
56.k odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.3.o.a 2
4.b odd 2 1 56.3.k.a 2
7.c even 3 1 224.3.o.b 2
7.c even 3 1 1568.3.g.c 2
7.d odd 6 1 1568.3.g.f 2
8.b even 2 1 56.3.k.b yes 2
8.d odd 2 1 224.3.o.b 2
28.d even 2 1 392.3.k.a 2
28.f even 6 1 392.3.g.d 2
28.f even 6 1 392.3.k.c 2
28.g odd 6 1 56.3.k.b yes 2
28.g odd 6 1 392.3.g.e 2
56.h odd 2 1 392.3.k.c 2
56.j odd 6 1 392.3.g.d 2
56.j odd 6 1 392.3.k.a 2
56.k odd 6 1 inner 224.3.o.a 2
56.k odd 6 1 1568.3.g.c 2
56.m even 6 1 1568.3.g.f 2
56.p even 6 1 56.3.k.a 2
56.p even 6 1 392.3.g.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.k.a 2 4.b odd 2 1
56.3.k.a 2 56.p even 6 1
56.3.k.b yes 2 8.b even 2 1
56.3.k.b yes 2 28.g odd 6 1
224.3.o.a 2 1.a even 1 1 trivial
224.3.o.a 2 56.k odd 6 1 inner
224.3.o.b 2 7.c even 3 1
224.3.o.b 2 8.d odd 2 1
392.3.g.d 2 28.f even 6 1
392.3.g.d 2 56.j odd 6 1
392.3.g.e 2 28.g odd 6 1
392.3.g.e 2 56.p even 6 1
392.3.k.a 2 28.d even 2 1
392.3.k.a 2 56.j odd 6 1
392.3.k.c 2 28.f even 6 1
392.3.k.c 2 56.h odd 2 1
1568.3.g.c 2 7.c even 3 1
1568.3.g.c 2 56.k odd 6 1
1568.3.g.f 2 7.d odd 6 1
1568.3.g.f 2 56.m even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(224, [\chi])\):

\( T_{3}^{2} - T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} + 9T_{5} + 27 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$7$ \( T^{2} - 2T + 49 \) Copy content Toggle raw display
$11$ \( T^{2} - 17T + 289 \) Copy content Toggle raw display
$13$ \( T^{2} + 192 \) Copy content Toggle raw display
$17$ \( T^{2} - 25T + 625 \) Copy content Toggle raw display
$19$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$23$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$29$ \( T^{2} + 192 \) Copy content Toggle raw display
$31$ \( T^{2} + 57T + 1083 \) Copy content Toggle raw display
$37$ \( T^{2} - 15T + 75 \) Copy content Toggle raw display
$41$ \( (T - 26)^{2} \) Copy content Toggle raw display
$43$ \( (T + 14)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 87T + 2523 \) Copy content Toggle raw display
$53$ \( T^{2} - 159T + 8427 \) Copy content Toggle raw display
$59$ \( T^{2} + 55T + 3025 \) Copy content Toggle raw display
$61$ \( T^{2} - 39T + 507 \) Copy content Toggle raw display
$67$ \( T^{2} - 17T + 289 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 119T + 14161 \) Copy content Toggle raw display
$79$ \( T^{2} + 129T + 5547 \) Copy content Toggle raw display
$83$ \( (T + 110)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 71T + 5041 \) Copy content Toggle raw display
$97$ \( (T + 22)^{2} \) Copy content Toggle raw display
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