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: \(\Q(\sqrt{-3}) \)
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.

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

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 \(S_{3}^{\mathrm{new}}(224, [\chi])\):

\( T_{3}^{2} - T_{3} + 1 \)
\( T_{5}^{2} + 9 T_{5} + 27 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - T - 8 T^{2} - 9 T^{3} + 81 T^{4} \)
$5$ \( 1 + 9 T + 52 T^{2} + 225 T^{3} + 625 T^{4} \)
$7$ \( 1 - 2 T + 49 T^{2} \)
$11$ \( 1 - 17 T + 168 T^{2} - 2057 T^{3} + 14641 T^{4} \)
$13$ \( ( 1 - 22 T + 169 T^{2} )( 1 + 22 T + 169 T^{2} ) \)
$17$ \( 1 - 25 T + 336 T^{2} - 7225 T^{3} + 83521 T^{4} \)
$19$ \( 1 + 7 T - 312 T^{2} + 2527 T^{3} + 130321 T^{4} \)
$23$ \( 1 + 9 T + 556 T^{2} + 4761 T^{3} + 279841 T^{4} \)
$29$ \( 1 - 1490 T^{2} + 707281 T^{4} \)
$31$ \( 1 + 57 T + 2044 T^{2} + 54777 T^{3} + 923521 T^{4} \)
$37$ \( 1 - 15 T + 1444 T^{2} - 20535 T^{3} + 1874161 T^{4} \)
$41$ \( ( 1 - 26 T + 1681 T^{2} )^{2} \)
$43$ \( ( 1 + 14 T + 1849 T^{2} )^{2} \)
$47$ \( 1 - 87 T + 4732 T^{2} - 192183 T^{3} + 4879681 T^{4} \)
$53$ \( ( 1 - 53 T )^{2}( 1 - 53 T + 2809 T^{2} ) \)
$59$ \( 1 + 55 T - 456 T^{2} + 191455 T^{3} + 12117361 T^{4} \)
$61$ \( 1 - 39 T + 4228 T^{2} - 145119 T^{3} + 13845841 T^{4} \)
$67$ \( 1 - 17 T - 4200 T^{2} - 76313 T^{3} + 20151121 T^{4} \)
$71$ \( ( 1 - 71 T )^{2}( 1 + 71 T )^{2} \)
$73$ \( 1 + 119 T + 8832 T^{2} + 634151 T^{3} + 28398241 T^{4} \)
$79$ \( 1 + 129 T + 11788 T^{2} + 805089 T^{3} + 38950081 T^{4} \)
$83$ \( ( 1 + 110 T + 6889 T^{2} )^{2} \)
$89$ \( 1 + 71 T - 2880 T^{2} + 562391 T^{3} + 62742241 T^{4} \)
$97$ \( ( 1 + 22 T + 9409 T^{2} )^{2} \)
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