Properties

Label 224.3.r.b
Level $224$
Weight $3$
Character orbit 224.r
Analytic conductor $6.104$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{3} - \zeta_{12}^{2} q^{5} - 7 \zeta_{12}^{3} q^{7} - 8 \zeta_{12}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12} q^{3} - \zeta_{12}^{2} q^{5} - 7 \zeta_{12}^{3} q^{7} - 8 \zeta_{12}^{2} q^{9} - 17 \zeta_{12} q^{11} + 24 q^{13} - \zeta_{12}^{3} q^{15} + (\zeta_{12}^{2} - 1) q^{17} + ( - 7 \zeta_{12}^{3} + 7 \zeta_{12}) q^{19} + ( - 7 \zeta_{12}^{2} + 7) q^{21} + (7 \zeta_{12}^{3} - 7 \zeta_{12}) q^{23} + ( - 24 \zeta_{12}^{2} + 24) q^{25} - 17 \zeta_{12}^{3} q^{27} + 24 q^{29} - 41 \zeta_{12} q^{31} - 17 \zeta_{12}^{2} q^{33} + (7 \zeta_{12}^{3} - 7 \zeta_{12}) q^{35} + 49 \zeta_{12}^{2} q^{37} + 24 \zeta_{12} q^{39} - 48 q^{41} + 24 \zeta_{12}^{3} q^{43} + (8 \zeta_{12}^{2} - 8) q^{45} + ( - 55 \zeta_{12}^{3} + 55 \zeta_{12}) q^{47} - 49 q^{49} + (\zeta_{12}^{3} - \zeta_{12}) q^{51} + ( - 25 \zeta_{12}^{2} + 25) q^{53} + 17 \zeta_{12}^{3} q^{55} + 7 q^{57} + 17 \zeta_{12} q^{59} + \zeta_{12}^{2} q^{61} + (56 \zeta_{12}^{3} - 56 \zeta_{12}) q^{63} - 24 \zeta_{12}^{2} q^{65} + 65 \zeta_{12} q^{67} - 7 q^{69} + 96 \zeta_{12}^{3} q^{71} + (95 \zeta_{12}^{2} - 95) q^{73} + ( - 24 \zeta_{12}^{3} + 24 \zeta_{12}) q^{75} + (119 \zeta_{12}^{2} - 119) q^{77} + ( - 41 \zeta_{12}^{3} + 41 \zeta_{12}) q^{79} + (55 \zeta_{12}^{2} - 55) q^{81} + 72 \zeta_{12}^{3} q^{83} + q^{85} + 24 \zeta_{12} q^{87} - 95 \zeta_{12}^{2} q^{89} - 168 \zeta_{12}^{3} q^{91} - 41 \zeta_{12}^{2} q^{93} - 7 \zeta_{12} q^{95} + 144 q^{97} + 136 \zeta_{12}^{3} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{5} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{5} - 16 q^{9} + 96 q^{13} - 2 q^{17} + 14 q^{21} + 48 q^{25} + 96 q^{29} - 34 q^{33} + 98 q^{37} - 192 q^{41} - 16 q^{45} - 196 q^{49} + 50 q^{53} + 28 q^{57} + 2 q^{61} - 48 q^{65} - 28 q^{69} - 190 q^{73} - 238 q^{77} - 110 q^{81} + 4 q^{85} - 190 q^{89} - 82 q^{93} + 576 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
95.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0 −0.866025 0.500000i 0 −0.500000 0.866025i 0 7.00000i 0 −4.00000 6.92820i 0
95.2 0 0.866025 + 0.500000i 0 −0.500000 0.866025i 0 7.00000i 0 −4.00000 6.92820i 0
191.1 0 −0.866025 + 0.500000i 0 −0.500000 + 0.866025i 0 7.00000i 0 −4.00000 + 6.92820i 0
191.2 0 0.866025 0.500000i 0 −0.500000 + 0.866025i 0 7.00000i 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
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 224.3.r.b 4
4.b odd 2 1 inner 224.3.r.b 4
7.c even 3 1 inner 224.3.r.b 4
7.c even 3 1 1568.3.d.d 2
7.d odd 6 1 1568.3.d.c 2
8.b even 2 1 448.3.r.b 4
8.d odd 2 1 448.3.r.b 4
28.f even 6 1 1568.3.d.c 2
28.g odd 6 1 inner 224.3.r.b 4
28.g odd 6 1 1568.3.d.d 2
56.k odd 6 1 448.3.r.b 4
56.p even 6 1 448.3.r.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.3.r.b 4 1.a even 1 1 trivial
224.3.r.b 4 4.b odd 2 1 inner
224.3.r.b 4 7.c even 3 1 inner
224.3.r.b 4 28.g odd 6 1 inner
448.3.r.b 4 8.b even 2 1
448.3.r.b 4 8.d odd 2 1
448.3.r.b 4 56.k odd 6 1
448.3.r.b 4 56.p even 6 1
1568.3.d.c 2 7.d odd 6 1
1568.3.d.c 2 28.f even 6 1
1568.3.d.d 2 7.c even 3 1
1568.3.d.d 2 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - T_{3}^{2} + 1 \) acting on \(S_{3}^{\mathrm{new}}(224, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 289 T^{2} + 83521 \) Copy content Toggle raw display
$13$ \( (T - 24)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 49T^{2} + 2401 \) Copy content Toggle raw display
$23$ \( T^{4} - 49T^{2} + 2401 \) Copy content Toggle raw display
$29$ \( (T - 24)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} - 1681 T^{2} + \cdots + 2825761 \) Copy content Toggle raw display
$37$ \( (T^{2} - 49 T + 2401)^{2} \) Copy content Toggle raw display
$41$ \( (T + 48)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 576)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 3025 T^{2} + \cdots + 9150625 \) Copy content Toggle raw display
$53$ \( (T^{2} - 25 T + 625)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 289 T^{2} + 83521 \) Copy content Toggle raw display
$61$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 4225 T^{2} + \cdots + 17850625 \) Copy content Toggle raw display
$71$ \( (T^{2} + 9216)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 95 T + 9025)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 1681 T^{2} + \cdots + 2825761 \) Copy content Toggle raw display
$83$ \( (T^{2} + 5184)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 95 T + 9025)^{2} \) Copy content Toggle raw display
$97$ \( (T - 144)^{4} \) Copy content Toggle raw display
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