Properties

Label 224.3.g.a
Level $224$
Weight $3$
Character orbit 224.g
Analytic conductor $6.104$
Analytic rank $0$
Dimension $4$
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.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-7})\)
Defining polynomial: \( x^{4} + 6x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 2) q^{3} + (2 \beta_{3} - \beta_{2}) q^{5} - \beta_{3} q^{7} + ( - 4 \beta_1 - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 2) q^{3} + (2 \beta_{3} - \beta_{2}) q^{5} - \beta_{3} q^{7} + ( - 4 \beta_1 - 3) q^{9} + ( - 6 \beta_1 - 4) q^{11} + ( - 2 \beta_{3} + \beta_{2}) q^{13} - 2 \beta_{3} q^{15} + ( - 4 \beta_1 + 18) q^{17} + ( - 19 \beta_1 - 2) q^{19} + (2 \beta_{3} + \beta_{2}) q^{21} + ( - 2 \beta_{3} - 8 \beta_{2}) q^{23} + ( - 28 \beta_1 - 17) q^{25} + ( - 4 \beta_1 + 16) q^{27} - 6 \beta_{2} q^{29} + (12 \beta_{3} + 4 \beta_{2}) q^{31} + (8 \beta_1 - 4) q^{33} + (7 \beta_1 + 14) q^{35} + ( - 8 \beta_{3} - 10 \beta_{2}) q^{37} + 2 \beta_{3} q^{39} + (12 \beta_1 - 10) q^{41} + (2 \beta_1 + 20) q^{43} + ( - 14 \beta_{3} + 11 \beta_{2}) q^{45} + (8 \beta_{3} + 4 \beta_{2}) q^{47} - 7 q^{49} + (26 \beta_1 - 44) q^{51} + (20 \beta_{3} + 12 \beta_{2}) q^{53} + ( - 20 \beta_{3} + 16 \beta_{2}) q^{55} + (36 \beta_1 - 34) q^{57} + (11 \beta_1 - 46) q^{59} + ( - 10 \beta_{3} - 3 \beta_{2}) q^{61} + (3 \beta_{3} - 4 \beta_{2}) q^{63} + (28 \beta_1 + 42) q^{65} + (16 \beta_1 + 56) q^{67} + (20 \beta_{3} + 18 \beta_{2}) q^{69} + ( - 16 \beta_{3} - 16 \beta_{2}) q^{71} + ( - 8 \beta_1 + 58) q^{73} + (39 \beta_1 - 22) q^{75} + (4 \beta_{3} - 6 \beta_{2}) q^{77} + (8 \beta_{3} - 16 \beta_{2}) q^{79} + (60 \beta_1 - 13) q^{81} + ( - 13 \beta_1 - 22) q^{83} + (28 \beta_{3} - 10 \beta_{2}) q^{85} + (12 \beta_{3} + 12 \beta_{2}) q^{87} + (24 \beta_1 + 78) q^{89} + ( - 7 \beta_1 - 14) q^{91} + ( - 32 \beta_{3} - 20 \beta_{2}) q^{93} + ( - 42 \beta_{3} + 40 \beta_{2}) q^{95} + ( - 92 \beta_1 - 34) q^{97} + (34 \beta_1 + 60) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{3} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{3} - 12 q^{9} - 16 q^{11} + 72 q^{17} - 8 q^{19} - 68 q^{25} + 64 q^{27} - 16 q^{33} + 56 q^{35} - 40 q^{41} + 80 q^{43} - 28 q^{49} - 176 q^{51} - 136 q^{57} - 184 q^{59} + 168 q^{65} + 224 q^{67} + 232 q^{73} - 88 q^{75} - 52 q^{81} - 88 q^{83} + 312 q^{89} - 56 q^{91} - 136 q^{97} + 240 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 6x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 2\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 10\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{2} + 5\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1
0.707107 1.87083i
0.707107 + 1.87083i
−0.707107 + 1.87083i
−0.707107 1.87083i
0 −3.41421 0 1.54985i 0 2.64575i 0 2.65685 0
15.2 0 −3.41421 0 1.54985i 0 2.64575i 0 2.65685 0
15.3 0 −0.585786 0 9.03316i 0 2.64575i 0 −8.65685 0
15.4 0 −0.585786 0 9.03316i 0 2.64575i 0 −8.65685 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
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.3.g.a 4
3.b odd 2 1 2016.3.g.a 4
4.b odd 2 1 56.3.g.a 4
7.b odd 2 1 1568.3.g.h 4
8.b even 2 1 56.3.g.a 4
8.d odd 2 1 inner 224.3.g.a 4
12.b even 2 1 504.3.g.a 4
16.e even 4 2 1792.3.d.g 8
16.f odd 4 2 1792.3.d.g 8
24.f even 2 1 2016.3.g.a 4
24.h odd 2 1 504.3.g.a 4
28.d even 2 1 392.3.g.h 4
28.f even 6 2 392.3.k.j 8
28.g odd 6 2 392.3.k.i 8
56.e even 2 1 1568.3.g.h 4
56.h odd 2 1 392.3.g.h 4
56.j odd 6 2 392.3.k.j 8
56.p even 6 2 392.3.k.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.g.a 4 4.b odd 2 1
56.3.g.a 4 8.b even 2 1
224.3.g.a 4 1.a even 1 1 trivial
224.3.g.a 4 8.d odd 2 1 inner
392.3.g.h 4 28.d even 2 1
392.3.g.h 4 56.h odd 2 1
392.3.k.i 8 28.g odd 6 2
392.3.k.i 8 56.p even 6 2
392.3.k.j 8 28.f even 6 2
392.3.k.j 8 56.j odd 6 2
504.3.g.a 4 12.b even 2 1
504.3.g.a 4 24.h odd 2 1
1568.3.g.h 4 7.b odd 2 1
1568.3.g.h 4 56.e even 2 1
1792.3.d.g 8 16.e even 4 2
1792.3.d.g 8 16.f odd 4 2
2016.3.g.a 4 3.b odd 2 1
2016.3.g.a 4 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 4T_{3} + 2 \) 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
$3$ \( (T^{2} + 4 T + 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 84T^{2} + 196 \) Copy content Toggle raw display
$7$ \( (T^{2} + 7)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 8 T - 56)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 84T^{2} + 196 \) Copy content Toggle raw display
$17$ \( (T^{2} - 36 T + 292)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 4 T - 718)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 1848 T^{2} + 753424 \) Copy content Toggle raw display
$29$ \( (T^{2} + 504)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 2464 T^{2} + 614656 \) Copy content Toggle raw display
$37$ \( T^{4} + 3696 T^{2} + 906304 \) Copy content Toggle raw display
$41$ \( (T^{2} + 20 T - 188)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 40 T + 392)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 1344 T^{2} + 50176 \) Copy content Toggle raw display
$53$ \( T^{4} + 9632 T^{2} + 614656 \) Copy content Toggle raw display
$59$ \( (T^{2} + 92 T + 1874)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 1652 T^{2} + 329476 \) Copy content Toggle raw display
$67$ \( (T^{2} - 112 T + 2624)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 10752 T^{2} + \cdots + 3211264 \) Copy content Toggle raw display
$73$ \( (T^{2} - 116 T + 3236)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 8064 T^{2} + \cdots + 9834496 \) Copy content Toggle raw display
$83$ \( (T^{2} + 44 T + 146)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 156 T + 4932)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 68 T - 15772)^{2} \) Copy content Toggle raw display
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