Properties

Label 224.3.c.a
Level $224$
Weight $3$
Character orbit 224.c
Analytic conductor $6.104$
Analytic rank $0$
Dimension $8$
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.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.4694952902656.3
Defining polynomial: \( x^{8} + 20x^{6} + 56x^{4} + 20x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{3} + \beta_{7} q^{5} + \beta_{2} q^{7} + ( - \beta_{5} - 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{3} + \beta_{7} q^{5} + \beta_{2} q^{7} + ( - \beta_{5} - 7) q^{9} + (\beta_{3} + \beta_{2} - 2 \beta_1) q^{11} + (\beta_{7} - \beta_{4}) q^{13} + ( - 2 \beta_{3} - 2 \beta_{2} + 5 \beta_1) q^{15} + ( - 2 \beta_{7} - 3 \beta_{4}) q^{17} + (\beta_{6} + \beta_{3} - \beta_{2} - \beta_1) q^{19} + (5 \beta_{7} - \beta_{5} + \beta_{4} - 4) q^{21} + ( - \beta_{3} - \beta_{2} - 4 \beta_1) q^{23} + ( - \beta_{5} + 1) q^{25} + (8 \beta_{6} + 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{27} + ( - 2 \beta_{5} - 14) q^{29} + (2 \beta_{6} - 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{31} + (12 \beta_{7} + 3 \beta_{4}) q^{33} + ( - 7 \beta_{6} + 7 \beta_1) q^{35} + (2 \beta_{5} + 26) q^{37} + ( - 2 \beta_{3} - 2 \beta_{2} - \beta_1) q^{39} + ( - 6 \beta_{7} - 2 \beta_{4}) q^{41} + (5 \beta_{3} + 5 \beta_{2} + 4 \beta_1) q^{43} + ( - 17 \beta_{7} - 7 \beta_{4}) q^{45} + ( - 2 \beta_{6} - 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{47} + (4 \beta_{7} + 2 \beta_{5} + 5 \beta_{4} + 1) q^{49} + (4 \beta_{3} + 4 \beta_{2} - 28 \beta_1) q^{51} - 30 q^{53} + ( - 16 \beta_{6} - \beta_{3} + \beta_{2} + \beta_1) q^{55} + (3 \beta_{5} + 24) q^{57} + ( - 17 \beta_{6} + 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{59} + ( - 5 \beta_{7} + 8 \beta_{4}) q^{61} + (14 \beta_{6} - 7 \beta_{3} - 4 \beta_{2} + 28 \beta_1) q^{63} + (\beta_{5} - 32) q^{65} + ( - 3 \beta_{3} - 3 \beta_{2}) q^{67} + 3 \beta_{4} q^{69} + (5 \beta_{3} + 5 \beta_{2} + 15 \beta_1) q^{71} + ( - 16 \beta_{7} + \beta_{4}) q^{73} + (9 \beta_{6} + 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{75} + (6 \beta_{7} + 3 \beta_{5} + 4 \beta_{4} + 54) q^{77} + ( - 3 \beta_{3} - 3 \beta_{2} - 39 \beta_1) q^{79} + (5 \beta_{5} + 89) q^{81} + (23 \beta_{6} - 6 \beta_{3} + 6 \beta_{2} + 6 \beta_1) q^{83} + (8 \beta_{5} + 24) q^{85} + (34 \beta_{6} + 6 \beta_{3} - 6 \beta_{2} - 6 \beta_1) q^{87} + (4 \beta_{7} + \beta_{4}) q^{89} + ( - 7 \beta_{6} + 7 \beta_{3} + \beta_{2} + 14 \beta_1) q^{91} + ( - 4 \beta_{5} + 8) q^{93} + (2 \beta_{3} + 2 \beta_{2} - 19 \beta_1) q^{95} + ( - 18 \beta_{7} + 3 \beta_{4}) q^{97} + ( - 15 \beta_{3} - 15 \beta_{2} + 60 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 56 q^{9} - 32 q^{21} + 8 q^{25} - 112 q^{29} + 208 q^{37} + 8 q^{49} - 240 q^{53} + 192 q^{57} - 256 q^{65} + 432 q^{77} + 712 q^{81} + 192 q^{85} + 64 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 20x^{6} + 56x^{4} + 20x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 10\nu^{6} + 196\nu^{4} + 490\nu^{2} + 88 ) / 21 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - \nu^{6} + 21\nu^{5} - 21\nu^{4} + 77\nu^{3} - 77\nu^{2} + 97\nu - 48 ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} + 7\nu^{6} - 63\nu^{5} + 133\nu^{4} - 231\nu^{3} + 259\nu^{2} - 291\nu - 56 ) / 21 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{7} + 38\nu^{5} + 74\nu^{3} - 34\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 2\nu^{6} + 40\nu^{4} + 110\nu^{2} + 20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -13\nu^{7} - 259\nu^{5} - 707\nu^{3} - 197\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -7\nu^{7} - 139\nu^{5} - 373\nu^{3} - 97\nu ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{7} - 2\beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} - 2\beta_{3} - 2\beta_{2} + 5\beta _1 - 20 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -38\beta_{7} + 42\beta_{6} - 13\beta_{4} + 7\beta_{3} - 7\beta_{2} - 7\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 10\beta_{5} + 15\beta_{3} + 15\beta_{2} - 48\beta _1 + 144 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 646\beta_{7} - 726\beta_{6} + 197\beta_{4} - 97\beta_{3} + 97\beta_{2} + 97\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -343\beta_{5} - 490\beta_{3} - 490\beta_{2} + 1645\beta _1 - 4700 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -10834\beta_{7} + 12206\beta_{6} - 3233\beta_{4} + 1567\beta_{3} - 1567\beta_{2} - 1567\beta_1 ) / 8 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
4.08932i
0.244539i
1.69385i
0.590371i
0.590371i
1.69385i
0.244539i
4.08932i
0 5.43734i 0 6.12900i 0 6.21005 3.23038i 0 −20.5647 0
97.2 0 5.43734i 0 6.12900i 0 −6.21005 3.23038i 0 −20.5647 0
97.3 0 1.56056i 0 3.23038i 0 3.38162 + 6.12900i 0 6.56466 0
97.4 0 1.56056i 0 3.23038i 0 −3.38162 + 6.12900i 0 6.56466 0
97.5 0 1.56056i 0 3.23038i 0 −3.38162 6.12900i 0 6.56466 0
97.6 0 1.56056i 0 3.23038i 0 3.38162 6.12900i 0 6.56466 0
97.7 0 5.43734i 0 6.12900i 0 −6.21005 + 3.23038i 0 −20.5647 0
97.8 0 5.43734i 0 6.12900i 0 6.21005 + 3.23038i 0 −20.5647 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.3.c.a 8
3.b odd 2 1 2016.3.f.c 8
4.b odd 2 1 inner 224.3.c.a 8
7.b odd 2 1 inner 224.3.c.a 8
8.b even 2 1 448.3.c.g 8
8.d odd 2 1 448.3.c.g 8
12.b even 2 1 2016.3.f.c 8
21.c even 2 1 2016.3.f.c 8
28.d even 2 1 inner 224.3.c.a 8
56.e even 2 1 448.3.c.g 8
56.h odd 2 1 448.3.c.g 8
84.h odd 2 1 2016.3.f.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.3.c.a 8 1.a even 1 1 trivial
224.3.c.a 8 4.b odd 2 1 inner
224.3.c.a 8 7.b odd 2 1 inner
224.3.c.a 8 28.d even 2 1 inner
448.3.c.g 8 8.b even 2 1
448.3.c.g 8 8.d odd 2 1
448.3.c.g 8 56.e even 2 1
448.3.c.g 8 56.h odd 2 1
2016.3.f.c 8 3.b odd 2 1
2016.3.f.c 8 12.b even 2 1
2016.3.f.c 8 21.c even 2 1
2016.3.f.c 8 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 32T_{3}^{2} + 72 \) acting on \(S_{3}^{\mathrm{new}}(224, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 32 T^{2} + 72)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 48 T^{2} + 392)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 4 T^{6} + 1862 T^{4} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{4} - 248 T^{2} + 3600)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 208 T^{2} + 1800)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 1152 T^{2} + 320000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 256 T^{2} + 16200)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 440 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 28 T - 540)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 1664 T^{2} + 115200)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 52 T - 60)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 1856 T^{2} + 10368)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 4856 T^{2} + 4717584)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 2048 T^{2} + 1036800)^{2} \) Copy content Toggle raw display
$53$ \( (T + 30)^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} + 9344 T^{2} + 21385800)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 10672 T^{2} + 342792)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 828)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 8200 T^{2} + 250000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 12928 T^{2} + 35280000)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 25992 T^{2} + \cdots + 128595600)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 19424 T^{2} + 89191368)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 768 T^{2} + 3200)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 18432 T^{2} + 83980800)^{2} \) Copy content Toggle raw display
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