Properties

Label 224.3.c.b
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: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \( x^{8} + 20x^{6} + 116x^{4} + 180x^{2} + 81 \) 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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_{5} q^{5} + \beta_{7} q^{7} + ( - \beta_{4} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + \beta_{5} q^{5} + \beta_{7} q^{7} + ( - \beta_{4} + 1) q^{9} + ( - \beta_{6} + \beta_{3} + \beta_{2}) q^{11} + ( - \beta_{5} - \beta_1) q^{13} + (\beta_{7} + \beta_{3}) q^{15} + \beta_1 q^{17} + (3 \beta_{7} - 3 \beta_{6} + 2 \beta_{2}) q^{19} + ( - \beta_{5} + 3 \beta_{4} + \beta_1 - 4) q^{21} + (2 \beta_{7} + \beta_{6} + \beta_{3} - \beta_{2}) q^{23} + ( - \beta_{4} - 7) q^{25} + (\beta_{7} - \beta_{6} + 9 \beta_{2}) q^{27} + (6 \beta_{4} + 2) q^{29} + ( - 5 \beta_{7} + 5 \beta_{6} + 9 \beta_{2}) q^{31} + ( - 6 \beta_{5} - \beta_1) q^{33} + (3 \beta_{7} - 6 \beta_{6} + \beta_{3} - 3 \beta_{2}) q^{35} + ( - 14 \beta_{4} - 6) q^{37} + (3 \beta_{7} + 4 \beta_{6} - \beta_{3} - 4 \beta_{2}) q^{39} + (6 \beta_{5} - 2 \beta_1) q^{41} + ( - 2 \beta_{7} - \beta_{6} - \beta_{3} + \beta_{2}) q^{43} + (\beta_{5} + \beta_1) q^{45} + (3 \beta_{7} - 3 \beta_{6} - 11 \beta_{2}) q^{47} + (6 \beta_{5} + 10 \beta_{4} + \beta_1 + 17) q^{49} + ( - 4 \beta_{7} - 4 \beta_{6} + 4 \beta_{2}) q^{51} + ( - 8 \beta_{4} + 34) q^{53} + ( - 7 \beta_{7} + 7 \beta_{6} - 23 \beta_{2}) q^{55} + (19 \beta_{4} - 16) q^{57} + ( - 7 \beta_{7} + 7 \beta_{6} - 6 \beta_{2}) q^{59} + 11 \beta_{5} q^{61} + (\beta_{7} - \beta_{6} - \beta_{3} + 3 \beta_{2}) q^{63} + ( - 31 \beta_{4} + 24) q^{65} + ( - 6 \beta_{7} - \beta_{6} - 5 \beta_{3} + \beta_{2}) q^{67} + ( - 10 \beta_{5} + 3 \beta_1) q^{69} + (\beta_{7} + 5 \beta_{6} - 4 \beta_{3} - 5 \beta_{2}) q^{71} + ( - 6 \beta_{5} - 3 \beta_1) q^{73} + (\beta_{7} - \beta_{6} - 8 \beta_{2}) q^{75} + ( - 10 \beta_{5} + 23 \beta_{4} - 4 \beta_1 - 26) q^{77} + ( - 5 \beta_{7} + \beta_{6} - 6 \beta_{3} - \beta_{2}) q^{79} + ( - 11 \beta_{4} - 63) q^{81} + ( - 2 \beta_{7} + 2 \beta_{6} + 23 \beta_{2}) q^{83} + (32 \beta_{4} + 8) q^{85} + ( - 6 \beta_{7} + 6 \beta_{6} + 8 \beta_{2}) q^{87} + ( - 2 \beta_{5} + 5 \beta_1) q^{89} + ( - 2 \beta_{7} + 9 \beta_{6} - 5 \beta_{3} - 34 \beta_{2}) q^{91} + ( - 44 \beta_{4} - 72) q^{93} + ( - 7 \beta_{7} - 12 \beta_{6} + 5 \beta_{3} + 12 \beta_{2}) q^{95} + ( - 4 \beta_{5} + 7 \beta_1) q^{97} + ( - 2 \beta_{7} - 5 \beta_{6} + 3 \beta_{3} + 5 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{9} - 32 q^{21} - 56 q^{25} + 16 q^{29} - 48 q^{37} + 136 q^{49} + 272 q^{53} - 128 q^{57} + 192 q^{65} - 208 q^{77} - 504 q^{81} + 64 q^{85} - 576 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 20x^{6} + 116x^{4} + 180x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4\nu^{7} + 80\nu^{5} + 464\nu^{3} + 828\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{7} + 131\nu^{5} + 659\nu^{3} + 567\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{7} - 9\nu^{6} - 91\nu^{5} - 153\nu^{4} - 427\nu^{3} - 585\nu^{2} - 261\nu - 54 ) / 27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{6} + 40\nu^{4} + 214\nu^{2} + 180 ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -13\nu^{7} - 251\nu^{5} - 1301\nu^{3} - 1107\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2\nu^{7} + 15\nu^{6} + 40\nu^{5} + 273\nu^{4} + 232\nu^{3} + 1335\nu^{2} + 306\nu + 1134 ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{7} + 15\nu^{6} + 91\nu^{5} + 273\nu^{4} + 427\nu^{3} + 1335\nu^{2} + 261\nu + 1134 ) / 27 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} - \beta_{2} + \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{4} + \beta_{3} - 20 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -13\beta_{7} + 13\beta_{6} + 4\beta_{5} + 17\beta_{2} - 7\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -6\beta_{7} - \beta_{6} + 10\beta_{4} - 5\beta_{3} + \beta_{2} + 84 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 123\beta_{7} - 123\beta_{6} - 68\beta_{5} - 215\beta_{2} + 63\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 133\beta_{7} + 40\beta_{6} - 275\beta_{4} + 93\beta_{3} - 40\beta_{2} - 1580 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -1159\beta_{7} + 1159\beta_{6} + 896\beta_{5} + 2535\beta_{2} - 601\beta_1 ) / 8 \) 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
3.24994i
0.923094i
2.71359i
1.10555i
1.10555i
2.71359i
0.923094i
3.24994i
0 3.29066i 0 5.90156i 0 −6.86601 + 1.36303i 0 −1.82843 0
97.2 0 3.29066i 0 5.90156i 0 6.86601 + 1.36303i 0 −1.82843 0
97.3 0 2.27411i 0 5.40107i 0 4.34256 5.49019i 0 3.82843 0
97.4 0 2.27411i 0 5.40107i 0 −4.34256 5.49019i 0 3.82843 0
97.5 0 2.27411i 0 5.40107i 0 −4.34256 + 5.49019i 0 3.82843 0
97.6 0 2.27411i 0 5.40107i 0 4.34256 + 5.49019i 0 3.82843 0
97.7 0 3.29066i 0 5.90156i 0 6.86601 1.36303i 0 −1.82843 0
97.8 0 3.29066i 0 5.90156i 0 −6.86601 1.36303i 0 −1.82843 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.b 8
3.b odd 2 1 2016.3.f.b 8
4.b odd 2 1 inner 224.3.c.b 8
7.b odd 2 1 inner 224.3.c.b 8
8.b even 2 1 448.3.c.h 8
8.d odd 2 1 448.3.c.h 8
12.b even 2 1 2016.3.f.b 8
21.c even 2 1 2016.3.f.b 8
28.d even 2 1 inner 224.3.c.b 8
56.e even 2 1 448.3.c.h 8
56.h odd 2 1 448.3.c.h 8
84.h odd 2 1 2016.3.f.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.3.c.b 8 1.a even 1 1 trivial
224.3.c.b 8 4.b odd 2 1 inner
224.3.c.b 8 7.b odd 2 1 inner
224.3.c.b 8 28.d even 2 1 inner
448.3.c.h 8 8.b even 2 1
448.3.c.h 8 8.d odd 2 1
448.3.c.h 8 56.e even 2 1
448.3.c.h 8 56.h odd 2 1
2016.3.f.b 8 3.b odd 2 1
2016.3.f.b 8 12.b even 2 1
2016.3.f.b 8 21.c even 2 1
2016.3.f.b 8 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 16T_{3}^{2} + 56 \) 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
$3$ \( (T^{4} + 16 T^{2} + 56)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 64 T^{2} + 1016)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 68 T^{6} + 2758 T^{4} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{4} - 472 T^{2} + 14224)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 544 T^{2} + 49784)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 512 T^{2} + 65024)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 1072 T^{2} + 123704)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 1112 T^{2} + 14224)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 4 T - 284)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 4096 T^{2} + 1895936)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 12 T - 1532)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 4736 T^{2} + 16256)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 1112 T^{2} + 14224)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 2944 T^{2} + 3584)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 68 T + 644)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + 6064 T^{2} + 2784824)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 7744 T^{2} + 14875256)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 15064 T^{2} + 34151824)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 8648 T^{2} + 696976)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 6336 T^{2} + 1316736)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 17352 T^{2} + 71703184)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 8912 T^{2} + 9367736)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 13376 T^{2} + 39030656)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 27008 T^{2} + \cdots + 143638016)^{2} \) Copy content Toggle raw display
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