Properties

Label 224.2.b.b
Level 224
Weight 2
Character orbit 224.b
Analytic conductor 1.789
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 224.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.78864900528\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.2312.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + \beta_{2} q^{5} + q^{7} + ( -2 + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + \beta_{2} q^{5} + q^{7} + ( -2 + \beta_{3} ) q^{9} + ( -\beta_{1} + \beta_{2} ) q^{11} -\beta_{2} q^{13} + ( -1 + \beta_{3} ) q^{15} + 2 q^{17} + \beta_{1} q^{19} + \beta_{1} q^{21} + ( -1 + \beta_{3} ) q^{23} + ( -2 - \beta_{3} ) q^{25} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{27} + 2 \beta_{1} q^{29} + ( 2 - 2 \beta_{3} ) q^{31} + 4 q^{33} + \beta_{2} q^{35} -2 \beta_{1} q^{37} + ( 1 - \beta_{3} ) q^{39} + ( -4 - 2 \beta_{3} ) q^{41} + ( -\beta_{1} + \beta_{2} ) q^{43} + ( -4 \beta_{1} + \beta_{2} ) q^{45} + q^{49} + 2 \beta_{1} q^{51} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{53} + ( -6 - 2 \beta_{3} ) q^{55} + ( -5 + \beta_{3} ) q^{57} + ( \beta_{1} - 2 \beta_{2} ) q^{59} -\beta_{2} q^{61} + ( -2 + \beta_{3} ) q^{63} + ( 7 + \beta_{3} ) q^{65} + ( -\beta_{1} - 5 \beta_{2} ) q^{67} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{69} + 8 q^{71} -6 q^{73} + ( \beta_{1} + 2 \beta_{2} ) q^{75} + ( -\beta_{1} + \beta_{2} ) q^{77} + ( 6 - \beta_{3} ) q^{81} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{83} + 2 \beta_{2} q^{85} + ( -10 + 2 \beta_{3} ) q^{87} + ( -8 + 2 \beta_{3} ) q^{89} -\beta_{2} q^{91} + ( 8 \beta_{1} + 4 \beta_{2} ) q^{93} + ( -1 + \beta_{3} ) q^{95} + ( 4 - 2 \beta_{3} ) q^{97} + ( \beta_{1} + 3 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{7} - 8q^{9} + O(q^{10}) \) \( 4q + 4q^{7} - 8q^{9} - 4q^{15} + 8q^{17} - 4q^{23} - 8q^{25} + 8q^{31} + 16q^{33} + 4q^{39} - 16q^{41} + 4q^{49} - 24q^{55} - 20q^{57} - 8q^{63} + 28q^{65} + 32q^{71} - 24q^{73} + 24q^{81} - 40q^{87} - 32q^{89} - 4q^{95} + 16q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( -\nu^{2} + \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} + \nu - 2 \)
\(\beta_{3}\)\(=\)\( -\nu^{3} + \nu^{2} + 2 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} + \beta_{1} + 1\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + \beta_{2} - 3 \beta_{1} + 1\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{3} + 3 \beta_{2} - \beta_{1} + 7\)\()/4\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1
−0.780776 1.17915i
1.28078 + 0.599676i
1.28078 0.599676i
−0.780776 + 1.17915i
0 3.02045i 0 1.69614i 0 1.00000 0 −6.12311 0
113.2 0 0.936426i 0 3.33513i 0 1.00000 0 2.12311 0
113.3 0 0.936426i 0 3.33513i 0 1.00000 0 2.12311 0
113.4 0 3.02045i 0 1.69614i 0 1.00000 0 −6.12311 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.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.2.b.b 4
3.b odd 2 1 2016.2.c.c 4
4.b odd 2 1 56.2.b.b 4
7.b odd 2 1 1568.2.b.d 4
7.c even 3 2 1568.2.t.d 8
7.d odd 6 2 1568.2.t.e 8
8.b even 2 1 inner 224.2.b.b 4
8.d odd 2 1 56.2.b.b 4
12.b even 2 1 504.2.c.d 4
16.e even 4 2 1792.2.a.v 4
16.f odd 4 2 1792.2.a.x 4
24.f even 2 1 504.2.c.d 4
24.h odd 2 1 2016.2.c.c 4
28.d even 2 1 392.2.b.c 4
28.f even 6 2 392.2.p.e 8
28.g odd 6 2 392.2.p.f 8
56.e even 2 1 392.2.b.c 4
56.h odd 2 1 1568.2.b.d 4
56.j odd 6 2 1568.2.t.e 8
56.k odd 6 2 392.2.p.f 8
56.m even 6 2 392.2.p.e 8
56.p even 6 2 1568.2.t.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.b.b 4 4.b odd 2 1
56.2.b.b 4 8.d odd 2 1
224.2.b.b 4 1.a even 1 1 trivial
224.2.b.b 4 8.b even 2 1 inner
392.2.b.c 4 28.d even 2 1
392.2.b.c 4 56.e even 2 1
392.2.p.e 8 28.f even 6 2
392.2.p.e 8 56.m even 6 2
392.2.p.f 8 28.g odd 6 2
392.2.p.f 8 56.k odd 6 2
504.2.c.d 4 12.b even 2 1
504.2.c.d 4 24.f even 2 1
1568.2.b.d 4 7.b odd 2 1
1568.2.b.d 4 56.h odd 2 1
1568.2.t.d 8 7.c even 3 2
1568.2.t.d 8 56.p even 6 2
1568.2.t.e 8 7.d odd 6 2
1568.2.t.e 8 56.j odd 6 2
1792.2.a.v 4 16.e even 4 2
1792.2.a.x 4 16.f odd 4 2
2016.2.c.c 4 3.b odd 2 1
2016.2.c.c 4 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 10 T_{3}^{2} + 8 \) acting on \(S_{2}^{\mathrm{new}}(224, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 - 2 T^{2} + 2 T^{4} - 18 T^{6} + 81 T^{8} \)
$5$ \( 1 - 6 T^{2} + 42 T^{4} - 150 T^{6} + 625 T^{8} \)
$7$ \( ( 1 - T )^{4} \)
$11$ \( 1 - 24 T^{2} + 318 T^{4} - 2904 T^{6} + 14641 T^{8} \)
$13$ \( 1 - 38 T^{2} + 682 T^{4} - 6422 T^{6} + 28561 T^{8} \)
$17$ \( ( 1 - 2 T + 17 T^{2} )^{4} \)
$19$ \( 1 - 66 T^{2} + 1794 T^{4} - 23826 T^{6} + 130321 T^{8} \)
$23$ \( ( 1 + 2 T + 30 T^{2} + 46 T^{3} + 529 T^{4} )^{2} \)
$29$ \( 1 - 76 T^{2} + 2854 T^{4} - 63916 T^{6} + 707281 T^{8} \)
$31$ \( ( 1 - 4 T - 2 T^{2} - 124 T^{3} + 961 T^{4} )^{2} \)
$37$ \( 1 - 108 T^{2} + 5382 T^{4} - 147852 T^{6} + 1874161 T^{8} \)
$41$ \( ( 1 + 8 T + 30 T^{2} + 328 T^{3} + 1681 T^{4} )^{2} \)
$43$ \( 1 - 152 T^{2} + 9406 T^{4} - 281048 T^{6} + 3418801 T^{8} \)
$47$ \( ( 1 + 47 T^{2} )^{4} \)
$53$ \( 1 - 132 T^{2} + 8886 T^{4} - 370788 T^{6} + 7890481 T^{8} \)
$59$ \( 1 - 178 T^{2} + 14050 T^{4} - 619618 T^{6} + 12117361 T^{8} \)
$61$ \( 1 - 230 T^{2} + 20650 T^{4} - 855830 T^{6} + 13845841 T^{8} \)
$67$ \( 1 + 112 T^{2} + 8782 T^{4} + 502768 T^{6} + 20151121 T^{8} \)
$71$ \( ( 1 - 8 T + 71 T^{2} )^{4} \)
$73$ \( ( 1 + 6 T + 73 T^{2} )^{4} \)
$79$ \( ( 1 + 79 T^{2} )^{4} \)
$83$ \( 1 - 210 T^{2} + 23970 T^{4} - 1446690 T^{6} + 47458321 T^{8} \)
$89$ \( ( 1 + 16 T + 174 T^{2} + 1424 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 8 T + 142 T^{2} - 776 T^{3} + 9409 T^{4} )^{2} \)
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