Properties

Label 224.3.d.a
Level 224
Weight 3
Character orbit 224.d
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.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{7})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( -2 + \beta_{2} ) q^{5} -\beta_{3} q^{7} + 5 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( -2 + \beta_{2} ) q^{5} -\beta_{3} q^{7} + 5 q^{9} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{11} + ( -2 + 3 \beta_{2} ) q^{13} + ( -2 \beta_{1} + 4 \beta_{3} ) q^{15} + ( -14 + 2 \beta_{2} ) q^{17} + ( -5 \beta_{1} + 8 \beta_{3} ) q^{19} + \beta_{2} q^{21} + ( 6 \beta_{1} + 4 \beta_{3} ) q^{23} + ( 7 - 4 \beta_{2} ) q^{25} + 14 \beta_{1} q^{27} + ( 10 + 6 \beta_{2} ) q^{29} -2 \beta_{1} q^{31} + ( 8 - 4 \beta_{2} ) q^{33} + ( -7 \beta_{1} + 2 \beta_{3} ) q^{35} + ( 34 + 2 \beta_{2} ) q^{37} + ( -2 \beta_{1} + 12 \beta_{3} ) q^{39} + ( -30 - 2 \beta_{2} ) q^{41} + ( 6 \beta_{1} - 20 \beta_{3} ) q^{43} + ( -10 + 5 \beta_{2} ) q^{45} + ( -18 \beta_{1} - 8 \beta_{3} ) q^{47} -7 q^{49} + ( -14 \beta_{1} + 8 \beta_{3} ) q^{51} + ( -70 + 4 \beta_{2} ) q^{53} + ( 32 \beta_{1} - 16 \beta_{3} ) q^{55} + ( 20 - 8 \beta_{2} ) q^{57} + ( -9 \beta_{1} - 16 \beta_{3} ) q^{59} + ( -2 - 3 \beta_{2} ) q^{61} -5 \beta_{3} q^{63} + ( 88 - 8 \beta_{2} ) q^{65} + ( 12 \beta_{1} - 20 \beta_{3} ) q^{67} + ( -24 - 4 \beta_{2} ) q^{69} + ( -32 \beta_{1} + 16 \beta_{3} ) q^{71} + ( -38 - 16 \beta_{2} ) q^{73} + ( 7 \beta_{1} - 16 \beta_{3} ) q^{75} + ( 28 - 2 \beta_{2} ) q^{77} -40 \beta_{3} q^{79} -11 q^{81} + ( 31 \beta_{1} + 24 \beta_{3} ) q^{83} + ( 84 - 18 \beta_{2} ) q^{85} + ( 10 \beta_{1} + 24 \beta_{3} ) q^{87} + ( 34 + 8 \beta_{2} ) q^{89} + ( -21 \beta_{1} + 2 \beta_{3} ) q^{91} + 8 q^{93} + ( 66 \beta_{1} - 36 \beta_{3} ) q^{95} + ( -150 - 2 \beta_{2} ) q^{97} + ( -10 \beta_{1} + 20 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{5} + 20q^{9} + O(q^{10}) \) \( 4q - 8q^{5} + 20q^{9} - 8q^{13} - 56q^{17} + 28q^{25} + 40q^{29} + 32q^{33} + 136q^{37} - 120q^{41} - 40q^{45} - 28q^{49} - 280q^{53} + 80q^{57} - 8q^{61} + 352q^{65} - 96q^{69} - 152q^{73} + 112q^{77} - 44q^{81} + 336q^{85} + 136q^{89} + 32q^{93} - 600q^{97} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} - \nu \)
\(\beta_{2}\)\(=\)\( -\nu^{3} + 5 \nu \)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 3\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{2} + 5 \beta_{1}\)\()/4\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
−1.32288 0.500000i
1.32288 0.500000i
−1.32288 + 0.500000i
1.32288 + 0.500000i
0 2.00000i 0 −7.29150 0 2.64575i 0 5.00000 0
127.2 0 2.00000i 0 3.29150 0 2.64575i 0 5.00000 0
127.3 0 2.00000i 0 −7.29150 0 2.64575i 0 5.00000 0
127.4 0 2.00000i 0 3.29150 0 2.64575i 0 5.00000 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.3.d.a 4
3.b odd 2 1 2016.3.m.a 4
4.b odd 2 1 inner 224.3.d.a 4
7.b odd 2 1 1568.3.d.h 4
8.b even 2 1 448.3.d.c 4
8.d odd 2 1 448.3.d.c 4
12.b even 2 1 2016.3.m.a 4
16.e even 4 1 1792.3.g.a 4
16.e even 4 1 1792.3.g.c 4
16.f odd 4 1 1792.3.g.a 4
16.f odd 4 1 1792.3.g.c 4
28.d even 2 1 1568.3.d.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.3.d.a 4 1.a even 1 1 trivial
224.3.d.a 4 4.b odd 2 1 inner
448.3.d.c 4 8.b even 2 1
448.3.d.c 4 8.d odd 2 1
1568.3.d.h 4 7.b odd 2 1
1568.3.d.h 4 28.d even 2 1
1792.3.g.a 4 16.e even 4 1
1792.3.g.a 4 16.f odd 4 1
1792.3.g.c 4 16.e even 4 1
1792.3.g.c 4 16.f odd 4 1
2016.3.m.a 4 3.b odd 2 1
2016.3.m.a 4 12.b even 2 1

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 14 T^{2} + 81 T^{4} )^{2} \)
$5$ \( ( 1 + 4 T + 26 T^{2} + 100 T^{3} + 625 T^{4} )^{2} \)
$7$ \( ( 1 + 7 T^{2} )^{2} \)
$11$ \( 1 - 228 T^{2} + 35110 T^{4} - 3338148 T^{6} + 214358881 T^{8} \)
$13$ \( ( 1 + 4 T + 90 T^{2} + 676 T^{3} + 28561 T^{4} )^{2} \)
$17$ \( ( 1 + 28 T + 662 T^{2} + 8092 T^{3} + 83521 T^{4} )^{2} \)
$19$ \( 1 - 348 T^{2} + 111718 T^{4} - 45351708 T^{6} + 16983563041 T^{8} \)
$23$ \( 1 - 1604 T^{2} + 1138374 T^{4} - 448864964 T^{6} + 78310985281 T^{8} \)
$29$ \( ( 1 - 20 T + 774 T^{2} - 16820 T^{3} + 707281 T^{4} )^{2} \)
$31$ \( ( 1 - 1906 T^{2} + 923521 T^{4} )^{2} \)
$37$ \( ( 1 - 68 T + 3782 T^{2} - 93092 T^{3} + 1874161 T^{4} )^{2} \)
$41$ \( ( 1 + 60 T + 4150 T^{2} + 100860 T^{3} + 2825761 T^{4} )^{2} \)
$43$ \( 1 - 1508 T^{2} + 5793318 T^{4} - 5155551908 T^{6} + 11688200277601 T^{8} \)
$47$ \( 1 - 5348 T^{2} + 14587206 T^{4} - 26096533988 T^{6} + 23811286661761 T^{8} \)
$53$ \( ( 1 + 140 T + 10070 T^{2} + 393260 T^{3} + 7890481 T^{4} )^{2} \)
$59$ \( 1 - 9692 T^{2} + 45396006 T^{4} - 117441462812 T^{6} + 146830437604321 T^{8} \)
$61$ \( ( 1 + 4 T + 7194 T^{2} + 14884 T^{3} + 13845841 T^{4} )^{2} \)
$67$ \( 1 - 11204 T^{2} + 65233446 T^{4} - 225773159684 T^{6} + 406067677556641 T^{8} \)
$71$ \( 1 - 8388 T^{2} + 39052870 T^{4} - 213153180228 T^{6} + 645753531245761 T^{8} \)
$73$ \( ( 1 + 76 T + 4934 T^{2} + 405004 T^{3} + 28398241 T^{4} )^{2} \)
$79$ \( ( 1 - 1282 T^{2} + 38950081 T^{4} )^{2} \)
$83$ \( 1 - 11804 T^{2} + 67754214 T^{4} - 560198021084 T^{6} + 2252292232139041 T^{8} \)
$89$ \( ( 1 - 68 T + 15206 T^{2} - 538628 T^{3} + 62742241 T^{4} )^{2} \)
$97$ \( ( 1 + 300 T + 41206 T^{2} + 2822700 T^{3} + 88529281 T^{4} )^{2} \)
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