Properties

Label 224.2.i.d
Level 224
Weight 2
Character orbit 224.i
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.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.78864900528\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{1} + \beta_{2} ) q^{3} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{5} + ( 1 - 2 \beta_{1} - \beta_{3} ) q^{7} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( 1 + \beta_{1} + \beta_{2} ) q^{3} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{5} + ( 1 - 2 \beta_{1} - \beta_{3} ) q^{7} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{9} + ( -1 + \beta_{1} - \beta_{2} ) q^{11} + 2 \beta_{3} q^{13} + ( -5 + 3 \beta_{3} ) q^{15} + ( 3 + 2 \beta_{1} + 3 \beta_{2} ) q^{17} + ( \beta_{1} - 5 \beta_{2} + \beta_{3} ) q^{19} + ( 3 - \beta_{2} - 2 \beta_{3} ) q^{21} + ( -3 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{23} + ( -4 - 4 \beta_{1} - 4 \beta_{2} ) q^{25} + ( -1 - \beta_{3} ) q^{27} -2 \beta_{3} q^{29} + ( 7 + \beta_{1} + 7 \beta_{2} ) q^{31} + \beta_{2} q^{33} + ( 8 + 3 \beta_{1} + 5 \beta_{2} + \beta_{3} ) q^{35} + ( -4 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{37} + ( -4 - 2 \beta_{1} - 4 \beta_{2} ) q^{39} + ( -4 - 2 \beta_{3} ) q^{41} + ( -4 - 4 \beta_{3} ) q^{43} + ( -8 - 2 \beta_{1} - 8 \beta_{2} ) q^{45} + ( \beta_{1} - 9 \beta_{2} + \beta_{3} ) q^{47} + ( -5 - 4 \beta_{1} - 2 \beta_{3} ) q^{49} + ( 5 \beta_{1} + 7 \beta_{2} + 5 \beta_{3} ) q^{51} + ( 1 + \beta_{2} ) q^{53} + ( -3 - \beta_{3} ) q^{55} + ( 3 - 4 \beta_{3} ) q^{57} + ( 1 - 7 \beta_{1} + \beta_{2} ) q^{59} + ( -4 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{61} + ( 8 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{63} + ( -2 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} ) q^{65} + ( -7 - 3 \beta_{1} - 7 \beta_{2} ) q^{67} + ( 5 - 2 \beta_{3} ) q^{69} + ( -8 - 4 \beta_{3} ) q^{71} + ( 9 - 4 \beta_{1} + 9 \beta_{2} ) q^{73} + ( -8 \beta_{1} - 12 \beta_{2} - 8 \beta_{3} ) q^{75} + ( 1 + 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{77} + ( 5 \beta_{1} + \beta_{2} + 5 \beta_{3} ) q^{79} + ( 1 + 6 \beta_{1} + \beta_{2} ) q^{81} + ( -4 + 8 \beta_{3} ) q^{83} + ( -11 + 8 \beta_{3} ) q^{85} + ( 4 + 2 \beta_{1} + 4 \beta_{2} ) q^{87} -9 \beta_{2} q^{89} + ( 4 + 8 \beta_{2} + 2 \beta_{3} ) q^{91} + ( 8 \beta_{1} + 9 \beta_{2} + 8 \beta_{3} ) q^{93} + ( 1 + 9 \beta_{1} + \beta_{2} ) q^{95} + ( -4 + 2 \beta_{3} ) q^{97} + ( -4 - 2 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} - 2q^{5} + 4q^{7} + O(q^{10}) \) \( 4q + 2q^{3} - 2q^{5} + 4q^{7} - 2q^{11} - 20q^{15} + 6q^{17} + 10q^{19} + 14q^{21} - 2q^{23} - 8q^{25} - 4q^{27} + 14q^{31} - 2q^{33} + 22q^{35} - 6q^{37} - 8q^{39} - 16q^{41} - 16q^{43} - 16q^{45} + 18q^{47} - 20q^{49} - 14q^{51} + 2q^{53} - 12q^{55} + 12q^{57} + 2q^{59} - 6q^{61} + 24q^{63} + 16q^{65} - 14q^{67} + 20q^{69} - 32q^{71} + 18q^{73} + 24q^{75} + 10q^{77} - 2q^{79} + 2q^{81} - 16q^{83} - 44q^{85} + 8q^{87} + 18q^{89} - 18q^{93} + 2q^{95} - 16q^{97} - 16q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{3}\)

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
0 −0.207107 0.358719i 0 0.914214 1.58346i 0 1.00000 + 2.44949i 0 1.41421 2.44949i 0
65.2 0 1.20711 + 2.09077i 0 −1.91421 + 3.31552i 0 1.00000 2.44949i 0 −1.41421 + 2.44949i 0
193.1 0 −0.207107 + 0.358719i 0 0.914214 + 1.58346i 0 1.00000 2.44949i 0 1.41421 + 2.44949i 0
193.2 0 1.20711 2.09077i 0 −1.91421 3.31552i 0 1.00000 + 2.44949i 0 −1.41421 2.44949i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.2.i.d yes 4
3.b odd 2 1 2016.2.s.s 4
4.b odd 2 1 224.2.i.a 4
7.b odd 2 1 1568.2.i.o 4
7.c even 3 1 inner 224.2.i.d yes 4
7.c even 3 1 1568.2.a.l 2
7.d odd 6 1 1568.2.a.u 2
7.d odd 6 1 1568.2.i.o 4
8.b even 2 1 448.2.i.g 4
8.d odd 2 1 448.2.i.j 4
12.b even 2 1 2016.2.s.q 4
21.h odd 6 1 2016.2.s.s 4
28.d even 2 1 1568.2.i.x 4
28.f even 6 1 1568.2.a.j 2
28.f even 6 1 1568.2.i.x 4
28.g odd 6 1 224.2.i.a 4
28.g odd 6 1 1568.2.a.w 2
56.j odd 6 1 3136.2.a.be 2
56.k odd 6 1 448.2.i.j 4
56.k odd 6 1 3136.2.a.bd 2
56.m even 6 1 3136.2.a.bx 2
56.p even 6 1 448.2.i.g 4
56.p even 6 1 3136.2.a.bw 2
84.n even 6 1 2016.2.s.q 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.i.a 4 4.b odd 2 1
224.2.i.a 4 28.g odd 6 1
224.2.i.d yes 4 1.a even 1 1 trivial
224.2.i.d yes 4 7.c even 3 1 inner
448.2.i.g 4 8.b even 2 1
448.2.i.g 4 56.p even 6 1
448.2.i.j 4 8.d odd 2 1
448.2.i.j 4 56.k odd 6 1
1568.2.a.j 2 28.f even 6 1
1568.2.a.l 2 7.c even 3 1
1568.2.a.u 2 7.d odd 6 1
1568.2.a.w 2 28.g odd 6 1
1568.2.i.o 4 7.b odd 2 1
1568.2.i.o 4 7.d odd 6 1
1568.2.i.x 4 28.d even 2 1
1568.2.i.x 4 28.f even 6 1
2016.2.s.q 4 12.b even 2 1
2016.2.s.q 4 84.n even 6 1
2016.2.s.s 4 3.b odd 2 1
2016.2.s.s 4 21.h odd 6 1
3136.2.a.bd 2 56.k odd 6 1
3136.2.a.be 2 56.j odd 6 1
3136.2.a.bw 2 56.p even 6 1
3136.2.a.bx 2 56.m even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 2 T_{3}^{3} + 5 T_{3}^{2} + 2 T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(224, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 - 2 T - T^{2} + 2 T^{3} + 4 T^{4} + 6 T^{5} - 9 T^{6} - 54 T^{7} + 81 T^{8} \)
$5$ \( 1 + 2 T + T^{2} - 14 T^{3} - 36 T^{4} - 70 T^{5} + 25 T^{6} + 250 T^{7} + 625 T^{8} \)
$7$ \( ( 1 - 2 T + 7 T^{2} )^{2} \)
$11$ \( 1 + 2 T - 17 T^{2} - 2 T^{3} + 276 T^{4} - 22 T^{5} - 2057 T^{6} + 2662 T^{7} + 14641 T^{8} \)
$13$ \( ( 1 + 18 T^{2} + 169 T^{4} )^{2} \)
$17$ \( 1 - 6 T + T^{2} - 6 T^{3} + 324 T^{4} - 102 T^{5} + 289 T^{6} - 29478 T^{7} + 83521 T^{8} \)
$19$ \( 1 - 10 T + 39 T^{2} - 230 T^{3} + 1460 T^{4} - 4370 T^{5} + 14079 T^{6} - 68590 T^{7} + 130321 T^{8} \)
$23$ \( 1 + 2 T - 25 T^{2} - 34 T^{3} + 220 T^{4} - 782 T^{5} - 13225 T^{6} + 24334 T^{7} + 279841 T^{8} \)
$29$ \( ( 1 + 50 T^{2} + 841 T^{4} )^{2} \)
$31$ \( 1 - 14 T + 87 T^{2} - 658 T^{3} + 4844 T^{4} - 20398 T^{5} + 83607 T^{6} - 417074 T^{7} + 923521 T^{8} \)
$37$ \( 1 + 6 T - 15 T^{2} - 138 T^{3} - 100 T^{4} - 5106 T^{5} - 20535 T^{6} + 303918 T^{7} + 1874161 T^{8} \)
$41$ \( ( 1 + 8 T + 90 T^{2} + 328 T^{3} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 + 8 T + 70 T^{2} + 344 T^{3} + 1849 T^{4} )^{2} \)
$47$ \( 1 - 18 T + 151 T^{2} - 1422 T^{3} + 12492 T^{4} - 66834 T^{5} + 333559 T^{6} - 1868814 T^{7} + 4879681 T^{8} \)
$53$ \( ( 1 - T - 52 T^{2} - 53 T^{3} + 2809 T^{4} )^{2} \)
$59$ \( 1 - 2 T - 17 T^{2} + 194 T^{3} - 3276 T^{4} + 11446 T^{5} - 59177 T^{6} - 410758 T^{7} + 12117361 T^{8} \)
$61$ \( 1 + 6 T - 63 T^{2} - 138 T^{3} + 3884 T^{4} - 8418 T^{5} - 234423 T^{6} + 1361886 T^{7} + 13845841 T^{8} \)
$67$ \( 1 + 14 T + 31 T^{2} + 434 T^{3} + 9604 T^{4} + 29078 T^{5} + 139159 T^{6} + 4210682 T^{7} + 20151121 T^{8} \)
$71$ \( ( 1 + 16 T + 174 T^{2} + 1136 T^{3} + 5041 T^{4} )^{2} \)
$73$ \( 1 - 18 T + 129 T^{2} - 882 T^{3} + 9044 T^{4} - 64386 T^{5} + 687441 T^{6} - 7002306 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 2 T - 105 T^{2} - 98 T^{3} + 5324 T^{4} - 7742 T^{5} - 655305 T^{6} + 986078 T^{7} + 38950081 T^{8} \)
$83$ \( ( 1 + 8 T + 54 T^{2} + 664 T^{3} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 + 8 T + 202 T^{2} + 776 T^{3} + 9409 T^{4} )^{2} \)
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