Properties

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

Related objects

Downloads

Learn more about

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: \(\Q(\sqrt{-3}, \sqrt{7})\)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} -3 \beta_{2} q^{5} -\beta_{3} q^{7} + 4 \beta_{2} q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} -3 \beta_{2} q^{5} -\beta_{3} q^{7} + 4 \beta_{2} q^{9} -\beta_{1} q^{11} -4 q^{13} -3 \beta_{3} q^{15} + ( -1 - \beta_{2} ) q^{17} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{19} + ( 7 + 7 \beta_{2} ) q^{21} + ( \beta_{1} + \beta_{3} ) q^{23} + ( -4 - 4 \beta_{2} ) q^{25} + \beta_{3} q^{27} -4 q^{29} -\beta_{1} q^{31} -7 \beta_{2} q^{33} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{35} -5 \beta_{2} q^{37} -4 \beta_{1} q^{39} + 8 q^{41} + 4 \beta_{3} q^{43} + ( 12 + 12 \beta_{2} ) q^{45} + ( -\beta_{1} - \beta_{3} ) q^{47} + 7 q^{49} + ( -\beta_{1} - \beta_{3} ) q^{51} + ( -7 - 7 \beta_{2} ) q^{53} + 3 \beta_{3} q^{55} -21 q^{57} + \beta_{1} q^{59} -5 \beta_{2} q^{61} + ( 4 \beta_{1} + 4 \beta_{3} ) q^{63} + 12 \beta_{2} q^{65} + \beta_{1} q^{67} -7 q^{69} + ( 9 + 9 \beta_{2} ) q^{73} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{75} + ( -7 - 7 \beta_{2} ) q^{77} + ( \beta_{1} + \beta_{3} ) q^{79} + ( 5 + 5 \beta_{2} ) q^{81} -4 \beta_{3} q^{83} -3 q^{85} -4 \beta_{1} q^{87} -9 \beta_{2} q^{89} + 4 \beta_{3} q^{91} -7 \beta_{2} q^{93} + 9 \beta_{1} q^{95} -8 q^{97} -4 \beta_{3} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 6q^{5} - 8q^{9} + O(q^{10}) \) \( 4q + 6q^{5} - 8q^{9} - 16q^{13} - 2q^{17} + 14q^{21} - 8q^{25} - 16q^{29} + 14q^{33} + 10q^{37} + 32q^{41} + 24q^{45} + 28q^{49} - 14q^{53} - 84q^{57} + 10q^{61} - 24q^{65} - 28q^{69} + 18q^{73} - 14q^{77} + 10q^{81} - 12q^{85} + 18q^{89} + 14q^{93} - 32q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 7 x^{2} + 49\):

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

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
−1.32288 2.29129i
1.32288 + 2.29129i
−1.32288 + 2.29129i
1.32288 2.29129i
0 −1.32288 2.29129i 0 1.50000 2.59808i 0 −2.64575 0 −2.00000 + 3.46410i 0
65.2 0 1.32288 + 2.29129i 0 1.50000 2.59808i 0 2.64575 0 −2.00000 + 3.46410i 0
193.1 0 −1.32288 + 2.29129i 0 1.50000 + 2.59808i 0 −2.64575 0 −2.00000 3.46410i 0
193.2 0 1.32288 2.29129i 0 1.50000 + 2.59808i 0 2.64575 0 −2.00000 3.46410i 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
7.c even 3 1 inner
28.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.2.i.c 4
3.b odd 2 1 2016.2.s.o 4
4.b odd 2 1 inner 224.2.i.c 4
7.b odd 2 1 1568.2.i.p 4
7.c even 3 1 inner 224.2.i.c 4
7.c even 3 1 1568.2.a.m 2
7.d odd 6 1 1568.2.a.t 2
7.d odd 6 1 1568.2.i.p 4
8.b even 2 1 448.2.i.h 4
8.d odd 2 1 448.2.i.h 4
12.b even 2 1 2016.2.s.o 4
21.h odd 6 1 2016.2.s.o 4
28.d even 2 1 1568.2.i.p 4
28.f even 6 1 1568.2.a.t 2
28.f even 6 1 1568.2.i.p 4
28.g odd 6 1 inner 224.2.i.c 4
28.g odd 6 1 1568.2.a.m 2
56.j odd 6 1 3136.2.a.bg 2
56.k odd 6 1 448.2.i.h 4
56.k odd 6 1 3136.2.a.bv 2
56.m even 6 1 3136.2.a.bg 2
56.p even 6 1 448.2.i.h 4
56.p even 6 1 3136.2.a.bv 2
84.n even 6 1 2016.2.s.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.i.c 4 1.a even 1 1 trivial
224.2.i.c 4 4.b odd 2 1 inner
224.2.i.c 4 7.c even 3 1 inner
224.2.i.c 4 28.g odd 6 1 inner
448.2.i.h 4 8.b even 2 1
448.2.i.h 4 8.d odd 2 1
448.2.i.h 4 56.k odd 6 1
448.2.i.h 4 56.p even 6 1
1568.2.a.m 2 7.c even 3 1
1568.2.a.m 2 28.g odd 6 1
1568.2.a.t 2 7.d odd 6 1
1568.2.a.t 2 28.f even 6 1
1568.2.i.p 4 7.b odd 2 1
1568.2.i.p 4 7.d odd 6 1
1568.2.i.p 4 28.d even 2 1
1568.2.i.p 4 28.f even 6 1
2016.2.s.o 4 3.b odd 2 1
2016.2.s.o 4 12.b even 2 1
2016.2.s.o 4 21.h odd 6 1
2016.2.s.o 4 84.n even 6 1
3136.2.a.bg 2 56.j odd 6 1
3136.2.a.bg 2 56.m even 6 1
3136.2.a.bv 2 56.k odd 6 1
3136.2.a.bv 2 56.p even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 7 T_{3}^{2} + 49 \) acting on \(S_{2}^{\mathrm{new}}(224, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + T^{2} - 8 T^{4} + 9 T^{6} + 81 T^{8} \)
$5$ \( ( 1 - 3 T + 4 T^{2} - 15 T^{3} + 25 T^{4} )^{2} \)
$7$ \( ( 1 - 7 T^{2} )^{2} \)
$11$ \( 1 - 15 T^{2} + 104 T^{4} - 1815 T^{6} + 14641 T^{8} \)
$13$ \( ( 1 + 4 T + 13 T^{2} )^{4} \)
$17$ \( ( 1 + T - 16 T^{2} + 17 T^{3} + 289 T^{4} )^{2} \)
$19$ \( 1 + 25 T^{2} + 264 T^{4} + 9025 T^{6} + 130321 T^{8} \)
$23$ \( 1 - 39 T^{2} + 992 T^{4} - 20631 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 + 4 T + 29 T^{2} )^{4} \)
$31$ \( 1 - 55 T^{2} + 2064 T^{4} - 52855 T^{6} + 923521 T^{8} \)
$37$ \( ( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 - 8 T + 41 T^{2} )^{4} \)
$43$ \( ( 1 - 26 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( 1 - 87 T^{2} + 5360 T^{4} - 192183 T^{6} + 4879681 T^{8} \)
$53$ \( ( 1 + 7 T - 4 T^{2} + 371 T^{3} + 2809 T^{4} )^{2} \)
$59$ \( 1 - 111 T^{2} + 8840 T^{4} - 386391 T^{6} + 12117361 T^{8} \)
$61$ \( ( 1 - 5 T - 36 T^{2} - 305 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( 1 - 127 T^{2} + 11640 T^{4} - 570103 T^{6} + 20151121 T^{8} \)
$71$ \( ( 1 + 71 T^{2} )^{4} \)
$73$ \( ( 1 - 9 T + 8 T^{2} - 657 T^{3} + 5329 T^{4} )^{2} \)
$79$ \( 1 - 151 T^{2} + 16560 T^{4} - 942391 T^{6} + 38950081 T^{8} \)
$83$ \( ( 1 + 54 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 9 T - 8 T^{2} - 801 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 + 8 T + 97 T^{2} )^{4} \)
show more
show less