Properties

Label 28.2.f.a
Level $28$
Weight $2$
Character orbit 28.f
Analytic conductor $0.224$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 28 = 2^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 28.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.223581125660\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{2} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12} q^{4} + ( -2 + \zeta_{12}^{2} ) q^{5} + ( -1 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{6} + ( -2 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{7} + ( -2 - 2 \zeta_{12}^{3} ) q^{8} +O(q^{10})\) \( q + ( -\zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{2} + ( \zeta_{12} - 2 \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12} q^{4} + ( -2 + \zeta_{12}^{2} ) q^{5} + ( -1 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{6} + ( -2 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{7} + ( -2 - 2 \zeta_{12}^{3} ) q^{8} + ( 1 + \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{10} + \zeta_{12} q^{11} + ( 4 - 2 \zeta_{12}^{2} ) q^{12} + ( 2 - 4 \zeta_{12}^{2} ) q^{13} + ( 2 + 3 \zeta_{12} - 3 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{14} + 3 \zeta_{12}^{3} q^{15} + 4 \zeta_{12}^{2} q^{16} + ( -1 - \zeta_{12}^{2} ) q^{17} + ( -3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{19} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{20} + ( -1 + 5 \zeta_{12}^{2} ) q^{21} + ( -1 - \zeta_{12}^{3} ) q^{22} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{23} + ( -2 - 2 \zeta_{12} - 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{24} + ( -2 + 2 \zeta_{12}^{2} ) q^{25} + ( -4 + 2 \zeta_{12} + 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{26} + ( 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{27} + ( -6 + 2 \zeta_{12}^{2} ) q^{28} + 4 q^{29} + ( 3 \zeta_{12} - 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{30} + ( -\zeta_{12} + 2 \zeta_{12}^{3} ) q^{31} + ( 4 - 4 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{32} + ( 2 - \zeta_{12}^{2} ) q^{33} + ( -1 + 2 \zeta_{12} + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{34} + ( \zeta_{12} - 5 \zeta_{12}^{3} ) q^{35} -3 \zeta_{12}^{2} q^{37} + ( 3 - 3 \zeta_{12} + 3 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{38} -6 \zeta_{12} q^{39} + ( 4 + 2 \zeta_{12} - 2 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{40} + ( -2 + 4 \zeta_{12}^{2} ) q^{41} + ( 5 - 4 \zeta_{12} - 4 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{42} + 2 \zeta_{12}^{3} q^{43} + 2 \zeta_{12}^{2} q^{44} + ( -1 - \zeta_{12} + \zeta_{12}^{2} ) q^{46} + ( 5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{47} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{48} + ( 3 - 8 \zeta_{12}^{2} ) q^{49} + ( 2 - 2 \zeta_{12}^{3} ) q^{50} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{51} + ( 4 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{52} + ( 1 - \zeta_{12}^{2} ) q^{53} + ( -6 - 3 \zeta_{12} + 3 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{54} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{55} + ( 2 + 4 \zeta_{12} + 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{56} -9 q^{57} + ( -4 \zeta_{12} - 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{58} + ( -3 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{59} + ( -6 + 6 \zeta_{12}^{2} ) q^{60} + ( -6 + 3 \zeta_{12}^{2} ) q^{61} + ( 1 + 2 \zeta_{12} - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{62} + 8 \zeta_{12}^{3} q^{64} + 6 \zeta_{12}^{2} q^{65} + ( -1 - \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{66} + 3 \zeta_{12} q^{67} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{68} + ( 1 - 2 \zeta_{12}^{2} ) q^{69} + ( -1 - 5 \zeta_{12} + 5 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{70} -14 \zeta_{12}^{3} q^{71} + ( 5 + 5 \zeta_{12}^{2} ) q^{73} + ( -3 + 3 \zeta_{12} + 3 \zeta_{12}^{2} ) q^{74} + ( 2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{75} + ( 6 - 12 \zeta_{12}^{2} ) q^{76} + ( -3 + \zeta_{12}^{2} ) q^{77} + ( 6 + 6 \zeta_{12}^{3} ) q^{78} + ( 9 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{79} + ( -4 - 4 \zeta_{12}^{2} ) q^{80} + ( 9 - 9 \zeta_{12}^{2} ) q^{81} + ( 4 - 2 \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{82} + ( -16 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{83} + ( -2 \zeta_{12} + 10 \zeta_{12}^{3} ) q^{84} + 3 q^{85} + ( 2 \zeta_{12} - 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{86} + ( 4 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{87} + ( 2 - 2 \zeta_{12} - 2 \zeta_{12}^{2} ) q^{88} + ( 18 - 9 \zeta_{12}^{2} ) q^{89} + ( 8 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{91} + 2 q^{92} + 3 \zeta_{12}^{2} q^{93} + ( -5 + 5 \zeta_{12} - 5 \zeta_{12}^{2} - 10 \zeta_{12}^{3} ) q^{94} + 9 \zeta_{12} q^{95} + ( -8 - 4 \zeta_{12} + 4 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{96} + ( -10 + 20 \zeta_{12}^{2} ) q^{97} + ( -8 + 5 \zeta_{12} + 5 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} - 6q^{5} - 8q^{8} + O(q^{10}) \) \( 4q - 2q^{2} - 6q^{5} - 8q^{8} + 6q^{10} + 12q^{12} + 2q^{14} + 8q^{16} - 6q^{17} + 6q^{21} - 4q^{22} - 12q^{24} - 4q^{25} - 12q^{26} - 20q^{28} + 16q^{29} - 6q^{30} + 8q^{32} + 6q^{33} - 6q^{37} + 18q^{38} + 12q^{40} + 12q^{42} + 4q^{44} - 2q^{46} - 4q^{49} + 8q^{50} + 2q^{53} - 18q^{54} + 16q^{56} - 36q^{57} - 8q^{58} - 12q^{60} - 18q^{61} + 12q^{65} - 6q^{66} + 6q^{70} + 30q^{73} - 6q^{74} - 10q^{77} + 24q^{78} - 24q^{80} + 18q^{81} + 12q^{82} + 12q^{85} - 4q^{86} + 4q^{88} + 54q^{89} + 8q^{92} + 6q^{93} - 30q^{94} - 24q^{96} - 22q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/28\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\)
\(\chi(n)\) \(-1\) \(\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−1.36603 0.366025i 0.866025 1.50000i 1.73205 + 1.00000i −1.50000 + 0.866025i −1.73205 + 1.73205i −1.73205 + 2.00000i −2.00000 2.00000i 0 2.36603 0.633975i
3.2 0.366025 1.36603i −0.866025 + 1.50000i −1.73205 1.00000i −1.50000 + 0.866025i 1.73205 + 1.73205i 1.73205 2.00000i −2.00000 + 2.00000i 0 0.633975 + 2.36603i
19.1 −1.36603 + 0.366025i 0.866025 + 1.50000i 1.73205 1.00000i −1.50000 0.866025i −1.73205 1.73205i −1.73205 2.00000i −2.00000 + 2.00000i 0 2.36603 + 0.633975i
19.2 0.366025 + 1.36603i −0.866025 1.50000i −1.73205 + 1.00000i −1.50000 0.866025i 1.73205 1.73205i 1.73205 + 2.00000i −2.00000 2.00000i 0 0.633975 2.36603i
\(n\): 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.d odd 6 1 inner
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 28.2.f.a 4
3.b odd 2 1 252.2.bf.e 4
4.b odd 2 1 inner 28.2.f.a 4
5.b even 2 1 700.2.p.a 4
5.c odd 4 1 700.2.t.a 4
5.c odd 4 1 700.2.t.b 4
7.b odd 2 1 196.2.f.a 4
7.c even 3 1 196.2.d.b 4
7.c even 3 1 196.2.f.a 4
7.d odd 6 1 inner 28.2.f.a 4
7.d odd 6 1 196.2.d.b 4
8.b even 2 1 448.2.p.d 4
8.d odd 2 1 448.2.p.d 4
12.b even 2 1 252.2.bf.e 4
20.d odd 2 1 700.2.p.a 4
20.e even 4 1 700.2.t.a 4
20.e even 4 1 700.2.t.b 4
21.g even 6 1 252.2.bf.e 4
21.g even 6 1 1764.2.b.a 4
21.h odd 6 1 1764.2.b.a 4
28.d even 2 1 196.2.f.a 4
28.f even 6 1 inner 28.2.f.a 4
28.f even 6 1 196.2.d.b 4
28.g odd 6 1 196.2.d.b 4
28.g odd 6 1 196.2.f.a 4
35.i odd 6 1 700.2.p.a 4
35.k even 12 1 700.2.t.a 4
35.k even 12 1 700.2.t.b 4
56.j odd 6 1 448.2.p.d 4
56.j odd 6 1 3136.2.f.e 4
56.k odd 6 1 3136.2.f.e 4
56.m even 6 1 448.2.p.d 4
56.m even 6 1 3136.2.f.e 4
56.p even 6 1 3136.2.f.e 4
84.j odd 6 1 252.2.bf.e 4
84.j odd 6 1 1764.2.b.a 4
84.n even 6 1 1764.2.b.a 4
140.s even 6 1 700.2.p.a 4
140.x odd 12 1 700.2.t.a 4
140.x odd 12 1 700.2.t.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.f.a 4 1.a even 1 1 trivial
28.2.f.a 4 4.b odd 2 1 inner
28.2.f.a 4 7.d odd 6 1 inner
28.2.f.a 4 28.f even 6 1 inner
196.2.d.b 4 7.c even 3 1
196.2.d.b 4 7.d odd 6 1
196.2.d.b 4 28.f even 6 1
196.2.d.b 4 28.g odd 6 1
196.2.f.a 4 7.b odd 2 1
196.2.f.a 4 7.c even 3 1
196.2.f.a 4 28.d even 2 1
196.2.f.a 4 28.g odd 6 1
252.2.bf.e 4 3.b odd 2 1
252.2.bf.e 4 12.b even 2 1
252.2.bf.e 4 21.g even 6 1
252.2.bf.e 4 84.j odd 6 1
448.2.p.d 4 8.b even 2 1
448.2.p.d 4 8.d odd 2 1
448.2.p.d 4 56.j odd 6 1
448.2.p.d 4 56.m even 6 1
700.2.p.a 4 5.b even 2 1
700.2.p.a 4 20.d odd 2 1
700.2.p.a 4 35.i odd 6 1
700.2.p.a 4 140.s even 6 1
700.2.t.a 4 5.c odd 4 1
700.2.t.a 4 20.e even 4 1
700.2.t.a 4 35.k even 12 1
700.2.t.a 4 140.x odd 12 1
700.2.t.b 4 5.c odd 4 1
700.2.t.b 4 20.e even 4 1
700.2.t.b 4 35.k even 12 1
700.2.t.b 4 140.x odd 12 1
1764.2.b.a 4 21.g even 6 1
1764.2.b.a 4 21.h odd 6 1
1764.2.b.a 4 84.j odd 6 1
1764.2.b.a 4 84.n even 6 1
3136.2.f.e 4 56.j odd 6 1
3136.2.f.e 4 56.k odd 6 1
3136.2.f.e 4 56.m even 6 1
3136.2.f.e 4 56.p even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(28, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} \)
$3$ \( ( 1 - 3 T^{2} )^{2}( 1 + 3 T^{2} + 9 T^{4} ) \)
$5$ \( ( 1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4} )^{2} \)
$7$ \( 1 + 2 T^{2} + 49 T^{4} \)
$11$ \( 1 + 21 T^{2} + 320 T^{4} + 2541 T^{6} + 14641 T^{8} \)
$13$ \( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 + 3 T + 20 T^{2} + 51 T^{3} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 37 T^{2} + 361 T^{4} )( 1 + 26 T^{2} + 361 T^{4} ) \)
$23$ \( 1 + 45 T^{2} + 1496 T^{4} + 23805 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 - 4 T + 29 T^{2} )^{4} \)
$31$ \( ( 1 - 46 T^{2} + 961 T^{4} )( 1 - 13 T^{2} + 961 T^{4} ) \)
$37$ \( ( 1 + 3 T - 28 T^{2} + 111 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 - 70 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 82 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( 1 - 19 T^{2} - 1848 T^{4} - 41971 T^{6} + 4879681 T^{8} \)
$53$ \( ( 1 - T - 52 T^{2} - 53 T^{3} + 2809 T^{4} )^{2} \)
$59$ \( 1 - 91 T^{2} + 4800 T^{4} - 316771 T^{6} + 12117361 T^{8} \)
$61$ \( ( 1 + 9 T + 88 T^{2} + 549 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( 1 + 125 T^{2} + 11136 T^{4} + 561125 T^{6} + 20151121 T^{8} \)
$71$ \( ( 1 + 54 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 15 T + 148 T^{2} - 1095 T^{3} + 5329 T^{4} )^{2} \)
$79$ \( 1 + 77 T^{2} - 312 T^{4} + 480557 T^{6} + 38950081 T^{8} \)
$83$ \( ( 1 - 26 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 27 T + 332 T^{2} - 2403 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 + 106 T^{2} + 9409 T^{4} )^{2} \)
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