Properties

Label 28.4.d.a
Level $28$
Weight $4$
Character orbit 28.d
Analytic conductor $1.652$
Analytic rank $0$
Dimension $2$
CM discriminant -7
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 28 = 2^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 28.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.65205348016\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-7}) \)
Defining polynomial: \(x^{2} - x + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{-7})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 3 - \beta ) q^{2} + ( 7 - 5 \beta ) q^{4} + ( -7 + 14 \beta ) q^{7} + ( 11 - 17 \beta ) q^{8} -27 q^{9} +O(q^{10})\) \( q + ( 3 - \beta ) q^{2} + ( 7 - 5 \beta ) q^{4} + ( -7 + 14 \beta ) q^{7} + ( 11 - 17 \beta ) q^{8} -27 q^{9} + ( -10 + 20 \beta ) q^{11} + ( 7 + 35 \beta ) q^{14} + ( -1 - 45 \beta ) q^{16} + ( -81 + 27 \beta ) q^{18} + ( 10 + 50 \beta ) q^{22} + ( 82 - 164 \beta ) q^{23} + 125 q^{25} + ( 91 + 63 \beta ) q^{28} + 166 q^{29} + ( -93 - 89 \beta ) q^{32} + ( -189 + 135 \beta ) q^{36} -450 q^{37} + ( -202 + 404 \beta ) q^{43} + ( 130 + 90 \beta ) q^{44} + ( -82 - 410 \beta ) q^{46} -343 q^{49} + ( 375 - 125 \beta ) q^{50} + 590 q^{53} + ( 399 + 35 \beta ) q^{56} + ( 498 - 166 \beta ) q^{58} + ( 189 - 378 \beta ) q^{63} + ( -457 - 85 \beta ) q^{64} + ( -306 + 612 \beta ) q^{67} + ( 370 - 740 \beta ) q^{71} + ( -297 + 459 \beta ) q^{72} + ( -1350 + 450 \beta ) q^{74} -490 q^{77} + ( 90 - 180 \beta ) q^{79} + 729 q^{81} + ( 202 + 1010 \beta ) q^{86} + ( 570 + 50 \beta ) q^{88} + ( -1066 - 738 \beta ) q^{92} + ( -1029 + 343 \beta ) q^{98} + ( 270 - 540 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 5q^{2} + 9q^{4} + 5q^{8} - 54q^{9} + O(q^{10}) \) \( 2q + 5q^{2} + 9q^{4} + 5q^{8} - 54q^{9} + 49q^{14} - 47q^{16} - 135q^{18} + 70q^{22} + 250q^{25} + 245q^{28} + 332q^{29} - 275q^{32} - 243q^{36} - 900q^{37} + 350q^{44} - 574q^{46} - 686q^{49} + 625q^{50} + 1180q^{53} + 833q^{56} + 830q^{58} - 999q^{64} - 135q^{72} - 2250q^{74} - 980q^{77} + 1458q^{81} + 1414q^{86} + 1190q^{88} - 2870q^{92} - 1715q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(15\) \(17\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1
0.500000 + 1.32288i
0.500000 1.32288i
2.50000 1.32288i 0 4.50000 6.61438i 0 0 18.5203i 2.50000 22.4889i −27.0000 0
27.2 2.50000 + 1.32288i 0 4.50000 + 6.61438i 0 0 18.5203i 2.50000 + 22.4889i −27.0000 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.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
4.b odd 2 1 inner
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 28.4.d.a 2
3.b odd 2 1 252.4.b.a 2
4.b odd 2 1 inner 28.4.d.a 2
7.b odd 2 1 CM 28.4.d.a 2
7.c even 3 2 196.4.f.a 4
7.d odd 6 2 196.4.f.a 4
8.b even 2 1 448.4.f.a 2
8.d odd 2 1 448.4.f.a 2
12.b even 2 1 252.4.b.a 2
21.c even 2 1 252.4.b.a 2
28.d even 2 1 inner 28.4.d.a 2
28.f even 6 2 196.4.f.a 4
28.g odd 6 2 196.4.f.a 4
56.e even 2 1 448.4.f.a 2
56.h odd 2 1 448.4.f.a 2
84.h odd 2 1 252.4.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.d.a 2 1.a even 1 1 trivial
28.4.d.a 2 4.b odd 2 1 inner
28.4.d.a 2 7.b odd 2 1 CM
28.4.d.a 2 28.d even 2 1 inner
196.4.f.a 4 7.c even 3 2
196.4.f.a 4 7.d odd 6 2
196.4.f.a 4 28.f even 6 2
196.4.f.a 4 28.g odd 6 2
252.4.b.a 2 3.b odd 2 1
252.4.b.a 2 12.b even 2 1
252.4.b.a 2 21.c even 2 1
252.4.b.a 2 84.h odd 2 1
448.4.f.a 2 8.b even 2 1
448.4.f.a 2 8.d odd 2 1
448.4.f.a 2 56.e even 2 1
448.4.f.a 2 56.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{4}^{\mathrm{new}}(28, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 8 - 5 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 343 + T^{2} \)
$11$ \( 700 + T^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( T^{2} \)
$23$ \( 47068 + T^{2} \)
$29$ \( ( -166 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( ( 450 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( 285628 + T^{2} \)
$47$ \( T^{2} \)
$53$ \( ( -590 + T )^{2} \)
$59$ \( T^{2} \)
$61$ \( T^{2} \)
$67$ \( 655452 + T^{2} \)
$71$ \( 958300 + T^{2} \)
$73$ \( T^{2} \)
$79$ \( 56700 + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( T^{2} \)
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