Properties

Label 124.4.d.a
Level $124$
Weight $4$
Character orbit 124.d
Analytic conductor $7.316$
Analytic rank $0$
Dimension $2$
CM discriminant -31
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 124 = 2^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 124.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.31623684071\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-31}) \)
Defining polynomial: \( x^{2} - x + 8 \) Copy content Toggle raw display
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 \(\beta = \frac{1}{2}(1 + \sqrt{-31})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + (\beta - 8) q^{4} - 2 q^{5} + ( - 12 \beta + 6) q^{7} + ( - 7 \beta - 8) q^{8} - 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + (\beta - 8) q^{4} - 2 q^{5} + ( - 12 \beta + 6) q^{7} + ( - 7 \beta - 8) q^{8} - 27 q^{9} - 2 \beta q^{10} + ( - 6 \beta + 96) q^{14} + ( - 15 \beta + 56) q^{16} - 27 \beta q^{18} + ( - 20 \beta + 10) q^{19} + ( - 2 \beta + 16) q^{20} - 121 q^{25} + (90 \beta + 48) q^{28} + ( - 62 \beta + 31) q^{31} + (41 \beta + 120) q^{32} + (24 \beta - 12) q^{35} + ( - 27 \beta + 216) q^{36} + ( - 10 \beta + 160) q^{38} + (14 \beta + 16) q^{40} - 278 q^{41} + 54 q^{45} + (68 \beta - 34) q^{47} - 773 q^{49} - 121 \beta q^{50} + (138 \beta - 720) q^{56} + (188 \beta - 94) q^{59} + ( - 31 \beta + 496) q^{62} + (324 \beta - 162) q^{63} + (161 \beta - 328) q^{64} + ( - 308 \beta + 154) q^{67} + (12 \beta - 192) q^{70} + ( - 236 \beta + 118) q^{71} + (189 \beta + 216) q^{72} + (150 \beta + 80) q^{76} + (30 \beta - 112) q^{80} + 729 q^{81} - 278 \beta q^{82} + 54 \beta q^{90} + (34 \beta - 544) q^{94} + (40 \beta - 20) q^{95} + 1906 q^{97} - 773 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 15 q^{4} - 4 q^{5} - 23 q^{8} - 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 15 q^{4} - 4 q^{5} - 23 q^{8} - 54 q^{9} - 2 q^{10} + 186 q^{14} + 97 q^{16} - 27 q^{18} + 30 q^{20} - 242 q^{25} + 186 q^{28} + 281 q^{32} + 405 q^{36} + 310 q^{38} + 46 q^{40} - 556 q^{41} + 108 q^{45} - 1546 q^{49} - 121 q^{50} - 1302 q^{56} + 961 q^{62} - 495 q^{64} - 372 q^{70} + 621 q^{72} + 310 q^{76} - 194 q^{80} + 1458 q^{81} - 278 q^{82} + 54 q^{90} - 1054 q^{94} + 3812 q^{97} - 773 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/124\mathbb{Z}\right)^\times\).

\(n\) \(63\) \(65\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
123.1
0.500000 2.78388i
0.500000 + 2.78388i
0.500000 2.78388i 0 −7.50000 2.78388i −2.00000 0 33.4066i −11.5000 + 19.4872i −27.0000 −1.00000 + 5.56776i
123.2 0.500000 + 2.78388i 0 −7.50000 + 2.78388i −2.00000 0 33.4066i −11.5000 19.4872i −27.0000 −1.00000 5.56776i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 CM by \(\Q(\sqrt{-31}) \)
4.b odd 2 1 inner
124.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 124.4.d.a 2
4.b odd 2 1 inner 124.4.d.a 2
31.b odd 2 1 CM 124.4.d.a 2
124.d even 2 1 inner 124.4.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.4.d.a 2 1.a even 1 1 trivial
124.4.d.a 2 4.b odd 2 1 inner
124.4.d.a 2 31.b odd 2 1 CM
124.4.d.a 2 124.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(124, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 8 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 1116 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 3100 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 29791 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( (T + 278)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 35836 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 273916 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 735196 \) Copy content Toggle raw display
$71$ \( T^{2} + 431644 \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T - 1906)^{2} \) Copy content Toggle raw display
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