Properties

Label 124.4.a.b
Level $124$
Weight $4$
Character orbit 124.a
Self dual yes
Analytic conductor $7.316$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 124 = 2^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 124.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.31623684071\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.4000044.1
Defining polynomial: \( x^{4} - x^{3} - 21x^{2} + 16x + 62 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{3} + \beta_1 + 1) q^{5} + (\beta_{2} + \beta_1 + 4) q^{7} + (2 \beta_{3} - 2 \beta_{2} + 15) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{3} + \beta_1 + 1) q^{5} + (\beta_{2} + \beta_1 + 4) q^{7} + (2 \beta_{3} - 2 \beta_{2} + 15) q^{9} + (\beta_{3} - \beta_{2} - 3 \beta_1 + 25) q^{11} + (2 \beta_{3} + 4 \beta_{2} + \beta_1 + 12) q^{13} + ( - 3 \beta_{3} - 7 \beta_{2} - 6 \beta_1 + 45) q^{15} + (\beta_{3} + 11 \beta_{2} + 4 \beta_1 + 3) q^{17} + ( - 8 \beta_{3} - 3 \beta_{2} - 9 \beta_1 + 48) q^{19} + (3 \beta_{3} - 5 \beta_{2} + 35) q^{21} + (3 \beta_{3} + 11 \beta_{2} - 4 \beta_1 + 43) q^{23} + ( - 13 \beta_{3} - 18 \beta_{2} - 3 \beta_1 + 30) q^{25} + (8 \beta_{3} + 16 \beta_{2} + 10 \beta_1 + 8) q^{27} + (8 \beta_{3} - 20 \beta_{2} - 5 \beta_1 + 58) q^{29} + 31 q^{31} + ( - 2 \beta_{3} + 14 \beta_{2} + 36 \beta_1 - 122) q^{33} + ( - 2 \beta_{3} - 5 \beta_{2} - 9 \beta_1 + 58) q^{35} + (15 \beta_{3} + 7 \beta_{2} + 7 \beta_1 - 71) q^{37} + (16 \beta_{3} - 4 \beta_{2} + 10 \beta_1 + 8) q^{39} + ( - 17 \beta_{3} + 18 \beta_{2} + \beta_1 - 123) q^{41} + ( - 16 \beta_{3} + 6 \beta_{2} + 3 \beta_1 - 80) q^{43} + ( - 7 \beta_{3} + 18 \beta_{2} + 25 \beta_1 - 221) q^{45} + ( - 4 \beta_{3} - 40 \beta_{2} - 22 \beta_1 + 108) q^{47} + (3 \beta_{3} + 6 \beta_{2} - 3 \beta_1 - 240) q^{49} + (24 \beta_{3} - 36 \beta_{2} - 34 \beta_1 + 88) q^{51} + (3 \beta_{3} - 35 \beta_{2} - 43 \beta_1 - 59) q^{53} + ( - 12 \beta_{3} + 30 \beta_{2} + 48 \beta_1 - 228) q^{55} + ( - 61 \beta_{3} - 13 \beta_{2} + 4 \beta_1 - 333) q^{57} + (26 \beta_{3} + 45 \beta_{2} - 25 \beta_1 - 6) q^{59} + (2 \beta_{3} + 20 \beta_{2} - 73 \beta_1 - 220) q^{61} + (10 \beta_{3} + 3 \beta_{2} + 49 \beta_1 - 82) q^{63} + (23 \beta_{3} + 23 \beta_{2} - 26 \beta_1 - 125) q^{65} + ( - 4 \beta_{3} + 16 \beta_{2} - 32 \beta_1 + 24) q^{67} + (18 \beta_{3} - 10 \beta_{2} + 20 \beta_1 - 254) q^{69} + (6 \beta_{3} - 15 \beta_{2} - 35 \beta_1 + 386) q^{71} + (54 \beta_{3} - 58 \beta_{2} - 40 \beta_1 - 192) q^{73} + ( - 89 \beta_{3} - 5 \beta_{2} + 11 \beta_1 + 39) q^{75} + ( - 4 \beta_{3} + 34 \beta_{2} + 42 \beta_1 - 76) q^{77} + ( - 15 \beta_{3} - 85 \beta_{2} + 68 \beta_1 + 129) q^{79} + (22 \beta_{3} + 26 \beta_{2} - 121) q^{81} + (41 \beta_{3} + 11 \beta_{2} + 57 \beta_1 + 329) q^{83} + (49 \beta_{3} + 5 \beta_{2} - 100 \beta_1 + 173) q^{85} + (10 \beta_{3} + 110 \beta_{2} + 194 \beta_1 - 94) q^{87} + ( - 39 \beta_{3} - 19 \beta_{2} + 68 \beta_1 + 19) q^{89} + (13 \beta_{3} + 37 \beta_{2} + 4 \beta_1 + 253) q^{91} + 31 \beta_1 q^{93} + ( - 108 \beta_{3} - 31 \beta_{2} + 139 \beta_1 + 488) q^{95} + ( - 27 \beta_{3} - 110 \beta_{2} - 29 \beta_1 + 91) q^{97} + (49 \beta_{3} - 97 \beta_{2} - 111 \beta_1 + 745) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 6 q^{5} + 16 q^{7} + 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 6 q^{5} + 16 q^{7} + 64 q^{9} + 96 q^{11} + 42 q^{13} + 182 q^{15} - 2 q^{17} + 180 q^{19} + 150 q^{21} + 142 q^{23} + 150 q^{25} + 20 q^{27} + 262 q^{29} + 124 q^{31} - 444 q^{33} + 224 q^{35} - 284 q^{37} + 60 q^{39} - 526 q^{41} - 326 q^{43} - 870 q^{45} + 468 q^{47} - 978 q^{49} + 356 q^{51} - 252 q^{53} - 876 q^{55} - 1298 q^{57} - 164 q^{59} - 1066 q^{61} - 236 q^{63} - 598 q^{65} - 956 q^{69} + 1504 q^{71} - 732 q^{73} + 188 q^{75} - 288 q^{77} + 822 q^{79} - 536 q^{81} + 1408 q^{83} + 482 q^{85} - 208 q^{87} + 250 q^{89} + 946 q^{91} + 62 q^{93} + 2292 q^{95} + 526 q^{97} + 2952 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 21x^{2} + 16x + 62 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 2\nu^{2} - 16\nu + 20 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 4\nu^{2} - 16\nu - 43 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - \beta_{2} + 21 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 8\beta _1 + 1 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−4.13619
−1.48616
2.37430
4.24805
0 −8.27238 0 −14.2221 0 −10.5389 0 41.4323 0
1.2 0 −2.97232 0 2.58408 0 13.0539 0 −18.1653 0
1.3 0 4.74860 0 20.7669 0 3.45568 0 −4.45080 0
1.4 0 8.49610 0 −3.12892 0 10.0292 0 45.1838 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(31\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 124.4.a.b 4
3.b odd 2 1 1116.4.a.f 4
4.b odd 2 1 496.4.a.f 4
8.b even 2 1 1984.4.a.m 4
8.d odd 2 1 1984.4.a.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.4.a.b 4 1.a even 1 1 trivial
496.4.a.f 4 4.b odd 2 1
1116.4.a.f 4 3.b odd 2 1
1984.4.a.m 4 8.b even 2 1
1984.4.a.o 4 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 2T_{3}^{3} - 84T_{3}^{2} + 128T_{3} + 992 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(124))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} - 84 T^{2} + 128 T + 992 \) Copy content Toggle raw display
$5$ \( T^{4} - 6 T^{3} - 307 T^{2} + \cdots + 2388 \) Copy content Toggle raw display
$7$ \( T^{4} - 16 T^{3} - 69 T^{2} + \cdots - 4768 \) Copy content Toggle raw display
$11$ \( T^{4} - 96 T^{3} + 2364 T^{2} + \cdots + 16416 \) Copy content Toggle raw display
$13$ \( T^{4} - 42 T^{3} - 1536 T^{2} + \cdots + 441648 \) Copy content Toggle raw display
$17$ \( T^{4} + 2 T^{3} - 12864 T^{2} + \cdots + 12601872 \) Copy content Toggle raw display
$19$ \( T^{4} - 180 T^{3} + \cdots - 110971856 \) Copy content Toggle raw display
$23$ \( T^{4} - 142 T^{3} - 8700 T^{2} + \cdots - 7610112 \) Copy content Toggle raw display
$29$ \( T^{4} - 262 T^{3} + \cdots - 189481680 \) Copy content Toggle raw display
$31$ \( (T - 31)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 284 T^{3} + \cdots - 250857520 \) Copy content Toggle raw display
$41$ \( T^{4} + 526 T^{3} + \cdots + 949605156 \) Copy content Toggle raw display
$43$ \( T^{4} + 326 T^{3} + \cdots + 551576736 \) Copy content Toggle raw display
$47$ \( T^{4} - 468 T^{3} + \cdots + 1256256000 \) Copy content Toggle raw display
$53$ \( T^{4} + 252 T^{3} + \cdots - 7248886080 \) Copy content Toggle raw display
$59$ \( T^{4} + 164 T^{3} + \cdots + 42747696 \) Copy content Toggle raw display
$61$ \( T^{4} + 1066 T^{3} + \cdots + 37784393520 \) Copy content Toggle raw display
$67$ \( T^{4} - 135984 T^{2} + \cdots + 2730691584 \) Copy content Toggle raw display
$71$ \( T^{4} - 1504 T^{3} + \cdots + 9642905280 \) Copy content Toggle raw display
$73$ \( T^{4} + 732 T^{3} + \cdots + 231605048496 \) Copy content Toggle raw display
$79$ \( T^{4} - 822 T^{3} + \cdots + 4306806912 \) Copy content Toggle raw display
$83$ \( T^{4} - 1408 T^{3} + \cdots - 747812064 \) Copy content Toggle raw display
$89$ \( T^{4} - 250 T^{3} + \cdots + 25296843024 \) Copy content Toggle raw display
$97$ \( T^{4} - 526 T^{3} + \cdots + 180198106052 \) Copy content Toggle raw display
show more
show less