Properties

Label 124.4.a.a
Level $124$
Weight $4$
Character orbit 124.a
Self dual yes
Analytic conductor $7.316$
Analytic rank $1$
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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.841724.1
Defining polynomial: \( x^{4} - x^{3} - 16x^{2} + 11x + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - 1) q^{3} + (2 \beta_{3} - \beta_{2} - \beta_1 - 3) q^{5} + (2 \beta_{3} + 3 \beta_{2} + 4 \beta_1 - 5) q^{7} + (6 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - 1) q^{3} + (2 \beta_{3} - \beta_{2} - \beta_1 - 3) q^{5} + (2 \beta_{3} + 3 \beta_{2} + 4 \beta_1 - 5) q^{7} + (6 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 5) q^{9} + ( - 3 \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 23) q^{11} + ( - 5 \beta_{3} - 10 \beta_{2} + 2 \beta_1 - 33) q^{13} + ( - 8 \beta_{3} - 4 \beta_{2} + 3 \beta_1 - 42) q^{15} + (4 \beta_{3} + 6 \beta_{2} - 15 \beta_1 - 14) q^{17} + ( - 10 \beta_{3} + 13 \beta_{2} - 29) q^{19} + ( - 4 \beta_{2} + 9 \beta_1 - 34) q^{21} + ( - 6 \beta_{3} + 4 \beta_{2} + 5 \beta_1 - 56) q^{23} + (4 \beta_{3} + 21 \beta_{2} + 15 \beta_1 + 18) q^{25} + ( - 4 \beta_{3} - 12 \beta_{2} + 6 \beta_1 - 64) q^{27} + ( - 7 \beta_{3} - 28 \beta_{2} + 6 \beta_1 - 45) q^{29} - 31 q^{31} + (34 \beta_{3} + 6 \beta_{2} - 10 \beta_1 + 86) q^{33} + (6 \beta_{3} - 15 \beta_{2} - 62 \beta_1 - 37) q^{35} + (31 \beta_{3} + 36 \beta_{2} - 19 \beta_1 + 159) q^{37} + (72 \beta_{3} + 10 \beta_{2} + 4 \beta_1 + 36) q^{39} + ( - 30 \beta_{3} - 49 \beta_{2} + 39 \beta_1 + 35) q^{41} + ( - 67 \beta_{3} - 62 \beta_{2} - 14 \beta_1 - 41) q^{43} + (38 \beta_{3} + 43 \beta_{2} + 21 \beta_1 + 237) q^{45} + (52 \beta_{3} + 46 \beta_{2} + 32 \beta_1 - 8) q^{47} + ( - 76 \beta_{3} + 33 \beta_{2} + 35 \beta_1 + 260) q^{49} + ( - 42 \beta_{3} - 8 \beta_{2} - 28 \beta_1 + 74) q^{51} + ( - 55 \beta_{3} - 14 \beta_{2} - 37 \beta_1 + 91) q^{53} + ( - 66 \beta_{3} + 30 \beta_{2} + 76 \beta_1 + 32) q^{55} + (66 \beta_{3} + 20 \beta_{2} - 33 \beta_1 + 356) q^{57} + (12 \beta_{3} + 55 \beta_{2} + 94 \beta_1 - 141) q^{59} + ( - 57 \beta_{3} - 82 \beta_{2} - 76 \beta_1 + 87) q^{61} + (2 \beta_{3} - 81 \beta_{2} - 86 \beta_1 + 79) q^{63} + ( - 140 \beta_{3} + 16 \beta_{2} + 23 \beta_1 + 194) q^{65} + (88 \beta_{3} - 40 \beta_{2} + 4 \beta_1 + 8) q^{67} + (92 \beta_{3} + 12 \beta_{2} - 6 \beta_1 + 188) q^{69} + (126 \beta_{3} + 95 \beta_{2} - 6 \beta_1 - 131) q^{71} + (4 \beta_{3} - 20 \beta_{2} - 70 \beta_1 + 394) q^{73} + ( - 29 \beta_{3} - 8 \beta_{2} + 17 \beta_1 - 3) q^{75} + (6 \beta_{3} - 110 \beta_{2} - 110 \beta_1 - 372) q^{77} + (24 \beta_{3} - 66 \beta_{2} - 33 \beta_1 - 236) q^{79} + ( - 54 \beta_{3} - 46 \beta_{2} + 70 \beta_1 + 139) q^{81} + (23 \beta_{3} + 110 \beta_{2} + 47 \beta_1 - 317) q^{83} + (50 \beta_{3} + 112 \beta_{2} + 225 \beta_1 + 168) q^{85} + (120 \beta_{3} + 14 \beta_{2} + 26 \beta_1 - 96) q^{87} + ( - 78 \beta_{3} + 108 \beta_{2} + 145 \beta_1 - 202) q^{89} + ( - 64 \beta_{3} + 80 \beta_{2} - 199 \beta_1 - 10) q^{91} + (31 \beta_{3} + 31) q^{93} + ( - 66 \beta_{3} - 47 \beta_{2} + 36 \beta_1 - 805) q^{95} + (82 \beta_{3} - 215 \beta_{2} - 83 \beta_1 - 75) q^{97} + ( - 201 \beta_{3} - 14 \beta_{2} + 123 \beta_1 - 65) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 14 q^{5} - 12 q^{7} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} - 14 q^{5} - 12 q^{7} - 24 q^{9} - 98 q^{11} - 128 q^{13} - 162 q^{15} - 86 q^{17} - 116 q^{19} - 118 q^{21} - 214 q^{23} + 102 q^{25} - 244 q^{27} - 168 q^{29} - 124 q^{31} + 324 q^{33} - 272 q^{35} + 598 q^{37} + 152 q^{39} + 218 q^{41} - 192 q^{43} + 990 q^{45} + 32 q^{47} + 1110 q^{49} + 240 q^{51} + 290 q^{53} + 280 q^{55} + 1358 q^{57} - 376 q^{59} + 196 q^{61} + 144 q^{63} + 822 q^{65} + 40 q^{67} + 740 q^{69} - 536 q^{71} + 1436 q^{73} + 22 q^{75} - 1708 q^{77} - 1010 q^{79} + 696 q^{81} - 1174 q^{83} + 1122 q^{85} - 332 q^{87} - 518 q^{89} - 438 q^{91} + 124 q^{93} - 3148 q^{95} - 466 q^{97} - 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 16x^{2} + 11x + 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + \nu^{2} - 17\nu - 8 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - 2\nu^{2} - 14\nu + 16 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{3} + 2\beta_{2} + \beta _1 + 16 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 8\beta_1 \) 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
−0.405600
1.09434
4.12989
−3.81863
0 −8.09422 0 12.3353 0 4.93663 0 38.5164 0
1.2 0 −0.864858 0 2.57292 0 −20.6112 0 −26.2520 0
1.3 0 0.830383 0 −18.0162 0 33.6654 0 −26.3105 0
1.4 0 4.12869 0 −10.8920 0 −29.9908 0 −9.95389 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.a 4
3.b odd 2 1 1116.4.a.g 4
4.b odd 2 1 496.4.a.h 4
8.b even 2 1 1984.4.a.p 4
8.d odd 2 1 1984.4.a.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.4.a.a 4 1.a even 1 1 trivial
496.4.a.h 4 4.b odd 2 1
1116.4.a.g 4 3.b odd 2 1
1984.4.a.k 4 8.d odd 2 1
1984.4.a.p 4 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 4T_{3}^{3} - 34T_{3}^{2} - 4T_{3} + 24 \) 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
$3$ \( T^{4} + 4 T^{3} - 34 T^{2} - 4 T + 24 \) Copy content Toggle raw display
$5$ \( T^{4} + 14 T^{3} - 203 T^{2} + \cdots + 6228 \) Copy content Toggle raw display
$7$ \( T^{4} + 12 T^{3} - 1169 T^{2} + \cdots + 102732 \) Copy content Toggle raw display
$11$ \( T^{4} + 98 T^{3} + 2682 T^{2} + \cdots - 124296 \) Copy content Toggle raw display
$13$ \( T^{4} + 128 T^{3} + \cdots - 10082792 \) Copy content Toggle raw display
$17$ \( T^{4} + 86 T^{3} - 15424 T^{2} + \cdots + 33154416 \) Copy content Toggle raw display
$19$ \( T^{4} + 116 T^{3} + \cdots + 36966128 \) Copy content Toggle raw display
$23$ \( T^{4} + 214 T^{3} + 13644 T^{2} + \cdots - 2911744 \) Copy content Toggle raw display
$29$ \( T^{4} + 168 T^{3} + \cdots - 245403688 \) Copy content Toggle raw display
$31$ \( (T + 31)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} - 598 T^{3} + \cdots - 1278468840 \) Copy content Toggle raw display
$41$ \( T^{4} - 218 T^{3} + \cdots + 9561006744 \) Copy content Toggle raw display
$43$ \( T^{4} + 192 T^{3} + \cdots + 3938013320 \) Copy content Toggle raw display
$47$ \( T^{4} - 32 T^{3} + \cdots + 8434470080 \) Copy content Toggle raw display
$53$ \( T^{4} - 290 T^{3} + \cdots + 3143843424 \) Copy content Toggle raw display
$59$ \( T^{4} + 376 T^{3} + \cdots - 28173754200 \) Copy content Toggle raw display
$61$ \( T^{4} - 196 T^{3} + \cdots + 48768423288 \) Copy content Toggle raw display
$67$ \( T^{4} - 40 T^{3} + \cdots + 34989096960 \) Copy content Toggle raw display
$71$ \( T^{4} + 536 T^{3} + \cdots - 18360511296 \) Copy content Toggle raw display
$73$ \( T^{4} - 1436 T^{3} + \cdots - 39620287696 \) Copy content Toggle raw display
$79$ \( T^{4} + 1010 T^{3} + \cdots - 1265961056 \) Copy content Toggle raw display
$83$ \( T^{4} + 1174 T^{3} + \cdots - 19929689864 \) Copy content Toggle raw display
$89$ \( T^{4} + 518 T^{3} + \cdots - 337618042704 \) Copy content Toggle raw display
$97$ \( T^{4} + 466 T^{3} + \cdots + 1478985763416 \) Copy content Toggle raw display
show more
show less