Properties

Label 24.5.b.a
Level $24$
Weight $5$
Character orbit 24.b
Analytic conductor $2.481$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [24,5,Mod(19,24)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(24, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("24.19");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 24.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.48087911401\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 20x^{6} - 6x^{5} + 121x^{4} + 18x^{3} - 114x^{2} + 72x + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - 1) q^{2} + \beta_{3} q^{3} + (\beta_{4} + \beta_{3} - \beta_{2} + 1) q^{4} + ( - \beta_{7} + 3 \beta_{2} + \beta_1 - 1) q^{5} + (\beta_{6} - \beta_{5} - \beta_{3} - \beta_1 + 2) q^{6} + ( - 2 \beta_{7} + \beta_{6} + \beta_{4} - 3 \beta_{2} - \beta_1) q^{7} + (\beta_{7} + 3 \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + \beta_1 - 22) q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 1) q^{2} + \beta_{3} q^{3} + (\beta_{4} + \beta_{3} - \beta_{2} + 1) q^{4} + ( - \beta_{7} + 3 \beta_{2} + \beta_1 - 1) q^{5} + (\beta_{6} - \beta_{5} - \beta_{3} - \beta_1 + 2) q^{6} + ( - 2 \beta_{7} + \beta_{6} + \beta_{4} - 3 \beta_{2} - \beta_1) q^{7} + (\beta_{7} + 3 \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + \beta_1 - 22) q^{8} + 27 q^{9} + (2 \beta_{7} - 4 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 4 \beta_{2} + \cdots - 40) q^{10}+ \cdots + (216 \beta_{5} + 216 \beta_{2} + 648) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{2} + 8 q^{4} + 18 q^{6} - 180 q^{8} + 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 6 q^{2} + 8 q^{4} + 18 q^{6} - 180 q^{8} + 216 q^{9} - 324 q^{10} + 192 q^{11} + 180 q^{12} + 420 q^{14} - 712 q^{16} + 240 q^{17} - 162 q^{18} - 704 q^{19} + 168 q^{20} + 592 q^{22} - 108 q^{24} - 664 q^{25} + 1008 q^{26} - 528 q^{28} + 468 q^{30} + 3624 q^{32} + 2716 q^{34} - 5184 q^{35} + 216 q^{36} - 6360 q^{38} + 408 q^{40} + 720 q^{41} - 2412 q^{42} + 10048 q^{43} - 6720 q^{44} + 2616 q^{46} - 3168 q^{48} - 1240 q^{49} + 5394 q^{50} - 4032 q^{51} + 2448 q^{52} + 486 q^{54} + 7512 q^{56} + 3744 q^{57} - 10740 q^{58} + 13056 q^{59} + 10656 q^{60} - 8724 q^{62} - 17632 q^{64} - 1344 q^{65} - 13680 q^{66} - 6656 q^{67} - 5616 q^{68} + 19800 q^{70} - 4860 q^{72} - 16880 q^{73} + 17400 q^{74} - 1152 q^{75} + 14320 q^{76} + 18720 q^{78} + 28512 q^{80} + 5832 q^{81} - 9740 q^{82} - 24000 q^{83} + 21960 q^{84} - 34344 q^{86} - 19616 q^{88} + 15600 q^{89} - 8748 q^{90} + 1344 q^{91} - 48096 q^{92} + 12120 q^{94} - 21528 q^{96} - 12176 q^{97} + 47778 q^{98} + 5184 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 20x^{6} - 6x^{5} + 121x^{4} + 18x^{3} - 114x^{2} + 72x + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 67\nu^{7} - 1405\nu^{6} + 1607\nu^{5} - 30251\nu^{4} + 21016\nu^{3} - 225826\nu^{2} + 8472\nu - 78750 ) / 16122 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 280\nu^{7} - 1019\nu^{6} + 5633\nu^{5} - 20827\nu^{4} + 39823\nu^{3} - 101396\nu^{2} - 61086\nu + 151668 ) / 32244 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1019\nu^{7} - 33\nu^{6} + 19147\nu^{5} - 5943\nu^{4} + 106436\nu^{3} + 29166\nu^{2} - 228240\nu + 40320 ) / 42992 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3533 \nu^{7} - 4787 \nu^{6} + 71585 \nu^{5} - 129277 \nu^{4} + 474872 \nu^{3} - 687838 \nu^{2} - 276424 \nu + 306784 ) / 42992 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2737\nu^{7} + 86\nu^{6} - 54458\nu^{5} + 22816\nu^{4} - 297979\nu^{3} + 40754\nu^{2} + 607998\nu - 25932 ) / 32244 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 12661 \nu^{7} - 10599 \nu^{6} + 267581 \nu^{5} - 290721 \nu^{4} + 1919812 \nu^{3} - 1325694 \nu^{2} + 878112 \nu + 967008 ) / 128976 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2373 \nu^{7} - 1717 \nu^{6} + 51703 \nu^{5} - 50287 \nu^{4} + 377066 \nu^{3} - 232454 \nu^{2} + 26548 \nu + 208168 ) / 21496 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -3\beta_{7} + 4\beta_{6} - \beta_{5} - 6\beta_{3} - \beta _1 - 1 ) / 36 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + \beta_{5} + 3\beta_{4} - 4\beta_{3} + 6\beta_{2} - 5\beta _1 - 62 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 30\beta_{7} - 31\beta_{6} + 37\beta_{5} + 9\beta_{4} + 108\beta_{3} - 54\beta_{2} + 10\beta _1 + 109 ) / 36 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -6\beta_{7} + 31\beta_{6} - 19\beta_{5} - 69\beta_{4} + 56\beta_{3} + 12\beta_{2} + 59\beta _1 + 482 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -156\beta_{7} + 211\beta_{6} - 541\beta_{5} - 153\beta_{4} - 1884\beta_{3} + 1008\beta_{2} - 226\beta _1 - 3493 ) / 36 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 66\beta_{7} - 187\beta_{6} + 105\beta_{5} + 333\beta_{4} - 116\beta_{3} - 442\beta_{2} - 191\beta _1 - 274 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 960 \beta_{7} + 743 \beta_{6} + 5689 \beta_{5} + 567 \beta_{4} + 25584 \beta_{3} - 13734 \beta_{2} + 4384 \beta _1 + 66277 ) / 36 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/24\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(13\) \(17\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.866025 3.62169i
−0.866025 + 3.62169i
0.866025 + 3.05404i
0.866025 3.05404i
0.866025 + 0.663939i
0.866025 0.663939i
−0.866025 0.339683i
−0.866025 + 0.339683i
−3.56731 1.80951i 5.19615 9.45137 + 12.9101i 38.1617i −18.5363 9.40247i 42.0542i −10.3550 63.1567i 27.0000 −69.0539 + 136.135i
19.2 −3.56731 + 1.80951i 5.19615 9.45137 12.9101i 38.1617i −18.5363 + 9.40247i 42.0542i −10.3550 + 63.1567i 27.0000 −69.0539 136.135i
19.3 −3.40459 2.09970i −5.19615 7.18250 + 14.2973i 13.7025i 17.6908 + 10.9104i 88.3075i 5.56649 63.7575i 27.0000 28.7711 46.6513i
19.4 −3.40459 + 2.09970i −5.19615 7.18250 14.2973i 13.7025i 17.6908 10.9104i 88.3075i 5.56649 + 63.7575i 27.0000 28.7711 + 46.6513i
19.5 1.03857 3.86282i −5.19615 −13.8428 8.02360i 34.2464i −5.39655 + 20.0718i 9.22331i −45.3703 + 45.1390i 27.0000 −132.288 35.5672i
19.6 1.03857 + 3.86282i −5.19615 −13.8428 + 8.02360i 34.2464i −5.39655 20.0718i 9.22331i −45.3703 45.1390i 27.0000 −132.288 + 35.5672i
19.7 2.93333 2.71948i 5.19615 1.20889 15.9543i 3.88698i 15.2420 14.1308i 23.9200i −39.8412 50.0867i 27.0000 10.5706 + 11.4018i
19.8 2.93333 + 2.71948i 5.19615 1.20889 + 15.9543i 3.88698i 15.2420 + 14.1308i 23.9200i −39.8412 + 50.0867i 27.0000 10.5706 11.4018i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 24.5.b.a 8
3.b odd 2 1 72.5.b.d 8
4.b odd 2 1 96.5.b.a 8
8.b even 2 1 96.5.b.a 8
8.d odd 2 1 inner 24.5.b.a 8
12.b even 2 1 288.5.b.d 8
16.e even 4 2 768.5.g.k 16
16.f odd 4 2 768.5.g.k 16
24.f even 2 1 72.5.b.d 8
24.h odd 2 1 288.5.b.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.5.b.a 8 1.a even 1 1 trivial
24.5.b.a 8 8.d odd 2 1 inner
72.5.b.d 8 3.b odd 2 1
72.5.b.d 8 24.f even 2 1
96.5.b.a 8 4.b odd 2 1
96.5.b.a 8 8.b even 2 1
288.5.b.d 8 12.b even 2 1
288.5.b.d 8 24.h odd 2 1
768.5.g.k 16 16.e even 4 2
768.5.g.k 16 16.f odd 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(24, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 6 T^{7} + 14 T^{6} + \cdots + 65536 \) Copy content Toggle raw display
$3$ \( (T^{2} - 27)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + 2832 T^{6} + \cdots + 4845166848 \) Copy content Toggle raw display
$7$ \( T^{8} + 10224 T^{6} + \cdots + 671288262912 \) Copy content Toggle raw display
$11$ \( (T^{4} - 96 T^{3} - 40064 T^{2} + \cdots + 193957888)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 155136 T^{6} + \cdots + 56\!\cdots\!12 \) Copy content Toggle raw display
$17$ \( (T^{4} - 120 T^{3} - 96872 T^{2} + \cdots - 1023349232)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 352 T^{3} - 220128 T^{2} + \cdots - 5712629504)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 1644480 T^{6} + \cdots + 64\!\cdots\!92 \) Copy content Toggle raw display
$29$ \( T^{8} + 4596240 T^{6} + \cdots + 22\!\cdots\!88 \) Copy content Toggle raw display
$31$ \( T^{8} + 2201712 T^{6} + \cdots + 15\!\cdots\!12 \) Copy content Toggle raw display
$37$ \( T^{8} + 6796224 T^{6} + \cdots + 78\!\cdots\!68 \) Copy content Toggle raw display
$41$ \( (T^{4} - 360 T^{3} + \cdots + 226033840912)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 5024 T^{3} + \cdots + 2240667904)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 14015424 T^{6} + \cdots + 13\!\cdots\!08 \) Copy content Toggle raw display
$53$ \( T^{8} + 27560592 T^{6} + \cdots + 86\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( (T^{4} - 6528 T^{3} + \cdots - 21853379868416)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 79948992 T^{6} + \cdots + 34\!\cdots\!12 \) Copy content Toggle raw display
$67$ \( (T^{4} + 3328 T^{3} + \cdots - 30681172039424)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 82031040 T^{6} + \cdots + 64\!\cdots\!92 \) Copy content Toggle raw display
$73$ \( (T^{4} + 8440 T^{3} + \cdots - 35664142829552)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 162047472 T^{6} + \cdots + 43\!\cdots\!52 \) Copy content Toggle raw display
$83$ \( (T^{4} + 12000 T^{3} + \cdots + 15292937408512)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 7800 T^{3} + \cdots + 219207569912848)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 6088 T^{3} + \cdots - 29\!\cdots\!68)^{2} \) Copy content Toggle raw display
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