Properties

Label 24.7.e.a
Level $24$
Weight $7$
Character orbit 24.e
Analytic conductor $5.521$
Analytic rank $0$
Dimension $6$
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] = [24,7,Mod(17,24)]
 
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([0, 0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("24.17");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 24.e (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: \(5.52129800688\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.1173604352.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 14x^{4} - 12x^{3} + 112x^{2} + 192x + 324 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 2) q^{3} + ( - \beta_{2} - \beta_1) q^{5} + (\beta_{5} - 3 \beta_1 + 27) q^{7} + (\beta_{5} - \beta_{4} - \beta_{3} - 5 \beta_{2} - 3 \beta_1 - 13) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 2) q^{3} + ( - \beta_{2} - \beta_1) q^{5} + (\beta_{5} - 3 \beta_1 + 27) q^{7} + (\beta_{5} - \beta_{4} - \beta_{3} - 5 \beta_{2} - 3 \beta_1 - 13) q^{9} + (3 \beta_{4} + 2 \beta_{3} - 12 \beta_{2} + 15 \beta_1 - 9) q^{11} + (2 \beta_{5} - 6 \beta_{4} + 48 \beta_1 + 10) q^{13} + (\beta_{5} + 8 \beta_{4} - 10 \beta_{3} + 4 \beta_{2} - \beta_1 + 487) q^{15} + ( - 12 \beta_{4} + 18 \beta_{3} + 16 \beta_{2} - 92 \beta_1 + 36) q^{17} + ( - 10 \beta_{5} + 21 \beta_{4} - 159 \beta_1 - 697) q^{19} + ( - 12 \beta_{5} - 24 \beta_{4} - 42 \beta_{3} + 33 \beta_{2} + \cdots - 2514) q^{21}+ \cdots + ( - 1670 \beta_{5} - 103 \beta_{4} - 2812 \beta_{3} + \cdots - 470245) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 10 q^{3} + 156 q^{7} - 74 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 10 q^{3} + 156 q^{7} - 74 q^{9} + 156 q^{13} + 2912 q^{15} - 4500 q^{19} - 15108 q^{21} + 21366 q^{25} + 37574 q^{27} - 74244 q^{31} - 83104 q^{33} + 171132 q^{37} + 200444 q^{39} - 291060 q^{43} - 355136 q^{45} + 517746 q^{49} + 452224 q^{51} - 748224 q^{55} - 650420 q^{57} + 592092 q^{61} + 1009788 q^{63} - 570900 q^{67} - 981184 q^{69} + 1119660 q^{73} + 521446 q^{75} - 1053636 q^{79} - 742874 q^{81} + 197376 q^{85} + 1251360 q^{87} + 839640 q^{91} + 354652 q^{93} - 798516 q^{97} - 2849600 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 14x^{4} - 12x^{3} + 112x^{2} + 192x + 324 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 17\nu^{4} - 40\nu^{3} - 152\nu^{2} + 138\nu + 1143 ) / 45 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -53\nu^{5} + 19\nu^{4} + 1160\nu^{3} + 896\nu^{2} - 8994\nu - 12069 ) / 135 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -128\nu^{5} + 64\nu^{4} + 1280\nu^{3} + 896\nu^{2} - 15744\nu - 13824 ) / 135 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\nu^{5} - 13\nu^{4} - 200\nu^{3} + 448\nu^{2} + 1758\nu - 792 ) / 15 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -23\nu^{5} + 49\nu^{4} + 200\nu^{3} - 304\nu^{2} - 54\nu + 216 ) / 15 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 6\beta_{5} + 2\beta_{4} - 9\beta_{3} - 36\beta _1 + 12 ) / 1152 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 16\beta_{4} + 3\beta_{3} + 24\beta_{2} + 24\beta _1 + 2688 ) / 576 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{5} - 2\beta_{4} - 18\beta_{3} + 27\beta_{2} - 18\beta _1 + 879 ) / 144 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9\beta_{5} + 59\beta_{4} - 30\beta_{3} + 192\beta_{2} + 786\beta _1 - 2886 ) / 288 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -240\beta_{5} - 32\beta_{4} - 783\beta_{3} + 1440\beta_{2} + 2448\beta _1 - 11856 ) / 576 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
3.42788 1.41421i
3.42788 + 1.41421i
−2.80354 + 1.41421i
−2.80354 1.41421i
−0.624336 1.41421i
−0.624336 + 1.41421i
0 −22.0014 15.6505i 0 161.417i 0 562.144 0 239.124 + 688.666i 0
17.2 0 −22.0014 + 15.6505i 0 161.417i 0 562.144 0 239.124 688.666i 0
17.3 0 −6.43940 26.2209i 0 10.3581i 0 −540.917 0 −646.068 + 337.693i 0
17.4 0 −6.43940 + 26.2209i 0 10.3581i 0 −540.917 0 −646.068 337.693i 0
17.5 0 23.4408 13.3988i 0 100.147i 0 56.7723 0 369.944 628.158i 0
17.6 0 23.4408 + 13.3988i 0 100.147i 0 56.7723 0 369.944 + 628.158i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 24.7.e.a 6
3.b odd 2 1 inner 24.7.e.a 6
4.b odd 2 1 48.7.e.d 6
8.b even 2 1 192.7.e.h 6
8.d odd 2 1 192.7.e.g 6
12.b even 2 1 48.7.e.d 6
24.f even 2 1 192.7.e.g 6
24.h odd 2 1 192.7.e.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.7.e.a 6 1.a even 1 1 trivial
24.7.e.a 6 3.b odd 2 1 inner
48.7.e.d 6 4.b odd 2 1
48.7.e.d 6 12.b even 2 1
192.7.e.g 6 8.d odd 2 1
192.7.e.g 6 24.f even 2 1
192.7.e.h 6 8.b even 2 1
192.7.e.h 6 24.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(24, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 10 T^{5} + \cdots + 387420489 \) Copy content Toggle raw display
$5$ \( T^{6} + 36192 T^{4} + \cdots + 28037120000 \) Copy content Toggle raw display
$7$ \( (T^{3} - 78 T^{2} - 302868 T + 17262936)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} + 7292256 T^{4} + \cdots + 14\!\cdots\!32 \) Copy content Toggle raw display
$13$ \( (T^{3} - 78 T^{2} - 5834772 T - 4988511400)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 81606144 T^{4} + \cdots + 49\!\cdots\!48 \) Copy content Toggle raw display
$19$ \( (T^{3} + 2250 T^{2} + \cdots + 111704435512)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 925695360 T^{4} + \cdots + 20\!\cdots\!92 \) Copy content Toggle raw display
$29$ \( T^{6} + 1907973216 T^{4} + \cdots + 49\!\cdots\!12 \) Copy content Toggle raw display
$31$ \( (T^{3} + 37122 T^{2} + \cdots + 242319962776)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 85566 T^{2} + \cdots + 58243421953944)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 20931520896 T^{4} + \cdots + 45\!\cdots\!68 \) Copy content Toggle raw display
$43$ \( (T^{3} + 145530 T^{2} + \cdots - 162635792076808)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 12335998464 T^{4} + \cdots + 86\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( T^{6} + 57878690400 T^{4} + \cdots + 21\!\cdots\!72 \) Copy content Toggle raw display
$59$ \( T^{6} + 72389698656 T^{4} + \cdots + 67\!\cdots\!68 \) Copy content Toggle raw display
$61$ \( (T^{3} - 296046 T^{2} + \cdots + 26\!\cdots\!84)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + 285450 T^{2} + \cdots - 14\!\cdots\!92)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + 365617852800 T^{4} + \cdots + 74\!\cdots\!92 \) Copy content Toggle raw display
$73$ \( (T^{3} - 559830 T^{2} + \cdots + 22\!\cdots\!12)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 526818 T^{2} + \cdots - 26\!\cdots\!20)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 182279086176 T^{4} + \cdots + 10\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{6} + 1517610584448 T^{4} + \cdots + 45\!\cdots\!28 \) Copy content Toggle raw display
$97$ \( (T^{3} + 399258 T^{2} + \cdots + 36\!\cdots\!16)^{2} \) Copy content Toggle raw display
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