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Properties

Label 24.11.e.a
Level $24$
Weight $11$
Character orbit 24.e
Analytic conductor $15.249$
Analytic rank $0$
Dimension $10$
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] = [24,11,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, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("24.17");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) \(=\) \( 11 \)
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: \(15.2485740642\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 532 x^{8} - 1350 x^{7} + 106101 x^{6} + 516780 x^{5} - 8879077 x^{4} - 65126430 x^{3} + \cdots + 6338653425 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{72}\cdot 3^{16} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 2) q^{3} + (\beta_{2} + \beta_1) q^{5} + (\beta_{3} - 7 \beta_1 - 545) q^{7} + (\beta_{4} - 2 \beta_1 - 2894) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 2) q^{3} + (\beta_{2} + \beta_1) q^{5} + (\beta_{3} - 7 \beta_1 - 545) q^{7} + (\beta_{4} - 2 \beta_1 - 2894) q^{9} + ( - \beta_{8} - \beta_{4} + \cdots + 4 \beta_1) q^{11}+ \cdots + (3204 \beta_{9} - 27510 \beta_{8} + \cdots + 4350990363) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 22 q^{3} - 5436 q^{7} - 28934 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 22 q^{3} - 5436 q^{7} - 28934 q^{9} - 124508 q^{13} - 627808 q^{15} - 4893484 q^{19} - 3929724 q^{21} - 17742214 q^{25} - 3536326 q^{27} - 4251484 q^{31} - 2965600 q^{33} + 89985156 q^{37} + 52569188 q^{39} + 159987316 q^{43} + 39125824 q^{45} + 301480958 q^{49} + 387377536 q^{51} - 852340544 q^{55} - 970086764 q^{57} - 101460764 q^{61} + 733153572 q^{63} - 3014528044 q^{67} - 3501669184 q^{69} + 4920922036 q^{73} + 5355440986 q^{75} - 7631690012 q^{79} - 7700105942 q^{81} + 18713636096 q^{85} + 19781179104 q^{87} - 17913072600 q^{91} - 24272938652 q^{93} + 37861379156 q^{97} + 43508497216 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 532 x^{8} - 1350 x^{7} + 106101 x^{6} + 516780 x^{5} - 8879077 x^{4} - 65126430 x^{3} + \cdots + 6338653425 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 433897493826154 \nu^{9} + \cdots - 31\!\cdots\!35 ) / 77\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 19\!\cdots\!98 \nu^{9} + \cdots - 96\!\cdots\!65 ) / 17\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 25\!\cdots\!58 \nu^{9} + \cdots - 10\!\cdots\!19 ) / 15\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 65\!\cdots\!56 \nu^{9} + \cdots - 64\!\cdots\!15 ) / 38\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 42\!\cdots\!90 \nu^{9} + \cdots + 36\!\cdots\!80 ) / 11\!\cdots\!35 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 47\!\cdots\!34 \nu^{9} + \cdots - 38\!\cdots\!05 ) / 77\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 27\!\cdots\!54 \nu^{9} + \cdots + 18\!\cdots\!95 ) / 23\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 13\!\cdots\!58 \nu^{9} + \cdots + 16\!\cdots\!55 ) / 59\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 10\!\cdots\!86 \nu^{9} + \cdots - 52\!\cdots\!95 ) / 23\!\cdots\!70 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( 12 \beta_{9} - 42 \beta_{8} - 9 \beta_{7} - 451 \beta_{6} + 171 \beta_{5} + 84 \beta_{4} + \cdots + 6628 ) / 1327104 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 180 \beta_{9} - 246 \beta_{8} - 27 \beta_{7} - 2273 \beta_{6} + 186 \beta_{5} + \cdots + 141238115 ) / 1327104 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 1800 \beta_{9} - 16632 \beta_{8} - 2088 \beta_{7} - 109376 \beta_{6} + 114003 \beta_{5} + \cdots + 1076878853 ) / 2654208 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 42372 \beta_{9} - 74706 \beta_{8} - 3789 \beta_{7} - 485231 \beta_{6} + 332157 \beta_{5} + \cdots + 18805481390 ) / 1327104 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 296592 \beta_{9} - 3084684 \beta_{8} - 150354 \beta_{7} - 13834718 \beta_{6} + \cdots + 267660949781 ) / 2654208 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 9279936 \beta_{9} - 18038340 \beta_{8} + 837234 \beta_{7} - 78976274 \beta_{6} + \cdots + 2815883834351 ) / 1327104 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 81294612 \beta_{9} - 296013030 \beta_{8} + 14208741 \beta_{7} - 788366929 \beta_{6} + \cdots + 27017186166007 ) / 1327104 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 2043726300 \beta_{9} - 3968373846 \beta_{8} + 606765033 \beta_{7} - 10095187613 \beta_{6} + \cdots + 442364104120742 ) / 1327104 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 48328109496 \beta_{9} - 118153732320 \beta_{8} + 19352204172 \beta_{7} - 107805032180 \beta_{6} + \cdots + 98\!\cdots\!23 ) / 2654208 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−5.15423 + 1.41421i
−5.15423 − 1.41421i
−11.0996 − 1.41421i
−11.0996 + 1.41421i
−8.61800 + 1.41421i
−8.61800 − 1.41421i
10.3554 + 1.41421i
10.3554 − 1.41421i
14.5165 − 1.41421i
14.5165 + 1.41421i
0 −230.417 − 77.1815i 0 3573.67i 0 7988.14 0 47135.0 + 35567.9i 0
17.2 0 −230.417 + 77.1815i 0 − 3573.67i 0 7988.14 0 47135.0 − 35567.9i 0
17.3 0 −126.767 − 207.314i 0 1673.71i 0 −12918.9 0 −26909.2 + 52561.2i 0
17.4 0 −126.767 + 207.314i 0 − 1673.71i 0 −12918.9 0 −26909.2 − 52561.2i 0
17.5 0 −27.8503 − 241.399i 0 − 5802.43i 0 13554.8 0 −57497.7 + 13446.0i 0
17.6 0 −27.8503 + 241.399i 0 5802.43i 0 13554.8 0 −57497.7 − 13446.0i 0
17.7 0 171.197 − 172.454i 0 2896.26i 0 −28955.7 0 −431.864 − 59047.4i 0
17.8 0 171.197 + 172.454i 0 − 2896.26i 0 −28955.7 0 −431.864 + 59047.4i 0
17.9 0 202.837 − 133.814i 0 − 265.153i 0 17613.7 0 23236.7 − 54284.8i 0
17.10 0 202.837 + 133.814i 0 265.153i 0 17613.7 0 23236.7 + 54284.8i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.10
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 24.11.e.a ✓ 10
3.b odd 2 1 inner 24.11.e.a ✓ 10
4.b odd 2 1 48.11.e.e 10
8.b even 2 1 192.11.e.j 10
8.d odd 2 1 192.11.e.i 10
12.b even 2 1 48.11.e.e 10
24.f even 2 1 192.11.e.i 10
24.h odd 2 1 192.11.e.j 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.11.e.a ✓ 10 1.a even 1 1 trivial
24.11.e.a ✓ 10 3.b odd 2 1 inner
48.11.e.e 10 4.b odd 2 1
48.11.e.e 10 12.b even 2 1
192.11.e.i 10 8.d odd 2 1
192.11.e.i 10 24.f even 2 1
192.11.e.j 10 8.b even 2 1
192.11.e.j 10 24.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(24, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 71\!\cdots\!49 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{5} + \cdots - 71\!\cdots\!44)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 10\!\cdots\!52 \) Copy content Toggle raw display
$13$ \( (T^{5} + \cdots + 68\!\cdots\!40)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 60\!\cdots\!92 \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots - 20\!\cdots\!32)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 51\!\cdots\!88 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 11\!\cdots\!08 \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots + 33\!\cdots\!08)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots - 93\!\cdots\!04)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 17\!\cdots\!72 \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots + 15\!\cdots\!88)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 43\!\cdots\!08 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 31\!\cdots\!52 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 31\!\cdots\!48 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 14\!\cdots\!36)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} + \cdots - 22\!\cdots\!12)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 13\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 23\!\cdots\!12)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 10\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 42\!\cdots\!32 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots - 54\!\cdots\!56)^{2} \) Copy content Toggle raw display
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