↑

Properties

Label 24.5.h.c
Level $24$
Weight $5$
Character orbit 24.h
Analytic conductor $2.481$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [24,5,Mod(5,24)]
 
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, 1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("24.5");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 24.h (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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 10x^{10} + 216x^{8} + 6848x^{6} + 55296x^{4} + 655360x^{2} + 16777216 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{17}\cdot 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + \beta_{2} q^{3} + (\beta_{5} - 2) q^{4} + ( - \beta_{4} + 3 \beta_{3} + \cdots + \beta_1) q^{5}+ \cdots + ( - \beta_{11} - \beta_{10} + \cdots - 13) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + \beta_{2} q^{3} + (\beta_{5} - 2) q^{4} + ( - \beta_{4} + 3 \beta_{3} + \cdots + \beta_1) q^{5}+ \cdots + ( - 24 \beta_{11} + 168 \beta_{10} + \cdots - 48) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 20 q^{4} - 56 q^{6} - 8 q^{7} - 164 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 20 q^{4} - 56 q^{6} - 8 q^{7} - 164 q^{9} + 360 q^{10} - 20 q^{12} + 664 q^{15} - 664 q^{16} - 232 q^{18} + 224 q^{22} - 88 q^{24} - 2236 q^{25} + 1184 q^{28} + 2000 q^{30} + 1080 q^{31} + 1896 q^{33} + 2320 q^{34} - 5580 q^{36} - 1184 q^{39} + 64 q^{40} - 5816 q^{42} - 7712 q^{46} + 9432 q^{48} + 3972 q^{49} + 11632 q^{52} + 8408 q^{54} - 9680 q^{55} - 400 q^{57} + 11816 q^{58} - 6960 q^{60} + 4792 q^{63} - 30128 q^{64} - 24120 q^{66} - 28336 q^{70} + 29840 q^{72} + 12632 q^{73} + 36392 q^{76} + 36560 q^{78} + 6712 q^{79} - 8692 q^{81} + 45920 q^{82} - 47984 q^{84} - 23784 q^{87} - 34768 q^{88} - 33016 q^{90} - 57984 q^{94} + 47408 q^{96} - 31304 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 10x^{10} + 216x^{8} + 6848x^{6} + 55296x^{4} + 655360x^{2} + 16777216 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{11} - 38 \nu^{10} + 6 \nu^{9} + 4 \nu^{8} - 2104 \nu^{7} + 3824 \nu^{6} + 8896 \nu^{5} + \cdots + 24379392 ) / 5505024 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{11} - 10\nu^{9} - 216\nu^{7} - 6848\nu^{5} - 55296\nu^{3} - 655360\nu ) / 1048576 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13 \nu^{11} - 152 \nu^{10} + 1086 \nu^{9} + 16 \nu^{8} + 25736 \nu^{7} + 15296 \nu^{6} + \cdots + 97517568 ) / 22020096 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{10} - 10\nu^{8} - 216\nu^{6} - 6848\nu^{4} - 55296\nu^{2} - 524288 ) / 65536 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{10} - 74\nu^{8} - 856\nu^{6} - 4288\nu^{4} - 264192\nu^{2} - 1638400 ) / 65536 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 25 \nu^{11} + 76 \nu^{10} + 486 \nu^{9} - 8 \nu^{8} - 6232 \nu^{7} - 7648 \nu^{6} + \cdots - 48758784 ) / 11010048 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 55 \nu^{11} + 32 \nu^{10} - 1126 \nu^{9} - 3776 \nu^{8} - 1256 \nu^{7} + 31488 \nu^{6} + \cdots + 66584576 ) / 11010048 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -3\nu^{11} + 34\nu^{9} - 8\nu^{7} - 6720\nu^{5} + 534528\nu^{3} + 2097152\nu ) / 524288 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 135 \nu^{11} - 64 \nu^{10} - 1094 \nu^{9} + 384 \nu^{8} - 2024 \nu^{7} - 134656 \nu^{6} + \cdots - 294649856 ) / 22020096 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 57 \nu^{11} - 212 \nu^{10} - 1114 \nu^{9} - 5448 \nu^{8} - 5464 \nu^{7} + 2848 \nu^{6} + \cdots + 93323264 ) / 11010048 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} - \beta_{8} - \beta_{5} - \beta_{2} - 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{9} - 2\beta_{7} - 2\beta_{4} - 6\beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 3 \beta_{11} + 4 \beta_{10} - \beta_{8} - 8 \beta_{7} + 4 \beta_{6} - 17 \beta_{5} + 4 \beta_{4} + \cdots - 106 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4 \beta_{11} + 12 \beta_{10} - 6 \beta_{9} + 8 \beta_{8} + 22 \beta_{7} - 4 \beta_{6} - 6 \beta_{4} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{11} - 68 \beta_{10} + 67 \beta_{8} + 24 \beta_{7} - 20 \beta_{6} - 13 \beta_{5} + 44 \beta_{4} + \cdots - 2490 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 24 \beta_{11} + 72 \beta_{10} - 92 \beta_{9} + 48 \beta_{8} - 196 \beta_{7} - 24 \beta_{6} + 532 \beta_{4} + \cdots - 24 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 886 \beta_{11} + 760 \beta_{10} + 126 \beta_{8} - 400 \beta_{7} - 744 \beta_{6} + 1630 \beta_{5} + \cdots + 10268 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1104 \beta_{11} - 3312 \beta_{10} - 536 \beta_{9} - 2208 \beta_{8} + 8152 \beta_{7} + 1104 \beta_{6} + \cdots + 1104 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 5092 \beta_{11} - 6608 \beta_{10} + 1516 \beta_{8} + 26208 \beta_{7} - 1936 \beta_{6} - 6996 \beta_{5} + \cdots + 356760 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 21536 \beta_{11} - 64608 \beta_{10} + 11024 \beta_{9} - 43072 \beta_{8} - 134544 \beta_{7} + \cdots + 21536 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−3.49180 + 1.95124i
−3.49180 − 1.95124i
−2.97547 + 2.67331i
−2.97547 − 2.67331i
−0.673742 + 3.94285i
−0.673742 − 3.94285i
0.673742 + 3.94285i
0.673742 − 3.94285i
2.97547 + 2.67331i
2.97547 − 2.67331i
3.49180 + 1.95124i
3.49180 − 1.95124i
−3.49180 − 1.95124i 0.370354 − 8.99238i 8.38535 + 13.6267i −31.6607 −18.8395 + 30.6769i −41.0115 −2.69111 − 63.9434i −80.7257 − 6.66073i 110.553 + 61.7775i
5.2 −3.49180 + 1.95124i 0.370354 + 8.99238i 8.38535 − 13.6267i −31.6607 −18.8395 − 30.6769i −41.0115 −2.69111 + 63.9434i −80.7257 + 6.66073i 110.553 − 61.7775i
5.3 −2.97547 − 2.67331i 6.08044 + 6.63538i 1.70680 + 15.9087i 10.1873 −0.353701 − 35.9983i 73.1393 37.4504 − 51.8986i −7.05649 + 80.6920i −30.3121 − 27.2339i
5.4 −2.97547 + 2.67331i 6.08044 − 6.63538i 1.70680 − 15.9087i 10.1873 −0.353701 + 35.9983i 73.1393 37.4504 + 51.8986i −7.05649 − 80.6920i −30.3121 + 27.2339i
5.5 −0.673742 − 3.94285i −7.99319 + 4.13629i −15.0921 + 5.31293i −14.4851 21.6941 + 28.7292i −34.1278 31.1163 + 55.9265i 46.7822 − 66.1243i 9.75922 + 57.1126i
5.6 −0.673742 + 3.94285i −7.99319 − 4.13629i −15.0921 − 5.31293i −14.4851 21.6941 − 28.7292i −34.1278 31.1163 − 55.9265i 46.7822 + 66.1243i 9.75922 − 57.1126i
5.7 0.673742 − 3.94285i 7.99319 − 4.13629i −15.0921 − 5.31293i 14.4851 −10.9234 − 34.3027i −34.1278 −31.1163 + 55.9265i 46.7822 − 66.1243i 9.75922 − 57.1126i
5.8 0.673742 + 3.94285i 7.99319 + 4.13629i −15.0921 + 5.31293i 14.4851 −10.9234 + 34.3027i −34.1278 −31.1163 − 55.9265i 46.7822 + 66.1243i 9.75922 + 57.1126i
5.9 2.97547 − 2.67331i −6.08044 − 6.63538i 1.70680 − 15.9087i −10.1873 −35.8306 − 3.48842i 73.1393 −37.4504 − 51.8986i −7.05649 + 80.6920i −30.3121 + 27.2339i
5.10 2.97547 + 2.67331i −6.08044 + 6.63538i 1.70680 + 15.9087i −10.1873 −35.8306 + 3.48842i 73.1393 −37.4504 + 51.8986i −7.05649 − 80.6920i −30.3121 − 27.2339i
5.11 3.49180 − 1.95124i −0.370354 + 8.99238i 8.38535 − 13.6267i 31.6607 16.2531 + 32.1222i −41.0115 2.69111 − 63.9434i −80.7257 − 6.66073i 110.553 − 61.7775i
5.12 3.49180 + 1.95124i −0.370354 − 8.99238i 8.38535 + 13.6267i 31.6607 16.2531 − 32.1222i −41.0115 2.69111 + 63.9434i −80.7257 + 6.66073i 110.553 + 61.7775i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h 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.h.c ✓ 12
3.b odd 2 1 inner 24.5.h.c ✓ 12
4.b odd 2 1 96.5.h.c 12
8.b even 2 1 inner 24.5.h.c ✓ 12
8.d odd 2 1 96.5.h.c 12
12.b even 2 1 96.5.h.c 12
24.f even 2 1 96.5.h.c 12
24.h odd 2 1 inner 24.5.h.c ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.5.h.c ✓ 12 1.a even 1 1 trivial
24.5.h.c ✓ 12 3.b odd 2 1 inner
24.5.h.c ✓ 12 8.b even 2 1 inner
24.5.h.c ✓ 12 24.h odd 2 1 inner
96.5.h.c 12 4.b odd 2 1
96.5.h.c 12 8.d odd 2 1
96.5.h.c 12 12.b even 2 1
96.5.h.c 12 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 1316T_{5}^{4} + 336128T_{5}^{2} - 21827584 \) acting on \(S_{5}^{\mathrm{new}}(24, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 10 T^{10} + \cdots + 16777216 \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 282429536481 \) Copy content Toggle raw display
$5$ \( (T^{6} - 1316 T^{4} + \cdots - 21827584)^{2} \) Copy content Toggle raw display
$7$ \( (T^{3} + 2 T^{2} + \cdots - 102368)^{4} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 2501306728704)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 1287403978752)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots + 469646435155968)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 21\!\cdots\!68)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 922297011535872)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 20\!\cdots\!64)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 270 T^{2} + \cdots + 248591392)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 11\!\cdots\!52)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 70\!\cdots\!08)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 71\!\cdots\!88)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 26\!\cdots\!28)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 32\!\cdots\!24)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 14\!\cdots\!16)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 74\!\cdots\!92)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 19\!\cdots\!48)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 30\!\cdots\!72)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 3158 T^{2} + \cdots + 55695205944)^{4} \) Copy content Toggle raw display
$79$ \( (T^{3} - 1678 T^{2} + \cdots + 2204573728)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 18\!\cdots\!16)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 14\!\cdots\!88)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 7826 T^{2} + \cdots + 286711096)^{4} \) Copy content Toggle raw display
show more
show less