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Properties

Label 24.11.b.a
Level $24$
Weight $11$
Character orbit 24.b
Analytic conductor $15.249$
Analytic rank $0$
Dimension $20$
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(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, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("24.19");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) \(=\) \( 11 \)
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: \(15.2485740642\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 6 x^{19} + 27 x^{18} + 30770 x^{17} - 1815282 x^{16} + 57539616 x^{15} - 4088960592 x^{14} + \cdots + 11\!\cdots\!48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{82}\cdot 3^{38} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 1) q^{2} - \beta_1 q^{3} + (\beta_{3} + \beta_{2} - \beta_1 - 93) q^{4} + (\beta_{6} - 6 \beta_{2} + 1) q^{5} + (\beta_{7} + \beta_1 - 122) q^{6} + ( - \beta_{11} - \beta_{7} + \beta_{3} + \cdots + 3) q^{7}+ \cdots + 19683 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 1) q^{2} - \beta_1 q^{3} + (\beta_{3} + \beta_{2} - \beta_1 - 93) q^{4} + (\beta_{6} - 6 \beta_{2} + 1) q^{5} + (\beta_{7} + \beta_1 - 122) q^{6} + ( - \beta_{11} - \beta_{7} + \beta_{3} + \cdots + 3) q^{7}+ \cdots + (19683 \beta_{17} + 19683 \beta_{15} + \cdots - 90777996) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 22 q^{2} - 1864 q^{4} - 2430 q^{6} + 118076 q^{8} + 393660 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 22 q^{2} - 1864 q^{4} - 2430 q^{6} + 118076 q^{8} + 393660 q^{9} - 113028 q^{10} - 91808 q^{11} + 310068 q^{12} - 503244 q^{14} - 545032 q^{16} - 905768 q^{17} - 433026 q^{18} - 5107040 q^{19} - 5944248 q^{20} - 2702288 q^{22} - 6248988 q^{24} - 40539484 q^{25} + 19400160 q^{26} - 19709616 q^{28} - 6556140 q^{30} + 118772648 q^{32} + 155305372 q^{34} - 68411424 q^{35} - 36689112 q^{36} - 119065400 q^{38} + 477325560 q^{40} + 51872200 q^{41} + 315008676 q^{42} - 392685920 q^{43} - 333411776 q^{44} - 1035567720 q^{46} + 520626528 q^{48} - 1130175964 q^{49} + 2330181842 q^{50} + 373209120 q^{51} - 2683888176 q^{52} - 47829690 q^{54} + 4216587096 q^{56} + 563025168 q^{57} + 2827093836 q^{58} - 1499085440 q^{59} - 574187616 q^{60} - 5504139012 q^{62} + 3240358112 q^{64} + 1717119456 q^{65} + 2366318448 q^{66} - 4570092800 q^{67} - 6509360144 q^{68} - 9179287176 q^{70} + 2324089908 q^{72} - 2312358680 q^{73} + 7298736360 q^{74} + 5958608832 q^{75} - 12863545616 q^{76} - 2062821168 q^{78} + 4230593184 q^{80} + 7748409780 q^{81} + 1201856596 q^{82} + 14192131360 q^{83} - 807928344 q^{84} + 11933558200 q^{86} - 12709901024 q^{88} + 8963359000 q^{89} - 2224730124 q^{90} - 401968224 q^{91} + 13243366752 q^{92} + 5851812792 q^{94} - 7654043160 q^{96} + 23275134232 q^{97} - 20615960782 q^{98} - 1807056864 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 6 x^{19} + 27 x^{18} + 30770 x^{17} - 1815282 x^{16} + 57539616 x^{15} - 4088960592 x^{14} + \cdots + 11\!\cdots\!48 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 17\!\cdots\!55 \nu^{19} + \cdots + 34\!\cdots\!04 ) / 24\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 65\!\cdots\!69 \nu^{19} + \cdots - 12\!\cdots\!80 ) / 20\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 26\!\cdots\!35 \nu^{19} + \cdots - 17\!\cdots\!20 ) / 20\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 43\!\cdots\!23 \nu^{19} + \cdots - 32\!\cdots\!40 ) / 68\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 46\!\cdots\!45 \nu^{19} + \cdots + 70\!\cdots\!52 ) / 17\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 32\!\cdots\!17 \nu^{19} + \cdots + 12\!\cdots\!56 ) / 88\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 37\!\cdots\!95 \nu^{19} + \cdots - 62\!\cdots\!20 ) / 84\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 18\!\cdots\!38 \nu^{19} + \cdots + 29\!\cdots\!92 ) / 29\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11\!\cdots\!43 \nu^{19} + \cdots + 74\!\cdots\!56 ) / 75\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 79\!\cdots\!03 \nu^{19} + \cdots + 40\!\cdots\!84 ) / 38\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 11\!\cdots\!09 \nu^{19} + \cdots + 43\!\cdots\!56 ) / 44\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 15\!\cdots\!67 \nu^{19} + \cdots - 25\!\cdots\!68 ) / 59\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 42\!\cdots\!69 \nu^{19} + \cdots - 25\!\cdots\!04 ) / 14\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 66\!\cdots\!25 \nu^{19} + \cdots - 20\!\cdots\!84 ) / 17\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 20\!\cdots\!53 \nu^{19} + \cdots - 32\!\cdots\!68 ) / 44\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 16\!\cdots\!54 \nu^{19} + \cdots + 44\!\cdots\!88 ) / 29\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 18\!\cdots\!01 \nu^{19} + \cdots - 21\!\cdots\!28 ) / 17\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 13\!\cdots\!91 \nu^{19} + \cdots - 49\!\cdots\!64 ) / 77\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 10\!\cdots\!27 \nu^{19} + \cdots - 16\!\cdots\!96 ) / 59\!\cdots\!56 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 3 \beta_{17} - 3 \beta_{16} - 3 \beta_{14} - 3 \beta_{13} - 9 \beta_{12} + 6 \beta_{11} + \cdots + 12274 ) / 41472 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 60 \beta_{19} - 84 \beta_{18} + 273 \beta_{17} + 21 \beta_{16} - 3 \beta_{14} + 177 \beta_{13} + \cdots - 20138 ) / 41472 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 1536 \beta_{19} - 1344 \beta_{18} + 93 \beta_{17} - 6243 \beta_{16} - 99 \beta_{14} + \cdots - 64328778 ) / 13824 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 86988 \beta_{19} - 107844 \beta_{18} + 40365 \beta_{17} - 690975 \beta_{16} + 456192 \beta_{15} + \cdots + 13565806054 ) / 41472 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2772912 \beta_{19} - 18336816 \beta_{18} - 4793319 \beta_{17} + 5650569 \beta_{16} + \cdots - 490707180698 ) / 41472 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 106096092 \beta_{19} - 42666228 \beta_{18} - 152385831 \beta_{17} + 18958077 \beta_{16} + \cdots + 17615038975662 ) / 13824 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 8728049616 \beta_{19} + 10817439408 \beta_{18} + 7590401283 \beta_{17} - 8855683053 \beta_{16} + \cdots - 19\!\cdots\!14 ) / 41472 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 714958225404 \beta_{19} + 315606886572 \beta_{18} + 1456436780673 \beta_{17} + \cdots + 18\!\cdots\!14 ) / 41472 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 4505190225648 \beta_{19} - 3020531536752 \beta_{18} - 994622267847 \beta_{17} - 2817935077143 \beta_{16} + \cdots + 27\!\cdots\!94 ) / 4608 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 15\!\cdots\!92 \beta_{19} + 405264160340772 \beta_{18} + \cdots - 27\!\cdots\!98 ) / 41472 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 40\!\cdots\!44 \beta_{19} + \cdots - 76\!\cdots\!54 ) / 41472 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 33\!\cdots\!00 \beta_{19} + \cdots + 37\!\cdots\!46 ) / 13824 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 56\!\cdots\!28 \beta_{19} + \cdots + 10\!\cdots\!34 ) / 41472 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 49\!\cdots\!24 \beta_{19} + \cdots - 70\!\cdots\!30 ) / 41472 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 56\!\cdots\!96 \beta_{19} + \cdots + 13\!\cdots\!58 ) / 13824 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 18\!\cdots\!96 \beta_{19} + \cdots - 31\!\cdots\!30 ) / 41472 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 20\!\cdots\!92 \beta_{19} + \cdots + 85\!\cdots\!70 ) / 41472 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 13\!\cdots\!32 \beta_{19} + \cdots - 14\!\cdots\!38 ) / 4608 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 79\!\cdots\!96 \beta_{19} + \cdots + 93\!\cdots\!58 ) / 41472 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
39.4938 − 11.3042i
39.4938 + 11.3042i
−7.81942 + 8.00390i
−7.81942 − 8.00390i
24.5187 − 34.1791i
24.5187 + 34.1791i
−6.56862 + 9.15916i
−6.56862 − 9.15916i
4.81653 − 43.0561i
4.81653 + 43.0561i
−0.0177468 + 11.6888i
−0.0177468 − 11.6888i
5.90667 + 10.5078i
5.90667 − 10.5078i
−23.2864 − 38.5789i
−23.2864 + 38.5789i
11.7312 + 4.02609i
11.7312 − 4.02609i
−45.7747 − 7.62525i
−45.7747 + 7.62525i
−30.9115 − 8.27527i 140.296 887.040 + 511.602i 4403.89i −4336.76 − 1160.99i 7529.34i −23186.1 − 23154.9i 19683.0 36443.4 − 136131.i
19.2 −30.9115 + 8.27527i 140.296 887.040 − 511.602i − 4403.89i −4336.76 + 1160.99i − 7529.34i −23186.1 + 23154.9i 19683.0 36443.4 + 136131.i
19.3 −23.3630 − 21.8671i −140.296 67.6636 + 1021.76i − 2621.93i 3277.74 + 3067.86i 22511.9i 20762.1 − 25351.1i 19683.0 −57334.0 + 61256.4i
19.4 −23.3630 + 21.8671i −140.296 67.6636 − 1021.76i 2621.93i 3277.74 − 3067.86i − 22511.9i 20762.1 + 25351.1i 19683.0 −57334.0 − 61256.4i
19.5 −19.9489 − 25.0208i 140.296 −228.083 + 998.276i − 2696.55i −2798.75 − 3510.32i 18441.7i 29527.7 − 14207.7i 19683.0 −67469.8 + 53793.1i
19.6 −19.9489 + 25.0208i 140.296 −228.083 − 998.276i 2696.55i −2798.75 + 3510.32i − 18441.7i 29527.7 + 14207.7i 19683.0 −67469.8 − 53793.1i
19.7 −19.9458 − 25.0233i −140.296 −228.330 + 998.219i 5785.54i 2798.32 + 3510.67i − 14762.1i 29533.0 − 14196.7i 19683.0 144773. − 115397.i
19.8 −19.9458 + 25.0233i −140.296 −228.330 − 998.219i − 5785.54i 2798.32 − 3510.67i 14762.1i 29533.0 + 14196.7i 19683.0 144773. + 115397.i
19.9 −5.52595 − 31.5193i 140.296 −962.928 + 348.347i 2370.40i −775.269 − 4422.03i − 10154.6i 16300.7 + 28425.8i 19683.0 74713.2 − 13098.7i
19.10 −5.52595 + 31.5193i 140.296 −962.928 − 348.347i − 2370.40i −775.269 + 4422.03i 10154.6i 16300.7 − 28425.8i 19683.0 74713.2 + 13098.7i
19.11 −2.04849 − 31.9344i −140.296 −1015.61 + 130.834i − 5061.04i 287.394 + 4480.27i − 31335.6i 6258.56 + 32164.8i 19683.0 −161621. + 10367.5i
19.12 −2.04849 + 31.9344i −140.296 −1015.61 − 130.834i 5061.04i 287.394 − 4480.27i 31335.6i 6258.56 − 32164.8i 19683.0 −161621. − 10367.5i
19.13 14.1373 − 28.7078i −140.296 −624.273 − 811.702i 1631.98i −1983.41 + 4027.59i 11799.6i −32127.7 + 6446.21i 19683.0 46850.7 + 23071.9i
19.14 14.1373 + 28.7078i −140.296 −624.273 + 811.702i − 1631.98i −1983.41 − 4027.59i − 11799.6i −32127.7 − 6446.21i 19683.0 46850.7 − 23071.9i
19.15 15.0468 − 28.2417i 140.296 −571.186 − 849.896i − 3385.10i 2111.01 − 3962.20i 2423.26i −32597.0 + 3343.01i 19683.0 −95601.0 − 50935.1i
19.16 15.0468 + 28.2417i 140.296 −571.186 + 849.896i 3385.10i 2111.01 + 3962.20i − 2423.26i −32597.0 − 3343.01i 19683.0 −95601.0 + 50935.1i
19.17 30.0501 − 10.9995i −140.296 782.022 − 661.073i 977.966i −4215.92 + 1543.19i − 13461.7i 16228.4 − 28467.2i 19683.0 10757.1 + 29388.0i
19.18 30.0501 + 10.9995i −140.296 782.022 + 661.073i − 977.966i −4215.92 − 1543.19i 13461.7i 16228.4 + 28467.2i 19683.0 10757.1 − 29388.0i
19.19 31.5094 − 5.58207i 140.296 961.681 − 351.775i 2145.17i 4420.64 − 783.143i 29274.4i 28338.3 − 16452.4i 19683.0 11974.5 + 67592.9i
19.20 31.5094 + 5.58207i 140.296 961.681 + 351.775i − 2145.17i 4420.64 + 783.143i − 29274.4i 28338.3 + 16452.4i 19683.0 11974.5 − 67592.9i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.20
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 24.11.b.a ✓ 20
3.b odd 2 1 72.11.b.d 20
4.b odd 2 1 96.11.b.a 20
8.b even 2 1 96.11.b.a 20
8.d odd 2 1 inner 24.11.b.a ✓ 20
12.b even 2 1 288.11.b.c 20
24.f even 2 1 72.11.b.d 20
24.h odd 2 1 288.11.b.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.11.b.a ✓ 20 1.a even 1 1 trivial
24.11.b.a ✓ 20 8.d odd 2 1 inner
72.11.b.d 20 3.b odd 2 1
72.11.b.d 20 24.f even 2 1
96.11.b.a 20 4.b odd 2 1
96.11.b.a 20 8.b even 2 1
288.11.b.c 20 12.b even 2 1
288.11.b.c 20 24.h odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$3$ \( (T^{2} - 19683)^{10} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 27\!\cdots\!12 \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots + 58\!\cdots\!96)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 16\!\cdots\!08 \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots - 78\!\cdots\!12)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 97\!\cdots\!16)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 76\!\cdots\!72 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 43\!\cdots\!52 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 32\!\cdots\!08 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 22\!\cdots\!68 \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 10\!\cdots\!08)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 89\!\cdots\!72)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 90\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 40\!\cdots\!92 \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 13\!\cdots\!52)^{2} \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 33\!\cdots\!52 \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots - 75\!\cdots\!56)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 10\!\cdots\!52 \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 32\!\cdots\!72)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 11\!\cdots\!88 \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 18\!\cdots\!72)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots - 97\!\cdots\!08)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 10\!\cdots\!56)^{2} \) Copy content Toggle raw display
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