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Properties

Label 24.8.d.a
Level $24$
Weight $8$
Character orbit 24.d
Analytic conductor $7.497$
Analytic rank $0$
Dimension $14$
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] = [24,8,Mod(13,24)]
 
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, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("24.13");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 24.d (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: \(7.49724061162\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 6 x^{13} - 52 x^{12} + 300 x^{11} - 1005 x^{10} - 23250 x^{9} + 349930 x^{8} + \cdots + 3813237677250 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{37}\cdot 3^{16} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} - \beta_{3} q^{3} + (\beta_{6} + \beta_{3} - \beta_1 - 15) q^{4} + (\beta_{7} - \beta_{6} + \cdots - 3 \beta_1) q^{5}+ \cdots - 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} - \beta_{3} q^{3} + (\beta_{6} + \beta_{3} - \beta_1 - 15) q^{4} + (\beta_{7} - \beta_{6} + \cdots - 3 \beta_1) q^{5}+ \cdots + ( - 1458 \beta_{13} + 729 \beta_{12} + \cdots - 1458) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 14 q^{2} - 208 q^{4} - 54 q^{6} + 1372 q^{7} - 428 q^{8} - 10206 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 14 q^{2} - 208 q^{4} - 54 q^{6} + 1372 q^{7} - 428 q^{8} - 10206 q^{9} + 5020 q^{10} + 7668 q^{12} + 4636 q^{14} - 13500 q^{15} - 43336 q^{16} - 2908 q^{17} + 10206 q^{18} + 175096 q^{20} - 128480 q^{22} - 143416 q^{23} - 29268 q^{24} - 202626 q^{25} + 424984 q^{26} + 567520 q^{28} - 250668 q^{30} - 89468 q^{31} - 893944 q^{32} + 1109820 q^{34} + 151632 q^{36} - 823816 q^{38} + 474552 q^{39} - 860888 q^{40} - 441284 q^{41} + 427788 q^{42} + 1275264 q^{44} - 2167992 q^{46} - 1056408 q^{47} - 233280 q^{48} + 2158134 q^{49} + 324610 q^{50} - 2059248 q^{52} + 39366 q^{54} + 4757504 q^{55} + 1643704 q^{56} + 1551096 q^{57} - 5494676 q^{58} - 3203712 q^{60} + 5767172 q^{62} - 1000188 q^{63} + 3852224 q^{64} - 2520464 q^{65} - 3615840 q^{66} - 3735840 q^{68} + 12890312 q^{70} + 5172696 q^{71} + 312012 q^{72} - 5446196 q^{73} - 6468800 q^{74} - 9084624 q^{76} + 3542184 q^{78} - 14373548 q^{79} + 14369088 q^{80} + 7440174 q^{81} - 7935708 q^{82} - 2775816 q^{84} + 4738312 q^{86} + 7902036 q^{87} + 12598720 q^{88} - 11952620 q^{89} - 3659580 q^{90} + 11004480 q^{92} - 15440088 q^{94} - 69327376 q^{95} + 1341576 q^{96} + 133732 q^{97} + 53030538 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 6 x^{13} - 52 x^{12} + 300 x^{11} - 1005 x^{10} - 23250 x^{9} + 349930 x^{8} + \cdots + 3813237677250 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1339739 \nu^{13} - 15980711 \nu^{12} + 21034557 \nu^{11} + 215590849 \nu^{10} + \cdots + 66\!\cdots\!06 ) / 26\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 24634993 \nu^{13} - 9286397141 \nu^{12} + 125608142679 \nu^{11} - 691247115421 \nu^{10} + \cdots + 85\!\cdots\!10 ) / 41\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 45486279 \nu^{13} + 613276227 \nu^{12} - 1623882033 \nu^{11} - 9211813317 \nu^{10} + \cdots - 18\!\cdots\!78 ) / 27\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 232216063 \nu^{13} + 10715039579 \nu^{12} - 173010009017 \nu^{11} + \cdots + 39\!\cdots\!54 ) / 83\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 179339189 \nu^{13} + 2135131735 \nu^{12} - 31681195725 \nu^{11} + 37724754799 \nu^{10} + \cdots - 13\!\cdots\!06 ) / 41\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 414718539 \nu^{13} + 4476417111 \nu^{12} - 7726634189 \nu^{11} - 119116174993 \nu^{10} + \cdots - 28\!\cdots\!98 ) / 83\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 629872933 \nu^{13} - 7758524121 \nu^{12} + 38268518211 \nu^{11} - 384199764289 \nu^{10} + \cdots + 39\!\cdots\!70 ) / 83\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 39005411 \nu^{13} - 428467791 \nu^{12} + 1109091573 \nu^{11} - 13635772967 \nu^{10} + \cdots + 31\!\cdots\!50 ) / 32\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 340358553 \nu^{13} + 3989168541 \nu^{12} - 4434070383 \nu^{11} - 52878351195 \nu^{10} + \cdots - 17\!\cdots\!54 ) / 27\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 19011601 \nu^{13} - 238222965 \nu^{12} + 740348087 \nu^{11} + 1084756419 \nu^{10} + \cdots + 81\!\cdots\!18 ) / 13\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 39027681 \nu^{13} - 536450053 \nu^{12} + 1433725287 \nu^{11} + 10887104691 \nu^{10} + \cdots + 15\!\cdots\!46 ) / 16\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 2625997723 \nu^{13} - 37720840423 \nu^{12} + 153136812541 \nu^{11} + 409146235777 \nu^{10} + \cdots + 10\!\cdots\!42 ) / 83\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 4393185439 \nu^{13} + 53314528379 \nu^{12} - 1504144921 \nu^{11} - 1197574289101 \nu^{10} + \cdots - 30\!\cdots\!54 ) / 83\!\cdots\!44 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} + \beta_{3} + 27\beta _1 + 23 ) / 54 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{9} - 3\beta_{8} + 3\beta_{7} + 16\beta_{3} + 54\beta _1 + 541 ) / 54 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 18 \beta_{13} + 3 \beta_{12} - 24 \beta_{11} - 36 \beta_{10} - 2 \beta_{9} - 12 \beta_{8} + \cdots + 1943 ) / 108 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 30 \beta_{13} + 103 \beta_{12} - 14 \beta_{11} - 138 \beta_{10} - 104 \beta_{9} - 22 \beta_{8} + \cdots + 28343 ) / 36 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 702 \beta_{13} + 51 \beta_{12} + 195 \beta_{11} + 243 \beta_{10} + 216 \beta_{9} - 114 \beta_{8} + \cdots + 615006 ) / 54 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2952 \beta_{13} + 4176 \beta_{12} - 5130 \beta_{11} + 21078 \beta_{10} + 15751 \beta_{9} + \cdots - 1406845 ) / 54 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 26082 \beta_{13} + 669 \beta_{12} - 98736 \beta_{11} - 59508 \beta_{10} - 142050 \beta_{9} + \cdots - 122247195 ) / 108 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 11706 \beta_{13} + 103325 \beta_{12} - 536842 \beta_{11} - 584718 \beta_{10} - 774740 \beta_{9} + \cdots - 122156023 ) / 36 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1105452 \beta_{13} + 968922 \beta_{12} - 2879379 \beta_{11} - 11779551 \beta_{10} + 3108490 \beta_{9} + \cdots - 930607885 ) / 54 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 18378774 \beta_{13} - 7602453 \beta_{12} - 3286872 \beta_{11} - 16841340 \beta_{10} + 117660149 \beta_{9} + \cdots + 7709620384 ) / 54 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 104200830 \beta_{13} - 238923429 \beta_{12} - 432406644 \beta_{11} + 372886416 \beta_{10} + \cdots - 86001757345 ) / 108 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 107793438 \beta_{13} - 712698473 \beta_{12} - 1777191854 \beta_{11} - 2574032586 \beta_{10} + \cdots - 262461379393 ) / 36 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 3338497530 \beta_{13} - 9980609295 \beta_{12} - 19026211413 \beta_{11} - 50017662477 \beta_{10} + \cdots + 2425490692320 ) / 54 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
−4.99225 + 5.30027i
−4.99225 − 5.30027i
−6.99438 − 0.299706i
−6.99438 + 0.299706i
2.71713 − 7.81354i
2.71713 + 7.81354i
−5.80663 − 4.20354i
−5.80663 + 4.20354i
7.97707 − 3.91414i
7.97707 + 3.91414i
1.24645 − 7.99620i
1.24645 + 7.99620i
8.85262 + 1.52851i
8.85262 − 1.52851i
−11.2925 − 0.691979i − 27.0000i 127.042 + 15.6284i 124.215i −18.6834 + 304.898i 646.373 −1423.81 − 264.394i −729.000 85.9543 − 1402.70i
13.2 −11.2925 + 0.691979i 27.0000i 127.042 − 15.6284i − 124.215i −18.6834 − 304.898i 646.373 −1423.81 + 264.394i −729.000 85.9543 + 1402.70i
13.3 −7.69468 − 8.29409i − 27.0000i −9.58387 + 127.641i − 455.347i −223.940 + 207.756i −743.502 1132.41 − 902.665i −729.000 −3776.69 + 3503.75i
13.4 −7.69468 + 8.29409i 27.0000i −9.58387 − 127.641i 455.347i −223.940 − 207.756i −743.502 1132.41 + 902.665i −729.000 −3776.69 − 3503.75i
13.5 −6.09641 − 9.53067i 27.0000i −53.6675 + 116.206i − 137.155i 257.328 − 164.603i 808.153 1434.70 − 196.951i −729.000 −1307.18 + 836.155i
13.6 −6.09641 + 9.53067i − 27.0000i −53.6675 − 116.206i 137.155i 257.328 + 164.603i 808.153 1434.70 + 196.951i −729.000 −1307.18 − 836.155i
13.7 −2.60309 − 11.0102i − 27.0000i −114.448 + 57.3210i 468.400i −297.275 + 70.2836i 81.2421 929.033 + 1110.88i −729.000 5157.17 − 1219.29i
13.8 −2.60309 + 11.0102i 27.0000i −114.448 − 57.3210i − 468.400i −297.275 − 70.2836i 81.2421 929.033 − 1110.88i −729.000 5157.17 + 1219.29i
13.9 3.06293 − 10.8912i 27.0000i −109.237 − 66.7181i 23.0228i 294.063 + 82.6992i −1547.56 −1061.23 + 985.368i −729.000 250.747 + 70.5175i
13.10 3.06293 + 10.8912i − 27.0000i −109.237 + 66.7181i − 23.0228i 294.063 − 82.6992i −1547.56 −1061.23 − 985.368i −729.000 250.747 − 70.5175i
13.11 8.24265 − 7.74976i − 27.0000i 7.88255 − 127.757i − 76.0929i −209.243 − 222.552i −222.735 −925.113 − 1114.14i −729.000 −589.701 − 627.207i
13.12 8.24265 + 7.74976i 27.0000i 7.88255 + 127.757i 76.0929i −209.243 + 222.552i −222.735 −925.113 + 1114.14i −729.000 −589.701 + 627.207i
13.13 9.38113 − 6.32411i 27.0000i 48.0111 − 118.655i 425.308i 170.751 + 253.290i 1664.03 −299.987 − 1416.74i −729.000 2689.70 + 3989.87i
13.14 9.38113 + 6.32411i − 27.0000i 48.0111 + 118.655i − 425.308i 170.751 − 253.290i 1664.03 −299.987 + 1416.74i −729.000 2689.70 − 3989.87i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.14
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 24.8.d.a ✓ 14
3.b odd 2 1 72.8.d.d 14
4.b odd 2 1 96.8.d.a 14
8.b even 2 1 inner 24.8.d.a ✓ 14
8.d odd 2 1 96.8.d.a 14
12.b even 2 1 288.8.d.d 14
24.f even 2 1 288.8.d.d 14
24.h odd 2 1 72.8.d.d 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.8.d.a ✓ 14 1.a even 1 1 trivial
24.8.d.a ✓ 14 8.b even 2 1 inner
72.8.d.d 14 3.b odd 2 1
72.8.d.d 14 24.h odd 2 1
96.8.d.a 14 4.b odd 2 1
96.8.d.a 14 8.d odd 2 1
288.8.d.d 14 12.b even 2 1
288.8.d.d 14 24.f even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(24, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots + 562949953421312 \) Copy content Toggle raw display
$3$ \( (T^{2} + 729)^{7} \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{7} + \cdots + 18\!\cdots\!56)^{2} \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 92\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 82\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( (T^{7} + \cdots + 20\!\cdots\!36)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 23\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( (T^{7} + \cdots - 23\!\cdots\!68)^{2} \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 30\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( (T^{7} + \cdots - 33\!\cdots\!12)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 88\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( (T^{7} + \cdots + 38\!\cdots\!28)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 14\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( (T^{7} + \cdots + 43\!\cdots\!56)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 98\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 33\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 45\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots - 12\!\cdots\!68)^{2} \) Copy content Toggle raw display
$73$ \( (T^{7} + \cdots - 49\!\cdots\!48)^{2} \) Copy content Toggle raw display
$79$ \( (T^{7} + \cdots + 64\!\cdots\!80)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 31\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{7} + \cdots - 78\!\cdots\!80)^{2} \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots - 18\!\cdots\!04)^{2} \) Copy content Toggle raw display
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