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Properties

Label 24.6.d.a
Level $24$
Weight $6$
Character orbit 24.d
Analytic conductor $3.849$
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] = [24,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("24.13");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(3.84921167551\)
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: \( x^{10} - 4 x^{9} + 20 x^{8} - 16 x^{7} + 290 x^{6} + 896 x^{5} + 416 x^{4} + 13952 x^{3} + 60997 x^{2} + \cdots + 750564 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{17}\cdot 3^{8} \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - \beta_1 q^{3} + ( - \beta_{3} + \beta_{2} - \beta_1 - 3) q^{4} + (\beta_{6} - \beta_{3} + 3 \beta_{2}) q^{5} + ( - \beta_{4} + 9) q^{6} + ( - \beta_{8} - \beta_{5} - \beta_{4} + \cdots - 23) q^{7}+ \cdots - 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - \beta_1 q^{3} + ( - \beta_{3} + \beta_{2} - \beta_1 - 3) q^{4} + (\beta_{6} - \beta_{3} + 3 \beta_{2}) q^{5} + ( - \beta_{4} + 9) q^{6} + ( - \beta_{8} - \beta_{5} - \beta_{4} + \cdots - 23) q^{7}+ \cdots + (162 \beta_{8} + 162 \beta_{7} + \cdots - 324) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2 q^{2} - 32 q^{4} + 90 q^{6} - 196 q^{7} + 68 q^{8} - 810 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 2 q^{2} - 32 q^{4} + 90 q^{6} - 196 q^{7} + 68 q^{8} - 810 q^{9} - 868 q^{10} - 396 q^{12} + 3692 q^{14} + 900 q^{15} + 3064 q^{16} - 404 q^{17} - 162 q^{18} - 6376 q^{20} + 5888 q^{22} + 1672 q^{23} + 2556 q^{24} - 9366 q^{25} - 6520 q^{26} - 10048 q^{28} - 684 q^{30} + 18692 q^{31} - 4088 q^{32} + 14844 q^{34} + 2592 q^{36} - 4712 q^{38} - 12168 q^{39} - 29560 q^{40} - 2476 q^{41} + 18396 q^{42} + 6912 q^{44} - 34776 q^{46} + 65256 q^{47} - 31104 q^{48} + 9378 q^{49} + 7394 q^{50} + 89808 q^{52} - 7290 q^{54} - 103424 q^{55} + 10808 q^{56} - 12888 q^{57} + 19436 q^{58} + 41472 q^{60} + 25300 q^{62} + 15876 q^{63} + 26752 q^{64} + 51536 q^{65} + 31104 q^{66} - 142656 q^{68} - 37784 q^{70} - 272232 q^{71} - 5508 q^{72} - 27388 q^{73} - 214864 q^{74} - 32208 q^{76} - 91944 q^{78} + 252308 q^{79} + 387456 q^{80} + 65610 q^{81} - 36156 q^{82} + 132120 q^{84} + 403688 q^{86} - 90828 q^{87} - 61312 q^{88} + 175580 q^{89} + 70308 q^{90} - 399744 q^{92} - 109560 q^{94} + 243184 q^{95} - 205560 q^{96} + 59372 q^{97} - 459846 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4 x^{9} + 20 x^{8} - 16 x^{7} + 290 x^{6} + 896 x^{5} + 416 x^{4} + 13952 x^{3} + 60997 x^{2} + \cdots + 750564 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 23 \nu^{9} - 469 \nu^{8} + 1703 \nu^{7} - 1369 \nu^{6} - 21211 \nu^{5} + 44789 \nu^{4} + \cdots - 17544636 ) / 1310720 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 25 \nu^{9} - 107 \nu^{8} + 169 \nu^{7} + 1657 \nu^{6} - 2837 \nu^{5} + 22027 \nu^{4} + \cdots + 719676 ) / 737280 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 277 \nu^{9} - 1553 \nu^{8} + 15931 \nu^{7} - 112325 \nu^{6} + 74785 \nu^{5} - 656207 \nu^{4} + \cdots - 155286924 ) / 3932160 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 25 \nu^{9} - 107 \nu^{8} + 169 \nu^{7} + 1657 \nu^{6} - 2837 \nu^{5} + 22027 \nu^{4} - 27509 \nu^{3} + \cdots - 17604 ) / 81920 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 41 \nu^{9} - 171 \nu^{8} + 57 \nu^{7} + 2745 \nu^{6} + 7515 \nu^{5} - 17589 \nu^{4} + \cdots + 5736252 ) / 122880 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4673 \nu^{9} + 30643 \nu^{8} - 128561 \nu^{7} + 3535 \nu^{6} - 427235 \nu^{5} - 663443 \nu^{4} + \cdots + 8512164 ) / 11796480 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1925 \nu^{9} + 13759 \nu^{8} - 61493 \nu^{7} + 57931 \nu^{6} - 263471 \nu^{5} - 871199 \nu^{4} + \cdots + 62529588 ) / 2949120 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1349 \nu^{9} + 4351 \nu^{8} - 7157 \nu^{7} - 55157 \nu^{6} - 277103 \nu^{5} - 723551 \nu^{4} + \cdots - 284590284 ) / 1966080 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 13849 \nu^{9} - 85691 \nu^{8} + 335497 \nu^{7} - 198263 \nu^{6} + 2829163 \nu^{5} + \cdots + 964864764 ) / 11796480 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - 9\beta_{2} + 9 ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} + 2\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - 8\beta_{2} + 2\beta _1 - 42 ) / 18 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 4 \beta_{9} - 8 \beta_{8} + 3 \beta_{7} - 10 \beta_{6} - 2 \beta_{5} - 8 \beta_{4} + 4 \beta_{3} + \cdots - 238 ) / 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 13 \beta_{9} - 50 \beta_{8} - 9 \beta_{7} + 11 \beta_{6} - 29 \beta_{5} - 52 \beta_{4} - 20 \beta_{3} + \cdots - 2041 ) / 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 14 \beta_{9} - 80 \beta_{8} - 60 \beta_{7} + 98 \beta_{6} + 34 \beta_{5} - 337 \beta_{4} - 212 \beta_{3} + \cdots - 14251 ) / 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 41 \beta_{9} + 420 \beta_{8} - 392 \beta_{7} - 53 \beta_{6} + 283 \beta_{5} - 1393 \beta_{4} + \cdots - 19044 ) / 18 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 8 \beta_{9} + 3304 \beta_{8} - 2547 \beta_{7} - 1930 \beta_{6} - 914 \beta_{5} - 1354 \beta_{4} + \cdots + 104864 ) / 18 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1951 \beta_{9} + 17114 \beta_{8} - 5655 \beta_{7} - 5333 \beta_{6} - 9037 \beta_{5} + 22066 \beta_{4} + \cdots + 391345 ) / 18 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 7890 \beta_{9} + 49248 \beta_{8} + 10758 \beta_{7} - 9510 \beta_{6} - 17718 \beta_{5} + 162925 \beta_{4} + \cdots + 1666251 ) / 18 \) 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
2.80405 + 3.57007i
2.80405 − 3.57007i
−0.700222 + 3.62067i
−0.700222 − 3.62067i
3.93225 + 2.72065i
3.93225 − 2.72065i
−2.89538 + 0.908872i
−2.89538 − 0.908872i
−1.14070 + 3.37896i
−1.14070 − 3.37896i
−5.37412 − 1.76602i 9.00000i 25.7624 + 18.9816i − 66.8717i 15.8941 − 48.3671i −230.960 −104.928 − 147.506i −81.0000 −118.097 + 359.377i
13.2 −5.37412 + 1.76602i − 9.00000i 25.7624 − 18.9816i 66.8717i 15.8941 + 48.3671i −230.960 −104.928 + 147.506i −81.0000 −118.097 − 359.377i
13.3 −1.92045 − 5.32089i 9.00000i −24.6238 + 20.4370i 67.1492i 47.8880 − 17.2840i 37.4101 156.032 + 91.7722i −81.0000 357.294 − 128.957i
13.4 −1.92045 + 5.32089i − 9.00000i −24.6238 − 20.4370i − 67.1492i 47.8880 + 17.2840i 37.4101 156.032 − 91.7722i −81.0000 357.294 + 128.957i
13.5 −0.211600 − 5.65290i − 9.00000i −31.9105 + 2.39231i − 29.4414i −50.8761 + 1.90440i −94.2792 20.2757 + 179.880i −81.0000 −166.429 + 6.22982i
13.6 −0.211600 + 5.65290i 9.00000i −31.9105 − 2.39231i 29.4414i −50.8761 − 1.90440i −94.2792 20.2757 − 179.880i −81.0000 −166.429 − 6.22982i
13.7 2.98650 − 4.80425i 9.00000i −14.1616 − 28.6958i − 100.204i 43.2382 + 26.8785i 154.395 −180.155 − 17.6644i −81.0000 −481.402 − 299.258i
13.8 2.98650 + 4.80425i − 9.00000i −14.1616 + 28.6958i 100.204i 43.2382 − 26.8785i 154.395 −180.155 + 17.6644i −81.0000 −481.402 + 299.258i
13.9 5.51967 − 1.23826i − 9.00000i 28.9334 − 13.6696i − 20.4846i −11.1443 − 49.6770i 35.4345 142.776 − 111.278i −81.0000 −25.3652 − 113.068i
13.10 5.51967 + 1.23826i 9.00000i 28.9334 + 13.6696i 20.4846i −11.1443 + 49.6770i 35.4345 142.776 + 111.278i −81.0000 −25.3652 + 113.068i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.10
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 24.6.d.a ✓ 10
3.b odd 2 1 72.6.d.d 10
4.b odd 2 1 96.6.d.a 10
8.b even 2 1 inner 24.6.d.a ✓ 10
8.d odd 2 1 96.6.d.a 10
12.b even 2 1 288.6.d.d 10
16.e even 4 1 768.6.a.ba 5
16.e even 4 1 768.6.a.bd 5
16.f odd 4 1 768.6.a.bb 5
16.f odd 4 1 768.6.a.bc 5
24.f even 2 1 288.6.d.d 10
24.h odd 2 1 72.6.d.d 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.6.d.a ✓ 10 1.a even 1 1 trivial
24.6.d.a ✓ 10 8.b even 2 1 inner
72.6.d.d 10 3.b odd 2 1
72.6.d.d 10 24.h odd 2 1
96.6.d.a 10 4.b odd 2 1
96.6.d.a 10 8.d odd 2 1
288.6.d.d 10 12.b even 2 1
288.6.d.d 10 24.f even 2 1
768.6.a.ba 5 16.e even 4 1
768.6.a.bb 5 16.f odd 4 1
768.6.a.bc 5 16.f odd 4 1
768.6.a.bd 5 16.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(24, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 2 T^{9} + \cdots + 33554432 \) Copy content Toggle raw display
$3$ \( (T^{2} + 81)^{5} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 73\!\cdots\!64 \) Copy content Toggle raw display
$7$ \( (T^{5} + 98 T^{4} + \cdots - 4456567264)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{5} + \cdots + 25\!\cdots\!24)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 30\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( (T^{5} + \cdots + 28\!\cdots\!56)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots + 17\!\cdots\!80)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots + 12\!\cdots\!20)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 15\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( (T^{5} + \cdots - 65\!\cdots\!12)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 15\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots - 11\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots + 17\!\cdots\!60)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 37\!\cdots\!60)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots - 85\!\cdots\!20)^{2} \) Copy content Toggle raw display
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