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Properties

Label 3024.2.t.j
Level $3024$
Weight $2$
Character orbit 3024.t
Analytic conductor $24.147$
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] = [3024,2,Mod(289,3024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3024, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3024.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.t (of order \(3\), degree \(2\), not 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: \(24.1467615712\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 5x^{12} - 3x^{11} + 7x^{10} + 30x^{9} - 117x^{7} + 270x^{5} + 189x^{4} - 243x^{3} - 1215x^{2} + 2187 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{7} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{7} + \beta_{3}) q^{5} + \beta_{5} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} + \beta_{3}) q^{5} + \beta_{5} q^{7} + (\beta_{13} - \beta_{9}) q^{11} - \beta_{2} q^{13} + (\beta_{12} + \beta_{9} - \beta_{6} + \cdots - \beta_1) q^{17}+ \cdots + (\beta_{13} + \beta_{12} + \beta_{11} + \cdots - 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 4 q^{5} + 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 4 q^{5} + 3 q^{7} - 4 q^{11} + 2 q^{13} - 2 q^{17} - 7 q^{19} - 22 q^{23} + 18 q^{25} - q^{29} + q^{31} - 19 q^{35} + 10 q^{37} + 33 q^{41} - 7 q^{43} - 3 q^{47} - 13 q^{49} + 15 q^{53} + 28 q^{55} - 14 q^{59} - 10 q^{61} - 15 q^{65} - 6 q^{67} + 2 q^{71} + 21 q^{73} - 19 q^{77} + 10 q^{79} - 25 q^{83} + 8 q^{85} + 6 q^{89} - 2 q^{91} - 28 q^{95} - 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 5x^{12} - 3x^{11} + 7x^{10} + 30x^{9} - 117x^{7} + 270x^{5} + 189x^{4} - 243x^{3} - 1215x^{2} + 2187 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 35 \nu^{13} + 72 \nu^{12} + 157 \nu^{11} + 312 \nu^{10} - 290 \nu^{9} - 1383 \nu^{8} + \cdots - 3645 ) / 43011 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10 \nu^{13} - 45 \nu^{12} + 196 \nu^{11} + 249 \nu^{10} + 136 \nu^{9} + 84 \nu^{8} - 2274 \nu^{7} + \cdots + 140940 ) / 14337 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 26 \nu^{13} + 7 \nu^{12} - 70 \nu^{11} - 266 \nu^{10} - 301 \nu^{9} + 955 \nu^{8} + 846 \nu^{7} + \cdots - 30618 ) / 14337 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 25 \nu^{13} + 294 \nu^{12} + 109 \nu^{11} - 924 \nu^{10} - 1715 \nu^{9} - 351 \nu^{8} + \cdots - 218700 ) / 43011 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 56 \nu^{13} - 441 \nu^{12} + 62 \nu^{11} + 1227 \nu^{10} + 761 \nu^{9} - 165 \nu^{8} - 8622 \nu^{7} + \cdots + 293058 ) / 43011 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 44 \nu^{13} + 78 \nu^{12} + 145 \nu^{11} - 150 \nu^{10} - 392 \nu^{9} - 387 \nu^{8} + \cdots - 48114 ) / 14337 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 73 \nu^{13} - 360 \nu^{12} - 40 \nu^{11} + 804 \nu^{10} + 1703 \nu^{9} - 417 \nu^{8} + \cdots + 239841 ) / 43011 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 73 \nu^{13} + 69 \nu^{12} - 28 \nu^{11} + 36 \nu^{10} - 373 \nu^{9} - 1446 \nu^{8} + \cdots - 121257 ) / 14337 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 142 \nu^{13} - 351 \nu^{12} + 386 \nu^{11} + 1398 \nu^{10} + 869 \nu^{9} - 2478 \nu^{8} + \cdots + 187353 ) / 43011 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 49 \nu^{13} + 125 \nu^{12} - 77 \nu^{11} - 313 \nu^{10} - 368 \nu^{9} + 140 \nu^{8} + 2352 \nu^{7} + \cdots - 68283 ) / 14337 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 281 \nu^{13} - 6 \nu^{12} + 208 \nu^{11} + 657 \nu^{10} - 500 \nu^{9} - 6555 \nu^{8} + \cdots - 334611 ) / 43011 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 179 \nu^{13} + 471 \nu^{12} - 769 \nu^{11} - 2352 \nu^{10} - 763 \nu^{9} + 5184 \nu^{8} + \cdots - 324405 ) / 43011 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 443 \nu^{13} + 546 \nu^{12} - 1099 \nu^{11} - 2736 \nu^{10} - 2092 \nu^{9} + 8283 \nu^{8} + \cdots - 414072 ) / 43011 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{12} - \beta_{10} + \beta_{5} + \beta_{4} - \beta_{3} + \beta _1 - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{13} - 2 \beta_{12} + 2 \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} - 2 \beta_{5} + \cdots + 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{12} + 2\beta_{10} + 3\beta_{9} + 3\beta_{7} + 3\beta_{6} + 4\beta_{5} + \beta_{4} - \beta_{3} + 4\beta _1 - 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2 \beta_{13} + 2 \beta_{11} - 3 \beta_{10} - \beta_{9} - 4 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + \cdots + 9 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3 \beta_{13} + 3 \beta_{12} + 3 \beta_{11} - 3 \beta_{10} + 3 \beta_{9} + 4 \beta_{8} + \beta_{7} + \cdots - 11 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 8 \beta_{13} - 19 \beta_{12} - 7 \beta_{11} + 7 \beta_{10} + 5 \beta_{9} + \beta_{8} + 9 \beta_{7} + \cdots + 23 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 21 \beta_{13} + 32 \beta_{12} - 12 \beta_{11} + 7 \beta_{10} + 39 \beta_{9} + 2 \beta_{8} - 34 \beta_{7} + \cdots + 18 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 17 \beta_{13} - 26 \beta_{12} - 22 \beta_{11} - 85 \beta_{10} - 13 \beta_{9} - \beta_{8} - 32 \beta_{7} + \cdots - 50 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 9 \beta_{13} + \beta_{12} + 18 \beta_{11} + 86 \beta_{10} + 75 \beta_{9} - 69 \beta_{8} + 42 \beta_{7} + \cdots - 115 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 16 \beta_{13} + 72 \beta_{12} - 79 \beta_{11} - 48 \beta_{10} + 62 \beta_{9} + 9 \beta_{8} + \cdots - 306 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 150 \beta_{13} + 57 \beta_{12} + 12 \beta_{11} + 105 \beta_{10} + 66 \beta_{9} - 122 \beta_{8} + \cdots - 650 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 134 \beta_{13} - 280 \beta_{12} - 97 \beta_{11} - 56 \beta_{10} - 328 \beta_{9} + 199 \beta_{8} + \cdots - 598 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 363 \beta_{13} - 247 \beta_{12} - 516 \beta_{11} + 1393 \beta_{10} + 426 \beta_{9} - 196 \beta_{8} + \cdots - 1089 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(-\beta_{1}\) \(1\) \(-1 + \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
−0.674693 + 1.59524i
1.68442 + 0.403398i
−1.73040 − 0.0755709i
1.64515 − 0.541745i
1.13119 + 1.31165i
−0.473632 − 1.66604i
−1.58203 + 0.705117i
−0.674693 − 1.59524i
1.68442 − 0.403398i
−1.73040 + 0.0755709i
1.64515 + 0.541745i
1.13119 − 1.31165i
−0.473632 + 1.66604i
−1.58203 − 0.705117i
0 0 0 −4.14520 0 −0.190437 + 2.63889i 0 0 0
289.2 0 0 0 −3.60346 0 1.60302 − 2.10483i 0 0 0
289.3 0 0 0 −0.967857 0 1.11482 − 2.39941i 0 0 0
289.4 0 0 0 0.763837 0 1.05641 + 2.42569i 0 0 0
289.5 0 0 0 1.52940 0 −2.53654 − 0.752299i 0 0 0
289.6 0 0 0 1.90301 0 2.43415 + 1.03677i 0 0 0
289.7 0 0 0 2.52026 0 −1.98143 + 1.75326i 0 0 0
1873.1 0 0 0 −4.14520 0 −0.190437 − 2.63889i 0 0 0
1873.2 0 0 0 −3.60346 0 1.60302 + 2.10483i 0 0 0
1873.3 0 0 0 −0.967857 0 1.11482 + 2.39941i 0 0 0
1873.4 0 0 0 0.763837 0 1.05641 − 2.42569i 0 0 0
1873.5 0 0 0 1.52940 0 −2.53654 + 0.752299i 0 0 0
1873.6 0 0 0 1.90301 0 2.43415 − 1.03677i 0 0 0
1873.7 0 0 0 2.52026 0 −1.98143 − 1.75326i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.2.t.j 14
3.b odd 2 1 1008.2.t.j 14
4.b odd 2 1 756.2.l.b 14
7.c even 3 1 3024.2.q.j 14
9.c even 3 1 3024.2.q.j 14
9.d odd 6 1 1008.2.q.j 14
12.b even 2 1 252.2.l.b yes 14
21.h odd 6 1 1008.2.q.j 14
28.d even 2 1 5292.2.l.i 14
28.f even 6 1 5292.2.i.i 14
28.f even 6 1 5292.2.j.g 14
28.g odd 6 1 756.2.i.b 14
28.g odd 6 1 5292.2.j.h 14
36.f odd 6 1 756.2.i.b 14
36.f odd 6 1 2268.2.k.f 14
36.h even 6 1 252.2.i.b ✓ 14
36.h even 6 1 2268.2.k.e 14
63.g even 3 1 inner 3024.2.t.j 14
63.n odd 6 1 1008.2.t.j 14
84.h odd 2 1 1764.2.l.i 14
84.j odd 6 1 1764.2.i.i 14
84.j odd 6 1 1764.2.j.h 14
84.n even 6 1 252.2.i.b ✓ 14
84.n even 6 1 1764.2.j.g 14
252.n even 6 1 5292.2.l.i 14
252.o even 6 1 252.2.l.b yes 14
252.r odd 6 1 1764.2.j.h 14
252.s odd 6 1 1764.2.i.i 14
252.u odd 6 1 2268.2.k.f 14
252.u odd 6 1 5292.2.j.h 14
252.bb even 6 1 1764.2.j.g 14
252.bb even 6 1 2268.2.k.e 14
252.bi even 6 1 5292.2.i.i 14
252.bj even 6 1 5292.2.j.g 14
252.bl odd 6 1 756.2.l.b 14
252.bn odd 6 1 1764.2.l.i 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.i.b ✓ 14 36.h even 6 1
252.2.i.b ✓ 14 84.n even 6 1
252.2.l.b yes 14 12.b even 2 1
252.2.l.b yes 14 252.o even 6 1
756.2.i.b 14 28.g odd 6 1
756.2.i.b 14 36.f odd 6 1
756.2.l.b 14 4.b odd 2 1
756.2.l.b 14 252.bl odd 6 1
1008.2.q.j 14 9.d odd 6 1
1008.2.q.j 14 21.h odd 6 1
1008.2.t.j 14 3.b odd 2 1
1008.2.t.j 14 63.n odd 6 1
1764.2.i.i 14 84.j odd 6 1
1764.2.i.i 14 252.s odd 6 1
1764.2.j.g 14 84.n even 6 1
1764.2.j.g 14 252.bb even 6 1
1764.2.j.h 14 84.j odd 6 1
1764.2.j.h 14 252.r odd 6 1
1764.2.l.i 14 84.h odd 2 1
1764.2.l.i 14 252.bn odd 6 1
2268.2.k.e 14 36.h even 6 1
2268.2.k.e 14 252.bb even 6 1
2268.2.k.f 14 36.f odd 6 1
2268.2.k.f 14 252.u odd 6 1
3024.2.q.j 14 7.c even 3 1
3024.2.q.j 14 9.c even 3 1
3024.2.t.j 14 1.a even 1 1 trivial
3024.2.t.j 14 63.g even 3 1 inner
5292.2.i.i 14 28.f even 6 1
5292.2.i.i 14 252.bi even 6 1
5292.2.j.g 14 28.f even 6 1
5292.2.j.g 14 252.bj even 6 1
5292.2.j.h 14 28.g odd 6 1
5292.2.j.h 14 252.u odd 6 1
5292.2.l.i 14 28.d even 2 1
5292.2.l.i 14 252.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3024, [\chi])\):

\( T_{5}^{7} + 2T_{5}^{6} - 20T_{5}^{5} - 12T_{5}^{4} + 129T_{5}^{3} - 81T_{5}^{2} - 108T_{5} + 81 \) Copy content Toggle raw display
\( T_{11}^{7} + 2T_{11}^{6} - 41T_{11}^{5} - 75T_{11}^{4} + 174T_{11}^{3} + 72T_{11}^{2} - 216T_{11} + 81 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( (T^{7} + 2 T^{6} - 20 T^{5} + \cdots + 81)^{2} \) Copy content Toggle raw display
$7$ \( T^{14} - 3 T^{13} + \cdots + 823543 \) Copy content Toggle raw display
$11$ \( (T^{7} + 2 T^{6} - 41 T^{5} + \cdots + 81)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 150626529 \) Copy content Toggle raw display
$17$ \( T^{14} + 2 T^{13} + \cdots + 6561 \) Copy content Toggle raw display
$19$ \( T^{14} + 7 T^{13} + \cdots + 4084441 \) Copy content Toggle raw display
$23$ \( (T^{7} + 11 T^{6} + \cdots + 10287)^{2} \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 145660761 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 13807190016 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 1566893056 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 1108290681 \) Copy content Toggle raw display
$43$ \( T^{14} + 7 T^{13} + \cdots + 4084441 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 136048896 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 952401321 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 688747536 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 148644864 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 116985856 \) Copy content Toggle raw display
$71$ \( (T^{7} - T^{6} - 116 T^{5} + \cdots - 972)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 2748590329 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 54397165824 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 901054679121 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 16524331209 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 767677849 \) Copy content Toggle raw display
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