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Properties

Label 3024.2.df.c
Level $3024$
Weight $2$
Character orbit 3024.df
Analytic conductor $24.147$
Analytic rank $0$
Dimension $16$
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(17,3024)]
 
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, 5, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3024.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3024.df (of order \(6\), 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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 23 x^{14} - 8 x^{13} - 131 x^{12} + 380 x^{11} - 289 x^{10} - 880 x^{9} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{15} - \beta_{11}) q^{5} + (\beta_{11} + \beta_{6} + \cdots + \beta_{2}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{15} - \beta_{11}) q^{5} + (\beta_{11} + \beta_{6} + \cdots + \beta_{2}) q^{7}+ \cdots + (\beta_{14} - 2 \beta_{12} + \cdots + 3 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{7} - 6 q^{13} - 18 q^{17} + 16 q^{25} - 6 q^{29} - 6 q^{31} - 30 q^{35} - 2 q^{37} - 6 q^{41} + 2 q^{43} - 18 q^{47} + 10 q^{49} - 36 q^{53} + 30 q^{59} - 60 q^{61} - 42 q^{65} - 14 q^{67} + 18 q^{77} + 16 q^{79} - 12 q^{85} - 24 q^{89} + 12 q^{91} - 66 q^{95} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 23 x^{14} - 8 x^{13} - 131 x^{12} + 380 x^{11} - 289 x^{10} - 880 x^{9} + \cdots + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 169 \nu^{15} + 866 \nu^{14} - 1319 \nu^{13} - 2308 \nu^{12} + 13199 \nu^{11} - 19055 \nu^{10} + \cdots + 244944 ) / 47385 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 154 \nu^{15} + 1325 \nu^{14} - 3608 \nu^{13} + 224 \nu^{12} + 22478 \nu^{11} - 55022 \nu^{10} + \cdots + 1285227 ) / 47385 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 50 \nu^{15} - 1352 \nu^{14} + 6827 \nu^{13} - 7676 \nu^{12} - 27422 \nu^{11} + 107246 \nu^{10} + \cdots - 2825604 ) / 47385 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 964 \nu^{15} + 5336 \nu^{14} - 8294 \nu^{13} - 15778 \nu^{12} + 90644 \nu^{11} - 137225 \nu^{10} + \cdots + 2924019 ) / 142155 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1445 \nu^{15} + 9836 \nu^{14} - 21081 \nu^{13} - 15627 \nu^{12} + 172766 \nu^{11} + \cdots + 7117227 ) / 47385 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2858 \nu^{15} - 19265 \nu^{14} + 40866 \nu^{13} + 31392 \nu^{12} - 338066 \nu^{11} + 649239 \nu^{10} + \cdots - 14041269 ) / 47385 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11192 \nu^{15} - 70123 \nu^{14} + 136087 \nu^{13} + 145859 \nu^{12} - 1215277 \nu^{11} + \cdots - 42998607 ) / 142155 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4120 \nu^{15} + 25571 \nu^{14} - 48788 \nu^{13} - 55006 \nu^{12} + 441224 \nu^{11} + \cdots + 14963454 ) / 28431 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 7903 \nu^{15} - 48406 \nu^{14} + 90897 \nu^{13} + 107784 \nu^{12} - 834172 \nu^{11} + \cdots - 27526311 ) / 47385 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 26068 \nu^{15} - 165707 \nu^{14} + 325403 \nu^{13} + 334861 \nu^{12} - 2873363 \nu^{11} + \cdots - 102032298 ) / 142155 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 33311 \nu^{15} - 207064 \nu^{14} + 396466 \nu^{13} + 441842 \nu^{12} - 3574291 \nu^{11} + \cdots - 121811526 ) / 142155 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 49417 \nu^{15} + 306332 \nu^{14} - 583301 \nu^{13} - 662377 \nu^{12} + 5287136 \nu^{11} + \cdots + 177999930 ) / 142155 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 16898 \nu^{15} + 108472 \nu^{14} - 217033 \nu^{13} - 208526 \nu^{12} + 1883968 \nu^{11} + \cdots + 69445998 ) / 47385 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 56329 \nu^{15} - 353180 \nu^{14} + 683693 \nu^{13} + 735946 \nu^{12} - 6112403 \nu^{11} + \cdots - 213650217 ) / 142155 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 62668 \nu^{15} + 397070 \nu^{14} - 778856 \nu^{13} - 802747 \nu^{12} + 6878696 \nu^{11} + \cdots + 244346949 ) / 142155 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{12} - 2\beta_{9} + \beta_{8} + 2\beta_{7} + \beta_{5} + \beta_{2} - 2\beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} - 2 \beta_{14} - \beta_{13} - 3 \beta_{12} + \beta_{11} - \beta_{10} - 3 \beta_{9} - \beta_{8} + \cdots + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2 \beta_{14} - \beta_{13} - 6 \beta_{12} - \beta_{11} - 3 \beta_{10} - 6 \beta_{9} + 4 \beta_{6} + \cdots - 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 3 \beta_{14} - \beta_{13} - 7 \beta_{12} - 4 \beta_{11} - 6 \beta_{10} - 2 \beta_{9} - 6 \beta_{8} + \cdots + 8 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 10 \beta_{15} + 3 \beta_{14} + 4 \beta_{13} - 6 \beta_{12} - 6 \beta_{11} - 5 \beta_{10} - 6 \beta_{9} + \cdots - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 2 \beta_{15} + 7 \beta_{14} - 4 \beta_{13} - \beta_{12} - 13 \beta_{11} + 2 \beta_{10} - 8 \beta_{9} + \cdots + 20 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3 \beta_{15} + 6 \beta_{14} - 11 \beta_{13} + 3 \beta_{12} + 13 \beta_{11} + 21 \beta_{10} - 33 \beta_{9} + \cdots - 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 68 \beta_{15} + 10 \beta_{14} - 29 \beta_{13} + 16 \beta_{11} + 25 \beta_{10} - 42 \beta_{9} - 121 \beta_{8} + \cdots + 94 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 32 \beta_{15} + 45 \beta_{13} + 4 \beta_{12} + 144 \beta_{11} - 14 \beta_{10} - 40 \beta_{9} + \cdots + 168 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 18 \beta_{15} + 90 \beta_{14} + 115 \beta_{13} - 4 \beta_{12} + 46 \beta_{11} - 162 \beta_{10} + \cdots + 188 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 351 \beta_{15} - 100 \beta_{14} + 207 \beta_{13} + 78 \beta_{12} + 218 \beta_{11} - 333 \beta_{10} + \cdots + 217 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 418 \beta_{15} - 251 \beta_{14} - 16 \beta_{13} - 92 \beta_{12} - 238 \beta_{11} - 416 \beta_{10} + \cdots - 97 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 609 \beta_{15} - 1440 \beta_{14} - 117 \beta_{13} - 230 \beta_{12} + 588 \beta_{11} - 414 \beta_{10} + \cdots + 1243 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 511 \beta_{15} - 835 \beta_{14} - 260 \beta_{13} - 1254 \beta_{12} - 82 \beta_{11} - 812 \beta_{10} + \cdots - 1641 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 1179 \beta_{15} - 1012 \beta_{14} - 431 \beta_{13} - 96 \beta_{12} + 1306 \beta_{11} - 2838 \beta_{10} + \cdots + 208 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(757\) \(785\) \(1135\) \(2593\)
\(\chi(n)\) \(1\) \(1 - \beta_{8}\) \(1\) \(\beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
1.27866 − 1.16834i
1.73109 − 0.0577511i
−1.68301 + 0.409224i
−1.70672 − 0.295146i
0.320287 + 1.70218i
1.58110 + 0.707199i
1.71298 + 0.256290i
0.765614 − 1.55365i
1.27866 + 1.16834i
1.73109 + 0.0577511i
−1.68301 − 0.409224i
−1.70672 + 0.295146i
0.320287 − 1.70218i
1.58110 − 0.707199i
1.71298 − 0.256290i
0.765614 + 1.55365i
0 0 0 −3.55225 0 1.49384 − 2.18368i 0 0 0
17.2 0 0 0 −2.28190 0 −1.21977 − 2.34780i 0 0 0
17.3 0 0 0 −1.42985 0 2.43739 + 1.02913i 0 0 0
17.4 0 0 0 −0.967324 0 −2.40137 + 1.11060i 0 0 0
17.5 0 0 0 0.0676069 0 2.64192 + 0.142361i 0 0 0
17.6 0 0 0 0.900258 0 −1.05755 − 2.42520i 0 0 0
17.7 0 0 0 3.61932 0 −0.266972 + 2.63225i 0 0 0
17.8 0 0 0 3.64414 0 −2.62749 + 0.310282i 0 0 0
1601.1 0 0 0 −3.55225 0 1.49384 + 2.18368i 0 0 0
1601.2 0 0 0 −2.28190 0 −1.21977 + 2.34780i 0 0 0
1601.3 0 0 0 −1.42985 0 2.43739 − 1.02913i 0 0 0
1601.4 0 0 0 −0.967324 0 −2.40137 − 1.11060i 0 0 0
1601.5 0 0 0 0.0676069 0 2.64192 − 0.142361i 0 0 0
1601.6 0 0 0 0.900258 0 −1.05755 + 2.42520i 0 0 0
1601.7 0 0 0 3.61932 0 −0.266972 − 2.63225i 0 0 0
1601.8 0 0 0 3.64414 0 −2.62749 − 0.310282i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3024.2.df.c 16
3.b odd 2 1 1008.2.df.c 16
4.b odd 2 1 378.2.t.a 16
7.d odd 6 1 3024.2.ca.c 16
9.c even 3 1 1008.2.ca.c 16
9.d odd 6 1 3024.2.ca.c 16
12.b even 2 1 126.2.t.a yes 16
21.g even 6 1 1008.2.ca.c 16
28.d even 2 1 2646.2.t.b 16
28.f even 6 1 378.2.l.a 16
28.f even 6 1 2646.2.m.b 16
28.g odd 6 1 2646.2.l.a 16
28.g odd 6 1 2646.2.m.a 16
36.f odd 6 1 126.2.l.a ✓ 16
36.f odd 6 1 1134.2.k.a 16
36.h even 6 1 378.2.l.a 16
36.h even 6 1 1134.2.k.b 16
63.k odd 6 1 1008.2.df.c 16
63.s even 6 1 inner 3024.2.df.c 16
84.h odd 2 1 882.2.t.a 16
84.j odd 6 1 126.2.l.a ✓ 16
84.j odd 6 1 882.2.m.b 16
84.n even 6 1 882.2.l.b 16
84.n even 6 1 882.2.m.a 16
252.n even 6 1 126.2.t.a yes 16
252.o even 6 1 2646.2.t.b 16
252.r odd 6 1 1134.2.k.a 16
252.r odd 6 1 2646.2.m.a 16
252.s odd 6 1 2646.2.l.a 16
252.u odd 6 1 882.2.m.b 16
252.bb even 6 1 2646.2.m.b 16
252.bi even 6 1 882.2.l.b 16
252.bj even 6 1 882.2.m.a 16
252.bj even 6 1 1134.2.k.b 16
252.bl odd 6 1 882.2.t.a 16
252.bn odd 6 1 378.2.t.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.l.a ✓ 16 36.f odd 6 1
126.2.l.a ✓ 16 84.j odd 6 1
126.2.t.a yes 16 12.b even 2 1
126.2.t.a yes 16 252.n even 6 1
378.2.l.a 16 28.f even 6 1
378.2.l.a 16 36.h even 6 1
378.2.t.a 16 4.b odd 2 1
378.2.t.a 16 252.bn odd 6 1
882.2.l.b 16 84.n even 6 1
882.2.l.b 16 252.bi even 6 1
882.2.m.a 16 84.n even 6 1
882.2.m.a 16 252.bj even 6 1
882.2.m.b 16 84.j odd 6 1
882.2.m.b 16 252.u odd 6 1
882.2.t.a 16 84.h odd 2 1
882.2.t.a 16 252.bl odd 6 1
1008.2.ca.c 16 9.c even 3 1
1008.2.ca.c 16 21.g even 6 1
1008.2.df.c 16 3.b odd 2 1
1008.2.df.c 16 63.k odd 6 1
1134.2.k.a 16 36.f odd 6 1
1134.2.k.a 16 252.r odd 6 1
1134.2.k.b 16 36.h even 6 1
1134.2.k.b 16 252.bj even 6 1
2646.2.l.a 16 28.g odd 6 1
2646.2.l.a 16 252.s odd 6 1
2646.2.m.a 16 28.g odd 6 1
2646.2.m.a 16 252.r odd 6 1
2646.2.m.b 16 28.f even 6 1
2646.2.m.b 16 252.bb even 6 1
2646.2.t.b 16 28.d even 2 1
2646.2.t.b 16 252.o even 6 1
3024.2.ca.c 16 7.d odd 6 1
3024.2.ca.c 16 9.d odd 6 1
3024.2.df.c 16 1.a even 1 1 trivial
3024.2.df.c 16 63.s even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 24T_{5}^{6} - 12T_{5}^{5} + 153T_{5}^{4} + 162T_{5}^{3} - 117T_{5}^{2} - 126T_{5} + 9 \) acting on \(S_{2}^{\mathrm{new}}(3024, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 24 T^{6} - 12 T^{5} + \cdots + 9)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + 2 T^{15} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{16} + 108 T^{14} + \cdots + 61732449 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 390971529 \) Copy content Toggle raw display
$17$ \( T^{16} + 18 T^{15} + \cdots + 56070144 \) Copy content Toggle raw display
$19$ \( T^{16} - 72 T^{14} + \cdots + 9199089 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 187388721 \) Copy content Toggle raw display
$29$ \( T^{16} + 6 T^{15} + \cdots + 1108809 \) Copy content Toggle raw display
$31$ \( T^{16} + 6 T^{15} + \cdots + 65610000 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 32746159681 \) Copy content Toggle raw display
$41$ \( T^{16} + 6 T^{15} + \cdots + 81 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 2999643361 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 588203099136 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 36759242529 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 216504090000 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 547560000 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 2603856784 \) Copy content Toggle raw display
$71$ \( T^{16} + 486 T^{14} + \cdots + 65610000 \) Copy content Toggle raw display
$73$ \( T^{16} - 150 T^{14} + \cdots + 71115489 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 970422010000 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 953512641 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 131145120363321 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 9120206721024 \) Copy content Toggle raw display
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