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Properties

Label 1440.3.j.d
Level $1440$
Weight $3$
Character orbit 1440.j
Analytic conductor $39.237$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1440,3,Mod(1279,1440)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1440.1279"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1440.j (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,-4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,80] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(39.2371580679\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 49 x^{10} - 190 x^{9} + 792 x^{8} - 2094 x^{7} + 5517 x^{6} - 9954 x^{5} + \cdots + 5584 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{25} \)
Twist minimal: no (minimal twist has level 480)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{8} q^{5} + \beta_{5} q^{7} + (\beta_{9} + \beta_{8} + \cdots - \beta_{6}) q^{11} + ( - \beta_{11} + \beta_{7} + \beta_1) q^{13} + ( - \beta_{7} - \beta_{6} + \beta_1) q^{17} + ( - \beta_{10} - \beta_{9} + \cdots + \beta_1) q^{19}+ \cdots + (10 \beta_{11} + 4 \beta_{10} + \cdots - 4 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{5} + 80 q^{23} + 28 q^{25} + 40 q^{29} + 144 q^{35} - 136 q^{41} - 224 q^{43} - 208 q^{47} + 212 q^{49} - 192 q^{55} + 40 q^{61} - 96 q^{65} + 352 q^{67} - 64 q^{85} + 8 q^{89} + 176 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 49 x^{10} - 190 x^{9} + 792 x^{8} - 2094 x^{7} + 5517 x^{6} - 9954 x^{5} + \cdots + 5584 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 154 \nu^{11} + 847 \nu^{10} - 5377 \nu^{9} + 17844 \nu^{8} - 43226 \nu^{7} + 73948 \nu^{6} + \cdots - 332836 ) / 34910 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{10} + 5 \nu^{9} - 39 \nu^{8} + 126 \nu^{7} - 468 \nu^{6} + 984 \nu^{5} - 2103 \nu^{4} + \cdots - 870 ) / 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{10} + 5 \nu^{9} - 41 \nu^{8} + 134 \nu^{7} - 533 \nu^{6} + 1151 \nu^{5} - 2707 \nu^{4} + \cdots - 1546 ) / 10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{10} + 5 \nu^{9} - 41 \nu^{8} + 134 \nu^{7} - 533 \nu^{6} + 1151 \nu^{5} - 2707 \nu^{4} + \cdots - 1776 ) / 10 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4 \nu^{10} - 20 \nu^{9} + 158 \nu^{8} - 512 \nu^{7} + 1947 \nu^{6} - 4133 \nu^{5} + 9236 \nu^{4} + \cdots + 5176 ) / 10 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 354 \nu^{11} - 1947 \nu^{10} + 14627 \nu^{9} - 51219 \nu^{8} + 186684 \nu^{7} - 428001 \nu^{6} + \cdots - 155264 ) / 6982 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2108 \nu^{11} - 11594 \nu^{10} + 87022 \nu^{9} - 304644 \nu^{8} + 1112258 \nu^{7} - 2552389 \nu^{6} + \cdots - 1604700 ) / 34910 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5240 \nu^{11} - 18347 \nu^{10} + 169039 \nu^{9} - 368230 \nu^{8} + 1595576 \nu^{7} - 1771935 \nu^{6} + \cdots + 8618592 ) / 69820 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5240 \nu^{11} - 39293 \nu^{10} + 273769 \nu^{9} - 1192106 \nu^{8} + 4262700 \nu^{7} + \cdots - 15734624 ) / 69820 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3292 \nu^{11} - 18106 \nu^{10} + 141782 \nu^{9} - 502224 \nu^{8} + 1930250 \nu^{7} - 4538905 \nu^{6} + \cdots - 2623076 ) / 34910 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2540 \nu^{11} + 13970 \nu^{10} - 107002 \nu^{9} + 376734 \nu^{8} - 1407884 \nu^{7} + \cdots + 1513948 ) / 17455 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} + \beta_{10} - 2\beta_{9} - 2\beta_{8} + \beta_{7} + 4 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{11} + \beta_{10} - 2\beta_{9} - 2\beta_{8} + \beta_{7} - 2\beta_{4} + 2\beta_{3} - 42 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 16 \beta_{11} - 5 \beta_{10} + 20 \beta_{9} + 20 \beta_{8} - 7 \beta_{7} + 4 \beta_{6} - 3 \beta_{4} + \cdots - 65 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 33 \beta_{11} - 11 \beta_{10} + 58 \beta_{9} + 26 \beta_{8} - 15 \beta_{7} + 8 \beta_{6} + 12 \beta_{5} + \cdots + 288 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 187 \beta_{11} + 26 \beta_{10} - 156 \beta_{9} - 236 \beta_{8} + 70 \beta_{7} - 84 \beta_{6} + \cdots + 829 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 322 \beta_{11} + 53 \beta_{10} - 463 \beta_{9} - 231 \beta_{8} + 124 \beta_{7} - 136 \beta_{6} + \cdots - 1131 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1849 \beta_{11} - 146 \beta_{10} + 844 \beta_{9} + 2748 \beta_{8} - 654 \beta_{7} + 1004 \beta_{6} + \cdots - 10895 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 10479 \beta_{11} - 1105 \beta_{10} + 12762 \beta_{9} + 8282 \beta_{8} - 3809 \beta_{7} + 5304 \beta_{6} + \cdots + 17104 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 13888 \beta_{11} + 731 \beta_{10} + 3052 \beta_{9} - 28868 \beta_{8} + 4753 \beta_{7} - 8436 \beta_{6} + \cdots + 145951 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 152707 \beta_{11} + 12741 \beta_{10} - 161750 \beta_{9} - 135302 \beta_{8} + 54145 \beta_{7} + \cdots - 81436 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 29185 \beta_{11} + 1310 \beta_{10} - 207636 \beta_{9} + 252252 \beta_{8} - 8574 \beta_{7} + \cdots - 1915107 ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(991\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1279.1
0.500000 − 0.723587i
0.500000 + 0.723587i
0.500000 + 2.16741i
0.500000 − 2.16741i
0.500000 + 3.59014i
0.500000 − 3.59014i
0.500000 + 1.80359i
0.500000 − 1.80359i
0.500000 − 2.02498i
0.500000 + 2.02498i
0.500000 + 2.65258i
0.500000 − 2.65258i
0 0 0 −4.94155 − 0.762320i 0 7.21051 0 0 0
1279.2 0 0 0 −4.94155 + 0.762320i 0 7.21051 0 0 0
1279.3 0 0 0 −3.65509 − 3.41179i 0 −13.0243 0 0 0
1279.4 0 0 0 −3.65509 + 3.41179i 0 −13.0243 0 0 0
1279.5 0 0 0 −1.77322 − 4.67501i 0 2.37003 0 0 0
1279.6 0 0 0 −1.77322 + 4.67501i 0 2.37003 0 0 0
1279.7 0 0 0 −0.621940 − 4.96117i 0 −9.43574 0 0 0
1279.8 0 0 0 −0.621940 + 4.96117i 0 −9.43574 0 0 0
1279.9 0 0 0 4.13253 − 2.81463i 0 5.81384 0 0 0
1279.10 0 0 0 4.13253 + 2.81463i 0 5.81384 0 0 0
1279.11 0 0 0 4.85927 − 1.17794i 0 7.06571 0 0 0
1279.12 0 0 0 4.85927 + 1.17794i 0 7.06571 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1279.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1440.3.j.d 12
3.b odd 2 1 480.3.j.a ✓ 12
4.b odd 2 1 1440.3.j.c 12
5.b even 2 1 1440.3.j.c 12
12.b even 2 1 480.3.j.b yes 12
15.d odd 2 1 480.3.j.b yes 12
15.e even 4 1 2400.3.e.h 12
15.e even 4 1 2400.3.e.i 12
20.d odd 2 1 inner 1440.3.j.d 12
24.f even 2 1 960.3.j.g 12
24.h odd 2 1 960.3.j.f 12
60.h even 2 1 480.3.j.a ✓ 12
60.l odd 4 1 2400.3.e.h 12
60.l odd 4 1 2400.3.e.i 12
120.i odd 2 1 960.3.j.g 12
120.m even 2 1 960.3.j.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.3.j.a ✓ 12 3.b odd 2 1
480.3.j.a ✓ 12 60.h even 2 1
480.3.j.b yes 12 12.b even 2 1
480.3.j.b yes 12 15.d odd 2 1
960.3.j.f 12 24.h odd 2 1
960.3.j.f 12 120.m even 2 1
960.3.j.g 12 24.f even 2 1
960.3.j.g 12 120.i odd 2 1
1440.3.j.c 12 4.b odd 2 1
1440.3.j.c 12 5.b even 2 1
1440.3.j.d 12 1.a even 1 1 trivial
1440.3.j.d 12 20.d odd 2 1 inner
2400.3.e.h 12 15.e even 4 1
2400.3.e.h 12 60.l odd 4 1
2400.3.e.i 12 15.e even 4 1
2400.3.e.i 12 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1440, [\chi])\):

\( T_{7}^{6} - 200T_{7}^{4} + 704T_{7}^{3} + 9232T_{7}^{2} - 59648T_{7} + 86272 \) Copy content Toggle raw display
\( T_{23}^{6} - 40T_{23}^{5} - 1104T_{23}^{4} + 49152T_{23}^{3} + 180992T_{23}^{2} - 14882816T_{23} + 67145728 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 244140625 \) Copy content Toggle raw display
$7$ \( (T^{6} - 200 T^{4} + \cdots + 86272)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 144769024 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 1071241560064 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 1963047583744 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 5879810228224 \) Copy content Toggle raw display
$23$ \( (T^{6} - 40 T^{5} + \cdots + 67145728)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 20 T^{5} + \cdots + 174739456)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 652427275534336 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 62\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( (T^{6} + 68 T^{5} + \cdots + 997804096)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 112 T^{5} + \cdots + 49254400)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 104 T^{5} + \cdots + 818692096)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 56\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 72\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( (T^{6} - 20 T^{5} + \cdots - 24763044800)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 176 T^{5} + \cdots - 85655552)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 24\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( (T^{6} - 11408 T^{4} + \cdots + 243552256)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 4 T^{5} + \cdots + 34204460608)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
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