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Properties

Label 1440.4.k.d
Level $1440$
Weight $4$
Character orbit 1440.k
Analytic conductor $84.963$
Analytic rank $0$
Dimension $14$
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] = [1440,4,Mod(721,1440)] 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("1440.721"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1440.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(84.9627504083\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 4 x^{13} + 7 x^{12} - 22 x^{11} + 70 x^{10} - 232 x^{9} + 1080 x^{8} - 4000 x^{7} + \cdots + 2097152 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{42} \)
Twist minimal: no (minimal twist has level 120)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 5 \beta_1 q^{5} + (\beta_{6} + 2) q^{7} + (\beta_{10} - 6 \beta_1) q^{11} + ( - \beta_{9} - \beta_{2}) q^{13} + (\beta_{6} + \beta_{3} - 15) q^{17} + (\beta_{11} + \beta_{9} + \cdots + 4 \beta_1) q^{19}+ \cdots + (7 \beta_{8} + 3 \beta_{7} + \cdots + 245) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 28 q^{7} - 204 q^{17} - 328 q^{23} - 350 q^{25} - 596 q^{31} + 820 q^{41} - 104 q^{47} + 1110 q^{49} + 440 q^{55} + 1592 q^{71} - 2260 q^{73} + 220 q^{79} + 2492 q^{89} - 280 q^{95} + 3508 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 4 x^{13} + 7 x^{12} - 22 x^{11} + 70 x^{10} - 232 x^{9} + 1080 x^{8} - 4000 x^{7} + \cdots + 2097152 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 405 \nu^{13} - 4300 \nu^{12} + 14835 \nu^{11} - 44822 \nu^{10} + 104462 \nu^{9} + \cdots + 484179968 ) / 231211008 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 261 \nu^{13} - 6388 \nu^{12} + 24189 \nu^{11} - 78794 \nu^{10} + 164306 \nu^{9} + \cdots + 1597767680 ) / 16515072 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 21 \nu^{13} + 20 \nu^{12} - 13 \nu^{11} + 10 \nu^{10} + 974 \nu^{9} - 14232 \nu^{8} + \cdots - 20971520 ) / 786432 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 25 \nu^{13} - 76 \nu^{12} - 945 \nu^{11} + 8066 \nu^{10} - 36666 \nu^{9} + 122600 \nu^{8} + \cdots + 182452224 ) / 786432 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 31 \nu^{13} - 164 \nu^{12} - 391 \nu^{11} + 3390 \nu^{10} - 14614 \nu^{9} + 48632 \nu^{8} + \cdots + 60030976 ) / 786432 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 37 \nu^{13} - 92 \nu^{12} + 35 \nu^{11} + 858 \nu^{10} - 4786 \nu^{9} + 12488 \nu^{8} + \cdots + 4980736 ) / 786432 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 197 \nu^{13} - 556 \nu^{12} - 3389 \nu^{11} + 29802 \nu^{10} - 116498 \nu^{9} + 388072 \nu^{8} + \cdots + 564396032 ) / 2359296 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 77 \nu^{13} - 108 \nu^{12} - 517 \nu^{11} + 2522 \nu^{10} - 13090 \nu^{9} + 47080 \nu^{8} + \cdots + 41156608 ) / 786432 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1485 \nu^{13} + 5150 \nu^{12} - 18723 \nu^{11} + 59716 \nu^{10} - 142666 \nu^{9} + \cdots + 422248448 ) / 14450688 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 855 \nu^{13} + 2980 \nu^{12} - 8769 \nu^{11} + 29042 \nu^{10} - 59450 \nu^{9} + \cdots + 361234432 ) / 8257536 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3555 \nu^{13} + 23732 \nu^{12} - 102741 \nu^{11} + 276634 \nu^{10} - 630274 \nu^{9} + \cdots - 1798045696 ) / 16515072 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 16923 \nu^{13} - 49300 \nu^{12} + 138333 \nu^{11} - 366698 \nu^{10} + 1429970 \nu^{9} + \cdots - 4346085376 ) / 77070336 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 54045 \nu^{13} + 489836 \nu^{12} - 1862571 \nu^{11} + 5187142 \nu^{10} - 12375358 \nu^{9} + \cdots - 32418562048 ) / 231211008 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{13} + \beta_{11} - \beta_{10} - 3\beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - 9\beta _1 + 18 ) / 64 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{13} + 2 \beta_{12} + \beta_{11} - 7 \beta_{10} + 2 \beta_{9} - 4 \beta_{8} - \beta_{6} + \cdots + 10 ) / 64 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3 \beta_{13} - 4 \beta_{12} + 5 \beta_{11} + 7 \beta_{10} - 44 \beta_{9} - 14 \beta_{8} + 6 \beta_{7} + \cdots + 212 ) / 64 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 13 \beta_{13} - 10 \beta_{12} - 15 \beta_{11} - 59 \beta_{10} + 6 \beta_{9} - 12 \beta_{8} + 24 \beta_{7} + \cdots - 102 ) / 64 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 15 \beta_{13} - 16 \beta_{12} - 19 \beta_{11} - 61 \beta_{10} + 8 \beta_{9} - 22 \beta_{8} + \cdots + 2404 ) / 64 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 85 \beta_{13} + 34 \beta_{12} - 19 \beta_{11} + 229 \beta_{10} - 414 \beta_{9} - 96 \beta_{8} + \cdots - 11194 ) / 64 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 279 \beta_{13} + 132 \beta_{12} - 383 \beta_{11} - 413 \beta_{10} + 44 \beta_{9} - 442 \beta_{8} + \cdots + 32192 ) / 64 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1627 \beta_{13} - 402 \beta_{12} + 753 \beta_{11} + 1901 \beta_{10} - 3906 \beta_{9} + 844 \beta_{8} + \cdots + 61890 ) / 64 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 2137 \beta_{13} + 6264 \beta_{12} + 1021 \beta_{11} - 1093 \beta_{10} + 7760 \beta_{9} + \cdots - 151708 ) / 64 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 3659 \beta_{13} + 12442 \beta_{12} + 4117 \beta_{11} + 22037 \beta_{10} - 9222 \beta_{9} + \cdots + 508046 ) / 64 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 7481 \beta_{13} - 11892 \beta_{12} - 22615 \beta_{11} + 35651 \beta_{10} - 64044 \beta_{9} + \cdots - 1180808 ) / 64 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 65605 \beta_{13} - 17946 \beta_{12} - 134055 \beta_{11} - 79747 \beta_{10} + 30550 \beta_{9} + \cdots + 5868858 ) / 64 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 58343 \beta_{13} + 60224 \beta_{12} + 38645 \beta_{11} + 532427 \beta_{10} - 1008136 \beta_{9} + \cdots + 17632268 ) / 64 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
721.1
2.82087 + 0.206677i
0.194895 − 2.82170i
1.74774 + 2.22383i
−2.34569 + 1.58043i
−0.594754 + 2.76519i
2.49456 − 1.33311i
−2.31762 − 1.62131i
2.82087 − 0.206677i
0.194895 + 2.82170i
1.74774 − 2.22383i
−2.34569 − 1.58043i
−0.594754 − 2.76519i
2.49456 + 1.33311i
−2.31762 + 1.62131i
0 0 0 − 5.00000i 0 −31.1865 0 0 0
721.2 0 0 0 − 5.00000i 0 −19.4906 0 0 0
721.3 0 0 0 − 5.00000i 0 −2.94025 0 0 0
721.4 0 0 0 − 5.00000i 0 3.68242 0 0 0
721.5 0 0 0 − 5.00000i 0 10.3416 0 0 0
721.6 0 0 0 − 5.00000i 0 22.4239 0 0 0
721.7 0 0 0 − 5.00000i 0 31.1694 0 0 0
721.8 0 0 0 5.00000i 0 −31.1865 0 0 0
721.9 0 0 0 5.00000i 0 −19.4906 0 0 0
721.10 0 0 0 5.00000i 0 −2.94025 0 0 0
721.11 0 0 0 5.00000i 0 3.68242 0 0 0
721.12 0 0 0 5.00000i 0 10.3416 0 0 0
721.13 0 0 0 5.00000i 0 22.4239 0 0 0
721.14 0 0 0 5.00000i 0 31.1694 0 0 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 721.14
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 1440.4.k.d 14
3.b odd 2 1 480.4.k.c 14
4.b odd 2 1 360.4.k.d 14
8.b even 2 1 inner 1440.4.k.d 14
8.d odd 2 1 360.4.k.d 14
12.b even 2 1 120.4.k.c ✓ 14
24.f even 2 1 120.4.k.c ✓ 14
24.h odd 2 1 480.4.k.c 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.k.c ✓ 14 12.b even 2 1
120.4.k.c ✓ 14 24.f even 2 1
360.4.k.d 14 4.b odd 2 1
360.4.k.d 14 8.d odd 2 1
480.4.k.c 14 3.b odd 2 1
480.4.k.c 14 24.h odd 2 1
1440.4.k.d 14 1.a even 1 1 trivial
1440.4.k.d 14 8.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{7} - 14T_{7}^{6} - 1380T_{7}^{5} + 18584T_{7}^{4} + 397440T_{7}^{3} - 4875648T_{7}^{2} - 1020672T_{7} + 47570432 \) acting on \(S_{4}^{\mathrm{new}}(1440, [\chi])\). 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
$5$ \( (T^{2} + 25)^{7} \) Copy content Toggle raw display
$7$ \( (T^{7} - 14 T^{6} + \cdots + 47570432)^{2} \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 87\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 25\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( (T^{7} + \cdots - 6976273176576)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( (T^{7} + \cdots + 94987172617216)^{2} \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 19\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( (T^{7} + \cdots + 3035229478912)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 50\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( (T^{7} + \cdots - 553103116856704)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 83\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( (T^{7} + \cdots + 12\!\cdots\!92)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 27\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 95\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots - 44\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( (T^{7} + \cdots + 18\!\cdots\!36)^{2} \) Copy content Toggle raw display
$79$ \( (T^{7} + \cdots + 37\!\cdots\!60)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 29\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{7} + \cdots + 33\!\cdots\!20)^{2} \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots + 34\!\cdots\!04)^{2} \) Copy content Toggle raw display
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