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Properties

Label 1040.4.k.d
Level $1040$
Weight $4$
Character orbit 1040.k
Analytic conductor $61.362$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1040,4,Mod(961,1040)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1040, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1040.961"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1040.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(61.3619864060\)
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: \( x^{14} + 84x^{12} + 2674x^{10} + 40048x^{8} + 278769x^{6} + 727552x^{4} + 339456x^{2} + 9216 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{9}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 65)
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} + \beta_{7} q^{5} + (\beta_{10} + 2 \beta_{7} - \beta_1) q^{7} + (\beta_{5} + 11) q^{9} + ( - \beta_{11} + \beta_{10} + \beta_1) q^{11} + (\beta_{12} - \beta_{9} - \beta_{7} + \beta_1) q^{13}+ \cdots + ( - 3 \beta_{13} - 31 \beta_{11} + \cdots + 27 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 158 q^{9} - 4 q^{13} - 100 q^{17} + 532 q^{23} - 350 q^{25} + 48 q^{27} + 588 q^{29} - 540 q^{35} + 260 q^{39} - 728 q^{43} - 1302 q^{49} - 176 q^{51} + 1040 q^{53} + 40 q^{55} - 1460 q^{61} - 30 q^{65}+ \cdots + 3120 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 84x^{12} + 2674x^{10} + 40048x^{8} + 278769x^{6} + 727552x^{4} + 339456x^{2} + 9216 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} + 24 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 14711 \nu^{12} + 1018260 \nu^{10} + 25200254 \nu^{8} + 275320352 \nu^{6} + 1298780391 \nu^{4} + \cdots + 291591744 ) / 32566080 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1069 \nu^{12} - 72444 \nu^{10} - 1716106 \nu^{8} - 17134688 \nu^{6} - 67868029 \nu^{4} + \cdots - 20876288 ) / 2171072 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1315 \nu^{12} + 90892 \nu^{10} + 2243030 \nu^{8} + 24552176 \nu^{6} + 121124275 \nu^{4} + \cdots + 79049344 ) / 2171072 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 37951 \nu^{12} - 2341860 \nu^{10} - 46164334 \nu^{8} - 290585632 \nu^{6} + 209579889 \nu^{4} + \cdots - 286093824 ) / 32566080 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 24787 \nu^{13} + 2086080 \nu^{11} + 66485638 \nu^{9} + 994293784 \nu^{7} + 6854947107 \nu^{5} + \cdots + 6236239488 \nu ) / 195396480 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 34171 \nu^{13} - 3646980 \nu^{11} - 142120294 \nu^{9} - 2493964192 \nu^{7} + \cdots - 12255022464 \nu ) / 130264320 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 38953 \nu^{13} - 2845590 \nu^{11} - 76077802 \nu^{9} - 929158516 \nu^{7} + \cdots - 6949574592 \nu ) / 97698240 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 118777 \nu^{13} + 8952540 \nu^{11} + 250798018 \nu^{9} + 3279360064 \nu^{7} + \cdots + 28011179328 \nu ) / 195396480 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 51665 \nu^{13} - 4464660 \nu^{11} - 146134418 \nu^{9} - 2239705328 \nu^{7} + \cdots - 16195334016 \nu ) / 78158592 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 208751 \nu^{13} - 972504 \nu^{12} - 18151140 \nu^{11} - 63080640 \nu^{10} + \cdots + 17517072384 ) / 390792960 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 4558 \nu^{13} + 320025 \nu^{11} + 8047632 \nu^{9} + 89292066 \nu^{7} + 429624478 \nu^{5} + \cdots + 75607072 \nu ) / 5427680 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} - 24 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{11} + \beta_{9} + \beta_{8} - 21\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2 \beta_{12} - \beta_{11} - \beta_{9} - \beta_{8} - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 6 \beta_{4} + \cdots + 490 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -6\beta_{13} + 40\beta_{11} - 4\beta_{10} - 50\beta_{9} - 32\beta_{8} + 44\beta_{7} + 487\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 80 \beta_{12} + 40 \beta_{11} + 40 \beta_{9} + 40 \beta_{8} + 80 \beta_{7} + 80 \beta_{6} + \cdots - 11234 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 278\beta_{13} - 1359\beta_{11} + 208\beta_{10} + 1757\beta_{9} + 919\beta_{8} - 2496\beta_{7} - 12095\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2602 \beta_{12} - 1301 \beta_{11} - 1301 \beta_{9} - 1301 \beta_{8} - 2602 \beta_{7} - 2630 \beta_{6} + \cdots + 276820 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 9888 \beta_{13} + 43310 \beta_{11} - 7388 \beta_{10} - 55286 \beta_{9} - 26038 \beta_{8} + \cdots + 315637 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 79240 \beta_{12} + 39620 \beta_{11} + 39620 \beta_{9} + 39620 \beta_{8} + 79240 \beta_{7} + \cdots - 7183244 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 319028 \beta_{13} - 1331417 \beta_{11} + 231760 \beta_{10} + 1663549 \beta_{9} + 739177 \beta_{8} + \cdots - 8529953 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 2348170 \beta_{12} - 1174085 \beta_{11} - 1174085 \beta_{9} - 1174085 \beta_{8} - 2348170 \beta_{7} + \cdots + 193357342 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 9819338 \beta_{13} + 40028576 \beta_{11} - 6925700 \beta_{10} - 49057558 \beta_{9} + \cdots + 236061139 \beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1040\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(417\) \(561\) \(911\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1
− 3.65500i
3.65500i
4.29153i
− 4.29153i
0.742335i
− 0.742335i
− 5.37688i
5.37688i
− 2.10835i
2.10835i
4.27643i
− 4.27643i
− 0.170066i
0.170066i
0 −8.59394 0 − 5.00000i 0 − 19.4590i 0 46.8559 0
961.2 0 −8.59394 0 5.00000i 0 19.4590i 0 46.8559 0
961.3 0 −5.89541 0 − 5.00000i 0 − 34.1818i 0 7.75582 0
961.4 0 −5.89541 0 5.00000i 0 34.1818i 0 7.75582 0
961.5 0 −5.13862 0 − 5.00000i 0 9.97127i 0 −0.594575 0
961.6 0 −5.13862 0 5.00000i 0 − 9.97127i 0 −0.594575 0
961.7 0 0.936724 0 − 5.00000i 0 − 0.201771i 0 −26.1225 0
961.8 0 0.936724 0 5.00000i 0 0.201771i 0 −26.1225 0
961.9 0 3.08338 0 − 5.00000i 0 17.4384i 0 −17.4928 0
961.10 0 3.08338 0 5.00000i 0 − 17.4384i 0 −17.4928 0
961.11 0 7.17327 0 − 5.00000i 0 5.20875i 0 24.4558 0
961.12 0 7.17327 0 5.00000i 0 − 5.20875i 0 24.4558 0
961.13 0 8.43460 0 − 5.00000i 0 − 32.7758i 0 44.1424 0
961.14 0 8.43460 0 5.00000i 0 32.7758i 0 44.1424 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 961.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1040.4.k.d 14
4.b odd 2 1 65.4.c.a ✓ 14
12.b even 2 1 585.4.b.e 14
13.b even 2 1 inner 1040.4.k.d 14
20.d odd 2 1 325.4.c.e 14
20.e even 4 1 325.4.d.c 14
20.e even 4 1 325.4.d.d 14
52.b odd 2 1 65.4.c.a ✓ 14
52.f even 4 1 845.4.a.i 7
52.f even 4 1 845.4.a.l 7
156.h even 2 1 585.4.b.e 14
260.g odd 2 1 325.4.c.e 14
260.p even 4 1 325.4.d.c 14
260.p even 4 1 325.4.d.d 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.4.c.a ✓ 14 4.b odd 2 1
65.4.c.a ✓ 14 52.b odd 2 1
325.4.c.e 14 20.d odd 2 1
325.4.c.e 14 260.g odd 2 1
325.4.d.c 14 20.e even 4 1
325.4.d.c 14 260.p even 4 1
325.4.d.d 14 20.e even 4 1
325.4.d.d 14 260.p even 4 1
585.4.b.e 14 12.b even 2 1
585.4.b.e 14 156.h even 2 1
845.4.a.i 7 52.f even 4 1
845.4.a.l 7 52.f even 4 1
1040.4.k.d 14 1.a even 1 1 trivial
1040.4.k.d 14 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{7} - 134T_{3}^{5} - 8T_{3}^{4} + 5188T_{3}^{3} + 196T_{3}^{2} - 53196T_{3} + 45496 \) acting on \(S_{4}^{\mathrm{new}}(1040, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( (T^{7} - 134 T^{5} + \cdots + 45496)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 25)^{7} \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 15872256000000 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 24\!\cdots\!13 \) Copy content Toggle raw display
$17$ \( (T^{7} + 50 T^{6} + \cdots + 44413747200)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( (T^{7} + \cdots + 5699551522752)^{2} \) Copy content Toggle raw display
$29$ \( (T^{7} + \cdots + 440579141474400)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 76\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{7} + \cdots + 97\!\cdots\!24)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 48\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( (T^{7} + \cdots - 405893642720256)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 71\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( (T^{7} + \cdots - 35\!\cdots\!44)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 22\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 42\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( (T^{7} + \cdots + 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 22\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 18\!\cdots\!84 \) Copy content Toggle raw display
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