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Properties

Label 40.11.l.a
Level $40$
Weight $11$
Character orbit 40.l
Analytic conductor $25.414$
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] = [40,11,Mod(17,40)] 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("40.17"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(40, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 40.l (of order \(4\), degree \(2\), 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: \(25.4142901069\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(i)\)
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: \( x^{14} - 57366 x^{11} + 226018348 x^{10} - 639855644 x^{9} + 1645428978 x^{8} - 3566916972676 x^{7} + \cdots + 89\!\cdots\!72 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{47}\cdot 3^{4}\cdot 5^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{2} + 14 \beta_1 - 14) q^{3} + ( - \beta_{4} + \beta_{3} - 2 \beta_{2} + \cdots - 36) q^{5} + (\beta_{11} + \beta_{4} - 14 \beta_{3} + \cdots - 326) q^{7} + ( - \beta_{12} - \beta_{11} + \cdots - 26135 \beta_1) q^{9}+ \cdots + (46023 \beta_{13} - 64258 \beta_{12} + \cdots - 36279) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 190 q^{3} - 510 q^{5} - 4646 q^{7} - 57004 q^{11} + 665006 q^{13} + 1478410 q^{15} + 665638 q^{17} - 17091900 q^{21} + 4904618 q^{23} - 10103850 q^{25} + 33966320 q^{27} + 11624148 q^{31} + 2326100 q^{33}+ \cdots + 1599867854 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 57366 x^{11} + 226018348 x^{10} - 639855644 x^{9} + 1645428978 x^{8} - 3566916972676 x^{7} + \cdots + 89\!\cdots\!72 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 46\!\cdots\!71 \nu^{13} + \cdots + 18\!\cdots\!84 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 83\!\cdots\!19 \nu^{13} + \cdots + 42\!\cdots\!56 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10\!\cdots\!63 \nu^{13} + \cdots - 88\!\cdots\!32 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 43\!\cdots\!46 \nu^{13} + \cdots - 50\!\cdots\!44 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 61\!\cdots\!48 \nu^{13} + \cdots - 44\!\cdots\!52 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 29\!\cdots\!93 \nu^{13} + \cdots - 25\!\cdots\!12 ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 19\!\cdots\!41 \nu^{13} + \cdots - 32\!\cdots\!56 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 18\!\cdots\!06 \nu^{13} + \cdots - 18\!\cdots\!04 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 19\!\cdots\!17 \nu^{13} + \cdots + 34\!\cdots\!92 ) / 91\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 10\!\cdots\!19 \nu^{13} + \cdots - 34\!\cdots\!76 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 26\!\cdots\!81 \nu^{13} + \cdots - 23\!\cdots\!84 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 76\!\cdots\!41 \nu^{13} + \cdots - 37\!\cdots\!64 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 31\!\cdots\!21 \nu^{13} + \cdots + 14\!\cdots\!84 ) / 41\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 7\beta_{10} - \beta_{8} + 7\beta_{5} - 21\beta_{4} - 4\beta_{3} + 142\beta_{2} + 68\beta _1 - 54 ) / 2000 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 25 \beta_{13} + 625 \beta_{12} - 1050 \beta_{11} + 247 \beta_{10} - 1050 \beta_{9} - 866 \beta_{8} + \cdots - 594 ) / 8000 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 85 \beta_{13} - 1420 \beta_{12} - 15900 \beta_{11} + 7908 \beta_{10} + 83302 \beta_{8} + \cdots + 20275580 ) / 1600 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 3800200 \beta_{11} + 3551772 \beta_{10} + 3800200 \beta_{9} + 835989 \beta_{8} + \cdots - 129137913354 ) / 2000 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5087375 \beta_{13} + 129507925 \beta_{12} - 6266583603 \beta_{10} + 1211178100 \beta_{9} + \cdots + 1859640983576 ) / 8000 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 372682370 \beta_{13} - 14757499495 \beta_{12} + 22533349820 \beta_{11} - 4102114118 \beta_{10} + \cdots + 17700918121 ) / 800 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 39477001300 \beta_{13} + 1079739614425 \beta_{12} + 8803477751400 \beta_{11} - 3968600756891 \beta_{10} + \cdots - 19\!\cdots\!88 ) / 4000 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 16\!\cdots\!50 \beta_{11} + \cdots + 57\!\cdots\!98 ) / 4000 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 125891398160045 \beta_{13} + \cdots - 73\!\cdots\!18 ) / 800 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 21\!\cdots\!75 \beta_{13} + \cdots - 11\!\cdots\!14 ) / 2000 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 10\!\cdots\!25 \beta_{13} + \cdots + 67\!\cdots\!48 ) / 4000 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 37\!\cdots\!00 \beta_{11} + \cdots - 13\!\cdots\!79 ) / 400 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 79\!\cdots\!75 \beta_{13} + \cdots + 59\!\cdots\!06 ) / 2000 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/40\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(\beta_{1}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−24.8690 + 24.8690i
8.01095 − 8.01095i
−87.3048 + 87.3048i
13.3337 − 13.3337i
85.9155 − 85.9155i
−4.46802 + 4.46802i
9.38170 − 9.38170i
−24.8690 − 24.8690i
8.01095 + 8.01095i
−87.3048 − 87.3048i
13.3337 + 13.3337i
85.9155 + 85.9155i
−4.46802 − 4.46802i
9.38170 + 9.38170i
0 −321.485 + 321.485i 0 −1288.25 − 2847.11i 0 20235.5 + 20235.5i 0 − 147656.i 0
17.2 0 −229.542 + 229.542i 0 2860.36 + 1258.56i 0 −12686.5 − 12686.5i 0 − 46329.8i 0
17.3 0 −82.5406 + 82.5406i 0 −1917.37 + 2467.66i 0 5641.09 + 5641.09i 0 45423.1i 0
17.4 0 −50.3214 + 50.3214i 0 −1124.35 − 2915.73i 0 −18991.2 − 18991.2i 0 53984.5i 0
17.5 0 90.7103 − 90.7103i 0 2851.94 − 1277.53i 0 11858.6 + 11858.6i 0 42592.3i 0
17.6 0 236.074 − 236.074i 0 −2741.95 − 1499.11i 0 1877.44 + 1877.44i 0 − 52413.3i 0
17.7 0 262.104 − 262.104i 0 1104.63 + 2923.26i 0 −10257.9 − 10257.9i 0 − 78347.8i 0
33.1 0 −321.485 − 321.485i 0 −1288.25 + 2847.11i 0 20235.5 − 20235.5i 0 147656.i 0
33.2 0 −229.542 − 229.542i 0 2860.36 − 1258.56i 0 −12686.5 + 12686.5i 0 46329.8i 0
33.3 0 −82.5406 − 82.5406i 0 −1917.37 − 2467.66i 0 5641.09 − 5641.09i 0 − 45423.1i 0
33.4 0 −50.3214 − 50.3214i 0 −1124.35 + 2915.73i 0 −18991.2 + 18991.2i 0 − 53984.5i 0
33.5 0 90.7103 + 90.7103i 0 2851.94 + 1277.53i 0 11858.6 − 11858.6i 0 − 42592.3i 0
33.6 0 236.074 + 236.074i 0 −2741.95 + 1499.11i 0 1877.44 − 1877.44i 0 52413.3i 0
33.7 0 262.104 + 262.104i 0 1104.63 − 2923.26i 0 −10257.9 + 10257.9i 0 78347.8i 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 17.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 40.11.l.a ✓ 14
4.b odd 2 1 80.11.p.f 14
5.b even 2 1 200.11.l.c 14
5.c odd 4 1 inner 40.11.l.a ✓ 14
5.c odd 4 1 200.11.l.c 14
20.e even 4 1 80.11.p.f 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.11.l.a ✓ 14 1.a even 1 1 trivial
40.11.l.a ✓ 14 5.c odd 4 1 inner
80.11.p.f 14 4.b odd 2 1
80.11.p.f 14 20.e even 4 1
200.11.l.c 14 5.b even 2 1
200.11.l.c 14 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} + 190 T_{3}^{13} + 18050 T_{3}^{12} - 17658480 T_{3}^{11} + 39289743720 T_{3}^{10} + \cdots + 37\!\cdots\!00 \) acting on \(S_{11}^{\mathrm{new}}(40, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 84\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 50\!\cdots\!92 \) Copy content Toggle raw display
$11$ \( (T^{7} + \cdots - 55\!\cdots\!16)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 15\!\cdots\!72 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 13\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 24\!\cdots\!88 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 37\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( (T^{7} + \cdots + 62\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 19\!\cdots\!88 \) Copy content Toggle raw display
$41$ \( (T^{7} + \cdots - 45\!\cdots\!72)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 26\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 55\!\cdots\!08 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 30\!\cdots\!52 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{7} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 24\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots - 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 14\!\cdots\!08 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 80\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 30\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 22\!\cdots\!48 \) Copy content Toggle raw display
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