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Properties

Label 40.4.f.a
Level $40$
Weight $4$
Character orbit 40.f
Analytic conductor $2.360$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [40,4,Mod(29,40)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(40, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("40.29");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 40.f (of order \(2\), degree \(1\), 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: \(2.36007640023\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2x^{14} + 44x^{12} + 400x^{10} - 3200x^{8} + 25600x^{6} + 180224x^{4} - 524288x^{2} + 16777216 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{13} q^{3} + \beta_{2} q^{4} + \beta_{12} q^{5} + (\beta_{8} - 1) q^{6} + \beta_{14} q^{7} + \beta_{3} q^{8} + ( - \beta_{8} + \beta_{4} + 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{13} q^{3} + \beta_{2} q^{4} + \beta_{12} q^{5} + (\beta_{8} - 1) q^{6} + \beta_{14} q^{7} + \beta_{3} q^{8} + ( - \beta_{8} + \beta_{4} + 7) q^{9} + ( - \beta_{13} + \beta_{7} - 3) q^{10} + (\beta_{10} + \beta_{8} - \beta_{2}) q^{11} + (\beta_{15} - \beta_{14} + \cdots - 2 \beta_1) q^{12}+ \cdots + ( - 56 \beta_{12} - 56 \beta_{11} + \cdots - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{4} - 12 q^{6} + 104 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{4} - 12 q^{6} + 104 q^{9} - 48 q^{10} + 28 q^{14} - 56 q^{15} - 168 q^{16} - 196 q^{20} - 112 q^{24} - 24 q^{25} + 200 q^{26} + 492 q^{30} - 112 q^{31} + 408 q^{34} - 84 q^{36} - 736 q^{39} + 672 q^{40} + 232 q^{41} + 920 q^{44} + 212 q^{46} - 200 q^{49} - 648 q^{50} - 2320 q^{54} + 392 q^{55} + 1120 q^{56} - 1208 q^{60} - 2912 q^{64} - 600 q^{65} - 2488 q^{66} + 1532 q^{70} + 2096 q^{71} + 4224 q^{74} - 3000 q^{76} + 2992 q^{79} + 2280 q^{80} - 728 q^{81} + 7304 q^{84} + 3076 q^{86} - 208 q^{89} - 4280 q^{90} - 7036 q^{94} - 1064 q^{95} + 3632 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2x^{14} + 44x^{12} + 400x^{10} - 3200x^{8} + 25600x^{6} + 180224x^{4} - 524288x^{2} + 16777216 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3 \nu^{14} - 138 \nu^{12} + 412 \nu^{10} + 4240 \nu^{8} - 44928 \nu^{6} - 54272 \nu^{4} + \cdots - 42074112 ) / 917504 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{14} - 2\nu^{12} + 44\nu^{10} + 400\nu^{8} - 3200\nu^{6} + 25600\nu^{4} + 114688\nu^{2} - 458752 ) / 65536 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11 \nu^{15} + 192 \nu^{14} + 842 \nu^{13} - 1920 \nu^{12} + 804 \nu^{11} - 4864 \nu^{10} + \cdots - 698351616 ) / 14680064 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11 \nu^{15} - 192 \nu^{14} + 842 \nu^{13} + 1920 \nu^{12} + 804 \nu^{11} + 4864 \nu^{10} + \cdots + 698351616 ) / 14680064 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 9 \nu^{14} - 146 \nu^{12} - 116 \nu^{10} + 5648 \nu^{8} - 60544 \nu^{6} + 177152 \nu^{4} + \cdots - 35717120 ) / 458752 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{15} - 2\nu^{13} + 44\nu^{11} + 400\nu^{9} - 3200\nu^{7} + 25600\nu^{5} + 180224\nu^{3} - 524288\nu ) / 524288 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 39 \nu^{14} - 114 \nu^{12} + 204 \nu^{10} - 10736 \nu^{8} - 48256 \nu^{6} - 31744 \nu^{4} + \cdots - 44171264 ) / 917504 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 39 \nu^{15} + 80 \nu^{14} + 114 \nu^{13} + 992 \nu^{12} - 204 \nu^{11} - 832 \nu^{10} + \cdots + 270532608 ) / 14680064 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 39 \nu^{15} + 80 \nu^{14} - 114 \nu^{13} + 992 \nu^{12} + 204 \nu^{11} - 832 \nu^{10} + \cdots + 270532608 ) / 14680064 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 69 \nu^{15} - 150 \nu^{13} + 1636 \nu^{11} - 23888 \nu^{9} + 40064 \nu^{7} + \cdots - 34603008 \nu ) / 14680064 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 11 \nu^{15} + 182 \nu^{13} + 220 \nu^{11} - 7600 \nu^{9} + 95104 \nu^{7} - 89088 \nu^{5} + \cdots + 50331648 \nu ) / 2097152 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 19 \nu^{15} - 518 \nu^{13} - 252 \nu^{11} + 15152 \nu^{9} - 195456 \nu^{7} + 842752 \nu^{5} + \cdots - 123731968 \nu ) / 2097152 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{12} - 2\beta_{11} + \beta_{5} - 2\beta_{4} + \beta_{2} - 11 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{15} + 6\beta_{14} - 8\beta_{13} + 6\beta_{12} - 6\beta_{11} + 4\beta_{9} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -12\beta_{12} - 12\beta_{11} - 8\beta_{10} + 4\beta_{7} - 4\beta_{6} - 6\beta_{5} + 4\beta_{4} - 2\beta_{2} - 182 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12 \beta_{15} + 36 \beta_{14} + 32 \beta_{13} - 60 \beta_{12} + 60 \beta_{11} - 40 \beta_{9} + \cdots - 232 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 128 \beta_{12} - 128 \beta_{11} - 48 \beta_{10} - 96 \beta_{8} - 56 \beta_{7} + 56 \beta_{6} + \cdots + 1672 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 192 \beta_{15} - 192 \beta_{14} + 32 \beta_{13} - 128 \beta_{12} + 128 \beta_{11} + 320 \beta_{9} + \cdots + 1632 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 304 \beta_{12} + 304 \beta_{11} - 192 \beta_{8} + 224 \beta_{7} - 224 \beta_{6} + 616 \beta_{5} + \cdots + 136 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 688 \beta_{15} - 1296 \beta_{14} + 5952 \beta_{13} - 3088 \beta_{12} + 3088 \beta_{11} + \cdots - 2336 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 544 \beta_{12} + 544 \beta_{11} + 1216 \beta_{10} - 4352 \beta_{8} - 2784 \beta_{7} + 2784 \beta_{6} + \cdots - 170864 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 7712 \beta_{15} - 5728 \beta_{14} - 27904 \beta_{13} + 26272 \beta_{12} - 26272 \beta_{11} + \cdots - 143936 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 51712 \beta_{12} + 51712 \beta_{11} - 3968 \beta_{10} + 38144 \beta_{8} + 19776 \beta_{7} + \cdots - 858560 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 78848 \beta_{15} + 83968 \beta_{14} - 23296 \beta_{13} - 105984 \beta_{12} + 105984 \beta_{11} + \cdots - 953600 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
−2.72041 − 0.774197i
−2.72041 + 0.774197i
−2.15123 − 1.83636i
−2.15123 + 1.83636i
−2.07496 − 1.92212i
−2.07496 + 1.92212i
−0.407618 − 2.79890i
−0.407618 + 2.79890i
0.407618 − 2.79890i
0.407618 + 2.79890i
2.07496 − 1.92212i
2.07496 + 1.92212i
2.15123 − 1.83636i
2.15123 + 1.83636i
2.72041 − 0.774197i
2.72041 + 0.774197i
−2.72041 − 0.774197i −0.0507699 6.80124 + 4.21226i −4.72745 − 10.1317i 0.138115 + 0.0393059i − 20.6415i −15.2410 − 16.7246i −26.9974 5.01667 + 31.2223i
29.2 −2.72041 + 0.774197i −0.0507699 6.80124 − 4.21226i −4.72745 + 10.1317i 0.138115 − 0.0393059i 20.6415i −15.2410 + 16.7246i −26.9974 5.01667 − 31.2223i
29.3 −2.15123 − 1.83636i 8.63004 1.25556 + 7.90086i 2.63740 + 10.8648i −18.5652 − 15.8479i − 4.74133i 11.8078 − 19.3022i 47.4777 14.2781 − 28.2159i
29.4 −2.15123 + 1.83636i 8.63004 1.25556 − 7.90086i 2.63740 − 10.8648i −18.5652 + 15.8479i 4.74133i 11.8078 + 19.3022i 47.4777 14.2781 + 28.2159i
29.5 −2.07496 − 1.92212i −6.67494 0.610901 + 7.97664i 11.1461 − 0.874848i 13.8502 + 12.8300i 29.8250i 14.0645 − 17.7254i 17.5548 −24.8092 − 19.6088i
29.6 −2.07496 + 1.92212i −6.67494 0.610901 − 7.97664i 11.1461 + 0.874848i 13.8502 − 12.8300i − 29.8250i 14.0645 + 17.7254i 17.5548 −24.8092 + 19.6088i
29.7 −0.407618 − 2.79890i −3.86846 −7.66769 + 2.28177i −9.66751 + 5.61599i 1.57685 + 10.8274i − 9.16068i 9.51193 + 20.5310i −12.0350 19.6592 + 24.7692i
29.8 −0.407618 + 2.79890i −3.86846 −7.66769 − 2.28177i −9.66751 − 5.61599i 1.57685 − 10.8274i 9.16068i 9.51193 − 20.5310i −12.0350 19.6592 − 24.7692i
29.9 0.407618 − 2.79890i 3.86846 −7.66769 − 2.28177i 9.66751 − 5.61599i 1.57685 − 10.8274i − 9.16068i −9.51193 + 20.5310i −12.0350 −11.7779 − 29.3476i
29.10 0.407618 + 2.79890i 3.86846 −7.66769 + 2.28177i 9.66751 + 5.61599i 1.57685 + 10.8274i 9.16068i −9.51193 − 20.5310i −12.0350 −11.7779 + 29.3476i
29.11 2.07496 − 1.92212i 6.67494 0.610901 − 7.97664i −11.1461 + 0.874848i 13.8502 − 12.8300i 29.8250i −14.0645 − 17.7254i 17.5548 −21.4460 + 23.2393i
29.12 2.07496 + 1.92212i 6.67494 0.610901 + 7.97664i −11.1461 − 0.874848i 13.8502 + 12.8300i − 29.8250i −14.0645 + 17.7254i 17.5548 −21.4460 − 23.2393i
29.13 2.15123 − 1.83636i −8.63004 1.25556 − 7.90086i −2.63740 − 10.8648i −18.5652 + 15.8479i − 4.74133i −11.8078 − 19.3022i 47.4777 −25.6254 − 18.5295i
29.14 2.15123 + 1.83636i −8.63004 1.25556 + 7.90086i −2.63740 + 10.8648i −18.5652 − 15.8479i 4.74133i −11.8078 + 19.3022i 47.4777 −25.6254 + 18.5295i
29.15 2.72041 − 0.774197i 0.0507699 6.80124 − 4.21226i 4.72745 + 10.1317i 0.138115 − 0.0393059i − 20.6415i 15.2410 − 16.7246i −26.9974 20.7045 + 23.9024i
29.16 2.72041 + 0.774197i 0.0507699 6.80124 + 4.21226i 4.72745 − 10.1317i 0.138115 + 0.0393059i 20.6415i 15.2410 + 16.7246i −26.9974 20.7045 − 23.9024i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.b even 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 40.4.f.a ✓ 16
3.b odd 2 1 360.4.d.d 16
4.b odd 2 1 160.4.f.a 16
5.b even 2 1 inner 40.4.f.a ✓ 16
5.c odd 4 2 200.4.d.e 16
8.b even 2 1 inner 40.4.f.a ✓ 16
8.d odd 2 1 160.4.f.a 16
12.b even 2 1 1440.4.d.d 16
15.d odd 2 1 360.4.d.d 16
20.d odd 2 1 160.4.f.a 16
20.e even 4 2 800.4.d.e 16
24.f even 2 1 1440.4.d.d 16
24.h odd 2 1 360.4.d.d 16
40.e odd 2 1 160.4.f.a 16
40.f even 2 1 inner 40.4.f.a ✓ 16
40.i odd 4 2 200.4.d.e 16
40.k even 4 2 800.4.d.e 16
60.h even 2 1 1440.4.d.d 16
120.i odd 2 1 360.4.d.d 16
120.m even 2 1 1440.4.d.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.4.f.a ✓ 16 1.a even 1 1 trivial
40.4.f.a ✓ 16 5.b even 2 1 inner
40.4.f.a ✓ 16 8.b even 2 1 inner
40.4.f.a ✓ 16 40.f even 2 1 inner
160.4.f.a 16 4.b odd 2 1
160.4.f.a 16 8.d odd 2 1
160.4.f.a 16 20.d odd 2 1
160.4.f.a 16 40.e odd 2 1
200.4.d.e 16 5.c odd 4 2
200.4.d.e 16 40.i odd 4 2
360.4.d.d 16 3.b odd 2 1
360.4.d.d 16 15.d odd 2 1
360.4.d.d 16 24.h odd 2 1
360.4.d.d 16 120.i odd 2 1
800.4.d.e 16 20.e even 4 2
800.4.d.e 16 40.k even 4 2
1440.4.d.d 16 12.b even 2 1
1440.4.d.d 16 24.f even 2 1
1440.4.d.d 16 60.h even 2 1
1440.4.d.d 16 120.m even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(40, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 2 T^{14} + \cdots + 16777216 \) Copy content Toggle raw display
$3$ \( (T^{8} - 134 T^{6} + \cdots + 128)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 59\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{8} + 1422 T^{6} + \cdots + 714987936)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 2238493420800)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 9784472371200)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 12119907631104)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 28704 T^{6} + \cdots + 38178047232)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 23\!\cdots\!96)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 58\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 28 T^{3} + \cdots + 400132096)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 930111561566208)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 58 T^{3} + \cdots + 2763694080)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} - 111062 T^{6} + \cdots + 542115358848)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 21\!\cdots\!64)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 27\!\cdots\!48)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 43\!\cdots\!48)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 13\!\cdots\!68)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 524 T^{3} + \cdots - 30115911168)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 68\!\cdots\!04)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 748 T^{3} + \cdots + 45513395200)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 39\!\cdots\!48)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 52 T^{3} + \cdots - 4266435600)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 22\!\cdots\!44)^{2} \) Copy content Toggle raw display
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