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Properties

Label 60.6.i.a
Level $60$
Weight $6$
Character orbit 60.i
Analytic conductor $9.623$
Analytic rank $0$
Dimension $20$
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] = [60,6,Mod(17,60)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(60, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("60.17");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 60.i (of order \(4\), degree \(2\), 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: \(9.62302918878\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{19} + 2 x^{18} - 382 x^{17} + 117610 x^{16} - 661518 x^{15} + 1160778 x^{14} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{23}\cdot 3^{14}\cdot 5^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{3} - \beta_{5} q^{5} + (\beta_{8} + 4 \beta_1 + 4) q^{7} + ( - \beta_{15} - 8 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{3} - \beta_{5} q^{5} + (\beta_{8} + 4 \beta_1 + 4) q^{7} + ( - \beta_{15} - 8 \beta_1) q^{9} + \beta_{17} q^{11} + ( - \beta_{19} + \beta_{15} + \cdots + 53) q^{13}+ \cdots + (79 \beta_{19} + 111 \beta_{18} + \cdots + 24864 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{3} + 76 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{3} + 76 q^{7} + 1068 q^{13} - 130 q^{15} + 2180 q^{21} + 4060 q^{25} + 1454 q^{27} - 4720 q^{31} - 460 q^{33} - 612 q^{37} - 24012 q^{43} - 18860 q^{45} - 31700 q^{51} + 19200 q^{55} + 33476 q^{57} + 59880 q^{61} + 67208 q^{63} - 80804 q^{67} - 56956 q^{73} - 102470 q^{75} - 9980 q^{81} + 239260 q^{85} + 71540 q^{87} + 218520 q^{91} + 307928 q^{93} - 151164 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 2 x^{19} + 2 x^{18} - 382 x^{17} + 117610 x^{16} - 661518 x^{15} + 1160778 x^{14} + \cdots + 48\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 68\!\cdots\!09 \nu^{19} + \cdots - 17\!\cdots\!00 ) / 65\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 32\!\cdots\!37 \nu^{19} + \cdots - 23\!\cdots\!00 ) / 42\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 21\!\cdots\!55 \nu^{19} + \cdots + 20\!\cdots\!00 ) / 26\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 53\!\cdots\!87 \nu^{19} + \cdots - 11\!\cdots\!00 ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10\!\cdots\!69 \nu^{19} + \cdots - 21\!\cdots\!00 ) / 42\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11\!\cdots\!95 \nu^{19} + \cdots + 67\!\cdots\!00 ) / 42\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12\!\cdots\!29 \nu^{19} + \cdots + 13\!\cdots\!00 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 23\!\cdots\!79 \nu^{19} + \cdots + 22\!\cdots\!00 ) / 26\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 98\!\cdots\!83 \nu^{19} + \cdots + 21\!\cdots\!00 ) / 96\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 65\!\cdots\!29 \nu^{19} + \cdots + 22\!\cdots\!00 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 60\!\cdots\!25 \nu^{19} + \cdots + 60\!\cdots\!00 ) / 42\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 66\!\cdots\!67 \nu^{19} + \cdots + 52\!\cdots\!00 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 11\!\cdots\!90 \nu^{19} + \cdots - 28\!\cdots\!00 ) / 70\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 14\!\cdots\!07 \nu^{19} + \cdots + 20\!\cdots\!00 ) / 87\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 10\!\cdots\!93 \nu^{19} + \cdots + 15\!\cdots\!00 ) / 38\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 13\!\cdots\!55 \nu^{19} + \cdots - 51\!\cdots\!00 ) / 42\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 30\!\cdots\!63 \nu^{19} + \cdots + 54\!\cdots\!00 ) / 43\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 11\!\cdots\!59 \nu^{19} + \cdots - 61\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 39\!\cdots\!05 \nu^{19} + \cdots + 40\!\cdots\!00 ) / 42\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 5 \beta_{19} + 5 \beta_{18} - 5 \beta_{17} - 5 \beta_{16} + 5 \beta_{13} + 3 \beta_{11} + 30 \beta_{10} + \cdots + 40 ) / 600 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 51 \beta_{19} + 48 \beta_{18} + 9 \beta_{16} + 96 \beta_{15} + 51 \beta_{13} + 107 \beta_{12} + \cdots - 39908 \beta_1 ) / 360 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 183 \beta_{19} - 999 \beta_{18} - 999 \beta_{17} - 927 \beta_{16} - 324 \beta_{15} + 324 \beta_{14} + \cdots + 15080 ) / 360 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 295 \beta_{19} - 1222 \beta_{17} - 5437 \beta_{16} + 5168 \beta_{14} + 295 \beta_{13} - 16181 \beta_{12} + \cdots - 2782044 ) / 120 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 90327 \beta_{19} - 103167 \beta_{18} + 103167 \beta_{17} - 72729 \beta_{16} - 122672 \beta_{15} + \cdots + 13127632 ) / 120 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2956113 \beta_{19} + 882192 \beta_{18} + 1237581 \beta_{16} + 4194072 \beta_{15} + \cdots - 75618124 \beta_1 ) / 360 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 95351517 \beta_{19} + 63335793 \beta_{18} + 63335793 \beta_{17} + 70737069 \beta_{16} + \cdots + 14582099368 ) / 360 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 275668645 \beta_{19} - 435594430 \beta_{17} + 41198711 \beta_{16} + 1561769784 \beta_{14} + \cdots - 269553198356 ) / 120 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 3675273565 \beta_{19} + 391886377 \beta_{18} - 391886377 \beta_{17} + 9660640931 \beta_{16} + \cdots - 765857074864 ) / 120 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 184806014037 \beta_{19} - 613510580664 \beta_{18} - 368149758327 \beta_{16} + \cdots + 452353482886300 \beta_1 ) / 360 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 6401045939703 \beta_{19} + 6476333444697 \beta_{18} + 6476333444697 \beta_{17} + \cdots + 10\!\cdots\!44 ) / 360 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 40462377172519 \beta_{19} + 62493239576538 \beta_{17} + 33684102976803 \beta_{16} + \cdots + 56\!\cdots\!76 ) / 120 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 835514983998889 \beta_{19} + \cdots - 35\!\cdots\!24 ) / 120 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 33\!\cdots\!91 \beta_{19} + \cdots - 45\!\cdots\!28 \beta_1 ) / 360 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 24\!\cdots\!99 \beta_{19} + \cdots - 53\!\cdots\!92 ) / 360 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 15\!\cdots\!63 \beta_{19} + \cdots - 23\!\cdots\!08 ) / 120 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 15\!\cdots\!31 \beta_{19} + \cdots + 64\!\cdots\!44 ) / 120 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 55\!\cdots\!45 \beta_{19} + \cdots - 11\!\cdots\!12 \beta_1 ) / 360 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 76\!\cdots\!05 \beta_{19} + \cdots + 50\!\cdots\!36 ) / 360 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/60\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(41\)
\(\chi(n)\) \(1\) \(\beta_{1}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
14.4102 − 9.21685i
2.77873 + 6.78762i
9.05032 − 16.3862i
6.78762 + 2.77873i
−5.69920 − 7.08001i
−9.21685 + 14.4102i
−4.98882 + 11.3442i
−16.3862 + 9.05032i
11.3442 − 4.98882i
−7.08001 − 5.69920i
14.4102 + 9.21685i
2.77873 − 6.78762i
9.05032 + 16.3862i
6.78762 − 2.77873i
−5.69920 + 7.08001i
−9.21685 − 14.4102i
−4.98882 − 11.3442i
−16.3862 − 9.05032i
11.3442 + 4.98882i
−7.08001 + 5.69920i
0 −15.0272 − 4.14525i 0 29.4180 + 47.5351i 0 −98.6056 − 98.6056i 0 208.634 + 124.583i 0
17.2 0 −12.0403 + 9.90111i 0 −51.4858 + 21.7765i 0 56.0116 + 56.0116i 0 46.9362 − 238.424i 0
17.3 0 −12.0317 − 9.91146i 0 −26.7203 − 49.1022i 0 1.70680 + 1.70680i 0 46.5259 + 238.504i 0
17.4 0 −9.90111 + 12.0403i 0 51.4858 − 21.7765i 0 56.0116 + 56.0116i 0 −46.9362 − 238.424i 0
17.5 0 0.876355 − 15.5638i 0 55.7692 − 3.84667i 0 151.287 + 151.287i 0 −241.464 − 27.2788i 0
17.6 0 4.14525 + 15.0272i 0 −29.4180 − 47.5351i 0 −98.6056 − 98.6056i 0 −208.634 + 124.583i 0
17.7 0 4.61326 − 14.8902i 0 −31.2988 + 46.3183i 0 −91.3999 − 91.3999i 0 −200.436 − 137.385i 0
17.8 0 9.91146 + 12.0317i 0 26.7203 + 49.1022i 0 1.70680 + 1.70680i 0 −46.5259 + 238.504i 0
17.9 0 14.8902 − 4.61326i 0 31.2988 − 46.3183i 0 −91.3999 − 91.3999i 0 200.436 − 137.385i 0
17.10 0 15.5638 − 0.876355i 0 −55.7692 + 3.84667i 0 151.287 + 151.287i 0 241.464 − 27.2788i 0
53.1 0 −15.0272 + 4.14525i 0 29.4180 − 47.5351i 0 −98.6056 + 98.6056i 0 208.634 − 124.583i 0
53.2 0 −12.0403 − 9.90111i 0 −51.4858 − 21.7765i 0 56.0116 − 56.0116i 0 46.9362 + 238.424i 0
53.3 0 −12.0317 + 9.91146i 0 −26.7203 + 49.1022i 0 1.70680 − 1.70680i 0 46.5259 − 238.504i 0
53.4 0 −9.90111 − 12.0403i 0 51.4858 + 21.7765i 0 56.0116 − 56.0116i 0 −46.9362 + 238.424i 0
53.5 0 0.876355 + 15.5638i 0 55.7692 + 3.84667i 0 151.287 − 151.287i 0 −241.464 + 27.2788i 0
53.6 0 4.14525 − 15.0272i 0 −29.4180 + 47.5351i 0 −98.6056 + 98.6056i 0 −208.634 − 124.583i 0
53.7 0 4.61326 + 14.8902i 0 −31.2988 − 46.3183i 0 −91.3999 + 91.3999i 0 −200.436 + 137.385i 0
53.8 0 9.91146 − 12.0317i 0 26.7203 − 49.1022i 0 1.70680 − 1.70680i 0 −46.5259 − 238.504i 0
53.9 0 14.8902 + 4.61326i 0 31.2988 + 46.3183i 0 −91.3999 + 91.3999i 0 200.436 + 137.385i 0
53.10 0 15.5638 + 0.876355i 0 −55.7692 − 3.84667i 0 151.287 − 151.287i 0 241.464 + 27.2788i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.6.i.a ✓ 20
3.b odd 2 1 inner 60.6.i.a ✓ 20
5.b even 2 1 300.6.i.d 20
5.c odd 4 1 inner 60.6.i.a ✓ 20
5.c odd 4 1 300.6.i.d 20
15.d odd 2 1 300.6.i.d 20
15.e even 4 1 inner 60.6.i.a ✓ 20
15.e even 4 1 300.6.i.d 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.6.i.a ✓ 20 1.a even 1 1 trivial
60.6.i.a ✓ 20 3.b odd 2 1 inner
60.6.i.a ✓ 20 5.c odd 4 1 inner
60.6.i.a ✓ 20 15.e even 4 1 inner
300.6.i.d 20 5.b even 2 1
300.6.i.d 20 5.c odd 4 1
300.6.i.d 20 15.d odd 2 1
300.6.i.d 20 15.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(60, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 71\!\cdots\!49 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 88\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 54\!\cdots\!32)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 18\!\cdots\!68)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 41\!\cdots\!24)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots - 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots - 21\!\cdots\!24)^{4} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 52\!\cdots\!68)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 36\!\cdots\!68)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots - 79\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 17\!\cdots\!76)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 17\!\cdots\!32)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 56\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 10\!\cdots\!32)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 13\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots - 91\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 40\!\cdots\!32)^{2} \) Copy content Toggle raw display
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