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Properties

Label 60.7.k.a
Level $60$
Weight $7$
Character orbit 60.k
Analytic conductor $13.803$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [60,7,Mod(13,60)]
 
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, 0, 3]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("60.13");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 60.k (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: \(13.8032450172\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 66x^{10} + 1601x^{8} + 17520x^{6} + 84208x^{4} + 136704x^{2} + 14400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{19}\cdot 3^{10}\cdot 5^{7} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + ( - \beta_{4} + 2 \beta_{3} + \cdots + 26) q^{5}+ \cdots - 243 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + ( - \beta_{4} + 2 \beta_{3} + \cdots + 26) q^{5}+ \cdots + ( - 729 \beta_{11} - 729 \beta_{10} + \cdots - 243) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 312 q^{5} + 120 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 312 q^{5} + 120 q^{7} - 3248 q^{11} - 2100 q^{13} + 4536 q^{15} - 5540 q^{17} - 15552 q^{21} - 23840 q^{23} + 10044 q^{25} - 127152 q^{31} - 35640 q^{33} + 102976 q^{35} + 282900 q^{37} - 320720 q^{41} - 62880 q^{43} - 10692 q^{45} + 381600 q^{47} - 145152 q^{51} - 400300 q^{53} + 502152 q^{55} - 38880 q^{57} + 807024 q^{61} + 29160 q^{63} + 124500 q^{65} + 752160 q^{67} + 202400 q^{71} - 322020 q^{73} - 645408 q^{75} - 2448400 q^{77} - 708588 q^{81} + 1894560 q^{83} - 857124 q^{85} - 1007640 q^{87} + 2294400 q^{91} + 835920 q^{93} - 2620000 q^{95} - 3161700 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 66x^{10} + 1601x^{8} + 17520x^{6} + 84208x^{4} + 136704x^{2} + 14400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -19\nu^{11} - 1185\nu^{9} - 26441\nu^{7} - 259203\nu^{5} - 1142320\nu^{3} - 2087592\nu ) / 659520 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 687 \nu^{11} + 1380 \nu^{10} + 53586 \nu^{9} + 79560 \nu^{8} + 1528575 \nu^{7} + 1638420 \nu^{6} + \cdots + 49000320 ) / 3810560 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 687 \nu^{11} + 1380 \nu^{10} - 53586 \nu^{9} + 79560 \nu^{8} - 1528575 \nu^{7} + \cdots + 49000320 ) / 3810560 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1169 \nu^{11} + 13980 \nu^{10} - 77103 \nu^{9} + 860340 \nu^{8} - 1891927 \nu^{7} + \cdots - 145394880 ) / 2857920 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7261 \nu^{11} + 13950 \nu^{10} + 478023 \nu^{9} + 1177020 \nu^{8} + 11695295 \nu^{7} + \cdots - 888773760 ) / 8573760 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10756 \nu^{11} - 114660 \nu^{10} - 737763 \nu^{9} - 6587100 \nu^{8} - 20154950 \nu^{7} + \cdots - 2593618560 ) / 8573760 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 11789 \nu^{11} + 31860 \nu^{10} - 645879 \nu^{9} + 1813500 \nu^{8} - 10996579 \nu^{7} + \cdots - 97761600 ) / 8573760 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 55495 \nu^{11} - 127440 \nu^{10} - 3057288 \nu^{9} - 7254000 \nu^{8} - 53619731 \nu^{7} + \cdots + 408193920 ) / 17147520 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 7777 \nu^{11} + 34800 \nu^{10} - 452244 \nu^{9} + 2037360 \nu^{8} - 9175341 \nu^{7} + \cdots + 479429760 ) / 1905280 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 380345 \nu^{11} - 239580 \nu^{10} - 24313458 \nu^{9} - 16328520 \nu^{8} - 565989601 \nu^{7} + \cdots - 11332494720 ) / 34295040 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 441529 \nu^{11} + 199260 \nu^{10} - 27822210 \nu^{9} + 13258440 \nu^{8} - 632679041 \nu^{7} + \cdots + 12686186880 ) / 34295040 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - \beta_{9} + 3 \beta_{8} - 4 \beta_{7} - \beta_{6} - 5 \beta_{5} + 11 \beta_{4} - 7 \beta_{3} + \cdots + 2 ) / 720 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{11} - 2 \beta_{10} + \beta_{9} - \beta_{8} - 2 \beta_{7} - \beta_{6} - 3 \beta_{5} + \cdots - 1978 ) / 180 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 9 \beta_{11} - 9 \beta_{10} + 28 \beta_{9} - 12 \beta_{8} + 49 \beta_{7} + 28 \beta_{6} + 32 \beta_{5} + \cdots - 20 ) / 360 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 6 \beta_{11} + 6 \beta_{10} - 6 \beta_{9} + 3 \beta_{8} + 12 \beta_{7} + 6 \beta_{6} + 9 \beta_{5} + \cdots + 5760 ) / 30 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 84 \beta_{11} + 84 \beta_{10} - 199 \beta_{9} + 24 \beta_{8} - 268 \beta_{7} - 199 \beta_{6} + \cdots + 92 ) / 90 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 332 \beta_{11} - 332 \beta_{10} + 514 \beta_{9} - 121 \beta_{8} - 1118 \beta_{7} - 514 \beta_{6} + \cdots - 345064 ) / 90 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2460 \beta_{11} - 2460 \beta_{10} + 4934 \beta_{9} + 27 \beta_{8} + 6320 \beta_{7} + 4934 \beta_{6} + \cdots - 1930 ) / 90 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1956 \beta_{11} + 1956 \beta_{10} - 4626 \beta_{9} + 351 \beta_{8} + 10506 \beta_{7} + \cdots + 2461728 ) / 30 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 66186 \beta_{11} + 66186 \beta_{10} - 117470 \beta_{9} - 9363 \beta_{8} - 154286 \beta_{7} + \cdots + 44050 ) / 90 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 96988 \beta_{11} - 96988 \beta_{10} + 365750 \beta_{9} + 6523 \beta_{8} - 841534 \beta_{7} + \cdots - 164691152 ) / 90 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1715178 \beta_{11} - 1715178 \beta_{10} + 2767798 \beta_{9} + 358131 \beta_{8} + 3802030 \beta_{7} + \cdots - 1043426 ) / 90 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
2.81042i
− 3.51909i
1.70867i
4.93976i
− 4.27595i
0.336188i
− 2.81042i
3.51909i
− 1.70867i
− 4.93976i
4.27595i
− 0.336188i
0 −11.0227 − 11.0227i 0 −102.872 − 71.0089i 0 −79.3849 + 79.3849i 0 243.000i 0
13.2 0 −11.0227 − 11.0227i 0 −23.5531 + 122.761i 0 312.733 − 312.733i 0 243.000i 0
13.3 0 −11.0227 − 11.0227i 0 123.592 − 18.7066i 0 −26.9845 + 26.9845i 0 243.000i 0
13.4 0 11.0227 + 11.0227i 0 −41.3794 − 117.952i 0 −49.0142 + 49.0142i 0 243.000i 0
13.5 0 11.0227 + 11.0227i 0 75.4215 + 99.6825i 0 337.242 − 337.242i 0 243.000i 0
13.6 0 11.0227 + 11.0227i 0 124.791 + 7.22440i 0 −434.591 + 434.591i 0 243.000i 0
37.1 0 −11.0227 + 11.0227i 0 −102.872 + 71.0089i 0 −79.3849 − 79.3849i 0 − 243.000i 0
37.2 0 −11.0227 + 11.0227i 0 −23.5531 − 122.761i 0 312.733 + 312.733i 0 − 243.000i 0
37.3 0 −11.0227 + 11.0227i 0 123.592 + 18.7066i 0 −26.9845 − 26.9845i 0 − 243.000i 0
37.4 0 11.0227 − 11.0227i 0 −41.3794 + 117.952i 0 −49.0142 − 49.0142i 0 − 243.000i 0
37.5 0 11.0227 − 11.0227i 0 75.4215 − 99.6825i 0 337.242 + 337.242i 0 − 243.000i 0
37.6 0 11.0227 − 11.0227i 0 124.791 − 7.22440i 0 −434.591 − 434.591i 0 − 243.000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.6
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 60.7.k.a ✓ 12
3.b odd 2 1 180.7.l.b 12
4.b odd 2 1 240.7.bg.d 12
5.b even 2 1 300.7.k.d 12
5.c odd 4 1 inner 60.7.k.a ✓ 12
5.c odd 4 1 300.7.k.d 12
15.e even 4 1 180.7.l.b 12
20.e even 4 1 240.7.bg.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.7.k.a ✓ 12 1.a even 1 1 trivial
60.7.k.a ✓ 12 5.c odd 4 1 inner
180.7.l.b 12 3.b odd 2 1
180.7.l.b 12 15.e even 4 1
240.7.bg.d 12 4.b odd 2 1
240.7.bg.d 12 20.e even 4 1
300.7.k.d 12 5.b even 2 1
300.7.k.d 12 5.c odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(60, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{4} + 59049)^{3} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 15\!\cdots\!04)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 26\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 61\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 20\!\cdots\!04)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 33\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 71\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 22\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 26\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 24\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 49\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 22\!\cdots\!36)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 18\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 34\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 30\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 56\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 91\!\cdots\!64 \) Copy content Toggle raw display
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