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Properties

Label 60.7.g.a
Level $60$
Weight $7$
Character orbit 60.g
Analytic conductor $13.803$
Analytic rank $0$
Dimension $8$
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(41,60)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(60, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("60.41");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 60.g (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: \(13.8032450172\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 202x^{6} + 620x^{5} + 12167x^{4} - 25372x^{3} - 177926x^{2} + 190716x + 977814 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{7}\cdot 5^{8} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 3) q^{3} + \beta_{2} q^{5} + (\beta_{5} - \beta_{4} - 71) q^{7} + ( - \beta_{7} - 2 \beta_{6} + \cdots + 187) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 3) q^{3} + \beta_{2} q^{5} + (\beta_{5} - \beta_{4} - 71) q^{7} + ( - \beta_{7} - 2 \beta_{6} + \cdots + 187) q^{9}+ \cdots + (1482 \beta_{7} - 1491 \beta_{6} + \cdots + 336369) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 20 q^{3} - 560 q^{7} + 1492 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 20 q^{3} - 560 q^{7} + 1492 q^{9} - 6440 q^{13} + 2000 q^{15} - 15272 q^{19} - 868 q^{21} - 25000 q^{25} - 18620 q^{27} + 35032 q^{31} - 111120 q^{33} + 99880 q^{37} + 39608 q^{39} - 161000 q^{43} - 5500 q^{45} + 202560 q^{49} + 429120 q^{51} - 33000 q^{55} - 27160 q^{57} - 135608 q^{61} + 377240 q^{63} + 404920 q^{67} - 254940 q^{69} - 356960 q^{73} + 62500 q^{75} + 707704 q^{79} - 1198112 q^{81} + 828000 q^{85} - 1528440 q^{87} - 2004112 q^{91} - 467920 q^{93} - 1326320 q^{97} + 2650080 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 202x^{6} + 620x^{5} + 12167x^{4} - 25372x^{3} - 177926x^{2} + 190716x + 977814 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 23087 \nu^{7} + 111002 \nu^{6} - 5186194 \nu^{5} - 22636834 \nu^{4} + 334471142 \nu^{3} + \cdots - 11257731684 ) / 281404449 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 200 \nu^{7} - 700 \nu^{6} - 40750 \nu^{5} + 103625 \nu^{4} + 2282600 \nu^{3} - 3527875 \nu^{2} + \cdots + 8831475 ) / 437643 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 157940 \nu^{7} - 1552345 \nu^{6} - 29586835 \nu^{5} + 339001955 \nu^{4} + 1585530710 \nu^{3} + \cdots + 153005351670 ) / 281404449 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 260806 \nu^{7} - 2593154 \nu^{6} - 41476847 \nu^{5} + 462577573 \nu^{4} + 1454099926 \nu^{3} + \cdots + 240704956512 ) / 281404449 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 273517 \nu^{7} - 3245480 \nu^{6} - 41161250 \nu^{5} + 558849883 \nu^{4} + 1242783286 \nu^{3} + \cdots + 149114143197 ) / 281404449 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 113413 \nu^{7} + 39797 \nu^{6} - 20577944 \nu^{5} - 34593850 \nu^{4} + 986989690 \nu^{3} + \cdots - 37919990457 ) / 93801483 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 152056 \nu^{7} - 721301 \nu^{6} - 28663523 \nu^{5} + 103151545 \nu^{4} + 1599227110 \nu^{3} + \cdots + 18159439041 ) / 93801483 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + 61\beta_{6} - 62\beta_{5} - 17\beta_{4} + 44\beta_{3} + 140\beta_{2} - 1073\beta _1 + 2186 ) / 5400 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 47 \beta_{7} + 103 \beta_{6} + 214 \beta_{5} - 371 \beta_{4} + 146 \beta_{3} + 152 \beta_{2} + \cdots + 141068 ) / 2700 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2803 \beta_{7} + 4663 \beta_{6} - 5171 \beta_{5} - 2831 \beta_{4} + 4910 \beta_{3} + 3593 \beta_{2} + \cdots + 382838 ) / 5400 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 3934 \beta_{7} + 7211 \beta_{6} + 9953 \beta_{5} - 19567 \beta_{4} + 13177 \beta_{3} + \cdots + 6072211 ) / 1350 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 164846 \beta_{7} + 267296 \beta_{6} - 207097 \beta_{5} - 169072 \beta_{4} + 280096 \beta_{3} + \cdots + 28297906 ) / 2700 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 547093 \beta_{7} + 1009802 \beta_{6} + 786311 \beta_{5} - 1894204 \beta_{4} + 1812889 \beta_{3} + \cdots + 560236357 ) / 1350 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 17051159 \beta_{7} + 31778429 \beta_{6} - 16276903 \beta_{5} - 18521053 \beta_{4} + 32474356 \beta_{3} + \cdots + 3669549064 ) / 2700 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(31\) \(37\) \(41\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1
−2.67163 − 0.707107i
−2.67163 + 0.707107i
−9.26938 − 0.707107i
−9.26938 + 0.707107i
3.67163 + 0.707107i
3.67163 − 0.707107i
10.2694 + 0.707107i
10.2694 − 0.707107i
0 −24.1542 − 12.0655i 0 − 55.9017i 0 −436.815 0 437.848 + 582.864i 0
41.2 0 −24.1542 + 12.0655i 0 55.9017i 0 −436.815 0 437.848 − 582.864i 0
41.3 0 −22.1779 − 15.3993i 0 55.9017i 0 437.053 0 254.720 + 683.051i 0
41.4 0 −22.1779 + 15.3993i 0 − 55.9017i 0 437.053 0 254.720 − 683.051i 0
41.5 0 11.2485 − 24.5453i 0 55.9017i 0 −414.697 0 −475.944 − 552.194i 0
41.6 0 11.2485 + 24.5453i 0 − 55.9017i 0 −414.697 0 −475.944 + 552.194i 0
41.7 0 25.0836 − 9.99060i 0 − 55.9017i 0 134.460 0 529.376 − 501.201i 0
41.8 0 25.0836 + 9.99060i 0 55.9017i 0 134.460 0 529.376 + 501.201i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.7.g.a ✓ 8
3.b odd 2 1 inner 60.7.g.a ✓ 8
4.b odd 2 1 240.7.l.c 8
5.b even 2 1 300.7.g.h 8
5.c odd 4 2 300.7.b.e 16
12.b even 2 1 240.7.l.c 8
15.d odd 2 1 300.7.g.h 8
15.e even 4 2 300.7.b.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.7.g.a ✓ 8 1.a even 1 1 trivial
60.7.g.a ✓ 8 3.b odd 2 1 inner
240.7.l.c 8 4.b odd 2 1
240.7.l.c 8 12.b even 2 1
300.7.b.e 16 5.c odd 4 2
300.7.b.e 16 15.e even 4 2
300.7.g.h 8 5.b even 2 1
300.7.g.h 8 15.d odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 282429536481 \) Copy content Toggle raw display
$5$ \( (T^{2} + 3125)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 280 T^{3} + \cdots + 10645216936)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{4} + 3220 T^{3} + \cdots - 261697247744)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 11\!\cdots\!36)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 43\!\cdots\!56)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots - 15\!\cdots\!44)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 26\!\cdots\!84)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 88\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 48\!\cdots\!76)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 93\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 89\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 23\!\cdots\!56)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 63\!\cdots\!36)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 98\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 19\!\cdots\!76)^{2} \) Copy content Toggle raw display
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