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Properties

Label 36.4.e.a
Level $36$
Weight $4$
Character orbit 36.e
Analytic conductor $2.124$
Analytic rank $0$
Dimension $6$
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] = [36,4,Mod(13,36)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(36, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("36.13");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 36.e (of order \(3\), 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: \(2.12406876021\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.6831243.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 13x^{4} + 49x^{2} + 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 1) q^{3} + (\beta_{5} - \beta_{3} - 2 \beta_{2} + \cdots + 1) q^{5}+ \cdots + (\beta_{5} + 3 \beta_{4} + \beta_{2} + \cdots + 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - 1) q^{3} + (\beta_{5} - \beta_{3} - 2 \beta_{2} + \cdots + 1) q^{5}+ \cdots + ( - 168 \beta_{5} + 9 \beta_{4} + \cdots - 15) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} + 6 q^{5} - 6 q^{7} + 39 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{3} + 6 q^{5} - 6 q^{7} + 39 q^{9} + 51 q^{11} + 12 q^{13} - 180 q^{15} - 222 q^{17} + 30 q^{19} - 120 q^{21} + 210 q^{23} - 3 q^{25} + 648 q^{27} + 456 q^{29} + 48 q^{31} - 603 q^{33} - 1104 q^{35} - 96 q^{37} - 36 q^{39} + 897 q^{41} + 129 q^{43} + 1494 q^{45} + 522 q^{47} - 225 q^{49} - 1647 q^{51} - 2208 q^{53} - 216 q^{55} - 645 q^{57} + 453 q^{59} - 402 q^{61} + 1896 q^{63} + 1110 q^{65} - 213 q^{67} - 198 q^{69} + 120 q^{71} + 750 q^{73} + 921 q^{75} + 1128 q^{77} + 552 q^{79} - 549 q^{81} - 612 q^{83} + 1188 q^{85} - 1386 q^{87} - 924 q^{89} - 264 q^{91} - 1998 q^{93} - 2184 q^{95} + 93 q^{97} - 1854 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 13x^{4} + 49x^{2} + 48 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 9\nu^{3} + 17\nu + 4 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} - 4\nu^{4} - 5\nu^{3} - 20\nu^{2} + 15\nu + 4 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} - 9\nu^{3} - 12\nu^{2} - 5\nu - 52 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{4} + 3\nu^{3} + 21\nu^{2} + 18\nu + 20 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} - 13\nu^{3} + 44\nu^{2} - 25\nu + 100 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{3} + \beta_{2} + 4\beta _1 - 2 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - \beta_{3} + \beta_{2} - 26 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{5} + \beta_{4} - 3\beta_{3} - \beta_{2} - 11\beta _1 + 7 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5\beta_{5} + 2\beta_{4} + 7\beta_{3} - 11\beta_{2} - 2\beta _1 + 140 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 55\beta_{5} - 18\beta_{4} + 37\beta_{3} + \beta_{2} + 178\beta _1 - 116 ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).

\(n\) \(19\) \(29\)
\(\chi(n)\) \(1\) \(-1 + \beta_{1}\)

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
1.23396i
− 2.13353i
2.63162i
− 1.23396i
2.13353i
− 2.63162i
0 −4.14739 − 3.13036i 0 6.92194 − 11.9892i 0 −15.3540 − 26.5939i 0 7.40171 + 25.9656i 0
13.2 0 −2.51979 + 4.54430i 0 −6.37096 + 11.0348i 0 7.02674 + 12.1707i 0 −14.3013 − 22.9014i 0
13.3 0 5.16718 − 0.547914i 0 2.44901 − 4.24182i 0 5.32725 + 9.22708i 0 26.3996 − 5.66234i 0
25.1 0 −4.14739 + 3.13036i 0 6.92194 + 11.9892i 0 −15.3540 + 26.5939i 0 7.40171 − 25.9656i 0
25.2 0 −2.51979 − 4.54430i 0 −6.37096 − 11.0348i 0 7.02674 − 12.1707i 0 −14.3013 + 22.9014i 0
25.3 0 5.16718 + 0.547914i 0 2.44901 + 4.24182i 0 5.32725 − 9.22708i 0 26.3996 + 5.66234i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 36.4.e.a ✓ 6
3.b odd 2 1 108.4.e.a 6
4.b odd 2 1 144.4.i.d 6
9.c even 3 1 inner 36.4.e.a ✓ 6
9.c even 3 1 324.4.a.c 3
9.d odd 6 1 108.4.e.a 6
9.d odd 6 1 324.4.a.d 3
12.b even 2 1 432.4.i.d 6
36.f odd 6 1 144.4.i.d 6
36.f odd 6 1 1296.4.a.v 3
36.h even 6 1 432.4.i.d 6
36.h even 6 1 1296.4.a.w 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.4.e.a ✓ 6 1.a even 1 1 trivial
36.4.e.a ✓ 6 9.c even 3 1 inner
108.4.e.a 6 3.b odd 2 1
108.4.e.a 6 9.d odd 6 1
144.4.i.d 6 4.b odd 2 1
144.4.i.d 6 36.f odd 6 1
324.4.a.c 3 9.c even 3 1
324.4.a.d 3 9.d odd 6 1
432.4.i.d 6 12.b even 2 1
432.4.i.d 6 36.h even 6 1
1296.4.a.v 3 36.f odd 6 1
1296.4.a.w 3 36.h even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(36, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 3 T^{5} + \cdots + 19683 \) Copy content Toggle raw display
$5$ \( T^{6} - 6 T^{5} + \cdots + 746496 \) Copy content Toggle raw display
$7$ \( T^{6} + 6 T^{5} + \cdots + 21141604 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 4386545361 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 4155865156 \) Copy content Toggle raw display
$17$ \( (T^{3} + 111 T^{2} + \cdots - 577476)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 15 T^{2} + \cdots - 216368)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} - 210 T^{5} + \cdots + 5391684 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 8292430196964 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 9331729724944 \) Copy content Toggle raw display
$37$ \( (T^{3} + 48 T^{2} + \cdots - 682352)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 139158426750849 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 2031049672201 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 41\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( (T^{3} + 1104 T^{2} + \cdots + 11853648)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 93\!\cdots\!69 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 1463461189696 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 9579414973969 \) Copy content Toggle raw display
$71$ \( (T^{3} - 60 T^{2} + \cdots + 113211648)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 375 T^{2} + \cdots + 158369284)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 318578661907984 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 12101965948944 \) Copy content Toggle raw display
$89$ \( (T^{3} + 462 T^{2} + \cdots + 170122248)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 74\!\cdots\!29 \) Copy content Toggle raw display
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