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Properties

Label 36.6.e.a
Level $36$
Weight $6$
Character orbit 36.e
Analytic conductor $5.774$
Analytic rank $0$
Dimension $10$
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] = [36,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("36.13");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
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: \(5.77381751327\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 175x^{8} + 8800x^{6} + 124623x^{4} + 498609x^{2} + 442368 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{10} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} - \beta_{3} + 2 \beta_1 + 2) q^{3} + (\beta_{7} + \beta_{4} + \cdots + 4 \beta_1) q^{5}+ \cdots + (\beta_{9} + \beta_{7} + \beta_{6} + \cdots + 21) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} - \beta_{3} + 2 \beta_1 + 2) q^{3} + (\beta_{7} + \beta_{4} + \cdots + 4 \beta_1) q^{5}+ \cdots + (12 \beta_{9} - 345 \beta_{8} + \cdots - 51981) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 12 q^{3} - 21 q^{5} + 29 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 12 q^{3} - 21 q^{5} + 29 q^{7} + 12 q^{9} + 177 q^{11} - 181 q^{13} + 117 q^{15} + 2280 q^{17} - 832 q^{19} - 207 q^{21} + 399 q^{23} - 4778 q^{25} - 7128 q^{27} - 6033 q^{29} + 2759 q^{31} + 9603 q^{33} + 37146 q^{35} - 15172 q^{37} + 5529 q^{39} - 18435 q^{41} + 1469 q^{43} - 64089 q^{45} - 25155 q^{47} - 4056 q^{49} + 90612 q^{51} + 116844 q^{53} + 14778 q^{55} + 26934 q^{57} - 90537 q^{59} + 1403 q^{61} - 198255 q^{63} - 148407 q^{65} + 13907 q^{67} + 214425 q^{69} + 229368 q^{71} + 15200 q^{73} + 44640 q^{75} - 211983 q^{77} + 29993 q^{79} - 404172 q^{81} - 228951 q^{83} - 49662 q^{85} + 397323 q^{87} + 598332 q^{89} + 124930 q^{91} + 250041 q^{93} - 394764 q^{95} + 40541 q^{97} - 697239 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 175x^{8} + 8800x^{6} + 124623x^{4} + 498609x^{2} + 442368 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 99\nu^{9} + 23021\nu^{7} + 1847072\nu^{5} + 56550029\nu^{3} + 389674035\nu - 207097728 ) / 414195456 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -47225\nu^{8} - 9472484\nu^{6} - 609253100\nu^{4} - 13835630451\nu^{2} - 61649294112 ) / 427139064 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 338269 \nu^{9} - 1280856 \nu^{8} + 58481419 \nu^{7} - 217698528 \nu^{6} + 2848921216 \nu^{5} + \cdots - 198138592512 ) / 6834225024 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1356343 \nu^{9} + 375936 \nu^{8} - 234685369 \nu^{7} + 64407552 \nu^{6} - 11456638240 \nu^{5} + \cdots + 3943792512 ) / 13668450048 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4752101 \nu^{9} + 5053584 \nu^{8} - 822538331 \nu^{7} + 855338304 \nu^{6} + \cdots + 1147556557440 ) / 13668450048 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5525341 \nu^{9} + 5651680 \nu^{8} - 929563411 \nu^{7} + 984567424 \nu^{6} + \cdots + 696632164992 ) / 13668450048 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5368449 \nu^{9} + 1430176 \nu^{8} + 927771087 \nu^{7} + 219429760 \nu^{6} + \cdots - 594055313280 ) / 13668450048 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5415571 \nu^{9} + 6627168 \nu^{8} + 936462397 \nu^{7} + 1128424320 \nu^{6} + \cdots + 815163765120 ) / 13668450048 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 6201879 \nu^{9} - 3841840 \nu^{8} - 1046526249 \nu^{7} - 677985472 \nu^{6} + \cdots - 308242564992 ) / 13668450048 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{8} + \beta_{5} + 3\beta_{4} + 5\beta_{3} + 7\beta _1 + 4 ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 9\beta_{9} + 6\beta_{8} - 9\beta_{6} - \beta_{5} + 17\beta_{4} - 12\beta_{3} - 9\beta_{2} + 6\beta _1 - 1261 ) / 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6 \beta_{9} - 83 \beta_{8} + 42 \beta_{7} + 6 \beta_{6} - 74 \beta_{5} - 132 \beta_{4} - 340 \beta_{3} + \cdots - 995 ) / 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 396 \beta_{9} - 329 \beta_{8} + 396 \beta_{6} + 147 \beta_{5} - 1393 \beta_{4} + 479 \beta_{3} + \cdots + 46848 ) / 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1851 \beta_{9} + 14124 \beta_{8} - 8790 \beta_{7} - 1851 \beta_{6} + 12955 \beta_{5} + 21697 \beta_{4} + \cdots + 278107 ) / 36 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 33687 \beta_{9} + 32633 \beta_{8} - 33687 \beta_{6} - 20593 \beta_{5} + 168231 \beta_{4} + \cdots - 3998422 ) / 18 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 111930 \beta_{9} - 625543 \beta_{8} + 426012 \beta_{7} + 111930 \beta_{6} - 595221 \beta_{5} + \cdots - 15701778 ) / 18 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 5933115 \beta_{9} - 6360114 \beta_{8} + 5933115 \beta_{6} + 4761227 \beta_{5} - 36526363 \beta_{4} + \cdots + 717686687 ) / 36 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 12187632 \beta_{9} + 57177589 \beta_{8} - 41054970 \beta_{7} - 12187632 \beta_{6} + \cdots + 1647122695 ) / 18 \) Copy content Toggle raw display

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

Character values

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

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

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
2.13639i
− 7.64342i
− 3.71922i
9.84603i
1.11227i
− 2.13639i
7.64342i
3.71922i
− 9.84603i
− 1.11227i
0 −12.2647 − 9.62174i 0 −14.0718 + 24.3731i 0 75.7039 + 131.123i 0 57.8441 + 236.015i 0
13.2 0 −11.7655 + 10.2260i 0 4.88422 − 8.45972i 0 −68.3340 − 118.358i 0 33.8560 − 240.630i 0
13.3 0 7.64564 + 13.5847i 0 40.7270 − 70.5412i 0 89.6312 + 155.246i 0 −126.088 + 207.727i 0
13.4 0 8.67637 − 12.9507i 0 13.1603 − 22.7942i 0 −31.6287 − 54.7826i 0 −92.4411 − 224.730i 0
13.5 0 13.7082 + 7.42194i 0 −55.1996 + 95.6086i 0 −50.8724 − 88.1135i 0 132.829 + 203.483i 0
25.1 0 −12.2647 + 9.62174i 0 −14.0718 − 24.3731i 0 75.7039 − 131.123i 0 57.8441 − 236.015i 0
25.2 0 −11.7655 − 10.2260i 0 4.88422 + 8.45972i 0 −68.3340 + 118.358i 0 33.8560 + 240.630i 0
25.3 0 7.64564 − 13.5847i 0 40.7270 + 70.5412i 0 89.6312 − 155.246i 0 −126.088 − 207.727i 0
25.4 0 8.67637 + 12.9507i 0 13.1603 + 22.7942i 0 −31.6287 + 54.7826i 0 −92.4411 + 224.730i 0
25.5 0 13.7082 − 7.42194i 0 −55.1996 − 95.6086i 0 −50.8724 + 88.1135i 0 132.829 − 203.483i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.5
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 36.6.e.a ✓ 10
3.b odd 2 1 108.6.e.a 10
4.b odd 2 1 144.6.i.d 10
9.c even 3 1 inner 36.6.e.a ✓ 10
9.c even 3 1 324.6.a.e 5
9.d odd 6 1 108.6.e.a 10
9.d odd 6 1 324.6.a.d 5
12.b even 2 1 432.6.i.d 10
36.f odd 6 1 144.6.i.d 10
36.h even 6 1 432.6.i.d 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.6.e.a ✓ 10 1.a even 1 1 trivial
36.6.e.a ✓ 10 9.c even 3 1 inner
108.6.e.a 10 3.b odd 2 1
108.6.e.a 10 9.d odd 6 1
144.6.i.d 10 4.b odd 2 1
144.6.i.d 10 36.f odd 6 1
324.6.a.d 5 9.d odd 6 1
324.6.a.e 5 9.c even 3 1
432.6.i.d 10 12.b even 2 1
432.6.i.d 10 36.h even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(36, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 847288609443 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 42\!\cdots\!24 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 56\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 17\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( (T^{5} + \cdots + 113704762586184)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots - 97\!\cdots\!92)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 46\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 96\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 85\!\cdots\!09 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 66\!\cdots\!21 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 18\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots - 86\!\cdots\!28)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 48\!\cdots\!81 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 45\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 20\!\cdots\!49 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 18\!\cdots\!64)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 17\!\cdots\!80)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 16\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots - 18\!\cdots\!72)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 17\!\cdots\!25 \) Copy content Toggle raw display
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