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Properties

Label 36.6.b.b
Level $36$
Weight $6$
Character orbit 36.b
Analytic conductor $5.774$
Analytic rank $0$
Dimension $8$
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] = [36,6,Mod(35,36)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(36, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("36.35");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 36.b (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: \(5.77381751327\)
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: \( x^{8} - 4x^{7} + 58x^{6} - 160x^{5} + 805x^{4} - 1348x^{3} + 3024x^{2} - 2376x + 972 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{6} \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{7} + 6) q^{4} + ( - \beta_{4} + 3 \beta_{2} + 6 \beta_1) q^{5} + (3 \beta_{7} - \beta_{6} + \beta_{5} + 1) q^{7} + ( - 5 \beta_{4} - \beta_{3} + \cdots + 17 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{7} + 6) q^{4} + ( - \beta_{4} + 3 \beta_{2} + 6 \beta_1) q^{5} + (3 \beta_{7} - \beta_{6} + \beta_{5} + 1) q^{7} + ( - 5 \beta_{4} - \beta_{3} + \cdots + 17 \beta_1) q^{8}+ \cdots + (1632 \beta_{4} - 544 \beta_{3} + \cdots + 13600 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 44 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 44 q^{4} - 596 q^{10} + 256 q^{13} - 4216 q^{16} + 9984 q^{22} + 15192 q^{25} - 23232 q^{28} + 23236 q^{34} - 26096 q^{37} - 36104 q^{40} + 84480 q^{46} + 4664 q^{49} - 96368 q^{52} + 126964 q^{58} - 102224 q^{61} - 14608 q^{64} - 28416 q^{70} + 110848 q^{73} + 50688 q^{76} - 240308 q^{82} + 100208 q^{85} + 349056 q^{88} - 287232 q^{94} + 229888 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 58x^{6} - 160x^{5} + 805x^{4} - 1348x^{3} + 3024x^{2} - 2376x + 972 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{7} - 7\nu^{6} + 167\nu^{5} - 400\nu^{4} + 3917\nu^{3} - 5479\nu^{2} + 21348\nu - 9774 ) / 1962 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11\nu^{7} + 16\nu^{6} + 428\nu^{5} + 1288\nu^{4} + 2087\nu^{3} + 14392\nu^{2} + 4926\nu + 17856 ) / 3924 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -11\nu^{7} - 16\nu^{6} - 428\nu^{5} - 3250\nu^{4} + 1837\nu^{3} - 83062\nu^{2} + 61782\nu - 194436 ) / 1962 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -59\nu^{7} + 370\nu^{6} - 3782\nu^{5} + 15724\nu^{4} - 58817\nu^{3} + 129094\nu^{2} - 237366\nu + 165708 ) / 3924 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -69\nu^{7} + 187\nu^{6} - 3636\nu^{5} + 6388\nu^{4} - 42105\nu^{3} + 46399\nu^{2} - 116514\nu + 79200 ) / 1962 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 50\nu^{7} - 284\nu^{6} + 3194\nu^{5} - 11744\nu^{4} + 48548\nu^{3} - 82148\nu^{2} + 142608\nu - 3024 ) / 981 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 107\nu^{7} - 429\nu^{6} + 5828\nu^{5} - 15732\nu^{4} + 67151\nu^{3} - 95673\nu^{2} + 166350\nu - 41238 ) / 1962 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{7} - \beta_{6} + 2\beta_{5} + 12\beta _1 + 18 ) / 36 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{7} + 2\beta_{6} + 11\beta_{5} + 6\beta_{4} + 2\beta_{3} + 34\beta_{2} + 18\beta _1 - 453 ) / 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -58\beta_{7} + 29\beta_{6} - 22\beta_{5} - 45\beta_{4} + 3\beta_{3} + 213\beta_{2} - 339\beta _1 - 702 ) / 36 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -223\beta_{7} - 46\beta_{6} - 361\beta_{5} - 300\beta_{4} - 100\beta_{3} - 836\beta_{2} - 900\beta _1 + 11823 ) / 36 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1754 \beta_{7} - 985 \beta_{6} + 206 \beta_{5} + 1665 \beta_{4} - 255 \beta_{3} - 9465 \beta_{2} + \cdots + 31302 ) / 36 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 9395 \beta_{7} + 734 \beta_{6} + 12245 \beta_{5} + 13218 \beta_{4} + 3542 \beta_{3} + 19174 \beta_{2} + \cdots - 365619 ) / 36 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 52234 \beta_{7} + 34973 \beta_{6} + 5330 \beta_{5} - 48195 \beta_{4} + 13293 \beta_{3} + \cdots - 1411110 ) / 36 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(19\) \(29\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1
0.500000 − 2.36350i
0.500000 + 2.36350i
0.500000 + 5.95281i
0.500000 − 5.95281i
0.500000 − 3.12438i
0.500000 + 3.12438i
0.500000 − 0.464928i
0.500000 + 0.464928i
−5.11240 − 2.42144i 0 20.2733 + 24.7587i − 49.0700i 0 12.2745i −43.6935 − 175.667i 0 −118.820 + 250.865i
35.2 −5.11240 + 2.42144i 0 20.2733 − 24.7587i 49.0700i 0 − 12.2745i −43.6935 + 175.667i 0 −118.820 − 250.865i
35.3 −3.37096 − 4.54276i 0 −9.27329 + 30.6269i − 6.64357i 0 179.715i 170.390 − 61.1157i 0 −30.1801 + 22.3952i
35.4 −3.37096 + 4.54276i 0 −9.27329 − 30.6269i 6.64357i 0 − 179.715i 170.390 + 61.1157i 0 −30.1801 − 22.3952i
35.5 3.37096 − 4.54276i 0 −9.27329 − 30.6269i − 6.64357i 0 − 179.715i −170.390 − 61.1157i 0 −30.1801 − 22.3952i
35.6 3.37096 + 4.54276i 0 −9.27329 + 30.6269i 6.64357i 0 179.715i −170.390 + 61.1157i 0 −30.1801 + 22.3952i
35.7 5.11240 − 2.42144i 0 20.2733 − 24.7587i − 49.0700i 0 − 12.2745i 43.6935 − 175.667i 0 −118.820 − 250.865i
35.8 5.11240 + 2.42144i 0 20.2733 + 24.7587i 49.0700i 0 12.2745i 43.6935 + 175.667i 0 −118.820 + 250.865i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 36.6.b.b ✓ 8
3.b odd 2 1 inner 36.6.b.b ✓ 8
4.b odd 2 1 inner 36.6.b.b ✓ 8
8.b even 2 1 576.6.c.c 8
8.d odd 2 1 576.6.c.c 8
12.b even 2 1 inner 36.6.b.b ✓ 8
24.f even 2 1 576.6.c.c 8
24.h odd 2 1 576.6.c.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.6.b.b ✓ 8 1.a even 1 1 trivial
36.6.b.b ✓ 8 3.b odd 2 1 inner
36.6.b.b ✓ 8 4.b odd 2 1 inner
36.6.b.b ✓ 8 12.b even 2 1 inner
576.6.c.c 8 8.b even 2 1
576.6.c.c 8 8.d odd 2 1
576.6.c.c 8 24.f even 2 1
576.6.c.c 8 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 2452T_{5}^{2} + 106276 \) acting on \(S_{6}^{\mathrm{new}}(36, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 22 T^{6} + \cdots + 1048576 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 2452 T^{2} + 106276)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 32448 T^{2} + 4866048)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 608640 T^{2} + 54674915328)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 64 T - 683408)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 1309252 T^{2} + 155650764676)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 25831007059968)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 24025060933632)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 464227616667364)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 128409016614912)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 6524 T + 5053444)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 55\!\cdots\!24)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 781881341902848)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 4354704211968)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 13\!\cdots\!16)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 10\!\cdots\!28)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 25556 T - 24676124)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 36\!\cdots\!12)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 65\!\cdots\!92)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 27712 T + 4035328)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 22\!\cdots\!08)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 11\!\cdots\!88)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 31\!\cdots\!96)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 57472 T - 14809518656)^{4} \) Copy content Toggle raw display
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