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Properties

Label 36.4.h.a
Level $36$
Weight $4$
Character orbit 36.h
Analytic conductor $2.124$
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] = [36,4,Mod(11,36)]
 
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([3, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("36.11");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 36.h (of order \(6\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.553553856144.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 11x^{6} + 96x^{4} + 704x^{2} + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} + (\beta_{5} - \beta_{2}) q^{3} + (\beta_{6} - 2 \beta_{4} + \cdots + 2 \beta_{2}) q^{4}+ \cdots + ( - 3 \beta_{7} - 6 \beta_{6} + \cdots + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} + (\beta_{5} - \beta_{2}) q^{3} + (\beta_{6} - 2 \beta_{4} + \cdots + 2 \beta_{2}) q^{4}+ \cdots + (243 \beta_{7} + 270 \beta_{6} + \cdots - 270 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 3 q^{2} + 11 q^{4} + 66 q^{5} - 39 q^{6} + 90 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 3 q^{2} + 11 q^{4} + 66 q^{5} - 39 q^{6} + 90 q^{9} - 116 q^{10} - 222 q^{12} + 214 q^{13} - 42 q^{14} + 71 q^{16} - 396 q^{18} + 306 q^{20} - 510 q^{21} + 207 q^{22} + 171 q^{24} - 54 q^{25} + 540 q^{28} - 498 q^{29} - 222 q^{30} + 327 q^{32} + 840 q^{33} + 469 q^{34} + 747 q^{36} - 1256 q^{37} + 1035 q^{38} - 602 q^{40} - 1272 q^{41} + 1020 q^{42} + 990 q^{45} - 912 q^{46} - 1083 q^{48} - 154 q^{49} - 1329 q^{50} - 464 q^{52} - 405 q^{54} - 1314 q^{56} + 2070 q^{57} - 830 q^{58} - 1110 q^{60} + 262 q^{61} - 550 q^{64} + 3282 q^{65} + 2046 q^{66} - 843 q^{68} - 1398 q^{69} - 480 q^{70} + 1251 q^{72} + 3940 q^{73} + 222 q^{74} + 105 q^{76} + 330 q^{77} - 1176 q^{78} - 1566 q^{81} + 4786 q^{82} - 978 q^{84} - 472 q^{85} + 1209 q^{86} - 1425 q^{88} - 3546 q^{90} - 1308 q^{92} - 5178 q^{93} + 1356 q^{94} + 1740 q^{96} - 572 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 11x^{6} + 96x^{4} + 704x^{2} + 4096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - 5\nu^{5} - \nu^{4} - 15\nu^{3} + 28\nu^{2} - 40\nu + 128 ) / 320 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} - 19\nu^{4} - 88\nu^{2} - 160\nu - 768 ) / 320 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{7} - 17\nu^{5} + 16\nu^{3} - 704\nu - 2560 ) / 5120 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{7} + 8\nu^{6} + 25\nu^{5} + 152\nu^{4} - 240\nu^{3} + 1984\nu^{2} - 320\nu + 8704 ) / 5120 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{7} - 24\nu^{6} + 55\nu^{5} - 136\nu^{4} + 480\nu^{3} + 128\nu^{2} + 3520\nu - 5632 ) / 5120 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{7} + 24\nu^{6} + 55\nu^{5} + 136\nu^{4} + 480\nu^{3} - 128\nu^{2} + 3520\nu + 5632 ) / 5120 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{6} + 2\beta_{5} + 2\beta_{2} - \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{7} + 6\beta_{6} - 2\beta_{5} + 20\beta_{4} - 2\beta_{3} + 2\beta_{2} - 5\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{7} - 6\beta_{6} - 14\beta_{5} - 24\beta_{3} - 14\beta_{2} - 5\beta _1 - 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 20\beta_{7} - 18\beta_{6} + 38\beta_{5} - 60\beta_{4} + 38\beta_{3} - 38\beta_{2} + 7\beta _1 - 30 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 152\beta_{7} - 62\beta_{6} + 90\beta_{5} + 136\beta_{3} + 90\beta_{2} + 23\beta _1 - 98 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -92\beta_{7} + 134\beta_{6} - 226\beta_{5} - 1260\beta_{4} - 226\beta_{3} + 226\beta_{2} - 301\beta _1 - 630 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(19\) \(29\)
\(\chi(n)\) \(-1\) \(-\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−0.807795 + 2.71062i
0.807795 + 2.71062i
−2.14417 − 1.84460i
2.14417 − 1.84460i
−0.807795 − 2.71062i
0.807795 − 2.71062i
−2.14417 + 1.84460i
2.14417 + 1.84460i
−2.75136 − 0.655739i −5.00424 + 1.39914i 7.14001 + 3.60836i 12.1949 − 7.04075i 14.6860 − 0.568075i 21.1499 + 12.2109i −17.2786 − 14.6099i 23.0848 − 14.0033i −38.1696 + 11.3750i
11.2 −1.94357 − 2.05488i 5.00424 − 1.39914i −0.445079 + 7.98761i 12.1949 − 7.04075i −12.6012 − 7.56379i −21.1499 − 12.2109i 17.2786 − 14.6099i 23.0848 − 14.0033i −38.1696 − 11.3750i
11.3 0.525382 + 2.77920i 3.63423 + 3.71381i −7.44795 + 2.92028i 4.30507 − 2.48553i −8.41209 + 12.0514i −3.07102 − 1.77306i −12.0291 − 19.1651i −0.584800 + 26.9937i 9.16960 + 10.6588i
11.4 2.66955 − 0.934608i −3.63423 − 3.71381i 6.25302 − 4.98997i 4.30507 − 2.48553i −13.1727 − 6.51764i 3.07102 + 1.77306i 12.0291 − 19.1651i −0.584800 + 26.9937i 9.16960 − 10.6588i
23.1 −2.75136 + 0.655739i −5.00424 − 1.39914i 7.14001 − 3.60836i 12.1949 + 7.04075i 14.6860 + 0.568075i 21.1499 − 12.2109i −17.2786 + 14.6099i 23.0848 + 14.0033i −38.1696 − 11.3750i
23.2 −1.94357 + 2.05488i 5.00424 + 1.39914i −0.445079 − 7.98761i 12.1949 + 7.04075i −12.6012 + 7.56379i −21.1499 + 12.2109i 17.2786 + 14.6099i 23.0848 + 14.0033i −38.1696 + 11.3750i
23.3 0.525382 − 2.77920i 3.63423 − 3.71381i −7.44795 − 2.92028i 4.30507 + 2.48553i −8.41209 − 12.0514i −3.07102 + 1.77306i −12.0291 + 19.1651i −0.584800 − 26.9937i 9.16960 − 10.6588i
23.4 2.66955 + 0.934608i −3.63423 + 3.71381i 6.25302 + 4.98997i 4.30507 + 2.48553i −13.1727 + 6.51764i 3.07102 − 1.77306i 12.0291 + 19.1651i −0.584800 − 26.9937i 9.16960 + 10.6588i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.d odd 6 1 inner
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 36.4.h.a ✓ 8
3.b odd 2 1 108.4.h.a 8
4.b odd 2 1 inner 36.4.h.a ✓ 8
9.c even 3 1 108.4.h.a 8
9.c even 3 1 324.4.b.b 8
9.d odd 6 1 inner 36.4.h.a ✓ 8
9.d odd 6 1 324.4.b.b 8
12.b even 2 1 108.4.h.a 8
36.f odd 6 1 108.4.h.a 8
36.f odd 6 1 324.4.b.b 8
36.h even 6 1 inner 36.4.h.a ✓ 8
36.h even 6 1 324.4.b.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.4.h.a ✓ 8 1.a even 1 1 trivial
36.4.h.a ✓ 8 4.b odd 2 1 inner
36.4.h.a ✓ 8 9.d odd 6 1 inner
36.4.h.a ✓ 8 36.h even 6 1 inner
108.4.h.a 8 3.b odd 2 1
108.4.h.a 8 9.c even 3 1
108.4.h.a 8 12.b even 2 1
108.4.h.a 8 36.f odd 6 1
324.4.b.b 8 9.c even 3 1
324.4.b.b 8 9.d odd 6 1
324.4.b.b 8 36.f odd 6 1
324.4.b.b 8 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 33T_{5}^{3} + 433T_{5}^{2} - 2310T_{5} + 4900 \) acting on \(S_{4}^{\mathrm{new}}(36, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 3 T^{7} + \cdots + 4096 \) Copy content Toggle raw display
$3$ \( T^{8} - 45 T^{6} + \cdots + 531441 \) Copy content Toggle raw display
$5$ \( (T^{4} - 33 T^{3} + \cdots + 4900)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 609 T^{6} + \cdots + 56250000 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 44383955625 \) Copy content Toggle raw display
$13$ \( (T^{4} - 107 T^{3} + \cdots + 7840000)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 10327 T^{2} + 3422500)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 27675 T^{2} + 149107500)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 732895782914304 \) Copy content Toggle raw display
$29$ \( (T^{4} + 249 T^{3} + \cdots + 33756100)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{2} + 314 T + 24400)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 636 T^{3} + \cdots + 330039889)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 94\!\cdots\!25 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 94\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( (T^{4} + 231628 T^{2} + 12418873600)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 22\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( (T^{4} - 131 T^{3} + \cdots + 95797678144)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 44\!\cdots\!29 \) Copy content Toggle raw display
$71$ \( (T^{4} - 171216 T^{2} + 7279642800)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 985 T - 207200)^{4} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{4} + 125260 T^{2} + 634233856)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 286 T^{3} + \cdots + 256476409225)^{2} \) Copy content Toggle raw display
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