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Properties

Label 756.2.b.e
Level $756$
Weight $2$
Character orbit 756.b
Analytic conductor $6.037$
Analytic rank $0$
Dimension $16$
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] = [756,2,Mod(55,756)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(756, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("756.55");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.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: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{12} - 4x^{10} - 4x^{8} - 16x^{6} - 16x^{4} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{4} - \beta_{11} q^{5} + \beta_{6} q^{7} + \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{2} q^{4} - \beta_{11} q^{5} + \beta_{6} q^{7} + \beta_{3} q^{8} + ( - \beta_{15} - \beta_{5}) q^{10} - \beta_{8} q^{11} + (\beta_{8} + \beta_{5} + \beta_{3}) q^{13} + \beta_{9} q^{14} + (\beta_{12} + \beta_{11}) q^{16} + (\beta_{12} + \beta_{4}) q^{17} + (\beta_{15} - \beta_{14} + \cdots - \beta_1) q^{19}+ \cdots + (\beta_{15} - \beta_{13} - \beta_{12} + \cdots - 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - q^{14} + 4 q^{16} - 26 q^{20} + 10 q^{22} - 20 q^{25} - 6 q^{26} - 11 q^{28} - 6 q^{35} + 8 q^{37} - 20 q^{38} - 6 q^{46} - 8 q^{47} - 14 q^{49} + 21 q^{56} + 14 q^{58} - 44 q^{59} + 48 q^{62} + 24 q^{64} - 2 q^{68} - 27 q^{70} + 54 q^{80} + 4 q^{83} + 8 q^{85} - 34 q^{88} - 47 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{12} - 4x^{10} - 4x^{8} - 16x^{6} - 16x^{4} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{12} - \nu^{8} - 4\nu^{6} - 4\nu^{4} - 16\nu^{2} - 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{13} - \nu^{9} - 4\nu^{7} - 4\nu^{5} - 16\nu^{3} - 16\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{15} - 6 \nu^{14} - 4 \nu^{13} - 24 \nu^{12} - 15 \nu^{11} - 90 \nu^{10} - 56 \nu^{9} + \cdots + 1920 ) / 1536 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - \nu^{15} + 6 \nu^{14} - 4 \nu^{13} + 24 \nu^{12} - 15 \nu^{11} + 90 \nu^{10} - 56 \nu^{9} + \cdots - 1920 ) / 1536 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{15} - 4\nu^{13} - 15\nu^{11} + 8\nu^{9} + 36\nu^{7} - 32\nu^{5} + 16\nu^{3} + 320\nu ) / 256 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3 \nu^{15} - 2 \nu^{14} - 12 \nu^{13} - 8 \nu^{12} - 45 \nu^{11} - 30 \nu^{10} + 24 \nu^{9} + \cdots + 128 ) / 768 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3 \nu^{15} - 2 \nu^{14} + 12 \nu^{13} - 8 \nu^{12} + 45 \nu^{11} - 30 \nu^{10} - 24 \nu^{9} + \cdots + 128 ) / 768 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -7\nu^{14} - 4\nu^{12} - 9\nu^{10} - 32\nu^{8} - 4\nu^{6} + 192\nu^{4} + 112\nu^{2} + 448 ) / 384 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 7\nu^{14} + 4\nu^{12} + 9\nu^{10} + 32\nu^{8} + 4\nu^{6} + 192\nu^{4} - 112\nu^{2} - 448 ) / 384 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 7 \nu^{15} + 22 \nu^{14} + 28 \nu^{13} - 8 \nu^{12} + 105 \nu^{11} - 54 \nu^{10} + 8 \nu^{9} + \cdots + 128 ) / 1536 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -\nu^{15} + \nu^{11} + 4\nu^{9} + 4\nu^{7} + 16\nu^{5} + 16\nu^{3} ) / 64 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -7\nu^{15} - 16\nu^{13} - 9\nu^{11} - 20\nu^{9} + 44\nu^{7} + 240\nu^{5} + 304\nu^{3} + 640\nu ) / 384 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} + \beta_{11} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{15} + \beta_{14} - \beta_{10} + \beta_{9} - 2\beta_{8} + 2\beta_{7} + 2\beta_{6} - \beta_{5} - 2\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{13} - \beta_{12} + \beta_{10} + 2\beta_{9} + 2\beta_{7} - \beta_{6} + \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -\beta_{15} + \beta_{14} + \beta_{10} - \beta_{9} + 2\beta_{8} + 2\beta_{7} + 2\beta_{6} - \beta_{5} + \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -3\beta_{13} + \beta_{12} - 2\beta_{11} + 3\beta_{10} - 2\beta_{7} - \beta_{6} + \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 7 \beta_{15} + 7 \beta_{14} - 3 \beta_{10} + 3 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + \cdots + 3 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 5 \beta_{13} + 3 \beta_{12} + 2 \beta_{11} - \beta_{10} - 6 \beta_{9} + 2 \beta_{7} - 7 \beta_{6} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3 \beta_{15} + \beta_{14} + 5 \beta_{10} - 5 \beta_{9} - 6 \beta_{8} - 6 \beta_{7} - 6 \beta_{6} + \cdots + 13 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( \beta_{13} + \beta_{12} + 2 \beta_{11} + 7 \beta_{10} + 8 \beta_{9} + 6 \beta_{7} - 5 \beta_{6} + \cdots + 26 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 15 \beta_{15} + 15 \beta_{14} - 3 \beta_{10} + 3 \beta_{9} - 2 \beta_{8} + 18 \beta_{7} + \cdots + 27 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 19 \beta_{13} + 19 \beta_{12} - 22 \beta_{11} - 17 \beta_{10} + 2 \beta_{9} + 2 \beta_{7} + 17 \beta_{6} + \cdots + 26 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 45 \beta_{15} - 15 \beta_{14} - 19 \beta_{10} + 19 \beta_{9} - 38 \beta_{8} + 42 \beta_{7} + \cdots + 45 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/756\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
−1.41109 − 0.0939769i
−1.41109 + 0.0939769i
−1.16433 − 0.802711i
−1.16433 + 0.802711i
−0.748450 − 1.19993i
−0.748450 + 1.19993i
−0.304958 − 1.38094i
−0.304958 + 1.38094i
0.304958 − 1.38094i
0.304958 + 1.38094i
0.748450 − 1.19993i
0.748450 + 1.19993i
1.16433 − 0.802711i
1.16433 + 0.802711i
1.41109 − 0.0939769i
1.41109 + 0.0939769i
−1.41109 − 0.0939769i 0 1.98234 + 0.265219i 2.91758i 0 −1.63542 − 2.07976i −2.77233 − 0.560541i 0 0.274185 − 4.11696i
55.2 −1.41109 + 0.0939769i 0 1.98234 − 0.265219i − 2.91758i 0 −1.63542 + 2.07976i −2.77233 + 0.560541i 0 0.274185 + 4.11696i
55.3 −1.16433 − 0.802711i 0 0.711311 + 1.86923i − 0.944421i 0 2.59468 + 0.517335i 0.672257 − 2.74738i 0 −0.758097 + 1.09961i
55.4 −1.16433 + 0.802711i 0 0.711311 − 1.86923i 0.944421i 0 2.59468 − 0.517335i 0.672257 + 2.74738i 0 −0.758097 − 1.09961i
55.5 −0.748450 − 1.19993i 0 −0.879646 + 1.79617i 3.90968i 0 −1.12924 + 2.39266i 2.81364 − 0.288831i 0 4.69133 − 2.92620i
55.6 −0.748450 + 1.19993i 0 −0.879646 − 1.79617i − 3.90968i 0 −1.12924 − 2.39266i 2.81364 + 0.288831i 0 4.69133 + 2.92620i
55.7 −0.304958 − 1.38094i 0 −1.81400 + 0.842259i 0.556957i 0 1.25214 − 2.33070i 1.71630 + 2.24818i 0 0.769125 − 0.169848i
55.8 −0.304958 + 1.38094i 0 −1.81400 − 0.842259i − 0.556957i 0 1.25214 + 2.33070i 1.71630 − 2.24818i 0 0.769125 + 0.169848i
55.9 0.304958 − 1.38094i 0 −1.81400 − 0.842259i − 0.556957i 0 −1.25214 + 2.33070i −1.71630 + 2.24818i 0 −0.769125 − 0.169848i
55.10 0.304958 + 1.38094i 0 −1.81400 + 0.842259i 0.556957i 0 −1.25214 − 2.33070i −1.71630 − 2.24818i 0 −0.769125 + 0.169848i
55.11 0.748450 − 1.19993i 0 −0.879646 − 1.79617i − 3.90968i 0 1.12924 − 2.39266i −2.81364 − 0.288831i 0 −4.69133 − 2.92620i
55.12 0.748450 + 1.19993i 0 −0.879646 + 1.79617i 3.90968i 0 1.12924 + 2.39266i −2.81364 + 0.288831i 0 −4.69133 + 2.92620i
55.13 1.16433 − 0.802711i 0 0.711311 − 1.86923i 0.944421i 0 −2.59468 − 0.517335i −0.672257 − 2.74738i 0 0.758097 + 1.09961i
55.14 1.16433 + 0.802711i 0 0.711311 + 1.86923i − 0.944421i 0 −2.59468 + 0.517335i −0.672257 + 2.74738i 0 0.758097 − 1.09961i
55.15 1.41109 − 0.0939769i 0 1.98234 − 0.265219i − 2.91758i 0 1.63542 + 2.07976i 2.77233 − 0.560541i 0 −0.274185 − 4.11696i
55.16 1.41109 + 0.0939769i 0 1.98234 + 0.265219i 2.91758i 0 1.63542 − 2.07976i 2.77233 + 0.560541i 0 −0.274185 + 4.11696i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.b.e ✓ 16
3.b odd 2 1 756.2.b.f yes 16
4.b odd 2 1 756.2.b.f yes 16
7.b odd 2 1 756.2.b.f yes 16
12.b even 2 1 inner 756.2.b.e ✓ 16
21.c even 2 1 inner 756.2.b.e ✓ 16
28.d even 2 1 inner 756.2.b.e ✓ 16
84.h odd 2 1 756.2.b.f yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.b.e ✓ 16 1.a even 1 1 trivial
756.2.b.e ✓ 16 12.b even 2 1 inner
756.2.b.e ✓ 16 21.c even 2 1 inner
756.2.b.e ✓ 16 28.d even 2 1 inner
756.2.b.f yes 16 3.b odd 2 1
756.2.b.f yes 16 4.b odd 2 1
756.2.b.f yes 16 7.b odd 2 1
756.2.b.f yes 16 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(756, [\chi])\):

\( T_{5}^{8} + 25T_{5}^{6} + 159T_{5}^{4} + 163T_{5}^{2} + 36 \) Copy content Toggle raw display
\( T_{19}^{8} - 68T_{19}^{6} + 1424T_{19}^{4} - 9280T_{19}^{2} + 9216 \) Copy content Toggle raw display
\( T_{47}^{4} + 2T_{47}^{3} - 115T_{47}^{2} - 408T_{47} - 324 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - T^{12} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + 25 T^{6} + \cdots + 36)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + 7 T^{14} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{8} + 43 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 61 T^{6} + \cdots + 5184)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 81 T^{6} + \cdots + 86436)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 68 T^{6} + \cdots + 9216)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 97 T^{6} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 99 T^{6} + \cdots + 166464)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 124 T^{6} + \cdots + 11664)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 2 T^{3} + \cdots + 1152)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + 118 T^{6} + \cdots + 36864)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 185 T^{6} + \cdots + 459684)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 2 T^{3} + \cdots - 324)^{4} \) Copy content Toggle raw display
$53$ \( (T^{8} - 76 T^{6} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 11 T^{3} + \cdots - 1044)^{4} \) Copy content Toggle raw display
$61$ \( (T^{8} + 220 T^{6} + \cdots + 2985984)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 211 T^{6} + \cdots + 419904)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 149 T^{6} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 479 T^{6} + \cdots + 15116544)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 226 T^{6} + \cdots + 3779136)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - T^{3} - 157 T^{2} + \cdots + 1764)^{4} \) Copy content Toggle raw display
$89$ \( (T^{8} + 619 T^{6} + \cdots + 244859904)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 823 T^{6} + \cdots + 1035809856)^{2} \) Copy content Toggle raw display
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