Properties

Label 1156.2.h.e
Level $1156$
Weight $2$
Character orbit 1156.h
Analytic conductor $9.231$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1156,2,Mod(733,1156)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1156, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1156.733");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1156 = 2^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1156.h (of order \(8\), degree \(4\), not 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: \(9.23070647366\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{8})\)
Coefficient field: 16.0.229607785695641627262976.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 799x^{8} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 68)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{10} q^{3} - \beta_{8} q^{5} - \beta_{15} q^{7} + (\beta_{11} + 4 \beta_{5}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{10} q^{3} - \beta_{8} q^{5} - \beta_{15} q^{7} + (\beta_{11} + 4 \beta_{5}) q^{9} + ( - \beta_{14} - 2 \beta_{12}) q^{11} + (\beta_{6} - \beta_{4}) q^{13} + (\beta_{3} + \beta_1) q^{15} + (\beta_{3} - 3 \beta_1) q^{19} + (\beta_{6} + 7 \beta_{4}) q^{21} + (\beta_{14} - 2 \beta_{12}) q^{23} + 3 \beta_{5} q^{25} + (2 \beta_{15} - 6 \beta_{13}) q^{27} + ( - 2 \beta_{9} - 3 \beta_{8}) q^{29} + ( - \beta_{10} + 2 \beta_{7}) q^{31} + ( - \beta_{2} - 5) q^{33} + (\beta_{2} - 1) q^{35} - 3 \beta_{7} q^{37} - 6 \beta_{8} q^{39} + \beta_{13} q^{41} + ( - \beta_{11} + 7 \beta_{5}) q^{43} + (2 \beta_{14} - 3 \beta_{12}) q^{45} - 4 \beta_{4} q^{47} - \beta_{3} q^{49} + ( - 2 \beta_{3} + 2 \beta_1) q^{53} + ( - \beta_{6} + 3 \beta_{4}) q^{55} + ( - 2 \beta_{14} - 6 \beta_{12}) q^{57} + ( - \beta_{11} - 5 \beta_{5}) q^{59} + ( - 4 \beta_{15} - 3 \beta_{13}) q^{61} + (5 \beta_{9} - 6 \beta_{8}) q^{63} + (2 \beta_{10} + 2 \beta_{7}) q^{65} + (2 \beta_{2} - 2) q^{67} + ( - 3 \beta_{2} + 9) q^{69} + (3 \beta_{10} + 4 \beta_{7}) q^{71} - 7 \beta_{8} q^{73} + 3 \beta_{15} q^{75} + ( - \beta_{11} + 5 \beta_{5}) q^{77} + ( - \beta_{14} - 2 \beta_{12}) q^{79} + ( - 5 \beta_{6} - 8 \beta_{4}) q^{81} + ( - 3 \beta_{3} - 7 \beta_1) q^{83} + (\beta_{3} - 11 \beta_1) q^{87} + ( - 3 \beta_{6} + 3 \beta_{4}) q^{89} + 6 \beta_{12} q^{91} + (3 \beta_{11} + 9 \beta_{5}) q^{93} + ( - 2 \beta_{15} - 4 \beta_{13}) q^{95} + ( - 2 \beta_{9} + 3 \beta_{8}) q^{97} + \beta_{10} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 80 q^{33} - 16 q^{35} - 32 q^{67} + 144 q^{69}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 799x^{8} + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{10} + 1159\nu^{2} ) / 1953 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{8} + 799 ) / 217 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{10} - 601\nu^{2} ) / 279 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{12} + 880\nu^{4} ) / 2511 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -40\nu^{14} - 32689\nu^{6} ) / 158193 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{12} - 718\nu^{4} ) / 567 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19\nu^{13} - 81\nu^{9} + 14209\nu^{5} - 41148\nu ) / 52731 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 19\nu^{13} + 81\nu^{9} + 14209\nu^{5} + 41148\nu ) / 52731 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{13} + 880\nu^{5} + 2511\nu ) / 2511 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{13} + 880\nu^{5} - 2511\nu ) / 2511 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 22\nu^{14} + 16849\nu^{6} ) / 22599 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -97\nu^{15} + 324\nu^{11} - 75316\nu^{7} + 217323\nu^{3} ) / 474579 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -97\nu^{15} - 324\nu^{11} - 75316\nu^{7} - 217323\nu^{3} ) / 474579 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -40\nu^{15} + 81\nu^{11} - 32689\nu^{7} + 93879\nu^{3} ) / 158193 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -40\nu^{15} - 81\nu^{11} - 32689\nu^{7} - 93879\nu^{3} ) / 158193 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{10} + \beta_{9} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 7\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{15} + 4\beta_{14} + 3\beta_{13} - 3\beta_{12} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 7\beta_{6} + 31\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 19\beta_{10} + 19\beta_{9} - 21\beta_{8} - 21\beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -20\beta_{11} - 77\beta_{5} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -97\beta_{15} - 97\beta_{14} + 120\beta_{13} + 120\beta_{12} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 217\beta_{2} - 799 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 508\beta_{10} - 508\beta_{9} + 651\beta_{8} - 651\beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -1159\beta_{3} - 4207\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 2683\beta_{15} - 2683\beta_{14} - 3477\beta_{13} + 3477\beta_{12} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -3080\beta_{6} - 11129\beta_{4} \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( -14209\beta_{10} - 14209\beta_{9} + 18480\beta_{8} + 18480\beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 32689\beta_{11} + 117943\beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 75316\beta_{15} + 75316\beta_{14} - 98067\beta_{13} - 98067\beta_{12} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1156\mathbb{Z}\right)^\times\).

\(n\) \(579\) \(581\)
\(\chi(n)\) \(1\) \(\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
733.1
−2.12749 0.881234i
−1.20361 0.498551i
1.20361 + 0.498551i
2.12749 + 0.881234i
−2.12749 + 0.881234i
−1.20361 + 0.498551i
1.20361 0.498551i
2.12749 0.881234i
0.881234 + 2.12749i
0.498551 + 1.20361i
−0.498551 1.20361i
−0.881234 2.12749i
0.881234 2.12749i
0.498551 1.20361i
−0.498551 + 1.20361i
−0.881234 + 2.12749i
0 −3.00872 + 1.24625i 0 −0.541196 1.30656i 0 1.24625 3.00872i 0 5.37794 5.37794i 0
733.2 0 −1.70216 + 0.705057i 0 0.541196 + 1.30656i 0 0.705057 1.70216i 0 0.278917 0.278917i 0
733.3 0 1.70216 0.705057i 0 −0.541196 1.30656i 0 −0.705057 + 1.70216i 0 0.278917 0.278917i 0
733.4 0 3.00872 1.24625i 0 0.541196 + 1.30656i 0 −1.24625 + 3.00872i 0 5.37794 5.37794i 0
757.1 0 −3.00872 1.24625i 0 −0.541196 + 1.30656i 0 1.24625 + 3.00872i 0 5.37794 + 5.37794i 0
757.2 0 −1.70216 0.705057i 0 0.541196 1.30656i 0 0.705057 + 1.70216i 0 0.278917 + 0.278917i 0
757.3 0 1.70216 + 0.705057i 0 −0.541196 + 1.30656i 0 −0.705057 1.70216i 0 0.278917 + 0.278917i 0
757.4 0 3.00872 + 1.24625i 0 0.541196 1.30656i 0 −1.24625 3.00872i 0 5.37794 + 5.37794i 0
977.1 0 −1.24625 + 3.00872i 0 1.30656 + 0.541196i 0 −3.00872 + 1.24625i 0 −5.37794 5.37794i 0
977.2 0 −0.705057 + 1.70216i 0 −1.30656 0.541196i 0 −1.70216 + 0.705057i 0 −0.278917 0.278917i 0
977.3 0 0.705057 1.70216i 0 1.30656 + 0.541196i 0 1.70216 0.705057i 0 −0.278917 0.278917i 0
977.4 0 1.24625 3.00872i 0 −1.30656 0.541196i 0 3.00872 1.24625i 0 −5.37794 5.37794i 0
1001.1 0 −1.24625 3.00872i 0 1.30656 0.541196i 0 −3.00872 1.24625i 0 −5.37794 + 5.37794i 0
1001.2 0 −0.705057 1.70216i 0 −1.30656 + 0.541196i 0 −1.70216 0.705057i 0 −0.278917 + 0.278917i 0
1001.3 0 0.705057 + 1.70216i 0 1.30656 0.541196i 0 1.70216 + 0.705057i 0 −0.278917 + 0.278917i 0
1001.4 0 1.24625 + 3.00872i 0 −1.30656 + 0.541196i 0 3.00872 + 1.24625i 0 −5.37794 + 5.37794i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 733.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner
17.c even 4 2 inner
17.d even 8 4 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1156.2.h.e 16
17.b even 2 1 inner 1156.2.h.e 16
17.c even 4 2 inner 1156.2.h.e 16
17.d even 8 4 inner 1156.2.h.e 16
17.e odd 16 2 68.2.e.a 4
17.e odd 16 2 1156.2.a.h 4
17.e odd 16 2 1156.2.b.a 4
17.e odd 16 2 1156.2.e.c 4
51.i even 16 2 612.2.k.e 4
68.i even 16 2 272.2.o.g 4
68.i even 16 2 4624.2.a.bq 4
85.o even 16 1 1700.2.m.a 4
85.o even 16 1 1700.2.m.b 4
85.p odd 16 2 1700.2.o.c 4
85.r even 16 1 1700.2.m.a 4
85.r even 16 1 1700.2.m.b 4
136.q odd 16 2 1088.2.o.t 4
136.s even 16 2 1088.2.o.s 4
204.t odd 16 2 2448.2.be.u 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.2.e.a 4 17.e odd 16 2
272.2.o.g 4 68.i even 16 2
612.2.k.e 4 51.i even 16 2
1088.2.o.s 4 136.s even 16 2
1088.2.o.t 4 136.q odd 16 2
1156.2.a.h 4 17.e odd 16 2
1156.2.b.a 4 17.e odd 16 2
1156.2.e.c 4 17.e odd 16 2
1156.2.h.e 16 1.a even 1 1 trivial
1156.2.h.e 16 17.b even 2 1 inner
1156.2.h.e 16 17.c even 4 2 inner
1156.2.h.e 16 17.d even 8 4 inner
1700.2.m.a 4 85.o even 16 1
1700.2.m.a 4 85.r even 16 1
1700.2.m.b 4 85.o even 16 1
1700.2.m.b 4 85.r even 16 1
1700.2.o.c 4 85.p odd 16 2
2448.2.be.u 4 204.t odd 16 2
4624.2.a.bq 4 68.i even 16 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 12784T_{3}^{8} + 1679616 \) acting on \(S_{2}^{\mathrm{new}}(1156, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 12784 T^{8} + \cdots + 1679616 \) Copy content Toggle raw display
$5$ \( (T^{8} + 16)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + 12784 T^{8} + \cdots + 1679616 \) Copy content Toggle raw display
$11$ \( T^{16} + 226544 T^{8} + 256 \) Copy content Toggle raw display
$13$ \( (T^{4} + 28 T^{2} + 144)^{4} \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( (T^{8} + 1904 T^{4} + 256)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 1879792 T^{8} + \cdots + 1679616 \) Copy content Toggle raw display
$29$ \( T^{16} + 15598624 T^{8} + \cdots + 11019960576 \) Copy content Toggle raw display
$31$ \( T^{16} + 1879792 T^{8} + \cdots + 1679616 \) Copy content Toggle raw display
$37$ \( (T^{8} + 104976)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 16)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 12784 T^{4} + 1679616)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 16)^{8} \) Copy content Toggle raw display
$53$ \( (T^{8} + 7936 T^{4} + 5308416)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 5488 T^{4} + 20736)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + 366060064 T^{8} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( (T^{2} + 4 T - 48)^{8} \) Copy content Toggle raw display
$71$ \( T^{16} + 244873712 T^{8} + \cdots + 20047612231936 \) Copy content Toggle raw display
$73$ \( (T^{8} + 92236816)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + 226544 T^{8} + 256 \) Copy content Toggle raw display
$83$ \( (T^{8} + 100976 T^{4} + \cdots + 21381376)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 252 T^{2} + 11664)^{4} \) Copy content Toggle raw display
$97$ \( T^{16} + 179128864 T^{8} + \cdots + 1679616 \) Copy content Toggle raw display
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