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Properties

Label 1360.2.bt.b
Level $1360$
Weight $2$
Character orbit 1360.bt
Analytic conductor $10.860$
Analytic rank $0$
Dimension $8$
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] = [1360,2,Mod(81,1360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1360, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("1360.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1360 = 2^{4} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1360.bt (of order \(4\), degree \(2\), 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: \(10.8596546749\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.23045668864.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 26x^{6} + 237x^{4} + 892x^{2} + 1156 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 170)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{7} + \beta_{4}) q^{3} + \beta_{4} q^{5} + (\beta_{5} + \beta_{2} + \beta_1) q^{7} + (\beta_{4} - \beta_{3} - 4 \beta_{2} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} + \beta_{4}) q^{3} + \beta_{4} q^{5} + (\beta_{5} + \beta_{2} + \beta_1) q^{7} + (\beta_{4} - \beta_{3} - 4 \beta_{2} - \beta_1) q^{9} + (4 \beta_{3} - \beta_{2} - 1) q^{11} + (\beta_{5} - \beta_{4} - \beta_{3} - 2) q^{13} + ( - \beta_{2} - \beta_1) q^{15} + (\beta_{7} - 2 \beta_{4} + \beta_{3} + \cdots - 1) q^{17}+ \cdots + ( - 4 \beta_{7} - \beta_{5} + 14 \beta_{4} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{7} - 8 q^{11} - 12 q^{13} - 8 q^{17} + 16 q^{21} + 16 q^{23} - 12 q^{27} + 24 q^{29} - 4 q^{31} + 16 q^{33} - 20 q^{37} + 4 q^{39} - 8 q^{45} - 20 q^{47} - 4 q^{51} + 32 q^{55} + 40 q^{57} - 16 q^{61} + 64 q^{63} - 8 q^{65} + 16 q^{67} + 8 q^{69} - 4 q^{71} + 28 q^{73} - 8 q^{79} - 8 q^{81} + 8 q^{85} - 44 q^{89} + 44 q^{91} - 4 q^{95} + 20 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 26x^{6} + 237x^{4} + 892x^{2} + 1156 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{7} - 96\nu^{5} - 505\nu^{3} - 618\nu ) / 272 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{7} - 17\nu^{6} - 52\nu^{5} - 340\nu^{4} - 406\nu^{3} - 2057\nu^{2} - 900\nu - 3706 ) / 272 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{7} - 17\nu^{6} + 52\nu^{5} - 340\nu^{4} + 406\nu^{3} - 2057\nu^{2} + 900\nu - 3706 ) / 272 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} - 20\nu^{4} - 113\nu^{2} - 162 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} - 20\nu^{5} - 4\nu^{4} - 121\nu^{3} - 52\nu^{2} - 218\nu - 136 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} - 20\nu^{5} + 4\nu^{4} - 121\nu^{3} + 52\nu^{2} - 218\nu + 136 ) / 16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} - \beta_{3} - 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} + 6\beta_{2} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{7} - 2\beta_{6} - 13\beta_{5} + 13\beta_{4} + 13\beta_{3} + 57 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 15\beta_{7} + 15\beta_{6} + 25\beta_{4} - 25\beta_{3} - 82\beta_{2} + 57\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -40\beta_{7} + 40\beta_{6} + 139\beta_{5} - 147\beta_{4} - 147\beta_{3} - 511 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -187\beta_{7} - 187\beta_{6} - 379\beta_{4} + 379\beta_{3} + 914\beta_{2} - 511\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1360\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(341\) \(511\) \(817\)
\(\chi(n)\) \(-\beta_{2}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
− 1.69904i
3.26843i
2.69904i
− 2.26843i
1.69904i
− 3.26843i
− 2.69904i
2.26843i
0 −1.90851 − 1.90851i 0 −0.707107 − 0.707107i 0 2.69904 − 2.69904i 0 4.28483i 0
81.2 0 −1.60402 − 1.60402i 0 0.707107 + 0.707107i 0 −2.26843 + 2.26843i 0 2.14578i 0
81.3 0 1.20140 + 1.20140i 0 −0.707107 − 0.707107i 0 −1.69904 + 1.69904i 0 − 0.113256i 0
81.4 0 2.31113 + 2.31113i 0 0.707107 + 0.707107i 0 3.26843 − 3.26843i 0 7.68265i 0
1041.1 0 −1.90851 + 1.90851i 0 −0.707107 + 0.707107i 0 2.69904 + 2.69904i 0 − 4.28483i 0
1041.2 0 −1.60402 + 1.60402i 0 0.707107 − 0.707107i 0 −2.26843 − 2.26843i 0 − 2.14578i 0
1041.3 0 1.20140 − 1.20140i 0 −0.707107 + 0.707107i 0 −1.69904 − 1.69904i 0 0.113256i 0
1041.4 0 2.31113 − 2.31113i 0 0.707107 − 0.707107i 0 3.26843 + 3.26843i 0 − 7.68265i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1360.2.bt.b 8
4.b odd 2 1 170.2.h.b ✓ 8
12.b even 2 1 1530.2.q.g 8
17.c even 4 1 inner 1360.2.bt.b 8
20.d odd 2 1 850.2.h.n 8
20.e even 4 1 850.2.g.i 8
20.e even 4 1 850.2.g.l 8
68.f odd 4 1 170.2.h.b ✓ 8
68.g odd 8 1 2890.2.a.bd 4
68.g odd 8 1 2890.2.a.be 4
68.g odd 8 2 2890.2.b.o 8
204.l even 4 1 1530.2.q.g 8
340.i even 4 1 850.2.g.i 8
340.n odd 4 1 850.2.h.n 8
340.s even 4 1 850.2.g.l 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
170.2.h.b ✓ 8 4.b odd 2 1
170.2.h.b ✓ 8 68.f odd 4 1
850.2.g.i 8 20.e even 4 1
850.2.g.i 8 340.i even 4 1
850.2.g.l 8 20.e even 4 1
850.2.g.l 8 340.s even 4 1
850.2.h.n 8 20.d odd 2 1
850.2.h.n 8 340.n odd 4 1
1360.2.bt.b 8 1.a even 1 1 trivial
1360.2.bt.b 8 17.c even 4 1 inner
1530.2.q.g 8 12.b even 2 1
1530.2.q.g 8 204.l even 4 1
2890.2.a.bd 4 68.g odd 8 1
2890.2.a.be 4 68.g odd 8 1
2890.2.b.o 8 68.g odd 8 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 4T_{3}^{5} + 101T_{3}^{4} + 52T_{3}^{3} + 8T_{3}^{2} - 136T_{3} + 1156 \) acting on \(S_{2}^{\mathrm{new}}(1360, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 4 T^{5} + \cdots + 1156 \) Copy content Toggle raw display
$5$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 4 T^{7} + \cdots + 18496 \) Copy content Toggle raw display
$11$ \( (T^{4} + 4 T^{3} + \cdots + 196)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 6 T^{3} - 3 T^{2} + \cdots - 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 8 T^{7} + \cdots + 83521 \) Copy content Toggle raw display
$19$ \( T^{8} + 86 T^{6} + \cdots + 50176 \) Copy content Toggle raw display
$23$ \( (T^{4} - 8 T^{3} + 32 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - 24 T^{7} + \cdots + 4624 \) Copy content Toggle raw display
$31$ \( T^{8} + 4 T^{7} + \cdots + 4624 \) Copy content Toggle raw display
$37$ \( T^{8} + 20 T^{7} + \cdots + 506944 \) Copy content Toggle raw display
$41$ \( T^{8} + 5256 T^{4} + 5626384 \) Copy content Toggle raw display
$43$ \( T^{8} + 116 T^{6} + \cdots + 3136 \) Copy content Toggle raw display
$47$ \( (T^{4} + 10 T^{3} + \cdots - 392)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 58 T^{6} + \cdots + 784 \) Copy content Toggle raw display
$59$ \( T^{8} + 222 T^{6} + \cdots + 80656 \) Copy content Toggle raw display
$61$ \( T^{8} + 16 T^{7} + \cdots + 1547536 \) Copy content Toggle raw display
$67$ \( (T^{4} - 8 T^{3} + \cdots + 752)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 4 T^{7} + \cdots + 1817104 \) Copy content Toggle raw display
$73$ \( T^{8} - 28 T^{7} + \cdots + 30316036 \) Copy content Toggle raw display
$79$ \( T^{8} + 8 T^{7} + \cdots + 3625216 \) Copy content Toggle raw display
$83$ \( T^{8} + 364 T^{6} + \cdots + 23116864 \) Copy content Toggle raw display
$89$ \( (T^{4} + 22 T^{3} + \cdots - 32368)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 20 T^{7} + \cdots + 15984004 \) Copy content Toggle raw display
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