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Properties

Label 2775.1.bi.a
Level $2775$
Weight $1$
Character orbit 2775.bi
Analytic conductor $1.385$
Analytic rank $0$
Dimension $32$
Projective image $D_{40}$
CM discriminant -111
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2775,1,Mod(554,2775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2775, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 1, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2775.554");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2775 = 3 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2775.bi (of order \(10\), degree \(4\), 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: \(1.38490541006\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{80})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{32} - x^{24} + x^{16} - x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{40}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{40} + \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + ( - \zeta_{80}^{31} + \zeta_{80}^{17}) q^{2} + \zeta_{80}^{28} q^{3} + (\zeta_{80}^{34} + \cdots + \zeta_{80}^{8}) q^{4}+ \cdots - \zeta_{80}^{16} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{80}^{31} + \zeta_{80}^{17}) q^{2} + \zeta_{80}^{28} q^{3} + (\zeta_{80}^{34} + \cdots + \zeta_{80}^{8}) q^{4}+ \cdots + (\zeta_{80}^{31} - \zeta_{80}^{17}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 8 q^{4} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 32 q + 8 q^{4} + 8 q^{9} - 8 q^{16} - 32 q^{30} - 16 q^{34} - 8 q^{36} + 8 q^{40} - 32 q^{49} + 8 q^{64} + 8 q^{70} - 8 q^{81} + 24 q^{84}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2775\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(926\) \(1777\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{80}^{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
554.1
−0.760406 + 0.649448i
−0.0784591 + 0.996917i
−0.996917 − 0.0784591i
0.649448 + 0.760406i
−0.649448 − 0.760406i
0.996917 + 0.0784591i
0.0784591 − 0.996917i
0.760406 − 0.649448i
0.972370 − 0.233445i
0.522499 − 0.852640i
0.852640 + 0.522499i
−0.233445 − 0.972370i
0.233445 + 0.972370i
−0.852640 − 0.522499i
−0.522499 + 0.852640i
−0.972370 + 0.233445i
0.972370 + 0.233445i
0.522499 + 0.852640i
0.852640 − 0.522499i
−0.233445 + 0.972370i
−1.84956 − 0.600958i 0.587785 − 0.809017i 2.25070 + 1.63523i 0.649448 + 0.760406i −1.57333 + 1.14309i 1.41421i −2.03700 − 2.80369i −0.309017 − 0.951057i −0.744220 − 1.79671i
554.2 −1.62182 − 0.526961i −0.587785 + 0.809017i 1.54359 + 1.12148i −0.996917 − 0.0784591i 1.37960 − 1.00234i 1.41421i −0.910104 − 1.25265i −0.309017 − 0.951057i 1.57547 + 0.652583i
554.3 −0.993851 − 0.322922i −0.587785 + 0.809017i 0.0744449 + 0.0540874i 0.0784591 − 0.996917i 0.845420 − 0.614234i − 1.41421i 0.557713 + 0.767626i −0.309017 − 0.951057i −0.399903 + 0.965451i
554.4 −0.444039 − 0.144277i 0.587785 − 0.809017i −0.632662 − 0.459656i 0.760406 − 0.649448i −0.377723 + 0.274431i − 1.41421i 0.489040 + 0.673106i −0.309017 − 0.951057i −0.431351 + 0.178671i
554.5 0.444039 + 0.144277i 0.587785 − 0.809017i −0.632662 − 0.459656i −0.760406 + 0.649448i 0.377723 − 0.274431i − 1.41421i −0.489040 − 0.673106i −0.309017 − 0.951057i −0.431351 + 0.178671i
554.6 0.993851 + 0.322922i −0.587785 + 0.809017i 0.0744449 + 0.0540874i −0.0784591 + 0.996917i −0.845420 + 0.614234i − 1.41421i −0.557713 − 0.767626i −0.309017 − 0.951057i −0.399903 + 0.965451i
554.7 1.62182 + 0.526961i −0.587785 + 0.809017i 1.54359 + 1.12148i 0.996917 + 0.0784591i −1.37960 + 1.00234i 1.41421i 0.910104 + 1.25265i −0.309017 − 0.951057i 1.57547 + 0.652583i
554.8 1.84956 + 0.600958i 0.587785 − 0.809017i 2.25070 + 1.63523i −0.649448 − 0.760406i 1.57333 − 1.14309i 1.41421i 2.03700 + 2.80369i −0.309017 − 0.951057i −0.744220 − 1.79671i
1109.1 −1.17195 + 1.61305i 0.951057 − 0.309017i −0.919442 − 2.82975i 0.233445 + 0.972370i −0.616129 + 1.89625i 1.41421i 3.74581 + 1.21709i 0.809017 − 0.587785i −1.84206 − 0.763007i
1109.2 −0.893911 + 1.23036i −0.951057 + 0.309017i −0.405699 − 1.24861i −0.852640 − 0.522499i 0.469957 − 1.44638i − 1.41421i 0.452527 + 0.147035i 0.809017 − 0.587785i 1.40505 − 0.581990i
1109.3 −0.763472 + 1.05083i −0.951057 + 0.309017i −0.212335 − 0.653500i 0.522499 − 0.852640i 0.401381 − 1.23532i 1.41421i −0.386494 − 0.125580i 0.809017 − 0.587785i 0.497066 + 1.20002i
1109.4 −0.0922342 + 0.126949i 0.951057 − 0.309017i 0.301408 + 0.927638i 0.972370 − 0.233445i −0.0484904 + 0.149238i − 1.41421i −0.294801 − 0.0957868i 0.809017 − 0.587785i −0.0600500 + 0.144974i
1109.5 0.0922342 − 0.126949i 0.951057 − 0.309017i 0.301408 + 0.927638i −0.972370 + 0.233445i 0.0484904 − 0.149238i − 1.41421i 0.294801 + 0.0957868i 0.809017 − 0.587785i −0.0600500 + 0.144974i
1109.6 0.763472 − 1.05083i −0.951057 + 0.309017i −0.212335 − 0.653500i −0.522499 + 0.852640i −0.401381 + 1.23532i 1.41421i 0.386494 + 0.125580i 0.809017 − 0.587785i 0.497066 + 1.20002i
1109.7 0.893911 − 1.23036i −0.951057 + 0.309017i −0.405699 − 1.24861i 0.852640 + 0.522499i −0.469957 + 1.44638i − 1.41421i −0.452527 − 0.147035i 0.809017 − 0.587785i 1.40505 − 0.581990i
1109.8 1.17195 − 1.61305i 0.951057 − 0.309017i −0.919442 − 2.82975i −0.233445 − 0.972370i 0.616129 − 1.89625i 1.41421i −3.74581 − 1.21709i 0.809017 − 0.587785i −1.84206 − 0.763007i
1664.1 −1.17195 − 1.61305i 0.951057 + 0.309017i −0.919442 + 2.82975i 0.233445 − 0.972370i −0.616129 − 1.89625i − 1.41421i 3.74581 − 1.21709i 0.809017 + 0.587785i −1.84206 + 0.763007i
1664.2 −0.893911 − 1.23036i −0.951057 − 0.309017i −0.405699 + 1.24861i −0.852640 + 0.522499i 0.469957 + 1.44638i 1.41421i 0.452527 − 0.147035i 0.809017 + 0.587785i 1.40505 + 0.581990i
1664.3 −0.763472 − 1.05083i −0.951057 − 0.309017i −0.212335 + 0.653500i 0.522499 + 0.852640i 0.401381 + 1.23532i − 1.41421i −0.386494 + 0.125580i 0.809017 + 0.587785i 0.497066 − 1.20002i
1664.4 −0.0922342 − 0.126949i 0.951057 + 0.309017i 0.301408 − 0.927638i 0.972370 + 0.233445i −0.0484904 − 0.149238i 1.41421i −0.294801 + 0.0957868i 0.809017 + 0.587785i −0.0600500 − 0.144974i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 554.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
111.d odd 2 1 CM by \(\Q(\sqrt{-111}) \)
3.b odd 2 1 inner
25.e even 10 1 inner
37.b even 2 1 inner
75.h odd 10 1 inner
925.q even 10 1 inner
2775.bi odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2775.1.bi.a ✓ 32
3.b odd 2 1 inner 2775.1.bi.a ✓ 32
25.e even 10 1 inner 2775.1.bi.a ✓ 32
37.b even 2 1 inner 2775.1.bi.a ✓ 32
75.h odd 10 1 inner 2775.1.bi.a ✓ 32
111.d odd 2 1 CM 2775.1.bi.a ✓ 32
925.q even 10 1 inner 2775.1.bi.a ✓ 32
2775.bi odd 10 1 inner 2775.1.bi.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2775.1.bi.a ✓ 32 1.a even 1 1 trivial
2775.1.bi.a ✓ 32 3.b odd 2 1 inner
2775.1.bi.a ✓ 32 25.e even 10 1 inner
2775.1.bi.a ✓ 32 37.b even 2 1 inner
2775.1.bi.a ✓ 32 75.h odd 10 1 inner
2775.1.bi.a ✓ 32 111.d odd 2 1 CM
2775.1.bi.a ✓ 32 925.q even 10 1 inner
2775.1.bi.a ✓ 32 2775.bi odd 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2775, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{32} - 8 T^{30} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{32} - T^{24} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( (T^{2} + 2)^{16} \) Copy content Toggle raw display
$11$ \( T^{32} \) Copy content Toggle raw display
$13$ \( T^{32} \) Copy content Toggle raw display
$17$ \( (T^{16} - 4 T^{14} + \cdots + 16)^{2} \) Copy content Toggle raw display
$19$ \( T^{32} \) Copy content Toggle raw display
$23$ \( T^{32} - 8 T^{30} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( T^{32} + 8 T^{30} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{32} \) Copy content Toggle raw display
$37$ \( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{4} \) Copy content Toggle raw display
$41$ \( T^{32} \) Copy content Toggle raw display
$43$ \( T^{32} \) Copy content Toggle raw display
$47$ \( T^{32} \) Copy content Toggle raw display
$53$ \( T^{32} \) Copy content Toggle raw display
$59$ \( T^{32} + 8 T^{30} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{32} \) Copy content Toggle raw display
$67$ \( (T^{16} - 4 T^{14} + \cdots + 1)^{2} \) Copy content Toggle raw display
$71$ \( T^{32} \) Copy content Toggle raw display
$73$ \( (T^{16} - 4 T^{14} + \cdots + 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{32} \) Copy content Toggle raw display
$83$ \( T^{32} \) Copy content Toggle raw display
$89$ \( T^{32} + 8 T^{30} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{32} \) Copy content Toggle raw display
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