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Properties

Label 2601.1.x.a
Level $2601$
Weight $1$
Character orbit 2601.x
Analytic conductor $1.298$
Analytic rank $0$
Dimension $32$
Projective image $A_{4}$
CM/RM no
Inner twists $32$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2601,1,Mod(40,2601)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2601, base_ring=CyclotomicField(48))
 
chi = DirichletCharacter(H, H._module([16, 45]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2601.40");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2601 = 3^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2601.x (of order \(48\), degree \(16\), 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: \(1.29806809786\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(2\) over \(\Q(\zeta_{48})\)
Coefficient field: \(\Q(\zeta_{96})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{32} - x^{16} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.23409.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{96}^{38} q^{2} - \zeta_{96}^{21} q^{3} + \zeta_{96}^{25} q^{5} + \zeta_{96}^{11} q^{6} - \zeta_{96}^{23} q^{7} - \zeta_{96}^{18} q^{8} + \zeta_{96}^{42} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{96}^{38} q^{2} - \zeta_{96}^{21} q^{3} + \zeta_{96}^{25} q^{5} + \zeta_{96}^{11} q^{6} - \zeta_{96}^{23} q^{7} - \zeta_{96}^{18} q^{8} + \zeta_{96}^{42} q^{9} - \zeta_{96}^{15} q^{10} + \zeta_{96}^{35} q^{11} + \zeta_{96}^{4} q^{13} + \zeta_{96}^{13} q^{14} - \zeta_{96}^{46} q^{15} + \zeta_{96}^{8} q^{16} - \zeta_{96}^{32} q^{18} + \zeta_{96}^{44} q^{21} - \zeta_{96}^{25} q^{22} + \zeta_{96}^{43} q^{23} + \zeta_{96}^{39} q^{24} + \zeta_{96}^{42} q^{26} + \zeta_{96}^{15} q^{27} - \zeta_{96}^{17} q^{29} + \zeta_{96}^{36} q^{30} - \zeta_{96}^{13} q^{31} + \zeta_{96}^{8} q^{33} + q^{35} + 2 \zeta_{96}^{21} q^{37} - \zeta_{96}^{25} q^{39} - \zeta_{96}^{43} q^{40} + \zeta_{96}^{7} q^{41} - \zeta_{96}^{34} q^{42} + \zeta_{96}^{26} q^{43} - \zeta_{96}^{19} q^{45} - \zeta_{96}^{33} q^{46} + \zeta_{96}^{20} q^{47} - \zeta_{96}^{29} q^{48} - \zeta_{96}^{5} q^{54} - \zeta_{96}^{12} q^{55} + \zeta_{96}^{41} q^{56} + \zeta_{96}^{7} q^{58} + \zeta_{96}^{34} q^{59} + \zeta_{96}^{47} q^{61} + \zeta_{96}^{3} q^{62} + \zeta_{96}^{17} q^{63} + \zeta_{96}^{36} q^{64} + \zeta_{96}^{29} q^{65} + \zeta_{96}^{46} q^{66} + \zeta_{96}^{40} q^{67} + \zeta_{96}^{16} q^{69} + \zeta_{96}^{38} q^{70} + \zeta_{96}^{12} q^{72} - 2 \zeta_{96}^{11} q^{74} + \zeta_{96}^{10} q^{77} + \zeta_{96}^{15} q^{78} - \zeta_{96}^{35} q^{79} + \zeta_{96}^{33} q^{80} - \zeta_{96}^{36} q^{81} + \zeta_{96}^{45} q^{82} + \zeta_{96}^{14} q^{83} - \zeta_{96}^{16} q^{86} + \zeta_{96}^{38} q^{87} + \zeta_{96}^{5} q^{88} + \zeta_{96}^{9} q^{90} - \zeta_{96}^{27} q^{91} + \zeta_{96}^{34} q^{93} - \zeta_{96}^{10} q^{94} + \zeta_{96}^{17} q^{97} - \zeta_{96}^{29} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q+O(q^{10}) \) Copy content Toggle raw display \( 32 q + 16 q^{18} + 32 q^{35} + 16 q^{69} - 16 q^{86}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2601\mathbb{Z}\right)^\times\).

\(n\) \(290\) \(2026\)
\(\chi(n)\) \(-\zeta_{96}^{16}\) \(\zeta_{96}^{42}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
40.1
−0.896873 + 0.442289i
0.896873 − 0.442289i
−0.751840 + 0.659346i
0.751840 − 0.659346i
0.321439 − 0.946930i
−0.321439 + 0.946930i
0.997859 + 0.0654031i
−0.997859 − 0.0654031i
0.997859 − 0.0654031i
−0.997859 + 0.0654031i
0.946930 − 0.321439i
−0.946930 + 0.321439i
0.321439 + 0.946930i
−0.321439 − 0.946930i
0.0654031 − 0.997859i
−0.0654031 + 0.997859i
0.659346 + 0.751840i
−0.659346 − 0.751840i
−0.442289 + 0.896873i
0.442289 − 0.896873i
0.130526 + 0.991445i −0.980785 + 0.195090i 0 −0.442289 − 0.896873i −0.321439 − 0.946930i −0.442289 + 0.896873i 0.382683 + 0.923880i 0.923880 − 0.382683i 0.831470 − 0.555570i
40.2 0.130526 + 0.991445i 0.980785 − 0.195090i 0 0.442289 + 0.896873i 0.321439 + 0.946930i 0.442289 − 0.896873i 0.382683 + 0.923880i 0.923880 − 0.382683i −0.831470 + 0.555570i
214.1 −0.608761 − 0.793353i −0.831470 − 0.555570i 0 −0.659346 − 0.751840i 0.0654031 + 0.997859i −0.659346 + 0.751840i −0.923880 + 0.382683i 0.382683 + 0.923880i −0.195090 + 0.980785i
214.2 −0.608761 − 0.793353i 0.831470 + 0.555570i 0 0.659346 + 0.751840i −0.0654031 − 0.997859i 0.659346 − 0.751840i −0.923880 + 0.382683i 0.382683 + 0.923880i 0.195090 − 0.980785i
364.1 −0.991445 + 0.130526i −0.555570 + 0.831470i 0 0.946930 + 0.321439i 0.442289 − 0.896873i 0.946930 − 0.321439i 0.923880 − 0.382683i −0.382683 − 0.923880i −0.980785 − 0.195090i
364.2 −0.991445 + 0.130526i 0.555570 − 0.831470i 0 −0.946930 − 0.321439i −0.442289 + 0.896873i −0.946930 + 0.321439i 0.923880 − 0.382683i −0.382683 − 0.923880i 0.980785 + 0.195090i
538.1 −0.793353 + 0.608761i −0.195090 − 0.980785i 0 −0.0654031 + 0.997859i 0.751840 + 0.659346i −0.0654031 − 0.997859i −0.382683 − 0.923880i −0.923880 + 0.382683i −0.555570 − 0.831470i
538.2 −0.793353 + 0.608761i 0.195090 + 0.980785i 0 0.0654031 − 0.997859i −0.751840 − 0.659346i 0.0654031 + 0.997859i −0.382683 − 0.923880i −0.923880 + 0.382683i 0.555570 + 0.831470i
643.1 −0.793353 − 0.608761i −0.195090 + 0.980785i 0 −0.0654031 − 0.997859i 0.751840 − 0.659346i −0.0654031 + 0.997859i −0.382683 + 0.923880i −0.923880 − 0.382683i −0.555570 + 0.831470i
643.2 −0.793353 − 0.608761i 0.195090 − 0.980785i 0 0.0654031 + 0.997859i −0.751840 + 0.659346i 0.0654031 − 0.997859i −0.382683 + 0.923880i −0.923880 − 0.382683i 0.555570 − 0.831470i
709.1 0.991445 + 0.130526i −0.831470 + 0.555570i 0 −0.321439 − 0.946930i −0.896873 + 0.442289i −0.321439 + 0.946930i −0.923880 − 0.382683i 0.382683 − 0.923880i −0.195090 − 0.980785i
709.2 0.991445 + 0.130526i 0.831470 − 0.555570i 0 0.321439 + 0.946930i 0.896873 − 0.442289i 0.321439 − 0.946930i −0.923880 − 0.382683i 0.382683 − 0.923880i 0.195090 + 0.980785i
736.1 −0.991445 − 0.130526i −0.555570 − 0.831470i 0 0.946930 − 0.321439i 0.442289 + 0.896873i 0.946930 + 0.321439i 0.923880 + 0.382683i −0.382683 + 0.923880i −0.980785 + 0.195090i
736.2 −0.991445 − 0.130526i 0.555570 + 0.831470i 0 −0.946930 + 0.321439i −0.442289 − 0.896873i −0.946930 − 0.321439i 0.923880 + 0.382683i −0.382683 + 0.923880i 0.980785 − 0.195090i
907.1 0.793353 − 0.608761i −0.980785 + 0.195090i 0 0.997859 + 0.0654031i −0.659346 + 0.751840i 0.997859 − 0.0654031i 0.382683 + 0.923880i 0.923880 − 0.382683i 0.831470 − 0.555570i
907.2 0.793353 − 0.608761i 0.980785 − 0.195090i 0 −0.997859 − 0.0654031i 0.659346 − 0.751840i −0.997859 + 0.0654031i 0.382683 + 0.923880i 0.923880 − 0.382683i −0.831470 + 0.555570i
1231.1 0.608761 + 0.793353i −0.555570 + 0.831470i 0 −0.751840 + 0.659346i −0.997859 + 0.0654031i −0.751840 − 0.659346i 0.923880 − 0.382683i −0.382683 − 0.923880i −0.980785 − 0.195090i
1231.2 0.608761 + 0.793353i 0.555570 − 0.831470i 0 0.751840 − 0.659346i 0.997859 − 0.0654031i 0.751840 + 0.659346i 0.923880 − 0.382683i −0.382683 − 0.923880i 0.980785 + 0.195090i
1510.1 −0.130526 + 0.991445i −0.195090 + 0.980785i 0 0.896873 + 0.442289i −0.946930 − 0.321439i 0.896873 − 0.442289i −0.382683 + 0.923880i −0.923880 − 0.382683i −0.555570 + 0.831470i
1510.2 −0.130526 + 0.991445i 0.195090 − 0.980785i 0 −0.896873 − 0.442289i 0.946930 + 0.321439i −0.896873 + 0.442289i −0.382683 + 0.923880i −0.923880 − 0.382683i 0.555570 − 0.831470i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 40.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
17.b even 2 1 inner
17.c even 4 2 inner
17.d even 8 4 inner
17.e odd 16 8 inner
153.h even 6 1 inner
153.n even 12 2 inner
153.r even 24 4 inner
153.t odd 48 8 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2601.1.x.a ✓ 32
9.c even 3 1 inner 2601.1.x.a ✓ 32
17.b even 2 1 inner 2601.1.x.a ✓ 32
17.c even 4 2 inner 2601.1.x.a ✓ 32
17.d even 8 4 inner 2601.1.x.a ✓ 32
17.e odd 16 8 inner 2601.1.x.a ✓ 32
153.h even 6 1 inner 2601.1.x.a ✓ 32
153.n even 12 2 inner 2601.1.x.a ✓ 32
153.r even 24 4 inner 2601.1.x.a ✓ 32
153.t odd 48 8 inner 2601.1.x.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2601.1.x.a ✓ 32 1.a even 1 1 trivial
2601.1.x.a ✓ 32 9.c even 3 1 inner
2601.1.x.a ✓ 32 17.b even 2 1 inner
2601.1.x.a ✓ 32 17.c even 4 2 inner
2601.1.x.a ✓ 32 17.d even 8 4 inner
2601.1.x.a ✓ 32 17.e odd 16 8 inner
2601.1.x.a ✓ 32 153.h even 6 1 inner
2601.1.x.a ✓ 32 153.n even 12 2 inner
2601.1.x.a ✓ 32 153.r even 24 4 inner
2601.1.x.a ✓ 32 153.t odd 48 8 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2601, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{16} - T^{8} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{16} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{32} - T^{16} + 1 \) Copy content Toggle raw display
$7$ \( T^{32} - T^{16} + 1 \) Copy content Toggle raw display
$11$ \( T^{32} - T^{16} + 1 \) Copy content Toggle raw display
$13$ \( (T^{8} - T^{4} + 1)^{4} \) Copy content Toggle raw display
$17$ \( T^{32} \) Copy content Toggle raw display
$19$ \( T^{32} \) Copy content Toggle raw display
$23$ \( T^{32} - T^{16} + 1 \) Copy content Toggle raw display
$29$ \( T^{32} - T^{16} + 1 \) Copy content Toggle raw display
$31$ \( T^{32} - T^{16} + 1 \) Copy content Toggle raw display
$37$ \( (T^{16} + 65536)^{2} \) Copy content Toggle raw display
$41$ \( T^{32} - T^{16} + 1 \) Copy content Toggle raw display
$43$ \( (T^{16} - T^{8} + 1)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - T^{4} + 1)^{4} \) Copy content Toggle raw display
$53$ \( T^{32} \) Copy content Toggle raw display
$59$ \( (T^{16} - T^{8} + 1)^{2} \) Copy content Toggle raw display
$61$ \( T^{32} - T^{16} + 1 \) Copy content Toggle raw display
$67$ \( (T^{4} - T^{2} + 1)^{8} \) Copy content Toggle raw display
$71$ \( T^{32} \) Copy content Toggle raw display
$73$ \( T^{32} \) Copy content Toggle raw display
$79$ \( T^{32} - T^{16} + 1 \) Copy content Toggle raw display
$83$ \( (T^{16} - T^{8} + 1)^{2} \) Copy content Toggle raw display
$89$ \( T^{32} \) Copy content Toggle raw display
$97$ \( T^{32} - T^{16} + 1 \) Copy content Toggle raw display
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