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Properties

Label 2299.1.w.a
Level $2299$
Weight $1$
Character orbit 2299.w
Analytic conductor $1.147$
Analytic rank $0$
Dimension $16$
Projective image $A_{4}$
CM/RM no
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2299,1,Mod(239,2299)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2299, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([9, 20]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2299.239");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2299 = 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2299.w (of order \(30\), degree \(8\), 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: \(1.14735046404\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{30})\)
Coefficient field: \(\Q(\zeta_{60})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + x^{14} - x^{10} - x^{8} - x^{6} + x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 209)
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.43681.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{60}^{17} q^{2} - \zeta_{60}^{26} q^{3} + \zeta_{60}^{8} q^{5} - \zeta_{60}^{13} q^{6} - \zeta_{60}^{21} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{60}^{17} q^{2} - \zeta_{60}^{26} q^{3} + \zeta_{60}^{8} q^{5} - \zeta_{60}^{13} q^{6} - \zeta_{60}^{21} q^{8} - \zeta_{60}^{25} q^{10} - \zeta_{60}^{7} q^{13} + \zeta_{60}^{4} q^{15} - \zeta_{60}^{8} q^{16} - \zeta_{60}^{23} q^{17} + \zeta_{60}^{21} q^{19} - \zeta_{60}^{10} q^{23} - \zeta_{60}^{17} q^{24} + \zeta_{60}^{24} q^{26} + \zeta_{60}^{18} q^{27} + \zeta_{60}^{19} q^{29} - \zeta_{60}^{21} q^{30} - \zeta_{60}^{10} q^{34} + \zeta_{60}^{8} q^{38} - \zeta_{60}^{3} q^{39} - \zeta_{60}^{29} q^{40} + \zeta_{60}^{11} q^{41} + \zeta_{60}^{5} q^{43} + \zeta_{60}^{27} q^{46} + \zeta_{60}^{16} q^{47} - \zeta_{60}^{4} q^{48} - \zeta_{60}^{18} q^{49} - \zeta_{60}^{19} q^{51} + \zeta_{60}^{22} q^{53} + \zeta_{60}^{5} q^{54} + \zeta_{60}^{17} q^{57} + \zeta_{60}^{6} q^{58} - \zeta_{60}^{14} q^{59} - \zeta_{60}^{13} q^{61} - \zeta_{60}^{12} q^{64} - \zeta_{60}^{15} q^{65} + \zeta_{60}^{10} q^{67} - \zeta_{60}^{6} q^{69} - \zeta_{60}^{8} q^{71} + \zeta_{60}^{29} q^{73} + \zeta_{60}^{20} q^{78} + \zeta_{60}^{17} q^{79} - \zeta_{60}^{16} q^{80} + \zeta_{60}^{14} q^{81} - \zeta_{60}^{28} q^{82} + \zeta_{60} q^{85} - \zeta_{60}^{22} q^{86} + \zeta_{60}^{15} q^{87} + \zeta_{60}^{10} q^{89} + \zeta_{60}^{3} q^{94} + \zeta_{60}^{29} q^{95} + \zeta_{60}^{2} q^{97} - \zeta_{60}^{5} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{3} + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{3} + 2 q^{5} + 2 q^{15} - 2 q^{16} - 8 q^{23} - 4 q^{26} + 4 q^{27} - 8 q^{34} + 2 q^{38} + 2 q^{47} - 2 q^{48} - 4 q^{49} - 2 q^{53} + 4 q^{58} + 2 q^{59} + 4 q^{64} + 8 q^{67} - 4 q^{69} - 2 q^{71} - 8 q^{78} - 2 q^{80} - 2 q^{81} - 2 q^{82} + 2 q^{86} + 8 q^{89} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2299\mathbb{Z}\right)^\times\).

\(n\) \(970\) \(1332\)
\(\chi(n)\) \(-\zeta_{60}^{24}\) \(-\zeta_{60}^{10}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1
0.743145 + 0.669131i
−0.743145 − 0.669131i
−0.207912 − 0.978148i
0.207912 + 0.978148i
−0.994522 + 0.104528i
0.994522 − 0.104528i
0.406737 − 0.913545i
−0.406737 + 0.913545i
0.406737 + 0.913545i
−0.406737 − 0.913545i
−0.207912 + 0.978148i
0.207912 − 0.978148i
0.743145 − 0.669131i
−0.743145 + 0.669131i
−0.994522 − 0.104528i
0.994522 + 0.104528i
−0.994522 + 0.104528i −0.978148 − 0.207912i 0 0.913545 − 0.406737i 0.994522 + 0.104528i 0 0.951057 − 0.309017i 0 −0.866025 + 0.500000i
239.2 0.994522 − 0.104528i −0.978148 − 0.207912i 0 0.913545 − 0.406737i −0.994522 − 0.104528i 0 −0.951057 + 0.309017i 0 0.866025 − 0.500000i
524.1 −0.406737 − 0.913545i 0.669131 + 0.743145i 0 −0.104528 − 0.994522i 0.406737 − 0.913545i 0 −0.951057 − 0.309017i 0 −0.866025 + 0.500000i
524.2 0.406737 + 0.913545i 0.669131 + 0.743145i 0 −0.104528 − 0.994522i −0.406737 + 0.913545i 0 0.951057 + 0.309017i 0 0.866025 − 0.500000i
596.1 −0.207912 − 0.978148i 0.913545 + 0.406737i 0 0.669131 − 0.743145i 0.207912 − 0.978148i 0 −0.587785 − 0.809017i 0 −0.866025 − 0.500000i
596.2 0.207912 + 0.978148i 0.913545 + 0.406737i 0 0.669131 − 0.743145i −0.207912 + 0.978148i 0 0.587785 + 0.809017i 0 0.866025 + 0.500000i
1322.1 −0.743145 + 0.669131i −0.104528 − 0.994522i 0 −0.978148 − 0.207912i 0.743145 + 0.669131i 0 −0.587785 − 0.809017i 0 0.866025 − 0.500000i
1322.2 0.743145 − 0.669131i −0.104528 − 0.994522i 0 −0.978148 − 0.207912i −0.743145 − 0.669131i 0 0.587785 + 0.809017i 0 −0.866025 + 0.500000i
1546.1 −0.743145 − 0.669131i −0.104528 + 0.994522i 0 −0.978148 + 0.207912i 0.743145 − 0.669131i 0 −0.587785 + 0.809017i 0 0.866025 + 0.500000i
1546.2 0.743145 + 0.669131i −0.104528 + 0.994522i 0 −0.978148 + 0.207912i −0.743145 + 0.669131i 0 0.587785 − 0.809017i 0 −0.866025 − 0.500000i
1812.1 −0.406737 + 0.913545i 0.669131 − 0.743145i 0 −0.104528 + 0.994522i 0.406737 + 0.913545i 0 −0.951057 + 0.309017i 0 −0.866025 − 0.500000i
1812.2 0.406737 − 0.913545i 0.669131 − 0.743145i 0 −0.104528 + 0.994522i −0.406737 − 0.913545i 0 0.951057 − 0.309017i 0 0.866025 + 0.500000i
2097.1 −0.994522 − 0.104528i −0.978148 + 0.207912i 0 0.913545 + 0.406737i 0.994522 − 0.104528i 0 0.951057 + 0.309017i 0 −0.866025 − 0.500000i
2097.2 0.994522 + 0.104528i −0.978148 + 0.207912i 0 0.913545 + 0.406737i −0.994522 + 0.104528i 0 −0.951057 − 0.309017i 0 0.866025 + 0.500000i
2272.1 −0.207912 + 0.978148i 0.913545 − 0.406737i 0 0.669131 + 0.743145i 0.207912 + 0.978148i 0 −0.587785 + 0.809017i 0 −0.866025 + 0.500000i
2272.2 0.207912 − 0.978148i 0.913545 − 0.406737i 0 0.669131 + 0.743145i −0.207912 − 0.978148i 0 0.587785 − 0.809017i 0 0.866025 − 0.500000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 239.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
19.c even 3 1 inner
209.h odd 6 1 inner
209.n even 15 3 inner
209.s odd 30 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2299.1.w.a 16
11.b odd 2 1 inner 2299.1.w.a 16
11.c even 5 1 209.1.h.a ✓ 4
11.c even 5 3 inner 2299.1.w.a 16
11.d odd 10 1 209.1.h.a ✓ 4
11.d odd 10 3 inner 2299.1.w.a 16
19.c even 3 1 inner 2299.1.w.a 16
33.f even 10 1 1881.1.bc.a 4
33.h odd 10 1 1881.1.bc.a 4
44.g even 10 1 3344.1.bb.a 4
44.h odd 10 1 3344.1.bb.a 4
209.h odd 6 1 inner 2299.1.w.a 16
209.k even 10 1 3971.1.h.d 4
209.m odd 10 1 3971.1.h.d 4
209.n even 15 1 209.1.h.a ✓ 4
209.n even 15 3 inner 2299.1.w.a 16
209.n even 15 1 3971.1.c.e 2
209.r odd 30 1 3971.1.c.b 2
209.r odd 30 1 3971.1.h.d 4
209.s odd 30 1 209.1.h.a ✓ 4
209.s odd 30 3 inner 2299.1.w.a 16
209.s odd 30 1 3971.1.c.e 2
209.t even 30 1 3971.1.c.b 2
209.t even 30 1 3971.1.h.d 4
209.u even 45 6 3971.1.q.e 12
209.v odd 90 6 3971.1.q.e 12
209.w even 90 6 3971.1.q.d 12
209.x odd 90 6 3971.1.q.d 12
627.bh odd 30 1 1881.1.bc.a 4
627.bm even 30 1 1881.1.bc.a 4
836.bh even 30 1 3344.1.bb.a 4
836.bk odd 30 1 3344.1.bb.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.1.h.a ✓ 4 11.c even 5 1
209.1.h.a ✓ 4 11.d odd 10 1
209.1.h.a ✓ 4 209.n even 15 1
209.1.h.a ✓ 4 209.s odd 30 1
1881.1.bc.a 4 33.f even 10 1
1881.1.bc.a 4 33.h odd 10 1
1881.1.bc.a 4 627.bh odd 30 1
1881.1.bc.a 4 627.bm even 30 1
2299.1.w.a 16 1.a even 1 1 trivial
2299.1.w.a 16 11.b odd 2 1 inner
2299.1.w.a 16 11.c even 5 3 inner
2299.1.w.a 16 11.d odd 10 3 inner
2299.1.w.a 16 19.c even 3 1 inner
2299.1.w.a 16 209.h odd 6 1 inner
2299.1.w.a 16 209.n even 15 3 inner
2299.1.w.a 16 209.s odd 30 3 inner
3344.1.bb.a 4 44.g even 10 1
3344.1.bb.a 4 44.h odd 10 1
3344.1.bb.a 4 836.bh even 30 1
3344.1.bb.a 4 836.bk odd 30 1
3971.1.c.b 2 209.r odd 30 1
3971.1.c.b 2 209.t even 30 1
3971.1.c.e 2 209.n even 15 1
3971.1.c.e 2 209.s odd 30 1
3971.1.h.d 4 209.k even 10 1
3971.1.h.d 4 209.m odd 10 1
3971.1.h.d 4 209.r odd 30 1
3971.1.h.d 4 209.t even 30 1
3971.1.q.d 12 209.w even 90 6
3971.1.q.d 12 209.x odd 90 6
3971.1.q.e 12 209.u even 45 6
3971.1.q.e 12 209.v odd 90 6

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2299, [\chi])\).

Hecke characteristic polynomials

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