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Properties

Label 2904.1.cj.b
Level $2904$
Weight $1$
Character orbit 2904.cj
Analytic conductor $1.449$
Analytic rank $0$
Dimension $40$
Projective image $D_{55}$
CM discriminant -24
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2904,1,Mod(5,2904)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2904, base_ring=CyclotomicField(110))
 
chi = DirichletCharacter(H, H._module([0, 55, 55, 74]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2904.5");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2904 = 2^{3} \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2904.cj (of order \(110\), degree \(40\), 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: \(1.44928479669\)
Analytic rank: \(0\)
Dimension: \(40\)
Coefficient field: \(\Q(\zeta_{55})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{40} - x^{39} + x^{35} - x^{34} + x^{30} - x^{28} + x^{25} - x^{23} + x^{20} - x^{17} + x^{15} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{55}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{55} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{110}^{53} q^{2} + \zeta_{110}^{44} q^{3} - \zeta_{110}^{51} q^{4} + (\zeta_{110}^{46} - \zeta_{110}^{43}) q^{5} + \zeta_{110}^{42} q^{6} + ( - \zeta_{110}^{19} + \zeta_{110}^{8}) q^{7} - \zeta_{110}^{49} q^{8} - \zeta_{110}^{33} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{110}^{53} q^{2} + \zeta_{110}^{44} q^{3} - \zeta_{110}^{51} q^{4} + (\zeta_{110}^{46} - \zeta_{110}^{43}) q^{5} + \zeta_{110}^{42} q^{6} + ( - \zeta_{110}^{19} + \zeta_{110}^{8}) q^{7} - \zeta_{110}^{49} q^{8} - \zeta_{110}^{33} q^{9} + (\zeta_{110}^{44} - \zeta_{110}^{41}) q^{10} - \zeta_{110}^{15} q^{11} + \zeta_{110}^{40} q^{12} + ( - \zeta_{110}^{17} + \zeta_{110}^{6}) q^{14} + ( - \zeta_{110}^{35} + \zeta_{110}^{32}) q^{15} - \zeta_{110}^{47} q^{16} - \zeta_{110}^{31} q^{18} + (\zeta_{110}^{42} - \zeta_{110}^{39}) q^{20} + (\zeta_{110}^{52} + \zeta_{110}^{8}) q^{21} - \zeta_{110}^{13} q^{22} + \zeta_{110}^{38} q^{24} + ( - \zeta_{110}^{37} + \cdots - \zeta_{110}^{31}) q^{25} + \cdots + \zeta_{110}^{48} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q + q^{2} - 10 q^{3} + q^{4} + 2 q^{5} + q^{6} + 2 q^{7} + q^{8} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 40 q + q^{2} - 10 q^{3} + q^{4} + 2 q^{5} + q^{6} + 2 q^{7} + q^{8} - 10 q^{9} - 9 q^{10} - 4 q^{11} - 4 q^{12} + 2 q^{14} - 3 q^{15} + q^{16} + q^{18} + 2 q^{20} + 2 q^{21} + q^{22} + q^{24} + 3 q^{25} - 10 q^{27} - 3 q^{28} - 3 q^{29} - 14 q^{30} - 3 q^{31} - 4 q^{32} + q^{33} - q^{35} + q^{36} - 3 q^{40} - 3 q^{42} - 10 q^{44} + 2 q^{45} + q^{48} + 3 q^{49} - 2 q^{50} + 41 q^{53} - 4 q^{54} + 2 q^{55} + 2 q^{56} - 3 q^{58} + 2 q^{59} + 2 q^{60} + 2 q^{62} + 2 q^{63} + q^{64} + q^{66} - q^{70} + q^{72} - 3 q^{73} - 2 q^{75} + 2 q^{77} + 2 q^{79} - 3 q^{80} - 10 q^{81} + 2 q^{83} + 2 q^{84} + 2 q^{87} + q^{88} - 9 q^{90} - 3 q^{93} + q^{96} + 2 q^{97} - 2 q^{98} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2904\mathbb{Z}\right)^\times\).

\(n\) \(727\) \(1453\) \(1937\) \(2785\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-\zeta_{110}^{51}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−0.870746 − 0.491733i
−0.998369 − 0.0570888i
−0.0285561 − 0.999592i
0.974012 − 0.226497i
0.993482 + 0.113991i
0.516397 + 0.856349i
0.198590 + 0.980083i
−0.870746 + 0.491733i
−0.254218 + 0.967147i
−0.985354 − 0.170522i
0.974012 + 0.226497i
0.696938 − 0.717132i
−0.921124 − 0.389270i
0.774142 − 0.633012i
0.610648 − 0.791902i
−0.466667 + 0.884433i
0.516397 − 0.856349i
−0.0285561 + 0.999592i
−0.466667 − 0.884433i
0.198590 − 0.980083i
0.516397 − 0.856349i −0.809017 − 0.587785i −0.466667 − 0.884433i 1.07906 − 0.882340i −0.921124 + 0.389270i 0.377401 − 0.489423i −0.998369 − 0.0570888i 0.309017 + 0.951057i −0.198369 − 1.37969i
53.1 0.993482 − 0.113991i −0.809017 + 0.587785i 0.974012 − 0.226497i −0.0966045 − 0.141280i −0.736741 + 0.676175i 0.430731 − 0.443212i 0.941844 − 0.336049i 0.309017 − 0.951057i −0.112079 − 0.129347i
125.1 −0.998369 − 0.0570888i 0.309017 − 0.951057i 0.993482 + 0.113991i 0.687626 + 1.30320i −0.362808 + 0.931864i 1.49041 + 0.629852i −0.985354 − 0.170522i −0.809017 − 0.587785i −0.612107 − 1.34033i
317.1 0.897398 + 0.441221i −0.809017 + 0.587785i 0.610648 + 0.791902i −1.38779 + 1.27370i −0.985354 + 0.170522i −0.617026 − 0.0352828i 0.198590 + 0.980083i 0.309017 − 0.951057i −1.80739 + 0.530696i
389.1 0.974012 − 0.226497i 0.309017 − 0.951057i 0.897398 − 0.441221i 0.714988 − 1.83643i 0.0855750 − 0.996332i 0.0462047 + 1.61737i 0.774142 − 0.633012i −0.809017 − 0.587785i 0.280461 − 1.95065i
509.1 −0.466667 − 0.884433i 0.309017 + 0.951057i −0.564443 + 0.825472i −0.0113419 + 0.0559746i 0.696938 − 0.717132i 0.411334 + 1.56488i 0.993482 + 0.113991i −0.809017 + 0.587785i 0.0547987 − 0.0160903i
533.1 −0.921124 − 0.389270i −0.809017 − 0.587785i 0.696938 + 0.717132i 0.237271 + 0.902672i 0.516397 + 0.856349i 0.582092 − 0.207690i −0.362808 − 0.931864i 0.309017 + 0.951057i 0.132827 − 0.923835i
581.1 0.516397 + 0.856349i −0.809017 + 0.587785i −0.466667 + 0.884433i 1.07906 + 0.882340i −0.921124 − 0.389270i 0.377401 + 0.489423i −0.998369 + 0.0570888i 0.309017 − 0.951057i −0.198369 + 1.37969i
653.1 −0.870746 + 0.491733i 0.309017 − 0.951057i 0.516397 − 0.856349i −1.73511 + 0.619086i 0.198590 + 0.980083i −1.45202 + 0.713911i −0.0285561 + 0.999592i −0.809017 − 0.587785i 1.20642 − 1.39228i
773.1 0.941844 − 0.336049i 0.309017 + 0.951057i 0.774142 − 0.633012i −0.495223 + 0.115159i 0.610648 + 0.791902i 1.19207 + 1.09407i 0.516397 − 0.856349i −0.809017 + 0.587785i −0.427724 + 0.274882i
797.1 0.897398 − 0.441221i −0.809017 − 0.587785i 0.610648 − 0.791902i −1.38779 − 1.27370i −0.985354 − 0.170522i −0.617026 + 0.0352828i 0.198590 − 0.980083i 0.309017 + 0.951057i −1.80739 − 0.530696i
845.1 −0.0285561 + 0.999592i −0.809017 + 0.587785i −0.998369 − 0.0570888i −0.374706 + 0.621380i −0.564443 − 0.825472i 0.122736 − 0.605724i 0.0855750 − 0.996332i 0.309017 − 0.951057i −0.610427 − 0.392297i
917.1 0.696938 − 0.717132i 0.309017 − 0.951057i −0.0285561 − 0.999592i 0.982973 + 0.555111i −0.466667 − 0.884433i −1.25259 − 1.02424i −0.736741 − 0.676175i −0.809017 − 0.587785i 1.08316 − 0.318044i
1037.1 0.198590 + 0.980083i 0.309017 + 0.951057i −0.921124 + 0.389270i 0.630674 + 0.817873i −0.870746 + 0.491733i 1.59434 + 0.275911i −0.564443 − 0.825472i −0.809017 + 0.587785i −0.676337 + 0.780534i
1061.1 −0.254218 + 0.967147i −0.809017 − 0.587785i −0.870746 − 0.491733i −0.391364 − 0.0677282i 0.774142 − 0.633012i 0.601972 + 0.139983i 0.696938 − 0.717132i 0.309017 + 0.951057i 0.164995 − 0.361288i
1109.1 −0.564443 + 0.825472i −0.809017 + 0.587785i −0.362808 − 0.931864i 1.83924 + 0.777270i −0.0285561 − 0.999592i −0.538151 + 0.303908i 0.974012 + 0.226497i 0.309017 − 0.951057i −1.67976 + 1.07952i
1181.1 −0.466667 + 0.884433i 0.309017 − 0.951057i −0.564443 − 0.825472i −0.0113419 − 0.0559746i 0.696938 + 0.717132i 0.411334 − 1.56488i 0.993482 − 0.113991i −0.809017 − 0.587785i 0.0547987 + 0.0160903i
1301.1 −0.998369 + 0.0570888i 0.309017 + 0.951057i 0.993482 − 0.113991i 0.687626 − 1.30320i −0.362808 − 0.931864i 1.49041 − 0.629852i −0.985354 + 0.170522i −0.809017 + 0.587785i −0.612107 + 1.34033i
1325.1 −0.564443 − 0.825472i −0.809017 − 0.587785i −0.362808 + 0.931864i 1.83924 − 0.777270i −0.0285561 + 0.999592i −0.538151 − 0.303908i 0.974012 − 0.226497i 0.309017 + 0.951057i −1.67976 − 1.07952i
1373.1 −0.921124 + 0.389270i −0.809017 + 0.587785i 0.696938 − 0.717132i 0.237271 − 0.902672i 0.516397 − 0.856349i 0.582092 + 0.207690i −0.362808 + 0.931864i 0.309017 − 0.951057i 0.132827 + 0.923835i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
121.g even 55 1 inner
2904.cj odd 110 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2904.1.cj.b yes 40
3.b odd 2 1 2904.1.cj.a ✓ 40
8.b even 2 1 2904.1.cj.a ✓ 40
24.h odd 2 1 CM 2904.1.cj.b yes 40
121.g even 55 1 inner 2904.1.cj.b yes 40
363.n odd 110 1 2904.1.cj.a ✓ 40
968.bd even 110 1 2904.1.cj.a ✓ 40
2904.cj odd 110 1 inner 2904.1.cj.b yes 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2904.1.cj.a ✓ 40 3.b odd 2 1
2904.1.cj.a ✓ 40 8.b even 2 1
2904.1.cj.a ✓ 40 363.n odd 110 1
2904.1.cj.a ✓ 40 968.bd even 110 1
2904.1.cj.b yes 40 1.a even 1 1 trivial
2904.1.cj.b yes 40 24.h odd 2 1 CM
2904.1.cj.b yes 40 121.g even 55 1 inner
2904.1.cj.b yes 40 2904.cj odd 110 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{40} - 2 T_{5}^{39} + 16 T_{5}^{37} - 27 T_{5}^{36} - 8 T_{5}^{35} + 164 T_{5}^{34} - 220 T_{5}^{33} + \cdots + 1 \) acting on \(S_{1}^{\mathrm{new}}(2904, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{40} - T^{39} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{10} \) Copy content Toggle raw display
$5$ \( T^{40} - 2 T^{39} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{40} - 2 T^{39} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( (T^{10} + T^{9} + T^{8} + \cdots + 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{40} \) Copy content Toggle raw display
$17$ \( T^{40} \) Copy content Toggle raw display
$19$ \( T^{40} \) Copy content Toggle raw display
$23$ \( T^{40} \) Copy content Toggle raw display
$29$ \( T^{40} + 3 T^{39} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{40} + 3 T^{39} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{40} \) Copy content Toggle raw display
$41$ \( T^{40} \) Copy content Toggle raw display
$43$ \( T^{40} \) Copy content Toggle raw display
$47$ \( T^{40} \) Copy content Toggle raw display
$53$ \( T^{40} - 41 T^{39} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{40} - 2 T^{39} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{40} \) Copy content Toggle raw display
$67$ \( T^{40} \) Copy content Toggle raw display
$71$ \( T^{40} \) Copy content Toggle raw display
$73$ \( T^{40} + 3 T^{39} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{40} - 2 T^{39} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{40} - 2 T^{39} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( T^{40} \) Copy content Toggle raw display
$97$ \( T^{40} - 2 T^{39} + \cdots + 1 \) Copy content Toggle raw display
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