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Properties

Label 845.2.c.e
Level $845$
Weight $2$
Character orbit 845.c
Analytic conductor $6.747$
Analytic rank $0$
Dimension $4$
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] = [845,2,Mod(506,845)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(845, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("845.506");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.c (of order \(2\), degree \(1\), 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: \(6.74735897080\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - \beta_{3} + 1) q^{3} - q^{4} + \beta_1 q^{5} + ( - \beta_{2} + 3 \beta_1) q^{6} + 2 \beta_1 q^{7} - \beta_{2} q^{8} + ( - 2 \beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + ( - \beta_{3} + 1) q^{3} - q^{4} + \beta_1 q^{5} + ( - \beta_{2} + 3 \beta_1) q^{6} + 2 \beta_1 q^{7} - \beta_{2} q^{8} + ( - 2 \beta_{3} + 1) q^{9} + \beta_{3} q^{10} + (\beta_{2} - 3 \beta_1) q^{11} + (\beta_{3} - 1) q^{12} + 2 \beta_{3} q^{14} + ( - \beta_{2} + \beta_1) q^{15} - 5 q^{16} - 2 \beta_{3} q^{17} + ( - \beta_{2} + 6 \beta_1) q^{18} + ( - 3 \beta_{2} + \beta_1) q^{19} - \beta_1 q^{20} + ( - 2 \beta_{2} + 2 \beta_1) q^{21} + ( - 3 \beta_{3} + 3) q^{22} + ( - \beta_{3} - 3) q^{23} + ( - \beta_{2} + 3 \beta_1) q^{24} - q^{25} + 4 q^{27} - 2 \beta_1 q^{28} + ( - 2 \beta_{3} - 6) q^{29} + (\beta_{3} - 3) q^{30} + (3 \beta_{2} - 5 \beta_1) q^{31} + 3 \beta_{2} q^{32} + (4 \beta_{2} - 6 \beta_1) q^{33} + 6 \beta_1 q^{34} - 2 q^{35} + (2 \beta_{3} - 1) q^{36} - 4 \beta_1 q^{37} + (\beta_{3} - 9) q^{38} + \beta_{3} q^{40} + 2 \beta_{2} q^{41} + (2 \beta_{3} - 6) q^{42} + ( - 3 \beta_{3} - 5) q^{43} + ( - \beta_{2} + 3 \beta_1) q^{44} + ( - 2 \beta_{2} + \beta_1) q^{45} + (3 \beta_{2} + 3 \beta_1) q^{46} + 6 \beta_1 q^{47} + (5 \beta_{3} - 5) q^{48} + 3 q^{49} + \beta_{2} q^{50} + ( - 2 \beta_{3} + 6) q^{51} - 6 \beta_{3} q^{53} - 4 \beta_{2} q^{54} + ( - \beta_{3} + 3) q^{55} + 2 \beta_{3} q^{56} + ( - 4 \beta_{2} + 10 \beta_1) q^{57} + (6 \beta_{2} + 6 \beta_1) q^{58} + ( - 7 \beta_{2} - 3 \beta_1) q^{59} + (\beta_{2} - \beta_1) q^{60} + (6 \beta_{3} + 2) q^{61} + ( - 5 \beta_{3} + 9) q^{62} + ( - 4 \beta_{2} + 2 \beta_1) q^{63} - q^{64} + ( - 6 \beta_{3} + 12) q^{66} + (6 \beta_{2} + 4 \beta_1) q^{67} + 2 \beta_{3} q^{68} + 2 \beta_{3} q^{69} + 2 \beta_{2} q^{70} + (\beta_{2} - 3 \beta_1) q^{71} + ( - \beta_{2} + 6 \beta_1) q^{72} - 4 \beta_1 q^{73} - 4 \beta_{3} q^{74} + (\beta_{3} - 1) q^{75} + (3 \beta_{2} - \beta_1) q^{76} + ( - 2 \beta_{3} + 6) q^{77} + (6 \beta_{3} + 2) q^{79} - 5 \beta_1 q^{80} + (2 \beta_{3} + 1) q^{81} + 6 q^{82} + 6 \beta_1 q^{83} + (2 \beta_{2} - 2 \beta_1) q^{84} - 2 \beta_{2} q^{85} + (5 \beta_{2} + 9 \beta_1) q^{86} + 4 \beta_{3} q^{87} + ( - 3 \beta_{3} + 3) q^{88} + (4 \beta_{2} - 6 \beta_1) q^{89} + (\beta_{3} - 6) q^{90} + (\beta_{3} + 3) q^{92} + (8 \beta_{2} - 14 \beta_1) q^{93} + 6 \beta_{3} q^{94} + (3 \beta_{3} - 1) q^{95} + (3 \beta_{2} - 9 \beta_1) q^{96} - 2 \beta_1 q^{97} - 3 \beta_{2} q^{98} + (7 \beta_{2} - 9 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9} - 4 q^{12} - 20 q^{16} + 12 q^{22} - 12 q^{23} - 4 q^{25} + 16 q^{27} - 24 q^{29} - 12 q^{30} - 8 q^{35} - 4 q^{36} - 36 q^{38} - 24 q^{42} - 20 q^{43} - 20 q^{48} + 12 q^{49} + 24 q^{51} + 12 q^{55} + 8 q^{61} + 36 q^{62} - 4 q^{64} + 48 q^{66} - 4 q^{75} + 24 q^{77} + 8 q^{79} + 4 q^{81} + 24 q^{82} + 12 q^{88} - 24 q^{90} + 12 q^{92} - 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\) \(171\) \(677\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
506.1
0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 − 0.500000i
−0.866025 + 0.500000i
− 1.73205i −0.732051 −1.00000 1.00000i 1.26795i 2.00000i − 1.73205i −2.46410 1.73205
506.2 − 1.73205i 2.73205 −1.00000 − 1.00000i − 4.73205i − 2.00000i − 1.73205i 4.46410 −1.73205
506.3 1.73205i −0.732051 −1.00000 − 1.00000i − 1.26795i − 2.00000i 1.73205i −2.46410 1.73205
506.4 1.73205i 2.73205 −1.00000 1.00000i 4.73205i 2.00000i 1.73205i 4.46410 −1.73205
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.c.e 4
13.b even 2 1 inner 845.2.c.e 4
13.c even 3 1 845.2.m.a 4
13.c even 3 1 845.2.m.c 4
13.d odd 4 1 65.2.a.c ✓ 2
13.d odd 4 1 845.2.a.d 2
13.e even 6 1 845.2.m.a 4
13.e even 6 1 845.2.m.c 4
13.f odd 12 2 845.2.e.e 4
13.f odd 12 2 845.2.e.f 4
39.f even 4 1 585.2.a.k 2
39.f even 4 1 7605.2.a.be 2
52.f even 4 1 1040.2.a.h 2
65.f even 4 1 325.2.b.e 4
65.g odd 4 1 325.2.a.g 2
65.g odd 4 1 4225.2.a.w 2
65.k even 4 1 325.2.b.e 4
91.i even 4 1 3185.2.a.k 2
104.j odd 4 1 4160.2.a.y 2
104.m even 4 1 4160.2.a.bj 2
143.g even 4 1 7865.2.a.h 2
156.l odd 4 1 9360.2.a.cm 2
195.j odd 4 1 2925.2.c.v 4
195.n even 4 1 2925.2.a.z 2
195.u odd 4 1 2925.2.c.v 4
260.u even 4 1 5200.2.a.ca 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.a.c ✓ 2 13.d odd 4 1
325.2.a.g 2 65.g odd 4 1
325.2.b.e 4 65.f even 4 1
325.2.b.e 4 65.k even 4 1
585.2.a.k 2 39.f even 4 1
845.2.a.d 2 13.d odd 4 1
845.2.c.e 4 1.a even 1 1 trivial
845.2.c.e 4 13.b even 2 1 inner
845.2.e.e 4 13.f odd 12 2
845.2.e.f 4 13.f odd 12 2
845.2.m.a 4 13.c even 3 1
845.2.m.a 4 13.e even 6 1
845.2.m.c 4 13.c even 3 1
845.2.m.c 4 13.e even 6 1
1040.2.a.h 2 52.f even 4 1
2925.2.a.z 2 195.n even 4 1
2925.2.c.v 4 195.j odd 4 1
2925.2.c.v 4 195.u odd 4 1
3185.2.a.k 2 91.i even 4 1
4160.2.a.y 2 104.j odd 4 1
4160.2.a.bj 2 104.m even 4 1
4225.2.a.w 2 65.g odd 4 1
5200.2.a.ca 2 260.u even 4 1
7605.2.a.be 2 39.f even 4 1
7865.2.a.h 2 143.g even 4 1
9360.2.a.cm 2 156.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 3 \) acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - 2 T - 2)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 56T^{2} + 676 \) Copy content Toggle raw display
$23$ \( (T^{2} + 6 T + 6)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 12 T + 24)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 104T^{2} + 4 \) Copy content Toggle raw display
$37$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 10 T - 2)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 312 T^{2} + 19044 \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T - 104)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 248T^{2} + 8464 \) Copy content Toggle raw display
$71$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$73$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T - 104)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 168T^{2} + 144 \) Copy content Toggle raw display
$97$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
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