Properties

Label 845.2.o.b
Level $845$
Weight $2$
Character orbit 845.o
Analytic conductor $6.747$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [845,2,Mod(258,845)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(845, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([9, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("845.258");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.o (of order \(12\), degree \(4\), 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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(171\) \(677\)
\(\chi(n)\) \(\zeta_{12}\) \(\zeta_{12}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
258.1
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
0.500000 + 0.866025i 0.366025 1.36603i 0.500000 0.866025i 1.00000 + 2.00000i 1.36603 0.366025i 1.73205 + 1.00000i 3.00000 0.866025 + 0.500000i −1.23205 + 1.86603i
357.1 0.500000 0.866025i 0.366025 + 1.36603i 0.500000 + 0.866025i 1.00000 2.00000i 1.36603 + 0.366025i 1.73205 1.00000i 3.00000 0.866025 0.500000i −1.23205 1.86603i
488.1 0.500000 0.866025i −1.36603 + 0.366025i 0.500000 + 0.866025i 1.00000 + 2.00000i −0.366025 + 1.36603i −1.73205 + 1.00000i 3.00000 −0.866025 + 0.500000i 2.23205 + 0.133975i
587.1 0.500000 + 0.866025i −1.36603 0.366025i 0.500000 0.866025i 1.00000 2.00000i −0.366025 1.36603i −1.73205 1.00000i 3.00000 −0.866025 0.500000i 2.23205 0.133975i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner
65.k even 4 1 inner
65.o even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.o.b 4
5.c odd 4 1 845.2.t.b 4
13.b even 2 1 845.2.o.a 4
13.c even 3 1 845.2.k.a 2
13.c even 3 1 inner 845.2.o.b 4
13.d odd 4 1 845.2.t.a 4
13.d odd 4 1 845.2.t.b 4
13.e even 6 1 65.2.k.a yes 2
13.e even 6 1 845.2.o.a 4
13.f odd 12 1 65.2.f.a 2
13.f odd 12 1 845.2.f.a 2
13.f odd 12 1 845.2.t.a 4
13.f odd 12 1 845.2.t.b 4
39.h odd 6 1 585.2.w.b 2
39.k even 12 1 585.2.n.c 2
52.i odd 6 1 1040.2.bg.a 2
52.l even 12 1 1040.2.cd.b 2
65.f even 4 1 845.2.o.a 4
65.h odd 4 1 845.2.t.a 4
65.k even 4 1 inner 845.2.o.b 4
65.l even 6 1 325.2.k.a 2
65.o even 12 1 325.2.k.a 2
65.o even 12 1 845.2.k.a 2
65.o even 12 1 inner 845.2.o.b 4
65.q odd 12 1 845.2.f.a 2
65.q odd 12 1 845.2.t.b 4
65.r odd 12 1 65.2.f.a 2
65.r odd 12 1 325.2.f.a 2
65.r odd 12 1 845.2.t.a 4
65.s odd 12 1 325.2.f.a 2
65.t even 12 1 65.2.k.a yes 2
65.t even 12 1 845.2.o.a 4
195.bc odd 12 1 585.2.w.b 2
195.bf even 12 1 585.2.n.c 2
260.bg even 12 1 1040.2.cd.b 2
260.bl odd 12 1 1040.2.bg.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.f.a 2 13.f odd 12 1
65.2.f.a 2 65.r odd 12 1
65.2.k.a yes 2 13.e even 6 1
65.2.k.a yes 2 65.t even 12 1
325.2.f.a 2 65.r odd 12 1
325.2.f.a 2 65.s odd 12 1
325.2.k.a 2 65.l even 6 1
325.2.k.a 2 65.o even 12 1
585.2.n.c 2 39.k even 12 1
585.2.n.c 2 195.bf even 12 1
585.2.w.b 2 39.h odd 6 1
585.2.w.b 2 195.bc odd 12 1
845.2.f.a 2 13.f odd 12 1
845.2.f.a 2 65.q odd 12 1
845.2.k.a 2 13.c even 3 1
845.2.k.a 2 65.o even 12 1
845.2.o.a 4 13.b even 2 1
845.2.o.a 4 13.e even 6 1
845.2.o.a 4 65.f even 4 1
845.2.o.a 4 65.t even 12 1
845.2.o.b 4 1.a even 1 1 trivial
845.2.o.b 4 13.c even 3 1 inner
845.2.o.b 4 65.k even 4 1 inner
845.2.o.b 4 65.o even 12 1 inner
845.2.t.a 4 13.d odd 4 1
845.2.t.a 4 13.f odd 12 1
845.2.t.a 4 65.h odd 4 1
845.2.t.a 4 65.r odd 12 1
845.2.t.b 4 5.c odd 4 1
845.2.t.b 4 13.d odd 4 1
845.2.t.b 4 13.f odd 12 1
845.2.t.b 4 65.q odd 12 1
1040.2.bg.a 2 52.i odd 6 1
1040.2.bg.a 2 260.bl odd 12 1
1040.2.cd.b 2 52.l even 12 1
1040.2.cd.b 2 260.bg even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\):

\( T_{2}^{2} - T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{4} + 2T_{3}^{3} + 2T_{3}^{2} + 4T_{3} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$5$ \( (T^{2} - 2 T + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4 \) Copy content Toggle raw display
$19$ \( T^{4} + 10 T^{3} + 50 T^{2} + \cdots + 2500 \) Copy content Toggle raw display
$23$ \( T^{4} + 6 T^{3} + 18 T^{2} + 108 T + 324 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 10 T + 50)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + 14 T^{3} + 98 T^{2} + \cdots + 9604 \) Copy content Toggle raw display
$43$ \( T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$47$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 10 T + 50)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 14 T^{3} + 98 T^{2} + \cdots + 9604 \) Copy content Toggle raw display
$61$ \( (T^{2} - 14 T + 196)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$73$ \( (T - 10)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 10 T^{3} + 50 T^{2} + \cdots + 2500 \) Copy content Toggle raw display
$97$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
show more
show less