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Properties

Label 845.2.m.c
Level $845$
Weight $2$
Character orbit 845.m
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(316,845)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(845, base_ring=CyclotomicField(6))
 
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("845.316");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.m (of order \(6\), degree \(2\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
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: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{2} + 2) q^{2} + (\zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{3} + ( - \zeta_{12}^{2} + 1) q^{4} - \zeta_{12}^{3} q^{5} + ( - \zeta_{12}^{2} + 3 \zeta_{12} - 1) q^{6} + 2 \zeta_{12} q^{7} + (2 \zeta_{12}^{2} - 1) q^{8} + ( - 4 \zeta_{12}^{3} + \zeta_{12}^{2} + \cdots - 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{2} + 2) q^{2} + (\zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{3} + ( - \zeta_{12}^{2} + 1) q^{4} - \zeta_{12}^{3} q^{5} + ( - \zeta_{12}^{2} + 3 \zeta_{12} - 1) q^{6} + 2 \zeta_{12} q^{7} + (2 \zeta_{12}^{2} - 1) q^{8} + ( - 4 \zeta_{12}^{3} + \zeta_{12}^{2} + \cdots - 1) q^{9} + \cdots + (9 \zeta_{12}^{3} - 14 \zeta_{12}^{2} + 7) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{2} - 2 q^{3} + 2 q^{4} - 6 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{2} - 2 q^{3} + 2 q^{4} - 6 q^{6} - 2 q^{9} - 6 q^{11} - 4 q^{12} + 6 q^{15} + 10 q^{16} - 18 q^{19} - 6 q^{22} + 6 q^{23} + 6 q^{24} - 4 q^{25} + 16 q^{27} + 12 q^{29} + 6 q^{30} + 18 q^{32} + 24 q^{33} + 4 q^{35} + 2 q^{36} - 36 q^{38} - 12 q^{41} + 12 q^{42} + 10 q^{43} - 12 q^{45} + 18 q^{46} + 10 q^{48} - 6 q^{49} - 6 q^{50} + 24 q^{51} + 24 q^{54} - 6 q^{55} + 36 q^{58} - 42 q^{59} - 4 q^{61} - 18 q^{62} + 24 q^{63} - 4 q^{64} + 48 q^{66} - 36 q^{67} + 6 q^{71} - 6 q^{72} + 2 q^{75} - 18 q^{76} + 24 q^{77} + 8 q^{79} - 2 q^{81} - 12 q^{82} + 12 q^{84} - 12 q^{85} - 6 q^{88} - 24 q^{89} - 24 q^{90} + 12 q^{92} - 48 q^{93} + 2 q^{95} - 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(171\) \(677\)
\(\chi(n)\) \(1 - \zeta_{12}^{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
316.1
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i
0.866025 + 0.500000i
1.50000 + 0.866025i −1.36603 + 2.36603i 0.500000 + 0.866025i − 1.00000i −4.09808 + 2.36603i −1.73205 + 1.00000i − 1.73205i −2.23205 − 3.86603i 0.866025 − 1.50000i
316.2 1.50000 + 0.866025i 0.366025 − 0.633975i 0.500000 + 0.866025i 1.00000i 1.09808 − 0.633975i 1.73205 − 1.00000i − 1.73205i 1.23205 + 2.13397i −0.866025 + 1.50000i
361.1 1.50000 − 0.866025i −1.36603 − 2.36603i 0.500000 − 0.866025i 1.00000i −4.09808 − 2.36603i −1.73205 − 1.00000i 1.73205i −2.23205 + 3.86603i 0.866025 + 1.50000i
361.2 1.50000 − 0.866025i 0.366025 + 0.633975i 0.500000 − 0.866025i − 1.00000i 1.09808 + 0.633975i 1.73205 + 1.00000i 1.73205i 1.23205 − 2.13397i −0.866025 − 1.50000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{4} + 6 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$19$ \( T^{4} + 18 T^{3} + \cdots + 676 \) Copy content Toggle raw display
$23$ \( T^{4} - 6 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$29$ \( T^{4} - 12 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$31$ \( T^{4} + 104T^{2} + 4 \) Copy content Toggle raw display
$37$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$41$ \( (T^{2} + 6 T + 12)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 10 T^{3} + \cdots + 4 \) 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
$59$ \( T^{4} + 42 T^{3} + \cdots + 19044 \) Copy content Toggle raw display
$61$ \( T^{4} + 4 T^{3} + \cdots + 10816 \) Copy content Toggle raw display
$67$ \( T^{4} + 36 T^{3} + \cdots + 8464 \) Copy content Toggle raw display
$71$ \( T^{4} - 6 T^{3} + \cdots + 36 \) 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
$89$ \( T^{4} + 24 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$97$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
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