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Properties

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

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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: \(\Z[a_1, a_2]\)
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.

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

Display 100 coefficients 10 coefficients

Character values

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

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

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
−0.866025 − 0.500000i 1.00000 − 1.73205i −0.500000 − 0.866025i 1.00000i −1.73205 + 1.00000i −3.46410 + 2.00000i 3.00000i −0.500000 − 0.866025i 0.500000 − 0.866025i
316.2 0.866025 + 0.500000i 1.00000 − 1.73205i −0.500000 − 0.866025i − 1.00000i 1.73205 − 1.00000i 3.46410 − 2.00000i − 3.00000i −0.500000 − 0.866025i 0.500000 − 0.866025i
361.1 −0.866025 + 0.500000i 1.00000 + 1.73205i −0.500000 + 0.866025i − 1.00000i −1.73205 − 1.00000i −3.46410 − 2.00000i − 3.00000i −0.500000 + 0.866025i 0.500000 + 0.866025i
361.2 0.866025 − 0.500000i 1.00000 + 1.73205i −0.500000 + 0.866025i 1.00000i 1.73205 + 1.00000i 3.46410 + 2.00000i 3.00000i −0.500000 + 0.866025i 0.500000 + 0.866025i
\(n\): 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
13.c even 3 1 inner
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.b 4
13.b even 2 1 inner 845.2.m.b 4
13.c even 3 1 845.2.c.a 2
13.c even 3 1 inner 845.2.m.b 4
13.d odd 4 1 845.2.e.a 2
13.d odd 4 1 845.2.e.b 2
13.e even 6 1 845.2.c.a 2
13.e even 6 1 inner 845.2.m.b 4
13.f odd 12 1 65.2.a.a ✓ 1
13.f odd 12 1 845.2.a.a 1
13.f odd 12 1 845.2.e.a 2
13.f odd 12 1 845.2.e.b 2
39.k even 12 1 585.2.a.h 1
39.k even 12 1 7605.2.a.f 1
52.l even 12 1 1040.2.a.f 1
65.o even 12 1 325.2.b.b 2
65.s odd 12 1 325.2.a.d 1
65.s odd 12 1 4225.2.a.g 1
65.t even 12 1 325.2.b.b 2
91.bc even 12 1 3185.2.a.e 1
104.u even 12 1 4160.2.a.f 1
104.x odd 12 1 4160.2.a.q 1
143.o even 12 1 7865.2.a.c 1
156.v odd 12 1 9360.2.a.ca 1
195.bc odd 12 1 2925.2.c.h 2
195.bh even 12 1 2925.2.a.f 1
195.bn odd 12 1 2925.2.c.h 2
260.bc even 12 1 5200.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.a.a ✓ 1 13.f odd 12 1
325.2.a.d 1 65.s odd 12 1
325.2.b.b 2 65.o even 12 1
325.2.b.b 2 65.t even 12 1
585.2.a.h 1 39.k even 12 1
845.2.a.a 1 13.f odd 12 1
845.2.c.a 2 13.c even 3 1
845.2.c.a 2 13.e even 6 1
845.2.e.a 2 13.d odd 4 1
845.2.e.a 2 13.f odd 12 1
845.2.e.b 2 13.d odd 4 1
845.2.e.b 2 13.f odd 12 1
845.2.m.b 4 1.a even 1 1 trivial
845.2.m.b 4 13.b even 2 1 inner
845.2.m.b 4 13.c even 3 1 inner
845.2.m.b 4 13.e even 6 1 inner
1040.2.a.f 1 52.l even 12 1
2925.2.a.f 1 195.bh even 12 1
2925.2.c.h 2 195.bc odd 12 1
2925.2.c.h 2 195.bn odd 12 1
3185.2.a.e 1 91.bc even 12 1
4160.2.a.f 1 104.u even 12 1
4160.2.a.q 1 104.x odd 12 1
4225.2.a.g 1 65.s odd 12 1
5200.2.a.d 1 260.bc even 12 1
7605.2.a.f 1 39.k even 12 1
7865.2.a.c 1 143.o even 12 1
9360.2.a.ca 1 156.v odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$11$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$23$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$41$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$43$ \( (T^{2} - 10 T + 100)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$53$ \( (T - 2)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$61$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$71$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$73$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$79$ \( (T + 12)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 256)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$97$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
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