Properties

Label 845.2.f.a
Level $845$
Weight $2$
Character orbit 845.f
Analytic conductor $6.747$
Analytic rank $0$
Dimension $2$
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(408,845)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(845, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("845.408");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.f (of order \(4\), degree \(2\), 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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) 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: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(171\) \(677\)
\(\chi(n)\) \(i\) \(i\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
408.1
1.00000i
1.00000i
1.00000i 1.00000 + 1.00000i 1.00000 2.00000 1.00000i −1.00000 + 1.00000i 2.00000 3.00000i 1.00000i 1.00000 + 2.00000i
437.1 1.00000i 1.00000 1.00000i 1.00000 2.00000 + 1.00000i −1.00000 1.00000i 2.00000 3.00000i 1.00000i 1.00000 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.f even 4 1 inner

Twists

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

Hecke kernels

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

\( T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{7} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$5$ \( T^{2} - 4T + 5 \) Copy content Toggle raw display
$7$ \( (T - 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$19$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 10T + 50 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 14T + 98 \) Copy content Toggle raw display
$43$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$47$ \( (T + 6)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
$59$ \( T^{2} + 14T + 98 \) Copy content Toggle raw display
$61$ \( (T + 14)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$73$ \( T^{2} + 100 \) Copy content Toggle raw display
$79$ \( T^{2} + 4 \) Copy content Toggle raw display
$83$ \( (T + 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
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