Properties

Label 34.2.c.b
Level $34$
Weight $2$
Character orbit 34.c
Analytic conductor $0.271$
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] = [34,2,Mod(13,34)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(34, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("34.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 34 = 2 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 34.c (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: \(0.271491366872\)
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: yes
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.

\(f(q)\) \(=\) \( q + i q^{2} - q^{4} + ( - i - 1) q^{5} - i q^{8} - 3 i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} - q^{4} + ( - i - 1) q^{5} - i q^{8} - 3 i q^{9} + ( - i + 1) q^{10} + (4 i - 4) q^{11} + 4 q^{13} + q^{16} + (4 i - 1) q^{17} + 3 q^{18} + 4 i q^{19} + (i + 1) q^{20} + ( - 4 i - 4) q^{22} + ( - 4 i + 4) q^{23} - 3 i q^{25} + 4 i q^{26} + ( - 3 i - 3) q^{29} + ( - 4 i - 4) q^{31} + i q^{32} + ( - i - 4) q^{34} + 3 i q^{36} + (3 i + 3) q^{37} - 4 q^{38} + (i - 1) q^{40} + ( - i + 1) q^{41} - 4 i q^{43} + ( - 4 i + 4) q^{44} + (3 i - 3) q^{45} + (4 i + 4) q^{46} + 8 q^{47} + 7 i q^{49} + 3 q^{50} - 4 q^{52} - 4 i q^{53} + 8 q^{55} + ( - 3 i + 3) q^{58} + 4 i q^{59} + (9 i - 9) q^{61} + ( - 4 i + 4) q^{62} - q^{64} + ( - 4 i - 4) q^{65} - 12 q^{67} + ( - 4 i + 1) q^{68} + ( - 4 i - 4) q^{71} - 3 q^{72} + (5 i + 5) q^{73} + (3 i - 3) q^{74} - 4 i q^{76} + ( - 8 i + 8) q^{79} + ( - i - 1) q^{80} - 9 q^{81} + (i + 1) q^{82} - 4 i q^{83} + ( - 3 i + 5) q^{85} + 4 q^{86} + (4 i + 4) q^{88} + ( - 3 i - 3) q^{90} + (4 i - 4) q^{92} + 8 i q^{94} + ( - 4 i + 4) q^{95} + (3 i + 3) q^{97} - 7 q^{98} + (12 i + 12) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 2 q^{5} + 2 q^{10} - 8 q^{11} + 8 q^{13} + 2 q^{16} - 2 q^{17} + 6 q^{18} + 2 q^{20} - 8 q^{22} + 8 q^{23} - 6 q^{29} - 8 q^{31} - 8 q^{34} + 6 q^{37} - 8 q^{38} - 2 q^{40} + 2 q^{41} + 8 q^{44} - 6 q^{45} + 8 q^{46} + 16 q^{47} + 6 q^{50} - 8 q^{52} + 16 q^{55} + 6 q^{58} - 18 q^{61} + 8 q^{62} - 2 q^{64} - 8 q^{65} - 24 q^{67} + 2 q^{68} - 8 q^{71} - 6 q^{72} + 10 q^{73} - 6 q^{74} + 16 q^{79} - 2 q^{80} - 18 q^{81} + 2 q^{82} + 10 q^{85} + 8 q^{86} + 8 q^{88} - 6 q^{90} - 8 q^{92} + 8 q^{95} + 6 q^{97} - 14 q^{98} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/34\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(i\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
1.00000i
1.00000i
1.00000i 0 −1.00000 −1.00000 1.00000i 0 0 1.00000i 3.00000i 1.00000 1.00000i
21.1 1.00000i 0 −1.00000 −1.00000 + 1.00000i 0 0 1.00000i 3.00000i 1.00000 + 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 34.2.c.b 2
3.b odd 2 1 306.2.g.d 2
4.b odd 2 1 272.2.o.c 2
5.b even 2 1 850.2.h.c 2
5.c odd 4 1 850.2.g.a 2
5.c odd 4 1 850.2.g.d 2
8.b even 2 1 1088.2.o.k 2
8.d odd 2 1 1088.2.o.i 2
12.b even 2 1 2448.2.be.j 2
17.b even 2 1 578.2.c.b 2
17.c even 4 1 inner 34.2.c.b 2
17.c even 4 1 578.2.c.b 2
17.d even 8 2 578.2.a.b 2
17.d even 8 2 578.2.b.c 2
17.e odd 16 8 578.2.d.f 8
51.f odd 4 1 306.2.g.d 2
51.g odd 8 2 5202.2.a.bb 2
68.f odd 4 1 272.2.o.c 2
68.g odd 8 2 4624.2.a.l 2
85.f odd 4 1 850.2.g.d 2
85.i odd 4 1 850.2.g.a 2
85.j even 4 1 850.2.h.c 2
136.i even 4 1 1088.2.o.k 2
136.j odd 4 1 1088.2.o.i 2
204.l even 4 1 2448.2.be.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
34.2.c.b 2 1.a even 1 1 trivial
34.2.c.b 2 17.c even 4 1 inner
272.2.o.c 2 4.b odd 2 1
272.2.o.c 2 68.f odd 4 1
306.2.g.d 2 3.b odd 2 1
306.2.g.d 2 51.f odd 4 1
578.2.a.b 2 17.d even 8 2
578.2.b.c 2 17.d even 8 2
578.2.c.b 2 17.b even 2 1
578.2.c.b 2 17.c even 4 1
578.2.d.f 8 17.e odd 16 8
850.2.g.a 2 5.c odd 4 1
850.2.g.a 2 85.i odd 4 1
850.2.g.d 2 5.c odd 4 1
850.2.g.d 2 85.f odd 4 1
850.2.h.c 2 5.b even 2 1
850.2.h.c 2 85.j even 4 1
1088.2.o.i 2 8.d odd 2 1
1088.2.o.i 2 136.j odd 4 1
1088.2.o.k 2 8.b even 2 1
1088.2.o.k 2 136.i even 4 1
2448.2.be.j 2 12.b even 2 1
2448.2.be.j 2 204.l even 4 1
4624.2.a.l 2 68.g odd 8 2
5202.2.a.bb 2 51.g odd 8 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{2}^{\mathrm{new}}(34, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2T + 2 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 8T + 32 \) Copy content Toggle raw display
$13$ \( (T - 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 17 \) Copy content Toggle raw display
$19$ \( T^{2} + 16 \) Copy content Toggle raw display
$23$ \( T^{2} - 8T + 32 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 18 \) Copy content Toggle raw display
$31$ \( T^{2} + 8T + 32 \) Copy content Toggle raw display
$37$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$41$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( (T - 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 16 \) Copy content Toggle raw display
$59$ \( T^{2} + 16 \) Copy content Toggle raw display
$61$ \( T^{2} + 18T + 162 \) Copy content Toggle raw display
$67$ \( (T + 12)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 8T + 32 \) Copy content Toggle raw display
$73$ \( T^{2} - 10T + 50 \) Copy content Toggle raw display
$79$ \( T^{2} - 16T + 128 \) Copy content Toggle raw display
$83$ \( T^{2} + 16 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
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