Properties

Label 35.2.b.a
Level $35$
Weight $2$
Character orbit 35.b
Analytic conductor $0.279$
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] = [35,2,Mod(29,35)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(35, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("35.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 35.b (of order \(2\), degree \(1\), 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: \(0.279476407074\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 2 i q^{2} - i q^{3} - 2 q^{4} + ( - i - 2) q^{5} + 2 q^{6} - i q^{7} + 2 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 i q^{2} - i q^{3} - 2 q^{4} + ( - i - 2) q^{5} + 2 q^{6} - i q^{7} + 2 q^{9} + ( - 4 i + 2) q^{10} - 3 q^{11} + 2 i q^{12} - i q^{13} + 2 q^{14} + (2 i - 1) q^{15} - 4 q^{16} + 7 i q^{17} + 4 i q^{18} + (2 i + 4) q^{20} - q^{21} - 6 i q^{22} - 6 i q^{23} + (4 i + 3) q^{25} + 2 q^{26} - 5 i q^{27} + 2 i q^{28} + 5 q^{29} + ( - 2 i - 4) q^{30} + 2 q^{31} - 8 i q^{32} + 3 i q^{33} - 14 q^{34} + (2 i - 1) q^{35} - 4 q^{36} + 2 i q^{37} - q^{39} + 2 q^{41} - 2 i q^{42} + 4 i q^{43} + 6 q^{44} + ( - 2 i - 4) q^{45} + 12 q^{46} - 3 i q^{47} + 4 i q^{48} - q^{49} + (6 i - 8) q^{50} + 7 q^{51} + 2 i q^{52} - 6 i q^{53} + 10 q^{54} + (3 i + 6) q^{55} + 10 i q^{58} - 10 q^{59} + ( - 4 i + 2) q^{60} - 8 q^{61} + 4 i q^{62} - 2 i q^{63} + 8 q^{64} + (2 i - 1) q^{65} - 6 q^{66} + 2 i q^{67} - 14 i q^{68} - 6 q^{69} + ( - 2 i - 4) q^{70} - 8 q^{71} - 6 i q^{73} - 4 q^{74} + ( - 3 i + 4) q^{75} + 3 i q^{77} - 2 i q^{78} + 5 q^{79} + (4 i + 8) q^{80} + q^{81} + 4 i q^{82} + 4 i q^{83} + 2 q^{84} + ( - 14 i + 7) q^{85} - 8 q^{86} - 5 i q^{87} + ( - 8 i + 4) q^{90} - q^{91} + 12 i q^{92} - 2 i q^{93} + 6 q^{94} - 8 q^{96} + 7 i q^{97} - 2 i q^{98} - 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} - 4 q^{5} + 4 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{4} - 4 q^{5} + 4 q^{6} + 4 q^{9} + 4 q^{10} - 6 q^{11} + 4 q^{14} - 2 q^{15} - 8 q^{16} + 8 q^{20} - 2 q^{21} + 6 q^{25} + 4 q^{26} + 10 q^{29} - 8 q^{30} + 4 q^{31} - 28 q^{34} - 2 q^{35} - 8 q^{36} - 2 q^{39} + 4 q^{41} + 12 q^{44} - 8 q^{45} + 24 q^{46} - 2 q^{49} - 16 q^{50} + 14 q^{51} + 20 q^{54} + 12 q^{55} - 20 q^{59} + 4 q^{60} - 16 q^{61} + 16 q^{64} - 2 q^{65} - 12 q^{66} - 12 q^{69} - 8 q^{70} - 16 q^{71} - 8 q^{74} + 8 q^{75} + 10 q^{79} + 16 q^{80} + 2 q^{81} + 4 q^{84} + 14 q^{85} - 16 q^{86} + 8 q^{90} - 2 q^{91} + 12 q^{94} - 16 q^{96} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
1.00000i
1.00000i
2.00000i 1.00000i −2.00000 −2.00000 + 1.00000i 2.00000 1.00000i 0 2.00000 2.00000 + 4.00000i
29.2 2.00000i 1.00000i −2.00000 −2.00000 1.00000i 2.00000 1.00000i 0 2.00000 2.00000 4.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.2.b.a 2
3.b odd 2 1 315.2.d.a 2
4.b odd 2 1 560.2.g.b 2
5.b even 2 1 inner 35.2.b.a 2
5.c odd 4 1 175.2.a.a 1
5.c odd 4 1 175.2.a.c 1
7.b odd 2 1 245.2.b.a 2
7.c even 3 2 245.2.j.e 4
7.d odd 6 2 245.2.j.d 4
8.b even 2 1 2240.2.g.h 2
8.d odd 2 1 2240.2.g.g 2
12.b even 2 1 5040.2.t.p 2
15.d odd 2 1 315.2.d.a 2
15.e even 4 1 1575.2.a.a 1
15.e even 4 1 1575.2.a.k 1
20.d odd 2 1 560.2.g.b 2
20.e even 4 1 2800.2.a.l 1
20.e even 4 1 2800.2.a.w 1
21.c even 2 1 2205.2.d.b 2
35.c odd 2 1 245.2.b.a 2
35.f even 4 1 1225.2.a.a 1
35.f even 4 1 1225.2.a.i 1
35.i odd 6 2 245.2.j.d 4
35.j even 6 2 245.2.j.e 4
40.e odd 2 1 2240.2.g.g 2
40.f even 2 1 2240.2.g.h 2
60.h even 2 1 5040.2.t.p 2
105.g even 2 1 2205.2.d.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.b.a 2 1.a even 1 1 trivial
35.2.b.a 2 5.b even 2 1 inner
175.2.a.a 1 5.c odd 4 1
175.2.a.c 1 5.c odd 4 1
245.2.b.a 2 7.b odd 2 1
245.2.b.a 2 35.c odd 2 1
245.2.j.d 4 7.d odd 6 2
245.2.j.d 4 35.i odd 6 2
245.2.j.e 4 7.c even 3 2
245.2.j.e 4 35.j even 6 2
315.2.d.a 2 3.b odd 2 1
315.2.d.a 2 15.d odd 2 1
560.2.g.b 2 4.b odd 2 1
560.2.g.b 2 20.d odd 2 1
1225.2.a.a 1 35.f even 4 1
1225.2.a.i 1 35.f even 4 1
1575.2.a.a 1 15.e even 4 1
1575.2.a.k 1 15.e even 4 1
2205.2.d.b 2 21.c even 2 1
2205.2.d.b 2 105.g even 2 1
2240.2.g.g 2 8.d odd 2 1
2240.2.g.g 2 40.e odd 2 1
2240.2.g.h 2 8.b even 2 1
2240.2.g.h 2 40.f even 2 1
2800.2.a.l 1 20.e even 4 1
2800.2.a.w 1 20.e even 4 1
5040.2.t.p 2 12.b even 2 1
5040.2.t.p 2 60.h even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(35, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + 4T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T + 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 49 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 36 \) Copy content Toggle raw display
$29$ \( (T - 5)^{2} \) Copy content Toggle raw display
$31$ \( (T - 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 9 \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( (T + 10)^{2} \) Copy content Toggle raw display
$61$ \( (T + 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 4 \) Copy content Toggle raw display
$71$ \( (T + 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 36 \) Copy content Toggle raw display
$79$ \( (T - 5)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 49 \) Copy content Toggle raw display
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