Properties

Label 735.2.q.b
Level $735$
Weight $2$
Character orbit 735.q
Analytic conductor $5.869$
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] = [735,2,Mod(79,735)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(735, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("735.79");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.q (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: \(5.86900454856\)
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 105)
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.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{3} - \zeta_{12}^{2} q^{4} + ( - \zeta_{12}^{2} + 2 \zeta_{12} + 1) q^{5} + q^{6} - 3 \zeta_{12}^{3} q^{8} + ( - \zeta_{12}^{2} + 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12} q^{2} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{3} - \zeta_{12}^{2} q^{4} + ( - \zeta_{12}^{2} + 2 \zeta_{12} + 1) q^{5} + q^{6} - 3 \zeta_{12}^{3} q^{8} + ( - \zeta_{12}^{2} + 1) q^{9} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \zeta_{12}) q^{10} + 6 \zeta_{12}^{2} q^{11} - \zeta_{12} q^{12} - 2 \zeta_{12}^{3} q^{13} + ( - \zeta_{12}^{3} + 2) q^{15} + ( - \zeta_{12}^{2} + 1) q^{16} + (4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{17} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{18} + ( - 6 \zeta_{12}^{2} + 6) q^{19} + ( - 2 \zeta_{12}^{3} - 1) q^{20} + 6 \zeta_{12}^{3} q^{22} - 3 \zeta_{12}^{2} q^{24} + ( - 4 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 4 \zeta_{12}) q^{25} + ( - 2 \zeta_{12}^{2} + 2) q^{26} - \zeta_{12}^{3} q^{27} + 2 q^{29} + ( - \zeta_{12}^{2} + 2 \zeta_{12} + 1) q^{30} - 10 \zeta_{12}^{2} q^{31} + (5 \zeta_{12}^{3} - 5 \zeta_{12}) q^{32} + 6 \zeta_{12} q^{33} - 4 q^{34} - q^{36} + 4 \zeta_{12} q^{37} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{38} - 2 \zeta_{12}^{2} q^{39} + ( - 6 \zeta_{12}^{2} - 3 \zeta_{12} + 6) q^{40} - 2 q^{41} + 4 \zeta_{12}^{3} q^{43} + ( - 6 \zeta_{12}^{2} + 6) q^{44} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} + 2 \zeta_{12}) q^{45} - \zeta_{12}^{3} q^{48} + (3 \zeta_{12}^{3} + 4) q^{50} + (4 \zeta_{12}^{2} - 4) q^{51} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{52} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{53} + ( - \zeta_{12}^{2} + 1) q^{54} + (12 \zeta_{12}^{3} + 6) q^{55} - 6 \zeta_{12}^{3} q^{57} + 2 \zeta_{12} q^{58} + 8 \zeta_{12}^{2} q^{59} + (\zeta_{12}^{3} - 2 \zeta_{12}^{2} - \zeta_{12}) q^{60} + (2 \zeta_{12}^{2} - 2) q^{61} - 10 \zeta_{12}^{3} q^{62} - 7 q^{64} + ( - 4 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{65} + 6 \zeta_{12}^{2} q^{66} + (16 \zeta_{12}^{3} - 16 \zeta_{12}) q^{67} + 4 \zeta_{12} q^{68} + 10 q^{71} - 3 \zeta_{12} q^{72} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{73} + 4 \zeta_{12}^{2} q^{74} + ( - 4 \zeta_{12}^{2} + 3 \zeta_{12} + 4) q^{75} - 6 q^{76} - 2 \zeta_{12}^{3} q^{78} + ( - 4 \zeta_{12}^{2} + 4) q^{79} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} + 2 \zeta_{12}) q^{80} - \zeta_{12}^{2} q^{81} - 2 \zeta_{12} q^{82} + 8 \zeta_{12}^{3} q^{83} + (4 \zeta_{12}^{3} - 8) q^{85} + (4 \zeta_{12}^{2} - 4) q^{86} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{87} + ( - 18 \zeta_{12}^{3} + 18 \zeta_{12}) q^{88} + (6 \zeta_{12}^{2} - 6) q^{89} + ( - \zeta_{12}^{3} + 2) q^{90} - 10 \zeta_{12} q^{93} + ( - 12 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 12 \zeta_{12}) q^{95} + (5 \zeta_{12}^{2} - 5) q^{96} + 2 \zeta_{12}^{3} q^{97} + 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{4} + 2 q^{5} + 4 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{4} + 2 q^{5} + 4 q^{6} + 2 q^{9} + 4 q^{10} + 12 q^{11} + 8 q^{15} + 2 q^{16} + 12 q^{19} - 4 q^{20} - 6 q^{24} + 6 q^{25} + 4 q^{26} + 8 q^{29} + 2 q^{30} - 20 q^{31} - 16 q^{34} - 4 q^{36} - 4 q^{39} + 12 q^{40} - 8 q^{41} + 12 q^{44} - 2 q^{45} + 16 q^{50} - 8 q^{51} + 2 q^{54} + 24 q^{55} + 16 q^{59} - 4 q^{60} - 4 q^{61} - 28 q^{64} + 8 q^{65} + 12 q^{66} + 40 q^{71} + 8 q^{74} + 8 q^{75} - 24 q^{76} + 8 q^{79} - 2 q^{80} - 2 q^{81} - 32 q^{85} - 8 q^{86} - 12 q^{89} + 8 q^{90} - 12 q^{95} - 10 q^{96} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/735\mathbb{Z}\right)^\times\).

\(n\) \(346\) \(442\) \(491\)
\(\chi(n)\) \(-1 + \zeta_{12}^{2}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i −0.866025 + 0.500000i −0.500000 0.866025i −1.23205 1.86603i 1.00000 0 3.00000i 0.500000 0.866025i 0.133975 + 2.23205i
79.2 0.866025 + 0.500000i 0.866025 0.500000i −0.500000 0.866025i 2.23205 + 0.133975i 1.00000 0 3.00000i 0.500000 0.866025i 1.86603 + 1.23205i
214.1 −0.866025 + 0.500000i −0.866025 0.500000i −0.500000 + 0.866025i −1.23205 + 1.86603i 1.00000 0 3.00000i 0.500000 + 0.866025i 0.133975 2.23205i
214.2 0.866025 0.500000i 0.866025 + 0.500000i −0.500000 + 0.866025i 2.23205 0.133975i 1.00000 0 3.00000i 0.500000 + 0.866025i 1.86603 1.23205i
\(n\): 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
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.2.q.b 4
5.b even 2 1 inner 735.2.q.b 4
7.b odd 2 1 735.2.q.a 4
7.c even 3 1 735.2.d.a 2
7.c even 3 1 inner 735.2.q.b 4
7.d odd 6 1 105.2.d.a 2
7.d odd 6 1 735.2.q.a 4
21.g even 6 1 315.2.d.c 2
21.h odd 6 1 2205.2.d.f 2
28.f even 6 1 1680.2.t.f 2
35.c odd 2 1 735.2.q.a 4
35.i odd 6 1 105.2.d.a 2
35.i odd 6 1 735.2.q.a 4
35.j even 6 1 735.2.d.a 2
35.j even 6 1 inner 735.2.q.b 4
35.k even 12 1 525.2.a.b 1
35.k even 12 1 525.2.a.c 1
35.l odd 12 1 3675.2.a.d 1
35.l odd 12 1 3675.2.a.l 1
84.j odd 6 1 5040.2.t.e 2
105.o odd 6 1 2205.2.d.f 2
105.p even 6 1 315.2.d.c 2
105.w odd 12 1 1575.2.a.e 1
105.w odd 12 1 1575.2.a.i 1
140.s even 6 1 1680.2.t.f 2
140.x odd 12 1 8400.2.a.bj 1
140.x odd 12 1 8400.2.a.ch 1
420.be odd 6 1 5040.2.t.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.d.a 2 7.d odd 6 1
105.2.d.a 2 35.i odd 6 1
315.2.d.c 2 21.g even 6 1
315.2.d.c 2 105.p even 6 1
525.2.a.b 1 35.k even 12 1
525.2.a.c 1 35.k even 12 1
735.2.d.a 2 7.c even 3 1
735.2.d.a 2 35.j even 6 1
735.2.q.a 4 7.b odd 2 1
735.2.q.a 4 7.d odd 6 1
735.2.q.a 4 35.c odd 2 1
735.2.q.a 4 35.i odd 6 1
735.2.q.b 4 1.a even 1 1 trivial
735.2.q.b 4 5.b even 2 1 inner
735.2.q.b 4 7.c even 3 1 inner
735.2.q.b 4 35.j even 6 1 inner
1575.2.a.e 1 105.w odd 12 1
1575.2.a.i 1 105.w odd 12 1
1680.2.t.f 2 28.f even 6 1
1680.2.t.f 2 140.s even 6 1
2205.2.d.f 2 21.h odd 6 1
2205.2.d.f 2 105.o odd 6 1
3675.2.a.d 1 35.l odd 12 1
3675.2.a.l 1 35.l odd 12 1
5040.2.t.e 2 84.j odd 6 1
5040.2.t.e 2 420.be odd 6 1
8400.2.a.bj 1 140.x odd 12 1
8400.2.a.ch 1 140.x odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(735, [\chi])\):

\( T_{2}^{4} - T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{19}^{2} - 6T_{19} + 36 \) Copy content Toggle raw display
\( T_{73}^{4} - 36T_{73}^{2} + 1296 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} - 2 T^{3} - T^{2} - 10 T + 25 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$19$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T - 2)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 10 T + 100)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$41$ \( (T + 2)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$59$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 256 T^{2} + 65536 \) Copy content Toggle raw display
$71$ \( (T - 10)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
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