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Properties

Label 70.2.c.a
Level $70$
Weight $2$
Character orbit 70.c
Analytic conductor $0.559$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [70,2,Mod(29,70)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(70, 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("70.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 70.c (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: \(0.558952814149\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{3} + \beta_1) q^{3} - q^{4} + ( - \beta_{2} - \beta_1 + 1) q^{5} + (\beta_{3} - \beta_1) q^{6} + \beta_{2} q^{7} - \beta_{2} q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{3} + \beta_1) q^{3} - q^{4} + ( - \beta_{2} - \beta_1 + 1) q^{5} + (\beta_{3} - \beta_1) q^{6} + \beta_{2} q^{7} - \beta_{2} q^{8} - 3 q^{9} + ( - \beta_{3} + \beta_{2} + 1) q^{10} + ( - 2 \beta_{3} + 2 \beta_1) q^{11} + ( - \beta_{3} - \beta_1) q^{12} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{13} - q^{14} + ( - 3 \beta_{2} + 2 \beta_1 + 3) q^{15} + q^{16} - 2 \beta_{2} q^{17} - 3 \beta_{2} q^{18} + (\beta_{3} - \beta_1 - 4) q^{19} + (\beta_{2} + \beta_1 - 1) q^{20} + (\beta_{3} - \beta_1) q^{21} + (2 \beta_{3} + 2 \beta_1) q^{22} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{23} + ( - \beta_{3} + \beta_1) q^{24} + (2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{25} + ( - \beta_{3} + \beta_1 - 2) q^{26} - \beta_{2} q^{28} + ( - 2 \beta_{3} + 2 \beta_1 - 2) q^{29} + (2 \beta_{3} + 3 \beta_{2} + 3) q^{30} + (2 \beta_{3} - 2 \beta_1 + 4) q^{31} + \beta_{2} q^{32} + 12 \beta_{2} q^{33} + 2 q^{34} + ( - \beta_{3} + \beta_{2} + 1) q^{35} + 3 q^{36} - 2 \beta_{2} q^{37} + ( - \beta_{3} - 4 \beta_{2} - \beta_1) q^{38} + (2 \beta_{3} - 2 \beta_1 + 6) q^{39} + (\beta_{3} - \beta_{2} - 1) q^{40} + (2 \beta_{3} - 2 \beta_1 - 6) q^{41} + ( - \beta_{3} - \beta_1) q^{42} + (2 \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{43} + (2 \beta_{3} - 2 \beta_1) q^{44} + (3 \beta_{2} + 3 \beta_1 - 3) q^{45} + ( - 2 \beta_{3} + 2 \beta_1 + 2) q^{46} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1) q^{47} + (\beta_{3} + \beta_1) q^{48} - q^{49} + ( - 2 \beta_{3} - 2 \beta_1 - 1) q^{50} + ( - 2 \beta_{3} + 2 \beta_1) q^{51} + (\beta_{3} - 2 \beta_{2} + \beta_1) q^{52} + (2 \beta_{3} - 6 \beta_{2} + 2 \beta_1) q^{53} + ( - 4 \beta_{3} - 6 \beta_{2} - 6) q^{55} + q^{56} + ( - 4 \beta_{3} - 6 \beta_{2} - 4 \beta_1) q^{57} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{58} + ( - \beta_{3} + \beta_1 + 4) q^{59} + (3 \beta_{2} - 2 \beta_1 - 3) q^{60} + ( - \beta_{3} + \beta_1 + 6) q^{61} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{62} - 3 \beta_{2} q^{63} - q^{64} + ( - 2 \beta_{3} + 5 \beta_{2} + \cdots - 1) q^{65}+ \cdots + (6 \beta_{3} - 6 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 4 q^{5} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 4 q^{5} - 12 q^{9} + 4 q^{10} - 4 q^{14} + 12 q^{15} + 4 q^{16} - 16 q^{19} - 4 q^{20} - 8 q^{26} - 8 q^{29} + 12 q^{30} + 16 q^{31} + 8 q^{34} + 4 q^{35} + 12 q^{36} + 24 q^{39} - 4 q^{40} - 24 q^{41} - 12 q^{45} + 8 q^{46} - 4 q^{49} - 4 q^{50} - 24 q^{55} + 4 q^{56} + 16 q^{59} - 12 q^{60} + 24 q^{61} - 4 q^{64} - 4 q^{65} - 48 q^{66} + 48 q^{69} - 4 q^{70} - 24 q^{71} + 8 q^{74} + 16 q^{76} - 8 q^{79} + 4 q^{80} - 36 q^{81} - 8 q^{85} - 16 q^{86} + 40 q^{89} - 12 q^{90} - 8 q^{91} + 16 q^{94} - 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 3 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/70\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(57\)
\(\chi(n)\) \(1\) \(-1\)

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
1.22474 − 1.22474i
−1.22474 + 1.22474i
−1.22474 − 1.22474i
1.22474 + 1.22474i
− 1.00000i − 2.44949i −1.00000 −0.224745 + 2.22474i −2.44949 − 1.00000i 1.00000i −3.00000 2.22474 + 0.224745i
29.2 − 1.00000i 2.44949i −1.00000 2.22474 − 0.224745i 2.44949 − 1.00000i 1.00000i −3.00000 −0.224745 − 2.22474i
29.3 1.00000i − 2.44949i −1.00000 2.22474 + 0.224745i 2.44949 1.00000i − 1.00000i −3.00000 −0.224745 + 2.22474i
29.4 1.00000i 2.44949i −1.00000 −0.224745 − 2.22474i −2.44949 1.00000i − 1.00000i −3.00000 2.22474 − 0.224745i
\(n\): 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 70.2.c.a ✓ 4
3.b odd 2 1 630.2.g.g 4
4.b odd 2 1 560.2.g.e 4
5.b even 2 1 inner 70.2.c.a ✓ 4
5.c odd 4 1 350.2.a.g 2
5.c odd 4 1 350.2.a.h 2
7.b odd 2 1 490.2.c.e 4
7.c even 3 2 490.2.i.c 8
7.d odd 6 2 490.2.i.f 8
8.b even 2 1 2240.2.g.j 4
8.d odd 2 1 2240.2.g.i 4
12.b even 2 1 5040.2.t.t 4
15.d odd 2 1 630.2.g.g 4
15.e even 4 1 3150.2.a.bs 2
15.e even 4 1 3150.2.a.bt 2
20.d odd 2 1 560.2.g.e 4
20.e even 4 1 2800.2.a.bl 2
20.e even 4 1 2800.2.a.bm 2
35.c odd 2 1 490.2.c.e 4
35.f even 4 1 2450.2.a.bl 2
35.f even 4 1 2450.2.a.bq 2
35.i odd 6 2 490.2.i.f 8
35.j even 6 2 490.2.i.c 8
40.e odd 2 1 2240.2.g.i 4
40.f even 2 1 2240.2.g.j 4
60.h even 2 1 5040.2.t.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.c.a ✓ 4 1.a even 1 1 trivial
70.2.c.a ✓ 4 5.b even 2 1 inner
350.2.a.g 2 5.c odd 4 1
350.2.a.h 2 5.c odd 4 1
490.2.c.e 4 7.b odd 2 1
490.2.c.e 4 35.c odd 2 1
490.2.i.c 8 7.c even 3 2
490.2.i.c 8 35.j even 6 2
490.2.i.f 8 7.d odd 6 2
490.2.i.f 8 35.i odd 6 2
560.2.g.e 4 4.b odd 2 1
560.2.g.e 4 20.d odd 2 1
630.2.g.g 4 3.b odd 2 1
630.2.g.g 4 15.d odd 2 1
2240.2.g.i 4 8.d odd 2 1
2240.2.g.i 4 40.e odd 2 1
2240.2.g.j 4 8.b even 2 1
2240.2.g.j 4 40.f even 2 1
2450.2.a.bl 2 35.f even 4 1
2450.2.a.bq 2 35.f even 4 1
2800.2.a.bl 2 20.e even 4 1
2800.2.a.bm 2 20.e even 4 1
3150.2.a.bs 2 15.e even 4 1
3150.2.a.bt 2 15.e even 4 1
5040.2.t.t 4 12.b even 2 1
5040.2.t.t 4 60.h even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(70, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - 4 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 20T^{2} + 4 \) Copy content Toggle raw display
$17$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 8 T + 10)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 56T^{2} + 400 \) Copy content Toggle raw display
$29$ \( (T^{2} + 4 T - 20)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 8 T - 8)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 12 T + 12)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 80T^{2} + 64 \) Copy content Toggle raw display
$47$ \( T^{4} + 80T^{2} + 64 \) Copy content Toggle raw display
$53$ \( T^{4} + 120T^{2} + 144 \) Copy content Toggle raw display
$59$ \( (T^{2} - 8 T + 10)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 12 T + 30)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 12 T + 12)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 56T^{2} + 400 \) Copy content Toggle raw display
$79$ \( (T^{2} + 4 T - 20)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$89$ \( (T - 10)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 264T^{2} + 3600 \) Copy content Toggle raw display
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