Properties

Label 70.4.e.b
Level $70$
Weight $4$
Character orbit 70.e
Analytic conductor $4.130$
Analytic rank $0$
Dimension $2$
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] = [70,4,Mod(11,70)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(70, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("70.11");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 70.e (of order \(3\), 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: \(4.13013370040\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{2} + (10 \zeta_{6} - 10) q^{3} + (4 \zeta_{6} - 4) q^{4} + 5 \zeta_{6} q^{5} - 20 q^{6} + ( - 14 \zeta_{6} + 21) q^{7} - 8 q^{8} - 73 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 \zeta_{6} q^{2} + (10 \zeta_{6} - 10) q^{3} + (4 \zeta_{6} - 4) q^{4} + 5 \zeta_{6} q^{5} - 20 q^{6} + ( - 14 \zeta_{6} + 21) q^{7} - 8 q^{8} - 73 \zeta_{6} q^{9} + (10 \zeta_{6} - 10) q^{10} + (53 \zeta_{6} - 53) q^{11} - 40 \zeta_{6} q^{12} + 25 q^{13} + (14 \zeta_{6} + 28) q^{14} - 50 q^{15} - 16 \zeta_{6} q^{16} + (14 \zeta_{6} - 14) q^{17} + ( - 146 \zeta_{6} + 146) q^{18} + 95 \zeta_{6} q^{19} - 20 q^{20} + (210 \zeta_{6} - 70) q^{21} - 106 q^{22} - \zeta_{6} q^{23} + ( - 80 \zeta_{6} + 80) q^{24} + (25 \zeta_{6} - 25) q^{25} + 50 \zeta_{6} q^{26} + 460 q^{27} + (84 \zeta_{6} - 28) q^{28} - 206 q^{29} - 100 \zeta_{6} q^{30} + (108 \zeta_{6} - 108) q^{31} + ( - 32 \zeta_{6} + 32) q^{32} - 530 \zeta_{6} q^{33} - 28 q^{34} + (35 \zeta_{6} + 70) q^{35} + 292 q^{36} + 57 \zeta_{6} q^{37} + (190 \zeta_{6} - 190) q^{38} + (250 \zeta_{6} - 250) q^{39} - 40 \zeta_{6} q^{40} + 243 q^{41} + (280 \zeta_{6} - 420) q^{42} + 434 q^{43} - 212 \zeta_{6} q^{44} + ( - 365 \zeta_{6} + 365) q^{45} + ( - 2 \zeta_{6} + 2) q^{46} + 231 \zeta_{6} q^{47} + 160 q^{48} + ( - 392 \zeta_{6} + 245) q^{49} - 50 q^{50} - 140 \zeta_{6} q^{51} + (100 \zeta_{6} - 100) q^{52} + (263 \zeta_{6} - 263) q^{53} + 920 \zeta_{6} q^{54} - 265 q^{55} + (112 \zeta_{6} - 168) q^{56} - 950 q^{57} - 412 \zeta_{6} q^{58} + (24 \zeta_{6} - 24) q^{59} + ( - 200 \zeta_{6} + 200) q^{60} - 116 \zeta_{6} q^{61} - 216 q^{62} + ( - 511 \zeta_{6} - 1022) q^{63} + 64 q^{64} + 125 \zeta_{6} q^{65} + ( - 1060 \zeta_{6} + 1060) q^{66} + ( - 204 \zeta_{6} + 204) q^{67} - 56 \zeta_{6} q^{68} + 10 q^{69} + (210 \zeta_{6} - 70) q^{70} + 484 q^{71} + 584 \zeta_{6} q^{72} + ( - 692 \zeta_{6} + 692) q^{73} + (114 \zeta_{6} - 114) q^{74} - 250 \zeta_{6} q^{75} - 380 q^{76} + (1113 \zeta_{6} - 371) q^{77} - 500 q^{78} - 466 \zeta_{6} q^{79} + ( - 80 \zeta_{6} + 80) q^{80} + (2629 \zeta_{6} - 2629) q^{81} + 486 \zeta_{6} q^{82} + 228 q^{83} + ( - 280 \zeta_{6} - 560) q^{84} - 70 q^{85} + 868 \zeta_{6} q^{86} + ( - 2060 \zeta_{6} + 2060) q^{87} + ( - 424 \zeta_{6} + 424) q^{88} + 362 \zeta_{6} q^{89} + 730 q^{90} + ( - 350 \zeta_{6} + 525) q^{91} + 4 q^{92} - 1080 \zeta_{6} q^{93} + (462 \zeta_{6} - 462) q^{94} + (475 \zeta_{6} - 475) q^{95} + 320 \zeta_{6} q^{96} + 854 q^{97} + ( - 294 \zeta_{6} + 784) q^{98} + 3869 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 10 q^{3} - 4 q^{4} + 5 q^{5} - 40 q^{6} + 28 q^{7} - 16 q^{8} - 73 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 10 q^{3} - 4 q^{4} + 5 q^{5} - 40 q^{6} + 28 q^{7} - 16 q^{8} - 73 q^{9} - 10 q^{10} - 53 q^{11} - 40 q^{12} + 50 q^{13} + 70 q^{14} - 100 q^{15} - 16 q^{16} - 14 q^{17} + 146 q^{18} + 95 q^{19} - 40 q^{20} + 70 q^{21} - 212 q^{22} - q^{23} + 80 q^{24} - 25 q^{25} + 50 q^{26} + 920 q^{27} + 28 q^{28} - 412 q^{29} - 100 q^{30} - 108 q^{31} + 32 q^{32} - 530 q^{33} - 56 q^{34} + 175 q^{35} + 584 q^{36} + 57 q^{37} - 190 q^{38} - 250 q^{39} - 40 q^{40} + 486 q^{41} - 560 q^{42} + 868 q^{43} - 212 q^{44} + 365 q^{45} + 2 q^{46} + 231 q^{47} + 320 q^{48} + 98 q^{49} - 100 q^{50} - 140 q^{51} - 100 q^{52} - 263 q^{53} + 920 q^{54} - 530 q^{55} - 224 q^{56} - 1900 q^{57} - 412 q^{58} - 24 q^{59} + 200 q^{60} - 116 q^{61} - 432 q^{62} - 2555 q^{63} + 128 q^{64} + 125 q^{65} + 1060 q^{66} + 204 q^{67} - 56 q^{68} + 20 q^{69} + 70 q^{70} + 968 q^{71} + 584 q^{72} + 692 q^{73} - 114 q^{74} - 250 q^{75} - 760 q^{76} + 371 q^{77} - 1000 q^{78} - 466 q^{79} + 80 q^{80} - 2629 q^{81} + 486 q^{82} + 456 q^{83} - 1400 q^{84} - 140 q^{85} + 868 q^{86} + 2060 q^{87} + 424 q^{88} + 362 q^{89} + 1460 q^{90} + 700 q^{91} + 8 q^{92} - 1080 q^{93} - 462 q^{94} - 475 q^{95} + 320 q^{96} + 1708 q^{97} + 1274 q^{98} + 7738 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(31\) \(57\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
0.500000 + 0.866025i
0.500000 0.866025i
1.00000 + 1.73205i −5.00000 + 8.66025i −2.00000 + 3.46410i 2.50000 + 4.33013i −20.0000 14.0000 12.1244i −8.00000 −36.5000 63.2199i −5.00000 + 8.66025i
51.1 1.00000 1.73205i −5.00000 8.66025i −2.00000 3.46410i 2.50000 4.33013i −20.0000 14.0000 + 12.1244i −8.00000 −36.5000 + 63.2199i −5.00000 8.66025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 70.4.e.b 2
3.b odd 2 1 630.4.k.c 2
4.b odd 2 1 560.4.q.g 2
5.b even 2 1 350.4.e.d 2
5.c odd 4 2 350.4.j.a 4
7.b odd 2 1 490.4.e.s 2
7.c even 3 1 inner 70.4.e.b 2
7.c even 3 1 490.4.a.h 1
7.d odd 6 1 490.4.a.a 1
7.d odd 6 1 490.4.e.s 2
21.h odd 6 1 630.4.k.c 2
28.g odd 6 1 560.4.q.g 2
35.i odd 6 1 2450.4.a.bq 1
35.j even 6 1 350.4.e.d 2
35.j even 6 1 2450.4.a.v 1
35.l odd 12 2 350.4.j.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.4.e.b 2 1.a even 1 1 trivial
70.4.e.b 2 7.c even 3 1 inner
350.4.e.d 2 5.b even 2 1
350.4.e.d 2 35.j even 6 1
350.4.j.a 4 5.c odd 4 2
350.4.j.a 4 35.l odd 12 2
490.4.a.a 1 7.d odd 6 1
490.4.a.h 1 7.c even 3 1
490.4.e.s 2 7.b odd 2 1
490.4.e.s 2 7.d odd 6 1
560.4.q.g 2 4.b odd 2 1
560.4.q.g 2 28.g odd 6 1
630.4.k.c 2 3.b odd 2 1
630.4.k.c 2 21.h odd 6 1
2450.4.a.v 1 35.j even 6 1
2450.4.a.bq 1 35.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 10T_{3} + 100 \) acting on \(S_{4}^{\mathrm{new}}(70, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$5$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$7$ \( T^{2} - 28T + 343 \) Copy content Toggle raw display
$11$ \( T^{2} + 53T + 2809 \) Copy content Toggle raw display
$13$ \( (T - 25)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$19$ \( T^{2} - 95T + 9025 \) Copy content Toggle raw display
$23$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$29$ \( (T + 206)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 108T + 11664 \) Copy content Toggle raw display
$37$ \( T^{2} - 57T + 3249 \) Copy content Toggle raw display
$41$ \( (T - 243)^{2} \) Copy content Toggle raw display
$43$ \( (T - 434)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 231T + 53361 \) Copy content Toggle raw display
$53$ \( T^{2} + 263T + 69169 \) Copy content Toggle raw display
$59$ \( T^{2} + 24T + 576 \) Copy content Toggle raw display
$61$ \( T^{2} + 116T + 13456 \) Copy content Toggle raw display
$67$ \( T^{2} - 204T + 41616 \) Copy content Toggle raw display
$71$ \( (T - 484)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 692T + 478864 \) Copy content Toggle raw display
$79$ \( T^{2} + 466T + 217156 \) Copy content Toggle raw display
$83$ \( (T - 228)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 362T + 131044 \) Copy content Toggle raw display
$97$ \( (T - 854)^{2} \) Copy content Toggle raw display
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