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Properties

Label 70.3.j.a
Level $70$
Weight $3$
Character orbit 70.j
Analytic conductor $1.907$
Analytic rank $0$
Dimension $8$
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] = [70,3,Mod(31,70)]
 
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, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("70.31");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 70.j (of order \(6\), 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: \(1.90736185052\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.3317760000.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_{3}) q^{2} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \cdots + 1) q^{3}+ \cdots + ( - 2 \beta_{7} + 2 \beta_{5} + \cdots + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - \beta_{3}) q^{2} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \cdots + 1) q^{3}+ \cdots + ( - 8 \beta_{7} + 8 \beta_{6} + \cdots + 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{3} - 8 q^{4} - 12 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{3} - 8 q^{4} - 12 q^{7} + 12 q^{9} - 8 q^{11} - 24 q^{12} + 40 q^{14} + 40 q^{15} - 16 q^{16} - 48 q^{17} + 16 q^{18} + 4 q^{21} + 48 q^{22} - 36 q^{23} - 24 q^{24} + 20 q^{25} - 96 q^{26} - 24 q^{28} - 120 q^{29} - 20 q^{30} - 72 q^{31} + 192 q^{33} - 40 q^{35} - 48 q^{36} + 96 q^{37} + 120 q^{38} + 24 q^{39} + 64 q^{42} + 104 q^{43} - 16 q^{44} + 60 q^{45} + 20 q^{46} - 192 q^{47} + 24 q^{49} + 56 q^{51} + 144 q^{52} - 176 q^{53} + 180 q^{54} - 40 q^{56} - 240 q^{57} - 16 q^{58} + 48 q^{59} - 40 q^{60} + 72 q^{61} - 152 q^{63} + 64 q^{64} + 40 q^{65} - 192 q^{66} + 132 q^{67} + 96 q^{68} - 60 q^{70} + 64 q^{71} + 32 q^{72} + 216 q^{73} - 80 q^{74} + 60 q^{75} - 24 q^{77} - 128 q^{78} - 288 q^{79} + 304 q^{81} - 144 q^{82} - 40 q^{84} + 4 q^{86} - 324 q^{87} - 48 q^{88} - 132 q^{89} + 616 q^{91} + 144 q^{92} - 112 q^{93} - 336 q^{94} - 80 q^{95} + 48 q^{96} - 240 q^{98} + 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 14\nu^{4} + 7\nu^{2} - 36 ) / 63 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{7} + 7\nu^{5} + 35\nu^{3} + 81\nu ) / 189 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{6} + 7\nu^{4} - 28\nu^{2} + 144 ) / 63 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{7} - 7\nu^{5} - 35\nu^{3} + 180\nu ) / 189 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 4\nu^{5} + 7\nu^{3} - 36\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -8\nu^{6} + 14\nu^{4} + 7\nu^{2} + 162 ) / 63 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - 2\beta_{4} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{5} + 3\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 4\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -5\beta_{6} - 7\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -7\beta_{7} + 7\beta_{2} + 22 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -21\beta_{5} - 21\beta_{3} + 29\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(31\) \(57\)
\(\chi(n)\) \(\beta_{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
1.72286 − 0.178197i
−1.01575 + 1.40294i
−1.72286 + 0.178197i
1.01575 − 1.40294i
1.72286 + 0.178197i
−1.01575 − 1.40294i
−1.72286 − 0.178197i
1.01575 + 1.40294i
−0.707107 − 1.22474i −2.86646 − 1.65495i −1.00000 + 1.73205i −1.93649 + 1.11803i 4.68091i −5.43919 + 4.40626i 2.82843 0.977722 + 1.69346i 2.73861 + 1.58114i
31.2 −0.707107 − 1.22474i 3.74514 + 2.16226i −1.00000 + 1.73205i 1.93649 − 1.11803i − 6.11578i −1.09634 + 6.91361i 2.82843 4.85071 + 8.40167i −2.73861 − 1.58114i
31.3 0.707107 + 1.22474i 1.99347 + 1.15093i −1.00000 + 1.73205i −1.93649 + 1.11803i 3.25533i 6.31218 + 3.02596i −2.82843 −1.85071 − 3.20552i −2.73861 − 1.58114i
31.4 0.707107 + 1.22474i 3.12785 + 1.80586i −1.00000 + 1.73205i 1.93649 − 1.11803i 5.10775i −5.77664 − 3.95353i −2.82843 2.02228 + 3.50269i 2.73861 + 1.58114i
61.1 −0.707107 + 1.22474i −2.86646 + 1.65495i −1.00000 − 1.73205i −1.93649 − 1.11803i − 4.68091i −5.43919 − 4.40626i 2.82843 0.977722 − 1.69346i 2.73861 − 1.58114i
61.2 −0.707107 + 1.22474i 3.74514 − 2.16226i −1.00000 − 1.73205i 1.93649 + 1.11803i 6.11578i −1.09634 − 6.91361i 2.82843 4.85071 − 8.40167i −2.73861 + 1.58114i
61.3 0.707107 − 1.22474i 1.99347 − 1.15093i −1.00000 − 1.73205i −1.93649 − 1.11803i − 3.25533i 6.31218 − 3.02596i −2.82843 −1.85071 + 3.20552i −2.73861 + 1.58114i
61.4 0.707107 − 1.22474i 3.12785 − 1.80586i −1.00000 − 1.73205i 1.93649 + 1.11803i − 5.10775i −5.77664 + 3.95353i −2.82843 2.02228 − 3.50269i 2.73861 − 1.58114i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 70.3.j.a ✓ 8
3.b odd 2 1 630.3.v.a 8
4.b odd 2 1 560.3.bx.a 8
5.b even 2 1 350.3.k.b 8
5.c odd 4 2 350.3.i.b 16
7.b odd 2 1 490.3.j.a 8
7.c even 3 1 490.3.b.b 8
7.c even 3 1 490.3.j.a 8
7.d odd 6 1 inner 70.3.j.a ✓ 8
7.d odd 6 1 490.3.b.b 8
21.g even 6 1 630.3.v.a 8
28.f even 6 1 560.3.bx.a 8
35.i odd 6 1 350.3.k.b 8
35.k even 12 2 350.3.i.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.3.j.a ✓ 8 1.a even 1 1 trivial
70.3.j.a ✓ 8 7.d odd 6 1 inner
350.3.i.b 16 5.c odd 4 2
350.3.i.b 16 35.k even 12 2
350.3.k.b 8 5.b even 2 1
350.3.k.b 8 35.i odd 6 1
490.3.b.b 8 7.c even 3 1
490.3.b.b 8 7.d odd 6 1
490.3.j.a 8 7.b odd 2 1
490.3.j.a 8 7.c even 3 1
560.3.bx.a 8 4.b odd 2 1
560.3.bx.a 8 28.f even 6 1
630.3.v.a 8 3.b odd 2 1
630.3.v.a 8 21.g even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(70, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 12 T^{7} + \cdots + 14161 \) Copy content Toggle raw display
$5$ \( (T^{4} - 5 T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 12 T^{7} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{8} + 8 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 2206744576 \) Copy content Toggle raw display
$17$ \( T^{8} + 48 T^{7} + \cdots + 9834496 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 2265760000 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 3249114001 \) Copy content Toggle raw display
$29$ \( (T^{4} + 60 T^{3} + \cdots + 5569)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 46214680576 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 6677386756096 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 4105968689761 \) Copy content Toggle raw display
$43$ \( (T^{4} - 52 T^{3} + \cdots + 742441)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 141408152778256 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 20618719334656 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 41420242452736 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 216605537281 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 2452384188081 \) Copy content Toggle raw display
$71$ \( (T^{4} - 32 T^{3} + \cdots - 4477424)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 1627604402176 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 102942614692096 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 2396523821041 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 35539413361 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 511556553523456 \) Copy content Toggle raw display
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