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Properties

Label 70.3.l.a
Level $70$
Weight $3$
Character orbit 70.l
Analytic conductor $1.907$
Analytic rank $0$
Dimension $8$
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] = [70,3,Mod(23,70)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(70, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([9, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("70.23");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 70.l (of order \(12\), degree \(4\), 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: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: 8.0.3317760000.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 25x^{4} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_{2} + 1) q^{2} + ( - \beta_{7} + \beta_{6} + \cdots - \beta_{2}) q^{3}+ \cdots + ( - 2 \beta_{7} + 2 \beta_{5} + \cdots - 4 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + \beta_{2} + 1) q^{2} + ( - \beta_{7} + \beta_{6} + \cdots - \beta_{2}) q^{3}+ \cdots + (8 \beta_{7} - 44 \beta_{6} + \cdots + 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 4 q^{3} + 12 q^{5} - 16 q^{6} - 4 q^{7} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} - 4 q^{3} + 12 q^{5} - 16 q^{6} - 4 q^{7} + 16 q^{8} - 28 q^{10} + 8 q^{11} - 8 q^{12} + 8 q^{15} + 16 q^{16} - 16 q^{17} + 32 q^{18} - 64 q^{20} + 100 q^{21} + 16 q^{22} - 4 q^{23} + 28 q^{25} - 232 q^{27} - 40 q^{28} - 24 q^{30} - 64 q^{31} - 16 q^{32} - 52 q^{33} + 112 q^{35} + 128 q^{36} + 144 q^{37} + 8 q^{38} - 8 q^{40} - 136 q^{41} + 188 q^{42} + 120 q^{43} - 128 q^{45} + 8 q^{46} + 72 q^{47} - 32 q^{48} - 136 q^{50} + 208 q^{51} + 76 q^{53} + 48 q^{55} - 16 q^{56} - 376 q^{57} + 68 q^{58} + 56 q^{60} - 324 q^{61} - 128 q^{62} + 112 q^{63} + 104 q^{66} + 124 q^{67} + 32 q^{68} + 160 q^{70} + 16 q^{71} + 64 q^{72} + 32 q^{73} + 124 q^{75} + 32 q^{76} + 20 q^{77} - 480 q^{78} - 48 q^{80} + 124 q^{81} - 68 q^{82} + 168 q^{83} - 224 q^{85} + 120 q^{86} + 428 q^{87} + 16 q^{88} - 64 q^{90} - 720 q^{91} + 16 q^{92} - 64 q^{93} - 32 q^{95} - 32 q^{96} - 72 q^{97} - 132 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 25x^{4} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 25 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 125 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 125 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 25\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 25\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 125\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 125\beta_{7} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(31\) \(57\)
\(\chi(n)\) \(-1 + \beta_{4}\) \(\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
0.578737 − 2.15988i
−0.578737 + 2.15988i
−2.15988 − 0.578737i
2.15988 + 0.578737i
−2.15988 + 0.578737i
2.15988 − 0.578737i
0.578737 + 2.15988i
−0.578737 − 2.15988i
−0.366025 − 1.36603i −0.636376 + 2.37499i −1.73205 + 1.00000i 4.96410 − 0.598076i 3.47723 6.99656 − 0.219274i 2.00000 + 2.00000i 2.55863 + 1.47723i −2.63397 − 6.56218i
23.2 −0.366025 − 1.36603i 1.36843 − 5.10704i −1.73205 + 1.00000i 4.96410 − 0.598076i −7.47723 −2.80041 + 6.41543i 2.00000 + 2.00000i −16.4150 − 9.47723i −2.63397 − 6.56218i
37.1 1.36603 − 0.366025i −5.10704 − 1.36843i 1.73205 − 1.00000i −1.96410 − 4.59808i −7.47723 −6.41543 − 2.80041i 2.00000 − 2.00000i 16.4150 + 9.47723i −4.36603 − 5.56218i
37.2 1.36603 − 0.366025i 2.37499 + 0.636376i 1.73205 − 1.00000i −1.96410 − 4.59808i 3.47723 0.219274 + 6.99656i 2.00000 − 2.00000i −2.55863 − 1.47723i −4.36603 − 5.56218i
53.1 1.36603 + 0.366025i −5.10704 + 1.36843i 1.73205 + 1.00000i −1.96410 + 4.59808i −7.47723 −6.41543 + 2.80041i 2.00000 + 2.00000i 16.4150 − 9.47723i −4.36603 + 5.56218i
53.2 1.36603 + 0.366025i 2.37499 − 0.636376i 1.73205 + 1.00000i −1.96410 + 4.59808i 3.47723 0.219274 − 6.99656i 2.00000 + 2.00000i −2.55863 + 1.47723i −4.36603 + 5.56218i
67.1 −0.366025 + 1.36603i −0.636376 − 2.37499i −1.73205 − 1.00000i 4.96410 + 0.598076i 3.47723 6.99656 + 0.219274i 2.00000 − 2.00000i 2.55863 − 1.47723i −2.63397 + 6.56218i
67.2 −0.366025 + 1.36603i 1.36843 + 5.10704i −1.73205 − 1.00000i 4.96410 + 0.598076i −7.47723 −2.80041 − 6.41543i 2.00000 − 2.00000i −16.4150 + 9.47723i −2.63397 + 6.56218i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.c even 3 1 inner
35.l odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 70.3.l.a ✓ 8
5.b even 2 1 350.3.p.c 8
5.c odd 4 1 inner 70.3.l.a ✓ 8
5.c odd 4 1 350.3.p.c 8
7.c even 3 1 inner 70.3.l.a ✓ 8
7.c even 3 1 490.3.f.l 4
7.d odd 6 1 490.3.f.e 4
35.j even 6 1 350.3.p.c 8
35.k even 12 1 490.3.f.e 4
35.l odd 12 1 inner 70.3.l.a ✓ 8
35.l odd 12 1 350.3.p.c 8
35.l odd 12 1 490.3.f.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.3.l.a ✓ 8 1.a even 1 1 trivial
70.3.l.a ✓ 8 5.c odd 4 1 inner
70.3.l.a ✓ 8 7.c even 3 1 inner
70.3.l.a ✓ 8 35.l odd 12 1 inner
350.3.p.c 8 5.b even 2 1
350.3.p.c 8 5.c odd 4 1
350.3.p.c 8 35.j even 6 1
350.3.p.c 8 35.l odd 12 1
490.3.f.e 4 7.d odd 6 1
490.3.f.e 4 35.k even 12 1
490.3.f.l 4 7.c even 3 1
490.3.f.l 4 35.l odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 4T_{3}^{7} + 8T_{3}^{6} + 136T_{3}^{5} + 103T_{3}^{4} - 1768T_{3}^{3} + 1352T_{3}^{2} - 8788T_{3} + 28561 \) acting on \(S_{3}^{\mathrm{new}}(70, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 4 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$5$ \( (T^{4} - 6 T^{3} + \cdots + 625)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 4 T^{7} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{4} - 4 T^{3} + \cdots + 676)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 57600)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 1871773696 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 5006411536 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 19356878641 \) Copy content Toggle raw display
$29$ \( (T^{4} + 2738 T^{2} + 625681)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 16 T + 256)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 30601961865216 \) Copy content Toggle raw display
$41$ \( (T^{2} + 34 T + 169)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 60 T^{3} + \cdots + 585225)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 5226454388736 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 53974440976 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 562491345610000 \) Copy content Toggle raw display
$61$ \( (T^{4} + 162 T^{3} + \cdots + 36978561)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 10197605570161 \) Copy content Toggle raw display
$71$ \( (T^{2} - 4 T - 1466)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 302373843210000 \) Copy content Toggle raw display
$83$ \( (T^{4} - 84 T^{3} + \cdots + 870489)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 264028733579521 \) Copy content Toggle raw display
$97$ \( (T^{4} + 36 T^{3} + \cdots + 34082244)^{2} \) Copy content Toggle raw display
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