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Properties

Label 5070.2.b.ba
Level $5070$
Weight $2$
Character orbit 5070.b
Analytic conductor $40.484$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5070,2,Mod(1351,5070)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5070.1351"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5070, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,8,-8,0,0,0,0,8,8,0,-8,0,4,0,8,-32,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(22)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.17284886784.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 30x^{5} + 185x^{4} + 36x^{3} + 8x^{2} + 208x + 2704 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + q^{3} - q^{4} + \beta_1 q^{5} - \beta_1 q^{6} + ( - \beta_{7} + \beta_1) q^{7} + \beta_1 q^{8} + q^{9} + q^{10} + ( - \beta_{3} - \beta_1) q^{11} - q^{12} + \beta_{6} q^{14} + \beta_1 q^{15}+ \cdots + ( - \beta_{3} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3} - 8 q^{4} + 8 q^{9} + 8 q^{10} - 8 q^{12} + 4 q^{14} + 8 q^{16} - 32 q^{17} - 4 q^{22} - 8 q^{23} - 8 q^{25} + 8 q^{27} + 16 q^{29} + 8 q^{30} - 4 q^{35} - 8 q^{36} - 8 q^{40} + 4 q^{42} - 28 q^{43}+ \cdots + 16 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 2x^{6} + 30x^{5} + 185x^{4} + 36x^{3} + 8x^{2} + 208x + 2704 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 2225 \nu^{7} + 27304 \nu^{6} - 47714 \nu^{5} - 21770 \nu^{4} + 204731 \nu^{3} + 5488658 \nu^{2} + \cdots - 320632 ) / 20561424 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 879 \nu^{7} + 1664 \nu^{6} - 1730 \nu^{5} - 23706 \nu^{4} - 214183 \nu^{3} - 10614 \nu^{2} + \cdots - 231400 ) / 790824 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 879 \nu^{7} + 1664 \nu^{6} - 1730 \nu^{5} - 23706 \nu^{4} - 214183 \nu^{3} - 10614 \nu^{2} + \cdots - 231400 ) / 790824 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 79\nu^{7} - 108\nu^{6} - 634\nu^{5} + 6190\nu^{4} + 7075\nu^{3} + 206\nu^{2} - 33852\nu + 174712 ) / 19056 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 151\nu^{7} - 822\nu^{6} + 2018\nu^{5} + 2554\nu^{4} + 7135\nu^{3} - 37828\nu^{2} + 46500\nu + 29224 ) / 25896 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3671 \nu^{7} - 5300 \nu^{6} - 24114 \nu^{5} + 242954 \nu^{4} + 411219 \nu^{3} + 14638 \nu^{2} + \cdots + 5391360 ) / 395412 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 242617 \nu^{7} - 1102864 \nu^{6} + 2820138 \nu^{5} + 4476178 \nu^{4} + 18741717 \nu^{3} + \cdots + 49538424 ) / 20561424 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + 2\beta_{5} - \beta_{3} + 9\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -6\beta_{7} - 6\beta_{6} + 10\beta_{5} + 10\beta_{4} - 13\beta_{3} - 13\beta_{2} + 20\beta _1 - 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -23\beta_{6} + 44\beta_{4} - 28\beta_{2} - 98 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 144\beta_{7} - 144\beta_{6} - 250\beta_{5} + 250\beta_{4} + 221\beta_{3} - 221\beta_{2} - 546\beta _1 - 402 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 471\beta_{7} - 874\beta_{5} + 619\beta_{3} - 2095\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2986 \beta_{7} + 2986 \beta_{6} - 5302 \beta_{5} - 5302 \beta_{4} + 4207 \beta_{3} + 4207 \beta_{2} + \cdots + 8962 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times\).

\(n\) \(1691\) \(1861\) \(4057\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1351.1
1.33404 + 1.33404i
−1.80668 − 1.80668i
3.17270 + 3.17270i
−1.70006 − 1.70006i
−1.70006 + 1.70006i
3.17270 − 3.17270i
−1.80668 + 1.80668i
1.33404 − 1.33404i
− 1.00000i 1.00000 −1.00000 1.00000i − 1.00000i − 3.64466i 1.00000i 1.00000 1.00000
1351.2 − 1.00000i 1.00000 −1.00000 1.00000i − 1.00000i − 1.32258i 1.00000i 1.00000 1.00000
1351.3 − 1.00000i 1.00000 −1.00000 1.00000i − 1.00000i 2.32258i 1.00000i 1.00000 1.00000
1351.4 − 1.00000i 1.00000 −1.00000 1.00000i − 1.00000i 4.64466i 1.00000i 1.00000 1.00000
1351.5 1.00000i 1.00000 −1.00000 − 1.00000i 1.00000i − 4.64466i − 1.00000i 1.00000 1.00000
1351.6 1.00000i 1.00000 −1.00000 − 1.00000i 1.00000i − 2.32258i − 1.00000i 1.00000 1.00000
1351.7 1.00000i 1.00000 −1.00000 − 1.00000i 1.00000i 1.32258i − 1.00000i 1.00000 1.00000
1351.8 1.00000i 1.00000 −1.00000 − 1.00000i 1.00000i 3.64466i − 1.00000i 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1351.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5070.2.b.ba 8
13.b even 2 1 inner 5070.2.b.ba 8
13.c even 3 1 390.2.bb.c ✓ 8
13.d odd 4 1 5070.2.a.bz 4
13.d odd 4 1 5070.2.a.ca 4
13.e even 6 1 390.2.bb.c ✓ 8
39.h odd 6 1 1170.2.bs.f 8
39.i odd 6 1 1170.2.bs.f 8
65.l even 6 1 1950.2.bc.g 8
65.n even 6 1 1950.2.bc.g 8
65.q odd 12 1 1950.2.y.j 8
65.q odd 12 1 1950.2.y.k 8
65.r odd 12 1 1950.2.y.j 8
65.r odd 12 1 1950.2.y.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bb.c ✓ 8 13.c even 3 1
390.2.bb.c ✓ 8 13.e even 6 1
1170.2.bs.f 8 39.h odd 6 1
1170.2.bs.f 8 39.i odd 6 1
1950.2.y.j 8 65.q odd 12 1
1950.2.y.j 8 65.r odd 12 1
1950.2.y.k 8 65.q odd 12 1
1950.2.y.k 8 65.r odd 12 1
1950.2.bc.g 8 65.l even 6 1
1950.2.bc.g 8 65.n even 6 1
5070.2.a.bz 4 13.d odd 4 1
5070.2.a.ca 4 13.d odd 4 1
5070.2.b.ba 8 1.a even 1 1 trivial
5070.2.b.ba 8 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(5070, [\chi])\):

\( T_{7}^{8} + 42T_{7}^{6} + 545T_{7}^{4} + 2376T_{7}^{2} + 2704 \) Copy content Toggle raw display
\( T_{11}^{8} + 72T_{11}^{6} + 1766T_{11}^{4} + 16152T_{11}^{2} + 32761 \) Copy content Toggle raw display
\( T_{17} + 4 \) Copy content Toggle raw display
\( T_{31}^{8} + 174T_{31}^{6} + 8777T_{31}^{4} + 158088T_{31}^{2} + 913936 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T - 1)^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} + 42 T^{6} + \cdots + 2704 \) Copy content Toggle raw display
$11$ \( T^{8} + 72 T^{6} + \cdots + 32761 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T + 4)^{8} \) Copy content Toggle raw display
$19$ \( T^{8} + 126 T^{6} + \cdots + 219024 \) Copy content Toggle raw display
$23$ \( (T^{4} + 4 T^{3} - 43 T^{2} + \cdots + 52)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 8 T^{3} + \cdots + 376)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 174 T^{6} + \cdots + 913936 \) Copy content Toggle raw display
$37$ \( T^{8} + 168 T^{6} + \cdots + 644809 \) Copy content Toggle raw display
$41$ \( T^{8} + 168 T^{6} + \cdots + 692224 \) Copy content Toggle raw display
$43$ \( (T^{4} + 14 T^{3} + \cdots - 572)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 228 T^{6} + \cdots + 1216609 \) Copy content Toggle raw display
$53$ \( (T^{4} - 8 T^{3} - 75 T^{2} + \cdots + 52)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 98 T^{6} + \cdots + 80656 \) Copy content Toggle raw display
$61$ \( (T^{2} - 8 T + 4)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} + 248 T^{6} + \cdots + 123904 \) Copy content Toggle raw display
$71$ \( T^{8} + 456 T^{6} + \cdots + 25240576 \) Copy content Toggle raw display
$73$ \( T^{8} + 392 T^{6} + \cdots + 20647936 \) Copy content Toggle raw display
$79$ \( (T^{4} + 10 T^{3} + \cdots + 3508)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 456 T^{6} + \cdots + 25240576 \) Copy content Toggle raw display
$89$ \( T^{8} + 246 T^{6} + \cdots + 1008016 \) Copy content Toggle raw display
$97$ \( T^{8} + 536 T^{6} + \cdots + 29246464 \) Copy content Toggle raw display
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