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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1665,2,Mod(334,1665)] 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("1665.334"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1665, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1665 = 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1665.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,4,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.2950919365\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.309760000.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 8x^{6} - 2x^{5} - x^{4} - 2x^{3} + 18x^{2} + 6x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 555)
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.

\(f(q)\) \(=\) \( q + (\beta_{5} - \beta_{4}) q^{2} + ( - \beta_1 + 1) q^{4} + (\beta_{4} + \beta_{2}) q^{5} + ( - \beta_{6} + 2 \beta_{5}) q^{7} + (\beta_{5} - 2 \beta_{4}) q^{8} + (\beta_{7} + \beta_{6} + 1) q^{10} + (\beta_{2} + 1) q^{11}+ \cdots + (4 \beta_{7} + 2 \beta_{5}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} + 8 q^{10} + 8 q^{11} - 8 q^{14} - 12 q^{16} + 8 q^{19} + 16 q^{25} + 12 q^{26} - 24 q^{29} - 4 q^{31} - 12 q^{34} - 8 q^{35} + 20 q^{40} + 52 q^{41} + 4 q^{44} - 20 q^{46} - 8 q^{49} + 28 q^{55}+ \cdots - 32 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 8x^{6} - 2x^{5} - x^{4} - 2x^{3} + 18x^{2} + 6x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -8\nu^{7} + 44\nu^{6} - 130\nu^{5} + 211\nu^{4} - 186\nu^{3} + 50\nu^{2} + 26\nu + 403 ) / 245 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11\nu^{7} + 13\nu^{6} - 152\nu^{5} + 451\nu^{4} - 222\nu^{3} - 32\nu^{2} + \nu + 530 ) / 245 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13\nu^{7} - 47\nu^{6} + 101\nu^{5} - 55\nu^{4} + 143\nu^{3} - 69\nu^{2} - 30\nu + 319 ) / 245 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -61\nu^{7} + 262\nu^{6} - 538\nu^{5} + 194\nu^{4} + 162\nu^{3} - 23\nu^{2} - 941\nu - 155 ) / 245 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 74\nu^{7} - 309\nu^{6} + 639\nu^{5} - 249\nu^{4} - 19\nu^{3} - 291\nu^{2} + 1401\nu + 229 ) / 245 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 87\nu^{7} - 356\nu^{6} + 740\nu^{5} - 304\nu^{4} + 124\nu^{3} - 360\nu^{2} + 1861\nu + 303 ) / 245 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 178\nu^{7} - 734\nu^{6} + 1545\nu^{5} - 591\nu^{4} - 149\nu^{3} - 10\nu^{2} + 3219\nu + 527 ) / 245 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} - \beta_{5} - \beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 2\beta_{5} - \beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} + 4\beta_{6} - 5\beta_{5} - 3\beta_{4} + 4\beta_{3} - \beta_{2} + 3\beta _1 - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{3} - 2\beta_{2} + 7\beta _1 - 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 9\beta_{7} - 21\beta_{6} + 24\beta_{5} + 25\beta_{4} + 21\beta_{3} - 9\beta_{2} + 25\beta _1 - 49 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 17\beta_{7} - 35\beta_{6} + 39\beta_{5} + 47\beta_{4} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 64\beta_{7} - 121\beta_{6} + 127\beta_{5} + 168\beta_{4} - 121\beta_{3} + 64\beta_{2} - 168\beta _1 + 295 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1665\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
334.1
1.16407 + 1.16407i
−0.164066 − 0.164066i
−0.748606 + 0.748606i
1.74861 − 1.74861i
−0.748606 − 0.748606i
1.74861 + 1.74861i
1.16407 − 1.16407i
−0.164066 + 0.164066i
− 1.61803i 0 −0.618034 −2.14896 + 0.618034i 0 − 3.32813i − 2.23607i 0 1.00000 + 3.47709i
334.2 − 1.61803i 0 −0.618034 2.14896 + 0.618034i 0 − 0.671869i − 2.23607i 0 1.00000 − 3.47709i
334.3 − 0.618034i 0 1.61803 −1.54336 + 1.61803i 0 − 0.497212i − 2.23607i 0 1.00000 + 0.953850i
334.4 − 0.618034i 0 1.61803 1.54336 + 1.61803i 0 4.49721i − 2.23607i 0 1.00000 − 0.953850i
334.5 0.618034i 0 1.61803 −1.54336 − 1.61803i 0 0.497212i 2.23607i 0 1.00000 − 0.953850i
334.6 0.618034i 0 1.61803 1.54336 − 1.61803i 0 − 4.49721i 2.23607i 0 1.00000 + 0.953850i
334.7 1.61803i 0 −0.618034 −2.14896 − 0.618034i 0 3.32813i 2.23607i 0 1.00000 − 3.47709i
334.8 1.61803i 0 −0.618034 2.14896 − 0.618034i 0 0.671869i 2.23607i 0 1.00000 + 3.47709i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 334.8
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 1665.2.c.d 8
3.b odd 2 1 555.2.c.b ✓ 8
5.b even 2 1 inner 1665.2.c.d 8
5.c odd 4 1 8325.2.a.bt 4
5.c odd 4 1 8325.2.a.bw 4
15.d odd 2 1 555.2.c.b ✓ 8
15.e even 4 1 2775.2.a.w 4
15.e even 4 1 2775.2.a.y 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
555.2.c.b ✓ 8 3.b odd 2 1
555.2.c.b ✓ 8 15.d odd 2 1
1665.2.c.d 8 1.a even 1 1 trivial
1665.2.c.d 8 5.b even 2 1 inner
2775.2.a.w 4 15.e even 4 1
2775.2.a.y 4 15.e even 4 1
8325.2.a.bt 4 5.c odd 4 1
8325.2.a.bw 4 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1665, [\chi])\):

\( T_{2}^{4} + 3T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{4} - 4T_{11}^{3} - T_{11}^{2} + 10T_{11} + 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 3 T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 8 T^{6} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} + 32 T^{6} + \cdots + 25 \) Copy content Toggle raw display
$11$ \( (T^{4} - 4 T^{3} - T^{2} + \cdots + 5)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 32 T^{6} + \cdots + 25 \) Copy content Toggle raw display
$17$ \( T^{8} + 62 T^{6} + \cdots + 21025 \) Copy content Toggle raw display
$19$ \( (T^{4} - 4 T^{3} - 11 T^{2} + \cdots - 5)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 84 T^{6} + \cdots + 1681 \) Copy content Toggle raw display
$29$ \( (T^{4} + 12 T^{3} + \cdots - 49)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2 T^{3} - 28 T^{2} + \cdots + 59)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 26 T^{3} + \cdots + 1289)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 78 T^{6} + \cdots + 25 \) Copy content Toggle raw display
$47$ \( T^{8} + 180 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$53$ \( (T^{4} + 68 T^{2} + 176)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 12 T^{3} + \cdots - 1949)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 6 T^{3} - 16 T^{2} + \cdots - 25)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 142 T^{6} + \cdots + 429025 \) Copy content Toggle raw display
$71$ \( (T^{4} - 34 T^{3} + \cdots + 2801)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 290 T^{6} + \cdots + 13140625 \) Copy content Toggle raw display
$79$ \( (T^{4} + 6 T^{3} + \cdots - 6245)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 126 T^{6} + \cdots + 121 \) Copy content Toggle raw display
$89$ \( (T^{4} + 24 T^{3} + \cdots - 725)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 200 T^{6} + \cdots + 398161 \) Copy content Toggle raw display
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