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Properties

Label 532.2.g.b
Level $532$
Weight $2$
Character orbit 532.g
Analytic conductor $4.248$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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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] = [532,2,Mod(265,532)] 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("532.265"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(532, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 532 = 2^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 532.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.24804138753\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.113164960000.5
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 17x^{6} - 30x^{5} + 174x^{4} - 208x^{3} + 962x^{2} - 382x + 2449 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} - \beta_{3} q^{5} + ( - \beta_{4} - \beta_{2} - 1) q^{7} + (\beta_{4} + 3) q^{9} - \beta_{4} q^{11} - \beta_1 q^{13} + ( - \beta_{7} - \beta_{5}) q^{15} + \beta_{2} q^{17} + (\beta_{6} + \beta_{3} - \beta_{2} + \beta_1) q^{19}+ \cdots + ( - 2 \beta_{4} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{7} + 20 q^{9} + 4 q^{11} + 24 q^{23} - 16 q^{25} - 12 q^{35} + 12 q^{39} + 8 q^{43} - 32 q^{49} + 32 q^{57} - 20 q^{63} + 8 q^{77} - 72 q^{81} + 12 q^{85} + 8 q^{93} + 44 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 17x^{6} - 30x^{5} + 174x^{4} - 208x^{3} + 962x^{2} - 382x + 2449 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 50780 \nu^{7} + 208046 \nu^{6} + 556076 \nu^{5} + 8943094 \nu^{4} - 11927624 \nu^{3} + \cdots + 410306309 ) / 106738385 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 92960 \nu^{7} + 5906 \nu^{6} - 1387924 \nu^{5} + 856199 \nu^{4} - 16202164 \nu^{3} + \cdots - 138071 ) / 106738385 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 118797 \nu^{7} + 1811169 \nu^{6} - 5355202 \nu^{5} + 20871312 \nu^{4} - 55150975 \nu^{3} + \cdots + 439947389 ) / 106738385 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1622 \nu^{7} - 3241 \nu^{6} + 14760 \nu^{5} - 23900 \nu^{4} + 117888 \nu^{3} - 35949 \nu^{2} + \cdots + 222753 ) / 716365 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -25\nu^{7} + 247\nu^{6} - 608\nu^{5} + 2896\nu^{4} - 6451\nu^{3} + 30906\nu^{2} - 28654\nu + 101769 ) / 11021 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 427598 \nu^{7} - 494721 \nu^{6} + 4975088 \nu^{5} - 5273498 \nu^{4} + 49969640 \nu^{3} + \cdots + 140204724 ) / 106738385 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 4698 \nu^{7} + 8504 \nu^{6} - 90450 \nu^{5} + 181405 \nu^{4} - 666512 \nu^{3} + 659726 \nu^{2} + \cdots + 390053 ) / 716365 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} + \beta_{4} - 2\beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + 2\beta_{5} + 3\beta_{4} - 2\beta_{3} - \beta _1 - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{7} + 12\beta_{6} + 3\beta_{5} - 10\beta_{4} - 5\beta_{3} + 3\beta_{2} - 4\beta _1 - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 8\beta_{7} + 19\beta_{6} - 8\beta_{5} - 27\beta_{4} - 8\beta_{2} + 26\beta _1 - 47 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -30\beta_{7} - 2\beta_{6} - 15\beta_{5} - 90\beta_{4} + 7\beta_{3} + 56\beta_{2} + 79\beta _1 - 147 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -90\beta_{7} + 67\beta_{6} - 52\beta_{5} - 291\beta_{4} + 204\beta_{3} + 38\beta_{2} - 121\beta _1 + 156 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -7\beta_{7} - 275\beta_{6} - 259\beta_{5} + 1329\beta_{4} + 663\beta_{3} - 395\beta_{2} - 309\beta _1 + 798 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/532\mathbb{Z}\right)^\times\).

\(n\) \(267\) \(381\) \(477\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
265.1
2.09529 − 2.09331i
2.09529 + 2.09331i
0.737640 + 2.57255i
0.737640 − 2.57255i
−1.35567 + 2.57255i
−1.35567 − 2.57255i
−0.477260 − 2.09331i
−0.477260 + 2.09331i
0 −2.57255 0 − 3.38705i 0 −1.61803 − 2.09331i 0 3.61803 0
265.2 0 −2.57255 0 3.38705i 0 −1.61803 + 2.09331i 0 3.61803 0
265.3 0 −2.09331 0 − 1.58993i 0 0.618034 + 2.57255i 0 1.38197 0
265.4 0 −2.09331 0 1.58993i 0 0.618034 − 2.57255i 0 1.38197 0
265.5 0 2.09331 0 − 1.58993i 0 0.618034 + 2.57255i 0 1.38197 0
265.6 0 2.09331 0 1.58993i 0 0.618034 − 2.57255i 0 1.38197 0
265.7 0 2.57255 0 − 3.38705i 0 −1.61803 − 2.09331i 0 3.61803 0
265.8 0 2.57255 0 3.38705i 0 −1.61803 + 2.09331i 0 3.61803 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 265.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
19.b odd 2 1 inner
133.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 532.2.g.b ✓ 8
3.b odd 2 1 4788.2.i.e 8
4.b odd 2 1 2128.2.m.e 8
7.b odd 2 1 inner 532.2.g.b ✓ 8
19.b odd 2 1 inner 532.2.g.b ✓ 8
21.c even 2 1 4788.2.i.e 8
28.d even 2 1 2128.2.m.e 8
57.d even 2 1 4788.2.i.e 8
76.d even 2 1 2128.2.m.e 8
133.c even 2 1 inner 532.2.g.b ✓ 8
399.h odd 2 1 4788.2.i.e 8
532.b odd 2 1 2128.2.m.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
532.2.g.b ✓ 8 1.a even 1 1 trivial
532.2.g.b ✓ 8 7.b odd 2 1 inner
532.2.g.b ✓ 8 19.b odd 2 1 inner
532.2.g.b ✓ 8 133.c even 2 1 inner
2128.2.m.e 8 4.b odd 2 1
2128.2.m.e 8 28.d even 2 1
2128.2.m.e 8 76.d even 2 1
2128.2.m.e 8 532.b odd 2 1
4788.2.i.e 8 3.b odd 2 1
4788.2.i.e 8 21.c even 2 1
4788.2.i.e 8 57.d even 2 1
4788.2.i.e 8 399.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 11T_{3}^{2} + 29 \) acting on \(S_{2}^{\mathrm{new}}(532, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 11 T^{2} + 29)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 14 T^{2} + 29)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 2 T^{3} + 10 T^{2} + \cdots + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - T - 1)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 14 T^{2} + 29)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 11 T^{2} + 29)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} - 258 T^{4} + 130321 \) Copy content Toggle raw display
$23$ \( (T^{2} - 6 T - 11)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 87 T^{2} + 841)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 119 T^{2} + 3509)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 29)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 119 T^{2} + 3509)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 2 T - 44)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 126 T^{2} + 2349)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 203 T^{2} + 841)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 46 T^{2} + 29)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 46 T^{2} + 29)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 203 T^{2} + 841)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 145)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 95 T^{2} + 725)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 116)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 279 T^{2} + 2349)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 176 T^{2} + 7424)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 126 T^{2} + 2349)^{2} \) Copy content Toggle raw display
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