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Properties

Label 528.6.o.a
Level $528$
Weight $6$
Character orbit 528.o
Analytic conductor $84.683$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [528,6,Mod(175,528)] 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("528.175"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(528, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 1])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 528.o (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20] 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: \(84.6826568613\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 10 x^{18} - 35668 x^{17} + 53174974 x^{16} - 285344 x^{15} + 636103232 x^{14} + \cdots + 10\!\cdots\!50 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{43}\cdot 3^{20} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + (\beta_{3} - 4) q^{5} - \beta_{7} q^{7} - 81 q^{9} + (\beta_{11} - 5 \beta_{4}) q^{11} + \beta_{9} q^{13} + (\beta_{13} - 4 \beta_{4}) q^{15} + \beta_{6} q^{17} + ( - \beta_{12} - \beta_{11} + \cdots + \beta_1) q^{19}+ \cdots + ( - 81 \beta_{11} + 405 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 88 q^{5} - 1620 q^{9} + 23852 q^{25} + 7308 q^{33} - 4576 q^{37} + 7128 q^{45} - 46436 q^{49} + 60264 q^{53} - 16632 q^{69} + 23464 q^{77} + 131220 q^{81} - 148008 q^{89} - 97920 q^{93} + 188096 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 10 x^{18} - 35668 x^{17} + 53174974 x^{16} - 285344 x^{15} + 636103232 x^{14} + \cdots + 10\!\cdots\!50 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 11\!\cdots\!98 \nu^{19} + \cdots + 35\!\cdots\!50 ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 40\!\cdots\!12 \nu^{19} + \cdots + 37\!\cdots\!50 ) / 77\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 77\!\cdots\!92 \nu^{19} + \cdots + 12\!\cdots\!50 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 18\!\cdots\!58 \nu^{19} + \cdots + 20\!\cdots\!50 ) / 72\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 53\!\cdots\!48 \nu^{19} + \cdots - 17\!\cdots\!50 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 19\!\cdots\!84 \nu^{19} + \cdots - 62\!\cdots\!50 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 33\!\cdots\!97 \nu^{19} + \cdots - 57\!\cdots\!50 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 12\!\cdots\!31 \nu^{19} + \cdots - 59\!\cdots\!50 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 77\!\cdots\!59 \nu^{19} + \cdots + 93\!\cdots\!50 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 50\!\cdots\!29 \nu^{19} + \cdots + 93\!\cdots\!50 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 82\!\cdots\!78 \nu^{19} + \cdots - 11\!\cdots\!50 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 83\!\cdots\!77 \nu^{19} + \cdots + 12\!\cdots\!50 ) / 32\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 12\!\cdots\!80 \nu^{19} + \cdots + 11\!\cdots\!50 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 11\!\cdots\!18 \nu^{19} + \cdots + 11\!\cdots\!50 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 53\!\cdots\!33 \nu^{19} + \cdots + 80\!\cdots\!50 ) / 77\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 14\!\cdots\!97 \nu^{19} + \cdots - 16\!\cdots\!50 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 24\!\cdots\!68 \nu^{19} + \cdots - 74\!\cdots\!50 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 25\!\cdots\!48 \nu^{19} + \cdots - 32\!\cdots\!50 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 12\!\cdots\!23 \nu^{19} + \cdots - 21\!\cdots\!50 ) / 64\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - \beta_{19} + \beta_{18} - \beta_{17} - \beta_{15} - \beta_{12} + 5 \beta_{11} + 2 \beta_{10} + \cdots - 5 \beta_1 ) / 432 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{19} - \beta_{18} + \beta_{17} + 27 \beta_{16} + \beta_{15} + 351 \beta_{14} + 117 \beta_{13} + \cdots - 216 ) / 216 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3903 \beta_{19} + 10487 \beta_{18} - 10883 \beta_{17} - 19019 \beta_{15} + 3915 \beta_{12} + \cdots + 2309778 ) / 432 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 3899 \beta_{19} + 116058 \beta_{18} + 94680 \beta_{17} - 81 \beta_{16} - 19023 \beta_{15} + \cdots - 1148061183 ) / 108 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 25126961 \beta_{19} - 82151741 \beta_{18} + 82153061 \beta_{17} + 3934980 \beta_{16} + 157352933 \beta_{15} + \cdots - 7699260 ) / 432 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 75536895 \beta_{19} + 242868625 \beta_{18} - 249190765 \beta_{17} - 3054849741 \beta_{16} + \cdots + 34441838514 ) / 216 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 178349758483 \beta_{19} - 574141040655 \beta_{18} + 614648815347 \beta_{17} + \cdots - 224875404901350 ) / 432 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 177645220643 \beta_{19} - 3170215198532 \beta_{18} - 1976830194562 \beta_{17} + 21383950833 \beta_{16} + \cdots + 28\!\cdots\!84 ) / 54 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 12\!\cdots\!77 \beta_{19} + \cdots + 26\!\cdots\!16 ) / 432 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 21\!\cdots\!23 \beta_{19} + \cdots - 17\!\cdots\!32 ) / 72 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 92\!\cdots\!39 \beta_{19} + \cdots + 18\!\cdots\!70 ) / 432 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 27\!\cdots\!33 \beta_{19} + \cdots - 29\!\cdots\!33 ) / 108 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 64\!\cdots\!09 \beta_{19} + \cdots - 47\!\cdots\!56 ) / 432 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 46\!\cdots\!19 \beta_{19} + \cdots + 53\!\cdots\!58 ) / 216 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 47\!\cdots\!87 \beta_{19} + \cdots - 12\!\cdots\!82 ) / 432 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 93\!\cdots\!54 \beta_{19} + \cdots + 76\!\cdots\!89 ) / 54 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 33\!\cdots\!01 \beta_{19} + \cdots + 57\!\cdots\!64 ) / 432 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 30\!\cdots\!93 \beta_{19} + \cdots - 46\!\cdots\!48 ) / 216 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 24\!\cdots\!55 \beta_{19} + \cdots + 83\!\cdots\!38 ) / 432 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/528\mathbb{Z}\right)^\times\).

\(n\) \(133\) \(145\) \(353\) \(463\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
175.1
−7.31817 − 8.31817i
7.31817 + 6.31817i
24.5470 + 23.5470i
−24.5470 − 25.5470i
−59.9168 − 60.9168i
59.9168 + 58.9168i
14.0862 + 13.0862i
−14.0862 − 15.0862i
−3.73531 − 4.73531i
3.73531 + 2.73531i
−7.31817 + 8.31817i
7.31817 − 6.31817i
24.5470 − 23.5470i
−24.5470 + 25.5470i
−59.9168 + 60.9168i
59.9168 − 58.9168i
14.0862 − 13.0862i
−14.0862 + 15.0862i
−3.73531 + 4.73531i
3.73531 − 2.73531i
0 − 9.00000i 0 −100.687 0 −26.3697 0 −81.0000 0
175.2 0 − 9.00000i 0 −100.687 0 26.3697 0 −81.0000 0
175.3 0 − 9.00000i 0 −40.5561 0 −180.397 0 −81.0000 0
175.4 0 − 9.00000i 0 −40.5561 0 180.397 0 −81.0000 0
175.5 0 − 9.00000i 0 −9.42653 0 −53.2538 0 −81.0000 0
175.6 0 − 9.00000i 0 −9.42653 0 53.2538 0 −81.0000 0
175.7 0 − 9.00000i 0 37.5173 0 −70.0633 0 −81.0000 0
175.8 0 − 9.00000i 0 37.5173 0 70.0633 0 −81.0000 0
175.9 0 − 9.00000i 0 91.1528 0 −177.321 0 −81.0000 0
175.10 0 − 9.00000i 0 91.1528 0 177.321 0 −81.0000 0
175.11 0 9.00000i 0 −100.687 0 −26.3697 0 −81.0000 0
175.12 0 9.00000i 0 −100.687 0 26.3697 0 −81.0000 0
175.13 0 9.00000i 0 −40.5561 0 −180.397 0 −81.0000 0
175.14 0 9.00000i 0 −40.5561 0 180.397 0 −81.0000 0
175.15 0 9.00000i 0 −9.42653 0 −53.2538 0 −81.0000 0
175.16 0 9.00000i 0 −9.42653 0 53.2538 0 −81.0000 0
175.17 0 9.00000i 0 37.5173 0 −70.0633 0 −81.0000 0
175.18 0 9.00000i 0 37.5173 0 70.0633 0 −81.0000 0
175.19 0 9.00000i 0 91.1528 0 −177.321 0 −81.0000 0
175.20 0 9.00000i 0 91.1528 0 177.321 0 −81.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 175.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
11.b odd 2 1 inner
44.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 528.6.o.a ✓ 20
4.b odd 2 1 inner 528.6.o.a ✓ 20
11.b odd 2 1 inner 528.6.o.a ✓ 20
44.c even 2 1 inner 528.6.o.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
528.6.o.a ✓ 20 1.a even 1 1 trivial
528.6.o.a ✓ 20 4.b odd 2 1 inner
528.6.o.a ✓ 20 11.b odd 2 1 inner
528.6.o.a ✓ 20 44.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} + 22T_{5}^{4} - 10552T_{5}^{3} - 142984T_{5}^{2} + 13565076T_{5} + 131639112 \) acting on \(S_{6}^{\mathrm{new}}(528, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{2} + 81)^{10} \) Copy content Toggle raw display
$5$ \( (T^{5} + 22 T^{4} + \cdots + 131639112)^{4} \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots - 99\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots + 49\!\cdots\!80)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots - 72\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots + 57\!\cdots\!64)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 85\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots - 85\!\cdots\!64)^{4} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 18\!\cdots\!80)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots - 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots - 11\!\cdots\!20)^{4} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 86\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 53\!\cdots\!24)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 81\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots - 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots - 55\!\cdots\!40)^{4} \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots - 67\!\cdots\!52)^{4} \) Copy content Toggle raw display
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