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Properties

Label 528.6.d.b
Level $528$
Weight $6$
Character orbit 528.d
Analytic conductor $84.683$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [528,6,Mod(287,528)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(528, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("528.287");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 528.d (of order \(2\), degree \(1\), 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: \(84.6826568613\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + 15 x^{14} + 440 x^{13} - 30788 x^{12} + 421206 x^{11} + 3494011 x^{10} + \cdots + 94\!\cdots\!91 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{31}\cdot 3^{10}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 2) q^{3} - \beta_{5} q^{5} + ( - \beta_{3} - \beta_{2}) q^{7} + ( - \beta_{5} + \beta_{4} - 2 \beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 2) q^{3} - \beta_{5} q^{5} + ( - \beta_{3} - \beta_{2}) q^{7} + ( - \beta_{5} + \beta_{4} - 2 \beta_{2} + 2) q^{9} - 121 q^{11} + ( - \beta_{7} + 2 \beta_{2} + 28) q^{13} + ( - \beta_{11} - 3 \beta_{5} + \cdots - 61) q^{15}+ \cdots + (121 \beta_{5} - 121 \beta_{4} + \cdots - 242) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 31 q^{3} + 31 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 31 q^{3} + 31 q^{9} - 1936 q^{11} + 444 q^{13} - 979 q^{15} - 3874 q^{21} + 4246 q^{23} - 4194 q^{25} + 1306 q^{27} - 3751 q^{33} - 13964 q^{35} + 35094 q^{37} - 7848 q^{39} - 43369 q^{45} + 18476 q^{47} - 6172 q^{49} + 40886 q^{51} - 27316 q^{57} - 59134 q^{59} + 93580 q^{61} + 52286 q^{63} + 23881 q^{69} + 129154 q^{71} + 63188 q^{73} - 30048 q^{75} + 132223 q^{81} - 46276 q^{83} + 141316 q^{85} - 86304 q^{87} + 228139 q^{93} - 238328 q^{95} - 244198 q^{97} - 3751 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} + 15 x^{14} + 440 x^{13} - 30788 x^{12} + 421206 x^{11} + 3494011 x^{10} + \cdots + 94\!\cdots\!91 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 9\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{15} - \nu^{14} - 17 \nu^{13} - 474 \nu^{12} + 29840 \nu^{11} + \cdots - 12\!\cdots\!41 ) / 50\!\cdots\!07 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 428015230 \nu^{15} + 19576051193 \nu^{14} - 659357483843 \nu^{13} + \cdots - 58\!\cdots\!49 ) / 14\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 108953608 \nu^{15} + 10650642799 \nu^{14} - 386397800203 \nu^{13} + 4784850426264 \nu^{12} + \cdots - 13\!\cdots\!41 ) / 28\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1531175960 \nu^{15} + 22753667887 \nu^{14} - 373050538891 \nu^{13} + \cdots - 11\!\cdots\!33 ) / 28\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 140545045 \nu^{15} - 2671218404 \nu^{14} + 50954309729 \nu^{13} - 578055263061 \nu^{12} + \cdots + 11\!\cdots\!07 ) / 14\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 21354065 \nu^{15} + 389395639 \nu^{14} - 4169440846 \nu^{13} + 18977179461 \nu^{12} + \cdots - 98\!\cdots\!54 ) / 12\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 16368997763 \nu^{15} + 260645897131 \nu^{14} - 3427912469536 \nu^{13} + \cdots - 85\!\cdots\!96 ) / 47\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 16846818149 \nu^{15} + 335223777535 \nu^{14} - 5299213789114 \nu^{13} + \cdots - 12\!\cdots\!82 ) / 28\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 31285718521 \nu^{15} - 415155856178 \nu^{14} + 7696659541205 \nu^{13} + \cdots + 24\!\cdots\!51 ) / 47\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 110939190125 \nu^{15} + 1425175076416 \nu^{14} - 21909769368331 \nu^{13} + \cdots - 82\!\cdots\!33 ) / 14\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 112403217719 \nu^{15} - 1788994665502 \nu^{14} + 26064655887679 \nu^{13} + \cdots + 75\!\cdots\!05 ) / 14\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 174314946001 \nu^{15} - 2611638724073 \nu^{14} + 33061661178728 \nu^{13} + \cdots + 11\!\cdots\!00 ) / 14\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 252894257120 \nu^{15} + 3722581333645 \nu^{14} - 54547405529461 \nu^{13} + \cdots - 17\!\cdots\!47 ) / 14\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 295393987795 \nu^{15} + 4091271114857 \nu^{14} - 59978323237622 \nu^{13} + \cdots - 20\!\cdots\!46 ) / 14\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 9 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + 2\beta_{5} + \beta_{4} - \beta_{3} - 2\beta_{2} + \beta _1 - 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 9 \beta_{14} + 9 \beta_{13} + 18 \beta_{12} - 9 \beta_{11} + 9 \beta_{8} + 27 \beta_{7} - 3 \beta_{6} + \cdots - 800 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 81 \beta_{15} - 42 \beta_{14} + 57 \beta_{13} + 105 \beta_{12} + 15 \beta_{11} + 54 \beta_{10} + \cdots + 22847 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4293 \beta_{15} - 4500 \beta_{14} - 1017 \beta_{13} + 4446 \beta_{12} - 1008 \beta_{11} - 1944 \beta_{10} + \cdots - 1072664 ) / 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4005 \beta_{15} - 5160 \beta_{14} - 7053 \beta_{13} + 6339 \beta_{12} - 9249 \beta_{11} - 846 \beta_{10} + \cdots - 1583198 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 301401 \beta_{15} - 289548 \beta_{14} - 385101 \beta_{13} - 98604 \beta_{12} - 1461510 \beta_{11} + \cdots - 128738609 ) / 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 107163 \beta_{15} + 5100858 \beta_{14} - 860265 \beta_{13} - 2788875 \beta_{12} - 9889281 \beta_{11} + \cdots - 412304324 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 45295281 \beta_{15} + 291560751 \beta_{14} - 19713960 \beta_{13} - 477450054 \beta_{12} + \cdots - 103625098424 ) / 9 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 490403376 \beta_{15} + 1113674376 \beta_{14} + 1256300832 \beta_{13} - 3785605896 \beta_{12} + \cdots - 259344273676 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 7439572656 \beta_{15} + 30730684752 \beta_{14} + 83012844420 \beta_{13} - 171914878422 \beta_{12} + \cdots + 23523406989814 ) / 9 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 28864555344 \beta_{15} + 15436013880 \beta_{14} + 119874027912 \beta_{13} - 226474214232 \beta_{12} + \cdots - 30021926191721 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 5568494685408 \beta_{15} + 2640226719492 \beta_{14} + 6963721259184 \beta_{13} + \cdots - 29\!\cdots\!99 ) / 9 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 22242518765748 \beta_{15} + 23723518541364 \beta_{14} + 43934851715688 \beta_{13} + \cdots - 18\!\cdots\!87 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 924689731682160 \beta_{15} + \cdots - 82\!\cdots\!20 ) / 9 \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
287.1
16.9817 − 4.30668i
16.9817 + 4.30668i
13.8159 − 10.1678i
13.8159 + 10.1678i
7.09175 − 14.7334i
7.09175 + 14.7334i
−0.415167 − 15.4002i
−0.415167 + 15.4002i
−2.36019 − 14.9663i
−2.36019 + 14.9663i
−8.41804 − 11.5959i
−8.41804 + 11.5959i
−12.6093 − 5.43771i
−12.6093 + 5.43771i
−13.5867 − 0.234175i
−13.5867 + 0.234175i
0 −14.9817 − 4.30668i 0 66.3793i 0 − 12.8824i 0 205.905 + 129.043i 0
287.2 0 −14.9817 + 4.30668i 0 − 66.3793i 0 12.8824i 0 205.905 − 129.043i 0
287.3 0 −11.8159 − 10.1678i 0 − 2.66669i 0 − 196.235i 0 36.2306 + 240.284i 0
287.4 0 −11.8159 + 10.1678i 0 2.66669i 0 196.235i 0 36.2306 − 240.284i 0
287.5 0 −5.09175 − 14.7334i 0 49.7957i 0 155.302i 0 −191.148 + 150.038i 0
287.6 0 −5.09175 + 14.7334i 0 − 49.7957i 0 − 155.302i 0 −191.148 − 150.038i 0
287.7 0 2.41517 − 15.4002i 0 − 109.050i 0 − 143.992i 0 −231.334 − 74.3882i 0
287.8 0 2.41517 + 15.4002i 0 109.050i 0 143.992i 0 −231.334 + 74.3882i 0
287.9 0 4.36019 − 14.9663i 0 − 2.85488i 0 − 4.23609i 0 −204.978 − 130.511i 0
287.10 0 4.36019 + 14.9663i 0 2.85488i 0 4.23609i 0 −204.978 + 130.511i 0
287.11 0 10.4180 − 11.5959i 0 49.3758i 0 − 90.4265i 0 −25.9288 − 241.613i 0
287.12 0 10.4180 + 11.5959i 0 − 49.3758i 0 90.4265i 0 −25.9288 + 241.613i 0
287.13 0 14.6093 − 5.43771i 0 − 59.0190i 0 213.234i 0 183.863 − 158.882i 0
287.14 0 14.6093 + 5.43771i 0 59.0190i 0 − 213.234i 0 183.863 + 158.882i 0
287.15 0 15.5867 − 0.234175i 0 − 48.8145i 0 − 18.7604i 0 242.890 − 7.30003i 0
287.16 0 15.5867 + 0.234175i 0 48.8145i 0 18.7604i 0 242.890 + 7.30003i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 287.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b 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.d.b yes 16
3.b odd 2 1 528.6.d.a ✓ 16
4.b odd 2 1 528.6.d.a ✓ 16
12.b even 2 1 inner 528.6.d.b yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
528.6.d.a ✓ 16 3.b odd 2 1
528.6.d.a ✓ 16 4.b odd 2 1
528.6.d.b yes 16 1.a even 1 1 trivial
528.6.d.b yes 16 12.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 \(S_{6}^{\mathrm{new}}(528, [\chi])\):

\( T_{5}^{16} + 27097 T_{5}^{14} + 271756675 T_{5}^{12} + 1349414221499 T_{5}^{10} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
\( T_{23}^{8} - 2123 T_{23}^{7} - 27004565 T_{23}^{6} + 50650389647 T_{23}^{5} + 178250549848280 T_{23}^{4} + \cdots - 17\!\cdots\!72 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 12\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 75\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( (T + 121)^{16} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots - 15\!\cdots\!32)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 24\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots - 17\!\cdots\!72)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 84\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 11\!\cdots\!80)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 47\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots - 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 93\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 10\!\cdots\!32)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots - 12\!\cdots\!20)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 61\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots - 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 61\!\cdots\!44)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 70\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 11\!\cdots\!48)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 62\!\cdots\!00)^{2} \) Copy content Toggle raw display
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