Properties

Label 528.7.j.c
Level $528$
Weight $7$
Character orbit 528.j
Analytic conductor $121.469$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [528,7,Mod(241,528)]
 
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([0, 0, 0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("528.241");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 528.j (of order \(2\), degree \(1\), not 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: \(121.468556151\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 486x^{10} + 82401x^{8} + 6062364x^{6} + 204706260x^{4} + 2964086784x^{2} + 15081209856 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{11} \)
Twist minimal: no (minimal twist has level 33)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + (\beta_{5} + \beta_{2} + 19) q^{5} + ( - \beta_{3} + \beta_1) q^{7} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + (\beta_{5} + \beta_{2} + 19) q^{5} + ( - \beta_{3} + \beta_1) q^{7} + 243 q^{9} + ( - 2 \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - 2 \beta_{4} - \beta_{3} - \beta_{2} + \cdots + 288) q^{11}+ \cdots + ( - 486 \beta_{10} - 243 \beta_{9} - 243 \beta_{8} - 243 \beta_{7} + \cdots + 69984) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 224 q^{5} + 2916 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 224 q^{5} + 2916 q^{9} + 3464 q^{11} - 1944 q^{15} + 15304 q^{23} + 95652 q^{25} + 58608 q^{31} + 4212 q^{33} - 202512 q^{37} + 54432 q^{45} - 516920 q^{47} + 157812 q^{49} - 1042192 q^{53} - 262656 q^{55} + 461008 q^{59} - 364752 q^{67} + 504144 q^{69} + 755176 q^{71} - 1364688 q^{75} - 102384 q^{77} + 708588 q^{81} - 3513544 q^{89} + 702768 q^{91} + 789264 q^{93} + 2370192 q^{97} + 841752 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 486x^{10} + 82401x^{8} + 6062364x^{6} + 204706260x^{4} + 2964086784x^{2} + 15081209856 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 8\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 117\nu^{10} + 56806\nu^{8} + 9541621\nu^{6} + 673291092\nu^{4} + 19439761332\nu^{2} + 162772955136 ) / 269967872 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 30253 \nu^{11} + 11912366 \nu^{9} + 1324309389 \nu^{7} + 34902969132 \nu^{5} + 335536481988 \nu^{3} + 30540405528576 \nu ) / 252689928192 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 66873 \nu^{10} + 30142826 \nu^{8} + 4473500265 \nu^{6} + 258149028432 \nu^{4} + 5933517838524 \nu^{2} + 63172482048 \nu + 46478367060480 ) / 15793120512 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 40281 \nu^{10} + 18394564 \nu^{8} + 2794934961 \nu^{6} + 168462043098 \nu^{4} + 4072398716160 \nu^{2} + 30202915882752 ) / 7896560256 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 788101 \nu^{10} + 368615102 \nu^{8} + 58224152805 \nu^{6} + 3726817193676 \nu^{4} + 95759332043940 \nu^{2} + \cdots + 732251158225920 ) / 31586241024 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 249277 \nu^{11} + 118253198 \nu^{9} + 19186223901 \nu^{7} + 1295303893356 \nu^{5} + 36474079767300 \nu^{3} + \cdots + 298611337955328 \nu ) / 252689928192 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 468301 \nu^{11} + 224594030 \nu^{9} + 37048138413 \nu^{7} + 2555704817580 \nu^{5} + 73118002908996 \nu^{3} + \cdots + 640973109270528 \nu ) / 252689928192 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 878169 \nu^{11} + 11130976 \nu^{10} - 409604982 \nu^{9} + 5240500032 \nu^{8} - 64509725241 \nu^{7} + 838488195168 \nu^{6} + \cdots + 12\!\cdots\!96 ) / 505379856384 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 292723 \nu^{11} - 136534994 \nu^{9} - 21503241747 \nu^{7} - 1376699073300 \nu^{5} - 35936281084476 \nu^{3} + \cdots - 285938693220864 \nu ) / 84229976064 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2002587 \nu^{11} - 959211394 \nu^{9} - 157677000507 \nu^{7} - 10780709053236 \nu^{5} - 302030780555484 \nu^{3} + \cdots - 25\!\cdots\!52 \nu ) / 252689928192 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{5} + 2\beta_{4} + 4\beta_{2} - \beta _1 - 648 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{8} - 8\beta_{7} + 4\beta_{3} - 147\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -20\beta_{10} + 40\beta_{9} - 4\beta_{6} + 362\beta_{5} - 438\beta_{4} - 1784\beta_{2} + 219\beta _1 + 95168 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -144\beta_{11} + 176\beta_{10} - 1288\beta_{8} + 1912\beta_{7} - 240\beta_{3} + 26177\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 6528 \beta_{10} - 13056 \beta_{9} - 552 \beta_{6} - 59946 \beta_{5} + 90354 \beta_{4} + 518092 \beta_{2} - 45177 \beta _1 - 17106600 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 58176\beta_{11} - 61248\beta_{10} + 345436\beta_{8} - 391320\beta_{7} - 49732\beta_{3} - 5014355\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1661964 \beta_{10} + 3323928 \beta_{9} + 450564 \beta_{6} + 9999330 \beta_{5} - 18730134 \beta_{4} - 129560352 \beta_{2} + 9365067 \beta _1 + 3308491824 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 16730352 \beta_{11} + 16390608 \beta_{10} - 84341712 \beta_{8} + 78754296 \beta_{7} + 24969816 \beta_{3} + 996075249 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 389638008 \beta_{10} - 779276016 \beta_{9} - 150723072 \beta_{6} - 1717023762 \beta_{5} + 3913524306 \beta_{4} + 30272819268 \beta_{2} - 1956762153 \beta _1 - 662380151256 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 4207221792 \beta_{11} - 3975888096 \beta_{10} + 19530561684 \beta_{8} - 15997469112 \beta_{7} - 7355731548 \beta_{3} - 202291577331 \beta_1 ) / 8 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
241.1
7.49503i
7.49503i
3.61107i
3.61107i
11.9419i
11.9419i
3.80817i
3.80817i
14.5918i
14.5918i
6.83773i
6.83773i
0 −15.5885 0 −189.722 0 392.912i 0 243.000 0
241.2 0 −15.5885 0 −189.722 0 392.912i 0 243.000 0
241.3 0 −15.5885 0 42.6809 0 392.632i 0 243.000 0
241.4 0 −15.5885 0 42.6809 0 392.632i 0 243.000 0
241.5 0 −15.5885 0 234.218 0 368.050i 0 243.000 0
241.6 0 −15.5885 0 234.218 0 368.050i 0 243.000 0
241.7 0 15.5885 0 −143.833 0 413.864i 0 243.000 0
241.8 0 15.5885 0 −143.833 0 413.864i 0 243.000 0
241.9 0 15.5885 0 0.687766 0 18.0081i 0 243.000 0
241.10 0 15.5885 0 0.687766 0 18.0081i 0 243.000 0
241.11 0 15.5885 0 167.968 0 106.676i 0 243.000 0
241.12 0 15.5885 0 167.968 0 106.676i 0 243.000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 241.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 528.7.j.c 12
4.b odd 2 1 33.7.c.a 12
11.b odd 2 1 inner 528.7.j.c 12
12.b even 2 1 99.7.c.d 12
44.c even 2 1 33.7.c.a 12
132.d odd 2 1 99.7.c.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.7.c.a 12 4.b odd 2 1
33.7.c.a 12 44.c even 2 1
99.7.c.d 12 12.b even 2 1
99.7.c.d 12 132.d odd 2 1
528.7.j.c 12 1.a even 1 1 trivial
528.7.j.c 12 11.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 112T_{5}^{5} - 64516T_{5}^{4} + 5073800T_{5}^{3} + 978439700T_{5}^{2} - 46495666000T_{5} + 31513690000 \) acting on \(S_{7}^{\mathrm{new}}(528, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{2} - 243)^{6} \) Copy content Toggle raw display
$5$ \( (T^{6} - 112 T^{5} + \cdots + 31513690000)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + 626988 T^{10} + \cdots + 20\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( T^{12} - 3464 T^{11} + \cdots + 30\!\cdots\!61 \) Copy content Toggle raw display
$13$ \( T^{12} + 46108704 T^{10} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{12} + 176680836 T^{10} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{12} + 265613868 T^{10} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( (T^{6} - 7652 T^{5} + \cdots + 11\!\cdots\!88)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 3585415380 T^{10} + \cdots + 86\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( (T^{6} - 29304 T^{5} + \cdots + 52\!\cdots\!52)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 101256 T^{5} + \cdots + 26\!\cdots\!68)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 38558024532 T^{10} + \cdots + 76\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( T^{12} + 52055166636 T^{10} + \cdots + 61\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( (T^{6} + 258460 T^{5} + \cdots - 25\!\cdots\!96)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 521096 T^{5} + \cdots - 89\!\cdots\!12)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 230504 T^{5} + \cdots + 59\!\cdots\!16)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + 273287036976 T^{10} + \cdots + 30\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( (T^{6} + 182376 T^{5} + \cdots + 95\!\cdots\!32)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 377588 T^{5} + \cdots - 79\!\cdots\!76)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 777459022416 T^{10} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{12} + 856808323212 T^{10} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{12} + 2879312245056 T^{10} + \cdots + 46\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{6} + 1756772 T^{5} + \cdots + 48\!\cdots\!64)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 1185096 T^{5} + \cdots + 71\!\cdots\!28)^{2} \) Copy content Toggle raw display
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