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Properties

Label 3528.2.k.b
Level $3528$
Weight $2$
Character orbit 3528.k
Analytic conductor $28.171$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3528,2,Mod(881,3528)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3528, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3528.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3528.k (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: \(28.1712218331\)
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: \( x^{16} + 10x^{14} + 61x^{12} + 266x^{10} + 852x^{8} + 1438x^{6} + 1933x^{4} + 3038x^{2} + 2401 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: no (minimal twist has level 504)
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.

\(f(q)\) \(=\) \( q + \beta_{13} q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{13} q^{5} - \beta_{6} q^{11} + (\beta_{8} - \beta_{7} + \cdots + 2 \beta_{3}) q^{13}+ \cdots + (3 \beta_{7} + \beta_{5} - 4 \beta_{3}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 24 q^{25} - 8 q^{37} + 8 q^{43} + 56 q^{67} + 64 q^{79} - 32 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 10x^{14} + 61x^{12} + 266x^{10} + 852x^{8} + 1438x^{6} + 1933x^{4} + 3038x^{2} + 2401 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 39 \nu^{14} - 589 \nu^{12} - 1808 \nu^{10} - 7082 \nu^{8} - 13930 \nu^{6} - 16292 \nu^{4} + \cdots - 1014081 ) / 348700 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 16193 \nu^{14} - 167663 \nu^{12} - 1074356 \nu^{10} - 4573114 \nu^{8} - 14837490 \nu^{6} + \cdots - 2048347 ) / 17086300 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1436747 \nu^{14} + 11493277 \nu^{12} + 59655364 \nu^{10} + 210639226 \nu^{8} + 476470070 \nu^{6} + \cdots + 170666853 ) / 785969800 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7053 \nu^{14} + 80183 \nu^{12} + 488396 \nu^{10} + 2172254 \nu^{8} + 6869890 \nu^{6} + \cdots + 19924527 ) / 3106600 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 623003 \nu^{14} - 5366258 \nu^{12} - 30938216 \nu^{10} - 125675214 \nu^{8} + \cdots - 1076511772 ) / 196492450 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 154544 \nu^{15} + 1700539 \nu^{13} + 10639878 \nu^{11} + 53193252 \nu^{9} + \cdots + 2058858151 \nu ) / 392984900 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2119573 \nu^{14} - 17608783 \nu^{12} - 99042676 \nu^{10} - 397519234 \nu^{8} + \cdots - 3079980407 ) / 392984900 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4624911 \nu^{14} - 38588401 \nu^{12} - 217244012 \nu^{10} - 865007178 \nu^{8} + \cdots - 6297464369 ) / 785969800 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 13169 \nu^{15} - 130759 \nu^{13} - 753868 \nu^{11} - 3141922 \nu^{9} - 9111910 \nu^{7} + \cdots - 21443331 \nu ) / 17355800 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 40317 \nu^{15} + 198497 \nu^{13} + 568672 \nu^{11} - 201110 \nu^{9} - 9668478 \nu^{7} + \cdots + 44011261 \nu ) / 47841640 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 5232309 \nu^{15} + 49250349 \nu^{13} + 269215888 \nu^{11} + 1049829662 \nu^{9} + \cdots - 2548006419 \nu ) / 5501788600 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 6467291 \nu^{15} - 64058891 \nu^{13} - 384048592 \nu^{11} - 1625328978 \nu^{9} + \cdots - 21129729579 \nu ) / 5501788600 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 104513 \nu^{15} - 887273 \nu^{13} - 5023936 \nu^{11} - 20069294 \nu^{9} - 58201830 \nu^{7} + \cdots - 147858697 \nu ) / 34172600 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 3107 \nu^{15} + 24945 \nu^{13} + 136852 \nu^{11} + 526582 \nu^{9} + 1437746 \nu^{7} + \cdots + 4147997 \nu ) / 869848 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 68856 \nu^{15} - 571303 \nu^{13} - 3213552 \nu^{11} - 12805303 \nu^{9} - 36798719 \nu^{7} + \cdots - 96590368 \nu ) / 11960410 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{15} + 3\beta_{13} - 3\beta_{12} - \beta_{11} + \beta_{10} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - 2\beta_{5} + \beta_{2} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{15} - 2\beta_{14} - 9\beta_{13} + 4\beta_{12} - 4\beta_{11} + \beta_{10} - 6\beta_{9} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -6\beta_{8} + \beta_{7} + 8\beta_{5} - 2\beta_{4} - 4\beta_{3} - 7\beta_{2} - 6\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 13 \beta_{15} + 56 \beta_{14} + 37 \beta_{13} - 7 \beta_{12} + 31 \beta_{11} - 25 \beta_{10} + \cdots + 12 \beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 43\beta_{8} - 29\beta_{7} - 22\beta_{5} + 15\beta_{3} + 5\beta _1 + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 83 \beta_{15} - 256 \beta_{14} - 119 \beta_{13} - 69 \beta_{12} + 105 \beta_{11} + 91 \beta_{10} + \cdots + 8 \beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -112\beta_{8} + 37\beta_{7} + 112\beta_{5} + 62\beta_{4} - 50\beta_{3} + 149\beta_{2} + 112\beta _1 - 37 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -206\beta_{15} + 150\beta_{14} + 542\beta_{13} + 329\beta_{12} - 329\beta_{11} - 94\beta_{10} - 942\beta_{9} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -67\beta_{8} + 404\beta_{7} - 486\beta_{5} - 198\beta_{4} + 131\beta_{3} - 471\beta_{2} - 67\beta _1 + 957 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 2285 \beta_{15} + 1328 \beta_{14} - 2891 \beta_{13} - 3803 \beta_{12} - 1373 \beta_{11} + \cdots - 2140 \beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 477\beta_{8} - 311\beta_{7} - 228\beta_{5} + 145\beta_{3} - 1265\beta _1 - 3575 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 7163 \beta_{15} - 5808 \beta_{14} - 18833 \beta_{13} + 14769 \beta_{12} + 11227 \beta_{11} + \cdots + 10784 \beta_{6} ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 3816 \beta_{8} - 6831 \beta_{7} + 15030 \beta_{5} - 1244 \beta_{4} - 5060 \beta_{3} - 3015 \beta_{2} + \cdots + 18045 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 28809 \beta_{15} + 21294 \beta_{14} + 73899 \beta_{13} - 15316 \beta_{12} + 15316 \beta_{11} + \cdots + 43306 \beta_{9} ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
−0.587308 − 2.01725i
−0.587308 + 2.01725i
−0.144868 + 1.25092i
−0.144868 − 1.25092i
−1.45333 − 1.51725i
−1.45333 + 1.51725i
−1.01089 + 0.750919i
−1.01089 − 0.750919i
1.01089 + 0.750919i
1.01089 − 0.750919i
1.45333 − 1.51725i
1.45333 + 1.51725i
0.144868 + 1.25092i
0.144868 − 1.25092i
0.587308 − 2.01725i
0.587308 + 2.01725i
0 0 0 −2.90667 0 0 0 0 0
881.2 0 0 0 −2.90667 0 0 0 0 0
881.3 0 0 0 −2.02179 0 0 0 0 0
881.4 0 0 0 −2.02179 0 0 0 0 0
881.5 0 0 0 −1.17462 0 0 0 0 0
881.6 0 0 0 −1.17462 0 0 0 0 0
881.7 0 0 0 −0.289737 0 0 0 0 0
881.8 0 0 0 −0.289737 0 0 0 0 0
881.9 0 0 0 0.289737 0 0 0 0 0
881.10 0 0 0 0.289737 0 0 0 0 0
881.11 0 0 0 1.17462 0 0 0 0 0
881.12 0 0 0 1.17462 0 0 0 0 0
881.13 0 0 0 2.02179 0 0 0 0 0
881.14 0 0 0 2.02179 0 0 0 0 0
881.15 0 0 0 2.90667 0 0 0 0 0
881.16 0 0 0 2.90667 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.2.k.b 16
3.b odd 2 1 inner 3528.2.k.b 16
4.b odd 2 1 7056.2.k.h 16
7.b odd 2 1 inner 3528.2.k.b 16
7.c even 3 1 504.2.bl.a ✓ 16
7.c even 3 1 3528.2.bl.a 16
7.d odd 6 1 504.2.bl.a ✓ 16
7.d odd 6 1 3528.2.bl.a 16
12.b even 2 1 7056.2.k.h 16
21.c even 2 1 inner 3528.2.k.b 16
21.g even 6 1 504.2.bl.a ✓ 16
21.g even 6 1 3528.2.bl.a 16
21.h odd 6 1 504.2.bl.a ✓ 16
21.h odd 6 1 3528.2.bl.a 16
28.d even 2 1 7056.2.k.h 16
28.f even 6 1 1008.2.bt.d 16
28.g odd 6 1 1008.2.bt.d 16
84.h odd 2 1 7056.2.k.h 16
84.j odd 6 1 1008.2.bt.d 16
84.n even 6 1 1008.2.bt.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.bl.a ✓ 16 7.c even 3 1
504.2.bl.a ✓ 16 7.d odd 6 1
504.2.bl.a ✓ 16 21.g even 6 1
504.2.bl.a ✓ 16 21.h odd 6 1
1008.2.bt.d 16 28.f even 6 1
1008.2.bt.d 16 28.g odd 6 1
1008.2.bt.d 16 84.j odd 6 1
1008.2.bt.d 16 84.n even 6 1
3528.2.k.b 16 1.a even 1 1 trivial
3528.2.k.b 16 3.b odd 2 1 inner
3528.2.k.b 16 7.b odd 2 1 inner
3528.2.k.b 16 21.c even 2 1 inner
3528.2.bl.a 16 7.c even 3 1
3528.2.bl.a 16 7.d odd 6 1
3528.2.bl.a 16 21.g even 6 1
3528.2.bl.a 16 21.h odd 6 1
7056.2.k.h 16 4.b odd 2 1
7056.2.k.h 16 12.b even 2 1
7056.2.k.h 16 28.d even 2 1
7056.2.k.h 16 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 14T_{5}^{6} + 53T_{5}^{4} - 52T_{5}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(3528, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 14 T^{6} + 53 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + 54 T^{6} + \cdots + 2116)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 82 T^{6} + \cdots + 24964)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 52 T^{6} + \cdots + 12544)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 106 T^{6} + \cdots + 111556)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 100 T^{6} + \cdots + 50176)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 150 T^{6} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 88 T^{6} + \cdots + 529)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2 T^{3} + \cdots + 766)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} - 124 T^{6} + \cdots + 160000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 2 T^{3} - 33 T^{2} + \cdots - 98)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} - 176 T^{6} + \cdots + 10000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 342 T^{6} + \cdots + 3013696)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 150 T^{6} + \cdots + 254016)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 348 T^{6} + \cdots + 254016)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 14 T^{3} + 53 T^{2} + \cdots + 4)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + 176 T^{6} + \cdots + 26896)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 302 T^{6} + \cdots + 446224)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 16 T^{3} + \cdots - 3353)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} - 226 T^{6} + \cdots + 3844)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 328 T^{6} + \cdots + 26543104)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 542 T^{6} + \cdots + 65480464)^{2} \) Copy content Toggle raw display
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