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Properties

Label 3528.3.f.i
Level $3528$
Weight $3$
Character orbit 3528.f
Analytic conductor $96.131$
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] = [3528,3,Mod(2449,3528)]
 
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, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3528.2449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 3528.f (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: \(96.1310372663\)
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: \( x^{16} - 2 x^{15} + 81 x^{14} - 118 x^{13} + 1960 x^{12} - 366 x^{11} + 37625 x^{10} - 83714 x^{9} + \cdots + 1148023744 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{26} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{5} + ( - \beta_{7} - \beta_{5}) q^{11} + (\beta_{6} - \beta_{2}) q^{13} + ( - \beta_{13} + \beta_1) q^{17} + (\beta_{11} + \beta_{2}) q^{19} + (\beta_{10} - \beta_{7} - \beta_{3}) q^{23} + (\beta_{12} - 2 \beta_{8} - \beta_{4} - 6) q^{25} + (2 \beta_{5} + \beta_{3}) q^{29} + ( - \beta_{11} + \beta_{9} + \cdots - 5 \beta_{2}) q^{31}+ \cdots + (12 \beta_{11} + 4 \beta_{9} + \cdots + 22 \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 72 q^{25} + 136 q^{37} + 80 q^{43} - 112 q^{67} - 56 q^{79} + 448 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 81 x^{14} - 118 x^{13} + 1960 x^{12} - 366 x^{11} + 37625 x^{10} - 83714 x^{9} + \cdots + 1148023744 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 11\!\cdots\!90 \nu^{15} + \cdots - 62\!\cdots\!96 ) / 12\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 40\!\cdots\!73 \nu^{15} + \cdots + 14\!\cdots\!44 ) / 64\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 40\!\cdots\!74 \nu^{15} + \cdots - 16\!\cdots\!12 ) / 50\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 21\!\cdots\!97 \nu^{15} + \cdots - 61\!\cdots\!88 ) / 24\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 23\!\cdots\!04 \nu^{15} + \cdots + 18\!\cdots\!28 ) / 25\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 12\!\cdots\!90 \nu^{15} + \cdots - 13\!\cdots\!80 ) / 12\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 31\!\cdots\!52 \nu^{15} + \cdots + 11\!\cdots\!16 ) / 21\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10\!\cdots\!06 \nu^{15} + \cdots - 45\!\cdots\!96 ) / 70\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 30\!\cdots\!34 \nu^{15} + \cdots + 15\!\cdots\!00 ) / 18\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 26\!\cdots\!47 \nu^{15} + \cdots - 22\!\cdots\!40 ) / 12\!\cdots\!38 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 29\!\cdots\!84 \nu^{15} + \cdots - 16\!\cdots\!12 ) / 12\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 14\!\cdots\!26 \nu^{15} + \cdots - 14\!\cdots\!56 ) / 49\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 43\!\cdots\!54 \nu^{15} + \cdots + 61\!\cdots\!16 ) / 12\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 48\!\cdots\!42 \nu^{15} + \cdots + 95\!\cdots\!40 ) / 12\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 62\!\cdots\!32 \nu^{15} + \cdots + 25\!\cdots\!12 ) / 12\!\cdots\!92 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{9} - \beta_{4} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{15} + \beta_{14} + \beta_{13} + 2 \beta_{12} + 3 \beta_{11} + \beta_{10} + \beta_{9} + \cdots - 40 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 19 \beta_{15} - 25 \beta_{14} + 7 \beta_{13} - \beta_{12} - 56 \beta_{11} - 27 \beta_{10} - 43 \beta_{9} + \cdots - 38 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 16 \beta_{15} - 54 \beta_{13} - 109 \beta_{12} - 55 \beta_{11} - 64 \beta_{10} + 20 \beta_{9} + \cdots + 1310 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 871 \beta_{15} + 941 \beta_{14} - 1031 \beta_{13} + 279 \beta_{12} + 2478 \beta_{11} + 545 \beta_{10} + \cdots - 878 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1485 \beta_{15} - 707 \beta_{14} + 1773 \beta_{13} + 5209 \beta_{12} - 1426 \beta_{11} + 5327 \beta_{10} + \cdots - 88550 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 45339 \beta_{15} - 39298 \beta_{14} + 60641 \beta_{13} - 24675 \beta_{12} - 120594 \beta_{11} + \cdots + 365890 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 70358 \beta_{15} + 58808 \beta_{14} - 80100 \beta_{13} - 281111 \beta_{12} + 179371 \beta_{11} + \cdots + 5199630 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 2500899 \beta_{15} + 2225317 \beta_{14} - 3154011 \beta_{13} + 1823989 \beta_{12} + 6959036 \beta_{11} + \cdots - 33921758 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 4194951 \beta_{15} - 4285624 \beta_{14} + 4901829 \beta_{13} + 15761336 \beta_{12} - 13694729 \beta_{11} + \cdots - 286768838 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 138044043 \beta_{15} - 131145571 \beta_{14} + 162928491 \beta_{13} - 127051415 \beta_{12} + \cdots + 2376789650 ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 145434292 \beta_{15} + 149302447 \beta_{14} - 164963539 \beta_{13} - 434792991 \beta_{12} + \cdots + 7835072875 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 7417316179 \beta_{15} + 7268635024 \beta_{14} - 8554784217 \beta_{13} + 8475803371 \beta_{12} + \cdots - 154827702582 ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 19502362417 \beta_{15} - 19560181311 \beta_{14} + 22128463729 \beta_{13} + 46926047268 \beta_{12} + \cdots - 851278011330 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 393537231435 \beta_{15} - 388861144781 \beta_{14} + 453707852703 \beta_{13} - 546882905353 \beta_{12} + \cdots + 9935970865146 ) / 8 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2449.1
−0.338813 + 1.51822i
−0.338813 + 7.55243i
−3.20903 + 1.64043i
−3.20903 + 3.57433i
1.75486 − 2.33310i
1.75486 + 4.85286i
2.29298 − 1.67052i
2.29298 + 3.14719i
2.29298 + 1.67052i
2.29298 − 3.14719i
1.75486 + 2.33310i
1.75486 − 4.85286i
−3.20903 − 1.64043i
−3.20903 − 3.57433i
−0.338813 − 1.51822i
−0.338813 − 7.55243i
0 0 0 − 9.07064i 0 0 0 0 0
2449.2 0 0 0 − 9.07064i 0 0 0 0 0
2449.3 0 0 0 − 5.21476i 0 0 0 0 0
2449.4 0 0 0 − 5.21476i 0 0 0 0 0
2449.5 0 0 0 − 2.51976i 0 0 0 0 0
2449.6 0 0 0 − 2.51976i 0 0 0 0 0
2449.7 0 0 0 − 1.47666i 0 0 0 0 0
2449.8 0 0 0 − 1.47666i 0 0 0 0 0
2449.9 0 0 0 1.47666i 0 0 0 0 0
2449.10 0 0 0 1.47666i 0 0 0 0 0
2449.11 0 0 0 2.51976i 0 0 0 0 0
2449.12 0 0 0 2.51976i 0 0 0 0 0
2449.13 0 0 0 5.21476i 0 0 0 0 0
2449.14 0 0 0 5.21476i 0 0 0 0 0
2449.15 0 0 0 9.07064i 0 0 0 0 0
2449.16 0 0 0 9.07064i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2449.16
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 3528.3.f.i 16
3.b odd 2 1 inner 3528.3.f.i 16
7.b odd 2 1 inner 3528.3.f.i 16
7.c even 3 1 504.3.by.d ✓ 16
7.d odd 6 1 504.3.by.d ✓ 16
21.c even 2 1 inner 3528.3.f.i 16
21.g even 6 1 504.3.by.d ✓ 16
21.h odd 6 1 504.3.by.d ✓ 16
28.f even 6 1 1008.3.cg.q 16
28.g odd 6 1 1008.3.cg.q 16
84.j odd 6 1 1008.3.cg.q 16
84.n even 6 1 1008.3.cg.q 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.3.by.d ✓ 16 7.c even 3 1
504.3.by.d ✓ 16 7.d odd 6 1
504.3.by.d ✓ 16 21.g even 6 1
504.3.by.d ✓ 16 21.h odd 6 1
1008.3.cg.q 16 28.f even 6 1
1008.3.cg.q 16 28.g odd 6 1
1008.3.cg.q 16 84.j odd 6 1
1008.3.cg.q 16 84.n even 6 1
3528.3.f.i 16 1.a even 1 1 trivial
3528.3.f.i 16 3.b odd 2 1 inner
3528.3.f.i 16 7.b odd 2 1 inner
3528.3.f.i 16 21.c 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 \(S_{3}^{\mathrm{new}}(3528, [\chi])\):

\( T_{5}^{8} + 118T_{5}^{6} + 3185T_{5}^{4} + 20600T_{5}^{2} + 30976 \) Copy content Toggle raw display
\( T_{11}^{8} - 666T_{11}^{6} + 140489T_{11}^{4} - 9972384T_{11}^{2} + 192876544 \) 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
$5$ \( (T^{8} + 118 T^{6} + \cdots + 30976)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} - 666 T^{6} + \cdots + 192876544)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 766 T^{6} + \cdots + 63043600)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 920 T^{6} + \cdots + 668532736)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 334 T^{6} + \cdots + 2458624)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 3152 T^{6} + \cdots + 60366524416)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 4554 T^{6} + \cdots + 142968684544)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 4852 T^{6} + \cdots + 74287318249)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 34 T^{3} + \cdots - 460448)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 12478867111936)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 20 T^{3} + \cdots + 3352240)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 68829987475456)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 2730531514624)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 10038 T^{6} + \cdots + 36185170176)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 2040555110400)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 28 T^{3} + \cdots - 54752)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} - 6544 T^{6} + \cdots + 21724401664)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 25766 T^{6} + \cdots + 999648030976)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 14 T^{3} + \cdots - 1759733)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 1728025927936)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 68068572135424)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
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