↑

Properties

Label 544.2.bb.d
Level $544$
Weight $2$
Character orbit 544.bb
Analytic conductor $4.344$
Analytic rank $0$
Dimension $16$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [544,2,Mod(161,544)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(544, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 0, 5])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("544.161"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 544 = 2^{5} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 544.bb (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,8,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.34386186996\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 28 x^{14} - 36 x^{13} - 28 x^{12} + 20 x^{11} + 644 x^{10} - 2244 x^{9} + \cdots + 1394 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

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

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{14} - \beta_{4}) q^{5} + (\beta_{6} - \beta_{5}) q^{7} + (\beta_{14} - \beta_{11} - \beta_{8} + \cdots - 1) q^{9} + ( - \beta_{15} - \beta_{9} + \beta_{7}) q^{11} + (2 \beta_{10} + 2 \beta_{8} + 2 \beta_{4}) q^{13}+ \cdots + ( - \beta_{12} + \beta_{9} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{5} - 8 q^{17} - 24 q^{25} + 8 q^{29} - 32 q^{33} + 56 q^{37} + 56 q^{41} + 72 q^{45} - 24 q^{49} - 56 q^{53} + 8 q^{57} - 24 q^{61} + 16 q^{65} + 64 q^{69} + 8 q^{73} - 88 q^{77} + 8 q^{85}+ \cdots - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 28 x^{14} - 36 x^{13} - 28 x^{12} + 20 x^{11} + 644 x^{10} - 2244 x^{9} + \cdots + 1394 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 55\!\cdots\!60 \nu^{15} + \cdots + 12\!\cdots\!58 ) / 11\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 75\!\cdots\!15 \nu^{15} + \cdots + 12\!\cdots\!72 ) / 80\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 61\!\cdots\!45 \nu^{15} + \cdots + 76\!\cdots\!76 ) / 56\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 14\!\cdots\!34 \nu^{15} + \cdots - 38\!\cdots\!98 ) / 11\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 50\!\cdots\!52 \nu^{15} + \cdots + 19\!\cdots\!10 ) / 16\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 51\!\cdots\!70 \nu^{15} + \cdots + 12\!\cdots\!94 ) / 11\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 53\!\cdots\!06 \nu^{15} + \cdots - 50\!\cdots\!98 ) / 11\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 691829692401 \nu^{15} + 5177700027237 \nu^{14} - 16618192183179 \nu^{13} + \cdots + 15\!\cdots\!34 ) / 8104259917558 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11\!\cdots\!25 \nu^{15} + \cdots - 17\!\cdots\!40 ) / 11\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 16\!\cdots\!65 \nu^{15} + \cdots + 25\!\cdots\!60 ) / 11\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 89\!\cdots\!16 \nu^{15} + \cdots + 12\!\cdots\!02 ) / 56\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 22\!\cdots\!05 \nu^{15} + \cdots + 28\!\cdots\!20 ) / 11\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 23\!\cdots\!69 \nu^{15} + \cdots + 42\!\cdots\!04 ) / 11\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 38\!\cdots\!18 \nu^{15} + \cdots + 71\!\cdots\!14 ) / 11\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 64\!\cdots\!29 \nu^{15} + \cdots - 11\!\cdots\!52 ) / 11\!\cdots\!04 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} + \beta_{9} - \beta_{8} + \beta_{5} - \beta_{4} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{14} - \beta_{11} + 2\beta_{10} + 2\beta_{9} - 2\beta_{8} + 5\beta_{4} - \beta_{3} - \beta_{2} - 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{15} + 6 \beta_{14} + 7 \beta_{13} - 3 \beta_{11} - 7 \beta_{10} + 3 \beta_{9} - 11 \beta_{8} + \cdots - 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 7 \beta_{14} + 8 \beta_{13} - \beta_{11} - 28 \beta_{10} + 4 \beta_{9} + 20 \beta_{8} + 4 \beta_{7} + \cdots - 42 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 5 \beta_{14} + 4 \beta_{13} - 15 \beta_{12} + 50 \beta_{11} - 131 \beta_{10} - 54 \beta_{9} + \cdots - 19 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 40 \beta_{15} - 149 \beta_{14} - 152 \beta_{13} - 30 \beta_{12} + 151 \beta_{11} - 70 \beta_{10} + \cdots + 176 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 136 \beta_{15} - 525 \beta_{14} - 568 \beta_{13} - 69 \beta_{12} + 266 \beta_{11} + 771 \beta_{10} + \cdots + 1331 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 104 \beta_{15} - 579 \beta_{14} - 600 \beta_{13} + 176 \beta_{12} - 643 \beta_{11} + 3888 \beta_{10} + \cdots + 2846 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 516 \beta_{15} + 2217 \beta_{14} + 2196 \beta_{13} + 1437 \beta_{12} - 5280 \beta_{11} + 8803 \beta_{10} + \cdots - 989 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 4690 \beta_{15} + 17073 \beta_{14} + 18202 \beta_{13} + 3340 \beta_{12} - 13773 \beta_{11} + \cdots - 31236 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 10446 \beta_{15} + 41481 \beta_{14} + 43490 \beta_{13} + 2585 \beta_{12} - 8206 \beta_{11} + \cdots - 123947 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 1484 \beta_{15} + 6013 \beta_{14} + 6748 \beta_{13} - 31624 \beta_{12} + 117177 \beta_{11} + \cdots - 157278 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 100420 \beta_{15} - 391963 \beta_{14} - 410500 \beta_{13} - 136003 \beta_{12} + 526500 \beta_{11} + \cdots + 528373 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 439102 \beta_{15} - 1665357 \beta_{14} - 1760634 \beta_{13} - 252152 \beta_{12} + 980013 \beta_{11} + \cdots + 3801268 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 666740 \beta_{15} - 2642575 \beta_{14} - 2764760 \beta_{13} + 312049 \beta_{12} - 1254412 \beta_{11} + \cdots + 9919447 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/544\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(511\) \(513\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
2.00611 − 2.42897i
1.51341 − 1.23949i
0.900800 + 0.239491i
0.408102 + 1.42897i
−2.03846 + 1.25857i
−1.16309 + 0.895981i
0.748875 + 0.104019i
1.62425 − 0.258573i
2.00611 + 2.42897i
1.51341 + 1.23949i
0.900800 − 0.239491i
0.408102 − 1.42897i
−2.03846 − 1.25857i
−1.16309 − 0.895981i
0.748875 − 0.104019i
1.62425 + 0.258573i
0 −1.12996 + 2.72797i 0 −0.461191 − 0.191032i 0 −1.77925 + 0.736988i 0 −4.04370 − 4.04370i 0
161.2 0 −0.433184 + 1.04580i 0 2.16830 + 0.898138i 0 3.20687 − 1.32833i 0 1.21528 + 1.21528i 0
161.3 0 0.433184 − 1.04580i 0 2.16830 + 0.898138i 0 −3.20687 + 1.32833i 0 1.21528 + 1.21528i 0
161.4 0 1.12996 − 2.72797i 0 −0.461191 − 0.191032i 0 1.77925 − 0.736988i 0 −4.04370 − 4.04370i 0
257.1 0 −2.58993 + 1.07278i 0 1.15711 + 2.79352i 0 1.75551 − 4.23817i 0 3.43555 − 3.43555i 0
257.2 0 −1.35196 + 0.560001i 0 −0.864220 − 2.08641i 0 −0.684429 + 1.65236i 0 −0.607119 + 0.607119i 0
257.3 0 1.35196 − 0.560001i 0 −0.864220 − 2.08641i 0 0.684429 − 1.65236i 0 −0.607119 + 0.607119i 0
257.4 0 2.58993 − 1.07278i 0 1.15711 + 2.79352i 0 −1.75551 + 4.23817i 0 3.43555 − 3.43555i 0
321.1 0 −1.12996 − 2.72797i 0 −0.461191 + 0.191032i 0 −1.77925 − 0.736988i 0 −4.04370 + 4.04370i 0
321.2 0 −0.433184 − 1.04580i 0 2.16830 − 0.898138i 0 3.20687 + 1.32833i 0 1.21528 − 1.21528i 0
321.3 0 0.433184 + 1.04580i 0 2.16830 − 0.898138i 0 −3.20687 − 1.32833i 0 1.21528 − 1.21528i 0
321.4 0 1.12996 + 2.72797i 0 −0.461191 + 0.191032i 0 1.77925 + 0.736988i 0 −4.04370 + 4.04370i 0
417.1 0 −2.58993 − 1.07278i 0 1.15711 − 2.79352i 0 1.75551 + 4.23817i 0 3.43555 + 3.43555i 0
417.2 0 −1.35196 − 0.560001i 0 −0.864220 + 2.08641i 0 −0.684429 − 1.65236i 0 −0.607119 − 0.607119i 0
417.3 0 1.35196 + 0.560001i 0 −0.864220 + 2.08641i 0 0.684429 + 1.65236i 0 −0.607119 − 0.607119i 0
417.4 0 2.58993 + 1.07278i 0 1.15711 − 2.79352i 0 −1.75551 − 4.23817i 0 3.43555 + 3.43555i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
17.d even 8 1 inner
68.g odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 544.2.bb.d ✓ 16
4.b odd 2 1 inner 544.2.bb.d ✓ 16
17.d even 8 1 inner 544.2.bb.d ✓ 16
17.e odd 16 2 9248.2.a.bx 16
68.g odd 8 1 inner 544.2.bb.d ✓ 16
68.i even 16 2 9248.2.a.bx 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
544.2.bb.d ✓ 16 1.a even 1 1 trivial
544.2.bb.d ✓ 16 4.b odd 2 1 inner
544.2.bb.d ✓ 16 17.d even 8 1 inner
544.2.bb.d ✓ 16 68.g odd 8 1 inner
9248.2.a.bx 16 17.e odd 16 2
9248.2.a.bx 16 68.i even 16 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(544, [\chi])\):

\( T_{3}^{16} - 80T_{3}^{10} + 4808T_{3}^{8} - 5760T_{3}^{6} + 3200T_{3}^{4} + 15040T_{3}^{2} + 35344 \) Copy content Toggle raw display
\( T_{5}^{8} - 4T_{5}^{7} + 14T_{5}^{6} - 28T_{5}^{5} + 50T_{5}^{4} - 112T_{5}^{3} + 112T_{5}^{2} + 192T_{5} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 80 T^{10} + \cdots + 35344 \) Copy content Toggle raw display
$5$ \( (T^{8} - 4 T^{7} + 14 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + 12 T^{14} + \cdots + 9048064 \) Copy content Toggle raw display
$11$ \( T^{16} - 8 T^{14} + \cdots + 35344 \) Copy content Toggle raw display
$13$ \( (T^{4} + 24 T^{2} + 16)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 4 T^{7} + \cdots + 83521)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + 3064 T^{12} + \cdots + 9048064 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 2316304384 \) Copy content Toggle raw display
$29$ \( (T^{8} - 4 T^{7} + \cdots + 322624)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + 124 T^{14} + \cdots + 9048064 \) Copy content Toggle raw display
$37$ \( (T^{8} - 28 T^{7} + \cdots + 614656)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 28 T^{7} + \cdots + 20164)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 2759479124224 \) Copy content Toggle raw display
$47$ \( (T^{8} + 184 T^{6} + \cdots + 192512)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 28 T^{7} + \cdots + 141376)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 752608554098944 \) Copy content Toggle raw display
$61$ \( (T^{8} + 12 T^{7} + \cdots + 2458624)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 524 T^{6} + \cdots + 12032)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 144769024 \) Copy content Toggle raw display
$73$ \( (T^{8} - 4 T^{7} + \cdots + 58564)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 12091253653504 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 87\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T^{8} + 220 T^{6} + \cdots + 33856)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 8 T^{7} + \cdots + 1012036)^{2} \) Copy content Toggle raw display
show more
show less