↑

Properties

Label 1445.2.b.f
Level $1445$
Weight $2$
Character orbit 1445.b
Analytic conductor $11.538$
Analytic rank $0$
Dimension $12$
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] = [1445,2,Mod(579,1445)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1445.579"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1445, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1445 = 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1445.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,-12,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.5383830921\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2x^{10} - 9x^{8} + 228x^{6} - 225x^{4} - 1250x^{2} + 15625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 85)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - \beta_{10} q^{3} + (\beta_{8} - 1) q^{4} - \beta_1 q^{5} + ( - \beta_{11} + \beta_{9} + \cdots + \beta_1) q^{6} + \beta_{7} q^{7} + ( - \beta_{6} + \beta_{3}) q^{8} + (\beta_{8} - \beta_{5} - 2) q^{9}+ \cdots + ( - 3 \beta_{11} + \beta_{9} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{4} - 28 q^{9} - 12 q^{15} + 4 q^{16} - 48 q^{19} + 24 q^{21} + 4 q^{25} + 24 q^{26} - 52 q^{30} + 20 q^{35} + 68 q^{36} + 28 q^{49} - 40 q^{50} + 12 q^{55} - 72 q^{59} + 76 q^{60} + 76 q^{64}+ \cdots - 96 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2x^{10} - 9x^{8} + 228x^{6} - 225x^{4} - 1250x^{2} + 15625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{10} + 123\nu^{8} + 366\nu^{6} + 978\nu^{4} + 17025\nu^{2} + 41875 ) / 30000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{10} - 27\nu^{8} + 666\nu^{6} - 1422\nu^{4} - 9675\nu^{2} + 75625 ) / 30000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{11} - 23\nu^{9} + 59\nu^{7} - 3\nu^{5} - 2350\nu^{3} + 3750\nu ) / 12500 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{10} + 2\nu^{8} + 9\nu^{6} - 228\nu^{4} + 850\nu^{2} + 1250 ) / 1250 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4\nu^{10} - 33\nu^{8} + 14\nu^{6} + 512\nu^{4} - 2850\nu^{2} + 625 ) / 5000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 23\nu^{11} - 171\nu^{9} - 582\nu^{7} + 4494\nu^{5} - 37425\nu^{3} - 11875\nu ) / 150000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -6\nu^{10} - 13\nu^{8} + 104\nu^{6} - 518\nu^{4} - 1850\nu^{2} + 6875 ) / 5000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{11} + 2\nu^{9} + 9\nu^{7} - 228\nu^{5} + 225\nu^{3} + 1250\nu ) / 3125 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 19\nu^{11} - 63\nu^{9} - 746\nu^{7} + 2682\nu^{5} - 1225\nu^{3} - 66875\nu ) / 50000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -29\nu^{11} + 83\nu^{9} - 414\nu^{7} - 2462\nu^{5} + 8475\nu^{3} - 43125\nu ) / 50000 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{10} - \beta_{9} - 6\beta_{7} + 2\beta_{4} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{8} + 2\beta_{6} - 4\beta_{5} - 6\beta_{3} + 6\beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{11} - 6\beta_{10} - 26\beta_{9} - 6\beta_{7} + 6\beta_{4} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{8} + 11\beta_{6} + \beta_{5} + 21\beta_{3} + 27\beta_{2} - 104 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -24\beta_{11} - 44\beta_{10} - 31\beta_{9} - 36\beta_{7} + 20\beta_{4} - 76\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -48\beta_{8} - 188\beta_{6} - 112\beta_{5} + 116\beta_{3} - 20\beta_{2} - 63 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 24\beta_{11} - 348\beta_{10} - 68\beta_{9} + 492\beta_{7} - 612\beta_{4} - 195\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -936\beta_{8} + 117\beta_{6} + 297\beta_{5} + 939\beta_{3} - 315\beta_{2} - 952 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -1992\beta_{11} + 726\beta_{10} + 2163\beta_{9} + 678\beta_{7} - 1962\beta_{4} - 970\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1445\mathbb{Z}\right)^\times\).

\(n\) \(581\) \(1157\)
\(\chi(n)\) \(1\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
579.1
−1.90494 + 1.17098i
1.90494 − 1.17098i
0.178191 + 2.22896i
−0.178191 − 2.22896i
2.08313 − 0.812746i
−2.08313 + 0.812746i
−2.08313 − 0.812746i
2.08313 + 0.812746i
−0.178191 + 2.22896i
0.178191 − 2.22896i
1.90494 + 1.17098i
−1.90494 − 1.17098i
− 2.38621i − 3.15462i −3.69399 1.90494 − 1.17098i −7.52757 − 0.219993i 4.04223i −6.95160 −2.79421 − 4.54559i
579.2 − 2.38621i 3.15462i −3.69399 −1.90494 + 1.17098i 7.52757 0.219993i 4.04223i −6.95160 2.79421 + 4.54559i
579.3 − 1.80333i − 0.561698i −1.25200 −0.178191 − 2.22896i −1.01293 3.11199i − 1.34889i 2.68450 −4.01955 + 0.321338i
579.4 − 1.80333i 0.561698i −1.25200 0.178191 + 2.22896i 1.01293 − 3.11199i − 1.34889i 2.68450 4.01955 − 0.321338i
579.5 − 0.232389i − 2.39435i 1.94600 −2.08313 + 0.812746i −0.556420 2.06570i − 0.917007i −2.73289 0.188874 + 0.484098i
579.6 − 0.232389i 2.39435i 1.94600 2.08313 − 0.812746i 0.556420 − 2.06570i − 0.917007i −2.73289 −0.188874 − 0.484098i
579.7 0.232389i − 2.39435i 1.94600 2.08313 + 0.812746i 0.556420 2.06570i 0.917007i −2.73289 −0.188874 + 0.484098i
579.8 0.232389i 2.39435i 1.94600 −2.08313 − 0.812746i −0.556420 − 2.06570i 0.917007i −2.73289 0.188874 − 0.484098i
579.9 1.80333i − 0.561698i −1.25200 0.178191 − 2.22896i 1.01293 3.11199i 1.34889i 2.68450 4.01955 + 0.321338i
579.10 1.80333i 0.561698i −1.25200 −0.178191 + 2.22896i −1.01293 − 3.11199i 1.34889i 2.68450 −4.01955 − 0.321338i
579.11 2.38621i − 3.15462i −3.69399 −1.90494 − 1.17098i 7.52757 − 0.219993i − 4.04223i −6.95160 2.79421 − 4.54559i
579.12 2.38621i 3.15462i −3.69399 1.90494 + 1.17098i −7.52757 0.219993i − 4.04223i −6.95160 −2.79421 + 4.54559i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 579.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
17.b even 2 1 inner
85.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1445.2.b.f 12
5.b even 2 1 inner 1445.2.b.f 12
5.c odd 4 2 7225.2.a.bp 12
17.b even 2 1 inner 1445.2.b.f 12
17.d even 8 2 85.2.j.c ✓ 12
51.g odd 8 2 765.2.t.e 12
85.c even 2 1 inner 1445.2.b.f 12
85.g odd 4 2 7225.2.a.bp 12
85.k odd 8 2 425.2.e.d 12
85.m even 8 2 85.2.j.c ✓ 12
85.n odd 8 2 425.2.e.d 12
255.y odd 8 2 765.2.t.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.2.j.c ✓ 12 17.d even 8 2
85.2.j.c ✓ 12 85.m even 8 2
425.2.e.d 12 85.k odd 8 2
425.2.e.d 12 85.n odd 8 2
765.2.t.e 12 51.g odd 8 2
765.2.t.e 12 255.y odd 8 2
1445.2.b.f 12 1.a even 1 1 trivial
1445.2.b.f 12 5.b even 2 1 inner
1445.2.b.f 12 17.b even 2 1 inner
1445.2.b.f 12 85.c even 2 1 inner
7225.2.a.bp 12 5.c odd 4 2
7225.2.a.bp 12 85.g odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1445, [\chi])\):

\( T_{2}^{6} + 9T_{2}^{4} + 19T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{6} - 32T_{11}^{4} + 16T_{11}^{2} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 9 T^{4} + 19 T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{6} + 16 T^{4} + \cdots + 18)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} - 2 T^{10} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( (T^{6} + 14 T^{4} + 42 T^{2} + 2)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 32 T^{4} + 16 T^{2} - 2)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 22 T^{4} + \cdots + 100)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( (T + 4)^{12} \) Copy content Toggle raw display
$23$ \( (T^{6} + 72 T^{4} + \cdots + 50)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 94 T^{4} + \cdots - 2312)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 66 T^{4} + \cdots - 6498)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 38 T^{4} + \cdots + 200)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 152 T^{4} + \cdots - 2592)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 66 T^{4} + \cdots + 1444)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 82 T^{4} + \cdots + 13924)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 156 T^{4} + \cdots + 110224)^{2} \) Copy content Toggle raw display
$59$ \( (T + 6)^{12} \) Copy content Toggle raw display
$61$ \( (T^{2} - 32)^{6} \) Copy content Toggle raw display
$67$ \( (T^{6} + 214 T^{4} + \cdots + 93636)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 132 T^{4} + \cdots - 13122)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 152 T^{4} + \cdots + 512)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 174 T^{4} + \cdots - 114242)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 58 T^{4} + \cdots + 2916)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} - 4 T^{2} + \cdots + 1590)^{4} \) Copy content Toggle raw display
$97$ \( (T^{6} + 518 T^{4} + \cdots + 3645000)^{2} \) Copy content Toggle raw display
show more
show less