↑

Properties

Label 1856.2.g.b
Level $1856$
Weight $2$
Character orbit 1856.g
Analytic conductor $14.820$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1856,2,Mod(289,1856)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1856, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1856.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1856.g (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: \(14.8202346151\)
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: \( x^{16} - 8x^{14} + 49x^{12} - 104x^{10} + 160x^{8} - 104x^{6} + 49x^{4} - 8x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + \beta_{8} q^{5} - \beta_{10} q^{7} + (\beta_{4} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{3} + \beta_{8} q^{5} - \beta_{10} q^{7} + (\beta_{4} + 3) q^{9} + \beta_{12} q^{11} + (\beta_{11} + \beta_{8}) q^{13} + (\beta_{13} + 5 \beta_{3}) q^{15} - \beta_{14} q^{17} + (\beta_{12} + \beta_{5}) q^{19} + ( - 6 \beta_{6} + \beta_{2}) q^{21} - \beta_{10} q^{23} + (4 \beta_{4} - 4) q^{25} - \beta_{12} q^{27} + ( - \beta_{11} - \beta_{8} + \cdots - \beta_{2}) q^{29}+ \cdots + (3 \beta_{12} - 3 \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 48 q^{9} - 64 q^{25} - 48 q^{33} + 80 q^{49} + 48 q^{57} - 96 q^{65} - 96 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8x^{14} + 49x^{12} - 104x^{10} + 160x^{8} - 104x^{6} + 49x^{4} - 8x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 116\nu^{14} - 791\nu^{12} + 4704\nu^{10} - 6208\nu^{8} + 9544\nu^{6} + 480\nu^{4} + 7316\nu^{2} - 599 ) / 1584 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{15} - 13\nu^{13} + 88\nu^{11} - 340\nu^{9} + 624\nu^{7} - 760\nu^{5} + 357\nu^{3} - 101\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 127\nu^{15} - 945\nu^{13} + 5760\nu^{11} - 10520\nu^{9} + 17640\nu^{7} - 10080\nu^{5} + 8911\nu^{3} - 225\nu ) / 1056 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 15\nu^{14} - 112\nu^{12} + 672\nu^{10} - 1176\nu^{8} + 1616\nu^{6} - 384\nu^{4} + 63\nu^{2} + 80 ) / 48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 179 \nu^{15} - 1853 \nu^{13} + 12032 \nu^{11} - 38392 \nu^{9} + 67208 \nu^{7} - 74912 \nu^{5} + \cdots - 10029 \nu ) / 1056 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -41\nu^{15} + 329\nu^{13} - 2016\nu^{11} + 4312\nu^{9} - 6664\nu^{7} + 4608\nu^{5} - 2345\nu^{3} + 665\nu ) / 144 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 140\nu^{14} - 1051\nu^{12} + 6272\nu^{10} - 10976\nu^{8} + 14104\nu^{6} - 3584\nu^{4} + 588\nu^{2} - 795 ) / 264 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -25\nu^{14} + 208\nu^{12} - 1280\nu^{10} + 2920\nu^{8} - 4400\nu^{6} + 3008\nu^{4} - 969\nu^{2} + 112 ) / 48 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 59\nu^{14} - 476\nu^{12} + 2916\nu^{10} - 6280\nu^{8} + 9538\nu^{6} - 6192\nu^{4} + 2351\nu^{2} - 260 ) / 99 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -340\nu^{14} + 2543\nu^{12} - 15232\nu^{10} + 26656\nu^{8} - 35912\nu^{6} + 8704\nu^{4} - 1428\nu^{2} - 2001 ) / 528 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -43\nu^{14} + 336\nu^{12} - 2048\nu^{10} + 4120\nu^{8} - 6288\nu^{6} + 3680\nu^{4} - 1851\nu^{2} + 192 ) / 48 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 887 \nu^{15} + 7257 \nu^{13} - 44736 \nu^{11} + 100024 \nu^{9} - 157992 \nu^{7} + 116832 \nu^{5} + \cdots + 16713 \nu ) / 1056 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 183\nu^{15} - 1449\nu^{13} + 8832\nu^{11} - 18184\nu^{9} + 27048\nu^{7} - 15456\nu^{5} + 5943\nu^{3} - 345\nu ) / 176 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 45\nu^{15} - 357\nu^{13} + 2176\nu^{11} - 4496\nu^{9} + 6664\nu^{7} - 3808\nu^{5} + 1301\nu^{3} - 85\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -147\nu^{15} + 1155\nu^{13} - 7040\nu^{11} + 14296\nu^{9} - 21560\nu^{7} + 12320\nu^{5} - 5635\nu^{3} + 275\nu ) / 48 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{15} - 2\beta_{13} + 2\beta_{12} - 3\beta_{6} - 2\beta_{3} + \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{11} + 2\beta_{10} + 7\beta_{9} + 2\beta_{8} + \beta_{7} + 2\beta_{4} + 8\beta _1 + 8 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{15} + \beta_{14} - 7\beta_{13} - 6\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 22\beta_{11} - 12\beta_{10} + 41\beta_{9} + 14\beta_{8} - 5\beta_{7} - 16\beta_{4} + 34\beta _1 - 34 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 21 \beta_{15} + 14 \beta_{14} - 80 \beta_{13} - 28 \beta_{12} + 93 \beta_{6} + 42 \beta_{5} + \cdots - 49 \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -34\beta_{10} - 13\beta_{7} - 48\beta_{4} - 88 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 115 \beta_{15} - 82 \beta_{14} + 448 \beta_{13} - 148 \beta_{12} + 501 \beta_{6} + 246 \beta_{5} + \cdots - 281 \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -662\beta_{11} - 380\beta_{10} - 1295\beta_{9} - 478\beta_{8} - 141\beta_{7} - 544\beta_{4} - 958\beta _1 - 958 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 318\beta_{15} - 231\beta_{14} + 1247\beta_{13} + 918\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 3668 \beta_{11} + 2114 \beta_{10} - 7201 \beta_{9} - 2674 \beta_{8} + 777 \beta_{7} + 3038 \beta_{4} + \cdots + 5288 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 3527 \beta_{15} - 2576 \beta_{14} + 13858 \beta_{13} + 4478 \beta_{12} - 15243 \beta_{6} + \cdots + 8777 \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2936\beta_{10} + 1076\beta_{7} + 4224\beta_{4} + 7327 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 19575 \beta_{15} + 14320 \beta_{14} - 76958 \beta_{13} + 24830 \beta_{12} - 84549 \beta_{6} + \cdots + 48775 \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 112972 \beta_{11} + 65214 \beta_{10} + 222097 \beta_{9} + 82670 \beta_{8} + 23879 \beta_{7} + \cdots + 162632 ) / 8 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( -54334\beta_{15} + 39767\beta_{14} - 213649\beta_{13} - 156426\beta_{3} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1856\mathbb{Z}\right)^\times\).

\(n\) \(321\) \(581\) \(639\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
2.04058 − 1.17813i
0.367543 + 0.212201i
2.04058 + 1.17813i
0.367543 − 0.212201i
−0.670418 + 0.387066i
−1.11871 − 0.645885i
−0.670418 − 0.387066i
−1.11871 + 0.645885i
0.670418 − 0.387066i
1.11871 + 0.645885i
0.670418 + 0.387066i
1.11871 − 0.645885i
−2.04058 + 1.17813i
−0.367543 − 0.212201i
−2.04058 − 1.17813i
−0.367543 + 0.212201i
0 −2.78066 0 − 1.43937i 0 −3.93244 0 4.73205 0
289.2 0 −2.78066 0 − 1.43937i 0 3.93244 0 4.73205 0
289.3 0 −2.78066 0 1.43937i 0 −3.93244 0 4.73205 0
289.4 0 −2.78066 0 1.43937i 0 3.93244 0 4.73205 0
289.5 0 −2.06590 0 − 3.99102i 0 −2.92163 0 1.26795 0
289.6 0 −2.06590 0 − 3.99102i 0 2.92163 0 1.26795 0
289.7 0 −2.06590 0 3.99102i 0 −2.92163 0 1.26795 0
289.8 0 −2.06590 0 3.99102i 0 2.92163 0 1.26795 0
289.9 0 2.06590 0 − 3.99102i 0 −2.92163 0 1.26795 0
289.10 0 2.06590 0 − 3.99102i 0 2.92163 0 1.26795 0
289.11 0 2.06590 0 3.99102i 0 −2.92163 0 1.26795 0
289.12 0 2.06590 0 3.99102i 0 2.92163 0 1.26795 0
289.13 0 2.78066 0 − 1.43937i 0 −3.93244 0 4.73205 0
289.14 0 2.78066 0 − 1.43937i 0 3.93244 0 4.73205 0
289.15 0 2.78066 0 1.43937i 0 −3.93244 0 4.73205 0
289.16 0 2.78066 0 1.43937i 0 3.93244 0 4.73205 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
29.b even 2 1 inner
116.d odd 2 1 inner
232.b odd 2 1 inner
232.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1856.2.g.b ✓ 16
4.b odd 2 1 inner 1856.2.g.b ✓ 16
8.b even 2 1 inner 1856.2.g.b ✓ 16
8.d odd 2 1 inner 1856.2.g.b ✓ 16
29.b even 2 1 inner 1856.2.g.b ✓ 16
116.d odd 2 1 inner 1856.2.g.b ✓ 16
232.b odd 2 1 inner 1856.2.g.b ✓ 16
232.g even 2 1 inner 1856.2.g.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1856.2.g.b ✓ 16 1.a even 1 1 trivial
1856.2.g.b ✓ 16 4.b odd 2 1 inner
1856.2.g.b ✓ 16 8.b even 2 1 inner
1856.2.g.b ✓ 16 8.d odd 2 1 inner
1856.2.g.b ✓ 16 29.b even 2 1 inner
1856.2.g.b ✓ 16 116.d odd 2 1 inner
1856.2.g.b ✓ 16 232.b odd 2 1 inner
1856.2.g.b ✓ 16 232.g even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 12T_{3}^{2} + 33 \) acting on \(S_{2}^{\mathrm{new}}(1856, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{4} - 12 T^{2} + 33)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} + 18 T^{2} + 33)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} - 24 T^{2} + 132)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 36 T^{2} + 297)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 30 T^{2} + 33)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 36 T^{2} + 132)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} - 36 T^{2} + 132)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 24 T^{2} + 132)^{4} \) Copy content Toggle raw display
$29$ \( (T^{8} + 4 T^{6} + \cdots + 707281)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 52 T^{2} + 529)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 112 T^{2} + 2704)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 60 T^{2} + 132)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 60 T^{2} + 33)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 28 T^{2} + 121)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + 150 T^{2} + 5577)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + 152 T^{2} + 5476)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} - 124 T^{2} + 2116)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 96 T^{2} + 576)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 216 T^{2} + 4752)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 192 T^{2} + 8448)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 28 T^{2} + 121)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 104 T^{2} + 4)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 372 T^{2} + 22308)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 300 T^{2} + 22308)^{4} \) Copy content Toggle raw display
show more
show less