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Properties

Label 890.2.b.a
Level $890$
Weight $2$
Character orbit 890.b
Analytic conductor $7.107$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [890,2,Mod(179,890)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(890, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("890.179");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 890 = 2 \cdot 5 \cdot 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 890.b (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: \(7.10668577989\)
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: \( x^{16} + 24x^{14} + 218x^{12} + 948x^{10} + 2061x^{8} + 2076x^{6} + 748x^{4} + 96x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
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.

\(f(q)\) \(=\) \( q - \beta_{10} q^{2} + ( - \beta_{10} + \beta_1) q^{3} - q^{4} + ( - \beta_{13} + \beta_{12} + \beta_{11}) q^{5} + (\beta_{3} - 1) q^{6} + ( - \beta_{14} - \beta_{13} + \cdots - \beta_{9}) q^{7} + \beta_{10} q^{8}+ \cdots + (\beta_{7} - 3 \beta_{6} - \beta_{5} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} - 2 q^{5} - 8 q^{6} - 2 q^{10} - 8 q^{11} + 4 q^{14} + 16 q^{16} + 16 q^{19} + 2 q^{20} - 20 q^{21} + 8 q^{24} - 4 q^{25} - 8 q^{26} + 20 q^{29} + 4 q^{30} + 28 q^{34} - 32 q^{35} + 20 q^{39}+ \cdots - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 24x^{14} + 218x^{12} + 948x^{10} + 2061x^{8} + 2076x^{6} + 748x^{4} + 96x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 45\nu^{14} + 1075\nu^{12} + 9691\nu^{10} + 41593\nu^{8} + 88200\nu^{6} + 83876\nu^{4} + 24692\nu^{2} + 1716 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -47\nu^{14} - 1123\nu^{12} - 10127\nu^{10} - 43489\nu^{8} - 92322\nu^{6} - 88024\nu^{4} - 26152\nu^{2} - 1844 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 151 \nu^{14} - 3609 \nu^{12} - 32559 \nu^{10} - 139903 \nu^{8} - 297230 \nu^{6} - 283664 \nu^{4} + \cdots - 5940 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 115 \nu^{14} - 2748 \nu^{12} - 24784 \nu^{10} - 106450 \nu^{8} - 226031 \nu^{6} - 215558 \nu^{4} + \cdots - 4520 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 135 \nu^{14} + 3226 \nu^{12} + 29096 \nu^{10} + 124974 \nu^{8} + 265357 \nu^{6} + 252990 \nu^{4} + \cdots + 5280 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 287 \nu^{14} - 6859 \nu^{12} - 61873 \nu^{10} - 265825 \nu^{8} - 564648 \nu^{6} - 538732 \nu^{4} + \cdots - 11308 ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 537 \nu^{15} + 363 \nu^{14} + 12833 \nu^{13} + 8675 \nu^{12} + 115751 \nu^{11} + 78249 \nu^{10} + \cdots + 14212 ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 639 \nu^{15} - 363 \nu^{14} - 15271 \nu^{13} - 8675 \nu^{12} - 137749 \nu^{11} - 78249 \nu^{10} + \cdots - 14212 ) / 16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 461 \nu^{15} - 11017 \nu^{13} - 99375 \nu^{11} - 426901 \nu^{9} - 906632 \nu^{7} - 864714 \nu^{5} + \cdots - 18104 \nu ) / 8 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 903 \nu^{15} + 363 \nu^{14} + 21579 \nu^{13} + 8675 \nu^{12} + 194633 \nu^{11} + 78249 \nu^{10} + \cdots + 14212 ) / 16 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 903 \nu^{15} + 363 \nu^{14} - 21579 \nu^{13} + 8675 \nu^{12} - 194633 \nu^{11} + 78249 \nu^{10} + \cdots + 14212 ) / 16 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1191 \nu^{15} + 363 \nu^{14} - 28463 \nu^{13} + 8675 \nu^{12} - 256745 \nu^{11} + 78249 \nu^{10} + \cdots + 14212 ) / 16 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 362 \nu^{15} + 8651 \nu^{13} + 78032 \nu^{11} + 335205 \nu^{9} + 711856 \nu^{7} + 678870 \nu^{5} + \cdots + 14232 \nu ) / 4 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1769 \nu^{15} + 363 \nu^{14} - 42277 \nu^{13} + 8675 \nu^{12} - 381363 \nu^{11} + 78249 \nu^{10} + \cdots + 14212 ) / 16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + \beta_{5} + \beta_{4} - 2\beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{14} + \beta_{12} - \beta_{11} - 3\beta_{10} + \beta_{9} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{7} - \beta_{6} - 8\beta_{5} - 6\beta_{4} + 17\beta_{3} + \beta_{2} + 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 11 \beta_{15} - 10 \beta_{14} - 2 \beta_{13} - 8 \beta_{12} + 15 \beta_{11} + 35 \beta_{10} + \cdots + 32 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{12} + 2\beta_{11} - 36\beta_{7} + 15\beta_{6} + 62\beta_{5} + 42\beta_{4} - 137\beta_{3} - 21\beta_{2} - 68 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 101 \beta_{15} + 94 \beta_{14} + 32 \beta_{13} + 54 \beta_{12} - 167 \beta_{11} - 323 \beta_{10} + \cdots - 232 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 34 \beta_{12} - 34 \beta_{11} + 220 \beta_{7} - 173 \beta_{6} - 488 \beta_{5} - 326 \beta_{4} + \cdots + 536 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 877 \beta_{15} - 866 \beta_{14} - 378 \beta_{13} - 348 \beta_{12} + 1657 \beta_{11} + \cdots + 1786 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 402 \beta_{12} + 402 \beta_{11} - 1354 \beta_{7} + 1785 \beta_{6} + 3900 \beta_{5} + 2652 \beta_{4} + \cdots - 4480 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 7465 \beta_{15} + 7848 \beta_{14} + 3948 \beta_{13} + 2200 \beta_{12} - 15543 \beta_{11} + \cdots - 14166 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 4130 \beta_{12} - 4130 \beta_{11} + 8288 \beta_{7} - 17309 \beta_{6} - 31544 \beta_{5} - 22014 \beta_{4} + \cdots + 38176 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 63121 \beta_{15} - 70110 \beta_{14} - 38566 \beta_{13} - 13688 \beta_{12} + 141281 \beta_{11} + \cdots + 114290 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 39594 \beta_{12} + 39594 \beta_{11} - 49818 \beta_{7} + 161449 \beta_{6} + 257560 \beta_{5} + \cdots - 326908 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 532857 \beta_{15} + 619024 \beta_{14} + 361464 \beta_{13} + 83384 \beta_{12} - 1259411 \beta_{11} + \cdots - 932534 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/890\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(357\)
\(\chi(n)\) \(1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
179.1
− 1.85072i
− 1.67435i
− 0.318854i
0.319587i
0.578580i
1.46161i
2.57103i
2.91313i
− 2.91313i
− 2.57103i
− 1.46161i
− 0.578580i
− 0.319587i
0.318854i
1.67435i
1.85072i
− 1.00000i − 2.85072i −1.00000 1.58682 − 1.57543i −2.85072 − 4.52360i 1.00000i −5.12662 −1.57543 − 1.58682i
179.2 − 1.00000i − 2.67435i −1.00000 −1.23338 + 1.86515i −2.67435 3.29849i 1.00000i −4.15217 1.86515 + 1.23338i
179.3 − 1.00000i − 1.31885i −1.00000 0.252328 − 2.22179i −1.31885 − 0.332784i 1.00000i 1.26063 −2.22179 − 0.252328i
179.4 − 1.00000i − 0.680413i −1.00000 −2.20729 − 0.357579i −0.680413 1.50187i 1.00000i 2.53704 −0.357579 + 2.20729i
179.5 − 1.00000i − 0.421420i −1.00000 0.968212 + 2.01558i −0.421420 − 0.408024i 1.00000i 2.82241 2.01558 − 0.968212i
179.6 − 1.00000i 0.461608i −1.00000 −2.08054 + 0.819371i 0.461608 − 0.960393i 1.00000i 2.78692 0.819371 + 2.08054i
179.7 − 1.00000i 1.57103i −1.00000 −0.425708 − 2.19517i 1.57103 − 0.668798i 1.00000i 0.531874 −2.19517 + 0.425708i
179.8 − 1.00000i 1.91313i −1.00000 2.13955 + 0.649868i 1.91313 4.09323i 1.00000i −0.660064 0.649868 − 2.13955i
179.9 1.00000i − 1.91313i −1.00000 2.13955 − 0.649868i 1.91313 − 4.09323i − 1.00000i −0.660064 0.649868 + 2.13955i
179.10 1.00000i − 1.57103i −1.00000 −0.425708 + 2.19517i 1.57103 0.668798i − 1.00000i 0.531874 −2.19517 − 0.425708i
179.11 1.00000i − 0.461608i −1.00000 −2.08054 − 0.819371i 0.461608 0.960393i − 1.00000i 2.78692 0.819371 − 2.08054i
179.12 1.00000i 0.421420i −1.00000 0.968212 − 2.01558i −0.421420 0.408024i − 1.00000i 2.82241 2.01558 + 0.968212i
179.13 1.00000i 0.680413i −1.00000 −2.20729 + 0.357579i −0.680413 − 1.50187i − 1.00000i 2.53704 −0.357579 − 2.20729i
179.14 1.00000i 1.31885i −1.00000 0.252328 + 2.22179i −1.31885 0.332784i − 1.00000i 1.26063 −2.22179 + 0.252328i
179.15 1.00000i 2.67435i −1.00000 −1.23338 − 1.86515i −2.67435 − 3.29849i − 1.00000i −4.15217 1.86515 − 1.23338i
179.16 1.00000i 2.85072i −1.00000 1.58682 + 1.57543i −2.85072 4.52360i − 1.00000i −5.12662 −1.57543 + 1.58682i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 179.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 890.2.b.a ✓ 16
5.b even 2 1 inner 890.2.b.a ✓ 16
5.c odd 4 1 4450.2.a.bi 8
5.c odd 4 1 4450.2.a.bj 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
890.2.b.a ✓ 16 1.a even 1 1 trivial
890.2.b.a ✓ 16 5.b even 2 1 inner
4450.2.a.bi 8 5.c odd 4 1
4450.2.a.bj 8 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 24T_{3}^{14} + 218T_{3}^{12} + 948T_{3}^{10} + 2089T_{3}^{8} + 2268T_{3}^{6} + 1096T_{3}^{4} + 224T_{3}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(890, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} + 24 T^{14} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{16} + 2 T^{15} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} + 52 T^{14} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( (T^{8} + 4 T^{7} - 30 T^{6} + \cdots + 28)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 72 T^{14} + \cdots + 419904 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 1246937344 \) Copy content Toggle raw display
$19$ \( (T^{8} - 8 T^{7} - 28 T^{6} + \cdots - 64)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 228130816 \) Copy content Toggle raw display
$29$ \( (T^{8} - 10 T^{7} + \cdots - 311752)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 94 T^{6} + \cdots + 200)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 6712197184 \) Copy content Toggle raw display
$41$ \( (T^{8} - 124 T^{6} + \cdots + 44288)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 4187125264 \) Copy content Toggle raw display
$47$ \( T^{16} + 468 T^{14} + \cdots + 19784704 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 788870817856 \) Copy content Toggle raw display
$59$ \( (T^{8} - 4 T^{7} + \cdots - 6891008)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 2 T^{7} + \cdots + 2243950)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + 572 T^{14} + \cdots + 12544 \) Copy content Toggle raw display
$71$ \( (T^{8} - 20 T^{7} + \cdots + 774400)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 75146983837696 \) Copy content Toggle raw display
$79$ \( (T^{8} + 2 T^{7} + \cdots + 259328)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 2007968352784 \) Copy content Toggle raw display
$89$ \( (T - 1)^{16} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 972310139244544 \) Copy content Toggle raw display
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