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Properties

Label 1890.2.b.c
Level $1890$
Weight $2$
Character orbit 1890.b
Analytic conductor $15.092$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1890,2,Mod(1511,1890)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1890, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1890.1511");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1890 = 2 \cdot 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1890.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: \(15.0917259820\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 34x^{10} + 413x^{8} + 2164x^{6} + 4688x^{4} + 3688x^{2} + 676 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} - q^{4} - q^{5} - \beta_{4} q^{7} + \beta_{5} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} - q^{4} - q^{5} - \beta_{4} q^{7} + \beta_{5} q^{8} + \beta_{5} q^{10} - \beta_{11} q^{11} + (\beta_{10} + \beta_{5}) q^{13} + \beta_{7} q^{14} + q^{16} + (\beta_{8} - \beta_{7} - \beta_{3}) q^{17} + (\beta_{8} + \beta_{7} - 2 \beta_{5}) q^{19} + q^{20} - \beta_{3} q^{22} + ( - \beta_{10} - \beta_{8} + \cdots + \beta_{4}) q^{23}+ \cdots + (\beta_{10} + 2 \beta_{6} + \beta_{3} + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{4} - 12 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{4} - 12 q^{5} - 4 q^{7} + 12 q^{16} + 12 q^{20} + 12 q^{25} + 8 q^{26} + 4 q^{28} + 4 q^{35} - 32 q^{37} - 16 q^{38} + 8 q^{41} - 24 q^{43} + 8 q^{46} + 28 q^{47} + 4 q^{49} + 8 q^{58} + 24 q^{59} - 28 q^{62} - 12 q^{64} + 8 q^{67} - 28 q^{77} + 32 q^{79} - 12 q^{80} - 16 q^{83} - 16 q^{89} - 8 q^{91} + 16 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 34x^{10} + 413x^{8} + 2164x^{6} + 4688x^{4} + 3688x^{2} + 676 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{8} + 27\nu^{6} + 232\nu^{4} + 702\nu^{2} + 496 ) / 54 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{10} + 31\nu^{8} + 340\nu^{6} + 1576\nu^{4} + 2548\nu^{2} + 364 ) / 216 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{10} - 33\nu^{8} - 376\nu^{6} - 1698\nu^{4} - 2404\nu^{2} - 348 ) / 108 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11 \nu^{11} - 39 \nu^{10} - 322 \nu^{9} - 1287 \nu^{8} - 2957 \nu^{7} - 14664 \nu^{6} + \cdots - 34632 ) / 8424 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -17\nu^{11} - 565\nu^{9} - 6644\nu^{7} - 33070\nu^{5} - 63836\nu^{3} - 30976\nu ) / 8424 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11 \nu^{11} + 39 \nu^{10} - 322 \nu^{9} + 1287 \nu^{8} - 2957 \nu^{7} + 14664 \nu^{6} + \cdots + 34632 ) / 8424 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 11 \nu^{11} + 52 \nu^{10} - 322 \nu^{9} + 1664 \nu^{8} - 2957 \nu^{7} + 18382 \nu^{6} + \cdots + 46124 ) / 8424 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11 \nu^{11} - 52 \nu^{10} - 322 \nu^{9} - 1664 \nu^{8} - 2957 \nu^{7} - 18382 \nu^{6} + \cdots - 46124 ) / 8424 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{11} + 35\nu^{9} + 430\nu^{7} + 2162\nu^{5} + 3808\nu^{3} + 1232\nu ) / 216 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -23\nu^{11} - 769\nu^{9} - 9044\nu^{7} - 44416\nu^{5} - 84164\nu^{3} - 52636\nu ) / 2808 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -47\nu^{11} - 1546\nu^{9} - 17708\nu^{7} - 82273\nu^{5} - 131312\nu^{3} - 41542\nu ) / 4212 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{4} + \beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} - \beta_{10} - \beta_{9} + 5\beta_{8} + 5\beta_{7} - 4\beta_{6} + 6\beta_{5} - 4\beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{8} + 3\beta_{7} - 15\beta_{6} + 15\beta_{4} - 12\beta_{3} - 2\beta_{2} + \beta _1 + 46 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 24\beta_{11} + 8\beta_{10} + 17\beta_{9} - 59\beta_{8} - 59\beta_{7} + 38\beta_{6} - 99\beta_{5} + 38\beta_{4} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 57\beta_{8} - 57\beta_{7} + 199\beta_{6} - 199\beta_{4} + 148\beta_{3} + 50\beta_{2} - 13\beta _1 - 500 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 406 \beta_{11} - 42 \beta_{10} - 261 \beta_{9} + 740 \beta_{8} + 740 \beta_{7} - 391 \beta_{6} + \cdots - 391 \beta_{4} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -843\beta_{8} + 843\beta_{7} - 2595\beta_{6} + 2595\beta_{4} - 1914\beta_{3} - 886\beta_{2} + 173\beta _1 + 5842 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 6096 \beta_{11} - 20 \beta_{10} + 3859 \beta_{9} - 9500 \beta_{8} - 9500 \beta_{7} + \cdots + 4301 \beta_{4} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 11481 \beta_{8} - 11481 \beta_{7} + 33877 \beta_{6} - 33877 \beta_{4} + 25378 \beta_{3} + 13834 \beta_{2} + \cdots - 71222 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 86860 \beta_{11} + 5272 \beta_{10} - 55565 \beta_{9} + 123434 \beta_{8} + 123434 \beta_{7} + \cdots - 49945 \beta_{4} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1890\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(1081\) \(1541\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1511.1
− 3.12844i
− 3.65370i
1.07252i
2.63707i
− 0.508629i
1.58118i
3.12844i
3.65370i
− 1.07252i
− 2.63707i
0.508629i
− 1.58118i
− 1.00000i 0 −1.00000 −1.00000 0 −2.59048 + 0.537960i 1.00000i 0 1.00000i
1511.2 − 1.00000i 0 −1.00000 −1.00000 0 −2.23010 + 1.42360i 1.00000i 0 1.00000i
1511.3 − 1.00000i 0 −1.00000 −1.00000 0 −1.25606 − 2.32859i 1.00000i 0 1.00000i
1511.4 − 1.00000i 0 −1.00000 −1.00000 0 −0.00866574 − 2.64574i 1.00000i 0 1.00000i
1511.5 − 1.00000i 0 −1.00000 −1.00000 0 1.59915 + 2.10778i 1.00000i 0 1.00000i
1511.6 − 1.00000i 0 −1.00000 −1.00000 0 2.48616 + 0.904982i 1.00000i 0 1.00000i
1511.7 1.00000i 0 −1.00000 −1.00000 0 −2.59048 − 0.537960i − 1.00000i 0 − 1.00000i
1511.8 1.00000i 0 −1.00000 −1.00000 0 −2.23010 − 1.42360i − 1.00000i 0 − 1.00000i
1511.9 1.00000i 0 −1.00000 −1.00000 0 −1.25606 + 2.32859i − 1.00000i 0 − 1.00000i
1511.10 1.00000i 0 −1.00000 −1.00000 0 −0.00866574 + 2.64574i − 1.00000i 0 − 1.00000i
1511.11 1.00000i 0 −1.00000 −1.00000 0 1.59915 − 2.10778i − 1.00000i 0 − 1.00000i
1511.12 1.00000i 0 −1.00000 −1.00000 0 2.48616 − 0.904982i − 1.00000i 0 − 1.00000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1511.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 1890.2.b.c ✓ 12
3.b odd 2 1 1890.2.b.d yes 12
7.b odd 2 1 1890.2.b.d yes 12
21.c even 2 1 inner 1890.2.b.c ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1890.2.b.c ✓ 12 1.a even 1 1 trivial
1890.2.b.c ✓ 12 21.c even 2 1 inner
1890.2.b.d yes 12 3.b odd 2 1
1890.2.b.d yes 12 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1890, [\chi])\):

\( T_{11}^{12} + 68T_{11}^{10} + 1350T_{11}^{8} + 10340T_{11}^{6} + 33025T_{11}^{4} + 38304T_{11}^{2} + 5184 \) Copy content Toggle raw display
\( T_{17}^{6} - 46T_{17}^{4} - 60T_{17}^{3} + 97T_{17}^{2} + 84T_{17} - 72 \) Copy content Toggle raw display
\( T_{41}^{6} - 4T_{41}^{5} - 175T_{41}^{4} + 664T_{41}^{3} + 6592T_{41}^{2} - 16320T_{41} - 70272 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T + 1)^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 4 T^{11} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{12} + 68 T^{10} + \cdots + 5184 \) Copy content Toggle raw display
$13$ \( T^{12} + 84 T^{10} + \cdots + 1296 \) Copy content Toggle raw display
$17$ \( (T^{6} - 46 T^{4} + \cdots - 72)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 96 T^{10} + \cdots + 82944 \) Copy content Toggle raw display
$23$ \( T^{12} + 172 T^{10} + \cdots + 219024 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 830995929 \) Copy content Toggle raw display
$31$ \( T^{12} + 174 T^{10} + \cdots + 9199089 \) Copy content Toggle raw display
$37$ \( (T^{6} + 16 T^{5} + \cdots + 58432)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 4 T^{5} + \cdots - 70272)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 12 T^{5} + \cdots - 73004)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 14 T^{5} + \cdots + 14913)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 1445216256 \) Copy content Toggle raw display
$59$ \( (T^{6} - 12 T^{5} + \cdots + 24336)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 379314576 \) Copy content Toggle raw display
$67$ \( (T^{6} - 4 T^{5} + \cdots - 329472)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + 462 T^{10} + \cdots + 40144896 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 1471242850704 \) Copy content Toggle raw display
$79$ \( (T^{6} - 16 T^{5} + \cdots + 200692)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 8 T^{5} + \cdots - 288)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 8 T^{5} + \cdots - 165888)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 464 T^{10} + \cdots + 331776 \) Copy content Toggle raw display
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