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Properties

Label 1840.2.i.b
Level $1840$
Weight $2$
Character orbit 1840.i
Analytic conductor $14.692$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1840,2,Mod(1471,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 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("1840.1471");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.i (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: \(14.6924739719\)
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: \( x^{16} - 8x^{14} + 33x^{12} - 98x^{10} + 272x^{8} - 882x^{6} + 2673x^{4} - 5832x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{10} \)
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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{3} + \beta_{2} q^{5} + \beta_{13} q^{7} + ( - \beta_{4} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{3} + \beta_{2} q^{5} + \beta_{13} q^{7} + ( - \beta_{4} - 1) q^{9} + ( - \beta_{6} - \beta_1) q^{11} + (\beta_{3} - 1) q^{13} + \beta_1 q^{15} + (\beta_{12} - \beta_{5} + \beta_{2}) q^{17} + ( - \beta_{6} + \beta_1) q^{19} + ( - \beta_{8} + \beta_{5} - 2 \beta_{2}) q^{21} + ( - \beta_{15} - \beta_{10} - \beta_{9}) q^{23} - q^{25} + (\beta_{14} - \beta_{10} - \beta_{7}) q^{27} + ( - \beta_{11} + \beta_{4} + \beta_{3} + 2) q^{29} + (\beta_{15} - \beta_{7}) q^{31} + (\beta_{8} - \beta_{5} + 4 \beta_{2}) q^{33} + \beta_{15} q^{35} + (\beta_{12} + \beta_{8} + \beta_{2}) q^{37} + ( - 2 \beta_{15} + \beta_{14} + \cdots - 2 \beta_{7}) q^{39}+ \cdots + (2 \beta_{13} - 2 \beta_{6} + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{9} - 8 q^{13} - 16 q^{25} + 36 q^{29} - 44 q^{41} + 20 q^{49} - 20 q^{69} - 48 q^{73} - 72 q^{77} + 40 q^{81} - 4 q^{85} + 88 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8x^{14} + 33x^{12} - 98x^{10} + 272x^{8} - 882x^{6} + 2673x^{4} - 5832x^{2} + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -5\nu^{15} - 41\nu^{13} + 78\nu^{11} + 328\nu^{9} + 1232\nu^{7} - 8550\nu^{5} + 23733\nu^{3} - 28431\nu ) / 69984 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{14} + 7\nu^{12} - 90\nu^{10} + 184\nu^{8} - 232\nu^{6} + 834\nu^{4} - 2619\nu^{2} + 8505 ) / 15552 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{14} + 5\nu^{12} + 3\nu^{10} + 149\nu^{8} - 386\nu^{6} + 477\nu^{4} + 567\nu^{2} + 729 ) / 2916 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{14} - 8\nu^{12} + 33\nu^{10} - 98\nu^{8} + 272\nu^{6} - 882\nu^{4} + 1944\nu^{2} - 4374 ) / 729 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{14} - 8\nu^{12} + 33\nu^{10} - 98\nu^{8} + 272\nu^{6} - 882\nu^{4} + 3402\nu^{2} - 5832 ) / 729 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{15} + 8\nu^{13} - 33\nu^{11} + 98\nu^{9} - 272\nu^{7} + 882\nu^{5} - 2673\nu^{3} + 8019\nu ) / 2187 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{15} - 8\nu^{13} + 33\nu^{11} - 98\nu^{9} + 272\nu^{7} - 882\nu^{5} + 2673\nu^{3} - 3645\nu ) / 2187 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 13\nu^{14} - 41\nu^{12} + 168\nu^{10} - 410\nu^{8} + 1736\nu^{6} - 6480\nu^{4} + 14661\nu^{2} - 22599 ) / 5832 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5\nu^{15} - 13\nu^{13} + 111\nu^{11} - 166\nu^{9} + 415\nu^{7} - 2007\nu^{5} + 5913\nu^{3} - 13122\nu ) / 8748 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 145 \nu^{15} + 1403 \nu^{13} - 4866 \nu^{11} + 9512 \nu^{9} - 25832 \nu^{7} + 85770 \nu^{5} + \cdots + 499365 \nu ) / 139968 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -14\nu^{14} + 67\nu^{12} - 183\nu^{10} + 535\nu^{8} - 1342\nu^{6} + 5859\nu^{4} - 14661\nu^{2} + 19683 ) / 2916 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 149 \nu^{14} - 895 \nu^{12} + 2946 \nu^{10} - 7312 \nu^{8} + 22600 \nu^{6} - 71370 \nu^{4} + \cdots - 333153 ) / 23328 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 229 \nu^{15} - 1175 \nu^{13} + 3354 \nu^{11} - 8456 \nu^{9} + 24632 \nu^{7} - 113346 \nu^{5} + \cdots - 257337 \nu ) / 139968 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 239 \nu^{15} - 1093 \nu^{13} + 3198 \nu^{11} - 9112 \nu^{9} + 22168 \nu^{7} - 96246 \nu^{5} + \cdots - 480411 \nu ) / 139968 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 2\nu^{15} - 10\nu^{13} + 27\nu^{11} - 70\nu^{9} + 253\nu^{7} - 771\nu^{5} + 2016\nu^{3} - 2835\nu ) / 972 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - \beta_{4} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} - \beta_{13} + \beta_{7} + \beta_{6} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{12} + 2\beta_{11} - 2\beta_{8} + \beta_{5} - 3\beta_{4} - 2\beta_{3} - 4\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4\beta_{15} + 2\beta_{14} - 8\beta_{13} - 2\beta_{10} + \beta_{7} + 3\beta_{6} - 4\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{12} + 5\beta_{11} + 3\beta_{8} - \beta_{5} - \beta_{4} - 3\beta_{3} + 10\beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 24\beta_{15} - 10\beta_{14} - 20\beta_{13} + 6\beta_{10} + 4\beta_{9} + 17\beta_{7} + 5\beta_{6} + 16\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 14\beta_{12} + 14\beta_{11} + 26\beta_{8} - 17\beta_{5} - 15\beta_{4} + 18\beta_{3} + 36\beta_{2} - 44 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 18\beta_{15} - 8\beta_{14} - 13\beta_{13} - 3\beta_{10} + 22\beta_{9} - 21\beta_{7} + 9\beta_{6} + 57\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 20\beta_{12} + 32\beta_{11} + 12\beta_{8} - 37\beta_{5} + 23\beta_{4} + 56\beta_{3} - 296\beta_{2} + 106 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -4\beta_{15} - 92\beta_{14} - 4\beta_{13} - 64\beta_{10} + 220\beta_{9} + 25\beta_{7} + 71\beta_{6} - 12\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -48\beta_{12} + 34\beta_{11} + 104\beta_{8} + 56\beta_{5} + 6\beta_{4} - 14\beta_{3} - 416\beta_{2} + 203 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 44 \beta_{15} + 32 \beta_{14} - 236 \beta_{13} + 332 \beta_{10} + 700 \beta_{9} + 613 \beta_{7} + \cdots + 76 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 76\beta_{12} - 96\beta_{11} + 420\beta_{8} - 67\beta_{5} - 833\beta_{4} - 440\beta_{3} - 2328\beta_{2} - 790 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 506 \beta_{15} + 431 \beta_{14} - 287 \beta_{13} + 392 \beta_{10} + 782 \beta_{9} - 553 \beta_{7} + \cdots - 525 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).

\(n\) \(737\) \(1151\) \(1201\) \(1381\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1471.1
0.739948 − 1.56604i
−0.739948 − 1.56604i
−1.38594 − 1.03883i
1.38594 − 1.03883i
1.59950 − 0.664536i
−1.59950 − 0.664536i
−1.72432 − 0.163515i
1.72432 − 0.163515i
1.72432 + 0.163515i
−1.72432 + 0.163515i
−1.59950 + 0.664536i
1.59950 + 0.664536i
1.38594 + 1.03883i
−1.38594 + 1.03883i
−0.739948 + 1.56604i
0.739948 + 1.56604i
0 − 3.13208i 0 − 1.00000i 0 −1.01363 0 −6.80991 0
1471.2 0 − 3.13208i 0 1.00000i 0 1.01363 0 −6.80991 0
1471.3 0 − 2.07767i 0 − 1.00000i 0 −3.88693 0 −1.31671 0
1471.4 0 − 2.07767i 0 1.00000i 0 3.88693 0 −1.31671 0
1471.5 0 − 1.32907i 0 − 1.00000i 0 3.37473 0 1.23357 0
1471.6 0 − 1.32907i 0 1.00000i 0 −3.37473 0 1.23357 0
1471.7 0 − 0.327030i 0 − 1.00000i 0 2.33999 0 2.89305 0
1471.8 0 − 0.327030i 0 1.00000i 0 −2.33999 0 2.89305 0
1471.9 0 0.327030i 0 − 1.00000i 0 −2.33999 0 2.89305 0
1471.10 0 0.327030i 0 1.00000i 0 2.33999 0 2.89305 0
1471.11 0 1.32907i 0 − 1.00000i 0 −3.37473 0 1.23357 0
1471.12 0 1.32907i 0 1.00000i 0 3.37473 0 1.23357 0
1471.13 0 2.07767i 0 − 1.00000i 0 3.88693 0 −1.31671 0
1471.14 0 2.07767i 0 1.00000i 0 −3.88693 0 −1.31671 0
1471.15 0 3.13208i 0 − 1.00000i 0 1.01363 0 −6.80991 0
1471.16 0 3.13208i 0 1.00000i 0 −1.01363 0 −6.80991 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1471.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
23.b odd 2 1 inner
92.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1840.2.i.b ✓ 16
4.b odd 2 1 inner 1840.2.i.b ✓ 16
23.b odd 2 1 inner 1840.2.i.b ✓ 16
92.b even 2 1 inner 1840.2.i.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1840.2.i.b ✓ 16 1.a even 1 1 trivial
1840.2.i.b ✓ 16 4.b odd 2 1 inner
1840.2.i.b ✓ 16 23.b odd 2 1 inner
1840.2.i.b ✓ 16 92.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 16T_{3}^{6} + 69T_{3}^{4} + 82T_{3}^{2} + 8 \) acting on \(S_{2}^{\mathrm{new}}(1840, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + 16 T^{6} + 69 T^{4} + \cdots + 8)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$7$ \( (T^{8} - 33 T^{6} + \cdots + 968)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 44 T^{6} + \cdots + 3200)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 2 T^{3} + \cdots + 128)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 117 T^{6} + \cdots + 179776)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 52 T^{6} + \cdots + 2048)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 78310985281 \) Copy content Toggle raw display
$29$ \( (T^{4} - 9 T^{3} + \cdots - 110)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 61 T^{6} + \cdots + 10368)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 101 T^{6} + \cdots + 55696)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 11 T^{3} + \cdots - 3244)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} - 186 T^{6} + \cdots + 850208)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 136 T^{6} + \cdots + 2312)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 189 T^{6} + \cdots + 331776)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 323 T^{6} + \cdots + 4147200)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 192 T^{6} + \cdots + 270400)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 217 T^{6} + \cdots + 1555848)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 109 T^{6} + \cdots + 2592)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 12 T^{3} + \cdots + 4244)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} - 52 T^{6} + \cdots + 2048)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 233 T^{6} + \cdots + 2562848)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 492 T^{6} + \cdots + 34857216)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 448 T^{6} + \cdots + 97298496)^{2} \) Copy content Toggle raw display
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