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Properties

Label 2898.2.g.i
Level $2898$
Weight $2$
Character orbit 2898.g
Analytic conductor $23.141$
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] = [2898,2,Mod(2575,2898)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2898, 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("2898.2575");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2898.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: \(23.1406465058\)
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: \( x^{16} - 8x^{14} + 128x^{12} - 760x^{10} + 7998x^{8} - 37240x^{6} + 307328x^{4} - 941192x^{2} + 5764801 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{7} \)
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - \beta_{12} q^{5} + \beta_1 q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} - \beta_{12} q^{5} + \beta_1 q^{7} - q^{8} + \beta_{12} q^{10} - \beta_{8} q^{11} + \beta_{4} q^{13} - \beta_1 q^{14} + q^{16} + \beta_{10} q^{17} - \beta_{3} q^{19} - \beta_{12} q^{20} + \beta_{8} q^{22} + (\beta_{14} - \beta_{9} - \beta_1) q^{23} + ( - \beta_{7} + 1) q^{25} - \beta_{4} q^{26} + \beta_1 q^{28} + \beta_{7} q^{29} + (\beta_{6} + \beta_{5} + \beta_{4}) q^{31} - q^{32} - \beta_{10} q^{34} + (\beta_{5} + \beta_{4} + \beta_{2} + 1) q^{35} + ( - \beta_{11} + \beta_{8}) q^{37} + \beta_{3} q^{38} + \beta_{12} q^{40} + ( - \beta_{15} - \beta_{14} + \cdots - \beta_{5}) q^{41}+ \cdots + ( - \beta_{14} + \beta_{13} - \beta_{6} + \cdots - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{2} + 16 q^{4} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{2} + 16 q^{4} - 16 q^{8} + 16 q^{16} + 20 q^{25} - 4 q^{29} - 16 q^{32} + 16 q^{35} + 16 q^{49} - 20 q^{50} + 4 q^{58} + 16 q^{64} - 16 q^{70} - 32 q^{71} - 20 q^{77} + 40 q^{85} + 40 q^{95} - 16 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8x^{14} + 128x^{12} - 760x^{10} + 7998x^{8} - 37240x^{6} + 307328x^{4} - 941192x^{2} + 5764801 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9 \nu^{14} - 569 \nu^{12} + 2041 \nu^{10} - 28953 \nu^{8} + 172215 \nu^{6} - 1409415 \nu^{4} + \cdots - 32286247 ) / 6453888 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 41 \nu^{15} - 2043 \nu^{13} + 30973 \nu^{11} - 229071 \nu^{9} + 1403223 \nu^{7} + \cdots - 237298033 \nu ) / 158120256 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 613 \nu^{14} - 7305 \nu^{12} + 71261 \nu^{10} - 679569 \nu^{8} + 2993979 \nu^{6} + \cdots - 801777935 ) / 135531648 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 331 \nu^{14} + 1521 \nu^{12} - 21347 \nu^{10} + 128913 \nu^{8} - 607125 \nu^{6} + \cdots - 11411953 ) / 67765824 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 809 \nu^{14} + 2691 \nu^{12} + 13441 \nu^{10} + 104235 \nu^{8} + 1237623 \nu^{6} + \cdots + 328123061 ) / 135531648 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -6\nu^{14} - \nu^{12} - 376\nu^{10} - 1712\nu^{8} - 10748\nu^{6} - 50813\nu^{4} - 254506\nu^{2} - 3529470 ) / 941192 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 151 \nu^{15} + 2923 \nu^{13} - 21043 \nu^{11} + 120591 \nu^{9} - 856809 \nu^{7} + \cdots + 79177777 \nu ) / 158120256 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{15} - 8\nu^{13} + 128\nu^{11} - 760\nu^{9} + 7998\nu^{7} - 37240\nu^{5} + 307328\nu^{3} - 941192\nu ) / 823543 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 631 \nu^{15} + 6663 \nu^{13} + 3887 \nu^{11} + 767343 \nu^{9} + 62409 \nu^{7} + \cdots + 1129548049 \nu ) / 474360768 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 425 \nu^{15} + 5801 \nu^{13} - 56801 \nu^{11} + 613521 \nu^{9} - 3190263 \nu^{7} + \cdots + 892132367 \nu ) / 316240512 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 415 \nu^{15} + 3075 \nu^{13} - 24749 \nu^{11} + 190401 \nu^{9} - 1517097 \nu^{7} + \cdots + 47883143 \nu ) / 237180384 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1563 \nu^{15} - 3346 \nu^{14} + 5595 \nu^{13} + 42546 \nu^{12} - 94371 \nu^{11} + \cdots + 159767342 ) / 948721536 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 521 \nu^{15} - 1344 \nu^{14} + 1865 \nu^{13} + 10752 \nu^{12} - 31457 \nu^{11} + \cdots + 948721536 ) / 316240512 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 521 \nu^{15} + 1344 \nu^{14} + 1865 \nu^{13} - 10752 \nu^{12} - 31457 \nu^{11} + \cdots - 948721536 ) / 316240512 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} - \beta_{13} + \beta_{6} + 2\beta_{5} + \beta_{4} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} - \beta_{14} + 2\beta_{12} + 2\beta_{11} - 2\beta_{10} + 5\beta_{9} - 2\beta_{3} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{15} + 5\beta_{14} - 8\beta_{13} + 4\beta_{7} + 4\beta_{6} - 4\beta_{4} - 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{15} - \beta_{14} - 18\beta_{12} + 18\beta_{11} - 6\beta_{10} + 2\beta_{9} + 8\beta_{8} + 10\beta_{3} - 19\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 23\beta_{15} - 40\beta_{14} + 17\beta_{13} + 16\beta_{7} + 7\beta_{6} - 18\beta_{5} - 57\beta_{4} + 16\beta_{2} - 31 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 120 \beta_{15} + 120 \beta_{14} - 160 \beta_{12} - 32 \beta_{11} + 88 \beta_{10} - 39 \beta_{9} + \cdots - 144 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 104 \beta_{15} - 104 \beta_{14} + 208 \beta_{13} - 384 \beta_{7} - 168 \beta_{6} + 160 \beta_{5} + \cdots - 543 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 328 \beta_{15} + 328 \beta_{14} + 864 \beta_{12} - 480 \beta_{11} + 792 \beta_{10} - 240 \beta_{9} + \cdots - 1103 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 856 \beta_{15} + 41 \beta_{14} - 897 \beta_{13} - 1296 \beta_{7} - 2935 \beta_{6} - 750 \beta_{5} + \cdots - 2383 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 3097 \beta_{15} - 3097 \beta_{14} + 9378 \beta_{12} - 1374 \beta_{11} + 3078 \beta_{10} + \cdots + 3378 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 6443 \beta_{15} - 12307 \beta_{14} + 5864 \beta_{13} + 9348 \beta_{7} - 1012 \beta_{6} - 4096 \beta_{5} + \cdots + 15177 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 21609 \beta_{15} - 21609 \beta_{14} + 27662 \beta_{12} - 9742 \beta_{11} - 2750 \beta_{10} + \cdots + 61917 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 91649 \beta_{15} + 16048 \beta_{14} + 75601 \beta_{13} - 30176 \beta_{7} + 143199 \beta_{6} + \cdots - 78607 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 216848 \beta_{15} - 216848 \beta_{14} - 327744 \beta_{12} + 44736 \beta_{11} - 126672 \beta_{10} + \cdots - 4640 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2898\mathbb{Z}\right)^\times\).

\(n\) \(829\) \(1289\) \(1891\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2575.1
−1.83258 − 1.90831i
−1.83258 + 1.90831i
2.21809 − 1.44224i
2.21809 + 1.44224i
−1.26577 − 2.32332i
−1.26577 + 2.32332i
−2.47377 − 0.938326i
−2.47377 + 0.938326i
2.47377 − 0.938326i
2.47377 + 0.938326i
1.26577 − 2.32332i
1.26577 + 2.32332i
−2.21809 − 1.44224i
−2.21809 + 1.44224i
1.83258 − 1.90831i
1.83258 + 1.90831i
−1.00000 0 1.00000 −3.43776 0 −1.83258 − 1.90831i −1.00000 0 3.43776
2575.2 −1.00000 0 1.00000 −3.43776 0 −1.83258 + 1.90831i −1.00000 0 3.43776
2575.3 −1.00000 0 1.00000 −3.06390 0 2.21809 − 1.44224i −1.00000 0 3.06390
2575.4 −1.00000 0 1.00000 −3.06390 0 2.21809 + 1.44224i −1.00000 0 3.06390
2575.5 −1.00000 0 1.00000 −1.70258 0 −1.26577 − 2.32332i −1.00000 0 1.70258
2575.6 −1.00000 0 1.00000 −1.70258 0 −1.26577 + 2.32332i −1.00000 0 1.70258
2575.7 −1.00000 0 1.00000 −0.946322 0 −2.47377 − 0.938326i −1.00000 0 0.946322
2575.8 −1.00000 0 1.00000 −0.946322 0 −2.47377 + 0.938326i −1.00000 0 0.946322
2575.9 −1.00000 0 1.00000 0.946322 0 2.47377 − 0.938326i −1.00000 0 −0.946322
2575.10 −1.00000 0 1.00000 0.946322 0 2.47377 + 0.938326i −1.00000 0 −0.946322
2575.11 −1.00000 0 1.00000 1.70258 0 1.26577 − 2.32332i −1.00000 0 −1.70258
2575.12 −1.00000 0 1.00000 1.70258 0 1.26577 + 2.32332i −1.00000 0 −1.70258
2575.13 −1.00000 0 1.00000 3.06390 0 −2.21809 − 1.44224i −1.00000 0 −3.06390
2575.14 −1.00000 0 1.00000 3.06390 0 −2.21809 + 1.44224i −1.00000 0 −3.06390
2575.15 −1.00000 0 1.00000 3.43776 0 1.83258 − 1.90831i −1.00000 0 −3.43776
2575.16 −1.00000 0 1.00000 3.43776 0 1.83258 + 1.90831i −1.00000 0 −3.43776
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2575.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
23.b odd 2 1 inner
161.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2898.2.g.i ✓ 16
3.b odd 2 1 2898.2.g.k yes 16
7.b odd 2 1 inner 2898.2.g.i ✓ 16
21.c even 2 1 2898.2.g.k yes 16
23.b odd 2 1 inner 2898.2.g.i ✓ 16
69.c even 2 1 2898.2.g.k yes 16
161.c even 2 1 inner 2898.2.g.i ✓ 16
483.c odd 2 1 2898.2.g.k yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2898.2.g.i ✓ 16 1.a even 1 1 trivial
2898.2.g.i ✓ 16 7.b odd 2 1 inner
2898.2.g.i ✓ 16 23.b odd 2 1 inner
2898.2.g.i ✓ 16 161.c even 2 1 inner
2898.2.g.k yes 16 3.b odd 2 1
2898.2.g.k yes 16 21.c even 2 1
2898.2.g.k yes 16 69.c even 2 1
2898.2.g.k yes 16 483.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2898, [\chi])\):

\( T_{5}^{8} - 25T_{5}^{6} + 194T_{5}^{4} - 476T_{5}^{2} + 288 \) Copy content Toggle raw display
\( T_{11}^{8} + 42T_{11}^{6} + 340T_{11}^{4} + 832T_{11}^{2} + 576 \) Copy content Toggle raw display
\( T_{13}^{8} + 55T_{13}^{6} + 666T_{13}^{4} + 1408T_{13}^{2} + 288 \) Copy content Toggle raw display
\( T_{29}^{4} + T_{29}^{3} - 40T_{29}^{2} - 16T_{29} + 312 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 25 T^{6} + \cdots + 288)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} - 8 T^{14} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{8} + 42 T^{6} + \cdots + 576)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 55 T^{6} + \cdots + 288)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 90 T^{6} + \cdots + 10368)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 96 T^{6} + \cdots + 18432)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 80 T^{5} + \cdots + 279841)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + T^{3} - 40 T^{2} + \cdots + 312)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 102 T^{6} + \cdots + 10368)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 177 T^{6} + \cdots + 197136)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 237 T^{6} + \cdots + 6948992)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 137 T^{6} + \cdots + 11664)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 75 T^{6} + \cdots + 32)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 118 T^{6} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 148 T^{6} + \cdots + 663552)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 114 T^{6} + \cdots + 2048)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 490 T^{6} + \cdots + 63872064)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 8 T^{3} + \cdots + 6912)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + 456 T^{6} + \cdots + 74322432)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 448 T^{6} + \cdots + 15116544)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 340 T^{6} + \cdots + 39214368)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 470 T^{6} + \cdots + 19618848)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 357 T^{6} + \cdots + 123008)^{2} \) Copy content Toggle raw display
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