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Properties

Label 1872.2.bx.d
Level $1872$
Weight $2$
Character orbit 1872.bx
Analytic conductor $14.948$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1872,2,Mod(575,1872)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1872, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1872.575");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1872.bx (of order \(6\), degree \(2\), 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: \(14.9479952584\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 34 x^{18} + 738 x^{16} - 9772 x^{14} + 94699 x^{12} - 620358 x^{10} + 2979018 x^{8} + \cdots + 2125764 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{7} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{13} + \beta_1) q^{5} - \beta_{6} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{13} + \beta_1) q^{5} - \beta_{6} q^{7} - \beta_{15} q^{11} + (\beta_{5} - 1) q^{13} + ( - \beta_{18} + \beta_{13} - \beta_{11}) q^{17} + ( - \beta_{14} - \beta_{8} + \cdots - \beta_{4}) q^{19}+ \cdots + ( - \beta_{14} + 2 \beta_{10} + \cdots - \beta_{4}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 6 q^{7} - 10 q^{13} - 36 q^{19} - 36 q^{25} + 4 q^{37} + 6 q^{43} + 16 q^{49} - 6 q^{61} - 30 q^{67} - 4 q^{73} + 36 q^{85} + 18 q^{91} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 34 x^{18} + 738 x^{16} - 9772 x^{14} + 94699 x^{12} - 620358 x^{10} + 2979018 x^{8} + \cdots + 2125764 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 199223642660125 \nu^{19} + \cdots + 52\!\cdots\!08 \nu ) / 85\!\cdots\!58 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 750906497983507 \nu^{18} + \cdots - 19\!\cdots\!84 ) / 56\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 581955103757713 \nu^{18} + \cdots + 34\!\cdots\!94 ) / 28\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12\!\cdots\!84 \nu^{18} + \cdots - 19\!\cdots\!40 ) / 56\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 13\!\cdots\!89 \nu^{18} + \cdots + 20\!\cdots\!16 ) / 56\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 21\!\cdots\!65 \nu^{18} + \cdots + 30\!\cdots\!20 ) / 56\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 804832003689317 \nu^{18} + \cdots + 46\!\cdots\!48 ) / 18\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 20\!\cdots\!62 \nu^{19} + \cdots - 19\!\cdots\!12 \nu ) / 17\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 28\!\cdots\!92 \nu^{18} + \cdots + 67\!\cdots\!84 ) / 56\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 28\!\cdots\!29 \nu^{19} + \cdots + 63\!\cdots\!44 \nu ) / 17\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 38\!\cdots\!35 \nu^{18} + \cdots + 21\!\cdots\!40 ) / 56\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 581955103757713 \nu^{19} + \cdots + 34\!\cdots\!94 \nu ) / 28\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 20\!\cdots\!50 \nu^{18} + \cdots - 67\!\cdots\!80 ) / 18\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 10\!\cdots\!53 \nu^{19} + \cdots + 42\!\cdots\!88 \nu ) / 17\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 62\!\cdots\!63 \nu^{19} + \cdots + 95\!\cdots\!92 \nu ) / 56\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 12\!\cdots\!92 \nu^{19} + \cdots + 63\!\cdots\!32 \nu ) / 85\!\cdots\!58 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 92\!\cdots\!49 \nu^{19} + \cdots - 73\!\cdots\!52 \nu ) / 56\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 11\!\cdots\!16 \nu^{19} + \cdots - 32\!\cdots\!76 \nu ) / 56\!\cdots\!72 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{14} + \beta_{8} - 8\beta_{4} + \beta_{3} + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} + \beta_{18} + \beta_{16} + \beta_{15} - 9\beta_{13} - \beta_{11} - \beta_{9} - \beta_{2} + 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -12\beta_{14} - \beta_{12} + 2\beta_{10} + \beta_{8} - \beta_{7} - 4\beta_{6} - 12\beta_{5} - 67\beta_{4} - \beta_{3} + 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 3 \beta_{19} + 17 \beta_{18} + 17 \beta_{17} + 11 \beta_{16} + 20 \beta_{15} - 93 \beta_{13} + \cdots - 17 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 26 \beta_{14} + 20 \beta_{12} - 34 \beta_{10} - 135 \beta_{8} - 40 \beta_{7} - 34 \beta_{6} - 161 \beta_{5} + \cdots - 566 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -163\beta_{19} + 229\beta_{17} - 66\beta_{16} - 84\beta_{15} + 608\beta_{11} + 211\beta_{9} - 1252\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1975 \beta_{14} + 626 \beta_{12} - 880 \beta_{10} - 1975 \beta_{8} - 313 \beta_{7} + 440 \beta_{6} + \cdots - 7717 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 677 \beta_{19} - 2369 \beta_{18} - 1802 \beta_{16} - 6077 \beta_{15} + 11667 \beta_{13} + \cdots - 11667 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 17073 \beta_{14} + 4502 \beta_{12} - 5188 \beta_{10} - 7094 \beta_{8} + 4502 \beta_{7} + 10376 \beta_{6} + \cdots - 17073 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 17322 \beta_{19} - 25543 \beta_{18} - 36991 \beta_{17} - 2347 \beta_{16} - 62089 \beta_{15} + \cdots + 36991 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 100837 \beta_{14} - 62089 \beta_{12} + 58862 \beta_{10} + 195084 \beta_{8} + 124178 \beta_{7} + \cdots + 824383 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 215198 \beta_{19} - 466781 \beta_{17} + 251583 \beta_{16} + 367827 \beta_{15} - 1473268 \beta_{11} + \cdots + 2078090 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 3631217 \beta_{14} - 1669216 \beta_{12} + 1315820 \beta_{10} + 3631217 \beta_{8} + 834608 \beta_{7} + \cdots + 12028685 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 1140023 \beta_{19} + 2856319 \beta_{18} + 2375107 \beta_{16} + 16158733 \beta_{15} + \cdots + 19291119 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 26416440 \beta_{14} - 11024485 \beta_{12} + 7331756 \beta_{10} + 18259063 \beta_{8} - 11024485 \beta_{7} + \cdots + 26416440 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 47790669 \beta_{19} + 30152645 \beta_{18} + 74304287 \beta_{17} - 21277051 \beta_{16} + \cdots - 74304287 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 238635854 \beta_{14} + 143798690 \beta_{12} - 82009024 \beta_{10} - 312467697 \beta_{8} + \cdots - 1416796622 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 299639557 \beta_{19} + 936623983 \beta_{17} - 636984426 \beta_{16} - 921495918 \beta_{15} + \cdots - 3768095404 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(469\) \(703\)
\(\chi(n)\) \(-1 + \beta_{4}\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
575.1
3.06327 − 1.76858i
2.73620 − 1.57974i
2.21243 − 1.27735i
1.85539 − 1.07121i
0.540623 − 0.312129i
−0.540623 + 0.312129i
−1.85539 + 1.07121i
−2.21243 + 1.27735i
−2.73620 + 1.57974i
−3.06327 + 1.76858i
−3.06327 − 1.76858i
−2.73620 − 1.57974i
−2.21243 − 1.27735i
−1.85539 − 1.07121i
−0.540623 − 0.312129i
0.540623 + 0.312129i
1.85539 + 1.07121i
2.21243 + 1.27735i
2.73620 + 1.57974i
3.06327 + 1.76858i
0 0 0 − 3.53716i 0 −0.101687 − 0.0587089i 0 0 0
575.2 0 0 0 − 3.15949i 0 −2.24308 − 1.29504i 0 0 0
575.3 0 0 0 − 2.55470i 0 4.01553 + 2.31837i 0 0 0
575.4 0 0 0 − 2.14242i 0 2.26719 + 1.30896i 0 0 0
575.5 0 0 0 − 0.624258i 0 −2.43795 − 1.40755i 0 0 0
575.6 0 0 0 0.624258i 0 −2.43795 − 1.40755i 0 0 0
575.7 0 0 0 2.14242i 0 2.26719 + 1.30896i 0 0 0
575.8 0 0 0 2.55470i 0 4.01553 + 2.31837i 0 0 0
575.9 0 0 0 3.15949i 0 −2.24308 − 1.29504i 0 0 0
575.10 0 0 0 3.53716i 0 −0.101687 − 0.0587089i 0 0 0
1439.1 0 0 0 − 3.53716i 0 −0.101687 + 0.0587089i 0 0 0
1439.2 0 0 0 − 3.15949i 0 −2.24308 + 1.29504i 0 0 0
1439.3 0 0 0 − 2.55470i 0 4.01553 − 2.31837i 0 0 0
1439.4 0 0 0 − 2.14242i 0 2.26719 − 1.30896i 0 0 0
1439.5 0 0 0 − 0.624258i 0 −2.43795 + 1.40755i 0 0 0
1439.6 0 0 0 0.624258i 0 −2.43795 + 1.40755i 0 0 0
1439.7 0 0 0 2.14242i 0 2.26719 − 1.30896i 0 0 0
1439.8 0 0 0 2.55470i 0 4.01553 − 2.31837i 0 0 0
1439.9 0 0 0 3.15949i 0 −2.24308 + 1.29504i 0 0 0
1439.10 0 0 0 3.53716i 0 −0.101687 + 0.0587089i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 575.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
52.j odd 6 1 inner
156.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1872.2.bx.d yes 20
3.b odd 2 1 inner 1872.2.bx.d yes 20
4.b odd 2 1 1872.2.bx.c ✓ 20
12.b even 2 1 1872.2.bx.c ✓ 20
13.c even 3 1 1872.2.bx.c ✓ 20
39.i odd 6 1 1872.2.bx.c ✓ 20
52.j odd 6 1 inner 1872.2.bx.d yes 20
156.p even 6 1 inner 1872.2.bx.d yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1872.2.bx.c ✓ 20 4.b odd 2 1
1872.2.bx.c ✓ 20 12.b even 2 1
1872.2.bx.c ✓ 20 13.c even 3 1
1872.2.bx.c ✓ 20 39.i odd 6 1
1872.2.bx.d yes 20 1.a even 1 1 trivial
1872.2.bx.d yes 20 3.b odd 2 1 inner
1872.2.bx.d yes 20 52.j odd 6 1 inner
1872.2.bx.d yes 20 156.p even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1872, [\chi])\):

\( T_{5}^{10} + 34T_{5}^{8} + 418T_{5}^{6} + 2220T_{5}^{4} + 4545T_{5}^{2} + 1458 \) Copy content Toggle raw display
\( T_{7}^{10} - 3 T_{7}^{9} - 17 T_{7}^{8} + 60 T_{7}^{7} + 292 T_{7}^{6} - 378 T_{7}^{5} - 1674 T_{7}^{4} + \cdots + 108 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( (T^{10} + 34 T^{8} + \cdots + 1458)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} - 3 T^{9} + \cdots + 108)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + 74 T^{18} + \cdots + 3779136 \) Copy content Toggle raw display
$13$ \( (T^{10} + 5 T^{9} + \cdots + 371293)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 8707129344 \) Copy content Toggle raw display
$19$ \( (T^{10} + 18 T^{9} + \cdots + 2239488)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 306110016 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 172186884 \) Copy content Toggle raw display
$31$ \( (T^{10} + 259 T^{8} + \cdots + 6912)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} - 2 T^{9} + \cdots + 929296)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 21\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( (T^{10} - 3 T^{9} + \cdots + 514188)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} - 338 T^{8} + \cdots - 364478616)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + 352 T^{8} + \cdots + 132388992)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 101559956668416 \) Copy content Toggle raw display
$61$ \( (T^{10} + 3 T^{9} + \cdots + 986049)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + 15 T^{9} + \cdots + 9526572)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 37\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( (T^{5} + T^{4} - 218 T^{3} + \cdots - 1447)^{4} \) Copy content Toggle raw display
$79$ \( (T^{10} + 475 T^{8} + \cdots + 181398528)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} - 588 T^{8} + \cdots - 90699264)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( (T^{10} + 3 T^{9} + \cdots + 144)^{2} \) Copy content Toggle raw display
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