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Properties

Label 1872.4.d.a
Level $1872$
Weight $4$
Character orbit 1872.d
Analytic conductor $110.452$
Analytic rank $0$
Dimension $12$
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] = [1872,4,Mod(287,1872)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1872, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1872.287");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1872.d (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: \(110.451575531\)
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: \( x^{12} - 156x^{10} + 10590x^{8} - 394520x^{6} + 8530257x^{4} - 102204060x^{2} + 554414116 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{13} \)
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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{5} - \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{5} - \beta_1 q^{7} - \beta_{7} q^{11} - 13 q^{13} + ( - \beta_{10} + 2 \beta_{8} + \beta_{6}) q^{17} + ( - \beta_{5} - \beta_{3}) q^{19} + (\beta_{9} + 2 \beta_{7}) q^{23} + ( - \beta_{2} - 55) q^{25} + (2 \beta_{10} + \beta_{8} - 2 \beta_{6}) q^{29} + (\beta_{5} - 2 \beta_{3} - 3 \beta_1) q^{31} + (\beta_{11} + 2 \beta_{9} + 6 \beta_{7}) q^{35} + ( - 2 \beta_{4} - \beta_{2} - 34) q^{37} + ( - 2 \beta_{10} - 12 \beta_{8} + 25 \beta_{6}) q^{41} + ( - 2 \beta_{5} + 3 \beta_{3} - 5 \beta_1) q^{43} + ( - 2 \beta_{11} + 2 \beta_{9} - \beta_{7}) q^{47} + ( - \beta_{4} - \beta_{2} - 161) q^{49} + (5 \beta_{10} - 12 \beta_{8} + 11 \beta_{6}) q^{53} + (4 \beta_{5} + 5 \beta_{3} - 13 \beta_1) q^{55} + ( - 2 \beta_{11} - \beta_{9} - 11 \beta_{7}) q^{59} + ( - 3 \beta_{4} + \beta_{2} - 160) q^{61} - 13 \beta_{6} q^{65} + (2 \beta_{5} - 9 \beta_{3} + 10 \beta_1) q^{67} + (3 \beta_{11} + 2 \beta_{9} + 3 \beta_{7}) q^{71} + ( - 2 \beta_{4} + \beta_{2} - 182) q^{73} + ( - 9 \beta_{10} - 17 \beta_{8} + 45 \beta_{6}) q^{77} + ( - 2 \beta_{5} + 10 \beta_{3} - 8 \beta_1) q^{79} + (4 \beta_{11} - 3 \beta_{9} + \beta_{7}) q^{83} + ( - 2 \beta_{4} - 9 \beta_{2} - 396) q^{85} + (2 \beta_{10} - 50 \beta_{8} + 21 \beta_{6}) q^{89} + 13 \beta_1 q^{91} + (\beta_{11} - 6 \beta_{9} - 42 \beta_{7}) q^{95} + (10 \beta_{4} - \beta_{2} - 722) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 156 q^{13} - 660 q^{25} - 408 q^{37} - 1932 q^{49} - 1920 q^{61} - 2184 q^{73} - 4752 q^{85} - 8664 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 156x^{10} + 10590x^{8} - 394520x^{6} + 8530257x^{4} - 102204060x^{2} + 554414116 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 10407 \nu^{10} - 1343304 \nu^{8} + 75078009 \nu^{6} - 2180920284 \nu^{4} + 34863051078 \nu^{2} - 251151599388 ) / 1593730060 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 6404 \nu^{10} + 137478 \nu^{8} + 18119252 \nu^{6} - 1064864212 \nu^{4} + 21198535104 \nu^{2} - 205554271244 ) / 398432515 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 31488 \nu^{10} - 4408941 \nu^{8} + 259319656 \nu^{6} - 8200598901 \nu^{4} + \cdots - 1103306464732 ) / 1593730060 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -1909\nu^{10} + 299418\nu^{8} - 18719483\nu^{6} + 585201928\nu^{4} - 9124300236\nu^{2} + 59426357876 ) / 30648655 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 51679 \nu^{10} - 6211148 \nu^{8} + 303380372 \nu^{6} - 7259793634 \nu^{4} + 84843645533 \nu^{2} - 379434861086 ) / 637492024 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 7140461 \nu^{11} + 1151609062 \nu^{9} - 76426746237 \nu^{7} + 2710395696822 \nu^{5} + \cdots + 398606103621784 \nu ) / 18762983996380 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 135283457 \nu^{11} - 32641924114 \nu^{9} + 2316122782574 \nu^{7} + \cdots - 99\!\cdots\!78 \nu ) / 225155807956560 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 16608807 \nu^{11} - 2223409059 \nu^{9} + 128443112154 \nu^{7} - 3900826337769 \nu^{5} + \cdots - 466159406132538 \nu ) / 18762983996380 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 16608807 \nu^{11} + 2223409059 \nu^{9} - 128443112154 \nu^{7} + \cdots + 635026262099958 \nu ) / 9381491998190 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 176794047 \nu^{11} - 20075690494 \nu^{9} + 917153948944 \nu^{7} + \cdots - 700171054083178 \nu ) / 18762983996380 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 896900357 \nu^{11} + 127553086834 \nu^{9} - 7442604130154 \nu^{7} + \cdots + 18\!\cdots\!78 \nu ) / 37525967992760 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} + 2\beta_{8} ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + \beta_{2} + 12\beta _1 + 468 ) / 18 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{11} + 24\beta_{9} + 140\beta_{8} + 18\beta_{7} + 54\beta_{6} ) / 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 12\beta_{4} - 24\beta_{3} + 15\beta_{2} + 224\beta _1 + 3156 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 84\beta_{11} - 90\beta_{10} + 383\beta_{9} + 5438\beta_{8} + 720\beta_{7} + 4500\beta_{6} ) / 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -144\beta_{5} + 137\beta_{4} - 1776\beta_{3} + 299\beta_{2} + 9204\beta _1 + 23856 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -255\beta_{11} - 9324\beta_{10} - 3920\beta_{9} + 175124\beta_{8} + 7542\beta_{7} + 222642\beta_{6} ) / 18 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -40320\beta_{5} - 25484\beta_{4} - 245808\beta_{3} - 21965\beta_{2} + 947136\beta _1 - 6829236 ) / 18 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 131508 \beta_{11} - 535086 \beta_{10} - 742563 \beta_{9} + 4829090 \beta_{8} + \cdots + 8203356 \beta_{6} ) / 18 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -695952\beta_{5} - 686709\beta_{4} - 2793168\beta_{3} - 1075311\beta_{2} + 8812868\beta _1 - 182349984 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 7581645 \beta_{11} - 20662884 \beta_{10} - 44490592 \beta_{9} + 100873676 \beta_{8} + \cdots + 223088382 \beta_{6} ) / 18 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(145\) \(209\) \(469\) \(703\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
287.1
−6.15833 − 1.41421i
6.15833 − 1.41421i
4.11641 + 1.41421i
−4.11641 + 1.41421i
−5.39723 − 1.41421i
5.39723 − 1.41421i
5.39723 + 1.41421i
−5.39723 + 1.41421i
−4.11641 − 1.41421i
4.11641 − 1.41421i
6.15833 + 1.41421i
−6.15833 + 1.41421i
0 0 0 − 19.6930i 0 − 26.1276i 0 0 0
287.2 0 0 0 − 19.6930i 0 26.1276i 0 0 0
287.3 0 0 0 − 9.97750i 0 − 17.4645i 0 0 0
287.4 0 0 0 − 9.97750i 0 17.4645i 0 0 0
287.5 0 0 0 − 7.25509i 0 − 22.8985i 0 0 0
287.6 0 0 0 − 7.25509i 0 22.8985i 0 0 0
287.7 0 0 0 7.25509i 0 − 22.8985i 0 0 0
287.8 0 0 0 7.25509i 0 22.8985i 0 0 0
287.9 0 0 0 9.97750i 0 − 17.4645i 0 0 0
287.10 0 0 0 9.97750i 0 17.4645i 0 0 0
287.11 0 0 0 19.6930i 0 − 26.1276i 0 0 0
287.12 0 0 0 19.6930i 0 26.1276i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 287.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1872.4.d.a ✓ 12
3.b odd 2 1 inner 1872.4.d.a ✓ 12
4.b odd 2 1 inner 1872.4.d.a ✓ 12
12.b even 2 1 inner 1872.4.d.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1872.4.d.a ✓ 12 1.a even 1 1 trivial
1872.4.d.a ✓ 12 3.b odd 2 1 inner
1872.4.d.a ✓ 12 4.b odd 2 1 inner
1872.4.d.a ✓ 12 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 540T_{5}^{4} + 64260T_{5}^{2} + 2032128 \) acting on \(S_{4}^{\mathrm{new}}(1872, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} + 540 T^{4} + \cdots + 2032128)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 1512 T^{4} + \cdots + 109175040)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 3726 T^{4} + \cdots - 722821320)^{2} \) Copy content Toggle raw display
$13$ \( (T + 13)^{12} \) Copy content Toggle raw display
$17$ \( (T^{6} + 19710 T^{4} + \cdots + 158527290888)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 36396 T^{4} + \cdots + 6278322960)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 46008 T^{4} + \cdots - 71182800000)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 2271236895432)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 83052 T^{4} + \cdots + 12228362640)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 102 T^{2} + \cdots - 10511648)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 603301239440928)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 435681787357440)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 129231020547720)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 26\!\cdots\!08)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 695571745943880)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 480 T^{2} + \cdots + 11383840)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 44\!\cdots\!60)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 15\!\cdots\!80)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 546 T^{2} + \cdots - 1290592)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 47\!\cdots\!40)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 12\!\cdots\!80)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 94\!\cdots\!28)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 2166 T^{2} + \cdots - 1969169824)^{4} \) Copy content Toggle raw display
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