Properties

Label 1872.4.d.b
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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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: \( x^{12} + 180x^{10} + 45168x^{8} + 667388x^{6} + 86783544x^{4} + 12732769404x^{2} + 502441386561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{4} \)
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.

\(f(q)\) \(=\) \( q - \beta_{7} q^{5} - \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{5} - \beta_1 q^{7} - \beta_{11} q^{11} + 13 q^{13} + (\beta_{8} + 2 \beta_{7} - \beta_{6}) q^{17} + ( - \beta_{5} + \beta_1) q^{19} + ( - \beta_{11} - 2 \beta_{10} + \beta_{9}) q^{23} + ( - \beta_{4} - 2 \beta_{2} - 23) q^{25} + ( - 4 \beta_{8} + 10 \beta_{7} - 15 \beta_{6}) q^{29} + (\beta_{3} + 3 \beta_1) q^{31} + ( - 3 \beta_{11} + 3 \beta_{10} + \beta_{9}) q^{35} + ( - 4 \beta_{4} + 3 \beta_{2} + 18) q^{37} + ( - 2 \beta_{8} + 9 \beta_{7} - 10 \beta_{6}) q^{41} + (\beta_{5} + 3 \beta_{3} + 2 \beta_1) q^{43} + ( - \beta_{11} + 8 \beta_{10} + 4 \beta_{9}) q^{47} + ( - 4 \beta_{4} + 3 \beta_{2} + 27) q^{49} + (15 \beta_{8} + 16 \beta_{7} + 5 \beta_{6}) q^{53} + ( - 5 \beta_{5} - 7 \beta_{3} + 20 \beta_1) q^{55} + (2 \beta_{11} - 10 \beta_{10} - \beta_{9}) q^{59} + ( - 10 \beta_{4} + 3 \beta_{2} - 32) q^{61} - 13 \beta_{7} q^{65} + (5 \beta_{5} + \beta_{3} - 13 \beta_1) q^{67} + (6 \beta_{11} - 13 \beta_{10} + 3 \beta_{9}) q^{71} + (2 \beta_{4} - 17 \beta_{2} - 46) q^{73} + ( - 15 \beta_{8} + 46 \beta_{7} - 84 \beta_{6}) q^{77} + (10 \beta_{5} - 2 \beta_{3} + 14 \beta_1) q^{79} + (2 \beta_{11} + 10 \beta_{10} - 9 \beta_{9}) q^{83} + ( - 2 \beta_{4} + 5 \beta_{2} + 308) q^{85} + (2 \beta_{8} - 3 \beta_{7} - 64 \beta_{6}) q^{89} - 13 \beta_1 q^{91} + (9 \beta_{11} + 11 \beta_{10} + \beta_{9}) q^{95} + (4 \beta_{4} + 7 \beta_{2} + 374) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 156 q^{13} - 276 q^{25} + 216 q^{37} + 324 q^{49} - 384 q^{61} - 552 q^{73} + 3696 q^{85} + 4488 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 180x^{10} + 45168x^{8} + 667388x^{6} + 86783544x^{4} + 12732769404x^{2} + 502441386561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1344269861 \nu^{10} - 250606322343 \nu^{8} - 76131758398098 \nu^{6} + \cdots - 18\!\cdots\!52 ) / 23\!\cdots\!61 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 6765538 \nu^{10} - 1743804036 \nu^{8} - 474007637436 \nu^{6} - 13733030333780 \nu^{4} + 814747851807606 \nu^{2} + \cdots + 29\!\cdots\!28 ) / 11\!\cdots\!67 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 631807564226 \nu^{10} + 3013984529298 \nu^{8} + \cdots + 11\!\cdots\!12 ) / 27\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 15964996 \nu^{10} - 3393969246 \nu^{8} - 763080668880 \nu^{6} - 23600413111778 \nu^{4} + \cdots - 15\!\cdots\!96 ) / 38\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2996287599784 \nu^{10} - 409443004063530 \nu^{8} + \cdots - 23\!\cdots\!64 ) / 27\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1148249752 \nu^{11} - 216062553213 \nu^{9} - 53768253413382 \nu^{7} + \cdots - 27\!\cdots\!48 \nu ) / 36\!\cdots\!93 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 73109282417633 \nu^{11} + \cdots - 71\!\cdots\!63 \nu ) / 55\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 20314422394 \nu^{11} - 3898109401794 \nu^{9} - 960883617950322 \nu^{7} + \cdots - 45\!\cdots\!40 \nu ) / 10\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1202226983648 \nu^{11} + 97887638037753 \nu^{9} + \cdots + 96\!\cdots\!80 \nu ) / 39\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 92968292414 \nu^{11} + 8540893578060 \nu^{9} + \cdots + 75\!\cdots\!50 \nu ) / 14\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3579014403746 \nu^{11} + 553966438047651 \nu^{9} + \cdots + 15\!\cdots\!04 \nu ) / 39\!\cdots\!63 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{11} + \beta_{9} - 3\beta_{8} ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -7\beta_{5} + 6\beta_{4} + 11\beta_{3} + 9\beta_{2} + 36\beta _1 - 360 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 20\beta_{11} + 171\beta_{10} - 290\beta_{9} - 84\beta_{8} + 378\beta_{7} + 198\beta_{6} ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 989\beta_{5} - 1023\beta_{4} - 1153\beta_{3} + 3366\beta_{2} - 10206\beta _1 - 115872 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2033\beta_{11} - 12798\beta_{10} + 16327\beta_{9} + 98457\beta_{8} - 28080\beta_{7} - 562476\beta_{6} ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 23875\beta_{5} + 92862\beta_{4} - 56522\beta_{3} - 523125\beta_{2} - 366462\beta _1 + 16556556 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 241277 \beta_{11} - 4695147 \beta_{10} + 9580675 \beta_{9} - 14334051 \beta_{8} + 4899690 \beta_{7} + 77134116 \beta_{6} ) / 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 43664515 \beta_{5} - 8165433 \beta_{4} + 82455095 \beta_{3} + 31895514 \beta_{2} + 527370786 \beta _1 - 1180662336 ) / 12 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 41264192 \beta_{11} + 1143031725 \beta_{10} - 2438125094 \beta_{9} - 1848832320 \beta_{8} + 612188982 \beta_{7} + 10103980338 \beta_{6} ) / 12 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 5058472931 \beta_{5} - 8108490750 \beta_{4} - 9667434835 \beta_{3} + 38868148557 \beta_{2} - 59526603972 \beta _1 - 1301887858200 ) / 12 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1125572761 \beta_{11} - 950125491 \beta_{10} + 4234822777 \beta_{9} + 945942587565 \beta_{8} - 357624354582 \beta_{7} - 5021516691840 \beta_{6} ) / 12 \) Copy content Toggle raw display

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
−7.41677 + 4.00305i
7.41677 + 4.00305i
1.67875 6.72010i
−1.67875 6.72010i
7.78930 + 12.1374i
−7.78930 + 12.1374i
−7.78930 12.1374i
7.78930 12.1374i
−1.67875 + 6.72010i
1.67875 + 6.72010i
7.41677 4.00305i
−7.41677 4.00305i
0 0 0 17.5993i 0 20.9778i 0 0 0
287.2 0 0 0 17.5993i 0 20.9778i 0 0 0
287.3 0 0 0 11.5817i 0 4.74823i 0 0 0
287.4 0 0 0 11.5817i 0 4.74823i 0 0 0
287.5 0 0 0 0.360788i 0 22.0315i 0 0 0
287.6 0 0 0 0.360788i 0 22.0315i 0 0 0
287.7 0 0 0 0.360788i 0 22.0315i 0 0 0
287.8 0 0 0 0.360788i 0 22.0315i 0 0 0
287.9 0 0 0 11.5817i 0 4.74823i 0 0 0
287.10 0 0 0 11.5817i 0 4.74823i 0 0 0
287.11 0 0 0 17.5993i 0 20.9778i 0 0 0
287.12 0 0 0 17.5993i 0 20.9778i 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.b 12
3.b odd 2 1 inner 1872.4.d.b 12
4.b odd 2 1 inner 1872.4.d.b 12
12.b even 2 1 inner 1872.4.d.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1872.4.d.b 12 1.a even 1 1 trivial
1872.4.d.b 12 3.b odd 2 1 inner
1872.4.d.b 12 4.b odd 2 1 inner
1872.4.d.b 12 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 444T_{5}^{4} + 41604T_{5}^{2} + 5408 \) 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
$5$ \( (T^{6} + 444 T^{4} + 41604 T^{2} + \cdots + 5408)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 948 T^{4} + 234468 T^{2} + \cdots + 4815824)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 7086 T^{4} + \cdots - 3187690792)^{2} \) Copy content Toggle raw display
$13$ \( (T - 13)^{12} \) Copy content Toggle raw display
$17$ \( (T^{6} + 5358 T^{4} + \cdots + 3122080200)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 14448 T^{4} + \cdots + 332377344)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 36936 T^{4} + \cdots - 24079120000)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 113286 T^{4} + \cdots + 5084822232072)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 32448 T^{4} + \cdots + 128314182464)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 54 T^{2} - 64128 T - 5001104)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + 57636 T^{4} + \cdots + 11779283072)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 208344 T^{4} + \cdots + 241274043518976)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 627486 T^{4} + \cdots - 49\!\cdots\!28)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 863934 T^{4} + \cdots + 74541512760968)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 685590 T^{4} + \cdots - 21\!\cdots\!92)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 96 T^{2} - 342972 T - 48454848)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + 501348 T^{4} + \cdots + 38\!\cdots\!76)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 1603686 T^{4} + \cdots - 67\!\cdots\!08)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 138 T^{2} - 1006608 T - 422788816)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + 1710672 T^{4} + \cdots + 95\!\cdots\!96)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 1601718 T^{4} + \cdots - 58\!\cdots\!88)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 235308 T^{4} + \cdots + 303816180020000)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 1122 T^{2} + 100608 T + 42862608)^{4} \) Copy content Toggle raw display
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