Properties

Label 1840.2.i.c
Level $1840$
Weight $2$
Character orbit 1840.i
Analytic conductor $14.692$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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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: 16.0.85667234766862611972096.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 8x^{14} + 44x^{12} + 162x^{10} + 404x^{8} - 84x^{6} - 79x^{4} - 4x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{8} \)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + \beta_{5} q^{5} + (\beta_{14} - \beta_{10}) q^{7} + ( - \beta_{4} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + \beta_{5} q^{5} + (\beta_{14} - \beta_{10}) q^{7} + ( - \beta_{4} - \beta_1 + 1) q^{9} + ( - \beta_{13} + \beta_{11}) q^{11} + (2 \beta_{8} - \beta_{4} - \beta_1) q^{13} - \beta_{10} q^{15} + ( - \beta_{15} + \beta_{12} + \beta_{6}) q^{17} + (\beta_{14} - 2 \beta_{11} + \beta_{10}) q^{19} + (\beta_{6} - 3 \beta_{5}) q^{21} + ( - \beta_{14} + \beta_{13} + \cdots + \beta_{7}) q^{23}+ \cdots + (\beta_{14} + 2 \beta_{11} + 3 \beta_{10}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{13} - 16 q^{25} - 84 q^{29} + 24 q^{41} + 36 q^{49} - 12 q^{69} + 68 q^{73} - 48 q^{77} + 32 q^{81} + 4 q^{85} - 52 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 8x^{14} + 44x^{12} + 162x^{10} + 404x^{8} - 84x^{6} - 79x^{4} - 4x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -35\nu^{14} - 828\nu^{12} - 1561\nu^{10} - 5803\nu^{8} + 1072\nu^{6} + 1099\nu^{4} + 84\nu^{2} - 1863616 ) / 1193292 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 293 \nu^{14} + 3459 \nu^{12} + 21907 \nu^{10} + 97195 \nu^{8} + 303239 \nu^{6} + 441599 \nu^{4} + \cdots - 48044 ) / 44196 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11062 \nu^{14} + 74178 \nu^{12} + 374036 \nu^{10} + 1177769 \nu^{8} + 2201194 \nu^{6} + \cdots + 511628 ) / 1193292 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4767 \nu^{14} + 38377 \nu^{12} + 211135 \nu^{10} + 774900 \nu^{8} + 1930469 \nu^{6} - 401601 \nu^{4} + \cdots + 204280 ) / 397764 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 85 \nu^{15} - 942 \nu^{13} - 5966 \nu^{11} - 26360 \nu^{9} - 82582 \nu^{7} - 120262 \nu^{5} + \cdots + 25120 \nu ) / 18792 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 8459 \nu^{15} - 93414 \nu^{13} - 591622 \nu^{11} - 2609053 \nu^{9} - 8189294 \nu^{7} + \cdots + 2491040 \nu ) / 1193292 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 23473 \nu^{14} - 177744 \nu^{12} - 956294 \nu^{10} - 3387989 \nu^{8} - 8017342 \nu^{6} + \cdots - 261674 ) / 596646 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 50557 \nu^{14} - 408714 \nu^{12} - 2246003 \nu^{10} - 8289539 \nu^{8} - 20627170 \nu^{6} + \cdots - 989684 ) / 1193292 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 59564 \nu^{14} + 502149 \nu^{12} + 2837758 \nu^{10} + 10883380 \nu^{8} + 28818305 \nu^{6} + \cdots - 1501916 ) / 1193292 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 30172 \nu^{15} - 227943 \nu^{13} - 1226342 \nu^{11} - 4346207 \nu^{9} - 10267249 \nu^{7} + \cdots + 428758 \nu ) / 1193292 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 65371 \nu^{15} + 523038 \nu^{13} + 2877980 \nu^{11} + 10593224 \nu^{9} + 26421490 \nu^{7} + \cdots - 2648236 \nu ) / 2386584 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 65371 \nu^{15} + 523038 \nu^{13} + 2877980 \nu^{11} + 10593224 \nu^{9} + 26421490 \nu^{7} + \cdots + 2124932 \nu ) / 2386584 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 88865 \nu^{15} + 643824 \nu^{13} + 3399880 \nu^{11} + 11651146 \nu^{9} + 26182376 \nu^{7} + \cdots + 450832 \nu ) / 2386584 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 107167 \nu^{15} + 882966 \nu^{13} + 4914446 \nu^{11} + 18437858 \nu^{9} + 47147158 \nu^{7} + \cdots - 5926288 \nu ) / 2386584 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 243817 \nu^{15} - 1984836 \nu^{13} - 11009036 \nu^{11} - 41094428 \nu^{9} - 104586772 \nu^{7} + \cdots - 4378520 \nu ) / 2386584 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{12} - \beta_{11} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{9} - \beta_{7} - \beta_{4} + \beta_{3} + 2\beta_{2} - \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{14} + \beta_{13} + 3\beta_{11} + 3\beta_{10} + 3\beta_{6} - 5\beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{9} + 2\beta_{8} + 4\beta_{7} + 7\beta_{4} + 7\beta_{3} - \beta_{2} - 4\beta _1 - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -2\beta_{15} - 9\beta_{14} - 11\beta_{13} - 7\beta_{12} + 12\beta_{11} - 19\beta_{10} - 11\beta_{6} + 19\beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 11\beta_{9} - 11\beta_{7} - 78\beta_{3} - 39\beta_{2} + 7\beta _1 + 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2 \beta_{15} + 48 \beta_{14} + 50 \beta_{13} - 7 \beta_{12} - 73 \beta_{11} + 80 \beta_{10} + \cdots + 80 \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 9\beta_{9} - 48\beta_{8} + 57\beta_{7} - 171\beta_{4} + 171\beta_{3} + 34\beta_{2} + 57\beta _1 + 137 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 57\beta_{14} - 57\beta_{13} - 203\beta_{11} - 203\beta_{10} + 381\beta_{6} - 595\beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -422\beta_{9} + 162\beta_{8} - 260\beta_{7} + 577\beta_{4} + 577\beta_{3} + 821\beta_{2} + 260\beta _1 + 244 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 422 \beta_{15} - 739 \beta_{14} - 317 \beta_{13} + 1503 \beta_{12} + 1576 \beta_{11} + \cdots + 73 \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 317\beta_{9} - 317\beta_{7} - 2258\beta_{3} - 1129\beta_{2} - 4167\beta _1 - 6507 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 2242 \beta_{15} - 796 \beta_{14} + 1446 \beta_{13} - 7985 \beta_{12} + 3485 \beta_{11} + \cdots + 4500 \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 10227 \beta_{9} + 796 \beta_{8} + 9431 \beta_{7} + 2835 \beta_{4} - 2835 \beta_{3} - 16766 \beta_{2} + \cdots + 13931 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 9431\beta_{14} - 9431\beta_{13} - 33589\beta_{11} - 33589\beta_{10} - 16693\beta_{6} + 26067\beta_{5} ) / 2 \) Copy content Toggle raw display

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
−0.605201 2.04824i
0.605201 + 2.04824i
0.215960 0.625946i
−0.215960 + 0.625946i
−1.47123 + 1.54824i
1.47123 1.54824i
0.650065 + 0.125946i
−0.650065 0.125946i
−0.650065 + 0.125946i
0.650065 0.125946i
1.47123 + 1.54824i
−1.47123 1.54824i
−0.215960 0.625946i
0.215960 + 0.625946i
0.605201 2.04824i
−0.605201 + 2.04824i
0 2.94245i 0 1.00000i 0 −1.89011 0 −5.65803 0
1471.2 0 2.94245i 0 1.00000i 0 1.89011 0 −5.65803 0
1471.3 0 1.30013i 0 1.00000i 0 −1.10639 0 1.30966 0
1471.4 0 1.30013i 0 1.00000i 0 1.10639 0 1.30966 0
1471.5 0 1.21040i 0 1.00000i 0 −4.59480 0 1.53493 0
1471.6 0 1.21040i 0 1.00000i 0 4.59480 0 1.53493 0
1471.7 0 0.431920i 0 1.00000i 0 −3.33035 0 2.81344 0
1471.8 0 0.431920i 0 1.00000i 0 3.33035 0 2.81344 0
1471.9 0 0.431920i 0 1.00000i 0 3.33035 0 2.81344 0
1471.10 0 0.431920i 0 1.00000i 0 −3.33035 0 2.81344 0
1471.11 0 1.21040i 0 1.00000i 0 4.59480 0 1.53493 0
1471.12 0 1.21040i 0 1.00000i 0 −4.59480 0 1.53493 0
1471.13 0 1.30013i 0 1.00000i 0 1.10639 0 1.30966 0
1471.14 0 1.30013i 0 1.00000i 0 −1.10639 0 1.30966 0
1471.15 0 2.94245i 0 1.00000i 0 1.89011 0 −5.65803 0
1471.16 0 2.94245i 0 1.00000i 0 −1.89011 0 −5.65803 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.c 16
4.b odd 2 1 inner 1840.2.i.c 16
23.b odd 2 1 inner 1840.2.i.c 16
92.b even 2 1 inner 1840.2.i.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1840.2.i.c 16 1.a even 1 1 trivial
1840.2.i.c 16 4.b odd 2 1 inner
1840.2.i.c 16 23.b odd 2 1 inner
1840.2.i.c 16 92.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 12T_{3}^{6} + 32T_{3}^{4} + 27T_{3}^{2} + 4 \) 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
$3$ \( (T^{8} + 12 T^{6} + 32 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$7$ \( (T^{8} - 37 T^{6} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 29 T^{6} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 2 T^{3} - 36 T^{2} + \cdots - 2)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 67 T^{6} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 73 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 16 T^{6} + \cdots + 279841)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 21 T^{3} + \cdots + 376)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 56 T^{6} + \cdots + 13924)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 176 T^{6} + \cdots + 262144)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 6 T^{3} + \cdots + 118)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} - 220 T^{6} + \cdots + 5683456)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 283 T^{6} + \cdots + 11999296)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 252 T^{6} + \cdots + 82944)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 12)^{8} \) Copy content Toggle raw display
$61$ \( (T^{8} + 395 T^{6} + \cdots + 19254544)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 348 T^{6} + \cdots + 5308416)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 384 T^{6} + \cdots + 69455556)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 17 T^{3} + \cdots + 376)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} - 112 T^{6} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 200 T^{6} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 144 T^{6} + \cdots + 82944)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 111 T^{6} + \cdots + 82944)^{2} \) Copy content Toggle raw display
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