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Properties

Label 1848.2.bg.k
Level $1848$
Weight $2$
Character orbit 1848.bg
Analytic conductor $14.756$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1848,2,Mod(529,1848)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1848, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1848.529");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1848.bg (of order \(3\), 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: \(14.7563542935\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + x^{10} - 24x^{9} - 28x^{8} + 96x^{7} + 247x^{6} + 672x^{5} - 1372x^{4} - 8232x^{3} + 2401x^{2} + 117649 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + 1) q^{3} + ( - \beta_{11} - \beta_{4}) q^{5} + (\beta_{7} + \beta_1) q^{7} - \beta_{5} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + 1) q^{3} + ( - \beta_{11} - \beta_{4}) q^{5} + (\beta_{7} + \beta_1) q^{7} - \beta_{5} q^{9} + (\beta_{5} - 1) q^{11} + ( - \beta_{8} + \beta_{4} + \cdots + \beta_1) q^{13}+ \cdots + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{3} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{3} - 6 q^{9} - 6 q^{11} - 2 q^{17} - 4 q^{23} - 16 q^{25} - 12 q^{27} + 24 q^{29} + 6 q^{33} + 10 q^{35} - 20 q^{37} + 20 q^{41} + 4 q^{47} + 10 q^{49} + 2 q^{51} - 4 q^{59} + 24 q^{61} - 24 q^{65} - 2 q^{67} - 8 q^{69} - 8 q^{71} - 12 q^{73} + 16 q^{75} - 6 q^{81} - 44 q^{83} + 12 q^{87} + 16 q^{91} - 20 q^{95} - 28 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + x^{10} - 24x^{9} - 28x^{8} + 96x^{7} + 247x^{6} + 672x^{5} - 1372x^{4} - 8232x^{3} + 2401x^{2} + 117649 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 248 \nu^{11} - 119 \nu^{10} + 2894 \nu^{9} + 1818 \nu^{8} - 20650 \nu^{7} - 28475 \nu^{6} + \cdots - 18756612 ) / 5747994 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} + \nu^{9} - 24 \nu^{8} - 28 \nu^{7} + 96 \nu^{6} + 247 \nu^{5} + 672 \nu^{4} + \cdots + 2401 \nu ) / 16807 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 373 \nu^{11} - 364 \nu^{10} - 1391 \nu^{9} - 5886 \nu^{8} - 5873 \nu^{7} + 91766 \nu^{6} + \cdots + 4840416 ) / 5747994 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 412 \nu^{11} - 203 \nu^{10} + 3508 \nu^{9} + 3854 \nu^{8} + 10724 \nu^{7} - 66551 \nu^{6} + \cdots - 20033944 ) / 5747994 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 415 \nu^{11} - 1113 \nu^{10} + 2721 \nu^{9} + 5417 \nu^{8} + 5453 \nu^{7} - 70563 \nu^{6} + \cdots - 32387089 ) / 5747994 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 29 \nu^{11} - 560 \nu^{10} + 862 \nu^{9} + 116 \nu^{8} + 3857 \nu^{7} + 9448 \nu^{6} + \cdots - 6924484 ) / 821142 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 165 \nu^{11} + 1148 \nu^{10} - 1047 \nu^{9} + 306 \nu^{8} - 2205 \nu^{7} - 31618 \nu^{6} + \cdots + 10084200 ) / 1915998 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 77 \nu^{11} + 680 \nu^{10} - 1057 \nu^{9} + 568 \nu^{8} - 395 \nu^{7} - 14476 \nu^{6} + \cdots + 8177806 ) / 821142 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1192 \nu^{11} + 2884 \nu^{10} + 229 \nu^{9} + 4052 \nu^{8} + 6398 \nu^{7} - 189500 \nu^{6} + \cdots + 9613604 ) / 5747994 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1805 \nu^{11} - 336 \nu^{10} + 11866 \nu^{9} + 16916 \nu^{8} + 33859 \nu^{7} - 249132 \nu^{6} + \cdots - 68942314 ) / 5747994 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{5} - \beta_{4} - \beta_{2} - \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - 2\beta_{7} + 12\beta_{5} + 3\beta_{4} - 3\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4 \beta_{11} - 8 \beta_{10} + 6 \beta_{9} - 9 \beta_{8} - 8 \beta_{7} + 4 \beta_{6} + 5 \beta_{5} + \cdots + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 9 \beta_{11} + 17 \beta_{10} - 19 \beta_{9} - 4 \beta_{8} - 6 \beta_{7} - 24 \beta_{6} + 12 \beta_{5} + \cdots - 60 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 54 \beta_{11} - 47 \beta_{10} + 18 \beta_{9} - 25 \beta_{8} - 43 \beta_{7} - 18 \beta_{6} - 18 \beta_{5} + \cdots + 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 63 \beta_{11} - \beta_{10} + 259 \beta_{9} - 161 \beta_{8} + 176 \beta_{7} - 48 \beta_{6} + \cdots - 120 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 326 \beta_{11} - 304 \beta_{10} + 282 \beta_{9} - 327 \beta_{8} - 82 \beta_{7} - 250 \beta_{6} + \cdots - 199 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 435 \beta_{11} + 199 \beta_{10} - 359 \beta_{9} + 574 \beta_{8} + 558 \beta_{7} - 1440 \beta_{6} + \cdots + 2820 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1566 \beta_{11} - 2736 \beta_{10} + 3679 \beta_{9} + 414 \beta_{8} + 824 \beta_{7} - 90 \beta_{6} + \cdots - 1980 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 4140 \beta_{11} - 8694 \beta_{10} + 3708 \beta_{9} + 3366 \beta_{8} + 2412 \beta_{7} - 936 \beta_{6} + \cdots - 9072 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1848\mathbb{Z}\right)^\times\).

\(n\) \(463\) \(617\) \(673\) \(925\) \(1585\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(1\) \(-\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
529.1
−1.15292 + 2.38134i
−1.05420 − 2.42666i
2.60004 + 0.489669i
2.19641 + 1.47506i
−2.54421 + 0.725954i
−0.0451257 − 2.64537i
−1.15292 − 2.38134i
−1.05420 + 2.42666i
2.60004 − 0.489669i
2.19641 − 1.47506i
−2.54421 − 0.725954i
−0.0451257 + 2.64537i
0 0.500000 − 0.866025i 0 −1.96925 − 3.41084i 0 1.48584 + 2.18913i 0 −0.500000 − 0.866025i 0
529.2 0 0.500000 − 0.866025i 0 −1.56712 − 2.71432i 0 −2.62865 − 0.300366i 0 −0.500000 − 0.866025i 0
529.3 0 0.500000 − 0.866025i 0 −0.0485602 − 0.0841088i 0 1.72409 − 2.00687i 0 −0.500000 − 0.866025i 0
529.4 0 0.500000 − 0.866025i 0 0.634542 + 1.09906i 0 2.37564 − 1.16461i 0 −0.500000 − 0.866025i 0
529.5 0 0.500000 − 0.866025i 0 1.02307 + 1.77202i 0 −0.643409 + 2.56633i 0 −0.500000 − 0.866025i 0
529.6 0 0.500000 − 0.866025i 0 1.92731 + 3.33820i 0 −2.31352 − 1.28360i 0 −0.500000 − 0.866025i 0
793.1 0 0.500000 + 0.866025i 0 −1.96925 + 3.41084i 0 1.48584 − 2.18913i 0 −0.500000 + 0.866025i 0
793.2 0 0.500000 + 0.866025i 0 −1.56712 + 2.71432i 0 −2.62865 + 0.300366i 0 −0.500000 + 0.866025i 0
793.3 0 0.500000 + 0.866025i 0 −0.0485602 + 0.0841088i 0 1.72409 + 2.00687i 0 −0.500000 + 0.866025i 0
793.4 0 0.500000 + 0.866025i 0 0.634542 − 1.09906i 0 2.37564 + 1.16461i 0 −0.500000 + 0.866025i 0
793.5 0 0.500000 + 0.866025i 0 1.02307 − 1.77202i 0 −0.643409 − 2.56633i 0 −0.500000 + 0.866025i 0
793.6 0 0.500000 + 0.866025i 0 1.92731 − 3.33820i 0 −2.31352 + 1.28360i 0 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 529.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1848.2.bg.k ✓ 12
7.c even 3 1 inner 1848.2.bg.k ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1848.2.bg.k ✓ 12 1.a even 1 1 trivial
1848.2.bg.k ✓ 12 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 23 T_{5}^{10} - 16 T_{5}^{9} + 409 T_{5}^{8} - 296 T_{5}^{7} + 2800 T_{5}^{6} - 4192 T_{5}^{5} + \cdots + 144 \) acting on \(S_{2}^{\mathrm{new}}(1848, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} + 23 T^{10} + \cdots + 144 \) Copy content Toggle raw display
$7$ \( T^{12} - 5 T^{10} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$13$ \( (T^{6} - 44 T^{4} + \cdots - 384)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 2 T^{11} + \cdots + 5349969 \) Copy content Toggle raw display
$19$ \( T^{12} + 38 T^{10} + \cdots + 32400 \) Copy content Toggle raw display
$23$ \( T^{12} + 4 T^{11} + \cdots + 1125721 \) Copy content Toggle raw display
$29$ \( (T^{6} - 12 T^{5} + \cdots - 17764)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 76 T^{10} + \cdots + 2985984 \) Copy content Toggle raw display
$37$ \( T^{12} + 20 T^{11} + \cdots + 64448784 \) Copy content Toggle raw display
$41$ \( (T^{6} - 10 T^{5} + \cdots - 612)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 66 T^{4} + \cdots + 2364)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} - 4 T^{11} + \cdots + 164025 \) Copy content Toggle raw display
$53$ \( T^{12} + 84 T^{10} + \cdots + 1327104 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 544195584 \) Copy content Toggle raw display
$61$ \( T^{12} - 24 T^{11} + \cdots + 32400 \) Copy content Toggle raw display
$67$ \( T^{12} + 2 T^{11} + \cdots + 24336 \) Copy content Toggle raw display
$71$ \( (T^{6} + 4 T^{5} + \cdots - 20844)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 12 T^{11} + \cdots + 9216 \) Copy content Toggle raw display
$79$ \( T^{12} + 143 T^{10} + \cdots + 944784 \) Copy content Toggle raw display
$83$ \( (T^{6} + 22 T^{5} + \cdots + 2592)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + 212 T^{10} + \cdots + 102400 \) Copy content Toggle raw display
$97$ \( (T^{6} + 14 T^{5} + \cdots + 2310073)^{2} \) Copy content Toggle raw display
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