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Properties

Label 1848.2.bg.j
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: \( x^{12} - 2 x^{11} + 22 x^{10} + 12 x^{9} + 291 x^{8} - 30 x^{7} + 1062 x^{6} - 804 x^{5} + 3537 x^{4} + \cdots + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{6} - 1) q^{3} + (\beta_{6} + \beta_1) q^{5} + ( - \beta_{11} - \beta_{9} - 1) q^{7} + \beta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} - 1) q^{3} + (\beta_{6} + \beta_1) q^{5} + ( - \beta_{11} - \beta_{9} - 1) q^{7} + \beta_{6} q^{9} + (\beta_{6} + 1) q^{11} + (\beta_{10} + \beta_{3} - \beta_{2}) q^{13} + ( - \beta_{3} + 1) q^{15} + ( - \beta_{10} - \beta_{8} - \beta_{5}) q^{17} + (\beta_{11} + \beta_{9} + 2 \beta_{6} + \cdots + 2) q^{19}+ \cdots - q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{3} - 4 q^{5} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{3} - 4 q^{5} - 6 q^{9} + 6 q^{11} + 8 q^{13} + 8 q^{15} - 10 q^{19} + 6 q^{21} - 8 q^{23} - 12 q^{25} + 12 q^{27} + 36 q^{29} - 4 q^{31} + 6 q^{33} + 22 q^{35} - 8 q^{37} - 4 q^{39} - 12 q^{41} + 4 q^{43} - 4 q^{45} - 4 q^{47} + 6 q^{49} - 8 q^{55} + 20 q^{57} - 24 q^{59} - 16 q^{61} - 6 q^{63} + 16 q^{65} + 6 q^{67} + 16 q^{69} + 24 q^{71} - 24 q^{73} - 12 q^{75} - 6 q^{77} + 12 q^{79} - 6 q^{81} - 4 q^{83} + 48 q^{85} - 18 q^{87} + 12 q^{89} + 28 q^{91} - 4 q^{93} + 48 q^{95} + 4 q^{97} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 22 x^{10} + 12 x^{9} + 291 x^{8} - 30 x^{7} + 1062 x^{6} - 804 x^{5} + 3537 x^{4} + \cdots + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 216193915 \nu^{11} + 9929656632 \nu^{10} - 21596941566 \nu^{9} + 215654121344 \nu^{8} + \cdots + 5017911834024 ) / 10577837526624 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 652097739 \nu^{11} + 722780584 \nu^{10} - 12996828834 \nu^{9} - 20475451960 \nu^{8} + \cdots - 391300635816 ) / 1762972921104 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4636316001 \nu^{11} + 31019314120 \nu^{10} - 106145171030 \nu^{9} + \cdots + 39091729210632 ) / 10577837526624 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2461166677 \nu^{11} - 4693240068 \nu^{10} + 59390181276 \nu^{9} + 28631814566 \nu^{8} + \cdots + 1018517458176 ) / 2644459381656 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5434731053 \nu^{11} - 12173657584 \nu^{10} + 121009644334 \nu^{9} + 39223114968 \nu^{8} + \cdots - 992896115592 ) / 3525945842208 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2127435021 \nu^{11} - 5675467583 \nu^{10} + 52831365961 \nu^{9} - 9105667838 \nu^{8} + \cdots + 3333755857644 ) / 1322229690828 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 4848095695 \nu^{11} - 13139534532 \nu^{10} + 121303509024 \nu^{9} - 9687097762 \nu^{8} + \cdots - 453443732064 ) / 2644459381656 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3465078972 \nu^{11} - 5069767829 \nu^{10} + 77387133514 \nu^{9} + 70242827923 \nu^{8} + \cdots - 2471983567056 ) / 1322229690828 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 33010524529 \nu^{11} - 14006502072 \nu^{10} + 611647011942 \nu^{9} + 1467363828512 \nu^{8} + \cdots + 35646349235256 ) / 10577837526624 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 39959642295 \nu^{11} - 78565545064 \nu^{10} + 832824057650 \nu^{9} + 587747484632 \nu^{8} + \cdots + 26673539222664 ) / 10577837526624 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{7} - 5\beta_{6} + 2\beta_{5} - \beta_{4} + \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} - \beta_{10} - 2\beta_{9} + 2\beta_{7} + 3\beta_{5} - \beta_{4} - 15\beta_{3} + 7\beta_{2} - 3\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 17 \beta_{11} - 13 \beta_{9} + 2 \beta_{8} + 4 \beta_{7} + 69 \beta_{6} - 23 \beta_{5} + 13 \beta_{4} + \cdots - 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 39 \beta_{11} + 19 \beta_{10} + 21 \beta_{9} + 19 \beta_{8} - 39 \beta_{7} + 216 \beta_{6} + \cdots + 177 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 102 \beta_{11} + 62 \beta_{10} + 345 \beta_{9} - 345 \beta_{7} - 455 \beta_{5} + 102 \beta_{4} + \cdots + 1572 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1494 \beta_{11} + 1017 \beta_{9} - 385 \beta_{8} - 477 \beta_{7} - 5586 \beta_{6} + 2444 \beta_{5} + \cdots + 2034 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 5253 \beta_{11} - 1586 \beta_{10} - 2352 \beta_{9} - 1586 \beta_{8} + 5253 \beta_{7} - 28881 \beta_{6} + \cdots - 23628 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 11049 \beta_{11} - 8371 \beta_{10} - 35280 \beta_{9} + 35280 \beta_{7} + 48133 \beta_{5} + \cdots - 157251 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 172581 \beta_{11} - 118755 \beta_{9} + 37958 \beta_{8} + 53826 \beta_{7} + 653013 \beta_{6} + \cdots - 237510 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 563337 \beta_{11} + 188449 \beta_{10} + 255657 \beta_{9} + 188449 \beta_{8} - 563337 \beta_{7} + \cdots + 2530485 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(463\) \(617\) \(673\) \(925\) \(1585\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(1\) \(\beta_{6}\)

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
−1.36040 − 2.35628i
−1.23188 − 2.13367i
−0.111313 − 0.192801i
0.571804 + 0.990393i
0.732704 + 1.26908i
2.39908 + 4.15533i
−1.36040 + 2.35628i
−1.23188 + 2.13367i
−0.111313 + 0.192801i
0.571804 − 0.990393i
0.732704 − 1.26908i
2.39908 − 4.15533i
0 −0.500000 + 0.866025i 0 −1.86040 − 3.22230i 0 −2.62928 + 0.294756i 0 −0.500000 − 0.866025i 0
529.2 0 −0.500000 + 0.866025i 0 −1.73188 − 2.99970i 0 2.48484 − 0.908600i 0 −0.500000 − 0.866025i 0
529.3 0 −0.500000 + 0.866025i 0 −0.611313 − 1.05883i 0 −0.882281 + 2.49431i 0 −0.500000 − 0.866025i 0
529.4 0 −0.500000 + 0.866025i 0 0.0718037 + 0.124368i 0 2.61247 − 0.418353i 0 −0.500000 − 0.866025i 0
529.5 0 −0.500000 + 0.866025i 0 0.232704 + 0.403054i 0 −1.31811 − 2.29404i 0 −0.500000 − 0.866025i 0
529.6 0 −0.500000 + 0.866025i 0 1.89908 + 3.28931i 0 −0.267643 − 2.63218i 0 −0.500000 − 0.866025i 0
793.1 0 −0.500000 − 0.866025i 0 −1.86040 + 3.22230i 0 −2.62928 − 0.294756i 0 −0.500000 + 0.866025i 0
793.2 0 −0.500000 − 0.866025i 0 −1.73188 + 2.99970i 0 2.48484 + 0.908600i 0 −0.500000 + 0.866025i 0
793.3 0 −0.500000 − 0.866025i 0 −0.611313 + 1.05883i 0 −0.882281 − 2.49431i 0 −0.500000 + 0.866025i 0
793.4 0 −0.500000 − 0.866025i 0 0.0718037 − 0.124368i 0 2.61247 + 0.418353i 0 −0.500000 + 0.866025i 0
793.5 0 −0.500000 − 0.866025i 0 0.232704 − 0.403054i 0 −1.31811 + 2.29404i 0 −0.500000 + 0.866025i 0
793.6 0 −0.500000 − 0.866025i 0 1.89908 − 3.28931i 0 −0.267643 + 2.63218i 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.j ✓ 12
7.c even 3 1 inner 1848.2.bg.j ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1848.2.bg.j ✓ 12 1.a even 1 1 trivial
1848.2.bg.j ✓ 12 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 4 T_{5}^{11} + 29 T_{5}^{10} + 68 T_{5}^{9} + 429 T_{5}^{8} + 972 T_{5}^{7} + 3204 T_{5}^{6} + \cdots + 16 \) 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
$5$ \( T^{12} + 4 T^{11} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{12} - 3 T^{10} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$13$ \( (T^{6} - 4 T^{5} + \cdots - 5424)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 69 T^{10} + \cdots + 66341025 \) Copy content Toggle raw display
$19$ \( T^{12} + 10 T^{11} + \cdots + 712336 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 367373889 \) Copy content Toggle raw display
$29$ \( (T^{6} - 18 T^{5} + \cdots - 36)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 671846400 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 376515216 \) Copy content Toggle raw display
$41$ \( (T^{6} + 6 T^{5} + \cdots + 1516)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 2 T^{5} + \cdots - 11052)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 23294390625 \) Copy content Toggle raw display
$53$ \( T^{12} + 180 T^{10} + \cdots + 1679616 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 9510150400 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 24118090000 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 91511090064 \) Copy content Toggle raw display
$71$ \( (T^{6} - 12 T^{5} + \cdots - 299840)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 9056948224 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 159715326736 \) Copy content Toggle raw display
$83$ \( (T^{6} + 2 T^{5} + \cdots - 266592)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 1973642858496 \) Copy content Toggle raw display
$97$ \( (T^{6} - 2 T^{5} + \cdots + 27617)^{2} \) Copy content Toggle raw display
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