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Properties

Label 1400.2.bh.j
Level $1400$
Weight $2$
Character orbit 1400.bh
Analytic conductor $11.179$
Analytic rank $0$
Dimension $16$
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] = [1400,2,Mod(249,1400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1400, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1400.249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1400.bh (of order \(6\), degree \(2\), not 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: \(11.1790562830\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 15x^{14} + 170x^{12} - 789x^{10} + 2754x^{8} - 960x^{6} + 269x^{4} - 18x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{9} + \beta_1) q^{3} + ( - \beta_{7} + \beta_{6}) q^{7} + (\beta_{10} + \beta_{5} - \beta_{4} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{9} + \beta_1) q^{3} + ( - \beta_{7} + \beta_{6}) q^{7} + (\beta_{10} + \beta_{5} - \beta_{4} + \cdots + 1) q^{9}+ \cdots + ( - 5 \beta_{3} + 2 \beta_{2} + 15) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 10 q^{9} - 20 q^{19} + 50 q^{21} - 8 q^{29} + 28 q^{31} - 20 q^{39} - 16 q^{41} + 26 q^{49} - 10 q^{51} - 2 q^{59} + 50 q^{61} + 64 q^{69} - 80 q^{71} + 4 q^{79} - 48 q^{81} - 38 q^{89} + 34 q^{91} + 204 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 15x^{14} + 170x^{12} - 789x^{10} + 2754x^{8} - 960x^{6} + 269x^{4} - 18x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4102340 \nu^{14} - 60216631 \nu^{12} + 677333628 \nu^{10} - 3009625800 \nu^{8} + \cdots - 1313504207 ) / 359036523 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 5600705 \nu^{14} + 82189681 \nu^{12} - 924727311 \nu^{10} + 4108880850 \nu^{8} + \cdots - 98128630 ) / 359036523 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 5297 \nu^{14} + 80335 \nu^{12} - 913449 \nu^{10} + 4324629 \nu^{8} - 15233538 \nu^{6} + \cdots + 99650 ) / 259983 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2723695 \nu^{14} - 40688940 \nu^{12} + 460586700 \nu^{10} - 2121507168 \nu^{8} + 7378916580 \nu^{6} + \cdots - 48212247 ) / 39892947 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2723695 \nu^{15} + 40688940 \nu^{13} - 460586700 \nu^{11} + 2121507168 \nu^{9} + \cdots + 48212247 \nu ) / 39892947 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 35385680 \nu^{15} - 519187807 \nu^{13} + 5842497456 \nu^{11} - 25960221600 \nu^{9} + \cdots + 3763758820 \nu ) / 359036523 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 39488020 \nu^{14} + 579404438 \nu^{12} - 6519831084 \nu^{10} + 28969847400 \nu^{8} + \cdots - 1014108521 ) / 359036523 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 39488020 \nu^{15} + 579404438 \nu^{13} - 6519831084 \nu^{11} + 28969847400 \nu^{9} + \cdots - 1373145044 \nu ) / 359036523 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 60623903 \nu^{14} + 905469064 \nu^{12} - 10249640520 \nu^{10} + 47199500037 \nu^{8} + \cdots + 185133080 ) / 359036523 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 73615375 \nu^{14} + 1111329695 \nu^{12} - 12618776481 \nu^{10} + 59254651869 \nu^{8} + \cdots + 1359797656 ) / 359036523 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 98128630 \nu^{15} + 1477530155 \nu^{13} - 16764056781 \nu^{11} + 78348216381 \nu^{9} + \cdots + 1793707879 \nu ) / 359036523 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 112863355 \nu^{15} - 1656023633 \nu^{13} + 18634765941 \nu^{11} - 82800661350 \nu^{9} + \cdots + 3662269979 \nu ) / 359036523 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( -\nu^{15} + 15\nu^{13} - 170\nu^{11} + 789\nu^{9} - 2754\nu^{7} + 960\nu^{5} - 269\nu^{3} + 18\nu \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 360686108 \nu^{15} - 5432977525 \nu^{13} + 61649473755 \nu^{11} - 288326988363 \nu^{9} + \cdots - 6603304643 \nu ) / 359036523 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + 3\beta_{5} + \beta_{4} + \beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{14} + \beta_{13} + 2\beta_{12} + 2\beta_{9} - \beta_{7} - 8\beta_{6} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} + 8\beta_{10} + 22\beta_{5} + 13\beta_{4} - 8\beta_{2} - 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 13\beta_{15} + 8\beta_{14} + 35\beta_{12} - 66\beta_{6} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 13\beta_{8} - 140\beta_{3} - 66\beta_{2} - 243 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -140\beta_{13} - 341\beta_{9} + 66\beta_{7} - 568\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 140\beta_{11} - 568\beta_{10} + 140\beta_{8} - 1498\beta_{5} - 1403\beta_{4} - 1403\beta_{3} - 1498 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1403 \beta_{15} - 568 \beta_{14} - 1403 \beta_{13} - 4336 \beta_{12} - 3501 \beta_{9} + \cdots - 5033 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1403\beta_{11} - 5033\beta_{10} - 13128\beta_{5} - 13578\beta_{4} + 5033\beta_{2} + 5033 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -13578\beta_{15} - 5033\beta_{14} - 42843\beta_{12} + 45435\beta_{6} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -13578\beta_{8} + 129012\beta_{3} + 45435\beta_{2} + 163129 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 129012\beta_{13} + 328023\beta_{9} - 45435\beta_{7} + 414868\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 129012 \beta_{11} + 414868 \beta_{10} - 129012 \beta_{8} + 1070157 \beta_{5} + 1213504 \beta_{4} + \cdots + 1070157 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1213504 \beta_{15} + 414868 \beta_{14} + 1213504 \beta_{13} + 3895268 \beta_{12} + 3096632 \beta_{9} + \cdots + 3814121 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{5}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
249.1
−2.00531 − 1.15777i
−2.63854 − 1.52336i
−0.229810 − 0.132681i
0.462601 + 0.267083i
−0.462601 − 0.267083i
0.229810 + 0.132681i
2.63854 + 1.52336i
2.00531 + 1.15777i
−2.00531 + 1.15777i
−2.63854 + 1.52336i
−0.229810 + 0.132681i
0.462601 − 0.267083i
−0.462601 + 0.267083i
0.229810 − 0.132681i
2.63854 − 1.52336i
2.00531 − 1.15777i
0 −2.87134 − 1.65777i 0 0 0 −2.49733 + 0.873699i 0 3.99638 + 6.92193i 0
249.2 0 −1.77252 − 1.02336i 0 0 0 −1.48827 + 2.18747i 0 0.594550 + 1.02979i 0
249.3 0 −1.09584 − 0.632681i 0 0 0 2.16772 + 1.51690i 0 −0.699429 − 1.21145i 0
249.4 0 −0.403424 − 0.232917i 0 0 0 −2.02469 − 1.70312i 0 −1.39150 − 2.41015i 0
249.5 0 0.403424 + 0.232917i 0 0 0 2.02469 + 1.70312i 0 −1.39150 − 2.41015i 0
249.6 0 1.09584 + 0.632681i 0 0 0 −2.16772 − 1.51690i 0 −0.699429 − 1.21145i 0
249.7 0 1.77252 + 1.02336i 0 0 0 1.48827 − 2.18747i 0 0.594550 + 1.02979i 0
249.8 0 2.87134 + 1.65777i 0 0 0 2.49733 − 0.873699i 0 3.99638 + 6.92193i 0
849.1 0 −2.87134 + 1.65777i 0 0 0 −2.49733 − 0.873699i 0 3.99638 − 6.92193i 0
849.2 0 −1.77252 + 1.02336i 0 0 0 −1.48827 − 2.18747i 0 0.594550 − 1.02979i 0
849.3 0 −1.09584 + 0.632681i 0 0 0 2.16772 − 1.51690i 0 −0.699429 + 1.21145i 0
849.4 0 −0.403424 + 0.232917i 0 0 0 −2.02469 + 1.70312i 0 −1.39150 + 2.41015i 0
849.5 0 0.403424 − 0.232917i 0 0 0 2.02469 − 1.70312i 0 −1.39150 + 2.41015i 0
849.6 0 1.09584 − 0.632681i 0 0 0 −2.16772 + 1.51690i 0 −0.699429 + 1.21145i 0
849.7 0 1.77252 − 1.02336i 0 0 0 1.48827 + 2.18747i 0 0.594550 − 1.02979i 0
849.8 0 2.87134 − 1.65777i 0 0 0 2.49733 + 0.873699i 0 3.99638 − 6.92193i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 249.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.2.bh.j 16
5.b even 2 1 inner 1400.2.bh.j 16
5.c odd 4 1 1400.2.q.l ✓ 8
5.c odd 4 1 1400.2.q.m yes 8
7.c even 3 1 inner 1400.2.bh.j 16
35.j even 6 1 inner 1400.2.bh.j 16
35.k even 12 1 9800.2.a.ck 4
35.k even 12 1 9800.2.a.cu 4
35.l odd 12 1 1400.2.q.l ✓ 8
35.l odd 12 1 1400.2.q.m yes 8
35.l odd 12 1 9800.2.a.cj 4
35.l odd 12 1 9800.2.a.ct 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.2.q.l ✓ 8 5.c odd 4 1
1400.2.q.l ✓ 8 35.l odd 12 1
1400.2.q.m yes 8 5.c odd 4 1
1400.2.q.m yes 8 35.l odd 12 1
1400.2.bh.j 16 1.a even 1 1 trivial
1400.2.bh.j 16 5.b even 2 1 inner
1400.2.bh.j 16 7.c even 3 1 inner
1400.2.bh.j 16 35.j even 6 1 inner
9800.2.a.cj 4 35.l odd 12 1
9800.2.a.ck 4 35.k even 12 1
9800.2.a.ct 4 35.l odd 12 1
9800.2.a.cu 4 35.k even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1400, [\chi])\):

\( T_{3}^{16} - 17T_{3}^{14} + 215T_{3}^{12} - 1080T_{3}^{10} + 3947T_{3}^{8} - 6042T_{3}^{6} + 6737T_{3}^{4} - 1424T_{3}^{2} + 256 \) Copy content Toggle raw display
\( T_{11}^{8} + 37T_{11}^{6} - 98T_{11}^{5} + 1235T_{11}^{4} - 1813T_{11}^{3} + 7359T_{11}^{2} + 6566T_{11} + 17956 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 17 T^{14} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - 13 T^{14} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{8} + 37 T^{6} + \cdots + 17956)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 62 T^{6} + \cdots + 7056)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} - 35 T^{14} + \cdots + 625 \) Copy content Toggle raw display
$19$ \( (T^{8} + 10 T^{7} + \cdots + 678976)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 824843587681 \) Copy content Toggle raw display
$29$ \( (T^{4} + 2 T^{3} + \cdots - 222)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} - 14 T^{7} + \cdots + 625)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} - 90 T^{14} + \cdots + 71639296 \) Copy content Toggle raw display
$41$ \( (T^{4} + 4 T^{3} + \cdots + 3319)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + 217 T^{6} + \cdots + 3825936)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 82632190550625 \) Copy content Toggle raw display
$53$ \( T^{16} - 42 T^{14} + \cdots + 1048576 \) Copy content Toggle raw display
$59$ \( (T^{8} + T^{7} + 67 T^{6} + \cdots + 4900)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 25 T^{7} + \cdots + 4536900)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 4694952902656 \) Copy content Toggle raw display
$71$ \( (T^{4} + 20 T^{3} + \cdots - 1035)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 1761205026816 \) Copy content Toggle raw display
$79$ \( (T^{8} - 2 T^{7} + \cdots + 22201)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 302 T^{6} + \cdots + 7986276)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 19 T^{7} + \cdots + 2064969)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 150 T^{6} + \cdots + 729)^{2} \) Copy content Toggle raw display
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