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Properties

Label 273.2.bd.b
Level $273$
Weight $2$
Character orbit 273.bd
Analytic conductor $2.180$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [273,2,Mod(43,273)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(273, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("273.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.bd (of order \(6\), 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: \(2.17991597518\)
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: \( x^{16} - 4 x^{15} + 10 x^{14} - 8 x^{13} - 3 x^{12} + 32 x^{11} - 5 x^{10} - 44 x^{9} + 214 x^{8} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{14} q^{2} - \beta_{5} q^{3} + (\beta_{13} + \beta_{7} + 2 \beta_{5} + 2) q^{4} + (\beta_{12} - \beta_{9} - 2 \beta_{5} + \cdots - 1) q^{5} + ( - \beta_{14} + \beta_{4}) q^{6} + (\beta_{15} - \beta_{3}) q^{7}+ \cdots + (\beta_{13} + \beta_{11} - \beta_{10} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{3} + 14 q^{4} - 8 q^{9} - 4 q^{10} + 28 q^{12} - 12 q^{13} - 4 q^{14} - 12 q^{15} - 10 q^{16} - 2 q^{17} + 18 q^{20} - 18 q^{22} - 6 q^{23} - 20 q^{25} + 20 q^{26} - 16 q^{27} - 12 q^{29} + 4 q^{30}+ \cdots - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 10 x^{14} - 8 x^{13} - 3 x^{12} + 32 x^{11} - 5 x^{10} - 44 x^{9} + 214 x^{8} + \cdots + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{15} - 4 \nu^{14} + 10 \nu^{13} - 8 \nu^{12} - 3 \nu^{11} + 32 \nu^{10} - 5 \nu^{9} - 44 \nu^{8} + \cdots - 8748 ) / 2187 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 57 \nu^{15} - 5518 \nu^{14} + 7381 \nu^{13} - 18694 \nu^{12} - 28900 \nu^{11} - 3540 \nu^{10} + \cdots - 11910402 ) / 64152 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2194 \nu^{15} + 19537 \nu^{14} - 30118 \nu^{13} + 44909 \nu^{12} + 84636 \nu^{11} + \cdots + 35549685 ) / 192456 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 686 \nu^{15} - 4953 \nu^{14} + 7898 \nu^{13} - 10141 \nu^{12} - 20180 \nu^{11} + 11796 \nu^{10} + \cdots - 8790525 ) / 42768 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2551 \nu^{15} - 30666 \nu^{14} + 35541 \nu^{13} - 130226 \nu^{12} - 178900 \nu^{11} + \cdots - 73435086 ) / 128304 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 515 \nu^{15} - 348 \nu^{14} + 1311 \nu^{13} + 3844 \nu^{12} + 2060 \nu^{11} + 9484 \nu^{10} + \cdots + 769824 ) / 11664 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 533 \nu^{15} - 2036 \nu^{14} + 3545 \nu^{13} - 1876 \nu^{12} - 6828 \nu^{11} + 8356 \nu^{10} + \cdots - 2881008 ) / 11664 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2209 \nu^{15} - 1038 \nu^{14} + 4653 \nu^{13} + 18122 \nu^{12} + 10156 \nu^{11} + 37244 \nu^{10} + \cdots + 4543614 ) / 42768 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 25793 \nu^{15} - 43421 \nu^{14} + 101321 \nu^{13} + 100655 \nu^{12} - 37320 \nu^{11} + \cdots - 21167973 ) / 384912 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 35555 \nu^{15} + 209918 \nu^{14} - 340439 \nu^{13} + 377302 \nu^{12} + 822876 \nu^{11} + \cdots + 358488666 ) / 384912 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 36175 \nu^{15} - 126178 \nu^{14} + 228019 \nu^{13} - 76154 \nu^{12} - 382332 \nu^{11} + \cdots - 166373838 ) / 384912 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 13197 \nu^{15} - 41905 \nu^{14} + 78133 \nu^{13} - 14485 \nu^{12} - 117448 \nu^{11} + \cdots - 53118585 ) / 128304 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 47866 \nu^{15} + 127777 \nu^{14} - 247654 \nu^{13} - 30859 \nu^{12} + 311940 \nu^{11} + \cdots + 134472069 ) / 384912 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 5481 \nu^{15} + 12006 \nu^{14} - 25157 \nu^{13} - 12274 \nu^{12} + 21460 \nu^{11} + \cdots + 10314378 ) / 42768 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} + \beta_{9} + \beta_{8} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{15} - \beta_{13} - \beta_{12} - 2\beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} - 2\beta_{4} + \beta_{3} - \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} - 3 \beta_{14} + 5 \beta_{13} - 4 \beta_{12} + 2 \beta_{11} - \beta_{10} - \beta_{9} + \cdots - \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7 \beta_{15} - 2 \beta_{14} + \beta_{12} - 5 \beta_{11} - 5 \beta_{10} + 5 \beta_{9} + \beta_{8} + \cdots - 2 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10 \beta_{15} - 5 \beta_{14} - 9 \beta_{13} + 13 \beta_{12} - 6 \beta_{11} + 12 \beta_{10} - 10 \beta_{9} + \cdots - 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 15 \beta_{15} - 2 \beta_{14} - 8 \beta_{13} - 2 \beta_{12} + 9 \beta_{11} + 9 \beta_{10} - 6 \beta_{9} + \cdots + 14 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 37 \beta_{15} + 22 \beta_{14} - 10 \beta_{13} + 26 \beta_{12} + 18 \beta_{11} - 12 \beta_{10} + \cdots - 10 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 22 \beta_{15} + 40 \beta_{14} + 46 \beta_{13} - 18 \beta_{12} + 14 \beta_{11} + 2 \beta_{10} + 26 \beta_{9} + \cdots + 36 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 11 \beta_{15} + 14 \beta_{14} - 24 \beta_{13} - 18 \beta_{12} - 44 \beta_{11} - 50 \beta_{10} + \cdots + 112 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 41 \beta_{15} - 78 \beta_{14} + 89 \beta_{13} - 90 \beta_{12} + 2 \beta_{11} + 68 \beta_{10} + \cdots + 153 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 30 \beta_{15} + 5 \beta_{14} + 24 \beta_{13} - 78 \beta_{12} - 28 \beta_{11} - 211 \beta_{10} + \cdots - 232 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 27 \beta_{15} + 238 \beta_{14} - 349 \beta_{13} + 305 \beta_{12} - 146 \beta_{11} + 220 \beta_{10} + \cdots - 643 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 289 \beta_{15} - 753 \beta_{14} + 145 \beta_{13} - 851 \beta_{12} + 332 \beta_{11} + 383 \beta_{10} + \cdots - 687 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 786 \beta_{15} - 652 \beta_{14} + 1048 \beta_{13} - 316 \beta_{12} + 169 \beta_{11} - 773 \beta_{10} + \cdots + 1595 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(1\) \(1 + \beta_{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
−1.67549 + 0.438998i
1.35201 − 1.08262i
1.33452 − 1.10411i
−1.36010 − 1.07244i
0.750089 + 1.56121i
−0.306536 − 1.70471i
0.590887 − 1.62814i
1.31463 + 1.12772i
−1.67549 − 0.438998i
1.35201 + 1.08262i
1.33452 + 1.10411i
−1.36010 + 1.07244i
0.750089 − 1.56121i
−0.306536 + 1.70471i
0.590887 + 1.62814i
1.31463 − 1.12772i
−2.20165 − 1.27113i 0.500000 − 0.866025i 2.23152 + 3.86511i − 4.07309i −2.20165 + 1.27113i 0.866025 − 0.500000i − 6.26168i −0.500000 − 0.866025i −5.17741 + 8.96754i
43.2 −1.98604 − 1.14664i 0.500000 − 0.866025i 1.62956 + 2.82249i 0.692320i −1.98604 + 1.14664i −0.866025 + 0.500000i − 2.88753i −0.500000 − 0.866025i 0.793841 − 1.37497i
43.3 −1.44724 − 0.835563i 0.500000 − 0.866025i 0.396329 + 0.686463i 2.68351i −1.44724 + 0.835563i 0.866025 − 0.500000i 2.01762i −0.500000 − 0.866025i 2.24224 − 3.88368i
43.4 −0.654865 − 0.378087i 0.500000 − 0.866025i −0.714101 − 1.23686i − 4.01537i −0.654865 + 0.378087i −0.866025 + 0.500000i 2.59231i −0.500000 − 0.866025i −1.51816 + 2.62953i
43.5 0.924500 + 0.533760i 0.500000 − 0.866025i −0.430200 − 0.745128i − 0.994065i 0.924500 − 0.533760i 0.866025 − 0.500000i − 3.05354i −0.500000 − 0.866025i 0.530592 − 0.919013i
43.6 1.15033 + 0.664145i 0.500000 − 0.866025i −0.117823 − 0.204075i − 1.55828i 1.15033 − 0.664145i −0.866025 + 0.500000i − 2.96959i −0.500000 − 0.866025i 1.03492 − 1.79254i
43.7 1.85837 + 1.07293i 0.500000 − 0.866025i 1.30235 + 2.25573i 1.91954i 1.85837 − 1.07293i 0.866025 − 0.500000i 1.29759i −0.500000 − 0.866025i −2.05953 + 3.56721i
43.8 2.35660 + 1.36058i 0.500000 − 0.866025i 2.70236 + 4.68063i − 1.58278i 2.35660 − 1.36058i −0.866025 + 0.500000i 9.26480i −0.500000 − 0.866025i 2.15350 − 3.72996i
127.1 −2.20165 + 1.27113i 0.500000 + 0.866025i 2.23152 − 3.86511i 4.07309i −2.20165 − 1.27113i 0.866025 + 0.500000i 6.26168i −0.500000 + 0.866025i −5.17741 − 8.96754i
127.2 −1.98604 + 1.14664i 0.500000 + 0.866025i 1.62956 − 2.82249i − 0.692320i −1.98604 − 1.14664i −0.866025 − 0.500000i 2.88753i −0.500000 + 0.866025i 0.793841 + 1.37497i
127.3 −1.44724 + 0.835563i 0.500000 + 0.866025i 0.396329 − 0.686463i − 2.68351i −1.44724 − 0.835563i 0.866025 + 0.500000i − 2.01762i −0.500000 + 0.866025i 2.24224 + 3.88368i
127.4 −0.654865 + 0.378087i 0.500000 + 0.866025i −0.714101 + 1.23686i 4.01537i −0.654865 − 0.378087i −0.866025 − 0.500000i − 2.59231i −0.500000 + 0.866025i −1.51816 − 2.62953i
127.5 0.924500 − 0.533760i 0.500000 + 0.866025i −0.430200 + 0.745128i 0.994065i 0.924500 + 0.533760i 0.866025 + 0.500000i 3.05354i −0.500000 + 0.866025i 0.530592 + 0.919013i
127.6 1.15033 − 0.664145i 0.500000 + 0.866025i −0.117823 + 0.204075i 1.55828i 1.15033 + 0.664145i −0.866025 − 0.500000i 2.96959i −0.500000 + 0.866025i 1.03492 + 1.79254i
127.7 1.85837 − 1.07293i 0.500000 + 0.866025i 1.30235 − 2.25573i − 1.91954i 1.85837 + 1.07293i 0.866025 + 0.500000i − 1.29759i −0.500000 + 0.866025i −2.05953 − 3.56721i
127.8 2.35660 − 1.36058i 0.500000 + 0.866025i 2.70236 − 4.68063i 1.58278i 2.35660 + 1.36058i −0.866025 − 0.500000i − 9.26480i −0.500000 + 0.866025i 2.15350 + 3.72996i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.bd.b ✓ 16
3.b odd 2 1 819.2.ct.c 16
13.e even 6 1 inner 273.2.bd.b ✓ 16
13.f odd 12 1 3549.2.a.ba 8
13.f odd 12 1 3549.2.a.bc 8
39.h odd 6 1 819.2.ct.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bd.b ✓ 16 1.a even 1 1 trivial
273.2.bd.b ✓ 16 13.e even 6 1 inner
819.2.ct.c 16 3.b odd 2 1
819.2.ct.c 16 39.h odd 6 1
3549.2.a.ba 8 13.f odd 12 1
3549.2.a.bc 8 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 15 T_{2}^{14} + 152 T_{2}^{12} + 6 T_{2}^{11} - 839 T_{2}^{10} - 24 T_{2}^{9} + 3348 T_{2}^{8} + \cdots + 3721 \) acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 15 T^{14} + \cdots + 3721 \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} + 50 T^{14} + \cdots + 20449 \) Copy content Toggle raw display
$7$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{16} - 68 T^{14} + \cdots + 583696 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 238177489 \) Copy content Toggle raw display
$19$ \( T^{16} - 79 T^{14} + \cdots + 9412624 \) Copy content Toggle raw display
$23$ \( T^{16} + 6 T^{15} + \cdots + 22886656 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 81511963009 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 1539149824 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 223550241721 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 7256313856 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 599528101264 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 1683953296 \) Copy content Toggle raw display
$53$ \( (T^{8} - 14 T^{7} + \cdots - 8807)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 2315546542864 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 114864732889 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 119295633664 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 672053776 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 9572078569 \) Copy content Toggle raw display
$79$ \( (T^{8} - 46 T^{7} + \cdots - 7916)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 572 T^{14} + \cdots + 1430416 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 29787842814976 \) Copy content Toggle raw display
$97$ \( T^{16} + 6 T^{15} + \cdots + 43264 \) Copy content Toggle raw display
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