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Properties

Label 273.2.bl.c
Level $273$
Weight $2$
Character orbit 273.bl
Analytic conductor $2.180$
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] = [273,2,Mod(88,273)]
 
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, 4, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("273.88");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.bl (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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.2346760387617129.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + x^{10} + 10 x^{9} - 15 x^{8} - 10 x^{7} + 45 x^{6} - 20 x^{5} - 60 x^{4} + 80 x^{3} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{8} - \beta_{6} - \beta_1) q^{2} + q^{3} + (\beta_{5} - \beta_{4} - \beta_{3} + \cdots - 1) q^{4}+ \cdots + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{8} - \beta_{6} - \beta_1) q^{2} + q^{3} + (\beta_{5} - \beta_{4} - \beta_{3} + \cdots - 1) q^{4}+ \cdots + (\beta_{9} - 2 \beta_{4} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{2} + 12 q^{3} + 5 q^{4} + 6 q^{5} - 3 q^{6} + 3 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 3 q^{2} + 12 q^{3} + 5 q^{4} + 6 q^{5} - 3 q^{6} + 3 q^{7} + 12 q^{9} + 14 q^{10} + 5 q^{12} - q^{13} - 16 q^{14} + 6 q^{15} + 3 q^{16} - 3 q^{18} - 27 q^{20} + 3 q^{21} + 7 q^{22} - 16 q^{23} + 10 q^{25} - q^{26} + 12 q^{27} + 24 q^{28} - 5 q^{29} + 14 q^{30} + 15 q^{31} - 6 q^{32} - 2 q^{35} + 5 q^{36} + 6 q^{37} + 24 q^{38} - q^{39} + 21 q^{40} - 15 q^{41} - 16 q^{42} - 13 q^{43} - 30 q^{44} + 6 q^{45} - 9 q^{46} - 9 q^{47} + 3 q^{48} + 9 q^{49} - 63 q^{50} - 55 q^{52} + 18 q^{53} - 3 q^{54} + 13 q^{55} - 21 q^{56} - 33 q^{59} - 27 q^{60} - 52 q^{61} - 13 q^{62} + 3 q^{63} - 4 q^{64} - 41 q^{65} + 7 q^{66} - 16 q^{69} - 42 q^{70} - 15 q^{71} - 18 q^{73} + 38 q^{74} + 10 q^{75} - 30 q^{76} + 20 q^{77} - q^{78} - 4 q^{79} + 12 q^{81} + 28 q^{82} + 24 q^{84} - 12 q^{85} - 15 q^{86} - 5 q^{87} - 32 q^{88} + 12 q^{89} + 14 q^{90} + 49 q^{91} - 40 q^{92} + 15 q^{93} + 6 q^{94} - 28 q^{95} - 6 q^{96} + 45 q^{97} + 48 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + x^{10} + 10 x^{9} - 15 x^{8} - 10 x^{7} + 45 x^{6} - 20 x^{5} - 60 x^{4} + 80 x^{3} + \cdots + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} - 13 \nu^{10} - 9 \nu^{9} + 72 \nu^{8} - 91 \nu^{7} - 164 \nu^{6} + 313 \nu^{5} + 42 \nu^{4} + \cdots - 800 ) / 224 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 9 \nu^{11} + 5 \nu^{10} + 25 \nu^{9} - 32 \nu^{8} - 21 \nu^{7} + 132 \nu^{6} - 73 \nu^{5} + \cdots + 256 ) / 224 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11 \nu^{11} + 17 \nu^{10} + 29 \nu^{9} - 78 \nu^{8} + 21 \nu^{7} + 166 \nu^{6} - 167 \nu^{5} + \cdots + 288 ) / 224 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 13 \nu^{11} + 29 \nu^{10} + 5 \nu^{9} - 96 \nu^{8} + 91 \nu^{7} + 200 \nu^{6} - 289 \nu^{5} + \cdots + 544 ) / 224 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8 \nu^{11} - 13 \nu^{10} - 9 \nu^{9} + 51 \nu^{8} - 42 \nu^{7} - 101 \nu^{6} + 194 \nu^{5} + \cdots - 464 ) / 112 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13 \nu^{11} - 57 \nu^{10} - 5 \nu^{9} + 208 \nu^{8} - 231 \nu^{7} - 396 \nu^{6} + 821 \nu^{5} + \cdots - 1664 ) / 224 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2 \nu^{11} - 5 \nu^{10} - 4 \nu^{9} + 18 \nu^{8} - 7 \nu^{7} - 41 \nu^{6} + 45 \nu^{5} + 35 \nu^{4} + \cdots - 88 ) / 28 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3 \nu^{11} - 4 \nu^{10} - 6 \nu^{9} + 20 \nu^{8} - 44 \nu^{6} + 43 \nu^{5} + 56 \nu^{4} - 82 \nu^{3} + \cdots - 48 ) / 28 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 15 \nu^{11} + 20 \nu^{10} + 30 \nu^{9} - 121 \nu^{8} + 21 \nu^{7} + 269 \nu^{6} - 271 \nu^{5} + \cdots + 464 ) / 112 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 17 \nu^{11} + 39 \nu^{10} + 13 \nu^{9} - 160 \nu^{8} + 133 \nu^{7} + 310 \nu^{6} - 547 \nu^{5} + \cdots + 1056 ) / 112 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{9} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} + \beta_{10} + \beta_{9} - \beta_{7} - \beta_{6} + \beta_{2} - \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{10} + 2\beta_{9} - 2\beta_{8} + 2\beta_{7} - \beta_{6} + \beta_{5} - 2\beta_{3} - \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 4 \beta_{11} + 2 \beta_{10} - 3 \beta_{8} + \beta_{7} - 5 \beta_{6} + 4 \beta_{5} - 7 \beta_{4} + \cdots - 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - \beta_{11} - \beta_{10} - \beta_{9} + 3 \beta_{8} + \beta_{7} + \beta_{6} + 6 \beta_{5} + \cdots + \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -4\beta_{10} - 2\beta_{9} - \beta_{8} + 2\beta_{5} - 4\beta_{4} + 8\beta_{3} - 2\beta_{2} + 3\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2 \beta_{11} - 6 \beta_{10} - 2 \beta_{9} + 6 \beta_{8} - 3 \beta_{7} + 7 \beta_{6} - 4 \beta_{5} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 5 \beta_{11} - 9 \beta_{10} + \beta_{9} - 16 \beta_{8} + \beta_{7} + 3 \beta_{6} - 8 \beta_{5} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2 \beta_{11} - \beta_{10} - 19 \beta_{8} + \beta_{7} + 4 \beta_{6} - 15 \beta_{5} - 5 \beta_{4} + \cdots - 5 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(1\) \(1 + \beta_{4}\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
88.1
0.655911 + 1.25291i
−1.18541 + 0.771231i
1.21245 + 0.727987i
−1.38488 − 0.286553i
1.32725 − 0.488273i
0.874681 − 1.11128i
0.655911 − 1.25291i
−1.18541 − 0.771231i
1.21245 − 0.727987i
−1.38488 + 0.286553i
1.32725 + 0.488273i
0.874681 + 1.11128i
−2.17010 − 1.25291i 1.00000 2.13956 + 3.70583i −2.61265 + 1.50841i −2.17010 − 1.25291i −0.393717 − 2.61629i − 5.71107i 1.00000 7.55962
88.2 −1.33581 − 0.771231i 1.00000 0.189594 + 0.328387i −1.27069 + 0.733632i −1.33581 − 0.771231i 1.52469 + 2.16225i 2.50004i 1.00000 2.26320
88.3 −1.26091 − 0.727987i 1.00000 0.0599314 + 0.103804i 3.67267 − 2.12042i −1.26091 − 0.727987i 2.09135 − 1.62057i 2.73743i 1.00000 −6.17455
88.4 0.496325 + 0.286553i 1.00000 −0.835774 − 1.44760i 2.74304 − 1.58369i 0.496325 + 0.286553i −2.25549 + 1.38302i − 2.10419i 1.00000 1.81525
88.5 0.845714 + 0.488273i 1.00000 −0.523178 − 0.906171i −0.233786 + 0.134976i 0.845714 + 0.488273i 2.62954 − 0.292422i − 2.97491i 1.00000 −0.263621
88.6 1.92478 + 1.11128i 1.00000 1.46986 + 2.54588i 0.701414 − 0.404962i 1.92478 + 1.11128i −2.09638 − 1.61406i 2.08860i 1.00000 1.80010
121.1 −2.17010 + 1.25291i 1.00000 2.13956 − 3.70583i −2.61265 − 1.50841i −2.17010 + 1.25291i −0.393717 + 2.61629i 5.71107i 1.00000 7.55962
121.2 −1.33581 + 0.771231i 1.00000 0.189594 − 0.328387i −1.27069 − 0.733632i −1.33581 + 0.771231i 1.52469 − 2.16225i − 2.50004i 1.00000 2.26320
121.3 −1.26091 + 0.727987i 1.00000 0.0599314 − 0.103804i 3.67267 + 2.12042i −1.26091 + 0.727987i 2.09135 + 1.62057i − 2.73743i 1.00000 −6.17455
121.4 0.496325 − 0.286553i 1.00000 −0.835774 + 1.44760i 2.74304 + 1.58369i 0.496325 − 0.286553i −2.25549 − 1.38302i 2.10419i 1.00000 1.81525
121.5 0.845714 − 0.488273i 1.00000 −0.523178 + 0.906171i −0.233786 − 0.134976i 0.845714 − 0.488273i 2.62954 + 0.292422i 2.97491i 1.00000 −0.263621
121.6 1.92478 − 1.11128i 1.00000 1.46986 − 2.54588i 0.701414 + 0.404962i 1.92478 − 1.11128i −2.09638 + 1.61406i − 2.08860i 1.00000 1.80010
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 88.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.u 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.bl.c yes 12
3.b odd 2 1 819.2.do.f 12
7.c even 3 1 273.2.t.c ✓ 12
13.e even 6 1 273.2.t.c ✓ 12
21.h odd 6 1 819.2.bm.e 12
39.h odd 6 1 819.2.bm.e 12
91.u even 6 1 inner 273.2.bl.c yes 12
273.x odd 6 1 819.2.do.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.t.c ✓ 12 7.c even 3 1
273.2.t.c ✓ 12 13.e even 6 1
273.2.bl.c yes 12 1.a even 1 1 trivial
273.2.bl.c yes 12 91.u even 6 1 inner
819.2.bm.e 12 21.h odd 6 1
819.2.bm.e 12 39.h odd 6 1
819.2.do.f 12 3.b odd 2 1
819.2.do.f 12 273.x odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 3 T_{2}^{11} - 4 T_{2}^{10} - 21 T_{2}^{9} + 20 T_{2}^{8} + 108 T_{2}^{7} + 27 T_{2}^{6} + \cdots + 49 \) acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 3 T^{11} + \cdots + 49 \) Copy content Toggle raw display
$3$ \( (T - 1)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 6 T^{11} + \cdots + 169 \) Copy content Toggle raw display
$7$ \( T^{12} - 3 T^{11} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{12} + 34 T^{10} + \cdots + 169 \) Copy content Toggle raw display
$13$ \( T^{12} + T^{11} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{12} + 11 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{12} + 95 T^{10} + \cdots + 17689 \) Copy content Toggle raw display
$23$ \( T^{12} + 16 T^{11} + \cdots + 2247001 \) Copy content Toggle raw display
$29$ \( T^{12} + 5 T^{11} + \cdots + 63001 \) Copy content Toggle raw display
$31$ \( T^{12} - 15 T^{11} + \cdots + 82369 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 1104964081 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 5098959649 \) Copy content Toggle raw display
$43$ \( T^{12} + 13 T^{11} + \cdots + 3455881 \) Copy content Toggle raw display
$47$ \( T^{12} + 9 T^{11} + \cdots + 14341369 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 823747401 \) Copy content Toggle raw display
$59$ \( T^{12} + 33 T^{11} + \cdots + 48177481 \) Copy content Toggle raw display
$61$ \( (T^{6} + 26 T^{5} + \cdots + 244009)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 31267787929 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 1583801209 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 1836722449 \) Copy content Toggle raw display
$79$ \( T^{12} + 4 T^{11} + \cdots + 59969536 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 184063489 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 1666190761 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 325035674161 \) Copy content Toggle raw display
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