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Properties

Label 273.3.w.c
Level $273$
Weight $3$
Character orbit 273.w
Analytic conductor $7.439$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [273,3,Mod(116,273)]
 
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([3, 4, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("273.116");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 273.w (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: \(7.43871121704\)
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: \( x^{16} - 16 x^{14} - 176 x^{13} + 344 x^{12} + 4576 x^{11} + 11040 x^{10} - 37664 x^{9} + \cdots + 97900608 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{10} - \beta_1) q^{2} + ( - \beta_{9} + \beta_{4}) q^{3} + ( - \beta_{9} - \beta_{7} + \cdots + \beta_{4}) q^{4}+ \cdots + ( - \beta_{15} - 2 \beta_{5} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{10} - \beta_1) q^{2} + ( - \beta_{9} + \beta_{4}) q^{3} + ( - \beta_{9} - \beta_{7} + \cdots + \beta_{4}) q^{4}+ \cdots + (2 \beta_{14} - 2 \beta_{13} + \cdots - 5 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 24 q^{4} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 24 q^{4} - 16 q^{9} + 96 q^{10} - 88 q^{12} - 208 q^{13} - 24 q^{16} + 144 q^{22} - 40 q^{25} + 264 q^{30} + 96 q^{36} + 432 q^{40} - 448 q^{42} - 128 q^{43} + 352 q^{48} - 504 q^{49} + 280 q^{51} + 312 q^{52} - 96 q^{55} + 184 q^{61} - 112 q^{64} - 448 q^{69} - 528 q^{75} + 80 q^{79} + 584 q^{81} + 544 q^{82} - 448 q^{87} + 72 q^{88} - 384 q^{90} - 88 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 16 x^{14} - 176 x^{13} + 344 x^{12} + 4576 x^{11} + 11040 x^{10} - 37664 x^{9} + \cdots + 97900608 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 27\!\cdots\!23 \nu^{15} + \cdots - 95\!\cdots\!28 ) / 31\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 242923652655874 \nu^{15} - 447584982412244 \nu^{14} + \cdots - 13\!\cdots\!12 ) / 82\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 23\!\cdots\!36 \nu^{15} + \cdots + 94\!\cdots\!60 ) / 52\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 18\!\cdots\!78 \nu^{15} + \cdots + 95\!\cdots\!52 ) / 31\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8720676811 \nu^{15} - 13202371445 \nu^{14} - 114600900799 \nu^{13} - 1350686662990 \nu^{12} + \cdots + 51\!\cdots\!60 ) / 11\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 50\!\cdots\!08 \nu^{15} + \cdots + 54\!\cdots\!72 ) / 52\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 37\!\cdots\!08 \nu^{15} + \cdots - 13\!\cdots\!16 ) / 31\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 25\!\cdots\!25 \nu^{15} + \cdots - 12\!\cdots\!08 ) / 15\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 50\!\cdots\!26 \nu^{15} + \cdots - 20\!\cdots\!80 ) / 31\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 37\!\cdots\!54 \nu^{15} + \cdots - 18\!\cdots\!20 ) / 15\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 76\!\cdots\!06 \nu^{15} + \cdots - 33\!\cdots\!64 ) / 31\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 79\!\cdots\!29 \nu^{15} + \cdots - 23\!\cdots\!36 ) / 31\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 17\!\cdots\!39 \nu^{15} + \cdots - 99\!\cdots\!52 ) / 52\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 18\!\cdots\!41 \nu^{15} + \cdots + 66\!\cdots\!68 ) / 52\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 76\!\cdots\!74 \nu^{15} + \cdots - 28\!\cdots\!56 ) / 82\!\cdots\!16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{12} + \beta_{11} - \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{15} - 2\beta_{12} + \beta_{11} + 6\beta_{10} - \beta_{9} + 2\beta_{5} - 2\beta_{3} - \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 3 \beta_{15} - 6 \beta_{13} - 2 \beta_{12} + 7 \beta_{11} + 2 \beta_{10} + 9 \beta_{9} - 3 \beta_{8} + \cdots + 33 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 10 \beta_{15} - 16 \beta_{13} - 48 \beta_{12} + 80 \beta_{11} - 32 \beta_{10} + 56 \beta_{9} + \cdots - 42 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 140 \beta_{15} - 100 \beta_{14} - 40 \beta_{13} + 282 \beta_{12} + 70 \beta_{11} + 90 \beta_{10} + \cdots - 440 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 280 \beta_{15} - 120 \beta_{14} - 1040 \beta_{13} + 1532 \beta_{12} - 32 \beta_{11} - 988 \beta_{10} + \cdots - 916 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 392 \beta_{15} - 1400 \beta_{14} - 1736 \beta_{13} + 6112 \beta_{12} + 1152 \beta_{11} - 11408 \beta_{10} + \cdots - 26180 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 4680 \beta_{15} - 14560 \beta_{14} + 5280 \beta_{13} + 63264 \beta_{12} - 31984 \beta_{11} + \cdots - 102080 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 34512 \beta_{15} + 31920 \beta_{14} - 52992 \beta_{13} + 193328 \beta_{12} - 151664 \beta_{11} + \cdots - 332112 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 225416 \beta_{15} + 138560 \beta_{14} + 324000 \beta_{13} + 586080 \beta_{12} - 630536 \beta_{11} + \cdots - 2529224 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 912120 \beta_{15} + 132000 \beta_{14} + 3524752 \beta_{13} + 3000592 \beta_{12} - 5249976 \beta_{11} + \cdots - 2419032 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 8715152 \beta_{15} + 16343360 \beta_{14} + 5499584 \beta_{13} - 11473984 \beta_{12} - 11912352 \beta_{11} + \cdots + 12213744 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 26048256 \beta_{15} + 60513440 \beta_{14} + 83049408 \beta_{13} - 74961680 \beta_{12} - 24939472 \beta_{11} + \cdots + 50642592 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 62006112 \beta_{15} + 137067840 \beta_{14} + 470008448 \beta_{13} - 402492768 \beta_{12} + \cdots + 1507589632 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 413475488 \beta_{15} + 2002436800 \beta_{14} + 36917888 \beta_{13} - 4235715200 \beta_{12} + \cdots + 6623300288 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
116.1
−2.73864 − 0.330355i
−2.73864 − 4.07201i
4.54639 + 2.26426i
4.54639 − 1.47739i
0.144025 + 1.47739i
0.144025 − 2.26426i
−1.95177 + 4.07201i
−1.95177 + 0.330355i
−2.73864 + 0.330355i
−2.73864 + 4.07201i
4.54639 − 2.26426i
4.54639 + 1.47739i
0.144025 − 1.47739i
0.144025 + 2.26426i
−1.95177 − 4.07201i
−1.95177 − 0.330355i
−1.70956 − 2.96105i −2.79279 + 1.09560i −3.84521 + 6.66010i 3.81256 + 6.60355i 8.01856 + 6.39660i −4.11804 + 5.66055i 12.6180 6.59934 − 6.11953i 13.0356 − 22.5784i
116.2 −1.70956 − 2.96105i 0.447581 + 2.96642i −3.84521 + 6.66010i 3.81256 + 6.60355i 8.01856 − 6.39660i 4.11804 − 5.66055i 12.6180 −8.59934 + 2.65543i 13.0356 − 22.5784i
116.3 −0.759866 − 1.31613i −0.447581 − 2.96642i 0.845208 − 1.46394i −0.681452 − 1.18031i −3.56409 + 2.84316i 0.736052 + 6.96119i −8.64790 −8.59934 + 2.65543i −1.03562 + 1.79375i
116.4 −0.759866 − 1.31613i 2.79279 − 1.09560i 0.845208 − 1.46394i −0.681452 − 1.18031i −3.56409 − 2.84316i −0.736052 − 6.96119i −8.64790 6.59934 − 6.11953i −1.03562 + 1.79375i
116.5 0.759866 + 1.31613i −0.447581 − 2.96642i 0.845208 − 1.46394i 0.681452 + 1.18031i 3.56409 − 2.84316i −0.736052 − 6.96119i 8.64790 −8.59934 + 2.65543i −1.03562 + 1.79375i
116.6 0.759866 + 1.31613i 2.79279 − 1.09560i 0.845208 − 1.46394i 0.681452 + 1.18031i 3.56409 + 2.84316i 0.736052 + 6.96119i 8.64790 6.59934 − 6.11953i −1.03562 + 1.79375i
116.7 1.70956 + 2.96105i −2.79279 + 1.09560i −3.84521 + 6.66010i −3.81256 − 6.60355i −8.01856 − 6.39660i 4.11804 − 5.66055i −12.6180 6.59934 − 6.11953i 13.0356 − 22.5784i
116.8 1.70956 + 2.96105i 0.447581 + 2.96642i −3.84521 + 6.66010i −3.81256 − 6.60355i −8.01856 + 6.39660i −4.11804 + 5.66055i −12.6180 −8.59934 + 2.65543i 13.0356 − 22.5784i
233.1 −1.70956 + 2.96105i −2.79279 − 1.09560i −3.84521 − 6.66010i 3.81256 − 6.60355i 8.01856 − 6.39660i −4.11804 − 5.66055i 12.6180 6.59934 + 6.11953i 13.0356 + 22.5784i
233.2 −1.70956 + 2.96105i 0.447581 − 2.96642i −3.84521 − 6.66010i 3.81256 − 6.60355i 8.01856 + 6.39660i 4.11804 + 5.66055i 12.6180 −8.59934 − 2.65543i 13.0356 + 22.5784i
233.3 −0.759866 + 1.31613i −0.447581 + 2.96642i 0.845208 + 1.46394i −0.681452 + 1.18031i −3.56409 − 2.84316i 0.736052 − 6.96119i −8.64790 −8.59934 − 2.65543i −1.03562 − 1.79375i
233.4 −0.759866 + 1.31613i 2.79279 + 1.09560i 0.845208 + 1.46394i −0.681452 + 1.18031i −3.56409 + 2.84316i −0.736052 + 6.96119i −8.64790 6.59934 + 6.11953i −1.03562 − 1.79375i
233.5 0.759866 − 1.31613i −0.447581 + 2.96642i 0.845208 + 1.46394i 0.681452 − 1.18031i 3.56409 + 2.84316i −0.736052 + 6.96119i 8.64790 −8.59934 − 2.65543i −1.03562 − 1.79375i
233.6 0.759866 − 1.31613i 2.79279 + 1.09560i 0.845208 + 1.46394i 0.681452 − 1.18031i 3.56409 − 2.84316i 0.736052 − 6.96119i 8.64790 6.59934 + 6.11953i −1.03562 − 1.79375i
233.7 1.70956 − 2.96105i −2.79279 − 1.09560i −3.84521 − 6.66010i −3.81256 + 6.60355i −8.01856 + 6.39660i 4.11804 + 5.66055i −12.6180 6.59934 + 6.11953i 13.0356 + 22.5784i
233.8 1.70956 − 2.96105i 0.447581 − 2.96642i −3.84521 − 6.66010i −3.81256 + 6.60355i −8.01856 − 6.39660i −4.11804 − 5.66055i −12.6180 −8.59934 − 2.65543i 13.0356 + 22.5784i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 116.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
13.b even 2 1 inner
21.h odd 6 1 inner
39.d odd 2 1 inner
91.r even 6 1 inner
273.w odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.3.w.c ✓ 16
3.b odd 2 1 inner 273.3.w.c ✓ 16
7.c even 3 1 inner 273.3.w.c ✓ 16
13.b even 2 1 inner 273.3.w.c ✓ 16
21.h odd 6 1 inner 273.3.w.c ✓ 16
39.d odd 2 1 inner 273.3.w.c ✓ 16
91.r even 6 1 inner 273.3.w.c ✓ 16
273.w odd 6 1 inner 273.3.w.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.3.w.c ✓ 16 1.a even 1 1 trivial
273.3.w.c ✓ 16 3.b odd 2 1 inner
273.3.w.c ✓ 16 7.c even 3 1 inner
273.3.w.c ✓ 16 13.b even 2 1 inner
273.3.w.c ✓ 16 21.h odd 6 1 inner
273.3.w.c ✓ 16 39.d odd 2 1 inner
273.3.w.c ✓ 16 91.r even 6 1 inner
273.3.w.c ✓ 16 273.w odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(273, [\chi])\):

\( T_{2}^{8} + 14T_{2}^{6} + 169T_{2}^{4} + 378T_{2}^{2} + 729 \) Copy content Toggle raw display
\( T_{19}^{8} - 70T_{19}^{6} + 4753T_{19}^{4} - 10290T_{19}^{2} + 21609 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 14 T^{6} + \cdots + 729)^{2} \) Copy content Toggle raw display
$3$ \( (T^{8} + 4 T^{6} + \cdots + 6561)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} + 60 T^{6} + \cdots + 11664)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 126 T^{6} + \cdots + 5764801)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 42 T^{6} + \cdots + 59049)^{2} \) Copy content Toggle raw display
$13$ \( (T + 13)^{16} \) Copy content Toggle raw display
$17$ \( (T^{8} - 854 T^{6} + \cdots + 5554571841)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 70 T^{6} + \cdots + 21609)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 1064 T^{6} + \cdots + 49787136)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 1946 T^{2} + 670761)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} - 364 T^{6} + \cdots + 830131344)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 1120 T^{6} + \cdots + 1416167424)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 2528 T^{2} + 645888)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 16 T + 42)^{8} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 5132413223289)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 13\!\cdots\!41)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 1610 T^{6} + \cdots + 149492809449)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 46 T^{3} + \cdots + 1570009)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} - 10486 T^{6} + \cdots + 841672300329)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 9738 T^{2} + 6940323)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 13\!\cdots\!84)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 20 T^{3} + \cdots + 4410000)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 11004 T^{2} + 1379052)^{4} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 33\!\cdots\!24)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 1288 T^{2} + 397488)^{4} \) Copy content Toggle raw display
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