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Properties

Label 270.3.j.b
Level $270$
Weight $3$
Character orbit 270.j
Analytic conductor $7.357$
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] = [270,3,Mod(89,270)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(270, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("270.89");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 270.j (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: \(7.35696713773\)
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: \( x^{16} + 2x^{14} - 230x^{12} - 96x^{10} + 25551x^{8} - 7776x^{6} - 1509030x^{4} + 1062882x^{2} + 43046721 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: no (minimal twist has level 90)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - 2 \beta_{2} - 2) q^{4} + (\beta_{13} + \beta_{6} + 2 \beta_{2} + \beta_1) q^{5} + ( - \beta_{15} + 2 \beta_{14} + \cdots + 1) q^{7}+ \cdots + ( - 2 \beta_{4} - 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - 2 \beta_{2} - 2) q^{4} + (\beta_{13} + \beta_{6} + 2 \beta_{2} + \beta_1) q^{5} + ( - \beta_{15} + 2 \beta_{14} + \cdots + 1) q^{7}+ \cdots + ( - 4 \beta_{15} - 4 \beta_{14} + \cdots + 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} - 30 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} - 30 q^{5} - 16 q^{10} - 60 q^{11} - 12 q^{14} - 32 q^{16} + 144 q^{19} + 60 q^{20} + 6 q^{25} - 72 q^{29} + 28 q^{31} - 136 q^{34} + 16 q^{40} + 180 q^{41} - 56 q^{46} + 12 q^{49} - 144 q^{50} + 20 q^{55} + 24 q^{56} + 228 q^{59} + 68 q^{61} + 128 q^{64} + 102 q^{65} - 112 q^{70} - 72 q^{74} - 144 q^{76} + 420 q^{79} - 136 q^{85} + 48 q^{86} - 168 q^{91} - 164 q^{94} - 276 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 2x^{14} - 230x^{12} - 96x^{10} + 25551x^{8} - 7776x^{6} - 1509030x^{4} + 1062882x^{2} + 43046721 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 256 \nu^{15} + 703 \nu^{13} + 72245 \nu^{11} - 469200 \nu^{9} - 2736405 \nu^{7} + \cdots - 1856323413 \nu ) / 4878628380 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 212 \nu^{14} + 1034 \nu^{12} + 27619 \nu^{10} - 245004 \nu^{8} - 1736091 \nu^{6} + \cdots - 865185948 ) / 271034910 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 647 \nu^{15} - 3144 \nu^{14} - 8183 \nu^{13} + 29676 \nu^{12} - 215878 \nu^{11} + \cdots - 3392719344 ) / 3252418920 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1225 \nu^{15} - 16666 \nu^{13} - 256559 \nu^{11} + 3638289 \nu^{9} + 19138311 \nu^{7} + \cdots + 12751926795 \nu ) / 4878628380 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 644 \nu^{15} + 5394 \nu^{14} + 13265 \nu^{13} - 7680 \nu^{12} + 167749 \nu^{11} + \cdots - 376260228 ) / 3252418920 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 644 \nu^{15} + 5394 \nu^{14} - 13265 \nu^{13} - 7680 \nu^{12} - 167749 \nu^{11} + \cdots - 376260228 ) / 3252418920 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1495 \nu^{15} - 4806 \nu^{14} + 74 \nu^{13} - 32211 \nu^{12} - 338747 \nu^{11} + \cdots - 4739922279 ) / 3252418920 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 4495 \nu^{15} - 15597 \nu^{14} - 29809 \nu^{13} + 304875 \nu^{12} - 779024 \nu^{11} + \cdots - 147946797108 ) / 9757256760 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1495 \nu^{15} - 4806 \nu^{14} - 74 \nu^{13} - 32211 \nu^{12} + 338747 \nu^{11} + \cdots - 4739922279 ) / 3252418920 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 4495 \nu^{15} - 15597 \nu^{14} + 29809 \nu^{13} + 304875 \nu^{12} + 779024 \nu^{11} + \cdots - 147946797108 ) / 9757256760 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2563 \nu^{15} - 47709 \nu^{14} - 9986 \nu^{13} + 342711 \nu^{12} + 275777 \nu^{11} + \cdots - 307726659522 ) / 9757256760 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 8980 \nu^{15} - 10989 \nu^{14} - 29587 \nu^{13} - 67176 \nu^{12} - 1795265 \nu^{11} + \cdots + 79392502431 ) / 9757256760 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 9061 \nu^{15} - 9432 \nu^{14} + 60367 \nu^{13} + 89028 \nu^{12} + 1497428 \nu^{11} + \cdots - 10178158032 ) / 9757256760 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 9061 \nu^{15} - 9432 \nu^{14} - 60367 \nu^{13} + 89028 \nu^{12} - 1497428 \nu^{11} + \cdots - 10178158032 ) / 9757256760 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 3187 \nu^{15} - 294 \nu^{14} - 28132 \nu^{13} + 34404 \nu^{12} - 406175 \nu^{11} + \cdots - 10898792028 ) / 3252418920 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} - 2\beta_{14} - \beta_{13} - \beta_{6} - \beta_{4} + 2\beta_{3} - \beta_{2} - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3 \beta_{14} + 3 \beta_{13} - \beta_{12} - 2 \beta_{11} + \beta_{10} + 2 \beta_{9} + 3 \beta_{7} + \cdots + 13 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2 \beta_{15} - \beta_{14} + 10 \beta_{13} - 3 \beta_{10} - 6 \beta_{9} + 3 \beta_{8} + 6 \beta_{7} + \cdots - 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 21 \beta_{14} - 21 \beta_{13} + 10 \beta_{12} + 2 \beta_{11} + 14 \beta_{10} + 10 \beta_{9} + \cdots + 131 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 91 \beta_{15} - 131 \beta_{14} - 85 \beta_{13} + 57 \beta_{10} - 21 \beta_{9} - 57 \beta_{8} + 21 \beta_{7} + \cdots - 91 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 138 \beta_{14} + 138 \beta_{13} - 283 \beta_{12} - 287 \beta_{11} + 97 \beta_{10} + 59 \beta_{9} + \cdots + 394 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 391 \beta_{15} + 578 \beta_{14} + 619 \beta_{13} - 147 \beta_{10} - 348 \beta_{9} + 147 \beta_{8} + \cdots + 391 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1605 \beta_{14} - 1605 \beta_{13} + 1798 \beta_{12} - 130 \beta_{11} + 2294 \beta_{10} + 1474 \beta_{9} + \cdots - 3160 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3238 \beta_{15} - 1259 \beta_{14} - 3106 \beta_{13} + 7203 \beta_{10} - 1713 \beta_{9} - 7203 \beta_{8} + \cdots - 3238 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 5853 \beta_{14} + 5853 \beta_{13} - 27838 \beta_{12} - 17129 \beta_{11} + 2020 \beta_{10} - 2827 \beta_{9} + \cdots - 55463 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 32581 \beta_{15} + 67271 \beta_{14} + 6169 \beta_{13} - 3270 \beta_{10} + 13440 \beta_{9} + \cdots + 32581 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 54510 \beta_{14} + 54510 \beta_{13} + 165640 \beta_{12} - 37360 \beta_{11} + 222320 \beta_{10} + \cdots - 701107 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 264389 \beta_{15} + 720358 \beta_{14} + 191369 \beta_{13} + 346530 \beta_{10} - 141510 \beta_{9} + \cdots + 264389 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 326523 \beta_{14} + 326523 \beta_{13} - 1824631 \beta_{12} - 78242 \beta_{11} - 620369 \beta_{10} + \cdots - 7997417 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 1216262 \beta_{15} + 3523319 \beta_{14} - 4368860 \beta_{13} - 1081533 \beta_{10} + 2884044 \beta_{9} + \cdots - 1216262 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/270\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(217\)
\(\chi(n)\) \(1 + \beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
89.1
0.633522 + 2.93235i
−2.97088 + 0.416995i
0.173499 − 2.99498i
2.87096 + 0.870383i
−0.173499 + 2.99498i
−0.633522 − 2.93235i
2.97088 − 0.416995i
−2.87096 − 0.870383i
0.633522 − 2.93235i
−2.97088 − 0.416995i
0.173499 + 2.99498i
2.87096 − 0.870383i
−0.173499 − 2.99498i
−0.633522 + 2.93235i
2.97088 + 0.416995i
−2.87096 + 0.870383i
−0.707107 + 1.22474i 0 −1.00000 − 1.73205i −4.70326 + 1.69687i 0 −7.59195 − 4.38322i 2.82843 0 1.24747 − 6.96016i
89.2 −0.707107 + 1.22474i 0 −1.00000 − 1.73205i −4.52070 − 2.13618i 0 8.30302 + 4.79375i 2.82843 0 5.81290 − 4.02619i
89.3 −0.707107 + 1.22474i 0 −1.00000 − 1.73205i −1.25903 + 4.83889i 0 −2.75375 − 1.58988i 2.82843 0 −5.03613 − 4.96360i
89.4 −0.707107 + 1.22474i 0 −1.00000 − 1.73205i 4.39720 + 2.38004i 0 4.16400 + 2.40409i 2.82843 0 −6.02424 + 3.70251i
89.5 0.707107 − 1.22474i 0 −1.00000 − 1.73205i −4.82012 − 1.32909i 0 2.75375 + 1.58988i −2.82843 0 −5.03613 + 4.96360i
89.6 0.707107 − 1.22474i 0 −1.00000 − 1.73205i −3.82116 + 3.22471i 0 7.59195 + 4.38322i −2.82843 0 1.24747 + 6.96016i
89.7 0.707107 − 1.22474i 0 −1.00000 − 1.73205i −0.410361 + 4.98313i 0 −8.30302 − 4.79375i −2.82843 0 5.81290 + 4.02619i
89.8 0.707107 − 1.22474i 0 −1.00000 − 1.73205i 0.137425 − 4.99811i 0 −4.16400 − 2.40409i −2.82843 0 −6.02424 − 3.70251i
179.1 −0.707107 − 1.22474i 0 −1.00000 + 1.73205i −4.70326 − 1.69687i 0 −7.59195 + 4.38322i 2.82843 0 1.24747 + 6.96016i
179.2 −0.707107 − 1.22474i 0 −1.00000 + 1.73205i −4.52070 + 2.13618i 0 8.30302 − 4.79375i 2.82843 0 5.81290 + 4.02619i
179.3 −0.707107 − 1.22474i 0 −1.00000 + 1.73205i −1.25903 − 4.83889i 0 −2.75375 + 1.58988i 2.82843 0 −5.03613 + 4.96360i
179.4 −0.707107 − 1.22474i 0 −1.00000 + 1.73205i 4.39720 − 2.38004i 0 4.16400 − 2.40409i 2.82843 0 −6.02424 − 3.70251i
179.5 0.707107 + 1.22474i 0 −1.00000 + 1.73205i −4.82012 + 1.32909i 0 2.75375 − 1.58988i −2.82843 0 −5.03613 − 4.96360i
179.6 0.707107 + 1.22474i 0 −1.00000 + 1.73205i −3.82116 − 3.22471i 0 7.59195 − 4.38322i −2.82843 0 1.24747 − 6.96016i
179.7 0.707107 + 1.22474i 0 −1.00000 + 1.73205i −0.410361 − 4.98313i 0 −8.30302 + 4.79375i −2.82843 0 5.81290 − 4.02619i
179.8 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 0.137425 + 4.99811i 0 −4.16400 + 2.40409i −2.82843 0 −6.02424 + 3.70251i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 89.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.d odd 6 1 inner
45.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 270.3.j.b 16
3.b odd 2 1 90.3.j.b ✓ 16
5.b even 2 1 inner 270.3.j.b 16
5.c odd 4 2 1350.3.i.e 16
9.c even 3 1 90.3.j.b ✓ 16
9.c even 3 1 810.3.b.b 16
9.d odd 6 1 inner 270.3.j.b 16
9.d odd 6 1 810.3.b.b 16
15.d odd 2 1 90.3.j.b ✓ 16
15.e even 4 2 450.3.i.e 16
45.h odd 6 1 inner 270.3.j.b 16
45.h odd 6 1 810.3.b.b 16
45.j even 6 1 90.3.j.b ✓ 16
45.j even 6 1 810.3.b.b 16
45.k odd 12 2 450.3.i.e 16
45.l even 12 2 1350.3.i.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.3.j.b ✓ 16 3.b odd 2 1
90.3.j.b ✓ 16 9.c even 3 1
90.3.j.b ✓ 16 15.d odd 2 1
90.3.j.b ✓ 16 45.j even 6 1
270.3.j.b 16 1.a even 1 1 trivial
270.3.j.b 16 5.b even 2 1 inner
270.3.j.b 16 9.d odd 6 1 inner
270.3.j.b 16 45.h odd 6 1 inner
450.3.i.e 16 15.e even 4 2
450.3.i.e 16 45.k odd 12 2
810.3.b.b 16 9.c even 3 1
810.3.b.b 16 9.d odd 6 1
810.3.b.b 16 45.h odd 6 1
810.3.b.b 16 45.j even 6 1
1350.3.i.e 16 5.c odd 4 2
1350.3.i.e 16 45.l even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} - 202 T_{7}^{14} + 27898 T_{7}^{12} - 2058640 T_{7}^{10} + 109528039 T_{7}^{8} + \cdots + 2726544000625 \) acting on \(S_{3}^{\mathrm{new}}(270, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 2 T^{2} + 4)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 152587890625 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 2726544000625 \) Copy content Toggle raw display
$11$ \( (T^{8} + 30 T^{7} + \cdots + 110923024)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 1991891886336 \) Copy content Toggle raw display
$17$ \( (T^{8} - 1520 T^{6} + \cdots + 640000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 36 T^{3} + \cdots - 118800)^{4} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 26\!\cdots\!61 \) Copy content Toggle raw display
$29$ \( (T^{8} + 36 T^{7} + \cdots + 334196141409)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 14 T^{7} + \cdots + 27544049296)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 3016 T^{6} + \cdots + 1414963456)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 90 T^{7} + \cdots + 15399072649)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 38\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 30\!\cdots\!01 \) Copy content Toggle raw display
$53$ \( (T^{8} - 3264 T^{6} + \cdots + 81293414400)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 28666857972736)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 34 T^{7} + \cdots + 116562836569)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 31\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{8} + 11328 T^{6} + \cdots + 39661519104)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 12985356390400)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 68\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 45\!\cdots\!21 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
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