Properties

Label 270.2.m.b
Level $270$
Weight $2$
Character orbit 270.m
Analytic conductor $2.156$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [270,2,Mod(17,270)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(270, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([10, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("270.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 270.m (of order \(12\), degree \(4\), 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: \(2.15596085457\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: 16.0.9349208943630483456.9
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + \cdots + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} - \beta_{6} q^{4} + (\beta_{15} + \beta_{14} + \cdots - \beta_{2}) q^{5}+ \cdots - \beta_{15} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} - \beta_{6} q^{4} + (\beta_{15} + \beta_{14} + \cdots - \beta_{2}) q^{5}+ \cdots + (\beta_{15} + 2 \beta_{13} - 2 \beta_{12} + \cdots - 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 12 q^{5} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 12 q^{5} + 8 q^{7} - 8 q^{10} + 8 q^{16} + 12 q^{20} + 8 q^{22} + 24 q^{23} - 16 q^{25} - 16 q^{28} - 8 q^{31} - 24 q^{38} - 4 q^{40} - 24 q^{41} - 32 q^{46} - 48 q^{47} - 24 q^{50} + 24 q^{55} - 24 q^{56} + 16 q^{58} - 24 q^{61} - 16 q^{67} + 24 q^{68} + 16 q^{70} + 16 q^{73} + 16 q^{76} + 72 q^{77} - 16 q^{82} - 48 q^{83} - 4 q^{85} + 48 q^{86} + 8 q^{88} + 24 q^{92} - 84 q^{95} - 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + \cdots + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 4341 \nu^{15} + 25062 \nu^{14} - 134821 \nu^{13} + 417198 \nu^{12} - 1075015 \nu^{11} + \cdots - 740270 ) / 17095 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3456 \nu^{15} - 25920 \nu^{14} + 151876 \nu^{13} - 594074 \nu^{12} + 1879372 \nu^{11} + \cdots - 142555 ) / 17095 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 5033 \nu^{15} + 21310 \nu^{14} - 104426 \nu^{13} + 187861 \nu^{12} - 172588 \nu^{11} + \cdots - 887140 ) / 17095 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2793 \nu^{15} + 28180 \nu^{14} - 172311 \nu^{13} + 766484 \nu^{12} - 2581266 \nu^{11} + \cdots + 618310 ) / 17095 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15977 \nu^{15} - 141262 \nu^{14} + 856169 \nu^{13} - 3642449 \nu^{12} + 12106306 \nu^{11} + \cdots - 1883725 ) / 17095 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 26630 \nu^{15} - 182630 \nu^{14} + 1056740 \nu^{13} - 3923413 \nu^{12} + 12097498 \nu^{11} + \cdots + 218885 ) / 17095 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 26630 \nu^{15} - 210508 \nu^{14} + 1251886 \nu^{13} - 5050631 \nu^{12} + 16323908 \nu^{11} + \cdots - 1160550 ) / 17095 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 26630 \nu^{15} + 212086 \nu^{14} - 1262932 \nu^{13} + 5122693 \nu^{12} - 16612682 \nu^{11} + \cdots + 2084995 ) / 17095 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 26630 \nu^{15} - 216820 \nu^{14} + 1296070 \nu^{13} - 5311527 \nu^{12} + 17314892 \nu^{11} + \cdots - 2071845 ) / 17095 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 36479 \nu^{15} - 258470 \nu^{14} + 1499462 \nu^{13} - 5670854 \nu^{12} + 17608509 \nu^{11} + \cdots + 18365 ) / 17095 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 38207 \nu^{15} + 272482 \nu^{14} - 1582764 \nu^{13} + 6010234 \nu^{12} - 18706521 \nu^{11} + \cdots + 68035 ) / 17095 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 35414 \nu^{15} + 286908 \nu^{14} - 1708695 \nu^{13} + 6974027 \nu^{12} - 22629771 \nu^{11} + \cdots + 2511045 ) / 17095 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 38207 \nu^{15} + 300623 \nu^{14} - 1779751 \nu^{13} + 7154021 \nu^{12} - 23008412 \nu^{11} + \cdots + 1923500 ) / 17095 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 39935 \nu^{15} - 314635 \nu^{14} + 1863053 \nu^{13} - 7493401 \nu^{12} + 24106424 \nu^{11} + \cdots - 2009900 ) / 17095 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 54184 \nu^{15} - 399016 \nu^{14} + 2335837 \nu^{13} - 9056462 \nu^{12} + 28575749 \nu^{11} + \cdots - 1312670 ) / 17095 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{14} + \beta_{13} + \beta_{11} + \beta_{10} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{8} + \beta_{7} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - 3 \beta_{14} - 5 \beta_{13} + 3 \beta_{12} - 3 \beta_{10} + 3 \beta_{9} + \beta_{7} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 8 \beta_{15} - 6 \beta_{14} - 8 \beta_{13} - 2 \beta_{12} - 6 \beta_{11} - 8 \beta_{10} + 5 \beta_{9} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 17 \beta_{15} + 13 \beta_{14} + 27 \beta_{13} - 20 \beta_{12} - 21 \beta_{11} + 8 \beta_{10} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 43 \beta_{15} + 40 \beta_{14} + 64 \beta_{13} - 2 \beta_{12} + 11 \beta_{11} + 59 \beta_{10} - 63 \beta_{9} + \cdots - 32 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 170 \beta_{15} - 68 \beta_{14} - 154 \beta_{13} + 133 \beta_{12} + 208 \beta_{11} + 16 \beta_{10} + \cdots + 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 160 \beta_{15} - 316 \beta_{14} - 556 \beta_{13} + 108 \beta_{12} + 196 \beta_{11} - 388 \beta_{10} + \cdots + 231 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1442 \beta_{15} + 303 \beta_{14} + 759 \beta_{13} - 888 \beta_{12} - 1474 \beta_{11} - 546 \beta_{10} + \cdots + 116 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 163 \beta_{15} + 2609 \beta_{14} + 4787 \beta_{13} - 1507 \beta_{12} - 3305 \beta_{11} + 2222 \beta_{10} + \cdots - 1611 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 10909 \beta_{15} - 253 \beta_{14} - 1879 \beta_{13} + 5489 \beta_{12} + 8068 \beta_{11} + 6380 \beta_{10} + \cdots - 2314 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 11866 \beta_{15} - 20444 \beta_{14} - 38548 \beta_{13} + 16364 \beta_{12} + 34826 \beta_{11} + \cdots + 10052 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 72687 \beta_{15} - 16328 \beta_{14} - 20114 \beta_{13} - 27664 \beta_{12} - 26597 \beta_{11} + \cdots + 27130 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 163323 \beta_{15} + 144445 \beta_{14} + 280994 \beta_{13} - 152817 \beta_{12} - 300247 \beta_{11} + \cdots - 51022 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 402584 \beta_{15} + 264336 \beta_{14} + 422112 \beta_{13} + 70486 \beta_{12} - 100959 \beta_{11} + \cdots - 258101 \) Copy content Toggle raw display

Character values

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

\(n\) \(191\) \(217\)
\(\chi(n)\) \(-\beta_{2}\) \(-\beta_{6} - \beta_{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
0.500000 + 0.589118i
0.500000 1.00333i
0.500000 + 2.00333i
0.500000 + 0.410882i
0.500000 0.589118i
0.500000 + 1.00333i
0.500000 2.00333i
0.500000 0.410882i
0.500000 0.331082i
0.500000 + 2.74530i
0.500000 1.74530i
0.500000 + 1.33108i
0.500000 + 0.331082i
0.500000 2.74530i
0.500000 + 1.74530i
0.500000 1.33108i
−0.965926 0.258819i 0 0.866025 + 0.500000i 0.847015 2.06944i 0 0.686453 2.56188i −0.707107 0.707107i 0 −1.35376 + 1.77970i
17.2 −0.965926 0.258819i 0 0.866025 + 0.500000i 2.22612 + 0.210717i 0 −0.521929 + 1.94786i −0.707107 0.707107i 0 −2.09573 0.779698i
17.3 0.965926 + 0.258819i 0 0.866025 + 0.500000i 0.139908 + 2.23169i 0 −0.622279 + 2.32238i 0.707107 + 0.707107i 0 −0.442462 + 2.19185i
17.4 0.965926 + 0.258819i 0 0.866025 + 0.500000i 1.51901 1.64092i 0 −1.00635 + 3.75574i 0.707107 + 0.707107i 0 1.89195 1.19185i
143.1 −0.965926 + 0.258819i 0 0.866025 0.500000i 0.847015 + 2.06944i 0 0.686453 + 2.56188i −0.707107 + 0.707107i 0 −1.35376 1.77970i
143.2 −0.965926 + 0.258819i 0 0.866025 0.500000i 2.22612 0.210717i 0 −0.521929 1.94786i −0.707107 + 0.707107i 0 −2.09573 + 0.779698i
143.3 0.965926 0.258819i 0 0.866025 0.500000i 0.139908 2.23169i 0 −0.622279 2.32238i 0.707107 0.707107i 0 −0.442462 2.19185i
143.4 0.965926 0.258819i 0 0.866025 0.500000i 1.51901 + 1.64092i 0 −1.00635 3.75574i 0.707107 0.707107i 0 1.89195 + 1.19185i
197.1 −0.258819 0.965926i 0 −0.866025 + 0.500000i −1.36868 1.76825i 0 −2.56188 + 0.686453i 0.707107 + 0.707107i 0 −1.35376 + 1.77970i
197.2 −0.258819 0.965926i 0 −0.866025 + 0.500000i 1.29554 1.82252i 0 1.94786 0.521929i 0.707107 + 0.707107i 0 −2.09573 0.779698i
197.3 0.258819 + 0.965926i 0 −0.866025 + 0.500000i −0.661570 2.13596i 0 3.75574 1.00635i −0.707107 0.707107i 0 1.89195 1.19185i
197.4 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 2.00265 + 0.994679i 0 2.32238 0.622279i −0.707107 0.707107i 0 −0.442462 + 2.19185i
233.1 −0.258819 + 0.965926i 0 −0.866025 0.500000i −1.36868 + 1.76825i 0 −2.56188 0.686453i 0.707107 0.707107i 0 −1.35376 1.77970i
233.2 −0.258819 + 0.965926i 0 −0.866025 0.500000i 1.29554 + 1.82252i 0 1.94786 + 0.521929i 0.707107 0.707107i 0 −2.09573 + 0.779698i
233.3 0.258819 0.965926i 0 −0.866025 0.500000i −0.661570 + 2.13596i 0 3.75574 + 1.00635i −0.707107 + 0.707107i 0 1.89195 + 1.19185i
233.4 0.258819 0.965926i 0 −0.866025 0.500000i 2.00265 0.994679i 0 2.32238 + 0.622279i −0.707107 + 0.707107i 0 −0.442462 2.19185i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
9.d odd 6 1 inner
45.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 270.2.m.b 16
3.b odd 2 1 90.2.l.b 16
5.b even 2 1 1350.2.q.h 16
5.c odd 4 1 inner 270.2.m.b 16
5.c odd 4 1 1350.2.q.h 16
9.c even 3 1 90.2.l.b 16
9.c even 3 1 810.2.f.c 16
9.d odd 6 1 inner 270.2.m.b 16
9.d odd 6 1 810.2.f.c 16
12.b even 2 1 720.2.cu.b 16
15.d odd 2 1 450.2.p.h 16
15.e even 4 1 90.2.l.b 16
15.e even 4 1 450.2.p.h 16
36.f odd 6 1 720.2.cu.b 16
45.h odd 6 1 1350.2.q.h 16
45.j even 6 1 450.2.p.h 16
45.k odd 12 1 90.2.l.b 16
45.k odd 12 1 450.2.p.h 16
45.k odd 12 1 810.2.f.c 16
45.l even 12 1 inner 270.2.m.b 16
45.l even 12 1 810.2.f.c 16
45.l even 12 1 1350.2.q.h 16
60.l odd 4 1 720.2.cu.b 16
180.x even 12 1 720.2.cu.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.l.b 16 3.b odd 2 1
90.2.l.b 16 9.c even 3 1
90.2.l.b 16 15.e even 4 1
90.2.l.b 16 45.k odd 12 1
270.2.m.b 16 1.a even 1 1 trivial
270.2.m.b 16 5.c odd 4 1 inner
270.2.m.b 16 9.d odd 6 1 inner
270.2.m.b 16 45.l even 12 1 inner
450.2.p.h 16 15.d odd 2 1
450.2.p.h 16 15.e even 4 1
450.2.p.h 16 45.j even 6 1
450.2.p.h 16 45.k odd 12 1
720.2.cu.b 16 12.b even 2 1
720.2.cu.b 16 36.f odd 6 1
720.2.cu.b 16 60.l odd 4 1
720.2.cu.b 16 180.x even 12 1
810.2.f.c 16 9.c even 3 1
810.2.f.c 16 9.d odd 6 1
810.2.f.c 16 45.k odd 12 1
810.2.f.c 16 45.l even 12 1
1350.2.q.h 16 5.b even 2 1
1350.2.q.h 16 5.c odd 4 1
1350.2.q.h 16 45.h odd 6 1
1350.2.q.h 16 45.l even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} - 8 T_{7}^{15} + 32 T_{7}^{14} - 144 T_{7}^{13} + 476 T_{7}^{12} - 592 T_{7}^{11} + \cdots + 6250000 \) acting on \(S_{2}^{\mathrm{new}}(270, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 12 T^{15} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} - 8 T^{15} + \cdots + 6250000 \) Copy content Toggle raw display
$11$ \( (T^{8} - 22 T^{6} + \cdots + 1849)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 48 T^{13} + \cdots + 1296 \) Copy content Toggle raw display
$17$ \( T^{16} + 4132 T^{12} + \cdots + 390625 \) Copy content Toggle raw display
$19$ \( (T^{8} + 52 T^{6} + \cdots + 10201)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 82538991616 \) Copy content Toggle raw display
$29$ \( T^{16} + 80 T^{14} + \cdots + 40960000 \) Copy content Toggle raw display
$31$ \( (T^{8} + 4 T^{7} + \cdots + 448900)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 576 T^{5} + \cdots + 82944)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 12 T^{7} + \cdots + 555025)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 35806100625 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 2981133747216 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 3906250000 \) Copy content Toggle raw display
$59$ \( T^{16} + 44 T^{14} + \cdots + 625 \) Copy content Toggle raw display
$61$ \( (T^{8} + 12 T^{7} + \cdots + 45887076)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + 16 T^{15} + \cdots + 3418801 \) Copy content Toggle raw display
$71$ \( (T^{8} + 272 T^{6} + \cdots + 14032516)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 8 T^{7} + \cdots + 966289)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 21743271936 \) Copy content Toggle raw display
$89$ \( (T^{4} - 28 T^{2} + 100)^{4} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 19559470366881 \) Copy content Toggle raw display
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