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Properties

Label 152.2.q.c
Level $152$
Weight $2$
Character orbit 152.q
Analytic conductor $1.214$
Analytic rank $0$
Dimension $18$
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] = [152,2,Mod(9,152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(152, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("152.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 152 = 2^{3} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 152.q (of order \(9\), degree \(6\), 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: \(1.21372611072\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 3 x^{17} + 21 x^{16} - 34 x^{15} + 204 x^{14} - 267 x^{13} + 1304 x^{12} - 972 x^{11} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_{2}) q^{3} + ( - \beta_{10} + \beta_{7} - \beta_{6}) q^{5} + (\beta_{17} - \beta_{16} + \cdots - \beta_{3}) q^{7}+ \cdots + (\beta_{16} + \beta_{14} + \beta_{13} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + \beta_{2}) q^{3} + ( - \beta_{10} + \beta_{7} - \beta_{6}) q^{5} + (\beta_{17} - \beta_{16} + \cdots - \beta_{3}) q^{7}+ \cdots + ( - \beta_{17} + \beta_{15} + \cdots + 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 9 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 9 q^{7} - 6 q^{9} - 3 q^{11} + 3 q^{13} + 33 q^{15} + 9 q^{17} - 24 q^{19} - 15 q^{21} + 6 q^{23} + 6 q^{25} - 12 q^{27} - 3 q^{29} - 6 q^{31} - 45 q^{33} - 15 q^{35} + 48 q^{37} + 12 q^{39} - 18 q^{41} - 39 q^{43} - 42 q^{45} - 27 q^{47} - 18 q^{49} + 48 q^{51} + 39 q^{53} - 27 q^{55} - 6 q^{57} + 9 q^{59} - 24 q^{61} + 3 q^{63} + 27 q^{65} + 39 q^{67} - 3 q^{69} + 12 q^{73} + 90 q^{75} + 60 q^{77} + 63 q^{79} - 6 q^{81} - 27 q^{83} - 30 q^{85} + 18 q^{87} + 66 q^{89} + 108 q^{91} + 60 q^{93} - 75 q^{95} - 81 q^{97} + 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 3 x^{17} + 21 x^{16} - 34 x^{15} + 204 x^{14} - 267 x^{13} + 1304 x^{12} - 972 x^{11} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 14\!\cdots\!07 \nu^{17} + \cdots - 22\!\cdots\!58 ) / 23\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 69\!\cdots\!18 \nu^{17} + \cdots - 32\!\cdots\!94 ) / 79\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 21\!\cdots\!82 \nu^{17} + \cdots - 44\!\cdots\!85 ) / 23\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 67\!\cdots\!01 \nu^{17} + \cdots + 80\!\cdots\!40 ) / 23\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 13\!\cdots\!81 \nu^{17} + \cdots + 14\!\cdots\!20 ) / 23\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 16\!\cdots\!08 \nu^{17} + \cdots - 94\!\cdots\!68 ) / 20\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 44\!\cdots\!85 \nu^{17} + \cdots + 14\!\cdots\!53 ) / 23\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 62\!\cdots\!80 \nu^{17} + \cdots + 15\!\cdots\!28 ) / 23\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 68\!\cdots\!44 \nu^{17} + \cdots - 24\!\cdots\!02 ) / 23\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 80\!\cdots\!40 \nu^{17} + \cdots + 80\!\cdots\!14 ) / 23\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 94\!\cdots\!68 \nu^{17} + \cdots + 14\!\cdots\!85 ) / 20\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 18\!\cdots\!73 \nu^{17} + \cdots + 40\!\cdots\!33 ) / 23\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 19\!\cdots\!93 \nu^{17} + \cdots - 34\!\cdots\!23 ) / 23\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 68\!\cdots\!99 \nu^{17} + \cdots - 10\!\cdots\!93 ) / 79\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 26\!\cdots\!52 \nu^{17} + \cdots + 12\!\cdots\!20 ) / 23\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 42\!\cdots\!94 \nu^{17} + \cdots - 22\!\cdots\!29 ) / 23\!\cdots\!46 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{13} + 3\beta_{12} + \beta_{11} + \beta_{8} - \beta_{7} - \beta_{3} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} - \beta_{13} - \beta_{11} - 6\beta_{7} - \beta_{5} + \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{17} - \beta_{16} - 6 \beta_{15} + 7 \beta_{13} - 18 \beta_{12} + \beta_{10} + 7 \beta_{9} + \cdots - 9 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{17} - \beta_{14} + 4 \beta_{13} - 13 \beta_{12} - 4 \beta_{11} + \beta_{10} - 11 \beta_{9} + \cdots + 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{17} + \beta_{16} + 39 \beta_{15} - 9 \beta_{14} + 2 \beta_{13} - 48 \beta_{11} - 9 \beta_{10} + \cdots + 118 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2 \beta_{17} + 16 \beta_{16} + 70 \beta_{15} + 14 \beta_{14} + 42 \beta_{13} + 123 \beta_{12} + \cdots + 266 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 86 \beta_{17} + 68 \beta_{16} + 86 \beta_{14} - 366 \beta_{13} + 806 \beta_{12} + 366 \beta_{11} + \cdots - 806 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 142 \beta_{17} - 142 \beta_{16} - 529 \beta_{15} + 30 \beta_{14} - 780 \beta_{13} - 241 \beta_{11} + \cdots - 1053 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 486 \beta_{17} - 710 \beta_{16} - 1867 \beta_{15} - 224 \beta_{14} + 2095 \beta_{13} - 5649 \beta_{12} + \cdots - 3978 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1593 \beta_{17} - 305 \beta_{16} - 1593 \beta_{14} + 4811 \beta_{13} - 8640 \beta_{12} - 4811 \beta_{11} + \cdots + 8640 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2369 \beta_{17} + 2369 \beta_{16} + 13361 \beta_{15} - 3372 \beta_{14} + 7035 \beta_{13} - 13525 \beta_{11} + \cdots + 40325 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 2619 \beta_{17} + 13738 \beta_{16} + 29957 \beta_{15} + 11119 \beta_{14} + 7774 \beta_{13} + \cdots + 97004 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 45780 \beta_{17} + 22888 \beta_{16} + 45780 \beta_{14} - 155988 \beta_{13} + 291909 \beta_{12} + \cdots - 291909 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 93604 \beta_{17} - 93604 \beta_{16} - 226308 \beta_{15} + 20352 \beta_{14} - 380348 \beta_{13} + \cdots - 550560 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 152252 \beta_{17} - 361504 \beta_{16} - 712225 \beta_{15} - 209252 \beta_{14} + 539661 \beta_{13} + \cdots - 1715401 \beta_1 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 923744 \beta_{17} - 146936 \beta_{16} - 923744 \beta_{14} + 2698412 \beta_{13} - 4335677 \beta_{12} + \cdots + 4335677 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/152\mathbb{Z}\right)^\times\).

\(n\) \(39\) \(77\) \(97\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{9} - \beta_{11}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−0.643662 − 1.11486i
−0.0372770 − 0.0645657i
1.18094 + 2.04545i
−0.643662 + 1.11486i
−0.0372770 + 0.0645657i
1.18094 − 2.04545i
1.40179 + 2.42798i
0.0744612 + 0.128971i
−0.976256 − 1.69092i
1.40179 − 2.42798i
0.0744612 − 0.128971i
−0.976256 + 1.69092i
−1.24611 + 2.15833i
0.404662 − 0.700896i
1.34145 − 2.32346i
−1.24611 − 2.15833i
0.404662 + 0.700896i
1.34145 + 2.32346i
0 −0.986147 + 0.827476i 0 −3.98739 − 1.45129i 0 −1.68618 + 2.92055i 0 −0.233174 + 1.32240i 0
9.2 0 −0.0571117 + 0.0479224i 0 0.846087 + 0.307950i 0 2.43770 − 4.22221i 0 −0.519979 + 2.94895i 0
9.3 0 1.80930 − 1.51819i 0 2.20161 + 0.801320i 0 −2.07787 + 3.59897i 0 0.447746 − 2.53930i 0
17.1 0 −0.986147 − 0.827476i 0 −3.98739 + 1.45129i 0 −1.68618 − 2.92055i 0 −0.233174 − 1.32240i 0
17.2 0 −0.0571117 − 0.0479224i 0 0.846087 − 0.307950i 0 2.43770 + 4.22221i 0 −0.519979 − 2.94895i 0
17.3 0 1.80930 + 1.51819i 0 2.20161 − 0.801320i 0 −2.07787 − 3.59897i 0 0.447746 + 2.53930i 0
25.1 0 −2.63451 − 0.958884i 0 0.465881 + 2.64214i 0 −0.731458 + 1.26692i 0 3.72306 + 3.12402i 0
25.2 0 −0.139941 − 0.0509345i 0 −0.594280 − 3.37033i 0 0.960111 − 1.66296i 0 −2.28114 − 1.91411i 0
25.3 0 1.83476 + 0.667798i 0 0.302047 + 1.71299i 0 −0.962609 + 1.66729i 0 0.622259 + 0.522138i 0
73.1 0 −2.63451 + 0.958884i 0 0.465881 − 2.64214i 0 −0.731458 − 1.26692i 0 3.72306 − 3.12402i 0
73.2 0 −0.139941 + 0.0509345i 0 −0.594280 + 3.37033i 0 0.960111 + 1.66296i 0 −2.28114 + 1.91411i 0
73.3 0 1.83476 − 0.667798i 0 0.302047 − 1.71299i 0 −0.962609 − 1.66729i 0 0.622259 − 0.522138i 0
81.1 0 −0.432771 + 2.45437i 0 0.270433 + 0.226920i 0 0.343717 + 0.595336i 0 −3.01754 − 1.09830i 0
81.2 0 0.140538 − 0.797029i 0 −2.64013 − 2.21534i 0 −2.01284 − 3.48634i 0 2.20357 + 0.802035i 0
81.3 0 0.465881 − 2.64214i 0 3.13575 + 2.63120i 0 −0.770573 − 1.33467i 0 −3.94480 − 1.43579i 0
137.1 0 −0.432771 − 2.45437i 0 0.270433 − 0.226920i 0 0.343717 − 0.595336i 0 −3.01754 + 1.09830i 0
137.2 0 0.140538 + 0.797029i 0 −2.64013 + 2.21534i 0 −2.01284 + 3.48634i 0 2.20357 − 0.802035i 0
137.3 0 0.465881 + 2.64214i 0 3.13575 − 2.63120i 0 −0.770573 + 1.33467i 0 −3.94480 + 1.43579i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 152.2.q.c ✓ 18
4.b odd 2 1 304.2.u.f 18
19.e even 9 1 inner 152.2.q.c ✓ 18
19.e even 9 1 2888.2.a.y 9
19.f odd 18 1 2888.2.a.x 9
76.k even 18 1 5776.2.a.ce 9
76.l odd 18 1 304.2.u.f 18
76.l odd 18 1 5776.2.a.cd 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.2.q.c ✓ 18 1.a even 1 1 trivial
152.2.q.c ✓ 18 19.e even 9 1 inner
304.2.u.f 18 4.b odd 2 1
304.2.u.f 18 76.l odd 18 1
2888.2.a.x 9 19.f odd 18 1
2888.2.a.y 9 19.e even 9 1
5776.2.a.cd 9 76.l odd 18 1
5776.2.a.ce 9 76.k even 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{18} + 3 T_{3}^{16} + 10 T_{3}^{15} - 21 T_{3}^{14} + 33 T_{3}^{13} + 200 T_{3}^{12} - 387 T_{3}^{11} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(152, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( T^{18} + 3 T^{16} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{18} - 3 T^{16} + \cdots + 506944 \) Copy content Toggle raw display
$7$ \( T^{18} + 9 T^{17} + \cdots + 2483776 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 321449041 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 958026304 \) Copy content Toggle raw display
$17$ \( T^{18} - 9 T^{17} + \cdots + 683929 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 322687697779 \) Copy content Toggle raw display
$23$ \( T^{18} - 6 T^{17} + \cdots + 20647936 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 224041024 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 13655324736 \) Copy content Toggle raw display
$37$ \( (T^{9} - 24 T^{8} + \cdots + 24066368)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + 18 T^{17} + \cdots + 2627641 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 46067324689 \) Copy content Toggle raw display
$47$ \( T^{18} + 27 T^{17} + \cdots + 93238336 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 8307228736 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 50856110034409 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 250264069696 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 276869601856 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 2926781894656 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 2212049391616 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 262136564559424 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 614174437546849 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 5372303459329 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 27903253216609 \) Copy content Toggle raw display
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