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Properties

Label 165.2.d.c
Level $165$
Weight $2$
Character orbit 165.d
Analytic conductor $1.318$
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] = [165,2,Mod(164,165)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(165, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("165.164");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 165.d (of order \(2\), degree \(1\), 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: \(1.31753163335\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 24x^{14} + 244x^{12} - 1224x^{10} + 2880x^{8} - 2208x^{6} + 3976x^{4} + 432x^{2} + 2116 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + \beta_{4} q^{3} + ( - \beta_{7} - 2) q^{4} + ( - \beta_{13} + \beta_{11} + \cdots - \beta_1) q^{5}+ \cdots + (\beta_{13} - \beta_{11} + \cdots + \beta_{6}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + \beta_{4} q^{3} + ( - \beta_{7} - 2) q^{4} + ( - \beta_{13} + \beta_{11} + \cdots - \beta_1) q^{5}+ \cdots + ( - 2 \beta_{15} - \beta_{13} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 32 q^{4} - 16 q^{16} + 16 q^{25} + 16 q^{34} + 48 q^{36} + 48 q^{45} + 16 q^{49} - 16 q^{55} - 48 q^{60} + 32 q^{64} - 48 q^{66} - 48 q^{69} + 16 q^{70} - 48 q^{81} - 128 q^{91} + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 24x^{14} + 244x^{12} - 1224x^{10} + 2880x^{8} - 2208x^{6} + 3976x^{4} + 432x^{2} + 2116 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 363 \nu^{15} - 3071 \nu^{13} - 28361 \nu^{11} + 634291 \nu^{9} - 3652348 \nu^{7} + \cdots - 6737122 \nu ) / 11947672 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 9845 \nu^{15} + 219416 \nu^{13} - 2309677 \nu^{11} + 15411692 \nu^{9} - 74815508 \nu^{7} + \cdots + 363771080 \nu ) / 155319736 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 8971 \nu^{15} + 56394 \nu^{14} + 43388 \nu^{13} - 1403038 \nu^{12} + 2069306 \nu^{11} + \cdots + 68700632 ) / 310639472 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 8971 \nu^{15} - 56394 \nu^{14} + 43388 \nu^{13} + 1403038 \nu^{12} + 2069306 \nu^{11} + \cdots - 68700632 ) / 310639472 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 19606 \nu^{14} - 567841 \nu^{12} + 6959163 \nu^{10} - 44547989 \nu^{8} + 149567906 \nu^{6} + \cdots - 106793094 ) / 77659868 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 60124 \nu^{15} - 1413405 \nu^{13} + 14012484 \nu^{11} - 67446266 \nu^{9} + 146255268 \nu^{7} + \cdots + 198491328 \nu ) / 155319736 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 232 \nu^{14} + 5459 \nu^{12} - 53458 \nu^{10} + 250659 \nu^{8} - 493628 \nu^{6} + 53274 \nu^{4} + \cdots - 340216 ) / 271538 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 58483 \nu^{15} - 29571 \nu^{14} - 1486262 \nu^{13} + 657772 \nu^{12} + 15956428 \nu^{11} + \cdots - 494056560 ) / 155319736 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 65291 \nu^{15} - 1609409 \nu^{13} + 17484637 \nu^{11} - 98553123 \nu^{9} + 279192184 \nu^{7} + \cdots - 240799422 \nu ) / 155319736 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 5315 \nu^{15} + 112061 \nu^{13} - 918706 \nu^{11} + 2610036 \nu^{9} + 4644706 \nu^{7} + \cdots - 27517724 \nu ) / 11947672 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 58483 \nu^{15} + 68783 \nu^{14} + 1486262 \nu^{13} - 1793454 \nu^{12} - 15956428 \nu^{11} + \cdots - 185488836 ) / 155319736 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 83727 \nu^{14} - 1962503 \nu^{12} + 19864914 \nu^{10} - 101806378 \nu^{8} + 261543794 \nu^{6} + \cdots - 75127752 ) / 77659868 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 58483 \nu^{15} + 68783 \nu^{14} - 1486262 \nu^{13} - 1793454 \nu^{12} + 15956428 \nu^{11} + \cdots - 185488836 ) / 155319736 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 210925 \nu^{15} - 303128 \nu^{14} - 4995848 \nu^{13} + 7315240 \nu^{12} + 49904426 \nu^{11} + \cdots - 190436872 ) / 310639472 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 210925 \nu^{15} + 303128 \nu^{14} - 4995848 \nu^{13} - 7315240 \nu^{12} + 49904426 \nu^{11} + \cdots + 190436872 ) / 310639472 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} + \beta_{6} - \beta_{4} - \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{8} - \beta_{5} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{15} - 3 \beta_{14} + 2 \beta_{13} - 2 \beta_{11} + 4 \beta_{10} + 10 \beta_{6} + \cdots + 2 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{13} - 2\beta_{12} + 8\beta_{11} + 4\beta_{8} - \beta_{7} - 10\beta_{5} + 2\beta_{4} - 2\beta_{3} + 11 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 30 \beta_{15} - 30 \beta_{14} + 9 \beta_{13} - 9 \beta_{11} + 8 \beta_{10} + 94 \beta_{6} + \cdots + 2 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{15} + \beta_{14} + 50 \beta_{13} - 19 \beta_{12} + 48 \beta_{11} - 2 \beta_{8} - 3 \beta_{7} + \cdots - 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 105 \beta_{15} - 105 \beta_{14} + 5 \beta_{13} - 5 \beta_{11} - 47 \beta_{10} - 7 \beta_{9} + \cdots - 23 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 404 \beta_{13} - 132 \beta_{12} + 180 \beta_{11} - 224 \beta_{8} + 38 \beta_{7} - 432 \beta_{5} + \cdots - 668 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 528 \beta_{15} - 528 \beta_{14} - 149 \beta_{13} + 149 \beta_{11} - 755 \beta_{10} - 84 \beta_{9} + \cdots - 234 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 122 \beta_{15} - 122 \beta_{14} + 2428 \beta_{13} - 622 \beta_{12} - 230 \beta_{11} - 2658 \beta_{8} + \cdots - 7632 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1133 \beta_{15} - 1133 \beta_{14} - 2134 \beta_{13} + 2134 \beta_{11} - 6698 \beta_{10} + \cdots - 1252 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1784 \beta_{15} - 1784 \beta_{14} + 9348 \beta_{13} - 384 \beta_{12} - 11972 \beta_{11} - 21320 \beta_{8} + \cdots - 59230 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 13130 \beta_{15} + 13130 \beta_{14} - 19969 \beta_{13} + 19969 \beta_{11} - 43310 \beta_{10} + \cdots - 2142 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 15994 \beta_{15} - 15994 \beta_{14} - 8480 \beta_{13} + 31638 \beta_{12} - 135328 \beta_{11} + \cdots - 339042 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 220728 \beta_{15} + 220728 \beta_{14} - 148962 \beta_{13} + 148962 \beta_{11} - 193454 \beta_{10} + \cdots + 32458 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/165\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\) \(67\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
164.1
2.48916 − 0.707107i
−0.896709 + 0.707107i
−2.48916 + 0.707107i
0.896709 − 0.707107i
0.473307 + 0.707107i
2.60307 − 0.707107i
−0.473307 − 0.707107i
−2.60307 + 0.707107i
2.60307 + 0.707107i
0.473307 − 0.707107i
−2.60307 − 0.707107i
−0.473307 + 0.707107i
−0.896709 − 0.707107i
2.48916 + 0.707107i
0.896709 + 0.707107i
−2.48916 − 0.707107i
− 2.39417i −0.796225 − 1.53819i −3.73205 −2.17533 − 0.517638i −3.68269 + 1.90630i 3.38587 4.14682i −1.73205 + 2.44949i −1.23931 + 5.20810i
164.2 − 2.39417i −0.796225 + 1.53819i −3.73205 −2.17533 + 0.517638i 3.68269 + 1.90630i −3.38587 4.14682i −1.73205 − 2.44949i 1.23931 + 5.20810i
164.3 − 2.39417i 0.796225 − 1.53819i −3.73205 2.17533 + 0.517638i −3.68269 − 1.90630i −3.38587 4.14682i −1.73205 − 2.44949i 1.23931 − 5.20810i
164.4 − 2.39417i 0.796225 + 1.53819i −3.73205 2.17533 − 0.517638i 3.68269 − 1.90630i 3.38587 4.14682i −1.73205 + 2.44949i −1.23931 − 5.20810i
164.5 − 1.50597i −1.53819 − 0.796225i −0.267949 1.12603 − 1.93185i −1.19909 + 2.31647i −2.12976 − 2.60842i 1.73205 + 2.44949i −2.90931 − 1.69577i
164.6 − 1.50597i −1.53819 + 0.796225i −0.267949 1.12603 + 1.93185i 1.19909 + 2.31647i 2.12976 − 2.60842i 1.73205 − 2.44949i 2.90931 − 1.69577i
164.7 − 1.50597i 1.53819 − 0.796225i −0.267949 −1.12603 + 1.93185i −1.19909 − 2.31647i 2.12976 − 2.60842i 1.73205 − 2.44949i 2.90931 + 1.69577i
164.8 − 1.50597i 1.53819 + 0.796225i −0.267949 −1.12603 − 1.93185i 1.19909 − 2.31647i −2.12976 − 2.60842i 1.73205 + 2.44949i −2.90931 + 1.69577i
164.9 1.50597i −1.53819 − 0.796225i −0.267949 1.12603 − 1.93185i 1.19909 − 2.31647i 2.12976 2.60842i 1.73205 + 2.44949i 2.90931 + 1.69577i
164.10 1.50597i −1.53819 + 0.796225i −0.267949 1.12603 + 1.93185i −1.19909 − 2.31647i −2.12976 2.60842i 1.73205 − 2.44949i −2.90931 + 1.69577i
164.11 1.50597i 1.53819 − 0.796225i −0.267949 −1.12603 + 1.93185i 1.19909 + 2.31647i −2.12976 2.60842i 1.73205 − 2.44949i −2.90931 − 1.69577i
164.12 1.50597i 1.53819 + 0.796225i −0.267949 −1.12603 − 1.93185i −1.19909 + 2.31647i 2.12976 2.60842i 1.73205 + 2.44949i 2.90931 − 1.69577i
164.13 2.39417i −0.796225 − 1.53819i −3.73205 −2.17533 − 0.517638i 3.68269 − 1.90630i −3.38587 − 4.14682i −1.73205 + 2.44949i 1.23931 − 5.20810i
164.14 2.39417i −0.796225 + 1.53819i −3.73205 −2.17533 + 0.517638i −3.68269 − 1.90630i 3.38587 − 4.14682i −1.73205 − 2.44949i −1.23931 − 5.20810i
164.15 2.39417i 0.796225 − 1.53819i −3.73205 2.17533 + 0.517638i 3.68269 + 1.90630i 3.38587 − 4.14682i −1.73205 − 2.44949i −1.23931 + 5.20810i
164.16 2.39417i 0.796225 + 1.53819i −3.73205 2.17533 − 0.517638i −3.68269 + 1.90630i −3.38587 − 4.14682i −1.73205 + 2.44949i 1.23931 + 5.20810i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 164.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
11.b odd 2 1 inner
15.d odd 2 1 inner
33.d even 2 1 inner
55.d odd 2 1 inner
165.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.2.d.c ✓ 16
3.b odd 2 1 inner 165.2.d.c ✓ 16
5.b even 2 1 inner 165.2.d.c ✓ 16
5.c odd 4 2 825.2.f.f 16
11.b odd 2 1 inner 165.2.d.c ✓ 16
15.d odd 2 1 inner 165.2.d.c ✓ 16
15.e even 4 2 825.2.f.f 16
33.d even 2 1 inner 165.2.d.c ✓ 16
55.d odd 2 1 inner 165.2.d.c ✓ 16
55.e even 4 2 825.2.f.f 16
165.d even 2 1 inner 165.2.d.c ✓ 16
165.l odd 4 2 825.2.f.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.d.c ✓ 16 1.a even 1 1 trivial
165.2.d.c ✓ 16 3.b odd 2 1 inner
165.2.d.c ✓ 16 5.b even 2 1 inner
165.2.d.c ✓ 16 11.b odd 2 1 inner
165.2.d.c ✓ 16 15.d odd 2 1 inner
165.2.d.c ✓ 16 33.d even 2 1 inner
165.2.d.c ✓ 16 55.d odd 2 1 inner
165.2.d.c ✓ 16 165.d even 2 1 inner
825.2.f.f 16 5.c odd 4 2
825.2.f.f 16 15.e even 4 2
825.2.f.f 16 55.e even 4 2
825.2.f.f 16 165.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(165, [\chi])\):

\( T_{2}^{4} + 8T_{2}^{2} + 13 \) Copy content Toggle raw display
\( T_{23}^{4} - 36T_{23}^{2} + 216 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 8 T^{2} + 13)^{4} \) Copy content Toggle raw display
$3$ \( (T^{8} + 6 T^{4} + 81)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} - 4 T^{6} + \cdots + 625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 16 T^{2} + 52)^{4} \) Copy content Toggle raw display
$11$ \( (T^{8} - 28 T^{6} + \cdots + 14641)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 16 T^{2} + 52)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 20 T^{2} + 52)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 36 T^{2} + 312)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 36 T^{2} + 216)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} - 84 T^{2} + 312)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 12)^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} + 72 T^{2} + 96)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 60 T^{2} + 312)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 160 T^{2} + 6292)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 12 T^{2} + 24)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} - 96 T^{2} + 1536)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 24)^{8} \) Copy content Toggle raw display
$61$ \( (T^{4} + 120 T^{2} + 1248)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 324 T^{2} + 26136)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 192 T^{2} + 2304)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} - 16 T^{2} + 52)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 324 T^{2} + 25272)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 188 T^{2} + 8788)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 112 T^{2} + 1936)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 72 T^{2} + 96)^{4} \) Copy content Toggle raw display
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