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Properties

Label 720.2.bw.c
Level $720$
Weight $2$
Character orbit 720.bw
Analytic conductor $5.749$
Analytic rank $0$
Dimension $16$
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] = [720,2,Mod(191,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.191");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.bw (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: \(5.74922894553\)
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: \( x^{16} - 2 x^{15} + 2 x^{14} + 14 x^{13} + 53 x^{12} - 28 x^{11} + 48 x^{10} + 288 x^{9} + 580 x^{8} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{10} - \beta_{6}) q^{3} + (\beta_{10} - \beta_{3}) q^{5} + (\beta_{13} - \beta_{11} + \cdots - \beta_1) q^{7}+ \cdots + (\beta_{14} + \beta_{13} + \beta_{11} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{10} - \beta_{6}) q^{3} + (\beta_{10} - \beta_{3}) q^{5} + (\beta_{13} - \beta_{11} + \cdots - \beta_1) q^{7}+ \cdots + ( - \beta_{15} + 3 \beta_{14} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 6 q^{7} + 6 q^{11} - 2 q^{13} - 6 q^{15} + 18 q^{23} + 8 q^{25} + 18 q^{27} - 6 q^{29} + 6 q^{33} - 4 q^{37} + 24 q^{39} + 24 q^{41} + 12 q^{45} + 10 q^{49} - 18 q^{51} - 36 q^{57} + 30 q^{59} + 14 q^{61} - 12 q^{63} + 6 q^{65} + 42 q^{67} - 54 q^{69} + 52 q^{73} - 66 q^{77} - 24 q^{79} + 6 q^{83} + 6 q^{85} - 54 q^{87} - 6 q^{93} - 12 q^{95} + 10 q^{97} - 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 2 x^{14} + 14 x^{13} + 53 x^{12} - 28 x^{11} + 48 x^{10} + 288 x^{9} + 580 x^{8} + \cdots + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1369139164006 \nu^{15} + 8531555022194 \nu^{14} - 16537417488508 \nu^{13} + \cdots + 280530153856978 ) / 22485707154228 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3916889397811 \nu^{15} - 9493688218037 \nu^{14} + 10826265165913 \nu^{13} + \cdots - 123972878157536 ) / 44971414308456 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1433583634206 \nu^{15} - 3195552950691 \nu^{14} + 3650869228227 \nu^{13} + \cdots + 26512664021108 ) / 11242853577114 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 8168713263899 \nu^{15} - 15479419601115 \nu^{14} + 14070946842255 \nu^{13} + \cdots + 146489526331676 ) / 44971414308456 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4637508235841 \nu^{15} + 9669135568473 \nu^{14} - 9851820955191 \nu^{13} + \cdots - 105568584226772 ) / 14990471436152 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 18924901004166 \nu^{15} + 19711955665913 \nu^{14} + 7588377978180 \nu^{13} + \cdots - 10\!\cdots\!52 ) / 44971414308456 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19219221508487 \nu^{15} - 24889497169514 \nu^{14} + 4370631404545 \nu^{13} + \cdots + 10\!\cdots\!80 ) / 44971414308456 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 20610486943375 \nu^{15} - 48814712509171 \nu^{14} + 59068644254011 \nu^{13} + \cdots + 159876025853768 ) / 44971414308456 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 25768912 \nu^{15} - 48879474 \nu^{14} + 42500935 \nu^{13} + 374893060 \nu^{12} + 1392599232 \nu^{11} + \cdots + 560556516 ) / 34159128 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 46107039278430 \nu^{15} + 109727910894028 \nu^{14} - 130118712721473 \nu^{13} + \cdots - 79440604929256 ) / 44971414308456 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 46852012095165 \nu^{15} + 82261565004920 \nu^{14} - 61365196001984 \nu^{13} + \cdots - 13\!\cdots\!04 ) / 44971414308456 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 58266175159214 \nu^{15} - 149488376111773 \nu^{14} + 192087052827152 \nu^{13} + \cdots - 242685027322168 ) / 44971414308456 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 29693036536951 \nu^{15} - 65227998238004 \nu^{14} + 71394926947815 \nu^{13} + \cdots + 318539981392102 ) / 22485707154228 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 59955279251884 \nu^{15} + 146923081305769 \nu^{14} - 179048915794153 \nu^{13} + \cdots + 112085077707340 ) / 44971414308456 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 81869476783629 \nu^{15} - 173528218315838 \nu^{14} + 180850382919753 \nu^{13} + \cdots + 11\!\cdots\!64 ) / 44971414308456 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{15} - 2\beta_{12} + 2\beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} - \beta _1 - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{15} + \beta_{14} - \beta_{13} + \beta_{12} + 2 \beta_{11} + 2 \beta_{10} + 5 \beta_{9} + \cdots + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 8 \beta_{14} - 7 \beta_{13} + 3 \beta_{12} + 8 \beta_{11} + 14 \beta_{10} + 14 \beta_{9} + 7 \beta_{8} + \cdots + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 7 \beta_{15} - 7 \beta_{14} + 7 \beta_{12} - 4 \beta_{11} + 20 \beta_{10} + 7 \beta_{9} + 10 \beta_{8} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 118 \beta_{15} - 51 \beta_{14} + 78 \beta_{13} + 46 \beta_{12} - 51 \beta_{11} + 63 \beta_{10} + \cdots - 100 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 223 \beta_{15} - 122 \beta_{14} + 272 \beta_{13} - 122 \beta_{12} - 217 \beta_{11} - 217 \beta_{10} + \cdots - 384 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 343 \beta_{14} + 569 \beta_{13} - 654 \beta_{12} - 343 \beta_{11} - 1849 \beta_{10} - 1849 \beta_{9} + \cdots - 221 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 807 \beta_{15} + 770 \beta_{14} - 250 \beta_{13} - 770 \beta_{12} + 417 \beta_{11} - 2319 \beta_{10} + \cdots + 880 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 11804 \beta_{15} + 7641 \beta_{14} - 9825 \beta_{13} - 3074 \beta_{12} + 7641 \beta_{11} - 10113 \beta_{10} + \cdots + 14933 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 26519 \beta_{15} + 13180 \beta_{14} - 35662 \beta_{13} + 13180 \beta_{12} + 25031 \beta_{11} + \cdots + 44364 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 30689 \beta_{14} - 63277 \beta_{13} + 86088 \beta_{12} + 30689 \beta_{11} + 231299 \beta_{10} + \cdots + 32947 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 97067 \beta_{15} - 91128 \beta_{14} + 34754 \beta_{13} + 91128 \beta_{12} - 47193 \beta_{11} + \cdots - 106136 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 1375924 \beta_{15} - 955545 \beta_{14} + 1184895 \beta_{13} + 323188 \beta_{12} - 955545 \beta_{11} + \cdots - 1857667 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 3198829 \beta_{15} - 1537622 \beta_{14} + 4360106 \beta_{13} - 1537622 \beta_{12} - 2995399 \beta_{11} + \cdots - 5285136 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 3485995 \beta_{14} + 7522913 \beta_{13} - 10539912 \beta_{12} - 3485995 \beta_{11} - 28078921 \beta_{10} + \cdots - 4120187 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1 + \beta_{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
1.43423 − 1.43423i
−0.205906 + 0.205906i
−0.233123 − 0.233123i
−0.922751 + 0.922751i
0.806564 + 0.806564i
−1.55038 − 1.55038i
2.34297 + 2.34297i
−0.671598 + 0.671598i
1.43423 + 1.43423i
−0.205906 − 0.205906i
−0.233123 + 0.233123i
−0.922751 − 0.922751i
0.806564 − 0.806564i
−1.55038 + 1.55038i
2.34297 − 2.34297i
−0.671598 − 0.671598i
0 −1.70636 + 0.297213i 0 0.866025 − 0.500000i 0 3.90729 + 2.25587i 0 2.82333 − 1.01431i 0
191.2 0 −1.28772 + 1.15835i 0 0.866025 − 0.500000i 0 −3.58054 − 2.06722i 0 0.316451 − 2.98326i 0
191.3 0 −0.794271 + 1.53920i 0 −0.866025 + 0.500000i 0 0.465734 + 0.268892i 0 −1.73827 − 2.44508i 0
191.4 0 −0.456597 − 1.67078i 0 0.866025 − 0.500000i 0 −0.178470 − 0.103039i 0 −2.58304 + 1.52575i 0
191.5 0 −0.157325 − 1.72489i 0 −0.866025 + 0.500000i 0 3.87443 + 2.23690i 0 −2.95050 + 0.542738i 0
191.6 0 1.26649 + 1.18152i 0 −0.866025 + 0.500000i 0 −1.03765 − 0.599087i 0 0.208014 + 2.99278i 0
191.7 0 1.41715 − 0.995829i 0 −0.866025 + 0.500000i 0 −1.80251 − 1.04068i 0 1.01665 − 2.82249i 0
191.8 0 1.71863 + 0.215221i 0 0.866025 − 0.500000i 0 1.35172 + 0.780414i 0 2.90736 + 0.739768i 0
671.1 0 −1.70636 − 0.297213i 0 0.866025 + 0.500000i 0 3.90729 − 2.25587i 0 2.82333 + 1.01431i 0
671.2 0 −1.28772 − 1.15835i 0 0.866025 + 0.500000i 0 −3.58054 + 2.06722i 0 0.316451 + 2.98326i 0
671.3 0 −0.794271 − 1.53920i 0 −0.866025 − 0.500000i 0 0.465734 − 0.268892i 0 −1.73827 + 2.44508i 0
671.4 0 −0.456597 + 1.67078i 0 0.866025 + 0.500000i 0 −0.178470 + 0.103039i 0 −2.58304 − 1.52575i 0
671.5 0 −0.157325 + 1.72489i 0 −0.866025 − 0.500000i 0 3.87443 − 2.23690i 0 −2.95050 − 0.542738i 0
671.6 0 1.26649 − 1.18152i 0 −0.866025 − 0.500000i 0 −1.03765 + 0.599087i 0 0.208014 − 2.99278i 0
671.7 0 1.41715 + 0.995829i 0 −0.866025 − 0.500000i 0 −1.80251 + 1.04068i 0 1.01665 + 2.82249i 0
671.8 0 1.71863 − 0.215221i 0 0.866025 + 0.500000i 0 1.35172 − 0.780414i 0 2.90736 − 0.739768i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.bw.c yes 16
3.b odd 2 1 2160.2.bw.c 16
4.b odd 2 1 720.2.bw.a ✓ 16
9.c even 3 1 2160.2.bw.a 16
9.c even 3 1 6480.2.h.a 16
9.d odd 6 1 720.2.bw.a ✓ 16
9.d odd 6 1 6480.2.h.f 16
12.b even 2 1 2160.2.bw.a 16
36.f odd 6 1 2160.2.bw.c 16
36.f odd 6 1 6480.2.h.f 16
36.h even 6 1 inner 720.2.bw.c yes 16
36.h even 6 1 6480.2.h.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.2.bw.a ✓ 16 4.b odd 2 1
720.2.bw.a ✓ 16 9.d odd 6 1
720.2.bw.c yes 16 1.a even 1 1 trivial
720.2.bw.c yes 16 36.h even 6 1 inner
2160.2.bw.a 16 9.c even 3 1
2160.2.bw.a 16 12.b even 2 1
2160.2.bw.c 16 3.b odd 2 1
2160.2.bw.c 16 36.f odd 6 1
6480.2.h.a 16 9.c even 3 1
6480.2.h.a 16 36.h even 6 1
6480.2.h.f 16 9.d odd 6 1
6480.2.h.f 16 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} - 6 T_{7}^{15} - 15 T_{7}^{14} + 162 T_{7}^{13} + 288 T_{7}^{12} - 3006 T_{7}^{11} + \cdots + 1296 \) acting on \(S_{2}^{\mathrm{new}}(720, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 6 T^{13} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{16} - 6 T^{15} + \cdots + 1296 \) Copy content Toggle raw display
$11$ \( T^{16} - 6 T^{15} + \cdots + 2862864 \) Copy content Toggle raw display
$13$ \( T^{16} + 2 T^{15} + \cdots + 1024 \) Copy content Toggle raw display
$17$ \( T^{16} + 126 T^{14} + \cdots + 104976 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 24715612944 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 288728064 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 60156391824 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 13072720896 \) Copy content Toggle raw display
$37$ \( (T^{8} + 2 T^{7} + \cdots - 508784)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 7144495980561 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 2699225556624 \) Copy content Toggle raw display
$47$ \( T^{16} + 135 T^{14} + \cdots + 5184 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 13006946304 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 989531541504 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 12186393664 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 62361576729 \) Copy content Toggle raw display
$71$ \( (T^{8} - 144 T^{6} + \cdots + 15696)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 26 T^{7} + \cdots - 23622848)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 105936461991936 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 136048896 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 107495424 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 4014156503296 \) Copy content Toggle raw display
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