Properties

Label 693.2.i.l
Level $693$
Weight $2$
Character orbit 693.i
Analytic conductor $5.534$
Analytic rank $0$
Dimension $12$
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] = [693,2,Mod(100,693)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(693, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("693.100");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.i (of order \(3\), 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: \(5.53363286007\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 11 x^{10} - 4 x^{9} + 88 x^{8} - 32 x^{7} + 325 x^{6} - 154 x^{5} + 878 x^{4} - 246 x^{3} + 793 x^{2} + 210 x + 441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + ( - \beta_{8} + 2 \beta_{4} - 2) q^{4} + ( - \beta_{8} + \beta_{4} - \beta_{2}) q^{5} + ( - \beta_{8} - \beta_{7} + \beta_{4}) q^{7} + ( - \beta_{10} + \beta_{7} + \beta_{6} - \beta_{2} - \beta_1 + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} + ( - \beta_{8} + 2 \beta_{4} - 2) q^{4} + ( - \beta_{8} + \beta_{4} - \beta_{2}) q^{5} + ( - \beta_{8} - \beta_{7} + \beta_{4}) q^{7} + ( - \beta_{10} + \beta_{7} + \beta_{6} - \beta_{2} - \beta_1 + 1) q^{8} + ( - \beta_{11} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{4} - 2 \beta_1 + 2) q^{10} + ( - \beta_{4} + 1) q^{11} + (\beta_{10} - \beta_{9} + \beta_{8} - \beta_{5} + 2 \beta_{2}) q^{13} + ( - \beta_{10} - 2 \beta_{8} + \beta_{5} + \beta_{4} - 2 \beta_{2} - \beta_1 - 1) q^{14} + (\beta_{9} + 2 \beta_{8} + \beta_{7} - \beta_{6} - 2 \beta_{4} - \beta_{3} + \beta_{2}) q^{16} + (\beta_{11} - \beta_{10} + 2 \beta_{9} - \beta_{8} - \beta_{7} + \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1) q^{17} + ( - \beta_{11} + \beta_{10} + \beta_{8} + \beta_{7} - 2 \beta_{4} - \beta_{3} + 1) q^{19} + ( - \beta_{11} + \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_{2} + \beta_1 - 6) q^{20} + ( - \beta_{6} + \beta_1) q^{22} + (\beta_{11} - \beta_{10} - \beta_{7} + \beta_{6} + \beta_{2} - 1) q^{23} + ( - \beta_{11} - \beta_{9} - \beta_{8} - \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_1) q^{25} + (2 \beta_{11} - 2 \beta_{10} - 3 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} + 3 \beta_{4} + \cdots - 2) q^{26}+ \cdots + (2 \beta_{10} - 3 \beta_{9} - 2 \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + 5 \beta_{4} + \beta_{3} + \cdots - 9) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 10 q^{4} + 4 q^{5} + 6 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 10 q^{4} + 4 q^{5} + 6 q^{7} + 12 q^{8} + 6 q^{10} + 6 q^{11} - 4 q^{13} - 6 q^{14} - 2 q^{16} + 8 q^{17} - 6 q^{19} - 72 q^{20} - 2 q^{23} - 2 q^{25} + 8 q^{26} - 28 q^{28} + 4 q^{29} - 4 q^{31} - 12 q^{32} + 28 q^{34} - 26 q^{35} + 6 q^{37} + 10 q^{38} + 18 q^{40} - 60 q^{41} - 24 q^{43} + 10 q^{44} + 4 q^{46} + 14 q^{47} - 12 q^{49} + 48 q^{50} + 22 q^{52} + 16 q^{53} + 8 q^{55} - 60 q^{56} - 2 q^{58} + 34 q^{59} + 2 q^{61} - 52 q^{62} + 4 q^{64} - 20 q^{65} + 20 q^{67} + 18 q^{68} - 24 q^{70} + 12 q^{71} - 26 q^{73} + 14 q^{74} - 8 q^{76} + 6 q^{77} - 2 q^{79} + 44 q^{80} + 4 q^{82} - 16 q^{83} + 20 q^{85} + 78 q^{86} + 6 q^{88} + 20 q^{89} + 20 q^{91} + 60 q^{92} + 10 q^{94} - 10 q^{95} - 28 q^{97} - 90 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 11 x^{10} - 4 x^{9} + 88 x^{8} - 32 x^{7} + 325 x^{6} - 154 x^{5} + 878 x^{4} - 246 x^{3} + 793 x^{2} + 210 x + 441 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9584894 \nu^{11} - 19466909 \nu^{10} + 96208552 \nu^{9} - 185361308 \nu^{8} + 829838862 \nu^{7} - 1454375956 \nu^{6} + 2798591972 \nu^{5} + \cdots + 42783471744 ) / 13064295921 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 83950323 \nu^{11} - 2544055360 \nu^{10} - 1987004452 \nu^{9} - 28133793045 \nu^{8} - 8032591776 \nu^{7} - 199447097536 \nu^{6} + \cdots - 429571070040 ) / 91450071447 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 89129676 \nu^{11} - 150376380 \nu^{10} + 913332178 \nu^{9} - 1874390521 \nu^{8} + 7771457144 \nu^{7} - 14787741916 \nu^{6} + \cdots - 5712921480 ) / 91450071447 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 428571998 \nu^{11} - 5483555728 \nu^{10} - 7640302548 \nu^{9} - 48482085067 \nu^{8} - 35859605275 \nu^{7} - 337244433571 \nu^{6} + \cdots + 287267184414 ) / 274350214341 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 21482340 \nu^{11} + 9584894 \nu^{10} + 216838831 \nu^{9} + 10279192 \nu^{8} + 1705084612 \nu^{7} + 142403982 \nu^{6} + 5527384544 \nu^{5} + \cdots + 5615169588 ) / 13064295921 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 524621182 \nu^{11} - 1552619726 \nu^{10} - 9905099250 \nu^{9} - 16737717755 \nu^{8} - 82755430433 \nu^{7} - 112809458051 \nu^{6} + \cdots - 358578680418 ) / 274350214341 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 289424446 \nu^{11} - 465237157 \nu^{10} + 2979868848 \nu^{9} - 6200032928 \nu^{8} + 25276956542 \nu^{7} - 48970335972 \nu^{6} + \cdots - 322335988128 ) / 91450071447 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1051275293 \nu^{11} - 3185176876 \nu^{10} - 14960396037 \nu^{9} - 26982671710 \nu^{8} - 101524433275 \nu^{7} + \cdots + 77341538421 ) / 274350214341 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1529741744 \nu^{11} - 137400767 \nu^{10} + 10842598413 \nu^{9} - 11765815127 \nu^{8} + 78851837725 \nu^{7} - 67315144865 \nu^{6} + \cdots - 393088565961 ) / 274350214341 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1536343510 \nu^{11} + 881705741 \nu^{10} + 14342363169 \nu^{9} - 2445705256 \nu^{8} + 95386347344 \nu^{7} - 19974546271 \nu^{6} + \cdots - 748323286767 ) / 274350214341 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - 4\beta_{4} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} + \beta_{7} + 5\beta_{6} - \beta_{2} - 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} - \beta_{9} - 7\beta_{8} - \beta_{5} + 22\beta_{4} + \beta_{3} + \beta _1 - 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 9 \beta_{11} + 9 \beta_{10} - 7 \beta_{9} + 10 \beta_{8} + 2 \beta_{7} - 29 \beta_{6} - 12 \beta_{4} - \beta_{3} + 8 \beta_{2} + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11\beta_{11} - 2\beta_{10} - 11\beta_{8} - 9\beta_{7} + 11\beta_{6} + 11\beta_{5} - 46\beta_{2} - 11\beta _1 + 111 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 42 \beta_{11} - 11 \beta_{10} + 53 \beta_{9} - 70 \beta_{8} - 66 \beta_{7} - 13 \beta_{5} + 103 \beta_{4} + 13 \beta_{3} - 11 \beta_{2} + 177 \beta _1 - 132 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 30 \beta_{11} + 30 \beta_{10} + 62 \beta_{9} + 394 \beta_{8} + 92 \beta_{7} - 96 \beta_{6} - 835 \beta_{4} - 90 \beta_{3} + 302 \beta_{2} + 30 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 210 \beta_{11} - 366 \beta_{10} - 88 \beta_{9} - 122 \beta_{8} + 244 \beta_{7} + 1107 \beta_{6} + 122 \beta_{5} - 340 \beta_{2} - 1107 \beta _1 + 472 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 392 \beta_{11} - 34 \beta_{10} - 358 \beta_{9} - 2023 \beta_{8} - 306 \beta_{7} - 664 \beta_{5} + 5344 \beta_{4} + 664 \beta_{3} - 34 \beta_{2} + 778 \beta _1 - 4288 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 3045 \beta_{11} + 3045 \beta_{10} - 1411 \beta_{9} + 4707 \beta_{8} + 1634 \beta_{7} - 7027 \beta_{6} - 6037 \beta_{4} - 1004 \beta_{3} + 3073 \beta_{2} + 3045 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/693\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(442\)
\(\chi(n)\) \(1\) \(-\beta_{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
100.1
−1.30499 2.26031i
−1.03585 1.79415i
−0.363272 0.629205i
0.670521 + 1.16138i
0.823543 + 1.42642i
1.21005 + 2.09587i
−1.30499 + 2.26031i
−1.03585 + 1.79415i
−0.363272 + 0.629205i
0.670521 1.16138i
0.823543 1.42642i
1.21005 2.09587i
−1.30499 + 2.26031i 0 −2.40599 4.16730i 1.90599 3.30128i 0 −1.36462 2.26667i 7.33922 0 4.97460 + 8.61626i
100.2 −1.03585 + 1.79415i 0 −1.14599 1.98490i 0.645985 1.11888i 0 1.73341 + 1.99882i 0.604878 0 1.33829 + 2.31799i
100.3 −0.363272 + 0.629205i 0 0.736067 + 1.27491i −1.23607 + 2.14093i 0 2.62460 + 0.333872i −2.52266 0 −0.898057 1.55548i
100.4 0.670521 1.16138i 0 0.100803 + 0.174595i −0.600803 + 1.04062i 0 −1.14674 + 2.38432i 2.95245 0 0.805702 + 1.39552i
100.5 0.823543 1.42642i 0 −0.356445 0.617380i −0.143555 + 0.248645i 0 −0.883533 2.49387i 2.11998 0 0.236448 + 0.409540i
100.6 1.21005 2.09587i 0 −1.92845 3.34017i 1.42845 2.47414i 0 2.03688 1.68852i −4.49387 0 −3.45699 5.98768i
298.1 −1.30499 2.26031i 0 −2.40599 + 4.16730i 1.90599 + 3.30128i 0 −1.36462 + 2.26667i 7.33922 0 4.97460 8.61626i
298.2 −1.03585 1.79415i 0 −1.14599 + 1.98490i 0.645985 + 1.11888i 0 1.73341 1.99882i 0.604878 0 1.33829 2.31799i
298.3 −0.363272 0.629205i 0 0.736067 1.27491i −1.23607 2.14093i 0 2.62460 0.333872i −2.52266 0 −0.898057 + 1.55548i
298.4 0.670521 + 1.16138i 0 0.100803 0.174595i −0.600803 1.04062i 0 −1.14674 2.38432i 2.95245 0 0.805702 1.39552i
298.5 0.823543 + 1.42642i 0 −0.356445 + 0.617380i −0.143555 0.248645i 0 −0.883533 + 2.49387i 2.11998 0 0.236448 0.409540i
298.6 1.21005 + 2.09587i 0 −1.92845 + 3.34017i 1.42845 + 2.47414i 0 2.03688 + 1.68852i −4.49387 0 −3.45699 + 5.98768i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 100.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 693.2.i.l yes 12
3.b odd 2 1 693.2.i.k 12
7.c even 3 1 inner 693.2.i.l yes 12
7.c even 3 1 4851.2.a.cc 6
7.d odd 6 1 4851.2.a.ce 6
21.g even 6 1 4851.2.a.cb 6
21.h odd 6 1 693.2.i.k 12
21.h odd 6 1 4851.2.a.cd 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.i.k 12 3.b odd 2 1
693.2.i.k 12 21.h odd 6 1
693.2.i.l yes 12 1.a even 1 1 trivial
693.2.i.l yes 12 7.c even 3 1 inner
4851.2.a.cb 6 21.g even 6 1
4851.2.a.cc 6 7.c even 3 1
4851.2.a.cd 6 21.h odd 6 1
4851.2.a.ce 6 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(693, [\chi])\):

\( T_{2}^{12} + 11 T_{2}^{10} - 4 T_{2}^{9} + 88 T_{2}^{8} - 32 T_{2}^{7} + 325 T_{2}^{6} - 154 T_{2}^{5} + 878 T_{2}^{4} - 246 T_{2}^{3} + 793 T_{2}^{2} + 210 T_{2} + 441 \) Copy content Toggle raw display
\( T_{5}^{12} - 4 T_{5}^{11} + 24 T_{5}^{10} - 32 T_{5}^{9} + 176 T_{5}^{8} - 168 T_{5}^{7} + 968 T_{5}^{6} - 80 T_{5}^{5} + 1440 T_{5}^{4} + 128 T_{5}^{3} + 1792 T_{5}^{2} + 480 T_{5} + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 11 T^{10} - 4 T^{9} + 88 T^{8} + \cdots + 441 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 4 T^{11} + 24 T^{10} - 32 T^{9} + \cdots + 144 \) Copy content Toggle raw display
$7$ \( T^{12} - 6 T^{11} + 24 T^{10} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$13$ \( (T^{6} + 2 T^{5} - 42 T^{4} - 80 T^{3} + \cdots - 356)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} - 8 T^{11} + 95 T^{10} - 308 T^{9} + \cdots + 225 \) Copy content Toggle raw display
$19$ \( T^{12} + 6 T^{11} + 89 T^{10} + \cdots + 3207681 \) Copy content Toggle raw display
$23$ \( T^{12} + 2 T^{11} + 53 T^{10} + \cdots + 378225 \) Copy content Toggle raw display
$29$ \( (T^{6} - 2 T^{5} - 97 T^{4} + 114 T^{3} + \cdots - 11853)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 4 T^{11} + 64 T^{10} + \cdots + 10000 \) Copy content Toggle raw display
$37$ \( T^{12} - 6 T^{11} + \cdots + 3080583009 \) Copy content Toggle raw display
$41$ \( (T^{6} + 30 T^{5} + 310 T^{4} + 1072 T^{3} + \cdots + 2988)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 12 T^{5} - 123 T^{4} + \cdots - 44001)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} - 14 T^{11} + \cdots + 916454529 \) Copy content Toggle raw display
$53$ \( T^{12} - 16 T^{11} + 260 T^{10} + \cdots + 8503056 \) Copy content Toggle raw display
$59$ \( T^{12} - 34 T^{11} + 725 T^{10} + \cdots + 1535121 \) Copy content Toggle raw display
$61$ \( T^{12} - 2 T^{11} + 106 T^{10} + \cdots + 242064 \) Copy content Toggle raw display
$67$ \( T^{12} - 20 T^{11} + \cdots + 5726856976 \) Copy content Toggle raw display
$71$ \( (T^{6} - 6 T^{5} - 341 T^{4} + \cdots + 561573)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 26 T^{11} + \cdots + 1755610000 \) Copy content Toggle raw display
$79$ \( T^{12} + 2 T^{11} + 282 T^{10} + \cdots + 1056784 \) Copy content Toggle raw display
$83$ \( (T^{6} + 8 T^{5} - 224 T^{4} - 1296 T^{3} + \cdots + 17280)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} - 20 T^{11} + \cdots + 3640432896 \) Copy content Toggle raw display
$97$ \( (T^{6} + 14 T^{5} + 35 T^{4} - 220 T^{3} + \cdots + 1909)^{2} \) Copy content Toggle raw display
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