↑

Properties

Label 1197.2.j.m
Level $1197$
Weight $2$
Character orbit 1197.j
Analytic conductor $9.558$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1197,2,Mod(172,1197)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1197, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1197.172");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1197 = 3^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1197.j (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: \(9.55809312195\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 13 x^{14} - 2 x^{13} + 118 x^{12} - 16 x^{11} + 534 x^{10} - 21 x^{9} + 1743 x^{8} - 101 x^{7} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 399)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{9} + \beta_{5} - 1) q^{4} + ( - \beta_{12} - \beta_{5}) q^{5} + (\beta_{9} + \beta_{8} - \beta_{3}) q^{7} + (\beta_{14} - \beta_{10} + \beta_{7} + \cdots + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{9} + \beta_{5} - 1) q^{4} + ( - \beta_{12} - \beta_{5}) q^{5} + (\beta_{9} + \beta_{8} - \beta_{3}) q^{7} + (\beta_{14} - \beta_{10} + \beta_{7} + \cdots + 1) q^{8}+ \cdots + (\beta_{14} + \beta_{13} - \beta_{11} + \cdots + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 10 q^{4} - 5 q^{5} + q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 10 q^{4} - 5 q^{5} + q^{7} + 6 q^{8} + 3 q^{10} - 7 q^{11} - 12 q^{13} + 12 q^{14} - 10 q^{16} + 8 q^{19} + 32 q^{20} + 36 q^{22} - 9 q^{23} - 15 q^{25} - 12 q^{26} - 40 q^{28} - 8 q^{29} + 11 q^{31} - 26 q^{32} - 32 q^{34} + 7 q^{35} - 17 q^{37} + 3 q^{40} + 34 q^{41} + 16 q^{43} - 31 q^{44} - q^{46} - 29 q^{47} + q^{49} - 60 q^{50} + 25 q^{52} - 6 q^{53} - 42 q^{55} + 54 q^{56} + 37 q^{58} - 7 q^{59} + 2 q^{61} + 78 q^{62} + 58 q^{64} - 13 q^{65} - 13 q^{67} + 14 q^{68} - 81 q^{70} - 36 q^{71} + 20 q^{73} - 26 q^{74} - 20 q^{76} - 19 q^{77} + 3 q^{79} - 35 q^{80} + 5 q^{82} + 72 q^{83} + 10 q^{85} - 51 q^{86} - 53 q^{88} - q^{89} - 9 q^{91} - 30 q^{92} + 30 q^{94} + 5 q^{95} + 6 q^{97} + 75 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 13 x^{14} - 2 x^{13} + 118 x^{12} - 16 x^{11} + 534 x^{10} - 21 x^{9} + 1743 x^{8} - 101 x^{7} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 58831967328673 \nu^{15} + \cdots - 48\!\cdots\!20 ) / 16\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 38192915263935 \nu^{15} - 121908751271206 \nu^{14} + 480754785462561 \nu^{13} + \cdots + 65\!\cdots\!06 ) / 24\!\cdots\!58 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 348452801576113 \nu^{15} - 228304043811896 \nu^{14} + \cdots + 28\!\cdots\!04 ) / 83\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 21\!\cdots\!69 \nu^{15} + \cdots + 47\!\cdots\!96 ) / 38\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 626142901716981 \nu^{15} + 76385830527870 \nu^{14} + \cdots + 68\!\cdots\!08 ) / 48\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10\!\cdots\!39 \nu^{15} + \cdots + 10\!\cdots\!76 ) / 83\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 83\!\cdots\!05 \nu^{15} + \cdots + 73\!\cdots\!40 ) / 58\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 21\!\cdots\!69 \nu^{15} + \cdots - 81\!\cdots\!68 ) / 12\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 84\!\cdots\!35 \nu^{15} + \cdots + 26\!\cdots\!76 ) / 38\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 23\!\cdots\!21 \nu^{15} + 495405685296776 \nu^{14} + \cdots + 93\!\cdots\!56 ) / 55\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 16\!\cdots\!73 \nu^{15} + \cdots + 20\!\cdots\!04 ) / 38\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 17\!\cdots\!59 \nu^{15} + \cdots + 38\!\cdots\!84 ) / 38\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 18\!\cdots\!57 \nu^{15} + \cdots - 98\!\cdots\!04 ) / 29\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 33\!\cdots\!57 \nu^{15} + \cdots - 67\!\cdots\!52 ) / 38\!\cdots\!28 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + 3\beta_{5} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} - \beta_{10} + \beta_{7} + 5\beta_{6} - \beta_{5} - \beta_{4} - 2\beta_{2} - 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{12} + \beta_{11} + \beta_{10} - 6 \beta_{9} + \beta_{8} - 2 \beta_{7} - 15 \beta_{5} + \cdots + 2 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{15} - 9 \beta_{14} + \beta_{13} + \beta_{9} - \beta_{8} - 31 \beta_{6} + 12 \beta_{5} + \cdots - 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 10 \beta_{15} + 12 \beta_{14} - 10 \beta_{13} + 10 \beta_{12} - 10 \beta_{11} - 12 \beta_{10} + \cdots + 101 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 14 \beta_{12} + 23 \beta_{11} + 69 \beta_{10} - 18 \beta_{9} + 12 \beta_{8} - 81 \beta_{7} + \cdots + 207 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 81 \beta_{15} - 111 \beta_{14} + 80 \beta_{13} + 228 \beta_{9} - 80 \beta_{8} + 13 \beta_{7} + \cdots - 685 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 204 \beta_{15} + 512 \beta_{14} - 138 \beta_{13} + 138 \beta_{12} - 204 \beta_{11} - 512 \beta_{10} + \cdots + 1058 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 606 \beta_{12} + 625 \beta_{11} + 944 \beta_{10} - 1635 \beta_{9} + 486 \beta_{8} - 1548 \beta_{7} + \cdots + 2302 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1666 \beta_{15} - 3787 \beta_{14} + 1201 \beta_{13} + 1994 \beta_{9} - 965 \beta_{8} + 23 \beta_{7} + \cdots - 8978 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 4775 \beta_{15} + 7734 \beta_{14} - 4534 \beta_{13} + 4534 \beta_{12} - 4775 \beta_{11} + \cdots + 34992 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 9895 \beta_{12} + 13159 \beta_{11} + 28143 \beta_{10} - 17901 \beta_{9} + 7688 \beta_{8} + \cdots + 74855 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 36497 \beta_{15} - 62086 \beta_{14} + 33947 \beta_{13} + 74806 \beta_{9} - 33281 \beta_{8} + \cdots - 257783 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 102379 \beta_{15} + 210466 \beta_{14} - 79326 \beta_{13} + 79326 \beta_{12} - 102379 \beta_{11} + \cdots + 590651 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1197\mathbb{Z}\right)^\times\).

\(n\) \(514\) \(533\) \(1009\)
\(\chi(n)\) \(-\beta_{5}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
172.1
−1.38622 − 2.40100i
−0.859129 − 1.48805i
−0.770547 − 1.33463i
−0.250893 − 0.434559i
0.323314 + 0.559997i
0.624863 + 1.08229i
1.14143 + 1.97701i
1.17719 + 2.03895i
−1.38622 + 2.40100i
−0.859129 + 1.48805i
−0.770547 + 1.33463i
−0.250893 + 0.434559i
0.323314 − 0.559997i
0.624863 − 1.08229i
1.14143 − 1.97701i
1.17719 − 2.03895i
−1.38622 − 2.40100i 0 −2.84321 + 4.92458i −0.602471 − 1.04351i 0 1.58377 + 2.11935i 10.2204 0 −1.67031 + 2.89307i
172.2 −0.859129 − 1.48805i 0 −0.476204 + 0.824810i 0.896598 + 1.55295i 0 −2.13097 − 1.56812i −1.80003 0 1.54059 − 2.66837i
172.3 −0.770547 − 1.33463i 0 −0.187486 + 0.324736i −1.29804 − 2.24827i 0 −1.69049 + 2.03525i −2.50432 0 −2.00040 + 3.46479i
172.4 −0.250893 − 0.434559i 0 0.874105 − 1.51400i 0.568593 + 0.984832i 0 0.630101 − 2.56963i −1.88080 0 0.285312 − 0.494175i
172.5 0.323314 + 0.559997i 0 0.790936 − 1.36994i −2.12692 − 3.68394i 0 −0.107079 − 2.64358i 2.31614 0 1.37533 − 2.38214i
172.6 0.624863 + 1.08229i 0 0.219092 − 0.379479i 1.99797 + 3.46058i 0 2.16069 + 1.52690i 3.04706 0 −2.49691 + 4.32477i
172.7 1.14143 + 1.97701i 0 −1.60570 + 2.78116i −1.27263 − 2.20425i 0 −2.54832 + 0.711380i −2.76546 0 2.90522 − 5.03198i
172.8 1.17719 + 2.03895i 0 −1.77153 + 3.06838i −0.663098 − 1.14852i 0 2.60229 − 0.477591i −3.63295 0 1.56118 − 2.70404i
856.1 −1.38622 + 2.40100i 0 −2.84321 − 4.92458i −0.602471 + 1.04351i 0 1.58377 − 2.11935i 10.2204 0 −1.67031 − 2.89307i
856.2 −0.859129 + 1.48805i 0 −0.476204 − 0.824810i 0.896598 − 1.55295i 0 −2.13097 + 1.56812i −1.80003 0 1.54059 + 2.66837i
856.3 −0.770547 + 1.33463i 0 −0.187486 − 0.324736i −1.29804 + 2.24827i 0 −1.69049 − 2.03525i −2.50432 0 −2.00040 − 3.46479i
856.4 −0.250893 + 0.434559i 0 0.874105 + 1.51400i 0.568593 − 0.984832i 0 0.630101 + 2.56963i −1.88080 0 0.285312 + 0.494175i
856.5 0.323314 − 0.559997i 0 0.790936 + 1.36994i −2.12692 + 3.68394i 0 −0.107079 + 2.64358i 2.31614 0 1.37533 + 2.38214i
856.6 0.624863 − 1.08229i 0 0.219092 + 0.379479i 1.99797 − 3.46058i 0 2.16069 − 1.52690i 3.04706 0 −2.49691 − 4.32477i
856.7 1.14143 − 1.97701i 0 −1.60570 − 2.78116i −1.27263 + 2.20425i 0 −2.54832 − 0.711380i −2.76546 0 2.90522 + 5.03198i
856.8 1.17719 − 2.03895i 0 −1.77153 − 3.06838i −0.663098 + 1.14852i 0 2.60229 + 0.477591i −3.63295 0 1.56118 + 2.70404i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 172.8
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 1197.2.j.m 16
3.b odd 2 1 399.2.j.g ✓ 16
7.c even 3 1 inner 1197.2.j.m 16
7.c even 3 1 8379.2.a.cr 8
7.d odd 6 1 8379.2.a.cq 8
21.g even 6 1 2793.2.a.bn 8
21.h odd 6 1 399.2.j.g ✓ 16
21.h odd 6 1 2793.2.a.bm 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
399.2.j.g ✓ 16 3.b odd 2 1
399.2.j.g ✓ 16 21.h odd 6 1
1197.2.j.m 16 1.a even 1 1 trivial
1197.2.j.m 16 7.c even 3 1 inner
2793.2.a.bm 8 21.h odd 6 1
2793.2.a.bn 8 21.g even 6 1
8379.2.a.cq 8 7.d odd 6 1
8379.2.a.cr 8 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1197, [\chi])\):

\( T_{2}^{16} + 13 T_{2}^{14} - 2 T_{2}^{13} + 118 T_{2}^{12} - 16 T_{2}^{11} + 534 T_{2}^{10} - 21 T_{2}^{9} + \cdots + 256 \) Copy content Toggle raw display
\( T_{5}^{16} + 5 T_{5}^{15} + 40 T_{5}^{14} + 129 T_{5}^{13} + 773 T_{5}^{12} + 2292 T_{5}^{11} + \cdots + 133956 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 13 T^{14} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 5 T^{15} + \cdots + 133956 \) Copy content Toggle raw display
$7$ \( T^{16} - T^{15} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{16} + 7 T^{15} + \cdots + 712336 \) Copy content Toggle raw display
$13$ \( (T^{8} + 6 T^{7} + \cdots + 448)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 192321424 \) Copy content Toggle raw display
$19$ \( (T^{2} - T + 1)^{8} \) Copy content Toggle raw display
$23$ \( T^{16} + 9 T^{15} + \cdots + 8219689 \) Copy content Toggle raw display
$29$ \( (T^{8} + 4 T^{7} + \cdots + 4016)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 920272896 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 1809433903104 \) Copy content Toggle raw display
$41$ \( (T^{8} - 17 T^{7} + \cdots - 11781168)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 8 T^{7} + \cdots - 33724)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 13435509040704 \) Copy content Toggle raw display
$53$ \( T^{16} + 6 T^{15} + \cdots + 4096 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 5616603136 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 174187204 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 5616603136 \) Copy content Toggle raw display
$71$ \( (T^{8} + 18 T^{7} + \cdots + 2744496)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 146845584696196 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 191409750016 \) Copy content Toggle raw display
$83$ \( (T^{8} - 36 T^{7} + \cdots - 2462397)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 681773884416 \) Copy content Toggle raw display
$97$ \( (T^{8} - 3 T^{7} + \cdots + 40753888)^{2} \) Copy content Toggle raw display
show more
show less