↑

Properties

Label 1134.2.k.c
Level $1134$
Weight $2$
Character orbit 1134.k
Analytic conductor $9.055$
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] = [1134,2,Mod(647,1134)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1134, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1134.647");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1134.k (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: \(9.05503558921\)
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: \( x^{16} - 8 x^{15} + 52 x^{14} - 224 x^{13} + 796 x^{12} - 2228 x^{11} + 5254 x^{10} - 10232 x^{9} + \cdots + 225 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{4} \)
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.

\(f(q)\) \(=\) \( q - \beta_{9} q^{2} + (\beta_{2} + 1) q^{4} + (\beta_{14} - \beta_{12} - \beta_1) q^{5} + (\beta_{14} - \beta_{12} + \cdots - \beta_{3}) q^{7}+ \cdots + ( - \beta_{9} - \beta_{4}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{9} q^{2} + (\beta_{2} + 1) q^{4} + (\beta_{14} - \beta_{12} - \beta_1) q^{5} + (\beta_{14} - \beta_{12} + \cdots - \beta_{3}) q^{7}+ \cdots + ( - \beta_{15} + 2 \beta_{13} + \cdots - 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} + 8 q^{7} - 12 q^{11} - 12 q^{14} - 8 q^{16} - 12 q^{23} - 8 q^{25} + 4 q^{28} + 12 q^{31} - 60 q^{35} + 4 q^{37} + 12 q^{38} + 48 q^{41} - 32 q^{43} - 12 q^{44} + 4 q^{49} - 12 q^{52} - 12 q^{56} - 12 q^{58} + 24 q^{59} - 12 q^{61} + 48 q^{62} - 16 q^{64} - 48 q^{65} - 4 q^{67} - 24 q^{70} + 36 q^{73} - 36 q^{74} - 84 q^{77} + 8 q^{79} + 72 q^{83} + 24 q^{85} - 24 q^{86} - 24 q^{89} - 12 q^{91} - 36 q^{94} - 12 q^{95} - 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 52 x^{14} - 224 x^{13} + 796 x^{12} - 2228 x^{11} + 5254 x^{10} - 10232 x^{9} + \cdots + 225 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 120 \nu^{15} + 451 \nu^{14} - 1737 \nu^{13} + 24265 \nu^{12} - 85444 \nu^{11} + 341872 \nu^{10} + \cdots + 357090 ) / 20265 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 976 \nu^{15} + 7320 \nu^{14} - 44776 \nu^{13} + 180024 \nu^{12} - 586524 \nu^{11} + 1489862 \nu^{10} + \cdots - 348240 ) / 20265 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1972 \nu^{15} + 12088 \nu^{14} - 76428 \nu^{13} + 280573 \nu^{12} - 939495 \nu^{11} + \cdots + 919245 ) / 20265 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1750 \nu^{15} - 5791 \nu^{14} + 29722 \nu^{13} - 8895 \nu^{12} - 164036 \nu^{11} + 1325903 \nu^{10} + \cdots + 1296900 ) / 20265 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1972 \nu^{15} + 17492 \nu^{14} - 114256 \nu^{13} + 510243 \nu^{12} - 1825751 \nu^{11} + \cdots + 1506930 ) / 20265 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 57 \nu^{14} - 399 \nu^{13} + 2470 \nu^{12} - 9633 \nu^{11} + 31634 \nu^{10} - 79377 \nu^{9} + \cdots + 8325 ) / 105 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 153 \nu^{15} + 1072 \nu^{14} - 9071 \nu^{13} + 71106 \nu^{12} - 302192 \nu^{11} + 1061863 \nu^{10} + \cdots + 381033 ) / 4053 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 153 \nu^{15} - 3367 \nu^{14} + 22002 \nu^{13} - 120350 \nu^{12} + 442595 \nu^{11} - 1368972 \nu^{10} + \cdots - 133119 ) / 4053 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1750 \nu^{15} - 20459 \nu^{14} + 132398 \nu^{13} - 646760 \nu^{12} + 2328366 \nu^{11} + \cdots - 987255 ) / 20265 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 342 \nu^{15} + 13952 \nu^{14} - 101711 \nu^{13} + 598673 \nu^{12} - 2388001 \nu^{11} + \cdots + 3095220 ) / 20265 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 8209 \nu^{15} + 42364 \nu^{14} - 250707 \nu^{13} + 739681 \nu^{12} - 2041947 \nu^{11} + \cdots - 2078580 ) / 20265 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 4986 \nu^{15} - 39711 \nu^{14} + 249163 \nu^{13} - 1047384 \nu^{12} + 3565403 \nu^{11} + \cdots - 1734180 ) / 20265 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 6117 \nu^{15} - 61028 \nu^{14} + 388123 \nu^{13} - 1791913 \nu^{12} + 6281745 \nu^{11} + \cdots - 1267515 ) / 20265 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 11817 \nu^{15} - 84092 \nu^{14} + 522451 \nu^{13} - 2059323 \nu^{12} + 6806326 \nu^{11} + \cdots - 332460 ) / 20265 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 11817 \nu^{15} + 93163 \nu^{14} - 585948 \nu^{13} + 2456903 \nu^{12} - 8366345 \nu^{11} + \cdots + 1800225 ) / 20265 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} + \beta_{14} - 2 \beta_{12} - \beta_{10} + 2 \beta_{8} + 2 \beta_{7} - \beta_{6} - \beta_{5} + \cdots + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{13} - 2 \beta_{12} - \beta_{11} + 2 \beta_{9} + 3 \beta_{8} + \beta_{7} + 2 \beta_{6} + \cdots - 7 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 7 \beta_{15} - 5 \beta_{14} - 6 \beta_{13} + 9 \beta_{12} + 3 \beta_{11} + 9 \beta_{10} + 12 \beta_{9} + \cdots - 18 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2 \beta_{15} + 2 \beta_{14} + \beta_{13} + 20 \beta_{12} + 19 \beta_{11} - 2 \beta_{9} - 28 \beta_{8} + \cdots + 25 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 66 \beta_{15} + 45 \beta_{14} + 74 \beta_{13} - 56 \beta_{12} - 19 \beta_{11} - 93 \beta_{10} + \cdots + 164 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 71 \beta_{15} - 2 \beta_{14} + 90 \beta_{13} - 219 \beta_{12} - 234 \beta_{11} - 69 \beta_{10} + \cdots - 39 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 584 \beta_{15} - 457 \beta_{14} - 647 \beta_{13} + 320 \beta_{12} - 53 \beta_{11} + 861 \beta_{10} + \cdots - 1487 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1236 \beta_{15} - 378 \beta_{14} - 1630 \beta_{13} + 2350 \beta_{12} + 2300 \beta_{11} + 1464 \beta_{10} + \cdots - 1132 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 4484 \beta_{15} + 4174 \beta_{14} + 4536 \beta_{13} - 783 \beta_{12} + 2916 \beta_{11} - 6966 \beta_{10} + \cdots + 12687 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 16333 \beta_{15} + 7817 \beta_{14} + 20545 \beta_{13} - 23176 \beta_{12} - 19388 \beta_{11} + \cdots + 23263 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 27156 \beta_{15} - 32829 \beta_{14} - 22339 \beta_{13} - 15398 \beta_{12} - 47929 \beta_{11} + \cdots - 96880 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 184180 \beta_{15} - 108713 \beta_{14} - 219897 \beta_{13} + 206481 \beta_{12} + 138096 \beta_{11} + \cdots - 317523 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 77602 \beta_{15} + 207089 \beta_{14} - 9023 \beta_{13} + 353180 \beta_{12} + 599431 \beta_{11} + \cdots + 606508 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 1845600 \beta_{15} + 1254318 \beta_{14} + 2095874 \beta_{13} - 1624184 \beta_{12} - 721732 \beta_{11} + \cdots + 3635045 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 1100006 \beta_{15} - 726293 \beta_{14} + 2193417 \beta_{13} - 5004327 \beta_{12} - 6467724 \beta_{11} + \cdots - 2166348 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(-\beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
647.1
0.500000 − 1.76390i
0.500000 − 0.911095i
0.500000 + 1.24626i
0.500000 + 1.42873i
0.500000 + 1.97090i
0.500000 − 3.05304i
0.500000 − 0.0390518i
0.500000 + 1.12119i
0.500000 + 1.76390i
0.500000 + 0.911095i
0.500000 − 1.24626i
0.500000 − 1.42873i
0.500000 − 1.97090i
0.500000 + 3.05304i
0.500000 + 0.0390518i
0.500000 − 1.12119i
−0.866025 + 0.500000i 0 0.500000 − 0.866025i −1.75446 − 3.03881i 0 2.30943 − 1.29094i 1.00000i 0 3.03881 + 1.75446i
647.2 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.330211 + 0.571943i 0 −2.34953 + 1.21644i 1.00000i 0 −0.571943 − 0.330211i
647.3 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.529713 + 0.917490i 0 1.22963 + 2.34265i 1.00000i 0 −0.917490 − 0.529713i
647.4 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.894533 + 1.54938i 0 2.54252 + 0.731847i 1.00000i 0 −1.54938 − 0.894533i
647.5 0.866025 − 0.500000i 0 0.500000 − 0.866025i −2.08560 − 3.61236i 0 1.94402 − 1.79465i − 1.00000i 0 −3.61236 − 2.08560i
647.6 0.866025 − 0.500000i 0 0.500000 − 0.866025i −0.360068 − 0.623656i 0 1.03982 − 2.43285i − 1.00000i 0 −0.623656 − 0.360068i
647.7 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0.860850 + 1.49104i 0 −2.25833 − 1.37838i − 1.00000i 0 1.49104 + 0.860850i
647.8 0.866025 − 0.500000i 0 0.500000 − 0.866025i 1.58481 + 2.74498i 0 −0.457557 + 2.60589i − 1.00000i 0 2.74498 + 1.58481i
971.1 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −1.75446 + 3.03881i 0 2.30943 + 1.29094i − 1.00000i 0 3.03881 − 1.75446i
971.2 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.330211 − 0.571943i 0 −2.34953 − 1.21644i − 1.00000i 0 −0.571943 + 0.330211i
971.3 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.529713 − 0.917490i 0 1.22963 − 2.34265i − 1.00000i 0 −0.917490 + 0.529713i
971.4 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.894533 − 1.54938i 0 2.54252 − 0.731847i − 1.00000i 0 −1.54938 + 0.894533i
971.5 0.866025 + 0.500000i 0 0.500000 + 0.866025i −2.08560 + 3.61236i 0 1.94402 + 1.79465i 1.00000i 0 −3.61236 + 2.08560i
971.6 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.360068 + 0.623656i 0 1.03982 + 2.43285i 1.00000i 0 −0.623656 + 0.360068i
971.7 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.860850 − 1.49104i 0 −2.25833 + 1.37838i 1.00000i 0 1.49104 − 0.860850i
971.8 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.58481 − 2.74498i 0 −0.457557 − 2.60589i 1.00000i 0 2.74498 − 1.58481i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 647.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1134.2.k.c ✓ 16
3.b odd 2 1 1134.2.k.d yes 16
7.d odd 6 1 1134.2.k.d yes 16
9.c even 3 1 1134.2.l.h 16
9.c even 3 1 1134.2.t.h 16
9.d odd 6 1 1134.2.l.g 16
9.d odd 6 1 1134.2.t.g 16
21.g even 6 1 inner 1134.2.k.c ✓ 16
63.i even 6 1 1134.2.t.h 16
63.k odd 6 1 1134.2.l.g 16
63.s even 6 1 1134.2.l.h 16
63.t odd 6 1 1134.2.t.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1134.2.k.c ✓ 16 1.a even 1 1 trivial
1134.2.k.c ✓ 16 21.g even 6 1 inner
1134.2.k.d yes 16 3.b odd 2 1
1134.2.k.d yes 16 7.d odd 6 1
1134.2.l.g 16 9.d odd 6 1
1134.2.l.g 16 63.k odd 6 1
1134.2.l.h 16 9.c even 3 1
1134.2.l.h 16 63.s even 6 1
1134.2.t.g 16 9.d odd 6 1
1134.2.t.g 16 63.t odd 6 1
1134.2.t.h 16 9.c even 3 1
1134.2.t.h 16 63.i even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 24 T_{5}^{14} - 48 T_{5}^{13} + 432 T_{5}^{12} - 864 T_{5}^{11} + 4176 T_{5}^{10} + \cdots + 5184 \) acting on \(S_{2}^{\mathrm{new}}(1134, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 24 T^{14} + \cdots + 5184 \) Copy content Toggle raw display
$7$ \( T^{16} - 8 T^{15} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{16} + 12 T^{15} + \cdots + 26244 \) Copy content Toggle raw display
$13$ \( T^{16} + 84 T^{14} + \cdots + 324 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 6324066576 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 2730480516 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 126617123889 \) Copy content Toggle raw display
$29$ \( T^{16} + 180 T^{14} + \cdots + 26244 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 48291381009 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 3838213866496 \) Copy content Toggle raw display
$41$ \( (T^{8} - 24 T^{7} + \cdots - 6471)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 16 T^{7} + \cdots - 2863688)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 218493609489 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 21321176960196 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 248817401856 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 4617126562500 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 1049630404 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 381069441 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 218493609489 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 1016646707521 \) Copy content Toggle raw display
$83$ \( (T^{8} - 36 T^{7} + \cdots + 320526)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 15178486401 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 74261306250000 \) Copy content Toggle raw display
show more
show less