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Properties

Label 1134.2.m.h
Level $1134$
Weight $2$
Character orbit 1134.m
Analytic conductor $9.055$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1134,2,Mod(377,1134)]
 
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([5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1134.377");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1134.m (of order \(6\), degree \(2\), not 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: \( x^{16} - 8 x^{15} + 56 x^{14} - 252 x^{13} + 962 x^{12} - 2860 x^{11} + 7240 x^{10} - 15036 x^{9} + \cdots + 457 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} + \beta_{2} q^{4} - \beta_{14} q^{5} + ( - \beta_{10} - \beta_{8} + \beta_{7} + \cdots - 1) q^{7}+ \cdots + ( - \beta_{8} + \beta_{7}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{2} + \beta_{2} q^{4} - \beta_{14} q^{5} + ( - \beta_{10} - \beta_{8} + \beta_{7} + \cdots - 1) q^{7}+ \cdots + (2 \beta_{11} + 2 \beta_{10} + \cdots + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} - 4 q^{7} - 24 q^{11} - 12 q^{14} - 8 q^{16} - 24 q^{23} - 8 q^{25} - 8 q^{28} + 48 q^{29} + 16 q^{37} - 32 q^{43} + 16 q^{49} - 24 q^{50} - 12 q^{56} + 24 q^{58} - 16 q^{64} + 96 q^{65} + 8 q^{67} + 12 q^{70} - 72 q^{74} - 24 q^{77} - 16 q^{79} + 24 q^{85} - 48 q^{86} + 24 q^{91} - 24 q^{92} + 168 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 56 x^{14} - 252 x^{13} + 962 x^{12} - 2860 x^{11} + 7240 x^{10} - 15036 x^{9} + \cdots + 457 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 849 \nu^{14} - 5943 \nu^{13} + 40084 \nu^{12} - 163245 \nu^{11} + 582011 \nu^{10} - 1555284 \nu^{9} + \cdots + 221216 ) / 1965 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 45568 \nu^{15} - 341760 \nu^{14} + 2354508 \nu^{13} - 10120942 \nu^{12} + 37543452 \nu^{11} + \cdots - 23435204 ) / 2595765 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 68799 \nu^{15} - 442677 \nu^{14} + 2983455 \nu^{13} - 11365449 \nu^{12} + 39544909 \nu^{11} + \cdots + 3912758 ) / 2595765 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 148419 \nu^{15} - 1665981 \nu^{14} + 11316639 \nu^{13} - 57523244 \nu^{12} + 218085836 \nu^{11} + \cdots - 126707833 ) / 2595765 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 179429 \nu^{15} + 1629072 \nu^{14} - 11088172 \nu^{13} + 52131694 \nu^{12} - 194853821 \nu^{11} + \cdots + 76329941 ) / 2595765 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 179429 \nu^{15} + 1062363 \nu^{14} - 7121209 \nu^{13} + 25409185 \nu^{12} - 86089286 \nu^{11} + \cdots - 6306535 ) / 2595765 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 244350 \nu^{15} - 2351778 \nu^{14} + 16014966 \nu^{13} - 77254247 \nu^{12} + 290700834 \nu^{11} + \cdots - 173876339 ) / 2595765 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 244350 \nu^{15} + 1313472 \nu^{14} - 8746824 \nu^{13} + 28107763 \nu^{12} - 90307776 \nu^{11} + \cdots - 123425279 ) / 2595765 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 244350 \nu^{15} + 436328 \nu^{14} - 2606816 \nu^{13} - 13226327 \nu^{12} + 77876660 \nu^{11} + \cdots - 301560808 ) / 2595765 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 261214 \nu^{15} - 2605074 \nu^{14} + 17737587 \nu^{13} - 86696147 \nu^{12} + 327072198 \nu^{11} + \cdots - 209400939 ) / 2595765 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 261214 \nu^{15} - 1313136 \nu^{14} + 8694021 \nu^{13} - 25683120 \nu^{12} + 78560394 \nu^{11} + \cdots + 145147535 ) / 2595765 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 262786 \nu^{15} - 2597049 \nu^{14} + 17681019 \nu^{13} - 86083590 \nu^{12} + 324274506 \nu^{11} + \cdots - 160485190 ) / 2595765 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 262786 \nu^{15} - 1344741 \nu^{14} + 8914863 \nu^{13} - 27005828 \nu^{12} + 83767962 \nu^{11} + \cdots + 144235164 ) / 2595765 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 841689 \nu^{15} - 6873432 \nu^{14} + 46840527 \nu^{13} - 209681999 \nu^{12} + 775547226 \nu^{11} + \cdots - 311828236 ) / 2595765 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 957908 \nu^{15} - 8161850 \nu^{14} + 55556621 \nu^{13} - 253885851 \nu^{12} + 942739159 \nu^{11} + \cdots - 421652127 ) / 2595765 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 4 \beta_{15} + 2 \beta_{14} + 3 \beta_{13} - \beta_{12} - 3 \beta_{11} + \beta_{10} + 2 \beta_{9} + \cdots + 2 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2 \beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} + 2 \beta_{11} - 3 \beta_{10} + \beta_{9} + \cdots - 10 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8 \beta_{15} - 5 \beta_{14} - 5 \beta_{13} + 6 \beta_{12} + 11 \beta_{11} - 4 \beta_{10} - 4 \beta_{9} + \cdots - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 26 \beta_{15} - 16 \beta_{14} - 2 \beta_{13} + 18 \beta_{12} - 10 \beta_{11} + 32 \beta_{10} - 4 \beta_{9} + \cdots + 54 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 142 \beta_{15} + 107 \beta_{14} + 114 \beta_{13} - 193 \beta_{12} - 354 \beta_{11} + 163 \beta_{10} + \cdots + 290 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 93 \beta_{15} + 67 \beta_{14} - 8 \beta_{13} - 92 \beta_{12} - 16 \beta_{11} - 98 \beta_{10} + \cdots - 89 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 762 \beta_{15} - 654 \beta_{14} - 1249 \beta_{13} + 1715 \beta_{12} + 3601 \beta_{11} - 2173 \beta_{10} + \cdots - 3028 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2888 \beta_{15} - 2284 \beta_{14} + 148 \beta_{13} + 3780 \beta_{12} + 2216 \beta_{11} + 2288 \beta_{10} + \cdots + 738 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 702 \beta_{15} + 729 \beta_{14} + 4821 \beta_{13} - 3900 \beta_{12} - 11085 \beta_{11} + 9060 \beta_{10} + \cdots + 9802 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 29011 \beta_{15} + 24086 \beta_{14} + 3575 \beta_{13} - 46145 \beta_{12} - 39458 \beta_{11} + \cdots + 8008 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 32220 \beta_{15} + 24093 \beta_{14} - 153892 \beta_{13} + 35351 \beta_{12} + 268468 \beta_{11} + \cdots - 269764 ) / 6 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 92230 \beta_{15} - 78348 \beta_{14} - 36246 \beta_{13} + 168646 \beta_{12} + 181512 \beta_{11} + \cdots - 71115 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 850318 \beta_{15} - 706472 \beta_{14} + 1449249 \beta_{13} + 628963 \beta_{12} - 1710081 \beta_{11} + \cdots + 2285374 ) / 6 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 2424790 \beta_{15} + 2080514 \beta_{14} + 1862285 \beta_{13} - 4999289 \beta_{12} - 6522056 \beta_{11} + \cdots + 3290077 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 4479556 \beta_{15} + 3804055 \beta_{14} - 3891197 \beta_{13} - 5521158 \beta_{12} + 1587617 \beta_{11} + \cdots - 5615374 ) / 2 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
377.1
0.500000 − 0.221083i
0.500000 − 2.18553i
0.500000 + 3.18553i
0.500000 + 1.22108i
0.500000 + 0.612384i
0.500000 + 1.24085i
0.500000 − 2.24085i
0.500000 − 1.61238i
0.500000 + 0.221083i
0.500000 + 2.18553i
0.500000 − 3.18553i
0.500000 − 1.22108i
0.500000 − 0.612384i
0.500000 − 1.24085i
0.500000 + 2.24085i
0.500000 + 1.61238i
−0.866025 + 0.500000i 0 0.500000 − 0.866025i −1.23097 + 2.13209i 0 −1.03549 + 2.43470i 1.00000i 0 − 2.46193i
377.2 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −0.786575 + 1.36239i 0 1.73789 − 1.99493i 1.00000i 0 − 1.57315i
377.3 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.786575 − 1.36239i 0 −2.59660 + 0.507587i 1.00000i 0 1.57315i
377.4 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 1.23097 − 2.13209i 0 2.62626 + 0.320594i 1.00000i 0 2.46193i
377.5 0.866025 − 0.500000i 0 0.500000 − 0.866025i −1.89896 + 3.28909i 0 −0.830700 − 2.51196i − 1.00000i 0 3.79792i
377.6 0.866025 − 0.500000i 0 0.500000 − 0.866025i −0.509882 + 0.883142i 0 2.21738 − 1.44334i − 1.00000i 0 1.01976i
377.7 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0.509882 − 0.883142i 0 −2.35866 + 1.19864i − 1.00000i 0 − 1.01976i
377.8 0.866025 − 0.500000i 0 0.500000 − 0.866025i 1.89896 − 3.28909i 0 −1.76007 − 1.97539i − 1.00000i 0 − 3.79792i
755.1 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −1.23097 − 2.13209i 0 −1.03549 − 2.43470i − 1.00000i 0 2.46193i
755.2 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −0.786575 − 1.36239i 0 1.73789 + 1.99493i − 1.00000i 0 1.57315i
755.3 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.786575 + 1.36239i 0 −2.59660 − 0.507587i − 1.00000i 0 − 1.57315i
755.4 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 1.23097 + 2.13209i 0 2.62626 − 0.320594i − 1.00000i 0 − 2.46193i
755.5 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.89896 − 3.28909i 0 −0.830700 + 2.51196i 1.00000i 0 − 3.79792i
755.6 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.509882 − 0.883142i 0 2.21738 + 1.44334i 1.00000i 0 − 1.01976i
755.7 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.509882 + 0.883142i 0 −2.35866 − 1.19864i 1.00000i 0 1.01976i
755.8 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.89896 + 3.28909i 0 −1.76007 + 1.97539i 1.00000i 0 3.79792i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 377.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
9.d odd 6 1 inner
63.o 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.m.h 16
3.b odd 2 1 1134.2.m.i 16
7.b odd 2 1 inner 1134.2.m.h 16
9.c even 3 1 1134.2.d.b ✓ 16
9.c even 3 1 1134.2.m.i 16
9.d odd 6 1 1134.2.d.b ✓ 16
9.d odd 6 1 inner 1134.2.m.h 16
21.c even 2 1 1134.2.m.i 16
63.l odd 6 1 1134.2.d.b ✓ 16
63.l odd 6 1 1134.2.m.i 16
63.o even 6 1 1134.2.d.b ✓ 16
63.o even 6 1 inner 1134.2.m.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1134.2.d.b ✓ 16 9.c even 3 1
1134.2.d.b ✓ 16 9.d odd 6 1
1134.2.d.b ✓ 16 63.l odd 6 1
1134.2.d.b ✓ 16 63.o even 6 1
1134.2.m.h 16 1.a even 1 1 trivial
1134.2.m.h 16 7.b odd 2 1 inner
1134.2.m.h 16 9.d odd 6 1 inner
1134.2.m.h 16 63.o even 6 1 inner
1134.2.m.i 16 3.b odd 2 1
1134.2.m.i 16 9.c even 3 1
1134.2.m.i 16 21.c even 2 1
1134.2.m.i 16 63.l odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1134, [\chi])\):

\( T_{5}^{16} + 24 T_{5}^{14} + 414 T_{5}^{12} + 3168 T_{5}^{10} + 17379 T_{5}^{8} + 47520 T_{5}^{6} + \cdots + 50625 \) Copy content Toggle raw display
\( T_{11}^{8} + 12T_{11}^{7} + 54T_{11}^{6} + 72T_{11}^{5} - 90T_{11}^{4} - 216T_{11}^{3} + 324T_{11}^{2} + 648T_{11} + 324 \) Copy content Toggle raw display
\( T_{13}^{16} - 84 T_{13}^{14} + 4554 T_{13}^{12} - 149328 T_{13}^{10} + 3585699 T_{13}^{8} + \cdots + 14166950625 \) 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
$5$ \( T^{16} + 24 T^{14} + \cdots + 50625 \) Copy content Toggle raw display
$7$ \( T^{16} + 4 T^{15} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{8} + 12 T^{7} + \cdots + 324)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 14166950625 \) Copy content Toggle raw display
$17$ \( (T^{8} - 60 T^{6} + \cdots + 225)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 156 T^{6} + \cdots + 476100)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 12 T^{7} + \cdots + 324)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 24 T^{7} + \cdots + 42849)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 5314410000 \) Copy content Toggle raw display
$37$ \( (T^{4} - 4 T^{3} - 84 T^{2} + \cdots + 73)^{4} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 74711820960000 \) Copy content Toggle raw display
$43$ \( (T^{8} + 16 T^{7} + \cdots + 5456896)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 226671210000 \) Copy content Toggle raw display
$53$ \( (T^{4} + 72 T^{2} + 324)^{4} \) Copy content Toggle raw display
$59$ \( T^{16} + 60 T^{14} + \cdots + 810000 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 247033850625 \) Copy content Toggle raw display
$67$ \( (T^{8} - 4 T^{7} + \cdots + 58564)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 360 T^{6} + \cdots + 11451456)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 168 T^{6} + \cdots + 225)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 8 T^{7} + \cdots + 20164)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 71708717610000 \) Copy content Toggle raw display
$89$ \( (T^{8} - 540 T^{6} + \cdots + 9641025)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} - 120 T^{14} + \cdots + 12960000 \) Copy content Toggle raw display
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