Properties

Label 927.2.o.a
Level $927$
Weight $2$
Character orbit 927.o
Analytic conductor $7.402$
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] = [927,2,Mod(665,927)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(927, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("927.665");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 927 = 3^{2} \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 927.o (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: \(7.40213226737\)
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: \( x^{16} - 12x^{14} + 97x^{12} - 444x^{10} + 1488x^{8} - 2796x^{6} + 3553x^{4} - 60x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{12} + \beta_{11} + \beta_{10} + 1) q^{4} + (\beta_{14} - \beta_{6}) q^{5} + ( - 2 \beta_{12} + \beta_{10} - \beta_{7} + 1) q^{7} - \beta_{13} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{12} + \beta_{11} + \beta_{10} + 1) q^{4} + (\beta_{14} - \beta_{6}) q^{5} + ( - 2 \beta_{12} + \beta_{10} - \beta_{7} + 1) q^{7} - \beta_{13} q^{8} + ( - 2 \beta_{11} - 2 \beta_{10} - \beta_{5} + \beta_{4} - 1) q^{10} + ( - \beta_{15} + \beta_{8} - \beta_{6} + \beta_{3}) q^{11} + ( - \beta_{5} - \beta_{2} + 3) q^{13} + (\beta_{15} - 3 \beta_{13} - \beta_{9}) q^{14} + ( - 2 \beta_{12} - \beta_{11} - \beta_{10} - \beta_{7} + \beta_{5} + \beta_{4} + \beta_{2}) q^{16} + (\beta_{15} + 2 \beta_{14} - 2 \beta_{13} - 2 \beta_{9} + \beta_{8} + \beta_{6} - \beta_{3} + 2 \beta_1) q^{17} + (3 \beta_{10} - \beta_{7} + \beta_{2}) q^{19} + ( - 2 \beta_{15} - 2 \beta_{14} + 2 \beta_{13} - 2 \beta_{9} - \beta_{8} + \beta_1) q^{20} + (2 \beta_{12} - 2 \beta_{11} - 2 \beta_{7} - 2 \beta_{5} + \beta_{4} + \beta_{2}) q^{22} + (3 \beta_{14} - 3 \beta_{13} + 2 \beta_{9}) q^{23} + ( - 3 \beta_{12} + \beta_{10} + 3 \beta_{5}) q^{25} + ( - \beta_{6} + 3 \beta_{3} + 2 \beta_1) q^{26} + (2 \beta_{12} + 3 \beta_{11} - 2 \beta_{10} + \beta_{5} - 3 \beta_{4}) q^{28} + ( - \beta_{8} + 2 \beta_{6} - 2 \beta_{3} - 2 \beta_1) q^{29} + ( - 4 \beta_{10} - 2) q^{31} + (\beta_{14} - 3 \beta_{9} + \beta_{6} + 3 \beta_1) q^{32} + (2 \beta_{5} + \beta_{4} + \beta_{2} + 6) q^{34} + (2 \beta_{15} + 8 \beta_{14} - 4 \beta_{13} + \beta_{8} + 3 \beta_{6} - \beta_{3}) q^{35} + (4 \beta_{12} + 4 \beta_{11} + 4 \beta_{10} - 2 \beta_{4} + 2) q^{37} + ( - \beta_{15} + 2 \beta_{14} - 3 \beta_{13} + 3 \beta_{9} - \beta_{8} + 3 \beta_{6} + \cdots - 3 \beta_1) q^{38}+ \cdots + ( - 2 \beta_{15} + 12 \beta_{13} + 2 \beta_{9} - 2 \beta_{8} + 2 \beta_{6} + \cdots - 2 \beta_1) q^{98}+O(q^{100}) \) 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 \( 16 q + 8 q^{4} + 4 q^{7} + 40 q^{13} + 12 q^{16} - 20 q^{19} - 8 q^{25} + 16 q^{28} + 104 q^{34} + 24 q^{40} + 36 q^{43} - 36 q^{46} - 40 q^{49} + 12 q^{52} - 12 q^{55} - 40 q^{58} - 56 q^{61} + 112 q^{64} + 48 q^{67} + 60 q^{70} - 48 q^{76} + 112 q^{79} - 24 q^{82} - 84 q^{85} - 12 q^{88} + 68 q^{91} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 12x^{14} + 97x^{12} - 444x^{10} + 1488x^{8} - 2796x^{6} + 3553x^{4} - 60x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1551 \nu^{14} - 31123 \nu^{12} + 126720 \nu^{10} - 496980 \nu^{8} - 99276 \nu^{6} - 1405008 \nu^{4} + 23727 \nu^{2} - 1498291 ) / 8150064 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3760 \nu^{15} + 39433 \nu^{13} - 307200 \nu^{11} + 1204800 \nu^{9} - 3772620 \nu^{7} + 3406080 \nu^{5} - 57520 \nu^{3} - 32461655 \nu ) / 8150064 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3760 \nu^{14} + 39433 \nu^{12} - 307200 \nu^{10} + 1204800 \nu^{8} - 3772620 \nu^{6} + 3406080 \nu^{4} - 57520 \nu^{2} - 24311591 ) / 8150064 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 141 \nu^{14} + 1642 \nu^{12} - 11520 \nu^{10} + 45180 \nu^{8} - 112086 \nu^{6} + 127728 \nu^{4} - 2157 \nu^{2} - 135698 ) / 78366 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 24393 \nu^{15} + 257944 \nu^{13} - 1992960 \nu^{11} + 7816140 \nu^{9} - 23074080 \nu^{7} + 22096944 \nu^{5} - 373161 \nu^{3} - 121145912 \nu ) / 8150064 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 29085 \nu^{14} + 315641 \nu^{12} - 2549712 \nu^{10} + 11400444 \nu^{8} - 38999004 \nu^{6} + 72993408 \nu^{4} - 89922237 \nu^{2} + 20633 ) / 8150064 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 59690 \nu^{15} + 647223 \nu^{13} - 4876800 \nu^{11} + 19126200 \nu^{9} - 54032484 \nu^{7} + 54071520 \nu^{5} - 913130 \nu^{3} - 232092825 \nu ) / 8150064 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 138601 \nu^{15} - 1659452 \nu^{13} + 13404864 \nu^{11} - 61231644 \nu^{9} + 205033488 \nu^{7} - 383755776 \nu^{5} + 489043273 \nu^{3} - 108476 \nu ) / 8150064 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 138601 \nu^{14} - 1659452 \nu^{12} + 13404864 \nu^{10} - 61231644 \nu^{8} + 205033488 \nu^{6} - 383755776 \nu^{4} + 489043273 \nu^{2} - 8258540 ) / 8150064 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 173373 \nu^{14} + 2032613 \nu^{12} - 16419216 \nu^{10} + 74454348 \nu^{8} - 251139372 \nu^{6} + 470050944 \nu^{4} - 611652765 \nu^{2} + \cdots + 132869 ) / 8150064 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 242430 \nu^{14} + 2945743 \nu^{12} - 23795376 \nu^{10} + 109240584 \nu^{8} - 363961092 \nu^{6} + 681216384 \nu^{4} - 847326990 \nu^{2} + \cdots + 192559 ) / 8150064 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 138601 \nu^{15} - 1659452 \nu^{13} + 13404864 \nu^{11} - 61231644 \nu^{9} + 205033488 \nu^{7} - 383755776 \nu^{5} + 487005757 \nu^{3} - 108476 \nu ) / 2037516 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 518332 \nu^{15} - 6208565 \nu^{13} + 50152080 \nu^{11} - 229163100 \nu^{9} + 767098860 \nu^{7} - 1435758720 \nu^{5} + 1817590396 \nu^{3} + \cdots - 405845 \nu ) / 2037516 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 2454359 \nu^{15} - 29439455 \nu^{13} + 237808560 \nu^{11} - 1087124628 \nu^{9} + 3637390020 \nu^{7} - 6808007040 \nu^{5} + 8606731847 \nu^{3} + \cdots - 1924415 \nu ) / 8150064 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} + \beta_{11} + 3\beta_{10} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{13} + 4\beta_{9} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{12} + 5\beta_{11} + 13\beta_{10} - \beta_{7} + \beta_{5} - 5\beta_{4} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{14} - 8\beta_{13} + 17\beta_{9} + \beta_{6} - 8\beta_{3} - 17\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7\beta_{5} - 24\beta_{4} + 8\beta_{2} - 58 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -\beta_{8} + 10\beta_{6} - 49\beta_{3} - 75\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -76\beta_{12} - 113\beta_{11} - 264\beta_{10} + 49\beta_{7} - 264 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 12\beta_{15} - 61\beta_{14} + 260\beta_{13} - 340\beta_{9} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -352\beta_{12} - 527\beta_{11} - 1219\beta_{10} + 272\beta_{7} - 175\beta_{5} + 527\beta_{4} - 272\beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 97 \beta_{15} - 369 \beta_{14} + 1343 \beta_{13} - 1571 \beta_{9} + 97 \beta_{8} - 466 \beta_{6} + 1440 \beta_{3} + 1571 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -780\beta_{5} + 2448\beta_{4} - 1440\beta_{2} + 5687 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 660\beta_{8} - 2760\beta_{6} + 7428\beta_{3} + 7355\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 8015\beta_{12} + 11363\beta_{11} + 26733\beta_{10} - 7428\beta_{7} + 26733 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( -4080\beta_{15} + 11508\beta_{14} - 33647\beta_{13} + 34748\beta_{9} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/927\mathbb{Z}\right)^\times\).

\(n\) \(722\) \(829\)
\(\chi(n)\) \(-1\) \(-\beta_{10}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
665.1
−1.91212 1.10397i
−1.78558 1.03090i
−1.46384 0.845147i
−0.112547 0.0649791i
0.112547 + 0.0649791i
1.46384 + 0.845147i
1.78558 + 1.03090i
1.91212 + 1.10397i
−1.91212 + 1.10397i
−1.78558 + 1.03090i
−1.46384 + 0.845147i
−0.112547 + 0.0649791i
0.112547 0.0649791i
1.46384 0.845147i
1.78558 1.03090i
1.91212 1.10397i
−1.91212 1.10397i 0 1.43748 + 2.48979i 0.448288 + 0.776457i 0 0.329192 + 0.570178i 1.93185i 0 1.97958i
665.2 −1.78558 1.03090i 0 1.12553 + 1.94948i 1.67303 + 2.89778i 0 0.774131 + 1.34083i 0.517638i 0 6.89895i
665.3 −1.46384 0.845147i 0 0.428545 + 0.742262i −0.448288 0.776457i 0 −2.42727 4.20415i 1.93185i 0 1.51548i
665.4 −0.112547 0.0649791i 0 −0.991555 1.71742i −1.67303 2.89778i 0 2.32395 + 4.02519i 0.517638i 0 0.434849i
665.5 0.112547 + 0.0649791i 0 −0.991555 1.71742i 1.67303 + 2.89778i 0 2.32395 + 4.02519i 0.517638i 0 0.434849i
665.6 1.46384 + 0.845147i 0 0.428545 + 0.742262i 0.448288 + 0.776457i 0 −2.42727 4.20415i 1.93185i 0 1.51548i
665.7 1.78558 + 1.03090i 0 1.12553 + 1.94948i −1.67303 2.89778i 0 0.774131 + 1.34083i 0.517638i 0 6.89895i
665.8 1.91212 + 1.10397i 0 1.43748 + 2.48979i −0.448288 0.776457i 0 0.329192 + 0.570178i 1.93185i 0 1.97958i
881.1 −1.91212 + 1.10397i 0 1.43748 2.48979i 0.448288 0.776457i 0 0.329192 0.570178i 1.93185i 0 1.97958i
881.2 −1.78558 + 1.03090i 0 1.12553 1.94948i 1.67303 2.89778i 0 0.774131 1.34083i 0.517638i 0 6.89895i
881.3 −1.46384 + 0.845147i 0 0.428545 0.742262i −0.448288 + 0.776457i 0 −2.42727 + 4.20415i 1.93185i 0 1.51548i
881.4 −0.112547 + 0.0649791i 0 −0.991555 + 1.71742i −1.67303 + 2.89778i 0 2.32395 4.02519i 0.517638i 0 0.434849i
881.5 0.112547 0.0649791i 0 −0.991555 + 1.71742i 1.67303 2.89778i 0 2.32395 4.02519i 0.517638i 0 0.434849i
881.6 1.46384 0.845147i 0 0.428545 0.742262i 0.448288 0.776457i 0 −2.42727 + 4.20415i 1.93185i 0 1.51548i
881.7 1.78558 1.03090i 0 1.12553 1.94948i −1.67303 + 2.89778i 0 0.774131 1.34083i 0.517638i 0 6.89895i
881.8 1.91212 1.10397i 0 1.43748 2.48979i −0.448288 + 0.776457i 0 0.329192 0.570178i 1.93185i 0 1.97958i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 665.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
103.d odd 6 1 inner
309.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 927.2.o.a 16
3.b odd 2 1 inner 927.2.o.a 16
103.d odd 6 1 inner 927.2.o.a 16
309.g even 6 1 inner 927.2.o.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
927.2.o.a 16 1.a even 1 1 trivial
927.2.o.a 16 3.b odd 2 1 inner
927.2.o.a 16 103.d odd 6 1 inner
927.2.o.a 16 309.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 12T_{2}^{14} + 97T_{2}^{12} - 444T_{2}^{10} + 1488T_{2}^{8} - 2796T_{2}^{6} + 3553T_{2}^{4} - 60T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(927, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 12 T^{14} + 97 T^{12} - 444 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + 12 T^{6} + 135 T^{4} + 108 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} - 2 T^{7} + 26 T^{6} - 56 T^{5} + \cdots + 529)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + 72 T^{14} + \cdots + 187388721 \) Copy content Toggle raw display
$13$ \( (T^{4} - 10 T^{3} + 26 T^{2} + 4 T - 44)^{4} \) Copy content Toggle raw display
$17$ \( T^{16} - 96 T^{14} + \cdots + 27439591201 \) Copy content Toggle raw display
$19$ \( (T^{8} + 10 T^{7} + 74 T^{6} + 232 T^{5} + \cdots + 1)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 96 T^{6} + 1592 T^{4} + 1152 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} - 72 T^{14} + 4630 T^{12} + \cdots + 28561 \) Copy content Toggle raw display
$31$ \( (T^{2} + 12)^{8} \) Copy content Toggle raw display
$37$ \( (T^{8} + 216 T^{6} + 15408 T^{4} + \cdots + 2509056)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} - 168 T^{14} + \cdots + 43617904801 \) Copy content Toggle raw display
$43$ \( (T^{8} - 18 T^{7} + 42 T^{6} + \cdots + 2259009)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 120 T^{14} + \cdots + 96059601 \) Copy content Toggle raw display
$53$ \( T^{16} + 192 T^{14} + \cdots + 5103121662081 \) Copy content Toggle raw display
$59$ \( T^{16} - 96 T^{14} + \cdots + 607573201 \) Copy content Toggle raw display
$61$ \( (T^{4} + 14 T^{3} + 50 T^{2} - 20 T - 236)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} - 24 T^{7} + 126 T^{6} + \cdots + 1390041)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 36 T^{6} + 1215 T^{4} + \cdots + 6561)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 432 T^{6} + 61416 T^{4} + \cdots + 41525136)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 28 T^{3} + 200 T^{2} + 160 T - 3776)^{4} \) Copy content Toggle raw display
$83$ \( T^{16} - 648 T^{14} + \cdots + 91\!\cdots\!41 \) Copy content Toggle raw display
$89$ \( (T^{4} - 144 T^{2} + 1296)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + 16 T^{7} + 242 T^{6} + 256 T^{5} + \cdots + 1)^{2} \) Copy content Toggle raw display
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