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Properties

Label 810.3.g.g
Level $810$
Weight $3$
Character orbit 810.g
Analytic conductor $22.071$
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] = [810,3,Mod(163,810)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(810, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("810.163");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 810.g (of order \(4\), 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: \(22.0709014132\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 2 x^{10} + 32 x^{9} + 295 x^{8} - 228 x^{7} + 378 x^{6} + 5598 x^{5} + 25425 x^{4} + \cdots + 1296 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{2} - 1) q^{2} - 2 \beta_{2} q^{4} - \beta_1 q^{5} + (\beta_{9} - \beta_{2} + 1) q^{7} + (2 \beta_{2} + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 1) q^{2} - 2 \beta_{2} q^{4} - \beta_1 q^{5} + (\beta_{9} - \beta_{2} + 1) q^{7} + (2 \beta_{2} + 2) q^{8} + (\beta_{4} + \beta_1) q^{10} + (\beta_{11} + \beta_{9} + \beta_{8} + \cdots + 2) q^{11}+ \cdots + ( - 4 \beta_{10} - 6 \beta_{8} + \cdots - 13) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} - 6 q^{5} + 12 q^{7} + 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} - 6 q^{5} + 12 q^{7} + 24 q^{8} + 6 q^{10} + 6 q^{13} - 48 q^{16} + 30 q^{17} - 24 q^{23} + 78 q^{25} - 12 q^{26} - 24 q^{28} - 24 q^{31} + 48 q^{32} - 132 q^{35} + 18 q^{37} - 24 q^{38} - 12 q^{40} + 120 q^{41} - 204 q^{43} + 48 q^{46} - 126 q^{50} + 12 q^{52} - 180 q^{53} + 264 q^{55} + 48 q^{56} - 60 q^{58} + 96 q^{61} + 24 q^{62} - 372 q^{65} + 48 q^{67} - 60 q^{68} + 72 q^{70} + 576 q^{71} - 402 q^{73} + 48 q^{76} + 96 q^{77} + 24 q^{80} - 120 q^{82} - 192 q^{83} + 294 q^{85} + 408 q^{86} + 120 q^{91} - 48 q^{92} - 252 q^{95} - 192 q^{97} - 84 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 2 x^{10} + 32 x^{9} + 295 x^{8} - 228 x^{7} + 378 x^{6} + 5598 x^{5} + 25425 x^{4} + \cdots + 1296 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 55404634633 \nu^{11} + 866123519310 \nu^{10} - 1993854811058 \nu^{9} + \cdots + 26\!\cdots\!92 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 44836058361 \nu^{11} - 56101527824 \nu^{10} + 10844080498 \nu^{9} + 1529900552876 \nu^{8} + \cdots - 94788890387784 ) / 279920465543088 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 335430520309 \nu^{11} + 410508512792 \nu^{10} - 212292277274 \nu^{9} + \cdots - 13\!\cdots\!76 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 964393359419 \nu^{11} + 1813433687090 \nu^{10} - 2129202108306 \nu^{9} + \cdots + 70\!\cdots\!24 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 130870637833 \nu^{11} + 166883624289 \nu^{10} - 33480599074 \nu^{9} + \cdots - 364483477379322 ) / 174950290964430 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1077749873791 \nu^{11} - 2292929173022 \nu^{10} + 2574370577594 \nu^{9} + \cdots + 33\!\cdots\!24 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 76233 \nu^{11} + 174748 \nu^{10} - 158922 \nu^{9} - 2508884 \nu^{8} - 21435927 \nu^{7} + \cdots + 351194616 ) / 90789360 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1260695390577 \nu^{11} - 1904259829474 \nu^{10} + 639125323598 \nu^{9} + \cdots - 57\!\cdots\!32 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1506307380905 \nu^{11} + 3143559073288 \nu^{10} - 3380292717058 \nu^{9} + \cdots + 47\!\cdots\!92 ) / 13\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 258887 \nu^{11} - 545014 \nu^{10} + 678626 \nu^{9} + 8013272 \nu^{8} + 75337273 \nu^{7} + \cdots - 451516248 ) / 205545720 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1811136339749 \nu^{11} + 3130200920052 \nu^{10} - 2453935907642 \nu^{9} + \cdots + 86\!\cdots\!68 ) / 13\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} + \beta_{8} + \beta_{7} - 2\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - 2\beta _1 + 3 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} - 2\beta_{9} - 5\beta_{8} - 3\beta_{6} + \beta_{5} + 3\beta_{4} - 2\beta_{3} + 53\beta_{2} - 5\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 8 \beta_{11} + 13 \beta_{10} - 8 \beta_{9} - 28 \beta_{8} - 13 \beta_{7} - 14 \beta_{6} + 28 \beta_{4} + \cdots - 75 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 14 \beta_{11} - 43 \beta_{9} - 72 \beta_{8} - 20 \beta_{7} + 70 \beta_{6} - 14 \beta_{5} + 70 \beta_{4} + \cdots - 780 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 216 \beta_{10} - 205 \beta_{8} - 216 \beta_{7} + 440 \beta_{6} + 61 \beta_{5} - 205 \beta_{4} + \cdots - 1422 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 125 \beta_{11} - 295 \beta_{10} + 370 \beta_{9} + 490 \beta_{8} + 705 \beta_{6} - 125 \beta_{5} + \cdots + 490 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 101 \beta_{11} - 3701 \beta_{10} + 2231 \beta_{9} + 7132 \beta_{8} + 3701 \beta_{7} + 3371 \beta_{6} + \cdots + 25323 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 4856 \beta_{11} + 12187 \beta_{9} + 25908 \beta_{8} + 13560 \beta_{7} - 14020 \beta_{6} + \cdots + 189594 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 64523 \beta_{10} + 58864 \beta_{8} + 64523 \beta_{7} - 117458 \beta_{6} + 10952 \beta_{5} + \cdots + 440835 ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 95809 \beta_{11} + 281470 \beta_{10} - 200018 \beta_{9} - 203375 \beta_{8} - 464607 \beta_{6} + \cdots - 203375 \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 365479 \beta_{11} + 1137756 \beta_{10} - 799691 \beta_{9} - 1955080 \beta_{8} - 1137756 \beta_{7} + \cdots - 7604892 ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/810\mathbb{Z}\right)^\times\).

\(n\) \(487\) \(731\)
\(\chi(n)\) \(-\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
−2.47258 − 2.47258i
−1.80850 − 1.80850i
0.204429 + 0.204429i
−0.610094 − 0.610094i
2.71665 + 2.71665i
2.97010 + 2.97010i
−2.47258 + 2.47258i
−1.80850 + 1.80850i
0.204429 − 0.204429i
−0.610094 + 0.610094i
2.71665 − 2.71665i
2.97010 − 2.97010i
−1.00000 + 1.00000i 0 − 2.00000i −4.98825 + 0.342590i 0 9.30550 − 9.30550i 2.00000 + 2.00000i 0 4.64566 − 5.33084i
163.2 −1.00000 + 1.00000i 0 − 2.00000i −4.26646 + 2.60717i 0 −4.48912 + 4.48912i 2.00000 + 2.00000i 0 1.65928 − 6.87363i
163.3 −1.00000 + 1.00000i 0 − 2.00000i −3.69234 − 3.37144i 0 3.88795 − 3.88795i 2.00000 + 2.00000i 0 7.06378 − 0.320906i
163.4 −1.00000 + 1.00000i 0 − 2.00000i 1.65794 − 4.71712i 0 −1.62535 + 1.62535i 2.00000 + 2.00000i 0 3.05918 + 6.37506i
163.5 −1.00000 + 1.00000i 0 − 2.00000i 3.56236 + 3.50851i 0 −1.21604 + 1.21604i 2.00000 + 2.00000i 0 −7.07086 + 0.0538519i
163.6 −1.00000 + 1.00000i 0 − 2.00000i 4.72675 + 1.63029i 0 0.137067 − 0.137067i 2.00000 + 2.00000i 0 −6.35704 + 3.09646i
487.1 −1.00000 − 1.00000i 0 2.00000i −4.98825 − 0.342590i 0 9.30550 + 9.30550i 2.00000 − 2.00000i 0 4.64566 + 5.33084i
487.2 −1.00000 − 1.00000i 0 2.00000i −4.26646 − 2.60717i 0 −4.48912 − 4.48912i 2.00000 − 2.00000i 0 1.65928 + 6.87363i
487.3 −1.00000 − 1.00000i 0 2.00000i −3.69234 + 3.37144i 0 3.88795 + 3.88795i 2.00000 − 2.00000i 0 7.06378 + 0.320906i
487.4 −1.00000 − 1.00000i 0 2.00000i 1.65794 + 4.71712i 0 −1.62535 − 1.62535i 2.00000 − 2.00000i 0 3.05918 − 6.37506i
487.5 −1.00000 − 1.00000i 0 2.00000i 3.56236 − 3.50851i 0 −1.21604 − 1.21604i 2.00000 − 2.00000i 0 −7.07086 − 0.0538519i
487.6 −1.00000 − 1.00000i 0 2.00000i 4.72675 − 1.63029i 0 0.137067 + 0.137067i 2.00000 − 2.00000i 0 −6.35704 − 3.09646i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 810.3.g.g ✓ 12
3.b odd 2 1 810.3.g.l yes 12
5.c odd 4 1 inner 810.3.g.g ✓ 12
15.e even 4 1 810.3.g.l yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
810.3.g.g ✓ 12 1.a even 1 1 trivial
810.3.g.g ✓ 12 5.c odd 4 1 inner
810.3.g.l yes 12 3.b odd 2 1
810.3.g.l yes 12 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(810, [\chi])\):

\( T_{7}^{12} - 12 T_{7}^{11} + 72 T_{7}^{10} + 736 T_{7}^{9} + 4860 T_{7}^{8} - 13632 T_{7}^{7} + \cdots + 123904 \) Copy content Toggle raw display
\( T_{11}^{6} - 426T_{11}^{4} + 820T_{11}^{3} + 37869T_{11}^{2} - 37260T_{11} - 461192 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 2)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 244140625 \) Copy content Toggle raw display
$7$ \( T^{12} - 12 T^{11} + \cdots + 123904 \) Copy content Toggle raw display
$11$ \( (T^{6} - 426 T^{4} + \cdots - 461192)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 21751702032384 \) Copy content Toggle raw display
$17$ \( T^{12} - 30 T^{11} + \cdots + 913936 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 169935221824 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 64\!\cdots\!89 \) Copy content Toggle raw display
$31$ \( (T^{6} + 12 T^{5} + \cdots - 75585968)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 50\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( (T^{6} - 60 T^{5} + \cdots + 3022665508)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 259675950007296 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 36\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 32\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 88\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{6} - 48 T^{5} + \cdots + 2512955664)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( (T^{6} - 288 T^{5} + \cdots - 18772302536)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 31\!\cdots\!21 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
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