Properties

Label 810.3.g.i
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: \( x^{12} - 54 x^{9} + 921 x^{8} - 1350 x^{7} + 1458 x^{6} - 18792 x^{5} + 231804 x^{4} - 552420 x^{3} + 583200 x^{2} + 874800 x + 656100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: no (minimal twist has level 90)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{2} + 2 \beta_1 q^{4} + (\beta_{7} - \beta_1) q^{5} + ( - \beta_{5} - \beta_{4} + \beta_{2} + 1) q^{7} + ( - 2 \beta_1 + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{2} + 2 \beta_1 q^{4} + (\beta_{7} - \beta_1) q^{5} + ( - \beta_{5} - \beta_{4} + \beta_{2} + 1) q^{7} + ( - 2 \beta_1 + 2) q^{8} + ( - \beta_{7} + \beta_{4} + \beta_1 - 1) q^{10} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_1 - 1) q^{11} + ( - \beta_{11} + \beta_{10} - \beta_{9} + \beta_{7} - \beta_{5} + \beta_{4} - \beta_1) q^{13} + (\beta_{10} - \beta_{9} + \beta_{7} + \beta_{5} + \beta_{4} - \beta_{2} - \beta_1) q^{14} - 4 q^{16} + ( - 2 \beta_{9} - \beta_{8} - 2 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \cdots + 2) q^{17}+ \cdots + ( - 4 \beta_{11} + 2 \beta_{10} - 6 \beta_{9} - \beta_{8} + 6 \beta_{7} + \beta_{6} + 2 \beta_{5} + \cdots + 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} + 6 q^{7} + 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} + 6 q^{7} + 24 q^{8} - 6 q^{10} - 12 q^{11} - 48 q^{16} + 18 q^{17} + 12 q^{20} + 12 q^{22} - 54 q^{23} - 54 q^{25} - 12 q^{28} + 72 q^{31} + 48 q^{32} + 168 q^{35} + 66 q^{37} - 36 q^{38} - 12 q^{40} + 24 q^{41} - 108 q^{43} + 108 q^{46} + 48 q^{47} + 54 q^{50} - 192 q^{53} - 276 q^{55} + 24 q^{56} - 60 q^{58} + 456 q^{61} - 72 q^{62} + 264 q^{65} - 12 q^{67} - 36 q^{68} - 174 q^{70} + 84 q^{71} - 216 q^{73} + 72 q^{76} - 48 q^{77} - 24 q^{82} - 246 q^{83} - 324 q^{85} + 216 q^{86} - 24 q^{88} + 612 q^{91} - 108 q^{92} + 432 q^{95} + 102 q^{97} - 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 54 x^{9} + 921 x^{8} - 1350 x^{7} + 1458 x^{6} - 18792 x^{5} + 231804 x^{4} - 552420 x^{3} + 583200 x^{2} + 874800 x + 656100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1875243545 \nu^{11} + 2732664981 \nu^{10} + 2774149290 \nu^{9} + 104197645485 \nu^{8} - 1873864921524 \nu^{7} + \cdots - 866523189915450 ) / 20\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 801101591 \nu^{11} - 10289055060 \nu^{10} - 84332616198 \nu^{9} - 293504336784 \nu^{8} + 1034227795947 \nu^{7} + \cdots - 514673191072440 ) / 452341418052060 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 801101591 \nu^{11} + 10289055060 \nu^{10} + 84332616198 \nu^{9} + 293504336784 \nu^{8} - 1034227795947 \nu^{7} + \cdots + 514673191072440 ) / 452341418052060 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10813781488 \nu^{11} - 18832079700 \nu^{10} - 253052027046 \nu^{9} - 1619639566146 \nu^{8} + 8313264642702 \nu^{7} + \cdots - 13\!\cdots\!70 ) / 20\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3622731026 \nu^{11} - 25387385657 \nu^{10} - 103969950084 \nu^{9} - 404288852031 \nu^{8} + 5012953239183 \nu^{7} + \cdots + 809857776786660 ) / 678512127078090 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 16157749007 \nu^{11} - 16169837742 \nu^{10} + 245744928891 \nu^{9} + 2957021682174 \nu^{8} - 7713673859184 \nu^{7} + \cdots + 42\!\cdots\!00 ) / 20\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 5903758925 \nu^{11} - 33865345891 \nu^{10} - 67696741206 \nu^{9} + 347221587906 \nu^{8} - 2763963071277 \nu^{7} + \cdots + 10\!\cdots\!90 ) / 678512127078090 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 24025140695 \nu^{11} + 169173040263 \nu^{10} + 438081208245 \nu^{9} + 1746363892779 \nu^{8} - 34924088358249 \nu^{7} + \cdots - 17\!\cdots\!00 ) / 20\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 28593720413 \nu^{11} + 60064310034 \nu^{10} - 33351976611 \nu^{9} - 2346815029725 \nu^{8} + 19824337365018 \nu^{7} + \cdots - 98\!\cdots\!70 ) / 20\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 25717796021 \nu^{11} + 60311648106 \nu^{10} + 62594041086 \nu^{9} - 1808354936952 \nu^{8} + 17343352267269 \nu^{7} + \cdots + 34\!\cdots\!80 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 31183125983 \nu^{11} + 65859946686 \nu^{10} + 68463029196 \nu^{9} - 2101886170110 \nu^{8} + 22243226512101 \nu^{7} + \cdots + 59\!\cdots\!80 ) / 13\!\cdots\!80 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{7} + \beta_{5} - \beta_{2} + 15\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 6\beta_{11} - 8\beta_{10} - 3\beta_{5} + 3\beta_{4} - 15\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 5 \beta_{11} + 29 \beta_{10} - 29 \beta_{9} - 4 \beta_{7} - 4 \beta_{6} + 4 \beta_{5} - 29 \beta_{4} + 5 \beta_{3} + 29 \beta_{2} + 4 \beta _1 - 278 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 126 \beta_{9} + 9 \beta_{8} + 126 \beta_{7} + 9 \beta_{6} + 9 \beta_{5} + 9 \beta_{4} - 128 \beta_{3} - 200 \beta_{2} + 504 \beta _1 + 495 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 207 \beta_{11} - 789 \beta_{10} - 261 \beta_{9} - 180 \beta_{8} - 834 \beta_{7} - 834 \beta_{5} + 261 \beta_{4} + 207 \beta_{3} + 789 \beta_{2} - 7074 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2988 \beta_{11} + 5340 \beta_{10} - 540 \beta_{9} + 513 \beta_{8} + 540 \beta_{7} - 513 \beta_{6} + 4365 \beta_{5} - 4365 \beta_{4} + 19962 \beta _1 - 15597 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 6771 \beta_{11} - 21387 \beta_{10} + 24357 \beta_{9} + 11076 \beta_{7} + 6378 \beta_{6} - 11076 \beta_{5} + 24357 \beta_{4} - 6771 \beta_{3} - 21387 \beta_{2} - 11076 \beta _1 + 152454 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 141048 \beta_{9} - 20817 \beta_{8} - 141048 \beta_{7} - 20817 \beta_{6} - 21303 \beta_{5} - 21303 \beta_{4} + 74442 \beta_{3} + 147666 \beta_{2} - 478008 \beta _1 - 456705 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 206037 \beta_{11} + 586557 \beta_{10} + 397683 \beta_{9} + 208872 \beta_{8} + 718722 \beta_{7} + 718722 \beta_{5} - 397683 \beta_{4} - 206037 \beta_{3} - 586557 \beta_{2} + 4667490 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1946268 \beta_{11} - 4162068 \beta_{10} + 717012 \beta_{9} - 735885 \beta_{8} - 717012 \beta_{7} + 735885 \beta_{6} - 4405131 \beta_{5} + 4405131 \beta_{4} - 17494920 \beta _1 + 13089789 \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
2.88670 2.88670i
−3.24958 + 3.24958i
−0.509217 + 0.509217i
2.49352 2.49352i
2.21548 2.21548i
−3.83690 + 3.83690i
2.88670 + 2.88670i
−3.24958 3.24958i
−0.509217 0.509217i
2.49352 + 2.49352i
2.21548 + 2.21548i
−3.83690 3.83690i
−1.00000 + 1.00000i 0 2.00000i −4.59534 1.97049i 0 −4.92840 + 4.92840i 2.00000 + 2.00000i 0 6.56583 2.62486i
163.2 −1.00000 + 1.00000i 0 2.00000i −1.76846 + 4.67681i 0 −4.32826 + 4.32826i 2.00000 + 2.00000i 0 −2.90835 6.44527i
163.3 −1.00000 + 1.00000i 0 2.00000i −1.18786 + 4.85685i 0 6.55284 6.55284i 2.00000 + 2.00000i 0 −3.66899 6.04471i
163.4 −1.00000 + 1.00000i 0 2.00000i −0.830091 4.93061i 0 0.952249 0.952249i 2.00000 + 2.00000i 0 5.76070 + 4.10052i
163.5 −1.00000 + 1.00000i 0 2.00000i 4.07172 + 2.90192i 0 7.16337 7.16337i 2.00000 + 2.00000i 0 −6.97363 + 1.16980i
163.6 −1.00000 + 1.00000i 0 2.00000i 4.31004 2.53448i 0 −2.41180 + 2.41180i 2.00000 + 2.00000i 0 −1.77556 + 6.84452i
487.1 −1.00000 1.00000i 0 2.00000i −4.59534 + 1.97049i 0 −4.92840 4.92840i 2.00000 2.00000i 0 6.56583 + 2.62486i
487.2 −1.00000 1.00000i 0 2.00000i −1.76846 4.67681i 0 −4.32826 4.32826i 2.00000 2.00000i 0 −2.90835 + 6.44527i
487.3 −1.00000 1.00000i 0 2.00000i −1.18786 4.85685i 0 6.55284 + 6.55284i 2.00000 2.00000i 0 −3.66899 + 6.04471i
487.4 −1.00000 1.00000i 0 2.00000i −0.830091 + 4.93061i 0 0.952249 + 0.952249i 2.00000 2.00000i 0 5.76070 4.10052i
487.5 −1.00000 1.00000i 0 2.00000i 4.07172 2.90192i 0 7.16337 + 7.16337i 2.00000 2.00000i 0 −6.97363 1.16980i
487.6 −1.00000 1.00000i 0 2.00000i 4.31004 + 2.53448i 0 −2.41180 2.41180i 2.00000 2.00000i 0 −1.77556 6.84452i
\(n\): 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.i 12
3.b odd 2 1 810.3.g.k 12
5.c odd 4 1 inner 810.3.g.i 12
9.c even 3 2 270.3.l.b 24
9.d odd 6 2 90.3.k.a 24
15.e even 4 1 810.3.g.k 12
45.k odd 12 2 270.3.l.b 24
45.l even 12 2 90.3.k.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.3.k.a 24 9.d odd 6 2
90.3.k.a 24 45.l even 12 2
270.3.l.b 24 9.c even 3 2
270.3.l.b 24 45.k odd 12 2
810.3.g.i 12 1.a even 1 1 trivial
810.3.g.i 12 5.c odd 4 1 inner
810.3.g.k 12 3.b odd 2 1
810.3.g.k 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 \(S_{3}^{\mathrm{new}}(810, [\chi])\):

\( T_{7}^{12} - 6 T_{7}^{11} + 18 T_{7}^{10} + 544 T_{7}^{9} + 7479 T_{7}^{8} - 12756 T_{7}^{7} + 89882 T_{7}^{6} + 2545176 T_{7}^{5} + 23901939 T_{7}^{4} + 53701934 T_{7}^{3} + 41259528 T_{7}^{2} + \cdots + 338449609 \) Copy content Toggle raw display
\( T_{11}^{6} + 6T_{11}^{5} - 621T_{11}^{4} - 5228T_{11}^{3} + 87432T_{11}^{2} + 913818T_{11} + 1018990 \) 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
$5$ \( T^{12} + 27 T^{10} + \cdots + 244140625 \) Copy content Toggle raw display
$7$ \( T^{12} - 6 T^{11} + 18 T^{10} + \cdots + 338449609 \) Copy content Toggle raw display
$11$ \( (T^{6} + 6 T^{5} - 621 T^{4} + \cdots + 1018990)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 1338 T^{9} + \cdots + 3000191076 \) Copy content Toggle raw display
$17$ \( T^{12} - 18 T^{11} + \cdots + 9106503184 \) Copy content Toggle raw display
$19$ \( T^{12} + 1632 T^{10} + \cdots + 87379360000 \) Copy content Toggle raw display
$23$ \( T^{12} + 54 T^{11} + \cdots + 34464707542225 \) Copy content Toggle raw display
$29$ \( T^{12} + 7770 T^{10} + \cdots + 22\!\cdots\!61 \) Copy content Toggle raw display
$31$ \( (T^{6} - 36 T^{5} - 597 T^{4} + \cdots - 18709310)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} - 66 T^{11} + \cdots + 88\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{6} - 12 T^{5} - 5673 T^{4} + \cdots + 2668453777)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 108 T^{11} + \cdots + 38\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{12} - 48 T^{11} + \cdots + 26\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{12} + 192 T^{11} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{12} + 18222 T^{10} + \cdots + 27\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( (T^{6} - 228 T^{5} + 20583 T^{4} + \cdots + 1299208905)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 12 T^{11} + \cdots + 37\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{6} - 42 T^{5} - 21156 T^{4} + \cdots + 910683616)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 216 T^{11} + \cdots + 87\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{12} + 19494 T^{10} + \cdots + 14\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( T^{12} + 246 T^{11} + \cdots + 56\!\cdots\!29 \) Copy content Toggle raw display
$89$ \( T^{12} + 25716 T^{10} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} - 102 T^{11} + \cdots + 52\!\cdots\!84 \) Copy content Toggle raw display
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