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Properties

Label 810.2.m.i
Level $810$
Weight $2$
Character orbit 810.m
Analytic conductor $6.468$
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] = [810,2,Mod(53,810)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(810, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([10, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("810.53");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 810.m (of order \(12\), degree \(4\), 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: \(6.46788256372\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 16x^{14} + 145x^{12} - 976x^{10} + 5296x^{8} - 24400x^{6} + 90625x^{4} - 250000x^{2} + 390625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{9} q^{2} + ( - \beta_{6} + \beta_{5}) q^{4} + (\beta_{15} - \beta_{13} + \cdots + \beta_1) q^{5}+ \cdots + ( - \beta_{15} + \beta_{7}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{2} + ( - \beta_{6} + \beta_{5}) q^{4} + (\beta_{15} - \beta_{13} + \cdots + \beta_1) q^{5}+ \cdots + ( - 2 \beta_{15} - \beta_{13} + \cdots + 2 \beta_{7}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{7} - 8 q^{10} - 24 q^{13} + 8 q^{16} - 4 q^{22} + 32 q^{25} - 4 q^{28} + 4 q^{31} - 24 q^{34} - 12 q^{37} + 8 q^{40} - 12 q^{43} + 40 q^{46} - 12 q^{49} + 24 q^{52} - 24 q^{55} - 68 q^{58} + 36 q^{61} + 56 q^{67} + 4 q^{70} - 68 q^{73} - 8 q^{76} - 84 q^{79} - 4 q^{82} + 68 q^{85} - 4 q^{88} + 168 q^{91} + 12 q^{94} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 16x^{14} + 145x^{12} - 976x^{10} + 5296x^{8} - 24400x^{6} + 90625x^{4} - 250000x^{2} + 390625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 44 \nu^{14} - 28729 \nu^{12} + 107280 \nu^{10} - 180944 \nu^{8} + 885424 \nu^{6} + \cdots + 336734375 ) / 186750000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1567 \nu^{14} + 242897 \nu^{12} - 1523040 \nu^{10} + 9443392 \nu^{8} - 53796032 \nu^{6} + \cdots + 1179640625 ) / 933750000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1363 \nu^{14} + 21983 \nu^{12} - 471060 \nu^{10} + 2888788 \nu^{8} - 16426748 \nu^{6} + \cdots + 783171875 ) / 233437500 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16099 \nu^{14} - 257984 \nu^{12} + 1958880 \nu^{10} - 12160624 \nu^{8} + 52950704 \nu^{6} + \cdots - 1301000000 ) / 933750000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 577 \nu^{14} + 4232 \nu^{12} - 30540 \nu^{10} + 143152 \nu^{8} - 744542 \nu^{6} + \cdots + 15875000 ) / 23343750 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 21551 \nu^{15} - 345916 \nu^{13} + 3843120 \nu^{11} - 23715776 \nu^{9} + 118657696 \nu^{7} + \cdots - 4433687500 \nu ) / 4668750000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10056 \nu^{14} - 121121 \nu^{12} + 1006720 \nu^{10} - 6116656 \nu^{8} + 31636176 \nu^{6} + \cdots - 1220640625 ) / 311250000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 37612 \nu^{15} - 184367 \nu^{13} + 1657440 \nu^{11} - 8443312 \nu^{9} + 28936352 \nu^{7} + \cdots - 756359375 \nu ) / 4668750000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 4912 \nu^{15} + 78667 \nu^{13} - 495940 \nu^{11} + 3090612 \nu^{9} - 12174652 \nu^{7} + \cdots + 172609375 \nu ) / 389062500 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{15} + 16\nu^{13} - 145\nu^{11} + 976\nu^{9} - 5296\nu^{7} + 24400\nu^{5} - 90625\nu^{3} + 250000\nu ) / 78125 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 5161 \nu^{15} + 14176 \nu^{13} - 110820 \nu^{11} + 304136 \nu^{9} - 1649256 \nu^{7} + \cdots + 46812500 \nu ) / 389062500 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 61972 \nu^{15} + 832877 \nu^{13} - 5954640 \nu^{11} + 37768672 \nu^{9} + \cdots + 4416828125 \nu ) / 4668750000 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 16697 \nu^{14} + 151852 \nu^{12} - 1130640 \nu^{10} + 6810272 \nu^{8} - 33044512 \nu^{6} + \cdots + 774437500 ) / 186750000 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 75019 \nu^{15} - 1022204 \nu^{13} + 8621280 \nu^{11} - 48696544 \nu^{9} + 247775024 \nu^{7} + \cdots - 7046937500 \nu ) / 4668750000 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} + 3\beta_{8} + \beta_{4} + \beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{15} + 5\beta_{13} + \beta_{12} - 3\beta_{11} - \beta_{10} + 3\beta_{9} - 6\beta_{7} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{14} + 8\beta_{8} + 10\beta_{6} - 5\beta_{5} + \beta_{4} + 6\beta_{3} + 6\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10\beta_{15} - 5\beta_{13} + 22\beta_{12} - 2\beta_{11} + 11\beta_{10} + 37\beta_{9} - 26\beta_{7} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -26\beta_{14} + 26\beta_{8} + 55\beta_{6} - 110\beta_{5} - 10\beta_{4} - 16\beta_{3} + 26\beta_{2} + 26 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 65\beta_{15} - 130\beta_{13} + 65\beta_{12} + 130\beta_{10} + 50\beta_{9} - 145\beta_{7} + 26\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 106\beta_{14} + 208\beta_{8} - 325\beta_{6} - 325\beta_{5} - 39\beta_{4} - 39\beta_{3} + 145\beta_{2} + 63 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1007 \beta_{15} + 530 \beta_{13} - 219 \beta_{12} - 63 \beta_{11} + 219 \beta_{10} - 662 \beta_{9} + \cdots + 63 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 944\beta_{14} + 228\beta_{8} - 2190\beta_{6} + 1095\beta_{5} - 944\beta_{4} + 1031\beta_{3} + 1031\beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 4480\beta_{15} + 4720\beta_{13} - 128\beta_{12} + 803\beta_{11} - 64\beta_{10} - 368\beta_{9} + 304\beta_{7} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 304\beta_{14} - 304\beta_{8} - 320\beta_{6} + 640\beta_{5} - 4480\beta_{4} + 4784\beta_{3} - 304\beta_{2} + 8191 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -5680\beta_{15} + 1520\beta_{13} + 4160\beta_{12} + 8320\beta_{10} + 22400\beta_{9} + 19760\beta_{7} + 8191\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 5889 \beta_{14} + 13213 \beta_{8} - 20800 \beta_{6} - 20800 \beta_{5} + 13871 \beta_{4} + \cdots + 32973 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 39138 \beta_{15} - 29445 \beta_{13} - 26689 \beta_{12} - 32973 \beta_{11} + 26689 \beta_{10} + \cdots + 32973 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(487\) \(731\)
\(\chi(n)\) \(-\beta_{6}\) \(1 + \beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−2.22946 + 0.171822i
1.26353 − 1.84485i
−1.26353 + 1.84485i
2.22946 − 0.171822i
−2.22946 − 0.171822i
1.26353 + 1.84485i
−1.26353 − 1.84485i
2.22946 + 0.171822i
−2.05289 + 0.886375i
1.79407 − 1.33466i
−1.79407 + 1.33466i
2.05289 − 0.886375i
−2.05289 − 0.886375i
1.79407 + 1.33466i
−1.79407 − 1.33466i
2.05289 + 0.886375i
−0.258819 + 0.965926i 0 −0.866025 − 0.500000i −1.26353 + 1.84485i 0 −2.19796 − 0.588942i 0.707107 − 0.707107i 0 −1.45497 − 1.69796i
53.2 −0.258819 + 0.965926i 0 −0.866025 − 0.500000i 2.22946 − 0.171822i 0 1.69796 + 0.454967i 0.707107 − 0.707107i 0 −0.411058 + 2.19796i
53.3 0.258819 − 0.965926i 0 −0.866025 − 0.500000i −2.22946 + 0.171822i 0 1.69796 + 0.454967i −0.707107 + 0.707107i 0 −0.411058 + 2.19796i
53.4 0.258819 − 0.965926i 0 −0.866025 − 0.500000i 1.26353 − 1.84485i 0 −2.19796 − 0.588942i −0.707107 + 0.707107i 0 −1.45497 − 1.69796i
107.1 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i −1.26353 − 1.84485i 0 −2.19796 + 0.588942i 0.707107 + 0.707107i 0 −1.45497 + 1.69796i
107.2 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i 2.22946 + 0.171822i 0 1.69796 − 0.454967i 0.707107 + 0.707107i 0 −0.411058 − 2.19796i
107.3 0.258819 + 0.965926i 0 −0.866025 + 0.500000i −2.22946 − 0.171822i 0 1.69796 − 0.454967i −0.707107 − 0.707107i 0 −0.411058 − 2.19796i
107.4 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 1.26353 + 1.84485i 0 −2.19796 + 0.588942i −0.707107 − 0.707107i 0 −1.45497 + 1.69796i
377.1 −0.965926 − 0.258819i 0 0.866025 + 0.500000i −1.79407 + 1.33466i 0 0.324847 − 1.21235i −0.707107 − 0.707107i 0 2.07837 − 0.824847i
377.2 −0.965926 − 0.258819i 0 0.866025 + 0.500000i 2.05289 − 0.886375i 0 −0.824847 + 3.07837i −0.707107 − 0.707107i 0 −2.21235 + 0.324847i
377.3 0.965926 + 0.258819i 0 0.866025 + 0.500000i −2.05289 + 0.886375i 0 −0.824847 + 3.07837i 0.707107 + 0.707107i 0 −2.21235 + 0.324847i
377.4 0.965926 + 0.258819i 0 0.866025 + 0.500000i 1.79407 − 1.33466i 0 0.324847 − 1.21235i 0.707107 + 0.707107i 0 2.07837 − 0.824847i
593.1 −0.965926 + 0.258819i 0 0.866025 − 0.500000i −1.79407 − 1.33466i 0 0.324847 + 1.21235i −0.707107 + 0.707107i 0 2.07837 + 0.824847i
593.2 −0.965926 + 0.258819i 0 0.866025 − 0.500000i 2.05289 + 0.886375i 0 −0.824847 − 3.07837i −0.707107 + 0.707107i 0 −2.21235 − 0.324847i
593.3 0.965926 − 0.258819i 0 0.866025 − 0.500000i −2.05289 − 0.886375i 0 −0.824847 − 3.07837i 0.707107 − 0.707107i 0 −2.21235 − 0.324847i
593.4 0.965926 − 0.258819i 0 0.866025 − 0.500000i 1.79407 + 1.33466i 0 0.324847 + 1.21235i 0.707107 − 0.707107i 0 2.07837 + 0.824847i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
45.k odd 12 1 inner
45.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 810.2.m.i 16
3.b odd 2 1 inner 810.2.m.i 16
5.c odd 4 1 810.2.m.j 16
9.c even 3 1 810.2.f.d ✓ 16
9.c even 3 1 810.2.m.j 16
9.d odd 6 1 810.2.f.d ✓ 16
9.d odd 6 1 810.2.m.j 16
15.e even 4 1 810.2.m.j 16
45.k odd 12 1 810.2.f.d ✓ 16
45.k odd 12 1 inner 810.2.m.i 16
45.l even 12 1 810.2.f.d ✓ 16
45.l even 12 1 inner 810.2.m.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
810.2.f.d ✓ 16 9.c even 3 1
810.2.f.d ✓ 16 9.d odd 6 1
810.2.f.d ✓ 16 45.k odd 12 1
810.2.f.d ✓ 16 45.l even 12 1
810.2.m.i 16 1.a even 1 1 trivial
810.2.m.i 16 3.b odd 2 1 inner
810.2.m.i 16 45.k odd 12 1 inner
810.2.m.i 16 45.l even 12 1 inner
810.2.m.j 16 5.c odd 4 1
810.2.m.j 16 9.c even 3 1
810.2.m.j 16 9.d odd 6 1
810.2.m.j 16 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 \(S_{2}^{\mathrm{new}}(810, [\chi])\):

\( T_{7}^{8} + 2T_{7}^{7} + 5T_{7}^{6} - 4T_{7}^{5} - 47T_{7}^{4} + 16T_{7}^{3} + 80T_{7}^{2} - 128T_{7} + 256 \) Copy content Toggle raw display
\( T_{11}^{16} - 68 T_{11}^{14} + 3340 T_{11}^{12} - 75536 T_{11}^{10} + 1244176 T_{11}^{8} + \cdots + 16777216 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 16 T^{14} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( (T^{8} + 2 T^{7} + \cdots + 256)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} - 68 T^{14} + \cdots + 16777216 \) Copy content Toggle raw display
$13$ \( (T^{8} + 12 T^{7} + \cdots + 19881)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 874 T^{12} + \cdots + 6250000 \) Copy content Toggle raw display
$19$ \( (T^{8} + 46 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 409600000000 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 4294967296 \) Copy content Toggle raw display
$31$ \( (T^{8} - 2 T^{7} + \cdots + 541696)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 6 T^{7} + \cdots + 324)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 237367737616 \) Copy content Toggle raw display
$43$ \( (T^{8} + 6 T^{7} + \cdots + 589824)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 429981696 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 260120641601536 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 245635219456 \) Copy content Toggle raw display
$61$ \( (T^{8} - 18 T^{7} + \cdots + 11410884)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 28 T^{7} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 140 T^{6} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 34 T^{7} + \cdots + 952576)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 42 T^{7} + \cdots + 147456)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 60 T^{14} + \cdots + 84934656 \) Copy content Toggle raw display
$89$ \( (T^{8} - 176 T^{6} + \cdots + 1865956)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 6 T^{7} + \cdots + 2304)^{2} \) Copy content Toggle raw display
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