↑

Properties

Label 3564.2.i.t
Level $3564$
Weight $2$
Character orbit 3564.i
Analytic conductor $28.459$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3564,2,Mod(1189,3564)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3564, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3564.1189");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3564 = 2^{2} \cdot 3^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3564.i (of order \(3\), 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: \(28.4586832804\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 18x^{9} + 152x^{8} - 204x^{7} + 162x^{6} - 408x^{5} + 2800x^{4} - 4422x^{3} + 3528x^{2} - 252x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{5} + ( - \beta_{10} - \beta_{3}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{5} + ( - \beta_{10} - \beta_{3}) q^{7} + ( - \beta_{8} + 1) q^{11} + (\beta_{9} + \beta_{8} + \beta_{5}) q^{13} + (\beta_{4} - \beta_{3} - 1) q^{17} + ( - \beta_{7} - 2 \beta_{5} + \cdots + 2 \beta_{2}) q^{19}+ \cdots + ( - 2 \beta_{10} + 4 \beta_{9} + \cdots - 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{7} + 6 q^{11} + 6 q^{13} - 8 q^{17} + 4 q^{23} - 10 q^{25} - 4 q^{29} - 6 q^{31} - 28 q^{35} - 12 q^{37} + 22 q^{41} + 10 q^{43} + 20 q^{47} - 6 q^{49} - 8 q^{53} + 24 q^{59} + 10 q^{61} + 40 q^{65} + 6 q^{67} - 80 q^{71} - 16 q^{73} - 2 q^{77} + 14 q^{79} + 12 q^{83} - 6 q^{85} - 48 q^{89} + 20 q^{91} + 44 q^{95} - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 18x^{9} + 152x^{8} - 204x^{7} + 162x^{6} - 408x^{5} + 2800x^{4} - 4422x^{3} + 3528x^{2} - 252x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4219095960 \nu^{11} + 150004431449 \nu^{10} + 281639173048 \nu^{9} + 151821416226 \nu^{8} + \cdots + 2796863125032 ) / 76706025554046 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 6484052088 \nu^{11} + 71705413468 \nu^{10} + 57113609088 \nu^{9} - 10874788028 \nu^{8} + \cdots + 11199188889288 ) / 76706025554046 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9299982465 \nu^{11} + 8133490914 \nu^{10} - 19668060275 \nu^{9} - 157262552999 \nu^{8} + \cdots + 252524494555971 ) / 76706025554046 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2750711188 \nu^{11} - 2069597445 \nu^{10} - 1650447072 \nu^{9} + 48776182110 \nu^{8} + \cdots + 355793955480 ) / 4512119150238 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 82821032344 \nu^{11} + 7932704851 \nu^{10} + 168410363512 \nu^{9} + 1821086395994 \nu^{8} + \cdots + 3694668722064 ) / 76706025554046 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 279432 \nu^{11} - 222972 \nu^{10} + 95747 \nu^{9} + 5714144 \nu^{8} - 37197626 \nu^{7} + \cdots - 247656519 ) / 143870286 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 42033703465 \nu^{11} - 33468459198 \nu^{10} - 17385401814 \nu^{9} + \cdots - 86245333972914 ) / 10958003650578 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 405744 \nu^{11} - 28556 \nu^{10} - 75781 \nu^{9} - 7363776 \nu^{8} + 62268674 \nu^{7} + \cdots - 24784878 ) / 54382422 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1082573996136 \nu^{11} + 129583122540 \nu^{10} + 27250451714 \nu^{9} + \cdots - 60402814067835 ) / 76706025554046 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1909077352544 \nu^{11} - 7566377254 \nu^{10} + 105634150883 \nu^{9} + \cdots + 114964945674135 ) / 76706025554046 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 636792327880 \nu^{11} + 30428609356 \nu^{10} + 32154593429 \nu^{9} - 11467900114482 \nu^{8} + \cdots - 125528950369608 ) / 10958003650578 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{5} + 2\beta_{4} - \beta_{2} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2 \beta_{11} - 2 \beta_{10} + 10 \beta_{9} - 8 \beta_{8} + \beta_{7} + 5 \beta_{6} + 2 \beta_{5} + \cdots + 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -9\beta_{9} + 15\beta_{8} + 3\beta_{6} - 14\beta_{5} - 8\beta_{4} - 2\beta_{2} + 14\beta _1 + 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{7} - 18\beta_{6} + 17\beta_{5} + 19\beta_{4} + 10\beta_{3} - 17\beta_{2} - 48 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 9 \beta_{11} - 24 \beta_{10} + 162 \beta_{9} - 285 \beta_{8} - 3 \beta_{7} + 219 \beta_{6} + \cdots + 393 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 152 \beta_{11} + 344 \beta_{10} - 1108 \beta_{9} + 1838 \beta_{8} - 76 \beta_{7} - 554 \beta_{6} + \cdots - 919 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 204 \beta_{11} - 552 \beta_{10} + 2532 \beta_{9} - 4560 \beta_{8} + 276 \beta_{7} - 912 \beta_{6} + \cdots - 1692 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -453\beta_{7} + 4175\beta_{6} - 3788\beta_{5} - 4532\beta_{4} - 1331\beta_{3} + 3788\beta_{2} + 7990 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3492 \beta_{11} + 9504 \beta_{10} - 38169 \beta_{9} + 69429 \beta_{8} + 1260 \beta_{7} - 52053 \beta_{6} + \cdots - 94974 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 23060 \beta_{11} - 61532 \beta_{10} + 215260 \beta_{9} - 391088 \beta_{8} + 11530 \beta_{7} + \cdots + 195544 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 54657 \beta_{11} + 149106 \beta_{10} - 567684 \beta_{9} + 1036947 \beta_{8} - 74553 \beta_{7} + \cdots + 380277 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3564\mathbb{Z}\right)^\times\).

\(n\) \(1541\) \(1783\) \(2917\)
\(\chi(n)\) \(-\beta_{8}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1189.1
−1.52436 − 1.52436i
1.81730 − 1.81730i
0.895540 − 0.895540i
0.0374740 + 0.0374740i
−2.71284 + 2.71284i
1.48688 + 1.48688i
−1.52436 + 1.52436i
1.81730 + 1.81730i
0.895540 + 0.895540i
0.0374740 − 0.0374740i
−2.71284 − 2.71284i
1.48688 − 1.48688i
0 0 0 −2.08231 − 3.60667i 0 1.24765 − 2.16099i 0 0 0
1189.2 0 0 0 −0.665179 − 1.15212i 0 2.41905 − 4.18992i 0 0 0
1189.3 0 0 0 −0.327790 − 0.567749i 0 −1.34070 + 2.32217i 0 0 0
1189.4 0 0 0 0.0511904 + 0.0886644i 0 −1.64629 + 2.85145i 0 0 0
1189.5 0 0 0 0.992970 + 1.71987i 0 0.287680 − 0.498276i 0 0 0
1189.6 0 0 0 2.03112 + 3.51801i 0 0.0326133 − 0.0564879i 0 0 0
2377.1 0 0 0 −2.08231 + 3.60667i 0 1.24765 + 2.16099i 0 0 0
2377.2 0 0 0 −0.665179 + 1.15212i 0 2.41905 + 4.18992i 0 0 0
2377.3 0 0 0 −0.327790 + 0.567749i 0 −1.34070 − 2.32217i 0 0 0
2377.4 0 0 0 0.0511904 − 0.0886644i 0 −1.64629 − 2.85145i 0 0 0
2377.5 0 0 0 0.992970 − 1.71987i 0 0.287680 + 0.498276i 0 0 0
2377.6 0 0 0 2.03112 − 3.51801i 0 0.0326133 + 0.0564879i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1189.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3564.2.i.t 12
3.b odd 2 1 3564.2.i.s 12
9.c even 3 1 3564.2.a.o ✓ 6
9.c even 3 1 inner 3564.2.i.t 12
9.d odd 6 1 3564.2.a.p yes 6
9.d odd 6 1 3564.2.i.s 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3564.2.a.o ✓ 6 9.c even 3 1
3564.2.a.p yes 6 9.d odd 6 1
3564.2.i.s 12 3.b odd 2 1
3564.2.i.s 12 9.d odd 6 1
3564.2.i.t 12 1.a even 1 1 trivial
3564.2.i.t 12 9.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3564, [\chi])\):

\( T_{5}^{12} + 20 T_{5}^{10} + 348 T_{5}^{8} + 24 T_{5}^{7} + 1034 T_{5}^{6} + 960 T_{5}^{5} + 2644 T_{5}^{4} + \cdots + 9 \) Copy content Toggle raw display
\( T_{7}^{12} - 2 T_{7}^{11} + 26 T_{7}^{10} + 427 T_{7}^{8} - 148 T_{7}^{7} + 2578 T_{7}^{6} - 778 T_{7}^{5} + \cdots + 16 \) Copy content Toggle raw display
\( T_{17}^{6} + 4T_{17}^{5} - 51T_{17}^{4} - 272T_{17}^{3} - 68T_{17}^{2} + 1056T_{17} + 1104 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 20 T^{10} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{12} - 2 T^{11} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{12} - 6 T^{11} + \cdots + 9 \) Copy content Toggle raw display
$17$ \( (T^{6} + 4 T^{5} + \cdots + 1104)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 95 T^{4} + \cdots - 108)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 4 T^{11} + \cdots + 1679616 \) Copy content Toggle raw display
$29$ \( T^{12} + 4 T^{11} + \cdots + 2712609 \) Copy content Toggle raw display
$31$ \( T^{12} + 6 T^{11} + \cdots + 4822416 \) Copy content Toggle raw display
$37$ \( (T^{6} + 6 T^{5} + \cdots - 33363)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 3041743104 \) Copy content Toggle raw display
$43$ \( T^{12} - 10 T^{11} + \cdots + 3810304 \) Copy content Toggle raw display
$47$ \( T^{12} - 20 T^{11} + \cdots + 77158656 \) Copy content Toggle raw display
$53$ \( (T^{6} + 4 T^{5} + \cdots - 1728)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 165867223824 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 64045437184 \) Copy content Toggle raw display
$67$ \( T^{12} - 6 T^{11} + \cdots + 1296 \) Copy content Toggle raw display
$71$ \( (T^{6} + 40 T^{5} + \cdots + 144)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 8 T^{5} + \cdots + 22653)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 1673791744 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 1925389357056 \) Copy content Toggle raw display
$89$ \( (T^{6} + 24 T^{5} + \cdots + 147921)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 207469318144 \) Copy content Toggle raw display
show more
show less