Properties

Label 1440.2.w.g
Level $1440$
Weight $2$
Character orbit 1440.w
Analytic conductor $11.498$
Analytic rank $0$
Dimension $12$
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] = [1440,2,Mod(737,1440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1440, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1440.737");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1440.w (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: \(11.4984578911\)
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: \( x^{12} + 27x^{8} + 107x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{9} \)
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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{5} + ( - \beta_{5} - \beta_{2} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{5} + ( - \beta_{5} - \beta_{2} + 1) q^{7} + (\beta_{10} + \beta_{8} + \beta_{6}) q^{11} + \beta_{4} q^{13} + ( - \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6}) q^{17} + ( - \beta_{5} - \beta_{4} + 2 \beta_{2}) q^{19} + ( - 2 \beta_{11} + 2 \beta_{10}) q^{23} + ( - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - 1) q^{25} + (\beta_{11} + 2 \beta_{9} - \beta_{7}) q^{29} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} + 4) q^{31} + (\beta_{11} + 2 \beta_{10} + \beta_{9} - 2 \beta_{8} + \beta_{7}) q^{35} + (3 \beta_{5} + \beta_{3} + 3 \beta_{2} + \beta_1 - 3) q^{37} + ( - \beta_{11} - \beta_{10} + \beta_{8} - \beta_{7} - \beta_{6}) q^{41} + (2 \beta_{4} - 2 \beta_{3} + 2 \beta_1) q^{43} + (3 \beta_{11} + 3 \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6}) q^{47} + ( - 2 \beta_{5} - 2 \beta_{4} - 3 \beta_{2} - 2 \beta_1) q^{49} + ( - 2 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} - \beta_{7} - \beta_{6}) q^{53} + ( - \beta_{5} - 2 \beta_{3} - \beta_{2} - 5) q^{55} + ( - \beta_{11} + 4 \beta_{10} - 5 \beta_{9} + \beta_{7} - 4 \beta_{6}) q^{59} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3}) q^{61} + (\beta_{11} - \beta_{10} + 2 \beta_{9} + \beta_{8} + 2 \beta_{7}) q^{65} + ( - 2 \beta_{5} - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 4) q^{67} + (2 \beta_{11} + 2 \beta_{10} + 2 \beta_{7} + 2 \beta_{6}) q^{71} + ( - \beta_{3} + 4 \beta_{2} + \beta_1 + 4) q^{73} + (2 \beta_{11} + 2 \beta_{10}) q^{77} + (\beta_{5} + \beta_{4} + 12 \beta_{2} + 2 \beta_1) q^{79} + ( - \beta_{11} + \beta_{10} + 3 \beta_{9} - 3 \beta_{8} + \beta_{7} + \beta_{6}) q^{83} + (2 \beta_{5} + 2 \beta_{3} - 3 \beta_{2} + \beta_1) q^{85} + ( - \beta_{11} - 3 \beta_{10} - 3 \beta_{9} + \beta_{7} + 3 \beta_{6}) q^{89} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} + 8) q^{91} + ( - 3 \beta_{11} + 3 \beta_{10} - \beta_{9} - 3 \beta_{8} - \beta_{7} - \beta_{6}) q^{95} + ( - 2 \beta_{5} - 3 \beta_{3} - 4 \beta_{2} - 3 \beta_1 + 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 8 q^{7} - 4 q^{13} - 16 q^{25} + 32 q^{31} - 20 q^{37} - 16 q^{43} - 72 q^{55} + 32 q^{67} + 44 q^{73} + 16 q^{85} + 80 q^{91} + 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 27x^{8} + 107x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{10} + 38\nu^{6} + 297\nu^{2} ) / 76 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -7\nu^{10} - 190\nu^{6} - 787\nu^{2} ) / 76 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{8} - 19\nu^{4} + 7 ) / 19 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -14\nu^{10} + \nu^{8} - 380\nu^{6} + 38\nu^{4} - 1498\nu^{2} + 145 ) / 76 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -14\nu^{10} - \nu^{8} - 380\nu^{6} - 38\nu^{4} - 1498\nu^{2} - 145 ) / 76 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7\nu^{11} + 4\nu^{9} + 190\nu^{7} + 114\nu^{5} + 787\nu^{3} + 542\nu ) / 76 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -7\nu^{11} + 4\nu^{9} - 190\nu^{7} + 114\nu^{5} - 787\nu^{3} + 542\nu ) / 76 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -24\nu^{11} - 3\nu^{9} - 646\nu^{7} - 76\nu^{5} - 2530\nu^{3} - 245\nu ) / 76 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -24\nu^{11} + 3\nu^{9} - 646\nu^{7} + 76\nu^{5} - 2530\nu^{3} + 245\nu ) / 76 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 26\nu^{11} + 703\nu^{7} + 2801\nu^{3} + 38\nu ) / 38 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 26\nu^{11} + 703\nu^{7} + 2801\nu^{3} - 38\nu ) / 38 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{11} + \beta_{10} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{4} - 4\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - 4\beta_{7} + 4\beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{5} + 2\beta_{4} + \beta_{3} - 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 17\beta_{11} - 17\beta_{10} - 8\beta_{9} + 8\beta_{8} + 6\beta_{7} + 6\beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -17\beta_{5} - 17\beta_{4} + 70\beta_{2} + 14\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 31\beta_{11} + 31\beta_{10} + 45\beta_{9} + 45\beta_{8} + 76\beta_{7} - 76\beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 38\beta_{5} - 38\beta_{4} - 38\beta_{3} + 159 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -349\beta_{11} + 349\beta_{10} + 228\beta_{9} - 228\beta_{8} - 152\beta_{7} - 152\beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 349\beta_{5} + 349\beta_{4} - 1472\beta_{2} - 380\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -729\beta_{11} - 729\beta_{10} - 1109\beta_{9} - 1109\beta_{8} - 1624\beta_{7} + 1624\beta_{6} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(991\)
\(\chi(n)\) \(-\beta_{2}\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
737.1
1.53448 + 1.53448i
−0.219986 0.219986i
−1.04736 1.04736i
1.04736 + 1.04736i
0.219986 + 0.219986i
−1.53448 1.53448i
1.53448 1.53448i
−0.219986 + 0.219986i
−1.04736 + 1.04736i
1.04736 1.04736i
0.219986 0.219986i
−1.53448 + 1.53448i
0 0 0 −1.91575 + 1.15322i 0 −1.70928 1.70928i 0 0 0
737.2 0 0 0 −1.34577 1.78575i 0 2.90321 + 2.90321i 0 0 0
737.3 0 0 0 −0.137134 2.23186i 0 0.806063 + 0.806063i 0 0 0
737.4 0 0 0 0.137134 + 2.23186i 0 0.806063 + 0.806063i 0 0 0
737.5 0 0 0 1.34577 + 1.78575i 0 2.90321 + 2.90321i 0 0 0
737.6 0 0 0 1.91575 1.15322i 0 −1.70928 1.70928i 0 0 0
1313.1 0 0 0 −1.91575 1.15322i 0 −1.70928 + 1.70928i 0 0 0
1313.2 0 0 0 −1.34577 + 1.78575i 0 2.90321 2.90321i 0 0 0
1313.3 0 0 0 −0.137134 + 2.23186i 0 0.806063 0.806063i 0 0 0
1313.4 0 0 0 0.137134 2.23186i 0 0.806063 0.806063i 0 0 0
1313.5 0 0 0 1.34577 1.78575i 0 2.90321 2.90321i 0 0 0
1313.6 0 0 0 1.91575 + 1.15322i 0 −1.70928 + 1.70928i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 737.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1440.2.w.g yes 12
3.b odd 2 1 inner 1440.2.w.g yes 12
4.b odd 2 1 1440.2.w.f 12
5.c odd 4 1 inner 1440.2.w.g yes 12
8.b even 2 1 2880.2.w.q 12
8.d odd 2 1 2880.2.w.p 12
12.b even 2 1 1440.2.w.f 12
15.e even 4 1 inner 1440.2.w.g yes 12
20.e even 4 1 1440.2.w.f 12
24.f even 2 1 2880.2.w.p 12
24.h odd 2 1 2880.2.w.q 12
40.i odd 4 1 2880.2.w.q 12
40.k even 4 1 2880.2.w.p 12
60.l odd 4 1 1440.2.w.f 12
120.q odd 4 1 2880.2.w.p 12
120.w even 4 1 2880.2.w.q 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1440.2.w.f 12 4.b odd 2 1
1440.2.w.f 12 12.b even 2 1
1440.2.w.f 12 20.e even 4 1
1440.2.w.f 12 60.l odd 4 1
1440.2.w.g yes 12 1.a even 1 1 trivial
1440.2.w.g yes 12 3.b odd 2 1 inner
1440.2.w.g yes 12 5.c odd 4 1 inner
1440.2.w.g yes 12 15.e even 4 1 inner
2880.2.w.p 12 8.d odd 2 1
2880.2.w.p 12 24.f even 2 1
2880.2.w.p 12 40.k even 4 1
2880.2.w.p 12 120.q odd 4 1
2880.2.w.q 12 8.b even 2 1
2880.2.w.q 12 24.h odd 2 1
2880.2.w.q 12 40.i odd 4 1
2880.2.w.q 12 120.w even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1440, [\chi])\):

\( T_{7}^{6} - 4T_{7}^{5} + 8T_{7}^{4} + 16T_{7}^{3} + 64T_{7}^{2} - 128T_{7} + 128 \) Copy content Toggle raw display
\( T_{13}^{6} + 2T_{13}^{5} + 2T_{13}^{4} - 16T_{13}^{3} + 100T_{13}^{2} + 40T_{13} + 8 \) 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
$5$ \( T^{12} + 8 T^{10} + 43 T^{8} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( (T^{6} - 4 T^{5} + 8 T^{4} + 16 T^{3} + \cdots + 128)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 40 T^{4} + 192 T^{2} + 128)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 2 T^{5} + 2 T^{4} - 16 T^{3} + 100 T^{2} + \cdots + 8)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 640 T^{8} + 20480 T^{4} + \cdots + 65536 \) Copy content Toggle raw display
$19$ \( (T^{6} + 64 T^{4} + 512 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 6912 T^{8} + \cdots + 16777216 \) Copy content Toggle raw display
$29$ \( (T^{6} - 54 T^{4} + 44 T^{2} - 8)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 8 T^{2} - 16 T + 160)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + 10 T^{5} + 50 T^{4} - 592 T^{3} + \cdots + 4232)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 70 T^{4} + 652 T^{2} + 8)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 8 T^{5} + 32 T^{4} - 640 T^{3} + \cdots + 8192)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 28160 T^{8} + \cdots + 741637881856 \) Copy content Toggle raw display
$53$ \( T^{12} + 5632 T^{8} + \cdots + 5473632256 \) Copy content Toggle raw display
$59$ \( (T^{6} - 424 T^{4} + 57024 T^{2} + \cdots - 2473088)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 112 T + 416)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} - 16 T^{5} + 128 T^{4} + \cdots + 204800)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 256 T^{4} + 16384 T^{2} + \cdots + 131072)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 22 T^{5} + 242 T^{4} - 1184 T^{3} + \cdots + 8)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 416 T^{4} + 45312 T^{2} + \cdots + 541696)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 22272 T^{8} + \cdots + 1048576 \) Copy content Toggle raw display
$89$ \( (T^{6} - 326 T^{4} + 31244 T^{2} + \cdots - 897800)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 14 T^{5} + 98 T^{4} + \cdots + 1805000)^{2} \) Copy content Toggle raw display
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