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Properties

Label 4080.2.m.t
Level $4080$
Weight $2$
Character orbit 4080.m
Analytic conductor $32.579$
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] = [4080,2,Mod(2449,4080)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4080, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4080.2449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4080.m (of order \(2\), degree \(1\), 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: \(32.5789640247\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 2 x^{10} + 6 x^{9} - 14 x^{8} - 34 x^{7} + 114 x^{6} - 290 x^{5} + 529 x^{4} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 2040)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} + \beta_{8} q^{5} - \beta_{7} q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{3} + \beta_{8} q^{5} - \beta_{7} q^{7} - q^{9} + (\beta_{4} + \beta_{3} - \beta_1) q^{11} + ( - \beta_{11} - \beta_{7}) q^{13} + \beta_{4} q^{15} + \beta_{6} q^{17} + (\beta_{4} + \beta_{3} - \beta_1 + 2) q^{19} - \beta_{2} q^{21} + ( - \beta_{10} - \beta_{7} + \cdots + \beta_{3}) q^{23}+ \cdots + ( - \beta_{4} - \beta_{3} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{9} - 4 q^{11} + 20 q^{19} + 4 q^{21} - 4 q^{29} - 16 q^{31} + 12 q^{35} + 8 q^{39} - 4 q^{41} + 8 q^{49} - 12 q^{51} + 20 q^{55} - 8 q^{59} + 8 q^{61} - 16 q^{65} - 8 q^{69} - 32 q^{71} + 4 q^{75} + 24 q^{79} + 12 q^{81} + 40 q^{89} - 56 q^{91} + 20 q^{95} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 2 x^{10} + 6 x^{9} - 14 x^{8} - 34 x^{7} + 114 x^{6} - 290 x^{5} + 529 x^{4} + \cdots + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 83893 \nu^{11} + 3401190 \nu^{10} - 1188382 \nu^{9} + 3269774 \nu^{8} + 24377002 \nu^{7} + \cdots + 514452192 ) / 98942464 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2146759 \nu^{11} + 5036078 \nu^{10} - 5035030 \nu^{9} - 13487482 \nu^{8} + 37345554 \nu^{7} + \cdots + 166092000 ) / 98942464 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1309133 \nu^{11} + 1781032 \nu^{10} - 1429110 \nu^{9} - 9990146 \nu^{8} + 12132298 \nu^{7} + \cdots + 125654288 ) / 49471232 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 741438 \nu^{11} - 193991 \nu^{10} + 354278 \nu^{9} + 6238926 \nu^{8} - 755550 \nu^{7} + \cdots + 30218728 ) / 24735616 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3594723 \nu^{11} - 8173070 \nu^{10} + 7489950 \nu^{9} + 23160210 \nu^{8} - 58174602 \nu^{7} + \cdots - 325179296 ) / 98942464 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 172247 \nu^{11} + 195886 \nu^{10} - 174560 \nu^{9} - 1213868 \nu^{8} + 1350616 \nu^{7} + \cdots + 12461856 ) / 3091952 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 6653465 \nu^{11} - 8105742 \nu^{10} + 9599090 \nu^{9} + 43822606 \nu^{8} - 54707446 \nu^{7} + \cdots - 729410816 ) / 98942464 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 4574911 \nu^{11} - 5647704 \nu^{10} + 3963986 \nu^{9} + 30405014 \nu^{8} - 40777454 \nu^{7} + \cdots - 451433648 ) / 49471232 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2487618 \nu^{11} - 3735141 \nu^{10} + 2867186 \nu^{9} + 16245514 \nu^{8} - 27578906 \nu^{7} + \cdots - 210836680 ) / 24735616 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 14695849 \nu^{11} - 18286918 \nu^{10} + 19386914 \nu^{9} + 99948926 \nu^{8} - 124206438 \nu^{7} + \cdots - 1469563840 ) / 98942464 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 27318205 \nu^{11} + 36514710 \nu^{10} - 28010394 \nu^{9} - 183405734 \nu^{8} + \cdots + 2394088320 ) / 98942464 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{11} - \beta_{9} - 3\beta_{8} - \beta_{7} - 6\beta_{6} - 5\beta_{4} - \beta_{3} + 2\beta_{2} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6 \beta_{11} + \beta_{10} + 13 \beta_{9} + 3 \beta_{8} + \beta_{7} + 5 \beta_{6} + \beta_{5} + 5 \beta_{4} + \cdots - 5 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 15 \beta_{11} - 10 \beta_{10} - 13 \beta_{9} - 23 \beta_{8} + 15 \beta_{7} + 3 \beta_{4} + 7 \beta_{3} + \cdots + 4 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 14 \beta_{11} + 5 \beta_{10} + 5 \beta_{9} + 47 \beta_{8} + 9 \beta_{7} + 73 \beta_{6} - 5 \beta_{5} + \cdots + 73 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 17 \beta_{11} - 13 \beta_{9} - 63 \beta_{8} - 17 \beta_{7} - 110 \beta_{6} - 64 \beta_{5} + \cdots - 78 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 62 \beta_{11} + 73 \beta_{10} - 95 \beta_{9} + 143 \beta_{8} - 151 \beta_{7} - 307 \beta_{6} + \cdots + 307 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 217 \beta_{11} + 166 \beta_{10} + 727 \beta_{9} - 235 \beta_{8} - 217 \beta_{7} + 535 \beta_{4} + \cdots - 1092 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 922 \beta_{11} - 595 \beta_{10} - 1159 \beta_{9} - 1093 \beta_{8} + 801 \beta_{7} - 207 \beta_{6} + \cdots - 207 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1599 \beta_{11} + 3143 \beta_{9} + 2757 \beta_{8} + 1599 \beta_{7} + 7194 \beta_{6} - 528 \beta_{5} + \cdots - 662 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 5562 \beta_{11} - 3039 \beta_{10} - 9523 \beta_{9} - 5725 \beta_{8} + 2849 \beta_{7} - 7163 \beta_{6} + \cdots + 7163 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4080\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(511\) \(817\) \(1361\) \(3061\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2449.1
−0.129607 − 0.702790i
1.64386 + 0.0924738i
0.702790 + 0.129607i
−2.16636 − 1.04179i
−0.0924738 − 1.64386i
1.04179 + 2.16636i
−0.129607 + 0.702790i
1.64386 − 0.0924738i
0.702790 − 0.129607i
−2.16636 + 1.04179i
−0.0924738 + 1.64386i
1.04179 − 2.16636i
0 − 1.00000i 0 −2.21977 + 0.269523i 0 3.52564i 0 −1.00000 0
2449.2 0 − 1.00000i 0 −1.71910 − 1.42993i 0 − 1.50289i 0 −1.00000 0
2449.3 0 − 1.00000i 0 −0.269523 + 2.21977i 0 − 1.32908i 0 −1.00000 0
2449.4 0 − 1.00000i 0 0.634216 − 2.14424i 0 − 0.237858i 0 −1.00000 0
2449.5 0 − 1.00000i 0 1.42993 + 1.71910i 0 − 2.41349i 0 −1.00000 0
2449.6 0 − 1.00000i 0 2.14424 − 0.634216i 0 3.95768i 0 −1.00000 0
2449.7 0 1.00000i 0 −2.21977 − 0.269523i 0 − 3.52564i 0 −1.00000 0
2449.8 0 1.00000i 0 −1.71910 + 1.42993i 0 1.50289i 0 −1.00000 0
2449.9 0 1.00000i 0 −0.269523 − 2.21977i 0 1.32908i 0 −1.00000 0
2449.10 0 1.00000i 0 0.634216 + 2.14424i 0 0.237858i 0 −1.00000 0
2449.11 0 1.00000i 0 1.42993 − 1.71910i 0 2.41349i 0 −1.00000 0
2449.12 0 1.00000i 0 2.14424 + 0.634216i 0 − 3.95768i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2449.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4080.2.m.t 12
4.b odd 2 1 2040.2.m.i ✓ 12
5.b even 2 1 inner 4080.2.m.t 12
20.d odd 2 1 2040.2.m.i ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2040.2.m.i ✓ 12 4.b odd 2 1
2040.2.m.i ✓ 12 20.d odd 2 1
4080.2.m.t 12 1.a even 1 1 trivial
4080.2.m.t 12 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4080, [\chi])\):

\( T_{7}^{12} + 38T_{7}^{10} + 501T_{7}^{8} + 2740T_{7}^{6} + 6148T_{7}^{4} + 4864T_{7}^{2} + 256 \) Copy content Toggle raw display
\( T_{11}^{6} + 2T_{11}^{5} - 27T_{11}^{4} - 36T_{11}^{3} + 60T_{11}^{2} + 96T_{11} + 32 \) Copy content Toggle raw display
\( T_{23}^{12} + 140T_{23}^{10} + 6116T_{23}^{8} + 110336T_{23}^{6} + 919808T_{23}^{4} + 3407872T_{23}^{2} + 4194304 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} + 8 T^{9} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( T^{12} + 38 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( (T^{6} + 2 T^{5} - 27 T^{4} + \cdots + 32)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 36 T^{4} + \cdots + 256)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$19$ \( (T^{6} - 10 T^{5} + \cdots - 64)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 140 T^{10} + \cdots + 4194304 \) Copy content Toggle raw display
$29$ \( (T^{6} + 2 T^{5} - 25 T^{4} + \cdots - 64)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 8 T^{5} + \cdots + 10496)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 214 T^{10} + \cdots + 3297856 \) Copy content Toggle raw display
$41$ \( (T^{6} + 2 T^{5} - 107 T^{4} + \cdots - 32)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 211062784 \) Copy content Toggle raw display
$47$ \( T^{12} + 282 T^{10} + \cdots + 16384 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 671017216 \) Copy content Toggle raw display
$59$ \( (T^{6} + 4 T^{5} + \cdots - 64000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 4 T^{5} - 182 T^{4} + \cdots - 64)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 200 T^{4} + \cdots + 246016)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 16 T^{5} + \cdots - 1508096)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 553943296 \) Copy content Toggle raw display
$79$ \( (T^{6} - 12 T^{5} + \cdots + 6784)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 4294967296 \) Copy content Toggle raw display
$89$ \( (T^{6} - 20 T^{5} + \cdots + 177664)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 264 T^{10} + \cdots + 6885376 \) Copy content Toggle raw display
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