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Properties

Label 1360.2.bt.d
Level $1360$
Weight $2$
Character orbit 1360.bt
Analytic conductor $10.860$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1360,2,Mod(81,1360)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1360.81"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1360, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1360 = 2^{4} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1360.bt (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.8596546749\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 18x^{10} + 83x^{8} + 152x^{6} + 111x^{4} + 22x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 85)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + ( - \beta_{6} + \beta_{3}) q^{3} + \beta_{3} q^{5} + (\beta_{8} - \beta_{6} + \beta_{5} + \cdots - \beta_1) q^{7} + ( - \beta_{11} + \beta_{10} + \cdots - \beta_1) q^{9} + (\beta_{9} - \beta_{8} + \beta_{7} + \cdots + 1) q^{11}+ \cdots + ( - 3 \beta_{11} - \beta_{8} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{3} + 4 q^{11} + 12 q^{17} - 16 q^{21} - 12 q^{23} + 4 q^{27} - 12 q^{29} - 16 q^{33} - 16 q^{35} + 12 q^{37} + 20 q^{39} - 24 q^{41} + 8 q^{45} + 48 q^{47} - 32 q^{51} + 40 q^{61} - 12 q^{63}+ \cdots + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 18x^{10} + 83x^{8} + 152x^{6} + 111x^{4} + 22x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} - \nu^{10} + 15 \nu^{9} - 19 \nu^{8} + 34 \nu^{7} - 98 \nu^{6} - 16 \nu^{5} - 188 \nu^{4} + \cdots - 13 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} + \nu^{10} + 15 \nu^{9} + 19 \nu^{8} + 34 \nu^{7} + 98 \nu^{6} - 16 \nu^{5} + 188 \nu^{4} + \cdots + 13 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{10} - 18\nu^{8} - 84\nu^{6} - 166\nu^{4} - 133\nu^{2} - 18 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{11} + 3 \nu^{10} - 19 \nu^{9} + 49 \nu^{8} - 98 \nu^{7} + 168 \nu^{6} - 188 \nu^{5} + 186 \nu^{4} + \cdots + 1 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{11} + 3 \nu^{10} + 19 \nu^{9} + 49 \nu^{8} + 98 \nu^{7} + 168 \nu^{6} + 188 \nu^{5} + 186 \nu^{4} + \cdots + 1 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{11} + 37\nu^{9} + 182\nu^{7} + 354\nu^{5} + 258\nu^{3} + 31\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{11} + 5 \nu^{10} + 19 \nu^{9} + 81 \nu^{8} + 98 \nu^{7} + 268 \nu^{6} + 188 \nu^{5} + 258 \nu^{4} + \cdots - 3 ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 9 \nu^{11} - \nu^{10} + 159 \nu^{9} - 15 \nu^{8} + 696 \nu^{7} - 34 \nu^{6} + 1170 \nu^{5} + 16 \nu^{4} + \cdots + 19 ) / 8 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -5\nu^{11} - 87\nu^{9} - 366\nu^{7} - 592\nu^{5} - 369\nu^{3} - 61\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 9 \nu^{11} - \nu^{10} - 159 \nu^{9} - 15 \nu^{8} - 696 \nu^{7} - 34 \nu^{6} - 1170 \nu^{5} + \cdots + 19 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{9} - \beta_{8} + 2\beta_{6} + \beta_{5} + \beta_{4} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{11} - 3\beta_{10} - 2\beta_{9} + 3\beta_{6} - 3\beta_{5} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -9\beta_{11} - 9\beta_{9} + 16\beta_{8} - 28\beta_{6} - 12\beta_{5} - 12\beta_{4} - \beta_{3} + \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 28 \beta_{11} + 40 \beta_{10} + 28 \beta_{9} - 4 \beta_{7} - 41 \beta_{6} + 41 \beta_{5} + \cdots + 61 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 102 \beta_{11} + 102 \beta_{9} - 203 \beta_{8} + 348 \beta_{6} + 145 \beta_{5} + 143 \beta_{4} + \cdots - 21 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 348 \beta_{11} - 493 \beta_{10} - 348 \beta_{9} + 60 \beta_{7} + 504 \beta_{6} - 504 \beta_{5} + \cdots - 718 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1222 \beta_{11} - 1222 \beta_{9} + 2482 \beta_{8} - 4240 \beta_{6} - 1758 \beta_{5} - 1726 \beta_{4} + \cdots + 225 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 4240 \beta_{11} + 5998 \beta_{10} + 4240 \beta_{9} - 756 \beta_{7} - 6122 \beta_{6} + \cdots + 8667 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 14789 \beta_{11} + 14789 \beta_{9} - 30147 \beta_{8} + 51470 \beta_{6} + 21323 \beta_{5} + 20911 \beta_{4} + \cdots - 2669 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 51470 \beta_{11} - 72793 \beta_{10} - 51470 \beta_{9} + 9236 \beta_{7} + 74263 \beta_{6} + \cdots - 105040 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1360\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(341\) \(511\) \(817\)
\(\chi(n)\) \(-\beta_{7}\) \(1\) \(1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
− 1.52346i
0.455023i
1.19804i
− 0.254679i
− 1.35757i
3.48265i
1.52346i
− 0.455023i
− 1.19804i
0.254679i
1.35757i
− 3.48265i
0 −1.78436 − 1.78436i 0 −0.707107 − 0.707107i 0 0.260895 − 0.260895i 0 3.36786i 0
81.2 0 −0.385357 − 0.385357i 0 −0.707107 − 0.707107i 0 0.840380 − 0.840380i 0 − 2.70300i 0
81.3 0 −0.140032 − 0.140032i 0 0.707107 + 0.707107i 0 1.33807 − 1.33807i 0 − 2.96078i 0
81.4 0 0.887192 + 0.887192i 0 0.707107 + 0.707107i 0 −1.14187 + 1.14187i 0 − 1.42578i 0
81.5 0 1.66705 + 1.66705i 0 0.707107 + 0.707107i 0 −3.02462 + 3.02462i 0 2.55814i 0
81.6 0 1.75550 + 1.75550i 0 −0.707107 − 0.707107i 0 1.72715 − 1.72715i 0 3.16356i 0
1041.1 0 −1.78436 + 1.78436i 0 −0.707107 + 0.707107i 0 0.260895 + 0.260895i 0 − 3.36786i 0
1041.2 0 −0.385357 + 0.385357i 0 −0.707107 + 0.707107i 0 0.840380 + 0.840380i 0 2.70300i 0
1041.3 0 −0.140032 + 0.140032i 0 0.707107 − 0.707107i 0 1.33807 + 1.33807i 0 2.96078i 0
1041.4 0 0.887192 − 0.887192i 0 0.707107 − 0.707107i 0 −1.14187 − 1.14187i 0 1.42578i 0
1041.5 0 1.66705 − 1.66705i 0 0.707107 − 0.707107i 0 −3.02462 − 3.02462i 0 − 2.55814i 0
1041.6 0 1.75550 − 1.75550i 0 −0.707107 + 0.707107i 0 1.72715 + 1.72715i 0 − 3.16356i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1360.2.bt.d 12
4.b odd 2 1 85.2.e.a ✓ 12
12.b even 2 1 765.2.k.b 12
17.c even 4 1 inner 1360.2.bt.d 12
20.d odd 2 1 425.2.e.f 12
20.e even 4 1 425.2.j.b 12
20.e even 4 1 425.2.j.c 12
68.f odd 4 1 85.2.e.a ✓ 12
68.g odd 8 1 1445.2.a.n 6
68.g odd 8 1 1445.2.a.o 6
68.g odd 8 2 1445.2.d.g 12
204.l even 4 1 765.2.k.b 12
340.i even 4 1 425.2.j.c 12
340.n odd 4 1 425.2.e.f 12
340.s even 4 1 425.2.j.b 12
340.ba odd 8 1 7225.2.a.z 6
340.ba odd 8 1 7225.2.a.bb 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.2.e.a ✓ 12 4.b odd 2 1
85.2.e.a ✓ 12 68.f odd 4 1
425.2.e.f 12 20.d odd 2 1
425.2.e.f 12 340.n odd 4 1
425.2.j.b 12 20.e even 4 1
425.2.j.b 12 340.s even 4 1
425.2.j.c 12 20.e even 4 1
425.2.j.c 12 340.i even 4 1
765.2.k.b 12 12.b even 2 1
765.2.k.b 12 204.l even 4 1
1360.2.bt.d 12 1.a even 1 1 trivial
1360.2.bt.d 12 17.c even 4 1 inner
1445.2.a.n 6 68.g odd 8 1
1445.2.a.o 6 68.g odd 8 1
1445.2.d.g 12 68.g odd 8 2
7225.2.a.z 6 340.ba odd 8 1
7225.2.a.bb 6 340.ba odd 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 4 T_{3}^{11} + 8 T_{3}^{10} - 4 T_{3}^{9} + 40 T_{3}^{8} - 160 T_{3}^{7} + 328 T_{3}^{6} + \cdots + 4 \) acting on \(S_{2}^{\mathrm{new}}(1360, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 4 T^{11} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( (T^{4} + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{12} - 28 T^{9} + \cdots + 196 \) Copy content Toggle raw display
$11$ \( T^{12} - 4 T^{11} + \cdots + 198916 \) Copy content Toggle raw display
$13$ \( (T^{6} - 58 T^{4} + \cdots - 316)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} - 12 T^{11} + \cdots + 24137569 \) Copy content Toggle raw display
$19$ \( T^{12} + 136 T^{10} + \cdots + 2166784 \) Copy content Toggle raw display
$23$ \( T^{12} + 12 T^{11} + \cdots + 196 \) Copy content Toggle raw display
$29$ \( T^{12} + 12 T^{11} + \cdots + 5345344 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 1645275844 \) Copy content Toggle raw display
$37$ \( T^{12} - 12 T^{11} + \cdots + 3655744 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 192876544 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 225120016 \) Copy content Toggle raw display
$47$ \( (T^{6} - 24 T^{5} + \cdots - 18076)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 17453580544 \) Copy content Toggle raw display
$59$ \( T^{12} + 240 T^{10} + \cdots + 802816 \) Copy content Toggle raw display
$61$ \( T^{12} - 40 T^{11} + \cdots + 984064 \) Copy content Toggle raw display
$67$ \( (T^{6} - 4 T^{5} - 46 T^{4} + \cdots - 92)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 29133710596 \) Copy content Toggle raw display
$73$ \( T^{12} + 48 T^{11} + \cdots + 541696 \) Copy content Toggle raw display
$79$ \( T^{12} - 8 T^{11} + \cdots + 15225604 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 125529616 \) Copy content Toggle raw display
$89$ \( (T^{6} - 12 T^{5} + \cdots - 31292)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 227086559296 \) Copy content Toggle raw display
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