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Properties

Label 1512.2.k.a
Level $1512$
Weight $2$
Character orbit 1512.k
Analytic conductor $12.073$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1512,2,Mod(377,1512)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1512, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1512.377"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1512.k (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.0733807856\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 24x^{14} + 230x^{12} - 1052x^{10} + 2139x^{8} - 1244x^{6} + 1134x^{4} - 104x^{2} + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{9} q^{5} - \beta_1 q^{7} - \beta_{10} q^{11} - \beta_{5} q^{13} + \beta_{11} q^{17} - \beta_{3} q^{19} + ( - \beta_{15} - \beta_{13}) q^{23} + (\beta_{8} - 1) q^{25} + ( - \beta_{15} - \beta_{10}) q^{29}+ \cdots + (3 \beta_{6} + 2 \beta_{5} + \cdots + 3 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{7} - 12 q^{25} - 8 q^{37} + 8 q^{43} + 2 q^{49} - 28 q^{67} - 44 q^{79} + 16 q^{85} + 18 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 24x^{14} + 230x^{12} - 1052x^{10} + 2139x^{8} - 1244x^{6} + 1134x^{4} - 104x^{2} + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 305389 \nu^{14} + 7316240 \nu^{12} - 69689225 \nu^{10} + 319235369 \nu^{8} - 695756145 \nu^{6} + \cdots + 237296969 ) / 192947092 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2694 \nu^{14} + 61469 \nu^{12} - 552649 \nu^{10} + 2285409 \nu^{8} - 3789335 \nu^{6} + \cdots - 10061233 ) / 1663337 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2694 \nu^{14} - 61469 \nu^{12} + 552649 \nu^{10} - 2285409 \nu^{8} + 3789335 \nu^{6} + \cdots + 81211 ) / 1663337 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11520 \nu^{14} + 293628 \nu^{12} - 3040296 \nu^{10} + 15624254 \nu^{8} - 39217560 \nu^{6} + \cdots + 6370193 ) / 3710521 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1139534 \nu^{14} + 28712659 \nu^{12} - 295900503 \nu^{10} + 1529685268 \nu^{8} + \cdots + 691999581 ) / 192947092 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1429545 \nu^{14} + 33944011 \nu^{12} - 320576374 \nu^{10} + 1427099575 \nu^{8} + \cdots - 44292248 ) / 192947092 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 458582 \nu^{15} + 16643678 \nu^{13} - 243643736 \nu^{11} + 1832626360 \nu^{9} + \cdots + 2564873142 \nu ) / 627078049 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 88700 \nu^{14} - 2103563 \nu^{12} + 19736021 \nu^{10} - 86408618 \nu^{8} + 154953525 \nu^{6} + \cdots + 15626169 ) / 6653348 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 4302859 \nu^{15} - 113264303 \nu^{13} + 1228806350 \nu^{11} - 6828867199 \nu^{9} + \cdots - 10467201680 \nu ) / 2508312196 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 403473 \nu^{15} + 10071623 \nu^{13} - 101747496 \nu^{11} + 504132793 \nu^{9} + \cdots + 225114682 \nu ) / 86493524 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3300067 \nu^{15} - 75367115 \nu^{13} + 666756659 \nu^{11} - 2588222763 \nu^{9} + \cdots + 1131065416 \nu ) / 627078049 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 6600134 \nu^{15} - 150734230 \nu^{13} + 1333513318 \nu^{11} - 5176445526 \nu^{9} + \cdots + 4770443028 \nu ) / 627078049 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 248063 \nu^{15} + 6021151 \nu^{13} - 58426783 \nu^{11} + 271352435 \nu^{9} + \cdots - 8003320 \nu ) / 21623381 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 10755599 \nu^{15} + 252302706 \nu^{13} - 2332684002 \nu^{11} + 9941641297 \nu^{9} + \cdots + 1098391515 \nu ) / 627078049 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1723809 \nu^{15} + 41505147 \nu^{13} - 399318104 \nu^{11} + 1835444329 \nu^{9} + \cdots - 140493626 \nu ) / 86493524 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{12} - 2\beta_{11} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{15} - \beta_{14} + 7\beta_{12} - 10\beta_{11} - 3\beta_{10} + 4\beta_{9} - \beta_{7} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{8} + 4\beta_{6} + \beta_{5} - 4\beta_{4} + 13\beta_{3} + 5\beta_{2} + 2\beta _1 + 29 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 30\beta_{15} - 2\beta_{14} - 5\beta_{13} + 51\beta_{12} - 43\beta_{11} - 30\beta_{10} + 24\beta_{9} - 20\beta_{7} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 13\beta_{8} + 78\beta_{6} + 35\beta_{5} - 113\beta_{4} + 231\beta_{3} + 35\beta_{2} + 36\beta _1 + 212 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 245 \beta_{15} + 7 \beta_{14} - 77 \beta_{13} + 314 \beta_{12} - 99 \beta_{11} - 217 \beta_{10} + \cdots - 217 \beta_{7} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 20\beta_{8} + 266\beta_{6} + 194\beta_{5} - 559\beta_{4} + 864\beta_{3} + 2\beta_{2} + 178\beta _1 + 29 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1728 \beta_{15} + 88 \beta_{14} - 771 \beta_{13} + 1556 \beta_{12} + 629 \beta_{11} + \cdots - 1868 \beta_{7} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 319 \beta_{8} + 2600 \beta_{6} + 3285 \beta_{5} - 8921 \beta_{4} + 11245 \beta_{3} - 1473 \beta_{2} + \cdots - 8758 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 10560 \beta_{15} + 286 \beta_{14} - 5995 \beta_{13} + 4951 \beta_{12} + 12093 \beta_{11} + \cdots - 14166 \beta_{7} ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 3605 \beta_{8} + 2573 \beta_{6} + 11182 \beta_{5} - 29241 \beta_{4} + 31239 \beta_{3} - 9280 \beta_{2} + \cdots - 56567 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 53248 \beta_{15} - 2884 \beta_{14} - 36868 \beta_{13} - 10003 \beta_{12} + 123074 \beta_{11} + \cdots - 97052 \beta_{7} ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 41422 \beta_{8} - 36017 \beta_{6} + 58447 \beta_{5} - 148066 \beta_{4} + 133427 \beta_{3} + \cdots - 517223 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 176793 \beta_{15} - 58799 \beta_{14} - 158604 \beta_{13} - 357861 \beta_{12} + 1003586 \beta_{11} + \cdots - 595917 \beta_{7} ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1081\) \(1135\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
377.1
2.62616 − 0.500000i
2.62616 + 0.500000i
−0.415570 + 0.500000i
−0.415570 − 0.500000i
−2.32849 + 0.500000i
−2.32849 − 0.500000i
−0.713245 + 0.500000i
−0.713245 − 0.500000i
0.713245 − 0.500000i
0.713245 + 0.500000i
2.32849 − 0.500000i
2.32849 + 0.500000i
0.415570 − 0.500000i
0.415570 + 0.500000i
−2.62616 + 0.500000i
−2.62616 − 0.500000i
0 0 0 −2.86833 0 2.43500 − 1.03478i 0 0 0
377.2 0 0 0 −2.86833 0 2.43500 + 1.03478i 0 0 0
377.3 0 0 0 −2.70790 0 −0.946562 − 2.47063i 0 0 0
377.4 0 0 0 −2.70790 0 −0.946562 + 2.47063i 0 0 0
377.5 0 0 0 −1.10598 0 −2.64465 − 0.0763047i 0 0 0
377.6 0 0 0 −1.10598 0 −2.64465 + 0.0763047i 0 0 0
377.7 0 0 0 −0.465643 0 0.656211 − 2.56308i 0 0 0
377.8 0 0 0 −0.465643 0 0.656211 + 2.56308i 0 0 0
377.9 0 0 0 0.465643 0 0.656211 − 2.56308i 0 0 0
377.10 0 0 0 0.465643 0 0.656211 + 2.56308i 0 0 0
377.11 0 0 0 1.10598 0 −2.64465 − 0.0763047i 0 0 0
377.12 0 0 0 1.10598 0 −2.64465 + 0.0763047i 0 0 0
377.13 0 0 0 2.70790 0 −0.946562 − 2.47063i 0 0 0
377.14 0 0 0 2.70790 0 −0.946562 + 2.47063i 0 0 0
377.15 0 0 0 2.86833 0 2.43500 − 1.03478i 0 0 0
377.16 0 0 0 2.86833 0 2.43500 + 1.03478i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 377.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1512.2.k.a ✓ 16
3.b odd 2 1 inner 1512.2.k.a ✓ 16
4.b odd 2 1 3024.2.k.k 16
7.b odd 2 1 inner 1512.2.k.a ✓ 16
12.b even 2 1 3024.2.k.k 16
21.c even 2 1 inner 1512.2.k.a ✓ 16
28.d even 2 1 3024.2.k.k 16
84.h odd 2 1 3024.2.k.k 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.k.a ✓ 16 1.a even 1 1 trivial
1512.2.k.a ✓ 16 3.b odd 2 1 inner
1512.2.k.a ✓ 16 7.b odd 2 1 inner
1512.2.k.a ✓ 16 21.c even 2 1 inner
3024.2.k.k 16 4.b odd 2 1
3024.2.k.k 16 12.b even 2 1
3024.2.k.k 16 28.d even 2 1
3024.2.k.k 16 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 17T_{5}^{6} + 83T_{5}^{4} - 91T_{5}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(1512, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 17 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + T^{7} + 5 T^{5} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 51 T^{6} + \cdots + 784)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 55 T^{6} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 64 T^{6} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 52 T^{6} + \cdots + 841)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 163 T^{6} + \cdots + 559504)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 60 T^{6} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 61 T^{6} + \cdots + 3844)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2 T^{3} + \cdots - 161)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} - 193 T^{6} + \cdots + 1827904)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 2 T^{3} - 132 T^{2} + \cdots + 64)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} - 356 T^{6} + \cdots + 30647296)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 240 T^{6} + \cdots + 1048576)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 324 T^{6} + \cdots + 4064256)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 171 T^{6} + \cdots + 331776)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 7 T^{3} + \cdots - 2744)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + 227 T^{6} + \cdots + 8479744)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 263 T^{6} + \cdots + 10291264)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 11 T^{3} + \cdots + 184)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} - 532 T^{6} + \cdots + 51380224)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 361 T^{6} + \cdots + 8737936)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 491 T^{6} + \cdots + 177209344)^{2} \) Copy content Toggle raw display
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