Properties

Label 1512.2.k.a
Level $1512$
Weight $2$
Character orbit 1512.k
Analytic conductor $12.073$
Analytic rank $0$
Dimension $16$
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] = [1512,2,Mod(377,1512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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

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: \(12.0733807856\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
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

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

\(f(q)\) \(=\) \( q - \beta_{9} q^{5} - \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{9} q^{5} - \beta_1 q^{7} - \beta_{10} q^{11} - \beta_{5} q^{13} + \beta_{11} q^{17} - \beta_{3} q^{19} + ( - \beta_{15} - \beta_{13}) q^{23} + (\beta_{8} - 1) q^{25} + ( - \beta_{15} - \beta_{10}) q^{29} + (\beta_{5} - \beta_{4}) q^{31} + ( - \beta_{15} + \beta_{11} - \beta_{7}) q^{35} + (\beta_{8} - \beta_{6} - \beta_{2} + \beta_1 - 1) q^{37} + (\beta_{14} + \beta_{11} - \beta_{9}) q^{41} + ( - 2 \beta_{8} + \beta_{2} + 1) q^{43} + (\beta_{14} - \beta_{12} + 2 \beta_{11} + 2 \beta_{9} - \beta_{7}) q^{47} + (\beta_{5} + \beta_{3} + \beta_{2} + \beta_1) q^{49} + ( - 2 \beta_{15} - 2 \beta_{10}) q^{53} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3}) q^{55} + (\beta_{14} + \beta_{12} - 2 \beta_{9} + \beta_{7}) q^{59} + (\beta_{6} + 2 \beta_{3} + \beta_1) q^{61} + ( - \beta_{15} + \beta_{12} - \beta_{10} - \beta_{7}) q^{65} + ( - 2 \beta_{8} + \beta_{6} - \beta_{2} - \beta_1 - 1) q^{67} + ( - \beta_{15} + \beta_{12} - 2 \beta_{10} - \beta_{7}) q^{71} + (\beta_{5} + 2 \beta_{4} + 2 \beta_{3}) q^{73} + (\beta_{14} - \beta_{13} - \beta_{12} - \beta_{9}) q^{77} + ( - \beta_{6} - \beta_{2} + \beta_1 - 3) q^{79} + ( - \beta_{14} + 2 \beta_{11} - 2 \beta_{9}) q^{83} + (\beta_{8} + \beta_{6} - \beta_1 + 1) q^{85} + ( - \beta_{14} - \beta_{12} + \beta_{11} - \beta_{9} - \beta_{7}) q^{89} + ( - 2 \beta_{8} + 2 \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_1 + 2) q^{91} + (\beta_{15} - \beta_{13}) q^{95} + (3 \beta_{6} + 2 \beta_{5} - 4 \beta_{4} + 2 \beta_{3} + 3 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{7} - 12 q^{25} - 8 q^{37} + 8 q^{43} + 2 q^{49} - 28 q^{67} - 44 q^{79} + 16 q^{85} + 18 q^{91}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( - 305389 \nu^{14} + 7316240 \nu^{12} - 69689225 \nu^{10} + 319235369 \nu^{8} - 695756145 \nu^{6} + 747498217 \nu^{4} - 1154216584 \nu^{2} + \cdots + 237296969 ) / 192947092 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2694 \nu^{14} + 61469 \nu^{12} - 552649 \nu^{10} + 2285409 \nu^{8} - 3789335 \nu^{6} + 779859 \nu^{4} - 572101 \nu^{2} - 10061233 ) / 1663337 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2694 \nu^{14} - 61469 \nu^{12} + 552649 \nu^{10} - 2285409 \nu^{8} + 3789335 \nu^{6} - 779859 \nu^{4} + 3898775 \nu^{2} + 81211 ) / 1663337 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11520 \nu^{14} + 293628 \nu^{12} - 3040296 \nu^{10} + 15624254 \nu^{8} - 39217560 \nu^{6} + 38849348 \nu^{4} - 16270592 \nu^{2} + 6370193 ) / 3710521 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1139534 \nu^{14} + 28712659 \nu^{12} - 295900503 \nu^{10} + 1529685268 \nu^{8} - 3939691239 \nu^{6} + 4125515812 \nu^{4} + \cdots + 691999581 ) / 192947092 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1429545 \nu^{14} + 33944011 \nu^{12} - 320576374 \nu^{10} + 1427099575 \nu^{8} - 2709051014 \nu^{6} + 1088455103 \nu^{4} - 1464953121 \nu^{2} + \cdots - 44292248 ) / 192947092 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 458582 \nu^{15} + 16643678 \nu^{13} - 243643736 \nu^{11} + 1832626360 \nu^{9} - 7303260208 \nu^{7} + 13830825760 \nu^{5} + \cdots + 2564873142 \nu ) / 627078049 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 88700 \nu^{14} - 2103563 \nu^{12} + 19736021 \nu^{10} - 86408618 \nu^{8} + 154953525 \nu^{6} - 27036722 \nu^{4} + 23842785 \nu^{2} + 15626169 ) / 6653348 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 4302859 \nu^{15} - 113264303 \nu^{13} + 1228806350 \nu^{11} - 6828867199 \nu^{9} + 19868045926 \nu^{7} - 27619863991 \nu^{5} + \cdots - 10467201680 \nu ) / 2508312196 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 403473 \nu^{15} + 10071623 \nu^{13} - 101747496 \nu^{11} + 504132793 \nu^{9} - 1172875960 \nu^{7} + 854627793 \nu^{5} + 21762817 \nu^{3} + \cdots + 225114682 \nu ) / 86493524 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3300067 \nu^{15} - 75367115 \nu^{13} + 666756659 \nu^{11} - 2588222763 \nu^{9} + 3036067465 \nu^{7} + 3953470823 \nu^{5} - 407547934 \nu^{3} + \cdots + 1131065416 \nu ) / 627078049 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 6600134 \nu^{15} - 150734230 \nu^{13} + 1333513318 \nu^{11} - 5176445526 \nu^{9} + 6072134930 \nu^{7} + 7906941646 \nu^{5} + \cdots + 4770443028 \nu ) / 627078049 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 248063 \nu^{15} + 6021151 \nu^{13} - 58426783 \nu^{11} + 271352435 \nu^{9} - 561779405 \nu^{7} + 316469269 \nu^{5} - 175583178 \nu^{3} + \cdots - 8003320 \nu ) / 21623381 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 10755599 \nu^{15} + 252302706 \nu^{13} - 2332684002 \nu^{11} + 9941641297 \nu^{9} - 16587964242 \nu^{7} + 138651525 \nu^{5} + \cdots + 1098391515 \nu ) / 627078049 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1723809 \nu^{15} + 41505147 \nu^{13} - 399318104 \nu^{11} + 1835444329 \nu^{9} - 3743153672 \nu^{7} + 2039706321 \nu^{5} + \cdots - 140493626 \nu ) / 86493524 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{12} - 2\beta_{11} ) / 4 \) 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
\(\nu^{7}\)\(=\) \( ( 245 \beta_{15} + 7 \beta_{14} - 77 \beta_{13} + 314 \beta_{12} - 99 \beta_{11} - 217 \beta_{10} + 96 \beta_{9} - 217 \beta_{7} ) / 4 \) 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
\(\nu^{9}\)\(=\) \( ( 1728 \beta_{15} + 88 \beta_{14} - 771 \beta_{13} + 1556 \beta_{12} + 629 \beta_{11} - 1344 \beta_{10} - 128 \beta_{9} - 1868 \beta_{7} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 319 \beta_{8} + 2600 \beta_{6} + 3285 \beta_{5} - 8921 \beta_{4} + 11245 \beta_{3} - 1473 \beta_{2} + 3398 \beta _1 - 8758 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 10560 \beta_{15} + 286 \beta_{14} - 5995 \beta_{13} + 4951 \beta_{12} + 12093 \beta_{11} - 7172 \beta_{10} - 7564 \beta_{9} - 14166 \beta_{7} ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 3605 \beta_{8} + 2573 \beta_{6} + 11182 \beta_{5} - 29241 \beta_{4} + 31239 \beta_{3} - 9280 \beta_{2} + 14687 \beta _1 - 56567 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 53248 \beta_{15} - 2884 \beta_{14} - 36868 \beta_{13} - 10003 \beta_{12} + 123074 \beta_{11} - 30784 \beta_{10} - 98032 \beta_{9} - 97052 \beta_{7} ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 41422 \beta_{8} - 36017 \beta_{6} + 58447 \beta_{5} - 148066 \beta_{4} + 133427 \beta_{3} - 83574 \beta_{2} + 112843 \beta _1 - 517223 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 176793 \beta_{15} - 58799 \beta_{14} - 158604 \beta_{13} - 357861 \beta_{12} + 1003586 \beta_{11} - 72873 \beta_{10} - 927572 \beta_{9} - 595917 \beta_{7} ) / 4 \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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
$5$ \( (T^{8} - 17 T^{6} + 83 T^{4} - 91 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + T^{7} + 5 T^{5} - 34 T^{4} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 51 T^{6} + 779 T^{4} + 3801 T^{2} + \cdots + 784)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 55 T^{6} + 956 T^{4} + 5264 T^{2} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 64 T^{6} + 1088 T^{4} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 52 T^{6} + 734 T^{4} + 1700 T^{2} + \cdots + 841)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 163 T^{6} + 8379 T^{4} + \cdots + 559504)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 60 T^{6} + 944 T^{4} + 3648 T^{2} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 61 T^{6} + 791 T^{4} + 3431 T^{2} + \cdots + 3844)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2 T^{3} - 96 T^{2} - 322 T - 161)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} - 193 T^{6} + 11699 T^{4} + \cdots + 1827904)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 2 T^{3} - 132 T^{2} + 376 T + 64)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} - 356 T^{6} + 42608 T^{4} + \cdots + 30647296)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 240 T^{6} + 15104 T^{4} + \cdots + 1048576)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 324 T^{6} + 27504 T^{4} + \cdots + 4064256)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 171 T^{6} + 8496 T^{4} + \cdots + 331776)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 7 T^{3} - 154 T^{2} - 1372 T - 2744)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + 227 T^{6} + 18747 T^{4} + \cdots + 8479744)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 263 T^{6} + 23708 T^{4} + \cdots + 10291264)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 11 T^{3} - 70 T^{2} - 92 T + 184)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} - 532 T^{6} + 85040 T^{4} + \cdots + 51380224)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 361 T^{6} + 28979 T^{4} + \cdots + 8737936)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 491 T^{6} + 87296 T^{4} + \cdots + 177209344)^{2} \) Copy content Toggle raw display
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