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Properties

Label 912.2.bo.k
Level $912$
Weight $2$
Character orbit 912.bo
Analytic conductor $7.282$
Analytic rank $0$
Dimension $18$
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] = [912,2,Mod(289,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.bo (of order \(9\), degree \(6\), 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: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 3 x^{17} + 30 x^{16} - 51 x^{15} + 501 x^{14} - 768 x^{13} + 4499 x^{12} - 2946 x^{11} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 456)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{3} + ( - \beta_{4} + \beta_{2}) q^{5} + (\beta_{16} - \beta_{6} - \beta_{5} + \cdots + \beta_1) q^{7}+ \cdots - \beta_{9} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{3} + ( - \beta_{4} + \beta_{2}) q^{5} + (\beta_{16} - \beta_{6} - \beta_{5} + \cdots + \beta_1) q^{7}+ \cdots + ( - \beta_{17} + \beta_{13} - \beta_{2} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 3 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 3 q^{5} + 3 q^{11} + 18 q^{13} - 3 q^{15} - 3 q^{17} + 3 q^{19} - 3 q^{21} - 12 q^{23} - 3 q^{25} - 9 q^{27} + 3 q^{29} - 21 q^{31} + 6 q^{33} - 12 q^{35} - 30 q^{37} - 6 q^{39} + 33 q^{41} + 6 q^{43} - 3 q^{45} + 6 q^{47} - 21 q^{49} - 12 q^{51} + 30 q^{53} + 24 q^{55} - 21 q^{57} + 9 q^{59} + 48 q^{61} + 6 q^{63} - 9 q^{65} + 18 q^{67} - 6 q^{69} + 51 q^{71} + 12 q^{75} + 30 q^{77} + 39 q^{79} + 30 q^{83} - 87 q^{85} + 21 q^{87} + 45 q^{89} - 72 q^{91} - 27 q^{93} + 78 q^{95} + 39 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 3 x^{17} + 30 x^{16} - 51 x^{15} + 501 x^{14} - 768 x^{13} + 4499 x^{12} - 2946 x^{11} + \cdots + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 30\!\cdots\!43 \nu^{17} + \cdots - 13\!\cdots\!16 ) / 10\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 77\!\cdots\!03 \nu^{17} + \cdots + 36\!\cdots\!04 ) / 10\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 66\!\cdots\!75 \nu^{17} + \cdots - 81\!\cdots\!04 ) / 54\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 10\!\cdots\!75 \nu^{17} + \cdots + 78\!\cdots\!80 ) / 57\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 30\!\cdots\!45 \nu^{17} + \cdots - 69\!\cdots\!64 ) / 10\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 78\!\cdots\!81 \nu^{17} + \cdots - 17\!\cdots\!08 ) / 21\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 32\!\cdots\!65 \nu^{17} + \cdots - 25\!\cdots\!76 ) / 54\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 39\!\cdots\!84 \nu^{17} + \cdots - 35\!\cdots\!76 ) / 54\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 37\!\cdots\!63 \nu^{17} + \cdots - 11\!\cdots\!36 ) / 21\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 21\!\cdots\!19 \nu^{17} + \cdots + 17\!\cdots\!08 ) / 10\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 12\!\cdots\!20 \nu^{17} + \cdots - 18\!\cdots\!88 ) / 57\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 65\!\cdots\!95 \nu^{17} + \cdots - 21\!\cdots\!84 ) / 21\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 76\!\cdots\!59 \nu^{17} + \cdots + 33\!\cdots\!84 ) / 21\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 91\!\cdots\!53 \nu^{17} + \cdots - 69\!\cdots\!08 ) / 21\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 60\!\cdots\!57 \nu^{17} + \cdots - 26\!\cdots\!36 ) / 10\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 12\!\cdots\!83 \nu^{17} + \cdots - 53\!\cdots\!60 ) / 21\!\cdots\!28 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( - \beta_{17} - \beta_{15} - \beta_{14} - 4 \beta_{12} - 2 \beta_{11} + \beta_{10} + \beta_{9} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} + 2 \beta_{16} - \beta_{15} - \beta_{14} + 2 \beta_{13} + \beta_{12} - 2 \beta_{11} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 13 \beta_{17} - 2 \beta_{15} + 11 \beta_{14} + \beta_{13} + 44 \beta_{12} - 4 \beta_{11} - 16 \beta_{10} + \cdots - 11 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 3 \beta_{17} - 39 \beta_{16} - 3 \beta_{15} + 19 \beta_{14} + 36 \beta_{12} - 75 \beta_{11} + \cdots + 36 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 47 \beta_{17} - 36 \beta_{16} + 185 \beta_{15} + 47 \beta_{14} - 36 \beta_{13} - 47 \beta_{12} + \cdots + 680 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 341 \beta_{17} + 392 \beta_{15} + 51 \beta_{14} - 650 \beta_{13} - 1024 \beta_{12} + 2426 \beta_{11} + \cdots - 51 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1876 \beta_{17} + 850 \beta_{16} - 1876 \beta_{15} - 2800 \beta_{14} - 7738 \beta_{12} - 3834 \beta_{11} + \cdots - 7738 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 6542 \beta_{17} + 10426 \beta_{16} - 6047 \beta_{15} - 6542 \beta_{14} + 10426 \beta_{13} + 6542 \beta_{12} + \cdots - 10929 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 43703 \beta_{17} - 16927 \beta_{15} + 26776 \beta_{14} + 17258 \beta_{13} + 133369 \beta_{12} + \cdots - 26776 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 565 \beta_{17} - 165645 \beta_{16} - 565 \beta_{15} + 106327 \beta_{14} + 207770 \beta_{12} + \cdots + 207770 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 299424 \beta_{17} - 326485 \beta_{16} + 693297 \beta_{15} + 299424 \beta_{14} - 326485 \beta_{13} + \cdots + 2195897 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1853037 \beta_{17} + 1736549 \beta_{15} - 116488 \beta_{14} - 2632308 \beta_{13} - 5481675 \beta_{12} + \cdots + 116488 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 5910389 \beta_{17} + 5940602 \beta_{16} - 5910389 \beta_{15} - 11106121 \beta_{14} - 28343588 \beta_{12} + \cdots - 28343588 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 28209650 \beta_{17} + 41984978 \beta_{16} - 32020532 \beta_{15} - 28209650 \beta_{14} + 41984978 \beta_{13} + \cdots - 70420660 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 179102021 \beta_{17} - 89119356 \beta_{15} + 89982665 \beta_{14} + 105527504 \beta_{13} + \cdots - 89982665 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 90062544 \beta_{17} - 672819662 \beta_{16} + 90062544 \beta_{15} + 549208681 \beta_{14} + \cdots + 1169446746 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(-\beta_{10}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
−0.459637 − 0.796115i
−0.214696 − 0.371865i
1.34798 + 2.33477i
−0.459637 + 0.796115i
−0.214696 + 0.371865i
1.34798 − 2.33477i
−1.84854 − 3.20177i
0.0657218 + 0.113834i
1.34313 + 2.32637i
−1.84854 + 3.20177i
0.0657218 − 0.113834i
1.34313 − 2.32637i
2.03693 − 3.52807i
0.323050 − 0.559539i
−1.09394 + 1.89476i
2.03693 + 3.52807i
0.323050 + 0.559539i
−1.09394 − 1.89476i
0 0.173648 + 0.984808i 0 −0.704205 + 0.590899i 0 0.0532867 − 0.0922952i 0 −0.939693 + 0.342020i 0
289.2 0 0.173648 + 0.984808i 0 −0.328934 + 0.276008i 0 1.95470 − 3.38563i 0 −0.939693 + 0.342020i 0
289.3 0 0.173648 + 0.984808i 0 2.06523 − 1.73293i 0 −2.18163 + 3.77870i 0 −0.939693 + 0.342020i 0
385.1 0 0.173648 − 0.984808i 0 −0.704205 − 0.590899i 0 0.0532867 + 0.0922952i 0 −0.939693 − 0.342020i 0
385.2 0 0.173648 − 0.984808i 0 −0.328934 − 0.276008i 0 1.95470 + 3.38563i 0 −0.939693 − 0.342020i 0
385.3 0 0.173648 − 0.984808i 0 2.06523 + 1.73293i 0 −2.18163 − 3.77870i 0 −0.939693 − 0.342020i 0
481.1 0 −0.939693 − 0.342020i 0 −0.641992 − 3.64092i 0 1.29967 − 2.25110i 0 0.766044 + 0.642788i 0
481.2 0 −0.939693 − 0.342020i 0 0.0228249 + 0.129447i 0 −1.43643 + 2.48797i 0 0.766044 + 0.642788i 0
481.3 0 −0.939693 − 0.342020i 0 0.466463 + 2.64544i 0 1.07645 − 1.86447i 0 0.766044 + 0.642788i 0
529.1 0 −0.939693 + 0.342020i 0 −0.641992 + 3.64092i 0 1.29967 + 2.25110i 0 0.766044 − 0.642788i 0
529.2 0 −0.939693 + 0.342020i 0 0.0228249 − 0.129447i 0 −1.43643 − 2.48797i 0 0.766044 − 0.642788i 0
529.3 0 −0.939693 + 0.342020i 0 0.466463 − 2.64544i 0 1.07645 + 1.86447i 0 0.766044 − 0.642788i 0
625.1 0 0.766044 + 0.642788i 0 −3.82818 + 1.39334i 0 −0.631069 − 1.09304i 0 0.173648 + 0.984808i 0
625.2 0 0.766044 + 0.642788i 0 −0.607135 + 0.220979i 0 1.81642 + 3.14613i 0 0.173648 + 0.984808i 0
625.3 0 0.766044 + 0.642788i 0 2.05593 − 0.748299i 0 −1.95140 − 3.37992i 0 0.173648 + 0.984808i 0
769.1 0 0.766044 − 0.642788i 0 −3.82818 − 1.39334i 0 −0.631069 + 1.09304i 0 0.173648 − 0.984808i 0
769.2 0 0.766044 − 0.642788i 0 −0.607135 − 0.220979i 0 1.81642 − 3.14613i 0 0.173648 − 0.984808i 0
769.3 0 0.766044 − 0.642788i 0 2.05593 + 0.748299i 0 −1.95140 + 3.37992i 0 0.173648 − 0.984808i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.bo.k 18
4.b odd 2 1 456.2.bg.c ✓ 18
19.e even 9 1 inner 912.2.bo.k 18
76.k even 18 1 8664.2.a.bq 9
76.l odd 18 1 456.2.bg.c ✓ 18
76.l odd 18 1 8664.2.a.bo 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.bg.c ✓ 18 4.b odd 2 1
456.2.bg.c ✓ 18 76.l odd 18 1
912.2.bo.k 18 1.a even 1 1 trivial
912.2.bo.k 18 19.e even 9 1 inner
8664.2.a.bo 9 76.l odd 18 1
8664.2.a.bq 9 76.k even 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{18} + 3 T_{5}^{17} + 6 T_{5}^{16} + 24 T_{5}^{15} - 36 T_{5}^{14} - 27 T_{5}^{13} + 1481 T_{5}^{12} + \cdots + 64 \) acting on \(S_{2}^{\mathrm{new}}(912, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( (T^{6} + T^{3} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{18} + 3 T^{17} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( T^{18} + 42 T^{16} + \cdots + 273529 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 1145687104 \) Copy content Toggle raw display
$13$ \( T^{18} - 18 T^{17} + \cdots + 2368521 \) Copy content Toggle raw display
$17$ \( T^{18} + 3 T^{17} + \cdots + 74235456 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 322687697779 \) Copy content Toggle raw display
$23$ \( T^{18} + 12 T^{17} + \cdots + 5184 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 551264670784 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 739388455129 \) Copy content Toggle raw display
$37$ \( (T^{9} + 15 T^{8} + \cdots - 52260283)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 17825454144 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 232537092841 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 42\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 29\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 591065740864 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 536862478681 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 982760804168256 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 186880492339776 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 58\!\cdots\!09 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 124037091721 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 312105363582016 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 8487806331456 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 104695686232281 \) Copy content Toggle raw display
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