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Properties

Label 1224.2.bq.e
Level $1224$
Weight $2$
Character orbit 1224.bq
Analytic conductor $9.774$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1224,2,Mod(145,1224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1224, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1224.145");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1224.bq (of order \(8\), degree \(4\), 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: \(9.77368920740\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 36x^{14} + 466x^{12} + 2956x^{10} + 10049x^{8} + 18032x^{6} + 14800x^{4} + 3200x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 408)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{14} + \beta_{7} + \beta_{4}) q^{5} + ( - \beta_{10} + \beta_{6} + \cdots + \beta_1) q^{7} + ( - \beta_{14} + \beta_{13} + \cdots - \beta_{2}) q^{11} + (\beta_{15} + \beta_{14} + \cdots + \beta_{3}) q^{13}+ \cdots + ( - \beta_{15} - 2 \beta_{14} + \cdots + 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{11} - 8 q^{19} + 8 q^{23} - 8 q^{25} - 32 q^{29} + 8 q^{31} - 16 q^{35} + 8 q^{37} - 40 q^{41} - 24 q^{43} + 24 q^{49} + 24 q^{53} + 16 q^{59} + 64 q^{65} + 16 q^{71} + 24 q^{73} - 96 q^{79}+ \cdots + 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 36x^{14} + 466x^{12} + 2956x^{10} + 10049x^{8} + 18032x^{6} + 14800x^{4} + 3200x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2 \nu^{15} + 12 \nu^{14} + 51 \nu^{13} + 393 \nu^{12} + 264 \nu^{11} + 4324 \nu^{10} - 1058 \nu^{9} + \cdots - 2168 ) / 1088 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2 \nu^{15} - 12 \nu^{14} + 51 \nu^{13} - 393 \nu^{12} + 264 \nu^{11} - 4324 \nu^{10} - 1058 \nu^{9} + \cdots + 1080 ) / 1088 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 23 \nu^{15} - 776 \nu^{13} - 9062 \nu^{11} - 50684 \nu^{9} - 150727 \nu^{7} - 238604 \nu^{5} + \cdots - 28576 \nu ) / 2176 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 47 \nu^{15} + 1492 \nu^{13} + 15446 \nu^{11} + 69732 \nu^{9} + 137983 \nu^{7} + 75336 \nu^{5} + \cdots - 30720 \nu ) / 4352 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 39 \nu^{14} - 1288 \nu^{12} - 14390 \nu^{10} - 73964 \nu^{8} - 187047 \nu^{6} - 214076 \nu^{4} + \cdots - 7456 ) / 2176 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 15 \nu^{15} + 124 \nu^{14} + 454 \nu^{13} + 4032 \nu^{12} + 4174 \nu^{11} + 43768 \nu^{10} + \cdots + 6016 ) / 4352 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 15 \nu^{15} - 124 \nu^{14} + 454 \nu^{13} - 4032 \nu^{12} + 4174 \nu^{11} - 43768 \nu^{10} + \cdots - 6016 ) / 4352 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 29 \nu^{15} + 3 \nu^{14} + 957 \nu^{13} + 127 \nu^{12} + 10678 \nu^{11} + 1938 \nu^{10} + 54822 \nu^{9} + \cdots + 1048 ) / 1088 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 35 \nu^{15} + 67 \nu^{14} + 1138 \nu^{13} + 2194 \nu^{12} + 12294 \nu^{11} + 24086 \nu^{10} + \cdots - 1680 ) / 2176 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 29 \nu^{15} + 3 \nu^{14} - 957 \nu^{13} + 127 \nu^{12} - 10678 \nu^{11} + 1938 \nu^{10} + \cdots + 1048 ) / 1088 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 81 \nu^{15} - 123 \nu^{14} - 2770 \nu^{13} - 3952 \nu^{12} - 32850 \nu^{11} - 41918 \nu^{10} + \cdots + 11680 ) / 4352 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 35 \nu^{15} + 67 \nu^{14} - 1138 \nu^{13} + 2194 \nu^{12} - 12294 \nu^{11} + 24086 \nu^{10} + \cdots - 1680 ) / 2176 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 81 \nu^{15} + 123 \nu^{14} - 2770 \nu^{13} + 3952 \nu^{12} - 32850 \nu^{11} + 41918 \nu^{10} + \cdots - 11680 ) / 4352 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 22 \nu^{15} + 245 \nu^{14} - 770 \nu^{13} + 8104 \nu^{12} - 9652 \nu^{11} + 90498 \nu^{10} + \cdots + 352 ) / 4352 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 22 \nu^{15} - 245 \nu^{14} - 770 \nu^{13} - 8104 \nu^{12} - 9652 \nu^{11} - 90498 \nu^{10} + \cdots - 352 ) / 4352 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{12} - \beta_{10} - \beta_{9} + \beta_{8} + 2\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( - \beta_{13} + 2 \beta_{12} + \beta_{11} - \beta_{10} + 2 \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 6 \beta_{15} - 6 \beta_{14} - 2 \beta_{13} - 11 \beta_{12} - 2 \beta_{11} + 15 \beta_{10} + 11 \beta_{9} + \cdots - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 5 \beta_{15} + 5 \beta_{14} + 16 \beta_{13} - 44 \beta_{12} - 16 \beta_{11} + 14 \beta_{10} + \cdots + 39 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 126 \beta_{15} + 126 \beta_{14} + 50 \beta_{13} + 167 \beta_{12} + 50 \beta_{11} - 257 \beta_{10} + \cdots + 108 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 108 \beta_{15} - 108 \beta_{14} - 275 \beta_{13} + 796 \beta_{12} + 275 \beta_{11} - 225 \beta_{10} + \cdots - 532 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2258 \beta_{15} - 2258 \beta_{14} - 926 \beta_{13} - 2785 \beta_{12} - 926 \beta_{11} + 4393 \beta_{10} + \cdots - 1806 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1933 \beta_{15} + 1933 \beta_{14} + 4718 \beta_{13} - 13740 \beta_{12} - 4718 \beta_{11} + \cdots + 8355 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 38890 \beta_{15} + 38890 \beta_{14} + 16070 \beta_{13} + 47145 \beta_{12} + 16070 \beta_{11} + \cdots + 30256 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 33206 \beta_{15} - 33206 \beta_{14} - 80351 \beta_{13} + 233944 \beta_{12} + 80351 \beta_{11} + \cdots - 137862 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 661806 \beta_{15} - 661806 \beta_{14} - 273970 \beta_{13} - 798691 \beta_{12} - 273970 \beta_{11} + \cdots - 509542 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 564289 \beta_{15} + 564289 \beta_{14} + 1362980 \beta_{13} - 3966012 \beta_{12} - 1362980 \beta_{11} + \cdots + 2312571 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 11218030 \beta_{15} + 11218030 \beta_{14} + 4646018 \beta_{13} + 13521263 \beta_{12} + 4646018 \beta_{11} + \cdots + 8604244 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 9559272 \beta_{15} - 9559272 \beta_{14} - 23080535 \beta_{13} + 67138788 \beta_{12} + 23080535 \beta_{11} + \cdots - 39006928 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 189898978 \beta_{15} - 189898978 \beta_{14} - 78656174 \beta_{13} - 228801361 \beta_{12} + \cdots - 145453174 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times\).

\(n\) \(137\) \(613\) \(649\) \(919\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
− 4.11290i
1.88319i
− 0.149074i
− 2.44964i
1.58886i
− 0.535137i
− 1.94001i
1.71472i
− 1.58886i
0.535137i
1.94001i
− 1.71472i
4.11290i
− 1.88319i
0.149074i
2.44964i
0 0 0 −1.19125 + 2.87594i 0 −0.497533 − 1.20115i 0 0 0
145.2 0 0 0 −1.10335 + 2.66372i 0 1.25868 + 3.03872i 0 0 0
145.3 0 0 0 0.325635 − 0.786153i 0 0.663444 + 1.60170i 0 0 0
145.4 0 0 0 0.554753 − 1.33929i 0 −0.0103761 − 0.0250502i 0 0 0
433.1 0 0 0 −2.39179 − 0.990712i 0 2.00524 − 0.830597i 0 0 0
433.2 0 0 0 0.429478 + 0.177896i 0 −1.62064 + 0.671292i 0 0 0
433.3 0 0 0 0.868455 + 0.359726i 0 −4.01891 + 1.66469i 0 0 0
433.4 0 0 0 2.50807 + 1.03888i 0 2.22010 − 0.919595i 0 0 0
865.1 0 0 0 −2.39179 + 0.990712i 0 2.00524 + 0.830597i 0 0 0
865.2 0 0 0 0.429478 − 0.177896i 0 −1.62064 − 0.671292i 0 0 0
865.3 0 0 0 0.868455 − 0.359726i 0 −4.01891 − 1.66469i 0 0 0
865.4 0 0 0 2.50807 − 1.03888i 0 2.22010 + 0.919595i 0 0 0
937.1 0 0 0 −1.19125 − 2.87594i 0 −0.497533 + 1.20115i 0 0 0
937.2 0 0 0 −1.10335 − 2.66372i 0 1.25868 − 3.03872i 0 0 0
937.3 0 0 0 0.325635 + 0.786153i 0 0.663444 − 1.60170i 0 0 0
937.4 0 0 0 0.554753 + 1.33929i 0 −0.0103761 + 0.0250502i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 145.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1224.2.bq.e 16
3.b odd 2 1 408.2.ba.a ✓ 16
12.b even 2 1 816.2.bq.f 16
17.d even 8 1 inner 1224.2.bq.e 16
51.g odd 8 1 408.2.ba.a ✓ 16
51.i even 16 1 6936.2.a.bl 8
51.i even 16 1 6936.2.a.bo 8
204.p even 8 1 816.2.bq.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
408.2.ba.a ✓ 16 3.b odd 2 1
408.2.ba.a ✓ 16 51.g odd 8 1
816.2.bq.f 16 12.b even 2 1
816.2.bq.f 16 204.p even 8 1
1224.2.bq.e 16 1.a even 1 1 trivial
1224.2.bq.e 16 17.d even 8 1 inner
6936.2.a.bl 8 51.i even 16 1
6936.2.a.bo 8 51.i even 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 4 T_{5}^{14} - 32 T_{5}^{13} + 8 T_{5}^{12} + 56 T_{5}^{11} + 336 T_{5}^{10} + 72 T_{5}^{9} + \cdots + 1156 \) acting on \(S_{2}^{\mathrm{new}}(1224, [\chi])\). 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
$5$ \( T^{16} + 4 T^{14} + \cdots + 1156 \) Copy content Toggle raw display
$7$ \( T^{16} - 12 T^{14} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( T^{16} - 8 T^{15} + \cdots + 73984 \) Copy content Toggle raw display
$13$ \( T^{16} + 100 T^{14} + \cdots + 150544 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 6975757441 \) Copy content Toggle raw display
$19$ \( T^{16} + 8 T^{15} + \cdots + 64 \) Copy content Toggle raw display
$23$ \( T^{16} - 8 T^{15} + \cdots + 4624 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 277155904 \) Copy content Toggle raw display
$31$ \( T^{16} - 8 T^{15} + \cdots + 75203584 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 8599223824 \) Copy content Toggle raw display
$41$ \( T^{16} + 40 T^{15} + \cdots + 6728836 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 4773151744 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 33926692864 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 143598555136 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 29302041354496 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 307387995210256 \) Copy content Toggle raw display
$67$ \( (T^{8} - 316 T^{6} + \cdots + 2236384)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 4226040064 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 130269021184 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 25305004646464 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 797549019136 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 47876546426944 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 343239946813696 \) Copy content Toggle raw display
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