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Properties

Label 1368.2.g.c
Level $1368$
Weight $2$
Character orbit 1368.g
Analytic conductor $10.924$
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] = [1368,2,Mod(685,1368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1368, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1368.685");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1368.g (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: \(10.9235349965\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - x^{16} - 8 x^{13} - 6 x^{12} + 12 x^{11} + 20 x^{10} - 16 x^{9} + 40 x^{8} + 48 x^{7} + \cdots + 512 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 456)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{14} q^{2} - \beta_{2} q^{4} - \beta_{13} q^{5} + ( - \beta_{16} - \beta_{10} + \cdots + \beta_{4}) q^{7}+ \cdots + ( - \beta_{16} - \beta_{13} - \beta_{11}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{14} q^{2} - \beta_{2} q^{4} - \beta_{13} q^{5} + ( - \beta_{16} - \beta_{10} + \cdots + \beta_{4}) q^{7}+ \cdots + (2 \beta_{17} - \beta_{16} + \beta_{15} + \cdots - 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 2 q^{2} - 2 q^{4} - 12 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 2 q^{2} - 2 q^{4} - 12 q^{7} - 2 q^{8} + 4 q^{10} + 10 q^{14} + 2 q^{16} - 4 q^{17} - 6 q^{20} - 26 q^{22} + 36 q^{23} - 22 q^{25} - 34 q^{26} + 24 q^{28} + 32 q^{31} + 18 q^{32} + 18 q^{34} - 50 q^{40} + 4 q^{41} - 32 q^{44} + 2 q^{46} - 52 q^{47} + 18 q^{49} - 48 q^{50} + 2 q^{52} - 40 q^{55} + 36 q^{56} + 32 q^{58} + 46 q^{62} - 38 q^{64} + 16 q^{65} - 60 q^{68} - 46 q^{70} + 64 q^{71} + 28 q^{73} + 2 q^{74} + 4 q^{76} + 32 q^{79} + 34 q^{80} + 48 q^{82} + 58 q^{86} - 40 q^{88} - 4 q^{89} - 22 q^{92} - 32 q^{94} - 4 q^{97} - 68 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - x^{16} - 8 x^{13} - 6 x^{12} + 12 x^{11} + 20 x^{10} - 16 x^{9} + 40 x^{8} + 48 x^{7} + \cdots + 512 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{17} + 4 \nu^{16} - \nu^{15} - 4 \nu^{14} + 6 \nu^{13} + 8 \nu^{12} - 6 \nu^{11} + \cdots + 512 \nu ) / 512 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{16} - 5 \nu^{14} + 2 \nu^{12} + 40 \nu^{11} - 14 \nu^{10} + 4 \nu^{9} + 24 \nu^{8} + \cdots + 256 ) / 256 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{17} + \nu^{16} - 3 \nu^{15} + \nu^{14} + 4 \nu^{13} + 2 \nu^{12} - 2 \nu^{11} - 6 \nu^{10} + \cdots - 512 ) / 256 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{17} - 3 \nu^{15} - 10 \nu^{13} - 8 \nu^{12} - 2 \nu^{11} + 28 \nu^{10} - 8 \nu^{9} + 56 \nu^{8} + \cdots + 1024 ) / 256 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - \nu^{16} + 2 \nu^{15} + 11 \nu^{14} + 2 \nu^{13} - 14 \nu^{12} - 4 \nu^{11} + 2 \nu^{10} + \cdots - 896 \nu ) / 256 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3 \nu^{16} - 2 \nu^{15} + 5 \nu^{14} - 2 \nu^{13} + 10 \nu^{12} + 4 \nu^{11} + 30 \nu^{10} + \cdots + 512 ) / 256 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - \nu^{17} + \nu^{16} + \nu^{15} - 3 \nu^{14} - 8 \nu^{13} + 14 \nu^{12} + 22 \nu^{11} + \cdots + 896 \nu ) / 256 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - \nu^{17} - \nu^{16} + 5 \nu^{15} - \nu^{14} - 4 \nu^{13} + 14 \nu^{12} + 14 \nu^{11} - 18 \nu^{10} + \cdots + 512 ) / 256 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{16} + \nu^{12} - 8 \nu^{11} - 4 \nu^{10} + 4 \nu^{9} + 2 \nu^{8} - 20 \nu^{7} + 40 \nu^{6} + \cdots - 128 ) / 64 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( \nu^{17} + 2 \nu^{16} + 3 \nu^{15} + 2 \nu^{14} - 4 \nu^{13} - 4 \nu^{12} - 22 \nu^{11} - 24 \nu^{10} + \cdots - 256 ) / 256 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( \nu^{17} + \nu^{13} - 8 \nu^{12} - 12 \nu^{11} + 20 \nu^{10} - 6 \nu^{9} - 4 \nu^{8} + 56 \nu^{7} + \cdots + 384 ) / 128 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 2 \nu^{17} + \nu^{16} + 2 \nu^{15} - 3 \nu^{14} + 6 \nu^{12} + 4 \nu^{11} - 26 \nu^{10} + \cdots - 256 ) / 256 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - \nu^{17} + 5 \nu^{16} + \nu^{15} - 3 \nu^{14} - 8 \nu^{13} + 18 \nu^{12} - 10 \nu^{11} - 46 \nu^{10} + \cdots - 512 ) / 256 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - \nu^{17} - \nu^{16} + 5 \nu^{14} - \nu^{13} + 4 \nu^{12} + 20 \nu^{11} + 2 \nu^{10} - 38 \nu^{9} + \cdots + 128 ) / 128 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - \nu^{17} + 3 \nu^{16} - 3 \nu^{15} - 5 \nu^{14} + 4 \nu^{13} + 30 \nu^{12} - 18 \nu^{11} + \cdots - 1024 ) / 256 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - \beta_{11} - \beta_{9} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{17} + \beta_{13} + \beta_{12} + \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} + 2\beta_{5} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{17} + \beta_{13} - \beta_{12} + \beta_{9} - \beta_{8} + \beta_{7} - 2\beta_{3} + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{17} - \beta_{16} + \beta_{15} + 2 \beta_{14} + 2 \beta_{13} - \beta_{12} + 2 \beta_{11} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{17} + 2 \beta_{14} + \beta_{13} + \beta_{12} - 2 \beta_{11} - 2 \beta_{10} + \beta_{9} + \beta_{8} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - \beta_{17} - 3 \beta_{16} + 3 \beta_{15} - 2 \beta_{14} - 3 \beta_{12} - 2 \beta_{11} + 2 \beta_{10} + \cdots - 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 3 \beta_{17} + 2 \beta_{15} - 2 \beta_{14} + \beta_{13} + \beta_{12} - 4 \beta_{11} - 2 \beta_{10} + \cdots + 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( \beta_{17} + 5 \beta_{16} - 5 \beta_{15} - 2 \beta_{14} + 10 \beta_{13} - 5 \beta_{12} + 6 \beta_{11} + \cdots - 14 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - \beta_{17} - 2 \beta_{15} + 6 \beta_{14} + 11 \beta_{13} - 9 \beta_{12} + 16 \beta_{11} - 10 \beta_{10} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 11 \beta_{17} - \beta_{16} + 9 \beta_{15} + 10 \beta_{14} + 6 \beta_{13} - 3 \beta_{12} - 6 \beta_{11} + \cdots - 10 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 11 \beta_{17} - 12 \beta_{16} - 14 \beta_{15} - 6 \beta_{14} + 5 \beta_{13} + 5 \beta_{12} + \cdots - 26 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 13 \beta_{17} + \beta_{16} + 7 \beta_{15} - 50 \beta_{14} - 30 \beta_{13} - 29 \beta_{12} - 18 \beta_{11} + \cdots + 18 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 37 \beta_{17} + 16 \beta_{16} - 14 \beta_{15} - 18 \beta_{14} + 15 \beta_{13} - 5 \beta_{12} + \cdots - 38 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 27 \beta_{17} + 67 \beta_{16} - 35 \beta_{15} + 2 \beta_{14} + 42 \beta_{13} + 5 \beta_{12} + 58 \beta_{11} + \cdots - 82 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 5 \beta_{17} - 52 \beta_{16} + 66 \beta_{15} - 46 \beta_{14} + 29 \beta_{13} - 3 \beta_{12} + 52 \beta_{11} + \cdots - 74 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times\).

\(n\) \(343\) \(685\) \(1009\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
685.1
0.476379 − 1.33156i
0.476379 + 1.33156i
0.719332 − 1.21760i
0.719332 + 1.21760i
−0.764203 + 1.18996i
−0.764203 − 1.18996i
−1.34058 + 0.450394i
−1.34058 − 0.450394i
1.39612 − 0.225498i
1.39612 + 0.225498i
1.35992 + 0.388110i
1.35992 − 0.388110i
−1.30899 − 0.535307i
−1.30899 + 0.535307i
−0.868070 − 1.11645i
−0.868070 + 1.11645i
0.330091 + 1.37515i
0.330091 − 1.37515i
−1.33156 − 0.476379i 0 1.54613 + 1.26866i − 1.06546i 0 −5.11575 −1.45441 − 2.42584i 0 −0.507563 + 1.41873i
685.2 −1.33156 + 0.476379i 0 1.54613 − 1.26866i 1.06546i 0 −5.11575 −1.45441 + 2.42584i 0 −0.507563 − 1.41873i
685.3 −1.21760 − 0.719332i 0 0.965124 + 1.75172i − 0.0253729i 0 3.47283 0.0849313 − 2.82715i 0 −0.0182515 + 0.0308942i
685.4 −1.21760 + 0.719332i 0 0.965124 − 1.75172i 0.0253729i 0 3.47283 0.0849313 + 2.82715i 0 −0.0182515 − 0.0308942i
685.5 −1.18996 − 0.764203i 0 0.831987 + 1.81874i − 2.21313i 0 −0.0905383 0.399857 − 2.80002i 0 −1.69128 + 2.63352i
685.6 −1.18996 + 0.764203i 0 0.831987 − 1.81874i 2.21313i 0 −0.0905383 0.399857 + 2.80002i 0 −1.69128 − 2.63352i
685.7 −0.450394 − 1.34058i 0 −1.59429 + 1.20757i 2.05835i 0 0.983157 2.33690 + 1.59339i 0 2.75937 − 0.927066i
685.8 −0.450394 + 1.34058i 0 −1.59429 − 1.20757i − 2.05835i 0 0.983157 2.33690 − 1.59339i 0 2.75937 + 0.927066i
685.9 −0.225498 − 1.39612i 0 −1.89830 + 0.629643i 2.54674i 0 −4.00545 1.30712 + 2.50827i 0 3.55555 − 0.574283i
685.10 −0.225498 + 1.39612i 0 −1.89830 − 0.629643i − 2.54674i 0 −4.00545 1.30712 − 2.50827i 0 3.55555 + 0.574283i
685.11 0.388110 − 1.35992i 0 −1.69874 − 1.05559i − 2.94870i 0 −1.28777 −2.09482 + 1.90046i 0 −4.00998 − 1.14442i
685.12 0.388110 + 1.35992i 0 −1.69874 + 1.05559i 2.94870i 0 −1.28777 −2.09482 − 1.90046i 0 −4.00998 + 1.14442i
685.13 0.535307 − 1.30899i 0 −1.42689 − 1.40142i 3.74133i 0 −3.14665 −2.59827 + 1.11759i 0 4.89736 + 2.00276i
685.14 0.535307 + 1.30899i 0 −1.42689 + 1.40142i − 3.74133i 0 −3.14665 −2.59827 − 1.11759i 0 4.89736 − 2.00276i
685.15 1.11645 − 0.868070i 0 0.492909 − 1.93831i − 3.89084i 0 1.36952 −1.13228 − 2.59190i 0 −3.37752 − 4.34391i
685.16 1.11645 + 0.868070i 0 0.492909 + 1.93831i 3.89084i 0 1.36952 −1.13228 + 2.59190i 0 −3.37752 + 4.34391i
685.17 1.37515 − 0.330091i 0 1.78208 − 0.907849i 1.18851i 0 1.82065 2.15096 − 1.83668i 0 0.392316 + 1.63438i
685.18 1.37515 + 0.330091i 0 1.78208 + 0.907849i − 1.18851i 0 1.82065 2.15096 + 1.83668i 0 0.392316 − 1.63438i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 685.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1368.2.g.c 18
3.b odd 2 1 456.2.g.a ✓ 18
4.b odd 2 1 5472.2.g.d 18
8.b even 2 1 inner 1368.2.g.c 18
8.d odd 2 1 5472.2.g.d 18
12.b even 2 1 1824.2.g.b 18
24.f even 2 1 1824.2.g.b 18
24.h odd 2 1 456.2.g.a ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.g.a ✓ 18 3.b odd 2 1
456.2.g.a ✓ 18 24.h odd 2 1
1368.2.g.c 18 1.a even 1 1 trivial
1368.2.g.c 18 8.b even 2 1 inner
1824.2.g.b 18 12.b even 2 1
1824.2.g.b 18 24.f even 2 1
5472.2.g.d 18 4.b odd 2 1
5472.2.g.d 18 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{18} + 56 T_{5}^{16} + 1274 T_{5}^{14} + 15252 T_{5}^{12} + 104185 T_{5}^{10} + 410876 T_{5}^{8} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(1368, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + 2 T^{17} + \cdots + 512 \) Copy content Toggle raw display
$3$ \( T^{18} \) Copy content Toggle raw display
$5$ \( T^{18} + 56 T^{16} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( (T^{9} + 6 T^{8} - 18 T^{7} + \cdots + 64)^{2} \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 507510784 \) Copy content Toggle raw display
$13$ \( T^{18} + 64 T^{16} + \cdots + 262144 \) Copy content Toggle raw display
$17$ \( (T^{9} + 2 T^{8} + \cdots + 13024)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 1)^{9} \) Copy content Toggle raw display
$23$ \( (T^{9} - 18 T^{8} + \cdots - 5632)^{2} \) Copy content Toggle raw display
$29$ \( T^{18} + 244 T^{16} + \cdots + 67108864 \) Copy content Toggle raw display
$31$ \( (T^{9} - 16 T^{8} + \cdots + 505088)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 1099511627776 \) Copy content Toggle raw display
$41$ \( (T^{9} - 2 T^{8} + \cdots + 1015808)^{2} \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 400112647929856 \) Copy content Toggle raw display
$47$ \( (T^{9} + 26 T^{8} + \cdots - 1888)^{2} \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 135890684870656 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 5346231844864 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 80759410917376 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 122668560941056 \) Copy content Toggle raw display
$71$ \( (T^{9} - 32 T^{8} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$73$ \( (T^{9} - 14 T^{8} + \cdots - 14963776)^{2} \) Copy content Toggle raw display
$79$ \( (T^{9} - 16 T^{8} + \cdots + 94714496)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{9} + 2 T^{8} + \cdots + 11264)^{2} \) Copy content Toggle raw display
$97$ \( (T^{9} + 2 T^{8} + \cdots + 6443264)^{2} \) Copy content Toggle raw display
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