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Properties

Label 368.2.j.c
Level $368$
Weight $2$
Character orbit 368.j
Analytic conductor $2.938$
Analytic rank $0$
Dimension $12$
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] = [368,2,Mod(93,368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(368, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("368.93");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 368 = 2^{4} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 368.j (of order \(4\), degree \(2\), 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: \(2.93849479438\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.221124989353984.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} - 2 x^{10} + 2 x^{9} + 12 x^{8} - 8 x^{7} - 14 x^{6} - 16 x^{5} + 48 x^{4} + 16 x^{3} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + \beta_{10} q^{3} + (\beta_{11} - \beta_{10} + \cdots + \beta_{2}) q^{4}+ \cdots + (\beta_{8} + \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + \beta_{10} q^{3} + (\beta_{11} - \beta_{10} + \cdots + \beta_{2}) q^{4}+ \cdots + (4 \beta_{11} - 4 \beta_{10} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} + 2 q^{3} - 4 q^{5} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{2} + 2 q^{3} - 4 q^{5} - 4 q^{8} - 6 q^{10} - 4 q^{11} - 8 q^{12} + 18 q^{13} - 2 q^{14} + 8 q^{16} - 8 q^{17} - 4 q^{18} - 8 q^{19} - 32 q^{20} + 8 q^{21} - 34 q^{22} + 12 q^{24} - 14 q^{26} + 14 q^{27} + 12 q^{28} + 2 q^{29} - 30 q^{30} + 20 q^{31} - 8 q^{32} - 36 q^{33} + 10 q^{34} + 4 q^{35} + 4 q^{36} - 4 q^{37} + 24 q^{38} - 14 q^{42} + 20 q^{43} + 4 q^{44} - 20 q^{45} - 2 q^{46} - 16 q^{47} - 12 q^{48} + 52 q^{49} + 6 q^{50} - 4 q^{51} - 16 q^{53} + 16 q^{54} + 28 q^{56} + 14 q^{58} + 8 q^{59} + 48 q^{60} + 12 q^{61} - 44 q^{62} - 4 q^{63} + 24 q^{64} - 52 q^{65} + 34 q^{66} - 4 q^{67} - 16 q^{68} + 2 q^{69} + 28 q^{70} + 8 q^{72} - 26 q^{74} - 46 q^{75} - 8 q^{76} - 12 q^{77} - 44 q^{78} - 4 q^{79} + 4 q^{80} + 48 q^{81} - 6 q^{82} + 28 q^{83} + 12 q^{84} - 8 q^{85} + 44 q^{86} - 36 q^{88} + 4 q^{90} - 4 q^{92} - 14 q^{93} + 48 q^{95} + 32 q^{96} + 36 q^{97} - 2 q^{98} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} - 2 x^{10} + 2 x^{9} + 12 x^{8} - 8 x^{7} - 14 x^{6} - 16 x^{5} + 48 x^{4} + 16 x^{3} + \cdots + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{11} + 2\nu^{10} - 2\nu^{9} + 2\nu^{8} - 12\nu^{7} + 8\nu^{6} + 2\nu^{5} + 24\nu^{4} - 32\nu^{3} - 64\nu + 96 ) / 32 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} - 2 \nu^{10} - 2 \nu^{9} + 2 \nu^{8} + 12 \nu^{7} - 8 \nu^{6} - 14 \nu^{5} - 16 \nu^{4} + \cdots - 64 ) / 32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} + 2 \nu^{10} - 10 \nu^{9} + 2 \nu^{8} + 4 \nu^{7} + 24 \nu^{6} - 30 \nu^{5} - 8 \nu^{4} + \cdots - 64 \nu ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{11} + 2\nu^{10} + 2\nu^{9} + 2\nu^{8} - 12\nu^{7} + 6\nu^{5} + 32\nu^{4} - 32\nu^{3} - 24\nu^{2} - 16\nu + 64 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3 \nu^{11} + \nu^{10} + 6 \nu^{9} + 12 \nu^{8} - 26 \nu^{7} - 16 \nu^{6} + 2 \nu^{5} + 86 \nu^{4} + \cdots + 128 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3 \nu^{11} - \nu^{10} - 6 \nu^{9} - 12 \nu^{8} + 26 \nu^{7} + 16 \nu^{6} - 2 \nu^{5} - 86 \nu^{4} + \cdots - 144 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2 \nu^{11} - 2 \nu^{10} - 5 \nu^{9} - 2 \nu^{8} + 18 \nu^{7} + 6 \nu^{6} - 16 \nu^{5} - 44 \nu^{4} + \cdots - 72 ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5 \nu^{11} - 3 \nu^{10} - 14 \nu^{9} - 12 \nu^{8} + 46 \nu^{7} + 24 \nu^{6} - 30 \nu^{5} - 130 \nu^{4} + \cdots - 224 ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3 \nu^{11} + \nu^{10} + 10 \nu^{9} + 6 \nu^{8} - 26 \nu^{7} - 20 \nu^{6} + 22 \nu^{5} + 78 \nu^{4} + \cdots + 128 ) / 8 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 13 \nu^{11} - 10 \nu^{10} - 30 \nu^{9} - 30 \nu^{8} + 116 \nu^{7} + 48 \nu^{6} - 70 \nu^{5} + \cdots - 544 ) / 32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 7 \nu^{11} - 3 \nu^{10} - 24 \nu^{9} - 12 \nu^{8} + 66 \nu^{7} + 44 \nu^{6} - 66 \nu^{5} - 186 \nu^{4} + \cdots - 336 ) / 16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} + \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} + \beta_{7} + 2\beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{3} + \beta_{2} - \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{10} - \beta_{9} + \beta_{7} + 2\beta_{6} - 3\beta_{4} - \beta_{3} + \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{11} - \beta_{9} - 5\beta_{8} + 3\beta_{7} + 3\beta_{6} - \beta_{5} - \beta_{4} + 4\beta_{3} - 2\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{11} - 5 \beta_{10} + \beta_{9} + \beta_{8} + 4 \beta_{7} + \beta_{6} - 3 \beta_{5} + \beta_{4} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 4 \beta_{10} - 6 \beta_{9} - 8 \beta_{8} + 4 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4 \beta_{11} + 2 \beta_{10} + 4 \beta_{9} - 10 \beta_{8} + 6 \beta_{7} + 6 \beta_{6} + 6 \beta_{4} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 12 \beta_{11} - 10 \beta_{10} + 6 \beta_{9} - 2 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} - 12 \beta_{5} + \cdots + 18 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 4 \beta_{11} + 10 \beta_{10} - 2 \beta_{9} - 12 \beta_{8} + 6 \beta_{7} + 12 \beta_{6} + 8 \beta_{5} + \cdots - 22 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/368\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(97\) \(277\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
93.1
−1.14441 − 0.830857i
1.35092 − 0.418349i
1.09121 + 0.899589i
1.41072 − 0.0993495i
−1.18970 + 0.764606i
−0.518742 − 1.31564i
−1.14441 + 0.830857i
1.35092 + 0.418349i
1.09121 − 0.899589i
1.41072 + 0.0993495i
−1.18970 − 0.764606i
−0.518742 + 1.31564i
−1.40983 − 0.111202i −1.70582 − 1.70582i 1.97527 + 0.313554i −2.61219 + 2.61219i 2.21524 + 2.59462i 1.66171i −2.74993 − 0.661714i 2.81967i 3.97323 − 3.39227i
93.2 −0.730558 + 1.21090i 1.49351 + 1.49351i −0.932570 − 1.76927i −1.17219 + 1.17219i −2.89958 + 0.717397i 0.836699i 2.82371 + 0.163301i 1.46112i −0.563056 − 2.27576i
93.3 0.0678262 − 1.41259i 1.19673 + 1.19673i −1.99080 − 0.191621i 0.672033 − 0.672033i 1.77166 − 1.60932i − 1.79918i −0.405709 + 2.79918i − 0.135652i −0.903723 − 0.994886i
93.4 0.586784 − 1.28673i −0.955624 − 0.955624i −1.31137 − 1.51007i 2.38359 − 2.38359i −1.79038 + 0.668889i 0.198699i −2.71255 + 0.801301i − 1.17357i −1.66839 − 4.46569i
93.5 1.10116 + 0.887386i 0.631541 + 0.631541i 0.425090 + 1.95430i −2.74053 + 2.74053i 0.135004 + 1.25585i − 1.52921i −1.26613 + 2.52921i − 2.20231i −5.44967 + 0.585843i
93.6 1.38463 − 0.287766i 0.339667 + 0.339667i 1.83438 − 0.796898i 1.46929 − 1.46929i 0.568056 + 0.372567i 2.63128i 2.31061 − 1.63128i − 2.76925i 1.61161 − 2.45723i
277.1 −1.40983 + 0.111202i −1.70582 + 1.70582i 1.97527 − 0.313554i −2.61219 − 2.61219i 2.21524 − 2.59462i − 1.66171i −2.74993 + 0.661714i − 2.81967i 3.97323 + 3.39227i
277.2 −0.730558 − 1.21090i 1.49351 − 1.49351i −0.932570 + 1.76927i −1.17219 − 1.17219i −2.89958 − 0.717397i − 0.836699i 2.82371 − 0.163301i − 1.46112i −0.563056 + 2.27576i
277.3 0.0678262 + 1.41259i 1.19673 − 1.19673i −1.99080 + 0.191621i 0.672033 + 0.672033i 1.77166 + 1.60932i 1.79918i −0.405709 − 2.79918i 0.135652i −0.903723 + 0.994886i
277.4 0.586784 + 1.28673i −0.955624 + 0.955624i −1.31137 + 1.51007i 2.38359 + 2.38359i −1.79038 − 0.668889i − 0.198699i −2.71255 − 0.801301i 1.17357i −1.66839 + 4.46569i
277.5 1.10116 − 0.887386i 0.631541 − 0.631541i 0.425090 − 1.95430i −2.74053 − 2.74053i 0.135004 − 1.25585i 1.52921i −1.26613 − 2.52921i 2.20231i −5.44967 − 0.585843i
277.6 1.38463 + 0.287766i 0.339667 − 0.339667i 1.83438 + 0.796898i 1.46929 + 1.46929i 0.568056 − 0.372567i − 2.63128i 2.31061 + 1.63128i 2.76925i 1.61161 + 2.45723i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 93.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 368.2.j.c ✓ 12
4.b odd 2 1 1472.2.j.c 12
16.e even 4 1 inner 368.2.j.c ✓ 12
16.f odd 4 1 1472.2.j.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
368.2.j.c ✓ 12 1.a even 1 1 trivial
368.2.j.c ✓ 12 16.e even 4 1 inner
1472.2.j.c 12 4.b odd 2 1
1472.2.j.c 12 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 2 T_{3}^{11} + 2 T_{3}^{10} - 2 T_{3}^{9} + 35 T_{3}^{8} - 76 T_{3}^{7} + 84 T_{3}^{6} + \cdots + 25 \) acting on \(S_{2}^{\mathrm{new}}(368, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 2 T^{11} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( T^{12} - 2 T^{11} + \cdots + 25 \) Copy content Toggle raw display
$5$ \( T^{12} + 4 T^{11} + \cdots + 24964 \) Copy content Toggle raw display
$7$ \( T^{12} + 16 T^{10} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( T^{12} + 4 T^{11} + \cdots + 14884 \) Copy content Toggle raw display
$13$ \( T^{12} - 18 T^{11} + \cdots + 25 \) Copy content Toggle raw display
$17$ \( (T^{6} + 4 T^{5} + \cdots + 202)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 8 T^{11} + \cdots + 576 \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$29$ \( T^{12} - 2 T^{11} + \cdots + 16641 \) Copy content Toggle raw display
$31$ \( (T^{6} - 10 T^{5} + \cdots - 3727)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 4 T^{11} + \cdots + 30140100 \) Copy content Toggle raw display
$41$ \( T^{12} + 238 T^{10} + \cdots + 4995225 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 2437989376 \) Copy content Toggle raw display
$47$ \( (T^{6} + 8 T^{5} + \cdots - 51211)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 10123579456 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 17761425984 \) Copy content Toggle raw display
$61$ \( T^{12} - 12 T^{11} + \cdots + 2304 \) Copy content Toggle raw display
$67$ \( T^{12} + 4 T^{11} + \cdots + 438244 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 138180025 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 1176284209 \) Copy content Toggle raw display
$79$ \( (T^{6} + 2 T^{5} + \cdots - 120)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 1492276900 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 3846976576 \) Copy content Toggle raw display
$97$ \( (T^{6} - 18 T^{5} + \cdots - 123118)^{2} \) Copy content Toggle raw display
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