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Properties

Label 4368.2.h.t
Level $4368$
Weight $2$
Character orbit 4368.h
Analytic conductor $34.879$
Analytic rank $0$
Dimension $10$
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] = [4368,2,Mod(337,4368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4368, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4368.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4368.h (of order \(2\), degree \(1\), 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: \(34.8786556029\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4x^{9} + 8x^{8} + 18x^{7} + 28x^{6} - 48x^{5} + 130x^{4} + 316x^{3} + 324x^{2} + 144x + 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 2184)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + (\beta_{5} + \beta_1) q^{5} + \beta_1 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + (\beta_{5} + \beta_1) q^{5} + \beta_1 q^{7} + q^{9} + ( - \beta_{6} + \beta_{5} - \beta_1) q^{11} + ( - \beta_{4} - \beta_{3} - \beta_{2}) q^{13} + (\beta_{5} + \beta_1) q^{15} - \beta_{8} q^{17} + ( - \beta_{9} + \beta_{6} + \cdots - 2 \beta_1) q^{19}+ \cdots + ( - \beta_{6} + \beta_{5} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{3} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 10 q^{3} + 10 q^{9} + 2 q^{13} - 2 q^{23} - 8 q^{25} + 10 q^{27} - 2 q^{29} - 6 q^{35} + 2 q^{39} + 34 q^{43} - 10 q^{49} - 2 q^{53} - 64 q^{55} + 4 q^{61} + 2 q^{65} - 2 q^{69} - 8 q^{75} + 16 q^{77} + 38 q^{79} + 10 q^{81} - 2 q^{87} + 34 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4x^{9} + 8x^{8} + 18x^{7} + 28x^{6} - 48x^{5} + 130x^{4} + 316x^{3} + 324x^{2} + 144x + 32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 2335639 \nu^{9} + 10122488 \nu^{8} - 22385964 \nu^{7} - 32822558 \nu^{6} - 58610700 \nu^{5} + \cdots - 176351608 ) / 105963352 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2199633 \nu^{9} - 11134171 \nu^{8} + 27719552 \nu^{7} + 17207430 \nu^{6} + 28767166 \nu^{5} + \cdots - 199458284 ) / 52981676 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2589599 \nu^{9} + 12984597 \nu^{8} - 32329024 \nu^{7} - 20601026 \nu^{6} - 37372866 \nu^{5} + \cdots + 109106384 ) / 52981676 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 154047 \nu^{9} - 438600 \nu^{8} + 335220 \nu^{7} + 5014126 \nu^{6} + 5698812 \nu^{5} + \cdots + 33487184 ) / 2464264 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 114626 \nu^{9} + 478857 \nu^{8} - 988670 \nu^{7} - 1948576 \nu^{6} - 2736774 \nu^{5} + \cdots - 8054672 ) / 1232132 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 12973407 \nu^{9} - 55005042 \nu^{8} + 116630528 \nu^{7} + 207187286 \nu^{6} + 311184664 \nu^{5} + \cdots + 943793848 ) / 105963352 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 7709981 \nu^{9} + 30449668 \nu^{8} - 55204652 \nu^{7} - 167118734 \nu^{6} - 159470092 \nu^{5} + \cdots - 494347560 ) / 52981676 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2071882 \nu^{9} - 10424657 \nu^{8} + 26161144 \nu^{7} + 16150117 \nu^{6} + 26504200 \nu^{5} + \cdots + 2304942 ) / 13245419 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 25921189 \nu^{9} - 115617736 \nu^{8} + 256644668 \nu^{7} + 367636130 \nu^{6} + 508354068 \nu^{5} + \cdots + 810078176 ) / 105963352 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} + \beta_{7} + 2\beta_{6} + 2\beta_{5} + \beta_{4} + 10\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{7} + 3\beta_{6} + 5\beta_{5} + 3\beta_{4} - 5\beta_{3} - 3\beta_{2} + 10\beta _1 - 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -10\beta_{9} + 8\beta_{8} + 10\beta_{4} - 21\beta_{3} - 11\beta_{2} - 59 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 114 \beta_{9} + 41 \beta_{8} - 41 \beta_{7} - 60 \beta_{6} - 132 \beta_{5} - 132 \beta_{3} + \cdots - 300 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -164\beta_{9} - 123\beta_{7} - 144\beta_{6} - 340\beta_{5} - 164\beta_{4} - 838\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 334 \beta_{8} - 334 \beta_{7} - 393 \beta_{6} - 967 \beta_{5} - 914 \beta_{4} + 967 \beta_{3} + \cdots + 2263 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2549\beta_{9} - 1881\beta_{8} - 2549\beta_{4} + 5246\beta_{3} + 2082\beta_{2} + 12514 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 14106 \beta_{9} - 5172 \beta_{8} + 5172 \beta_{7} + 5716 \beta_{6} + 14576 \beta_{5} + 14576 \beta_{3} + \cdots + 34330 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4368\mathbb{Z}\right)^\times\).

\(n\) \(1093\) \(1249\) \(1457\) \(2017\) \(3823\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
−1.35191 − 1.35191i
−0.340491 + 0.340491i
2.75844 − 2.75844i
−0.535829 + 0.535829i
1.46979 + 1.46979i
1.46979 − 1.46979i
−0.535829 − 0.535829i
2.75844 + 2.75844i
−0.340491 − 0.340491i
−1.35191 + 1.35191i
0 1.00000 0 − 3.22444i 0 1.00000i 0 1.00000 0
337.2 0 1.00000 0 − 3.19289i 0 − 1.00000i 0 1.00000 0
337.3 0 1.00000 0 − 2.79184i 0 − 1.00000i 0 1.00000 0
337.4 0 1.00000 0 − 0.660872i 0 − 1.00000i 0 1.00000 0
337.5 0 1.00000 0 − 0.421158i 0 1.00000i 0 1.00000 0
337.6 0 1.00000 0 0.421158i 0 − 1.00000i 0 1.00000 0
337.7 0 1.00000 0 0.660872i 0 1.00000i 0 1.00000 0
337.8 0 1.00000 0 2.79184i 0 1.00000i 0 1.00000 0
337.9 0 1.00000 0 3.19289i 0 1.00000i 0 1.00000 0
337.10 0 1.00000 0 3.22444i 0 − 1.00000i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4368.2.h.t 10
4.b odd 2 1 2184.2.h.e ✓ 10
13.b even 2 1 inner 4368.2.h.t 10
52.b odd 2 1 2184.2.h.e ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2184.2.h.e ✓ 10 4.b odd 2 1
2184.2.h.e ✓ 10 52.b odd 2 1
4368.2.h.t 10 1.a even 1 1 trivial
4368.2.h.t 10 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4368, [\chi])\):

\( T_{5}^{10} + 29T_{5}^{8} + 284T_{5}^{6} + 992T_{5}^{4} + 528T_{5}^{2} + 64 \) Copy content Toggle raw display
\( T_{11}^{10} + 96T_{11}^{8} + 3168T_{11}^{6} + 39312T_{11}^{4} + 111872T_{11}^{2} + 16384 \) Copy content Toggle raw display
\( T_{17}^{5} - 52T_{17}^{3} + 64T_{17}^{2} + 384T_{17} - 128 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( (T - 1)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + 29 T^{8} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$11$ \( T^{10} + 96 T^{8} + \cdots + 16384 \) Copy content Toggle raw display
$13$ \( T^{10} - 2 T^{9} + \cdots + 371293 \) Copy content Toggle raw display
$17$ \( (T^{5} - 52 T^{3} + \cdots - 128)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + 97 T^{8} + \cdots + 16384 \) Copy content Toggle raw display
$23$ \( (T^{5} + T^{4} - 52 T^{3} + \cdots - 1072)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} + T^{4} - 90 T^{3} + \cdots + 2144)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + 241 T^{8} + \cdots + 4194304 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 409010176 \) Copy content Toggle raw display
$41$ \( T^{10} + 268 T^{8} + \cdots + 5456896 \) Copy content Toggle raw display
$43$ \( (T^{5} - 17 T^{4} + \cdots - 1024)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + 269 T^{8} + \cdots + 2027776 \) Copy content Toggle raw display
$53$ \( (T^{5} + T^{4} - 94 T^{3} + \cdots - 256)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + 48 T^{8} + \cdots + 1024 \) Copy content Toggle raw display
$61$ \( (T^{5} - 2 T^{4} + \cdots + 2176)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 635846656 \) Copy content Toggle raw display
$71$ \( T^{10} + 288 T^{8} + \cdots + 11075584 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 1628283904 \) Copy content Toggle raw display
$79$ \( (T^{5} - 19 T^{4} + \cdots - 23168)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 281 T^{8} + \cdots + 66455104 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 1514143744 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 2252071936 \) Copy content Toggle raw display
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