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Properties

Label 4338.2.a.w
Level $4338$
Weight $2$
Character orbit 4338.a
Self dual yes
Analytic conductor $34.639$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4338,2,Mod(1,4338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4338, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4338.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4338 = 2 \cdot 3^{2} \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4338.a (trivial)

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: yes
Analytic conductor: \(34.6391043968\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 20x^{5} + 26x^{4} + 95x^{3} - 121x^{2} - 126x + 158 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1446)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + (\beta_{3} - 1) q^{5} + ( - \beta_{5} + 1) q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + (\beta_{3} - 1) q^{5} + ( - \beta_{5} + 1) q^{7} + q^{8} + (\beta_{3} - 1) q^{10} + ( - \beta_{5} + \beta_{4} - \beta_{2} - 1) q^{11} + ( - \beta_1 + 2) q^{13} + ( - \beta_{5} + 1) q^{14} + q^{16} + ( - \beta_{6} + \beta_{3} - 1) q^{17} + ( - \beta_{5} + \beta_1 + 1) q^{19} + (\beta_{3} - 1) q^{20} + ( - \beta_{5} + \beta_{4} - \beta_{2} - 1) q^{22} + (\beta_{6} + \beta_{5} + \beta_{3}) q^{23} + ( - \beta_{6} - \beta_{5} + \beta_{4} + \cdots + 2) q^{25}+ \cdots + (\beta_{6} + \beta_{5} - \beta_{4} + \cdots + 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 7 q^{2} + 7 q^{4} - 5 q^{5} + 7 q^{7} + 7 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 7 q^{2} + 7 q^{4} - 5 q^{5} + 7 q^{7} + 7 q^{8} - 5 q^{10} - 4 q^{11} + 14 q^{13} + 7 q^{14} + 7 q^{16} - 4 q^{17} + 7 q^{19} - 5 q^{20} - 4 q^{22} + q^{23} + 14 q^{25} + 14 q^{26} + 7 q^{28} - 3 q^{29} + 8 q^{31} + 7 q^{32} - 4 q^{34} + 17 q^{35} + 18 q^{37} + 7 q^{38} - 5 q^{40} - 3 q^{41} + 11 q^{43} - 4 q^{44} + q^{46} + 18 q^{47} + 20 q^{49} + 14 q^{50} + 14 q^{52} + 9 q^{53} - 4 q^{55} + 7 q^{56} - 3 q^{58} - 6 q^{59} + 31 q^{61} + 8 q^{62} + 7 q^{64} + 6 q^{65} + 4 q^{67} - 4 q^{68} + 17 q^{70} + 3 q^{71} + 16 q^{73} + 18 q^{74} + 7 q^{76} + 34 q^{79} - 5 q^{80} - 3 q^{82} - 10 q^{83} + 34 q^{85} + 11 q^{86} - 4 q^{88} - 24 q^{89} + 40 q^{91} + q^{92} + 18 q^{94} + q^{95} + 27 q^{97} + 20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 20x^{5} + 26x^{4} + 95x^{3} - 121x^{2} - 126x + 158 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + \nu - 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} - 10\nu^{5} + 24\nu^{4} + 162\nu^{3} - 251\nu^{2} - 436\nu + 536 ) / 38 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15\nu^{6} - 2\nu^{5} - 284\nu^{4} + 116\nu^{3} + 1219\nu^{2} - 224\nu - 1276 ) / 38 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12\nu^{6} + 6\nu^{5} - 231\nu^{4} - 25\nu^{3} + 1112\nu^{2} + 102\nu - 1397 ) / 19 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -27\nu^{6} - 4\nu^{5} + 534\nu^{4} - 72\nu^{3} - 2597\nu^{2} + 46\nu + 3224 ) / 38 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 43\nu^{6} + 12\nu^{5} - 842\nu^{4} + 26\nu^{3} + 4067\nu^{2} + 242\nu - 5074 ) / 38 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} - \beta_{5} + \beta_{3} - \beta_{2} + 2\beta _1 + 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{6} + 7\beta_{5} + 4\beta_{4} - 13\beta_{3} + 13\beta_{2} - 2\beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -17\beta_{6} - 13\beta_{5} - 2\beta_{4} + 27\beta_{3} - 23\beta_{2} + 30\beta _1 + 97 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 81\beta_{6} + 83\beta_{5} + 66\beta_{4} - 177\beta_{3} + 171\beta_{2} - 52\beta _1 - 41 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -269\beta_{6} - 193\beta_{5} - 60\beta_{4} + 497\beta_{3} - 417\beta_{2} + 414\beta _1 + 1099 ) / 2 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.85662
3.18498
2.49848
1.28093
1.36253
−3.89858
−1.57172
1.00000 0 1.00000 −4.06778 0 1.66241 1.00000 0 −4.06778
1.2 1.00000 0 1.00000 −3.57869 0 −3.07991 1.00000 0 −3.57869
1.3 1.00000 0 1.00000 −1.78201 0 4.79149 1.00000 0 −1.78201
1.4 1.00000 0 1.00000 −1.63765 0 −3.60660 1.00000 0 −1.63765
1.5 1.00000 0 1.00000 1.18526 0 2.78499 1.00000 0 1.18526
1.6 1.00000 0 1.00000 1.95448 0 0.971244 1.00000 0 1.95448
1.7 1.00000 0 1.00000 2.92640 0 3.47637 1.00000 0 2.92640
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(241\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4338.2.a.w 7
3.b odd 2 1 1446.2.a.n ✓ 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1446.2.a.n ✓ 7 3.b odd 2 1
4338.2.a.w 7 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4338))\):

\( T_{5}^{7} + 5T_{5}^{6} - 12T_{5}^{5} - 72T_{5}^{4} + 32T_{5}^{3} + 276T_{5}^{2} - 288 \) Copy content Toggle raw display
\( T_{7}^{7} - 7T_{7}^{6} - 10T_{7}^{5} + 150T_{7}^{4} - 160T_{7}^{3} - 704T_{7}^{2} + 1568T_{7} - 832 \) Copy content Toggle raw display
\( T_{11}^{7} + 4T_{11}^{6} - 54T_{11}^{5} - 176T_{11}^{4} + 816T_{11}^{3} + 1440T_{11}^{2} - 4256T_{11} + 896 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{7} \) Copy content Toggle raw display
$3$ \( T^{7} \) Copy content Toggle raw display
$5$ \( T^{7} + 5 T^{6} + \cdots - 288 \) Copy content Toggle raw display
$7$ \( T^{7} - 7 T^{6} + \cdots - 832 \) Copy content Toggle raw display
$11$ \( T^{7} + 4 T^{6} + \cdots + 896 \) Copy content Toggle raw display
$13$ \( T^{7} - 14 T^{6} + \cdots + 14400 \) Copy content Toggle raw display
$17$ \( T^{7} + 4 T^{6} + \cdots + 400 \) Copy content Toggle raw display
$19$ \( T^{7} - 7 T^{6} + \cdots + 24 \) Copy content Toggle raw display
$23$ \( T^{7} - T^{6} + \cdots + 96 \) Copy content Toggle raw display
$29$ \( T^{7} + 3 T^{6} + \cdots - 2816 \) Copy content Toggle raw display
$31$ \( T^{7} - 8 T^{6} + \cdots - 1280 \) Copy content Toggle raw display
$37$ \( T^{7} - 18 T^{6} + \cdots - 6248 \) Copy content Toggle raw display
$41$ \( T^{7} + 3 T^{6} + \cdots - 3520 \) Copy content Toggle raw display
$43$ \( T^{7} - 11 T^{6} + \cdots - 10136 \) Copy content Toggle raw display
$47$ \( T^{7} - 18 T^{6} + \cdots - 111360 \) Copy content Toggle raw display
$53$ \( T^{7} - 9 T^{6} + \cdots + 93616 \) Copy content Toggle raw display
$59$ \( T^{7} + 6 T^{6} + \cdots - 1856 \) Copy content Toggle raw display
$61$ \( T^{7} - 31 T^{6} + \cdots + 92224 \) Copy content Toggle raw display
$67$ \( T^{7} - 4 T^{6} + \cdots - 406016 \) Copy content Toggle raw display
$71$ \( T^{7} - 3 T^{6} + \cdots + 23536 \) Copy content Toggle raw display
$73$ \( T^{7} - 16 T^{6} + \cdots - 945920 \) Copy content Toggle raw display
$79$ \( T^{7} - 34 T^{6} + \cdots - 12800 \) Copy content Toggle raw display
$83$ \( T^{7} + 10 T^{6} + \cdots - 1769472 \) Copy content Toggle raw display
$89$ \( T^{7} + 24 T^{6} + \cdots + 10661400 \) Copy content Toggle raw display
$97$ \( T^{7} - 27 T^{6} + \cdots - 4060352 \) Copy content Toggle raw display
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