Properties

Label 3376.2.a.s
Level $3376$
Weight $2$
Character orbit 3376.a
Self dual yes
Analytic conductor $26.957$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3376,2,Mod(1,3376)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3376, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3376.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3376 = 2^{4} \cdot 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3376.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: \(26.9574957224\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - x^{8} - 14x^{7} + 11x^{6} + 66x^{5} - 36x^{4} - 123x^{3} + 38x^{2} + 72x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 211)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} + (\beta_{2} + 2) q^{5} + ( - \beta_{6} + \beta_{4} + \beta_1) q^{7} + (\beta_{7} - \beta_{5} - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{3} + (\beta_{2} + 2) q^{5} + ( - \beta_{6} + \beta_{4} + \beta_1) q^{7} + (\beta_{7} - \beta_{5} - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{9} + ( - \beta_{8} - \beta_{3} + \beta_{2} - 1) q^{11} + ( - \beta_{6} - \beta_{5} - 1) q^{13} + (\beta_{8} - 2 \beta_{5} - \beta_{4} + \beta_{3} - 1) q^{15} + (\beta_{8} + \beta_{6} - \beta_{5} - \beta_{2} + \beta_1) q^{17} + (\beta_{7} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2}) q^{19} + (\beta_{7} - \beta_{6} + \beta_{4} - 2 \beta_{2} + 2 \beta_1) q^{21} + ( - \beta_{8} - \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 1) q^{23} + (\beta_{4} + 3 \beta_{2} + 3) q^{25} + (\beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} - 2 \beta_1 + 1) q^{27} + (\beta_{6} + \beta_{4} - \beta_1 + 4) q^{29} + ( - 2 \beta_{8} + \beta_{5}) q^{31} + ( - \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + \beta_{3} - 3 \beta_{2} + \beta_1 - 3) q^{33} + ( - \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 3 \beta_1 + 2) q^{35} + (\beta_{8} - \beta_{7} + \beta_1) q^{37} + (\beta_{7} + 2 \beta_{6} - \beta_{5} - 2 \beta_{3} + 2 \beta_{2} - \beta_1 + 6) q^{39} + ( - 2 \beta_{8} + \beta_{7} - 3 \beta_{6} + \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{41} + ( - \beta_{7} + \beta_{5} - \beta_{2} - \beta_1 + 4) q^{43} + (\beta_{8} + 2 \beta_{7} + 3 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + \cdots + 6) q^{45}+ \cdots + (3 \beta_{6} + 4 \beta_{5} + 2 \beta_{4} - \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + q^{3} + 15 q^{5} + 2 q^{7} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + q^{3} + 15 q^{5} + 2 q^{7} + 14 q^{9} - 13 q^{11} - 4 q^{13} - 2 q^{15} + 4 q^{17} + 2 q^{19} + 9 q^{21} + 3 q^{23} + 14 q^{25} + 4 q^{27} + 26 q^{29} - 5 q^{31} - 10 q^{33} + 21 q^{35} + 5 q^{37} + 40 q^{39} + 20 q^{41} + 37 q^{43} + 36 q^{45} - 4 q^{47} + 11 q^{49} + 42 q^{51} + 13 q^{53} - 6 q^{55} + 4 q^{57} - 14 q^{59} + 23 q^{61} - q^{63} - 13 q^{65} + 3 q^{67} + 16 q^{69} - 19 q^{71} + 17 q^{73} - 4 q^{75} + 13 q^{77} - 7 q^{79} + 13 q^{81} - 6 q^{83} - 4 q^{85} + 5 q^{87} + 33 q^{89} + 27 q^{91} - 27 q^{93} + 23 q^{95} - 11 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - x^{8} - 14x^{7} + 11x^{6} + 66x^{5} - 36x^{4} - 123x^{3} + 38x^{2} + 72x - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{8} - 31\nu^{7} - 58\nu^{6} + 309\nu^{5} + 82\nu^{4} - 732\nu^{3} + 91\nu^{2} + 186\nu - 200 ) / 116 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{8} + 31\nu^{7} + 58\nu^{6} - 309\nu^{5} - 82\nu^{4} + 848\nu^{3} - 91\nu^{2} - 766\nu + 84 ) / 116 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{8} - 47\nu^{7} + 519\nu^{5} - 364\nu^{4} - 1600\nu^{3} + 1109\nu^{2} + 1384\nu - 524 ) / 116 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9\nu^{8} - 15\nu^{7} - 116\nu^{6} + 157\nu^{5} + 470\nu^{4} - 444\nu^{3} - 637\nu^{2} + 322\nu + 124 ) / 58 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -21\nu^{8} + 35\nu^{7} + 232\nu^{6} - 347\nu^{5} - 768\nu^{4} + 920\nu^{3} + 1003\nu^{2} - 616\nu - 560 ) / 116 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -6\nu^{8} + 10\nu^{7} + 58\nu^{6} - 95\nu^{5} - 120\nu^{4} + 238\nu^{3} - 49\nu^{2} - 147\nu + 101 ) / 29 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 13\nu^{8} - 12\nu^{7} - 145\nu^{6} + 85\nu^{5} + 463\nu^{4} - 42\nu^{3} - 469\nu^{2} - 189\nu + 134 ) / 58 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{8} + \beta_{7} + \beta_{4} - \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{8} + 5\beta_{7} + 5\beta_{4} + 2\beta_{3} - 3\beta_{2} + 5\beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - 2\beta_{6} - \beta_{5} + 8\beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 27\beta_{8} + 29\beta_{7} - 4\beta_{6} + 31\beta_{4} + 20\beta_{3} - 15\beta_{2} + 33\beta _1 + 16 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{8} + 10\beta_{7} - 21\beta_{6} - 12\beta_{5} + 2\beta_{4} + 2\beta_{3} - \beta_{2} + 58\beta _1 + 67 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 161 \beta_{8} + 181 \beta_{7} - 50 \beta_{6} - 6 \beta_{5} + 203 \beta_{4} + 160 \beta_{3} - 97 \beta_{2} + 235 \beta _1 + 108 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 17\beta_{8} + 80\beta_{7} - 173\beta_{6} - 101\beta_{5} + 30\beta_{4} + 34\beta_{3} - 19\beta_{2} + 414\beta _1 + 383 \) Copy content Toggle raw display

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
2.16500
0.107203
−1.54891
1.76769
−1.79023
−2.50967
2.67629
1.09167
−0.959048
0 −3.06548 0 3.67947 0 −0.600695 0 6.39720 0
1.2 0 −2.59855 0 0.449128 0 −0.845532 0 3.75245 0
1.3 0 −1.20044 0 4.25492 0 4.34069 0 −1.55895 0
1.4 0 −1.01743 0 −0.330474 0 −1.14456 0 −1.96484 0
1.5 0 −0.155219 0 0.0597430 0 −4.81622 0 −2.97591 0
1.6 0 1.34369 0 1.39234 0 0.671123 0 −1.19451 0
1.7 0 1.87149 0 3.51131 0 4.24372 0 0.502488 0
1.8 0 2.69353 0 −1.33282 0 2.43218 0 4.25511 0
1.9 0 3.12841 0 3.31637 0 −2.28069 0 6.78696 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(211\) \(1\)

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 3376.2.a.s 9
4.b odd 2 1 211.2.a.d 9
12.b even 2 1 1899.2.a.j 9
20.d odd 2 1 5275.2.a.o 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
211.2.a.d 9 4.b odd 2 1
1899.2.a.j 9 12.b even 2 1
3376.2.a.s 9 1.a even 1 1 trivial
5275.2.a.o 9 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3376))\):

\( T_{3}^{9} - T_{3}^{8} - 20T_{3}^{7} + 17T_{3}^{6} + 128T_{3}^{5} - 80T_{3}^{4} - 292T_{3}^{3} + 72T_{3}^{2} + 224T_{3} + 32 \) Copy content Toggle raw display
\( T_{7}^{9} - 2T_{7}^{8} - 35T_{7}^{7} + 57T_{7}^{6} + 322T_{7}^{5} - 200T_{7}^{4} - 984T_{7}^{3} - 352T_{7}^{2} + 384T_{7} + 192 \) Copy content Toggle raw display
\( T_{11}^{9} + 13 T_{11}^{8} + 31 T_{11}^{7} - 235 T_{11}^{6} - 1233 T_{11}^{5} - 671 T_{11}^{4} + 5452 T_{11}^{3} + 9568 T_{11}^{2} + 3705 T_{11} + 333 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( T^{9} - T^{8} - 20 T^{7} + 17 T^{6} + \cdots + 32 \) Copy content Toggle raw display
$5$ \( T^{9} - 15 T^{8} + 83 T^{7} - 189 T^{6} + \cdots - 3 \) Copy content Toggle raw display
$7$ \( T^{9} - 2 T^{8} - 35 T^{7} + 57 T^{6} + \cdots + 192 \) Copy content Toggle raw display
$11$ \( T^{9} + 13 T^{8} + 31 T^{7} + \cdots + 333 \) Copy content Toggle raw display
$13$ \( T^{9} + 4 T^{8} - 37 T^{7} - 52 T^{6} + \cdots - 931 \) Copy content Toggle raw display
$17$ \( T^{9} - 4 T^{8} - 69 T^{7} + 345 T^{6} + \cdots - 768 \) Copy content Toggle raw display
$19$ \( T^{9} - 2 T^{8} - 77 T^{7} + \cdots - 92579 \) Copy content Toggle raw display
$23$ \( T^{9} - 3 T^{8} - 72 T^{7} + 217 T^{6} + \cdots + 512 \) Copy content Toggle raw display
$29$ \( T^{9} - 26 T^{8} + 221 T^{7} + \cdots + 102912 \) Copy content Toggle raw display
$31$ \( T^{9} + 5 T^{8} - 118 T^{7} + \cdots - 4064 \) Copy content Toggle raw display
$37$ \( T^{9} - 5 T^{8} - 89 T^{7} + \cdots - 70173 \) Copy content Toggle raw display
$41$ \( T^{9} - 20 T^{8} - 56 T^{7} + \cdots + 34048 \) Copy content Toggle raw display
$43$ \( T^{9} - 37 T^{8} + 507 T^{7} + \cdots - 385587 \) Copy content Toggle raw display
$47$ \( T^{9} + 4 T^{8} - 319 T^{7} + \cdots - 4961361 \) Copy content Toggle raw display
$53$ \( T^{9} - 13 T^{8} - 54 T^{7} + \cdots - 101352 \) Copy content Toggle raw display
$59$ \( T^{9} + 14 T^{8} - 258 T^{7} + \cdots + 48901984 \) Copy content Toggle raw display
$61$ \( T^{9} - 23 T^{8} + 97 T^{7} + \cdots - 1016544 \) Copy content Toggle raw display
$67$ \( T^{9} - 3 T^{8} - 364 T^{7} + \cdots - 45391648 \) Copy content Toggle raw display
$71$ \( T^{9} + 19 T^{8} - 105 T^{7} + \cdots - 10015233 \) Copy content Toggle raw display
$73$ \( T^{9} - 17 T^{8} - 201 T^{7} + \cdots - 35767104 \) Copy content Toggle raw display
$79$ \( T^{9} + 7 T^{8} - 210 T^{7} + \cdots - 6812632 \) Copy content Toggle raw display
$83$ \( T^{9} + 6 T^{8} - 322 T^{7} + \cdots + 81789792 \) Copy content Toggle raw display
$89$ \( T^{9} - 33 T^{8} + 54 T^{7} + \cdots + 45488928 \) Copy content Toggle raw display
$97$ \( T^{9} + 11 T^{8} - 168 T^{7} + \cdots - 1454432 \) Copy content Toggle raw display
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