Properties

Label 3332.2.b.e
Level $3332$
Weight $2$
Character orbit 3332.b
Analytic conductor $26.606$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3332,2,Mod(2549,3332)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3332, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("3332.2549");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3332.b (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: \(26.6061539535\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 22x^{10} + 174x^{8} + 585x^{6} + 756x^{4} + 325x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 476)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{8} q^{5} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{8} q^{5} + (\beta_{2} - 1) q^{9} - \beta_{3} q^{11} - \beta_{9} q^{13} + ( - \beta_{9} + \beta_{7} + \beta_{5} + \beta_{4} - \beta_{2}) q^{15} - \beta_{5} q^{17} + \beta_{7} q^{19} + ( - \beta_{8} + \beta_{6} + \beta_{3}) q^{23} + (\beta_{11} + \beta_{9} - \beta_{7} - 2) q^{25} + \beta_{3} q^{27} + ( - \beta_{10} + \beta_{8} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_1) q^{29} + ( - \beta_{10} - \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_1) q^{31} + ( - \beta_{9} + \beta_{7} + \beta_{2} - 1) q^{33} + (\beta_{8} + \beta_{6} + \beta_1) q^{37} + ( - \beta_{8} + \beta_{3}) q^{39} + ( - \beta_{10} - \beta_{3}) q^{41} + (\beta_{7} + \beta_{5} + \beta_{4}) q^{43} + (\beta_{10} + \beta_{6} + \beta_{3} + \beta_1) q^{45} + (\beta_{7} - 2 \beta_{2} + 2) q^{47} + (\beta_{9} + \beta_{8} - 2 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{2} - 1) q^{51} + ( - \beta_{11} + \beta_{9} + 2) q^{53} + ( - \beta_{11} + \beta_{7} - \beta_{5} - \beta_{4} + \beta_{2} + 3) q^{55} + (\beta_{10} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_1) q^{57} + (\beta_{11} + \beta_{7} + \beta_{5} + \beta_{4} + \beta_{2} - 1) q^{59} + ( - \beta_{8} + 2 \beta_{3}) q^{61} + ( - \beta_{10} + \beta_{8} - \beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_1) q^{65} + ( - \beta_{11} - 2 \beta_{9} - \beta_{2} + 3) q^{67} + (\beta_{11} + 3 \beta_{9} - 3 \beta_{7} - 2 \beta_{5} - 2 \beta_{4} + \beta_{2}) q^{69} + (2 \beta_{10} + \beta_{8} + 3 \beta_1) q^{71} + ( - \beta_{10} + 2 \beta_{8} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_1) q^{73} + (\beta_{8} - 3 \beta_{6} - 2 \beta_{3}) q^{75} + (\beta_{10} - 2 \beta_{8} - \beta_{6} + \beta_1) q^{79} + (\beta_{9} - \beta_{7} + 2 \beta_{2} - 2) q^{81} + ( - \beta_{9} + 3 \beta_{7} + 2 \beta_{5} + 2 \beta_{4} - 3 \beta_{2} - 2) q^{83} + ( - \beta_{10} + \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{85} + ( - \beta_{11} - 2 \beta_{9} + \beta_{7} + 2 \beta_{5} + 2 \beta_{4} - \beta_{2} + 1) q^{87} + ( - \beta_{11} - \beta_{9} - \beta_{2} + 3) q^{89} + ( - \beta_{11} - \beta_{7} + \beta_{5} + \beta_{4} - 2 \beta_{2} + 6) q^{93} + (\beta_{8} + 2 \beta_{3}) q^{95} + (\beta_{8} + \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_1) q^{97} + (\beta_{10} - \beta_{8} - \beta_{6} - \beta_{5} + \beta_{4} - 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8 q^{9} + 4 q^{13} + 2 q^{15} - 3 q^{17} - 4 q^{19} - 18 q^{25} - 8 q^{33} + 2 q^{43} + 12 q^{47} - 7 q^{51} + 14 q^{53} + 24 q^{55} + 34 q^{67} - 2 q^{69} - 16 q^{81} - 32 q^{83} + 9 q^{85} + 18 q^{87} + 30 q^{89} + 68 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 22x^{10} + 174x^{8} + 585x^{6} + 756x^{4} + 325x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 6\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} + \nu^{10} + 19 \nu^{9} + 11 \nu^{8} + 117 \nu^{7} + 5 \nu^{6} + 210 \nu^{5} - 166 \nu^{4} - 210 \nu^{3} - 82 \nu^{2} - 365 \nu + 75 ) / 24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{11} + \nu^{10} - 19 \nu^{9} + 11 \nu^{8} - 117 \nu^{7} + 5 \nu^{6} - 210 \nu^{5} - 166 \nu^{4} + 210 \nu^{3} - 82 \nu^{2} + 365 \nu + 75 ) / 24 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{11} + 19\nu^{9} + 123\nu^{7} + 306\nu^{5} + 216\nu^{3} + \nu ) / 6 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{10} + 23\nu^{8} + 185\nu^{6} + 578\nu^{4} + 482\nu^{2} + 39 ) / 12 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{11} + 23\nu^{9} + 185\nu^{7} + 590\nu^{5} + 578\nu^{3} + 147\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{10} + 23\nu^{8} + 185\nu^{6} + 590\nu^{4} + 566\nu^{2} + 75 ) / 12 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{11} + 21\nu^{9} + 157\nu^{7} + 490\nu^{5} + 556\nu^{3} + 173\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -5\nu^{10} - 91\nu^{8} - 553\nu^{6} - 1258\nu^{4} - 826\nu^{2} - 15 ) / 12 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} - \beta_{7} - 7\beta_{2} + 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{10} + \beta_{8} + \beta_{6} + \beta_{5} - \beta_{4} - 9\beta_{3} + 40\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{11} - 12\beta_{9} + 15\beta_{7} + 2\beta_{5} + 2\beta_{4} + 46\beta_{2} - 169 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 16\beta_{10} - 16\beta_{8} - 15\beta_{6} - 14\beta_{5} + 14\beta_{4} + 73\beta_{3} - 275\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -15\beta_{11} + 118\beta_{9} - 162\beta_{7} - 31\beta_{5} - 31\beta_{4} - 303\beta_{2} + 1176 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -177\beta_{10} + 180\beta_{8} + 160\beta_{6} + 146\beta_{5} - 146\beta_{4} - 583\beta_{3} + 1929\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 160\beta_{11} - 1072\beta_{9} + 1541\beta_{7} + 343\beta_{5} + 343\beta_{4} + 2023\beta_{2} - 8344 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1701\beta_{10} - 1758\beta_{8} - 1495\beta_{6} - 1358\beta_{5} + 1358\beta_{4} + 4636\beta_{3} - 13771\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2549.1
2.80235i
2.47053i
2.42743i
1.13505i
0.912626i
0.172328i
0.172328i
0.912626i
1.13505i
2.42743i
2.47053i
2.80235i
0 2.80235i 0 1.23560i 0 0 0 −4.85318 0
2549.2 0 2.47053i 0 3.53416i 0 0 0 −3.10351 0
2549.3 0 2.42743i 0 0.208554i 0 0 0 −2.89240 0
2549.4 0 1.13505i 0 4.33588i 0 0 0 1.71167 0
2549.5 0 0.912626i 0 1.62343i 0 0 0 2.16711 0
2549.6 0 0.172328i 0 1.87193i 0 0 0 2.97030 0
2549.7 0 0.172328i 0 1.87193i 0 0 0 2.97030 0
2549.8 0 0.912626i 0 1.62343i 0 0 0 2.16711 0
2549.9 0 1.13505i 0 4.33588i 0 0 0 1.71167 0
2549.10 0 2.42743i 0 0.208554i 0 0 0 −2.89240 0
2549.11 0 2.47053i 0 3.53416i 0 0 0 −3.10351 0
2549.12 0 2.80235i 0 1.23560i 0 0 0 −4.85318 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2549.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.2.b.e 12
7.b odd 2 1 3332.2.b.d 12
7.c even 3 2 476.2.t.a 24
17.b even 2 1 inner 3332.2.b.e 12
119.d odd 2 1 3332.2.b.d 12
119.j even 6 2 476.2.t.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
476.2.t.a 24 7.c even 3 2
476.2.t.a 24 119.j even 6 2
3332.2.b.d 12 7.b odd 2 1
3332.2.b.d 12 119.d odd 2 1
3332.2.b.e 12 1.a even 1 1 trivial
3332.2.b.e 12 17.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 \(S_{2}^{\mathrm{new}}(3332, [\chi])\):

\( T_{3}^{12} + 22T_{3}^{10} + 174T_{3}^{8} + 585T_{3}^{6} + 756T_{3}^{4} + 325T_{3}^{2} + 9 \) Copy content Toggle raw display
\( T_{13}^{6} - 2T_{13}^{5} - 38T_{13}^{4} + 9T_{13}^{3} + 127T_{13}^{2} - 95T_{13} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 22 T^{10} + 174 T^{8} + 585 T^{6} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( T^{12} + 39 T^{10} + 495 T^{8} + \cdots + 144 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 79 T^{10} + 2100 T^{8} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( (T^{6} - 2 T^{5} - 38 T^{4} + 9 T^{3} + 127 T^{2} + \cdots + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 3 T^{11} + 32 T^{10} + \cdots + 24137569 \) Copy content Toggle raw display
$19$ \( (T^{6} + 2 T^{5} - 52 T^{4} + 34 T^{3} + \cdots + 1040)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 143 T^{10} + 6836 T^{8} + \cdots + 2458624 \) Copy content Toggle raw display
$29$ \( T^{12} + 215 T^{10} + \cdots + 10240000 \) Copy content Toggle raw display
$31$ \( T^{12} + 259 T^{10} + 22949 T^{8} + \cdots + 6411024 \) Copy content Toggle raw display
$37$ \( T^{12} + 144 T^{10} + 6939 T^{8} + \cdots + 5531904 \) Copy content Toggle raw display
$41$ \( T^{12} + 142 T^{10} + 5547 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$43$ \( (T^{6} - T^{5} - 73 T^{4} - 123 T^{3} + \cdots + 3920)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 6 T^{5} - 104 T^{4} + 566 T^{3} + \cdots + 240)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 7 T^{5} - 176 T^{4} + \cdots + 162045)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 247 T^{4} + 425 T^{3} + \cdots - 41148)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + 307 T^{10} + 29675 T^{8} + \cdots + 20736 \) Copy content Toggle raw display
$67$ \( (T^{6} - 17 T^{5} - 87 T^{4} + \cdots + 305552)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + 419 T^{10} + \cdots + 1492276900 \) Copy content Toggle raw display
$73$ \( T^{12} + 358 T^{10} + \cdots + 19306546704 \) Copy content Toggle raw display
$79$ \( T^{12} + 312 T^{10} + \cdots + 177023025 \) Copy content Toggle raw display
$83$ \( (T^{6} + 16 T^{5} - 247 T^{4} + \cdots - 26304)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 15 T^{5} - 45 T^{4} + 1231 T^{3} + \cdots + 7272)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 497 T^{10} + \cdots + 6635079936 \) Copy content Toggle raw display
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