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Properties

Label 2576.2.e.a
Level $2576$
Weight $2$
Character orbit 2576.e
Analytic conductor $20.569$
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] = [2576,2,Mod(1471,2576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2576, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("2576.1471");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2576 = 2^{4} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2576.e (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: \(20.5694635607\)
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: \( x^{12} + 22x^{10} + 165x^{8} + 508x^{6} + 582x^{4} + 136x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{9} q^{5} - q^{7} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{9} q^{5} - q^{7} + (\beta_{2} - 1) q^{9} + \beta_{3} q^{11} + ( - \beta_{5} - 1) q^{13} + ( - \beta_{4} + \beta_{2} - 1) q^{15} + (\beta_{11} + \beta_{10} - \beta_{9}) q^{17} + (\beta_{5} + \beta_{3} + \beta_{2} - 1) q^{19} - \beta_1 q^{21} + ( - \beta_{8} + \beta_{4} + \beta_1) q^{23} + ( - \beta_{6} + \beta_{4} + \beta_{3}) q^{25} + ( - \beta_{11} - \beta_{10} + \cdots - \beta_1) q^{27}+ \cdots + (\beta_{6} - \beta_{5} - \beta_{3} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{7} - 8 q^{9} - 12 q^{13} - 8 q^{15} - 8 q^{19} - 4 q^{25} + 12 q^{29} - 12 q^{41} + 16 q^{43} + 12 q^{49} - 8 q^{51} + 8 q^{63} - 40 q^{67} - 28 q^{69} + 4 q^{73} - 16 q^{79} + 20 q^{81} - 48 q^{83} - 32 q^{85} + 12 q^{91} + 4 q^{93} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 22x^{10} + 165x^{8} + 508x^{6} + 582x^{4} + 136x^{2} + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 17\nu^{10} + 416\nu^{8} + 3568\nu^{6} + 12534\nu^{4} + 14573\nu^{2} + 568 ) / 643 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -59\nu^{10} - 1179\nu^{8} - 7466\nu^{6} - 16570\nu^{4} - 7042\nu^{2} + 3324 ) / 1286 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -81\nu^{10} - 1793\nu^{8} - 13672\nu^{6} - 43608\nu^{4} - 52302\nu^{2} - 9212 ) / 1286 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 139\nu^{10} + 3061\nu^{8} + 22668\nu^{6} + 66022\nu^{4} + 63366\nu^{2} + 6838 ) / 1286 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 104\nu^{11} + 2318\nu^{9} + 17705\nu^{7} + 55800\nu^{5} + 65891\nu^{3} + 16297\nu ) / 643 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -255\nu^{11} - 5597\nu^{9} - 41946\nu^{7} - 130140\nu^{5} - 154938\nu^{3} - 41956\nu ) / 1286 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 267\nu^{11} + 5815\nu^{9} + 42876\nu^{7} + 128170\nu^{5} + 138824\nu^{3} + 27984\nu ) / 1286 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -289\nu^{11} - 6429\nu^{9} - 49082\nu^{7} - 155208\nu^{5} - 184084\nu^{3} - 43092\nu ) / 1286 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 284\nu^{11} + 6231\nu^{9} + 46444\nu^{7} + 140704\nu^{5} + 152754\nu^{3} + 24051\nu ) / 643 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} - \beta_{10} + 2\beta_{9} + \beta_{8} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} - \beta_{5} + 2\beta_{4} - 3\beta_{3} - 11\beta_{2} + 29 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 11\beta_{11} + 14\beta_{10} - 23\beta_{9} - 13\beta_{8} + 3\beta_{7} + 58\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -17\beta_{6} + 10\beta_{5} - 29\beta_{4} + 43\beta_{3} + 111\beta_{2} - 245 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -111\beta_{11} - 161\beta_{10} + 234\beta_{9} + 137\beta_{8} - 53\beta_{7} - 522\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 214\beta_{6} - 84\beta_{5} + 340\beta_{4} - 485\beta_{3} - 1107\beta_{2} + 2238 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1107\beta_{11} + 1722\beta_{10} - 2340\beta_{9} - 1378\beta_{8} + 684\beta_{7} + 4921\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -2406\beta_{6} + 694\beta_{5} - 3708\beta_{4} + 5093\beta_{3} + 11045\beta_{2} - 21330 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -11045\beta_{11} - 17850\beta_{10} + 23392\beta_{9} + 13732\beta_{8} - 7826\beta_{7} - 47657\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2576\mathbb{Z}\right)^\times\).

\(n\) \(645\) \(1473\) \(1569\) \(2255\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1471.1
− 3.17144i
− 2.42172i
− 1.95079i
− 1.40307i
− 0.535992i
− 0.125512i
0.125512i
0.535992i
1.40307i
1.95079i
2.42172i
3.17144i
0 − 3.17144i 0 − 2.64852i 0 −1.00000 0 −7.05804 0
1471.2 0 − 2.42172i 0 0.329897i 0 −1.00000 0 −2.86474 0
1471.3 0 − 1.95079i 0 − 0.986486i 0 −1.00000 0 −0.805571 0
1471.4 0 − 1.40307i 0 4.07819i 0 −1.00000 0 1.03140 0
1471.5 0 − 0.535992i 0 − 0.957587i 0 −1.00000 0 2.71271 0
1471.6 0 − 0.125512i 0 2.52086i 0 −1.00000 0 2.98425 0
1471.7 0 0.125512i 0 − 2.52086i 0 −1.00000 0 2.98425 0
1471.8 0 0.535992i 0 0.957587i 0 −1.00000 0 2.71271 0
1471.9 0 1.40307i 0 − 4.07819i 0 −1.00000 0 1.03140 0
1471.10 0 1.95079i 0 0.986486i 0 −1.00000 0 −0.805571 0
1471.11 0 2.42172i 0 − 0.329897i 0 −1.00000 0 −2.86474 0
1471.12 0 3.17144i 0 2.64852i 0 −1.00000 0 −7.05804 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1471.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2576.2.e.a ✓ 12
4.b odd 2 1 2576.2.e.b yes 12
23.b odd 2 1 2576.2.e.b yes 12
92.b even 2 1 inner 2576.2.e.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2576.2.e.a ✓ 12 1.a even 1 1 trivial
2576.2.e.a ✓ 12 92.b even 2 1 inner
2576.2.e.b yes 12 4.b odd 2 1
2576.2.e.b yes 12 23.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2576, [\chi])\):

\( T_{3}^{12} + 22T_{3}^{10} + 165T_{3}^{8} + 508T_{3}^{6} + 582T_{3}^{4} + 136T_{3}^{2} + 2 \) Copy content Toggle raw display
\( T_{11}^{6} - 28T_{11}^{4} - 36T_{11}^{3} + 50T_{11}^{2} + 48T_{11} - 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 22 T^{10} + \cdots + 2 \) Copy content Toggle raw display
$5$ \( T^{12} + 32 T^{10} + \cdots + 72 \) Copy content Toggle raw display
$7$ \( (T + 1)^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} - 28 T^{4} + \cdots - 32)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 6 T^{5} + \cdots + 816)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 104 T^{10} + \cdots + 10368 \) Copy content Toggle raw display
$19$ \( (T^{6} + 4 T^{5} + \cdots - 576)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 148035889 \) Copy content Toggle raw display
$29$ \( (T^{6} - 6 T^{5} + \cdots - 2822)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 146 T^{10} + \cdots + 167042 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 3013363712 \) Copy content Toggle raw display
$41$ \( (T^{6} + 6 T^{5} + \cdots - 10128)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 8 T^{5} + \cdots - 4832)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 298 T^{10} + \cdots + 7091378 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 460136448 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 1038950528 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 3109765248 \) Copy content Toggle raw display
$67$ \( (T^{6} + 20 T^{5} + \cdots - 7508)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 85757626368 \) Copy content Toggle raw display
$73$ \( (T^{6} - 2 T^{5} + \cdots - 77328)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 8 T^{5} + \cdots - 425844)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 24 T^{5} + \cdots + 703488)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + 296 T^{10} + \cdots + 27915392 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 528385032 \) Copy content Toggle raw display
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