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Properties

Label 7569.2.a.bp
Level $7569$
Weight $2$
Character orbit 7569.a
Self dual yes
Analytic conductor $60.439$
Analytic rank $1$
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] = [7569,2,Mod(1,7569)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7569, 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("7569.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7569 = 3^{2} \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7569.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: \(60.4387692899\)
Analytic rank: \(1\)
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: \( x^{12} - 15x^{10} + 78x^{8} - 169x^{6} + 148x^{4} - 36x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 29)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{11} q^{2} + ( - \beta_{10} + \beta_{7} - \beta_{2} + \cdots + 1) q^{4}+ \cdots + (\beta_{11} + \beta_{9} + \cdots + 2 \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{11} q^{2} + ( - \beta_{10} + \beta_{7} - \beta_{2} + \cdots + 1) q^{4}+ \cdots + ( - 5 \beta_{11} - \beta_{9} + \cdots - 4 \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 8 q^{4} - 8 q^{5} + 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 8 q^{4} - 8 q^{5} + 10 q^{7} + 12 q^{13} + 16 q^{16} - 24 q^{20} - 38 q^{22} - 30 q^{23} - 8 q^{25} - 12 q^{28} + 6 q^{34} - 44 q^{35} + 6 q^{49} - 6 q^{52} - 32 q^{53} - 44 q^{59} + 16 q^{62} - 2 q^{64} - 8 q^{65} + 30 q^{67} - 70 q^{71} - 56 q^{74} - 34 q^{80} - 62 q^{82} - 82 q^{83} - 44 q^{86} - 66 q^{88} - 32 q^{91} + 22 q^{92} + 10 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 15x^{10} + 78x^{8} - 169x^{6} + 148x^{4} - 36x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{10} - 12\nu^{8} + 42\nu^{6} - 43\nu^{4} + 35\nu^{2} - 43 ) / 16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{8} - 13\nu^{6} + 51\nu^{4} - 58\nu^{2} + 9 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} - 13\nu^{7} + 51\nu^{5} - 58\nu^{3} + 9\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{10} + 16\nu^{8} - 94\nu^{6} + 247\nu^{4} - 251\nu^{2} + 31 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{11} + 13\nu^{9} - 51\nu^{7} + 54\nu^{5} + 15\nu^{3} - 20\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{11} - 64\nu^{9} + 246\nu^{7} - 259\nu^{5} - 25\nu^{3} + 37\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{10} + 68\nu^{8} - 298\nu^{6} + 463\nu^{4} - 207\nu^{2} - 1 ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -5\nu^{11} + 76\nu^{9} - 402\nu^{7} + 887\nu^{5} - 783\nu^{3} + 199\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5\nu^{11} - 80\nu^{9} + 454\nu^{7} - 1091\nu^{5} + 999\nu^{3} - 155\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -7\nu^{10} + 94\nu^{8} - 400\nu^{6} + 579\nu^{4} - 233\nu^{2} + 23 ) / 8 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -5\nu^{11} + 72\nu^{9} - 350\nu^{7} + 675\nu^{5} - 487\nu^{3} + 67\nu ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{11} + \beta_{9} + \beta_{6} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{11} + 4\beta_{9} - \beta_{8} + 5\beta_{6} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} - 3\beta_{7} + 7\beta_{4} - 6\beta_{2} + 6\beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 27\beta_{11} + 20\beta_{9} - 6\beta_{8} + 28\beta_{6} - 10\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10\beta_{10} - 29\beta_{7} + 43\beta_{4} - 35\beta_{2} + 38\beta _1 + 67 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 153\beta_{11} + 110\beta_{9} - 33\beta_{8} + 167\beta_{6} + 5\beta_{5} - 74\beta_{3} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 79\beta_{10} - 224\beta_{7} + 260\beta_{4} - 203\beta_{2} + 246\beta _1 + 373 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 893\beta_{11} + 633\beta_{9} - 181\beta_{8} + 1024\beta_{6} + 65\beta_{5} - 506\beta_{3} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 571\beta_{10} - 1599\beta_{7} + 1580\beta_{4} - 1189\beta_{2} + 1595\beta _1 + 2159 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 5319\beta_{11} + 3739\beta_{9} - 1009\beta_{8} + 6362\beta_{6} + 586\beta_{5} - 3359\beta_{3} \) 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
−2.16310
0.572821
1.38539
−0.178275
−2.52268
−1.29533
1.29533
2.52268
0.178275
−1.38539
−0.572821
2.16310
−2.60244 0 4.77269 −2.58042 0 0.0751311 −7.21577 0 6.71540
1.2 −2.26775 0 3.14268 −0.0578828 0 −1.56196 −2.59131 0 0.131264
1.3 −1.54928 0 0.400257 0.454328 0 3.41326 2.47844 0 −0.703879
1.4 −1.16308 0 −0.647237 −1.89937 0 1.52574 3.07896 0 2.20913
1.5 −0.549876 0 −1.69764 −2.74405 0 4.47381 2.03324 0 1.50889
1.6 −0.171009 0 −1.97076 2.82740 0 −2.92599 0.679037 0 −0.483513
1.7 0.171009 0 −1.97076 2.82740 0 −2.92599 −0.679037 0 0.483513
1.8 0.549876 0 −1.69764 −2.74405 0 4.47381 −2.03324 0 −1.50889
1.9 1.16308 0 −0.647237 −1.89937 0 1.52574 −3.07896 0 −2.20913
1.10 1.54928 0 0.400257 0.454328 0 3.41326 −2.47844 0 0.703879
1.11 2.26775 0 3.14268 −0.0578828 0 −1.56196 2.59131 0 −0.131264
1.12 2.60244 0 4.77269 −2.58042 0 0.0751311 7.21577 0 −6.71540
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(29\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7569.2.a.bp 12
3.b odd 2 1 841.2.a.k 12
29.b even 2 1 inner 7569.2.a.bp 12
29.f odd 28 2 261.2.o.a 12
87.d odd 2 1 841.2.a.k 12
87.f even 4 2 841.2.b.e 12
87.h odd 14 2 841.2.d.k 24
87.h odd 14 2 841.2.d.l 24
87.h odd 14 2 841.2.d.m 24
87.j odd 14 2 841.2.d.k 24
87.j odd 14 2 841.2.d.l 24
87.j odd 14 2 841.2.d.m 24
87.k even 28 2 29.2.e.a ✓ 12
87.k even 28 2 841.2.e.a 12
87.k even 28 2 841.2.e.e 12
87.k even 28 2 841.2.e.f 12
87.k even 28 2 841.2.e.h 12
87.k even 28 2 841.2.e.i 12
348.v odd 28 2 464.2.y.d 12
435.bc odd 28 2 725.2.p.a 24
435.bk even 28 2 725.2.q.a 12
435.bn odd 28 2 725.2.p.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.e.a ✓ 12 87.k even 28 2
261.2.o.a 12 29.f odd 28 2
464.2.y.d 12 348.v odd 28 2
725.2.p.a 24 435.bc odd 28 2
725.2.p.a 24 435.bn odd 28 2
725.2.q.a 12 435.bk even 28 2
841.2.a.k 12 3.b odd 2 1
841.2.a.k 12 87.d odd 2 1
841.2.b.e 12 87.f even 4 2
841.2.d.k 24 87.h odd 14 2
841.2.d.k 24 87.j odd 14 2
841.2.d.l 24 87.h odd 14 2
841.2.d.l 24 87.j odd 14 2
841.2.d.m 24 87.h odd 14 2
841.2.d.m 24 87.j odd 14 2
841.2.e.a 12 87.k even 28 2
841.2.e.e 12 87.k even 28 2
841.2.e.f 12 87.k even 28 2
841.2.e.h 12 87.k even 28 2
841.2.e.i 12 87.k even 28 2
7569.2.a.bp 12 1.a even 1 1 trivial
7569.2.a.bp 12 29.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7569))\):

\( T_{2}^{12} - 16T_{2}^{10} + 88T_{2}^{8} - 197T_{2}^{6} + 170T_{2}^{4} - 39T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{6} + 4T_{5}^{5} - 5T_{5}^{4} - 34T_{5}^{3} - 24T_{5}^{2} + 16T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{6} - 5T_{7}^{5} - 10T_{7}^{4} + 57T_{7}^{3} + 16T_{7}^{2} - 108T_{7} + 8 \) Copy content Toggle raw display
\( T_{19}^{12} - 63T_{19}^{10} + 1218T_{19}^{8} - 9471T_{19}^{6} + 27832T_{19}^{4} - 18032T_{19}^{2} + 3136 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 16 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} + 4 T^{5} - 5 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} - 5 T^{5} - 10 T^{4} + \cdots + 8)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} - 51 T^{10} + \cdots + 10816 \) Copy content Toggle raw display
$13$ \( (T^{6} - 6 T^{5} - 13 T^{4} + \cdots + 29)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} - 71 T^{10} + \cdots + 53824 \) Copy content Toggle raw display
$19$ \( T^{12} - 63 T^{10} + \cdots + 3136 \) Copy content Toggle raw display
$23$ \( (T^{6} + 15 T^{5} + 78 T^{4} + \cdots + 8)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} - 95 T^{10} + \cdots + 817216 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 110103049 \) Copy content Toggle raw display
$41$ \( T^{12} - 99 T^{10} + \cdots + 107584 \) Copy content Toggle raw display
$43$ \( T^{12} - 263 T^{10} + \cdots + 24364096 \) Copy content Toggle raw display
$47$ \( T^{12} - 387 T^{10} + \cdots + 11343424 \) Copy content Toggle raw display
$53$ \( (T^{6} + 16 T^{5} + \cdots - 97)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 22 T^{5} + \cdots + 1856)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 325694209 \) Copy content Toggle raw display
$67$ \( (T^{6} - 15 T^{5} + \cdots + 20392)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 35 T^{5} + \cdots - 51688)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} - 210 T^{10} + \cdots + 625681 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 30056463424 \) Copy content Toggle raw display
$83$ \( (T^{6} + 41 T^{5} + \cdots - 20488)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 203946961 \) Copy content Toggle raw display
$97$ \( T^{12} - 478 T^{10} + \cdots + 1697809 \) Copy content Toggle raw display
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