Properties

Label 3549.2.a.bc
Level $3549$
Weight $2$
Character orbit 3549.a
Self dual yes
Analytic conductor $28.339$
Analytic rank $0$
Dimension $8$
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] = [3549,2,Mod(1,3549)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3549, 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("3549.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3549.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: \(28.3389076774\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 13x^{6} + 22x^{5} + 57x^{4} - 72x^{3} - 96x^{2} + 64x + 61 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 273)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} + (\beta_{5} + \beta_{4} + \beta_1 + 1) q^{4} + ( - \beta_{5} - \beta_{2} + 1) q^{5} - \beta_1 q^{6} - q^{7} + (\beta_{4} + \beta_{2} + \beta_1 + 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{3} + (\beta_{5} + \beta_{4} + \beta_1 + 1) q^{4} + ( - \beta_{5} - \beta_{2} + 1) q^{5} - \beta_1 q^{6} - q^{7} + (\beta_{4} + \beta_{2} + \beta_1 + 1) q^{8} + q^{9} + (2 \beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} + 1) q^{10} + (\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_1 + 2) q^{11} + ( - \beta_{5} - \beta_{4} - \beta_1 - 1) q^{12} - \beta_1 q^{14} + (\beta_{5} + \beta_{2} - 1) q^{15} + (\beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1) q^{16} + (\beta_{7} - 2 \beta_{5} - \beta_{4} + \beta_{2} + 1) q^{17} + \beta_1 q^{18} + (\beta_{7} - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_1 + 3) q^{19} + (2 \beta_{7} + \beta_{6} - 2 \beta_{5} - \beta_{2}) q^{20} + q^{21} + ( - \beta_{5} + 2 \beta_{4} - 3 \beta_{3} - \beta_{2} + 4 \beta_1 + 1) q^{22} + ( - \beta_{6} - \beta_{5} - \beta_{2} - \beta_1) q^{23} + ( - \beta_{4} - \beta_{2} - \beta_1 - 1) q^{24} + ( - 2 \beta_{7} - \beta_{6} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1) q^{25} - q^{27} + ( - \beta_{5} - \beta_{4} - \beta_1 - 1) q^{28} + (\beta_{7} + 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 2) q^{29} + ( - 2 \beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} - 1) q^{30} + ( - \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{31} + (\beta_{6} + 2 \beta_{5} + 3 \beta_{4} + \beta_{3} + 3 \beta_1 + 1) q^{32} + ( - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_1 - 2) q^{33} + (\beta_{7} + 2 \beta_{5} + 3 \beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_1 + 3) q^{34} + (\beta_{5} + \beta_{2} - 1) q^{35} + (\beta_{5} + \beta_{4} + \beta_1 + 1) q^{36} + (\beta_{7} - 2 \beta_{6} - \beta_{2}) q^{37} + ( - 3 \beta_{7} + \beta_{6} + \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 1) q^{38} + (2 \beta_{7} - 3 \beta_{5} + \beta_{4} + 3 \beta_{3} - \beta_{2} - \beta_1) q^{40} + ( - \beta_{7} + 3 \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - 2) q^{41} + \beta_1 q^{42} + ( - 2 \beta_{7} - \beta_{5} + \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + 3 \beta_1) q^{43} + (2 \beta_{7} - \beta_{6} + 4 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 7) q^{44} + ( - \beta_{5} - \beta_{2} + 1) q^{45} + (\beta_{7} - \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{46} + ( - 3 \beta_{7} + \beta_{5} - \beta_{4} - 2 \beta_{2} + \beta_1 - 1) q^{47} + ( - \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1) q^{48} + q^{49} + ( - \beta_{7} - \beta_{6} + 5 \beta_{5} + \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{50} + ( - \beta_{7} + 2 \beta_{5} + \beta_{4} - \beta_{2} - 1) q^{51} + ( - \beta_{7} + 3 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2}) q^{53} - \beta_1 q^{54} + (3 \beta_{7} - \beta_{6} - 3 \beta_{5} - \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{55} + ( - \beta_{4} - \beta_{2} - \beta_1 - 1) q^{56} + ( - \beta_{7} + 2 \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_1 - 3) q^{57} + ( - 5 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 4 \beta_{3} - 3 \beta_{2} + 2 \beta_1) q^{58} + ( - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + 6) q^{59} + ( - 2 \beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_{2}) q^{60} + (2 \beta_{7} - \beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 2) q^{61} + ( - 3 \beta_{7} + \beta_{6} + 2 \beta_{5} + 2 \beta_{3} + \beta_{2} + \beta_1 - 5) q^{62} - q^{63} + (2 \beta_{7} + \beta_{6} - 3 \beta_{5} - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 4) q^{64} + (\beta_{5} - 2 \beta_{4} + 3 \beta_{3} + \beta_{2} - 4 \beta_1 - 1) q^{66} + ( - \beta_{5} + 3 \beta_{4} - \beta_{2} + 2 \beta_1 - 3) q^{67} + (2 \beta_{7} + \beta_{6} - 4 \beta_{5} + \beta_{3} + 5 \beta_1 - 4) q^{68} + (\beta_{6} + \beta_{5} + \beta_{2} + \beta_1) q^{69} + ( - 2 \beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} - 1) q^{70} + (\beta_{6} + \beta_{3} + 3 \beta_{2} + 4) q^{71} + (\beta_{4} + \beta_{2} + \beta_1 + 1) q^{72} + (\beta_{7} + \beta_{4} - 3 \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{73} + ( - 2 \beta_{5} - 6 \beta_{3} - 3 \beta_{2} - \beta_1) q^{74} + (2 \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{75} + ( - 2 \beta_{6} + 3 \beta_{5} + 2 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{76} + ( - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_1 - 2) q^{77} + ( - 2 \beta_{7} - \beta_{5} + \beta_{3} - 3 \beta_{2} + 5) q^{79} + (3 \beta_{7} + \beta_{6} - 3 \beta_{5} + \beta_{4} + 4 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{80} + q^{81} + ( - 2 \beta_{7} - \beta_{6} - 2 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \cdots - 4) q^{82}+ \cdots + (\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_1 + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} - 8 q^{3} + 14 q^{4} + 6 q^{5} - 2 q^{6} - 8 q^{7} + 12 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} - 8 q^{3} + 14 q^{4} + 6 q^{5} - 2 q^{6} - 8 q^{7} + 12 q^{8} + 8 q^{9} - 4 q^{10} + 16 q^{11} - 14 q^{12} - 2 q^{14} - 6 q^{15} + 10 q^{16} - 2 q^{17} + 2 q^{18} + 14 q^{19} - 10 q^{20} + 8 q^{21} + 18 q^{22} - 6 q^{23} - 12 q^{24} + 10 q^{25} - 8 q^{27} - 14 q^{28} + 12 q^{29} + 4 q^{30} + 16 q^{31} + 26 q^{32} - 16 q^{33} + 32 q^{34} - 6 q^{35} + 14 q^{36} - 8 q^{37} + 12 q^{38} - 14 q^{40} - 4 q^{41} + 2 q^{42} + 14 q^{43} + 68 q^{44} + 6 q^{45} - 28 q^{46} + 6 q^{47} - 10 q^{48} + 8 q^{49} + 32 q^{50} + 2 q^{51} + 14 q^{53} - 2 q^{54} - 2 q^{55} - 12 q^{56} - 14 q^{57} + 24 q^{58} + 50 q^{59} + 10 q^{60} - 2 q^{61} - 20 q^{62} - 8 q^{63} + 24 q^{64} - 18 q^{66} - 16 q^{67} - 36 q^{68} + 6 q^{69} + 4 q^{70} + 34 q^{71} + 12 q^{72} - 8 q^{73} - 6 q^{74} - 10 q^{75} - 4 q^{76} - 16 q^{77} + 46 q^{79} - 24 q^{80} + 8 q^{81} - 42 q^{82} + 42 q^{83} + 14 q^{84} + 20 q^{85} + 50 q^{86} - 12 q^{87} + 62 q^{88} + 28 q^{89} - 4 q^{90} - 58 q^{92} - 16 q^{93} + 24 q^{94} - 24 q^{95} - 26 q^{96} + 16 q^{97} + 2 q^{98} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 13x^{6} + 22x^{5} + 57x^{4} - 72x^{3} - 96x^{2} + 64x + 61 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{7} + 3\nu^{6} + 21\nu^{5} - 27\nu^{4} - 56\nu^{3} + 64\nu^{2} + 16\nu - 29 ) / 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - \nu^{3} - 7\nu^{2} + 4\nu + 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{7} - 3\nu^{6} - 21\nu^{5} + 27\nu^{4} + 69\nu^{3} - 64\nu^{2} - 81\nu + 16 ) / 13 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{7} + 3\nu^{6} + 21\nu^{5} - 27\nu^{4} - 69\nu^{3} + 77\nu^{2} + 68\nu - 55 ) / 13 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2\nu^{7} + 3\nu^{6} + 34\nu^{5} - 40\nu^{4} - 160\nu^{3} + 129\nu^{2} + 172\nu - 55 ) / 13 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 5\nu^{6} - 17\nu^{5} - 58\nu^{4} + 80\nu^{3} + 202\nu^{2} - 112\nu - 200 ) / 13 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{5} + 8\beta_{4} + \beta_{3} + \beta_{2} + 8\beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{6} + 2\beta_{5} + 11\beta_{4} + \beta_{3} + 8\beta_{2} + 31\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{7} + \beta_{6} + 43\beta_{5} + 55\beta_{4} + 12\beta_{3} + 12\beta_{2} + 59\beta _1 + 80 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3\beta_{7} + 12\beta_{6} + 23\beta_{5} + 94\beta_{4} + 15\beta_{3} + 54\beta_{2} + 206\beta _1 + 79 \) 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
−2.29328
−2.14586
−1.06752
−0.756173
1.32829
1.67113
2.54225
2.72116
−2.29328 −1.00000 3.25913 −0.692320 2.29328 −1.00000 −2.88753 1.00000 1.58768
1.2 −2.14586 −1.00000 2.60470 1.91954 2.14586 −1.00000 −1.29759 1.00000 −4.11906
1.3 −1.06752 −1.00000 −0.860399 −0.994065 1.06752 −1.00000 3.05354 1.00000 1.06118
1.4 −0.756173 −1.00000 −1.42820 4.01537 0.756173 −1.00000 2.59231 1.00000 −3.03631
1.5 1.32829 −1.00000 −0.235645 1.55828 −1.32829 −1.00000 −2.96959 1.00000 2.06984
1.6 1.67113 −1.00000 0.792659 2.68351 −1.67113 −1.00000 −2.01762 1.00000 4.48449
1.7 2.54225 −1.00000 4.46304 −4.07309 −2.54225 −1.00000 6.26168 1.00000 −10.3548
1.8 2.72116 −1.00000 5.40472 1.58278 −2.72116 −1.00000 9.26480 1.00000 4.30699
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(1\)
\(13\) \(-1\)

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 3549.2.a.bc 8
13.b even 2 1 3549.2.a.ba 8
13.f odd 12 2 273.2.bd.b 16
39.k even 12 2 819.2.ct.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bd.b 16 13.f odd 12 2
819.2.ct.c 16 39.k even 12 2
3549.2.a.ba 8 13.b even 2 1
3549.2.a.bc 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3549))\):

\( T_{2}^{8} - 2T_{2}^{7} - 13T_{2}^{6} + 22T_{2}^{5} + 57T_{2}^{4} - 72T_{2}^{3} - 96T_{2}^{2} + 64T_{2} + 61 \) Copy content Toggle raw display
\( T_{5}^{8} - 6T_{5}^{7} - 7T_{5}^{6} + 104T_{5}^{5} - 177T_{5}^{4} - 70T_{5}^{3} + 312T_{5}^{2} - 40T_{5} - 143 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 2 T^{7} - 13 T^{6} + 22 T^{5} + \cdots + 61 \) Copy content Toggle raw display
$3$ \( (T + 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 6 T^{7} - 7 T^{6} + 104 T^{5} + \cdots - 143 \) Copy content Toggle raw display
$7$ \( (T + 1)^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 16 T^{7} + 60 T^{6} + \cdots - 764 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} + 2 T^{7} - 91 T^{6} + \cdots + 15433 \) Copy content Toggle raw display
$19$ \( T^{8} - 14 T^{7} + 19 T^{6} + \cdots - 3068 \) Copy content Toggle raw display
$23$ \( T^{8} + 6 T^{7} - 57 T^{6} + \cdots - 4784 \) Copy content Toggle raw display
$29$ \( T^{8} - 12 T^{7} - 106 T^{6} + \cdots - 285503 \) Copy content Toggle raw display
$31$ \( T^{8} - 16 T^{7} + 9 T^{6} + \cdots + 39232 \) Copy content Toggle raw display
$37$ \( T^{8} + 8 T^{7} - 105 T^{6} + \cdots - 472811 \) Copy content Toggle raw display
$41$ \( T^{8} + 4 T^{7} - 195 T^{6} + \cdots - 85184 \) Copy content Toggle raw display
$43$ \( T^{8} - 14 T^{7} - 123 T^{6} + \cdots + 774292 \) Copy content Toggle raw display
$47$ \( T^{8} - 6 T^{7} - 265 T^{6} + \cdots - 41036 \) Copy content Toggle raw display
$53$ \( T^{8} - 14 T^{7} - 124 T^{6} + \cdots - 8807 \) Copy content Toggle raw display
$59$ \( T^{8} - 50 T^{7} + 1029 T^{6} + \cdots - 1521692 \) Copy content Toggle raw display
$61$ \( T^{8} + 2 T^{7} - 136 T^{6} + \cdots + 338917 \) Copy content Toggle raw display
$67$ \( T^{8} + 16 T^{7} - 123 T^{6} + \cdots - 345392 \) Copy content Toggle raw display
$71$ \( T^{8} - 34 T^{7} + 336 T^{6} + \cdots + 25924 \) Copy content Toggle raw display
$73$ \( T^{8} + 8 T^{7} - 243 T^{6} + \cdots + 97837 \) Copy content Toggle raw display
$79$ \( T^{8} - 46 T^{7} + 741 T^{6} + \cdots - 7916 \) Copy content Toggle raw display
$83$ \( T^{8} - 42 T^{7} + 596 T^{6} + \cdots - 1196 \) Copy content Toggle raw display
$89$ \( T^{8} - 28 T^{7} + 19 T^{6} + \cdots - 5457824 \) Copy content Toggle raw display
$97$ \( T^{8} - 16 T^{7} - 119 T^{6} + \cdots + 208 \) Copy content Toggle raw display
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