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Properties

Label 3549.2.a.bh
Level $3549$
Weight $2$
Character orbit 3549.a
Self dual yes
Analytic conductor $28.339$
Analytic rank $0$
Dimension $15$
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: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 2 x^{14} - 27 x^{13} + 51 x^{12} + 290 x^{11} - 510 x^{10} - 1575 x^{9} + 2522 x^{8} + \cdots + 281 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 2) q^{4} + (\beta_{13} - 1) q^{5} + \beta_1 q^{6} - q^{7} + (\beta_{14} - \beta_{12} + \cdots + \beta_1) q^{8}+ \cdots + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 2) q^{4} + (\beta_{13} - 1) q^{5} + \beta_1 q^{6} - q^{7} + (\beta_{14} - \beta_{12} + \cdots + \beta_1) q^{8}+ \cdots + (\beta_{10} - \beta_{8} - \beta_{6} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 2 q^{2} + 15 q^{3} + 28 q^{4} - 9 q^{5} + 2 q^{6} - 15 q^{7} + 9 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 2 q^{2} + 15 q^{3} + 28 q^{4} - 9 q^{5} + 2 q^{6} - 15 q^{7} + 9 q^{8} + 15 q^{9} + 21 q^{10} + 5 q^{11} + 28 q^{12} - 2 q^{14} - 9 q^{15} + 50 q^{16} - q^{17} + 2 q^{18} + 3 q^{19} - 23 q^{20} - 15 q^{21} + 21 q^{22} + 4 q^{23} + 9 q^{24} + 50 q^{25} + 15 q^{27} - 28 q^{28} + 9 q^{29} + 21 q^{30} + 7 q^{31} + 35 q^{32} + 5 q^{33} - 2 q^{34} + 9 q^{35} + 28 q^{36} + 17 q^{37} - 12 q^{38} + 46 q^{40} - 22 q^{41} - 2 q^{42} + 36 q^{43} + 29 q^{44} - 9 q^{45} - q^{46} - 12 q^{47} + 50 q^{48} + 15 q^{49} + 53 q^{50} - q^{51} - 5 q^{53} + 2 q^{54} + 43 q^{55} - 9 q^{56} + 3 q^{57} + 29 q^{58} - 29 q^{59} - 23 q^{60} + 12 q^{61} + 14 q^{62} - 15 q^{63} + 95 q^{64} + 21 q^{66} - 12 q^{67} - 16 q^{68} + 4 q^{69} - 21 q^{70} + 36 q^{71} + 9 q^{72} - 29 q^{73} - 5 q^{74} + 50 q^{75} + 25 q^{76} - 5 q^{77} + 35 q^{79} - 89 q^{80} + 15 q^{81} + 51 q^{82} - 10 q^{83} - 28 q^{84} + 23 q^{85} + 19 q^{86} + 9 q^{87} + 73 q^{88} + 25 q^{89} + 21 q^{90} - 31 q^{92} + 7 q^{93} - 19 q^{94} - 7 q^{95} + 35 q^{96} - 26 q^{97} + 2 q^{98} + 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 2 x^{14} - 27 x^{13} + 51 x^{12} + 290 x^{11} - 510 x^{10} - 1575 x^{9} + 2522 x^{8} + \cdots + 281 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 4840 \nu^{14} - 207205 \nu^{13} + 946938 \nu^{12} + 3734994 \nu^{11} - 17154925 \nu^{10} + \cdots + 17099693 ) / 6033052 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 26527 \nu^{14} + 348794 \nu^{13} - 1761470 \nu^{12} - 7125065 \nu^{11} + 28878896 \nu^{10} + \cdots - 38588056 ) / 6033052 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 38567 \nu^{14} + 8311 \nu^{13} + 1175340 \nu^{12} - 610919 \nu^{11} - 13591371 \nu^{10} + \cdots + 5587509 ) / 6033052 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 39159 \nu^{14} + 79358 \nu^{13} - 1156542 \nu^{12} - 1536175 \nu^{11} + 12934898 \nu^{10} + \cdots - 41386850 ) / 6033052 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 21755 \nu^{14} + 123497 \nu^{13} + 139769 \nu^{12} - 2498265 \nu^{11} + 3036617 \nu^{10} + \cdots - 20411763 ) / 3016526 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 60853 \nu^{14} + 116866 \nu^{13} + 1435826 \nu^{12} - 2156565 \nu^{11} - 13912376 \nu^{10} + \cdots + 9987024 ) / 6033052 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 73663 \nu^{14} + 73174 \nu^{13} - 2302936 \nu^{12} - 1328053 \nu^{11} + 27645810 \nu^{10} + \cdots - 27937020 ) / 6033052 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 177180 \nu^{14} + 186673 \nu^{13} + 5359798 \nu^{12} - 4786396 \nu^{11} - 63240615 \nu^{10} + \cdots + 17466621 ) / 6033052 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 121202 \nu^{14} + 43404 \nu^{13} + 3702933 \nu^{12} - 1127106 \nu^{11} - 44424222 \nu^{10} + \cdots + 58774419 ) / 3016526 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 131583 \nu^{14} - 276767 \nu^{13} - 3276487 \nu^{12} + 5917076 \nu^{11} + 33124120 \nu^{10} + \cdots - 37166976 ) / 3016526 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 286506 \nu^{14} + 246133 \nu^{13} + 7666606 \nu^{12} - 5103648 \nu^{11} - 82118737 \nu^{10} + \cdots + 64260237 ) / 6033052 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 328859 \nu^{14} - 463195 \nu^{13} - 8935738 \nu^{12} + 10255723 \nu^{11} + 97315541 \nu^{10} + \cdots - 101420669 ) / 6033052 \) 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
\(\nu^{3}\)\(=\) \( \beta_{14} - \beta_{12} + \beta_{8} + \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{13} + \beta_{11} + \beta_{7} - \beta_{6} - \beta_{5} + 8\beta_{2} + 2\beta _1 + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10 \beta_{14} - 9 \beta_{12} + 2 \beta_{11} - \beta_{10} + 9 \beta_{8} + \beta_{7} + \beta_{6} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3 \beta_{14} - 11 \beta_{13} + 13 \beta_{11} + \beta_{10} - 2 \beta_{9} + 3 \beta_{8} + 12 \beta_{7} + \cdots + 157 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 82 \beta_{14} - 3 \beta_{13} - 69 \beta_{12} + 27 \beta_{11} - 14 \beta_{10} + 78 \beta_{8} + 14 \beta_{7} + \cdots + 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 46 \beta_{14} - 97 \beta_{13} - 5 \beta_{12} + 123 \beta_{11} + 12 \beta_{10} - 34 \beta_{9} + \cdots + 1063 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 636 \beta_{14} - 53 \beta_{13} - 509 \beta_{12} + 267 \beta_{11} - 140 \beta_{10} - 4 \beta_{9} + \cdots + 189 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 503 \beta_{14} - 795 \beta_{13} - 94 \beta_{12} + 1045 \beta_{11} + 100 \beta_{10} - 386 \beta_{9} + \cdots + 7341 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 4838 \beta_{14} - 634 \beta_{13} - 3711 \beta_{12} + 2366 \beta_{11} - 1232 \beta_{10} - 86 \beta_{9} + \cdots + 1979 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 4826 \beta_{14} - 6311 \beta_{13} - 1180 \beta_{12} + 8491 \beta_{11} + 705 \beta_{10} - 3702 \beta_{9} + \cdots + 51420 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 36569 \beta_{14} - 6458 \beta_{13} - 26997 \beta_{12} + 19948 \beta_{11} - 10167 \beta_{10} + \cdots + 19338 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 43352 \beta_{14} - 49362 \beta_{13} - 12504 \beta_{12} + 67626 \beta_{11} + 4396 \beta_{10} + \cdots + 364313 \) 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
−2.61511
−2.60515
−2.41695
−1.89376
−1.29981
−1.08127
−0.415563
0.182130
1.02667
1.09372
1.86754
2.07185
2.49813
2.78394
2.80363
−2.61511 1.00000 4.83880 −4.09685 −2.61511 −1.00000 −7.42376 1.00000 10.7137
1.2 −2.60515 1.00000 4.78683 −1.51368 −2.60515 −1.00000 −7.26012 1.00000 3.94338
1.3 −2.41695 1.00000 3.84166 −1.98263 −2.41695 −1.00000 −4.45120 1.00000 4.79192
1.4 −1.89376 1.00000 1.58633 0.892462 −1.89376 −1.00000 0.783390 1.00000 −1.69011
1.5 −1.29981 1.00000 −0.310501 1.99349 −1.29981 −1.00000 3.00321 1.00000 −2.59115
1.6 −1.08127 1.00000 −0.830866 −1.66920 −1.08127 −1.00000 3.06092 1.00000 1.80485
1.7 −0.415563 1.00000 −1.82731 −4.32224 −0.415563 −1.00000 1.59049 1.00000 1.79616
1.8 0.182130 1.00000 −1.96683 1.12886 0.182130 −1.00000 −0.722479 1.00000 0.205599
1.9 1.02667 1.00000 −0.945953 −3.27646 1.02667 −1.00000 −3.02452 1.00000 −3.36384
1.10 1.09372 1.00000 −0.803784 3.36980 1.09372 −1.00000 −3.06655 1.00000 3.68561
1.11 1.86754 1.00000 1.48770 −4.30941 1.86754 −1.00000 −0.956744 1.00000 −8.04798
1.12 2.07185 1.00000 2.29258 3.98491 2.07185 −1.00000 0.606176 1.00000 8.25616
1.13 2.49813 1.00000 4.24068 2.35199 2.49813 −1.00000 5.59751 1.00000 5.87558
1.14 2.78394 1.00000 5.75034 1.59144 2.78394 −1.00000 10.4407 1.00000 4.43048
1.15 2.80363 1.00000 5.86034 −3.14248 2.80363 −1.00000 10.8230 1.00000 −8.81036
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
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.bh yes 15
13.b even 2 1 3549.2.a.bg ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3549.2.a.bg ✓ 15 13.b even 2 1
3549.2.a.bh yes 15 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3549))\):

\( T_{2}^{15} - 2 T_{2}^{14} - 27 T_{2}^{13} + 51 T_{2}^{12} + 290 T_{2}^{11} - 510 T_{2}^{10} - 1575 T_{2}^{9} + \cdots + 281 \) Copy content Toggle raw display
\( T_{5}^{15} + 9 T_{5}^{14} - 22 T_{5}^{13} - 380 T_{5}^{12} - 169 T_{5}^{11} + 6081 T_{5}^{10} + \cdots - 397312 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} - 2 T^{14} + \cdots + 281 \) Copy content Toggle raw display
$3$ \( (T - 1)^{15} \) Copy content Toggle raw display
$5$ \( T^{15} + 9 T^{14} + \cdots - 397312 \) Copy content Toggle raw display
$7$ \( (T + 1)^{15} \) Copy content Toggle raw display
$11$ \( T^{15} - 5 T^{14} + \cdots + 695227 \) Copy content Toggle raw display
$13$ \( T^{15} \) Copy content Toggle raw display
$17$ \( T^{15} + T^{14} + \cdots + 13071808 \) Copy content Toggle raw display
$19$ \( T^{15} - 3 T^{14} + \cdots + 5898752 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots + 900147443 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots + 2666274624 \) Copy content Toggle raw display
$31$ \( T^{15} - 7 T^{14} + \cdots + 59072 \) Copy content Toggle raw display
$37$ \( T^{15} - 17 T^{14} + \cdots - 1666363 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots + 85163678208 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 2841571328 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 21639614464 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 156125119424 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots + 144963371008 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots + 28606050304 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 1364176809472 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 140599367987 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 15834251264 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 150736832 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots - 10802692096 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots + 150573032217152 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 19521302528 \) Copy content Toggle raw display
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