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Properties

Label 5290.2.a.bk
Level $5290$
Weight $2$
Character orbit 5290.a
Self dual yes
Analytic conductor $42.241$
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] = [5290,2,Mod(1,5290)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5290, 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("5290.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5290 = 2 \cdot 5 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5290.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: \(42.2408626693\)
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: \( x^{15} - 5 x^{14} - 24 x^{13} + 142 x^{12} + 184 x^{11} - 1488 x^{10} - 426 x^{9} + 7165 x^{8} + \cdots - 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
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.

\(f(q)\) \(=\) \( q + q^{2} + \beta_1 q^{3} + q^{4} - q^{5} + \beta_1 q^{6} - \beta_{13} q^{7} + q^{8} + (\beta_{12} + \beta_{11} + \beta_{10} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + \beta_1 q^{3} + q^{4} - q^{5} + \beta_1 q^{6} - \beta_{13} q^{7} + q^{8} + (\beta_{12} + \beta_{11} + \beta_{10} + \cdots + 2) q^{9}+ \cdots + (\beta_{13} - 2 \beta_{12} - 5 \beta_{11} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 15 q^{2} + 5 q^{3} + 15 q^{4} - 15 q^{5} + 5 q^{6} + 4 q^{7} + 15 q^{8} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 15 q^{2} + 5 q^{3} + 15 q^{4} - 15 q^{5} + 5 q^{6} + 4 q^{7} + 15 q^{8} + 28 q^{9} - 15 q^{10} - 7 q^{11} + 5 q^{12} + 17 q^{13} + 4 q^{14} - 5 q^{15} + 15 q^{16} - 2 q^{17} + 28 q^{18} - 18 q^{19} - 15 q^{20} - 7 q^{22} + 5 q^{24} + 15 q^{25} + 17 q^{26} + 29 q^{27} + 4 q^{28} + 35 q^{29} - 5 q^{30} + 19 q^{31} + 15 q^{32} + 21 q^{33} - 2 q^{34} - 4 q^{35} + 28 q^{36} + 12 q^{37} - 18 q^{38} + 26 q^{39} - 15 q^{40} + 27 q^{41} - 12 q^{43} - 7 q^{44} - 28 q^{45} + 40 q^{47} + 5 q^{48} + 29 q^{49} + 15 q^{50} + 27 q^{51} + 17 q^{52} + 20 q^{53} + 29 q^{54} + 7 q^{55} + 4 q^{56} + 11 q^{57} + 35 q^{58} + 15 q^{59} - 5 q^{60} - 28 q^{61} + 19 q^{62} + 51 q^{63} + 15 q^{64} - 17 q^{65} + 21 q^{66} - 4 q^{67} - 2 q^{68} - 4 q^{70} + 22 q^{71} + 28 q^{72} + 48 q^{73} + 12 q^{74} + 5 q^{75} - 18 q^{76} + 45 q^{77} + 26 q^{78} + 2 q^{79} - 15 q^{80} + 79 q^{81} + 27 q^{82} + 29 q^{83} + 2 q^{85} - 12 q^{86} - 7 q^{87} - 7 q^{88} - 20 q^{89} - 28 q^{90} - 6 q^{91} + 63 q^{93} + 40 q^{94} + 18 q^{95} + 5 q^{96} + 22 q^{97} + 29 q^{98} - 23 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 5 x^{14} - 24 x^{13} + 142 x^{12} + 184 x^{11} - 1488 x^{10} - 426 x^{9} + 7165 x^{8} + \cdots - 32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 22531579107 \nu^{14} + 556623134808 \nu^{13} - 513706547783 \nu^{12} + \cdots + 68485306833296 ) / 15304491342104 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 76352132097 \nu^{14} + 735952770633 \nu^{13} + 927884133644 \nu^{12} + \cdots + 112584696092304 ) / 30608982684208 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 73466274029 \nu^{14} - 247894962788 \nu^{13} - 2020084509641 \nu^{12} + \cdots - 37229452420768 ) / 15304491342104 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 80463787143 \nu^{14} + 78303641679 \nu^{13} + 2688805524524 \nu^{12} + \cdots - 9518919964928 ) / 15304491342104 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 119436407357 \nu^{14} - 256893932945 \nu^{13} - 3628595839476 \nu^{12} + \cdots + 2350920768928 ) / 15304491342104 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 269906479373 \nu^{14} + 210748358911 \nu^{13} + 9570441126478 \nu^{12} + \cdots + 109034969279600 ) / 30608982684208 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 146370896995 \nu^{14} - 1009256102743 \nu^{13} - 2990860565408 \nu^{12} + \cdots - 110034142918232 ) / 15304491342104 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 88548027537 \nu^{14} - 226141759171 \nu^{13} - 2615628668494 \nu^{12} + 5977316474866 \nu^{11} + \cdots - 610817056776 ) / 7652245671052 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 192123799646 \nu^{14} + 247377081423 \nu^{13} + 6046884060359 \nu^{12} + \cdots - 99770993854232 ) / 15304491342104 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 201468395768 \nu^{14} + 602349359781 \nu^{13} + 5711115653517 \nu^{12} + \cdots + 19650164010968 ) / 15304491342104 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 443965239273 \nu^{14} - 1054464446351 \nu^{13} - 13242482317426 \nu^{12} + \cdots - 721010531424 ) / 15304491342104 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 951978577767 \nu^{14} - 3418982812591 \nu^{13} - 25664032257004 \nu^{12} + \cdots - 248662477203472 ) / 30608982684208 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 973663711765 \nu^{14} + 5441244054933 \nu^{13} + 21689399001464 \nu^{12} + \cdots + 389574100354272 ) / 30608982684208 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} + \beta_{11} + \beta_{10} - \beta_{8} + \beta_{4} - \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{12} - 3 \beta_{11} + \beta_{10} - \beta_{8} + 2 \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{14} - 2 \beta_{13} + 9 \beta_{12} + 10 \beta_{11} + 9 \beta_{10} + 2 \beta_{9} - 10 \beta_{8} + \cdots + 42 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 14 \beta_{12} - 35 \beta_{11} + 11 \beta_{10} + \beta_{9} - 15 \beta_{8} - 2 \beta_{7} + 28 \beta_{6} + \cdots + 41 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 18 \beta_{14} - 28 \beta_{13} + 77 \beta_{12} + 100 \beta_{11} + 83 \beta_{10} + 28 \beta_{9} + \cdots + 397 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 11 \beta_{14} - 10 \beta_{13} - 163 \beta_{12} - 347 \beta_{11} + 109 \beta_{10} + 25 \beta_{9} + \cdots + 472 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 251 \beta_{14} - 332 \beta_{13} + 666 \beta_{12} + 1001 \beta_{11} + 795 \beta_{10} + 317 \beta_{9} + \cdots + 3928 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 276 \beta_{14} - 238 \beta_{13} - 1772 \beta_{12} - 3255 \beta_{11} + 1107 \beta_{10} + 415 \beta_{9} + \cdots + 5249 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 3196 \beta_{14} - 3781 \beta_{13} + 5851 \beta_{12} + 10083 \beta_{11} + 7850 \beta_{10} + \cdots + 39784 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 4757 \beta_{14} - 3963 \beta_{13} - 18549 \beta_{12} - 29488 \beta_{11} + 11704 \beta_{10} + \cdots + 58161 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 39031 \beta_{14} - 42519 \beta_{13} + 52187 \beta_{12} + 102693 \beta_{11} + 79403 \beta_{10} + \cdots + 408818 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 70188 \beta_{14} - 57175 \beta_{13} - 189526 \beta_{12} - 258510 \beta_{11} + 127918 \beta_{10} + \cdots + 646832 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 465996 \beta_{14} - 476269 \beta_{13} + 472132 \beta_{12} + 1059540 \beta_{11} + 818817 \beta_{10} + \cdots + 4244841 \) 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
−3.12386
−3.06629
−2.03401
−1.53931
−1.10043
−0.687865
−0.117629
−0.0566801
0.763450
1.31120
2.59599
2.75508
2.78829
3.15164
3.36041
1.00000 −3.12386 1.00000 −1.00000 −3.12386 2.47855 1.00000 6.75851 −1.00000
1.2 1.00000 −3.06629 1.00000 −1.00000 −3.06629 0.998427 1.00000 6.40210 −1.00000
1.3 1.00000 −2.03401 1.00000 −1.00000 −2.03401 1.56763 1.00000 1.13722 −1.00000
1.4 1.00000 −1.53931 1.00000 −1.00000 −1.53931 −3.71769 1.00000 −0.630528 −1.00000
1.5 1.00000 −1.10043 1.00000 −1.00000 −1.10043 −4.03086 1.00000 −1.78906 −1.00000
1.6 1.00000 −0.687865 1.00000 −1.00000 −0.687865 3.00706 1.00000 −2.52684 −1.00000
1.7 1.00000 −0.117629 1.00000 −1.00000 −0.117629 1.70098 1.00000 −2.98616 −1.00000
1.8 1.00000 −0.0566801 1.00000 −1.00000 −0.0566801 5.00659 1.00000 −2.99679 −1.00000
1.9 1.00000 0.763450 1.00000 −1.00000 0.763450 −4.04396 1.00000 −2.41714 −1.00000
1.10 1.00000 1.31120 1.00000 −1.00000 1.31120 −3.61614 1.00000 −1.28075 −1.00000
1.11 1.00000 2.59599 1.00000 −1.00000 2.59599 −1.59863 1.00000 3.73916 −1.00000
1.12 1.00000 2.75508 1.00000 −1.00000 2.75508 4.44855 1.00000 4.59049 −1.00000
1.13 1.00000 2.78829 1.00000 −1.00000 2.78829 0.104808 1.00000 4.77457 −1.00000
1.14 1.00000 3.15164 1.00000 −1.00000 3.15164 −0.586101 1.00000 6.93286 −1.00000
1.15 1.00000 3.36041 1.00000 −1.00000 3.36041 2.28078 1.00000 8.29236 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(23\) \(1\)

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 5290.2.a.bk 15
23.b odd 2 1 5290.2.a.bl 15
23.c even 11 2 230.2.g.d ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.g.d ✓ 30 23.c even 11 2
5290.2.a.bk 15 1.a even 1 1 trivial
5290.2.a.bl 15 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5290))\):

\( T_{3}^{15} - 5 T_{3}^{14} - 24 T_{3}^{13} + 142 T_{3}^{12} + 184 T_{3}^{11} - 1488 T_{3}^{10} + \cdots - 32 \) Copy content Toggle raw display
\( T_{7}^{15} - 4 T_{7}^{14} - 59 T_{7}^{13} + 243 T_{7}^{12} + 1256 T_{7}^{11} - 5710 T_{7}^{10} + \cdots - 21691 \) Copy content Toggle raw display
\( T_{11}^{15} + 7 T_{11}^{14} - 72 T_{11}^{13} - 510 T_{11}^{12} + 1934 T_{11}^{11} + 13514 T_{11}^{10} + \cdots + 563872 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{15} \) Copy content Toggle raw display
$3$ \( T^{15} - 5 T^{14} + \cdots - 32 \) Copy content Toggle raw display
$5$ \( (T + 1)^{15} \) Copy content Toggle raw display
$7$ \( T^{15} - 4 T^{14} + \cdots - 21691 \) Copy content Toggle raw display
$11$ \( T^{15} + 7 T^{14} + \cdots + 563872 \) Copy content Toggle raw display
$13$ \( T^{15} - 17 T^{14} + \cdots - 34700192 \) Copy content Toggle raw display
$17$ \( T^{15} + 2 T^{14} + \cdots - 13087744 \) Copy content Toggle raw display
$19$ \( T^{15} + 18 T^{14} + \cdots + 82821344 \) Copy content Toggle raw display
$23$ \( T^{15} \) Copy content Toggle raw display
$29$ \( T^{15} - 35 T^{14} + \cdots - 28157152 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 1768741888 \) Copy content Toggle raw display
$37$ \( T^{15} - 12 T^{14} + \cdots + 180896 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots + 5052544199 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 40533032992 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 926946641567 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 709550368 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots - 6952492832 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots + 2359770656 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 41308152544 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots + 105724928 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots + 396062989312 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots - 4511302872064 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 1226113860256 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 44903107363643 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 19390524416 \) Copy content Toggle raw display
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