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Properties

Label 5290.2.a.bi
Level $5290$
Weight $2$
Character orbit 5290.a
Self dual yes
Analytic conductor $42.241$
Analytic rank $0$
Dimension $10$
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 3x^{9} - 15x^{8} + 35x^{7} + 78x^{6} - 123x^{5} - 185x^{4} + 140x^{3} + 177x^{2} - 15x - 23 \) 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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + (\beta_{9} + \beta_{7} + 1) q^{3} + q^{4} - q^{5} + ( - \beta_{9} - \beta_{7} - 1) q^{6} + ( - \beta_{8} + \beta_{7} + \beta_{6} + \cdots + 1) q^{7}+ \cdots + (\beta_{9} + 2 \beta_{8} - \beta_{7} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + (\beta_{9} + \beta_{7} + 1) q^{3} + q^{4} - q^{5} + ( - \beta_{9} - \beta_{7} - 1) q^{6} + ( - \beta_{8} + \beta_{7} + \beta_{6} + \cdots + 1) q^{7}+ \cdots + ( - 3 \beta_{9} + \beta_{7} + 4 \beta_{5} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{2} + 4 q^{3} + 10 q^{4} - 10 q^{5} - 4 q^{6} + 7 q^{7} - 10 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 10 q^{2} + 4 q^{3} + 10 q^{4} - 10 q^{5} - 4 q^{6} + 7 q^{7} - 10 q^{8} + 14 q^{9} + 10 q^{10} - 9 q^{11} + 4 q^{12} - 7 q^{13} - 7 q^{14} - 4 q^{15} + 10 q^{16} + 18 q^{17} - 14 q^{18} - 16 q^{19} - 10 q^{20} - 12 q^{21} + 9 q^{22} - 4 q^{24} + 10 q^{25} + 7 q^{26} + 13 q^{27} + 7 q^{28} + 10 q^{29} + 4 q^{30} - 3 q^{31} - 10 q^{32} - 25 q^{33} - 18 q^{34} - 7 q^{35} + 14 q^{36} + 8 q^{37} + 16 q^{38} + 12 q^{39} + 10 q^{40} + 10 q^{41} + 12 q^{42} + 9 q^{43} - 9 q^{44} - 14 q^{45} + 21 q^{47} + 4 q^{48} + 7 q^{49} - 10 q^{50} + 9 q^{51} - 7 q^{52} + 40 q^{53} - 13 q^{54} + 9 q^{55} - 7 q^{56} - 9 q^{57} - 10 q^{58} + 29 q^{59} - 4 q^{60} - 25 q^{61} + 3 q^{62} - 6 q^{63} + 10 q^{64} + 7 q^{65} + 25 q^{66} + 7 q^{67} + 18 q^{68} + 7 q^{70} + 64 q^{71} - 14 q^{72} - 16 q^{73} - 8 q^{74} + 4 q^{75} - 16 q^{76} + 57 q^{77} - 12 q^{78} - 44 q^{79} - 10 q^{80} + 14 q^{81} - 10 q^{82} + 26 q^{83} - 12 q^{84} - 18 q^{85} - 9 q^{86} + 25 q^{87} + 9 q^{88} - 11 q^{89} + 14 q^{90} + 5 q^{93} - 21 q^{94} + 16 q^{95} - 4 q^{96} + 10 q^{97} - 7 q^{98} - 13 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 3x^{9} - 15x^{8} + 35x^{7} + 78x^{6} - 123x^{5} - 185x^{4} + 140x^{3} + 177x^{2} - 15x - 23 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2208 \nu^{9} + 55689 \nu^{8} - 154322 \nu^{7} - 545688 \nu^{6} + 1652926 \nu^{5} + 1112130 \nu^{4} + \cdots + 112859 ) / 249061 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2856 \nu^{9} - 20596 \nu^{8} - 22377 \nu^{7} + 272686 \nu^{6} + 130597 \nu^{5} - 1043267 \nu^{4} + \cdots - 121616 ) / 249061 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5316 \nu^{9} - 48801 \nu^{8} + 49392 \nu^{7} + 490819 \nu^{6} - 814900 \nu^{5} - 1229230 \nu^{4} + \cdots - 611472 ) / 249061 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10640 \nu^{9} - 57196 \nu^{8} - 54064 \nu^{7} + 615438 \nu^{6} - 294830 \nu^{5} - 1796522 \nu^{4} + \cdots - 33094 ) / 249061 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 12028 \nu^{9} - 20463 \nu^{8} - 172726 \nu^{7} + 92171 \nu^{6} + 691979 \nu^{5} + 225088 \nu^{4} + \cdots - 65688 ) / 249061 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12627 \nu^{9} - 46061 \nu^{8} - 160414 \nu^{7} + 567778 \nu^{6} + 587601 \nu^{5} - 2203788 \nu^{4} + \cdots - 824425 ) / 249061 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 12901 \nu^{9} + 31991 \nu^{8} + 165177 \nu^{7} - 229417 \nu^{6} - 607630 \nu^{5} + 79944 \nu^{4} + \cdots - 202708 ) / 249061 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 38772 \nu^{9} - 183328 \nu^{8} - 319479 \nu^{7} + 2012874 \nu^{6} + 323569 \nu^{5} - 6049688 \nu^{4} + \cdots - 1146613 ) / 249061 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + \beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} - 2\beta_{3} + \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{9} + 5\beta_{8} - 5\beta_{7} + 2\beta_{6} + \beta_{4} - 4\beta_{3} + 3\beta_{2} + 7\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 15 \beta_{9} + 18 \beta_{8} - 22 \beta_{7} + 2 \beta_{6} - 4 \beta_{5} + 12 \beta_{4} - 27 \beta_{3} + \cdots + 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 52 \beta_{9} + 82 \beta_{8} - 90 \beta_{7} + 27 \beta_{6} + 5 \beta_{5} + 29 \beta_{4} - 76 \beta_{3} + \cdots + 22 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 221 \beta_{9} + 299 \beta_{8} - 359 \beta_{7} + 56 \beta_{6} - 3 \beta_{5} + 166 \beta_{4} - 378 \beta_{3} + \cdots + 182 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 819 \beta_{9} + 1252 \beta_{8} - 1424 \beta_{7} + 361 \beta_{6} + 105 \beta_{5} + 535 \beta_{4} + \cdots + 407 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3324 \beta_{9} + 4738 \beta_{8} - 5602 \beta_{7} + 1058 \beta_{6} + 253 \beta_{5} + 2438 \beta_{4} + \cdots + 2245 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 12724 \beta_{9} + 19118 \beta_{8} - 22063 \beta_{7} + 5135 \beta_{6} + 1700 \beta_{5} + 8789 \beta_{4} + \cdots + 6874 \) 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
−1.41654
−0.414299
0.380178
1.49412
2.00869
3.92764
−2.60304
−1.58379
−0.973013
2.18006
−1.00000 −2.66797 1.00000 −1.00000 2.66797 1.17277 −1.00000 4.11808 1.00000
1.2 −1.00000 −2.26320 1.00000 −1.00000 2.26320 0.287092 −1.00000 2.12206 1.00000
1.3 −1.00000 −2.03928 1.00000 −1.00000 2.03928 3.23485 −1.00000 1.15865 1.00000
1.4 −1.00000 −0.274376 1.00000 −1.00000 0.274376 −0.557477 −1.00000 −2.92472 1.00000
1.5 −1.00000 0.259097 1.00000 −1.00000 −0.259097 4.66427 −1.00000 −2.93287 1.00000
1.6 −1.00000 1.34421 1.00000 −1.00000 −1.34421 −0.948752 −1.00000 −1.19309 1.00000
1.7 −1.00000 1.57173 1.00000 −1.00000 −1.57173 −3.09501 −1.00000 −0.529655 1.00000
1.8 −1.00000 1.72956 1.00000 −1.00000 −1.72956 1.60312 −1.00000 −0.00863819 1.00000
1.9 −1.00000 2.95688 1.00000 −1.00000 −2.95688 4.17692 −1.00000 5.74316 1.00000
1.10 −1.00000 3.38334 1.00000 −1.00000 −3.38334 −3.53778 −1.00000 8.44702 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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.bi 10
23.b odd 2 1 5290.2.a.bj 10
23.d odd 22 2 230.2.g.b ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.g.b ✓ 20 23.d odd 22 2
5290.2.a.bi 10 1.a even 1 1 trivial
5290.2.a.bj 10 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5290))\):

\( T_{3}^{10} - 4T_{3}^{9} - 14T_{3}^{8} + 65T_{3}^{7} + 46T_{3}^{6} - 343T_{3}^{5} + 86T_{3}^{4} + 604T_{3}^{3} - 448T_{3}^{2} - 48T_{3} + 32 \) Copy content Toggle raw display
\( T_{7}^{10} - 7 T_{7}^{9} - 14 T_{7}^{8} + 166 T_{7}^{7} - 64 T_{7}^{6} - 1076 T_{7}^{5} + 1142 T_{7}^{4} + \cdots + 197 \) Copy content Toggle raw display
\( T_{11}^{10} + 9 T_{11}^{9} - 4 T_{11}^{8} - 202 T_{11}^{7} - 212 T_{11}^{6} + 1496 T_{11}^{5} + \cdots + 1541 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 4 T^{9} + \cdots + 32 \) Copy content Toggle raw display
$5$ \( (T + 1)^{10} \) Copy content Toggle raw display
$7$ \( T^{10} - 7 T^{9} + \cdots + 197 \) Copy content Toggle raw display
$11$ \( T^{10} + 9 T^{9} + \cdots + 1541 \) Copy content Toggle raw display
$13$ \( T^{10} + 7 T^{9} + \cdots - 115831 \) Copy content Toggle raw display
$17$ \( T^{10} - 18 T^{9} + \cdots - 24608 \) Copy content Toggle raw display
$19$ \( T^{10} + 16 T^{9} + \cdots + 19823 \) Copy content Toggle raw display
$23$ \( T^{10} \) Copy content Toggle raw display
$29$ \( T^{10} - 10 T^{9} + \cdots + 1992352 \) Copy content Toggle raw display
$31$ \( T^{10} + 3 T^{9} + \cdots + 32 \) Copy content Toggle raw display
$37$ \( T^{10} - 8 T^{9} + \cdots + 2765929 \) Copy content Toggle raw display
$41$ \( T^{10} - 10 T^{9} + \cdots + 42481 \) Copy content Toggle raw display
$43$ \( T^{10} - 9 T^{9} + \cdots - 154144 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 130098649 \) Copy content Toggle raw display
$53$ \( T^{10} - 40 T^{9} + \cdots - 2729 \) Copy content Toggle raw display
$59$ \( T^{10} - 29 T^{9} + \cdots - 5224207 \) Copy content Toggle raw display
$61$ \( T^{10} + 25 T^{9} + \cdots + 47470048 \) Copy content Toggle raw display
$67$ \( T^{10} - 7 T^{9} + \cdots - 15690784 \) Copy content Toggle raw display
$71$ \( T^{10} - 64 T^{9} + \cdots + 9795424 \) Copy content Toggle raw display
$73$ \( T^{10} + 16 T^{9} + \cdots - 29322976 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 3014060576 \) Copy content Toggle raw display
$83$ \( T^{10} - 26 T^{9} + \cdots - 400544 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 143444653 \) Copy content Toggle raw display
$97$ \( T^{10} - 10 T^{9} + \cdots + 16912928 \) Copy content Toggle raw display
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