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Properties

Label 1840.4.a.y
Level $1840$
Weight $4$
Character orbit 1840.a
Self dual yes
Analytic conductor $108.564$
Analytic rank $1$
Dimension $9$
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] = [1840,4,Mod(1,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1840.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1840.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: \(108.563514411\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 148x^{7} + 278x^{6} + 6790x^{5} - 5460x^{4} - 105651x^{3} + 62613x^{2} + 508436x - 434240 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 920)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + 5 q^{5} + (\beta_{3} - 3) q^{7} + (\beta_{2} + \beta_1 + 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + 5 q^{5} + (\beta_{3} - 3) q^{7} + (\beta_{2} + \beta_1 + 7) q^{9} + (\beta_{6} - 7) q^{11} + (\beta_{4} - \beta_{3} + 2 \beta_1 + 4) q^{13} - 5 \beta_1 q^{15} + (\beta_{8} + \beta_{7} - \beta_{6} + \cdots + 5) q^{17}+ \cdots + ( - 2 \beta_{8} + 26 \beta_{7} + \cdots - 401) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 3 q^{3} + 45 q^{5} - 25 q^{7} + 62 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 3 q^{3} + 45 q^{5} - 25 q^{7} + 62 q^{9} - 64 q^{11} + 37 q^{13} - 15 q^{15} + 47 q^{17} - 232 q^{19} + 60 q^{21} - 207 q^{23} + 225 q^{25} - 363 q^{27} - 176 q^{29} - 388 q^{31} - 6 q^{33} - 125 q^{35} + 345 q^{37} - 539 q^{39} - 226 q^{41} - 684 q^{43} + 310 q^{45} - 813 q^{47} + 216 q^{49} - 72 q^{51} + 161 q^{53} - 320 q^{55} - 582 q^{57} + 253 q^{59} + 6 q^{61} - 1435 q^{63} + 185 q^{65} - 1221 q^{67} + 69 q^{69} - 2026 q^{71} + 219 q^{73} - 75 q^{75} + 1566 q^{77} - 1378 q^{79} + 577 q^{81} - 2207 q^{83} + 235 q^{85} - 2517 q^{87} + 1194 q^{89} - 2066 q^{91} + 1143 q^{93} - 1160 q^{95} + 256 q^{97} - 3354 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3x^{8} - 148x^{7} + 278x^{6} + 6790x^{5} - 5460x^{4} - 105651x^{3} + 62613x^{2} + 508436x - 434240 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 34 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1689031 \nu^{8} + 6348642 \nu^{7} - 278279136 \nu^{6} - 1137833854 \nu^{5} + \cdots + 1023078218880 ) / 14698407768 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2028883 \nu^{8} + 23457369 \nu^{7} + 166459578 \nu^{6} - 2515209170 \nu^{5} + \cdots + 1162999598520 ) / 7349203884 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5428937 \nu^{8} + 48391258 \nu^{7} + 592045308 \nu^{6} - 5263298494 \nu^{5} + \cdots + 292947844280 ) / 14698407768 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 7148237 \nu^{8} + 31652184 \nu^{7} + 974943780 \nu^{6} - 3059186650 \nu^{5} + \cdots - 756612798312 ) / 14698407768 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 609097 \nu^{8} - 1925721 \nu^{7} - 85633267 \nu^{6} + 177186708 \nu^{5} + 3479907019 \nu^{4} + \cdots + 38323393630 ) / 1224867314 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 10320561 \nu^{8} + 59957372 \nu^{7} + 1318138380 \nu^{6} - 6134999598 \nu^{5} + \cdots - 357907505096 ) / 14698407768 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 34 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{8} + 4\beta_{6} + \beta_{4} + \beta_{3} + 4\beta_{2} + 61\beta _1 + 41 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{8} + \beta_{7} + 2\beta_{6} - 15\beta_{5} + 6\beta_{4} + 7\beta_{3} + 88\beta_{2} + 130\beta _1 + 2072 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 254 \beta_{8} + 63 \beta_{7} + 432 \beta_{6} - 65 \beta_{5} + 120 \beta_{4} + 83 \beta_{3} + \cdots + 5060 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 775 \beta_{8} + 427 \beta_{7} + 682 \beta_{6} - 2003 \beta_{5} + 691 \beta_{4} + 996 \beta_{3} + \cdots + 151015 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 17952 \beta_{8} + 9634 \beta_{7} + 39460 \beta_{6} - 10842 \beta_{5} + 11444 \beta_{4} + \cdots + 489632 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 75424 \beta_{8} + 68462 \beta_{7} + 105392 \beta_{6} - 211274 \beta_{5} + 69604 \beta_{4} + \cdots + 11701066 \) 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
9.40101
8.00856
3.75726
2.51493
0.906617
−3.43544
−4.39603
−5.22516
−8.53174
0 −9.40101 0 5.00000 0 4.64335 0 61.3789 0
1.2 0 −8.00856 0 5.00000 0 −20.0185 0 37.1370 0
1.3 0 −3.75726 0 5.00000 0 6.90943 0 −12.8830 0
1.4 0 −2.51493 0 5.00000 0 −36.6508 0 −20.6751 0
1.5 0 −0.906617 0 5.00000 0 21.6841 0 −26.1780 0
1.6 0 3.43544 0 5.00000 0 26.0624 0 −15.1977 0
1.7 0 4.39603 0 5.00000 0 −2.21142 0 −7.67494 0
1.8 0 5.22516 0 5.00000 0 −10.2305 0 0.302313 0
1.9 0 8.53174 0 5.00000 0 −15.1880 0 45.7906 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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 1840.4.a.y 9
4.b odd 2 1 920.4.a.f ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.4.a.f ✓ 9 4.b odd 2 1
1840.4.a.y 9 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1840))\):

\( T_{3}^{9} + 3 T_{3}^{8} - 148 T_{3}^{7} - 278 T_{3}^{6} + 6790 T_{3}^{5} + 5460 T_{3}^{4} + \cdots + 434240 \) Copy content Toggle raw display
\( T_{7}^{9} + 25 T_{7}^{8} - 1339 T_{7}^{7} - 27513 T_{7}^{6} + 464567 T_{7}^{5} + 8637971 T_{7}^{4} + \cdots + 4571025984 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( T^{9} + 3 T^{8} + \cdots + 434240 \) Copy content Toggle raw display
$5$ \( (T - 5)^{9} \) Copy content Toggle raw display
$7$ \( T^{9} + \cdots + 4571025984 \) Copy content Toggle raw display
$11$ \( T^{9} + \cdots + 263549928000 \) Copy content Toggle raw display
$13$ \( T^{9} + \cdots + 17640339460820 \) Copy content Toggle raw display
$17$ \( T^{9} + \cdots + 899473765732992 \) Copy content Toggle raw display
$19$ \( T^{9} + \cdots - 14\!\cdots\!32 \) Copy content Toggle raw display
$23$ \( (T + 23)^{9} \) Copy content Toggle raw display
$29$ \( T^{9} + \cdots + 20\!\cdots\!92 \) Copy content Toggle raw display
$31$ \( T^{9} + \cdots - 33\!\cdots\!40 \) Copy content Toggle raw display
$37$ \( T^{9} + \cdots - 30\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{9} + \cdots + 82\!\cdots\!82 \) Copy content Toggle raw display
$43$ \( T^{9} + \cdots - 63\!\cdots\!12 \) Copy content Toggle raw display
$47$ \( T^{9} + \cdots + 53\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( T^{9} + \cdots + 30\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{9} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{9} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{9} + \cdots - 36\!\cdots\!08 \) Copy content Toggle raw display
$71$ \( T^{9} + \cdots - 12\!\cdots\!08 \) Copy content Toggle raw display
$73$ \( T^{9} + \cdots - 29\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{9} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{9} + \cdots + 14\!\cdots\!80 \) Copy content Toggle raw display
$89$ \( T^{9} + \cdots + 30\!\cdots\!80 \) Copy content Toggle raw display
$97$ \( T^{9} + \cdots + 12\!\cdots\!64 \) Copy content Toggle raw display
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