Properties

Label 810.4.a.v
Level $810$
Weight $4$
Character orbit 810.a
Self dual yes
Analytic conductor $47.792$
Analytic rank $0$
Dimension $4$
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] = [810,4,Mod(1,810)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(810, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("810.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 810.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: \(47.7915471046\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 101x^{2} + 51x + 2520 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: no (minimal twist has level 90)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + 5 q^{5} + (\beta_1 + 6) q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 4 q^{4} + 5 q^{5} + (\beta_1 + 6) q^{7} + 8 q^{8} + 10 q^{10} + (\beta_{2} + \beta_1 + 10) q^{11} + ( - \beta_{3} + \beta_{2} + 14) q^{13} + (2 \beta_1 + 12) q^{14} + 16 q^{16} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 + 20) q^{17} + (2 \beta_{3} - \beta_{2} - 3 \beta_1 + 20) q^{19} + 20 q^{20} + (2 \beta_{2} + 2 \beta_1 + 20) q^{22} + (4 \beta_{3} - 4 \beta_{2} + 5 \beta_1 + 20) q^{23} + 25 q^{25} + ( - 2 \beta_{3} + 2 \beta_{2} + 28) q^{26} + (4 \beta_1 + 24) q^{28} + (\beta_{3} + 4 \beta_{2} - 3 \beta_1 + 45) q^{29} + (3 \beta_{3} - 3 \beta_{2} - 10 \beta_1 + 52) q^{31} + 32 q^{32} + ( - 4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 40) q^{34} + (5 \beta_1 + 30) q^{35} + ( - \beta_{3} - 3 \beta_{2} + 6 \beta_1 + 38) q^{37} + (4 \beta_{3} - 2 \beta_{2} - 6 \beta_1 + 40) q^{38} + 40 q^{40} + (\beta_{3} - 7 \beta_{2} - 8 \beta_1 + 49) q^{41} + ( - 4 \beta_{3} - \beta_{2} - 7 \beta_1 + 40) q^{43} + (4 \beta_{2} + 4 \beta_1 + 40) q^{44} + (8 \beta_{3} - 8 \beta_{2} + 10 \beta_1 + 40) q^{46} + ( - 5 \beta_{3} + \beta_{2} - 11 \beta_1 + 112) q^{47} + ( - 4 \beta_{3} + 7 \beta_{2} - 3 \beta_1 + 12) q^{49} + 50 q^{50} + ( - 4 \beta_{3} + 4 \beta_{2} + 56) q^{52} + ( - 12 \beta_{3} + 2 \beta_{2} - 12 \beta_1 + 168) q^{53} + (5 \beta_{2} + 5 \beta_1 + 50) q^{55} + (8 \beta_1 + 48) q^{56} + (2 \beta_{3} + 8 \beta_{2} - 6 \beta_1 + 90) q^{58} + (5 \beta_{3} + 10 \beta_{2} + 3 \beta_1 + 108) q^{59} + ( - 6 \beta_{3} + 11 \beta_{2} + \beta_1 + 123) q^{61} + (6 \beta_{3} - 6 \beta_{2} - 20 \beta_1 + 104) q^{62} + 64 q^{64} + ( - 5 \beta_{3} + 5 \beta_{2} + 70) q^{65} + (10 \beta_{3} - 3 \beta_{2} + 36 \beta_1 + 44) q^{67} + ( - 8 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 80) q^{68} + (10 \beta_1 + 60) q^{70} + (11 \beta_{3} - 3 \beta_{2} + 24 \beta_1 + 36) q^{71} + ( - \beta_{3} - 2 \beta_{2} - 25 \beta_1 - 32) q^{73} + ( - 2 \beta_{3} - 6 \beta_{2} + 12 \beta_1 + 76) q^{74} + (8 \beta_{3} - 4 \beta_{2} - 12 \beta_1 + 80) q^{76} + ( - \beta_{3} + 11 \beta_{2} + 20 \beta_1 + 356) q^{77} + (22 \beta_{3} - 10 \beta_{2} + 4 \beta_1 - 102) q^{79} + 80 q^{80} + (2 \beta_{3} - 14 \beta_{2} - 16 \beta_1 + 98) q^{82} + (6 \beta_{3} + 14 \beta_{2} - 7 \beta_1 + 374) q^{83} + ( - 10 \beta_{3} - 5 \beta_{2} - 5 \beta_1 + 100) q^{85} + ( - 8 \beta_{3} - 2 \beta_{2} - 14 \beta_1 + 80) q^{86} + (8 \beta_{2} + 8 \beta_1 + 80) q^{88} + (7 \beta_{3} - 22 \beta_{2} - 17 \beta_1 + 289) q^{89} + ( - 19 \beta_{3} + 11 \beta_{2} + 26 \beta_1 + 210) q^{91} + (16 \beta_{3} - 16 \beta_{2} + 20 \beta_1 + 80) q^{92} + ( - 10 \beta_{3} + 2 \beta_{2} - 22 \beta_1 + 224) q^{94} + (10 \beta_{3} - 5 \beta_{2} - 15 \beta_1 + 100) q^{95} + (3 \beta_{3} - 4 \beta_{2} - 9 \beta_1 - 88) q^{97} + ( - 8 \beta_{3} + 14 \beta_{2} - 6 \beta_1 + 24) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{2} + 16 q^{4} + 20 q^{5} + 23 q^{7} + 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{2} + 16 q^{4} + 20 q^{5} + 23 q^{7} + 32 q^{8} + 40 q^{10} + 39 q^{11} + 56 q^{13} + 46 q^{14} + 64 q^{16} + 81 q^{17} + 83 q^{19} + 80 q^{20} + 78 q^{22} + 75 q^{23} + 100 q^{25} + 112 q^{26} + 92 q^{28} + 183 q^{29} + 218 q^{31} + 128 q^{32} + 162 q^{34} + 115 q^{35} + 146 q^{37} + 166 q^{38} + 160 q^{40} + 204 q^{41} + 167 q^{43} + 156 q^{44} + 150 q^{46} + 459 q^{47} + 51 q^{49} + 200 q^{50} + 224 q^{52} + 684 q^{53} + 195 q^{55} + 184 q^{56} + 366 q^{58} + 429 q^{59} + 491 q^{61} + 436 q^{62} + 256 q^{64} + 280 q^{65} + 140 q^{67} + 324 q^{68} + 230 q^{70} + 120 q^{71} - 103 q^{73} + 292 q^{74} + 332 q^{76} + 1404 q^{77} - 412 q^{79} + 320 q^{80} + 408 q^{82} + 1503 q^{83} + 405 q^{85} + 334 q^{86} + 312 q^{88} + 1173 q^{89} + 814 q^{91} + 300 q^{92} + 918 q^{94} + 415 q^{95} - 343 q^{97} + 102 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 101x^{2} + 51x + 2520 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 2\nu^{2} - 56\nu - 126 ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 7\nu^{2} - 38\nu + 327 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 3\nu^{2} + 3\nu - 153 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - \beta _1 + 2 ) / 9 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{3} - \beta_{2} + \beta _1 + 457 ) / 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 52\beta_{3} + 58\beta_{2} - 31\beta _1 + 332 ) / 9 \) 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
6.90878
−7.32913
7.80045
−6.38010
2.00000 0 4.00000 5.00000 0 −23.2217 8.00000 0 10.0000
1.2 2.00000 0 4.00000 5.00000 0 5.39039 8.00000 0 10.0000
1.3 2.00000 0 4.00000 5.00000 0 17.1678 8.00000 0 10.0000
1.4 2.00000 0 4.00000 5.00000 0 23.6636 8.00000 0 10.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(-1\)

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 810.4.a.v 4
3.b odd 2 1 810.4.a.s 4
9.c even 3 2 90.4.e.e 8
9.d odd 6 2 270.4.e.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.4.e.e 8 9.c even 3 2
270.4.e.e 8 9.d odd 6 2
810.4.a.s 4 3.b odd 2 1
810.4.a.v 4 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(810))\):

\( T_{7}^{4} - 23T_{7}^{3} - 447T_{7}^{2} + 12355T_{7} - 50852 \) Copy content Toggle raw display
\( T_{11}^{4} - 39T_{11}^{3} - 2046T_{11}^{2} + 79236T_{11} - 63252 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T - 5)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 23 T^{3} - 447 T^{2} + \cdots - 50852 \) Copy content Toggle raw display
$11$ \( T^{4} - 39 T^{3} - 2046 T^{2} + \cdots - 63252 \) Copy content Toggle raw display
$13$ \( T^{4} - 56 T^{3} - 1752 T^{2} + \cdots + 562384 \) Copy content Toggle raw display
$17$ \( T^{4} - 81 T^{3} - 14106 T^{2} + \cdots - 8793540 \) Copy content Toggle raw display
$19$ \( T^{4} - 83 T^{3} - 15900 T^{2} + \cdots - 49868756 \) Copy content Toggle raw display
$23$ \( T^{4} - 75 T^{3} + \cdots + 666450486 \) Copy content Toggle raw display
$29$ \( T^{4} - 183 T^{3} + \cdots + 176616378 \) Copy content Toggle raw display
$31$ \( T^{4} - 218 T^{3} + \cdots - 340753256 \) Copy content Toggle raw display
$37$ \( T^{4} - 146 T^{3} + \cdots + 203845072 \) Copy content Toggle raw display
$41$ \( T^{4} - 204 T^{3} + \cdots + 2240913033 \) Copy content Toggle raw display
$43$ \( T^{4} - 167 T^{3} + \cdots - 834564188 \) Copy content Toggle raw display
$47$ \( T^{4} - 459 T^{3} + \cdots - 2330137008 \) Copy content Toggle raw display
$53$ \( T^{4} - 684 T^{3} + \cdots - 31164536880 \) Copy content Toggle raw display
$59$ \( T^{4} - 429 T^{3} + \cdots + 609186888 \) Copy content Toggle raw display
$61$ \( T^{4} - 491 T^{3} + \cdots - 5623974566 \) Copy content Toggle raw display
$67$ \( T^{4} - 140 T^{3} + \cdots - 6369165845 \) Copy content Toggle raw display
$71$ \( T^{4} - 120 T^{3} + \cdots + 1875123432 \) Copy content Toggle raw display
$73$ \( T^{4} + 103 T^{3} + \cdots + 12661457536 \) Copy content Toggle raw display
$79$ \( T^{4} + 412 T^{3} + \cdots + 25376329840 \) Copy content Toggle raw display
$83$ \( T^{4} - 1503 T^{3} + \cdots - 28242634788 \) Copy content Toggle raw display
$89$ \( T^{4} - 1173 T^{3} + \cdots + 35969800734 \) Copy content Toggle raw display
$97$ \( T^{4} + 343 T^{3} + \cdots + 705224200 \) Copy content Toggle raw display
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