Properties

Label 819.4.a.h
Level $819$
Weight $4$
Character orbit 819.a
Self dual yes
Analytic conductor $48.323$
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] = [819,4,Mod(1,819)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(819, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("819.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 819.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: \(48.3225642947\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.5364412.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 27x^{2} - 24x + 76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + (\beta_{3} - \beta_1 + 7) q^{4} + (\beta_{3} - 2 \beta_{2} + \beta_1 + 10) q^{5} + 7 q^{7} + (\beta_{3} - 4 \beta_{2} - 5 \beta_1 + 9) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + (\beta_{3} - \beta_1 + 7) q^{4} + (\beta_{3} - 2 \beta_{2} + \beta_1 + 10) q^{5} + 7 q^{7} + (\beta_{3} - 4 \beta_{2} - 5 \beta_1 + 9) q^{8} + (\beta_{3} - 10 \beta_{2} - 16 \beta_1 - 8) q^{10} + ( - 2 \beta_{3} - \beta_{2} + 5 \beta_1 + 23) q^{11} - 13 q^{13} + ( - 7 \beta_1 + 7) q^{14} + (\beta_{3} - 16 \beta_{2} - 7 \beta_1 + 19) q^{16} + (2 \beta_{2} - 20 \beta_1 + 36) q^{17} + ( - 3 \beta_{3} + 10 \beta_{2} + 9 \beta_1 - 16) q^{19} + (18 \beta_{3} - 18 \beta_{2} - 6 \beta_1 + 132) q^{20} + ( - 4 \beta_{3} + 5 \beta_{2} - 11 \beta_1 - 39) q^{22} + (\beta_{3} + 7 \beta_{2} + 18 \beta_1 + 29) q^{23} + (19 \beta_{3} - 4 \beta_{2} + 11 \beta_1 + 137) q^{25} + (13 \beta_1 - 13) q^{26} + (7 \beta_{3} - 7 \beta_1 + 49) q^{28} + (9 \beta_{3} + 18 \beta_{2} - 29 \beta_1 + 110) q^{29} + ( - 9 \beta_{3} - \beta_{2} + 30 \beta_1 - 75) q^{31} + (15 \beta_{3} - 20 \beta_{2} + 15 \beta_1 + 41) q^{32} + (18 \beta_{3} + 6 \beta_{2} - 36 \beta_1 + 316) q^{34} + (7 \beta_{3} - 14 \beta_{2} + 7 \beta_1 + 70) q^{35} + ( - 18 \beta_{3} + 7 \beta_{2} - 39 \beta_1 - 63) q^{37} + ( - 19 \beta_{3} + 42 \beta_{2} + 34 \beta_1 - 130) q^{38} + (16 \beta_{3} - 46 \beta_{2} - 112 \beta_1 + 208) q^{40} + ( - 2 \beta_{3} + 77 \beta_{2} + 37 \beta_1 + 3) q^{41} + (9 \beta_{3} + 8 \beta_{2} + 21 \beta_1 + 134) q^{43} + (22 \beta_{3} + 39 \beta_{2} + 23 \beta_1 - 53) q^{44} + ( - 25 \beta_{3} + 17 \beta_{2} - 35 \beta_1 - 227) q^{46} + (7 \beta_{3} - 31 \beta_{2} + 4 \beta_1 + 207) q^{47} + 49 q^{49} + ( - 7 \beta_{3} - 88 \beta_{2} - 251 \beta_1 - 93) q^{50} + ( - 13 \beta_{3} + 13 \beta_1 - 91) q^{52} + ( - 27 \beta_{3} + 54 \beta_{2} + 59 \beta_1 + 58) q^{53} + ( - 12 \beta_{2} + 90 \beta_1 + 192) q^{55} + (7 \beta_{3} - 28 \beta_{2} - 35 \beta_1 + 63) q^{56} + (11 \beta_{3} + 18 \beta_{2} - 164 \beta_1 + 480) q^{58} + ( - 48 \beta_{3} + 50 \beta_{2} + 26 \beta_1 + 194) q^{59} + ( - 6 \beta_{3} - 13 \beta_{2} - 9 \beta_1 - 35) q^{61} + ( - 29 \beta_{3} + 33 \beta_{2} + 129 \beta_1 - 459) q^{62} + ( - 3 \beta_{3} + 8 \beta_{2} - 75 \beta_1 - 381) q^{64} + ( - 13 \beta_{3} + 26 \beta_{2} - 13 \beta_1 - 130) q^{65} + ( - 8 \beta_{3} - 31 \beta_{2} + 11 \beta_1 - 127) q^{67} + (30 \beta_{3} - 70 \beta_{2} - 264 \beta_1 + 460) q^{68} + (7 \beta_{3} - 70 \beta_{2} - 112 \beta_1 - 56) q^{70} + ( - 18 \beta_{3} - 32 \beta_{2} - 32 \beta_1 - 368) q^{71} + (75 \beta_{3} - 5 \beta_{2} - 78 \beta_1 + 27) q^{73} + (32 \beta_{3} + 93 \beta_{2} + 171 \beta_1 + 555) q^{74} + ( - 52 \beta_{3} + 122 \beta_{2} + 172 \beta_1 - 402) q^{76} + ( - 14 \beta_{3} - 7 \beta_{2} + 35 \beta_1 + 161) q^{77} + ( - 37 \beta_{3} + 3 \beta_{2} + 64 \beta_1 + 237) q^{79} + (14 \beta_{3} - 58 \beta_{2} - 256 \beta_1 + 656) q^{80} + ( - 114 \beta_{3} + 239 \beta_{2} + 9 \beta_1 - 507) q^{82} + ( - 47 \beta_{3} - 48 \beta_{2} - 13 \beta_1 + 286) q^{83} + (38 \beta_{3} - 236 \beta_{2} - 284 \beta_1 - 64) q^{85} + ( - 29 \beta_{3} - 12 \beta_{2} - 188 \beta_1 - 196) q^{86} + ( - 30 \beta_{3} - 11 \beta_{2} + 9 \beta_1 - 151) q^{88} + ( - 33 \beta_{3} + 16 \beta_{2} - 7 \beta_1 + 720) q^{89} - 91 q^{91} + (10 \beta_{3} + 95 \beta_{2} + 233 \beta_1 + 131) q^{92} + (27 \beta_{3} - 121 \beta_{2} - 249 \beta_1 + 123) q^{94} + ( - 39 \beta_{3} + 72 \beta_{2} + 225 \beta_1 - 558) q^{95} + ( - 13 \beta_{3} - 83 \beta_{2} - 224 \beta_1 + 693) q^{97} + ( - 49 \beta_1 + 49) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 26 q^{4} + 36 q^{5} + 28 q^{7} + 30 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 26 q^{4} + 36 q^{5} + 28 q^{7} + 30 q^{8} - 44 q^{10} + 95 q^{11} - 52 q^{13} + 28 q^{14} + 58 q^{16} + 146 q^{17} - 48 q^{19} + 474 q^{20} - 143 q^{22} + 121 q^{23} + 506 q^{25} - 52 q^{26} + 182 q^{28} + 440 q^{29} - 283 q^{31} + 114 q^{32} + 1234 q^{34} + 252 q^{35} - 209 q^{37} - 440 q^{38} + 754 q^{40} + 93 q^{41} + 526 q^{43} - 217 q^{44} - 841 q^{46} + 783 q^{47} + 196 q^{49} - 446 q^{50} - 338 q^{52} + 340 q^{53} + 756 q^{55} + 210 q^{56} + 1916 q^{58} + 922 q^{59} - 141 q^{61} - 1745 q^{62} - 1510 q^{64} - 468 q^{65} - 523 q^{67} + 1710 q^{68} - 308 q^{70} - 1468 q^{71} - 47 q^{73} + 2249 q^{74} - 1382 q^{76} + 665 q^{77} + 1025 q^{79} + 2538 q^{80} - 1561 q^{82} + 1190 q^{83} - 568 q^{85} - 738 q^{86} - 555 q^{88} + 2962 q^{89} - 364 q^{91} + 599 q^{92} + 317 q^{94} - 2082 q^{95} + 2715 q^{97} + 196 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 27x^{2} - 24x + 76 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 2\nu^{2} - 19\nu + 10 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - \nu - 14 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + 4\beta_{2} + 21\beta _1 + 18 \) 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
5.36970
1.32361
−2.63459
−4.05873
−4.36970 0 11.0943 22.2654 0 7.00000 −13.5212 0 −97.2930
1.2 −0.323612 0 −7.89528 5.91876 0 7.00000 5.14391 0 −1.91538
1.3 3.63459 0 5.21021 −11.0031 0 7.00000 −10.1397 0 −39.9917
1.4 5.05873 0 17.5908 18.8190 0 7.00000 48.5170 0 95.2001
\(n\): e.g. 2-40 or 990-1000
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 819.4.a.h 4
3.b odd 2 1 91.4.a.b 4
12.b even 2 1 1456.4.a.s 4
15.d odd 2 1 2275.4.a.h 4
21.c even 2 1 637.4.a.d 4
39.d odd 2 1 1183.4.a.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.4.a.b 4 3.b odd 2 1
637.4.a.d 4 21.c even 2 1
819.4.a.h 4 1.a even 1 1 trivial
1183.4.a.e 4 39.d odd 2 1
1456.4.a.s 4 12.b even 2 1
2275.4.a.h 4 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(819))\):

\( T_{2}^{4} - 4T_{2}^{3} - 21T_{2}^{2} + 74T_{2} + 26 \) Copy content Toggle raw display
\( T_{5}^{4} - 36T_{5}^{3} + 145T_{5}^{2} + 4806T_{5} - 27288 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 4 T^{3} - 21 T^{2} + 74 T + 26 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 36 T^{3} + 145 T^{2} + \cdots - 27288 \) Copy content Toggle raw display
$7$ \( (T - 7)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 95 T^{3} + 2128 T^{2} + \cdots - 151632 \) Copy content Toggle raw display
$13$ \( (T + 13)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 146 T^{3} - 3120 T^{2} + \cdots - 1065472 \) Copy content Toggle raw display
$19$ \( T^{4} + 48 T^{3} - 5327 T^{2} + \cdots + 317232 \) Copy content Toggle raw display
$23$ \( T^{4} - 121 T^{3} - 5241 T^{2} + \cdots - 2384104 \) Copy content Toggle raw display
$29$ \( T^{4} - 440 T^{3} + \cdots - 484339768 \) Copy content Toggle raw display
$31$ \( T^{4} + 283 T^{3} - 3281 T^{2} + \cdots - 1026856 \) Copy content Toggle raw display
$37$ \( T^{4} + 209 T^{3} + \cdots + 328158128 \) Copy content Toggle raw display
$41$ \( T^{4} - 93 T^{3} + \cdots + 12096773224 \) Copy content Toggle raw display
$43$ \( T^{4} - 526 T^{3} + \cdots + 18583856 \) Copy content Toggle raw display
$47$ \( T^{4} - 783 T^{3} + \cdots - 1054241384 \) Copy content Toggle raw display
$53$ \( T^{4} - 340 T^{3} + \cdots - 11218230832 \) Copy content Toggle raw display
$59$ \( T^{4} - 922 T^{3} + \cdots + 10047112192 \) Copy content Toggle raw display
$61$ \( T^{4} + 141 T^{3} - 9038 T^{2} + \cdots + 3710376 \) Copy content Toggle raw display
$67$ \( T^{4} + 523 T^{3} + \cdots - 951710544 \) Copy content Toggle raw display
$71$ \( T^{4} + 1468 T^{3} + \cdots + 2887158784 \) Copy content Toggle raw display
$73$ \( T^{4} + 47 T^{3} + \cdots + 38124898514 \) Copy content Toggle raw display
$79$ \( T^{4} - 1025 T^{3} + \cdots - 13183278632 \) Copy content Toggle raw display
$83$ \( T^{4} - 1190 T^{3} + \cdots - 11400717312 \) Copy content Toggle raw display
$89$ \( T^{4} - 2962 T^{3} + \cdots + 205066944356 \) Copy content Toggle raw display
$97$ \( T^{4} - 2715 T^{3} + \cdots - 914822530202 \) Copy content Toggle raw display
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