Properties

Label 153.4.a.f
Level $153$
Weight $4$
Character orbit 153.a
Self dual yes
Analytic conductor $9.027$
Analytic rank $1$
Dimension $3$
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] = [153,4,Mod(1,153)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(153, 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("153.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 153 = 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 153.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: \(9.02729223088\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.5912.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 14x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 51)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 2) q^{2} + (\beta_{2} - 2 \beta_1 + 5) q^{4} + (\beta_{2} - 2 \beta_1 - 2) q^{5} + ( - 4 \beta_{2} + 4 \beta_1 - 4) q^{7} + ( - 5 \beta_{2} - 11) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 2) q^{2} + (\beta_{2} - 2 \beta_1 + 5) q^{4} + (\beta_{2} - 2 \beta_1 - 2) q^{5} + ( - 4 \beta_{2} + 4 \beta_1 - 4) q^{7} + ( - 5 \beta_{2} - 11) q^{8} + ( - 5 \beta_{2} + \beta_1 - 13) q^{10} + ( - 3 \beta_{2} - 10 \beta_1 - 8) q^{11} + (13 \beta_{2} - 6 \beta_1 + 14) q^{13} + (16 \beta_{2} - 16 \beta_1 + 40) q^{14} + (7 \beta_{2} - 10 \beta_1 - 23) q^{16} - 17 q^{17} + ( - 5 \beta_{2} - 10 \beta_1 - 44) q^{19} + (8 \beta_{2} - 12 \beta_1 + 46) q^{20} + ( - \beta_{2} - 17 \beta_1 - 77) q^{22} + ( - 11 \beta_{2} + 22 \beta_1 - 44) q^{23} + (\beta_{2} + 2 \beta_1 - 65) q^{25} + ( - 45 \beta_{2} + 53 \beta_1 - 69) q^{26} + ( - 32 \beta_{2} + 56 \beta_1 - 176) q^{28} + (28 \beta_{2} + 24 \beta_1 - 38) q^{29} + (14 \beta_{2} + 20 \beta_1 - 56) q^{31} + (9 \beta_{2} - 2 \beta_1 + 51) q^{32} + ( - 17 \beta_1 + 34) q^{34} + ( - 4 \beta_{2} + 28 \beta_1 - 148) q^{35} + (18 \beta_{2} + 68 \beta_1 + 14) q^{37} + (5 \beta_{2} - 59 \beta_1 - 7) q^{38} + (4 \beta_{2} + 62 \beta_1 - 88) q^{40} + (7 \beta_{2} - 30 \beta_1 - 230) q^{41} + ( - 91 \beta_{2} + 10 \beta_1 - 52) q^{43} + (10 \beta_{2} + 64) q^{44} + (55 \beta_{2} - 77 \beta_1 + 275) q^{46} + ( - 14 \beta_{2} + 112 \beta_1 - 204) q^{47} + (32 \beta_{2} - 128 \beta_1 + 169) q^{49} + ( - \beta_{2} - 62 \beta_1 + 149) q^{50} + (84 \beta_{2} - 156 \beta_1 + 458) q^{52} + (24 \beta_{2} + 116 \beta_1 - 242) q^{53} + (31 \beta_{2} + 70 \beta_1 + 120) q^{55} + (24 \beta_{2} - 144 \beta_1 + 504) q^{56} + ( - 60 \beta_{2} + 46 \beta_1 + 320) q^{58} + ( - 38 \beta_{2} - 12 \beta_1 + 76) q^{59} + ( - 38 \beta_{2} - 72 \beta_1 + 18) q^{61} + ( - 22 \beta_{2} - 14 \beta_1 + 306) q^{62} + ( - 85 \beta_{2} + 158 \beta_1 + 73) q^{64} + ( - 7 \beta_{2} - 114 \beta_1 + 360) q^{65} + ( - 48 \beta_{2} + 24 \beta_1 - 476) q^{67} + ( - 17 \beta_{2} + 34 \beta_1 - 85) q^{68} + (40 \beta_{2} - 160 \beta_1 + 544) q^{70} + (4 \beta_{2} - 192 \beta_1 + 384) q^{71} + (24 \beta_{2} + 172 \beta_1 - 322) q^{73} + (14 \beta_{2} + 68 \beta_1 + 602) q^{74} + ( - 34 \beta_{2} + 88 \beta_1 - 160) q^{76} + ( - 60 \beta_{2} - 12 \beta_1 - 12) q^{77} + (174 \beta_{2} - 132 \beta_1 - 48) q^{79} + ( - 14 \beta_{2} + 20 \beta_1 + 370) q^{80} + ( - 51 \beta_{2} - 209 \beta_1 + 197) q^{82} + ( - 18 \beta_{2} + 136 \beta_1 + 472) q^{83} + ( - 17 \beta_{2} + 34 \beta_1 + 34) q^{85} + (283 \beta_{2} - 325 \beta_1 + 103) q^{86} + ( - 22 \beta_{2} + 230 \beta_1 + 498) q^{88} + (106 \beta_{2} - 128 \beta_1 - 422) q^{89} + ( - 52 \beta_{2} + 364 \beta_1 - 1444) q^{91} + ( - 154 \beta_{2} + 264 \beta_1 - 836) q^{92} + (154 \beta_{2} - 246 \beta_1 + 1402) q^{94} + (\beta_{2} + 158 \beta_1 + 148) q^{95} + ( - 62 \beta_{2} - 408 \beta_1 + 270) q^{97} + ( - 224 \beta_{2} + 265 \beta_1 - 1458) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 5 q^{2} + 13 q^{4} - 8 q^{5} - 8 q^{7} - 33 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 5 q^{2} + 13 q^{4} - 8 q^{5} - 8 q^{7} - 33 q^{8} - 38 q^{10} - 34 q^{11} + 36 q^{13} + 104 q^{14} - 79 q^{16} - 51 q^{17} - 142 q^{19} + 126 q^{20} - 248 q^{22} - 110 q^{23} - 193 q^{25} - 154 q^{26} - 472 q^{28} - 90 q^{29} - 148 q^{31} + 151 q^{32} + 85 q^{34} - 416 q^{35} + 110 q^{37} - 80 q^{38} - 202 q^{40} - 720 q^{41} - 146 q^{43} + 192 q^{44} + 748 q^{46} - 500 q^{47} + 379 q^{49} + 385 q^{50} + 1218 q^{52} - 610 q^{53} + 430 q^{55} + 1368 q^{56} + 1006 q^{58} + 216 q^{59} - 18 q^{61} + 904 q^{62} + 377 q^{64} + 966 q^{65} - 1404 q^{67} - 221 q^{68} + 1472 q^{70} + 960 q^{71} - 794 q^{73} + 1874 q^{74} - 392 q^{76} - 48 q^{77} - 276 q^{79} + 1130 q^{80} + 382 q^{82} + 1552 q^{83} + 136 q^{85} - 16 q^{86} + 1724 q^{88} - 1394 q^{89} - 3968 q^{91} - 2244 q^{92} + 3960 q^{94} + 602 q^{95} + 402 q^{97} - 4109 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 14x - 10 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 9 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 9 \) 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
−2.75985
−0.795427
4.55528
−4.75985 0 14.6562 7.65616 0 −31.5852 −31.6823 0 −36.4422
1.2 −2.79543 0 −0.185590 −7.18559 0 19.9241 22.8822 0 20.0868
1.3 2.55528 0 −1.47057 −8.47057 0 3.66117 −24.1999 0 −21.6446
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(17\) \(1\)

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 153.4.a.f 3
3.b odd 2 1 51.4.a.e 3
4.b odd 2 1 2448.4.a.bd 3
12.b even 2 1 816.4.a.s 3
15.d odd 2 1 1275.4.a.q 3
21.c even 2 1 2499.4.a.n 3
51.c odd 2 1 867.4.a.k 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.4.a.e 3 3.b odd 2 1
153.4.a.f 3 1.a even 1 1 trivial
816.4.a.s 3 12.b even 2 1
867.4.a.k 3 51.c odd 2 1
1275.4.a.q 3 15.d odd 2 1
2448.4.a.bd 3 4.b odd 2 1
2499.4.a.n 3 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(153))\):

\( T_{2}^{3} + 5T_{2}^{2} - 6T_{2} - 34 \) Copy content Toggle raw display
\( T_{5}^{3} + 8T_{5}^{2} - 59T_{5} - 466 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 5 T^{2} - 6 T - 34 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 8 T^{2} - 59 T - 466 \) Copy content Toggle raw display
$7$ \( T^{3} + 8 T^{2} - 672 T + 2304 \) Copy content Toggle raw display
$11$ \( T^{3} + 34 T^{2} - 1543 T + 8964 \) Copy content Toggle raw display
$13$ \( T^{3} - 36 T^{2} - 5531 T + 122698 \) Copy content Toggle raw display
$17$ \( (T + 17)^{3} \) Copy content Toggle raw display
$19$ \( T^{3} + 142 T^{2} + 4113 T + 8244 \) Copy content Toggle raw display
$23$ \( T^{3} + 110 T^{2} - 5687 T + 53240 \) Copy content Toggle raw display
$29$ \( T^{3} + 90 T^{2} - 37028 T + 415320 \) Copy content Toggle raw display
$31$ \( T^{3} + 148 T^{2} - 6972 T - 640448 \) Copy content Toggle raw display
$37$ \( T^{3} - 110 T^{2} - 80928 T - 5969792 \) Copy content Toggle raw display
$41$ \( T^{3} + 720 T^{2} + \cdots + 10440042 \) Copy content Toggle raw display
$43$ \( T^{3} + 146 T^{2} + \cdots - 62624916 \) Copy content Toggle raw display
$47$ \( T^{3} + 500 T^{2} + \cdots - 30472896 \) Copy content Toggle raw display
$53$ \( T^{3} + 610 T^{2} + \cdots - 80447688 \) Copy content Toggle raw display
$59$ \( T^{3} - 216 T^{2} - 39788 T - 1302384 \) Copy content Toggle raw display
$61$ \( T^{3} + 18 T^{2} - 141152 T + 8127200 \) Copy content Toggle raw display
$67$ \( T^{3} + 1404 T^{2} + \cdots + 62069312 \) Copy content Toggle raw display
$71$ \( T^{3} - 960 T^{2} + \cdots + 227624576 \) Copy content Toggle raw display
$73$ \( T^{3} + 794 T^{2} + \cdots - 227482344 \) Copy content Toggle raw display
$79$ \( T^{3} + 276 T^{2} + \cdots - 220814208 \) Copy content Toggle raw display
$83$ \( T^{3} - 1552 T^{2} + \cdots - 11261392 \) Copy content Toggle raw display
$89$ \( T^{3} + 1394 T^{2} + \cdots - 278458912 \) Copy content Toggle raw display
$97$ \( T^{3} - 402 T^{2} + \cdots + 2026068032 \) Copy content Toggle raw display
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