Properties

Label 51.4.a.e
Level $51$
Weight $4$
Character orbit 51.a
Self dual yes
Analytic conductor $3.009$
Analytic rank $0$
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] = [51,4,Mod(1,51)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(51, 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("51.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 51 = 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 51.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: \(3.00909741029\)
Analytic rank: \(0\)
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: yes
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 2) q^{2} + 3 q^{3} + (\beta_{2} - 2 \beta_1 + 5) q^{4} + ( - \beta_{2} + 2 \beta_1 + 2) q^{5} + ( - 3 \beta_1 + 6) q^{6} + ( - 4 \beta_{2} + 4 \beta_1 - 4) q^{7} + (5 \beta_{2} + 11) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 2) q^{2} + 3 q^{3} + (\beta_{2} - 2 \beta_1 + 5) q^{4} + ( - \beta_{2} + 2 \beta_1 + 2) q^{5} + ( - 3 \beta_1 + 6) q^{6} + ( - 4 \beta_{2} + 4 \beta_1 - 4) q^{7} + (5 \beta_{2} + 11) q^{8} + 9 q^{9} + ( - 5 \beta_{2} + \beta_1 - 13) q^{10} + (3 \beta_{2} + 10 \beta_1 + 8) q^{11} + (3 \beta_{2} - 6 \beta_1 + 15) q^{12} + (13 \beta_{2} - 6 \beta_1 + 14) q^{13} + ( - 16 \beta_{2} + 16 \beta_1 - 40) q^{14} + ( - 3 \beta_{2} + 6 \beta_1 + 6) q^{15} + (7 \beta_{2} - 10 \beta_1 - 23) q^{16} + 17 q^{17} + ( - 9 \beta_1 + 18) q^{18} + ( - 5 \beta_{2} - 10 \beta_1 - 44) q^{19} + ( - 8 \beta_{2} + 12 \beta_1 - 46) q^{20} + ( - 12 \beta_{2} + 12 \beta_1 - 12) q^{21} + ( - \beta_{2} - 17 \beta_1 - 77) q^{22} + (11 \beta_{2} - 22 \beta_1 + 44) q^{23} + (15 \beta_{2} + 33) q^{24} + (\beta_{2} + 2 \beta_1 - 65) q^{25} + (45 \beta_{2} - 53 \beta_1 + 69) q^{26} + 27 q^{27} + ( - 32 \beta_{2} + 56 \beta_1 - 176) q^{28} + ( - 28 \beta_{2} - 24 \beta_1 + 38) q^{29} + ( - 15 \beta_{2} + 3 \beta_1 - 39) q^{30} + (14 \beta_{2} + 20 \beta_1 - 56) q^{31} + ( - 9 \beta_{2} + 2 \beta_1 - 51) q^{32} + (9 \beta_{2} + 30 \beta_1 + 24) q^{33} + ( - 17 \beta_1 + 34) q^{34} + (4 \beta_{2} - 28 \beta_1 + 148) q^{35} + (9 \beta_{2} - 18 \beta_1 + 45) q^{36} + (18 \beta_{2} + 68 \beta_1 + 14) q^{37} + ( - 5 \beta_{2} + 59 \beta_1 + 7) q^{38} + (39 \beta_{2} - 18 \beta_1 + 42) q^{39} + (4 \beta_{2} + 62 \beta_1 - 88) q^{40} + ( - 7 \beta_{2} + 30 \beta_1 + 230) q^{41} + ( - 48 \beta_{2} + 48 \beta_1 - 120) q^{42} + ( - 91 \beta_{2} + 10 \beta_1 - 52) q^{43} + ( - 10 \beta_{2} - 64) q^{44} + ( - 9 \beta_{2} + 18 \beta_1 + 18) q^{45} + (55 \beta_{2} - 77 \beta_1 + 275) q^{46} + (14 \beta_{2} - 112 \beta_1 + 204) q^{47} + (21 \beta_{2} - 30 \beta_1 - 69) q^{48} + (32 \beta_{2} - 128 \beta_1 + 169) q^{49} + (\beta_{2} + 62 \beta_1 - 149) q^{50} + 51 q^{51} + (84 \beta_{2} - 156 \beta_1 + 458) q^{52} + ( - 24 \beta_{2} - 116 \beta_1 + 242) q^{53} + ( - 27 \beta_1 + 54) q^{54} + (31 \beta_{2} + 70 \beta_1 + 120) q^{55} + ( - 24 \beta_{2} + 144 \beta_1 - 504) q^{56} + ( - 15 \beta_{2} - 30 \beta_1 - 132) q^{57} + ( - 60 \beta_{2} + 46 \beta_1 + 320) q^{58} + (38 \beta_{2} + 12 \beta_1 - 76) q^{59} + ( - 24 \beta_{2} + 36 \beta_1 - 138) q^{60} + ( - 38 \beta_{2} - 72 \beta_1 + 18) q^{61} + (22 \beta_{2} + 14 \beta_1 - 306) q^{62} + ( - 36 \beta_{2} + 36 \beta_1 - 36) q^{63} + ( - 85 \beta_{2} + 158 \beta_1 + 73) q^{64} + (7 \beta_{2} + 114 \beta_1 - 360) q^{65} + ( - 3 \beta_{2} - 51 \beta_1 - 231) q^{66} + ( - 48 \beta_{2} + 24 \beta_1 - 476) q^{67} + (17 \beta_{2} - 34 \beta_1 + 85) q^{68} + (33 \beta_{2} - 66 \beta_1 + 132) q^{69} + (40 \beta_{2} - 160 \beta_1 + 544) q^{70} + ( - 4 \beta_{2} + 192 \beta_1 - 384) q^{71} + (45 \beta_{2} + 99) q^{72} + (24 \beta_{2} + 172 \beta_1 - 322) q^{73} + ( - 14 \beta_{2} - 68 \beta_1 - 602) q^{74} + (3 \beta_{2} + 6 \beta_1 - 195) q^{75} + ( - 34 \beta_{2} + 88 \beta_1 - 160) q^{76} + (60 \beta_{2} + 12 \beta_1 + 12) q^{77} + (135 \beta_{2} - 159 \beta_1 + 207) q^{78} + (174 \beta_{2} - 132 \beta_1 - 48) q^{79} + (14 \beta_{2} - 20 \beta_1 - 370) q^{80} + 81 q^{81} + ( - 51 \beta_{2} - 209 \beta_1 + 197) q^{82} + (18 \beta_{2} - 136 \beta_1 - 472) q^{83} + ( - 96 \beta_{2} + 168 \beta_1 - 528) q^{84} + ( - 17 \beta_{2} + 34 \beta_1 + 34) q^{85} + ( - 283 \beta_{2} + 325 \beta_1 - 103) q^{86} + ( - 84 \beta_{2} - 72 \beta_1 + 114) q^{87} + ( - 22 \beta_{2} + 230 \beta_1 + 498) q^{88} + ( - 106 \beta_{2} + 128 \beta_1 + 422) q^{89} + ( - 45 \beta_{2} + 9 \beta_1 - 117) q^{90} + ( - 52 \beta_{2} + 364 \beta_1 - 1444) q^{91} + (154 \beta_{2} - 264 \beta_1 + 836) q^{92} + (42 \beta_{2} + 60 \beta_1 - 168) q^{93} + (154 \beta_{2} - 246 \beta_1 + 1402) q^{94} + ( - \beta_{2} - 158 \beta_1 - 148) q^{95} + ( - 27 \beta_{2} + 6 \beta_1 - 153) q^{96} + ( - 62 \beta_{2} - 408 \beta_1 + 270) q^{97} + (224 \beta_{2} - 265 \beta_1 + 1458) q^{98} + (27 \beta_{2} + 90 \beta_1 + 72) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 5 q^{2} + 9 q^{3} + 13 q^{4} + 8 q^{5} + 15 q^{6} - 8 q^{7} + 33 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 5 q^{2} + 9 q^{3} + 13 q^{4} + 8 q^{5} + 15 q^{6} - 8 q^{7} + 33 q^{8} + 27 q^{9} - 38 q^{10} + 34 q^{11} + 39 q^{12} + 36 q^{13} - 104 q^{14} + 24 q^{15} - 79 q^{16} + 51 q^{17} + 45 q^{18} - 142 q^{19} - 126 q^{20} - 24 q^{21} - 248 q^{22} + 110 q^{23} + 99 q^{24} - 193 q^{25} + 154 q^{26} + 81 q^{27} - 472 q^{28} + 90 q^{29} - 114 q^{30} - 148 q^{31} - 151 q^{32} + 102 q^{33} + 85 q^{34} + 416 q^{35} + 117 q^{36} + 110 q^{37} + 80 q^{38} + 108 q^{39} - 202 q^{40} + 720 q^{41} - 312 q^{42} - 146 q^{43} - 192 q^{44} + 72 q^{45} + 748 q^{46} + 500 q^{47} - 237 q^{48} + 379 q^{49} - 385 q^{50} + 153 q^{51} + 1218 q^{52} + 610 q^{53} + 135 q^{54} + 430 q^{55} - 1368 q^{56} - 426 q^{57} + 1006 q^{58} - 216 q^{59} - 378 q^{60} - 18 q^{61} - 904 q^{62} - 72 q^{63} + 377 q^{64} - 966 q^{65} - 744 q^{66} - 1404 q^{67} + 221 q^{68} + 330 q^{69} + 1472 q^{70} - 960 q^{71} + 297 q^{72} - 794 q^{73} - 1874 q^{74} - 579 q^{75} - 392 q^{76} + 48 q^{77} + 462 q^{78} - 276 q^{79} - 1130 q^{80} + 243 q^{81} + 382 q^{82} - 1552 q^{83} - 1416 q^{84} + 136 q^{85} + 16 q^{86} + 270 q^{87} + 1724 q^{88} + 1394 q^{89} - 342 q^{90} - 3968 q^{91} + 2244 q^{92} - 444 q^{93} + 3960 q^{94} - 602 q^{95} - 453 q^{96} + 402 q^{97} + 4109 q^{98} + 306 q^{99}+O(q^{100}) \) 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
4.55528
−0.795427
−2.75985
−2.55528 3.00000 −1.47057 8.47057 −7.66583 3.66117 24.1999 9.00000 −21.6446
1.2 2.79543 3.00000 −0.185590 7.18559 8.38628 19.9241 −22.8822 9.00000 20.0868
1.3 4.75985 3.00000 14.6562 −7.65616 14.2795 −31.5852 31.6823 9.00000 −36.4422
\(n\): 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 51.4.a.e 3
3.b odd 2 1 153.4.a.f 3
4.b odd 2 1 816.4.a.s 3
5.b even 2 1 1275.4.a.q 3
7.b odd 2 1 2499.4.a.n 3
12.b even 2 1 2448.4.a.bd 3
17.b even 2 1 867.4.a.k 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.4.a.e 3 1.a even 1 1 trivial
153.4.a.f 3 3.b odd 2 1
816.4.a.s 3 4.b odd 2 1
867.4.a.k 3 17.b even 2 1
1275.4.a.q 3 5.b even 2 1
2448.4.a.bd 3 12.b even 2 1
2499.4.a.n 3 7.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(51))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 5 T^{2} + \cdots + 34 \) Copy content Toggle raw display
$3$ \( (T - 3)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 8 T^{2} + \cdots + 466 \) Copy content Toggle raw display
$7$ \( T^{3} + 8 T^{2} + \cdots + 2304 \) Copy content Toggle raw display
$11$ \( T^{3} - 34 T^{2} + \cdots - 8964 \) Copy content Toggle raw display
$13$ \( T^{3} - 36 T^{2} + \cdots + 122698 \) Copy content Toggle raw display
$17$ \( (T - 17)^{3} \) Copy content Toggle raw display
$19$ \( T^{3} + 142 T^{2} + \cdots + 8244 \) Copy content Toggle raw display
$23$ \( T^{3} - 110 T^{2} + \cdots - 53240 \) Copy content Toggle raw display
$29$ \( T^{3} - 90 T^{2} + \cdots - 415320 \) Copy content Toggle raw display
$31$ \( T^{3} + 148 T^{2} + \cdots - 640448 \) Copy content Toggle raw display
$37$ \( T^{3} - 110 T^{2} + \cdots - 5969792 \) 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
$59$ \( T^{3} + 216 T^{2} + \cdots + 1302384 \) Copy content Toggle raw display
$61$ \( T^{3} + 18 T^{2} + \cdots + 8127200 \) 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
$97$ \( T^{3} + \cdots + 2026068032 \) Copy content Toggle raw display
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