Properties

Label 154.4.a.i
Level $154$
Weight $4$
Character orbit 154.a
Self dual yes
Analytic conductor $9.086$
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] = [154,4,Mod(1,154)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(154, 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("154.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 154 = 2 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 154.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: \(9.08629414088\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.7636.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 16x - 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + 2 q^{2} + ( - \beta_1 + 2) q^{3} + 4 q^{4} + ( - \beta_{2} + 9) q^{5} + ( - 2 \beta_1 + 4) q^{6} - 7 q^{7} + 8 q^{8} + (2 \beta_{2} + 19) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + ( - \beta_1 + 2) q^{3} + 4 q^{4} + ( - \beta_{2} + 9) q^{5} + ( - 2 \beta_1 + 4) q^{6} - 7 q^{7} + 8 q^{8} + (2 \beta_{2} + 19) q^{9} + ( - 2 \beta_{2} + 18) q^{10} + 11 q^{11} + ( - 4 \beta_1 + 8) q^{12} + (6 \beta_{2} + 5 \beta_1 + 22) q^{13} - 14 q^{14} + ( - 6 \beta_{2} - 6 \beta_1 + 6) q^{15} + 16 q^{16} + (5 \beta_{2} + 11 \beta_1 + 17) q^{17} + (4 \beta_{2} + 38) q^{18} + ( - 3 \beta_{2} + 6 \beta_1 - 15) q^{19} + ( - 4 \beta_{2} + 36) q^{20} + (7 \beta_1 - 14) q^{21} + 22 q^{22} + (2 \beta_{2} - 6 \beta_1 - 82) q^{23} + ( - 8 \beta_1 + 16) q^{24} + ( - 20 \beta_{2} - 12 \beta_1 + 67) q^{25} + (12 \beta_{2} + 10 \beta_1 + 44) q^{26} + (12 \beta_{2} + 2 \beta_1 + 8) q^{27} - 28 q^{28} + ( - 6 \beta_{2} + 22 \beta_1 + 68) q^{29} + ( - 12 \beta_{2} - 12 \beta_1 + 12) q^{30} + ( - 13 \beta_{2} + 25 \beta_1 + 129) q^{31} + 32 q^{32} + ( - 11 \beta_1 + 22) q^{33} + (10 \beta_{2} + 22 \beta_1 + 34) q^{34} + (7 \beta_{2} - 63) q^{35} + (8 \beta_{2} + 76) q^{36} + (2 \beta_{2} - 32 \beta_1 + 36) q^{37} + ( - 6 \beta_{2} + 12 \beta_1 - 30) q^{38} + (26 \beta_{2} - 50 \beta_1 - 94) q^{39} + ( - 8 \beta_{2} + 72) q^{40} + ( - 27 \beta_{2} - 13 \beta_1 - 207) q^{41} + (14 \beta_1 - 28) q^{42} + (10 \beta_{2} + 22 \beta_1 - 170) q^{43} + 44 q^{44} + (3 \beta_{2} + 24 \beta_1 - 51) q^{45} + (4 \beta_{2} - 12 \beta_1 - 164) q^{46} + (37 \beta_{2} - 41 \beta_1 - 121) q^{47} + ( - 16 \beta_1 + 32) q^{48} + 49 q^{49} + ( - 40 \beta_{2} - 24 \beta_1 + 134) q^{50} + (8 \beta_{2} - 54 \beta_1 - 368) q^{51} + (24 \beta_{2} + 20 \beta_1 + 88) q^{52} + (8 \beta_{2} + 20 \beta_1 + 66) q^{53} + (24 \beta_{2} + 4 \beta_1 + 16) q^{54} + ( - 11 \beta_{2} + 99) q^{55} - 56 q^{56} + ( - 30 \beta_{2} + 12 \beta_1 - 318) q^{57} + ( - 12 \beta_{2} + 44 \beta_1 + 136) q^{58} + ( - 26 \beta_{2} - 61 \beta_1 - 80) q^{59} + ( - 24 \beta_{2} - 24 \beta_1 + 24) q^{60} + ( - 34 \beta_{2} - 77 \beta_1 + 114) q^{61} + ( - 26 \beta_{2} + 50 \beta_1 + 258) q^{62} + ( - 14 \beta_{2} - 133) q^{63} + 64 q^{64} + (64 \beta_{2} + 102 \beta_1 - 408) q^{65} + ( - 22 \beta_1 + 44) q^{66} + ( - 20 \beta_{2} - 18 \beta_1 + 92) q^{67} + (20 \beta_{2} + 44 \beta_1 + 68) q^{68} + (24 \beta_{2} + 88 \beta_1 + 112) q^{69} + (14 \beta_{2} - 126) q^{70} + (38 \beta_{2} + 58 \beta_1 - 174) q^{71} + (16 \beta_{2} + 152) q^{72} + (25 \beta_{2} + 117 \beta_1 - 351) q^{73} + (4 \beta_{2} - 64 \beta_1 + 72) q^{74} + ( - 96 \beta_{2} + 17 \beta_1 + 398) q^{75} + ( - 12 \beta_{2} + 24 \beta_1 - 60) q^{76} - 77 q^{77} + (52 \beta_{2} - 100 \beta_1 - 188) q^{78} + (8 \beta_{2} + 74 \beta_1 - 360) q^{79} + ( - 16 \beta_{2} + 144) q^{80} + (14 \beta_{2} - 48 \beta_1 - 437) q^{81} + ( - 54 \beta_{2} - 26 \beta_1 - 414) q^{82} + (33 \beta_{2} + 2 \beta_1 + 293) q^{83} + (28 \beta_1 - 56) q^{84} + (82 \beta_{2} + 126 \beta_1 - 270) q^{85} + (20 \beta_{2} + 44 \beta_1 - 340) q^{86} + ( - 80 \beta_{2} - 94 \beta_1 - 860) q^{87} + 88 q^{88} + ( - 8 \beta_{2} - 4 \beta_1 + 1138) q^{89} + (6 \beta_{2} + 48 \beta_1 - 102) q^{90} + ( - 42 \beta_{2} - 35 \beta_1 - 154) q^{91} + (8 \beta_{2} - 24 \beta_1 - 328) q^{92} + ( - 128 \beta_{2} - 140 \beta_1 - 948) q^{93} + (74 \beta_{2} - 82 \beta_1 - 242) q^{94} + (6 \beta_{2} + 270) q^{95} + ( - 32 \beta_1 + 64) q^{96} + ( - 4 \beta_{2} - 34 \beta_1 + 682) q^{97} + 98 q^{98} + (22 \beta_{2} + 209) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 6 q^{2} + 6 q^{3} + 12 q^{4} + 26 q^{5} + 12 q^{6} - 21 q^{7} + 24 q^{8} + 59 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 6 q^{2} + 6 q^{3} + 12 q^{4} + 26 q^{5} + 12 q^{6} - 21 q^{7} + 24 q^{8} + 59 q^{9} + 52 q^{10} + 33 q^{11} + 24 q^{12} + 72 q^{13} - 42 q^{14} + 12 q^{15} + 48 q^{16} + 56 q^{17} + 118 q^{18} - 48 q^{19} + 104 q^{20} - 42 q^{21} + 66 q^{22} - 244 q^{23} + 48 q^{24} + 181 q^{25} + 144 q^{26} + 36 q^{27} - 84 q^{28} + 198 q^{29} + 24 q^{30} + 374 q^{31} + 96 q^{32} + 66 q^{33} + 112 q^{34} - 182 q^{35} + 236 q^{36} + 110 q^{37} - 96 q^{38} - 256 q^{39} + 208 q^{40} - 648 q^{41} - 84 q^{42} - 500 q^{43} + 132 q^{44} - 150 q^{45} - 488 q^{46} - 326 q^{47} + 96 q^{48} + 147 q^{49} + 362 q^{50} - 1096 q^{51} + 288 q^{52} + 206 q^{53} + 72 q^{54} + 286 q^{55} - 168 q^{56} - 984 q^{57} + 396 q^{58} - 266 q^{59} + 48 q^{60} + 308 q^{61} + 748 q^{62} - 413 q^{63} + 192 q^{64} - 1160 q^{65} + 132 q^{66} + 256 q^{67} + 224 q^{68} + 360 q^{69} - 364 q^{70} - 484 q^{71} + 472 q^{72} - 1028 q^{73} + 220 q^{74} + 1098 q^{75} - 192 q^{76} - 231 q^{77} - 512 q^{78} - 1072 q^{79} + 416 q^{80} - 1297 q^{81} - 1296 q^{82} + 912 q^{83} - 168 q^{84} - 728 q^{85} - 1000 q^{86} - 2660 q^{87} + 264 q^{88} + 3406 q^{89} - 300 q^{90} - 504 q^{91} - 976 q^{92} - 2972 q^{93} - 652 q^{94} + 816 q^{95} + 192 q^{96} + 2042 q^{97} + 294 q^{98} + 649 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 16x - 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} - 4\nu - 21 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 2\beta _1 + 21 ) / 2 \) 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
4.47467
−1.24586
−3.22881
2.00000 −6.94933 4.00000 7.85338 −13.8987 −7.00000 8.00000 21.2932 15.7068
1.2 2.00000 4.49172 4.00000 21.9122 8.98345 −7.00000 8.00000 −6.82442 43.8244
1.3 2.00000 8.45761 4.00000 −3.76558 16.9152 −7.00000 8.00000 44.5312 −7.53117
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( +1 \)
\(11\) \( -1 \)

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 154.4.a.i 3
3.b odd 2 1 1386.4.a.bc 3
4.b odd 2 1 1232.4.a.q 3
7.b odd 2 1 1078.4.a.p 3
11.b odd 2 1 1694.4.a.s 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.4.a.i 3 1.a even 1 1 trivial
1078.4.a.p 3 7.b odd 2 1
1232.4.a.q 3 4.b odd 2 1
1386.4.a.bc 3 3.b odd 2 1
1694.4.a.s 3 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 6T_{3}^{2} - 52T_{3} + 264 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(154))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 6 T^{2} + \cdots + 264 \) Copy content Toggle raw display
$5$ \( T^{3} - 26 T^{2} + \cdots + 648 \) Copy content Toggle raw display
$7$ \( (T + 7)^{3} \) Copy content Toggle raw display
$11$ \( (T - 11)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 72 T^{2} + \cdots + 331632 \) Copy content Toggle raw display
$17$ \( T^{3} - 56 T^{2} + \cdots + 88976 \) Copy content Toggle raw display
$19$ \( T^{3} + 48 T^{2} + \cdots + 28512 \) Copy content Toggle raw display
$23$ \( T^{3} + 244 T^{2} + \cdots + 219584 \) Copy content Toggle raw display
$29$ \( T^{3} - 198 T^{2} + \cdots + 3523896 \) Copy content Toggle raw display
$31$ \( T^{3} - 374 T^{2} + \cdots + 15721176 \) Copy content Toggle raw display
$37$ \( T^{3} - 110 T^{2} + \cdots + 5981784 \) Copy content Toggle raw display
$41$ \( T^{3} + 648 T^{2} + \cdots - 28837872 \) Copy content Toggle raw display
$43$ \( T^{3} + 500 T^{2} + \cdots - 2503232 \) Copy content Toggle raw display
$47$ \( T^{3} + 326 T^{2} + \cdots - 136297368 \) Copy content Toggle raw display
$53$ \( T^{3} - 206 T^{2} + \cdots + 863064 \) Copy content Toggle raw display
$59$ \( T^{3} + 266 T^{2} + \cdots - 4809816 \) Copy content Toggle raw display
$61$ \( T^{3} - 308 T^{2} + \cdots + 81057168 \) Copy content Toggle raw display
$67$ \( T^{3} - 256 T^{2} + \cdots - 1711488 \) Copy content Toggle raw display
$71$ \( T^{3} + 484 T^{2} + \cdots - 19953216 \) Copy content Toggle raw display
$73$ \( T^{3} + 1028 T^{2} + \cdots - 550878192 \) Copy content Toggle raw display
$79$ \( T^{3} + 1072 T^{2} + \cdots - 148401792 \) Copy content Toggle raw display
$83$ \( T^{3} - 912 T^{2} + \cdots + 33774048 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots - 1452062376 \) Copy content Toggle raw display
$97$ \( T^{3} - 2042 T^{2} + \cdots - 259709688 \) Copy content Toggle raw display
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