Properties

Label 154.4.a.f
Level $154$
Weight $4$
Character orbit 154.a
Self dual yes
Analytic conductor $9.086$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{137}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 34 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{137})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + ( - \beta - 2) q^{3} + 4 q^{4} + (\beta - 4) q^{5} + (2 \beta + 4) q^{6} + 7 q^{7} - 8 q^{8} + (5 \beta + 11) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + ( - \beta - 2) q^{3} + 4 q^{4} + (\beta - 4) q^{5} + (2 \beta + 4) q^{6} + 7 q^{7} - 8 q^{8} + (5 \beta + 11) q^{9} + ( - 2 \beta + 8) q^{10} - 11 q^{11} + ( - 4 \beta - 8) q^{12} + (10 \beta + 18) q^{13} - 14 q^{14} + (\beta - 26) q^{15} + 16 q^{16} + (4 \beta - 46) q^{17} + ( - 10 \beta - 22) q^{18} + ( - 14 \beta - 16) q^{19} + (4 \beta - 16) q^{20} + ( - 7 \beta - 14) q^{21} + 22 q^{22} + ( - 15 \beta - 50) q^{23} + (8 \beta + 16) q^{24} + ( - 7 \beta - 75) q^{25} + ( - 20 \beta - 36) q^{26} + (\beta - 138) q^{27} + 28 q^{28} + (16 \beta - 194) q^{29} + ( - 2 \beta + 52) q^{30} + (3 \beta - 70) q^{31} - 32 q^{32} + (11 \beta + 22) q^{33} + ( - 8 \beta + 92) q^{34} + (7 \beta - 28) q^{35} + (20 \beta + 44) q^{36} + (5 \beta - 12) q^{37} + (28 \beta + 32) q^{38} + ( - 48 \beta - 376) q^{39} + ( - 8 \beta + 32) q^{40} + (20 \beta - 230) q^{41} + (14 \beta + 28) q^{42} + (8 \beta + 92) q^{43} - 44 q^{44} + ( - 4 \beta + 126) q^{45} + (30 \beta + 100) q^{46} + (24 \beta + 352) q^{47} + ( - 16 \beta - 32) q^{48} + 49 q^{49} + (14 \beta + 150) q^{50} + (34 \beta - 44) q^{51} + (40 \beta + 72) q^{52} + ( - 20 \beta - 394) q^{53} + ( - 2 \beta + 276) q^{54} + ( - 11 \beta + 44) q^{55} - 56 q^{56} + (58 \beta + 508) q^{57} + ( - 32 \beta + 388) q^{58} + ( - 71 \beta + 18) q^{59} + (4 \beta - 104) q^{60} + ( - 100 \beta + 206) q^{61} + ( - 6 \beta + 140) q^{62} + (35 \beta + 77) q^{63} + 64 q^{64} + ( - 12 \beta + 268) q^{65} + ( - 22 \beta - 44) q^{66} + ( - 153 \beta - 74) q^{67} + (16 \beta - 184) q^{68} + (95 \beta + 610) q^{69} + ( - 14 \beta + 56) q^{70} + ( - 125 \beta - 6) q^{71} + ( - 40 \beta - 88) q^{72} + (80 \beta + 354) q^{73} + ( - 10 \beta + 24) q^{74} + (96 \beta + 388) q^{75} + ( - 56 \beta - 64) q^{76} - 77 q^{77} + (96 \beta + 752) q^{78} + ( - 6 \beta - 180) q^{79} + (16 \beta - 64) q^{80} - 55 q^{81} + ( - 40 \beta + 460) q^{82} + (20 \beta - 1172) q^{83} + ( - 28 \beta - 56) q^{84} + ( - 58 \beta + 320) q^{85} + ( - 16 \beta - 184) q^{86} + (146 \beta - 156) q^{87} + 88 q^{88} + ( - 21 \beta + 628) q^{89} + (8 \beta - 252) q^{90} + (70 \beta + 126) q^{91} + ( - 60 \beta - 200) q^{92} + (61 \beta + 38) q^{93} + ( - 48 \beta - 704) q^{94} + (26 \beta - 412) q^{95} + (32 \beta + 64) q^{96} + (181 \beta + 264) q^{97} - 98 q^{98} + ( - 55 \beta - 121) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} - 5 q^{3} + 8 q^{4} - 7 q^{5} + 10 q^{6} + 14 q^{7} - 16 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} - 5 q^{3} + 8 q^{4} - 7 q^{5} + 10 q^{6} + 14 q^{7} - 16 q^{8} + 27 q^{9} + 14 q^{10} - 22 q^{11} - 20 q^{12} + 46 q^{13} - 28 q^{14} - 51 q^{15} + 32 q^{16} - 88 q^{17} - 54 q^{18} - 46 q^{19} - 28 q^{20} - 35 q^{21} + 44 q^{22} - 115 q^{23} + 40 q^{24} - 157 q^{25} - 92 q^{26} - 275 q^{27} + 56 q^{28} - 372 q^{29} + 102 q^{30} - 137 q^{31} - 64 q^{32} + 55 q^{33} + 176 q^{34} - 49 q^{35} + 108 q^{36} - 19 q^{37} + 92 q^{38} - 800 q^{39} + 56 q^{40} - 440 q^{41} + 70 q^{42} + 192 q^{43} - 88 q^{44} + 248 q^{45} + 230 q^{46} + 728 q^{47} - 80 q^{48} + 98 q^{49} + 314 q^{50} - 54 q^{51} + 184 q^{52} - 808 q^{53} + 550 q^{54} + 77 q^{55} - 112 q^{56} + 1074 q^{57} + 744 q^{58} - 35 q^{59} - 204 q^{60} + 312 q^{61} + 274 q^{62} + 189 q^{63} + 128 q^{64} + 524 q^{65} - 110 q^{66} - 301 q^{67} - 352 q^{68} + 1315 q^{69} + 98 q^{70} - 137 q^{71} - 216 q^{72} + 788 q^{73} + 38 q^{74} + 872 q^{75} - 184 q^{76} - 154 q^{77} + 1600 q^{78} - 366 q^{79} - 112 q^{80} - 110 q^{81} + 880 q^{82} - 2324 q^{83} - 140 q^{84} + 582 q^{85} - 384 q^{86} - 166 q^{87} + 176 q^{88} + 1235 q^{89} - 496 q^{90} + 322 q^{91} - 460 q^{92} + 137 q^{93} - 1456 q^{94} - 798 q^{95} + 160 q^{96} + 709 q^{97} - 196 q^{98} - 297 q^{99}+O(q^{100}) \) 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
6.35235
−5.35235
−2.00000 −8.35235 4.00000 2.35235 16.7047 7.00000 −8.00000 42.7617 −4.70470
1.2 −2.00000 3.35235 4.00000 −9.35235 −6.70470 7.00000 −8.00000 −15.7617 18.7047
\(n\): 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.f 2
3.b odd 2 1 1386.4.a.ba 2
4.b odd 2 1 1232.4.a.p 2
7.b odd 2 1 1078.4.a.j 2
11.b odd 2 1 1694.4.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.4.a.f 2 1.a even 1 1 trivial
1078.4.a.j 2 7.b odd 2 1
1232.4.a.p 2 4.b odd 2 1
1386.4.a.ba 2 3.b odd 2 1
1694.4.a.l 2 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 5T_{3} - 28 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(154))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 5T - 28 \) Copy content Toggle raw display
$5$ \( T^{2} + 7T - 22 \) Copy content Toggle raw display
$7$ \( (T - 7)^{2} \) Copy content Toggle raw display
$11$ \( (T + 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 46T - 2896 \) Copy content Toggle raw display
$17$ \( T^{2} + 88T + 1388 \) Copy content Toggle raw display
$19$ \( T^{2} + 46T - 6184 \) Copy content Toggle raw display
$23$ \( T^{2} + 115T - 4400 \) Copy content Toggle raw display
$29$ \( T^{2} + 372T + 25828 \) Copy content Toggle raw display
$31$ \( T^{2} + 137T + 4384 \) Copy content Toggle raw display
$37$ \( T^{2} + 19T - 766 \) Copy content Toggle raw display
$41$ \( T^{2} + 440T + 34700 \) Copy content Toggle raw display
$43$ \( T^{2} - 192T + 7024 \) Copy content Toggle raw display
$47$ \( T^{2} - 728T + 112768 \) Copy content Toggle raw display
$53$ \( T^{2} + 808T + 149516 \) Copy content Toggle raw display
$59$ \( T^{2} + 35T - 172348 \) Copy content Toggle raw display
$61$ \( T^{2} - 312T - 318164 \) Copy content Toggle raw display
$67$ \( T^{2} + 301T - 779108 \) Copy content Toggle raw display
$71$ \( T^{2} + 137T - 530464 \) Copy content Toggle raw display
$73$ \( T^{2} - 788T - 63964 \) Copy content Toggle raw display
$79$ \( T^{2} + 366T + 32256 \) Copy content Toggle raw display
$83$ \( T^{2} + 2324 T + 1336544 \) Copy content Toggle raw display
$89$ \( T^{2} - 1235 T + 366202 \) Copy content Toggle raw display
$97$ \( T^{2} - 709T - 996394 \) Copy content Toggle raw display
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