Properties

Label 154.4.a.h
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: \(\Q(\sqrt{57}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 14 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + ( - \beta - 2) q^{3} + 4 q^{4} + (3 \beta - 10) q^{5} + ( - 2 \beta - 4) q^{6} - 7 q^{7} + 8 q^{8} + (5 \beta - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + ( - \beta - 2) q^{3} + 4 q^{4} + (3 \beta - 10) q^{5} + ( - 2 \beta - 4) q^{6} - 7 q^{7} + 8 q^{8} + (5 \beta - 9) q^{9} + (6 \beta - 20) q^{10} - 11 q^{11} + ( - 4 \beta - 8) q^{12} + ( - 2 \beta - 42) q^{13} - 14 q^{14} + (\beta - 22) q^{15} + 16 q^{16} + ( - 26 \beta - 20) q^{17} + (10 \beta - 18) q^{18} + (4 \beta - 70) q^{19} + (12 \beta - 40) q^{20} + (7 \beta + 14) q^{21} - 22 q^{22} + (21 \beta - 2) q^{23} + ( - 8 \beta - 16) q^{24} + ( - 51 \beta + 101) q^{25} + ( - 4 \beta - 84) q^{26} + (21 \beta + 2) q^{27} - 28 q^{28} + ( - 48 \beta + 86) q^{29} + (2 \beta - 44) q^{30} + (9 \beta - 40) q^{31} + 32 q^{32} + (11 \beta + 22) q^{33} + ( - 52 \beta - 40) q^{34} + ( - 21 \beta + 70) q^{35} + (20 \beta - 36) q^{36} + (17 \beta - 76) q^{37} + (8 \beta - 140) q^{38} + (48 \beta + 112) q^{39} + (24 \beta - 80) q^{40} + (46 \beta - 332) q^{41} + (14 \beta + 28) q^{42} + ( - 52 \beta + 328) q^{43} - 44 q^{44} + ( - 62 \beta + 300) q^{45} + (42 \beta - 4) q^{46} + (134 \beta - 66) q^{47} + ( - 16 \beta - 32) q^{48} + 49 q^{49} + ( - 102 \beta + 202) q^{50} + (98 \beta + 404) q^{51} + ( - 8 \beta - 168) q^{52} + (172 \beta - 10) q^{53} + (42 \beta + 4) q^{54} + ( - 33 \beta + 110) q^{55} - 56 q^{56} + (58 \beta + 84) q^{57} + ( - 96 \beta + 172) q^{58} + ( - 115 \beta - 114) q^{59} + (4 \beta - 88) q^{60} + ( - 184 \beta + 282) q^{61} + (18 \beta - 80) q^{62} + ( - 35 \beta + 63) q^{63} + 64 q^{64} + ( - 112 \beta + 336) q^{65} + (22 \beta + 44) q^{66} + (31 \beta + 498) q^{67} + ( - 104 \beta - 80) q^{68} + ( - 61 \beta - 290) q^{69} + ( - 42 \beta + 140) q^{70} + ( - 45 \beta - 562) q^{71} + (40 \beta - 72) q^{72} + (42 \beta - 68) q^{73} + (34 \beta - 152) q^{74} + (52 \beta + 512) q^{75} + (16 \beta - 280) q^{76} + 77 q^{77} + (96 \beta + 224) q^{78} + (50 \beta + 396) q^{79} + (48 \beta - 160) q^{80} + ( - 200 \beta - 55) q^{81} + (92 \beta - 664) q^{82} + ( - 42 \beta - 722) q^{83} + (28 \beta + 56) q^{84} + (122 \beta - 892) q^{85} + ( - 104 \beta + 656) q^{86} + (58 \beta + 500) q^{87} - 88 q^{88} + (119 \beta - 240) q^{89} + ( - 124 \beta + 600) q^{90} + (14 \beta + 294) q^{91} + (84 \beta - 8) q^{92} + (13 \beta - 46) q^{93} + (268 \beta - 132) q^{94} + ( - 238 \beta + 868) q^{95} + ( - 32 \beta - 64) q^{96} + ( - 55 \beta + 140) q^{97} + 98 q^{98} + ( - 55 \beta + 99) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} - 5 q^{3} + 8 q^{4} - 17 q^{5} - 10 q^{6} - 14 q^{7} + 16 q^{8} - 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} - 5 q^{3} + 8 q^{4} - 17 q^{5} - 10 q^{6} - 14 q^{7} + 16 q^{8} - 13 q^{9} - 34 q^{10} - 22 q^{11} - 20 q^{12} - 86 q^{13} - 28 q^{14} - 43 q^{15} + 32 q^{16} - 66 q^{17} - 26 q^{18} - 136 q^{19} - 68 q^{20} + 35 q^{21} - 44 q^{22} + 17 q^{23} - 40 q^{24} + 151 q^{25} - 172 q^{26} + 25 q^{27} - 56 q^{28} + 124 q^{29} - 86 q^{30} - 71 q^{31} + 64 q^{32} + 55 q^{33} - 132 q^{34} + 119 q^{35} - 52 q^{36} - 135 q^{37} - 272 q^{38} + 272 q^{39} - 136 q^{40} - 618 q^{41} + 70 q^{42} + 604 q^{43} - 88 q^{44} + 538 q^{45} + 34 q^{46} + 2 q^{47} - 80 q^{48} + 98 q^{49} + 302 q^{50} + 906 q^{51} - 344 q^{52} + 152 q^{53} + 50 q^{54} + 187 q^{55} - 112 q^{56} + 226 q^{57} + 248 q^{58} - 343 q^{59} - 172 q^{60} + 380 q^{61} - 142 q^{62} + 91 q^{63} + 128 q^{64} + 560 q^{65} + 110 q^{66} + 1027 q^{67} - 264 q^{68} - 641 q^{69} + 238 q^{70} - 1169 q^{71} - 104 q^{72} - 94 q^{73} - 270 q^{74} + 1076 q^{75} - 544 q^{76} + 154 q^{77} + 544 q^{78} + 842 q^{79} - 272 q^{80} - 310 q^{81} - 1236 q^{82} - 1486 q^{83} + 140 q^{84} - 1662 q^{85} + 1208 q^{86} + 1058 q^{87} - 176 q^{88} - 361 q^{89} + 1076 q^{90} + 602 q^{91} + 68 q^{92} - 79 q^{93} + 4 q^{94} + 1498 q^{95} - 160 q^{96} + 225 q^{97} + 196 q^{98} + 143 q^{99}+O(q^{100}) \) 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
4.27492
−3.27492
2.00000 −6.27492 4.00000 2.82475 −12.5498 −7.00000 8.00000 12.3746 5.64950
1.2 2.00000 1.27492 4.00000 −19.8248 2.54983 −7.00000 8.00000 −25.3746 −39.6495
\(n\): 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.h 2
3.b odd 2 1 1386.4.a.s 2
4.b odd 2 1 1232.4.a.o 2
7.b odd 2 1 1078.4.a.o 2
11.b odd 2 1 1694.4.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.4.a.h 2 1.a even 1 1 trivial
1078.4.a.o 2 7.b odd 2 1
1232.4.a.o 2 4.b odd 2 1
1386.4.a.s 2 3.b odd 2 1
1694.4.a.h 2 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 5T_{3} - 8 \) 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
$3$ \( T^{2} + 5T - 8 \) Copy content Toggle raw display
$5$ \( T^{2} + 17T - 56 \) Copy content Toggle raw display
$7$ \( (T + 7)^{2} \) Copy content Toggle raw display
$11$ \( (T + 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 86T + 1792 \) Copy content Toggle raw display
$17$ \( T^{2} + 66T - 8544 \) Copy content Toggle raw display
$19$ \( T^{2} + 136T + 4396 \) Copy content Toggle raw display
$23$ \( T^{2} - 17T - 6212 \) Copy content Toggle raw display
$29$ \( T^{2} - 124T - 28988 \) Copy content Toggle raw display
$31$ \( T^{2} + 71T + 106 \) Copy content Toggle raw display
$37$ \( T^{2} + 135T + 438 \) Copy content Toggle raw display
$41$ \( T^{2} + 618T + 65328 \) Copy content Toggle raw display
$43$ \( T^{2} - 604T + 52672 \) Copy content Toggle raw display
$47$ \( T^{2} - 2T - 255872 \) Copy content Toggle raw display
$53$ \( T^{2} - 152T - 415796 \) Copy content Toggle raw display
$59$ \( T^{2} + 343T - 159044 \) Copy content Toggle raw display
$61$ \( T^{2} - 380T - 446348 \) Copy content Toggle raw display
$67$ \( T^{2} - 1027 T + 249988 \) Copy content Toggle raw display
$71$ \( T^{2} + 1169 T + 312784 \) Copy content Toggle raw display
$73$ \( T^{2} + 94T - 22928 \) Copy content Toggle raw display
$79$ \( T^{2} - 842T + 141616 \) Copy content Toggle raw display
$83$ \( T^{2} + 1486 T + 526912 \) Copy content Toggle raw display
$89$ \( T^{2} + 361T - 169214 \) Copy content Toggle raw display
$97$ \( T^{2} - 225T - 30450 \) Copy content Toggle raw display
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