Properties

Label 1110.4.a.g
Level $1110$
Weight $4$
Character orbit 1110.a
Self dual yes
Analytic conductor $65.492$
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] = [1110,4,Mod(1,1110)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1110, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1110.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1110.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: \(65.4921201064\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.837.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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.

\(f(q)\) \(=\) \( q - 2 q^{2} + 3 q^{3} + 4 q^{4} + 5 q^{5} - 6 q^{6} + (3 \beta_{2} + \beta_1 - 4) q^{7} - 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 3 q^{3} + 4 q^{4} + 5 q^{5} - 6 q^{6} + (3 \beta_{2} + \beta_1 - 4) q^{7} - 8 q^{8} + 9 q^{9} - 10 q^{10} + (10 \beta_{2} + 5 \beta_1 + 3) q^{11} + 12 q^{12} + ( - 2 \beta_{2} - 11) q^{13} + ( - 6 \beta_{2} - 2 \beta_1 + 8) q^{14} + 15 q^{15} + 16 q^{16} + ( - 11 \beta_{2} + 12 \beta_1 - 35) q^{17} - 18 q^{18} + ( - 17 \beta_{2} - 26 \beta_1 - 27) q^{19} + 20 q^{20} + (9 \beta_{2} + 3 \beta_1 - 12) q^{21} + ( - 20 \beta_{2} - 10 \beta_1 - 6) q^{22} + ( - 29 \beta_{2} - 47 \beta_1 - 20) q^{23} - 24 q^{24} + 25 q^{25} + (4 \beta_{2} + 22) q^{26} + 27 q^{27} + (12 \beta_{2} + 4 \beta_1 - 16) q^{28} + ( - 50 \beta_{2} + 5 \beta_1 - 36) q^{29} - 30 q^{30} + ( - 21 \beta_{2} + 47 \beta_1 - 114) q^{31} - 32 q^{32} + (30 \beta_{2} + 15 \beta_1 + 9) q^{33} + (22 \beta_{2} - 24 \beta_1 + 70) q^{34} + (15 \beta_{2} + 5 \beta_1 - 20) q^{35} + 36 q^{36} + 37 q^{37} + (34 \beta_{2} + 52 \beta_1 + 54) q^{38} + ( - 6 \beta_{2} - 33) q^{39} - 40 q^{40} + (31 \beta_{2} - 125 \beta_1 - 200) q^{41} + ( - 18 \beta_{2} - 6 \beta_1 + 24) q^{42} + ( - 26 \beta_{2} + 99 \beta_1 + 30) q^{43} + (40 \beta_{2} + 20 \beta_1 + 12) q^{44} + 45 q^{45} + (58 \beta_{2} + 94 \beta_1 + 40) q^{46} + (86 \beta_{2} - 140 \beta_1 - 79) q^{47} + 48 q^{48} + ( - 41 \beta_{2} + 13 \beta_1 - 245) q^{49} - 50 q^{50} + ( - 33 \beta_{2} + 36 \beta_1 - 105) q^{51} + ( - 8 \beta_{2} - 44) q^{52} + (39 \beta_{2} + 153 \beta_1 + 21) q^{53} - 54 q^{54} + (50 \beta_{2} + 25 \beta_1 + 15) q^{55} + ( - 24 \beta_{2} - 8 \beta_1 + 32) q^{56} + ( - 51 \beta_{2} - 78 \beta_1 - 81) q^{57} + (100 \beta_{2} - 10 \beta_1 + 72) q^{58} + ( - 165 \beta_{2} + 156 \beta_1 - 88) q^{59} + 60 q^{60} + (29 \beta_{2} - 33 \beta_1 - 244) q^{61} + (42 \beta_{2} - 94 \beta_1 + 228) q^{62} + (27 \beta_{2} + 9 \beta_1 - 36) q^{63} + 64 q^{64} + ( - 10 \beta_{2} - 55) q^{65} + ( - 60 \beta_{2} - 30 \beta_1 - 18) q^{66} + (49 \beta_{2} - 48 \beta_1 - 289) q^{67} + ( - 44 \beta_{2} + 48 \beta_1 - 140) q^{68} + ( - 87 \beta_{2} - 141 \beta_1 - 60) q^{69} + ( - 30 \beta_{2} - 10 \beta_1 + 40) q^{70} + (135 \beta_{2} + 220 \beta_1 - 411) q^{71} - 72 q^{72} + (152 \beta_{2} - 429 \beta_1 - 279) q^{73} - 74 q^{74} + 75 q^{75} + ( - 68 \beta_{2} - 104 \beta_1 - 108) q^{76} + ( - 86 \beta_{2} + 63 \beta_1 + 273) q^{77} + (12 \beta_{2} + 66) q^{78} + (200 \beta_{2} - 191 \beta_1 - 351) q^{79} + 80 q^{80} + 81 q^{81} + ( - 62 \beta_{2} + 250 \beta_1 + 400) q^{82} + ( - 63 \beta_{2} + 284 \beta_1 + 104) q^{83} + (36 \beta_{2} + 12 \beta_1 - 48) q^{84} + ( - 55 \beta_{2} + 60 \beta_1 - 175) q^{85} + (52 \beta_{2} - 198 \beta_1 - 60) q^{86} + ( - 150 \beta_{2} + 15 \beta_1 - 108) q^{87} + ( - 80 \beta_{2} - 40 \beta_1 - 24) q^{88} + (134 \beta_{2} - 388 \beta_1 + 165) q^{89} - 90 q^{90} + ( - 13 \beta_{2} - 21 \beta_1 - 6) q^{91} + ( - 116 \beta_{2} - 188 \beta_1 - 80) q^{92} + ( - 63 \beta_{2} + 141 \beta_1 - 342) q^{93} + ( - 172 \beta_{2} + 280 \beta_1 + 158) q^{94} + ( - 85 \beta_{2} - 130 \beta_1 - 135) q^{95} - 96 q^{96} + ( - 13 \beta_{2} + 401 \beta_1 - 364) q^{97} + (82 \beta_{2} - 26 \beta_1 + 490) q^{98} + (90 \beta_{2} + 45 \beta_1 + 27) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 6 q^{2} + 9 q^{3} + 12 q^{4} + 15 q^{5} - 18 q^{6} - 12 q^{7} - 24 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 6 q^{2} + 9 q^{3} + 12 q^{4} + 15 q^{5} - 18 q^{6} - 12 q^{7} - 24 q^{8} + 27 q^{9} - 30 q^{10} + 9 q^{11} + 36 q^{12} - 33 q^{13} + 24 q^{14} + 45 q^{15} + 48 q^{16} - 105 q^{17} - 54 q^{18} - 81 q^{19} + 60 q^{20} - 36 q^{21} - 18 q^{22} - 60 q^{23} - 72 q^{24} + 75 q^{25} + 66 q^{26} + 81 q^{27} - 48 q^{28} - 108 q^{29} - 90 q^{30} - 342 q^{31} - 96 q^{32} + 27 q^{33} + 210 q^{34} - 60 q^{35} + 108 q^{36} + 111 q^{37} + 162 q^{38} - 99 q^{39} - 120 q^{40} - 600 q^{41} + 72 q^{42} + 90 q^{43} + 36 q^{44} + 135 q^{45} + 120 q^{46} - 237 q^{47} + 144 q^{48} - 735 q^{49} - 150 q^{50} - 315 q^{51} - 132 q^{52} + 63 q^{53} - 162 q^{54} + 45 q^{55} + 96 q^{56} - 243 q^{57} + 216 q^{58} - 264 q^{59} + 180 q^{60} - 732 q^{61} + 684 q^{62} - 108 q^{63} + 192 q^{64} - 165 q^{65} - 54 q^{66} - 867 q^{67} - 420 q^{68} - 180 q^{69} + 120 q^{70} - 1233 q^{71} - 216 q^{72} - 837 q^{73} - 222 q^{74} + 225 q^{75} - 324 q^{76} + 819 q^{77} + 198 q^{78} - 1053 q^{79} + 240 q^{80} + 243 q^{81} + 1200 q^{82} + 312 q^{83} - 144 q^{84} - 525 q^{85} - 180 q^{86} - 324 q^{87} - 72 q^{88} + 495 q^{89} - 270 q^{90} - 18 q^{91} - 240 q^{92} - 1026 q^{93} + 474 q^{94} - 405 q^{95} - 288 q^{96} - 1092 q^{97} + 1470 q^{98} + 81 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 6x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) 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
−0.167449
−2.36147
2.52892
−2.00000 3.00000 4.00000 5.00000 −6.00000 −16.0833 −8.00000 9.00000 −10.0000
1.2 −2.00000 3.00000 4.00000 5.00000 −6.00000 −1.63186 −8.00000 9.00000 −10.0000
1.3 −2.00000 3.00000 4.00000 5.00000 −6.00000 5.71520 −8.00000 9.00000 −10.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(37\) \(-1\)

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 1110.4.a.g 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.4.a.g 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{3} + 12T_{7}^{2} - 75T_{7} - 150 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1110))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{3} \) Copy content Toggle raw display
$3$ \( (T - 3)^{3} \) Copy content Toggle raw display
$5$ \( (T - 5)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 12 T^{2} - 75 T - 150 \) Copy content Toggle raw display
$11$ \( T^{3} - 9 T^{2} - 1473 T + 10348 \) Copy content Toggle raw display
$13$ \( T^{3} + 33 T^{2} + 315 T + 683 \) Copy content Toggle raw display
$17$ \( T^{3} + 105 T^{2} + 1755 T - 16714 \) Copy content Toggle raw display
$19$ \( T^{3} + 81 T^{2} - 6663 T + 45506 \) Copy content Toggle raw display
$23$ \( T^{3} + 60 T^{2} - 26235 T + 971914 \) Copy content Toggle raw display
$29$ \( T^{3} + 108 T^{2} - 25512 T - 2931869 \) Copy content Toggle raw display
$31$ \( T^{3} + 342 T^{2} + 23403 T + 451090 \) Copy content Toggle raw display
$37$ \( (T - 37)^{3} \) Copy content Toggle raw display
$41$ \( T^{3} + 600 T^{2} + \cdots - 19235660 \) Copy content Toggle raw display
$43$ \( T^{3} - 90 T^{2} - 56496 T + 6229221 \) Copy content Toggle raw display
$47$ \( T^{3} + 237 T^{2} + \cdots - 34912209 \) Copy content Toggle raw display
$53$ \( T^{3} - 63 T^{2} - 175284 T - 22299408 \) Copy content Toggle raw display
$59$ \( T^{3} + 264 T^{2} + \cdots - 34412507 \) Copy content Toggle raw display
$61$ \( T^{3} + 732 T^{2} + \cdots + 10980910 \) Copy content Toggle raw display
$67$ \( T^{3} + 867 T^{2} + \cdots + 13712260 \) Copy content Toggle raw display
$71$ \( T^{3} + 1233 T^{2} + \cdots - 331049044 \) Copy content Toggle raw display
$73$ \( T^{3} + 837 T^{2} + \cdots - 789421254 \) Copy content Toggle raw display
$79$ \( T^{3} + 1053 T^{2} + \cdots - 164141764 \) Copy content Toggle raw display
$83$ \( T^{3} - 312 T^{2} + \cdots + 137107175 \) Copy content Toggle raw display
$89$ \( T^{3} - 495 T^{2} + \cdots - 193482029 \) Copy content Toggle raw display
$97$ \( T^{3} + 1092 T^{2} + \cdots - 312757294 \) Copy content Toggle raw display
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