Properties

Label 1110.4.a.l
Level $1110$
Weight $4$
Character orbit 1110.a
Self dual yes
Analytic conductor $65.492$
Analytic rank $1$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.8827413.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 25x^{2} + 6x + 120 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + 3 q^{3} + 4 q^{4} - 5 q^{5} + 6 q^{6} + (\beta_{3} + 3 \beta_{2} + \beta_1 - 4) q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 3 q^{3} + 4 q^{4} - 5 q^{5} + 6 q^{6} + (\beta_{3} + 3 \beta_{2} + \beta_1 - 4) q^{7} + 8 q^{8} + 9 q^{9} - 10 q^{10} + ( - \beta_{3} - 5 \beta_{2} + 6 \beta_1 - 6) q^{11} + 12 q^{12} + ( - 5 \beta_{3} - 5 \beta_{2} + \beta_1 - 20) q^{13} + (2 \beta_{3} + 6 \beta_{2} + 2 \beta_1 - 8) q^{14} - 15 q^{15} + 16 q^{16} + ( - \beta_{3} + 2 \beta_{2} - 22 \beta_1 - 37) q^{17} + 18 q^{18} + (5 \beta_{3} - 6 \beta_{2} - 14 \beta_1 - 61) q^{19} - 20 q^{20} + (3 \beta_{3} + 9 \beta_{2} + 3 \beta_1 - 12) q^{21} + ( - 2 \beta_{3} - 10 \beta_{2} + 12 \beta_1 - 12) q^{22} + (11 \beta_{3} - 3 \beta_{2} - 23 \beta_1 - 16) q^{23} + 24 q^{24} + 25 q^{25} + ( - 10 \beta_{3} - 10 \beta_{2} + 2 \beta_1 - 40) q^{26} + 27 q^{27} + (4 \beta_{3} + 12 \beta_{2} + 4 \beta_1 - 16) q^{28} + ( - 10 \beta_{3} + 35 \beta_1 - 100) q^{29} - 30 q^{30} + ( - 4 \beta_{3} + \beta_{2} - 24 \beta_1 - 167) q^{31} + 32 q^{32} + ( - 3 \beta_{3} - 15 \beta_{2} + 18 \beta_1 - 18) q^{33} + ( - 2 \beta_{3} + 4 \beta_{2} - 44 \beta_1 - 74) q^{34} + ( - 5 \beta_{3} - 15 \beta_{2} - 5 \beta_1 + 20) q^{35} + 36 q^{36} + 37 q^{37} + (10 \beta_{3} - 12 \beta_{2} - 28 \beta_1 - 122) q^{38} + ( - 15 \beta_{3} - 15 \beta_{2} + 3 \beta_1 - 60) q^{39} - 40 q^{40} + (4 \beta_{3} + 23 \beta_{2} + 34 \beta_1 - 163) q^{41} + (6 \beta_{3} + 18 \beta_{2} + 6 \beta_1 - 24) q^{42} + (14 \beta_{3} + 12 \beta_{2} + 23 \beta_1 - 98) q^{43} + ( - 4 \beta_{3} - 20 \beta_{2} + 24 \beta_1 - 24) q^{44} - 45 q^{45} + (22 \beta_{3} - 6 \beta_{2} - 46 \beta_1 - 32) q^{46} + ( - 14 \beta_{3} + 27 \beta_{2} + 46 \beta_1 - 91) q^{47} + 48 q^{48} + ( - 23 \beta_{3} - 49 \beta_{2} + 91 \beta_1 + 9) q^{49} + 50 q^{50} + ( - 3 \beta_{3} + 6 \beta_{2} - 66 \beta_1 - 111) q^{51} + ( - 20 \beta_{3} - 20 \beta_{2} + 4 \beta_1 - 80) q^{52} + ( - 39 \beta_{3} - 26 \beta_{2} - 23 \beta_1 - 11) q^{53} + 54 q^{54} + (5 \beta_{3} + 25 \beta_{2} - 30 \beta_1 + 30) q^{55} + (8 \beta_{3} + 24 \beta_{2} + 8 \beta_1 - 32) q^{56} + (15 \beta_{3} - 18 \beta_{2} - 42 \beta_1 - 183) q^{57} + ( - 20 \beta_{3} + 70 \beta_1 - 200) q^{58} + ( - 6 \beta_{3} + 7 \beta_{2} + 151 \beta_1 - 7) q^{59} - 60 q^{60} + (10 \beta_{3} + 71 \beta_{2} + 96 \beta_1 - 81) q^{61} + ( - 8 \beta_{3} + 2 \beta_{2} - 48 \beta_1 - 334) q^{62} + (9 \beta_{3} + 27 \beta_{2} + 9 \beta_1 - 36) q^{63} + 64 q^{64} + (25 \beta_{3} + 25 \beta_{2} - 5 \beta_1 + 100) q^{65} + ( - 6 \beta_{3} - 30 \beta_{2} + 36 \beta_1 - 36) q^{66} + (8 \beta_{3} - 56 \beta_{2} + 51 \beta_1 - 248) q^{67} + ( - 4 \beta_{3} + 8 \beta_{2} - 88 \beta_1 - 148) q^{68} + (33 \beta_{3} - 9 \beta_{2} - 69 \beta_1 - 48) q^{69} + ( - 10 \beta_{3} - 30 \beta_{2} - 10 \beta_1 + 40) q^{70} + (104 \beta_{3} + 44 \beta_{2} + 19 \beta_1 + 56) q^{71} + 72 q^{72} + ( - 49 \beta_{3} + 51 \beta_{2} - 24 \beta_1 + 136) q^{73} + 74 q^{74} + 75 q^{75} + (20 \beta_{3} - 24 \beta_{2} - 56 \beta_1 - 244) q^{76} + (51 \beta_{3} + 117 \beta_{2} - 130 \beta_1 - 372) q^{77} + ( - 30 \beta_{3} - 30 \beta_{2} + 6 \beta_1 - 120) q^{78} + (72 \beta_{3} - 101 \beta_{2} - 49 \beta_1 - 61) q^{79} - 80 q^{80} + 81 q^{81} + (8 \beta_{3} + 46 \beta_{2} + 68 \beta_1 - 326) q^{82} + (29 \beta_{3} + 39 \beta_{2} - 216 \beta_1 - 252) q^{83} + (12 \beta_{3} + 36 \beta_{2} + 12 \beta_1 - 48) q^{84} + (5 \beta_{3} - 10 \beta_{2} + 110 \beta_1 + 185) q^{85} + (28 \beta_{3} + 24 \beta_{2} + 46 \beta_1 - 196) q^{86} + ( - 30 \beta_{3} + 105 \beta_1 - 300) q^{87} + ( - 8 \beta_{3} - 40 \beta_{2} + 48 \beta_1 - 48) q^{88} + ( - 47 \beta_{3} + 125 \beta_{2} - 53 \beta_1 - 580) q^{89} - 90 q^{90} + (49 \beta_{3} + 45 \beta_{2} - 227 \beta_1 - 562) q^{91} + (44 \beta_{3} - 12 \beta_{2} - 92 \beta_1 - 64) q^{92} + ( - 12 \beta_{3} + 3 \beta_{2} - 72 \beta_1 - 501) q^{93} + ( - 28 \beta_{3} + 54 \beta_{2} + 92 \beta_1 - 182) q^{94} + ( - 25 \beta_{3} + 30 \beta_{2} + 70 \beta_1 + 305) q^{95} + 96 q^{96} + ( - 8 \beta_{3} + 59 \beta_{2} - 188 \beta_1 - 787) q^{97} + ( - 46 \beta_{3} - 98 \beta_{2} + 182 \beta_1 + 18) q^{98} + ( - 9 \beta_{3} - 45 \beta_{2} + 54 \beta_1 - 54) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{2} + 12 q^{3} + 16 q^{4} - 20 q^{5} + 24 q^{6} - 23 q^{7} + 32 q^{8} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{2} + 12 q^{3} + 16 q^{4} - 20 q^{5} + 24 q^{6} - 23 q^{7} + 32 q^{8} + 36 q^{9} - 40 q^{10} - 6 q^{11} + 48 q^{12} - 59 q^{13} - 46 q^{14} - 60 q^{15} + 64 q^{16} - 172 q^{17} + 72 q^{18} - 256 q^{19} - 80 q^{20} - 69 q^{21} - 12 q^{22} - 103 q^{23} + 96 q^{24} + 100 q^{25} - 118 q^{26} + 108 q^{27} - 92 q^{28} - 345 q^{29} - 120 q^{30} - 686 q^{31} + 128 q^{32} - 18 q^{33} - 344 q^{34} + 115 q^{35} + 144 q^{36} + 148 q^{37} - 512 q^{38} - 177 q^{39} - 160 q^{40} - 672 q^{41} - 138 q^{42} - 421 q^{43} - 24 q^{44} - 180 q^{45} - 206 q^{46} - 344 q^{47} + 192 q^{48} + 271 q^{49} + 200 q^{50} - 516 q^{51} - 236 q^{52} + 63 q^{53} + 216 q^{54} + 30 q^{55} - 184 q^{56} - 768 q^{57} - 690 q^{58} + 121 q^{59} - 240 q^{60} - 390 q^{61} - 1372 q^{62} - 207 q^{63} + 256 q^{64} + 295 q^{65} - 36 q^{66} - 845 q^{67} - 688 q^{68} - 309 q^{69} + 230 q^{70} - 53 q^{71} + 288 q^{72} + 516 q^{73} + 296 q^{74} + 300 q^{75} - 1024 q^{76} - 1954 q^{77} - 354 q^{78} - 235 q^{79} - 320 q^{80} + 324 q^{81} - 1344 q^{82} - 1360 q^{83} - 276 q^{84} + 860 q^{85} - 842 q^{86} - 1035 q^{87} - 48 q^{88} - 2529 q^{89} - 360 q^{90} - 2663 q^{91} - 412 q^{92} - 2058 q^{93} - 688 q^{94} + 1280 q^{95} + 384 q^{96} - 3438 q^{97} + 542 q^{98} - 54 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 25x^{2} + 6x + 120 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - 3\nu^{2} - 13\nu + 26 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + 3\beta_{2} + 16\beta _1 + 13 \) 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
2.52913
−2.57121
−3.80616
4.84825
2.00000 3.00000 4.00000 −5.00000 6.00000 −33.8141 8.00000 9.00000 −10.0000
1.2 2.00000 3.00000 4.00000 −5.00000 6.00000 −6.72729 8.00000 9.00000 −10.0000
1.3 2.00000 3.00000 4.00000 −5.00000 6.00000 −3.48700 8.00000 9.00000 −10.0000
1.4 2.00000 3.00000 4.00000 −5.00000 6.00000 21.0284 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.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.4.a.l 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 23T_{7}^{3} - 557T_{7}^{2} - 6963T_{7} - 16680 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1110))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{4} \) Copy content Toggle raw display
$3$ \( (T - 3)^{4} \) Copy content Toggle raw display
$5$ \( (T + 5)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 23 T^{3} - 557 T^{2} + \cdots - 16680 \) Copy content Toggle raw display
$11$ \( T^{4} + 6 T^{3} - 2854 T^{2} + \cdots - 299716 \) Copy content Toggle raw display
$13$ \( T^{4} + 59 T^{3} - 3255 T^{2} + \cdots + 1420990 \) Copy content Toggle raw display
$17$ \( T^{4} + 172 T^{3} - 2382 T^{2} + \cdots + 627862 \) Copy content Toggle raw display
$19$ \( T^{4} + 256 T^{3} + \cdots - 51511592 \) Copy content Toggle raw display
$23$ \( T^{4} + 103 T^{3} + \cdots - 141260816 \) Copy content Toggle raw display
$29$ \( T^{4} + 345 T^{3} + \cdots + 50882500 \) Copy content Toggle raw display
$31$ \( T^{4} + 686 T^{3} + \cdots + 234681832 \) Copy content Toggle raw display
$37$ \( (T - 37)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + 672 T^{3} + \cdots - 2823967280 \) Copy content Toggle raw display
$43$ \( T^{4} + 421 T^{3} + \cdots - 351189408 \) Copy content Toggle raw display
$47$ \( T^{4} + 344 T^{3} + \cdots + 663296928 \) Copy content Toggle raw display
$53$ \( T^{4} - 63 T^{3} + \cdots + 16289516160 \) Copy content Toggle raw display
$59$ \( T^{4} - 121 T^{3} + \cdots + 59111981960 \) Copy content Toggle raw display
$61$ \( T^{4} + 390 T^{3} + \cdots - 39681273100 \) Copy content Toggle raw display
$67$ \( T^{4} + 845 T^{3} + \cdots - 6416221016 \) Copy content Toggle raw display
$71$ \( T^{4} + 53 T^{3} + \cdots + 593103510304 \) Copy content Toggle raw display
$73$ \( T^{4} - 516 T^{3} + \cdots + 13095941550 \) Copy content Toggle raw display
$79$ \( T^{4} + 235 T^{3} + \cdots + 382378425184 \) Copy content Toggle raw display
$83$ \( T^{4} + 1360 T^{3} + \cdots + 272148028388 \) Copy content Toggle raw display
$89$ \( T^{4} + 2529 T^{3} + \cdots - 546655237958 \) Copy content Toggle raw display
$97$ \( T^{4} + 3438 T^{3} + \cdots - 505578013232 \) Copy content Toggle raw display
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