Properties

Label 105.4.a.d
Level $105$
Weight $4$
Character orbit 105.a
Self dual yes
Analytic conductor $6.195$
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] = [105,4,Mod(1,105)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(105, 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("105.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 105.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: \(6.19520055060\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 2) q^{2} + 3 q^{3} + (4 \beta + 1) q^{4} - 5 q^{5} + ( - 3 \beta - 6) q^{6} - 7 q^{7} + ( - \beta - 6) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 2) q^{2} + 3 q^{3} + (4 \beta + 1) q^{4} - 5 q^{5} + ( - 3 \beta - 6) q^{6} - 7 q^{7} + ( - \beta - 6) q^{8} + 9 q^{9} + (5 \beta + 10) q^{10} + (2 \beta - 46) q^{11} + (12 \beta + 3) q^{12} + (38 \beta + 4) q^{13} + (7 \beta + 14) q^{14} - 15 q^{15} + ( - 24 \beta + 9) q^{16} + ( - 44 \beta - 22) q^{17} + ( - 9 \beta - 18) q^{18} + ( - 26 \beta - 54) q^{19} + ( - 20 \beta - 5) q^{20} - 21 q^{21} + (42 \beta + 82) q^{22} + (20 \beta - 160) q^{23} + ( - 3 \beta - 18) q^{24} + 25 q^{25} + ( - 80 \beta - 198) q^{26} + 27 q^{27} + ( - 28 \beta - 7) q^{28} + ( - 12 \beta - 118) q^{29} + (15 \beta + 30) q^{30} + ( - 102 \beta - 30) q^{31} + (47 \beta + 150) q^{32} + (6 \beta - 138) q^{33} + (110 \beta + 264) q^{34} + 35 q^{35} + (36 \beta + 9) q^{36} + ( - 24 \beta + 102) q^{37} + (106 \beta + 238) q^{38} + (114 \beta + 12) q^{39} + (5 \beta + 30) q^{40} + (80 \beta + 22) q^{41} + (21 \beta + 42) q^{42} + ( - 128 \beta + 68) q^{43} + ( - 182 \beta - 6) q^{44} - 45 q^{45} + (120 \beta + 220) q^{46} + (168 \beta + 200) q^{47} + ( - 72 \beta + 27) q^{48} + 49 q^{49} + ( - 25 \beta - 50) q^{50} + ( - 132 \beta - 66) q^{51} + (54 \beta + 764) q^{52} + ( - 86 \beta + 8) q^{53} + ( - 27 \beta - 54) q^{54} + ( - 10 \beta + 230) q^{55} + (7 \beta + 42) q^{56} + ( - 78 \beta - 162) q^{57} + (142 \beta + 296) q^{58} + (36 \beta - 232) q^{59} + ( - 60 \beta - 15) q^{60} + ( - 84 \beta - 342) q^{61} + (234 \beta + 570) q^{62} - 63 q^{63} + ( - 52 \beta - 607) q^{64} + ( - 190 \beta - 20) q^{65} + (126 \beta + 246) q^{66} + ( - 164 \beta + 368) q^{67} + ( - 132 \beta - 902) q^{68} + (60 \beta - 480) q^{69} + ( - 35 \beta - 70) q^{70} + (138 \beta - 370) q^{71} + ( - 9 \beta - 54) q^{72} + (122 \beta + 212) q^{73} + ( - 54 \beta - 84) q^{74} + 75 q^{75} + ( - 242 \beta - 574) q^{76} + ( - 14 \beta + 322) q^{77} + ( - 240 \beta - 594) q^{78} + (484 \beta - 204) q^{79} + (120 \beta - 45) q^{80} + 81 q^{81} + ( - 182 \beta - 444) q^{82} + (84 \beta + 304) q^{83} + ( - 84 \beta - 21) q^{84} + (220 \beta + 110) q^{85} + (188 \beta + 504) q^{86} + ( - 36 \beta - 354) q^{87} + (34 \beta + 266) q^{88} + (112 \beta - 666) q^{89} + (45 \beta + 90) q^{90} + ( - 266 \beta - 28) q^{91} + ( - 620 \beta + 240) q^{92} + ( - 306 \beta - 90) q^{93} + ( - 536 \beta - 1240) q^{94} + (130 \beta + 270) q^{95} + (141 \beta + 450) q^{96} + (86 \beta - 1224) q^{97} + ( - 49 \beta - 98) q^{98} + (18 \beta - 414) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 6 q^{3} + 2 q^{4} - 10 q^{5} - 12 q^{6} - 14 q^{7} - 12 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} + 6 q^{3} + 2 q^{4} - 10 q^{5} - 12 q^{6} - 14 q^{7} - 12 q^{8} + 18 q^{9} + 20 q^{10} - 92 q^{11} + 6 q^{12} + 8 q^{13} + 28 q^{14} - 30 q^{15} + 18 q^{16} - 44 q^{17} - 36 q^{18} - 108 q^{19} - 10 q^{20} - 42 q^{21} + 164 q^{22} - 320 q^{23} - 36 q^{24} + 50 q^{25} - 396 q^{26} + 54 q^{27} - 14 q^{28} - 236 q^{29} + 60 q^{30} - 60 q^{31} + 300 q^{32} - 276 q^{33} + 528 q^{34} + 70 q^{35} + 18 q^{36} + 204 q^{37} + 476 q^{38} + 24 q^{39} + 60 q^{40} + 44 q^{41} + 84 q^{42} + 136 q^{43} - 12 q^{44} - 90 q^{45} + 440 q^{46} + 400 q^{47} + 54 q^{48} + 98 q^{49} - 100 q^{50} - 132 q^{51} + 1528 q^{52} + 16 q^{53} - 108 q^{54} + 460 q^{55} + 84 q^{56} - 324 q^{57} + 592 q^{58} - 464 q^{59} - 30 q^{60} - 684 q^{61} + 1140 q^{62} - 126 q^{63} - 1214 q^{64} - 40 q^{65} + 492 q^{66} + 736 q^{67} - 1804 q^{68} - 960 q^{69} - 140 q^{70} - 740 q^{71} - 108 q^{72} + 424 q^{73} - 168 q^{74} + 150 q^{75} - 1148 q^{76} + 644 q^{77} - 1188 q^{78} - 408 q^{79} - 90 q^{80} + 162 q^{81} - 888 q^{82} + 608 q^{83} - 42 q^{84} + 220 q^{85} + 1008 q^{86} - 708 q^{87} + 532 q^{88} - 1332 q^{89} + 180 q^{90} - 56 q^{91} + 480 q^{92} - 180 q^{93} - 2480 q^{94} + 540 q^{95} + 900 q^{96} - 2448 q^{97} - 196 q^{98} - 828 q^{99}+O(q^{100}) \) 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
1.61803
−0.618034
−4.23607 3.00000 9.94427 −5.00000 −12.7082 −7.00000 −8.23607 9.00000 21.1803
1.2 0.236068 3.00000 −7.94427 −5.00000 0.708204 −7.00000 −3.76393 9.00000 −1.18034
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( +1 \)
\(7\) \( +1 \)

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 105.4.a.d 2
3.b odd 2 1 315.4.a.l 2
4.b odd 2 1 1680.4.a.bd 2
5.b even 2 1 525.4.a.o 2
5.c odd 4 2 525.4.d.k 4
7.b odd 2 1 735.4.a.m 2
15.d odd 2 1 1575.4.a.n 2
21.c even 2 1 2205.4.a.be 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.d 2 1.a even 1 1 trivial
315.4.a.l 2 3.b odd 2 1
525.4.a.o 2 5.b even 2 1
525.4.d.k 4 5.c odd 4 2
735.4.a.m 2 7.b odd 2 1
1575.4.a.n 2 15.d odd 2 1
1680.4.a.bd 2 4.b odd 2 1
2205.4.a.be 2 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 4T_{2} - 1 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(105))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$3$ \( (T - 3)^{2} \) Copy content Toggle raw display
$5$ \( (T + 5)^{2} \) Copy content Toggle raw display
$7$ \( (T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 92T + 2096 \) Copy content Toggle raw display
$13$ \( T^{2} - 8T - 7204 \) Copy content Toggle raw display
$17$ \( T^{2} + 44T - 9196 \) Copy content Toggle raw display
$19$ \( T^{2} + 108T - 464 \) Copy content Toggle raw display
$23$ \( T^{2} + 320T + 23600 \) Copy content Toggle raw display
$29$ \( T^{2} + 236T + 13204 \) Copy content Toggle raw display
$31$ \( T^{2} + 60T - 51120 \) Copy content Toggle raw display
$37$ \( T^{2} - 204T + 7524 \) Copy content Toggle raw display
$41$ \( T^{2} - 44T - 31516 \) Copy content Toggle raw display
$43$ \( T^{2} - 136T - 77296 \) Copy content Toggle raw display
$47$ \( T^{2} - 400T - 101120 \) Copy content Toggle raw display
$53$ \( T^{2} - 16T - 36916 \) Copy content Toggle raw display
$59$ \( T^{2} + 464T + 47344 \) Copy content Toggle raw display
$61$ \( T^{2} + 684T + 81684 \) Copy content Toggle raw display
$67$ \( T^{2} - 736T + 944 \) Copy content Toggle raw display
$71$ \( T^{2} + 740T + 41680 \) Copy content Toggle raw display
$73$ \( T^{2} - 424T - 29476 \) Copy content Toggle raw display
$79$ \( T^{2} + 408 T - 1129664 \) Copy content Toggle raw display
$83$ \( T^{2} - 608T + 57136 \) Copy content Toggle raw display
$89$ \( T^{2} + 1332 T + 380836 \) Copy content Toggle raw display
$97$ \( T^{2} + 2448 T + 1461196 \) Copy content Toggle raw display
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