Properties

Label 105.4.a.c
Level $105$
Weight $4$
Character orbit 105.a
Self dual yes
Analytic conductor $6.195$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 3) q^{2} - 3 q^{3} + (7 \beta + 5) q^{4} - 5 q^{5} + (3 \beta + 9) q^{6} - 7 q^{7} + ( - 25 \beta - 19) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 3) q^{2} - 3 q^{3} + (7 \beta + 5) q^{4} - 5 q^{5} + (3 \beta + 9) q^{6} - 7 q^{7} + ( - 25 \beta - 19) q^{8} + 9 q^{9} + (5 \beta + 15) q^{10} + ( - 10 \beta - 8) q^{11} + ( - 21 \beta - 15) q^{12} + ( - 22 \beta + 18) q^{13} + (7 \beta + 21) q^{14} + 15 q^{15} + (63 \beta + 117) q^{16} + (28 \beta - 6) q^{17} + ( - 9 \beta - 27) q^{18} + ( - 26 \beta + 100) q^{19} + ( - 35 \beta - 25) q^{20} + 21 q^{21} + (48 \beta + 64) q^{22} + (56 \beta + 64) q^{23} + (75 \beta + 57) q^{24} + 25 q^{25} + (70 \beta + 34) q^{26} - 27 q^{27} + ( - 49 \beta - 35) q^{28} + ( - 84 \beta + 26) q^{29} + ( - 15 \beta - 45) q^{30} + (18 \beta + 156) q^{31} + ( - 169 \beta - 451) q^{32} + (30 \beta + 24) q^{33} + ( - 106 \beta - 94) q^{34} + 35 q^{35} + (63 \beta + 45) q^{36} + (24 \beta - 78) q^{37} + (4 \beta - 196) q^{38} + (66 \beta - 54) q^{39} + (125 \beta + 95) q^{40} + (140 \beta + 30) q^{41} + ( - 21 \beta - 63) q^{42} + ( - 68 \beta + 216) q^{43} + ( - 176 \beta - 320) q^{44} - 45 q^{45} + ( - 288 \beta - 416) q^{46} + (108 \beta + 92) q^{47} + ( - 189 \beta - 351) q^{48} + 49 q^{49} + ( - 25 \beta - 75) q^{50} + ( - 84 \beta + 18) q^{51} + ( - 138 \beta - 526) q^{52} + (214 \beta - 90) q^{53} + (27 \beta + 81) q^{54} + (50 \beta + 40) q^{55} + (175 \beta + 133) q^{56} + (78 \beta - 300) q^{57} + (310 \beta + 258) q^{58} + (36 \beta + 164) q^{59} + (105 \beta + 75) q^{60} + ( - 252 \beta + 522) q^{61} + ( - 228 \beta - 540) q^{62} - 63 q^{63} + (623 \beta + 1093) q^{64} + (110 \beta - 90) q^{65} + ( - 144 \beta - 192) q^{66} + (28 \beta - 408) q^{67} + (294 \beta + 754) q^{68} + ( - 168 \beta - 192) q^{69} + ( - 35 \beta - 105) q^{70} + ( - 330 \beta + 392) q^{71} + ( - 225 \beta - 171) q^{72} + ( - 178 \beta + 478) q^{73} + ( - 18 \beta + 138) q^{74} - 75 q^{75} + (388 \beta - 228) q^{76} + (70 \beta + 56) q^{77} + ( - 210 \beta - 102) q^{78} + (88 \beta + 160) q^{79} + ( - 315 \beta - 585) q^{80} + 81 q^{81} + ( - 590 \beta - 650) q^{82} + ( - 264 \beta + 700) q^{83} + (147 \beta + 105) q^{84} + ( - 140 \beta + 30) q^{85} + (56 \beta - 376) q^{86} + (252 \beta - 78) q^{87} + (640 \beta + 1152) q^{88} + ( - 728 \beta + 382) q^{89} + (45 \beta + 135) q^{90} + (154 \beta - 126) q^{91} + (1120 \beta + 1888) q^{92} + ( - 54 \beta - 468) q^{93} + ( - 524 \beta - 708) q^{94} + (130 \beta - 500) q^{95} + (507 \beta + 1353) q^{96} + (146 \beta - 322) q^{97} + ( - 49 \beta - 147) q^{98} + ( - 90 \beta - 72) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 7 q^{2} - 6 q^{3} + 17 q^{4} - 10 q^{5} + 21 q^{6} - 14 q^{7} - 63 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 7 q^{2} - 6 q^{3} + 17 q^{4} - 10 q^{5} + 21 q^{6} - 14 q^{7} - 63 q^{8} + 18 q^{9} + 35 q^{10} - 26 q^{11} - 51 q^{12} + 14 q^{13} + 49 q^{14} + 30 q^{15} + 297 q^{16} + 16 q^{17} - 63 q^{18} + 174 q^{19} - 85 q^{20} + 42 q^{21} + 176 q^{22} + 184 q^{23} + 189 q^{24} + 50 q^{25} + 138 q^{26} - 54 q^{27} - 119 q^{28} - 32 q^{29} - 105 q^{30} + 330 q^{31} - 1071 q^{32} + 78 q^{33} - 294 q^{34} + 70 q^{35} + 153 q^{36} - 132 q^{37} - 388 q^{38} - 42 q^{39} + 315 q^{40} + 200 q^{41} - 147 q^{42} + 364 q^{43} - 816 q^{44} - 90 q^{45} - 1120 q^{46} + 292 q^{47} - 891 q^{48} + 98 q^{49} - 175 q^{50} - 48 q^{51} - 1190 q^{52} + 34 q^{53} + 189 q^{54} + 130 q^{55} + 441 q^{56} - 522 q^{57} + 826 q^{58} + 364 q^{59} + 255 q^{60} + 792 q^{61} - 1308 q^{62} - 126 q^{63} + 2809 q^{64} - 70 q^{65} - 528 q^{66} - 788 q^{67} + 1802 q^{68} - 552 q^{69} - 245 q^{70} + 454 q^{71} - 567 q^{72} + 778 q^{73} + 258 q^{74} - 150 q^{75} - 68 q^{76} + 182 q^{77} - 414 q^{78} + 408 q^{79} - 1485 q^{80} + 162 q^{81} - 1890 q^{82} + 1136 q^{83} + 357 q^{84} - 80 q^{85} - 696 q^{86} + 96 q^{87} + 2944 q^{88} + 36 q^{89} + 315 q^{90} - 98 q^{91} + 4896 q^{92} - 990 q^{93} - 1940 q^{94} - 870 q^{95} + 3213 q^{96} - 498 q^{97} - 343 q^{98} - 234 q^{99}+O(q^{100}) \) 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
2.56155
−1.56155
−5.56155 −3.00000 22.9309 −5.00000 16.6847 −7.00000 −83.0388 9.00000 27.8078
1.2 −1.43845 −3.00000 −5.93087 −5.00000 4.31534 −7.00000 20.0388 9.00000 7.19224
\(n\): 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.c 2
3.b odd 2 1 315.4.a.m 2
4.b odd 2 1 1680.4.a.bk 2
5.b even 2 1 525.4.a.p 2
5.c odd 4 2 525.4.d.i 4
7.b odd 2 1 735.4.a.k 2
15.d odd 2 1 1575.4.a.m 2
21.c even 2 1 2205.4.a.bh 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.c 2 1.a even 1 1 trivial
315.4.a.m 2 3.b odd 2 1
525.4.a.p 2 5.b even 2 1
525.4.d.i 4 5.c odd 4 2
735.4.a.k 2 7.b odd 2 1
1575.4.a.m 2 15.d odd 2 1
1680.4.a.bk 2 4.b odd 2 1
2205.4.a.bh 2 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 7T_{2} + 8 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(105))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 7T + 8 \) 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
$11$ \( T^{2} + 26T - 256 \) Copy content Toggle raw display
$13$ \( T^{2} - 14T - 2008 \) Copy content Toggle raw display
$17$ \( T^{2} - 16T - 3268 \) Copy content Toggle raw display
$19$ \( T^{2} - 174T + 4696 \) Copy content Toggle raw display
$23$ \( T^{2} - 184T - 4864 \) Copy content Toggle raw display
$29$ \( T^{2} + 32T - 29732 \) Copy content Toggle raw display
$31$ \( T^{2} - 330T + 25848 \) Copy content Toggle raw display
$37$ \( T^{2} + 132T + 1908 \) Copy content Toggle raw display
$41$ \( T^{2} - 200T - 73300 \) Copy content Toggle raw display
$43$ \( T^{2} - 364T + 13472 \) Copy content Toggle raw display
$47$ \( T^{2} - 292T - 28256 \) Copy content Toggle raw display
$53$ \( T^{2} - 34T - 194344 \) Copy content Toggle raw display
$59$ \( T^{2} - 364T + 27616 \) Copy content Toggle raw display
$61$ \( T^{2} - 792T - 113076 \) Copy content Toggle raw display
$67$ \( T^{2} + 788T + 151904 \) Copy content Toggle raw display
$71$ \( T^{2} - 454T - 411296 \) Copy content Toggle raw display
$73$ \( T^{2} - 778T + 16664 \) Copy content Toggle raw display
$79$ \( T^{2} - 408T + 8704 \) Copy content Toggle raw display
$83$ \( T^{2} - 1136T + 26416 \) Copy content Toggle raw display
$89$ \( T^{2} - 36T - 2252108 \) Copy content Toggle raw display
$97$ \( T^{2} + 498T - 28592 \) Copy content Toggle raw display
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