Properties

Label 105.10.a.b
Level $105$
Weight $10$
Character orbit 105.a
Self dual yes
Analytic conductor $54.079$
Analytic rank $0$
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] = [105,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("105.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(54.0787627972\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 295x^{2} + 188x + 11500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
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 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.

\(f(q)\) \(=\) \( q + (\beta_1 - 5) q^{2} - 81 q^{3} + (2 \beta_{2} - 8 \beta_1 + 106) q^{4} + 625 q^{5} + ( - 81 \beta_1 + 405) q^{6} - 2401 q^{7} + (8 \beta_{3} - 24 \beta_{2} + \cdots - 3252) q^{8}+ \cdots + 6561 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 5) q^{2} - 81 q^{3} + (2 \beta_{2} - 8 \beta_1 + 106) q^{4} + 625 q^{5} + ( - 81 \beta_1 + 405) q^{6} - 2401 q^{7} + (8 \beta_{3} - 24 \beta_{2} + \cdots - 3252) q^{8}+ \cdots + (269001 \beta_{3} + 413343 \beta_{2} + \cdots + 120059739) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 20 q^{2} - 324 q^{3} + 424 q^{4} + 2500 q^{5} + 1620 q^{6} - 9604 q^{7} - 13008 q^{8} + 26244 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 20 q^{2} - 324 q^{3} + 424 q^{4} + 2500 q^{5} + 1620 q^{6} - 9604 q^{7} - 13008 q^{8} + 26244 q^{9} - 12500 q^{10} + 73196 q^{11} - 34344 q^{12} + 16660 q^{13} + 48020 q^{14} - 202500 q^{15} - 366048 q^{16} + 534488 q^{17} - 131220 q^{18} + 346908 q^{19} + 265000 q^{20} + 777924 q^{21} - 1908368 q^{22} + 2063936 q^{23} + 1053648 q^{24} + 1562500 q^{25} - 5185656 q^{26} - 2125764 q^{27} - 1018024 q^{28} - 9002440 q^{29} + 1012500 q^{30} - 8181204 q^{31} + 3751104 q^{32} - 5928876 q^{33} - 21806280 q^{34} - 6002500 q^{35} + 2781864 q^{36} - 196992 q^{37} - 7497968 q^{38} - 1349460 q^{39} - 8130000 q^{40} + 18191656 q^{41} - 3889620 q^{42} + 46050920 q^{43} + 38999040 q^{44} + 16402500 q^{45} - 29598080 q^{46} + 65551400 q^{47} + 29649888 q^{48} + 23059204 q^{49} - 7812500 q^{50} - 43293528 q^{51} + 203344976 q^{52} + 115512044 q^{53} + 10628820 q^{54} + 45747500 q^{55} + 31232208 q^{56} - 28099548 q^{57} + 270146024 q^{58} + 126534680 q^{59} - 21465000 q^{60} - 52474008 q^{61} + 328289856 q^{62} - 63011844 q^{63} - 79976320 q^{64} + 10412500 q^{65} + 154577808 q^{66} + 122378144 q^{67} + 86352400 q^{68} - 167178816 q^{69} + 30012500 q^{70} + 280662164 q^{71} - 85345488 q^{72} + 607309724 q^{73} + 375379464 q^{74} - 126562500 q^{75} + 1145351360 q^{76} - 175743596 q^{77} + 420038136 q^{78} + 718313400 q^{79} - 228780000 q^{80} + 172186884 q^{81} - 11553960 q^{82} + 412007752 q^{83} + 82459944 q^{84} + 334055000 q^{85} - 344822640 q^{86} + 729197640 q^{87} + 728924288 q^{88} - 22182312 q^{89} - 82012500 q^{90} - 40000660 q^{91} + 1839984768 q^{92} + 662677524 q^{93} + 1645627712 q^{94} + 216817500 q^{95} - 303839424 q^{96} - 409915884 q^{97} - 115296020 q^{98} + 480238956 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 295x^{2} + 188x + 11500 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} - 4\nu - 295 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu^{2} - 216\nu + 250 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 2\beta _1 + 297 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{3} + 3\beta_{2} + 222\beta _1 + 607 ) / 2 \) 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
−15.0545
−6.55562
7.02653
16.5836
−36.1089 −81.0000 791.856 625.000 2924.82 −2401.00 −10105.3 6561.00 −22568.1
1.2 −19.1112 −81.0000 −146.761 625.000 1548.01 −2401.00 12589.7 6561.00 −11944.5
1.3 8.05305 −81.0000 −447.148 625.000 −652.297 −2401.00 −7724.07 6561.00 5033.16
1.4 27.1671 −81.0000 226.053 625.000 −2200.54 −2401.00 −7768.36 6561.00 16979.5
\(n\): 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 105.10.a.b 4
3.b odd 2 1 315.10.a.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.10.a.b 4 1.a even 1 1 trivial
315.10.a.h 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 20T_{2}^{3} - 1036T_{2}^{2} - 12224T_{2} + 150976 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(105))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 20 T^{3} + \cdots + 150976 \) Copy content Toggle raw display
$3$ \( (T + 81)^{4} \) Copy content Toggle raw display
$5$ \( (T - 625)^{4} \) Copy content Toggle raw display
$7$ \( (T + 2401)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots - 23\!\cdots\!60 \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 11\!\cdots\!08 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 69\!\cdots\!68 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 16\!\cdots\!60 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 22\!\cdots\!88 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots - 90\!\cdots\!40 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 73\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 31\!\cdots\!08 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 63\!\cdots\!88 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots - 33\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 21\!\cdots\!80 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots - 55\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 45\!\cdots\!12 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 33\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots - 76\!\cdots\!88 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots - 18\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 26\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots - 13\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots - 15\!\cdots\!48 \) Copy content Toggle raw display
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