Properties

Label 105.6.a.d
Level $105$
Weight $6$
Character orbit 105.a
Self dual yes
Analytic conductor $16.840$
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] = [105,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("105.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(16.8403010804\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{73}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 18 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{73})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + 9 q^{3} + (\beta - 14) q^{4} - 25 q^{5} - 9 \beta q^{6} + 49 q^{7} + (45 \beta - 18) q^{8} + 81 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + 9 q^{3} + (\beta - 14) q^{4} - 25 q^{5} - 9 \beta q^{6} + 49 q^{7} + (45 \beta - 18) q^{8} + 81 q^{9} + 25 \beta q^{10} + (60 \beta - 228) q^{11} + (9 \beta - 126) q^{12} + ( - 68 \beta - 82) q^{13} - 49 \beta q^{14} - 225 q^{15} + ( - 59 \beta - 362) q^{16} + (88 \beta - 1230) q^{17} - 81 \beta q^{18} + (572 \beta - 808) q^{19} + ( - 25 \beta + 350) q^{20} + 441 q^{21} + (168 \beta - 1080) q^{22} + ( - 176 \beta - 1968) q^{23} + (405 \beta - 162) q^{24} + 625 q^{25} + (150 \beta + 1224) q^{26} + 729 q^{27} + (49 \beta - 686) q^{28} + ( - 832 \beta - 1158) q^{29} + 225 \beta q^{30} + ( - 1084 \beta - 1648) q^{31} + ( - 1019 \beta + 1638) q^{32} + (540 \beta - 2052) q^{33} + (1142 \beta - 1584) q^{34} - 1225 q^{35} + (81 \beta - 1134) q^{36} + ( - 976 \beta - 1870) q^{37} + (236 \beta - 10296) q^{38} + ( - 612 \beta - 738) q^{39} + ( - 1125 \beta + 450) q^{40} + ( - 3120 \beta - 6330) q^{41} - 441 \beta q^{42} + (1448 \beta - 7600) q^{43} + ( - 1008 \beta + 4272) q^{44} - 2025 q^{45} + (2144 \beta + 3168) q^{46} + (2968 \beta - 36) q^{47} + ( - 531 \beta - 3258) q^{48} + 2401 q^{49} - 625 \beta q^{50} + (792 \beta - 11070) q^{51} + (802 \beta - 76) q^{52} + ( - 1684 \beta - 17214) q^{53} - 729 \beta q^{54} + ( - 1500 \beta + 5700) q^{55} + (2205 \beta - 882) q^{56} + (5148 \beta - 7272) q^{57} + (1990 \beta + 14976) q^{58} + ( - 2872 \beta - 4356) q^{59} + ( - 225 \beta + 3150) q^{60} + ( - 5824 \beta + 23378) q^{61} + (2732 \beta + 19512) q^{62} + 3969 q^{63} + (1269 \beta + 29926) q^{64} + (1700 \beta + 2050) q^{65} + (1512 \beta - 9720) q^{66} + (848 \beta + 44504) q^{67} + ( - 2374 \beta + 18804) q^{68} + ( - 1584 \beta - 17712) q^{69} + 1225 \beta q^{70} + ( - 9780 \beta - 3348) q^{71} + (3645 \beta - 1458) q^{72} + (9668 \beta - 6046) q^{73} + (2846 \beta + 17568) q^{74} + 5625 q^{75} + ( - 8244 \beta + 21608) q^{76} + (2940 \beta - 11172) q^{77} + (1350 \beta + 11016) q^{78} + (13224 \beta + 23864) q^{79} + (1475 \beta + 9050) q^{80} + 6561 q^{81} + (9450 \beta + 56160) q^{82} + ( - 5096 \beta - 38076) q^{83} + (441 \beta - 6174) q^{84} + ( - 2200 \beta + 30750) q^{85} + (6152 \beta - 26064) q^{86} + ( - 7488 \beta - 10422) q^{87} + ( - 8640 \beta + 52704) q^{88} + (17136 \beta - 21306) q^{89} + 2025 \beta q^{90} + ( - 3332 \beta - 4018) q^{91} + (320 \beta + 24384) q^{92} + ( - 9756 \beta - 14832) q^{93} + ( - 2932 \beta - 53424) q^{94} + ( - 14300 \beta + 20200) q^{95} + ( - 9171 \beta + 14742) q^{96} + ( - 20084 \beta + 71690) q^{97} - 2401 \beta q^{98} + (4860 \beta - 18468) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 18 q^{3} - 27 q^{4} - 50 q^{5} - 9 q^{6} + 98 q^{7} + 9 q^{8} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 18 q^{3} - 27 q^{4} - 50 q^{5} - 9 q^{6} + 98 q^{7} + 9 q^{8} + 162 q^{9} + 25 q^{10} - 396 q^{11} - 243 q^{12} - 232 q^{13} - 49 q^{14} - 450 q^{15} - 783 q^{16} - 2372 q^{17} - 81 q^{18} - 1044 q^{19} + 675 q^{20} + 882 q^{21} - 1992 q^{22} - 4112 q^{23} + 81 q^{24} + 1250 q^{25} + 2598 q^{26} + 1458 q^{27} - 1323 q^{28} - 3148 q^{29} + 225 q^{30} - 4380 q^{31} + 2257 q^{32} - 3564 q^{33} - 2026 q^{34} - 2450 q^{35} - 2187 q^{36} - 4716 q^{37} - 20356 q^{38} - 2088 q^{39} - 225 q^{40} - 15780 q^{41} - 441 q^{42} - 13752 q^{43} + 7536 q^{44} - 4050 q^{45} + 8480 q^{46} + 2896 q^{47} - 7047 q^{48} + 4802 q^{49} - 625 q^{50} - 21348 q^{51} + 650 q^{52} - 36112 q^{53} - 729 q^{54} + 9900 q^{55} + 441 q^{56} - 9396 q^{57} + 31942 q^{58} - 11584 q^{59} + 6075 q^{60} + 40932 q^{61} + 41756 q^{62} + 7938 q^{63} + 61121 q^{64} + 5800 q^{65} - 17928 q^{66} + 89856 q^{67} + 35234 q^{68} - 37008 q^{69} + 1225 q^{70} - 16476 q^{71} + 729 q^{72} - 2424 q^{73} + 37982 q^{74} + 11250 q^{75} + 34972 q^{76} - 19404 q^{77} + 23382 q^{78} + 60952 q^{79} + 19575 q^{80} + 13122 q^{81} + 121770 q^{82} - 81248 q^{83} - 11907 q^{84} + 59300 q^{85} - 45976 q^{86} - 28332 q^{87} + 96768 q^{88} - 25476 q^{89} + 2025 q^{90} - 11368 q^{91} + 49088 q^{92} - 39420 q^{93} - 109780 q^{94} + 26100 q^{95} + 20313 q^{96} + 123296 q^{97} - 2401 q^{98} - 32076 q^{99}+O(q^{100}) \) 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
4.77200
−3.77200
−4.77200 9.00000 −9.22800 −25.0000 −42.9480 49.0000 196.740 81.0000 119.300
1.2 3.77200 9.00000 −17.7720 −25.0000 33.9480 49.0000 −187.740 81.0000 −94.3000
\(n\): 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 105.6.a.d 2
3.b odd 2 1 315.6.a.e 2
5.b even 2 1 525.6.a.g 2
5.c odd 4 2 525.6.d.j 4
7.b odd 2 1 735.6.a.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.6.a.d 2 1.a even 1 1 trivial
315.6.a.e 2 3.b odd 2 1
525.6.a.g 2 5.b even 2 1
525.6.d.j 4 5.c odd 4 2
735.6.a.f 2 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + T_{2} - 18 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(105))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 18 \) Copy content Toggle raw display
$3$ \( (T - 9)^{2} \) Copy content Toggle raw display
$5$ \( (T + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T - 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 396T - 26496 \) Copy content Toggle raw display
$13$ \( T^{2} + 232T - 70932 \) Copy content Toggle raw display
$17$ \( T^{2} + 2372 T + 1265268 \) Copy content Toggle raw display
$19$ \( T^{2} + 1044 T - 5698624 \) Copy content Toggle raw display
$23$ \( T^{2} + 4112 T + 3661824 \) Copy content Toggle raw display
$29$ \( T^{2} + 3148 T - 10155612 \) Copy content Toggle raw display
$31$ \( T^{2} + 4380 T - 16648672 \) Copy content Toggle raw display
$37$ \( T^{2} + 4716 T - 11824348 \) Copy content Toggle raw display
$41$ \( T^{2} + 15780 T - 115400700 \) Copy content Toggle raw display
$43$ \( T^{2} + 13752 T + 9014528 \) Copy content Toggle raw display
$47$ \( T^{2} - 2896 T - 158667984 \) Copy content Toggle raw display
$53$ \( T^{2} + 36112 T + 274264764 \) Copy content Toggle raw display
$59$ \( T^{2} + 11584 T - 116985744 \) Copy content Toggle raw display
$61$ \( T^{2} - 40932 T - 200164156 \) Copy content Toggle raw display
$67$ \( T^{2} - 89856 T + 2005401536 \) Copy content Toggle raw display
$71$ \( T^{2} + 16476 T - 1677718656 \) Copy content Toggle raw display
$73$ \( T^{2} + 2424 T - 1704362644 \) Copy content Toggle raw display
$79$ \( T^{2} - 60952 T - 2262667136 \) Copy content Toggle raw display
$83$ \( T^{2} + 81248 T + 1176371184 \) Copy content Toggle raw display
$89$ \( T^{2} + 25476 T - 5196718908 \) Copy content Toggle raw display
$97$ \( T^{2} - 123296 T - 3560972868 \) Copy content Toggle raw display
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