Properties

Label 102.6.a.h
Level $102$
Weight $6$
Character orbit 102.a
Self dual yes
Analytic conductor $16.359$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [102,6,Mod(1,102)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(102, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("102.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 102 = 2 \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 102.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: \(16.3591496209\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 1027x - 7656 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 q^{2} + 9 q^{3} + 16 q^{4} + (\beta_{2} + 18) q^{5} + 36 q^{6} + (\beta_1 + 56) q^{7} + 64 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} + 9 q^{3} + 16 q^{4} + (\beta_{2} + 18) q^{5} + 36 q^{6} + (\beta_1 + 56) q^{7} + 64 q^{8} + 81 q^{9} + (4 \beta_{2} + 72) q^{10} + ( - 5 \beta_{2} - 5 \beta_1 + 120) q^{11} + 144 q^{12} + ( - 9 \beta_{2} - \beta_1 + 218) q^{13} + (4 \beta_1 + 224) q^{14} + (9 \beta_{2} + 162) q^{15} + 256 q^{16} + 289 q^{17} + 324 q^{18} + ( - 25 \beta_{2} + 15 \beta_1 + 584) q^{19} + (16 \beta_{2} + 288) q^{20} + (9 \beta_1 + 504) q^{21} + ( - 20 \beta_{2} - 20 \beta_1 + 480) q^{22} + (39 \beta_{2} + 204) q^{23} + 576 q^{24} + (7 \beta_{2} - 15 \beta_1 + 235) q^{25} + ( - 36 \beta_{2} - 4 \beta_1 + 872) q^{26} + 729 q^{27} + (16 \beta_1 + 896) q^{28} + (36 \beta_{2} + 5 \beta_1 + 438) q^{29} + (36 \beta_{2} + 648) q^{30} + (86 \beta_{2} - 35 \beta_1 - 664) q^{31} + 1024 q^{32} + ( - 45 \beta_{2} - 45 \beta_1 + 1080) q^{33} + 1156 q^{34} + (8 \beta_{2} + 46 \beta_1 + 672) q^{35} + 1296 q^{36} + ( - 74 \beta_{2} + 115 \beta_1 + 1118) q^{37} + ( - 100 \beta_{2} + 60 \beta_1 + 2336) q^{38} + ( - 81 \beta_{2} - 9 \beta_1 + 1962) q^{39} + (64 \beta_{2} + 1152) q^{40} + ( - 149 \beta_{2} - 99 \beta_1 - 90) q^{41} + (36 \beta_1 + 2016) q^{42} + ( - 79 \beta_{2} - 121 \beta_1 - 2200) q^{43} + ( - 80 \beta_{2} - 80 \beta_1 + 1920) q^{44} + (81 \beta_{2} + 1458) q^{45} + (156 \beta_{2} + 816) q^{46} + (250 \beta_{2} + 68 \beta_1 - 3672) q^{47} + 2304 q^{48} + (112 \beta_{2} + 160 \beta_1 - 2679) q^{49} + (28 \beta_{2} - 60 \beta_1 + 940) q^{50} + 2601 q^{51} + ( - 144 \beta_{2} - 16 \beta_1 + 3488) q^{52} + ( - 260 \beta_{2} + 2 \beta_1 - 4386) q^{53} + 2916 q^{54} + (415 \beta_{2} - 155 \beta_1 - 11340) q^{55} + (64 \beta_1 + 3584) q^{56} + ( - 225 \beta_{2} + 135 \beta_1 + 5256) q^{57} + (144 \beta_{2} + 20 \beta_1 + 1752) q^{58} + (54 \beta_{2} + 24 \beta_1 - 13188) q^{59} + (144 \beta_{2} + 2592) q^{60} + ( - 414 \beta_{2} - 117 \beta_1 - 5818) q^{61} + (344 \beta_{2} - 140 \beta_1 - 2656) q^{62} + (81 \beta_1 + 4536) q^{63} + 4096 q^{64} + (365 \beta_{2} + 89 \beta_1 - 23064) q^{65} + ( - 180 \beta_{2} - 180 \beta_1 + 4320) q^{66} + ( - 944 \beta_{2} - 126 \beta_1 - 6388) q^{67} + 4624 q^{68} + (351 \beta_{2} + 1836) q^{69} + (32 \beta_{2} + 184 \beta_1 + 2688) q^{70} + (60 \beta_{2} + 21 \beta_1 - 32688) q^{71} + 5184 q^{72} + (724 \beta_{2} + 76 \beta_1 - 15622) q^{73} + ( - 296 \beta_{2} + 460 \beta_1 + 4472) q^{74} + (63 \beta_{2} - 135 \beta_1 + 2115) q^{75} + ( - 400 \beta_{2} + 240 \beta_1 + 9344) q^{76} + ( - 600 \beta_{2} - 540 \beta_1 - 46560) q^{77} + ( - 324 \beta_{2} - 36 \beta_1 + 7848) q^{78} + (254 \beta_{2} + 597 \beta_1 - 9016) q^{79} + (256 \beta_{2} + 4608) q^{80} + 6561 q^{81} + ( - 596 \beta_{2} - 396 \beta_1 - 360) q^{82} + (302 \beta_{2} - 172 \beta_1 - 60684) q^{83} + (144 \beta_1 + 8064) q^{84} + (289 \beta_{2} + 5202) q^{85} + ( - 316 \beta_{2} - 484 \beta_1 - 8800) q^{86} + (324 \beta_{2} + 45 \beta_1 + 3942) q^{87} + ( - 320 \beta_{2} - 320 \beta_1 + 7680) q^{88} + ( - 1010 \beta_{2} + 182 \beta_1 - 66846) q^{89} + (324 \beta_{2} + 5832) q^{90} + ( - 184 \beta_{2} - 138 \beta_1 + 4240) q^{91} + (624 \beta_{2} + 3264) q^{92} + (774 \beta_{2} - 315 \beta_1 - 5976) q^{93} + (1000 \beta_{2} + 272 \beta_1 - 14688) q^{94} + (139 \beta_{2} + 1065 \beta_1 - 70428) q^{95} + 9216 q^{96} + (594 \beta_{2} - 1012 \beta_1 + 30698) q^{97} + (448 \beta_{2} + 640 \beta_1 - 10716) q^{98} + ( - 405 \beta_{2} - 405 \beta_1 + 9720) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 12 q^{2} + 27 q^{3} + 48 q^{4} + 53 q^{5} + 108 q^{6} + 168 q^{7} + 192 q^{8} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 12 q^{2} + 27 q^{3} + 48 q^{4} + 53 q^{5} + 108 q^{6} + 168 q^{7} + 192 q^{8} + 243 q^{9} + 212 q^{10} + 365 q^{11} + 432 q^{12} + 663 q^{13} + 672 q^{14} + 477 q^{15} + 768 q^{16} + 867 q^{17} + 972 q^{18} + 1777 q^{19} + 848 q^{20} + 1512 q^{21} + 1460 q^{22} + 573 q^{23} + 1728 q^{24} + 698 q^{25} + 2652 q^{26} + 2187 q^{27} + 2688 q^{28} + 1278 q^{29} + 1908 q^{30} - 2078 q^{31} + 3072 q^{32} + 3285 q^{33} + 3468 q^{34} + 2008 q^{35} + 3888 q^{36} + 3428 q^{37} + 7108 q^{38} + 5967 q^{39} + 3392 q^{40} - 121 q^{41} + 6048 q^{42} - 6521 q^{43} + 5840 q^{44} + 4293 q^{45} + 2292 q^{46} - 11266 q^{47} + 6912 q^{48} - 8149 q^{49} + 2792 q^{50} + 7803 q^{51} + 10608 q^{52} - 12898 q^{53} + 8748 q^{54} - 34435 q^{55} + 10752 q^{56} + 15993 q^{57} + 5112 q^{58} - 39618 q^{59} + 7632 q^{60} - 17040 q^{61} - 8312 q^{62} + 13608 q^{63} + 12288 q^{64} - 69557 q^{65} + 13140 q^{66} - 18220 q^{67} + 13872 q^{68} + 5157 q^{69} + 8032 q^{70} - 98124 q^{71} + 15552 q^{72} - 47590 q^{73} + 13712 q^{74} + 6282 q^{75} + 28432 q^{76} - 139080 q^{77} + 23868 q^{78} - 27302 q^{79} + 13568 q^{80} + 19683 q^{81} - 484 q^{82} - 182354 q^{83} + 24192 q^{84} + 15317 q^{85} - 26084 q^{86} + 11502 q^{87} + 23360 q^{88} - 199528 q^{89} + 17172 q^{90} + 12904 q^{91} + 9168 q^{92} - 18702 q^{93} - 45064 q^{94} - 211423 q^{95} + 27648 q^{96} + 91500 q^{97} - 32596 q^{98} + 29565 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 1027x - 7656 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - 12\nu - 687 ) / 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 7\beta_{2} + 3\beta _1 + 687 \) 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
−7.94261
35.2712
−27.3286
4.00000 9.00000 16.0000 −57.5148 36.0000 24.2296 64.0000 81.0000 −230.059
1.2 4.00000 9.00000 16.0000 37.1151 36.0000 197.085 64.0000 81.0000 148.460
1.3 4.00000 9.00000 16.0000 73.3997 36.0000 −53.3145 64.0000 81.0000 293.599
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(17\) \(-1\)

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 102.6.a.h 3
3.b odd 2 1 306.6.a.p 3
4.b odd 2 1 816.6.a.j 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.6.a.h 3 1.a even 1 1 trivial
306.6.a.p 3 3.b odd 2 1
816.6.a.j 3 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} - 53T_{5}^{2} - 3632T_{5} + 156684 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(102))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 4)^{3} \) Copy content Toggle raw display
$3$ \( (T - 9)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 53 T^{2} - 3632 T + 156684 \) Copy content Toggle raw display
$7$ \( T^{3} - 168 T^{2} - 7024 T + 254592 \) Copy content Toggle raw display
$11$ \( T^{3} - 365 T^{2} + \cdots + 174150000 \) Copy content Toggle raw display
$13$ \( T^{3} - 663 T^{2} + \cdots - 15142380 \) Copy content Toggle raw display
$17$ \( (T - 289)^{3} \) Copy content Toggle raw display
$19$ \( T^{3} - 1777 T^{2} + \cdots + 10823035696 \) Copy content Toggle raw display
$23$ \( T^{3} - 573 T^{2} + \cdots + 6154132464 \) Copy content Toggle raw display
$29$ \( T^{3} - 1278 T^{2} + \cdots + 8425642584 \) Copy content Toggle raw display
$31$ \( T^{3} + 2078 T^{2} + \cdots - 189692148384 \) Copy content Toggle raw display
$37$ \( T^{3} - 3428 T^{2} + \cdots + 756189044384 \) Copy content Toggle raw display
$41$ \( T^{3} + 121 T^{2} + \cdots + 599247187860 \) Copy content Toggle raw display
$43$ \( T^{3} + 6521 T^{2} + \cdots + 1051734223248 \) Copy content Toggle raw display
$47$ \( T^{3} + 11266 T^{2} + \cdots + 725716865664 \) Copy content Toggle raw display
$53$ \( T^{3} + 12898 T^{2} + \cdots - 2618974153224 \) Copy content Toggle raw display
$59$ \( T^{3} + 39618 T^{2} + \cdots + 2026881400608 \) Copy content Toggle raw display
$61$ \( T^{3} + 17040 T^{2} + \cdots - 14077185611408 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots - 130615297281216 \) Copy content Toggle raw display
$71$ \( T^{3} + 98124 T^{2} + \cdots + 34278499449792 \) Copy content Toggle raw display
$73$ \( T^{3} + 47590 T^{2} + \cdots + 10516757566376 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots - 227350425704480 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots + 155380242235680 \) Copy content Toggle raw display
$89$ \( T^{3} + 199528 T^{2} + \cdots - 31426154645904 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots + 315785543467872 \) Copy content Toggle raw display
show more
show less