Properties

Label 165.6.a.e
Level $165$
Weight $6$
Character orbit 165.a
Self dual yes
Analytic conductor $26.463$
Analytic rank $0$
Dimension $3$
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] = [165,6,Mod(1,165)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(165, 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("165.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 165.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: \(26.4633302691\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.307532.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 76x + 168 \) 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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 2) q^{2} - 9 q^{3} + (\beta_{2} + 2 \beta_1 + 24) q^{4} - 25 q^{5} + ( - 9 \beta_1 - 18) q^{6} + (4 \beta_{2} - 6 \beta_1 + 34) q^{7} + (7 \beta_{2} + 10 \beta_1 + 76) q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 2) q^{2} - 9 q^{3} + (\beta_{2} + 2 \beta_1 + 24) q^{4} - 25 q^{5} + ( - 9 \beta_1 - 18) q^{6} + (4 \beta_{2} - 6 \beta_1 + 34) q^{7} + (7 \beta_{2} + 10 \beta_1 + 76) q^{8} + 81 q^{9} + ( - 25 \beta_1 - 50) q^{10} + 121 q^{11} + ( - 9 \beta_{2} - 18 \beta_1 - 216) q^{12} + ( - \beta_{2} + 113 \beta_1 - 68) q^{13} + (14 \beta_{2} + 106 \beta_1 - 292) q^{14} + 225 q^{15} + (13 \beta_{2} + 138 \beta_1 - 180) q^{16} + ( - 12 \beta_{2} - 58 \beta_1 + 660) q^{17} + (81 \beta_1 + 162) q^{18} + ( - 11 \beta_{2} + 57 \beta_1 + 672) q^{19} + ( - 25 \beta_{2} - 50 \beta_1 - 600) q^{20} + ( - 36 \beta_{2} + 54 \beta_1 - 306) q^{21} + (121 \beta_1 + 242) q^{22} + (14 \beta_{2} + 286 \beta_1 + 316) q^{23} + ( - 63 \beta_{2} - 90 \beta_1 - 684) q^{24} + 625 q^{25} + (108 \beta_{2} - 86 \beta_1 + 5752) q^{26} - 729 q^{27} + (48 \beta_{2} + 152 \beta_1 + 3672) q^{28} + (145 \beta_{2} + 53 \beta_1 + 1498) q^{29} + (225 \beta_1 + 450) q^{30} + ( - 34 \beta_{2} + 582 \beta_1 - 3768) q^{31} + ( - 21 \beta_{2} - 266 \beta_1 + 4228) q^{32} - 1089 q^{33} + ( - 118 \beta_{2} + 444 \beta_1 - 1552) q^{34} + ( - 100 \beta_{2} + 150 \beta_1 - 850) q^{35} + (81 \beta_{2} + 162 \beta_1 + 1944) q^{36} + ( - 202 \beta_{2} - 274 \beta_1 - 2706) q^{37} + (2 \beta_{2} + 474 \beta_1 + 4440) q^{38} + (9 \beta_{2} - 1017 \beta_1 + 612) q^{39} + ( - 175 \beta_{2} - 250 \beta_1 - 1900) q^{40} + (35 \beta_{2} + 879 \beta_1 + 1710) q^{41} + ( - 126 \beta_{2} - 954 \beta_1 + 2628) q^{42} + ( - 322 \beta_{2} - 1992 \beta_1 + 6846) q^{43} + (121 \beta_{2} + 242 \beta_1 + 2904) q^{44} - 2025 q^{45} + (356 \beta_{2} + 568 \beta_1 + 15336) q^{46} + (134 \beta_{2} - 442 \beta_1 + 18692) q^{47} + ( - 117 \beta_{2} - 1242 \beta_1 + 1620) q^{48} + ( - 140 \beta_{2} - 672 \beta_1 + 813) q^{49} + (625 \beta_1 + 1250) q^{50} + (108 \beta_{2} + 522 \beta_1 - 5940) q^{51} + (486 \beta_{2} + 4080 \beta_1 + 7912) q^{52} + (74 \beta_{2} - 1946 \beta_1 + 3742) q^{53} + ( - 729 \beta_1 - 1458) q^{54} - 3025 q^{55} + ( - 56 \beta_{2} + 1144 \beta_1 + 24016) q^{56} + (99 \beta_{2} - 513 \beta_1 - 6048) q^{57} + (778 \beta_{2} + 4108 \beta_1 + 4012) q^{58} + ( - 14 \beta_{2} - 566 \beta_1 - 19548) q^{59} + (225 \beta_{2} + 450 \beta_1 + 5400) q^{60} + (544 \beta_{2} - 968 \beta_1 + 27138) q^{61} + (412 \beta_{2} - 4380 \beta_1 + 23136) q^{62} + (324 \beta_{2} - 486 \beta_1 + 2754) q^{63} + ( - 787 \beta_{2} - 566 \beta_1 + 636) q^{64} + (25 \beta_{2} - 2825 \beta_1 + 1700) q^{65} + ( - 1089 \beta_1 - 2178) q^{66} + ( - 844 \beta_{2} + 2136 \beta_1 + 496) q^{67} + (238 \beta_{2} - 1820 \beta_1 + 280) q^{68} + ( - 126 \beta_{2} - 2574 \beta_1 - 2844) q^{69} + ( - 350 \beta_{2} - 2650 \beta_1 + 7300) q^{70} + ( - 1064 \beta_{2} - 1228 \beta_1 - 25000) q^{71} + (567 \beta_{2} + 810 \beta_1 + 6156) q^{72} + (563 \beta_{2} - 6451 \beta_1 - 18092) q^{73} + ( - 1284 \beta_{2} - 6342 \beta_1 - 17236) q^{74} - 5625 q^{75} + (836 \beta_{2} + 2652 \beta_1 + 12000) q^{76} + (484 \beta_{2} - 726 \beta_1 + 4114) q^{77} + ( - 972 \beta_{2} + 774 \beta_1 - 51768) q^{78} + ( - 1927 \beta_{2} + 2985 \beta_1 - 29492) q^{79} + ( - 325 \beta_{2} - 3450 \beta_1 + 4500) q^{80} + 6561 q^{81} + (1054 \beta_{2} + 2340 \beta_1 + 48708) q^{82} + (1715 \beta_{2} - 2811 \beta_1 + 26430) q^{83} + ( - 432 \beta_{2} - 1368 \beta_1 - 33048) q^{84} + (300 \beta_{2} + 1450 \beta_1 - 16500) q^{85} + ( - 3602 \beta_{2} + 1050 \beta_1 - 86028) q^{86} + ( - 1305 \beta_{2} - 477 \beta_1 - 13482) q^{87} + (847 \beta_{2} + 1210 \beta_1 + 9196) q^{88} + ( - 1410 \beta_{2} - 8246 \beta_1 + 14726) q^{89} + ( - 2025 \beta_1 - 4050) q^{90} + (466 \beta_{2} + 13682 \beta_1 - 46568) q^{91} + (1900 \beta_{2} + 12592 \beta_1 + 45824) q^{92} + (306 \beta_{2} - 5238 \beta_1 + 33912) q^{93} + (228 \beta_{2} + 21104 \beta_1 + 12792) q^{94} + (275 \beta_{2} - 1425 \beta_1 - 16800) q^{95} + (189 \beta_{2} + 2394 \beta_1 - 38052) q^{96} + ( - 2108 \beta_{2} + 88 \beta_1 + 10430) q^{97} + ( - 1372 \beta_{2} - 1707 \beta_1 - 31638) q^{98} + 9801 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 7 q^{2} - 27 q^{3} + 73 q^{4} - 75 q^{5} - 63 q^{6} + 92 q^{7} + 231 q^{8} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 7 q^{2} - 27 q^{3} + 73 q^{4} - 75 q^{5} - 63 q^{6} + 92 q^{7} + 231 q^{8} + 243 q^{9} - 175 q^{10} + 363 q^{11} - 657 q^{12} - 90 q^{13} - 784 q^{14} + 675 q^{15} - 415 q^{16} + 1934 q^{17} + 567 q^{18} + 2084 q^{19} - 1825 q^{20} - 828 q^{21} + 847 q^{22} + 1220 q^{23} - 2079 q^{24} + 1875 q^{25} + 17062 q^{26} - 2187 q^{27} + 11120 q^{28} + 4402 q^{29} + 1575 q^{30} - 10688 q^{31} + 12439 q^{32} - 3267 q^{33} - 4094 q^{34} - 2300 q^{35} + 5913 q^{36} - 8190 q^{37} + 13792 q^{38} + 810 q^{39} - 5775 q^{40} + 5974 q^{41} + 7056 q^{42} + 18868 q^{43} + 8833 q^{44} - 6075 q^{45} + 46220 q^{46} + 55500 q^{47} + 3735 q^{48} + 1907 q^{49} + 4375 q^{50} - 17406 q^{51} + 27330 q^{52} + 9206 q^{53} - 5103 q^{54} - 9075 q^{55} + 73248 q^{56} - 18756 q^{57} + 15366 q^{58} - 59196 q^{59} + 16425 q^{60} + 79902 q^{61} + 64616 q^{62} + 7452 q^{63} + 2129 q^{64} + 2250 q^{65} - 7623 q^{66} + 4468 q^{67} - 1218 q^{68} - 10980 q^{69} + 19600 q^{70} - 75164 q^{71} + 18711 q^{72} - 61290 q^{73} - 56766 q^{74} - 16875 q^{75} + 37816 q^{76} + 11132 q^{77} - 153558 q^{78} - 83564 q^{79} + 10375 q^{80} + 19683 q^{81} + 147410 q^{82} + 74764 q^{83} - 100080 q^{84} - 48350 q^{85} - 253432 q^{86} - 39618 q^{87} + 27951 q^{88} + 37342 q^{89} - 14175 q^{90} - 126488 q^{91} + 148164 q^{92} + 96192 q^{93} + 59252 q^{94} - 52100 q^{95} - 111951 q^{96} + 33486 q^{97} - 95249 q^{98} + 29403 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 76x + 168 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2\nu - 52 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2\beta _1 + 52 \) 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
−9.21967
2.30119
7.91848
−7.21967 −9.00000 20.1236 −25.0000 64.9770 147.570 85.7438 81.0000 180.492
1.2 4.30119 −9.00000 −13.4998 −25.0000 −38.7107 −148.216 −195.703 81.0000 −107.530
1.3 9.91848 −9.00000 66.3762 −25.0000 −89.2663 92.6461 340.959 81.0000 −247.962
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(11\) \(-1\)

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 165.6.a.e 3
3.b odd 2 1 495.6.a.a 3
5.b even 2 1 825.6.a.f 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.6.a.e 3 1.a even 1 1 trivial
495.6.a.a 3 3.b odd 2 1
825.6.a.f 3 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 7T_{2}^{2} - 60T_{2} + 308 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(165))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 7 T^{2} + \cdots + 308 \) Copy content Toggle raw display
$3$ \( (T + 9)^{3} \) Copy content Toggle raw display
$5$ \( (T + 25)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 92 T^{2} + \cdots + 2026368 \) Copy content Toggle raw display
$11$ \( (T - 121)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 90 T^{2} + \cdots + 210673232 \) Copy content Toggle raw display
$17$ \( T^{3} - 1934 T^{2} + \cdots + 123909408 \) Copy content Toggle raw display
$19$ \( T^{3} - 2084 T^{2} + \cdots + 14437440 \) Copy content Toggle raw display
$23$ \( T^{3} + \cdots + 2404328128 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots + 80705567064 \) Copy content Toggle raw display
$31$ \( T^{3} + 10688 T^{2} + \cdots + 593236224 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots - 165140256344 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots + 127610311752 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots + 5672527691040 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots - 5576540180928 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots - 850686588776 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots + 7185357358784 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots - 2993338614376 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots - 6432661987328 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots - 31171869026560 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots - 152306824713328 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots - 283704612543488 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots + 200710881230832 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots + 340522035911288 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots - 93893760682568 \) Copy content Toggle raw display
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