Properties

Label 165.6.a.f
Level $165$
Weight $6$
Character orbit 165.a
Self dual yes
Analytic conductor $26.463$
Analytic rank $0$
Dimension $5$
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: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 119x^{3} + 206x^{2} + 1428x - 1320 \) 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + 9 q^{3} + (\beta_{2} + 16) q^{4} - 25 q^{5} - 9 \beta_1 q^{6} + ( - \beta_{4} + \beta_{2} - 2 \beta_1 + 37) q^{7} + ( - \beta_{3} - \beta_{2} - 31 \beta_1 + 18) q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + 9 q^{3} + (\beta_{2} + 16) q^{4} - 25 q^{5} - 9 \beta_1 q^{6} + ( - \beta_{4} + \beta_{2} - 2 \beta_1 + 37) q^{7} + ( - \beta_{3} - \beta_{2} - 31 \beta_1 + 18) q^{8} + 81 q^{9} + 25 \beta_1 q^{10} - 121 q^{11} + (9 \beta_{2} + 144) q^{12} + ( - \beta_{4} + 3 \beta_{3} + 4 \beta_{2} - 24 \beta_1 + 223) q^{13} + (2 \beta_{4} + 6 \beta_{2} - 94 \beta_1 + 122) q^{14} - 225 q^{15} + (4 \beta_{4} + 19 \beta_{2} + 8 \beta_1 + 944) q^{16} + (7 \beta_{4} + 9 \beta_{2} + 14 \beta_1 + 427) q^{17} - 81 \beta_1 q^{18} + ( - 6 \beta_{4} + 3 \beta_{3} - 7 \beta_{2} + 62 \beta_1 + 302) q^{19} + ( - 25 \beta_{2} - 400) q^{20} + ( - 9 \beta_{4} + 9 \beta_{2} - 18 \beta_1 + 333) q^{21} + 121 \beta_1 q^{22} + (4 \beta_{4} - 2 \beta_{3} - 38 \beta_{2} - 52 \beta_1 + 280) q^{23} + ( - 9 \beta_{3} - 9 \beta_{2} - 279 \beta_1 + 162) q^{24} + 625 q^{25} + ( - 10 \beta_{4} - 32 \beta_{2} - 358 \beta_1 + 1274) q^{26} + 729 q^{27} + (28 \beta_{4} - 8 \beta_{3} + 46 \beta_{2} - 320 \beta_1 + 3420) q^{28} + (2 \beta_{4} - 3 \beta_{3} - 85 \beta_{2} - 310 \beta_1 + 176) q^{29} + 225 \beta_1 q^{30} + ( - 24 \beta_{4} + 2 \beta_{3} + 26 \beta_{2} - 420 \beta_1 + 2576) q^{31} + ( - 8 \beta_{4} + 9 \beta_{3} - 15 \beta_{2} - 805 \beta_1 - 650) q^{32} - 1089 q^{33} + ( - 14 \beta_{4} - 16 \beta_{3} - 58 \beta_{2} - 780 \beta_1 - 566) q^{34} + (25 \beta_{4} - 25 \beta_{2} + 50 \beta_1 - 925) q^{35} + (81 \beta_{2} + 1296) q^{36} + ( - 60 \beta_{4} + 26 \beta_{3} - 74 \beta_{2} + 228 \beta_1 - 1390) q^{37} + (16 \beta_{3} - 82 \beta_{2} + 30 \beta_1 - 3012) q^{38} + ( - 9 \beta_{4} + 27 \beta_{3} + 36 \beta_{2} - 216 \beta_1 + 2007) q^{39} + (25 \beta_{3} + 25 \beta_{2} + 775 \beta_1 - 450) q^{40} + (6 \beta_{4} - 29 \beta_{3} - 19 \beta_{2} + 1222 \beta_1 + 888) q^{41} + (18 \beta_{4} + 54 \beta_{2} - 846 \beta_1 + 1098) q^{42} + (73 \beta_{4} - 46 \beta_{3} + 137 \beta_{2} + 590 \beta_1 + 8979) q^{43} + ( - 121 \beta_{2} - 1936) q^{44} - 2025 q^{45} + (32 \beta_{3} + 108 \beta_{2} + 1504 \beta_1 + 1752) q^{46} + ( - 28 \beta_{4} + 66 \beta_{3} - 18 \beta_{2} + 1332 \beta_1 - 2896) q^{47} + (36 \beta_{4} + 171 \beta_{2} + 72 \beta_1 + 8496) q^{48} + (78 \beta_{4} - 88 \beta_{3} - 278 \beta_{2} - 20 \beta_1 + 7059) q^{49} - 625 \beta_1 q^{50} + (63 \beta_{4} + 81 \beta_{2} + 126 \beta_1 + 3843) q^{51} + (52 \beta_{4} - 54 \beta_{3} + 312 \beta_{2} + 898 \beta_1 + 9552) q^{52} + ( - 94 \beta_{4} - 26 \beta_{3} + 132 \beta_{2} + 904 \beta_1 + 1640) q^{53} - 729 \beta_1 q^{54} + 3025 q^{55} + ( - 88 \beta_{4} - 82 \beta_{3} + 94 \beta_{2} - 2462 \beta_1 + 11948) q^{56} + ( - 54 \beta_{4} + 27 \beta_{3} - 63 \beta_{2} + 558 \beta_1 + 2718) q^{57} + (8 \beta_{4} + 80 \beta_{3} + 442 \beta_{2} + 3776 \beta_1 + 13292) q^{58} + (40 \beta_{4} + 18 \beta_{3} - 342 \beta_{2} + 1900 \beta_1 + 17956) q^{59} + ( - 225 \beta_{2} - 3600) q^{60} + ( - 96 \beta_{4} + 104 \beta_{3} - 120 \beta_{2} + 736 \beta_1 - 286) q^{61} + (40 \beta_{4} + 476 \beta_{2} - 3996 \beta_1 + 20848) q^{62} + ( - 81 \beta_{4} + 81 \beta_{2} - 162 \beta_1 + 2997) q^{63} + ( - 148 \beta_{4} + 32 \beta_{3} + 81 \beta_{2} + 1208 \beta_1 + 8352) q^{64} + (25 \beta_{4} - 75 \beta_{3} - 100 \beta_{2} + 600 \beta_1 - 5575) q^{65} + 1089 \beta_1 q^{66} + (82 \beta_{4} - 160 \beta_{3} - 210 \beta_{2} + 564 \beta_1 + 20034) q^{67} + ( - 132 \beta_{4} + 56 \beta_{3} + 924 \beta_{2} + 2368 \beta_1 + 22620) q^{68} + (36 \beta_{4} - 18 \beta_{3} - 342 \beta_{2} - 468 \beta_1 + 2520) q^{69} + ( - 50 \beta_{4} - 150 \beta_{2} + 2350 \beta_1 - 3050) q^{70} + (126 \beta_{4} + 44 \beta_{3} + 126 \beta_{2} - 4428 \beta_1 - 1026) q^{71} + ( - 81 \beta_{3} - 81 \beta_{2} - 2511 \beta_1 + 1458) q^{72} + (91 \beta_{4} - 17 \beta_{3} - 140 \beta_{2} + 5992 \beta_1 + 16515) q^{73} + (16 \beta_{4} + 160 \beta_{3} - 348 \beta_{2} + 4814 \beta_1 - 11432) q^{74} + 5625 q^{75} + (128 \beta_{4} + 2 \beta_{3} - 28 \beta_{2} + 5218 \beta_1 - 12356) q^{76} + (121 \beta_{4} - 121 \beta_{2} + 242 \beta_1 - 4477) q^{77} + ( - 90 \beta_{4} - 288 \beta_{2} - 3222 \beta_1 + 11466) q^{78} + ( - 188 \beta_{4} + 203 \beta_{3} - 393 \beta_{2} + 6098 \beta_1 + 33348) q^{79} + ( - 100 \beta_{4} - 475 \beta_{2} - 200 \beta_1 - 23600) q^{80} + 6561 q^{81} + (104 \beta_{4} - 16 \beta_{3} - 682 \beta_{2} - 544 \beta_1 - 59452) q^{82} + (105 \beta_{4} - 35 \beta_{3} - 316 \beta_{2} + 3288 \beta_1 - 21177) q^{83} + (252 \beta_{4} - 72 \beta_{3} + 414 \beta_{2} - 2880 \beta_1 + 30780) q^{84} + ( - 175 \beta_{4} - 225 \beta_{2} - 350 \beta_1 - 10675) q^{85} + (38 \beta_{4} - 256 \beta_{3} - 218 \beta_{2} - 15654 \beta_1 - 27082) q^{86} + (18 \beta_{4} - 27 \beta_{3} - 765 \beta_{2} - 2790 \beta_1 + 1584) q^{87} + (121 \beta_{3} + 121 \beta_{2} + 3751 \beta_1 - 2178) q^{88} + (54 \beta_{4} + 110 \beta_{3} + 88 \beta_{2} - 3136 \beta_1 + 25068) q^{89} + 2025 \beta_1 q^{90} + (332 \beta_{4} + 158 \beta_{3} + 370 \beta_{2} - 3924 \beta_1 + 10932) q^{91} + ( - 256 \beta_{4} - 12 \beta_{3} - 1004 \beta_{2} - 4492 \beta_1 - 78760) q^{92} + ( - 216 \beta_{4} + 18 \beta_{3} + 234 \beta_{2} - 3780 \beta_1 + 23184) q^{93} + ( - 208 \beta_{4} + 112 \beta_{3} - 2428 \beta_{2} + 4848 \beta_1 - 63112) q^{94} + (150 \beta_{4} - 75 \beta_{3} + 175 \beta_{2} - 1550 \beta_1 - 7550) q^{95} + ( - 72 \beta_{4} + 81 \beta_{3} - 135 \beta_{2} - 7245 \beta_1 - 5850) q^{96} + ( - 2 \beta_{4} - 256 \beta_{3} + 1522 \beta_{2} - 4116 \beta_1 + 21276) q^{97} + (196 \beta_{4} + 112 \beta_{3} + 1580 \beta_{2} + 4939 \beta_1 - 5900) q^{98} - 9801 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 2 q^{2} + 45 q^{3} + 82 q^{4} - 125 q^{5} - 18 q^{6} + 184 q^{7} + 24 q^{8} + 405 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 2 q^{2} + 45 q^{3} + 82 q^{4} - 125 q^{5} - 18 q^{6} + 184 q^{7} + 24 q^{8} + 405 q^{9} + 50 q^{10} - 605 q^{11} + 738 q^{12} + 1082 q^{13} + 432 q^{14} - 1125 q^{15} + 4770 q^{16} + 2174 q^{17} - 162 q^{18} + 1632 q^{19} - 2050 q^{20} + 1656 q^{21} + 242 q^{22} + 1212 q^{23} + 216 q^{24} + 3125 q^{25} + 5600 q^{26} + 3645 q^{27} + 16508 q^{28} + 82 q^{29} + 450 q^{30} + 12120 q^{31} - 4864 q^{32} - 5445 q^{33} - 4524 q^{34} - 4600 q^{35} + 6642 q^{36} - 6530 q^{37} - 15132 q^{38} + 9738 q^{39} - 600 q^{40} + 6782 q^{41} + 3888 q^{42} + 46184 q^{43} - 9922 q^{44} - 10125 q^{45} + 12048 q^{46} - 11692 q^{47} + 42930 q^{48} + 34445 q^{49} - 1250 q^{50} + 19566 q^{51} + 50020 q^{52} + 10314 q^{53} - 1458 q^{54} + 15125 q^{55} + 54928 q^{56} + 14688 q^{57} + 75048 q^{58} + 92892 q^{59} - 18450 q^{60} + 106 q^{61} + 97160 q^{62} + 14904 q^{63} + 44550 q^{64} - 27050 q^{65} + 2178 q^{66} + 100476 q^{67} + 119928 q^{68} + 10908 q^{69} - 10800 q^{70} - 13772 q^{71} + 1944 q^{72} + 94154 q^{73} - 47924 q^{74} + 28125 q^{75} - 51524 q^{76} - 22264 q^{77} + 50400 q^{78} + 178744 q^{79} - 119250 q^{80} + 32805 q^{81} - 299848 q^{82} - 100116 q^{83} + 148572 q^{84} - 54350 q^{85} - 167704 q^{86} + 738 q^{87} - 2904 q^{88} + 119410 q^{89} + 4050 q^{90} + 47536 q^{91} - 404560 q^{92} + 109080 q^{93} - 310288 q^{94} - 40800 q^{95} - 43776 q^{96} + 100682 q^{97} - 16434 q^{98} - 49005 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 119x^{3} + 206x^{2} + 1428x - 1320 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 48 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 95\nu + 66 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} - 115\nu^{2} - 8\nu + 992 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 48 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 95\beta _1 - 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{4} + 115\beta_{2} + 8\beta _1 + 4528 \) 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
10.3866
4.28026
0.870481
−3.34733
−10.1900
−10.3866 9.00000 75.8811 −25.0000 −93.4793 40.8791 −455.775 81.0000 259.665
1.2 −4.28026 9.00000 −13.6794 −25.0000 −38.5223 202.127 195.520 81.0000 107.006
1.3 −0.870481 9.00000 −31.2423 −25.0000 −7.83433 −236.601 55.0512 81.0000 21.7620
1.4 3.34733 9.00000 −20.7954 −25.0000 30.1260 42.9514 −176.724 81.0000 −83.6832
1.5 10.1900 9.00000 71.8359 −25.0000 91.7099 134.644 405.928 81.0000 −254.750
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.f 5
3.b odd 2 1 495.6.a.j 5
5.b even 2 1 825.6.a.l 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.6.a.f 5 1.a even 1 1 trivial
495.6.a.j 5 3.b odd 2 1
825.6.a.l 5 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} + 2T_{2}^{4} - 119T_{2}^{3} - 206T_{2}^{2} + 1428T_{2} + 1320 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(165))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + 2 T^{4} - 119 T^{3} + \cdots + 1320 \) Copy content Toggle raw display
$3$ \( (T - 9)^{5} \) Copy content Toggle raw display
$5$ \( (T + 25)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} - 184 T^{4} + \cdots + 11305890304 \) Copy content Toggle raw display
$11$ \( (T + 121)^{5} \) Copy content Toggle raw display
$13$ \( T^{5} - 1082 T^{4} + \cdots - 29790890509184 \) Copy content Toggle raw display
$17$ \( T^{5} + \cdots - 335880301363200 \) Copy content Toggle raw display
$19$ \( T^{5} + \cdots - 745265742345728 \) Copy content Toggle raw display
$23$ \( T^{5} - 1212 T^{4} + \cdots - 84\!\cdots\!08 \) Copy content Toggle raw display
$29$ \( T^{5} - 82 T^{4} + \cdots - 23\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( T^{5} - 12120 T^{4} + \cdots - 66\!\cdots\!12 \) Copy content Toggle raw display
$37$ \( T^{5} + 6530 T^{4} + \cdots + 27\!\cdots\!92 \) Copy content Toggle raw display
$41$ \( T^{5} - 6782 T^{4} + \cdots - 71\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( T^{5} - 46184 T^{4} + \cdots + 57\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{5} + 11692 T^{4} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{5} - 10314 T^{4} + \cdots - 53\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{5} - 92892 T^{4} + \cdots + 14\!\cdots\!40 \) Copy content Toggle raw display
$61$ \( T^{5} - 106 T^{4} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{5} - 100476 T^{4} + \cdots + 28\!\cdots\!60 \) Copy content Toggle raw display
$71$ \( T^{5} + 13772 T^{4} + \cdots - 26\!\cdots\!88 \) Copy content Toggle raw display
$73$ \( T^{5} - 94154 T^{4} + \cdots - 44\!\cdots\!88 \) Copy content Toggle raw display
$79$ \( T^{5} - 178744 T^{4} + \cdots + 38\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{5} + 100116 T^{4} + \cdots - 26\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( T^{5} - 119410 T^{4} + \cdots + 28\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{5} - 100682 T^{4} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
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