Properties

Label 165.6.a.g
Level $165$
Weight $6$
Character orbit 165.a
Self dual yes
Analytic conductor $26.463$
Analytic rank $1$
Dimension $5$
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] = [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: \(1\)
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: \( x^{5} - x^{4} - 143x^{3} + 71x^{2} + 4216x - 3740 \) 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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - 9 q^{3} + (\beta_{3} + \beta_1 + 25) q^{4} - 25 q^{5} + 9 \beta_1 q^{6} + ( - \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 24) q^{7} + ( - 3 \beta_{3} - 5 \beta_{2} - 20 \beta_1 - 26) q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - 9 q^{3} + (\beta_{3} + \beta_1 + 25) q^{4} - 25 q^{5} + 9 \beta_1 q^{6} + ( - \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 24) q^{7} + ( - 3 \beta_{3} - 5 \beta_{2} - 20 \beta_1 - 26) q^{8} + 81 q^{9} + 25 \beta_1 q^{10} - 121 q^{11} + ( - 9 \beta_{3} - 9 \beta_1 - 225) q^{12} + (\beta_{4} - 10 \beta_{3} - 2 \beta_{2} + 28 \beta_1 - 189) q^{13} + ( - 3 \beta_{4} - 10 \beta_{3} + 8 \beta_{2} - 4 \beta_1 + 77) q^{14} + 225 q^{15} + (5 \beta_{4} + 19 \beta_{3} + 10 \beta_{2} + 107 \beta_1 + 352) q^{16} + ( - 9 \beta_{3} - 37 \beta_{2} - 27 \beta_1 - 42) q^{17} - 81 \beta_1 q^{18} + ( - 7 \beta_{4} - 3 \beta_{3} + 25 \beta_{2} + 145 \beta_1 + 657) q^{19} + ( - 25 \beta_{3} - 25 \beta_1 - 625) q^{20} + (9 \beta_{3} - 27 \beta_{2} + 27 \beta_1 - 216) q^{21} + 121 \beta_1 q^{22} + (8 \beta_{4} + 16 \beta_{3} + 32 \beta_{2} + 352 \beta_1 - 924) q^{23} + (27 \beta_{3} + 45 \beta_{2} + 180 \beta_1 + 234) q^{24} + 625 q^{25} + (3 \beta_{4} + 4 \beta_{3} + 16 \beta_{2} + 446 \beta_1 - 1863) q^{26} - 729 q^{27} + ( - 11 \beta_{4} + 10 \beta_{3} + 58 \beta_{2} + 202 \beta_1 - 1021) q^{28} + (9 \beta_{4} + 85 \beta_{3} - 79 \beta_{2} + 169 \beta_1 - 637) q^{29} - 225 \beta_1 q^{30} + ( - 24 \beta_{4} - 32 \beta_{2} + 512 \beta_1 - 1320) q^{31} + ( - 5 \beta_{4} - 89 \beta_{3} - 85 \beta_{2} - 248 \beta_1 - 4883) q^{32} + 1089 q^{33} + (37 \beta_{4} + 230 \beta_{3} + 8 \beta_{2} + 414 \beta_1 + 2037) q^{34} + (25 \beta_{3} - 75 \beta_{2} + 75 \beta_1 - 600) q^{35} + (81 \beta_{3} + 81 \beta_1 + 2025) q^{36} + (6 \beta_{4} + 110 \beta_{3} - 170 \beta_{2} - 122 \beta_1 + 1332) q^{37} + ( - 32 \beta_{4} - 278 \beta_{3} + 264 \beta_{2} - 932 \beta_1 - 8890) q^{38} + ( - 9 \beta_{4} + 90 \beta_{3} + 18 \beta_{2} - 252 \beta_1 + 1701) q^{39} + (75 \beta_{3} + 125 \beta_{2} + 500 \beta_1 + 650) q^{40} + (7 \beta_{4} - 69 \beta_{3} - 161 \beta_{2} - 57 \beta_1 - 2963) q^{41} + (27 \beta_{4} + 90 \beta_{3} - 72 \beta_{2} + 36 \beta_1 - 693) q^{42} + (48 \beta_{4} + 25 \beta_{3} + 101 \beta_{2} + 91 \beta_1 + 2128) q^{43} + ( - 121 \beta_{3} - 121 \beta_1 - 3025) q^{44} - 2025 q^{45} + ( - 24 \beta_{4} - 528 \beta_{3} - 304 \beta_{2} + 212 \beta_1 - 20232) q^{46} + ( - 16 \beta_{4} - 40 \beta_{3} + 280 \beta_{2} - 1304 \beta_1 - 8020) q^{47} + ( - 45 \beta_{4} - 171 \beta_{3} - 90 \beta_{2} - 963 \beta_1 - 3168) q^{48} + ( - 22 \beta_{4} - 100 \beta_{3} + 108 \beta_{2} - 1032 \beta_1 - 2189) q^{49} - 625 \beta_1 q^{50} + (81 \beta_{3} + 333 \beta_{2} + 243 \beta_1 + 378) q^{51} + ( - 45 \beta_{4} - 208 \beta_{3} - 36 \beta_{2} + 426 \beta_1 - 19583) q^{52} + ( - 2 \beta_{4} - 60 \beta_{3} + 324 \beta_{2} - 536 \beta_1 - 12360) q^{53} + 729 \beta_1 q^{54} + 3025 q^{55} + (27 \beta_{4} - 214 \beta_{3} + 104 \beta_{2} + 304 \beta_1 - 14897) q^{56} + (63 \beta_{4} + 27 \beta_{3} - 225 \beta_{2} - 1305 \beta_1 - 5913) q^{57} + (88 \beta_{4} + 74 \beta_{3} - 792 \beta_{2} - 1334 \beta_1 - 5330) q^{58} + (616 \beta_{3} + 40 \beta_{2} - 152 \beta_1 + 1612) q^{59} + (225 \beta_{3} + 225 \beta_1 + 5625) q^{60} + (82 \beta_{4} + 258 \beta_{3} - 246 \beta_{2} + 1210 \beta_1 + 11492) q^{61} + (8 \beta_{4} - 400 \beta_{3} + 736 \beta_{2} + 448 \beta_1 - 28536) q^{62} + ( - 81 \beta_{3} + 243 \beta_{2} - 243 \beta_1 + 1944) q^{63} + ( - 80 \beta_{4} + 233 \beta_{3} + 200 \beta_{2} + 4181 \beta_1 + 1893) q^{64} + ( - 25 \beta_{4} + 250 \beta_{3} + 50 \beta_{2} - 700 \beta_1 + 4725) q^{65} - 1089 \beta_1 q^{66} + ( - 144 \beta_{4} + 322 \beta_{3} + 234 \beta_{2} - 650 \beta_1 - 3060) q^{67} + (29 \beta_{4} - 552 \beta_{3} - 1142 \beta_{2} - 6888 \beta_1 - 15255) q^{68} + ( - 72 \beta_{4} - 144 \beta_{3} - 288 \beta_{2} - 3168 \beta_1 + 8316) q^{69} + (75 \beta_{4} + 250 \beta_{3} - 200 \beta_{2} + 100 \beta_1 - 1925) q^{70} + ( - 8 \beta_{4} + 74 \beta_{3} + 274 \beta_{2} - 3602 \beta_1 + 18332) q^{71} + ( - 243 \beta_{3} - 405 \beta_{2} - 1620 \beta_1 - 2106) q^{72} + ( - 145 \beta_{4} - 1088 \beta_{3} + 176 \beta_{2} - 810 \beta_1 - 23263) q^{73} + (176 \beta_{4} + 764 \beta_{3} - 912 \beta_{2} - 3446 \beta_1 + 13940) q^{74} - 5625 q^{75} + ( - 72 \beta_{4} + 200 \beta_{3} + 1878 \beta_{2} + 11010 \beta_1 + 17906) q^{76} + (121 \beta_{3} - 363 \beta_{2} + 363 \beta_1 - 2904) q^{77} + ( - 27 \beta_{4} - 36 \beta_{3} - 144 \beta_{2} - 4014 \beta_1 + 16767) q^{78} + (39 \beta_{4} - 487 \beta_{3} + 53 \beta_{2} - 6223 \beta_1 + 12799) q^{79} + ( - 125 \beta_{4} - 475 \beta_{3} - 250 \beta_{2} - 2675 \beta_1 - 8800) q^{80} + 6561 q^{81} + (168 \beta_{4} + 1014 \beta_{3} - 40 \beta_{2} + 5430 \beta_1 + 4498) q^{82} + (97 \beta_{4} + 820 \beta_{3} - 348 \beta_{2} - 6170 \beta_1 - 403) q^{83} + (99 \beta_{4} - 90 \beta_{3} - 522 \beta_{2} - 1818 \beta_1 + 9189) q^{84} + (225 \beta_{3} + 925 \beta_{2} + 675 \beta_1 + 1050) q^{85} + ( - 53 \beta_{4} - 550 \beta_{3} - 1560 \beta_{2} - 2260 \beta_1 - 6485) q^{86} + ( - 81 \beta_{4} - 765 \beta_{3} + 711 \beta_{2} - 1521 \beta_1 + 5733) q^{87} + (363 \beta_{3} + 605 \beta_{2} + 2420 \beta_1 + 3146) q^{88} + ( - 160 \beta_{4} + 1638 \beta_{3} - 130 \beta_{2} + 3090 \beta_1 + 13682) q^{89} + 2025 \beta_1 q^{90} + (304 \beta_{4} - 124 \beta_{3} - 1020 \beta_{2} + 716 \beta_1 - 472) q^{91} + (24 \beta_{4} + 1804 \beta_{3} + 2080 \beta_{2} + 22940 \beta_1 + 7476) q^{92} + (216 \beta_{4} + 288 \beta_{2} - 4608 \beta_1 + 11880) q^{93} + ( - 296 \beta_{4} - 48 \beta_{3} + 992 \beta_{2} + 9220 \beta_1 + 67192) q^{94} + (175 \beta_{4} + 75 \beta_{3} - 625 \beta_{2} - 3625 \beta_1 - 16425) q^{95} + (45 \beta_{4} + 801 \beta_{3} + 765 \beta_{2} + 2232 \beta_1 + 43947) q^{96} + (360 \beta_{4} + 374 \beta_{3} - 114 \beta_{2} + 7922 \beta_1 - 13566) q^{97} + ( - 130 \beta_{4} + 648 \beta_{3} + 1312 \beta_{2} + 5079 \beta_1 + 53434) q^{98} - 9801 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - q^{2} - 45 q^{3} + 127 q^{4} - 125 q^{5} + 9 q^{6} + 116 q^{7} - 153 q^{8} + 405 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - q^{2} - 45 q^{3} + 127 q^{4} - 125 q^{5} + 9 q^{6} + 116 q^{7} - 153 q^{8} + 405 q^{9} + 25 q^{10} - 605 q^{11} - 1143 q^{12} - 926 q^{13} + 368 q^{14} + 1125 q^{15} + 1891 q^{16} - 246 q^{17} - 81 q^{18} + 3420 q^{19} - 3175 q^{20} - 1044 q^{21} + 121 q^{22} - 4244 q^{23} + 1377 q^{24} + 3125 q^{25} - 8862 q^{26} - 3645 q^{27} - 4904 q^{28} - 2922 q^{29} - 225 q^{30} - 6112 q^{31} - 24757 q^{32} + 5445 q^{33} + 10866 q^{34} - 2900 q^{35} + 10287 q^{36} + 6654 q^{37} - 45692 q^{38} + 8334 q^{39} + 3825 q^{40} - 14934 q^{41} - 3312 q^{42} + 10804 q^{43} - 15367 q^{44} - 10125 q^{45} - 101500 q^{46} - 41460 q^{47} - 17019 q^{48} - 12099 q^{49} - 625 q^{50} + 2214 q^{51} - 97742 q^{52} - 62398 q^{53} + 729 q^{54} + 15125 q^{55} - 74368 q^{56} - 30780 q^{57} - 27822 q^{58} + 8524 q^{59} + 28575 q^{60} + 59010 q^{61} - 142624 q^{62} + 9396 q^{63} + 13799 q^{64} + 23150 q^{65} - 1089 q^{66} - 15772 q^{67} - 83686 q^{68} + 38196 q^{69} - 9200 q^{70} + 88124 q^{71} - 12393 q^{72} - 118358 q^{73} + 67194 q^{74} - 28125 q^{75} + 100668 q^{76} - 14036 q^{77} + 79758 q^{78} + 57324 q^{79} - 47275 q^{80} + 32805 q^{81} + 29102 q^{82} - 7268 q^{83} + 44136 q^{84} + 6150 q^{85} - 35288 q^{86} + 26298 q^{87} + 18513 q^{88} + 72978 q^{89} + 2025 q^{90} - 1464 q^{91} + 62148 q^{92} + 55008 q^{93} + 344836 q^{94} - 85500 q^{95} + 222813 q^{96} - 59174 q^{97} + 272767 q^{98} - 49005 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 143x^{3} + 71x^{2} + 4216x - 3740 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 3\nu^{2} - 81\nu + 145 ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - \nu - 57 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} - 2\nu^{3} - 109\nu^{2} + 74\nu + 1465 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta _1 + 57 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} + 5\beta_{2} + 84\beta _1 + 26 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{4} + 115\beta_{3} + 10\beta_{2} + 203\beta _1 + 4800 \) 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
10.6119
5.93734
0.898099
−6.98466
−9.46271
−10.6119 −9.00000 80.6130 −25.0000 95.5073 32.7446 −515.877 81.0000 265.298
1.2 −5.93734 −9.00000 3.25206 −25.0000 53.4361 −105.553 170.686 81.0000 148.434
1.3 −0.898099 −9.00000 −31.1934 −25.0000 8.08289 120.732 56.7540 81.0000 22.4525
1.4 6.98466 −9.00000 16.7854 −25.0000 −62.8619 180.375 −106.269 81.0000 −174.616
1.5 9.46271 −9.00000 57.5429 −25.0000 −85.1644 −112.299 241.706 81.0000 −236.568
\(n\): 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.g 5
3.b odd 2 1 495.6.a.i 5
5.b even 2 1 825.6.a.k 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.6.a.g 5 1.a even 1 1 trivial
495.6.a.i 5 3.b odd 2 1
825.6.a.k 5 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} + T_{2}^{4} - 143T_{2}^{3} - 71T_{2}^{2} + 4216T_{2} + 3740 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(165))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + T^{4} - 143 T^{3} - 71 T^{2} + \cdots + 3740 \) Copy content Toggle raw display
$3$ \( (T + 9)^{5} \) Copy content Toggle raw display
$5$ \( (T + 25)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} - 116 T^{4} + \cdots - 8452488256 \) Copy content Toggle raw display
$11$ \( (T + 121)^{5} \) Copy content Toggle raw display
$13$ \( T^{5} + 926 T^{4} + \cdots + 13445413206016 \) Copy content Toggle raw display
$17$ \( T^{5} + \cdots + 475773584554112 \) Copy content Toggle raw display
$19$ \( T^{5} - 3420 T^{4} + \cdots - 25\!\cdots\!92 \) Copy content Toggle raw display
$23$ \( T^{5} + 4244 T^{4} + \cdots + 23\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( T^{5} + 2922 T^{4} + \cdots + 18\!\cdots\!88 \) Copy content Toggle raw display
$31$ \( T^{5} + 6112 T^{4} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{5} - 6654 T^{4} + \cdots - 11\!\cdots\!48 \) Copy content Toggle raw display
$41$ \( T^{5} + 14934 T^{4} + \cdots - 86\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( T^{5} - 10804 T^{4} + \cdots - 69\!\cdots\!08 \) Copy content Toggle raw display
$47$ \( T^{5} + 41460 T^{4} + \cdots - 59\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{5} + 62398 T^{4} + \cdots - 27\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{5} - 8524 T^{4} + \cdots - 17\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{5} - 59010 T^{4} + \cdots + 27\!\cdots\!80 \) Copy content Toggle raw display
$67$ \( T^{5} + 15772 T^{4} + \cdots - 76\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{5} - 88124 T^{4} + \cdots - 45\!\cdots\!12 \) Copy content Toggle raw display
$73$ \( T^{5} + 118358 T^{4} + \cdots + 70\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{5} - 57324 T^{4} + \cdots - 19\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( T^{5} + 7268 T^{4} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{5} - 72978 T^{4} + \cdots + 35\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{5} + 59174 T^{4} + \cdots + 11\!\cdots\!12 \) Copy content Toggle raw display
show more
show less