Properties

Label 165.6.a.h
Level $165$
Weight $6$
Character orbit 165.a
Self dual yes
Analytic conductor $26.463$
Analytic rank $0$
Dimension $7$
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 209x^{5} + 137x^{4} + 12724x^{3} - 1040x^{2} - 218208x - 8784 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3\cdot 5 \)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 9 q^{3} + (\beta_{2} + 28) q^{4} + 25 q^{5} + 9 \beta_1 q^{6} + (\beta_{4} - \beta_{2} + 6 \beta_1 - 1) q^{7} + (\beta_{3} - \beta_{2} + 24 \beta_1 + 18) q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 9 q^{3} + (\beta_{2} + 28) q^{4} + 25 q^{5} + 9 \beta_1 q^{6} + (\beta_{4} - \beta_{2} + 6 \beta_1 - 1) q^{7} + (\beta_{3} - \beta_{2} + 24 \beta_1 + 18) q^{8} + 81 q^{9} + 25 \beta_1 q^{10} + 121 q^{11} + (9 \beta_{2} + 252) q^{12} + (\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 7 \beta_{2} - 22 \beta_1 + 207) q^{13} + ( - 2 \beta_{6} - \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + 8 \beta_{2} - 29 \beta_1 + 370) q^{14} + 225 q^{15} + (\beta_{6} - 3 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 28 \beta_{2} - 17 \beta_1 + 536) q^{16} + (\beta_{6} + 2 \beta_{5} - 4 \beta_{4} + 3 \beta_{3} - 14 \beta_{2} + 56 \beta_1 + 79) q^{17} + 81 \beta_1 q^{18} + ( - 2 \beta_{6} + 3 \beta_{5} - \beta_{4} - 2 \beta_{3} - 9 \beta_{2} - 42 \beta_1 + 372) q^{19} + (25 \beta_{2} + 700) q^{20} + (9 \beta_{4} - 9 \beta_{2} + 54 \beta_1 - 9) q^{21} + 121 \beta_1 q^{22} + (3 \beta_{6} - 4 \beta_{5} + 3 \beta_{4} + \beta_{3} - 5 \beta_{2} + 74 \beta_1 + 66) q^{23} + (9 \beta_{3} - 9 \beta_{2} + 216 \beta_1 + 162) q^{24} + 625 q^{25} + (2 \beta_{6} + 7 \beta_{5} - 23 \beta_{4} + 9 \beta_{3} - 74 \beta_{2} + \cdots - 1162) q^{26}+ \cdots + 9801 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + q^{2} + 63 q^{3} + 195 q^{4} + 175 q^{5} + 9 q^{6} + 153 q^{8} + 567 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + q^{2} + 63 q^{3} + 195 q^{4} + 175 q^{5} + 9 q^{6} + 153 q^{8} + 567 q^{9} + 25 q^{10} + 847 q^{11} + 1755 q^{12} + 1418 q^{13} + 2548 q^{14} + 1575 q^{15} + 3699 q^{16} + 630 q^{17} + 81 q^{18} + 2572 q^{19} + 4875 q^{20} + 121 q^{22} + 536 q^{23} + 1377 q^{24} + 4375 q^{25} - 7626 q^{26} + 5103 q^{27} - 11368 q^{28} - 1038 q^{29} + 225 q^{30} + 1872 q^{31} - 7523 q^{32} + 7623 q^{33} + 20790 q^{34} + 15795 q^{36} + 24298 q^{37} - 18952 q^{38} + 12762 q^{39} + 3825 q^{40} - 17658 q^{41} + 22932 q^{42} + 7244 q^{43} + 23595 q^{44} + 14175 q^{45} + 31016 q^{46} + 34560 q^{47} + 33291 q^{48} + 78735 q^{49} + 625 q^{50} + 5670 q^{51} + 110222 q^{52} - 10214 q^{53} + 729 q^{54} + 21175 q^{55} + 81124 q^{56} + 23148 q^{57} - 5718 q^{58} + 94676 q^{59} + 43875 q^{60} + 69538 q^{61} - 4208 q^{62} + 112339 q^{64} + 35450 q^{65} + 1089 q^{66} + 64908 q^{67} - 136010 q^{68} + 4824 q^{69} + 63700 q^{70} + 61816 q^{71} + 12393 q^{72} - 11890 q^{73} - 124050 q^{74} + 39375 q^{75} - 47216 q^{76} - 68634 q^{78} + 18928 q^{79} + 92475 q^{80} + 45927 q^{81} + 36398 q^{82} + 17492 q^{83} - 102312 q^{84} + 15750 q^{85} - 216688 q^{86} - 9342 q^{87} + 18513 q^{88} + 25302 q^{89} + 2025 q^{90} + 3392 q^{91} - 27408 q^{92} + 16848 q^{93} - 30800 q^{94} + 64300 q^{95} - 67707 q^{96} - 172546 q^{97} - 615271 q^{98} + 68607 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 209x^{5} + 137x^{4} + 12724x^{3} - 1040x^{2} - 218208x - 8784 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + \nu^{2} - 88\nu - 78 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -15\nu^{6} + 71\nu^{5} + 2323\nu^{4} - 11587\nu^{3} - 70328\nu^{2} + 346212\nu - 34776 ) / 2344 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7\nu^{6} + 45\nu^{5} - 1631\nu^{4} - 7641\nu^{3} + 104468\nu^{2} + 291764\nu - 1252344 ) / 2344 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -9\nu^{6} + 277\nu^{5} + 2097\nu^{4} - 41409\nu^{3} - 113220\nu^{2} + 1195020\nu + 889544 ) / 2344 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 60 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - \beta_{2} + 88\beta _1 + 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} - 3\beta_{5} - 2\beta_{4} - 2\beta_{3} + 124\beta_{2} - 17\beta _1 + 5272 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7\beta_{6} + 9\beta_{5} + 153\beta_{3} - 216\beta_{2} + 8775\beta _1 + 1126 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 188\beta_{6} - 422\beta_{5} - 466\beta_{4} - 358\beta_{3} + 14265\beta_{2} - 5994\beta _1 + 524252 \) 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
−10.8333
−6.96291
−5.88025
−0.0402667
5.17374
9.12102
10.4220
−10.8333 9.00000 85.3606 25.0000 −97.4998 −188.476 −578.072 81.0000 −270.833
1.2 −6.96291 9.00000 16.4822 25.0000 −62.6662 244.038 108.049 81.0000 −174.073
1.3 −5.88025 9.00000 2.57733 25.0000 −52.9222 −219.194 173.013 81.0000 −147.006
1.4 −0.0402667 9.00000 −31.9984 25.0000 −0.362400 37.9248 2.57700 81.0000 −1.00667
1.5 5.17374 9.00000 −5.23241 25.0000 46.5637 24.5405 −192.631 81.0000 129.343
1.6 9.12102 9.00000 51.1931 25.0000 82.0892 202.409 175.061 81.0000 228.026
1.7 10.4220 9.00000 76.6176 25.0000 93.7978 −101.242 465.003 81.0000 260.549
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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.h 7
3.b odd 2 1 495.6.a.n 7
5.b even 2 1 825.6.a.n 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.6.a.h 7 1.a even 1 1 trivial
495.6.a.n 7 3.b odd 2 1
825.6.a.n 7 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} - T_{2}^{6} - 209T_{2}^{5} + 137T_{2}^{4} + 12724T_{2}^{3} - 1040T_{2}^{2} - 218208T_{2} - 8784 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(165))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} - T^{6} - 209 T^{5} + 137 T^{4} + \cdots - 8784 \) Copy content Toggle raw display
$3$ \( (T - 9)^{7} \) Copy content Toggle raw display
$5$ \( (T - 25)^{7} \) Copy content Toggle raw display
$7$ \( T^{7} + \cdots + 192283411073024 \) Copy content Toggle raw display
$11$ \( (T - 121)^{7} \) Copy content Toggle raw display
$13$ \( T^{7} - 1418 T^{6} + \cdots + 62\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{7} - 630 T^{6} + \cdots - 73\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{7} - 2572 T^{6} + \cdots - 20\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{7} - 536 T^{6} + \cdots - 52\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{7} + 1038 T^{6} + \cdots - 26\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{7} - 1872 T^{6} + \cdots - 15\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( T^{7} - 24298 T^{6} + \cdots - 38\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{7} + 17658 T^{6} + \cdots + 71\!\cdots\!68 \) Copy content Toggle raw display
$43$ \( T^{7} - 7244 T^{6} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{7} - 34560 T^{6} + \cdots + 24\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( T^{7} + 10214 T^{6} + \cdots + 47\!\cdots\!08 \) Copy content Toggle raw display
$59$ \( T^{7} - 94676 T^{6} + \cdots - 29\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{7} - 69538 T^{6} + \cdots + 55\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{7} - 64908 T^{6} + \cdots - 12\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{7} - 61816 T^{6} + \cdots + 83\!\cdots\!40 \) Copy content Toggle raw display
$73$ \( T^{7} + 11890 T^{6} + \cdots + 45\!\cdots\!28 \) Copy content Toggle raw display
$79$ \( T^{7} - 18928 T^{6} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{7} - 17492 T^{6} + \cdots - 39\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{7} - 25302 T^{6} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{7} + 172546 T^{6} + \cdots - 29\!\cdots\!68 \) Copy content Toggle raw display
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