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Properties

Label 65.3.j.a
Level $65$
Weight $3$
Character orbit 65.j
Analytic conductor $1.771$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [65,3,Mod(21,65)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(65, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("65.21");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 65 = 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 65.j (of order \(4\), degree \(2\), minimal)

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: no
Analytic conductor: \(1.77112171834\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 12 x^{13} + 168 x^{12} - 120 x^{11} + 72 x^{10} - 852 x^{9} + 6994 x^{8} - 8856 x^{7} + \cdots + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + \beta_{6} q^{3} + (\beta_{15} + \beta_{12} + \cdots + \beta_1) q^{4}+ \cdots + ( - \beta_{11} + \beta_{7} - \beta_{6} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + \beta_{6} q^{3} + (\beta_{15} + \beta_{12} + \cdots + \beta_1) q^{4}+ \cdots + (11 \beta_{15} - 3 \beta_{13} + \cdots - 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 12 q^{6} + 20 q^{7} - 36 q^{8} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 12 q^{6} + 20 q^{7} - 36 q^{8} + 24 q^{9} + 12 q^{13} - 20 q^{15} + 32 q^{16} - 116 q^{18} - 28 q^{19} - 40 q^{20} - 108 q^{21} + 48 q^{22} + 8 q^{24} + 144 q^{26} - 24 q^{27} + 36 q^{28} - 136 q^{29} - 12 q^{31} - 24 q^{32} + 68 q^{33} + 148 q^{34} + 80 q^{35} - 84 q^{37} - 80 q^{39} + 4 q^{41} - 40 q^{42} + 108 q^{44} + 40 q^{45} - 68 q^{46} - 20 q^{47} + 8 q^{48} - 40 q^{52} + 432 q^{53} + 172 q^{54} + 40 q^{55} + 276 q^{57} + 180 q^{58} - 108 q^{59} + 160 q^{60} + 64 q^{61} - 76 q^{63} - 120 q^{65} - 768 q^{66} - 204 q^{67} - 72 q^{68} - 120 q^{70} - 360 q^{71} + 356 q^{73} - 536 q^{74} + 408 q^{76} - 516 q^{78} + 80 q^{79} + 120 q^{81} - 364 q^{83} - 20 q^{84} - 60 q^{85} - 48 q^{86} - 32 q^{87} + 168 q^{89} + 308 q^{91} - 184 q^{92} - 276 q^{93} + 704 q^{94} + 668 q^{96} + 272 q^{97} - 384 q^{98} - 596 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 12 x^{13} + 168 x^{12} - 120 x^{11} + 72 x^{10} - 852 x^{9} + 6994 x^{8} - 8856 x^{7} + \cdots + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 36\!\cdots\!97 \nu^{15} + \cdots - 30\!\cdots\!34 ) / 32\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 78\!\cdots\!52 \nu^{15} + \cdots - 52\!\cdots\!46 ) / 21\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 81\!\cdots\!64 \nu^{15} + \cdots + 51\!\cdots\!86 ) / 21\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 23\!\cdots\!02 \nu^{15} + \cdots - 40\!\cdots\!97 ) / 57\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 14\!\cdots\!75 \nu^{15} + \cdots - 35\!\cdots\!62 ) / 19\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 18\!\cdots\!35 \nu^{15} + \cdots - 13\!\cdots\!44 ) / 19\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 33\!\cdots\!26 \nu^{15} + \cdots + 23\!\cdots\!02 ) / 28\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10\!\cdots\!10 \nu^{15} + \cdots + 16\!\cdots\!89 ) / 57\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 10\!\cdots\!30 \nu^{15} + \cdots + 17\!\cdots\!11 ) / 57\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 11\!\cdots\!22 \nu^{15} + \cdots + 58\!\cdots\!11 ) / 57\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 13\!\cdots\!45 \nu^{15} + \cdots + 93\!\cdots\!08 ) / 57\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 58\!\cdots\!76 \nu^{15} + \cdots - 93\!\cdots\!97 ) / 19\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 88\!\cdots\!38 \nu^{15} + \cdots + 11\!\cdots\!65 ) / 19\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 47\!\cdots\!51 \nu^{15} + \cdots - 33\!\cdots\!40 ) / 57\!\cdots\!12 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{15} - \beta_{12} - 5\beta_{8} + \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{14} + \beta_{9} + 2\beta_{8} + \beta_{7} - \beta_{4} + \beta_{3} - 10\beta_{2} - \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{14} + \beta_{13} + \beta_{11} - 10\beta_{7} + 12\beta_{6} + \beta_{5} + 2\beta_{4} + 14\beta_{2} + 14\beta _1 - 41 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 14 \beta_{15} - 14 \beta_{13} - 3 \beta_{12} + 8 \beta_{10} - 34 \beta_{8} + 14 \beta_{7} - 3 \beta_{6} + \cdots + 34 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 103 \beta_{15} + 19 \beta_{14} + 19 \beta_{13} + 129 \beta_{12} - 16 \beta_{11} + 8 \beta_{10} + \cdots + 135 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 173 \beta_{15} - 173 \beta_{14} - 72 \beta_{12} + 12 \beta_{11} - 41 \beta_{9} - 463 \beta_{8} + \cdots - 463 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 288 \beta_{14} - 288 \beta_{13} - 204 \beta_{11} - 172 \beta_{10} - 172 \beta_{9} + 1110 \beta_{7} + \cdots + 4033 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2094 \beta_{15} + 2106 \beta_{13} + 1224 \beta_{12} + 42 \beta_{10} + 5916 \beta_{8} - 2094 \beta_{7} + \cdots - 5916 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 12415 \beta_{15} - 3994 \beta_{14} - 3994 \beta_{13} - 15195 \beta_{12} + 2458 \beta_{11} + \cdots - 17343 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 25331 \beta_{15} + 25643 \beta_{14} + 18126 \beta_{12} - 5532 \beta_{11} - 5413 \beta_{9} + 74038 \beta_{8} + \cdots + 74038 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 52889 \beta_{14} + 52889 \beta_{13} + 29231 \beta_{11} + 36792 \beta_{10} + 36792 \beta_{9} + \cdots - 489661 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 307540 \beta_{15} - 312796 \beta_{13} - 250245 \beta_{12} - 109442 \beta_{10} - 919856 \beta_{8} + \cdots + 919856 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 1672193 \beta_{15} + 681953 \beta_{14} + 681953 \beta_{13} + 1951215 \beta_{12} - 347804 \beta_{11} + \cdots + 2383977 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 3747883 \beta_{15} - 3821263 \beta_{14} - 3318912 \beta_{12} + 1174440 \beta_{11} + 1716281 \beta_{9} + \cdots - 11394137 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/65\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(41\)
\(\chi(n)\) \(1\) \(\beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1
2.10114 − 2.10114i
1.55497 − 1.55497i
1.31421 − 1.31421i
0.530829 − 0.530829i
−0.0522614 + 0.0522614i
−0.936959 + 0.936959i
−2.03354 + 2.03354i
−2.47839 + 2.47839i
2.10114 + 2.10114i
1.55497 + 1.55497i
1.31421 + 1.31421i
0.530829 + 0.530829i
−0.0522614 − 0.0522614i
−0.936959 − 0.936959i
−2.03354 − 2.03354i
−2.47839 − 2.47839i
−2.10114 − 2.10114i −5.32747 4.82957i 1.58114 + 1.58114i 11.1938 + 11.1938i 3.75600 − 3.75600i 1.74304 − 1.74304i 19.3819 − 6.64438i
21.2 −1.55497 − 1.55497i −1.67644 0.835865i −1.58114 − 1.58114i 2.60681 + 2.60681i −5.14316 + 5.14316i −4.92014 + 4.92014i −6.18956 4.91725i
21.3 −1.31421 − 1.31421i 2.64218 − 0.545679i 1.58114 + 1.58114i −3.47239 − 3.47239i 6.15655 − 6.15655i −5.97400 + 5.97400i −2.01891 − 4.15591i
21.4 −0.530829 − 0.530829i 5.16385 − 3.43644i −1.58114 − 1.58114i −2.74112 − 2.74112i −4.94074 + 4.94074i −3.94748 + 3.94748i 17.6653 1.67863i
21.5 0.0522614 + 0.0522614i −1.91568 − 3.99454i −1.58114 − 1.58114i −0.100116 − 0.100116i 9.56240 − 9.56240i 0.417805 − 0.417805i −5.33017 − 0.165265i
21.6 0.936959 + 0.936959i 2.29605 − 2.24421i 1.58114 + 1.58114i 2.15130 + 2.15130i −1.94401 + 1.94401i 5.85057 − 5.85057i −3.72817 2.96293i
21.7 2.03354 + 2.03354i 1.59054 4.27056i −1.58114 − 1.58114i 3.23443 + 3.23443i −0.803062 + 0.803062i −0.550188 + 0.550188i −6.47017 − 6.43061i
21.8 2.47839 + 2.47839i −2.77303 8.28488i 1.58114 + 1.58114i −6.87267 − 6.87267i 3.35602 − 3.35602i −10.6196 + 10.6196i −1.31029 7.83737i
31.1 −2.10114 + 2.10114i −5.32747 − 4.82957i 1.58114 − 1.58114i 11.1938 − 11.1938i 3.75600 + 3.75600i 1.74304 + 1.74304i 19.3819 6.64438i
31.2 −1.55497 + 1.55497i −1.67644 − 0.835865i −1.58114 + 1.58114i 2.60681 − 2.60681i −5.14316 − 5.14316i −4.92014 − 4.92014i −6.18956 − 4.91725i
31.3 −1.31421 + 1.31421i 2.64218 0.545679i 1.58114 − 1.58114i −3.47239 + 3.47239i 6.15655 + 6.15655i −5.97400 − 5.97400i −2.01891 4.15591i
31.4 −0.530829 + 0.530829i 5.16385 3.43644i −1.58114 + 1.58114i −2.74112 + 2.74112i −4.94074 − 4.94074i −3.94748 − 3.94748i 17.6653 − 1.67863i
31.5 0.0522614 − 0.0522614i −1.91568 3.99454i −1.58114 + 1.58114i −0.100116 + 0.100116i 9.56240 + 9.56240i 0.417805 + 0.417805i −5.33017 0.165265i
31.6 0.936959 − 0.936959i 2.29605 2.24421i 1.58114 − 1.58114i 2.15130 − 2.15130i −1.94401 − 1.94401i 5.85057 + 5.85057i −3.72817 − 2.96293i
31.7 2.03354 − 2.03354i 1.59054 − 4.27056i −1.58114 + 1.58114i 3.23443 − 3.23443i −0.803062 − 0.803062i −0.550188 − 0.550188i −6.47017 6.43061i
31.8 2.47839 − 2.47839i −2.77303 − 8.28488i 1.58114 − 1.58114i −6.87267 + 6.87267i 3.35602 + 3.35602i −10.6196 − 10.6196i −1.31029 − 7.83737i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 65.3.j.a ✓ 16
3.b odd 2 1 585.3.s.a 16
5.b even 2 1 325.3.j.b 16
5.c odd 4 1 325.3.g.c 16
5.c odd 4 1 325.3.g.d 16
13.d odd 4 1 inner 65.3.j.a ✓ 16
39.f even 4 1 585.3.s.a 16
65.f even 4 1 325.3.g.d 16
65.g odd 4 1 325.3.j.b 16
65.k even 4 1 325.3.g.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.3.j.a ✓ 16 1.a even 1 1 trivial
65.3.j.a ✓ 16 13.d odd 4 1 inner
325.3.g.c 16 5.c odd 4 1
325.3.g.c 16 65.k even 4 1
325.3.g.d 16 5.c odd 4 1
325.3.g.d 16 65.f even 4 1
325.3.j.b 16 5.b even 2 1
325.3.j.b 16 65.g odd 4 1
585.3.s.a 16 3.b odd 2 1
585.3.s.a 16 39.f even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(65, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 12 T^{13} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( (T^{8} - 42 T^{6} + \cdots + 2364)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 25)^{4} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 221863608576 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 1753124291136 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 66\!\cdots\!41 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 37\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{8} + 68 T^{7} + \cdots - 261146435856)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 44\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 50\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 75\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 2684009717184)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 1856471260176)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 37\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 47\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots - 62456990009856)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
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