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Properties

Label 260.2.z.a
Level $260$
Weight $2$
Character orbit 260.z
Analytic conductor $2.076$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [260,2,Mod(49,260)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(260, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("260.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 260 = 2^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 260.z (of order \(6\), 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: \(2.07611045255\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 7x^{14} + 21x^{12} - 22x^{10} - 26x^{8} - 198x^{6} + 1701x^{4} - 5103x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + \beta_{14} q^{5} - \beta_{8} q^{7} + (\beta_{15} - \beta_{14} + \beta_{12} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + \beta_{14} q^{5} - \beta_{8} q^{7} + (\beta_{15} - \beta_{14} + \beta_{12} + \cdots + 1) q^{9}+ \cdots + ( - 2 \beta_{15} + 4 \beta_{14} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 10 q^{9} - 6 q^{11} + 6 q^{15} - 18 q^{19} - 14 q^{25} + 12 q^{29} + 18 q^{39} - 48 q^{41} + 45 q^{45} - 6 q^{49} + 44 q^{51} + 2 q^{55} - 30 q^{59} - 28 q^{61} - 15 q^{65} - 34 q^{69} - 18 q^{71} - 42 q^{75} - 16 q^{79} - 44 q^{81} - 45 q^{85} + 30 q^{89} - 10 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 7x^{14} + 21x^{12} - 22x^{10} - 26x^{8} - 198x^{6} + 1701x^{4} - 5103x^{2} + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 5\nu^{14} - 2\nu^{12} + 63\nu^{10} - 11\nu^{8} - 289\nu^{6} + 69\nu^{4} + 1674\nu^{2} - 24057 ) / 15552 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{15} - 46\nu^{13} + 57\nu^{11} + 515\nu^{9} + 697\nu^{7} - 7677\nu^{5} + 23166\nu^{3} - 28431\nu ) / 69984 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{15} - 10\nu^{13} + 27\nu^{11} - 7\nu^{9} - 5\nu^{7} - 519\nu^{5} + 2874\nu^{3} - 4509\nu ) / 2592 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{15} + 7\nu^{13} - 21\nu^{11} + 22\nu^{9} + 26\nu^{7} + 198\nu^{5} - 1701\nu^{3} + 7290\nu ) / 2187 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 43 \nu^{15} - 15 \nu^{14} - 142 \nu^{13} + 438 \nu^{12} + 465 \nu^{11} - 1917 \nu^{10} + \cdots + 433755 ) / 93312 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 43 \nu^{15} + 15 \nu^{14} - 142 \nu^{13} - 438 \nu^{12} + 465 \nu^{11} + 1917 \nu^{10} + \cdots - 433755 ) / 93312 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 41 \nu^{15} - 79 \nu^{14} + 218 \nu^{13} + 22 \nu^{12} - 459 \nu^{11} + 195 \nu^{10} + \cdots - 9477 ) / 93312 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -5\nu^{15} + 8\nu^{13} - 78\nu^{11} - 52\nu^{9} + 238\nu^{7} + 882\nu^{5} - 4779\nu^{3} + 13122\nu ) / 8748 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 179 \nu^{15} - 237 \nu^{14} - 610 \nu^{13} + 66 \nu^{12} + 1911 \nu^{11} + 585 \nu^{10} + \cdots - 28431 ) / 279936 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 47 \nu^{15} - 229 \nu^{14} - 38 \nu^{13} + 946 \nu^{12} - 51 \nu^{11} - 1263 \nu^{10} + \cdots + 373977 ) / 93312 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 47 \nu^{15} + 229 \nu^{14} - 38 \nu^{13} - 946 \nu^{12} - 51 \nu^{11} + 1263 \nu^{10} + \cdots - 373977 ) / 93312 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 107 \nu^{15} + 399 \nu^{14} + 686 \nu^{13} - 1902 \nu^{12} - 1401 \nu^{11} + 3357 \nu^{10} + \cdots - 789507 ) / 139968 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 47 \nu^{15} + 357 \nu^{14} - 38 \nu^{13} - 1842 \nu^{12} - 51 \nu^{11} + 3951 \nu^{10} + \cdots - 840537 ) / 93312 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 107 \nu^{15} - 399 \nu^{14} + 686 \nu^{13} + 1902 \nu^{12} - 1401 \nu^{11} - 3357 \nu^{10} + \cdots + 789507 ) / 139968 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 47 \nu^{15} - 511 \nu^{14} + 38 \nu^{13} + 2038 \nu^{12} + 51 \nu^{11} - 4413 \nu^{10} + \cdots + 1040283 ) / 93312 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{13} - \beta_{10} - \beta_{9} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{2} - 2\beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2 \beta_{15} - 6 \beta_{14} - 3 \beta_{13} + 6 \beta_{12} - 4 \beta_{11} + 3 \beta_{10} - 3 \beta_{9} + \cdots - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -4\beta_{11} - 4\beta_{10} + 6\beta_{8} + 8\beta_{6} + 8\beta_{5} - 5\beta_{4} - 7\beta_{3} - 7\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 4 \beta_{15} - 2 \beta_{14} - \beta_{13} + 2 \beta_{12} - 8 \beta_{11} + 3 \beta_{10} - \beta_{9} + \cdots - 14 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 12 \beta_{14} + 12 \beta_{12} + 12 \beta_{11} + 12 \beta_{10} + 8 \beta_{9} + 6 \beta_{8} + \cdots + 11 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 10 \beta_{15} - 22 \beta_{14} - 17 \beta_{13} + 22 \beta_{12} + 22 \beta_{11} + 51 \beta_{10} + \cdots + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 6 \beta_{14} - 6 \beta_{12} - \beta_{11} - \beta_{10} + 15 \beta_{9} - 11 \beta_{8} + \cdots + 51 \beta_{2} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 56 \beta_{15} + 32 \beta_{14} + 47 \beta_{13} - 32 \beta_{12} - 30 \beta_{11} + 73 \beta_{10} + \cdots + 443 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 96 \beta_{14} + 96 \beta_{12} + 80 \beta_{11} + 80 \beta_{10} + 152 \beta_{9} - 312 \beta_{8} + \cdots - 85 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 72 \beta_{15} + 140 \beta_{14} - 60 \beta_{13} - 140 \beta_{12} + 84 \beta_{11} - 112 \beta_{10} + \cdots + 445 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 216 \beta_{14} + 216 \beta_{12} - 120 \beta_{11} - 120 \beta_{10} - 272 \beta_{9} - 552 \beta_{8} + \cdots + 239 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 136 \beta_{15} - 488 \beta_{14} - 417 \beta_{13} + 488 \beta_{12} - 376 \beta_{11} - 257 \beta_{10} + \cdots + 2473 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 228 \beta_{14} - 228 \beta_{12} + 179 \beta_{11} + 179 \beta_{10} - 413 \beta_{9} + \cdots - 1344 \beta_{2} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/260\mathbb{Z}\right)^\times\).

\(n\) \(41\) \(131\) \(157\)
\(\chi(n)\) \(1 + \beta_{1}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.517063 − 1.65307i
1.42836 − 0.979681i
−1.56631 − 0.739379i
1.72890 − 0.104392i
−1.72890 + 0.104392i
1.56631 + 0.739379i
−1.42836 + 0.979681i
0.517063 + 1.65307i
−0.517063 + 1.65307i
1.42836 + 0.979681i
−1.56631 + 0.739379i
1.72890 + 0.104392i
−1.72890 − 0.104392i
1.56631 − 0.739379i
−1.42836 − 0.979681i
0.517063 − 1.65307i
0 −2.86320 + 1.65307i 0 0.877236 + 2.05681i 0 −0.517063 + 0.895580i 0 3.96529 − 6.86809i 0
49.2 0 −1.69686 + 0.979681i 0 −2.16188 − 0.571200i 0 1.42836 − 2.47400i 0 0.419550 − 0.726682i 0
49.3 0 −1.28064 + 0.739379i 0 −0.494086 − 2.18080i 0 −1.56631 + 2.71292i 0 −0.406637 + 0.704315i 0
49.4 0 −0.180812 + 0.104392i 0 1.60081 + 1.56122i 0 1.72890 − 2.99455i 0 −1.47820 + 2.56033i 0
49.5 0 0.180812 − 0.104392i 0 −1.60081 + 1.56122i 0 −1.72890 + 2.99455i 0 −1.47820 + 2.56033i 0
49.6 0 1.28064 − 0.739379i 0 0.494086 − 2.18080i 0 1.56631 − 2.71292i 0 −0.406637 + 0.704315i 0
49.7 0 1.69686 − 0.979681i 0 2.16188 − 0.571200i 0 −1.42836 + 2.47400i 0 0.419550 − 0.726682i 0
49.8 0 2.86320 − 1.65307i 0 −0.877236 + 2.05681i 0 0.517063 − 0.895580i 0 3.96529 − 6.86809i 0
69.1 0 −2.86320 − 1.65307i 0 0.877236 − 2.05681i 0 −0.517063 − 0.895580i 0 3.96529 + 6.86809i 0
69.2 0 −1.69686 − 0.979681i 0 −2.16188 + 0.571200i 0 1.42836 + 2.47400i 0 0.419550 + 0.726682i 0
69.3 0 −1.28064 − 0.739379i 0 −0.494086 + 2.18080i 0 −1.56631 − 2.71292i 0 −0.406637 − 0.704315i 0
69.4 0 −0.180812 − 0.104392i 0 1.60081 − 1.56122i 0 1.72890 + 2.99455i 0 −1.47820 − 2.56033i 0
69.5 0 0.180812 + 0.104392i 0 −1.60081 − 1.56122i 0 −1.72890 − 2.99455i 0 −1.47820 − 2.56033i 0
69.6 0 1.28064 + 0.739379i 0 0.494086 + 2.18080i 0 1.56631 + 2.71292i 0 −0.406637 − 0.704315i 0
69.7 0 1.69686 + 0.979681i 0 2.16188 + 0.571200i 0 −1.42836 − 2.47400i 0 0.419550 + 0.726682i 0
69.8 0 2.86320 + 1.65307i 0 −0.877236 − 2.05681i 0 0.517063 + 0.895580i 0 3.96529 + 6.86809i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.e even 6 1 inner
65.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 260.2.z.a ✓ 16
3.b odd 2 1 2340.2.cr.a 16
4.b odd 2 1 1040.2.df.d 16
5.b even 2 1 inner 260.2.z.a ✓ 16
5.c odd 4 2 1300.2.y.e 16
13.c even 3 1 3380.2.d.d 16
13.e even 6 1 inner 260.2.z.a ✓ 16
13.e even 6 1 3380.2.d.d 16
13.f odd 12 2 3380.2.c.e 16
15.d odd 2 1 2340.2.cr.a 16
20.d odd 2 1 1040.2.df.d 16
39.h odd 6 1 2340.2.cr.a 16
52.i odd 6 1 1040.2.df.d 16
65.l even 6 1 inner 260.2.z.a ✓ 16
65.l even 6 1 3380.2.d.d 16
65.n even 6 1 3380.2.d.d 16
65.r odd 12 2 1300.2.y.e 16
65.s odd 12 2 3380.2.c.e 16
195.y odd 6 1 2340.2.cr.a 16
260.w odd 6 1 1040.2.df.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.z.a ✓ 16 1.a even 1 1 trivial
260.2.z.a ✓ 16 5.b even 2 1 inner
260.2.z.a ✓ 16 13.e even 6 1 inner
260.2.z.a ✓ 16 65.l even 6 1 inner
1040.2.df.d 16 4.b odd 2 1
1040.2.df.d 16 20.d odd 2 1
1040.2.df.d 16 52.i odd 6 1
1040.2.df.d 16 260.w odd 6 1
1300.2.y.e 16 5.c odd 4 2
1300.2.y.e 16 65.r odd 12 2
2340.2.cr.a 16 3.b odd 2 1
2340.2.cr.a 16 15.d odd 2 1
2340.2.cr.a 16 39.h odd 6 1
2340.2.cr.a 16 195.y odd 6 1
3380.2.c.e 16 13.f odd 12 2
3380.2.c.e 16 65.s odd 12 2
3380.2.d.d 16 13.c even 3 1
3380.2.d.d 16 13.e even 6 1
3380.2.d.d 16 65.l even 6 1
3380.2.d.d 16 65.n even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(260, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 17 T^{14} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{16} + 7 T^{14} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} + 31 T^{14} + \cdots + 1048576 \) Copy content Toggle raw display
$11$ \( (T^{8} + 3 T^{7} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 2750058481 \) Copy content Toggle raw display
$19$ \( (T^{8} + 9 T^{7} + \cdots + 412164)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} - 89 T^{14} + \cdots + 1336336 \) Copy content Toggle raw display
$29$ \( (T^{8} - 6 T^{7} + \cdots + 154449)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 108 T^{6} + \cdots + 2304)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 85662167761 \) Copy content Toggle raw display
$41$ \( (T^{8} + 24 T^{7} + \cdots + 1771561)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 287107358976 \) Copy content Toggle raw display
$47$ \( (T^{8} - 132 T^{6} + \cdots + 147456)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 191 T^{6} + \cdots + 30976)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 15 T^{7} + \cdots + 1936)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 14 T^{7} + \cdots + 1338649)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 722642807056 \) Copy content Toggle raw display
$71$ \( (T^{8} + 9 T^{7} + \cdots + 3104644)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 133 T^{6} + \cdots + 861184)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 4 T^{3} + \cdots + 4096)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} - 228 T^{6} + \cdots + 36864)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 15 T^{7} + \cdots + 1936)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 9124290174736 \) Copy content Toggle raw display
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