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Properties

Label 1260.2.bm.c
Level $1260$
Weight $2$
Character orbit 1260.bm
Analytic conductor $10.061$
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] = [1260,2,Mod(109,1260)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1260, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1260.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1260.bm (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: \(10.0611506547\)
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: \( x^{16} - 2 x^{15} + 2 x^{14} - 20 x^{13} - 12 x^{12} + 124 x^{11} - 24 x^{10} + 328 x^{9} + 1132 x^{8} + \cdots + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 420)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{14} - \beta_{12} - \beta_{8}) q^{5} + ( - \beta_{10} - \beta_{8} + \cdots - \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{14} - \beta_{12} - \beta_{8}) q^{5} + ( - \beta_{10} - \beta_{8} + \cdots - \beta_1) q^{7}+ \cdots + (\beta_{14} + \beta_{12} + \cdots + \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{5} + 8 q^{11} + 8 q^{19} + 12 q^{25} - 24 q^{29} - 10 q^{35} - 48 q^{41} + 8 q^{49} - 40 q^{55} + 28 q^{59} - 32 q^{61} + 26 q^{65} + 56 q^{71} - 16 q^{79} + 32 q^{85} + 16 q^{89} - 40 q^{91} - 22 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 2 x^{14} - 20 x^{13} - 12 x^{12} + 124 x^{11} - 24 x^{10} + 328 x^{9} + 1132 x^{8} + \cdots + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 815800393 \nu^{15} + 2673831092 \nu^{14} - 3883997324 \nu^{13} + 18884645174 \nu^{12} + \cdots + 44850913600 ) / 368869396200 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3666156550 \nu^{15} - 10571649399 \nu^{14} + 20258320131 \nu^{13} - 92689454057 \nu^{12} + \cdots + 243050063688 ) / 1106608188600 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1061747 \nu^{15} - 2719244 \nu^{14} + 4300990 \nu^{13} - 24319714 \nu^{12} + 1049764 \nu^{11} + \cdots + 102201712 ) / 299650200 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1481542323 \nu^{15} - 2022904404 \nu^{14} + 4258233612 \nu^{13} - 33231243395 \nu^{12} + \cdots + 801890042704 ) / 368869396200 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2803182100 \nu^{15} - 4790563807 \nu^{14} + 2932533108 \nu^{13} - 52179644676 \nu^{12} + \cdots - 13292202416 ) / 368869396200 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 54930425 \nu^{15} - 180505122 \nu^{14} + 263220555 \nu^{13} - 1267478870 \nu^{12} + \cdots + 1801462888 ) / 4918258616 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4482520007 \nu^{15} + 8828146783 \nu^{14} - 8781280426 \nu^{13} + 89255496976 \nu^{12} + \cdots - 110767423600 ) / 368869396200 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 15307095029 \nu^{15} + 34943564467 \nu^{14} - 36725109876 \nu^{13} + 317157640960 \nu^{12} + \cdots + 177423565208 ) / 1106608188600 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3873653420 \nu^{15} + 7448890281 \nu^{14} - 7238166579 \nu^{13} + 76870493803 \nu^{12} + \cdots - 33709778712 ) / 221321637720 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 6387607 \nu^{15} - 13836961 \nu^{14} + 15494458 \nu^{13} - 132053130 \nu^{12} - 52331570 \nu^{11} + \cdots + 379351936 ) / 299650200 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 9625865075 \nu^{15} + 28739517942 \nu^{14} - 43111499223 \nu^{13} + 221599776281 \nu^{12} + \cdots - 349833032704 ) / 368869396200 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 6481120706 \nu^{15} + 14315113753 \nu^{14} - 15529250754 \nu^{13} + 133043040655 \nu^{12} + \cdots - 304504925368 ) / 221321637720 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 4286532590 \nu^{15} - 9809826442 \nu^{14} + 11172208383 \nu^{13} - 88375637896 \nu^{12} + \cdots + 58776615024 ) / 73773879240 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2215400801 \nu^{15} + 4806096758 \nu^{14} - 5320336184 \nu^{13} + 45392678620 \nu^{12} + \cdots - 27584746888 ) / 36886939620 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{8} + 3\beta_{6} + \beta_{4} + \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{15} + \beta_{14} + 2 \beta_{13} + 2 \beta_{10} - \beta_{9} + 2 \beta_{8} - 2 \beta_{7} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 6 \beta_{15} + 8 \beta_{13} - 20 \beta_{11} - 16 \beta_{9} - 4 \beta_{8} + 8 \beta_{5} - 8 \beta_{4} + \cdots + 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8 \beta_{15} + 8 \beta_{14} - 12 \beta_{13} + 8 \beta_{12} - 26 \beta_{11} + 12 \beta_{10} + \cdots + 66 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 66 \beta_{14} - 42 \beta_{12} + 132 \beta_{10} + 66 \beta_{8} - 158 \beta_{7} + 54 \beta_{4} + \cdots + 66 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 54 \beta_{15} + 54 \beta_{14} + 120 \beta_{13} - 120 \beta_{12} - 270 \beta_{11} + 120 \beta_{10} + \cdots + 270 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 240 \beta_{15} - 572 \beta_{13} - 1344 \beta_{11} - 572 \beta_{9} - 1136 \beta_{8} + 332 \beta_{5} + \cdots + 1136 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1136 \beta_{15} + 1136 \beta_{14} - 2272 \beta_{13} - 564 \beta_{12} + 2272 \beta_{10} + 1136 \beta_{9} + \cdots - 2596 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2820 \beta_{14} - 5092 \beta_{12} + 5092 \beta_{10} - 5432 \beta_{8} - 11860 \beta_{7} + \cdots - 5432 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 5092 \beta_{15} - 5092 \beta_{14} - 10524 \beta_{13} - 5092 \beta_{12} - 24208 \beta_{11} + \cdots - 10524 \beta_{2} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 45912 \beta_{15} - 91824 \beta_{13} + 45912 \beta_{9} - 45912 \beta_{8} - 21048 \beta_{5} + \cdots - 106504 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 50688 \beta_{15} + 50688 \beta_{14} - 96600 \beta_{13} - 96600 \beta_{12} + 222840 \beta_{11} + \cdots - 222840 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 193200 \beta_{14} - 193200 \beta_{12} - 416392 \beta_{10} - 883120 \beta_{8} - 964152 \beta_{6} + \cdots - 883120 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 883120 \beta_{15} - 883120 \beta_{14} - 1766240 \beta_{13} + 466728 \beta_{12} - 1766240 \beta_{10} + \cdots - 2039768 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(757\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-\beta_{11}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
−0.210782 − 0.786650i
−0.389934 + 0.104482i
2.91586 − 0.781303i
0.521577 + 1.94655i
−0.781303 − 2.91586i
0.786650 − 0.210782i
0.104482 + 0.389934i
−1.94655 + 0.521577i
−0.210782 + 0.786650i
−0.389934 − 0.104482i
2.91586 + 0.781303i
0.521577 − 1.94655i
−0.781303 + 2.91586i
0.786650 + 0.210782i
0.104482 − 0.389934i
−1.94655 − 0.521577i
0 0 0 −2.15911 + 0.581605i 0 −0.418148 + 2.61250i 0 0 0
109.2 0 0 0 −1.77792 + 1.35609i 0 2.57243 + 0.618546i 0 0 0
109.3 0 0 0 −1.64412 − 1.51554i 0 −2.55486 + 0.687541i 0 0 0
109.4 0 0 0 −0.779855 + 2.09567i 0 −1.29633 − 2.30641i 0 0 0
109.5 0 0 0 −0.490439 − 2.18162i 0 2.55486 − 0.687541i 0 0 0
109.6 0 0 0 1.58324 − 1.57904i 0 0.418148 − 2.61250i 0 0 0
109.7 0 0 0 2.06337 − 0.861678i 0 −2.57243 − 0.618546i 0 0 0
109.8 0 0 0 2.20483 + 0.372460i 0 1.29633 + 2.30641i 0 0 0
289.1 0 0 0 −2.15911 − 0.581605i 0 −0.418148 − 2.61250i 0 0 0
289.2 0 0 0 −1.77792 − 1.35609i 0 2.57243 − 0.618546i 0 0 0
289.3 0 0 0 −1.64412 + 1.51554i 0 −2.55486 − 0.687541i 0 0 0
289.4 0 0 0 −0.779855 − 2.09567i 0 −1.29633 + 2.30641i 0 0 0
289.5 0 0 0 −0.490439 + 2.18162i 0 2.55486 + 0.687541i 0 0 0
289.6 0 0 0 1.58324 + 1.57904i 0 0.418148 + 2.61250i 0 0 0
289.7 0 0 0 2.06337 + 0.861678i 0 −2.57243 + 0.618546i 0 0 0
289.8 0 0 0 2.20483 − 0.372460i 0 1.29633 − 2.30641i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1260.2.bm.c 16
3.b odd 2 1 420.2.bb.a ✓ 16
5.b even 2 1 inner 1260.2.bm.c 16
7.c even 3 1 inner 1260.2.bm.c 16
12.b even 2 1 1680.2.di.e 16
15.d odd 2 1 420.2.bb.a ✓ 16
15.e even 4 1 2100.2.q.l 8
15.e even 4 1 2100.2.q.m 8
21.c even 2 1 2940.2.bb.i 16
21.g even 6 1 2940.2.k.g 8
21.g even 6 1 2940.2.bb.i 16
21.h odd 6 1 420.2.bb.a ✓ 16
21.h odd 6 1 2940.2.k.f 8
35.j even 6 1 inner 1260.2.bm.c 16
60.h even 2 1 1680.2.di.e 16
84.n even 6 1 1680.2.di.e 16
105.g even 2 1 2940.2.bb.i 16
105.o odd 6 1 420.2.bb.a ✓ 16
105.o odd 6 1 2940.2.k.f 8
105.p even 6 1 2940.2.k.g 8
105.p even 6 1 2940.2.bb.i 16
105.x even 12 1 2100.2.q.l 8
105.x even 12 1 2100.2.q.m 8
420.ba even 6 1 1680.2.di.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.bb.a ✓ 16 3.b odd 2 1
420.2.bb.a ✓ 16 15.d odd 2 1
420.2.bb.a ✓ 16 21.h odd 6 1
420.2.bb.a ✓ 16 105.o odd 6 1
1260.2.bm.c 16 1.a even 1 1 trivial
1260.2.bm.c 16 5.b even 2 1 inner
1260.2.bm.c 16 7.c even 3 1 inner
1260.2.bm.c 16 35.j even 6 1 inner
1680.2.di.e 16 12.b even 2 1
1680.2.di.e 16 60.h even 2 1
1680.2.di.e 16 84.n even 6 1
1680.2.di.e 16 420.ba even 6 1
2100.2.q.l 8 15.e even 4 1
2100.2.q.l 8 105.x even 12 1
2100.2.q.m 8 15.e even 4 1
2100.2.q.m 8 105.x even 12 1
2940.2.k.f 8 21.h odd 6 1
2940.2.k.f 8 105.o odd 6 1
2940.2.k.g 8 21.g even 6 1
2940.2.k.g 8 105.p even 6 1
2940.2.bb.i 16 21.c even 2 1
2940.2.bb.i 16 21.g even 6 1
2940.2.bb.i 16 105.g even 2 1
2940.2.bb.i 16 105.p even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1260, [\chi])\):

\( T_{11}^{8} - 4T_{11}^{7} + 22T_{11}^{6} - 20T_{11}^{5} + 110T_{11}^{4} - 20T_{11}^{3} + 568T_{11}^{2} + 308T_{11} + 196 \) Copy content Toggle raw display
\( T_{17}^{16} - 52 T_{17}^{14} + 2188 T_{17}^{12} - 26632 T_{17}^{10} + 261052 T_{17}^{8} - 51184 T_{17}^{6} + \cdots + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 2 T^{15} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} - 4 T^{14} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{8} - 4 T^{7} + \cdots + 196)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 52 T^{6} + \cdots + 5625)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} - 52 T^{14} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( (T^{8} - 4 T^{7} + \cdots + 25281)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} - 60 T^{14} + \cdots + 8503056 \) Copy content Toggle raw display
$29$ \( (T^{4} + 6 T^{3} + \cdots + 1098)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 102 T^{6} + \cdots + 1212201)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 937890625 \) Copy content Toggle raw display
$41$ \( (T^{4} + 12 T^{3} + 42 T^{2} + \cdots + 6)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + 280 T^{6} + \cdots + 249001)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 143986855936 \) Copy content Toggle raw display
$53$ \( T^{16} - 160 T^{14} + \cdots + 84934656 \) Copy content Toggle raw display
$59$ \( (T^{8} - 14 T^{7} + \cdots + 142611364)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 16 T^{7} + \cdots + 3984016)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} - 252 T^{14} + \cdots + 81 \) Copy content Toggle raw display
$71$ \( (T^{4} - 14 T^{3} + \cdots - 8802)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 12465425870881 \) Copy content Toggle raw display
$79$ \( (T^{8} + 8 T^{7} + \cdots + 1089)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 364 T^{6} + \cdots + 27899524)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 8 T^{7} + \cdots + 36264484)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 196 T^{6} + \cdots + 712336)^{2} \) Copy content Toggle raw display
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