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Properties

Label 1050.2.s.g
Level $1050$
Weight $2$
Character orbit 1050.s
Analytic conductor $8.384$
Analytic rank $0$
Dimension $12$
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] = [1050,2,Mod(101,1050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1050, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1050.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1050.s (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: \(8.38429221223\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 11 x^{10} - 32 x^{9} + 64 x^{8} - 120 x^{7} + 237 x^{6} - 360 x^{5} + 576 x^{4} + \cdots + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + ( - \beta_{9} + \beta_{3}) q^{3} - \beta_{6} q^{4} + (\beta_{11} - \beta_1) q^{6} + (\beta_{11} - \beta_{9} + \beta_{8} + \cdots - \beta_1) q^{7}+ \cdots + (\beta_{10} - \beta_{9} + \beta_{6} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + ( - \beta_{9} + \beta_{3}) q^{3} - \beta_{6} q^{4} + (\beta_{11} - \beta_1) q^{6} + (\beta_{11} - \beta_{9} + \beta_{8} + \cdots - \beta_1) q^{7}+ \cdots + ( - 3 \beta_{11} - \beta_{10} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{3} + 6 q^{4} - 2 q^{6} - 8 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{3} + 6 q^{4} - 2 q^{6} - 8 q^{7} + 6 q^{9} - 12 q^{11} - 2 q^{12} + 12 q^{14} - 6 q^{16} - 12 q^{17} + 4 q^{18} - 18 q^{21} + 24 q^{23} - 4 q^{24} - 4 q^{26} + 8 q^{27} - 4 q^{28} + 12 q^{31} + 22 q^{33} + 6 q^{36} + 8 q^{37} - 8 q^{38} + 30 q^{39} - 4 q^{41} + 20 q^{42} - 12 q^{44} + 2 q^{46} - 16 q^{47} - 4 q^{48} - 14 q^{49} + 4 q^{51} + 12 q^{52} + 48 q^{53} - 4 q^{54} + 6 q^{56} + 36 q^{57} - 8 q^{58} + 12 q^{59} - 30 q^{61} - 8 q^{62} + 4 q^{63} - 12 q^{64} - 34 q^{66} + 4 q^{67} + 12 q^{68} + 50 q^{69} - 4 q^{72} - 20 q^{77} - 32 q^{78} - 4 q^{79} + 50 q^{81} + 40 q^{83} - 12 q^{84} - 54 q^{86} - 8 q^{87} + 26 q^{89} + 28 q^{91} + 32 q^{93} + 24 q^{94} - 2 q^{96} - 16 q^{98} + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 11 x^{10} - 32 x^{9} + 64 x^{8} - 120 x^{7} + 237 x^{6} - 360 x^{5} + 576 x^{4} + \cdots + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{11} + 4 \nu^{10} - 11 \nu^{9} + 32 \nu^{8} - 64 \nu^{7} + 120 \nu^{6} - 237 \nu^{5} + \cdots + 972 ) / 243 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 4 \nu^{11} + \nu^{10} - 11 \nu^{9} + 35 \nu^{8} - 10 \nu^{7} + 123 \nu^{6} - 183 \nu^{5} + \cdots + 1701 ) / 324 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7 \nu^{11} - 16 \nu^{10} + 74 \nu^{9} - 191 \nu^{8} + 343 \nu^{7} - 810 \nu^{6} + 1290 \nu^{5} + \cdots - 6075 ) / 972 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 28 \nu^{11} - 67 \nu^{10} + 191 \nu^{9} - 545 \nu^{8} + 802 \nu^{7} - 1821 \nu^{6} + 3243 \nu^{5} + \cdots - 10935 ) / 972 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 16 \nu^{11} + 31 \nu^{10} - 98 \nu^{9} + 284 \nu^{8} - 400 \nu^{7} + 969 \nu^{6} - 1668 \nu^{5} + \cdots + 5832 ) / 486 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 25 \nu^{11} - 79 \nu^{10} + 227 \nu^{9} - 578 \nu^{8} + 1027 \nu^{7} - 1971 \nu^{6} + 3495 \nu^{5} + \cdots - 8748 ) / 972 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 15 \nu^{11} + 32 \nu^{10} - 98 \nu^{9} + 289 \nu^{8} - 415 \nu^{7} + 998 \nu^{6} - 1734 \nu^{5} + \cdots + 6885 ) / 324 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5 \nu^{11} - 13 \nu^{10} + 39 \nu^{9} - 110 \nu^{8} + 171 \nu^{7} - 377 \nu^{6} + 675 \nu^{5} + \cdots - 2268 ) / 108 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 26 \nu^{11} - 56 \nu^{10} + 193 \nu^{9} - 538 \nu^{8} + 812 \nu^{7} - 1920 \nu^{6} + 3255 \nu^{5} + \cdots - 12636 ) / 486 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 11 \nu^{11} - 26 \nu^{10} + 76 \nu^{9} - 208 \nu^{8} + 317 \nu^{7} - 708 \nu^{6} + 1230 \nu^{5} + \cdots - 3888 ) / 162 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{10} + \beta_{9} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} + \beta_{9} - 2\beta_{8} + \beta_{7} + \beta_{6} - 3\beta_{5} - \beta_{3} + \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{11} - 2\beta_{9} - \beta_{8} + 2\beta_{7} + \beta_{6} + 5\beta_{5} + \beta_{4} - \beta_{3} + 2\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{11} - 4 \beta_{10} + 5 \beta_{9} + 5 \beta_{8} - \beta_{7} - 3 \beta_{6} + 4 \beta_{5} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 3 \beta_{11} - 5 \beta_{10} + 13 \beta_{9} - 6 \beta_{8} + 9 \beta_{7} - 2 \beta_{6} - 26 \beta_{5} + \cdots - 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 14 \beta_{11} - 11 \beta_{10} - 5 \beta_{9} - 16 \beta_{8} + 23 \beta_{7} - 19 \beta_{6} - 9 \beta_{5} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 10 \beta_{11} - 18 \beta_{10} - 10 \beta_{9} + 13 \beta_{8} - 2 \beta_{7} + 17 \beta_{6} + 43 \beta_{5} + \cdots + 58 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3 \beta_{11} + 22 \beta_{10} + 21 \beta_{9} + 43 \beta_{8} + 13 \beta_{7} + 63 \beta_{6} + 26 \beta_{5} + \cdots + 39 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 81 \beta_{11} - 10 \beta_{10} + 60 \beta_{9} + 18 \beta_{8} + 81 \beta_{7} - 157 \beta_{6} + \cdots - 53 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 16 \beta_{11} - 94 \beta_{10} - 48 \beta_{9} - 74 \beta_{8} + 106 \beta_{7} - 170 \beta_{6} - 222 \beta_{5} + \cdots + 68 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(1\) \(1 + \beta_{6}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
−1.21252 + 1.23685i
1.73138 − 0.0481063i
−0.384890 − 1.68874i
0.312065 − 1.70371i
1.66557 + 0.475255i
−0.111613 + 1.72845i
−1.21252 − 1.23685i
1.73138 + 0.0481063i
−0.384890 + 1.68874i
0.312065 + 1.70371i
1.66557 − 0.475255i
−0.111613 − 1.72845i
−0.866025 + 0.500000i −1.23685 − 1.21252i 0.500000 − 0.866025i 0 1.67740 + 0.431645i 0.311378 − 2.62736i 1.00000i 0.0596020 + 2.99941i 0
101.2 −0.866025 + 0.500000i 0.0481063 + 1.73138i 0.500000 − 0.866025i 0 −0.907353 − 1.47537i −2.27338 + 1.35342i 1.00000i −2.99537 + 0.166581i 0
101.3 −0.866025 + 0.500000i 1.68874 − 0.384890i 0.500000 − 0.866025i 0 −1.27005 + 1.17770i −2.63608 − 0.226058i 1.00000i 2.70372 − 1.29996i 0
101.4 0.866025 − 0.500000i −1.70371 − 0.312065i 0.500000 − 0.866025i 0 −1.63149 + 0.581597i −1.26546 + 2.32349i − 1.00000i 2.80523 + 1.06334i 0
101.5 0.866025 − 0.500000i 0.475255 − 1.66557i 0.500000 − 0.866025i 0 −0.421203 − 1.68006i −0.0551777 − 2.64518i − 1.00000i −2.54826 − 1.58314i 0
101.6 0.866025 − 0.500000i 1.72845 + 0.111613i 0.500000 − 0.866025i 0 1.55269 − 0.767566i 1.91871 + 1.82168i − 1.00000i 2.97509 + 0.385834i 0
551.1 −0.866025 − 0.500000i −1.23685 + 1.21252i 0.500000 + 0.866025i 0 1.67740 − 0.431645i 0.311378 + 2.62736i − 1.00000i 0.0596020 − 2.99941i 0
551.2 −0.866025 − 0.500000i 0.0481063 − 1.73138i 0.500000 + 0.866025i 0 −0.907353 + 1.47537i −2.27338 − 1.35342i − 1.00000i −2.99537 − 0.166581i 0
551.3 −0.866025 − 0.500000i 1.68874 + 0.384890i 0.500000 + 0.866025i 0 −1.27005 − 1.17770i −2.63608 + 0.226058i − 1.00000i 2.70372 + 1.29996i 0
551.4 0.866025 + 0.500000i −1.70371 + 0.312065i 0.500000 + 0.866025i 0 −1.63149 − 0.581597i −1.26546 − 2.32349i 1.00000i 2.80523 − 1.06334i 0
551.5 0.866025 + 0.500000i 0.475255 + 1.66557i 0.500000 + 0.866025i 0 −0.421203 + 1.68006i −0.0551777 + 2.64518i 1.00000i −2.54826 + 1.58314i 0
551.6 0.866025 + 0.500000i 1.72845 − 0.111613i 0.500000 + 0.866025i 0 1.55269 + 0.767566i 1.91871 − 1.82168i 1.00000i 2.97509 − 0.385834i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.2.s.g 12
3.b odd 2 1 1050.2.s.f 12
5.b even 2 1 210.2.r.a ✓ 12
5.c odd 4 1 1050.2.u.f 12
5.c odd 4 1 1050.2.u.g 12
7.d odd 6 1 1050.2.s.f 12
15.d odd 2 1 210.2.r.b yes 12
15.e even 4 1 1050.2.u.e 12
15.e even 4 1 1050.2.u.h 12
21.g even 6 1 inner 1050.2.s.g 12
35.i odd 6 1 210.2.r.b yes 12
35.i odd 6 1 1470.2.b.a 12
35.j even 6 1 1470.2.b.b 12
35.k even 12 1 1050.2.u.e 12
35.k even 12 1 1050.2.u.h 12
105.o odd 6 1 1470.2.b.a 12
105.p even 6 1 210.2.r.a ✓ 12
105.p even 6 1 1470.2.b.b 12
105.w odd 12 1 1050.2.u.f 12
105.w odd 12 1 1050.2.u.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.r.a ✓ 12 5.b even 2 1
210.2.r.a ✓ 12 105.p even 6 1
210.2.r.b yes 12 15.d odd 2 1
210.2.r.b yes 12 35.i odd 6 1
1050.2.s.f 12 3.b odd 2 1
1050.2.s.f 12 7.d odd 6 1
1050.2.s.g 12 1.a even 1 1 trivial
1050.2.s.g 12 21.g even 6 1 inner
1050.2.u.e 12 15.e even 4 1
1050.2.u.e 12 35.k even 12 1
1050.2.u.f 12 5.c odd 4 1
1050.2.u.f 12 105.w odd 12 1
1050.2.u.g 12 5.c odd 4 1
1050.2.u.g 12 105.w odd 12 1
1050.2.u.h 12 15.e even 4 1
1050.2.u.h 12 35.k even 12 1
1470.2.b.a 12 35.i odd 6 1
1470.2.b.a 12 105.o odd 6 1
1470.2.b.b 12 35.j even 6 1
1470.2.b.b 12 105.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 \(S_{2}^{\mathrm{new}}(1050, [\chi])\):

\( T_{11}^{12} + 12 T_{11}^{11} + 38 T_{11}^{10} - 120 T_{11}^{9} - 645 T_{11}^{8} + 3192 T_{11}^{7} + \cdots + 1296 \) Copy content Toggle raw display
\( T_{37}^{12} - 8 T_{37}^{11} + 118 T_{37}^{10} - 352 T_{37}^{9} + 5863 T_{37}^{8} - 19376 T_{37}^{7} + \cdots + 4096 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{12} - 2 T^{11} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 8 T^{11} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{12} + 12 T^{11} + \cdots + 1296 \) Copy content Toggle raw display
$13$ \( T^{12} + 132 T^{10} + \cdots + 8433216 \) Copy content Toggle raw display
$17$ \( T^{12} + 12 T^{11} + \cdots + 82944 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 242985744 \) Copy content Toggle raw display
$23$ \( T^{12} - 24 T^{11} + \cdots + 5103081 \) Copy content Toggle raw display
$29$ \( T^{12} + 190 T^{10} + \cdots + 12194064 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 301925376 \) Copy content Toggle raw display
$37$ \( T^{12} - 8 T^{11} + \cdots + 4096 \) Copy content Toggle raw display
$41$ \( (T^{6} + 2 T^{5} + \cdots + 549)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 127 T^{4} + \cdots - 2972)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 16 T^{11} + \cdots + 15397776 \) Copy content Toggle raw display
$53$ \( T^{12} - 48 T^{11} + \cdots + 11664 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 454201344 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 2158903296 \) Copy content Toggle raw display
$67$ \( T^{12} - 4 T^{11} + \cdots + 331776 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 8680276224 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 89884836864 \) Copy content Toggle raw display
$79$ \( T^{12} + 4 T^{11} + \cdots + 8386816 \) Copy content Toggle raw display
$83$ \( (T^{6} - 20 T^{5} + \cdots - 1152)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 15803506944 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 2449062144 \) Copy content Toggle raw display
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