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Properties

Label 1050.2.j.d
Level $1050$
Weight $2$
Character orbit 1050.j
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(407,1050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1050, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1050.407");
 
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.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: \(8.38429221223\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 16x^{10} + 86x^{8} + 196x^{6} + 185x^{4} + 60x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 210)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_{4} q^{3} - \beta_{7} q^{4} - \beta_{5} q^{6} + \beta_1 q^{7} - \beta_1 q^{8} + (\beta_{11} - \beta_{9} + \cdots + 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + \beta_{4} q^{3} - \beta_{7} q^{4} - \beta_{5} q^{6} + \beta_1 q^{7} - \beta_1 q^{8} + (\beta_{11} - \beta_{9} + \cdots + 2 \beta_1) q^{9}+ \cdots + ( - \beta_{11} - \beta_{9} - 2 \beta_{8} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{3} + 4 q^{12} + 12 q^{14} - 12 q^{16} + 28 q^{17} + 8 q^{18} - 4 q^{21} - 4 q^{22} - 24 q^{23} + 4 q^{24} - 28 q^{27} - 8 q^{29} - 8 q^{31} + 36 q^{33} + 4 q^{36} + 20 q^{37} - 4 q^{38} + 40 q^{39} - 4 q^{42} - 8 q^{43} - 8 q^{44} + 8 q^{46} + 16 q^{47} + 4 q^{48} + 8 q^{51} - 24 q^{53} + 4 q^{54} - 44 q^{57} + 8 q^{58} - 32 q^{59} + 28 q^{62} - 8 q^{66} - 28 q^{68} + 32 q^{69} + 24 q^{73} - 8 q^{74} + 4 q^{77} + 8 q^{78} - 36 q^{81} - 32 q^{82} - 24 q^{83} + 16 q^{87} - 4 q^{88} - 48 q^{89} + 24 q^{91} - 24 q^{92} + 20 q^{93} - 8 q^{97} + 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 16x^{10} + 86x^{8} + 196x^{6} + 185x^{4} + 60x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{8} + \nu^{7} + 13\nu^{6} + 12\nu^{5} + 47\nu^{4} + 35\nu^{3} + 53\nu^{2} + 22\nu + 10 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{8} - \nu^{7} + 13\nu^{6} - 12\nu^{5} + 47\nu^{4} - 35\nu^{3} + 53\nu^{2} - 22\nu + 10 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{11} + \nu^{10} - 15 \nu^{9} + 15 \nu^{8} - 73 \nu^{7} + 71 \nu^{6} - 151 \nu^{5} + 127 \nu^{4} + \cdots + 24 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} + \nu^{10} + 15 \nu^{9} + 15 \nu^{8} + 73 \nu^{7} + 71 \nu^{6} + 151 \nu^{5} + 127 \nu^{4} + \cdots + 24 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3 \nu^{11} + \nu^{10} - 45 \nu^{9} + 15 \nu^{8} - 217 \nu^{7} + 69 \nu^{6} - 421 \nu^{5} + 103 \nu^{4} + \cdots - 28 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3 \nu^{11} + \nu^{10} + 45 \nu^{9} + 15 \nu^{8} + 217 \nu^{7} + 69 \nu^{6} + 421 \nu^{5} + 103 \nu^{4} + \cdots - 28 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{11} + 31\nu^{9} + 157\nu^{7} + 321\nu^{5} + 243\nu^{3} + 46\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3 \nu^{11} - 3 \nu^{10} + 49 \nu^{9} - 43 \nu^{8} + 271 \nu^{7} - 191 \nu^{6} + 633 \nu^{5} - 327 \nu^{4} + \cdots - 12 ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3 \nu^{11} + 3 \nu^{10} - 47 \nu^{9} + 47 \nu^{8} - 243 \nu^{7} + 241 \nu^{6} - 515 \nu^{5} + \cdots + 48 ) / 16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3 \nu^{11} + 3 \nu^{10} + 49 \nu^{9} + 43 \nu^{8} + 271 \nu^{7} + 191 \nu^{6} + 633 \nu^{5} + 327 \nu^{4} + \cdots + 12 ) / 16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3 \nu^{11} + 3 \nu^{10} + 47 \nu^{9} + 47 \nu^{8} + 243 \nu^{7} + 241 \nu^{6} + 515 \nu^{5} + 491 \nu^{4} + \cdots + 48 ) / 16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} + 2 \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \cdots + 2 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - 2\beta_{2} - 2\beta _1 - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{11} + 5 \beta_{10} + 3 \beta_{9} + 5 \beta_{8} - 14 \beta_{7} + 5 \beta_{6} + \cdots - 12 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 13 \beta_{11} + 11 \beta_{10} - 13 \beta_{9} - 11 \beta_{8} + 9 \beta_{6} + 9 \beta_{5} - 3 \beta_{4} + \cdots + 42 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 11 \beta_{11} - 33 \beta_{10} - 11 \beta_{9} - 33 \beta_{8} + 110 \beta_{7} - 35 \beta_{6} + \cdots + 86 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 117 \beta_{11} - 93 \beta_{10} + 117 \beta_{9} + 93 \beta_{8} - 77 \beta_{6} - 77 \beta_{5} + 5 \beta_{4} + \cdots - 322 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 49 \beta_{11} + 243 \beta_{10} + 49 \beta_{9} + 243 \beta_{8} - 874 \beta_{7} + 267 \beta_{6} + \cdots - 648 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 963 \beta_{11} + 745 \beta_{10} - 963 \beta_{9} - 745 \beta_{8} + 631 \beta_{6} + 631 \beta_{5} + \cdots + 2510 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 269 \beta_{11} - 1863 \beta_{10} - 269 \beta_{9} - 1863 \beta_{8} + 6930 \beta_{7} - 2089 \beta_{6} + \cdots + 5006 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 7707 \beta_{11} - 5887 \beta_{10} + 7707 \beta_{9} + 5887 \beta_{8} - 5059 \beta_{6} - 5059 \beta_{5} + \cdots - 19678 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1747 \beta_{11} + 14513 \beta_{10} + 1747 \beta_{9} + 14513 \beta_{8} - 54798 \beta_{7} + \cdots - 39116 \beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(\beta_{7}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
407.1
− 1.85804i
− 0.678294i
1.12212i
2.80721i
− 1.69093i
0.297931i
1.85804i
0.678294i
− 1.12212i
− 2.80721i
1.69093i
− 0.297931i
−0.707107 + 0.707107i −1.65519 + 0.510256i − 1.00000i 0 0.809587 − 1.53120i −0.707107 − 0.707107i 0.707107 + 0.707107i 2.47928 − 1.68914i 0
407.2 −0.707107 + 0.707107i 0.430811 − 1.67762i − 1.00000i 0 0.881625 + 1.49088i −0.707107 − 0.707107i 0.707107 + 0.707107i −2.62880 − 1.44547i 0
407.3 −0.707107 + 0.707107i 0.931481 + 1.46025i − 1.00000i 0 −1.69121 − 0.373900i −0.707107 − 0.707107i 0.707107 + 0.707107i −1.26469 + 2.72040i 0
407.4 0.707107 − 0.707107i −1.53661 − 0.799269i − 1.00000i 0 −1.65172 + 0.521378i 0.707107 + 0.707107i −0.707107 − 0.707107i 1.72234 + 2.45633i 0
407.5 0.707107 − 0.707107i −1.27508 + 1.17225i − 1.00000i 0 −0.0727133 + 1.73052i 0.707107 + 0.707107i −0.707107 − 0.707107i 0.251664 − 2.98943i 0
407.6 0.707107 − 0.707107i 1.10458 + 1.33413i − 1.00000i 0 1.72443 + 0.162311i 0.707107 + 0.707107i −0.707107 − 0.707107i −0.559788 + 2.94731i 0
743.1 −0.707107 − 0.707107i −1.65519 − 0.510256i 1.00000i 0 0.809587 + 1.53120i −0.707107 + 0.707107i 0.707107 − 0.707107i 2.47928 + 1.68914i 0
743.2 −0.707107 − 0.707107i 0.430811 + 1.67762i 1.00000i 0 0.881625 − 1.49088i −0.707107 + 0.707107i 0.707107 − 0.707107i −2.62880 + 1.44547i 0
743.3 −0.707107 − 0.707107i 0.931481 − 1.46025i 1.00000i 0 −1.69121 + 0.373900i −0.707107 + 0.707107i 0.707107 − 0.707107i −1.26469 − 2.72040i 0
743.4 0.707107 + 0.707107i −1.53661 + 0.799269i 1.00000i 0 −1.65172 − 0.521378i 0.707107 − 0.707107i −0.707107 + 0.707107i 1.72234 − 2.45633i 0
743.5 0.707107 + 0.707107i −1.27508 − 1.17225i 1.00000i 0 −0.0727133 − 1.73052i 0.707107 − 0.707107i −0.707107 + 0.707107i 0.251664 + 2.98943i 0
743.6 0.707107 + 0.707107i 1.10458 − 1.33413i 1.00000i 0 1.72443 − 0.162311i 0.707107 − 0.707107i −0.707107 + 0.707107i −0.559788 − 2.94731i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 407.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.2.j.d 12
3.b odd 2 1 1050.2.j.c 12
5.b even 2 1 210.2.j.b yes 12
5.c odd 4 1 210.2.j.a ✓ 12
5.c odd 4 1 1050.2.j.c 12
15.d odd 2 1 210.2.j.a ✓ 12
15.e even 4 1 210.2.j.b yes 12
15.e even 4 1 inner 1050.2.j.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.j.a ✓ 12 5.c odd 4 1
210.2.j.a ✓ 12 15.d odd 2 1
210.2.j.b yes 12 5.b even 2 1
210.2.j.b yes 12 15.e even 4 1
1050.2.j.c 12 3.b odd 2 1
1050.2.j.c 12 5.c odd 4 1
1050.2.j.d 12 1.a even 1 1 trivial
1050.2.j.d 12 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1050, [\chi])\):

\( T_{11}^{12} + 76T_{11}^{10} + 1732T_{11}^{8} + 12288T_{11}^{6} + 29440T_{11}^{4} + 20480T_{11}^{2} + 4096 \) Copy content Toggle raw display
\( T_{17}^{12} - 28 T_{17}^{11} + 392 T_{17}^{10} - 3240 T_{17}^{9} + 16868 T_{17}^{8} - 52032 T_{17}^{7} + \cdots + 10863616 \) Copy content Toggle raw display
\( T_{29}^{6} + 4T_{29}^{5} - 60T_{29}^{4} - 32T_{29}^{3} + 976T_{29}^{2} - 1600T_{29} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{12} + 4 T^{11} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{4} + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{12} + 76 T^{10} + \cdots + 4096 \) Copy content Toggle raw display
$13$ \( T^{12} + 32 T^{9} + \cdots + 61504 \) Copy content Toggle raw display
$17$ \( T^{12} - 28 T^{11} + \cdots + 10863616 \) Copy content Toggle raw display
$19$ \( T^{12} + 124 T^{10} + \cdots + 104976 \) Copy content Toggle raw display
$23$ \( T^{12} + 24 T^{11} + \cdots + 1364224 \) Copy content Toggle raw display
$29$ \( (T^{6} + 4 T^{5} - 60 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 4 T^{5} + \cdots - 23616)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} - 20 T^{11} + \cdots + 640000 \) Copy content Toggle raw display
$41$ \( T^{12} + 280 T^{10} + \cdots + 9048064 \) Copy content Toggle raw display
$43$ \( T^{12} + 8 T^{11} + \cdots + 16384 \) Copy content Toggle raw display
$47$ \( T^{12} - 16 T^{11} + \cdots + 15745024 \) Copy content Toggle raw display
$53$ \( T^{12} + 24 T^{11} + \cdots + 93392896 \) Copy content Toggle raw display
$59$ \( (T^{6} + 16 T^{5} + \cdots - 2416)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 188 T^{4} + \cdots - 3568)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 110166016 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 296666176 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 317699067904 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 14369296384 \) Copy content Toggle raw display
$83$ \( T^{12} + 24 T^{11} + \cdots + 2560000 \) Copy content Toggle raw display
$89$ \( (T^{6} + 24 T^{5} + \cdots + 449408)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 600838144 \) Copy content Toggle raw display
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