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Properties

Label 1950.2.j.a
Level $1950$
Weight $2$
Character orbit 1950.j
Analytic conductor $15.571$
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] = [1950,2,Mod(1243,1950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1950, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1950.1243");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.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: \(15.5708283941\)
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: \( x^{12} + 26x^{10} + 227x^{8} + 726x^{6} + 565x^{4} + 108x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_{6} q^{3} + q^{4} - \beta_{6} q^{6} + ( - \beta_{3} - \beta_{2}) q^{7} - q^{8} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta_{6} q^{3} + q^{4} - \beta_{6} q^{6} + ( - \beta_{3} - \beta_{2}) q^{7} - q^{8} + \beta_{2} q^{9} + ( - \beta_{11} + \beta_{10}) q^{11} + \beta_{6} q^{12} + ( - \beta_{10} - \beta_{6} - \beta_{5} + \cdots + 1) q^{13}+ \cdots + ( - \beta_{9} - \beta_{8}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} + 12 q^{4} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} + 12 q^{4} - 12 q^{8} + 4 q^{11} + 8 q^{13} + 12 q^{16} - 4 q^{17} + 4 q^{21} - 4 q^{22} - 8 q^{26} - 8 q^{31} - 12 q^{32} - 8 q^{33} + 4 q^{34} - 4 q^{39} + 12 q^{41} - 4 q^{42} + 8 q^{43} + 4 q^{44} - 20 q^{49} + 8 q^{52} + 4 q^{53} - 8 q^{57} - 4 q^{59} + 8 q^{62} + 8 q^{63} + 12 q^{64} + 8 q^{66} - 48 q^{67} - 4 q^{68} - 8 q^{69} - 8 q^{71} - 40 q^{73} - 32 q^{77} + 4 q^{78} - 12 q^{81} - 12 q^{82} + 4 q^{84} - 8 q^{86} + 4 q^{87} - 4 q^{88} - 12 q^{89} - 4 q^{91} - 72 q^{97} + 20 q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 26x^{10} + 227x^{8} + 726x^{6} + 565x^{4} + 108x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -8\nu^{10} - 203\nu^{8} - 1724\nu^{6} - 5428\nu^{4} - 4615\nu^{2} + 172 ) / 558 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9\nu^{11} + 240\nu^{9} + 2203\nu^{7} + 7920\nu^{5} + 9125\nu^{3} + 2674\nu ) / 496 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 229\nu^{11} + 5776\nu^{9} + 47815\nu^{7} + 134312\nu^{5} + 45161\nu^{3} - 12038\nu ) / 4464 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 386 \nu^{11} + 155 \nu^{10} - 10004 \nu^{9} + 3968 \nu^{8} - 86810 \nu^{7} + 33821 \nu^{6} + \cdots + 6014 ) / 4464 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 386 \nu^{11} - 209 \nu^{10} + 10004 \nu^{9} - 5408 \nu^{8} + 86810 \nu^{7} - 46295 \nu^{6} + \cdots - 5690 ) / 4464 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 386 \nu^{11} + 155 \nu^{10} + 10004 \nu^{9} + 3968 \nu^{8} + 86810 \nu^{7} + 33821 \nu^{6} + \cdots + 6014 ) / 4464 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 386 \nu^{11} - 209 \nu^{10} - 10004 \nu^{9} - 5408 \nu^{8} - 86810 \nu^{7} - 46295 \nu^{6} + \cdots - 5690 ) / 4464 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 345 \nu^{11} - 127 \nu^{10} + 8952 \nu^{9} - 3304 \nu^{8} + 77835 \nu^{7} - 28717 \nu^{6} + \cdots - 4942 ) / 1488 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 400 \nu^{11} + 27 \nu^{10} - 10336 \nu^{9} + 720 \nu^{8} - 89176 \nu^{7} + 6609 \nu^{6} + \cdots + 6534 ) / 1488 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 400 \nu^{11} - 27 \nu^{10} - 10336 \nu^{9} - 720 \nu^{8} - 89176 \nu^{7} - 6609 \nu^{6} + \cdots - 6534 ) / 1488 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 345 \nu^{11} - 127 \nu^{10} - 8952 \nu^{9} - 3304 \nu^{8} - 77835 \nu^{7} - 28717 \nu^{6} + \cdots - 4942 ) / 1488 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + 2\beta _1 - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} - 8\beta_{7} - 14\beta_{6} + 8\beta_{5} + 14\beta_{4} + 2\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 8 \beta_{11} + 8 \beta_{10} - 8 \beta_{9} - 8 \beta_{8} + 7 \beta_{7} - 11 \beta_{6} + 7 \beta_{5} + \cdots + 72 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 7 \beta_{11} + 17 \beta_{10} + 17 \beta_{9} - 7 \beta_{8} + 71 \beta_{7} + 145 \beta_{6} + \cdots - 34 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 67 \beta_{11} - 71 \beta_{10} + 71 \beta_{9} + 67 \beta_{8} - 46 \beta_{7} + 118 \beta_{6} - 46 \beta_{5} + \cdots - 692 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 46 \beta_{11} - 208 \beta_{10} - 208 \beta_{9} + 46 \beta_{8} - 657 \beta_{7} - 1461 \beta_{6} + \cdots + 460 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 609 \beta_{11} + 657 \beta_{10} - 657 \beta_{9} - 609 \beta_{8} + 289 \beta_{7} - 1297 \beta_{6} + \cdots + 6752 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 289 \beta_{11} + 2293 \beta_{10} + 2293 \beta_{9} - 289 \beta_{8} + 6222 \beta_{7} + \cdots - 5606 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 5866 \beta_{11} - 6222 \beta_{10} + 6222 \beta_{9} + 5866 \beta_{8} - 1593 \beta_{7} + 14369 \beta_{6} + \cdots - 66400 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1593 \beta_{11} - 24179 \beta_{10} - 24179 \beta_{9} + 1593 \beta_{8} - 59767 \beta_{7} + \cdots + 64898 \beta_{2} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1243.1
− 3.19697i
0.459724i
2.32303i
3.06111i
0.220596i
− 0.867489i
− 2.32303i
− 0.459724i
3.19697i
0.867489i
− 0.220596i
− 3.06111i
−1.00000 −0.707107 − 0.707107i 1.00000 0 0.707107 + 0.707107i − 4.52120i −1.00000 1.00000i 0
1243.2 −1.00000 −0.707107 − 0.707107i 1.00000 0 0.707107 + 0.707107i 0.650148i −1.00000 1.00000i 0
1243.3 −1.00000 −0.707107 − 0.707107i 1.00000 0 0.707107 + 0.707107i 3.28526i −1.00000 1.00000i 0
1243.4 −1.00000 0.707107 + 0.707107i 1.00000 0 −0.707107 − 0.707107i − 4.32906i −1.00000 1.00000i 0
1243.5 −1.00000 0.707107 + 0.707107i 1.00000 0 −0.707107 − 0.707107i − 0.311969i −1.00000 1.00000i 0
1243.6 −1.00000 0.707107 + 0.707107i 1.00000 0 −0.707107 − 0.707107i 1.22681i −1.00000 1.00000i 0
1357.1 −1.00000 −0.707107 + 0.707107i 1.00000 0 0.707107 − 0.707107i − 3.28526i −1.00000 − 1.00000i 0
1357.2 −1.00000 −0.707107 + 0.707107i 1.00000 0 0.707107 − 0.707107i − 0.650148i −1.00000 − 1.00000i 0
1357.3 −1.00000 −0.707107 + 0.707107i 1.00000 0 0.707107 − 0.707107i 4.52120i −1.00000 − 1.00000i 0
1357.4 −1.00000 0.707107 − 0.707107i 1.00000 0 −0.707107 + 0.707107i − 1.22681i −1.00000 − 1.00000i 0
1357.5 −1.00000 0.707107 − 0.707107i 1.00000 0 −0.707107 + 0.707107i 0.311969i −1.00000 − 1.00000i 0
1357.6 −1.00000 0.707107 − 0.707107i 1.00000 0 −0.707107 + 0.707107i 4.32906i −1.00000 − 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1243.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1950.2.j.a ✓ 12
5.b even 2 1 1950.2.j.b yes 12
5.c odd 4 1 1950.2.t.a yes 12
5.c odd 4 1 1950.2.t.c yes 12
13.d odd 4 1 1950.2.t.c yes 12
65.f even 4 1 1950.2.j.b yes 12
65.g odd 4 1 1950.2.t.a yes 12
65.k even 4 1 inner 1950.2.j.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1950.2.j.a ✓ 12 1.a even 1 1 trivial
1950.2.j.a ✓ 12 65.k even 4 1 inner
1950.2.j.b yes 12 5.b even 2 1
1950.2.j.b yes 12 65.f even 4 1
1950.2.t.a yes 12 5.c odd 4 1
1950.2.t.a yes 12 65.g odd 4 1
1950.2.t.c yes 12 5.c odd 4 1
1950.2.t.c yes 12 13.d odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1950, [\chi])\):

\( T_{7}^{12} + 52T_{7}^{10} + 908T_{7}^{8} + 5808T_{7}^{6} + 9040T_{7}^{4} + 3456T_{7}^{2} + 256 \) Copy content Toggle raw display
\( T_{17}^{12} + 4 T_{17}^{11} + 8 T_{17}^{10} - 64 T_{17}^{9} + 1660 T_{17}^{8} + 4640 T_{17}^{7} + \cdots + 4064256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{12} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 52 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( T^{12} - 4 T^{11} + \cdots + 82944 \) Copy content Toggle raw display
$13$ \( T^{12} - 8 T^{11} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{12} + 4 T^{11} + \cdots + 4064256 \) Copy content Toggle raw display
$19$ \( T^{12} + 200 T^{9} + \cdots + 29506624 \) Copy content Toggle raw display
$23$ \( T^{12} + 40 T^{9} + \cdots + 5184 \) Copy content Toggle raw display
$29$ \( T^{12} + 196 T^{10} + \cdots + 1016064 \) Copy content Toggle raw display
$31$ \( T^{12} + 8 T^{11} + \cdots + 2458624 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 286015744 \) Copy content Toggle raw display
$41$ \( T^{12} - 12 T^{11} + \cdots + 6718464 \) Copy content Toggle raw display
$43$ \( T^{12} - 8 T^{11} + \cdots + 802816 \) Copy content Toggle raw display
$47$ \( T^{12} + 344 T^{10} + \cdots + 84934656 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 656999424 \) Copy content Toggle raw display
$59$ \( T^{12} + 4 T^{11} + \cdots + 4064256 \) Copy content Toggle raw display
$61$ \( (T^{6} - 160 T^{4} + \cdots + 40896)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 24 T^{5} + \cdots + 273344)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 260112384 \) Copy content Toggle raw display
$73$ \( (T^{6} + 20 T^{5} + \cdots - 187376)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 197733187584 \) Copy content Toggle raw display
$83$ \( T^{12} + 480 T^{10} + \cdots + 5308416 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 18793119744 \) Copy content Toggle raw display
$97$ \( (T^{6} + 36 T^{5} + \cdots - 214384)^{2} \) Copy content Toggle raw display
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