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Properties

Label 1200.3.c.l
Level $1200$
Weight $3$
Character orbit 1200.c
Analytic conductor $32.698$
Analytic rank $0$
Dimension $12$
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] = [1200,3,Mod(449,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.c (of order \(2\), degree \(1\), not 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: \(32.6976317232\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 18x^{10} + 75x^{8} + 1270x^{6} + 14397x^{4} - 7740x^{2} + 39204 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{37}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 600)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{10} - \beta_{6}) q^{3} + ( - \beta_{11} + \beta_{10} + \cdots - 2 \beta_{6}) q^{7}+ \cdots + ( - \beta_{4} + \beta_{3} + \beta_1 - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{10} - \beta_{6}) q^{3} + ( - \beta_{11} + \beta_{10} + \cdots - 2 \beta_{6}) q^{7}+ \cdots + (3 \beta_{5} - \beta_{4} - 8 \beta_{3} + \cdots - 42) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 32 q^{9} + 100 q^{19} - 36 q^{21} + 228 q^{31} - 12 q^{39} - 152 q^{49} + 12 q^{51} + 124 q^{61} - 312 q^{69} - 152 q^{79} - 448 q^{81} + 620 q^{91} - 500 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 18x^{10} + 75x^{8} + 1270x^{6} + 14397x^{4} - 7740x^{2} + 39204 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -39\nu^{10} - 1068\nu^{8} - 5305\nu^{6} - 45816\nu^{4} - 1125197\nu^{2} - 291510 ) / 745170 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -39\nu^{10} - 1068\nu^{8} - 5305\nu^{6} - 45816\nu^{4} - 752612\nu^{2} + 453660 ) / 372585 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -339\nu^{10} - 5462\nu^{8} - 19363\nu^{6} - 589316\nu^{4} - 5294253\nu^{2} + 6304968 ) / 2235510 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 111\nu^{10} + 1129\nu^{8} + 1724\nu^{6} + 101739\nu^{4} + 462551\nu^{2} - 1354239 ) / 372585 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -1766\nu^{10} - 31165\nu^{8} - 161246\nu^{6} - 2437674\nu^{4} - 24014925\nu^{2} - 9938619 ) / 3353265 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -617\nu^{11} - 12492\nu^{9} - 67131\nu^{7} - 852428\nu^{5} - 11366133\nu^{3} - 12637134\nu ) / 30008880 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13087\nu^{11} + 174956\nu^{9} + 330589\nu^{7} + 19080948\nu^{5} + 121414299\nu^{3} - 2193841134\nu ) / 590174640 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 136583 \nu^{11} + 1982436 \nu^{9} + 16375125 \nu^{7} + 221022452 \nu^{5} + \cdots + 5815099890 \nu ) / 1770523920 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 35393\nu^{11} + 495372\nu^{9} + 917817\nu^{7} + 29751422\nu^{5} + 293274123\nu^{3} - 1096750422\nu ) / 442630980 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -37019\nu^{11} - 585756\nu^{9} - 1956771\nu^{7} - 45762506\nu^{5} - 442706289\nu^{3} + 540370386\nu ) / 442630980 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 616375 \nu^{11} - 11391684 \nu^{9} - 44271093 \nu^{7} - 730709380 \nu^{5} + \cdots + 11124243918 \nu ) / 1770523920 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -3\beta_{10} - \beta_{9} - 4\beta_{7} + 4\beta_{6} ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{11} - \beta_{10} + 5\beta_{9} - 3\beta_{8} + 6\beta_{7} - 49\beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{4} - 9\beta_{3} - 4\beta_{2} + 17\beta _1 + 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -13\beta_{11} - 110\beta_{10} - 96\beta_{9} + 6\beta_{8} - 22\beta_{7} + 293\beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -81\beta_{5} - 36\beta_{4} + 135\beta_{3} + 115\beta_{2} - 11\beta _1 - 896 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 91\beta_{11} + 1328\beta_{10} + 654\beta_{9} + 261\beta_{8} + 688\beta_{7} - 2678\beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 567\beta_{5} + 102\beta_{4} - 810\beta_{3} - 2179\beta_{2} + 1580\beta _1 + 7607 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -6568\beta_{11} - 2348\beta_{10} - 11100\beta_{9} + 885\beta_{8} - 11458\beta_{7} + 68555\beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -4509\beta_{5} + 5628\beta_{4} + 14391\beta_{3} + 19876\beta_{2} - 23147\beta _1 - 66752 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 67351\beta_{11} + 83684\beta_{10} + 197514\beta_{9} + 660\beta_{8} + 90646\beta_{7} - 709685\beta_{6} ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
0.994180 − 0.788754i
0.994180 + 0.788754i
0.139571 − 3.56627i
0.139571 + 3.56627i
2.54797 − 1.77752i
2.54797 + 1.77752i
−2.54797 − 1.77752i
−2.54797 + 1.77752i
−0.139571 − 3.56627i
−0.139571 + 3.56627i
−0.994180 − 0.788754i
−0.994180 + 0.788754i
0 −2.40839 − 1.78875i 0 0 0 − 12.2014i 0 2.60072 + 8.61605i 0
449.2 0 −2.40839 + 1.78875i 0 0 0 12.2014i 0 2.60072 − 8.61605i 0
449.3 0 −1.55378 − 2.56627i 0 0 0 − 1.34301i 0 −4.17150 + 7.97487i 0
449.4 0 −1.55378 + 2.56627i 0 0 0 1.34301i 0 −4.17150 − 7.97487i 0
449.5 0 −1.13375 − 2.77752i 0 0 0 5.85843i 0 −6.42922 + 6.29803i 0
449.6 0 −1.13375 + 2.77752i 0 0 0 − 5.85843i 0 −6.42922 − 6.29803i 0
449.7 0 1.13375 − 2.77752i 0 0 0 5.85843i 0 −6.42922 − 6.29803i 0
449.8 0 1.13375 + 2.77752i 0 0 0 − 5.85843i 0 −6.42922 + 6.29803i 0
449.9 0 1.55378 − 2.56627i 0 0 0 − 1.34301i 0 −4.17150 − 7.97487i 0
449.10 0 1.55378 + 2.56627i 0 0 0 1.34301i 0 −4.17150 + 7.97487i 0
449.11 0 2.40839 − 1.78875i 0 0 0 − 12.2014i 0 2.60072 − 8.61605i 0
449.12 0 2.40839 + 1.78875i 0 0 0 12.2014i 0 2.60072 + 8.61605i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.3.c.l 12
3.b odd 2 1 inner 1200.3.c.l 12
4.b odd 2 1 600.3.c.c 12
5.b even 2 1 inner 1200.3.c.l 12
5.c odd 4 1 1200.3.l.v 6
5.c odd 4 1 1200.3.l.w 6
12.b even 2 1 600.3.c.c 12
15.d odd 2 1 inner 1200.3.c.l 12
15.e even 4 1 1200.3.l.v 6
15.e even 4 1 1200.3.l.w 6
20.d odd 2 1 600.3.c.c 12
20.e even 4 1 600.3.l.d ✓ 6
20.e even 4 1 600.3.l.e yes 6
60.h even 2 1 600.3.c.c 12
60.l odd 4 1 600.3.l.d ✓ 6
60.l odd 4 1 600.3.l.e yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.3.c.c 12 4.b odd 2 1
600.3.c.c 12 12.b even 2 1
600.3.c.c 12 20.d odd 2 1
600.3.c.c 12 60.h even 2 1
600.3.l.d ✓ 6 20.e even 4 1
600.3.l.d ✓ 6 60.l odd 4 1
600.3.l.e yes 6 20.e even 4 1
600.3.l.e yes 6 60.l odd 4 1
1200.3.c.l 12 1.a even 1 1 trivial
1200.3.c.l 12 3.b odd 2 1 inner
1200.3.c.l 12 5.b even 2 1 inner
1200.3.c.l 12 15.d odd 2 1 inner
1200.3.l.v 6 5.c odd 4 1
1200.3.l.v 6 15.e even 4 1
1200.3.l.w 6 5.c odd 4 1
1200.3.l.w 6 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7}^{6} + 185T_{7}^{4} + 5440T_{7}^{2} + 9216 \) Copy content Toggle raw display
\( T_{11}^{6} + 246T_{11}^{4} + 15129T_{11}^{2} + 161312 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 16 T^{10} + \cdots + 531441 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} + 185 T^{4} + \cdots + 9216)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 246 T^{4} + \cdots + 161312)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 233 T^{4} + \cdots + 291600)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 1174 T^{4} + \cdots - 9435168)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 25 T^{2} + \cdots + 877)^{4} \) Copy content Toggle raw display
$23$ \( (T^{6} - 2680 T^{4} + \cdots - 423055872)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 2616 T^{4} + \cdots + 37601792)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 57 T^{2} + \cdots + 1804)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + 5092 T^{4} + \cdots + 4857532416)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 4806 T^{4} + \cdots + 22418208)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 6065 T^{4} + \cdots + 3555498384)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 17304 T^{4} + \cdots - 187644280832)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 9720 T^{4} + \cdots - 24652657152)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 17056 T^{4} + \cdots + 91824979968)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 31 T^{2} + \cdots + 117664)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + 15531 T^{4} + \cdots + 105555461449)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 20600 T^{4} + \cdots + 15890452992)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 12150 T^{4} + \cdots + 44511716484)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 38 T^{2} + \cdots + 370080)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} - 5206 T^{4} + \cdots - 1152)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 13478 T^{4} + \cdots + 5739061248)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 39129 T^{4} + \cdots + 376549140496)^{2} \) Copy content Toggle raw display
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