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Properties

Label 1200.3.l.y
Level $1200$
Weight $3$
Character orbit 1200.l
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(401,1200)]
 
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, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.401");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.l (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: \( x^{12} - 4x^{10} + 30x^{8} - 216x^{6} + 1080x^{4} - 5184x^{2} + 46656 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 120)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} - \beta_{5} q^{7} + ( - \beta_{9} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} - \beta_{5} q^{7} + ( - \beta_{9} - 1) q^{9} + ( - \beta_{9} + \beta_{7}) q^{11} + (\beta_{8} - \beta_{4}) q^{13} + (\beta_{6} + \beta_{2}) q^{17} + ( - \beta_{11} + \beta_{3}) q^{19} + ( - \beta_{9} + 2 \beta_{7} - \beta_{3} + \cdots + 1) q^{21}+ \cdots + (2 \beta_{11} + \beta_{9} - 5 \beta_{7} + \cdots - 50) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8 q^{9} + 4 q^{21} + 48 q^{31} - 128 q^{39} + 252 q^{49} - 48 q^{51} + 144 q^{61} - 268 q^{69} + 432 q^{79} - 188 q^{81} - 560 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4x^{10} + 30x^{8} - 216x^{6} + 1080x^{4} - 5184x^{2} + 46656 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{11} + 22\nu^{9} + 150\nu^{7} + 1260\nu^{5} + 432\nu^{3} - 7776\nu ) / 23328 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{10} + 10\nu^{8} + 126\nu^{6} - 108\nu^{4} + 864\nu^{2} - 7776 ) / 5832 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} - 22 \nu^{9} - 72 \nu^{8} + 138 \nu^{7} + 288 \nu^{6} - 684 \nu^{5} + 432 \nu^{4} + \cdots - 23328 ) / 7776 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} - 11 \nu^{10} - 10 \nu^{9} + 2 \nu^{8} + 90 \nu^{7} - 126 \nu^{6} - 756 \nu^{5} + \cdots + 7776 ) / 46656 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{11} + 50\nu^{9} - 18\nu^{7} - 540\nu^{5} - 4752\nu^{3} + 23328\nu ) / 23328 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{10} + 10\nu^{8} - 36\nu^{6} + 540\nu^{4} - 3996\nu^{2} + 9720 ) / 2916 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13 \nu^{11} - 24 \nu^{10} - 70 \nu^{9} + 168 \nu^{8} + 642 \nu^{7} - 1008 \nu^{6} - 2988 \nu^{5} + \cdots + 93312 ) / 46656 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 25 \nu^{11} - 11 \nu^{10} + 38 \nu^{9} + 2 \nu^{8} + 234 \nu^{7} - 126 \nu^{6} - 1620 \nu^{5} + \cdots + 7776 ) / 46656 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 7 \nu^{11} - 24 \nu^{10} - 62 \nu^{9} + 168 \nu^{8} + 186 \nu^{7} - 1008 \nu^{6} - 1116 \nu^{5} + \cdots + 93312 ) / 46656 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{11} + 11 \nu^{10} - 10 \nu^{9} - 2 \nu^{8} + 90 \nu^{7} + 126 \nu^{6} - 756 \nu^{5} + \cdots - 7776 ) / 15552 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{11} + 8 \nu^{10} - 22 \nu^{9} + 16 \nu^{8} + 138 \nu^{7} + 48 \nu^{6} - 684 \nu^{5} + \cdots - 7776 ) / 7776 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - 7\beta_{10} - 5\beta_{9} - 3\beta_{8} + 7\beta_{7} + 3\beta_{5} - 18\beta_{4} + \beta_{3} - 3\beta_1 ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -4\beta_{11} + 4\beta_{10} + 5\beta_{9} + 5\beta_{7} - 12\beta_{6} - 12\beta_{4} - \beta_{3} + 3\beta_{2} + 21 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{11} + 7\beta_{10} - 22\beta_{9} - 15\beta_{8} + 14\beta_{7} - 3\beta_{5} + 36\beta_{4} - 4\beta_{3} ) / 24 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 14 \beta_{11} + 20 \beta_{10} + 4 \beta_{9} + 4 \beta_{7} + 30 \beta_{6} - 60 \beta_{4} + 10 \beta_{3} + \cdots - 84 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 13 \beta_{11} - 47 \beta_{10} + 5 \beta_{9} - 15 \beta_{8} - 31 \beta_{7} - 165 \beta_{5} + \cdots + 207 \beta_1 ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 4 \beta_{11} + 20 \beta_{10} - 59 \beta_{9} - 59 \beta_{7} + 84 \beta_{6} - 60 \beta_{4} + 55 \beta_{3} + \cdots + 645 ) / 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 142 \beta_{11} - 37 \beta_{10} + 124 \beta_{9} + 33 \beta_{8} + 160 \beta_{7} + 201 \beta_{5} + \cdots + 738 \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 110 \beta_{11} + 172 \beta_{10} + 200 \beta_{9} + 200 \beta_{7} + 42 \beta_{6} - 516 \beta_{4} + \cdots - 528 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 587 \beta_{11} + 935 \beta_{10} - 1901 \beta_{9} + 519 \beta_{8} + 727 \beta_{7} + 3117 \beta_{5} + \cdots + 1305 \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1244 \beta_{11} + 2764 \beta_{10} - 853 \beta_{9} - 853 \beta_{7} + 1500 \beta_{6} - 8292 \beta_{4} + \cdots - 2229 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1306 \beta_{11} + 265 \beta_{10} - 7552 \beta_{9} + 4755 \beta_{8} + 4940 \beta_{7} + \cdots - 2142 \beta_1 ) / 6 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
1.05281 − 2.21170i
1.05281 + 2.21170i
1.79523 + 1.66648i
1.79523 − 1.66648i
2.38091 + 0.575548i
2.38091 − 0.575548i
−2.38091 − 0.575548i
−2.38091 + 0.575548i
−1.79523 − 1.66648i
−1.79523 + 1.66648i
−1.05281 + 2.21170i
−1.05281 − 2.21170i
0 −2.84952 − 0.938195i 0 0 0 6.81219 0 7.23958 + 5.34682i 0
401.2 0 −2.84952 + 0.938195i 0 0 0 6.81219 0 7.23958 − 5.34682i 0
401.3 0 −1.67109 − 2.49147i 0 0 0 −12.7692 0 −3.41489 + 8.32698i 0
401.4 0 −1.67109 + 2.49147i 0 0 0 −12.7692 0 −3.41489 − 8.32698i 0
401.5 0 −1.26002 − 2.72256i 0 0 0 0.735748 0 −5.82469 + 6.86097i 0
401.6 0 −1.26002 + 2.72256i 0 0 0 0.735748 0 −5.82469 − 6.86097i 0
401.7 0 1.26002 − 2.72256i 0 0 0 −0.735748 0 −5.82469 − 6.86097i 0
401.8 0 1.26002 + 2.72256i 0 0 0 −0.735748 0 −5.82469 + 6.86097i 0
401.9 0 1.67109 − 2.49147i 0 0 0 12.7692 0 −3.41489 − 8.32698i 0
401.10 0 1.67109 + 2.49147i 0 0 0 12.7692 0 −3.41489 + 8.32698i 0
401.11 0 2.84952 − 0.938195i 0 0 0 −6.81219 0 7.23958 − 5.34682i 0
401.12 0 2.84952 + 0.938195i 0 0 0 −6.81219 0 7.23958 + 5.34682i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 401.12
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.l.y 12
3.b odd 2 1 inner 1200.3.l.y 12
4.b odd 2 1 600.3.l.g 12
5.b even 2 1 inner 1200.3.l.y 12
5.c odd 4 2 240.3.c.e 12
12.b even 2 1 600.3.l.g 12
15.d odd 2 1 inner 1200.3.l.y 12
15.e even 4 2 240.3.c.e 12
20.d odd 2 1 600.3.l.g 12
20.e even 4 2 120.3.c.a ✓ 12
40.i odd 4 2 960.3.c.j 12
40.k even 4 2 960.3.c.k 12
60.h even 2 1 600.3.l.g 12
60.l odd 4 2 120.3.c.a ✓ 12
120.q odd 4 2 960.3.c.k 12
120.w even 4 2 960.3.c.j 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.3.c.a ✓ 12 20.e even 4 2
120.3.c.a ✓ 12 60.l odd 4 2
240.3.c.e 12 5.c odd 4 2
240.3.c.e 12 15.e even 4 2
600.3.l.g 12 4.b odd 2 1
600.3.l.g 12 12.b even 2 1
600.3.l.g 12 20.d odd 2 1
600.3.l.g 12 60.h even 2 1
960.3.c.j 12 40.i odd 4 2
960.3.c.j 12 120.w even 4 2
960.3.c.k 12 40.k even 4 2
960.3.c.k 12 120.q odd 4 2
1200.3.l.y 12 1.a even 1 1 trivial
1200.3.l.y 12 3.b odd 2 1 inner
1200.3.l.y 12 5.b even 2 1 inner
1200.3.l.y 12 15.d odd 2 1 inner

Hecke kernels

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

\( T_{7}^{6} - 210T_{7}^{4} + 7680T_{7}^{2} - 4096 \) Copy content Toggle raw display
\( T_{11}^{6} + 336T_{11}^{4} + 34944T_{11}^{2} + 1083392 \) Copy content Toggle raw display
\( T_{13}^{6} - 768T_{13}^{4} + 157824T_{13}^{2} - 6553600 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 4 T^{10} + \cdots + 531441 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} - 210 T^{4} + \cdots - 4096)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 336 T^{4} + \cdots + 1083392)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} - 768 T^{4} + \cdots - 6553600)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 264 T^{4} + \cdots + 165888)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 780 T + 5648)^{4} \) Copy content Toggle raw display
$23$ \( (T^{6} + 1050 T^{4} + \cdots + 13148192)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 4416 T^{4} + \cdots + 807698432)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 12 T^{2} + \cdots + 31104)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} - 2592 T^{4} + \cdots - 237899776)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 4356 T^{4} + \cdots + 772087808)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 4050 T^{4} + \cdots - 1224440064)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 5754 T^{4} + \cdots + 2393766432)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 11400 T^{4} + \cdots + 21525635072)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 11856 T^{4} + \cdots + 1313998848)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 36 T^{2} + \cdots + 1984)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} - 5106 T^{4} + \cdots - 1763584)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 19200 T^{4} + \cdots + 124856041472)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 26400 T^{4} + \cdots - 238331428864)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 108 T^{2} + \cdots + 234400)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + 14586 T^{4} + \cdots + 387755552)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 27648 T^{4} + \cdots + 278628139008)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 13344 T^{4} + \cdots - 16777216)^{2} \) Copy content Toggle raw display
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