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Properties

Label 1200.3.l.x
Level $1200$
Weight $3$
Character orbit 1200.l
Analytic conductor $32.698$
Analytic rank $0$
Dimension $8$
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] = [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: \(8\)
Coefficient field: 8.0.681615360000.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 2x^{6} + 20x^{5} + 49x^{4} - 136x^{3} + 168x^{2} - 96x + 864 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{7}\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + ( - \beta_{7} - \beta_{4} + \beta_{3} + 2) q^{7} + ( - \beta_{6} + \beta_{5} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + ( - \beta_{7} - \beta_{4} + \beta_{3} + 2) q^{7} + ( - \beta_{6} + \beta_{5} + 2) q^{9} + (\beta_{7} + \beta_{6} + \beta_{4} + \cdots - 1) q^{11}+ \cdots + (23 \beta_{7} - 15 \beta_{6} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} + 16 q^{7} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} + 16 q^{7} + 20 q^{9} + 8 q^{13} + 8 q^{19} + 28 q^{21} + 20 q^{27} - 120 q^{31} + 112 q^{33} - 8 q^{37} + 72 q^{39} - 328 q^{43} + 64 q^{49} - 64 q^{51} - 72 q^{57} + 8 q^{61} + 88 q^{63} + 152 q^{67} + 100 q^{69} - 32 q^{73} - 88 q^{79} + 224 q^{81} - 152 q^{87} - 560 q^{91} + 368 q^{93} - 144 q^{97} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 2x^{6} + 20x^{5} + 49x^{4} - 136x^{3} + 168x^{2} - 96x + 864 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -9\nu^{7} - 511\nu^{6} + 1235\nu^{5} + 2685\nu^{4} - 3206\nu^{3} - 34634\nu^{2} + 25656\nu - 66288 ) / 18600 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{7} + 7\nu^{6} + 85\nu^{5} - 230\nu^{4} - 523\nu^{3} + 243\nu^{2} + 4668\nu - 2124 ) / 3100 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -26\nu^{7} - 64\nu^{6} + 795\nu^{5} - 45\nu^{4} - 4939\nu^{3} - 1181\nu^{2} + 23484\nu - 23892 ) / 9300 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -59\nu^{7} + 129\nu^{6} + 415\nu^{5} + 1275\nu^{4} - 7136\nu^{3} - 11664\nu^{2} - 15744\nu + 64752 ) / 18600 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -37\nu^{7} + 362\nu^{6} - 675\nu^{5} - 1310\nu^{4} + 322\nu^{3} + 16818\nu^{2} - 1992\nu + 24876 ) / 9300 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8\nu^{7} - 28\nu^{6} - 30\nu^{5} + 145\nu^{4} + 232\nu^{3} - 507\nu^{2} + 1788\nu - 804 ) / 930 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 82\nu^{7} - 287\nu^{6} - 385\nu^{5} + 1680\nu^{4} + 5943\nu^{3} - 8413\nu^{2} + 13212\nu - 5916 ) / 9300 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{7} + 3\beta_{5} + \beta_{3} + 3\beta_{2} + 2\beta _1 + 5 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{4} + 2\beta_{3} - 3\beta_{2} - \beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6\beta_{7} - 6\beta_{6} - \beta_{4} + 2\beta_{3} - \beta_{2} - \beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 16\beta_{7} - 8\beta_{6} + 2\beta_{5} + 11\beta_{4} + 16\beta_{3} - 17\beta_{2} - 9\beta _1 - 43 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 54\beta_{7} - 50\beta_{6} - 28\beta_{5} + 23\beta_{4} + 38\beta_{3} + 3\beta_{2} - 49\beta _1 - 89 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 188\beta_{7} - 124\beta_{6} - 26\beta_{5} + 179\beta_{4} + 36\beta_{3} + 143\beta_{2} - 137\beta _1 - 839 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 322\beta_{7} - 70\beta_{6} - 344\beta_{5} + 479\beta_{4} + 10\beta_{3} + 547\beta_{2} - 609\beta _1 - 2373 ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
3.22255 + 1.41421i
3.22255 − 1.41421i
−0.542939 − 1.41421i
−0.542939 + 1.41421i
−2.22255 − 1.41421i
−2.22255 + 1.41421i
1.54294 − 1.41421i
1.54294 + 1.41421i
0 −2.87275 − 0.864473i 0 0 0 9.02416 0 7.50537 + 4.96683i 0
401.2 0 −2.87275 + 0.864473i 0 0 0 9.02416 0 7.50537 − 4.96683i 0
401.3 0 −2.40140 − 1.79813i 0 0 0 −10.2132 0 2.53346 + 8.63606i 0
401.4 0 −2.40140 + 1.79813i 0 0 0 −10.2132 0 2.53346 − 8.63606i 0
401.5 0 0.291610 − 2.98579i 0 0 0 4.46268 0 −8.82993 − 1.74137i 0
401.6 0 0.291610 + 2.98579i 0 0 0 4.46268 0 −8.82993 + 1.74137i 0
401.7 0 2.98254 − 0.323191i 0 0 0 4.72640 0 8.79110 − 1.92786i 0
401.8 0 2.98254 + 0.323191i 0 0 0 4.72640 0 8.79110 + 1.92786i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 401.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b 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.x 8
3.b odd 2 1 inner 1200.3.l.x 8
4.b odd 2 1 600.3.l.f 8
5.b even 2 1 240.3.l.d 8
5.c odd 4 2 1200.3.c.m 16
12.b even 2 1 600.3.l.f 8
15.d odd 2 1 240.3.l.d 8
15.e even 4 2 1200.3.c.m 16
20.d odd 2 1 120.3.l.a ✓ 8
20.e even 4 2 600.3.c.d 16
40.e odd 2 1 960.3.l.h 8
40.f even 2 1 960.3.l.g 8
60.h even 2 1 120.3.l.a ✓ 8
60.l odd 4 2 600.3.c.d 16
120.i odd 2 1 960.3.l.g 8
120.m even 2 1 960.3.l.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.3.l.a ✓ 8 20.d odd 2 1
120.3.l.a ✓ 8 60.h even 2 1
240.3.l.d 8 5.b even 2 1
240.3.l.d 8 15.d odd 2 1
600.3.c.d 16 20.e even 4 2
600.3.c.d 16 60.l odd 4 2
600.3.l.f 8 4.b odd 2 1
600.3.l.f 8 12.b even 2 1
960.3.l.g 8 40.f even 2 1
960.3.l.g 8 120.i odd 2 1
960.3.l.h 8 40.e odd 2 1
960.3.l.h 8 120.m even 2 1
1200.3.c.m 16 5.c odd 4 2
1200.3.c.m 16 15.e even 4 2
1200.3.l.x 8 1.a even 1 1 trivial
1200.3.l.x 8 3.b 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 \(S_{3}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7}^{4} - 8T_{7}^{3} - 82T_{7}^{2} + 872T_{7} - 1944 \) Copy content Toggle raw display
\( T_{11}^{8} + 888T_{11}^{6} + 226128T_{11}^{4} + 14951168T_{11}^{2} + 232989696 \) Copy content Toggle raw display
\( T_{13}^{4} - 4T_{13}^{3} - 380T_{13}^{2} + 3648T_{13} - 3456 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 4 T^{7} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 8 T^{3} + \cdots - 1944)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 888 T^{6} + \cdots + 232989696 \) Copy content Toggle raw display
$13$ \( (T^{4} - 4 T^{3} + \cdots - 3456)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 1104 T^{6} + \cdots + 15872256 \) Copy content Toggle raw display
$19$ \( (T^{4} - 4 T^{3} + \cdots + 16736)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 93650688576 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 3474395136 \) Copy content Toggle raw display
$31$ \( (T^{4} + 60 T^{3} + \cdots - 151296)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 4 T^{3} + \cdots - 31104)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 43961355472896 \) Copy content Toggle raw display
$43$ \( (T^{4} + 164 T^{3} + \cdots + 1582656)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 13517317696 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 6801580544256 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 15563214360576 \) Copy content Toggle raw display
$61$ \( (T^{4} - 4 T^{3} + \cdots + 30631296)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 76 T^{3} + \cdots - 668224)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 35499479924736 \) Copy content Toggle raw display
$73$ \( (T^{4} + 16 T^{3} + \cdots + 2938896)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 44 T^{3} + \cdots - 12384)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 336130569170496 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( (T^{4} + 72 T^{3} + \cdots + 10270096)^{2} \) Copy content Toggle raw display
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