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Properties

Label 150.11.b.a
Level $150$
Weight $11$
Character orbit 150.b
Analytic conductor $95.304$
Analytic rank $0$
Dimension $8$
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] = [150,11,Mod(149,150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(150, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("150.149");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 150.b (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: \(95.3035879011\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3421020160000.10
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 967x^{4} + 194481 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 6)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + ( - \beta_{4} + 3 \beta_{3} + 10 \beta_1) q^{3} + 512 q^{4} + ( - 4 \beta_{7} + 21 \beta_{2} + 1344) q^{6} + (3 \beta_{6} + 48 \beta_{4} + \cdots + 5663 \beta_1) q^{7}+ \cdots + ( - 21 \beta_{7} - 21 \beta_{5} + \cdots - 39753) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + ( - \beta_{4} + 3 \beta_{3} + 10 \beta_1) q^{3} + 512 q^{4} + ( - 4 \beta_{7} + 21 \beta_{2} + 1344) q^{6} + (3 \beta_{6} + 48 \beta_{4} + \cdots + 5663 \beta_1) q^{7}+ \cdots + ( - 6000813 \beta_{7} + \cdots + 656728128) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4096 q^{4} + 10752 q^{6} - 318024 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4096 q^{4} + 10752 q^{6} - 318024 q^{9} + 2097152 q^{16} + 3137456 q^{19} + 19256016 q^{21} + 5505024 q^{24} - 43571696 q^{31} - 302174208 q^{34} - 162828288 q^{36} - 434574480 q^{39} + 377628672 q^{46} + 100116840 q^{49} - 1417153536 q^{51} - 963325440 q^{54} - 2368077488 q^{61} + 1073741824 q^{64} + 6246890496 q^{66} - 1192536576 q^{69} + 1606377472 q^{76} - 398565136 q^{79} + 2917929096 q^{81} + 9859080192 q^{84} + 16634464160 q^{91} - 17010954240 q^{94} + 2818572288 q^{96} + 5253825024 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 967x^{4} + 194481 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{6} + 2816\nu^{2} ) / 18963 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -352\nu^{7} - 7056\nu^{5} - 192208\nu^{3} - 3563280\nu ) / 398223 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 352\nu^{7} - 7056\nu^{5} + 192208\nu^{3} - 3563280\nu ) / 398223 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 172\nu^{7} - 7231\nu^{6} - 1764\nu^{5} + 166324\nu^{3} - 3809680\nu^{2} - 2483712\nu ) / 132741 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -680\nu^{7} - 3528\nu^{5} + 889056\nu^{4} - 805736\nu^{3} - 11338992\nu + 429858576 ) / 398223 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2048\nu^{7} - 14448\nu^{6} + 14112\nu^{5} - 2276768\nu^{3} - 7599648\nu^{2} + 32612832\nu ) / 132741 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2048\nu^{7} + 14112\nu^{5} + 296352\nu^{4} + 2276768\nu^{3} + 32612832\nu + 143286192 ) / 132741 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{7} + 2\beta_{6} - 2\beta_{5} - 4\beta_{4} + 15\beta_{3} + 13\beta_{2} - 2\beta_1 ) / 864 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + 16\beta_{4} - 6\beta_{3} + 9296\beta_1 ) / 864 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 44\beta_{7} - 44\beta_{6} - 44\beta_{5} + 88\beta_{4} - 897\beta_{3} + 853\beta_{2} + 44\beta_1 ) / 864 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 43\beta_{7} + 344\beta_{5} + 86\beta_{2} - 417744 ) / 864 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -505\beta_{7} - 505\beta_{6} + 505\beta_{5} + 1010\beta_{4} - 15978\beta_{3} - 15473\beta_{2} + 505\beta_1 ) / 432 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -88\beta_{6} - 1408\beta_{4} + 528\beta_{3} - 306047\beta_1 ) / 54 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 24026 \beta_{7} + 24026 \beta_{6} + 24026 \beta_{5} - 48052 \beta_{4} + 978531 \beta_{3} + \cdots - 24026 \beta_1 ) / 864 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
−3.61315 − 3.61315i
−3.61315 + 3.61315i
2.90605 − 2.90605i
2.90605 + 2.90605i
−2.90605 + 2.90605i
−2.90605 − 2.90605i
3.61315 + 3.61315i
3.61315 − 3.61315i
−22.6274 −137.627 − 200.269i 512.000 0 3114.15 + 4531.57i 23226.5i −11585.2 −21166.4 + 55125.0i 0
149.2 −22.6274 −137.627 + 200.269i 512.000 0 3114.15 − 4531.57i − 23226.5i −11585.2 −21166.4 − 55125.0i 0
149.3 −22.6274 18.8335 − 242.269i 512.000 0 −426.153 + 5481.92i 670.530i −11585.2 −58339.6 − 9125.53i 0
149.4 −22.6274 18.8335 + 242.269i 512.000 0 −426.153 − 5481.92i − 670.530i −11585.2 −58339.6 + 9125.53i 0
149.5 22.6274 −18.8335 − 242.269i 512.000 0 −426.153 − 5481.92i 670.530i 11585.2 −58339.6 + 9125.53i 0
149.6 22.6274 −18.8335 + 242.269i 512.000 0 −426.153 + 5481.92i − 670.530i 11585.2 −58339.6 − 9125.53i 0
149.7 22.6274 137.627 − 200.269i 512.000 0 3114.15 − 4531.57i 23226.5i 11585.2 −21166.4 − 55125.0i 0
149.8 22.6274 137.627 + 200.269i 512.000 0 3114.15 + 4531.57i − 23226.5i 11585.2 −21166.4 + 55125.0i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.8
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 150.11.b.a 8
3.b odd 2 1 inner 150.11.b.a 8
5.b even 2 1 inner 150.11.b.a 8
5.c odd 4 1 6.11.b.a ✓ 4
5.c odd 4 1 150.11.d.a 4
15.d odd 2 1 inner 150.11.b.a 8
15.e even 4 1 6.11.b.a ✓ 4
15.e even 4 1 150.11.d.a 4
20.e even 4 1 48.11.e.d 4
40.i odd 4 1 192.11.e.g 4
40.k even 4 1 192.11.e.h 4
45.k odd 12 2 162.11.d.d 8
45.l even 12 2 162.11.d.d 8
60.l odd 4 1 48.11.e.d 4
120.q odd 4 1 192.11.e.h 4
120.w even 4 1 192.11.e.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.11.b.a ✓ 4 5.c odd 4 1
6.11.b.a ✓ 4 15.e even 4 1
48.11.e.d 4 20.e even 4 1
48.11.e.d 4 60.l odd 4 1
150.11.b.a 8 1.a even 1 1 trivial
150.11.b.a 8 3.b odd 2 1 inner
150.11.b.a 8 5.b even 2 1 inner
150.11.b.a 8 15.d odd 2 1 inner
150.11.d.a 4 5.c odd 4 1
150.11.d.a 4 15.e even 4 1
162.11.d.d 8 45.k odd 12 2
162.11.d.d 8 45.l even 12 2
192.11.e.g 4 40.i odd 4 1
192.11.e.g 4 120.w even 4 1
192.11.e.h 4 40.k even 4 1
192.11.e.h 4 120.q odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 539921288T_{7}^{2} + 242551843253776 \) acting on \(S_{11}^{\mathrm{new}}(150, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 512)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 12\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + \cdots + 242551843253776)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 21\!\cdots\!24)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 32\!\cdots\!84)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + \cdots - 1189491369116)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 61\!\cdots\!24)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + \cdots - 12\!\cdots\!56)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 18\!\cdots\!16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 28\!\cdots\!04)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 52\!\cdots\!56)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 37\!\cdots\!04)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 20\!\cdots\!04)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots + 32\!\cdots\!36)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 12\!\cdots\!16)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 49\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 55\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots - 36\!\cdots\!76)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 57\!\cdots\!44)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 15\!\cdots\!84)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 90\!\cdots\!16)^{2} \) Copy content Toggle raw display
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