Properties

Label 150.15.b.a
Level $150$
Weight $15$
Character orbit 150.b
Analytic conductor $186.493$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [150,15,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, 15, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("150.149");
 
S:= CuspForms(chi, 15);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 15 \)
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: \(186.493452228\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.98344960000.9
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 127x^{4} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{34}\cdot 3^{12} \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{7} + 7 \beta_{2} - 410 \beta_1) q^{3} + 8192 q^{4} + (16 \beta_{5} - 56 \beta_{4} - 811 \beta_{3} + 59904) q^{6} + (148 \beta_{7} - 259 \beta_{6} - 111 \beta_{2} + 206685 \beta_1) q^{7} + 8192 \beta_{2} q^{8} + ( - 585 \beta_{5} - 1638 \beta_{4} + 28341 \beta_{3} + \cdots + 538407) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{7} + 7 \beta_{2} - 410 \beta_1) q^{3} + 8192 q^{4} + (16 \beta_{5} - 56 \beta_{4} - 811 \beta_{3} + 59904) q^{6} + (148 \beta_{7} - 259 \beta_{6} - 111 \beta_{2} + 206685 \beta_1) q^{7} + 8192 \beta_{2} q^{8} + ( - 585 \beta_{5} - 1638 \beta_{4} + 28341 \beta_{3} + \cdots + 538407) q^{9}+ \cdots + (2024145153 \beta_{5} + 28230095484 \beta_{4} + \cdots + 39602152594368) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 65536 q^{4} + 479232 q^{6} + 4307256 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 65536 q^{4} + 479232 q^{6} + 4307256 q^{9} + 536870912 q^{16} + 2504135216 q^{19} - 5024328624 q^{21} + 3925868544 q^{24} + 97088971024 q^{31} + 146362171392 q^{34} + 35285041152 q^{36} - 352226543760 q^{39} + 2814157996032 q^{46} - 1092835748760 q^{49} + 660779265024 q^{51} + 6745173626880 q^{54} - 16017013867568 q^{61} + 4398046511104 q^{64} - 25890677858304 q^{66} - 13533755844096 q^{69} + 20513875689472 q^{76} + 160584105446384 q^{79} + 70871263643016 q^{81} - 41159300087808 q^{84} + 241474068137120 q^{91} + 438992555704320 q^{94} + 32160715112448 q^{96} + 316817220754944 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 127x^{4} + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{6} - 416\nu^{2} ) / 1377 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 512\nu^{7} + 5184\nu^{5} + 18368\nu^{3} + 285120\nu ) / 12393 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -512\nu^{7} + 5184\nu^{5} - 18368\nu^{3} + 285120\nu ) / 12393 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -7616\nu^{7} + 24624\nu^{5} + 209952\nu^{4} - 1165520\nu^{3} + 8492688\nu + 13331952 ) / 12393 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -64\nu^{7} + 648\nu^{5} - 314928\nu^{4} - 2296\nu^{3} + 35640\nu - 19997928 ) / 4131 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -7616\nu^{7} - 22032\nu^{6} - 24624\nu^{5} - 1165520\nu^{3} - 1013472\nu^{2} - 8492688\nu ) / 12393 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3904\nu^{7} - 38565\nu^{6} + 13284\nu^{5} + 586204\nu^{3} - 1775448\nu^{2} + 4299804\nu ) / 12393 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 8\beta_{7} - 14\beta_{6} + 4\beta_{5} + 18\beta_{4} - 87\beta_{3} - 87\beta_{2} - 4\beta_1 ) / 20736 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 16\beta_{7} + 8\beta_{6} - 3\beta_{2} - 44072\beta_1 ) / 10368 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 32\beta_{7} - 56\beta_{6} - 16\beta_{5} - 72\beta_{4} + 1077\beta_{3} - 1077\beta_{2} - 16\beta_1 ) / 10368 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -136\beta_{5} + 51\beta_{3} - 658368 ) / 10368 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -440\beta_{7} + 770\beta_{6} - 220\beta_{5} - 990\beta_{4} + 29571\beta_{3} + 29571\beta_{2} + 220\beta_1 ) / 20736 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -208\beta_{7} - 104\beta_{6} + 39\beta_{2} + 126788\beta_1 ) / 648 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2296 \beta_{7} + 4018 \beta_{6} + 1148 \beta_{5} + 5166 \beta_{4} - 328233 \beta_{3} + 328233 \beta_{2} + 1148 \beta_1 ) / 20736 \) Copy content Toggle raw display

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
−1.73810 2.44520i
−1.73810 + 2.44520i
2.44520 1.73810i
2.44520 + 1.73810i
1.73810 2.44520i
1.73810 + 2.44520i
−2.44520 1.73810i
−2.44520 + 1.73810i
−90.5097 −2152.70 385.790i 8192.00 0 194841. + 34917.8i 1.21591e6i −741455. 4.48530e6 + 1.66099e6i 0
149.2 −90.5097 −2152.70 + 385.790i 8192.00 0 194841. 34917.8i 1.21591e6i −741455. 4.48530e6 1.66099e6i 0
149.3 −90.5097 829.000 2023.79i 8192.00 0 −75032.5 + 183173.i 388872.i −741455. −3.40849e6 3.35545e6i 0
149.4 −90.5097 829.000 + 2023.79i 8192.00 0 −75032.5 183173.i 388872.i −741455. −3.40849e6 + 3.35545e6i 0
149.5 90.5097 −829.000 2023.79i 8192.00 0 −75032.5 183173.i 388872.i 741455. −3.40849e6 + 3.35545e6i 0
149.6 90.5097 −829.000 + 2023.79i 8192.00 0 −75032.5 + 183173.i 388872.i 741455. −3.40849e6 3.35545e6i 0
149.7 90.5097 2152.70 385.790i 8192.00 0 194841. 34917.8i 1.21591e6i 741455. 4.48530e6 1.66099e6i 0
149.8 90.5097 2152.70 + 385.790i 8192.00 0 194841. + 34917.8i 1.21591e6i 741455. 4.48530e6 + 1.66099e6i 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 150.15.b.a 8
3.b odd 2 1 inner 150.15.b.a 8
5.b even 2 1 inner 150.15.b.a 8
5.c odd 4 1 6.15.b.a 4
5.c odd 4 1 150.15.d.a 4
15.d odd 2 1 inner 150.15.b.a 8
15.e even 4 1 6.15.b.a 4
15.e even 4 1 150.15.d.a 4
20.e even 4 1 48.15.e.d 4
60.l odd 4 1 48.15.e.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.15.b.a 4 5.c odd 4 1
6.15.b.a 4 15.e even 4 1
48.15.e.d 4 20.e even 4 1
48.15.e.d 4 60.l odd 4 1
150.15.b.a 8 1.a even 1 1 trivial
150.15.b.a 8 3.b odd 2 1 inner
150.15.b.a 8 5.b even 2 1 inner
150.15.b.a 8 15.d odd 2 1 inner
150.15.d.a 4 5.c odd 4 1
150.15.d.a 4 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 1629655082888T_{7}^{2} + 223571300080310387289616 \) acting on \(S_{15}^{\mathrm{new}}(150, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 8192)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} - 2153628 T^{6} + \cdots + 52\!\cdots\!21 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 1629655082888 T^{2} + \cdots + 22\!\cdots\!16)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 30\!\cdots\!44)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 15\!\cdots\!44)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 626033804 T - 11\!\cdots\!76)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 47\!\cdots\!64)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 24272242756 T - 22\!\cdots\!16)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 14\!\cdots\!56)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 11\!\cdots\!64)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 19\!\cdots\!36)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 88\!\cdots\!24)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 10\!\cdots\!64)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 4004253466892 T - 99\!\cdots\!04)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 12\!\cdots\!76)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 40\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 40146026361596 T + 39\!\cdots\!24)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 20\!\cdots\!04)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 19\!\cdots\!64)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 19\!\cdots\!76)^{2} \) Copy content Toggle raw display
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