Properties

Label 150.4.e.a
Level $150$
Weight $4$
Character orbit 150.e
Analytic conductor $8.850$
Analytic rank $0$
Dimension $4$
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] = [150,4,Mod(107,150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(150, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("150.107");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 150.e (of order \(4\), degree \(2\), 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: \(8.85028650086\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \zeta_{8} q^{2} + (3 \zeta_{8}^{3} - 3 \zeta_{8}^{2} - 3) q^{3} + 4 \zeta_{8}^{2} q^{4} + ( - 6 \zeta_{8}^{3} - 6 \zeta_{8} - 6) q^{6} + ( - 18 \zeta_{8}^{2} + 18) q^{7} + 8 \zeta_{8}^{3} q^{8} + ( - 18 \zeta_{8}^{3} + 9 \zeta_{8}^{2} + 18 \zeta_{8}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \zeta_{8} q^{2} + (3 \zeta_{8}^{3} - 3 \zeta_{8}^{2} - 3) q^{3} + 4 \zeta_{8}^{2} q^{4} + ( - 6 \zeta_{8}^{3} - 6 \zeta_{8} - 6) q^{6} + ( - 18 \zeta_{8}^{2} + 18) q^{7} + 8 \zeta_{8}^{3} q^{8} + ( - 18 \zeta_{8}^{3} + 9 \zeta_{8}^{2} + 18 \zeta_{8}) q^{9} + (36 \zeta_{8}^{3} + 36 \zeta_{8}) q^{11} + ( - 12 \zeta_{8}^{2} - 12 \zeta_{8} + 12) q^{12} + ( - 36 \zeta_{8}^{3} + 36 \zeta_{8}) q^{14} - 16 q^{16} + 84 \zeta_{8} q^{17} + (18 \zeta_{8}^{3} + 36 \zeta_{8}^{2} + 36) q^{18} + 124 \zeta_{8}^{2} q^{19} + (54 \zeta_{8}^{3} + 54 \zeta_{8} - 108) q^{21} + (72 \zeta_{8}^{2} - 72) q^{22} - 42 \zeta_{8}^{3} q^{23} + ( - 24 \zeta_{8}^{3} - 24 \zeta_{8}^{2} + 24 \zeta_{8}) q^{24} + (27 \zeta_{8}^{2} - 135 \zeta_{8} - 27) q^{27} + (72 \zeta_{8}^{2} + 72) q^{28} + ( - 180 \zeta_{8}^{3} + 180 \zeta_{8}) q^{29} + 88 q^{31} - 32 \zeta_{8} q^{32} + ( - 216 \zeta_{8}^{3} - 108 \zeta_{8}^{2} - 108) q^{33} + 168 \zeta_{8}^{2} q^{34} + (72 \zeta_{8}^{3} + 72 \zeta_{8} - 36) q^{36} + ( - 72 \zeta_{8}^{2} + 72) q^{37} + 248 \zeta_{8}^{3} q^{38} + ( - 36 \zeta_{8}^{3} - 36 \zeta_{8}) q^{41} + (108 \zeta_{8}^{2} - 216 \zeta_{8} - 108) q^{42} + ( - 342 \zeta_{8}^{2} - 342) q^{43} + (144 \zeta_{8}^{3} - 144 \zeta_{8}) q^{44} + 84 q^{46} + 450 \zeta_{8} q^{47} + ( - 48 \zeta_{8}^{3} + 48 \zeta_{8}^{2} + 48) q^{48} - 305 \zeta_{8}^{2} q^{49} + ( - 252 \zeta_{8}^{3} - 252 \zeta_{8} - 252) q^{51} + 312 \zeta_{8}^{3} q^{53} + (54 \zeta_{8}^{3} - 270 \zeta_{8}^{2} - 54 \zeta_{8}) q^{54} + (144 \zeta_{8}^{3} + 144 \zeta_{8}) q^{56} + ( - 372 \zeta_{8}^{2} - 372 \zeta_{8} + 372) q^{57} + (360 \zeta_{8}^{2} + 360) q^{58} + (612 \zeta_{8}^{3} - 612 \zeta_{8}) q^{59} + 434 q^{61} + 176 \zeta_{8} q^{62} + ( - 648 \zeta_{8}^{3} + 162 \zeta_{8}^{2} + 162) q^{63} - 64 \zeta_{8}^{2} q^{64} + ( - 216 \zeta_{8}^{3} - 216 \zeta_{8} + 432) q^{66} + (18 \zeta_{8}^{2} - 18) q^{67} + 336 \zeta_{8}^{3} q^{68} + (126 \zeta_{8}^{3} + 126 \zeta_{8}^{2} - 126 \zeta_{8}) q^{69} + ( - 360 \zeta_{8}^{3} - 360 \zeta_{8}) q^{71} + (144 \zeta_{8}^{2} - 72 \zeta_{8} - 144) q^{72} + (360 \zeta_{8}^{2} + 360) q^{73} + ( - 144 \zeta_{8}^{3} + 144 \zeta_{8}) q^{74} - 496 q^{76} + 1296 \zeta_{8} q^{77} - 1024 \zeta_{8}^{2} q^{79} + (324 \zeta_{8}^{3} + 324 \zeta_{8} + 567) q^{81} + ( - 72 \zeta_{8}^{2} + 72) q^{82} + 246 \zeta_{8}^{3} q^{83} + (216 \zeta_{8}^{3} - 432 \zeta_{8}^{2} - 216 \zeta_{8}) q^{84} + ( - 684 \zeta_{8}^{3} - 684 \zeta_{8}) q^{86} + (540 \zeta_{8}^{2} - 1080 \zeta_{8} - 540) q^{87} + ( - 288 \zeta_{8}^{2} - 288) q^{88} + (72 \zeta_{8}^{3} - 72 \zeta_{8}) q^{89} + 168 \zeta_{8} q^{92} + (264 \zeta_{8}^{3} - 264 \zeta_{8}^{2} - 264) q^{93} + 900 \zeta_{8}^{2} q^{94} + (96 \zeta_{8}^{3} + 96 \zeta_{8} + 96) q^{96} + ( - 216 \zeta_{8}^{2} + 216) q^{97} - 610 \zeta_{8}^{3} q^{98} + (324 \zeta_{8}^{3} + 1296 \zeta_{8}^{2} - 324 \zeta_{8}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{3} - 24 q^{6} + 72 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{3} - 24 q^{6} + 72 q^{7} + 48 q^{12} - 64 q^{16} + 144 q^{18} - 432 q^{21} - 288 q^{22} - 108 q^{27} + 288 q^{28} + 352 q^{31} - 432 q^{33} - 144 q^{36} + 288 q^{37} - 432 q^{42} - 1368 q^{43} + 336 q^{46} + 192 q^{48} - 1008 q^{51} + 1488 q^{57} + 1440 q^{58} + 1736 q^{61} + 648 q^{63} + 1728 q^{66} - 72 q^{67} - 576 q^{72} + 1440 q^{73} - 1984 q^{76} + 2268 q^{81} + 288 q^{82} - 2160 q^{87} - 1152 q^{88} - 1056 q^{93} + 384 q^{96} + 864 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(-\zeta_{8}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
−1.41421 + 1.41421i −0.878680 + 5.12132i 4.00000i 0 −6.00000 8.48528i 18.0000 + 18.0000i 5.65685 + 5.65685i −25.4558 9.00000i 0
107.2 1.41421 1.41421i −5.12132 + 0.878680i 4.00000i 0 −6.00000 + 8.48528i 18.0000 + 18.0000i −5.65685 5.65685i 25.4558 9.00000i 0
143.1 −1.41421 1.41421i −0.878680 5.12132i 4.00000i 0 −6.00000 + 8.48528i 18.0000 18.0000i 5.65685 5.65685i −25.4558 + 9.00000i 0
143.2 1.41421 + 1.41421i −5.12132 0.878680i 4.00000i 0 −6.00000 8.48528i 18.0000 18.0000i −5.65685 + 5.65685i 25.4558 + 9.00000i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 150.4.e.a 4
3.b odd 2 1 inner 150.4.e.a 4
5.b even 2 1 150.4.e.b yes 4
5.c odd 4 1 inner 150.4.e.a 4
5.c odd 4 1 150.4.e.b yes 4
15.d odd 2 1 150.4.e.b yes 4
15.e even 4 1 inner 150.4.e.a 4
15.e even 4 1 150.4.e.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.4.e.a 4 1.a even 1 1 trivial
150.4.e.a 4 3.b odd 2 1 inner
150.4.e.a 4 5.c odd 4 1 inner
150.4.e.a 4 15.e even 4 1 inner
150.4.e.b yes 4 5.b even 2 1
150.4.e.b yes 4 5.c odd 4 1
150.4.e.b yes 4 15.d odd 2 1
150.4.e.b yes 4 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} - 36T_{7} + 648 \) acting on \(S_{4}^{\mathrm{new}}(150, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 16 \) Copy content Toggle raw display
$3$ \( T^{4} + 12 T^{3} + 72 T^{2} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 36 T + 648)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2592)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 49787136 \) Copy content Toggle raw display
$19$ \( (T^{2} + 15376)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 3111696 \) Copy content Toggle raw display
$29$ \( (T^{2} - 64800)^{2} \) Copy content Toggle raw display
$31$ \( (T - 88)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 144 T + 10368)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 2592)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 684 T + 233928)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 41006250000 \) Copy content Toggle raw display
$53$ \( T^{4} + 9475854336 \) Copy content Toggle raw display
$59$ \( (T^{2} - 749088)^{2} \) Copy content Toggle raw display
$61$ \( (T - 434)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 36 T + 648)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 259200)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 720 T + 259200)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 1048576)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 3662186256 \) Copy content Toggle raw display
$89$ \( (T^{2} - 10368)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 432 T + 93312)^{2} \) Copy content Toggle raw display
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