Properties

Label 150.4.c.c
Level $150$
Weight $4$
Character orbit 150.c
Analytic conductor $8.850$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [150,4,Mod(49,150)]
 
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([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("150.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 150.c (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: \(8.85028650086\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 i q^{2} - 3 i q^{3} - 4 q^{4} + 6 q^{6} - 4 i q^{7} - 8 i q^{8} - 9 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 i q^{2} - 3 i q^{3} - 4 q^{4} + 6 q^{6} - 4 i q^{7} - 8 i q^{8} - 9 q^{9} - 48 q^{11} + 12 i q^{12} - 2 i q^{13} + 8 q^{14} + 16 q^{16} - 114 i q^{17} - 18 i q^{18} - 140 q^{19} - 12 q^{21} - 96 i q^{22} - 72 i q^{23} - 24 q^{24} + 4 q^{26} + 27 i q^{27} + 16 i q^{28} - 210 q^{29} + 272 q^{31} + 32 i q^{32} + 144 i q^{33} + 228 q^{34} + 36 q^{36} - 334 i q^{37} - 280 i q^{38} - 6 q^{39} - 198 q^{41} - 24 i q^{42} + 268 i q^{43} + 192 q^{44} + 144 q^{46} + 216 i q^{47} - 48 i q^{48} + 327 q^{49} - 342 q^{51} + 8 i q^{52} + 78 i q^{53} - 54 q^{54} - 32 q^{56} + 420 i q^{57} - 420 i q^{58} - 240 q^{59} + 302 q^{61} + 544 i q^{62} + 36 i q^{63} - 64 q^{64} - 288 q^{66} + 596 i q^{67} + 456 i q^{68} - 216 q^{69} - 768 q^{71} + 72 i q^{72} + 478 i q^{73} + 668 q^{74} + 560 q^{76} + 192 i q^{77} - 12 i q^{78} + 640 q^{79} + 81 q^{81} - 396 i q^{82} + 348 i q^{83} + 48 q^{84} - 536 q^{86} + 630 i q^{87} + 384 i q^{88} - 210 q^{89} - 8 q^{91} + 288 i q^{92} - 816 i q^{93} - 432 q^{94} + 96 q^{96} - 1534 i q^{97} + 654 i q^{98} + 432 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4} + 12 q^{6} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} + 12 q^{6} - 18 q^{9} - 96 q^{11} + 16 q^{14} + 32 q^{16} - 280 q^{19} - 24 q^{21} - 48 q^{24} + 8 q^{26} - 420 q^{29} + 544 q^{31} + 456 q^{34} + 72 q^{36} - 12 q^{39} - 396 q^{41} + 384 q^{44} + 288 q^{46} + 654 q^{49} - 684 q^{51} - 108 q^{54} - 64 q^{56} - 480 q^{59} + 604 q^{61} - 128 q^{64} - 576 q^{66} - 432 q^{69} - 1536 q^{71} + 1336 q^{74} + 1120 q^{76} + 1280 q^{79} + 162 q^{81} + 96 q^{84} - 1072 q^{86} - 420 q^{89} - 16 q^{91} - 864 q^{94} + 192 q^{96} + 864 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.00000i
1.00000i
2.00000i 3.00000i −4.00000 0 6.00000 4.00000i 8.00000i −9.00000 0
49.2 2.00000i 3.00000i −4.00000 0 6.00000 4.00000i 8.00000i −9.00000 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 150.4.c.c 2
3.b odd 2 1 450.4.c.j 2
4.b odd 2 1 1200.4.f.r 2
5.b even 2 1 inner 150.4.c.c 2
5.c odd 4 1 30.4.a.b 1
5.c odd 4 1 150.4.a.b 1
15.d odd 2 1 450.4.c.j 2
15.e even 4 1 90.4.a.c 1
15.e even 4 1 450.4.a.r 1
20.d odd 2 1 1200.4.f.r 2
20.e even 4 1 240.4.a.b 1
20.e even 4 1 1200.4.a.ba 1
35.f even 4 1 1470.4.a.r 1
40.i odd 4 1 960.4.a.n 1
40.k even 4 1 960.4.a.bg 1
45.k odd 12 2 810.4.e.i 2
45.l even 12 2 810.4.e.p 2
60.l odd 4 1 720.4.a.y 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.4.a.b 1 5.c odd 4 1
90.4.a.c 1 15.e even 4 1
150.4.a.b 1 5.c odd 4 1
150.4.c.c 2 1.a even 1 1 trivial
150.4.c.c 2 5.b even 2 1 inner
240.4.a.b 1 20.e even 4 1
450.4.a.r 1 15.e even 4 1
450.4.c.j 2 3.b odd 2 1
450.4.c.j 2 15.d odd 2 1
720.4.a.y 1 60.l odd 4 1
810.4.e.i 2 45.k odd 12 2
810.4.e.p 2 45.l even 12 2
960.4.a.n 1 40.i odd 4 1
960.4.a.bg 1 40.k even 4 1
1200.4.a.ba 1 20.e even 4 1
1200.4.f.r 2 4.b odd 2 1
1200.4.f.r 2 20.d odd 2 1
1470.4.a.r 1 35.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 16 \) acting on \(S_{4}^{\mathrm{new}}(150, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T + 48)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 12996 \) Copy content Toggle raw display
$19$ \( (T + 140)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 5184 \) Copy content Toggle raw display
$29$ \( (T + 210)^{2} \) Copy content Toggle raw display
$31$ \( (T - 272)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 111556 \) Copy content Toggle raw display
$41$ \( (T + 198)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 71824 \) Copy content Toggle raw display
$47$ \( T^{2} + 46656 \) Copy content Toggle raw display
$53$ \( T^{2} + 6084 \) Copy content Toggle raw display
$59$ \( (T + 240)^{2} \) Copy content Toggle raw display
$61$ \( (T - 302)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 355216 \) Copy content Toggle raw display
$71$ \( (T + 768)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 228484 \) Copy content Toggle raw display
$79$ \( (T - 640)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 121104 \) Copy content Toggle raw display
$89$ \( (T + 210)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 2353156 \) Copy content Toggle raw display
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