Properties

Label 1150.4.b.a
Level $1150$
Weight $4$
Character orbit 1150.b
Analytic conductor $67.852$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1150,4,Mod(599,1150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1150, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1150.599");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1150.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: \(67.8521965066\)
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 46)
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.

\(f(q)\) \(=\) \( q + 2 i q^{2} + 9 i q^{3} - 4 q^{4} - 18 q^{6} + 2 i q^{7} - 8 i q^{8} - 54 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 i q^{2} + 9 i q^{3} - 4 q^{4} - 18 q^{6} + 2 i q^{7} - 8 i q^{8} - 54 q^{9} - 52 q^{11} - 36 i q^{12} - 43 i q^{13} - 4 q^{14} + 16 q^{16} - 50 i q^{17} - 108 i q^{18} + 74 q^{19} - 18 q^{21} - 104 i q^{22} + 23 i q^{23} + 72 q^{24} + 86 q^{26} - 243 i q^{27} - 8 i q^{28} + 7 q^{29} - 273 q^{31} + 32 i q^{32} - 468 i q^{33} + 100 q^{34} + 216 q^{36} - 4 i q^{37} + 148 i q^{38} + 387 q^{39} + 123 q^{41} - 36 i q^{42} + 152 i q^{43} + 208 q^{44} - 46 q^{46} + 75 i q^{47} + 144 i q^{48} + 339 q^{49} + 450 q^{51} + 172 i q^{52} - 86 i q^{53} + 486 q^{54} + 16 q^{56} + 666 i q^{57} + 14 i q^{58} + 444 q^{59} + 262 q^{61} - 546 i q^{62} - 108 i q^{63} - 64 q^{64} + 936 q^{66} + 764 i q^{67} + 200 i q^{68} - 207 q^{69} - 21 q^{71} + 432 i q^{72} - 681 i q^{73} + 8 q^{74} - 296 q^{76} - 104 i q^{77} + 774 i q^{78} - 426 q^{79} + 729 q^{81} + 246 i q^{82} - 902 i q^{83} + 72 q^{84} - 304 q^{86} + 63 i q^{87} + 416 i q^{88} + 1272 q^{89} + 86 q^{91} - 92 i q^{92} - 2457 i q^{93} - 150 q^{94} - 288 q^{96} - 342 i q^{97} + 678 i q^{98} + 2808 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4} - 36 q^{6} - 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} - 36 q^{6} - 108 q^{9} - 104 q^{11} - 8 q^{14} + 32 q^{16} + 148 q^{19} - 36 q^{21} + 144 q^{24} + 172 q^{26} + 14 q^{29} - 546 q^{31} + 200 q^{34} + 432 q^{36} + 774 q^{39} + 246 q^{41} + 416 q^{44} - 92 q^{46} + 678 q^{49} + 900 q^{51} + 972 q^{54} + 32 q^{56} + 888 q^{59} + 524 q^{61} - 128 q^{64} + 1872 q^{66} - 414 q^{69} - 42 q^{71} + 16 q^{74} - 592 q^{76} - 852 q^{79} + 1458 q^{81} + 144 q^{84} - 608 q^{86} + 2544 q^{89} + 172 q^{91} - 300 q^{94} - 576 q^{96} + 5616 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
599.1
1.00000i
1.00000i
2.00000i 9.00000i −4.00000 0 −18.0000 2.00000i 8.00000i −54.0000 0
599.2 2.00000i 9.00000i −4.00000 0 −18.0000 2.00000i 8.00000i −54.0000 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 1150.4.b.a 2
5.b even 2 1 inner 1150.4.b.a 2
5.c odd 4 1 46.4.a.b 1
5.c odd 4 1 1150.4.a.d 1
15.e even 4 1 414.4.a.b 1
20.e even 4 1 368.4.a.e 1
35.f even 4 1 2254.4.a.b 1
40.i odd 4 1 1472.4.a.j 1
40.k even 4 1 1472.4.a.a 1
115.e even 4 1 1058.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
46.4.a.b 1 5.c odd 4 1
368.4.a.e 1 20.e even 4 1
414.4.a.b 1 15.e even 4 1
1058.4.a.b 1 115.e even 4 1
1150.4.a.d 1 5.c odd 4 1
1150.4.b.a 2 1.a even 1 1 trivial
1150.4.b.a 2 5.b even 2 1 inner
1472.4.a.a 1 40.k even 4 1
1472.4.a.j 1 40.i odd 4 1
2254.4.a.b 1 35.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1150, [\chi])\):

\( T_{3}^{2} + 81 \) Copy content Toggle raw display
\( T_{7}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + 81 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T + 52)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1849 \) Copy content Toggle raw display
$17$ \( T^{2} + 2500 \) Copy content Toggle raw display
$19$ \( (T - 74)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 529 \) Copy content Toggle raw display
$29$ \( (T - 7)^{2} \) Copy content Toggle raw display
$31$ \( (T + 273)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 16 \) Copy content Toggle raw display
$41$ \( (T - 123)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 23104 \) Copy content Toggle raw display
$47$ \( T^{2} + 5625 \) Copy content Toggle raw display
$53$ \( T^{2} + 7396 \) Copy content Toggle raw display
$59$ \( (T - 444)^{2} \) Copy content Toggle raw display
$61$ \( (T - 262)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 583696 \) Copy content Toggle raw display
$71$ \( (T + 21)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 463761 \) Copy content Toggle raw display
$79$ \( (T + 426)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 813604 \) Copy content Toggle raw display
$89$ \( (T - 1272)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 116964 \) Copy content Toggle raw display
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