Properties

Label 1150.2.b.f
Level $1150$
Weight $2$
Character orbit 1150.b
Analytic conductor $9.183$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1150,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1150.599");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
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: \(9.18279623245\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) 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 230)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - \beta_{2} + \beta_1) q^{3} - q^{4} + (\beta_{3} - 2) q^{6} + (2 \beta_{2} + \beta_1) q^{7} + \beta_{2} q^{8} + (3 \beta_{3} - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + ( - \beta_{2} + \beta_1) q^{3} - q^{4} + (\beta_{3} - 2) q^{6} + (2 \beta_{2} + \beta_1) q^{7} + \beta_{2} q^{8} + (3 \beta_{3} - 4) q^{9} + (\beta_{3} - 4) q^{11} + (\beta_{2} - \beta_1) q^{12} + ( - 2 \beta_{2} - \beta_1) q^{13} + (\beta_{3} + 1) q^{14} + q^{16} + (3 \beta_{2} + 3 \beta_1) q^{17} + (\beta_{2} - 3 \beta_1) q^{18} + ( - 3 \beta_{3} + 1) q^{19} - q^{21} + (3 \beta_{2} - \beta_1) q^{22} + \beta_{2} q^{23} + ( - \beta_{3} + 2) q^{24} + ( - \beta_{3} - 1) q^{26} + (7 \beta_{2} - 4 \beta_1) q^{27} + ( - 2 \beta_{2} - \beta_1) q^{28} + (2 \beta_{3} - 2) q^{29} + ( - 3 \beta_{3} - 1) q^{31} - \beta_{2} q^{32} + (6 \beta_{2} - 5 \beta_1) q^{33} + 3 \beta_{3} q^{34} + ( - 3 \beta_{3} + 4) q^{36} + 8 \beta_{2} q^{37} + (2 \beta_{2} + 3 \beta_1) q^{38} + q^{39} + (3 \beta_{3} - 6) q^{41} + \beta_{2} q^{42} + (4 \beta_{2} + 4 \beta_1) q^{43} + ( - \beta_{3} + 4) q^{44} + q^{46} - 2 \beta_1 q^{47} + ( - \beta_{2} + \beta_1) q^{48} + ( - 3 \beta_{3} + 3) q^{49} + (3 \beta_{3} - 9) q^{51} + (2 \beta_{2} + \beta_1) q^{52} + (6 \beta_{2} + 4 \beta_1) q^{53} + ( - 4 \beta_{3} + 11) q^{54} + ( - \beta_{3} - 1) q^{56} + ( - 7 \beta_{2} + 4 \beta_1) q^{57} - 2 \beta_1 q^{58} + ( - 2 \beta_{3} + 8) q^{59} + 5 \beta_{3} q^{61} + (4 \beta_{2} + 3 \beta_1) q^{62} + (7 \beta_{2} + 2 \beta_1) q^{63} - q^{64} + ( - 5 \beta_{3} + 11) q^{66} - 4 \beta_{2} q^{67} + ( - 3 \beta_{2} - 3 \beta_1) q^{68} + ( - \beta_{3} + 2) q^{69} + ( - \beta_{3} - 14) q^{71} + ( - \beta_{2} + 3 \beta_1) q^{72} + ( - 2 \beta_{2} + 6 \beta_1) q^{73} + 8 q^{74} + (3 \beta_{3} - 1) q^{76} + ( - 3 \beta_{2} - 2 \beta_1) q^{77} - \beta_{2} q^{78} + (8 \beta_{3} - 4) q^{79} + ( - 6 \beta_{3} + 22) q^{81} + (3 \beta_{2} - 3 \beta_1) q^{82} + ( - 6 \beta_{2} - 4 \beta_1) q^{83} + q^{84} + 4 \beta_{3} q^{86} + (6 \beta_{2} - 4 \beta_1) q^{87} + ( - 3 \beta_{2} + \beta_1) q^{88} + (3 \beta_{3} + 4) q^{91} - \beta_{2} q^{92} + ( - 5 \beta_{2} + 2 \beta_1) q^{93} + ( - 2 \beta_{3} + 2) q^{94} + (\beta_{3} - 2) q^{96} + (5 \beta_{2} + \beta_1) q^{97} + 3 \beta_1 q^{98} + ( - 13 \beta_{3} + 25) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 6 q^{6} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 6 q^{6} - 10 q^{9} - 14 q^{11} + 6 q^{14} + 4 q^{16} - 2 q^{19} - 4 q^{21} + 6 q^{24} - 6 q^{26} - 4 q^{29} - 10 q^{31} + 6 q^{34} + 10 q^{36} + 4 q^{39} - 18 q^{41} + 14 q^{44} + 4 q^{46} + 6 q^{49} - 30 q^{51} + 36 q^{54} - 6 q^{56} + 28 q^{59} + 10 q^{61} - 4 q^{64} + 34 q^{66} + 6 q^{69} - 58 q^{71} + 32 q^{74} + 2 q^{76} + 76 q^{81} + 4 q^{84} + 8 q^{86} + 22 q^{91} + 4 q^{94} - 6 q^{96} + 74 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 4\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{2} - 4\beta_1 \) 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
2.30278i
1.30278i
1.30278i
2.30278i
1.00000i 3.30278i −1.00000 0 −3.30278 0.302776i 1.00000i −7.90833 0
599.2 1.00000i 0.302776i −1.00000 0 0.302776 3.30278i 1.00000i 2.90833 0
599.3 1.00000i 0.302776i −1.00000 0 0.302776 3.30278i 1.00000i 2.90833 0
599.4 1.00000i 3.30278i −1.00000 0 −3.30278 0.302776i 1.00000i −7.90833 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 1150.2.b.f 4
5.b even 2 1 inner 1150.2.b.f 4
5.c odd 4 1 230.2.a.b 2
5.c odd 4 1 1150.2.a.m 2
15.e even 4 1 2070.2.a.w 2
20.e even 4 1 1840.2.a.j 2
20.e even 4 1 9200.2.a.ca 2
40.i odd 4 1 7360.2.a.bc 2
40.k even 4 1 7360.2.a.bu 2
115.e even 4 1 5290.2.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.b 2 5.c odd 4 1
1150.2.a.m 2 5.c odd 4 1
1150.2.b.f 4 1.a even 1 1 trivial
1150.2.b.f 4 5.b even 2 1 inner
1840.2.a.j 2 20.e even 4 1
2070.2.a.w 2 15.e even 4 1
5290.2.a.j 2 115.e even 4 1
7360.2.a.bc 2 40.i odd 4 1
7360.2.a.bu 2 40.k even 4 1
9200.2.a.ca 2 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1150, [\chi])\):

\( T_{3}^{4} + 11T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} + 11T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} + 7T_{11} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 11T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 11T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T^{2} + 7 T + 9)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 11T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{4} + 63T^{2} + 729 \) Copy content Toggle raw display
$19$ \( (T^{2} + T - 29)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2 T - 12)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 5 T - 23)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 9 T - 9)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 112T^{2} + 2304 \) Copy content Toggle raw display
$47$ \( T^{4} + 28T^{2} + 144 \) Copy content Toggle raw display
$53$ \( T^{4} + 136T^{2} + 1296 \) Copy content Toggle raw display
$59$ \( (T^{2} - 14 T + 36)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 5 T - 75)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 29 T + 207)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 284T^{2} + 8464 \) Copy content Toggle raw display
$79$ \( (T^{2} - 208)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 136T^{2} + 1296 \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 47T^{2} + 289 \) Copy content Toggle raw display
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