Properties

Label 130.3.h.b
Level $130$
Weight $3$
Character orbit 130.h
Analytic conductor $3.542$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [130,3,Mod(77,130)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(130, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("130.77");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 130 = 2 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 130.h (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: \(3.54224343668\)
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: \(\Z[a_1, a_2]\)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - i + 1) q^{2} + (2 i - 2) q^{3} - 2 i q^{4} + 5 i q^{5} + 4 i q^{6} + (2 i - 2) q^{7} + ( - 2 i - 2) q^{8} + i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - i + 1) q^{2} + (2 i - 2) q^{3} - 2 i q^{4} + 5 i q^{5} + 4 i q^{6} + (2 i - 2) q^{7} + ( - 2 i - 2) q^{8} + i q^{9} + (5 i + 5) q^{10} + 20 i q^{11} + (4 i + 4) q^{12} - 13 i q^{13} + 4 i q^{14} + ( - 10 i - 10) q^{15} - 4 q^{16} + (17 i + 17) q^{17} + (i + 1) q^{18} - 8 q^{19} + 10 q^{20} - 8 i q^{21} + (20 i + 20) q^{22} + ( - 18 i + 18) q^{23} + 8 q^{24} - 25 q^{25} + ( - 13 i - 13) q^{26} + ( - 20 i - 20) q^{27} + (4 i + 4) q^{28} - 40 i q^{29} - 20 q^{30} + 40 i q^{31} + (4 i - 4) q^{32} + ( - 40 i - 40) q^{33} + 34 q^{34} + ( - 10 i - 10) q^{35} + 2 q^{36} + ( - 43 i + 43) q^{37} + (8 i - 8) q^{38} + (26 i + 26) q^{39} + ( - 10 i + 10) q^{40} + 10 i q^{41} + ( - 8 i - 8) q^{42} + (2 i - 2) q^{43} + 40 q^{44} - 5 q^{45} - 36 i q^{46} + (22 i - 22) q^{47} + ( - 8 i + 8) q^{48} + 41 i q^{49} + (25 i - 25) q^{50} - 68 q^{51} - 26 q^{52} + ( - 13 i + 13) q^{53} - 40 q^{54} - 100 q^{55} + 8 q^{56} + ( - 16 i + 16) q^{57} + ( - 40 i - 40) q^{58} + 32 q^{59} + (20 i - 20) q^{60} + 112 q^{61} + (40 i + 40) q^{62} + ( - 2 i - 2) q^{63} + 8 i q^{64} + 65 q^{65} - 80 q^{66} + ( - 18 i + 18) q^{67} + ( - 34 i + 34) q^{68} + 72 i q^{69} - 20 q^{70} - 40 i q^{71} + ( - 2 i + 2) q^{72} + (7 i + 7) q^{73} - 86 i q^{74} + ( - 50 i + 50) q^{75} + 16 i q^{76} + ( - 40 i - 40) q^{77} + 52 q^{78} + 120 i q^{79} - 20 i q^{80} + 71 q^{81} + (10 i + 10) q^{82} + (22 i + 22) q^{83} - 16 q^{84} + (85 i - 85) q^{85} + 4 i q^{86} + (80 i + 80) q^{87} + ( - 40 i + 40) q^{88} + 82 q^{89} + (5 i - 5) q^{90} + (26 i + 26) q^{91} + ( - 36 i - 36) q^{92} + ( - 80 i - 80) q^{93} + 44 i q^{94} - 40 i q^{95} - 16 i q^{96} + ( - 33 i + 33) q^{97} + (41 i + 41) q^{98} - 20 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 4 q^{3} - 4 q^{7} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 4 q^{3} - 4 q^{7} - 4 q^{8} + 10 q^{10} + 8 q^{12} - 20 q^{15} - 8 q^{16} + 34 q^{17} + 2 q^{18} - 16 q^{19} + 20 q^{20} + 40 q^{22} + 36 q^{23} + 16 q^{24} - 50 q^{25} - 26 q^{26} - 40 q^{27} + 8 q^{28} - 40 q^{30} - 8 q^{32} - 80 q^{33} + 68 q^{34} - 20 q^{35} + 4 q^{36} + 86 q^{37} - 16 q^{38} + 52 q^{39} + 20 q^{40} - 16 q^{42} - 4 q^{43} + 80 q^{44} - 10 q^{45} - 44 q^{47} + 16 q^{48} - 50 q^{50} - 136 q^{51} - 52 q^{52} + 26 q^{53} - 80 q^{54} - 200 q^{55} + 16 q^{56} + 32 q^{57} - 80 q^{58} + 64 q^{59} - 40 q^{60} + 224 q^{61} + 80 q^{62} - 4 q^{63} + 130 q^{65} - 160 q^{66} + 36 q^{67} + 68 q^{68} - 40 q^{70} + 4 q^{72} + 14 q^{73} + 100 q^{75} - 80 q^{77} + 104 q^{78} + 142 q^{81} + 20 q^{82} + 44 q^{83} - 32 q^{84} - 170 q^{85} + 160 q^{87} + 80 q^{88} + 164 q^{89} - 10 q^{90} + 52 q^{91} - 72 q^{92} - 160 q^{93} + 66 q^{97} + 82 q^{98} - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/130\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(41\)
\(\chi(n)\) \(i\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
77.1
1.00000i
1.00000i
1.00000 1.00000i −2.00000 + 2.00000i 2.00000i 5.00000i 4.00000i −2.00000 + 2.00000i −2.00000 2.00000i 1.00000i 5.00000 + 5.00000i
103.1 1.00000 + 1.00000i −2.00000 2.00000i 2.00000i 5.00000i 4.00000i −2.00000 2.00000i −2.00000 + 2.00000i 1.00000i 5.00000 5.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.h odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 130.3.h.b yes 2
5.b even 2 1 650.3.h.a 2
5.c odd 4 1 130.3.h.a 2
5.c odd 4 1 650.3.h.b 2
13.b even 2 1 130.3.h.a 2
65.d even 2 1 650.3.h.b 2
65.h odd 4 1 inner 130.3.h.b yes 2
65.h odd 4 1 650.3.h.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.3.h.a 2 5.c odd 4 1
130.3.h.a 2 13.b even 2 1
130.3.h.b yes 2 1.a even 1 1 trivial
130.3.h.b yes 2 65.h odd 4 1 inner
650.3.h.a 2 5.b even 2 1
650.3.h.a 2 65.h odd 4 1
650.3.h.b 2 5.c odd 4 1
650.3.h.b 2 65.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(130, [\chi])\):

\( T_{3}^{2} + 4T_{3} + 8 \) Copy content Toggle raw display
\( T_{7}^{2} + 4T_{7} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$3$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$5$ \( T^{2} + 25 \) Copy content Toggle raw display
$7$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$11$ \( T^{2} + 400 \) Copy content Toggle raw display
$13$ \( T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{2} - 34T + 578 \) Copy content Toggle raw display
$19$ \( (T + 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 36T + 648 \) Copy content Toggle raw display
$29$ \( T^{2} + 1600 \) Copy content Toggle raw display
$31$ \( T^{2} + 1600 \) Copy content Toggle raw display
$37$ \( T^{2} - 86T + 3698 \) Copy content Toggle raw display
$41$ \( T^{2} + 100 \) Copy content Toggle raw display
$43$ \( T^{2} + 4T + 8 \) Copy content Toggle raw display
$47$ \( T^{2} + 44T + 968 \) Copy content Toggle raw display
$53$ \( T^{2} - 26T + 338 \) Copy content Toggle raw display
$59$ \( (T - 32)^{2} \) Copy content Toggle raw display
$61$ \( (T - 112)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 36T + 648 \) Copy content Toggle raw display
$71$ \( T^{2} + 1600 \) Copy content Toggle raw display
$73$ \( T^{2} - 14T + 98 \) Copy content Toggle raw display
$79$ \( T^{2} + 14400 \) Copy content Toggle raw display
$83$ \( T^{2} - 44T + 968 \) Copy content Toggle raw display
$89$ \( (T - 82)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 66T + 2178 \) Copy content Toggle raw display
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