Properties

Label 10.4.b.a
Level $10$
Weight $4$
Character orbit 10.b
Analytic conductor $0.590$
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] = [10,4,Mod(9,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.9");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 10.b (of order \(2\), degree \(1\), 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: \(0.590019100057\)
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: \( 2 \)
Twist minimal: yes
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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} - \beta q^{3} - 4 q^{4} + ( - 5 \beta - 5) q^{5} + 4 q^{6} + 13 \beta q^{7} - 4 \beta q^{8} + 23 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} - \beta q^{3} - 4 q^{4} + ( - 5 \beta - 5) q^{5} + 4 q^{6} + 13 \beta q^{7} - 4 \beta q^{8} + 23 q^{9} + ( - 5 \beta + 20) q^{10} - 28 q^{11} + 4 \beta q^{12} - 6 \beta q^{13} - 52 q^{14} + (5 \beta - 20) q^{15} + 16 q^{16} - 32 \beta q^{17} + 23 \beta q^{18} + 60 q^{19} + (20 \beta + 20) q^{20} + 52 q^{21} - 28 \beta q^{22} + 29 \beta q^{23} - 16 q^{24} + (50 \beta - 75) q^{25} + 24 q^{26} - 50 \beta q^{27} - 52 \beta q^{28} - 90 q^{29} + ( - 20 \beta - 20) q^{30} - 128 q^{31} + 16 \beta q^{32} + 28 \beta q^{33} + 128 q^{34} + ( - 65 \beta + 260) q^{35} - 92 q^{36} + 118 \beta q^{37} + 60 \beta q^{38} - 24 q^{39} + (20 \beta - 80) q^{40} + 242 q^{41} + 52 \beta q^{42} - 181 \beta q^{43} + 112 q^{44} + ( - 115 \beta - 115) q^{45} - 116 q^{46} + 113 \beta q^{47} - 16 \beta q^{48} - 333 q^{49} + ( - 75 \beta - 200) q^{50} - 128 q^{51} + 24 \beta q^{52} + 54 \beta q^{53} + 200 q^{54} + (140 \beta + 140) q^{55} + 208 q^{56} - 60 \beta q^{57} - 90 \beta q^{58} + 20 q^{59} + ( - 20 \beta + 80) q^{60} + 542 q^{61} - 128 \beta q^{62} + 299 \beta q^{63} - 64 q^{64} + (30 \beta - 120) q^{65} - 112 q^{66} - 217 \beta q^{67} + 128 \beta q^{68} + 116 q^{69} + (260 \beta + 260) q^{70} - 1128 q^{71} - 92 \beta q^{72} - 316 \beta q^{73} - 472 q^{74} + (75 \beta + 200) q^{75} - 240 q^{76} - 364 \beta q^{77} - 24 \beta q^{78} + 720 q^{79} + ( - 80 \beta - 80) q^{80} + 421 q^{81} + 242 \beta q^{82} + 239 \beta q^{83} - 208 q^{84} + (160 \beta - 640) q^{85} + 724 q^{86} + 90 \beta q^{87} + 112 \beta q^{88} + 490 q^{89} + ( - 115 \beta + 460) q^{90} + 312 q^{91} - 116 \beta q^{92} + 128 \beta q^{93} - 452 q^{94} + ( - 300 \beta - 300) q^{95} + 64 q^{96} + 728 \beta q^{97} - 333 \beta q^{98} - 644 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4} - 10 q^{5} + 8 q^{6} + 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} - 10 q^{5} + 8 q^{6} + 46 q^{9} + 40 q^{10} - 56 q^{11} - 104 q^{14} - 40 q^{15} + 32 q^{16} + 120 q^{19} + 40 q^{20} + 104 q^{21} - 32 q^{24} - 150 q^{25} + 48 q^{26} - 180 q^{29} - 40 q^{30} - 256 q^{31} + 256 q^{34} + 520 q^{35} - 184 q^{36} - 48 q^{39} - 160 q^{40} + 484 q^{41} + 224 q^{44} - 230 q^{45} - 232 q^{46} - 666 q^{49} - 400 q^{50} - 256 q^{51} + 400 q^{54} + 280 q^{55} + 416 q^{56} + 40 q^{59} + 160 q^{60} + 1084 q^{61} - 128 q^{64} - 240 q^{65} - 224 q^{66} + 232 q^{69} + 520 q^{70} - 2256 q^{71} - 944 q^{74} + 400 q^{75} - 480 q^{76} + 1440 q^{79} - 160 q^{80} + 842 q^{81} - 416 q^{84} - 1280 q^{85} + 1448 q^{86} + 980 q^{89} + 920 q^{90} + 624 q^{91} - 904 q^{94} - 600 q^{95} + 128 q^{96} - 1288 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).

\(n\) \(7\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
1.00000i
1.00000i
2.00000i 2.00000i −4.00000 −5.00000 + 10.0000i 4.00000 26.0000i 8.00000i 23.0000 20.0000 + 10.0000i
9.2 2.00000i 2.00000i −4.00000 −5.00000 10.0000i 4.00000 26.0000i 8.00000i 23.0000 20.0000 10.0000i
\(n\): 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 10.4.b.a 2
3.b odd 2 1 90.4.c.b 2
4.b odd 2 1 80.4.c.a 2
5.b even 2 1 inner 10.4.b.a 2
5.c odd 4 1 50.4.a.b 1
5.c odd 4 1 50.4.a.d 1
7.b odd 2 1 490.4.c.b 2
8.b even 2 1 320.4.c.d 2
8.d odd 2 1 320.4.c.c 2
12.b even 2 1 720.4.f.f 2
15.d odd 2 1 90.4.c.b 2
15.e even 4 1 450.4.a.j 1
15.e even 4 1 450.4.a.k 1
20.d odd 2 1 80.4.c.a 2
20.e even 4 1 400.4.a.h 1
20.e even 4 1 400.4.a.n 1
35.c odd 2 1 490.4.c.b 2
35.f even 4 1 2450.4.a.o 1
35.f even 4 1 2450.4.a.bb 1
40.e odd 2 1 320.4.c.c 2
40.f even 2 1 320.4.c.d 2
40.i odd 4 1 1600.4.a.u 1
40.i odd 4 1 1600.4.a.bh 1
40.k even 4 1 1600.4.a.t 1
40.k even 4 1 1600.4.a.bg 1
60.h even 2 1 720.4.f.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.b.a 2 1.a even 1 1 trivial
10.4.b.a 2 5.b even 2 1 inner
50.4.a.b 1 5.c odd 4 1
50.4.a.d 1 5.c odd 4 1
80.4.c.a 2 4.b odd 2 1
80.4.c.a 2 20.d odd 2 1
90.4.c.b 2 3.b odd 2 1
90.4.c.b 2 15.d odd 2 1
320.4.c.c 2 8.d odd 2 1
320.4.c.c 2 40.e odd 2 1
320.4.c.d 2 8.b even 2 1
320.4.c.d 2 40.f even 2 1
400.4.a.h 1 20.e even 4 1
400.4.a.n 1 20.e even 4 1
450.4.a.j 1 15.e even 4 1
450.4.a.k 1 15.e even 4 1
490.4.c.b 2 7.b odd 2 1
490.4.c.b 2 35.c odd 2 1
720.4.f.f 2 12.b even 2 1
720.4.f.f 2 60.h even 2 1
1600.4.a.t 1 40.k even 4 1
1600.4.a.u 1 40.i odd 4 1
1600.4.a.bg 1 40.k even 4 1
1600.4.a.bh 1 40.i odd 4 1
2450.4.a.o 1 35.f even 4 1
2450.4.a.bb 1 35.f even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(10, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{2} + 10T + 125 \) Copy content Toggle raw display
$7$ \( T^{2} + 676 \) Copy content Toggle raw display
$11$ \( (T + 28)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 144 \) Copy content Toggle raw display
$17$ \( T^{2} + 4096 \) Copy content Toggle raw display
$19$ \( (T - 60)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 3364 \) Copy content Toggle raw display
$29$ \( (T + 90)^{2} \) Copy content Toggle raw display
$31$ \( (T + 128)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 55696 \) Copy content Toggle raw display
$41$ \( (T - 242)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 131044 \) Copy content Toggle raw display
$47$ \( T^{2} + 51076 \) Copy content Toggle raw display
$53$ \( T^{2} + 11664 \) Copy content Toggle raw display
$59$ \( (T - 20)^{2} \) Copy content Toggle raw display
$61$ \( (T - 542)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 188356 \) Copy content Toggle raw display
$71$ \( (T + 1128)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 399424 \) Copy content Toggle raw display
$79$ \( (T - 720)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 228484 \) Copy content Toggle raw display
$89$ \( (T - 490)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 2119936 \) Copy content Toggle raw display
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