Properties

Label 6.5.b.a
Level $6$
Weight $5$
Character orbit 6.b
Analytic conductor $0.620$
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] = [6,5,Mod(5,6)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6.5");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 6.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: \(0.620219778503\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) 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 \(\beta = 2\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( - 3 \beta - 3) q^{3} - 8 q^{4} + 6 \beta q^{5} + ( - 3 \beta + 24) q^{6} + 26 q^{7} - 8 \beta q^{8} + (18 \beta - 63) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + ( - 3 \beta - 3) q^{3} - 8 q^{4} + 6 \beta q^{5} + ( - 3 \beta + 24) q^{6} + 26 q^{7} - 8 \beta q^{8} + (18 \beta - 63) q^{9} - 48 q^{10} - 42 \beta q^{11} + (24 \beta + 24) q^{12} + 50 q^{13} + 26 \beta q^{14} + ( - 18 \beta + 144) q^{15} + 64 q^{16} + 72 \beta q^{17} + ( - 63 \beta - 144) q^{18} - 358 q^{19} - 48 \beta q^{20} + ( - 78 \beta - 78) q^{21} + 336 q^{22} + 132 \beta q^{23} + (24 \beta - 192) q^{24} + 337 q^{25} + 50 \beta q^{26} + (135 \beta + 621) q^{27} - 208 q^{28} - 510 \beta q^{29} + (144 \beta + 144) q^{30} - 742 q^{31} + 64 \beta q^{32} + (126 \beta - 1008) q^{33} - 576 q^{34} + 156 \beta q^{35} + ( - 144 \beta + 504) q^{36} + 1874 q^{37} - 358 \beta q^{38} + ( - 150 \beta - 150) q^{39} + 384 q^{40} + 852 \beta q^{41} + ( - 78 \beta + 624) q^{42} - 262 q^{43} + 336 \beta q^{44} + ( - 378 \beta - 864) q^{45} - 1056 q^{46} - 600 \beta q^{47} + ( - 192 \beta - 192) q^{48} - 1725 q^{49} + 337 \beta q^{50} + ( - 216 \beta + 1728) q^{51} - 400 q^{52} - 162 \beta q^{53} + (621 \beta - 1080) q^{54} + 2016 q^{55} - 208 \beta q^{56} + (1074 \beta + 1074) q^{57} + 4080 q^{58} - 642 \beta q^{59} + (144 \beta - 1152) q^{60} - 1486 q^{61} - 742 \beta q^{62} + (468 \beta - 1638) q^{63} - 512 q^{64} + 300 \beta q^{65} + ( - 1008 \beta - 1008) q^{66} - 4486 q^{67} - 576 \beta q^{68} + ( - 396 \beta + 3168) q^{69} - 1248 q^{70} + 1260 \beta q^{71} + (504 \beta + 1152) q^{72} + 290 q^{73} + 1874 \beta q^{74} + ( - 1011 \beta - 1011) q^{75} + 2864 q^{76} - 1092 \beta q^{77} + ( - 150 \beta + 1200) q^{78} + 9818 q^{79} + 384 \beta q^{80} + ( - 2268 \beta + 1377) q^{81} - 6816 q^{82} + 2514 \beta q^{83} + (624 \beta + 624) q^{84} - 3456 q^{85} - 262 \beta q^{86} + (1530 \beta - 12240) q^{87} - 2688 q^{88} - 2772 \beta q^{89} + ( - 864 \beta + 3024) q^{90} + 1300 q^{91} - 1056 \beta q^{92} + (2226 \beta + 2226) q^{93} + 4800 q^{94} - 2148 \beta q^{95} + ( - 192 \beta + 1536) q^{96} - 478 q^{97} - 1725 \beta q^{98} + (2646 \beta + 6048) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 16 q^{4} + 48 q^{6} + 52 q^{7} - 126 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 16 q^{4} + 48 q^{6} + 52 q^{7} - 126 q^{9} - 96 q^{10} + 48 q^{12} + 100 q^{13} + 288 q^{15} + 128 q^{16} - 288 q^{18} - 716 q^{19} - 156 q^{21} + 672 q^{22} - 384 q^{24} + 674 q^{25} + 1242 q^{27} - 416 q^{28} + 288 q^{30} - 1484 q^{31} - 2016 q^{33} - 1152 q^{34} + 1008 q^{36} + 3748 q^{37} - 300 q^{39} + 768 q^{40} + 1248 q^{42} - 524 q^{43} - 1728 q^{45} - 2112 q^{46} - 384 q^{48} - 3450 q^{49} + 3456 q^{51} - 800 q^{52} - 2160 q^{54} + 4032 q^{55} + 2148 q^{57} + 8160 q^{58} - 2304 q^{60} - 2972 q^{61} - 3276 q^{63} - 1024 q^{64} - 2016 q^{66} - 8972 q^{67} + 6336 q^{69} - 2496 q^{70} + 2304 q^{72} + 580 q^{73} - 2022 q^{75} + 5728 q^{76} + 2400 q^{78} + 19636 q^{79} + 2754 q^{81} - 13632 q^{82} + 1248 q^{84} - 6912 q^{85} - 24480 q^{87} - 5376 q^{88} + 6048 q^{90} + 2600 q^{91} + 4452 q^{93} + 9600 q^{94} + 3072 q^{96} - 956 q^{97} + 12096 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
1.41421i
1.41421i
2.82843i −3.00000 + 8.48528i −8.00000 16.9706i 24.0000 + 8.48528i 26.0000 22.6274i −63.0000 50.9117i −48.0000
5.2 2.82843i −3.00000 8.48528i −8.00000 16.9706i 24.0000 8.48528i 26.0000 22.6274i −63.0000 + 50.9117i −48.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6.5.b.a 2
3.b odd 2 1 inner 6.5.b.a 2
4.b odd 2 1 48.5.e.b 2
5.b even 2 1 150.5.d.a 2
5.c odd 4 2 150.5.b.a 4
7.b odd 2 1 294.5.b.a 2
8.b even 2 1 192.5.e.d 2
8.d odd 2 1 192.5.e.c 2
9.c even 3 2 162.5.d.a 4
9.d odd 6 2 162.5.d.a 4
12.b even 2 1 48.5.e.b 2
15.d odd 2 1 150.5.d.a 2
15.e even 4 2 150.5.b.a 4
21.c even 2 1 294.5.b.a 2
24.f even 2 1 192.5.e.c 2
24.h odd 2 1 192.5.e.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.5.b.a 2 1.a even 1 1 trivial
6.5.b.a 2 3.b odd 2 1 inner
48.5.e.b 2 4.b odd 2 1
48.5.e.b 2 12.b even 2 1
150.5.b.a 4 5.c odd 4 2
150.5.b.a 4 15.e even 4 2
150.5.d.a 2 5.b even 2 1
150.5.d.a 2 15.d odd 2 1
162.5.d.a 4 9.c even 3 2
162.5.d.a 4 9.d odd 6 2
192.5.e.c 2 8.d odd 2 1
192.5.e.c 2 24.f even 2 1
192.5.e.d 2 8.b even 2 1
192.5.e.d 2 24.h odd 2 1
294.5.b.a 2 7.b odd 2 1
294.5.b.a 2 21.c even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(6, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 8 \) Copy content Toggle raw display
$3$ \( T^{2} + 6T + 81 \) Copy content Toggle raw display
$5$ \( T^{2} + 288 \) Copy content Toggle raw display
$7$ \( (T - 26)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 14112 \) Copy content Toggle raw display
$13$ \( (T - 50)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 41472 \) Copy content Toggle raw display
$19$ \( (T + 358)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 139392 \) Copy content Toggle raw display
$29$ \( T^{2} + 2080800 \) Copy content Toggle raw display
$31$ \( (T + 742)^{2} \) Copy content Toggle raw display
$37$ \( (T - 1874)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 5807232 \) Copy content Toggle raw display
$43$ \( (T + 262)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 2880000 \) Copy content Toggle raw display
$53$ \( T^{2} + 209952 \) Copy content Toggle raw display
$59$ \( T^{2} + 3297312 \) Copy content Toggle raw display
$61$ \( (T + 1486)^{2} \) Copy content Toggle raw display
$67$ \( (T + 4486)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 12700800 \) Copy content Toggle raw display
$73$ \( (T - 290)^{2} \) Copy content Toggle raw display
$79$ \( (T - 9818)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 50561568 \) Copy content Toggle raw display
$89$ \( T^{2} + 61471872 \) Copy content Toggle raw display
$97$ \( (T + 478)^{2} \) Copy content Toggle raw display
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