Properties

Label 5.8.b.a
Level $5$
Weight $8$
Character orbit 5.b
Analytic conductor $1.562$
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] = [5,8,Mod(4,5)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5.4");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 5.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: \(1.56192512742\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-29}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 29 \) 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 \(\beta = 2\sqrt{-29}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 3 \beta q^{3} + 12 q^{4} + ( - 25 \beta + 75) q^{5} - 348 q^{6} - 39 \beta q^{7} + 140 \beta q^{8} + 1143 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + 3 \beta q^{3} + 12 q^{4} + ( - 25 \beta + 75) q^{5} - 348 q^{6} - 39 \beta q^{7} + 140 \beta q^{8} + 1143 q^{9} + (75 \beta + 2900) q^{10} - 6828 q^{11} + 36 \beta q^{12} - 942 \beta q^{13} + 4524 q^{14} + (225 \beta + 8700) q^{15} - 14704 q^{16} + 1456 \beta q^{17} + 1143 \beta q^{18} + 6860 q^{19} + ( - 300 \beta + 900) q^{20} + 13572 q^{21} - 6828 \beta q^{22} + 2713 \beta q^{23} - 48720 q^{24} + ( - 3750 \beta - 66875) q^{25} + 109272 q^{26} + 9990 \beta q^{27} - 468 \beta q^{28} + 25590 q^{29} + (8700 \beta - 26100) q^{30} + 82112 q^{31} + 3216 \beta q^{32} - 20484 \beta q^{33} - 168896 q^{34} + ( - 2925 \beta - 113100) q^{35} + 13716 q^{36} - 20754 \beta q^{37} + 6860 \beta q^{38} + 327816 q^{39} + (10500 \beta + 406000) q^{40} - 533118 q^{41} + 13572 \beta q^{42} + 65823 \beta q^{43} - 81936 q^{44} + ( - 28575 \beta + 85725) q^{45} - 314708 q^{46} + 541 \beta q^{47} - 44112 \beta q^{48} + 647107 q^{49} + ( - 66875 \beta + 435000) q^{50} - 506688 q^{51} - 11304 \beta q^{52} - 54722 \beta q^{53} - 1158840 q^{54} + (170700 \beta - 512100) q^{55} + 633360 q^{56} + 20580 \beta q^{57} + 25590 \beta q^{58} + 1438980 q^{59} + (2700 \beta + 104400) q^{60} + 1381022 q^{61} + 82112 \beta q^{62} - 44577 \beta q^{63} - 2255168 q^{64} + ( - 70650 \beta - 2731800) q^{65} + 2376144 q^{66} - 252069 \beta q^{67} + 17472 \beta q^{68} - 944124 q^{69} + ( - 113100 \beta + 339300) q^{70} - 481608 q^{71} + 160020 \beta q^{72} + 137988 \beta q^{73} + 2407464 q^{74} + ( - 200625 \beta + 1305000) q^{75} + 82320 q^{76} + 266292 \beta q^{77} + 327816 \beta q^{78} - 1059760 q^{79} + (367600 \beta - 1102800) q^{80} - 976779 q^{81} - 533118 \beta q^{82} - 241757 \beta q^{83} + 162864 q^{84} + (109200 \beta + 4222400) q^{85} - 7635468 q^{86} + 76770 \beta q^{87} - 955920 \beta q^{88} + 5644170 q^{89} + (85725 \beta + 3314700) q^{90} - 4261608 q^{91} + 32556 \beta q^{92} + 246336 \beta q^{93} - 62756 q^{94} + ( - 171500 \beta + 514500) q^{95} - 1119168 q^{96} + 1115016 \beta q^{97} + 647107 \beta q^{98} - 7804404 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 24 q^{4} + 150 q^{5} - 696 q^{6} + 2286 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 24 q^{4} + 150 q^{5} - 696 q^{6} + 2286 q^{9} + 5800 q^{10} - 13656 q^{11} + 9048 q^{14} + 17400 q^{15} - 29408 q^{16} + 13720 q^{19} + 1800 q^{20} + 27144 q^{21} - 97440 q^{24} - 133750 q^{25} + 218544 q^{26} + 51180 q^{29} - 52200 q^{30} + 164224 q^{31} - 337792 q^{34} - 226200 q^{35} + 27432 q^{36} + 655632 q^{39} + 812000 q^{40} - 1066236 q^{41} - 163872 q^{44} + 171450 q^{45} - 629416 q^{46} + 1294214 q^{49} + 870000 q^{50} - 1013376 q^{51} - 2317680 q^{54} - 1024200 q^{55} + 1266720 q^{56} + 2877960 q^{59} + 208800 q^{60} + 2762044 q^{61} - 4510336 q^{64} - 5463600 q^{65} + 4752288 q^{66} - 1888248 q^{69} + 678600 q^{70} - 963216 q^{71} + 4814928 q^{74} + 2610000 q^{75} + 164640 q^{76} - 2119520 q^{79} - 2205600 q^{80} - 1953558 q^{81} + 325728 q^{84} + 8444800 q^{85} - 15270936 q^{86} + 11288340 q^{89} + 6629400 q^{90} - 8523216 q^{91} - 125512 q^{94} + 1029000 q^{95} - 2238336 q^{96} - 15608808 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
5.38516i
5.38516i
10.7703i 32.3110i 12.0000 75.0000 + 269.258i −348.000 420.043i 1507.85i 1143.00 2900.00 807.775i
4.2 10.7703i 32.3110i 12.0000 75.0000 269.258i −348.000 420.043i 1507.85i 1143.00 2900.00 + 807.775i
\(n\): 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 5.8.b.a 2
3.b odd 2 1 45.8.b.a 2
4.b odd 2 1 80.8.c.a 2
5.b even 2 1 inner 5.8.b.a 2
5.c odd 4 2 25.8.a.d 2
8.b even 2 1 320.8.c.d 2
8.d odd 2 1 320.8.c.c 2
15.d odd 2 1 45.8.b.a 2
15.e even 4 2 225.8.a.n 2
20.d odd 2 1 80.8.c.a 2
20.e even 4 2 400.8.a.y 2
40.e odd 2 1 320.8.c.c 2
40.f even 2 1 320.8.c.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.8.b.a 2 1.a even 1 1 trivial
5.8.b.a 2 5.b even 2 1 inner
25.8.a.d 2 5.c odd 4 2
45.8.b.a 2 3.b odd 2 1
45.8.b.a 2 15.d odd 2 1
80.8.c.a 2 4.b odd 2 1
80.8.c.a 2 20.d odd 2 1
225.8.a.n 2 15.e even 4 2
320.8.c.c 2 8.d odd 2 1
320.8.c.c 2 40.e odd 2 1
320.8.c.d 2 8.b even 2 1
320.8.c.d 2 40.f even 2 1
400.8.a.y 2 20.e even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(5, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 116 \) Copy content Toggle raw display
$3$ \( T^{2} + 1044 \) Copy content Toggle raw display
$5$ \( T^{2} - 150T + 78125 \) Copy content Toggle raw display
$7$ \( T^{2} + 176436 \) Copy content Toggle raw display
$11$ \( (T + 6828)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 102934224 \) Copy content Toggle raw display
$17$ \( T^{2} + 245912576 \) Copy content Toggle raw display
$19$ \( (T - 6860)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 853802804 \) Copy content Toggle raw display
$29$ \( (T - 25590)^{2} \) Copy content Toggle raw display
$31$ \( (T - 82112)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 49964507856 \) Copy content Toggle raw display
$41$ \( (T + 533118)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 502589410164 \) Copy content Toggle raw display
$47$ \( T^{2} + 33950996 \) Copy content Toggle raw display
$53$ \( T^{2} + 347361684944 \) Copy content Toggle raw display
$59$ \( (T - 1438980)^{2} \) Copy content Toggle raw display
$61$ \( (T - 1381022)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 7370498568276 \) Copy content Toggle raw display
$71$ \( (T + 481608)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 2208719824704 \) Copy content Toggle raw display
$79$ \( (T + 1059760)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 6779787857684 \) Copy content Toggle raw display
$89$ \( (T - 5644170)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 144218238909696 \) Copy content Toggle raw display
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