Properties

Label 605.2.b.g
Level $605$
Weight $2$
Character orbit 605.b
Analytic conductor $4.831$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [605,2,Mod(364,605)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(605, 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("605.364");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.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: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1480160000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 9x^{6} + 27x^{4} + 31x^{2} + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{7} + \beta_{6} - \beta_1) q^{3} + \beta_{2} q^{4} + (\beta_{6} + \beta_{4} - \beta_{3}) q^{5} + ( - \beta_{5} + 2 \beta_{3} - \beta_{2} + 2) q^{6} + \beta_{4} q^{7} + (\beta_{6} + \beta_{4} + \beta_1) q^{8} + ( - \beta_{5} - \beta_{3} + 2 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{7} + \beta_{6} - \beta_1) q^{3} + \beta_{2} q^{4} + (\beta_{6} + \beta_{4} - \beta_{3}) q^{5} + ( - \beta_{5} + 2 \beta_{3} - \beta_{2} + 2) q^{6} + \beta_{4} q^{7} + (\beta_{6} + \beta_{4} + \beta_1) q^{8} + ( - \beta_{5} - \beta_{3} + 2 \beta_{2}) q^{9} + ( - \beta_{7} + \beta_{6} + \beta_{3} - 2 \beta_{2}) q^{10} + ( - 2 \beta_{6} - \beta_{4} + \beta_1) q^{12} + (2 \beta_{6} - \beta_1) q^{13} + (2 \beta_{5} - 2 \beta_{2} - 1) q^{14} + (\beta_{6} + \beta_{5} + 2 \beta_{2} + \beta_1 - 1) q^{15} + (\beta_{3} + \beta_{2} - 2) q^{16} + (3 \beta_{7} - \beta_{6} + 2 \beta_1) q^{17} + ( - \beta_{7} + 2 \beta_{6} + 2 \beta_{4} - 2 \beta_1) q^{18} + ( - \beta_{5} + \beta_{3} + 2 \beta_{2}) q^{19} + (\beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_1) q^{20} + (2 \beta_{5} - \beta_{3} - 1) q^{21} + (\beta_{6} - 2 \beta_{4} - 2 \beta_1) q^{23} + (2 \beta_{3} + \beta_{2} + 1) q^{24} + ( - 2 \beta_{7} + 2 \beta_{5} - 2 \beta_{3} - 3) q^{25} + ( - 4 \beta_{5} + 2 \beta_{3} - \beta_{2} + 4) q^{26} + ( - 2 \beta_{7} - \beta_{4}) q^{27} + \beta_1 q^{28} + (\beta_{5} - \beta_{3} - 4) q^{29} + (3 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} - 3 \beta_1 - 1) q^{30} + (\beta_{5} + \beta_{3} + \beta_{2} + 2) q^{31} + (\beta_{7} + 2 \beta_{6} + 3 \beta_{4} - \beta_1) q^{32} + ( - \beta_{5} - 4 \beta_{3} + 2 \beta_{2} - 2) q^{34} + ( - \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{2} - 3) q^{35} + ( - \beta_{5} + \beta_{3} - 2 \beta_{2} + 3) q^{36} + (\beta_{7} + 3 \beta_{6} + 2 \beta_{4}) q^{37} + (\beta_{7} + 2 \beta_{4} - 2 \beta_1) q^{38} + ( - \beta_{5} + 5 \beta_{2} - 2) q^{39} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} - 2 \beta_{2} - 4) q^{40} + (5 \beta_{5} + 2 \beta_{3} - 3 \beta_{2} + 2) q^{41} + ( - \beta_{7} + 3 \beta_{6} - \beta_1) q^{42} + ( - 2 \beta_{7} - 2 \beta_{6} - 3 \beta_1) q^{43} + ( - 2 \beta_{6} - 2 \beta_{5} + \beta_{2} + 3 \beta_1 + 2) q^{45} + ( - 6 \beta_{5} + \beta_{3} + 2 \beta_{2} + 7) q^{46} + (\beta_{7} - \beta_{6} + 3 \beta_{4} - \beta_1) q^{47} + (2 \beta_{7} - 5 \beta_{6} - \beta_{4} + 2 \beta_1) q^{48} + (\beta_{5} + 2 \beta_{3} - \beta_{2} + 4) q^{49} + ( - 2 \beta_{7} + 4 \beta_{6} + 2 \beta_{5} + 2 \beta_{3} - 3 \beta_1 - 2) q^{50} + (2 \beta_{5} + 3 \beta_{3} - \beta_{2} + 7) q^{51} + (2 \beta_{7} - 3 \beta_{6} - \beta_{4} + 3 \beta_1) q^{52} + (2 \beta_{7} - 3 \beta_{6} - \beta_{4} - 3 \beta_1) q^{53} + (2 \beta_{3} + 2 \beta_{2} - 1) q^{54} + (4 \beta_{5} - 3 \beta_{2} - 4) q^{56} + (\beta_{7} - 5 \beta_{6} - \beta_{4} + \beta_1) q^{57} + ( - \beta_{7} + 2 \beta_{6} - 4 \beta_1) q^{58} + (2 \beta_{5} + 3 \beta_{3} - 5 \beta_{2}) q^{59} + (\beta_{7} + \beta_{4} + 3 \beta_{3} - 3 \beta_{2} + 5) q^{60} + ( - 5 \beta_{5} - 4 \beta_{3} + 4 \beta_{2} + 4) q^{61} + (\beta_{7} + \beta_{6} + \beta_{4} + \beta_1) q^{62} + ( - \beta_{7} + \beta_{4} + 2 \beta_1) q^{63} + (\beta_{5} + 3 \beta_{3} - 5 \beta_{2} - 2) q^{64} + ( - \beta_{7} + \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} + 4 \beta_{2} - 2) q^{65} + (3 \beta_{7} - 2 \beta_{6} - \beta_{4} + 4 \beta_1) q^{67} + (2 \beta_{7} + 3 \beta_{6} + 2 \beta_{4}) q^{68} + ( - 3 \beta_{5} - \beta_{3} + 4 \beta_{2} - 2) q^{69} + (\beta_{6} + 4 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{70} + (3 \beta_{5} - 3 \beta_{3} - 3 \beta_{2} - 9) q^{71} + ( - \beta_{7} + 2 \beta_{4} + \beta_1) q^{72} + ( - \beta_{7} - 5 \beta_{4} - 2 \beta_1) q^{73} + ( - 3 \beta_{5} + 2 \beta_{3} - 4 \beta_{2} + 2) q^{74} + (\beta_{7} - \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{2} + 5 \beta_1 - 2) q^{75} + (\beta_{5} + \beta_{3} - 2 \beta_{2} + 3) q^{76} + (4 \beta_{6} + 5 \beta_{4} - 7 \beta_1) q^{78} + (5 \beta_{5} - \beta_{3} - 4 \beta_{2} - 6) q^{79} + (2 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} + 2 \beta_1 - 1) q^{80} + ( - 7 \beta_{5} - 2 \beta_{3} + 4 \beta_{2} - 1) q^{81} + (2 \beta_{7} - 3 \beta_{4} + 5 \beta_1) q^{82} + ( - \beta_{7} - 3 \beta_{6} - \beta_{4} + \beta_1) q^{83} + ( - \beta_{5} + 2 \beta_{3} - \beta_{2} + 2) q^{84} + ( - \beta_{7} - 2 \beta_{6} - 5 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - 2 \beta_{2} + \cdots + 1) q^{85}+ \cdots + (2 \beta_{7} - 2 \beta_{6} - \beta_{4} + 5 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{4} + 4 q^{5} + 6 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{4} + 4 q^{5} + 6 q^{6} - 4 q^{9} + 4 q^{14} - 8 q^{15} - 22 q^{16} - 12 q^{19} - 4 q^{20} + 4 q^{21} - 2 q^{24} - 8 q^{25} + 10 q^{26} - 24 q^{29} - 22 q^{30} + 14 q^{31} - 8 q^{34} - 14 q^{35} + 20 q^{36} - 30 q^{39} - 24 q^{40} + 34 q^{41} + 6 q^{45} + 24 q^{46} + 30 q^{49} - 16 q^{50} + 54 q^{51} - 20 q^{54} - 10 q^{56} + 6 q^{59} + 34 q^{60} + 20 q^{61} - 14 q^{64} - 20 q^{65} - 32 q^{69} + 8 q^{70} - 42 q^{71} + 4 q^{74} - 20 q^{75} + 28 q^{76} - 16 q^{79} - 28 q^{80} - 36 q^{81} + 6 q^{84} + 4 q^{85} + 46 q^{86} + 12 q^{89} - 46 q^{90} + 20 q^{91} + 42 q^{94} - 26 q^{95} - 8 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 9x^{6} + 27x^{4} + 31x^{2} + 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} + 5\nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{7} - 7\nu^{5} - 13\nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{6} + 7\nu^{4} + 14\nu^{2} + 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{7} + 7\nu^{5} + 14\nu^{3} + 8\nu \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{7} + 8\nu^{5} + 19\nu^{3} + 12\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{4} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{3} - 5\beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} - 6\beta_{6} - 5\beta_{4} + 11\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{5} - 7\beta_{3} + 21\beta_{2} - 22 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -7\beta_{7} + 29\beta_{6} + 21\beta_{4} - 43\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(486\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
364.1
2.02368i
1.65458i
1.23399i
0.802699i
0.802699i
1.23399i
1.65458i
2.02368i
2.02368i 2.62059i −2.09529 −0.294963 + 2.21653i 5.30325 0.965823i 0.192845i −3.86752 4.48555 + 0.596911i
364.2 1.65458i 1.97479i −0.737640 2.19353 0.434096i −3.26745 2.24307i 2.08868i −0.899788 −0.718246 3.62937i
364.3 1.23399i 0.363982i 0.477260 1.29496 1.82293i −0.449152 2.58558i 3.05692i 2.86752 −2.24948 1.59798i
364.4 0.802699i 1.76074i 1.35567 −1.19353 1.89090i 1.41335 0.592103i 2.69360i −0.100212 −1.51782 + 0.958043i
364.5 0.802699i 1.76074i 1.35567 −1.19353 + 1.89090i 1.41335 0.592103i 2.69360i −0.100212 −1.51782 0.958043i
364.6 1.23399i 0.363982i 0.477260 1.29496 + 1.82293i −0.449152 2.58558i 3.05692i 2.86752 −2.24948 + 1.59798i
364.7 1.65458i 1.97479i −0.737640 2.19353 + 0.434096i −3.26745 2.24307i 2.08868i −0.899788 −0.718246 + 3.62937i
364.8 2.02368i 2.62059i −2.09529 −0.294963 2.21653i 5.30325 0.965823i 0.192845i −3.86752 4.48555 0.596911i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 364.8
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 605.2.b.g 8
5.b even 2 1 inner 605.2.b.g 8
5.c odd 4 2 3025.2.a.bl 8
11.b odd 2 1 605.2.b.f 8
11.c even 5 2 55.2.j.a 16
11.c even 5 2 605.2.j.h 16
11.d odd 10 2 605.2.j.d 16
11.d odd 10 2 605.2.j.g 16
33.h odd 10 2 495.2.ba.a 16
44.h odd 10 2 880.2.cd.c 16
55.d odd 2 1 605.2.b.f 8
55.e even 4 2 3025.2.a.bk 8
55.h odd 10 2 605.2.j.d 16
55.h odd 10 2 605.2.j.g 16
55.j even 10 2 55.2.j.a 16
55.j even 10 2 605.2.j.h 16
55.k odd 20 4 275.2.h.d 16
165.o odd 10 2 495.2.ba.a 16
220.n odd 10 2 880.2.cd.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.j.a 16 11.c even 5 2
55.2.j.a 16 55.j even 10 2
275.2.h.d 16 55.k odd 20 4
495.2.ba.a 16 33.h odd 10 2
495.2.ba.a 16 165.o odd 10 2
605.2.b.f 8 11.b odd 2 1
605.2.b.f 8 55.d odd 2 1
605.2.b.g 8 1.a even 1 1 trivial
605.2.b.g 8 5.b even 2 1 inner
605.2.j.d 16 11.d odd 10 2
605.2.j.d 16 55.h odd 10 2
605.2.j.g 16 11.d odd 10 2
605.2.j.g 16 55.h odd 10 2
605.2.j.h 16 11.c even 5 2
605.2.j.h 16 55.j even 10 2
880.2.cd.c 16 44.h odd 10 2
880.2.cd.c 16 220.n odd 10 2
3025.2.a.bk 8 55.e even 4 2
3025.2.a.bl 8 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(605, [\chi])\):

\( T_{2}^{8} + 9T_{2}^{6} + 27T_{2}^{4} + 31T_{2}^{2} + 11 \) Copy content Toggle raw display
\( T_{19}^{4} + 6T_{19}^{3} - 4T_{19}^{2} - 39T_{19} + 11 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 9 T^{6} + 27 T^{4} + 31 T^{2} + \cdots + 11 \) Copy content Toggle raw display
$3$ \( T^{8} + 14 T^{6} + 62 T^{4} + 91 T^{2} + \cdots + 11 \) Copy content Toggle raw display
$5$ \( T^{8} - 4 T^{7} + 12 T^{6} - 36 T^{5} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} + 13 T^{6} + 49 T^{4} + 47 T^{2} + \cdots + 11 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + 45 T^{6} + 675 T^{4} + \cdots + 6875 \) Copy content Toggle raw display
$17$ \( T^{8} + 81 T^{6} + 1842 T^{4} + \cdots + 40931 \) Copy content Toggle raw display
$19$ \( (T^{4} + 6 T^{3} - 4 T^{2} - 39 T + 11)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 99 T^{6} + 2232 T^{4} + \cdots + 26411 \) Copy content Toggle raw display
$29$ \( (T^{4} + 12 T^{3} + 48 T^{2} + 67 T + 11)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 7 T^{3} + 5 T^{2} + T - 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 101 T^{6} + 2822 T^{4} + \cdots + 3971 \) Copy content Toggle raw display
$41$ \( (T^{4} - 17 T^{3} + 48 T^{2} + 438 T - 1969)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 173 T^{6} + 8319 T^{4} + \cdots + 212531 \) Copy content Toggle raw display
$47$ \( T^{8} + 191 T^{6} + 7351 T^{4} + \cdots + 244211 \) Copy content Toggle raw display
$53$ \( T^{8} + 249 T^{6} + 15237 T^{4} + \cdots + 489731 \) Copy content Toggle raw display
$59$ \( (T^{4} - 3 T^{3} - 75 T^{2} + 29 T - 1)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 10 T^{3} - 61 T^{2} + 10 T + 209)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 149 T^{6} + 5736 T^{4} + \cdots + 18491 \) Copy content Toggle raw display
$71$ \( (T^{4} + 21 T^{3} + 90 T^{2} - 432 T - 2511)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 297 T^{6} + 25504 T^{4} + \cdots + 1279091 \) Copy content Toggle raw display
$79$ \( (T^{4} + 8 T^{3} - 52 T^{2} - 377 T - 319)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 111 T^{6} + 3931 T^{4} + \cdots + 212531 \) Copy content Toggle raw display
$89$ \( (T^{4} - 6 T^{3} - 128 T^{2} + 486 T + 1871)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 162 T^{6} + 5588 T^{4} + \cdots + 161051 \) Copy content Toggle raw display
show more
show less