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Properties

Label 605.2.g.m
Level $605$
Weight $2$
Character orbit 605.g
Analytic conductor $4.831$
Analytic rank $0$
Dimension $8$
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] = [605,2,Mod(81,605)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(605, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("605.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.g (of order \(5\), degree \(4\), not 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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.13140625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{7} + \beta_{5} - \beta_{2} + 1) q^{3} + (\beta_{7} - \beta_{6} + \beta_{4} + \cdots + 1) q^{4}+ \cdots + ( - \beta_{7} - \beta_{6} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{7} + \beta_{5} - \beta_{2} + 1) q^{3} + (\beta_{7} - \beta_{6} + \beta_{4} + \cdots + 1) q^{4}+ \cdots + (3 \beta_{7} - 2 \beta_{5} - 3 \beta_{3} + 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 3 q^{2} + 5 q^{3} + 3 q^{4} + 2 q^{5} + 8 q^{6} + 4 q^{7} - q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 3 q^{2} + 5 q^{3} + 3 q^{4} + 2 q^{5} + 8 q^{6} + 4 q^{7} - q^{8} + 5 q^{9} + 2 q^{10} + 16 q^{12} + 3 q^{13} + 14 q^{14} - 5 q^{15} - q^{16} + 12 q^{17} + 10 q^{18} + 5 q^{19} + 2 q^{20} - 20 q^{21} + 10 q^{23} - 2 q^{24} - 2 q^{25} + 5 q^{26} + 5 q^{27} + 19 q^{28} + 21 q^{29} + 7 q^{30} + 15 q^{31} + 16 q^{32} + 4 q^{34} + q^{35} + 15 q^{36} - 31 q^{37} - 20 q^{38} - 14 q^{39} + q^{40} + 3 q^{41} - 21 q^{42} - 38 q^{43} - 7 q^{46} - 5 q^{47} + 5 q^{48} - 4 q^{49} + 3 q^{50} + 6 q^{51} + 17 q^{52} - 2 q^{53} + 16 q^{54} + 22 q^{56} + 40 q^{57} + 2 q^{58} + 18 q^{59} + 4 q^{60} + 6 q^{61} - 5 q^{62} - 30 q^{63} + 29 q^{64} + 2 q^{65} - 38 q^{67} - 14 q^{68} + 9 q^{69} - 14 q^{70} + 15 q^{71} + 5 q^{72} - 2 q^{73} - 20 q^{74} - 5 q^{75} - 16 q^{78} - 3 q^{79} - 4 q^{80} - 12 q^{81} - 22 q^{82} - 38 q^{83} - 17 q^{84} + 13 q^{85} + 2 q^{86} + 38 q^{87} - 16 q^{89} - 36 q^{91} + q^{92} + 40 q^{93} - 18 q^{94} - 5 q^{95} - 17 q^{96} - 56 q^{97} + 16 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} - 3\nu^{5} - 4\nu^{3} - 7\nu^{2} - 12\nu - 7 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 7\nu^{5} + 20\nu^{4} - 16\nu^{3} + 19\nu^{2} + 6\nu + 9 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 4\nu^{6} - 9\nu^{5} + 12\nu^{4} - 16\nu^{3} + 13\nu^{2} - 10\nu - 1 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} + 10\nu^{6} - 17\nu^{5} + 8\nu^{4} - 4\nu^{3} - 13\nu^{2} - 8\nu - 5 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{7} - 12\nu^{6} + 23\nu^{5} - 20\nu^{4} + 16\nu^{3} + \nu^{2} + 6\nu - 1 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{7} + 18\nu^{6} - 35\nu^{5} + 32\nu^{4} - 28\nu^{3} - 11\nu^{2} - 12\nu - 7 ) / 8 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 3\beta_{6} + 2\beta_{5} + \beta_{4} - 3\beta_{2} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{7} + 4\beta_{6} + \beta_{4} - \beta_{3} - 5\beta_{2} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{7} - 6\beta_{5} - 4\beta_{3} - 6\beta_{2} - 6\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -16\beta_{6} - 16\beta_{5} - 6\beta_{4} - 6\beta_{3} - 7\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -16\beta_{7} - 51\beta_{6} - 29\beta_{5} - 23\beta_{4} + 29\beta_{2} - 16 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(122\) \(486\)
\(\chi(n)\) \(1\) \(-1 + \beta_{3} - \beta_{4} - \beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
−0.386111 + 0.280526i
1.69513 − 1.23158i
−0.227943 − 0.701538i
0.418926 + 1.28932i
−0.386111 − 0.280526i
1.69513 + 1.23158i
−0.227943 + 0.701538i
0.418926 − 1.28932i
−0.386111 − 0.280526i 0.0998345 − 0.307259i −0.547647 − 1.68548i 0.809017 − 0.587785i −0.124741 + 0.0906300i 0.829779 + 2.55380i −0.556333 + 1.71222i 2.34261 + 1.70201i −0.477260
81.2 1.69513 + 1.23158i 0.591149 − 1.81937i 0.738630 + 2.27327i 0.809017 − 0.587785i 3.24278 − 2.35601i −0.947813 − 2.91707i −0.252684 + 0.777682i −0.533593 − 0.387678i 2.09529
251.1 −0.227943 + 0.701538i 2.27460 − 1.65259i 1.17784 + 0.855749i −0.309017 − 0.951057i 0.640877 + 1.97242i −0.834404 − 0.606230i −2.06235 + 1.49838i 1.51569 − 4.66481i 0.737640
251.2 0.418926 − 1.28932i −0.465584 + 0.338266i 0.131180 + 0.0953077i −0.309017 − 0.951057i 0.241089 + 0.741996i 2.95244 + 2.14507i 2.37136 − 1.72290i −0.824707 + 2.53819i −1.35567
366.1 −0.386111 + 0.280526i 0.0998345 + 0.307259i −0.547647 + 1.68548i 0.809017 + 0.587785i −0.124741 − 0.0906300i 0.829779 − 2.55380i −0.556333 − 1.71222i 2.34261 − 1.70201i −0.477260
366.2 1.69513 − 1.23158i 0.591149 + 1.81937i 0.738630 − 2.27327i 0.809017 + 0.587785i 3.24278 + 2.35601i −0.947813 + 2.91707i −0.252684 − 0.777682i −0.533593 + 0.387678i 2.09529
511.1 −0.227943 − 0.701538i 2.27460 + 1.65259i 1.17784 − 0.855749i −0.309017 + 0.951057i 0.640877 − 1.97242i −0.834404 + 0.606230i −2.06235 − 1.49838i 1.51569 + 4.66481i 0.737640
511.2 0.418926 + 1.28932i −0.465584 − 0.338266i 0.131180 − 0.0953077i −0.309017 + 0.951057i 0.241089 − 0.741996i 2.95244 − 2.14507i 2.37136 + 1.72290i −0.824707 − 2.53819i −1.35567
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.g.m 8
11.b odd 2 1 605.2.g.e 8
11.c even 5 2 55.2.g.b ✓ 8
11.c even 5 1 605.2.a.j 4
11.c even 5 1 inner 605.2.g.m 8
11.d odd 10 1 605.2.a.k 4
11.d odd 10 1 605.2.g.e 8
11.d odd 10 2 605.2.g.k 8
33.f even 10 1 5445.2.a.bi 4
33.h odd 10 2 495.2.n.e 8
33.h odd 10 1 5445.2.a.bp 4
44.g even 10 1 9680.2.a.cm 4
44.h odd 10 2 880.2.bo.h 8
44.h odd 10 1 9680.2.a.cn 4
55.h odd 10 1 3025.2.a.w 4
55.j even 10 2 275.2.h.a 8
55.j even 10 1 3025.2.a.bd 4
55.k odd 20 4 275.2.z.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.g.b ✓ 8 11.c even 5 2
275.2.h.a 8 55.j even 10 2
275.2.z.a 16 55.k odd 20 4
495.2.n.e 8 33.h odd 10 2
605.2.a.j 4 11.c even 5 1
605.2.a.k 4 11.d odd 10 1
605.2.g.e 8 11.b odd 2 1
605.2.g.e 8 11.d odd 10 1
605.2.g.k 8 11.d odd 10 2
605.2.g.m 8 1.a even 1 1 trivial
605.2.g.m 8 11.c even 5 1 inner
880.2.bo.h 8 44.h odd 10 2
3025.2.a.w 4 55.h odd 10 1
3025.2.a.bd 4 55.j even 10 1
5445.2.a.bi 4 33.f even 10 1
5445.2.a.bp 4 33.h odd 10 1
9680.2.a.cm 4 44.g even 10 1
9680.2.a.cn 4 44.h odd 10 1

Hecke kernels

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

\( T_{2}^{8} - 3T_{2}^{7} + 5T_{2}^{6} - 3T_{2}^{5} + 4T_{2}^{4} + 3T_{2}^{3} + 5T_{2}^{2} + 3T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{8} - 5T_{3}^{7} + 13T_{3}^{6} - 15T_{3}^{5} + 14T_{3}^{4} + 15T_{3}^{3} + 7T_{3}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} - 5 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 4 T^{7} + \cdots + 961 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 3 T^{7} + \cdots + 19321 \) Copy content Toggle raw display
$17$ \( T^{8} - 12 T^{7} + \cdots + 361 \) Copy content Toggle raw display
$19$ \( T^{8} - 5 T^{7} + \cdots + 625 \) Copy content Toggle raw display
$23$ \( (T^{4} - 5 T^{3} + 4 T^{2} + \cdots - 11)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - 21 T^{7} + \cdots + 203401 \) Copy content Toggle raw display
$31$ \( T^{8} - 15 T^{7} + \cdots + 390625 \) Copy content Toggle raw display
$37$ \( T^{8} + 31 T^{7} + \cdots + 1324801 \) Copy content Toggle raw display
$41$ \( T^{8} - 3 T^{7} + \cdots + 101761 \) Copy content Toggle raw display
$43$ \( (T^{4} + 19 T^{3} + \cdots + 211)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 5 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$53$ \( T^{8} + 2 T^{7} + \cdots + 885481 \) Copy content Toggle raw display
$59$ \( T^{8} - 18 T^{7} + \cdots + 687241 \) Copy content Toggle raw display
$61$ \( T^{8} - 6 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$67$ \( (T^{4} + 19 T^{3} + \cdots - 4079)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 15 T^{7} + \cdots + 17161 \) Copy content Toggle raw display
$73$ \( T^{8} + 2 T^{7} + \cdots + 121 \) Copy content Toggle raw display
$79$ \( T^{8} + 3 T^{7} + \cdots + 45954841 \) Copy content Toggle raw display
$83$ \( T^{8} + 38 T^{7} + \cdots + 2886601 \) Copy content Toggle raw display
$89$ \( (T^{4} + 8 T^{3} + \cdots + 1861)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 56 T^{7} + \cdots + 9066121 \) Copy content Toggle raw display
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