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Properties

Label 605.2.g.g
Level $605$
Weight $2$
Character orbit 605.g
Analytic conductor $4.831$
Analytic rank $0$
Dimension $8$
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: 8.0.159390625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 6x^{6} - 11x^{5} + 21x^{4} - 5x^{3} + 10x^{2} + 25x + 25 \) 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.

\(f(q)\) \(=\) \( q + ( - \beta_{6} - \beta_{5} - \beta_{3} + \cdots + 1) q^{2}+ \cdots + (\beta_{7} - \beta_{6} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} - \beta_{5} - \beta_{3} + \cdots + 1) q^{2}+ \cdots + ( - 7 \beta_{6} + 2 \beta_{5} + \cdots + 13) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{2} + q^{3} - q^{4} - 2 q^{5} + 12 q^{6} + 8 q^{7} + 7 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - q^{2} + q^{3} - q^{4} - 2 q^{5} + 12 q^{6} + 8 q^{7} + 7 q^{8} + 5 q^{9} - 6 q^{10} - 28 q^{12} + 11 q^{13} - 14 q^{14} + q^{15} + 15 q^{16} + 4 q^{17} + 16 q^{18} + 11 q^{19} - 6 q^{20} + 12 q^{21} - 18 q^{23} + 10 q^{24} - 2 q^{25} + q^{26} - 5 q^{27} + 11 q^{28} + 11 q^{29} - 13 q^{30} - 9 q^{31} - 12 q^{32} - 20 q^{34} + 3 q^{35} - 29 q^{36} - q^{37} - 6 q^{38} + 6 q^{39} + 7 q^{40} - 11 q^{41} - 31 q^{42} - 42 q^{43} - 29 q^{46} - q^{47} + 21 q^{48} - q^{50} + 22 q^{51} + q^{52} + 8 q^{53} + 40 q^{54} + 30 q^{56} + 4 q^{58} + 26 q^{59} - 8 q^{60} + 2 q^{61} - 27 q^{62} + 4 q^{63} + 21 q^{64} - 14 q^{65} - 2 q^{67} + 20 q^{68} + 49 q^{69} - 14 q^{70} - 25 q^{71} - 21 q^{72} - 32 q^{73} + 12 q^{74} + q^{75} - 16 q^{76} + 12 q^{78} + 23 q^{79} - 20 q^{80} + 20 q^{81} + 42 q^{82} - 10 q^{83} - 51 q^{84} - q^{85} + 34 q^{86} - 30 q^{87} - 14 q^{90} + 8 q^{91} - 99 q^{92} - 8 q^{93} + 22 q^{94} + 11 q^{95} - 3 q^{96} - 18 q^{97} + 84 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 6x^{6} - 11x^{5} + 21x^{4} - 5x^{3} + 10x^{2} + 25x + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 555\nu^{7} - 2159\nu^{6} + 7489\nu^{5} - 18164\nu^{4} + 40069\nu^{3} - 84434\nu^{2} + 43855\nu + 375 ) / 94655 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -970\nu^{7} - 1002\nu^{6} - 6608\nu^{5} + 9063\nu^{4} - 14943\nu^{3} + 27673\nu^{2} - 68120\nu + 35160 ) / 94655 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -1604\nu^{7} + 4159\nu^{6} - 12059\nu^{5} + 28414\nu^{4} - 81659\nu^{3} + 38305\nu^{2} - 13500\nu - 13875 ) / 94655 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2052\nu^{7} + 2252\nu^{6} - 19912\nu^{5} + 21007\nu^{4} - 82042\nu^{3} + 35785\nu^{2} - 19395\nu - 90925 ) / 94655 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2667\nu^{7} + 6691\nu^{6} - 17466\nu^{5} + 50856\nu^{4} - 82441\nu^{3} + 72554\nu^{2} - 4035\nu - 12035 ) / 94655 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4024\nu^{7} - 1464\nu^{6} + 21519\nu^{5} - 26434\nu^{4} + 59219\nu^{3} + 22635\nu^{2} + 54640\nu + 66675 ) / 94655 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - 3\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{6} + \beta_{5} - 4\beta_{4} - \beta_{3} - \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{7} + 7\beta_{6} + 2\beta_{5} + 13\beta_{3} + 13\beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{7} - 11\beta_{6} - 20\beta_{5} + 20\beta_{4} - 11\beta_{2} + 8\beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -19\beta_{7} - 19\beta_{4} - 68\beta_{3} - 36\beta_{2} - 24\beta _1 + 36 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 111\beta_{7} + 81\beta_{6} + 111\beta_{5} - 55\beta_{4} + 81\beta_{3} + 148\beta_{2} - 56\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(122\) \(486\)
\(\chi(n)\) \(1\) \(-\beta_{6}\)

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
1.43801 − 1.04478i
−0.628998 + 0.456994i
−0.762262 − 2.34600i
0.453245 + 1.39494i
1.43801 + 1.04478i
−0.628998 − 0.456994i
−0.762262 + 2.34600i
0.453245 − 1.39494i
−1.51774 − 1.10270i −0.549273 + 1.69049i 0.469548 + 1.44512i −0.809017 + 0.587785i 2.69776 − 1.96004i −1.31579 − 4.04959i −0.278565 + 0.857334i −0.128998 − 0.0937225i 1.87603
81.2 1.82676 + 1.32722i 0.240256 − 0.739431i 0.957503 + 2.94689i −0.809017 + 0.587785i 1.42027 − 1.03189i −0.0383089 − 0.117903i −0.766520 + 2.35911i 1.93801 + 1.40805i −2.25800
251.1 −0.780121 + 2.40097i 1.99563 − 1.44991i −3.53801 − 2.57052i 0.309017 + 0.951057i 1.92435 + 5.92254i 1.69369 + 1.23053i 4.84703 − 3.52157i 0.953245 − 2.93379i −2.52452
251.2 −0.0288961 + 0.0889332i −1.18661 + 0.862123i 1.61096 + 1.17043i 0.309017 + 0.951057i −0.0423829 − 0.130441i 3.66042 + 2.65945i −0.301943 + 0.219374i −0.262262 + 0.807160i −0.0935099
366.1 −1.51774 + 1.10270i −0.549273 − 1.69049i 0.469548 − 1.44512i −0.809017 − 0.587785i 2.69776 + 1.96004i −1.31579 + 4.04959i −0.278565 − 0.857334i −0.128998 + 0.0937225i 1.87603
366.2 1.82676 − 1.32722i 0.240256 + 0.739431i 0.957503 − 2.94689i −0.809017 − 0.587785i 1.42027 + 1.03189i −0.0383089 + 0.117903i −0.766520 − 2.35911i 1.93801 − 1.40805i −2.25800
511.1 −0.780121 − 2.40097i 1.99563 + 1.44991i −3.53801 + 2.57052i 0.309017 − 0.951057i 1.92435 − 5.92254i 1.69369 − 1.23053i 4.84703 + 3.52157i 0.953245 + 2.93379i −2.52452
511.2 −0.0288961 − 0.0889332i −1.18661 − 0.862123i 1.61096 − 1.17043i 0.309017 − 0.951057i −0.0423829 + 0.130441i 3.66042 − 2.65945i −0.301943 − 0.219374i −0.262262 − 0.807160i −0.0935099
\(n\): 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.g 8
11.b odd 2 1 605.2.g.j 8
11.c even 5 1 605.2.a.i 4
11.c even 5 1 inner 605.2.g.g 8
11.c even 5 2 605.2.g.n 8
11.d odd 10 2 55.2.g.a ✓ 8
11.d odd 10 1 605.2.a.l 4
11.d odd 10 1 605.2.g.j 8
33.f even 10 2 495.2.n.f 8
33.f even 10 1 5445.2.a.bg 4
33.h odd 10 1 5445.2.a.bu 4
44.g even 10 2 880.2.bo.e 8
44.g even 10 1 9680.2.a.cs 4
44.h odd 10 1 9680.2.a.cv 4
55.h odd 10 2 275.2.h.b 8
55.h odd 10 1 3025.2.a.v 4
55.j even 10 1 3025.2.a.be 4
55.l even 20 4 275.2.z.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.g.a ✓ 8 11.d odd 10 2
275.2.h.b 8 55.h odd 10 2
275.2.z.b 16 55.l even 20 4
495.2.n.f 8 33.f even 10 2
605.2.a.i 4 11.c even 5 1
605.2.a.l 4 11.d odd 10 1
605.2.g.g 8 1.a even 1 1 trivial
605.2.g.g 8 11.c even 5 1 inner
605.2.g.j 8 11.b odd 2 1
605.2.g.j 8 11.d odd 10 1
605.2.g.n 8 11.c even 5 2
880.2.bo.e 8 44.g even 10 2
3025.2.a.v 4 55.h odd 10 1
3025.2.a.be 4 55.j even 10 1
5445.2.a.bg 4 33.f even 10 1
5445.2.a.bu 4 33.h odd 10 1
9680.2.a.cs 4 44.g even 10 1
9680.2.a.cv 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 \(S_{2}^{\mathrm{new}}(605, [\chi])\):

\( T_{2}^{8} + T_{2}^{7} + 3T_{2}^{6} - 5T_{2}^{5} + 6T_{2}^{4} + 45T_{2}^{3} + 117T_{2}^{2} + 7T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{8} - T_{3}^{7} + T_{3}^{6} - T_{3}^{5} + 16T_{3}^{4} + 25T_{3}^{3} + 35T_{3}^{2} + 25 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{7} + 3 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} - T^{7} + \cdots + 25 \) Copy content Toggle raw display
$5$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 8 T^{7} + \cdots + 25 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 11 T^{7} + \cdots + 121 \) Copy content Toggle raw display
$17$ \( T^{8} - 4 T^{7} + \cdots + 121 \) Copy content Toggle raw display
$19$ \( T^{8} - 11 T^{7} + \cdots + 625 \) Copy content Toggle raw display
$23$ \( (T^{4} + 9 T^{3} + \cdots - 1669)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - 11 T^{7} + \cdots + 3025 \) Copy content Toggle raw display
$31$ \( T^{8} + 9 T^{7} + \cdots + 10201 \) Copy content Toggle raw display
$37$ \( T^{8} + T^{7} + \cdots + 22801 \) Copy content Toggle raw display
$41$ \( T^{8} + 11 T^{7} + \cdots + 249001 \) Copy content Toggle raw display
$43$ \( (T^{4} + 21 T^{3} + \cdots - 59)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + T^{7} + \cdots + 5041 \) Copy content Toggle raw display
$53$ \( T^{8} - 8 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{8} - 26 T^{7} + \cdots + 9150625 \) Copy content Toggle raw display
$61$ \( T^{8} - 2 T^{7} + \cdots + 3025 \) Copy content Toggle raw display
$67$ \( (T^{4} + T^{3} - 82 T^{2} + \cdots - 101)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 25 T^{7} + \cdots + 60824401 \) Copy content Toggle raw display
$73$ \( T^{8} + 32 T^{7} + \cdots + 151321 \) Copy content Toggle raw display
$79$ \( T^{8} - 23 T^{7} + \cdots + 4644025 \) Copy content Toggle raw display
$83$ \( T^{8} + 10 T^{7} + \cdots + 841 \) Copy content Toggle raw display
$89$ \( (T^{4} - 150 T^{2} + \cdots + 725)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 18 T^{7} + \cdots + 625 \) Copy content Toggle raw display
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