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Properties

Label 845.2.e.l
Level $845$
Weight $2$
Character orbit 845.e
Analytic conductor $6.747$
Analytic rank $0$
Dimension $6$
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] = [845,2,Mod(146,845)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(845, base_ring=CyclotomicField(6))
 
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("845.146");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.e (of order \(3\), degree \(2\), 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: \(6.74735897080\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.64827.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 3x^{4} + 5x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - 2 \beta_{5} - \beta_{4} + 2) q^{3} + (\beta_{5} + \beta_{4} + \beta_{3} + \cdots + \beta_1) q^{4}+ \cdots + ( - 3 \beta_{5} - 4 \beta_{4} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - 2 \beta_{5} - \beta_{4} + 2) q^{3} + (\beta_{5} + \beta_{4} + \beta_{3} + \cdots + \beta_1) q^{4}+ \cdots + ( - 4 \beta_{2} + 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} + 5 q^{3} + q^{4} + 6 q^{5} - 4 q^{6} + 5 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{2} + 5 q^{3} + q^{4} + 6 q^{5} - 4 q^{6} + 5 q^{7} - 4 q^{9} + q^{10} + 3 q^{11} + 8 q^{12} + 8 q^{14} + 5 q^{15} + 5 q^{16} + 3 q^{17} - 12 q^{18} - 8 q^{19} + q^{20} + 26 q^{21} + 6 q^{22} + 2 q^{23} + 7 q^{24} + 6 q^{25} - 16 q^{27} - 4 q^{28} + 6 q^{29} - 4 q^{30} + 6 q^{31} - 4 q^{32} + 2 q^{33} - 12 q^{34} + 5 q^{35} + 13 q^{36} - 5 q^{37} - 38 q^{38} - 20 q^{41} + 9 q^{42} + 23 q^{43} + 2 q^{44} - 4 q^{45} - 24 q^{46} - 32 q^{47} + q^{48} + 8 q^{49} + q^{50} - 18 q^{51} - 4 q^{53} + 2 q^{54} + 3 q^{55} + 7 q^{56} - 36 q^{57} + 19 q^{58} + 11 q^{59} + 8 q^{60} + 17 q^{61} - 6 q^{62} + 23 q^{63} + 8 q^{64} + 6 q^{66} - 15 q^{67} + 6 q^{68} - 15 q^{69} + 8 q^{70} - 13 q^{71} - 21 q^{72} - 24 q^{73} + 11 q^{74} + 5 q^{75} - 9 q^{76} - 4 q^{77} - 74 q^{79} + 5 q^{80} - 27 q^{81} + 2 q^{82} - 58 q^{83} + 16 q^{84} + 3 q^{85} + 20 q^{86} - 10 q^{87} - 14 q^{88} + 3 q^{89} - 12 q^{90} - 22 q^{92} + 19 q^{93} - 10 q^{94} - 8 q^{95} + 10 q^{96} + 12 q^{97} + 2 q^{98} + 34 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 3\nu^{4} - 9\nu^{3} + 5\nu^{2} - 2\nu + 6 ) / 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{5} + 9\nu^{4} - 14\nu^{3} + 15\nu^{2} - 6\nu + 18 ) / 13 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{5} - \nu^{4} - 10\nu^{3} - 6\nu^{2} - 34\nu - 2 ) / 13 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -6\nu^{5} + 5\nu^{4} - 15\nu^{3} - 9\nu^{2} - 25\nu + 10 ) / 13 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} - 3\beta_{4} - 4\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 4\beta_{4} - 4\beta_{3} + 9\beta_{2} - 9\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\) \(171\) \(677\)
\(\chi(n)\) \(-\beta_{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
146.1
−0.623490 − 1.07992i
0.222521 + 0.385418i
0.900969 + 1.56052i
−0.623490 + 1.07992i
0.222521 − 0.385418i
0.900969 − 1.56052i
−0.623490 − 1.07992i 0.0990311 + 0.171527i 0.222521 − 0.385418i 1.00000 0.123490 − 0.213891i 0.0990311 − 0.171527i −3.04892 1.48039 − 2.56410i −0.623490 − 1.07992i
146.2 0.222521 + 0.385418i 1.62349 + 2.81197i 0.900969 − 1.56052i 1.00000 −0.722521 + 1.25144i 1.62349 − 2.81197i 1.69202 −3.77144 + 6.53232i 0.222521 + 0.385418i
146.3 0.900969 + 1.56052i 0.777479 + 1.34663i −0.623490 + 1.07992i 1.00000 −1.40097 + 2.42655i 0.777479 − 1.34663i 1.35690 0.291053 − 0.504118i 0.900969 + 1.56052i
191.1 −0.623490 + 1.07992i 0.0990311 − 0.171527i 0.222521 + 0.385418i 1.00000 0.123490 + 0.213891i 0.0990311 + 0.171527i −3.04892 1.48039 + 2.56410i −0.623490 + 1.07992i
191.2 0.222521 − 0.385418i 1.62349 − 2.81197i 0.900969 + 1.56052i 1.00000 −0.722521 − 1.25144i 1.62349 + 2.81197i 1.69202 −3.77144 − 6.53232i 0.222521 − 0.385418i
191.3 0.900969 − 1.56052i 0.777479 − 1.34663i −0.623490 − 1.07992i 1.00000 −1.40097 − 2.42655i 0.777479 + 1.34663i 1.35690 0.291053 + 0.504118i 0.900969 − 1.56052i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 146.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.e.l 6
13.b even 2 1 845.2.e.j 6
13.c even 3 1 845.2.a.h ✓ 3
13.c even 3 1 inner 845.2.e.l 6
13.d odd 4 2 845.2.m.i 12
13.e even 6 1 845.2.a.j yes 3
13.e even 6 1 845.2.e.j 6
13.f odd 12 2 845.2.c.f 6
13.f odd 12 2 845.2.m.i 12
39.h odd 6 1 7605.2.a.br 3
39.i odd 6 1 7605.2.a.by 3
65.l even 6 1 4225.2.a.bd 3
65.n even 6 1 4225.2.a.bf 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
845.2.a.h ✓ 3 13.c even 3 1
845.2.a.j yes 3 13.e even 6 1
845.2.c.f 6 13.f odd 12 2
845.2.e.j 6 13.b even 2 1
845.2.e.j 6 13.e even 6 1
845.2.e.l 6 1.a even 1 1 trivial
845.2.e.l 6 13.c even 3 1 inner
845.2.m.i 12 13.d odd 4 2
845.2.m.i 12 13.f odd 12 2
4225.2.a.bd 3 65.l even 6 1
4225.2.a.bf 3 65.n even 6 1
7605.2.a.br 3 39.h odd 6 1
7605.2.a.by 3 39.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\):

\( T_{2}^{6} - T_{2}^{5} + 3T_{2}^{4} + 5T_{2}^{2} - 2T_{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{6} - 5T_{7}^{5} + 19T_{7}^{4} - 28T_{7}^{3} + 31T_{7}^{2} - 6T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{5} + 3 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{6} - 5 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T - 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - 5 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{6} - 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} - 3 T^{5} + \cdots + 729 \) Copy content Toggle raw display
$19$ \( T^{6} + 8 T^{5} + \cdots + 841 \) Copy content Toggle raw display
$23$ \( T^{6} - 2 T^{5} + \cdots + 5041 \) Copy content Toggle raw display
$29$ \( T^{6} - 6 T^{5} + \cdots + 94249 \) Copy content Toggle raw display
$31$ \( (T^{3} - 3 T^{2} - 46 T - 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 5 T^{5} + \cdots + 169 \) Copy content Toggle raw display
$41$ \( T^{6} + 20 T^{5} + \cdots + 1849 \) Copy content Toggle raw display
$43$ \( T^{6} - 23 T^{5} + \cdots + 187489 \) Copy content Toggle raw display
$47$ \( (T^{3} + 16 T^{2} + \cdots + 41)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} + 2 T^{2} - T - 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 11 T^{5} + \cdots + 169 \) Copy content Toggle raw display
$61$ \( T^{6} - 17 T^{5} + \cdots + 28561 \) Copy content Toggle raw display
$67$ \( T^{6} + 15 T^{5} + \cdots + 123201 \) Copy content Toggle raw display
$71$ \( T^{6} + 13 T^{5} + \cdots + 361201 \) Copy content Toggle raw display
$73$ \( (T^{3} + 12 T^{2} + \cdots - 1448)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 37 T^{2} + \cdots + 1217)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} + 29 T^{2} + \cdots + 587)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 3 T^{5} + \cdots + 284089 \) Copy content Toggle raw display
$97$ \( T^{6} - 12 T^{5} + \cdots + 32761 \) Copy content Toggle raw display
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