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Properties

Label 845.2.e.i
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: 6.0.954288.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 65)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_{4}) q^{2} + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{3} + (2 \beta_{3} - \beta_{2} - \beta_1 - 2) q^{4} - q^{5} + ( - \beta_{5} + 4 \beta_{3} - \beta_{2} + \cdots - 4) q^{6}+ \cdots + ( - 2 \beta_{5} + 2 \beta_{3} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_{4}) q^{2} + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{3} + (2 \beta_{3} - \beta_{2} - \beta_1 - 2) q^{4} - q^{5} + ( - \beta_{5} + 4 \beta_{3} - \beta_{2} + \cdots - 4) q^{6}+ \cdots + (7 \beta_{4} + 5 \beta_{2} + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{2} + 2 q^{3} - 5 q^{4} - 6 q^{5} - 10 q^{6} - 2 q^{7} + 6 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - q^{2} + 2 q^{3} - 5 q^{4} - 6 q^{5} - 10 q^{6} - 2 q^{7} + 6 q^{8} - 3 q^{9} + q^{10} - 6 q^{11} + 8 q^{14} - 2 q^{15} - 5 q^{16} + 4 q^{17} - 34 q^{18} - 8 q^{19} + 5 q^{20} + 4 q^{21} + 12 q^{22} + 8 q^{23} - 12 q^{24} + 6 q^{25} - 28 q^{27} - 22 q^{28} + 2 q^{29} + 10 q^{30} + 20 q^{31} - 17 q^{32} + 18 q^{33} - 16 q^{34} + 2 q^{35} - 17 q^{36} - 20 q^{37} + 20 q^{38} - 6 q^{40} + 2 q^{41} - 22 q^{42} + 12 q^{43} - 24 q^{44} + 3 q^{45} - 8 q^{46} - 4 q^{47} - 18 q^{48} + q^{49} - q^{50} - 8 q^{51} + 12 q^{53} - 10 q^{54} + 6 q^{55} - 24 q^{56} + 4 q^{57} + 4 q^{58} + 12 q^{59} + 6 q^{61} + 4 q^{62} - 14 q^{63} - 30 q^{64} + 24 q^{66} - 18 q^{67} + 44 q^{68} - 16 q^{69} - 8 q^{70} - 8 q^{71} + 3 q^{72} + 40 q^{73} - 10 q^{74} + 2 q^{75} + 2 q^{76} - 36 q^{77} + 24 q^{79} + 5 q^{80} - 15 q^{81} - 14 q^{82} + 36 q^{83} - 10 q^{84} - 4 q^{85} - 16 q^{86} + 2 q^{87} + 30 q^{88} + 34 q^{90} - 20 q^{92} + 14 q^{93} + 4 q^{94} + 8 q^{95} - 44 q^{96} + 14 q^{97} - 21 q^{98} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} - \nu^{3} + 9\nu^{2} - 21\nu - 9 ) / 27 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} - \nu^{3} - 18\nu^{2} + 33\nu - 9 ) / 27 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{5} - \nu^{4} - 2\nu^{3} + 12\nu + 36 ) / 27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{5} - \nu^{4} + 7\nu^{3} + 9\nu^{2} + 12\nu + 9 ) / 27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4\nu^{5} + 2\nu^{4} - 5\nu^{3} + 18\nu^{2} + 3\nu - 72 ) / 27 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} + 2\beta_{4} + 2\beta_{3} - \beta_{2} + \beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{5} + 7\beta_{4} - 11\beta_{3} + \beta_{2} - \beta _1 + 7 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{5} + 2\beta_{4} - 7\beta_{3} + 8\beta_{2} + 10\beta _1 + 20 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 7\beta_{5} - 2\beta_{4} - 20\beta_{3} + \beta_{2} - 10\beta _1 + 43 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(171\) \(677\)
\(\chi(n)\) \(-1 + \beta_{3}\) \(1\)

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
0.403374 + 1.68443i
−1.62241 − 0.606458i
1.71903 − 0.211943i
0.403374 − 1.68443i
−1.62241 + 0.606458i
1.71903 + 0.211943i
−1.25707 − 2.17731i −0.757068 − 1.31128i −2.16044 + 3.74200i −1.00000 −1.90337 + 3.29674i −1.66044 + 2.87597i 5.83502 0.353695 − 0.612617i 1.25707 + 2.17731i
146.2 −0.285997 − 0.495361i 0.214003 + 0.370665i 0.836412 − 1.44871i −1.00000 0.122408 − 0.212018i 1.33641 − 2.31473i −2.10083 1.40841 − 2.43943i 0.285997 + 0.495361i
146.3 1.04307 + 1.80664i 1.54307 + 2.67267i −1.17597 + 2.03684i −1.00000 −3.21903 + 5.57553i −0.675970 + 1.17081i −0.734191 −3.26210 + 5.65012i −1.04307 − 1.80664i
191.1 −1.25707 + 2.17731i −0.757068 + 1.31128i −2.16044 − 3.74200i −1.00000 −1.90337 − 3.29674i −1.66044 − 2.87597i 5.83502 0.353695 + 0.612617i 1.25707 − 2.17731i
191.2 −0.285997 + 0.495361i 0.214003 − 0.370665i 0.836412 + 1.44871i −1.00000 0.122408 + 0.212018i 1.33641 + 2.31473i −2.10083 1.40841 + 2.43943i 0.285997 − 0.495361i
191.3 1.04307 − 1.80664i 1.54307 − 2.67267i −1.17597 − 2.03684i −1.00000 −3.21903 − 5.57553i −0.675970 − 1.17081i −0.734191 −3.26210 − 5.65012i −1.04307 + 1.80664i
\(n\): 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.i 6
13.b even 2 1 845.2.e.k 6
13.c even 3 1 845.2.a.k 3
13.c even 3 1 inner 845.2.e.i 6
13.d odd 4 2 845.2.m.h 12
13.e even 6 1 845.2.a.i 3
13.e even 6 1 845.2.e.k 6
13.f odd 12 2 65.2.c.a ✓ 6
13.f odd 12 2 845.2.m.h 12
39.h odd 6 1 7605.2.a.cc 3
39.i odd 6 1 7605.2.a.bs 3
39.k even 12 2 585.2.b.g 6
52.l even 12 2 1040.2.k.d 6
65.l even 6 1 4225.2.a.be 3
65.n even 6 1 4225.2.a.bc 3
65.o even 12 2 325.2.d.f 6
65.s odd 12 2 325.2.c.g 6
65.t even 12 2 325.2.d.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.c.a ✓ 6 13.f odd 12 2
325.2.c.g 6 65.s odd 12 2
325.2.d.e 6 65.t even 12 2
325.2.d.f 6 65.o even 12 2
585.2.b.g 6 39.k even 12 2
845.2.a.i 3 13.e even 6 1
845.2.a.k 3 13.c even 3 1
845.2.e.i 6 1.a even 1 1 trivial
845.2.e.i 6 13.c even 3 1 inner
845.2.e.k 6 13.b even 2 1
845.2.e.k 6 13.e even 6 1
845.2.m.h 12 13.d odd 4 2
845.2.m.h 12 13.f odd 12 2
1040.2.k.d 6 52.l even 12 2
4225.2.a.bc 3 65.n even 6 1
4225.2.a.be 3 65.l even 6 1
7605.2.a.bs 3 39.i odd 6 1
7605.2.a.cc 3 39.h odd 6 1

Hecke kernels

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

\( T_{2}^{6} + T_{2}^{5} + 6T_{2}^{4} + T_{2}^{3} + 28T_{2}^{2} + 15T_{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{6} + 2T_{7}^{5} + 12T_{7}^{4} + 8T_{7}^{3} + 88T_{7}^{2} + 96T_{7} + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + T^{5} + 6 T^{4} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{6} - 2 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( (T + 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 2 T^{5} + \cdots + 144 \) Copy content Toggle raw display
$11$ \( T^{6} + 6 T^{5} + \cdots + 2916 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} - 4 T^{5} + \cdots + 9216 \) Copy content Toggle raw display
$19$ \( T^{6} + 8 T^{5} + \cdots + 36 \) Copy content Toggle raw display
$23$ \( T^{6} - 8 T^{5} + \cdots + 36 \) Copy content Toggle raw display
$29$ \( T^{6} - 2 T^{5} + \cdots + 144 \) Copy content Toggle raw display
$31$ \( (T^{3} - 10 T^{2} + 22 T + 6)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 20 T^{5} + \cdots + 51984 \) Copy content Toggle raw display
$41$ \( T^{6} - 2 T^{5} + \cdots + 5184 \) Copy content Toggle raw display
$43$ \( T^{6} - 12 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$47$ \( (T^{3} + 2 T^{2} - 8 T - 12)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - 6 T^{2} - 60 T - 72)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 12 T^{5} + \cdots + 324 \) Copy content Toggle raw display
$61$ \( T^{6} - 6 T^{5} + \cdots + 5776 \) Copy content Toggle raw display
$67$ \( T^{6} + 18 T^{5} + \cdots + 11664 \) Copy content Toggle raw display
$71$ \( T^{6} + 8 T^{5} + \cdots + 36 \) Copy content Toggle raw display
$73$ \( (T^{3} - 20 T^{2} + \cdots + 516)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 12 T^{2} + \cdots + 32)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} - 18 T^{2} + \cdots - 36)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 96 T^{4} + \cdots + 82944 \) Copy content Toggle raw display
$97$ \( T^{6} - 14 T^{5} + \cdots + 576 \) Copy content Toggle raw display
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