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Properties

Label 845.2.n.f
Level $845$
Weight $2$
Character orbit 845.n
Analytic conductor $6.747$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [845,2,Mod(484,845)]
 
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([3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("845.484");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.n (of order \(6\), 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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.89539436150784.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2x^{11} + 2x^{10} - 8x^{9} + 4x^{8} + 16x^{7} - 8x^{6} + 20x^{5} + 20x^{4} - 24x^{3} + 8x^{2} - 8x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{9} - \beta_{5} + \cdots + \beta_{3}) q^{2}+ \cdots + (\beta_{11} + 3 \beta_{10}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{9} - \beta_{5} + \cdots + \beta_{3}) q^{2}+ \cdots + ( - 8 \beta_{2} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 10 q^{4} - 4 q^{6} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 10 q^{4} - 4 q^{6} + 6 q^{9} - 2 q^{10} + 12 q^{11} + 16 q^{14} - 16 q^{15} - 10 q^{16} + 20 q^{20} - 8 q^{21} - 16 q^{24} + 4 q^{25} + 12 q^{29} - 8 q^{30} - 40 q^{31} + 40 q^{34} - 8 q^{35} + 22 q^{36} - 68 q^{40} + 8 q^{41} + 80 q^{44} + 4 q^{45} - 32 q^{46} - 18 q^{49} - 16 q^{50} + 48 q^{51} + 68 q^{54} + 16 q^{55} - 40 q^{56} - 16 q^{59} - 24 q^{60} - 12 q^{61} - 132 q^{64} - 32 q^{66} + 24 q^{69} - 40 q^{70} + 24 q^{71} - 4 q^{74} - 16 q^{75} + 20 q^{76} + 64 q^{79} - 48 q^{80} - 46 q^{81} - 12 q^{84} + 12 q^{85} - 64 q^{86} + 20 q^{89} + 140 q^{90} + 32 q^{94} - 16 q^{95} - 72 q^{96} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2x^{11} + 2x^{10} - 8x^{9} + 4x^{8} + 16x^{7} - 8x^{6} + 20x^{5} + 20x^{4} - 24x^{3} + 8x^{2} - 8x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{11} + \nu^{10} - 4 \nu^{9} + 28 \nu^{8} - 18 \nu^{7} + 22 \nu^{6} - 94 \nu^{5} - 146 \nu^{4} + \cdots + 748 ) / 460 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5 \nu^{11} + 5 \nu^{10} - 20 \nu^{9} + 94 \nu^{8} - 44 \nu^{7} + 64 \nu^{6} - 286 \nu^{5} + \cdots + 612 ) / 460 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12 \nu^{11} - 43 \nu^{10} + 43 \nu^{9} - 103 \nu^{8} + 166 \nu^{7} + 264 \nu^{6} - 414 \nu^{5} + \cdots - 12 ) / 460 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 31 \nu^{11} - 31 \nu^{10} + 9 \nu^{9} - 201 \nu^{8} - 109 \nu^{7} + 560 \nu^{6} + 246 \nu^{5} + \cdots - 142 ) / 230 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12 \nu^{11} + 43 \nu^{10} - 43 \nu^{9} + 103 \nu^{8} - 166 \nu^{7} - 264 \nu^{6} + 414 \nu^{5} + \cdots + 12 ) / 230 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 32 \nu^{11} - 107 \nu^{10} + 107 \nu^{9} - 267 \nu^{8} + 412 \nu^{7} + 658 \nu^{6} - 1058 \nu^{5} + \cdots - 32 ) / 460 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 81 \nu^{11} - 81 \nu^{10} + 48 \nu^{9} - 566 \nu^{8} - 244 \nu^{7} + 1208 \nu^{6} + 806 \nu^{5} + \cdots - 420 ) / 460 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3 \nu^{11} - 5 \nu^{10} + 5 \nu^{9} - 23 \nu^{8} + 4 \nu^{7} + 46 \nu^{6} - 12 \nu^{5} + 74 \nu^{4} + \cdots - 12 ) / 20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 101 \nu^{11} + 101 \nu^{10} - 36 \nu^{9} + 666 \nu^{8} + 344 \nu^{7} - 1734 \nu^{6} - 846 \nu^{5} + \cdots + 476 ) / 460 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 117 \nu^{11} - 195 \nu^{10} + 195 \nu^{9} - 941 \nu^{8} + 250 \nu^{7} + 1700 \nu^{6} - 92 \nu^{5} + \cdots - 1060 ) / 460 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 60 \nu^{11} + 100 \nu^{10} - 100 \nu^{9} + 469 \nu^{8} - 94 \nu^{7} - 906 \nu^{6} + 184 \nu^{5} + \cdots + 612 ) / 230 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} - 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{9} + \beta_{7} + 2\beta_{4} + \beta_{2} - 2\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{11} + \beta_{10} + 7\beta_{8} + \beta_{2} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{11} + 3\beta_{10} + 9\beta_{8} - 3\beta_{6} - 8\beta_{5} - 9\beta_{3} - 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 22\beta_{9} + 6\beta_{7} + 28\beta_{4} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 33 \beta_{11} + 11 \beta_{10} + 33 \beta_{9} + 39 \beta_{8} + 11 \beta_{7} + 11 \beta_{6} + \cdots - 33 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 94\beta_{11} + 28\beta_{10} + 116\beta_{8} - 116 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 138\beta_{9} + 44\beta_{7} + 166\beta_{4} - 44\beta_{2} + 138\beta _1 - 166 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 398\beta_{9} + 122\beta_{7} + 122\beta_{6} + 398\beta_{5} + 486\beta_{4} + 486\beta_{3} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 580\beta_{11} + 182\beta_{10} + 702\beta_{8} + 182\beta_{6} + 580\beta_{5} + 702\beta_{3} - 702 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(171\) \(677\)
\(\chi(n)\) \(-1 + \beta_{8}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
484.1
0.550552 + 0.147520i
−1.16746 − 0.312819i
−0.531325 + 1.98293i
1.98293 + 0.531325i
0.312819 − 1.16746i
−0.147520 + 0.550552i
0.550552 − 0.147520i
−1.16746 + 0.312819i
−0.531325 − 1.98293i
1.98293 − 0.531325i
0.312819 + 1.16746i
−0.147520 − 0.550552i
−2.31673 + 1.33757i −0.416726 + 0.240597i 2.57816 − 4.46551i 1.67513 + 1.48119i 0.643629 − 1.11480i 0.698071 + 0.403032i 8.44358i −1.38423 + 2.39755i −5.86202 − 1.19093i
484.2 −1.33298 + 0.769594i 2.74538 − 1.58504i 0.184551 − 0.319652i 0.539189 − 2.17009i −2.43968 + 4.22565i −1.48028 − 0.854638i − 2.51026i 3.52472 − 6.10500i 0.951360 + 3.30763i
484.3 −1.05163 + 0.607160i −1.13545 + 0.655554i −0.262714 + 0.455034i −2.21432 + 0.311108i 0.796052 − 1.37880i −2.51426 − 1.45161i − 3.06668i −0.640498 + 1.10938i 2.13976 − 1.67162i
484.4 1.05163 − 0.607160i 1.13545 − 0.655554i −0.262714 + 0.455034i −2.21432 − 0.311108i 0.796052 − 1.37880i 2.51426 + 1.45161i 3.06668i −0.640498 + 1.10938i −2.51754 + 1.01728i
484.5 1.33298 − 0.769594i −2.74538 + 1.58504i 0.184551 − 0.319652i 0.539189 + 2.17009i −2.43968 + 4.22565i 1.48028 + 0.854638i 2.51026i 3.52472 − 6.10500i 2.38881 + 2.47772i
484.6 2.31673 − 1.33757i 0.416726 − 0.240597i 2.57816 − 4.46551i 1.67513 − 1.48119i 0.643629 − 1.11480i −0.698071 − 0.403032i − 8.44358i −1.38423 + 2.39755i 1.89963 − 5.67213i
529.1 −2.31673 − 1.33757i −0.416726 − 0.240597i 2.57816 + 4.46551i 1.67513 − 1.48119i 0.643629 + 1.11480i 0.698071 − 0.403032i − 8.44358i −1.38423 − 2.39755i −5.86202 + 1.19093i
529.2 −1.33298 − 0.769594i 2.74538 + 1.58504i 0.184551 + 0.319652i 0.539189 + 2.17009i −2.43968 − 4.22565i −1.48028 + 0.854638i 2.51026i 3.52472 + 6.10500i 0.951360 − 3.30763i
529.3 −1.05163 − 0.607160i −1.13545 − 0.655554i −0.262714 − 0.455034i −2.21432 − 0.311108i 0.796052 + 1.37880i −2.51426 + 1.45161i 3.06668i −0.640498 − 1.10938i 2.13976 + 1.67162i
529.4 1.05163 + 0.607160i 1.13545 + 0.655554i −0.262714 − 0.455034i −2.21432 + 0.311108i 0.796052 + 1.37880i 2.51426 − 1.45161i − 3.06668i −0.640498 − 1.10938i −2.51754 − 1.01728i
529.5 1.33298 + 0.769594i −2.74538 − 1.58504i 0.184551 + 0.319652i 0.539189 − 2.17009i −2.43968 − 4.22565i 1.48028 − 0.854638i − 2.51026i 3.52472 + 6.10500i 2.38881 − 2.47772i
529.6 2.31673 + 1.33757i 0.416726 + 0.240597i 2.57816 + 4.46551i 1.67513 + 1.48119i 0.643629 + 1.11480i −0.698071 + 0.403032i 8.44358i −1.38423 − 2.39755i 1.89963 + 5.67213i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 484.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.c even 3 1 inner
65.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.n.f 12
5.b even 2 1 inner 845.2.n.f 12
13.b even 2 1 845.2.n.g 12
13.c even 3 1 65.2.b.a ✓ 6
13.c even 3 1 inner 845.2.n.f 12
13.d odd 4 1 845.2.l.d 12
13.d odd 4 1 845.2.l.e 12
13.e even 6 1 845.2.b.c 6
13.e even 6 1 845.2.n.g 12
13.f odd 12 1 845.2.d.a 6
13.f odd 12 1 845.2.d.b 6
13.f odd 12 1 845.2.l.d 12
13.f odd 12 1 845.2.l.e 12
39.i odd 6 1 585.2.c.b 6
52.j odd 6 1 1040.2.d.c 6
65.d even 2 1 845.2.n.g 12
65.g odd 4 1 845.2.l.d 12
65.g odd 4 1 845.2.l.e 12
65.l even 6 1 845.2.b.c 6
65.l even 6 1 845.2.n.g 12
65.n even 6 1 65.2.b.a ✓ 6
65.n even 6 1 inner 845.2.n.f 12
65.q odd 12 1 325.2.a.j 3
65.q odd 12 1 325.2.a.k 3
65.r odd 12 1 4225.2.a.ba 3
65.r odd 12 1 4225.2.a.bh 3
65.s odd 12 1 845.2.d.a 6
65.s odd 12 1 845.2.d.b 6
65.s odd 12 1 845.2.l.d 12
65.s odd 12 1 845.2.l.e 12
195.x odd 6 1 585.2.c.b 6
195.bl even 12 1 2925.2.a.bf 3
195.bl even 12 1 2925.2.a.bj 3
260.v odd 6 1 1040.2.d.c 6
260.bj even 12 1 5200.2.a.cb 3
260.bj even 12 1 5200.2.a.cj 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.b.a ✓ 6 13.c even 3 1
65.2.b.a ✓ 6 65.n even 6 1
325.2.a.j 3 65.q odd 12 1
325.2.a.k 3 65.q odd 12 1
585.2.c.b 6 39.i odd 6 1
585.2.c.b 6 195.x odd 6 1
845.2.b.c 6 13.e even 6 1
845.2.b.c 6 65.l even 6 1
845.2.d.a 6 13.f odd 12 1
845.2.d.a 6 65.s odd 12 1
845.2.d.b 6 13.f odd 12 1
845.2.d.b 6 65.s odd 12 1
845.2.l.d 12 13.d odd 4 1
845.2.l.d 12 13.f odd 12 1
845.2.l.d 12 65.g odd 4 1
845.2.l.d 12 65.s odd 12 1
845.2.l.e 12 13.d odd 4 1
845.2.l.e 12 13.f odd 12 1
845.2.l.e 12 65.g odd 4 1
845.2.l.e 12 65.s odd 12 1
845.2.n.f 12 1.a even 1 1 trivial
845.2.n.f 12 5.b even 2 1 inner
845.2.n.f 12 13.c even 3 1 inner
845.2.n.f 12 65.n even 6 1 inner
845.2.n.g 12 13.b even 2 1
845.2.n.g 12 13.e even 6 1
845.2.n.g 12 65.d even 2 1
845.2.n.g 12 65.l even 6 1
1040.2.d.c 6 52.j odd 6 1
1040.2.d.c 6 260.v odd 6 1
2925.2.a.bf 3 195.bl even 12 1
2925.2.a.bj 3 195.bl even 12 1
4225.2.a.ba 3 65.r odd 12 1
4225.2.a.bh 3 65.r odd 12 1
5200.2.a.cb 3 260.bj even 12 1
5200.2.a.cj 3 260.bj even 12 1

Hecke kernels

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

\( T_{2}^{12} - 11T_{2}^{10} + 90T_{2}^{8} - 291T_{2}^{6} + 686T_{2}^{4} - 775T_{2}^{2} + 625 \) Copy content Toggle raw display
\( T_{11}^{6} - 6T_{11}^{5} + 28T_{11}^{4} - 52T_{11}^{3} + 76T_{11}^{2} + 16T_{11} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 11 T^{10} + \cdots + 625 \) Copy content Toggle raw display
$3$ \( T^{12} - 12 T^{10} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( (T^{6} - T^{4} + 16 T^{3} + \cdots + 125)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} - 12 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( (T^{6} - 6 T^{5} + 28 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( T^{12} - 44 T^{10} + \cdots + 4096 \) Copy content Toggle raw display
$19$ \( (T^{6} + 4 T^{4} + 4 T^{3} + \cdots + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 72 T^{10} + \cdots + 54700816 \) Copy content Toggle raw display
$29$ \( (T^{6} - 6 T^{5} + \cdots + 11664)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 10 T^{2} + \cdots - 26)^{4} \) Copy content Toggle raw display
$37$ \( T^{12} - 56 T^{10} + \cdots + 7311616 \) Copy content Toggle raw display
$41$ \( (T^{6} - 4 T^{5} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 5972816656 \) Copy content Toggle raw display
$47$ \( (T^{6} + 44 T^{4} + \cdots + 400)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 144 T^{4} + \cdots + 92416)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 8 T^{5} + \cdots + 68644)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 6 T^{5} + 52 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 133090713856 \) Copy content Toggle raw display
$71$ \( (T^{6} - 12 T^{5} + \cdots + 568516)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 248 T^{4} + \cdots + 55696)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 16 T^{2} + \cdots + 16)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + 180 T^{4} + \cdots + 99856)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 10 T^{5} + \cdots + 40000)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 1600000000 \) Copy content Toggle raw display
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