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Properties

Label 574.2.e.f
Level $574$
Weight $2$
Character orbit 574.e
Analytic conductor $4.583$
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] = [574,2,Mod(165,574)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(574, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("574.165");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 574 = 2 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 574.e (of order \(3\), degree \(2\), 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: \(4.58341307602\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 13x^{6} + 12x^{5} + 75x^{4} + 21x^{3} + 117x^{2} + 36x + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{2} - 1) q^{3} + (\beta_{2} - 1) q^{4} + (\beta_{5} - \beta_{3} - \beta_1) q^{5} - q^{6} + (\beta_{6} + \beta_{3} - \beta_{2} + 1) q^{7} - q^{8} + 2 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{2} - 1) q^{3} + (\beta_{2} - 1) q^{4} + (\beta_{5} - \beta_{3} - \beta_1) q^{5} - q^{6} + (\beta_{6} + \beta_{3} - \beta_{2} + 1) q^{7} - q^{8} + 2 \beta_{2} q^{9} + (\beta_{7} - \beta_{6} + \beta_{4} + \cdots - \beta_1) q^{10}+ \cdots + (2 \beta_{7} - 2 \beta_{4} + 2 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 4 q^{3} - 4 q^{4} - 3 q^{5} - 8 q^{6} + 5 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} - 4 q^{3} - 4 q^{4} - 3 q^{5} - 8 q^{6} + 5 q^{7} - 8 q^{8} + 8 q^{9} + 3 q^{10} - 4 q^{12} - 2 q^{13} + 7 q^{14} + 6 q^{15} - 4 q^{16} + 3 q^{17} - 8 q^{18} - 2 q^{19} + 6 q^{20} + 2 q^{21} + 9 q^{23} + 4 q^{24} - 19 q^{25} - q^{26} - 40 q^{27} + 2 q^{28} - 36 q^{29} + 3 q^{30} + 19 q^{31} + 4 q^{32} + 6 q^{34} + 24 q^{35} - 16 q^{36} - 2 q^{37} + 2 q^{38} + q^{39} + 3 q^{40} - 8 q^{41} - 5 q^{42} + 10 q^{43} + 6 q^{45} - 9 q^{46} + 8 q^{48} + 11 q^{49} - 38 q^{50} + 3 q^{51} + q^{52} - 6 q^{53} - 20 q^{54} - 12 q^{55} - 5 q^{56} + 4 q^{57} - 18 q^{58} + 3 q^{59} - 3 q^{60} + q^{61} + 38 q^{62} + 14 q^{63} + 8 q^{64} + 24 q^{65} - 11 q^{67} + 3 q^{68} - 18 q^{69} + 12 q^{70} - 18 q^{71} - 8 q^{72} + 22 q^{73} + 2 q^{74} - 19 q^{75} + 4 q^{76} + 45 q^{77} + 2 q^{78} + 28 q^{79} - 3 q^{80} - 4 q^{81} - 4 q^{82} - 24 q^{83} - 7 q^{84} - 48 q^{85} + 5 q^{86} + 18 q^{87} - 6 q^{89} + 12 q^{90} + 46 q^{91} - 18 q^{92} + 19 q^{93} + 27 q^{95} + 4 q^{96} + 22 q^{97} + 10 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 13x^{6} + 12x^{5} + 75x^{4} + 21x^{3} + 117x^{2} + 36x + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -47\nu^{7} + 226\nu^{6} - 791\nu^{5} + 888\nu^{4} - 1017\nu^{3} + 10509\nu^{2} - 1539\nu + 16272 ) / 14796 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 115\nu^{7} - 1066\nu^{6} + 3731\nu^{5} - 8912\nu^{4} - 135\nu^{3} - 24909\nu^{2} + 21255\nu - 2772 ) / 14796 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{7} + 15\nu^{6} - 121\nu^{5} - 209\nu^{4} - 958\nu^{3} - 330\nu^{2} - 264\nu - 564 ) / 1233 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 57\nu^{7} - 140\nu^{6} + 490\nu^{5} + 1357\nu^{4} - 603\nu^{3} - 345\nu^{2} - 4386\nu + 8415 ) / 3699 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 257\nu^{7} + 210\nu^{6} + 909\nu^{5} + 14884\nu^{4} + 15495\nu^{3} + 44289\nu^{2} + 10689\nu + 54576 ) / 14796 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -295\nu^{7} + 1162\nu^{6} - 5711\nu^{5} + 5492\nu^{4} - 21669\nu^{3} + 19509\nu^{2} - 25575\nu + 44208 ) / 14796 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + \beta_{5} - 2\beta_{4} + 5\beta_{2} + 2\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{7} - \beta_{6} - \beta_{5} - 10\beta_{4} + 3\beta_{3} - 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 15\beta_{7} + 15\beta_{6} - 17\beta_{5} + 17\beta_{3} - 53\beta_{2} - 35\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -13\beta_{7} + 67\beta_{6} - 54\beta_{5} + 142\beta_{4} + 13\beta_{3} - 181\beta_{2} - 142\beta _1 + 181 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -263\beta_{7} + 41\beta_{6} + 41\beta_{5} + 545\beta_{4} - 222\beta_{3} + 749 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -849\beta_{7} - 849\beta_{6} + 1030\beta_{5} - 1030\beta_{3} + 2848\beta_{2} + 2143\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/574\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(493\)
\(\chi(n)\) \(1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
165.1
0.630267 + 1.09165i
−0.923724 − 1.59994i
1.95306 + 3.38279i
−0.659599 − 1.14246i
0.630267 − 1.09165i
−0.923724 + 1.59994i
1.95306 − 3.38279i
−0.659599 + 1.14246i
0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −2.03329 − 3.52177i −1.00000 2.06303 + 1.65647i −1.00000 1.00000 + 1.73205i 2.03329 − 3.52177i
165.2 0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −1.15370 − 1.99827i −1.00000 −2.63141 − 0.275138i −1.00000 1.00000 + 1.73205i 1.15370 − 1.99827i
165.3 0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.352685 − 0.610869i −1.00000 0.877628 − 2.49595i −1.00000 1.00000 + 1.73205i 0.352685 − 0.610869i
165.4 0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 2.03968 + 3.53283i −1.00000 2.19074 − 1.48346i −1.00000 1.00000 + 1.73205i −2.03968 + 3.53283i
247.1 0.500000 − 0.866025i −0.500000 − 0.866025i −0.500000 − 0.866025i −2.03329 + 3.52177i −1.00000 2.06303 − 1.65647i −1.00000 1.00000 − 1.73205i 2.03329 + 3.52177i
247.2 0.500000 − 0.866025i −0.500000 − 0.866025i −0.500000 − 0.866025i −1.15370 + 1.99827i −1.00000 −2.63141 + 0.275138i −1.00000 1.00000 − 1.73205i 1.15370 + 1.99827i
247.3 0.500000 − 0.866025i −0.500000 − 0.866025i −0.500000 − 0.866025i −0.352685 + 0.610869i −1.00000 0.877628 + 2.49595i −1.00000 1.00000 − 1.73205i 0.352685 + 0.610869i
247.4 0.500000 − 0.866025i −0.500000 − 0.866025i −0.500000 − 0.866025i 2.03968 − 3.53283i −1.00000 2.19074 + 1.48346i −1.00000 1.00000 − 1.73205i −2.03968 − 3.53283i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 165.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 574.2.e.f ✓ 8
7.c even 3 1 inner 574.2.e.f ✓ 8
7.c even 3 1 4018.2.a.bk 4
7.d odd 6 1 4018.2.a.bi 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.e.f ✓ 8 1.a even 1 1 trivial
574.2.e.f ✓ 8 7.c even 3 1 inner
4018.2.a.bi 4 7.d odd 6 1
4018.2.a.bk 4 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(574, [\chi])\):

\( T_{3}^{2} + T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{8} + 3T_{5}^{7} + 24T_{5}^{6} + 55T_{5}^{5} + 402T_{5}^{4} + 912T_{5}^{3} + 2095T_{5}^{2} + 1350T_{5} + 729 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + 3 T^{7} + \cdots + 729 \) Copy content Toggle raw display
$7$ \( T^{8} - 5 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} + 33 T^{6} + \cdots + 225 \) Copy content Toggle raw display
$13$ \( (T^{4} + T^{3} - 27 T^{2} + \cdots + 122)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 3 T^{7} + \cdots + 2304 \) Copy content Toggle raw display
$19$ \( T^{8} + 2 T^{7} + \cdots + 463761 \) Copy content Toggle raw display
$23$ \( T^{8} - 9 T^{7} + \cdots + 51984 \) Copy content Toggle raw display
$29$ \( (T^{4} + 18 T^{3} + \cdots - 216)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 19 T^{7} + \cdots + 3276100 \) Copy content Toggle raw display
$37$ \( T^{8} + 2 T^{7} + \cdots + 605284 \) Copy content Toggle raw display
$41$ \( (T + 1)^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} - 5 T^{3} - 45 T^{2} + \cdots + 12)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 33 T^{6} + \cdots + 576 \) Copy content Toggle raw display
$53$ \( T^{8} + 6 T^{7} + \cdots + 324 \) Copy content Toggle raw display
$59$ \( T^{8} - 3 T^{7} + \cdots + 42224004 \) Copy content Toggle raw display
$61$ \( T^{8} - T^{7} + \cdots + 80089 \) Copy content Toggle raw display
$67$ \( T^{8} + 11 T^{7} + \cdots + 53824 \) Copy content Toggle raw display
$71$ \( (T^{4} + 9 T^{3} + \cdots - 3459)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 22 T^{7} + \cdots + 75076 \) Copy content Toggle raw display
$79$ \( T^{8} - 28 T^{7} + \cdots + 265225 \) Copy content Toggle raw display
$83$ \( (T^{4} + 12 T^{3} + \cdots - 3582)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 6 T^{7} + \cdots + 324900 \) Copy content Toggle raw display
$97$ \( (T^{4} - 11 T^{3} + \cdots + 360)^{2} \) Copy content Toggle raw display
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