Properties

Label 574.2.c.c
Level $574$
Weight $2$
Character orbit 574.c
Analytic conductor $4.583$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [574,2,Mod(491,574)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(574, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("574.491");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 574 = 2 \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 574.c (of order \(2\), degree \(1\), 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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 22x^{8} + 153x^{6} + 340x^{4} + 144x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_1 q^{3} + q^{4} + ( - \beta_{8} + \beta_{4}) q^{5} - \beta_1 q^{6} + \beta_{6} q^{7} - q^{8} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta_1 q^{3} + q^{4} + ( - \beta_{8} + \beta_{4}) q^{5} - \beta_1 q^{6} + \beta_{6} q^{7} - q^{8} + (\beta_{2} - 1) q^{9} + (\beta_{8} - \beta_{4}) q^{10} - \beta_{7} q^{11} + \beta_1 q^{12} + ( - \beta_{7} + \beta_{5}) q^{13} - \beta_{6} q^{14} + (\beta_{9} + 2 \beta_{6}) q^{15} + q^{16} + (2 \beta_{6} + \beta_1) q^{17} + ( - \beta_{2} + 1) q^{18} + (\beta_{7} + 2 \beta_{6} + \beta_{5}) q^{19} + ( - \beta_{8} + \beta_{4}) q^{20} - \beta_{4} q^{21} + \beta_{7} q^{22} + (\beta_{8} - \beta_{3}) q^{23} - \beta_1 q^{24} + (\beta_{8} + \beta_{3} + \beta_{2} + 3) q^{25} + (\beta_{7} - \beta_{5}) q^{26} + (\beta_{7} - 2 \beta_{6} + \beta_{5} - 3 \beta_1) q^{27} + \beta_{6} q^{28} + ( - \beta_{9} + \beta_{7}) q^{29} + ( - \beta_{9} - 2 \beta_{6}) q^{30} + (\beta_{8} + \beta_{4} + \beta_{3}) q^{31} - q^{32} + 2 q^{33} + ( - 2 \beta_{6} - \beta_1) q^{34} + ( - \beta_{7} + \beta_1) q^{35} + (\beta_{2} - 1) q^{36} + ( - 2 \beta_{4} - 2) q^{37} + ( - \beta_{7} - 2 \beta_{6} - \beta_{5}) q^{38} + ( - 2 \beta_{4} + 2 \beta_{3}) q^{39} + (\beta_{8} - \beta_{4}) q^{40} + ( - \beta_{9} + \beta_{6} + \beta_{5} + \beta_{3} - \beta_1 + 2) q^{41} + \beta_{4} q^{42} + (2 \beta_{4} + \beta_{2} + 4) q^{43} - \beta_{7} q^{44} + ( - 2 \beta_{8} - 4 \beta_{4} + \beta_{3} - 2) q^{45} + ( - \beta_{8} + \beta_{3}) q^{46} + (2 \beta_{9} + \beta_{7} + 2 \beta_{6} - \beta_{5}) q^{47} + \beta_1 q^{48} - q^{49} + ( - \beta_{8} - \beta_{3} - \beta_{2} - 3) q^{50} + ( - 2 \beta_{4} + \beta_{2} - 4) q^{51} + ( - \beta_{7} + \beta_{5}) q^{52} + (\beta_{9} - \beta_{7} - 6 \beta_{6}) q^{53} + ( - \beta_{7} + 2 \beta_{6} - \beta_{5} + 3 \beta_1) q^{54} + ( - 2 \beta_{9} + \beta_{7} + 6 \beta_{6} + \beta_{5}) q^{55} - \beta_{6} q^{56} + ( - 4 \beta_{4} + 2 \beta_{3} - 4) q^{57} + (\beta_{9} - \beta_{7}) q^{58} + ( - \beta_{3} + 6) q^{59} + (\beta_{9} + 2 \beta_{6}) q^{60} + ( - \beta_{8} - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 2) q^{61} + ( - \beta_{8} - \beta_{4} - \beta_{3}) q^{62} + ( - \beta_{9} - \beta_{6}) q^{63} + q^{64} + ( - 2 \beta_{9} + 6 \beta_{6} + 2 \beta_1) q^{65} - 2 q^{66} + (2 \beta_{9} + \beta_{7} - 2 \beta_{6} - 2 \beta_1) q^{67} + (2 \beta_{6} + \beta_1) q^{68} + (2 \beta_{5} - 2 \beta_1) q^{69} + (\beta_{7} - \beta_1) q^{70} + ( - \beta_{9} - 4 \beta_{6}) q^{71} + ( - \beta_{2} + 1) q^{72} + (2 \beta_{4} + 2 \beta_{2} + 2) q^{73} + (2 \beta_{4} + 2) q^{74} + (\beta_{7} + 2 \beta_{6} - \beta_{5}) q^{75} + (\beta_{7} + 2 \beta_{6} + \beta_{5}) q^{76} + \beta_{8} q^{77} + (2 \beta_{4} - 2 \beta_{3}) q^{78} + ( - \beta_{9} + 2 \beta_{5}) q^{79} + ( - \beta_{8} + \beta_{4}) q^{80} + (2 \beta_{3} + 5) q^{81} + (\beta_{9} - \beta_{6} - \beta_{5} - \beta_{3} + \beta_1 - 2) q^{82} + (4 \beta_{4} - 3 \beta_{3} - 2 \beta_{2} - 2) q^{83} - \beta_{4} q^{84} + (\beta_{9} - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_1) q^{85} + ( - 2 \beta_{4} - \beta_{2} - 4) q^{86} + ( - \beta_{8} + 5 \beta_{4} - \beta_{3}) q^{87} + \beta_{7} q^{88} + (\beta_{7} - 8 \beta_{6} + \beta_{5} - 3 \beta_1) q^{89} + (2 \beta_{8} + 4 \beta_{4} - \beta_{3} + 2) q^{90} + (\beta_{8} - \beta_{3}) q^{91} + (\beta_{8} - \beta_{3}) q^{92} + (\beta_{9} + 8 \beta_{6} - 2 \beta_{5} + 2 \beta_1) q^{93} + ( - 2 \beta_{9} - \beta_{7} - 2 \beta_{6} + \beta_{5}) q^{94} + (2 \beta_{9} - 4 \beta_{7} - 6 \beta_{6} - 2 \beta_{5} + 4 \beta_1) q^{95} - \beta_1 q^{96} + ( - \beta_{7} + 4 \beta_{6} + \beta_{5} - \beta_1) q^{97} + q^{98} + ( - 3 \beta_{7} + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{2} + 10 q^{4} - 4 q^{5} - 10 q^{8} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 10 q^{2} + 10 q^{4} - 4 q^{5} - 10 q^{8} - 14 q^{9} + 4 q^{10} + 10 q^{16} + 14 q^{18} - 4 q^{20} + 4 q^{21} + 26 q^{25} - 4 q^{31} - 10 q^{32} + 20 q^{33} - 14 q^{36} - 12 q^{37} + 8 q^{39} + 4 q^{40} + 20 q^{41} - 4 q^{42} + 28 q^{43} - 4 q^{45} - 10 q^{49} - 26 q^{50} - 36 q^{51} - 24 q^{57} + 60 q^{59} - 24 q^{61} + 4 q^{62} + 10 q^{64} - 20 q^{66} + 14 q^{72} + 4 q^{73} + 12 q^{74} - 8 q^{78} - 4 q^{80} + 50 q^{81} - 20 q^{82} - 28 q^{83} + 4 q^{84} - 28 q^{86} - 20 q^{87} + 4 q^{90} + 10 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 22x^{8} + 153x^{6} + 340x^{4} + 144x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + 9\nu^{2} + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 13\nu^{4} + 36\nu^{2} + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{9} + 22\nu^{7} + 149\nu^{5} + 296\nu^{3} + 72\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{9} + 22\nu^{7} + 151\nu^{5} + 314\nu^{3} + 72\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{9} + 22\nu^{7} + 153\nu^{5} + 340\nu^{3} + 144\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{8} + 22\nu^{6} + 151\nu^{4} + 314\nu^{2} + 72 ) / 4 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -2\nu^{9} - 43\nu^{7} - 289\nu^{5} - 592\nu^{3} - 136\nu ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - 2\beta_{6} + \beta_{5} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{3} - 9\beta_{2} + 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -9\beta_{7} + 22\beta_{6} - 13\beta_{5} + 81\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{4} - 26\beta_{3} + 81\beta_{2} - 280 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4\beta_{9} + 81\beta_{7} - 198\beta_{6} + 133\beta_{5} - 737\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 4\beta_{8} - 88\beta_{4} + 270\beta_{3} - 737\beta_{2} + 2512 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -88\beta_{9} - 737\beta_{7} + 1670\beta_{6} - 1277\beta_{5} + 6737\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(211\) \(493\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
491.1
3.06582i
2.98495i
1.78395i
0.567173i
0.431994i
0.431994i
0.567173i
1.78395i
2.98495i
3.06582i
−1.00000 3.06582i 1.00000 −2.41347 3.06582i 1.00000i −1.00000 −6.39928 2.41347
491.2 −1.00000 2.98495i 1.00000 2.31493 2.98495i 1.00000i −1.00000 −5.90994 −2.31493
491.3 −1.00000 1.78395i 1.00000 −0.662840 1.78395i 1.00000i −1.00000 −0.182473 0.662840
491.4 −1.00000 0.567173i 1.00000 2.95909 0.567173i 1.00000i −1.00000 2.67832 −2.95909
491.5 −1.00000 0.431994i 1.00000 −4.19770 0.431994i 1.00000i −1.00000 2.81338 4.19770
491.6 −1.00000 0.431994i 1.00000 −4.19770 0.431994i 1.00000i −1.00000 2.81338 4.19770
491.7 −1.00000 0.567173i 1.00000 2.95909 0.567173i 1.00000i −1.00000 2.67832 −2.95909
491.8 −1.00000 1.78395i 1.00000 −0.662840 1.78395i 1.00000i −1.00000 −0.182473 0.662840
491.9 −1.00000 2.98495i 1.00000 2.31493 2.98495i 1.00000i −1.00000 −5.90994 −2.31493
491.10 −1.00000 3.06582i 1.00000 −2.41347 3.06582i 1.00000i −1.00000 −6.39928 2.41347
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 491.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 574.2.c.c 10
41.b even 2 1 inner 574.2.c.c 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.c.c 10 1.a even 1 1 trivial
574.2.c.c 10 41.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 22T_{3}^{8} + 153T_{3}^{6} + 340T_{3}^{4} + 144T_{3}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(574, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 22 T^{8} + 153 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( (T^{5} + 2 T^{4} - 17 T^{3} - 20 T^{2} + \cdots + 46)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$11$ \( T^{10} + 36 T^{8} + 340 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( T^{10} + 88 T^{8} + 2736 T^{6} + \cdots + 135424 \) Copy content Toggle raw display
$17$ \( T^{10} + 34 T^{8} + 249 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( T^{10} + 148 T^{8} + 7728 T^{6} + \cdots + 1048576 \) Copy content Toggle raw display
$23$ \( (T^{5} - 44 T^{3} - 36 T^{2} + 400 T + 368)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} + 126 T^{8} + 4565 T^{6} + \cdots + 100 \) Copy content Toggle raw display
$31$ \( (T^{5} + 2 T^{4} - 89 T^{3} + 54 T^{2} + \cdots - 184)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} + 6 T^{4} - 28 T^{3} - 88 T^{2} + \cdots + 128)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} - 20 T^{9} + \cdots + 115856201 \) Copy content Toggle raw display
$43$ \( (T^{5} - 14 T^{4} - 3 T^{3} + 744 T^{2} + \cdots + 2704)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + 324 T^{8} + \cdots + 145926400 \) Copy content Toggle raw display
$53$ \( T^{10} + 282 T^{8} + 16757 T^{6} + \cdots + 2116 \) Copy content Toggle raw display
$59$ \( (T^{5} - 30 T^{4} + 324 T^{3} - 1618 T^{2} + \cdots - 3224)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + 12 T^{4} - 137 T^{3} - 1442 T^{2} + \cdots + 6074)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 384 T^{8} + 42812 T^{6} + \cdots + 8667136 \) Copy content Toggle raw display
$71$ \( T^{10} + 178 T^{8} + 8737 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$73$ \( (T^{5} - 2 T^{4} - 200 T^{3} + 944 T^{2} + \cdots - 160)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + 258 T^{8} + 14657 T^{6} + \cdots + 2768896 \) Copy content Toggle raw display
$83$ \( (T^{5} + 14 T^{4} - 344 T^{3} + \cdots + 381664)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + 454 T^{8} + \cdots + 530473024 \) Copy content Toggle raw display
$97$ \( T^{10} + 214 T^{8} + 10993 T^{6} + \cdots + 64 \) Copy content Toggle raw display
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