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Properties

Label 574.2.e.g
Level $574$
Weight $2$
Character orbit 574.e
Analytic conductor $4.583$
Analytic rank $0$
Dimension $12$
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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + 11 x^{10} - 8 x^{9} + 85 x^{8} - 60 x^{7} + 305 x^{6} - 145 x^{5} + 748 x^{4} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 11 x^{10} - 8 x^{9} + 85 x^{8} - 60 x^{7} + 305 x^{6} - 145 x^{5} + 748 x^{4} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2784122 \nu^{11} + 63371845 \nu^{10} - 234642065 \nu^{9} + 642874797 \nu^{8} + \cdots + 93911170 ) / 37839303531 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 93911170 \nu^{11} - 91127048 \nu^{10} + 969651025 \nu^{9} - 516647295 \nu^{8} + \cdots + 113418387268 ) / 37839303531 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 99523325 \nu^{11} + 5612155 \nu^{10} - 1003629527 \nu^{9} - 173464425 \nu^{8} + \cdots + 2233062046 ) / 37839303531 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 243789473 \nu^{11} - 160858145 \nu^{10} - 2801657816 \nu^{9} - 1808899065 \nu^{8} + \cdots - 58253923718 ) / 37839303531 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 497616625 \nu^{11} + 28060775 \nu^{10} - 5018147635 \nu^{9} - 867322125 \nu^{8} + \cdots + 11165310230 ) / 37839303531 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1000495922 \nu^{11} + 370806340 \nu^{10} - 10215451964 \nu^{9} + 748134765 \nu^{8} + \cdots - 101752676291 ) / 37839303531 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2233062046 \nu^{11} - 2133538721 \nu^{10} + 24558070351 \nu^{9} - 16860866841 \nu^{8} + \cdots - 40869047138 ) / 37839303531 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3833650388 \nu^{11} + 3413702419 \nu^{10} - 41625666026 \nu^{9} + 27315999558 \nu^{8} + \cdots + 4965080122 ) / 37839303531 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 6259373114 \nu^{11} + 6513200266 \nu^{10} - 69042023174 \nu^{9} + 52291474731 \nu^{8} + \cdots + 111345135892 ) / 37839303531 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2233062046 \nu^{11} - 2133538721 \nu^{10} + 24558070351 \nu^{9} - 16860866841 \nu^{8} + \cdots - 40869047138 ) / 12613101177 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} + 3\beta_{8} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - 5\beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{11} + \beta_{10} - 15\beta_{8} + \beta_{7} + 6\beta_{3} + \beta_{2} - 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{11} - \beta_{10} - \beta_{9} - 2\beta_{8} - 8\beta_{6} + \beta_{5} + 27\beta_{4} - 8\beta_{2} - 27\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -9\beta_{7} + 11\beta_{6} + \beta_{5} - 35\beta_{3} + 79 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11\beta_{11} + 10\beta_{10} + 11\beta_{9} + 18\beta_{8} + 10\beta_{7} + 11\beta_{3} + 54\beta_{2} + 149\beta _1 + 18 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 203 \beta_{11} - 65 \beta_{10} + 12 \beta_{9} + 427 \beta_{8} - 86 \beta_{6} - 12 \beta_{5} + \cdots - 4 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -74\beta_{7} + 342\beta_{6} - 89\beta_{5} - 833\beta_{4} - 90\beta_{3} - 118 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1175 \beta_{11} + 431 \beta_{10} - 104 \beta_{9} - 2351 \beta_{8} + 431 \beta_{7} + 1175 \beta_{3} + \cdots - 2351 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 657 \beta_{11} - 491 \beta_{10} - 639 \beta_{9} - 676 \beta_{8} - 2097 \beta_{6} + 639 \beta_{5} + \cdots - 4701 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(211\) \(493\)
\(\chi(n)\) \(1\) \(\beta_{8}\)

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
0.321774 − 0.557329i
−1.21929 + 2.11188i
0.0302735 − 0.0524352i
1.14160 − 1.97731i
1.19221 − 2.06497i
−0.966566 + 1.67414i
0.321774 + 0.557329i
−1.21929 − 2.11188i
0.0302735 + 0.0524352i
1.14160 + 1.97731i
1.19221 + 2.06497i
−0.966566 − 1.67414i
0.500000 + 0.866025i −1.47561 + 2.55582i −0.500000 + 0.866025i −1.13231 − 1.96121i −2.95121 0.810531 − 2.51854i −1.00000 −2.85482 − 4.94470i 1.13231 − 1.96121i
165.2 0.500000 + 0.866025i −1.15431 + 1.99932i −0.500000 + 0.866025i 0.187218 + 0.324272i −2.30861 1.03207 + 2.43615i −1.00000 −1.16485 − 2.01757i −0.187218 + 0.324272i
165.3 0.500000 + 0.866025i −0.151256 + 0.261984i −0.500000 + 0.866025i 1.33775 + 2.31706i −0.302513 −1.36803 + 2.26462i −1.00000 1.45424 + 2.51882i −1.33775 + 2.31706i
165.4 0.500000 + 0.866025i 0.243160 − 0.421166i −0.500000 + 0.866025i −0.308156 − 0.533741i 0.486321 −0.833444 − 2.51105i −1.00000 1.38175 + 2.39325i 0.308156 − 0.533741i
165.5 0.500000 + 0.866025i 0.817236 − 1.41549i −0.500000 + 0.866025i 1.42313 + 2.46494i 1.63447 −2.61534 + 0.399966i −1.00000 0.164252 + 0.284493i −1.42313 + 2.46494i
165.6 0.500000 + 0.866025i 1.22077 − 2.11444i −0.500000 + 0.866025i −1.50764 − 2.61131i 2.44154 2.47421 − 0.937172i −1.00000 −1.48057 − 2.56442i 1.50764 − 2.61131i
247.1 0.500000 − 0.866025i −1.47561 − 2.55582i −0.500000 − 0.866025i −1.13231 + 1.96121i −2.95121 0.810531 + 2.51854i −1.00000 −2.85482 + 4.94470i 1.13231 + 1.96121i
247.2 0.500000 − 0.866025i −1.15431 − 1.99932i −0.500000 − 0.866025i 0.187218 − 0.324272i −2.30861 1.03207 − 2.43615i −1.00000 −1.16485 + 2.01757i −0.187218 − 0.324272i
247.3 0.500000 − 0.866025i −0.151256 − 0.261984i −0.500000 − 0.866025i 1.33775 − 2.31706i −0.302513 −1.36803 − 2.26462i −1.00000 1.45424 − 2.51882i −1.33775 − 2.31706i
247.4 0.500000 − 0.866025i 0.243160 + 0.421166i −0.500000 − 0.866025i −0.308156 + 0.533741i 0.486321 −0.833444 + 2.51105i −1.00000 1.38175 − 2.39325i 0.308156 + 0.533741i
247.5 0.500000 − 0.866025i 0.817236 + 1.41549i −0.500000 − 0.866025i 1.42313 − 2.46494i 1.63447 −2.61534 − 0.399966i −1.00000 0.164252 − 0.284493i −1.42313 − 2.46494i
247.6 0.500000 − 0.866025i 1.22077 + 2.11444i −0.500000 − 0.866025i −1.50764 + 2.61131i 2.44154 2.47421 + 0.937172i −1.00000 −1.48057 + 2.56442i 1.50764 + 2.61131i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 165.6
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.g ✓ 12
7.c even 3 1 inner 574.2.e.g ✓ 12
7.c even 3 1 4018.2.a.bo 6
7.d odd 6 1 4018.2.a.bn 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.e.g ✓ 12 1.a even 1 1 trivial
574.2.e.g ✓ 12 7.c even 3 1 inner
4018.2.a.bn 6 7.d odd 6 1
4018.2.a.bo 6 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 \(S_{2}^{\mathrm{new}}(574, [\chi])\):

\( T_{3}^{12} + T_{3}^{11} + 12 T_{3}^{10} - T_{3}^{9} + 96 T_{3}^{8} - 9 T_{3}^{7} + 351 T_{3}^{6} + \cdots + 16 \) Copy content Toggle raw display
\( T_{5}^{12} + 15 T_{5}^{10} + 2 T_{5}^{9} + 169 T_{5}^{8} + 27 T_{5}^{7} + 817 T_{5}^{6} + 304 T_{5}^{5} + \cdots + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} + T^{11} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{12} + 15 T^{10} + \cdots + 144 \) Copy content Toggle raw display
$7$ \( T^{12} + T^{11} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{12} + T^{11} + \cdots + 467856 \) Copy content Toggle raw display
$13$ \( (T^{6} - 4 T^{5} - 25 T^{4} + \cdots - 5)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} - T^{11} + \cdots + 1296 \) Copy content Toggle raw display
$19$ \( T^{12} - 3 T^{11} + \cdots + 602176 \) Copy content Toggle raw display
$23$ \( T^{12} + 21 T^{11} + \cdots + 360000 \) Copy content Toggle raw display
$29$ \( (T^{6} - 5 T^{5} + \cdots + 375)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 3 T^{11} + \cdots + 92416 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 742889536 \) Copy content Toggle raw display
$41$ \( (T - 1)^{12} \) Copy content Toggle raw display
$43$ \( (T^{6} - 12 T^{5} + \cdots + 367)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} - 18 T^{11} + \cdots + 3600 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 941507856 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 451520001 \) Copy content Toggle raw display
$61$ \( T^{12} - 20 T^{11} + \cdots + 399424 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 2423395984 \) Copy content Toggle raw display
$71$ \( (T^{6} - 11 T^{5} + \cdots + 1053807)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + T^{11} + \cdots + 34445161 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 4451024656 \) Copy content Toggle raw display
$83$ \( (T^{6} - 15 T^{5} + \cdots + 2307)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 1531313424 \) Copy content Toggle raw display
$97$ \( (T^{6} - T^{5} - 214 T^{4} + \cdots + 3824)^{2} \) Copy content Toggle raw display
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