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Properties

Label 430.2.e.g
Level $430$
Weight $2$
Character orbit 430.e
Analytic conductor $3.434$
Analytic rank $0$
Dimension $10$
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] = [430,2,Mod(221,430)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(430, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("430.221");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 430 = 2 \cdot 5 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 430.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: \(3.43356728692\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 12x^{8} + x^{7} + 106x^{6} - 27x^{5} + 233x^{4} - 164x^{3} + 460x^{2} - 240x + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 12x^{8} + x^{7} + 106x^{6} - 27x^{5} + 233x^{4} - 164x^{3} + 460x^{2} - 240x + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4541 \nu^{9} + 13275 \nu^{8} - 36108 \nu^{7} + 250973 \nu^{6} - 194346 \nu^{5} + 1203777 \nu^{4} + \cdots + 8999376 ) / 15118180 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2655 \nu^{9} - 20775 \nu^{8} + 56508 \nu^{7} - 185333 \nu^{6} + 304146 \nu^{5} - 1883877 \nu^{4} + \cdots - 14025200 ) / 3023636 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 115659 \nu^{9} - 648775 \nu^{8} + 1764668 \nu^{7} - 6351673 \nu^{6} + 9498066 \nu^{5} + \cdots - 125349056 ) / 30236360 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 187487 \nu^{9} + 201110 \nu^{8} - 2210019 \nu^{7} - 295811 \nu^{6} - 19120703 \nu^{5} + \cdots + 40561968 ) / 45354540 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 250497 \nu^{9} - 825325 \nu^{8} + 2244884 \nu^{7} - 5540819 \nu^{6} + 12082758 \nu^{5} + \cdots - 213627928 ) / 30236360 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 309304 \nu^{9} - 184850 \nu^{8} - 3276753 \nu^{7} - 5574127 \nu^{6} - 35089246 \nu^{5} + \cdots - 15923124 ) / 22677270 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 34825 \nu^{9} + 26632 \nu^{8} - 405039 \nu^{7} - 160516 \nu^{6} - 3625183 \nu^{5} + \cdots - 1056600 ) / 2267727 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 145979 \nu^{9} - 36050 \nu^{8} + 1678593 \nu^{7} + 1416977 \nu^{6} + 15714671 \nu^{5} + \cdots + 5601924 ) / 4123140 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - 4\beta_{5} - \beta_{3} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{4} - 2\beta_{3} - 9\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} - 11\beta_{8} + \beta_{7} + 29\beta_{5} - 17\beta _1 - 29 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 12 \beta_{9} - 28 \beta_{8} - 10 \beta_{7} - 12 \beta_{6} + 32 \beta_{5} + 10 \beta_{4} + \cdots - 91 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -18\beta_{6} - 6\beta_{4} + 119\beta_{3} + 225\beta_{2} + 266 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 125\beta_{9} + 344\beta_{8} + 89\beta_{7} - 501\beta_{5} + 973\beta _1 + 501 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 255 \beta_{9} + 1317 \beta_{8} + 5 \beta_{7} + 255 \beta_{6} - 2699 \beta_{5} - 5 \beta_{4} + \cdots + 2761 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1312\beta_{6} - 802\beta_{4} - 4078\beta_{3} - 10723\beta_{2} - 6588 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/430\mathbb{Z}\right)^\times\).

\(n\) \(87\) \(261\)
\(\chi(n)\) \(1\) \(-\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
221.1
−1.28976 + 2.23393i
−0.850932 + 1.47386i
0.323824 − 0.560879i
0.622887 − 1.07887i
1.69398 − 2.93406i
−1.28976 − 2.23393i
−0.850932 − 1.47386i
0.323824 + 0.560879i
0.622887 + 1.07887i
1.69398 + 2.93406i
−1.00000 −1.28976 + 2.23393i 1.00000 0.500000 − 0.866025i 1.28976 − 2.23393i 2.24349 + 3.88584i −1.00000 −1.82697 − 3.16441i −0.500000 + 0.866025i
221.2 −1.00000 −0.850932 + 1.47386i 1.00000 0.500000 − 0.866025i 0.850932 − 1.47386i −1.11274 − 1.92732i −1.00000 0.0518303 + 0.0897728i −0.500000 + 0.866025i
221.3 −1.00000 0.323824 − 0.560879i 1.00000 0.500000 − 0.866025i −0.323824 + 0.560879i 1.69423 + 2.93449i −1.00000 1.29028 + 2.23482i −0.500000 + 0.866025i
221.4 −1.00000 0.622887 − 1.07887i 1.00000 0.500000 − 0.866025i −0.622887 + 1.07887i −0.561656 − 0.972816i −1.00000 0.724024 + 1.25405i −0.500000 + 0.866025i
221.5 −1.00000 1.69398 − 2.93406i 1.00000 0.500000 − 0.866025i −1.69398 + 2.93406i 0.736679 + 1.27596i −1.00000 −4.23916 − 7.34244i −0.500000 + 0.866025i
251.1 −1.00000 −1.28976 − 2.23393i 1.00000 0.500000 + 0.866025i 1.28976 + 2.23393i 2.24349 − 3.88584i −1.00000 −1.82697 + 3.16441i −0.500000 − 0.866025i
251.2 −1.00000 −0.850932 − 1.47386i 1.00000 0.500000 + 0.866025i 0.850932 + 1.47386i −1.11274 + 1.92732i −1.00000 0.0518303 − 0.0897728i −0.500000 − 0.866025i
251.3 −1.00000 0.323824 + 0.560879i 1.00000 0.500000 + 0.866025i −0.323824 − 0.560879i 1.69423 − 2.93449i −1.00000 1.29028 − 2.23482i −0.500000 − 0.866025i
251.4 −1.00000 0.622887 + 1.07887i 1.00000 0.500000 + 0.866025i −0.622887 − 1.07887i −0.561656 + 0.972816i −1.00000 0.724024 − 1.25405i −0.500000 − 0.866025i
251.5 −1.00000 1.69398 + 2.93406i 1.00000 0.500000 + 0.866025i −1.69398 − 2.93406i 0.736679 − 1.27596i −1.00000 −4.23916 + 7.34244i −0.500000 − 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 221.5
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.e.g ✓ 10
43.c even 3 1 inner 430.2.e.g ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.e.g ✓ 10 1.a even 1 1 trivial
430.2.e.g ✓ 10 43.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} - T_{3}^{9} + 12T_{3}^{8} + T_{3}^{7} + 106T_{3}^{6} - 27T_{3}^{5} + 233T_{3}^{4} - 164T_{3}^{3} + 460T_{3}^{2} - 240T_{3} + 144 \) acting on \(S_{2}^{\mathrm{new}}(430, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - T^{9} + \cdots + 144 \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{5} \) Copy content Toggle raw display
$7$ \( T^{10} - 6 T^{9} + \cdots + 3136 \) Copy content Toggle raw display
$11$ \( (T^{5} - 2 T^{4} + \cdots + 486)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 6 T^{9} + \cdots + 38416 \) Copy content Toggle raw display
$17$ \( T^{10} - 4 T^{9} + \cdots + 1327104 \) Copy content Toggle raw display
$19$ \( T^{10} - 4 T^{9} + \cdots + 31684 \) Copy content Toggle raw display
$23$ \( T^{10} - 8 T^{9} + \cdots + 147456 \) Copy content Toggle raw display
$29$ \( T^{10} + T^{9} + \cdots + 46656 \) Copy content Toggle raw display
$31$ \( T^{10} + 8 T^{9} + \cdots + 4210704 \) Copy content Toggle raw display
$37$ \( T^{10} - 14 T^{9} + \cdots + 27793984 \) Copy content Toggle raw display
$41$ \( (T^{5} + 3 T^{4} - 42 T^{3} + \cdots - 69)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 147008443 \) Copy content Toggle raw display
$47$ \( (T^{5} + 11 T^{4} + \cdots - 7452)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 1040449536 \) Copy content Toggle raw display
$59$ \( (T^{5} - 4 T^{4} + \cdots + 1152)^{2} \) Copy content Toggle raw display
$61$ \( T^{10} + 14 T^{9} + \cdots + 3136 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 209699361 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 3776348304 \) Copy content Toggle raw display
$73$ \( T^{10} - 28 T^{9} + \cdots + 5885476 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 20736000000 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 1211875344 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 6357032361 \) Copy content Toggle raw display
$97$ \( (T^{5} - 10 T^{4} + \cdots - 9368)^{2} \) Copy content Toggle raw display
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