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Properties

Label 430.2.k.a
Level $430$
Weight $2$
Character orbit 430.k
Analytic conductor $3.434$
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] = [430,2,Mod(11,430)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(430, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("430.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 430 = 2 \cdot 5 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 430.k (of order \(7\), degree \(6\), 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: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{7})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 6 x^{10} - 16 x^{9} + 44 x^{8} - 70 x^{7} + 141 x^{6} - 182 x^{5} + 270 x^{4} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 6 x^{10} - 16 x^{9} + 44 x^{8} - 70 x^{7} + 141 x^{6} - 182 x^{5} + 270 x^{4} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2557946812 \nu^{11} - 3680607848 \nu^{10} + 5871717403 \nu^{9} - 19049524293 \nu^{8} + \cdots + 99844959362 ) / 636866963069 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 6911224142 \nu^{11} + 17158748797 \nu^{10} - 23379170647 \nu^{9} + 77839170172 \nu^{8} + \cdots - 54865843774 ) / 636866963069 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 7145459624 \nu^{11} + 30723699477 \nu^{10} - 67845292380 \nu^{9} + 160567456914 \nu^{8} + \cdots - 101355141854 ) / 636866963069 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13015530208 \nu^{11} - 40728566222 \nu^{10} + 80752773771 \nu^{9} - 210033213720 \nu^{8} + \cdots + 81088467803 ) / 636866963069 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13027070856 \nu^{11} - 32077498270 \nu^{10} + 53757650992 \nu^{9} - 169355218836 \nu^{8} + \cdots + 47560487414 ) / 636866963069 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 47560487414 \nu^{11} + 155708533098 \nu^{10} - 317440422754 \nu^{9} + 814725449616 \nu^{8} + \cdots + 77037234579 ) / 636866963069 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 54865843774 \nu^{11} - 171508755464 \nu^{10} + 346353811441 \nu^{9} - 901232671031 \nu^{8} + \cdots + 1074027111334 ) / 636866963069 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 81088467803 \nu^{11} + 256280933617 \nu^{10} - 527259373040 \nu^{9} + 1378168258619 \nu^{8} + \cdots - 508460209612 ) / 636866963069 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 99844959362 \nu^{11} + 302092824898 \nu^{10} - 602750364020 \nu^{9} + 1603391067195 \nu^{8} + \cdots - 1307183435441 ) / 636866963069 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 112806277053 \nu^{11} + 341222554101 \nu^{10} - 685432152894 \nu^{9} + \cdots - 1284433711294 ) / 636866963069 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} + \beta_{6} - \beta_{4} + \beta_{3} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} + \beta_{6} + 3\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{11} - 3\beta_{9} + 3\beta_{8} + 4\beta_{7} + 5\beta_{6} - 6\beta_{4} + 5\beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{11} + 2 \beta_{10} + 5 \beta_{9} + \beta_{8} - 5 \beta_{7} + 18 \beta_{6} - 11 \beta_{5} + \cdots - 18 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 6 \beta_{11} - 16 \beta_{10} - 27 \beta_{8} - 6 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 26 \beta_{4} + \cdots + 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 22 \beta_{10} - 22 \beta_{9} + 29 \beta_{8} + 29 \beta_{7} - 41 \beta_{5} - 41 \beta_{4} - 45 \beta_{3} + \cdots - 38 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 31 \beta_{11} + 31 \beta_{10} + 35 \beta_{9} - 80 \beta_{7} - 22 \beta_{6} - 118 \beta_{5} + \cdots + 35 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 35 \beta_{11} - 131 \beta_{10} - 57 \beta_{9} - 131 \beta_{8} - 426 \beta_{6} - 74 \beta_{5} + 228 \beta_{4} + \cdots + 35 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 198 \beta_{11} + 322 \beta_{10} - 198 \beta_{9} + 476 \beta_{8} + 322 \beta_{7} - 767 \beta_{6} + \cdots - 476 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 320 \beta_{11} + 149 \beta_{9} - 149 \beta_{8} - 574 \beta_{7} - 1730 \beta_{6} + 2648 \beta_{4} + \cdots + 574 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(87\) \(261\)
\(\chi(n)\) \(1\) \(-\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
0.202247 + 0.886104i
−0.380695 − 1.66794i
0.963727 − 1.20848i
−1.30974 + 1.64236i
0.963727 + 1.20848i
−1.30974 − 1.64236i
−0.0765242 + 0.0368521i
2.10098 − 1.01178i
−0.0765242 − 0.0368521i
2.10098 + 1.01178i
0.202247 − 0.886104i
−0.380695 + 1.66794i
0.623490 + 0.781831i −1.69017 + 2.11941i −0.222521 + 0.974928i −0.900969 + 0.433884i −2.71083 −1.61896 −0.900969 + 0.433884i −0.967653 − 4.23957i −0.900969 − 0.433884i
11.2 0.623490 + 0.781831i −0.0568052 + 0.0712315i −0.222521 + 0.974928i −0.900969 + 0.433884i −0.0911085 0.481895 −0.900969 + 0.433884i 0.665716 + 2.91669i −0.900969 − 0.433884i
21.1 −0.900969 − 0.433884i −0.991657 + 0.477557i 0.623490 + 0.781831i −0.222521 − 0.974928i 1.10066 1.63616 −0.222521 − 0.974928i −1.11515 + 1.39835i −0.222521 + 0.974928i
21.2 −0.900969 − 0.433884i 2.29359 − 1.10454i 0.623490 + 0.781831i −0.222521 − 0.974928i −2.54570 3.65974 −0.222521 − 0.974928i 2.17010 − 2.72123i −0.222521 + 0.974928i
41.1 −0.900969 + 0.433884i −0.991657 − 0.477557i 0.623490 − 0.781831i −0.222521 + 0.974928i 1.10066 1.63616 −0.222521 + 0.974928i −1.11515 − 1.39835i −0.222521 − 0.974928i
41.2 −0.900969 + 0.433884i 2.29359 + 1.10454i 0.623490 − 0.781831i −0.222521 + 0.974928i −2.54570 3.65974 −0.222521 + 0.974928i 2.17010 + 2.72123i −0.222521 − 0.974928i
121.1 −0.222521 + 0.974928i −0.296379 − 1.29852i −0.900969 − 0.433884i 0.623490 − 0.781831i 1.33192 −3.79472 0.623490 − 0.781831i 1.10459 − 0.531942i 0.623490 + 0.781831i
121.2 −0.222521 + 0.974928i 0.241421 + 1.05773i −0.900969 − 0.433884i 0.623490 − 0.781831i −1.08494 1.63589 0.623490 − 0.781831i 1.64239 − 0.790933i 0.623490 + 0.781831i
231.1 −0.222521 − 0.974928i −0.296379 + 1.29852i −0.900969 + 0.433884i 0.623490 + 0.781831i 1.33192 −3.79472 0.623490 + 0.781831i 1.10459 + 0.531942i 0.623490 − 0.781831i
231.2 −0.222521 − 0.974928i 0.241421 − 1.05773i −0.900969 + 0.433884i 0.623490 + 0.781831i −1.08494 1.63589 0.623490 + 0.781831i 1.64239 + 0.790933i 0.623490 − 0.781831i
391.1 0.623490 − 0.781831i −1.69017 − 2.11941i −0.222521 − 0.974928i −0.900969 − 0.433884i −2.71083 −1.61896 −0.900969 − 0.433884i −0.967653 + 4.23957i −0.900969 + 0.433884i
391.2 0.623490 − 0.781831i −0.0568052 − 0.0712315i −0.222521 − 0.974928i −0.900969 − 0.433884i −0.0911085 0.481895 −0.900969 − 0.433884i 0.665716 − 2.91669i −0.900969 + 0.433884i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.k.a ✓ 12
43.e even 7 1 inner 430.2.k.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.k.a ✓ 12 1.a even 1 1 trivial
430.2.k.a ✓ 12 43.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + T_{3}^{11} - 15 T_{3}^{9} + 13 T_{3}^{8} + 42 T_{3}^{7} + 127 T_{3}^{6} + 196 T_{3}^{5} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(430, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + T^{5} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} + T^{11} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{6} + T^{5} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} - 2 T^{5} - 16 T^{4} + \cdots + 29)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} - 3 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( (T^{6} + 2 T^{5} + 4 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 5 T^{11} + \cdots + 169 \) Copy content Toggle raw display
$19$ \( T^{12} + 2 T^{11} + \cdots + 121801 \) Copy content Toggle raw display
$23$ \( (T^{6} + 8 T^{5} + \cdots + 169)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} - 2 T^{11} + \cdots + 9665881 \) Copy content Toggle raw display
$31$ \( T^{12} - 27 T^{11} + \cdots + 63001 \) Copy content Toggle raw display
$37$ \( (T^{6} + 9 T^{5} + \cdots + 1673)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} - 14 T^{11} + \cdots + 32761 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 6321363049 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 8977752001 \) Copy content Toggle raw display
$53$ \( T^{12} - 2 T^{11} + \cdots + 877969 \) Copy content Toggle raw display
$59$ \( T^{12} - 4 T^{11} + \cdots + 841 \) Copy content Toggle raw display
$61$ \( T^{12} - 16 T^{11} + \cdots + 1104601 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 242829889 \) Copy content Toggle raw display
$71$ \( T^{12} - 27 T^{11} + \cdots + 6497401 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 137147521 \) Copy content Toggle raw display
$79$ \( (T^{6} - 5 T^{5} + \cdots - 1421)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 1002418921 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 3683985035641 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 8139107089 \) Copy content Toggle raw display
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