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Properties

Label 430.2.k.c
Level $430$
Weight $2$
Character orbit 430.k
Analytic conductor $3.434$
Analytic rank $0$
Dimension $18$
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: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{7})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 5 x^{17} + 20 x^{16} - 61 x^{15} + 142 x^{14} - 195 x^{13} + 244 x^{12} + 320 x^{11} + \cdots + 49 \) 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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 5 x^{17} + 20 x^{16} - 61 x^{15} + 142 x^{14} - 195 x^{13} + 244 x^{12} + 320 x^{11} + \cdots + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 82\!\cdots\!53 \nu^{17} + \cdots + 17\!\cdots\!38 ) / 14\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 19\!\cdots\!05 \nu^{17} + \cdots + 12\!\cdots\!36 ) / 14\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 27\!\cdots\!98 \nu^{17} + \cdots - 18\!\cdots\!83 ) / 14\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 10\!\cdots\!44 \nu^{17} + \cdots - 78\!\cdots\!18 ) / 14\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 58\!\cdots\!99 \nu^{17} + \cdots - 30\!\cdots\!09 ) / 73\!\cdots\!54 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 11\!\cdots\!40 \nu^{17} + \cdots + 19\!\cdots\!91 ) / 14\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 15\!\cdots\!34 \nu^{17} + \cdots - 64\!\cdots\!78 ) / 14\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 19\!\cdots\!61 \nu^{17} + \cdots - 15\!\cdots\!50 ) / 14\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 22\!\cdots\!40 \nu^{17} + \cdots + 10\!\cdots\!72 ) / 14\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 29\!\cdots\!71 \nu^{17} + \cdots + 14\!\cdots\!42 ) / 14\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 31\!\cdots\!50 \nu^{17} + \cdots - 30\!\cdots\!30 ) / 14\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 32\!\cdots\!24 \nu^{17} + \cdots + 12\!\cdots\!70 ) / 14\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 34\!\cdots\!35 \nu^{17} + \cdots + 53\!\cdots\!59 ) / 14\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 37\!\cdots\!67 \nu^{17} + \cdots - 57\!\cdots\!15 ) / 14\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 52\!\cdots\!74 \nu^{17} + \cdots + 53\!\cdots\!74 ) / 14\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 12\!\cdots\!18 \nu^{17} + \cdots + 92\!\cdots\!83 ) / 14\!\cdots\!08 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( - \beta_{16} - \beta_{15} + \beta_{14} - \beta_{13} + \beta_{11} + \beta_{10} - \beta_{7} + \beta_{5} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{16} + \beta_{14} + 2 \beta_{12} - 2 \beta_{10} - \beta_{8} + 2 \beta_{7} - \beta_{6} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{17} + \beta_{16} + 9 \beta_{15} + 8 \beta_{13} - 8 \beta_{11} + 3 \beta_{9} - 10 \beta_{8} + \cdots + 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10 \beta_{17} + 5 \beta_{15} - 7 \beta_{14} + 10 \beta_{13} + 4 \beta_{12} - 7 \beta_{11} + 18 \beta_{10} + \cdots - 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 53 \beta_{17} - 8 \beta_{16} + 3 \beta_{15} - 53 \beta_{14} + 5 \beta_{13} + 143 \beta_{12} + \cdots - 33 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 79 \beta_{17} + 79 \beta_{16} + 202 \beta_{15} - 120 \beta_{14} + 120 \beta_{13} + 202 \beta_{12} + \cdots + 157 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 48 \beta_{17} + 423 \beta_{16} + 537 \beta_{15} - 88 \beta_{14} + 423 \beta_{13} - 538 \beta_{12} + \cdots - 601 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 737 \beta_{16} - 182 \beta_{15} - 737 \beta_{14} + 308 \beta_{13} + 389 \beta_{11} + 182 \beta_{10} + \cdots - 1965 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 224 \beta_{17} + 1317 \beta_{16} - 3690 \beta_{14} + 4028 \beta_{12} - 224 \beta_{11} - 8414 \beta_{10} + \cdots + 4028 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 6151 \beta_{17} + 4100 \beta_{16} + 1643 \beta_{15} - 2145 \beta_{13} - 15533 \beta_{12} + \cdots + 1643 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 28821 \beta_{17} - 36579 \beta_{15} + 29295 \beta_{14} - 28821 \beta_{13} - 76402 \beta_{12} + \cdots - 66641 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 53293 \beta_{17} - 37575 \beta_{16} - 134492 \beta_{15} + 53293 \beta_{14} - 91706 \beta_{13} + \cdots - 122901 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 112448 \beta_{17} - 112448 \beta_{16} - 102208 \beta_{15} + 119041 \beta_{14} - 119041 \beta_{13} + \cdots + 532296 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 336477 \beta_{17} - 458294 \beta_{16} - 430535 \beta_{15} + 811210 \beta_{14} - 458294 \beta_{13} + \cdots - 104770 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 2044822 \beta_{16} - 2686846 \beta_{15} + 2044822 \beta_{14} - 1910513 \beta_{13} + 894493 \beta_{11} + \cdots - 1668184 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 2970492 \beta_{17} - 4150650 \beta_{16} + 996226 \beta_{14} + 10750085 \beta_{12} - 2970492 \beta_{11} + \cdots + 10750085 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(87\) \(261\)
\(\chi(n)\) \(1\) \(-\beta_{3}\)

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
−0.583263 + 0.731388i
0.219976 − 0.275841i
1.14077 − 1.43048i
−1.43931 + 0.693135i
0.403095 − 0.194120i
2.65970 − 1.28085i
−1.43931 − 0.693135i
0.403095 + 0.194120i
2.65970 + 1.28085i
−0.635189 − 2.78294i
0.168441 + 0.737990i
0.565778 + 2.47884i
−0.635189 + 2.78294i
0.168441 − 0.737990i
0.565778 − 2.47884i
−0.583263 − 0.731388i
0.219976 + 0.275841i
1.14077 + 1.43048i
−0.623490 − 0.781831i −0.583263 + 0.731388i −0.222521 + 0.974928i −0.900969 + 0.433884i 0.935481 2.01178 0.900969 − 0.433884i 0.472829 + 2.07160i 0.900969 + 0.433884i
11.2 −0.623490 − 0.781831i 0.219976 − 0.275841i −0.222521 + 0.974928i −0.900969 + 0.433884i −0.352814 −0.963199 0.900969 − 0.433884i 0.639864 + 2.80343i 0.900969 + 0.433884i
11.3 −0.623490 − 0.781831i 1.14077 − 1.43048i −0.222521 + 0.974928i −0.900969 + 0.433884i −1.82965 4.80227 0.900969 − 0.433884i −0.0773496 − 0.338891i 0.900969 + 0.433884i
21.1 0.900969 + 0.433884i −1.43931 + 0.693135i 0.623490 + 0.781831i −0.222521 − 0.974928i −1.59751 −1.15014 0.222521 + 0.974928i −0.279294 + 0.350223i 0.222521 − 0.974928i
21.2 0.900969 + 0.433884i 0.403095 − 0.194120i 0.623490 + 0.781831i −0.222521 − 0.974928i 0.447401 2.78084 0.222521 + 0.974928i −1.74567 + 2.18900i 0.222521 − 0.974928i
21.3 0.900969 + 0.433884i 2.65970 − 1.28085i 0.623490 + 0.781831i −0.222521 − 0.974928i 2.95205 −1.54255 0.222521 + 0.974928i 3.56299 − 4.46785i 0.222521 − 0.974928i
41.1 0.900969 − 0.433884i −1.43931 − 0.693135i 0.623490 − 0.781831i −0.222521 + 0.974928i −1.59751 −1.15014 0.222521 − 0.974928i −0.279294 − 0.350223i 0.222521 + 0.974928i
41.2 0.900969 − 0.433884i 0.403095 + 0.194120i 0.623490 − 0.781831i −0.222521 + 0.974928i 0.447401 2.78084 0.222521 − 0.974928i −1.74567 − 2.18900i 0.222521 + 0.974928i
41.3 0.900969 − 0.433884i 2.65970 + 1.28085i 0.623490 − 0.781831i −0.222521 + 0.974928i 2.95205 −1.54255 0.222521 − 0.974928i 3.56299 + 4.46785i 0.222521 + 0.974928i
121.1 0.222521 − 0.974928i −0.635189 − 2.78294i −0.900969 − 0.433884i 0.623490 − 0.781831i −2.85451 −1.93646 −0.623490 + 0.781831i −4.63840 + 2.23374i −0.623490 − 0.781831i
121.2 0.222521 − 0.974928i 0.168441 + 0.737990i −0.900969 − 0.433884i 0.623490 − 0.781831i 0.756969 −2.52891 −0.623490 + 0.781831i 2.18665 − 1.05304i −0.623490 − 0.781831i
121.3 0.222521 − 0.974928i 0.565778 + 2.47884i −0.900969 − 0.433884i 0.623490 − 0.781831i 2.54259 2.52637 −0.623490 + 0.781831i −3.12162 + 1.50329i −0.623490 − 0.781831i
231.1 0.222521 + 0.974928i −0.635189 + 2.78294i −0.900969 + 0.433884i 0.623490 + 0.781831i −2.85451 −1.93646 −0.623490 − 0.781831i −4.63840 − 2.23374i −0.623490 + 0.781831i
231.2 0.222521 + 0.974928i 0.168441 − 0.737990i −0.900969 + 0.433884i 0.623490 + 0.781831i 0.756969 −2.52891 −0.623490 − 0.781831i 2.18665 + 1.05304i −0.623490 + 0.781831i
231.3 0.222521 + 0.974928i 0.565778 − 2.47884i −0.900969 + 0.433884i 0.623490 + 0.781831i 2.54259 2.52637 −0.623490 − 0.781831i −3.12162 − 1.50329i −0.623490 + 0.781831i
391.1 −0.623490 + 0.781831i −0.583263 − 0.731388i −0.222521 − 0.974928i −0.900969 − 0.433884i 0.935481 2.01178 0.900969 + 0.433884i 0.472829 − 2.07160i 0.900969 − 0.433884i
391.2 −0.623490 + 0.781831i 0.219976 + 0.275841i −0.222521 − 0.974928i −0.900969 − 0.433884i −0.352814 −0.963199 0.900969 + 0.433884i 0.639864 − 2.80343i 0.900969 − 0.433884i
391.3 −0.623490 + 0.781831i 1.14077 + 1.43048i −0.222521 − 0.974928i −0.900969 − 0.433884i −1.82965 4.80227 0.900969 + 0.433884i −0.0773496 + 0.338891i 0.900969 − 0.433884i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.3
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.c ✓ 18
43.e even 7 1 inner 430.2.k.c ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.k.c ✓ 18 1.a even 1 1 trivial
430.2.k.c ✓ 18 43.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{18} - 5 T_{3}^{17} + 20 T_{3}^{16} - 61 T_{3}^{15} + 142 T_{3}^{14} - 195 T_{3}^{13} + 244 T_{3}^{12} + \cdots + 49 \) acting on \(S_{2}^{\mathrm{new}}(430, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - T^{5} + T^{4} + \cdots + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{18} - 5 T^{17} + \cdots + 49 \) Copy content Toggle raw display
$5$ \( (T^{6} + T^{5} + T^{4} + \cdots + 1)^{3} \) Copy content Toggle raw display
$7$ \( (T^{9} - 4 T^{8} + \cdots + 568)^{2} \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 1078268569 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 3300272704 \) Copy content Toggle raw display
$17$ \( T^{18} - T^{17} + \cdots + 175561 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 2500900081 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 587965504 \) Copy content Toggle raw display
$29$ \( T^{18} + 6 T^{17} + \cdots + 27793984 \) Copy content Toggle raw display
$31$ \( T^{18} + 5 T^{17} + \cdots + 153664 \) Copy content Toggle raw display
$37$ \( (T^{9} + 9 T^{8} + \cdots + 1319704)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 970758649 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 502592611936843 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 32431687744 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 1476186120256 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 43452578708449 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 80\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 41\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 41183020096 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 52\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( (T^{9} + 13 T^{8} + \cdots - 1052136)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 797011550720449 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 50317415949529 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 106845692351161 \) Copy content Toggle raw display
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