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Properties

Label 429.2.n.a
Level $429$
Weight $2$
Character orbit 429.n
Analytic conductor $3.426$
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] = [429,2,Mod(157,429)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(429, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 8, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("429.157");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 429 = 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 429.n (of order \(5\), degree \(4\), 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: \(3.42558224671\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 9 x^{10} - 15 x^{9} + 29 x^{8} - 26 x^{7} + 43 x^{6} + 24 x^{5} + 16 x^{4} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 9 x^{10} - 15 x^{9} + 29 x^{8} - 26 x^{7} + 43 x^{6} + 24 x^{5} + 16 x^{4} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 19891 \nu^{11} - 114974 \nu^{10} + 362529 \nu^{9} - 673717 \nu^{8} + 1202532 \nu^{7} + \cdots + 95570 ) / 3960529 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 147786 \nu^{11} - 379469 \nu^{10} + 771926 \nu^{9} - 4107793 \nu^{8} + 4845187 \nu^{7} + \cdots - 5509490 ) / 3960529 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 279535 \nu^{11} - 503769 \nu^{10} + 1828536 \nu^{9} - 1989137 \nu^{8} + 5491685 \nu^{7} + \cdots - 299426 ) / 3960529 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 334836 \nu^{11} + 687279 \nu^{10} - 2203888 \nu^{9} + 2614830 \nu^{8} - 6289582 \nu^{7} + \cdots + 279535 ) / 3960529 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 351124 \nu^{11} - 1148942 \nu^{10} + 3466717 \nu^{9} - 6241964 \nu^{8} + 11978675 \nu^{7} + \cdots - 2574616 ) / 3960529 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1268916 \nu^{11} + 3238544 \nu^{10} - 9630840 \nu^{9} + 13682648 \nu^{8} - 27496876 \nu^{7} + \cdots + 1339344 ) / 3960529 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1848387 \nu^{11} - 5413412 \nu^{10} + 15752245 \nu^{9} - 25125343 \nu^{8} + 47506293 \nu^{7} + \cdots + 573314 ) / 3960529 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2442867 \nu^{11} - 6489486 \nu^{10} + 19422140 \nu^{9} - 29055593 \nu^{8} + 58085028 \nu^{7} + \cdots - 4061746 ) / 3960529 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3187731 \nu^{11} + 8162528 \nu^{10} - 24567797 \nu^{9} + 35584663 \nu^{8} - 72664621 \nu^{7} + \cdots + 544826 ) / 3960529 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3214409 \nu^{11} - 9807301 \nu^{10} + 29011875 \nu^{9} - 48707038 \nu^{8} + 92737202 \nu^{7} + \cdots - 9454239 ) / 3960529 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} + \beta_{8} - \beta_{5} + \beta_{4} - \beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + \beta_{8} - 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{10} + 8\beta_{9} + 5\beta_{7} - 4\beta_{4} - 4\beta_{3} + 4\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -5\beta_{11} + 17\beta_{9} + 16\beta_{7} + \beta_{6} + 16\beta_{5} + \beta_{3} - \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -16\beta_{11} - 16\beta_{10} + 23\beta_{6} + 23\beta_{5} + 15\beta_{4} + 26\beta_{3} - 15\beta_{2} - 23\beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -23\beta_{10} - 72\beta_{9} - 65\beta_{7} + 65\beta_{6} + 9\beta_{5} + \beta_{4} - 74\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 65 \beta_{11} - 9 \beta_{10} - 146 \beta_{9} - 9 \beta_{8} - 104 \beta_{7} + \beta_{5} - 42 \beta_{4} + \cdots - 98 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 104 \beta_{11} - \beta_{10} - 50 \beta_{9} - 104 \beta_{8} - 58 \beta_{7} - 267 \beta_{6} + \cdots + 103 \beta_{2} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 58 \beta_{11} + 197 \beta_{9} - 267 \beta_{8} - 13 \beta_{7} - 467 \beta_{6} - 13 \beta_{5} + 210 \beta_{3} + \cdots + 373 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 13 \beta_{11} + 13 \beta_{10} - 328 \beta_{6} - 328 \beta_{5} + 187 \beta_{4} + 232 \beta_{3} + \cdots + 187 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/429\mathbb{Z}\right)^\times\).

\(n\) \(67\) \(79\) \(287\)
\(\chi(n)\) \(1\) \(-1 - \beta_{3} - \beta_{4} - \beta_{9}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1
−0.689843 − 0.501200i
0.275227 + 0.199964i
1.72363 + 1.25229i
−0.566948 − 1.74489i
0.135246 + 0.416243i
0.622685 + 1.91643i
−0.689843 + 0.501200i
0.275227 − 0.199964i
1.72363 − 1.25229i
−0.566948 + 1.74489i
0.135246 − 0.416243i
0.622685 − 1.91643i
−0.689843 + 0.501200i 0.309017 + 0.951057i −0.393353 + 1.21061i −0.139763 − 0.101543i −0.689843 − 0.501200i −0.572513 + 1.76202i −0.862402 − 2.65420i −0.809017 + 0.587785i 0.147308
157.2 0.275227 − 0.199964i 0.309017 + 0.951057i −0.582270 + 1.79204i 3.18709 + 2.31555i 0.275227 + 0.199964i −0.203890 + 0.627508i 0.408342 + 1.25675i −0.809017 + 0.587785i 1.34020
157.3 1.72363 − 1.25229i 0.309017 + 0.951057i 0.784639 − 2.41487i 1.18874 + 0.863672i 1.72363 + 1.25229i 0.349352 − 1.07520i −0.354958 − 1.09245i −0.809017 + 0.587785i 3.13053
196.1 −0.566948 + 1.74489i −0.809017 + 0.587785i −1.10516 − 0.802947i −0.477448 − 1.46943i −0.566948 − 1.74489i −0.675271 − 0.490613i −0.940958 + 0.683646i 0.309017 − 0.951057i 2.83468
196.2 0.135246 − 0.416243i −0.809017 + 0.587785i 1.46307 + 1.06298i 0.397042 + 1.22197i 0.135246 + 0.416243i 1.16309 + 0.845038i 1.34849 − 0.979734i 0.309017 − 0.951057i 0.562336
196.3 0.622685 − 1.91643i −0.809017 + 0.587785i −1.66692 − 1.21109i −0.155663 − 0.479080i 0.622685 + 1.91643i 2.43923 + 1.77220i −0.0985128 + 0.0715738i 0.309017 − 0.951057i −1.01505
235.1 −0.689843 − 0.501200i 0.309017 − 0.951057i −0.393353 − 1.21061i −0.139763 + 0.101543i −0.689843 + 0.501200i −0.572513 − 1.76202i −0.862402 + 2.65420i −0.809017 − 0.587785i 0.147308
235.2 0.275227 + 0.199964i 0.309017 − 0.951057i −0.582270 − 1.79204i 3.18709 − 2.31555i 0.275227 − 0.199964i −0.203890 − 0.627508i 0.408342 − 1.25675i −0.809017 − 0.587785i 1.34020
235.3 1.72363 + 1.25229i 0.309017 − 0.951057i 0.784639 + 2.41487i 1.18874 − 0.863672i 1.72363 − 1.25229i 0.349352 + 1.07520i −0.354958 + 1.09245i −0.809017 − 0.587785i 3.13053
313.1 −0.566948 − 1.74489i −0.809017 − 0.587785i −1.10516 + 0.802947i −0.477448 + 1.46943i −0.566948 + 1.74489i −0.675271 + 0.490613i −0.940958 − 0.683646i 0.309017 + 0.951057i 2.83468
313.2 0.135246 + 0.416243i −0.809017 − 0.587785i 1.46307 − 1.06298i 0.397042 − 1.22197i 0.135246 − 0.416243i 1.16309 − 0.845038i 1.34849 + 0.979734i 0.309017 + 0.951057i 0.562336
313.3 0.622685 + 1.91643i −0.809017 − 0.587785i −1.66692 + 1.21109i −0.155663 + 0.479080i 0.622685 − 1.91643i 2.43923 − 1.77220i −0.0985128 − 0.0715738i 0.309017 + 0.951057i −1.01505
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 157.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 429.2.n.a ✓ 12
11.c even 5 1 inner 429.2.n.a ✓ 12
11.c even 5 1 4719.2.a.bg 6
11.d odd 10 1 4719.2.a.bh 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.n.a ✓ 12 1.a even 1 1 trivial
429.2.n.a ✓ 12 11.c even 5 1 inner
4719.2.a.bg 6 11.c even 5 1
4719.2.a.bh 6 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 3 T_{2}^{11} + 9 T_{2}^{10} - 15 T_{2}^{9} + 29 T_{2}^{8} - 26 T_{2}^{7} + 43 T_{2}^{6} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 3 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{12} - 8 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{12} - 5 T^{11} + \cdots + 25 \) Copy content Toggle raw display
$11$ \( T^{12} + 6 T^{11} + \cdots + 1771561 \) Copy content Toggle raw display
$13$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{12} + 14 T^{11} + \cdots + 10201 \) Copy content Toggle raw display
$19$ \( T^{12} + 2 T^{11} + \cdots + 10201 \) Copy content Toggle raw display
$23$ \( (T^{6} + 3 T^{5} + \cdots - 181)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 12 T^{11} + \cdots + 24025 \) Copy content Toggle raw display
$31$ \( T^{12} + 12 T^{11} + \cdots + 83631025 \) Copy content Toggle raw display
$37$ \( T^{12} - 4 T^{11} + \cdots + 28561 \) Copy content Toggle raw display
$41$ \( T^{12} + 10 T^{11} + \cdots + 15625 \) Copy content Toggle raw display
$43$ \( (T^{6} - 14 T^{5} + \cdots + 9001)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 171583801 \) Copy content Toggle raw display
$53$ \( T^{12} + 29 T^{11} + \cdots + 24025 \) Copy content Toggle raw display
$59$ \( T^{12} + 11 T^{11} + \cdots + 5041 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 10730680921 \) Copy content Toggle raw display
$67$ \( (T^{6} + 36 T^{5} + \cdots + 631)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 38023050025 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 608658241 \) Copy content Toggle raw display
$79$ \( T^{12} + 7 T^{11} + \cdots + 39300361 \) Copy content Toggle raw display
$83$ \( T^{12} - 16 T^{11} + \cdots + 52113961 \) Copy content Toggle raw display
$89$ \( (T^{6} + 31 T^{5} + \cdots - 65869)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 252083322241 \) Copy content Toggle raw display
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