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Properties

Label 435.2.f.f
Level $435$
Weight $2$
Character orbit 435.f
Analytic conductor $3.473$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [435,2,Mod(289,435)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(435, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("435.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 435 = 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 435.f (of order \(2\), degree \(1\), 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: \(3.47349248793\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} - 11 x^{9} + 55 x^{8} - 66 x^{7} + 328 x^{6} - 214 x^{5} + 207 x^{4} + 383 x^{3} + \cdots + 209 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + q^{3} + ( - \beta_{3} + 1) q^{4} + (\beta_{2} - 1) q^{5} - \beta_{5} q^{6} + \beta_{9} q^{7} + ( - \beta_{8} - 2 \beta_{5} + \beta_{3} + \cdots + 1) q^{8}+ \cdots + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + q^{3} + ( - \beta_{3} + 1) q^{4} + (\beta_{2} - 1) q^{5} - \beta_{5} q^{6} + \beta_{9} q^{7} + ( - \beta_{8} - 2 \beta_{5} + \beta_{3} + \cdots + 1) q^{8}+ \cdots + ( - \beta_{11} - \beta_{2} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{3} + 16 q^{4} - 6 q^{5} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{3} + 16 q^{4} - 6 q^{5} + 12 q^{9} + 2 q^{10} + 16 q^{12} - 6 q^{15} + 32 q^{16} - 8 q^{17} - 20 q^{20} + 12 q^{25} + 12 q^{27} + 8 q^{29} + 2 q^{30} - 40 q^{32} - 52 q^{34} + 14 q^{35} + 16 q^{36} + 20 q^{37} - 22 q^{40} + 44 q^{43} - 6 q^{45} - 32 q^{47} + 32 q^{48} - 4 q^{49} + 24 q^{50} - 8 q^{51} + 42 q^{55} - 24 q^{58} - 28 q^{59} - 20 q^{60} - 4 q^{64} + 22 q^{65} + 16 q^{68} - 26 q^{70} - 16 q^{71} - 36 q^{73} - 28 q^{74} + 12 q^{75} - 22 q^{80} + 12 q^{81} + 30 q^{85} - 16 q^{86} + 8 q^{87} + 2 q^{90} + 24 q^{91} - 28 q^{94} + 16 q^{95} - 40 q^{96} - 40 q^{97} - 112 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} - 11 x^{9} + 55 x^{8} - 66 x^{7} + 328 x^{6} - 214 x^{5} + 207 x^{4} + 383 x^{3} + \cdots + 209 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1397998659 \nu^{11} + 2925295595 \nu^{10} - 7051159320 \nu^{9} - 16202543074 \nu^{8} + \cdots - 4008235081411 ) / 3213459622340 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 438845871 \nu^{11} - 1612092568 \nu^{10} + 93438063 \nu^{9} - 3651448133 \nu^{8} + \cdots - 852415781250 ) / 642691924468 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 6774498506 \nu^{11} + 11302364650 \nu^{10} - 14173819980 \nu^{9} + 62634263811 \nu^{8} + \cdots - 3000675757541 ) / 3213459622340 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 14834961346 \nu^{11} - 19830017805 \nu^{10} + 8761728055 \nu^{9} - 165086515566 \nu^{8} + \cdots - 802809213514 ) / 3213459622340 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 15940407777 \nu^{11} + 23169062590 \nu^{10} + 3022410 \nu^{9} + 175231640332 \nu^{8} + \cdots + 2271334684498 ) / 3213459622340 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 9287830936 \nu^{11} + 18791211275 \nu^{10} - 7074036000 \nu^{9} + 97560593321 \nu^{8} + \cdots + 1390587069839 ) / 1606729811170 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1814963223 \nu^{11} + 1826616705 \nu^{10} + 1608454605 \nu^{9} + 19032062228 \nu^{8} + \cdots + 285575480212 ) / 292132692940 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 20053088776 \nu^{11} - 23152421330 \nu^{10} - 35040242445 \nu^{9} - 226992732701 \nu^{8} + \cdots - 11720209576834 ) / 3213459622340 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 26830096707 \nu^{11} - 18124014990 \nu^{10} + 14433023630 \nu^{9} - 344432223437 \nu^{8} + \cdots + 409628743007 ) / 3213459622340 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 42580783394 \nu^{11} - 6474185680 \nu^{10} + 34133697850 \nu^{9} + 503681618779 \nu^{8} + \cdots + 1277078933251 ) / 3213459622340 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 47064798624 \nu^{11} - 16749257055 \nu^{10} - 16699358825 \nu^{9} - 544078396489 \nu^{8} + \cdots - 1811594761361 ) / 3213459622340 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{11} + 2\beta_{10} - \beta_{9} + 2\beta_{8} + 3\beta_{7} + \beta_{6} - 2\beta_{3} - 2\beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 5 \beta_{11} - 2 \beta_{10} - \beta_{9} - 2 \beta_{8} - \beta_{7} + 5 \beta_{6} - 8 \beta_{5} + \cdots + 8 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5 \beta_{11} + 5 \beta_{10} + \beta_{9} - 2 \beta_{8} - 3 \beta_{7} + \beta_{6} - 3 \beta_{4} + 2 \beta_{3} + \cdots + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 7 \beta_{11} + 10 \beta_{10} + 5 \beta_{9} + 38 \beta_{8} - 7 \beta_{7} + 27 \beta_{6} + 40 \beta_{5} + \cdots - 96 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 85 \beta_{11} - 30 \beta_{10} + 7 \beta_{9} - 130 \beta_{8} - 145 \beta_{7} - 83 \beta_{6} + \cdots + 128 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 68 \beta_{11} + 48 \beta_{10} - 2 \beta_{8} - 12 \beta_{7} - 20 \beta_{6} + 22 \beta_{5} - 77 \beta_{4} + \cdots - 75 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1577 \beta_{11} - 1166 \beta_{10} + 43 \beta_{9} + 226 \beta_{8} - 401 \beta_{7} - 155 \beta_{6} + \cdots - 1768 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1631 \beta_{11} + 310 \beta_{10} - 381 \beta_{9} - 3146 \beta_{8} - 1533 \beta_{7} - 3951 \beta_{6} + \cdots + 5088 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 929 \beta_{11} - 85 \beta_{10} - 305 \beta_{9} + 1956 \beta_{8} + 2047 \beta_{7} + 59 \beta_{6} + \cdots - 5514 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 42507 \beta_{11} - 35934 \beta_{10} - 911 \beta_{9} - 11842 \beta_{8} - 1675 \beta_{7} - 7345 \beta_{6} + \cdots + 34784 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 122527 \beta_{11} + 65570 \beta_{10} - 9181 \beta_{9} - 6818 \beta_{8} + 45979 \beta_{7} - 22831 \beta_{6} + \cdots + 83272 ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/435\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(146\) \(262\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
−0.177521 + 2.06715i
−0.177521 − 2.06715i
0.469890 − 0.575682i
0.469890 + 0.575682i
−2.21342 + 2.00212i
−2.21342 − 2.00212i
0.730544 − 1.10073i
0.730544 + 1.10073i
2.44773 + 1.33046i
2.44773 − 1.33046i
−0.757215 − 0.394074i
−0.757215 + 0.394074i
−2.67451 1.00000 5.15300 −0.347696 − 2.20887i −2.67451 1.33684i −8.43272 1.00000 0.929917 + 5.90764i
289.2 −2.67451 1.00000 5.15300 −0.347696 + 2.20887i −2.67451 − 1.33684i −8.43272 1.00000 0.929917 − 5.90764i
289.3 −1.60465 1.00000 0.574897 −2.22390 − 0.232983i −1.60465 − 1.04058i 2.28679 1.00000 3.56857 + 0.373856i
289.4 −1.60465 1.00000 0.574897 −2.22390 + 0.232983i −1.60465 1.04058i 2.28679 1.00000 3.56857 − 0.373856i
289.5 −0.334522 1.00000 −1.88809 1.95387 − 1.08739i −0.334522 0.275019i 1.30066 1.00000 −0.653612 + 0.363756i
289.6 −0.334522 1.00000 −1.88809 1.95387 + 1.08739i −0.334522 − 0.275019i 1.30066 1.00000 −0.653612 − 0.363756i
289.7 0.141254 1.00000 −1.98005 −1.62735 − 1.53353i 0.141254 4.11523i −0.562197 1.00000 −0.229870 − 0.216617i
289.8 0.141254 1.00000 −1.98005 −1.62735 + 1.53353i 0.141254 − 4.11523i −0.562197 1.00000 −0.229870 + 0.216617i
289.9 1.97264 1.00000 1.89129 1.38075 − 1.75885i 1.97264 0.520254i −0.214447 1.00000 2.72371 − 3.46956i
289.10 1.97264 1.00000 1.89129 1.38075 + 1.75885i 1.97264 − 0.520254i −0.214447 1.00000 2.72371 + 3.46956i
289.11 2.49979 1.00000 4.24896 −2.13567 − 0.662521i 2.49979 − 4.88350i 5.62192 1.00000 −5.33872 − 1.65616i
289.12 2.49979 1.00000 4.24896 −2.13567 + 0.662521i 2.49979 4.88350i 5.62192 1.00000 −5.33872 + 1.65616i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
145.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 435.2.f.f yes 12
3.b odd 2 1 1305.2.f.l 12
5.b even 2 1 435.2.f.e ✓ 12
5.c odd 4 2 2175.2.d.j 24
15.d odd 2 1 1305.2.f.k 12
29.b even 2 1 435.2.f.e ✓ 12
87.d odd 2 1 1305.2.f.k 12
145.d even 2 1 inner 435.2.f.f yes 12
145.h odd 4 2 2175.2.d.j 24
435.b odd 2 1 1305.2.f.l 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.f.e ✓ 12 5.b even 2 1
435.2.f.e ✓ 12 29.b even 2 1
435.2.f.f yes 12 1.a even 1 1 trivial
435.2.f.f yes 12 145.d even 2 1 inner
1305.2.f.k 12 15.d odd 2 1
1305.2.f.k 12 87.d odd 2 1
1305.2.f.l 12 3.b odd 2 1
1305.2.f.l 12 435.b odd 2 1
2175.2.d.j 24 5.c odd 4 2
2175.2.d.j 24 145.h odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 10T_{2}^{4} + 22T_{2}^{2} + 4T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(435, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 10 T^{4} + 22 T^{2} + \cdots - 1)^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 6 T^{11} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( T^{12} + 44 T^{10} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{12} + 76 T^{10} + \cdots + 85264 \) Copy content Toggle raw display
$13$ \( T^{12} + 100 T^{10} + \cdots + 43264 \) Copy content Toggle raw display
$17$ \( (T^{6} + 4 T^{5} + \cdots - 1172)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 160 T^{10} + \cdots + 6801664 \) Copy content Toggle raw display
$23$ \( T^{12} + 96 T^{10} + \cdots + 43264 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 594823321 \) Copy content Toggle raw display
$31$ \( T^{12} + 248 T^{10} + \cdots + 63744256 \) Copy content Toggle raw display
$37$ \( (T^{6} - 10 T^{5} + \cdots + 9328)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 881852416 \) Copy content Toggle raw display
$43$ \( (T^{6} - 22 T^{5} + \cdots - 704)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 16 T^{5} + \cdots + 1408)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 7721488384 \) Copy content Toggle raw display
$59$ \( (T^{6} + 14 T^{5} + \cdots + 135232)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 104748027904 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 19142382736 \) Copy content Toggle raw display
$71$ \( (T^{6} + 8 T^{5} + \cdots - 43264)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 18 T^{5} + \cdots + 2864)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 342694144 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 405780736 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 1392185344 \) Copy content Toggle raw display
$97$ \( (T^{6} + 20 T^{5} + \cdots - 23504)^{2} \) Copy content Toggle raw display
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