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Properties

Label 285.2.c.b
Level $285$
Weight $2$
Character orbit 285.c
Analytic conductor $2.276$
Analytic rank $0$
Dimension $14$
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] = [285,2,Mod(229,285)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(285, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("285.229");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 285 = 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 285.c (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: \(2.27573645761\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 25x^{12} + 242x^{10} + 1134x^{8} + 2605x^{6} + 2545x^{4} + 552x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{6} q^{3} + (\beta_{2} - 2) q^{4} + \beta_{10} q^{5} + \beta_{3} q^{6} + (\beta_{6} + \beta_{5}) q^{7} + (\beta_{13} - \beta_{8} - \beta_{7} + \cdots - \beta_1) q^{8}+ \cdots - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{6} q^{3} + (\beta_{2} - 2) q^{4} + \beta_{10} q^{5} + \beta_{3} q^{6} + (\beta_{6} + \beta_{5}) q^{7} + (\beta_{13} - \beta_{8} - \beta_{7} + \cdots - \beta_1) q^{8}+ \cdots + (\beta_{12} + \beta_{11} + \cdots - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 22 q^{4} + 2 q^{5} + 6 q^{6} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 22 q^{4} + 2 q^{5} + 6 q^{6} - 14 q^{9} + 2 q^{10} - 8 q^{11} + 8 q^{14} - 2 q^{15} + 38 q^{16} - 14 q^{19} + 12 q^{20} + 16 q^{21} - 18 q^{24} - 4 q^{25} - 40 q^{26} + 12 q^{29} + 4 q^{30} + 8 q^{31} - 4 q^{34} - 14 q^{35} + 22 q^{36} - 16 q^{39} + 18 q^{40} + 4 q^{41} + 64 q^{44} - 2 q^{45} - 8 q^{46} - 34 q^{49} + 4 q^{50} + 8 q^{51} - 6 q^{54} - 2 q^{55} - 44 q^{56} + 36 q^{59} - 18 q^{60} + 24 q^{61} - 22 q^{64} - 12 q^{65} + 24 q^{66} - 20 q^{69} - 60 q^{70} - 36 q^{71} - 12 q^{74} + 22 q^{76} + 8 q^{79} + 36 q^{80} + 14 q^{81} - 28 q^{84} - 18 q^{85} - 92 q^{86} + 16 q^{89} - 2 q^{90} + 24 q^{91} + 40 q^{94} - 2 q^{95} + 62 q^{96} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 25x^{12} + 242x^{10} + 1134x^{8} + 2605x^{6} + 2545x^{4} + 552x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{10} + 16\nu^{8} + 78\nu^{6} + 104\nu^{4} - 27\nu^{2} - 4 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{12} + 17\nu^{10} + 86\nu^{8} + 46\nu^{6} - 635\nu^{4} - 1159\nu^{2} - 132 ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{13} + 25\nu^{11} + 246\nu^{9} + 1198\nu^{7} + 2949\nu^{5} + 3281\nu^{3} + 1116\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{13} - 25\nu^{11} - 246\nu^{9} - 1198\nu^{7} - 2917\nu^{5} - 2961\nu^{3} - 444\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3 \nu^{13} - 2 \nu^{12} - 79 \nu^{11} - 42 \nu^{10} - 802 \nu^{9} - 332 \nu^{8} - 3906 \nu^{7} + \cdots + 104 ) / 128 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3 \nu^{13} + 2 \nu^{12} - 79 \nu^{11} + 42 \nu^{10} - 802 \nu^{9} + 332 \nu^{8} - 3906 \nu^{7} + \cdots - 104 ) / 128 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3 \nu^{13} - 2 \nu^{12} - 71 \nu^{11} - 42 \nu^{10} - 642 \nu^{9} - 316 \nu^{8} - 2754 \nu^{7} + \cdots + 136 ) / 64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3 \nu^{13} - 2 \nu^{12} + 71 \nu^{11} - 42 \nu^{10} + 642 \nu^{9} - 316 \nu^{8} + 2754 \nu^{7} + \cdots + 136 ) / 64 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 5 \nu^{13} + 6 \nu^{12} + 121 \nu^{11} + 134 \nu^{10} + 1134 \nu^{9} + 1124 \nu^{8} + 5166 \nu^{7} + \cdots + 616 ) / 128 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 5 \nu^{13} + 6 \nu^{12} - 121 \nu^{11} + 134 \nu^{10} - 1134 \nu^{9} + 1124 \nu^{8} - 5166 \nu^{7} + \cdots + 616 ) / 128 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -\nu^{13} - 26\nu^{11} - 262\nu^{9} - 1276\nu^{7} - 3021\nu^{5} - 2934\nu^{3} - 440\nu ) / 16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} - \beta_{8} - \beta_{7} - \beta_{6} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} + \beta_{11} + 2\beta_{10} + 2\beta_{9} - \beta_{8} + \beta_{7} + 2\beta_{4} + 2\beta_{3} - 8\beta_{2} + 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -10\beta_{13} + 10\beta_{8} + 10\beta_{7} + 12\beta_{6} + 2\beta_{5} + 29\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 10 \beta_{12} - 10 \beta_{11} - 22 \beta_{10} - 22 \beta_{9} + 8 \beta_{8} - 8 \beta_{7} - 22 \beta_{4} + \cdots - 122 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 83 \beta_{13} - 4 \beta_{12} + 4 \beta_{11} - 2 \beta_{10} + 2 \beta_{9} - 83 \beta_{8} + \cdots - 185 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 81 \beta_{12} + 81 \beta_{11} + 194 \beta_{10} + 194 \beta_{9} - 47 \beta_{8} + 47 \beta_{7} + \cdots + 751 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 652 \beta_{13} + 66 \beta_{12} - 66 \beta_{11} + 32 \beta_{10} - 32 \beta_{9} + 654 \beta_{8} + \cdots + 1245 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 620 \beta_{12} - 620 \beta_{11} - 1596 \beta_{10} - 1596 \beta_{9} + 232 \beta_{8} - 232 \beta_{7} + \cdots - 4788 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 5009 \beta_{13} - 744 \beta_{12} + 744 \beta_{11} - 356 \beta_{10} + 356 \beta_{9} - 5057 \beta_{8} + \cdots - 8637 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 4669 \beta_{12} + 4669 \beta_{11} + 12730 \beta_{10} + 12730 \beta_{9} - 905 \beta_{8} + 905 \beta_{7} + \cdots + 31253 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 38058 \beta_{13} + 7156 \beta_{12} - 7156 \beta_{11} + 3424 \beta_{10} - 3424 \beta_{9} + \cdots + 61053 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/285\mathbb{Z}\right)^\times\).

\(n\) \(172\) \(191\) \(211\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
229.1
− 2.73532i
− 2.47637i
− 2.39336i
− 1.70383i
− 1.57229i
− 0.498112i
− 0.184902i
0.184902i
0.498112i
1.57229i
1.70383i
2.39336i
2.47637i
2.73532i
− 2.73532i 1.00000i −5.48197 1.15351 − 1.91557i 2.73532 − 2.95440i 9.52431i −1.00000 −5.23970 − 3.15521i
229.2 − 2.47637i 1.00000i −4.13242 −0.164064 + 2.23004i 2.47637 3.36493i 5.28066i −1.00000 5.52241 + 0.406282i
229.3 − 2.39336i − 1.00000i −3.72816 −2.23546 − 0.0520645i −2.39336 4.15221i 4.13612i −1.00000 −0.124609 + 5.35026i
229.4 − 1.70383i − 1.00000i −0.903045 1.83491 + 1.27793i −1.70383 − 0.338398i − 1.86903i −1.00000 2.17738 − 3.12637i
229.5 − 1.57229i 1.00000i −0.472094 −1.72335 − 1.42481i 1.57229 − 1.87913i − 2.40231i −1.00000 −2.24021 + 2.70960i
229.6 − 0.498112i 1.00000i 1.75188 0.192457 + 2.22777i 0.498112 − 4.62756i − 1.86886i −1.00000 1.10968 − 0.0958652i
229.7 − 0.184902i − 1.00000i 1.96581 1.94200 − 1.10844i −0.184902 − 1.90997i − 0.733287i −1.00000 −0.204953 − 0.359080i
229.8 0.184902i 1.00000i 1.96581 1.94200 + 1.10844i −0.184902 1.90997i 0.733287i −1.00000 −0.204953 + 0.359080i
229.9 0.498112i − 1.00000i 1.75188 0.192457 − 2.22777i 0.498112 4.62756i 1.86886i −1.00000 1.10968 + 0.0958652i
229.10 1.57229i − 1.00000i −0.472094 −1.72335 + 1.42481i 1.57229 1.87913i 2.40231i −1.00000 −2.24021 − 2.70960i
229.11 1.70383i 1.00000i −0.903045 1.83491 − 1.27793i −1.70383 0.338398i 1.86903i −1.00000 2.17738 + 3.12637i
229.12 2.39336i 1.00000i −3.72816 −2.23546 + 0.0520645i −2.39336 − 4.15221i − 4.13612i −1.00000 −0.124609 − 5.35026i
229.13 2.47637i − 1.00000i −4.13242 −0.164064 − 2.23004i 2.47637 − 3.36493i − 5.28066i −1.00000 5.52241 − 0.406282i
229.14 2.73532i − 1.00000i −5.48197 1.15351 + 1.91557i 2.73532 2.95440i − 9.52431i −1.00000 −5.23970 + 3.15521i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 229.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 285.2.c.b ✓ 14
3.b odd 2 1 855.2.c.g 14
5.b even 2 1 inner 285.2.c.b ✓ 14
5.c odd 4 1 1425.2.a.y 7
5.c odd 4 1 1425.2.a.z 7
15.d odd 2 1 855.2.c.g 14
15.e even 4 1 4275.2.a.bv 7
15.e even 4 1 4275.2.a.bw 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.c.b ✓ 14 1.a even 1 1 trivial
285.2.c.b ✓ 14 5.b even 2 1 inner
855.2.c.g 14 3.b odd 2 1
855.2.c.g 14 15.d odd 2 1
1425.2.a.y 7 5.c odd 4 1
1425.2.a.z 7 5.c odd 4 1
4275.2.a.bv 7 15.e even 4 1
4275.2.a.bw 7 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 25T_{2}^{12} + 242T_{2}^{10} + 1134T_{2}^{8} + 2605T_{2}^{6} + 2545T_{2}^{4} + 552T_{2}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(285, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 25 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{7} \) Copy content Toggle raw display
$5$ \( T^{14} - 2 T^{13} + \cdots + 78125 \) Copy content Toggle raw display
$7$ \( T^{14} + 66 T^{12} + \cdots + 53824 \) Copy content Toggle raw display
$11$ \( (T^{7} + 4 T^{6} + \cdots + 3232)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + 108 T^{12} + \cdots + 640000 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 1600000000 \) Copy content Toggle raw display
$19$ \( (T + 1)^{14} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 203689984 \) Copy content Toggle raw display
$29$ \( (T^{7} - 6 T^{6} + \cdots - 20000)^{2} \) Copy content Toggle raw display
$31$ \( (T^{7} - 4 T^{6} + \cdots - 13568)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 456933376 \) Copy content Toggle raw display
$41$ \( (T^{7} - 2 T^{6} + \cdots + 128)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 163226176 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 16626555136 \) Copy content Toggle raw display
$53$ \( T^{14} + 128 T^{12} + \cdots + 3444736 \) Copy content Toggle raw display
$59$ \( (T^{7} - 18 T^{6} + \cdots - 40960)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} - 12 T^{6} + \cdots + 5641360)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 104857600 \) Copy content Toggle raw display
$71$ \( (T^{7} + 18 T^{6} + \cdots - 10240)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 18722996224 \) Copy content Toggle raw display
$79$ \( (T^{7} - 4 T^{6} + \cdots + 81920)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 21273972736 \) Copy content Toggle raw display
$89$ \( (T^{7} - 8 T^{6} + \cdots + 652000)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 333865984 \) Copy content Toggle raw display
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