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Properties

Label 588.3.g.g
Level $588$
Weight $3$
Character orbit 588.g
Analytic conductor $16.022$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [588,3,Mod(295,588)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(588, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("588.295");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 588.g (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: \(16.0218395444\)
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: \( x^{14} - 3 x^{13} + x^{12} + 10 x^{11} - 10 x^{10} - 24 x^{9} + 36 x^{8} + 48 x^{7} + 144 x^{6} + \cdots + 16384 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 84)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - \beta_{4} q^{3} + \beta_{6} q^{4} + ( - \beta_{7} - \beta_{6} + 1) q^{5} + \beta_{2} q^{6} + (\beta_1 - 1) q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} - \beta_{4} q^{3} + \beta_{6} q^{4} + ( - \beta_{7} - \beta_{6} + 1) q^{5} + \beta_{2} q^{6} + (\beta_1 - 1) q^{8} - 3 q^{9} + (\beta_{11} + \beta_{10} - \beta_{4} + \cdots - 2) q^{10}+ \cdots + (3 \beta_{13} + 3 \beta_{11} + \cdots - 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 2 q^{4} + 10 q^{5} + 6 q^{6} - 12 q^{8} - 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 2 q^{4} + 10 q^{5} + 6 q^{6} - 12 q^{8} - 42 q^{9} - 28 q^{10} - 12 q^{12} + 40 q^{13} + 42 q^{16} - 8 q^{17} - 144 q^{20} - 4 q^{22} + 18 q^{24} + 72 q^{25} + 90 q^{26} - 106 q^{29} - 54 q^{30} - 40 q^{32} + 42 q^{33} + 204 q^{34} + 6 q^{36} - 22 q^{38} - 42 q^{40} - 96 q^{41} + 92 q^{44} - 30 q^{45} + 132 q^{46} - 24 q^{48} + 26 q^{50} - 84 q^{52} - 290 q^{53} - 18 q^{54} - 24 q^{57} - 64 q^{58} - 66 q^{60} + 228 q^{61} + 388 q^{62} + 58 q^{64} - 240 q^{65} - 54 q^{66} - 332 q^{68} - 72 q^{69} + 36 q^{72} - 44 q^{73} + 366 q^{74} - 132 q^{76} - 294 q^{78} - 292 q^{80} + 126 q^{81} + 156 q^{82} - 120 q^{85} - 146 q^{86} - 446 q^{88} - 340 q^{89} + 84 q^{90} + 4 q^{92} + 186 q^{93} + 240 q^{94} + 54 q^{96} - 470 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 3 x^{13} + x^{12} + 10 x^{11} - 10 x^{10} - 24 x^{9} + 36 x^{8} + 48 x^{7} + 144 x^{6} + \cdots + 16384 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5 \nu^{13} + 15 \nu^{12} - 49 \nu^{11} - 228 \nu^{10} - 122 \nu^{9} - 84 \nu^{8} + 604 \nu^{7} + \cdots - 40960 ) / 94208 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 5 \nu^{13} - 15 \nu^{12} + 49 \nu^{11} + 228 \nu^{10} + 122 \nu^{9} + 84 \nu^{8} - 604 \nu^{7} + \cdots + 40960 ) / 94208 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5 \nu^{13} - 5 \nu^{12} + 35 \nu^{11} - 48 \nu^{10} - 506 \nu^{9} - 364 \nu^{8} + 12 \nu^{7} + \cdots - 217088 ) / 94208 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 15 \nu^{13} + 27 \nu^{12} + 139 \nu^{11} + 36 \nu^{10} - 18 \nu^{9} - 212 \nu^{8} + 908 \nu^{7} + \cdots - 6144 ) / 47104 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 15 \nu^{13} + 27 \nu^{12} + 139 \nu^{11} + 36 \nu^{10} - 18 \nu^{9} - 212 \nu^{8} + 908 \nu^{7} + \cdots + 40960 ) / 47104 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 23 \nu^{13} - 67 \nu^{12} - 59 \nu^{11} - 18 \nu^{10} - 302 \nu^{9} - 616 \nu^{8} + 188 \nu^{7} + \cdots - 86016 ) / 47104 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{13} + 3 \nu^{12} - \nu^{11} - 10 \nu^{10} + 10 \nu^{9} + 24 \nu^{8} - 36 \nu^{7} + \cdots + 12288 ) / 2048 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 69 \nu^{13} - 25 \nu^{12} - 33 \nu^{11} + 156 \nu^{10} + 670 \nu^{9} - 580 \nu^{8} - 2580 \nu^{7} + \cdots - 102400 ) / 94208 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 45 \nu^{13} - 437 \nu^{12} + 11 \nu^{11} + 1808 \nu^{10} - 1298 \nu^{9} - 7292 \nu^{8} + \cdots - 1863680 ) / 94208 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 105 \nu^{13} + 357 \nu^{12} + 157 \nu^{11} - 1444 \nu^{10} + 266 \nu^{9} + 5284 \nu^{8} + \cdots + 1445888 ) / 94208 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 23 \nu^{13} - 12 \nu^{12} - 37 \nu^{11} + 200 \nu^{10} + 133 \nu^{9} - 490 \nu^{8} + 22 \nu^{7} + \cdots - 155136 ) / 11776 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 307 \nu^{13} + 583 \nu^{12} + 887 \nu^{11} - 2260 \nu^{10} - 3450 \nu^{9} + 8844 \nu^{8} + \cdots + 2080768 ) / 94208 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + \beta_{5} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{13} + \beta_{12} + 4\beta_{9} + 3\beta_{7} - 3\beta_{4} - \beta _1 - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - \beta_{13} - \beta_{12} + 8 \beta_{11} + 6 \beta_{10} - 2 \beta_{8} - 5 \beta_{7} + 2 \beta_{6} + \cdots + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{13} + 3 \beta_{12} + 4 \beta_{11} - 3 \beta_{10} + 2 \beta_{9} - 3 \beta_{8} + 10 \beta_{7} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( \beta_{13} + 9 \beta_{12} + 16 \beta_{11} + 4 \beta_{10} - 12 \beta_{9} - 36 \beta_{8} + 7 \beta_{7} + \cdots + 23 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( \beta_{13} + \beta_{12} + 8 \beta_{11} - 26 \beta_{10} - 16 \beta_{9} - 66 \beta_{8} + 9 \beta_{7} + \cdots - 241 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 14 \beta_{13} - 10 \beta_{12} - 12 \beta_{11} - 21 \beta_{10} - 6 \beta_{9} - 5 \beta_{8} + \cdots - 254 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 81 \beta_{13} + 105 \beta_{12} - 96 \beta_{11} + 8 \beta_{10} - 4 \beta_{9} + 96 \beta_{8} + 11 \beta_{7} + \cdots - 145 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 45 \beta_{13} - 221 \beta_{12} - 24 \beta_{11} + 6 \beta_{10} + 200 \beta_{9} - 242 \beta_{8} + \cdots - 107 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 37 \beta_{13} + 185 \beta_{12} + 28 \beta_{11} + 97 \beta_{10} - 222 \beta_{9} + 569 \beta_{8} + \cdots + 151 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 211 \beta_{13} + 21 \beta_{12} - 1008 \beta_{11} - 1268 \beta_{10} - 220 \beta_{9} - 1212 \beta_{8} + \cdots + 9155 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
295.1
0.930442 + 1.77039i
0.930442 − 1.77039i
1.80755 − 0.856011i
1.80755 + 0.856011i
1.89408 + 0.642246i
1.89408 − 0.642246i
−1.61263 + 1.18297i
−1.61263 − 1.18297i
1.09671 + 1.67249i
1.09671 − 1.67249i
−1.94212 − 0.477665i
−1.94212 + 0.477665i
−0.674031 + 1.88300i
−0.674031 − 1.88300i
−1.99842 − 0.0794078i − 1.73205i 3.98739 + 0.317381i −4.10779 −0.137538 + 3.46137i 0 −7.94329 − 0.950890i −3.00000 8.20911 + 0.326191i
295.2 −1.99842 + 0.0794078i 1.73205i 3.98739 − 0.317381i −4.10779 −0.137538 − 3.46137i 0 −7.94329 + 0.950890i −3.00000 8.20911 − 0.326191i
295.3 −1.64510 − 1.13738i 1.73205i 1.41273 + 3.74222i 5.87872 1.97000 − 2.84940i 0 1.93223 − 7.76315i −3.00000 −9.67111 − 6.68634i
295.4 −1.64510 + 1.13738i − 1.73205i 1.41273 − 3.74222i 5.87872 1.97000 + 2.84940i 0 1.93223 + 7.76315i −3.00000 −9.67111 + 6.68634i
295.5 −0.390836 − 1.96144i 1.73205i −3.69449 + 1.53320i −2.11281 3.39731 − 0.676948i 0 4.45123 + 6.64730i −3.00000 0.825762 + 4.14415i
295.6 −0.390836 + 1.96144i − 1.73205i −3.69449 − 1.53320i −2.11281 3.39731 + 0.676948i 0 4.45123 − 6.64730i −3.00000 0.825762 − 4.14415i
295.7 −0.218169 − 1.98807i − 1.73205i −3.90480 + 0.867467i 9.37678 −3.44343 + 0.377879i 0 2.57649 + 7.57375i −3.00000 −2.04572 − 18.6416i
295.8 −0.218169 + 1.98807i 1.73205i −3.90480 − 0.867467i 9.37678 −3.44343 − 0.377879i 0 2.57649 − 7.57375i −3.00000 −2.04572 + 18.6416i
295.9 0.900061 − 1.78603i 1.73205i −2.37978 − 3.21507i 2.55257 3.09349 + 1.55895i 0 −7.88414 + 1.35660i −3.00000 2.29746 − 4.55895i
295.10 0.900061 + 1.78603i − 1.73205i −2.37978 + 3.21507i 2.55257 3.09349 − 1.55895i 0 −7.88414 − 1.35660i −3.00000 2.29746 + 4.55895i
295.11 1.38473 − 1.44309i − 1.73205i −0.165041 − 3.99659i 1.12021 −2.49951 − 2.39842i 0 −5.99600 − 5.29604i −3.00000 1.55119 − 1.61657i
295.12 1.38473 + 1.44309i 1.73205i −0.165041 + 3.99659i 1.12021 −2.49951 + 2.39842i 0 −5.99600 + 5.29604i −3.00000 1.55119 + 1.61657i
295.13 1.96774 − 0.357771i 1.73205i 3.74400 − 1.40800i −7.70767 0.619677 + 3.40823i 0 6.86348 − 4.11007i −3.00000 −15.1667 + 2.75758i
295.14 1.96774 + 0.357771i − 1.73205i 3.74400 + 1.40800i −7.70767 0.619677 − 3.40823i 0 6.86348 + 4.11007i −3.00000 −15.1667 − 2.75758i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 295.14
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.3.g.g 14
4.b odd 2 1 inner 588.3.g.g 14
7.b odd 2 1 588.3.g.f 14
7.c even 3 1 84.3.l.c ✓ 14
7.c even 3 1 84.3.l.d yes 14
21.h odd 6 1 252.3.y.d 14
21.h odd 6 1 252.3.y.e 14
28.d even 2 1 588.3.g.f 14
28.g odd 6 1 84.3.l.c ✓ 14
28.g odd 6 1 84.3.l.d yes 14
84.n even 6 1 252.3.y.d 14
84.n even 6 1 252.3.y.e 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.3.l.c ✓ 14 7.c even 3 1
84.3.l.c ✓ 14 28.g odd 6 1
84.3.l.d yes 14 7.c even 3 1
84.3.l.d yes 14 28.g odd 6 1
252.3.y.d 14 21.h odd 6 1
252.3.y.d 14 84.n even 6 1
252.3.y.e 14 21.h odd 6 1
252.3.y.e 14 84.n even 6 1
588.3.g.f 14 7.b odd 2 1
588.3.g.f 14 28.d even 2 1
588.3.g.g 14 1.a even 1 1 trivial
588.3.g.g 14 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{7} - 5T_{5}^{6} - 93T_{5}^{5} + 337T_{5}^{4} + 1920T_{5}^{3} - 4112T_{5}^{2} - 7536T_{5} + 10544 \) acting on \(S_{3}^{\mathrm{new}}(588, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + T^{12} + \cdots + 16384 \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{7} \) Copy content Toggle raw display
$5$ \( (T^{7} - 5 T^{6} + \cdots + 10544)^{2} \) Copy content Toggle raw display
$7$ \( T^{14} \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 540293064966912 \) Copy content Toggle raw display
$13$ \( (T^{7} - 20 T^{6} + \cdots - 38439872)^{2} \) Copy content Toggle raw display
$17$ \( (T^{7} + 4 T^{6} + \cdots + 20799488)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 2501189087232 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 10\!\cdots\!88 \) Copy content Toggle raw display
$29$ \( (T^{7} + 53 T^{6} + \cdots - 125760256)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 36\!\cdots\!23 \) Copy content Toggle raw display
$37$ \( (T^{7} - 4130 T^{5} + \cdots + 108059776)^{2} \) Copy content Toggle raw display
$41$ \( (T^{7} + 48 T^{6} + \cdots - 15211042816)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 95\!\cdots\!92 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 32\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( (T^{7} + 145 T^{6} + \cdots + 344468573888)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 27\!\cdots\!48 \) Copy content Toggle raw display
$61$ \( (T^{7} - 114 T^{6} + \cdots - 419393053568)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 17\!\cdots\!88 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 79\!\cdots\!48 \) Copy content Toggle raw display
$73$ \( (T^{7} + \cdots - 1520097222536)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 23\!\cdots\!23 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 14\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( (T^{7} + 170 T^{6} + \cdots + 509596295168)^{2} \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots + 5504261016976)^{2} \) Copy content Toggle raw display
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