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Properties

Label 588.3.g.e
Level $588$
Weight $3$
Character orbit 588.g
Analytic conductor $16.022$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 8 x^{10} - 15 x^{9} + 37 x^{8} - 14 x^{7} - 38 x^{6} + 112 x^{5} + 67 x^{4} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{18}\cdot 3 \)
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.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} + \beta_{6} q^{3} + ( - \beta_{5} + 1) q^{4} + (\beta_{3} + \beta_{2}) q^{5} - \beta_{2} q^{6} + (\beta_{9} - \beta_{7} + \beta_{4} - 1) q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} + \beta_{6} q^{3} + ( - \beta_{5} + 1) q^{4} + (\beta_{3} + \beta_{2}) q^{5} - \beta_{2} q^{6} + (\beta_{9} - \beta_{7} + \beta_{4} - 1) q^{8} - 3 q^{9} + ( - \beta_{11} + \beta_{10} + \cdots + \beta_{3}) q^{10}+ \cdots + ( - 6 \beta_{5} + 3 \beta_{4} - 6 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} + 10 q^{4} - 10 q^{8} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{2} + 10 q^{4} - 10 q^{8} - 36 q^{9} + 66 q^{16} - 6 q^{18} + 32 q^{22} - 60 q^{25} - 8 q^{29} - 72 q^{30} + 182 q^{32} - 30 q^{36} - 104 q^{37} + 240 q^{44} - 160 q^{46} + 94 q^{50} + 104 q^{53} - 356 q^{58} - 24 q^{60} - 302 q^{64} + 384 q^{65} + 30 q^{72} - 580 q^{74} + 168 q^{78} + 108 q^{81} - 432 q^{85} - 240 q^{86} + 160 q^{88} + 384 q^{92} - 48 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 8 x^{10} - 15 x^{9} + 37 x^{8} - 14 x^{7} - 38 x^{6} + 112 x^{5} + 67 x^{4} + \cdots + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3413147 \nu^{11} - 10139764 \nu^{10} + 26569152 \nu^{9} - 50193231 \nu^{8} + 125089835 \nu^{7} + \cdots - 289820015 ) / 41035869 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2579651 \nu^{11} - 7839183 \nu^{10} + 20930237 \nu^{9} - 39240946 \nu^{8} + 96372196 \nu^{7} + \cdots - 116729128 ) / 13678623 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 8240656 \nu^{11} + 24539699 \nu^{10} - 65063823 \nu^{9} + 122076888 \nu^{8} - 302830021 \nu^{7} + \cdots + 274623172 ) / 41035869 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1333499 \nu^{11} + 3725380 \nu^{10} - 10058658 \nu^{9} + 18155502 \nu^{8} - 45937001 \nu^{7} + \cdots + 28244696 ) / 5862267 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 18513815 \nu^{11} + 52661629 \nu^{10} - 140318385 \nu^{9} + 258878601 \nu^{8} + \cdots + 652818209 ) / 41035869 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1870213 \nu^{11} + 5339045 \nu^{10} - 14154870 \nu^{9} + 25889703 \nu^{8} - 65283331 \nu^{7} + \cdots + 46209352 ) / 3908178 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 21305845 \nu^{11} + 60487529 \nu^{10} - 161455677 \nu^{9} + 295244970 \nu^{8} + \cdots + 526603612 ) / 41035869 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 144053501 \nu^{11} - 425583769 \nu^{10} + 1126277676 \nu^{9} - 2087562795 \nu^{8} + \cdots - 4923788336 ) / 82071738 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 81643325 \nu^{11} + 231615892 \nu^{10} - 615906996 \nu^{9} + 1125694737 \nu^{8} + \cdots + 1957924013 ) / 41035869 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 61404289 \nu^{11} + 173955153 \nu^{10} - 461191804 \nu^{9} + 842041247 \nu^{8} + \cdots + 1418627216 ) / 27357246 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 66759459 \nu^{11} + 189819541 \nu^{10} - 502597738 \nu^{9} + 919240997 \nu^{8} + \cdots + 1623954904 ) / 27357246 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} + \beta_{10} - 3\beta_{7} + 2\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta _1 + 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} + \beta_{9} + \beta_{8} - 5\beta_{7} - 2\beta_{6} - \beta_{5} + \beta_{4} - 6\beta_{2} - 2\beta _1 - 6 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6 \beta_{11} - 5 \beta_{10} - \beta_{9} + \beta_{8} + 7 \beta_{7} - 6 \beta_{6} - 5 \beta_{5} + 3 \beta_{4} + \cdots + 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4 \beta_{11} - 13 \beta_{10} - \beta_{9} - \beta_{8} - 9 \beta_{7} + 42 \beta_{6} + 15 \beta_{5} + \cdots - 30 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 9 \beta_{11} + 10 \beta_{10} + \beta_{9} + \beta_{8} - 14 \beta_{7} + 8 \beta_{6} + 10 \beta_{5} + \cdots - 208 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 45 \beta_{11} - 7 \beta_{10} - 38 \beta_{9} + 2 \beta_{8} + 231 \beta_{7} - 186 \beta_{6} - 77 \beta_{5} + \cdots + 58 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 23 \beta_{11} - 76 \beta_{10} - 51 \beta_{9} - 53 \beta_{8} + 138 \beta_{7} + 156 \beta_{6} + 134 \beta_{5} + \cdots + 744 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 399 \beta_{11} + 355 \beta_{10} + 158 \beta_{9} - 52 \beta_{8} - 979 \beta_{7} + 214 \beta_{6} + \cdots - 1022 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 54 \beta_{11} + 615 \beta_{10} + 115 \beta_{9} + 165 \beta_{8} + 577 \beta_{7} - 3150 \beta_{6} + \cdots + 374 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1243 \beta_{11} - 1632 \beta_{10} - 263 \beta_{9} - 185 \beta_{8} + 1856 \beta_{7} + 752 \beta_{6} + \cdots + 13384 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1955 \beta_{11} + 156 \beta_{10} + 1169 \beta_{9} - 237 \beta_{8} - 7785 \beta_{7} + 8854 \beta_{6} + \cdots - 5092 ) / 4 \) 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
−0.774016 − 2.30803i
1.27402 − 1.44200i
1.27402 + 1.44200i
−0.774016 + 2.30803i
0.344094 − 0.890356i
0.155906 − 0.0243308i
0.155906 + 0.0243308i
0.344094 + 0.890356i
1.70946 − 1.34824i
−1.20946 − 0.482219i
−1.20946 + 0.482219i
1.70946 + 1.34824i
−1.84208 − 0.778937i − 1.73205i 2.78651 + 2.86973i 4.44722 −1.34916 + 3.19058i 0 −2.89764 − 7.45679i −3.00000 −8.19213 − 3.46410i
295.2 −1.84208 − 0.778937i 1.73205i 2.78651 + 2.86973i −4.44722 1.34916 − 3.19058i 0 −2.89764 − 7.45679i −3.00000 8.19213 + 3.46410i
295.3 −1.84208 + 0.778937i − 1.73205i 2.78651 − 2.86973i −4.44722 1.34916 + 3.19058i 0 −2.89764 + 7.45679i −3.00000 8.19213 − 3.46410i
295.4 −1.84208 + 0.778937i 1.73205i 2.78651 − 2.86973i 4.44722 −1.34916 − 3.19058i 0 −2.89764 + 7.45679i −3.00000 −8.19213 + 3.46410i
295.5 0.424691 − 1.95439i − 1.73205i −3.63928 − 1.66002i 1.77247 −3.38510 − 0.735586i 0 −4.78989 + 6.40757i −3.00000 0.752752 − 3.46410i
295.6 0.424691 − 1.95439i 1.73205i −3.63928 − 1.66002i −1.77247 3.38510 + 0.735586i 0 −4.78989 + 6.40757i −3.00000 −0.752752 + 3.46410i
295.7 0.424691 + 1.95439i − 1.73205i −3.63928 + 1.66002i −1.77247 3.38510 − 0.735586i 0 −4.78989 − 6.40757i −3.00000 −0.752752 − 3.46410i
295.8 0.424691 + 1.95439i 1.73205i −3.63928 + 1.66002i 1.77247 −3.38510 + 0.735586i 0 −4.78989 − 6.40757i −3.00000 0.752752 + 3.46410i
295.9 1.91739 − 0.568876i − 1.73205i 3.35276 − 2.18151i 6.08938 −0.985321 − 3.32102i 0 5.18754 − 6.09011i −3.00000 11.6757 − 3.46410i
295.10 1.91739 − 0.568876i 1.73205i 3.35276 − 2.18151i −6.08938 0.985321 + 3.32102i 0 5.18754 − 6.09011i −3.00000 −11.6757 + 3.46410i
295.11 1.91739 + 0.568876i − 1.73205i 3.35276 + 2.18151i −6.08938 0.985321 − 3.32102i 0 5.18754 + 6.09011i −3.00000 −11.6757 − 3.46410i
295.12 1.91739 + 0.568876i 1.73205i 3.35276 + 2.18151i 6.08938 −0.985321 + 3.32102i 0 5.18754 + 6.09011i −3.00000 11.6757 + 3.46410i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 295.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d even 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.e ✓ 12
4.b odd 2 1 inner 588.3.g.e ✓ 12
7.b odd 2 1 inner 588.3.g.e ✓ 12
28.d even 2 1 inner 588.3.g.e ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.3.g.e ✓ 12 1.a even 1 1 trivial
588.3.g.e ✓ 12 4.b odd 2 1 inner
588.3.g.e ✓ 12 7.b odd 2 1 inner
588.3.g.e ✓ 12 28.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 60T_{5}^{4} + 912T_{5}^{2} - 2304 \) acting on \(S_{3}^{\mathrm{new}}(588, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - T^{5} - 2 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{6} \) Copy content Toggle raw display
$5$ \( (T^{6} - 60 T^{4} + \cdots - 2304)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} + 340 T^{4} + \cdots + 49152)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} - 720 T^{4} + \cdots - 5308416)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 924 T^{4} + \cdots - 331776)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 1728 T^{4} + \cdots + 110592)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 1348 T^{4} + \cdots + 61074432)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + 2 T^{2} + \cdots - 7848)^{4} \) Copy content Toggle raw display
$31$ \( (T^{6} + 4752 T^{4} + \cdots + 1062125568)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 26 T^{2} + \cdots + 5176)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} - 9084 T^{4} + \cdots - 21743271936)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 3648 T^{4} + \cdots + 346816512)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 9216 T^{4} + \cdots + 5159780352)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - 26 T^{2} + \cdots - 21048)^{4} \) Copy content Toggle raw display
$59$ \( (T^{6} + 18960 T^{4} + \cdots + 106429427712)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 15936 T^{4} + \cdots - 136320454656)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 17232 T^{4} + \cdots + 110150074368)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 12196 T^{4} + \cdots + 2131627008)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 19056 T^{4} + \cdots - 764411904)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 23952 T^{4} + \cdots + 62540218368)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 1562673917952)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 23484 T^{4} + \cdots - 3328828416)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 1997993558016)^{2} \) Copy content Toggle raw display
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