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Properties

Label 608.3.d.a
Level $608$
Weight $3$
Character orbit 608.d
Analytic conductor $16.567$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [608,3,Mod(191,608)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(608, 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("608.191");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 608 = 2^{5} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 608.d (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: \(16.5668000731\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 30 x^{14} + 116 x^{13} + 707 x^{12} - 2372 x^{11} - 7342 x^{10} + 12048 x^{9} + \cdots + 19859428 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{26} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + (\beta_{5} - 1) q^{5} + (\beta_{12} + \beta_{4} + \beta_1) q^{7} + (\beta_{3} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} + (\beta_{5} - 1) q^{5} + (\beta_{12} + \beta_{4} + \beta_1) q^{7} + (\beta_{3} - 3) q^{9} + (\beta_{15} - \beta_{6}) q^{11} + (\beta_{11} + \beta_{5} + 2) q^{13} + (\beta_{14} - \beta_{12} + \cdots - 2 \beta_1) q^{15}+ \cdots + ( - 3 \beta_{15} - 3 \beta_{14} + \cdots + 7 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{5} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{5} - 40 q^{9} + 40 q^{13} - 32 q^{17} - 96 q^{21} + 88 q^{25} + 144 q^{29} + 88 q^{33} - 56 q^{37} - 104 q^{41} + 40 q^{45} - 144 q^{49} - 320 q^{53} + 8 q^{61} + 336 q^{65} + 392 q^{69} - 72 q^{77} - 272 q^{81} + 552 q^{85} - 136 q^{89} - 752 q^{93} - 240 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 30 x^{14} + 116 x^{13} + 707 x^{12} - 2372 x^{11} - 7342 x^{10} + 12048 x^{9} + \cdots + 19859428 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 45\!\cdots\!75 \nu^{15} + \cdots - 61\!\cdots\!36 ) / 72\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 10\!\cdots\!61 \nu^{15} + \cdots + 15\!\cdots\!32 ) / 11\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 20\!\cdots\!61 \nu^{15} + \cdots + 76\!\cdots\!48 ) / 11\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 50\!\cdots\!65 \nu^{15} + \cdots - 40\!\cdots\!76 ) / 23\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 50\!\cdots\!65 \nu^{15} + \cdots - 40\!\cdots\!76 ) / 23\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 55\!\cdots\!05 \nu^{15} + \cdots - 11\!\cdots\!96 ) / 23\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10\!\cdots\!63 \nu^{15} + \cdots - 21\!\cdots\!44 ) / 39\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 16\!\cdots\!78 \nu^{15} + \cdots - 38\!\cdots\!68 ) / 47\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 82\!\cdots\!82 \nu^{15} + \cdots + 11\!\cdots\!00 ) / 23\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 18\!\cdots\!26 \nu^{15} + \cdots - 19\!\cdots\!60 ) / 47\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 24\!\cdots\!72 \nu^{15} + \cdots + 13\!\cdots\!04 ) / 52\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 27\!\cdots\!98 \nu^{15} + \cdots + 39\!\cdots\!64 ) / 50\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 61\!\cdots\!73 \nu^{15} + \cdots - 66\!\cdots\!88 ) / 78\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 98\!\cdots\!48 \nu^{15} + \cdots + 37\!\cdots\!20 ) / 11\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 38\!\cdots\!46 \nu^{15} + \cdots - 30\!\cdots\!60 ) / 39\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2 \beta_{14} + \beta_{13} + 2 \beta_{12} + \beta_{11} - 2 \beta_{9} + \beta_{8} + 4 \beta_{6} + \cdots + 17 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7 \beta_{15} + 2 \beta_{13} - 2 \beta_{11} - 7 \beta_{10} - 2 \beta_{9} - \beta_{8} + 3 \beta_{7} + \cdots + 13 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 3 \beta_{15} - 15 \beta_{14} + 3 \beta_{13} + 17 \beta_{12} - 9 \beta_{11} - 14 \beta_{10} + \cdots - 103 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 47 \beta_{15} + 14 \beta_{14} + 52 \beta_{13} - 48 \beta_{12} - 124 \beta_{11} - 215 \beta_{10} + \cdots - 13 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 86 \beta_{15} + 30 \beta_{14} - 15 \beta_{13} - 150 \beta_{12} - 247 \beta_{11} - 266 \beta_{10} + \cdots - 4241 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1092 \beta_{15} + 44 \beta_{14} + 471 \beta_{13} - 968 \beta_{12} - 853 \beta_{11} - 1608 \beta_{10} + \cdots + 3041 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1423 \beta_{15} + 5743 \beta_{14} - 55 \beta_{13} - 11725 \beta_{12} - 7 \beta_{11} - 1098 \beta_{10} + \cdots - 26885 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 60597 \beta_{15} - 9598 \beta_{14} + 4818 \beta_{13} - 21832 \beta_{12} + 28842 \beta_{11} + \cdots + 178173 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 12638 \beta_{15} + 93079 \beta_{14} + 23261 \beta_{13} - 152935 \beta_{12} + 110609 \beta_{11} + \cdots + 1243252 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 240575 \beta_{15} - 250454 \beta_{14} - 160368 \beta_{13} + 502296 \beta_{12} + 1247264 \beta_{11} + \cdots + 810977 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 466622 \beta_{15} - 198978 \beta_{14} + 336891 \beta_{13} + 1007882 \beta_{12} + 967767 \beta_{11} + \cdots + 16564165 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 9770549 \beta_{15} + 127426 \beta_{14} - 1871767 \beta_{13} + 9688114 \beta_{12} + 4704717 \beta_{11} + \cdots - 38959874 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 9845310 \beta_{15} - 40842985 \beta_{14} + 6290219 \beta_{13} + 96262061 \beta_{12} - 14302037 \beta_{11} + \cdots + 122463234 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 432261451 \beta_{15} + 151460718 \beta_{14} + 4863026 \beta_{13} + 87400756 \beta_{12} + \cdots - 1983524703 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/608\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
3.77456 + 2.45390i
−0.838009 + 2.37810i
−3.87493 + 2.24516i
3.11916 + 1.63987i
0.311336 + 1.38012i
−1.59249 + 1.11835i
−2.45546 + 0.645214i
3.55583 + 0.146051i
3.55583 − 0.146051i
−2.45546 − 0.645214i
−1.59249 − 1.11835i
0.311336 − 1.38012i
3.11916 − 1.63987i
−3.87493 − 2.24516i
−0.838009 − 2.37810i
3.77456 − 2.45390i
0 − 4.90780i 0 6.54911 0 5.90118i 0 −15.0865 0
191.2 0 − 4.75620i 0 −2.67602 0 − 8.34820i 0 −13.6215 0
191.3 0 − 4.49033i 0 −8.74985 0 − 7.73458i 0 −11.1630 0
191.4 0 − 3.27974i 0 5.23833 0 − 12.4244i 0 −1.75671 0
191.5 0 − 2.76024i 0 −0.377328 0 10.4601i 0 1.38105 0
191.6 0 − 2.23671i 0 −4.18498 0 4.53629i 0 3.99714 0
191.7 0 − 1.29043i 0 −5.91093 0 0.267056i 0 7.33480 0
191.8 0 − 0.292102i 0 6.11167 0 − 3.90281i 0 8.91468 0
191.9 0 0.292102i 0 6.11167 0 3.90281i 0 8.91468 0
191.10 0 1.29043i 0 −5.91093 0 − 0.267056i 0 7.33480 0
191.11 0 2.23671i 0 −4.18498 0 − 4.53629i 0 3.99714 0
191.12 0 2.76024i 0 −0.377328 0 − 10.4601i 0 1.38105 0
191.13 0 3.27974i 0 5.23833 0 12.4244i 0 −1.75671 0
191.14 0 4.49033i 0 −8.74985 0 7.73458i 0 −11.1630 0
191.15 0 4.75620i 0 −2.67602 0 8.34820i 0 −13.6215 0
191.16 0 4.90780i 0 6.54911 0 − 5.90118i 0 −15.0865 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.16
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 608.3.d.a ✓ 16
4.b odd 2 1 inner 608.3.d.a ✓ 16
8.b even 2 1 1216.3.d.e 16
8.d odd 2 1 1216.3.d.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.3.d.a ✓ 16 1.a even 1 1 trivial
608.3.d.a ✓ 16 4.b odd 2 1 inner
1216.3.d.e 16 8.b even 2 1
1216.3.d.e 16 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 92 T_{3}^{14} + 3382 T_{3}^{12} + 63436 T_{3}^{10} + 644369 T_{3}^{8} + 3478224 T_{3}^{6} + \cdots + 640000 \) acting on \(S_{3}^{\mathrm{new}}(608, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 92 T^{14} + \cdots + 640000 \) Copy content Toggle raw display
$5$ \( (T^{8} + 4 T^{7} + \cdots - 45824)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 54817325161 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{8} - 20 T^{7} + \cdots - 9027328)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 16 T^{7} + \cdots + 1205216865)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 19)^{8} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 42\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( (T^{8} - 72 T^{7} + \cdots - 265382816)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( (T^{8} + 28 T^{7} + \cdots - 130411942400)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 6714843430912)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 95\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{8} + 160 T^{7} + \cdots - 7376152496)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 38\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( (T^{8} - 4 T^{7} + \cdots - 420263934512)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 95\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 10\!\cdots\!61)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 38\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 12505460494336)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 2285057945600)^{2} \) Copy content Toggle raw display
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