↑

Properties

Label 608.3.e.a
Level $608$
Weight $3$
Character orbit 608.e
Analytic conductor $16.567$
Analytic rank $0$
Dimension $20$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [608,3,Mod(417,608)]
 
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([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("608.417");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 608 = 2^{5} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 608.e (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: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 136 x^{18} + 7362 x^{16} + 207796 x^{14} + 3349113 x^{12} + 31463740 x^{10} + 166967228 x^{8} + \cdots + 5494336 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{27} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{11} q^{3} - \beta_{2} q^{5} - \beta_{3} q^{7} + (\beta_{5} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{11} q^{3} - \beta_{2} q^{5} - \beta_{3} q^{7} + (\beta_{5} - 3) q^{9} - \beta_1 q^{11} - \beta_{12} q^{13} + \beta_{15} q^{15} + (\beta_{7} - \beta_{2}) q^{17} + \beta_{9} q^{19} + (\beta_{14} - \beta_{12}) q^{21} + ( - \beta_{6} + \beta_{4} + \beta_{3}) q^{23} + ( - \beta_{8} - \beta_{2} + 8) q^{25} + ( - \beta_{17} - 3 \beta_{11} + \cdots + \beta_{9}) q^{27}+ \cdots + ( - 2 \beta_{10} - 2 \beta_{9} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 52 q^{9} - 8 q^{17} + 164 q^{25} - 64 q^{45} - 116 q^{49} - 72 q^{57} - 96 q^{61} + 40 q^{73} - 152 q^{77} + 36 q^{81} + 472 q^{85} + 144 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 136 x^{18} + 7362 x^{16} + 207796 x^{14} + 3349113 x^{12} + 31463740 x^{10} + 166967228 x^{8} + \cdots + 5494336 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1314716133181 \nu^{18} - 167679742190618 \nu^{16} + \cdots - 55\!\cdots\!64 ) / 52\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8046875201 \nu^{18} + 1052701146830 \nu^{16} + 53686313166694 \nu^{14} + \cdots - 74\!\cdots\!04 ) / 22\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 794010605629 \nu^{18} + 100265203352826 \nu^{16} + \cdots - 18\!\cdots\!80 ) / 17\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1644861043 \nu^{18} - 217108544250 \nu^{16} - 11186466132978 \nu^{14} + \cdots + 59\!\cdots\!20 ) / 23\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1644861043 \nu^{18} + 217108544250 \nu^{16} + 11186466132978 \nu^{14} + \cdots + 26\!\cdots\!04 ) / 23\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 224144467295 \nu^{18} + 27958601966878 \nu^{16} + \cdots - 50\!\cdots\!76 ) / 27\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 924011461793 \nu^{18} - 120969927912470 \nu^{16} + \cdots - 13\!\cdots\!24 ) / 43\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 89761251541 \nu^{18} + 11396039530734 \nu^{16} + 557903406455550 \nu^{14} + \cdots + 60\!\cdots\!92 ) / 22\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2511441539020 \nu^{19} + 126760880079669 \nu^{18} - 104598472143352 \nu^{17} + \cdots + 19\!\cdots\!48 ) / 80\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2511441539020 \nu^{19} + 126760880079669 \nu^{18} + 104598472143352 \nu^{17} + \cdots + 19\!\cdots\!48 ) / 80\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 360088395119 \nu^{19} + 48008133164986 \nu^{17} + \cdots + 21\!\cdots\!76 \nu ) / 27\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 14573251002071 \nu^{19} + \cdots + 22\!\cdots\!84 \nu ) / 31\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 360088395119 \nu^{19} + 48008133164986 \nu^{17} + \cdots + 23\!\cdots\!28 \nu ) / 67\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 59967369012304 \nu^{19} + \cdots + 83\!\cdots\!28 \nu ) / 63\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 32705756506270 \nu^{19} + \cdots + 77\!\cdots\!84 \nu ) / 31\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 20\!\cdots\!20 \nu^{19} + \cdots - 36\!\cdots\!12 ) / 15\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 533093250286357 \nu^{19} + \cdots - 14\!\cdots\!24 \nu ) / 38\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 93117915851462 \nu^{19} + \cdots - 77\!\cdots\!28 \nu ) / 63\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 51475113175853 \nu^{19} + \cdots - 23\!\cdots\!60 \nu ) / 31\!\cdots\!36 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{13} - 4\beta_{11} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} - 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{18} + 2\beta_{17} + 3\beta_{14} - 19\beta_{13} - 3\beta_{12} + 54\beta_{11} + 2\beta_{10} - 2\beta_{9} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 4 \beta_{10} - 4 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} + 4 \beta_{6} - 43 \beta_{5} - 46 \beta_{4} + \cdots + 394 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 20 \beta_{19} + 165 \beta_{18} - 106 \beta_{17} + 4 \beta_{16} + 20 \beta_{15} - 165 \beta_{14} + \cdots + 118 \beta_{9} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 286 \beta_{10} + 286 \beta_{9} - 148 \beta_{8} + 140 \beta_{7} - 238 \beta_{6} + 1879 \beta_{5} + \cdots - 14240 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1988 \beta_{19} - 7665 \beta_{18} + 5006 \beta_{17} - 360 \beta_{16} - 1856 \beta_{15} + \cdots - 6038 \beta_{9} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 15928 \beta_{10} - 15928 \beta_{9} + 8478 \beta_{8} - 7646 \beta_{7} + 11128 \beta_{6} + \cdots + 579730 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 129588 \beta_{19} + 346605 \beta_{18} - 232130 \beta_{17} + 22588 \beta_{16} + 120412 \beta_{15} + \cdots + 294134 \beta_{9} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 811978 \beta_{10} + 811978 \beta_{9} - 439744 \beta_{8} + 384968 \beta_{7} - 494394 \beta_{6} + \cdots - 24982676 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 7191844 \beta_{19} - 15678421 \beta_{18} + 10727598 \beta_{17} - 1224224 \beta_{16} + \cdots - 14002494 \beta_{9} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 39704148 \beta_{10} - 39704148 \beta_{9} + 21721474 \beta_{8} - 18693106 \beta_{7} + \cdots + 1108081174 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 369215652 \beta_{19} + 712890945 \beta_{18} - 495453138 \beta_{17} + 61834404 \beta_{16} + \cdots + 658511070 \beta_{9} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 1898201822 \beta_{10} + 1898201822 \beta_{9} - 1044658444 \beta_{8} + 890263460 \beta_{7} + \cdots - 49930999184 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 18170040884 \beta_{19} - 32573032049 \beta_{18} + 22880537310 \beta_{17} + \cdots - 30753434742 \beta_{9} ) / 2 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 89567413488 \beta_{10} - 89567413488 \beta_{9} + 49467654598 \beta_{8} - 41917058406 \beta_{7} + \cdots + 2270846035018 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 872599688276 \beta_{19} + 1493704746853 \beta_{18} - 1056664292994 \beta_{17} + \cdots + 1430263642662 \beta_{9} ) / 2 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 4192779556562 \beta_{10} + 4192779556562 \beta_{9} - 2320653913432 \beta_{8} + 1959754151936 \beta_{7} + \cdots - 103862809828268 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 41295420118660 \beta_{19} - 68667302826637 \beta_{18} + 48799995568014 \beta_{17} + \cdots - 66348037652142 \beta_{9} ) / 2 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
417.1
3.96761i
6.79604i
5.43911i
2.61068i
4.00473i
1.17630i
0.735487i
3.56391i
2.64068i
− 0.187750i
− 2.64068i
0.187750i
− 0.735487i
− 3.56391i
− 4.00473i
− 1.17630i
− 5.43911i
− 2.61068i
− 3.96761i
− 6.79604i
0 − 5.38182i 0 −2.50364 0 −4.45673 0 −19.9640 0
417.2 0 − 5.38182i 0 −2.50364 0 4.45673 0 −19.9640 0
417.3 0 − 4.02489i 0 8.98105 0 −2.53328 0 −7.19976 0
417.4 0 − 4.02489i 0 8.98105 0 2.53328 0 −7.19976 0
417.5 0 − 2.59052i 0 −8.14452 0 −9.14882 0 2.28921 0
417.6 0 − 2.59052i 0 −8.14452 0 9.14882 0 2.28921 0
417.7 0 − 2.14970i 0 −1.54864 0 −5.59483 0 4.37879 0
417.8 0 − 2.14970i 0 −1.54864 0 5.59483 0 4.37879 0
417.9 0 − 1.22646i 0 3.21574 0 −8.64391 0 7.49579 0
417.10 0 − 1.22646i 0 3.21574 0 8.64391 0 7.49579 0
417.11 0 1.22646i 0 3.21574 0 −8.64391 0 7.49579 0
417.12 0 1.22646i 0 3.21574 0 8.64391 0 7.49579 0
417.13 0 2.14970i 0 −1.54864 0 −5.59483 0 4.37879 0
417.14 0 2.14970i 0 −1.54864 0 5.59483 0 4.37879 0
417.15 0 2.59052i 0 −8.14452 0 −9.14882 0 2.28921 0
417.16 0 2.59052i 0 −8.14452 0 9.14882 0 2.28921 0
417.17 0 4.02489i 0 8.98105 0 −2.53328 0 −7.19976 0
417.18 0 4.02489i 0 8.98105 0 2.53328 0 −7.19976 0
417.19 0 5.38182i 0 −2.50364 0 −4.45673 0 −19.9640 0
417.20 0 5.38182i 0 −2.50364 0 4.45673 0 −19.9640 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 417.20
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 608.3.e.a ✓ 20
4.b odd 2 1 inner 608.3.e.a ✓ 20
8.b even 2 1 1216.3.e.o 20
8.d odd 2 1 1216.3.e.o 20
19.b odd 2 1 inner 608.3.e.a ✓ 20
76.d even 2 1 inner 608.3.e.a ✓ 20
152.b even 2 1 1216.3.e.o 20
152.g odd 2 1 1216.3.e.o 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.3.e.a ✓ 20 1.a even 1 1 trivial
608.3.e.a ✓ 20 4.b odd 2 1 inner
608.3.e.a ✓ 20 19.b odd 2 1 inner
608.3.e.a ✓ 20 76.d even 2 1 inner
1216.3.e.o 20 8.b even 2 1
1216.3.e.o 20 8.d odd 2 1
1216.3.e.o 20 152.b even 2 1
1216.3.e.o 20 152.g odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} + 58T_{3}^{8} + 1097T_{3}^{6} + 8240T_{3}^{4} + 24656T_{3}^{2} + 21888 \) acting on \(S_{3}^{\mathrm{new}}(608, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{10} + 58 T^{8} + \cdots + 21888)^{2} \) Copy content Toggle raw display
$5$ \( (T^{5} - 83 T^{3} + \cdots + 912)^{4} \) Copy content Toggle raw display
$7$ \( (T^{10} - 216 T^{8} + \cdots - 24953004)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} - 566 T^{8} + \cdots - 232218304)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + 794 T^{8} + \cdots + 21618690048)^{2} \) Copy content Toggle raw display
$17$ \( (T^{5} + 2 T^{4} + \cdots + 480990)^{4} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 37\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 964612945600)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 6708403553408)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 2964237713408)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 83714760787968)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 57\!\cdots\!88)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots - 66\!\cdots\!04)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots - 80\!\cdots\!16)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 59\!\cdots\!32)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 57\!\cdots\!92)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + 24 T^{4} + \cdots + 84783360)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 73\!\cdots\!28)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 13\!\cdots\!12)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} - 10 T^{4} + \cdots - 13962426)^{4} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 16\!\cdots\!84)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 45\!\cdots\!08)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 10\!\cdots\!92)^{2} \) Copy content Toggle raw display
show more
show less