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Properties

Label 630.4.m.a
Level $630$
Weight $4$
Character orbit 630.m
Analytic conductor $37.171$
Analytic rank $0$
Dimension $16$
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] = [630,4,Mod(197,630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(630, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.197");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 630.m (of order \(4\), degree \(2\), 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: \(37.1712033036\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} - 62 x^{14} + 184 x^{13} + 5442 x^{12} + 68448 x^{11} + 1829094 x^{10} + \cdots + 101023536964 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + 2 \beta_{2} q^{2} - 4 \beta_{3} q^{4} + (\beta_{10} - \beta_{4} + \beta_{3} + \cdots - 3) q^{5}+ \cdots - 8 \beta_{4} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_{2} q^{2} - 4 \beta_{3} q^{4} + (\beta_{10} - \beta_{4} + \beta_{3} + \cdots - 3) q^{5}+ \cdots + 98 \beta_{4} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 44 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 44 q^{5} - 24 q^{10} - 212 q^{13} + 224 q^{14} - 256 q^{16} + 32 q^{17} + 32 q^{20} + 72 q^{22} - 128 q^{23} - 268 q^{25} + 232 q^{29} - 200 q^{31} + 112 q^{35} + 900 q^{37} - 368 q^{38} - 128 q^{40} - 908 q^{43} + 352 q^{44} + 208 q^{46} - 604 q^{47} + 896 q^{50} + 848 q^{52} - 440 q^{53} - 1232 q^{55} + 536 q^{58} + 224 q^{59} + 1200 q^{61} - 1304 q^{62} + 1936 q^{65} + 652 q^{67} - 128 q^{68} - 616 q^{70} - 2148 q^{73} + 1328 q^{74} - 252 q^{77} + 704 q^{80} + 616 q^{82} - 2624 q^{83} - 1712 q^{85} + 288 q^{88} + 2400 q^{89} - 512 q^{92} + 4200 q^{95} - 2668 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} - 62 x^{14} + 184 x^{13} + 5442 x^{12} + 68448 x^{11} + 1829094 x^{10} + \cdots + 101023536964 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 39\!\cdots\!18 \nu^{15} + \cdots - 34\!\cdots\!94 ) / 29\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 79\!\cdots\!38 \nu^{15} + \cdots - 64\!\cdots\!66 ) / 58\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 26\!\cdots\!16 \nu^{15} + \cdots - 70\!\cdots\!26 ) / 11\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 14\!\cdots\!91 \nu^{15} + \cdots - 13\!\cdots\!98 ) / 58\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 10\!\cdots\!34 \nu^{15} + \cdots - 32\!\cdots\!53 ) / 29\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 39\!\cdots\!63 \nu^{15} + \cdots + 22\!\cdots\!64 ) / 29\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 41\!\cdots\!54 \nu^{15} + \cdots - 18\!\cdots\!92 ) / 29\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 37\!\cdots\!66 \nu^{15} + \cdots - 75\!\cdots\!12 ) / 25\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 38\!\cdots\!06 \nu^{15} + \cdots - 31\!\cdots\!78 ) / 25\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 11\!\cdots\!46 \nu^{15} + \cdots - 10\!\cdots\!62 ) / 58\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 69\!\cdots\!79 \nu^{15} + \cdots - 71\!\cdots\!78 ) / 29\!\cdots\!85 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 14\!\cdots\!00 \nu^{15} + \cdots + 39\!\cdots\!34 ) / 58\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 36\!\cdots\!14 \nu^{15} + \cdots - 13\!\cdots\!88 ) / 12\!\cdots\!95 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 17\!\cdots\!67 \nu^{15} + \cdots - 15\!\cdots\!42 ) / 58\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 34\!\cdots\!00 \nu^{15} + \cdots + 10\!\cdots\!86 ) / 11\!\cdots\!74 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} + \beta_{13} + \beta_{7} + \beta_{6} + \beta_{2} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2 \beta_{15} - 7 \beta_{13} - 5 \beta_{12} + 2 \beta_{11} - 5 \beta_{10} - 10 \beta_{9} + 5 \beta_{8} + \cdots + 28 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 44 \beta_{15} + 15 \beta_{14} + 44 \beta_{13} - 15 \beta_{12} - 134 \beta_{11} + 156 \beta_{10} + \cdots + 143 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 1371 \beta_{15} + 1180 \beta_{14} - 701 \beta_{13} - 2030 \beta_{12} + 1371 \beta_{11} - 701 \beta_{10} + \cdots + 847 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 23143 \beta_{15} + 2035 \beta_{14} + 27891 \beta_{13} - 2035 \beta_{12} - 14521 \beta_{11} + \cdots - 50698 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 91888 \beta_{15} + 196765 \beta_{14} - 115611 \beta_{13} - 120655 \beta_{12} + 92802 \beta_{11} + \cdots - 1011666 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 912491 \beta_{15} + 45705 \beta_{14} + 912491 \beta_{13} - 183810 \beta_{12} - 2688788 \beta_{11} + \cdots + 7887147 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 22908453 \beta_{15} + 24172905 \beta_{14} + 17995361 \beta_{13} + 22908453 \beta_{11} - 17995361 \beta_{10} + \cdots - 494895365 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 536480265 \beta_{15} + 17626360 \beta_{14} - 421627223 \beta_{13} + 17626360 \beta_{12} + \cdots + 1756058866 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 5306434339 \beta_{15} + 6204869983 \beta_{13} + 4837216655 \beta_{12} - 3458646562 \beta_{11} + \cdots - 48993320136 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 96655931598 \beta_{15} + 163524670 \beta_{14} - 96655931598 \beta_{13} - 11166724155 \beta_{12} + \cdots + 18719650177 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1323776563456 \beta_{15} - 970605670675 \beta_{14} + 1209717393658 \beta_{13} + 1477765755660 \beta_{12} + \cdots - 667186206912 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 16816369019597 \beta_{15} + 3515660482315 \beta_{14} - 22308153830147 \beta_{13} + \cdots - 5330036191529 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 130043565677754 \beta_{15} - 293454760138300 \beta_{14} + 185752329037981 \beta_{13} + \cdots + 20\!\cdots\!32 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 12\!\cdots\!08 \beta_{15} + 958208597290185 \beta_{14} + \cdots - 22\!\cdots\!47 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/630\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(\beta_{3}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
3.20578 + 7.73944i
3.04417 + 7.34926i
−0.0858842 − 0.207343i
−4.87117 − 11.7600i
5.08631 − 2.10682i
12.5082 − 5.18107i
−13.4677 + 5.57851i
−1.41968 + 0.588052i
3.20578 − 7.73944i
3.04417 − 7.34926i
−0.0858842 + 0.207343i
−4.87117 + 11.7600i
5.08631 + 2.10682i
12.5082 + 5.18107i
−13.4677 − 5.57851i
−1.41968 − 0.588052i
−1.41421 + 1.41421i 0 − 4.00000i −10.7228 − 3.16564i 0 −4.94975 − 4.94975i 5.65685 + 5.65685i 0 19.6412 − 10.6875i
197.2 −1.41421 + 1.41421i 0 − 4.00000i −4.93229 + 10.0336i 0 −4.94975 − 4.94975i 5.65685 + 5.65685i 0 −7.21430 − 21.1649i
197.3 −1.41421 + 1.41421i 0 − 4.00000i 1.74600 + 11.0432i 0 −4.94975 − 4.94975i 5.65685 + 5.65685i 0 −18.0866 − 13.1482i
197.4 −1.41421 + 1.41421i 0 − 4.00000i 2.20200 − 10.9614i 0 −4.94975 − 4.94975i 5.65685 + 5.65685i 0 12.3876 + 18.6158i
197.5 1.41421 − 1.41421i 0 − 4.00000i −11.1489 − 0.838262i 0 4.94975 + 4.94975i −5.65685 − 5.65685i 0 −16.9524 + 14.5814i
197.6 1.41421 − 1.41421i 0 − 4.00000i −7.03196 − 8.69204i 0 4.94975 + 4.94975i −5.65685 − 5.65685i 0 −22.2371 − 2.34770i
197.7 1.41421 − 1.41421i 0 − 4.00000i −2.41767 + 10.9158i 0 4.94975 + 4.94975i −5.65685 − 5.65685i 0 12.0182 + 18.8564i
197.8 1.41421 − 1.41421i 0 − 4.00000i 10.3056 − 4.33526i 0 4.94975 + 4.94975i −5.65685 − 5.65685i 0 8.44335 − 20.7053i
323.1 −1.41421 − 1.41421i 0 4.00000i −10.7228 + 3.16564i 0 −4.94975 + 4.94975i 5.65685 − 5.65685i 0 19.6412 + 10.6875i
323.2 −1.41421 − 1.41421i 0 4.00000i −4.93229 − 10.0336i 0 −4.94975 + 4.94975i 5.65685 − 5.65685i 0 −7.21430 + 21.1649i
323.3 −1.41421 − 1.41421i 0 4.00000i 1.74600 − 11.0432i 0 −4.94975 + 4.94975i 5.65685 − 5.65685i 0 −18.0866 + 13.1482i
323.4 −1.41421 − 1.41421i 0 4.00000i 2.20200 + 10.9614i 0 −4.94975 + 4.94975i 5.65685 − 5.65685i 0 12.3876 − 18.6158i
323.5 1.41421 + 1.41421i 0 4.00000i −11.1489 + 0.838262i 0 4.94975 − 4.94975i −5.65685 + 5.65685i 0 −16.9524 − 14.5814i
323.6 1.41421 + 1.41421i 0 4.00000i −7.03196 + 8.69204i 0 4.94975 − 4.94975i −5.65685 + 5.65685i 0 −22.2371 + 2.34770i
323.7 1.41421 + 1.41421i 0 4.00000i −2.41767 − 10.9158i 0 4.94975 − 4.94975i −5.65685 + 5.65685i 0 12.0182 − 18.8564i
323.8 1.41421 + 1.41421i 0 4.00000i 10.3056 + 4.33526i 0 4.94975 − 4.94975i −5.65685 + 5.65685i 0 8.44335 + 20.7053i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.4.m.a ✓ 16
3.b odd 2 1 630.4.m.b yes 16
5.c odd 4 1 630.4.m.b yes 16
15.e even 4 1 inner 630.4.m.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.4.m.a ✓ 16 1.a even 1 1 trivial
630.4.m.a ✓ 16 15.e even 4 1 inner
630.4.m.b yes 16 3.b odd 2 1
630.4.m.b yes 16 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(630, [\chi])\):

\( T_{11}^{16} + 13420 T_{11}^{14} + 71202404 T_{11}^{12} + 188175590848 T_{11}^{10} + \cdots + 52\!\cdots\!04 \) Copy content Toggle raw display
\( T_{17}^{16} - 32 T_{17}^{15} + 512 T_{17}^{14} + 99424 T_{17}^{13} + 185601584 T_{17}^{12} + \cdots + 71\!\cdots\!84 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 16)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 59\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{4} + 2401)^{4} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 52\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 71\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 51\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots - 26\!\cdots\!44)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 59\!\cdots\!68)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 34\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 24\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 15\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 96\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 64\!\cdots\!32)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots - 95\!\cdots\!32)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 22\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 27\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 36\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 71\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 28\!\cdots\!88)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 46\!\cdots\!36 \) Copy content Toggle raw display
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