↑

Properties

Label 630.2.i.g
Level $630$
Weight $2$
Character orbit 630.i
Analytic conductor $5.031$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [630,2,Mod(121,630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(630, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 630.i (of order \(3\), 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: \(5.03057532734\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 7 x^{10} - 3 x^{9} - 2 x^{8} + 24 x^{7} - 21 x^{6} + 72 x^{5} - 18 x^{4} + \cdots + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + q^{2} - \beta_{8} q^{3} + q^{4} + ( - \beta_{6} - 1) q^{5} - \beta_{8} q^{6} + (\beta_{9} - \beta_{8} + \beta_{6} + \cdots + 1) q^{7}+ \cdots + ( - \beta_{11} + \beta_{9} + \beta_{5} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - \beta_{8} q^{3} + q^{4} + ( - \beta_{6} - 1) q^{5} - \beta_{8} q^{6} + (\beta_{9} - \beta_{8} + \beta_{6} + \cdots + 1) q^{7}+ \cdots + ( - \beta_{11} - \beta_{10} - 2 \beta_{9} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{2} + 12 q^{4} - 6 q^{5} + 4 q^{7} + 12 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{2} + 12 q^{4} - 6 q^{5} + 4 q^{7} + 12 q^{8} + 4 q^{9} - 6 q^{10} + 3 q^{11} - 2 q^{13} + 4 q^{14} + 3 q^{15} + 12 q^{16} + q^{17} + 4 q^{18} + 8 q^{19} - 6 q^{20} + 5 q^{21} + 3 q^{22} + 11 q^{23} - 6 q^{25} - 2 q^{26} - 27 q^{27} + 4 q^{28} + 13 q^{29} + 3 q^{30} - 42 q^{31} + 12 q^{32} + 17 q^{33} + q^{34} + 4 q^{35} + 4 q^{36} + 18 q^{37} + 8 q^{38} - 24 q^{39} - 6 q^{40} + 5 q^{41} + 5 q^{42} - 11 q^{43} + 3 q^{44} + q^{45} + 11 q^{46} + 46 q^{47} - 6 q^{50} - 27 q^{51} - 2 q^{52} + 2 q^{53} - 27 q^{54} - 6 q^{55} + 4 q^{56} - 44 q^{57} + 13 q^{58} - 2 q^{59} + 3 q^{60} + 2 q^{61} - 42 q^{62} + 9 q^{63} + 12 q^{64} + 4 q^{65} + 17 q^{66} - 4 q^{67} + q^{68} - 24 q^{69} + 4 q^{70} - 30 q^{71} + 4 q^{72} + 22 q^{73} + 18 q^{74} - 3 q^{75} + 8 q^{76} - 31 q^{77} - 24 q^{78} - 54 q^{79} - 6 q^{80} + 52 q^{81} + 5 q^{82} + 6 q^{83} + 5 q^{84} + q^{85} - 11 q^{86} - 28 q^{87} + 3 q^{88} - 18 q^{89} + q^{90} + 14 q^{91} + 11 q^{92} - 38 q^{93} + 46 q^{94} - 16 q^{95} - 4 q^{97} + 63 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 7 x^{10} - 3 x^{9} - 2 x^{8} + 24 x^{7} - 21 x^{6} + 72 x^{5} - 18 x^{4} + \cdots + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5 \nu^{11} - 1173 \nu^{10} + 1963 \nu^{9} - 2091 \nu^{8} - 5201 \nu^{7} + 1905 \nu^{6} + \cdots - 282366 ) / 11421 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 12 \nu^{11} + 221 \nu^{10} - 252 \nu^{9} + 161 \nu^{8} + 828 \nu^{7} - 316 \nu^{6} + 1677 \nu^{5} + \cdots + 34992 ) / 3807 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} - 3 \nu^{10} + 7 \nu^{9} - 3 \nu^{8} - 2 \nu^{7} + 24 \nu^{6} - 21 \nu^{5} + 72 \nu^{4} + \cdots - 729 ) / 243 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 46 \nu^{11} - 132 \nu^{10} + 209 \nu^{9} - 456 \nu^{8} - 727 \nu^{7} - 240 \nu^{6} - 1068 \nu^{5} + \cdots - 37179 ) / 3807 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 51 \nu^{11} - 199 \nu^{10} + 225 \nu^{9} + 56 \nu^{8} - 558 \nu^{7} + 497 \nu^{6} - 1311 \nu^{5} + \cdots - 23085 ) / 3807 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 196 \nu^{11} - 633 \nu^{10} + 967 \nu^{9} + 582 \nu^{8} - 1445 \nu^{7} + 2310 \nu^{6} - 3468 \nu^{5} + \cdots - 69012 ) / 11421 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 95 \nu^{11} + 132 \nu^{10} - 68 \nu^{9} - 390 \nu^{8} + 22 \nu^{7} - 606 \nu^{6} + 504 \nu^{5} + \cdots - 8505 ) / 3807 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 365 \nu^{11} + 1227 \nu^{10} - 1367 \nu^{9} - 222 \nu^{8} + 3565 \nu^{7} - 2922 \nu^{6} + \cdots + 150660 ) / 11421 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 151 \nu^{11} + 43 \nu^{10} - 119 \nu^{9} + 1174 \nu^{8} + 1096 \nu^{7} + 1219 \nu^{6} + 2574 \nu^{5} + \cdots + 66015 ) / 3807 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 713 \nu^{11} + 1620 \nu^{10} - 2471 \nu^{9} - 1710 \nu^{8} + 3325 \nu^{7} - 8514 \nu^{6} + \cdots + 183222 ) / 11421 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{5} + \beta_{4} + \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - 2\beta_{8} - \beta_{6} - \beta_{4} + \beta_{2} - \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{8} - \beta_{7} + 6\beta_{6} + 2\beta_{3} - 3\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{11} + \beta_{9} - 2\beta_{8} - 3\beta_{7} - 4\beta_{6} + 7\beta_{5} - 4\beta_{4} - 2\beta_{2} + 4\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{11} + 2 \beta_{10} + 10 \beta_{8} - 8 \beta_{7} + 6 \beta_{6} - 8 \beta_{5} + 8 \beta_{4} + \cdots + 14 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 6 \beta_{11} - 12 \beta_{10} + 4 \beta_{9} - 11 \beta_{8} + 12 \beta_{7} - 10 \beta_{6} + 17 \beta_{5} + \cdots - 11 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 10 \beta_{11} + 7 \beta_{10} - 10 \beta_{9} + 25 \beta_{8} + 18 \beta_{7} + 6 \beta_{6} - 19 \beta_{5} + \cdots - 24 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 9 \beta_{11} - 15 \beta_{10} + 13 \beta_{9} - 41 \beta_{8} + 45 \beta_{7} - 67 \beta_{6} + 21 \beta_{5} + \cdots + 25 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 27 \beta_{11} - 27 \beta_{10} + 28 \beta_{9} + \beta_{8} - \beta_{7} + 33 \beta_{6} - 36 \beta_{5} + \cdots + 37 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 42 \beta_{11} + 54 \beta_{10} - 143 \beta_{9} + 34 \beta_{8} - 39 \beta_{7} - 112 \beta_{6} - 20 \beta_{5} + \cdots + 22 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(1\) \(\beta_{6}\) \(\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
0.628063 − 1.61417i
1.22323 − 1.22626i
1.39898 + 1.02120i
−1.67391 − 0.444996i
0.702501 + 1.58319i
−0.778860 + 1.54705i
0.628063 + 1.61417i
1.22323 + 1.22626i
1.39898 − 1.02120i
−1.67391 + 0.444996i
0.702501 − 1.58319i
−0.778860 − 1.54705i
1.00000 −1.71194 − 0.263165i 1.00000 −0.500000 − 0.866025i −1.71194 − 0.263165i −2.25729 + 1.38008i 1.00000 2.86149 + 0.901046i −0.500000 − 0.866025i
121.2 1.00000 −1.67359 + 0.446216i 1.00000 −0.500000 − 0.866025i −1.67359 + 0.446216i 1.40545 + 2.24159i 1.00000 2.60178 − 1.49356i −0.500000 − 0.866025i
121.3 1.00000 0.184900 + 1.72215i 1.00000 −0.500000 − 0.866025i 0.184900 + 1.72215i −2.25729 + 1.38008i 1.00000 −2.93162 + 0.636851i −0.500000 − 0.866025i
121.4 1.00000 0.451577 − 1.67215i 1.00000 −0.500000 − 0.866025i 0.451577 − 1.67215i 1.85185 − 1.88962i 1.00000 −2.59216 − 1.51021i −0.500000 − 0.866025i
121.5 1.00000 1.01983 + 1.39998i 1.00000 −0.500000 − 0.866025i 1.01983 + 1.39998i 1.85185 − 1.88962i 1.00000 −0.919882 + 2.85549i −0.500000 − 0.866025i
121.6 1.00000 1.72922 + 0.0990147i 1.00000 −0.500000 − 0.866025i 1.72922 + 0.0990147i 1.40545 + 2.24159i 1.00000 2.98039 + 0.342436i −0.500000 − 0.866025i
151.1 1.00000 −1.71194 + 0.263165i 1.00000 −0.500000 + 0.866025i −1.71194 + 0.263165i −2.25729 − 1.38008i 1.00000 2.86149 − 0.901046i −0.500000 + 0.866025i
151.2 1.00000 −1.67359 − 0.446216i 1.00000 −0.500000 + 0.866025i −1.67359 − 0.446216i 1.40545 − 2.24159i 1.00000 2.60178 + 1.49356i −0.500000 + 0.866025i
151.3 1.00000 0.184900 − 1.72215i 1.00000 −0.500000 + 0.866025i 0.184900 − 1.72215i −2.25729 − 1.38008i 1.00000 −2.93162 − 0.636851i −0.500000 + 0.866025i
151.4 1.00000 0.451577 + 1.67215i 1.00000 −0.500000 + 0.866025i 0.451577 + 1.67215i 1.85185 + 1.88962i 1.00000 −2.59216 + 1.51021i −0.500000 + 0.866025i
151.5 1.00000 1.01983 − 1.39998i 1.00000 −0.500000 + 0.866025i 1.01983 − 1.39998i 1.85185 + 1.88962i 1.00000 −0.919882 − 2.85549i −0.500000 + 0.866025i
151.6 1.00000 1.72922 − 0.0990147i 1.00000 −0.500000 + 0.866025i 1.72922 − 0.0990147i 1.40545 − 2.24159i 1.00000 2.98039 − 0.342436i −0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.i.g ✓ 12
3.b odd 2 1 1890.2.i.g 12
7.c even 3 1 630.2.l.g yes 12
9.c even 3 1 630.2.l.g yes 12
9.d odd 6 1 1890.2.l.g 12
21.h odd 6 1 1890.2.l.g 12
63.h even 3 1 inner 630.2.i.g ✓ 12
63.j odd 6 1 1890.2.i.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.i.g ✓ 12 1.a even 1 1 trivial
630.2.i.g ✓ 12 63.h even 3 1 inner
630.2.l.g yes 12 7.c even 3 1
630.2.l.g yes 12 9.c even 3 1
1890.2.i.g 12 3.b odd 2 1
1890.2.i.g 12 63.j odd 6 1
1890.2.l.g 12 9.d odd 6 1
1890.2.l.g 12 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\):

\( T_{11}^{12} - 3 T_{11}^{11} + 53 T_{11}^{10} + 60 T_{11}^{9} + 1494 T_{11}^{8} + 2466 T_{11}^{7} + \cdots + 77841 \) Copy content Toggle raw display
\( T_{13}^{12} + 2 T_{13}^{11} + 38 T_{13}^{10} + 24 T_{13}^{9} + 1108 T_{13}^{8} + 1129 T_{13}^{7} + \cdots + 15625 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 2 T^{10} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$7$ \( (T^{6} - 2 T^{5} + \cdots + 343)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} - 3 T^{11} + \cdots + 77841 \) Copy content Toggle raw display
$13$ \( T^{12} + 2 T^{11} + \cdots + 15625 \) Copy content Toggle raw display
$17$ \( T^{12} - T^{11} + \cdots + 9801 \) Copy content Toggle raw display
$19$ \( T^{12} - 8 T^{11} + \cdots + 5929 \) Copy content Toggle raw display
$23$ \( T^{12} - 11 T^{11} + \cdots + 35390601 \) Copy content Toggle raw display
$29$ \( T^{12} - 13 T^{11} + \cdots + 729 \) Copy content Toggle raw display
$31$ \( (T^{6} + 21 T^{5} + \cdots + 34617)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 313396209 \) Copy content Toggle raw display
$41$ \( T^{12} - 5 T^{11} + \cdots + 531441 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 3018293721 \) Copy content Toggle raw display
$47$ \( (T^{6} - 23 T^{5} + \cdots + 59373)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} - 2 T^{11} + \cdots + 88209 \) Copy content Toggle raw display
$59$ \( (T^{6} + T^{5} + \cdots - 20547)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - T^{5} + \cdots + 95821)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 2 T^{5} + \cdots - 83727)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 15 T^{5} + \cdots + 4887)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} - 22 T^{11} + \cdots + 30614089 \) Copy content Toggle raw display
$79$ \( (T^{6} + 27 T^{5} + \cdots + 1983)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 265070961 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 18815334561 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 1888367924041 \) Copy content Toggle raw display
show more
show less