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Properties

Label 630.4.k.r
Level $630$
Weight $4$
Character orbit 630.k
Analytic conductor $37.171$
Analytic rank $0$
Dimension $6$
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(361,630)]
 
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([0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.361");
 
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.k (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: \(37.1712033036\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.362560708800.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 67x^{4} - 114x^{3} + 4446x^{2} - 5940x + 8100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 70)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{3} + 2) q^{2} - 4 \beta_{3} q^{4} + ( - 5 \beta_{3} + 5) q^{5} + (\beta_{5} + \beta_{4} + 7 \beta_{3} + \cdots - 1) q^{7}+ \cdots - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{3} + 2) q^{2} - 4 \beta_{3} q^{4} + ( - 5 \beta_{3} + 5) q^{5} + (\beta_{5} + \beta_{4} + 7 \beta_{3} + \cdots - 1) q^{7}+ \cdots + ( - 2 \beta_{5} + 6 \beta_{4} + \cdots - 566) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} - 12 q^{4} + 15 q^{5} + 14 q^{7} - 48 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{2} - 12 q^{4} + 15 q^{5} + 14 q^{7} - 48 q^{8} - 30 q^{10} + 41 q^{11} - 2 q^{13} + 74 q^{14} - 48 q^{16} + 30 q^{17} - 49 q^{19} - 120 q^{20} + 164 q^{22} + 145 q^{23} - 75 q^{25} - 2 q^{26} + 92 q^{28} - 536 q^{29} + 28 q^{31} + 96 q^{32} + 120 q^{34} + 185 q^{35} - 813 q^{37} + 98 q^{38} - 120 q^{40} - 626 q^{41} + 720 q^{43} + 164 q^{44} - 290 q^{46} - 977 q^{47} - 1860 q^{49} - 300 q^{50} + 4 q^{52} + 325 q^{53} + 410 q^{55} - 112 q^{56} - 536 q^{58} - 272 q^{59} + 902 q^{61} + 112 q^{62} + 384 q^{64} - 5 q^{65} + 170 q^{67} + 120 q^{68} + 230 q^{70} + 2160 q^{71} + 584 q^{73} + 1626 q^{74} + 392 q^{76} + 2395 q^{77} + 310 q^{79} + 240 q^{80} - 626 q^{82} - 1252 q^{83} + 300 q^{85} + 720 q^{86} - 328 q^{88} - 400 q^{89} - 120 q^{91} - 1160 q^{92} + 1954 q^{94} + 245 q^{95} + 3252 q^{97} - 1482 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 67x^{4} - 114x^{3} + 4446x^{2} - 5940x + 8100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 67\nu^{4} - 4489\nu^{3} + 4446\nu^{2} - 5940\nu + 397980 ) / 291942 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 737\nu^{5} - 722\nu^{4} + 48374\nu^{3} - 16683\nu^{2} + 3210012\nu + 90450 ) / 4379130 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2078\nu^{5} - 6745\nu^{4} - 131969\nu^{3} - 249327\nu^{2} - 8256132\nu - 13790520 ) / 875826 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2276\nu^{5} + 6521\nu^{4} - 144965\nu^{3} + 630981\nu^{2} - 9432252\nu + 25597350 ) / 875826 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{5} + \beta_{4} + 45\beta_{3} + \beta_{2} - 2\beta _1 - 45 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - \beta_{4} - 66\beta_{2} + 45 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -67\beta_{5} - 134\beta_{4} - 2925\beta_{3} - 67\beta_{2} + 158\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -86\beta_{5} - 43\beta_{4} + 4095\beta_{3} + 4289\beta_{2} - 4246\beta _1 - 4095 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
3.95365 − 6.84792i
0.687175 − 1.19022i
−4.14083 + 7.17212i
3.95365 + 6.84792i
0.687175 + 1.19022i
−4.14083 − 7.17212i
1.00000 + 1.73205i 0 −2.00000 + 3.46410i 2.50000 + 4.33013i 0 −1.45365 − 18.4631i −8.00000 0 −5.00000 + 8.66025i
361.2 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 2.50000 + 4.33013i 0 1.81283 + 18.4313i −8.00000 0 −5.00000 + 8.66025i
361.3 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 2.50000 + 4.33013i 0 6.64083 − 17.2887i −8.00000 0 −5.00000 + 8.66025i
541.1 1.00000 − 1.73205i 0 −2.00000 − 3.46410i 2.50000 − 4.33013i 0 −1.45365 + 18.4631i −8.00000 0 −5.00000 − 8.66025i
541.2 1.00000 − 1.73205i 0 −2.00000 − 3.46410i 2.50000 − 4.33013i 0 1.81283 − 18.4313i −8.00000 0 −5.00000 − 8.66025i
541.3 1.00000 − 1.73205i 0 −2.00000 − 3.46410i 2.50000 − 4.33013i 0 6.64083 + 17.2887i −8.00000 0 −5.00000 − 8.66025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.4.k.r 6
3.b odd 2 1 70.4.e.e ✓ 6
7.c even 3 1 inner 630.4.k.r 6
12.b even 2 1 560.4.q.m 6
15.d odd 2 1 350.4.e.k 6
15.e even 4 2 350.4.j.i 12
21.c even 2 1 490.4.e.y 6
21.g even 6 1 490.4.a.v 3
21.g even 6 1 490.4.e.y 6
21.h odd 6 1 70.4.e.e ✓ 6
21.h odd 6 1 490.4.a.w 3
84.n even 6 1 560.4.q.m 6
105.o odd 6 1 350.4.e.k 6
105.o odd 6 1 2450.4.a.cb 3
105.p even 6 1 2450.4.a.ce 3
105.x even 12 2 350.4.j.i 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.4.e.e ✓ 6 3.b odd 2 1
70.4.e.e ✓ 6 21.h odd 6 1
350.4.e.k 6 15.d odd 2 1
350.4.e.k 6 105.o odd 6 1
350.4.j.i 12 15.e even 4 2
350.4.j.i 12 105.x even 12 2
490.4.a.v 3 21.g even 6 1
490.4.a.w 3 21.h odd 6 1
490.4.e.y 6 21.c even 2 1
490.4.e.y 6 21.g even 6 1
560.4.q.m 6 12.b even 2 1
560.4.q.m 6 84.n even 6 1
630.4.k.r 6 1.a even 1 1 trivial
630.4.k.r 6 7.c even 3 1 inner
2450.4.a.cb 3 105.o odd 6 1
2450.4.a.ce 3 105.p even 6 1

Hecke kernels

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

\( T_{11}^{6} - 41T_{11}^{5} + 4177T_{11}^{4} - 102504T_{11}^{3} + 10429236T_{11}^{2} - 255640320T_{11} + 10489856400 \) Copy content Toggle raw display
\( T_{13}^{3} + T_{13}^{2} - 1360T_{13} - 19180 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 4)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T^{2} - 5 T + 25)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} - 14 T^{5} + \cdots + 40353607 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 10489856400 \) Copy content Toggle raw display
$13$ \( (T^{3} + T^{2} + \cdots - 19180)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 44899914816 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 348912313344 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 5665366321209 \) Copy content Toggle raw display
$29$ \( (T^{3} + 268 T^{2} + \cdots - 562914)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 1155367014400 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 364355408905216 \) Copy content Toggle raw display
$41$ \( (T^{3} + 313 T^{2} + \cdots - 15201837)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 360 T^{2} + \cdots + 21473350)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 89\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 252484981330176 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 212379790441536 \) Copy content Toggle raw display
$71$ \( (T^{3} - 1080 T^{2} + \cdots + 45563904)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 696242104960000 \) Copy content Toggle raw display
$83$ \( (T^{3} + 626 T^{2} + \cdots - 46561032)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 22\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( (T^{3} - 1626 T^{2} + \cdots + 35869960)^{2} \) Copy content Toggle raw display
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