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Properties

Label 630.3.o.f
Level $630$
Weight $3$
Character orbit 630.o
Analytic conductor $17.166$
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] = [630,3,Mod(127,630)]
 
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([0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.127");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 630.o (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: \(17.1662566547\)
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: \( x^{16} - 8 x^{15} + 32 x^{14} + 152 x^{13} + 1954 x^{12} - 12664 x^{11} + 50336 x^{10} + 231896 x^{9} + \cdots + 2560000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{14}\cdot 5 \)
Twist minimal: no (minimal twist has level 210)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 1) q^{2} - 2 \beta_{3} q^{4} + ( - \beta_{11} + 1) q^{5} + \beta_{5} q^{7} + ( - 2 \beta_{3} - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + 1) q^{2} - 2 \beta_{3} q^{4} + ( - \beta_{11} + 1) q^{5} + \beta_{5} q^{7} + ( - 2 \beta_{3} - 2) q^{8} + (\beta_{12} - \beta_{11} - \beta_{3} + 1) q^{10} + (\beta_{15} - \beta_{14} + \cdots + 2 \beta_{4}) q^{11}+ \cdots + ( - 7 \beta_{3} - 7) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{2} + 16 q^{5} - 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{2} + 16 q^{5} - 32 q^{8} + 24 q^{10} - 8 q^{11} - 32 q^{13} - 64 q^{16} - 56 q^{17} + 16 q^{20} - 8 q^{22} - 24 q^{23} + 40 q^{25} - 64 q^{26} - 112 q^{31} - 64 q^{32} - 28 q^{35} - 152 q^{37} - 16 q^{40} - 48 q^{46} - 80 q^{47} + 72 q^{50} - 64 q^{52} - 48 q^{53} - 24 q^{55} + 96 q^{58} + 96 q^{61} - 112 q^{62} - 16 q^{65} - 80 q^{67} + 112 q^{68} - 536 q^{71} + 168 q^{77} - 64 q^{80} + 256 q^{83} + 40 q^{85} + 16 q^{88} - 48 q^{92} - 360 q^{95} + 688 q^{97} - 112 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 32 x^{14} + 152 x^{13} + 1954 x^{12} - 12664 x^{11} + 50336 x^{10} + 231896 x^{9} + \cdots + 2560000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 498284030129 \nu^{15} - 3107145446008 \nu^{14} + 4536459111192 \nu^{13} + \cdots + 53\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 15\!\cdots\!25 \nu^{15} + \cdots + 53\!\cdots\!00 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 50\!\cdots\!87 \nu^{15} + \cdots + 23\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 12\!\cdots\!47 \nu^{15} + \cdots + 33\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 60\!\cdots\!14 \nu^{15} + \cdots + 11\!\cdots\!00 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 87\!\cdots\!15 \nu^{15} + \cdots + 10\!\cdots\!00 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2256059372367 \nu^{15} - 15491795991004 \nu^{14} + 51193155741176 \nu^{13} + \cdots - 11\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 53\!\cdots\!91 \nu^{15} + \cdots + 75\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 34\!\cdots\!29 \nu^{15} + \cdots - 10\!\cdots\!00 ) / 94\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 17\!\cdots\!92 \nu^{15} + \cdots - 34\!\cdots\!00 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 29\!\cdots\!57 \nu^{15} + \cdots - 15\!\cdots\!00 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 33\!\cdots\!41 \nu^{15} + \cdots - 13\!\cdots\!00 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 49\!\cdots\!60 \nu^{15} + \cdots - 36\!\cdots\!00 ) / 94\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 43\!\cdots\!17 \nu^{15} + \cdots - 30\!\cdots\!00 ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 65\!\cdots\!13 \nu^{15} + \cdots - 23\!\cdots\!00 ) / 75\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3 \beta_{15} + \beta_{13} - 5 \beta_{12} - 5 \beta_{11} - 3 \beta_{8} - \beta_{5} + 3 \beta_{4} + \cdots + 9 ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 6 \beta_{15} + 5 \beta_{14} - 3 \beta_{13} - 5 \beta_{12} - 15 \beta_{11} - \beta_{8} - 10 \beta_{7} + \cdots + 3 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{15} + 40 \beta_{14} - 123 \beta_{13} + 205 \beta_{12} - 205 \beta_{11} + 20 \beta_{9} + \cdots - 547 ) / 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 164 \beta_{15} + 121 \beta_{14} - 45 \beta_{13} + 247 \beta_{12} - 41 \beta_{11} + 24 \beta_{10} + \cdots - 2075 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 9647 \beta_{15} + 3000 \beta_{14} - 1749 \beta_{13} + 10445 \beta_{12} + 10145 \beta_{11} + \cdots - 39641 ) / 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 36214 \beta_{15} - 16245 \beta_{14} + 14507 \beta_{13} + 9045 \beta_{12} + 86935 \beta_{11} + \cdots - 14507 ) / 10 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 120511 \beta_{15} - 316960 \beta_{14} + 288947 \beta_{13} - 569245 \beta_{12} + 606645 \beta_{11} + \cdots + 2350683 ) / 10 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 834588 \beta_{15} - 641969 \beta_{14} + 293269 \beta_{13} - 1169783 \beta_{12} + 91969 \beta_{11} + \cdots + 6263171 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 38994703 \beta_{15} - 13557400 \beta_{14} + 4516101 \beta_{13} - 37620805 \beta_{12} - 34469505 \beta_{11} + \cdots + 166491609 ) / 10 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 161696566 \beta_{15} + 60895005 \beta_{14} - 63463483 \beta_{13} - 26125405 \beta_{12} + \cdots + 63463483 ) / 10 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 625863679 \beta_{15} + 1506954040 \beta_{14} - 1025775883 \beta_{13} + 2171613805 \beta_{12} + \cdots - 10117175187 ) / 10 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 3629592340 \beta_{15} + 2796620041 \beta_{14} - 1348312477 \beta_{13} + 5081296087 \beta_{12} + \cdots - 24624971723 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 165052824847 \beta_{15} + 58282606200 \beta_{14} - 16340771349 \beta_{13} + 155464748445 \beta_{12} + \cdots - 711061238841 ) / 10 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 695324153654 \beta_{15} - 250038606245 \beta_{14} + 272692226027 \beta_{13} + 100098904645 \beta_{12} + \cdots - 272692226027 ) / 10 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 2771173982431 \beta_{15} - 6581449626960 \beta_{14} + 4187571520387 \beta_{13} - 9035112495645 \beta_{12} + \cdots + 43316921316843 ) / 10 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
−0.394902 − 0.394902i
3.60306 + 3.60306i
−3.99135 − 3.99135i
5.71348 + 5.71348i
−3.48873 − 3.48873i
3.76660 + 3.76660i
−1.37832 − 1.37832i
0.170157 + 0.170157i
−0.394902 + 0.394902i
3.60306 − 3.60306i
−3.99135 + 3.99135i
5.71348 − 5.71348i
−3.48873 + 3.48873i
3.76660 − 3.76660i
−1.37832 + 1.37832i
0.170157 − 0.170157i
1.00000 + 1.00000i 0 2.00000i −4.80148 − 1.39490i 0 1.87083 + 1.87083i −2.00000 + 2.00000i 0 −3.40658 − 6.19639i
127.2 1.00000 + 1.00000i 0 2.00000i −4.26896 + 2.60306i 0 1.87083 + 1.87083i −2.00000 + 2.00000i 0 −6.87203 − 1.66590i
127.3 1.00000 + 1.00000i 0 2.00000i −0.294013 − 4.99135i 0 −1.87083 − 1.87083i −2.00000 + 2.00000i 0 4.69734 − 5.28536i
127.4 1.00000 + 1.00000i 0 2.00000i 1.66827 + 4.71348i 0 −1.87083 − 1.87083i −2.00000 + 2.00000i 0 −3.04521 + 6.38175i
127.5 1.00000 + 1.00000i 0 2.00000i 2.20256 − 4.48873i 0 −1.87083 − 1.87083i −2.00000 + 2.00000i 0 6.69129 − 2.28617i
127.6 1.00000 + 1.00000i 0 2.00000i 4.16484 + 2.76660i 0 −1.87083 − 1.87083i −2.00000 + 2.00000i 0 1.39824 + 6.93144i
127.7 1.00000 + 1.00000i 0 2.00000i 4.39814 − 2.37832i 0 1.87083 + 1.87083i −2.00000 + 2.00000i 0 6.77646 + 2.01982i
127.8 1.00000 + 1.00000i 0 2.00000i 4.93066 − 0.829843i 0 1.87083 + 1.87083i −2.00000 + 2.00000i 0 5.76050 + 4.10081i
253.1 1.00000 − 1.00000i 0 − 2.00000i −4.80148 + 1.39490i 0 1.87083 − 1.87083i −2.00000 − 2.00000i 0 −3.40658 + 6.19639i
253.2 1.00000 − 1.00000i 0 − 2.00000i −4.26896 − 2.60306i 0 1.87083 − 1.87083i −2.00000 − 2.00000i 0 −6.87203 + 1.66590i
253.3 1.00000 − 1.00000i 0 − 2.00000i −0.294013 + 4.99135i 0 −1.87083 + 1.87083i −2.00000 − 2.00000i 0 4.69734 + 5.28536i
253.4 1.00000 − 1.00000i 0 − 2.00000i 1.66827 − 4.71348i 0 −1.87083 + 1.87083i −2.00000 − 2.00000i 0 −3.04521 − 6.38175i
253.5 1.00000 − 1.00000i 0 − 2.00000i 2.20256 + 4.48873i 0 −1.87083 + 1.87083i −2.00000 − 2.00000i 0 6.69129 + 2.28617i
253.6 1.00000 − 1.00000i 0 − 2.00000i 4.16484 − 2.76660i 0 −1.87083 + 1.87083i −2.00000 − 2.00000i 0 1.39824 − 6.93144i
253.7 1.00000 − 1.00000i 0 − 2.00000i 4.39814 + 2.37832i 0 1.87083 − 1.87083i −2.00000 − 2.00000i 0 6.77646 − 2.01982i
253.8 1.00000 − 1.00000i 0 − 2.00000i 4.93066 + 0.829843i 0 1.87083 − 1.87083i −2.00000 − 2.00000i 0 5.76050 − 4.10081i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.3.o.f 16
3.b odd 2 1 210.3.l.b ✓ 16
5.c odd 4 1 inner 630.3.o.f 16
15.d odd 2 1 1050.3.l.h 16
15.e even 4 1 210.3.l.b ✓ 16
15.e even 4 1 1050.3.l.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.l.b ✓ 16 3.b odd 2 1
210.3.l.b ✓ 16 15.e even 4 1
630.3.o.f 16 1.a even 1 1 trivial
630.3.o.f 16 5.c odd 4 1 inner
1050.3.l.h 16 15.d odd 2 1
1050.3.l.h 16 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{8} + 4 T_{11}^{7} - 540 T_{11}^{6} - 368 T_{11}^{5} + 94224 T_{11}^{4} - 257024 T_{11}^{3} + \cdots - 50723840 \) acting on \(S_{3}^{\mathrm{new}}(630, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 2)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 152587890625 \) Copy content Toggle raw display
$7$ \( (T^{4} + 49)^{4} \) Copy content Toggle raw display
$11$ \( (T^{8} + 4 T^{7} + \cdots - 50723840)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 29\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{8} + 56 T^{7} + \cdots - 58191200000)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{8} - 5196 T^{6} + \cdots + 262151483392)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{8} - 48 T^{7} + \cdots - 3514265600)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 39499497333760)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 53\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
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