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Properties

Label 630.2.bo.a
Level $630$
Weight $2$
Character orbit 630.bo
Analytic conductor $5.031$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [630,2,Mod(89,630)]
 
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([3, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.89");
 
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.bo (of order \(6\), 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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6 x^{15} + 21 x^{14} - 54 x^{13} + 113 x^{12} - 168 x^{11} + 186 x^{10} - 84 x^{9} + \cdots + 390625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{8} q^{2} + (\beta_{8} - 1) q^{4} - \beta_{7} q^{5} + ( - \beta_{13} - \beta_{10}) q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{2} + (\beta_{8} - 1) q^{4} - \beta_{7} q^{5} + ( - \beta_{13} - \beta_{10}) q^{7} + q^{8} + (\beta_{8} + \beta_{7} - \beta_{4} - 1) q^{10} + (\beta_{13} - \beta_{5} + \beta_{3}) q^{11} + ( - \beta_{15} + \beta_{10} + \cdots + \beta_{3}) q^{13}+ \cdots + (\beta_{12} + \beta_{11} - 2 \beta_{9} + \cdots + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{2} - 8 q^{4} - 6 q^{5} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{2} - 8 q^{4} - 6 q^{5} + 16 q^{8} + 6 q^{10} - 8 q^{16} + 24 q^{19} - 8 q^{23} - 6 q^{25} + 12 q^{31} - 8 q^{32} + 4 q^{35} - 24 q^{38} - 6 q^{40} - 8 q^{46} + 60 q^{47} - 28 q^{49} + 12 q^{50} + 16 q^{53} + 24 q^{61} + 16 q^{64} - 20 q^{65} - 14 q^{70} - 88 q^{77} + 4 q^{79} + 6 q^{80} + 64 q^{85} - 28 q^{91} + 16 q^{92} - 60 q^{94} - 12 q^{95} + 20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{15} + 21 x^{14} - 54 x^{13} + 113 x^{12} - 168 x^{11} + 186 x^{10} - 84 x^{9} + \cdots + 390625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 384 \nu^{15} + 11789 \nu^{14} - 67399 \nu^{13} + 175846 \nu^{12} - 241632 \nu^{11} + \cdots + 608281250 ) / 568125000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 8097 \nu^{15} + 5902 \nu^{14} - 24132 \nu^{13} + 294133 \nu^{12} - 550416 \nu^{11} + \cdots + 3815703125 ) / 3408750000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6137 \nu^{15} + 20678 \nu^{14} - 42523 \nu^{13} + 210377 \nu^{12} - 519294 \nu^{11} + \cdots + 2128046875 ) / 1704375000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 6613 \nu^{15} + 7932 \nu^{14} - 14237 \nu^{13} + 93033 \nu^{12} + 60504 \nu^{11} + \cdots + 91640625 ) / 1704375000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 17122 \nu^{15} - 4902 \nu^{14} + 66757 \nu^{13} - 47583 \nu^{12} + 118266 \nu^{11} + \cdots + 1565390625 ) / 1704375000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - \nu^{15} + 6 \nu^{14} - 21 \nu^{13} + 54 \nu^{12} - 113 \nu^{11} + 168 \nu^{10} - 186 \nu^{9} + \cdots + 468750 ) / 78125 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 22146 \nu^{15} + 79646 \nu^{14} - 238236 \nu^{13} + 546479 \nu^{12} - 982878 \nu^{11} + \cdots + 3913984375 ) / 1704375000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 31084 \nu^{15} + 147184 \nu^{14} - 411169 \nu^{13} + 973141 \nu^{12} - 1435662 \nu^{11} + \cdots + 6441171875 ) / 1704375000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 29284 \nu^{15} - 117899 \nu^{14} + 303959 \nu^{13} - 699881 \nu^{12} + 1218072 \nu^{11} + \cdots - 2969453125 ) / 1136250000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 52692 \nu^{15} + 298312 \nu^{14} - 764517 \nu^{13} + 2102878 \nu^{12} - 3542736 \nu^{11} + \cdots + 14823593750 ) / 1704375000 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 53838 \nu^{15} - 173323 \nu^{14} + 537918 \nu^{13} - 1115872 \nu^{12} + 1770924 \nu^{11} + \cdots - 6022343750 ) / 1704375000 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 115983 \nu^{15} - 499393 \nu^{14} + 1698663 \nu^{13} - 4301152 \nu^{12} + 7691484 \nu^{11} + \cdots - 27148437500 ) / 3408750000 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 74084 \nu^{15} + 358884 \nu^{14} - 1026644 \nu^{13} + 2840991 \nu^{12} - 4526112 \nu^{11} + \cdots + 16796953125 ) / 1704375000 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 182946 \nu^{15} + 901171 \nu^{14} - 2551911 \nu^{13} + 6076429 \nu^{12} - 10614828 \nu^{11} + \cdots + 36593515625 ) / 3408750000 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{15} - \beta_{12} - \beta_{10} + \beta_{7} + 2\beta_{6} - \beta_{4} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{14} + 4 \beta_{13} - 2 \beta_{12} + \beta_{10} - 3 \beta_{9} + 3 \beta_{8} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{15} - 3 \beta_{14} + 2 \beta_{13} + 3 \beta_{12} + 6 \beta_{11} + \beta_{10} + 4 \beta_{9} + \cdots - 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6 \beta_{15} + 8 \beta_{14} - 2 \beta_{13} - 4 \beta_{12} - 6 \beta_{11} + 12 \beta_{10} - 8 \beta_{9} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 24 \beta_{15} + 16 \beta_{14} - 22 \beta_{13} - 48 \beta_{11} + 2 \beta_{10} - 6 \beta_{9} - 28 \beta_{8} + \cdots + 39 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8 \beta_{15} - 28 \beta_{14} - 40 \beta_{13} - 28 \beta_{12} + 6 \beta_{11} + 8 \beta_{10} - 42 \beta_{8} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 55 \beta_{15} + 58 \beta_{13} - 3 \beta_{12} + 42 \beta_{11} + 113 \beta_{10} + 20 \beta_{9} + \cdots - 190 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 33 \beta_{15} - 77 \beta_{14} - 142 \beta_{13} + 154 \beta_{12} + 45 \beta_{10} - 55 \beta_{9} + \cdots - 403 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 311 \beta_{15} - 121 \beta_{14} + 334 \beta_{13} + 121 \beta_{12} - 108 \beta_{11} + 311 \beta_{10} + \cdots - 2293 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 564 \beta_{15} + 384 \beta_{14} + 756 \beta_{13} - 192 \beta_{12} - 252 \beta_{11} + 768 \beta_{10} + \cdots + 1752 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1152 \beta_{15} + 3456 \beta_{14} + 2196 \beta_{13} + 1728 \beta_{11} + 1044 \beta_{10} - 2004 \beta_{9} + \cdots + 4945 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 15024 \beta_{15} - 96 \beta_{14} - 2304 \beta_{13} - 96 \beta_{12} + 14220 \beta_{11} - 15024 \beta_{10} + \cdots - 7908 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 6947 \beta_{15} + 10116 \beta_{13} - 3169 \beta_{12} + 9612 \beta_{11} - 10873 \beta_{10} - 8472 \beta_{9} + \cdots + 21406 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 39073 \beta_{15} - 25031 \beta_{14} - 11048 \beta_{13} + 50062 \beta_{12} - 59 \beta_{10} + 23433 \beta_{9} + \cdots + 47975 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(-1\) \(-1\) \(\beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
89.1
2.11423 − 0.728019i
1.98669 + 1.02619i
1.68760 − 1.46697i
0.948234 + 2.02506i
0.104634 − 2.23362i
−0.442358 + 2.19188i
−1.27963 − 1.83372i
−2.11940 − 0.712845i
2.11423 + 0.728019i
1.98669 − 1.02619i
1.68760 + 1.46697i
0.948234 − 2.02506i
0.104634 + 2.23362i
−0.442358 − 2.19188i
−1.27963 + 1.83372i
−2.11940 + 0.712845i
−0.500000 + 0.866025i 0 −0.500000 − 0.866025i −2.11423 − 0.728019i 0 2.30608 − 1.29693i 1.00000 0 1.68760 − 1.46697i
89.2 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −1.98669 + 1.02619i 0 1.39924 − 2.24547i 1.00000 0 0.104634 − 2.23362i
89.3 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −1.68760 − 1.46697i 0 −2.30608 + 1.29693i 1.00000 0 2.11423 − 0.728019i
89.4 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −0.948234 + 2.02506i 0 0.732536 + 2.54232i 1.00000 0 −1.27963 − 1.83372i
89.5 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −0.104634 − 2.23362i 0 −1.39924 + 2.24547i 1.00000 0 1.98669 + 1.02619i
89.6 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0.442358 + 2.19188i 0 −1.63937 − 2.07665i 1.00000 0 −2.11940 − 0.712845i
89.7 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 1.27963 − 1.83372i 0 −0.732536 − 2.54232i 1.00000 0 0.948234 + 2.02506i
89.8 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 2.11940 − 0.712845i 0 1.63937 + 2.07665i 1.00000 0 −0.442358 + 2.19188i
269.1 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −2.11423 + 0.728019i 0 2.30608 + 1.29693i 1.00000 0 1.68760 + 1.46697i
269.2 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −1.98669 − 1.02619i 0 1.39924 + 2.24547i 1.00000 0 0.104634 + 2.23362i
269.3 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −1.68760 + 1.46697i 0 −2.30608 − 1.29693i 1.00000 0 2.11423 + 0.728019i
269.4 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −0.948234 − 2.02506i 0 0.732536 − 2.54232i 1.00000 0 −1.27963 + 1.83372i
269.5 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −0.104634 + 2.23362i 0 −1.39924 − 2.24547i 1.00000 0 1.98669 − 1.02619i
269.6 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 0.442358 − 2.19188i 0 −1.63937 + 2.07665i 1.00000 0 −2.11940 + 0.712845i
269.7 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 1.27963 + 1.83372i 0 −0.732536 + 2.54232i 1.00000 0 0.948234 − 2.02506i
269.8 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 2.11940 + 0.712845i 0 1.63937 − 2.07665i 1.00000 0 −0.442358 − 2.19188i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 89.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
15.d odd 2 1 inner
105.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.bo.a ✓ 16
3.b odd 2 1 630.2.bo.b yes 16
5.b even 2 1 630.2.bo.b yes 16
5.c odd 4 2 3150.2.bf.f 32
7.c even 3 1 4410.2.d.b 16
7.d odd 6 1 inner 630.2.bo.a ✓ 16
7.d odd 6 1 4410.2.d.b 16
15.d odd 2 1 inner 630.2.bo.a ✓ 16
15.e even 4 2 3150.2.bf.f 32
21.g even 6 1 630.2.bo.b yes 16
21.g even 6 1 4410.2.d.a 16
21.h odd 6 1 4410.2.d.a 16
35.i odd 6 1 630.2.bo.b yes 16
35.i odd 6 1 4410.2.d.a 16
35.j even 6 1 4410.2.d.a 16
35.k even 12 2 3150.2.bf.f 32
105.o odd 6 1 4410.2.d.b 16
105.p even 6 1 inner 630.2.bo.a ✓ 16
105.p even 6 1 4410.2.d.b 16
105.w odd 12 2 3150.2.bf.f 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.bo.a ✓ 16 1.a even 1 1 trivial
630.2.bo.a ✓ 16 7.d odd 6 1 inner
630.2.bo.a ✓ 16 15.d odd 2 1 inner
630.2.bo.a ✓ 16 105.p even 6 1 inner
630.2.bo.b yes 16 3.b odd 2 1
630.2.bo.b yes 16 5.b even 2 1
630.2.bo.b yes 16 21.g even 6 1
630.2.bo.b yes 16 35.i odd 6 1
3150.2.bf.f 32 5.c odd 4 2
3150.2.bf.f 32 15.e even 4 2
3150.2.bf.f 32 35.k even 12 2
3150.2.bf.f 32 105.w odd 12 2
4410.2.d.a 16 21.g even 6 1
4410.2.d.a 16 21.h odd 6 1
4410.2.d.a 16 35.i odd 6 1
4410.2.d.a 16 35.j even 6 1
4410.2.d.b 16 7.c even 3 1
4410.2.d.b 16 7.d odd 6 1
4410.2.d.b 16 105.o odd 6 1
4410.2.d.b 16 105.p even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{8} - 42T_{17}^{6} + 1516T_{17}^{4} - 3024T_{17}^{3} - 8688T_{17}^{2} + 17856T_{17} + 61504 \) acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 6 T^{15} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} + 14 T^{14} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{16} - 50 T^{14} + \cdots + 923521 \) Copy content Toggle raw display
$13$ \( (T^{8} - 72 T^{6} + \cdots + 1156)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 42 T^{6} + \cdots + 61504)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 12 T^{7} + \cdots + 19044)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 4 T^{7} + \cdots + 86436)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 122 T^{6} + \cdots + 33856)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 6 T^{7} + \cdots + 6533136)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} - 112 T^{14} + \cdots + 3111696 \) Copy content Toggle raw display
$41$ \( (T^{8} - 244 T^{6} + \cdots + 12787776)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 148 T^{6} + \cdots + 553536)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 30 T^{7} + \cdots + 10404)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 8 T^{7} + \cdots + 690561)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + 46 T^{14} + \cdots + 104976 \) Copy content Toggle raw display
$61$ \( (T^{8} - 12 T^{7} + \cdots + 78428736)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 429981696 \) Copy content Toggle raw display
$71$ \( (T^{8} + 620 T^{6} + \cdots + 352538176)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 347892350976 \) Copy content Toggle raw display
$79$ \( (T^{8} - 2 T^{7} + \cdots + 86044176)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 414 T^{6} + \cdots + 364816)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 306402103296 \) Copy content Toggle raw display
$97$ \( (T^{8} - 338 T^{6} + \cdots + 27123264)^{2} \) Copy content Toggle raw display
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