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Properties

Label 630.3.h.e
Level $630$
Weight $3$
Character orbit 630.h
Analytic conductor $17.166$
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] = [630,3,Mod(559,630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(630, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.559");
 
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.h (of order \(2\), degree \(1\), 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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 96 x^{14} - 532 x^{13} + 3236 x^{12} - 12864 x^{11} + 49526 x^{10} - 141436 x^{9} + \cdots + 33750 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} - 2 q^{4} + \beta_{9} q^{5} + (\beta_{7} - \beta_{5} + \beta_{2} - \beta_1) q^{7} + 2 \beta_{5} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} - 2 q^{4} + \beta_{9} q^{5} + (\beta_{7} - \beta_{5} + \beta_{2} - \beta_1) q^{7} + 2 \beta_{5} q^{8} + ( - \beta_{10} - \beta_{7} + \beta_{5}) q^{10} + (\beta_{12} - \beta_{11} + \beta_{3} - 6) q^{11} + ( - \beta_{9} - \beta_{8} + \cdots - \beta_1) q^{13}+ \cdots + (2 \beta_{14} + 2 \beta_{13} + \cdots - 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 32 q^{4} - 96 q^{11} - 16 q^{14} + 64 q^{16} + 24 q^{25} - 64 q^{29} + 8 q^{35} + 192 q^{44} - 176 q^{46} + 224 q^{49} + 96 q^{50} + 32 q^{56} - 128 q^{64} - 368 q^{65} - 56 q^{70} + 384 q^{71} - 224 q^{74} - 608 q^{79} - 440 q^{85} - 416 q^{86} + 224 q^{91} + 560 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 96 x^{14} - 532 x^{13} + 3236 x^{12} - 12864 x^{11} + 49526 x^{10} - 141436 x^{9} + \cdots + 33750 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 61322 \nu^{14} + 429254 \nu^{13} - 5428353 \nu^{12} + 26989816 \nu^{11} - 168847606 \nu^{10} + \cdots - 4356184500 ) / 15136875 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2608 \nu^{14} - 18256 \nu^{13} + 230892 \nu^{12} - 1148024 \nu^{11} + 7183684 \nu^{10} + \cdots + 198141750 ) / 560625 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 181 \nu^{14} + 1267 \nu^{13} - 16034 \nu^{12} + 79733 \nu^{11} - 499338 \nu^{10} + \cdots - 14271525 ) / 26325 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1984 \nu^{15} - 14880 \nu^{14} + 171958 \nu^{13} - 892047 \nu^{12} + 4997626 \nu^{11} + \cdots + 575687250 ) / 85899825 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5163308 \nu^{15} - 38724810 \nu^{14} + 475369020 \nu^{13} - 2502572345 \nu^{12} + \cdots - 200352316500 ) / 1213801875 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 399728914 \nu^{15} - 3427703899 \nu^{14} + 39809532293 \nu^{13} - 231784568841 \nu^{12} + \cdots - 48079812922875 ) / 83752329375 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 399728914 \nu^{15} + 2568229811 \nu^{14} - 33793213677 \nu^{13} + 155694952029 \nu^{12} + \cdots - 16014851920125 ) / 83752329375 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 447494447 \nu^{15} - 3483298596 \nu^{14} + 42073363272 \nu^{13} - 228052000574 \nu^{12} + \cdots - 27118230861375 ) / 83752329375 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 447494447 \nu^{15} + 3229118109 \nu^{14} - 40294099863 \nu^{13} + 205531523111 \nu^{12} + \cdots + 5890964889375 ) / 83752329375 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 567713312 \nu^{15} + 4257849840 \nu^{14} - 52266459230 \nu^{13} + 275154595755 \nu^{12} + \cdots + 21472387601625 ) / 83752329375 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 785739041 \nu^{15} + 6191755645 \nu^{14} - 74430945615 \nu^{13} + 407289941690 \nu^{12} + \cdots + 53163415249875 ) / 83752329375 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 785739041 \nu^{15} + 5983941698 \nu^{14} - 72976247986 \nu^{13} + 388867038037 \nu^{12} + \cdots + 36667642356000 ) / 83752329375 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1154058879 \nu^{15} - 8813843083 \nu^{14} + 107361432781 \nu^{13} - 573407815907 \nu^{12} + \cdots - 56954262763125 ) / 83752329375 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1154058879 \nu^{15} - 8497040102 \nu^{14} + 105143811914 \nu^{13} - 545327879638 \nu^{12} + \cdots - 32182844707125 ) / 83752329375 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1607525398 \nu^{15} - 12056440485 \nu^{14} + 147968387895 \nu^{13} - 778938507295 \nu^{12} + \cdots - 60825320086500 ) / 83752329375 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} + \beta_{9} - \beta_{8} - 2\beta_{5} + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} - 2 \beta_{14} + 2 \beta_{13} + \beta_{9} - \beta_{8} - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \cdots - 32 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 13 \beta_{15} - 6 \beta_{14} + 3 \beta_{12} + 3 \beta_{11} - 2 \beta_{10} - 19 \beta_{9} + 19 \beta_{8} + \cdots - 52 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 27 \beta_{15} + 28 \beta_{14} - 40 \beta_{13} - 4 \beta_{12} + 16 \beta_{11} - 4 \beta_{10} + \cdots + 502 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 204 \beta_{15} + 140 \beta_{14} - 40 \beta_{13} - 125 \beta_{12} - 75 \beta_{11} - 4 \beta_{10} + \cdots + 1452 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 680 \beta_{15} - 440 \beta_{14} + 770 \beta_{13} + 40 \beta_{12} - 670 \beta_{11} - 2 \beta_{10} + \cdots - 8714 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 3395 \beta_{15} - 2975 \beta_{14} + 1897 \beta_{13} + 4004 \beta_{12} + 1344 \beta_{11} + \cdots - 39065 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 16817 \beta_{15} + 8014 \beta_{14} - 13894 \beta_{13} + 2910 \beta_{12} + 21450 \beta_{11} + \cdots + 151624 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 56252 \beta_{15} + 66195 \beta_{14} - 62391 \beta_{13} - 110352 \beta_{12} - 10752 \beta_{11} + \cdots + 1021793 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 412284 \beta_{15} - 158003 \beta_{14} + 223493 \beta_{13} - 191755 \beta_{12} - 600935 \beta_{11} + \cdots - 2461277 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 867390 \beta_{15} - 1574848 \beta_{14} + 1757690 \beta_{13} + 2770009 \beta_{12} - 423291 \beta_{11} + \cdots - 26117100 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 10032355 \beta_{15} + 3065077 \beta_{14} - 2792893 \beta_{13} + 7836073 \beta_{12} + 15379823 \beta_{11} + \cdots + 32343451 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 10645249 \beta_{15} + 39271219 \beta_{14} - 45400121 \beta_{13} - 64529374 \beta_{12} + 28502526 \beta_{11} + \cdots + 652421629 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 241703224 \beta_{15} - 52320569 \beta_{14} + 7929677 \beta_{13} - 266941479 \beta_{12} + \cdots - 136937987 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 26124292 \beta_{15} - 997078605 \beta_{14} + 1104587019 \beta_{13} + 1392676158 \beta_{12} + \cdots - 15888620797 ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1
0.500000 + 0.422343i
0.500000 − 4.10071i
0.500000 − 0.442923i
0.500000 + 3.55177i
0.500000 − 4.96598i
0.500000 − 0.971291i
0.500000 + 2.68650i
0.500000 − 1.83656i
0.500000 − 0.422343i
0.500000 + 4.10071i
0.500000 + 0.442923i
0.500000 − 3.55177i
0.500000 + 4.96598i
0.500000 + 0.971291i
0.500000 − 2.68650i
0.500000 + 1.83656i
− 1.41421i 0 −2.00000 −4.91728 + 0.905717i 0 −1.91369 − 6.73333i 2.82843i 0 1.28088 + 6.95409i
559.2 − 1.41421i 0 −2.00000 −4.59769 − 1.96501i 0 6.81963 − 1.57881i 2.82843i 0 −2.77894 + 6.50212i
559.3 − 1.41421i 0 −2.00000 −2.40341 − 4.38447i 0 −6.94781 + 0.853218i 2.82843i 0 −6.20058 + 3.39894i
559.4 − 1.41421i 0 −2.00000 −1.38028 + 4.80571i 0 −5.24961 + 4.63050i 2.82843i 0 6.79630 + 1.95201i
559.5 − 1.41421i 0 −2.00000 1.38028 − 4.80571i 0 5.24961 + 4.63050i 2.82843i 0 −6.79630 − 1.95201i
559.6 − 1.41421i 0 −2.00000 2.40341 + 4.38447i 0 6.94781 + 0.853218i 2.82843i 0 6.20058 − 3.39894i
559.7 − 1.41421i 0 −2.00000 4.59769 + 1.96501i 0 −6.81963 − 1.57881i 2.82843i 0 2.77894 − 6.50212i
559.8 − 1.41421i 0 −2.00000 4.91728 − 0.905717i 0 1.91369 − 6.73333i 2.82843i 0 −1.28088 − 6.95409i
559.9 1.41421i 0 −2.00000 −4.91728 − 0.905717i 0 −1.91369 + 6.73333i − 2.82843i 0 1.28088 − 6.95409i
559.10 1.41421i 0 −2.00000 −4.59769 + 1.96501i 0 6.81963 + 1.57881i − 2.82843i 0 −2.77894 − 6.50212i
559.11 1.41421i 0 −2.00000 −2.40341 + 4.38447i 0 −6.94781 − 0.853218i − 2.82843i 0 −6.20058 − 3.39894i
559.12 1.41421i 0 −2.00000 −1.38028 − 4.80571i 0 −5.24961 − 4.63050i − 2.82843i 0 6.79630 − 1.95201i
559.13 1.41421i 0 −2.00000 1.38028 + 4.80571i 0 5.24961 − 4.63050i − 2.82843i 0 −6.79630 + 1.95201i
559.14 1.41421i 0 −2.00000 2.40341 − 4.38447i 0 6.94781 − 0.853218i − 2.82843i 0 6.20058 + 3.39894i
559.15 1.41421i 0 −2.00000 4.59769 − 1.96501i 0 −6.81963 + 1.57881i − 2.82843i 0 2.77894 + 6.50212i
559.16 1.41421i 0 −2.00000 4.91728 + 0.905717i 0 1.91369 + 6.73333i − 2.82843i 0 −1.28088 + 6.95409i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 559.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.b odd 2 1 inner
35.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.3.h.e 16
3.b odd 2 1 210.3.h.a ✓ 16
5.b even 2 1 inner 630.3.h.e 16
7.b odd 2 1 inner 630.3.h.e 16
12.b even 2 1 1680.3.bd.a 16
15.d odd 2 1 210.3.h.a ✓ 16
15.e even 4 2 1050.3.f.e 16
21.c even 2 1 210.3.h.a ✓ 16
35.c odd 2 1 inner 630.3.h.e 16
60.h even 2 1 1680.3.bd.a 16
84.h odd 2 1 1680.3.bd.a 16
105.g even 2 1 210.3.h.a ✓ 16
105.k odd 4 2 1050.3.f.e 16
420.o odd 2 1 1680.3.bd.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.h.a ✓ 16 3.b odd 2 1
210.3.h.a ✓ 16 15.d odd 2 1
210.3.h.a ✓ 16 21.c even 2 1
210.3.h.a ✓ 16 105.g even 2 1
630.3.h.e 16 1.a even 1 1 trivial
630.3.h.e 16 5.b even 2 1 inner
630.3.h.e 16 7.b odd 2 1 inner
630.3.h.e 16 35.c odd 2 1 inner
1050.3.f.e 16 15.e even 4 2
1050.3.f.e 16 105.k odd 4 2
1680.3.bd.a 16 12.b even 2 1
1680.3.bd.a 16 60.h even 2 1
1680.3.bd.a 16 84.h odd 2 1
1680.3.bd.a 16 420.o odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(630, [\chi])\):

\( T_{11}^{4} + 24T_{11}^{3} + 28T_{11}^{2} - 1776T_{11} - 4844 \) Copy content Toggle raw display
\( T_{13}^{8} - 1044T_{13}^{6} + 313732T_{13}^{4} - 31875456T_{13}^{2} + 586802176 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 152587890625 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 33232930569601 \) Copy content Toggle raw display
$11$ \( (T^{4} + 24 T^{3} + \cdots - 4844)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} - 1044 T^{6} + \cdots + 586802176)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 996 T^{6} + \cdots + 338560000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 1796 T^{6} + \cdots + 2149991424)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 2932 T^{6} + \cdots + 11186869824)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 16 T^{3} + \cdots + 19600)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 3740 T^{6} + \cdots + 747256896)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 3617786594304)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 932 T^{6} + \cdots + 1141899264)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 9151544623104)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 14545741676544)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 38389325511744)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 16112 T^{6} + \cdots + 203235065856)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 13824 T^{6} + \cdots + 72666906624)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 17224 T^{6} + \cdots + 404129746944)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 96 T^{3} + \cdots - 18715436)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 105841627140096)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 152 T^{3} + \cdots - 12612096)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 13129665286144)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 135150669186624)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 5898136817664)^{2} \) Copy content Toggle raw display
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