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Properties

Label 210.3.h.a
Level $210$
Weight $3$
Character orbit 210.h
Analytic conductor $5.722$
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] = [210,3,Mod(139,210)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(210, 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("210.139");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 210.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: \(5.72208555157\)
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: \( 2^{12} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} - \beta_1 q^{3} - 2 q^{4} + \beta_{8} q^{5} - \beta_{4} q^{6} + ( - \beta_{6} + \beta_{5} - \beta_{2}) q^{7} + 2 \beta_{5} q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} - \beta_1 q^{3} - 2 q^{4} + \beta_{8} q^{5} - \beta_{4} q^{6} + ( - \beta_{6} + \beta_{5} - \beta_{2}) q^{7} + 2 \beta_{5} q^{8} + 3 q^{9} + (\beta_{11} + \beta_{6} - \beta_{5} - \beta_1) q^{10} + (\beta_{13} - \beta_{12} - \beta_{3} + 5) q^{11} + 2 \beta_1 q^{12} + (\beta_{9} + \beta_{8} + \cdots + 3 \beta_1) q^{13}+ \cdots + (3 \beta_{13} - 3 \beta_{12} + \cdots + 15) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 32 q^{4} + 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 32 q^{4} + 48 q^{9} + 96 q^{11} + 16 q^{14} - 24 q^{15} + 64 q^{16} + 24 q^{21} + 24 q^{25} + 64 q^{29} + 24 q^{30} - 8 q^{35} - 96 q^{36} - 144 q^{39} - 192 q^{44} - 176 q^{46} + 224 q^{49} - 96 q^{50} - 48 q^{51} - 32 q^{56} + 48 q^{60} - 128 q^{64} + 368 q^{65} - 56 q^{70} - 384 q^{71} + 224 q^{74} - 608 q^{79} + 144 q^{81} - 48 q^{84} - 440 q^{85} + 416 q^{86} + 224 q^{91} - 560 q^{95} + 288 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( ( - 1304 \nu^{14} + 9128 \nu^{13} - 115446 \nu^{12} + 574012 \nu^{11} - 3591842 \nu^{10} + \cdots - 99070875 ) / 560625 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 19306 \nu^{14} + 135142 \nu^{13} - 1709079 \nu^{12} + 8497628 \nu^{11} - 53165468 \nu^{10} + \cdots - 1406219625 ) / 3027375 \) 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
\(\beta_{6}\)\(=\) \( ( - 399728914 \nu^{15} + 3232898035 \nu^{14} - 38445891245 \nu^{13} + 214537975455 \nu^{12} + \cdots + 33279515835750 ) / 83752329375 \) 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
\(\beta_{9}\)\(=\) \( ( 447494447 \nu^{15} - 3034312245 \nu^{14} + 38930458815 \nu^{13} - 188284929725 \nu^{12} + \cdots + 8909332197750 ) / 83752329375 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 483208394 \nu^{15} + 3624062955 \nu^{14} - 44453885935 \nu^{13} + 233985303760 \nu^{12} + \cdots + 16389579233250 ) / 83752329375 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 567713312 \nu^{15} + 4257849840 \nu^{14} - 52266459230 \nu^{13} + 275154595755 \nu^{12} + \cdots + 21472387601625 ) / 83752329375 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 34162567 \nu^{15} - 260736947 \nu^{14} + 3176839329 \nu^{13} - 16958406448 \nu^{12} + \cdots - 1667961659250 ) / 3641405625 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 34162567 \nu^{15} - 251701558 \nu^{14} + 3113591606 \nu^{13} - 16157410637 \nu^{12} + \cdots - 947112736500 ) / 3641405625 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1650986060 \nu^{15} + 13415356753 \nu^{14} - 159200027621 \nu^{13} + 891510559422 \nu^{12} + \cdots + 143830007146125 ) / 83752329375 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 2019305898 \nu^{15} - 15144794235 \nu^{14} + 185881965095 \nu^{13} - 978536727220 \nu^{12} + \cdots - 77444936604000 ) / 83752329375 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} - \beta_{10} - 2\beta_{9} + 2\beta_{8} - 2\beta_{5} - 2\beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{15} - 4 \beta_{14} + 2 \beta_{13} - 2 \beta_{12} + \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + \cdots - 32 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 19 \beta_{15} - 6 \beta_{14} - 3 \beta_{13} - 9 \beta_{12} - 2 \beta_{11} + 13 \beta_{10} + 32 \beta_{9} + \cdots - 49 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{15} + 68 \beta_{14} - 36 \beta_{13} + 12 \beta_{12} - 4 \beta_{11} - 13 \beta_{10} + 70 \beta_{9} + \cdots + 498 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 344 \beta_{15} + 180 \beta_{14} + 85 \beta_{13} + 215 \beta_{12} - 4 \beta_{11} - 244 \beta_{10} + \cdots + 1327 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 120 \beta_{15} - 605 \beta_{14} + 365 \beta_{13} + 115 \beta_{12} - \beta_{11} + 45 \beta_{10} + \cdots - 4337 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 6370 \beta_{15} - 4872 \beta_{14} - 2107 \beta_{13} - 4319 \beta_{12} + 846 \beta_{11} + 5292 \beta_{10} + \cdots - 35061 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 8803 \beta_{15} + 21908 \beta_{14} - 16804 \beta_{13} - 13436 \beta_{12} + 3384 \beta_{11} + \cdots + 154534 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 122447 \beta_{15} + 128586 \beta_{14} + 47961 \beta_{13} + 76947 \beta_{12} - 27482 \beta_{11} + \cdots + 911441 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 254281 \beta_{15} - 381496 \beta_{14} + 415248 \beta_{13} + 442932 \beta_{12} - 162784 \beta_{11} + \cdots - 2653032 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 2442238 \beta_{15} - 3332538 \beta_{14} - 1012319 \beta_{13} - 1151557 \beta_{12} + 638966 \beta_{11} + \cdots - 23347091 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 3483639 \beta_{15} + 2928985 \beta_{14} - 5314483 \beta_{13} - 6157373 \beta_{12} + 2840117 \beta_{11} + \cdots + 20089762 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 49916468 \beta_{15} + 84671340 \beta_{14} + 19129253 \beta_{13} + 10768693 \beta_{12} - 11306262 \beta_{11} + \cdots + 587892255 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 189382655 \beta_{15} - 60250246 \beta_{14} + 274871156 \beta_{13} + 313288780 \beta_{12} + \cdots - 403879466 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 1023202897 \beta_{15} - 2101665624 \beta_{14} - 288089139 \beta_{13} + 136058607 \beta_{12} + \cdots - 14495944639 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/210\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(71\) \(127\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
139.1
0.500000 − 1.83656i
0.500000 + 3.55177i
0.500000 − 0.442923i
0.500000 − 4.10071i
0.500000 + 2.68650i
0.500000 − 0.971291i
0.500000 − 4.96598i
0.500000 + 0.422343i
0.500000 + 1.83656i
0.500000 − 3.55177i
0.500000 + 0.442923i
0.500000 + 4.10071i
0.500000 − 2.68650i
0.500000 + 0.971291i
0.500000 + 4.96598i
0.500000 − 0.422343i
− 1.41421i −1.73205 −2.00000 −4.91728 − 0.905717i 2.44949i 1.91369 + 6.73333i 2.82843i 3.00000 −1.28088 + 6.95409i
139.2 − 1.41421i −1.73205 −2.00000 1.38028 + 4.80571i 2.44949i −5.24961 − 4.63050i 2.82843i 3.00000 6.79630 − 1.95201i
139.3 − 1.41421i −1.73205 −2.00000 2.40341 − 4.38447i 2.44949i −6.94781 − 0.853218i 2.82843i 3.00000 −6.20058 − 3.39894i
139.4 − 1.41421i −1.73205 −2.00000 4.59769 − 1.96501i 2.44949i 6.81963 + 1.57881i 2.82843i 3.00000 −2.77894 − 6.50212i
139.5 − 1.41421i 1.73205 −2.00000 −4.59769 + 1.96501i − 2.44949i −6.81963 + 1.57881i 2.82843i 3.00000 2.77894 + 6.50212i
139.6 − 1.41421i 1.73205 −2.00000 −2.40341 + 4.38447i − 2.44949i 6.94781 − 0.853218i 2.82843i 3.00000 6.20058 + 3.39894i
139.7 − 1.41421i 1.73205 −2.00000 −1.38028 − 4.80571i − 2.44949i 5.24961 − 4.63050i 2.82843i 3.00000 −6.79630 + 1.95201i
139.8 − 1.41421i 1.73205 −2.00000 4.91728 + 0.905717i − 2.44949i −1.91369 + 6.73333i 2.82843i 3.00000 1.28088 − 6.95409i
139.9 1.41421i −1.73205 −2.00000 −4.91728 + 0.905717i − 2.44949i 1.91369 − 6.73333i − 2.82843i 3.00000 −1.28088 − 6.95409i
139.10 1.41421i −1.73205 −2.00000 1.38028 − 4.80571i − 2.44949i −5.24961 + 4.63050i − 2.82843i 3.00000 6.79630 + 1.95201i
139.11 1.41421i −1.73205 −2.00000 2.40341 + 4.38447i − 2.44949i −6.94781 + 0.853218i − 2.82843i 3.00000 −6.20058 + 3.39894i
139.12 1.41421i −1.73205 −2.00000 4.59769 + 1.96501i − 2.44949i 6.81963 − 1.57881i − 2.82843i 3.00000 −2.77894 + 6.50212i
139.13 1.41421i 1.73205 −2.00000 −4.59769 − 1.96501i 2.44949i −6.81963 − 1.57881i − 2.82843i 3.00000 2.77894 − 6.50212i
139.14 1.41421i 1.73205 −2.00000 −2.40341 − 4.38447i 2.44949i 6.94781 + 0.853218i − 2.82843i 3.00000 6.20058 − 3.39894i
139.15 1.41421i 1.73205 −2.00000 −1.38028 + 4.80571i 2.44949i 5.24961 + 4.63050i − 2.82843i 3.00000 −6.79630 − 1.95201i
139.16 1.41421i 1.73205 −2.00000 4.91728 − 0.905717i 2.44949i −1.91369 − 6.73333i − 2.82843i 3.00000 1.28088 + 6.95409i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 139.16
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 210.3.h.a ✓ 16
3.b odd 2 1 630.3.h.e 16
4.b odd 2 1 1680.3.bd.a 16
5.b even 2 1 inner 210.3.h.a ✓ 16
5.c odd 4 2 1050.3.f.e 16
7.b odd 2 1 inner 210.3.h.a ✓ 16
15.d odd 2 1 630.3.h.e 16
20.d odd 2 1 1680.3.bd.a 16
21.c even 2 1 630.3.h.e 16
28.d even 2 1 1680.3.bd.a 16
35.c odd 2 1 inner 210.3.h.a ✓ 16
35.f even 4 2 1050.3.f.e 16
105.g even 2 1 630.3.h.e 16
140.c even 2 1 1680.3.bd.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.h.a ✓ 16 1.a even 1 1 trivial
210.3.h.a ✓ 16 5.b even 2 1 inner
210.3.h.a ✓ 16 7.b odd 2 1 inner
210.3.h.a ✓ 16 35.c odd 2 1 inner
630.3.h.e 16 3.b odd 2 1
630.3.h.e 16 15.d odd 2 1
630.3.h.e 16 21.c even 2 1
630.3.h.e 16 105.g even 2 1
1050.3.f.e 16 5.c odd 4 2
1050.3.f.e 16 35.f even 4 2
1680.3.bd.a 16 4.b odd 2 1
1680.3.bd.a 16 20.d odd 2 1
1680.3.bd.a 16 28.d even 2 1
1680.3.bd.a 16 140.c even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(210, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{8} \) 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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