↑

Properties

Label 210.4.t.b
Level $210$
Weight $4$
Character orbit 210.t
Analytic conductor $12.390$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [210,4,Mod(59,210)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(210, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("210.59");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 210.t (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: \(12.3904011012\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 48 q^{2} - 3 q^{3} - 96 q^{4} - 9 q^{5} - 384 q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 48 q^{2} - 3 q^{3} - 96 q^{4} - 9 q^{5} - 384 q^{8} - 9 q^{9} - 18 q^{10} + 12 q^{12} - 154 q^{15} - 384 q^{16} + 18 q^{18} + 216 q^{19} - 79 q^{21} - 234 q^{23} + 24 q^{24} - 39 q^{25} - 154 q^{30} - 414 q^{31} + 768 q^{32} + 408 q^{33} - 156 q^{35} + 72 q^{36} + 432 q^{38} + 174 q^{39} + 72 q^{40} - 58 q^{42} - 1005 q^{45} + 468 q^{46} - 864 q^{47} - 384 q^{49} - 156 q^{50} - 32 q^{51} - 78 q^{53} - 1900 q^{57} + 308 q^{60} - 1350 q^{61} + 578 q^{63} + 3072 q^{64} + 810 q^{65} + 816 q^{66} - 210 q^{70} + 72 q^{72} - 1143 q^{75} + 714 q^{77} + 696 q^{78} + 1062 q^{79} + 144 q^{80} - 1465 q^{81} + 200 q^{84} - 528 q^{85} + 1491 q^{87} + 2760 q^{91} + 1872 q^{92} + 40 q^{93} - 1728 q^{94} + 1386 q^{95} - 96 q^{96} - 2688 q^{98} - 380 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1 1.00000 + 1.73205i −5.19394 − 0.151669i −2.00000 + 3.46410i −1.04304 − 11.1316i −4.93124 − 9.14783i −18.2739 + 3.01087i −8.00000 26.9540 + 1.57551i 18.2374 − 12.9382i
59.2 1.00000 + 1.73205i −5.18799 − 0.291064i −2.00000 + 3.46410i −8.96208 + 6.68439i −4.68386 − 9.27693i 8.12061 + 16.6450i −8.00000 26.8306 + 3.02007i −20.5398 − 8.83839i
59.3 1.00000 + 1.73205i −4.84077 − 1.88864i −2.00000 + 3.46410i 7.17822 + 8.57165i −1.56955 − 10.2731i −18.5063 + 0.718450i −8.00000 19.8661 + 18.2849i −7.66831 + 21.0047i
59.4 1.00000 + 1.73205i −4.78496 + 2.02588i −2.00000 + 3.46410i 10.8778 − 2.58344i −8.29388 − 6.26191i 12.1791 + 13.9524i −8.00000 18.7916 − 19.3875i 15.3524 + 16.2574i
59.5 1.00000 + 1.73205i −4.05599 − 3.24791i −2.00000 + 3.46410i −3.83415 − 10.5023i 1.56955 − 10.2731i 18.5063 − 0.718450i −8.00000 5.90215 + 26.3470i 14.3564 − 17.1433i
59.6 1.00000 + 1.73205i −3.52166 + 3.82073i −2.00000 + 3.46410i 10.7413 − 3.10220i −10.1393 − 2.27896i −4.70499 − 17.9127i −8.00000 −2.19589 − 26.9106i 16.1145 + 15.5023i
59.7 1.00000 + 1.73205i −3.48147 + 3.85738i −2.00000 + 3.46410i −9.30371 − 6.20008i −10.1626 − 2.17271i 12.5582 − 13.6121i −8.00000 −2.75870 − 26.8587i 1.43515 − 22.3146i
59.8 1.00000 + 1.73205i −2.84607 − 4.34740i −2.00000 + 3.46410i −10.2699 + 4.41919i 4.68386 − 9.27693i −8.12061 − 16.6450i −8.00000 −10.7998 + 24.7460i −17.9242 − 13.3688i
59.9 1.00000 + 1.73205i −2.79514 + 4.38032i −2.00000 + 3.46410i −8.40295 + 7.37498i −10.3821 − 0.460994i −14.0123 + 12.1101i −8.00000 −11.3744 − 24.4872i −21.1768 − 7.17936i
59.10 1.00000 + 1.73205i −2.72832 − 4.42225i −2.00000 + 3.46410i 9.11871 + 6.46909i 4.93124 − 9.14783i 18.2739 − 3.01087i −8.00000 −12.1126 + 24.1306i −2.08608 + 22.2632i
59.11 1.00000 + 1.73205i −0.638015 − 5.15683i −2.00000 + 3.46410i 7.67621 − 8.12870i 8.29388 − 6.26191i −12.1791 − 13.9524i −8.00000 −26.1859 + 6.58028i 21.7555 + 5.16688i
59.12 1.00000 + 1.73205i 0.198815 + 5.19235i −2.00000 + 3.46410i −1.56664 − 11.0700i −8.79460 + 5.53670i 0.387230 + 18.5162i −8.00000 −26.9209 + 2.06463i 17.6072 − 13.7835i
59.13 1.00000 + 1.73205i 0.460196 + 5.17573i −2.00000 + 3.46410i 5.98487 + 9.44358i −8.50444 + 5.97282i 18.3045 + 2.81859i −8.00000 −26.5764 + 4.76370i −10.3719 + 19.8097i
59.14 1.00000 + 1.73205i 0.533915 + 5.16865i −2.00000 + 3.46410i −1.79022 + 11.0361i −8.41845 + 6.09342i −7.57047 − 16.9023i −8.00000 −26.4299 + 5.51924i −20.9053 + 7.93533i
59.15 1.00000 + 1.73205i 1.54802 − 4.96021i −2.00000 + 3.46410i 8.05725 − 7.75117i 10.1393 − 2.27896i 4.70499 + 17.9127i −8.00000 −22.2073 − 15.3570i 21.4827 + 6.20440i
59.16 1.00000 + 1.73205i 1.59985 − 4.94373i −2.00000 + 3.46410i 0.717576 + 11.1573i 10.1626 − 2.17271i −12.5582 + 13.6121i −8.00000 −21.8810 − 15.8185i −18.6074 + 12.4002i
59.17 1.00000 + 1.73205i 2.39590 − 4.61082i −2.00000 + 3.46410i −10.5884 + 3.58968i 10.3821 − 0.460994i 14.0123 − 12.1101i −8.00000 −15.5193 − 22.0941i −16.8059 − 14.7500i
59.18 1.00000 + 1.73205i 3.83767 + 3.50319i −2.00000 + 3.46410i −11.1418 − 0.927147i −2.23003 + 10.1502i −13.9545 − 12.1767i −8.00000 2.45538 + 26.8881i −9.53597 − 20.2254i
59.19 1.00000 + 1.73205i 4.41659 + 2.73746i −2.00000 + 3.46410i 10.7911 + 2.92449i −0.324827 + 10.3872i −12.4849 + 13.6795i −8.00000 12.0126 + 24.1805i 5.72571 + 21.6152i
59.20 1.00000 + 1.73205i 4.57901 + 2.45615i −2.00000 + 3.46410i 2.86286 − 10.8076i 0.324827 + 10.3872i 12.4849 − 13.6795i −8.00000 14.9346 + 22.4935i 21.5822 − 5.84898i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.24
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 210.4.t.b yes 48
3.b odd 2 1 210.4.t.a ✓ 48
5.b even 2 1 210.4.t.a ✓ 48
7.d odd 6 1 inner 210.4.t.b yes 48
15.d odd 2 1 inner 210.4.t.b yes 48
21.g even 6 1 210.4.t.a ✓ 48
35.i odd 6 1 210.4.t.a ✓ 48
105.p even 6 1 inner 210.4.t.b yes 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.4.t.a ✓ 48 3.b odd 2 1
210.4.t.a ✓ 48 5.b even 2 1
210.4.t.a ✓ 48 21.g even 6 1
210.4.t.a ✓ 48 35.i odd 6 1
210.4.t.b yes 48 1.a even 1 1 trivial
210.4.t.b yes 48 7.d odd 6 1 inner
210.4.t.b yes 48 15.d odd 2 1 inner
210.4.t.b yes 48 105.p even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{24} - 38169 T_{17}^{22} + 956730201 T_{17}^{20} + 3399116616 T_{17}^{19} - 13529945555340 T_{17}^{18} + \cdots + 45\!\cdots\!00 \) acting on \(S_{4}^{\mathrm{new}}(210, [\chi])\). Copy content Toggle raw display