↑

Properties

Label 210.4.x.a
Level $210$
Weight $4$
Character orbit 210.x
Analytic conductor $12.390$
Analytic rank $0$
Dimension $192$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [210,4,Mod(23,210)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(210, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 9, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("210.23");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 210.x (of order \(12\), degree \(4\), 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: \(192\)
Relative dimension: \(48\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 192 q - 16 q^{6} - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 192 q - 16 q^{6} - 12 q^{7} + 24 q^{10} + 336 q^{15} + 1536 q^{16} + 16 q^{18} + 276 q^{21} - 48 q^{22} + 72 q^{25} - 504 q^{27} - 48 q^{28} + 168 q^{30} + 576 q^{31} + 640 q^{33} + 288 q^{36} - 216 q^{37} + 80 q^{42} + 2688 q^{43} - 448 q^{45} - 1008 q^{46} + 840 q^{51} - 2256 q^{55} - 3312 q^{57} + 1176 q^{58} - 352 q^{60} + 1272 q^{61} + 2876 q^{63} + 1008 q^{67} + 1608 q^{70} - 64 q^{72} - 1608 q^{73} - 5272 q^{75} + 2304 q^{76} - 2848 q^{78} + 1972 q^{81} - 288 q^{82} - 6768 q^{85} - 8 q^{87} + 96 q^{88} + 2800 q^{90} + 2976 q^{91} - 4008 q^{93} - 128 q^{96} - 5352 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −1.93185 + 0.517638i −5.19136 − 0.223217i 3.46410 − 2.00000i 2.78143 − 10.8288i 10.1445 − 2.25602i 10.7800 − 15.0596i −5.65685 + 5.65685i 26.9003 + 2.31760i 0.232100 + 22.3595i
23.2 −1.93185 + 0.517638i −5.05340 + 1.20963i 3.46410 − 2.00000i −3.75207 + 10.5319i 9.13626 − 4.95265i −10.6795 + 15.1310i −5.65685 + 5.65685i 24.0736 − 12.2254i 1.79671 − 22.2884i
23.3 −1.93185 + 0.517638i −5.04416 − 1.24758i 3.46410 − 2.00000i −10.7870 − 2.93934i 10.3904 − 0.200907i 10.8944 + 14.9771i −5.65685 + 5.65685i 23.8871 + 12.5860i 22.3605 + 0.0945827i
23.4 −1.93185 + 0.517638i −4.88797 − 1.76287i 3.46410 − 2.00000i 1.58073 + 11.0680i 10.3554 + 0.875411i −1.98473 − 18.4136i −5.65685 + 5.65685i 20.7845 + 17.2338i −8.78297 − 20.5635i
23.5 −1.93185 + 0.517638i −4.50336 + 2.59225i 3.46410 − 2.00000i 11.1400 − 0.948869i 7.35798 − 7.33895i −2.43768 + 18.3591i −5.65685 + 5.65685i 13.5605 − 23.3477i −21.0297 + 7.59956i
23.6 −1.93185 + 0.517638i −3.15320 + 4.13005i 3.46410 − 2.00000i −2.85804 − 10.8089i 3.95364 − 9.61086i −18.4889 − 1.07775i −5.65685 + 5.65685i −7.11466 − 26.0458i 11.1164 + 19.4017i
23.7 −1.93185 + 0.517638i −2.83671 − 4.35352i 3.46410 − 2.00000i 10.7403 + 3.10581i 7.73364 + 6.94196i 18.3462 + 2.53334i −5.65685 + 5.65685i −10.9062 + 24.6993i −22.3563 − 0.440378i
23.8 −1.93185 + 0.517638i −2.83411 + 4.35520i 3.46410 − 2.00000i −11.1451 + 0.886677i 3.22066 − 9.88065i 18.5114 − 0.571733i −5.65685 + 5.65685i −10.9356 − 24.6863i 21.0717 − 7.48207i
23.9 −1.93185 + 0.517638i −2.60909 − 4.49362i 3.46410 − 2.00000i −11.1613 + 0.652311i 7.36644 + 7.33045i −16.9437 − 7.47742i −5.65685 + 5.65685i −13.3853 + 23.4485i 21.2243 − 7.03768i
23.10 −1.93185 + 0.517638i −2.07435 + 4.76415i 3.46410 − 2.00000i 10.6423 + 3.42660i 1.54123 − 10.2774i −2.00328 − 18.4116i −5.65685 + 5.65685i −18.3942 − 19.7650i −22.3331 − 1.11082i
23.11 −1.93185 + 0.517638i −1.77750 − 4.88267i 3.46410 − 2.00000i −2.39757 − 10.9202i 5.96132 + 8.51250i 2.18097 + 18.3914i −5.65685 + 5.65685i −20.6810 + 17.3579i 10.2845 + 19.8552i
23.12 −1.93185 + 0.517638i 0.594359 + 5.16205i 3.46410 − 2.00000i −5.03931 + 9.98025i −3.82029 − 9.66465i −18.4260 + 1.86663i −5.65685 + 5.65685i −26.2935 + 6.13622i 4.56905 − 21.8889i
23.13 −1.93185 + 0.517638i 0.834437 − 5.12871i 3.46410 − 2.00000i 7.68158 − 8.12363i 1.04281 + 10.3399i −14.5705 − 11.4325i −5.65685 + 5.65685i −25.6074 − 8.55918i −10.6346 + 19.6699i
23.14 −1.93185 + 0.517638i 0.984834 − 5.10197i 3.46410 − 2.00000i −6.72942 + 8.92832i 0.738421 + 10.3660i 16.7118 − 7.98220i −5.65685 + 5.65685i −25.0602 − 10.0492i 8.37859 − 20.7316i
23.15 −1.93185 + 0.517638i 1.73537 + 4.89780i 3.46410 − 2.00000i 8.08159 + 7.72580i −5.88777 − 8.56353i 12.6291 + 13.5465i −5.65685 + 5.65685i −20.9770 + 16.9990i −19.6116 − 10.7418i
23.16 −1.93185 + 0.517638i 2.06931 − 4.76634i 3.46410 − 2.00000i 6.11077 + 9.36261i −1.53035 + 10.2790i −11.0378 + 14.8717i −5.65685 + 5.65685i −18.4360 − 19.7260i −16.6515 − 14.9240i
23.17 −1.93185 + 0.517638i 2.25671 + 4.68052i 3.46410 − 2.00000i −10.3588 − 4.20646i −6.78244 − 7.87392i 7.77698 − 16.8083i −5.65685 + 5.65685i −16.8146 + 21.1251i 22.1892 + 2.76412i
23.18 −1.93185 + 0.517638i 4.23915 − 3.00493i 3.46410 − 2.00000i 5.93324 − 9.47611i −6.63395 + 7.99942i 9.89205 + 15.6572i −5.65685 + 5.65685i 8.94085 − 25.4767i −6.55694 + 21.3777i
23.19 −1.93185 + 0.517638i 4.24351 + 2.99876i 3.46410 − 2.00000i 7.21425 − 8.54135i −9.75012 − 3.59656i 14.8312 − 11.0921i −5.65685 + 5.65685i 9.01484 + 25.4506i −9.51553 + 20.2350i
23.20 −1.93185 + 0.517638i 4.27861 − 2.94848i 3.46410 − 2.00000i −8.08113 − 7.72627i −6.73939 + 7.91079i 1.18666 − 18.4822i −5.65685 + 5.65685i 9.61295 − 25.2308i 19.6110 + 10.7429i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
7.c even 3 1 inner
15.e even 4 1 inner
21.h odd 6 1 inner
35.l odd 12 1 inner
105.x even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.4.x.a ✓ 192
3.b odd 2 1 inner 210.4.x.a ✓ 192
5.c odd 4 1 inner 210.4.x.a ✓ 192
7.c even 3 1 inner 210.4.x.a ✓ 192
15.e even 4 1 inner 210.4.x.a ✓ 192
21.h odd 6 1 inner 210.4.x.a ✓ 192
35.l odd 12 1 inner 210.4.x.a ✓ 192
105.x even 12 1 inner 210.4.x.a ✓ 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.4.x.a ✓ 192 1.a even 1 1 trivial
210.4.x.a ✓ 192 3.b odd 2 1 inner
210.4.x.a ✓ 192 5.c odd 4 1 inner
210.4.x.a ✓ 192 7.c even 3 1 inner
210.4.x.a ✓ 192 15.e even 4 1 inner
210.4.x.a ✓ 192 21.h odd 6 1 inner
210.4.x.a ✓ 192 35.l odd 12 1 inner
210.4.x.a ✓ 192 105.x even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(210, [\chi])\).