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Properties

Label 210.4.u.a
Level $210$
Weight $4$
Character orbit 210.u
Analytic conductor $12.390$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [210,4,Mod(73,210)]
 
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([0, 9, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("210.73");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 210.u (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: \(48\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 48 q^{5} - 36 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 48 q^{5} - 36 q^{7} + 64 q^{10} - 28 q^{11} + 112 q^{13} - 128 q^{14} + 84 q^{15} + 384 q^{16} - 204 q^{17} + 104 q^{19} - 32 q^{20} - 96 q^{21} - 312 q^{22} - 464 q^{23} - 576 q^{24} - 884 q^{25} - 456 q^{26} - 224 q^{28} - 24 q^{30} + 528 q^{31} - 216 q^{33} - 448 q^{34} + 1256 q^{35} - 1728 q^{36} + 444 q^{37} + 296 q^{38} - 684 q^{39} + 192 q^{42} - 832 q^{43} + 36 q^{45} - 392 q^{46} - 1548 q^{47} - 436 q^{49} - 336 q^{51} + 224 q^{52} + 412 q^{53} - 1296 q^{54} + 1736 q^{55} + 64 q^{56} - 696 q^{57} - 3256 q^{58} + 296 q^{59} + 2592 q^{61} - 304 q^{62} + 792 q^{63} + 236 q^{65} + 652 q^{67} - 816 q^{68} - 1176 q^{69} - 296 q^{70} - 496 q^{71} + 1300 q^{73} - 528 q^{74} + 936 q^{75} + 5392 q^{77} + 672 q^{78} + 11724 q^{79} - 768 q^{80} + 1944 q^{81} - 96 q^{82} + 6272 q^{83} + 48 q^{84} + 128 q^{85} + 176 q^{86} - 3576 q^{87} - 576 q^{88} + 4448 q^{89} - 5856 q^{91} - 1184 q^{92} - 3360 q^{93} + 16 q^{94} + 4572 q^{95} - 4652 q^{97} + 2256 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1 −1.93185 − 0.517638i −0.776457 − 2.89778i 3.46410 + 2.00000i −10.4494 + 3.97626i 6.00000i 1.73803 − 18.4385i −5.65685 − 5.65685i −7.79423 + 4.50000i 22.2449 − 2.27255i
73.2 −1.93185 − 0.517638i −0.776457 − 2.89778i 3.46410 + 2.00000i −7.34820 + 8.42638i 6.00000i −9.11575 + 16.1215i −5.65685 − 5.65685i −7.79423 + 4.50000i 18.5575 − 12.4748i
73.3 −1.93185 − 0.517638i −0.776457 − 2.89778i 3.46410 + 2.00000i −5.14619 − 9.92556i 6.00000i −17.1582 − 6.97110i −5.65685 − 5.65685i −7.79423 + 4.50000i 4.80383 + 21.8386i
73.4 −1.93185 − 0.517638i −0.776457 − 2.89778i 3.46410 + 2.00000i −3.79550 − 10.5164i 6.00000i 11.2035 + 14.7472i −5.65685 − 5.65685i −7.79423 + 4.50000i 1.88867 + 22.2808i
73.5 −1.93185 − 0.517638i −0.776457 − 2.89778i 3.46410 + 2.00000i 7.04113 + 8.68461i 6.00000i 16.5272 + 8.35764i −5.65685 − 5.65685i −7.79423 + 4.50000i −9.10694 − 20.4221i
73.6 −1.93185 − 0.517638i −0.776457 − 2.89778i 3.46410 + 2.00000i 9.41669 + 6.02709i 6.00000i −11.0668 − 14.8501i −5.65685 − 5.65685i −7.79423 + 4.50000i −15.0718 − 16.5179i
73.7 1.93185 + 0.517638i 0.776457 + 2.89778i 3.46410 + 2.00000i −10.7870 + 2.93963i 6.00000i −17.5314 + 5.97088i 5.65685 + 5.65685i −7.79423 + 4.50000i −22.3605 + 0.0951823i
73.8 1.93185 + 0.517638i 0.776457 + 2.89778i 3.46410 + 2.00000i −5.82019 + 9.54596i 6.00000i −10.6715 − 15.1367i 5.65685 + 5.65685i −7.79423 + 4.50000i −16.1851 + 15.4286i
73.9 1.93185 + 0.517638i 0.776457 + 2.89778i 3.46410 + 2.00000i −3.50106 − 10.6180i 6.00000i −12.4475 + 13.7135i 5.65685 + 5.65685i −7.79423 + 4.50000i −1.26722 − 22.3247i
73.10 1.93185 + 0.517638i 0.776457 + 2.89778i 3.46410 + 2.00000i 2.10469 − 10.9804i 6.00000i −2.05221 − 18.4062i 5.65685 + 5.65685i −7.79423 + 4.50000i 9.74985 − 20.1231i
73.11 1.93185 + 0.517638i 0.776457 + 2.89778i 3.46410 + 2.00000i 3.37632 + 10.6584i 6.00000i 18.3072 − 2.80098i 5.65685 + 5.65685i −7.79423 + 4.50000i 1.00538 + 22.3381i
73.12 1.93185 + 0.517638i 0.776457 + 2.89778i 3.46410 + 2.00000i 11.1766 − 0.289668i 6.00000i 2.48280 + 18.3531i 5.65685 + 5.65685i −7.79423 + 4.50000i 21.7415 + 5.22583i
103.1 −0.517638 − 1.93185i −2.89778 − 0.776457i −3.46410 + 2.00000i −11.0455 − 1.73102i 6.00000i 18.4708 + 1.35210i 5.65685 + 5.65685i 7.79423 + 4.50000i 2.37350 + 22.2344i
103.2 −0.517638 − 1.93185i −2.89778 − 0.776457i −3.46410 + 2.00000i −7.07673 + 8.65563i 6.00000i −6.72410 + 17.2565i 5.65685 + 5.65685i 7.79423 + 4.50000i 20.3846 + 9.19070i
103.3 −0.517638 − 1.93185i −2.89778 − 0.776457i −3.46410 + 2.00000i −6.33815 − 9.21021i 6.00000i −16.2530 − 8.87913i 5.65685 + 5.65685i 7.79423 + 4.50000i −14.5119 + 17.0119i
103.4 −0.517638 − 1.93185i −2.89778 − 0.776457i −3.46410 + 2.00000i 2.74303 − 10.8386i 6.00000i 18.5028 + 0.803538i 5.65685 + 5.65685i 7.79423 + 4.50000i −22.3585 + 0.311351i
103.5 −0.517638 − 1.93185i −2.89778 − 0.776457i −3.46410 + 2.00000i 3.05651 + 10.7544i 6.00000i 9.34209 − 15.9914i 5.65685 + 5.65685i 7.79423 + 4.50000i 19.1938 − 11.4716i
103.6 −0.517638 − 1.93185i −2.89778 − 0.776457i −3.46410 + 2.00000i 10.8186 + 2.82111i 6.00000i −14.6686 − 11.3062i 5.65685 + 5.65685i 7.79423 + 4.50000i −0.150143 − 22.3602i
103.7 0.517638 + 1.93185i 2.89778 + 0.776457i −3.46410 + 2.00000i −9.27185 − 6.24763i 6.00000i 14.3278 − 11.7352i −5.65685 − 5.65685i 7.79423 + 4.50000i 7.27003 − 21.1458i
103.8 0.517638 + 1.93185i 2.89778 + 0.776457i −3.46410 + 2.00000i −5.84803 + 9.52893i 6.00000i −17.0181 + 7.30642i −5.65685 − 5.65685i 7.79423 + 4.50000i −21.4356 − 6.36499i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.k 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.u.a ✓ 48
5.c odd 4 1 210.4.u.b yes 48
7.d odd 6 1 210.4.u.b yes 48
35.k even 12 1 inner 210.4.u.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.4.u.a ✓ 48 1.a even 1 1 trivial
210.4.u.a ✓ 48 35.k even 12 1 inner
210.4.u.b yes 48 5.c odd 4 1
210.4.u.b yes 48 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{48} - 112 T_{13}^{47} + 6272 T_{13}^{46} - 387608 T_{13}^{45} + 156225578 T_{13}^{44} + \cdots + 16\!\cdots\!04 \) acting on \(S_{4}^{\mathrm{new}}(210, [\chi])\). Copy content Toggle raw display