Properties

Label 210.4.i.e
Level $210$
Weight $4$
Character orbit 210.i
Analytic conductor $12.390$
Analytic rank $0$
Dimension $2$
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(121,210)]
 
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([0, 0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("210.121");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 210.i (of order \(3\), 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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \zeta_{6} q^{2} + (3 \zeta_{6} - 3) q^{3} + (4 \zeta_{6} - 4) q^{4} - 5 \zeta_{6} q^{5} - 6 q^{6} + ( - 21 \zeta_{6} + 14) q^{7} - 8 q^{8} - 9 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 2 \zeta_{6} q^{2} + (3 \zeta_{6} - 3) q^{3} + (4 \zeta_{6} - 4) q^{4} - 5 \zeta_{6} q^{5} - 6 q^{6} + ( - 21 \zeta_{6} + 14) q^{7} - 8 q^{8} - 9 \zeta_{6} q^{9} + ( - 10 \zeta_{6} + 10) q^{10} + ( - 32 \zeta_{6} + 32) q^{11} - 12 \zeta_{6} q^{12} + 15 q^{13} + ( - 14 \zeta_{6} + 42) q^{14} + 15 q^{15} - 16 \zeta_{6} q^{16} + ( - 70 \zeta_{6} + 70) q^{17} + ( - 18 \zeta_{6} + 18) q^{18} - 15 \zeta_{6} q^{19} + 20 q^{20} + (42 \zeta_{6} + 21) q^{21} + 64 q^{22} + 42 \zeta_{6} q^{23} + ( - 24 \zeta_{6} + 24) q^{24} + (25 \zeta_{6} - 25) q^{25} + 30 \zeta_{6} q^{26} + 27 q^{27} + (56 \zeta_{6} + 28) q^{28} + 90 q^{29} + 30 \zeta_{6} q^{30} + ( - 85 \zeta_{6} + 85) q^{31} + ( - 32 \zeta_{6} + 32) q^{32} + 96 \zeta_{6} q^{33} + 140 q^{34} + (35 \zeta_{6} - 105) q^{35} + 36 q^{36} - 113 \zeta_{6} q^{37} + ( - 30 \zeta_{6} + 30) q^{38} + (45 \zeta_{6} - 45) q^{39} + 40 \zeta_{6} q^{40} + 164 q^{41} + (126 \zeta_{6} - 84) q^{42} + 169 q^{43} + 128 \zeta_{6} q^{44} + (45 \zeta_{6} - 45) q^{45} + (84 \zeta_{6} - 84) q^{46} - 326 \zeta_{6} q^{47} + 48 q^{48} + ( - 147 \zeta_{6} - 245) q^{49} - 50 q^{50} + 210 \zeta_{6} q^{51} + (60 \zeta_{6} - 60) q^{52} + ( - 44 \zeta_{6} + 44) q^{53} + 54 \zeta_{6} q^{54} - 160 q^{55} + (168 \zeta_{6} - 112) q^{56} + 45 q^{57} + 180 \zeta_{6} q^{58} + ( - 782 \zeta_{6} + 782) q^{59} + (60 \zeta_{6} - 60) q^{60} - 658 \zeta_{6} q^{61} + 170 q^{62} + (63 \zeta_{6} - 189) q^{63} + 64 q^{64} - 75 \zeta_{6} q^{65} + (192 \zeta_{6} - 192) q^{66} + (1071 \zeta_{6} - 1071) q^{67} + 280 \zeta_{6} q^{68} - 126 q^{69} + ( - 140 \zeta_{6} - 70) q^{70} + 344 q^{71} + 72 \zeta_{6} q^{72} + (431 \zeta_{6} - 431) q^{73} + ( - 226 \zeta_{6} + 226) q^{74} - 75 \zeta_{6} q^{75} + 60 q^{76} + ( - 448 \zeta_{6} - 224) q^{77} - 90 q^{78} - 397 \zeta_{6} q^{79} + (80 \zeta_{6} - 80) q^{80} + (81 \zeta_{6} - 81) q^{81} + 328 \zeta_{6} q^{82} + 680 q^{83} + (84 \zeta_{6} - 252) q^{84} - 350 q^{85} + 338 \zeta_{6} q^{86} + (270 \zeta_{6} - 270) q^{87} + (256 \zeta_{6} - 256) q^{88} - 1534 \zeta_{6} q^{89} - 90 q^{90} + ( - 315 \zeta_{6} + 210) q^{91} - 168 q^{92} + 255 \zeta_{6} q^{93} + ( - 652 \zeta_{6} + 652) q^{94} + (75 \zeta_{6} - 75) q^{95} + 96 \zeta_{6} q^{96} - 1234 q^{97} + ( - 784 \zeta_{6} + 294) q^{98} - 288 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 3 q^{3} - 4 q^{4} - 5 q^{5} - 12 q^{6} + 7 q^{7} - 16 q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 3 q^{3} - 4 q^{4} - 5 q^{5} - 12 q^{6} + 7 q^{7} - 16 q^{8} - 9 q^{9} + 10 q^{10} + 32 q^{11} - 12 q^{12} + 30 q^{13} + 70 q^{14} + 30 q^{15} - 16 q^{16} + 70 q^{17} + 18 q^{18} - 15 q^{19} + 40 q^{20} + 84 q^{21} + 128 q^{22} + 42 q^{23} + 24 q^{24} - 25 q^{25} + 30 q^{26} + 54 q^{27} + 112 q^{28} + 180 q^{29} + 30 q^{30} + 85 q^{31} + 32 q^{32} + 96 q^{33} + 280 q^{34} - 175 q^{35} + 72 q^{36} - 113 q^{37} + 30 q^{38} - 45 q^{39} + 40 q^{40} + 328 q^{41} - 42 q^{42} + 338 q^{43} + 128 q^{44} - 45 q^{45} - 84 q^{46} - 326 q^{47} + 96 q^{48} - 637 q^{49} - 100 q^{50} + 210 q^{51} - 60 q^{52} + 44 q^{53} + 54 q^{54} - 320 q^{55} - 56 q^{56} + 90 q^{57} + 180 q^{58} + 782 q^{59} - 60 q^{60} - 658 q^{61} + 340 q^{62} - 315 q^{63} + 128 q^{64} - 75 q^{65} - 192 q^{66} - 1071 q^{67} + 280 q^{68} - 252 q^{69} - 280 q^{70} + 688 q^{71} + 72 q^{72} - 431 q^{73} + 226 q^{74} - 75 q^{75} + 120 q^{76} - 896 q^{77} - 180 q^{78} - 397 q^{79} - 80 q^{80} - 81 q^{81} + 328 q^{82} + 1360 q^{83} - 420 q^{84} - 700 q^{85} + 338 q^{86} - 270 q^{87} - 256 q^{88} - 1534 q^{89} - 180 q^{90} + 105 q^{91} - 336 q^{92} + 255 q^{93} + 652 q^{94} - 75 q^{95} + 96 q^{96} - 2468 q^{97} - 196 q^{98} - 576 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(31\) \(71\) \(127\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
0.500000 0.866025i
0.500000 + 0.866025i
1.00000 1.73205i −1.50000 2.59808i −2.00000 3.46410i −2.50000 + 4.33013i −6.00000 3.50000 + 18.1865i −8.00000 −4.50000 + 7.79423i 5.00000 + 8.66025i
151.1 1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i −2.50000 4.33013i −6.00000 3.50000 18.1865i −8.00000 −4.50000 7.79423i 5.00000 8.66025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 210.4.i.e 2
3.b odd 2 1 630.4.k.d 2
7.c even 3 1 inner 210.4.i.e 2
7.c even 3 1 1470.4.a.l 1
7.d odd 6 1 1470.4.a.b 1
21.h odd 6 1 630.4.k.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.4.i.e 2 1.a even 1 1 trivial
210.4.i.e 2 7.c even 3 1 inner
630.4.k.d 2 3.b odd 2 1
630.4.k.d 2 21.h odd 6 1
1470.4.a.b 1 7.d odd 6 1
1470.4.a.l 1 7.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{2} - 32T_{11} + 1024 \) acting on \(S_{4}^{\mathrm{new}}(210, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$7$ \( T^{2} - 7T + 343 \) Copy content Toggle raw display
$11$ \( T^{2} - 32T + 1024 \) Copy content Toggle raw display
$13$ \( (T - 15)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 70T + 4900 \) Copy content Toggle raw display
$19$ \( T^{2} + 15T + 225 \) Copy content Toggle raw display
$23$ \( T^{2} - 42T + 1764 \) Copy content Toggle raw display
$29$ \( (T - 90)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 85T + 7225 \) Copy content Toggle raw display
$37$ \( T^{2} + 113T + 12769 \) Copy content Toggle raw display
$41$ \( (T - 164)^{2} \) Copy content Toggle raw display
$43$ \( (T - 169)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 326T + 106276 \) Copy content Toggle raw display
$53$ \( T^{2} - 44T + 1936 \) Copy content Toggle raw display
$59$ \( T^{2} - 782T + 611524 \) Copy content Toggle raw display
$61$ \( T^{2} + 658T + 432964 \) Copy content Toggle raw display
$67$ \( T^{2} + 1071 T + 1147041 \) Copy content Toggle raw display
$71$ \( (T - 344)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 431T + 185761 \) Copy content Toggle raw display
$79$ \( T^{2} + 397T + 157609 \) Copy content Toggle raw display
$83$ \( (T - 680)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 1534 T + 2353156 \) Copy content Toggle raw display
$97$ \( (T + 1234)^{2} \) Copy content Toggle raw display
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