Properties

Label 210.4.i.j
Level $210$
Weight $4$
Character orbit 210.i
Analytic conductor $12.390$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{2} \)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{2} + 2) q^{2} + 3 \beta_{2} q^{3} - 4 \beta_{2} q^{4} + ( - 5 \beta_{2} + 5) q^{5} + 6 q^{6} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 - 6) q^{7} - 8 q^{8} + (9 \beta_{2} - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{2} + 2) q^{2} + 3 \beta_{2} q^{3} - 4 \beta_{2} q^{4} + ( - 5 \beta_{2} + 5) q^{5} + 6 q^{6} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 - 6) q^{7} - 8 q^{8} + (9 \beta_{2} - 9) q^{9} - 10 \beta_{2} q^{10} + ( - 3 \beta_{3} + 9 \beta_{2} - 3 \beta_1) q^{11} + ( - 12 \beta_{2} + 12) q^{12} + (3 \beta_{3} - 6 \beta_1 + 32) q^{13} + (2 \beta_{3} + 12 \beta_{2} - 6 \beta_1 - 8) q^{14} + 15 q^{15} + (16 \beta_{2} - 16) q^{16} + ( - 3 \beta_{3} - 51 \beta_{2} - 3 \beta_1) q^{17} + 18 \beta_{2} q^{18} + (8 \beta_{3} - 31 \beta_{2} - 4 \beta_1 + 31) q^{19} - 20 q^{20} + ( - 9 \beta_{3} - 12 \beta_{2} + 6 \beta_1 - 6) q^{21} + (6 \beta_{3} - 12 \beta_1 + 18) q^{22} + ( - 14 \beta_{3} + 21 \beta_{2} + 7 \beta_1 - 21) q^{23} - 24 \beta_{2} q^{24} - 25 \beta_{2} q^{25} + (12 \beta_{3} - 64 \beta_{2} - 6 \beta_1 + 64) q^{26} - 27 q^{27} + (12 \beta_{3} + 16 \beta_{2} - 8 \beta_1 + 8) q^{28} + (7 \beta_{3} - 14 \beta_1 + 111) q^{29} + ( - 30 \beta_{2} + 30) q^{30} + ( - 24 \beta_{3} + 25 \beta_{2} - 24 \beta_1) q^{31} + 32 \beta_{2} q^{32} + ( - 18 \beta_{3} + 27 \beta_{2} + 9 \beta_1 - 27) q^{33} + (6 \beta_{3} - 12 \beta_1 - 102) q^{34} + (5 \beta_{3} + 30 \beta_{2} - 15 \beta_1 - 20) q^{35} + 36 q^{36} + (14 \beta_{3} - 286 \beta_{2} - 7 \beta_1 + 286) q^{37} + (8 \beta_{3} - 62 \beta_{2} + 8 \beta_1) q^{38} + ( - 9 \beta_{3} + 96 \beta_{2} - 9 \beta_1) q^{39} + (40 \beta_{2} - 40) q^{40} + ( - 3 \beta_{3} + 6 \beta_1 + 27) q^{41} + ( - 12 \beta_{3} + 12 \beta_{2} - 6 \beta_1 - 36) q^{42} + ( - 5 \beta_{3} + 10 \beta_1 - 124) q^{43} + (24 \beta_{3} - 36 \beta_{2} - 12 \beta_1 + 36) q^{44} + 45 \beta_{2} q^{45} + ( - 14 \beta_{3} + 42 \beta_{2} - 14 \beta_1) q^{46} + ( - 8 \beta_{3} + 222 \beta_{2} + 4 \beta_1 - 222) q^{47} - 48 q^{48} + (12 \beta_{3} + 205 \beta_{2} + 20 \beta_1 - 328) q^{49} - 50 q^{50} + ( - 18 \beta_{3} - 153 \beta_{2} + 9 \beta_1 + 153) q^{51} + (12 \beta_{3} - 128 \beta_{2} + 12 \beta_1) q^{52} + (34 \beta_{3} + 6 \beta_{2} + 34 \beta_1) q^{53} + (54 \beta_{2} - 54) q^{54} + (15 \beta_{3} - 30 \beta_1 + 45) q^{55} + (16 \beta_{3} - 16 \beta_{2} + 8 \beta_1 + 48) q^{56} + (12 \beta_{3} - 24 \beta_1 + 93) q^{57} + (28 \beta_{3} - 222 \beta_{2} - 14 \beta_1 + 222) q^{58} + (45 \beta_{3} - 129 \beta_{2} + 45 \beta_1) q^{59} - 60 \beta_{2} q^{60} + ( - 16 \beta_{3} + 32 \beta_{2} + 8 \beta_1 - 32) q^{61} + (48 \beta_{3} - 96 \beta_1 + 50) q^{62} + ( - 9 \beta_{3} - 54 \beta_{2} + 27 \beta_1 + 36) q^{63} + 64 q^{64} + (30 \beta_{3} - 160 \beta_{2} - 15 \beta_1 + 160) q^{65} + ( - 18 \beta_{3} + 54 \beta_{2} - 18 \beta_1) q^{66} + ( - 37 \beta_{3} + 316 \beta_{2} - 37 \beta_1) q^{67} + (24 \beta_{3} + 204 \beta_{2} - 12 \beta_1 - 204) q^{68} + ( - 21 \beta_{3} + 42 \beta_1 - 63) q^{69} + (30 \beta_{3} + 40 \beta_{2} - 20 \beta_1 + 20) q^{70} + ( - 45 \beta_{3} + 90 \beta_1 - 39) q^{71} + ( - 72 \beta_{2} + 72) q^{72} + ( - 15 \beta_{3} + 232 \beta_{2} - 15 \beta_1) q^{73} + (14 \beta_{3} - 572 \beta_{2} + 14 \beta_1) q^{74} + ( - 75 \beta_{2} + 75) q^{75} + ( - 16 \beta_{3} + 32 \beta_1 - 124) q^{76} + ( - 21 \beta_{3} + 504 \beta_{2} + 42 \beta_1 - 693) q^{77} + (18 \beta_{3} - 36 \beta_1 + 192) q^{78} + ( - 8 \beta_{3} - 973 \beta_{2} + 4 \beta_1 + 973) q^{79} + 80 \beta_{2} q^{80} - 81 \beta_{2} q^{81} + ( - 12 \beta_{3} - 54 \beta_{2} + 6 \beta_1 + 54) q^{82} + ( - 85 \beta_{3} + 170 \beta_1 - 117) q^{83} + (12 \beta_{3} + 72 \beta_{2} - 36 \beta_1 - 48) q^{84} + (15 \beta_{3} - 30 \beta_1 - 255) q^{85} + ( - 20 \beta_{3} + 248 \beta_{2} + 10 \beta_1 - 248) q^{86} + ( - 21 \beta_{3} + 333 \beta_{2} - 21 \beta_1) q^{87} + (24 \beta_{3} - 72 \beta_{2} + 24 \beta_1) q^{88} + (22 \beta_{3} + 837 \beta_{2} - 11 \beta_1 - 837) q^{89} + 90 q^{90} + ( - 88 \beta_{3} + 739 \beta_{2} - 2 \beta_1 - 327) q^{91} + (28 \beta_{3} - 56 \beta_1 + 84) q^{92} + ( - 144 \beta_{3} + 75 \beta_{2} + 72 \beta_1 - 75) q^{93} + ( - 8 \beta_{3} + 444 \beta_{2} - 8 \beta_1) q^{94} + (20 \beta_{3} - 155 \beta_{2} + 20 \beta_1) q^{95} + (96 \beta_{2} - 96) q^{96} + ( - 40 \beta_{3} + 80 \beta_1 + 416) q^{97} + ( - 40 \beta_{3} + 656 \beta_{2} + 64 \beta_1 - 246) q^{98} + ( - 27 \beta_{3} + 54 \beta_1 - 81) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 6 q^{3} - 8 q^{4} + 10 q^{5} + 24 q^{6} - 20 q^{7} - 32 q^{8} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 6 q^{3} - 8 q^{4} + 10 q^{5} + 24 q^{6} - 20 q^{7} - 32 q^{8} - 18 q^{9} - 20 q^{10} + 18 q^{11} + 24 q^{12} + 128 q^{13} - 8 q^{14} + 60 q^{15} - 32 q^{16} - 102 q^{17} + 36 q^{18} + 62 q^{19} - 80 q^{20} - 48 q^{21} + 72 q^{22} - 42 q^{23} - 48 q^{24} - 50 q^{25} + 128 q^{26} - 108 q^{27} + 64 q^{28} + 444 q^{29} + 60 q^{30} + 50 q^{31} + 64 q^{32} - 54 q^{33} - 408 q^{34} - 20 q^{35} + 144 q^{36} + 572 q^{37} - 124 q^{38} + 192 q^{39} - 80 q^{40} + 108 q^{41} - 120 q^{42} - 496 q^{43} + 72 q^{44} + 90 q^{45} + 84 q^{46} - 444 q^{47} - 192 q^{48} - 902 q^{49} - 200 q^{50} + 306 q^{51} - 256 q^{52} + 12 q^{53} - 108 q^{54} + 180 q^{55} + 160 q^{56} + 372 q^{57} + 444 q^{58} - 258 q^{59} - 120 q^{60} - 64 q^{61} + 200 q^{62} + 36 q^{63} + 256 q^{64} + 320 q^{65} + 108 q^{66} + 632 q^{67} - 408 q^{68} - 252 q^{69} + 160 q^{70} - 156 q^{71} + 144 q^{72} + 464 q^{73} - 1144 q^{74} + 150 q^{75} - 496 q^{76} - 1764 q^{77} + 768 q^{78} + 1946 q^{79} + 160 q^{80} - 162 q^{81} + 108 q^{82} - 468 q^{83} - 48 q^{84} - 1020 q^{85} - 496 q^{86} + 666 q^{87} - 144 q^{88} - 1674 q^{89} + 360 q^{90} + 170 q^{91} + 336 q^{92} - 150 q^{93} + 888 q^{94} - 310 q^{95} - 192 q^{96} + 1664 q^{97} + 328 q^{98} - 324 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 3\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{3} ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{3} ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(31\) \(71\) \(127\)
\(\chi(n)\) \(-1 + \beta_{2}\) \(1\) \(1\)

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
1.93649 + 1.11803i
−1.93649 1.11803i
1.93649 1.11803i
−1.93649 + 1.11803i
1.00000 1.73205i 1.50000 + 2.59808i −2.00000 3.46410i 2.50000 4.33013i 6.00000 −10.8095 15.0385i −8.00000 −4.50000 + 7.79423i −5.00000 8.66025i
121.2 1.00000 1.73205i 1.50000 + 2.59808i −2.00000 3.46410i 2.50000 4.33013i 6.00000 0.809475 + 18.5026i −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 −10.8095 + 15.0385i −8.00000 −4.50000 7.79423i −5.00000 + 8.66025i
151.2 1.00000 + 1.73205i 1.50000 2.59808i −2.00000 + 3.46410i 2.50000 + 4.33013i 6.00000 0.809475 18.5026i −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.j 4
3.b odd 2 1 630.4.k.k 4
7.c even 3 1 inner 210.4.i.j 4
7.c even 3 1 1470.4.a.be 2
7.d odd 6 1 1470.4.a.bj 2
21.h odd 6 1 630.4.k.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.4.i.j 4 1.a even 1 1 trivial
210.4.i.j 4 7.c even 3 1 inner
630.4.k.k 4 3.b odd 2 1
630.4.k.k 4 21.h odd 6 1
1470.4.a.be 2 7.c even 3 1
1470.4.a.bj 2 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{4} - 18T_{11}^{3} + 1458T_{11}^{2} + 20412T_{11} + 1285956 \) acting on \(S_{4}^{\mathrm{new}}(210, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 20 T^{3} + 651 T^{2} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{4} - 18 T^{3} + 1458 T^{2} + \cdots + 1285956 \) Copy content Toggle raw display
$13$ \( (T^{2} - 64 T - 191)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 102 T^{3} + 9018 T^{2} + \cdots + 1920996 \) Copy content Toggle raw display
$19$ \( T^{4} - 62 T^{3} + 5043 T^{2} + \cdots + 1437601 \) Copy content Toggle raw display
$23$ \( T^{4} + 42 T^{3} + 7938 T^{2} + \cdots + 38118276 \) Copy content Toggle raw display
$29$ \( (T^{2} - 222 T + 5706)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 50 T^{3} + \cdots + 5949808225 \) Copy content Toggle raw display
$37$ \( T^{4} - 572 T^{3} + \cdots + 5652182761 \) Copy content Toggle raw display
$41$ \( (T^{2} - 54 T - 486)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 248 T + 12001)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 444 T^{3} + \cdots + 2220671376 \) Copy content Toggle raw display
$53$ \( T^{4} - 12 T^{3} + \cdots + 24343488576 \) Copy content Toggle raw display
$59$ \( T^{4} + 258 T^{3} + \cdots + 65912346756 \) Copy content Toggle raw display
$61$ \( T^{4} + 64 T^{3} + 11712 T^{2} + \cdots + 58003456 \) Copy content Toggle raw display
$67$ \( T^{4} - 632 T^{3} + \cdots + 7218031681 \) Copy content Toggle raw display
$71$ \( (T^{2} + 78 T - 271854)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 464 T^{3} + \cdots + 549855601 \) Copy content Toggle raw display
$79$ \( T^{4} - 1946 T^{3} + \cdots + 892210595761 \) Copy content Toggle raw display
$83$ \( (T^{2} + 234 T - 961686)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 1674 T^{3} + \cdots + 468176166756 \) Copy content Toggle raw display
$97$ \( (T^{2} - 832 T - 42944)^{2} \) Copy content Toggle raw display
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