Properties

Label 1890.2.l.h
Level $1890$
Weight $2$
Character orbit 1890.l
Analytic conductor $15.092$
Analytic rank $0$
Dimension $12$
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] = [1890,2,Mod(361,1890)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1890, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1890.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1890 = 2 \cdot 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1890.l (of order \(3\), degree \(2\), not 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: \(15.0917259820\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5 x^{11} + 14 x^{10} - 28 x^{9} + 36 x^{8} - 24 x^{7} + 33 x^{6} + 42 x^{5} + 114 x^{4} + 104 x^{3} + 197 x^{2} + 166 x + 79 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{5} \)
Twist minimal: no (minimal twist has level 630)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + 1) q^{2} + \beta_{3} q^{4} + q^{5} + ( - \beta_{9} - \beta_{7} - \beta_{5} - \beta_{3} - 1) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 1) q^{2} + \beta_{3} q^{4} + q^{5} + ( - \beta_{9} - \beta_{7} - \beta_{5} - \beta_{3} - 1) q^{7} - q^{8} + (\beta_{3} + 1) q^{10} + (\beta_{8} + \beta_{5} - \beta_{4} - \beta_1 - 2) q^{11} + (\beta_{7} + \beta_{6} + \beta_{3} + \beta_{2} + 1) q^{13} + ( - \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3}) q^{14} + ( - \beta_{3} - 1) q^{16} + ( - \beta_{7} - \beta_{6} + 2 \beta_{4} + \beta_{2} + \beta_1) q^{17} + (\beta_{11} + \beta_{10} - \beta_{9} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 1) q^{19} + \beta_{3} q^{20} + ( - \beta_{4} - 2 \beta_{3} - \beta_1 - 2) q^{22} + ( - 2 \beta_{9} - \beta_{8} - 2 \beta_{7} - 2 \beta_{5} + 2 \beta_{4} + \beta_1 - 1) q^{23} + q^{25} + ( - \beta_{11} + \beta_{10} - \beta_{9} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 1) q^{26} + (\beta_{9} + \beta_{4} + 1) q^{28} + ( - \beta_{11} - \beta_{9} - \beta_{8} - \beta_{5} - 2 \beta_{3}) q^{29} + (2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + \beta_{8} + \beta_{5} + 2 \beta_{4} + 2 \beta_{2} + \cdots + 2) q^{31}+ \cdots + ( - 3 \beta_{9} - 3 \beta_{7} + 2 \beta_{5} - \beta_{4} - \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{2} - 6 q^{4} + 12 q^{5} - 4 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{2} - 6 q^{4} + 12 q^{5} - 4 q^{7} - 12 q^{8} + 6 q^{10} - 14 q^{11} + 2 q^{13} + 4 q^{14} - 6 q^{16} - 7 q^{17} + 14 q^{19} - 6 q^{20} - 7 q^{22} - 18 q^{23} + 12 q^{25} - 2 q^{26} + 8 q^{28} + 9 q^{29} + 9 q^{31} + 6 q^{32} + 7 q^{34} - 4 q^{35} - 12 q^{37} + 28 q^{38} - 12 q^{40} - q^{41} + 7 q^{43} + 7 q^{44} - 9 q^{46} - 7 q^{47} + 6 q^{50} - 4 q^{52} - 2 q^{53} - 14 q^{55} + 4 q^{56} + 18 q^{58} - 29 q^{59} - 11 q^{61} + 18 q^{62} + 12 q^{64} + 2 q^{65} - 22 q^{67} + 14 q^{68} + 4 q^{70} - 10 q^{71} + 6 q^{73} - 24 q^{74} + 14 q^{76} + 11 q^{77} + q^{79} - 6 q^{80} + q^{82} + 26 q^{83} - 7 q^{85} + 14 q^{86} + 14 q^{88} - 2 q^{89} - 4 q^{91} + 9 q^{92} + 7 q^{94} + 14 q^{95} + 6 q^{97} + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5 x^{11} + 14 x^{10} - 28 x^{9} + 36 x^{8} - 24 x^{7} + 33 x^{6} + 42 x^{5} + 114 x^{4} + 104 x^{3} + 197 x^{2} + 166 x + 79 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 5 \nu^{11} + 337 \nu^{10} - 1489 \nu^{9} + 4209 \nu^{8} - 10929 \nu^{7} + 20640 \nu^{6} - 28191 \nu^{5} + 44748 \nu^{4} - 2661 \nu^{3} + 54649 \nu^{2} + 51731 \nu + 65239 ) / 5589 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{11} - 14 \nu^{10} + 107 \nu^{9} - 423 \nu^{8} + 1170 \nu^{7} - 2226 \nu^{6} + 2826 \nu^{5} - 2931 \nu^{4} + 501 \nu^{3} - 1727 \nu^{2} - 817 \nu - 3512 ) / 243 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 35 \nu^{11} + 82 \nu^{10} - 1108 \nu^{9} + 4209 \nu^{8} - 10470 \nu^{7} + 18003 \nu^{6} - 18903 \nu^{5} + 22194 \nu^{4} + 210 \nu^{3} + 18844 \nu^{2} + 10010 \nu + 20317 ) / 5589 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 53 \nu^{11} + 347 \nu^{10} - 1646 \nu^{9} + 5589 \nu^{8} - 12879 \nu^{7} + 21405 \nu^{6} - 26469 \nu^{5} + 13929 \nu^{4} - 10806 \nu^{3} - 7849 \nu^{2} - 17435 \nu + 14240 ) / 5589 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 85 \nu^{11} - 343 \nu^{10} + 562 \nu^{9} + 69 \nu^{8} - 3702 \nu^{7} + 12159 \nu^{6} - 18120 \nu^{5} + 25383 \nu^{4} - 2664 \nu^{3} + 10919 \nu^{2} + 16030 \nu + 27419 ) / 5589 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 212 \nu^{11} + 1526 \nu^{10} - 6170 \nu^{9} + 16698 \nu^{8} - 30885 \nu^{7} + 39390 \nu^{6} - 37221 \nu^{5} + 8451 \nu^{4} - 13692 \nu^{3} - 15526 \nu^{2} - 37034 \nu + 4934 ) / 5589 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 424 \nu^{11} + 2293 \nu^{10} - 6544 \nu^{9} + 12972 \nu^{8} - 16575 \nu^{7} + 8883 \nu^{6} - 5235 \nu^{5} - 28362 \nu^{4} - 14688 \nu^{3} - 35744 \nu^{2} - 40327 \nu - 32498 ) / 5589 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 16 \nu^{11} + 90 \nu^{10} - 286 \nu^{9} + 690 \nu^{8} - 1220 \nu^{7} + 1503 \nu^{6} - 1821 \nu^{5} + 469 \nu^{4} - 1545 \nu^{3} - 569 \nu^{2} - 1586 \nu + 250 ) / 207 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 473 \nu^{11} + 2882 \nu^{10} - 9464 \nu^{9} + 21321 \nu^{8} - 32760 \nu^{7} + 31206 \nu^{6} - 26505 \nu^{5} - 11037 \nu^{4} - 29679 \nu^{3} - 21181 \nu^{2} - 67382 \nu - 30430 ) / 5589 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 23 \nu^{11} - 116 \nu^{10} + 329 \nu^{9} - 690 \nu^{8} + 948 \nu^{7} - 684 \nu^{6} + 813 \nu^{5} + 1185 \nu^{4} + 1746 \nu^{3} + 2284 \nu^{2} + 2897 \nu + 1714 ) / 243 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 685 \nu^{11} + 3304 \nu^{10} - 8389 \nu^{9} + 14973 \nu^{8} - 15069 \nu^{7} - 1716 \nu^{6} + 2445 \nu^{5} - 55440 \nu^{4} - 41784 \nu^{3} - 64307 \nu^{2} - 89098 \nu - 65723 ) / 5589 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{11} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{11} + \beta_{9} + \beta_{7} - 3\beta_{6} - 2\beta_{5} + 6\beta_{4} + 3\beta_{3} + 2\beta_{2} + \beta _1 + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{10} - 3\beta_{8} + 3\beta_{7} - 6\beta_{6} - 2\beta_{5} + 9\beta_{4} + 13\beta_{3} + 2\beta_{2} + 9 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 7 \beta_{11} - 4 \beta_{9} - 3 \beta_{8} + 5 \beta_{7} - 12 \beta_{6} + 5 \beta_{5} + 12 \beta_{4} + 45 \beta_{3} + 7 \beta_{2} - 4 \beta _1 + 33 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 32 \beta_{11} + 22 \beta_{10} - 17 \beta_{9} + 3 \beta_{8} + 10 \beta_{7} - 18 \beta_{6} + 26 \beta_{5} + 30 \beta_{4} + 115 \beta_{3} + 28 \beta_{2} - 8 \beta _1 + 75 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 110 \beta_{11} + 82 \beta_{10} - 52 \beta_{9} + 2 \beta_{8} + 7 \beta_{7} - 5 \beta_{6} + 87 \beta_{5} + 77 \beta_{4} + 246 \beta_{3} + 77 \beta_{2} - \beta _1 + 82 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 337 \beta_{11} + 208 \beta_{10} - 157 \beta_{9} - 42 \beta_{8} - 55 \beta_{7} + 111 \beta_{6} + 243 \beta_{5} + 102 \beta_{4} + 478 \beta_{3} + 153 \beta_{2} + 32 \beta _1 - 78 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 913 \beta_{11} + 493 \beta_{10} - 450 \beta_{9} - 148 \beta_{8} - 304 \beta_{7} + 583 \beta_{6} + 610 \beta_{5} - 163 \beta_{4} + 747 \beta_{3} + 199 \beta_{2} + 93 \beta _1 - 797 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2153 \beta_{11} + 1178 \beta_{10} - 1096 \beta_{9} - 280 \beta_{8} - 1043 \beta_{7} + 2077 \beta_{6} + 1426 \beta_{5} - 1489 \beta_{4} + 250 \beta_{3} - 20 \beta_{2} + 206 \beta _1 - 3323 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 4272 \beta_{11} + 2536 \beta_{10} - 2138 \beta_{9} - 394 \beta_{8} - 3024 \beta_{7} + 6235 \beta_{6} + 2984 \beta_{5} - 5824 \beta_{4} - 4338 \beta_{3} - 1287 \beta_{2} + 571 \beta _1 - 11201 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 6253 \beta_{11} + 4001 \beta_{10} - 2973 \beta_{9} - 481 \beta_{8} - 8041 \beta_{7} + 16636 \beta_{6} + 4697 \beta_{5} - 17962 \beta_{4} - 23225 \beta_{3} - 6078 \beta_{2} + 1764 \beta _1 - 33023 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1890\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(1081\) \(1541\)
\(\chi(n)\) \(1\) \(\beta_{3}\) \(-1 - \beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.142686 1.50500i
0.737753 + 1.71208i
−0.633121 0.576989i
2.48293 0.894932i
−0.659665 + 0.495491i
0.429413 + 1.63537i
0.142686 + 1.50500i
0.737753 1.71208i
−0.633121 + 0.576989i
2.48293 + 0.894932i
−0.659665 0.495491i
0.429413 1.63537i
0.500000 0.866025i 0 −0.500000 0.866025i 1.00000 0 −1.85185 1.88962i −1.00000 0 0.500000 0.866025i
361.2 0.500000 0.866025i 0 −0.500000 0.866025i 1.00000 0 −1.85185 1.88962i −1.00000 0 0.500000 0.866025i
361.3 0.500000 0.866025i 0 −0.500000 0.866025i 1.00000 0 −1.40545 + 2.24159i −1.00000 0 0.500000 0.866025i
361.4 0.500000 0.866025i 0 −0.500000 0.866025i 1.00000 0 −1.40545 + 2.24159i −1.00000 0 0.500000 0.866025i
361.5 0.500000 0.866025i 0 −0.500000 0.866025i 1.00000 0 2.25729 + 1.38008i −1.00000 0 0.500000 0.866025i
361.6 0.500000 0.866025i 0 −0.500000 0.866025i 1.00000 0 2.25729 + 1.38008i −1.00000 0 0.500000 0.866025i
1801.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.00000 0 −1.85185 + 1.88962i −1.00000 0 0.500000 + 0.866025i
1801.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.00000 0 −1.85185 + 1.88962i −1.00000 0 0.500000 + 0.866025i
1801.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.00000 0 −1.40545 2.24159i −1.00000 0 0.500000 + 0.866025i
1801.4 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.00000 0 −1.40545 2.24159i −1.00000 0 0.500000 + 0.866025i
1801.5 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.00000 0 2.25729 1.38008i −1.00000 0 0.500000 + 0.866025i
1801.6 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.00000 0 2.25729 1.38008i −1.00000 0 0.500000 + 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1890.2.l.h 12
3.b odd 2 1 630.2.l.f yes 12
7.c even 3 1 1890.2.i.f 12
9.c even 3 1 1890.2.i.f 12
9.d odd 6 1 630.2.i.h 12
21.h odd 6 1 630.2.i.h 12
63.g even 3 1 inner 1890.2.l.h 12
63.n odd 6 1 630.2.l.f yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.i.h 12 9.d odd 6 1
630.2.i.h 12 21.h odd 6 1
630.2.l.f yes 12 3.b odd 2 1
630.2.l.f yes 12 63.n odd 6 1
1890.2.i.f 12 7.c even 3 1
1890.2.i.f 12 9.c even 3 1
1890.2.l.h 12 1.a even 1 1 trivial
1890.2.l.h 12 63.g even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1890, [\chi])\):

\( T_{11}^{6} + 7T_{11}^{5} - 12T_{11}^{4} - 146T_{11}^{3} - 140T_{11}^{2} + 324T_{11} + 363 \) Copy content Toggle raw display
\( T_{13}^{12} - 2 T_{13}^{11} + 78 T_{13}^{10} - 140 T_{13}^{9} + 4390 T_{13}^{8} - 8067 T_{13}^{7} + 111972 T_{13}^{6} - 227754 T_{13}^{5} + 2135322 T_{13}^{4} - 3328074 T_{13}^{3} + 11628711 T_{13}^{2} + \cdots + 5349969 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T - 1)^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} + 2 T^{5} + 2 T^{4} - 19 T^{3} + \cdots + 343)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 7 T^{5} - 12 T^{4} - 146 T^{3} + \cdots + 363)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} - 2 T^{11} + 78 T^{10} + \cdots + 5349969 \) Copy content Toggle raw display
$17$ \( T^{12} + 7 T^{11} + 76 T^{10} + \cdots + 1046529 \) Copy content Toggle raw display
$19$ \( T^{12} - 14 T^{11} + 180 T^{10} + \cdots + 62869041 \) Copy content Toggle raw display
$23$ \( (T^{6} + 9 T^{5} - 33 T^{4} - 540 T^{3} + \cdots - 891)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} - 9 T^{11} + 105 T^{10} + \cdots + 59049 \) Copy content Toggle raw display
$31$ \( T^{12} - 9 T^{11} + 171 T^{10} + \cdots + 296080849 \) Copy content Toggle raw display
$37$ \( T^{12} + 12 T^{11} + 147 T^{10} + \cdots + 185761 \) Copy content Toggle raw display
$41$ \( T^{12} + T^{11} + 49 T^{10} + \cdots + 349281 \) Copy content Toggle raw display
$43$ \( T^{12} - 7 T^{11} + \cdots + 5930694121 \) Copy content Toggle raw display
$47$ \( T^{12} + 7 T^{11} + 139 T^{10} + \cdots + 1447209 \) Copy content Toggle raw display
$53$ \( T^{12} + 2 T^{11} + 151 T^{10} + \cdots + 41177889 \) Copy content Toggle raw display
$59$ \( T^{12} + 29 T^{11} + 598 T^{10} + \cdots + 4173849 \) Copy content Toggle raw display
$61$ \( T^{12} + 11 T^{11} + 125 T^{10} + \cdots + 537289 \) Copy content Toggle raw display
$67$ \( T^{12} + 22 T^{11} + 321 T^{10} + \cdots + 3470769 \) Copy content Toggle raw display
$71$ \( (T^{6} + 5 T^{5} - 144 T^{4} - 238 T^{3} + \cdots - 43461)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} - 6 T^{11} + 174 T^{10} + \cdots + 131079601 \) Copy content Toggle raw display
$79$ \( T^{12} - T^{11} + 422 T^{10} + \cdots + 9983806561 \) Copy content Toggle raw display
$83$ \( T^{12} - 26 T^{11} + \cdots + 1938758266449 \) Copy content Toggle raw display
$89$ \( T^{12} + 2 T^{11} + \cdots + 851631819921 \) Copy content Toggle raw display
$97$ \( T^{12} - 6 T^{11} + 414 T^{10} + \cdots + 212955649 \) Copy content Toggle raw display
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