Properties

Label 1470.2.b.a
Level $1470$
Weight $2$
Character orbit 1470.b
Analytic conductor $11.738$
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] = [1470,2,Mod(881,1470)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1470, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1470.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1470.b (of order \(2\), degree \(1\), 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: \(11.7380090971\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 11 x^{10} - 32 x^{9} + 64 x^{8} - 120 x^{7} + 237 x^{6} - 360 x^{5} + 576 x^{4} - 864 x^{3} + 891 x^{2} - 972 x + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} - \beta_1 q^{3} - q^{4} + q^{5} - \beta_{9} q^{6} - \beta_{4} q^{8} + (\beta_{9} - \beta_{8} - \beta_{7} + 2 \beta_{6} - 2 \beta_{4} + \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} - \beta_1 q^{3} - q^{4} + q^{5} - \beta_{9} q^{6} - \beta_{4} q^{8} + (\beta_{9} - \beta_{8} - \beta_{7} + 2 \beta_{6} - 2 \beta_{4} + \beta_1 - 1) q^{9} + \beta_{4} q^{10} + (\beta_{11} - \beta_{10} - \beta_{9} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} - 1) q^{11} + \beta_1 q^{12} + ( - \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} + \beta_{6} + \beta_{5}) q^{13} - \beta_1 q^{15} + q^{16} + (\beta_{9} - \beta_{6} + \beta_{5} - \beta_{2} - \beta_1 + 3) q^{17} + (\beta_{11} + \beta_{9} + \beta_{5} - \beta_{4} + 1) q^{18} + ( - 2 \beta_{11} + 2 \beta_{10} + \beta_{9} - \beta_{8} + \beta_{6} - 2 \beta_{4} - 2 \beta_{2} + \cdots + 2) q^{19}+ \cdots + (2 \beta_{10} - \beta_{9} + \beta_{8} + \beta_{6} - \beta_{4} + 3 \beta_{2} + 2 \beta_1 + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{3} - 12 q^{4} + 12 q^{5} + 2 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{3} - 12 q^{4} + 12 q^{5} + 2 q^{6} - 6 q^{9} + 4 q^{12} - 4 q^{15} + 12 q^{16} + 24 q^{17} + 8 q^{18} - 12 q^{20} - 2 q^{24} + 12 q^{25} - 8 q^{26} + 8 q^{27} + 2 q^{30} - 20 q^{33} + 6 q^{36} + 16 q^{37} + 16 q^{38} + 12 q^{39} + 4 q^{41} - 6 q^{45} - 4 q^{46} + 32 q^{47} - 4 q^{48} + 4 q^{51} + 28 q^{54} - 36 q^{57} - 16 q^{58} + 24 q^{59} + 4 q^{60} - 8 q^{62} - 12 q^{64} - 20 q^{66} + 8 q^{67} - 24 q^{68} - 50 q^{69} - 8 q^{72} - 4 q^{75} + 32 q^{78} + 8 q^{79} + 12 q^{80} - 10 q^{81} + 40 q^{83} + 24 q^{85} - 56 q^{87} + 52 q^{89} + 8 q^{90} + 28 q^{93} + 2 q^{96} + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 11 x^{10} - 32 x^{9} + 64 x^{8} - 120 x^{7} + 237 x^{6} - 360 x^{5} + 576 x^{4} - 864 x^{3} + 891 x^{2} - 972 x + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 4 \nu^{11} + \nu^{10} - 11 \nu^{9} + 35 \nu^{8} - 10 \nu^{7} + 123 \nu^{6} - 183 \nu^{5} + 63 \nu^{4} - 630 \nu^{3} + 351 \nu^{2} - 243 \nu + 1701 ) / 324 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 7 \nu^{11} + 17 \nu^{10} - 39 \nu^{9} + 118 \nu^{8} - 153 \nu^{7} + 337 \nu^{6} - 651 \nu^{5} + 678 \nu^{4} - 1413 \nu^{3} + 1521 \nu^{2} - 837 \nu + 1620 ) / 324 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8 \nu^{11} - 11 \nu^{10} + 43 \nu^{9} - 145 \nu^{8} + 170 \nu^{7} - 513 \nu^{6} + 891 \nu^{5} - 1017 \nu^{4} + 2502 \nu^{3} - 2457 \nu^{2} + 2403 \nu - 4455 ) / 324 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 35 \nu^{11} + 83 \nu^{10} - 265 \nu^{9} + 736 \nu^{8} - 1145 \nu^{7} + 2631 \nu^{6} - 4533 \nu^{5} + 5976 \nu^{4} - 11925 \nu^{3} + 12879 \nu^{2} - 12879 \nu + 17010 ) / 972 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 15 \nu^{11} + 32 \nu^{10} - 98 \nu^{9} + 289 \nu^{8} - 415 \nu^{7} + 998 \nu^{6} - 1734 \nu^{5} + 2157 \nu^{4} - 4635 \nu^{3} + 4878 \nu^{2} - 4644 \nu + 6885 ) / 324 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8 \nu^{11} - 16 \nu^{10} + 57 \nu^{9} - 158 \nu^{8} + 228 \nu^{7} - 560 \nu^{6} + 927 \nu^{5} - 1212 \nu^{4} + 2538 \nu^{3} - 2610 \nu^{2} + 2673 \nu - 3564 ) / 162 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 16 \nu^{11} - 39 \nu^{10} + 127 \nu^{9} - 351 \nu^{8} + 542 \nu^{7} - 1241 \nu^{6} + 2139 \nu^{5} - 2859 \nu^{4} + 5670 \nu^{3} - 6309 \nu^{2} + 5967 \nu - 7857 ) / 324 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 16 \nu^{11} - 31 \nu^{10} + 98 \nu^{9} - 284 \nu^{8} + 400 \nu^{7} - 969 \nu^{6} + 1668 \nu^{5} - 2070 \nu^{4} + 4302 \nu^{3} - 4455 \nu^{2} + 4212 \nu - 6075 ) / 243 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11 \nu^{11} - 26 \nu^{10} + 76 \nu^{9} - 208 \nu^{8} + 317 \nu^{7} - 708 \nu^{6} + 1230 \nu^{5} - 1638 \nu^{4} + 3123 \nu^{3} - 3348 \nu^{2} + 3078 \nu - 3888 ) / 162 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 38 \nu^{11} + 91 \nu^{10} - 279 \nu^{9} + 767 \nu^{8} - 1176 \nu^{7} + 2657 \nu^{6} - 4599 \nu^{5} + 6135 \nu^{4} - 11916 \nu^{3} + 13005 \nu^{2} - 12771 \nu + 15633 ) / 324 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 119 \nu^{11} + 287 \nu^{10} - 877 \nu^{9} + 2458 \nu^{8} - 3809 \nu^{7} + 8475 \nu^{6} - 14865 \nu^{5} + 19782 \nu^{4} - 38385 \nu^{3} + 42147 \nu^{2} - 40743 \nu + 51516 ) / 972 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{10} - \beta_{9} - \beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{8} - 3 \beta_{7} + 2 \beta_{6} + \beta_{5} - 3 \beta_{4} + \beta_{3} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + \beta_{10} + \beta_{9} - 2\beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4\beta_{9} + \beta_{8} - 2\beta_{5} - 4\beta_{4} - 4\beta_{3} - 2\beta_{2} - 2\beta _1 + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 4 \beta_{11} - \beta_{10} - 3 \beta_{9} - 4 \beta_{8} - 5 \beta_{7} - 4 \beta_{6} + 14 \beta_{5} - 16 \beta_{4} + 6 \beta_{3} - 6 \beta_{2} - 8 \beta _1 + 16 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 11 \beta_{11} + \beta_{10} - 2 \beta_{9} - 9 \beta_{8} - 6 \beta_{7} + 2 \beta_{6} + 5 \beta_{5} + 8 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 10 \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 18 \beta_{11} + 23 \beta_{10} - 5 \beta_{9} + 11 \beta_{8} - 3 \beta_{7} + 4 \beta_{6} - 14 \beta_{5} - 5 \beta_{4} - 28 \beta_{3} + 7 \beta_{2} - 6 \beta _1 + 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 12 \beta_{11} - 4 \beta_{10} + 16 \beta_{9} - \beta_{8} + 8 \beta_{7} - 48 \beta_{6} + 14 \beta_{5} - 74 \beta_{4} - 8 \beta_{3} + 20 \beta_{2} - 46 \beta _1 + 83 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 17 \beta_{11} - 15 \beta_{10} - 12 \beta_{9} - 35 \beta_{8} - \beta_{7} + 8 \beta_{6} + 60 \beta_{5} - 11 \beta_{4} + 4 \beta_{3} - 22 \beta_{2} - 60 \beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 141 \beta_{11} + 97 \beta_{10} - 65 \beta_{9} + 68 \beta_{8} + 39 \beta_{7} + 38 \beta_{6} + 177 \beta_{5} + 55 \beta_{4} - 21 \beta_{3} + 9 \beta_{2} - 92 \beta _1 + 140 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 58 \beta_{11} - 45 \beta_{10} - 13 \beta_{9} + 273 \beta_{8} + 65 \beta_{7} - 298 \beta_{6} - 90 \beta_{5} + 337 \beta_{4} - 154 \beta_{3} + 313 \beta_{2} - 208 \beta _1 + 203 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times\).

\(n\) \(491\) \(1081\) \(1177\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
1.66557 + 0.475255i
−0.384890 + 1.68874i
−0.111613 + 1.72845i
−1.21252 1.23685i
0.312065 1.70371i
1.73138 + 0.0481063i
1.66557 0.475255i
−0.384890 1.68874i
−0.111613 1.72845i
−1.21252 + 1.23685i
0.312065 + 1.70371i
1.73138 0.0481063i
1.00000i −1.68006 + 0.421203i −1.00000 1.00000 0.421203 + 1.68006i 0 1.00000i 2.64518 1.41529i 1.00000i
881.2 1.00000i −1.17770 + 1.27005i −1.00000 1.00000 1.27005 + 1.17770i 0 1.00000i −0.226058 2.99147i 1.00000i
881.3 1.00000i −0.767566 1.55269i −1.00000 1.00000 −1.55269 + 0.767566i 0 1.00000i −1.82168 + 2.38358i 1.00000i
881.4 1.00000i −0.431645 1.67740i −1.00000 1.00000 −1.67740 + 0.431645i 0 1.00000i −2.62736 + 1.44809i 1.00000i
881.5 1.00000i 0.581597 + 1.63149i −1.00000 1.00000 1.63149 0.581597i 0 1.00000i −2.32349 + 1.89773i 1.00000i
881.6 1.00000i 1.47537 + 0.907353i −1.00000 1.00000 0.907353 1.47537i 0 1.00000i 1.35342 + 2.67736i 1.00000i
881.7 1.00000i −1.68006 0.421203i −1.00000 1.00000 0.421203 1.68006i 0 1.00000i 2.64518 + 1.41529i 1.00000i
881.8 1.00000i −1.17770 1.27005i −1.00000 1.00000 1.27005 1.17770i 0 1.00000i −0.226058 + 2.99147i 1.00000i
881.9 1.00000i −0.767566 + 1.55269i −1.00000 1.00000 −1.55269 0.767566i 0 1.00000i −1.82168 2.38358i 1.00000i
881.10 1.00000i −0.431645 + 1.67740i −1.00000 1.00000 −1.67740 0.431645i 0 1.00000i −2.62736 1.44809i 1.00000i
881.11 1.00000i 0.581597 1.63149i −1.00000 1.00000 1.63149 + 0.581597i 0 1.00000i −2.32349 1.89773i 1.00000i
881.12 1.00000i 1.47537 0.907353i −1.00000 1.00000 0.907353 + 1.47537i 0 1.00000i 1.35342 2.67736i 1.00000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.2.b.a 12
3.b odd 2 1 1470.2.b.b 12
7.b odd 2 1 1470.2.b.b 12
7.c even 3 1 210.2.r.b yes 12
7.d odd 6 1 210.2.r.a 12
21.c even 2 1 inner 1470.2.b.a 12
21.g even 6 1 210.2.r.b yes 12
21.h odd 6 1 210.2.r.a 12
35.i odd 6 1 1050.2.s.g 12
35.j even 6 1 1050.2.s.f 12
35.k even 12 1 1050.2.u.f 12
35.k even 12 1 1050.2.u.g 12
35.l odd 12 1 1050.2.u.e 12
35.l odd 12 1 1050.2.u.h 12
105.o odd 6 1 1050.2.s.g 12
105.p even 6 1 1050.2.s.f 12
105.w odd 12 1 1050.2.u.e 12
105.w odd 12 1 1050.2.u.h 12
105.x even 12 1 1050.2.u.f 12
105.x even 12 1 1050.2.u.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.r.a 12 7.d odd 6 1
210.2.r.a 12 21.h odd 6 1
210.2.r.b yes 12 7.c even 3 1
210.2.r.b yes 12 21.g even 6 1
1050.2.s.f 12 35.j even 6 1
1050.2.s.f 12 105.p even 6 1
1050.2.s.g 12 35.i odd 6 1
1050.2.s.g 12 105.o odd 6 1
1050.2.u.e 12 35.l odd 12 1
1050.2.u.e 12 105.w odd 12 1
1050.2.u.f 12 35.k even 12 1
1050.2.u.f 12 105.x even 12 1
1050.2.u.g 12 35.k even 12 1
1050.2.u.g 12 105.x even 12 1
1050.2.u.h 12 35.l odd 12 1
1050.2.u.h 12 105.w odd 12 1
1470.2.b.a 12 1.a even 1 1 trivial
1470.2.b.a 12 21.c even 2 1 inner
1470.2.b.b 12 3.b odd 2 1
1470.2.b.b 12 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1470, [\chi])\):

\( T_{11}^{12} + 68T_{11}^{10} + 1590T_{11}^{8} + 14828T_{11}^{6} + 50113T_{11}^{4} + 36360T_{11}^{2} + 1296 \) Copy content Toggle raw display
\( T_{17}^{6} - 12T_{17}^{5} + 32T_{17}^{4} + 96T_{17}^{3} - 572T_{17}^{2} + 816T_{17} - 288 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} + 4 T^{11} + 11 T^{10} + 20 T^{9} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( (T - 1)^{12} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 68 T^{10} + 1590 T^{8} + \cdots + 1296 \) Copy content Toggle raw display
$13$ \( T^{12} + 132 T^{10} + 6638 T^{8} + \cdots + 8433216 \) Copy content Toggle raw display
$17$ \( (T^{6} - 12 T^{5} + 32 T^{4} + 96 T^{3} + \cdots - 288)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 180 T^{10} + \cdots + 242985744 \) Copy content Toggle raw display
$23$ \( T^{12} + 118 T^{10} + 5031 T^{8} + \cdots + 5103081 \) Copy content Toggle raw display
$29$ \( T^{12} + 190 T^{10} + \cdots + 12194064 \) Copy content Toggle raw display
$31$ \( T^{12} + 192 T^{10} + \cdots + 301925376 \) Copy content Toggle raw display
$37$ \( (T^{6} - 8 T^{5} - 54 T^{4} + 392 T^{3} + \cdots + 64)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 2 T^{5} - 145 T^{4} + 260 T^{3} + \cdots + 549)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 127 T^{4} - 196 T^{3} + \cdots - 2972)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 16 T^{5} - 58 T^{4} + 1696 T^{3} + \cdots + 3924)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + 396 T^{10} + 42326 T^{8} + \cdots + 11664 \) Copy content Toggle raw display
$59$ \( (T^{6} - 12 T^{5} - 148 T^{4} + \cdots + 21312)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + 366 T^{10} + \cdots + 2158903296 \) Copy content Toggle raw display
$67$ \( (T^{6} - 4 T^{5} - 47 T^{4} + 24 T^{3} + \cdots + 576)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + 552 T^{10} + \cdots + 8680276224 \) Copy content Toggle raw display
$73$ \( T^{12} + 560 T^{10} + \cdots + 89884836864 \) Copy content Toggle raw display
$79$ \( (T^{6} - 4 T^{5} - 204 T^{4} + 2032 T^{3} + \cdots + 2896)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 20 T^{5} + 137 T^{4} - 328 T^{3} + \cdots - 1152)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 26 T^{5} + 41 T^{4} + \cdots + 125712)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 680 T^{10} + \cdots + 2449062144 \) Copy content Toggle raw display
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