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Properties

Label 1170.2.m.i
Level $1170$
Weight $2$
Character orbit 1170.m
Analytic conductor $9.342$
Analytic rank $0$
Dimension $16$
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] = [1170,2,Mod(73,1170)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1170, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1170.73");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1170.m (of order \(4\), 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: \(9.34249703649\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 32x^{14} + 396x^{12} + 2412x^{10} + 7716x^{8} + 12984x^{6} + 10756x^{4} + 3648x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + \beta_{11} q^{5} + ( - \beta_{12} + \beta_{9} + \cdots - \beta_{2}) q^{7}+ \cdots - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + \beta_{11} q^{5} + ( - \beta_{12} + \beta_{9} + \cdots - \beta_{2}) q^{7}+ \cdots + ( - \beta_{15} + 2 \beta_{14} + \cdots + 5) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{2} + 16 q^{4} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{2} + 16 q^{4} - 16 q^{8} - 4 q^{11} + 4 q^{13} + 16 q^{16} - 4 q^{17} + 4 q^{19} + 4 q^{22} - 16 q^{23} - 16 q^{25} - 4 q^{26} + 12 q^{31} - 16 q^{32} + 4 q^{34} + 12 q^{35} - 4 q^{38} - 4 q^{41} - 16 q^{43} - 4 q^{44} + 16 q^{46} - 80 q^{49} + 16 q^{50} + 4 q^{52} + 44 q^{53} - 20 q^{55} - 12 q^{59} - 32 q^{61} - 12 q^{62} + 16 q^{64} + 44 q^{65} - 32 q^{67} - 4 q^{68} - 12 q^{70} - 16 q^{71} + 4 q^{76} + 32 q^{77} + 4 q^{82} - 64 q^{85} + 16 q^{86} + 4 q^{88} - 4 q^{89} + 76 q^{91} - 16 q^{92} - 40 q^{95} - 8 q^{97} + 80 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 32x^{14} + 396x^{12} + 2412x^{10} + 7716x^{8} + 12984x^{6} + 10756x^{4} + 3648x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 201 \nu^{15} + 1232 \nu^{14} + 7384 \nu^{13} + 37440 \nu^{12} + 108716 \nu^{11} + 429632 \nu^{10} + \cdots + 906752 ) / 113152 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 319 \nu^{15} + 2136 \nu^{14} - 10920 \nu^{13} + 63232 \nu^{12} - 147956 \nu^{11} + 694432 \nu^{10} + \cdots - 211712 ) / 113152 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 319 \nu^{15} - 2136 \nu^{14} - 10920 \nu^{13} - 63232 \nu^{12} - 147956 \nu^{11} - 694432 \nu^{10} + \cdots + 211712 ) / 113152 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 235 \nu^{15} + 44 \nu^{14} + 7176 \nu^{13} + 832 \nu^{12} + 83076 \nu^{11} + 1200 \nu^{10} + \cdots + 88960 ) / 56576 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 235 \nu^{15} - 44 \nu^{14} + 7176 \nu^{13} - 832 \nu^{12} + 83076 \nu^{11} - 1200 \nu^{10} + \cdots - 88960 ) / 56576 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 725 \nu^{15} + 22776 \nu^{13} + 273788 \nu^{11} + 1589116 \nu^{9} + 4674452 \nu^{7} + \cdots + 697504 \nu ) / 113152 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 707 \nu^{15} + 1040 \nu^{14} - 21320 \nu^{13} + 30784 \nu^{12} - 240036 \nu^{11} + 337792 \nu^{10} + \cdots + 126464 ) / 113152 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 527 \nu^{15} - 144 \nu^{14} + 15912 \nu^{13} - 4160 \nu^{12} + 180404 \nu^{11} - 43296 \nu^{10} + \cdots + 190208 ) / 56576 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 527 \nu^{15} - 144 \nu^{14} - 15912 \nu^{13} - 4160 \nu^{12} - 180404 \nu^{11} - 43296 \nu^{10} + \cdots + 190208 ) / 56576 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 527 \nu^{15} - 1288 \nu^{14} + 15912 \nu^{13} - 39936 \nu^{12} + 180404 \nu^{11} - 470528 \nu^{10} + \cdots - 312320 ) / 56576 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 87 \nu^{15} + 80 \nu^{14} - 2664 \nu^{13} + 2368 \nu^{12} - 30740 \nu^{11} + 25984 \nu^{10} + \cdots + 9728 ) / 8704 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 87 \nu^{15} - 80 \nu^{14} - 2664 \nu^{13} - 2368 \nu^{12} - 30740 \nu^{11} - 25984 \nu^{10} + \cdots - 9728 ) / 8704 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1323 \nu^{15} + 1856 \nu^{14} - 38792 \nu^{13} + 57408 \nu^{12} - 418244 \nu^{11} + 673408 \nu^{10} + \cdots + 81408 ) / 113152 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 2655 \nu^{15} - 1840 \nu^{14} + 80808 \nu^{13} - 54080 \nu^{12} + 926580 \nu^{11} - 586368 \nu^{10} + \cdots - 109568 ) / 113152 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 2655 \nu^{15} + 1840 \nu^{14} + 80808 \nu^{13} + 54080 \nu^{12} + 926580 \nu^{11} + 586368 \nu^{10} + \cdots + 109568 ) / 113152 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{15} + \beta_{14} + \beta_{12} + \beta_{11} + \beta_{9} - \beta_{8} - \beta_{5} - \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{14} - \beta_{13} - \beta_{11} + \beta_{5} - \beta_{4} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 4 \beta_{15} - 4 \beta_{14} - 3 \beta_{12} - 3 \beta_{11} - 4 \beta_{9} + 4 \beta_{8} + \cdots + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{15} + 10 \beta_{14} + 9 \beta_{13} - 2 \beta_{12} + 11 \beta_{11} - \beta_{10} - 2 \beta_{9} + \cdots + 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 37 \beta_{15} + 37 \beta_{14} + 22 \beta_{12} + 28 \beta_{11} + 38 \beta_{9} - 38 \beta_{8} + \cdots - 14 \beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 20 \beta_{15} - 99 \beta_{14} - 79 \beta_{13} + 33 \beta_{12} - 112 \beta_{11} + 13 \beta_{10} + \cdots - 238 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 354 \beta_{15} - 346 \beta_{14} + 8 \beta_{13} - 185 \beta_{12} - 275 \beta_{11} + 8 \beta_{10} + \cdots + 8 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 258 \beta_{15} + 974 \beta_{14} + 716 \beta_{13} - 408 \beta_{12} + 1124 \beta_{11} - 134 \beta_{10} + \cdots + 2168 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3422 \beta_{15} + 3262 \beta_{14} - 160 \beta_{13} + 1662 \beta_{12} + 2706 \beta_{11} + \cdots - 160 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2842 \beta_{15} - 9518 \beta_{14} - 6676 \beta_{13} + 4546 \beta_{12} - 11222 \beta_{11} + 1280 \beta_{10} + \cdots - 20324 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 33218 \beta_{15} - 30982 \beta_{14} + 2236 \beta_{13} - 15394 \beta_{12} - 26542 \beta_{11} + \cdots + 2236 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 29068 \beta_{15} + 92566 \beta_{14} + 63498 \beta_{13} - 48226 \beta_{12} + 111724 \beta_{11} + \cdots + 193260 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 323184 \beta_{15} + 295960 \beta_{14} - 27224 \beta_{13} + 144802 \beta_{12} + 259686 \beta_{11} + \cdots - 27224 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 285864 \beta_{15} - 897744 \beta_{14} - 611880 \beta_{13} + 498468 \beta_{12} - 1110348 \beta_{11} + \cdots - 1852744 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 3149120 \beta_{15} - 2839152 \beta_{14} + 309968 \beta_{13} - 1374128 \beta_{12} + \cdots + 309968 \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times\).

\(n\) \(911\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(\beta_{6}\) \(-\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1
3.03462i
− 2.42515i
− 1.54171i
− 0.304860i
0.821039i
− 1.09662i
− 1.63314i
3.14581i
− 3.03462i
2.42515i
1.54171i
0.304860i
− 0.821039i
1.09662i
1.63314i
− 3.14581i
−1.00000 0 1.00000 −2.14580 + 0.628922i 0 − 5.15192i −1.00000 0 2.14580 − 0.628922i
73.2 −1.00000 0 1.00000 −1.71484 + 1.43504i 0 3.30195i −1.00000 0 1.71484 − 1.43504i
73.3 −1.00000 0 1.00000 −1.09015 − 1.95232i 0 − 0.720033i −1.00000 0 1.09015 + 1.95232i
73.4 −1.00000 0 1.00000 0.215569 − 2.22565i 0 − 2.86620i −1.00000 0 −0.215569 + 2.22565i
73.5 −1.00000 0 1.00000 0.580562 + 2.15939i 0 − 3.80685i −1.00000 0 −0.580562 − 2.15939i
73.6 −1.00000 0 1.00000 0.775429 + 2.09731i 0 0.946681i −1.00000 0 −0.775429 − 2.09731i
73.7 −1.00000 0 1.00000 1.15480 − 1.91479i 0 4.24302i −1.00000 0 −1.15480 + 1.91479i
73.8 −1.00000 0 1.00000 2.22443 − 0.227886i 0 4.05336i −1.00000 0 −2.22443 + 0.227886i
577.1 −1.00000 0 1.00000 −2.14580 − 0.628922i 0 5.15192i −1.00000 0 2.14580 + 0.628922i
577.2 −1.00000 0 1.00000 −1.71484 − 1.43504i 0 − 3.30195i −1.00000 0 1.71484 + 1.43504i
577.3 −1.00000 0 1.00000 −1.09015 + 1.95232i 0 0.720033i −1.00000 0 1.09015 − 1.95232i
577.4 −1.00000 0 1.00000 0.215569 + 2.22565i 0 2.86620i −1.00000 0 −0.215569 − 2.22565i
577.5 −1.00000 0 1.00000 0.580562 − 2.15939i 0 3.80685i −1.00000 0 −0.580562 + 2.15939i
577.6 −1.00000 0 1.00000 0.775429 − 2.09731i 0 − 0.946681i −1.00000 0 −0.775429 + 2.09731i
577.7 −1.00000 0 1.00000 1.15480 + 1.91479i 0 − 4.24302i −1.00000 0 −1.15480 − 1.91479i
577.8 −1.00000 0 1.00000 2.22443 + 0.227886i 0 − 4.05336i −1.00000 0 −2.22443 − 0.227886i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1170.2.m.i 16
3.b odd 2 1 390.2.j.b ✓ 16
5.c odd 4 1 1170.2.w.i 16
13.d odd 4 1 1170.2.w.i 16
15.d odd 2 1 1950.2.j.e 16
15.e even 4 1 390.2.t.b yes 16
15.e even 4 1 1950.2.t.e 16
39.f even 4 1 390.2.t.b yes 16
65.k even 4 1 inner 1170.2.m.i 16
195.j odd 4 1 390.2.j.b ✓ 16
195.n even 4 1 1950.2.t.e 16
195.u odd 4 1 1950.2.j.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.j.b ✓ 16 3.b odd 2 1
390.2.j.b ✓ 16 195.j odd 4 1
390.2.t.b yes 16 15.e even 4 1
390.2.t.b yes 16 39.f even 4 1
1170.2.m.i 16 1.a even 1 1 trivial
1170.2.m.i 16 65.k even 4 1 inner
1170.2.w.i 16 5.c odd 4 1
1170.2.w.i 16 13.d odd 4 1
1950.2.j.e 16 15.d odd 2 1
1950.2.j.e 16 195.u odd 4 1
1950.2.t.e 16 15.e even 4 1
1950.2.t.e 16 195.n even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1170, [\chi])\):

\( T_{7}^{16} + 96 T_{7}^{14} + 3760 T_{7}^{12} + 77336 T_{7}^{10} + 890304 T_{7}^{8} + 5594816 T_{7}^{6} + \cdots + 4734976 \) Copy content Toggle raw display
\( T_{11}^{16} + 4 T_{11}^{15} + 8 T_{11}^{14} - 8 T_{11}^{13} + 988 T_{11}^{12} + 3552 T_{11}^{11} + \cdots + 58003456 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 8 T^{14} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} + 96 T^{14} + \cdots + 4734976 \) Copy content Toggle raw display
$11$ \( T^{16} + 4 T^{15} + \cdots + 58003456 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{16} + 4 T^{15} + \cdots + 2534464 \) Copy content Toggle raw display
$19$ \( T^{16} - 4 T^{15} + \cdots + 1024 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 77275104256 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 21484523776 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 837825347584 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 1092810981376 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 79370665984 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 349241344 \) Copy content Toggle raw display
$47$ \( T^{16} + 176 T^{14} + \cdots + 1048576 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 50182602625024 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 48729781571584 \) Copy content Toggle raw display
$61$ \( (T^{8} + 16 T^{7} + \cdots - 3982336)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 16 T^{7} + \cdots + 220928)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 846046756864 \) Copy content Toggle raw display
$73$ \( (T^{8} - 384 T^{6} + \cdots + 2967568)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 604860841984 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 509407657984 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 17644340224 \) Copy content Toggle raw display
$97$ \( (T^{8} + 4 T^{7} + \cdots + 12563008)^{2} \) Copy content Toggle raw display
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