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Properties

Label 360.2.m.c
Level $360$
Weight $2$
Character orbit 360.m
Analytic conductor $2.875$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [360,2,Mod(179,360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(360, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("360.179");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.m (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: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 17x^{12} - 104x^{10} + 713x^{8} + 238x^{6} + 1004x^{4} - 152x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{2} + 1) q^{4} - \beta_{12} q^{5} + ( - \beta_{8} - \beta_{6}) q^{7} + (\beta_{15} + \beta_{13} - \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{2} + 1) q^{4} - \beta_{12} q^{5} + ( - \beta_{8} - \beta_{6}) q^{7} + (\beta_{15} + \beta_{13} - \beta_{3}) q^{8} + ( - \beta_{14} - \beta_{10} + \cdots - \beta_{6}) q^{10}+ \cdots + (2 \beta_{4} + 2 \beta_{3} + \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 12 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 12 q^{4} - 8 q^{10} - 12 q^{16} + 16 q^{19} + 40 q^{25} - 32 q^{34} - 28 q^{40} - 48 q^{46} - 32 q^{49} + 36 q^{64} + 32 q^{70} - 56 q^{76} - 16 q^{91} + 24 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 17x^{12} - 104x^{10} + 713x^{8} + 238x^{6} + 1004x^{4} - 152x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 49720141 \nu^{14} - 25545398 \nu^{12} + 871193385 \nu^{10} + 5556489734 \nu^{8} + \cdots - 35294863008 ) / 26277512192 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 107882115 \nu^{14} - 77610858 \nu^{12} + 1877392743 \nu^{10} + 12516347930 \nu^{8} + \cdots - 17971784544 ) / 26277512192 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 172551585 \nu^{14} - 95959474 \nu^{12} - 2917951085 \nu^{10} - 16219192030 \nu^{8} + \cdots - 68095836896 ) / 26277512192 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 174161853 \nu^{14} - 18352086 \nu^{12} + 2890619609 \nu^{10} + 18336673830 \nu^{8} + \cdots - 21115026592 ) / 26277512192 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 74983 \nu^{15} + 31874 \nu^{13} - 1281915 \nu^{11} - 8349682 \nu^{9} + 50321507 \nu^{7} + \cdots + 9807584 \nu ) / 8515072 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 281117679 \nu^{15} - 123908562 \nu^{13} + 4827183075 \nu^{11} + 31323608386 \nu^{9} + \cdots - 65561067232 \nu ) / 26277512192 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 294063601 \nu^{15} + 120583890 \nu^{13} + 5015106685 \nu^{11} + 28521843902 \nu^{9} + \cdots + 167281353952 \nu ) / 26277512192 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 354912961 \nu^{15} + 119115470 \nu^{13} - 6034586637 \nu^{11} - 39017889374 \nu^{9} + \cdots + 16283111712 \nu ) / 26277512192 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 234112727 \nu^{15} + 231790677 \nu^{14} + 51115902 \nu^{13} + 29428326 \nu^{12} + \cdots - 26297259104 ) / 26277512192 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 293670462 \nu^{15} + 157920899 \nu^{14} + 51649052 \nu^{13} - 22764694 \nu^{12} + \cdots - 26166657184 ) / 26277512192 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 234112727 \nu^{15} - 231790677 \nu^{14} + 51115902 \nu^{13} - 29428326 \nu^{12} + \cdots + 19746912 ) / 26277512192 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 70157797 \nu^{15} - 42874683 \nu^{14} + 14792042 \nu^{13} - 5353898 \nu^{12} + \cdots + 2357445536 ) / 6569378048 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 73637523 \nu^{15} + 42874683 \nu^{14} - 15623210 \nu^{13} + 5353898 \nu^{12} + \cdots - 2357445536 ) / 6569378048 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 293670462 \nu^{15} - 157920899 \nu^{14} + 51649052 \nu^{13} + 22764694 \nu^{12} + \cdots + 26166657184 ) / 26277512192 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 73637523 \nu^{15} + 42874683 \nu^{14} + 15623210 \nu^{13} + 5353898 \nu^{12} + \cdots - 2357445536 ) / 6569378048 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{13} - \beta_{12} + \beta_{7} + \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{15} - \beta_{14} - \beta_{13} - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{4} - 2 \beta_{3} + \cdots - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 4 \beta_{15} + 5 \beta_{14} - 4 \beta_{12} - 2 \beta_{11} + 5 \beta_{10} - 2 \beta_{9} + 6 \beta_{8} + \cdots - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 10 \beta_{15} - 5 \beta_{14} + 10 \beta_{13} + 11 \beta_{11} + 5 \beta_{10} - 11 \beta_{9} - 6 \beta_{4} + \cdots + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 14 \beta_{15} - 12 \beta_{14} + 10 \beta_{13} - 4 \beta_{12} - 11 \beta_{11} - 12 \beta_{10} + \cdots - 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 44 \beta_{15} - \beta_{14} - 44 \beta_{13} - 26 \beta_{11} + \beta_{10} + 26 \beta_{9} + 4 \beta_{4} + \cdots + 88 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 10 \beta_{15} + 59 \beta_{14} - 40 \beta_{13} - 30 \beta_{12} + 3 \beta_{11} + 59 \beta_{10} + 3 \beta_{9} + \cdots + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 120 \beta_{15} - 206 \beta_{14} - 120 \beta_{13} + 109 \beta_{11} + 206 \beta_{10} - 109 \beta_{9} + \cdots - 693 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 588 \beta_{15} + 393 \beta_{14} + 686 \beta_{13} + 98 \beta_{12} - 534 \beta_{11} + 393 \beta_{10} + \cdots - 534 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 816 \beta_{15} + 357 \beta_{14} + 816 \beta_{13} + 1241 \beta_{11} - 357 \beta_{10} - 1241 \beta_{9} + \cdots + 2701 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1974 \beta_{15} - 3674 \beta_{14} + 622 \beta_{13} + 2596 \beta_{12} + 225 \beta_{11} - 3674 \beta_{10} + \cdots + 225 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 13520 \beta_{15} + 293 \beta_{14} - 13520 \beta_{13} - 8456 \beta_{11} - 293 \beta_{10} + 8456 \beta_{9} + \cdots - 8774 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 1986 \beta_{15} + 20283 \beta_{14} + 1324 \beta_{13} + 3310 \beta_{12} - 475 \beta_{11} + 20283 \beta_{10} + \cdots - 475 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 31448 \beta_{15} - 12444 \beta_{14} + 31448 \beta_{13} + 49285 \beta_{11} + 12444 \beta_{10} + \cdots - 89305 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 27344 \beta_{15} - 65833 \beta_{14} + 107890 \beta_{13} + 80546 \beta_{12} - 49240 \beta_{11} + \cdots - 49240 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
179.1
0.877859 + 2.23141i
−0.877859 − 2.23141i
0.877859 − 2.23141i
−0.877859 + 2.23141i
−2.15532 − 0.457057i
2.15532 + 0.457057i
−2.15532 + 0.457057i
2.15532 − 0.457057i
−0.645096 − 0.854135i
0.645096 + 0.854135i
−0.645096 + 0.854135i
0.645096 − 0.854135i
0.409646 − 0.286988i
−0.409646 + 0.286988i
0.409646 + 0.286988i
−0.409646 − 0.286988i
−1.37491 − 0.331077i 0 1.78078 + 0.910404i −1.64901 − 1.51022i 0 −0.936426 −2.14700 − 1.84130i 0 1.76724 + 2.62238i
179.2 −1.37491 − 0.331077i 0 1.78078 + 0.910404i 1.64901 − 1.51022i 0 0.936426 −2.14700 − 1.84130i 0 −2.76724 + 1.53048i
179.3 −1.37491 + 0.331077i 0 1.78078 − 0.910404i −1.64901 + 1.51022i 0 −0.936426 −2.14700 + 1.84130i 0 1.76724 − 2.62238i
179.4 −1.37491 + 0.331077i 0 1.78078 − 0.910404i 1.64901 + 1.51022i 0 0.936426 −2.14700 + 1.84130i 0 −2.76724 − 1.53048i
179.5 −0.927153 − 1.06789i 0 −0.280776 + 1.98019i −2.18650 − 0.468213i 0 3.02045 2.37495 − 1.53610i 0 1.52722 + 2.76904i
179.6 −0.927153 − 1.06789i 0 −0.280776 + 1.98019i 2.18650 − 0.468213i 0 −3.02045 2.37495 − 1.53610i 0 −2.52722 − 1.90083i
179.7 −0.927153 + 1.06789i 0 −0.280776 − 1.98019i −2.18650 + 0.468213i 0 3.02045 2.37495 + 1.53610i 0 1.52722 − 2.76904i
179.8 −0.927153 + 1.06789i 0 −0.280776 − 1.98019i 2.18650 + 0.468213i 0 −3.02045 2.37495 + 1.53610i 0 −2.52722 + 1.90083i
179.9 0.927153 − 1.06789i 0 −0.280776 − 1.98019i −2.18650 − 0.468213i 0 −3.02045 −2.37495 − 1.53610i 0 −2.52722 + 1.90083i
179.10 0.927153 − 1.06789i 0 −0.280776 − 1.98019i 2.18650 − 0.468213i 0 3.02045 −2.37495 − 1.53610i 0 1.52722 − 2.76904i
179.11 0.927153 + 1.06789i 0 −0.280776 + 1.98019i −2.18650 + 0.468213i 0 −3.02045 −2.37495 + 1.53610i 0 −2.52722 − 1.90083i
179.12 0.927153 + 1.06789i 0 −0.280776 + 1.98019i 2.18650 + 0.468213i 0 3.02045 −2.37495 + 1.53610i 0 1.52722 + 2.76904i
179.13 1.37491 − 0.331077i 0 1.78078 − 0.910404i −1.64901 − 1.51022i 0 0.936426 2.14700 − 1.84130i 0 −2.76724 − 1.53048i
179.14 1.37491 − 0.331077i 0 1.78078 − 0.910404i 1.64901 − 1.51022i 0 −0.936426 2.14700 − 1.84130i 0 1.76724 − 2.62238i
179.15 1.37491 + 0.331077i 0 1.78078 + 0.910404i −1.64901 + 1.51022i 0 0.936426 2.14700 + 1.84130i 0 −2.76724 + 1.53048i
179.16 1.37491 + 0.331077i 0 1.78078 + 0.910404i 1.64901 + 1.51022i 0 −0.936426 2.14700 + 1.84130i 0 1.76724 + 2.62238i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 179.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
8.d odd 2 1 inner
15.d odd 2 1 inner
24.f even 2 1 inner
40.e odd 2 1 inner
120.m even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.2.m.c ✓ 16
3.b odd 2 1 inner 360.2.m.c ✓ 16
4.b odd 2 1 1440.2.m.c 16
5.b even 2 1 inner 360.2.m.c ✓ 16
5.c odd 4 2 1800.2.b.g 16
8.b even 2 1 1440.2.m.c 16
8.d odd 2 1 inner 360.2.m.c ✓ 16
12.b even 2 1 1440.2.m.c 16
15.d odd 2 1 inner 360.2.m.c ✓ 16
15.e even 4 2 1800.2.b.g 16
20.d odd 2 1 1440.2.m.c 16
20.e even 4 2 7200.2.b.i 16
24.f even 2 1 inner 360.2.m.c ✓ 16
24.h odd 2 1 1440.2.m.c 16
40.e odd 2 1 inner 360.2.m.c ✓ 16
40.f even 2 1 1440.2.m.c 16
40.i odd 4 2 7200.2.b.i 16
40.k even 4 2 1800.2.b.g 16
60.h even 2 1 1440.2.m.c 16
60.l odd 4 2 7200.2.b.i 16
120.i odd 2 1 1440.2.m.c 16
120.m even 2 1 inner 360.2.m.c ✓ 16
120.q odd 4 2 1800.2.b.g 16
120.w even 4 2 7200.2.b.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.m.c ✓ 16 1.a even 1 1 trivial
360.2.m.c ✓ 16 3.b odd 2 1 inner
360.2.m.c ✓ 16 5.b even 2 1 inner
360.2.m.c ✓ 16 8.d odd 2 1 inner
360.2.m.c ✓ 16 15.d odd 2 1 inner
360.2.m.c ✓ 16 24.f even 2 1 inner
360.2.m.c ✓ 16 40.e odd 2 1 inner
360.2.m.c ✓ 16 120.m even 2 1 inner
1440.2.m.c 16 4.b odd 2 1
1440.2.m.c 16 8.b even 2 1
1440.2.m.c 16 12.b even 2 1
1440.2.m.c 16 20.d odd 2 1
1440.2.m.c 16 24.h odd 2 1
1440.2.m.c 16 40.f even 2 1
1440.2.m.c 16 60.h even 2 1
1440.2.m.c 16 120.i odd 2 1
1800.2.b.g 16 5.c odd 4 2
1800.2.b.g 16 15.e even 4 2
1800.2.b.g 16 40.k even 4 2
1800.2.b.g 16 120.q odd 4 2
7200.2.b.i 16 20.e even 4 2
7200.2.b.i 16 40.i odd 4 2
7200.2.b.i 16 60.l odd 4 2
7200.2.b.i 16 120.w even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 10T_{7}^{2} + 8 \) acting on \(S_{2}^{\mathrm{new}}(360, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 3 T^{6} + 6 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 10 T^{6} + \cdots + 625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 10 T^{2} + 8)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 18 T^{2} + 64)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 14 T^{2} + 32)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 46 T^{2} + 104)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 2 T - 16)^{8} \) Copy content Toggle raw display
$23$ \( (T^{4} + 56 T^{2} + 512)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} - 118 T^{2} + 3328)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 72 T^{2} + 208)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 58 T^{2} + 8)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 34)^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} + 88 T^{2} + 1664)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 116 T^{2} + 32)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + 126 T^{2} + 2592)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + 138 T^{2} + 2704)^{4} \) Copy content Toggle raw display
$61$ \( T^{16} \) Copy content Toggle raw display
$67$ \( (T^{4} + 328 T^{2} + 26624)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 144 T^{2} + 832)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 244 T^{2} + 6656)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 200 T^{2} + 208)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 232 T^{2} + 6656)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 276 T^{2} + 17956)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 92 T^{2} + 416)^{4} \) Copy content Toggle raw display
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