Properties

Label 360.4.k.c
Level $360$
Weight $4$
Character orbit 360.k
Analytic conductor $21.241$
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] = [360,4,Mod(181,360)]
 
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([0, 1, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("360.181");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 360.k (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: \(21.2406876021\)
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: \( x^{12} - 4x^{11} + 7x^{10} - 12x^{9} + 21x^{8} - 68x^{6} + 336x^{4} - 768x^{3} + 1792x^{2} - 4096x + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 40)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{4} + 1) q^{4} - \beta_{7} q^{5} + ( - \beta_{6} + \beta_{5} - \beta_{4} - \beta_1 + 2) q^{7} + (\beta_{9} + \beta_{5} - \beta_{3} + 2 \beta_1 + 3) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{4} + 1) q^{4} - \beta_{7} q^{5} + ( - \beta_{6} + \beta_{5} - \beta_{4} - \beta_1 + 2) q^{7} + (\beta_{9} + \beta_{5} - \beta_{3} + 2 \beta_1 + 3) q^{8} + (\beta_{8} + \beta_{5} - \beta_{4} + 3) q^{10} + ( - \beta_{10} - \beta_{9} + 2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{11} + (\beta_{11} + \beta_{10} + \beta_{8} + \beta_{5} - 4 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 5 \beta_1) q^{13} + (\beta_{11} - \beta_{10} - \beta_{9} + 4 \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} + \cdots - 4) q^{14}+ \cdots + ( - 12 \beta_{10} + 24 \beta_{9} + 8 \beta_{8} - 48 \beta_{7} + 28 \beta_{6} + \cdots + 590) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} + 16 q^{4} + 28 q^{7} + 40 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{2} + 16 q^{4} + 28 q^{7} + 40 q^{8} + 30 q^{10} - 68 q^{14} - 56 q^{16} - 20 q^{20} - 164 q^{22} - 604 q^{23} - 300 q^{25} + 308 q^{26} - 436 q^{28} - 264 q^{31} - 72 q^{32} - 180 q^{34} - 820 q^{38} + 120 q^{40} - 40 q^{41} + 472 q^{44} - 1268 q^{46} + 940 q^{47} + 1308 q^{49} + 50 q^{50} + 1024 q^{52} + 440 q^{55} + 728 q^{56} - 360 q^{58} - 592 q^{62} - 2048 q^{64} + 2344 q^{68} + 1160 q^{70} + 1592 q^{71} + 432 q^{73} + 420 q^{74} + 2256 q^{76} + 2016 q^{79} - 1600 q^{80} + 88 q^{82} + 244 q^{86} + 4080 q^{88} + 424 q^{89} + 900 q^{92} + 292 q^{94} + 1520 q^{95} - 1584 q^{97} + 7266 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4x^{11} + 7x^{10} - 12x^{9} + 21x^{8} - 68x^{6} + 336x^{4} - 768x^{3} + 1792x^{2} - 4096x + 4096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2 \nu^{11} - 5 \nu^{10} + 10 \nu^{9} - 27 \nu^{8} + 30 \nu^{7} + 55 \nu^{6} - 64 \nu^{5} - 132 \nu^{4} + 384 \nu^{3} - 1168 \nu^{2} + 3008 \nu - 5888 ) / 960 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 35 \nu^{11} + 72 \nu^{10} - 203 \nu^{9} + 168 \nu^{8} + 223 \nu^{7} + 484 \nu^{6} + 644 \nu^{5} - 6288 \nu^{4} + 944 \nu^{3} + 24128 \nu^{2} - 22528 \nu - 16384 ) / 7680 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 93 \nu^{11} + 204 \nu^{10} - 379 \nu^{9} + 388 \nu^{8} - 817 \nu^{7} - 1672 \nu^{6} + 4708 \nu^{5} + 4640 \nu^{4} - 20048 \nu^{3} + 34432 \nu^{2} - 96256 \nu + 191488 ) / 15360 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3 \nu^{11} + 4 \nu^{10} + 3 \nu^{9} + 12 \nu^{8} - 55 \nu^{7} - 72 \nu^{6} + 132 \nu^{5} + 288 \nu^{4} - 368 \nu^{3} - 384 \nu^{2} - 2304 \nu + 7712 ) / 480 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 107 \nu^{11} + 204 \nu^{10} - 29 \nu^{9} + 292 \nu^{8} - 407 \nu^{7} - 1952 \nu^{6} + 5756 \nu^{5} + 15104 \nu^{4} - 14896 \nu^{3} - 1792 \nu^{2} - 33792 \nu + 134144 ) / 15360 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 15 \nu^{11} - 4 \nu^{10} - 35 \nu^{9} - 28 \nu^{8} + 263 \nu^{7} - 248 \nu^{6} + 200 \nu^{5} - 1088 \nu^{4} - 128 \nu^{3} + 7232 \nu^{2} - 2240 \nu - 23424 ) / 1920 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 23 \nu^{11} - 60 \nu^{10} + 81 \nu^{9} - 116 \nu^{8} + 51 \nu^{7} + 480 \nu^{6} - 684 \nu^{5} - 1024 \nu^{4} + 5616 \nu^{3} - 11520 \nu^{2} + 22528 \nu - 30720 ) / 3072 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 171 \nu^{11} - 476 \nu^{10} + 925 \nu^{9} - 2068 \nu^{8} + 1047 \nu^{7} + 4048 \nu^{6} - 6652 \nu^{5} - 16448 \nu^{4} + 33840 \nu^{3} - 103936 \nu^{2} + 354304 \nu - 404480 ) / 15360 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 103 \nu^{11} - 304 \nu^{10} + 49 \nu^{9} + 352 \nu^{8} + 467 \nu^{7} + 412 \nu^{6} - 4108 \nu^{5} - 13360 \nu^{4} + 49648 \nu^{3} - 12352 \nu^{2} - 13824 \nu - 96768 ) / 7680 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 59 \nu^{11} + 134 \nu^{10} - 189 \nu^{9} + 314 \nu^{8} - 407 \nu^{7} - 1182 \nu^{6} + 2580 \nu^{5} + 4216 \nu^{4} - 11920 \nu^{3} + 39456 \nu^{2} - 73600 \nu + 109312 ) / 3840 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 173 \nu^{11} - 352 \nu^{10} + 459 \nu^{9} - 1424 \nu^{8} + 1633 \nu^{7} + 1956 \nu^{6} - 9108 \nu^{5} - 9616 \nu^{4} + 42512 \nu^{3} - 55488 \nu^{2} + 207616 \nu - 421376 ) / 7680 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{8} + \beta_{5} - \beta_{4} - 5\beta _1 + 3 ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{10} + \beta_{8} + \beta_{7} - \beta_{4} - 2\beta_{3} - 3\beta _1 + 2 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{11} + 3\beta_{10} + 3\beta_{9} - 4\beta_{6} + 8\beta_{5} + 2\beta_{3} + 5\beta_{2} + 7\beta _1 + 27 ) / 20 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{11} + \beta_{10} - \beta_{9} - 2\beta_{8} + 6\beta_{7} + 2\beta_{5} - 2\beta_{4} - \beta_{2} - 5\beta _1 + 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 7 \beta_{11} + 3 \beta_{10} + 3 \beta_{9} + 8 \beta_{8} + 18 \beta_{7} + 6 \beta_{6} + 16 \beta_{5} - 3 \beta_{4} + 12 \beta_{3} + 5 \beta_{2} + 17 \beta _1 - 94 ) / 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 17 \beta_{11} + 23 \beta_{10} - 7 \beta_{9} - 16 \beta_{8} + 72 \beta_{7} - 44 \beta_{6} + 2 \beta_{5} - 64 \beta_{4} - 8 \beta_{3} + 25 \beta_{2} + 127 \beta _1 + 29 ) / 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 33 \beta_{11} - 37 \beta_{10} + 27 \beta_{9} - 24 \beta_{8} - 164 \beta_{7} + 24 \beta_{6} + 80 \beta_{5} - 86 \beta_{4} - 98 \beta_{3} - 35 \beta_{2} + 339 \beta _1 + 1513 ) / 20 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 11 \beta_{11} - 7 \beta_{10} + 3 \beta_{9} + 9 \beta_{8} - 7 \beta_{7} + 2 \beta_{6} + 16 \beta_{5} - 15 \beta_{4} - 40 \beta_{3} - \beta_{2} - 88 \beta _1 - 117 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 139 \beta_{11} + 237 \beta_{10} + 21 \beta_{9} + 96 \beta_{8} - 432 \beta_{7} + 12 \beta_{6} - 56 \beta_{5} + 544 \beta_{4} - 1442 \beta_{3} + 155 \beta_{2} + 1025 \beta _1 + 589 ) / 20 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 387 \beta_{11} - 109 \beta_{10} - 123 \beta_{9} - 270 \beta_{8} - 1046 \beta_{7} - 416 \beta_{6} + 510 \beta_{5} + 970 \beta_{4} - 1096 \beta_{3} + 1325 \beta_{2} + 1097 \beta _1 + 3321 ) / 20 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 223 \beta_{11} - 99 \beta_{10} - 467 \beta_{9} - 564 \beta_{8} + 1690 \beta_{7} + 586 \beta_{6} - 200 \beta_{5} + 859 \beta_{4} - 596 \beta_{3} + 635 \beta_{2} - 1805 \beta _1 + 4026 ) / 10 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
1.71681 1.02595i
1.71681 + 1.02595i
1.98839 0.215211i
1.98839 + 0.215211i
−0.428316 + 1.95360i
−0.428316 1.95360i
1.23537 + 1.57285i
1.23537 1.57285i
−0.650488 + 1.89126i
−0.650488 1.89126i
−1.86176 0.730647i
−1.86176 + 0.730647i
−2.74276 0.690860i 0 7.04543 + 3.78972i 5.00000i 0 −14.6308 −16.7057 15.2617i 0 3.45430 13.7138i
181.2 −2.74276 + 0.690860i 0 7.04543 3.78972i 5.00000i 0 −14.6308 −16.7057 + 15.2617i 0 3.45430 + 13.7138i
181.3 −2.20360 1.77318i 0 1.71169 + 7.81474i 5.00000i 0 21.5703 10.0850 20.2557i 0 8.86588 11.0180i
181.4 −2.20360 + 1.77318i 0 1.71169 7.81474i 5.00000i 0 21.5703 10.0850 + 20.2557i 0 8.86588 + 11.0180i
181.5 −1.52528 2.38191i 0 −3.34703 + 7.26618i 5.00000i 0 5.13620 22.4126 3.11063i 0 −11.9096 + 7.62641i
181.6 −1.52528 + 2.38191i 0 −3.34703 7.26618i 5.00000i 0 5.13620 22.4126 + 3.11063i 0 −11.9096 7.62641i
181.7 0.337480 2.80822i 0 −7.77221 1.89544i 5.00000i 0 9.93501 −7.94578 + 21.1864i 0 14.0411 + 1.68740i
181.8 0.337480 + 2.80822i 0 −7.77221 + 1.89544i 5.00000i 0 9.93501 −7.94578 21.1864i 0 14.0411 1.68740i
181.9 2.54175 1.24077i 0 4.92097 6.30746i 5.00000i 0 26.6173 4.68175 22.1378i 0 6.20386 + 12.7087i
181.10 2.54175 + 1.24077i 0 4.92097 + 6.30746i 5.00000i 0 26.6173 4.68175 + 22.1378i 0 6.20386 12.7087i
181.11 2.59241 1.13111i 0 5.44116 5.86462i 5.00000i 0 −34.6280 7.47214 21.3581i 0 −5.65557 12.9620i
181.12 2.59241 + 1.13111i 0 5.44116 + 5.86462i 5.00000i 0 −34.6280 7.47214 + 21.3581i 0 −5.65557 + 12.9620i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 360.4.k.c 12
3.b odd 2 1 40.4.d.a 12
4.b odd 2 1 1440.4.k.c 12
8.b even 2 1 inner 360.4.k.c 12
8.d odd 2 1 1440.4.k.c 12
12.b even 2 1 160.4.d.a 12
15.d odd 2 1 200.4.d.b 12
15.e even 4 1 200.4.f.b 12
15.e even 4 1 200.4.f.c 12
24.f even 2 1 160.4.d.a 12
24.h odd 2 1 40.4.d.a 12
48.i odd 4 1 1280.4.a.bb 6
48.i odd 4 1 1280.4.a.bc 6
48.k even 4 1 1280.4.a.ba 6
48.k even 4 1 1280.4.a.bd 6
60.h even 2 1 800.4.d.d 12
60.l odd 4 1 800.4.f.b 12
60.l odd 4 1 800.4.f.c 12
120.i odd 2 1 200.4.d.b 12
120.m even 2 1 800.4.d.d 12
120.q odd 4 1 800.4.f.b 12
120.q odd 4 1 800.4.f.c 12
120.w even 4 1 200.4.f.b 12
120.w even 4 1 200.4.f.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.4.d.a 12 3.b odd 2 1
40.4.d.a 12 24.h odd 2 1
160.4.d.a 12 12.b even 2 1
160.4.d.a 12 24.f even 2 1
200.4.d.b 12 15.d odd 2 1
200.4.d.b 12 120.i odd 2 1
200.4.f.b 12 15.e even 4 1
200.4.f.b 12 120.w even 4 1
200.4.f.c 12 15.e even 4 1
200.4.f.c 12 120.w even 4 1
360.4.k.c 12 1.a even 1 1 trivial
360.4.k.c 12 8.b even 2 1 inner
800.4.d.d 12 60.h even 2 1
800.4.d.d 12 120.m even 2 1
800.4.f.b 12 60.l odd 4 1
800.4.f.b 12 120.q odd 4 1
800.4.f.c 12 60.l odd 4 1
800.4.f.c 12 120.q odd 4 1
1280.4.a.ba 6 48.k even 4 1
1280.4.a.bb 6 48.i odd 4 1
1280.4.a.bc 6 48.i odd 4 1
1280.4.a.bd 6 48.k even 4 1
1440.4.k.c 12 4.b odd 2 1
1440.4.k.c 12 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} - 14T_{7}^{5} - 1258T_{7}^{4} + 23408T_{7}^{3} + 166612T_{7}^{2} - 4186552T_{7} + 14843128 \) acting on \(S_{4}^{\mathrm{new}}(360, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 2 T^{11} - 6 T^{10} + \cdots + 262144 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{2} + 25)^{6} \) Copy content Toggle raw display
$7$ \( (T^{6} - 14 T^{5} - 1258 T^{4} + \cdots + 14843128)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 10072 T^{10} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{12} + 11144 T^{10} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{6} - 14500 T^{4} + \cdots - 7473839808)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 46312 T^{10} + \cdots + 17\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( (T^{6} + 302 T^{5} + 12942 T^{4} + \cdots + 881168216)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 92448 T^{10} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{6} + 132 T^{5} + \cdots - 1437816300032)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 310360 T^{10} + \cdots + 21\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( (T^{6} + 20 T^{5} + \cdots - 71667547865600)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 338808 T^{10} + \cdots + 25\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( (T^{6} - 470 T^{5} + \cdots - 72048375466472)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + 939912 T^{10} + \cdots + 86\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{12} + 1889384 T^{10} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{12} + 627448 T^{10} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{12} + 1320888 T^{10} + \cdots + 87\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( (T^{6} - 796 T^{5} + \cdots - 36\!\cdots\!48)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 216 T^{5} + \cdots - 378730163491776)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 1008 T^{5} + \cdots - 44\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 4256648 T^{10} + \cdots + 78\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T^{6} - 212 T^{5} + \cdots - 62\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 792 T^{5} + \cdots - 33\!\cdots\!28)^{2} \) Copy content Toggle raw display
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