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Properties

Label 360.4.k.d
Level $360$
Weight $4$
Character orbit 360.k
Analytic conductor $21.241$
Analytic rank $0$
Dimension $14$
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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 4 x^{13} + 7 x^{12} - 22 x^{11} + 70 x^{10} - 232 x^{9} + 1080 x^{8} - 4000 x^{7} + \cdots + 2097152 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{15} \)
Twist minimal: no (minimal twist has level 120)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + \beta_{6} q^{4} + 5 \beta_1 q^{5} + ( - \beta_{8} - 2 \beta_{3} - 3) q^{7} + (\beta_{8} - \beta_{7} - \beta_{3} + \cdots + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + \beta_{6} q^{4} + 5 \beta_1 q^{5} + ( - \beta_{8} - 2 \beta_{3} - 3) q^{7} + (\beta_{8} - \beta_{7} - \beta_{3} + \cdots + 1) q^{8}+ \cdots + ( - 26 \beta_{13} + 6 \beta_{12} + \cdots - 442) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 2 q^{2} - 2 q^{4} - 28 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 2 q^{2} - 2 q^{4} - 28 q^{7} + 8 q^{8} - 20 q^{10} + 8 q^{14} - 22 q^{16} - 204 q^{17} + 20 q^{20} - 84 q^{22} + 328 q^{23} - 350 q^{25} + 4 q^{26} + 68 q^{28} + 596 q^{31} - 588 q^{32} + 756 q^{34} + 1144 q^{38} + 230 q^{40} + 820 q^{41} + 2084 q^{44} - 1060 q^{46} + 104 q^{47} + 1110 q^{49} - 50 q^{50} - 1736 q^{52} - 440 q^{55} - 3812 q^{56} + 2664 q^{58} + 772 q^{62} + 2470 q^{64} + 2864 q^{68} - 1280 q^{70} - 1592 q^{71} - 2260 q^{73} - 1020 q^{74} - 2468 q^{76} - 220 q^{79} - 80 q^{80} + 3444 q^{82} + 1184 q^{86} - 500 q^{88} + 2492 q^{89} + 4536 q^{92} - 4300 q^{94} + 280 q^{95} + 3508 q^{97} - 6246 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 4 x^{13} + 7 x^{12} - 22 x^{11} + 70 x^{10} - 232 x^{9} + 1080 x^{8} - 4000 x^{7} + \cdots + 2097152 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 405 \nu^{13} + 4300 \nu^{12} - 14835 \nu^{11} + 44822 \nu^{10} - 104462 \nu^{9} + \cdots - 484179968 ) / 231211008 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 17 \nu^{13} - 720 \nu^{12} + 41 \nu^{11} + 4330 \nu^{10} - 41150 \nu^{9} + 144592 \nu^{8} + \cdots + 268959744 ) / 8257536 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{13} - 4 \nu^{12} + 7 \nu^{11} - 22 \nu^{10} + 70 \nu^{9} - 232 \nu^{8} + 1080 \nu^{7} + \cdots - 1048576 ) / 262144 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1847 \nu^{13} - 10628 \nu^{12} + 47329 \nu^{11} - 159314 \nu^{10} + 487866 \nu^{9} + \cdots - 833355776 ) / 231211008 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3179 \nu^{13} + 2060 \nu^{12} - 62771 \nu^{11} + 308694 \nu^{10} - 1051982 \nu^{9} + \cdots + 1174667264 ) / 231211008 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{13} - 2 \nu^{12} - \nu^{11} - 8 \nu^{10} + 26 \nu^{9} - 92 \nu^{8} + 616 \nu^{7} + \cdots - 589824 ) / 65536 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2483 \nu^{13} + 6996 \nu^{12} + 10693 \nu^{11} - 113794 \nu^{10} + 438482 \nu^{9} + \cdots - 4951375872 ) / 115605504 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 31 \nu^{13} - 68 \nu^{12} - 7 \nu^{11} + 990 \nu^{10} - 5206 \nu^{9} + 13880 \nu^{8} + \cdots + 10485760 ) / 786432 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 4645 \nu^{13} + 34852 \nu^{12} - 81059 \nu^{11} + 117022 \nu^{10} + 261090 \nu^{9} + \cdots - 6873153536 ) / 115605504 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 10281 \nu^{13} + 46988 \nu^{12} - 158463 \nu^{11} + 274654 \nu^{10} - 489190 \nu^{9} + \cdots - 1986789376 ) / 231211008 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 11125 \nu^{13} + 52300 \nu^{12} - 158483 \nu^{11} + 349270 \nu^{10} - 649038 \nu^{9} + \cdots - 4343988224 ) / 231211008 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 5765 \nu^{13} - 28300 \nu^{12} + 86659 \nu^{11} - 197046 \nu^{10} + 376750 \nu^{9} + \cdots + 2067267584 ) / 115605504 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 19661 \nu^{13} + 56844 \nu^{12} - 190075 \nu^{11} + 592006 \nu^{10} - 1202462 \nu^{9} + \cdots + 5573443584 ) / 231211008 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{12} + \beta_{11} + \beta _1 + 3 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{12} - \beta_{11} + 2\beta_{9} - 4\beta_{3} + 2\beta_{2} - \beta _1 - 1 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 8 \beta_{13} - 7 \beta_{12} - \beta_{11} + 2 \beta_{9} - 8 \beta_{7} - 8 \beta_{6} - 4 \beta_{4} + \cdots + 27 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 5 \beta_{12} + 13 \beta_{11} - 24 \beta_{10} - 6 \beta_{9} - 8 \beta_{8} + 32 \beta_{6} - 8 \beta_{5} + \cdots - 15 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 7 \beta_{12} - 5 \beta_{11} - 24 \beta_{10} + 6 \beta_{9} - 40 \beta_{8} + 32 \beta_{7} - 64 \beta_{6} + \cdots + 355 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 64 \beta_{13} - 73 \beta_{12} + 97 \beta_{11} - 8 \beta_{10} + 2 \beta_{9} - 56 \beta_{8} + \cdots - 1523 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 48 \beta_{13} - 327 \beta_{12} + 11 \beta_{11} - 200 \beta_{10} - 66 \beta_{9} - 120 \beta_{8} + \cdots + 3331 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 320 \beta_{13} + 439 \beta_{12} - 7 \beta_{11} + 280 \beta_{10} - 526 \beta_{9} - 792 \beta_{8} + \cdots + 8413 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2032 \beta_{13} + 3153 \beta_{12} - 2581 \beta_{11} - 8 \beta_{10} + 1406 \beta_{9} - 2104 \beta_{8} + \cdots - 22005 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 288 \beta_{13} - 6465 \beta_{12} - 8439 \beta_{11} - 1864 \beta_{10} + 6306 \beta_{9} + 1864 \beta_{8} + \cdots + 63349 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1968 \beta_{13} - 5271 \beta_{12} + 21243 \beta_{11} - 26408 \beta_{10} - 9170 \beta_{9} + \cdots - 170445 ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 18688 \beta_{13} + 31319 \beta_{12} - 35671 \beta_{11} - 38760 \beta_{10} + 3314 \beta_{9} + \cdots + 641053 ) / 8 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 142736 \beta_{13} + 60097 \beta_{12} + 30699 \beta_{11} + 89336 \beta_{10} + 42622 \beta_{9} + \cdots + 2306459 ) / 8 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
0.194895 + 2.82170i
0.194895 − 2.82170i
−2.31762 − 1.62131i
−2.31762 + 1.62131i
2.49456 + 1.33311i
2.49456 − 1.33311i
2.82087 − 0.206677i
2.82087 + 0.206677i
−2.34569 + 1.58043i
−2.34569 − 1.58043i
1.74774 − 2.22383i
1.74774 + 2.22383i
−0.594754 + 2.76519i
−0.594754 − 2.76519i
−2.82170 − 0.194895i 0 7.92403 + 1.09987i − 5.00000i 0 19.4906 −22.1449 − 4.64786i 0 −0.974474 + 14.1085i
181.2 −2.82170 + 0.194895i 0 7.92403 − 1.09987i 5.00000i 0 19.4906 −22.1449 + 4.64786i 0 −0.974474 − 14.1085i
181.3 −1.62131 − 2.31762i 0 −2.74270 + 7.51516i 5.00000i 0 −31.1694 21.8640 − 5.82786i 0 11.5881 − 8.10655i
181.4 −1.62131 + 2.31762i 0 −2.74270 − 7.51516i − 5.00000i 0 −31.1694 21.8640 + 5.82786i 0 11.5881 + 8.10655i
181.5 −1.33311 − 2.49456i 0 −4.44564 + 6.65104i − 5.00000i 0 −22.4239 22.5179 + 2.22335i 0 −12.4728 + 6.66555i
181.6 −1.33311 + 2.49456i 0 −4.44564 − 6.65104i 5.00000i 0 −22.4239 22.5179 − 2.22335i 0 −12.4728 − 6.66555i
181.7 0.206677 − 2.82087i 0 −7.91457 − 1.16601i − 5.00000i 0 31.1865 −4.92492 + 22.0850i 0 −14.1043 − 1.03338i
181.8 0.206677 + 2.82087i 0 −7.91457 + 1.16601i 5.00000i 0 31.1865 −4.92492 − 22.0850i 0 −14.1043 + 1.03338i
181.9 1.58043 − 2.34569i 0 −3.00448 − 7.41439i 5.00000i 0 −3.68242 −22.1402 − 4.67037i 0 11.7284 + 7.90215i
181.10 1.58043 + 2.34569i 0 −3.00448 + 7.41439i − 5.00000i 0 −3.68242 −22.1402 + 4.67037i 0 11.7284 − 7.90215i
181.11 2.22383 − 1.74774i 0 1.89082 − 7.77334i − 5.00000i 0 2.94025 −9.38089 − 20.5912i 0 −8.73869 − 11.1191i
181.12 2.22383 + 1.74774i 0 1.89082 + 7.77334i 5.00000i 0 2.94025 −9.38089 + 20.5912i 0 −8.73869 + 11.1191i
181.13 2.76519 − 0.594754i 0 7.29254 − 3.28921i 5.00000i 0 −10.3416 18.2090 − 13.4326i 0 2.97377 + 13.8259i
181.14 2.76519 + 0.594754i 0 7.29254 + 3.28921i − 5.00000i 0 −10.3416 18.2090 + 13.4326i 0 2.97377 − 13.8259i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.14
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.d 14
3.b odd 2 1 120.4.k.c ✓ 14
4.b odd 2 1 1440.4.k.d 14
8.b even 2 1 inner 360.4.k.d 14
8.d odd 2 1 1440.4.k.d 14
12.b even 2 1 480.4.k.c 14
24.f even 2 1 480.4.k.c 14
24.h odd 2 1 120.4.k.c ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.k.c ✓ 14 3.b odd 2 1
120.4.k.c ✓ 14 24.h odd 2 1
360.4.k.d 14 1.a even 1 1 trivial
360.4.k.d 14 8.b even 2 1 inner
480.4.k.c 14 12.b even 2 1
480.4.k.c 14 24.f even 2 1
1440.4.k.d 14 4.b odd 2 1
1440.4.k.d 14 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{7} + 14T_{7}^{6} - 1380T_{7}^{5} - 18584T_{7}^{4} + 397440T_{7}^{3} + 4875648T_{7}^{2} - 1020672T_{7} - 47570432 \) acting on \(S_{4}^{\mathrm{new}}(360, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} - 2 T^{13} + \cdots + 2097152 \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( (T^{2} + 25)^{7} \) Copy content Toggle raw display
$7$ \( (T^{7} + 14 T^{6} + \cdots - 47570432)^{2} \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 87\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 25\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( (T^{7} + \cdots - 6976273176576)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( (T^{7} + \cdots - 94987172617216)^{2} \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 19\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( (T^{7} + \cdots - 3035229478912)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 50\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( (T^{7} + \cdots - 553103116856704)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 83\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( (T^{7} + \cdots - 12\!\cdots\!92)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 27\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 95\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots + 44\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( (T^{7} + \cdots + 18\!\cdots\!36)^{2} \) Copy content Toggle raw display
$79$ \( (T^{7} + \cdots - 37\!\cdots\!60)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 29\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{7} + \cdots + 33\!\cdots\!20)^{2} \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots + 34\!\cdots\!04)^{2} \) Copy content Toggle raw display
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