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Properties

Label 360.4.d.f
Level $360$
Weight $4$
Character orbit 360.d
Analytic conductor $21.241$
Analytic rank $0$
Dimension $18$
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(109,360)]
 
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, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("360.109");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 360.d (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: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - x^{17} + 2 x^{16} - 8 x^{15} - 16 x^{14} - 12 x^{13} - 272 x^{12} + 368 x^{11} + \cdots + 134217728 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{25}\cdot 3^{4}\cdot 5 \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{4} - \beta_{9} q^{5} - \beta_{8} q^{7} + (\beta_{3} + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{2} q^{4} - \beta_{9} q^{5} - \beta_{8} q^{7} + (\beta_{3} + 1) q^{8} + (\beta_{16} + 1) q^{10} + (\beta_{11} - \beta_{2} - \beta_1) q^{11} + ( - \beta_{6} - \beta_{5} - \beta_{3} + 6) q^{13} + ( - \beta_{13} - \beta_{12} + \beta_{11} + \cdots + 2) q^{14}+ \cdots + (2 \beta_{17} - 4 \beta_{16} + \cdots - 50) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + q^{2} - 3 q^{4} + 19 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + q^{2} - 3 q^{4} + 19 q^{8} + 27 q^{10} + 104 q^{13} + 14 q^{14} + 97 q^{16} - 249 q^{20} + 98 q^{22} + 22 q^{25} + 82 q^{26} - 382 q^{28} + 240 q^{31} + 111 q^{32} - 4 q^{34} - 92 q^{35} + 592 q^{37} + 674 q^{38} - 1019 q^{40} + 236 q^{41} + 1090 q^{44} + 114 q^{46} - 882 q^{49} + 777 q^{50} - 122 q^{52} + 504 q^{53} - 72 q^{55} + 1094 q^{56} + 1116 q^{58} + 116 q^{62} + 1665 q^{64} - 348 q^{65} + 1392 q^{68} + 866 q^{70} + 112 q^{71} + 2226 q^{74} + 1302 q^{76} + 1848 q^{77} - 2288 q^{79} + 2031 q^{80} + 782 q^{82} - 1136 q^{83} - 1712 q^{85} + 1700 q^{86} + 1446 q^{88} + 220 q^{89} + 230 q^{92} + 3666 q^{94} - 1312 q^{95} - 837 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - x^{17} + 2 x^{16} - 8 x^{15} - 16 x^{14} - 12 x^{13} - 272 x^{12} + 368 x^{11} + \cdots + 134217728 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 101 \nu^{17} - 2947 \nu^{16} - 18178 \nu^{15} + 8824 \nu^{14} + 7248 \nu^{13} + \cdots + 13438550016 ) / 1761607680 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 177 \nu^{17} - 1841 \nu^{16} - 13534 \nu^{15} - 32968 \nu^{14} - 4496 \nu^{13} + \cdots - 11777605632 ) / 880803840 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 177 \nu^{17} + 1701 \nu^{16} + 3594 \nu^{15} - 6512 \nu^{14} - 72784 \nu^{13} + \cdots + 50826575872 ) / 880803840 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 54 \nu^{17} + 273 \nu^{16} + 997 \nu^{15} + 3134 \nu^{14} + 968 \nu^{13} + 35656 \nu^{12} + \cdots + 6580862976 ) / 220200960 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 51 \nu^{17} + 49 \nu^{16} + 578 \nu^{15} - 656 \nu^{14} - 6736 \nu^{13} - 20900 \nu^{12} + \cdots - 3791650816 ) / 176160768 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 521 \nu^{17} + 2527 \nu^{16} + 5578 \nu^{15} + 1256 \nu^{14} - 40848 \nu^{13} + \cdots + 4177526784 ) / 1761607680 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 521 \nu^{17} - 5887 \nu^{16} - 2218 \nu^{15} - 7976 \nu^{14} + 67728 \nu^{13} + \cdots - 16508780544 ) / 1761607680 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 81 \nu^{17} - 693 \nu^{16} - 1882 \nu^{15} - 3440 \nu^{14} + 11728 \nu^{13} + 86132 \nu^{12} + \cdots - 10166992896 ) / 176160768 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - \nu^{17} + \nu^{16} - 2 \nu^{15} + 8 \nu^{14} + 16 \nu^{13} + 12 \nu^{12} + 272 \nu^{11} + \cdots + 16777216 ) / 2097152 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 253 \nu^{17} - 469 \nu^{16} - 782 \nu^{15} - 5816 \nu^{14} + 3632 \nu^{13} + 60068 \nu^{12} + \cdots - 24679284736 ) / 352321536 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1119 \nu^{17} - 2807 \nu^{16} - 8458 \nu^{15} - 36296 \nu^{14} - 32752 \nu^{13} + \cdots - 120359747584 ) / 880803840 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1203 \nu^{17} + 259 \nu^{16} + 1066 \nu^{15} - 1448 \nu^{14} + 56944 \nu^{13} + \cdots + 72444018688 ) / 880803840 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 381 \nu^{17} - 567 \nu^{16} - 678 \nu^{15} + 4064 \nu^{14} + 8368 \nu^{13} + \cdots + 8520728576 ) / 220200960 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 111 \nu^{17} + 57 \nu^{16} - 682 \nu^{15} - 3656 \nu^{14} - 13040 \nu^{13} - 5812 \nu^{12} + \cdots - 1560281088 ) / 50331648 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + 2\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{14} + \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} - 5 \beta_{8} + \beta_{7} + \beta_{6} + \cdots + 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{17} + 2 \beta_{16} - 2 \beta_{15} - 6 \beta_{13} - 7 \beta_{12} + 3 \beta_{11} - 7 \beta_{10} + \cdots + 104 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 6 \beta_{17} + 2 \beta_{16} - 2 \beta_{15} + 4 \beta_{14} - 6 \beta_{13} - 29 \beta_{12} + 9 \beta_{11} + \cdots + 28 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 6 \beta_{17} + 22 \beta_{16} - 6 \beta_{15} - 20 \beta_{14} + 62 \beta_{13} - 55 \beta_{12} - 69 \beta_{11} + \cdots - 552 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 34 \beta_{17} + 34 \beta_{16} + 110 \beta_{15} + 68 \beta_{14} - 166 \beta_{13} - 233 \beta_{12} + \cdots - 6240 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 122 \beta_{17} + 38 \beta_{16} - 310 \beta_{15} + 36 \beta_{14} - 178 \beta_{13} - 247 \beta_{12} + \cdots - 7256 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 298 \beta_{17} - 566 \beta_{16} + 134 \beta_{15} - 164 \beta_{14} + 482 \beta_{13} + 799 \beta_{12} + \cdots - 12376 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 358 \beta_{17} + 3622 \beta_{16} - 54 \beta_{15} + 324 \beta_{14} - 178 \beta_{13} - 3495 \beta_{12} + \cdots + 40712 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 2966 \beta_{17} - 6614 \beta_{16} + 3174 \beta_{15} + 956 \beta_{14} - 3902 \beta_{13} + \cdots - 167720 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 922 \beta_{17} + 2342 \beta_{16} - 14646 \beta_{15} - 18812 \beta_{14} + 3918 \beta_{13} + \cdots - 546488 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 4694 \beta_{17} + 6506 \beta_{16} + 39718 \beta_{15} - 13668 \beta_{14} + 39682 \beta_{13} + \cdots - 1348424 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 1926 \beta_{17} - 67258 \beta_{16} + 20138 \beta_{15} - 65660 \beta_{14} + 88814 \beta_{13} + \cdots - 4646968 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 32950 \beta_{17} - 702326 \beta_{16} + 67078 \beta_{15} + 241756 \beta_{14} - 369246 \beta_{13} + \cdots + 20360056 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
−2.81291 − 0.295858i
−2.81291 + 0.295858i
−2.33755 − 1.59244i
−2.33755 + 1.59244i
−1.63965 − 2.30468i
−1.63965 + 2.30468i
−0.785894 − 2.71705i
−0.785894 + 2.71705i
−0.113641 − 2.82614i
−0.113641 + 2.82614i
1.33786 − 2.49201i
1.33786 + 2.49201i
1.42182 − 2.44508i
1.42182 + 2.44508i
2.69604 − 0.855210i
2.69604 + 0.855210i
2.73393 − 0.725009i
2.73393 + 0.725009i
−2.81291 − 0.295858i 0 7.82494 + 1.66445i 5.06199 − 9.96877i 0 12.3358i −21.5184 − 6.99702i 0 −17.1883 + 26.5436i
109.2 −2.81291 + 0.295858i 0 7.82494 − 1.66445i 5.06199 + 9.96877i 0 − 12.3358i −21.5184 + 6.99702i 0 −17.1883 − 26.5436i
109.3 −2.33755 − 1.59244i 0 2.92827 + 7.44481i −10.6259 − 3.47701i 0 − 10.7467i 5.01042 − 22.0657i 0 19.3017 + 25.0488i
109.4 −2.33755 + 1.59244i 0 2.92827 − 7.44481i −10.6259 + 3.47701i 0 10.7467i 5.01042 + 22.0657i 0 19.3017 − 25.0488i
109.5 −1.63965 − 2.30468i 0 −2.62312 + 7.55773i −1.42811 + 11.0888i 0 30.7524i 21.7191 − 6.34656i 0 27.8976 − 14.8903i
109.6 −1.63965 + 2.30468i 0 −2.62312 − 7.55773i −1.42811 − 11.0888i 0 − 30.7524i 21.7191 + 6.34656i 0 27.8976 + 14.8903i
109.7 −0.785894 − 2.71705i 0 −6.76474 + 4.27063i 8.33359 − 7.45328i 0 12.3087i 16.9199 + 15.0239i 0 −26.8003 − 16.7853i
109.8 −0.785894 + 2.71705i 0 −6.76474 − 4.27063i 8.33359 + 7.45328i 0 − 12.3087i 16.9199 − 15.0239i 0 −26.8003 + 16.7853i
109.9 −0.113641 − 2.82614i 0 −7.97417 + 0.642332i −5.10528 + 9.94666i 0 − 31.6787i 2.72152 + 22.4632i 0 28.6909 + 13.2979i
109.10 −0.113641 + 2.82614i 0 −7.97417 − 0.642332i −5.10528 − 9.94666i 0 31.6787i 2.72152 − 22.4632i 0 28.6909 − 13.2979i
109.11 1.33786 − 2.49201i 0 −4.42027 − 6.66793i −4.64605 − 10.1693i 0 − 3.99372i −22.5303 + 2.09462i 0 −31.5577 − 2.02703i
109.12 1.33786 + 2.49201i 0 −4.42027 + 6.66793i −4.64605 + 10.1693i 0 3.99372i −22.5303 − 2.09462i 0 −31.5577 + 2.02703i
109.13 1.42182 − 2.44508i 0 −3.95687 − 6.95293i 10.4145 + 4.06677i 0 6.74900i −22.6264 − 0.210922i 0 24.7511 − 19.6821i
109.14 1.42182 + 2.44508i 0 −3.95687 + 6.95293i 10.4145 − 4.06677i 0 − 6.74900i −22.6264 + 0.210922i 0 24.7511 + 19.6821i
109.15 2.69604 − 0.855210i 0 6.53723 − 4.61136i −10.9988 − 2.00675i 0 20.8271i 13.6809 − 18.0231i 0 −31.3693 + 3.99599i
109.16 2.69604 + 0.855210i 0 6.53723 + 4.61136i −10.9988 + 2.00675i 0 − 20.8271i 13.6809 + 18.0231i 0 −31.3693 − 3.99599i
109.17 2.73393 − 0.725009i 0 6.94872 − 3.96425i 8.99409 − 6.64126i 0 − 25.7743i 16.1232 − 15.8758i 0 19.7742 − 24.6775i
109.18 2.73393 + 0.725009i 0 6.94872 + 3.96425i 8.99409 + 6.64126i 0 25.7743i 16.1232 + 15.8758i 0 19.7742 + 24.6775i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.f 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.d.f 18
3.b odd 2 1 120.4.d.a ✓ 18
4.b odd 2 1 1440.4.d.f 18
5.b even 2 1 360.4.d.e 18
8.b even 2 1 360.4.d.e 18
8.d odd 2 1 1440.4.d.e 18
12.b even 2 1 480.4.d.a 18
15.d odd 2 1 120.4.d.b yes 18
20.d odd 2 1 1440.4.d.e 18
24.f even 2 1 480.4.d.b 18
24.h odd 2 1 120.4.d.b yes 18
40.e odd 2 1 1440.4.d.f 18
40.f even 2 1 inner 360.4.d.f 18
60.h even 2 1 480.4.d.b 18
120.i odd 2 1 120.4.d.a ✓ 18
120.m even 2 1 480.4.d.a 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.d.a ✓ 18 3.b odd 2 1
120.4.d.a ✓ 18 120.i odd 2 1
120.4.d.b yes 18 15.d odd 2 1
120.4.d.b yes 18 24.h odd 2 1
360.4.d.e 18 5.b even 2 1
360.4.d.e 18 8.b even 2 1
360.4.d.f 18 1.a even 1 1 trivial
360.4.d.f 18 40.f even 2 1 inner
480.4.d.a 18 12.b even 2 1
480.4.d.a 18 120.m even 2 1
480.4.d.b 18 24.f even 2 1
480.4.d.b 18 60.h even 2 1
1440.4.d.e 18 8.d odd 2 1
1440.4.d.e 18 20.d odd 2 1
1440.4.d.f 18 4.b odd 2 1
1440.4.d.f 18 40.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(360, [\chi])\):

\( T_{7}^{18} + 3528 T_{7}^{16} + 4927040 T_{7}^{14} + 3491809984 T_{7}^{12} + 1350390727680 T_{7}^{10} + \cdots + 52\!\cdots\!84 \) Copy content Toggle raw display
\( T_{11}^{18} + 13152 T_{11}^{16} + 67975968 T_{11}^{14} + 173332385472 T_{11}^{12} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
\( T_{13}^{9} - 52 T_{13}^{8} - 8960 T_{13}^{7} + 320088 T_{13}^{6} + 27415744 T_{13}^{5} + \cdots - 63920793392640 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + \cdots + 134217728 \) Copy content Toggle raw display
$3$ \( T^{18} \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 74\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 52\!\cdots\!84 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{9} + \cdots - 63920793392640)^{2} \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 37\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 71\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 55\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$31$ \( (T^{9} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{9} + \cdots + 21\!\cdots\!04)^{2} \) Copy content Toggle raw display
$41$ \( (T^{9} + \cdots - 33\!\cdots\!20)^{2} \) Copy content Toggle raw display
$43$ \( (T^{9} + \cdots - 80\!\cdots\!04)^{2} \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 27\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{9} + \cdots - 23\!\cdots\!20)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 15\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{9} + \cdots - 71\!\cdots\!56)^{2} \) Copy content Toggle raw display
$71$ \( (T^{9} + \cdots - 48\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{9} + \cdots - 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{9} + \cdots + 48\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{9} + \cdots - 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
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