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Properties

Label 3720.2.k.f
Level $3720$
Weight $2$
Character orbit 3720.k
Analytic conductor $29.704$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3720,2,Mod(1489,3720)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3720.1489"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3720.k (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,0,0,0,-4,0,0,0,-18,0,-14,0,0,0,-4,0,0,0,22,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(29.7043495519\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 4 x^{17} + 14 x^{16} - 28 x^{15} + 43 x^{14} + 8 x^{13} - 155 x^{12} + 316 x^{11} + \cdots + 1953125 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} - \beta_{16} q^{5} + (\beta_{9} + \beta_{3}) q^{7} - q^{9} + (\beta_{14} - \beta_{5} - 1) q^{11} + (\beta_{11} - \beta_{7} + \beta_{3}) q^{13} - \beta_{6} q^{15} + ( - \beta_{9} + \beta_{8} + \cdots + \beta_1) q^{17}+ \cdots + ( - \beta_{14} + \beta_{5} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 4 q^{5} - 18 q^{9} - 14 q^{11} - 4 q^{15} + 22 q^{19} - 14 q^{21} + 12 q^{25} + 8 q^{29} + 18 q^{31} + 6 q^{35} - 12 q^{39} - 12 q^{41} + 4 q^{45} - 16 q^{49} - 4 q^{51} - 10 q^{55} + 52 q^{59} - 60 q^{61}+ \cdots + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 4 x^{17} + 14 x^{16} - 28 x^{15} + 43 x^{14} + 8 x^{13} - 155 x^{12} + 316 x^{11} + \cdots + 1953125 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 184 \nu^{17} - 709 \nu^{16} + 6758 \nu^{15} - 409 \nu^{14} - 1384 \nu^{13} + 33238 \nu^{12} + \cdots + 426796875 ) / 205000000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 53981 \nu^{17} + 648559 \nu^{16} - 3109499 \nu^{15} + 8321633 \nu^{14} + \cdots + 270176953125 ) / 45100000000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 65023 \nu^{17} - 924577 \nu^{16} + 3345137 \nu^{15} - 6834559 \nu^{14} + 8086694 \nu^{13} + \cdots - 150460546875 ) / 45100000000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1848 \nu^{17} + 1527 \nu^{16} - 13162 \nu^{15} - 23241 \nu^{14} + 80256 \nu^{13} + \cdots - 2766015625 ) / 1025000000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{17} + 4 \nu^{16} - 14 \nu^{15} + 28 \nu^{14} - 43 \nu^{13} - 8 \nu^{12} + 155 \nu^{11} + \cdots + 1562500 ) / 390625 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 12641 \nu^{17} - 90481 \nu^{16} + 386701 \nu^{15} - 1171087 \nu^{14} + 1560702 \nu^{13} + \cdots - 39449296875 ) / 4510000000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 6561 \nu^{17} + 42549 \nu^{16} - 233499 \nu^{15} + 619803 \nu^{14} - 712738 \nu^{13} + \cdots + 21468359375 ) / 2050000000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 30139 \nu^{17} + 177125 \nu^{16} - 471557 \nu^{15} + 825723 \nu^{14} - 552574 \nu^{13} + \cdots + 24267734375 ) / 9020000000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 76 \nu^{17} - 759 \nu^{16} + 2007 \nu^{15} - 3145 \nu^{14} - 700 \nu^{13} + 10504 \nu^{12} + \cdots - 66859375 ) / 20500000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 41455 \nu^{17} + 192367 \nu^{16} - 1188943 \nu^{15} + 4094113 \nu^{14} - 5166106 \nu^{13} + \cdots + 149713203125 ) / 9020000000 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 343549 \nu^{17} + 509269 \nu^{16} - 5937949 \nu^{15} + 19692363 \nu^{14} - 26906638 \nu^{13} + \cdots + 574646484375 ) / 45100000000 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 86527 \nu^{17} - 470753 \nu^{16} + 1362033 \nu^{15} - 1324711 \nu^{14} - 761354 \nu^{13} + \cdots + 21086328125 ) / 9020000000 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 2136 \nu^{17} + 8637 \nu^{16} - 25886 \nu^{15} + 34325 \nu^{14} + 408 \nu^{13} + \cdots + 701328125 ) / 205000000 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 11928 \nu^{17} - 67107 \nu^{16} + 168222 \nu^{15} - 223239 \nu^{14} - 112936 \nu^{13} + \cdots - 2669921875 ) / 1025000000 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 691653 \nu^{17} - 2496707 \nu^{16} + 6440347 \nu^{15} - 3818789 \nu^{14} - 11867086 \nu^{13} + \cdots + 180052734375 ) / 45100000000 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 4552 \nu^{17} - 20968 \nu^{16} + 58193 \nu^{15} - 58796 \nu^{14} - 55534 \nu^{13} + \cdots + 446875000 ) / 256250000 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{17} + \beta_{14} - \beta_{13} - \beta_{8} + \beta_{4} + 2\beta_{3} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 2 \beta_{16} - \beta_{15} + 2 \beta_{13} + \beta_{12} + 2 \beta_{10} + \beta_{9} - 2 \beta_{8} + \cdots - 3 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{17} + \beta_{16} - 2 \beta_{15} - 3 \beta_{14} - 4 \beta_{12} + 2 \beta_{11} + \beta_{8} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4 \beta_{17} - 2 \beta_{15} - 8 \beta_{12} + 8 \beta_{11} - 4 \beta_{10} - 4 \beta_{9} + 8 \beta_{8} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 26 \beta_{17} - 32 \beta_{16} - 2 \beta_{15} - 2 \beta_{14} + 4 \beta_{12} - 10 \beta_{11} - 16 \beta_{10} + \cdots - 17 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8 \beta_{17} - 8 \beta_{16} - 14 \beta_{15} - 24 \beta_{14} - 16 \beta_{13} + 28 \beta_{12} - 16 \beta_{11} + \cdots + 96 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 55 \beta_{17} + 56 \beta_{16} - 10 \beta_{15} + 61 \beta_{14} + 57 \beta_{13} + 20 \beta_{12} + \cdots + 217 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 100 \beta_{17} + 106 \beta_{16} + 47 \beta_{15} + 232 \beta_{14} + 94 \beta_{13} + 31 \beta_{12} + \cdots - 552 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 167 \beta_{17} + 119 \beta_{16} + 144 \beta_{15} + 297 \beta_{14} - 336 \beta_{13} - 296 \beta_{12} + \cdots + 303 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 60 \beta_{17} - 1704 \beta_{16} + 604 \beta_{15} - 760 \beta_{14} + 976 \beta_{13} - 396 \beta_{12} + \cdots + 1672 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1016 \beta_{17} + 1064 \beta_{16} + 1336 \beta_{15} - 2320 \beta_{14} + 664 \beta_{13} - 3008 \beta_{12} + \cdots - 4087 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 956 \beta_{17} - 5184 \beta_{16} - 44 \beta_{15} - 264 \beta_{14} + 5496 \beta_{13} + 1076 \beta_{12} + \cdots - 616 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 751 \beta_{17} - 7024 \beta_{16} - 8992 \beta_{15} - 10927 \beta_{14} + 4303 \beta_{13} + \cdots - 18825 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 11428 \beta_{17} + 17062 \beta_{16} - 13541 \beta_{15} + 12248 \beta_{14} - 16846 \beta_{13} + \cdots - 952 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 58455 \beta_{17} + 33897 \beta_{16} - 5194 \beta_{15} + 40061 \beta_{14} + 69000 \beta_{13} + \cdots - 154935 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 101968 \beta_{17} + 133984 \beta_{16} + 71330 \beta_{15} - 5480 \beta_{14} - 79848 \beta_{13} + \cdots + 43088 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3720\mathbb{Z}\right)^\times\).

\(n\) \(1241\) \(1801\) \(1861\) \(2791\) \(2977\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(1\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1489.1
0.125526 + 2.23254i
−0.378465 + 2.20381i
1.51384 + 1.64569i
−1.70589 + 1.44566i
−2.19929 + 0.403880i
2.20257 − 0.385590i
1.71540 − 1.43437i
1.15204 − 1.91645i
−0.425730 − 2.19517i
0.125526 − 2.23254i
−0.378465 − 2.20381i
1.51384 − 1.64569i
−1.70589 − 1.44566i
−2.19929 − 0.403880i
2.20257 + 0.385590i
1.71540 + 1.43437i
1.15204 + 1.91645i
−0.425730 + 2.19517i
0 − 1.00000i 0 −2.23254 − 0.125526i 0 − 4.58075i 0 −1.00000 0
1489.2 0 − 1.00000i 0 −2.20381 + 0.378465i 0 1.84344i 0 −1.00000 0
1489.3 0 − 1.00000i 0 −1.64569 − 1.51384i 0 0.450807i 0 −1.00000 0
1489.4 0 − 1.00000i 0 −1.44566 + 1.70589i 0 0.190753i 0 −1.00000 0
1489.5 0 − 1.00000i 0 −0.403880 + 2.19929i 0 − 0.206504i 0 −1.00000 0
1489.6 0 − 1.00000i 0 0.385590 − 2.20257i 0 3.73454i 0 −1.00000 0
1489.7 0 − 1.00000i 0 1.43437 − 1.71540i 0 − 3.56720i 0 −1.00000 0
1489.8 0 − 1.00000i 0 1.91645 − 1.15204i 0 − 0.453926i 0 −1.00000 0
1489.9 0 − 1.00000i 0 2.19517 + 0.425730i 0 − 4.41118i 0 −1.00000 0
1489.10 0 1.00000i 0 −2.23254 + 0.125526i 0 4.58075i 0 −1.00000 0
1489.11 0 1.00000i 0 −2.20381 − 0.378465i 0 − 1.84344i 0 −1.00000 0
1489.12 0 1.00000i 0 −1.64569 + 1.51384i 0 − 0.450807i 0 −1.00000 0
1489.13 0 1.00000i 0 −1.44566 − 1.70589i 0 − 0.190753i 0 −1.00000 0
1489.14 0 1.00000i 0 −0.403880 − 2.19929i 0 0.206504i 0 −1.00000 0
1489.15 0 1.00000i 0 0.385590 + 2.20257i 0 − 3.73454i 0 −1.00000 0
1489.16 0 1.00000i 0 1.43437 + 1.71540i 0 3.56720i 0 −1.00000 0
1489.17 0 1.00000i 0 1.91645 + 1.15204i 0 0.453926i 0 −1.00000 0
1489.18 0 1.00000i 0 2.19517 − 0.425730i 0 4.41118i 0 −1.00000 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1489.18
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3720.2.k.f ✓ 18
5.b even 2 1 inner 3720.2.k.f ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3720.2.k.f ✓ 18 1.a even 1 1 trivial
3720.2.k.f ✓ 18 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3720, [\chi])\):

\( T_{7}^{18} + 71 T_{7}^{16} + 1927 T_{7}^{14} + 24653 T_{7}^{12} + 145588 T_{7}^{10} + 313416 T_{7}^{8} + \cdots + 16 \) Copy content Toggle raw display
\( T_{11}^{9} + 7 T_{11}^{8} - 34 T_{11}^{7} - 256 T_{11}^{6} + 212 T_{11}^{5} + 2240 T_{11}^{4} + \cdots - 1408 \) Copy content Toggle raw display
\( T_{29}^{9} - 4 T_{29}^{8} - 94 T_{29}^{7} + 172 T_{29}^{6} + 2846 T_{29}^{5} + 410 T_{29}^{4} + \cdots + 10496 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{9} \) Copy content Toggle raw display
$5$ \( T^{18} + 4 T^{17} + \cdots + 1953125 \) Copy content Toggle raw display
$7$ \( T^{18} + 71 T^{16} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{9} + 7 T^{8} + \cdots - 1408)^{2} \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 481890304 \) Copy content Toggle raw display
$17$ \( T^{18} + 120 T^{16} + \cdots + 67108864 \) Copy content Toggle raw display
$19$ \( (T^{9} - 11 T^{8} + \cdots - 201088)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 963481600 \) Copy content Toggle raw display
$29$ \( (T^{9} - 4 T^{8} + \cdots + 10496)^{2} \) Copy content Toggle raw display
$31$ \( (T - 1)^{18} \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 1124663296 \) Copy content Toggle raw display
$41$ \( (T^{9} + 6 T^{8} + \cdots + 27808)^{2} \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 263334482477056 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 28894560256 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 1026671509504 \) Copy content Toggle raw display
$59$ \( (T^{9} - 26 T^{8} + \cdots - 36104)^{2} \) Copy content Toggle raw display
$61$ \( (T^{9} + 30 T^{8} + \cdots - 11264000)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 122556006400 \) Copy content Toggle raw display
$71$ \( (T^{9} + 3 T^{8} + \cdots + 3542440)^{2} \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 34852409344 \) Copy content Toggle raw display
$79$ \( (T^{9} - 11 T^{8} + \cdots + 7284352)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 383197224779776 \) Copy content Toggle raw display
$89$ \( (T^{9} - 27 T^{8} + \cdots + 82189360)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} + 242 T^{16} + \cdots + 87310336 \) Copy content Toggle raw display
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