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Properties

Label 560.4.g.h
Level $560$
Weight $4$
Character orbit 560.g
Analytic conductor $33.041$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [560,4,Mod(449,560)] 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("560.449"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(560, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 560.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(33.0410696032\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 315 x^{14} + 37955 x^{12} + 2158801 x^{10} + 57185968 x^{8} + 585896424 x^{6} + \cdots + 256000000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 5^{2}\cdot 11^{2} \)
Twist minimal: no (minimal twist has level 280)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{4} + \beta_{3} + 1) q^{5} - 7 \beta_{3} q^{7} + (\beta_{5} + \beta_{4} - \beta_{2} - 12) q^{9} + (\beta_{6} + 2) q^{11} + ( - \beta_{14} + \beta_{11} + \cdots - \beta_1) q^{13}+ \cdots + ( - 6 \beta_{15} - 26 \beta_{14} + \cdots + 266) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 22 q^{5} - 198 q^{9} + 34 q^{11} - 90 q^{15} + 264 q^{19} + 42 q^{21} - 52 q^{25} - 662 q^{29} - 332 q^{31} + 98 q^{35} + 342 q^{39} + 1160 q^{41} - 1490 q^{45} - 784 q^{49} + 198 q^{51} + 780 q^{55}+ \cdots + 3124 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 315 x^{14} + 37955 x^{12} + 2158801 x^{10} + 57185968 x^{8} + 585896424 x^{6} + \cdots + 256000000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 221899 \nu^{14} + 68873075 \nu^{12} + 8296109835 \nu^{10} + 481888952149 \nu^{8} + \cdots + 205490031616000 ) / 226826411028480 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 200673859 \nu^{15} + 63156790835 \nu^{13} + 7599358049595 \nu^{11} + \cdots + 18\!\cdots\!00 \nu ) / 56\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 274648437389 \nu^{15} - 48934178550 \nu^{14} - 86072343362785 \nu^{13} + \cdots + 39\!\cdots\!00 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 274648437389 \nu^{15} + 76338705050 \nu^{14} + 86072343362785 \nu^{13} + \cdots + 72\!\cdots\!00 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 19068761329 \nu^{14} + 5973873579245 \nu^{12} + 713393599514925 \nu^{10} + \cdots + 43\!\cdots\!20 ) / 56\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 587509553016 \nu^{15} + 630874156825 \nu^{14} + 192310095173040 \nu^{13} + \cdots - 58\!\cdots\!00 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1077413013226 \nu^{15} + 505601273225 \nu^{14} - 338577261452690 \nu^{13} + \cdots + 20\!\cdots\!00 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2452460041916 \nu^{15} - 12240715722925 \nu^{14} + 771063489775540 \nu^{13} + \cdots - 18\!\cdots\!00 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2761365027768 \nu^{15} - 505601273225 \nu^{14} - 869695646011920 \nu^{13} + \cdots - 20\!\cdots\!00 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 321893314629 \nu^{15} - 177788768375 \nu^{14} - 101099633561385 \nu^{13} + \cdots + 15\!\cdots\!00 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 5183838596961 \nu^{15} + 518253119800 \nu^{14} + \cdots - 46\!\cdots\!00 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1362470665342 \nu^{15} - 2309651108195 \nu^{14} - 427984202382230 \nu^{13} + \cdots - 48\!\cdots\!00 ) / 56\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1460451357384 \nu^{15} - 452463962065 \nu^{14} + 457237635638160 \nu^{13} + \cdots + 32\!\cdots\!00 ) / 56\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 8972662276477 \nu^{15} + 1023854393025 \nu^{14} + \cdots - 26\!\cdots\!00 ) / 28\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} - \beta_{2} - 39 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - 2 \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 9 \beta_{15} - 2 \beta_{14} - 10 \beta_{13} + \beta_{12} + 9 \beta_{10} + 10 \beta_{9} - 29 \beta_{8} + \cdots + 2900 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 85 \beta_{15} + 28 \beta_{14} + 42 \beta_{13} + 178 \beta_{12} + 107 \beta_{11} - 151 \beta_{10} + \cdots - 166 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1101 \beta_{15} + 380 \beta_{14} + 1432 \beta_{13} - 87 \beta_{12} - 38 \beta_{11} - 1101 \beta_{10} + \cdots - 232862 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 6123 \beta_{15} - 3784 \beta_{14} - 6504 \beta_{13} - 13910 \beta_{12} - 10507 \beta_{11} + 18491 \beta_{10} + \cdots + 12974 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 109083 \beta_{15} - 50326 \beta_{14} - 159302 \beta_{13} + 8691 \beta_{12} + 8584 \beta_{11} + \cdots + 19285038 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 411837 \beta_{15} + 383972 \beta_{14} + 738846 \beta_{13} + 1072306 \beta_{12} + 1015343 \beta_{11} + \cdots - 1034590 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 10112997 \beta_{15} + 5713280 \beta_{14} + 16125556 \beta_{13} - 928131 \beta_{12} - 1227410 \beta_{11} + \cdots - 1626206802 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 26272487 \beta_{15} - 35481760 \beta_{14} - 75096276 \beta_{13} - 83208222 \beta_{12} - 96550251 \beta_{11} + \cdots + 84432798 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 912534375 \beta_{15} - 595240154 \beta_{14} - 1557959218 \beta_{13} + 96139211 \beta_{12} + \cdots + 138758135662 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1584959121 \beta_{15} + 3173671612 \beta_{14} + 7253051610 \beta_{13} + 6530305322 \beta_{12} + \cdots - 7004389310 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 81406304361 \beta_{15} + 58737658724 \beta_{14} + 146575039936 \beta_{13} - 9426487939 \beta_{12} + \cdots - 11936963187258 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 89130714027 \beta_{15} - 281928443512 \beta_{14} - 682124371152 \beta_{13} - 519233311286 \beta_{12} + \cdots + 587168603870 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
− 9.47732i
− 9.27983i
− 8.54486i
− 6.91475i
− 3.79543i
− 1.45794i
− 1.22452i
− 0.454408i
0.454408i
1.22452i
1.45794i
3.79543i
6.91475i
8.54486i
9.27983i
9.47732i
0 − 9.47732i 0 10.7282 + 3.14732i 0 − 7.00000i 0 −62.8195 0
449.2 0 − 9.27983i 0 −9.02002 − 6.60599i 0 7.00000i 0 −59.1153 0
449.3 0 − 8.54486i 0 11.1589 − 0.692313i 0 7.00000i 0 −46.0147 0
449.4 0 − 6.91475i 0 −0.00172249 − 11.1803i 0 − 7.00000i 0 −20.8138 0
449.5 0 − 3.79543i 0 1.94260 + 11.0103i 0 7.00000i 0 12.5947 0
449.6 0 − 1.45794i 0 6.24604 + 9.27292i 0 − 7.00000i 0 24.8744 0
449.7 0 − 1.22452i 0 1.00194 + 11.1354i 0 − 7.00000i 0 25.5006 0
449.8 0 − 0.454408i 0 −11.0559 + 1.66328i 0 7.00000i 0 26.7935 0
449.9 0 0.454408i 0 −11.0559 − 1.66328i 0 − 7.00000i 0 26.7935 0
449.10 0 1.22452i 0 1.00194 − 11.1354i 0 7.00000i 0 25.5006 0
449.11 0 1.45794i 0 6.24604 − 9.27292i 0 7.00000i 0 24.8744 0
449.12 0 3.79543i 0 1.94260 − 11.0103i 0 − 7.00000i 0 12.5947 0
449.13 0 6.91475i 0 −0.00172249 + 11.1803i 0 7.00000i 0 −20.8138 0
449.14 0 8.54486i 0 11.1589 + 0.692313i 0 − 7.00000i 0 −46.0147 0
449.15 0 9.27983i 0 −9.02002 + 6.60599i 0 − 7.00000i 0 −59.1153 0
449.16 0 9.47732i 0 10.7282 − 3.14732i 0 7.00000i 0 −62.8195 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.16
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 560.4.g.h 16
4.b odd 2 1 280.4.g.b ✓ 16
5.b even 2 1 inner 560.4.g.h 16
20.d odd 2 1 280.4.g.b ✓ 16
20.e even 4 1 1400.4.a.y 8
20.e even 4 1 1400.4.a.z 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.4.g.b ✓ 16 4.b odd 2 1
280.4.g.b ✓ 16 20.d odd 2 1
560.4.g.h 16 1.a even 1 1 trivial
560.4.g.h 16 5.b even 2 1 inner
1400.4.a.y 8 20.e even 4 1
1400.4.a.z 8 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(560, [\chi])\):

\( T_{3}^{16} + 315 T_{3}^{14} + 37955 T_{3}^{12} + 2158801 T_{3}^{10} + 57185968 T_{3}^{8} + \cdots + 256000000 \) Copy content Toggle raw display
\( T_{11}^{8} - 17 T_{11}^{7} - 7945 T_{11}^{6} + 89641 T_{11}^{5} + 22214464 T_{11}^{4} + \cdots + 9629409996800 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 256000000 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 59\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{2} + 49)^{8} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 9629409996800)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 17\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots - 494067063151616)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 14\!\cdots\!68)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 14\!\cdots\!96)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 53\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 11\!\cdots\!60)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 26\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots - 31\!\cdots\!56)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots - 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 20\!\cdots\!36)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 24\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots - 97\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 21\!\cdots\!04 \) Copy content Toggle raw display
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