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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [320,5,Mod(193,320)] 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("320.193"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(320, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 3])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 320.p (of order \(4\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,12,0,12,0,44] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(33.0783881868\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{29})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 15x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 5 \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 193.1
Root \(3.19258i\) of defining polynomial
Character \(\chi\) \(=\) 320.193
Dual form 320.5.p.n.257.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(3.00000 + 3.00000i) q^{3} +(-18.5407 - 16.7703i) q^{5} +(64.8516 - 64.8516i) q^{7} -63.0000i q^{9} -33.7033 q^{11} +(-132.852 - 132.852i) q^{13} +(-5.31099 - 105.933i) q^{15} +(-270.407 + 270.407i) q^{17} +295.703i q^{19} +389.110 q^{21} +(1.44506 + 1.44506i) q^{23} +(62.5121 + 621.866i) q^{25} +(432.000 - 432.000i) q^{27} +297.484i q^{29} -1228.52 q^{31} +(-101.110 - 101.110i) q^{33} +(-2289.98 + 114.809i) q^{35} +(-402.478 + 402.478i) q^{37} -797.110i q^{39} +860.659 q^{41} +(-753.736 - 753.736i) q^{43} +(-1056.53 + 1168.06i) q^{45} +(-2228.48 + 2228.48i) q^{47} -6010.47i q^{49} -1622.44 q^{51} +(-2538.77 - 2538.77i) q^{53} +(624.881 + 565.215i) q^{55} +(-887.110 + 887.110i) q^{57} -3421.02i q^{59} +6691.91 q^{61} +(-4085.65 - 4085.65i) q^{63} +(235.191 + 4691.12i) q^{65} +(318.264 - 318.264i) q^{67} +8.67033i q^{69} -3427.02 q^{71} +(3052.86 + 3052.86i) q^{73} +(-1678.06 + 2053.13i) q^{75} +(-2185.71 + 2185.71i) q^{77} -2639.38i q^{79} -2511.00 q^{81} +(-225.132 - 225.132i) q^{83} +(9548.32 - 478.709i) q^{85} +(-892.451 + 892.451i) q^{87} +4989.60i q^{89} -17231.3 q^{91} +(-3685.55 - 3685.55i) q^{93} +(4959.04 - 5482.53i) q^{95} +(4891.66 - 4891.66i) q^{97} +2123.31i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{3} + 12 q^{5} + 44 q^{7} + 296 q^{11} - 316 q^{13} + 108 q^{15} - 220 q^{17} + 264 q^{21} + 652 q^{23} + 1284 q^{25} + 1728 q^{27} - 2760 q^{31} + 888 q^{33} - 7092 q^{35} + 2052 q^{37} - 4312 q^{41}+ \cdots + 11812 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.00000 + 3.00000i 0.333333 + 0.333333i 0.853851 − 0.520518i \(-0.174261\pi\)
−0.520518 + 0.853851i \(0.674261\pi\)
\(4\) 0 0
\(5\) −18.5407 − 16.7703i −0.741626 − 0.670813i
\(6\) 0 0
\(7\) 64.8516 − 64.8516i 1.32350 − 1.32350i 0.412583 − 0.910920i \(-0.364627\pi\)
0.910920 − 0.412583i \(-0.135373\pi\)
\(8\) 0 0
\(9\) − 63.0000i − 0.777778i
\(10\) 0 0
\(11\) −33.7033 −0.278540 −0.139270 − 0.990254i \(-0.544476\pi\)
−0.139270 + 0.990254i \(0.544476\pi\)
\(12\) 0 0
\(13\) −132.852 − 132.852i −0.786104 − 0.786104i 0.194749 − 0.980853i \(-0.437611\pi\)
−0.980853 + 0.194749i \(0.937611\pi\)
\(14\) 0 0
\(15\) −5.31099 − 105.933i −0.0236044 − 0.470813i
\(16\) 0 0
\(17\) −270.407 + 270.407i −0.935663 + 0.935663i −0.998052 − 0.0623890i \(-0.980128\pi\)
0.0623890 + 0.998052i \(0.480128\pi\)
\(18\) 0 0
\(19\) 295.703i 0.819123i 0.912283 + 0.409561i \(0.134318\pi\)
−0.912283 + 0.409561i \(0.865682\pi\)
\(20\) 0 0
\(21\) 389.110 0.882335
\(22\) 0 0
\(23\) 1.44506 + 1.44506i 0.00273167 + 0.00273167i 0.708471 − 0.705740i \(-0.249385\pi\)
−0.705740 + 0.708471i \(0.749385\pi\)
\(24\) 0 0
\(25\) 62.5121 + 621.866i 0.100019 + 0.994985i
\(26\) 0 0
\(27\) 432.000 − 432.000i 0.592593 − 0.592593i
\(28\) 0 0
\(29\) 297.484i 0.353726i 0.984235 + 0.176863i \(0.0565949\pi\)
−0.984235 + 0.176863i \(0.943405\pi\)
\(30\) 0 0
\(31\) −1228.52 −1.27837 −0.639187 − 0.769052i \(-0.720729\pi\)
−0.639187 + 0.769052i \(0.720729\pi\)
\(32\) 0 0
\(33\) −101.110 − 101.110i −0.0928465 − 0.0928465i
\(34\) 0 0
\(35\) −2289.98 + 114.809i −1.86937 + 0.0937215i
\(36\) 0 0
\(37\) −402.478 + 402.478i −0.293994 + 0.293994i −0.838656 − 0.544662i \(-0.816658\pi\)
0.544662 + 0.838656i \(0.316658\pi\)
\(38\) 0 0
\(39\) − 797.110i − 0.524070i
\(40\) 0 0
\(41\) 860.659 0.511992 0.255996 − 0.966678i \(-0.417597\pi\)
0.255996 + 0.966678i \(0.417597\pi\)
\(42\) 0 0
\(43\) −753.736 − 753.736i −0.407645 − 0.407645i 0.473271 − 0.880917i \(-0.343073\pi\)
−0.880917 + 0.473271i \(0.843073\pi\)
\(44\) 0 0
\(45\) −1056.53 + 1168.06i −0.521744 + 0.576821i
\(46\) 0 0
\(47\) −2228.48 + 2228.48i −1.00882 + 1.00882i −0.00885679 + 0.999961i \(0.502819\pi\)
−0.999961 + 0.00885679i \(0.997181\pi\)
\(48\) 0 0
\(49\) − 6010.47i − 2.50332i
\(50\) 0 0
\(51\) −1622.44 −0.623775
\(52\) 0 0
\(53\) −2538.77 − 2538.77i −0.903800 − 0.903800i 0.0919623 − 0.995762i \(-0.470686\pi\)
−0.995762 + 0.0919623i \(0.970686\pi\)
\(54\) 0 0
\(55\) 624.881 + 565.215i 0.206572 + 0.186848i
\(56\) 0 0
\(57\) −887.110 + 887.110i −0.273041 + 0.273041i
\(58\) 0 0
\(59\) − 3421.02i − 0.982770i −0.870943 − 0.491385i \(-0.836491\pi\)
0.870943 − 0.491385i \(-0.163509\pi\)
\(60\) 0 0
\(61\) 6691.91 1.79842 0.899209 − 0.437520i \(-0.144143\pi\)
0.899209 + 0.437520i \(0.144143\pi\)
\(62\) 0 0
\(63\) −4085.65 − 4085.65i −1.02939 − 1.02939i
\(64\) 0 0
\(65\) 235.191 + 4691.12i 0.0556666 + 1.11032i
\(66\) 0 0
\(67\) 318.264 − 318.264i 0.0708986 − 0.0708986i −0.670768 − 0.741667i \(-0.734036\pi\)
0.741667 + 0.670768i \(0.234036\pi\)
\(68\) 0 0
\(69\) 8.67033i 0.00182112i
\(70\) 0 0
\(71\) −3427.02 −0.679830 −0.339915 − 0.940456i \(-0.610398\pi\)
−0.339915 + 0.940456i \(0.610398\pi\)
\(72\) 0 0
\(73\) 3052.86 + 3052.86i 0.572876 + 0.572876i 0.932931 − 0.360055i \(-0.117242\pi\)
−0.360055 + 0.932931i \(0.617242\pi\)
\(74\) 0 0
\(75\) −1678.06 + 2053.13i −0.298322 + 0.365002i
\(76\) 0 0
\(77\) −2185.71 + 2185.71i −0.368648 + 0.368648i
\(78\) 0 0
\(79\) − 2639.38i − 0.422911i −0.977388 − 0.211455i \(-0.932180\pi\)
0.977388 − 0.211455i \(-0.0678203\pi\)
\(80\) 0 0
\(81\) −2511.00 −0.382716
\(82\) 0 0
\(83\) −225.132 − 225.132i −0.0326799 − 0.0326799i 0.690578 − 0.723258i \(-0.257356\pi\)
−0.723258 + 0.690578i \(0.757356\pi\)
\(84\) 0 0
\(85\) 9548.32 − 478.709i 1.32157 − 0.0662573i
\(86\) 0 0
\(87\) −892.451 + 892.451i −0.117909 + 0.117909i
\(88\) 0 0
\(89\) 4989.60i 0.629921i 0.949105 + 0.314961i \(0.101991\pi\)
−0.949105 + 0.314961i \(0.898009\pi\)
\(90\) 0 0
\(91\) −17231.3 −2.08082
\(92\) 0 0
\(93\) −3685.55 − 3685.55i −0.426124 − 0.426124i
\(94\) 0 0
\(95\) 4959.04 − 5482.53i 0.549478 − 0.607483i
\(96\) 0 0
\(97\) 4891.66 − 4891.66i 0.519892 − 0.519892i −0.397647 − 0.917538i \(-0.630173\pi\)
0.917538 + 0.397647i \(0.130173\pi\)
\(98\) 0 0
\(99\) 2123.31i 0.216642i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 320.5.p.n.193.1 4
4.3 odd 2 320.5.p.k.193.1 4
5.2 odd 4 inner 320.5.p.n.257.1 4
8.3 odd 2 80.5.p.f.33.2 4
8.5 even 2 40.5.l.b.33.2 yes 4
20.7 even 4 320.5.p.k.257.1 4
24.5 odd 2 360.5.v.b.73.1 4
40.3 even 4 400.5.p.g.257.2 4
40.13 odd 4 200.5.l.c.57.1 4
40.19 odd 2 400.5.p.g.193.2 4
40.27 even 4 80.5.p.f.17.2 4
40.29 even 2 200.5.l.c.193.1 4
40.37 odd 4 40.5.l.b.17.2 ✓ 4
120.77 even 4 360.5.v.b.217.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.5.l.b.17.2 ✓ 4 40.37 odd 4
40.5.l.b.33.2 yes 4 8.5 even 2
80.5.p.f.17.2 4 40.27 even 4
80.5.p.f.33.2 4 8.3 odd 2
200.5.l.c.57.1 4 40.13 odd 4
200.5.l.c.193.1 4 40.29 even 2
320.5.p.k.193.1 4 4.3 odd 2
320.5.p.k.257.1 4 20.7 even 4
320.5.p.n.193.1 4 1.1 even 1 trivial
320.5.p.n.257.1 4 5.2 odd 4 inner
360.5.v.b.73.1 4 24.5 odd 2
360.5.v.b.217.1 4 120.77 even 4
400.5.p.g.193.2 4 40.19 odd 2
400.5.p.g.257.2 4 40.3 even 4